id	sid	tid	token	lemma	pos
ejpam-6574	1	1	european	european	PROPN
ejpam-6574	1	2	journal	journal	PROPN
ejpam-6574	1	3	of	of	ADP
ejpam-6574	1	4	pure	pure	ADJ
ejpam-6574	1	5	and	and	CCONJ
ejpam-6574	1	6	applied	applied	ADJ
ejpam-6574	1	7	mathematics	mathematic	NOUN
ejpam-6574	1	8	2025	2025	NUM
ejpam-6574	1	9	,	,	PUNCT
ejpam-6574	1	10	vol	vol	NOUN
ejpam-6574	1	11	.	.	PROPN
ejpam-6574	1	12	18	18	NUM
ejpam-6574	1	13	,	,	PUNCT
ejpam-6574	1	14	issue	issue	NOUN
ejpam-6574	1	15	3	3	NUM
ejpam-6574	1	16	,	,	PUNCT
ejpam-6574	1	17	article	article	NOUN
ejpam-6574	1	18	number	number	NOUN
ejpam-6574	1	19	6574	6574	NUM
ejpam-6574	1	20	issn	issn	VERB
ejpam-6574	1	21	1307	1307	NUM
ejpam-6574	1	22	-	-	SYM
ejpam-6574	1	23	5543	5543	NUM
ejpam-6574	1	24	–	–	PUNCT
ejpam-6574	1	25	ejpam.com	ejpam.com	X
ejpam-6574	1	26	published	publish	VERB
ejpam-6574	1	27	by	by	ADP
ejpam-6574	1	28	new	new	PROPN
ejpam-6574	1	29	york	york	PROPN
ejpam-6574	1	30	business	business	PROPN
ejpam-6574	1	31	global	global	ADJ
ejpam-6574	1	32	on	on	ADP
ejpam-6574	1	33	weak	weak	ADJ
ejpam-6574	1	34	forms	form	NOUN
ejpam-6574	1	35	of	of	ADP
ejpam-6574	1	36	open	open	ADJ
ejpam-6574	1	37	and	and	CCONJ
ejpam-6574	1	38	closed	closed	ADJ
ejpam-6574	1	39	functions	function	NOUN
ejpam-6574	1	40	between	between	ADP
ejpam-6574	1	41	bitopological	bitopological	ADJ
ejpam-6574	1	42	spaces	space	NOUN
ejpam-6574	1	43	nipaporn	nipaporn	ADV
ejpam-6574	1	44	chutiman1	chutiman1	ADJ
ejpam-6574	1	45	,	,	PUNCT
ejpam-6574	1	46	areeyuth	areeyuth	NOUN
ejpam-6574	1	47	sama	sama	NOUN
ejpam-6574	1	48	-	-	PUNCT
ejpam-6574	1	49	ae2	ae2	PROPN
ejpam-6574	1	50	,	,	PUNCT
ejpam-6574	1	51	chawalit	chawalit	VERB
ejpam-6574	1	52	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6574	1	53	1	1	NUM
ejpam-6574	1	54	mathematics	mathematic	NOUN
ejpam-6574	1	55	and	and	CCONJ
ejpam-6574	1	56	applied	apply	VERB
ejpam-6574	1	57	mathematics	mathematics	PROPN
ejpam-6574	1	58	research	research	NOUN
ejpam-6574	1	59	unit	unit	NOUN
ejpam-6574	1	60	,	,	PUNCT
ejpam-6574	1	61	department	department	NOUN
ejpam-6574	1	62	of	of	ADP
ejpam-6574	1	63	mathematics	mathematic	NOUN
ejpam-6574	1	64	,	,	PUNCT
ejpam-6574	1	65	faculty	faculty	NOUN
ejpam-6574	1	66	of	of	ADP
ejpam-6574	1	67	science	science	NOUN
ejpam-6574	1	68	,	,	PUNCT
ejpam-6574	1	69	mahasarakham	mahasarakham	PROPN
ejpam-6574	1	70	university	university	PROPN
ejpam-6574	1	71	,	,	PUNCT
ejpam-6574	1	72	maha	maha	PROPN
ejpam-6574	1	73	sarakham	sarakham	PROPN
ejpam-6574	1	74	,	,	PUNCT
ejpam-6574	1	75	44150	44150	NUM
ejpam-6574	1	76	,	,	PUNCT
ejpam-6574	1	77	thailand	thailand	PROPN
ejpam-6574	1	78	2	2	NUM
ejpam-6574	1	79	department	department	NOUN
ejpam-6574	1	80	of	of	ADP
ejpam-6574	1	81	mathematics	mathematic	NOUN
ejpam-6574	1	82	and	and	CCONJ
ejpam-6574	1	83	computer	computer	NOUN
ejpam-6574	1	84	science	science	NOUN
ejpam-6574	1	85	,	,	PUNCT
ejpam-6574	1	86	faculty	faculty	NOUN
ejpam-6574	1	87	of	of	ADP
ejpam-6574	1	88	science	science	NOUN
ejpam-6574	1	89	and	and	CCONJ
ejpam-6574	1	90	technology	technology	NOUN
ejpam-6574	1	91	,	,	PUNCT
ejpam-6574	1	92	prince	prince	NOUN
ejpam-6574	1	93	of	of	ADP
ejpam-6574	1	94	songkla	songkla	PROPN
ejpam-6574	1	95	university	university	PROPN
ejpam-6574	1	96	,	,	PUNCT
ejpam-6574	1	97	pattani	pattani	NOUN
ejpam-6574	1	98	campus	campus	NOUN
ejpam-6574	1	99	,	,	PUNCT
ejpam-6574	1	100	pattani	pattani	NOUN
ejpam-6574	1	101	,	,	PUNCT
ejpam-6574	1	102	94000	94000	NUM
ejpam-6574	1	103	,	,	PUNCT
ejpam-6574	1	104	thailand	thailand	PROPN
ejpam-6574	1	105	abstract	abstract	PROPN
ejpam-6574	1	106	.	.	PUNCT
ejpam-6574	2	1	this	this	DET
ejpam-6574	2	2	paper	paper	NOUN
ejpam-6574	2	3	presents	present	VERB
ejpam-6574	2	4	new	new	ADJ
ejpam-6574	2	5	classes	class	NOUN
ejpam-6574	2	6	of	of	ADP
ejpam-6574	2	7	open	open	ADJ
ejpam-6574	2	8	and	and	CCONJ
ejpam-6574	2	9	closed	closed	ADJ
ejpam-6574	2	10	functions	function	NOUN
ejpam-6574	2	11	defined	define	VERB
ejpam-6574	2	12	between	between	ADP
ejpam-6574	2	13	bitopological	bitopological	ADJ
ejpam-6574	2	14	spaces	space	NOUN
ejpam-6574	2	15	,	,	PUNCT
ejpam-6574	2	16	called	call	VERB
ejpam-6574	2	17	weakly	weakly	ADJ
ejpam-6574	2	18	θ(τ1	θ(τ1	NOUN
ejpam-6574	2	19	,	,	PUNCT
ejpam-6574	2	20	τ2)b	τ2)b	ADJ
ejpam-6574	2	21	-	-	PUNCT
ejpam-6574	2	22	open	open	ADJ
ejpam-6574	2	23	functions	function	NOUN
ejpam-6574	2	24	and	and	CCONJ
ejpam-6574	2	25	weakly	weakly	ADJ
ejpam-6574	2	26	θ(τ1	θ(τ1	NOUN
ejpam-6574	2	27	,	,	PUNCT
ejpam-6574	2	28	τ2)b	τ2)b	NOUN
ejpam-6574	2	29	-	-	PUNCT
ejpam-6574	2	30	closed	close	VERB
ejpam-6574	2	31	functions	function	NOUN
ejpam-6574	2	32	.	.	PUNCT
ejpam-6574	3	1	moreover	moreover	ADV
ejpam-6574	3	2	,	,	PUNCT
ejpam-6574	3	3	several	several	ADJ
ejpam-6574	3	4	characterizations	characterization	NOUN
ejpam-6574	3	5	and	and	CCONJ
ejpam-6574	3	6	some	some	DET
ejpam-6574	3	7	properties	property	NOUN
ejpam-6574	3	8	concerning	concern	VERB
ejpam-6574	3	9	weakly	weakly	ADJ
ejpam-6574	3	10	θ(τ1	θ(τ1	NOUN
ejpam-6574	3	11	,	,	PUNCT
ejpam-6574	3	12	τ2)b	τ2)b	ADJ
ejpam-6574	3	13	-	-	PUNCT
ejpam-6574	3	14	open	open	ADJ
ejpam-6574	3	15	functions	function	NOUN
ejpam-6574	3	16	and	and	CCONJ
ejpam-6574	3	17	weakly	weakly	ADJ
ejpam-6574	3	18	θ(τ1	θ(τ1	NOUN
ejpam-6574	3	19	,	,	PUNCT
ejpam-6574	3	20	τ2)b	τ2)b	ADJ
ejpam-6574	3	21	-	-	PUNCT
ejpam-6574	3	22	closed	close	VERB
ejpam-6574	3	23	functions	function	NOUN
ejpam-6574	3	24	are	be	AUX
ejpam-6574	3	25	established	establish	VERB
ejpam-6574	3	26	.	.	PUNCT
ejpam-6574	4	1	2020	2020	NUM
ejpam-6574	4	2	mathematics	mathematics	PROPN
ejpam-6574	4	3	subject	subject	NOUN
ejpam-6574	4	4	classifications	classification	NOUN
ejpam-6574	4	5	:	:	PUNCT
ejpam-6574	4	6	54c10	54c10	NUM
ejpam-6574	4	7	,	,	PUNCT
ejpam-6574	4	8	54e55	54e55	NUM
ejpam-6574	4	9	key	key	ADJ
ejpam-6574	4	10	words	word	NOUN
ejpam-6574	4	11	and	and	CCONJ
ejpam-6574	4	12	phrases	phrase	NOUN
ejpam-6574	4	13	:	:	PUNCT
ejpam-6574	4	14	weakly	weakly	ADJ
ejpam-6574	4	15	θ(τ1	θ(τ1	NOUN
ejpam-6574	4	16	,	,	PUNCT
ejpam-6574	4	17	τ2)b	τ2)b	ADJ
ejpam-6574	4	18	-	-	PUNCT
ejpam-6574	4	19	open	open	ADJ
ejpam-6574	4	20	function	function	NOUN
ejpam-6574	4	21	,	,	PUNCT
ejpam-6574	4	22	weakly	weakly	ADJ
ejpam-6574	4	23	θ(τ1	θ(τ1	NOUN
ejpam-6574	4	24	,	,	PUNCT
ejpam-6574	4	25	τ2)b	τ2)b	NOUN
ejpam-6574	4	26	-	-	PUNCT
ejpam-6574	4	27	closed	close	VERB
ejpam-6574	4	28	function	function	NOUN
ejpam-6574	4	29	1	1	NUM
ejpam-6574	4	30	.	.	PUNCT
ejpam-6574	5	1	introduction	introduction	NOUN
ejpam-6574	5	2	topology	topology	NOUN
ejpam-6574	5	3	is	be	AUX
ejpam-6574	5	4	concerned	concern	VERB
ejpam-6574	5	5	with	with	ADP
ejpam-6574	5	6	all	all	DET
ejpam-6574	5	7	questions	question	NOUN
ejpam-6574	5	8	directly	directly	ADV
ejpam-6574	5	9	or	or	CCONJ
ejpam-6574	5	10	indirectly	indirectly	ADV
ejpam-6574	5	11	related	relate	VERB
ejpam-6574	5	12	to	to	ADP
ejpam-6574	5	13	openness	openness	NOUN
ejpam-6574	5	14	and	and	CCONJ
ejpam-6574	5	15	closedness	closedness	NOUN
ejpam-6574	5	16	.	.	PUNCT
ejpam-6574	6	1	semi	semi	ADJ
ejpam-6574	6	2	-	-	ADJ
ejpam-6574	6	3	open	open	ADJ
ejpam-6574	6	4	sets	set	NOUN
ejpam-6574	6	5	,	,	PUNCT
ejpam-6574	6	6	preopen	preopen	ADJ
ejpam-6574	6	7	sets	set	NOUN
ejpam-6574	6	8	,	,	PUNCT
ejpam-6574	6	9	α	α	NOUN
ejpam-6574	6	10	-	-	ADJ
ejpam-6574	6	11	open	open	ADJ
ejpam-6574	6	12	sets	set	NOUN
ejpam-6574	6	13	,	,	PUNCT
ejpam-6574	6	14	β	β	ADJ
ejpam-6574	6	15	-	-	ADJ
ejpam-6574	6	16	open	open	ADJ
ejpam-6574	6	17	sets	set	NOUN
ejpam-6574	6	18	,	,	PUNCT
ejpam-6574	6	19	b	b	X
ejpam-6574	6	20	-	-	PUNCT
ejpam-6574	6	21	open	open	ADJ
ejpam-6574	6	22	sets	set	NOUN
ejpam-6574	6	23	,	,	PUNCT
ejpam-6574	6	24	δ	δ	NOUN
ejpam-6574	6	25	-	-	PUNCT
ejpam-6574	6	26	open	open	ADJ
ejpam-6574	6	27	sets	set	NOUN
ejpam-6574	6	28	,	,	PUNCT
ejpam-6574	6	29	θ	θ	ADJ
ejpam-6574	6	30	-	-	ADJ
ejpam-6574	6	31	open	open	ADJ
ejpam-6574	6	32	sets	set	NOUN
ejpam-6574	6	33	and	and	CCONJ
ejpam-6574	6	34	b	b	X
ejpam-6574	6	35	-	-	PUNCT
ejpam-6574	6	36	θ	θ	ADJ
ejpam-6574	6	37	-	-	ADJ
ejpam-6574	6	38	open	open	ADJ
ejpam-6574	6	39	sets	set	NOUN
ejpam-6574	6	40	play	play	VERB
ejpam-6574	6	41	an	an	DET
ejpam-6574	6	42	important	important	ADJ
ejpam-6574	6	43	role	role	NOUN
ejpam-6574	6	44	in	in	ADP
ejpam-6574	6	45	the	the	DET
ejpam-6574	6	46	researches	research	NOUN
ejpam-6574	6	47	of	of	ADP
ejpam-6574	6	48	generalizations	generalization	NOUN
ejpam-6574	6	49	of	of	ADP
ejpam-6574	6	50	open	open	ADJ
ejpam-6574	6	51	functions	function	NOUN
ejpam-6574	6	52	and	and	CCONJ
ejpam-6574	6	53	closed	closed	ADJ
ejpam-6574	6	54	functions	function	NOUN
ejpam-6574	6	55	.	.	PUNCT
ejpam-6574	7	1	by	by	ADP
ejpam-6574	7	2	using	use	VERB
ejpam-6574	7	3	these	these	DET
ejpam-6574	7	4	sets	set	NOUN
ejpam-6574	7	5	,	,	PUNCT
ejpam-6574	7	6	many	many	ADJ
ejpam-6574	7	7	authors	author	NOUN
ejpam-6574	7	8	introduced	introduce	VERB
ejpam-6574	7	9	and	and	CCONJ
ejpam-6574	7	10	studied	study	VERB
ejpam-6574	7	11	various	various	ADJ
ejpam-6574	7	12	types	type	NOUN
ejpam-6574	7	13	of	of	ADP
ejpam-6574	7	14	open	open	ADJ
ejpam-6574	7	15	functions	function	NOUN
ejpam-6574	7	16	and	and	CCONJ
ejpam-6574	7	17	closed	closed	ADJ
ejpam-6574	7	18	functions	function	NOUN
ejpam-6574	7	19	.	.	PUNCT
ejpam-6574	8	1	in	in	ADP
ejpam-6574	8	2	1983	1983	NUM
ejpam-6574	8	3	,	,	PUNCT
ejpam-6574	8	4	rose	rise	VERB
ejpam-6574	8	5	[	[	X
ejpam-6574	8	6	1	1	X
ejpam-6574	8	7	]	]	PUNCT
ejpam-6574	8	8	introduced	introduce	VERB
ejpam-6574	8	9	and	and	CCONJ
ejpam-6574	8	10	studied	study	VERB
ejpam-6574	8	11	the	the	DET
ejpam-6574	8	12	notions	notion	NOUN
ejpam-6574	8	13	of	of	ADP
ejpam-6574	8	14	weakly	weakly	ADJ
ejpam-6574	8	15	open	open	ADJ
ejpam-6574	8	16	functions	function	NOUN
ejpam-6574	8	17	and	and	CCONJ
ejpam-6574	8	18	almost	almost	ADV
ejpam-6574	8	19	open	open	ADJ
ejpam-6574	8	20	functions	function	NOUN
ejpam-6574	8	21	.	.	PUNCT
ejpam-6574	9	1	in	in	ADP
ejpam-6574	9	2	1987	1987	NUM
ejpam-6574	9	3	,	,	PUNCT
ejpam-6574	9	4	rose	rise	VERB
ejpam-6574	9	5	and	and	CCONJ
ejpam-6574	9	6	janković	janković	ADJ
ejpam-6574	10	1	[	[	X
ejpam-6574	10	2	2	2	X
ejpam-6574	10	3	]	]	PUNCT
ejpam-6574	10	4	investigated	investigate	VERB
ejpam-6574	10	5	some	some	PRON
ejpam-6574	10	6	of	of	ADP
ejpam-6574	10	7	the	the	DET
ejpam-6574	10	8	fundamental	fundamental	ADJ
ejpam-6574	10	9	properties	property	NOUN
ejpam-6574	10	10	of	of	ADP
ejpam-6574	10	11	weakly	weakly	ADJ
ejpam-6574	10	12	closed	closed	ADJ
ejpam-6574	10	13	functions	function	NOUN
ejpam-6574	10	14	.	.	PUNCT
ejpam-6574	11	1	in	in	ADP
ejpam-6574	11	2	2006	2006	NUM
ejpam-6574	11	3	,	,	PUNCT
ejpam-6574	11	4	caldas	caldas	PROPN
ejpam-6574	11	5	et	et	PROPN
ejpam-6574	11	6	al	al	PROPN
ejpam-6574	11	7	.	.	PUNCT
ejpam-6574	12	1	[	[	X
ejpam-6574	12	2	3	3	NUM
ejpam-6574	12	3	]	]	PUNCT
ejpam-6574	12	4	introduced	introduce	VERB
ejpam-6574	12	5	and	and	CCONJ
ejpam-6574	12	6	studied	study	VERB
ejpam-6574	12	7	the	the	DET
ejpam-6574	12	8	concepts	concept	NOUN
ejpam-6574	12	9	of	of	ADP
ejpam-6574	12	10	θpreopen	θpreopen	ADJ
ejpam-6574	12	11	functions	function	NOUN
ejpam-6574	12	12	and	and	CCONJ
ejpam-6574	12	13	θ	θ	NOUN
ejpam-6574	12	14	-	-	PUNCT
ejpam-6574	12	15	preclosed	preclose	VERB
ejpam-6574	12	16	functions	function	NOUN
ejpam-6574	12	17	by	by	ADP
ejpam-6574	12	18	using	use	VERB
ejpam-6574	12	19	the	the	DET
ejpam-6574	12	20	notions	notion	NOUN
ejpam-6574	12	21	of	of	ADP
ejpam-6574	12	22	pre	pre	ADJ
ejpam-6574	12	23	-	-	ADJ
ejpam-6574	12	24	θ	θ	ADJ
ejpam-6574	12	25	-	-	ADJ
ejpam-6574	12	26	interior	interior	ADJ
ejpam-6574	12	27	and	and	CCONJ
ejpam-6574	12	28	pre	pre	ADJ
ejpam-6574	12	29	-	-	ADJ
ejpam-6574	12	30	θ	θ	ADJ
ejpam-6574	12	31	-	-	NOUN
ejpam-6574	12	32	closure	closure	NOUN
ejpam-6574	12	33	.	.	PUNCT
ejpam-6574	13	1	moreover	moreover	ADV
ejpam-6574	13	2	,	,	PUNCT
ejpam-6574	13	3	caldas	caldas	PROPN
ejpam-6574	13	4	et	et	PROPN
ejpam-6574	13	5	al	al	PROPN
ejpam-6574	13	6	.	.	PUNCT
ejpam-6574	14	1	[	[	X
ejpam-6574	14	2	4	4	X
ejpam-6574	14	3	]	]	PUNCT
ejpam-6574	14	4	introduced	introduce	VERB
ejpam-6574	14	5	and	and	CCONJ
ejpam-6574	14	6	investigated	investigate	VERB
ejpam-6574	14	7	the	the	DET
ejpam-6574	14	8	concepts	concept	NOUN
ejpam-6574	14	9	of	of	ADP
ejpam-6574	14	10	weakly	weakly	ADJ
ejpam-6574	14	11	semi	semi	ADJ
ejpam-6574	14	12	-	-	ADJ
ejpam-6574	14	13	θ	θ	ADJ
ejpam-6574	14	14	-	-	PUNCT
ejpam-6574	14	15	open	open	ADJ
ejpam-6574	14	16	functions	function	NOUN
ejpam-6574	14	17	and	and	CCONJ
ejpam-6574	14	18	weakly	weakly	ADJ
ejpam-6574	14	19	semi	semi	ADJ
ejpam-6574	14	20	-	-	ADJ
ejpam-6574	14	21	θ	θ	ADJ
ejpam-6574	14	22	-	-	PUNCT
ejpam-6574	14	23	closed	closed	ADJ
ejpam-6574	14	24	functions	function	NOUN
ejpam-6574	14	25	.	.	PUNCT
ejpam-6574	15	1	in	in	ADP
ejpam-6574	15	2	2009	2009	NUM
ejpam-6574	15	3	,	,	PUNCT
ejpam-6574	15	4	noiri	noiri	ADV
ejpam-6574	15	5	et	et	PROPN
ejpam-6574	15	6	al	al	PROPN
ejpam-6574	15	7	.	.	PUNCT
ejpam-6574	16	1	[	[	X
ejpam-6574	16	2	5	5	NUM
ejpam-6574	16	3	]	]	PUNCT
ejpam-6574	16	4	introduced	introduce	VERB
ejpam-6574	16	5	and	and	CCONJ
ejpam-6574	16	6	studied	study	VERB
ejpam-6574	16	7	two	two	NUM
ejpam-6574	16	8	new	new	ADJ
ejpam-6574	16	9	classes	class	NOUN
ejpam-6574	16	10	of	of	ADP
ejpam-6574	16	11	functions	function	NOUN
ejpam-6574	16	12	called	call	VERB
ejpam-6574	16	13	weakly	weakly	ADJ
ejpam-6574	16	14	b	b	NOUN
ejpam-6574	16	15	-	-	PUNCT
ejpam-6574	16	16	θ	θ	ADJ
ejpam-6574	16	17	-	-	PUNCT
ejpam-6574	16	18	open	open	ADJ
ejpam-6574	16	19	functions	function	NOUN
ejpam-6574	16	20	and	and	CCONJ
ejpam-6574	16	21	weakly	weakly	ADJ
ejpam-6574	16	22	b	b	NOUN
ejpam-6574	16	23	-	-	PUNCT
ejpam-6574	16	24	θ	θ	ADJ
ejpam-6574	16	25	-	-	PUNCT
ejpam-6574	16	26	open	open	ADJ
ejpam-6574	16	27	functions	function	NOUN
ejpam-6574	16	28	by	by	ADP
ejpam-6574	16	29	utilizing	utilize	VERB
ejpam-6574	16	30	the	the	DET
ejpam-6574	16	31	notions	notion	NOUN
ejpam-6574	16	32	of	of	ADP
ejpam-6574	16	33	b	b	NOUN
ejpam-6574	16	34	-	-	PUNCT
ejpam-6574	16	35	θ	θ	ADJ
ejpam-6574	16	36	-	-	ADJ
ejpam-6574	16	37	open	open	ADJ
ejpam-6574	16	38	sets	set	NOUN
ejpam-6574	16	39	and	and	CCONJ
ejpam-6574	16	40	the	the	DET
ejpam-6574	16	41	b	b	PROPN
ejpam-6574	16	42	-	-	PUNCT
ejpam-6574	16	43	θ	θ	NOUN
ejpam-6574	16	44	-	-	PUNCT
ejpam-6574	16	45	closure	closure	NOUN
ejpam-6574	16	46	operator	operator	NOUN
ejpam-6574	16	47	.	.	PUNCT
ejpam-6574	17	1	weak	weak	ADJ
ejpam-6574	17	2	b	b	X
ejpam-6574	17	3	-	-	PUNCT
ejpam-6574	17	4	θ	θ	NOUN
ejpam-6574	17	5	-	-	PUNCT
ejpam-6574	17	6	openness	openness	NOUN
ejpam-6574	17	7	(	(	PUNCT
ejpam-6574	17	8	resp	resp	NOUN
ejpam-6574	17	9	.	.	PUNCT
ejpam-6574	18	1	b	b	X
ejpam-6574	18	2	-	-	PUNCT
ejpam-6574	18	3	θ	θ	NOUN
ejpam-6574	18	4	-	-	PUNCT
ejpam-6574	18	5	closedness	closedness	NOUN
ejpam-6574	18	6	)	)	PUNCT
ejpam-6574	18	7	is	be	AUX
ejpam-6574	18	8	a	a	DET
ejpam-6574	18	9	generalization	generalization	NOUN
ejpam-6574	18	10	of	of	ADP
ejpam-6574	18	11	both	both	DET
ejpam-6574	18	12	θ	θ	NOUN
ejpam-6574	18	13	-	-	PUNCT
ejpam-6574	18	14	preopenness	preopenness	NOUN
ejpam-6574	18	15	∗corresponding	∗corresponde	VERB
ejpam-6574	18	16	author	author	NOUN
ejpam-6574	18	17	.	.	PUNCT
ejpam-6574	19	1	doi	doi	NOUN
ejpam-6574	19	2	:	:	PUNCT
ejpam-6574	19	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6574	https://doi.org/10.29020/nybg.ejpam.v18i3.6574	NOUN
ejpam-6574	19	4	email	email	NOUN
ejpam-6574	19	5	addresses	address	NOUN
ejpam-6574	19	6	:	:	PUNCT
ejpam-6574	19	7	nipaporn.c@msu.ac.th	nipaporn.c@msu.ac.th	PROPN
ejpam-6574	19	8	(	(	PUNCT
ejpam-6574	19	9	n.	n.	NOUN
ejpam-6574	19	10	chutiman	chutiman	NOUN
ejpam-6574	19	11	)	)	PUNCT
ejpam-6574	19	12	,	,	PUNCT
ejpam-6574	19	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6574	19	14	(	(	PUNCT
ejpam-6574	19	15	a.	a.	PROPN
ejpam-6574	19	16	sama	sama	PROPN
ejpam-6574	19	17	-	-	PUNCT
ejpam-6574	19	18	ae	ae	PROPN
ejpam-6574	19	19	)	)	PUNCT
ejpam-6574	19	20	,	,	PUNCT
ejpam-6574	19	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6574	19	22	(	(	PUNCT
ejpam-6574	19	23	c.	c.	PROPN
ejpam-6574	19	24	boonpok	boonpok	PROPN
ejpam-6574	19	25	)	)	PUNCT
ejpam-6574	19	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6574	20	1	1	1	NUM
ejpam-6574	20	2	copyright	copyright	NOUN
ejpam-6574	20	3	:	:	PUNCT
ejpam-6574	20	4	©	©	PROPN
ejpam-6574	20	5	2025	2025	NUM
ejpam-6574	20	6	the	the	DET
ejpam-6574	20	7	author(s	author(s	NOUN
ejpam-6574	20	8	)	)	PUNCT
ejpam-6574	20	9	.	.	PUNCT
ejpam-6574	21	1	(	(	PUNCT
ejpam-6574	21	2	cc	cc	NOUN
ejpam-6574	21	3	by	by	ADP
ejpam-6574	21	4	-	-	PUNCT
ejpam-6574	21	5	nc	nc	PROPN
ejpam-6574	21	6	4.0	4.0	NUM
ejpam-6574	21	7	)	)	PUNCT
ejpam-6574	21	8	n.	n.	NOUN
ejpam-6574	21	9	chutiman	chutiman	NOUN
ejpam-6574	21	10	,	,	PUNCT
ejpam-6574	21	11	a.	a.	PROPN
ejpam-6574	21	12	sama	sama	PROPN
ejpam-6574	21	13	-	-	PUNCT
ejpam-6574	21	14	ae	ae	PROPN
ejpam-6574	21	15	,	,	PUNCT
ejpam-6574	21	16	c.	c.	PROPN
ejpam-6574	21	17	boonpok	boonpok	PROPN
ejpam-6574	21	18	/	/	SYM
ejpam-6574	21	19	eur	eur	PROPN
ejpam-6574	21	20	.	.	PUNCT
ejpam-6574	22	1	j.	j.	PROPN
ejpam-6574	22	2	pure	pure	PROPN
ejpam-6574	22	3	appl	appl	PROPN
ejpam-6574	22	4	.	.	PROPN
ejpam-6574	22	5	math	math	PROPN
ejpam-6574	22	6	,	,	PUNCT
ejpam-6574	22	7	18	18	NUM
ejpam-6574	22	8	(	(	PUNCT
ejpam-6574	22	9	3	3	NUM
ejpam-6574	22	10	)	)	PUNCT
ejpam-6574	22	11	(	(	PUNCT
ejpam-6574	22	12	2025	2025	NUM
ejpam-6574	22	13	)	)	PUNCT
ejpam-6574	22	14	,	,	PUNCT
ejpam-6574	22	15	6574	6574	NUM
ejpam-6574	22	16	2	2	NUM
ejpam-6574	22	17	of	of	ADP
ejpam-6574	22	18	10	10	NUM
ejpam-6574	22	19	and	and	CCONJ
ejpam-6574	22	20	weak	weak	ADJ
ejpam-6574	22	21	semi	semi	ADJ
ejpam-6574	22	22	-	-	ADJ
ejpam-6574	22	23	θ	θ	ADJ
ejpam-6574	22	24	-	-	PUNCT
ejpam-6574	22	25	openness	openness	NOUN
ejpam-6574	22	26	(	(	PUNCT
ejpam-6574	22	27	resp	resp	NOUN
ejpam-6574	22	28	.	.	PUNCT
ejpam-6574	23	1	θ	θ	NOUN
ejpam-6574	23	2	-	-	PUNCT
ejpam-6574	23	3	preclosedness	preclosedness	NOUN
ejpam-6574	23	4	and	and	CCONJ
ejpam-6574	23	5	weak	weak	ADJ
ejpam-6574	23	6	semi	semi	ADJ
ejpam-6574	23	7	-	-	ADJ
ejpam-6574	23	8	θ	θ	ADJ
ejpam-6574	23	9	-	-	PUNCT
ejpam-6574	23	10	closedness	closedness	NOUN
ejpam-6574	23	11	)	)	PUNCT
ejpam-6574	23	12	.	.	PUNCT
ejpam-6574	24	1	quite	quite	ADV
ejpam-6574	24	2	recently	recently	ADV
ejpam-6574	24	3	,	,	PUNCT
ejpam-6574	24	4	klanarong	klanarong	NOUN
ejpam-6574	24	5	and	and	CCONJ
ejpam-6574	24	6	boonpok	boonpok	PROPN
ejpam-6574	24	7	[	[	X
ejpam-6574	24	8	6	6	NUM
ejpam-6574	24	9	]	]	PUNCT
ejpam-6574	24	10	studied	study	VERB
ejpam-6574	24	11	the	the	DET
ejpam-6574	24	12	notions	notion	NOUN
ejpam-6574	24	13	of	of	ADP
ejpam-6574	24	14	weakly	weakly	ADJ
ejpam-6574	24	15	s(λ	s(λ	NOUN
ejpam-6574	24	16	,	,	PUNCT
ejpam-6574	24	17	p)-open	p)-open	NOUN
ejpam-6574	24	18	functions	function	NOUN
ejpam-6574	24	19	and	and	CCONJ
ejpam-6574	24	20	weakly	weakly	ADJ
ejpam-6574	24	21	s(λ	s(λ	PROPN
ejpam-6574	24	22	,	,	PUNCT
ejpam-6574	24	23	p)-closed	p)-close	VERB
ejpam-6574	24	24	functions	function	NOUN
ejpam-6574	24	25	by	by	ADP
ejpam-6574	24	26	utilizing	utilize	VERB
ejpam-6574	24	27	s(λ	s(λ	NOUN
ejpam-6574	24	28	,	,	PUNCT
ejpam-6574	24	29	p)-open	p)-open	VERB
ejpam-6574	24	30	sets	set	NOUN
ejpam-6574	24	31	and	and	CCONJ
ejpam-6574	24	32	the	the	DET
ejpam-6574	24	33	s(λ	s(λ	PROPN
ejpam-6574	24	34	,	,	PUNCT
ejpam-6574	24	35	p)-closure	p)-closure	NOUN
ejpam-6574	24	36	operator	operator	NOUN
ejpam-6574	24	37	.	.	PUNCT
ejpam-6574	25	1	on	on	ADP
ejpam-6574	25	2	the	the	DET
ejpam-6574	25	3	other	other	ADJ
ejpam-6574	25	4	hand	hand	NOUN
ejpam-6574	25	5	,	,	PUNCT
ejpam-6574	25	6	the	the	DET
ejpam-6574	25	7	present	present	ADJ
ejpam-6574	25	8	authors	author	NOUN
ejpam-6574	25	9	introduced	introduce	VERB
ejpam-6574	25	10	and	and	CCONJ
ejpam-6574	25	11	studied	study	VERB
ejpam-6574	25	12	the	the	DET
ejpam-6574	25	13	concepts	concept	NOUN
ejpam-6574	25	14	of	of	ADP
ejpam-6574	25	15	θp(λ	θp(λ	NOUN
ejpam-6574	25	16	,	,	PUNCT
ejpam-6574	25	17	p)-open	p)-open	VERB
ejpam-6574	25	18	functions	function	NOUN
ejpam-6574	25	19	[	[	X
ejpam-6574	25	20	7	7	NUM
ejpam-6574	25	21	]	]	NUM
ejpam-6574	25	22	,	,	PUNCT
ejpam-6574	25	23	θp(λ	θp(λ	NOUN
ejpam-6574	25	24	,	,	PUNCT
ejpam-6574	25	25	p)-closed	p)-close	VERB
ejpam-6574	25	26	functions	function	NOUN
ejpam-6574	25	27	[	[	X
ejpam-6574	25	28	7	7	NUM
ejpam-6574	25	29	]	]	PUNCT
ejpam-6574	25	30	,	,	PUNCT
ejpam-6574	25	31	semi-(i	semi-(i	PROPN
ejpam-6574	25	32	,	,	PUNCT
ejpam-6574	25	33	j	j	NOUN
ejpam-6574	25	34	)	)	PUNCT
ejpam-6574	25	35	-open	-open	NOUN
ejpam-6574	25	36	functions	function	NOUN
ejpam-6574	25	37	[	[	X
ejpam-6574	25	38	8	8	NUM
ejpam-6574	25	39	]	]	PUNCT
ejpam-6574	25	40	,	,	PUNCT
ejpam-6574	25	41	semi-(i	semi-(i	PROPN
ejpam-6574	25	42	,	,	PUNCT
ejpam-6574	25	43	j	j	PROPN
ejpam-6574	25	44	)	)	PUNCT
ejpam-6574	25	45	-closed	-close	VERB
ejpam-6574	25	46	functions	function	NOUN
ejpam-6574	25	47	[	[	X
ejpam-6574	25	48	8	8	NUM
ejpam-6574	25	49	]	]	PUNCT
ejpam-6574	25	50	,	,	PUNCT
ejpam-6574	25	51	weakly	weakly	ADJ
ejpam-6574	25	52	δ(λ	δ(λ	PROPN
ejpam-6574	25	53	,	,	PUNCT
ejpam-6574	25	54	p)-open	p)-open	VERB
ejpam-6574	25	55	functions	function	NOUN
ejpam-6574	25	56	[	[	X
ejpam-6574	25	57	9	9	NUM
ejpam-6574	25	58	]	]	PUNCT
ejpam-6574	25	59	,	,	PUNCT
ejpam-6574	25	60	weakly	weakly	ADJ
ejpam-6574	25	61	δ(λ	δ(λ	PROPN
ejpam-6574	25	62	,	,	PUNCT
ejpam-6574	25	63	p)-closed	p)-close	VERB
ejpam-6574	25	64	functions	function	NOUN
ejpam-6574	25	65	[	[	X
ejpam-6574	25	66	10	10	NUM
ejpam-6574	25	67	]	]	PUNCT
ejpam-6574	25	68	,	,	PUNCT
ejpam-6574	25	69	weakly	weakly	ADJ
ejpam-6574	25	70	θs(λ	θs(λ	NOUN
ejpam-6574	25	71	,	,	PUNCT
ejpam-6574	25	72	p)-open	p)-open	VERB
ejpam-6574	25	73	functions	function	NOUN
ejpam-6574	25	74	[	[	X
ejpam-6574	25	75	11	11	NUM
ejpam-6574	25	76	]	]	PUNCT
ejpam-6574	25	77	,	,	PUNCT
ejpam-6574	25	78	weakly	weakly	ADJ
ejpam-6574	25	79	θs(λ	θs(λ	NOUN
ejpam-6574	25	80	,	,	PUNCT
ejpam-6574	25	81	p)-closed	p)-close	VERB
ejpam-6574	25	82	functions	function	NOUN
ejpam-6574	25	83	[	[	X
ejpam-6574	25	84	11	11	NUM
ejpam-6574	25	85	]	]	PUNCT
ejpam-6574	25	86	and	and	CCONJ
ejpam-6574	25	87	weakly	weakly	ADJ
ejpam-6574	25	88	b(λ	b(λ	NOUN
ejpam-6574	25	89	,	,	PUNCT
ejpam-6574	25	90	p)-open	p)-open	ADJ
ejpam-6574	25	91	functions	function	NOUN
ejpam-6574	25	92	[	[	X
ejpam-6574	25	93	12	12	NUM
ejpam-6574	25	94	]	]	PUNCT
ejpam-6574	25	95	.	.	PUNCT
ejpam-6574	26	1	in	in	ADP
ejpam-6574	26	2	this	this	DET
ejpam-6574	26	3	paper	paper	NOUN
ejpam-6574	26	4	,	,	PUNCT
ejpam-6574	26	5	we	we	PRON
ejpam-6574	26	6	introduce	introduce	VERB
ejpam-6574	26	7	the	the	DET
ejpam-6574	26	8	notions	notion	NOUN
ejpam-6574	26	9	of	of	ADP
ejpam-6574	26	10	weakly	weakly	ADJ
ejpam-6574	26	11	θ(τ1	θ(τ1	NOUN
ejpam-6574	26	12	,	,	PUNCT
ejpam-6574	26	13	τ2)b	τ2)b	ADJ
ejpam-6574	26	14	-	-	PUNCT
ejpam-6574	26	15	open	open	ADJ
ejpam-6574	26	16	functions	function	NOUN
ejpam-6574	26	17	and	and	CCONJ
ejpam-6574	26	18	θ(τ1	θ(τ1	NOUN
ejpam-6574	26	19	,	,	PUNCT
ejpam-6574	26	20	τ2)b	τ2)b	NOUN
ejpam-6574	26	21	-	-	PUNCT
ejpam-6574	26	22	closed	close	VERB
ejpam-6574	26	23	functions	function	NOUN
ejpam-6574	26	24	.	.	PUNCT
