id	sid	tid	token	lemma	pos
ejpam-4733	1	1	european	european	PROPN
ejpam-4733	1	2	journal	journal	PROPN
ejpam-4733	1	3	of	of	ADP
ejpam-4733	1	4	pure	pure	ADJ
ejpam-4733	1	5	and	and	CCONJ
ejpam-4733	1	6	applied	apply	VERB
ejpam-4733	1	7	mathematics	mathematic	NOUN
ejpam-4733	1	8	vol	vol	NOUN
ejpam-4733	1	9	.	.	PUNCT
ejpam-4733	2	1	16	16	NUM
ejpam-4733	2	2	,	,	PUNCT
ejpam-4733	2	3	no	no	INTJ
ejpam-4733	2	4	.	.	NOUN
ejpam-4733	2	5	3	3	NUM
ejpam-4733	2	6	,	,	PUNCT
ejpam-4733	2	7	2023	2023	NUM
ejpam-4733	2	8	,	,	PUNCT
ejpam-4733	2	9	1533	1533	NUM
ejpam-4733	2	10	-	-	SYM
ejpam-4733	2	11	1542	1542	NUM
ejpam-4733	2	12	issn	issn	PROPN
ejpam-4733	2	13	1307	1307	NUM
ejpam-4733	2	14	-	-	SYM
ejpam-4733	2	15	5543	5543	NUM
ejpam-4733	2	16	–	–	PUNCT
ejpam-4733	2	17	ejpam.com	ejpam.com	X
ejpam-4733	2	18	published	publish	VERB
ejpam-4733	2	19	by	by	ADP
ejpam-4733	2	20	new	new	PROPN
ejpam-4733	2	21	york	york	PROPN
ejpam-4733	2	22	business	business	PROPN
ejpam-4733	2	23	global	global	PROPN
ejpam-4733	2	24	δp(λ	δp(λ	NOUN
ejpam-4733	2	25	,	,	PUNCT
ejpam-4733	2	26	p)-open	p)-open	VERB
ejpam-4733	2	27	sets	set	NOUN
ejpam-4733	2	28	in	in	ADP
ejpam-4733	2	29	topological	topological	ADJ
ejpam-4733	2	30	spaces	space	NOUN
ejpam-4733	2	31	chawalit	chawalit	VERB
ejpam-4733	2	32	boonpok1	boonpok1	PROPN
ejpam-4733	2	33	,	,	PUNCT
ejpam-4733	2	34	montri	montri	PROPN
ejpam-4733	2	35	thongmoon1,∗	thongmoon1,∗	NOUN
ejpam-4733	2	36	1	1	NUM
ejpam-4733	2	37	mathematics	mathematic	NOUN
ejpam-4733	2	38	and	and	CCONJ
ejpam-4733	2	39	applied	apply	VERB
ejpam-4733	2	40	mathematics	mathematics	PROPN
ejpam-4733	2	41	research	research	NOUN
ejpam-4733	2	42	unit	unit	NOUN
ejpam-4733	2	43	,	,	PUNCT
ejpam-4733	2	44	department	department	NOUN
ejpam-4733	2	45	of	of	ADP
ejpam-4733	2	46	mathematics	mathematic	NOUN
ejpam-4733	2	47	,	,	PUNCT
ejpam-4733	2	48	faculty	faculty	NOUN
ejpam-4733	2	49	of	of	ADP
ejpam-4733	2	50	science	science	NOUN
ejpam-4733	2	51	,	,	PUNCT
ejpam-4733	2	52	mahasarakham	mahasarakham	PROPN
ejpam-4733	2	53	university	university	PROPN
ejpam-4733	2	54	,	,	PUNCT
ejpam-4733	2	55	maha	maha	PROPN
ejpam-4733	2	56	sarakham	sarakham	PROPN
ejpam-4733	2	57	,	,	PUNCT
ejpam-4733	2	58	44150	44150	NUM
ejpam-4733	2	59	,	,	PUNCT
ejpam-4733	2	60	thailand	thailand	PROPN
ejpam-4733	2	61	abstract	abstract	PROPN
ejpam-4733	2	62	.	.	PUNCT
ejpam-4733	3	1	this	this	DET
ejpam-4733	3	2	paper	paper	NOUN
ejpam-4733	3	3	deals	deal	NOUN
ejpam-4733	3	4	with	with	ADP
ejpam-4733	3	5	the	the	DET
ejpam-4733	3	6	notion	notion	NOUN
ejpam-4733	3	7	of	of	ADP
ejpam-4733	3	8	δp(λ	δp(λ	NOUN
ejpam-4733	3	9	,	,	PUNCT
ejpam-4733	3	10	p)-open	p)-open	VERB
ejpam-4733	3	11	sets	set	NOUN
ejpam-4733	3	12	.	.	PUNCT
ejpam-4733	4	1	some	some	DET
ejpam-4733	4	2	properties	property	NOUN
ejpam-4733	4	3	of	of	ADP
ejpam-4733	4	4	δp(λ	δp(λ	NOUN
ejpam-4733	4	5	,	,	PUNCT
ejpam-4733	4	6	p)open	p)open	ADJ
ejpam-4733	4	7	sets	set	NOUN
ejpam-4733	4	8	and	and	CCONJ
ejpam-4733	4	9	δp(λ	δp(λ	NOUN
ejpam-4733	4	10	,	,	PUNCT
ejpam-4733	4	11	p)-closed	p)-close	VERB
ejpam-4733	4	12	sets	set	NOUN
ejpam-4733	4	13	are	be	AUX
ejpam-4733	4	14	investigated	investigate	VERB
ejpam-4733	4	15	.	.	PUNCT
ejpam-4733	5	1	moreover	moreover	ADV
ejpam-4733	5	2	,	,	PUNCT
ejpam-4733	5	3	several	several	ADJ
ejpam-4733	5	4	characterizations	characterization	NOUN
ejpam-4733	5	5	of	of	ADP
ejpam-4733	5	6	δp(λ	δp(λ	NOUN
ejpam-4733	5	7	,	,	PUNCT
ejpam-4733	5	8	p)d1	p)d1	NOUN
ejpam-4733	5	9	spaces	space	NOUN
ejpam-4733	5	10	and	and	CCONJ
ejpam-4733	5	11	δp(λ	δp(λ	NOUN
ejpam-4733	5	12	,	,	PUNCT
ejpam-4733	5	13	p)-r0	p)-r0	PROPN
ejpam-4733	5	14	spaces	space	NOUN
ejpam-4733	5	15	are	be	AUX
ejpam-4733	5	16	established	establish	VERB
ejpam-4733	5	17	.	.	PUNCT
ejpam-4733	6	1	2020	2020	NUM
ejpam-4733	6	2	mathematics	mathematics	PROPN
ejpam-4733	6	3	subject	subject	NOUN
ejpam-4733	6	4	classifications	classification	NOUN
ejpam-4733	6	5	:	:	PUNCT
ejpam-4733	6	6	54a05	54a05	NUM
ejpam-4733	6	7	,	,	PUNCT
ejpam-4733	6	8	54d10	54d10	NUM
ejpam-4733	6	9	key	key	ADJ
ejpam-4733	6	10	words	word	NOUN
ejpam-4733	6	11	and	and	CCONJ
ejpam-4733	6	12	phrases	phrase	NOUN
ejpam-4733	6	13	:	:	PUNCT
ejpam-4733	6	14	δp(λ	δp(λ	NOUN
ejpam-4733	6	15	,	,	PUNCT
ejpam-4733	6	16	p)-open	p)-open	VERB
ejpam-4733	6	17	set	set	VERB
ejpam-4733	6	18	,	,	PUNCT
ejpam-4733	6	19	δp(λ	δp(λ	NOUN
ejpam-4733	6	20	,	,	PUNCT
ejpam-4733	6	21	p)-d1	p)-d1	NUM
ejpam-4733	6	22	space	space	NOUN
ejpam-4733	6	23	,	,	PUNCT
ejpam-4733	6	24	δp(λ	δp(λ	NOUN
ejpam-4733	6	25	,	,	PUNCT
ejpam-4733	6	26	p)-r0	p)-r0	NOUN
ejpam-4733	6	27	space	space	NOUN
ejpam-4733	6	28	1	1	NUM
ejpam-4733	6	29	.	.	PUNCT
ejpam-4733	7	1	introduction	introduction	NOUN
ejpam-4733	7	2	the	the	DET
ejpam-4733	7	3	concept	concept	NOUN
ejpam-4733	7	4	of	of	ADP
ejpam-4733	7	5	δ	δ	NOUN
ejpam-4733	7	6	-	-	PUNCT
ejpam-4733	7	7	open	open	ADJ
ejpam-4733	7	8	sets	set	NOUN
ejpam-4733	7	9	was	be	AUX
ejpam-4733	7	10	first	first	ADV
ejpam-4733	7	11	introduced	introduce	VERB
ejpam-4733	7	12	by	by	ADP
ejpam-4733	7	13	veličko	veličko	PROPN
ejpam-4733	8	1	[	[	X
ejpam-4733	8	2	10	10	NUM
ejpam-4733	8	3	]	]	PUNCT
ejpam-4733	8	4	.	.	PUNCT
ejpam-4733	9	1	in	in	ADP
ejpam-4733	9	2	1982	1982	NUM
ejpam-4733	9	3	,	,	PUNCT
ejpam-4733	9	4	mashhour	mashhour	PROPN
ejpam-4733	9	5	et	et	PROPN
ejpam-4733	9	6	al	al	PROPN
ejpam-4733	9	7	.	.	PUNCT
ejpam-4733	10	1	[	[	X
ejpam-4733	10	2	7	7	X
ejpam-4733	10	3	]	]	PUNCT
ejpam-4733	10	4	introduced	introduce	VERB
ejpam-4733	10	5	and	and	CCONJ
ejpam-4733	10	6	investigated	investigate	VERB
ejpam-4733	10	7	the	the	DET
ejpam-4733	10	8	notion	notion	NOUN
ejpam-4733	10	9	of	of	ADP
ejpam-4733	10	10	preopen	preopen	ADJ
ejpam-4733	10	11	sets	set	NOUN
ejpam-4733	10	12	which	which	PRON
ejpam-4733	10	13	is	be	AUX
ejpam-4733	10	14	weaker	weak	ADJ
ejpam-4733	10	15	than	than	ADP
ejpam-4733	10	16	the	the	DET
ejpam-4733	10	17	notion	notion	NOUN
ejpam-4733	10	18	of	of	ADP
ejpam-4733	10	19	open	open	ADJ
ejpam-4733	10	20	sets	set	NOUN
ejpam-4733	10	21	in	in	ADP
ejpam-4733	10	22	topological	topological	ADJ
ejpam-4733	10	23	spaces	space	NOUN
ejpam-4733	10	24	.	.	PUNCT
ejpam-4733	11	1	raychaudhuri	raychaudhuri	PROPN
ejpam-4733	11	2	and	and	CCONJ
ejpam-4733	11	3	mukherjee	mukherjee	NOUN
ejpam-4733	12	1	[	[	X
ejpam-4733	12	2	8	8	NUM
ejpam-4733	12	3	]	]	PUNCT
ejpam-4733	12	4	introduced	introduce	VERB
ejpam-4733	12	5	and	and	CCONJ
ejpam-4733	12	6	studied	study	VERB
ejpam-4733	12	7	the	the	DET
ejpam-4733	12	8	notions	notion	NOUN
ejpam-4733	12	9	of	of	ADP
ejpam-4733	12	10	δ	δ	PROPN
ejpam-4733	12	11	-	-	PUNCT
ejpam-4733	12	12	preopen	preopen	ADJ
ejpam-4733	12	13	sets	set	NOUN
ejpam-4733	12	14	and	and	CCONJ
ejpam-4733	12	15	δ	δ	NOUN
ejpam-4733	12	16	-	-	NOUN
ejpam-4733	12	17	closure	closure	NOUN
ejpam-4733	12	18	.	.	PUNCT
ejpam-4733	13	1	the	the	DET
ejpam-4733	13	2	class	class	NOUN
ejpam-4733	13	3	of	of	ADP
ejpam-4733	13	4	δ	δ	PROPN
ejpam-4733	13	5	-	-	PUNCT
ejpam-4733	13	6	preopen	preopen	ADJ
ejpam-4733	13	7	sets	set	NOUN
ejpam-4733	13	8	is	be	AUX
ejpam-4733	13	9	larger	large	ADJ
ejpam-4733	13	10	than	than	ADP
ejpam-4733	13	11	that	that	PRON
ejpam-4733	13	12	of	of	ADP
ejpam-4733	13	13	preopen	preopen	ADJ
ejpam-4733	13	14	sets	set	NOUN
ejpam-4733	13	15	.	.	PUNCT
ejpam-4733	14	1	in	in	ADP
ejpam-4733	14	2	1996	1996	NUM
ejpam-4733	14	3	,	,	PUNCT
ejpam-4733	14	4	raychaudhuri	raychaudhuri	NOUN
ejpam-4733	14	5	and	and	CCONJ
ejpam-4733	14	6	mukherjee	mukherjee	NOUN
ejpam-4733	14	7	[	[	X
ejpam-4733	14	8	9	9	NUM
ejpam-4733	14	9	]	]	PUNCT
ejpam-4733	14	10	introduced	introduce	VERB
ejpam-4733	14	11	and	and	CCONJ
ejpam-4733	14	12	investigated	investigate	VERB
ejpam-4733	14	13	the	the	DET
ejpam-4733	14	14	concept	concept	NOUN
ejpam-4733	14	15	of	of	ADP
ejpam-4733	14	16	δp	δp	ADV
ejpam-4733	14	17	-	-	PUNCT
ejpam-4733	14	18	closed	closed	ADJ
ejpam-4733	14	19	spaces	space	NOUN
ejpam-4733	14	20	.	.	PUNCT
ejpam-4733	15	1	caldas	caldas	PROPN
ejpam-4733	15	2	et	et	PROPN
ejpam-4733	15	3	al	al	PROPN
ejpam-4733	15	4	.	.	PUNCT
ejpam-4733	16	1	[	[	X
ejpam-4733	16	2	3	3	X
ejpam-4733	16	3	]	]	PUNCT
ejpam-4733	16	4	introduced	introduce	VERB
ejpam-4733	16	5	some	some	DET
ejpam-4733	16	6	weak	weak	ADJ
ejpam-4733	16	7	separation	separation	NOUN
ejpam-4733	16	8	axioms	axiom	NOUN
ejpam-4733	16	9	by	by	ADP
ejpam-4733	16	10	utilizing	utilize	VERB
ejpam-4733	16	11	the	the	DET
ejpam-4733	16	12	notions	notion	NOUN
ejpam-4733	16	13	of	of	ADP
ejpam-4733	16	14	δ	δ	PROPN
ejpam-4733	16	15	-	-	PUNCT
ejpam-4733	16	16	preopen	preopen	ADJ
ejpam-4733	16	17	sets	set	NOUN
ejpam-4733	16	18	and	and	CCONJ
ejpam-4733	16	19	the	the	DET
ejpam-4733	16	20	δ	δ	NOUN
ejpam-4733	16	21	-	-	PUNCT
ejpam-4733	16	22	preclosure	preclosure	ADJ
ejpam-4733	16	23	operator	operator	NOUN
ejpam-4733	16	24	.	.	PUNCT
ejpam-4733	17	1	furthermore	furthermore	ADV
ejpam-4733	17	2	,	,	PUNCT
ejpam-4733	17	3	caldas	caldas	PROPN
ejpam-4733	17	4	et	et	PROPN
ejpam-4733	17	5	al	al	PROPN
ejpam-4733	17	6	.	.	PUNCT
ejpam-4733	18	1	[	[	X
ejpam-4733	18	2	3	3	X
ejpam-4733	18	3	]	]	PUNCT
ejpam-4733	18	4	showed	show	VERB
ejpam-4733	18	5	that	that	SCONJ
ejpam-4733	18	6	(	(	PUNCT
ejpam-4733	18	7	δ	δ	PROPN
ejpam-4733	18	8	,	,	PUNCT
ejpam-4733	18	9	p)-t1	p)-t1	VERB
ejpam-4733	18	10	spaces	space	NOUN
ejpam-4733	18	11	,	,	PUNCT
ejpam-4733	18	12	(	(	PUNCT
ejpam-4733	18	13	δ	δ	PROPN
ejpam-4733	18	14	,	,	PUNCT
ejpam-4733	18	15	p)-r0	p)-r0	PROPN
ejpam-4733	18	16	spaces	space	VERB
ejpam-4733	18	17	and	and	CCONJ
ejpam-4733	18	18	(	(	PUNCT
ejpam-4733	18	19	δ	δ	PROPN
ejpam-4733	18	20	,	,	PUNCT
ejpam-4733	18	21	p)-symmetric	p)-symmetric	ADJ
ejpam-4733	18	22	spaces	space	NOUN
ejpam-4733	18	23	are	be	AUX
ejpam-4733	18	24	all	all	ADV
ejpam-4733	18	25	equivalent	equivalent	ADJ
ejpam-4733	18	26	.	.	PUNCT
ejpam-4733	19	1	in	in	ADP
ejpam-4733	19	2	2003	2003	NUM
ejpam-4733	19	3	,	,	PUNCT
ejpam-4733	19	4	caldas	caldas	PROPN
ejpam-4733	19	5	et	et	PROPN
ejpam-4733	19	6	al	al	PROPN
ejpam-4733	19	7	.	.	PUNCT
ejpam-4733	20	1	[	[	X
ejpam-4733	20	2	5	5	NUM
ejpam-4733	20	3	]	]	PUNCT
ejpam-4733	20	4	investigated	investigate	VERB
