id	sid	tid	token	lemma	pos
ejpam-4974	1	1	european	european	PROPN
ejpam-4974	1	2	journal	journal	PROPN
ejpam-4974	1	3	of	of	ADP
ejpam-4974	1	4	pure	pure	ADJ
ejpam-4974	1	5	and	and	CCONJ
ejpam-4974	1	6	applied	apply	VERB
ejpam-4974	1	7	mathematics	mathematic	NOUN
ejpam-4974	1	8	vol	vol	NOUN
ejpam-4974	1	9	.	.	PROPN
ejpam-4974	2	1	17	17	NUM
ejpam-4974	2	2	,	,	PUNCT
ejpam-4974	2	3	no	no	INTJ
ejpam-4974	2	4	.	.	NOUN
ejpam-4974	2	5	1	1	NUM
ejpam-4974	2	6	,	,	PUNCT
ejpam-4974	2	7	2024	2024	NUM
ejpam-4974	2	8	,	,	PUNCT
ejpam-4974	2	9	248	248	NUM
ejpam-4974	2	10	-	-	SYM
ejpam-4974	2	11	255	255	NUM
ejpam-4974	2	12	issn	issn	PROPN
ejpam-4974	2	13	1307	1307	NUM
ejpam-4974	2	14	-	-	SYM
ejpam-4974	2	15	5543	5543	NUM
ejpam-4974	2	16	–	–	PUNCT
ejpam-4974	2	17	ejpam.com	ejpam.com	X
ejpam-4974	2	18	published	publish	VERB
ejpam-4974	2	19	by	by	ADP
ejpam-4974	2	20	new	new	PROPN
ejpam-4974	2	21	york	york	PROPN
ejpam-4974	2	22	business	business	PROPN
ejpam-4974	2	23	global	global	ADJ
ejpam-4974	2	24	properties	property	NOUN
ejpam-4974	2	25	of	of	ADP
ejpam-4974	2	26	weakly	weakly	ADJ
ejpam-4974	2	27	β(λ	β(λ	NOUN
ejpam-4974	2	28	,	,	PUNCT
ejpam-4974	2	29	p)-open	p)-open	NOUN
ejpam-4974	2	30	functions	function	NOUN
ejpam-4974	2	31	and	and	CCONJ
ejpam-4974	2	32	weakly	weakly	ADJ
ejpam-4974	2	33	β(λ	β(λ	NOUN
ejpam-4974	2	34	,	,	PUNCT
ejpam-4974	2	35	p)-closed	p)-close	VERB
ejpam-4974	2	36	functions	function	NOUN
ejpam-4974	2	37	chawalit	chawalit	VERB
ejpam-4974	2	38	boonpok1	boonpok1	PROPN
ejpam-4974	2	39	,	,	PUNCT
ejpam-4974	2	40	montri	montri	PROPN
ejpam-4974	2	41	thongmoon1,∗	thongmoon1,∗	NOUN
ejpam-4974	2	42	1	1	NUM
ejpam-4974	2	43	mathematics	mathematic	NOUN
ejpam-4974	2	44	and	and	CCONJ
ejpam-4974	2	45	applied	apply	VERB
ejpam-4974	2	46	mathematics	mathematics	PROPN
ejpam-4974	2	47	research	research	NOUN
ejpam-4974	2	48	unit	unit	NOUN
ejpam-4974	2	49	,	,	PUNCT
ejpam-4974	2	50	department	department	NOUN
ejpam-4974	2	51	of	of	ADP
ejpam-4974	2	52	mathematics	mathematic	NOUN
ejpam-4974	2	53	,	,	PUNCT
ejpam-4974	2	54	faculty	faculty	NOUN
ejpam-4974	2	55	of	of	ADP
ejpam-4974	2	56	science	science	NOUN
ejpam-4974	2	57	,	,	PUNCT
ejpam-4974	2	58	mahasarakham	mahasarakham	PROPN
ejpam-4974	2	59	university	university	PROPN
ejpam-4974	2	60	,	,	PUNCT
ejpam-4974	2	61	maha	maha	PROPN
ejpam-4974	2	62	sarakham	sarakham	PROPN
ejpam-4974	2	63	,	,	PUNCT
ejpam-4974	2	64	44150	44150	NUM
ejpam-4974	2	65	,	,	PUNCT
ejpam-4974	2	66	thailand	thailand	PROPN
ejpam-4974	2	67	abstract	abstract	PROPN
ejpam-4974	2	68	.	.	PUNCT
ejpam-4974	3	1	this	this	DET
ejpam-4974	3	2	paper	paper	NOUN
ejpam-4974	3	3	deals	deal	NOUN
ejpam-4974	3	4	with	with	ADP
ejpam-4974	3	5	the	the	DET
ejpam-4974	3	6	concepts	concept	NOUN
ejpam-4974	3	7	of	of	ADP
ejpam-4974	3	8	weakly	weakly	ADJ
ejpam-4974	3	9	β(λ	β(λ	NOUN
ejpam-4974	3	10	,	,	PUNCT
ejpam-4974	3	11	p)-open	p)-open	NOUN
ejpam-4974	3	12	functions	function	NOUN
ejpam-4974	3	13	and	and	CCONJ
ejpam-4974	3	14	weakly	weakly	ADJ
ejpam-4974	3	15	β(λ	β(λ	NOUN
ejpam-4974	3	16	,	,	PUNCT
ejpam-4974	3	17	p)-closed	p)-close	VERB
ejpam-4974	3	18	functions	function	NOUN
ejpam-4974	3	19	.	.	PUNCT
ejpam-4974	4	1	moreover	moreover	ADV
ejpam-4974	4	2	,	,	PUNCT
ejpam-4974	4	3	some	some	DET
ejpam-4974	4	4	properties	property	NOUN
ejpam-4974	4	5	of	of	ADP
ejpam-4974	4	6	weakly	weakly	ADJ
ejpam-4974	4	7	β(λ	β(λ	NOUN
ejpam-4974	4	8	,	,	PUNCT
ejpam-4974	4	9	p)-open	p)-open	NOUN
ejpam-4974	4	10	functions	function	NOUN
ejpam-4974	4	11	and	and	CCONJ
ejpam-4974	4	12	weakly	weakly	ADJ
ejpam-4974	4	13	β(λ	β(λ	NOUN
ejpam-4974	4	14	,	,	PUNCT
ejpam-4974	4	15	p)-closed	p)-close	VERB
ejpam-4974	4	16	functions	function	NOUN
ejpam-4974	4	17	are	be	AUX
ejpam-4974	4	18	investigated	investigate	VERB
ejpam-4974	4	19	.	.	PUNCT
ejpam-4974	5	1	2020	2020	NUM
ejpam-4974	5	2	mathematics	mathematic	NOUN
ejpam-4974	5	3	subject	subject	NOUN
ejpam-4974	5	4	classifications	classification	NOUN
ejpam-4974	5	5	:	:	PUNCT
ejpam-4974	5	6	54a05	54a05	NUM
ejpam-4974	5	7	;	;	PUNCT
ejpam-4974	5	8	54c10	54c10	NUM
ejpam-4974	5	9	key	key	ADJ
ejpam-4974	5	10	words	word	NOUN
ejpam-4974	5	11	and	and	CCONJ
ejpam-4974	5	12	phrases	phrase	NOUN
ejpam-4974	5	13	:	:	PUNCT
ejpam-4974	5	14	weakly	weakly	ADJ
ejpam-4974	5	15	β(λ	β(λ	X
ejpam-4974	5	16	,	,	PUNCT
ejpam-4974	5	17	p)-open	p)-open	VERB
ejpam-4974	5	18	function	function	NOUN
ejpam-4974	5	19	,	,	PUNCT
ejpam-4974	5	20	weakly	weakly	ADJ
ejpam-4974	5	21	β(λ	β(λ	NOUN
ejpam-4974	5	22	,	,	PUNCT
ejpam-4974	5	23	p)-closed	p)-close	VERB
ejpam-4974	5	24	function	function	NOUN
ejpam-4974	5	25	1	1	NUM
ejpam-4974	5	26	.	.	PUNCT
ejpam-4974	6	1	introduction	introduction	NOUN
ejpam-4974	6	2	it	it	PRON
ejpam-4974	6	3	is	be	AUX
ejpam-4974	6	4	well	well	ADV
ejpam-4974	6	5	-	-	PUNCT
ejpam-4974	6	6	known	know	VERB
ejpam-4974	6	7	that	that	SCONJ
ejpam-4974	6	8	the	the	DET
ejpam-4974	6	9	branch	branch	NOUN
ejpam-4974	6	10	of	of	ADP
ejpam-4974	6	11	mathematics	mathematic	NOUN
ejpam-4974	6	12	called	call	VERB
ejpam-4974	6	13	topology	topology	NOUN
ejpam-4974	6	14	is	be	AUX
ejpam-4974	6	15	related	relate	VERB
ejpam-4974	6	16	to	to	ADP
ejpam-4974	6	17	all	all	DET
ejpam-4974	6	18	questions	question	NOUN
ejpam-4974	6	19	directly	directly	ADV
ejpam-4974	6	20	or	or	CCONJ
ejpam-4974	6	21	indirectly	indirectly	ADV
ejpam-4974	6	22	concerned	concerned	ADJ
ejpam-4974	6	23	with	with	ADP
ejpam-4974	6	24	openness	openness	NOUN
ejpam-4974	6	25	and	and	CCONJ
ejpam-4974	6	26	closedness	closedness	NOUN
ejpam-4974	6	27	.	.	PUNCT
ejpam-4974	7	1	semi	semi	ADJ
ejpam-4974	7	2	-	-	ADJ
ejpam-4974	7	3	open	open	ADJ
ejpam-4974	7	4	sets	set	NOUN
ejpam-4974	7	5	,	,	PUNCT
ejpam-4974	7	6	preopen	preopen	ADJ
ejpam-4974	7	7	sets	set	NOUN
ejpam-4974	7	8	,	,	PUNCT
ejpam-4974	7	9	α	α	NOUN
ejpam-4974	7	10	-	-	ADJ
ejpam-4974	7	11	open	open	ADJ
ejpam-4974	7	12	sets	set	NOUN
ejpam-4974	7	13	,	,	PUNCT
ejpam-4974	7	14	β	β	ADJ
ejpam-4974	7	15	-	-	ADJ
ejpam-4974	7	16	open	open	ADJ
ejpam-4974	7	17	sets	set	NOUN
ejpam-4974	7	18	,	,	PUNCT
ejpam-4974	7	19	b	b	X
ejpam-4974	7	20	-	-	PUNCT
ejpam-4974	7	21	open	open	ADJ
ejpam-4974	7	22	sets	set	NOUN
ejpam-4974	7	23	,	,	PUNCT
ejpam-4974	7	24	δ	δ	NOUN
ejpam-4974	7	25	-	-	ADJ
ejpam-4974	7	26	open	open	ADJ
ejpam-4974	7	27	sets	set	NOUN
ejpam-4974	7	28	and	and	CCONJ
ejpam-4974	7	29	θ	θ	ADJ
ejpam-4974	7	30	-	-	ADJ
ejpam-4974	7	31	open	open	ADJ
ejpam-4974	7	32	sets	set	NOUN
ejpam-4974	7	33	play	play	VERB
ejpam-4974	7	34	an	an	DET
ejpam-4974	7	35	important	important	ADJ
ejpam-4974	7	36	role	role	NOUN
ejpam-4974	7	37	in	in	ADP
ejpam-4974	7	38	the	the	DET
ejpam-4974	7	39	researches	research	NOUN
ejpam-4974	7	40	of	of	ADP
ejpam-4974	7	41	generalizations	generalization	NOUN
ejpam-4974	7	42	of	of	ADP
ejpam-4974	7	43	open	open	ADJ
ejpam-4974	7	44	functions	function	NOUN
ejpam-4974	7	45	and	and	CCONJ
ejpam-4974	7	46	closed	closed	ADJ
ejpam-4974	7	47	functions	function	NOUN
ejpam-4974	7	48	.	.	PUNCT
ejpam-4974	8	1	by	by	ADP
ejpam-4974	8	2	using	use	VERB
ejpam-4974	8	3	these	these	DET
ejpam-4974	8	4	sets	set	NOUN
ejpam-4974	8	5	,	,	PUNCT
ejpam-4974	8	6	many	many	ADJ
ejpam-4974	8	7	authors	author	NOUN
ejpam-4974	8	8	introduced	introduce	VERB
ejpam-4974	8	9	and	and	CCONJ
ejpam-4974	8	10	studied	study	VERB
ejpam-4974	8	11	various	various	ADJ
ejpam-4974	8	12	types	type	NOUN
ejpam-4974	8	13	of	of	ADP
ejpam-4974	8	14	open	open	ADJ
ejpam-4974	8	15	functions	function	NOUN
ejpam-4974	8	16	and	and	CCONJ
ejpam-4974	8	17	closed	closed	ADJ
ejpam-4974	8	18	functions	function	NOUN
ejpam-4974	8	19	.	.	PUNCT
ejpam-4974	9	1	the	the	DET
ejpam-4974	9	2	concept	concept	NOUN
ejpam-4974	9	3	of	of	ADP
ejpam-4974	9	4	weakly	weakly	ADJ
ejpam-4974	9	5	open	open	ADJ
ejpam-4974	9	6	functions	function	NOUN
ejpam-4974	9	7	was	be	AUX
ejpam-4974	9	8	first	first	ADV
ejpam-4974	9	9	introduced	introduce	VERB
ejpam-4974	9	10	by	by	ADP
ejpam-4974	9	11	rose	rose	NOUN
ejpam-4974	9	12	[	[	X
ejpam-4974	9	13	15	15	NUM
ejpam-4974	9	14	]	]	PUNCT
ejpam-4974	9	15	.	.	PUNCT
ejpam-4974	10	1	rose	rise	VERB
ejpam-4974	10	2	and	and	CCONJ
ejpam-4974	10	3	janković	janković	ADJ
ejpam-4974	11	1	[	[	X
ejpam-4974	11	2	16	16	NUM
ejpam-4974	11	3	]	]	PUNCT
ejpam-4974	11	4	investigated	investigate	VERB
ejpam-4974	11	5	some	some	PRON
ejpam-4974	11	6	of	of	ADP
ejpam-4974	11	7	the	the	DET
ejpam-4974	11	8	fundamental	fundamental	ADJ
ejpam-4974	11	9	properties	property	NOUN
ejpam-4974	11	10	of	of	ADP
ejpam-4974	11	11	weakly	weakly	ADJ
ejpam-4974	11	12	closed	closed	ADJ
ejpam-4974	11	13	functions	function	NOUN
ejpam-4974	11	14	.	.	PUNCT
ejpam-4974	12	1	caldas	calda	NOUN
ejpam-4974	12	2	and	and	CCONJ
ejpam-4974	12	3	navalagi	navalagi	ADJ
ejpam-4974	12	4	[	[	X
ejpam-4974	12	5	7	7	NUM
ejpam-4974	12	6	]	]	PUNCT
ejpam-4974	12	7	introduced	introduce	VERB
ejpam-4974	12	8	two	two	NUM
ejpam-4974	12	9	new	new	ADJ
ejpam-4974	12	10	classes	class	NOUN
ejpam-4974	12	11	of	of	ADP
ejpam-4974	12	12	functions	function	NOUN
ejpam-4974	12	13	called	call	VERB
ejpam-4974	12	14	weakly	weakly	ADJ
ejpam-4974	12	15	preopen	preopen	ADJ
ejpam-4974	12	16	functions	function	NOUN
ejpam-4974	12	17	and	and	CCONJ
ejpam-4974	12	18	weakly	weakly	ADJ
ejpam-4974	12	19	preclosed	preclose	VERB
ejpam-4974	12	20	functions	function	NOUN
ejpam-4974	12	21	as	as	ADP
ejpam-4974	12	22	a	a	DET
ejpam-4974	12	23	generalization	generalization	NOUN
ejpam-4974	12	24	of	of	ADP
ejpam-4974	12	25	weak	weak	ADJ
ejpam-4974	12	26	openness	openness	NOUN
ejpam-4974	12	27	and	and	CCONJ
ejpam-4974	12	28	weak	weak	ADJ
ejpam-4974	12	29	closedness	closedness	NOUN
ejpam-4974	12	30	due	due	ADP
ejpam-4974	12	31	to	to	ADP
ejpam-4974	12	32	[	[	X
ejpam-4974	12	33	15	15	NUM
ejpam-4974	12	34	]	]	PUNCT
ejpam-4974	12	35	and	and	CCONJ
ejpam-4974	13	1	[	[	X
ejpam-4974	13	2	16	16	NUM
ejpam-4974	13	3	]	]	X
ejpam-4974	13	4	,	,	PUNCT
ejpam-4974	13	5	respectively	respectively	ADV
ejpam-4974	13	6	.	.	PUNCT
ejpam-4974	14	1	moreover	moreover	ADV
ejpam-4974	14	2	,	,	PUNCT
ejpam-4974	14	3	caldas	calda	NOUN
ejpam-4974	14	4	and	and	CCONJ
ejpam-4974	14	5	navalagi	navalagi	ADJ
ejpam-4974	14	6	[	[	X
ejpam-4974	14	7	8	8	NUM
ejpam-4974	14	8	]	]	PUNCT
ejpam-4974	14	9	introduced	introduce	VERB
ejpam-4974	14	10	and	and	CCONJ
ejpam-4974	14	11	investigated	investigate	VERB
ejpam-4974	14	12	the	the	DET
ejpam-4974	14	13	concepts	concept	NOUN
ejpam-4974	14	14	of	of	ADP
ejpam-4974	14	15	weakly	weakly	ADJ
ejpam-4974	14	16	semi	semi	ADJ
ejpam-4974	14	17	-	-	ADJ
ejpam-4974	14	18	open	open	ADJ
ejpam-4974	14	19	functions	function	NOUN
ejpam-4974	14	20	and	and	CCONJ
ejpam-4974	14	21	weakly	weakly	ADJ
ejpam-4974	14	22	semi	semi	ADJ
ejpam-4974	14	23	-	-	ADJ
ejpam-4974	14	24	closed	closed	ADJ
ejpam-4974	14	25	functions	function	NOUN
ejpam-4974	14	26	as	as	ADP
ejpam-4974	14	27	a	a	DET
ejpam-4974	14	28	new	new	ADJ
ejpam-4974	14	29	generalization	generalization	NOUN
ejpam-4974	14	30	of	of	ADP
ejpam-4974	14	31	weakly	weakly	ADJ
ejpam-4974	14	32	open	open	ADJ
ejpam-4974	14	33	functions	function	NOUN
ejpam-4974	14	34	and	and	CCONJ
ejpam-4974	14	35	weakly	weakly	ADJ
ejpam-4974	14	36	closed	closed	ADJ
ejpam-4974	14	37	functions	function	NOUN
ejpam-4974	14	38	,	,	PUNCT
ejpam-4974	14	39	respectively	respectively	ADV
ejpam-4974	14	40	.	.	PUNCT
ejpam-4974	15	1	noiri	noiri	PROPN
ejpam-4974	15	2	et	et	PROPN
ejpam-4974	15	3	al	al	PROPN
ejpam-4974	15	4	.	.	PUNCT
ejpam-4974	16	1	[	[	X
ejpam-4974	16	2	14	14	NUM
ejpam-4974	16	3	]	]	PUNCT
ejpam-4974	16	4	introduced	introduce	VERB
ejpam-4974	16	5	and	and	CCONJ
ejpam-4974	16	6	studied	study	VERB
ejpam-4974	16	7	two	two	NUM
ejpam-4974	16	8	new	new	ADJ
ejpam-4974	16	9	classes	class	NOUN
ejpam-4974	16	10	of	of	ADP
ejpam-4974	16	11	functions	function	NOUN
ejpam-4974	16	12	called	call	VERB
ejpam-4974	16	13	weakly	weakly	ADJ
ejpam-4974	16	14	b	b	NOUN
ejpam-4974	16	15	-	-	PUNCT
ejpam-4974	16	16	θ	θ	ADJ
ejpam-4974	16	17	-	-	PUNCT
ejpam-4974	16	18	open	open	ADJ
ejpam-4974	16	19	functions	function	NOUN
ejpam-4974	16	20	and	and	CCONJ
ejpam-4974	16	21	weakly	weakly	ADJ
ejpam-4974	16	22	b	b	NOUN
ejpam-4974	16	23	-	-	PUNCT
ejpam-4974	16	24	θ	θ	ADJ
ejpam-4974	16	25	-	-	PUNCT
ejpam-4974	16	26	open	open	ADJ
ejpam-4974	16	27	functions	function	NOUN
ejpam-4974	16	28	by	by	ADP
ejpam-4974	16	29	utilizing	utilize	VERB
ejpam-4974	16	30	the	the	DET
ejpam-4974	16	31	notions	notion	NOUN
ejpam-4974	16	32	of	of	ADP
ejpam-4974	16	33	b	b	NOUN
ejpam-4974	16	34	-	-	PUNCT
ejpam-4974	16	35	θ	θ	ADJ
ejpam-4974	16	36	-	-	ADJ
ejpam-4974	16	37	open	open	ADJ
ejpam-4974	16	38	sets	set	NOUN
ejpam-4974	16	39	and	and	CCONJ
ejpam-4974	16	40	the	the	DET
ejpam-4974	16	41	b	b	PROPN
ejpam-4974	16	42	-	-	PUNCT
ejpam-4974	16	43	θ	θ	NOUN
ejpam-4974	16	44	-	-	PUNCT
ejpam-4974	16	45	closure	closure	NOUN
ejpam-4974	16	46	operator	operator	NOUN
ejpam-4974	16	47	.	.	PUNCT
ejpam-4974	17	1	weak	weak	ADJ
ejpam-4974	17	2	b	b	X
ejpam-4974	17	3	-	-	PUNCT
ejpam-4974	17	4	θ	θ	NOUN
ejpam-4974	17	5	-	-	PUNCT
ejpam-4974	17	6	openness	openness	NOUN
ejpam-4974	17	7	(	(	PUNCT
ejpam-4974	17	8	resp	resp	NOUN
ejpam-4974	17	9	.	.	PUNCT
ejpam-4974	18	1	b	b	X
ejpam-4974	18	2	-	-	PUNCT
ejpam-4974	18	3	θ	θ	NOUN
ejpam-4974	18	4	-	-	PUNCT
ejpam-4974	18	5	closedness	closedness	NOUN
ejpam-4974	18	6	)	)	PUNCT
ejpam-4974	18	7	is	be	AUX
ejpam-4974	18	8	a	a	DET
ejpam-4974	18	9	generalization	generalization	NOUN
ejpam-4974	18	10	of	of	ADP
ejpam-4974	18	11	both	both	DET
ejpam-4974	18	12	θ	θ	NOUN
ejpam-4974	18	13	-	-	NOUN
ejpam-4974	18	14	preopenness	preopenness	NOUN
ejpam-4974	18	15	and	and	CCONJ
ejpam-4974	18	16	weak	weak	ADJ
ejpam-4974	18	17	semi	semi	ADJ
ejpam-4974	18	18	-	-	ADJ
ejpam-4974	18	19	θ	θ	ADJ
ejpam-4974	18	20	-	-	PUNCT
ejpam-4974	18	21	openness	openness	NOUN
ejpam-4974	18	22	(	(	PUNCT
ejpam-4974	18	23	resp	resp	NOUN
ejpam-4974	18	24	.	.	PUNCT
ejpam-4974	19	1	θ	θ	NOUN
ejpam-4974	19	2	-	-	PUNCT
ejpam-4974	19	3	preclosedness	preclosedness	NOUN
ejpam-4974	19	4	and	and	CCONJ
ejpam-4974	19	5	weak	weak	ADJ
ejpam-4974	19	6	semi	semi	ADJ
ejpam-4974	19	7	-	-	ADJ
ejpam-4974	19	8	θ	θ	ADJ
ejpam-4974	19	9	-	-	PUNCT
ejpam-4974	19	10	closedness	closedness	ADJ
ejpam-4974	19	11	)	)	PUNCT
ejpam-4974	19	12	.	.	PUNCT
ejpam-4974	20	1	caldas	caldas	PROPN
ejpam-4974	20	2	and	and	CCONJ
ejpam-4974	20	3	navalagi	navalagi	ADJ
ejpam-4974	20	4	[	[	X
ejpam-4974	20	5	6	6	NUM
ejpam-4974	20	6	]	]	PUNCT
ejpam-4974	20	7	introduced	introduce	VERB
ejpam-4974	20	8	and	and	CCONJ
ejpam-4974	20	9	investigated	investigate	VERB
ejpam-4974	20	10	the	the	DET
ejpam-4974	20	11	notions	notion	NOUN
ejpam-4974	20	12	of	of	ADP
ejpam-4974	20	13	weakly	weakly	ADJ
ejpam-4974	20	14	β	β	X
ejpam-4974	20	15	-	-	ADJ
ejpam-4974	20	16	open	open	ADJ
ejpam-4974	20	17	functions	function	NOUN
ejpam-4974	20	18	and	and	CCONJ
ejpam-4974	20	19	weakly	weakly	ADJ
ejpam-4974	20	20	β	β	NOUN
ejpam-4974	20	21	-	-	ADJ
ejpam-4974	20	22	closed	closed	ADJ
ejpam-4974	20	23	functions	function	NOUN
ejpam-4974	20	24	.	.	PUNCT
ejpam-4974	21	1	∗corresponding	∗corresponde	VERB
ejpam-4974	21	2	author	author	NOUN
ejpam-4974	21	3	.	.	PUNCT
ejpam-4974	22	1	doi	doi	NOUN
ejpam-4974	22	2	:	:	PUNCT
ejpam-4974	22	3	https://doi.org/10.29020/nybg.ejpam.v17i1.4974	https://doi.org/10.29020/nybg.ejpam.v17i1.4974	NOUN
ejpam-4974	22	4	email	email	NOUN
ejpam-4974	22	5	addresses	address	NOUN
ejpam-4974	22	6	:	:	PUNCT
ejpam-4974	23	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4974	23	2	(	(	PUNCT
ejpam-4974	23	3	c.	c.	PROPN
ejpam-4974	23	4	boonpok	boonpok	PROPN
ejpam-4974	23	5	)	)	PUNCT
ejpam-4974	23	6	,	,	PUNCT
ejpam-4974	23	7	montri.t@msu.ac.th	montri.t@msu.ac.th	PROPN
ejpam-4974	23	8	(	(	PUNCT
ejpam-4974	23	9	m.	m.	NOUN
ejpam-4974	23	10	thongmoon	thongmoon	PROPN
ejpam-4974	23	11	)	)	PUNCT
ejpam-4974	23	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4974	24	1	248	248	NUM
ejpam-4974	24	2	©	©	ADP
ejpam-4974	24	3	2024	2024	NUM
ejpam-4974	24	4	ejpam	ejpam	NOUN
ejpam-4974	24	5	all	all	DET
ejpam-4974	24	6	rights	right	NOUN
ejpam-4974	24	7	reserved	reserve	VERB
ejpam-4974	24	8	.	.	PUNCT
ejpam-4974	25	1	c.	c.	PROPN
ejpam-4974	25	2	boonpok	boonpok	PROPN
ejpam-4974	25	3	,	,	PUNCT
ejpam-4974	25	4	m.	m.	NOUN
ejpam-4974	25	5	thongmoon	thongmoon	PROPN
ejpam-4974	25	6	/	/	SYM
ejpam-4974	25	7	eur	eur	PROPN
ejpam-4974	25	8	.	.	PUNCT
ejpam-4974	26	1	j.	j.	PROPN
ejpam-4974	26	2	pure	pure	PROPN
ejpam-4974	26	3	appl	appl	PROPN
ejpam-4974	26	4	.	.	PROPN
ejpam-4974	26	5	math	math	PROPN
ejpam-4974	26	6	,	,	PUNCT
ejpam-4974	26	7	17	17	NUM
ejpam-4974	26	8	(	(	PUNCT
ejpam-4974	26	9	1	1	NUM
ejpam-4974	26	10	)	)	PUNCT
ejpam-4974	26	11	(	(	PUNCT
ejpam-4974	26	12	2024	2024	NUM
ejpam-4974	26	13	)	)	PUNCT
ejpam-4974	26	14	,	,	PUNCT
ejpam-4974	26	15	248	248	NUM
ejpam-4974	26	16	-	-	SYM
ejpam-4974	26	17	255	255	NUM
ejpam-4974	26	18	249	249	NUM
ejpam-4974	26	19	the	the	DET
ejpam-4974	26	20	concepts	concept	NOUN
ejpam-4974	26	21	of	of	ADP
ejpam-4974	26	22	(	(	PUNCT
ejpam-4974	26	23	λ	λ	PROPN
ejpam-4974	26	24	,	,	PUNCT
ejpam-4974	26	25	s)-open	s)-open	PUNCT
ejpam-4974	26	26	sets	set	NOUN
ejpam-4974	26	27	,	,	PUNCT
ejpam-4974	26	28	s(λ	s(λ	PROPN
ejpam-4974	26	29	,	,	PUNCT
ejpam-4974	26	30	s)-open	s)-open	PUNCT
ejpam-4974	26	31	sets	set	NOUN
ejpam-4974	26	32	,	,	PUNCT
ejpam-4974	26	33	p(λ	p(λ	NOUN
ejpam-4974	26	34	,	,	PUNCT
ejpam-4974	26	35	s)-open	s)-open	PUNCT
ejpam-4974	26	36	sets	set	NOUN
ejpam-4974	26	37	,	,	PUNCT
ejpam-4974	26	38	α(λ	α(λ	PROPN
ejpam-4974	26	39	,	,	PUNCT
ejpam-4974	26	40	s)-open	s)-open	PUNCT
ejpam-4974	26	41	sets	set	NOUN
ejpam-4974	26	42	,	,	PUNCT
ejpam-4974	26	43	β(λ	β(λ	X
ejpam-4974	26	44	,	,	PUNCT
ejpam-4974	26	45	s)-open	s)-open	PUNCT
ejpam-4974	26	46	sets	set	NOUN
ejpam-4974	26	47	and	and	CCONJ
ejpam-4974	26	48	b(λ	b(λ	NOUN
ejpam-4974	26	49	,	,	PUNCT
ejpam-4974	26	50	s)-open	s)-open	PUNCT
ejpam-4974	26	51	sets	set	NOUN
ejpam-4974	26	52	were	be	AUX
ejpam-4974	26	53	studied	study	VERB
ejpam-4974	26	54	in	in	ADP
ejpam-4974	26	55	[	[	X
ejpam-4974	26	56	5	5	NUM
ejpam-4974	26	57	]	]	PUNCT
ejpam-4974	26	58	.	.	PUNCT
ejpam-4974	27	1	in	in	ADP
ejpam-4974	27	2	[	[	X
ejpam-4974	27	3	1	1	NUM
ejpam-4974	27	4	]	]	PUNCT
ejpam-4974	27	5	,	,	PUNCT
ejpam-4974	27	6	the	the	DET
ejpam-4974	27	7	present	present	ADJ
ejpam-4974	27	8	authors	author	NOUN