ejpam-6574	27	1	furthermore	furthermore	ADV
ejpam-6574	27	2	,	,	PUNCT
ejpam-6574	27	3	several	several	ADJ
ejpam-6574	27	4	characterizations	characterization	NOUN
ejpam-6574	27	5	of	of	ADP
ejpam-6574	27	6	weakly	weakly	ADJ
ejpam-6574	27	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	27	8	,	,	PUNCT
ejpam-6574	27	9	τ2)b	τ2)b	ADJ
ejpam-6574	27	10	-	-	PUNCT
ejpam-6574	27	11	open	open	ADJ
ejpam-6574	27	12	functions	function	NOUN
ejpam-6574	27	13	and	and	CCONJ
ejpam-6574	27	14	θ(τ1	θ(τ1	NOUN
ejpam-6574	27	15	,	,	PUNCT
ejpam-6574	27	16	τ2)b	τ2)b	ADJ
ejpam-6574	27	17	-	-	PUNCT
ejpam-6574	27	18	closed	close	VERB
ejpam-6574	27	19	functions	function	NOUN
ejpam-6574	27	20	are	be	AUX
ejpam-6574	27	21	investigated	investigate	VERB
ejpam-6574	27	22	.	.	PUNCT
ejpam-6574	28	1	2	2	X
ejpam-6574	28	2	.	.	X
ejpam-6574	28	3	preliminaries	preliminary	NOUN
ejpam-6574	28	4	throughout	throughout	ADP
ejpam-6574	28	5	the	the	DET
ejpam-6574	28	6	present	present	ADJ
ejpam-6574	28	7	paper	paper	NOUN
ejpam-6574	28	8	,	,	PUNCT
ejpam-6574	28	9	spaces	space	NOUN
ejpam-6574	28	10	(	(	PUNCT
ejpam-6574	28	11	x	x	NOUN
ejpam-6574	28	12	,	,	PUNCT
ejpam-6574	28	13	τ1	τ1	NOUN
ejpam-6574	28	14	,	,	PUNCT
ejpam-6574	28	15	τ2	τ2	NOUN
ejpam-6574	28	16	)	)	PUNCT
ejpam-6574	28	17	and	and	CCONJ
ejpam-6574	28	18	(	(	PUNCT
ejpam-6574	28	19	y	y	PROPN
ejpam-6574	28	20	,	,	PUNCT
ejpam-6574	28	21	σ1	σ1	PROPN
ejpam-6574	28	22	,	,	PUNCT
ejpam-6574	28	23	σ2	σ2	NOUN
ejpam-6574	28	24	)	)	PUNCT
ejpam-6574	28	25	(	(	PUNCT
ejpam-6574	28	26	or	or	CCONJ
ejpam-6574	28	27	simply	simply	ADV
ejpam-6574	28	28	x	x	X
ejpam-6574	28	29	and	and	CCONJ
ejpam-6574	28	30	y	y	PROPN
ejpam-6574	28	31	)	)	PUNCT
ejpam-6574	28	32	always	always	ADV
ejpam-6574	28	33	mean	mean	VERB
ejpam-6574	28	34	bitopological	bitopological	ADJ
ejpam-6574	28	35	spaces	space	NOUN
ejpam-6574	28	36	on	on	ADP
ejpam-6574	28	37	which	which	PRON
ejpam-6574	28	38	no	no	DET
ejpam-6574	28	39	separation	separation	NOUN
ejpam-6574	28	40	axioms	axiom	NOUN
ejpam-6574	28	41	are	be	AUX
ejpam-6574	28	42	assumed	assume	VERB
ejpam-6574	28	43	unless	unless	SCONJ
ejpam-6574	28	44	explicitly	explicitly	ADV
ejpam-6574	28	45	stated	state	VERB
ejpam-6574	28	46	.	.	PUNCT
ejpam-6574	29	1	let	let	VERB
ejpam-6574	29	2	a	a	DET
ejpam-6574	29	3	be	be	AUX
ejpam-6574	29	4	a	a	DET
ejpam-6574	29	5	subset	subset	NOUN
ejpam-6574	29	6	of	of	ADP
ejpam-6574	29	7	a	a	DET
ejpam-6574	29	8	bitopological	bitopological	ADJ
ejpam-6574	29	9	space	space	NOUN
ejpam-6574	29	10	(	(	PUNCT
ejpam-6574	29	11	x	x	NOUN
ejpam-6574	29	12	,	,	PUNCT
ejpam-6574	29	13	τ1	τ1	NOUN
ejpam-6574	29	14	,	,	PUNCT
ejpam-6574	29	15	τ2	τ2	NOUN
ejpam-6574	29	16	)	)	PUNCT
ejpam-6574	29	17	.	.	PUNCT
ejpam-6574	30	1	the	the	DET
ejpam-6574	30	2	closure	closure	NOUN
ejpam-6574	30	3	of	of	ADP
ejpam-6574	30	4	a	a	PRON
ejpam-6574	30	5	and	and	CCONJ
ejpam-6574	30	6	the	the	DET
ejpam-6574	30	7	interior	interior	NOUN
ejpam-6574	30	8	of	of	ADP
ejpam-6574	30	9	a	a	PRON
ejpam-6574	30	10	with	with	ADP
ejpam-6574	30	11	respect	respect	NOUN
ejpam-6574	30	12	to	to	ADP
ejpam-6574	30	13	τi	τi	PROPN
ejpam-6574	30	14	are	be	AUX
ejpam-6574	30	15	denoted	denote	VERB
ejpam-6574	30	16	by	by	ADP
ejpam-6574	30	17	τi	τi	NOUN
ejpam-6574	30	18	-	-	PUNCT
ejpam-6574	30	19	cl(a	cl(a	NUM
ejpam-6574	30	20	)	)	PUNCT
ejpam-6574	30	21	and	and	CCONJ
ejpam-6574	30	22	τi	τi	NOUN
ejpam-6574	30	23	-	-	PUNCT
ejpam-6574	30	24	int(a	int(a	NOUN
ejpam-6574	30	25	)	)	PUNCT
ejpam-6574	30	26	,	,	PUNCT
ejpam-6574	30	27	respectively	respectively	ADV
ejpam-6574	30	28	,	,	PUNCT
ejpam-6574	30	29	for	for	ADP
ejpam-6574	30	30	i	i	PROPN
ejpam-6574	30	31	=	=	SYM
ejpam-6574	30	32	1	1	NUM
ejpam-6574	30	33	,	,	PUNCT
ejpam-6574	30	34	2	2	NUM
ejpam-6574	30	35	.	.	X
ejpam-6574	30	36	a	a	DET
ejpam-6574	30	37	subset	subset	NOUN
ejpam-6574	30	38	a	a	PRON
ejpam-6574	30	39	of	of	ADP
ejpam-6574	30	40	a	a	DET
ejpam-6574	30	41	bitopological	bitopological	ADJ
ejpam-6574	30	42	space	space	NOUN
ejpam-6574	30	43	(	(	PUNCT
ejpam-6574	30	44	x	x	NOUN
ejpam-6574	30	45	,	,	PUNCT
ejpam-6574	30	46	τ1	τ1	NOUN
ejpam-6574	30	47	,	,	PUNCT
ejpam-6574	30	48	τ2	τ2	NOUN
ejpam-6574	30	49	)	)	PUNCT
ejpam-6574	30	50	is	be	AUX
ejpam-6574	30	51	called	call	VERB
ejpam-6574	30	52	τ1τ2	τ1τ2	VERB
ejpam-6574	30	53	-	-	ADJ
ejpam-6574	30	54	closed	closed	ADJ
ejpam-6574	30	55	[	[	X
ejpam-6574	30	56	13	13	NUM
ejpam-6574	30	57	]	]	PUNCT
ejpam-6574	30	58	if	if	SCONJ
ejpam-6574	30	59	a	a	DET
ejpam-6574	30	60	=	=	NOUN
ejpam-6574	30	61	τ1	τ1	NOUN
ejpam-6574	30	62	-	-	PUNCT
ejpam-6574	30	63	cl(τ2	cl(τ2	NOUN
ejpam-6574	30	64	-	-	PUNCT
ejpam-6574	30	65	cl(a	cl(a	NUM
ejpam-6574	30	66	)	)	PUNCT
ejpam-6574	30	67	)	)	PUNCT
ejpam-6574	30	68	.	.	PUNCT
ejpam-6574	31	1	the	the	DET
ejpam-6574	31	2	complement	complement	NOUN
ejpam-6574	31	3	of	of	ADP
ejpam-6574	31	4	a	a	DET
ejpam-6574	31	5	τ1τ2	τ1τ2	ADJ
ejpam-6574	31	6	-	-	ADJ
ejpam-6574	31	7	closed	closed	ADJ
ejpam-6574	31	8	set	set	NOUN
ejpam-6574	31	9	is	be	AUX
ejpam-6574	31	10	called	call	VERB
ejpam-6574	31	11	τ1τ2	τ1τ2	NOUN
ejpam-6574	31	12	-	-	ADJ
ejpam-6574	31	13	open	open	ADJ
ejpam-6574	31	14	.	.	PUNCT
ejpam-6574	32	1	let	let	VERB
ejpam-6574	32	2	a	a	DET
ejpam-6574	32	3	be	be	AUX
ejpam-6574	32	4	a	a	DET
ejpam-6574	32	5	subset	subset	NOUN
ejpam-6574	32	6	of	of	ADP
ejpam-6574	32	7	a	a	DET
ejpam-6574	32	8	bitopological	bitopological	ADJ
ejpam-6574	32	9	space	space	NOUN
ejpam-6574	32	10	(	(	PUNCT
ejpam-6574	32	11	x	x	NOUN
ejpam-6574	32	12	,	,	PUNCT
ejpam-6574	32	13	τ1	τ1	NOUN
ejpam-6574	32	14	,	,	PUNCT
ejpam-6574	32	15	τ2	τ2	NOUN
ejpam-6574	32	16	)	)	PUNCT
ejpam-6574	32	17	.	.	PUNCT
ejpam-6574	33	1	the	the	DET
ejpam-6574	33	2	intersection	intersection	NOUN
ejpam-6574	33	3	of	of	ADP
ejpam-6574	33	4	all	all	DET
ejpam-6574	33	5	τ1τ2	τ1τ2	ADJ
ejpam-6574	33	6	-	-	ADJ
ejpam-6574	33	7	closed	closed	ADJ
ejpam-6574	33	8	sets	set	NOUN
ejpam-6574	33	9	of	of	ADP
ejpam-6574	33	10	x	x	PUNCT
ejpam-6574	33	11	containing	contain	VERB
ejpam-6574	33	12	a	a	PRON
ejpam-6574	33	13	is	be	AUX
ejpam-6574	33	14	called	call	VERB
ejpam-6574	33	15	the	the	DET
ejpam-6574	33	16	τ1τ2	τ1τ2	NOUN
ejpam-6574	33	17	-	-	NOUN
ejpam-6574	33	18	closure	closure	NOUN
ejpam-6574	33	19	[	[	X
ejpam-6574	33	20	13	13	NUM
ejpam-6574	33	21	]	]	PUNCT
ejpam-6574	33	22	of	of	ADP
ejpam-6574	33	23	a	a	PRON
ejpam-6574	33	24	and	and	CCONJ
ejpam-6574	33	25	is	be	AUX
ejpam-6574	33	26	denoted	denote	VERB
ejpam-6574	33	27	by	by	ADP
ejpam-6574	33	28	τ1τ2	τ1τ2	NOUN
ejpam-6574	33	29	-	-	NUM
ejpam-6574	33	30	cl(a	cl(a	NUM
ejpam-6574	33	31	)	)	PUNCT
ejpam-6574	33	32	.	.	PUNCT
ejpam-6574	34	1	the	the	DET
ejpam-6574	34	2	union	union	NOUN
ejpam-6574	34	3	of	of	ADP
ejpam-6574	34	4	all	all	DET
ejpam-6574	34	5	τ1τ2	τ1τ2	ADJ
ejpam-6574	34	6	-	-	ADJ
ejpam-6574	34	7	open	open	ADJ
ejpam-6574	34	8	sets	set	NOUN
ejpam-6574	34	9	of	of	ADP
ejpam-6574	34	10	x	x	PUNCT
ejpam-6574	34	11	contained	contain	VERB
ejpam-6574	34	12	in	in	ADP
ejpam-6574	34	13	a	a	PRON
ejpam-6574	34	14	is	be	AUX
ejpam-6574	34	15	called	call	VERB
ejpam-6574	34	16	the	the	DET
ejpam-6574	34	17	τ1τ2	τ1τ2	NOUN
ejpam-6574	34	18	-	-	ADJ
ejpam-6574	34	19	interior	interior	ADJ
ejpam-6574	34	20	[	[	X
ejpam-6574	34	21	13	13	NUM
ejpam-6574	34	22	]	]	PUNCT
ejpam-6574	34	23	of	of	ADP
ejpam-6574	34	24	a	a	PRON
ejpam-6574	34	25	and	and	CCONJ
ejpam-6574	34	26	is	be	AUX
ejpam-6574	34	27	denoted	denote	VERB
ejpam-6574	34	28	by	by	ADP
ejpam-6574	34	29	τ1τ2	τ1τ2	NOUN
ejpam-6574	34	30	-	-	ADJ
ejpam-6574	34	31	int(a	int(a	NOUN
ejpam-6574	34	32	)	)	PUNCT
ejpam-6574	34	33	.	.	PUNCT
ejpam-6574	35	1	lemma	lemma	PROPN
ejpam-6574	35	2	1	1	NUM
ejpam-6574	35	3	.	.	PUNCT
ejpam-6574	36	1	[	[	X
ejpam-6574	36	2	13	13	NUM
ejpam-6574	36	3	]	]	PUNCT
ejpam-6574	36	4	let	let	VERB
ejpam-6574	36	5	a	a	PRON
ejpam-6574	36	6	and	and	CCONJ
ejpam-6574	36	7	b	b	NOUN
ejpam-6574	36	8	be	be	AUX
ejpam-6574	36	9	subsets	subset	NOUN
ejpam-6574	36	10	of	of	ADP
ejpam-6574	36	11	a	a	DET
ejpam-6574	36	12	bitopological	bitopological	ADJ
ejpam-6574	36	13	space	space	NOUN
ejpam-6574	36	14	(	(	PUNCT
ejpam-6574	36	15	x	x	NOUN
ejpam-6574	36	16	,	,	PUNCT
ejpam-6574	36	17	τ1	τ1	NOUN
ejpam-6574	36	18	,	,	PUNCT
ejpam-6574	36	19	τ2	τ2	NOUN
ejpam-6574	36	20	)	)	PUNCT
ejpam-6574	36	21	.	.	PUNCT
ejpam-6574	37	1	for	for	ADP
ejpam-6574	37	2	the	the	DET
ejpam-6574	37	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6574	37	4	,	,	PUNCT
ejpam-6574	37	5	the	the	DET
ejpam-6574	37	6	following	follow	VERB
ejpam-6574	37	7	properties	property	NOUN
ejpam-6574	37	8	hold	hold	VERB
ejpam-6574	37	9	:	:	PUNCT
ejpam-6574	37	10	(	(	PUNCT
ejpam-6574	37	11	1	1	X
ejpam-6574	37	12	)	)	PUNCT
ejpam-6574	37	13	a	a	DET
ejpam-6574	37	14	⊆	⊆	NUM
ejpam-6574	37	15	τ1τ2	τ1τ2	NOUN
ejpam-6574	37	16	-	-	NUM
ejpam-6574	37	17	cl(a	cl(a	NUM
ejpam-6574	37	18	)	)	PUNCT
ejpam-6574	37	19	and	and	CCONJ
ejpam-6574	37	20	τ1τ2	τ1τ2	NOUN
ejpam-6574	37	21	-	-	ADJ
ejpam-6574	37	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	37	23	-	-	PUNCT
ejpam-6574	37	24	cl(a	cl(a	NUM
ejpam-6574	37	25	)	)	PUNCT
ejpam-6574	37	26	)	)	PUNCT
ejpam-6574	38	1	=	=	PUNCT
ejpam-6574	38	2	τ1τ2	τ1τ2	NOUN
ejpam-6574	38	3	-	-	NUM
ejpam-6574	38	4	cl(a	cl(a	NUM
ejpam-6574	38	5	)	)	PUNCT
ejpam-6574	38	6	.	.	PUNCT
ejpam-6574	39	1	(	(	PUNCT
ejpam-6574	39	2	2	2	X
ejpam-6574	39	3	)	)	PUNCT
ejpam-6574	39	4	if	if	SCONJ
ejpam-6574	39	5	a	a	DET
ejpam-6574	39	6	⊆	⊆	NUM
ejpam-6574	39	7	b	b	NOUN
ejpam-6574	39	8	,	,	PUNCT
ejpam-6574	39	9	then	then	ADV
ejpam-6574	39	10	τ1τ2	τ1τ2	NOUN
ejpam-6574	39	11	-	-	NUM
ejpam-6574	39	12	cl(a	cl(a	NUM
ejpam-6574	39	13	)	)	PUNCT
ejpam-6574	39	14	⊆	⊆	NUM
ejpam-6574	39	15	τ1τ2	τ1τ2	NOUN
ejpam-6574	39	16	-	-	NOUN
ejpam-6574	39	17	cl(b	cl(b	NOUN
ejpam-6574	39	18	)	)	PUNCT
ejpam-6574	39	19	.	.	PUNCT
ejpam-6574	40	1	(	(	PUNCT
ejpam-6574	40	2	3	3	X
ejpam-6574	40	3	)	)	PUNCT
ejpam-6574	40	4	τ1τ2	τ1τ2	NOUN
ejpam-6574	40	5	-	-	NUM
ejpam-6574	40	6	cl(a	cl(a	NUM
ejpam-6574	40	7	)	)	PUNCT
ejpam-6574	40	8	is	be	AUX
ejpam-6574	40	9	τ1τ2	τ1τ2	NOUN
ejpam-6574	40	10	-	-	ADJ
ejpam-6574	40	11	closed	closed	ADJ
ejpam-6574	40	12	.	.	PUNCT
ejpam-6574	41	1	(	(	PUNCT
ejpam-6574	41	2	4	4	X
ejpam-6574	41	3	)	)	PUNCT
ejpam-6574	41	4	a	a	PRON
ejpam-6574	41	5	is	be	AUX
ejpam-6574	41	6	τ1τ2	τ1τ2	NOUN
ejpam-6574	41	7	-	-	ADJ
ejpam-6574	41	8	closed	closed	ADJ
ejpam-6574	41	9	if	if	SCONJ
ejpam-6574	41	10	and	and	CCONJ
ejpam-6574	41	11	only	only	ADV
ejpam-6574	41	12	if	if	SCONJ
ejpam-6574	41	13	a	a	DET
ejpam-6574	41	14	=	=	PUNCT
ejpam-6574	41	15	τ1τ2	τ1τ2	NOUN
ejpam-6574	41	16	-	-	NUM
ejpam-6574	41	17	cl(a	cl(a	NUM
ejpam-6574	41	18	)	)	PUNCT
ejpam-6574	41	19	.	.	PUNCT
ejpam-6574	42	1	(	(	PUNCT
ejpam-6574	42	2	5	5	X
ejpam-6574	42	3	)	)	PUNCT
ejpam-6574	42	4	τ1τ2	τ1τ2	NOUN
ejpam-6574	42	5	-	-	NOUN
ejpam-6574	42	6	cl(x	cl(x	X
ejpam-6574	42	7	−a	−a	NOUN
ejpam-6574	42	8	)	)	PUNCT
ejpam-6574	43	1	=	=	PUNCT
ejpam-6574	43	2	x	x	X
ejpam-6574	44	1	−	−	ADP
ejpam-6574	44	2	τ1τ2	τ1τ2	NOUN
ejpam-6574	44	3	-	-	PUNCT
ejpam-6574	44	4	int(a	int(a	NOUN
ejpam-6574	44	5	)	)	PUNCT
ejpam-6574	44	6	.	.	PUNCT
ejpam-6574	45	1	a	a	DET
ejpam-6574	45	2	subset	subset	NOUN
ejpam-6574	45	3	a	a	PRON
ejpam-6574	45	4	of	of	ADP
ejpam-6574	45	5	a	a	DET
ejpam-6574	45	6	bitopological	bitopological	ADJ
ejpam-6574	45	7	space	space	NOUN
ejpam-6574	45	8	(	(	PUNCT
ejpam-6574	45	9	x	x	NOUN
ejpam-6574	45	10	,	,	PUNCT
ejpam-6574	45	11	τ1	τ1	NOUN
ejpam-6574	45	12	,	,	PUNCT
ejpam-6574	45	13	τ2	τ2	NOUN
ejpam-6574	45	14	)	)	PUNCT
ejpam-6574	45	15	is	be	AUX
ejpam-6574	45	16	said	say	VERB
ejpam-6574	45	17	to	to	PART
ejpam-6574	45	18	be	be	AUX
ejpam-6574	45	19	(	(	PUNCT
ejpam-6574	45	20	τ1	τ1	NOUN
ejpam-6574	45	21	,	,	PUNCT
ejpam-6574	45	22	τ2)r	τ2)r	NOUN
ejpam-6574	45	23	-	-	PUNCT
ejpam-6574	45	24	open	open	NOUN
ejpam-6574	46	1	[	[	X
ejpam-6574	46	2	14	14	NUM
ejpam-6574	46	3	]	]	X
ejpam-6574	46	4	(	(	PUNCT
ejpam-6574	46	5	resp	resp	NOUN
ejpam-6574	46	6	.	.	PUNCT
ejpam-6574	47	1	(	(	PUNCT
ejpam-6574	47	2	τ1	τ1	NOUN
ejpam-6574	47	3	,	,	PUNCT
ejpam-6574	47	4	τ2)s	τ2)s	NOUN
ejpam-6574	47	5	-	-	PUNCT
ejpam-6574	47	6	open	open	ADJ
ejpam-6574	47	7	[	[	X
ejpam-6574	47	8	15	15	NUM
ejpam-6574	47	9	]	]	PUNCT
ejpam-6574	47	10	,	,	PUNCT
ejpam-6574	47	11	(	(	PUNCT
ejpam-6574	47	12	τ1	τ1	NOUN
ejpam-6574	47	13	,	,	PUNCT
ejpam-6574	47	14	τ2)p	τ2)p	NOUN
ejpam-6574	47	15	-	-	ADJ
ejpam-6574	47	16	open	open	ADJ
ejpam-6574	48	1	[	[	X
ejpam-6574	48	2	15	15	NUM
ejpam-6574	48	3	]	]	PUNCT
ejpam-6574	48	4	,	,	PUNCT
ejpam-6574	48	5	(	(	PUNCT
ejpam-6574	48	6	τ1	τ1	NOUN
ejpam-6574	48	7	,	,	PUNCT
ejpam-6574	48	8	τ2)β	τ2)β	ADJ
ejpam-6574	48	9	-	-	PUNCT
ejpam-6574	48	10	open	open	NOUN
ejpam-6574	49	1	[	[	X
ejpam-6574	49	2	15	15	NUM
ejpam-6574	49	3	]	]	PUNCT
ejpam-6574	49	4	)	)	PUNCT
ejpam-6574	49	5	if	if	SCONJ
ejpam-6574	49	6	a	a	DET
ejpam-6574	49	7	=	=	PUNCT
ejpam-6574	49	8	τ1τ2	τ1τ2	NOUN
ejpam-6574	49	9	-	-	NOUN
ejpam-6574	49	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	49	11	-	-	PUNCT
ejpam-6574	49	12	cl(a	cl(a	NUM
ejpam-6574	49	13	)	)	PUNCT
ejpam-6574	49	14	)	)	PUNCT
ejpam-6574	49	15	(	(	PUNCT
ejpam-6574	49	16	resp	resp	NOUN
ejpam-6574	49	17	.	.	PUNCT
ejpam-6574	50	1	a	a	DET
ejpam-6574	50	2	⊆	⊆	NUM
ejpam-6574	50	3	τ1τ2	τ1τ2	NOUN
ejpam-6574	50	4	-	-	ADJ
ejpam-6574	50	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	50	6	-	-	PUNCT
ejpam-6574	50	7	int(a	int(a	NOUN
ejpam-6574	50	8	)	)	PUNCT
ejpam-6574	50	9	)	)	PUNCT
ejpam-6574	50	10	,	,	PUNCT
ejpam-6574	50	11	a	a	DET
ejpam-6574	50	12	⊆	⊆	NUM
ejpam-6574	50	13	τ1τ2	τ1τ2	NOUN
ejpam-6574	50	14	-	-	NOUN
ejpam-6574	50	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	50	16	-	-	PUNCT
ejpam-6574	50	17	cl(a	cl(a	NUM
ejpam-6574	50	18	)	)	PUNCT
ejpam-6574	50	19	)	)	PUNCT
ejpam-6574	50	20	,	,	PUNCT
ejpam-6574	50	21	a	a	DET
ejpam-6574	50	22	⊆	⊆	NUM
ejpam-6574	50	23	τ1τ2	τ1τ2	NOUN
ejpam-6574	50	24	-	-	PUNCT
ejpam-6574	50	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	50	26	-	-	PUNCT
ejpam-6574	50	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	50	28	-	-	PUNCT
ejpam-6574	50	29	cl(a	cl(a	NUM
ejpam-6574	50	30	)	)	PUNCT
ejpam-6574	50	31	)	)	PUNCT
ejpam-6574	50	32	)	)	PUNCT
ejpam-6574	50	33	)	)	PUNCT
ejpam-6574	50	34	.	.	PUNCT
ejpam-6574	51	1	the	the	DET
ejpam-6574	51	2	complement	complement	NOUN
ejpam-6574	51	3	of	of	ADP
ejpam-6574	51	4	a	a	DET
ejpam-6574	51	5	(	(	PUNCT
ejpam-6574	51	6	τ1	τ1	NOUN
ejpam-6574	51	7	,	,	PUNCT
ejpam-6574	51	8	τ2)r	τ2)r	NOUN
ejpam-6574	51	9	-	-	PUNCT
ejpam-6574	51	10	open	open	ADJ
ejpam-6574	51	11	(	(	PUNCT
ejpam-6574	51	12	resp	resp	NOUN
ejpam-6574	51	13	.	.	PUNCT
ejpam-6574	52	1	(	(	PUNCT
ejpam-6574	52	2	τ1	τ1	NOUN
ejpam-6574	52	3	,	,	PUNCT
ejpam-6574	52	4	τ2)s	τ2)s	NOUN
ejpam-6574	52	5	-	-	PUNCT
ejpam-6574	52	6	open	open	ADJ
ejpam-6574	52	7	,	,	PUNCT
ejpam-6574	52	8	(	(	PUNCT
ejpam-6574	52	9	τ1	τ1	NOUN
ejpam-6574	52	10	,	,	PUNCT
ejpam-6574	52	11	τ2)p	τ2)p	NOUN
ejpam-6574	52	12	-	-	ADJ
ejpam-6574	52	13	open	open	ADJ
ejpam-6574	52	14	,	,	PUNCT
ejpam-6574	52	15	(	(	PUNCT
ejpam-6574	52	16	τ1	τ1	NOUN
ejpam-6574	52	17	,	,	PUNCT
ejpam-6574	52	18	τ2)β	τ2)β	ADJ
ejpam-6574	52	19	-	-	PUNCT
ejpam-6574	52	20	open	open	ADJ
ejpam-6574	52	21	)	)	PUNCT
ejpam-6574	52	22	set	set	NOUN
ejpam-6574	52	23	is	be	AUX
ejpam-6574	52	24	called	call	VERB
ejpam-6574	52	25	(	(	PUNCT
ejpam-6574	52	26	τ1	τ1	NOUN
ejpam-6574	52	27	,	,	PUNCT
ejpam-6574	52	28	τ2)r	τ2)r	NOUN
ejpam-6574	52	29	-	-	PUNCT
ejpam-6574	52	30	closed	closed	ADJ
ejpam-6574	52	31	(	(	PUNCT
ejpam-6574	52	32	resp	resp	NOUN
ejpam-6574	52	33	.	.	PUNCT
ejpam-6574	53	1	(	(	PUNCT
ejpam-6574	53	2	τ1	τ1	NOUN
ejpam-6574	53	3	,	,	PUNCT
ejpam-6574	53	4	τ2)s	τ2)s	NOUN
ejpam-6574	53	5	-	-	PUNCT
ejpam-6574	53	6	closed	closed	ADJ
ejpam-6574	53	7	,	,	PUNCT
ejpam-6574	53	8	(	(	PUNCT
ejpam-6574	53	9	τ1	τ1	NOUN
ejpam-6574	53	10	,	,	PUNCT
ejpam-6574	53	11	τ2)p	τ2)p	NOUN
ejpam-6574	53	12	-	-	PUNCT
ejpam-6574	53	13	closed	closed	ADJ
ejpam-6574	53	14	,	,	PUNCT
ejpam-6574	53	15	(	(	PUNCT
ejpam-6574	53	16	τ1	τ1	NOUN
ejpam-6574	53	17	,	,	PUNCT
ejpam-6574	53	18	τ2)β	τ2)β	ADJ
ejpam-6574	53	19	-	-	PUNCT
ejpam-6574	53	20	closed	closed	ADJ
ejpam-6574	53	21	)	)	PUNCT
ejpam-6574	53	22	.	.	PUNCT
ejpam-6574	54	1	a	a	DET
ejpam-6574	54	2	subset	subset	NOUN
ejpam-6574	54	3	a	a	PRON
ejpam-6574	54	4	of	of	ADP
ejpam-6574	54	5	a	a	DET
ejpam-6574	54	6	bitopological	bitopological	ADJ
ejpam-6574	54	7	space	space	NOUN
ejpam-6574	54	8	(	(	PUNCT
ejpam-6574	54	9	x	x	NOUN
ejpam-6574	54	10	,	,	PUNCT
ejpam-6574	54	11	τ1	τ1	NOUN
ejpam-6574	54	12	,	,	PUNCT
ejpam-6574	54	13	τ2	τ2	NOUN
ejpam-6574	54	14	)	)	PUNCT
ejpam-6574	54	15	is	be	AUX
ejpam-6574	54	16	said	say	VERB
ejpam-6574	54	17	to	to	PART
ejpam-6574	54	18	be	be	AUX
ejpam-6574	54	19	α(τ1	α(τ1	NOUN
ejpam-6574	54	20	,	,	PUNCT
ejpam-6574	54	21	τ2)-open	τ2)-open	ADJ
ejpam-6574	54	22	[	[	X
ejpam-6574	54	23	16	16	NUM
ejpam-6574	54	24	]	]	X
ejpam-6574	54	25	if	if	SCONJ
ejpam-6574	54	26	a	a	DET
ejpam-6574	54	27	⊆	⊆	NUM
ejpam-6574	54	28	τ1τ2	τ1τ2	NOUN
ejpam-6574	54	29	-	-	PUNCT
ejpam-6574	54	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	54	31	-	-	PUNCT
ejpam-6574	54	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	54	33	-	-	PUNCT
ejpam-6574	54	34	int(a	int(a	NOUN
ejpam-6574	54	35	)	)	PUNCT
ejpam-6574	54	36	)	)	PUNCT
ejpam-6574	54	37	)	)	PUNCT
ejpam-6574	54	38	.	.	PUNCT
ejpam-6574	55	1	the	the	DET
ejpam-6574	55	2	complement	complement	NOUN
ejpam-6574	55	3	of	of	ADP
ejpam-6574	55	4	an	an	DET
ejpam-6574	55	5	α(τ1	α(τ1	NOUN
ejpam-6574	55	6	,	,	PUNCT
ejpam-6574	55	7	τ2)-open	τ2)-open	ADJ
ejpam-6574	55	8	set	set	NOUN
ejpam-6574	55	9	is	be	AUX
ejpam-6574	55	10	said	say	VERB
ejpam-6574	55	11	to	to	PART
ejpam-6574	55	12	be	be	AUX
ejpam-6574	55	13	α(τ1	α(τ1	NOUN
ejpam-6574	55	14	,	,	PUNCT
ejpam-6574	55	15	τ2)-closed	τ2)-close	VERB
ejpam-6574	55	16	.	.	PUNCT
ejpam-6574	56	1	let	let	VERB
ejpam-6574	56	2	a	a	DET
ejpam-6574	56	3	be	be	AUX
ejpam-6574	56	4	a	a	DET
ejpam-6574	56	5	subset	subset	NOUN
ejpam-6574	56	6	of	of	ADP
ejpam-6574	56	7	a	a	DET
ejpam-6574	56	8	bitopological	bitopological	ADJ
ejpam-6574	56	9	space	space	NOUN
ejpam-6574	56	10	(	(	PUNCT
ejpam-6574	56	11	x	x	NOUN
ejpam-6574	56	12	,	,	PUNCT
ejpam-6574	56	13	τ1	τ1	NOUN
ejpam-6574	56	14	,	,	PUNCT
ejpam-6574	56	15	τ2	τ2	NOUN
ejpam-6574	56	16	)	)	PUNCT
ejpam-6574	56	17	.	.	PUNCT
ejpam-6574	57	1	the	the	DET
ejpam-6574	57	2	intersection	intersection	NOUN
ejpam-6574	57	3	of	of	ADP
ejpam-6574	57	4	all	all	DET
ejpam-6574	57	5	(	(	PUNCT
ejpam-6574	57	6	τ1	τ1	NOUN
ejpam-6574	57	7	,	,	PUNCT
ejpam-6574	57	8	τ2)p	τ2)p	NOUN
ejpam-6574	57	9	-	-	PUNCT
ejpam-6574	57	10	closed	closed	ADJ
ejpam-6574	57	11	(	(	PUNCT
ejpam-6574	57	12	resp	resp	NOUN
ejpam-6574	57	13	.	.	PUNCT
ejpam-6574	58	1	(	(	PUNCT
ejpam-6574	58	2	τ1	τ1	NOUN
ejpam-6574	58	3	,	,	PUNCT
ejpam-6574	58	4	τ2)s	τ2)s	NOUN
ejpam-6574	58	5	-	-	PUNCT
ejpam-6574	58	6	closed	closed	ADJ
ejpam-6574	58	7	,	,	PUNCT
ejpam-6574	58	8	α(τ1	α(τ1	NOUN
ejpam-6574	58	9	,	,	PUNCT
ejpam-6574	58	10	τ2)-closed	τ2)-closed	ADJ
ejpam-6574	58	11	)	)	PUNCT
ejpam-6574	58	12	sets	set	NOUN
ejpam-6574	58	13	of	of	ADP
ejpam-6574	58	14	x	x	PUNCT
ejpam-6574	58	15	containing	contain	VERB
ejpam-6574	58	16	a	a	PRON
ejpam-6574	58	17	is	be	AUX
ejpam-6574	58	18	n.	n.	NOUN
ejpam-6574	58	19	chutiman	chutiman	NOUN
ejpam-6574	58	20	,	,	PUNCT
ejpam-6574	58	21	a.	a.	PROPN
ejpam-6574	58	22	sama	sama	PROPN
ejpam-6574	58	23	-	-	PUNCT
ejpam-6574	58	24	ae	ae	PROPN
ejpam-6574	58	25	,	,	PUNCT
ejpam-6574	58	26	c.	c.	PROPN
ejpam-6574	58	27	boonpok	boonpok	PROPN
ejpam-6574	58	28	/	/	SYM
ejpam-6574	58	29	eur	eur	PROPN
ejpam-6574	58	30	.	.	PUNCT
ejpam-6574	59	1	j.	j.	PROPN
ejpam-6574	59	2	pure	pure	PROPN
ejpam-6574	59	3	appl	appl	PROPN
ejpam-6574	59	4	.	.	PROPN
ejpam-6574	59	5	math	math	PROPN
ejpam-6574	59	6	,	,	PUNCT
ejpam-6574	59	7	18	18	NUM
ejpam-6574	59	8	(	(	PUNCT
ejpam-6574	59	9	3	3	NUM
ejpam-6574	59	10	)	)	PUNCT
ejpam-6574	59	11	(	(	PUNCT
ejpam-6574	59	12	2025	2025	NUM
ejpam-6574	59	13	)	)	PUNCT
ejpam-6574	59	14	,	,	PUNCT
ejpam-6574	59	15	6574	6574	NUM
ejpam-6574	59	16	3	3	NUM
ejpam-6574	59	17	of	of	ADP
ejpam-6574	59	18	10	10	NUM
ejpam-6574	59	19	called	call	VERB
ejpam-6574	59	20	the	the	DET
ejpam-6574	59	21	(	(	PUNCT
ejpam-6574	59	22	τ1	τ1	NOUN
ejpam-6574	59	23	,	,	PUNCT
ejpam-6574	59	24	τ2)p	τ2)p	NOUN
ejpam-6574	59	25	-	-	NOUN
ejpam-6574	59	26	closure	closure	NOUN
ejpam-6574	59	27	[	[	X
ejpam-6574	59	28	17	17	NUM
ejpam-6574	59	29	]	]	X
ejpam-6574	59	30	(	(	PUNCT
ejpam-6574	59	31	resp	resp	NOUN
ejpam-6574	59	32	.	.	PUNCT
ejpam-6574	60	1	(	(	PUNCT
ejpam-6574	60	2	τ1	τ1	NOUN
ejpam-6574	60	3	,	,	PUNCT
ejpam-6574	60	4	τ2)s	τ2)s	NOUN
ejpam-6574	60	5	-	-	PUNCT
ejpam-6574	60	6	closure	closure	NOUN
ejpam-6574	60	7	[	[	X
ejpam-6574	60	8	15	15	NUM
ejpam-6574	60	9	]	]	PUNCT
ejpam-6574	60	10	,	,	PUNCT
ejpam-6574	60	11	α(τ1	α(τ1	NOUN
ejpam-6574	60	12	,	,	PUNCT
ejpam-6574	60	13	τ2)-closure	τ2)-closure	NOUN
ejpam-6574	60	14	[	[	X
ejpam-6574	60	15	18	18	NUM
ejpam-6574	60	16	]	]	PUNCT
ejpam-6574	60	17	)	)	PUNCT
ejpam-6574	60	18	of	of	ADP
ejpam-6574	60	19	a	a	PRON
ejpam-6574	60	20	and	and	CCONJ
ejpam-6574	60	21	is	be	AUX
ejpam-6574	60	22	denoted	denote	VERB
ejpam-6574	60	23	by	by	ADP
ejpam-6574	60	24	(	(	PUNCT
ejpam-6574	60	25	τ1	τ1	NOUN
ejpam-6574	60	26	,	,	PUNCT
ejpam-6574	60	27	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6574	60	28	)	)	PUNCT
ejpam-6574	60	29	(	(	PUNCT
ejpam-6574	60	30	resp	resp	NOUN
ejpam-6574	60	31	.	.	PUNCT
ejpam-6574	61	1	(	(	PUNCT
ejpam-6574	61	2	τ1	τ1	NOUN
ejpam-6574	61	3	,	,	PUNCT
ejpam-6574	61	4	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6574	61	5	)	)	PUNCT
ejpam-6574	61	6	,	,	PUNCT
ejpam-6574	61	7	α(τ1	α(τ1	NOUN
ejpam-6574	61	8	,	,	PUNCT
ejpam-6574	61	9	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-6574	61	10	)	)	PUNCT
ejpam-6574	61	11	)	)	PUNCT
ejpam-6574	61	12	.	.	PUNCT
ejpam-6574	62	1	the	the	DET
ejpam-6574	62	2	union	union	NOUN
ejpam-6574	62	3	of	of	ADP
ejpam-6574	62	4	all	all	DET
ejpam-6574	62	5	(	(	PUNCT
ejpam-6574	62	6	τ1	τ1	NOUN
ejpam-6574	62	7	,	,	PUNCT
ejpam-6574	62	8	τ2)p	τ2)p	NOUN
ejpam-6574	62	9	-	-	ADJ
ejpam-6574	62	10	open	open	ADJ
ejpam-6574	62	11	(	(	PUNCT
ejpam-6574	62	12	resp	resp	NOUN
ejpam-6574	62	13	.	.	PUNCT
ejpam-6574	63	1	(	(	PUNCT
ejpam-6574	63	2	τ1	τ1	NOUN
ejpam-6574	63	3	,	,	PUNCT
ejpam-6574	63	4	τ2)s	τ2)s	NOUN
ejpam-6574	63	5	-	-	PUNCT
ejpam-6574	63	6	open	open	ADJ
ejpam-6574	63	7	,	,	PUNCT
ejpam-6574	63	8	α(τ1	α(τ1	NOUN
ejpam-6574	63	9	,	,	PUNCT
ejpam-6574	63	10	τ2)-open	τ2)-open	ADJ
ejpam-6574	63	11	)	)	PUNCT
ejpam-6574	63	12	sets	set	NOUN
ejpam-6574	63	13	of	of	ADP
ejpam-6574	63	14	x	x	PUNCT
ejpam-6574	63	15	contained	contain	VERB
ejpam-6574	63	16	in	in	ADP
ejpam-6574	63	17	a	a	PRON
ejpam-6574	63	18	is	be	AUX
ejpam-6574	63	19	called	call	VERB
ejpam-6574	63	20	the	the	DET
ejpam-6574	63	21	(	(	PUNCT
ejpam-6574	63	22	τ1	τ1	NOUN
ejpam-6574	63	23	,	,	PUNCT
ejpam-6574	63	24	τ2)p	τ2)p	ADJ
ejpam-6574	63	25	-	-	NOUN
ejpam-6574	63	26	interior	interior	ADJ
ejpam-6574	63	27	[	[	X
ejpam-6574	63	28	17	17	NUM
ejpam-6574	63	29	]	]	X
ejpam-6574	63	30	(	(	PUNCT
ejpam-6574	63	31	resp	resp	NOUN