ejpam-4733	20	5	some	some	DET
ejpam-4733	20	6	weak	weak	ADJ
ejpam-4733	20	7	separation	separation	NOUN
ejpam-4733	20	8	axioms	axiom	NOUN
ejpam-4733	20	9	by	by	ADP
ejpam-4733	20	10	utilizing	utilize	VERB
ejpam-4733	20	11	δ	δ	PROPN
ejpam-4733	20	12	-	-	PUNCT
ejpam-4733	20	13	semiopen	semiopen	ADJ
ejpam-4733	20	14	sets	set	NOUN
ejpam-4733	20	15	and	and	CCONJ
ejpam-4733	20	16	the	the	DET
ejpam-4733	20	17	δ	δ	PROPN
ejpam-4733	20	18	-	-	PUNCT
ejpam-4733	20	19	semiclosure	semiclosure	NOUN
ejpam-4733	20	20	operator	operator	NOUN
ejpam-4733	20	21	.	.	PUNCT
ejpam-4733	21	1	in	in	ADP
ejpam-4733	21	2	2005	2005	NUM
ejpam-4733	21	3	,	,	PUNCT
ejpam-4733	21	4	caldas	caldas	PROPN
ejpam-4733	21	5	et	et	PROPN
ejpam-4733	21	6	al	al	PROPN
ejpam-4733	21	7	.	.	PUNCT
ejpam-4733	22	1	[	[	X
ejpam-4733	22	2	4	4	NUM
ejpam-4733	22	3	]	]	PUNCT
ejpam-4733	22	4	investigated	investigate	VERB
ejpam-4733	22	5	the	the	DET
ejpam-4733	22	6	notion	notion	NOUN
ejpam-4733	22	7	of	of	ADP
ejpam-4733	22	8	δ	δ	PROPN
ejpam-4733	22	9	-	-	PUNCT
ejpam-4733	22	10	λs	λs	ADV
ejpam-4733	22	11	-	-	PUNCT
ejpam-4733	22	12	semiclosed	semiclose	VERB
ejpam-4733	22	13	sets	set	NOUN
ejpam-4733	22	14	which	which	PRON
ejpam-4733	22	15	is	be	AUX
ejpam-4733	22	16	defined	define	VERB
ejpam-4733	22	17	as	as	ADP
ejpam-4733	22	18	the	the	DET
ejpam-4733	22	19	intersection	intersection	NOUN
ejpam-4733	22	20	of	of	ADP
ejpam-4733	22	21	a	a	DET
ejpam-4733	22	22	δ	δ	PROPN
ejpam-4733	22	23	-	-	PUNCT
ejpam-4733	22	24	λs	λs	NOUN
ejpam-4733	22	25	-	-	PUNCT
ejpam-4733	22	26	set	set	NOUN
ejpam-4733	22	27	and	and	CCONJ
ejpam-4733	22	28	a	a	DET
ejpam-4733	22	29	δ	δ	NOUN
ejpam-4733	22	30	-	-	PUNCT
ejpam-4733	22	31	semiclosed	semiclose	VERB
ejpam-4733	22	32	set	set	NOUN
ejpam-4733	22	33	.	.	PUNCT
ejpam-4733	23	1	in	in	ADP
ejpam-4733	23	2	[	[	X
ejpam-4733	23	3	2	2	NUM
ejpam-4733	23	4	]	]	PUNCT
ejpam-4733	23	5	,	,	PUNCT
ejpam-4733	23	6	the	the	DET
ejpam-4733	23	7	present	present	ADJ
ejpam-4733	23	8	authors	author	NOUN
ejpam-4733	23	9	introduced	introduce	VERB
ejpam-4733	23	10	the	the	DET
ejpam-4733	23	11	notions	notion	NOUN
ejpam-4733	23	12	of	of	ADP
ejpam-4733	23	13	(	(	PUNCT
ejpam-4733	23	14	λ	λ	INTJ
ejpam-4733	23	15	,	,	PUNCT
ejpam-4733	23	16	p)-open	p)-open	VERB
ejpam-4733	23	17	sets	set	NOUN
ejpam-4733	23	18	and	and	CCONJ
ejpam-4733	23	19	(	(	PUNCT
ejpam-4733	23	20	λ	λ	PROPN
ejpam-4733	23	21	,	,	PUNCT
ejpam-4733	23	22	p)-closed	p)-close	VERB
ejpam-4733	23	23	sets	set	NOUN
ejpam-4733	23	24	which	which	PRON
ejpam-4733	23	25	are	be	AUX
ejpam-4733	23	26	defined	define	VERB
ejpam-4733	23	27	by	by	ADP
ejpam-4733	23	28	utilizing	utilize	VERB
ejpam-4733	23	29	the	the	DET
ejpam-4733	23	30	notions	notion	NOUN
ejpam-4733	23	31	of	of	ADP
ejpam-4733	23	32	λp	λp	NOUN
ejpam-4733	23	33	-	-	PUNCT
ejpam-4733	23	34	sets	set	NOUN
ejpam-4733	23	35	and	and	CCONJ
ejpam-4733	23	36	preclosed	preclose	VERB
ejpam-4733	23	37	sets	set	NOUN
ejpam-4733	23	38	.	.	PUNCT
ejpam-4733	24	1	quite	quite	ADV
ejpam-4733	24	2	recently	recently	ADV
ejpam-4733	24	3	,	,	PUNCT
ejpam-4733	24	4	boonpok	boonpok	PROPN
ejpam-4733	24	5	and	and	CCONJ
ejpam-4733	24	6	viriyapong	viriyapong	ADJ
ejpam-4733	25	1	[	[	X
ejpam-4733	25	2	1	1	X
ejpam-4733	25	3	]	]	PUNCT
ejpam-4733	25	4	investigated	investigate	VERB
ejpam-4733	25	5	some	some	DET
ejpam-4733	25	6	characterizations	characterization	NOUN
ejpam-4733	25	7	of	of	ADP
ejpam-4733	25	8	(	(	PUNCT
ejpam-4733	25	9	λ	λ	PROPN
ejpam-4733	25	10	,	,	PUNCT
ejpam-4733	25	11	s)r0	s)r0	PROPN
ejpam-4733	25	12	topological	topological	ADJ
ejpam-4733	25	13	spaces	space	NOUN
ejpam-4733	25	14	.	.	PUNCT
ejpam-4733	26	1	in	in	ADP
ejpam-4733	26	2	this	this	DET
ejpam-4733	26	3	paper	paper	NOUN
ejpam-4733	26	4	,	,	PUNCT
ejpam-4733	26	5	we	we	PRON
ejpam-4733	26	6	introduced	introduce	VERB
ejpam-4733	26	7	the	the	DET
ejpam-4733	26	8	concept	concept	NOUN
ejpam-4733	26	9	of	of	ADP
ejpam-4733	26	10	δp(λ	δp(λ	NOUN
ejpam-4733	26	11	,	,	PUNCT
ejpam-4733	26	12	p)-open	p)-open	VERB
ejpam-4733	26	13	sets	set	NOUN
ejpam-4733	26	14	.	.	PUNCT
ejpam-4733	27	1	moreover	moreover	ADV
ejpam-4733	27	2	,	,	PUNCT
ejpam-4733	27	3	some	some	DET
ejpam-4733	27	4	properties	property	NOUN
ejpam-4733	27	5	of	of	ADP
ejpam-4733	27	6	δp(λ	δp(λ	NOUN
ejpam-4733	27	7	,	,	PUNCT
ejpam-4733	27	8	p)-open	p)-open	VERB
ejpam-4733	27	9	sets	set	NOUN
ejpam-4733	27	10	and	and	CCONJ
ejpam-4733	27	11	δp(λ	δp(λ	NOUN
ejpam-4733	27	12	,	,	PUNCT
ejpam-4733	27	13	p)-closed	p)-close	VERB
ejpam-4733	27	14	sets	set	NOUN
ejpam-4733	27	15	are	be	AUX
ejpam-4733	27	16	discussed	discuss	VERB
ejpam-4733	27	17	.	.	PUNCT
ejpam-4733	28	1	in	in	ADP
ejpam-4733	28	2	particular	particular	ADJ
ejpam-4733	28	3	,	,	PUNCT
ejpam-4733	28	4	several	several	ADJ
ejpam-4733	28	5	characterizations	characterization	NOUN
ejpam-4733	28	6	of	of	ADP
ejpam-4733	28	7	δp(λ	δp(λ	NOUN
ejpam-4733	28	8	,	,	PUNCT
ejpam-4733	28	9	p)-d1	p)-d1	NOUN
ejpam-4733	28	10	spaces	space	NOUN
ejpam-4733	28	11	and	and	CCONJ
ejpam-4733	28	12	δp(λ	δp(λ	NOUN
ejpam-4733	28	13	,	,	PUNCT
ejpam-4733	28	14	p)-r0	p)-r0	PROPN
ejpam-4733	28	15	spaces	space	NOUN
ejpam-4733	28	16	are	be	AUX
ejpam-4733	28	17	investigated	investigate	VERB
ejpam-4733	28	18	.	.	PUNCT
ejpam-4733	29	1	∗corresponding	∗corresponde	VERB
ejpam-4733	29	2	author	author	NOUN
ejpam-4733	29	3	.	.	PUNCT
ejpam-4733	30	1	doi	doi	NOUN
ejpam-4733	30	2	:	:	PUNCT
ejpam-4733	30	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4733	https://doi.org/10.29020/nybg.ejpam.v16i3.4733	PROPN
ejpam-4733	30	4	email	email	NOUN
ejpam-4733	30	5	addresses	address	NOUN
ejpam-4733	30	6	:	:	PUNCT
ejpam-4733	30	7	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4733	30	8	(	(	PUNCT
ejpam-4733	30	9	c.	c.	PROPN
ejpam-4733	30	10	boonpok	boonpok	PROPN
ejpam-4733	30	11	)	)	PUNCT
ejpam-4733	30	12	,	,	PUNCT
ejpam-4733	30	13	montri.t@msu.ac.th	montri.t@msu.ac.th	PROPN
ejpam-4733	30	14	(	(	PUNCT
ejpam-4733	30	15	m.	m.	NOUN
ejpam-4733	30	16	thongmoon	thongmoon	PROPN
ejpam-4733	30	17	)	)	PUNCT
ejpam-4733	30	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4733	30	19	1533	1533	NUM
ejpam-4733	31	1	©	©	PROPN
ejpam-4733	31	2	2023	2023	NUM
ejpam-4733	31	3	ejpam	ejpam	NOUN
ejpam-4733	31	4	all	all	DET
ejpam-4733	31	5	rights	right	NOUN
ejpam-4733	31	6	reserved	reserve	VERB
ejpam-4733	31	7	.	.	PUNCT
ejpam-4733	32	1	c.	c.	PROPN
ejpam-4733	32	2	boonpok	boonpok	PROPN
ejpam-4733	32	3	,	,	PUNCT
ejpam-4733	32	4	m.	m.	NOUN
ejpam-4733	32	5	thongmoon	thongmoon	PROPN
ejpam-4733	32	6	/	/	SYM
ejpam-4733	32	7	eur	eur	PROPN
ejpam-4733	32	8	.	.	PUNCT
ejpam-4733	33	1	j.	j.	PROPN
ejpam-4733	33	2	pure	pure	PROPN
ejpam-4733	33	3	appl	appl	PROPN
ejpam-4733	33	4	.	.	PROPN
ejpam-4733	33	5	math	math	PROPN
ejpam-4733	33	6	,	,	PUNCT
ejpam-4733	33	7	16	16	NUM
ejpam-4733	33	8	(	(	PUNCT
ejpam-4733	33	9	3	3	NUM
ejpam-4733	33	10	)	)	PUNCT
ejpam-4733	33	11	(	(	PUNCT
ejpam-4733	33	12	2023	2023	NUM
ejpam-4733	33	13	)	)	PUNCT
ejpam-4733	33	14	,	,	PUNCT
ejpam-4733	33	15	1533	1533	NUM
ejpam-4733	33	16	-	-	SYM
ejpam-4733	33	17	1542	1542	NUM
ejpam-4733	33	18	1534	1534	NUM
ejpam-4733	33	19	2	2	NUM
ejpam-4733	33	20	.	.	PUNCT
ejpam-4733	33	21	preliminaries	preliminary	NOUN
ejpam-4733	33	22	throughout	throughout	ADP
ejpam-4733	33	23	the	the	DET
ejpam-4733	33	24	present	present	ADJ
ejpam-4733	33	25	paper	paper	NOUN
ejpam-4733	33	26	,	,	PUNCT
ejpam-4733	33	27	spaces	space	NOUN
ejpam-4733	33	28	(	(	PUNCT
ejpam-4733	33	29	x	x	X
ejpam-4733	33	30	,	,	PUNCT
ejpam-4733	33	31	τ	τ	X
ejpam-4733	33	32	)	)	PUNCT
ejpam-4733	33	33	and	and	CCONJ
ejpam-4733	33	34	(	(	PUNCT
ejpam-4733	33	35	y	y	PROPN
ejpam-4733	33	36	,	,	PUNCT
ejpam-4733	33	37	σ	σ	PROPN
ejpam-4733	33	38	)	)	PUNCT
ejpam-4733	33	39	(	(	PUNCT
ejpam-4733	33	40	or	or	CCONJ
ejpam-4733	33	41	simply	simply	ADV
ejpam-4733	33	42	x	x	X
ejpam-4733	33	43	and	and	CCONJ
ejpam-4733	33	44	y	y	PROPN
ejpam-4733	33	45	)	)	PUNCT
ejpam-4733	33	46	always	always	ADV
ejpam-4733	33	47	mean	mean	VERB
ejpam-4733	33	48	topological	topological	ADJ
ejpam-4733	33	49	spaces	space	NOUN
ejpam-4733	33	50	on	on	ADP
ejpam-4733	33	51	which	which	PRON
ejpam-4733	33	52	no	no	DET
ejpam-4733	33	53	separation	separation	NOUN
ejpam-4733	33	54	axioms	axiom	NOUN
ejpam-4733	33	55	are	be	AUX
ejpam-4733	33	56	assumed	assume	VERB
ejpam-4733	33	57	unless	unless	SCONJ
ejpam-4733	33	58	explicitly	explicitly	ADV
ejpam-4733	33	59	stated	state	VERB
ejpam-4733	33	60	.	.	PUNCT
ejpam-4733	34	1	for	for	ADP
ejpam-4733	34	2	a	a	DET
ejpam-4733	34	3	subset	subset	NOUN
ejpam-4733	34	4	a	a	PRON
ejpam-4733	34	5	of	of	ADP
ejpam-4733	34	6	a	a	DET
ejpam-4733	34	7	topological	topological	ADJ
ejpam-4733	34	8	space	space	NOUN
ejpam-4733	34	9	(	(	PUNCT
ejpam-4733	34	10	x	x	X
ejpam-4733	34	11	,	,	PUNCT
ejpam-4733	34	12	τ	τ	PROPN
ejpam-4733	34	13	)	)	PUNCT
ejpam-4733	34	14	,	,	PUNCT
ejpam-4733	34	15	cl(a	cl(a	NUM
ejpam-4733	34	16	)	)	PUNCT
ejpam-4733	34	17	and	and	CCONJ
ejpam-4733	34	18	int(a	int(a	PROPN
ejpam-4733	34	19	)	)	PUNCT
ejpam-4733	34	20	,	,	PUNCT
ejpam-4733	34	21	represent	represent	VERB
ejpam-4733	34	22	the	the	DET
ejpam-4733	34	23	closure	closure	NOUN
ejpam-4733	34	24	and	and	CCONJ
ejpam-4733	34	25	the	the	DET
ejpam-4733	34	26	interior	interior	NOUN
ejpam-4733	34	27	of	of	ADP
ejpam-4733	34	28	a	a	PRON
ejpam-4733	34	29	,	,	PUNCT
ejpam-4733	34	30	respectively	respectively	ADV
ejpam-4733	34	31	.	.	PUNCT
ejpam-4733	35	1	a	a	DET
ejpam-4733	35	2	subset	subset	NOUN
ejpam-4733	35	3	a	a	PRON
ejpam-4733	35	4	of	of	ADP
ejpam-4733	35	5	a	a	DET
ejpam-4733	35	6	topological	topological	ADJ
ejpam-4733	35	7	space	space	NOUN
ejpam-4733	35	8	(	(	PUNCT
ejpam-4733	35	9	x	x	X
ejpam-4733	35	10	,	,	PUNCT
ejpam-4733	35	11	τ	τ	X
ejpam-4733	35	12	)	)	PUNCT
ejpam-4733	35	13	is	be	AUX
ejpam-4733	35	14	said	say	VERB
ejpam-4733	35	15	to	to	PART
ejpam-4733	35	16	be	be	AUX
ejpam-4733	35	17	preopen	preopen	ADJ
ejpam-4733	35	18	[	[	X
ejpam-4733	35	19	7	7	X
ejpam-4733	35	20	]	]	X
ejpam-4733	35	21	if	if	SCONJ
ejpam-4733	35	22	a	a	DET
ejpam-4733	35	23	⊆	⊆	NUM
ejpam-4733	35	24	int(cl(a	int(cl(a	PROPN
