id	sid	tid	token	lemma	pos
ejpam-5094	1	1	european	european	PROPN
ejpam-5094	1	2	journal	journal	PROPN
ejpam-5094	1	3	of	of	ADP
ejpam-5094	1	4	pure	pure	ADJ
ejpam-5094	1	5	and	and	CCONJ
ejpam-5094	1	6	applied	apply	VERB
ejpam-5094	1	7	mathematics	mathematic	NOUN
ejpam-5094	1	8	vol	vol	NOUN
ejpam-5094	1	9	.	.	PROPN
ejpam-5094	2	1	17	17	NUM
ejpam-5094	2	2	,	,	PUNCT
ejpam-5094	2	3	no	no	INTJ
ejpam-5094	2	4	.	.	NOUN
ejpam-5094	2	5	2	2	NUM
ejpam-5094	2	6	,	,	PUNCT
ejpam-5094	2	7	2024	2024	NUM
ejpam-5094	2	8	,	,	PUNCT
ejpam-5094	2	9	582	582	NUM
ejpam-5094	2	10	-	-	SYM
ejpam-5094	2	11	590	590	NUM
ejpam-5094	2	12	issn	issn	PROPN
ejpam-5094	2	13	1307	1307	NUM
ejpam-5094	2	14	-	-	SYM
ejpam-5094	2	15	5543	5543	NUM
ejpam-5094	2	16	–	–	PUNCT
ejpam-5094	2	17	ejpam.com	ejpam.com	X
ejpam-5094	2	18	published	publish	VERB
ejpam-5094	2	19	by	by	ADP
ejpam-5094	2	20	new	new	PROPN
ejpam-5094	2	21	york	york	PROPN
ejpam-5094	2	22	business	business	PROPN
ejpam-5094	2	23	global	global	ADJ
ejpam-5094	2	24	weakly	weakly	ADJ
ejpam-5094	2	25	θb(λ	θb(λ	NUM
ejpam-5094	2	26	,	,	PUNCT
ejpam-5094	2	27	p)-open	p)-open	VERB
ejpam-5094	2	28	functions	function	NOUN
ejpam-5094	2	29	and	and	CCONJ
ejpam-5094	2	30	weakly	weakly	ADJ
ejpam-5094	2	31	θb(λ	θb(λ	NUM
ejpam-5094	2	32	,	,	PUNCT
ejpam-5094	2	33	p)-closed	p)-close	VERB
ejpam-5094	2	34	functions	function	NOUN
ejpam-5094	2	35	chawalit	chawalit	VERB
ejpam-5094	2	36	boonpok1	boonpok1	PROPN
ejpam-5094	2	37	,	,	PUNCT
ejpam-5094	2	38	jeeranunt	jeeranunt	PROPN
ejpam-5094	2	39	khampakdee1,∗	khampakdee1,∗	PROPN
ejpam-5094	2	40	1	1	NUM
ejpam-5094	2	41	mathematics	mathematic	NOUN
ejpam-5094	2	42	and	and	CCONJ
ejpam-5094	2	43	applied	apply	VERB
ejpam-5094	2	44	mathematics	mathematics	PROPN
ejpam-5094	2	45	research	research	NOUN
ejpam-5094	2	46	unit	unit	NOUN
ejpam-5094	2	47	,	,	PUNCT
ejpam-5094	2	48	department	department	NOUN
ejpam-5094	2	49	of	of	ADP
ejpam-5094	2	50	mathematics	mathematic	NOUN
ejpam-5094	2	51	,	,	PUNCT
ejpam-5094	2	52	faculty	faculty	NOUN
ejpam-5094	2	53	of	of	ADP
ejpam-5094	2	54	science	science	NOUN
ejpam-5094	2	55	,	,	PUNCT
ejpam-5094	2	56	mahasarakham	mahasarakham	PROPN
ejpam-5094	2	57	university	university	PROPN
ejpam-5094	2	58	,	,	PUNCT
ejpam-5094	2	59	maha	maha	PROPN
ejpam-5094	2	60	sarakham	sarakham	PROPN
ejpam-5094	2	61	,	,	PUNCT
ejpam-5094	2	62	44150	44150	NUM
ejpam-5094	2	63	,	,	PUNCT
ejpam-5094	2	64	thailand	thailand	PROPN
ejpam-5094	2	65	abstract	abstract	PROPN
ejpam-5094	2	66	.	.	PUNCT
ejpam-5094	3	1	this	this	DET
ejpam-5094	3	2	article	article	NOUN
ejpam-5094	3	3	is	be	AUX
ejpam-5094	3	4	concerned	concern	VERB
ejpam-5094	3	5	with	with	ADP
ejpam-5094	3	6	the	the	DET
ejpam-5094	3	7	concepts	concept	NOUN
ejpam-5094	3	8	of	of	ADP
ejpam-5094	3	9	weakly	weakly	ADJ
ejpam-5094	3	10	θb(λ	θb(λ	NUM
ejpam-5094	3	11	,	,	PUNCT
ejpam-5094	3	12	p)-open	p)-open	VERB
ejpam-5094	3	13	functions	function	NOUN
ejpam-5094	3	14	and	and	CCONJ
ejpam-5094	3	15	weakly	weakly	ADJ
ejpam-5094	3	16	θb(λ	θb(λ	NUM
ejpam-5094	3	17	,	,	PUNCT
ejpam-5094	3	18	p)-closed	p)-close	VERB
ejpam-5094	3	19	functions	function	NOUN
ejpam-5094	3	20	.	.	PUNCT
ejpam-5094	4	1	moreover	moreover	ADV
ejpam-5094	4	2	,	,	PUNCT
ejpam-5094	4	3	some	some	DET
ejpam-5094	4	4	characterizations	characterization	NOUN
ejpam-5094	4	5	of	of	ADP
ejpam-5094	4	6	weakly	weakly	ADJ
ejpam-5094	4	7	θb(λ	θb(λ	NUM
ejpam-5094	4	8	,	,	PUNCT
ejpam-5094	4	9	p)-open	p)-open	VERB
ejpam-5094	4	10	functions	function	NOUN
ejpam-5094	4	11	and	and	CCONJ
ejpam-5094	4	12	weakly	weakly	ADJ
ejpam-5094	4	13	θb(λ	θb(λ	NUM
ejpam-5094	4	14	,	,	PUNCT
ejpam-5094	4	15	p)-closed	p)-close	VERB
ejpam-5094	4	16	functions	function	NOUN
ejpam-5094	4	17	are	be	AUX
ejpam-5094	4	18	established	establish	VERB
ejpam-5094	4	19	.	.	PUNCT
ejpam-5094	5	1	2020	2020	NUM
ejpam-5094	5	2	mathematics	mathematics	PROPN
ejpam-5094	5	3	subject	subject	NOUN
ejpam-5094	5	4	classifications	classification	NOUN
ejpam-5094	5	5	:	:	PUNCT
ejpam-5094	5	6	54a05	54a05	NUM
ejpam-5094	5	7	,	,	PUNCT
ejpam-5094	5	8	54c10	54c10	NUM
ejpam-5094	5	9	key	key	ADJ
ejpam-5094	5	10	words	word	NOUN
ejpam-5094	5	11	and	and	CCONJ
ejpam-5094	5	12	phrases	phrase	NOUN
ejpam-5094	5	13	:	:	PUNCT
ejpam-5094	5	14	weakly	weakly	ADJ
ejpam-5094	5	15	θb(λ	θb(λ	NUM
ejpam-5094	5	16	,	,	PUNCT
ejpam-5094	5	17	p)-open	p)-open	VERB
ejpam-5094	5	18	function	function	NOUN
ejpam-5094	5	19	,	,	PUNCT
ejpam-5094	5	20	weakly	weakly	ADJ
ejpam-5094	5	21	θb(λ	θb(λ	NUM
ejpam-5094	5	22	,	,	PUNCT
ejpam-5094	5	23	p)-closed	p)-close	VERB
ejpam-5094	5	24	function	function	NOUN
ejpam-5094	5	25	1	1	NUM
ejpam-5094	5	26	.	.	PUNCT
ejpam-5094	6	1	introduction	introduction	NOUN
ejpam-5094	6	2	topology	topology	NOUN
ejpam-5094	6	3	is	be	AUX
ejpam-5094	6	4	concerned	concern	VERB
ejpam-5094	6	5	with	with	ADP
ejpam-5094	6	6	all	all	DET
ejpam-5094	6	7	questions	question	NOUN
ejpam-5094	6	8	directly	directly	ADV
ejpam-5094	6	9	or	or	CCONJ
ejpam-5094	6	10	indirectly	indirectly	ADV
ejpam-5094	6	11	related	relate	VERB
ejpam-5094	6	12	to	to	ADP
ejpam-5094	6	13	openness	openness	NOUN
ejpam-5094	6	14	and	and	CCONJ
ejpam-5094	6	15	closedness	closedness	NOUN
ejpam-5094	6	16	.	.	PUNCT
ejpam-5094	7	1	semi	semi	ADJ
ejpam-5094	7	2	-	-	ADJ
ejpam-5094	7	3	open	open	ADJ
ejpam-5094	7	4	sets	set	NOUN
ejpam-5094	7	5	,	,	PUNCT
ejpam-5094	7	6	preopen	preopen	ADJ
ejpam-5094	7	7	sets	set	NOUN
ejpam-5094	7	8	,	,	PUNCT
ejpam-5094	7	9	α	α	NOUN
ejpam-5094	7	10	-	-	ADJ
ejpam-5094	7	11	open	open	ADJ
ejpam-5094	7	12	sets	set	NOUN
ejpam-5094	7	13	,	,	PUNCT
ejpam-5094	7	14	β	β	ADJ
ejpam-5094	7	15	-	-	ADJ
ejpam-5094	7	16	open	open	ADJ
ejpam-5094	7	17	sets	set	NOUN
ejpam-5094	7	18	,	,	PUNCT
ejpam-5094	7	19	b	b	X
ejpam-5094	7	20	-	-	PUNCT
ejpam-5094	7	21	open	open	ADJ
ejpam-5094	7	22	sets	set	NOUN
ejpam-5094	7	23	,	,	PUNCT
ejpam-5094	7	24	δ	δ	NOUN
ejpam-5094	7	25	-	-	ADJ
ejpam-5094	7	26	open	open	ADJ
ejpam-5094	7	27	sets	set	NOUN
ejpam-5094	7	28	and	and	CCONJ
ejpam-5094	7	29	θ	θ	ADJ
ejpam-5094	7	30	-	-	ADJ
ejpam-5094	7	31	open	open	ADJ
ejpam-5094	7	32	sets	set	NOUN
ejpam-5094	7	33	play	play	VERB
ejpam-5094	7	34	an	an	DET
ejpam-5094	7	35	important	important	ADJ
ejpam-5094	7	36	role	role	NOUN
ejpam-5094	7	37	in	in	ADP
ejpam-5094	7	38	the	the	DET
ejpam-5094	7	39	researches	research	NOUN
ejpam-5094	7	40	of	of	ADP
ejpam-5094	7	41	generalizations	generalization	NOUN
ejpam-5094	7	42	of	of	ADP
ejpam-5094	7	43	open	open	ADJ
ejpam-5094	7	44	functions	function	NOUN
ejpam-5094	7	45	and	and	CCONJ
ejpam-5094	7	46	closed	closed	ADJ
ejpam-5094	7	47	functions	function	NOUN
ejpam-5094	7	48	.	.	PUNCT
ejpam-5094	8	1	by	by	ADP
ejpam-5094	8	2	using	use	VERB
ejpam-5094	8	3	these	these	DET
ejpam-5094	8	4	sets	set	NOUN
ejpam-5094	8	5	,	,	PUNCT
ejpam-5094	8	6	many	many	ADJ
ejpam-5094	8	7	authors	author	NOUN
ejpam-5094	8	8	introduced	introduce	VERB
ejpam-5094	8	9	and	and	CCONJ
ejpam-5094	8	10	studied	study	VERB
ejpam-5094	8	11	various	various	ADJ
ejpam-5094	8	12	types	type	NOUN
ejpam-5094	8	13	of	of	ADP
ejpam-5094	8	14	open	open	ADJ
ejpam-5094	8	15	functions	function	NOUN
ejpam-5094	8	16	and	and	CCONJ
ejpam-5094	8	17	closed	closed	ADJ
ejpam-5094	8	18	functions	function	NOUN
ejpam-5094	8	19	.	.	PUNCT
ejpam-5094	9	1	in	in	ADP
ejpam-5094	9	2	1996	1996	NUM
ejpam-5094	9	3	,	,	PUNCT
ejpam-5094	9	4	andrijević	andrijević	VERB
ejpam-5094	9	5	[	[	X
ejpam-5094	9	6	1	1	X
ejpam-5094	9	7	]	]	PUNCT
ejpam-5094	9	8	introduced	introduce	VERB
ejpam-5094	9	9	a	a	DET
ejpam-5094	9	10	new	new	ADJ
ejpam-5094	9	11	class	class	NOUN
ejpam-5094	9	12	of	of	ADP
ejpam-5094	9	13	generalized	generalized	ADJ
ejpam-5094	9	14	open	open	ADJ
ejpam-5094	9	15	sets	set	NOUN
ejpam-5094	9	16	called	call	VERB
ejpam-5094	9	17	b	b	NOUN
ejpam-5094	9	18	-	-	PUNCT
ejpam-5094	9	19	open	open	ADJ
ejpam-5094	9	20	sets	set	NOUN
ejpam-5094	9	21	in	in	ADP
ejpam-5094	9	22	a	a	DET
ejpam-5094	9	23	topological	topological	ADJ
ejpam-5094	9	24	space	space	NOUN
ejpam-5094	9	25	.	.	PUNCT
ejpam-5094	10	1	park	park	NOUN
ejpam-5094	11	1	[	[	X
ejpam-5094	11	2	16	16	NUM
ejpam-5094	11	3	]	]	PUNCT
ejpam-5094	11	4	introduced	introduce	VERB
ejpam-5094	11	5	the	the	DET
ejpam-5094	11	6	notion	notion	NOUN
ejpam-5094	11	7	of	of	ADP
ejpam-5094	11	8	b	b	NOUN
ejpam-5094	11	9	-	-	PUNCT
ejpam-5094	11	10	θ	θ	ADJ
ejpam-5094	11	11	-	-	ADJ
ejpam-5094	11	12	open	open	ADJ
ejpam-5094	11	13	sets	set	NOUN
ejpam-5094	11	14	and	and	CCONJ
ejpam-5094	11	15	showed	show	VERB
ejpam-5094	11	16	that	that	SCONJ
ejpam-5094	11	17	b	b	X
ejpam-5094	11	18	-	-	PUNCT
ejpam-5094	11	19	θ	θ	NOUN
ejpam-5094	11	20	-	-	PUNCT
ejpam-5094	11	21	cluster	cluster	NOUN
ejpam-5094	11	22	points	point	NOUN
ejpam-5094	11	23	can	can	AUX
ejpam-5094	11	24	be	be	AUX
ejpam-5094	11	25	characterized	characterize	VERB
ejpam-5094	11	26	by	by	ADP
ejpam-5094	11	27	b	b	NOUN
ejpam-5094	11	28	-	-	PUNCT
ejpam-5094	11	29	regular	regular	ADJ
ejpam-5094	11	30	sets	set	NOUN
ejpam-5094	11	31	.	.	PUNCT
ejpam-5094	12	1	in	in	ADP
ejpam-5094	12	2	1983	1983	NUM
ejpam-5094	12	3	,	,	PUNCT
ejpam-5094	12	4	rose	rise	VERB
ejpam-5094	12	5	[	[	X
ejpam-5094	12	6	17	17	NUM
ejpam-5094	12	7	]	]	PUNCT
ejpam-5094	12	8	introduced	introduce	VERB
ejpam-5094	12	9	and	and	CCONJ
ejpam-5094	12	10	studied	study	VERB
ejpam-5094	12	11	the	the	DET
ejpam-5094	12	12	notions	notion	NOUN
ejpam-5094	12	13	of	of	ADP
ejpam-5094	12	14	weakly	weakly	ADJ
ejpam-5094	12	15	open	open	ADJ
ejpam-5094	12	16	functions	function	NOUN
ejpam-5094	12	17	and	and	CCONJ
ejpam-5094	12	18	almost	almost	ADV
ejpam-5094	12	19	open	open	ADJ
ejpam-5094	12	20	functions	function	NOUN
ejpam-5094	12	21	.	.	PUNCT
ejpam-5094	13	1	in	in	ADP
ejpam-5094	13	2	1987	1987	NUM
ejpam-5094	13	3	,	,	PUNCT
ejpam-5094	13	4	rose	rise	VERB
ejpam-5094	13	5	and	and	CCONJ
ejpam-5094	13	6	janković	janković	ADJ
ejpam-5094	14	1	[	[	X
ejpam-5094	14	2	18	18	NUM
ejpam-5094	14	3	]	]	PUNCT
ejpam-5094	14	4	investigated	investigate	VERB
ejpam-5094	14	5	some	some	PRON
ejpam-5094	14	6	of	of	ADP
ejpam-5094	14	7	the	the	DET
ejpam-5094	14	8	fundamental	fundamental	ADJ
ejpam-5094	14	9	properties	property	NOUN
ejpam-5094	14	10	of	of	ADP
ejpam-5094	14	11	weakly	weakly	ADJ
ejpam-5094	14	12	closed	closed	ADJ
ejpam-5094	14	13	functions	function	NOUN
ejpam-5094	14	14	.	.	PUNCT
ejpam-5094	15	1	in	in	ADP
ejpam-5094	15	2	2006	2006	NUM
ejpam-5094	15	3	,	,	PUNCT
ejpam-5094	15	4	caldas	caldas	PROPN
ejpam-5094	15	5	et	et	PROPN
ejpam-5094	15	6	al	al	PROPN
ejpam-5094	15	7	.	.	PUNCT
ejpam-5094	16	1	[	[	X
ejpam-5094	16	2	8	8	NUM
ejpam-5094	16	3	]	]	PUNCT
ejpam-5094	16	4	introduced	introduce	VERB
ejpam-5094	16	5	and	and	CCONJ
ejpam-5094	16	6	studied	study	VERB
ejpam-5094	16	7	the	the	DET
ejpam-5094	16	8	concepts	concept	NOUN
ejpam-5094	16	9	of	of	ADP
ejpam-5094	16	10	θ	θ	PROPN
ejpam-5094	16	11	-	-	PUNCT
ejpam-5094	16	12	preopen	preopen	ADJ
ejpam-5094	16	13	functions	function	NOUN
ejpam-5094	16	14	and	and	CCONJ
ejpam-5094	16	15	θ	θ	NOUN
ejpam-5094	16	16	-	-	PUNCT
ejpam-5094	16	17	preclosed	preclose	VERB
ejpam-5094	16	18	functions	function	NOUN
ejpam-5094	16	19	by	by	ADP
ejpam-5094	16	20	using	use	VERB
ejpam-5094	16	21	the	the	DET
ejpam-5094	16	22	notions	notion	NOUN
ejpam-5094	16	23	of	of	ADP
ejpam-5094	16	24	pre	pre	ADJ
ejpam-5094	16	25	-	-	ADJ
ejpam-5094	16	26	θ	θ	ADJ
ejpam-5094	16	27	-	-	ADJ
ejpam-5094	16	28	interior	interior	ADJ
ejpam-5094	16	29	and	and	CCONJ
ejpam-5094	16	30	pre	pre	ADJ
ejpam-5094	16	31	-	-	ADJ
ejpam-5094	16	32	θ	θ	ADJ
ejpam-5094	16	33	-	-	NOUN
ejpam-5094	16	34	closure	closure	NOUN
ejpam-5094	16	35	.	.	PUNCT
ejpam-5094	17	1	moreover	moreover	ADV
ejpam-5094	17	2	,	,	PUNCT
ejpam-5094	17	3	caldas	caldas	PROPN
ejpam-5094	17	4	et	et	PROPN
ejpam-5094	17	5	al	al	PROPN
ejpam-5094	17	6	.	.	PUNCT
ejpam-5094	18	1	[	[	X
ejpam-5094	18	2	7	7	X
ejpam-5094	18	3	]	]	PUNCT
ejpam-5094	18	4	introduced	introduce	VERB
ejpam-5094	18	5	and	and	CCONJ
ejpam-5094	18	6	investigated	investigate	VERB
ejpam-5094	18	7	the	the	DET
ejpam-5094	18	8	concepts	concept	NOUN
ejpam-5094	18	9	of	of	ADP
ejpam-5094	18	10	weakly	weakly	ADJ
ejpam-5094	18	11	semi	semi	ADJ
ejpam-5094	18	12	-	-	ADJ
ejpam-5094	18	13	θ	θ	ADJ
ejpam-5094	18	14	-	-	PUNCT
ejpam-5094	18	15	open	open	ADJ
ejpam-5094	18	16	functions	function	NOUN
ejpam-5094	18	17	and	and	CCONJ
ejpam-5094	18	18	weakly	weakly	ADJ
ejpam-5094	18	19	semi	semi	ADJ
ejpam-5094	18	20	-	-	ADJ
ejpam-5094	18	21	θ	θ	ADJ
ejpam-5094	18	22	-	-	PUNCT
ejpam-5094	18	23	closed	closed	ADJ
ejpam-5094	18	24	functions	function	NOUN
ejpam-5094	18	25	.	.	PUNCT
ejpam-5094	19	1	in	in	ADP
ejpam-5094	19	2	2009	2009	NUM
ejpam-5094	19	3	,	,	PUNCT
ejpam-5094	19	4	noiri	noiri	ADV
ejpam-5094	19	5	et	et	PROPN
ejpam-5094	19	6	al	al	PROPN
ejpam-5094	19	7	.	.	PUNCT
ejpam-5094	20	1	[	[	X
ejpam-5094	20	2	15	15	NUM
ejpam-5094	20	3	]	]	PUNCT
ejpam-5094	20	4	introduced	introduce	VERB
ejpam-5094	20	5	and	and	CCONJ
ejpam-5094	20	6	studied	study	VERB
ejpam-5094	20	7	two	two	NUM
ejpam-5094	20	8	new	new	ADJ
ejpam-5094	20	9	classes	class	NOUN
ejpam-5094	20	10	of	of	ADP
ejpam-5094	20	11	functions	function	NOUN
ejpam-5094	20	12	called	call	VERB
ejpam-5094	20	13	weakly	weakly	ADJ
ejpam-5094	20	14	b	b	NOUN
ejpam-5094	20	15	-	-	PUNCT
ejpam-5094	20	16	θ	θ	ADJ
ejpam-5094	20	17	-	-	PUNCT
ejpam-5094	20	18	open	open	ADJ
ejpam-5094	20	19	functions	function	NOUN
ejpam-5094	20	20	and	and	CCONJ
ejpam-5094	20	21	weakly	weakly	ADJ
ejpam-5094	20	22	b	b	NOUN
ejpam-5094	20	23	-	-	PUNCT
ejpam-5094	20	24	θ	θ	ADJ
ejpam-5094	20	25	-	-	PUNCT
ejpam-5094	20	26	closed	close	VERB
ejpam-5094	20	27	functions	function	NOUN
ejpam-5094	20	28	by	by	ADP
ejpam-5094	20	29	utilizing	utilize	VERB
ejpam-5094	20	30	the	the	DET
ejpam-5094	20	31	notions	notion	NOUN
ejpam-5094	20	32	of	of	ADP
ejpam-5094	20	33	b	b	NOUN
ejpam-5094	20	34	-	-	PUNCT
ejpam-5094	20	35	θ	θ	ADJ
ejpam-5094	20	36	-	-	ADJ
ejpam-5094	20	37	open	open	ADJ
ejpam-5094	20	38	sets	set	NOUN
ejpam-5094	20	39	and	and	CCONJ
ejpam-5094	20	40	the	the	DET
ejpam-5094	20	41	b	b	PROPN
ejpam-5094	20	42	-	-	PUNCT
ejpam-5094	20	43	θ	θ	NOUN
ejpam-5094	20	44	-	-	PUNCT
ejpam-5094	20	45	closure	closure	NOUN
ejpam-5094	20	46	operator	operator	NOUN
ejpam-5094	20	47	.	.	PUNCT
ejpam-5094	21	1	weak	weak	ADJ
ejpam-5094	21	2	b	b	X
ejpam-5094	21	3	-	-	PUNCT
ejpam-5094	21	4	θ	θ	NOUN
ejpam-5094	21	5	-	-	PUNCT
ejpam-5094	21	6	openness	openness	NOUN
ejpam-5094	21	7	(	(	PUNCT
ejpam-5094	21	8	resp	resp	NOUN
ejpam-5094	21	9	.	.	PUNCT
ejpam-5094	22	1	b	b	X
ejpam-5094	22	2	-	-	PUNCT
ejpam-5094	22	3	θ	θ	NOUN
ejpam-5094	22	4	-	-	PUNCT
ejpam-5094	22	5	closedness	closedness	NOUN
ejpam-5094	22	6	)	)	PUNCT
ejpam-5094	22	7	is	be	AUX
ejpam-5094	22	8	a	a	DET
ejpam-5094	22	9	generalization	generalization	NOUN
ejpam-5094	22	10	of	of	ADP
ejpam-5094	22	11	both	both	DET
ejpam-5094	22	12	θ	θ	NOUN
ejpam-5094	22	13	-	-	NOUN
ejpam-5094	22	14	preopenness	preopenness	NOUN
ejpam-5094	22	15	and	and	CCONJ
ejpam-5094	22	16	weak	weak	ADJ
ejpam-5094	22	17	semi	semi	ADJ
ejpam-5094	22	18	-	-	ADJ
ejpam-5094	22	19	θ	θ	ADJ
ejpam-5094	22	20	-	-	PUNCT
ejpam-5094	22	21	openness	openness	NOUN
ejpam-5094	22	22	(	(	PUNCT
ejpam-5094	22	23	resp	resp	NOUN
ejpam-5094	22	24	.	.	PUNCT
ejpam-5094	23	1	θ	θ	NOUN
ejpam-5094	23	2	-	-	PUNCT
ejpam-5094	23	3	preclosedness	preclosedness	NOUN
ejpam-5094	23	4	and	and	CCONJ
ejpam-5094	23	5	weak	weak	ADJ
ejpam-5094	23	6	semi	semi	ADJ
ejpam-5094	23	7	-	-	ADJ
ejpam-5094	23	8	θ	θ	ADJ
ejpam-5094	23	9	-	-	PUNCT
ejpam-5094	23	10	closedness	closedness	ADJ
ejpam-5094	23	11	)	)	PUNCT
ejpam-5094	23	12	.	.	PUNCT
ejpam-5094	24	1	in	in	ADP
ejpam-5094	24	2	[	[	X
ejpam-5094	24	3	3	3	NUM
ejpam-5094	24	4	]	]	PUNCT
ejpam-5094	24	5	,	,	PUNCT
ejpam-5094	24	6	the	the	DET
ejpam-5094	24	7	present	present	ADJ
ejpam-5094	24	8	authors	author	NOUN
ejpam-5094	24	9	introduced	introduce	VERB
ejpam-5094	24	10	and	and	CCONJ
ejpam-5094	24	11	studied	study	VERB
ejpam-5094	24	12	the	the	DET
ejpam-5094	24	13	notions	notion	NOUN
ejpam-5094	24	14	of	of	ADP
ejpam-5094	24	15	∗corresponding	∗corresponde	VERB
ejpam-5094	24	16	author	author	NOUN
ejpam-5094	24	17	.	.	PUNCT
ejpam-5094	25	1	doi	doi	PROPN
ejpam-5094	25	2	:	:	PUNCT
ejpam-5094	25	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5094	https://doi.org/10.29020/nybg.ejpam.v17i2.5094	ADJ
ejpam-5094	25	4	email	email	NOUN
ejpam-5094	25	5	addresses	address	VERB
ejpam-5094	25	6	:	:	PUNCT
ejpam-5094	25	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-5094	25	8	(	(	PUNCT
ejpam-5094	25	9	c.	c.	PROPN
ejpam-5094	25	10	boonpok	boonpok	PROPN
ejpam-5094	25	11	)	)	PUNCT
ejpam-5094	25	12	,	,	PUNCT
ejpam-5094	26	1	jeeranunt.k@msu.ac.th	jeeranunt.k@msu.ac.th	INTJ
ejpam-5094	26	2	(	(	PUNCT
ejpam-5094	26	3	j.	j.	PROPN
ejpam-5094	26	4	khampakdee	khampakdee	PROPN
ejpam-5094	26	5	)	)	PUNCT
ejpam-5094	26	6	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5094	27	1	582	582	NUM
ejpam-5094	27	2	©	©	PROPN
ejpam-5094	27	3	2024	2024	NUM
ejpam-5094	27	4	ejpam	ejpam	NOUN
ejpam-5094	27	5	all	all	DET
ejpam-5094	27	6	rights	right	NOUN
ejpam-5094	27	7	reserved	reserve	VERB
ejpam-5094	27	8	.	.	PUNCT
ejpam-5094	28	1	c.	c.	PROPN
ejpam-5094	28	2	boonpok	boonpok	PROPN
ejpam-5094	28	3	,	,	PUNCT
ejpam-5094	28	4	j.	j.	PROPN
ejpam-5094	28	5	khampakdee	khampakdee	PROPN
ejpam-5094	28	6	/	/	PUNCT
ejpam-5094	28	7	eur	eur	PROPN
ejpam-5094	28	8	.	.	PUNCT
ejpam-5094	29	1	j.	j.	PROPN
ejpam-5094	29	2	pure	pure	PROPN
ejpam-5094	29	3	appl	appl	PROPN
ejpam-5094	29	4	.	.	PROPN
ejpam-5094	29	5	math	math	PROPN
ejpam-5094	29	6	,	,	PUNCT
ejpam-5094	29	7	17	17	NUM
ejpam-5094	29	8	(	(	PUNCT
ejpam-5094	29	9	2	2	NUM
ejpam-5094	29	10	)	)	PUNCT
ejpam-5094	29	11	(	(	PUNCT
ejpam-5094	29	12	2024	2024	NUM
ejpam-5094	29	13	)	)	PUNCT
ejpam-5094	29	14	,	,	PUNCT
ejpam-5094	29	15	582	582	NUM
ejpam-5094	29	16	-	-	SYM
ejpam-5094	29	17	590	590	NUM
ejpam-5094	29	18	583	583	NUM
ejpam-5094	29	19	θp(λ	θp(λ	NOUN
ejpam-5094	29	20	,	,	PUNCT
ejpam-5094	29	21	p)-open	p)-open	NOUN
ejpam-5094	29	22	functions	function	NOUN
ejpam-5094	29	23	and	and	CCONJ
ejpam-5094	29	24	θp(λ	θp(λ	NOUN
ejpam-5094	29	25	,	,	PUNCT
ejpam-5094	29	26	p)-closed	p)-close	VERB
ejpam-5094	29	27	functions	function	NOUN
ejpam-5094	29	28	.	.	PUNCT
ejpam-5094	30	1	khampakdee	khampakdee	NOUN
ejpam-5094	30	2	and	and	CCONJ
ejpam-5094	30	3	boonpok	boonpok	PRON
ejpam-5094	30	4	[	[	X
ejpam-5094	30	5	11	11	NUM
ejpam-5094	30	6	]	]	PUNCT
ejpam-5094	30	7	investigated	investigate	VERB
ejpam-5094	30	8	some	some	DET
ejpam-5094	30	9	properties	property	NOUN
ejpam-5094	30	10	of	of	ADP
ejpam-5094	30	11	(	(	PUNCT
ejpam-5094	30	12	λ	λ	PROPN
ejpam-5094	30	13	,	,	PUNCT
ejpam-5094	30	14	p)-closed	p)-close	VERB
ejpam-5094	30	15	functions	function	NOUN
ejpam-5094	30	16	.	.	PUNCT
ejpam-5094	31	1	chutiman	chutiman	NOUN
ejpam-5094	31	2	and	and	CCONJ
ejpam-5094	31	3	boonpok	boonpok	NOUN
ejpam-5094	32	1	[	[	X
ejpam-5094	32	2	9	9	NUM
ejpam-5094	32	3	]	]	PUNCT
ejpam-5094	32	4	obtained	obtain	VERB
ejpam-5094	32	5	several	several	ADJ
ejpam-5094	32	6	properties	property	NOUN
ejpam-5094	32	7	of	of	ADP
ejpam-5094	32	8	weakly	weakly	ADJ
ejpam-5094	32	9	b(λ	b(λ	NOUN
ejpam-5094	32	10	,	,	PUNCT
ejpam-5094	32	11	p)-open	p)-open	NOUN
ejpam-5094	32	12	functions	function	NOUN
ejpam-5094	32	13	.	.	PUNCT
ejpam-5094	33	1	furthermore	furthermore	ADV
ejpam-5094	33	2	,	,	PUNCT
ejpam-5094	33	3	some	some	DET
ejpam-5094	33	4	characterizations	characterization	NOUN
ejpam-5094	33	5	of	of	ADP
ejpam-5094	33	6	weakly	weakly	ADJ
ejpam-5094	33	7	δ(λ	δ(λ	PROPN
ejpam-5094	33	8	,	,	PUNCT
ejpam-5094	33	9	p)-open	p)-open	VERB
ejpam-5094	33	10	functions	function	NOUN
ejpam-5094	33	11	and	and	CCONJ
ejpam-5094	33	12	weakly	weakly	ADJ
ejpam-5094	33	13	δ(λ	δ(λ	PROPN
ejpam-5094	33	14	,	,	PUNCT
ejpam-5094	33	15	p)-closed	p)-close	VERB
ejpam-5094	33	16	functions	function	NOUN
ejpam-5094	33	17	were	be	AUX
ejpam-5094	33	18	presented	present	VERB
ejpam-5094	33	19	in	in	ADP
ejpam-5094	33	20	[	[	X
ejpam-5094	33	21	19	19	NUM
ejpam-5094	33	22	]	]	PUNCT
ejpam-5094	33	23	and	and	CCONJ
ejpam-5094	33	24	[	[	X
ejpam-5094	33	25	12	12	NUM
ejpam-5094	33	26	]	]	PUNCT
ejpam-5094	33	27	,	,	PUNCT
ejpam-5094	33	28	respectively	respectively	ADV
ejpam-5094	33	29	.	.	PUNCT
ejpam-5094	34	1	klanarong	klanarong	NOUN
ejpam-5094	34	2	and	and	CCONJ
ejpam-5094	34	3	boonpok	boonpok	PROPN
ejpam-5094	35	1	[	[	X
ejpam-5094	35	2	13	13	NUM
ejpam-5094	35	3	]	]	PUNCT
ejpam-5094	35	4	studied	study	VERB
ejpam-5094	35	5	the	the	DET
ejpam-5094	35	6	notions	notion	NOUN
ejpam-5094	35	7	of	of	ADP
ejpam-5094	35	8	weakly	weakly	ADJ
ejpam-5094	35	9	s(λ	s(λ	NOUN
ejpam-5094	35	10	,	,	PUNCT
ejpam-5094	35	11	p)-open	p)-open	NOUN
ejpam-5094	35	12	functions	function	NOUN
ejpam-5094	35	13	and	and	CCONJ
ejpam-5094	35	14	weakly	weakly	ADJ
ejpam-5094	35	15	s(λ	s(λ	PROPN
ejpam-5094	35	16	,	,	PUNCT
ejpam-5094	35	17	p)-closed	p)-close	VERB
ejpam-5094	35	18	functions	function	NOUN
ejpam-5094	35	19	by	by	ADP
ejpam-5094	35	20	utilizing	utilize	VERB
ejpam-5094	35	21	s(λ	s(λ	NOUN
ejpam-5094	35	22	,	,	PUNCT
ejpam-5094	35	23	p)-open	p)-open	VERB
ejpam-5094	35	24	sets	set	NOUN
ejpam-5094	35	25	and	and	CCONJ
ejpam-5094	35	26	the	the	DET
ejpam-5094	35	27	s(λ	s(λ	PROPN
ejpam-5094	35	28	,	,	PUNCT
ejpam-5094	35	29	p)-closure	p)-closure	NOUN
ejpam-5094	35	30	operator	operator	NOUN
ejpam-5094	35	31	.	.	PUNCT
ejpam-5094	36	1	in	in	ADP
ejpam-5094	36	2	[	[	X
ejpam-5094	36	3	4	4	NUM
ejpam-5094	36	4	]	]	PUNCT
ejpam-5094	36	5	,	,	PUNCT
ejpam-5094	36	6	the	the	DET
ejpam-5094	36	7	authors	author	NOUN
ejpam-5094	36	8	introduced	introduce	VERB
ejpam-5094	36	9	and	and	CCONJ
ejpam-5094	36	10	investigated	investigate	VERB
ejpam-5094	36	11	the	the	DET
ejpam-5094	36	12	concepts	concept	NOUN
ejpam-5094	36	13	of	of	ADP
ejpam-5094	36	14	weakly	weakly	ADJ
ejpam-5094	36	15	β(λ	β(λ	NOUN
ejpam-5094	36	16	,	,	PUNCT
ejpam-5094	36	17	p)-open	p)-open	NOUN
ejpam-5094	36	18	functions	function	NOUN
ejpam-5094	36	19	and	and	CCONJ
ejpam-5094	36	20	weakly	weakly	ADJ
ejpam-5094	36	21	β(λ	β(λ	NOUN
ejpam-5094	36	22	,	,	PUNCT
ejpam-5094	36	23	p)-closed	p)-close	VERB
ejpam-5094	36	24	functions	function	NOUN
ejpam-5094	36	25	.	.	PUNCT
ejpam-5094	37	1	moreover	moreover	ADV
ejpam-5094	37	2	,	,	PUNCT
ejpam-5094	37	3	several	several	ADJ
ejpam-5094	37	4	characterizations	characterization	NOUN
ejpam-5094	37	5	of	of	ADP
ejpam-5094	37	6	weakly	weakly	ADJ
ejpam-5094	37	7	p(λ	p(λ	NOUN
ejpam-5094	37	8	,	,	PUNCT
ejpam-5094	37	9	p)-open	p)-open	NOUN
ejpam-5094	37	10	(	(	PUNCT
ejpam-5094	37	11	resp	resp	NOUN
ejpam-5094	37	12	.	.	PUNCT
ejpam-5094	38	1	weakly	weakly	ADJ