ejpam-4974	27	9	investigated	investigate	VERB
ejpam-4974	27	10	some	some	DET
ejpam-4974	27	11	properties	property	NOUN
ejpam-4974	27	12	of	of	ADP
ejpam-4974	27	13	(	(	PUNCT
ejpam-4974	27	14	λ	λ	PROPN
ejpam-4974	27	15	,	,	PUNCT
ejpam-4974	27	16	sp)-open	sp)-open	ADJ
ejpam-4974	27	17	sets	set	NOUN
ejpam-4974	27	18	,	,	PUNCT
ejpam-4974	27	19	s(λ	s(λ	PROPN
ejpam-4974	27	20	,	,	PUNCT
ejpam-4974	27	21	sp)-open	sp)-open	ADJ
ejpam-4974	27	22	sets	set	NOUN
ejpam-4974	27	23	,	,	PUNCT
ejpam-4974	27	24	p(λ	p(λ	NOUN
ejpam-4974	27	25	,	,	PUNCT
ejpam-4974	27	26	sp)-open	sp)-open	ADJ
ejpam-4974	27	27	sets	set	NOUN
ejpam-4974	27	28	,	,	PUNCT
ejpam-4974	27	29	α(λ	α(λ	PROPN
ejpam-4974	27	30	,	,	PUNCT
ejpam-4974	27	31	sp)-open	sp)-open	ADJ
ejpam-4974	27	32	sets	set	NOUN
ejpam-4974	27	33	,	,	PUNCT
ejpam-4974	27	34	β(λ	β(λ	X
ejpam-4974	27	35	,	,	PUNCT
ejpam-4974	27	36	sp)-open	sp)-open	ADJ
ejpam-4974	27	37	sets	set	NOUN
ejpam-4974	27	38	and	and	CCONJ
ejpam-4974	27	39	b(λ	b(λ	NOUN
ejpam-4974	27	40	,	,	PUNCT
ejpam-4974	27	41	sp)-open	sp)-open	ADJ
ejpam-4974	27	42	sets	set	NOUN
ejpam-4974	27	43	.	.	PUNCT
ejpam-4974	28	1	srisarakham	srisarakham	PROPN
ejpam-4974	28	2	and	and	CCONJ
ejpam-4974	28	3	boonpok	boonpok	VERB
ejpam-4974	29	1	[	[	X
ejpam-4974	29	2	17	17	NUM
ejpam-4974	29	3	]	]	PUNCT
ejpam-4974	29	4	studied	study	VERB
ejpam-4974	29	5	several	several	ADJ
ejpam-4974	29	6	properties	property	NOUN
ejpam-4974	29	7	of	of	ADP
ejpam-4974	29	8	δp(λ	δp(λ	NOUN
ejpam-4974	29	9	,	,	PUNCT
ejpam-4974	29	10	s)-closed	s)-close	VERB
ejpam-4974	29	11	sets	set	NOUN
ejpam-4974	29	12	and	and	CCONJ
ejpam-4974	29	13	the	the	DET
ejpam-4974	29	14	δp(λ	δp(λ	NOUN
ejpam-4974	29	15	,	,	PUNCT
ejpam-4974	29	16	s)-closure	s)-closure	NOUN
ejpam-4974	29	17	operator	operator	NOUN
ejpam-4974	29	18	.	.	PUNCT
ejpam-4974	30	1	khampakdee	khampakdee	NOUN
ejpam-4974	30	2	and	and	CCONJ
ejpam-4974	30	3	boonpok	boonpok	PRON
ejpam-4974	31	1	[	[	X
ejpam-4974	31	2	11	11	NUM
ejpam-4974	31	3	]	]	PUNCT
ejpam-4974	31	4	introduced	introduce	VERB
ejpam-4974	31	5	and	and	CCONJ
ejpam-4974	31	6	investigated	investigate	VERB
ejpam-4974	31	7	the	the	DET
ejpam-4974	31	8	concept	concept	NOUN
ejpam-4974	31	9	of	of	ADP
ejpam-4974	31	10	(	(	PUNCT
ejpam-4974	31	11	λ	λ	PROPN
ejpam-4974	31	12	,	,	PUNCT
ejpam-4974	31	13	p)-closed	p)-close	VERB
ejpam-4974	31	14	functions	function	NOUN
ejpam-4974	31	15	.	.	PUNCT
ejpam-4974	32	1	the	the	DET
ejpam-4974	32	2	notion	notion	NOUN
ejpam-4974	32	3	of	of	ADP
ejpam-4974	32	4	weakly	weakly	ADJ
ejpam-4974	32	5	b(λ	b(λ	NOUN
ejpam-4974	32	6	,	,	PUNCT
ejpam-4974	32	7	p)-open	p)-open	ADJ
ejpam-4974	32	8	functions	function	NOUN
ejpam-4974	32	9	was	be	AUX
ejpam-4974	32	10	studied	study	VERB
ejpam-4974	32	11	by	by	ADP
ejpam-4974	32	12	chutiman	chutiman	NOUN
ejpam-4974	32	13	and	and	CCONJ
ejpam-4974	32	14	boonpok	boonpok	NOUN
ejpam-4974	33	1	[	[	X
ejpam-4974	33	2	9	9	NUM
ejpam-4974	33	3	]	]	PUNCT
ejpam-4974	33	4	.	.	PUNCT
ejpam-4974	34	1	some	some	DET
ejpam-4974	34	2	characterizations	characterization	NOUN
ejpam-4974	34	3	of	of	ADP
ejpam-4974	34	4	weakly	weakly	ADJ
ejpam-4974	34	5	δ(λ	δ(λ	PROPN
ejpam-4974	34	6	,	,	PUNCT
ejpam-4974	34	7	p)-open	p)-open	VERB
ejpam-4974	34	8	functions	function	NOUN
ejpam-4974	34	9	and	and	CCONJ
ejpam-4974	34	10	weakly	weakly	ADJ
ejpam-4974	34	11	δ(λ	δ(λ	PROPN
ejpam-4974	34	12	,	,	PUNCT
ejpam-4974	34	13	p)closed	p)close	VERB
ejpam-4974	34	14	functions	function	NOUN
ejpam-4974	34	15	were	be	AUX
ejpam-4974	34	16	presented	present	VERB
ejpam-4974	34	17	in	in	ADP
ejpam-4974	34	18	[	[	X
ejpam-4974	34	19	18	18	NUM
ejpam-4974	34	20	]	]	PUNCT
ejpam-4974	34	21	and	and	CCONJ
ejpam-4974	34	22	[	[	X
ejpam-4974	34	23	12	12	NUM
ejpam-4974	34	24	]	]	PUNCT
ejpam-4974	34	25	,	,	PUNCT
ejpam-4974	34	26	respectively	respectively	ADV
ejpam-4974	34	27	.	.	PUNCT
ejpam-4974	35	1	moreover	moreover	ADV
ejpam-4974	35	2	,	,	PUNCT
ejpam-4974	35	3	several	several	ADJ
ejpam-4974	35	4	characterizations	characterization	NOUN
ejpam-4974	35	5	θp(λ	θp(λ	NOUN
ejpam-4974	35	6	,	,	PUNCT
ejpam-4974	35	7	p)-open	p)-open	NOUN
ejpam-4974	35	8	functions	function	NOUN
ejpam-4974	35	9	and	and	CCONJ
ejpam-4974	35	10	θp(λ	θp(λ	NOUN
ejpam-4974	35	11	,	,	PUNCT
ejpam-4974	35	12	p)-closed	p)-close	VERB
ejpam-4974	35	13	functions	function	NOUN
ejpam-4974	35	14	were	be	AUX
ejpam-4974	35	15	established	establish	VERB
ejpam-4974	35	16	in	in	ADP
ejpam-4974	35	17	[	[	X
ejpam-4974	35	18	2	2	NUM
ejpam-4974	35	19	]	]	PUNCT
ejpam-4974	35	20	.	.	PUNCT
ejpam-4974	36	1	in	in	ADP
ejpam-4974	36	2	[	[	X
ejpam-4974	36	3	3	3	NUM
ejpam-4974	36	4	]	]	PUNCT
ejpam-4974	36	5	,	,	PUNCT
ejpam-4974	36	6	the	the	DET
ejpam-4974	36	7	authors	author	NOUN
ejpam-4974	36	8	introduced	introduce	VERB
ejpam-4974	36	9	and	and	CCONJ
ejpam-4974	36	10	investigated	investigate	VERB
ejpam-4974	36	11	the	the	DET
ejpam-4974	36	12	concepts	concept	NOUN
ejpam-4974	36	13	of	of	ADP
ejpam-4974	36	14	weakly	weakly	ADJ
ejpam-4974	36	15	p(λ	p(λ	NOUN
ejpam-4974	36	16	,	,	PUNCT
ejpam-4974	36	17	p)-open	p)-open	NOUN
ejpam-4974	36	18	functions	function	NOUN
ejpam-4974	36	19	and	and	CCONJ
ejpam-4974	36	20	weakly	weakly	ADJ
ejpam-4974	36	21	p(λ	p(λ	NOUN
ejpam-4974	36	22	,	,	PUNCT
ejpam-4974	36	23	p)-closed	p)-close	VERB
ejpam-4974	36	24	functions	function	NOUN
ejpam-4974	36	25	.	.	PUNCT
ejpam-4974	37	1	in	in	ADP
ejpam-4974	37	2	this	this	DET
ejpam-4974	37	3	paper	paper	NOUN
ejpam-4974	37	4	,	,	PUNCT
ejpam-4974	37	5	we	we	PRON
ejpam-4974	37	6	introduce	introduce	VERB
ejpam-4974	37	7	the	the	DET
ejpam-4974	37	8	notions	notion	NOUN
ejpam-4974	37	9	of	of	ADP
ejpam-4974	37	10	weakly	weakly	ADJ
ejpam-4974	37	11	β(λ	β(λ	NOUN
ejpam-4974	37	12	,	,	PUNCT
ejpam-4974	37	13	p)open	p)open	ADJ
ejpam-4974	37	14	functions	function	NOUN
ejpam-4974	37	15	and	and	CCONJ
ejpam-4974	37	16	weakly	weakly	ADJ
ejpam-4974	37	17	β(λ	β(λ	NOUN
ejpam-4974	37	18	,	,	PUNCT
ejpam-4974	37	19	p)-closed	p)-close	VERB
ejpam-4974	37	20	functions	function	NOUN
ejpam-4974	37	21	.	.	PUNCT
ejpam-4974	38	1	furthermore	furthermore	ADV
ejpam-4974	38	2	,	,	PUNCT
ejpam-4974	38	3	some	some	DET
ejpam-4974	38	4	characterizations	characterization	NOUN
ejpam-4974	38	5	of	of	ADP
ejpam-4974	38	6	weakly	weakly	ADJ
ejpam-4974	38	7	β(λ	β(λ	NOUN
ejpam-4974	38	8	,	,	PUNCT
ejpam-4974	38	9	p)-open	p)-open	NOUN
ejpam-4974	38	10	functions	function	NOUN
ejpam-4974	38	11	and	and	CCONJ
ejpam-4974	38	12	weakly	weakly	ADJ
ejpam-4974	38	13	β(λ	β(λ	NOUN
ejpam-4974	38	14	,	,	PUNCT
ejpam-4974	38	15	p)-closed	p)-close	VERB
ejpam-4974	38	16	functions	function	NOUN
ejpam-4974	38	17	are	be	AUX
ejpam-4974	38	18	discussed	discuss	VERB
ejpam-4974	38	19	.	.	PUNCT
ejpam-4974	39	1	2	2	X
ejpam-4974	39	2	.	.	X
ejpam-4974	39	3	preliminaries	preliminary	NOUN
ejpam-4974	39	4	throughout	throughout	ADP
ejpam-4974	39	5	the	the	DET
ejpam-4974	39	6	present	present	ADJ
ejpam-4974	39	7	paper	paper	NOUN
ejpam-4974	39	8	,	,	PUNCT
ejpam-4974	39	9	spaces	space	NOUN
ejpam-4974	39	10	(	(	PUNCT
ejpam-4974	39	11	x	x	X
ejpam-4974	39	12	,	,	PUNCT
ejpam-4974	39	13	τ	τ	X
ejpam-4974	39	14	)	)	PUNCT
ejpam-4974	39	15	and	and	CCONJ
ejpam-4974	39	16	(	(	PUNCT
ejpam-4974	39	17	y	y	PROPN
ejpam-4974	39	18	,	,	PUNCT
ejpam-4974	39	19	σ	σ	PROPN
ejpam-4974	39	20	)	)	PUNCT
ejpam-4974	39	21	(	(	PUNCT
ejpam-4974	39	22	or	or	CCONJ
ejpam-4974	39	23	simply	simply	ADV
ejpam-4974	39	24	x	x	X
ejpam-4974	39	25	and	and	CCONJ
ejpam-4974	39	26	y	y	PROPN
ejpam-4974	39	27	)	)	PUNCT
ejpam-4974	39	28	always	always	ADV
ejpam-4974	39	29	mean	mean	VERB
ejpam-4974	39	30	topological	topological	ADJ
ejpam-4974	39	31	spaces	space	NOUN
ejpam-4974	39	32	on	on	ADP
ejpam-4974	39	33	which	which	PRON
ejpam-4974	39	34	no	no	DET
ejpam-4974	39	35	separation	separation	NOUN
ejpam-4974	39	36	axioms	axiom	NOUN
ejpam-4974	39	37	are	be	AUX
ejpam-4974	39	38	assumed	assume	VERB
ejpam-4974	39	39	unless	unless	SCONJ
ejpam-4974	39	40	explicitly	explicitly	ADV
ejpam-4974	39	41	stated	state	VERB
ejpam-4974	39	42	.	.	PUNCT
ejpam-4974	40	1	for	for	ADP
ejpam-4974	40	2	a	a	DET
ejpam-4974	40	3	subset	subset	NOUN
ejpam-4974	40	4	a	a	PRON
ejpam-4974	40	5	of	of	ADP
ejpam-4974	40	6	a	a	DET
ejpam-4974	40	7	topological	topological	ADJ
ejpam-4974	40	8	space	space	NOUN
ejpam-4974	40	9	(	(	PUNCT
ejpam-4974	40	10	x	x	X
ejpam-4974	40	11	,	,	PUNCT
ejpam-4974	40	12	τ	τ	PROPN
ejpam-4974	40	13	)	)	PUNCT
ejpam-4974	40	14	,	,	PUNCT
ejpam-4974	40	15	cl(a	cl(a	NUM
ejpam-4974	40	16	)	)	PUNCT
ejpam-4974	40	17	and	and	CCONJ
ejpam-4974	40	18	int(a	int(a	PROPN
ejpam-4974	40	19	)	)	PUNCT
ejpam-4974	40	20	,	,	PUNCT
ejpam-4974	40	21	represent	represent	VERB
ejpam-4974	40	22	the	the	DET
ejpam-4974	40	23	closure	closure	NOUN
ejpam-4974	40	24	and	and	CCONJ
ejpam-4974	40	25	the	the	DET
ejpam-4974	40	26	interior	interior	NOUN
ejpam-4974	40	27	of	of	ADP
ejpam-4974	40	28	a	a	PRON
ejpam-4974	40	29	,	,	PUNCT
ejpam-4974	40	30	respectively	respectively	ADV
ejpam-4974	40	31	.	.	PUNCT
ejpam-4974	41	1	a	a	DET
ejpam-4974	41	2	subset	subset	NOUN
ejpam-4974	41	3	a	a	PRON
ejpam-4974	41	4	of	of	ADP
ejpam-4974	41	5	a	a	DET
ejpam-4974	41	6	topological	topological	ADJ
ejpam-4974	41	7	space	space	NOUN
ejpam-4974	41	8	(	(	PUNCT
ejpam-4974	41	9	x	x	X
ejpam-4974	41	10	,	,	PUNCT
ejpam-4974	41	11	τ	τ	X
ejpam-4974	41	12	)	)	PUNCT
ejpam-4974	41	13	is	be	AUX
ejpam-4974	41	14	said	say	VERB
ejpam-4974	41	15	to	to	PART
ejpam-4974	41	16	be	be	AUX
ejpam-4974	41	17	preopen	preopen	ADJ
ejpam-4974	41	18	[	[	X
ejpam-4974	41	19	13	13	NUM
ejpam-4974	41	20	]	]	PUNCT
ejpam-4974	41	21	if	if	SCONJ
ejpam-4974	41	22	a	a	DET
ejpam-4974	41	23	⊆	⊆	NUM
ejpam-4974	41	24	int(cl(a	int(cl(a	PROPN
ejpam-4974	41	25	)	)	PUNCT
ejpam-4974	41	26	)	)	PUNCT
ejpam-4974	41	27	.	.	PUNCT
ejpam-4974	42	1	the	the	DET
ejpam-4974	42	2	complement	complement	NOUN
ejpam-4974	42	3	of	of	ADP
ejpam-4974	42	4	a	a	DET
ejpam-4974	42	5	preopen	preopen	ADJ
ejpam-4974	42	6	set	set	NOUN
ejpam-4974	42	7	is	be	AUX
ejpam-4974	42	8	called	call	VERB
ejpam-4974	42	9	preclosed	preclose	VERB
ejpam-4974	42	10	.	.	PUNCT
ejpam-4974	43	1	the	the	DET
ejpam-4974	43	2	family	family	NOUN
ejpam-4974	43	3	of	of	ADP
ejpam-4974	43	4	all	all	DET
ejpam-4974	43	5	preopen	preopen	ADJ
ejpam-4974	43	6	sets	set	NOUN
ejpam-4974	43	7	of	of	ADP
ejpam-4974	43	8	a	a	DET
ejpam-4974	43	9	topological	topological	ADJ
ejpam-4974	43	10	space	space	NOUN
ejpam-4974	43	11	(	(	PUNCT
ejpam-4974	43	12	x	x	X
ejpam-4974	43	13	,	,	PUNCT
ejpam-4974	43	14	τ	τ	X
ejpam-4974	43	15	)	)	PUNCT
ejpam-4974	43	16	is	be	AUX
ejpam-4974	43	17	denoted	denote	VERB
ejpam-4974	43	18	by	by	ADP
ejpam-4974	43	19	po(x	po(x	NUM
ejpam-4974	43	20	,	,	PUNCT
ejpam-4974	43	21	τ	τ	PROPN
ejpam-4974	43	22	)	)	PUNCT
ejpam-4974	43	23	.	.	PUNCT
ejpam-4974	44	1	a	a	DET
ejpam-4974	44	2	subset	subset	NOUN
ejpam-4974	44	3	λp(a	λp(a	NOUN
ejpam-4974	44	4	)	)	PUNCT
ejpam-4974	45	1	[	[	X
ejpam-4974	45	2	10	10	NUM
ejpam-4974	45	3	]	]	PUNCT
ejpam-4974	45	4	is	be	AUX
ejpam-4974	45	5	defined	define	VERB
ejpam-4974	45	6	as	as	SCONJ
ejpam-4974	45	7	follows	follow	VERB
ejpam-4974	45	8	:	:	PUNCT
ejpam-4974	45	9	λp(a	λp(a	NUM
ejpam-4974	45	10	)	)	PUNCT
ejpam-4974	46	1	=	=	PUNCT
ejpam-4974	47	1	∩{u	∩{u	PROPN
ejpam-4974	47	2	|	|	ADV
ejpam-4974	47	3	a	a	DET
ejpam-4974	47	4	⊆	⊆	NUM
ejpam-4974	47	5	u	u	NOUN
ejpam-4974	47	6	,	,	PUNCT
ejpam-4974	47	7	u	u	PROPN
ejpam-4974	47	8	∈	∈	PROPN
ejpam-4974	47	9	po(x	po(x	NOUN
ejpam-4974	47	10	,	,	PUNCT
ejpam-4974	47	11	τ	τ	NOUN
ejpam-4974	47	12	)	)	PUNCT
ejpam-4974	47	13	}	}	PUNCT
ejpam-4974	47	14	.	.	PUNCT
ejpam-4974	48	1	a	a	DET
ejpam-4974	48	2	subset	subset	NOUN
ejpam-4974	48	3	a	a	PRON
ejpam-4974	48	4	of	of	ADP
ejpam-4974	48	5	a	a	DET
ejpam-4974	48	6	topological	topological	ADJ
ejpam-4974	48	7	space	space	NOUN
ejpam-4974	48	8	(	(	PUNCT
ejpam-4974	48	9	x	x	X
ejpam-4974	48	10	,	,	PUNCT
ejpam-4974	48	11	τ	τ	X
ejpam-4974	48	12	)	)	PUNCT
ejpam-4974	48	13	is	be	AUX
ejpam-4974	48	14	called	call	VERB
ejpam-4974	48	15	a	a	DET
ejpam-4974	48	16	λp	λp	NOUN
ejpam-4974	48	17	-	-	PUNCT
ejpam-4974	48	18	set	set	NOUN
ejpam-4974	48	19	[	[	X
ejpam-4974	48	20	4	4	NUM
ejpam-4974	48	21	]	]	X
ejpam-4974	48	22	(	(	PUNCT
ejpam-4974	48	23	pre	pre	ADJ
ejpam-4974	48	24	-	-	ADJ
ejpam-4974	48	25	λ	λ	NOUN
ejpam-4974	48	26	-	-	NOUN
ejpam-4974	48	27	set	set	NOUN
ejpam-4974	48	28	[	[	X
ejpam-4974	48	29	10	10	NUM
ejpam-4974	48	30	]	]	SYM
ejpam-4974	48	31	)	)	PUNCT
ejpam-4974	48	32	if	if	SCONJ
ejpam-4974	48	33	a	a	DET
ejpam-4974	48	34	=	=	NOUN
ejpam-4974	48	35	λp(a	λp(a	NOUN
ejpam-4974	48	36	)	)	PUNCT
ejpam-4974	48	37	.	.	PUNCT
ejpam-4974	49	1	a	a	DET
ejpam-4974	49	2	subset	subset	NOUN
ejpam-4974	49	3	a	a	PRON
ejpam-4974	49	4	of	of	ADP
ejpam-4974	49	5	a	a	DET
ejpam-4974	49	6	topological	topological	ADJ
ejpam-4974	49	7	space	space	NOUN
ejpam-4974	49	8	(	(	PUNCT
ejpam-4974	49	9	x	x	X
ejpam-4974	49	10	,	,	PUNCT
ejpam-4974	49	11	τ	τ	X
ejpam-4974	49	12	)	)	PUNCT
ejpam-4974	49	13	is	be	AUX
ejpam-4974	49	14	called	call	VERB
ejpam-4974	49	15	(	(	PUNCT
ejpam-4974	49	16	λ	λ	X
ejpam-4974	49	17	,	,	PUNCT
ejpam-4974	49	18	p)-closed	p)-close	VERB
ejpam-4974	49	19	[	[	X
ejpam-4974	49	20	4	4	X
ejpam-4974	49	21	]	]	X
ejpam-4974	49	22	if	if	SCONJ
ejpam-4974	49	23	a	a	DET
ejpam-4974	49	24	=	=	X
ejpam-4974	49	25	t	t	PROPN
ejpam-4974	49	26	∩	∩	ADJ
ejpam-4974	49	27	c	c	NOUN
ejpam-4974	49	28	,	,	PUNCT
ejpam-4974	49	29	where	where	SCONJ
ejpam-4974	49	30	t	t	PROPN
ejpam-4974	49	31	is	be	AUX
ejpam-4974	49	32	a	a	DET
ejpam-4974	49	33	λp	λp	ADV
ejpam-4974	49	34	-	-	PUNCT
ejpam-4974	49	35	set	set	NOUN
ejpam-4974	49	36	and	and	CCONJ
ejpam-4974	49	37	c	c	NOUN
ejpam-4974	49	38	is	be	AUX
ejpam-4974	49	39	a	a	DET
ejpam-4974	49	40	preclosed	preclose	VERB
ejpam-4974	49	41	set	set	NOUN
ejpam-4974	49	42	.	.	PUNCT
ejpam-4974	50	1	the	the	DET
ejpam-4974	50	2	complement	complement	NOUN
ejpam-4974	50	3	of	of	ADP
ejpam-4974	50	4	a	a	DET
ejpam-4974	50	5	(	(	PUNCT
ejpam-4974	50	6	λ	λ	PROPN
ejpam-4974	50	7	,	,	PUNCT
ejpam-4974	50	8	p)-closed	p)-close	VERB
ejpam-4974	50	9	set	set	NOUN
ejpam-4974	50	10	is	be	AUX
ejpam-4974	50	11	called	call	VERB
ejpam-4974	50	12	(	(	PUNCT
ejpam-4974	50	13	λ	λ	X
ejpam-4974	50	14	,	,	PUNCT
ejpam-4974	50	15	p)-open	p)-open	ADJ
ejpam-4974	50	16	.	.	PUNCT
ejpam-4974	51	1	the	the	DET
ejpam-4974	51	2	family	family	NOUN
ejpam-4974	51	3	of	of	ADP
ejpam-4974	51	4	all	all	DET
ejpam-4974	51	5	(	(	PUNCT
ejpam-4974	51	6	λ	λ	X
ejpam-4974	51	7	,	,	PUNCT
ejpam-4974	51	8	p)-open	p)-open	ADJ
ejpam-4974	51	9	(	(	PUNCT
ejpam-4974	51	10	resp	resp	NOUN
ejpam-4974	51	11	.	.	PUNCT
ejpam-4974	52	1	(	(	PUNCT
ejpam-4974	52	2	λ	λ	X
ejpam-4974	52	3	,	,	PUNCT
ejpam-4974	52	4	p)-closed	p)-close	VERB
ejpam-4974	52	5	)	)	PUNCT
ejpam-4974	52	6	sets	set	NOUN
ejpam-4974	52	7	in	in	ADP
ejpam-4974	52	8	a	a	DET
ejpam-4974	52	9	topological	topological	ADJ
ejpam-4974	52	10	space	space	NOUN
ejpam-4974	52	11	(	(	PUNCT
ejpam-4974	52	12	x	x	X
ejpam-4974	52	13	,	,	PUNCT
ejpam-4974	52	14	τ	τ	X
ejpam-4974	52	15	)	)	PUNCT
ejpam-4974	52	16	is	be	AUX
ejpam-4974	52	17	denoted	denote	VERB
ejpam-4974	52	18	by	by	ADP
ejpam-4974	52	19	λpo(x	λpo(x	PROPN
ejpam-4974	52	20	,	,	PUNCT
ejpam-4974	52	21	τ	τ	X
ejpam-4974	52	22	)	)	PUNCT
ejpam-4974	52	23	(	(	PUNCT
ejpam-4974	52	24	resp	resp	NOUN
ejpam-4974	52	25	.	.	PUNCT
ejpam-4974	53	1	λpc(x	λpc(x	PROPN
ejpam-4974	53	2	,	,	PUNCT
ejpam-4974	53	3	τ	τ	PROPN
ejpam-4974	53	4	)	)	PUNCT
ejpam-4974	53	5	)	)	PUNCT
ejpam-4974	53	6	.	.	PUNCT
ejpam-4974	54	1	let	let	VERB
ejpam-4974	54	2	a	a	DET
ejpam-4974	54	3	be	be	AUX
ejpam-4974	54	4	a	a	DET
ejpam-4974	54	5	subset	subset	NOUN
ejpam-4974	54	6	of	of	ADP
ejpam-4974	54	7	a	a	DET
ejpam-4974	54	8	topological	topological	ADJ
ejpam-4974	54	9	space	space	NOUN
ejpam-4974	54	10	(	(	PUNCT
ejpam-4974	54	11	x	x	X
ejpam-4974	54	12	,	,	PUNCT
ejpam-4974	54	13	τ	τ	PROPN
ejpam-4974	54	14	)	)	PUNCT
ejpam-4974	54	15	.	.	PUNCT
ejpam-4974	55	1	a	a	DET
ejpam-4974	55	2	point	point	NOUN
ejpam-4974	55	3	x	x	X
ejpam-4974	55	4	∈	∈	NOUN
ejpam-4974	55	5	x	x	PUNCT
ejpam-4974	55	6	is	be	AUX
ejpam-4974	55	7	called	call	VERB
ejpam-4974	55	8	a	a	DET
ejpam-4974	55	9	(	(	PUNCT
ejpam-4974	55	10	λ	λ	NOUN
ejpam-4974	55	11	,	,	PUNCT
ejpam-4974	55	12	p)-cluster	p)-cluster	NOUN
ejpam-4974	55	13	point	point	NOUN
ejpam-4974	55	14	[	[	X
ejpam-4974	55	15	4	4	X
ejpam-4974	55	16	]	]	PUNCT
ejpam-4974	55	17	of	of	ADP
ejpam-4974	55	18	a	a	PRON
ejpam-4974	55	19	if	if	SCONJ
ejpam-4974	55	20	a	a	DET
ejpam-4974	55	21	∩	∩	ADJ
ejpam-4974	55	22	u	u	ADJ
ejpam-4974	55	23	̸=	̸=	PROPN
ejpam-4974	55	24	∅	∅	NOUN
ejpam-4974	55	25	for	for	ADP
ejpam-4974	55	26	every	every	DET
ejpam-4974	55	27	(	(	PUNCT
ejpam-4974	55	28	λ	λ	NOUN
ejpam-4974	55	29	,	,	PUNCT
ejpam-4974	55	30	p)-open	p)-open	VERB
ejpam-4974	55	31	set	set	VERB
ejpam-4974	55	32	u	u	NOUN
ejpam-4974	55	33	of	of	ADP
ejpam-4974	55	34	x	x	SYM
ejpam-4974	55	35	containing	contain	VERB
ejpam-4974	55	36	x.	x.	NOUN
ejpam-4974	55	37	the	the	DET
ejpam-4974	55	38	set	set	NOUN
ejpam-4974	55	39	of	of	ADP
ejpam-4974	55	40	all	all	DET
ejpam-4974	55	41	(	(	PUNCT
ejpam-4974	55	42	λ	λ	NOUN
ejpam-4974	55	43	,	,	PUNCT
ejpam-4974	55	44	p)-cluster	p)-cluster	VERB
ejpam-4974	55	45	points	point	NOUN
ejpam-4974	55	46	of	of	ADP
ejpam-4974	55	47	a	a	PRON
ejpam-4974	55	48	is	be	AUX
ejpam-4974	55	49	called	call	VERB
ejpam-4974	55	50	the	the	DET
ejpam-4974	55	51	(	(	PUNCT
ejpam-4974	55	52	λ	λ	PROPN
ejpam-4974	55	53	,	,	PUNCT
ejpam-4974	55	54	p)-closure	p)-closure	X
ejpam-4974	56	1	[	[	X
ejpam-4974	56	2	4	4	X
ejpam-4974	56	3	]	]	PUNCT
ejpam-4974	56	4	of	of	ADP
ejpam-4974	56	5	a	a	PRON
ejpam-4974	56	6	and	and	CCONJ
ejpam-4974	56	7	is	be	AUX
ejpam-4974	56	8	denoted	denote	VERB
ejpam-4974	56	9	by	by	ADP
ejpam-4974	56	10	a(λ	a(λ	PROPN
ejpam-4974	56	11	,	,	PUNCT
ejpam-4974	56	12	p	p	NOUN
ejpam-4974	56	13	)	)	PUNCT
ejpam-4974	56	14	.	.	PUNCT
ejpam-4974	57	1	the	the	DET
ejpam-4974	57	2	union	union	NOUN
ejpam-4974	57	3	of	of	ADP
ejpam-4974	57	4	all	all	PRON
ejpam-4974	57	5	(	(	PUNCT
ejpam-4974	57	6	λ	λ	NOUN
ejpam-4974	57	7	,	,	PUNCT
ejpam-4974	57	8	p)-open	p)-open	VERB
ejpam-4974	57	9	sets	set	NOUN
ejpam-4974	57	10	of	of	ADP
ejpam-4974	57	11	x	x	PUNCT
ejpam-4974	57	12	contained	contain	VERB
ejpam-4974	57	13	in	in	ADP
ejpam-4974	57	14	a	a	PRON
ejpam-4974	57	15	is	be	AUX
ejpam-4974	57	16	called	call	VERB