ejpam-6574	63	32	.	.	PUNCT
ejpam-6574	64	1	(	(	PUNCT
ejpam-6574	64	2	τ1	τ1	NOUN
ejpam-6574	64	3	,	,	PUNCT
ejpam-6574	64	4	τ2)s	τ2)s	NOUN
ejpam-6574	64	5	-	-	ADJ
ejpam-6574	64	6	interior	interior	NOUN
ejpam-6574	65	1	[	[	X
ejpam-6574	65	2	15	15	NUM
ejpam-6574	65	3	]	]	PUNCT
ejpam-6574	65	4	,	,	PUNCT
ejpam-6574	65	5	α(τ1	α(τ1	NOUN
ejpam-6574	65	6	,	,	PUNCT
ejpam-6574	65	7	τ2)-interior	τ2)-interior	PROPN
ejpam-6574	65	8	[	[	X
ejpam-6574	65	9	18	18	NUM
ejpam-6574	65	10	]	]	PUNCT
ejpam-6574	65	11	)	)	PUNCT
ejpam-6574	65	12	of	of	ADP
ejpam-6574	65	13	a	a	PRON
ejpam-6574	65	14	and	and	CCONJ
ejpam-6574	65	15	is	be	AUX
ejpam-6574	65	16	denoted	denote	VERB
ejpam-6574	65	17	by	by	ADP
ejpam-6574	65	18	(	(	PUNCT
ejpam-6574	65	19	τ1	τ1	NOUN
ejpam-6574	65	20	,	,	PUNCT
ejpam-6574	65	21	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6574	65	22	)	)	PUNCT
ejpam-6574	65	23	(	(	PUNCT
ejpam-6574	65	24	resp	resp	NOUN
ejpam-6574	65	25	.	.	PUNCT
ejpam-6574	66	1	(	(	PUNCT
ejpam-6574	66	2	τ1	τ1	NOUN
ejpam-6574	66	3	,	,	PUNCT
ejpam-6574	66	4	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-6574	66	5	)	)	PUNCT
ejpam-6574	66	6	,	,	PUNCT
ejpam-6574	66	7	α(τ1	α(τ1	NOUN
ejpam-6574	66	8	,	,	PUNCT
ejpam-6574	66	9	τ2)-int(a	τ2)-int(a	NOUN
ejpam-6574	66	10	)	)	PUNCT
ejpam-6574	66	11	)	)	PUNCT
ejpam-6574	66	12	.	.	PUNCT
ejpam-6574	67	1	lemma	lemma	PROPN
ejpam-6574	67	2	2	2	NUM
ejpam-6574	67	3	.	.	X
ejpam-6574	68	1	for	for	ADP
ejpam-6574	68	2	subsets	subset	NOUN
ejpam-6574	68	3	a	a	PRON
ejpam-6574	68	4	and	and	CCONJ
ejpam-6574	68	5	b	b	NOUN
ejpam-6574	68	6	of	of	ADP
ejpam-6574	68	7	a	a	DET
ejpam-6574	68	8	bitopological	bitopological	ADJ
ejpam-6574	68	9	space	space	NOUN
ejpam-6574	68	10	(	(	PUNCT
ejpam-6574	68	11	x	x	NOUN
ejpam-6574	68	12	,	,	PUNCT
ejpam-6574	68	13	τ1	τ1	NOUN
ejpam-6574	68	14	,	,	PUNCT
ejpam-6574	68	15	τ2	τ2	NOUN
ejpam-6574	68	16	)	)	PUNCT
ejpam-6574	68	17	,	,	PUNCT
ejpam-6574	68	18	the	the	DET
ejpam-6574	68	19	following	follow	VERB
ejpam-6574	68	20	properties	property	NOUN
ejpam-6574	68	21	hold	hold	VERB
ejpam-6574	68	22	:	:	PUNCT
ejpam-6574	68	23	(	(	PUNCT
ejpam-6574	68	24	1	1	X
ejpam-6574	68	25	)	)	PUNCT
ejpam-6574	68	26	α(τ1	α(τ1	NOUN
ejpam-6574	68	27	,	,	PUNCT
ejpam-6574	68	28	τ2)-int(a	τ2)-int(a	NOUN
ejpam-6574	68	29	)	)	PUNCT
ejpam-6574	68	30	=	=	PUNCT
ejpam-6574	69	1	τ1τ2	τ1τ2	NOUN
ejpam-6574	69	2	-	-	NOUN
ejpam-6574	69	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	69	4	-	-	PUNCT
ejpam-6574	69	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	69	6	-	-	PUNCT
ejpam-6574	69	7	int(a	int(a	NOUN
ejpam-6574	69	8	)	)	PUNCT
ejpam-6574	69	9	)	)	PUNCT
ejpam-6574	69	10	)	)	PUNCT
ejpam-6574	70	1	∩a	∩a	PROPN
ejpam-6574	70	2	;	;	PUNCT
ejpam-6574	70	3	(	(	PUNCT
ejpam-6574	70	4	2	2	X
ejpam-6574	70	5	)	)	PUNCT
ejpam-6574	70	6	(	(	PUNCT
ejpam-6574	70	7	τ1	τ1	NOUN
ejpam-6574	70	8	,	,	PUNCT
ejpam-6574	70	9	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-6574	70	10	)	)	PUNCT
ejpam-6574	70	11	=	=	PUNCT
ejpam-6574	71	1	τ1τ2	τ1τ2	NOUN
ejpam-6574	71	2	-	-	ADJ
ejpam-6574	71	3	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	71	4	-	-	PUNCT
ejpam-6574	71	5	int(a	int(a	NOUN
ejpam-6574	71	6	)	)	PUNCT
ejpam-6574	71	7	)	)	PUNCT
ejpam-6574	72	1	∩a	∩a	PROPN
ejpam-6574	73	1	[	[	X
ejpam-6574	73	2	19	19	NUM
ejpam-6574	73	3	]	]	X
ejpam-6574	73	4	;	;	PUNCT
ejpam-6574	73	5	(	(	PUNCT
ejpam-6574	73	6	3	3	X
ejpam-6574	73	7	)	)	PUNCT
ejpam-6574	73	8	(	(	PUNCT
ejpam-6574	73	9	τ1	τ1	NOUN
ejpam-6574	73	10	,	,	PUNCT
ejpam-6574	73	11	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6574	73	12	)	)	PUNCT
ejpam-6574	73	13	=	=	PUNCT
ejpam-6574	73	14	τ1τ2	τ1τ2	NOUN
ejpam-6574	73	15	-	-	NOUN
ejpam-6574	73	16	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	73	17	-	-	PUNCT
ejpam-6574	73	18	cl(a	cl(a	NUM
ejpam-6574	73	19	)	)	PUNCT
ejpam-6574	73	20	)	)	PUNCT
ejpam-6574	74	1	∩a	∩a	PROPN
ejpam-6574	75	1	[	[	X
ejpam-6574	75	2	20	20	NUM
ejpam-6574	75	3	]	]	PUNCT
ejpam-6574	75	4	.	.	PUNCT
ejpam-6574	76	1	for	for	ADP
ejpam-6574	76	2	a	a	DET
ejpam-6574	76	3	subset	subset	NOUN
ejpam-6574	76	4	a	a	PRON
ejpam-6574	76	5	of	of	ADP
ejpam-6574	76	6	a	a	DET
ejpam-6574	76	7	bitopological	bitopological	ADJ
ejpam-6574	76	8	space	space	NOUN
ejpam-6574	76	9	(	(	PUNCT
ejpam-6574	76	10	x	x	NOUN
ejpam-6574	76	11	,	,	PUNCT
ejpam-6574	76	12	τ1	τ1	NOUN
ejpam-6574	76	13	,	,	PUNCT
ejpam-6574	76	14	τ2	τ2	PROPN
ejpam-6574	76	15	)	)	PUNCT
ejpam-6574	76	16	,	,	PUNCT
ejpam-6574	76	17	a	a	DET
ejpam-6574	76	18	point	point	NOUN
ejpam-6574	76	19	x	x	X
ejpam-6574	76	20	∈	∈	NOUN
ejpam-6574	76	21	x	x	PUNCT
ejpam-6574	76	22	is	be	AUX
ejpam-6574	76	23	called	call	VERB
ejpam-6574	76	24	(	(	PUNCT
ejpam-6574	76	25	τ1	τ1	NOUN
ejpam-6574	76	26	,	,	PUNCT
ejpam-6574	76	27	τ2)θcluster	τ2)θcluster	PROPN
ejpam-6574	76	28	point	point	NOUN
ejpam-6574	76	29	[	[	X
ejpam-6574	76	30	14	14	NUM
ejpam-6574	76	31	]	]	PUNCT
ejpam-6574	76	32	of	of	ADP
ejpam-6574	76	33	a	a	DET
ejpam-6574	76	34	if	if	SCONJ
ejpam-6574	76	35	τ1τ2	τ1τ2	ADJ
ejpam-6574	76	36	-	-	ADJ
ejpam-6574	76	37	cl(u)∩a	cl(u)∩a	ADJ
ejpam-6574	76	38	̸=	̸=	PROPN
ejpam-6574	76	39	∅	∅	NOUN
ejpam-6574	76	40	for	for	ADP
ejpam-6574	76	41	every	every	DET
ejpam-6574	76	42	τ1τ2	τ1τ2	ADJ
ejpam-6574	76	43	-	-	ADJ
ejpam-6574	76	44	open	open	ADJ
ejpam-6574	76	45	set	set	NOUN
ejpam-6574	76	46	u	u	NOUN
ejpam-6574	76	47	containing	contain	VERB
ejpam-6574	76	48	x.	x.	NOUN
ejpam-6574	76	49	the	the	DET
ejpam-6574	76	50	set	set	NOUN
ejpam-6574	76	51	of	of	ADP
ejpam-6574	76	52	all	all	DET
ejpam-6574	76	53	(	(	PUNCT
ejpam-6574	76	54	τ1	τ1	NOUN
ejpam-6574	76	55	,	,	PUNCT
ejpam-6574	76	56	τ2)θ	τ2)θ	ADJ
ejpam-6574	76	57	-	-	PUNCT
ejpam-6574	76	58	cluster	cluster	NOUN
ejpam-6574	76	59	points	point	NOUN
ejpam-6574	76	60	of	of	ADP
ejpam-6574	76	61	a	a	PRON
ejpam-6574	76	62	is	be	AUX
ejpam-6574	76	63	called	call	VERB
ejpam-6574	76	64	the	the	DET
ejpam-6574	76	65	(	(	PUNCT
ejpam-6574	76	66	τ1	τ1	NOUN
ejpam-6574	76	67	,	,	PUNCT
ejpam-6574	76	68	τ2)θ	τ2)θ	ADJ
ejpam-6574	76	69	-	-	PUNCT
ejpam-6574	76	70	closure	closure	NOUN
ejpam-6574	76	71	[	[	X
ejpam-6574	76	72	14	14	NUM
ejpam-6574	76	73	]	]	PUNCT
ejpam-6574	76	74	of	of	ADP
ejpam-6574	76	75	a	a	PRON
ejpam-6574	76	76	and	and	CCONJ
ejpam-6574	76	77	is	be	AUX
ejpam-6574	76	78	denoted	denote	VERB
ejpam-6574	76	79	by	by	ADP
ejpam-6574	76	80	(	(	PUNCT
ejpam-6574	76	81	τ1	τ1	NOUN
ejpam-6574	76	82	,	,	PUNCT
ejpam-6574	76	83	τ2)θ	τ2)θ	NOUN
ejpam-6574	76	84	-	-	PUNCT
ejpam-6574	76	85	cl(a	cl(a	NUM
ejpam-6574	76	86	)	)	PUNCT
ejpam-6574	76	87	.	.	PUNCT
ejpam-6574	77	1	a	a	DET
ejpam-6574	77	2	subset	subset	NOUN
ejpam-6574	77	3	a	a	PRON
ejpam-6574	77	4	of	of	ADP
ejpam-6574	77	5	a	a	DET
ejpam-6574	77	6	bitopological	bitopological	ADJ
ejpam-6574	77	7	space	space	NOUN
ejpam-6574	77	8	(	(	PUNCT
ejpam-6574	77	9	x	x	NOUN
ejpam-6574	77	10	,	,	PUNCT
ejpam-6574	77	11	τ1	τ1	NOUN
ejpam-6574	77	12	,	,	PUNCT
ejpam-6574	77	13	τ2	τ2	NOUN
ejpam-6574	77	14	)	)	PUNCT
ejpam-6574	77	15	is	be	AUX
ejpam-6574	77	16	said	say	VERB
ejpam-6574	77	17	to	to	PART
ejpam-6574	77	18	be	be	AUX
ejpam-6574	77	19	(	(	PUNCT
ejpam-6574	77	20	τ1	τ1	NOUN
ejpam-6574	77	21	,	,	PUNCT
ejpam-6574	77	22	τ2)θ	τ2)θ	NOUN
ejpam-6574	77	23	-	-	PUNCT
ejpam-6574	77	24	closed	closed	ADJ
ejpam-6574	77	25	[	[	X
ejpam-6574	77	26	14	14	NUM
ejpam-6574	77	27	]	]	X
ejpam-6574	77	28	if	if	SCONJ
ejpam-6574	77	29	a	a	PRON
ejpam-6574	77	30	=	=	X
ejpam-6574	77	31	(	(	PUNCT
ejpam-6574	77	32	τ1	τ1	NOUN
ejpam-6574	77	33	,	,	PUNCT
ejpam-6574	77	34	τ2)θ	τ2)θ	NOUN
ejpam-6574	77	35	-	-	PUNCT
ejpam-6574	77	36	cl(a	cl(a	NUM
ejpam-6574	77	37	)	)	PUNCT
ejpam-6574	77	38	.	.	PUNCT
ejpam-6574	78	1	the	the	DET
ejpam-6574	78	2	complement	complement	NOUN
ejpam-6574	78	3	of	of	ADP
ejpam-6574	78	4	a	a	DET
ejpam-6574	78	5	(	(	PUNCT
ejpam-6574	78	6	τ1	τ1	NOUN
ejpam-6574	78	7	,	,	PUNCT
ejpam-6574	78	8	τ2)θ	τ2)θ	ADJ
ejpam-6574	78	9	-	-	PUNCT
ejpam-6574	78	10	closed	close	VERB
ejpam-6574	78	11	set	set	NOUN
ejpam-6574	78	12	is	be	AUX
ejpam-6574	78	13	said	say	VERB
ejpam-6574	78	14	to	to	PART
ejpam-6574	78	15	be	be	AUX
ejpam-6574	78	16	(	(	PUNCT
ejpam-6574	78	17	τ1	τ1	NOUN
ejpam-6574	78	18	,	,	PUNCT
ejpam-6574	78	19	τ2)θopen	τ2)θopen	PROPN
ejpam-6574	78	20	.	.	PUNCT
ejpam-6574	79	1	the	the	DET
ejpam-6574	79	2	union	union	NOUN
ejpam-6574	79	3	of	of	ADP
ejpam-6574	79	4	all	all	DET
ejpam-6574	79	5	(	(	PUNCT
ejpam-6574	79	6	τ1	τ1	NOUN
ejpam-6574	79	7	,	,	PUNCT
ejpam-6574	79	8	τ2)θ	τ2)θ	ADJ
ejpam-6574	79	9	-	-	PUNCT
ejpam-6574	79	10	open	open	ADJ
ejpam-6574	79	11	sets	set	NOUN
ejpam-6574	79	12	contained	contain	VERB
ejpam-6574	79	13	in	in	ADP
ejpam-6574	79	14	a	a	PRON
ejpam-6574	79	15	is	be	AUX
ejpam-6574	79	16	called	call	VERB
ejpam-6574	79	17	the	the	DET
ejpam-6574	79	18	(	(	PUNCT
ejpam-6574	79	19	τ1	τ1	NOUN
ejpam-6574	79	20	,	,	PUNCT
ejpam-6574	79	21	τ2)θ	τ2)θ	ADJ
ejpam-6574	79	22	-	-	PUNCT
ejpam-6574	79	23	interior	interior	NOUN
ejpam-6574	79	24	[	[	X
ejpam-6574	79	25	14	14	NUM
ejpam-6574	79	26	]	]	PUNCT
ejpam-6574	79	27	of	of	ADP
ejpam-6574	79	28	a	a	PRON
ejpam-6574	79	29	and	and	CCONJ
ejpam-6574	79	30	is	be	AUX
ejpam-6574	79	31	denoted	denote	VERB
ejpam-6574	79	32	by	by	ADP
ejpam-6574	79	33	(	(	PUNCT
ejpam-6574	79	34	τ1	τ1	NOUN
ejpam-6574	79	35	,	,	PUNCT
ejpam-6574	79	36	τ2)θ	τ2)θ	NOUN
ejpam-6574	79	37	-	-	PUNCT
ejpam-6574	79	38	int(a	int(a	NOUN
ejpam-6574	79	39	)	)	PUNCT
ejpam-6574	79	40	.	.	PUNCT
ejpam-6574	80	1	lemma	lemma	PROPN
ejpam-6574	80	2	3	3	X
ejpam-6574	80	3	.	.	PUNCT
ejpam-6574	81	1	[	[	X
ejpam-6574	81	2	14	14	NUM
ejpam-6574	81	3	]	]	PUNCT
ejpam-6574	81	4	for	for	ADP
ejpam-6574	81	5	a	a	DET
ejpam-6574	81	6	subset	subset	NOUN
ejpam-6574	81	7	a	a	PRON
ejpam-6574	81	8	of	of	ADP
ejpam-6574	81	9	a	a	DET
ejpam-6574	81	10	bitopological	bitopological	ADJ
ejpam-6574	81	11	space	space	NOUN
ejpam-6574	81	12	(	(	PUNCT
ejpam-6574	81	13	x	x	NOUN
ejpam-6574	81	14	,	,	PUNCT
ejpam-6574	81	15	τ1	τ1	NOUN
ejpam-6574	81	16	,	,	PUNCT
ejpam-6574	81	17	τ2	τ2	NOUN
ejpam-6574	81	18	)	)	PUNCT
ejpam-6574	81	19	,	,	PUNCT
ejpam-6574	81	20	the	the	DET
ejpam-6574	81	21	following	follow	VERB
ejpam-6574	81	22	properties	property	NOUN
ejpam-6574	81	23	hold	hold	VERB
ejpam-6574	81	24	:	:	PUNCT
ejpam-6574	81	25	(	(	PUNCT
ejpam-6574	81	26	1	1	X
ejpam-6574	81	27	)	)	PUNCT
ejpam-6574	81	28	if	if	SCONJ
ejpam-6574	81	29	a	a	PRON
ejpam-6574	81	30	is	be	AUX
ejpam-6574	81	31	τ2τ2	τ2τ2	VERB
ejpam-6574	81	32	-	-	VERB
ejpam-6574	81	33	open	open	ADJ
ejpam-6574	81	34	in	in	ADP
ejpam-6574	81	35	x	x	NOUN
ejpam-6574	81	36	,	,	PUNCT
ejpam-6574	81	37	then	then	ADV
ejpam-6574	81	38	τ1τ2	τ1τ2	NOUN
ejpam-6574	81	39	-	-	NUM
ejpam-6574	81	40	cl(a	cl(a	NUM
ejpam-6574	81	41	)	)	PUNCT
ejpam-6574	81	42	=	=	PUNCT
ejpam-6574	81	43	(	(	PUNCT
ejpam-6574	81	44	τ1	τ1	NOUN
ejpam-6574	81	45	,	,	PUNCT
ejpam-6574	81	46	τ2)θ	τ2)θ	NOUN
ejpam-6574	81	47	-	-	PUNCT
ejpam-6574	81	48	cl(a	cl(a	NUM
ejpam-6574	81	49	)	)	PUNCT
ejpam-6574	81	50	.	.	PUNCT
ejpam-6574	82	1	(	(	PUNCT
ejpam-6574	82	2	2	2	X
ejpam-6574	82	3	)	)	PUNCT
ejpam-6574	82	4	(	(	PUNCT
ejpam-6574	82	5	τ1	τ1	NOUN
ejpam-6574	82	6	,	,	PUNCT
ejpam-6574	82	7	τ2)θ	τ2)θ	NOUN
ejpam-6574	82	8	-	-	PUNCT
ejpam-6574	82	9	cl(a	cl(a	NUM
ejpam-6574	82	10	)	)	PUNCT
ejpam-6574	82	11	is	be	AUX
ejpam-6574	82	12	τ1τ2	τ1τ2	NOUN
ejpam-6574	82	13	-	-	ADJ
ejpam-6574	82	14	closed	closed	ADJ
ejpam-6574	82	15	in	in	ADP
ejpam-6574	82	16	x.	x.	PROPN
ejpam-6574	82	17	a	a	DET
ejpam-6574	82	18	subset	subset	NOUN
ejpam-6574	82	19	a	a	PRON
ejpam-6574	82	20	of	of	ADP
ejpam-6574	82	21	a	a	DET
ejpam-6574	82	22	bitopological	bitopological	ADJ
ejpam-6574	82	23	space	space	NOUN
ejpam-6574	82	24	(	(	PUNCT
ejpam-6574	82	25	x	x	NOUN
ejpam-6574	82	26	,	,	PUNCT
ejpam-6574	82	27	τ1	τ1	NOUN
ejpam-6574	82	28	,	,	PUNCT
ejpam-6574	82	29	τ2	τ2	NOUN
ejpam-6574	82	30	)	)	PUNCT
ejpam-6574	82	31	is	be	AUX
ejpam-6574	82	32	said	say	VERB
ejpam-6574	82	33	to	to	PART
ejpam-6574	82	34	be	be	AUX
ejpam-6574	82	35	(	(	PUNCT
ejpam-6574	82	36	τ1	τ1	NOUN
ejpam-6574	82	37	,	,	PUNCT
ejpam-6574	82	38	τ2)b	τ2)b	NOUN
ejpam-6574	82	39	-	-	PUNCT
ejpam-6574	82	40	open	open	ADJ
ejpam-6574	82	41	if	if	SCONJ
ejpam-6574	82	42	a	a	DET
ejpam-6574	82	43	⊆	⊆	NUM
ejpam-6574	82	44	τ1τ2	τ1τ2	NOUN
ejpam-6574	82	45	-	-	NOUN
ejpam-6574	82	46	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	82	47	-	-	PUNCT
ejpam-6574	82	48	cl(a	cl(a	NUM
ejpam-6574	82	49	)	)	PUNCT
ejpam-6574	82	50	)	)	PUNCT
ejpam-6574	82	51	∪	∪	ADP
ejpam-6574	82	52	τ1τ2	τ1τ2	NOUN
ejpam-6574	82	53	-	-	ADJ
ejpam-6574	82	54	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	82	55	-	-	PUNCT
ejpam-6574	82	56	int(a	int(a	NOUN
ejpam-6574	82	57	)	)	PUNCT
ejpam-6574	82	58	)	)	PUNCT
ejpam-6574	82	59	.	.	PUNCT
ejpam-6574	83	1	the	the	DET
ejpam-6574	83	2	complement	complement	NOUN
ejpam-6574	83	3	of	of	ADP
ejpam-6574	83	4	a	a	PRON
ejpam-6574	83	5	(	(	PUNCT
ejpam-6574	83	6	τ1	τ1	NOUN
ejpam-6574	83	7	,	,	PUNCT
ejpam-6574	83	8	τ2)b	τ2)b	NOUN
ejpam-6574	83	9	-	-	PUNCT
ejpam-6574	83	10	open	open	ADJ
ejpam-6574	83	11	set	set	NOUN
ejpam-6574	83	12	is	be	AUX
ejpam-6574	83	13	called	call	VERB
ejpam-6574	83	14	(	(	PUNCT
ejpam-6574	83	15	τ1	τ1	NOUN
ejpam-6574	83	16	,	,	PUNCT
ejpam-6574	83	17	τ2)b	τ2)b	NOUN
ejpam-6574	83	18	-	-	PUNCT
ejpam-6574	83	19	closed	closed	ADJ
ejpam-6574	83	20	.	.	PUNCT
ejpam-6574	84	1	let	let	VERB
ejpam-6574	84	2	a	a	DET
ejpam-6574	84	3	be	be	AUX
ejpam-6574	84	4	a	a	DET
ejpam-6574	84	5	subset	subset	NOUN
ejpam-6574	84	6	of	of	ADP
ejpam-6574	84	7	a	a	DET
ejpam-6574	84	8	bitopological	bitopological	ADJ
ejpam-6574	84	9	space	space	NOUN
ejpam-6574	84	10	(	(	PUNCT
ejpam-6574	84	11	x	x	NOUN
ejpam-6574	84	12	,	,	PUNCT
ejpam-6574	84	13	τ1	τ1	NOUN
ejpam-6574	84	14	,	,	PUNCT
ejpam-6574	84	15	τ2	τ2	NOUN
ejpam-6574	84	16	)	)	PUNCT
ejpam-6574	84	17	.	.	PUNCT
ejpam-6574	85	1	the	the	DET
ejpam-6574	85	2	union	union	NOUN
ejpam-6574	85	3	of	of	ADP
ejpam-6574	85	4	all	all	DET
ejpam-6574	85	5	(	(	PUNCT
ejpam-6574	85	6	τ1	τ1	NOUN
ejpam-6574	85	7	,	,	PUNCT
ejpam-6574	85	8	τ2)b	τ2)b	NOUN
ejpam-6574	85	9	-	-	PUNCT
ejpam-6574	85	10	open	open	ADJ
ejpam-6574	85	11	sets	set	NOUN
ejpam-6574	85	12	of	of	ADP
ejpam-6574	85	13	x	x	PUNCT
ejpam-6574	85	14	contained	contain	VERB
ejpam-6574	85	15	in	in	ADP
ejpam-6574	85	16	a	a	PRON
ejpam-6574	85	17	is	be	AUX
ejpam-6574	85	18	called	call	VERB
ejpam-6574	85	19	the	the	DET
ejpam-6574	85	20	(	(	PUNCT
ejpam-6574	85	21	τ1	τ1	NOUN
ejpam-6574	85	22	,	,	PUNCT
ejpam-6574	85	23	τ2)b	τ2)b	ADJ
ejpam-6574	85	24	-	-	PUNCT
ejpam-6574	85	25	interior	interior	NOUN
ejpam-6574	85	26	of	of	ADP
ejpam-6574	85	27	a	a	PRON
ejpam-6574	85	28	and	and	CCONJ
ejpam-6574	85	29	is	be	AUX
ejpam-6574	85	30	denoted	denote	VERB
ejpam-6574	85	31	by	by	ADP
ejpam-6574	85	32	(	(	PUNCT
ejpam-6574	85	33	τ1	τ1	NOUN
ejpam-6574	85	34	,	,	PUNCT
ejpam-6574	85	35	τ2)-bint(a	τ2)-bint(a	NUM
ejpam-6574	85	36	)	)	PUNCT
ejpam-6574	85	37	.	.	PUNCT
ejpam-6574	86	1	the	the	DET
ejpam-6574	86	2	intersection	intersection	NOUN
ejpam-6574	86	3	of	of	ADP
ejpam-6574	86	4	all	all	DET
ejpam-6574	86	5	(	(	PUNCT
ejpam-6574	86	6	τ1	τ1	NOUN
ejpam-6574	86	7	,	,	PUNCT
ejpam-6574	86	8	τ2)b	τ2)b	NOUN
ejpam-6574	86	9	-	-	PUNCT
ejpam-6574	86	10	closed	close	VERB
ejpam-6574	86	11	sets	set	NOUN
ejpam-6574	86	12	of	of	ADP
ejpam-6574	86	13	x	x	PUNCT
ejpam-6574	86	14	containing	contain	VERB
ejpam-6574	86	15	a	a	PRON
ejpam-6574	86	16	is	be	AUX
ejpam-6574	86	17	called	call	VERB
ejpam-6574	86	18	the	the	DET
ejpam-6574	86	19	(	(	PUNCT
ejpam-6574	86	20	τ1	τ1	NOUN
ejpam-6574	86	21	,	,	PUNCT
ejpam-6574	86	22	τ2)b	τ2)b	NOUN
ejpam-6574	86	23	-	-	PUNCT
ejpam-6574	86	24	closure	closure	NOUN
ejpam-6574	86	25	of	of	ADP
ejpam-6574	86	26	a	a	PRON
ejpam-6574	86	27	and	and	CCONJ
ejpam-6574	86	28	is	be	AUX
ejpam-6574	86	29	denoted	denote	VERB
ejpam-6574	86	30	by	by	ADP
ejpam-6574	86	31	(	(	PUNCT
ejpam-6574	86	32	τ1	τ1	NOUN
ejpam-6574	86	33	,	,	PUNCT
ejpam-6574	86	34	τ2)-bcl(a	τ2)-bcl(a	NOUN
ejpam-6574	86	35	)	)	PUNCT
ejpam-6574	86	36	.	.	PUNCT
ejpam-6574	87	1	lemma	lemma	PROPN
ejpam-6574	87	2	4	4	NUM
ejpam-6574	87	3	.	.	X
ejpam-6574	88	1	for	for	ADP
ejpam-6574	88	2	subsets	subset	NOUN
ejpam-6574	88	3	a	a	PRON
ejpam-6574	88	4	and	and	CCONJ
ejpam-6574	88	5	b	b	NOUN
ejpam-6574	88	6	of	of	ADP
ejpam-6574	88	7	a	a	DET
ejpam-6574	88	8	bitopological	bitopological	ADJ
ejpam-6574	88	9	space	space	NOUN
ejpam-6574	88	10	(	(	PUNCT
ejpam-6574	88	11	x	x	NOUN
ejpam-6574	88	12	,	,	PUNCT
ejpam-6574	88	13	τ1	τ1	NOUN
ejpam-6574	88	14	,	,	PUNCT
ejpam-6574	88	15	τ2	τ2	NOUN
ejpam-6574	88	16	)	)	PUNCT
ejpam-6574	88	17	,	,	PUNCT
ejpam-6574	88	18	the	the	DET
ejpam-6574	88	19	following	follow	VERB
ejpam-6574	88	20	properties	property	NOUN
ejpam-6574	88	21	hold	hold	VERB
ejpam-6574	88	22	:	:	PUNCT
ejpam-6574	88	23	(	(	PUNCT
ejpam-6574	88	24	1	1	X
ejpam-6574	88	25	)	)	PUNCT
ejpam-6574	88	26	(	(	PUNCT
ejpam-6574	88	27	τ1	τ1	NOUN
ejpam-6574	88	28	,	,	PUNCT
ejpam-6574	88	29	τ2)-bint(a	τ2)-bint(a	NUM
ejpam-6574	88	30	)	)	PUNCT
ejpam-6574	88	31	=	=	PUNCT
ejpam-6574	88	32	(	(	PUNCT
ejpam-6574	88	33	τ1	τ1	NOUN
ejpam-6574	88	34	,	,	PUNCT
ejpam-6574	88	35	τ2)-sint(a	τ2)-sint(a	PROPN
ejpam-6574	88	36	)	)	PUNCT
ejpam-6574	88	37	∪	∪	NOUN
ejpam-6574	88	38	(	(	PUNCT
ejpam-6574	88	39	τ1	τ1	NOUN
ejpam-6574	88	40	,	,	PUNCT
ejpam-6574	88	41	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6574	88	42	)	)	PUNCT
ejpam-6574	88	43	;	;	PUNCT
ejpam-6574	88	44	(	(	PUNCT
ejpam-6574	88	45	2	2	X
ejpam-6574	88	46	)	)	PUNCT
ejpam-6574	88	47	(	(	PUNCT
ejpam-6574	88	48	τ1	τ1	NOUN
ejpam-6574	88	49	,	,	PUNCT
ejpam-6574	88	50	τ2)-bcl(a	τ2)-bcl(a	NOUN
ejpam-6574	88	51	)	)	PUNCT
ejpam-6574	88	52	=	=	PUNCT
ejpam-6574	88	53	(	(	PUNCT
ejpam-6574	88	54	τ1	τ1	NOUN
ejpam-6574	88	55	,	,	PUNCT
ejpam-6574	88	56	τ2)-scl(a	τ2)-scl(a	NOUN
ejpam-6574	88	57	)	)	PUNCT
ejpam-6574	88	58	∩	∩	NOUN
ejpam-6574	88	59	(	(	PUNCT
ejpam-6574	88	60	τ1	τ1	NOUN
ejpam-6574	88	61	,	,	PUNCT
ejpam-6574	88	62	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6574	88	63	)	)	PUNCT
ejpam-6574	88	64	;	;	PUNCT
ejpam-6574	88	65	(	(	PUNCT
ejpam-6574	88	66	3	3	X
ejpam-6574	88	67	)	)	PUNCT
ejpam-6574	88	68	(	(	PUNCT
ejpam-6574	88	69	τ1	τ1	NOUN
ejpam-6574	88	70	,	,	PUNCT
ejpam-6574	88	71	τ2)-bcl(x	τ2)-bcl(x	NOUN
ejpam-6574	88	72	−a	−a	NOUN
ejpam-6574	88	73	)	)	PUNCT
ejpam-6574	88	74	=	=	PUNCT
ejpam-6574	89	1	x	x	X
ejpam-6574	89	2	−	−	PROPN
ejpam-6574	89	3	(	(	PUNCT
ejpam-6574	89	4	τ1	τ1	NOUN
ejpam-6574	89	5	,	,	PUNCT
ejpam-6574	89	6	τ2)-bint(a	τ2)-bint(a	NUM
ejpam-6574	89	7	)	)	PUNCT
ejpam-6574	89	8	;	;	PUNCT
ejpam-6574	89	9	(	(	PUNCT
ejpam-6574	89	10	4	4	X
ejpam-6574	89	11	)	)	PUNCT
ejpam-6574	89	12	x	x	SYM
ejpam-6574	89	13	∈	∈	PROPN
ejpam-6574	89	14	(	(	PUNCT
ejpam-6574	89	15	τ1	τ1	NOUN
ejpam-6574	89	16	,	,	PUNCT
ejpam-6574	89	17	τ2)-bint(a	τ2)-bint(a	ADJ
ejpam-6574	89	18	)	)	PUNCT
ejpam-6574	89	19	if	if	SCONJ
ejpam-6574	89	20	and	and	CCONJ
ejpam-6574	89	21	only	only	ADV
ejpam-6574	89	22	if	if	SCONJ
ejpam-6574	89	23	a	a	DET
ejpam-6574	89	24	∩	∩	ADJ
ejpam-6574	89	25	u	u	ADJ
ejpam-6574	89	26	̸=	̸=	PROPN
ejpam-6574	89	27	∅	∅	NOUN
ejpam-6574	89	28	for	for	ADP
ejpam-6574	89	29	every	every	DET
ejpam-6574	89	30	(	(	PUNCT
ejpam-6574	89	31	τ1	τ1	NOUN
ejpam-6574	89	32	,	,	PUNCT
ejpam-6574	89	33	τ2)b	τ2)b	NOUN
ejpam-6574	89	34	-	-	PUNCT
ejpam-6574	89	35	open	open	ADJ
ejpam-6574	89	36	set	set	NOUN
ejpam-6574	89	37	u	u	NOUN
ejpam-6574	89	38	of	of	ADP
ejpam-6574	89	39	x	x	PUNCT
ejpam-6574	89	40	containing	contain	VERB
ejpam-6574	89	41	x	x	NOUN
ejpam-6574	89	42	;	;	PUNCT
ejpam-6574	89	43	n.	n.	NOUN
ejpam-6574	89	44	chutiman	chutiman	NOUN
ejpam-6574	89	45	,	,	PUNCT
ejpam-6574	89	46	a.	a.	PROPN
ejpam-6574	89	47	sama	sama	PROPN
ejpam-6574	89	48	-	-	PUNCT
ejpam-6574	89	49	ae	ae	PROPN
ejpam-6574	89	50	,	,	PUNCT
ejpam-6574	89	51	c.	c.	PROPN
ejpam-6574	89	52	boonpok	boonpok	PROPN
ejpam-6574	89	53	/	/	SYM
ejpam-6574	89	54	eur	eur	PROPN
ejpam-6574	89	55	.	.	PUNCT
ejpam-6574	90	1	j.	j.	PROPN
ejpam-6574	90	2	pure	pure	PROPN
ejpam-6574	90	3	appl	appl	PROPN
ejpam-6574	90	4	.	.	PROPN
ejpam-6574	90	5	math	math	PROPN
ejpam-6574	90	6	,	,	PUNCT
ejpam-6574	90	7	18	18	NUM
ejpam-6574	90	8	(	(	PUNCT
ejpam-6574	90	9	3	3	NUM
ejpam-6574	90	10	)	)	PUNCT
ejpam-6574	90	11	(	(	PUNCT
ejpam-6574	90	12	2025	2025	NUM
ejpam-6574	90	13	)	)	PUNCT
ejpam-6574	90	14	,	,	PUNCT
ejpam-6574	90	15	6574	6574	NUM
ejpam-6574	90	16	4	4	NUM
ejpam-6574	90	17	of	of	ADP
ejpam-6574	90	18	10	10	NUM
ejpam-6574	90	19	(	(	PUNCT
ejpam-6574	90	20	5	5	NUM
ejpam-6574	90	21	)	)	PUNCT
ejpam-6574	90	22	a	a	PRON
ejpam-6574	90	23	is	is	NOUN
ejpam-6574	90	24	(	(	PUNCT
ejpam-6574	90	25	τ1	τ1	NOUN
ejpam-6574	90	26	,	,	PUNCT
ejpam-6574	90	27	τ2)b	τ2)b	NOUN
ejpam-6574	90	28	-	-	PUNCT
ejpam-6574	90	29	closed	close	VERB
ejpam-6574	90	30	if	if	SCONJ
ejpam-6574	90	31	and	and	CCONJ
ejpam-6574	90	32	only	only	ADV
ejpam-6574	90	33	if	if	SCONJ
ejpam-6574	90	34	a	a	PRON
ejpam-6574	90	35	=	=	X
ejpam-6574	90	36	(	(	PUNCT
ejpam-6574	90	37	τ1	τ1	PROPN
ejpam-6574	90	38	,	,	PUNCT
ejpam-6574	90	39	τ2)-bcl(a	τ2)-bcl(a	NOUN
ejpam-6574	90	40	)	)	PUNCT
ejpam-6574	90	41	;	;	PUNCT
ejpam-6574	90	42	(	(	PUNCT
ejpam-6574	90	43	6	6	NUM
ejpam-6574	90	44	)	)	PUNCT
ejpam-6574	90	45	(	(	PUNCT
ejpam-6574	90	46	τ1	τ1	NOUN
ejpam-6574	90	47	,	,	PUNCT
ejpam-6574	90	48	τ2)-pint((τ1	τ2)-pint((τ1	PROPN
ejpam-6574	90	49	,	,	PUNCT
ejpam-6574	90	50	τ2)-bcl(a	τ2)-bcl(a	NOUN
ejpam-6574	90	51	)	)	PUNCT
ejpam-6574	90	52	)	)	PUNCT
ejpam-6574	91	1	=	=	PRON
ejpam-6574	91	2	(	(	PUNCT
ejpam-6574	91	3	τ1	τ1	PROPN
ejpam-6574	91	4	,	,	PUNCT
ejpam-6574	91	5	τ2)-bcl((τ1	τ2)-bcl((τ1	NUM
ejpam-6574	91	6	,	,	PUNCT
ejpam-6574	91	7	τ2)-pint(a	τ2)-pint(a	NOUN
ejpam-6574	91	8	)	)	PUNCT
ejpam-6574	91	9	)	)	PUNCT
ejpam-6574	91	10	.	.	PUNCT
ejpam-6574	92	1	for	for	ADP
ejpam-6574	92	2	a	a	DET
ejpam-6574	92	3	subset	subset	NOUN
ejpam-6574	92	4	a	a	PRON
ejpam-6574	92	5	of	of	ADP
ejpam-6574	92	6	a	a	DET
ejpam-6574	92	7	bitopological	bitopological	ADJ
ejpam-6574	92	8	space	space	NOUN
ejpam-6574	92	9	(	(	PUNCT
ejpam-6574	92	10	x	x	NOUN
ejpam-6574	92	11	,	,	PUNCT
ejpam-6574	92	12	τ1	τ1	NOUN
ejpam-6574	92	13	,	,	PUNCT
ejpam-6574	92	14	τ2	τ2	PROPN
ejpam-6574	92	15	)	)	PUNCT
ejpam-6574	92	16	,	,	PUNCT
ejpam-6574	92	17	a	a	DET
ejpam-6574	92	18	point	point	NOUN
ejpam-6574	92	19	x	x	X
ejpam-6574	92	20	∈	∈	NOUN
ejpam-6574	92	21	x	x	PUNCT
ejpam-6574	92	22	is	be	AUX
ejpam-6574	92	23	called	call	VERB
ejpam-6574	92	24	a	a	DET
ejpam-6574	92	25	θ(τ1	θ(τ1	NOUN
ejpam-6574	92	26	,	,	PUNCT
ejpam-6574	92	27	τ2)bcluster	τ2)bcluster	NOUN
ejpam-6574	92	28	point	point	NOUN
ejpam-6574	92	29	of	of	ADP
ejpam-6574	92	30	a	a	DET
ejpam-6574	92	31	if	if	NOUN
ejpam-6574	92	32	(	(	PUNCT
ejpam-6574	92	33	τ1	τ1	NOUN
ejpam-6574	92	34	,	,	PUNCT
ejpam-6574	92	35	τ2)-bcl(u)∩a	τ2)-bcl(u)∩a	PUNCT
ejpam-6574	92	36	̸=	̸=	NOUN
ejpam-6574	92	37	∅	∅	NOUN
ejpam-6574	92	38	for	for	ADP
ejpam-6574	92	39	every	every	DET
ejpam-6574	92	40	(	(	PUNCT
ejpam-6574	92	41	τ1	τ1	NOUN
ejpam-6574	92	42	,	,	PUNCT
ejpam-6574	92	43	τ2)b	τ2)b	NOUN
ejpam-6574	92	44	-	-	PUNCT
ejpam-6574	92	45	open	open	ADJ
ejpam-6574	92	46	set	set	NOUN
ejpam-6574	92	47	u	u	NOUN
ejpam-6574	92	48	of	of	ADP