ejpam-5094	38	2	p(λ	p(λ	NOUN
ejpam-5094	38	3	,	,	PUNCT
ejpam-5094	38	4	p)-closed	p)-close	VERB
ejpam-5094	38	5	)	)	PUNCT
ejpam-5094	38	6	functions	function	NOUN
ejpam-5094	38	7	and	and	CCONJ
ejpam-5094	38	8	weakly	weakly	ADJ
ejpam-5094	38	9	θs(λ	θs(λ	NOUN
ejpam-5094	38	10	,	,	PUNCT
ejpam-5094	38	11	p)-open	p)-open	NOUN
ejpam-5094	38	12	(	(	PUNCT
ejpam-5094	38	13	resp	resp	NOUN
ejpam-5094	38	14	.	.	PUNCT
ejpam-5094	39	1	weakly	weakly	ADJ
ejpam-5094	39	2	θs(λ	θs(λ	NOUN
ejpam-5094	39	3	,	,	PUNCT
ejpam-5094	39	4	p)-closed	p)-close	VERB
ejpam-5094	39	5	)	)	PUNCT
ejpam-5094	39	6	functions	function	NOUN
ejpam-5094	39	7	were	be	AUX
ejpam-5094	39	8	established	establish	VERB
ejpam-5094	39	9	in	in	ADP
ejpam-5094	39	10	[	[	X
ejpam-5094	39	11	5	5	NUM
ejpam-5094	39	12	]	]	PUNCT
ejpam-5094	39	13	and	and	CCONJ
ejpam-5094	39	14	[	[	X
ejpam-5094	39	15	2	2	NUM
ejpam-5094	39	16	]	]	PUNCT
ejpam-5094	39	17	,	,	PUNCT
ejpam-5094	39	18	respectively	respectively	ADV
ejpam-5094	39	19	.	.	PUNCT
ejpam-5094	40	1	in	in	ADP
ejpam-5094	40	2	this	this	DET
ejpam-5094	40	3	article	article	NOUN
ejpam-5094	40	4	,	,	PUNCT
ejpam-5094	40	5	we	we	PRON
ejpam-5094	40	6	introduce	introduce	VERB
ejpam-5094	40	7	the	the	DET
ejpam-5094	40	8	notions	notion	NOUN
ejpam-5094	40	9	of	of	ADP
ejpam-5094	40	10	weakly	weakly	ADJ
ejpam-5094	40	11	θb(λ	θb(λ	NUM
ejpam-5094	40	12	,	,	PUNCT
ejpam-5094	40	13	p)-open	p)-open	VERB
ejpam-5094	40	14	functions	function	NOUN
ejpam-5094	40	15	and	and	CCONJ
ejpam-5094	40	16	θb(λ	θb(λ	NUM
ejpam-5094	40	17	,	,	PUNCT
ejpam-5094	40	18	p)-closed	p)-close	VERB
ejpam-5094	40	19	functions	function	NOUN
ejpam-5094	40	20	.	.	PUNCT
ejpam-5094	41	1	in	in	ADP
ejpam-5094	41	2	particular	particular	ADJ
ejpam-5094	41	3	,	,	PUNCT
ejpam-5094	41	4	several	several	ADJ
ejpam-5094	41	5	characterizations	characterization	NOUN
ejpam-5094	41	6	of	of	ADP
ejpam-5094	41	7	weakly	weakly	ADJ
ejpam-5094	41	8	θb(λ	θb(λ	NUM
ejpam-5094	41	9	,	,	PUNCT
ejpam-5094	41	10	p)-open	p)-open	VERB
ejpam-5094	41	11	functions	function	NOUN
ejpam-5094	41	12	and	and	CCONJ
ejpam-5094	41	13	θb(λ	θb(λ	NUM
ejpam-5094	41	14	,	,	PUNCT
ejpam-5094	41	15	p)-closed	p)-close	VERB
ejpam-5094	41	16	functions	function	NOUN
ejpam-5094	41	17	are	be	AUX
ejpam-5094	41	18	investigated	investigate	VERB
ejpam-5094	41	19	.	.	PUNCT
ejpam-5094	42	1	2	2	X
ejpam-5094	42	2	.	.	X
ejpam-5094	42	3	preliminaries	preliminary	NOUN
ejpam-5094	42	4	throughout	throughout	ADP
ejpam-5094	42	5	the	the	DET
ejpam-5094	42	6	present	present	ADJ
ejpam-5094	42	7	paper	paper	NOUN
ejpam-5094	42	8	,	,	PUNCT
ejpam-5094	42	9	spaces	space	NOUN
ejpam-5094	42	10	(	(	PUNCT
ejpam-5094	42	11	x	x	X
ejpam-5094	42	12	,	,	PUNCT
ejpam-5094	42	13	τ	τ	X
ejpam-5094	42	14	)	)	PUNCT
ejpam-5094	42	15	and	and	CCONJ
ejpam-5094	42	16	(	(	PUNCT
ejpam-5094	42	17	y	y	PROPN
ejpam-5094	42	18	,	,	PUNCT
ejpam-5094	42	19	σ	σ	PROPN
ejpam-5094	42	20	)	)	PUNCT
ejpam-5094	42	21	(	(	PUNCT
ejpam-5094	42	22	or	or	CCONJ
ejpam-5094	42	23	simply	simply	ADV
ejpam-5094	42	24	x	x	X
ejpam-5094	42	25	and	and	CCONJ
ejpam-5094	42	26	y	y	PROPN
ejpam-5094	42	27	)	)	PUNCT
ejpam-5094	42	28	always	always	ADV
ejpam-5094	42	29	mean	mean	VERB
ejpam-5094	42	30	topological	topological	ADJ
ejpam-5094	42	31	spaces	space	NOUN
ejpam-5094	42	32	on	on	ADP
ejpam-5094	42	33	which	which	PRON
ejpam-5094	42	34	no	no	DET
ejpam-5094	42	35	separation	separation	NOUN
ejpam-5094	42	36	axioms	axiom	NOUN
ejpam-5094	42	37	are	be	AUX
ejpam-5094	42	38	assumed	assume	VERB
ejpam-5094	42	39	unless	unless	SCONJ
ejpam-5094	42	40	explicitly	explicitly	ADV
ejpam-5094	42	41	stated	state	VERB
ejpam-5094	42	42	.	.	PUNCT
ejpam-5094	43	1	for	for	ADP
ejpam-5094	43	2	a	a	DET
ejpam-5094	43	3	subset	subset	NOUN
ejpam-5094	43	4	a	a	PRON
ejpam-5094	43	5	of	of	ADP
ejpam-5094	43	6	a	a	DET
ejpam-5094	43	7	topological	topological	ADJ
ejpam-5094	43	8	space	space	NOUN
ejpam-5094	43	9	(	(	PUNCT
ejpam-5094	43	10	x	x	X
ejpam-5094	43	11	,	,	PUNCT
ejpam-5094	43	12	τ	τ	PROPN
ejpam-5094	43	13	)	)	PUNCT
ejpam-5094	43	14	,	,	PUNCT
ejpam-5094	43	15	cl(a	cl(a	NUM
ejpam-5094	43	16	)	)	PUNCT
ejpam-5094	43	17	and	and	CCONJ
ejpam-5094	43	18	int(a	int(a	PROPN
ejpam-5094	43	19	)	)	PUNCT
ejpam-5094	43	20	,	,	PUNCT
ejpam-5094	43	21	represent	represent	VERB
ejpam-5094	43	22	the	the	DET
ejpam-5094	43	23	closure	closure	NOUN
ejpam-5094	43	24	and	and	CCONJ
ejpam-5094	43	25	the	the	DET
ejpam-5094	43	26	interior	interior	NOUN
ejpam-5094	43	27	of	of	ADP
ejpam-5094	43	28	a	a	PRON
ejpam-5094	43	29	,	,	PUNCT
ejpam-5094	43	30	respectively	respectively	ADV
ejpam-5094	43	31	.	.	PUNCT
ejpam-5094	44	1	a	a	DET
ejpam-5094	44	2	subset	subset	NOUN
ejpam-5094	44	3	a	a	PRON
ejpam-5094	44	4	of	of	ADP
ejpam-5094	44	5	a	a	DET
ejpam-5094	44	6	topological	topological	ADJ
ejpam-5094	44	7	space	space	NOUN
ejpam-5094	44	8	(	(	PUNCT
ejpam-5094	44	9	x	x	X
ejpam-5094	44	10	,	,	PUNCT
ejpam-5094	44	11	τ	τ	X
ejpam-5094	44	12	)	)	PUNCT
ejpam-5094	44	13	is	be	AUX
ejpam-5094	44	14	said	say	VERB
ejpam-5094	44	15	to	to	PART
ejpam-5094	44	16	be	be	AUX
ejpam-5094	44	17	preopen	preopen	ADJ
ejpam-5094	44	18	[	[	X
ejpam-5094	44	19	14	14	NUM
ejpam-5094	44	20	]	]	X
ejpam-5094	44	21	if	if	SCONJ
ejpam-5094	44	22	a	a	DET
ejpam-5094	44	23	⊆	⊆	NUM
ejpam-5094	44	24	int(cl(a	int(cl(a	PROPN
ejpam-5094	44	25	)	)	PUNCT
ejpam-5094	44	26	)	)	PUNCT
ejpam-5094	44	27	.	.	PUNCT
ejpam-5094	45	1	the	the	DET
ejpam-5094	45	2	complement	complement	NOUN
ejpam-5094	45	3	of	of	ADP
ejpam-5094	45	4	a	a	DET
ejpam-5094	45	5	preopen	preopen	ADJ
ejpam-5094	45	6	set	set	NOUN
ejpam-5094	45	7	is	be	AUX
ejpam-5094	45	8	called	call	VERB
ejpam-5094	45	9	preclosed	preclose	VERB
ejpam-5094	45	10	.	.	PUNCT
ejpam-5094	46	1	the	the	DET
ejpam-5094	46	2	family	family	NOUN
ejpam-5094	46	3	of	of	ADP
ejpam-5094	46	4	all	all	DET
ejpam-5094	46	5	preopen	preopen	ADJ
ejpam-5094	46	6	sets	set	NOUN
ejpam-5094	46	7	of	of	ADP
ejpam-5094	46	8	a	a	DET
ejpam-5094	46	9	topological	topological	ADJ
ejpam-5094	46	10	space	space	NOUN
ejpam-5094	46	11	(	(	PUNCT
ejpam-5094	46	12	x	x	X
ejpam-5094	46	13	,	,	PUNCT
ejpam-5094	46	14	τ	τ	X
ejpam-5094	46	15	)	)	PUNCT
ejpam-5094	46	16	is	be	AUX
ejpam-5094	46	17	denoted	denote	VERB
ejpam-5094	46	18	by	by	ADP
ejpam-5094	46	19	po(x	po(x	NUM
ejpam-5094	46	20	,	,	PUNCT
ejpam-5094	46	21	τ	τ	PROPN
ejpam-5094	46	22	)	)	PUNCT
ejpam-5094	46	23	.	.	PUNCT
ejpam-5094	47	1	a	a	DET
ejpam-5094	47	2	subset	subset	NOUN
ejpam-5094	47	3	λp(a	λp(a	NOUN
ejpam-5094	47	4	)	)	PUNCT
ejpam-5094	48	1	[	[	X
ejpam-5094	48	2	10	10	NUM
ejpam-5094	48	3	]	]	PUNCT
ejpam-5094	48	4	is	be	AUX
ejpam-5094	48	5	defined	define	VERB
ejpam-5094	48	6	as	as	SCONJ
ejpam-5094	48	7	follows	follow	VERB
ejpam-5094	48	8	:	:	PUNCT
ejpam-5094	48	9	λp(a	λp(a	NUM
ejpam-5094	48	10	)	)	PUNCT
ejpam-5094	49	1	=	=	PUNCT
ejpam-5094	50	1	∩{u	∩{u	PROPN
ejpam-5094	50	2	|	|	ADV
ejpam-5094	50	3	a	a	DET
ejpam-5094	50	4	⊆	⊆	NUM
ejpam-5094	50	5	u	u	NOUN
ejpam-5094	50	6	,	,	PUNCT
ejpam-5094	50	7	u	u	PROPN
ejpam-5094	50	8	∈	∈	PROPN
ejpam-5094	50	9	po(x	po(x	NOUN
ejpam-5094	50	10	,	,	PUNCT
ejpam-5094	50	11	τ	τ	NOUN
ejpam-5094	50	12	)	)	PUNCT
ejpam-5094	50	13	}	}	PUNCT
ejpam-5094	50	14	.	.	PUNCT
ejpam-5094	51	1	a	a	DET
ejpam-5094	51	2	subset	subset	NOUN
ejpam-5094	51	3	a	a	PRON
ejpam-5094	51	4	of	of	ADP
ejpam-5094	51	5	a	a	DET
ejpam-5094	51	6	topological	topological	ADJ
ejpam-5094	51	7	space	space	NOUN
ejpam-5094	51	8	(	(	PUNCT
ejpam-5094	51	9	x	x	X
ejpam-5094	51	10	,	,	PUNCT
ejpam-5094	51	11	τ	τ	X
ejpam-5094	51	12	)	)	PUNCT
ejpam-5094	51	13	is	be	AUX
ejpam-5094	51	14	called	call	VERB
ejpam-5094	51	15	a	a	DET
ejpam-5094	51	16	λp	λp	NOUN
ejpam-5094	51	17	-	-	PUNCT
ejpam-5094	51	18	set	set	VERB
ejpam-5094	51	19	[	[	X
ejpam-5094	51	20	6	6	NUM
ejpam-5094	51	21	]	]	PUNCT
ejpam-5094	51	22	(	(	PUNCT
ejpam-5094	51	23	pre	pre	ADJ
ejpam-5094	51	24	-	-	ADJ
ejpam-5094	51	25	λ	λ	NOUN
ejpam-5094	51	26	-	-	NOUN
ejpam-5094	51	27	set	set	NOUN
ejpam-5094	51	28	[	[	X
ejpam-5094	51	29	10	10	NUM
ejpam-5094	51	30	]	]	SYM
ejpam-5094	51	31	)	)	PUNCT
ejpam-5094	51	32	if	if	SCONJ
ejpam-5094	51	33	a	a	DET
ejpam-5094	51	34	=	=	NOUN
ejpam-5094	51	35	λp(a	λp(a	NOUN
ejpam-5094	51	36	)	)	PUNCT
ejpam-5094	51	37	.	.	PUNCT
ejpam-5094	52	1	a	a	DET
ejpam-5094	52	2	subset	subset	NOUN
ejpam-5094	52	3	a	a	PRON
ejpam-5094	52	4	of	of	ADP
ejpam-5094	52	5	a	a	DET
ejpam-5094	52	6	topological	topological	ADJ
ejpam-5094	52	7	space	space	NOUN
ejpam-5094	52	8	(	(	PUNCT
ejpam-5094	52	9	x	x	X
ejpam-5094	52	10	,	,	PUNCT
ejpam-5094	52	11	τ	τ	X
ejpam-5094	52	12	)	)	PUNCT
ejpam-5094	52	13	is	be	AUX
ejpam-5094	52	14	called	call	VERB
ejpam-5094	52	15	(	(	PUNCT
ejpam-5094	52	16	λ	λ	X
ejpam-5094	52	17	,	,	PUNCT
ejpam-5094	52	18	p)-closed	p)-close	VERB
ejpam-5094	52	19	[	[	X
ejpam-5094	52	20	6	6	NUM
ejpam-5094	52	21	]	]	PUNCT
ejpam-5094	52	22	if	if	SCONJ
ejpam-5094	52	23	a	a	DET
ejpam-5094	52	24	=	=	X
ejpam-5094	52	25	t	t	PROPN
ejpam-5094	52	26	∩	∩	ADJ
ejpam-5094	52	27	c	c	NOUN
ejpam-5094	52	28	,	,	PUNCT
ejpam-5094	52	29	where	where	SCONJ
ejpam-5094	52	30	t	t	PROPN
ejpam-5094	52	31	is	be	AUX
ejpam-5094	52	32	a	a	DET
ejpam-5094	52	33	λp	λp	ADV
ejpam-5094	52	34	-	-	PUNCT
ejpam-5094	52	35	set	set	NOUN
ejpam-5094	52	36	and	and	CCONJ
ejpam-5094	52	37	c	c	NOUN
ejpam-5094	52	38	is	be	AUX
ejpam-5094	52	39	a	a	DET
ejpam-5094	52	40	preclosed	preclose	VERB
ejpam-5094	52	41	set	set	NOUN
ejpam-5094	52	42	.	.	PUNCT
ejpam-5094	53	1	the	the	DET
ejpam-5094	53	2	complement	complement	NOUN
ejpam-5094	53	3	of	of	ADP
ejpam-5094	53	4	a	a	DET
ejpam-5094	53	5	(	(	PUNCT
ejpam-5094	53	6	λ	λ	PROPN
ejpam-5094	53	7	,	,	PUNCT
ejpam-5094	53	8	p)-closed	p)-close	VERB
ejpam-5094	53	9	set	set	NOUN
ejpam-5094	53	10	is	be	AUX
ejpam-5094	53	11	called	call	VERB
ejpam-5094	53	12	(	(	PUNCT
ejpam-5094	53	13	λ	λ	X
ejpam-5094	53	14	,	,	PUNCT
ejpam-5094	53	15	p)-open	p)-open	ADJ
ejpam-5094	53	16	.	.	PUNCT
ejpam-5094	54	1	the	the	DET
ejpam-5094	54	2	family	family	NOUN
ejpam-5094	54	3	of	of	ADP
ejpam-5094	54	4	all	all	DET
ejpam-5094	54	5	(	(	PUNCT
ejpam-5094	54	6	λ	λ	X
ejpam-5094	54	7	,	,	PUNCT
ejpam-5094	54	8	p)-open	p)-open	ADJ
ejpam-5094	54	9	(	(	PUNCT
ejpam-5094	54	10	resp	resp	NOUN
ejpam-5094	54	11	.	.	PUNCT
ejpam-5094	55	1	(	(	PUNCT
ejpam-5094	55	2	λ	λ	X
ejpam-5094	55	3	,	,	PUNCT
ejpam-5094	55	4	p)-closed	p)-close	VERB
ejpam-5094	55	5	)	)	PUNCT
ejpam-5094	55	6	sets	set	NOUN
ejpam-5094	55	7	in	in	ADP
ejpam-5094	55	8	a	a	DET
ejpam-5094	55	9	topological	topological	ADJ
ejpam-5094	55	10	space	space	NOUN
ejpam-5094	55	11	(	(	PUNCT
ejpam-5094	55	12	x	x	X
ejpam-5094	55	13	,	,	PUNCT
ejpam-5094	55	14	τ	τ	X
ejpam-5094	55	15	)	)	PUNCT
ejpam-5094	55	16	is	be	AUX
ejpam-5094	55	17	denoted	denote	VERB
ejpam-5094	55	18	by	by	ADP
ejpam-5094	55	19	λpo(x	λpo(x	PROPN
ejpam-5094	55	20	,	,	PUNCT
ejpam-5094	55	21	τ	τ	X
ejpam-5094	55	22	)	)	PUNCT
ejpam-5094	55	23	(	(	PUNCT
ejpam-5094	55	24	resp	resp	NOUN
ejpam-5094	55	25	.	.	PUNCT
ejpam-5094	56	1	λpc(x	λpc(x	PROPN
ejpam-5094	56	2	,	,	PUNCT
ejpam-5094	56	3	τ	τ	PROPN
ejpam-5094	56	4	)	)	PUNCT
ejpam-5094	56	5	)	)	PUNCT
ejpam-5094	56	6	.	.	PUNCT
ejpam-5094	57	1	let	let	VERB
ejpam-5094	57	2	a	a	DET
ejpam-5094	57	3	be	be	AUX
ejpam-5094	57	4	a	a	DET
ejpam-5094	57	5	subset	subset	NOUN
ejpam-5094	57	6	of	of	ADP
ejpam-5094	57	7	a	a	DET
ejpam-5094	57	8	topological	topological	ADJ
ejpam-5094	57	9	space	space	NOUN
ejpam-5094	57	10	(	(	PUNCT
ejpam-5094	57	11	x	x	X
ejpam-5094	57	12	,	,	PUNCT
ejpam-5094	57	13	τ	τ	PROPN
ejpam-5094	57	14	)	)	PUNCT
ejpam-5094	57	15	.	.	PUNCT
ejpam-5094	58	1	a	a	DET
ejpam-5094	58	2	point	point	NOUN
ejpam-5094	58	3	x	x	X
ejpam-5094	58	4	∈	∈	NOUN
ejpam-5094	58	5	x	x	PUNCT
ejpam-5094	58	6	is	be	AUX
ejpam-5094	58	7	called	call	VERB
ejpam-5094	58	8	a	a	DET
ejpam-5094	58	9	(	(	PUNCT
ejpam-5094	58	10	λ	λ	NOUN
ejpam-5094	58	11	,	,	PUNCT
ejpam-5094	58	12	p)-cluster	p)-cluster	NOUN
ejpam-5094	58	13	point	point	NOUN
ejpam-5094	58	14	[	[	X
ejpam-5094	58	15	6	6	NUM
ejpam-5094	58	16	]	]	PUNCT
ejpam-5094	58	17	of	of	ADP
ejpam-5094	58	18	a	a	DET
ejpam-5094	58	19	if	if	SCONJ
ejpam-5094	58	20	a∩u	a∩u	VERB
ejpam-5094	58	21	̸=	̸=	NOUN
ejpam-5094	58	22	∅	∅	NOUN
ejpam-5094	58	23	for	for	ADP
ejpam-5094	58	24	every	every	DET
ejpam-5094	58	25	(	(	PUNCT
ejpam-5094	58	26	λ	λ	NOUN
ejpam-5094	58	27	,	,	PUNCT
ejpam-5094	58	28	p)-open	p)-open	VERB
ejpam-5094	58	29	set	set	VERB
ejpam-5094	58	30	u	u	NOUN
ejpam-5094	58	31	of	of	ADP
ejpam-5094	58	32	x	x	SYM
ejpam-5094	58	33	containing	contain	VERB
ejpam-5094	58	34	x.	x.	NOUN
ejpam-5094	58	35	the	the	DET
ejpam-5094	58	36	set	set	NOUN
ejpam-5094	58	37	of	of	ADP
ejpam-5094	58	38	all	all	DET
ejpam-5094	58	39	(	(	PUNCT
ejpam-5094	58	40	λ	λ	NOUN
ejpam-5094	58	41	,	,	PUNCT
ejpam-5094	58	42	p)-cluster	p)-cluster	VERB
ejpam-5094	58	43	points	point	NOUN
ejpam-5094	58	44	of	of	ADP
ejpam-5094	58	45	a	a	PRON
ejpam-5094	58	46	is	be	AUX
ejpam-5094	58	47	called	call	VERB
ejpam-5094	58	48	the	the	DET
ejpam-5094	58	49	(	(	PUNCT
ejpam-5094	58	50	λ	λ	PROPN
ejpam-5094	58	51	,	,	PUNCT
ejpam-5094	58	52	p)-closure	p)-closure	PUNCT
ejpam-5094	59	1	[	[	X
ejpam-5094	59	2	6	6	NUM
ejpam-5094	59	3	]	]	PUNCT
ejpam-5094	59	4	of	of	ADP
ejpam-5094	59	5	a	a	PRON
ejpam-5094	59	6	and	and	CCONJ
ejpam-5094	59	7	is	be	AUX
ejpam-5094	59	8	denoted	denote	VERB
ejpam-5094	59	9	by	by	ADP
ejpam-5094	59	10	a(λ	a(λ	PROPN
ejpam-5094	59	11	,	,	PUNCT
ejpam-5094	59	12	p	p	NOUN
ejpam-5094	59	13	)	)	PUNCT
ejpam-5094	59	14	.	.	PUNCT
ejpam-5094	60	1	the	the	DET
ejpam-5094	60	2	union	union	NOUN
ejpam-5094	60	3	of	of	ADP
ejpam-5094	60	4	all	all	PRON
ejpam-5094	60	5	(	(	PUNCT
ejpam-5094	60	6	λ	λ	NOUN
ejpam-5094	60	7	,	,	PUNCT
ejpam-5094	60	8	p)-open	p)-open	VERB
ejpam-5094	60	9	sets	set	NOUN
ejpam-5094	60	10	of	of	ADP
ejpam-5094	60	11	x	x	PUNCT
ejpam-5094	60	12	contained	contain	VERB
ejpam-5094	60	13	in	in	ADP
ejpam-5094	60	14	a	a	PRON
ejpam-5094	60	15	is	be	AUX
ejpam-5094	60	16	called	call	VERB
ejpam-5094	60	17	the	the	DET
ejpam-5094	60	18	(	(	PUNCT
ejpam-5094	60	19	λ	λ	PROPN
ejpam-5094	60	20	,	,	PUNCT
ejpam-5094	60	21	p)-interior	p)-interior	ADJ
ejpam-5094	61	1	[	[	X
ejpam-5094	61	2	6	6	NUM
ejpam-5094	61	3	]	]	PUNCT
ejpam-5094	61	4	of	of	ADP
ejpam-5094	61	5	a	a	PRON
ejpam-5094	61	6	and	and	CCONJ
ejpam-5094	61	7	is	be	AUX
ejpam-5094	61	8	denoted	denote	VERB
ejpam-5094	61	9	by	by	ADP
ejpam-5094	61	10	a(λ	a(λ	PROPN
ejpam-5094	61	11	,	,	PUNCT
ejpam-5094	61	12	p	p	NOUN
ejpam-5094	61	13	)	)	PUNCT
ejpam-5094	61	14	.	.	PUNCT
ejpam-5094	62	1	let	let	VERB
ejpam-5094	62	2	a	a	DET
ejpam-5094	62	3	be	be	AUX
ejpam-5094	62	4	a	a	DET
ejpam-5094	62	5	subset	subset	NOUN
ejpam-5094	62	6	of	of	ADP
ejpam-5094	62	7	a	a	DET
ejpam-5094	62	8	topological	topological	ADJ
ejpam-5094	62	9	space	space	NOUN
ejpam-5094	62	10	(	(	PUNCT
ejpam-5094	62	11	x	x	X
ejpam-5094	62	12	,	,	PUNCT
ejpam-5094	62	13	τ	τ	PROPN
ejpam-5094	62	14	)	)	PUNCT
ejpam-5094	62	15	.	.	PUNCT
ejpam-5094	63	1	the	the	DET
ejpam-5094	63	2	θ(λ	θ(λ	PROPN
ejpam-5094	63	3	,	,	PUNCT
ejpam-5094	63	4	p)-closure	p)-closure	X
ejpam-5094	64	1	[	[	X
ejpam-5094	64	2	6	6	NUM
ejpam-5094	64	3	]	]	PUNCT
ejpam-5094	64	4	of	of	ADP
ejpam-5094	64	5	a	a	DET
ejpam-5094	64	6	,	,	PUNCT
ejpam-5094	64	7	aθ(λ	aθ(λ	ADJ
ejpam-5094	64	8	,	,	PUNCT
ejpam-5094	64	9	p	p	NOUN
ejpam-5094	64	10	)	)	PUNCT
ejpam-5094	64	11	,	,	PUNCT
ejpam-5094	64	12	is	be	AUX
ejpam-5094	64	13	defined	define	VERB
ejpam-5094	64	14	as	as	SCONJ
ejpam-5094	64	15	follows	follow	VERB
ejpam-5094	64	16	:	:	PUNCT
ejpam-5094	65	1	aθ(λ	aθ(λ	NOUN
ejpam-5094	65	2	,	,	PUNCT
ejpam-5094	65	3	p	p	NOUN
ejpam-5094	65	4	)	)	PUNCT
ejpam-5094	65	5	=	=	SYM
ejpam-5094	65	6	{	{	PUNCT
ejpam-5094	65	7	x	x	PUNCT
ejpam-5094	65	8	∈	∈	NOUN
ejpam-5094	65	9	x	x	PUNCT
ejpam-5094	65	10	|	|	ADV
ejpam-5094	65	11	a	a	DET
ejpam-5094	65	12	∩	∩	ADJ
ejpam-5094	65	13	u	u	NOUN
ejpam-5094	65	14	(	(	PUNCT
ejpam-5094	65	15	λ	λ	PROPN
ejpam-5094	65	16	,	,	PUNCT
ejpam-5094	65	17	p	p	NOUN
ejpam-5094	65	18	)	)	PUNCT
ejpam-5094	65	19	̸=	̸=	PROPN
ejpam-5094	65	20	∅	∅	NOUN
ejpam-5094	65	21	for	for	ADP
ejpam-5094	65	22	each	each	DET
ejpam-5094	65	23	(	(	PUNCT
ejpam-5094	65	24	λ	λ	PROPN
ejpam-5094	65	25	,	,	PUNCT
ejpam-5094	65	26	p)-open	p)-open	VERB
ejpam-5094	65	27	set	set	VERB
ejpam-5094	65	28	u	u	NOUN
ejpam-5094	65	29	containing	contain	VERB
ejpam-5094	65	30	x	x	X
ejpam-5094	65	31	}	}	PUNCT
ejpam-5094	65	32	.	.	PUNCT
ejpam-5094	66	1	a	a	DET
ejpam-5094	66	2	subset	subset	NOUN
ejpam-5094	66	3	a	a	PRON
ejpam-5094	66	4	of	of	ADP
ejpam-5094	66	5	a	a	DET
ejpam-5094	66	6	topological	topological	ADJ
ejpam-5094	66	7	space	space	NOUN
ejpam-5094	66	8	(	(	PUNCT
ejpam-5094	66	9	x	x	X
ejpam-5094	66	10	,	,	PUNCT
ejpam-5094	66	11	τ	τ	X
ejpam-5094	66	12	)	)	PUNCT
ejpam-5094	66	13	is	be	AUX
ejpam-5094	66	14	called	call	VERB
ejpam-5094	66	15	θ(λ	θ(λ	PROPN
ejpam-5094	66	16	,	,	PUNCT
ejpam-5094	66	17	p)-closed	p)-close	VERB
ejpam-5094	66	18	[	[	X
ejpam-5094	66	19	6	6	NUM
ejpam-5094	66	20	]	]	PUNCT
ejpam-5094	66	21	if	if	SCONJ
ejpam-5094	66	22	a	a	PRON
ejpam-5094	66	23	=	=	NOUN
ejpam-5094	66	24	aθ(λ	aθ(λ	NOUN
ejpam-5094	66	25	,	,	PUNCT
ejpam-5094	66	26	p	p	NOUN
ejpam-5094	66	27	)	)	PUNCT
ejpam-5094	66	28	.	.	PUNCT
ejpam-5094	67	1	the	the	DET
ejpam-5094	67	2	complement	complement	NOUN
ejpam-5094	67	3	of	of	ADP
ejpam-5094	67	4	a	a	DET
ejpam-5094	67	5	θ(λ	θ(λ	PROPN
ejpam-5094	67	6	,	,	PUNCT
ejpam-5094	67	7	p)-closed	p)-close	VERB
ejpam-5094	67	8	set	set	NOUN
ejpam-5094	67	9	is	be	AUX
ejpam-5094	67	10	said	say	VERB
ejpam-5094	67	11	to	to	PART
ejpam-5094	67	12	be	be	AUX
ejpam-5094	67	13	θ(λ	θ(λ	PROPN
ejpam-5094	67	14	,	,	PUNCT
ejpam-5094	67	15	p)-open	p)-open	NOUN
ejpam-5094	67	16	.	.	PUNCT
ejpam-5094	68	1	a	a	DET
ejpam-5094	68	2	point	point	NOUN
ejpam-5094	68	3	x	x	X
ejpam-5094	68	4	∈	∈	NOUN
ejpam-5094	68	5	x	x	PUNCT
ejpam-5094	68	6	is	be	AUX
ejpam-5094	68	7	called	call	VERB
ejpam-5094	68	8	a	a	DET
ejpam-5094	68	9	θ(λ	θ(λ	PROPN
ejpam-5094	68	10	,	,	PUNCT
ejpam-5094	68	11	p)-interior	p)-interior	ADJ
ejpam-5094	68	12	point	point	NOUN
ejpam-5094	68	13	[	[	X
ejpam-5094	68	14	20	20	NUM
ejpam-5094	68	15	]	]	PUNCT
ejpam-5094	68	16	of	of	ADP
ejpam-5094	68	17	a	a	DET
ejpam-5094	68	18	if	if	NOUN
ejpam-5094	68	19	x	x	SYM
ejpam-5094	68	20	∈	∈	PROPN
ejpam-5094	68	21	u	u	NOUN
ejpam-5094	68	22	⊆	⊆	NUM
ejpam-5094	68	23	u	u	PROPN
ejpam-5094	68	24	(	(	PUNCT
ejpam-5094	68	25	λ	λ	PROPN
ejpam-5094	68	26	,	,	PUNCT
ejpam-5094	68	27	p	p	NOUN
ejpam-5094	68	28	)	)	PUNCT
ejpam-5094	68	29	⊆	⊆	NUM
ejpam-5094	68	30	a	a	PRON
ejpam-5094	68	31	for	for	ADP
ejpam-5094	68	32	some	some	DET
ejpam-5094	68	33	u	u	NOUN
ejpam-5094	68	34	∈	∈	PROPN
ejpam-5094	68	35	λpo(x	λpo(x	PROPN
ejpam-5094	68	36	,	,	PUNCT
ejpam-5094	68	37	τ	τ	PROPN
ejpam-5094	68	38	)	)	PUNCT
ejpam-5094	68	39	.	.	PUNCT
ejpam-5094	69	1	the	the	DET
ejpam-5094	69	2	set	set	NOUN
ejpam-5094	69	3	of	of	ADP
ejpam-5094	69	4	all	all	DET
ejpam-5094	69	5	θ(λ	θ(λ	PROPN
ejpam-5094	69	6	,	,	PUNCT
ejpam-5094	69	7	p)-interior	p)-interior	ADJ
ejpam-5094	69	8	points	point	NOUN
ejpam-5094	69	9	of	of	ADP
ejpam-5094	69	10	a	a	PRON
ejpam-5094	69	11	is	be	AUX
ejpam-5094	69	12	called	call	VERB
ejpam-5094	69	13	the	the	DET
ejpam-5094	69	14	θ(λ	θ(λ	PROPN
ejpam-5094	69	15	,	,	PUNCT
ejpam-5094	69	16	p)-interior	p)-interior	ADJ
ejpam-5094	69	17	[	[	X
ejpam-5094	69	18	20	20	NUM
ejpam-5094	69	19	]	]	PUNCT
ejpam-5094	69	20	of	of	ADP
ejpam-5094	69	21	a	a	PRON
ejpam-5094	69	22	and	and	CCONJ
ejpam-5094	69	23	is	be	AUX
ejpam-5094	69	24	denoted	denote	VERB
ejpam-5094	69	25	by	by	ADP
ejpam-5094	69	26	aθ(λ	aθ(λ	NOUN
ejpam-5094	69	27	,	,	PUNCT
ejpam-5094	69	28	p	p	NOUN
ejpam-5094	69	29	)	)	PUNCT
ejpam-5094	69	30	.	.	PUNCT
ejpam-5094	70	1	lemma	lemma	PROPN
ejpam-5094	70	2	1	1	NUM
ejpam-5094	70	3	.	.	PUNCT
ejpam-5094	71	1	[	[	X
ejpam-5094	71	2	20	20	NUM
ejpam-5094	71	3	]	]	PUNCT
ejpam-5094	71	4	for	for	ADP
ejpam-5094	71	5	subsets	subset	NOUN
ejpam-5094	71	6	a	a	PRON
ejpam-5094	71	7	and	and	CCONJ
ejpam-5094	71	8	b	b	NOUN
ejpam-5094	71	9	of	of	ADP
ejpam-5094	71	10	a	a	DET
ejpam-5094	71	11	topological	topological	ADJ
ejpam-5094	71	12	space	space	NOUN
ejpam-5094	71	13	(	(	PUNCT
ejpam-5094	71	14	x	x	X
ejpam-5094	71	15	,	,	PUNCT
ejpam-5094	71	16	τ	τ	PROPN
ejpam-5094	71	17	)	)	PUNCT
ejpam-5094	71	18	,	,	PUNCT
ejpam-5094	71	19	the	the	DET
ejpam-5094	71	20	following	follow	VERB
ejpam-5094	71	21	properties	property	NOUN
ejpam-5094	71	22	hold	hold	VERB
ejpam-5094	71	23	:	:	PUNCT
ejpam-5094	71	24	(	(	PUNCT
ejpam-5094	71	25	1	1	X
ejpam-5094	71	26	)	)	PUNCT
ejpam-5094	71	27	x	x	PUNCT
ejpam-5094	72	1	−aθ(λ	−aθ(λ	NOUN
ejpam-5094	72	2	,	,	PUNCT
ejpam-5094	72	3	p	p	NOUN
ejpam-5094	72	4	)	)	PUNCT
ejpam-5094	72	5	=	=	PUNCT
ejpam-5094	73	1	[	[	X
ejpam-5094	73	2	x	x	X
ejpam-5094	73	3	−a]θ(λ	−a]θ(λ	NOUN
ejpam-5094	73	4	,	,	PUNCT
ejpam-5094	73	5	p	p	NOUN
ejpam-5094	73	6	)	)	PUNCT
ejpam-5094	73	7	and	and	CCONJ
ejpam-5094	73	8	x	x	PUNCT
ejpam-5094	73	9	−aθ(λ	−aθ(λ	NOUN
ejpam-5094	73	10	,	,	PUNCT
ejpam-5094	73	11	p	p	NOUN
ejpam-5094	73	12	)	)	PUNCT
ejpam-5094	73	13	=	=	PUNCT
ejpam-5094	74	1	[	[	X
ejpam-5094	74	2	x	x	X
ejpam-5094	74	3	−a]θ(λ	−a]θ(λ	NOUN
ejpam-5094	74	4	,	,	PUNCT
ejpam-5094	74	5	p	p	NOUN
ejpam-5094	74	6	)	)	PUNCT
ejpam-5094	74	7	.	.	PUNCT
ejpam-5094	75	1	c.	c.	PROPN
ejpam-5094	75	2	boonpok	boonpok	PROPN
ejpam-5094	75	3	,	,	PUNCT
ejpam-5094	75	4	j.	j.	PROPN
ejpam-5094	75	5	khampakdee	khampakdee	PROPN
ejpam-5094	75	6	/	/	PUNCT
ejpam-5094	75	7	eur	eur	PROPN
ejpam-5094	75	8	.	.	PUNCT
ejpam-5094	76	1	j.	j.	PROPN
ejpam-5094	76	2	pure	pure	PROPN
ejpam-5094	76	3	appl	appl	PROPN
ejpam-5094	76	4	.	.	PROPN
ejpam-5094	76	5	math	math	PROPN
ejpam-5094	76	6	,	,	PUNCT
ejpam-5094	76	7	17	17	NUM
ejpam-5094	76	8	(	(	PUNCT
ejpam-5094	76	9	2	2	NUM
ejpam-5094	76	10	)	)	PUNCT
ejpam-5094	76	11	(	(	PUNCT
ejpam-5094	76	12	2024	2024	NUM
ejpam-5094	76	13	)	)	PUNCT
ejpam-5094	76	14	,	,	PUNCT
ejpam-5094	76	15	582	582	NUM
ejpam-5094	76	16	-	-	SYM
ejpam-5094	76	17	590	590	NUM
ejpam-5094	76	18	584	584	NUM
ejpam-5094	76	19	(	(	PUNCT
ejpam-5094	76	20	2	2	NUM
ejpam-5094	76	21	)	)	PUNCT
ejpam-5094	76	22	a	a	PRON
ejpam-5094	76	23	is	be	AUX
ejpam-5094	76	24	θ(λ	θ(λ	PROPN
ejpam-5094	76	25	,	,	PUNCT
ejpam-5094	76	26	p)-open	p)-open	VERB
ejpam-5094	76	27	if	if	SCONJ
ejpam-5094	76	28	and	and	CCONJ
ejpam-5094	76	29	only	only	ADV
ejpam-5094	76	30	if	if	SCONJ
ejpam-5094	76	31	a	a	PRON
ejpam-5094	76	32	=	=	NOUN
ejpam-5094	76	33	aθ(λ	aθ(λ	NOUN
ejpam-5094	76	34	,	,	PUNCT
ejpam-5094	76	35	p	p	NOUN
ejpam-5094	76	36	)	)	PUNCT
ejpam-5094	76	37	.	.	PUNCT
ejpam-5094	77	1	(	(	PUNCT
ejpam-5094	77	2	3	3	X
ejpam-5094	77	3	)	)	PUNCT