ejpam-4974	57	17	the	the	DET
ejpam-4974	57	18	(	(	PUNCT
ejpam-4974	57	19	λ	λ	PROPN
ejpam-4974	57	20	,	,	PUNCT
ejpam-4974	57	21	p)-interior	p)-interior	ADJ
ejpam-4974	57	22	[	[	X
ejpam-4974	57	23	4	4	X
ejpam-4974	57	24	]	]	PUNCT
ejpam-4974	57	25	of	of	ADP
ejpam-4974	57	26	a	a	PRON
ejpam-4974	57	27	and	and	CCONJ
ejpam-4974	57	28	is	be	AUX
ejpam-4974	57	29	denoted	denote	VERB
ejpam-4974	57	30	by	by	ADP
ejpam-4974	57	31	a(λ	a(λ	PROPN
ejpam-4974	57	32	,	,	PUNCT
ejpam-4974	57	33	p	p	NOUN
ejpam-4974	57	34	)	)	PUNCT
ejpam-4974	57	35	.	.	PUNCT
ejpam-4974	58	1	a	a	DET
ejpam-4974	58	2	subset	subset	NOUN
ejpam-4974	58	3	a	a	PRON
ejpam-4974	58	4	of	of	ADP
ejpam-4974	58	5	a	a	DET
ejpam-4974	58	6	topological	topological	ADJ
ejpam-4974	58	7	space	space	NOUN
ejpam-4974	58	8	(	(	PUNCT
ejpam-4974	58	9	x	x	X
ejpam-4974	58	10	,	,	PUNCT
ejpam-4974	58	11	τ	τ	X
ejpam-4974	58	12	)	)	PUNCT
ejpam-4974	58	13	is	be	AUX
ejpam-4974	58	14	said	say	VERB
ejpam-4974	58	15	to	to	PART
ejpam-4974	58	16	be	be	AUX
ejpam-4974	58	17	s(λ	s(λ	NOUN
ejpam-4974	58	18	,	,	PUNCT
ejpam-4974	58	19	p)-open	p)-open	VERB
ejpam-4974	58	20	[	[	X
ejpam-4974	58	21	4	4	NUM
ejpam-4974	58	22	]	]	PUNCT
ejpam-4974	58	23	(	(	PUNCT
ejpam-4974	58	24	resp	resp	NOUN
ejpam-4974	58	25	.	.	PUNCT
ejpam-4974	59	1	p(λ	p(λ	PROPN
ejpam-4974	59	2	,	,	PUNCT
ejpam-4974	59	3	p)open	p)open	PROPN
ejpam-4974	60	1	[	[	X
ejpam-4974	60	2	4	4	NUM
ejpam-4974	60	3	]	]	PUNCT
ejpam-4974	60	4	,	,	PUNCT
ejpam-4974	60	5	β(λ	β(λ	X
ejpam-4974	60	6	,	,	PUNCT
ejpam-4974	60	7	p)-open	p)-open	VERB
ejpam-4974	60	8	[	[	X
ejpam-4974	60	9	4	4	NUM
ejpam-4974	60	10	]	]	PUNCT
ejpam-4974	60	11	,	,	PUNCT
ejpam-4974	60	12	α(λ	α(λ	PROPN
ejpam-4974	60	13	,	,	PUNCT
ejpam-4974	60	14	p)-open	p)-open	VERB
ejpam-4974	60	15	[	[	X
ejpam-4974	60	16	20	20	NUM
ejpam-4974	60	17	]	]	PUNCT
ejpam-4974	60	18	,	,	PUNCT
ejpam-4974	60	19	r(λ	r(λ	PROPN
ejpam-4974	60	20	,	,	PUNCT
ejpam-4974	60	21	p)-open	p)-open	VERB
ejpam-4974	60	22	[	[	X
ejpam-4974	60	23	4	4	NUM
ejpam-4974	60	24	]	]	PUNCT
ejpam-4974	60	25	)	)	PUNCT
ejpam-4974	60	26	if	if	SCONJ
ejpam-4974	60	27	a	a	DET
ejpam-4974	60	28	⊆	⊆	NUM
ejpam-4974	60	29	[	[	X
ejpam-4974	60	30	a(λ	a(λ	ADV
ejpam-4974	60	31	,	,	PUNCT
ejpam-4974	60	32	p	p	NOUN
ejpam-4974	60	33	)	)	PUNCT
ejpam-4974	60	34	]	]	PUNCT
ejpam-4974	60	35	(	(	PUNCT
ejpam-4974	60	36	λ	λ	X
ejpam-4974	60	37	,	,	PUNCT
ejpam-4974	60	38	p	p	NOUN
ejpam-4974	60	39	)	)	PUNCT
ejpam-4974	60	40	(	(	PUNCT
ejpam-4974	60	41	resp	resp	NOUN
ejpam-4974	60	42	.	.	PUNCT
ejpam-4974	61	1	a	a	DET
ejpam-4974	61	2	⊆	⊆	NUM
ejpam-4974	61	3	[	[	X
ejpam-4974	61	4	a(λ	a(λ	ADV
ejpam-4974	61	5	,	,	PUNCT
ejpam-4974	61	6	p)](λ	p)](λ	X
ejpam-4974	61	7	,	,	PUNCT
ejpam-4974	61	8	p	p	NOUN
ejpam-4974	61	9	)	)	PUNCT
ejpam-4974	61	10	,	,	PUNCT
ejpam-4974	61	11	a	a	DET
ejpam-4974	61	12	⊆	⊆	NUM
ejpam-4974	61	13	[	[	X
ejpam-4974	61	14	[	[	X
ejpam-4974	61	15	a(λ	a(λ	ADJ
ejpam-4974	61	16	,	,	PUNCT
ejpam-4974	61	17	p)](λ	p)](λ	X
ejpam-4974	61	18	,	,	PUNCT
ejpam-4974	61	19	p	p	NOUN
ejpam-4974	61	20	)	)	PUNCT
ejpam-4974	61	21	]	]	PUNCT
ejpam-4974	61	22	(	(	PUNCT
ejpam-4974	61	23	λ	λ	X
ejpam-4974	61	24	,	,	PUNCT
ejpam-4974	61	25	p	p	NOUN
ejpam-4974	61	26	)	)	PUNCT
ejpam-4974	61	27	,	,	PUNCT
ejpam-4974	61	28	a	a	DET
ejpam-4974	61	29	⊆	⊆	NUM
ejpam-4974	61	30	[	[	X
ejpam-4974	61	31	[	[	X
ejpam-4974	61	32	a(λ	a(λ	ADJ
ejpam-4974	61	33	,	,	PUNCT
ejpam-4974	61	34	p	p	NOUN
ejpam-4974	61	35	)	)	PUNCT
ejpam-4974	61	36	]	]	PUNCT
ejpam-4974	61	37	(	(	PUNCT
ejpam-4974	61	38	λ	λ	X
ejpam-4974	61	39	,	,	PUNCT
ejpam-4974	61	40	p)](λ	p)](λ	ADJ
ejpam-4974	61	41	,	,	PUNCT
ejpam-4974	61	42	p	p	NOUN
ejpam-4974	61	43	)	)	PUNCT
ejpam-4974	61	44	,	,	PUNCT
ejpam-4974	61	45	a	a	PRON
ejpam-4974	61	46	=	=	X
ejpam-4974	62	1	[	[	X
ejpam-4974	62	2	a(λ	a(λ	ADV
ejpam-4974	62	3	,	,	PUNCT
ejpam-4974	62	4	p)](λ	p)](λ	X
ejpam-4974	62	5	,	,	PUNCT
ejpam-4974	62	6	p	p	NOUN
ejpam-4974	62	7	)	)	PUNCT
ejpam-4974	62	8	)	)	PUNCT
ejpam-4974	62	9	.	.	PUNCT
ejpam-4974	63	1	the	the	DET
ejpam-4974	63	2	union	union	NOUN
ejpam-4974	63	3	of	of	ADP
ejpam-4974	63	4	all	all	DET
ejpam-4974	63	5	β(λ	β(λ	NOUN
ejpam-4974	63	6	,	,	PUNCT
ejpam-4974	63	7	p)-open	p)-open	VERB
ejpam-4974	63	8	sets	set	NOUN
ejpam-4974	63	9	of	of	ADP
ejpam-4974	63	10	x	x	PUNCT
ejpam-4974	63	11	contained	contain	VERB
ejpam-4974	63	12	in	in	ADP
ejpam-4974	63	13	a	a	PRON
ejpam-4974	63	14	is	be	AUX
ejpam-4974	63	15	called	call	VERB
ejpam-4974	63	16	the	the	DET
ejpam-4974	63	17	β(λ	β(λ	NOUN
ejpam-4974	63	18	,	,	PUNCT
ejpam-4974	63	19	p)-interior	p)-interior	NOUN
ejpam-4974	63	20	of	of	ADP
ejpam-4974	63	21	a	a	PRON
ejpam-4974	63	22	and	and	CCONJ
ejpam-4974	63	23	is	be	AUX
ejpam-4974	63	24	denoted	denote	VERB
ejpam-4974	63	25	by	by	ADP
ejpam-4974	63	26	aβ(λ	aβ(λ	NOUN
ejpam-4974	63	27	,	,	PUNCT
ejpam-4974	63	28	p	p	NOUN
ejpam-4974	63	29	)	)	PUNCT
ejpam-4974	63	30	.	.	PUNCT
ejpam-4974	64	1	the	the	DET
ejpam-4974	64	2	complement	complement	NOUN
ejpam-4974	64	3	of	of	ADP
ejpam-4974	64	4	a	a	DET
ejpam-4974	64	5	s(λ	s(λ	PROPN
ejpam-4974	64	6	,	,	PUNCT
ejpam-4974	64	7	p)-open	p)-open	ADJ
ejpam-4974	64	8	(	(	PUNCT
ejpam-4974	64	9	resp	resp	NOUN
ejpam-4974	64	10	.	.	PUNCT
ejpam-4974	65	1	p(λ	p(λ	NOUN
ejpam-4974	65	2	,	,	PUNCT
ejpam-4974	65	3	p)-open	p)-open	ADJ
ejpam-4974	65	4	,	,	PUNCT
ejpam-4974	65	5	β(λ	β(λ	X
ejpam-4974	65	6	,	,	PUNCT
ejpam-4974	65	7	p)-open	p)-open	NOUN
ejpam-4974	65	8	,	,	PUNCT
ejpam-4974	65	9	α(λ	α(λ	PROPN
ejpam-4974	65	10	,	,	PUNCT
ejpam-4974	65	11	p)-open	p)-open	NOUN
ejpam-4974	65	12	,	,	PUNCT
ejpam-4974	65	13	r(λ	r(λ	NOUN
ejpam-4974	65	14	,	,	PUNCT
ejpam-4974	65	15	p)-open	p)-open	ADJ
ejpam-4974	65	16	)	)	PUNCT
ejpam-4974	65	17	set	set	NOUN
ejpam-4974	65	18	is	be	AUX
ejpam-4974	65	19	called	call	VERB
ejpam-4974	65	20	s(λ	s(λ	PROPN
ejpam-4974	65	21	,	,	PUNCT
ejpam-4974	65	22	p)-closed	p)-close	VERB
ejpam-4974	65	23	(	(	PUNCT
ejpam-4974	65	24	resp	resp	NOUN
ejpam-4974	65	25	.	.	PUNCT
ejpam-4974	66	1	p(λ	p(λ	NOUN
ejpam-4974	66	2	,	,	PUNCT
ejpam-4974	66	3	p)-closed	p)-close	VERB
ejpam-4974	66	4	,	,	PUNCT
ejpam-4974	66	5	β(λ	β(λ	PROPN
ejpam-4974	66	6	,	,	PUNCT
ejpam-4974	66	7	p)-closed	p)-close	VERB
ejpam-4974	66	8	,	,	PUNCT
ejpam-4974	66	9	α(λ	α(λ	PROPN
ejpam-4974	66	10	,	,	PUNCT
ejpam-4974	66	11	p)-closed	p)-close	VERB
ejpam-4974	66	12	,	,	PUNCT
ejpam-4974	66	13	r(λ	r(λ	PROPN
ejpam-4974	66	14	,	,	PUNCT
ejpam-4974	66	15	p)-closed	p)-close	VERB
ejpam-4974	66	16	)	)	PUNCT
ejpam-4974	66	17	.	.	PUNCT
ejpam-4974	67	1	the	the	DET
ejpam-4974	67	2	intersection	intersection	NOUN
ejpam-4974	67	3	of	of	ADP
ejpam-4974	67	4	all	all	DET
ejpam-4974	67	5	β(λ	β(λ	NOUN
ejpam-4974	67	6	,	,	PUNCT
ejpam-4974	67	7	p)-closed	p)-close	VERB
ejpam-4974	67	8	sets	set	NOUN
ejpam-4974	67	9	of	of	ADP
ejpam-4974	67	10	x	x	PUNCT
ejpam-4974	67	11	containing	contain	VERB
ejpam-4974	67	12	a	a	PRON
ejpam-4974	67	13	is	be	AUX
ejpam-4974	67	14	called	call	VERB
ejpam-4974	67	15	the	the	DET
ejpam-4974	67	16	β(λ	β(λ	NOUN
ejpam-4974	67	17	,	,	PUNCT
ejpam-4974	67	18	p)-closure	p)-closure	NOUN
ejpam-4974	67	19	of	of	ADP
ejpam-4974	67	20	a	a	PRON
ejpam-4974	67	21	and	and	CCONJ
ejpam-4974	67	22	is	be	AUX
ejpam-4974	67	23	denoted	denote	VERB
ejpam-4974	67	24	by	by	ADP
ejpam-4974	67	25	aβ(λ	aβ(λ	NOUN
ejpam-4974	67	26	,	,	PUNCT
ejpam-4974	67	27	p	p	NOUN
ejpam-4974	67	28	)	)	PUNCT
ejpam-4974	67	29	.	.	PUNCT
ejpam-4974	68	1	let	let	VERB
ejpam-4974	68	2	a	a	DET
ejpam-4974	68	3	be	be	AUX
ejpam-4974	68	4	a	a	DET
ejpam-4974	68	5	subset	subset	NOUN
ejpam-4974	68	6	of	of	ADP
ejpam-4974	68	7	a	a	DET
ejpam-4974	68	8	c.	c.	PROPN
ejpam-4974	68	9	boonpok	boonpok	PROPN
ejpam-4974	68	10	,	,	PUNCT
ejpam-4974	68	11	m.	m.	NOUN
ejpam-4974	68	12	thongmoon	thongmoon	PROPN
ejpam-4974	68	13	/	/	SYM
ejpam-4974	68	14	eur	eur	PROPN
ejpam-4974	68	15	.	.	PUNCT
ejpam-4974	69	1	j.	j.	PROPN
ejpam-4974	69	2	pure	pure	PROPN
ejpam-4974	69	3	appl	appl	PROPN
ejpam-4974	69	4	.	.	PROPN
ejpam-4974	69	5	math	math	PROPN
ejpam-4974	69	6	,	,	PUNCT
ejpam-4974	69	7	17	17	NUM
ejpam-4974	69	8	(	(	PUNCT
ejpam-4974	69	9	1	1	NUM
ejpam-4974	69	10	)	)	PUNCT
ejpam-4974	69	11	(	(	PUNCT
ejpam-4974	69	12	2024	2024	NUM
ejpam-4974	69	13	)	)	PUNCT
ejpam-4974	69	14	,	,	PUNCT
ejpam-4974	69	15	248	248	NUM
ejpam-4974	69	16	-	-	SYM
ejpam-4974	69	17	255	255	NUM
ejpam-4974	69	18	250	250	NUM
ejpam-4974	69	19	topological	topological	ADJ
ejpam-4974	69	20	space	space	NOUN
ejpam-4974	69	21	(	(	PUNCT
ejpam-4974	69	22	x	x	X
ejpam-4974	69	23	,	,	PUNCT
ejpam-4974	69	24	τ	τ	PROPN
ejpam-4974	69	25	)	)	PUNCT
ejpam-4974	69	26	.	.	PUNCT
ejpam-4974	70	1	the	the	DET
ejpam-4974	70	2	θ(λ	θ(λ	PROPN
ejpam-4974	70	3	,	,	PUNCT
ejpam-4974	70	4	p)-closure	p)-closure	X
ejpam-4974	71	1	[	[	X
ejpam-4974	71	2	4	4	X
ejpam-4974	71	3	]	]	PUNCT
ejpam-4974	71	4	of	of	ADP
ejpam-4974	71	5	a	a	DET
ejpam-4974	71	6	,	,	PUNCT
ejpam-4974	71	7	aθ(λ	aθ(λ	ADJ
ejpam-4974	71	8	,	,	PUNCT
ejpam-4974	71	9	p	p	NOUN
ejpam-4974	71	10	)	)	PUNCT
ejpam-4974	71	11	,	,	PUNCT
ejpam-4974	71	12	is	be	AUX
ejpam-4974	71	13	defined	define	VERB
ejpam-4974	71	14	as	as	SCONJ
ejpam-4974	71	15	follows	follow	VERB
ejpam-4974	71	16	:	:	PUNCT
ejpam-4974	72	1	aθ(λ	aθ(λ	NOUN
ejpam-4974	72	2	,	,	PUNCT
ejpam-4974	72	3	p	p	NOUN
ejpam-4974	72	4	)	)	PUNCT
ejpam-4974	72	5	=	=	SYM
ejpam-4974	72	6	{	{	PUNCT
ejpam-4974	72	7	x	x	PUNCT
ejpam-4974	72	8	∈	∈	NOUN
ejpam-4974	72	9	x	x	PUNCT
ejpam-4974	72	10	|	|	ADV
ejpam-4974	72	11	a	a	DET
ejpam-4974	72	12	∩	∩	ADJ
ejpam-4974	72	13	u	u	NOUN
ejpam-4974	72	14	(	(	PUNCT
ejpam-4974	72	15	λ	λ	PROPN
ejpam-4974	72	16	,	,	PUNCT
ejpam-4974	72	17	p	p	NOUN
ejpam-4974	72	18	)	)	PUNCT
ejpam-4974	72	19	̸=	̸=	PROPN
ejpam-4974	72	20	∅	∅	NOUN
ejpam-4974	72	21	for	for	ADP
ejpam-4974	72	22	each	each	DET
ejpam-4974	72	23	(	(	PUNCT
ejpam-4974	72	24	λ	λ	PROPN
ejpam-4974	72	25	,	,	PUNCT
ejpam-4974	72	26	p)-open	p)-open	VERB
ejpam-4974	72	27	set	set	VERB
ejpam-4974	72	28	u	u	NOUN
ejpam-4974	72	29	containing	contain	VERB
ejpam-4974	72	30	x	x	X
ejpam-4974	72	31	}	}	PUNCT
ejpam-4974	72	32	.	.	PUNCT
ejpam-4974	73	1	a	a	DET
ejpam-4974	73	2	subset	subset	NOUN
ejpam-4974	73	3	a	a	PRON
ejpam-4974	73	4	of	of	ADP
ejpam-4974	73	5	a	a	DET
ejpam-4974	73	6	topological	topological	ADJ
ejpam-4974	73	7	space	space	NOUN
ejpam-4974	73	8	(	(	PUNCT
ejpam-4974	73	9	x	x	X
ejpam-4974	73	10	,	,	PUNCT
ejpam-4974	73	11	τ	τ	X
ejpam-4974	73	12	)	)	PUNCT
ejpam-4974	73	13	is	be	AUX
ejpam-4974	73	14	called	call	VERB
ejpam-4974	73	15	θ(λ	θ(λ	PROPN
ejpam-4974	73	16	,	,	PUNCT
ejpam-4974	73	17	p)-closed	p)-close	VERB
ejpam-4974	73	18	[	[	X
ejpam-4974	73	19	4	4	X
ejpam-4974	73	20	]	]	X
ejpam-4974	73	21	if	if	SCONJ
ejpam-4974	73	22	a	a	PRON
ejpam-4974	73	23	=	=	NOUN
ejpam-4974	73	24	aθ(λ	aθ(λ	NOUN
ejpam-4974	73	25	,	,	PUNCT
ejpam-4974	73	26	p	p	NOUN
ejpam-4974	73	27	)	)	PUNCT
ejpam-4974	73	28	.	.	PUNCT
ejpam-4974	74	1	the	the	DET
ejpam-4974	74	2	complement	complement	NOUN
ejpam-4974	74	3	of	of	ADP
ejpam-4974	74	4	a	a	DET
ejpam-4974	74	5	θ(λ	θ(λ	PROPN
ejpam-4974	74	6	,	,	PUNCT
ejpam-4974	74	7	p)-closed	p)-close	VERB
ejpam-4974	74	8	set	set	NOUN
ejpam-4974	74	9	is	be	AUX
ejpam-4974	74	10	said	say	VERB
ejpam-4974	74	11	to	to	PART
ejpam-4974	74	12	be	be	AUX
ejpam-4974	74	13	θ(λ	θ(λ	PROPN
ejpam-4974	74	14	,	,	PUNCT
ejpam-4974	74	15	p)-open	p)-open	VERB
ejpam-4974	74	16	.	.	PUNCT
ejpam-4974	75	1	let	let	VERB
ejpam-4974	75	2	a	a	DET
ejpam-4974	75	3	be	be	AUX
ejpam-4974	75	4	a	a	DET
ejpam-4974	75	5	subset	subset	NOUN
ejpam-4974	75	6	of	of	ADP
ejpam-4974	75	7	a	a	DET
ejpam-4974	75	8	topological	topological	ADJ
ejpam-4974	75	9	space	space	NOUN
ejpam-4974	75	10	(	(	PUNCT
ejpam-4974	75	11	x	x	X
ejpam-4974	75	12	,	,	PUNCT
ejpam-4974	75	13	τ	τ	PROPN
ejpam-4974	75	14	)	)	PUNCT
ejpam-4974	75	15	.	.	PUNCT
ejpam-4974	76	1	a	a	DET
ejpam-4974	76	2	point	point	NOUN
ejpam-4974	76	3	x	x	X
ejpam-4974	76	4	∈	∈	NOUN
ejpam-4974	76	5	x	x	PUNCT
ejpam-4974	76	6	is	be	AUX
ejpam-4974	76	7	called	call	VERB
ejpam-4974	76	8	a	a	DET
ejpam-4974	76	9	θ(λ	θ(λ	PROPN
ejpam-4974	76	10	,	,	PUNCT
ejpam-4974	76	11	p)-interior	p)-interior	ADJ
ejpam-4974	76	12	point	point	NOUN
ejpam-4974	76	13	[	[	X
ejpam-4974	76	14	19	19	NUM
ejpam-4974	76	15	]	]	PUNCT
ejpam-4974	76	16	of	of	ADP
ejpam-4974	76	17	a	a	DET
ejpam-4974	76	18	if	if	NOUN
ejpam-4974	76	19	x	x	SYM
ejpam-4974	76	20	∈	∈	PROPN
ejpam-4974	76	21	u	u	NOUN
ejpam-4974	76	22	⊆	⊆	NUM
ejpam-4974	76	23	u	u	PROPN
ejpam-4974	76	24	(	(	PUNCT
ejpam-4974	76	25	λ	λ	PROPN
ejpam-4974	76	26	,	,	PUNCT
ejpam-4974	76	27	p	p	NOUN
ejpam-4974	76	28	)	)	PUNCT
ejpam-4974	76	29	⊆	⊆	NUM
ejpam-4974	76	30	a	a	PRON
ejpam-4974	76	31	for	for	ADP
ejpam-4974	76	32	some	some	DET
ejpam-4974	76	33	u	u	NOUN
ejpam-4974	76	34	∈	∈	PROPN
ejpam-4974	76	35	λpo(x	λpo(x	PROPN
ejpam-4974	76	36	,	,	PUNCT
ejpam-4974	76	37	τ	τ	PROPN
ejpam-4974	76	38	)	)	PUNCT
ejpam-4974	76	39	.	.	PUNCT
ejpam-4974	77	1	the	the	DET
ejpam-4974	77	2	set	set	NOUN
ejpam-4974	77	3	of	of	ADP
ejpam-4974	77	4	all	all	DET
ejpam-4974	77	5	θ(λ	θ(λ	PROPN
ejpam-4974	77	6	,	,	PUNCT
ejpam-4974	77	7	p)-interior	p)-interior	ADJ
ejpam-4974	77	8	points	point	NOUN
ejpam-4974	77	9	of	of	ADP
ejpam-4974	77	10	a	a	PRON
ejpam-4974	77	11	is	be	AUX
ejpam-4974	77	12	called	call	VERB
ejpam-4974	77	13	the	the	DET
ejpam-4974	77	14	θ(λ	θ(λ	PROPN
ejpam-4974	77	15	,	,	PUNCT
ejpam-4974	77	16	p)-interior	p)-interior	ADJ
ejpam-4974	77	17	[	[	X
ejpam-4974	77	18	19	19	NUM
ejpam-4974	77	19	]	]	PUNCT
ejpam-4974	77	20	of	of	ADP
ejpam-4974	77	21	a	a	PRON
ejpam-4974	77	22	and	and	CCONJ
ejpam-4974	77	23	is	be	AUX
ejpam-4974	77	24	denoted	denote	VERB
ejpam-4974	77	25	by	by	ADP
ejpam-4974	77	26	aθ(λ	aθ(λ	NOUN
ejpam-4974	77	27	,	,	PUNCT
ejpam-4974	77	28	p	p	NOUN
ejpam-4974	77	29	)	)	PUNCT
ejpam-4974	77	30	.	.	PUNCT
ejpam-4974	78	1	lemma	lemma	PROPN
ejpam-4974	78	2	1	1	NUM
ejpam-4974	78	3	.	.	PUNCT
ejpam-4974	79	1	[	[	X
ejpam-4974	79	2	19	19	NUM
ejpam-4974	79	3	]	]	PUNCT
ejpam-4974	79	4	for	for	ADP
ejpam-4974	79	5	subsets	subset	NOUN
ejpam-4974	79	6	a	a	PRON
ejpam-4974	79	7	and	and	CCONJ
ejpam-4974	79	8	b	b	NOUN
ejpam-4974	79	9	of	of	ADP
ejpam-4974	79	10	a	a	DET
ejpam-4974	79	11	topological	topological	ADJ
ejpam-4974	79	12	space	space	NOUN
ejpam-4974	79	13	(	(	PUNCT
ejpam-4974	79	14	x	x	X
ejpam-4974	79	15	,	,	PUNCT
ejpam-4974	79	16	τ	τ	PROPN
ejpam-4974	79	17	)	)	PUNCT
ejpam-4974	79	18	,	,	PUNCT
ejpam-4974	79	19	the	the	DET
ejpam-4974	79	20	following	follow	VERB
ejpam-4974	79	21	properties	property	NOUN
ejpam-4974	79	22	hold	hold	VERB
ejpam-4974	79	23	:	:	PUNCT
ejpam-4974	79	24	(	(	PUNCT
ejpam-4974	79	25	1	1	X
ejpam-4974	79	26	)	)	PUNCT
ejpam-4974	79	27	x	x	PUNCT
ejpam-4974	80	1	−aθ(λ	−aθ(λ	NOUN
ejpam-4974	80	2	,	,	PUNCT
ejpam-4974	80	3	p	p	NOUN
ejpam-4974	80	4	)	)	PUNCT
ejpam-4974	80	5	=	=	PUNCT
ejpam-4974	81	1	[	[	X
ejpam-4974	81	2	x	x	X
ejpam-4974	81	3	−a]θ(λ	−a]θ(λ	NOUN
ejpam-4974	81	4	,	,	PUNCT
ejpam-4974	81	5	p	p	NOUN
ejpam-4974	81	6	)	)	PUNCT
ejpam-4974	81	7	and	and	CCONJ
ejpam-4974	81	8	x	x	PUNCT
ejpam-4974	81	9	−aθ(λ	−aθ(λ	NOUN
ejpam-4974	81	10	,	,	PUNCT
ejpam-4974	81	11	p	p	NOUN
ejpam-4974	81	12	)	)	PUNCT
ejpam-4974	81	13	=	=	PUNCT
ejpam-4974	82	1	[	[	X
ejpam-4974	82	2	x	x	X
ejpam-4974	82	3	−a]θ(λ	−a]θ(λ	NOUN
ejpam-4974	82	4	,	,	PUNCT
ejpam-4974	82	5	p	p	NOUN
ejpam-4974	82	6	)	)	PUNCT
ejpam-4974	82	7	.	.	PUNCT
ejpam-4974	83	1	(	(	PUNCT
ejpam-4974	83	2	2	2	X
ejpam-4974	83	3	)	)	PUNCT
ejpam-4974	83	4	a	a	PRON
ejpam-4974	83	5	is	be	AUX
ejpam-4974	83	6	θ(λ	θ(λ	PROPN
ejpam-4974	83	7	,	,	PUNCT
ejpam-4974	83	8	p)-open	p)-open	VERB
ejpam-4974	83	9	if	if	SCONJ
ejpam-4974	83	10	and	and	CCONJ
ejpam-4974	83	11	only	only	ADV
ejpam-4974	83	12	if	if	SCONJ
ejpam-4974	83	13	a	a	PRON
ejpam-4974	83	14	=	=	NOUN
ejpam-4974	83	15	aθ(λ	aθ(λ	NOUN
ejpam-4974	83	16	,	,	PUNCT
ejpam-4974	83	17	p	p	NOUN
ejpam-4974	83	18	)	)	PUNCT
ejpam-4974	83	19	.	.	PUNCT
ejpam-4974	84	1	(	(	PUNCT
ejpam-4974	84	2	3	3	X
ejpam-4974	84	3	)	)	PUNCT
ejpam-4974	84	4	a	a	DET
ejpam-4974	84	5	⊆	⊆	NUM
ejpam-4974	84	6	a(λ	a(λ	ADJ
ejpam-4974	84	7	,	,	PUNCT
ejpam-4974	84	8	p	p	X
ejpam-4974	84	9	)	)	PUNCT
ejpam-4974	84	10	⊆	⊆	NUM
ejpam-4974	84	11	aθ(λ	aθ(λ	NOUN
ejpam-4974	84	12	,	,	PUNCT
ejpam-4974	84	13	p	p	NOUN
ejpam-4974	84	14	)	)	PUNCT
ejpam-4974	84	15	and	and	CCONJ
ejpam-4974	84	16	aθ(λ	aθ(λ	NOUN
ejpam-4974	84	17	,	,	PUNCT
ejpam-4974	84	18	p	p	NOUN
ejpam-4974	84	19	)	)	PUNCT
ejpam-4974	84	20	⊆	⊆	NUM
ejpam-4974	84	21	a(λ	a(λ	ADV
ejpam-4974	84	22	,	,	PUNCT
ejpam-4974	84	23	p	p	NOUN
ejpam-4974	84	24	)	)	PUNCT
ejpam-4974	84	25	⊆	⊆	NUM
ejpam-4974	84	26	a.	a.	NOUN
ejpam-4974	84	27	(	(	PUNCT
ejpam-4974	84	28	4	4	NUM
ejpam-4974	84	29	)	)	PUNCT
ejpam-4974	84	30	if	if	SCONJ
ejpam-4974	84	31	a	a	DET
ejpam-4974	84	32	⊆	⊆	NUM
ejpam-4974	84	33	b	b	NOUN
ejpam-4974	84	34	,	,	PUNCT
ejpam-4974	84	35	then	then	ADV
ejpam-4974	84	36	aθ(λ	aθ(λ	NOUN
ejpam-4974	84	37	,	,	PUNCT
ejpam-4974	84	38	p	p	NOUN
ejpam-4974	84	39	)	)	PUNCT
ejpam-4974	84	40	⊆	⊆	NUM
ejpam-4974	84	41	bθ(λ	bθ(λ	NOUN
ejpam-4974	84	42	,	,	PUNCT
ejpam-4974	84	43	p	p	NOUN
ejpam-4974	84	44	)	)	PUNCT
ejpam-4974	84	45	and	and	CCONJ
ejpam-4974	84	46	aθ(λ	aθ(λ	NOUN
ejpam-4974	84	47	,	,	PUNCT
ejpam-4974	84	48	p	p	NOUN
ejpam-4974	84	49	)	)	PUNCT
ejpam-4974	84	50	⊆	⊆	NUM
ejpam-4974	84	51	bθ(λ	bθ(λ	NOUN
ejpam-4974	84	52	,	,	PUNCT
ejpam-4974	84	53	p	p	NOUN
ejpam-4974	84	54	)	)	PUNCT
ejpam-4974	84	55	.	.	PUNCT
ejpam-4974	85	1	(	(	PUNCT
ejpam-4974	85	2	5	5	X
ejpam-4974	85	3	)	)	PUNCT
ejpam-4974	85	4	if	if	SCONJ
ejpam-4974	85	5	a	a	PRON
ejpam-4974	85	6	is	be	AUX
ejpam-4974	85	7	(	(	PUNCT
ejpam-4974	85	8	λ	λ	NOUN
ejpam-4974	85	9	,	,	PUNCT
ejpam-4974	85	10	p)-open	p)-open	ADJ
ejpam-4974	85	11	,	,	PUNCT
ejpam-4974	85	12	then	then	ADV
ejpam-4974	85	13	a(λ	a(λ	ADV
ejpam-4974	85	14	,	,	PUNCT
ejpam-4974	85	15	p	p	X
ejpam-4974	85	16	)	)	PUNCT
ejpam-4974	85	17	=	=	PUNCT
ejpam-4974	85	18	aθ(λ	aθ(λ	NOUN
ejpam-4974	85	19	,	,	PUNCT
ejpam-4974	85	20	p	p	NOUN
ejpam-4974	85	21	)	)	PUNCT
ejpam-4974	85	22	.	.	PUNCT
ejpam-4974	86	1	3	3	X
ejpam-4974	86	2	.	.	X
ejpam-4974	86	3	properties	property	NOUN
ejpam-4974	86	4	of	of	ADP
ejpam-4974	86	5	weakly	weakly	ADJ
ejpam-4974	86	6	β(λ	β(λ	NOUN
ejpam-4974	86	7	,	,	PUNCT
ejpam-4974	86	8	p)-open	p)-open	VERB
ejpam-4974	86	9	functions	function	NOUN
ejpam-4974	86	10	in	in	ADP