ejpam-4733	35	25	)	)	PUNCT
ejpam-4733	35	26	)	)	PUNCT
ejpam-4733	35	27	.	.	PUNCT
ejpam-4733	36	1	the	the	DET
ejpam-4733	36	2	complement	complement	NOUN
ejpam-4733	36	3	of	of	ADP
ejpam-4733	36	4	a	a	DET
ejpam-4733	36	5	preopen	preopen	ADJ
ejpam-4733	36	6	set	set	NOUN
ejpam-4733	36	7	is	be	AUX
ejpam-4733	36	8	called	call	VERB
ejpam-4733	36	9	preclosed	preclose	VERB
ejpam-4733	36	10	.	.	PUNCT
ejpam-4733	37	1	the	the	DET
ejpam-4733	37	2	family	family	NOUN
ejpam-4733	37	3	of	of	ADP
ejpam-4733	37	4	all	all	DET
ejpam-4733	37	5	preopen	preopen	ADJ
ejpam-4733	37	6	sets	set	NOUN
ejpam-4733	37	7	of	of	ADP
ejpam-4733	37	8	a	a	DET
ejpam-4733	37	9	topological	topological	ADJ
ejpam-4733	37	10	space	space	NOUN
ejpam-4733	37	11	(	(	PUNCT
ejpam-4733	37	12	x	x	X
ejpam-4733	37	13	,	,	PUNCT
ejpam-4733	37	14	τ	τ	X
ejpam-4733	37	15	)	)	PUNCT
ejpam-4733	37	16	is	be	AUX
ejpam-4733	37	17	denoted	denote	VERB
ejpam-4733	37	18	by	by	ADP
ejpam-4733	37	19	po(x	po(x	NUM
ejpam-4733	37	20	,	,	PUNCT
ejpam-4733	37	21	τ	τ	PROPN
ejpam-4733	37	22	)	)	PUNCT
ejpam-4733	37	23	.	.	PUNCT
ejpam-4733	38	1	a	a	DET
ejpam-4733	38	2	subset	subset	NOUN
ejpam-4733	38	3	λp(a	λp(a	NOUN
ejpam-4733	38	4	)	)	PUNCT
ejpam-4733	39	1	[	[	X
ejpam-4733	39	2	6	6	NUM
ejpam-4733	39	3	]	]	PUNCT
ejpam-4733	39	4	is	be	AUX
ejpam-4733	39	5	defined	define	VERB
ejpam-4733	39	6	as	as	SCONJ
ejpam-4733	39	7	follows	follow	VERB
ejpam-4733	39	8	:	:	PUNCT
ejpam-4733	39	9	λp(a	λp(a	NUM
ejpam-4733	39	10	)	)	PUNCT
ejpam-4733	40	1	=	=	PUNCT
ejpam-4733	41	1	∩{u	∩{u	PROPN
ejpam-4733	41	2	|	|	ADV
ejpam-4733	41	3	a	a	DET
ejpam-4733	41	4	⊆	⊆	NUM
ejpam-4733	41	5	u	u	NOUN
ejpam-4733	41	6	,	,	PUNCT
ejpam-4733	41	7	u	u	PROPN
ejpam-4733	41	8	∈	∈	PROPN
ejpam-4733	41	9	po(x	po(x	NOUN
ejpam-4733	41	10	,	,	PUNCT
ejpam-4733	41	11	τ	τ	NOUN
ejpam-4733	41	12	)	)	PUNCT
ejpam-4733	41	13	}	}	PUNCT
ejpam-4733	41	14	.	.	PUNCT
ejpam-4733	42	1	a	a	DET
ejpam-4733	42	2	subset	subset	NOUN
ejpam-4733	42	3	a	a	PRON
ejpam-4733	42	4	of	of	ADP
ejpam-4733	42	5	a	a	DET
ejpam-4733	42	6	topological	topological	ADJ
ejpam-4733	42	7	space	space	NOUN
ejpam-4733	42	8	(	(	PUNCT
ejpam-4733	42	9	x	x	X
ejpam-4733	42	10	,	,	PUNCT
ejpam-4733	42	11	τ	τ	X
ejpam-4733	42	12	)	)	PUNCT
ejpam-4733	42	13	is	be	AUX
ejpam-4733	42	14	called	call	VERB
ejpam-4733	42	15	a	a	PRON
ejpam-4733	42	16	λp	λp	NOUN
ejpam-4733	42	17	-	-	PUNCT
ejpam-4733	42	18	set	set	VERB
ejpam-4733	42	19	[	[	X
ejpam-4733	42	20	1	1	NUM
ejpam-4733	42	21	]	]	PUNCT
ejpam-4733	42	22	(	(	PUNCT
ejpam-4733	42	23	pre	pre	ADJ
ejpam-4733	42	24	-	-	ADJ
ejpam-4733	42	25	λ	λ	NOUN
ejpam-4733	42	26	-	-	NOUN
ejpam-4733	42	27	set	set	ADJ
ejpam-4733	42	28	[	[	X
ejpam-4733	42	29	6	6	NUM
ejpam-4733	42	30	]	]	PUNCT
ejpam-4733	42	31	)	)	PUNCT
ejpam-4733	42	32	if	if	SCONJ
ejpam-4733	42	33	a	a	DET
ejpam-4733	42	34	=	=	NOUN
ejpam-4733	42	35	λp(a	λp(a	NOUN
ejpam-4733	42	36	)	)	PUNCT
ejpam-4733	42	37	.	.	PUNCT
ejpam-4733	43	1	a	a	DET
ejpam-4733	43	2	subset	subset	NOUN
ejpam-4733	43	3	a	a	PRON
ejpam-4733	43	4	of	of	ADP
ejpam-4733	43	5	a	a	DET
ejpam-4733	43	6	topological	topological	ADJ
ejpam-4733	43	7	space	space	NOUN
ejpam-4733	43	8	(	(	PUNCT
ejpam-4733	43	9	x	x	X
ejpam-4733	43	10	,	,	PUNCT
ejpam-4733	43	11	τ	τ	X
ejpam-4733	43	12	)	)	PUNCT
ejpam-4733	43	13	is	be	AUX
ejpam-4733	43	14	called	call	VERB
ejpam-4733	43	15	(	(	PUNCT
ejpam-4733	43	16	λ	λ	X
ejpam-4733	43	17	,	,	PUNCT
ejpam-4733	43	18	p)-closed	p)-close	VERB
ejpam-4733	43	19	[	[	X
ejpam-4733	43	20	1	1	X
ejpam-4733	43	21	]	]	X
ejpam-4733	43	22	if	if	SCONJ
ejpam-4733	43	23	a	a	DET
ejpam-4733	43	24	=	=	X
ejpam-4733	43	25	t	t	NOUN
ejpam-4733	43	26	∩c	∩c	NOUN
ejpam-4733	43	27	,	,	PUNCT
ejpam-4733	43	28	where	where	SCONJ
ejpam-4733	43	29	t	t	PROPN
ejpam-4733	43	30	is	be	AUX
ejpam-4733	43	31	a	a	DET
ejpam-4733	43	32	λp	λp	ADV
ejpam-4733	43	33	-	-	PUNCT
ejpam-4733	43	34	set	set	NOUN
ejpam-4733	43	35	and	and	CCONJ
ejpam-4733	43	36	c	c	NOUN
ejpam-4733	43	37	is	be	AUX
ejpam-4733	43	38	a	a	DET
ejpam-4733	43	39	preclosed	preclose	VERB
ejpam-4733	43	40	set	set	NOUN
ejpam-4733	43	41	.	.	PUNCT
ejpam-4733	44	1	the	the	DET
ejpam-4733	44	2	complement	complement	NOUN
ejpam-4733	44	3	of	of	ADP
ejpam-4733	44	4	a	a	DET
ejpam-4733	44	5	(	(	PUNCT
ejpam-4733	44	6	λ	λ	PROPN
ejpam-4733	44	7	,	,	PUNCT
ejpam-4733	44	8	p)-closed	p)-close	VERB
ejpam-4733	44	9	set	set	NOUN
ejpam-4733	44	10	is	be	AUX
ejpam-4733	44	11	called	call	VERB
ejpam-4733	44	12	(	(	PUNCT
ejpam-4733	44	13	λ	λ	X
ejpam-4733	44	14	,	,	PUNCT
ejpam-4733	44	15	p)-open	p)-open	ADJ
ejpam-4733	44	16	.	.	PUNCT
ejpam-4733	45	1	the	the	DET
ejpam-4733	45	2	family	family	NOUN
ejpam-4733	45	3	of	of	ADP
ejpam-4733	45	4	all	all	DET
ejpam-4733	45	5	(	(	PUNCT
ejpam-4733	45	6	λ	λ	X
ejpam-4733	45	7	,	,	PUNCT
ejpam-4733	45	8	p)-open	p)-open	ADJ
ejpam-4733	45	9	(	(	PUNCT
ejpam-4733	45	10	resp	resp	NOUN
ejpam-4733	45	11	.	.	PUNCT
ejpam-4733	46	1	(	(	PUNCT
ejpam-4733	46	2	λ	λ	X
ejpam-4733	46	3	,	,	PUNCT
ejpam-4733	46	4	p)-closed	p)-close	VERB
ejpam-4733	46	5	)	)	PUNCT
ejpam-4733	46	6	sets	set	NOUN
ejpam-4733	46	7	in	in	ADP
ejpam-4733	46	8	a	a	DET
ejpam-4733	46	9	topological	topological	ADJ
ejpam-4733	46	10	space	space	NOUN
ejpam-4733	46	11	(	(	PUNCT
ejpam-4733	46	12	x	x	X
ejpam-4733	46	13	,	,	PUNCT
ejpam-4733	46	14	τ	τ	X
ejpam-4733	46	15	)	)	PUNCT
ejpam-4733	46	16	is	be	AUX
ejpam-4733	46	17	denoted	denote	VERB
ejpam-4733	46	18	by	by	ADP
ejpam-4733	46	19	λpo(x	λpo(x	PROPN
ejpam-4733	46	20	,	,	PUNCT
ejpam-4733	46	21	τ	τ	X
ejpam-4733	46	22	)	)	PUNCT
ejpam-4733	46	23	(	(	PUNCT
ejpam-4733	46	24	resp	resp	NOUN
ejpam-4733	46	25	.	.	PUNCT
ejpam-4733	47	1	λpc(x	λpc(x	PROPN
ejpam-4733	47	2	,	,	PUNCT
ejpam-4733	47	3	τ	τ	PROPN
ejpam-4733	47	4	)	)	PUNCT
ejpam-4733	47	5	)	)	PUNCT
ejpam-4733	47	6	.	.	PUNCT
ejpam-4733	48	1	let	let	VERB
ejpam-4733	48	2	a	a	DET
ejpam-4733	48	3	be	be	AUX
ejpam-4733	48	4	a	a	DET
ejpam-4733	48	5	subset	subset	NOUN
ejpam-4733	48	6	of	of	ADP
ejpam-4733	48	7	a	a	DET
ejpam-4733	48	8	topological	topological	ADJ
ejpam-4733	48	9	space	space	NOUN
ejpam-4733	48	10	(	(	PUNCT
ejpam-4733	48	11	x	x	X
ejpam-4733	48	12	,	,	PUNCT
ejpam-4733	48	13	τ	τ	PROPN
ejpam-4733	48	14	)	)	PUNCT
ejpam-4733	48	15	.	.	PUNCT
ejpam-4733	49	1	a	a	DET
ejpam-4733	49	2	point	point	NOUN
ejpam-4733	49	3	x	x	X
ejpam-4733	49	4	∈	∈	NOUN
ejpam-4733	49	5	x	x	PUNCT
ejpam-4733	49	6	is	be	AUX
ejpam-4733	49	7	called	call	VERB
ejpam-4733	49	8	a	a	DET
ejpam-4733	49	9	(	(	PUNCT
ejpam-4733	49	10	λ	λ	NOUN
ejpam-4733	49	11	,	,	PUNCT
ejpam-4733	49	12	p)-cluster	p)-cluster	NOUN
ejpam-4733	49	13	point	point	NOUN
ejpam-4733	49	14	[	[	X
ejpam-4733	49	15	1	1	X
ejpam-4733	49	16	]	]	PUNCT
ejpam-4733	49	17	of	of	ADP
ejpam-4733	49	18	a	a	PRON
ejpam-4733	49	19	if	if	SCONJ
ejpam-4733	49	20	a	a	DET
ejpam-4733	49	21	∩	∩	ADJ
ejpam-4733	49	22	u	u	ADJ
ejpam-4733	49	23	̸=	̸=	PROPN
ejpam-4733	49	24	∅	∅	NOUN
ejpam-4733	49	25	for	for	ADP
ejpam-4733	49	26	every	every	DET
ejpam-4733	49	27	(	(	PUNCT
ejpam-4733	49	28	λ	λ	NOUN
ejpam-4733	49	29	,	,	PUNCT
ejpam-4733	49	30	p)-open	p)-open	VERB
ejpam-4733	49	31	set	set	VERB
ejpam-4733	49	32	u	u	NOUN
ejpam-4733	49	33	of	of	ADP
ejpam-4733	49	34	x	x	SYM
ejpam-4733	49	35	containing	contain	VERB
ejpam-4733	49	36	x.	x.	NOUN
ejpam-4733	49	37	the	the	DET
ejpam-4733	49	38	set	set	NOUN
ejpam-4733	49	39	of	of	ADP
ejpam-4733	49	40	all	all	DET
ejpam-4733	49	41	(	(	PUNCT
ejpam-4733	49	42	λ	λ	NOUN
ejpam-4733	49	43	,	,	PUNCT
ejpam-4733	49	44	p)-cluster	p)-cluster	VERB
ejpam-4733	49	45	points	point	NOUN
ejpam-4733	49	46	of	of	ADP
ejpam-4733	49	47	a	a	PRON
ejpam-4733	49	48	is	be	AUX
ejpam-4733	49	49	called	call	VERB
ejpam-4733	49	50	the	the	DET
ejpam-4733	49	51	(	(	PUNCT
ejpam-4733	49	52	λ	λ	PROPN
ejpam-4733	49	53	,	,	PUNCT
ejpam-4733	49	54	p)-closure	p)-closure	PUNCT
ejpam-4733	50	1	[	[	X
ejpam-4733	50	2	1	1	X
ejpam-4733	50	3	]	]	PUNCT
ejpam-4733	50	4	of	of	ADP
ejpam-4733	50	5	a	a	PRON
ejpam-4733	50	6	and	and	CCONJ
ejpam-4733	50	7	is	be	AUX
ejpam-4733	50	8	denoted	denote	VERB
ejpam-4733	50	9	by	by	ADP
ejpam-4733	50	10	a(λ	a(λ	PROPN
ejpam-4733	50	11	,	,	PUNCT
ejpam-4733	50	12	p	p	NOUN
ejpam-4733	50	13	)	)	PUNCT
ejpam-4733	50	14	.	.	PUNCT
ejpam-4733	51	1	the	the	DET
ejpam-4733	51	2	union	union	NOUN
ejpam-4733	51	3	of	of	ADP
ejpam-4733	51	4	all	all	DET
ejpam-4733	51	5	(	(	PUNCT
ejpam-4733	51	6	λ	λ	X
ejpam-4733	51	7	,	,	PUNCT
ejpam-4733	51	8	p)-open	p)-open	VERB
ejpam-4733	51	9	sets	set	NOUN
ejpam-4733	51	10	contained	contain	VERB
ejpam-4733	51	11	in	in	ADP
ejpam-4733	51	12	a	a	PRON
ejpam-4733	51	13	is	be	AUX
ejpam-4733	51	14	called	call	VERB
ejpam-4733	51	15	the	the	DET
ejpam-4733	51	16	(	(	PUNCT
ejpam-4733	51	17	λ	λ	PROPN
ejpam-4733	51	18	,	,	PUNCT
ejpam-4733	51	19	p)-interior	p)-interior	ADJ
ejpam-4733	51	20	[	[	X
ejpam-4733	51	21	1	1	X
ejpam-4733	51	22	]	]	PUNCT
ejpam-4733	51	23	of	of	ADP
ejpam-4733	51	24	a	a	PRON
ejpam-4733	51	25	and	and	CCONJ
ejpam-4733	51	26	is	be	AUX
ejpam-4733	51	27	denoted	denote	VERB
ejpam-4733	51	28	by	by	ADP
ejpam-4733	51	29	a(λ	a(λ	PROPN
ejpam-4733	51	30	,	,	PUNCT
ejpam-4733	51	31	p	p	NOUN
ejpam-4733	51	32	)	)	PUNCT
ejpam-4733	51	33	.	.	PUNCT
ejpam-4733	52	1	a	a	DET
ejpam-4733	52	2	subset	subset	NOUN
ejpam-4733	52	3	a	a	PRON
ejpam-4733	52	4	of	of	ADP
ejpam-4733	52	5	a	a	DET
ejpam-4733	52	6	topological	topological	ADJ
ejpam-4733	52	7	space	space	NOUN
ejpam-4733	52	8	(	(	PUNCT
ejpam-4733	52	9	x	x	X
ejpam-4733	52	10	,	,	PUNCT
ejpam-4733	52	11	τ	τ	X
ejpam-4733	52	12	)	)	PUNCT
ejpam-4733	52	13	is	be	AUX
ejpam-4733	52	14	said	say	VERB
ejpam-4733	52	15	to	to	PART
ejpam-4733	52	16	be	be	AUX
ejpam-4733	52	17	p(λ	p(λ	NOUN
ejpam-4733	52	18	,	,	PUNCT
ejpam-4733	52	19	p)-open	p)-open	VERB
ejpam-4733	52	20	[	[	X
ejpam-4733	52	21	1	1	X
ejpam-4733	52	22	]	]	X
ejpam-4733	52	23	if	if	SCONJ
ejpam-4733	52	24	a	a	DET
ejpam-4733	52	25	⊆	⊆	NUM
ejpam-4733	52	26	[	[	X
ejpam-4733	52	27	a(λ	a(λ	ADV
ejpam-4733	52	28	,	,	PUNCT
ejpam-4733	52	29	p)](λ	p)](λ	X
ejpam-4733	52	30	,	,	PUNCT
ejpam-4733	52	31	p	p	NOUN
ejpam-4733	52	32	)	)	PUNCT
ejpam-4733	52	33	.	.	PUNCT
ejpam-4733	53	1	the	the	DET