ejpam-5094	77	4	a	a	DET
ejpam-5094	77	5	⊆	⊆	NUM
ejpam-5094	77	6	a(λ	a(λ	ADJ
ejpam-5094	77	7	,	,	PUNCT
ejpam-5094	77	8	p	p	X
ejpam-5094	77	9	)	)	PUNCT
ejpam-5094	77	10	⊆	⊆	NUM
ejpam-5094	77	11	aθ(λ	aθ(λ	NOUN
ejpam-5094	77	12	,	,	PUNCT
ejpam-5094	77	13	p	p	NOUN
ejpam-5094	77	14	)	)	PUNCT
ejpam-5094	77	15	and	and	CCONJ
ejpam-5094	77	16	aθ(λ	aθ(λ	NOUN
ejpam-5094	77	17	,	,	PUNCT
ejpam-5094	77	18	p	p	NOUN
ejpam-5094	77	19	)	)	PUNCT
ejpam-5094	77	20	⊆	⊆	NUM
ejpam-5094	77	21	a(λ	a(λ	ADV
ejpam-5094	77	22	,	,	PUNCT
ejpam-5094	77	23	p	p	NOUN
ejpam-5094	77	24	)	)	PUNCT
ejpam-5094	77	25	⊆	⊆	NUM
ejpam-5094	77	26	a.	a.	NOUN
ejpam-5094	77	27	(	(	PUNCT
ejpam-5094	77	28	4	4	NUM
ejpam-5094	77	29	)	)	PUNCT
ejpam-5094	77	30	if	if	SCONJ
ejpam-5094	77	31	a	a	DET
ejpam-5094	77	32	⊆	⊆	NUM
ejpam-5094	77	33	b	b	NOUN
ejpam-5094	77	34	,	,	PUNCT
ejpam-5094	77	35	then	then	ADV
ejpam-5094	77	36	aθ(λ	aθ(λ	NOUN
ejpam-5094	77	37	,	,	PUNCT
ejpam-5094	77	38	p	p	NOUN
ejpam-5094	77	39	)	)	PUNCT
ejpam-5094	77	40	⊆	⊆	NUM
ejpam-5094	77	41	bθ(λ	bθ(λ	NOUN
ejpam-5094	77	42	,	,	PUNCT
ejpam-5094	77	43	p	p	NOUN
ejpam-5094	77	44	)	)	PUNCT
ejpam-5094	77	45	and	and	CCONJ
ejpam-5094	77	46	aθ(λ	aθ(λ	NOUN
ejpam-5094	77	47	,	,	PUNCT
ejpam-5094	77	48	p	p	NOUN
ejpam-5094	77	49	)	)	PUNCT
ejpam-5094	77	50	⊆	⊆	NUM
ejpam-5094	77	51	bθ(λ	bθ(λ	NOUN
ejpam-5094	77	52	,	,	PUNCT
ejpam-5094	77	53	p	p	NOUN
ejpam-5094	77	54	)	)	PUNCT
ejpam-5094	77	55	.	.	PUNCT
ejpam-5094	78	1	(	(	PUNCT
ejpam-5094	78	2	5	5	X
ejpam-5094	78	3	)	)	PUNCT
ejpam-5094	78	4	if	if	SCONJ
ejpam-5094	78	5	a	a	PRON
ejpam-5094	78	6	is	be	AUX
ejpam-5094	78	7	(	(	PUNCT
ejpam-5094	78	8	λ	λ	NOUN
ejpam-5094	78	9	,	,	PUNCT
ejpam-5094	78	10	p)-open	p)-open	ADJ
ejpam-5094	78	11	,	,	PUNCT
ejpam-5094	78	12	then	then	ADV
ejpam-5094	78	13	a(λ	a(λ	ADV
ejpam-5094	78	14	,	,	PUNCT
ejpam-5094	78	15	p	p	X
ejpam-5094	78	16	)	)	PUNCT
ejpam-5094	78	17	=	=	PUNCT
ejpam-5094	78	18	aθ(λ	aθ(λ	NOUN
ejpam-5094	78	19	,	,	PUNCT
ejpam-5094	78	20	p	p	NOUN
ejpam-5094	78	21	)	)	PUNCT
ejpam-5094	78	22	.	.	PUNCT
ejpam-5094	79	1	a	a	DET
ejpam-5094	79	2	subset	subset	NOUN
ejpam-5094	79	3	a	a	PRON
ejpam-5094	79	4	of	of	ADP
ejpam-5094	79	5	a	a	DET
ejpam-5094	79	6	topological	topological	ADJ
ejpam-5094	79	7	space	space	NOUN
ejpam-5094	79	8	(	(	PUNCT
ejpam-5094	79	9	x	x	X
ejpam-5094	79	10	,	,	PUNCT
ejpam-5094	79	11	τ	τ	X
ejpam-5094	79	12	)	)	PUNCT
ejpam-5094	79	13	is	be	AUX
ejpam-5094	79	14	said	say	VERB
ejpam-5094	79	15	to	to	PART
ejpam-5094	79	16	be	be	AUX
ejpam-5094	79	17	s(λ	s(λ	NOUN
ejpam-5094	79	18	,	,	PUNCT
ejpam-5094	79	19	p)-open	p)-open	VERB
ejpam-5094	80	1	[	[	X
ejpam-5094	80	2	6	6	NUM
ejpam-5094	80	3	]	]	PUNCT
ejpam-5094	80	4	(	(	PUNCT
ejpam-5094	80	5	resp	resp	NOUN
ejpam-5094	80	6	.	.	PUNCT
ejpam-5094	81	1	r(λ	r(λ	NOUN
ejpam-5094	81	2	,	,	PUNCT
ejpam-5094	81	3	p)-open	p)-open	VERB
ejpam-5094	81	4	[	[	X
ejpam-5094	81	5	6	6	NUM
ejpam-5094	81	6	]	]	PUNCT
ejpam-5094	81	7	,	,	PUNCT
ejpam-5094	81	8	p(λ	p(λ	NOUN
ejpam-5094	81	9	,	,	PUNCT
ejpam-5094	81	10	p)-open	p)-open	VERB
ejpam-5094	81	11	[	[	X
ejpam-5094	81	12	6	6	NUM
ejpam-5094	81	13	]	]	PUNCT
ejpam-5094	81	14	,	,	PUNCT
ejpam-5094	81	15	α(λ	α(λ	PROPN
ejpam-5094	81	16	,	,	PUNCT
ejpam-5094	81	17	p)-open	p)-open	VERB
ejpam-5094	81	18	[	[	X
ejpam-5094	81	19	21	21	NUM
ejpam-5094	81	20	]	]	PUNCT
ejpam-5094	81	21	)	)	PUNCT
ejpam-5094	81	22	if	if	SCONJ
ejpam-5094	81	23	a	a	DET
ejpam-5094	81	24	⊆	⊆	NUM
ejpam-5094	81	25	[	[	X
ejpam-5094	81	26	a(λ	a(λ	ADV
ejpam-5094	81	27	,	,	PUNCT
ejpam-5094	81	28	p	p	NOUN
ejpam-5094	81	29	)	)	PUNCT
ejpam-5094	81	30	]	]	PUNCT
ejpam-5094	81	31	(	(	PUNCT
ejpam-5094	81	32	λ	λ	X
ejpam-5094	81	33	,	,	PUNCT
ejpam-5094	81	34	p	p	NOUN
ejpam-5094	81	35	)	)	PUNCT
ejpam-5094	81	36	(	(	PUNCT
ejpam-5094	81	37	resp	resp	NOUN
ejpam-5094	81	38	.	.	PUNCT
ejpam-5094	82	1	a	a	PRON
ejpam-5094	82	2	=	=	X
ejpam-5094	83	1	[	[	X
ejpam-5094	83	2	a(λ	a(λ	ADV
ejpam-5094	83	3	,	,	PUNCT
ejpam-5094	83	4	p)](λ	p)](λ	X
ejpam-5094	83	5	,	,	PUNCT
ejpam-5094	83	6	p	p	NOUN
ejpam-5094	83	7	)	)	PUNCT
ejpam-5094	83	8	,	,	PUNCT
ejpam-5094	83	9	a	a	DET
ejpam-5094	83	10	⊆	⊆	NUM
ejpam-5094	83	11	[	[	X
ejpam-5094	83	12	a(λ	a(λ	ADV
ejpam-5094	83	13	,	,	PUNCT
ejpam-5094	83	14	p)](λ	p)](λ	X
ejpam-5094	83	15	,	,	PUNCT
ejpam-5094	83	16	p	p	NOUN
ejpam-5094	83	17	)	)	PUNCT
ejpam-5094	83	18	,	,	PUNCT
ejpam-5094	84	1	a	a	DET
ejpam-5094	84	2	⊆	⊆	NUM
ejpam-5094	84	3	[	[	X
ejpam-5094	84	4	[	[	X
ejpam-5094	84	5	a(λ	a(λ	ADJ
ejpam-5094	84	6	,	,	PUNCT
ejpam-5094	84	7	p	p	NOUN
ejpam-5094	84	8	)	)	PUNCT
ejpam-5094	84	9	]	]	PUNCT
ejpam-5094	84	10	(	(	PUNCT
ejpam-5094	84	11	λ	λ	X
ejpam-5094	84	12	,	,	PUNCT
ejpam-5094	84	13	p)](λ	p)](λ	ADJ
ejpam-5094	84	14	,	,	PUNCT
ejpam-5094	84	15	p	p	NOUN
ejpam-5094	84	16	)	)	PUNCT
ejpam-5094	84	17	)	)	PUNCT
ejpam-5094	84	18	.	.	PUNCT
ejpam-5094	85	1	the	the	DET
ejpam-5094	85	2	family	family	NOUN
ejpam-5094	85	3	of	of	ADP
ejpam-5094	85	4	all	all	DET
ejpam-5094	85	5	s(λ	s(λ	NOUN
ejpam-5094	85	6	,	,	PUNCT
ejpam-5094	85	7	p)-open	p)-open	ADJ
ejpam-5094	85	8	(	(	PUNCT
ejpam-5094	85	9	resp	resp	NOUN
ejpam-5094	85	10	.	.	PUNCT
ejpam-5094	86	1	r(λ	r(λ	NOUN
ejpam-5094	86	2	,	,	PUNCT
ejpam-5094	86	3	p)-open	p)-open	ADJ
ejpam-5094	86	4	,	,	PUNCT
ejpam-5094	86	5	p(λ	p(λ	NOUN
ejpam-5094	86	6	,	,	PUNCT
ejpam-5094	86	7	p)-open	p)-open	NOUN
ejpam-5094	86	8	,	,	PUNCT
ejpam-5094	86	9	α(λ	α(λ	PROPN
ejpam-5094	86	10	,	,	PUNCT
ejpam-5094	86	11	p)-open	p)-open	NOUN
ejpam-5094	86	12	)	)	PUNCT
ejpam-5094	86	13	sets	set	NOUN
ejpam-5094	86	14	in	in	ADP
ejpam-5094	86	15	a	a	DET
ejpam-5094	86	16	topological	topological	ADJ
ejpam-5094	86	17	space	space	NOUN
ejpam-5094	86	18	(	(	PUNCT
ejpam-5094	86	19	x	x	X
ejpam-5094	86	20	,	,	PUNCT
ejpam-5094	86	21	τ	τ	X
ejpam-5094	86	22	)	)	PUNCT
ejpam-5094	86	23	is	be	AUX
ejpam-5094	86	24	denoted	denote	VERB
ejpam-5094	86	25	by	by	ADP
ejpam-5094	86	26	s(λ	s(λ	PROPN
ejpam-5094	86	27	,	,	PUNCT
ejpam-5094	86	28	p)o(x	p)o(x	ADJ
ejpam-5094	86	29	,	,	PUNCT
ejpam-5094	86	30	τ	τ	PROPN
ejpam-5094	86	31	)	)	PUNCT
ejpam-5094	86	32	(	(	PUNCT
ejpam-5094	86	33	resp	resp	NOUN
ejpam-5094	86	34	.	.	PUNCT
ejpam-5094	87	1	r(λ	r(λ	NOUN
ejpam-5094	87	2	,	,	PUNCT
ejpam-5094	87	3	p)o(x	p)o(x	ADJ
ejpam-5094	87	4	,	,	PUNCT
ejpam-5094	87	5	τ	τ	PROPN
ejpam-5094	87	6	)	)	PUNCT
ejpam-5094	87	7	,	,	PUNCT
ejpam-5094	87	8	p(λ	p(λ	PROPN
ejpam-5094	87	9	,	,	PUNCT
ejpam-5094	87	10	p)o(x	p)o(x	ADJ
ejpam-5094	87	11	,	,	PUNCT
ejpam-5094	87	12	τ	τ	PROPN
ejpam-5094	87	13	)	)	PUNCT
ejpam-5094	87	14	,	,	PUNCT
ejpam-5094	87	15	α(λ	α(λ	PROPN
ejpam-5094	87	16	,	,	PUNCT
ejpam-5094	87	17	p)o(x	p)o(x	ADJ
ejpam-5094	87	18	,	,	PUNCT
ejpam-5094	87	19	τ	τ	PROPN
ejpam-5094	87	20	)	)	PUNCT
ejpam-5094	87	21	)	)	PUNCT
ejpam-5094	87	22	.	.	PUNCT
ejpam-5094	88	1	the	the	DET
ejpam-5094	88	2	union	union	NOUN
ejpam-5094	88	3	of	of	ADP
ejpam-5094	88	4	all	all	DET
ejpam-5094	88	5	s(λ	s(λ	NOUN
ejpam-5094	88	6	,	,	PUNCT
ejpam-5094	88	7	p)-open	p)-open	ADJ
ejpam-5094	88	8	(	(	PUNCT
ejpam-5094	88	9	resp	resp	NOUN
ejpam-5094	88	10	.	.	PUNCT
ejpam-5094	89	1	p(λ	p(λ	NOUN
ejpam-5094	89	2	,	,	PUNCT
ejpam-5094	89	3	p)-open	p)-open	NOUN
ejpam-5094	89	4	,	,	PUNCT
ejpam-5094	89	5	α(λ	α(λ	PROPN
ejpam-5094	89	6	,	,	PUNCT
ejpam-5094	89	7	p)-open	p)-open	NOUN
ejpam-5094	89	8	)	)	PUNCT
ejpam-5094	89	9	sets	set	NOUN
ejpam-5094	89	10	of	of	ADP
ejpam-5094	89	11	x	x	PUNCT
ejpam-5094	89	12	contained	contain	VERB
ejpam-5094	89	13	in	in	ADP
ejpam-5094	89	14	a	a	PRON
ejpam-5094	89	15	is	be	AUX
ejpam-5094	89	16	called	call	VERB
ejpam-5094	89	17	the	the	DET
ejpam-5094	89	18	s(λ	s(λ	NOUN
ejpam-5094	89	19	,	,	PUNCT
ejpam-5094	89	20	p)-interior	p)-interior	ADJ
ejpam-5094	89	21	(	(	PUNCT
ejpam-5094	89	22	resp	resp	NOUN
ejpam-5094	89	23	.	.	PUNCT
ejpam-5094	90	1	p(λ	p(λ	NOUN
ejpam-5094	90	2	,	,	PUNCT
ejpam-5094	90	3	p)-interior	p)-interior	PROPN
ejpam-5094	90	4	,	,	PUNCT
ejpam-5094	90	5	α(λ	α(λ	PROPN
ejpam-5094	90	6	,	,	PUNCT
ejpam-5094	90	7	p)-interior	p)-interior	ADJ
ejpam-5094	90	8	)	)	PUNCT
ejpam-5094	90	9	of	of	ADP
ejpam-5094	90	10	a	a	PRON
ejpam-5094	90	11	and	and	CCONJ
ejpam-5094	90	12	is	be	AUX
ejpam-5094	90	13	denoted	denote	VERB
ejpam-5094	90	14	by	by	ADP
ejpam-5094	90	15	as(λ	as(λ	NOUN
ejpam-5094	90	16	,	,	PUNCT
ejpam-5094	90	17	p	p	NOUN
ejpam-5094	90	18	)	)	PUNCT
ejpam-5094	90	19	(	(	PUNCT
ejpam-5094	90	20	resp	resp	NOUN
ejpam-5094	90	21	.	.	PUNCT
ejpam-5094	91	1	ap(λ	ap(λ	PROPN
ejpam-5094	91	2	,	,	PUNCT
ejpam-5094	91	3	p	p	NOUN
ejpam-5094	91	4	)	)	PUNCT
ejpam-5094	91	5	,	,	PUNCT
ejpam-5094	91	6	aα(λ	aα(λ	PROPN
ejpam-5094	91	7	,	,	PUNCT
ejpam-5094	91	8	p	p	NOUN
ejpam-5094	91	9	)	)	PUNCT
ejpam-5094	91	10	)	)	PUNCT
ejpam-5094	91	11	.	.	PUNCT
ejpam-5094	92	1	the	the	DET
ejpam-5094	92	2	complement	complement	NOUN
ejpam-5094	92	3	of	of	ADP
ejpam-5094	92	4	a	a	DET
ejpam-5094	92	5	s(λ	s(λ	PROPN
ejpam-5094	92	6	,	,	PUNCT
ejpam-5094	92	7	p)-open	p)-open	ADJ
ejpam-5094	92	8	(	(	PUNCT
ejpam-5094	92	9	resp	resp	NOUN
ejpam-5094	92	10	.	.	PUNCT
ejpam-5094	93	1	r(λ	r(λ	NOUN
ejpam-5094	93	2	,	,	PUNCT
ejpam-5094	93	3	p)-open	p)-open	ADJ
ejpam-5094	93	4	,	,	PUNCT
ejpam-5094	93	5	p(λ	p(λ	NOUN
ejpam-5094	93	6	,	,	PUNCT
ejpam-5094	93	7	p)-open	p)-open	NOUN
ejpam-5094	93	8	,	,	PUNCT
ejpam-5094	93	9	α(λ	α(λ	PROPN
ejpam-5094	93	10	,	,	PUNCT
ejpam-5094	93	11	p)open	p)open	ADJ
ejpam-5094	93	12	)	)	PUNCT
ejpam-5094	93	13	set	set	NOUN
ejpam-5094	93	14	is	be	AUX
ejpam-5094	93	15	called	call	VERB
ejpam-5094	93	16	s(λ	s(λ	PROPN
ejpam-5094	93	17	,	,	PUNCT
ejpam-5094	93	18	p)-closed	p)-close	VERB
ejpam-5094	93	19	(	(	PUNCT
ejpam-5094	93	20	resp	resp	NOUN
ejpam-5094	93	21	.	.	PUNCT
ejpam-5094	94	1	r(λ	r(λ	NOUN
ejpam-5094	94	2	,	,	PUNCT
ejpam-5094	94	3	p)-closed	p)-close	VERB
ejpam-5094	94	4	,	,	PUNCT
ejpam-5094	94	5	p(λ	p(λ	NOUN
ejpam-5094	94	6	,	,	PUNCT
ejpam-5094	94	7	p)-closed	p)-close	VERB
ejpam-5094	94	8	,	,	PUNCT
ejpam-5094	94	9	α(λ	α(λ	PROPN
ejpam-5094	94	10	,	,	PUNCT
ejpam-5094	94	11	p)-closed	p)-close	VERB
ejpam-5094	94	12	)	)	PUNCT
ejpam-5094	94	13	.	.	PUNCT
ejpam-5094	95	1	the	the	DET
ejpam-5094	95	2	family	family	NOUN
ejpam-5094	95	3	of	of	ADP
ejpam-5094	95	4	all	all	DET
ejpam-5094	95	5	s(λ	s(λ	PROPN
ejpam-5094	95	6	,	,	PUNCT
ejpam-5094	95	7	p)-closed	p)-close	VERB
ejpam-5094	95	8	(	(	PUNCT
ejpam-5094	95	9	resp	resp	NOUN
ejpam-5094	95	10	.	.	PUNCT
ejpam-5094	96	1	r(λ	r(λ	NOUN
ejpam-5094	96	2	,	,	PUNCT
ejpam-5094	96	3	p)-closed	p)-close	VERB
ejpam-5094	96	4	,	,	PUNCT
ejpam-5094	96	5	p(λ	p(λ	NOUN
ejpam-5094	96	6	,	,	PUNCT
ejpam-5094	96	7	p)-closed	p)-close	VERB
ejpam-5094	96	8	,	,	PUNCT
ejpam-5094	96	9	α(λ	α(λ	PROPN
ejpam-5094	96	10	,	,	PUNCT
ejpam-5094	96	11	p)-closed	p)-close	VERB
ejpam-5094	96	12	)	)	PUNCT
ejpam-5094	96	13	sets	set	NOUN
ejpam-5094	96	14	in	in	ADP
ejpam-5094	96	15	a	a	DET
ejpam-5094	96	16	topological	topological	ADJ
ejpam-5094	96	17	space	space	NOUN
ejpam-5094	96	18	(	(	PUNCT
ejpam-5094	96	19	x	x	X
ejpam-5094	96	20	,	,	PUNCT
ejpam-5094	96	21	τ	τ	X
ejpam-5094	96	22	)	)	PUNCT
ejpam-5094	96	23	is	be	AUX
ejpam-5094	96	24	denoted	denote	VERB
ejpam-5094	96	25	by	by	ADP
ejpam-5094	96	26	s(λ	s(λ	PROPN
ejpam-5094	96	27	,	,	PUNCT
ejpam-5094	96	28	p)c(x	p)c(x	NOUN
ejpam-5094	96	29	,	,	PUNCT
ejpam-5094	96	30	τ	τ	X
ejpam-5094	96	31	)	)	PUNCT
ejpam-5094	96	32	(	(	PUNCT
ejpam-5094	96	33	resp	resp	NOUN
ejpam-5094	96	34	.	.	PUNCT
ejpam-5094	97	1	r(λ	r(λ	NOUN
ejpam-5094	97	2	,	,	PUNCT
ejpam-5094	97	3	p)c(x	p)c(x	NOUN
ejpam-5094	97	4	,	,	PUNCT
ejpam-5094	97	5	τ	τ	PROPN
ejpam-5094	97	6	)	)	PUNCT
ejpam-5094	97	7	,	,	PUNCT
ejpam-5094	97	8	p(λ	p(λ	NOUN
ejpam-5094	97	9	,	,	PUNCT
ejpam-5094	97	10	p)c(x	p)c(x	NOUN
ejpam-5094	97	11	,	,	PUNCT
ejpam-5094	97	12	τ	τ	PROPN
ejpam-5094	97	13	)	)	PUNCT
ejpam-5094	97	14	,	,	PUNCT
ejpam-5094	97	15	α(λ	α(λ	PROPN
ejpam-5094	97	16	,	,	PUNCT
ejpam-5094	97	17	p)c(x	p)c(x	PROPN
ejpam-5094	97	18	,	,	PUNCT
ejpam-5094	97	19	τ	τ	NOUN
ejpam-5094	97	20	)	)	PUNCT
ejpam-5094	97	21	)	)	PUNCT
ejpam-5094	97	22	.	.	PUNCT
ejpam-5094	98	1	the	the	DET
ejpam-5094	98	2	intersection	intersection	NOUN
ejpam-5094	98	3	of	of	ADP
ejpam-5094	98	4	all	all	DET
ejpam-5094	98	5	s(λ	s(λ	NOUN
ejpam-5094	98	6	,	,	PUNCT
ejpam-5094	98	7	p)-closed	p)-close	VERB
ejpam-5094	98	8	(	(	PUNCT
ejpam-5094	98	9	resp	resp	NOUN
ejpam-5094	98	10	.	.	PUNCT
ejpam-5094	99	1	p(λ	p(λ	NOUN
ejpam-5094	99	2	,	,	PUNCT
ejpam-5094	99	3	p)-closed	p)-close	VERB
ejpam-5094	99	4	,	,	PUNCT
ejpam-5094	99	5	α(λ	α(λ	PROPN
ejpam-5094	99	6	,	,	PUNCT
ejpam-5094	99	7	p)-closed	p)-close	VERB
ejpam-5094	99	8	)	)	PUNCT
ejpam-5094	99	9	sets	set	NOUN
ejpam-5094	99	10	of	of	ADP
ejpam-5094	99	11	x	x	PUNCT
ejpam-5094	99	12	containing	contain	VERB
ejpam-5094	99	13	a	a	PRON
ejpam-5094	99	14	is	be	AUX
ejpam-5094	99	15	called	call	VERB
ejpam-5094	99	16	the	the	DET
ejpam-5094	99	17	s(λ	s(λ	NOUN
ejpam-5094	99	18	,	,	PUNCT
ejpam-5094	99	19	p)-closure	p)-closure	X
ejpam-5094	99	20	(	(	PUNCT
ejpam-5094	99	21	resp	resp	NOUN
ejpam-5094	99	22	.	.	PUNCT
ejpam-5094	100	1	p(λ	p(λ	NOUN
ejpam-5094	100	2	,	,	PUNCT
ejpam-5094	100	3	p)-closure	p)-closure	NOUN
ejpam-5094	100	4	,	,	PUNCT
ejpam-5094	100	5	α(λ	α(λ	PROPN
ejpam-5094	100	6	,	,	PUNCT
ejpam-5094	100	7	p)-closure	p)-closure	NOUN
ejpam-5094	100	8	)	)	PUNCT
ejpam-5094	100	9	of	of	ADP
ejpam-5094	100	10	a	a	PRON
ejpam-5094	100	11	and	and	CCONJ
ejpam-5094	100	12	is	be	AUX
ejpam-5094	100	13	denoted	denote	VERB
ejpam-5094	100	14	by	by	ADP
ejpam-5094	100	15	as(λ	as(λ	NOUN
ejpam-5094	100	16	,	,	PUNCT
ejpam-5094	100	17	p	p	NOUN
ejpam-5094	100	18	)	)	PUNCT
ejpam-5094	100	19	(	(	PUNCT
ejpam-5094	100	20	resp	resp	NOUN
ejpam-5094	100	21	.	.	PUNCT
ejpam-5094	101	1	ap(λ	ap(λ	PROPN
ejpam-5094	101	2	,	,	PUNCT
ejpam-5094	101	3	p	p	NOUN
ejpam-5094	101	4	)	)	PUNCT
ejpam-5094	101	5	,	,	PUNCT
ejpam-5094	101	6	aα(λ	aα(λ	PROPN
ejpam-5094	101	7	,	,	PUNCT
ejpam-5094	101	8	p	p	NOUN
ejpam-5094	101	9	)	)	PUNCT
ejpam-5094	101	10	)	)	PUNCT
ejpam-5094	101	11	.	.	PUNCT
ejpam-5094	102	1	lemma	lemma	PROPN
ejpam-5094	102	2	2	2	NUM
ejpam-5094	102	3	.	.	X
ejpam-5094	103	1	for	for	ADP
ejpam-5094	103	2	subsets	subset	NOUN
ejpam-5094	103	3	a	a	PRON
ejpam-5094	103	4	and	and	CCONJ
ejpam-5094	103	5	b	b	NOUN
ejpam-5094	103	6	of	of	ADP
ejpam-5094	103	7	a	a	DET
ejpam-5094	103	8	topological	topological	ADJ
ejpam-5094	103	9	space	space	NOUN
ejpam-5094	103	10	(	(	PUNCT
ejpam-5094	103	11	x	x	X
ejpam-5094	103	12	,	,	PUNCT
ejpam-5094	103	13	τ	τ	PROPN
ejpam-5094	103	14	)	)	PUNCT
ejpam-5094	103	15	,	,	PUNCT
ejpam-5094	103	16	the	the	DET
ejpam-5094	103	17	following	follow	VERB
ejpam-5094	103	18	properties	property	NOUN
ejpam-5094	103	19	hold	hold	VERB
ejpam-5094	103	20	:	:	PUNCT
ejpam-5094	103	21	(	(	PUNCT
ejpam-5094	103	22	1	1	X
ejpam-5094	103	23	)	)	PUNCT
ejpam-5094	103	24	aα(λ	aα(λ	NOUN
ejpam-5094	103	25	,	,	PUNCT
ejpam-5094	103	26	p	p	NOUN
ejpam-5094	103	27	)	)	PUNCT
ejpam-5094	103	28	=	=	PUNCT
ejpam-5094	103	29	a	a	DET
ejpam-5094	103	30	∩	∩	NOUN
ejpam-5094	104	1	[	[	X
ejpam-5094	104	2	[	[	X
ejpam-5094	104	3	a(λ	a(λ	ADV
ejpam-5094	104	4	,	,	PUNCT
ejpam-5094	104	5	p	p	NOUN
ejpam-5094	104	6	)	)	PUNCT
ejpam-5094	104	7	]	]	PUNCT
ejpam-5094	104	8	(	(	PUNCT
ejpam-5094	104	9	λ	λ	X
ejpam-5094	104	10	,	,	PUNCT
ejpam-5094	104	11	p)](λ	p)](λ	ADJ
ejpam-5094	104	12	,	,	PUNCT
ejpam-5094	104	13	p	p	NOUN
ejpam-5094	104	14	)	)	PUNCT
ejpam-5094	104	15	;	;	PUNCT
ejpam-5094	104	16	(	(	PUNCT
ejpam-5094	104	17	2	2	X
ejpam-5094	104	18	)	)	PUNCT
ejpam-5094	104	19	as(λ	as(λ	NUM
ejpam-5094	104	20	,	,	PUNCT
ejpam-5094	104	21	p	p	NOUN
ejpam-5094	104	22	)	)	PUNCT
ejpam-5094	104	23	=	=	PUNCT
ejpam-5094	105	1	a	a	DET
ejpam-5094	105	2	∩	∩	NOUN
ejpam-5094	105	3	[	[	X
ejpam-5094	105	4	a(λ	a(λ	ADJ
ejpam-5094	105	5	,	,	PUNCT
ejpam-5094	105	6	p	p	NOUN
ejpam-5094	105	7	)	)	PUNCT
ejpam-5094	105	8	]	]	PUNCT
ejpam-5094	105	9	(	(	PUNCT
ejpam-5094	105	10	λ	λ	X
ejpam-5094	105	11	,	,	PUNCT
ejpam-5094	105	12	p	p	NOUN
ejpam-5094	105	13	)	)	PUNCT
ejpam-5094	105	14	;	;	PUNCT
ejpam-5094	105	15	(	(	PUNCT
ejpam-5094	105	16	3	3	X
ejpam-5094	105	17	)	)	PUNCT
ejpam-5094	105	18	ap(λ	ap(λ	NOUN
ejpam-5094	105	19	,	,	PUNCT
ejpam-5094	105	20	p	p	NOUN
ejpam-5094	105	21	)	)	PUNCT
ejpam-5094	105	22	=	=	PUNCT
ejpam-5094	105	23	a	a	DET
ejpam-5094	105	24	∩	∩	ADJ
ejpam-5094	105	25	[	[	X
ejpam-5094	105	26	a(λ	a(λ	ADJ
ejpam-5094	105	27	,	,	PUNCT
ejpam-5094	105	28	p)](λ	p)](λ	X
ejpam-5094	105	29	,	,	PUNCT
ejpam-5094	105	30	p	p	NOUN
ejpam-5094	105	31	)	)	PUNCT
ejpam-5094	105	32	.	.	PUNCT
ejpam-5094	106	1	a	a	DET
ejpam-5094	106	2	subset	subset	NOUN
ejpam-5094	106	3	a	a	PRON
ejpam-5094	106	4	of	of	ADP
ejpam-5094	106	5	a	a	DET
ejpam-5094	106	6	topological	topological	ADJ
ejpam-5094	106	7	space	space	NOUN
ejpam-5094	106	8	(	(	PUNCT
ejpam-5094	106	9	x	x	X
ejpam-5094	106	10	,	,	PUNCT
ejpam-5094	106	11	τ	τ	X
ejpam-5094	106	12	)	)	PUNCT
ejpam-5094	106	13	is	be	AUX
ejpam-5094	106	14	said	say	VERB
ejpam-5094	106	15	to	to	PART
ejpam-5094	106	16	be	be	AUX
ejpam-5094	106	17	b(λ	b(λ	NOUN
ejpam-5094	106	18	,	,	PUNCT
ejpam-5094	106	19	p)-open	p)-open	VERB
ejpam-5094	106	20	if	if	SCONJ
ejpam-5094	106	21	a	a	DET
ejpam-5094	106	22	⊆	⊆	NUM
ejpam-5094	106	23	[	[	X
ejpam-5094	106	24	a(λ	a(λ	ADV
ejpam-5094	106	25	,	,	PUNCT
ejpam-5094	106	26	p)](λ	p)](λ	X
ejpam-5094	106	27	,	,	PUNCT
ejpam-5094	106	28	p	p	NOUN
ejpam-5094	106	29	)	)	PUNCT
ejpam-5094	106	30	∪	∪	ADP
ejpam-5094	106	31	[	[	X
ejpam-5094	106	32	a(λ	a(λ	ADJ
ejpam-5094	106	33	,	,	PUNCT
ejpam-5094	106	34	p	p	NOUN
ejpam-5094	106	35	)	)	PUNCT
ejpam-5094	106	36	]	]	PUNCT
ejpam-5094	107	1	(	(	PUNCT
ejpam-5094	107	2	λ	λ	X
ejpam-5094	107	3	,	,	PUNCT
ejpam-5094	107	4	p	p	NOUN
ejpam-5094	107	5	)	)	PUNCT
ejpam-5094	107	6	.	.	PUNCT
ejpam-5094	108	1	the	the	DET
ejpam-5094	108	2	family	family	NOUN
ejpam-5094	108	3	of	of	ADP
ejpam-5094	108	4	all	all	DET
ejpam-5094	108	5	b(λ	b(λ	NOUN
ejpam-5094	108	6	,	,	PUNCT
ejpam-5094	108	7	p)-open	p)-open	VERB
ejpam-5094	108	8	sets	set	NOUN
ejpam-5094	108	9	in	in	ADP
ejpam-5094	108	10	a	a	DET
ejpam-5094	108	11	topological	topological	ADJ
ejpam-5094	108	12	space	space	NOUN
ejpam-5094	108	13	(	(	PUNCT
ejpam-5094	108	14	x	x	X
ejpam-5094	108	15	,	,	PUNCT
ejpam-5094	108	16	τ	τ	X
ejpam-5094	108	17	)	)	PUNCT
ejpam-5094	108	18	is	be	AUX
ejpam-5094	108	19	denoted	denote	VERB
ejpam-5094	108	20	by	by	ADP
ejpam-5094	108	21	b(λ	b(λ	PROPN
ejpam-5094	108	22	,	,	PUNCT
ejpam-5094	108	23	p)o(x	p)o(x	ADJ
ejpam-5094	108	24	,	,	PUNCT
ejpam-5094	108	25	τ	τ	PROPN
ejpam-5094	108	26	)	)	PUNCT
ejpam-5094	108	27	.	.	PUNCT
ejpam-5094	109	1	the	the	DET
ejpam-5094	109	2	union	union	NOUN
ejpam-5094	109	3	of	of	ADP
ejpam-5094	109	4	all	all	DET
ejpam-5094	109	5	b(λ	b(λ	NOUN
ejpam-5094	109	6	,	,	PUNCT
ejpam-5094	109	7	p)-open	p)-open	VERB
ejpam-5094	109	8	sets	set	VERB
ejpam-5094	109	9	ofx	ofx	NOUN
ejpam-5094	109	10	contained	contain	VERB
ejpam-5094	109	11	in	in	ADP
ejpam-5094	109	12	a	a	PRON
ejpam-5094	109	13	is	be	AUX
ejpam-5094	109	14	called	call	VERB
ejpam-5094	109	15	the	the	DET
ejpam-5094	109	16	b(λ	b(λ	NOUN
ejpam-5094	109	17	,	,	PUNCT
ejpam-5094	109	18	p)-interior	p)-interior	ADJ
ejpam-5094	109	19	of	of	ADP
ejpam-5094	109	20	a	a	PRON
ejpam-5094	109	21	and	and	CCONJ
ejpam-5094	109	22	is	be	AUX
ejpam-5094	109	23	denoted	denote	VERB
ejpam-5094	109	24	by	by	ADP
ejpam-5094	109	25	ab(λ	ab(λ	ADP
ejpam-5094	109	26	,	,	PUNCT
ejpam-5094	109	27	p	p	NOUN
ejpam-5094	109	28	)	)	PUNCT
ejpam-5094	109	29	.	.	PUNCT
ejpam-5094	110	1	the	the	DET
ejpam-5094	110	2	complement	complement	NOUN
ejpam-5094	110	3	of	of	ADP
ejpam-5094	110	4	a	a	DET
ejpam-5094	110	5	b(λ	b(λ	NOUN
ejpam-5094	110	6	,	,	PUNCT
ejpam-5094	110	7	p)-open	p)-open	VERB
ejpam-5094	110	8	set	set	VERB
ejpam-5094	110	9	is	be	AUX
ejpam-5094	110	10	called	call	VERB
ejpam-5094	110	11	b(λ	b(λ	NOUN
ejpam-5094	110	12	,	,	PUNCT
ejpam-5094	110	13	p)-closed	p)-close	VERB
ejpam-5094	110	14	.	.	PUNCT
ejpam-5094	111	1	the	the	DET
ejpam-5094	111	2	family	family	NOUN
ejpam-5094	111	3	of	of	ADP
ejpam-5094	111	4	all	all	DET
ejpam-5094	111	5	b(λ	b(λ	PROPN
ejpam-5094	111	6	,	,	PUNCT
ejpam-5094	111	7	p)-closed	p)-close	VERB
ejpam-5094	111	8	sets	set	NOUN
ejpam-5094	111	9	in	in	ADP
ejpam-5094	111	10	a	a	DET
ejpam-5094	111	11	topological	topological	ADJ
ejpam-5094	111	12	space	space	NOUN
ejpam-5094	111	13	(	(	PUNCT
ejpam-5094	111	14	x	x	X
ejpam-5094	111	15	,	,	PUNCT
ejpam-5094	111	16	τ	τ	X
ejpam-5094	111	17	)	)	PUNCT
ejpam-5094	111	18	is	be	AUX
ejpam-5094	111	19	denoted	denote	VERB
ejpam-5094	111	20	by	by	ADP
ejpam-5094	111	21	b(λ	b(λ	PROPN
ejpam-5094	111	22	,	,	PUNCT
ejpam-5094	111	23	p)c(x	p)c(x	PROPN
ejpam-5094	111	24	,	,	PUNCT
ejpam-5094	111	25	τ	τ	PROPN
ejpam-5094	111	26	)	)	PUNCT
ejpam-5094	111	27	.	.	PUNCT
ejpam-5094	112	1	the	the	DET
ejpam-5094	112	2	intersection	intersection	NOUN
ejpam-5094	112	3	of	of	ADP
ejpam-5094	112	4	all	all	DET
ejpam-5094	112	5	b(λ	b(λ	NOUN
ejpam-5094	112	6	,	,	PUNCT
ejpam-5094	112	7	p)-closed	p)-close	VERB
ejpam-5094	112	8	sets	set	NOUN
ejpam-5094	112	9	of	of	ADP
ejpam-5094	112	10	x	x	PUNCT
ejpam-5094	112	11	containing	contain	VERB
ejpam-5094	112	12	a	a	PRON
ejpam-5094	112	13	is	be	AUX
ejpam-5094	112	14	called	call	VERB
ejpam-5094	112	15	the	the	DET
ejpam-5094	112	16	b(λ	b(λ	NOUN
ejpam-5094	112	17	,	,	PUNCT
ejpam-5094	112	18	p)-closure	p)-closure	NOUN
ejpam-5094	112	19	of	of	ADP
ejpam-5094	112	20	a	a	PRON
ejpam-5094	112	21	and	and	CCONJ
ejpam-5094	112	22	is	be	AUX
ejpam-5094	112	23	denoted	denote	VERB
ejpam-5094	112	24	by	by	ADP
ejpam-5094	112	25	ab(λ	ab(λ	ADP
ejpam-5094	112	26	,	,	PUNCT
ejpam-5094	112	27	p	p	NOUN
ejpam-5094	112	28	)	)	PUNCT
ejpam-5094	112	29	.	.	PUNCT
ejpam-5094	113	1	lemma	lemma	PROPN
ejpam-5094	113	2	3	3	X
ejpam-5094	113	3	.	.	X
ejpam-5094	114	1	for	for	ADP
ejpam-5094	114	2	subsets	subset	NOUN
ejpam-5094	114	3	a	a	PRON
ejpam-5094	114	4	and	and	CCONJ
ejpam-5094	114	5	b	b	NOUN
ejpam-5094	114	6	of	of	ADP
ejpam-5094	114	7	a	a	DET
ejpam-5094	114	8	topological	topological	ADJ
ejpam-5094	114	9	space	space	NOUN
ejpam-5094	114	10	(	(	PUNCT
ejpam-5094	114	11	x	x	X
ejpam-5094	114	12	,	,	PUNCT
ejpam-5094	114	13	τ	τ	PROPN
ejpam-5094	114	14	)	)	PUNCT
ejpam-5094	114	15	,	,	PUNCT
ejpam-5094	114	16	the	the	DET
ejpam-5094	114	17	following	follow	VERB
ejpam-5094	114	18	properties	property	NOUN
ejpam-5094	114	19	hold	hold	VERB
ejpam-5094	114	20	:	:	PUNCT
ejpam-5094	114	21	(	(	PUNCT
ejpam-5094	114	22	1	1	NUM
ejpam-5094	114	23	)	)	PUNCT
ejpam-5094	114	24	ab(λ	ab(λ	NOUN
ejpam-5094	114	25	,	,	PUNCT
ejpam-5094	114	26	p	p	NOUN
ejpam-5094	114	27	)	)	PUNCT
ejpam-5094	114	28	=	=	SYM
ejpam-5094	114	29	as(λ	as(λ	NOUN