ejpam-6574	92	49	x	x	SYM
ejpam-6574	92	50	containing	contain	VERB
ejpam-6574	92	51	x.	x.	NOUN
ejpam-6574	92	52	the	the	DET
ejpam-6574	92	53	set	set	NOUN
ejpam-6574	92	54	of	of	ADP
ejpam-6574	92	55	all	all	DET
ejpam-6574	92	56	θ(τ1	θ(τ1	NOUN
ejpam-6574	92	57	,	,	PUNCT
ejpam-6574	92	58	τ2)b	τ2)b	ADJ
ejpam-6574	92	59	-	-	PUNCT
ejpam-6574	92	60	cluster	cluster	NOUN
ejpam-6574	92	61	points	point	NOUN
ejpam-6574	92	62	of	of	ADP
ejpam-6574	92	63	a	a	PRON
ejpam-6574	92	64	is	be	AUX
ejpam-6574	92	65	called	call	VERB
ejpam-6574	92	66	the	the	DET
ejpam-6574	92	67	θ(τ1	θ(τ1	NOUN
ejpam-6574	92	68	,	,	PUNCT
ejpam-6574	92	69	τ2)b	τ2)b	NOUN
ejpam-6574	92	70	-	-	PUNCT
ejpam-6574	92	71	closure	closure	NOUN
ejpam-6574	92	72	of	of	ADP
ejpam-6574	92	73	a	a	PRON
ejpam-6574	92	74	and	and	CCONJ
ejpam-6574	92	75	is	be	AUX
ejpam-6574	92	76	denoted	denote	VERB
ejpam-6574	92	77	by	by	ADP
ejpam-6574	92	78	θ(τ1	θ(τ1	NOUN
ejpam-6574	92	79	,	,	PUNCT
ejpam-6574	92	80	τ2)b	τ2)b	NOUN
ejpam-6574	92	81	-	-	PUNCT
ejpam-6574	92	82	cl(a	cl(a	NUM
ejpam-6574	92	83	)	)	PUNCT
ejpam-6574	92	84	.	.	PUNCT
ejpam-6574	93	1	if	if	SCONJ
ejpam-6574	93	2	a	a	DET
ejpam-6574	93	3	=	=	NOUN
ejpam-6574	93	4	θ(τ1	θ(τ1	NOUN
ejpam-6574	93	5	,	,	PUNCT
ejpam-6574	93	6	τ2)b	τ2)b	NOUN
ejpam-6574	93	7	-	-	PUNCT
ejpam-6574	93	8	cl(a	cl(a	NUM
ejpam-6574	93	9	)	)	PUNCT
ejpam-6574	93	10	,	,	PUNCT
ejpam-6574	93	11	then	then	ADV
ejpam-6574	93	12	a	a	PRON
ejpam-6574	93	13	is	be	AUX
ejpam-6574	93	14	called	call	VERB
ejpam-6574	93	15	θ(τ1	θ(τ1	NOUN
ejpam-6574	93	16	,	,	PUNCT
ejpam-6574	93	17	τ2)b	τ2)b	NOUN
ejpam-6574	93	18	-	-	PUNCT
ejpam-6574	93	19	closed	closed	ADJ
ejpam-6574	93	20	.	.	PUNCT
ejpam-6574	94	1	the	the	DET
ejpam-6574	94	2	complement	complement	NOUN
ejpam-6574	94	3	of	of	ADP
ejpam-6574	94	4	a	a	DET
ejpam-6574	94	5	θ(τ1	θ(τ1	NOUN
ejpam-6574	94	6	,	,	PUNCT
ejpam-6574	94	7	τ2)b	τ2)b	NOUN
ejpam-6574	94	8	-	-	PUNCT
ejpam-6574	94	9	closed	close	VERB
ejpam-6574	94	10	set	set	NOUN
ejpam-6574	94	11	is	be	AUX
ejpam-6574	94	12	called	call	VERB
ejpam-6574	94	13	θ(τ1	θ(τ1	NOUN
ejpam-6574	94	14	,	,	PUNCT
ejpam-6574	94	15	τ2)b	τ2)b	NOUN
ejpam-6574	94	16	-	-	PUNCT
ejpam-6574	94	17	open	open	ADJ
ejpam-6574	94	18	.	.	PUNCT
ejpam-6574	95	1	the	the	DET
ejpam-6574	95	2	θ(τ1	θ(τ1	NOUN
ejpam-6574	95	3	,	,	PUNCT
ejpam-6574	95	4	τ2)b	τ2)b	ADJ
ejpam-6574	95	5	-	-	PUNCT
ejpam-6574	95	6	interior	interior	NOUN
ejpam-6574	95	7	of	of	ADP
ejpam-6574	95	8	a	a	PRON
ejpam-6574	95	9	is	be	AUX
ejpam-6574	95	10	defined	define	VERB
ejpam-6574	95	11	by	by	ADP
ejpam-6574	95	12	the	the	DET
ejpam-6574	95	13	union	union	NOUN
ejpam-6574	95	14	of	of	ADP
ejpam-6574	95	15	all	all	DET
ejpam-6574	95	16	θ(τ2	θ(τ2	ADJ
ejpam-6574	95	17	,	,	PUNCT
ejpam-6574	95	18	τ1)b	τ1)b	NOUN
ejpam-6574	95	19	-	-	PUNCT
ejpam-6574	95	20	open	open	ADJ
ejpam-6574	95	21	sets	set	NOUN
ejpam-6574	95	22	of	of	ADP
ejpam-6574	95	23	x	x	PUNCT
ejpam-6574	95	24	contained	contain	VERB
ejpam-6574	95	25	in	in	ADP
ejpam-6574	95	26	a	a	PRON
ejpam-6574	95	27	and	and	CCONJ
ejpam-6574	95	28	is	be	AUX
ejpam-6574	95	29	denoted	denote	VERB
ejpam-6574	95	30	by	by	ADP
ejpam-6574	95	31	θ(τ1	θ(τ1	NOUN
ejpam-6574	95	32	,	,	PUNCT
ejpam-6574	95	33	τ2)b	τ2)b	NOUN
ejpam-6574	95	34	-	-	PUNCT
ejpam-6574	95	35	int(a	int(a	NOUN
ejpam-6574	95	36	)	)	PUNCT
ejpam-6574	95	37	.	.	PUNCT
ejpam-6574	96	1	lemma	lemma	PROPN
ejpam-6574	96	2	5	5	NUM
ejpam-6574	96	3	.	.	PUNCT
ejpam-6574	97	1	for	for	ADP
ejpam-6574	97	2	subsets	subset	NOUN
ejpam-6574	97	3	a	a	PRON
ejpam-6574	97	4	and	and	CCONJ
ejpam-6574	97	5	cγ(γ	cγ(γ	CCONJ
ejpam-6574	97	6	∈	∈	PROPN
ejpam-6574	97	7	∇	∇	PROPN
ejpam-6574	97	8	)	)	PUNCT
ejpam-6574	97	9	of	of	ADP
ejpam-6574	97	10	a	a	DET
ejpam-6574	97	11	bitopological	bitopological	ADJ
ejpam-6574	97	12	space	space	NOUN
ejpam-6574	97	13	(	(	PUNCT
ejpam-6574	97	14	x	x	NOUN
ejpam-6574	97	15	,	,	PUNCT
ejpam-6574	97	16	τ1	τ1	NOUN
ejpam-6574	97	17	,	,	PUNCT
ejpam-6574	97	18	τ2	τ2	NOUN
ejpam-6574	97	19	)	)	PUNCT
ejpam-6574	97	20	,	,	PUNCT
ejpam-6574	97	21	the	the	DET
ejpam-6574	97	22	following	follow	VERB
ejpam-6574	97	23	properties	property	NOUN
ejpam-6574	97	24	hold	hold	VERB
ejpam-6574	97	25	:	:	PUNCT
ejpam-6574	97	26	(	(	PUNCT
ejpam-6574	97	27	1	1	X
ejpam-6574	97	28	)	)	PUNCT
ejpam-6574	97	29	if	if	SCONJ
ejpam-6574	97	30	cγ	cγ	PROPN
ejpam-6574	97	31	is	be	AUX
ejpam-6574	97	32	θ(τ1	θ(τ1	NOUN
ejpam-6574	97	33	,	,	PUNCT
ejpam-6574	97	34	τ2)b	τ2)b	NOUN
ejpam-6574	97	35	-	-	PUNCT
ejpam-6574	97	36	open	open	ADJ
ejpam-6574	97	37	for	for	ADP
ejpam-6574	97	38	each	each	DET
ejpam-6574	97	39	γ	γ	PROPN
ejpam-6574	97	40	∈	∈	PROPN
ejpam-6574	97	41	∇	∇	X
ejpam-6574	97	42	,	,	PUNCT
ejpam-6574	97	43	then	then	ADV
ejpam-6574	97	44	∪γ∈∇cγ	∪γ∈∇cγ	PROPN
ejpam-6574	97	45	is	be	AUX
ejpam-6574	97	46	θ(τ1	θ(τ1	NOUN
ejpam-6574	97	47	,	,	PUNCT
ejpam-6574	97	48	τ2)b	τ2)b	NOUN
ejpam-6574	97	49	-	-	PUNCT
ejpam-6574	97	50	open	open	ADJ
ejpam-6574	97	51	.	.	PUNCT
ejpam-6574	98	1	(	(	PUNCT
ejpam-6574	98	2	2	2	X
ejpam-6574	98	3	)	)	PUNCT
ejpam-6574	98	4	if	if	SCONJ
ejpam-6574	98	5	a	a	PRON
ejpam-6574	98	6	is	be	AUX
ejpam-6574	98	7	(	(	PUNCT
ejpam-6574	98	8	τ1	τ1	NOUN
ejpam-6574	98	9	,	,	PUNCT
ejpam-6574	98	10	τ2)b	τ2)b	NOUN
ejpam-6574	98	11	-	-	PUNCT
ejpam-6574	98	12	closed	closed	ADJ
ejpam-6574	98	13	,	,	PUNCT
ejpam-6574	98	14	then	then	ADV
ejpam-6574	98	15	(	(	PUNCT
ejpam-6574	98	16	τ1	τ1	NOUN
ejpam-6574	98	17	,	,	PUNCT
ejpam-6574	98	18	τ2)-bint(a	τ2)-bint(a	NUM
ejpam-6574	98	19	)	)	PUNCT
ejpam-6574	98	20	=	=	SYM
ejpam-6574	98	21	θ(τ1	θ(τ1	NOUN
ejpam-6574	98	22	,	,	PUNCT
ejpam-6574	98	23	τ2)b	τ2)b	NOUN
ejpam-6574	98	24	-	-	PUNCT
ejpam-6574	98	25	int(a	int(a	NOUN
ejpam-6574	98	26	)	)	PUNCT
ejpam-6574	98	27	.	.	PUNCT
ejpam-6574	99	1	(	(	PUNCT
ejpam-6574	99	2	3	3	X
ejpam-6574	99	3	)	)	PUNCT
ejpam-6574	99	4	θ(τ1	θ(τ1	NOUN
ejpam-6574	99	5	,	,	PUNCT
ejpam-6574	99	6	τ2)b	τ2)b	NOUN
ejpam-6574	99	7	-	-	PUNCT
ejpam-6574	99	8	cl(a	cl(a	NUM
ejpam-6574	99	9	)	)	PUNCT
ejpam-6574	99	10	is	be	AUX
ejpam-6574	99	11	θ(τ1	θ(τ1	NOUN
ejpam-6574	99	12	,	,	PUNCT
ejpam-6574	99	13	τ2)b	τ2)b	NOUN
ejpam-6574	99	14	-	-	PUNCT
ejpam-6574	99	15	closed	closed	ADJ
ejpam-6574	99	16	.	.	PUNCT
ejpam-6574	100	1	3	3	X
ejpam-6574	100	2	.	.	X
ejpam-6574	100	3	weakly	weakly	ADJ
ejpam-6574	100	4	θ(τ1	θ(τ1	NOUN
ejpam-6574	100	5	,	,	PUNCT
ejpam-6574	100	6	τ2)b	τ2)b	ADJ
ejpam-6574	100	7	-	-	PUNCT
ejpam-6574	100	8	open	open	ADJ
ejpam-6574	100	9	functions	function	NOUN
ejpam-6574	100	10	in	in	ADP
ejpam-6574	100	11	this	this	DET
ejpam-6574	100	12	section	section	NOUN
ejpam-6574	100	13	,	,	PUNCT
ejpam-6574	100	14	we	we	PRON
ejpam-6574	100	15	introduce	introduce	VERB
ejpam-6574	100	16	the	the	DET
ejpam-6574	100	17	concept	concept	NOUN
ejpam-6574	100	18	of	of	ADP
ejpam-6574	100	19	weakly	weakly	ADJ
ejpam-6574	100	20	θ(τ1	θ(τ1	NOUN
ejpam-6574	100	21	,	,	PUNCT
ejpam-6574	100	22	τ2)b	τ2)b	ADJ
ejpam-6574	100	23	-	-	PUNCT
ejpam-6574	100	24	open	open	ADJ
ejpam-6574	100	25	functions	function	NOUN
ejpam-6574	100	26	.	.	PUNCT
ejpam-6574	101	1	moreover	moreover	ADV
ejpam-6574	101	2	,	,	PUNCT
ejpam-6574	101	3	some	some	DET
ejpam-6574	101	4	characterizations	characterization	NOUN
ejpam-6574	101	5	of	of	ADP
ejpam-6574	101	6	weakly	weakly	ADJ
ejpam-6574	101	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	101	8	,	,	PUNCT
ejpam-6574	101	9	τ2)b	τ2)b	ADJ
ejpam-6574	101	10	-	-	PUNCT
ejpam-6574	101	11	open	open	ADJ
ejpam-6574	101	12	functions	function	NOUN
ejpam-6574	101	13	are	be	AUX
ejpam-6574	101	14	discussed	discuss	VERB
ejpam-6574	101	15	.	.	PUNCT
ejpam-6574	102	1	definition	definition	NOUN
ejpam-6574	102	2	1	1	NUM
ejpam-6574	102	3	.	.	PUNCT
ejpam-6574	103	1	a	a	DET
ejpam-6574	103	2	function	function	NOUN
ejpam-6574	103	3	f	f	NOUN
ejpam-6574	103	4	:	:	PUNCT
ejpam-6574	103	5	(	(	PUNCT
ejpam-6574	103	6	x	x	NOUN
ejpam-6574	103	7	,	,	PUNCT
ejpam-6574	103	8	τ1	τ1	NOUN
ejpam-6574	103	9	,	,	PUNCT
ejpam-6574	103	10	τ2	τ2	NOUN
ejpam-6574	103	11	)	)	PUNCT
ejpam-6574	103	12	→	→	SYM
ejpam-6574	103	13	(	(	PUNCT
ejpam-6574	103	14	y	y	PROPN
ejpam-6574	103	15	,	,	PUNCT
ejpam-6574	103	16	σ1	σ1	PROPN
ejpam-6574	103	17	,	,	PUNCT
ejpam-6574	103	18	σ2	σ2	PROPN
ejpam-6574	103	19	)	)	PUNCT
ejpam-6574	103	20	is	be	AUX
ejpam-6574	103	21	said	say	VERB
ejpam-6574	103	22	to	to	PART
ejpam-6574	103	23	be	be	AUX
ejpam-6574	103	24	weakly	weakly	ADJ
ejpam-6574	103	25	θ(τ1	θ(τ1	NOUN
ejpam-6574	103	26	,	,	PUNCT
ejpam-6574	103	27	τ2)b	τ2)b	NOUN
ejpam-6574	103	28	-	-	PUNCT
ejpam-6574	103	29	open	open	ADJ
ejpam-6574	103	30	if	if	SCONJ
ejpam-6574	103	31	f(u	f(u	PROPN
ejpam-6574	103	32	)	)	PUNCT
ejpam-6574	103	33	⊆	⊆	NUM
ejpam-6574	103	34	θ(σ1	θ(σ1	NOUN
ejpam-6574	103	35	,	,	PUNCT
ejpam-6574	103	36	σ2)b	σ2)b	NOUN
ejpam-6574	103	37	-	-	PUNCT
ejpam-6574	103	38	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	103	39	-	-	NOUN
ejpam-6574	103	40	cl(u	cl(u	NOUN
ejpam-6574	103	41	)	)	PUNCT
ejpam-6574	103	42	)	)	PUNCT
ejpam-6574	103	43	)	)	PUNCT
ejpam-6574	103	44	for	for	ADP
ejpam-6574	103	45	every	every	DET
ejpam-6574	103	46	τ1τ2	τ1τ2	ADJ
ejpam-6574	103	47	-	-	ADJ
ejpam-6574	103	48	open	open	ADJ
ejpam-6574	103	49	set	set	ADJ
ejpam-6574	103	50	u	u	NOUN
ejpam-6574	103	51	of	of	ADP
ejpam-6574	103	52	x.	x.	PROPN
ejpam-6574	103	53	theorem	theorem	VERB
ejpam-6574	103	54	1	1	NUM
ejpam-6574	103	55	.	.	PUNCT
ejpam-6574	103	56	for	for	ADP
ejpam-6574	103	57	a	a	DET
ejpam-6574	103	58	function	function	NOUN
ejpam-6574	103	59	f	f	NOUN
ejpam-6574	103	60	:	:	PUNCT
ejpam-6574	103	61	(	(	PUNCT
ejpam-6574	103	62	x	x	NOUN
ejpam-6574	103	63	,	,	PUNCT
ejpam-6574	103	64	τ1	τ1	NOUN
ejpam-6574	103	65	,	,	PUNCT
ejpam-6574	103	66	τ2	τ2	NOUN
ejpam-6574	103	67	)	)	PUNCT
ejpam-6574	103	68	→	→	SYM
ejpam-6574	103	69	(	(	PUNCT
ejpam-6574	103	70	y	y	PROPN
ejpam-6574	103	71	,	,	PUNCT
ejpam-6574	103	72	σ1	σ1	PROPN
ejpam-6574	103	73	,	,	PUNCT
ejpam-6574	103	74	σ2	σ2	NOUN
ejpam-6574	103	75	)	)	PUNCT
ejpam-6574	103	76	,	,	PUNCT
ejpam-6574	103	77	the	the	DET
ejpam-6574	103	78	following	follow	VERB
ejpam-6574	103	79	properties	property	NOUN
ejpam-6574	103	80	are	be	AUX
ejpam-6574	103	81	equivalent	equivalent	ADJ
ejpam-6574	103	82	:	:	PUNCT
ejpam-6574	103	83	(	(	PUNCT
ejpam-6574	103	84	1	1	X
ejpam-6574	103	85	)	)	PUNCT
ejpam-6574	103	86	f	f	PROPN
ejpam-6574	103	87	is	be	AUX
ejpam-6574	103	88	weakly	weakly	ADJ
ejpam-6574	103	89	θ(τ1	θ(τ1	NOUN
ejpam-6574	103	90	,	,	PUNCT
ejpam-6574	103	91	τ2)b	τ2)b	NOUN
ejpam-6574	103	92	-	-	PUNCT
ejpam-6574	103	93	open	open	ADJ
ejpam-6574	103	94	;	;	PUNCT
ejpam-6574	103	95	(	(	PUNCT
ejpam-6574	103	96	2	2	X
ejpam-6574	103	97	)	)	PUNCT
ejpam-6574	103	98	f((τ1	f((τ1	PROPN
ejpam-6574	103	99	,	,	PUNCT
ejpam-6574	103	100	τ2)θ	τ2)θ	NOUN
ejpam-6574	103	101	-	-	PUNCT
ejpam-6574	103	102	int(a	int(a	NOUN
ejpam-6574	103	103	)	)	PUNCT
ejpam-6574	103	104	)	)	PUNCT
ejpam-6574	104	1	⊆	⊆	NUM
ejpam-6574	104	2	θ(σ1	θ(σ1	NOUN
ejpam-6574	104	3	,	,	PUNCT
ejpam-6574	104	4	σ2)b	σ2)b	NOUN
ejpam-6574	104	5	-	-	PUNCT
ejpam-6574	104	6	int(f(a	int(f(a	NOUN
ejpam-6574	104	7	)	)	PUNCT
ejpam-6574	104	8	)	)	PUNCT
ejpam-6574	104	9	for	for	ADP
ejpam-6574	104	10	every	every	DET
ejpam-6574	104	11	subset	subset	NOUN
ejpam-6574	104	12	a	a	PRON
ejpam-6574	104	13	of	of	ADP
ejpam-6574	104	14	x	x	PRON
ejpam-6574	104	15	;	;	PUNCT
ejpam-6574	104	16	(	(	PUNCT
ejpam-6574	104	17	3	3	X
ejpam-6574	104	18	)	)	PUNCT
ejpam-6574	104	19	(	(	PUNCT
ejpam-6574	104	20	τ1	τ1	NOUN
ejpam-6574	104	21	,	,	PUNCT
ejpam-6574	104	22	τ2)θ	τ2)θ	NOUN
ejpam-6574	104	23	-	-	PUNCT
ejpam-6574	104	24	int(f	int(f	PROPN
ejpam-6574	104	25	−1(b	−1(b	NOUN
ejpam-6574	104	26	)	)	PUNCT
ejpam-6574	104	27	)	)	PUNCT
ejpam-6574	104	28	⊆	⊆	NUM
ejpam-6574	104	29	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	104	30	,	,	PUNCT
ejpam-6574	104	31	σ2)b	σ2)b	NOUN
ejpam-6574	104	32	-	-	PUNCT
ejpam-6574	104	33	int(b	int(b	NOUN
ejpam-6574	104	34	)	)	PUNCT
ejpam-6574	104	35	)	)	PUNCT
ejpam-6574	104	36	for	for	ADP
ejpam-6574	104	37	every	every	DET
ejpam-6574	104	38	subset	subset	NOUN
ejpam-6574	104	39	b	b	PROPN
ejpam-6574	104	40	of	of	ADP
ejpam-6574	104	41	y	y	PROPN
ejpam-6574	104	42	;	;	PUNCT
ejpam-6574	104	43	(	(	PUNCT
ejpam-6574	104	44	4	4	X
ejpam-6574	104	45	)	)	PUNCT
ejpam-6574	104	46	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	104	47	,	,	PUNCT
ejpam-6574	104	48	σ2)b	σ2)b	NOUN
ejpam-6574	104	49	-	-	PUNCT
ejpam-6574	104	50	cl(b	cl(b	NOUN
ejpam-6574	104	51	)	)	PUNCT
ejpam-6574	104	52	)	)	PUNCT
ejpam-6574	105	1	⊆	⊆	NUM
ejpam-6574	105	2	(	(	PUNCT
ejpam-6574	105	3	τ1	τ1	NOUN
ejpam-6574	105	4	,	,	PUNCT
ejpam-6574	105	5	τ2)θ	τ2)θ	PROPN
ejpam-6574	105	6	-	-	PUNCT
ejpam-6574	105	7	cl(f	cl(f	NOUN
ejpam-6574	105	8	−1(b	−1(b	NOUN
ejpam-6574	105	9	)	)	PUNCT
ejpam-6574	105	10	)	)	PUNCT
ejpam-6574	105	11	for	for	ADP
ejpam-6574	105	12	every	every	DET
ejpam-6574	105	13	subset	subset	NOUN
ejpam-6574	105	14	b	b	PROPN
ejpam-6574	105	15	of	of	ADP
ejpam-6574	105	16	y	y	PROPN
ejpam-6574	105	17	.	.	PUNCT
ejpam-6574	106	1	proof	proof	NOUN
ejpam-6574	106	2	.	.	PUNCT
ejpam-6574	107	1	(	(	PUNCT
ejpam-6574	107	2	1	1	X
ejpam-6574	107	3	)	)	PUNCT
ejpam-6574	107	4	⇒	⇒	NOUN
ejpam-6574	107	5	(	(	PUNCT
ejpam-6574	107	6	2	2	NUM
ejpam-6574	107	7	):	):	PUNCT
ejpam-6574	107	8	let	let	VERB
ejpam-6574	107	9	a	a	PRON
ejpam-6574	107	10	be	be	AUX
ejpam-6574	107	11	any	any	DET
ejpam-6574	107	12	subset	subset	NOUN
ejpam-6574	107	13	of	of	ADP
ejpam-6574	107	14	x	x	PUNCT
ejpam-6574	107	15	and	and	CCONJ
ejpam-6574	107	16	x	x	PROPN
ejpam-6574	107	17	∈	∈	PROPN
ejpam-6574	107	18	(	(	PUNCT
ejpam-6574	107	19	τ1	τ1	NOUN
ejpam-6574	107	20	,	,	PUNCT
ejpam-6574	107	21	τ2)θ	τ2)θ	NOUN
ejpam-6574	107	22	-	-	PUNCT
ejpam-6574	107	23	int(a	int(a	NOUN
ejpam-6574	107	24	)	)	PUNCT
ejpam-6574	107	25	.	.	PUNCT
ejpam-6574	108	1	then	then	ADV
ejpam-6574	108	2	,	,	PUNCT
ejpam-6574	108	3	there	there	PRON
ejpam-6574	108	4	exists	exist	VERB
ejpam-6574	108	5	a	a	DET
ejpam-6574	108	6	τ1τ2	τ1τ2	NOUN
ejpam-6574	108	7	-	-	ADJ
ejpam-6574	108	8	open	open	ADJ
ejpam-6574	108	9	set	set	ADJ
ejpam-6574	108	10	u	u	NOUN
ejpam-6574	108	11	of	of	ADP
ejpam-6574	108	12	x	x	SYM
ejpam-6574	108	13	such	such	ADJ
ejpam-6574	108	14	that	that	SCONJ
ejpam-6574	108	15	x	x	SYM
ejpam-6574	108	16	∈	∈	PROPN
ejpam-6574	108	17	u	u	NOUN
ejpam-6574	108	18	⊆	⊆	NUM
ejpam-6574	108	19	τ1τ2	τ1τ2	NOUN
ejpam-6574	108	20	-	-	NOUN
ejpam-6574	108	21	cl(u	cl(u	ADJ
ejpam-6574	108	22	)	)	PUNCT
ejpam-6574	108	23	⊆	⊆	NUM
ejpam-6574	108	24	a.	a.	NOUN
ejpam-6574	108	25	therefore	therefore	ADV
ejpam-6574	108	26	,	,	PUNCT
ejpam-6574	108	27	f(x	f(x	PROPN
ejpam-6574	108	28	)	)	PUNCT
ejpam-6574	108	29	∈	∈	PROPN
ejpam-6574	108	30	f(u	f(u	PROPN
ejpam-6574	108	31	)	)	PUNCT
ejpam-6574	108	32	⊆	⊆	NUM
ejpam-6574	108	33	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	108	34	-	-	PUNCT
ejpam-6574	108	35	cl(u	cl(u	NOUN
ejpam-6574	108	36	)	)	PUNCT
ejpam-6574	108	37	)	)	PUNCT
ejpam-6574	109	1	⊆	⊆	NUM
ejpam-6574	109	2	f(a	f(a	NOUN
ejpam-6574	109	3	)	)	PUNCT
ejpam-6574	109	4	.	.	PUNCT
ejpam-6574	110	1	since	since	SCONJ
ejpam-6574	110	2	f	f	PROPN
ejpam-6574	110	3	is	be	AUX
ejpam-6574	110	4	weakly	weakly	ADJ
ejpam-6574	110	5	θ(τ1	θ(τ1	NOUN
ejpam-6574	110	6	,	,	PUNCT
ejpam-6574	110	7	τ2)b	τ2)b	NOUN
ejpam-6574	110	8	-	-	PUNCT
ejpam-6574	110	9	open	open	ADJ
ejpam-6574	110	10	,	,	PUNCT
ejpam-6574	110	11	we	we	PRON
ejpam-6574	110	12	have	have	VERB
ejpam-6574	110	13	f(u	f(u	PROPN
ejpam-6574	110	14	)	)	PUNCT
ejpam-6574	110	15	⊆	⊆	NUM
ejpam-6574	110	16	θ(σ1	θ(σ1	NOUN
ejpam-6574	110	17	,	,	PUNCT
ejpam-6574	110	18	σ2)b	σ2)b	NOUN
ejpam-6574	110	19	-	-	PUNCT
ejpam-6574	110	20	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	110	21	-	-	NOUN
ejpam-6574	110	22	cl(u	cl(u	NOUN
ejpam-6574	110	23	)	)	PUNCT
ejpam-6574	110	24	)	)	PUNCT
ejpam-6574	110	25	)	)	PUNCT
ejpam-6574	110	26	⊆	⊆	NUM
ejpam-6574	110	27	θ(σ1	θ(σ1	NOUN
ejpam-6574	110	28	,	,	PUNCT
ejpam-6574	110	29	σ2)b	σ2)b	NOUN
ejpam-6574	110	30	-	-	PUNCT
ejpam-6574	110	31	int(f(a	int(f(a	NOUN
ejpam-6574	110	32	)	)	PUNCT
ejpam-6574	110	33	)	)	PUNCT
ejpam-6574	110	34	.	.	PUNCT
ejpam-6574	111	1	n.	n.	PROPN
ejpam-6574	111	2	chutiman	chutiman	PROPN
ejpam-6574	111	3	,	,	PUNCT
ejpam-6574	111	4	a.	a.	PROPN
ejpam-6574	111	5	sama	sama	PROPN
ejpam-6574	111	6	-	-	PUNCT
ejpam-6574	111	7	ae	ae	PROPN
ejpam-6574	111	8	,	,	PUNCT
ejpam-6574	111	9	c.	c.	PROPN
ejpam-6574	111	10	boonpok	boonpok	PROPN
ejpam-6574	111	11	/	/	SYM
ejpam-6574	111	12	eur	eur	PROPN
ejpam-6574	111	13	.	.	PUNCT
ejpam-6574	112	1	j.	j.	PROPN
ejpam-6574	112	2	pure	pure	PROPN
ejpam-6574	112	3	appl	appl	PROPN
ejpam-6574	112	4	.	.	PROPN
ejpam-6574	112	5	math	math	PROPN
ejpam-6574	112	6	,	,	PUNCT
ejpam-6574	112	7	18	18	NUM
ejpam-6574	112	8	(	(	PUNCT
ejpam-6574	112	9	3	3	NUM
ejpam-6574	112	10	)	)	PUNCT
ejpam-6574	112	11	(	(	PUNCT
ejpam-6574	112	12	2025	2025	NUM
ejpam-6574	112	13	)	)	PUNCT
ejpam-6574	112	14	,	,	PUNCT
ejpam-6574	112	15	6574	6574	NUM
ejpam-6574	112	16	5	5	NUM
ejpam-6574	112	17	of	of	ADP
ejpam-6574	112	18	10	10	NUM
ejpam-6574	112	19	this	this	PRON
ejpam-6574	112	20	implies	imply	VERB
ejpam-6574	112	21	that	that	SCONJ
ejpam-6574	112	22	f(x	f(x	PROPN
ejpam-6574	112	23	)	)	PUNCT
ejpam-6574	112	24	∈	∈	PROPN
ejpam-6574	112	25	θ(σ1	θ(σ1	NOUN
ejpam-6574	112	26	,	,	PUNCT
ejpam-6574	112	27	σ2)b	σ2)b	NOUN
ejpam-6574	112	28	-	-	PUNCT
ejpam-6574	112	29	int(f(a	int(f(a	NOUN
ejpam-6574	112	30	)	)	PUNCT
ejpam-6574	112	31	)	)	PUNCT
ejpam-6574	112	32	.	.	PUNCT
ejpam-6574	113	1	thus	thus	ADV
ejpam-6574	113	2	,	,	PUNCT
ejpam-6574	113	3	x	x	SYM
ejpam-6574	113	4	∈	∈	NOUN
ejpam-6574	113	5	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	113	6	,	,	PUNCT
ejpam-6574	113	7	σ2)b	σ2)b	NOUN
ejpam-6574	113	8	-	-	PUNCT
ejpam-6574	113	9	int(f(a	int(f(a	NOUN
ejpam-6574	113	10	)	)	PUNCT
ejpam-6574	113	11	)	)	PUNCT
ejpam-6574	113	12	)	)	PUNCT
ejpam-6574	113	13	and	and	CCONJ
ejpam-6574	113	14	hence	hence	ADV
ejpam-6574	113	15	(	(	PUNCT
ejpam-6574	113	16	τ1	τ1	NOUN
ejpam-6574	113	17	,	,	PUNCT
ejpam-6574	113	18	τ2)θ	τ2)θ	ADJ
ejpam-6574	113	19	-	-	PUNCT
ejpam-6574	113	20	int(a	int(a	PROPN
ejpam-6574	113	21	)	)	PUNCT
ejpam-6574	113	22	⊆	⊆	NUM
ejpam-6574	113	23	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	113	24	,	,	PUNCT
ejpam-6574	113	25	σ2)b	σ2)b	NOUN
ejpam-6574	113	26	-	-	PUNCT
ejpam-6574	113	27	int(f(a	int(f(a	NOUN
ejpam-6574	113	28	)	)	PUNCT
ejpam-6574	113	29	)	)	PUNCT
ejpam-6574	113	30	)	)	PUNCT
ejpam-6574	113	31	.	.	PUNCT
ejpam-6574	114	1	this	this	PRON
ejpam-6574	114	2	shows	show	VERB
ejpam-6574	114	3	that	that	SCONJ
ejpam-6574	114	4	f((τ1	f((τ1	PROPN
ejpam-6574	114	5	,	,	PUNCT
ejpam-6574	114	6	τ2)θ	τ2)θ	NOUN
ejpam-6574	114	7	-	-	PUNCT
ejpam-6574	114	8	int(a	int(a	NOUN
ejpam-6574	114	9	)	)	PUNCT
ejpam-6574	114	10	)	)	PUNCT
ejpam-6574	114	11	⊆	⊆	NUM
ejpam-6574	114	12	θ(σ1	θ(σ1	NOUN
ejpam-6574	114	13	,	,	PUNCT
ejpam-6574	114	14	σ2)b	σ2)b	NOUN
ejpam-6574	114	15	-	-	PUNCT
ejpam-6574	114	16	int(f(a	int(f(a	NOUN
ejpam-6574	114	17	)	)	PUNCT
ejpam-6574	114	18	)	)	PUNCT
ejpam-6574	114	19	.	.	PUNCT
ejpam-6574	115	1	(	(	PUNCT
ejpam-6574	115	2	2	2	X
ejpam-6574	115	3	)	)	PUNCT
ejpam-6574	115	4	⇒	⇒	NOUN
ejpam-6574	115	5	(	(	PUNCT
ejpam-6574	115	6	3	3	NUM
ejpam-6574	115	7	):	):	PUNCT
ejpam-6574	115	8	let	let	VERB
ejpam-6574	115	9	b	b	X
ejpam-6574	115	10	be	be	AUX
ejpam-6574	115	11	any	any	DET
ejpam-6574	115	12	subset	subset	NOUN
ejpam-6574	115	13	of	of	ADP
ejpam-6574	115	14	y	y	PROPN
ejpam-6574	115	15	.	.	PUNCT
ejpam-6574	116	1	thus	thus	ADV
ejpam-6574	116	2	by	by	ADP
ejpam-6574	116	3	(	(	PUNCT
ejpam-6574	116	4	2	2	NUM
ejpam-6574	116	5	)	)	PUNCT
ejpam-6574	116	6	,	,	PUNCT
ejpam-6574	116	7	f((τ1	f((τ1	PROPN
ejpam-6574	116	8	,	,	PUNCT
ejpam-6574	116	9	τ2)θ	τ2)θ	ADJ
ejpam-6574	116	10	-	-	PUNCT
ejpam-6574	116	11	int(f	int(f	PROPN
ejpam-6574	116	12	−1(b	−1(b	NOUN
ejpam-6574	116	13	)	)	PUNCT
ejpam-6574	116	14	)	)	PUNCT
ejpam-6574	116	15	)	)	PUNCT
ejpam-6574	117	1	⊆	⊆	NUM
ejpam-6574	117	2	θ(σ1	θ(σ1	NOUN
ejpam-6574	117	3	,	,	PUNCT
ejpam-6574	117	4	σ2)b	σ2)b	NOUN
ejpam-6574	117	5	-	-	PUNCT
ejpam-6574	117	6	int(b	int(b	NOUN
ejpam-6574	117	7	)	)	PUNCT
ejpam-6574	117	8	and	and	CCONJ
ejpam-6574	117	9	so	so	ADV
ejpam-6574	117	10	(	(	PUNCT
ejpam-6574	117	11	τ1	τ1	NOUN
ejpam-6574	117	12	,	,	PUNCT
ejpam-6574	117	13	τ2)θ	τ2)θ	ADJ
ejpam-6574	117	14	-	-	PUNCT
ejpam-6574	117	15	int(f	int(f	PROPN
ejpam-6574	117	16	−1(b	−1(b	NOUN
ejpam-6574	117	17	)	)	PUNCT
ejpam-6574	117	18	)	)	PUNCT
ejpam-6574	117	19	⊆	⊆	NUM
ejpam-6574	117	20	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	117	21	,	,	PUNCT
ejpam-6574	117	22	σ2)b	σ2)b	NOUN
ejpam-6574	117	23	-	-	PUNCT
ejpam-6574	117	24	int(b	int(b	NOUN
ejpam-6574	117	25	)	)	PUNCT
ejpam-6574	117	26	)	)	PUNCT
ejpam-6574	117	27	.	.	PUNCT
ejpam-6574	118	1	(	(	PUNCT
ejpam-6574	118	2	3	3	X
ejpam-6574	118	3	)	)	PUNCT
ejpam-6574	118	4	⇒	⇒	NOUN
ejpam-6574	118	5	(	(	PUNCT
ejpam-6574	118	6	4	4	NUM
ejpam-6574	118	7	):	):	PUNCT
ejpam-6574	118	8	let	let	VERB
ejpam-6574	118	9	b	b	X
ejpam-6574	118	10	be	be	AUX
ejpam-6574	118	11	any	any	DET
ejpam-6574	118	12	subset	subset	NOUN
ejpam-6574	118	13	of	of	ADP
ejpam-6574	118	14	y	y	PROPN
ejpam-6574	118	15	.	.	PUNCT
ejpam-6574	119	1	using	use	VERB
ejpam-6574	119	2	(	(	PUNCT
ejpam-6574	119	3	3	3	NUM
ejpam-6574	119	4	)	)	PUNCT
ejpam-6574	119	5	,	,	PUNCT
ejpam-6574	119	6	we	we	PRON
ejpam-6574	119	7	have	have	VERB
ejpam-6574	119	8	x	x	X
ejpam-6574	119	9	−	−	PROPN
ejpam-6574	119	10	(	(	PUNCT
ejpam-6574	119	11	τ1	τ1	NOUN
ejpam-6574	119	12	,	,	PUNCT
ejpam-6574	119	13	τ2)θ	τ2)θ	PROPN
ejpam-6574	119	14	-	-	PUNCT
ejpam-6574	119	15	cl(f	cl(f	NOUN
ejpam-6574	119	16	−1(b	−1(b	NOUN
ejpam-6574	119	17	)	)	PUNCT
ejpam-6574	119	18	)	)	PUNCT
ejpam-6574	120	1	=	=	PRON
ejpam-6574	120	2	(	(	PUNCT
ejpam-6574	120	3	τ1	τ1	NOUN
ejpam-6574	120	4	,	,	PUNCT
ejpam-6574	120	5	τ2)θ	τ2)θ	ADJ
ejpam-6574	120	6	-	-	PUNCT
ejpam-6574	120	7	int(x	int(x	PROPN
ejpam-6574	120	8	−	−	NOUN
ejpam-6574	120	9	f−1(b	f−1(b	PROPN
ejpam-6574	120	10	)	)	PUNCT
ejpam-6574	120	11	)	)	PUNCT
ejpam-6574	121	1	=	=	PRON
ejpam-6574	121	2	(	(	PUNCT
ejpam-6574	121	3	τ1	τ1	NOUN
ejpam-6574	121	4	,	,	PUNCT
ejpam-6574	121	5	τ2)θ	τ2)θ	NOUN
ejpam-6574	121	6	-	-	PUNCT
ejpam-6574	121	7	int(f	int(f	PROPN
ejpam-6574	121	8	−1(y	−1(y	VERB
ejpam-6574	121	9	−b	−b	ADJ
ejpam-6574	121	10	)	)	PUNCT
ejpam-6574	121	11	)	)	PUNCT
ejpam-6574	122	1	⊆	⊆	NUM
ejpam-6574	122	2	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	122	3	,	,	PUNCT
ejpam-6574	122	4	σ2)b	σ2)b	NOUN
ejpam-6574	122	5	-	-	PUNCT
ejpam-6574	122	6	int(y	int(y	PROPN
ejpam-6574	122	7	−b	−b	NOUN
ejpam-6574	122	8	)	)	PUNCT
ejpam-6574	122	9	)	)	PUNCT
ejpam-6574	123	1	=	=	PUNCT
ejpam-6574	123	2	f−1(y	f−1(y	PROPN
ejpam-6574	123	3	−	−	NOUN
ejpam-6574	123	4	θ(σ1	θ(σ1	NOUN
ejpam-6574	123	5	,	,	PUNCT
ejpam-6574	123	6	σ2)b	σ2)b	NOUN
ejpam-6574	123	7	-	-	PUNCT
ejpam-6574	123	8	cl(b	cl(b	NOUN
ejpam-6574	123	9	)	)	PUNCT
ejpam-6574	123	10	)	)	PUNCT
ejpam-6574	124	1	=	=	PUNCT
ejpam-6574	124	2	x	x	PUNCT
ejpam-6574	125	1	−	−	NOUN
ejpam-6574	125	2	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	125	3	,	,	PUNCT
ejpam-6574	125	4	σ2)b	σ2)b	NOUN
ejpam-6574	125	5	-	-	PUNCT
ejpam-6574	125	6	cl(b	cl(b	NOUN