ejpam-4974	86	11	this	this	DET
ejpam-4974	86	12	section	section	NOUN
ejpam-4974	86	13	,	,	PUNCT
ejpam-4974	86	14	we	we	PRON
ejpam-4974	86	15	introduce	introduce	VERB
ejpam-4974	86	16	the	the	DET
ejpam-4974	86	17	notion	notion	NOUN
ejpam-4974	86	18	of	of	ADP
ejpam-4974	86	19	weakly	weakly	ADJ
ejpam-4974	86	20	β(λ	β(λ	NOUN
ejpam-4974	86	21	,	,	PUNCT
ejpam-4974	86	22	p)-open	p)-open	ADJ
ejpam-4974	86	23	functions	function	NOUN
ejpam-4974	86	24	.	.	PUNCT
ejpam-4974	87	1	moreover	moreover	ADV
ejpam-4974	87	2	,	,	PUNCT
ejpam-4974	87	3	some	some	DET
ejpam-4974	87	4	properties	property	NOUN
ejpam-4974	87	5	of	of	ADP
ejpam-4974	87	6	weakly	weakly	ADJ
ejpam-4974	87	7	β(λ	β(λ	NOUN
ejpam-4974	87	8	,	,	PUNCT
ejpam-4974	87	9	p)-open	p)-open	VERB
ejpam-4974	87	10	functions	function	NOUN
ejpam-4974	87	11	are	be	AUX
ejpam-4974	87	12	discussed	discuss	VERB
ejpam-4974	87	13	.	.	PUNCT
ejpam-4974	88	1	definition	definition	NOUN
ejpam-4974	88	2	1	1	NUM
ejpam-4974	88	3	.	.	PUNCT
ejpam-4974	89	1	a	a	DET
ejpam-4974	89	2	functions	function	NOUN
ejpam-4974	89	3	f	f	X
ejpam-4974	89	4	:	:	PUNCT
ejpam-4974	89	5	(	(	PUNCT
ejpam-4974	89	6	x	x	X
ejpam-4974	89	7	,	,	PUNCT
ejpam-4974	89	8	τ	τ	X
ejpam-4974	89	9	)	)	PUNCT
ejpam-4974	89	10	→	→	SYM
ejpam-4974	89	11	(	(	PUNCT
ejpam-4974	89	12	y	y	PROPN
ejpam-4974	89	13	,	,	PUNCT
ejpam-4974	89	14	σ	σ	PROPN
ejpam-4974	89	15	)	)	PUNCT
ejpam-4974	89	16	is	be	AUX
ejpam-4974	89	17	said	say	VERB
ejpam-4974	89	18	to	to	PART
ejpam-4974	89	19	be	be	AUX
ejpam-4974	89	20	weakly	weakly	ADJ
ejpam-4974	89	21	β(λ	β(λ	NOUN
ejpam-4974	89	22	,	,	PUNCT
ejpam-4974	89	23	p)-open	p)-open	VERB
ejpam-4974	89	24	if	if	SCONJ
ejpam-4974	89	25	f(u	f(u	PROPN
ejpam-4974	89	26	)	)	PUNCT
ejpam-4974	89	27	⊆	⊆	NUM
ejpam-4974	90	1	[	[	X
ejpam-4974	90	2	f(u	f(u	PROPN
ejpam-4974	90	3	(	(	PUNCT
ejpam-4974	90	4	λ	λ	PROPN
ejpam-4974	90	5	,	,	PUNCT
ejpam-4974	90	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	90	7	,	,	PUNCT
ejpam-4974	90	8	p	p	NOUN
ejpam-4974	90	9	)	)	PUNCT
ejpam-4974	90	10	for	for	ADP
ejpam-4974	90	11	each	each	DET
ejpam-4974	90	12	(	(	PUNCT
ejpam-4974	90	13	λ	λ	PROPN
ejpam-4974	90	14	,	,	PUNCT
ejpam-4974	90	15	p)-open	p)-open	VERB
ejpam-4974	90	16	set	set	VERB
ejpam-4974	90	17	u	u	NOUN
ejpam-4974	90	18	of	of	ADP
ejpam-4974	90	19	x.	x.	PROPN
ejpam-4974	90	20	theorem	theorem	VERB
ejpam-4974	90	21	1	1	NUM
ejpam-4974	90	22	.	.	PUNCT
ejpam-4974	90	23	for	for	ADP
ejpam-4974	90	24	a	a	DET
ejpam-4974	90	25	function	function	NOUN
ejpam-4974	90	26	f	f	NOUN
ejpam-4974	90	27	:	:	PUNCT
ejpam-4974	90	28	(	(	PUNCT
ejpam-4974	90	29	x	x	X
ejpam-4974	90	30	,	,	PUNCT
ejpam-4974	90	31	τ	τ	X
ejpam-4974	90	32	)	)	PUNCT
ejpam-4974	90	33	→	→	SYM
ejpam-4974	90	34	(	(	PUNCT
ejpam-4974	90	35	y	y	PROPN
ejpam-4974	90	36	,	,	PUNCT
ejpam-4974	90	37	σ	σ	PROPN
ejpam-4974	90	38	)	)	PUNCT
ejpam-4974	90	39	,	,	PUNCT
ejpam-4974	90	40	the	the	DET
ejpam-4974	90	41	following	follow	VERB
ejpam-4974	90	42	properties	property	NOUN
ejpam-4974	90	43	are	be	AUX
ejpam-4974	90	44	equivalent	equivalent	ADJ
ejpam-4974	90	45	:	:	PUNCT
ejpam-4974	90	46	(	(	PUNCT
ejpam-4974	90	47	1	1	X
ejpam-4974	90	48	)	)	PUNCT
ejpam-4974	90	49	f	f	PROPN
ejpam-4974	90	50	is	be	AUX
ejpam-4974	90	51	weakly	weakly	ADJ
ejpam-4974	90	52	β(λ	β(λ	NOUN
ejpam-4974	90	53	,	,	PUNCT
ejpam-4974	90	54	p)-open	p)-open	VERB
ejpam-4974	90	55	;	;	PUNCT
ejpam-4974	90	56	(	(	PUNCT
ejpam-4974	90	57	2	2	X
ejpam-4974	90	58	)	)	PUNCT
ejpam-4974	90	59	f(aθ(λ	f(aθ(λ	PROPN
ejpam-4974	90	60	,	,	PUNCT
ejpam-4974	90	61	p	p	NOUN
ejpam-4974	90	62	)	)	PUNCT
ejpam-4974	90	63	)	)	PUNCT
ejpam-4974	91	1	⊆	⊆	NUM
ejpam-4974	91	2	[	[	X
ejpam-4974	91	3	f(a)]β(λ	f(a)]β(λ	PROPN
ejpam-4974	91	4	,	,	PUNCT
ejpam-4974	91	5	p	p	NOUN
ejpam-4974	91	6	)	)	PUNCT
ejpam-4974	91	7	for	for	ADP
ejpam-4974	91	8	every	every	DET
ejpam-4974	91	9	subset	subset	NOUN
ejpam-4974	91	10	a	a	PRON
ejpam-4974	91	11	of	of	ADP
ejpam-4974	91	12	x	x	PRON
ejpam-4974	91	13	;	;	PUNCT
ejpam-4974	91	14	(	(	PUNCT
ejpam-4974	91	15	3	3	X
ejpam-4974	91	16	)	)	PUNCT
ejpam-4974	92	1	[	[	X
ejpam-4974	92	2	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-4974	92	3	,	,	PUNCT
ejpam-4974	92	4	p	p	NOUN
ejpam-4974	92	5	)	)	PUNCT
ejpam-4974	92	6	⊆	⊆	NUM
ejpam-4974	92	7	f−1(bβ(λ	f−1(bβ(λ	NOUN
ejpam-4974	92	8	,	,	PUNCT
ejpam-4974	92	9	p	p	NOUN
ejpam-4974	92	10	)	)	PUNCT
ejpam-4974	92	11	)	)	PUNCT
ejpam-4974	92	12	for	for	ADP
ejpam-4974	92	13	every	every	DET
ejpam-4974	92	14	subset	subset	NOUN
ejpam-4974	92	15	b	b	PROPN
ejpam-4974	92	16	of	of	ADP
ejpam-4974	92	17	y	y	PROPN
ejpam-4974	92	18	;	;	PUNCT
ejpam-4974	92	19	(	(	PUNCT
ejpam-4974	92	20	4	4	X
ejpam-4974	92	21	)	)	PUNCT
ejpam-4974	92	22	f−1(bβ(λ	f−1(bβ(λ	NOUN
ejpam-4974	92	23	,	,	PUNCT
ejpam-4974	92	24	p	p	NOUN
ejpam-4974	92	25	)	)	PUNCT
ejpam-4974	92	26	)	)	PUNCT
ejpam-4974	93	1	⊆	⊆	NUM
ejpam-4974	93	2	[	[	X
ejpam-4974	93	3	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-4974	93	4	,	,	PUNCT
ejpam-4974	93	5	p	p	NOUN
ejpam-4974	93	6	)	)	PUNCT
ejpam-4974	93	7	for	for	ADP
ejpam-4974	93	8	every	every	DET
ejpam-4974	93	9	subset	subset	NOUN
ejpam-4974	93	10	b	b	PROPN
ejpam-4974	93	11	of	of	ADP
ejpam-4974	93	12	y	y	PROPN
ejpam-4974	93	13	;	;	PUNCT
ejpam-4974	93	14	(	(	PUNCT
ejpam-4974	93	15	5	5	X
ejpam-4974	93	16	)	)	PUNCT
ejpam-4974	93	17	for	for	ADP
ejpam-4974	93	18	each	each	DET
ejpam-4974	93	19	x	x	SYM
ejpam-4974	93	20	∈	∈	PROPN
ejpam-4974	93	21	x	x	X
ejpam-4974	93	22	and	and	CCONJ
ejpam-4974	93	23	each	each	PRON
ejpam-4974	93	24	(	(	PUNCT
ejpam-4974	93	25	λ	λ	NOUN
ejpam-4974	93	26	,	,	PUNCT
ejpam-4974	93	27	p)-open	p)-open	VERB
ejpam-4974	93	28	set	set	VERB
ejpam-4974	93	29	u	u	NOUN
ejpam-4974	93	30	of	of	ADP
ejpam-4974	93	31	x	x	SYM
ejpam-4974	93	32	containing	contain	VERB
ejpam-4974	93	33	x	x	PRON
ejpam-4974	93	34	,	,	PUNCT
ejpam-4974	93	35	there	there	PRON
ejpam-4974	93	36	exists	exist	VERB
ejpam-4974	93	37	a	a	DET
ejpam-4974	93	38	β(λ	β(λ	NOUN
ejpam-4974	93	39	,	,	PUNCT
ejpam-4974	93	40	p)open	p)open	PROPN
ejpam-4974	93	41	set	set	VERB
ejpam-4974	93	42	v	v	NOUN
ejpam-4974	93	43	of	of	ADP
ejpam-4974	93	44	y	y	NOUN
ejpam-4974	93	45	containing	contain	VERB
ejpam-4974	93	46	f(x	f(x	PROPN
ejpam-4974	93	47	)	)	PUNCT
ejpam-4974	93	48	such	such	ADJ
ejpam-4974	93	49	that	that	PRON
ejpam-4974	93	50	v	v	ADP
ejpam-4974	93	51	⊆	⊆	NUM
ejpam-4974	93	52	f(u	f(u	PROPN
ejpam-4974	93	53	(	(	PUNCT
ejpam-4974	93	54	λ	λ	PROPN
ejpam-4974	93	55	,	,	PUNCT
ejpam-4974	93	56	p	p	NOUN
ejpam-4974	93	57	)	)	PUNCT
ejpam-4974	93	58	)	)	PUNCT
ejpam-4974	93	59	;	;	PUNCT
ejpam-4974	93	60	(	(	PUNCT
ejpam-4974	93	61	6	6	X
ejpam-4974	93	62	)	)	PUNCT
ejpam-4974	93	63	f(k(λ	f(k(λ	NOUN
ejpam-4974	93	64	,	,	PUNCT
ejpam-4974	93	65	p	p	NOUN
ejpam-4974	93	66	)	)	PUNCT
ejpam-4974	93	67	)	)	PUNCT
ejpam-4974	94	1	⊆	⊆	NUM
ejpam-4974	94	2	[	[	X
ejpam-4974	94	3	f(k)]β(λ	f(k)]β(λ	PROPN
ejpam-4974	94	4	,	,	PUNCT
ejpam-4974	94	5	p	p	NOUN
ejpam-4974	94	6	)	)	PUNCT
ejpam-4974	94	7	for	for	ADP
ejpam-4974	94	8	every	every	DET
ejpam-4974	94	9	(	(	PUNCT
ejpam-4974	94	10	λ	λ	PROPN
ejpam-4974	94	11	,	,	PUNCT
ejpam-4974	94	12	p)-closed	p)-close	VERB
ejpam-4974	94	13	set	set	NOUN
ejpam-4974	94	14	k	k	PROPN
ejpam-4974	94	15	of	of	ADP
ejpam-4974	94	16	x	x	PROPN
ejpam-4974	94	17	;	;	PUNCT
ejpam-4974	94	18	c.	c.	PROPN
ejpam-4974	94	19	boonpok	boonpok	PROPN
ejpam-4974	94	20	,	,	PUNCT
ejpam-4974	94	21	m.	m.	NOUN
ejpam-4974	94	22	thongmoon	thongmoon	PROPN
ejpam-4974	94	23	/	/	SYM
ejpam-4974	94	24	eur	eur	PROPN
ejpam-4974	94	25	.	.	PUNCT
ejpam-4974	95	1	j.	j.	PROPN
ejpam-4974	95	2	pure	pure	PROPN
ejpam-4974	95	3	appl	appl	PROPN
ejpam-4974	95	4	.	.	PROPN
ejpam-4974	95	5	math	math	PROPN
ejpam-4974	95	6	,	,	PUNCT
ejpam-4974	95	7	17	17	NUM
ejpam-4974	95	8	(	(	PUNCT
ejpam-4974	95	9	1	1	NUM
ejpam-4974	95	10	)	)	PUNCT
ejpam-4974	95	11	(	(	PUNCT
ejpam-4974	95	12	2024	2024	NUM
ejpam-4974	95	13	)	)	PUNCT
ejpam-4974	95	14	,	,	PUNCT
ejpam-4974	95	15	248	248	NUM
ejpam-4974	95	16	-	-	SYM
ejpam-4974	95	17	255	255	NUM
ejpam-4974	95	18	251	251	NUM
ejpam-4974	95	19	(	(	PUNCT
ejpam-4974	95	20	7	7	NUM
ejpam-4974	95	21	)	)	PUNCT
ejpam-4974	95	22	f([u	f([u	NOUN
ejpam-4974	95	23	(	(	PUNCT
ejpam-4974	95	24	λ	λ	NOUN
ejpam-4974	95	25	,	,	PUNCT
ejpam-4974	95	26	p)](λ	p)](λ	ADJ
ejpam-4974	95	27	,	,	PUNCT
ejpam-4974	95	28	p	p	NOUN
ejpam-4974	95	29	)	)	PUNCT
ejpam-4974	95	30	)	)	PUNCT
ejpam-4974	96	1	⊆	⊆	NUM
ejpam-4974	96	2	[	[	X
ejpam-4974	96	3	f(u	f(u	PROPN
ejpam-4974	96	4	(	(	PUNCT
ejpam-4974	96	5	λ	λ	PROPN
ejpam-4974	96	6	,	,	PUNCT
ejpam-4974	96	7	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	96	8	,	,	PUNCT
ejpam-4974	96	9	p	p	NOUN
ejpam-4974	96	10	)	)	PUNCT
ejpam-4974	96	11	for	for	ADP
ejpam-4974	96	12	every	every	DET
ejpam-4974	96	13	(	(	PUNCT
ejpam-4974	96	14	λ	λ	NOUN
ejpam-4974	96	15	,	,	PUNCT
ejpam-4974	96	16	p)-open	p)-open	VERB
ejpam-4974	96	17	set	set	VERB
ejpam-4974	96	18	u	u	NOUN
ejpam-4974	96	19	of	of	ADP
ejpam-4974	96	20	x	x	PRON
ejpam-4974	96	21	;	;	PUNCT
ejpam-4974	96	22	(	(	PUNCT
ejpam-4974	96	23	8)	8)	NUM
ejpam-4974	96	24	f(u	f(u	NOUN
ejpam-4974	96	25	)	)	PUNCT
ejpam-4974	96	26	⊆	⊆	NUM
ejpam-4974	97	1	[	[	X
ejpam-4974	97	2	f(u	f(u	PROPN
ejpam-4974	97	3	(	(	PUNCT
ejpam-4974	97	4	λ	λ	PROPN
ejpam-4974	97	5	,	,	PUNCT
ejpam-4974	97	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	97	7	,	,	PUNCT
ejpam-4974	97	8	p	p	NOUN
ejpam-4974	97	9	)	)	PUNCT
ejpam-4974	97	10	for	for	ADP
ejpam-4974	97	11	every	every	DET
ejpam-4974	97	12	p(λ	p(λ	NOUN
ejpam-4974	97	13	,	,	PUNCT
ejpam-4974	97	14	p)-open	p)-open	VERB
ejpam-4974	97	15	set	set	VERB
ejpam-4974	97	16	u	u	NOUN
ejpam-4974	97	17	of	of	ADP
ejpam-4974	97	18	x	x	PRON
ejpam-4974	97	19	;	;	PUNCT
ejpam-4974	97	20	(	(	PUNCT
ejpam-4974	97	21	9	9	X
ejpam-4974	97	22	)	)	PUNCT
ejpam-4974	97	23	f(u	f(u	PROPN
ejpam-4974	97	24	)	)	PUNCT
ejpam-4974	97	25	⊆	⊆	NUM
ejpam-4974	98	1	[	[	X
ejpam-4974	98	2	f(u	f(u	PROPN
ejpam-4974	98	3	(	(	PUNCT
ejpam-4974	98	4	λ	λ	PROPN
ejpam-4974	98	5	,	,	PUNCT
ejpam-4974	98	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	98	7	,	,	PUNCT
ejpam-4974	98	8	p	p	NOUN
ejpam-4974	98	9	)	)	PUNCT
ejpam-4974	98	10	for	for	ADP
ejpam-4974	98	11	every	every	DET
ejpam-4974	98	12	α(λ	α(λ	PROPN
ejpam-4974	98	13	,	,	PUNCT
ejpam-4974	98	14	p)-open	p)-open	VERB
ejpam-4974	98	15	set	set	VERB
ejpam-4974	98	16	u	u	NOUN
ejpam-4974	98	17	of	of	ADP
ejpam-4974	98	18	x.	x.	NOUN
ejpam-4974	98	19	proof	proof	NOUN
ejpam-4974	98	20	.	.	PUNCT
ejpam-4974	99	1	(	(	PUNCT
ejpam-4974	99	2	1	1	X
ejpam-4974	99	3	)	)	PUNCT
ejpam-4974	99	4	⇒	⇒	NOUN
ejpam-4974	99	5	(	(	PUNCT
ejpam-4974	99	6	2	2	NUM
ejpam-4974	99	7	):	):	PUNCT
ejpam-4974	99	8	let	let	VERB
ejpam-4974	99	9	a	a	PRON
ejpam-4974	99	10	be	be	AUX
ejpam-4974	99	11	any	any	DET
ejpam-4974	99	12	subset	subset	NOUN
ejpam-4974	99	13	ofx	ofx	NOUN
ejpam-4974	99	14	and	and	CCONJ
ejpam-4974	99	15	x	x	PUNCT
ejpam-4974	99	16	∈	∈	PROPN
ejpam-4974	99	17	aθ(λ	aθ(λ	NOUN
ejpam-4974	99	18	,	,	PUNCT
ejpam-4974	99	19	p	p	NOUN
ejpam-4974	99	20	)	)	PUNCT
ejpam-4974	99	21	.	.	PUNCT
ejpam-4974	100	1	then	then	ADV
ejpam-4974	100	2	,	,	PUNCT
ejpam-4974	100	3	there	there	PRON
ejpam-4974	100	4	exists	exist	VERB
ejpam-4974	100	5	a	a	DET
ejpam-4974	100	6	(	(	PUNCT
ejpam-4974	100	7	λ	λ	PROPN
ejpam-4974	100	8	,	,	PUNCT
ejpam-4974	100	9	p)open	p)open	PROPN
ejpam-4974	100	10	set	set	ADJ
ejpam-4974	100	11	u	u	NOUN
ejpam-4974	100	12	of	of	ADP
ejpam-4974	100	13	x	x	SYM
ejpam-4974	100	14	such	such	ADJ
ejpam-4974	100	15	that	that	SCONJ
ejpam-4974	100	16	x	x	SYM
ejpam-4974	100	17	∈	∈	NOUN
ejpam-4974	100	18	u	u	NOUN
ejpam-4974	100	19	⊆	⊆	NUM
ejpam-4974	100	20	u	u	PROPN
ejpam-4974	100	21	(	(	PUNCT
ejpam-4974	100	22	λ	λ	PROPN
ejpam-4974	100	23	,	,	PUNCT
ejpam-4974	100	24	p	p	NOUN
ejpam-4974	100	25	)	)	PUNCT
ejpam-4974	100	26	⊆	⊆	NUM
ejpam-4974	100	27	a.	a.	NOUN
ejpam-4974	100	28	then	then	ADV
ejpam-4974	100	29	,	,	PUNCT
ejpam-4974	100	30	f(x	f(x	PROPN
ejpam-4974	100	31	)	)	PUNCT
ejpam-4974	100	32	∈	∈	PROPN
ejpam-4974	100	33	f(u	f(u	PROPN
ejpam-4974	100	34	)	)	PUNCT
ejpam-4974	100	35	⊆	⊆	NUM
ejpam-4974	100	36	f(u	f(u	PROPN
ejpam-4974	100	37	(	(	PUNCT
ejpam-4974	100	38	λ	λ	PROPN
ejpam-4974	100	39	,	,	PUNCT
ejpam-4974	100	40	p	p	NOUN
ejpam-4974	100	41	)	)	PUNCT
ejpam-4974	100	42	)	)	PUNCT
ejpam-4974	100	43	⊆	⊆	NUM
ejpam-4974	100	44	f(a	f(a	NOUN
ejpam-4974	100	45	)	)	PUNCT
ejpam-4974	100	46	.	.	PUNCT
ejpam-4974	101	1	since	since	SCONJ
ejpam-4974	101	2	f	f	PROPN
ejpam-4974	101	3	is	be	AUX
ejpam-4974	101	4	weakly	weakly	ADJ
ejpam-4974	101	5	β(λ	β(λ	NOUN
ejpam-4974	101	6	,	,	PUNCT
ejpam-4974	101	7	p)-open	p)-open	ADJ
ejpam-4974	101	8	,	,	PUNCT
ejpam-4974	101	9	f(u	f(u	PROPN
ejpam-4974	101	10	)	)	PUNCT
ejpam-4974	102	1	⊆	⊆	NUM
ejpam-4974	103	1	[	[	X
ejpam-4974	103	2	f(u	f(u	PROPN
ejpam-4974	103	3	(	(	PUNCT
ejpam-4974	103	4	λ	λ	PROPN
ejpam-4974	103	5	,	,	PUNCT
ejpam-4974	103	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	103	7	,	,	PUNCT
ejpam-4974	103	8	p	p	NOUN
ejpam-4974	103	9	)	)	PUNCT
ejpam-4974	103	10	⊆	⊆	NUM
ejpam-4974	104	1	[	[	X
ejpam-4974	104	2	f(a)]β(λ	f(a)]β(λ	PROPN
ejpam-4974	104	3	,	,	PUNCT
ejpam-4974	104	4	p	p	NOUN
ejpam-4974	104	5	)	)	PUNCT
ejpam-4974	104	6	.	.	PUNCT
ejpam-4974	105	1	it	it	PRON
ejpam-4974	105	2	implies	imply	VERB
ejpam-4974	105	3	that	that	SCONJ
ejpam-4974	105	4	f(x	f(x	PROPN
ejpam-4974	105	5	)	)	PUNCT
ejpam-4974	105	6	∈	∈	PROPN
ejpam-4974	106	1	[	[	X
ejpam-4974	106	2	f(a)]β(λ	f(a)]β(λ	PROPN
ejpam-4974	106	3	,	,	PUNCT
ejpam-4974	106	4	p	p	NOUN
ejpam-4974	106	5	)	)	PUNCT
ejpam-4974	106	6	.	.	PUNCT
ejpam-4974	107	1	thus	thus	ADV
ejpam-4974	107	2	,	,	PUNCT
ejpam-4974	107	3	x	x	SYM
ejpam-4974	107	4	∈	∈	PROPN
ejpam-4974	107	5	f−1([f(a)]β(λ	f−1([f(a)]β(λ	PROPN
ejpam-4974	107	6	,	,	PUNCT
ejpam-4974	107	7	p	p	NOUN
ejpam-4974	107	8	)	)	PUNCT
ejpam-4974	107	9	)	)	PUNCT
ejpam-4974	107	10	and	and	CCONJ
ejpam-4974	107	11	hence	hence	ADV
ejpam-4974	107	12	aθ(λ	aθ(λ	NOUN
ejpam-4974	107	13	,	,	PUNCT
ejpam-4974	107	14	p	p	NOUN
ejpam-4974	107	15	)	)	PUNCT
ejpam-4974	107	16	⊆	⊆	NUM
ejpam-4974	107	17	f−1([f(a)]β(λ	f−1([f(a)]β(λ	NOUN
ejpam-4974	107	18	,	,	PUNCT
ejpam-4974	107	19	p	p	NOUN
ejpam-4974	107	20	)	)	PUNCT
ejpam-4974	107	21	)	)	PUNCT
ejpam-4974	107	22	.	.	PUNCT
ejpam-4974	108	1	this	this	PRON
ejpam-4974	108	2	shows	show	VERB
ejpam-4974	108	3	that	that	SCONJ
ejpam-4974	108	4	f(aθ(λ	f(aθ(λ	PROPN
ejpam-4974	108	5	,	,	PUNCT
ejpam-4974	108	6	p	p	NOUN
ejpam-4974	108	7	)	)	PUNCT
ejpam-4974	108	8	)	)	PUNCT
ejpam-4974	108	9	⊆	⊆	NUM
ejpam-4974	109	1	[	[	X
ejpam-4974	109	2	f(a)]β(λ	f(a)]β(λ	PROPN
ejpam-4974	109	3	,	,	PUNCT
ejpam-4974	109	4	p	p	NOUN
ejpam-4974	109	5	)	)	PUNCT
ejpam-4974	109	6	.	.	PUNCT
ejpam-4974	110	1	(	(	PUNCT
ejpam-4974	110	2	2	2	X
ejpam-4974	110	3	)	)	PUNCT
ejpam-4974	110	4	⇒	⇒	NOUN
ejpam-4974	110	5	(	(	PUNCT
ejpam-4974	110	6	1	1	NUM
ejpam-4974	110	7	):	):	PUNCT
ejpam-4974	110	8	let	let	VERB
ejpam-4974	110	9	u	u	PRON
ejpam-4974	110	10	be	be	AUX
ejpam-4974	110	11	any	any	DET
ejpam-4974	110	12	(	(	PUNCT
ejpam-4974	110	13	λ	λ	NOUN
ejpam-4974	110	14	,	,	PUNCT
ejpam-4974	110	15	p)-open	p)-open	VERB
ejpam-4974	110	16	set	set	VERB
ejpam-4974	110	17	of	of	ADP
ejpam-4974	110	18	x.	x.	NOUN
ejpam-4974	110	19	as	as	ADP
ejpam-4974	110	20	u	u	NOUN
ejpam-4974	110	21	⊆	⊆	NUM
ejpam-4974	110	22	[	[	X
ejpam-4974	110	23	u	u	X
ejpam-4974	110	24	(	(	PUNCT
ejpam-4974	110	25	λ	λ	PROPN
ejpam-4974	110	26	,	,	PUNCT
ejpam-4974	110	27	p)]θ(λ	p)]θ(λ	NOUN
ejpam-4974	110	28	,	,	PUNCT
ejpam-4974	110	29	p	p	NOUN
ejpam-4974	110	30	)	)	PUNCT
ejpam-4974	110	31	implies	imply	VERB
ejpam-4974	110	32	f(u	f(u	PROPN
ejpam-4974	110	33	)	)	PUNCT
ejpam-4974	110	34	⊆	⊆	NUM
ejpam-4974	110	35	f([u	f([u	NUM
ejpam-4974	110	36	(	(	PUNCT
ejpam-4974	110	37	λ	λ	PROPN
ejpam-4974	110	38	,	,	PUNCT
ejpam-4974	110	39	p)]θ(λ	p)]θ(λ	NOUN
ejpam-4974	110	40	,	,	PUNCT
ejpam-4974	110	41	p	p	NOUN
ejpam-4974	110	42	)	)	PUNCT
ejpam-4974	110	43	)	)	PUNCT
ejpam-4974	110	44	⊆	⊆	NUM
ejpam-4974	111	1	[	[	X
ejpam-4974	111	2	f(u	f(u	PROPN
ejpam-4974	111	3	(	(	PUNCT
ejpam-4974	111	4	λ	λ	PROPN
ejpam-4974	111	5	,	,	PUNCT
ejpam-4974	111	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	111	7	,	,	PUNCT
ejpam-4974	111	8	p	p	NOUN
ejpam-4974	111	9	)	)	PUNCT
ejpam-4974	111	10	.	.	PUNCT
ejpam-4974	112	1	thus	thus	ADV
ejpam-4974	112	2	,	,	PUNCT
ejpam-4974	112	3	f	f	PROPN
ejpam-4974	112	4	is	be	AUX
ejpam-4974	112	5	weakly	weakly	ADJ
ejpam-4974	112	6	β(λ	β(λ	NOUN
ejpam-4974	112	7	,	,	PUNCT
ejpam-4974	112	8	p)-open	p)-open	ADJ
ejpam-4974	112	9	.	.	PUNCT
ejpam-4974	113	1	(	(	PUNCT
ejpam-4974	113	2	2	2	X
ejpam-4974	113	3	)	)	PUNCT
ejpam-4974	113	4	⇒	⇒	NOUN
ejpam-4974	113	5	(	(	PUNCT
ejpam-4974	113	6	3	3	NUM
ejpam-4974	113	7	):	):	PUNCT
ejpam-4974	113	8	let	let	VERB
ejpam-4974	113	9	b	b	X
ejpam-4974	113	10	be	be	AUX
ejpam-4974	113	11	any	any	DET
ejpam-4974	113	12	subset	subset	NOUN
ejpam-4974	113	13	of	of	ADP
ejpam-4974	113	14	y	y	PROPN
ejpam-4974	113	15	.	.	PUNCT
ejpam-4974	114	1	then	then	ADV
ejpam-4974	114	2	by	by	ADP
ejpam-4974	114	3	(	(	PUNCT
ejpam-4974	114	4	2	2	NUM
ejpam-4974	114	5	)	)	PUNCT
ejpam-4974	114	6	,	,	PUNCT
ejpam-4974	114	7	f([f−1(b)]θ(λ	f([f−1(b)]θ(λ	PROPN
ejpam-4974	114	8	,	,	PUNCT
ejpam-4974	114	9	p	p	NOUN
ejpam-4974	114	10	)	)	PUNCT
ejpam-4974	114	11	)	)	PUNCT
ejpam-4974	114	12	⊆	⊆	NUM
ejpam-4974	114	13	bβ(λ	bβ(λ	NUM
ejpam-4974	114	14	,	,	PUNCT
ejpam-4974	114	15	p	p	NOUN
ejpam-4974	114	16	)	)	PUNCT
ejpam-4974	114	17	.	.	PUNCT
ejpam-4974	115	1	thus	thus	ADV
ejpam-4974	115	2	,	,	PUNCT
ejpam-4974	115	3	[	[	X
ejpam-4974	115	4	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-4974	115	5	,	,	PUNCT
ejpam-4974	115	6	p	p	NOUN
ejpam-4974	115	7	)	)	PUNCT
ejpam-4974	115	8	⊆	⊆	NUM
ejpam-4974	115	9	f−1(bβ(λ	f−1(bβ(λ	NOUN
ejpam-4974	115	10	,	,	PUNCT
ejpam-4974	115	11	p	p	NOUN
ejpam-4974	115	12	)	)	PUNCT
ejpam-4974	115	13	)	)	PUNCT
ejpam-4974	115	14	.	.	PUNCT
ejpam-4974	116	1	(	(	PUNCT
ejpam-4974	116	2	3	3	X
ejpam-4974	116	3	)	)	PUNCT
ejpam-4974	116	4	⇒	⇒	NOUN
ejpam-4974	116	5	(	(	PUNCT
ejpam-4974	116	6	2	2	NUM
ejpam-4974	116	7	):	):	PUNCT
ejpam-4974	116	8	this	this	PRON
ejpam-4974	116	9	is	be	AUX
ejpam-4974	116	10	obvious	obvious	ADJ
ejpam-4974	116	11	.	.	PUNCT
ejpam-4974	117	1	(	(	PUNCT
ejpam-4974	117	2	3	3	X
ejpam-4974	117	3	)	)	PUNCT