ejpam-5094	114	30	,	,	PUNCT
ejpam-5094	114	31	p	p	NOUN
ejpam-5094	114	32	)	)	PUNCT
ejpam-5094	114	33	∪ap(λ	∪ap(λ	PROPN
ejpam-5094	114	34	,	,	PUNCT
ejpam-5094	114	35	p	p	NOUN
ejpam-5094	114	36	)	)	PUNCT
ejpam-5094	114	37	;	;	PUNCT
ejpam-5094	115	1	c.	c.	PROPN
ejpam-5094	115	2	boonpok	boonpok	PROPN
ejpam-5094	115	3	,	,	PUNCT
ejpam-5094	115	4	j.	j.	PROPN
ejpam-5094	115	5	khampakdee	khampakdee	PROPN
ejpam-5094	115	6	/	/	PUNCT
ejpam-5094	115	7	eur	eur	PROPN
ejpam-5094	115	8	.	.	PUNCT
ejpam-5094	116	1	j.	j.	PROPN
ejpam-5094	116	2	pure	pure	PROPN
ejpam-5094	116	3	appl	appl	PROPN
ejpam-5094	116	4	.	.	PROPN
ejpam-5094	116	5	math	math	PROPN
ejpam-5094	116	6	,	,	PUNCT
ejpam-5094	116	7	17	17	NUM
ejpam-5094	116	8	(	(	PUNCT
ejpam-5094	116	9	2	2	NUM
ejpam-5094	116	10	)	)	PUNCT
ejpam-5094	116	11	(	(	PUNCT
ejpam-5094	116	12	2024	2024	NUM
ejpam-5094	116	13	)	)	PUNCT
ejpam-5094	116	14	,	,	PUNCT
ejpam-5094	116	15	582	582	NUM
ejpam-5094	116	16	-	-	SYM
ejpam-5094	116	17	590	590	NUM
ejpam-5094	116	18	585	585	NUM
ejpam-5094	116	19	(	(	PUNCT
ejpam-5094	116	20	2	2	NUM
ejpam-5094	116	21	)	)	PUNCT
ejpam-5094	116	22	ab(λ	ab(λ	NOUN
ejpam-5094	116	23	,	,	PUNCT
ejpam-5094	116	24	p	p	NOUN
ejpam-5094	116	25	)	)	PUNCT
ejpam-5094	116	26	=	=	SYM
ejpam-5094	116	27	as(λ	as(λ	NOUN
ejpam-5094	116	28	,	,	PUNCT
ejpam-5094	116	29	p	p	NOUN
ejpam-5094	116	30	)	)	PUNCT
ejpam-5094	116	31	∩ap(λ	∩ap(λ	PROPN
ejpam-5094	116	32	,	,	PUNCT
ejpam-5094	116	33	p	p	NOUN
ejpam-5094	116	34	)	)	PUNCT
ejpam-5094	116	35	;	;	PUNCT
ejpam-5094	116	36	(	(	PUNCT
ejpam-5094	116	37	3	3	X
ejpam-5094	116	38	)	)	PUNCT
ejpam-5094	117	1	[	[	X
ejpam-5094	117	2	x	x	X
ejpam-5094	117	3	−a]b(λ	−a]b(λ	PROPN
ejpam-5094	117	4	,	,	PUNCT
ejpam-5094	117	5	p	p	NOUN
ejpam-5094	117	6	)	)	PUNCT
ejpam-5094	117	7	=	=	PUNCT
ejpam-5094	117	8	x	x	SYM
ejpam-5094	117	9	−ab(λ	−ab(λ	NOUN
ejpam-5094	117	10	,	,	PUNCT
ejpam-5094	117	11	p	p	NOUN
ejpam-5094	117	12	)	)	PUNCT
ejpam-5094	117	13	;	;	PUNCT
ejpam-5094	117	14	(	(	PUNCT
ejpam-5094	117	15	4	4	X
ejpam-5094	117	16	)	)	PUNCT
ejpam-5094	117	17	x	x	SYM
ejpam-5094	117	18	∈	∈	PROPN
ejpam-5094	117	19	ab(λ	ab(λ	NOUN
ejpam-5094	117	20	,	,	PUNCT
ejpam-5094	117	21	p	p	NOUN
ejpam-5094	117	22	)	)	PUNCT
ejpam-5094	117	23	if	if	SCONJ
ejpam-5094	117	24	and	and	CCONJ
ejpam-5094	117	25	only	only	ADV
ejpam-5094	117	26	if	if	SCONJ
ejpam-5094	117	27	a	a	DET
ejpam-5094	117	28	∩	∩	ADJ
ejpam-5094	117	29	u	u	ADJ
ejpam-5094	117	30	̸=	̸=	PROPN
ejpam-5094	117	31	∅	∅	NOUN
ejpam-5094	117	32	for	for	ADP
ejpam-5094	117	33	every	every	DET
ejpam-5094	117	34	u	u	PROPN
ejpam-5094	117	35	∈	∈	PROPN
ejpam-5094	117	36	b(λ	b(λ	NOUN
ejpam-5094	117	37	,	,	PUNCT
ejpam-5094	117	38	p)o(x	p)o(x	ADJ
ejpam-5094	117	39	,	,	PUNCT
ejpam-5094	117	40	τ	τ	X
ejpam-5094	117	41	)	)	PUNCT
ejpam-5094	117	42	containing	contain	VERB
ejpam-5094	117	43	x	x	PRON
ejpam-5094	117	44	;	;	PUNCT
ejpam-5094	117	45	(	(	PUNCT
ejpam-5094	117	46	5	5	X
ejpam-5094	117	47	)	)	PUNCT
ejpam-5094	117	48	a	a	DET
ejpam-5094	117	49	∈	∈	PROPN
ejpam-5094	117	50	b(λ	b(λ	NOUN
ejpam-5094	117	51	,	,	PUNCT
ejpam-5094	117	52	p)c(x	p)c(x	PROPN
ejpam-5094	117	53	,	,	PUNCT
ejpam-5094	117	54	τ	τ	X
ejpam-5094	117	55	)	)	PUNCT
ejpam-5094	117	56	if	if	SCONJ
ejpam-5094	117	57	and	and	CCONJ
ejpam-5094	117	58	only	only	ADV
ejpam-5094	117	59	if	if	SCONJ
ejpam-5094	117	60	a	a	PRON
ejpam-5094	117	61	=	=	PUNCT
ejpam-5094	117	62	ab(λ	ab(λ	NOUN
ejpam-5094	117	63	,	,	PUNCT
ejpam-5094	117	64	p	p	NOUN
ejpam-5094	117	65	)	)	PUNCT
ejpam-5094	117	66	;	;	PUNCT
ejpam-5094	117	67	(	(	PUNCT
ejpam-5094	117	68	6	6	X
ejpam-5094	117	69	)	)	PUNCT
ejpam-5094	118	1	[	[	X
ejpam-5094	118	2	ab(λ	ab(λ	NOUN
ejpam-5094	118	3	,	,	PUNCT
ejpam-5094	118	4	p)]p(λ	p)]p(λ	NOUN
ejpam-5094	118	5	,	,	PUNCT
ejpam-5094	118	6	p	p	NOUN
ejpam-5094	118	7	)	)	PUNCT
ejpam-5094	118	8	=	=	NOUN
ejpam-5094	119	1	[	[	X
ejpam-5094	119	2	ap(λ	ap(λ	ADP
ejpam-5094	119	3	,	,	PUNCT
ejpam-5094	119	4	p	p	NOUN
ejpam-5094	119	5	)	)	PUNCT
ejpam-5094	119	6	]	]	PUNCT
ejpam-5094	120	1	b(λ	b(λ	PROPN
ejpam-5094	120	2	,	,	PUNCT
ejpam-5094	120	3	p	p	NOUN
ejpam-5094	120	4	)	)	PUNCT
ejpam-5094	120	5	.	.	PUNCT
ejpam-5094	121	1	let	let	VERB
ejpam-5094	121	2	a	a	DET
ejpam-5094	121	3	be	be	AUX
ejpam-5094	121	4	a	a	DET
ejpam-5094	121	5	subset	subset	NOUN
ejpam-5094	121	6	of	of	ADP
ejpam-5094	121	7	a	a	DET
ejpam-5094	121	8	topological	topological	ADJ
ejpam-5094	121	9	space	space	NOUN
ejpam-5094	121	10	(	(	PUNCT
ejpam-5094	121	11	x	x	X
ejpam-5094	121	12	,	,	PUNCT
ejpam-5094	121	13	τ	τ	PROPN
ejpam-5094	121	14	)	)	PUNCT
ejpam-5094	121	15	.	.	PUNCT
ejpam-5094	122	1	a	a	DET
ejpam-5094	122	2	point	point	NOUN
ejpam-5094	122	3	x	x	X
ejpam-5094	122	4	∈	∈	NOUN
ejpam-5094	122	5	x	x	PUNCT
ejpam-5094	122	6	is	be	AUX
ejpam-5094	122	7	called	call	VERB
ejpam-5094	122	8	a	a	DET
ejpam-5094	122	9	θb(λ	θb(λ	ADJ
ejpam-5094	122	10	,	,	PUNCT
ejpam-5094	122	11	p)cluster	p)cluster	NOUN
ejpam-5094	122	12	point	point	NOUN
ejpam-5094	122	13	of	of	ADP
ejpam-5094	122	14	a	a	PRON
ejpam-5094	122	15	if	if	SCONJ
ejpam-5094	122	16	a	a	DET
ejpam-5094	122	17	∩	∩	ADJ
ejpam-5094	122	18	u	u	NOUN
ejpam-5094	122	19	b(λ	b(λ	PROPN
ejpam-5094	122	20	,	,	PUNCT
ejpam-5094	122	21	p	p	NOUN
ejpam-5094	122	22	)	)	PUNCT
ejpam-5094	122	23	̸=	̸=	PROPN
ejpam-5094	122	24	∅	∅	NOUN
ejpam-5094	122	25	for	for	ADP
ejpam-5094	122	26	every	every	DET
ejpam-5094	122	27	b(λ	b(λ	NOUN
ejpam-5094	122	28	,	,	PUNCT
ejpam-5094	122	29	p)-open	p)-open	VERB
ejpam-5094	122	30	set	set	VERB
ejpam-5094	122	31	u	u	NOUN
ejpam-5094	122	32	of	of	ADP
ejpam-5094	122	33	x	x	SYM
ejpam-5094	122	34	containing	contain	VERB
ejpam-5094	122	35	x.	x.	NOUN
ejpam-5094	122	36	the	the	DET
ejpam-5094	122	37	set	set	NOUN
ejpam-5094	122	38	of	of	ADP
ejpam-5094	122	39	all	all	DET
ejpam-5094	122	40	θb(λ	θb(λ	NUM
ejpam-5094	122	41	,	,	PUNCT
ejpam-5094	122	42	p)-cluster	p)-cluster	VERB
ejpam-5094	122	43	points	point	NOUN
ejpam-5094	122	44	of	of	ADP
ejpam-5094	122	45	a	a	PRON
ejpam-5094	122	46	is	be	AUX
ejpam-5094	122	47	called	call	VERB
ejpam-5094	122	48	the	the	DET
ejpam-5094	122	49	θb(λ	θb(λ	NUM
ejpam-5094	122	50	,	,	PUNCT
ejpam-5094	122	51	p)-closure	p)-closure	NOUN
ejpam-5094	122	52	of	of	ADP
ejpam-5094	122	53	a	a	PRON
ejpam-5094	122	54	and	and	CCONJ
ejpam-5094	122	55	is	be	AUX
ejpam-5094	122	56	denoted	denote	VERB
ejpam-5094	122	57	by	by	ADP
ejpam-5094	122	58	aθb(λ	aθb(λ	PROPN
ejpam-5094	122	59	,	,	PUNCT
ejpam-5094	122	60	p	p	NOUN
ejpam-5094	122	61	)	)	PUNCT
ejpam-5094	122	62	.	.	PUNCT
ejpam-5094	123	1	if	if	SCONJ
ejpam-5094	123	2	a	a	DET
ejpam-5094	123	3	=	=	SYM
ejpam-5094	123	4	aθb(λ	aθb(λ	PROPN
ejpam-5094	123	5	,	,	PUNCT
ejpam-5094	123	6	p	p	NOUN
ejpam-5094	123	7	)	)	PUNCT
ejpam-5094	123	8	,	,	PUNCT
ejpam-5094	123	9	then	then	ADV
ejpam-5094	123	10	a	a	PRON
ejpam-5094	123	11	is	be	AUX
ejpam-5094	123	12	called	call	VERB
ejpam-5094	123	13	θb(λ	θb(λ	VERB
ejpam-5094	123	14	,	,	PUNCT
ejpam-5094	123	15	p)-closed	p)-close	VERB
ejpam-5094	123	16	.	.	PUNCT
ejpam-5094	124	1	the	the	DET
ejpam-5094	124	2	complement	complement	NOUN
ejpam-5094	124	3	of	of	ADP
ejpam-5094	124	4	a	a	DET
ejpam-5094	124	5	θb(λ	θb(λ	NUM
ejpam-5094	124	6	,	,	PUNCT
ejpam-5094	124	7	p)closed	p)close	VERB
ejpam-5094	124	8	set	set	NOUN
ejpam-5094	124	9	is	be	AUX
ejpam-5094	124	10	called	call	VERB
ejpam-5094	124	11	θb(λ	θb(λ	VERB
ejpam-5094	124	12	,	,	PUNCT
ejpam-5094	124	13	p)-open	p)-open	PROPN
ejpam-5094	124	14	.	.	PUNCT
ejpam-5094	125	1	the	the	DET
ejpam-5094	125	2	θb(λ	θb(λ	NUM
ejpam-5094	125	3	,	,	PUNCT
ejpam-5094	125	4	p)-interior	p)-interior	ADJ
ejpam-5094	125	5	of	of	ADP
ejpam-5094	125	6	a	a	PRON
ejpam-5094	125	7	is	be	AUX
ejpam-5094	125	8	defined	define	VERB
ejpam-5094	125	9	by	by	ADP
ejpam-5094	125	10	the	the	DET
ejpam-5094	125	11	union	union	NOUN
ejpam-5094	125	12	of	of	ADP
ejpam-5094	125	13	all	all	PRON
ejpam-5094	125	14	θb(λ	θb(λ	NUM
ejpam-5094	125	15	,	,	PUNCT
ejpam-5094	125	16	p)-open	p)-open	VERB
ejpam-5094	125	17	sets	set	NOUN
ejpam-5094	125	18	of	of	ADP
ejpam-5094	125	19	x	x	PUNCT
ejpam-5094	125	20	contained	contain	VERB
ejpam-5094	125	21	in	in	ADP
ejpam-5094	125	22	a	a	PRON
ejpam-5094	125	23	and	and	CCONJ
ejpam-5094	125	24	is	be	AUX
ejpam-5094	125	25	denoted	denote	VERB
ejpam-5094	125	26	by	by	ADP
ejpam-5094	125	27	aθb(λ	aθb(λ	PROPN
ejpam-5094	125	28	,	,	PUNCT
ejpam-5094	125	29	p	p	NOUN
ejpam-5094	125	30	)	)	PUNCT
ejpam-5094	125	31	.	.	PUNCT
ejpam-5094	126	1	the	the	DET
ejpam-5094	126	2	family	family	NOUN
ejpam-5094	126	3	of	of	ADP
ejpam-5094	126	4	all	all	PRON
ejpam-5094	126	5	θb(λ	θb(λ	NUM
ejpam-5094	126	6	,	,	PUNCT
ejpam-5094	126	7	p)-open	p)-open	VERB
ejpam-5094	126	8	(	(	PUNCT
ejpam-5094	126	9	resp	resp	NOUN
ejpam-5094	126	10	.	.	PUNCT
ejpam-5094	127	1	θb(λ	θb(λ	NUM
ejpam-5094	127	2	,	,	PUNCT
ejpam-5094	127	3	p)-closed	p)-close	VERB
ejpam-5094	127	4	)	)	PUNCT
ejpam-5094	128	1	sets	set	NOUN
ejpam-5094	128	2	in	in	ADP
ejpam-5094	128	3	a	a	DET
ejpam-5094	128	4	topological	topological	ADJ
ejpam-5094	128	5	space	space	NOUN
ejpam-5094	128	6	(	(	PUNCT
ejpam-5094	128	7	x	x	X
ejpam-5094	128	8	,	,	PUNCT
ejpam-5094	128	9	τ	τ	X
ejpam-5094	128	10	)	)	PUNCT
ejpam-5094	128	11	is	be	AUX
ejpam-5094	128	12	denoted	denote	VERB
ejpam-5094	128	13	by	by	ADP
ejpam-5094	128	14	θb(λ	θb(λ	PROPN
ejpam-5094	128	15	,	,	PUNCT
ejpam-5094	128	16	p)o(x	p)o(x	ADJ
ejpam-5094	128	17	,	,	PUNCT
ejpam-5094	128	18	τ	τ	PROPN
ejpam-5094	128	19	)	)	PUNCT
ejpam-5094	128	20	(	(	PUNCT
ejpam-5094	128	21	resp	resp	NOUN
ejpam-5094	128	22	.	.	PUNCT
ejpam-5094	129	1	θb(λ	θb(λ	NUM
ejpam-5094	129	2	,	,	PUNCT
ejpam-5094	129	3	p)c(x	p)c(x	PROPN
ejpam-5094	129	4	,	,	PUNCT
ejpam-5094	129	5	τ	τ	NOUN
ejpam-5094	129	6	)	)	PUNCT
ejpam-5094	129	7	)	)	PUNCT
ejpam-5094	129	8	.	.	PUNCT
ejpam-5094	130	1	lemma	lemma	PROPN
ejpam-5094	130	2	4	4	NUM
ejpam-5094	130	3	.	.	X
ejpam-5094	131	1	for	for	ADP
ejpam-5094	131	2	subsets	subset	NOUN
ejpam-5094	131	3	a	a	PRON
ejpam-5094	131	4	and	and	CCONJ
ejpam-5094	131	5	cγ(γ	cγ(γ	CCONJ
ejpam-5094	131	6	∈	∈	PROPN
ejpam-5094	131	7	γ	γ	PROPN
ejpam-5094	131	8	)	)	PUNCT
ejpam-5094	131	9	of	of	ADP
ejpam-5094	131	10	a	a	DET
ejpam-5094	131	11	topological	topological	ADJ
ejpam-5094	131	12	space	space	NOUN
ejpam-5094	131	13	(	(	PUNCT
ejpam-5094	131	14	x	x	X
ejpam-5094	131	15	,	,	PUNCT
ejpam-5094	131	16	τ	τ	PROPN
ejpam-5094	131	17	)	)	PUNCT
ejpam-5094	131	18	,	,	PUNCT
ejpam-5094	131	19	the	the	DET
ejpam-5094	131	20	following	follow	VERB
ejpam-5094	131	21	properties	property	NOUN
ejpam-5094	131	22	hold	hold	VERB
ejpam-5094	131	23	:	:	PUNCT
ejpam-5094	131	24	(	(	PUNCT
ejpam-5094	131	25	1	1	X
ejpam-5094	131	26	)	)	PUNCT
ejpam-5094	131	27	if	if	SCONJ
ejpam-5094	131	28	cγ	cγ	PROPN
ejpam-5094	131	29	∈	∈	PROPN
ejpam-5094	131	30	θb(λ	θb(λ	NUM
ejpam-5094	131	31	,	,	PUNCT
ejpam-5094	131	32	p)o(x	p)o(x	ADJ
ejpam-5094	131	33	,	,	PUNCT
ejpam-5094	131	34	τ	τ	PROPN
ejpam-5094	131	35	)	)	PUNCT
ejpam-5094	131	36	for	for	ADP
ejpam-5094	131	37	each	each	DET
ejpam-5094	131	38	γ	γ	PROPN
ejpam-5094	131	39	∈	∈	PROPN
ejpam-5094	131	40	γ	γ	X
ejpam-5094	131	41	,	,	PUNCT
ejpam-5094	131	42	then	then	ADV
ejpam-5094	131	43	∪γ∈γcγ	∪γ∈γcγ	PROPN
ejpam-5094	131	44	∈	∈	PROPN
ejpam-5094	131	45	θb(λ	θb(λ	NUM
ejpam-5094	131	46	,	,	PUNCT
ejpam-5094	131	47	p)o(x	p)o(x	ADJ
ejpam-5094	131	48	,	,	PUNCT
ejpam-5094	131	49	τ	τ	PROPN
ejpam-5094	131	50	)	)	PUNCT
ejpam-5094	131	51	.	.	PUNCT
ejpam-5094	132	1	(	(	PUNCT
ejpam-5094	132	2	2	2	X
ejpam-5094	132	3	)	)	PUNCT
ejpam-5094	132	4	if	if	SCONJ
ejpam-5094	132	5	a	a	DET
ejpam-5094	132	6	∈	∈	PROPN
ejpam-5094	132	7	b(λ	b(λ	NOUN
ejpam-5094	132	8	,	,	PUNCT
ejpam-5094	132	9	p)c(x	p)c(x	PROPN
ejpam-5094	132	10	,	,	PUNCT
ejpam-5094	132	11	τ	τ	PROPN
ejpam-5094	132	12	)	)	PUNCT
ejpam-5094	132	13	,	,	PUNCT
ejpam-5094	132	14	then	then	ADV
ejpam-5094	132	15	ab(λ	ab(λ	ADP
ejpam-5094	132	16	,	,	PUNCT
ejpam-5094	132	17	p	p	NOUN
ejpam-5094	132	18	)	)	PUNCT
ejpam-5094	132	19	=	=	SYM
ejpam-5094	132	20	aθb(λ	aθb(λ	PROPN
ejpam-5094	132	21	,	,	PUNCT
ejpam-5094	132	22	p	p	NOUN
ejpam-5094	132	23	)	)	PUNCT
ejpam-5094	132	24	.	.	PUNCT
ejpam-5094	133	1	(	(	PUNCT
ejpam-5094	133	2	3	3	X
ejpam-5094	133	3	)	)	PUNCT
ejpam-5094	133	4	aθb(λ	aθb(λ	PROPN
ejpam-5094	133	5	,	,	PUNCT
ejpam-5094	133	6	p	p	NOUN
ejpam-5094	133	7	)	)	PUNCT
ejpam-5094	133	8	∈	∈	PROPN
ejpam-5094	133	9	θb(λ	θb(λ	NUM
ejpam-5094	133	10	,	,	PUNCT
ejpam-5094	133	11	p)c(x	p)c(x	PROPN
ejpam-5094	133	12	,	,	PUNCT
ejpam-5094	133	13	τ	τ	PROPN
ejpam-5094	133	14	)	)	PUNCT
ejpam-5094	133	15	.	.	PUNCT
ejpam-5094	134	1	3	3	X
ejpam-5094	134	2	.	.	X
ejpam-5094	134	3	weakly	weakly	ADJ
ejpam-5094	134	4	θb(λ	θb(λ	NUM
ejpam-5094	134	5	,	,	PUNCT
ejpam-5094	134	6	p)-open	p)-open	VERB
ejpam-5094	134	7	functions	function	NOUN
ejpam-5094	134	8	in	in	ADP
ejpam-5094	134	9	this	this	DET
ejpam-5094	134	10	section	section	NOUN
ejpam-5094	134	11	,	,	PUNCT
ejpam-5094	134	12	we	we	PRON
ejpam-5094	134	13	introduce	introduce	VERB
ejpam-5094	134	14	the	the	DET
ejpam-5094	134	15	concept	concept	NOUN
ejpam-5094	134	16	of	of	ADP
ejpam-5094	134	17	weakly	weakly	ADJ
ejpam-5094	134	18	θb(λ	θb(λ	NUM
ejpam-5094	134	19	,	,	PUNCT
ejpam-5094	134	20	p)-open	p)-open	ADJ
ejpam-5094	134	21	functions	function	NOUN
ejpam-5094	134	22	.	.	PUNCT
ejpam-5094	135	1	moreover	moreover	ADV
ejpam-5094	135	2	,	,	PUNCT
ejpam-5094	135	3	some	some	DET
ejpam-5094	135	4	characterizations	characterization	NOUN
ejpam-5094	135	5	of	of	ADP
ejpam-5094	135	6	weakly	weakly	ADJ
ejpam-5094	135	7	θb(λ	θb(λ	NUM
ejpam-5094	135	8	,	,	PUNCT
ejpam-5094	135	9	p)-open	p)-open	VERB
ejpam-5094	135	10	functions	function	NOUN
ejpam-5094	135	11	are	be	AUX
ejpam-5094	135	12	discussed	discuss	VERB
ejpam-5094	135	13	.	.	PUNCT
ejpam-5094	136	1	definition	definition	NOUN
ejpam-5094	136	2	1	1	NUM
ejpam-5094	136	3	.	.	PUNCT
ejpam-5094	137	1	a	a	DET
ejpam-5094	137	2	function	function	NOUN
ejpam-5094	137	3	f	f	NOUN
ejpam-5094	137	4	:	:	PUNCT
ejpam-5094	137	5	(	(	PUNCT
ejpam-5094	137	6	x	x	X
ejpam-5094	137	7	,	,	PUNCT
ejpam-5094	137	8	τ	τ	X
ejpam-5094	137	9	)	)	PUNCT
ejpam-5094	137	10	→	→	SYM
ejpam-5094	137	11	(	(	PUNCT
ejpam-5094	137	12	y	y	PROPN
ejpam-5094	137	13	,	,	PUNCT
ejpam-5094	137	14	σ	σ	PROPN
ejpam-5094	137	15	)	)	PUNCT
ejpam-5094	137	16	is	be	AUX
ejpam-5094	137	17	said	say	VERB
ejpam-5094	137	18	to	to	PART
ejpam-5094	137	19	be	be	AUX
ejpam-5094	137	20	weakly	weakly	ADJ
ejpam-5094	137	21	θb(λ	θb(λ	NUM
ejpam-5094	137	22	,	,	PUNCT
ejpam-5094	137	23	p)-open	p)-open	VERB
ejpam-5094	137	24	if	if	SCONJ
ejpam-5094	137	25	f(u	f(u	PROPN
ejpam-5094	137	26	)	)	PUNCT
ejpam-5094	137	27	⊆	⊆	NUM
ejpam-5094	138	1	[	[	X
ejpam-5094	138	2	f(u	f(u	PROPN
ejpam-5094	138	3	(	(	PUNCT
ejpam-5094	138	4	λ	λ	PROPN
ejpam-5094	138	5	,	,	PUNCT
ejpam-5094	138	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	138	7	,	,	PUNCT
ejpam-5094	138	8	p	p	NOUN
ejpam-5094	138	9	)	)	PUNCT
ejpam-5094	138	10	for	for	ADP
ejpam-5094	138	11	each	each	DET
ejpam-5094	138	12	(	(	PUNCT
ejpam-5094	138	13	λ	λ	PROPN
ejpam-5094	138	14	,	,	PUNCT
ejpam-5094	138	15	p)-open	p)-open	VERB
ejpam-5094	138	16	set	set	VERB
ejpam-5094	138	17	u	u	NOUN
ejpam-5094	138	18	of	of	ADP
ejpam-5094	138	19	x.	x.	PROPN
ejpam-5094	138	20	theorem	theorem	VERB
ejpam-5094	138	21	1	1	NUM
ejpam-5094	138	22	.	.	PUNCT
ejpam-5094	138	23	for	for	ADP
ejpam-5094	138	24	a	a	DET
ejpam-5094	138	25	function	function	NOUN
ejpam-5094	138	26	f	f	NOUN
ejpam-5094	138	27	:	:	PUNCT
ejpam-5094	138	28	(	(	PUNCT
ejpam-5094	138	29	x	x	X
ejpam-5094	138	30	,	,	PUNCT
ejpam-5094	138	31	τ	τ	X
ejpam-5094	138	32	)	)	PUNCT
ejpam-5094	138	33	→	→	SYM
ejpam-5094	138	34	(	(	PUNCT
ejpam-5094	138	35	y	y	PROPN
ejpam-5094	138	36	,	,	PUNCT
ejpam-5094	138	37	σ	σ	PROPN
ejpam-5094	138	38	)	)	PUNCT
ejpam-5094	138	39	,	,	PUNCT
ejpam-5094	138	40	the	the	DET
ejpam-5094	138	41	following	follow	VERB
ejpam-5094	138	42	properties	property	NOUN
ejpam-5094	138	43	are	be	AUX
ejpam-5094	138	44	equivalent	equivalent	ADJ
ejpam-5094	138	45	:	:	PUNCT
ejpam-5094	138	46	(	(	PUNCT
ejpam-5094	138	47	1	1	X
ejpam-5094	138	48	)	)	PUNCT
ejpam-5094	138	49	f	f	PROPN
ejpam-5094	138	50	is	be	AUX
ejpam-5094	138	51	weakly	weakly	ADJ
ejpam-5094	138	52	θb(λ	θb(λ	NOUN
ejpam-5094	138	53	,	,	PUNCT
ejpam-5094	138	54	p)-open	p)-open	VERB
ejpam-5094	138	55	;	;	PUNCT
ejpam-5094	138	56	(	(	PUNCT
ejpam-5094	138	57	2	2	X
ejpam-5094	138	58	)	)	PUNCT
ejpam-5094	138	59	f(aθ(λ	f(aθ(λ	PROPN
ejpam-5094	138	60	,	,	PUNCT
ejpam-5094	138	61	p	p	NOUN
ejpam-5094	138	62	)	)	PUNCT
ejpam-5094	138	63	)	)	PUNCT
ejpam-5094	139	1	⊆	⊆	NUM
ejpam-5094	139	2	[	[	X
ejpam-5094	139	3	f(a)]θb(λ	f(a)]θb(λ	PROPN
ejpam-5094	139	4	,	,	PUNCT
ejpam-5094	139	5	p	p	NOUN
ejpam-5094	139	6	)	)	PUNCT
ejpam-5094	139	7	for	for	ADP
ejpam-5094	139	8	every	every	DET
ejpam-5094	139	9	subset	subset	NOUN
ejpam-5094	139	10	a	a	PRON
ejpam-5094	139	11	of	of	ADP
ejpam-5094	139	12	x	x	PRON
ejpam-5094	139	13	;	;	PUNCT
ejpam-5094	139	14	(	(	PUNCT
ejpam-5094	139	15	3	3	X
ejpam-5094	139	16	)	)	PUNCT
ejpam-5094	140	1	[	[	X
ejpam-5094	140	2	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-5094	140	3	,	,	PUNCT
ejpam-5094	140	4	p	p	NOUN
ejpam-5094	140	5	)	)	PUNCT
ejpam-5094	140	6	⊆	⊆	NUM
ejpam-5094	140	7	f−1(bθb(λ	f−1(bθb(λ	PROPN
ejpam-5094	140	8	,	,	PUNCT
ejpam-5094	140	9	p	p	NOUN
ejpam-5094	140	10	)	)	PUNCT
ejpam-5094	140	11	)	)	PUNCT
ejpam-5094	140	12	for	for	ADP
ejpam-5094	140	13	every	every	DET
ejpam-5094	140	14	subset	subset	NOUN
ejpam-5094	140	15	b	b	PROPN
ejpam-5094	140	16	of	of	ADP
ejpam-5094	140	17	y	y	PROPN
ejpam-5094	140	18	;	;	PUNCT
ejpam-5094	140	19	(	(	PUNCT
ejpam-5094	140	20	4	4	X
ejpam-5094	140	21	)	)	PUNCT
ejpam-5094	140	22	f−1(bθb(λ	f−1(bθb(λ	PROPN
ejpam-5094	140	23	,	,	PUNCT
ejpam-5094	140	24	p	p	NOUN
ejpam-5094	140	25	)	)	PUNCT
ejpam-5094	140	26	)	)	PUNCT
ejpam-5094	141	1	⊆	⊆	NUM
ejpam-5094	141	2	[	[	X
ejpam-5094	141	3	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-5094	141	4	,	,	PUNCT
ejpam-5094	141	5	p	p	NOUN
ejpam-5094	141	6	)	)	PUNCT
ejpam-5094	141	7	for	for	ADP
ejpam-5094	141	8	every	every	DET
ejpam-5094	141	9	subset	subset	NOUN
ejpam-5094	141	10	b	b	PROPN
ejpam-5094	141	11	of	of	ADP
ejpam-5094	141	12	y	y	PROPN
ejpam-5094	141	13	.	.	PUNCT
ejpam-5094	142	1	c.	c.	PROPN
ejpam-5094	142	2	boonpok	boonpok	PROPN
ejpam-5094	142	3	,	,	PUNCT
ejpam-5094	142	4	j.	j.	PROPN
ejpam-5094	142	5	khampakdee	khampakdee	PROPN
ejpam-5094	142	6	/	/	PUNCT
ejpam-5094	142	7	eur	eur	PROPN
ejpam-5094	142	8	.	.	PUNCT
ejpam-5094	143	1	j.	j.	PROPN
ejpam-5094	143	2	pure	pure	PROPN
ejpam-5094	143	3	appl	appl	PROPN
ejpam-5094	143	4	.	.	PROPN
ejpam-5094	143	5	math	math	PROPN
ejpam-5094	143	6	,	,	PUNCT
ejpam-5094	143	7	17	17	NUM
ejpam-5094	143	8	(	(	PUNCT
ejpam-5094	143	9	2	2	NUM
ejpam-5094	143	10	)	)	PUNCT
ejpam-5094	143	11	(	(	PUNCT
ejpam-5094	143	12	2024	2024	NUM
ejpam-5094	143	13	)	)	PUNCT
ejpam-5094	143	14	,	,	PUNCT
ejpam-5094	143	15	582	582	NUM
ejpam-5094	143	16	-	-	SYM
ejpam-5094	143	17	590	590	NUM
ejpam-5094	143	18	586	586	NUM
ejpam-5094	143	19	proof	proof	NOUN
ejpam-5094	143	20	.	.	PUNCT
ejpam-5094	144	1	(	(	PUNCT
ejpam-5094	144	2	1	1	X
ejpam-5094	144	3	)	)	PUNCT
ejpam-5094	144	4	⇒	⇒	NOUN
ejpam-5094	144	5	(	(	PUNCT
ejpam-5094	144	6	2	2	NUM
ejpam-5094	144	7	):	):	PUNCT
ejpam-5094	144	8	let	let	VERB
ejpam-5094	144	9	a	a	PRON
ejpam-5094	144	10	be	be	AUX
ejpam-5094	144	11	any	any	DET
ejpam-5094	144	12	subset	subset	NOUN
ejpam-5094	144	13	ofx	ofx	NOUN
ejpam-5094	144	14	and	and	CCONJ
ejpam-5094	144	15	x	x	PUNCT
ejpam-5094	144	16	∈	∈	PROPN
ejpam-5094	144	17	aθ(λ	aθ(λ	NOUN
ejpam-5094	144	18	,	,	PUNCT
ejpam-5094	144	19	p	p	NOUN
ejpam-5094	144	20	)	)	PUNCT
ejpam-5094	144	21	.	.	PUNCT
ejpam-5094	145	1	then	then	ADV
ejpam-5094	145	2	,	,	PUNCT
ejpam-5094	145	3	there	there	PRON
ejpam-5094	145	4	exists	exist	VERB
ejpam-5094	145	5	a	a	DET
ejpam-5094	145	6	(	(	PUNCT
ejpam-5094	145	7	λ	λ	PROPN
ejpam-5094	145	8	,	,	PUNCT
ejpam-5094	145	9	p)open	p)open	PROPN
ejpam-5094	145	10	set	set	ADJ
ejpam-5094	145	11	u	u	NOUN
ejpam-5094	145	12	of	of	ADP
ejpam-5094	145	13	x	x	SYM
ejpam-5094	145	14	such	such	ADJ
ejpam-5094	145	15	that	that	SCONJ
ejpam-5094	145	16	x	x	SYM
ejpam-5094	145	17	∈	∈	NOUN
ejpam-5094	145	18	u	u	NOUN
ejpam-5094	145	19	⊆	⊆	NUM
ejpam-5094	145	20	u	u	PROPN
ejpam-5094	145	21	(	(	PUNCT
ejpam-5094	145	22	λ	λ	PROPN
ejpam-5094	145	23	,	,	PUNCT
ejpam-5094	145	24	p	p	NOUN
ejpam-5094	145	25	)	)	PUNCT
ejpam-5094	145	26	⊆	⊆	NUM
ejpam-5094	145	27	a.	a.	NOUN
ejpam-5094	145	28	then	then	ADV
ejpam-5094	145	29	,	,	PUNCT
ejpam-5094	145	30	f(x	f(x	PROPN
ejpam-5094	145	31	)	)	PUNCT
ejpam-5094	145	32	∈	∈	PROPN
ejpam-5094	145	33	f(u	f(u	PROPN
ejpam-5094	145	34	)	)	PUNCT
ejpam-5094	145	35	⊆	⊆	NUM
ejpam-5094	145	36	f(u	f(u	PROPN
ejpam-5094	145	37	(	(	PUNCT
ejpam-5094	145	38	λ	λ	PROPN
ejpam-5094	145	39	,	,	PUNCT
ejpam-5094	145	40	p	p	NOUN
ejpam-5094	145	41	)	)	PUNCT
ejpam-5094	145	42	)	)	PUNCT
ejpam-5094	145	43	⊆	⊆	NUM
ejpam-5094	145	44	f(a	f(a	NOUN
ejpam-5094	145	45	)	)	PUNCT
ejpam-5094	145	46	.	.	PUNCT
ejpam-5094	146	1	since	since	SCONJ
ejpam-5094	146	2	f	f	PROPN
ejpam-5094	146	3	is	be	AUX
ejpam-5094	146	4	weakly	weakly	ADJ
ejpam-5094	146	5	θb(λ	θb(λ	NUM
ejpam-5094	146	6	,	,	PUNCT
ejpam-5094	146	7	p)-open	p)-open	ADJ
ejpam-5094	146	8	,	,	PUNCT
ejpam-5094	146	9	f(u	f(u	PROPN
ejpam-5094	146	10	)	)	PUNCT
ejpam-5094	147	1	⊆	⊆	NUM
ejpam-5094	148	1	[	[	X
ejpam-5094	148	2	f(u	f(u	PROPN
ejpam-5094	148	3	(	(	PUNCT
ejpam-5094	148	4	λ	λ	PROPN
ejpam-5094	148	5	,	,	PUNCT
ejpam-5094	148	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	148	7	,	,	PUNCT
ejpam-5094	148	8	p	p	NOUN
ejpam-5094	148	9	)	)	PUNCT
ejpam-5094	148	10	⊆	⊆	NUM
ejpam-5094	148	11	[	[	X
ejpam-5094	148	12	f(a)]θb(λ	f(a)]θb(λ	PROPN
ejpam-5094	148	13	,	,	PUNCT
ejpam-5094	148	14	p	p	NOUN
ejpam-5094	148	15	)	)	PUNCT
ejpam-5094	148	16	.	.	PUNCT
ejpam-5094	149	1	this	this	PRON
ejpam-5094	149	2	implies	imply	VERB
ejpam-5094	149	3	that	that	SCONJ
ejpam-5094	149	4	f(x	f(x	PROPN
ejpam-5094	149	5	)	)	PUNCT
ejpam-5094	149	6	∈	∈	PROPN
ejpam-5094	150	1	[	[	X
ejpam-5094	150	2	f(a)]θb(λ	f(a)]θb(λ	PROPN
ejpam-5094	150	3	,	,	PUNCT
ejpam-5094	150	4	p	p	NOUN
ejpam-5094	150	5	)	)	PUNCT
ejpam-5094	150	6	.	.	PUNCT
ejpam-5094	151	1	thus	thus	ADV
ejpam-5094	151	2	,	,	PUNCT
ejpam-5094	151	3	x	x	PROPN
ejpam-5094	151	4	∈	∈	PROPN
ejpam-5094	151	5	f−1([f(a)]θb(λ	f−1([f(a)]θb(λ	PROPN
ejpam-5094	151	6	,	,	PUNCT
ejpam-5094	151	7	p	p	NOUN
ejpam-5094	151	8	)	)	PUNCT
ejpam-5094	151	9	)	)	PUNCT
ejpam-5094	151	10	and	and	CCONJ
ejpam-5094	151	11	hence	hence	ADV
ejpam-5094	151	12	aθ(λ	aθ(λ	NOUN
ejpam-5094	151	13	,	,	PUNCT