ejpam-6574	125	7	)	)	PUNCT
ejpam-6574	125	8	)	)	PUNCT
ejpam-6574	125	9	.	.	PUNCT
ejpam-6574	126	1	thus	thus	ADV
ejpam-6574	126	2	,	,	PUNCT
ejpam-6574	126	3	f−1(θ(σ1	f−1(θ(σ1	ADJ
ejpam-6574	126	4	,	,	PUNCT
ejpam-6574	126	5	σ2)b	σ2)b	NOUN
ejpam-6574	126	6	-	-	PUNCT
ejpam-6574	126	7	cl(b	cl(b	NOUN
ejpam-6574	126	8	)	)	PUNCT
ejpam-6574	126	9	)	)	PUNCT
ejpam-6574	126	10	⊆	⊆	NUM
ejpam-6574	126	11	(	(	PUNCT
ejpam-6574	126	12	τ1	τ1	NOUN
ejpam-6574	126	13	,	,	PUNCT
ejpam-6574	126	14	τ2)θ	τ2)θ	PROPN
ejpam-6574	126	15	-	-	PUNCT
ejpam-6574	126	16	cl(f	cl(f	NOUN
ejpam-6574	126	17	−1(b	−1(b	NOUN
ejpam-6574	126	18	)	)	PUNCT
ejpam-6574	126	19	)	)	PUNCT
ejpam-6574	126	20	.	.	PUNCT
ejpam-6574	127	1	(	(	PUNCT
ejpam-6574	127	2	4	4	X
ejpam-6574	127	3	)	)	PUNCT
ejpam-6574	127	4	⇒	⇒	NOUN
ejpam-6574	127	5	(	(	PUNCT
ejpam-6574	127	6	1	1	NUM
ejpam-6574	127	7	):	):	PUNCT
ejpam-6574	127	8	let	let	VERB
ejpam-6574	127	9	u	u	PRON
ejpam-6574	127	10	be	be	AUX
ejpam-6574	127	11	any	any	DET
ejpam-6574	127	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	127	13	-	-	ADJ
ejpam-6574	127	14	open	open	ADJ
ejpam-6574	127	15	set	set	NOUN
ejpam-6574	127	16	of	of	ADP
ejpam-6574	127	17	x.	x.	NOUN
ejpam-6574	127	18	by	by	ADP
ejpam-6574	127	19	(	(	PUNCT
ejpam-6574	127	20	4	4	NUM
ejpam-6574	127	21	)	)	PUNCT
ejpam-6574	127	22	,	,	PUNCT
ejpam-6574	127	23	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	127	24	,	,	PUNCT
ejpam-6574	127	25	σ2)b	σ2)b	NOUN
ejpam-6574	127	26	-	-	PUNCT
ejpam-6574	127	27	cl(y	cl(y	NOUN
ejpam-6574	127	28	−	−	NOUN
ejpam-6574	127	29	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	127	30	-	-	PUNCT
ejpam-6574	127	31	cl(u	cl(u	NOUN
ejpam-6574	127	32	)	)	PUNCT
ejpam-6574	127	33	)	)	PUNCT
ejpam-6574	127	34	)	)	PUNCT
ejpam-6574	127	35	)	)	PUNCT
ejpam-6574	128	1	⊆	⊆	X
ejpam-6574	128	2	(	(	PUNCT
ejpam-6574	128	3	τ1	τ1	NOUN
ejpam-6574	128	4	,	,	PUNCT
ejpam-6574	128	5	τ2)θ	τ2)θ	PROPN
ejpam-6574	128	6	-	-	PUNCT
ejpam-6574	128	7	cl(f	cl(f	NOUN
ejpam-6574	128	8	−1(y	−1(y	PUNCT
ejpam-6574	128	9	−	−	PROPN
ejpam-6574	128	10	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	128	11	-	-	PUNCT
ejpam-6574	128	12	cl(u	cl(u	NOUN
ejpam-6574	128	13	)	)	PUNCT
ejpam-6574	128	14	)	)	PUNCT
ejpam-6574	128	15	)	)	PUNCT
ejpam-6574	128	16	)	)	PUNCT
ejpam-6574	128	17	.	.	PUNCT
ejpam-6574	129	1	thus	thus	ADV
ejpam-6574	129	2	,	,	PUNCT
ejpam-6574	129	3	f−1(y	f−1(y	PROPN
ejpam-6574	129	4	−	−	PROPN
ejpam-6574	129	5	θ(σ1	θ(σ1	NOUN
ejpam-6574	129	6	,	,	PUNCT
ejpam-6574	129	7	σ2)b	σ2)b	NOUN
ejpam-6574	129	8	-	-	PUNCT
ejpam-6574	129	9	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	129	10	-	-	NOUN
ejpam-6574	129	11	cl(u	cl(u	NOUN
ejpam-6574	129	12	)	)	PUNCT
ejpam-6574	129	13	)	)	PUNCT
ejpam-6574	129	14	)	)	PUNCT
ejpam-6574	129	15	)	)	PUNCT
ejpam-6574	130	1	⊆	⊆	X
ejpam-6574	130	2	(	(	PUNCT
ejpam-6574	130	3	τ1	τ1	NOUN
ejpam-6574	130	4	,	,	PUNCT
ejpam-6574	130	5	τ2)θ	τ2)θ	ADJ
ejpam-6574	130	6	-	-	PUNCT
ejpam-6574	130	7	cl(x	cl(x	PUNCT
ejpam-6574	130	8	−	−	PROPN
ejpam-6574	130	9	f−1(f(τ1τ2	f−1(f(τ1τ2	NOUN
ejpam-6574	130	10	-	-	NOUN
ejpam-6574	130	11	cl(u	cl(u	NOUN
ejpam-6574	130	12	)	)	PUNCT
ejpam-6574	130	13	)	)	PUNCT
ejpam-6574	130	14	)	)	PUNCT
ejpam-6574	130	15	)	)	PUNCT
ejpam-6574	131	1	⊆	⊆	X
ejpam-6574	131	2	(	(	PUNCT
ejpam-6574	131	3	τ1	τ1	NOUN
ejpam-6574	131	4	,	,	PUNCT
ejpam-6574	131	5	τ2)θ	τ2)θ	ADJ
ejpam-6574	131	6	-	-	PUNCT
ejpam-6574	131	7	cl(x	cl(x	NOUN
ejpam-6574	131	8	−	−	PRON
ejpam-6574	131	9	τ1τ2	τ1τ2	NOUN
ejpam-6574	131	10	-	-	NOUN
ejpam-6574	131	11	cl(u	cl(u	NOUN
ejpam-6574	131	12	)	)	PUNCT
ejpam-6574	131	13	)	)	PUNCT
ejpam-6574	131	14	and	and	CCONJ
ejpam-6574	131	15	hence	hence	ADV
ejpam-6574	131	16	u	u	NOUN
ejpam-6574	131	17	⊆	⊆	NUM
ejpam-6574	131	18	(	(	PUNCT
ejpam-6574	131	19	τ1	τ1	NOUN
ejpam-6574	131	20	,	,	PUNCT
ejpam-6574	131	21	τ2)θ	τ2)θ	ADJ
ejpam-6574	131	22	-	-	PUNCT
ejpam-6574	131	23	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	131	24	-	-	PUNCT
ejpam-6574	131	25	cl(u	cl(u	NOUN
ejpam-6574	131	26	)	)	PUNCT
ejpam-6574	131	27	)	)	PUNCT
ejpam-6574	131	28	⊆	⊆	NUM
ejpam-6574	131	29	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	131	30	,	,	PUNCT
ejpam-6574	131	31	σ2)b	σ2)b	NOUN
ejpam-6574	131	32	-	-	PUNCT
ejpam-6574	131	33	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	131	34	-	-	NOUN
ejpam-6574	131	35	cl(u	cl(u	NOUN
ejpam-6574	131	36	)	)	PUNCT
ejpam-6574	131	37	)	)	PUNCT
ejpam-6574	131	38	)	)	PUNCT
ejpam-6574	131	39	)	)	PUNCT
ejpam-6574	131	40	.	.	PUNCT
ejpam-6574	132	1	therefore	therefore	ADV
ejpam-6574	132	2	,	,	PUNCT
ejpam-6574	132	3	f(u	f(u	PROPN
ejpam-6574	132	4	)	)	PUNCT
ejpam-6574	132	5	⊆	⊆	NUM
ejpam-6574	132	6	θ(σ1	θ(σ1	NOUN
ejpam-6574	132	7	,	,	PUNCT
ejpam-6574	132	8	σ2)b	σ2)b	NOUN
ejpam-6574	132	9	-	-	PUNCT
ejpam-6574	132	10	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	132	11	-	-	NOUN
ejpam-6574	132	12	cl(u	cl(u	NOUN
ejpam-6574	132	13	)	)	PUNCT
ejpam-6574	132	14	)	)	PUNCT
ejpam-6574	132	15	)	)	PUNCT
ejpam-6574	132	16	.	.	PUNCT
ejpam-6574	133	1	this	this	PRON
ejpam-6574	133	2	shows	show	VERB
ejpam-6574	133	3	that	that	SCONJ
ejpam-6574	133	4	f	f	PROPN
ejpam-6574	133	5	is	be	AUX
ejpam-6574	133	6	weakly	weakly	ADJ
ejpam-6574	133	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	133	8	,	,	PUNCT
ejpam-6574	133	9	τ2)b	τ2)b	NOUN
ejpam-6574	133	10	-	-	PUNCT
ejpam-6574	133	11	open	open	ADJ
ejpam-6574	133	12	.	.	PUNCT
ejpam-6574	134	1	theorem	theorem	NOUN
ejpam-6574	134	2	2	2	NUM
ejpam-6574	134	3	.	.	X
ejpam-6574	134	4	for	for	ADP
ejpam-6574	134	5	a	a	DET
ejpam-6574	134	6	function	function	NOUN
ejpam-6574	134	7	f	f	NOUN
ejpam-6574	134	8	:	:	PUNCT
ejpam-6574	134	9	(	(	PUNCT
ejpam-6574	134	10	x	x	NOUN
ejpam-6574	134	11	,	,	PUNCT
ejpam-6574	134	12	τ1	τ1	NOUN
ejpam-6574	134	13	,	,	PUNCT
ejpam-6574	134	14	τ2	τ2	NOUN
ejpam-6574	134	15	)	)	PUNCT
ejpam-6574	134	16	→	→	SYM
ejpam-6574	134	17	(	(	PUNCT
ejpam-6574	134	18	y	y	PROPN
ejpam-6574	134	19	,	,	PUNCT
ejpam-6574	134	20	σ1	σ1	PROPN
ejpam-6574	134	21	,	,	PUNCT
ejpam-6574	134	22	σ2	σ2	NOUN
ejpam-6574	134	23	)	)	PUNCT
ejpam-6574	134	24	,	,	PUNCT
ejpam-6574	134	25	the	the	DET
ejpam-6574	134	26	following	follow	VERB
ejpam-6574	134	27	properties	property	NOUN
ejpam-6574	134	28	are	be	AUX
ejpam-6574	134	29	equivalent	equivalent	ADJ
ejpam-6574	134	30	:	:	PUNCT
ejpam-6574	134	31	(	(	PUNCT
ejpam-6574	134	32	1	1	X
ejpam-6574	134	33	)	)	PUNCT
ejpam-6574	134	34	f	f	PROPN
ejpam-6574	134	35	is	be	AUX
ejpam-6574	134	36	weakly	weakly	ADJ
ejpam-6574	134	37	θ(τ1	θ(τ1	NOUN
ejpam-6574	134	38	,	,	PUNCT
ejpam-6574	134	39	τ2)b	τ2)b	NOUN
ejpam-6574	134	40	-	-	PUNCT
ejpam-6574	134	41	open	open	ADJ
ejpam-6574	134	42	;	;	PUNCT
ejpam-6574	134	43	(	(	PUNCT
ejpam-6574	134	44	2	2	X
ejpam-6574	134	45	)	)	PUNCT
ejpam-6574	134	46	for	for	ADP
ejpam-6574	134	47	each	each	DET
ejpam-6574	134	48	x	x	SYM
ejpam-6574	134	49	∈	∈	PROPN
ejpam-6574	134	50	x	x	X
ejpam-6574	134	51	and	and	CCONJ
ejpam-6574	134	52	each	each	DET
ejpam-6574	134	53	τ1τ2	τ1τ2	ADJ
ejpam-6574	134	54	-	-	ADJ
ejpam-6574	134	55	open	open	ADJ
ejpam-6574	134	56	set	set	ADJ
ejpam-6574	134	57	u	u	NOUN
ejpam-6574	134	58	of	of	ADP
ejpam-6574	134	59	x	x	PUNCT
ejpam-6574	134	60	containing	contain	VERB
ejpam-6574	134	61	x	x	PRON
ejpam-6574	134	62	,	,	PUNCT
ejpam-6574	134	63	there	there	PRON
ejpam-6574	134	64	exists	exist	VERB
ejpam-6574	134	65	a	a	DET
ejpam-6574	134	66	θ(σ1	θ(σ1	NOUN
ejpam-6574	134	67	,	,	PUNCT
ejpam-6574	134	68	σ2)bopen	σ2)bopen	NOUN
ejpam-6574	134	69	set	set	VERB
ejpam-6574	134	70	v	v	NOUN
ejpam-6574	134	71	of	of	ADP
ejpam-6574	134	72	y	y	NOUN
ejpam-6574	134	73	containing	contain	VERB
ejpam-6574	134	74	f(x	f(x	PROPN
ejpam-6574	134	75	)	)	PUNCT
ejpam-6574	134	76	such	such	ADJ
ejpam-6574	134	77	that	that	SCONJ
ejpam-6574	134	78	v	v	ADP
ejpam-6574	134	79	⊆	⊆	NUM
ejpam-6574	134	80	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	134	81	-	-	PUNCT
ejpam-6574	134	82	cl(u	cl(u	NOUN
ejpam-6574	134	83	)	)	PUNCT
ejpam-6574	134	84	)	)	PUNCT
ejpam-6574	134	85	.	.	PUNCT
ejpam-6574	135	1	proof	proof	NOUN
ejpam-6574	135	2	.	.	PUNCT
ejpam-6574	136	1	(	(	PUNCT
ejpam-6574	136	2	1	1	X
ejpam-6574	136	3	)	)	PUNCT
ejpam-6574	136	4	⇒	⇒	NOUN
ejpam-6574	136	5	(	(	PUNCT
ejpam-6574	136	6	2	2	NUM
ejpam-6574	136	7	):	):	PUNCT
ejpam-6574	136	8	let	let	VERB
ejpam-6574	136	9	x	x	PUNCT
ejpam-6574	136	10	∈	∈	PROPN
ejpam-6574	136	11	x	x	X
ejpam-6574	136	12	and	and	CCONJ
ejpam-6574	136	13	u	u	PRON
ejpam-6574	136	14	be	be	VERB
ejpam-6574	136	15	any	any	DET
ejpam-6574	136	16	τ1τ2	τ1τ2	ADJ
ejpam-6574	136	17	-	-	ADJ
ejpam-6574	136	18	open	open	ADJ
ejpam-6574	136	19	set	set	NOUN
ejpam-6574	136	20	of	of	ADP
ejpam-6574	136	21	x	x	PUNCT
ejpam-6574	136	22	containing	contain	VERB
ejpam-6574	136	23	x.	x.	NOUN
ejpam-6574	136	24	since	since	SCONJ
ejpam-6574	136	25	f	f	PROPN
ejpam-6574	136	26	is	be	AUX
ejpam-6574	136	27	weakly	weakly	ADJ
ejpam-6574	136	28	θ(τ1	θ(τ1	NOUN
ejpam-6574	136	29	,	,	PUNCT
ejpam-6574	136	30	τ2)b	τ2)b	NOUN
ejpam-6574	136	31	-	-	PUNCT
ejpam-6574	136	32	open	open	ADJ
ejpam-6574	136	33	,	,	PUNCT
ejpam-6574	136	34	f(x	f(x	PROPN
ejpam-6574	136	35	)	)	PUNCT
ejpam-6574	136	36	∈	∈	PROPN
ejpam-6574	136	37	f(u	f(u	PROPN
ejpam-6574	136	38	)	)	PUNCT
ejpam-6574	136	39	⊆	⊆	NUM
ejpam-6574	136	40	θ(σ1	θ(σ1	NOUN
ejpam-6574	136	41	,	,	PUNCT
ejpam-6574	136	42	σ2)b	σ2)b	NOUN
ejpam-6574	136	43	-	-	PUNCT
ejpam-6574	136	44	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	136	45	-	-	NOUN
ejpam-6574	136	46	cl(u	cl(u	NOUN
ejpam-6574	136	47	)	)	PUNCT
ejpam-6574	136	48	)	)	PUNCT
ejpam-6574	136	49	)	)	PUNCT
ejpam-6574	136	50	.	.	PUNCT
ejpam-6574	137	1	let	let	VERB
ejpam-6574	137	2	v	v	NOUN
ejpam-6574	137	3	=	=	VERB
ejpam-6574	137	4	θ(σ1	θ(σ1	NOUN
ejpam-6574	137	5	,	,	PUNCT
ejpam-6574	137	6	σ2)b	σ2)b	NOUN
ejpam-6574	137	7	-	-	PUNCT
ejpam-6574	137	8	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	137	9	-	-	NOUN
ejpam-6574	137	10	cl(u	cl(u	NOUN
ejpam-6574	137	11	)	)	PUNCT
ejpam-6574	137	12	)	)	PUNCT
ejpam-6574	137	13	)	)	PUNCT
ejpam-6574	137	14	.	.	PUNCT
ejpam-6574	138	1	then	then	ADV
ejpam-6574	138	2	,	,	PUNCT
ejpam-6574	138	3	v	v	NOUN
ejpam-6574	138	4	is	be	AUX
ejpam-6574	138	5	θ(σ1	θ(σ1	NOUN
ejpam-6574	138	6	,	,	PUNCT
ejpam-6574	138	7	σ2)b	σ2)b	NOUN
ejpam-6574	138	8	-	-	PUNCT
ejpam-6574	138	9	open	open	ADJ
ejpam-6574	138	10	in	in	ADP
ejpam-6574	138	11	y	y	PROPN
ejpam-6574	138	12	and	and	CCONJ
ejpam-6574	138	13	f(x	f(x	PROPN
ejpam-6574	138	14	)	)	PUNCT
ejpam-6574	138	15	∈	∈	PROPN
ejpam-6574	138	16	v	v	ADP
ejpam-6574	138	17	⊆	⊆	NUM
ejpam-6574	138	18	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	138	19	-	-	PUNCT
ejpam-6574	138	20	cl(u	cl(u	NOUN
ejpam-6574	138	21	)	)	PUNCT
ejpam-6574	138	22	)	)	PUNCT
ejpam-6574	138	23	.	.	PUNCT
ejpam-6574	139	1	n.	n.	PROPN
ejpam-6574	139	2	chutiman	chutiman	PROPN
ejpam-6574	139	3	,	,	PUNCT
ejpam-6574	139	4	a.	a.	PROPN
ejpam-6574	139	5	sama	sama	PROPN
ejpam-6574	139	6	-	-	PUNCT
ejpam-6574	139	7	ae	ae	PROPN
ejpam-6574	139	8	,	,	PUNCT
ejpam-6574	139	9	c.	c.	PROPN
ejpam-6574	139	10	boonpok	boonpok	PROPN
ejpam-6574	139	11	/	/	SYM
ejpam-6574	139	12	eur	eur	PROPN
ejpam-6574	139	13	.	.	PUNCT
ejpam-6574	140	1	j.	j.	PROPN
ejpam-6574	140	2	pure	pure	PROPN
ejpam-6574	140	3	appl	appl	PROPN
ejpam-6574	140	4	.	.	PROPN
ejpam-6574	140	5	math	math	PROPN
ejpam-6574	140	6	,	,	PUNCT
ejpam-6574	140	7	18	18	NUM
ejpam-6574	140	8	(	(	PUNCT
ejpam-6574	140	9	3	3	NUM
ejpam-6574	140	10	)	)	PUNCT
ejpam-6574	140	11	(	(	PUNCT
ejpam-6574	140	12	2025	2025	NUM
ejpam-6574	140	13	)	)	PUNCT
ejpam-6574	140	14	,	,	PUNCT
ejpam-6574	140	15	6574	6574	NUM
ejpam-6574	140	16	6	6	NUM
ejpam-6574	140	17	of	of	ADP
ejpam-6574	140	18	10	10	NUM
ejpam-6574	140	19	(	(	PUNCT
ejpam-6574	140	20	2	2	NUM
ejpam-6574	140	21	)	)	PUNCT
ejpam-6574	140	22	⇒	⇒	NOUN
ejpam-6574	140	23	(	(	PUNCT
ejpam-6574	140	24	1	1	NUM
ejpam-6574	140	25	):	):	PUNCT
ejpam-6574	140	26	let	let	VERB
ejpam-6574	140	27	u	u	PRON
ejpam-6574	140	28	be	be	AUX
ejpam-6574	140	29	any	any	DET
ejpam-6574	140	30	τ1τ2	τ1τ2	ADJ
ejpam-6574	140	31	-	-	ADJ
ejpam-6574	140	32	open	open	ADJ
ejpam-6574	140	33	set	set	NOUN
ejpam-6574	140	34	of	of	ADP
ejpam-6574	140	35	x	x	PUNCT
ejpam-6574	140	36	and	and	CCONJ
ejpam-6574	140	37	y	y	PROPN
ejpam-6574	140	38	∈	∈	PROPN
ejpam-6574	140	39	f(u	f(u	PROPN
ejpam-6574	140	40	)	)	PUNCT
ejpam-6574	140	41	.	.	PUNCT
ejpam-6574	141	1	it	it	PRON
ejpam-6574	141	2	follows	follow	VERB
ejpam-6574	141	3	from	from	ADP
ejpam-6574	141	4	(	(	PUNCT
ejpam-6574	141	5	2	2	NUM
ejpam-6574	141	6	)	)	PUNCT
ejpam-6574	142	1	that	that	PRON
ejpam-6574	142	2	v	v	ADP
ejpam-6574	142	3	⊆	⊆	NUM
ejpam-6574	142	4	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	142	5	-	-	PUNCT
ejpam-6574	142	6	cl(u	cl(u	NOUN
ejpam-6574	142	7	)	)	PUNCT
ejpam-6574	142	8	)	)	PUNCT
ejpam-6574	142	9	for	for	ADP
ejpam-6574	142	10	some	some	DET
ejpam-6574	142	11	θ(σ1	θ(σ1	NOUN
ejpam-6574	142	12	,	,	PUNCT
ejpam-6574	142	13	σ2)b	σ2)b	NOUN
ejpam-6574	142	14	-	-	PUNCT
ejpam-6574	142	15	open	open	NOUN
ejpam-6574	142	16	set	set	NOUN
ejpam-6574	142	17	v	v	NOUN
ejpam-6574	142	18	of	of	ADP
ejpam-6574	142	19	y	y	PROPN
ejpam-6574	142	20	containing	contain	VERB
ejpam-6574	142	21	y.	y.	PROPN
ejpam-6574	142	22	thus	thus	ADV
ejpam-6574	142	23	,	,	PUNCT
ejpam-6574	142	24	y	y	PROPN
ejpam-6574	142	25	∈	∈	PROPN
ejpam-6574	142	26	v	v	ADP
ejpam-6574	142	27	⊆	⊆	NUM
ejpam-6574	142	28	θ(σ1	θ(σ1	NOUN
ejpam-6574	142	29	,	,	PUNCT
ejpam-6574	142	30	σ2)b	σ2)b	NOUN
ejpam-6574	142	31	-	-	PUNCT
ejpam-6574	142	32	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	142	33	-	-	NOUN
ejpam-6574	142	34	cl(u	cl(u	NOUN
ejpam-6574	142	35	)	)	PUNCT
ejpam-6574	142	36	)	)	PUNCT
ejpam-6574	142	37	)	)	PUNCT
ejpam-6574	142	38	and	and	CCONJ
ejpam-6574	142	39	hence	hence	ADV
ejpam-6574	142	40	f(u	f(u	PROPN
ejpam-6574	142	41	)	)	PUNCT
ejpam-6574	142	42	⊆	⊆	NUM
ejpam-6574	142	43	θ(σ1	θ(σ1	NOUN
ejpam-6574	142	44	,	,	PUNCT
ejpam-6574	142	45	σ2)b	σ2)b	NOUN
ejpam-6574	142	46	-	-	PUNCT
ejpam-6574	142	47	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	142	48	-	-	NOUN
ejpam-6574	142	49	cl(u	cl(u	NOUN
ejpam-6574	142	50	)	)	PUNCT
ejpam-6574	142	51	)	)	PUNCT
ejpam-6574	142	52	)	)	PUNCT
ejpam-6574	142	53	.	.	PUNCT
ejpam-6574	143	1	this	this	PRON
ejpam-6574	143	2	shows	show	VERB
ejpam-6574	143	3	that	that	SCONJ
ejpam-6574	143	4	f	f	PROPN
ejpam-6574	143	5	is	be	AUX
ejpam-6574	143	6	weakly	weakly	ADJ
ejpam-6574	143	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	143	8	,	,	PUNCT
ejpam-6574	143	9	τ2)bopen	τ2)bopen	ADJ
ejpam-6574	143	10	.	.	PUNCT
ejpam-6574	144	1	theorem	theorem	NOUN
ejpam-6574	144	2	3	3	NUM
ejpam-6574	144	3	.	.	X
ejpam-6574	144	4	for	for	ADP
ejpam-6574	144	5	a	a	DET
ejpam-6574	144	6	bijective	bijective	ADJ
ejpam-6574	144	7	function	function	NOUN
ejpam-6574	145	1	f	f	NOUN
ejpam-6574	145	2	:	:	PUNCT
ejpam-6574	145	3	(	(	PUNCT
ejpam-6574	145	4	x	x	NOUN
ejpam-6574	145	5	,	,	PUNCT
ejpam-6574	145	6	τ1	τ1	NOUN
ejpam-6574	145	7	,	,	PUNCT
ejpam-6574	145	8	τ2	τ2	NOUN
ejpam-6574	145	9	)	)	PUNCT
ejpam-6574	145	10	→	→	SYM
ejpam-6574	145	11	(	(	PUNCT
ejpam-6574	145	12	y	y	PROPN
ejpam-6574	145	13	,	,	PUNCT
ejpam-6574	145	14	σ1	σ1	PROPN
ejpam-6574	145	15	,	,	PUNCT
ejpam-6574	145	16	σ2	σ2	NOUN
ejpam-6574	145	17	)	)	PUNCT
ejpam-6574	145	18	,	,	PUNCT
ejpam-6574	145	19	the	the	DET
ejpam-6574	145	20	following	follow	VERB
ejpam-6574	145	21	properties	property	NOUN
ejpam-6574	145	22	are	be	AUX
ejpam-6574	145	23	equivalent	equivalent	ADJ
ejpam-6574	145	24	:	:	PUNCT
ejpam-6574	145	25	(	(	PUNCT
ejpam-6574	145	26	1	1	X
ejpam-6574	145	27	)	)	PUNCT
ejpam-6574	145	28	f	f	PROPN
ejpam-6574	145	29	is	be	AUX
ejpam-6574	145	30	weakly	weakly	ADJ
ejpam-6574	145	31	θ(τ1	θ(τ1	NOUN
ejpam-6574	145	32	,	,	PUNCT
ejpam-6574	145	33	τ2)b	τ2)b	NOUN
ejpam-6574	145	34	-	-	PUNCT
ejpam-6574	145	35	open	open	ADJ
ejpam-6574	145	36	;	;	PUNCT
ejpam-6574	145	37	(	(	PUNCT
ejpam-6574	145	38	2	2	X
ejpam-6574	145	39	)	)	PUNCT
ejpam-6574	145	40	θ(σ1	θ(σ1	NOUN
ejpam-6574	145	41	,	,	PUNCT
ejpam-6574	145	42	σ2)b	σ2)b	NOUN
ejpam-6574	145	43	-	-	PUNCT
ejpam-6574	145	44	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	145	45	-	-	PUNCT
ejpam-6574	145	46	int(k	int(k	NUM
ejpam-6574	145	47	)	)	PUNCT
ejpam-6574	145	48	)	)	PUNCT
ejpam-6574	145	49	)	)	PUNCT
ejpam-6574	146	1	⊆	⊆	NUM
ejpam-6574	146	2	f(k	f(k	VERB
ejpam-6574	146	3	)	)	PUNCT
ejpam-6574	146	4	for	for	ADP
ejpam-6574	146	5	each	each	DET
ejpam-6574	146	6	τ1τ2	τ1τ2	ADJ
ejpam-6574	146	7	-	-	ADJ
ejpam-6574	146	8	closed	closed	ADJ
ejpam-6574	146	9	set	set	ADJ
ejpam-6574	146	10	k	k	PROPN
ejpam-6574	146	11	of	of	ADP
ejpam-6574	146	12	x	x	PROPN
ejpam-6574	146	13	;	;	PUNCT
ejpam-6574	146	14	(	(	PUNCT
ejpam-6574	146	15	3	3	X
ejpam-6574	146	16	)	)	PUNCT
ejpam-6574	146	17	θ(σ1	θ(σ1	NOUN
ejpam-6574	146	18	,	,	PUNCT
ejpam-6574	146	19	σ2)b	σ2)b	NOUN
ejpam-6574	146	20	-	-	PUNCT
ejpam-6574	146	21	cl(f(u	cl(f(u	NOUN
ejpam-6574	146	22	)	)	PUNCT
ejpam-6574	146	23	)	)	PUNCT
ejpam-6574	147	1	⊆	⊆	NUM
ejpam-6574	147	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	147	3	-	-	PUNCT
ejpam-6574	147	4	cl(u	cl(u	NOUN
ejpam-6574	147	5	)	)	PUNCT
ejpam-6574	147	6	)	)	PUNCT
ejpam-6574	147	7	for	for	ADP
ejpam-6574	147	8	each	each	DET
ejpam-6574	147	9	τ1τ2	τ1τ2	ADJ
ejpam-6574	147	10	-	-	ADJ
ejpam-6574	147	11	open	open	ADJ
ejpam-6574	147	12	set	set	ADJ
ejpam-6574	147	13	u	u	NOUN
ejpam-6574	147	14	of	of	ADP
ejpam-6574	147	15	x.	x.	NOUN
ejpam-6574	147	16	proof	proof	NOUN
ejpam-6574	147	17	.	.	PUNCT
ejpam-6574	148	1	(	(	PUNCT
ejpam-6574	148	2	1	1	X
ejpam-6574	148	3	)	)	PUNCT
ejpam-6574	148	4	⇒	⇒	NOUN
ejpam-6574	148	5	(	(	PUNCT
ejpam-6574	148	6	2	2	NUM
ejpam-6574	148	7	):	):	PUNCT
ejpam-6574	148	8	let	let	VERB
ejpam-6574	148	9	k	k	PRON
ejpam-6574	148	10	be	be	AUX
ejpam-6574	148	11	any	any	DET
ejpam-6574	148	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	148	13	-	-	ADJ
ejpam-6574	148	14	closed	closed	ADJ
ejpam-6574	148	15	set	set	NOUN
ejpam-6574	148	16	of	of	ADP
ejpam-6574	148	17	x.	x.	NOUN
ejpam-6574	148	18	then	then	ADV
ejpam-6574	148	19	,	,	PUNCT
ejpam-6574	148	20	we	we	PRON
ejpam-6574	148	21	have	have	VERB
ejpam-6574	148	22	f(x	f(x	PROPN
ejpam-6574	148	23	−k	−k	ADV
ejpam-6574	148	24	)	)	PUNCT
ejpam-6574	149	1	=	=	SYM
ejpam-6574	149	2	y	y	PROPN
ejpam-6574	149	3	−	−	PROPN
ejpam-6574	149	4	f(k	f(k	PROPN
ejpam-6574	149	5	)	)	PUNCT
ejpam-6574	149	6	⊆	⊆	NUM
ejpam-6574	149	7	θ(σ1	θ(σ1	NOUN
ejpam-6574	149	8	,	,	PUNCT
ejpam-6574	149	9	σ2)b	σ2)b	NOUN
ejpam-6574	149	10	-	-	PUNCT
ejpam-6574	149	11	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	149	12	-	-	PUNCT
ejpam-6574	149	13	cl(x	cl(x	NOUN
ejpam-6574	149	14	−k	−k	NOUN
ejpam-6574	149	15	)	)	PUNCT
ejpam-6574	149	16	)	)	PUNCT
ejpam-6574	149	17	)	)	PUNCT
ejpam-6574	149	18	and	and	CCONJ
ejpam-6574	149	19	hence	hence	ADV
ejpam-6574	149	20	y	y	PROPN
ejpam-6574	149	21	−	−	PROPN
ejpam-6574	149	22	f(k	f(k	PROPN
ejpam-6574	149	23	)	)	PUNCT
ejpam-6574	149	24	⊆	⊆	NUM
ejpam-6574	149	25	y	y	NOUN
ejpam-6574	149	26	−	−	PROPN
ejpam-6574	149	27	θ(σ1	θ(σ1	NOUN
ejpam-6574	149	28	,	,	PUNCT
ejpam-6574	149	29	σ2)b	σ2)b	NOUN
ejpam-6574	149	30	-	-	PUNCT
ejpam-6574	149	31	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	149	32	-	-	PUNCT
ejpam-6574	149	33	int(k	int(k	NOUN
ejpam-6574	149	34	)	)	PUNCT
ejpam-6574	149	35	)	)	PUNCT
ejpam-6574	149	36	)	)	PUNCT
ejpam-6574	149	37	.	.	PUNCT
ejpam-6574	150	1	thus	thus	ADV
ejpam-6574	150	2	,	,	PUNCT
ejpam-6574	150	3	θ(σ1	θ(σ1	NOUN
ejpam-6574	150	4	,	,	PUNCT
ejpam-6574	150	5	σ2)b	σ2)b	NOUN
ejpam-6574	150	6	-	-	PUNCT
ejpam-6574	150	7	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	150	8	-	-	PUNCT
ejpam-6574	150	9	int(k	int(k	NUM
ejpam-6574	150	10	)	)	PUNCT
ejpam-6574	150	11	)	)	PUNCT
ejpam-6574	150	12	)	)	PUNCT
ejpam-6574	151	1	⊆	⊆	NUM
ejpam-6574	151	2	f(k	f(k	VERB
ejpam-6574	151	3	)	)	PUNCT
ejpam-6574	151	4	.	.	PUNCT
ejpam-6574	152	1	(	(	PUNCT
ejpam-6574	152	2	2	2	X
ejpam-6574	152	3	)	)	PUNCT
ejpam-6574	152	4	⇒	⇒	NOUN
ejpam-6574	152	5	(	(	PUNCT
ejpam-6574	152	6	3	3	NUM
ejpam-6574	152	7	):	):	PUNCT
ejpam-6574	152	8	let	let	VERB
ejpam-6574	152	9	u	u	PRON
ejpam-6574	152	10	be	be	AUX
ejpam-6574	152	11	any	any	DET
ejpam-6574	152	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	152	13	-	-	ADJ
ejpam-6574	152	14	open	open	ADJ
ejpam-6574	152	15	set	set	NOUN
ejpam-6574	152	16	of	of	ADP
ejpam-6574	152	17	x.	x.	NOUN
ejpam-6574	152	18	since	since	SCONJ
ejpam-6574	152	19	τ1τ2	τ1τ2	NOUN
ejpam-6574	152	20	-	-	NOUN
ejpam-6574	152	21	cl(u	cl(u	PRON
ejpam-6574	152	22	)	)	PUNCT
ejpam-6574	152	23	is	be	AUX
ejpam-6574	152	24	a	a	DET
ejpam-6574	152	25	τ1τ2	τ1τ2	ADJ
ejpam-6574	152	26	-	-	ADJ
ejpam-6574	152	27	closed	closed	ADJ
ejpam-6574	152	28	set	set	NOUN
ejpam-6574	152	29	and	and	CCONJ
ejpam-6574	152	30	u	u	NOUN
ejpam-6574	152	31	⊆	⊆	NUM
ejpam-6574	152	32	τ1τ2	τ1τ2	NOUN
ejpam-6574	152	33	-	-	NOUN
ejpam-6574	152	34	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	152	35	-	-	PUNCT
ejpam-6574	152	36	cl(u	cl(u	NOUN
ejpam-6574	152	37	)	)	PUNCT
ejpam-6574	152	38	)	)	PUNCT
ejpam-6574	152	39	,	,	PUNCT
ejpam-6574	152	40	by	by	ADP
ejpam-6574	152	41	(	(	PUNCT
ejpam-6574	152	42	2	2	X
ejpam-6574	152	43	)	)	PUNCT
ejpam-6574	152	44	we	we	PRON
ejpam-6574	152	45	have	have	AUX
ejpam-6574	152	46	θ(σ1	θ(σ1	NOUN
ejpam-6574	152	47	,	,	PUNCT
ejpam-6574	152	48	σ2)b	σ2)b	NOUN
ejpam-6574	152	49	-	-	PUNCT
ejpam-6574	152	50	cl(f(u	cl(f(u	NOUN
ejpam-6574	152	51	)	)	PUNCT
ejpam-6574	152	52	)	)	PUNCT
ejpam-6574	153	1	⊆	⊆	NUM
ejpam-6574	153	2	θ(σ1	θ(σ1	NOUN
ejpam-6574	153	3	,	,	PUNCT
ejpam-6574	153	4	σ2)b	σ2)b	NOUN
ejpam-6574	153	5	-	-	PUNCT
ejpam-6574	153	6	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	153	7	-	-	PUNCT
ejpam-6574	153	8	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	153	9	-	-	PUNCT
ejpam-6574	153	10	cl(u	cl(u	NOUN
ejpam-6574	153	11	)	)	PUNCT
ejpam-6574	153	12	)	)	PUNCT
ejpam-6574	153	13	)	)	PUNCT
ejpam-6574	153	14	)	)	PUNCT
ejpam-6574	154	1	⊆	⊆	NUM
ejpam-6574	154	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	154	3	-	-	PUNCT
ejpam-6574	154	4	cl(u	cl(u	NOUN
ejpam-6574	154	5	)	)	PUNCT
ejpam-6574	154	6	)	)	PUNCT
ejpam-6574	154	7	.	.	PUNCT
ejpam-6574	155	1	(	(	PUNCT
ejpam-6574	155	2	3	3	X
ejpam-6574	155	3	)	)	PUNCT
ejpam-6574	155	4	⇒	⇒	NOUN
ejpam-6574	155	5	(	(	PUNCT
ejpam-6574	155	6	1	1	NUM
ejpam-6574	155	7	):	):	PUNCT
ejpam-6574	155	8	let	let	VERB
ejpam-6574	155	9	u	u	PRON
ejpam-6574	155	10	be	be	AUX
ejpam-6574	155	11	any	any	DET
ejpam-6574	155	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	155	13	-	-	ADJ
ejpam-6574	155	14	open	open	ADJ
ejpam-6574	155	15	set	set	NOUN
ejpam-6574	155	16	of	of	ADP
ejpam-6574	155	17	x.	x.	NOUN
ejpam-6574	155	18	using	use	VERB
ejpam-6574	155	19	(	(	PUNCT
ejpam-6574	155	20	3	3	NUM
ejpam-6574	155	21	)	)	PUNCT
ejpam-6574	155	22	,	,	PUNCT
ejpam-6574	155	23	we	we	PRON
ejpam-6574	155	24	have	have	VERB
ejpam-6574	155	25	y	y	NOUN
ejpam-6574	155	26	−	−	NOUN
ejpam-6574	155	27	θ(σ1	θ(σ1	NOUN
ejpam-6574	155	28	,	,	PUNCT
ejpam-6574	155	29	σ2)b	σ2)b	NOUN
ejpam-6574	155	30	-	-	PUNCT
ejpam-6574	155	31	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	155	32	-	-	NOUN