ejpam-4733	53	2	complement	complement	NOUN
ejpam-4733	53	3	of	of	ADP
ejpam-4733	53	4	a	a	DET
ejpam-4733	53	5	p(λ	p(λ	NOUN
ejpam-4733	53	6	,	,	PUNCT
ejpam-4733	53	7	p)-open	p)-open	VERB
ejpam-4733	53	8	set	set	NOUN
ejpam-4733	53	9	is	be	AUX
ejpam-4733	53	10	said	say	VERB
ejpam-4733	53	11	to	to	PART
ejpam-4733	53	12	be	be	AUX
ejpam-4733	53	13	p(λ	p(λ	NOUN
ejpam-4733	53	14	,	,	PUNCT
ejpam-4733	53	15	p)-closed	p)-close	VERB
ejpam-4733	53	16	.	.	PUNCT
ejpam-4733	54	1	3	3	X
ejpam-4733	54	2	.	.	NUM
ejpam-4733	54	3	δp(λ	δp(λ	NOUN
ejpam-4733	54	4	,	,	PUNCT
ejpam-4733	54	5	p)-open	p)-open	VERB
ejpam-4733	54	6	sets	set	NOUN
ejpam-4733	54	7	in	in	ADP
ejpam-4733	54	8	this	this	DET
ejpam-4733	54	9	section	section	NOUN
ejpam-4733	54	10	,	,	PUNCT
ejpam-4733	54	11	we	we	PRON
ejpam-4733	54	12	introduced	introduce	VERB
ejpam-4733	54	13	the	the	DET
ejpam-4733	54	14	concept	concept	NOUN
ejpam-4733	54	15	of	of	ADP
ejpam-4733	54	16	δp(λ	δp(λ	NOUN
ejpam-4733	54	17	,	,	PUNCT
ejpam-4733	54	18	p)-open	p)-open	VERB
ejpam-4733	54	19	sets	set	NOUN
ejpam-4733	54	20	.	.	PUNCT
ejpam-4733	55	1	moreover	moreover	ADV
ejpam-4733	55	2	,	,	PUNCT
ejpam-4733	55	3	some	some	DET
ejpam-4733	55	4	properties	property	NOUN
ejpam-4733	55	5	of	of	ADP
ejpam-4733	55	6	δp(λ	δp(λ	NOUN
ejpam-4733	55	7	,	,	PUNCT
ejpam-4733	55	8	p)-open	p)-open	VERB
ejpam-4733	55	9	sets	set	NOUN
ejpam-4733	55	10	and	and	CCONJ
ejpam-4733	55	11	δp(λ	δp(λ	NOUN
ejpam-4733	55	12	,	,	PUNCT
ejpam-4733	55	13	p)-closed	p)-close	VERB
ejpam-4733	55	14	sets	set	NOUN
ejpam-4733	55	15	are	be	AUX
ejpam-4733	55	16	investigated	investigate	VERB
ejpam-4733	55	17	.	.	PUNCT
ejpam-4733	56	1	furthermore	furthermore	ADV
ejpam-4733	56	2	,	,	PUNCT
ejpam-4733	56	3	several	several	ADJ
ejpam-4733	56	4	characterizations	characterization	NOUN
ejpam-4733	56	5	of	of	ADP
ejpam-4733	56	6	δp(λ	δp(λ	NOUN
ejpam-4733	56	7	,	,	PUNCT
ejpam-4733	56	8	p)-d1	p)-d1	NOUN
ejpam-4733	56	9	spaces	space	NOUN
ejpam-4733	56	10	and	and	CCONJ
ejpam-4733	56	11	δp(λ	δp(λ	NOUN
ejpam-4733	56	12	,	,	PUNCT
ejpam-4733	56	13	p)-r0	p)-r0	PROPN
ejpam-4733	56	14	spaces	space	NOUN
ejpam-4733	56	15	are	be	AUX
ejpam-4733	56	16	discussed	discuss	VERB
ejpam-4733	56	17	.	.	PUNCT
ejpam-4733	57	1	definition	definition	NOUN
ejpam-4733	57	2	1	1	NUM
ejpam-4733	57	3	.	.	PUNCT
ejpam-4733	58	1	let	let	VERB
ejpam-4733	58	2	a	a	DET
ejpam-4733	58	3	be	be	AUX
ejpam-4733	58	4	a	a	DET
ejpam-4733	58	5	subset	subset	NOUN
ejpam-4733	58	6	of	of	ADP
ejpam-4733	58	7	a	a	DET
ejpam-4733	58	8	topological	topological	ADJ
ejpam-4733	58	9	space	space	NOUN
ejpam-4733	58	10	(	(	PUNCT
ejpam-4733	58	11	x	x	X
ejpam-4733	58	12	,	,	PUNCT
ejpam-4733	58	13	τ	τ	PROPN
ejpam-4733	58	14	)	)	PUNCT
ejpam-4733	58	15	.	.	PUNCT
ejpam-4733	59	1	a	a	DET
ejpam-4733	59	2	point	point	NOUN
ejpam-4733	59	3	x	x	PUNCT
ejpam-4733	59	4	of	of	ADP
ejpam-4733	59	5	x	x	PROPN
ejpam-4733	59	6	is	be	AUX
ejpam-4733	59	7	called	call	VERB
ejpam-4733	59	8	a	a	DET
ejpam-4733	59	9	δ(λ	δ(λ	PROPN
ejpam-4733	59	10	,	,	PUNCT
ejpam-4733	59	11	p)-cluster	p)-cluster	NOUN
ejpam-4733	59	12	point	point	NOUN
ejpam-4733	59	13	of	of	ADP
ejpam-4733	59	14	a	a	DET
ejpam-4733	59	15	if	if	SCONJ
ejpam-4733	59	16	a∩[v	a∩[v	PROPN
ejpam-4733	59	17	(	(	PUNCT
ejpam-4733	59	18	λ	λ	PROPN
ejpam-4733	59	19	,	,	PUNCT
ejpam-4733	59	20	p)](λ	p)](λ	ADJ
ejpam-4733	59	21	,	,	PUNCT
ejpam-4733	59	22	p	p	NOUN
ejpam-4733	59	23	)	)	PUNCT
ejpam-4733	59	24	̸=	̸=	PROPN
ejpam-4733	59	25	∅	∅	NOUN
ejpam-4733	59	26	for	for	ADP
ejpam-4733	59	27	every	every	DET
ejpam-4733	59	28	(	(	PUNCT
ejpam-4733	59	29	λ	λ	NOUN
ejpam-4733	59	30	,	,	PUNCT
ejpam-4733	59	31	p)-open	p)-open	VERB
ejpam-4733	59	32	set	set	VERB
ejpam-4733	59	33	v	v	NOUN
ejpam-4733	59	34	of	of	ADP
ejpam-4733	59	35	x	x	PUNCT
ejpam-4733	59	36	containing	contain	VERB
ejpam-4733	59	37	x.	x.	NOUN
ejpam-4733	59	38	the	the	DET
ejpam-4733	59	39	set	set	NOUN
ejpam-4733	59	40	of	of	ADP
ejpam-4733	59	41	all	all	DET
ejpam-4733	59	42	δ(λ	δ(λ	PROPN
ejpam-4733	59	43	,	,	PUNCT
ejpam-4733	59	44	p)-cluster	p)-cluster	VERB
ejpam-4733	59	45	points	point	NOUN
ejpam-4733	59	46	of	of	ADP
ejpam-4733	59	47	a	a	PRON
ejpam-4733	59	48	is	be	AUX
ejpam-4733	59	49	called	call	VERB
ejpam-4733	59	50	the	the	DET
ejpam-4733	59	51	δ(λ	δ(λ	PROPN
ejpam-4733	59	52	,	,	PUNCT
ejpam-4733	59	53	p)-closure	p)-closure	NOUN
ejpam-4733	59	54	of	of	ADP
ejpam-4733	59	55	a	a	PRON
ejpam-4733	59	56	and	and	CCONJ
ejpam-4733	59	57	is	be	AUX
ejpam-4733	59	58	denoted	denote	VERB
ejpam-4733	59	59	by	by	ADP
ejpam-4733	59	60	aδ(λ	aδ(λ	NUM
ejpam-4733	59	61	,	,	PUNCT
ejpam-4733	59	62	p	p	NOUN
ejpam-4733	59	63	)	)	PUNCT
ejpam-4733	59	64	.	.	PUNCT
ejpam-4733	60	1	if	if	SCONJ
ejpam-4733	60	2	a	a	DET
ejpam-4733	60	3	=	=	NOUN
ejpam-4733	60	4	aδ(λ	aδ(λ	NUM
ejpam-4733	60	5	,	,	PUNCT
ejpam-4733	60	6	p	p	NOUN
ejpam-4733	60	7	)	)	PUNCT
ejpam-4733	60	8	,	,	PUNCT
ejpam-4733	60	9	then	then	ADV
ejpam-4733	60	10	a	a	PRON
ejpam-4733	60	11	is	be	AUX
ejpam-4733	60	12	said	say	VERB
ejpam-4733	60	13	to	to	PART
ejpam-4733	60	14	be	be	AUX
ejpam-4733	60	15	δ(λ	δ(λ	PROPN
ejpam-4733	60	16	,	,	PUNCT
ejpam-4733	60	17	p)-closed	p)-close	VERB
ejpam-4733	60	18	.	.	PUNCT
ejpam-4733	61	1	the	the	DET
ejpam-4733	61	2	complement	complement	NOUN
ejpam-4733	61	3	of	of	ADP
ejpam-4733	61	4	a	a	DET
ejpam-4733	61	5	δ(λ	δ(λ	PROPN
ejpam-4733	61	6	,	,	PUNCT
ejpam-4733	61	7	p)closed	p)close	VERB
ejpam-4733	61	8	set	set	NOUN
ejpam-4733	61	9	is	be	AUX
ejpam-4733	61	10	said	say	VERB
ejpam-4733	61	11	to	to	PART
ejpam-4733	61	12	be	be	AUX
ejpam-4733	61	13	δ(λ	δ(λ	PROPN
ejpam-4733	61	14	,	,	PUNCT
ejpam-4733	61	15	p)-open	p)-open	NOUN
ejpam-4733	61	16	.	.	PUNCT
ejpam-4733	62	1	the	the	DET
ejpam-4733	62	2	union	union	NOUN
ejpam-4733	62	3	of	of	ADP
ejpam-4733	62	4	all	all	DET
ejpam-4733	62	5	δ(λ	δ(λ	PROPN
ejpam-4733	62	6	,	,	PUNCT
ejpam-4733	62	7	p)-open	p)-open	VERB
ejpam-4733	62	8	sets	set	NOUN
ejpam-4733	62	9	contained	contain	VERB
ejpam-4733	62	10	in	in	ADP
ejpam-4733	62	11	a	a	PRON
ejpam-4733	62	12	is	be	AUX
ejpam-4733	62	13	called	call	VERB
ejpam-4733	62	14	the	the	DET
ejpam-4733	62	15	δ(λ	δ(λ	PROPN
ejpam-4733	62	16	,	,	PUNCT
ejpam-4733	62	17	p)-interior	p)-interior	ADJ
ejpam-4733	62	18	of	of	ADP
ejpam-4733	62	19	a	a	PRON
ejpam-4733	62	20	and	and	CCONJ
ejpam-4733	62	21	is	be	AUX
ejpam-4733	62	22	denoted	denote	VERB
ejpam-4733	62	23	by	by	ADP
ejpam-4733	62	24	aδ(λ	aδ(λ	NUM
ejpam-4733	62	25	,	,	PUNCT
ejpam-4733	62	26	p	p	NOUN
ejpam-4733	62	27	)	)	PUNCT
ejpam-4733	62	28	.	.	PUNCT
ejpam-4733	63	1	definition	definition	NOUN
ejpam-4733	63	2	2	2	NUM
ejpam-4733	63	3	.	.	PUNCT
ejpam-4733	64	1	a	a	DET
ejpam-4733	64	2	subset	subset	NOUN
ejpam-4733	64	3	a	a	PRON
ejpam-4733	64	4	of	of	ADP
ejpam-4733	64	5	a	a	DET
ejpam-4733	64	6	topological	topological	ADJ
ejpam-4733	64	7	space	space	NOUN
ejpam-4733	64	8	(	(	PUNCT
ejpam-4733	64	9	x	x	X
ejpam-4733	64	10	,	,	PUNCT
ejpam-4733	64	11	τ	τ	X
ejpam-4733	64	12	)	)	PUNCT
ejpam-4733	64	13	is	be	AUX
ejpam-4733	64	14	said	say	VERB
ejpam-4733	64	15	to	to	PART
ejpam-4733	64	16	be	be	AUX
ejpam-4733	64	17	δp(λ	δp(λ	NOUN
ejpam-4733	64	18	,	,	PUNCT
ejpam-4733	64	19	p)-open	p)-open	VERB
ejpam-4733	64	20	if	if	SCONJ
ejpam-4733	64	21	a	a	DET
ejpam-4733	64	22	⊆	⊆	NUM
ejpam-4733	64	23	[	[	X
ejpam-4733	64	24	a(λ	a(λ	ADV
ejpam-4733	64	25	,	,	PUNCT
ejpam-4733	64	26	p)]δ(λ	p)]δ(λ	PROPN
ejpam-4733	64	27	,	,	PUNCT
ejpam-4733	64	28	p	p	NOUN
ejpam-4733	64	29	)	)	PUNCT
ejpam-4733	64	30	.	.	PUNCT
ejpam-4733	65	1	the	the	DET
ejpam-4733	65	2	complement	complement	NOUN
ejpam-4733	65	3	of	of	ADP
ejpam-4733	65	4	a	a	DET
ejpam-4733	65	5	δp(λ	δp(λ	NOUN
ejpam-4733	65	6	,	,	PUNCT
ejpam-4733	65	7	p)-open	p)-open	VERB
ejpam-4733	65	8	set	set	NOUN
ejpam-4733	65	9	is	be	AUX
ejpam-4733	65	10	said	say	VERB
ejpam-4733	65	11	to	to	PART
ejpam-4733	65	12	be	be	AUX
ejpam-4733	65	13	δp(λ	δp(λ	NOUN
ejpam-4733	65	14	,	,	PUNCT
ejpam-4733	65	15	p)-closed	p)-close	VERB
ejpam-4733	65	16	.	.	PUNCT
ejpam-4733	66	1	the	the	DET
ejpam-4733	66	2	family	family	NOUN
ejpam-4733	66	3	of	of	ADP
ejpam-4733	66	4	all	all	PRON
ejpam-4733	66	5	δp(λ	δp(λ	NOUN
ejpam-4733	66	6	,	,	PUNCT
ejpam-4733	66	7	p)-open	p)-open	ADJ
ejpam-4733	66	8	(	(	PUNCT
ejpam-4733	66	9	resp	resp	NOUN
ejpam-4733	66	10	.	.	PUNCT
ejpam-4733	67	1	δp(λ	δp(λ	NOUN
ejpam-4733	67	2	,	,	PUNCT
ejpam-4733	67	3	p)-closed	p)-close	VERB
ejpam-4733	67	4	)	)	PUNCT
ejpam-4733	67	5	sets	set	NOUN
ejpam-4733	67	6	in	in	ADP
ejpam-4733	67	7	a	a	DET
ejpam-4733	67	8	topological	topological	ADJ
ejpam-4733	67	9	space	space	NOUN
ejpam-4733	67	10	(	(	PUNCT
ejpam-4733	67	11	x	x	X
ejpam-4733	67	12	,	,	PUNCT
ejpam-4733	67	13	τ	τ	X
ejpam-4733	67	14	)	)	PUNCT
ejpam-4733	67	15	is	be	AUX
ejpam-4733	67	16	denoted	denote	VERB
ejpam-4733	67	17	by	by	ADP
ejpam-4733	67	18	δp(λ	δp(λ	NOUN
ejpam-4733	67	19	,	,	PUNCT
ejpam-4733	67	20	p)o(x	p)o(x	ADJ
ejpam-4733	67	21	,	,	PUNCT
ejpam-4733	67	22	τ	τ	PROPN
ejpam-4733	67	23	)	)	PUNCT
ejpam-4733	67	24	(	(	PUNCT
ejpam-4733	67	25	resp	resp	NOUN
ejpam-4733	67	26	.	.	PUNCT
ejpam-4733	68	1	δp(λ	δp(λ	PROPN
ejpam-4733	68	2	,	,	PUNCT
ejpam-4733	68	3	p)c(x	p)c(x	PROPN
ejpam-4733	68	4	,	,	PUNCT
ejpam-4733	68	5	τ	τ	NOUN
ejpam-4733	68	6	)	)	PUNCT
ejpam-4733	68	7	)	)	PUNCT
ejpam-4733	68	8	.	.	PUNCT
ejpam-4733	69	1	let	let	VERB
ejpam-4733	69	2	a	a	DET
ejpam-4733	69	3	be	be	AUX
ejpam-4733	69	4	a	a	DET
ejpam-4733	69	5	subset	subset	NOUN
ejpam-4733	69	6	of	of	ADP
ejpam-4733	69	7	a	a	DET
ejpam-4733	69	8	topological	topological	ADJ
ejpam-4733	69	9	space	space	NOUN
ejpam-4733	69	10	(	(	PUNCT
ejpam-4733	69	11	x	x	X
ejpam-4733	69	12	,	,	PUNCT
ejpam-4733	69	13	τ	τ	PROPN
ejpam-4733	69	14	)	)	PUNCT
ejpam-4733	69	15	.	.	PUNCT
ejpam-4733	70	1	the	the	DET
ejpam-4733	70	2	intersection	intersection	NOUN
ejpam-4733	70	3	of	of	ADP
ejpam-4733	70	4	all	all	DET
ejpam-4733	70	5	δp(λ	δp(λ	NOUN
ejpam-4733	70	6	,	,	PUNCT
ejpam-4733	70	7	p)-closed	p)-close	VERB
ejpam-4733	70	8	sets	set	NOUN
ejpam-4733	70	9	containing	contain	VERB
ejpam-4733	70	10	a	a	PRON
ejpam-4733	70	11	is	be	AUX
ejpam-4733	70	12	called	call	VERB
ejpam-4733	70	13	the	the	DET