ejpam-5094	151	14	p	p	NOUN
ejpam-5094	151	15	)	)	PUNCT
ejpam-5094	151	16	⊆	⊆	NUM
ejpam-5094	151	17	f−1([f(a)]θb(λ	f−1([f(a)]θb(λ	PROPN
ejpam-5094	151	18	,	,	PUNCT
ejpam-5094	151	19	p	p	NOUN
ejpam-5094	151	20	)	)	PUNCT
ejpam-5094	151	21	)	)	PUNCT
ejpam-5094	151	22	.	.	PUNCT
ejpam-5094	152	1	this	this	PRON
ejpam-5094	152	2	shows	show	VERB
ejpam-5094	152	3	that	that	SCONJ
ejpam-5094	152	4	f(aθ(λ	f(aθ(λ	PROPN
ejpam-5094	152	5	,	,	PUNCT
ejpam-5094	152	6	p	p	NOUN
ejpam-5094	152	7	)	)	PUNCT
ejpam-5094	152	8	)	)	PUNCT
ejpam-5094	152	9	⊆	⊆	NUM
ejpam-5094	153	1	[	[	X
ejpam-5094	153	2	f(a)]θb(λ	f(a)]θb(λ	PROPN
ejpam-5094	153	3	,	,	PUNCT
ejpam-5094	153	4	p	p	NOUN
ejpam-5094	153	5	)	)	PUNCT
ejpam-5094	153	6	.	.	PUNCT
ejpam-5094	154	1	(	(	PUNCT
ejpam-5094	154	2	2	2	X
ejpam-5094	154	3	)	)	PUNCT
ejpam-5094	154	4	⇒	⇒	NOUN
ejpam-5094	154	5	(	(	PUNCT
ejpam-5094	154	6	3	3	NUM
ejpam-5094	154	7	):	):	PUNCT
ejpam-5094	154	8	let	let	VERB
ejpam-5094	154	9	b	b	X
ejpam-5094	154	10	be	be	AUX
ejpam-5094	154	11	any	any	DET
ejpam-5094	154	12	subset	subset	NOUN
ejpam-5094	154	13	of	of	ADP
ejpam-5094	154	14	y	y	PROPN
ejpam-5094	154	15	.	.	PUNCT
ejpam-5094	155	1	then	then	ADV
ejpam-5094	155	2	by	by	ADP
ejpam-5094	155	3	(	(	PUNCT
ejpam-5094	155	4	2	2	NUM
ejpam-5094	155	5	)	)	PUNCT
ejpam-5094	155	6	,	,	PUNCT
ejpam-5094	155	7	f([f−1(b)]θ(λ	f([f−1(b)]θ(λ	PROPN
ejpam-5094	155	8	,	,	PUNCT
ejpam-5094	155	9	p	p	NOUN
ejpam-5094	155	10	)	)	PUNCT
ejpam-5094	155	11	)	)	PUNCT
ejpam-5094	156	1	⊆	⊆	NUM
ejpam-5094	156	2	bθb(λ	bθb(λ	NOUN
ejpam-5094	156	3	,	,	PUNCT
ejpam-5094	156	4	p	p	NOUN
ejpam-5094	156	5	)	)	PUNCT
ejpam-5094	156	6	.	.	PUNCT
ejpam-5094	157	1	thus	thus	ADV
ejpam-5094	157	2	,	,	PUNCT
ejpam-5094	157	3	[	[	X
ejpam-5094	157	4	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-5094	157	5	,	,	PUNCT
ejpam-5094	157	6	p	p	NOUN
ejpam-5094	157	7	)	)	PUNCT
ejpam-5094	157	8	⊆	⊆	NUM
ejpam-5094	157	9	f−1(bθb(λ	f−1(bθb(λ	PROPN
ejpam-5094	157	10	,	,	PUNCT
ejpam-5094	157	11	p	p	NOUN
ejpam-5094	157	12	)	)	PUNCT
ejpam-5094	157	13	)	)	PUNCT
ejpam-5094	157	14	.	.	PUNCT
ejpam-5094	158	1	(	(	PUNCT
ejpam-5094	158	2	3	3	X
ejpam-5094	158	3	)	)	PUNCT
ejpam-5094	158	4	⇒	⇒	NOUN
ejpam-5094	158	5	(	(	PUNCT
ejpam-5094	158	6	4	4	NUM
ejpam-5094	158	7	):	):	PUNCT
ejpam-5094	158	8	let	let	VERB
ejpam-5094	158	9	b	b	X
ejpam-5094	158	10	be	be	AUX
ejpam-5094	158	11	any	any	DET
ejpam-5094	158	12	subset	subset	NOUN
ejpam-5094	158	13	of	of	ADP
ejpam-5094	158	14	y	y	PROPN
ejpam-5094	158	15	.	.	PUNCT
ejpam-5094	159	1	using	use	VERB
ejpam-5094	159	2	(	(	PUNCT
ejpam-5094	159	3	3	3	NUM
ejpam-5094	159	4	)	)	PUNCT
ejpam-5094	159	5	,	,	PUNCT
ejpam-5094	159	6	we	we	PRON
ejpam-5094	159	7	have	have	VERB
ejpam-5094	159	8	x	x	X
ejpam-5094	159	9	−	−	PROPN
ejpam-5094	160	1	[	[	X
ejpam-5094	160	2	f−1(b)]θ(λ	f−1(b)]θ(λ	PROPN
ejpam-5094	160	3	,	,	PUNCT
ejpam-5094	160	4	p	p	NOUN
ejpam-5094	160	5	)	)	PUNCT
ejpam-5094	160	6	=	=	PUNCT
ejpam-5094	161	1	[	[	X
ejpam-5094	161	2	x	x	X
ejpam-5094	161	3	−	−	ADP
ejpam-5094	161	4	f−1(b)]θ(λ	f−1(b)]θ(λ	PROPN
ejpam-5094	161	5	,	,	PUNCT
ejpam-5094	161	6	p	p	NOUN
ejpam-5094	161	7	)	)	PUNCT
ejpam-5094	161	8	=	=	PUNCT
ejpam-5094	162	1	[	[	X
ejpam-5094	162	2	f−1(y	f−1(y	PROPN
ejpam-5094	162	3	−b)]θ(λ	−b)]θ(λ	PROPN
ejpam-5094	162	4	,	,	PUNCT
ejpam-5094	162	5	p	p	NOUN
ejpam-5094	162	6	)	)	PUNCT
ejpam-5094	163	1	⊆	⊆	NUM
ejpam-5094	163	2	f−1([y	f−1([y	NOUN
ejpam-5094	163	3	−b]θb(λ	−b]θb(λ	NOUN
ejpam-5094	163	4	,	,	PUNCT
ejpam-5094	163	5	p	p	NOUN
ejpam-5094	163	6	)	)	PUNCT
ejpam-5094	163	7	)	)	PUNCT
ejpam-5094	164	1	=	=	PUNCT
ejpam-5094	164	2	f−(y	f−(y	PROPN
ejpam-5094	164	3	−bθb(λ	−bθb(λ	PROPN
ejpam-5094	164	4	,	,	PUNCT
ejpam-5094	164	5	p	p	NOUN
ejpam-5094	164	6	)	)	PUNCT
ejpam-5094	164	7	)	)	PUNCT
ejpam-5094	165	1	=	=	PUNCT
ejpam-5094	165	2	x	x	PUNCT
ejpam-5094	166	1	−	−	NUM
ejpam-5094	166	2	f−1(bθb(λ	f−1(bθb(λ	PROPN
ejpam-5094	166	3	,	,	PUNCT
ejpam-5094	166	4	p	p	NOUN
ejpam-5094	166	5	)	)	PUNCT
ejpam-5094	166	6	)	)	PUNCT
ejpam-5094	166	7	.	.	PUNCT
ejpam-5094	167	1	thus	thus	ADV
ejpam-5094	167	2	,	,	PUNCT
ejpam-5094	167	3	f−1(bθb(λ	f−1(bθb(λ	PROPN
ejpam-5094	167	4	,	,	PUNCT
ejpam-5094	167	5	p	p	NOUN
ejpam-5094	167	6	)	)	PUNCT
ejpam-5094	167	7	)	)	PUNCT
ejpam-5094	168	1	⊆	⊆	NUM
ejpam-5094	168	2	[	[	X
ejpam-5094	168	3	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-5094	168	4	,	,	PUNCT
ejpam-5094	168	5	p	p	NOUN
ejpam-5094	168	6	)	)	PUNCT
ejpam-5094	168	7	.	.	PUNCT
ejpam-5094	169	1	(	(	PUNCT
ejpam-5094	169	2	4	4	X
ejpam-5094	169	3	)	)	PUNCT
ejpam-5094	169	4	⇒	⇒	NOUN
ejpam-5094	169	5	(	(	PUNCT
ejpam-5094	169	6	1	1	NUM
ejpam-5094	169	7	):	):	PUNCT
ejpam-5094	169	8	let	let	VERB
ejpam-5094	169	9	u	u	PRON
ejpam-5094	169	10	be	be	AUX
ejpam-5094	169	11	any	any	DET
ejpam-5094	169	12	(	(	PUNCT
ejpam-5094	169	13	λ	λ	NOUN
ejpam-5094	169	14	,	,	PUNCT
ejpam-5094	169	15	p)-open	p)-open	VERB
ejpam-5094	169	16	set	set	VERB
ejpam-5094	169	17	of	of	ADP
ejpam-5094	169	18	x.	x.	NOUN
ejpam-5094	169	19	by	by	ADP
ejpam-5094	169	20	(	(	PUNCT
ejpam-5094	169	21	4	4	NUM
ejpam-5094	169	22	)	)	PUNCT
ejpam-5094	169	23	,	,	PUNCT
ejpam-5094	169	24	f−1([y	f−1([y	NOUN
ejpam-5094	169	25	−	−	ADP
ejpam-5094	169	26	f(u	f(u	PROPN
ejpam-5094	169	27	(	(	PUNCT
ejpam-5094	169	28	λ	λ	PROPN
ejpam-5094	169	29	,	,	PUNCT
ejpam-5094	169	30	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	169	31	,	,	PUNCT
ejpam-5094	169	32	p	p	NOUN
ejpam-5094	169	33	)	)	PUNCT
ejpam-5094	169	34	)	)	PUNCT
ejpam-5094	170	1	⊆	⊆	NUM
ejpam-5094	170	2	[	[	X
ejpam-5094	170	3	f−1(y	f−1(y	NOUN
ejpam-5094	170	4	−	−	PROPN
ejpam-5094	170	5	f(u	f(u	PROPN
ejpam-5094	170	6	(	(	PUNCT
ejpam-5094	170	7	λ	λ	PROPN
ejpam-5094	170	8	,	,	PUNCT
ejpam-5094	170	9	p)))]θ(λ	p)))]θ(λ	PROPN
ejpam-5094	170	10	,	,	PUNCT
ejpam-5094	170	11	p	p	NOUN
ejpam-5094	170	12	)	)	PUNCT
ejpam-5094	170	13	.	.	PUNCT
ejpam-5094	171	1	thus	thus	ADV
ejpam-5094	171	2	,	,	PUNCT
ejpam-5094	171	3	f−1(y	f−1(y	PROPN
ejpam-5094	171	4	−	−	PROPN
ejpam-5094	172	1	[	[	X
ejpam-5094	172	2	f(u	f(u	PROPN
ejpam-5094	172	3	(	(	PUNCT
ejpam-5094	172	4	λ	λ	PROPN
ejpam-5094	172	5	,	,	PUNCT
ejpam-5094	172	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	172	7	,	,	PUNCT
ejpam-5094	172	8	p	p	NOUN
ejpam-5094	172	9	)	)	PUNCT
ejpam-5094	172	10	)	)	PUNCT
ejpam-5094	172	11	⊆	⊆	NUM
ejpam-5094	172	12	[	[	X
ejpam-5094	172	13	x	x	X
ejpam-5094	172	14	−	−	PROPN
ejpam-5094	172	15	f−1(f(u	f−1(f(u	ADJ
ejpam-5094	172	16	(	(	PUNCT
ejpam-5094	172	17	λ	λ	PROPN
ejpam-5094	172	18	,	,	PUNCT
ejpam-5094	172	19	p)))]θ(λ	p)))]θ(λ	PROPN
ejpam-5094	172	20	,	,	PUNCT
ejpam-5094	172	21	p	p	NOUN
ejpam-5094	172	22	)	)	PUNCT
ejpam-5094	172	23	⊆	⊆	NUM
ejpam-5094	173	1	[	[	X
ejpam-5094	173	2	x	x	X
ejpam-5094	173	3	−	−	PROPN
ejpam-5094	173	4	u	u	NOUN
ejpam-5094	173	5	(	(	PUNCT
ejpam-5094	173	6	λ	λ	PROPN
ejpam-5094	173	7	,	,	PUNCT
ejpam-5094	173	8	p)]θ(λ	p)]θ(λ	NOUN
ejpam-5094	173	9	,	,	PUNCT
ejpam-5094	173	10	p	p	NOUN
ejpam-5094	173	11	)	)	PUNCT
ejpam-5094	173	12	and	and	CCONJ
ejpam-5094	173	13	hence	hence	ADV
ejpam-5094	173	14	u	u	NOUN
ejpam-5094	173	15	⊆	⊆	NUM
ejpam-5094	173	16	[	[	X
ejpam-5094	173	17	u	u	X
ejpam-5094	173	18	(	(	PUNCT
ejpam-5094	173	19	λ	λ	PROPN
ejpam-5094	173	20	,	,	PUNCT
ejpam-5094	173	21	p)]θ(λ	p)]θ(λ	NOUN
ejpam-5094	173	22	,	,	PUNCT
ejpam-5094	173	23	p	p	NOUN
ejpam-5094	173	24	)	)	PUNCT
ejpam-5094	173	25	⊆	⊆	NUM
ejpam-5094	173	26	f−1([f(u	f−1([f(u	NOUN
ejpam-5094	173	27	(	(	PUNCT
ejpam-5094	173	28	λ	λ	PROPN
ejpam-5094	173	29	,	,	PUNCT
ejpam-5094	173	30	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	173	31	,	,	PUNCT
ejpam-5094	173	32	p	p	NOUN
ejpam-5094	173	33	)	)	PUNCT
ejpam-5094	173	34	)	)	PUNCT
ejpam-5094	173	35	.	.	PUNCT
ejpam-5094	174	1	therefore	therefore	ADV
ejpam-5094	174	2	,	,	PUNCT
ejpam-5094	174	3	f(u	f(u	PROPN
ejpam-5094	174	4	)	)	PUNCT
ejpam-5094	174	5	⊆	⊆	NUM
ejpam-5094	174	6	[	[	X
ejpam-5094	174	7	f(u	f(u	PROPN
ejpam-5094	174	8	(	(	PUNCT
ejpam-5094	174	9	λ	λ	PROPN
ejpam-5094	174	10	,	,	PUNCT
ejpam-5094	174	11	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	174	12	,	,	PUNCT
ejpam-5094	174	13	p	p	NOUN
ejpam-5094	174	14	)	)	PUNCT
ejpam-5094	174	15	.	.	PUNCT
ejpam-5094	175	1	this	this	PRON
ejpam-5094	175	2	shows	show	VERB
ejpam-5094	175	3	that	that	SCONJ
ejpam-5094	175	4	f	f	PROPN
ejpam-5094	175	5	is	be	AUX
ejpam-5094	175	6	weakly	weakly	ADJ
ejpam-5094	175	7	θb(λ	θb(λ	NOUN
ejpam-5094	175	8	,	,	PUNCT
ejpam-5094	175	9	p)-open	p)-open	ADJ
ejpam-5094	175	10	.	.	PUNCT
ejpam-5094	176	1	theorem	theorem	NOUN
ejpam-5094	176	2	2	2	NUM
ejpam-5094	176	3	.	.	X
ejpam-5094	176	4	for	for	ADP
ejpam-5094	176	5	a	a	DET
ejpam-5094	176	6	function	function	NOUN
ejpam-5094	176	7	f	f	NOUN
ejpam-5094	176	8	:	:	PUNCT
ejpam-5094	176	9	(	(	PUNCT
ejpam-5094	176	10	x	x	X
ejpam-5094	176	11	,	,	PUNCT
ejpam-5094	176	12	τ	τ	X
ejpam-5094	176	13	)	)	PUNCT
ejpam-5094	176	14	→	→	SYM
ejpam-5094	176	15	(	(	PUNCT
ejpam-5094	176	16	y	y	PROPN
ejpam-5094	176	17	,	,	PUNCT
ejpam-5094	176	18	σ	σ	PROPN
ejpam-5094	176	19	)	)	PUNCT
ejpam-5094	176	20	,	,	PUNCT
ejpam-5094	176	21	the	the	DET
ejpam-5094	176	22	following	follow	VERB
ejpam-5094	176	23	properties	property	NOUN
ejpam-5094	176	24	are	be	AUX
ejpam-5094	176	25	equivalent	equivalent	ADJ
ejpam-5094	176	26	:	:	PUNCT
ejpam-5094	176	27	(	(	PUNCT
ejpam-5094	176	28	1	1	X
ejpam-5094	176	29	)	)	PUNCT
ejpam-5094	176	30	f	f	PROPN
ejpam-5094	176	31	is	be	AUX
ejpam-5094	176	32	weakly	weakly	ADJ
ejpam-5094	176	33	θb(λ	θb(λ	NOUN
ejpam-5094	176	34	,	,	PUNCT
ejpam-5094	176	35	p)-open	p)-open	VERB
ejpam-5094	176	36	;	;	PUNCT
ejpam-5094	176	37	(	(	PUNCT
ejpam-5094	176	38	2	2	X
ejpam-5094	176	39	)	)	PUNCT
ejpam-5094	176	40	for	for	ADP
ejpam-5094	176	41	each	each	DET
ejpam-5094	176	42	x	x	SYM
ejpam-5094	176	43	∈	∈	PROPN
ejpam-5094	176	44	x	x	X
ejpam-5094	176	45	and	and	CCONJ
ejpam-5094	176	46	each	each	PRON
ejpam-5094	176	47	(	(	PUNCT
ejpam-5094	176	48	λ	λ	NOUN
ejpam-5094	176	49	,	,	PUNCT
ejpam-5094	176	50	p)-open	p)-open	VERB
ejpam-5094	176	51	set	set	VERB
ejpam-5094	176	52	u	u	NOUN
ejpam-5094	176	53	of	of	ADP
ejpam-5094	176	54	x	x	SYM
ejpam-5094	176	55	containing	contain	VERB
ejpam-5094	176	56	x	x	PRON
ejpam-5094	176	57	,	,	PUNCT
ejpam-5094	176	58	there	there	PRON
ejpam-5094	176	59	exists	exist	VERB
ejpam-5094	176	60	a	a	DET
ejpam-5094	176	61	θb(λ	θb(λ	PROPN
ejpam-5094	176	62	,	,	PUNCT
ejpam-5094	176	63	p)open	p)open	PROPN
ejpam-5094	176	64	set	set	NOUN
ejpam-5094	176	65	v	v	NOUN
ejpam-5094	176	66	of	of	ADP
ejpam-5094	176	67	y	y	NOUN
ejpam-5094	176	68	containing	contain	VERB
ejpam-5094	176	69	f(x	f(x	PROPN
ejpam-5094	176	70	)	)	PUNCT
ejpam-5094	176	71	such	such	ADJ
ejpam-5094	176	72	that	that	PRON
ejpam-5094	176	73	v	v	ADP
ejpam-5094	176	74	⊆	⊆	NUM
ejpam-5094	176	75	f(u	f(u	PROPN
ejpam-5094	176	76	(	(	PUNCT
ejpam-5094	176	77	λ	λ	PROPN
ejpam-5094	176	78	,	,	PUNCT
ejpam-5094	176	79	p	p	NOUN
ejpam-5094	176	80	)	)	PUNCT
ejpam-5094	176	81	)	)	PUNCT
ejpam-5094	176	82	.	.	PUNCT
ejpam-5094	177	1	proof	proof	NOUN
ejpam-5094	177	2	.	.	PUNCT
ejpam-5094	178	1	(	(	PUNCT
ejpam-5094	178	2	1	1	X
ejpam-5094	178	3	)	)	PUNCT
ejpam-5094	178	4	⇒	⇒	NOUN
ejpam-5094	178	5	(	(	PUNCT
ejpam-5094	178	6	2	2	NUM
ejpam-5094	178	7	):	):	PUNCT
ejpam-5094	178	8	let	let	VERB
ejpam-5094	178	9	x	x	PUNCT
ejpam-5094	178	10	∈	∈	PROPN
ejpam-5094	178	11	x	x	X
ejpam-5094	178	12	and	and	CCONJ
ejpam-5094	178	13	u	u	PRON
ejpam-5094	178	14	be	be	VERB
ejpam-5094	178	15	any	any	DET
ejpam-5094	178	16	(	(	PUNCT
ejpam-5094	178	17	λ	λ	NOUN
ejpam-5094	178	18	,	,	PUNCT
ejpam-5094	178	19	p)-open	p)-open	VERB
ejpam-5094	178	20	set	set	VERB
ejpam-5094	178	21	of	of	ADP
ejpam-5094	178	22	x	x	PUNCT
ejpam-5094	178	23	containing	contain	VERB
ejpam-5094	178	24	x.	x.	NOUN
ejpam-5094	178	25	since	since	SCONJ
ejpam-5094	178	26	f	f	PROPN
ejpam-5094	178	27	is	be	AUX
ejpam-5094	178	28	weakly	weakly	ADJ
ejpam-5094	178	29	θb(λ	θb(λ	NUM
ejpam-5094	178	30	,	,	PUNCT
ejpam-5094	178	31	p)-open	p)-open	ADJ
ejpam-5094	178	32	,	,	PUNCT
ejpam-5094	178	33	f(x	f(x	PROPN
ejpam-5094	178	34	)	)	PUNCT
ejpam-5094	178	35	∈	∈	PROPN
ejpam-5094	178	36	f(u	f(u	PROPN
ejpam-5094	178	37	)	)	PUNCT
ejpam-5094	178	38	⊆	⊆	NUM
ejpam-5094	179	1	[	[	X
ejpam-5094	179	2	f(u	f(u	PROPN
ejpam-5094	179	3	(	(	PUNCT
ejpam-5094	179	4	λ	λ	PROPN
ejpam-5094	179	5	,	,	PUNCT
ejpam-5094	179	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	179	7	,	,	PUNCT
ejpam-5094	179	8	p	p	NOUN
ejpam-5094	179	9	)	)	PUNCT
ejpam-5094	179	10	.	.	PUNCT
ejpam-5094	180	1	let	let	VERB
ejpam-5094	180	2	v	v	NOUN
ejpam-5094	180	3	=	=	SYM
ejpam-5094	181	1	[	[	X
ejpam-5094	181	2	f(u	f(u	PROPN
ejpam-5094	181	3	(	(	PUNCT
ejpam-5094	181	4	λ	λ	PROPN
ejpam-5094	181	5	,	,	PUNCT
ejpam-5094	181	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	181	7	,	,	PUNCT
ejpam-5094	181	8	p	p	NOUN
ejpam-5094	181	9	)	)	PUNCT
ejpam-5094	181	10	.	.	PUNCT
ejpam-5094	182	1	then	then	ADV
ejpam-5094	182	2	,	,	PUNCT
ejpam-5094	182	3	v	v	NOUN
ejpam-5094	182	4	is	be	AUX
ejpam-5094	182	5	θb(λ	θb(λ	NUM
ejpam-5094	182	6	,	,	PUNCT
ejpam-5094	182	7	p)-open	p)-open	VERB
ejpam-5094	182	8	and	and	CCONJ
ejpam-5094	182	9	f(x	f(x	PROPN
ejpam-5094	182	10	)	)	PUNCT
ejpam-5094	182	11	∈	∈	PROPN
ejpam-5094	182	12	v	v	ADP
ejpam-5094	182	13	⊆	⊆	NUM
ejpam-5094	182	14	f(u	f(u	PROPN
ejpam-5094	182	15	(	(	PUNCT
ejpam-5094	182	16	λ	λ	PROPN
ejpam-5094	182	17	,	,	PUNCT
ejpam-5094	182	18	p	p	NOUN
ejpam-5094	182	19	)	)	PUNCT
ejpam-5094	182	20	)	)	PUNCT
ejpam-5094	182	21	.	.	PUNCT
ejpam-5094	183	1	(	(	PUNCT
ejpam-5094	183	2	2	2	X
ejpam-5094	183	3	)	)	PUNCT
ejpam-5094	183	4	⇒	⇒	NOUN
ejpam-5094	183	5	(	(	PUNCT
ejpam-5094	183	6	1	1	NUM
ejpam-5094	183	7	):	):	PUNCT
ejpam-5094	183	8	let	let	VERB
ejpam-5094	183	9	u	u	PRON
ejpam-5094	183	10	be	be	AUX
ejpam-5094	183	11	any	any	DET
ejpam-5094	183	12	(	(	PUNCT
ejpam-5094	183	13	λ	λ	NOUN
ejpam-5094	183	14	,	,	PUNCT
ejpam-5094	183	15	p)-open	p)-open	VERB
ejpam-5094	183	16	set	set	VERB
ejpam-5094	183	17	ofx	ofx	PROPN
ejpam-5094	183	18	and	and	CCONJ
ejpam-5094	183	19	y	y	PROPN
ejpam-5094	183	20	∈	∈	PROPN
ejpam-5094	183	21	f(u	f(u	PROPN
ejpam-5094	183	22	)	)	PUNCT
ejpam-5094	183	23	.	.	PUNCT
ejpam-5094	184	1	it	it	PRON
ejpam-5094	184	2	follows	follow	VERB
ejpam-5094	184	3	from	from	ADP
ejpam-5094	184	4	(	(	PUNCT
ejpam-5094	184	5	2	2	NUM
ejpam-5094	184	6	)	)	PUNCT
ejpam-5094	185	1	that	that	PRON
ejpam-5094	185	2	v	v	ADP
ejpam-5094	185	3	⊆	⊆	NUM
ejpam-5094	185	4	f(u	f(u	PROPN
ejpam-5094	185	5	(	(	PUNCT
ejpam-5094	185	6	λ	λ	PROPN
ejpam-5094	185	7	,	,	PUNCT
ejpam-5094	185	8	p	p	NOUN
ejpam-5094	185	9	)	)	PUNCT
ejpam-5094	185	10	)	)	PUNCT
ejpam-5094	185	11	for	for	ADP
ejpam-5094	185	12	some	some	PRON
ejpam-5094	185	13	θb(λ	θb(λ	NUM
ejpam-5094	185	14	,	,	PUNCT
ejpam-5094	185	15	p)-open	p)-open	VERB
ejpam-5094	185	16	set	set	VERB
ejpam-5094	185	17	v	v	NOUN
ejpam-5094	185	18	of	of	ADP
ejpam-5094	185	19	y	y	PROPN
ejpam-5094	185	20	containing	contain	VERB
ejpam-5094	185	21	y.	y.	PROPN
ejpam-5094	185	22	thus	thus	ADV
ejpam-5094	185	23	,	,	PUNCT
ejpam-5094	185	24	y	y	PROPN
ejpam-5094	185	25	∈	∈	PROPN
ejpam-5094	185	26	v	v	ADP
ejpam-5094	185	27	⊆	⊆	NUM
ejpam-5094	185	28	[	[	X
ejpam-5094	185	29	f(u	f(u	PROPN
ejpam-5094	185	30	(	(	PUNCT
ejpam-5094	185	31	λ	λ	PROPN
ejpam-5094	185	32	,	,	PUNCT
ejpam-5094	185	33	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	185	34	,	,	PUNCT
ejpam-5094	185	35	p	p	NOUN
ejpam-5094	185	36	)	)	PUNCT
ejpam-5094	185	37	and	and	CCONJ
ejpam-5094	185	38	hence	hence	ADV
ejpam-5094	185	39	f(u	f(u	PROPN
ejpam-5094	185	40	)	)	PUNCT
ejpam-5094	185	41	⊆	⊆	NUM
ejpam-5094	186	1	[	[	X
ejpam-5094	186	2	f(u	f(u	PROPN
ejpam-5094	186	3	(	(	PUNCT
ejpam-5094	186	4	λ	λ	PROPN
ejpam-5094	186	5	,	,	PUNCT
ejpam-5094	186	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	186	7	,	,	PUNCT
ejpam-5094	186	8	p	p	NOUN
ejpam-5094	186	9	)	)	PUNCT
ejpam-5094	186	10	.	.	PUNCT
ejpam-5094	187	1	this	this	PRON
ejpam-5094	187	2	shows	show	VERB
ejpam-5094	187	3	that	that	SCONJ
ejpam-5094	187	4	f	f	PROPN
ejpam-5094	187	5	is	be	AUX
ejpam-5094	187	6	weakly	weakly	ADJ
ejpam-5094	187	7	θb(λ	θb(λ	NOUN
ejpam-5094	187	8	,	,	PUNCT
ejpam-5094	187	9	p)-open	p)-open	ADJ
ejpam-5094	187	10	.	.	PUNCT
ejpam-5094	188	1	theorem	theorem	NOUN
ejpam-5094	188	2	3	3	NUM
ejpam-5094	188	3	.	.	X
ejpam-5094	188	4	for	for	ADP
ejpam-5094	188	5	a	a	DET
ejpam-5094	188	6	bijective	bijective	ADJ
ejpam-5094	188	7	function	function	NOUN
ejpam-5094	188	8	f	f	NOUN
ejpam-5094	188	9	:	:	PUNCT
ejpam-5094	188	10	(	(	PUNCT
ejpam-5094	188	11	x	x	X
ejpam-5094	188	12	,	,	PUNCT
ejpam-5094	188	13	τ	τ	X
ejpam-5094	188	14	)	)	PUNCT
ejpam-5094	188	15	→	→	SYM
ejpam-5094	188	16	(	(	PUNCT
ejpam-5094	188	17	y	y	PROPN
ejpam-5094	188	18	,	,	PUNCT
ejpam-5094	188	19	σ	σ	PROPN
ejpam-5094	188	20	)	)	PUNCT
ejpam-5094	188	21	,	,	PUNCT
ejpam-5094	188	22	the	the	DET
ejpam-5094	188	23	following	follow	VERB
ejpam-5094	188	24	properties	property	NOUN
ejpam-5094	188	25	are	be	AUX
ejpam-5094	188	26	equivalent	equivalent	ADJ
ejpam-5094	188	27	:	:	PUNCT
ejpam-5094	188	28	(	(	PUNCT
ejpam-5094	188	29	1	1	X
ejpam-5094	188	30	)	)	PUNCT
ejpam-5094	188	31	f	f	PROPN
ejpam-5094	188	32	is	be	AUX
ejpam-5094	188	33	weakly	weakly	ADJ
ejpam-5094	188	34	θb(λ	θb(λ	NOUN
ejpam-5094	188	35	,	,	PUNCT
ejpam-5094	188	36	p)-open	p)-open	VERB
ejpam-5094	188	37	;	;	PUNCT
ejpam-5094	188	38	(	(	PUNCT
ejpam-5094	188	39	2	2	X
ejpam-5094	188	40	)	)	PUNCT
ejpam-5094	189	1	[	[	X
ejpam-5094	189	2	f(k(λ	f(k(λ	NOUN
ejpam-5094	189	3	,	,	PUNCT
ejpam-5094	189	4	p	p	NOUN
ejpam-5094	189	5	)	)	PUNCT
ejpam-5094	189	6	)	)	PUNCT
ejpam-5094	189	7	]	]	PUNCT
ejpam-5094	190	1	θb(λ	θb(λ	X
ejpam-5094	190	2	,	,	PUNCT
ejpam-5094	190	3	p	p	NOUN
ejpam-5094	190	4	)	)	PUNCT
ejpam-5094	190	5	⊆	⊆	NUM
ejpam-5094	190	6	f(k	f(k	VERB
ejpam-5094	190	7	)	)	PUNCT
ejpam-5094	190	8	for	for	SCONJ
ejpam-5094	190	9	each	each	PRON
ejpam-5094	190	10	(	(	PUNCT
ejpam-5094	190	11	λ	λ	PROPN
ejpam-5094	190	12	,	,	PUNCT
ejpam-5094	190	13	p)-closed	p)-close	VERB
ejpam-5094	190	14	set	set	NOUN
ejpam-5094	190	15	k	k	PROPN
ejpam-5094	190	16	of	of	ADP
ejpam-5094	190	17	x	x	PROPN
ejpam-5094	190	18	;	;	PUNCT
ejpam-5094	190	19	c.	c.	PROPN
ejpam-5094	190	20	boonpok	boonpok	PROPN
ejpam-5094	190	21	,	,	PUNCT
ejpam-5094	190	22	j.	j.	PROPN
ejpam-5094	190	23	khampakdee	khampakdee	PROPN
ejpam-5094	190	24	/	/	PUNCT
ejpam-5094	190	25	eur	eur	PROPN
ejpam-5094	190	26	.	.	PUNCT
ejpam-5094	191	1	j.	j.	PROPN
ejpam-5094	191	2	pure	pure	PROPN
ejpam-5094	191	3	appl	appl	PROPN
ejpam-5094	191	4	.	.	PROPN
ejpam-5094	191	5	math	math	PROPN
ejpam-5094	191	6	,	,	PUNCT
ejpam-5094	191	7	17	17	NUM
ejpam-5094	191	8	(	(	PUNCT
ejpam-5094	191	9	2	2	NUM
ejpam-5094	191	10	)	)	PUNCT
ejpam-5094	191	11	(	(	PUNCT
ejpam-5094	191	12	2024	2024	NUM
ejpam-5094	191	13	)	)	PUNCT
ejpam-5094	191	14	,	,	PUNCT
ejpam-5094	191	15	582	582	NUM
ejpam-5094	191	16	-	-	SYM
ejpam-5094	191	17	590	590	NUM
ejpam-5094	191	18	587	587	NUM
ejpam-5094	191	19	(	(	PUNCT
ejpam-5094	191	20	3	3	NUM
ejpam-5094	191	21	)	)	PUNCT
ejpam-5094	192	1	[	[	X
ejpam-5094	192	2	f(u)]θb(λ	f(u)]θb(λ	X
ejpam-5094	192	3	,	,	PUNCT
ejpam-5094	192	4	p	p	NOUN
ejpam-5094	192	5	)	)	PUNCT
ejpam-5094	192	6	⊆	⊆	NUM
ejpam-5094	192	7	f(u	f(u	PROPN
ejpam-5094	192	8	(	(	PUNCT
ejpam-5094	192	9	λ	λ	PROPN
ejpam-5094	192	10	,	,	PUNCT
ejpam-5094	192	11	p	p	NOUN
ejpam-5094	192	12	)	)	PUNCT
ejpam-5094	192	13	)	)	PUNCT
ejpam-5094	192	14	for	for	SCONJ
ejpam-5094	192	15	each	each	DET
ejpam-5094	192	16	(	(	PUNCT
ejpam-5094	192	17	λ	λ	PROPN
ejpam-5094	192	18	,	,	PUNCT
ejpam-5094	192	19	p)-open	p)-open	VERB
ejpam-5094	192	20	set	set	VERB
ejpam-5094	192	21	u	u	NOUN
ejpam-5094	192	22	of	of	ADP
ejpam-5094	192	23	x.	x.	NOUN
ejpam-5094	192	24	proof	proof	NOUN
ejpam-5094	192	25	.	.	PUNCT
ejpam-5094	193	1	(	(	PUNCT
ejpam-5094	193	2	1	1	X
ejpam-5094	193	3	)	)	PUNCT
ejpam-5094	193	4	⇒	⇒	NOUN
ejpam-5094	193	5	(	(	PUNCT
ejpam-5094	193	6	2	2	NUM
ejpam-5094	193	7	):	):	PUNCT
ejpam-5094	193	8	let	let	VERB
ejpam-5094	193	9	k	k	PRON
ejpam-5094	193	10	be	be	AUX
ejpam-5094	193	11	any	any	DET
ejpam-5094	193	12	(	(	PUNCT
ejpam-5094	193	13	λ	λ	PROPN
ejpam-5094	193	14	,	,	PUNCT
ejpam-5094	193	15	p)-closed	p)-close	VERB
ejpam-5094	193	16	set	set	NOUN
ejpam-5094	193	17	of	of	ADP
ejpam-5094	193	18	x.	x.	NOUN
ejpam-5094	193	19	then	then	ADV
ejpam-5094	193	20	,	,	PUNCT
ejpam-5094	193	21	we	we	PRON
ejpam-5094	193	22	have	have	VERB
ejpam-5094	193	23	f(x	f(x	PROPN
ejpam-5094	193	24	−k	−k	ADV
ejpam-5094	193	25	)	)	PUNCT
ejpam-5094	194	1	=	=	SYM
ejpam-5094	194	2	y	y	PROPN
ejpam-5094	194	3	−	−	PROPN
ejpam-5094	194	4	f(k	f(k	PROPN
ejpam-5094	194	5	)	)	PUNCT
ejpam-5094	194	6	⊆	⊆	NUM
ejpam-5094	195	1	[	[	X
ejpam-5094	195	2	f([x	f([x	PROPN
ejpam-5094	195	3	−k](λ	−k](λ	NUM
ejpam-5094	195	4	,	,	PUNCT
ejpam-5094	195	5	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	195	6	,	,	PUNCT
ejpam-5094	195	7	p	p	NOUN
ejpam-5094	195	8	)	)	PUNCT
ejpam-5094	195	9	and	and	CCONJ
ejpam-5094	195	10	hence	hence	ADV
ejpam-5094	195	11	y	y	PROPN
ejpam-5094	195	12	−	−	PROPN
ejpam-5094	195	13	f(k	f(k	PROPN
ejpam-5094	195	14	)	)	PUNCT
ejpam-5094	195	15	⊆	⊆	NUM
ejpam-5094	195	16	y	y	NOUN
ejpam-5094	195	17	−	−	PROPN
ejpam-5094	196	1	[	[	X
ejpam-5094	196	2	f(k(λ	f(k(λ	NOUN
ejpam-5094	196	3	,	,	PUNCT
ejpam-5094	196	4	p	p	NOUN
ejpam-5094	196	5	)	)	PUNCT
ejpam-5094	196	6	)	)	PUNCT
ejpam-5094	196	7	]	]	PUNCT
ejpam-5094	196	8	θb(λ	θb(λ	X
ejpam-5094	196	9	,	,	PUNCT
ejpam-5094	196	10	p	p	NOUN
ejpam-5094	196	11	)	)	PUNCT
ejpam-5094	196	12	.	.	PUNCT
ejpam-5094	197	1	thus	thus	ADV
ejpam-5094	197	2	,	,	PUNCT
ejpam-5094	197	3	[	[	X
ejpam-5094	197	4	f(k(λ	f(k(λ	NOUN
ejpam-5094	197	5	,	,	PUNCT
ejpam-5094	197	6	p	p	NOUN
ejpam-5094	197	7	)	)	PUNCT
ejpam-5094	197	8	)	)	PUNCT
ejpam-5094	197	9	]	]	PUNCT
ejpam-5094	197	10	θb(λ	θb(λ	X
ejpam-5094	197	11	,	,	PUNCT
ejpam-5094	197	12	p	p	NOUN
ejpam-5094	197	13	)	)	PUNCT
ejpam-5094	197	14	⊆	⊆	NUM
ejpam-5094	197	15	f(k	f(k	VERB
ejpam-5094	197	16	)	)	PUNCT
ejpam-5094	197	17	.	.	PUNCT
ejpam-5094	198	1	(	(	PUNCT
ejpam-5094	198	2	2	2	X
ejpam-5094	198	3	)	)	PUNCT
ejpam-5094	198	4	⇒	⇒	NOUN
ejpam-5094	198	5	(	(	PUNCT
ejpam-5094	198	6	3	3	NUM
ejpam-5094	198	7	):	):	PUNCT
ejpam-5094	198	8	let	let	VERB
ejpam-5094	198	9	u	u	PRON
ejpam-5094	198	10	be	be	AUX
ejpam-5094	198	11	any	any	DET
ejpam-5094	198	12	(	(	PUNCT
ejpam-5094	198	13	λ	λ	NOUN
ejpam-5094	198	14	,	,	PUNCT
ejpam-5094	198	15	p)-open	p)-open	VERB
ejpam-5094	198	16	set	set	VERB
ejpam-5094	198	17	of	of	ADP
ejpam-5094	198	18	x.	x.	NOUN
ejpam-5094	198	19	since	since	SCONJ
ejpam-5094	198	20	u	u	PROPN