ejpam-6574	155	33	cl(u	cl(u	NOUN
ejpam-6574	155	34	)	)	PUNCT
ejpam-6574	155	35	)	)	PUNCT
ejpam-6574	155	36	)	)	PUNCT
ejpam-6574	156	1	=	=	SYM
ejpam-6574	156	2	θ(σ1	θ(σ1	NOUN
ejpam-6574	156	3	,	,	PUNCT
ejpam-6574	156	4	σ2)b	σ2)b	NOUN
ejpam-6574	156	5	-	-	PUNCT
ejpam-6574	156	6	cl(y	cl(y	NOUN
ejpam-6574	156	7	−	−	NOUN
ejpam-6574	156	8	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	156	9	-	-	NOUN
ejpam-6574	156	10	cl(u	cl(u	NOUN
ejpam-6574	156	11	)	)	PUNCT
ejpam-6574	156	12	)	)	PUNCT
ejpam-6574	156	13	)	)	PUNCT
ejpam-6574	157	1	=	=	SYM
ejpam-6574	157	2	θ(σ1	θ(σ1	NOUN
ejpam-6574	157	3	,	,	PUNCT
ejpam-6574	157	4	σ2)b	σ2)b	NOUN
ejpam-6574	157	5	-	-	PUNCT
ejpam-6574	157	6	cl(f(x	cl(f(x	NOUN
ejpam-6574	157	7	−	−	NOUN
ejpam-6574	157	8	τ1τ2	τ1τ2	NOUN
ejpam-6574	157	9	-	-	NOUN
ejpam-6574	157	10	cl(u	cl(u	NOUN
ejpam-6574	157	11	)	)	PUNCT
ejpam-6574	157	12	)	)	PUNCT
ejpam-6574	157	13	)	)	PUNCT
ejpam-6574	158	1	⊆	⊆	NUM
ejpam-6574	158	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	158	3	-	-	PUNCT
ejpam-6574	158	4	cl(x	cl(x	NOUN
ejpam-6574	158	5	−	−	PRON
ejpam-6574	158	6	τ1τ2	τ1τ2	NOUN
ejpam-6574	158	7	-	-	NOUN
ejpam-6574	158	8	cl(u	cl(u	NOUN
ejpam-6574	158	9	)	)	PUNCT
ejpam-6574	158	10	)	)	PUNCT
ejpam-6574	158	11	)	)	PUNCT
ejpam-6574	159	1	=	=	SYM
ejpam-6574	159	2	f(x	f(x	PROPN
ejpam-6574	159	3	−	−	PROPN
ejpam-6574	160	1	τ1τ2	τ1τ2	NOUN
ejpam-6574	160	2	-	-	NOUN
ejpam-6574	160	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	160	4	-	-	PUNCT
ejpam-6574	160	5	cl(u	cl(u	NOUN
ejpam-6574	160	6	)	)	PUNCT
ejpam-6574	160	7	)	)	PUNCT
ejpam-6574	160	8	)	)	PUNCT
ejpam-6574	161	1	⊆	⊆	NUM
ejpam-6574	161	2	f(x	f(x	PROPN
ejpam-6574	161	3	−	−	PROPN
ejpam-6574	161	4	u	u	NOUN
ejpam-6574	161	5	)	)	PUNCT
ejpam-6574	161	6	=	=	SYM
ejpam-6574	161	7	y	y	PROPN
ejpam-6574	161	8	−	−	PROPN
ejpam-6574	161	9	f(u	f(u	PROPN
ejpam-6574	161	10	)	)	PUNCT
ejpam-6574	161	11	and	and	CCONJ
ejpam-6574	161	12	so	so	ADV
ejpam-6574	161	13	f(u	f(u	PROPN
ejpam-6574	161	14	)	)	PUNCT
ejpam-6574	161	15	⊆	⊆	NUM
ejpam-6574	161	16	θ(σ1	θ(σ1	NOUN
ejpam-6574	161	17	,	,	PUNCT
ejpam-6574	161	18	σ2)b	σ2)b	NOUN
ejpam-6574	161	19	-	-	PUNCT
ejpam-6574	161	20	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	161	21	-	-	NOUN
ejpam-6574	161	22	cl(u	cl(u	NOUN
ejpam-6574	161	23	)	)	PUNCT
ejpam-6574	161	24	)	)	PUNCT
ejpam-6574	161	25	)	)	PUNCT
ejpam-6574	161	26	.	.	PUNCT
ejpam-6574	162	1	this	this	PRON
ejpam-6574	162	2	shows	show	VERB
ejpam-6574	162	3	that	that	SCONJ
ejpam-6574	162	4	f	f	PROPN
ejpam-6574	162	5	is	be	AUX
ejpam-6574	162	6	weakly	weakly	ADJ
ejpam-6574	162	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	162	8	,	,	PUNCT
ejpam-6574	162	9	τ2)b	τ2)b	NOUN
ejpam-6574	162	10	-	-	PUNCT
ejpam-6574	162	11	open	open	ADJ
ejpam-6574	162	12	.	.	PUNCT
ejpam-6574	163	1	the	the	DET
ejpam-6574	163	2	proof	proof	NOUN
ejpam-6574	163	3	of	of	ADP
ejpam-6574	163	4	the	the	DET
ejpam-6574	163	5	following	follow	VERB
ejpam-6574	163	6	theorem	theorem	NOUN
ejpam-6574	163	7	is	be	AUX
ejpam-6574	163	8	straightforward	straightforward	ADJ
ejpam-6574	163	9	and	and	CCONJ
ejpam-6574	163	10	thus	thus	ADV
ejpam-6574	163	11	is	be	AUX
ejpam-6574	163	12	omitted	omit	VERB
ejpam-6574	163	13	.	.	PUNCT
ejpam-6574	164	1	theorem	theorem	VERB
ejpam-6574	164	2	4	4	NUM
ejpam-6574	164	3	.	.	X
ejpam-6574	164	4	for	for	ADP
ejpam-6574	164	5	a	a	DET
ejpam-6574	164	6	function	function	NOUN
ejpam-6574	164	7	f	f	NOUN
ejpam-6574	164	8	:	:	PUNCT
ejpam-6574	164	9	(	(	PUNCT
ejpam-6574	164	10	x	x	NOUN
ejpam-6574	164	11	,	,	PUNCT
ejpam-6574	164	12	τ1	τ1	NOUN
ejpam-6574	164	13	,	,	PUNCT
ejpam-6574	164	14	τ2	τ2	NOUN
ejpam-6574	164	15	)	)	PUNCT
ejpam-6574	164	16	→	→	SYM
ejpam-6574	164	17	(	(	PUNCT
ejpam-6574	164	18	y	y	PROPN
ejpam-6574	164	19	,	,	PUNCT
ejpam-6574	164	20	σ1	σ1	PROPN
ejpam-6574	164	21	,	,	PUNCT
ejpam-6574	164	22	σ2	σ2	NOUN
ejpam-6574	164	23	)	)	PUNCT
ejpam-6574	164	24	,	,	PUNCT
ejpam-6574	164	25	the	the	DET
ejpam-6574	164	26	following	follow	VERB
ejpam-6574	164	27	properties	property	NOUN
ejpam-6574	164	28	are	be	AUX
ejpam-6574	164	29	equivalent	equivalent	ADJ
ejpam-6574	164	30	:	:	PUNCT
ejpam-6574	164	31	(	(	PUNCT
ejpam-6574	164	32	1	1	X
ejpam-6574	164	33	)	)	PUNCT
ejpam-6574	164	34	f	f	PROPN
ejpam-6574	164	35	is	be	AUX
ejpam-6574	164	36	weakly	weakly	ADJ
ejpam-6574	164	37	θ(τ1	θ(τ1	NOUN
ejpam-6574	164	38	,	,	PUNCT
ejpam-6574	164	39	τ2)b	τ2)b	NOUN
ejpam-6574	164	40	-	-	PUNCT
ejpam-6574	164	41	open	open	ADJ
ejpam-6574	164	42	;	;	PUNCT
ejpam-6574	164	43	n.	n.	NOUN
ejpam-6574	164	44	chutiman	chutiman	NOUN
ejpam-6574	164	45	,	,	PUNCT
ejpam-6574	164	46	a.	a.	PROPN
ejpam-6574	164	47	sama	sama	PROPN
ejpam-6574	164	48	-	-	PUNCT
ejpam-6574	164	49	ae	ae	PROPN
ejpam-6574	164	50	,	,	PUNCT
ejpam-6574	164	51	c.	c.	PROPN
ejpam-6574	164	52	boonpok	boonpok	PROPN
ejpam-6574	164	53	/	/	SYM
ejpam-6574	164	54	eur	eur	PROPN
ejpam-6574	164	55	.	.	PUNCT
ejpam-6574	165	1	j.	j.	PROPN
ejpam-6574	165	2	pure	pure	PROPN
ejpam-6574	165	3	appl	appl	PROPN
ejpam-6574	165	4	.	.	PROPN
ejpam-6574	165	5	math	math	PROPN
ejpam-6574	165	6	,	,	PUNCT
ejpam-6574	165	7	18	18	NUM
ejpam-6574	165	8	(	(	PUNCT
ejpam-6574	165	9	3	3	NUM
ejpam-6574	165	10	)	)	PUNCT
ejpam-6574	165	11	(	(	PUNCT
ejpam-6574	165	12	2025	2025	NUM
ejpam-6574	165	13	)	)	PUNCT
ejpam-6574	165	14	,	,	PUNCT
ejpam-6574	165	15	6574	6574	NUM
ejpam-6574	165	16	7	7	NUM
ejpam-6574	165	17	of	of	ADP
ejpam-6574	165	18	10	10	NUM
ejpam-6574	165	19	(	(	PUNCT
ejpam-6574	165	20	2	2	NUM
ejpam-6574	165	21	)	)	PUNCT
ejpam-6574	165	22	f(u	f(u	PROPN
ejpam-6574	165	23	)	)	PUNCT
ejpam-6574	165	24	⊆	⊆	NUM
ejpam-6574	165	25	θ(σ1	θ(σ1	NOUN
ejpam-6574	165	26	,	,	PUNCT
ejpam-6574	165	27	σ2)b	σ2)b	NOUN
ejpam-6574	165	28	-	-	PUNCT
ejpam-6574	165	29	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	165	30	-	-	NOUN
ejpam-6574	165	31	cl(u	cl(u	NOUN
ejpam-6574	165	32	)	)	PUNCT
ejpam-6574	165	33	)	)	PUNCT
ejpam-6574	165	34	)	)	PUNCT
ejpam-6574	165	35	for	for	ADP
ejpam-6574	165	36	each	each	DET
ejpam-6574	165	37	(	(	PUNCT
ejpam-6574	165	38	τ1	τ1	NOUN
ejpam-6574	165	39	,	,	PUNCT
ejpam-6574	165	40	τ2)p	τ2)p	ADJ
ejpam-6574	165	41	-	-	PUNCT
ejpam-6574	165	42	open	open	ADJ
ejpam-6574	165	43	set	set	NOUN
ejpam-6574	165	44	u	u	NOUN
ejpam-6574	165	45	of	of	ADP
ejpam-6574	165	46	x	x	PRON
ejpam-6574	165	47	;	;	PUNCT
ejpam-6574	165	48	(	(	PUNCT
ejpam-6574	165	49	3	3	X
ejpam-6574	165	50	)	)	PUNCT
ejpam-6574	165	51	f(u	f(u	PROPN
ejpam-6574	165	52	)	)	PUNCT
ejpam-6574	165	53	⊆	⊆	NUM
ejpam-6574	165	54	θ(σ1	θ(σ1	NOUN
ejpam-6574	165	55	,	,	PUNCT
ejpam-6574	165	56	σ2)b	σ2)b	NOUN
ejpam-6574	165	57	-	-	PUNCT
ejpam-6574	165	58	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	165	59	-	-	NOUN
ejpam-6574	165	60	cl(u	cl(u	NOUN
ejpam-6574	165	61	)	)	PUNCT
ejpam-6574	165	62	)	)	PUNCT
ejpam-6574	165	63	)	)	PUNCT
ejpam-6574	165	64	for	for	ADP
ejpam-6574	165	65	each	each	DET
ejpam-6574	165	66	α(τ1	α(τ1	NOUN
ejpam-6574	165	67	,	,	PUNCT
ejpam-6574	165	68	τ2)-open	τ2)-open	ADJ
ejpam-6574	165	69	set	set	VERB
ejpam-6574	165	70	u	u	NOUN
ejpam-6574	165	71	of	of	ADP
ejpam-6574	165	72	x	x	PRON
ejpam-6574	165	73	;	;	PUNCT
ejpam-6574	165	74	(	(	PUNCT
ejpam-6574	165	75	4	4	X
ejpam-6574	165	76	)	)	PUNCT
ejpam-6574	165	77	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	165	78	-	-	PUNCT
ejpam-6574	165	79	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	165	80	-	-	PUNCT
ejpam-6574	165	81	cl(u	cl(u	NOUN
ejpam-6574	165	82	)	)	PUNCT
ejpam-6574	165	83	)	)	PUNCT
ejpam-6574	165	84	)	)	PUNCT
ejpam-6574	166	1	⊆	⊆	NUM
ejpam-6574	166	2	θ(σ1	θ(σ1	NOUN
ejpam-6574	166	3	,	,	PUNCT
ejpam-6574	166	4	σ2)b	σ2)b	NOUN
ejpam-6574	166	5	-	-	PUNCT
ejpam-6574	166	6	int(f(τ1τ2	int(f(τ1τ2	NOUN
ejpam-6574	166	7	-	-	NOUN
ejpam-6574	166	8	cl(u	cl(u	NOUN
ejpam-6574	166	9	)	)	PUNCT
ejpam-6574	166	10	)	)	PUNCT
ejpam-6574	166	11	)	)	PUNCT
ejpam-6574	166	12	for	for	ADP
ejpam-6574	166	13	each	each	DET
ejpam-6574	166	14	τ1τ2	τ1τ2	ADJ
ejpam-6574	166	15	-	-	ADJ
ejpam-6574	166	16	open	open	ADJ
ejpam-6574	166	17	set	set	ADJ
ejpam-6574	166	18	u	u	NOUN
ejpam-6574	166	19	of	of	ADP
ejpam-6574	166	20	x	x	PRON
ejpam-6574	166	21	;	;	PUNCT
ejpam-6574	166	22	(	(	PUNCT
ejpam-6574	166	23	5	5	X
ejpam-6574	166	24	)	)	PUNCT
ejpam-6574	166	25	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	166	26	-	-	PUNCT
ejpam-6574	166	27	int(k	int(k	NOUN
ejpam-6574	166	28	)	)	PUNCT
ejpam-6574	166	29	)	)	PUNCT
ejpam-6574	166	30	⊆	⊆	NUM
ejpam-6574	166	31	θ(σ1	θ(σ1	NOUN
ejpam-6574	166	32	,	,	PUNCT
ejpam-6574	166	33	σ2)b	σ2)b	NOUN
ejpam-6574	166	34	-	-	PUNCT
ejpam-6574	166	35	int(f(k	int(f(k	NOUN
ejpam-6574	166	36	)	)	PUNCT
ejpam-6574	166	37	)	)	PUNCT
ejpam-6574	166	38	for	for	ADP
ejpam-6574	166	39	each	each	DET
ejpam-6574	166	40	τ1τ2	τ1τ2	ADJ
ejpam-6574	166	41	-	-	ADJ
ejpam-6574	166	42	closed	closed	ADJ
ejpam-6574	166	43	set	set	NOUN
ejpam-6574	166	44	k	k	PROPN
ejpam-6574	166	45	of	of	ADP
ejpam-6574	166	46	x.	x.	NOUN
ejpam-6574	166	47	4	4	NUM
ejpam-6574	166	48	.	.	PUNCT
ejpam-6574	166	49	weakly	weakly	ADJ
ejpam-6574	166	50	θ(τ1	θ(τ1	NOUN
ejpam-6574	166	51	,	,	PUNCT
ejpam-6574	166	52	τ2)b	τ2)b	NOUN
ejpam-6574	166	53	-	-	PUNCT
ejpam-6574	166	54	closed	close	VERB
ejpam-6574	166	55	functions	function	NOUN
ejpam-6574	166	56	in	in	ADP
ejpam-6574	166	57	this	this	DET
ejpam-6574	166	58	section	section	NOUN
ejpam-6574	166	59	,	,	PUNCT
ejpam-6574	166	60	we	we	PRON
ejpam-6574	166	61	introduce	introduce	VERB
ejpam-6574	166	62	the	the	DET
ejpam-6574	166	63	concept	concept	NOUN
ejpam-6574	166	64	of	of	ADP
ejpam-6574	166	65	weakly	weakly	ADJ
ejpam-6574	166	66	θ(τ1	θ(τ1	NOUN
ejpam-6574	166	67	,	,	PUNCT
ejpam-6574	166	68	τ2)b	τ2)b	NOUN
ejpam-6574	166	69	-	-	PUNCT
ejpam-6574	166	70	closed	close	VERB
ejpam-6574	166	71	functions	function	NOUN
ejpam-6574	166	72	.	.	PUNCT
ejpam-6574	167	1	furthermore	furthermore	ADV
ejpam-6574	167	2	,	,	PUNCT
ejpam-6574	167	3	some	some	DET
ejpam-6574	167	4	characterizations	characterization	NOUN
ejpam-6574	167	5	of	of	ADP
ejpam-6574	167	6	weakly	weakly	ADJ
ejpam-6574	167	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	167	8	,	,	PUNCT
ejpam-6574	167	9	τ2)b	τ2)b	ADJ
ejpam-6574	167	10	-	-	PUNCT
ejpam-6574	167	11	closed	close	VERB
ejpam-6574	167	12	functions	function	NOUN
ejpam-6574	167	13	are	be	AUX
ejpam-6574	167	14	investigated	investigate	VERB
ejpam-6574	167	15	.	.	PUNCT
ejpam-6574	168	1	definition	definition	NOUN
ejpam-6574	168	2	2	2	NUM
ejpam-6574	168	3	.	.	PUNCT
ejpam-6574	169	1	a	a	DET
ejpam-6574	169	2	function	function	NOUN
ejpam-6574	169	3	f	f	NOUN
ejpam-6574	169	4	:	:	PUNCT
ejpam-6574	169	5	(	(	PUNCT
ejpam-6574	169	6	x	x	NOUN
ejpam-6574	169	7	,	,	PUNCT
ejpam-6574	169	8	τ1	τ1	NOUN
ejpam-6574	169	9	,	,	PUNCT
ejpam-6574	169	10	τ2	τ2	NOUN
ejpam-6574	169	11	)	)	PUNCT
ejpam-6574	169	12	→	→	SYM
ejpam-6574	169	13	(	(	PUNCT
ejpam-6574	169	14	y	y	PROPN
ejpam-6574	169	15	,	,	PUNCT
ejpam-6574	169	16	σ1	σ1	PROPN
ejpam-6574	169	17	,	,	PUNCT
ejpam-6574	169	18	σ2	σ2	PROPN
ejpam-6574	169	19	)	)	PUNCT
ejpam-6574	169	20	is	be	AUX
ejpam-6574	169	21	said	say	VERB
ejpam-6574	169	22	to	to	PART
ejpam-6574	169	23	be	be	AUX
ejpam-6574	169	24	weakly	weakly	ADJ
ejpam-6574	169	25	θ(τ1	θ(τ1	NOUN
ejpam-6574	169	26	,	,	PUNCT
ejpam-6574	169	27	τ2)b	τ2)b	NOUN
ejpam-6574	169	28	-	-	PUNCT
ejpam-6574	169	29	closed	close	VERB
ejpam-6574	169	30	if	if	SCONJ
ejpam-6574	169	31	θ(σ1	θ(σ1	NOUN
ejpam-6574	169	32	,	,	PUNCT
ejpam-6574	169	33	σ2)b	σ2)b	NOUN
ejpam-6574	169	34	-	-	PUNCT
ejpam-6574	169	35	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	169	36	-	-	PUNCT
ejpam-6574	169	37	int(k	int(k	NUM
ejpam-6574	169	38	)	)	PUNCT
ejpam-6574	169	39	)	)	PUNCT
ejpam-6574	169	40	)	)	PUNCT
ejpam-6574	170	1	⊆	⊆	NUM
ejpam-6574	170	2	f(k	f(k	VERB
ejpam-6574	170	3	)	)	PUNCT
ejpam-6574	170	4	for	for	ADP
ejpam-6574	170	5	every	every	DET
ejpam-6574	170	6	τ1τ2	τ1τ2	ADJ
ejpam-6574	170	7	-	-	ADJ
ejpam-6574	170	8	closed	closed	ADJ
ejpam-6574	170	9	set	set	NOUN
ejpam-6574	170	10	k	k	PROPN
ejpam-6574	170	11	of	of	ADP
ejpam-6574	170	12	x.	x.	PROPN
ejpam-6574	170	13	theorem	theorem	VERB
ejpam-6574	170	14	5	5	NUM
ejpam-6574	170	15	.	.	PUNCT
ejpam-6574	170	16	for	for	ADP
ejpam-6574	170	17	a	a	DET
ejpam-6574	170	18	function	function	NOUN
ejpam-6574	170	19	f	f	NOUN
ejpam-6574	170	20	:	:	PUNCT
ejpam-6574	170	21	(	(	PUNCT
ejpam-6574	170	22	x	x	NOUN
ejpam-6574	170	23	,	,	PUNCT
ejpam-6574	170	24	τ1	τ1	NOUN
ejpam-6574	170	25	,	,	PUNCT
ejpam-6574	170	26	τ2	τ2	NOUN
ejpam-6574	170	27	)	)	PUNCT
ejpam-6574	170	28	→	→	SYM
ejpam-6574	170	29	(	(	PUNCT
ejpam-6574	170	30	y	y	PROPN
ejpam-6574	170	31	,	,	PUNCT
ejpam-6574	170	32	σ1	σ1	PROPN
ejpam-6574	170	33	,	,	PUNCT
ejpam-6574	170	34	σ2	σ2	NOUN
ejpam-6574	170	35	)	)	PUNCT
ejpam-6574	170	36	,	,	PUNCT
ejpam-6574	170	37	the	the	DET
ejpam-6574	170	38	following	follow	VERB
ejpam-6574	170	39	properties	property	NOUN
ejpam-6574	170	40	are	be	AUX
ejpam-6574	170	41	equivalent	equivalent	ADJ
ejpam-6574	170	42	:	:	PUNCT
ejpam-6574	170	43	(	(	PUNCT
ejpam-6574	170	44	1	1	X
ejpam-6574	170	45	)	)	PUNCT
ejpam-6574	170	46	f	f	PROPN
ejpam-6574	170	47	is	be	AUX
ejpam-6574	170	48	weakly	weakly	ADJ
ejpam-6574	170	49	θ(τ1	θ(τ1	NOUN
ejpam-6574	170	50	,	,	PUNCT
ejpam-6574	170	51	τ2)b	τ2)b	NOUN
ejpam-6574	170	52	-	-	PUNCT
ejpam-6574	170	53	closed	closed	ADJ
ejpam-6574	170	54	;	;	PUNCT
ejpam-6574	170	55	(	(	PUNCT
ejpam-6574	170	56	2	2	X
ejpam-6574	170	57	)	)	PUNCT
ejpam-6574	170	58	θ(σ1	θ(σ1	NOUN
ejpam-6574	170	59	,	,	PUNCT
ejpam-6574	170	60	σ2)b	σ2)b	NOUN
ejpam-6574	170	61	-	-	PUNCT
ejpam-6574	170	62	cl(f(u	cl(f(u	NOUN
ejpam-6574	170	63	)	)	PUNCT
ejpam-6574	170	64	)	)	PUNCT
ejpam-6574	171	1	⊆	⊆	NUM
ejpam-6574	171	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	171	3	-	-	PUNCT
ejpam-6574	171	4	cl(u	cl(u	NOUN
ejpam-6574	171	5	)	)	PUNCT
ejpam-6574	171	6	)	)	PUNCT
ejpam-6574	171	7	for	for	ADP
ejpam-6574	171	8	every	every	DET
ejpam-6574	171	9	τ1τ2	τ1τ2	ADJ
ejpam-6574	171	10	-	-	ADJ
ejpam-6574	171	11	open	open	ADJ
ejpam-6574	171	12	set	set	ADJ
ejpam-6574	171	13	u	u	NOUN
ejpam-6574	171	14	of	of	ADP
ejpam-6574	171	15	x.	x.	NOUN
ejpam-6574	171	16	proof	proof	NOUN
ejpam-6574	171	17	.	.	PUNCT
ejpam-6574	172	1	(	(	PUNCT
ejpam-6574	172	2	1	1	X
ejpam-6574	172	3	)	)	PUNCT
ejpam-6574	172	4	⇒	⇒	NOUN
ejpam-6574	172	5	(	(	PUNCT
ejpam-6574	172	6	2	2	NUM
ejpam-6574	172	7	):	):	PUNCT
ejpam-6574	172	8	let	let	VERB
ejpam-6574	172	9	u	u	PRON
ejpam-6574	172	10	be	be	AUX
ejpam-6574	172	11	any	any	DET
ejpam-6574	172	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	172	13	-	-	ADJ
ejpam-6574	172	14	open	open	ADJ
ejpam-6574	172	15	set	set	NOUN
ejpam-6574	172	16	of	of	ADP
ejpam-6574	172	17	x.	x.	NOUN
ejpam-6574	172	18	since	since	SCONJ
ejpam-6574	172	19	τ1τ2	τ1τ2	NOUN
ejpam-6574	172	20	-	-	NOUN
ejpam-6574	172	21	cl(u	cl(u	PRON
ejpam-6574	172	22	)	)	PUNCT
ejpam-6574	172	23	is	be	AUX
ejpam-6574	172	24	a	a	DET
ejpam-6574	172	25	τ1τ2	τ1τ2	ADJ
ejpam-6574	172	26	-	-	ADJ
ejpam-6574	172	27	closed	closed	ADJ
ejpam-6574	172	28	set	set	NOUN
ejpam-6574	172	29	and	and	CCONJ
ejpam-6574	172	30	u	u	NOUN
ejpam-6574	172	31	⊆	⊆	NUM
ejpam-6574	172	32	τ1τ2	τ1τ2	NOUN
ejpam-6574	172	33	-	-	NOUN
ejpam-6574	172	34	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	172	35	-	-	PUNCT
ejpam-6574	172	36	cl(u	cl(u	NOUN
ejpam-6574	172	37	)	)	PUNCT
ejpam-6574	172	38	)	)	PUNCT
ejpam-6574	172	39	,	,	PUNCT
ejpam-6574	172	40	we	we	PRON
ejpam-6574	172	41	have	have	VERB
ejpam-6574	172	42	θ(σ1	θ(σ1	NOUN
ejpam-6574	172	43	,	,	PUNCT
ejpam-6574	172	44	σ2)b	σ2)b	NOUN
ejpam-6574	172	45	-	-	PUNCT
ejpam-6574	172	46	cl(f(u	cl(f(u	NOUN
ejpam-6574	172	47	)	)	PUNCT
ejpam-6574	172	48	)	)	PUNCT
ejpam-6574	172	49	⊆	⊆	NUM
ejpam-6574	172	50	θ(σ1	θ(σ1	NOUN
ejpam-6574	172	51	,	,	PUNCT
ejpam-6574	172	52	σ2)b	σ2)b	NOUN
ejpam-6574	172	53	-	-	PUNCT
ejpam-6574	172	54	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	172	55	-	-	PUNCT
ejpam-6574	172	56	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	172	57	-	-	PUNCT
ejpam-6574	172	58	cl(u	cl(u	NOUN
ejpam-6574	172	59	)	)	PUNCT
ejpam-6574	172	60	)	)	PUNCT
ejpam-6574	172	61	)	)	PUNCT
ejpam-6574	172	62	)	)	PUNCT
ejpam-6574	173	1	⊆	⊆	NUM
ejpam-6574	173	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	173	3	-	-	PUNCT
ejpam-6574	173	4	cl(u	cl(u	NOUN
ejpam-6574	173	5	)	)	PUNCT
ejpam-6574	173	6	)	)	PUNCT
ejpam-6574	173	7	.	.	PUNCT
ejpam-6574	174	1	(	(	PUNCT
ejpam-6574	174	2	2	2	X
ejpam-6574	174	3	)	)	PUNCT
ejpam-6574	174	4	⇒	⇒	NOUN
ejpam-6574	174	5	(	(	PUNCT
ejpam-6574	174	6	1	1	NUM
ejpam-6574	174	7	):	):	PUNCT
ejpam-6574	174	8	let	let	VERB
ejpam-6574	174	9	k	k	PRON
ejpam-6574	174	10	be	be	AUX
ejpam-6574	174	11	any	any	DET
ejpam-6574	174	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	174	13	-	-	ADJ
ejpam-6574	174	14	closed	closed	ADJ
ejpam-6574	174	15	set	set	NOUN
ejpam-6574	174	16	of	of	ADP
ejpam-6574	174	17	x.	x.	NOUN
ejpam-6574	174	18	then	then	ADV
ejpam-6574	174	19	,	,	PUNCT
ejpam-6574	174	20	we	we	PRON
ejpam-6574	174	21	have	have	VERB
ejpam-6574	174	22	θ(σ1	θ(σ1	NOUN
ejpam-6574	174	23	,	,	PUNCT
ejpam-6574	174	24	σ2)b	σ2)b	NOUN
ejpam-6574	174	25	-	-	PUNCT
ejpam-6574	174	26	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	174	27	-	-	PUNCT
ejpam-6574	174	28	int(k	int(k	NUM
ejpam-6574	174	29	)	)	PUNCT
ejpam-6574	174	30	)	)	PUNCT
ejpam-6574	174	31	)	)	PUNCT
ejpam-6574	175	1	⊆	⊆	NUM
ejpam-6574	175	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	175	3	-	-	PUNCT
ejpam-6574	175	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6574	175	5	-	-	PUNCT
ejpam-6574	175	6	int(k	int(k	NOUN
ejpam-6574	175	7	)	)	PUNCT
ejpam-6574	175	8	)	)	PUNCT
ejpam-6574	175	9	)	)	PUNCT
ejpam-6574	176	1	⊆	⊆	NUM
ejpam-6574	176	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	176	3	-	-	PUNCT
ejpam-6574	176	4	cl(k	cl(k	NOUN
ejpam-6574	176	5	)	)	PUNCT
ejpam-6574	176	6	)	)	PUNCT
ejpam-6574	176	7	=	=	PUNCT
ejpam-6574	176	8	f(k	f(k	VERB
ejpam-6574	176	9	)	)	PUNCT
ejpam-6574	176	10	and	and	CCONJ
ejpam-6574	176	11	hence	hence	ADV
ejpam-6574	176	12	f	f	PROPN
ejpam-6574	176	13	is	be	AUX
ejpam-6574	176	14	weakly	weakly	ADJ
ejpam-6574	176	15	θ(τ1	θ(τ1	NOUN
ejpam-6574	176	16	,	,	PUNCT
ejpam-6574	176	17	τ2)b	τ2)b	NOUN
ejpam-6574	176	18	-	-	PUNCT
ejpam-6574	176	19	closed	closed	ADJ
ejpam-6574	176	20	.	.	PUNCT
ejpam-6574	177	1	corollary	corollary	ADJ
ejpam-6574	177	2	1	1	NUM
ejpam-6574	177	3	.	.	PUNCT
ejpam-6574	178	1	a	a	DET
ejpam-6574	178	2	bijective	bijective	ADJ
ejpam-6574	178	3	function	function	NOUN
ejpam-6574	178	4	f	f	NOUN
ejpam-6574	178	5	:	:	PUNCT
ejpam-6574	178	6	(	(	PUNCT
ejpam-6574	178	7	x	x	NOUN
ejpam-6574	178	8	,	,	PUNCT
ejpam-6574	178	9	τ1	τ1	NOUN
ejpam-6574	178	10	,	,	PUNCT
ejpam-6574	178	11	τ2	τ2	NOUN
ejpam-6574	178	12	)	)	PUNCT
ejpam-6574	178	13	→	→	SYM
ejpam-6574	178	14	(	(	PUNCT
ejpam-6574	178	15	y	y	PROPN
ejpam-6574	178	16	,	,	PUNCT
ejpam-6574	178	17	σ1	σ1	PROPN
ejpam-6574	178	18	,	,	PUNCT
ejpam-6574	178	19	σ2	σ2	NOUN
ejpam-6574	178	20	)	)	PUNCT
ejpam-6574	178	21	is	be	AUX
ejpam-6574	178	22	weakly	weakly	ADJ
ejpam-6574	178	23	θ(τ1	θ(τ1	NOUN
ejpam-6574	178	24	,	,	PUNCT
ejpam-6574	178	25	τ2)b	τ2)b	NOUN
ejpam-6574	178	26	-	-	PUNCT
ejpam-6574	178	27	open	open	ADJ
ejpam-6574	178	28	if	if	SCONJ
ejpam-6574	178	29	and	and	CCONJ
ejpam-6574	178	30	only	only	ADV
ejpam-6574	178	31	if	if	SCONJ
ejpam-6574	178	32	f	f	PROPN
ejpam-6574	178	33	is	be	AUX
ejpam-6574	178	34	weakly	weakly	ADJ
ejpam-6574	178	35	θ(τ1	θ(τ1	NOUN
ejpam-6574	178	36	,	,	PUNCT
ejpam-6574	178	37	τ2)b	τ2)b	NOUN
ejpam-6574	178	38	-	-	PUNCT
ejpam-6574	178	39	closed	closed	ADJ
ejpam-6574	178	40	.	.	PUNCT
ejpam-6574	179	1	proof	proof	NOUN
ejpam-6574	179	2	.	.	PUNCT
ejpam-6574	180	1	this	this	PRON
ejpam-6574	180	2	is	be	AUX
ejpam-6574	180	3	an	an	DET
ejpam-6574	180	4	immediate	immediate	ADJ
ejpam-6574	180	5	consequence	consequence	NOUN
ejpam-6574	180	6	of	of	ADP
ejpam-6574	180	7	theorem	theorem	ADJ
ejpam-6574	180	8	3	3	NUM
ejpam-6574	180	9	and	and	CCONJ
ejpam-6574	180	10	5	5	NUM
ejpam-6574	180	11	.	.	PUNCT
ejpam-6574	181	1	the	the	DET
ejpam-6574	181	2	proof	proof	NOUN
ejpam-6574	181	3	of	of	ADP
ejpam-6574	181	4	the	the	DET
ejpam-6574	181	5	following	follow	VERB
ejpam-6574	181	6	theorem	theorem	NOUN
ejpam-6574	181	7	is	be	AUX
ejpam-6574	181	8	straightforward	straightforward	ADJ
ejpam-6574	181	9	and	and	CCONJ
ejpam-6574	181	10	thus	thus	ADV
ejpam-6574	181	11	is	be	AUX
ejpam-6574	181	12	omitted	omit	VERB
ejpam-6574	181	13	.	.	PUNCT
ejpam-6574	182	1	theorem	theorem	VERB
ejpam-6574	182	2	6	6	NUM
ejpam-6574	182	3	.	.	PUNCT
ejpam-6574	182	4	for	for	ADP
ejpam-6574	182	5	a	a	DET
ejpam-6574	182	6	function	function	NOUN
ejpam-6574	182	7	f	f	NOUN
ejpam-6574	182	8	:	:	PUNCT
ejpam-6574	182	9	(	(	PUNCT
ejpam-6574	182	10	x	x	NOUN
ejpam-6574	182	11	,	,	PUNCT
ejpam-6574	182	12	τ1	τ1	NOUN
ejpam-6574	182	13	,	,	PUNCT
ejpam-6574	182	14	τ2	τ2	NOUN
ejpam-6574	182	15	)	)	PUNCT
ejpam-6574	182	16	→	→	SYM
ejpam-6574	182	17	(	(	PUNCT
ejpam-6574	182	18	y	y	PROPN
ejpam-6574	182	19	,	,	PUNCT
ejpam-6574	182	20	σ1	σ1	PROPN
ejpam-6574	182	21	,	,	PUNCT
ejpam-6574	182	22	σ2	σ2	NOUN
ejpam-6574	182	23	)	)	PUNCT
ejpam-6574	182	24	,	,	PUNCT
ejpam-6574	182	25	the	the	DET
ejpam-6574	182	26	following	follow	VERB
ejpam-6574	182	27	properties	property	NOUN
ejpam-6574	182	28	are	be	AUX
ejpam-6574	182	29	equivalent	equivalent	ADJ
ejpam-6574	182	30	:	:	PUNCT
ejpam-6574	182	31	n.	n.	NOUN
ejpam-6574	182	32	chutiman	chutiman	NOUN
ejpam-6574	182	33	,	,	PUNCT
ejpam-6574	182	34	a.	a.	PROPN
ejpam-6574	182	35	sama	sama	PROPN
ejpam-6574	182	36	-	-	PUNCT
ejpam-6574	182	37	ae	ae	PROPN
ejpam-6574	182	38	,	,	PUNCT
ejpam-6574	182	39	c.	c.	PROPN
ejpam-6574	182	40	boonpok	boonpok	PROPN
ejpam-6574	182	41	/	/	SYM
ejpam-6574	182	42	eur	eur	PROPN
ejpam-6574	182	43	.	.	PUNCT
ejpam-6574	183	1	j.	j.	PROPN
ejpam-6574	183	2	pure	pure	PROPN
ejpam-6574	183	3	appl	appl	PROPN
ejpam-6574	183	4	.	.	PROPN
ejpam-6574	183	5	math	math	PROPN
ejpam-6574	183	6	,	,	PUNCT
ejpam-6574	183	7	18	18	NUM
ejpam-6574	183	8	(	(	PUNCT
ejpam-6574	183	9	3	3	NUM
ejpam-6574	183	10	)	)	PUNCT
ejpam-6574	183	11	(	(	PUNCT
ejpam-6574	183	12	2025	2025	NUM
ejpam-6574	183	13	)	)	PUNCT
ejpam-6574	183	14	,	,	PUNCT
ejpam-6574	183	15	6574	6574	NUM
ejpam-6574	183	16	8	8	NUM
ejpam-6574	183	17	of	of	ADP
ejpam-6574	183	18	10	10	NUM
ejpam-6574	183	19	(	(	PUNCT
ejpam-6574	183	20	1	1	NUM
ejpam-6574	183	21	)	)	PUNCT
ejpam-6574	183	22	f	f	PROPN
ejpam-6574	183	23	is	be	AUX
ejpam-6574	183	24	weakly	weakly	ADJ
ejpam-6574	183	25	θ(τ1	θ(τ1	NOUN
ejpam-6574	183	26	,	,	PUNCT
ejpam-6574	183	27	τ2)b	τ2)b	NOUN
ejpam-6574	183	28	-	-	PUNCT
ejpam-6574	183	29	closed	closed	ADJ
ejpam-6574	183	30	;	;	PUNCT
ejpam-6574	183	31	(	(	PUNCT
ejpam-6574	183	32	2	2	X
ejpam-6574	183	33	)	)	PUNCT
ejpam-6574	183	34	θ(σ1	θ(σ1	NOUN
ejpam-6574	183	35	,	,	PUNCT
ejpam-6574	183	36	σ2)b	σ2)b	NOUN
ejpam-6574	183	37	-	-	PUNCT
ejpam-6574	183	38	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	183	39	-	-	PUNCT
ejpam-6574	183	40	int(k	int(k	NUM
ejpam-6574	183	41	)	)	PUNCT
ejpam-6574	183	42	)	)	PUNCT
ejpam-6574	183	43	)	)	PUNCT
ejpam-6574	184	1	⊆	⊆	NUM
ejpam-6574	184	2	f(k	f(k	VERB