ejpam-4974	117	4	⇒	⇒	NOUN
ejpam-4974	117	5	(	(	PUNCT
ejpam-4974	117	6	4	4	NUM
ejpam-4974	117	7	):	):	PUNCT
ejpam-4974	117	8	let	let	VERB
ejpam-4974	117	9	b	b	X
ejpam-4974	117	10	be	be	AUX
ejpam-4974	117	11	any	any	DET
ejpam-4974	117	12	subset	subset	NOUN
ejpam-4974	117	13	of	of	ADP
ejpam-4974	117	14	y	y	PROPN
ejpam-4974	117	15	.	.	PUNCT
ejpam-4974	118	1	using	use	VERB
ejpam-4974	118	2	(	(	PUNCT
ejpam-4974	118	3	3	3	NUM
ejpam-4974	118	4	)	)	PUNCT
ejpam-4974	118	5	,	,	PUNCT
ejpam-4974	118	6	we	we	PRON
ejpam-4974	118	7	have	have	VERB
ejpam-4974	118	8	x	x	X
ejpam-4974	118	9	−	−	PROPN
ejpam-4974	119	1	[	[	X
ejpam-4974	119	2	f−1(b)]θ(λ	f−1(b)]θ(λ	PROPN
ejpam-4974	119	3	,	,	PUNCT
ejpam-4974	119	4	p	p	NOUN
ejpam-4974	119	5	)	)	PUNCT
ejpam-4974	119	6	=	=	PUNCT
ejpam-4974	120	1	[	[	X
ejpam-4974	120	2	x	x	X
ejpam-4974	120	3	−	−	ADP
ejpam-4974	120	4	f−1(b)]θ(λ	f−1(b)]θ(λ	PROPN
ejpam-4974	120	5	,	,	PUNCT
ejpam-4974	120	6	p	p	NOUN
ejpam-4974	120	7	)	)	PUNCT
ejpam-4974	120	8	=	=	PUNCT
ejpam-4974	121	1	[	[	X
ejpam-4974	121	2	f−1(y	f−1(y	PROPN
ejpam-4974	121	3	−b)]θ(λ	−b)]θ(λ	PROPN
ejpam-4974	121	4	,	,	PUNCT
ejpam-4974	121	5	p	p	NOUN
ejpam-4974	121	6	)	)	PUNCT
ejpam-4974	121	7	⊆	⊆	NUM
ejpam-4974	121	8	f−1([y	f−1([y	NOUN
ejpam-4974	121	9	−b]β(λ	−b]β(λ	NOUN
ejpam-4974	121	10	,	,	PUNCT
ejpam-4974	121	11	p	p	NOUN
ejpam-4974	121	12	)	)	PUNCT
ejpam-4974	121	13	)	)	PUNCT
ejpam-4974	122	1	=	=	SYM
ejpam-4974	122	2	f−1(y	f−1(y	PROPN
ejpam-4974	123	1	−bβ(λ	−bβ(λ	NOUN
ejpam-4974	123	2	,	,	PUNCT
ejpam-4974	123	3	p	p	NOUN
ejpam-4974	123	4	)	)	PUNCT
ejpam-4974	123	5	)	)	PUNCT
ejpam-4974	124	1	=	=	PUNCT
ejpam-4974	124	2	x	x	X
ejpam-4974	125	1	−	−	NOUN
ejpam-4974	125	2	f−1(bβ(λ	f−1(bβ(λ	ADJ
ejpam-4974	125	3	,	,	PUNCT
ejpam-4974	125	4	p	p	NOUN
ejpam-4974	125	5	)	)	PUNCT
ejpam-4974	125	6	)	)	PUNCT
ejpam-4974	125	7	and	and	CCONJ
ejpam-4974	125	8	hence	hence	ADV
ejpam-4974	125	9	f−1(bβ(λ	f−1(bβ(λ	ADV
ejpam-4974	125	10	,	,	PUNCT
ejpam-4974	125	11	p	p	NOUN
ejpam-4974	125	12	)	)	PUNCT
ejpam-4974	125	13	)	)	PUNCT
ejpam-4974	125	14	⊆	⊆	NUM
ejpam-4974	126	1	[	[	X
ejpam-4974	126	2	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-4974	126	3	,	,	PUNCT
ejpam-4974	126	4	p	p	NOUN
ejpam-4974	126	5	)	)	PUNCT
ejpam-4974	126	6	.	.	PUNCT
ejpam-4974	127	1	(	(	PUNCT
ejpam-4974	127	2	4	4	X
ejpam-4974	127	3	)	)	PUNCT
ejpam-4974	127	4	⇒	⇒	NOUN
ejpam-4974	127	5	(	(	PUNCT
ejpam-4974	127	6	3	3	NUM
ejpam-4974	127	7	):	):	PUNCT
ejpam-4974	127	8	let	let	VERB
ejpam-4974	127	9	b	b	X
ejpam-4974	127	10	be	be	AUX
ejpam-4974	127	11	any	any	DET
ejpam-4974	127	12	subset	subset	NOUN
ejpam-4974	127	13	of	of	ADP
ejpam-4974	127	14	y	y	PROPN
ejpam-4974	127	15	.	.	PUNCT
ejpam-4974	128	1	by	by	ADP
ejpam-4974	128	2	(	(	PUNCT
ejpam-4974	128	3	4	4	NUM
ejpam-4974	128	4	)	)	PUNCT
ejpam-4974	128	5	,	,	PUNCT
ejpam-4974	128	6	x	x	PUNCT
ejpam-4974	128	7	−	−	NOUN
ejpam-4974	128	8	f−1(bβ(λ	f−1(bβ(λ	ADJ
ejpam-4974	128	9	,	,	PUNCT
ejpam-4974	128	10	p	p	NOUN
ejpam-4974	128	11	)	)	PUNCT
ejpam-4974	128	12	)	)	PUNCT
ejpam-4974	129	1	⊆	⊆	NUM
ejpam-4974	129	2	x	x	SYM
ejpam-4974	129	3	−	−	PROPN
ejpam-4974	130	1	[	[	X
ejpam-4974	130	2	f−1(b)]θ(λ	f−1(b)]θ(λ	PROPN
ejpam-4974	130	3	,	,	PUNCT
ejpam-4974	130	4	p	p	NOUN
ejpam-4974	130	5	)	)	PUNCT
ejpam-4974	130	6	.	.	PUNCT
ejpam-4974	131	1	thus	thus	ADV
ejpam-4974	131	2	,	,	PUNCT
ejpam-4974	131	3	[	[	X
ejpam-4974	131	4	f−1(b)]θ(λ	f−1(b)]θ(λ	NOUN
ejpam-4974	131	5	,	,	PUNCT
ejpam-4974	131	6	p	p	NOUN
ejpam-4974	131	7	)	)	PUNCT
ejpam-4974	131	8	⊆	⊆	NUM
ejpam-4974	131	9	f−1(bβ(λ	f−1(bβ(λ	NOUN
ejpam-4974	131	10	,	,	PUNCT
ejpam-4974	131	11	p	p	NOUN
ejpam-4974	131	12	)	)	PUNCT
ejpam-4974	131	13	)	)	PUNCT
ejpam-4974	131	14	.	.	PUNCT
ejpam-4974	132	1	(	(	PUNCT
ejpam-4974	132	2	1	1	X
ejpam-4974	132	3	)	)	PUNCT
ejpam-4974	132	4	⇒	⇒	NOUN
ejpam-4974	132	5	(	(	PUNCT
ejpam-4974	132	6	5	5	NUM
ejpam-4974	132	7	):	):	PUNCT
ejpam-4974	132	8	let	let	VERB
ejpam-4974	132	9	x	x	PUNCT
ejpam-4974	132	10	∈	∈	PROPN
ejpam-4974	132	11	x	x	X
ejpam-4974	132	12	and	and	CCONJ
ejpam-4974	132	13	u	u	PRON
ejpam-4974	132	14	be	be	VERB
ejpam-4974	132	15	any	any	DET
ejpam-4974	132	16	(	(	PUNCT
ejpam-4974	132	17	λ	λ	NOUN
ejpam-4974	132	18	,	,	PUNCT
ejpam-4974	132	19	p)-open	p)-open	VERB
ejpam-4974	132	20	set	set	VERB
ejpam-4974	132	21	of	of	ADP
ejpam-4974	132	22	x	x	PUNCT
ejpam-4974	132	23	containing	contain	VERB
ejpam-4974	132	24	x.	x.	NOUN
ejpam-4974	132	25	since	since	SCONJ
ejpam-4974	132	26	f	f	PROPN
ejpam-4974	132	27	is	be	AUX
ejpam-4974	132	28	weakly	weakly	ADJ
ejpam-4974	132	29	β(λ	β(λ	NOUN
ejpam-4974	132	30	,	,	PUNCT
ejpam-4974	132	31	p)-open	p)-open	ADJ
ejpam-4974	132	32	,	,	PUNCT
ejpam-4974	132	33	f(x	f(x	PROPN
ejpam-4974	132	34	)	)	PUNCT
ejpam-4974	132	35	∈	∈	PROPN
ejpam-4974	132	36	f(u	f(u	PROPN
ejpam-4974	132	37	)	)	PUNCT
ejpam-4974	132	38	⊆	⊆	NUM
ejpam-4974	133	1	[	[	X
ejpam-4974	133	2	f(u	f(u	PROPN
ejpam-4974	133	3	(	(	PUNCT
ejpam-4974	133	4	λ	λ	PROPN
ejpam-4974	133	5	,	,	PUNCT
ejpam-4974	133	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	133	7	,	,	PUNCT
ejpam-4974	133	8	p	p	NOUN
ejpam-4974	133	9	)	)	PUNCT
ejpam-4974	133	10	.	.	PUNCT
ejpam-4974	134	1	put	put	VERB
ejpam-4974	134	2	v	v	NOUN
ejpam-4974	134	3	=	=	SYM
ejpam-4974	135	1	[	[	X
ejpam-4974	135	2	f(u	f(u	PROPN
ejpam-4974	135	3	(	(	PUNCT
ejpam-4974	135	4	λ	λ	PROPN
ejpam-4974	135	5	,	,	PUNCT
ejpam-4974	135	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	135	7	,	,	PUNCT
ejpam-4974	135	8	p	p	NOUN
ejpam-4974	135	9	)	)	PUNCT
ejpam-4974	135	10	.	.	PUNCT
ejpam-4974	136	1	then	then	ADV
ejpam-4974	136	2	,	,	PUNCT
ejpam-4974	136	3	v	v	NOUN
ejpam-4974	136	4	is	be	AUX
ejpam-4974	136	5	a	a	DET
ejpam-4974	136	6	β(λ	β(λ	NOUN
ejpam-4974	136	7	,	,	PUNCT
ejpam-4974	136	8	p)-open	p)-open	VERB
ejpam-4974	136	9	set	set	VERB
ejpam-4974	136	10	of	of	ADP
ejpam-4974	136	11	y	y	PROPN
ejpam-4974	136	12	containing	contain	VERB
ejpam-4974	136	13	f(x	f(x	PROPN
ejpam-4974	136	14	)	)	PUNCT
ejpam-4974	136	15	such	such	ADJ
ejpam-4974	136	16	that	that	PRON
ejpam-4974	136	17	v	v	ADP
ejpam-4974	136	18	⊆	⊆	NUM
ejpam-4974	136	19	f(u	f(u	PROPN
ejpam-4974	136	20	(	(	PUNCT
ejpam-4974	136	21	λ	λ	PROPN
ejpam-4974	136	22	,	,	PUNCT
ejpam-4974	136	23	p	p	NOUN
ejpam-4974	136	24	)	)	PUNCT
ejpam-4974	136	25	)	)	PUNCT
ejpam-4974	136	26	.	.	PUNCT
ejpam-4974	137	1	(	(	PUNCT
ejpam-4974	137	2	5	5	X
ejpam-4974	137	3	)	)	PUNCT
ejpam-4974	137	4	⇒	⇒	NOUN
ejpam-4974	137	5	(	(	PUNCT
ejpam-4974	137	6	1	1	NUM
ejpam-4974	137	7	):	):	PUNCT
ejpam-4974	137	8	let	let	VERB
ejpam-4974	137	9	u	u	PRON
ejpam-4974	137	10	be	be	AUX
ejpam-4974	137	11	any	any	DET
ejpam-4974	137	12	(	(	PUNCT
ejpam-4974	137	13	λ	λ	NOUN
ejpam-4974	137	14	,	,	PUNCT
ejpam-4974	137	15	p)-open	p)-open	VERB
ejpam-4974	137	16	set	set	VERB
ejpam-4974	137	17	of	of	ADP
ejpam-4974	137	18	x	x	PUNCT
ejpam-4974	137	19	and	and	CCONJ
ejpam-4974	137	20	y	y	PROPN
ejpam-4974	137	21	∈	∈	PROPN
ejpam-4974	137	22	f(u	f(u	PROPN
ejpam-4974	137	23	)	)	PUNCT
ejpam-4974	137	24	.	.	PUNCT
ejpam-4974	138	1	it	it	PRON
ejpam-4974	138	2	following	follow	VERB
ejpam-4974	138	3	from	from	ADP
ejpam-4974	138	4	(	(	PUNCT
ejpam-4974	138	5	5	5	NUM
ejpam-4974	138	6	)	)	PUNCT
ejpam-4974	138	7	v	v	ADP
ejpam-4974	138	8	⊆	⊆	NUM
ejpam-4974	138	9	f(u	f(u	PROPN
ejpam-4974	138	10	(	(	PUNCT
ejpam-4974	138	11	λ	λ	PROPN
ejpam-4974	138	12	,	,	PUNCT
ejpam-4974	138	13	p	p	NOUN
ejpam-4974	138	14	)	)	PUNCT
ejpam-4974	138	15	)	)	PUNCT
ejpam-4974	138	16	for	for	ADP
ejpam-4974	138	17	some	some	DET
ejpam-4974	138	18	β(λ	β(λ	NOUN
ejpam-4974	138	19	,	,	PUNCT
ejpam-4974	138	20	p)-open	p)-open	VERB
ejpam-4974	138	21	set	set	VERB
ejpam-4974	138	22	v	v	NOUN
ejpam-4974	138	23	of	of	ADP
ejpam-4974	138	24	y	y	PROPN
ejpam-4974	138	25	containing	contain	VERB
ejpam-4974	138	26	y.	y.	PROPN
ejpam-4974	138	27	thus	thus	ADV
ejpam-4974	138	28	,	,	PUNCT
ejpam-4974	138	29	y	y	PROPN
ejpam-4974	138	30	∈	∈	PROPN
ejpam-4974	138	31	v	v	ADP
ejpam-4974	138	32	⊆	⊆	NUM
ejpam-4974	138	33	[	[	X
ejpam-4974	138	34	f(u	f(u	PROPN
ejpam-4974	138	35	(	(	PUNCT
ejpam-4974	138	36	λ	λ	PROPN
ejpam-4974	138	37	,	,	PUNCT
ejpam-4974	138	38	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	138	39	,	,	PUNCT
ejpam-4974	138	40	p	p	NOUN
ejpam-4974	138	41	)	)	PUNCT
ejpam-4974	138	42	and	and	CCONJ
ejpam-4974	138	43	hence	hence	ADV
ejpam-4974	138	44	f(u	f(u	PROPN
ejpam-4974	138	45	)	)	PUNCT
ejpam-4974	138	46	⊆	⊆	NUM
ejpam-4974	139	1	[	[	X
ejpam-4974	139	2	f(u	f(u	PROPN
ejpam-4974	139	3	(	(	PUNCT
ejpam-4974	139	4	λ	λ	PROPN
ejpam-4974	139	5	,	,	PUNCT
ejpam-4974	139	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	139	7	,	,	PUNCT
ejpam-4974	139	8	p	p	NOUN
ejpam-4974	139	9	)	)	PUNCT
ejpam-4974	139	10	.	.	PUNCT
ejpam-4974	140	1	this	this	PRON
ejpam-4974	140	2	shows	show	VERB
ejpam-4974	140	3	that	that	SCONJ
ejpam-4974	140	4	f	f	PROPN
ejpam-4974	140	5	is	be	AUX
ejpam-4974	140	6	weakly	weakly	ADJ
ejpam-4974	140	7	β(λ	β(λ	NOUN
ejpam-4974	140	8	,	,	PUNCT
ejpam-4974	140	9	p)-open	p)-open	ADJ
ejpam-4974	140	10	.	.	PUNCT
ejpam-4974	141	1	(	(	PUNCT
ejpam-4974	141	2	1	1	X
ejpam-4974	141	3	)	)	PUNCT
ejpam-4974	141	4	⇒	⇒	NOUN
ejpam-4974	141	5	(	(	PUNCT
ejpam-4974	141	6	6	6	NUM
ejpam-4974	141	7	)	)	PUNCT
ejpam-4974	141	8	⇒	⇒	NOUN
ejpam-4974	141	9	(	(	PUNCT
ejpam-4974	141	10	7	7	NUM
ejpam-4974	141	11	)	)	PUNCT
ejpam-4974	141	12	⇒	⇒	NOUN
ejpam-4974	141	13	(	(	PUNCT
ejpam-4974	141	14	8)	8)	NUM
ejpam-4974	141	15	⇒	⇒	NOUN
ejpam-4974	141	16	(	(	PUNCT
ejpam-4974	141	17	9	9	NUM
ejpam-4974	141	18	)	)	PUNCT
ejpam-4974	141	19	⇒	⇒	NOUN
ejpam-4974	141	20	(	(	PUNCT
ejpam-4974	141	21	1	1	NUM
ejpam-4974	141	22	):	):	PUNCT
ejpam-4974	141	23	this	this	PRON
ejpam-4974	141	24	is	be	AUX
ejpam-4974	141	25	obvious	obvious	ADJ
ejpam-4974	141	26	.	.	PUNCT
ejpam-4974	142	1	c.	c.	PROPN
ejpam-4974	142	2	boonpok	boonpok	PROPN
ejpam-4974	142	3	,	,	PUNCT
ejpam-4974	142	4	m.	m.	NOUN
ejpam-4974	142	5	thongmoon	thongmoon	PROPN
ejpam-4974	142	6	/	/	SYM
ejpam-4974	142	7	eur	eur	PROPN
ejpam-4974	142	8	.	.	PUNCT
ejpam-4974	143	1	j.	j.	PROPN
ejpam-4974	143	2	pure	pure	PROPN
ejpam-4974	143	3	appl	appl	PROPN
ejpam-4974	143	4	.	.	PROPN
ejpam-4974	143	5	math	math	PROPN
ejpam-4974	143	6	,	,	PUNCT
ejpam-4974	143	7	17	17	NUM
ejpam-4974	143	8	(	(	PUNCT
ejpam-4974	143	9	1	1	NUM
ejpam-4974	143	10	)	)	PUNCT
ejpam-4974	143	11	(	(	PUNCT
ejpam-4974	143	12	2024	2024	NUM
ejpam-4974	143	13	)	)	PUNCT
ejpam-4974	143	14	,	,	PUNCT
ejpam-4974	143	15	248	248	NUM
ejpam-4974	143	16	-	-	SYM
ejpam-4974	143	17	255	255	NUM
ejpam-4974	143	18	252	252	NUM
ejpam-4974	143	19	theorem	theorem	NOUN
ejpam-4974	143	20	2	2	NUM
ejpam-4974	143	21	.	.	PUNCT
ejpam-4974	144	1	let	let	VERB
ejpam-4974	144	2	f	f	NOUN
ejpam-4974	144	3	:	:	PUNCT
ejpam-4974	144	4	(	(	PUNCT
ejpam-4974	144	5	x	x	X
ejpam-4974	144	6	,	,	PUNCT
ejpam-4974	144	7	τ	τ	X
ejpam-4974	144	8	)	)	PUNCT
ejpam-4974	144	9	→	→	SYM
ejpam-4974	144	10	(	(	PUNCT
ejpam-4974	144	11	y	y	PROPN
ejpam-4974	144	12	,	,	PUNCT
ejpam-4974	144	13	σ	σ	PROPN
ejpam-4974	144	14	)	)	PUNCT
ejpam-4974	144	15	be	be	VERB
ejpam-4974	144	16	a	a	DET
ejpam-4974	144	17	bijective	bijective	ADJ
ejpam-4974	144	18	function	function	NOUN
ejpam-4974	144	19	.	.	PUNCT
ejpam-4974	145	1	then	then	ADV
ejpam-4974	145	2	,	,	PUNCT
ejpam-4974	145	3	the	the	DET
ejpam-4974	145	4	following	follow	VERB
ejpam-4974	145	5	properties	property	NOUN
ejpam-4974	145	6	are	be	AUX
ejpam-4974	145	7	equivalent	equivalent	ADJ
ejpam-4974	145	8	:	:	PUNCT
ejpam-4974	145	9	(	(	PUNCT
ejpam-4974	145	10	1	1	X
ejpam-4974	145	11	)	)	PUNCT
ejpam-4974	145	12	f	f	PROPN
ejpam-4974	145	13	is	be	AUX
ejpam-4974	145	14	weakly	weakly	ADJ
ejpam-4974	145	15	β(λ	β(λ	NOUN
ejpam-4974	145	16	,	,	PUNCT
ejpam-4974	145	17	p)-open	p)-open	VERB
ejpam-4974	145	18	;	;	PUNCT
ejpam-4974	145	19	(	(	PUNCT
ejpam-4974	145	20	2	2	X
ejpam-4974	145	21	)	)	PUNCT
ejpam-4974	146	1	[	[	X
ejpam-4974	146	2	f(u)]β(λ	f(u)]β(λ	X
ejpam-4974	146	3	,	,	PUNCT
ejpam-4974	146	4	p	p	NOUN
ejpam-4974	146	5	)	)	PUNCT
ejpam-4974	146	6	⊆	⊆	NUM
ejpam-4974	146	7	f(u	f(u	PROPN
ejpam-4974	146	8	(	(	PUNCT
ejpam-4974	146	9	λ	λ	PROPN
ejpam-4974	146	10	,	,	PUNCT
ejpam-4974	146	11	p	p	NOUN
ejpam-4974	146	12	)	)	PUNCT
ejpam-4974	146	13	)	)	PUNCT
ejpam-4974	146	14	for	for	SCONJ
ejpam-4974	146	15	every	every	DET
ejpam-4974	146	16	(	(	PUNCT
ejpam-4974	146	17	λ	λ	NOUN
ejpam-4974	146	18	,	,	PUNCT
ejpam-4974	146	19	p)-open	p)-open	VERB
ejpam-4974	146	20	set	set	VERB
ejpam-4974	146	21	u	u	NOUN
ejpam-4974	146	22	of	of	ADP
ejpam-4974	146	23	x	x	PRON
ejpam-4974	146	24	;	;	PUNCT
ejpam-4974	146	25	(	(	PUNCT
ejpam-4974	146	26	3	3	X
ejpam-4974	146	27	)	)	PUNCT
ejpam-4974	147	1	[	[	X
ejpam-4974	147	2	f(k(λ	f(k(λ	NOUN
ejpam-4974	147	3	,	,	PUNCT
ejpam-4974	147	4	p	p	NOUN
ejpam-4974	147	5	)	)	PUNCT
ejpam-4974	147	6	)	)	PUNCT
ejpam-4974	147	7	]	]	PUNCT
ejpam-4974	148	1	β(λ	β(λ	X
ejpam-4974	148	2	,	,	PUNCT
ejpam-4974	148	3	p	p	NOUN
ejpam-4974	148	4	)	)	PUNCT
ejpam-4974	148	5	⊆	⊆	NUM
ejpam-4974	148	6	f(k	f(k	VERB
ejpam-4974	148	7	)	)	PUNCT
ejpam-4974	148	8	for	for	SCONJ
ejpam-4974	148	9	every	every	DET
ejpam-4974	148	10	(	(	PUNCT
ejpam-4974	148	11	λ	λ	PROPN
ejpam-4974	148	12	,	,	PUNCT
ejpam-4974	148	13	p)-closed	p)-close	VERB
ejpam-4974	148	14	set	set	NOUN
ejpam-4974	148	15	k	k	PROPN
ejpam-4974	148	16	of	of	ADP
ejpam-4974	148	17	x.	x.	NOUN
ejpam-4974	148	18	proof	proof	NOUN
ejpam-4974	148	19	.	.	PUNCT
ejpam-4974	149	1	(	(	PUNCT
ejpam-4974	149	2	1	1	X
ejpam-4974	149	3	)	)	PUNCT
ejpam-4974	149	4	⇒	⇒	NOUN
ejpam-4974	149	5	(	(	PUNCT
ejpam-4974	149	6	3	3	NUM
ejpam-4974	149	7	):	):	PUNCT
ejpam-4974	149	8	let	let	VERB
ejpam-4974	149	9	k	k	PRON
ejpam-4974	149	10	be	be	AUX
ejpam-4974	149	11	any	any	DET
ejpam-4974	149	12	(	(	PUNCT
ejpam-4974	149	13	λ	λ	PROPN
ejpam-4974	149	14	,	,	PUNCT
ejpam-4974	149	15	p)-closed	p)-close	VERB
ejpam-4974	149	16	set	set	NOUN
ejpam-4974	149	17	of	of	ADP
ejpam-4974	149	18	x.	x.	NOUN
ejpam-4974	149	19	then	then	ADV
ejpam-4974	149	20	,	,	PUNCT
ejpam-4974	149	21	we	we	PRON
ejpam-4974	149	22	have	have	VERB
ejpam-4974	149	23	f(x	f(x	PROPN
ejpam-4974	149	24	−k	−k	ADV
ejpam-4974	149	25	)	)	PUNCT
ejpam-4974	150	1	=	=	SYM
ejpam-4974	150	2	y	y	PROPN
ejpam-4974	150	3	−	−	PROPN
ejpam-4974	150	4	f(k	f(k	PROPN
ejpam-4974	150	5	)	)	PUNCT
ejpam-4974	150	6	⊆	⊆	NUM
ejpam-4974	151	1	[	[	X
ejpam-4974	151	2	f([x	f([x	PROPN
ejpam-4974	151	3	−k](λ	−k](λ	NUM
ejpam-4974	151	4	,	,	PUNCT
ejpam-4974	151	5	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	151	6	,	,	PUNCT
ejpam-4974	151	7	p	p	NOUN
ejpam-4974	151	8	)	)	PUNCT
ejpam-4974	151	9	and	and	CCONJ
ejpam-4974	151	10	hence	hence	ADV
ejpam-4974	151	11	y	y	PROPN
ejpam-4974	151	12	−	−	PROPN
ejpam-4974	151	13	f(k	f(k	PROPN
ejpam-4974	151	14	)	)	PUNCT
ejpam-4974	152	1	⊆	⊆	NUM
ejpam-4974	152	2	y	y	NOUN
ejpam-4974	152	3	−	−	PROPN
ejpam-4974	152	4	[	[	X
ejpam-4974	152	5	f(k(λ	f(k(λ	NOUN
ejpam-4974	152	6	,	,	PUNCT
ejpam-4974	152	7	p	p	NOUN
ejpam-4974	152	8	)	)	PUNCT
ejpam-4974	152	9	)	)	PUNCT
ejpam-4974	152	10	]	]	PUNCT
ejpam-4974	153	1	β(λ	β(λ	X
ejpam-4974	153	2	,	,	PUNCT
ejpam-4974	153	3	p	p	NOUN
ejpam-4974	153	4	)	)	PUNCT
ejpam-4974	153	5	.	.	PUNCT
ejpam-4974	154	1	thus	thus	ADV
ejpam-4974	154	2	,	,	PUNCT
ejpam-4974	154	3	[	[	X
ejpam-4974	154	4	f(k(λ	f(k(λ	NOUN
ejpam-4974	154	5	,	,	PUNCT
ejpam-4974	154	6	p	p	NOUN
ejpam-4974	154	7	)	)	PUNCT
ejpam-4974	154	8	)	)	PUNCT
ejpam-4974	154	9	]	]	PUNCT
ejpam-4974	154	10	β(λ	β(λ	X
ejpam-4974	154	11	,	,	PUNCT
ejpam-4974	154	12	p	p	NOUN
ejpam-4974	154	13	)	)	PUNCT
ejpam-4974	154	14	⊆	⊆	NUM
ejpam-4974	154	15	f(k	f(k	VERB
ejpam-4974	154	16	)	)	PUNCT
ejpam-4974	154	17	.	.	PUNCT
ejpam-4974	155	1	(	(	PUNCT
ejpam-4974	155	2	3	3	X
ejpam-4974	155	3	)	)	PUNCT
ejpam-4974	155	4	⇒	⇒	NOUN
ejpam-4974	155	5	(	(	PUNCT
ejpam-4974	155	6	2	2	NUM
ejpam-4974	155	7	):	):	PUNCT
ejpam-4974	155	8	let	let	VERB
ejpam-4974	155	9	u	u	PRON
ejpam-4974	155	10	be	be	AUX
ejpam-4974	155	11	any	any	DET
ejpam-4974	155	12	(	(	PUNCT
ejpam-4974	155	13	λ	λ	NOUN
ejpam-4974	155	14	,	,	PUNCT
ejpam-4974	155	15	p)-open	p)-open	VERB
ejpam-4974	155	16	set	set	VERB
ejpam-4974	155	17	of	of	ADP
ejpam-4974	155	18	x.	x.	NOUN
ejpam-4974	155	19	since	since	SCONJ
ejpam-4974	155	20	u	u	PROPN
ejpam-4974	155	21	(	(	PUNCT
ejpam-4974	155	22	λ	λ	PROPN
ejpam-4974	155	23	,	,	PUNCT
ejpam-4974	155	24	p	p	NOUN
ejpam-4974	155	25	)	)	PUNCT
ejpam-4974	155	26	is	be	AUX
ejpam-4974	155	27	(	(	PUNCT
ejpam-4974	155	28	λ	λ	X
ejpam-4974	155	29	,	,	PUNCT
ejpam-4974	155	30	p)-closed	p)-close	VERB
ejpam-4974	155	31	and	and	CCONJ
ejpam-4974	155	32	u	u	NOUN
ejpam-4974	155	33	⊆	⊆	NUM
ejpam-4974	155	34	[	[	X
ejpam-4974	155	35	u	u	X
ejpam-4974	155	36	(	(	PUNCT
ejpam-4974	155	37	λ	λ	PROPN
ejpam-4974	155	38	,	,	PUNCT
ejpam-4974	155	39	p)](λ	p)](λ	ADJ
ejpam-4974	155	40	,	,	PUNCT
ejpam-4974	155	41	p	p	NOUN
ejpam-4974	155	42	)	)	PUNCT
ejpam-4974	155	43	,	,	PUNCT
ejpam-4974	155	44	by	by	ADP
ejpam-4974	155	45	(	(	PUNCT
ejpam-4974	155	46	3	3	X
ejpam-4974	155	47	)	)	PUNCT
ejpam-4974	155	48	we	we	PRON
ejpam-4974	155	49	have	have	VERB
ejpam-4974	155	50	[	[	X
ejpam-4974	155	51	f(u)]β(λ	f(u)]β(λ	X
ejpam-4974	155	52	,	,	PUNCT
ejpam-4974	155	53	p	p	NOUN
ejpam-4974	155	54	)	)	PUNCT
ejpam-4974	155	55	⊆	⊆	NUM
ejpam-4974	156	1	[	[	X
ejpam-4974	156	2	f([u	f([u	INTJ
ejpam-4974	156	3	(	(	PUNCT
ejpam-4974	156	4	λ	λ	PROPN
ejpam-4974	156	5	,	,	PUNCT
ejpam-4974	156	6	p)](λ	p)](λ	ADJ
ejpam-4974	156	7	,	,	PUNCT
ejpam-4974	156	8	p	p	NOUN
ejpam-4974	156	9	)	)	PUNCT
ejpam-4974	156	10	)	)	PUNCT
ejpam-4974	156	11	]	]	PUNCT
ejpam-4974	157	1	β(λ	β(λ	X
ejpam-4974	157	2	,	,	PUNCT
ejpam-4974	157	3	p	p	NOUN
ejpam-4974	157	4	)	)	PUNCT
ejpam-4974	157	5	⊆	⊆	NUM
ejpam-4974	157	6	f(u	f(u	PROPN
ejpam-4974	157	7	(	(	PUNCT
ejpam-4974	157	8	λ	λ	PROPN
ejpam-4974	157	9	,	,	PUNCT
ejpam-4974	157	10	p	p	NOUN
ejpam-4974	157	11	)	)	PUNCT
ejpam-4974	157	12	)	)	PUNCT
ejpam-4974	157	13	.	.	PUNCT
ejpam-4974	158	1	(	(	PUNCT
ejpam-4974	158	2	2	2	X
ejpam-4974	158	3	)	)	PUNCT
ejpam-4974	158	4	⇒	⇒	NOUN
ejpam-4974	158	5	(	(	PUNCT
ejpam-4974	158	6	1	1	NUM
ejpam-4974	158	7	):	):	PUNCT
ejpam-4974	158	8	let	let	VERB
ejpam-4974	158	9	u	u	PRON
ejpam-4974	158	10	be	be	AUX
ejpam-4974	158	11	any	any	DET
ejpam-4974	158	12	(	(	PUNCT
ejpam-4974	158	13	λ	λ	NOUN
ejpam-4974	158	14	,	,	PUNCT
ejpam-4974	158	15	p)-open	p)-open	VERB