ejpam-5094	198	21	(	(	PUNCT
ejpam-5094	198	22	λ	λ	PROPN
ejpam-5094	198	23	,	,	PUNCT
ejpam-5094	198	24	p	p	NOUN
ejpam-5094	198	25	)	)	PUNCT
ejpam-5094	198	26	is	be	AUX
ejpam-5094	198	27	a	a	DET
ejpam-5094	198	28	(	(	PUNCT
ejpam-5094	198	29	λ	λ	PROPN
ejpam-5094	198	30	,	,	PUNCT
ejpam-5094	198	31	p)-closed	p)-close	VERB
ejpam-5094	198	32	set	set	NOUN
ejpam-5094	198	33	and	and	CCONJ
ejpam-5094	198	34	u	u	NOUN
ejpam-5094	198	35	⊆	⊆	NUM
ejpam-5094	198	36	[	[	X
ejpam-5094	198	37	u	u	X
ejpam-5094	198	38	(	(	PUNCT
ejpam-5094	198	39	λ	λ	PROPN
ejpam-5094	198	40	,	,	PUNCT
ejpam-5094	198	41	p)](λ	p)](λ	ADJ
ejpam-5094	198	42	,	,	PUNCT
ejpam-5094	198	43	p	p	NOUN
ejpam-5094	198	44	)	)	PUNCT
ejpam-5094	198	45	,	,	PUNCT
ejpam-5094	198	46	by	by	ADP
ejpam-5094	198	47	(	(	PUNCT
ejpam-5094	198	48	2	2	X
ejpam-5094	198	49	)	)	PUNCT
ejpam-5094	198	50	we	we	PRON
ejpam-5094	198	51	have	have	VERB
ejpam-5094	198	52	[	[	X
ejpam-5094	198	53	f(u)]θb(λ	f(u)]θb(λ	X
ejpam-5094	198	54	,	,	PUNCT
ejpam-5094	198	55	p	p	NOUN
ejpam-5094	198	56	)	)	PUNCT
ejpam-5094	198	57	⊆	⊆	NUM
ejpam-5094	199	1	[	[	X
ejpam-5094	199	2	f([u	f([u	INTJ
ejpam-5094	199	3	(	(	PUNCT
ejpam-5094	199	4	λ	λ	PROPN
ejpam-5094	199	5	,	,	PUNCT
ejpam-5094	199	6	p)](λ	p)](λ	ADJ
ejpam-5094	199	7	,	,	PUNCT
ejpam-5094	199	8	p	p	NOUN
ejpam-5094	199	9	)	)	PUNCT
ejpam-5094	199	10	)	)	PUNCT
ejpam-5094	199	11	]	]	PUNCT
ejpam-5094	200	1	θb(λ	θb(λ	X
ejpam-5094	200	2	,	,	PUNCT
ejpam-5094	200	3	p	p	NOUN
ejpam-5094	200	4	)	)	PUNCT
ejpam-5094	200	5	⊆	⊆	NUM
ejpam-5094	200	6	f(u	f(u	PROPN
ejpam-5094	200	7	(	(	PUNCT
ejpam-5094	200	8	λ	λ	PROPN
ejpam-5094	200	9	,	,	PUNCT
ejpam-5094	200	10	p	p	NOUN
ejpam-5094	200	11	)	)	PUNCT
ejpam-5094	200	12	)	)	PUNCT
ejpam-5094	200	13	.	.	PUNCT
ejpam-5094	201	1	(	(	PUNCT
ejpam-5094	201	2	3	3	X
ejpam-5094	201	3	)	)	PUNCT
ejpam-5094	201	4	⇒	⇒	NOUN
ejpam-5094	201	5	(	(	PUNCT
ejpam-5094	201	6	1	1	NUM
ejpam-5094	201	7	):	):	PUNCT
ejpam-5094	201	8	let	let	VERB
ejpam-5094	201	9	u	u	PRON
ejpam-5094	201	10	be	be	AUX
ejpam-5094	201	11	any	any	DET
ejpam-5094	201	12	(	(	PUNCT
ejpam-5094	201	13	λ	λ	NOUN
ejpam-5094	201	14	,	,	PUNCT
ejpam-5094	201	15	p)-open	p)-open	VERB
ejpam-5094	201	16	set	set	VERB
ejpam-5094	201	17	of	of	ADP
ejpam-5094	201	18	x.	x.	NOUN
ejpam-5094	201	19	using	use	VERB
ejpam-5094	201	20	(	(	PUNCT
ejpam-5094	201	21	3	3	NUM
ejpam-5094	201	22	)	)	PUNCT
ejpam-5094	201	23	,	,	PUNCT
ejpam-5094	201	24	we	we	PRON
ejpam-5094	201	25	have	have	VERB
ejpam-5094	201	26	y	y	PROPN
ejpam-5094	201	27	−	−	PROPN
ejpam-5094	202	1	[	[	X
ejpam-5094	202	2	f(u	f(u	PROPN
ejpam-5094	202	3	(	(	PUNCT
ejpam-5094	202	4	λ	λ	PROPN
ejpam-5094	202	5	,	,	PUNCT
ejpam-5094	202	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	202	7	,	,	PUNCT
ejpam-5094	202	8	p	p	NOUN
ejpam-5094	202	9	)	)	PUNCT
ejpam-5094	202	10	=	=	PUNCT
ejpam-5094	203	1	[	[	X
ejpam-5094	203	2	y	y	X
ejpam-5094	203	3	−	−	PROPN
ejpam-5094	203	4	f(u	f(u	PROPN
ejpam-5094	203	5	(	(	PUNCT
ejpam-5094	203	6	λ	λ	PROPN
ejpam-5094	203	7	,	,	PUNCT
ejpam-5094	203	8	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	203	9	,	,	PUNCT
ejpam-5094	203	10	p	p	NOUN
ejpam-5094	203	11	)	)	PUNCT
ejpam-5094	203	12	=	=	NOUN
ejpam-5094	204	1	[	[	X
ejpam-5094	204	2	f(x	f(x	PROPN
ejpam-5094	204	3	−	−	PROPN
ejpam-5094	204	4	u	u	PROPN
ejpam-5094	204	5	(	(	PUNCT
ejpam-5094	204	6	λ	λ	PROPN
ejpam-5094	204	7	,	,	PUNCT
ejpam-5094	204	8	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	204	9	,	,	PUNCT
ejpam-5094	204	10	p	p	NOUN
ejpam-5094	204	11	)	)	PUNCT
ejpam-5094	204	12	⊆	⊆	NUM
ejpam-5094	204	13	f([x	f([x	PROPN
ejpam-5094	204	14	−	−	PROPN
ejpam-5094	204	15	u	u	PROPN
ejpam-5094	204	16	(	(	PUNCT
ejpam-5094	204	17	λ	λ	PROPN
ejpam-5094	204	18	,	,	PUNCT
ejpam-5094	204	19	p)](λ	p)](λ	ADJ
ejpam-5094	204	20	,	,	PUNCT
ejpam-5094	204	21	p	p	NOUN
ejpam-5094	204	22	)	)	PUNCT
ejpam-5094	204	23	)	)	PUNCT
ejpam-5094	205	1	=	=	SYM
ejpam-5094	205	2	f(x	f(x	PROPN
ejpam-5094	205	3	−	−	PROPN
ejpam-5094	206	1	[	[	X
ejpam-5094	206	2	u	u	X
ejpam-5094	206	3	(	(	PUNCT
ejpam-5094	206	4	λ	λ	PROPN
ejpam-5094	206	5	,	,	PUNCT
ejpam-5094	206	6	p)](λ	p)](λ	ADJ
ejpam-5094	206	7	,	,	PUNCT
ejpam-5094	206	8	p	p	NOUN
ejpam-5094	206	9	)	)	PUNCT
ejpam-5094	206	10	)	)	PUNCT
ejpam-5094	206	11	⊆	⊆	NUM
ejpam-5094	206	12	f(x	f(x	PROPN
ejpam-5094	206	13	−	−	PROPN
ejpam-5094	206	14	u	u	NOUN
ejpam-5094	206	15	)	)	PUNCT
ejpam-5094	206	16	=	=	SYM
ejpam-5094	206	17	y	y	PROPN
ejpam-5094	206	18	−	−	PROPN
ejpam-5094	206	19	f(u	f(u	PROPN
ejpam-5094	206	20	)	)	PUNCT
ejpam-5094	206	21	and	and	CCONJ
ejpam-5094	206	22	hence	hence	ADV
ejpam-5094	206	23	f(u	f(u	PROPN
ejpam-5094	206	24	)	)	PUNCT
ejpam-5094	206	25	⊆	⊆	NUM
ejpam-5094	207	1	[	[	X
ejpam-5094	207	2	f(u	f(u	PROPN
ejpam-5094	207	3	(	(	PUNCT
ejpam-5094	207	4	λ	λ	PROPN
ejpam-5094	207	5	,	,	PUNCT
ejpam-5094	207	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	207	7	,	,	PUNCT
ejpam-5094	207	8	p	p	NOUN
ejpam-5094	207	9	)	)	PUNCT
ejpam-5094	207	10	.	.	PUNCT
ejpam-5094	208	1	this	this	PRON
ejpam-5094	208	2	shows	show	VERB
ejpam-5094	208	3	that	that	SCONJ
ejpam-5094	208	4	f	f	PROPN
ejpam-5094	208	5	is	be	AUX
ejpam-5094	208	6	weakly	weakly	ADJ
ejpam-5094	208	7	θb(λ	θb(λ	NOUN
ejpam-5094	208	8	,	,	PUNCT
ejpam-5094	208	9	p)-open	p)-open	ADJ
ejpam-5094	208	10	.	.	PUNCT
ejpam-5094	209	1	the	the	DET
ejpam-5094	209	2	proof	proof	NOUN
ejpam-5094	209	3	of	of	ADP
ejpam-5094	209	4	the	the	DET
ejpam-5094	209	5	following	follow	VERB
ejpam-5094	209	6	theorem	theorem	NOUN
ejpam-5094	209	7	is	be	AUX
ejpam-5094	209	8	straightforward	straightforward	ADJ
ejpam-5094	209	9	and	and	CCONJ
ejpam-5094	209	10	thus	thus	ADV
ejpam-5094	209	11	is	be	AUX
ejpam-5094	209	12	omitted	omit	VERB
ejpam-5094	209	13	.	.	PUNCT
ejpam-5094	210	1	theorem	theorem	VERB
ejpam-5094	210	2	4	4	NUM
ejpam-5094	210	3	.	.	X
ejpam-5094	210	4	for	for	ADP
ejpam-5094	210	5	a	a	DET
ejpam-5094	210	6	function	function	NOUN
ejpam-5094	210	7	f	f	NOUN
ejpam-5094	210	8	:	:	PUNCT
ejpam-5094	210	9	(	(	PUNCT
ejpam-5094	210	10	x	x	X
ejpam-5094	210	11	,	,	PUNCT
ejpam-5094	210	12	τ	τ	X
ejpam-5094	210	13	)	)	PUNCT
ejpam-5094	210	14	→	→	SYM
ejpam-5094	210	15	(	(	PUNCT
ejpam-5094	210	16	y	y	PROPN
ejpam-5094	210	17	,	,	PUNCT
ejpam-5094	210	18	σ	σ	PROPN
ejpam-5094	210	19	)	)	PUNCT
ejpam-5094	210	20	,	,	PUNCT
ejpam-5094	210	21	the	the	DET
ejpam-5094	210	22	following	follow	VERB
ejpam-5094	210	23	properties	property	NOUN
ejpam-5094	210	24	are	be	AUX
ejpam-5094	210	25	equivalent	equivalent	ADJ
ejpam-5094	210	26	:	:	PUNCT
ejpam-5094	210	27	(	(	PUNCT
ejpam-5094	210	28	1	1	X
ejpam-5094	210	29	)	)	PUNCT
ejpam-5094	210	30	f	f	PROPN
ejpam-5094	210	31	is	be	AUX
ejpam-5094	210	32	weakly	weakly	ADJ
ejpam-5094	210	33	θb(λ	θb(λ	NOUN
ejpam-5094	210	34	,	,	PUNCT
ejpam-5094	210	35	p)-open	p)-open	VERB
ejpam-5094	210	36	;	;	PUNCT
ejpam-5094	210	37	(	(	PUNCT
ejpam-5094	210	38	2	2	X
ejpam-5094	210	39	)	)	PUNCT
ejpam-5094	210	40	f(u	f(u	PROPN
ejpam-5094	210	41	)	)	PUNCT
ejpam-5094	211	1	⊆	⊆	NUM
ejpam-5094	212	1	[	[	X
ejpam-5094	212	2	f(u	f(u	PROPN
ejpam-5094	212	3	(	(	PUNCT
ejpam-5094	212	4	λ	λ	PROPN
ejpam-5094	212	5	,	,	PUNCT
ejpam-5094	212	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	212	7	,	,	PUNCT
ejpam-5094	212	8	p	p	NOUN
ejpam-5094	212	9	)	)	PUNCT
ejpam-5094	212	10	for	for	ADP
ejpam-5094	212	11	each	each	DET
ejpam-5094	212	12	p(λ	p(λ	NOUN
ejpam-5094	212	13	,	,	PUNCT
ejpam-5094	212	14	p)-open	p)-open	VERB
ejpam-5094	212	15	set	set	VERB
ejpam-5094	212	16	u	u	NOUN
ejpam-5094	212	17	of	of	ADP
ejpam-5094	212	18	x	x	PRON
ejpam-5094	212	19	;	;	PUNCT
ejpam-5094	212	20	(	(	PUNCT
ejpam-5094	212	21	3	3	X
ejpam-5094	212	22	)	)	PUNCT
ejpam-5094	212	23	f(u	f(u	PROPN
ejpam-5094	212	24	)	)	PUNCT
ejpam-5094	212	25	⊆	⊆	NUM
ejpam-5094	213	1	[	[	X
ejpam-5094	213	2	f(u	f(u	PROPN
ejpam-5094	213	3	(	(	PUNCT
ejpam-5094	213	4	λ	λ	PROPN
ejpam-5094	213	5	,	,	PUNCT
ejpam-5094	213	6	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	213	7	,	,	PUNCT
ejpam-5094	213	8	p	p	NOUN
ejpam-5094	213	9	)	)	PUNCT
ejpam-5094	213	10	for	for	ADP
ejpam-5094	213	11	each	each	DET
ejpam-5094	213	12	α(λ	α(λ	PROPN
ejpam-5094	213	13	,	,	PUNCT
ejpam-5094	213	14	p)-open	p)-open	VERB
ejpam-5094	213	15	set	set	VERB
ejpam-5094	213	16	u	u	NOUN
ejpam-5094	213	17	of	of	ADP
ejpam-5094	213	18	x	x	PRON
ejpam-5094	213	19	;	;	PUNCT
ejpam-5094	213	20	(	(	PUNCT
ejpam-5094	213	21	4	4	X
ejpam-5094	213	22	)	)	PUNCT
ejpam-5094	213	23	f([u	f([u	NOUN
ejpam-5094	213	24	(	(	PUNCT
ejpam-5094	213	25	λ	λ	NOUN
ejpam-5094	213	26	,	,	PUNCT
ejpam-5094	213	27	p)](λ	p)](λ	ADJ
ejpam-5094	213	28	,	,	PUNCT
ejpam-5094	213	29	p	p	NOUN
ejpam-5094	213	30	)	)	PUNCT
ejpam-5094	213	31	)	)	PUNCT
ejpam-5094	214	1	⊆	⊆	NUM
ejpam-5094	214	2	[	[	X
ejpam-5094	214	3	f(u	f(u	PROPN
ejpam-5094	214	4	(	(	PUNCT
ejpam-5094	214	5	λ	λ	PROPN
ejpam-5094	214	6	,	,	PUNCT
ejpam-5094	214	7	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	214	8	,	,	PUNCT
ejpam-5094	214	9	p	p	NOUN
ejpam-5094	214	10	)	)	PUNCT
ejpam-5094	214	11	for	for	ADP
ejpam-5094	214	12	each	each	DET
ejpam-5094	214	13	(	(	PUNCT
ejpam-5094	214	14	λ	λ	PROPN
ejpam-5094	214	15	,	,	PUNCT
ejpam-5094	214	16	p)-open	p)-open	VERB
ejpam-5094	214	17	set	set	VERB
ejpam-5094	214	18	u	u	NOUN
ejpam-5094	214	19	of	of	ADP
ejpam-5094	214	20	x	x	PRON
ejpam-5094	214	21	;	;	PUNCT
ejpam-5094	214	22	(	(	PUNCT
ejpam-5094	214	23	5	5	X
ejpam-5094	214	24	)	)	PUNCT
ejpam-5094	214	25	f(k(λ	f(k(λ	NOUN
ejpam-5094	214	26	,	,	PUNCT
ejpam-5094	214	27	p	p	NOUN
ejpam-5094	214	28	)	)	PUNCT
ejpam-5094	214	29	)	)	PUNCT
ejpam-5094	215	1	⊆	⊆	NUM
ejpam-5094	215	2	[	[	X
ejpam-5094	215	3	f(k)]θb(λ	f(k)]θb(λ	ADJ
ejpam-5094	215	4	,	,	PUNCT
ejpam-5094	215	5	p	p	NOUN
ejpam-5094	215	6	)	)	PUNCT
ejpam-5094	215	7	for	for	ADP
ejpam-5094	215	8	each	each	DET
ejpam-5094	215	9	(	(	PUNCT
ejpam-5094	215	10	λ	λ	PROPN
ejpam-5094	215	11	,	,	PUNCT
ejpam-5094	215	12	p)-closed	p)-close	VERB
ejpam-5094	215	13	set	set	NOUN
ejpam-5094	215	14	k	k	PROPN
ejpam-5094	215	15	of	of	ADP
ejpam-5094	215	16	x.	x.	NOUN
ejpam-5094	215	17	4	4	NUM
ejpam-5094	215	18	.	.	PUNCT
ejpam-5094	216	1	weakly	weakly	ADJ
ejpam-5094	216	2	θb(λ	θb(λ	NUM
ejpam-5094	216	3	,	,	PUNCT
ejpam-5094	217	1	p)-closed	p)-close	VERB
ejpam-5094	217	2	functions	function	NOUN
ejpam-5094	217	3	in	in	ADP
ejpam-5094	217	4	this	this	DET
ejpam-5094	217	5	section	section	NOUN
ejpam-5094	217	6	,	,	PUNCT
ejpam-5094	217	7	we	we	PRON
ejpam-5094	217	8	introduce	introduce	VERB
ejpam-5094	217	9	the	the	DET
ejpam-5094	217	10	concept	concept	NOUN
ejpam-5094	217	11	of	of	ADP
ejpam-5094	217	12	weakly	weakly	ADJ
ejpam-5094	217	13	θb(λ	θb(λ	NUM
ejpam-5094	217	14	,	,	PUNCT
ejpam-5094	217	15	p)-closed	p)-close	VERB
ejpam-5094	217	16	functions	function	NOUN
ejpam-5094	217	17	.	.	PUNCT
ejpam-5094	218	1	furthermore	furthermore	ADV
ejpam-5094	218	2	,	,	PUNCT
ejpam-5094	218	3	some	some	DET
ejpam-5094	218	4	characterizations	characterization	NOUN
ejpam-5094	218	5	of	of	ADP
ejpam-5094	218	6	weakly	weakly	ADJ
ejpam-5094	218	7	θb(λ	θb(λ	NUM
ejpam-5094	218	8	,	,	PUNCT
ejpam-5094	218	9	p)-closed	p)-close	VERB
ejpam-5094	218	10	functions	function	NOUN
ejpam-5094	218	11	are	be	AUX
ejpam-5094	218	12	investigated	investigate	VERB
ejpam-5094	218	13	.	.	PUNCT
ejpam-5094	219	1	definition	definition	NOUN
ejpam-5094	219	2	2	2	NUM
ejpam-5094	219	3	.	.	PUNCT
ejpam-5094	220	1	a	a	DET
ejpam-5094	220	2	function	function	NOUN
ejpam-5094	220	3	f	f	NOUN
ejpam-5094	220	4	:	:	PUNCT
ejpam-5094	220	5	(	(	PUNCT
ejpam-5094	220	6	x	x	X
ejpam-5094	220	7	,	,	PUNCT
ejpam-5094	220	8	τ	τ	X
ejpam-5094	220	9	)	)	PUNCT
ejpam-5094	220	10	→	→	SYM
ejpam-5094	220	11	(	(	PUNCT
ejpam-5094	220	12	y	y	PROPN
ejpam-5094	220	13	,	,	PUNCT
ejpam-5094	220	14	σ	σ	PROPN
ejpam-5094	220	15	)	)	PUNCT
ejpam-5094	220	16	is	be	AUX
ejpam-5094	220	17	said	say	VERB
ejpam-5094	220	18	to	to	PART
ejpam-5094	220	19	be	be	AUX
ejpam-5094	220	20	weakly	weakly	ADJ
ejpam-5094	220	21	θb(λ	θb(λ	VERB
ejpam-5094	220	22	,	,	PUNCT
ejpam-5094	220	23	p)-closed	p)-close	VERB
ejpam-5094	220	24	if	if	SCONJ
ejpam-5094	220	25	[	[	X
ejpam-5094	220	26	f(k(λ	f(k(λ	NOUN
ejpam-5094	220	27	,	,	PUNCT
ejpam-5094	220	28	p	p	NOUN
ejpam-5094	220	29	)	)	PUNCT
ejpam-5094	220	30	)	)	PUNCT
ejpam-5094	220	31	]	]	PUNCT
ejpam-5094	221	1	θb(λ	θb(λ	X
ejpam-5094	221	2	,	,	PUNCT
ejpam-5094	221	3	p	p	NOUN
ejpam-5094	221	4	)	)	PUNCT
ejpam-5094	221	5	⊆	⊆	NUM
ejpam-5094	221	6	f(k	f(k	VERB
ejpam-5094	221	7	)	)	PUNCT
ejpam-5094	221	8	for	for	SCONJ
ejpam-5094	221	9	each	each	PRON
ejpam-5094	221	10	(	(	PUNCT
ejpam-5094	221	11	λ	λ	PROPN
ejpam-5094	221	12	,	,	PUNCT
ejpam-5094	221	13	p)-closed	p)-close	VERB
ejpam-5094	221	14	set	set	NOUN
ejpam-5094	221	15	k	k	PROPN
ejpam-5094	221	16	of	of	ADP
ejpam-5094	221	17	x.	x.	PROPN
ejpam-5094	221	18	theorem	theorem	VERB
ejpam-5094	221	19	5	5	NUM
ejpam-5094	221	20	.	.	PUNCT
ejpam-5094	222	1	for	for	ADP
ejpam-5094	222	2	a	a	DET
ejpam-5094	222	3	function	function	NOUN
ejpam-5094	222	4	f	f	NOUN
ejpam-5094	222	5	:	:	PUNCT
ejpam-5094	222	6	(	(	PUNCT
ejpam-5094	222	7	x	x	X
ejpam-5094	222	8	,	,	PUNCT
ejpam-5094	222	9	τ	τ	X
ejpam-5094	222	10	)	)	PUNCT
ejpam-5094	222	11	→	→	SYM
ejpam-5094	222	12	(	(	PUNCT
ejpam-5094	222	13	y	y	PROPN
ejpam-5094	222	14	,	,	PUNCT
ejpam-5094	222	15	σ	σ	PROPN
ejpam-5094	222	16	)	)	PUNCT
ejpam-5094	222	17	,	,	PUNCT
ejpam-5094	222	18	the	the	DET
ejpam-5094	222	19	following	follow	VERB
ejpam-5094	222	20	properties	property	NOUN
ejpam-5094	222	21	are	be	AUX
ejpam-5094	222	22	equivalent	equivalent	ADJ
ejpam-5094	222	23	:	:	PUNCT
ejpam-5094	222	24	(	(	PUNCT
ejpam-5094	222	25	1	1	X
ejpam-5094	222	26	)	)	PUNCT
ejpam-5094	222	27	f	f	PROPN
ejpam-5094	222	28	is	be	AUX
ejpam-5094	222	29	weakly	weakly	ADJ
ejpam-5094	222	30	θb(λ	θb(λ	NUM
ejpam-5094	222	31	,	,	PUNCT
ejpam-5094	222	32	p)-closed	p)-close	VERB
ejpam-5094	222	33	;	;	PUNCT
ejpam-5094	222	34	c.	c.	PROPN
ejpam-5094	222	35	boonpok	boonpok	PROPN
ejpam-5094	222	36	,	,	PUNCT
ejpam-5094	222	37	j.	j.	PROPN
ejpam-5094	222	38	khampakdee	khampakdee	PROPN
ejpam-5094	222	39	/	/	PUNCT
ejpam-5094	222	40	eur	eur	PROPN
ejpam-5094	222	41	.	.	PUNCT
ejpam-5094	223	1	j.	j.	PROPN
ejpam-5094	223	2	pure	pure	PROPN
ejpam-5094	223	3	appl	appl	PROPN
ejpam-5094	223	4	.	.	PROPN
ejpam-5094	223	5	math	math	PROPN
ejpam-5094	223	6	,	,	PUNCT
ejpam-5094	223	7	17	17	NUM
ejpam-5094	223	8	(	(	PUNCT
ejpam-5094	223	9	2	2	NUM
ejpam-5094	223	10	)	)	PUNCT
ejpam-5094	223	11	(	(	PUNCT
ejpam-5094	223	12	2024	2024	NUM
ejpam-5094	223	13	)	)	PUNCT
ejpam-5094	223	14	,	,	PUNCT
ejpam-5094	223	15	582	582	NUM
ejpam-5094	223	16	-	-	SYM
ejpam-5094	223	17	590	590	NUM
ejpam-5094	223	18	588	588	NUM
ejpam-5094	223	19	(	(	PUNCT
ejpam-5094	223	20	2	2	NUM
ejpam-5094	223	21	)	)	PUNCT
ejpam-5094	224	1	[	[	X
ejpam-5094	224	2	f(u)]θb(λ	f(u)]θb(λ	X
ejpam-5094	224	3	,	,	PUNCT
ejpam-5094	224	4	p	p	NOUN
ejpam-5094	224	5	)	)	PUNCT
ejpam-5094	224	6	⊆	⊆	NUM
ejpam-5094	224	7	f(u	f(u	PROPN
ejpam-5094	224	8	(	(	PUNCT
ejpam-5094	224	9	λ	λ	PROPN
ejpam-5094	224	10	,	,	PUNCT
ejpam-5094	224	11	p	p	NOUN
ejpam-5094	224	12	)	)	PUNCT
ejpam-5094	224	13	)	)	PUNCT
ejpam-5094	224	14	for	for	SCONJ
ejpam-5094	224	15	every	every	DET
ejpam-5094	224	16	(	(	PUNCT
ejpam-5094	224	17	λ	λ	NOUN
ejpam-5094	224	18	,	,	PUNCT
ejpam-5094	224	19	p)-open	p)-open	VERB
ejpam-5094	224	20	set	set	VERB
ejpam-5094	224	21	u	u	NOUN
ejpam-5094	224	22	of	of	ADP
ejpam-5094	224	23	x.	x.	NOUN
ejpam-5094	224	24	proof	proof	NOUN
ejpam-5094	224	25	.	.	PUNCT
ejpam-5094	225	1	(	(	PUNCT
ejpam-5094	225	2	1	1	X
ejpam-5094	225	3	)	)	PUNCT
ejpam-5094	225	4	⇒	⇒	NOUN
ejpam-5094	225	5	(	(	PUNCT
ejpam-5094	225	6	2	2	NUM
ejpam-5094	225	7	):	):	PUNCT
ejpam-5094	225	8	let	let	VERB
ejpam-5094	225	9	u	u	PRON
ejpam-5094	225	10	be	be	AUX
ejpam-5094	225	11	any	any	DET
ejpam-5094	225	12	(	(	PUNCT
ejpam-5094	225	13	λ	λ	NOUN
ejpam-5094	225	14	,	,	PUNCT
ejpam-5094	225	15	p)-open	p)-open	VERB
ejpam-5094	225	16	set	set	VERB
ejpam-5094	225	17	of	of	ADP
ejpam-5094	225	18	x.	x.	NOUN
ejpam-5094	225	19	since	since	SCONJ
ejpam-5094	225	20	u	u	PROPN
ejpam-5094	225	21	(	(	PUNCT
ejpam-5094	225	22	λ	λ	PROPN
ejpam-5094	225	23	,	,	PUNCT
ejpam-5094	225	24	p	p	NOUN
ejpam-5094	225	25	)	)	PUNCT
ejpam-5094	225	26	is	be	AUX
ejpam-5094	225	27	a	a	DET
ejpam-5094	225	28	(	(	PUNCT
ejpam-5094	225	29	λ	λ	PROPN
ejpam-5094	225	30	,	,	PUNCT
ejpam-5094	225	31	p)-closed	p)-close	VERB
ejpam-5094	225	32	set	set	NOUN
ejpam-5094	225	33	and	and	CCONJ
ejpam-5094	225	34	u	u	NOUN
ejpam-5094	225	35	⊆	⊆	NUM
ejpam-5094	225	36	[	[	X
ejpam-5094	225	37	u	u	X
ejpam-5094	225	38	(	(	PUNCT
ejpam-5094	225	39	λ	λ	PROPN
ejpam-5094	225	40	,	,	PUNCT
ejpam-5094	225	41	p)](λ	p)](λ	ADJ
ejpam-5094	225	42	,	,	PUNCT
ejpam-5094	225	43	p	p	NOUN
ejpam-5094	225	44	)	)	PUNCT
ejpam-5094	225	45	,	,	PUNCT
ejpam-5094	225	46	we	we	PRON
ejpam-5094	225	47	have	have	VERB
ejpam-5094	225	48	[	[	X
ejpam-5094	225	49	f(u)]θb(λ	f(u)]θb(λ	X
ejpam-5094	225	50	,	,	PUNCT
ejpam-5094	225	51	p	p	NOUN
ejpam-5094	225	52	)	)	PUNCT
ejpam-5094	225	53	⊆	⊆	NUM
ejpam-5094	226	1	[	[	X
ejpam-5094	226	2	f([u	f([u	INTJ
ejpam-5094	226	3	(	(	PUNCT
ejpam-5094	226	4	λ	λ	PROPN
ejpam-5094	226	5	,	,	PUNCT
ejpam-5094	226	6	p)](λ	p)](λ	ADJ
ejpam-5094	226	7	,	,	PUNCT
ejpam-5094	226	8	p	p	NOUN
ejpam-5094	226	9	)	)	PUNCT
ejpam-5094	226	10	)	)	PUNCT
ejpam-5094	226	11	]	]	PUNCT
ejpam-5094	227	1	θb(λ	θb(λ	X
ejpam-5094	227	2	,	,	PUNCT
ejpam-5094	227	3	p	p	NOUN
ejpam-5094	227	4	)	)	PUNCT
ejpam-5094	227	5	⊆	⊆	NUM
ejpam-5094	227	6	f(u	f(u	PROPN
ejpam-5094	227	7	(	(	PUNCT
ejpam-5094	227	8	λ	λ	PROPN
ejpam-5094	227	9	,	,	PUNCT
ejpam-5094	227	10	p	p	NOUN
ejpam-5094	227	11	)	)	PUNCT
ejpam-5094	227	12	)	)	PUNCT
ejpam-5094	227	13	.	.	PUNCT
ejpam-5094	228	1	(	(	PUNCT
ejpam-5094	228	2	2	2	X
ejpam-5094	228	3	)	)	PUNCT
ejpam-5094	228	4	⇒	⇒	NOUN
ejpam-5094	228	5	(	(	PUNCT
ejpam-5094	228	6	1	1	NUM
ejpam-5094	228	7	):	):	PUNCT
ejpam-5094	228	8	let	let	VERB
ejpam-5094	228	9	k	k	PRON
ejpam-5094	228	10	be	be	AUX
ejpam-5094	228	11	any	any	DET
ejpam-5094	228	12	(	(	PUNCT
ejpam-5094	228	13	λ	λ	PROPN
ejpam-5094	228	14	,	,	PUNCT
ejpam-5094	228	15	p)-closed	p)-close	VERB
ejpam-5094	228	16	set	set	NOUN
ejpam-5094	228	17	of	of	ADP
ejpam-5094	228	18	x.	x.	NOUN
ejpam-5094	228	19	then	then	ADV
ejpam-5094	228	20	,	,	PUNCT
ejpam-5094	228	21	we	we	PRON
ejpam-5094	228	22	have	have	VERB
ejpam-5094	228	23	[	[	X
ejpam-5094	228	24	f(k(λ	f(k(λ	NOUN
ejpam-5094	228	25	,	,	PUNCT
ejpam-5094	228	26	p	p	NOUN
ejpam-5094	228	27	)	)	PUNCT
ejpam-5094	228	28	)	)	PUNCT
ejpam-5094	228	29	]	]	PUNCT
ejpam-5094	229	1	θb(λ	θb(λ	X
ejpam-5094	229	2	,	,	PUNCT
ejpam-5094	229	3	p	p	NOUN
ejpam-5094	229	4	)	)	PUNCT
ejpam-5094	229	5	⊆	⊆	NUM
ejpam-5094	229	6	f([k(λ	f([k(λ	NOUN
ejpam-5094	229	7	,	,	PUNCT
ejpam-5094	229	8	p	p	NOUN
ejpam-5094	229	9	)	)	PUNCT
ejpam-5094	229	10	]	]	PUNCT
ejpam-5094	229	11	(	(	PUNCT
ejpam-5094	229	12	λ	λ	X
ejpam-5094	229	13	,	,	PUNCT
ejpam-5094	229	14	p	p	NOUN
ejpam-5094	229	15	)	)	PUNCT
ejpam-5094	229	16	)	)	PUNCT
ejpam-5094	229	17	⊆	⊆	NUM
ejpam-5094	230	1	f(k(λ	f(k(λ	NOUN
ejpam-5094	230	2	,	,	PUNCT
ejpam-5094	230	3	p	p	NOUN
ejpam-5094	230	4	)	)	PUNCT
ejpam-5094	230	5	)	)	PUNCT
ejpam-5094	230	6	=	=	PUNCT
ejpam-5094	230	7	f(k	f(k	VERB
ejpam-5094	230	8	)	)	PUNCT
ejpam-5094	230	9	and	and	CCONJ
ejpam-5094	230	10	hence	hence	ADV
ejpam-5094	230	11	f	f	PROPN
ejpam-5094	230	12	is	be	AUX
ejpam-5094	230	13	weakly	weakly	ADJ
ejpam-5094	230	14	θb(λ	θb(λ	NUM
ejpam-5094	230	15	,	,	PUNCT
ejpam-5094	230	16	p)-closed	p)-close	VERB
ejpam-5094	230	17	.	.	PUNCT
ejpam-5094	231	1	corollary	corollary	ADJ
ejpam-5094	231	2	1	1	NUM
ejpam-5094	231	3	.	.	PUNCT
ejpam-5094	232	1	a	a	DET
ejpam-5094	232	2	bijective	bijective	ADJ
ejpam-5094	232	3	function	function	NOUN
ejpam-5094	232	4	f	f	NOUN
ejpam-5094	232	5	:	:	PUNCT
ejpam-5094	232	6	(	(	PUNCT
ejpam-5094	232	7	x	x	X
ejpam-5094	232	8	,	,	PUNCT
ejpam-5094	232	9	τ	τ	X
ejpam-5094	232	10	)	)	PUNCT
ejpam-5094	232	11	→	→	SYM
ejpam-5094	232	12	(	(	PUNCT
ejpam-5094	232	13	y	y	PROPN
ejpam-5094	232	14	,	,	PUNCT
ejpam-5094	232	15	σ	σ	PROPN
ejpam-5094	232	16	)	)	PUNCT
ejpam-5094	232	17	is	be	AUX
ejpam-5094	232	18	weakly	weakly	ADJ
ejpam-5094	232	19	θb(λ	θb(λ	NUM
ejpam-5094	232	20	,	,	PUNCT
ejpam-5094	232	21	p)-open	p)-open	VERB
ejpam-5094	232	22	if	if	SCONJ
ejpam-5094	232	23	and	and	CCONJ
ejpam-5094	232	24	only	only	ADV
ejpam-5094	232	25	if	if	SCONJ
ejpam-5094	232	26	f	f	PROPN
ejpam-5094	232	27	is	be	AUX
ejpam-5094	232	28	weakly	weakly	ADJ
ejpam-5094	232	29	θb(λ	θb(λ	NUM
ejpam-5094	232	30	,	,	PUNCT
ejpam-5094	232	31	p)-closed	p)-close	VERB
ejpam-5094	232	32	.	.	PUNCT
ejpam-5094	233	1	proof	proof	NOUN
ejpam-5094	233	2	.	.	PUNCT
ejpam-5094	234	1	this	this	PRON
ejpam-5094	234	2	is	be	AUX
ejpam-5094	234	3	an	an	DET
ejpam-5094	234	4	immediate	immediate	ADJ
ejpam-5094	234	5	consequence	consequence	NOUN
ejpam-5094	234	6	of	of	ADP
ejpam-5094	234	7	theorem	theorem	ADJ
ejpam-5094	234	8	3	3	NUM
ejpam-5094	234	9	and	and	CCONJ
ejpam-5094	234	10	5	5	NUM
ejpam-5094	234	11	.	.	PUNCT
ejpam-5094	235	1	the	the	DET
ejpam-5094	235	2	proof	proof	NOUN
ejpam-5094	235	3	of	of	ADP
ejpam-5094	235	4	the	the	DET
ejpam-5094	235	5	following	follow	VERB
ejpam-5094	235	6	theorem	theorem	NOUN
ejpam-5094	235	7	is	be	AUX
ejpam-5094	235	8	straightforward	straightforward	ADJ
ejpam-5094	235	9	and	and	CCONJ
ejpam-5094	235	10	thus	thus	ADV
ejpam-5094	235	11	is	be	AUX
ejpam-5094	235	12	omitted	omit	VERB
ejpam-5094	235	13	.	.	PUNCT
ejpam-5094	236	1	theorem	theorem	VERB
ejpam-5094	236	2	6	6	NUM
ejpam-5094	236	3	.	.	PUNCT
ejpam-5094	236	4	for	for	ADP
ejpam-5094	236	5	a	a	DET
ejpam-5094	236	6	function	function	NOUN
ejpam-5094	236	7	f	f	NOUN
ejpam-5094	236	8	:	:	PUNCT
ejpam-5094	236	9	(	(	PUNCT
ejpam-5094	236	10	x	x	X
ejpam-5094	236	11	,	,	PUNCT
ejpam-5094	236	12	τ	τ	X
ejpam-5094	236	13	)	)	PUNCT
ejpam-5094	236	14	→	→	SYM
ejpam-5094	236	15	(	(	PUNCT
ejpam-5094	236	16	y	y	PROPN
ejpam-5094	236	17	,	,	PUNCT
ejpam-5094	236	18	σ	σ	PROPN
ejpam-5094	236	19	)	)	PUNCT
ejpam-5094	236	20	,	,	PUNCT
ejpam-5094	236	21	the	the	DET
ejpam-5094	236	22	following	follow	VERB
ejpam-5094	236	23	properties	property	NOUN
ejpam-5094	236	24	are	be	AUX
ejpam-5094	236	25	equivalent	equivalent	ADJ
ejpam-5094	236	26	:	:	PUNCT
ejpam-5094	236	27	(	(	PUNCT
ejpam-5094	236	28	1	1	X
ejpam-5094	236	29	)	)	PUNCT
ejpam-5094	236	30	f	f	PROPN
ejpam-5094	236	31	is	be	AUX
ejpam-5094	236	32	weakly	weakly	ADJ
ejpam-5094	236	33	θb(λ	θb(λ	NUM
ejpam-5094	236	34	,	,	PUNCT
ejpam-5094	236	35	p)-closed	p)-close	VERB
ejpam-5094	236	36	;	;	PUNCT