ejpam-4733	70	14	δp(λ	δp(λ	NOUN
ejpam-4733	70	15	,	,	PUNCT
ejpam-4733	70	16	p)closure	p)closure	NOUN
ejpam-4733	70	17	of	of	ADP
ejpam-4733	70	18	a	a	PRON
ejpam-4733	70	19	and	and	CCONJ
ejpam-4733	70	20	is	be	AUX
ejpam-4733	70	21	denoted	denote	VERB
ejpam-4733	70	22	by	by	ADP
ejpam-4733	70	23	aδp(λ	aδp(λ	PROPN
ejpam-4733	70	24	,	,	PUNCT
ejpam-4733	70	25	p	p	NOUN
ejpam-4733	70	26	)	)	PUNCT
ejpam-4733	70	27	.	.	PUNCT
ejpam-4733	71	1	lemma	lemma	PROPN
ejpam-4733	71	2	1	1	NUM
ejpam-4733	71	3	.	.	PUNCT
ejpam-4733	72	1	for	for	ADP
ejpam-4733	72	2	the	the	DET
ejpam-4733	72	3	δp(λ	δp(λ	NOUN
ejpam-4733	72	4	,	,	PUNCT
ejpam-4733	72	5	p)-closure	p)-closure	NOUN
ejpam-4733	72	6	of	of	ADP
ejpam-4733	72	7	subsets	subset	NOUN
ejpam-4733	72	8	a	a	PRON
ejpam-4733	72	9	,	,	PUNCT
ejpam-4733	72	10	b	b	NOUN
ejpam-4733	72	11	in	in	ADP
ejpam-4733	72	12	a	a	DET
ejpam-4733	72	13	topological	topological	ADJ
ejpam-4733	72	14	space	space	NOUN
ejpam-4733	72	15	(	(	PUNCT
ejpam-4733	72	16	x	x	X
ejpam-4733	72	17	,	,	PUNCT
ejpam-4733	72	18	τ	τ	PROPN
ejpam-4733	72	19	)	)	PUNCT
ejpam-4733	72	20	,	,	PUNCT
ejpam-4733	72	21	the	the	DET
ejpam-4733	72	22	following	follow	VERB
ejpam-4733	72	23	properties	property	NOUN
ejpam-4733	72	24	hold	hold	VERB
ejpam-4733	72	25	:	:	PUNCT
ejpam-4733	72	26	c.	c.	PROPN
ejpam-4733	72	27	boonpok	boonpok	PROPN
ejpam-4733	72	28	,	,	PUNCT
ejpam-4733	72	29	m.	m.	NOUN
ejpam-4733	72	30	thongmoon	thongmoon	PROPN
ejpam-4733	72	31	/	/	SYM
ejpam-4733	72	32	eur	eur	PROPN
ejpam-4733	72	33	.	.	PUNCT
ejpam-4733	73	1	j.	j.	PROPN
ejpam-4733	73	2	pure	pure	PROPN
ejpam-4733	73	3	appl	appl	PROPN
ejpam-4733	73	4	.	.	PROPN
ejpam-4733	73	5	math	math	PROPN
ejpam-4733	73	6	,	,	PUNCT
ejpam-4733	73	7	16	16	NUM
ejpam-4733	73	8	(	(	PUNCT
ejpam-4733	73	9	3	3	NUM
ejpam-4733	73	10	)	)	PUNCT
ejpam-4733	73	11	(	(	PUNCT
ejpam-4733	73	12	2023	2023	NUM
ejpam-4733	73	13	)	)	PUNCT
ejpam-4733	73	14	,	,	PUNCT
ejpam-4733	73	15	1533	1533	NUM
ejpam-4733	73	16	-	-	SYM
ejpam-4733	73	17	1542	1542	NUM
ejpam-4733	73	18	1535	1535	NUM
ejpam-4733	73	19	(	(	PUNCT
ejpam-4733	73	20	1	1	NUM
ejpam-4733	73	21	)	)	PUNCT
ejpam-4733	73	22	if	if	SCONJ
ejpam-4733	73	23	a	a	DET
ejpam-4733	73	24	⊆	⊆	NUM
ejpam-4733	73	25	b	b	NOUN
ejpam-4733	73	26	,	,	PUNCT
ejpam-4733	73	27	then	then	ADV
ejpam-4733	73	28	aδp(λ	aδp(λ	PROPN
ejpam-4733	73	29	,	,	PUNCT
ejpam-4733	73	30	p	p	NOUN
ejpam-4733	73	31	)	)	PUNCT
ejpam-4733	73	32	⊆	⊆	NUM
ejpam-4733	73	33	bδp(λ	bδp(λ	PROPN
ejpam-4733	73	34	,	,	PUNCT
ejpam-4733	73	35	p	p	NOUN
ejpam-4733	73	36	)	)	PUNCT
ejpam-4733	73	37	.	.	PUNCT
ejpam-4733	74	1	(	(	PUNCT
ejpam-4733	74	2	2	2	X
ejpam-4733	74	3	)	)	PUNCT
ejpam-4733	74	4	a	a	PRON
ejpam-4733	74	5	is	is	NOUN
ejpam-4733	74	6	δp(λ	δp(λ	NOUN
ejpam-4733	74	7	,	,	PUNCT
ejpam-4733	74	8	p)-closed	p)-close	VERB
ejpam-4733	74	9	in	in	ADP
ejpam-4733	74	10	(	(	PUNCT
ejpam-4733	74	11	x	x	X
ejpam-4733	74	12	,	,	PUNCT
ejpam-4733	74	13	τ	τ	X
ejpam-4733	74	14	)	)	PUNCT
ejpam-4733	74	15	if	if	SCONJ
ejpam-4733	74	16	and	and	CCONJ
ejpam-4733	74	17	only	only	ADV
ejpam-4733	74	18	if	if	SCONJ
ejpam-4733	74	19	a	a	DET
ejpam-4733	74	20	=	=	X
ejpam-4733	74	21	aδp(λ	aδp(λ	PROPN
ejpam-4733	74	22	,	,	PUNCT
ejpam-4733	74	23	p	p	NOUN
ejpam-4733	74	24	)	)	PUNCT
ejpam-4733	74	25	.	.	PUNCT
ejpam-4733	75	1	(	(	PUNCT
ejpam-4733	75	2	3	3	X
ejpam-4733	75	3	)	)	PUNCT
ejpam-4733	75	4	aδp(λ	aδp(λ	PROPN
ejpam-4733	75	5	,	,	PUNCT
ejpam-4733	75	6	p	p	NOUN
ejpam-4733	75	7	)	)	PUNCT
ejpam-4733	75	8	is	be	AUX
ejpam-4733	75	9	δp(λ	δp(λ	NOUN
ejpam-4733	75	10	,	,	PUNCT
ejpam-4733	75	11	p)-closed	p)-close	VERB
ejpam-4733	75	12	,	,	PUNCT
ejpam-4733	75	13	that	that	ADV
ejpam-4733	75	14	is	is	ADV
ejpam-4733	75	15	,	,	PUNCT
ejpam-4733	75	16	aδp(λ	aδp(λ	PROPN
ejpam-4733	75	17	,	,	PUNCT
ejpam-4733	75	18	p	p	NOUN
ejpam-4733	75	19	)	)	PUNCT
ejpam-4733	75	20	=	=	NOUN
ejpam-4733	76	1	[	[	X
ejpam-4733	76	2	aδp(λ	aδp(λ	PROPN
ejpam-4733	76	3	,	,	PUNCT
ejpam-4733	76	4	p)]δp(λ	p)]δp(λ	PROPN
ejpam-4733	76	5	,	,	PUNCT
ejpam-4733	76	6	p	p	NOUN
ejpam-4733	76	7	)	)	PUNCT
ejpam-4733	76	8	.	.	PUNCT
ejpam-4733	77	1	(	(	PUNCT
ejpam-4733	77	2	4	4	X
ejpam-4733	77	3	)	)	PUNCT
ejpam-4733	77	4	x	x	SYM
ejpam-4733	77	5	∈	∈	PROPN
ejpam-4733	77	6	aδp(λ	aδp(λ	PROPN
ejpam-4733	77	7	,	,	PUNCT
ejpam-4733	77	8	p	p	NOUN
ejpam-4733	77	9	)	)	PUNCT
ejpam-4733	77	10	if	if	SCONJ
ejpam-4733	77	11	and	and	CCONJ
ejpam-4733	77	12	only	only	ADV
ejpam-4733	77	13	if	if	SCONJ
ejpam-4733	77	14	a	a	DET
ejpam-4733	77	15	∩	∩	NOUN
ejpam-4733	77	16	v	v	ADP
ejpam-4733	77	17	̸=	̸=	PROPN
ejpam-4733	77	18	∅	∅	NOUN
ejpam-4733	77	19	for	for	ADP
ejpam-4733	77	20	every	every	DET
ejpam-4733	77	21	v	v	NOUN
ejpam-4733	77	22	∈	∈	PROPN
ejpam-4733	77	23	δp(λ	δp(λ	NOUN
ejpam-4733	77	24	,	,	PUNCT
ejpam-4733	77	25	p)o(x	p)o(x	ADJ
ejpam-4733	77	26	,	,	PUNCT
ejpam-4733	77	27	τ	τ	X
ejpam-4733	77	28	)	)	PUNCT
ejpam-4733	77	29	containing	contain	VERB
ejpam-4733	77	30	x.	x.	PROPN
ejpam-4733	77	31	lemma	lemma	PROPN
ejpam-4733	77	32	2	2	NUM
ejpam-4733	77	33	.	.	PUNCT
ejpam-4733	78	1	for	for	ADP
ejpam-4733	78	2	a	a	DET
ejpam-4733	78	3	family	family	NOUN
ejpam-4733	78	4	{	{	PUNCT
ejpam-4733	78	5	aγ	aγ	INTJ
ejpam-4733	78	6	|	|	ADV
ejpam-4733	78	7	γ	γ	X
ejpam-4733	78	8	∈	∈	PROPN
ejpam-4733	78	9	∇	∇	X
ejpam-4733	78	10	}	}	PUNCT
ejpam-4733	78	11	of	of	ADP
ejpam-4733	78	12	a	a	DET
ejpam-4733	78	13	topological	topological	ADJ
ejpam-4733	78	14	space	space	NOUN
ejpam-4733	78	15	(	(	PUNCT
ejpam-4733	78	16	x	x	X
ejpam-4733	78	17	,	,	PUNCT
ejpam-4733	78	18	τ	τ	PROPN
ejpam-4733	78	19	)	)	PUNCT
ejpam-4733	78	20	,	,	PUNCT
ejpam-4733	78	21	the	the	DET
ejpam-4733	78	22	following	follow	VERB
ejpam-4733	78	23	properties	property	NOUN
ejpam-4733	78	24	hold	hold	VERB
ejpam-4733	78	25	:	:	PUNCT
ejpam-4733	78	26	(	(	PUNCT
ejpam-4733	78	27	1	1	X
ejpam-4733	78	28	)	)	PUNCT
ejpam-4733	79	1	[	[	X
ejpam-4733	79	2	∩{aγ	∩{aγ	VERB
ejpam-4733	79	3	|	|	ADV
ejpam-4733	79	4	γ	γ	X
ejpam-4733	79	5	∈	∈	PROPN
ejpam-4733	79	6	∇}]δp(λ	∇}]δp(λ	X
ejpam-4733	79	7	,	,	PUNCT
ejpam-4733	79	8	p	p	NOUN
ejpam-4733	79	9	)	)	PUNCT
ejpam-4733	79	10	⊆	⊆	NUM
ejpam-4733	79	11	∩{aδp(λ	∩{aδp(λ	NOUN
ejpam-4733	79	12	,	,	PUNCT
ejpam-4733	79	13	p	p	NOUN
ejpam-4733	79	14	)	)	PUNCT
ejpam-4733	79	15	γ	γ	PROPN
ejpam-4733	79	16	|	|	ADV
ejpam-4733	79	17	γ	γ	X
ejpam-4733	79	18	∈	∈	NOUN
ejpam-4733	79	19	∇	∇	X
ejpam-4733	79	20	}	}	PUNCT
ejpam-4733	79	21	.	.	PUNCT
ejpam-4733	80	1	(	(	PUNCT
ejpam-4733	80	2	2	2	X
ejpam-4733	80	3	)	)	PUNCT
ejpam-4733	80	4	[	[	X
ejpam-4733	80	5	∪{aγ	∪{aγ	PROPN
ejpam-4733	80	6	|	|	ADV
ejpam-4733	80	7	γ	γ	X
ejpam-4733	80	8	∈	∈	PROPN
ejpam-4733	80	9	∇}]δp(λ	∇}]δp(λ	X
ejpam-4733	80	10	,	,	PUNCT
ejpam-4733	80	11	p	p	NOUN
ejpam-4733	80	12	)	)	PUNCT
ejpam-4733	80	13	⊇	⊇	NOUN
ejpam-4733	80	14	∪{aδp(λ	∪{aδp(λ	NOUN
ejpam-4733	80	15	,	,	PUNCT
ejpam-4733	80	16	p	p	NOUN
ejpam-4733	80	17	)	)	PUNCT
ejpam-4733	80	18	γ	γ	PROPN
ejpam-4733	80	19	|	|	ADV
ejpam-4733	80	20	γ	γ	X
ejpam-4733	80	21	∈	∈	NOUN
ejpam-4733	80	22	∇	∇	X
ejpam-4733	80	23	}	}	PUNCT
ejpam-4733	80	24	.	.	PUNCT
ejpam-4733	81	1	definition	definition	NOUN
ejpam-4733	81	2	3	3	NUM
ejpam-4733	81	3	.	.	PUNCT
ejpam-4733	82	1	a	a	DET
ejpam-4733	82	2	subset	subset	NOUN
ejpam-4733	82	3	a	a	PRON
ejpam-4733	82	4	of	of	ADP
ejpam-4733	82	5	a	a	DET
ejpam-4733	82	6	topological	topological	ADJ
ejpam-4733	82	7	space	space	NOUN
ejpam-4733	82	8	(	(	PUNCT
ejpam-4733	82	9	x	x	X
ejpam-4733	82	10	,	,	PUNCT
ejpam-4733	82	11	τ	τ	X
ejpam-4733	82	12	)	)	PUNCT
ejpam-4733	82	13	is	be	AUX
ejpam-4733	82	14	called	call	VERB
ejpam-4733	82	15	a	a	DET
ejpam-4733	82	16	δp(λ	δp(λ	NOUN
ejpam-4733	82	17	,	,	PUNCT
ejpam-4733	82	18	p)d	p)d	NOUN
ejpam-4733	82	19	-	-	PUNCT
ejpam-4733	82	20	set	set	ADJ
ejpam-4733	82	21	if	if	SCONJ
ejpam-4733	82	22	there	there	PRON
ejpam-4733	82	23	exist	exist	VERB
ejpam-4733	82	24	δp(λ	δp(λ	NOUN
ejpam-4733	82	25	,	,	PUNCT
ejpam-4733	82	26	p)-open	p)-open	VERB
ejpam-4733	82	27	sets	set	VERB
ejpam-4733	82	28	u	u	NOUN
ejpam-4733	82	29	and	and	CCONJ
ejpam-4733	82	30	v	v	ADP
ejpam-4733	82	31	such	such	ADJ
ejpam-4733	82	32	that	that	SCONJ
ejpam-4733	82	33	u	u	NOUN
ejpam-4733	82	34	̸=	̸=	PROPN
ejpam-4733	82	35	x	x	PUNCT
ejpam-4733	82	36	and	and	CCONJ
ejpam-4733	82	37	a	a	DET
ejpam-4733	82	38	=	=	X
ejpam-4733	82	39	u	u	NOUN
ejpam-4733	82	40	−	−	PROPN
ejpam-4733	82	41	v	v	NOUN
ejpam-4733	82	42	.	.	PUNCT
ejpam-4733	83	1	definition	definition	NOUN
ejpam-4733	83	2	4	4	NUM
ejpam-4733	83	3	.	.	PUNCT
ejpam-4733	84	1	a	a	DET
ejpam-4733	84	2	topological	topological	ADJ
ejpam-4733	84	3	space	space	NOUN
ejpam-4733	84	4	(	(	PUNCT
ejpam-4733	84	5	x	x	X
ejpam-4733	84	6	,	,	PUNCT
ejpam-4733	84	7	τ	τ	X
ejpam-4733	84	8	)	)	PUNCT
ejpam-4733	84	9	is	be	AUX
ejpam-4733	84	10	said	say	VERB
ejpam-4733	84	11	to	to	PART
ejpam-4733	84	12	be	be	AUX
ejpam-4733	84	13	:	:	PUNCT
ejpam-4733	84	14	(	(	PUNCT
ejpam-4733	84	15	i	i	NOUN
ejpam-4733	84	16	)	)	PUNCT
ejpam-4733	84	17	δp(λ	δp(λ	NOUN
ejpam-4733	84	18	,	,	PUNCT
ejpam-4733	84	19	p)-t1	p)-t1	VERB
ejpam-4733	84	20	if	if	SCONJ
ejpam-4733	84	21	for	for	ADP
ejpam-4733	84	22	any	any	DET
ejpam-4733	84	23	distinct	distinct	ADJ
ejpam-4733	84	24	pair	pair	NOUN
ejpam-4733	84	25	of	of	ADP
ejpam-4733	84	26	points	point	NOUN
ejpam-4733	84	27	x	x	PUNCT
ejpam-4733	84	28	and	and	CCONJ
ejpam-4733	84	29	y	y	PROPN
ejpam-4733	84	30	of	of	ADP
ejpam-4733	84	31	x	x	PRON
ejpam-4733	84	32	,	,	PUNCT
ejpam-4733	84	33	there	there	PRON
ejpam-4733	84	34	exist	exist	VERB
ejpam-4733	84	35	a	a	DET
ejpam-4733	84	36	δp(λ	δp(λ	NOUN
ejpam-4733	84	37	,	,	PUNCT
ejpam-4733	84	38	p)-open	p)-open	VERB
ejpam-4733	84	39	set	set	VERB
ejpam-4733	84	40	u	u	NOUN
ejpam-4733	84	41	of	of	ADP
ejpam-4733	84	42	x	x	PUNCT
ejpam-4733	84	43	containing	contain	VERB
ejpam-4733	84	44	x	x	NOUN
ejpam-4733	84	45	but	but	CCONJ
ejpam-4733	84	46	not	not	PART
ejpam-4733	84	47	y	y	PROPN
ejpam-4733	84	48	and	and	CCONJ
ejpam-4733	84	49	a	a	DET
ejpam-4733	84	50	δp(λ	δp(λ	NOUN
ejpam-4733	84	51	,	,	PUNCT
ejpam-4733	84	52	p)-open	p)-open	VERB
ejpam-4733	84	53	set	set	VERB
ejpam-4733	84	54	v	v	NOUN
ejpam-4733	84	55	of	of	ADP