ejpam-6574	184	3	)	)	PUNCT
ejpam-6574	184	4	for	for	ADP
ejpam-6574	184	5	every	every	DET
ejpam-6574	184	6	(	(	PUNCT
ejpam-6574	184	7	τ1	τ1	NOUN
ejpam-6574	184	8	,	,	PUNCT
ejpam-6574	184	9	τ2)p	τ2)p	NOUN
ejpam-6574	184	10	-	-	PUNCT
ejpam-6574	184	11	closed	closed	ADJ
ejpam-6574	184	12	set	set	NOUN
ejpam-6574	184	13	k	k	PROPN
ejpam-6574	184	14	of	of	ADP
ejpam-6574	184	15	x	x	PROPN
ejpam-6574	184	16	;	;	PUNCT
ejpam-6574	184	17	(	(	PUNCT
ejpam-6574	184	18	3	3	X
ejpam-6574	184	19	)	)	PUNCT
ejpam-6574	184	20	θ(σ1	θ(σ1	NOUN
ejpam-6574	184	21	,	,	PUNCT
ejpam-6574	184	22	σ2)b	σ2)b	NOUN
ejpam-6574	184	23	-	-	PUNCT
ejpam-6574	184	24	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	184	25	-	-	PUNCT
ejpam-6574	184	26	int(k	int(k	NUM
ejpam-6574	184	27	)	)	PUNCT
ejpam-6574	184	28	)	)	PUNCT
ejpam-6574	184	29	)	)	PUNCT
ejpam-6574	185	1	⊆	⊆	NUM
ejpam-6574	185	2	f(k	f(k	VERB
ejpam-6574	185	3	)	)	PUNCT
ejpam-6574	185	4	for	for	ADP
ejpam-6574	185	5	every	every	DET
ejpam-6574	185	6	α(τ1	α(τ1	NOUN
ejpam-6574	185	7	,	,	PUNCT
ejpam-6574	185	8	τ2)-closed	τ2)-close	VERB
ejpam-6574	185	9	set	set	NOUN
ejpam-6574	185	10	k	k	PROPN
ejpam-6574	185	11	of	of	ADP
ejpam-6574	185	12	x	x	PROPN
ejpam-6574	185	13	;	;	PUNCT
ejpam-6574	185	14	(	(	PUNCT
ejpam-6574	185	15	4	4	X
ejpam-6574	185	16	)	)	PUNCT
ejpam-6574	185	17	θ(σ1	θ(σ1	NOUN
ejpam-6574	185	18	,	,	PUNCT
ejpam-6574	185	19	σ2)b	σ2)b	NOUN
ejpam-6574	185	20	-	-	PUNCT
ejpam-6574	185	21	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	185	22	-	-	PUNCT
ejpam-6574	185	23	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	185	24	-	-	PUNCT
ejpam-6574	185	25	cl(a	cl(a	NUM
ejpam-6574	185	26	)	)	PUNCT
ejpam-6574	185	27	)	)	PUNCT
ejpam-6574	185	28	)	)	PUNCT
ejpam-6574	185	29	)	)	PUNCT
ejpam-6574	186	1	⊆	⊆	NUM
ejpam-6574	186	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	186	3	-	-	PUNCT
ejpam-6574	186	4	cl(a	cl(a	NUM
ejpam-6574	186	5	)	)	PUNCT
ejpam-6574	186	6	)	)	PUNCT
ejpam-6574	186	7	for	for	ADP
ejpam-6574	186	8	every	every	DET
ejpam-6574	186	9	subset	subset	NOUN
ejpam-6574	186	10	a	a	PRON
ejpam-6574	186	11	of	of	ADP
ejpam-6574	186	12	x	x	PRON
ejpam-6574	186	13	;	;	PUNCT
ejpam-6574	186	14	(	(	PUNCT
ejpam-6574	186	15	5	5	X
ejpam-6574	186	16	)	)	PUNCT
ejpam-6574	186	17	θ(σ1	θ(σ1	NOUN
ejpam-6574	186	18	,	,	PUNCT
ejpam-6574	186	19	σ2)b	σ2)b	NOUN
ejpam-6574	186	20	-	-	PUNCT
ejpam-6574	186	21	cl(f(u	cl(f(u	NOUN
ejpam-6574	186	22	)	)	PUNCT
ejpam-6574	186	23	)	)	PUNCT
ejpam-6574	187	1	⊆	⊆	NUM
ejpam-6574	187	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	187	3	-	-	PUNCT
ejpam-6574	187	4	cl(u	cl(u	NOUN
ejpam-6574	187	5	)	)	PUNCT
ejpam-6574	187	6	)	)	PUNCT
ejpam-6574	187	7	for	for	ADP
ejpam-6574	187	8	every	every	DET
ejpam-6574	187	9	(	(	PUNCT
ejpam-6574	187	10	τ1	τ1	NOUN
ejpam-6574	187	11	,	,	PUNCT
ejpam-6574	187	12	τ2)p	τ2)p	ADJ
ejpam-6574	187	13	-	-	PUNCT
ejpam-6574	187	14	open	open	ADJ
ejpam-6574	187	15	set	set	NOUN
ejpam-6574	187	16	u	u	PROPN
ejpam-6574	187	17	of	of	ADP
ejpam-6574	187	18	x.	x.	PROPN
ejpam-6574	187	19	theorem	theorem	VERB
ejpam-6574	187	20	7	7	NUM
ejpam-6574	187	21	.	.	X
ejpam-6574	187	22	for	for	ADP
ejpam-6574	187	23	a	a	DET
ejpam-6574	187	24	function	function	NOUN
ejpam-6574	187	25	f	f	NOUN
ejpam-6574	187	26	:	:	PUNCT
ejpam-6574	187	27	(	(	PUNCT
ejpam-6574	187	28	x	x	NOUN
ejpam-6574	187	29	,	,	PUNCT
ejpam-6574	187	30	τ1	τ1	NOUN
ejpam-6574	187	31	,	,	PUNCT
ejpam-6574	187	32	τ2	τ2	NOUN
ejpam-6574	187	33	)	)	PUNCT
ejpam-6574	187	34	→	→	SYM
ejpam-6574	187	35	(	(	PUNCT
ejpam-6574	187	36	y	y	PROPN
ejpam-6574	187	37	,	,	PUNCT
ejpam-6574	187	38	σ1	σ1	PROPN
ejpam-6574	187	39	,	,	PUNCT
ejpam-6574	187	40	σ2	σ2	NOUN
ejpam-6574	187	41	)	)	PUNCT
ejpam-6574	187	42	,	,	PUNCT
ejpam-6574	187	43	the	the	DET
ejpam-6574	187	44	following	follow	VERB
ejpam-6574	187	45	properties	property	NOUN
ejpam-6574	187	46	are	be	AUX
ejpam-6574	187	47	equivalent	equivalent	ADJ
ejpam-6574	187	48	:	:	PUNCT
ejpam-6574	187	49	(	(	PUNCT
ejpam-6574	187	50	1	1	X
ejpam-6574	187	51	)	)	PUNCT
ejpam-6574	187	52	f	f	PROPN
ejpam-6574	187	53	is	be	AUX
ejpam-6574	187	54	weakly	weakly	ADJ
ejpam-6574	187	55	θ(τ1	θ(τ1	NOUN
ejpam-6574	187	56	,	,	PUNCT
ejpam-6574	187	57	τ2)b	τ2)b	NOUN
ejpam-6574	187	58	-	-	PUNCT
ejpam-6574	187	59	closed	closed	ADJ
ejpam-6574	187	60	;	;	PUNCT
ejpam-6574	187	61	(	(	PUNCT
ejpam-6574	187	62	2	2	X
ejpam-6574	187	63	)	)	PUNCT
ejpam-6574	187	64	θ(σ1	θ(σ1	NOUN
ejpam-6574	187	65	,	,	PUNCT
ejpam-6574	187	66	σ2)b	σ2)b	NOUN
ejpam-6574	187	67	-	-	PUNCT
ejpam-6574	187	68	cl(f(u	cl(f(u	NOUN
ejpam-6574	187	69	)	)	PUNCT
ejpam-6574	187	70	)	)	PUNCT
ejpam-6574	188	1	⊆	⊆	NUM
ejpam-6574	188	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	188	3	-	-	PUNCT
ejpam-6574	188	4	cl(u	cl(u	NOUN
ejpam-6574	188	5	)	)	PUNCT
ejpam-6574	188	6	)	)	PUNCT
ejpam-6574	188	7	for	for	ADP
ejpam-6574	188	8	every	every	DET
ejpam-6574	188	9	(	(	PUNCT
ejpam-6574	188	10	τ1	τ1	NOUN
ejpam-6574	188	11	,	,	PUNCT
ejpam-6574	188	12	τ2)r	τ2)r	ADJ
ejpam-6574	188	13	-	-	PUNCT
ejpam-6574	188	14	open	open	NOUN
ejpam-6574	188	15	set	set	NOUN
ejpam-6574	188	16	u	u	NOUN
ejpam-6574	188	17	of	of	ADP
ejpam-6574	188	18	x	x	PRON
ejpam-6574	188	19	;	;	PUNCT
ejpam-6574	188	20	(	(	PUNCT
ejpam-6574	188	21	3	3	X
ejpam-6574	188	22	)	)	PUNCT
ejpam-6574	188	23	for	for	ADP
ejpam-6574	188	24	each	each	DET
ejpam-6574	188	25	subset	subset	NOUN
ejpam-6574	188	26	b	b	PROPN
ejpam-6574	188	27	of	of	ADP
ejpam-6574	188	28	y	y	PROPN
ejpam-6574	188	29	and	and	CCONJ
ejpam-6574	188	30	each	each	DET
ejpam-6574	188	31	τ1τ2	τ1τ2	ADJ
ejpam-6574	188	32	-	-	ADJ
ejpam-6574	188	33	open	open	ADJ
ejpam-6574	188	34	set	set	ADJ
ejpam-6574	188	35	u	u	NOUN
ejpam-6574	188	36	of	of	ADP
ejpam-6574	188	37	x	x	PUNCT
ejpam-6574	188	38	with	with	ADP
ejpam-6574	188	39	f−1(b	f−1(b	PROPN
ejpam-6574	188	40	)	)	PUNCT
ejpam-6574	188	41	⊆	⊆	NUM
ejpam-6574	188	42	u	u	NOUN
ejpam-6574	188	43	,	,	PUNCT
ejpam-6574	188	44	there	there	PRON
ejpam-6574	188	45	exists	exist	VERB
ejpam-6574	188	46	a	a	DET
ejpam-6574	188	47	θ(σ1	θ(σ1	NOUN
ejpam-6574	188	48	,	,	PUNCT
ejpam-6574	188	49	σ2)b	σ2)b	NOUN
ejpam-6574	188	50	-	-	PUNCT
ejpam-6574	188	51	open	open	NOUN
ejpam-6574	188	52	set	set	NOUN
ejpam-6574	188	53	v	v	NOUN
ejpam-6574	188	54	of	of	ADP
ejpam-6574	188	55	y	y	PRON
ejpam-6574	188	56	such	such	ADJ
ejpam-6574	188	57	that	that	DET
ejpam-6574	188	58	b	b	PROPN
ejpam-6574	188	59	⊆	⊆	NUM
ejpam-6574	188	60	v	v	NOUN
ejpam-6574	188	61	and	and	CCONJ
ejpam-6574	188	62	f−1(v	f−1(v	NOUN
ejpam-6574	188	63	)	)	PUNCT
ejpam-6574	189	1	⊆	⊆	NUM
ejpam-6574	189	2	τ1τ2	τ1τ2	NOUN
ejpam-6574	189	3	-	-	NOUN
ejpam-6574	189	4	cl(u	cl(u	NUM
ejpam-6574	189	5	)	)	PUNCT
ejpam-6574	189	6	;	;	PUNCT
ejpam-6574	189	7	(	(	PUNCT
ejpam-6574	189	8	4	4	X
ejpam-6574	189	9	)	)	PUNCT
ejpam-6574	189	10	for	for	ADP
ejpam-6574	189	11	each	each	DET
ejpam-6574	189	12	point	point	NOUN
ejpam-6574	189	13	y	y	PROPN
ejpam-6574	189	14	∈	∈	PROPN
ejpam-6574	189	15	y	y	PROPN
ejpam-6574	189	16	and	and	CCONJ
ejpam-6574	189	17	each	each	DET
ejpam-6574	189	18	τ1τ2	τ1τ2	ADJ
ejpam-6574	189	19	-	-	ADJ
ejpam-6574	189	20	open	open	ADJ
ejpam-6574	189	21	set	set	ADJ
ejpam-6574	189	22	u	u	NOUN
ejpam-6574	189	23	of	of	ADP
ejpam-6574	189	24	x	x	PUNCT
ejpam-6574	189	25	with	with	ADP
ejpam-6574	189	26	f−1(y	f−1(y	PROPN
ejpam-6574	189	27	)	)	PUNCT
ejpam-6574	189	28	⊆	⊆	NUM
ejpam-6574	189	29	u	u	NOUN
ejpam-6574	189	30	,	,	PUNCT
ejpam-6574	189	31	there	there	PRON
ejpam-6574	189	32	exists	exist	VERB
ejpam-6574	189	33	a	a	DET
ejpam-6574	189	34	θ(σ1	θ(σ1	NOUN
ejpam-6574	189	35	,	,	PUNCT
ejpam-6574	189	36	σ2)b	σ2)b	NOUN
ejpam-6574	189	37	-	-	PUNCT
ejpam-6574	189	38	open	open	NOUN
ejpam-6574	189	39	set	set	NOUN
ejpam-6574	189	40	v	v	NOUN
ejpam-6574	189	41	of	of	ADP
ejpam-6574	189	42	y	y	NOUN
ejpam-6574	189	43	containing	contain	VERB
ejpam-6574	189	44	y	y	PROPN
ejpam-6574	189	45	and	and	CCONJ
ejpam-6574	189	46	f−1(v	f−1(v	PROPN
ejpam-6574	189	47	)	)	PUNCT
ejpam-6574	190	1	⊆	⊆	NUM
ejpam-6574	190	2	τ1τ2	τ1τ2	NOUN
ejpam-6574	190	3	-	-	NOUN
ejpam-6574	190	4	cl(u	cl(u	NOUN
ejpam-6574	190	5	)	)	PUNCT
ejpam-6574	190	6	.	.	PUNCT
ejpam-6574	191	1	proof	proof	NOUN
ejpam-6574	191	2	.	.	PUNCT
ejpam-6574	192	1	(	(	PUNCT
ejpam-6574	192	2	1	1	X
ejpam-6574	192	3	)	)	PUNCT
ejpam-6574	192	4	⇒	⇒	NOUN
ejpam-6574	192	5	(	(	PUNCT
ejpam-6574	192	6	2	2	NUM
ejpam-6574	192	7	):	):	PUNCT
ejpam-6574	192	8	it	it	PRON
ejpam-6574	192	9	follows	follow	VERB
ejpam-6574	192	10	from	from	ADP
ejpam-6574	192	11	theorem	theorem	ADJ
ejpam-6574	192	12	5	5	NUM
ejpam-6574	192	13	.	.	PUNCT
ejpam-6574	192	14	(	(	PUNCT
ejpam-6574	192	15	2	2	X
ejpam-6574	192	16	)	)	PUNCT
ejpam-6574	192	17	⇒	⇒	NOUN
ejpam-6574	192	18	(	(	PUNCT
ejpam-6574	192	19	3	3	NUM
ejpam-6574	192	20	):	):	PUNCT
ejpam-6574	192	21	let	let	VERB
ejpam-6574	192	22	b	b	X
ejpam-6574	192	23	be	be	AUX
ejpam-6574	192	24	any	any	DET
ejpam-6574	192	25	subset	subset	NOUN
ejpam-6574	192	26	of	of	ADP
ejpam-6574	192	27	y	y	PROPN
ejpam-6574	192	28	and	and	CCONJ
ejpam-6574	192	29	u	u	NOUN
ejpam-6574	192	30	be	be	VERB
ejpam-6574	192	31	any	any	DET
ejpam-6574	192	32	τ1τ2	τ1τ2	ADJ
ejpam-6574	192	33	-	-	ADJ
ejpam-6574	192	34	open	open	ADJ
ejpam-6574	192	35	set	set	ADJ
ejpam-6574	192	36	ofx	ofx	NOUN
ejpam-6574	192	37	with	with	ADP
ejpam-6574	192	38	f−1(b	f−1(b	PROPN
ejpam-6574	192	39	)	)	PUNCT
ejpam-6574	192	40	⊆	⊆	NUM
ejpam-6574	192	41	u	u	NOUN
ejpam-6574	192	42	.	.	PUNCT
ejpam-6574	193	1	then	then	ADV
ejpam-6574	193	2	,	,	PUNCT
ejpam-6574	193	3	we	we	PRON
ejpam-6574	193	4	have	have	VERB
ejpam-6574	193	5	f−1(b	f−1(b	PROPN
ejpam-6574	193	6	)	)	PUNCT
ejpam-6574	193	7	∩	∩	NOUN
ejpam-6574	193	8	τ1τ2	τ1τ2	NOUN
ejpam-6574	193	9	-	-	NOUN
ejpam-6574	193	10	cl(x	cl(x	SYM
ejpam-6574	193	11	−	−	PRON
ejpam-6574	193	12	τ1τ2	τ1τ2	NOUN
ejpam-6574	193	13	-	-	NOUN
ejpam-6574	193	14	cl(u	cl(u	NOUN
ejpam-6574	193	15	)	)	PUNCT
ejpam-6574	193	16	)	)	PUNCT
ejpam-6574	194	1	=	=	NOUN
ejpam-6574	194	2	∅	∅	NOUN
ejpam-6574	194	3	and	and	CCONJ
ejpam-6574	194	4	hence	hence	ADV
ejpam-6574	194	5	b	b	NOUN
ejpam-6574	194	6	∩	∩	ADJ
ejpam-6574	194	7	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	194	8	-	-	NOUN
ejpam-6574	194	9	cl(x	cl(x	NUM
ejpam-6574	194	10	−	−	PRON
ejpam-6574	194	11	τ1τ2	τ1τ2	NOUN
ejpam-6574	194	12	-	-	NOUN
ejpam-6574	194	13	cl(u	cl(u	NOUN
ejpam-6574	194	14	)	)	PUNCT
ejpam-6574	194	15	)	)	PUNCT
ejpam-6574	194	16	)	)	PUNCT
ejpam-6574	195	1	=	=	PUNCT
ejpam-6574	195	2	∅.	∅.	VERB
ejpam-6574	195	3	since	since	SCONJ
ejpam-6574	195	4	x	x	INTJ
ejpam-6574	195	5	−	−	NUM
ejpam-6574	195	6	τ1τ2	τ1τ2	NOUN
ejpam-6574	195	7	-	-	NOUN
ejpam-6574	195	8	cl(u	cl(u	NOUN
ejpam-6574	195	9	)	)	PUNCT
ejpam-6574	195	10	is	be	AUX
ejpam-6574	195	11	(	(	PUNCT
ejpam-6574	195	12	τ1	τ1	NOUN
ejpam-6574	195	13	,	,	PUNCT
ejpam-6574	195	14	τ2)r	τ2)r	NOUN
ejpam-6574	195	15	-	-	PUNCT
ejpam-6574	195	16	open	open	ADJ
ejpam-6574	195	17	,	,	PUNCT
ejpam-6574	195	18	b	b	NOUN
ejpam-6574	195	19	∩	∩	ADJ
ejpam-6574	195	20	θ(σ1	θ(σ1	NOUN
ejpam-6574	195	21	,	,	PUNCT
ejpam-6574	195	22	σ2)b	σ2)b	NOUN
ejpam-6574	195	23	-	-	PUNCT
ejpam-6574	195	24	cl(f(x	cl(f(x	NOUN
ejpam-6574	195	25	−	−	NOUN
ejpam-6574	195	26	τ1τ2	τ1τ2	NOUN
ejpam-6574	195	27	-	-	NOUN
ejpam-6574	195	28	cl(u	cl(u	NOUN
ejpam-6574	195	29	)	)	PUNCT
ejpam-6574	195	30	)	)	PUNCT
ejpam-6574	195	31	)	)	PUNCT
ejpam-6574	196	1	=	=	PUNCT
ejpam-6574	196	2	∅.	∅.	AUX
ejpam-6574	196	3	let	let	VERB
ejpam-6574	196	4	v	v	VERB
ejpam-6574	196	5	=	=	SYM
ejpam-6574	196	6	y	y	PROPN
ejpam-6574	196	7	−	−	PROPN
ejpam-6574	196	8	θ(σ1	θ(σ1	NOUN
ejpam-6574	196	9	,	,	PUNCT
ejpam-6574	196	10	σ2)b	σ2)b	NOUN
ejpam-6574	196	11	-	-	PUNCT
ejpam-6574	196	12	cl(f(x	cl(f(x	NOUN
ejpam-6574	196	13	−	−	NOUN
ejpam-6574	196	14	τ1τ2	τ1τ2	NOUN
ejpam-6574	196	15	-	-	NOUN
ejpam-6574	196	16	cl(u	cl(u	NOUN
ejpam-6574	196	17	)	)	PUNCT
ejpam-6574	196	18	)	)	PUNCT
ejpam-6574	196	19	)	)	PUNCT
ejpam-6574	196	20	.	.	PUNCT
ejpam-6574	197	1	then	then	ADV
ejpam-6574	197	2	,	,	PUNCT
ejpam-6574	197	3	v	v	NOUN
ejpam-6574	197	4	is	be	AUX
ejpam-6574	197	5	a	a	DET
ejpam-6574	197	6	θ(σ1	θ(σ1	NOUN
ejpam-6574	197	7	,	,	PUNCT
ejpam-6574	197	8	σ2)b	σ2)b	NOUN
ejpam-6574	197	9	-	-	PUNCT
ejpam-6574	197	10	open	open	NOUN
ejpam-6574	197	11	set	set	VERB
ejpam-6574	197	12	with	with	ADP
ejpam-6574	197	13	b	b	PROPN
ejpam-6574	197	14	⊆	⊆	NUM
ejpam-6574	197	15	v	v	NOUN
ejpam-6574	197	16	and	and	CCONJ
ejpam-6574	197	17	f−1(v	f−1(v	NOUN
ejpam-6574	197	18	)	)	PUNCT
ejpam-6574	198	1	⊆	⊆	NUM
ejpam-6574	198	2	x	x	SYM
ejpam-6574	198	3	−	−	NOUN
ejpam-6574	198	4	f−1(θ(σ1	f−1(θ(σ1	NOUN
ejpam-6574	198	5	,	,	PUNCT
ejpam-6574	198	6	σ2)b	σ2)b	NOUN
ejpam-6574	198	7	-	-	PUNCT
ejpam-6574	198	8	cl(f(x	cl(f(x	NOUN
ejpam-6574	198	9	−	−	NOUN
ejpam-6574	198	10	τ1τ2	τ1τ2	NOUN
ejpam-6574	198	11	-	-	NOUN
ejpam-6574	198	12	cl(u	cl(u	NOUN
ejpam-6574	198	13	)	)	PUNCT
ejpam-6574	198	14	)	)	PUNCT
ejpam-6574	198	15	)	)	PUNCT
ejpam-6574	198	16	)	)	PUNCT
ejpam-6574	199	1	⊆	⊆	NUM
ejpam-6574	199	2	x	x	SYM
ejpam-6574	199	3	−	−	NOUN
ejpam-6574	199	4	f−1(f(x	f−1(f(x	NOUN
ejpam-6574	199	5	−	−	ADP
ejpam-6574	199	6	τ1τ2	τ1τ2	NOUN
ejpam-6574	199	7	-	-	NOUN
ejpam-6574	199	8	cl(u	cl(u	NOUN
ejpam-6574	199	9	)	)	PUNCT
ejpam-6574	199	10	)	)	PUNCT
ejpam-6574	199	11	)	)	PUNCT
ejpam-6574	200	1	⊆	⊆	X
ejpam-6574	200	2	τ1τ2	τ1τ2	NOUN
ejpam-6574	200	3	-	-	NOUN
ejpam-6574	200	4	cl(u	cl(u	NUM
ejpam-6574	200	5	)	)	PUNCT
ejpam-6574	200	6	.	.	PUNCT
ejpam-6574	201	1	(	(	PUNCT
ejpam-6574	201	2	3	3	X
ejpam-6574	201	3	)	)	PUNCT
ejpam-6574	201	4	⇒	⇒	NOUN
ejpam-6574	201	5	(	(	PUNCT
ejpam-6574	201	6	4	4	NUM
ejpam-6574	201	7	):	):	PUNCT
ejpam-6574	201	8	the	the	DET
ejpam-6574	201	9	proof	proof	NOUN
ejpam-6574	201	10	is	be	AUX
ejpam-6574	201	11	obvious	obvious	ADJ
ejpam-6574	201	12	.	.	PUNCT
ejpam-6574	202	1	(	(	PUNCT
ejpam-6574	202	2	4	4	X
ejpam-6574	202	3	)	)	PUNCT
ejpam-6574	202	4	⇒	⇒	NOUN
ejpam-6574	202	5	(	(	PUNCT
ejpam-6574	202	6	1	1	NUM
ejpam-6574	202	7	):	):	PUNCT
ejpam-6574	202	8	let	let	VERB
ejpam-6574	202	9	k	k	PRON
ejpam-6574	202	10	be	be	AUX
ejpam-6574	202	11	any	any	DET
ejpam-6574	202	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	202	13	-	-	ADJ
ejpam-6574	202	14	closed	closed	ADJ
ejpam-6574	202	15	set	set	NOUN
ejpam-6574	202	16	of	of	ADP
ejpam-6574	202	17	y	y	PROPN
ejpam-6574	202	18	and	and	CCONJ
ejpam-6574	202	19	y	y	PROPN
ejpam-6574	202	20	∈	∈	PROPN
ejpam-6574	202	21	y	y	PROPN
ejpam-6574	202	22	−f(k	−f(k	ADJ
ejpam-6574	202	23	)	)	PUNCT
ejpam-6574	202	24	.	.	PUNCT
ejpam-6574	203	1	since	since	SCONJ
ejpam-6574	203	2	f−1(y	f−1(y	PROPN
ejpam-6574	203	3	)	)	PUNCT
ejpam-6574	203	4	⊆	⊆	NUM
ejpam-6574	203	5	x−k	x−k	X
ejpam-6574	203	6	,	,	PUNCT
ejpam-6574	203	7	by	by	ADP
ejpam-6574	203	8	(	(	PUNCT
ejpam-6574	203	9	4	4	X
ejpam-6574	203	10	)	)	PUNCT
ejpam-6574	203	11	there	there	PRON
ejpam-6574	203	12	exists	exist	VERB
ejpam-6574	203	13	a	a	DET
ejpam-6574	203	14	θ(σ1	θ(σ1	NOUN
ejpam-6574	203	15	,	,	PUNCT
ejpam-6574	203	16	σ2)b	σ2)b	NOUN
ejpam-6574	203	17	-	-	PUNCT
ejpam-6574	203	18	open	open	NOUN
ejpam-6574	203	19	set	set	NOUN
ejpam-6574	203	20	v	v	NOUN
ejpam-6574	203	21	of	of	ADP
ejpam-6574	203	22	y	y	PRON
ejpam-6574	203	23	such	such	ADJ
ejpam-6574	203	24	that	that	SCONJ
ejpam-6574	203	25	y	y	PROPN
ejpam-6574	203	26	∈	∈	PROPN
ejpam-6574	203	27	v	v	NOUN
ejpam-6574	203	28	and	and	CCONJ
ejpam-6574	203	29	f−1(v	f−1(v	NOUN
ejpam-6574	203	30	)	)	PUNCT
ejpam-6574	204	1	⊆	⊆	NUM
ejpam-6574	204	2	τ1τ2	τ1τ2	NOUN
ejpam-6574	204	3	-	-	ADJ
ejpam-6574	204	4	cl(x	cl(x	SYM
ejpam-6574	204	5	−k	−k	NOUN
ejpam-6574	204	6	)	)	PUNCT
ejpam-6574	204	7	=	=	PUNCT
ejpam-6574	205	1	x	x	X
ejpam-6574	205	2	−	−	ADP
ejpam-6574	205	3	τ1τ2	τ1τ2	NOUN
ejpam-6574	205	4	-	-	PUNCT
ejpam-6574	205	5	int(k	int(k	NOUN
ejpam-6574	205	6	)	)	PUNCT
ejpam-6574	205	7	.	.	PUNCT
ejpam-6574	206	1	thus	thus	ADV
ejpam-6574	206	2	,	,	PUNCT
ejpam-6574	206	3	v	v	ADP
ejpam-6574	206	4	∩	∩	ADJ
ejpam-6574	206	5	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	206	6	-	-	PUNCT
ejpam-6574	206	7	int(k	int(k	NOUN
ejpam-6574	206	8	)	)	PUNCT
ejpam-6574	206	9	)	)	PUNCT
ejpam-6574	207	1	=	=	NOUN
ejpam-6574	207	2	∅	∅	NOUN
ejpam-6574	207	3	and	and	CCONJ
ejpam-6574	207	4	hence	hence	ADV
ejpam-6574	207	5	y	y	PROPN
ejpam-6574	207	6	̸∈	̸∈	PROPN
ejpam-6574	207	7	θ(σ1	θ(σ1	VERB
ejpam-6574	207	8	,	,	PUNCT
ejpam-6574	207	9	σ2)b	σ2)b	NOUN
ejpam-6574	207	10	-	-	PUNCT
ejpam-6574	207	11	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	207	12	-	-	PUNCT
ejpam-6574	207	13	int(k	int(k	NOUN
ejpam-6574	207	14	)	)	PUNCT
ejpam-6574	207	15	)	)	PUNCT
ejpam-6574	207	16	)	)	PUNCT
ejpam-6574	207	17	.	.	PUNCT
ejpam-6574	208	1	therefore	therefore	ADV
ejpam-6574	208	2	,	,	PUNCT
ejpam-6574	208	3	θ(σ1	θ(σ1	NOUN
ejpam-6574	208	4	,	,	PUNCT
ejpam-6574	208	5	σ2)b	σ2)b	NOUN
ejpam-6574	208	6	-	-	PUNCT
ejpam-6574	208	7	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	208	8	-	-	PUNCT
ejpam-6574	208	9	int(k	int(k	NUM
ejpam-6574	208	10	)	)	PUNCT
ejpam-6574	208	11	)	)	PUNCT
ejpam-6574	208	12	)	)	PUNCT
ejpam-6574	209	1	⊆	⊆	NUM
ejpam-6574	209	2	f(k	f(k	VERB
ejpam-6574	209	3	)	)	PUNCT
ejpam-6574	209	4	.	.	PUNCT
ejpam-6574	210	1	this	this	PRON
ejpam-6574	210	2	shows	show	VERB
ejpam-6574	210	3	that	that	SCONJ
ejpam-6574	210	4	f	f	PROPN
ejpam-6574	210	5	is	be	AUX
ejpam-6574	210	6	weakly	weakly	ADJ
ejpam-6574	210	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	210	8	,	,	PUNCT
ejpam-6574	210	9	τ2)b	τ2)b	NOUN
ejpam-6574	210	10	-	-	PUNCT
ejpam-6574	210	11	closed	closed	ADJ
ejpam-6574	210	12	.	.	PUNCT
ejpam-6574	211	1	theorem	theorem	VERB
ejpam-6574	211	2	8	8	NUM
ejpam-6574	211	3	.	.	PUNCT
ejpam-6574	212	1	if	if	SCONJ
ejpam-6574	212	2	f	f	PROPN
ejpam-6574	212	3	:	:	PUNCT
ejpam-6574	212	4	(	(	PUNCT
ejpam-6574	212	5	x	x	NOUN
ejpam-6574	212	6	,	,	PUNCT
ejpam-6574	212	7	τ1	τ1	NOUN
ejpam-6574	212	8	,	,	PUNCT
ejpam-6574	212	9	τ2	τ2	NOUN
ejpam-6574	212	10	)	)	PUNCT
ejpam-6574	212	11	→	→	SYM
ejpam-6574	212	12	(	(	PUNCT
ejpam-6574	212	13	y	y	PROPN
ejpam-6574	212	14	,	,	PUNCT
ejpam-6574	212	15	σ1	σ1	PROPN
ejpam-6574	212	16	,	,	PUNCT
ejpam-6574	212	17	σ2	σ2	PROPN
ejpam-6574	212	18	)	)	PUNCT
ejpam-6574	212	19	is	be	AUX
ejpam-6574	212	20	a	a	DET
ejpam-6574	212	21	bijective	bijective	ADJ
ejpam-6574	212	22	weakly	weakly	ADJ
ejpam-6574	212	23	θ(τ1	θ(τ1	NOUN
ejpam-6574	212	24	,	,	PUNCT
ejpam-6574	212	25	τ2)b	τ2)b	NOUN
ejpam-6574	212	26	-	-	PUNCT
ejpam-6574	212	27	closed	close	VERB
ejpam-6574	212	28	function	function	NOUN
ejpam-6574	212	29	,	,	PUNCT
ejpam-6574	212	30	then	then	ADV
ejpam-6574	212	31	for	for	ADP
ejpam-6574	212	32	every	every	DET
ejpam-6574	212	33	subset	subset	NOUN
ejpam-6574	212	34	b	b	PROPN
ejpam-6574	212	35	of	of	ADP
ejpam-6574	212	36	y	y	PROPN
ejpam-6574	212	37	and	and	CCONJ
ejpam-6574	212	38	every	every	DET
ejpam-6574	212	39	τ1τ2	τ1τ2	ADJ
ejpam-6574	212	40	-	-	ADJ
ejpam-6574	212	41	open	open	ADJ
ejpam-6574	212	42	set	set	ADJ
ejpam-6574	212	43	u	u	NOUN
ejpam-6574	212	44	of	of	ADP
ejpam-6574	212	45	x	x	PUNCT
ejpam-6574	212	46	with	with	ADP
ejpam-6574	212	47	f−1(b	f−1(b	PROPN
ejpam-6574	212	48	)	)	PUNCT
ejpam-6574	212	49	⊆	⊆	NUM
ejpam-6574	212	50	u	u	NOUN
ejpam-6574	212	51	,	,	PUNCT
ejpam-6574	212	52	there	there	PRON
ejpam-6574	212	53	exists	exist	VERB
ejpam-6574	212	54	a	a	DET
ejpam-6574	212	55	θ(σ1	θ(σ1	NOUN
ejpam-6574	212	56	,	,	PUNCT
ejpam-6574	212	57	σ2)b	σ2)b	NOUN
ejpam-6574	212	58	-	-	PUNCT
ejpam-6574	212	59	closed	close	VERB
ejpam-6574	212	60	set	set	NOUN
ejpam-6574	212	61	k	k	PROPN
ejpam-6574	212	62	of	of	ADP
ejpam-6574	212	63	y	y	PRON
ejpam-6574	213	1	such	such	ADJ
ejpam-6574	213	2	that	that	PRON
ejpam-6574	213	3	b	b	PROPN
ejpam-6574	213	4	⊆	⊆	NUM
ejpam-6574	213	5	k	k	PROPN
ejpam-6574	213	6	and	and	CCONJ
ejpam-6574	213	7	f−1(k	f−1(k	PROPN
ejpam-6574	213	8	)	)	PUNCT
ejpam-6574	213	9	⊆	⊆	NUM
ejpam-6574	213	10	τ1τ2	τ1τ2	NOUN
ejpam-6574	213	11	-	-	NOUN
ejpam-6574	213	12	cl(u	cl(u	NOUN
ejpam-6574	213	13	)	)	PUNCT
ejpam-6574	213	14	.	.	PUNCT
ejpam-6574	214	1	n.	n.	PROPN
ejpam-6574	214	2	chutiman	chutiman	PROPN
ejpam-6574	214	3	,	,	PUNCT
ejpam-6574	214	4	a.	a.	PROPN
ejpam-6574	214	5	sama	sama	PROPN
ejpam-6574	214	6	-	-	PUNCT
ejpam-6574	214	7	ae	ae	PROPN
ejpam-6574	214	8	,	,	PUNCT
ejpam-6574	214	9	c.	c.	PROPN
ejpam-6574	214	10	boonpok	boonpok	PROPN
ejpam-6574	214	11	/	/	SYM
ejpam-6574	214	12	eur	eur	PROPN
ejpam-6574	214	13	.	.	PUNCT
ejpam-6574	215	1	j.	j.	PROPN
ejpam-6574	215	2	pure	pure	PROPN
ejpam-6574	215	3	appl	appl	PROPN
ejpam-6574	215	4	.	.	PROPN
ejpam-6574	215	5	math	math	PROPN
ejpam-6574	215	6	,	,	PUNCT
ejpam-6574	215	7	18	18	NUM
ejpam-6574	215	8	(	(	PUNCT
ejpam-6574	215	9	3	3	NUM
ejpam-6574	215	10	)	)	PUNCT
ejpam-6574	215	11	(	(	PUNCT
ejpam-6574	215	12	2025	2025	NUM
ejpam-6574	215	13	)	)	PUNCT
ejpam-6574	215	14	,	,	PUNCT
ejpam-6574	215	15	6574	6574	NUM
ejpam-6574	215	16	9	9	NUM
ejpam-6574	215	17	of	of	ADP
ejpam-6574	215	18	10	10	NUM
ejpam-6574	215	19	proof	proof	NOUN
ejpam-6574	215	20	.	.	PUNCT
ejpam-6574	216	1	let	let	VERB
ejpam-6574	216	2	b	b	X
ejpam-6574	216	3	be	be	AUX
ejpam-6574	216	4	any	any	DET
ejpam-6574	216	5	subset	subset	NOUN
ejpam-6574	216	6	of	of	ADP
ejpam-6574	216	7	y	y	PROPN
ejpam-6574	216	8	and	and	CCONJ
ejpam-6574	216	9	u	u	NOUN
ejpam-6574	216	10	be	be	VERB
ejpam-6574	216	11	any	any	DET
ejpam-6574	216	12	τ1τ2	τ1τ2	ADJ
ejpam-6574	216	13	-	-	ADJ
ejpam-6574	216	14	open	open	ADJ
ejpam-6574	216	15	set	set	NOUN
ejpam-6574	216	16	of	of	ADP
ejpam-6574	216	17	x	x	PUNCT
ejpam-6574	216	18	with	with	ADP
ejpam-6574	216	19	f−1(b	f−1(b	PROPN
ejpam-6574	216	20	)	)	PUNCT
ejpam-6574	216	21	⊆	⊆	NUM
ejpam-6574	216	22	u	u	NOUN
ejpam-6574	216	23	.	.	PUNCT
ejpam-6574	217	1	put	put	VERB
ejpam-6574	217	2	k	k	NOUN
ejpam-6574	217	3	=	=	PUNCT
ejpam-6574	217	4	θ(σ1	θ(σ1	NOUN
ejpam-6574	217	5	,	,	PUNCT
ejpam-6574	217	6	σ2)b	σ2)b	NOUN
ejpam-6574	217	7	-	-	PUNCT
ejpam-6574	217	8	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	217	9	-	-	PUNCT
ejpam-6574	217	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	217	11	-	-	PUNCT
ejpam-6574	217	12	cl(u	cl(u	NOUN
ejpam-6574	217	13	)	)	PUNCT
ejpam-6574	217	14	)	)	PUNCT
ejpam-6574	217	15	)	)	PUNCT
ejpam-6574	217	16	)	)	PUNCT
ejpam-6574	217	17	.	.	PUNCT
ejpam-6574	218	1	then	then	ADV
ejpam-6574	218	2	,	,	PUNCT
ejpam-6574	218	3	k	k	PROPN
ejpam-6574	218	4	is	be	AUX
ejpam-6574	218	5	a	a	DET
ejpam-6574	218	6	θ(σ1	θ(σ1	NOUN
ejpam-6574	218	7	,	,	PUNCT
ejpam-6574	218	8	σ2)b	σ2)b	NOUN
ejpam-6574	218	9	-	-	PUNCT
ejpam-6574	218	10	closed	close	VERB
ejpam-6574	218	11	set	set	NOUN
ejpam-6574	218	12	of	of	ADP
ejpam-6574	218	13	y	y	PRON
ejpam-6574	218	14	such	such	ADJ
ejpam-6574	218	15	that	that	PRON
ejpam-6574	218	16	b	b	PROPN
ejpam-6574	218	17	⊆	⊆	NUM
ejpam-6574	218	18	k	k	NOUN
ejpam-6574	218	19	,	,	PUNCT
ejpam-6574	218	20	since	since	SCONJ
ejpam-6574	218	21	b	b	PROPN
ejpam-6574	218	22	⊆	⊆	NUM
ejpam-6574	218	23	f(u	f(u	PROPN
ejpam-6574	218	24	)	)	PUNCT
ejpam-6574	218	25	⊆	⊆	NUM
ejpam-6574	218	26	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6574	218	27	-	-	PUNCT
ejpam-6574	218	28	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	218	29	-	-	PUNCT
ejpam-6574	218	30	cl(u	cl(u	NOUN
ejpam-6574	218	31	)	)	PUNCT