ejpam-5094	236	37	(	(	PUNCT
ejpam-5094	236	38	2	2	X
ejpam-5094	236	39	)	)	PUNCT
ejpam-5094	237	1	[	[	X
ejpam-5094	237	2	f(k(λ	f(k(λ	NOUN
ejpam-5094	237	3	,	,	PUNCT
ejpam-5094	237	4	p	p	NOUN
ejpam-5094	237	5	)	)	PUNCT
ejpam-5094	237	6	)	)	PUNCT
ejpam-5094	237	7	]	]	PUNCT
ejpam-5094	238	1	θb(λ	θb(λ	X
ejpam-5094	238	2	,	,	PUNCT
ejpam-5094	238	3	p	p	NOUN
ejpam-5094	238	4	)	)	PUNCT
ejpam-5094	238	5	⊆	⊆	NUM
ejpam-5094	238	6	f(k	f(k	VERB
ejpam-5094	238	7	)	)	PUNCT
ejpam-5094	238	8	for	for	ADP
ejpam-5094	238	9	every	every	DET
ejpam-5094	238	10	p(λ	p(λ	NOUN
ejpam-5094	238	11	,	,	PUNCT
ejpam-5094	238	12	p)-closed	p)-close	VERB
ejpam-5094	238	13	set	set	NOUN
ejpam-5094	238	14	k	k	PROPN
ejpam-5094	238	15	of	of	ADP
ejpam-5094	238	16	x	x	PROPN
ejpam-5094	238	17	;	;	PUNCT
ejpam-5094	238	18	(	(	PUNCT
ejpam-5094	238	19	3	3	X
ejpam-5094	238	20	)	)	PUNCT
ejpam-5094	239	1	[	[	X
ejpam-5094	239	2	f(k(λ	f(k(λ	NOUN
ejpam-5094	239	3	,	,	PUNCT
ejpam-5094	239	4	p	p	NOUN
ejpam-5094	239	5	)	)	PUNCT
ejpam-5094	239	6	)	)	PUNCT
ejpam-5094	239	7	]	]	PUNCT
ejpam-5094	240	1	θb(λ	θb(λ	X
ejpam-5094	240	2	,	,	PUNCT
ejpam-5094	240	3	p	p	NOUN
ejpam-5094	240	4	)	)	PUNCT
ejpam-5094	240	5	⊆	⊆	NUM
ejpam-5094	240	6	f(k	f(k	VERB
ejpam-5094	240	7	)	)	PUNCT
ejpam-5094	240	8	for	for	ADP
ejpam-5094	240	9	every	every	DET
ejpam-5094	240	10	α(λ	α(λ	PROPN
ejpam-5094	240	11	,	,	PUNCT
ejpam-5094	240	12	p)-closed	p)-close	VERB
ejpam-5094	240	13	set	set	NOUN
ejpam-5094	240	14	k	k	PROPN
ejpam-5094	240	15	of	of	ADP
ejpam-5094	240	16	x	x	PROPN
ejpam-5094	240	17	;	;	PUNCT
ejpam-5094	240	18	(	(	PUNCT
ejpam-5094	240	19	4	4	X
ejpam-5094	240	20	)	)	PUNCT
ejpam-5094	241	1	[	[	X
ejpam-5094	241	2	f([a(λ	f([a(λ	NOUN
ejpam-5094	241	3	,	,	PUNCT
ejpam-5094	241	4	p)](λ	p)](λ	X
ejpam-5094	241	5	,	,	PUNCT
ejpam-5094	241	6	p	p	NOUN
ejpam-5094	241	7	)	)	PUNCT
ejpam-5094	241	8	)	)	PUNCT
ejpam-5094	241	9	]	]	PUNCT
ejpam-5094	242	1	θb(λ	θb(λ	X
ejpam-5094	242	2	,	,	PUNCT
ejpam-5094	242	3	p	p	NOUN
ejpam-5094	242	4	)	)	PUNCT
ejpam-5094	242	5	⊆	⊆	NUM
ejpam-5094	242	6	f(a(λ	f(a(λ	NOUN
ejpam-5094	242	7	,	,	PUNCT
ejpam-5094	242	8	p	p	NOUN
ejpam-5094	242	9	)	)	PUNCT
ejpam-5094	242	10	)	)	PUNCT
ejpam-5094	242	11	for	for	ADP
ejpam-5094	242	12	every	every	DET
ejpam-5094	242	13	subset	subset	NOUN
ejpam-5094	242	14	a	a	PRON
ejpam-5094	242	15	of	of	ADP
ejpam-5094	242	16	x	x	PRON
ejpam-5094	242	17	;	;	PUNCT
ejpam-5094	242	18	(	(	PUNCT
ejpam-5094	242	19	5	5	X
ejpam-5094	242	20	)	)	PUNCT
ejpam-5094	243	1	[	[	X
ejpam-5094	243	2	f(u)]θb(λ	f(u)]θb(λ	X
ejpam-5094	243	3	,	,	PUNCT
ejpam-5094	243	4	p	p	NOUN
ejpam-5094	243	5	)	)	PUNCT
ejpam-5094	243	6	⊆	⊆	NUM
ejpam-5094	243	7	f(u	f(u	PROPN
ejpam-5094	243	8	(	(	PUNCT
ejpam-5094	243	9	λ	λ	PROPN
ejpam-5094	243	10	,	,	PUNCT
ejpam-5094	243	11	p	p	NOUN
ejpam-5094	243	12	)	)	PUNCT
ejpam-5094	243	13	)	)	PUNCT
ejpam-5094	243	14	for	for	ADP
ejpam-5094	243	15	every	every	DET
ejpam-5094	243	16	p(λ	p(λ	NOUN
ejpam-5094	243	17	,	,	PUNCT
ejpam-5094	243	18	p)-open	p)-open	VERB
ejpam-5094	243	19	set	set	VERB
ejpam-5094	243	20	u	u	NOUN
ejpam-5094	243	21	of	of	ADP
ejpam-5094	243	22	x.	x.	PROPN
ejpam-5094	243	23	theorem	theorem	VERB
ejpam-5094	243	24	7	7	NUM
ejpam-5094	243	25	.	.	X
ejpam-5094	243	26	for	for	ADP
ejpam-5094	243	27	a	a	DET
ejpam-5094	243	28	function	function	NOUN
ejpam-5094	243	29	f	f	NOUN
ejpam-5094	243	30	:	:	PUNCT
ejpam-5094	243	31	(	(	PUNCT
ejpam-5094	243	32	x	x	X
ejpam-5094	243	33	,	,	PUNCT
ejpam-5094	243	34	τ	τ	X
ejpam-5094	243	35	)	)	PUNCT
ejpam-5094	243	36	→	→	SYM
ejpam-5094	243	37	(	(	PUNCT
ejpam-5094	243	38	y	y	PROPN
ejpam-5094	243	39	,	,	PUNCT
ejpam-5094	243	40	σ	σ	PROPN
ejpam-5094	243	41	)	)	PUNCT
ejpam-5094	243	42	,	,	PUNCT
ejpam-5094	243	43	the	the	DET
ejpam-5094	243	44	following	follow	VERB
ejpam-5094	243	45	properties	property	NOUN
ejpam-5094	243	46	are	be	AUX
ejpam-5094	243	47	equivalent	equivalent	ADJ
ejpam-5094	243	48	:	:	PUNCT
ejpam-5094	243	49	(	(	PUNCT
ejpam-5094	243	50	1	1	X
ejpam-5094	243	51	)	)	PUNCT
ejpam-5094	243	52	f	f	PROPN
ejpam-5094	243	53	is	be	AUX
ejpam-5094	243	54	weakly	weakly	ADJ
ejpam-5094	243	55	θb(λ	θb(λ	NUM
ejpam-5094	243	56	,	,	PUNCT
ejpam-5094	243	57	p)-closed	p)-close	VERB
ejpam-5094	243	58	;	;	PUNCT
ejpam-5094	243	59	(	(	PUNCT
ejpam-5094	243	60	2	2	X
ejpam-5094	243	61	)	)	PUNCT
ejpam-5094	244	1	[	[	X
ejpam-5094	244	2	f(u)]θb(λ	f(u)]θb(λ	X
ejpam-5094	244	3	,	,	PUNCT
ejpam-5094	244	4	p	p	NOUN
ejpam-5094	244	5	)	)	PUNCT
ejpam-5094	244	6	⊆	⊆	NUM
ejpam-5094	244	7	f(u	f(u	PROPN
ejpam-5094	244	8	(	(	PUNCT
ejpam-5094	244	9	λ	λ	PROPN
ejpam-5094	244	10	,	,	PUNCT
ejpam-5094	244	11	p	p	NOUN
ejpam-5094	244	12	)	)	PUNCT
ejpam-5094	244	13	)	)	PUNCT
ejpam-5094	244	14	for	for	ADP
ejpam-5094	244	15	every	every	DET
ejpam-5094	244	16	r(λ	r(λ	NOUN
ejpam-5094	244	17	,	,	PUNCT
ejpam-5094	244	18	p)-open	p)-open	VERB
ejpam-5094	244	19	set	set	VERB
ejpam-5094	244	20	u	u	NOUN
ejpam-5094	244	21	of	of	ADP
ejpam-5094	244	22	x	x	PRON
ejpam-5094	244	23	;	;	PUNCT
ejpam-5094	244	24	(	(	PUNCT
ejpam-5094	244	25	3	3	X
ejpam-5094	244	26	)	)	PUNCT
ejpam-5094	244	27	for	for	ADP
ejpam-5094	244	28	each	each	DET
ejpam-5094	244	29	subset	subset	NOUN
ejpam-5094	244	30	b	b	PROPN
ejpam-5094	244	31	of	of	ADP
ejpam-5094	244	32	y	y	PROPN
ejpam-5094	244	33	and	and	CCONJ
ejpam-5094	244	34	each	each	PRON
ejpam-5094	244	35	(	(	PUNCT
ejpam-5094	244	36	λ	λ	NOUN
ejpam-5094	244	37	,	,	PUNCT
ejpam-5094	244	38	p)-open	p)-open	VERB
ejpam-5094	244	39	set	set	VERB
ejpam-5094	244	40	u	u	NOUN
ejpam-5094	244	41	of	of	ADP
ejpam-5094	244	42	x	x	PUNCT
ejpam-5094	244	43	with	with	ADP
ejpam-5094	244	44	f−1(b	f−1(b	PROPN
ejpam-5094	244	45	)	)	PUNCT
ejpam-5094	244	46	⊆	⊆	NUM
ejpam-5094	244	47	u	u	NOUN
ejpam-5094	244	48	,	,	PUNCT
ejpam-5094	244	49	there	there	PRON
ejpam-5094	244	50	exists	exist	VERB
ejpam-5094	244	51	a	a	DET
ejpam-5094	244	52	θb(λ	θb(λ	NUM
ejpam-5094	244	53	,	,	PUNCT
ejpam-5094	244	54	p)-open	p)-open	VERB
ejpam-5094	244	55	set	set	VERB
ejpam-5094	244	56	v	v	NOUN
ejpam-5094	244	57	of	of	ADP
ejpam-5094	244	58	y	y	PRON
ejpam-5094	244	59	such	such	ADJ
ejpam-5094	244	60	that	that	DET
ejpam-5094	244	61	b	b	PROPN
ejpam-5094	244	62	⊆	⊆	NUM
ejpam-5094	244	63	v	v	NOUN
ejpam-5094	244	64	and	and	CCONJ
ejpam-5094	244	65	f−1(v	f−1(v	NOUN
ejpam-5094	244	66	)	)	PUNCT
ejpam-5094	245	1	⊆	⊆	NUM
ejpam-5094	245	2	u	u	NOUN
ejpam-5094	245	3	(	(	PUNCT
ejpam-5094	245	4	λ	λ	PROPN
ejpam-5094	245	5	,	,	PUNCT
ejpam-5094	245	6	p	p	NOUN
ejpam-5094	245	7	)	)	PUNCT
ejpam-5094	245	8	;	;	PUNCT
ejpam-5094	245	9	(	(	PUNCT
ejpam-5094	245	10	4	4	X
ejpam-5094	245	11	)	)	PUNCT
ejpam-5094	245	12	for	for	ADP
ejpam-5094	245	13	each	each	DET
ejpam-5094	245	14	point	point	NOUN
ejpam-5094	245	15	y	y	PROPN
ejpam-5094	245	16	∈	∈	PROPN
ejpam-5094	245	17	y	y	PROPN
ejpam-5094	245	18	and	and	CCONJ
ejpam-5094	245	19	each	each	DET
ejpam-5094	245	20	(	(	PUNCT
ejpam-5094	245	21	λ	λ	NOUN
ejpam-5094	245	22	,	,	PUNCT
ejpam-5094	245	23	p)-open	p)-open	VERB
ejpam-5094	245	24	set	set	VERB
ejpam-5094	245	25	u	u	NOUN
ejpam-5094	245	26	of	of	ADP
ejpam-5094	245	27	x	x	PUNCT
ejpam-5094	245	28	with	with	ADP
ejpam-5094	245	29	f−1(y	f−1(y	PROPN
ejpam-5094	245	30	)	)	PUNCT
ejpam-5094	245	31	⊆	⊆	NUM
ejpam-5094	245	32	u	u	NOUN
ejpam-5094	245	33	,	,	PUNCT
ejpam-5094	245	34	there	there	PRON
ejpam-5094	245	35	exists	exist	VERB
ejpam-5094	245	36	a	a	DET
ejpam-5094	245	37	θb(λ	θb(λ	NUM
ejpam-5094	245	38	,	,	PUNCT
ejpam-5094	245	39	p)-open	p)-open	VERB
ejpam-5094	245	40	set	set	VERB
ejpam-5094	245	41	v	v	NOUN
ejpam-5094	245	42	of	of	ADP
ejpam-5094	245	43	y	y	NOUN
ejpam-5094	245	44	containing	contain	VERB
ejpam-5094	245	45	y	y	PROPN
ejpam-5094	245	46	and	and	CCONJ
ejpam-5094	245	47	f−1(v	f−1(v	PROPN
ejpam-5094	245	48	)	)	PUNCT
ejpam-5094	246	1	⊆	⊆	NUM
ejpam-5094	246	2	u	u	NOUN
ejpam-5094	246	3	(	(	PUNCT
ejpam-5094	246	4	λ	λ	PROPN
ejpam-5094	246	5	,	,	PUNCT
ejpam-5094	246	6	p	p	NOUN
ejpam-5094	246	7	)	)	PUNCT
ejpam-5094	246	8	.	.	PUNCT
ejpam-5094	247	1	proof	proof	NOUN
ejpam-5094	247	2	.	.	PUNCT
ejpam-5094	248	1	(	(	PUNCT
ejpam-5094	248	2	1	1	X
ejpam-5094	248	3	)	)	PUNCT
ejpam-5094	248	4	⇒	⇒	NOUN
ejpam-5094	248	5	(	(	PUNCT
ejpam-5094	248	6	2	2	NUM
ejpam-5094	248	7	):	):	PUNCT
ejpam-5094	248	8	by	by	ADP
ejpam-5094	248	9	theorem	theorem	NOUN
ejpam-5094	248	10	5	5	NUM
ejpam-5094	248	11	.	.	PUNCT
ejpam-5094	248	12	(	(	PUNCT
ejpam-5094	248	13	2	2	X
ejpam-5094	248	14	)	)	PUNCT
ejpam-5094	248	15	⇒	⇒	NOUN
ejpam-5094	248	16	(	(	PUNCT
ejpam-5094	248	17	3	3	NUM
ejpam-5094	248	18	):	):	PUNCT
ejpam-5094	248	19	let	let	VERB
ejpam-5094	248	20	b	b	X
ejpam-5094	248	21	be	be	AUX
ejpam-5094	248	22	any	any	DET
ejpam-5094	248	23	subset	subset	NOUN
ejpam-5094	248	24	of	of	ADP
ejpam-5094	248	25	y	y	PROPN
ejpam-5094	248	26	and	and	CCONJ
ejpam-5094	248	27	u	u	NOUN
ejpam-5094	248	28	be	be	VERB
ejpam-5094	248	29	any	any	DET
ejpam-5094	248	30	(	(	PUNCT
ejpam-5094	248	31	λ	λ	NOUN
ejpam-5094	248	32	,	,	PUNCT
ejpam-5094	248	33	p)-open	p)-open	VERB
ejpam-5094	248	34	set	set	VERB
ejpam-5094	248	35	of	of	ADP
ejpam-5094	248	36	x	x	PUNCT
ejpam-5094	248	37	with	with	ADP
ejpam-5094	248	38	f−1(b	f−1(b	PROPN
ejpam-5094	248	39	)	)	PUNCT
ejpam-5094	248	40	⊆	⊆	NUM
ejpam-5094	248	41	u	u	NOUN
ejpam-5094	248	42	.	.	PUNCT
ejpam-5094	249	1	then	then	ADV
ejpam-5094	249	2	,	,	PUNCT
ejpam-5094	249	3	we	we	PRON
ejpam-5094	249	4	have	have	VERB
ejpam-5094	249	5	f−1(b	f−1(b	PROPN
ejpam-5094	249	6	)	)	PUNCT
ejpam-5094	249	7	∩	∩	NOUN
ejpam-5094	250	1	[	[	X
ejpam-5094	250	2	x	x	X
ejpam-5094	250	3	−	−	PROPN
ejpam-5094	250	4	u	u	NOUN
ejpam-5094	250	5	(	(	PUNCT
ejpam-5094	250	6	λ	λ	PROPN
ejpam-5094	250	7	,	,	PUNCT
ejpam-5094	250	8	p)](λ	p)](λ	ADJ
ejpam-5094	250	9	,	,	PUNCT
ejpam-5094	250	10	p	p	NOUN
ejpam-5094	250	11	)	)	PUNCT
ejpam-5094	250	12	=	=	NOUN
ejpam-5094	250	13	∅	∅	NOUN
ejpam-5094	250	14	and	and	CCONJ
ejpam-5094	250	15	hence	hence	ADV
ejpam-5094	250	16	b	b	NOUN
ejpam-5094	250	17	∩	∩	NOUN
ejpam-5094	250	18	f([x	f([x	NOUN
ejpam-5094	250	19	−	−	PROPN
ejpam-5094	250	20	u	u	PROPN
ejpam-5094	250	21	(	(	PUNCT
ejpam-5094	250	22	λ	λ	PROPN
ejpam-5094	250	23	,	,	PUNCT
ejpam-5094	250	24	p)](λ	p)](λ	ADJ
ejpam-5094	250	25	,	,	PUNCT
ejpam-5094	250	26	p	p	NOUN
ejpam-5094	250	27	)	)	PUNCT
ejpam-5094	250	28	)	)	PUNCT
ejpam-5094	251	1	=	=	PUNCT
ejpam-5094	251	2	∅.	∅.	VERB
ejpam-5094	251	3	since	since	SCONJ
ejpam-5094	251	4	x	x	INTJ
ejpam-5094	251	5	−	−	PROPN
ejpam-5094	251	6	u	u	NOUN
ejpam-5094	251	7	(	(	PUNCT
ejpam-5094	251	8	λ	λ	PROPN
ejpam-5094	251	9	,	,	PUNCT
ejpam-5094	251	10	p	p	NOUN
ejpam-5094	251	11	)	)	PUNCT
ejpam-5094	251	12	is	be	AUX
ejpam-5094	251	13	r(λ	r(λ	NOUN
ejpam-5094	251	14	,	,	PUNCT
ejpam-5094	251	15	p)-open	p)-open	ADJ
ejpam-5094	251	16	,	,	PUNCT
ejpam-5094	251	17	b	b	NOUN
ejpam-5094	251	18	∩	∩	NOUN
ejpam-5094	251	19	[	[	X
ejpam-5094	251	20	f(x	f(x	PROPN
ejpam-5094	251	21	−	−	PROPN
ejpam-5094	251	22	u	u	PROPN
ejpam-5094	251	23	(	(	PUNCT
ejpam-5094	251	24	λ	λ	PROPN
ejpam-5094	251	25	,	,	PUNCT
ejpam-5094	251	26	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	251	27	,	,	PUNCT
ejpam-5094	251	28	p	p	NOUN
ejpam-5094	251	29	)	)	PUNCT
ejpam-5094	251	30	=	=	VERB
ejpam-5094	251	31	∅.	∅.	AUX
ejpam-5094	251	32	let	let	VERB
ejpam-5094	251	33	v	v	VERB
ejpam-5094	251	34	=	=	SYM
ejpam-5094	251	35	y	y	NOUN
ejpam-5094	251	36	−	−	PROPN
ejpam-5094	252	1	[	[	X
ejpam-5094	252	2	f(x	f(x	PROPN
ejpam-5094	252	3	−	−	PROPN
ejpam-5094	252	4	u	u	PROPN
ejpam-5094	252	5	(	(	PUNCT
ejpam-5094	252	6	λ	λ	PROPN
ejpam-5094	252	7	,	,	PUNCT
ejpam-5094	252	8	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	252	9	,	,	PUNCT
ejpam-5094	252	10	p	p	NOUN
ejpam-5094	252	11	)	)	PUNCT
ejpam-5094	252	12	.	.	PUNCT
ejpam-5094	253	1	references	reference	NOUN
ejpam-5094	253	2	589	589	NUM
ejpam-5094	253	3	then	then	ADV
ejpam-5094	253	4	,	,	PUNCT
ejpam-5094	253	5	v	v	NOUN
ejpam-5094	253	6	is	be	AUX
ejpam-5094	253	7	a	a	DET
ejpam-5094	253	8	θb(λ	θb(λ	NUM
ejpam-5094	253	9	,	,	PUNCT
ejpam-5094	253	10	p)-open	p)-open	VERB
ejpam-5094	253	11	set	set	VERB
ejpam-5094	253	12	with	with	ADP
ejpam-5094	253	13	b	b	PROPN
ejpam-5094	253	14	⊆	⊆	NUM
ejpam-5094	253	15	v	v	NOUN
ejpam-5094	253	16	and	and	CCONJ
ejpam-5094	253	17	f−1(v	f−1(v	NOUN
ejpam-5094	253	18	)	)	PUNCT
ejpam-5094	254	1	⊆	⊆	NUM
ejpam-5094	254	2	x	x	SYM
ejpam-5094	254	3	−	−	NOUN
ejpam-5094	254	4	f−1([f(x	f−1([f(x	SYM
ejpam-5094	254	5	−	−	PROPN
ejpam-5094	254	6	u	u	NOUN
ejpam-5094	254	7	(	(	PUNCT
ejpam-5094	254	8	λ	λ	PROPN
ejpam-5094	254	9	,	,	PUNCT
ejpam-5094	254	10	p))]θb(λ	p))]θb(λ	PROPN
ejpam-5094	254	11	,	,	PUNCT
ejpam-5094	254	12	p	p	NOUN
ejpam-5094	254	13	)	)	PUNCT
ejpam-5094	254	14	)	)	PUNCT
ejpam-5094	255	1	⊆	⊆	NUM
ejpam-5094	255	2	x	x	SYM
ejpam-5094	255	3	−	−	NOUN
ejpam-5094	255	4	f−1(f(x	f−1(f(x	NOUN
ejpam-5094	255	5	−	−	PROPN
ejpam-5094	255	6	u	u	PROPN
ejpam-5094	255	7	(	(	PUNCT
ejpam-5094	255	8	λ	λ	PROPN
ejpam-5094	255	9	,	,	PUNCT
ejpam-5094	255	10	p	p	NOUN
ejpam-5094	255	11	)	)	PUNCT
ejpam-5094	255	12	)	)	PUNCT
ejpam-5094	255	13	)	)	PUNCT
ejpam-5094	256	1	⊆	⊆	X
ejpam-5094	256	2	u	u	NOUN
ejpam-5094	256	3	(	(	PUNCT
ejpam-5094	256	4	λ	λ	PROPN
ejpam-5094	256	5	,	,	PUNCT
ejpam-5094	256	6	p	p	NOUN
ejpam-5094	256	7	)	)	PUNCT
ejpam-5094	256	8	.	.	PUNCT
ejpam-5094	257	1	(	(	PUNCT
ejpam-5094	257	2	3	3	X
ejpam-5094	257	3	)	)	PUNCT
ejpam-5094	257	4	⇒	⇒	NOUN
ejpam-5094	257	5	(	(	PUNCT
ejpam-5094	257	6	4	4	NUM
ejpam-5094	257	7	):	):	PUNCT
ejpam-5094	257	8	this	this	PRON
ejpam-5094	257	9	is	be	AUX
ejpam-5094	257	10	obvious	obvious	ADJ
ejpam-5094	257	11	.	.	PUNCT
ejpam-5094	258	1	(	(	PUNCT
ejpam-5094	258	2	4	4	X
ejpam-5094	258	3	)	)	PUNCT
ejpam-5094	258	4	⇒	⇒	NOUN
ejpam-5094	258	5	(	(	PUNCT
ejpam-5094	258	6	1	1	NUM
ejpam-5094	258	7	):	):	PUNCT
ejpam-5094	258	8	let	let	VERB
ejpam-5094	258	9	k	k	PRON
ejpam-5094	258	10	be	be	AUX
ejpam-5094	258	11	any	any	DET
ejpam-5094	258	12	(	(	PUNCT
ejpam-5094	258	13	λ	λ	PROPN
ejpam-5094	258	14	,	,	PUNCT
ejpam-5094	258	15	p)-closed	p)-close	VERB
ejpam-5094	258	16	set	set	NOUN
ejpam-5094	258	17	of	of	ADP
ejpam-5094	258	18	y	y	PROPN
ejpam-5094	258	19	and	and	CCONJ
ejpam-5094	258	20	y	y	PROPN
ejpam-5094	258	21	∈	∈	PROPN
ejpam-5094	258	22	y	y	PROPN
ejpam-5094	258	23	−	−	PROPN
ejpam-5094	258	24	f(k	f(k	PROPN
ejpam-5094	258	25	)	)	PUNCT
ejpam-5094	258	26	.	.	PUNCT
ejpam-5094	259	1	since	since	SCONJ
ejpam-5094	259	2	f−1(y	f−1(y	PROPN
ejpam-5094	259	3	)	)	PUNCT
ejpam-5094	259	4	⊆	⊆	NUM
ejpam-5094	259	5	x	x	PUNCT
ejpam-5094	259	6	−k	−k	PROPN
ejpam-5094	259	7	,	,	PUNCT
ejpam-5094	259	8	by	by	ADP
ejpam-5094	259	9	(	(	PUNCT
ejpam-5094	259	10	4	4	X
ejpam-5094	259	11	)	)	PUNCT
ejpam-5094	259	12	there	there	PRON
ejpam-5094	259	13	exists	exist	VERB
ejpam-5094	259	14	a	a	DET
ejpam-5094	259	15	θb(λ	θb(λ	NUM
ejpam-5094	259	16	,	,	PUNCT
ejpam-5094	259	17	p)-open	p)-open	VERB
ejpam-5094	259	18	set	set	VERB
ejpam-5094	259	19	v	v	NOUN
ejpam-5094	259	20	of	of	ADP
ejpam-5094	259	21	y	y	PRON
ejpam-5094	259	22	such	such	ADJ
ejpam-5094	259	23	that	that	SCONJ
ejpam-5094	259	24	y	y	PROPN
ejpam-5094	259	25	∈	∈	PROPN
ejpam-5094	259	26	v	v	NOUN
ejpam-5094	259	27	and	and	CCONJ
ejpam-5094	259	28	f−1(v	f−1(v	NOUN
ejpam-5094	259	29	)	)	PUNCT
ejpam-5094	260	1	⊆	⊆	NUM
ejpam-5094	261	1	[	[	X
ejpam-5094	261	2	x	x	SYM
ejpam-5094	261	3	−k](λ	−k](λ	NUM
ejpam-5094	261	4	,	,	PUNCT
ejpam-5094	261	5	p	p	NOUN
ejpam-5094	261	6	)	)	PUNCT
ejpam-5094	261	7	=	=	PUNCT
ejpam-5094	261	8	x	x	SYM
ejpam-5094	261	9	−k(λ	−k(λ	NOUN
ejpam-5094	261	10	,	,	PUNCT
ejpam-5094	261	11	p	p	NOUN
ejpam-5094	261	12	)	)	PUNCT
ejpam-5094	261	13	.	.	PUNCT
ejpam-5094	262	1	thus	thus	ADV
ejpam-5094	262	2	,	,	PUNCT
ejpam-5094	262	3	v	v	ADP
ejpam-5094	262	4	∩	∩	ADJ
ejpam-5094	262	5	f(k(λ	f(k(λ	NOUN
ejpam-5094	262	6	,	,	PUNCT
ejpam-5094	262	7	p	p	NOUN
ejpam-5094	262	8	)	)	PUNCT
ejpam-5094	262	9	)	)	PUNCT
ejpam-5094	263	1	=	=	NOUN
ejpam-5094	263	2	∅	∅	NOUN
ejpam-5094	263	3	and	and	CCONJ
ejpam-5094	263	4	hence	hence	ADV
ejpam-5094	263	5	y	y	PROPN
ejpam-5094	263	6	̸∈	̸∈	PROPN
ejpam-5094	263	7	[	[	X
ejpam-5094	263	8	f(k(λ	f(k(λ	PROPN
ejpam-5094	263	9	,	,	PUNCT
ejpam-5094	263	10	p	p	NOUN
ejpam-5094	263	11	)	)	PUNCT
ejpam-5094	263	12	)	)	PUNCT
ejpam-5094	263	13	]	]	PUNCT
ejpam-5094	264	1	θb(λ	θb(λ	X
ejpam-5094	264	2	,	,	PUNCT
ejpam-5094	264	3	p	p	NOUN
ejpam-5094	264	4	)	)	PUNCT
ejpam-5094	264	5	.	.	PUNCT
ejpam-5094	265	1	therefore	therefore	ADV
ejpam-5094	265	2	,	,	PUNCT
ejpam-5094	265	3	[	[	X
ejpam-5094	265	4	f(k(λ	f(k(λ	NOUN
ejpam-5094	265	5	,	,	PUNCT
ejpam-5094	265	6	p	p	NOUN
ejpam-5094	265	7	)	)	PUNCT
ejpam-5094	265	8	)	)	PUNCT
ejpam-5094	265	9	]	]	PUNCT
ejpam-5094	266	1	θb(λ	θb(λ	X
ejpam-5094	266	2	,	,	PUNCT
ejpam-5094	266	3	p	p	NOUN
ejpam-5094	266	4	)	)	PUNCT
ejpam-5094	266	5	⊆	⊆	NUM
ejpam-5094	266	6	f(k	f(k	VERB
ejpam-5094	266	7	)	)	PUNCT
ejpam-5094	266	8	.	.	PUNCT
ejpam-5094	267	1	this	this	PRON
ejpam-5094	267	2	shows	show	VERB
ejpam-5094	267	3	that	that	SCONJ
ejpam-5094	267	4	f	f	PROPN
ejpam-5094	267	5	is	be	AUX
ejpam-5094	267	6	weakly	weakly	ADJ
ejpam-5094	267	7	θb(λ	θb(λ	NUM
ejpam-5094	267	8	,	,	PUNCT
ejpam-5094	267	9	p)-closed	p)-close	VERB
ejpam-5094	267	10	.	.	PUNCT
ejpam-5094	268	1	theorem	theorem	VERB
ejpam-5094	268	2	8	8	NUM
ejpam-5094	268	3	.	.	PUNCT
ejpam-5094	269	1	if	if	SCONJ
ejpam-5094	269	2	f	f	PROPN
ejpam-5094	269	3	:	:	PUNCT
ejpam-5094	269	4	(	(	PUNCT
ejpam-5094	269	5	x	x	X
ejpam-5094	269	6	,	,	PUNCT
ejpam-5094	269	7	τ	τ	X
ejpam-5094	269	8	)	)	PUNCT
ejpam-5094	269	9	→	→	SYM
ejpam-5094	269	10	(	(	PUNCT
ejpam-5094	269	11	y	y	PROPN
ejpam-5094	269	12	,	,	PUNCT
ejpam-5094	269	13	σ	σ	PROPN
ejpam-5094	269	14	)	)	PUNCT
ejpam-5094	269	15	is	be	AUX
ejpam-5094	269	16	a	a	DET
ejpam-5094	269	17	bijective	bijective	ADJ
ejpam-5094	269	18	weakly	weakly	ADJ
ejpam-5094	269	19	θb(λ	θb(λ	NUM
ejpam-5094	269	20	,	,	PUNCT
ejpam-5094	269	21	p)-closed	p)-close	VERB
ejpam-5094	269	22	function	function	NOUN
ejpam-5094	269	23	,	,	PUNCT
ejpam-5094	269	24	then	then	ADV
ejpam-5094	269	25	for	for	SCONJ
ejpam-5094	269	26	every	every	DET
ejpam-5094	269	27	subset	subset	NOUN
ejpam-5094	269	28	b	b	PROPN
ejpam-5094	269	29	of	of	ADP
ejpam-5094	269	30	y	y	PROPN
ejpam-5094	269	31	and	and	CCONJ
ejpam-5094	269	32	every	every	DET
ejpam-5094	269	33	(	(	PUNCT
ejpam-5094	269	34	λ	λ	NOUN
ejpam-5094	269	35	,	,	PUNCT
ejpam-5094	269	36	p)-open	p)-open	VERB
ejpam-5094	269	37	set	set	VERB
ejpam-5094	269	38	u	u	NOUN
ejpam-5094	269	39	of	of	ADP
ejpam-5094	269	40	x	x	PUNCT
ejpam-5094	269	41	with	with	ADP
ejpam-5094	269	42	f−1(b	f−1(b	PROPN
ejpam-5094	269	43	)	)	PUNCT
ejpam-5094	269	44	⊆	⊆	NUM
ejpam-5094	269	45	u	u	NOUN
ejpam-5094	269	46	,	,	PUNCT
ejpam-5094	269	47	there	there	PRON
ejpam-5094	269	48	exists	exist	VERB
ejpam-5094	269	49	a	a	DET
ejpam-5094	269	50	θb(λ	θb(λ	NUM
ejpam-5094	269	51	,	,	PUNCT
ejpam-5094	269	52	p)-closed	p)-close	VERB
ejpam-5094	269	53	set	set	NOUN
ejpam-5094	269	54	k	k	PROPN
ejpam-5094	269	55	of	of	ADP
ejpam-5094	269	56	y	y	PRON
ejpam-5094	270	1	such	such	ADJ
ejpam-5094	270	2	that	that	PRON
ejpam-5094	270	3	b	b	PROPN
ejpam-5094	270	4	⊆	⊆	NUM
ejpam-5094	270	5	k	k	PROPN
ejpam-5094	270	6	and	and	CCONJ
ejpam-5094	270	7	f−1(k	f−1(k	PROPN
ejpam-5094	270	8	)	)	PUNCT
ejpam-5094	270	9	⊆	⊆	NUM
ejpam-5094	270	10	u	u	NOUN
ejpam-5094	270	11	(	(	PUNCT
ejpam-5094	270	12	λ	λ	PROPN
ejpam-5094	270	13	,	,	PUNCT
ejpam-5094	270	14	p	p	NOUN
ejpam-5094	270	15	)	)	PUNCT
ejpam-5094	270	16	.	.	PUNCT
ejpam-5094	270	17	proof	proof	NOUN
ejpam-5094	270	18	.	.	PUNCT
ejpam-5094	271	1	let	let	VERB
ejpam-5094	271	2	b	b	X
ejpam-5094	271	3	be	be	AUX
ejpam-5094	271	4	any	any	DET
ejpam-5094	271	5	subset	subset	NOUN
ejpam-5094	271	6	of	of	ADP
ejpam-5094	271	7	y	y	PROPN
ejpam-5094	271	8	and	and	CCONJ
ejpam-5094	271	9	u	u	NOUN
ejpam-5094	271	10	be	be	VERB
ejpam-5094	271	11	any	any	DET
ejpam-5094	271	12	(	(	PUNCT
ejpam-5094	271	13	λ	λ	NOUN
ejpam-5094	271	14	,	,	PUNCT
ejpam-5094	271	15	p)-open	p)-open	VERB
ejpam-5094	271	16	set	set	VERB
ejpam-5094	271	17	of	of	ADP
ejpam-5094	271	18	x	x	PUNCT
ejpam-5094	271	19	with	with	ADP
ejpam-5094	271	20	f−1(b	f−1(b	PROPN
ejpam-5094	271	21	)	)	PUNCT
ejpam-5094	271	22	⊆	⊆	NUM
ejpam-5094	271	23	u	u	NOUN
ejpam-5094	271	24	.	.	PUNCT
ejpam-5094	272	1	put	put	VERB
ejpam-5094	272	2	k	k	NOUN
ejpam-5094	273	1	=	=	PUNCT
ejpam-5094	274	1	[	[	X
ejpam-5094	274	2	f([u	f([u	ADJ
ejpam-5094	274	3	(	(	PUNCT
ejpam-5094	274	4	λ	λ	PROPN
ejpam-5094	274	5	,	,	PUNCT
ejpam-5094	274	6	p)](λ	p)](λ	ADJ
ejpam-5094	274	7	,	,	PUNCT
ejpam-5094	274	8	p	p	NOUN
ejpam-5094	274	9	)	)	PUNCT
ejpam-5094	274	10	)	)	PUNCT
ejpam-5094	274	11	]	]	PUNCT
ejpam-5094	275	1	θb(λ	θb(λ	X
ejpam-5094	275	2	,	,	PUNCT
ejpam-5094	275	3	p	p	NOUN
ejpam-5094	275	4	)	)	PUNCT
ejpam-5094	275	5	.	.	PUNCT
ejpam-5094	276	1	then	then	ADV
ejpam-5094	276	2	,	,	PUNCT
ejpam-5094	276	3	k	k	PROPN
ejpam-5094	276	4	is	be	AUX
ejpam-5094	276	5	a	a	DET
ejpam-5094	276	6	θb(λ	θb(λ	NUM
ejpam-5094	276	7	,	,	PUNCT
ejpam-5094	276	8	p)-closed	p)-close	VERB
ejpam-5094	276	9	set	set	NOUN
ejpam-5094	276	10	of	of	ADP
ejpam-5094	276	11	y	y	PRON
ejpam-5094	276	12	such	such	ADJ
ejpam-5094	276	13	that	that	PRON
ejpam-5094	276	14	b	b	PROPN
ejpam-5094	276	15	⊆	⊆	NUM
ejpam-5094	276	16	k	k	NOUN
ejpam-5094	276	17	,	,	PUNCT
ejpam-5094	276	18	since	since	SCONJ
ejpam-5094	276	19	b	b	PROPN
ejpam-5094	276	20	⊆	⊆	NUM
ejpam-5094	276	21	f(u	f(u	PROPN
ejpam-5094	276	22	)	)	PUNCT
ejpam-5094	276	23	⊆	⊆	NUM
ejpam-5094	276	24	f([u	f([u	NUM
ejpam-5094	276	25	(	(	PUNCT
ejpam-5094	276	26	λ	λ	PROPN
ejpam-5094	276	27	,	,	PUNCT
ejpam-5094	276	28	p)](λ	p)](λ	ADJ
ejpam-5094	276	29	,	,	PUNCT
ejpam-5094	276	30	p	p	NOUN
ejpam-5094	276	31	)	)	PUNCT
ejpam-5094	276	32	)	)	PUNCT
ejpam-5094	277	1	⊆	⊆	NUM
ejpam-5094	277	2	[	[	X
ejpam-5094	277	3	f([u	f([u	INTJ
ejpam-5094	277	4	(	(	PUNCT
ejpam-5094	277	5	λ	λ	PROPN
ejpam-5094	277	6	,	,	PUNCT
ejpam-5094	277	7	p)](λ	p)](λ	ADJ
ejpam-5094	277	8	,	,	PUNCT
ejpam-5094	277	9	p	p	NOUN
ejpam-5094	277	10	)	)	PUNCT
ejpam-5094	277	11	)	)	PUNCT
ejpam-5094	277	12	]	]	PUNCT
ejpam-5094	278	1	θb(λ	θb(λ	X
ejpam-5094	278	2	,	,	PUNCT
ejpam-5094	278	3	p	p	NOUN
ejpam-5094	278	4	)	)	PUNCT
ejpam-5094	278	5	=	=	SYM
ejpam-5094	279	1	k.	k.	PROPN
ejpam-5094	279	2	since	since	SCONJ
ejpam-5094	279	3	f	f	PROPN
ejpam-5094	279	4	is	be	AUX
ejpam-5094	279	5	weakly	weakly	ADJ
ejpam-5094	279	6	θb(λ	θb(λ	NUM
ejpam-5094	279	7	,	,	PUNCT
ejpam-5094	279	8	p)-closed	p)-close	VERB
ejpam-5094	279	9	,	,	PUNCT
ejpam-5094	279	10	by	by	ADP
ejpam-5094	279	11	theorem	theorem	NOUN
ejpam-5094	279	12	6	6	NUM
ejpam-5094	279	13	we	we	PRON
ejpam-5094	279	14	have	have	VERB