ejpam-4974	158	16	set	set	VERB
ejpam-4974	158	17	of	of	ADP
ejpam-4974	158	18	x.	x.	NOUN
ejpam-4974	158	19	by	by	ADP
ejpam-4974	158	20	(	(	PUNCT
ejpam-4974	158	21	2	2	NUM
ejpam-4974	158	22	)	)	PUNCT
ejpam-4974	158	23	,	,	PUNCT
ejpam-4974	158	24	we	we	PRON
ejpam-4974	158	25	have	have	VERB
ejpam-4974	158	26	[	[	X
ejpam-4974	158	27	f(x	f(x	PROPN
ejpam-4974	158	28	−	−	PROPN
ejpam-4974	158	29	u	u	PROPN
ejpam-4974	158	30	(	(	PUNCT
ejpam-4974	158	31	λ	λ	PROPN
ejpam-4974	158	32	,	,	PUNCT
ejpam-4974	158	33	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	158	34	,	,	PUNCT
ejpam-4974	158	35	p	p	NOUN
ejpam-4974	158	36	)	)	PUNCT
ejpam-4974	158	37	⊆	⊆	NUM
ejpam-4974	158	38	f([x	f([x	PROPN
ejpam-4974	158	39	−	−	PROPN
ejpam-4974	158	40	u	u	PROPN
ejpam-4974	158	41	(	(	PUNCT
ejpam-4974	158	42	λ	λ	PROPN
ejpam-4974	158	43	,	,	PUNCT
ejpam-4974	158	44	p)](λ	p)](λ	ADJ
ejpam-4974	158	45	,	,	PUNCT
ejpam-4974	158	46	p	p	NOUN
ejpam-4974	158	47	)	)	PUNCT
ejpam-4974	158	48	)	)	PUNCT
ejpam-4974	158	49	.	.	PUNCT
ejpam-4974	159	1	since	since	SCONJ
ejpam-4974	159	2	f	f	PROPN
ejpam-4974	159	3	is	be	AUX
ejpam-4974	159	4	bijective	bijective	ADJ
ejpam-4974	159	5	,	,	PUNCT
ejpam-4974	159	6	[	[	X
ejpam-4974	159	7	f(x	f(x	PROPN
ejpam-4974	159	8	−	−	PROPN
ejpam-4974	159	9	u	u	PROPN
ejpam-4974	159	10	(	(	PUNCT
ejpam-4974	159	11	λ	λ	PROPN
ejpam-4974	159	12	,	,	PUNCT
ejpam-4974	159	13	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	159	14	,	,	PUNCT
ejpam-4974	159	15	p	p	NOUN
ejpam-4974	159	16	)	)	PUNCT
ejpam-4974	159	17	=	=	PUNCT
ejpam-4974	160	1	y	y	PROPN
ejpam-4974	160	2	−	−	PROPN
ejpam-4974	161	1	[	[	X
ejpam-4974	161	2	f(u	f(u	PROPN
ejpam-4974	161	3	(	(	PUNCT
ejpam-4974	161	4	λ	λ	PROPN
ejpam-4974	161	5	,	,	PUNCT
ejpam-4974	161	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	161	7	,	,	PUNCT
ejpam-4974	161	8	p	p	NOUN
ejpam-4974	161	9	)	)	PUNCT
ejpam-4974	161	10	and	and	CCONJ
ejpam-4974	161	11	f([x	f([x	NOUN
ejpam-4974	161	12	−	−	PROPN
ejpam-4974	161	13	u	u	PROPN
ejpam-4974	161	14	(	(	PUNCT
ejpam-4974	161	15	λ	λ	PROPN
ejpam-4974	161	16	,	,	PUNCT
ejpam-4974	161	17	p)](λ	p)](λ	ADJ
ejpam-4974	161	18	,	,	PUNCT
ejpam-4974	161	19	p	p	NOUN
ejpam-4974	161	20	)	)	PUNCT
ejpam-4974	161	21	)	)	PUNCT
ejpam-4974	162	1	=	=	SYM
ejpam-4974	162	2	f(x	f(x	PROPN
ejpam-4974	162	3	−	−	PROPN
ejpam-4974	163	1	[	[	X
ejpam-4974	163	2	u	u	X
ejpam-4974	163	3	(	(	PUNCT
ejpam-4974	163	4	λ	λ	PROPN
ejpam-4974	163	5	,	,	PUNCT
ejpam-4974	163	6	p)](λ	p)](λ	ADJ
ejpam-4974	163	7	,	,	PUNCT
ejpam-4974	163	8	p	p	NOUN
ejpam-4974	163	9	)	)	PUNCT
ejpam-4974	163	10	)	)	PUNCT
ejpam-4974	163	11	⊆	⊆	NUM
ejpam-4974	163	12	f(x	f(x	PROPN
ejpam-4974	163	13	−	−	PROPN
ejpam-4974	163	14	u	u	NOUN
ejpam-4974	163	15	)	)	PUNCT
ejpam-4974	163	16	=	=	SYM
ejpam-4974	163	17	y	y	PROPN
ejpam-4974	163	18	−	−	PROPN
ejpam-4974	163	19	f(u	f(u	PROPN
ejpam-4974	163	20	)	)	PUNCT
ejpam-4974	163	21	.	.	PUNCT
ejpam-4974	164	1	thus	thus	ADV
ejpam-4974	164	2	,	,	PUNCT
ejpam-4974	164	3	f(u	f(u	PROPN
ejpam-4974	164	4	)	)	PUNCT
ejpam-4974	165	1	⊆	⊆	NUM
ejpam-4974	166	1	[	[	X
ejpam-4974	166	2	f(u	f(u	PROPN
ejpam-4974	166	3	(	(	PUNCT
ejpam-4974	166	4	λ	λ	PROPN
ejpam-4974	166	5	,	,	PUNCT
ejpam-4974	166	6	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	166	7	,	,	PUNCT
ejpam-4974	166	8	p	p	NOUN
ejpam-4974	166	9	)	)	PUNCT
ejpam-4974	166	10	and	and	CCONJ
ejpam-4974	166	11	hence	hence	ADV
ejpam-4974	166	12	f	f	PROPN
ejpam-4974	166	13	is	be	AUX
ejpam-4974	166	14	weakly	weakly	ADJ
ejpam-4974	166	15	β(λ	β(λ	NOUN
ejpam-4974	166	16	,	,	PUNCT
ejpam-4974	166	17	p)-open	p)-open	NOUN
ejpam-4974	166	18	.	.	PUNCT
ejpam-4974	167	1	4	4	X
ejpam-4974	167	2	.	.	X
ejpam-4974	167	3	properties	property	NOUN
ejpam-4974	167	4	of	of	ADP
ejpam-4974	167	5	weakly	weakly	ADJ
ejpam-4974	167	6	β(λ	β(λ	NOUN
ejpam-4974	167	7	,	,	PUNCT
ejpam-4974	167	8	p)-closed	p)-close	VERB
ejpam-4974	167	9	functions	function	NOUN
ejpam-4974	167	10	we	we	PRON
ejpam-4974	167	11	begin	begin	VERB
ejpam-4974	167	12	this	this	DET
ejpam-4974	167	13	section	section	NOUN
ejpam-4974	167	14	by	by	ADP
ejpam-4974	167	15	introducing	introduce	VERB
ejpam-4974	167	16	the	the	DET
ejpam-4974	167	17	concept	concept	NOUN
ejpam-4974	167	18	of	of	ADP
ejpam-4974	167	19	weakly	weakly	ADJ
ejpam-4974	167	20	β(λ	β(λ	NOUN
ejpam-4974	167	21	,	,	PUNCT
ejpam-4974	167	22	p)-closed	p)-close	VERB
ejpam-4974	167	23	functions	function	NOUN
ejpam-4974	167	24	.	.	PUNCT
ejpam-4974	168	1	definition	definition	NOUN
ejpam-4974	168	2	2	2	NUM
ejpam-4974	168	3	.	.	PUNCT
ejpam-4974	169	1	a	a	DET
ejpam-4974	169	2	functions	function	NOUN
ejpam-4974	169	3	f	f	X
ejpam-4974	169	4	:	:	PUNCT
ejpam-4974	169	5	(	(	PUNCT
ejpam-4974	169	6	x	x	X
ejpam-4974	169	7	,	,	PUNCT
ejpam-4974	169	8	τ	τ	X
ejpam-4974	169	9	)	)	PUNCT
ejpam-4974	169	10	→	→	SYM
ejpam-4974	169	11	(	(	PUNCT
ejpam-4974	169	12	y	y	PROPN
ejpam-4974	169	13	,	,	PUNCT
ejpam-4974	169	14	σ	σ	PROPN
ejpam-4974	169	15	)	)	PUNCT
ejpam-4974	169	16	is	be	AUX
ejpam-4974	169	17	said	say	VERB
ejpam-4974	169	18	to	to	PART
ejpam-4974	169	19	be	be	AUX
ejpam-4974	169	20	weakly	weakly	ADJ
ejpam-4974	169	21	β(λ	β(λ	NOUN
ejpam-4974	169	22	,	,	PUNCT
ejpam-4974	169	23	p)-closed	p)-close	VERB
ejpam-4974	169	24	if	if	SCONJ
ejpam-4974	169	25	[	[	X
ejpam-4974	169	26	f(k(λ	f(k(λ	NOUN
ejpam-4974	169	27	,	,	PUNCT
ejpam-4974	169	28	p	p	NOUN
ejpam-4974	169	29	)	)	PUNCT
ejpam-4974	169	30	)	)	PUNCT
ejpam-4974	169	31	]	]	PUNCT
ejpam-4974	170	1	β(λ	β(λ	X
ejpam-4974	170	2	,	,	PUNCT
ejpam-4974	170	3	p	p	NOUN
ejpam-4974	170	4	)	)	PUNCT
ejpam-4974	170	5	⊆	⊆	NUM
ejpam-4974	170	6	f(k	f(k	VERB
ejpam-4974	170	7	)	)	PUNCT
ejpam-4974	170	8	for	for	SCONJ
ejpam-4974	170	9	each	each	DET
ejpam-4974	170	10	(	(	PUNCT
ejpam-4974	170	11	λ	λ	PROPN
ejpam-4974	170	12	,	,	PUNCT
ejpam-4974	170	13	p)-closed	p)-close	VERB
ejpam-4974	170	14	set	set	NOUN
ejpam-4974	170	15	k	k	PROPN
ejpam-4974	170	16	of	of	ADP
ejpam-4974	170	17	x.	x.	PROPN
ejpam-4974	170	18	theorem	theorem	VERB
ejpam-4974	170	19	3	3	NUM
ejpam-4974	170	20	.	.	X
ejpam-4974	171	1	for	for	ADP
ejpam-4974	171	2	a	a	DET
ejpam-4974	171	3	function	function	NOUN
ejpam-4974	171	4	f	f	NOUN
ejpam-4974	171	5	:	:	PUNCT
ejpam-4974	171	6	(	(	PUNCT
ejpam-4974	171	7	x	x	X
ejpam-4974	171	8	,	,	PUNCT
ejpam-4974	171	9	τ	τ	X
ejpam-4974	171	10	)	)	PUNCT
ejpam-4974	171	11	→	→	SYM
ejpam-4974	171	12	(	(	PUNCT
ejpam-4974	171	13	y	y	PROPN
ejpam-4974	171	14	,	,	PUNCT
ejpam-4974	171	15	σ	σ	PROPN
ejpam-4974	171	16	)	)	PUNCT
ejpam-4974	171	17	,	,	PUNCT
ejpam-4974	171	18	the	the	DET
ejpam-4974	171	19	following	follow	VERB
ejpam-4974	171	20	properties	property	NOUN
ejpam-4974	171	21	are	be	AUX
ejpam-4974	171	22	equivalent	equivalent	ADJ
ejpam-4974	171	23	:	:	PUNCT
ejpam-4974	171	24	(	(	PUNCT
ejpam-4974	171	25	1	1	X
ejpam-4974	171	26	)	)	PUNCT
ejpam-4974	171	27	f	f	PROPN
ejpam-4974	171	28	is	be	AUX
ejpam-4974	171	29	weakly	weakly	ADJ
ejpam-4974	171	30	β(λ	β(λ	NOUN
ejpam-4974	171	31	,	,	PUNCT
ejpam-4974	171	32	p)-closed	p)-close	VERB
ejpam-4974	171	33	;	;	PUNCT
ejpam-4974	171	34	(	(	PUNCT
ejpam-4974	171	35	2	2	X
ejpam-4974	171	36	)	)	PUNCT
ejpam-4974	172	1	[	[	X
ejpam-4974	172	2	f(u)]β(λ	f(u)]β(λ	X
ejpam-4974	172	3	,	,	PUNCT
ejpam-4974	172	4	p	p	NOUN
ejpam-4974	172	5	)	)	PUNCT
ejpam-4974	172	6	⊆	⊆	NUM
ejpam-4974	172	7	f(u	f(u	PROPN
ejpam-4974	172	8	(	(	PUNCT
ejpam-4974	172	9	λ	λ	PROPN
ejpam-4974	172	10	,	,	PUNCT
ejpam-4974	172	11	p	p	NOUN
ejpam-4974	172	12	)	)	PUNCT
ejpam-4974	172	13	)	)	PUNCT
ejpam-4974	172	14	for	for	SCONJ
ejpam-4974	172	15	every	every	DET
ejpam-4974	172	16	(	(	PUNCT
ejpam-4974	172	17	λ	λ	NOUN
ejpam-4974	172	18	,	,	PUNCT
ejpam-4974	172	19	p)-open	p)-open	VERB
ejpam-4974	172	20	set	set	VERB
ejpam-4974	172	21	u	u	NOUN
ejpam-4974	172	22	of	of	ADP
ejpam-4974	172	23	x	x	PRON
ejpam-4974	172	24	;	;	PUNCT
ejpam-4974	172	25	(	(	PUNCT
ejpam-4974	172	26	3	3	X
ejpam-4974	172	27	)	)	PUNCT
ejpam-4974	172	28	[	[	X
ejpam-4974	172	29	f(u)]β(λ	f(u)]β(λ	X
ejpam-4974	172	30	,	,	PUNCT
ejpam-4974	172	31	p	p	NOUN
ejpam-4974	172	32	)	)	PUNCT
ejpam-4974	172	33	⊆	⊆	NUM
ejpam-4974	172	34	f(u	f(u	PROPN
ejpam-4974	172	35	(	(	PUNCT
ejpam-4974	172	36	λ	λ	PROPN
ejpam-4974	172	37	,	,	PUNCT
ejpam-4974	172	38	p	p	NOUN
ejpam-4974	172	39	)	)	PUNCT
ejpam-4974	172	40	)	)	PUNCT
ejpam-4974	172	41	for	for	ADP
ejpam-4974	172	42	every	every	DET
ejpam-4974	172	43	r(λ	r(λ	NOUN
ejpam-4974	172	44	,	,	PUNCT
ejpam-4974	172	45	p)-open	p)-open	VERB
ejpam-4974	172	46	set	set	VERB
ejpam-4974	172	47	u	u	NOUN
ejpam-4974	172	48	of	of	ADP
ejpam-4974	172	49	x	x	PRON
ejpam-4974	172	50	;	;	PUNCT
ejpam-4974	172	51	(	(	PUNCT
ejpam-4974	172	52	4	4	X
ejpam-4974	172	53	)	)	PUNCT
ejpam-4974	172	54	for	for	ADP
ejpam-4974	172	55	each	each	DET
ejpam-4974	172	56	subset	subset	NOUN
ejpam-4974	172	57	b	b	PROPN
ejpam-4974	172	58	of	of	ADP
ejpam-4974	172	59	y	y	PROPN
ejpam-4974	172	60	and	and	CCONJ
ejpam-4974	172	61	each	each	PRON
ejpam-4974	172	62	(	(	PUNCT
ejpam-4974	172	63	λ	λ	NOUN
ejpam-4974	172	64	,	,	PUNCT
ejpam-4974	172	65	p)-open	p)-open	VERB
ejpam-4974	172	66	set	set	VERB
ejpam-4974	172	67	u	u	NOUN
ejpam-4974	172	68	of	of	ADP
ejpam-4974	172	69	x	x	PUNCT
ejpam-4974	172	70	with	with	ADP
ejpam-4974	172	71	f−1(b	f−1(b	PROPN
ejpam-4974	172	72	)	)	PUNCT
ejpam-4974	172	73	⊆	⊆	NUM
ejpam-4974	172	74	u	u	NOUN
ejpam-4974	172	75	,	,	PUNCT
ejpam-4974	172	76	there	there	PRON
ejpam-4974	172	77	exists	exist	VERB
ejpam-4974	172	78	a	a	DET
ejpam-4974	172	79	β(λ	β(λ	NOUN
ejpam-4974	172	80	,	,	PUNCT
ejpam-4974	172	81	p)-open	p)-open	VERB
ejpam-4974	172	82	set	set	VERB
ejpam-4974	172	83	v	v	NOUN
ejpam-4974	172	84	of	of	ADP
ejpam-4974	172	85	y	y	PRON
ejpam-4974	172	86	such	such	ADJ
ejpam-4974	172	87	that	that	DET
ejpam-4974	172	88	b	b	PROPN
ejpam-4974	172	89	⊆	⊆	NUM
ejpam-4974	172	90	v	v	NOUN
ejpam-4974	172	91	and	and	CCONJ
ejpam-4974	172	92	f−1(v	f−1(v	NOUN
ejpam-4974	172	93	)	)	PUNCT
ejpam-4974	173	1	⊆	⊆	NUM
ejpam-4974	173	2	u	u	NOUN
ejpam-4974	173	3	(	(	PUNCT
ejpam-4974	173	4	λ	λ	PROPN
ejpam-4974	173	5	,	,	PUNCT
ejpam-4974	173	6	p	p	NOUN
ejpam-4974	173	7	)	)	PUNCT
ejpam-4974	173	8	;	;	PUNCT
ejpam-4974	173	9	c.	c.	PROPN
ejpam-4974	173	10	boonpok	boonpok	PROPN
ejpam-4974	173	11	,	,	PUNCT
ejpam-4974	173	12	m.	m.	NOUN
ejpam-4974	173	13	thongmoon	thongmoon	PROPN
ejpam-4974	173	14	/	/	SYM
ejpam-4974	173	15	eur	eur	PROPN
ejpam-4974	173	16	.	.	PUNCT
ejpam-4974	174	1	j.	j.	PROPN
ejpam-4974	174	2	pure	pure	PROPN
ejpam-4974	174	3	appl	appl	PROPN
ejpam-4974	174	4	.	.	PROPN
ejpam-4974	174	5	math	math	PROPN
ejpam-4974	174	6	,	,	PUNCT
ejpam-4974	174	7	17	17	NUM
ejpam-4974	174	8	(	(	PUNCT
ejpam-4974	174	9	1	1	NUM
ejpam-4974	174	10	)	)	PUNCT
ejpam-4974	174	11	(	(	PUNCT
ejpam-4974	174	12	2024	2024	NUM
ejpam-4974	174	13	)	)	PUNCT
ejpam-4974	174	14	,	,	PUNCT
ejpam-4974	174	15	248	248	NUM
ejpam-4974	174	16	-	-	SYM
ejpam-4974	174	17	255	255	NUM
ejpam-4974	174	18	253	253	NUM
ejpam-4974	174	19	(	(	PUNCT
ejpam-4974	174	20	5	5	NUM
ejpam-4974	174	21	)	)	PUNCT
ejpam-4974	174	22	for	for	ADP
ejpam-4974	174	23	each	each	DET
ejpam-4974	174	24	point	point	NOUN
ejpam-4974	174	25	y	y	PROPN
ejpam-4974	174	26	∈	∈	PROPN
ejpam-4974	174	27	y	y	PROPN
ejpam-4974	174	28	and	and	CCONJ
ejpam-4974	174	29	each	each	DET
ejpam-4974	174	30	(	(	PUNCT
ejpam-4974	174	31	λ	λ	NOUN
ejpam-4974	174	32	,	,	PUNCT
ejpam-4974	174	33	p)-open	p)-open	VERB
ejpam-4974	174	34	set	set	VERB
ejpam-4974	174	35	u	u	NOUN
ejpam-4974	174	36	of	of	ADP
ejpam-4974	174	37	x	x	PUNCT
ejpam-4974	174	38	with	with	ADP
ejpam-4974	174	39	f−1(y	f−1(y	PROPN
ejpam-4974	174	40	)	)	PUNCT
ejpam-4974	175	1	⊆	⊆	NUM
ejpam-4974	175	2	u	u	NOUN
ejpam-4974	175	3	,	,	PUNCT
ejpam-4974	175	4	there	there	PRON
ejpam-4974	175	5	exists	exist	VERB
ejpam-4974	175	6	a	a	DET
ejpam-4974	175	7	β(λ	β(λ	NOUN
ejpam-4974	175	8	,	,	PUNCT
ejpam-4974	175	9	p)-open	p)-open	VERB
ejpam-4974	175	10	set	set	VERB
ejpam-4974	175	11	v	v	NOUN
ejpam-4974	175	12	of	of	ADP
ejpam-4974	175	13	y	y	PROPN
ejpam-4974	175	14	containing	contain	VERB
ejpam-4974	175	15	y	y	PRON
ejpam-4974	175	16	such	such	ADJ
ejpam-4974	175	17	that	that	DET
ejpam-4974	175	18	f−1(v	f−1(v	NOUN
ejpam-4974	175	19	)	)	PUNCT
ejpam-4974	176	1	⊆	⊆	NUM
ejpam-4974	176	2	u	u	NOUN
ejpam-4974	176	3	(	(	PUNCT
ejpam-4974	176	4	λ	λ	PROPN
ejpam-4974	176	5	,	,	PUNCT
ejpam-4974	176	6	p	p	NOUN
ejpam-4974	176	7	)	)	PUNCT
ejpam-4974	176	8	;	;	PUNCT
ejpam-4974	176	9	(	(	PUNCT
ejpam-4974	176	10	6	6	X
ejpam-4974	176	11	)	)	PUNCT
ejpam-4974	176	12	[	[	X
ejpam-4974	176	13	f([u	f([u	INTJ
ejpam-4974	176	14	(	(	PUNCT
ejpam-4974	176	15	λ	λ	PROPN
ejpam-4974	176	16	,	,	PUNCT
ejpam-4974	176	17	p)](λ	p)](λ	ADJ
ejpam-4974	176	18	,	,	PUNCT
ejpam-4974	176	19	p	p	NOUN
ejpam-4974	176	20	)	)	PUNCT
ejpam-4974	176	21	)	)	PUNCT
ejpam-4974	176	22	]	]	PUNCT
ejpam-4974	177	1	β(λ	β(λ	X
ejpam-4974	177	2	,	,	PUNCT
ejpam-4974	177	3	p	p	NOUN
ejpam-4974	177	4	)	)	PUNCT
ejpam-4974	177	5	⊆	⊆	NUM
ejpam-4974	177	6	f(u	f(u	PROPN
ejpam-4974	177	7	(	(	PUNCT
ejpam-4974	177	8	λ	λ	PROPN
ejpam-4974	177	9	,	,	PUNCT
ejpam-4974	177	10	p	p	NOUN
ejpam-4974	177	11	)	)	PUNCT
ejpam-4974	177	12	)	)	PUNCT
ejpam-4974	177	13	for	for	SCONJ
ejpam-4974	177	14	every	every	DET
ejpam-4974	177	15	(	(	PUNCT
ejpam-4974	177	16	λ	λ	NOUN
ejpam-4974	177	17	,	,	PUNCT
ejpam-4974	177	18	p)-open	p)-open	AUX
ejpam-4974	177	19	set	set	VERB
ejpam-4974	177	20	u	u	NOUN
ejpam-4974	177	21	of	of	ADP
ejpam-4974	177	22	x	x	PRON
ejpam-4974	177	23	;	;	PUNCT
ejpam-4974	177	24	(	(	PUNCT
ejpam-4974	177	25	7	7	X
ejpam-4974	177	26	)	)	PUNCT
ejpam-4974	178	1	[	[	X
ejpam-4974	178	2	f([u	f([u	X
ejpam-4974	178	3	θ(λ	θ(λ	PROPN
ejpam-4974	178	4	,	,	PUNCT
ejpam-4974	178	5	p)](λ	p)](λ	X
ejpam-4974	178	6	,	,	PUNCT
ejpam-4974	178	7	p	p	NOUN
ejpam-4974	178	8	)	)	PUNCT
ejpam-4974	178	9	)	)	PUNCT
ejpam-4974	178	10	]	]	PUNCT
ejpam-4974	178	11	β(λ	β(λ	X
ejpam-4974	178	12	,	,	PUNCT
ejpam-4974	178	13	p	p	NOUN
ejpam-4974	178	14	)	)	PUNCT
ejpam-4974	178	15	⊆	⊆	NUM
ejpam-4974	178	16	f(u	f(u	PROPN
ejpam-4974	178	17	θ(λ	θ(λ	PROPN
ejpam-4974	178	18	,	,	PUNCT
ejpam-4974	178	19	p	p	NOUN
ejpam-4974	178	20	)	)	PUNCT
ejpam-4974	178	21	)	)	PUNCT
ejpam-4974	178	22	for	for	ADP
ejpam-4974	178	23	every	every	DET
ejpam-4974	178	24	(	(	PUNCT
ejpam-4974	178	25	λ	λ	NOUN
ejpam-4974	178	26	,	,	PUNCT
ejpam-4974	178	27	p)-open	p)-open	VERB
ejpam-4974	178	28	set	set	VERB
ejpam-4974	178	29	u	u	NOUN
ejpam-4974	178	30	of	of	ADP
ejpam-4974	178	31	x	x	PRON
ejpam-4974	178	32	;	;	PUNCT
ejpam-4974	178	33	(	(	PUNCT
ejpam-4974	178	34	8)	8)	NUM
ejpam-4974	178	35	[	[	X
ejpam-4974	178	36	f(u)]β(λ	f(u)]β(λ	X
ejpam-4974	178	37	,	,	PUNCT
ejpam-4974	178	38	p	p	NOUN
ejpam-4974	178	39	)	)	PUNCT
ejpam-4974	178	40	⊆	⊆	NUM
ejpam-4974	178	41	f(u	f(u	PROPN
ejpam-4974	178	42	(	(	PUNCT
ejpam-4974	178	43	λ	λ	PROPN
ejpam-4974	178	44	,	,	PUNCT
ejpam-4974	178	45	p	p	NOUN
ejpam-4974	178	46	)	)	PUNCT
ejpam-4974	178	47	)	)	PUNCT
ejpam-4974	178	48	for	for	ADP
ejpam-4974	178	49	every	every	DET
ejpam-4974	178	50	β(λ	β(λ	NOUN
ejpam-4974	178	51	,	,	PUNCT
ejpam-4974	178	52	p)-open	p)-open	VERB
ejpam-4974	178	53	set	set	VERB
ejpam-4974	178	54	u	u	NOUN
ejpam-4974	178	55	of	of	ADP
ejpam-4974	178	56	x.	x.	NOUN
ejpam-4974	178	57	proof	proof	NOUN
ejpam-4974	178	58	.	.	PUNCT
ejpam-4974	179	1	(	(	PUNCT
ejpam-4974	179	2	1	1	X
ejpam-4974	179	3	)	)	PUNCT
ejpam-4974	179	4	⇒	⇒	NOUN
ejpam-4974	179	5	(	(	PUNCT
ejpam-4974	179	6	2	2	NUM
ejpam-4974	179	7	):	):	PUNCT
ejpam-4974	179	8	let	let	VERB
ejpam-4974	179	9	u	u	PRON
ejpam-4974	179	10	be	be	AUX
ejpam-4974	179	11	any	any	DET
ejpam-4974	179	12	(	(	PUNCT
ejpam-4974	179	13	λ	λ	NOUN
ejpam-4974	179	14	,	,	PUNCT
ejpam-4974	179	15	p)-open	p)-open	VERB
ejpam-4974	179	16	set	set	VERB
ejpam-4974	179	17	of	of	ADP
ejpam-4974	179	18	x.	x.	NOUN
ejpam-4974	179	19	then	then	ADV
ejpam-4974	179	20	by	by	ADP
ejpam-4974	179	21	(	(	PUNCT
ejpam-4974	179	22	1	1	NUM
ejpam-4974	179	23	)	)	PUNCT
ejpam-4974	179	24	,	,	PUNCT
ejpam-4974	179	25	[	[	X
ejpam-4974	179	26	f(u)]β(λ	f(u)]β(λ	X
ejpam-4974	179	27	,	,	PUNCT
ejpam-4974	179	28	p	p	NOUN
ejpam-4974	179	29	)	)	PUNCT
ejpam-4974	179	30	=	=	PUNCT
ejpam-4974	180	1	[	[	X
ejpam-4974	180	2	f(u(λ	f(u(λ	PROPN
ejpam-4974	180	3	,	,	PUNCT
ejpam-4974	180	4	p	p	NOUN
ejpam-4974	180	5	)	)	PUNCT
ejpam-4974	180	6	)	)	PUNCT
ejpam-4974	180	7	]	]	PUNCT
ejpam-4974	181	1	β(λ	β(λ	X
ejpam-4974	181	2	,	,	PUNCT
ejpam-4974	181	3	p	p	NOUN
ejpam-4974	181	4	)	)	PUNCT
ejpam-4974	181	5	⊆	⊆	NUM
ejpam-4974	181	6	[	[	X
ejpam-4974	181	7	f([u	f([u	INTJ
ejpam-4974	181	8	(	(	PUNCT
ejpam-4974	181	9	λ	λ	PROPN
ejpam-4974	181	10	,	,	PUNCT
ejpam-4974	181	11	p)](λ	p)](λ	ADJ
ejpam-4974	181	12	,	,	PUNCT
ejpam-4974	181	13	p	p	NOUN
ejpam-4974	181	14	)	)	PUNCT
ejpam-4974	181	15	)	)	PUNCT
ejpam-4974	181	16	]	]	PUNCT
ejpam-4974	181	17	β(λ	β(λ	X
ejpam-4974	181	18	,	,	PUNCT
ejpam-4974	181	19	p	p	NOUN
ejpam-4974	181	20	)	)	PUNCT
ejpam-4974	181	21	⊆	⊆	NUM
ejpam-4974	181	22	f(u	f(u	PROPN
ejpam-4974	181	23	(	(	PUNCT
ejpam-4974	181	24	λ	λ	PROPN
ejpam-4974	181	25	,	,	PUNCT
ejpam-4974	181	26	p	p	NOUN
ejpam-4974	181	27	)	)	PUNCT
ejpam-4974	181	28	)	)	PUNCT
ejpam-4974	181	29	.	.	PUNCT
ejpam-4974	182	1	(	(	PUNCT
ejpam-4974	182	2	2	2	X
ejpam-4974	182	3	)	)	PUNCT
ejpam-4974	182	4	⇒	⇒	NOUN
ejpam-4974	182	5	(	(	PUNCT
ejpam-4974	182	6	1	1	NUM
ejpam-4974	182	7	):	):	PUNCT
ejpam-4974	182	8	let	let	VERB
ejpam-4974	182	9	k	k	PRON
ejpam-4974	182	10	be	be	AUX
ejpam-4974	182	11	any	any	DET
ejpam-4974	182	12	(	(	PUNCT
ejpam-4974	182	13	λ	λ	PROPN
ejpam-4974	182	14	,	,	PUNCT
ejpam-4974	182	15	p)-closed	p)-close	VERB
ejpam-4974	182	16	set	set	NOUN
ejpam-4974	182	17	of	of	ADP
ejpam-4974	182	18	x.	x.	NOUN
ejpam-4974	182	19	using	use	VERB
ejpam-4974	182	20	(	(	PUNCT
ejpam-4974	182	21	2	2	NUM
ejpam-4974	182	22	)	)	PUNCT
ejpam-4974	182	23	,	,	PUNCT
ejpam-4974	182	24	we	we	PRON
ejpam-4974	182	25	have	have	VERB
ejpam-4974	182	26	[	[	X
ejpam-4974	182	27	f(k(λ	f(k(λ	NOUN
ejpam-4974	182	28	,	,	PUNCT
ejpam-4974	182	29	p	p	NOUN
ejpam-4974	182	30	)	)	PUNCT
ejpam-4974	182	31	)	)	PUNCT
ejpam-4974	182	32	]	]	PUNCT
ejpam-4974	183	1	β(λ	β(λ	X
ejpam-4974	183	2	,	,	PUNCT
ejpam-4974	183	3	p	p	NOUN
ejpam-4974	183	4	)	)	PUNCT
ejpam-4974	183	5	⊆	⊆	NUM
ejpam-4974	183	6	f([k(λ	f([k(λ	NOUN
ejpam-4974	183	7	,	,	PUNCT
ejpam-4974	183	8	p	p	NOUN