ejpam-4733	84	56	x	x	PUNCT
ejpam-4733	84	57	containing	contain	VERB
ejpam-4733	84	58	y	y	NOUN
ejpam-4733	84	59	but	but	CCONJ
ejpam-4733	84	60	not	not	PART
ejpam-4733	84	61	x	x	ADP
ejpam-4733	84	62	;	;	PUNCT
ejpam-4733	84	63	(	(	PUNCT
ejpam-4733	84	64	ii	ii	NOUN
ejpam-4733	84	65	)	)	PUNCT
ejpam-4733	84	66	δp(λ	δp(λ	NOUN
ejpam-4733	84	67	,	,	PUNCT
ejpam-4733	84	68	p)-d1	p)-d1	NOUN
ejpam-4733	84	69	if	if	SCONJ
ejpam-4733	84	70	for	for	ADP
ejpam-4733	84	71	any	any	DET
ejpam-4733	84	72	distinct	distinct	ADJ
ejpam-4733	84	73	pair	pair	NOUN
ejpam-4733	84	74	of	of	ADP
ejpam-4733	84	75	points	point	NOUN
ejpam-4733	84	76	x	x	PUNCT
ejpam-4733	84	77	and	and	CCONJ
ejpam-4733	84	78	y	y	PROPN
ejpam-4733	84	79	of	of	ADP
ejpam-4733	84	80	x	x	PRON
ejpam-4733	84	81	,	,	PUNCT
ejpam-4733	84	82	there	there	PRON
ejpam-4733	84	83	exist	exist	VERB
ejpam-4733	84	84	a	a	DET
ejpam-4733	84	85	δp(λ	δp(λ	NOUN
ejpam-4733	84	86	,	,	PUNCT
ejpam-4733	84	87	p)d	p)d	ADJ
ejpam-4733	84	88	-	-	PUNCT
ejpam-4733	84	89	set	set	ADJ
ejpam-4733	84	90	u	u	NOUN
ejpam-4733	84	91	of	of	ADP
ejpam-4733	84	92	x	x	PUNCT
ejpam-4733	84	93	containing	contain	VERB
ejpam-4733	84	94	x	x	NOUN
ejpam-4733	84	95	but	but	CCONJ
ejpam-4733	84	96	not	not	PART
ejpam-4733	84	97	y	y	PROPN
ejpam-4733	84	98	and	and	CCONJ
ejpam-4733	84	99	a	a	DET
ejpam-4733	84	100	δp(λ	δp(λ	NOUN
ejpam-4733	84	101	,	,	PUNCT
ejpam-4733	84	102	p)d	p)d	ADJ
ejpam-4733	84	103	-	-	PUNCT
ejpam-4733	84	104	set	set	VERB
ejpam-4733	84	105	v	v	NOUN
ejpam-4733	84	106	of	of	ADP
ejpam-4733	84	107	x	x	PUNCT
ejpam-4733	84	108	containing	contain	VERB
ejpam-4733	84	109	y	y	NOUN
ejpam-4733	84	110	but	but	CCONJ
ejpam-4733	84	111	not	not	PART
ejpam-4733	84	112	x.	x.	NOUN
ejpam-4733	84	113	definition	definition	NOUN
ejpam-4733	84	114	5	5	NUM
ejpam-4733	84	115	.	.	PUNCT
ejpam-4733	84	116	a	a	DET
ejpam-4733	84	117	subset	subset	NOUN
ejpam-4733	84	118	n	n	NOUN
ejpam-4733	84	119	of	of	ADP
ejpam-4733	84	120	a	a	DET
ejpam-4733	84	121	topological	topological	ADJ
ejpam-4733	84	122	space	space	NOUN
ejpam-4733	84	123	(	(	PUNCT
ejpam-4733	84	124	x	x	X
ejpam-4733	84	125	,	,	PUNCT
ejpam-4733	84	126	τ	τ	X
ejpam-4733	84	127	)	)	PUNCT
ejpam-4733	84	128	is	be	AUX
ejpam-4733	84	129	called	call	VERB
ejpam-4733	84	130	a	a	DET
ejpam-4733	84	131	δp(λ	δp(λ	NOUN
ejpam-4733	84	132	,	,	PUNCT
ejpam-4733	84	133	p)-neighborhood	p)-neighborhood	NOUN
ejpam-4733	84	134	of	of	ADP
ejpam-4733	84	135	a	a	DET
ejpam-4733	84	136	point	point	NOUN
ejpam-4733	84	137	x	x	X
ejpam-4733	84	138	∈	∈	NOUN
ejpam-4733	84	139	x	x	INTJ
ejpam-4733	84	140	if	if	SCONJ
ejpam-4733	84	141	there	there	PRON
ejpam-4733	84	142	exists	exist	VERB
ejpam-4733	84	143	a	a	DET
ejpam-4733	84	144	δp(λ	δp(λ	NOUN
ejpam-4733	84	145	,	,	PUNCT
ejpam-4733	84	146	p)-open	p)-open	VERB
ejpam-4733	84	147	set	set	VERB
ejpam-4733	84	148	u	u	PRON
ejpam-4733	84	149	such	such	ADJ
ejpam-4733	84	150	that	that	SCONJ
ejpam-4733	84	151	x	x	SYM
ejpam-4733	84	152	∈	∈	PROPN
ejpam-4733	84	153	u	u	NOUN
ejpam-4733	84	154	⊆	⊆	NUM
ejpam-4733	84	155	n	n	NOUN
ejpam-4733	84	156	.	.	PUNCT
ejpam-4733	85	1	definition	definition	NOUN
ejpam-4733	85	2	6	6	NUM
ejpam-4733	85	3	.	.	PUNCT
ejpam-4733	86	1	let	let	VERB
ejpam-4733	86	2	(	(	PUNCT
ejpam-4733	86	3	x	x	NOUN
ejpam-4733	86	4	,	,	PUNCT
ejpam-4733	86	5	τ	τ	X
ejpam-4733	86	6	)	)	PUNCT
ejpam-4733	86	7	be	be	VERB
ejpam-4733	86	8	a	a	DET
ejpam-4733	86	9	topological	topological	ADJ
ejpam-4733	86	10	space	space	NOUN
ejpam-4733	86	11	.	.	PUNCT
ejpam-4733	87	1	a	a	DET
ejpam-4733	87	2	point	point	NOUN
ejpam-4733	87	3	x	x	X
ejpam-4733	87	4	∈	∈	NOUN
ejpam-4733	87	5	x	x	PUNCT
ejpam-4733	87	6	which	which	PRON
ejpam-4733	87	7	has	have	VERB
ejpam-4733	87	8	only	only	ADV
ejpam-4733	87	9	x	x	PART
ejpam-4733	87	10	as	as	ADP
ejpam-4733	87	11	the	the	DET
ejpam-4733	87	12	δp(λ	δp(λ	NOUN
ejpam-4733	87	13	,	,	PUNCT
ejpam-4733	87	14	p)-neighbourhood	p)-neighbourhood	PRON
ejpam-4733	87	15	is	be	AUX
ejpam-4733	87	16	called	call	VERB
ejpam-4733	87	17	a	a	DET
ejpam-4733	87	18	δp(λ	δp(λ	NOUN
ejpam-4733	87	19	,	,	PUNCT
ejpam-4733	87	20	p)-neat	p)-neat	NOUN
ejpam-4733	87	21	point	point	NOUN
ejpam-4733	87	22	.	.	PUNCT
ejpam-4733	88	1	lemma	lemma	PROPN
ejpam-4733	88	2	3	3	X
ejpam-4733	88	3	.	.	PUNCT
ejpam-4733	89	1	let	let	AUX
ejpam-4733	89	2	(	(	PUNCT
ejpam-4733	89	3	x	x	NOUN
ejpam-4733	89	4	,	,	PUNCT
ejpam-4733	89	5	τ	τ	X
ejpam-4733	89	6	)	)	PUNCT
ejpam-4733	89	7	be	be	VERB
ejpam-4733	89	8	a	a	DET
ejpam-4733	89	9	topological	topological	ADJ
ejpam-4733	89	10	space	space	NOUN
ejpam-4733	89	11	.	.	PUNCT
ejpam-4733	90	1	for	for	ADP
ejpam-4733	90	2	each	each	DET
ejpam-4733	90	3	point	point	NOUN
ejpam-4733	90	4	x	x	X
ejpam-4733	90	5	∈	∈	NOUN
ejpam-4733	90	6	x	x	X
ejpam-4733	90	7	,	,	PUNCT
ejpam-4733	90	8	{	{	PUNCT
ejpam-4733	90	9	x	x	NOUN
ejpam-4733	90	10	}	}	PUNCT
ejpam-4733	90	11	is	be	AUX
ejpam-4733	90	12	p(λ	p(λ	NOUN
ejpam-4733	90	13	,	,	PUNCT
ejpam-4733	90	14	p)-open	p)-open	NOUN
ejpam-4733	90	15	or	or	CCONJ
ejpam-4733	90	16	p(λ	p(λ	NOUN
ejpam-4733	90	17	,	,	PUNCT
ejpam-4733	90	18	p)-closed	p)-close	VERB
ejpam-4733	90	19	.	.	PUNCT
ejpam-4733	91	1	theorem	theorem	NOUN
ejpam-4733	91	2	1	1	NUM
ejpam-4733	91	3	.	.	X
ejpam-4733	91	4	for	for	ADP
ejpam-4733	91	5	a	a	DET
ejpam-4733	91	6	topological	topological	ADJ
ejpam-4733	91	7	space	space	NOUN
ejpam-4733	91	8	(	(	PUNCT
ejpam-4733	91	9	x	x	X
ejpam-4733	91	10	,	,	PUNCT
ejpam-4733	91	11	τ	τ	PROPN
ejpam-4733	91	12	)	)	PUNCT
ejpam-4733	91	13	,	,	PUNCT
ejpam-4733	91	14	the	the	DET
ejpam-4733	91	15	following	follow	VERB
ejpam-4733	91	16	properties	property	NOUN
ejpam-4733	91	17	are	be	AUX
ejpam-4733	91	18	equivalent	equivalent	ADJ
ejpam-4733	91	19	:	:	PUNCT
ejpam-4733	91	20	(	(	PUNCT
ejpam-4733	91	21	1	1	X
ejpam-4733	91	22	)	)	PUNCT
ejpam-4733	91	23	(	(	PUNCT
ejpam-4733	91	24	x	x	X
ejpam-4733	91	25	,	,	PUNCT
ejpam-4733	91	26	τ	τ	X
ejpam-4733	91	27	)	)	PUNCT
ejpam-4733	91	28	is	be	AUX
ejpam-4733	91	29	δp(λ	δp(λ	NOUN
ejpam-4733	91	30	,	,	PUNCT
ejpam-4733	91	31	p)-d1	p)-d1	NUM
ejpam-4733	91	32	;	;	PUNCT
ejpam-4733	91	33	(	(	PUNCT
ejpam-4733	91	34	2	2	X
ejpam-4733	91	35	)	)	PUNCT
ejpam-4733	91	36	(	(	PUNCT
ejpam-4733	91	37	x	x	X
ejpam-4733	91	38	,	,	PUNCT
ejpam-4733	91	39	τ	τ	X
ejpam-4733	91	40	)	)	PUNCT
ejpam-4733	91	41	has	have	VERB
ejpam-4733	91	42	no	no	DET
ejpam-4733	91	43	δp(λ	δp(λ	NOUN
ejpam-4733	91	44	,	,	PUNCT
ejpam-4733	91	45	p)-neat	p)-neat	NOUN
ejpam-4733	91	46	point	point	NOUN
ejpam-4733	91	47	.	.	PUNCT
ejpam-4733	92	1	proof	proof	NOUN
ejpam-4733	92	2	.	.	PUNCT
ejpam-4733	93	1	(	(	PUNCT
ejpam-4733	93	2	1	1	X
ejpam-4733	93	3	)	)	PUNCT
ejpam-4733	93	4	⇒	⇒	NOUN
ejpam-4733	93	5	(	(	PUNCT
ejpam-4733	93	6	2	2	NUM
ejpam-4733	93	7	):	):	PUNCT
ejpam-4733	93	8	since	since	SCONJ
ejpam-4733	93	9	(	(	PUNCT
ejpam-4733	93	10	x	x	X
ejpam-4733	93	11	,	,	PUNCT
ejpam-4733	93	12	τ	τ	X
ejpam-4733	93	13	)	)	PUNCT
ejpam-4733	93	14	is	be	AUX
ejpam-4733	93	15	δp(λ	δp(λ	NOUN
ejpam-4733	93	16	,	,	PUNCT
ejpam-4733	93	17	p)-d1	p)-d1	NOUN
ejpam-4733	93	18	,	,	PUNCT
ejpam-4733	93	19	so	so	ADV
ejpam-4733	93	20	each	each	DET
ejpam-4733	93	21	point	point	NOUN
ejpam-4733	93	22	x	x	PUNCT
ejpam-4733	93	23	of	of	ADP
ejpam-4733	93	24	x	x	PRON
ejpam-4733	93	25	is	be	AUX
ejpam-4733	93	26	contained	contain	VERB
ejpam-4733	93	27	in	in	ADP
ejpam-4733	93	28	a	a	DET
ejpam-4733	93	29	δp(λ	δp(λ	NOUN
ejpam-4733	93	30	,	,	PUNCT
ejpam-4733	93	31	p)d	p)d	NOUN
ejpam-4733	93	32	-	-	PUNCT
ejpam-4733	93	33	set	set	VERB
ejpam-4733	93	34	g	g	NOUN
ejpam-4733	93	35	=	=	PUNCT
ejpam-4733	93	36	u	u	PROPN
ejpam-4733	93	37	−	−	PROPN
ejpam-4733	93	38	v	v	NOUN
ejpam-4733	93	39	and	and	CCONJ
ejpam-4733	93	40	thus	thus	ADV
ejpam-4733	93	41	in	in	ADP
ejpam-4733	93	42	u	u	PROPN
ejpam-4733	93	43	,	,	PUNCT
ejpam-4733	93	44	where	where	SCONJ
ejpam-4733	93	45	u	u	NOUN
ejpam-4733	93	46	and	and	CCONJ
ejpam-4733	93	47	v	v	NOUN
ejpam-4733	93	48	are	be	AUX
ejpam-4733	93	49	δp(λ	δp(λ	NOUN
ejpam-4733	93	50	,	,	PUNCT
ejpam-4733	93	51	p)-open	p)-open	ADJ
ejpam-4733	93	52	sets	set	NOUN
ejpam-4733	93	53	.	.	PUNCT
ejpam-4733	94	1	by	by	ADP
ejpam-4733	94	2	definition	definition	NOUN
ejpam-4733	94	3	u	u	NOUN
ejpam-4733	94	4	̸=	̸=	PROPN
ejpam-4733	94	5	x.	x.	NOUN
ejpam-4733	94	6	this	this	PRON
ejpam-4733	94	7	implies	imply	VERB
ejpam-4733	94	8	that	that	SCONJ
ejpam-4733	94	9	x	x	PRON
ejpam-4733	94	10	is	be	AUX
ejpam-4733	94	11	not	not	PART
ejpam-4733	94	12	a	a	DET
ejpam-4733	94	13	δp(λ	δp(λ	NOUN
ejpam-4733	94	14	,	,	PUNCT
ejpam-4733	94	15	p)-neat	p)-neat	NOUN
ejpam-4733	94	16	point	point	NOUN
ejpam-4733	94	17	.	.	PUNCT
ejpam-4733	95	1	(	(	PUNCT
ejpam-4733	95	2	2	2	X
ejpam-4733	95	3	)	)	PUNCT
ejpam-4733	95	4	⇒	⇒	NOUN
ejpam-4733	95	5	(	(	PUNCT
ejpam-4733	95	6	1	1	NUM
ejpam-4733	95	7	):	):	PUNCT
ejpam-4733	95	8	by	by	ADP
ejpam-4733	95	9	lemma	lemma	PROPN
ejpam-4733	95	10	3	3	NUM
ejpam-4733	95	11	for	for	ADP
ejpam-4733	95	12	each	each	DET
ejpam-4733	95	13	distinct	distinct	ADJ
ejpam-4733	95	14	pair	pair	NOUN
ejpam-4733	95	15	of	of	ADP
ejpam-4733	95	16	points	point	NOUN
ejpam-4733	95	17	x	x	X
ejpam-4733	95	18	,	,	PUNCT
ejpam-4733	95	19	y	y	PROPN
ejpam-4733	95	20	∈	∈	PROPN
ejpam-4733	95	21	x	x	X
ejpam-4733	95	22	,	,	PUNCT
ejpam-4733	95	23	at	at	ADP
ejpam-4733	95	24	least	least	ADJ
ejpam-4733	95	25	one	one	NUM
ejpam-4733	95	26	of	of	ADP
ejpam-4733	95	27	them	they	PRON
ejpam-4733	95	28	,	,	PUNCT
ejpam-4733	95	29	x(say	x(say	PROPN
ejpam-4733	95	30	)	)	PUNCT
ejpam-4733	95	31	has	have	VERB
ejpam-4733	95	32	a	a	DET
ejpam-4733	95	33	δp(λ	δp(λ	NOUN
ejpam-4733	95	34	,	,	PUNCT
ejpam-4733	95	35	p)-neighborhood	p)-neighborhood	PUNCT
ejpam-4733	95	36	u	u	NOUN
ejpam-4733	95	37	containing	contain	VERB
ejpam-4733	95	38	x	x	X
ejpam-4733	95	39	and	and	CCONJ
ejpam-4733	95	40	not	not	PART
ejpam-4733	95	41	y.	y.	PROPN
ejpam-4733	95	42	thus	thus	ADV
ejpam-4733	95	43	,	,	PUNCT
ejpam-4733	95	44	u	u	NOUN
ejpam-4733	95	45	which	which	PRON
ejpam-4733	95	46	is	be	AUX
ejpam-4733	95	47	different	different	ADJ
ejpam-4733	95	48	from	from	ADP
ejpam-4733	95	49	x	x	SYM
ejpam-4733	95	50	is	be	AUX
ejpam-4733	95	51	a	a	DET