ejpam-5094	279	15	f−1(k	f−1(k	PROPN
ejpam-5094	279	16	)	)	PUNCT
ejpam-5094	280	1	⊆	⊆	NUM
ejpam-5094	280	2	u	u	NOUN
ejpam-5094	280	3	(	(	PUNCT
ejpam-5094	280	4	λ	λ	PROPN
ejpam-5094	280	5	,	,	PUNCT
ejpam-5094	280	6	p	p	NOUN
ejpam-5094	280	7	)	)	PUNCT
ejpam-5094	280	8	.	.	PUNCT
ejpam-5094	281	1	acknowledgements	acknowledgement	NOUN
ejpam-5094	281	2	this	this	DET
ejpam-5094	281	3	research	research	NOUN
ejpam-5094	281	4	project	project	NOUN
ejpam-5094	281	5	was	be	AUX
ejpam-5094	281	6	financially	financially	ADV
ejpam-5094	281	7	supported	support	VERB
ejpam-5094	281	8	by	by	ADP
ejpam-5094	281	9	mahasarakham	mahasarakham	PROPN
ejpam-5094	281	10	university	university	PROPN
ejpam-5094	281	11	.	.	PUNCT
ejpam-5094	282	1	references	reference	NOUN
ejpam-5094	282	2	[	[	X
ejpam-5094	282	3	1	1	NUM
ejpam-5094	282	4	]	]	PUNCT
ejpam-5094	282	5	d.	d.	PROPN
ejpam-5094	282	6	andrijević.	andrijević.	PROPN
ejpam-5094	282	7	semi	semi	ADJ
ejpam-5094	282	8	-	-	ADJ
ejpam-5094	282	9	preopen	preopen	ADJ
ejpam-5094	282	10	sets	set	NOUN
ejpam-5094	282	11	.	.	PUNCT
ejpam-5094	283	1	matematički	matematički	PROPN
ejpam-5094	283	2	vesnik	vesnik	PROPN
ejpam-5094	283	3	,	,	PUNCT
ejpam-5094	283	4	38(1):24–32	38(1):24–32	NUM
ejpam-5094	283	5	,	,	PUNCT
ejpam-5094	283	6	1986	1986	NUM
ejpam-5094	283	7	.	.	PUNCT
ejpam-5094	284	1	[	[	X
ejpam-5094	284	2	2	2	NUM
ejpam-5094	284	3	]	]	PUNCT
ejpam-5094	284	4	c.	c.	NOUN
ejpam-5094	284	5	boonpok	boonpok	PROPN
ejpam-5094	284	6	and	and	CCONJ
ejpam-5094	284	7	p.	p.	NOUN
ejpam-5094	284	8	pue	pue	NOUN
ejpam-5094	284	9	-	-	PUNCT
ejpam-5094	284	10	on	on	ADP
ejpam-5094	284	11	.	.	PUNCT
ejpam-5094	285	1	weakly	weakly	ADJ
ejpam-5094	285	2	θs(λ	θs(λ	NOUN
ejpam-5094	285	3	,	,	PUNCT
ejpam-5094	285	4	p)-open	p)-open	VERB
ejpam-5094	285	5	functions	function	NOUN
ejpam-5094	285	6	and	and	CCONJ
ejpam-5094	285	7	weakly	weakly	ADJ
ejpam-5094	285	8	θs(λ	θs(λ	NOUN
ejpam-5094	285	9	,	,	PUNCT
ejpam-5094	285	10	p)closed	p)close	VERB
ejpam-5094	285	11	functions	function	NOUN
ejpam-5094	285	12	.	.	PUNCT
ejpam-5094	286	1	asia	asia	PROPN
ejpam-5094	286	2	pacific	pacific	PROPN
ejpam-5094	286	3	journal	journal	PROPN
ejpam-5094	286	4	of	of	ADP
ejpam-5094	286	5	mathematics	mathematic	NOUN
ejpam-5094	286	6	,	,	PUNCT
ejpam-5094	286	7	11:13	11:13	NUM
ejpam-5094	286	8	,	,	PUNCT
ejpam-5094	286	9	2024	2024	NUM
ejpam-5094	286	10	.	.	PUNCT
ejpam-5094	287	1	[	[	X
ejpam-5094	287	2	3	3	X
ejpam-5094	287	3	]	]	PUNCT
ejpam-5094	287	4	c.	c.	PROPN
ejpam-5094	287	5	boonpok	boonpok	PROPN
ejpam-5094	287	6	and	and	CCONJ
ejpam-5094	287	7	n.	n.	PROPN
ejpam-5094	287	8	srisarakham	srisarakham	PROPN
ejpam-5094	287	9	.	.	PUNCT
ejpam-5094	288	1	θp(λ	θp(λ	NOUN
ejpam-5094	288	2	,	,	PUNCT
ejpam-5094	288	3	p)-open	p)-open	NOUN
ejpam-5094	288	4	functions	function	NOUN
ejpam-5094	288	5	and	and	CCONJ
ejpam-5094	288	6	θp(λ	θp(λ	NOUN
ejpam-5094	288	7	,	,	PUNCT
ejpam-5094	288	8	p)-closed	p)-close	VERB
ejpam-5094	288	9	functions	function	NOUN
ejpam-5094	288	10	.	.	PUNCT
ejpam-5094	289	1	asia	asia	PROPN
ejpam-5094	289	2	pacific	pacific	PROPN
ejpam-5094	289	3	journal	journal	PROPN
ejpam-5094	289	4	of	of	ADP
ejpam-5094	289	5	mathematics	mathematic	NOUN
ejpam-5094	289	6	,	,	PUNCT
ejpam-5094	289	7	10:48	10:48	NUM
ejpam-5094	289	8	,	,	PUNCT
ejpam-5094	289	9	2023	2023	NUM
ejpam-5094	289	10	.	.	PUNCT
ejpam-5094	290	1	[	[	X
ejpam-5094	290	2	4	4	NUM
ejpam-5094	290	3	]	]	PUNCT
ejpam-5094	290	4	c.	c.	PROPN
ejpam-5094	290	5	boonpok	boonpok	PROPN
ejpam-5094	290	6	and	and	CCONJ
ejpam-5094	290	7	m.	m.	NOUN
ejpam-5094	290	8	thongmoon	thongmoon	NOUN
ejpam-5094	290	9	.	.	PUNCT
ejpam-5094	291	1	properties	property	NOUN
ejpam-5094	291	2	of	of	ADP
ejpam-5094	291	3	weakly	weakly	ADJ
ejpam-5094	291	4	β(λ	β(λ	NOUN
ejpam-5094	291	5	,	,	PUNCT
ejpam-5094	291	6	p)-open	p)-open	NOUN
ejpam-5094	291	7	functions	function	NOUN
ejpam-5094	291	8	and	and	CCONJ
ejpam-5094	291	9	weakly	weakly	ADJ
ejpam-5094	291	10	β(λ	β(λ	NOUN
ejpam-5094	291	11	,	,	PUNCT
ejpam-5094	291	12	p)-closed	p)-close	VERB
ejpam-5094	291	13	functions	function	NOUN
ejpam-5094	291	14	.	.	PUNCT
ejpam-5094	292	1	european	european	ADJ
ejpam-5094	292	2	journal	journal	PROPN
ejpam-5094	292	3	of	of	ADP
ejpam-5094	292	4	pure	pure	ADJ
ejpam-5094	292	5	and	and	CCONJ
ejpam-5094	292	6	applied	applied	ADJ
ejpam-5094	292	7	mathematics	mathematic	NOUN
ejpam-5094	292	8	,	,	PUNCT
ejpam-5094	292	9	17(1):248–255	17(1):248–255	PROPN
ejpam-5094	292	10	,	,	PUNCT
ejpam-5094	292	11	2024	2024	NUM
ejpam-5094	292	12	.	.	PUNCT
ejpam-5094	293	1	[	[	X
ejpam-5094	293	2	5	5	X
ejpam-5094	293	3	]	]	PUNCT
ejpam-5094	293	4	c.	c.	PROPN
ejpam-5094	293	5	boonpok	boonpok	PROPN
ejpam-5094	293	6	and	and	CCONJ
ejpam-5094	293	7	m.	m.	NOUN
ejpam-5094	293	8	thongmoon	thongmoon	NOUN
ejpam-5094	293	9	.	.	PUNCT
ejpam-5094	294	1	weakly	weakly	ADJ
ejpam-5094	294	2	p(λ	p(λ	NOUN
ejpam-5094	294	3	,	,	PUNCT
ejpam-5094	294	4	p)-open	p)-open	NOUN
ejpam-5094	294	5	functions	function	NOUN
ejpam-5094	294	6	and	and	CCONJ
ejpam-5094	294	7	weakly	weakly	ADJ
ejpam-5094	294	8	p(λ	p(λ	NOUN
ejpam-5094	294	9	,	,	PUNCT
ejpam-5094	294	10	p)closed	p)close	VERB
ejpam-5094	294	11	functions	function	NOUN
ejpam-5094	294	12	.	.	PUNCT
ejpam-5094	295	1	international	international	ADJ
ejpam-5094	295	2	journal	journal	NOUN
ejpam-5094	295	3	of	of	ADP
ejpam-5094	295	4	analysis	analysis	NOUN
ejpam-5094	295	5	and	and	CCONJ
ejpam-5094	295	6	applications	application	NOUN
ejpam-5094	295	7	,	,	PUNCT
ejpam-5094	295	8	22:10	22:10	NUM
ejpam-5094	295	9	,	,	PUNCT
ejpam-5094	295	10	2024	2024	NUM
ejpam-5094	295	11	.	.	PUNCT
ejpam-5094	296	1	references	reference	NOUN
ejpam-5094	296	2	590	590	NUM
ejpam-5094	296	3	[	[	SYM
ejpam-5094	296	4	6	6	NUM
ejpam-5094	296	5	]	]	PUNCT
ejpam-5094	296	6	c.	c.	PROPN
ejpam-5094	296	7	boonpok	boonpok	PROPN
ejpam-5094	296	8	and	and	CCONJ
ejpam-5094	296	9	c.	c.	PROPN
ejpam-5094	296	10	viriyapong	viriyapong	PROPN
ejpam-5094	296	11	.	.	PUNCT
ejpam-5094	297	1	on	on	ADP
ejpam-5094	297	2	(	(	PUNCT
ejpam-5094	297	3	λ	λ	PROPN
ejpam-5094	297	4	,	,	PUNCT
ejpam-5094	297	5	p)-closed	p)-close	VERB
ejpam-5094	297	6	sets	set	NOUN
ejpam-5094	297	7	and	and	CCONJ
ejpam-5094	297	8	the	the	DET
ejpam-5094	297	9	related	related	ADJ
ejpam-5094	297	10	notions	notion	NOUN
ejpam-5094	297	11	in	in	ADP
ejpam-5094	297	12	topological	topological	ADJ
ejpam-5094	297	13	spaces	space	NOUN
ejpam-5094	297	14	.	.	PUNCT
ejpam-5094	298	1	european	european	ADJ
ejpam-5094	298	2	journal	journal	PROPN
ejpam-5094	298	3	of	of	ADP
ejpam-5094	298	4	pure	pure	ADJ
ejpam-5094	298	5	and	and	CCONJ
ejpam-5094	298	6	applied	applied	ADJ
ejpam-5094	298	7	mathematics	mathematic	NOUN
ejpam-5094	298	8	,	,	PUNCT
ejpam-5094	298	9	15(2):415	15(2):415	PROPN
ejpam-5094	298	10	–	–	PUNCT
ejpam-5094	298	11	436	436	NUM
ejpam-5094	298	12	,	,	PUNCT
ejpam-5094	298	13	2022	2022	NUM
ejpam-5094	298	14	.	.	PUNCT
ejpam-5094	299	1	[	[	X
ejpam-5094	299	2	7	7	X
ejpam-5094	299	3	]	]	X
ejpam-5094	299	4	m.	m.	NOUN
ejpam-5094	299	5	caldas	caldas	PROPN
ejpam-5094	299	6	,	,	PUNCT
ejpam-5094	299	7	s.	s.	PROPN
ejpam-5094	299	8	jafari	jafari	PROPN
ejpam-5094	299	9	,	,	PUNCT
ejpam-5094	299	10	and	and	CCONJ
ejpam-5094	299	11	g.	g.	PROPN
ejpam-5094	299	12	navalagi	navalagi	PROPN
ejpam-5094	299	13	.	.	PUNCT
ejpam-5094	300	1	weak	weak	ADJ
ejpam-5094	300	2	forms	form	NOUN
ejpam-5094	300	3	of	of	ADP
ejpam-5094	300	4	open	open	ADJ
ejpam-5094	300	5	and	and	CCONJ
ejpam-5094	300	6	closed	closed	ADJ
ejpam-5094	300	7	functions	function	NOUN
ejpam-5094	300	8	via	via	ADP
ejpam-5094	300	9	semi	semi	ADJ
ejpam-5094	300	10	-	-	ADJ
ejpam-5094	300	11	θ	θ	ADJ
ejpam-5094	300	12	-	-	ADJ
ejpam-5094	300	13	open	open	ADJ
ejpam-5094	300	14	sets	set	NOUN
ejpam-5094	300	15	.	.	PUNCT
ejpam-5094	301	1	carpathian	carpathian	ADJ
ejpam-5094	301	2	journal	journal	PROPN
ejpam-5094	301	3	of	of	ADP
ejpam-5094	301	4	mathematics	mathematic	NOUN
ejpam-5094	301	5	,	,	PUNCT
ejpam-5094	301	6	22(1	22(1	NUM
ejpam-5094	301	7	-	-	PUNCT
ejpam-5094	301	8	2):21–31	2):21–31	NUM
ejpam-5094	301	9	,	,	PUNCT
ejpam-5094	301	10	2006	2006	NUM
ejpam-5094	301	11	.	.	PUNCT
ejpam-5094	302	1	[	[	X
ejpam-5094	302	2	8	8	NUM
ejpam-5094	302	3	]	]	PUNCT
ejpam-5094	302	4	m.	m.	NOUN
ejpam-5094	302	5	caldas	caldas	PROPN
ejpam-5094	302	6	,	,	PUNCT
ejpam-5094	302	7	s.	s.	PROPN
ejpam-5094	302	8	jafari	jafari	PROPN
ejpam-5094	302	9	,	,	PUNCT
ejpam-5094	302	10	g.	g.	PROPN
ejpam-5094	302	11	navalagi	navalagi	PROPN
ejpam-5094	302	12	,	,	PUNCT
ejpam-5094	302	13	and	and	CCONJ
ejpam-5094	302	14	t.	t.	PROPN
ejpam-5094	302	15	noiri	noiri	PROPN
ejpam-5094	302	16	.	.	PUNCT
ejpam-5094	303	1	on	on	ADP
ejpam-5094	303	2	pre	pre	ADJ
ejpam-5094	303	3	-	-	ADJ
ejpam-5094	303	4	θ	θ	ADJ
ejpam-5094	303	5	-	-	ADJ
ejpam-5094	303	6	open	open	ADJ
ejpam-5094	303	7	sets	set	NOUN
ejpam-5094	303	8	and	and	CCONJ
ejpam-5094	303	9	two	two	NUM
ejpam-5094	303	10	classes	class	NOUN
ejpam-5094	303	11	of	of	ADP
ejpam-5094	303	12	functions	function	NOUN
ejpam-5094	303	13	.	.	PUNCT
ejpam-5094	304	1	bulletin	bulletin	NOUN
ejpam-5094	304	2	of	of	ADP
ejpam-5094	304	3	the	the	DET
ejpam-5094	304	4	iranian	iranian	PROPN
ejpam-5094	304	5	mathematical	mathematical	ADJ
ejpam-5094	304	6	society	society	NOUN
ejpam-5094	304	7	,	,	PUNCT
ejpam-5094	304	8	32(1):45–63	32(1):45–63	NUM
ejpam-5094	304	9	,	,	PUNCT
ejpam-5094	304	10	2006	2006	NUM
ejpam-5094	304	11	.	.	PUNCT
ejpam-5094	305	1	[	[	X
ejpam-5094	305	2	9	9	NUM
ejpam-5094	305	3	]	]	X
ejpam-5094	305	4	n.	n.	NOUN
ejpam-5094	305	5	chutiman	chutiman	NOUN
ejpam-5094	305	6	and	and	CCONJ
ejpam-5094	305	7	c.	c.	PROPN
ejpam-5094	305	8	boonpok	boonpok	PROPN
ejpam-5094	305	9	.	.	PUNCT
ejpam-5094	306	1	some	some	DET
ejpam-5094	306	2	properties	property	NOUN
ejpam-5094	306	3	of	of	ADP
ejpam-5094	306	4	weakly	weakly	ADJ
ejpam-5094	306	5	b(λ	b(λ	NOUN
ejpam-5094	306	6	,	,	PUNCT
ejpam-5094	306	7	p)-open	p)-open	NOUN
ejpam-5094	306	8	functions	function	NOUN
ejpam-5094	306	9	.	.	PUNCT
ejpam-5094	307	1	international	international	ADJ
ejpam-5094	307	2	journal	journal	PROPN
ejpam-5094	307	3	of	of	ADP
ejpam-5094	307	4	mathematics	mathematic	NOUN
ejpam-5094	307	5	and	and	CCONJ
ejpam-5094	307	6	computer	computer	NOUN
ejpam-5094	307	7	science	science	NOUN
ejpam-5094	307	8	,	,	PUNCT
ejpam-5094	307	9	19(2):497–501	19(2):497–501	NUM
ejpam-5094	307	10	,	,	PUNCT
ejpam-5094	307	11	2024	2024	NUM
ejpam-5094	307	12	.	.	PUNCT
ejpam-5094	308	1	[	[	X
ejpam-5094	308	2	10	10	NUM
ejpam-5094	308	3	]	]	PUNCT
ejpam-5094	308	4	m.	m.	NOUN
ejpam-5094	308	5	ganster	ganster	NOUN
ejpam-5094	308	6	,	,	PUNCT
ejpam-5094	308	7	s.	s.	PROPN
ejpam-5094	308	8	jafari	jafari	PROPN
ejpam-5094	308	9	,	,	PUNCT
ejpam-5094	308	10	and	and	CCONJ
ejpam-5094	308	11	t.	t.	PROPN
ejpam-5094	308	12	noiri	noiri	PROPN
ejpam-5094	308	13	.	.	PUNCT
ejpam-5094	309	1	on	on	ADP
ejpam-5094	309	2	pre	pre	ADJ
ejpam-5094	309	3	-	-	ADJ
ejpam-5094	309	4	λ	λ	NOUN
ejpam-5094	309	5	-	-	NOUN
ejpam-5094	309	6	sets	set	NOUN
ejpam-5094	309	7	and	and	CCONJ
ejpam-5094	309	8	pre	pre	ADJ
ejpam-5094	309	9	-	-	ADJ
ejpam-5094	309	10	v	v	ADJ
ejpam-5094	309	11	-sets	-set	NOUN
ejpam-5094	309	12	.	.	PUNCT
ejpam-5094	310	1	acta	acta	PROPN
ejpam-5094	310	2	mathematica	mathematica	PROPN
ejpam-5094	310	3	hungarica	hungarica	PROPN
ejpam-5094	310	4	,	,	PUNCT
ejpam-5094	310	5	95:337–343	95:337–343	PROPN
ejpam-5094	310	6	,	,	PUNCT
ejpam-5094	310	7	2002	2002	NUM
ejpam-5094	310	8	.	.	PUNCT
ejpam-5094	311	1	[	[	X
ejpam-5094	311	2	11	11	NUM
ejpam-5094	311	3	]	]	X
ejpam-5094	311	4	j.	j.	PROPN
ejpam-5094	311	5	khampakdee	khampakdee	PROPN
ejpam-5094	311	6	and	and	CCONJ
ejpam-5094	311	7	c.	c.	PROPN
ejpam-5094	311	8	boonpok	boonpok	PROPN
ejpam-5094	311	9	.	.	PUNCT
ejpam-5094	312	1	properties	property	NOUN
ejpam-5094	312	2	of	of	ADP
ejpam-5094	312	3	(	(	PUNCT
ejpam-5094	312	4	λ	λ	PROPN
ejpam-5094	312	5	,	,	PUNCT
ejpam-5094	312	6	p)-closed	p)-close	VERB
ejpam-5094	312	7	functions	function	NOUN
ejpam-5094	312	8	.	.	PUNCT
ejpam-5094	313	1	international	international	ADJ
ejpam-5094	313	2	journal	journal	PROPN
ejpam-5094	313	3	of	of	ADP
ejpam-5094	313	4	mathematics	mathematic	NOUN
ejpam-5094	313	5	and	and	CCONJ
ejpam-5094	313	6	computer	computer	NOUN
ejpam-5094	313	7	science	science	NOUN
ejpam-5094	313	8	,	,	PUNCT
ejpam-5094	313	9	19(2):481–484	19(2):481–484	NUM
ejpam-5094	313	10	,	,	PUNCT
ejpam-5094	313	11	2024	2024	NUM
ejpam-5094	313	12	.	.	PUNCT
ejpam-5094	314	1	[	[	X
ejpam-5094	314	2	12	12	NUM
ejpam-5094	314	3	]	]	X
ejpam-5094	314	4	c.	c.	PROPN
ejpam-5094	314	5	klanarong	klanarong	PROPN
ejpam-5094	314	6	and	and	CCONJ
ejpam-5094	314	7	c.	c.	PROPN
ejpam-5094	314	8	boonpok	boonpok	PROPN
ejpam-5094	314	9	.	.	PUNCT
ejpam-5094	315	1	characterizations	characterization	NOUN
ejpam-5094	315	2	of	of	ADP
ejpam-5094	315	3	weakly	weakly	ADJ
ejpam-5094	315	4	δ(λ	δ(λ	PROPN
ejpam-5094	315	5	,	,	PUNCT
ejpam-5094	315	6	p)-closed	p)-close	VERB
ejpam-5094	315	7	functions	function	NOUN
ejpam-5094	315	8	.	.	PUNCT
ejpam-5094	316	1	international	international	ADJ
ejpam-5094	316	2	journal	journal	PROPN
ejpam-5094	316	3	of	of	ADP
ejpam-5094	316	4	mathematics	mathematic	NOUN
ejpam-5094	316	5	and	and	CCONJ
ejpam-5094	316	6	computer	computer	NOUN
ejpam-5094	316	7	science	science	NOUN
ejpam-5094	316	8	,	,	PUNCT
ejpam-5094	316	9	19(2):503–507	19(2):503–507	PROPN
ejpam-5094	316	10	,	,	PUNCT
ejpam-5094	316	11	2024	2024	NUM
ejpam-5094	316	12	.	.	PUNCT
ejpam-5094	317	1	[	[	X
ejpam-5094	317	2	13	13	NUM
ejpam-5094	317	3	]	]	X
ejpam-5094	317	4	c.	c.	PROPN
ejpam-5094	317	5	klanarong	klanarong	PROPN
ejpam-5094	317	6	and	and	CCONJ
ejpam-5094	317	7	c.	c.	PROPN
ejpam-5094	317	8	boonpok	boonpok	PROPN
ejpam-5094	317	9	.	.	PUNCT
ejpam-5094	318	1	characterizations	characterization	NOUN
ejpam-5094	318	2	of	of	ADP
ejpam-5094	318	3	weakly	weakly	ADJ
ejpam-5094	318	4	s(λ	s(λ	NOUN
ejpam-5094	318	5	,	,	PUNCT
ejpam-5094	318	6	p)-open	p)-open	NOUN
ejpam-5094	318	7	functions	function	NOUN
ejpam-5094	318	8	and	and	CCONJ
ejpam-5094	318	9	weakly	weakly	ADJ
ejpam-5094	318	10	s(λ	s(λ	PROPN
ejpam-5094	318	11	,	,	PUNCT
ejpam-5094	318	12	p)-closed	p)-close	VERB
ejpam-5094	318	13	functions	function	NOUN
ejpam-5094	318	14	.	.	PUNCT
ejpam-5094	319	1	international	international	ADJ
ejpam-5094	319	2	journal	journal	PROPN
ejpam-5094	319	3	of	of	ADP
ejpam-5094	319	4	mathematics	mathematic	NOUN
ejpam-5094	319	5	and	and	CCONJ
ejpam-5094	319	6	computer	computer	NOUN
ejpam-5094	319	7	science	science	NOUN
ejpam-5094	319	8	,	,	PUNCT
ejpam-5094	319	9	19(3):809–814	19(3):809–814	PROPN
ejpam-5094	319	10	,	,	PUNCT
ejpam-5094	319	11	2024	2024	NUM
ejpam-5094	319	12	.	.	PUNCT
ejpam-5094	320	1	[	[	X
ejpam-5094	320	2	14	14	NUM
ejpam-5094	320	3	]	]	PUNCT
ejpam-5094	320	4	a.	a.	NOUN
ejpam-5094	320	5	s.	s.	PROPN
ejpam-5094	320	6	mashhour	mashhour	PROPN
ejpam-5094	320	7	,	,	PUNCT
ejpam-5094	320	8	m.	m.	PROPN
ejpam-5094	320	9	e.	e.	PROPN
ejpam-5094	320	10	abd	abd	PROPN
ejpam-5094	320	11	el	el	PROPN
ejpam-5094	320	12	-	-	PROPN
ejpam-5094	320	13	monsef	monsef	ADJ
ejpam-5094	320	14	,	,	PUNCT
ejpam-5094	320	15	and	and	CCONJ
ejpam-5094	320	16	s.	s.	PROPN
ejpam-5094	320	17	n.	n.	PROPN
ejpam-5094	320	18	el	el	PROPN
ejpam-5094	320	19	-	-	PROPN
ejpam-5094	320	20	deeb	deeb	PROPN
ejpam-5094	320	21	.	.	PUNCT
ejpam-5094	321	1	on	on	ADP
ejpam-5094	321	2	precontinuous	precontinuous	ADJ
ejpam-5094	321	3	and	and	CCONJ
ejpam-5094	321	4	weak	weak	ADJ
ejpam-5094	321	5	precontinuous	precontinuous	ADJ
ejpam-5094	321	6	mappings	mapping	NOUN
ejpam-5094	321	7	.	.	PUNCT
ejpam-5094	322	1	proceedings	proceeding	NOUN
ejpam-5094	322	2	of	of	ADP
ejpam-5094	322	3	the	the	DET
ejpam-5094	322	4	mathematical	mathematical	ADJ
ejpam-5094	322	5	and	and	CCONJ
ejpam-5094	322	6	physical	physical	ADJ
ejpam-5094	322	7	society	society	NOUN
ejpam-5094	322	8	of	of	ADP
ejpam-5094	322	9	egypt	egypt	PROPN
ejpam-5094	322	10	,	,	PUNCT
ejpam-5094	322	11	53:47–53	53:47–53	NUM
ejpam-5094	322	12	,	,	PUNCT
ejpam-5094	322	13	1982	1982	NUM
ejpam-5094	322	14	.	.	PUNCT
ejpam-5094	323	1	[	[	X
ejpam-5094	323	2	15	15	NUM
ejpam-5094	323	3	]	]	X
ejpam-5094	323	4	t.	t.	PROPN
ejpam-5094	323	5	noiri	noiri	PROPN
ejpam-5094	323	6	,	,	PUNCT
ejpam-5094	323	7	a.	a.	PROPN
ejpam-5094	323	8	al	al	PROPN
ejpam-5094	323	9	-	-	PUNCT
ejpam-5094	323	10	omari	omari	PROPN
ejpam-5094	323	11	,	,	PUNCT
ejpam-5094	323	12	and	and	CCONJ
ejpam-5094	323	13	m.	m.	PROPN
ejpam-5094	323	14	s.	s.	PROPN
ejpam-5094	323	15	m.	m.	PROPN
ejpam-5094	323	16	noorani	noorani	PROPN
ejpam-5094	323	17	.	.	PUNCT
ejpam-5094	324	1	weak	weak	ADJ
ejpam-5094	324	2	forms	form	NOUN
ejpam-5094	324	3	of	of	ADP
ejpam-5094	324	4	open	open	ADJ
ejpam-5094	324	5	and	and	CCONJ
ejpam-5094	324	6	closed	closed	ADJ
ejpam-5094	324	7	functions	function	NOUN
ejpam-5094	324	8	via	via	ADP
ejpam-5094	324	9	b	b	X
ejpam-5094	324	10	-	-	PUNCT
ejpam-5094	324	11	θ	θ	ADJ
ejpam-5094	324	12	-	-	PUNCT
ejpam-5094	324	13	open	open	ADJ
ejpam-5094	324	14	sets	set	NOUN
ejpam-5094	324	15	.	.	PUNCT
ejpam-5094	325	1	demonstratio	demonstratio	PROPN
ejpam-5094	325	2	mathematica	mathematica	PROPN
ejpam-5094	325	3	,	,	PUNCT
ejpam-5094	325	4	42(1):193–203	42(1):193–203	PROPN
ejpam-5094	325	5	,	,	PUNCT
ejpam-5094	325	6	2009	2009	NUM
ejpam-5094	325	7	.	.	PUNCT
ejpam-5094	326	1	[	[	X
ejpam-5094	326	2	16	16	NUM
ejpam-5094	326	3	]	]	PUNCT
ejpam-5094	326	4	j.	j.	PROPN
ejpam-5094	326	5	h.	h.	PROPN
ejpam-5094	326	6	park	park	PROPN
ejpam-5094	326	7	.	.	PUNCT
ejpam-5094	327	1	strongly	strongly	ADV
ejpam-5094	327	2	θ	θ	PROPN
ejpam-5094	327	3	-	-	PUNCT
ejpam-5094	327	4	b	b	NOUN
ejpam-5094	327	5	-	-	PUNCT
ejpam-5094	327	6	continuous	continuous	ADJ
ejpam-5094	327	7	functions	function	NOUN
ejpam-5094	327	8	.	.	PUNCT
ejpam-5094	328	1	acta	acta	PROPN
ejpam-5094	328	2	mathematica	mathematica	PROPN
ejpam-5094	328	3	hungarica	hungarica	PROPN
ejpam-5094	328	4	,	,	PUNCT
ejpam-5094	328	5	110:347	110:347	NOUN
ejpam-5094	328	6	–	–	PUNCT
ejpam-5094	328	7	359	359	NUM
ejpam-5094	328	8	,	,	PUNCT
ejpam-5094	328	9	2006	2006	NUM
ejpam-5094	328	10	.	.	PUNCT
ejpam-5094	329	1	[	[	X
ejpam-5094	329	2	17	17	NUM
ejpam-5094	329	3	]	]	X
ejpam-5094	329	4	d.	d.	PROPN
ejpam-5094	329	5	a.	a.	PROPN
ejpam-5094	329	6	rose	rise	VERB
ejpam-5094	329	7	.	.	PUNCT
ejpam-5094	330	1	on	on	ADP
ejpam-5094	330	2	weak	weak	ADJ
ejpam-5094	330	3	openness	openness	NOUN
ejpam-5094	330	4	and	and	CCONJ
ejpam-5094	330	5	almost	almost	ADV
ejpam-5094	330	6	openness	openness	NOUN
ejpam-5094	330	7	.	.	PUNCT
ejpam-5094	331	1	international	international	ADJ
ejpam-5094	331	2	journal	journal	NOUN
ejpam-5094	331	3	of	of	ADP
ejpam-5094	331	4	mathematics	mathematics	PROPN
ejpam-5094	331	5	and	and	CCONJ
ejpam-5094	331	6	mathematical	mathematical	ADJ
ejpam-5094	331	7	sciences	science	NOUN
ejpam-5094	331	8	,	,	PUNCT
ejpam-5094	331	9	7:35–40	7:35–40	NUM
ejpam-5094	331	10	,	,	PUNCT
ejpam-5094	331	11	1984	1984	NUM
ejpam-5094	331	12	.	.	PUNCT
ejpam-5094	332	1	[	[	X
ejpam-5094	332	2	18	18	NUM
ejpam-5094	332	3	]	]	X
ejpam-5094	332	4	d.	d.	PROPN
ejpam-5094	332	5	a.	a.	PROPN
ejpam-5094	332	6	rose	rise	VERB
ejpam-5094	332	7	and	and	CCONJ
ejpam-5094	332	8	d.	d.	PROPN
ejpam-5094	332	9	s.	s.	PROPN
ejpam-5094	333	1	janković.	janković.	PROPN
ejpam-5094	333	2	weakly	weakly	ADJ
ejpam-5094	333	3	closed	close	VERB
ejpam-5094	333	4	functions	function	NOUN
ejpam-5094	333	5	and	and	CCONJ
ejpam-5094	333	6	hausdorff	hausdorff	NOUN
ejpam-5094	333	7	spaces	space	NOUN
ejpam-5094	333	8	.	.	PUNCT
ejpam-5094	334	1	mathematische	mathematische	PROPN
ejpam-5094	334	2	nachrichten	nachrichten	PROPN
ejpam-5094	334	3	,	,	PUNCT
ejpam-5094	334	4	130:105–110	130:105–110	NUM
ejpam-5094	334	5	,	,	PUNCT
ejpam-5094	334	6	1987	1987	NUM
ejpam-5094	334	7	.	.	PUNCT
ejpam-5094	335	1	[	[	X
ejpam-5094	335	2	19	19	NUM
ejpam-5094	335	3	]	]	X
ejpam-5094	335	4	n.	n.	NOUN
ejpam-5094	335	5	srisarakham	srisarakham	PROPN
ejpam-5094	335	6	and	and	CCONJ
ejpam-5094	335	7	c.	c.	PROPN
ejpam-5094	335	8	boonpok	boonpok	PROPN
ejpam-5094	335	9	.	.	PUNCT
ejpam-5094	336	1	on	on	ADP
ejpam-5094	336	2	weakly	weakly	ADJ
ejpam-5094	336	3	δ(λ	δ(λ	PROPN
ejpam-5094	336	4	,	,	PUNCT
ejpam-5094	336	5	p)-open	p)-open	VERB
ejpam-5094	336	6	functions	function	NOUN
ejpam-5094	336	7	.	.	PUNCT
ejpam-5094	337	1	international	international	ADJ
ejpam-5094	337	2	journal	journal	PROPN
ejpam-5094	337	3	of	of	ADP
ejpam-5094	337	4	mathematics	mathematic	NOUN
ejpam-5094	337	5	and	and	CCONJ
ejpam-5094	337	6	computer	computer	NOUN
ejpam-5094	337	7	science	science	NOUN
ejpam-5094	337	8	,	,	PUNCT
ejpam-5094	337	9	19(2):485–489	19(2):485–489	PROPN
ejpam-5094	337	10	,	,	PUNCT
ejpam-5094	337	11	2024	2024	NUM
ejpam-5094	337	12	.	.	PUNCT
ejpam-5094	338	1	[	[	X
ejpam-5094	338	2	20	20	NUM
ejpam-5094	338	3	]	]	PUNCT
ejpam-5094	338	4	m.	m.	NOUN
ejpam-5094	338	5	thongmoon	thongmoon	NOUN
ejpam-5094	338	6	and	and	CCONJ
ejpam-5094	338	7	c.	c.	PROPN
ejpam-5094	338	8	boonpok	boonpok	PROPN
ejpam-5094	338	9	.	.	PUNCT
ejpam-5094	339	1	strongly	strongly	ADV
ejpam-5094	339	2	θ(λ	θ(λ	PROPN
ejpam-5094	339	3	,	,	PUNCT
ejpam-5094	339	4	p)-continuous	p)-continuous	ADJ
ejpam-5094	339	5	functions	function	NOUN
ejpam-5094	339	6	.	.	PUNCT
ejpam-5094	340	1	international	international	ADJ
ejpam-5094	340	2	journal	journal	PROPN
ejpam-5094	340	3	of	of	ADP
ejpam-5094	340	4	mathematics	mathematic	NOUN
ejpam-5094	340	5	and	and	CCONJ
ejpam-5094	340	6	computer	computer	NOUN
ejpam-5094	340	7	science	science	NOUN
ejpam-5094	340	8	,	,	PUNCT
ejpam-5094	340	9	19(2):475–479	19(2):475–479	PROPN
ejpam-5094	340	10	,	,	PUNCT
ejpam-5094	340	11	2024	2024	NUM
ejpam-5094	340	12	.	.	PUNCT
ejpam-5094	341	1	[	[	X
ejpam-5094	341	2	21	21	NUM
ejpam-5094	341	3	]	]	X
ejpam-5094	341	4	n.	n.	PROPN
ejpam-5094	341	5	viriyapong	viriyapong	PROPN
ejpam-5094	341	6	and	and	CCONJ
ejpam-5094	341	7	c.	c.	PROPN
ejpam-5094	341	8	boonpok	boonpok	PROPN
ejpam-5094	341	9	.	.	PUNCT
ejpam-5094	342	1	on	on	ADP
ejpam-5094	342	2	(	(	PUNCT
ejpam-5094	342	3	λ	λ	INTJ
ejpam-5094	342	4	,	,	PUNCT
ejpam-5094	342	5	p)-extremally	p)-extremally	ADV
ejpam-5094	342	6	disconnected	disconnected	ADJ
ejpam-5094	342	7	spaces	space	NOUN
ejpam-5094	342	8	.	.	PUNCT
ejpam-5094	343	1	international	international	ADJ
ejpam-5094	343	2	journal	journal	PROPN
ejpam-5094	343	3	of	of	ADP
ejpam-5094	343	4	mathematics	mathematic	NOUN
ejpam-5094	343	5	and	and	CCONJ
ejpam-5094	343	6	computer	computer	NOUN
ejpam-5094	343	7	science	science	NOUN
ejpam-5094	343	8	,	,	PUNCT
ejpam-5094	343	9	18(2):289–293	18(2):289–293	NUM
ejpam-5094	343	10	,	,	PUNCT
ejpam-5094	343	11	2023	2023	NUM
ejpam-5094	343	12	.	.	PUNCT