ejpam-4974	183	9	)	)	PUNCT
ejpam-4974	183	10	]	]	PUNCT
ejpam-4974	183	11	(	(	PUNCT
ejpam-4974	183	12	λ	λ	X
ejpam-4974	183	13	,	,	PUNCT
ejpam-4974	183	14	p	p	NOUN
ejpam-4974	183	15	)	)	PUNCT
ejpam-4974	183	16	)	)	PUNCT
ejpam-4974	183	17	⊆	⊆	NUM
ejpam-4974	183	18	f(k(λ	f(k(λ	NOUN
ejpam-4974	183	19	,	,	PUNCT
ejpam-4974	183	20	p	p	NOUN
ejpam-4974	183	21	)	)	PUNCT
ejpam-4974	183	22	)	)	PUNCT
ejpam-4974	183	23	=	=	PUNCT
ejpam-4974	183	24	f(k	f(k	VERB
ejpam-4974	183	25	)	)	PUNCT
ejpam-4974	183	26	.	.	PUNCT
ejpam-4974	184	1	this	this	PRON
ejpam-4974	184	2	shows	show	VERB
ejpam-4974	184	3	that	that	SCONJ
ejpam-4974	184	4	f	f	PROPN
ejpam-4974	184	5	is	be	AUX
ejpam-4974	184	6	weakly	weakly	ADJ
ejpam-4974	184	7	β(λ	β(λ	NOUN
ejpam-4974	184	8	,	,	PUNCT
ejpam-4974	184	9	p)-closed	p)-close	VERB
ejpam-4974	184	10	.	.	PUNCT
ejpam-4974	185	1	it	it	PRON
ejpam-4974	185	2	is	be	AUX
ejpam-4974	185	3	clear	clear	ADJ
ejpam-4974	185	4	that	that	SCONJ
ejpam-4974	185	5	(	(	PUNCT
ejpam-4974	185	6	1	1	X
ejpam-4974	185	7	)	)	PUNCT
ejpam-4974	185	8	⇒	⇒	NOUN
ejpam-4974	185	9	(	(	PUNCT
ejpam-4974	185	10	7	7	NUM
ejpam-4974	185	11	)	)	PUNCT
ejpam-4974	185	12	,	,	PUNCT
ejpam-4974	185	13	(	(	PUNCT
ejpam-4974	185	14	4	4	X
ejpam-4974	185	15	)	)	PUNCT
ejpam-4974	185	16	⇒	⇒	NOUN
ejpam-4974	185	17	(	(	PUNCT
ejpam-4974	185	18	5	5	NUM
ejpam-4974	185	19	)	)	PUNCT
ejpam-4974	185	20	and	and	CCONJ
ejpam-4974	185	21	(	(	PUNCT
ejpam-4974	185	22	1	1	X
ejpam-4974	185	23	)	)	PUNCT
ejpam-4974	185	24	⇒	⇒	NOUN
ejpam-4974	185	25	(	(	PUNCT
ejpam-4974	185	26	6	6	NUM
ejpam-4974	185	27	)	)	PUNCT
ejpam-4974	185	28	⇒	⇒	NOUN
ejpam-4974	185	29	(	(	PUNCT
ejpam-4974	185	30	8)	8)	NUM
ejpam-4974	185	31	⇒	⇒	NOUN
ejpam-4974	185	32	(	(	PUNCT
ejpam-4974	185	33	3	3	NUM
ejpam-4974	185	34	)	)	PUNCT
ejpam-4974	185	35	⇒	⇒	NOUN
ejpam-4974	185	36	(	(	PUNCT
ejpam-4974	185	37	1	1	NUM
ejpam-4974	185	38	)	)	PUNCT
ejpam-4974	185	39	.	.	PUNCT
ejpam-4974	186	1	to	to	PART
ejpam-4974	186	2	show	show	VERB
ejpam-4974	186	3	that	that	SCONJ
ejpam-4974	186	4	(	(	PUNCT
ejpam-4974	186	5	3	3	X
ejpam-4974	186	6	)	)	PUNCT
ejpam-4974	186	7	⇒	⇒	NOUN
ejpam-4974	186	8	(	(	PUNCT
ejpam-4974	186	9	4	4	NUM
ejpam-4974	186	10	):	):	PUNCT
ejpam-4974	186	11	let	let	VERB
ejpam-4974	186	12	b	b	X
ejpam-4974	186	13	be	be	AUX
ejpam-4974	186	14	any	any	DET
ejpam-4974	186	15	subset	subset	NOUN
ejpam-4974	186	16	of	of	ADP
ejpam-4974	186	17	y	y	PROPN
ejpam-4974	186	18	and	and	CCONJ
ejpam-4974	186	19	u	u	NOUN
ejpam-4974	186	20	be	be	VERB
ejpam-4974	186	21	any	any	DET
ejpam-4974	186	22	(	(	PUNCT
ejpam-4974	186	23	λ	λ	NOUN
ejpam-4974	186	24	,	,	PUNCT
ejpam-4974	186	25	p)-open	p)-open	VERB
ejpam-4974	186	26	set	set	VERB
ejpam-4974	186	27	of	of	ADP
ejpam-4974	186	28	x	x	PUNCT
ejpam-4974	186	29	with	with	ADP
ejpam-4974	186	30	f−1(b	f−1(b	PROPN
ejpam-4974	186	31	)	)	PUNCT
ejpam-4974	186	32	⊆	⊆	NUM
ejpam-4974	186	33	u	u	NOUN
ejpam-4974	186	34	.	.	PUNCT
ejpam-4974	187	1	then	then	ADV
ejpam-4974	187	2	,	,	PUNCT
ejpam-4974	187	3	f−1(b	f−1(b	PROPN
ejpam-4974	187	4	)	)	PUNCT
ejpam-4974	187	5	∩	∩	NOUN
ejpam-4974	188	1	[	[	X
ejpam-4974	188	2	x	x	X
ejpam-4974	188	3	−	−	PROPN
ejpam-4974	188	4	u	u	NOUN
ejpam-4974	188	5	(	(	PUNCT
ejpam-4974	188	6	λ	λ	PROPN
ejpam-4974	188	7	,	,	PUNCT
ejpam-4974	188	8	p)](λ	p)](λ	ADJ
ejpam-4974	188	9	,	,	PUNCT
ejpam-4974	188	10	p	p	NOUN
ejpam-4974	188	11	)	)	PUNCT
ejpam-4974	188	12	=	=	NOUN
ejpam-4974	188	13	∅	∅	NOUN
ejpam-4974	188	14	and	and	CCONJ
ejpam-4974	188	15	b	b	NOUN
ejpam-4974	188	16	∩	∩	NOUN
ejpam-4974	188	17	f([x	f([x	NOUN
ejpam-4974	188	18	−	−	PROPN
ejpam-4974	188	19	u	u	PROPN
ejpam-4974	188	20	(	(	PUNCT
ejpam-4974	188	21	λ	λ	PROPN
ejpam-4974	188	22	,	,	PUNCT
ejpam-4974	188	23	p)](λ	p)](λ	ADJ
ejpam-4974	188	24	,	,	PUNCT
ejpam-4974	188	25	p	p	NOUN
ejpam-4974	188	26	)	)	PUNCT
ejpam-4974	188	27	)	)	PUNCT
ejpam-4974	189	1	=	=	PUNCT
ejpam-4974	189	2	∅.	∅.	VERB
ejpam-4974	189	3	since	since	SCONJ
ejpam-4974	189	4	x	x	INTJ
ejpam-4974	189	5	−	−	PROPN
ejpam-4974	189	6	u	u	NOUN
ejpam-4974	189	7	(	(	PUNCT
ejpam-4974	189	8	λ	λ	PROPN
ejpam-4974	189	9	,	,	PUNCT
ejpam-4974	189	10	p	p	NOUN
ejpam-4974	189	11	)	)	PUNCT
ejpam-4974	189	12	is	be	AUX
ejpam-4974	189	13	r(λ	r(λ	NOUN
ejpam-4974	189	14	,	,	PUNCT
ejpam-4974	189	15	p)-open	p)-open	ADJ
ejpam-4974	189	16	,	,	PUNCT
ejpam-4974	189	17	b	b	NOUN
ejpam-4974	189	18	∩	∩	NOUN
ejpam-4974	189	19	[	[	X
ejpam-4974	189	20	f(x	f(x	PROPN
ejpam-4974	189	21	−	−	PROPN
ejpam-4974	189	22	u	u	PROPN
ejpam-4974	189	23	(	(	PUNCT
ejpam-4974	189	24	λ	λ	PROPN
ejpam-4974	189	25	,	,	PUNCT
ejpam-4974	189	26	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	189	27	,	,	PUNCT
ejpam-4974	189	28	p	p	NOUN
ejpam-4974	189	29	)	)	PUNCT
ejpam-4974	189	30	=	=	NOUN
ejpam-4974	189	31	∅	∅	NOUN
ejpam-4974	189	32	by	by	ADP
ejpam-4974	189	33	(	(	PUNCT
ejpam-4974	189	34	3	3	NUM
ejpam-4974	189	35	)	)	PUNCT
ejpam-4974	189	36	.	.	PUNCT
ejpam-4974	190	1	put	put	VERB
ejpam-4974	190	2	v	v	NUM
ejpam-4974	190	3	=	=	SYM
ejpam-4974	190	4	y	y	NOUN
ejpam-4974	190	5	−	−	PROPN
ejpam-4974	191	1	[	[	X
ejpam-4974	191	2	f(x	f(x	PROPN
ejpam-4974	191	3	−	−	PROPN
ejpam-4974	191	4	u	u	PROPN
ejpam-4974	191	5	(	(	PUNCT
ejpam-4974	191	6	λ	λ	PROPN
ejpam-4974	191	7	,	,	PUNCT
ejpam-4974	191	8	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	191	9	,	,	PUNCT
ejpam-4974	191	10	p	p	NOUN
ejpam-4974	191	11	)	)	PUNCT
ejpam-4974	191	12	.	.	PUNCT
ejpam-4974	192	1	then	then	ADV
ejpam-4974	192	2	,	,	PUNCT
ejpam-4974	192	3	v	v	NOUN
ejpam-4974	192	4	is	be	AUX
ejpam-4974	192	5	β(λ	β(λ	NOUN
ejpam-4974	192	6	,	,	PUNCT
ejpam-4974	192	7	p)-open	p)-open	VERB
ejpam-4974	192	8	set	set	VERB
ejpam-4974	192	9	of	of	ADP
ejpam-4974	192	10	y	y	PRON
ejpam-4974	192	11	such	such	ADJ
ejpam-4974	192	12	that	that	PRON
ejpam-4974	192	13	b	b	PROPN
ejpam-4974	192	14	⊆	⊆	NUM
ejpam-4974	192	15	v	v	NOUN
ejpam-4974	192	16	and	and	CCONJ
ejpam-4974	192	17	f−1(v	f−1(v	NOUN
ejpam-4974	192	18	)	)	PUNCT
ejpam-4974	193	1	⊆	⊆	NUM
ejpam-4974	193	2	x	x	SYM
ejpam-4974	193	3	−	−	NOUN
ejpam-4974	193	4	f−1([f(x	f−1([f(x	SYM
ejpam-4974	193	5	−	−	PROPN
ejpam-4974	193	6	u	u	NOUN
ejpam-4974	193	7	(	(	PUNCT
ejpam-4974	193	8	λ	λ	PROPN
ejpam-4974	193	9	,	,	PUNCT
ejpam-4974	193	10	p))]β(λ	p))]β(λ	PROPN
ejpam-4974	193	11	,	,	PUNCT
ejpam-4974	193	12	p	p	NOUN
ejpam-4974	193	13	)	)	PUNCT
ejpam-4974	193	14	)	)	PUNCT
ejpam-4974	194	1	⊆	⊆	NUM
ejpam-4974	194	2	x	x	SYM
ejpam-4974	194	3	−	−	NOUN
ejpam-4974	194	4	f−1(f(x	f−1(f(x	NOUN
ejpam-4974	194	5	−	−	PROPN
ejpam-4974	194	6	u	u	PROPN
ejpam-4974	194	7	(	(	PUNCT
ejpam-4974	194	8	λ	λ	PROPN
ejpam-4974	194	9	,	,	PUNCT
ejpam-4974	194	10	p	p	NOUN
ejpam-4974	194	11	)	)	PUNCT
ejpam-4974	194	12	)	)	PUNCT
ejpam-4974	194	13	)	)	PUNCT
ejpam-4974	195	1	⊆	⊆	X
ejpam-4974	195	2	u	u	NOUN
ejpam-4974	195	3	(	(	PUNCT
ejpam-4974	195	4	λ	λ	PROPN
ejpam-4974	195	5	,	,	PUNCT
ejpam-4974	195	6	p	p	NOUN
ejpam-4974	195	7	)	)	PUNCT
ejpam-4974	195	8	.	.	PUNCT
ejpam-4974	196	1	(	(	PUNCT
ejpam-4974	196	2	7	7	X
ejpam-4974	196	3	)	)	PUNCT
ejpam-4974	196	4	⇒	⇒	NOUN
ejpam-4974	196	5	(	(	PUNCT
ejpam-4974	196	6	1	1	NUM
ejpam-4974	196	7	):	):	PUNCT
ejpam-4974	196	8	it	it	PRON
ejpam-4974	196	9	is	be	AUX
ejpam-4974	196	10	suffices	suffice	NOUN
ejpam-4974	196	11	see	see	VERB
ejpam-4974	196	12	that	that	SCONJ
ejpam-4974	196	13	u	u	PROPN
ejpam-4974	196	14	θ(λ	θ(λ	PROPN
ejpam-4974	196	15	,	,	PUNCT
ejpam-4974	196	16	p	p	NOUN
ejpam-4974	196	17	)	)	PUNCT
ejpam-4974	196	18	=	=	SYM
ejpam-4974	196	19	u	u	NOUN
ejpam-4974	196	20	(	(	PUNCT
ejpam-4974	196	21	λ	λ	PROPN
ejpam-4974	196	22	,	,	PUNCT
ejpam-4974	196	23	p	p	NOUN
ejpam-4974	196	24	)	)	PUNCT
ejpam-4974	196	25	for	for	ADP
ejpam-4974	196	26	every	every	DET
ejpam-4974	196	27	(	(	PUNCT
ejpam-4974	196	28	λ	λ	NOUN
ejpam-4974	196	29	,	,	PUNCT
ejpam-4974	196	30	p)-open	p)-open	VERB
ejpam-4974	196	31	set	set	VERB
ejpam-4974	196	32	u	u	NOUN
ejpam-4974	196	33	of	of	ADP
ejpam-4974	196	34	x.	x.	PROPN
ejpam-4974	196	35	(	(	PUNCT
ejpam-4974	196	36	5	5	NUM
ejpam-4974	196	37	)	)	PUNCT
ejpam-4974	196	38	⇒	⇒	NOUN
ejpam-4974	196	39	(	(	PUNCT
ejpam-4974	196	40	1	1	NUM
ejpam-4974	196	41	):	):	PUNCT
ejpam-4974	196	42	let	let	VERB
ejpam-4974	196	43	k	k	PRON
ejpam-4974	196	44	be	be	AUX
ejpam-4974	196	45	any	any	DET
ejpam-4974	196	46	(	(	PUNCT
ejpam-4974	196	47	λ	λ	PROPN
ejpam-4974	196	48	,	,	PUNCT
ejpam-4974	196	49	p)-closed	p)-close	VERB
ejpam-4974	196	50	set	set	VERB
ejpam-4974	196	51	u	u	NOUN
ejpam-4974	196	52	of	of	ADP
ejpam-4974	196	53	x	x	PUNCT
ejpam-4974	196	54	and	and	CCONJ
ejpam-4974	196	55	y	y	PROPN
ejpam-4974	196	56	∈	∈	PROPN
ejpam-4974	196	57	y	y	PROPN
ejpam-4974	196	58	−	−	PROPN
ejpam-4974	196	59	f(k	f(k	PROPN
ejpam-4974	196	60	)	)	PUNCT
ejpam-4974	196	61	.	.	PUNCT
ejpam-4974	197	1	since	since	SCONJ
ejpam-4974	197	2	f−1(y	f−1(y	PROPN
ejpam-4974	197	3	)	)	PUNCT
ejpam-4974	197	4	⊆	⊆	NUM
ejpam-4974	197	5	x	x	PUNCT
ejpam-4974	197	6	−k	−k	PROPN
ejpam-4974	197	7	,	,	PUNCT
ejpam-4974	197	8	there	there	PRON
ejpam-4974	197	9	exists	exist	VERB
ejpam-4974	197	10	a	a	DET
ejpam-4974	197	11	β(λ	β(λ	NOUN
ejpam-4974	197	12	,	,	PUNCT
ejpam-4974	197	13	p)-open	p)-open	VERB
ejpam-4974	197	14	set	set	VERB
ejpam-4974	197	15	v	v	NOUN
ejpam-4974	197	16	of	of	ADP
ejpam-4974	197	17	y	y	PRON
ejpam-4974	197	18	such	such	ADJ
ejpam-4974	197	19	that	that	SCONJ
ejpam-4974	197	20	y	y	PROPN
ejpam-4974	197	21	∈	∈	PROPN
ejpam-4974	197	22	v	v	NOUN
ejpam-4974	197	23	and	and	CCONJ
ejpam-4974	197	24	f−1(v	f−1(v	NOUN
ejpam-4974	197	25	)	)	PUNCT
ejpam-4974	198	1	⊆	⊆	NUM
ejpam-4974	199	1	[	[	X
ejpam-4974	199	2	x	x	SYM
ejpam-4974	199	3	−k](λ	−k](λ	NUM
ejpam-4974	199	4	,	,	PUNCT
ejpam-4974	199	5	p	p	NOUN
ejpam-4974	199	6	)	)	PUNCT
ejpam-4974	199	7	=	=	PUNCT
ejpam-4974	199	8	x	x	SYM
ejpam-4974	199	9	−k(λ	−k(λ	NOUN
ejpam-4974	199	10	,	,	PUNCT
ejpam-4974	199	11	p	p	NOUN
ejpam-4974	199	12	)	)	PUNCT
ejpam-4974	199	13	by	by	ADP
ejpam-4974	199	14	(	(	PUNCT
ejpam-4974	199	15	5	5	NUM
ejpam-4974	199	16	)	)	PUNCT
ejpam-4974	199	17	.	.	PUNCT
ejpam-4974	200	1	thus	thus	ADV
ejpam-4974	200	2	,	,	PUNCT
ejpam-4974	200	3	v	v	ADP
ejpam-4974	200	4	∩	∩	ADJ
ejpam-4974	200	5	f(k(λ	f(k(λ	NOUN
ejpam-4974	200	6	,	,	PUNCT
ejpam-4974	200	7	p	p	NOUN
ejpam-4974	200	8	)	)	PUNCT
ejpam-4974	200	9	)	)	PUNCT
ejpam-4974	201	1	=	=	NOUN
ejpam-4974	201	2	∅	∅	NOUN
ejpam-4974	201	3	and	and	CCONJ
ejpam-4974	201	4	hence	hence	ADV
ejpam-4974	201	5	y	y	PROPN
ejpam-4974	201	6	∈	∈	PROPN
ejpam-4974	201	7	y	y	NOUN
ejpam-4974	201	8	−	−	PROPN
ejpam-4974	202	1	[	[	X
ejpam-4974	202	2	f(k(λ	f(k(λ	NOUN
ejpam-4974	202	3	,	,	PUNCT
ejpam-4974	202	4	p	p	NOUN
ejpam-4974	202	5	)	)	PUNCT
ejpam-4974	202	6	)	)	PUNCT
ejpam-4974	202	7	]	]	PUNCT
ejpam-4974	202	8	β(λ	β(λ	X
ejpam-4974	202	9	,	,	PUNCT
ejpam-4974	202	10	p	p	NOUN
ejpam-4974	202	11	)	)	PUNCT
ejpam-4974	202	12	.	.	PUNCT
ejpam-4974	203	1	therefore	therefore	ADV
ejpam-4974	203	2	,	,	PUNCT
ejpam-4974	203	3	[	[	X
ejpam-4974	203	4	f(k(λ	f(k(λ	NOUN
ejpam-4974	203	5	,	,	PUNCT
ejpam-4974	203	6	p	p	NOUN
ejpam-4974	203	7	)	)	PUNCT
ejpam-4974	203	8	)	)	PUNCT
ejpam-4974	203	9	]	]	PUNCT
ejpam-4974	204	1	β(λ	β(λ	X
ejpam-4974	204	2	,	,	PUNCT
ejpam-4974	204	3	p	p	NOUN
ejpam-4974	204	4	)	)	PUNCT
ejpam-4974	204	5	⊆	⊆	NUM
ejpam-4974	204	6	f(k	f(k	VERB
ejpam-4974	204	7	)	)	PUNCT
ejpam-4974	204	8	.	.	PUNCT
ejpam-4974	205	1	this	this	PRON
ejpam-4974	205	2	shows	show	VERB
ejpam-4974	205	3	that	that	SCONJ
ejpam-4974	205	4	f	f	PROPN
ejpam-4974	205	5	is	be	AUX
ejpam-4974	205	6	weakly	weakly	ADJ
ejpam-4974	205	7	β(λ	β(λ	NOUN
ejpam-4974	205	8	,	,	PUNCT
ejpam-4974	205	9	p)-closed	p)-close	VERB
ejpam-4974	205	10	.	.	PUNCT
ejpam-4974	206	1	(	(	PUNCT
ejpam-4974	206	2	7	7	X
ejpam-4974	206	3	)	)	PUNCT
ejpam-4974	206	4	⇒	⇒	NOUN
ejpam-4974	206	5	(	(	PUNCT
ejpam-4974	206	6	8)	8)	NUM
ejpam-4974	206	7	:	:	PUNCT
ejpam-4974	206	8	this	this	PRON
ejpam-4974	206	9	is	be	AUX
ejpam-4974	206	10	obvious	obvious	ADJ
ejpam-4974	206	11	since	since	SCONJ
ejpam-4974	206	12	u	u	NOUN
ejpam-4974	206	13	θ(λ	θ(λ	PROPN
ejpam-4974	206	14	,	,	PUNCT
ejpam-4974	206	15	p	p	NOUN
ejpam-4974	206	16	)	)	PUNCT
ejpam-4974	206	17	=	=	SYM
ejpam-4974	206	18	u	u	NOUN
ejpam-4974	206	19	(	(	PUNCT
ejpam-4974	206	20	λ	λ	PROPN
ejpam-4974	206	21	,	,	PUNCT
ejpam-4974	206	22	p	p	NOUN
ejpam-4974	206	23	)	)	PUNCT
ejpam-4974	206	24	for	for	ADP
ejpam-4974	206	25	every	every	DET
ejpam-4974	206	26	β(λ	β(λ	NOUN
ejpam-4974	206	27	,	,	PUNCT
ejpam-4974	206	28	p)-open	p)-open	VERB
ejpam-4974	206	29	set	set	VERB
ejpam-4974	206	30	u	u	NOUN
ejpam-4974	206	31	of	of	ADP
ejpam-4974	206	32	x.	x.	NOUN
ejpam-4974	206	33	the	the	DET
ejpam-4974	206	34	following	follow	VERB
ejpam-4974	206	35	theorem	theorem	VERB
ejpam-4974	206	36	the	the	DET
ejpam-4974	206	37	proof	proof	NOUN
ejpam-4974	206	38	is	be	AUX
ejpam-4974	206	39	mostly	mostly	ADV
ejpam-4974	206	40	straightforward	straightforward	ADJ
ejpam-4974	206	41	and	and	CCONJ
ejpam-4974	206	42	is	be	AUX
ejpam-4974	206	43	omitted	omit	VERB
ejpam-4974	206	44	.	.	PUNCT
ejpam-4974	207	1	references	reference	NOUN
ejpam-4974	207	2	254	254	NUM
ejpam-4974	207	3	theorem	theorem	NOUN
ejpam-4974	207	4	4	4	NUM
ejpam-4974	207	5	.	.	X
ejpam-4974	207	6	for	for	ADP
ejpam-4974	207	7	a	a	DET
ejpam-4974	207	8	function	function	NOUN
ejpam-4974	207	9	f	f	NOUN
ejpam-4974	207	10	:	:	PUNCT
ejpam-4974	207	11	(	(	PUNCT
ejpam-4974	207	12	x	x	X
ejpam-4974	207	13	,	,	PUNCT
ejpam-4974	207	14	τ	τ	X
ejpam-4974	207	15	)	)	PUNCT
ejpam-4974	207	16	→	→	SYM
ejpam-4974	207	17	(	(	PUNCT
ejpam-4974	207	18	y	y	PROPN
ejpam-4974	207	19	,	,	PUNCT
ejpam-4974	207	20	σ	σ	PROPN
ejpam-4974	207	21	)	)	PUNCT
ejpam-4974	207	22	,	,	PUNCT
ejpam-4974	207	23	the	the	DET
ejpam-4974	207	24	following	follow	VERB
ejpam-4974	207	25	properties	property	NOUN
ejpam-4974	207	26	are	be	AUX
ejpam-4974	207	27	equivalent	equivalent	ADJ
ejpam-4974	207	28	:	:	PUNCT
ejpam-4974	207	29	(	(	PUNCT
ejpam-4974	207	30	1	1	X
ejpam-4974	207	31	)	)	PUNCT
ejpam-4974	207	32	f	f	PROPN
ejpam-4974	207	33	is	be	AUX
ejpam-4974	207	34	weakly	weakly	ADJ
ejpam-4974	207	35	β(λ	β(λ	NOUN
ejpam-4974	207	36	,	,	PUNCT
ejpam-4974	207	37	p)-closed	p)-close	VERB
ejpam-4974	207	38	;	;	PUNCT
ejpam-4974	207	39	(	(	PUNCT
ejpam-4974	207	40	2	2	X
ejpam-4974	207	41	)	)	PUNCT
ejpam-4974	208	1	[	[	X
ejpam-4974	208	2	f(k(λ	f(k(λ	NOUN
ejpam-4974	208	3	,	,	PUNCT
ejpam-4974	208	4	p	p	NOUN
ejpam-4974	208	5	)	)	PUNCT
ejpam-4974	208	6	)	)	PUNCT
ejpam-4974	208	7	]	]	PUNCT
ejpam-4974	209	1	β(λ	β(λ	X
ejpam-4974	209	2	,	,	PUNCT
ejpam-4974	209	3	p	p	NOUN
ejpam-4974	209	4	)	)	PUNCT
ejpam-4974	209	5	⊆	⊆	NUM
ejpam-4974	209	6	f(k	f(k	VERB
ejpam-4974	209	7	)	)	PUNCT
ejpam-4974	209	8	every	every	DET
ejpam-4974	209	9	β(λ	β(λ	PROPN
ejpam-4974	209	10	,	,	PUNCT
ejpam-4974	209	11	p)-closed	p)-close	VERB
ejpam-4974	209	12	set	set	NOUN
ejpam-4974	209	13	k	k	PROPN
ejpam-4974	209	14	of	of	ADP
ejpam-4974	209	15	x	x	PROPN
ejpam-4974	209	16	;	;	PUNCT
ejpam-4974	209	17	(	(	PUNCT
ejpam-4974	209	18	3	3	X
ejpam-4974	209	19	)	)	PUNCT
ejpam-4974	210	1	[	[	X
ejpam-4974	210	2	f(k(λ	f(k(λ	NOUN
ejpam-4974	210	3	,	,	PUNCT
ejpam-4974	210	4	p	p	NOUN
ejpam-4974	210	5	)	)	PUNCT
ejpam-4974	210	6	)	)	PUNCT
ejpam-4974	210	7	]	]	PUNCT
ejpam-4974	211	1	β(λ	β(λ	X
ejpam-4974	211	2	,	,	PUNCT
ejpam-4974	211	3	p	p	NOUN
ejpam-4974	211	4	)	)	PUNCT
ejpam-4974	211	5	⊆	⊆	NUM
ejpam-4974	211	6	f(k	f(k	VERB
ejpam-4974	211	7	)	)	PUNCT
ejpam-4974	211	8	every	every	DET
ejpam-4974	211	9	α(λ	α(λ	PROPN
ejpam-4974	211	10	,	,	PUNCT
ejpam-4974	211	11	p)-closed	p)-close	VERB
ejpam-4974	211	12	set	set	NOUN
ejpam-4974	211	13	k	k	PROPN
ejpam-4974	211	14	of	of	ADP
ejpam-4974	211	15	x.	x.	PROPN
ejpam-4974	211	16	acknowledgements	acknowledgement	NOUN
ejpam-4974	211	17	this	this	DET
ejpam-4974	211	18	research	research	NOUN
ejpam-4974	211	19	project	project	NOUN
ejpam-4974	211	20	was	be	AUX
ejpam-4974	211	21	financially	financially	ADV
ejpam-4974	211	22	supported	support	VERB
ejpam-4974	211	23	by	by	ADP
ejpam-4974	211	24	mahasarakham	mahasarakham	PROPN
ejpam-4974	211	25	university	university	PROPN
ejpam-4974	211	26	.	.	PUNCT
ejpam-4974	212	1	references	reference	NOUN
ejpam-4974	212	2	[	[	X
ejpam-4974	212	3	1	1	NUM
ejpam-4974	212	4	]	]	PUNCT
ejpam-4974	212	5	c.	c.	PROPN
ejpam-4974	212	6	boonpok	boonpok	PROPN
ejpam-4974	212	7	and	and	CCONJ
ejpam-4974	212	8	j.	j.	PROPN
ejpam-4974	212	9	khampakdee	khampakdee	PROPN
ejpam-4974	212	10	.	.	PUNCT
ejpam-4974	213	1	(	(	PUNCT
ejpam-4974	213	2	λ	λ	NOUN
ejpam-4974	213	3	,	,	PUNCT
ejpam-4974	213	4	sp)-open	sp)-open	ADJ
ejpam-4974	213	5	sets	set	NOUN
ejpam-4974	213	6	in	in	ADP
ejpam-4974	213	7	topological	topological	ADJ
ejpam-4974	213	8	spaces	space	NOUN
ejpam-4974	213	9	.	.	PUNCT
ejpam-4974	214	1	european	european	ADJ
ejpam-4974	214	2	journal	journal	PROPN
ejpam-4974	214	3	of	of	ADP
ejpam-4974	214	4	pure	pure	ADJ
ejpam-4974	214	5	and	and	CCONJ
ejpam-4974	214	6	applied	applied	ADJ
ejpam-4974	214	7	mathematics	mathematic	NOUN
ejpam-4974	214	8	,	,	PUNCT
ejpam-4974	214	9	15(2):572–588	15(2):572–588	NUM
ejpam-4974	214	10	,	,	PUNCT
ejpam-4974	214	11	2022	2022	NUM
ejpam-4974	214	12	.	.	PUNCT
ejpam-4974	215	1	[	[	X
ejpam-4974	215	2	2	2	NUM
ejpam-4974	215	3	]	]	PUNCT
ejpam-4974	215	4	c.	c.	PROPN
ejpam-4974	215	5	boonpok	boonpok	PROPN
ejpam-4974	215	6	and	and	CCONJ
ejpam-4974	215	7	n.	n.	PROPN
ejpam-4974	215	8	srisarakham	srisarakham	PROPN
ejpam-4974	215	9	.	.	PUNCT
ejpam-4974	216	1	θp(λ	θp(λ	NOUN
ejpam-4974	216	2	,	,	PUNCT
ejpam-4974	216	3	p)-open	p)-open	NOUN
ejpam-4974	216	4	functions	function	NOUN
ejpam-4974	216	5	and	and	CCONJ
ejpam-4974	216	6	θp(λ	θp(λ	NOUN
ejpam-4974	216	7	,	,	PUNCT
ejpam-4974	216	8	p)-closed	p)-close	VERB
ejpam-4974	216	9	functions	function	NOUN
ejpam-4974	216	10	.	.	PUNCT
ejpam-4974	217	1	asia	asia	PROPN
ejpam-4974	217	2	pacific	pacific	PROPN
ejpam-4974	217	3	journal	journal	PROPN
ejpam-4974	217	4	of	of	ADP
ejpam-4974	217	5	mathematics	mathematic	NOUN
ejpam-4974	217	6	,	,	PUNCT
ejpam-4974	217	7	10:48	10:48	NUM
ejpam-4974	217	8	,	,	PUNCT
ejpam-4974	217	9	2023	2023	NUM
ejpam-4974	217	10	.	.	PUNCT
ejpam-4974	218	1	[	[	X
ejpam-4974	218	2	3	3	X
ejpam-4974	218	3	]	]	PUNCT
ejpam-4974	218	4	c.	c.	PROPN
ejpam-4974	218	5	boonpok	boonpok	PROPN
ejpam-4974	218	6	and	and	CCONJ
ejpam-4974	218	7	m.	m.	NOUN
ejpam-4974	218	8	thongmoon	thongmoon	NOUN