ejpam-6574	218	32	)	)	PUNCT
ejpam-6574	218	33	)	)	PUNCT
ejpam-6574	218	34	⊆	⊆	NUM
ejpam-6574	218	35	θ(σ1	θ(σ1	NOUN
ejpam-6574	218	36	,	,	PUNCT
ejpam-6574	218	37	σ2)b	σ2)b	NOUN
ejpam-6574	218	38	-	-	PUNCT
ejpam-6574	218	39	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6574	218	40	-	-	PUNCT
ejpam-6574	218	41	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6574	218	42	-	-	PUNCT
ejpam-6574	218	43	cl(u	cl(u	NOUN
ejpam-6574	218	44	)	)	PUNCT
ejpam-6574	218	45	)	)	PUNCT
ejpam-6574	218	46	)	)	PUNCT
ejpam-6574	218	47	)	)	PUNCT
ejpam-6574	219	1	=	=	PUNCT
ejpam-6574	219	2	k.	k.	PROPN
ejpam-6574	219	3	since	since	SCONJ
ejpam-6574	219	4	f	f	PROPN
ejpam-6574	219	5	is	be	AUX
ejpam-6574	219	6	weakly	weakly	ADJ
ejpam-6574	219	7	θ(τ1	θ(τ1	NOUN
ejpam-6574	219	8	,	,	PUNCT
ejpam-6574	219	9	τ2)b	τ2)b	NOUN
ejpam-6574	219	10	-	-	PUNCT
ejpam-6574	219	11	closed	closed	ADJ
ejpam-6574	219	12	,	,	PUNCT
ejpam-6574	219	13	by	by	ADP
ejpam-6574	219	14	theorem	theorem	NOUN
ejpam-6574	219	15	6	6	NUM
ejpam-6574	219	16	we	we	PRON
ejpam-6574	219	17	have	have	VERB
ejpam-6574	219	18	f−1(k	f−1(k	PROPN
ejpam-6574	219	19	)	)	PUNCT
ejpam-6574	220	1	⊆	⊆	NUM
ejpam-6574	220	2	τ1τ2	τ1τ2	NOUN
ejpam-6574	220	3	-	-	NOUN
ejpam-6574	220	4	cl(u	cl(u	NOUN
ejpam-6574	220	5	)	)	PUNCT
ejpam-6574	220	6	.	.	PUNCT
ejpam-6574	221	1	acknowledgements	acknowledgement	NOUN
ejpam-6574	221	2	this	this	DET
ejpam-6574	221	3	research	research	NOUN
ejpam-6574	221	4	project	project	NOUN
ejpam-6574	221	5	was	be	AUX
ejpam-6574	221	6	financially	financially	ADV
ejpam-6574	221	7	supported	support	VERB
ejpam-6574	221	8	by	by	ADP
ejpam-6574	221	9	mahasarakham	mahasarakham	PROPN
ejpam-6574	221	10	university	university	PROPN
ejpam-6574	221	11	.	.	PUNCT
ejpam-6574	222	1	references	reference	NOUN
ejpam-6574	222	2	[	[	X
ejpam-6574	222	3	1	1	X
ejpam-6574	222	4	]	]	PUNCT
ejpam-6574	222	5	d.	d.	PROPN
ejpam-6574	222	6	a.	a.	PROPN
ejpam-6574	222	7	rose	rise	VERB
ejpam-6574	222	8	.	.	PUNCT
ejpam-6574	223	1	on	on	ADP
ejpam-6574	223	2	weak	weak	ADJ
ejpam-6574	223	3	openness	openness	NOUN
ejpam-6574	223	4	and	and	CCONJ
ejpam-6574	223	5	almost	almost	ADV
ejpam-6574	223	6	openness	openness	NOUN
ejpam-6574	223	7	.	.	PUNCT
ejpam-6574	224	1	international	international	ADJ
ejpam-6574	224	2	journal	journal	NOUN
ejpam-6574	224	3	of	of	ADP
ejpam-6574	224	4	mathematics	mathematics	PROPN
ejpam-6574	224	5	and	and	CCONJ
ejpam-6574	224	6	mathematical	mathematical	ADJ
ejpam-6574	224	7	sciences	science	NOUN
ejpam-6574	224	8	,	,	PUNCT
ejpam-6574	224	9	7:35–40	7:35–40	NUM
ejpam-6574	224	10	,	,	PUNCT
ejpam-6574	224	11	1984	1984	NUM
ejpam-6574	224	12	.	.	PUNCT
ejpam-6574	225	1	[	[	X
ejpam-6574	225	2	2	2	X
ejpam-6574	225	3	]	]	PUNCT
ejpam-6574	225	4	d.	d.	PROPN
ejpam-6574	225	5	a.	a.	PROPN
ejpam-6574	225	6	rose	rise	VERB
ejpam-6574	225	7	and	and	CCONJ
ejpam-6574	225	8	d.	d.	PROPN
ejpam-6574	225	9	s.	s.	PROPN
ejpam-6574	226	1	janković.	janković.	PROPN
ejpam-6574	226	2	weakly	weakly	ADJ
ejpam-6574	226	3	closed	close	VERB
ejpam-6574	226	4	functions	function	NOUN
ejpam-6574	226	5	and	and	CCONJ
ejpam-6574	226	6	hausdorff	hausdorff	NOUN
ejpam-6574	226	7	spaces	space	NOUN
ejpam-6574	226	8	.	.	PUNCT
ejpam-6574	227	1	mathematische	mathematische	PROPN
ejpam-6574	227	2	nachrichten	nachrichten	PROPN
ejpam-6574	227	3	,	,	PUNCT
ejpam-6574	227	4	130:105–110	130:105–110	NUM
ejpam-6574	227	5	,	,	PUNCT
ejpam-6574	227	6	1987	1987	NUM
ejpam-6574	227	7	.	.	PUNCT
ejpam-6574	228	1	[	[	X
ejpam-6574	228	2	3	3	NUM
ejpam-6574	228	3	]	]	X
ejpam-6574	228	4	m.	m.	NOUN
ejpam-6574	228	5	caldas	caldas	PROPN
ejpam-6574	228	6	,	,	PUNCT
ejpam-6574	228	7	s.	s.	PROPN
ejpam-6574	228	8	jafari	jafari	PROPN
ejpam-6574	228	9	,	,	PUNCT
ejpam-6574	228	10	g.	g.	PROPN
ejpam-6574	228	11	navalagi	navalagi	PROPN
ejpam-6574	228	12	,	,	PUNCT
ejpam-6574	228	13	and	and	CCONJ
ejpam-6574	228	14	t.	t.	PROPN
ejpam-6574	228	15	noiri	noiri	PROPN
ejpam-6574	228	16	.	.	PUNCT
ejpam-6574	229	1	on	on	ADP
ejpam-6574	229	2	pre	pre	ADJ
ejpam-6574	229	3	-	-	ADJ
ejpam-6574	229	4	θ	θ	ADJ
ejpam-6574	229	5	-	-	ADJ
ejpam-6574	229	6	open	open	ADJ
ejpam-6574	229	7	sets	set	NOUN
ejpam-6574	229	8	and	and	CCONJ
ejpam-6574	229	9	two	two	NUM
ejpam-6574	229	10	classes	class	NOUN
ejpam-6574	229	11	of	of	ADP
ejpam-6574	229	12	functions	function	NOUN
ejpam-6574	229	13	.	.	PUNCT
ejpam-6574	230	1	bulletin	bulletin	NOUN
ejpam-6574	230	2	of	of	ADP
ejpam-6574	230	3	the	the	DET
ejpam-6574	230	4	iranian	iranian	PROPN
ejpam-6574	230	5	mathematical	mathematical	ADJ
ejpam-6574	230	6	society	society	NOUN
ejpam-6574	230	7	,	,	PUNCT
ejpam-6574	230	8	32(1):45–63	32(1):45–63	NUM
ejpam-6574	230	9	,	,	PUNCT
ejpam-6574	230	10	2006	2006	NUM
ejpam-6574	230	11	.	.	PUNCT
ejpam-6574	231	1	[	[	X
ejpam-6574	231	2	4	4	NUM
ejpam-6574	231	3	]	]	PUNCT
ejpam-6574	231	4	m.	m.	NOUN
ejpam-6574	231	5	caldas	caldas	PROPN
ejpam-6574	231	6	,	,	PUNCT
ejpam-6574	231	7	s.	s.	PROPN
ejpam-6574	231	8	jafari	jafari	PROPN
ejpam-6574	231	9	,	,	PUNCT
ejpam-6574	231	10	and	and	CCONJ
ejpam-6574	231	11	g.	g.	PROPN
ejpam-6574	231	12	navalagi	navalagi	PROPN
ejpam-6574	231	13	.	.	PUNCT
ejpam-6574	232	1	weak	weak	ADJ
ejpam-6574	232	2	forms	form	NOUN
ejpam-6574	232	3	of	of	ADP
ejpam-6574	232	4	open	open	ADJ
ejpam-6574	232	5	and	and	CCONJ
ejpam-6574	232	6	closed	closed	ADJ
ejpam-6574	232	7	functions	function	NOUN
ejpam-6574	232	8	via	via	ADP
ejpam-6574	232	9	semi	semi	ADJ
ejpam-6574	232	10	-	-	ADJ
ejpam-6574	232	11	θ	θ	ADJ
ejpam-6574	232	12	-	-	ADJ
ejpam-6574	232	13	open	open	ADJ
ejpam-6574	232	14	sets	set	NOUN
ejpam-6574	232	15	.	.	PUNCT
ejpam-6574	233	1	carpathian	carpathian	ADJ
ejpam-6574	233	2	journal	journal	PROPN
ejpam-6574	233	3	of	of	ADP
ejpam-6574	233	4	mathematics	mathematic	NOUN
ejpam-6574	233	5	,	,	PUNCT
ejpam-6574	233	6	22(1	22(1	NUM
ejpam-6574	233	7	-	-	PUNCT
ejpam-6574	233	8	2):21–31	2):21–31	NUM
ejpam-6574	233	9	,	,	PUNCT
ejpam-6574	233	10	2006	2006	NUM
ejpam-6574	233	11	.	.	PUNCT
ejpam-6574	234	1	[	[	X
ejpam-6574	234	2	5	5	X
ejpam-6574	234	3	]	]	PUNCT
ejpam-6574	234	4	t.	t.	PROPN
ejpam-6574	234	5	noiri	noiri	PROPN
ejpam-6574	234	6	.	.	PUNCT
ejpam-6574	235	1	weak	weak	ADJ
ejpam-6574	235	2	forms	form	NOUN
ejpam-6574	235	3	of	of	ADP
ejpam-6574	235	4	open	open	ADJ
ejpam-6574	235	5	and	and	CCONJ
ejpam-6574	235	6	closed	closed	ADJ
ejpam-6574	235	7	functions	function	NOUN
ejpam-6574	235	8	via	via	ADP
ejpam-6574	235	9	b	b	X
ejpam-6574	235	10	-	-	PUNCT
ejpam-6574	235	11	θ	θ	ADJ
ejpam-6574	235	12	-	-	PUNCT
ejpam-6574	235	13	open	open	ADJ
ejpam-6574	235	14	sets	set	NOUN
ejpam-6574	235	15	.	.	PUNCT
ejpam-6574	236	1	demonstratio	demonstratio	PROPN
ejpam-6574	236	2	mathematica	mathematica	PROPN
ejpam-6574	236	3	,	,	PUNCT
ejpam-6574	236	4	42(1):193–203	42(1):193–203	PROPN
ejpam-6574	236	5	,	,	PUNCT
ejpam-6574	236	6	2009	2009	NUM
ejpam-6574	236	7	.	.	PUNCT
ejpam-6574	237	1	[	[	X
ejpam-6574	237	2	6	6	NUM
ejpam-6574	237	3	]	]	X
ejpam-6574	237	4	c.	c.	PROPN
ejpam-6574	237	5	klanarong	klanarong	PROPN
ejpam-6574	237	6	and	and	CCONJ
ejpam-6574	237	7	c.	c.	PROPN
ejpam-6574	237	8	boonpok	boonpok	PROPN
ejpam-6574	237	9	.	.	PUNCT
ejpam-6574	238	1	characterizations	characterization	NOUN
ejpam-6574	238	2	of	of	ADP
ejpam-6574	238	3	weakly	weakly	ADJ
ejpam-6574	238	4	s(λ	s(λ	NOUN
ejpam-6574	238	5	,	,	PUNCT
ejpam-6574	238	6	p)-open	p)-open	NOUN
ejpam-6574	238	7	functions	function	NOUN
ejpam-6574	238	8	and	and	CCONJ
ejpam-6574	238	9	weakly	weakly	ADJ
ejpam-6574	238	10	s(λ	s(λ	PROPN
ejpam-6574	238	11	,	,	PUNCT
ejpam-6574	238	12	p)-closed	p)-close	VERB
ejpam-6574	238	13	functions	function	NOUN
ejpam-6574	238	14	.	.	PUNCT
ejpam-6574	239	1	international	international	ADJ
ejpam-6574	239	2	journal	journal	PROPN
ejpam-6574	239	3	of	of	ADP
ejpam-6574	239	4	mathematics	mathematic	NOUN
ejpam-6574	239	5	and	and	CCONJ
ejpam-6574	239	6	computer	computer	NOUN
ejpam-6574	239	7	science	science	NOUN
ejpam-6574	239	8	,	,	PUNCT
ejpam-6574	239	9	19(3):809–814	19(3):809–814	PROPN
ejpam-6574	239	10	,	,	PUNCT
ejpam-6574	239	11	2024	2024	NUM
ejpam-6574	239	12	.	.	PUNCT
ejpam-6574	240	1	[	[	X
ejpam-6574	240	2	7	7	X
ejpam-6574	240	3	]	]	X
ejpam-6574	240	4	c.	c.	PROPN
ejpam-6574	240	5	boonpok	boonpok	PROPN
ejpam-6574	240	6	and	and	CCONJ
ejpam-6574	240	7	n.	n.	PROPN
ejpam-6574	240	8	srisarakham	srisarakham	PROPN
ejpam-6574	240	9	.	.	PUNCT
ejpam-6574	241	1	θp(λ	θp(λ	NOUN
ejpam-6574	241	2	,	,	PUNCT
ejpam-6574	241	3	p)-open	p)-open	NOUN
ejpam-6574	241	4	functions	function	NOUN
ejpam-6574	241	5	and	and	CCONJ
ejpam-6574	241	6	θp(λ	θp(λ	NOUN
ejpam-6574	241	7	,	,	PUNCT
ejpam-6574	241	8	p)-closed	p)-close	VERB
ejpam-6574	241	9	functions	function	NOUN
ejpam-6574	241	10	.	.	PUNCT
ejpam-6574	242	1	asia	asia	PROPN
ejpam-6574	242	2	pacific	pacific	PROPN
ejpam-6574	242	3	journal	journal	PROPN
ejpam-6574	242	4	of	of	ADP
ejpam-6574	242	5	mathematics	mathematic	NOUN
ejpam-6574	242	6	,	,	PUNCT
ejpam-6574	242	7	10:48	10:48	NUM
ejpam-6574	242	8	,	,	PUNCT
ejpam-6574	242	9	2023	2023	NUM
ejpam-6574	242	10	.	.	PUNCT
ejpam-6574	243	1	[	[	X
ejpam-6574	243	2	8	8	NUM
ejpam-6574	243	3	]	]	X
ejpam-6574	243	4	c.	c.	PROPN
ejpam-6574	243	5	boonpok	boonpok	PROPN
ejpam-6574	243	6	.	.	PUNCT
ejpam-6574	244	1	semi	semi	ADJ
ejpam-6574	244	2	-	-	ADJ
ejpam-6574	244	3	i	i	ADV
ejpam-6574	244	4	-	-	PUNCT
ejpam-6574	244	5	expandable	expandable	ADJ
ejpam-6574	244	6	ideal	ideal	ADJ
ejpam-6574	244	7	topological	topological	ADJ
ejpam-6574	244	8	spaces	space	NOUN
ejpam-6574	244	9	.	.	PUNCT
ejpam-6574	245	1	journal	journal	NOUN
ejpam-6574	245	2	of	of	ADP
ejpam-6574	245	3	mathematics	mathematic	NOUN
ejpam-6574	245	4	,	,	PUNCT
ejpam-6574	245	5	2021:9272335	2021:9272335	NUM
ejpam-6574	245	6	,	,	PUNCT
ejpam-6574	245	7	2021	2021	NUM
ejpam-6574	245	8	.	.	PUNCT
ejpam-6574	246	1	[	[	X
ejpam-6574	246	2	9	9	NUM
ejpam-6574	246	3	]	]	X
ejpam-6574	246	4	n.	n.	NOUN
ejpam-6574	246	5	srisarakham	srisarakham	PROPN
ejpam-6574	246	6	and	and	CCONJ
ejpam-6574	246	7	c.	c.	PROPN
ejpam-6574	246	8	boonpok	boonpok	PROPN
ejpam-6574	246	9	.	.	PUNCT
ejpam-6574	247	1	on	on	ADP
ejpam-6574	247	2	weakly	weakly	ADJ
ejpam-6574	247	3	δ(λ	δ(λ	PROPN
ejpam-6574	247	4	,	,	PUNCT
ejpam-6574	247	5	p)-open	p)-open	VERB
ejpam-6574	247	6	functions	function	NOUN
ejpam-6574	247	7	.	.	PUNCT
ejpam-6574	248	1	international	international	ADJ
ejpam-6574	248	2	journal	journal	PROPN
ejpam-6574	248	3	of	of	ADP
ejpam-6574	248	4	mathematics	mathematic	NOUN
ejpam-6574	248	5	and	and	CCONJ
ejpam-6574	248	6	computer	computer	NOUN
ejpam-6574	248	7	science	science	NOUN
ejpam-6574	248	8	,	,	PUNCT
ejpam-6574	248	9	19(2):485–489	19(2):485–489	PROPN
ejpam-6574	248	10	,	,	PUNCT
ejpam-6574	248	11	2024	2024	NUM
ejpam-6574	248	12	.	.	PUNCT
ejpam-6574	249	1	[	[	X
ejpam-6574	249	2	10	10	NUM
ejpam-6574	249	3	]	]	X
ejpam-6574	249	4	c.	c.	PROPN
ejpam-6574	249	5	klanarong	klanarong	PROPN
ejpam-6574	249	6	and	and	CCONJ
ejpam-6574	249	7	c.	c.	PROPN
ejpam-6574	249	8	boonpok	boonpok	PROPN
ejpam-6574	249	9	.	.	PUNCT
ejpam-6574	250	1	characterizations	characterization	NOUN
ejpam-6574	250	2	of	of	ADP
ejpam-6574	250	3	weakly	weakly	ADJ
ejpam-6574	250	4	δ(λ	δ(λ	PROPN
ejpam-6574	250	5	,	,	PUNCT
ejpam-6574	250	6	p)-closed	p)-close	VERB
ejpam-6574	250	7	functions	function	NOUN
ejpam-6574	250	8	.	.	PUNCT
ejpam-6574	251	1	international	international	ADJ
ejpam-6574	251	2	journal	journal	PROPN
ejpam-6574	251	3	of	of	ADP
ejpam-6574	251	4	mathematics	mathematic	NOUN
ejpam-6574	251	5	and	and	CCONJ
ejpam-6574	251	6	computer	computer	NOUN
ejpam-6574	251	7	science	science	NOUN
ejpam-6574	251	8	,	,	PUNCT
ejpam-6574	251	9	19(2):503–507	19(2):503–507	PROPN
ejpam-6574	251	10	,	,	PUNCT
ejpam-6574	251	11	2024	2024	NUM
ejpam-6574	251	12	.	.	PUNCT
ejpam-6574	252	1	[	[	X
ejpam-6574	252	2	11	11	NUM
ejpam-6574	252	3	]	]	X
ejpam-6574	252	4	c.	c.	PROPN
ejpam-6574	252	5	boonpok	boonpok	PROPN
ejpam-6574	252	6	and	and	CCONJ
ejpam-6574	252	7	p.	p.	NOUN
ejpam-6574	252	8	pue	pue	NOUN
ejpam-6574	252	9	-	-	PUNCT
ejpam-6574	252	10	on	on	ADP
ejpam-6574	252	11	.	.	PUNCT
ejpam-6574	253	1	weakly	weakly	ADJ
ejpam-6574	253	2	θs(λ	θs(λ	NOUN
ejpam-6574	253	3	,	,	PUNCT
ejpam-6574	253	4	p)-open	p)-open	VERB
ejpam-6574	253	5	functions	function	NOUN
ejpam-6574	253	6	and	and	CCONJ
ejpam-6574	253	7	weakly	weakly	ADJ
ejpam-6574	253	8	θs(λ	θs(λ	NOUN
ejpam-6574	253	9	,	,	PUNCT
ejpam-6574	253	10	p)closed	p)close	VERB
ejpam-6574	253	11	functions	function	NOUN
ejpam-6574	253	12	.	.	PUNCT
ejpam-6574	254	1	asia	asia	PROPN
ejpam-6574	254	2	pacific	pacific	PROPN
ejpam-6574	254	3	journal	journal	PROPN
ejpam-6574	254	4	of	of	ADP
ejpam-6574	254	5	mathematics	mathematic	NOUN
ejpam-6574	254	6	,	,	PUNCT
ejpam-6574	254	7	11:13	11:13	NUM
ejpam-6574	254	8	,	,	PUNCT
ejpam-6574	254	9	2024	2024	NUM
ejpam-6574	254	10	.	.	PUNCT
ejpam-6574	255	1	[	[	X
ejpam-6574	255	2	12	12	NUM
ejpam-6574	255	3	]	]	X
ejpam-6574	255	4	n.	n.	NOUN
ejpam-6574	255	5	chutiman	chutiman	NOUN
ejpam-6574	255	6	and	and	CCONJ
ejpam-6574	255	7	c.	c.	PROPN
ejpam-6574	255	8	boonpok	boonpok	PROPN
ejpam-6574	255	9	.	.	PUNCT
ejpam-6574	256	1	some	some	DET
ejpam-6574	256	2	properties	property	NOUN
ejpam-6574	256	3	of	of	ADP
ejpam-6574	256	4	weakly	weakly	ADJ
ejpam-6574	256	5	b(λ	b(λ	NOUN
ejpam-6574	256	6	,	,	PUNCT
ejpam-6574	256	7	p)-open	p)-open	NOUN
ejpam-6574	256	8	functions	function	NOUN
ejpam-6574	256	9	.	.	PUNCT
ejpam-6574	257	1	international	international	ADJ
ejpam-6574	257	2	journal	journal	PROPN
ejpam-6574	257	3	of	of	ADP
ejpam-6574	257	4	mathematics	mathematic	NOUN
ejpam-6574	257	5	and	and	CCONJ
ejpam-6574	257	6	computer	computer	NOUN
ejpam-6574	257	7	science	science	NOUN
ejpam-6574	257	8	,	,	PUNCT
ejpam-6574	257	9	19(2):497–501	19(2):497–501	NUM
ejpam-6574	257	10	,	,	PUNCT
ejpam-6574	257	11	2024	2024	NUM
ejpam-6574	257	12	.	.	PUNCT
ejpam-6574	258	1	[	[	X
ejpam-6574	258	2	13	13	NUM
ejpam-6574	258	3	]	]	PUNCT
ejpam-6574	258	4	c.	c.	PROPN
ejpam-6574	258	5	boonpok	boonpok	PROPN
ejpam-6574	258	6	,	,	PUNCT
ejpam-6574	258	7	c.	c.	PROPN
ejpam-6574	258	8	viriyapong	viriyapong	PROPN
ejpam-6574	258	9	,	,	PUNCT
ejpam-6574	258	10	and	and	CCONJ
ejpam-6574	258	11	m.	m.	NOUN
ejpam-6574	258	12	thongmoon	thongmoon	NOUN
ejpam-6574	258	13	.	.	PUNCT
ejpam-6574	259	1	on	on	ADP
ejpam-6574	259	2	upper	upper	ADJ
ejpam-6574	259	3	and	and	CCONJ
ejpam-6574	259	4	lower	low	ADJ
ejpam-6574	259	5	(	(	PUNCT
ejpam-6574	259	6	τ1	τ1	NOUN
ejpam-6574	259	7	,	,	PUNCT
ejpam-6574	259	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6574	259	9	multifunctions	multifunction	NOUN
ejpam-6574	259	10	.	.	PUNCT
ejpam-6574	260	1	journal	journal	PROPN
ejpam-6574	260	2	of	of	ADP
ejpam-6574	260	3	mathematics	mathematics	PROPN
ejpam-6574	260	4	and	and	CCONJ
ejpam-6574	260	5	computer	computer	NOUN
ejpam-6574	260	6	science	science	NOUN
ejpam-6574	260	7	,	,	PUNCT
ejpam-6574	260	8	18:282–293	18:282–293	NUM
ejpam-6574	260	9	,	,	PUNCT
ejpam-6574	260	10	2018	2018	NUM
ejpam-6574	260	11	.	.	PUNCT
ejpam-6574	261	1	n.	n.	NOUN
ejpam-6574	261	2	chutiman	chutiman	PROPN
ejpam-6574	261	3	,	,	PUNCT
ejpam-6574	261	4	a.	a.	PROPN
ejpam-6574	261	5	sama	sama	PROPN
ejpam-6574	261	6	-	-	PUNCT
ejpam-6574	261	7	ae	ae	PROPN
ejpam-6574	261	8	,	,	PUNCT
ejpam-6574	261	9	c.	c.	PROPN
ejpam-6574	261	10	boonpok	boonpok	PROPN
ejpam-6574	261	11	/	/	SYM
ejpam-6574	261	12	eur	eur	PROPN
ejpam-6574	261	13	.	.	PUNCT
ejpam-6574	262	1	j.	j.	PROPN
ejpam-6574	262	2	pure	pure	PROPN
ejpam-6574	262	3	appl	appl	PROPN
ejpam-6574	262	4	.	.	PROPN
ejpam-6574	262	5	math	math	PROPN
ejpam-6574	262	6	,	,	PUNCT
ejpam-6574	262	7	18	18	NUM
ejpam-6574	262	8	(	(	PUNCT
ejpam-6574	262	9	3	3	NUM
ejpam-6574	262	10	)	)	PUNCT
ejpam-6574	262	11	(	(	PUNCT
ejpam-6574	262	12	2025	2025	NUM
ejpam-6574	262	13	)	)	PUNCT
ejpam-6574	262	14	,	,	PUNCT
ejpam-6574	262	15	6574	6574	NUM
ejpam-6574	262	16	10	10	NUM
ejpam-6574	262	17	of	of	ADP
ejpam-6574	262	18	10	10	NUM
ejpam-6574	263	1	[	[	SYM
ejpam-6574	263	2	14	14	NUM
ejpam-6574	263	3	]	]	X
ejpam-6574	263	4	c.	c.	PROPN
ejpam-6574	263	5	viriyapong	viriyapong	PROPN
ejpam-6574	263	6	and	and	CCONJ
ejpam-6574	263	7	c.	c.	PROPN
ejpam-6574	263	8	boonpok	boonpok	PROPN
ejpam-6574	263	9	.	.	PUNCT
ejpam-6574	264	1	(	(	PUNCT
ejpam-6574	264	2	τ1	τ1	NOUN
ejpam-6574	264	3	,	,	PUNCT
ejpam-6574	264	4	τ2)α	τ2)α	NOUN
ejpam-6574	264	5	-	-	PUNCT
ejpam-6574	264	6	continuity	continuity	NOUN
ejpam-6574	264	7	for	for	ADP
ejpam-6574	264	8	multifunctions	multifunction	NOUN
ejpam-6574	264	9	.	.	PUNCT
ejpam-6574	265	1	journal	journal	PROPN
ejpam-6574	265	2	of	of	ADP
ejpam-6574	265	3	mathematics	mathematic	NOUN
ejpam-6574	265	4	,	,	PUNCT
ejpam-6574	265	5	2020:6285763	2020:6285763	NUM
ejpam-6574	265	6	,	,	PUNCT
ejpam-6574	265	7	2020	2020	NUM
ejpam-6574	265	8	.	.	PUNCT
ejpam-6574	266	1	[	[	X
ejpam-6574	266	2	15	15	NUM
ejpam-6574	266	3	]	]	X
ejpam-6574	266	4	c.	c.	PROPN
ejpam-6574	266	5	boonpok	boonpok	PROPN
ejpam-6574	266	6	.	.	PUNCT
ejpam-6574	267	1	(	(	PUNCT
ejpam-6574	267	2	τ1	τ1	NOUN
ejpam-6574	267	3	,	,	PUNCT
ejpam-6574	267	4	τ2)δ	τ2)δ	ADJ
ejpam-6574	267	5	-	-	PUNCT
ejpam-6574	267	6	semicontinuous	semicontinuous	ADJ
ejpam-6574	267	7	multifunctions	multifunction	NOUN
ejpam-6574	267	8	.	.	PUNCT
ejpam-6574	268	1	heliyon	heliyon	NOUN
ejpam-6574	268	2	,	,	PUNCT
ejpam-6574	268	3	6	6	NUM
ejpam-6574	268	4	:	:	SYM
ejpam-6574	268	5	e05367	e05367	PROPN
ejpam-6574	268	6	,	,	PUNCT
ejpam-6574	268	7	2020	2020	NUM
ejpam-6574	268	8	.	.	PUNCT
ejpam-6574	269	1	[	[	X
ejpam-6574	269	2	16	16	NUM
ejpam-6574	269	3	]	]	X
ejpam-6574	269	4	n.	n.	PROPN
ejpam-6574	269	5	viriyapong	viriyapong	PROPN
ejpam-6574	269	6	,	,	PUNCT
ejpam-6574	269	7	s.	s.	PROPN
ejpam-6574	269	8	sompong	sompong	PROPN
ejpam-6574	269	9	,	,	PUNCT
ejpam-6574	269	10	and	and	CCONJ
ejpam-6574	269	11	c.	c.	PROPN
ejpam-6574	269	12	boonpok	boonpok	PROPN
ejpam-6574	269	13	.	.	PUNCT
ejpam-6574	270	1	(	(	PUNCT
ejpam-6574	270	2	τ1	τ1	NOUN
ejpam-6574	270	3	,	,	PUNCT
ejpam-6574	270	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6574	270	5	disconnectedness	disconnectedness	NOUN
ejpam-6574	270	6	in	in	ADP
ejpam-6574	270	7	bitopological	bitopological	ADJ
ejpam-6574	270	8	spaces	space	NOUN
ejpam-6574	270	9	.	.	PUNCT
ejpam-6574	271	1	international	international	ADJ
ejpam-6574	271	2	journal	journal	PROPN
ejpam-6574	271	3	of	of	ADP
ejpam-6574	271	4	mathematics	mathematic	NOUN
ejpam-6574	271	5	and	and	CCONJ
ejpam-6574	271	6	computer	computer	NOUN
ejpam-6574	271	7	science	science	NOUN
ejpam-6574	271	8	,	,	PUNCT
ejpam-6574	271	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6574	271	10	,	,	PUNCT
ejpam-6574	271	11	2024	2024	NUM
ejpam-6574	271	12	.	.	PUNCT
ejpam-6574	272	1	[	[	X
ejpam-6574	272	2	17	17	NUM
ejpam-6574	272	3	]	]	X
ejpam-6574	272	4	n.	n.	PROPN
ejpam-6574	272	5	viriyapong	viriyapong	PROPN
ejpam-6574	272	6	,	,	PUNCT
ejpam-6574	272	7	s.	s.	PROPN
ejpam-6574	272	8	sompong	sompong	PROPN
ejpam-6574	272	9	,	,	PUNCT
ejpam-6574	272	10	and	and	CCONJ
ejpam-6574	272	11	c.	c.	PROPN
ejpam-6574	272	12	boonpok	boonpok	PROPN
ejpam-6574	272	13	.	.	PUNCT
ejpam-6574	273	1	upper	upper	ADJ
ejpam-6574	273	2	and	and	CCONJ
ejpam-6574	273	3	lower	low	ADJ
ejpam-6574	273	4	s-(τ1	s-(τ1	NOUN
ejpam-6574	273	5	,	,	PUNCT
ejpam-6574	273	6	τ2)p	τ2)p	ADJ
ejpam-6574	273	7	-	-	PUNCT
ejpam-6574	273	8	continuous	continuous	ADJ
ejpam-6574	273	9	multifunctions	multifunction	NOUN
ejpam-6574	273	10	.	.	PUNCT
ejpam-6574	274	1	european	european	ADJ
ejpam-6574	274	2	journal	journal	PROPN
ejpam-6574	274	3	of	of	ADP
ejpam-6574	274	4	pure	pure	ADJ
ejpam-6574	274	5	and	and	CCONJ
ejpam-6574	274	6	applied	applied	ADJ
ejpam-6574	274	7	mathematics	mathematic	NOUN
ejpam-6574	274	8	,	,	PUNCT
ejpam-6574	274	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-6574	274	10	,	,	PUNCT
ejpam-6574	274	11	2024	2024	NUM
ejpam-6574	274	12	.	.	PUNCT
ejpam-6574	275	1	[	[	X
ejpam-6574	275	2	18	18	NUM
ejpam-6574	275	3	]	]	X
ejpam-6574	275	4	c.	c.	PROPN
ejpam-6574	275	5	viriyapong	viriyapong	PROPN
ejpam-6574	275	6	,	,	PUNCT
ejpam-6574	275	7	s.	s.	PROPN
ejpam-6574	275	8	sompong	sompong	PROPN
ejpam-6574	275	9	,	,	PUNCT
ejpam-6574	275	10	and	and	CCONJ
ejpam-6574	275	11	c.	c.	PROPN
ejpam-6574	275	12	boonpok	boonpok	PROPN
ejpam-6574	275	13	.	.	PUNCT
ejpam-6574	276	1	upper	upper	ADJ
ejpam-6574	276	2	and	and	CCONJ
ejpam-6574	276	3	lower	low	ADJ
ejpam-6574	276	4	slight	slight	ADJ
ejpam-6574	276	5	α(τ1	α(τ1	NOUN
ejpam-6574	276	6	,	,	PUNCT
ejpam-6574	276	7	τ2)continuity	τ2)continuity	PROPN
ejpam-6574	276	8	.	.	PUNCT
ejpam-6574	277	1	european	european	PROPN
ejpam-6574	277	2	journal	journal	PROPN
ejpam-6574	277	3	of	of	ADP
ejpam-6574	277	4	pure	pure	ADJ
ejpam-6574	277	5	and	and	CCONJ
ejpam-6574	277	6	applied	applied	ADJ
ejpam-6574	277	7	mathematics	mathematic	NOUN
ejpam-6574	277	8	,	,	PUNCT
ejpam-6574	277	9	17(3):2142–2154	17(3):2142–2154	NUM
ejpam-6574	277	10	,	,	PUNCT
ejpam-6574	277	11	2024	2024	NUM
ejpam-6574	277	12	.	.	PUNCT
ejpam-6574	278	1	[	[	X
ejpam-6574	278	2	19	19	NUM
ejpam-6574	278	3	]	]	X
ejpam-6574	278	4	p.	p.	NOUN
ejpam-6574	278	5	pue	pue	NOUN
ejpam-6574	278	6	-	-	PUNCT
ejpam-6574	278	7	on	on	ADP
ejpam-6574	278	8	,	,	PUNCT
ejpam-6574	278	9	s.	s.	PROPN
ejpam-6574	278	10	sompong	sompong	PROPN
ejpam-6574	278	11	,	,	PUNCT
ejpam-6574	278	12	and	and	CCONJ
ejpam-6574	278	13	c.	c.	PROPN
ejpam-6574	278	14	boonpok	boonpok	PROPN
ejpam-6574	278	15	.	.	PUNCT
ejpam-6574	279	1	almost	almost	ADV
ejpam-6574	279	2	quasi	quasi	X
ejpam-6574	279	3	(	(	PUNCT
ejpam-6574	279	4	τ1	τ1	NOUN
ejpam-6574	279	5	,	,	PUNCT
ejpam-6574	279	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6574	279	7	for	for	ADP
ejpam-6574	279	8	multifunctions	multifunction	NOUN
ejpam-6574	279	9	.	.	PUNCT
ejpam-6574	280	1	international	international	ADJ
ejpam-6574	280	2	journal	journal	NOUN
ejpam-6574	280	3	of	of	ADP
ejpam-6574	280	4	analysis	analysis	NOUN
ejpam-6574	280	5	and	and	CCONJ
ejpam-6574	280	6	applications	application	NOUN
ejpam-6574	280	7	,	,	PUNCT
ejpam-6574	280	8	22:97	22:97	NUM
ejpam-6574	280	9	,	,	PUNCT
ejpam-6574	280	10	2024	2024	NUM
ejpam-6574	280	11	.	.	PUNCT
ejpam-6574	281	1	[	[	X
ejpam-6574	281	2	20	20	NUM
ejpam-6574	281	3	]	]	X
ejpam-6574	281	4	j.	j.	PROPN
ejpam-6574	281	5	khampakdee	khampakdee	PROPN
ejpam-6574	281	6	,	,	PUNCT
ejpam-6574	281	7	s.	s.	PROPN
ejpam-6574	281	8	sompong	sompong	PROPN
ejpam-6574	281	9	,	,	PUNCT
ejpam-6574	281	10	and	and	CCONJ
ejpam-6574	281	11	c.	c.	PROPN
ejpam-6574	281	12	boonpok	boonpok	PROPN
ejpam-6574	281	13	.	.	PUNCT
ejpam-6574	282	1	almost	almost	ADV
ejpam-6574	282	2	weakly	weakly	ADJ
ejpam-6574	282	3	(	(	PUNCT
ejpam-6574	282	4	τ1	τ1	NOUN
ejpam-6574	282	5	,	,	PUNCT
ejpam-6574	282	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6574	282	7	functions	function	NOUN
ejpam-6574	282	8	.	.	PUNCT
ejpam-6574	283	1	european	european	ADJ
ejpam-6574	283	2	journal	journal	PROPN
ejpam-6574	283	3	of	of	ADP
ejpam-6574	283	4	pure	pure	ADJ
ejpam-6574	283	5	and	and	CCONJ
ejpam-6574	283	6	applied	applied	ADJ
ejpam-6574	283	7	mathematics	mathematic	NOUN
ejpam-6574	283	8	,	,	PUNCT
ejpam-6574	283	9	18(1):5721	18(1):5721	NUM
ejpam-6574	283	10	,	,	PUNCT
ejpam-6574	283	11	2025	2025	NUM
ejpam-6574	283	12	.	.	PUNCT