ejpam-4733	95	52	δp(λ	δp(λ	NOUN
ejpam-4733	95	53	,	,	PUNCT
ejpam-4733	95	54	p)d	p)d	NOUN
ejpam-4733	95	55	-	-	PUNCT
ejpam-4733	95	56	set	set	NOUN
ejpam-4733	95	57	.	.	PUNCT
ejpam-4733	96	1	if	if	SCONJ
ejpam-4733	96	2	x	x	PRON
ejpam-4733	96	3	has	have	VERB
ejpam-4733	96	4	no	no	DET
ejpam-4733	96	5	δp(λ	δp(λ	NOUN
ejpam-4733	96	6	,	,	PUNCT
ejpam-4733	96	7	p)-neat	p)-neat	NOUN
ejpam-4733	96	8	point	point	NOUN
ejpam-4733	96	9	,	,	PUNCT
ejpam-4733	96	10	then	then	ADV
ejpam-4733	96	11	y	y	PROPN
ejpam-4733	96	12	is	be	AUX
ejpam-4733	96	13	not	not	PART
ejpam-4733	96	14	a	a	DET
ejpam-4733	96	15	δp(λ	δp(λ	NOUN
ejpam-4733	96	16	,	,	PUNCT
ejpam-4733	96	17	p)-neat	p)-neat	NOUN
ejpam-4733	96	18	point	point	NOUN
ejpam-4733	96	19	.	.	PUNCT
ejpam-4733	97	1	this	this	PRON
ejpam-4733	97	2	means	mean	VERB
ejpam-4733	97	3	that	that	SCONJ
ejpam-4733	97	4	there	there	PRON
ejpam-4733	97	5	exists	exist	VERB
ejpam-4733	97	6	a	a	DET
ejpam-4733	97	7	δp(λ	δp(λ	NOUN
ejpam-4733	97	8	,	,	PUNCT
ejpam-4733	97	9	p)-neighborhood	p)-neighborhood	PUNCT
ejpam-4733	97	10	v	v	NOUN
ejpam-4733	97	11	of	of	ADP
ejpam-4733	97	12	y	y	PRON
ejpam-4733	97	13	such	such	ADJ
ejpam-4733	97	14	that	that	PRON
ejpam-4733	97	15	v	v	ADP
ejpam-4733	97	16	̸=	̸=	PROPN
ejpam-4733	97	17	x.	x.	PUNCT
ejpam-4733	97	18	thus	thus	ADV
ejpam-4733	97	19	,	,	PUNCT
ejpam-4733	97	20	y	y	PROPN
ejpam-4733	97	21	∈	∈	PROPN
ejpam-4733	97	22	v	v	ADP
ejpam-4733	97	23	−	−	PROPN
ejpam-4733	97	24	u	u	NOUN
ejpam-4733	97	25	but	but	CCONJ
ejpam-4733	97	26	not	not	PART
ejpam-4733	97	27	y	y	PROPN
ejpam-4733	97	28	and	and	CCONJ
ejpam-4733	97	29	v	v	ADP
ejpam-4733	97	30	−	−	PROPN
ejpam-4733	97	31	u	u	NOUN
ejpam-4733	97	32	is	be	AUX
ejpam-4733	97	33	a	a	DET
ejpam-4733	97	34	δp(λ	δp(λ	NOUN
ejpam-4733	97	35	,	,	PUNCT
ejpam-4733	97	36	p)d	p)d	NOUN
ejpam-4733	97	37	-	-	PUNCT
ejpam-4733	97	38	set	set	NOUN
ejpam-4733	97	39	.	.	PUNCT
ejpam-4733	98	1	this	this	PRON
ejpam-4733	98	2	shows	show	VERB
ejpam-4733	98	3	that	that	SCONJ
ejpam-4733	98	4	(	(	PUNCT
ejpam-4733	98	5	x	x	X
ejpam-4733	98	6	,	,	PUNCT
ejpam-4733	98	7	τ	τ	X
ejpam-4733	98	8	)	)	PUNCT
ejpam-4733	98	9	is	be	AUX
ejpam-4733	98	10	δp(λ	δp(λ	NOUN
ejpam-4733	98	11	,	,	PUNCT
ejpam-4733	98	12	p)-d1	p)-d1	NOUN
ejpam-4733	98	13	.	.	PUNCT
ejpam-4733	99	1	c.	c.	PROPN
ejpam-4733	99	2	boonpok	boonpok	PROPN
ejpam-4733	99	3	,	,	PUNCT
ejpam-4733	99	4	m.	m.	NOUN
ejpam-4733	99	5	thongmoon	thongmoon	PROPN
ejpam-4733	99	6	/	/	SYM
ejpam-4733	99	7	eur	eur	PROPN
ejpam-4733	99	8	.	.	PUNCT
ejpam-4733	100	1	j.	j.	PROPN
ejpam-4733	100	2	pure	pure	PROPN
ejpam-4733	100	3	appl	appl	PROPN
ejpam-4733	100	4	.	.	PROPN
ejpam-4733	100	5	math	math	PROPN
ejpam-4733	100	6	,	,	PUNCT
ejpam-4733	100	7	16	16	NUM
ejpam-4733	100	8	(	(	PUNCT
ejpam-4733	100	9	3	3	NUM
ejpam-4733	100	10	)	)	PUNCT
ejpam-4733	100	11	(	(	PUNCT
ejpam-4733	100	12	2023	2023	NUM
ejpam-4733	100	13	)	)	PUNCT
ejpam-4733	100	14	,	,	PUNCT
ejpam-4733	100	15	1533	1533	NUM
ejpam-4733	100	16	-	-	SYM
ejpam-4733	100	17	1542	1542	NUM
ejpam-4733	100	18	1536	1536	NUM
ejpam-4733	100	19	definition	definition	NOUN
ejpam-4733	100	20	7	7	NUM
ejpam-4733	100	21	.	.	PUNCT
ejpam-4733	101	1	a	a	DET
ejpam-4733	101	2	function	function	NOUN
ejpam-4733	101	3	f	f	NOUN
ejpam-4733	101	4	:	:	PUNCT
ejpam-4733	101	5	(	(	PUNCT
ejpam-4733	101	6	x	x	X
ejpam-4733	101	7	,	,	PUNCT
ejpam-4733	101	8	τ	τ	X
ejpam-4733	101	9	)	)	PUNCT
ejpam-4733	101	10	→	→	SYM
ejpam-4733	101	11	(	(	PUNCT
ejpam-4733	101	12	y	y	PROPN
ejpam-4733	101	13	,	,	PUNCT
ejpam-4733	101	14	σ	σ	PROPN
ejpam-4733	101	15	)	)	PUNCT
ejpam-4733	101	16	is	be	AUX
ejpam-4733	101	17	called	call	VERB
ejpam-4733	101	18	δp(λ	δp(λ	NOUN
ejpam-4733	101	19	,	,	PUNCT
ejpam-4733	101	20	p)-continuous	p)-continuous	ADJ
ejpam-4733	101	21	if	if	SCONJ
ejpam-4733	101	22	,	,	PUNCT
ejpam-4733	101	23	for	for	ADP
ejpam-4733	101	24	each	each	DET
ejpam-4733	101	25	x	x	SYM
ejpam-4733	101	26	∈	∈	PROPN
ejpam-4733	101	27	x	x	X
ejpam-4733	101	28	and	and	CCONJ
ejpam-4733	101	29	each	each	DET
ejpam-4733	101	30	δp(λ	δp(λ	NOUN
ejpam-4733	101	31	,	,	PUNCT
ejpam-4733	101	32	p)-open	p)-open	VERB
ejpam-4733	101	33	set	set	VERB
ejpam-4733	101	34	v	v	NOUN
ejpam-4733	101	35	of	of	ADP
ejpam-4733	101	36	y	y	NOUN
ejpam-4733	101	37	containing	contain	VERB
ejpam-4733	101	38	f(x	f(x	PROPN
ejpam-4733	101	39	)	)	PUNCT
ejpam-4733	101	40	,	,	PUNCT
ejpam-4733	101	41	there	there	PRON
ejpam-4733	101	42	exists	exist	VERB
ejpam-4733	101	43	a	a	DET
ejpam-4733	101	44	δp(λ	δp(λ	NOUN
ejpam-4733	101	45	,	,	PUNCT
ejpam-4733	101	46	p)-open	p)-open	VERB
ejpam-4733	101	47	set	set	VERB
ejpam-4733	101	48	u	u	NOUN
ejpam-4733	101	49	of	of	ADP
ejpam-4733	101	50	x	x	PUNCT
ejpam-4733	101	51	containing	contain	VERB
ejpam-4733	101	52	x	x	PUNCT
ejpam-4733	101	53	such	such	ADJ
ejpam-4733	101	54	that	that	DET
ejpam-4733	101	55	f(u	f(u	PROPN
ejpam-4733	101	56	)	)	PUNCT
ejpam-4733	101	57	⊆	⊆	NUM
ejpam-4733	101	58	v	v	NOUN
ejpam-4733	101	59	.	.	PUNCT
ejpam-4733	102	1	lemma	lemma	PROPN
ejpam-4733	102	2	4	4	NUM
ejpam-4733	102	3	.	.	PUNCT
ejpam-4733	103	1	a	a	DET
ejpam-4733	103	2	function	function	NOUN
ejpam-4733	103	3	f	f	NOUN
ejpam-4733	103	4	:	:	PUNCT
ejpam-4733	103	5	(	(	PUNCT
ejpam-4733	103	6	x	x	X
ejpam-4733	103	7	,	,	PUNCT
ejpam-4733	103	8	τ	τ	X
ejpam-4733	103	9	)	)	PUNCT
ejpam-4733	103	10	→	→	SYM
ejpam-4733	103	11	(	(	PUNCT
ejpam-4733	103	12	y	y	PROPN
ejpam-4733	103	13	,	,	PUNCT
ejpam-4733	103	14	σ	σ	PROPN
ejpam-4733	103	15	)	)	PUNCT
ejpam-4733	103	16	is	be	AUX
ejpam-4733	103	17	δp(λ	δp(λ	NOUN
ejpam-4733	103	18	,	,	PUNCT
ejpam-4733	103	19	p)-continuous	p)-continuous	ADJ
ejpam-4733	103	20	if	if	SCONJ
ejpam-4733	103	21	and	and	CCONJ
ejpam-4733	103	22	only	only	ADV
ejpam-4733	103	23	if	if	SCONJ
ejpam-4733	103	24	f−1(v	f−1(v	PROPN
ejpam-4733	103	25	)	)	PUNCT
ejpam-4733	103	26	is	be	AUX
ejpam-4733	103	27	δp(λ	δp(λ	NOUN
ejpam-4733	103	28	,	,	PUNCT
ejpam-4733	103	29	p)-open	p)-open	VERB
ejpam-4733	103	30	in	in	ADP
ejpam-4733	103	31	x	x	PUNCT
ejpam-4733	103	32	for	for	ADP
ejpam-4733	103	33	every	every	DET
ejpam-4733	103	34	δp(λ	δp(λ	NOUN
ejpam-4733	103	35	,	,	PUNCT
ejpam-4733	103	36	p)-open	p)-open	VERB
ejpam-4733	103	37	set	set	VERB
ejpam-4733	103	38	v	v	NOUN
ejpam-4733	103	39	of	of	ADP
ejpam-4733	103	40	y	y	PROPN
ejpam-4733	103	41	.	.	PUNCT
ejpam-4733	104	1	theorem	theorem	NOUN
ejpam-4733	104	2	2	2	NUM
ejpam-4733	104	3	.	.	PUNCT
ejpam-4733	105	1	if	if	SCONJ
ejpam-4733	105	2	f	f	PROPN
ejpam-4733	105	3	:	:	PUNCT
ejpam-4733	105	4	(	(	PUNCT
ejpam-4733	105	5	x	x	X
ejpam-4733	105	6	,	,	PUNCT
ejpam-4733	105	7	τ	τ	X
ejpam-4733	105	8	)	)	PUNCT
ejpam-4733	105	9	→	→	SYM
ejpam-4733	105	10	(	(	PUNCT
ejpam-4733	105	11	y	y	PROPN
ejpam-4733	105	12	,	,	PUNCT
ejpam-4733	105	13	σ	σ	PROPN
ejpam-4733	105	14	)	)	PUNCT
ejpam-4733	105	15	is	be	AUX
ejpam-4733	105	16	a	a	DET
ejpam-4733	105	17	δp(λ	δp(λ	NOUN
ejpam-4733	105	18	,	,	PUNCT
ejpam-4733	105	19	p)-continuous	p)-continuous	ADJ
ejpam-4733	105	20	surjective	surjective	ADJ
ejpam-4733	105	21	function	function	NOUN
ejpam-4733	105	22	and	and	CCONJ
ejpam-4733	105	23	b	b	NOUN
ejpam-4733	105	24	is	be	AUX
ejpam-4733	105	25	a	a	DET
ejpam-4733	105	26	δp(λ	δp(λ	NOUN
ejpam-4733	105	27	,	,	PUNCT
ejpam-4733	105	28	p)d	p)d	NOUN
ejpam-4733	105	29	-	-	PUNCT
ejpam-4733	105	30	set	set	NOUN
ejpam-4733	105	31	in	in	ADP
ejpam-4733	105	32	y	y	PROPN
ejpam-4733	105	33	,	,	PUNCT
ejpam-4733	105	34	then	then	ADV
ejpam-4733	105	35	f−1(b	f−1(b	PROPN
ejpam-4733	105	36	)	)	PUNCT
ejpam-4733	105	37	is	be	AUX
ejpam-4733	105	38	a	a	DET
ejpam-4733	105	39	δp(λ	δp(λ	NOUN
ejpam-4733	105	40	,	,	PUNCT
ejpam-4733	105	41	p)d	p)d	NOUN
ejpam-4733	105	42	-	-	PUNCT
ejpam-4733	105	43	set	set	ADJ
ejpam-4733	105	44	in	in	ADP
ejpam-4733	105	45	x.	x.	NOUN
ejpam-4733	105	46	proof	proof	NOUN
ejpam-4733	105	47	.	.	PUNCT
ejpam-4733	106	1	let	let	VERB
ejpam-4733	106	2	b	b	X
ejpam-4733	106	3	be	be	AUX
ejpam-4733	106	4	a	a	DET
ejpam-4733	106	5	δp(λ	δp(λ	NOUN
ejpam-4733	106	6	,	,	PUNCT
ejpam-4733	106	7	p)d	p)d	NOUN
ejpam-4733	106	8	-	-	PUNCT
ejpam-4733	106	9	set	set	NOUN
ejpam-4733	106	10	in	in	ADP
ejpam-4733	106	11	y	y	PROPN
ejpam-4733	106	12	.	.	PUNCT
ejpam-4733	107	1	then	then	ADV
ejpam-4733	107	2	,	,	PUNCT
ejpam-4733	107	3	there	there	PRON
ejpam-4733	107	4	exist	exist	VERB
ejpam-4733	107	5	δp(λ	δp(λ	NOUN
ejpam-4733	107	6	,	,	PUNCT
ejpam-4733	107	7	p)-open	p)-open	VERB
ejpam-4733	107	8	sets	set	VERB
ejpam-4733	107	9	u	u	NOUN
ejpam-4733	107	10	and	and	CCONJ
ejpam-4733	107	11	v	v	NOUN
ejpam-4733	107	12	in	in	ADP
ejpam-4733	107	13	y	y	PRON
ejpam-4733	107	14	such	such	ADJ
ejpam-4733	107	15	that	that	PRON
ejpam-4733	107	16	b	b	X
ejpam-4733	107	17	=	=	SYM
ejpam-4733	107	18	u	u	NOUN
ejpam-4733	107	19	−	−	PROPN
ejpam-4733	107	20	v	v	NOUN
ejpam-4733	107	21	and	and	CCONJ
ejpam-4733	107	22	u	u	NOUN
ejpam-4733	107	23	̸=	̸=	PROPN
ejpam-4733	107	24	y	y	PROPN
ejpam-4733	107	25	.	.	PUNCT
ejpam-4733	108	1	by	by	ADP
ejpam-4733	108	2	the	the	DET
ejpam-4733	108	3	δp(λ	δp(λ	NOUN
ejpam-4733	108	4	,	,	PUNCT
ejpam-4733	108	5	p)-continuity	p)-continuity	NOUN
ejpam-4733	108	6	of	of	ADP
ejpam-4733	108	7	f	f	PROPN
ejpam-4733	108	8	,	,	PUNCT
ejpam-4733	108	9	f−1(u	f−1(u	PROPN
ejpam-4733	108	10	)	)	PUNCT
ejpam-4733	108	11	and	and	CCONJ
ejpam-4733	108	12	f−1(v	f−1(v	PROPN
ejpam-4733	108	13	)	)	PUNCT
ejpam-4733	108	14	are	be	AUX
ejpam-4733	108	15	δp(λ	δp(λ	NOUN
ejpam-4733	108	16	,	,	PUNCT
ejpam-4733	108	17	p)-open	p)-open	VERB
ejpam-4733	108	18	in	in	ADP
ejpam-4733	108	19	x.	x.	NOUN
ejpam-4733	108	20	since	since	SCONJ
ejpam-4733	108	21	u	u	PROPN
ejpam-4733	108	22	̸=	̸=	PROPN
ejpam-4733	108	23	y	y	PROPN
ejpam-4733	108	24	,	,	PUNCT
ejpam-4733	108	25	we	we	PRON
ejpam-4733	108	26	have	have	VERB
ejpam-4733	108	27	f−1(u	f−1(u	NOUN
ejpam-4733	108	28	)	)	PUNCT
ejpam-4733	108	29	̸=	̸=	PROPN
ejpam-4733	108	30	x.	x.	NOUN
ejpam-4733	108	31	thus	thus	ADV
ejpam-4733	108	32	,	,	PUNCT
ejpam-4733	108	33	f−1(b	f−1(b	PROPN
ejpam-4733	108	34	)	)	PUNCT
ejpam-4733	109	1	=	=	PUNCT
ejpam-4733	109	2	f−1(u)−	f−1(u)−	NOUN
ejpam-4733	109	3	f−1(v	f−1(v	PROPN
ejpam-4733	109	4	)	)	PUNCT