ejpam-4974	218	9	.	.	PUNCT
ejpam-4974	219	1	weakly	weakly	ADJ
ejpam-4974	219	2	p(λ	p(λ	NOUN
ejpam-4974	219	3	,	,	PUNCT
ejpam-4974	219	4	p)-open	p)-open	NOUN
ejpam-4974	219	5	functions	function	NOUN
ejpam-4974	219	6	and	and	CCONJ
ejpam-4974	219	7	weakly	weakly	ADJ
ejpam-4974	219	8	p(λ	p(λ	NOUN
ejpam-4974	219	9	,	,	PUNCT
ejpam-4974	219	10	p)closed	p)close	VERB
ejpam-4974	219	11	functions	function	NOUN
ejpam-4974	219	12	.	.	PUNCT
ejpam-4974	220	1	international	international	ADJ
ejpam-4974	220	2	journal	journal	NOUN
ejpam-4974	220	3	of	of	ADP
ejpam-4974	220	4	analysis	analysis	NOUN
ejpam-4974	220	5	and	and	CCONJ
ejpam-4974	220	6	applications	application	NOUN
ejpam-4974	220	7	,	,	PUNCT
ejpam-4974	220	8	22:10	22:10	NUM
ejpam-4974	220	9	,	,	PUNCT
ejpam-4974	220	10	2024	2024	NUM
ejpam-4974	220	11	.	.	PUNCT
ejpam-4974	221	1	[	[	X
ejpam-4974	221	2	4	4	NUM
ejpam-4974	221	3	]	]	PUNCT
ejpam-4974	221	4	c.	c.	PROPN
ejpam-4974	221	5	boonpok	boonpok	PROPN
ejpam-4974	221	6	and	and	CCONJ
ejpam-4974	221	7	c.	c.	PROPN
ejpam-4974	221	8	viriyapong	viriyapong	PROPN
ejpam-4974	221	9	.	.	PUNCT
ejpam-4974	222	1	on	on	ADP
ejpam-4974	222	2	(	(	PUNCT
ejpam-4974	222	3	λ	λ	PROPN
ejpam-4974	222	4	,	,	PUNCT
ejpam-4974	222	5	p)-closed	p)-close	VERB
ejpam-4974	222	6	sets	set	NOUN
ejpam-4974	222	7	and	and	CCONJ
ejpam-4974	222	8	the	the	DET
ejpam-4974	222	9	related	related	ADJ
ejpam-4974	222	10	notions	notion	NOUN
ejpam-4974	222	11	in	in	ADP
ejpam-4974	222	12	topological	topological	ADJ
ejpam-4974	222	13	spaces	space	NOUN
ejpam-4974	222	14	.	.	PUNCT
ejpam-4974	223	1	european	european	ADJ
ejpam-4974	223	2	journal	journal	PROPN
ejpam-4974	223	3	of	of	ADP
ejpam-4974	223	4	pure	pure	ADJ
ejpam-4974	223	5	and	and	CCONJ
ejpam-4974	223	6	applied	applied	ADJ
ejpam-4974	223	7	mathematics	mathematic	NOUN
ejpam-4974	223	8	,	,	PUNCT
ejpam-4974	223	9	15(2):415	15(2):415	PROPN
ejpam-4974	223	10	–	–	PUNCT
ejpam-4974	223	11	436	436	NUM
ejpam-4974	223	12	,	,	PUNCT
ejpam-4974	223	13	2022	2022	NUM
ejpam-4974	223	14	.	.	PUNCT
ejpam-4974	224	1	[	[	X
ejpam-4974	224	2	5	5	X
ejpam-4974	224	3	]	]	PUNCT
ejpam-4974	224	4	c.	c.	PROPN
ejpam-4974	224	5	boonpok	boonpok	PROPN
ejpam-4974	224	6	and	and	CCONJ
ejpam-4974	224	7	c.	c.	PROPN
ejpam-4974	224	8	viriyapong	viriyapong	PROPN
ejpam-4974	224	9	.	.	PUNCT
ejpam-4974	225	1	on	on	ADP
ejpam-4974	225	2	some	some	DET
ejpam-4974	225	3	forms	form	NOUN
ejpam-4974	225	4	of	of	ADP
ejpam-4974	225	5	closed	closed	ADJ
ejpam-4974	225	6	sets	set	NOUN
ejpam-4974	225	7	and	and	CCONJ
ejpam-4974	225	8	related	related	ADJ
ejpam-4974	225	9	topics	topic	NOUN
ejpam-4974	225	10	.	.	PUNCT
ejpam-4974	226	1	european	european	ADJ
ejpam-4974	226	2	journal	journal	PROPN
ejpam-4974	226	3	of	of	ADP
ejpam-4974	226	4	pure	pure	ADJ
ejpam-4974	226	5	and	and	CCONJ
ejpam-4974	226	6	applied	applied	ADJ
ejpam-4974	226	7	mathematics	mathematic	NOUN
ejpam-4974	226	8	,	,	PUNCT
ejpam-4974	226	9	16(1):336–362	16(1):336–362	NUM
ejpam-4974	226	10	,	,	PUNCT
ejpam-4974	226	11	2023	2023	NUM
ejpam-4974	226	12	.	.	PUNCT
ejpam-4974	227	1	[	[	X
ejpam-4974	227	2	6	6	NUM
ejpam-4974	227	3	]	]	PUNCT
ejpam-4974	227	4	m.	m.	NOUN
ejpam-4974	227	5	caldas	caldas	PROPN
ejpam-4974	227	6	and	and	CCONJ
ejpam-4974	227	7	g.	g.	PROPN
ejpam-4974	227	8	navalagi	navalagi	PROPN
ejpam-4974	227	9	.	.	PUNCT
ejpam-4974	228	1	on	on	ADP
ejpam-4974	228	2	weak	weak	ADJ
ejpam-4974	228	3	forms	form	NOUN
ejpam-4974	228	4	of	of	ADP
ejpam-4974	228	5	β	β	NOUN
ejpam-4974	228	6	-	-	ADJ
ejpam-4974	228	7	open	open	ADJ
ejpam-4974	228	8	and	and	CCONJ
ejpam-4974	228	9	β	β	NOUN
ejpam-4974	228	10	-	-	PUNCT
ejpam-4974	228	11	closed	closed	ADJ
ejpam-4974	228	12	functions	function	NOUN
ejpam-4974	228	13	.	.	PUNCT
ejpam-4974	229	1	annals	annal	NOUN
ejpam-4974	229	2	of	of	ADP
ejpam-4974	229	3	the	the	DET
ejpam-4974	229	4	alexandru	alexandru	PROPN
ejpam-4974	229	5	ioan	ioan	PROPN
ejpam-4974	229	6	cuza	cuza	PROPN
ejpam-4974	229	7	university	university	PROPN
ejpam-4974	229	8	.	.	PUNCT
ejpam-4974	230	1	mathematics	mathematic	NOUN
ejpam-4974	230	2	,	,	PUNCT
ejpam-4974	230	3	49:115–128	49:115–128	PROPN
ejpam-4974	230	4	,	,	PUNCT
ejpam-4974	230	5	2003	2003	NUM
ejpam-4974	230	6	.	.	PUNCT
ejpam-4974	231	1	[	[	X
ejpam-4974	231	2	7	7	X
ejpam-4974	231	3	]	]	PUNCT
ejpam-4974	231	4	m.	m.	NOUN
ejpam-4974	231	5	caldas	caldas	PROPN
ejpam-4974	231	6	and	and	CCONJ
ejpam-4974	231	7	g.	g.	PROPN
ejpam-4974	231	8	navalagi	navalagi	PROPN
ejpam-4974	231	9	.	.	PUNCT
ejpam-4974	232	1	on	on	ADP
ejpam-4974	232	2	weak	weak	ADJ
ejpam-4974	232	3	forms	form	NOUN
ejpam-4974	232	4	of	of	ADP
ejpam-4974	232	5	preopen	preopen	ADJ
ejpam-4974	232	6	and	and	CCONJ
ejpam-4974	232	7	preclosed	preclose	VERB
ejpam-4974	232	8	functions	function	NOUN
ejpam-4974	232	9	.	.	PUNCT
ejpam-4974	233	1	archivum	archivum	PROPN
ejpam-4974	233	2	mathematicum	mathematicum	PROPN
ejpam-4974	233	3	,	,	PUNCT
ejpam-4974	233	4	40:119–128	40:119–128	PROPN
ejpam-4974	233	5	,	,	PUNCT
ejpam-4974	233	6	2004	2004	NUM
ejpam-4974	233	7	.	.	PUNCT
ejpam-4974	234	1	[	[	X
ejpam-4974	234	2	8	8	NUM
ejpam-4974	234	3	]	]	PUNCT
ejpam-4974	234	4	m.	m.	NOUN
ejpam-4974	234	5	caldas	caldas	PROPN
ejpam-4974	234	6	and	and	CCONJ
ejpam-4974	234	7	g.	g.	PROPN
ejpam-4974	234	8	navalagi	navalagi	PROPN
ejpam-4974	234	9	.	.	PUNCT
ejpam-4974	235	1	on	on	ADP
ejpam-4974	235	2	weak	weak	ADJ
ejpam-4974	235	3	forms	form	NOUN
ejpam-4974	235	4	of	of	ADP
ejpam-4974	235	5	semi	semi	ADJ
ejpam-4974	235	6	-	-	ADJ
ejpam-4974	235	7	open	open	ADJ
ejpam-4974	235	8	and	and	CCONJ
ejpam-4974	235	9	semi	semi	ADJ
ejpam-4974	235	10	-	-	ADJ
ejpam-4974	235	11	closed	closed	ADJ
ejpam-4974	235	12	functions	function	NOUN
ejpam-4974	235	13	.	.	PUNCT
ejpam-4974	236	1	missouri	missouri	PROPN
ejpam-4974	236	2	journal	journal	PROPN
ejpam-4974	236	3	of	of	ADP
ejpam-4974	236	4	mathematical	mathematical	ADJ
ejpam-4974	236	5	sciences	science	NOUN
ejpam-4974	236	6	,	,	PUNCT
ejpam-4974	236	7	18(3):165–178	18(3):165–178	PROPN
ejpam-4974	236	8	,	,	PUNCT
ejpam-4974	236	9	2006	2006	NUM
ejpam-4974	236	10	.	.	PUNCT
ejpam-4974	237	1	[	[	X
ejpam-4974	237	2	9	9	NUM
ejpam-4974	237	3	]	]	X
ejpam-4974	237	4	n.	n.	NOUN
ejpam-4974	237	5	chutiman	chutiman	NOUN
ejpam-4974	237	6	and	and	CCONJ
ejpam-4974	237	7	c.	c.	PROPN
ejpam-4974	237	8	boonpok	boonpok	PROPN
ejpam-4974	237	9	.	.	PUNCT
ejpam-4974	238	1	some	some	DET
ejpam-4974	238	2	properties	property	NOUN
ejpam-4974	238	3	of	of	ADP
ejpam-4974	238	4	weakly	weakly	ADJ
ejpam-4974	238	5	b(λ	b(λ	NOUN
ejpam-4974	238	6	,	,	PUNCT
ejpam-4974	238	7	p)-open	p)-open	NOUN
ejpam-4974	238	8	functions	function	NOUN
ejpam-4974	238	9	.	.	PUNCT
ejpam-4974	239	1	international	international	ADJ
ejpam-4974	239	2	journal	journal	PROPN
ejpam-4974	239	3	of	of	ADP
ejpam-4974	239	4	mathematics	mathematic	NOUN
ejpam-4974	239	5	and	and	CCONJ
ejpam-4974	239	6	computer	computer	NOUN
ejpam-4974	239	7	science	science	NOUN
ejpam-4974	239	8	,	,	PUNCT
ejpam-4974	239	9	19(2):497–501	19(2):497–501	NUM
ejpam-4974	239	10	,	,	PUNCT
ejpam-4974	239	11	2024	2024	NUM
ejpam-4974	239	12	.	.	PUNCT
ejpam-4974	240	1	[	[	X
ejpam-4974	240	2	10	10	NUM
ejpam-4974	240	3	]	]	PUNCT
ejpam-4974	240	4	m.	m.	NOUN
ejpam-4974	240	5	ganster	ganster	NOUN
ejpam-4974	240	6	,	,	PUNCT
ejpam-4974	240	7	s.	s.	PROPN
ejpam-4974	240	8	jafari	jafari	PROPN
ejpam-4974	240	9	,	,	PUNCT
ejpam-4974	240	10	and	and	CCONJ
ejpam-4974	240	11	t.	t.	PROPN
ejpam-4974	240	12	noiri	noiri	PROPN
ejpam-4974	240	13	.	.	PUNCT
ejpam-4974	241	1	on	on	ADP
ejpam-4974	241	2	pre	pre	ADJ
ejpam-4974	241	3	-	-	ADJ
ejpam-4974	241	4	λ	λ	NOUN
ejpam-4974	241	5	-	-	NOUN
ejpam-4974	241	6	sets	set	NOUN
ejpam-4974	241	7	and	and	CCONJ
ejpam-4974	241	8	pre	pre	ADJ
ejpam-4974	241	9	-	-	ADJ
ejpam-4974	241	10	v	v	ADJ
ejpam-4974	241	11	-sets	-set	NOUN
ejpam-4974	241	12	.	.	PUNCT
ejpam-4974	242	1	acta	acta	PROPN
ejpam-4974	242	2	mathematica	mathematica	PROPN
ejpam-4974	242	3	hungarica	hungarica	PROPN
ejpam-4974	242	4	,	,	PUNCT
ejpam-4974	242	5	95:337–343	95:337–343	PROPN
ejpam-4974	242	6	,	,	PUNCT
ejpam-4974	242	7	2002	2002	NUM
ejpam-4974	242	8	.	.	PUNCT
ejpam-4974	243	1	[	[	X
ejpam-4974	243	2	11	11	NUM
ejpam-4974	243	3	]	]	X
ejpam-4974	243	4	j.	j.	PROPN
ejpam-4974	243	5	khampakdee	khampakdee	PROPN
ejpam-4974	243	6	and	and	CCONJ
ejpam-4974	243	7	c.	c.	PROPN
ejpam-4974	243	8	boonpok	boonpok	PROPN
ejpam-4974	243	9	.	.	PUNCT
ejpam-4974	244	1	properties	property	NOUN
ejpam-4974	244	2	of	of	ADP
ejpam-4974	244	3	(	(	PUNCT
ejpam-4974	244	4	λ	λ	PROPN
ejpam-4974	244	5	,	,	PUNCT
ejpam-4974	244	6	p)-closed	p)-close	VERB
ejpam-4974	244	7	functions	function	NOUN
ejpam-4974	244	8	.	.	PUNCT
ejpam-4974	245	1	international	international	ADJ
ejpam-4974	245	2	journal	journal	PROPN
ejpam-4974	245	3	of	of	ADP
ejpam-4974	245	4	mathematics	mathematic	NOUN
ejpam-4974	245	5	and	and	CCONJ
ejpam-4974	245	6	computer	computer	NOUN
ejpam-4974	245	7	science	science	NOUN
ejpam-4974	245	8	,	,	PUNCT
ejpam-4974	245	9	19(2):481–484	19(2):481–484	NUM
ejpam-4974	245	10	,	,	PUNCT
ejpam-4974	245	11	2024	2024	NUM
ejpam-4974	245	12	.	.	PUNCT
ejpam-4974	246	1	references	reference	NOUN
ejpam-4974	246	2	255	255	NUM
ejpam-4974	247	1	[	[	X
ejpam-4974	247	2	12	12	NUM
ejpam-4974	247	3	]	]	X
ejpam-4974	247	4	c.	c.	PROPN
ejpam-4974	247	5	khanarong	khanarong	PROPN
ejpam-4974	247	6	and	and	CCONJ
ejpam-4974	247	7	c.	c.	PROPN
ejpam-4974	247	8	boonpok	boonpok	PROPN
ejpam-4974	247	9	.	.	PUNCT
ejpam-4974	248	1	characterizations	characterization	NOUN
ejpam-4974	248	2	of	of	ADP
ejpam-4974	248	3	weakly	weakly	ADJ
ejpam-4974	248	4	δ(λ	δ(λ	PROPN
ejpam-4974	248	5	,	,	PUNCT
ejpam-4974	248	6	p)-closed	p)-close	VERB
ejpam-4974	248	7	functions	function	NOUN
ejpam-4974	248	8	.	.	PUNCT
ejpam-4974	249	1	international	international	ADJ
ejpam-4974	249	2	journal	journal	PROPN
ejpam-4974	249	3	of	of	ADP
ejpam-4974	249	4	mathematics	mathematic	NOUN
ejpam-4974	249	5	and	and	CCONJ
ejpam-4974	249	6	computer	computer	NOUN
ejpam-4974	249	7	science	science	NOUN
ejpam-4974	249	8	,	,	PUNCT
ejpam-4974	249	9	19(2):503–507	19(2):503–507	PROPN
ejpam-4974	249	10	,	,	PUNCT
ejpam-4974	249	11	2024	2024	NUM
ejpam-4974	249	12	.	.	PUNCT
ejpam-4974	250	1	[	[	X
ejpam-4974	250	2	13	13	NUM
ejpam-4974	250	3	]	]	PUNCT
ejpam-4974	250	4	a.	a.	NOUN
ejpam-4974	250	5	s.	s.	PROPN
ejpam-4974	250	6	mashhour	mashhour	PROPN
ejpam-4974	250	7	,	,	PUNCT
ejpam-4974	250	8	m.	m.	PROPN
ejpam-4974	250	9	e.	e.	PROPN
ejpam-4974	250	10	el	el	PROPN
ejpam-4974	250	11	-	-	PROPN
ejpam-4974	250	12	monsef	monsef	ADJ
ejpam-4974	250	13	,	,	PUNCT
ejpam-4974	250	14	and	and	CCONJ
ejpam-4974	250	15	s.	s.	PROPN
ejpam-4974	250	16	n.	n.	PROPN
ejpam-4974	250	17	el	el	PROPN
ejpam-4974	250	18	-	-	PROPN
ejpam-4974	250	19	deeb	deeb	PROPN
ejpam-4974	250	20	.	.	PUNCT
ejpam-4974	251	1	on	on	ADP
ejpam-4974	251	2	precontinuous	precontinuous	ADJ
ejpam-4974	251	3	and	and	CCONJ
ejpam-4974	251	4	weak	weak	ADJ
ejpam-4974	251	5	precontinuous	precontinuous	ADJ
ejpam-4974	251	6	mappings	mapping	NOUN
ejpam-4974	251	7	.	.	PUNCT
ejpam-4974	252	1	proceedings	proceeding	NOUN
ejpam-4974	252	2	of	of	ADP
ejpam-4974	252	3	the	the	DET
ejpam-4974	252	4	mathematical	mathematical	ADJ
ejpam-4974	252	5	and	and	CCONJ
ejpam-4974	252	6	physical	physical	ADJ
ejpam-4974	252	7	society	society	NOUN
ejpam-4974	252	8	of	of	ADP
ejpam-4974	252	9	egypt	egypt	PROPN
ejpam-4974	252	10	,	,	PUNCT
ejpam-4974	252	11	53:47–53	53:47–53	NUM
ejpam-4974	252	12	,	,	PUNCT
ejpam-4974	252	13	1982	1982	NUM
ejpam-4974	252	14	.	.	PUNCT
ejpam-4974	253	1	[	[	X
ejpam-4974	253	2	14	14	NUM
ejpam-4974	253	3	]	]	X
ejpam-4974	253	4	t.	t.	PROPN
ejpam-4974	253	5	noiri	noiri	PROPN
ejpam-4974	253	6	,	,	PUNCT
ejpam-4974	253	7	a.	a.	PROPN
ejpam-4974	253	8	al	al	PROPN
ejpam-4974	253	9	-	-	PUNCT
ejpam-4974	253	10	omari	omari	PROPN
ejpam-4974	253	11	,	,	PUNCT
ejpam-4974	253	12	and	and	CCONJ
ejpam-4974	253	13	m.	m.	PROPN
ejpam-4974	253	14	s.	s.	PROPN
ejpam-4974	253	15	m.	m.	PROPN
ejpam-4974	253	16	noorani	noorani	PROPN
ejpam-4974	253	17	.	.	PUNCT
ejpam-4974	254	1	weak	weak	ADJ
ejpam-4974	254	2	forms	form	NOUN
ejpam-4974	254	3	of	of	ADP
ejpam-4974	254	4	open	open	ADJ
ejpam-4974	254	5	and	and	CCONJ
ejpam-4974	254	6	closed	closed	ADJ
ejpam-4974	254	7	functions	function	NOUN
ejpam-4974	254	8	via	via	ADP
ejpam-4974	254	9	b	b	X
ejpam-4974	254	10	-	-	PUNCT
ejpam-4974	254	11	θ	θ	ADJ
ejpam-4974	254	12	-	-	PUNCT
ejpam-4974	254	13	open	open	ADJ
ejpam-4974	254	14	sets	set	NOUN
ejpam-4974	254	15	.	.	PUNCT
ejpam-4974	255	1	demonstratio	demonstratio	PROPN
ejpam-4974	255	2	mathematica	mathematica	PROPN
ejpam-4974	255	3	,	,	PUNCT
ejpam-4974	255	4	42(1):193–203	42(1):193–203	PROPN
ejpam-4974	255	5	,	,	PUNCT
ejpam-4974	255	6	2009	2009	NUM
ejpam-4974	255	7	.	.	PUNCT
ejpam-4974	256	1	[	[	X
ejpam-4974	256	2	15	15	NUM
ejpam-4974	256	3	]	]	X
ejpam-4974	256	4	d.	d.	PROPN
ejpam-4974	256	5	a.	a.	PROPN
ejpam-4974	256	6	rose	rise	VERB
ejpam-4974	256	7	.	.	PUNCT
ejpam-4974	257	1	on	on	ADP
ejpam-4974	257	2	weak	weak	ADJ
ejpam-4974	257	3	openness	openness	NOUN
ejpam-4974	257	4	and	and	CCONJ
ejpam-4974	257	5	almost	almost	ADV
ejpam-4974	257	6	openness	openness	NOUN
ejpam-4974	257	7	.	.	PUNCT
ejpam-4974	258	1	international	international	ADJ
ejpam-4974	258	2	journal	journal	NOUN
ejpam-4974	258	3	of	of	ADP
ejpam-4974	258	4	mathematics	mathematics	PROPN
ejpam-4974	258	5	and	and	CCONJ
ejpam-4974	258	6	mathematical	mathematical	ADJ
ejpam-4974	258	7	sciences	science	NOUN
ejpam-4974	258	8	,	,	PUNCT
ejpam-4974	258	9	7:35–40	7:35–40	NUM
ejpam-4974	258	10	,	,	PUNCT
ejpam-4974	258	11	1984	1984	NUM
ejpam-4974	258	12	.	.	PUNCT
ejpam-4974	259	1	[	[	X
ejpam-4974	259	2	16	16	NUM
ejpam-4974	259	3	]	]	X
ejpam-4974	259	4	d.	d.	PROPN
ejpam-4974	259	5	a.	a.	PROPN
ejpam-4974	259	6	rose	rise	VERB
ejpam-4974	259	7	and	and	CCONJ
ejpam-4974	259	8	d.	d.	PROPN
ejpam-4974	259	9	s.	s.	PROPN
ejpam-4974	260	1	janković.	janković.	PROPN
ejpam-4974	260	2	weakly	weakly	ADJ
ejpam-4974	260	3	closed	close	VERB
ejpam-4974	260	4	functions	function	NOUN
ejpam-4974	260	5	and	and	CCONJ
ejpam-4974	260	6	hausdorff	hausdorff	NOUN
ejpam-4974	260	7	spaces	space	NOUN
ejpam-4974	260	8	.	.	PUNCT
ejpam-4974	261	1	mathematische	mathematische	PROPN
ejpam-4974	261	2	nachrichten	nachrichten	PROPN
ejpam-4974	261	3	,	,	PUNCT
ejpam-4974	261	4	130:105–110	130:105–110	NUM
ejpam-4974	261	5	,	,	PUNCT
ejpam-4974	261	6	1987	1987	NUM
ejpam-4974	261	7	.	.	PUNCT
ejpam-4974	262	1	[	[	X
ejpam-4974	262	2	17	17	NUM
ejpam-4974	262	3	]	]	X
ejpam-4974	262	4	n.	n.	NOUN
ejpam-4974	262	5	srisarakham	srisarakham	PROPN
ejpam-4974	262	6	and	and	CCONJ
ejpam-4974	262	7	c.	c.	PROPN
ejpam-4974	262	8	boonpok	boonpok	PROPN
ejpam-4974	262	9	.	.	PUNCT
ejpam-4974	263	1	on	on	ADP
ejpam-4974	263	2	characterizations	characterization	NOUN
ejpam-4974	263	3	of	of	ADP
ejpam-4974	263	4	δp(λ	δp(λ	NOUN
ejpam-4974	263	5	,	,	PUNCT
ejpam-4974	263	6	s)-d1	s)-d1	NOUN
ejpam-4974	263	7	spaces	space	NOUN
ejpam-4974	263	8	.	.	PUNCT
ejpam-4974	264	1	international	international	ADJ
ejpam-4974	264	2	journal	journal	PROPN
ejpam-4974	264	3	of	of	ADP
ejpam-4974	264	4	mathematics	mathematic	NOUN
ejpam-4974	264	5	and	and	CCONJ
ejpam-4974	264	6	computer	computer	NOUN
ejpam-4974	264	7	science	science	NOUN
ejpam-4974	264	8	,	,	PUNCT
ejpam-4974	264	9	18(4):743–747	18(4):743–747	PROPN
ejpam-4974	264	10	,	,	PUNCT
ejpam-4974	264	11	2023	2023	NUM
ejpam-4974	264	12	.	.	PUNCT
ejpam-4974	265	1	[	[	X
ejpam-4974	265	2	18	18	NUM
ejpam-4974	265	3	]	]	X
ejpam-4974	265	4	n.	n.	NOUN
ejpam-4974	265	5	srisarakham	srisarakham	PROPN
ejpam-4974	265	6	and	and	CCONJ
ejpam-4974	265	7	c.	c.	PROPN
ejpam-4974	265	8	boonpok	boonpok	PROPN
ejpam-4974	265	9	.	.	PUNCT
ejpam-4974	266	1	on	on	ADP
ejpam-4974	266	2	weakly	weakly	ADJ
ejpam-4974	266	3	δ(λ	δ(λ	PROPN
ejpam-4974	266	4	,	,	PUNCT
ejpam-4974	266	5	p)-open	p)-open	VERB
ejpam-4974	266	6	functions	function	NOUN
ejpam-4974	266	7	.	.	PUNCT
ejpam-4974	267	1	international	international	ADJ
ejpam-4974	267	2	journal	journal	PROPN
ejpam-4974	267	3	of	of	ADP
ejpam-4974	267	4	mathematics	mathematic	NOUN
ejpam-4974	267	5	and	and	CCONJ
ejpam-4974	267	6	computer	computer	NOUN
ejpam-4974	267	7	science	science	NOUN
ejpam-4974	267	8	,	,	PUNCT
ejpam-4974	267	9	19(2):485–489	19(2):485–489	PROPN
ejpam-4974	267	10	,	,	PUNCT
ejpam-4974	267	11	2024	2024	NUM
ejpam-4974	267	12	.	.	PUNCT
ejpam-4974	268	1	[	[	X
ejpam-4974	268	2	19	19	NUM
ejpam-4974	268	3	]	]	PUNCT
ejpam-4974	268	4	m.	m.	NOUN
ejpam-4974	268	5	thongmoon	thongmoon	NOUN
ejpam-4974	268	6	and	and	CCONJ
ejpam-4974	268	7	c.	c.	PROPN
ejpam-4974	268	8	boonpok	boonpok	PROPN
ejpam-4974	268	9	.	.	PUNCT
ejpam-4974	269	1	strongly	strongly	ADV
ejpam-4974	269	2	θ(λ	θ(λ	PROPN
ejpam-4974	269	3	,	,	PUNCT
ejpam-4974	269	4	p)-continuous	p)-continuous	ADJ
ejpam-4974	269	5	functions	function	NOUN
ejpam-4974	269	6	.	.	PUNCT
ejpam-4974	270	1	international	international	ADJ
ejpam-4974	270	2	journal	journal	PROPN
ejpam-4974	270	3	of	of	ADP
ejpam-4974	270	4	mathematics	mathematic	NOUN
ejpam-4974	270	5	and	and	CCONJ
ejpam-4974	270	6	computer	computer	NOUN
ejpam-4974	270	7	science	science	NOUN
ejpam-4974	270	8	,	,	PUNCT
ejpam-4974	270	9	19(2):475–479	19(2):475–479	PROPN
ejpam-4974	270	10	,	,	PUNCT
ejpam-4974	270	11	2024	2024	NUM
ejpam-4974	270	12	.	.	PUNCT
ejpam-4974	271	1	[	[	X
ejpam-4974	271	2	20	20	NUM
ejpam-4974	271	3	]	]	X
ejpam-4974	271	4	n.	n.	PROPN
ejpam-4974	271	5	viriyapong	viriyapong	PROPN
ejpam-4974	272	1	and	and	CCONJ
ejpam-4974	272	2	c.	c.	PROPN
ejpam-4974	272	3	boonpok	boonpok	PROPN
ejpam-4974	272	4	.	.	PUNCT
ejpam-4974	273	1	on	on	ADP
ejpam-4974	273	2	(	(	PUNCT
ejpam-4974	273	3	λ	λ	INTJ
ejpam-4974	273	4	,	,	PUNCT
ejpam-4974	273	5	p)-extremally	p)-extremally	ADV
ejpam-4974	273	6	disconnected	disconnected	ADJ
ejpam-4974	273	7	spaces	space	NOUN
ejpam-4974	273	8	.	.	PUNCT
ejpam-4974	274	1	international	international	ADJ
ejpam-4974	274	2	journal	journal	PROPN
ejpam-4974	274	3	of	of	ADP
ejpam-4974	274	4	mathematics	mathematic	NOUN
ejpam-4974	274	5	and	and	CCONJ
ejpam-4974	274	6	computer	computer	NOUN
ejpam-4974	274	7	science	science	NOUN
ejpam-4974	274	8	,	,	PUNCT
ejpam-4974	274	9	18(2):289–293	18(2):289–293	NUM
ejpam-4974	274	10	,	,	PUNCT
ejpam-4974	274	11	2023	2023	NUM
ejpam-4974	274	12	.	.	PUNCT