ejpam-4733	109	5	is	be	AUX
ejpam-4733	109	6	a	a	DET
ejpam-4733	109	7	δp(λ	δp(λ	NOUN
ejpam-4733	109	8	,	,	PUNCT
ejpam-4733	109	9	p)d	p)d	NOUN
ejpam-4733	109	10	-	-	PUNCT
ejpam-4733	109	11	set	set	NOUN
ejpam-4733	109	12	.	.	PUNCT
ejpam-4733	110	1	theorem	theorem	NOUN
ejpam-4733	110	2	3	3	NUM
ejpam-4733	110	3	.	.	PUNCT
ejpam-4733	111	1	if	if	SCONJ
ejpam-4733	111	2	(	(	PUNCT
ejpam-4733	111	3	y	y	PROPN
ejpam-4733	111	4	,	,	PUNCT
ejpam-4733	111	5	σ	σ	PROPN
ejpam-4733	111	6	)	)	PUNCT
ejpam-4733	111	7	is	be	AUX
ejpam-4733	111	8	a	a	DET
ejpam-4733	111	9	δp(λ	δp(λ	NOUN
ejpam-4733	111	10	,	,	PUNCT
ejpam-4733	111	11	p)-d1	p)-d1	NUM
ejpam-4733	111	12	space	space	NOUN
ejpam-4733	111	13	and	and	CCONJ
ejpam-4733	111	14	f	f	NOUN
ejpam-4733	111	15	:	:	PUNCT
ejpam-4733	111	16	(	(	PUNCT
ejpam-4733	111	17	x	x	X
ejpam-4733	111	18	,	,	PUNCT
ejpam-4733	111	19	τ	τ	X
ejpam-4733	111	20	)	)	PUNCT
ejpam-4733	111	21	→	→	SYM
ejpam-4733	111	22	(	(	PUNCT
ejpam-4733	111	23	y	y	PROPN
ejpam-4733	111	24	,	,	PUNCT
ejpam-4733	111	25	σ	σ	PROPN
ejpam-4733	111	26	)	)	PUNCT
ejpam-4733	111	27	is	be	AUX
ejpam-4733	111	28	a	a	DET
ejpam-4733	111	29	δp(λ	δp(λ	NOUN
ejpam-4733	111	30	,	,	PUNCT
ejpam-4733	111	31	p)continuous	p)continuous	ADJ
ejpam-4733	111	32	bijection	bijection	NOUN
ejpam-4733	111	33	,	,	PUNCT
ejpam-4733	111	34	then	then	ADV
ejpam-4733	111	35	(	(	PUNCT
ejpam-4733	111	36	x	x	X
ejpam-4733	111	37	,	,	PUNCT
ejpam-4733	111	38	τ	τ	X
ejpam-4733	111	39	)	)	PUNCT
ejpam-4733	111	40	is	be	AUX
ejpam-4733	111	41	δp(λ	δp(λ	NOUN
ejpam-4733	111	42	,	,	PUNCT
ejpam-4733	111	43	p)-d1	p)-d1	NOUN
ejpam-4733	111	44	.	.	PUNCT
ejpam-4733	112	1	proof	proof	NOUN
ejpam-4733	112	2	.	.	PUNCT
ejpam-4733	113	1	suppose	suppose	VERB
ejpam-4733	113	2	that	that	SCONJ
ejpam-4733	113	3	(	(	PUNCT
ejpam-4733	113	4	y	y	PROPN
ejpam-4733	113	5	,	,	PUNCT
ejpam-4733	113	6	σ	σ	PROPN
ejpam-4733	113	7	)	)	PUNCT
ejpam-4733	113	8	is	be	AUX
ejpam-4733	113	9	a	a	DET
ejpam-4733	113	10	δp(λ	δp(λ	NOUN
ejpam-4733	113	11	,	,	PUNCT
ejpam-4733	113	12	p)-d1	p)-d1	NUM
ejpam-4733	113	13	space	space	NOUN
ejpam-4733	113	14	.	.	PUNCT
ejpam-4733	114	1	let	let	VERB
ejpam-4733	114	2	x	x	PRON
ejpam-4733	114	3	and	and	CCONJ
ejpam-4733	114	4	y	y	PROPN
ejpam-4733	114	5	be	be	AUX
ejpam-4733	114	6	any	any	DET
ejpam-4733	114	7	pair	pair	NOUN
ejpam-4733	114	8	of	of	ADP
ejpam-4733	114	9	distinct	distinct	ADJ
ejpam-4733	114	10	points	point	NOUN
ejpam-4733	114	11	in	in	ADP
ejpam-4733	114	12	x.	x.	NOUN
ejpam-4733	114	13	since	since	SCONJ
ejpam-4733	114	14	f	f	PROPN
ejpam-4733	114	15	is	be	AUX
ejpam-4733	114	16	injective	injective	ADJ
ejpam-4733	114	17	and	and	CCONJ
ejpam-4733	114	18	(	(	PUNCT
ejpam-4733	114	19	y	y	PROPN
ejpam-4733	114	20	,	,	PUNCT
ejpam-4733	114	21	σ	σ	PROPN
ejpam-4733	114	22	)	)	PUNCT
ejpam-4733	114	23	is	be	AUX
ejpam-4733	114	24	δp(λ	δp(λ	NOUN
ejpam-4733	114	25	,	,	PUNCT
ejpam-4733	114	26	p)-d1	p)-d1	NUM
ejpam-4733	114	27	,	,	PUNCT
ejpam-4733	114	28	there	there	PRON
ejpam-4733	114	29	exist	exist	VERB
ejpam-4733	114	30	δp(λ	δp(λ	NOUN
ejpam-4733	114	31	,	,	PUNCT
ejpam-4733	114	32	p)d	p)d	ADJ
ejpam-4733	114	33	-	-	PUNCT
ejpam-4733	114	34	sets	set	NOUN
ejpam-4733	114	35	u	u	NOUN
ejpam-4733	114	36	and	and	CCONJ
ejpam-4733	114	37	v	v	NOUN
ejpam-4733	114	38	of	of	ADP
ejpam-4733	114	39	y	y	NOUN
ejpam-4733	114	40	containing	contain	VERB
ejpam-4733	114	41	f(x	f(x	PROPN
ejpam-4733	114	42	)	)	PUNCT
ejpam-4733	114	43	and	and	CCONJ
ejpam-4733	114	44	f(y	f(y	NOUN
ejpam-4733	114	45	)	)	PUNCT
ejpam-4733	114	46	,	,	PUNCT
ejpam-4733	114	47	respectively	respectively	ADV
ejpam-4733	114	48	,	,	PUNCT
ejpam-4733	114	49	such	such	ADJ
ejpam-4733	114	50	that	that	DET
ejpam-4733	114	51	f(y	f(y	NOUN
ejpam-4733	114	52	)	)	PUNCT
ejpam-4733	114	53	̸∈	̸∈	PROPN
ejpam-4733	114	54	u	u	PROPN
ejpam-4733	114	55	and	and	CCONJ
ejpam-4733	114	56	f(x	f(x	PROPN
ejpam-4733	114	57	)	)	PUNCT
ejpam-4733	115	1	̸∈	̸∈	PROPN
ejpam-4733	115	2	v	v	PROPN
ejpam-4733	115	3	.	.	PUNCT
ejpam-4733	116	1	by	by	ADP
ejpam-4733	116	2	theorem	theorem	ADJ
ejpam-4733	116	3	2	2	NUM
ejpam-4733	116	4	,	,	PUNCT
ejpam-4733	116	5	f−1(u	f−1(u	PROPN
ejpam-4733	116	6	)	)	PUNCT
ejpam-4733	116	7	and	and	CCONJ
ejpam-4733	116	8	f−1(v	f−1(v	PROPN
ejpam-4733	116	9	)	)	PUNCT
ejpam-4733	116	10	are	be	AUX
ejpam-4733	116	11	δp(λ	δp(λ	NOUN
ejpam-4733	116	12	,	,	PUNCT
ejpam-4733	116	13	p)d	p)d	NOUN
ejpam-4733	116	14	-	-	PUNCT
ejpam-4733	116	15	sets	set	NOUN
ejpam-4733	116	16	in	in	ADP
ejpam-4733	116	17	x	x	PUNCT
ejpam-4733	116	18	containing	contain	VERB
ejpam-4733	116	19	x	x	PROPN
ejpam-4733	116	20	and	and	CCONJ
ejpam-4733	116	21	y	y	PROPN
ejpam-4733	116	22	,	,	PUNCT
ejpam-4733	116	23	respectively	respectively	ADV
ejpam-4733	116	24	,	,	PUNCT
ejpam-4733	116	25	such	such	ADJ
ejpam-4733	116	26	that	that	SCONJ
ejpam-4733	116	27	y	y	PROPN
ejpam-4733	116	28	̸∈	̸∈	PROPN
ejpam-4733	116	29	f−1(u	f−1(u	PROPN
ejpam-4733	116	30	)	)	PUNCT
ejpam-4733	116	31	and	and	CCONJ
ejpam-4733	116	32	x	x	PUNCT
ejpam-4733	116	33	̸∈	̸∈	PROPN
ejpam-4733	116	34	f−1(v	f−1(v	PROPN
ejpam-4733	116	35	)	)	PUNCT
ejpam-4733	116	36	.	.	PUNCT
ejpam-4733	117	1	this	this	PRON
ejpam-4733	117	2	shows	show	VERB
ejpam-4733	117	3	that	that	SCONJ
ejpam-4733	117	4	(	(	PUNCT
ejpam-4733	117	5	x	x	X
ejpam-4733	117	6	,	,	PUNCT
ejpam-4733	117	7	τ	τ	X
ejpam-4733	117	8	)	)	PUNCT
ejpam-4733	117	9	is	be	AUX
ejpam-4733	117	10	δp(λ	δp(λ	NOUN
ejpam-4733	117	11	,	,	PUNCT
ejpam-4733	117	12	p)-d1	p)-d1	NOUN
ejpam-4733	117	13	.	.	PUNCT
ejpam-4733	118	1	definition	definition	NOUN
ejpam-4733	118	2	8	8	NUM
ejpam-4733	118	3	.	.	PUNCT
ejpam-4733	119	1	a	a	DET
ejpam-4733	119	2	topological	topological	ADJ
ejpam-4733	119	3	space	space	NOUN
ejpam-4733	119	4	(	(	PUNCT
ejpam-4733	119	5	x	x	X
ejpam-4733	119	6	,	,	PUNCT
ejpam-4733	119	7	τ	τ	X
ejpam-4733	119	8	)	)	PUNCT
ejpam-4733	119	9	is	be	AUX
ejpam-4733	119	10	called	call	VERB
ejpam-4733	119	11	δp(λ	δp(λ	NOUN
ejpam-4733	119	12	,	,	PUNCT
ejpam-4733	119	13	p)-r0	p)-r0	NOUN
ejpam-4733	119	14	if	if	SCONJ
ejpam-4733	119	15	for	for	ADP
ejpam-4733	119	16	each	each	DET
ejpam-4733	119	17	δp(λ	δp(λ	NOUN
ejpam-4733	119	18	,	,	PUNCT
ejpam-4733	119	19	p)-open	p)-open	VERB
ejpam-4733	119	20	set	set	VERB
ejpam-4733	119	21	u	u	NOUN
ejpam-4733	119	22	and	and	CCONJ
ejpam-4733	119	23	each	each	DET
ejpam-4733	119	24	x	x	SYM
ejpam-4733	119	25	∈	∈	PROPN
ejpam-4733	119	26	u	u	NOUN
ejpam-4733	119	27	,	,	PUNCT
ejpam-4733	119	28	{	{	PUNCT
ejpam-4733	119	29	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	119	30	,	,	PUNCT
ejpam-4733	119	31	p	p	NOUN
ejpam-4733	119	32	)	)	PUNCT
ejpam-4733	119	33	⊆	⊆	NUM
ejpam-4733	119	34	u	u	NOUN
ejpam-4733	119	35	.	.	PUNCT
ejpam-4733	120	1	theorem	theorem	VERB
ejpam-4733	120	2	4	4	NUM
ejpam-4733	120	3	.	.	X
ejpam-4733	121	1	for	for	ADP
ejpam-4733	121	2	a	a	DET
ejpam-4733	121	3	topological	topological	ADJ
ejpam-4733	121	4	space	space	NOUN
ejpam-4733	121	5	(	(	PUNCT
ejpam-4733	121	6	x	x	X
ejpam-4733	121	7	,	,	PUNCT
ejpam-4733	121	8	τ	τ	PROPN
ejpam-4733	121	9	)	)	PUNCT
ejpam-4733	121	10	,	,	PUNCT
ejpam-4733	121	11	the	the	DET
ejpam-4733	121	12	following	follow	VERB
ejpam-4733	121	13	properties	property	NOUN
ejpam-4733	121	14	are	be	AUX
ejpam-4733	121	15	equivalent	equivalent	ADJ
ejpam-4733	121	16	:	:	PUNCT
ejpam-4733	121	17	(	(	PUNCT
ejpam-4733	121	18	1	1	X
ejpam-4733	121	19	)	)	PUNCT
ejpam-4733	121	20	(	(	PUNCT
ejpam-4733	121	21	x	x	X
ejpam-4733	121	22	,	,	PUNCT
ejpam-4733	121	23	τ	τ	X
ejpam-4733	121	24	)	)	PUNCT
ejpam-4733	121	25	is	be	AUX
ejpam-4733	121	26	δp(λ	δp(λ	NOUN
ejpam-4733	121	27	,	,	PUNCT
ejpam-4733	121	28	p)-r0	p)-r0	X
ejpam-4733	121	29	.	.	PUNCT
ejpam-4733	122	1	(	(	PUNCT
ejpam-4733	122	2	2	2	X
ejpam-4733	122	3	)	)	PUNCT
ejpam-4733	122	4	for	for	ADP
ejpam-4733	122	5	each	each	DET
ejpam-4733	122	6	δp(λ	δp(λ	NOUN
ejpam-4733	122	7	,	,	PUNCT
ejpam-4733	122	8	p)-closed	p)-close	VERB
ejpam-4733	122	9	set	set	VERB
ejpam-4733	122	10	f	f	PROPN
ejpam-4733	122	11	and	and	CCONJ
ejpam-4733	122	12	each	each	DET
ejpam-4733	122	13	x	x	SYM
ejpam-4733	122	14	∈	∈	PROPN
ejpam-4733	122	15	x−f	x−f	PROPN
ejpam-4733	122	16	,	,	PUNCT
ejpam-4733	122	17	there	there	PRON
ejpam-4733	122	18	exists	exist	VERB
ejpam-4733	122	19	u	u	PROPN
ejpam-4733	122	20	∈	∈	PROPN
ejpam-4733	122	21	δp(λ	δp(λ	NOUN
ejpam-4733	122	22	,	,	PUNCT
ejpam-4733	122	23	p)o(x	p)o(x	ADJ
ejpam-4733	122	24	,	,	PUNCT
ejpam-4733	122	25	τ	τ	PROPN
ejpam-4733	122	26	)	)	PUNCT
ejpam-4733	122	27	such	such	ADJ
ejpam-4733	122	28	that	that	SCONJ
ejpam-4733	122	29	f	f	PROPN
ejpam-4733	122	30	⊆	⊆	NUM
ejpam-4733	122	31	u	u	NOUN
ejpam-4733	122	32	and	and	CCONJ
ejpam-4733	122	33	x	x	PUNCT
ejpam-4733	122	34	̸∈	̸∈	PROPN
ejpam-4733	122	35	u	u	PROPN
ejpam-4733	122	36	.	.	PUNCT
ejpam-4733	123	1	(	(	PUNCT
ejpam-4733	123	2	3	3	X
ejpam-4733	123	3	)	)	PUNCT
ejpam-4733	123	4	for	for	ADP
ejpam-4733	123	5	each	each	DET
ejpam-4733	123	6	δp(λ	δp(λ	NOUN
ejpam-4733	123	7	,	,	PUNCT
ejpam-4733	123	8	p)-closed	p)-close	VERB
ejpam-4733	123	9	set	set	VERB
ejpam-4733	123	10	f	f	PROPN
ejpam-4733	123	11	and	and	CCONJ
ejpam-4733	123	12	each	each	DET
ejpam-4733	123	13	x	x	SYM
ejpam-4733	123	14	∈	∈	PROPN
ejpam-4733	123	15	x	x	X
ejpam-4733	124	1	−	−	PROPN
ejpam-4733	124	2	f	f	PROPN
ejpam-4733	124	3	,	,	PUNCT
ejpam-4733	124	4	f	f	PROPN
ejpam-4733	124	5	∩	∩	X
ejpam-4733	124	6	{	{	PUNCT
ejpam-4733	124	7	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	124	8	,	,	PUNCT
ejpam-4733	124	9	p	p	NOUN
ejpam-4733	124	10	)	)	PUNCT
ejpam-4733	124	11	=	=	SYM
ejpam-4733	124	12	∅.	∅.	X
ejpam-4733	124	13	(	(	PUNCT
ejpam-4733	124	14	4	4	NUM
ejpam-4733	124	15	)	)	PUNCT
ejpam-4733	124	16	for	for	ADP
ejpam-4733	124	17	any	any	DET
ejpam-4733	124	18	distinct	distinct	ADJ
ejpam-4733	124	19	points	point	NOUN
ejpam-4733	125	1	x	x	NOUN
ejpam-4733	125	2	,	,	PUNCT
ejpam-4733	125	3	y	y	PROPN
ejpam-4733	125	4	in	in	ADP
ejpam-4733	125	5	x	x	PROPN
ejpam-4733	125	6	,	,	PUNCT
ejpam-4733	125	7	{	{	PUNCT
ejpam-4733	125	8	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	125	9	,	,	PUNCT
ejpam-4733	125	10	p	p	NOUN
ejpam-4733	125	11	)	)	PUNCT
ejpam-4733	125	12	=	=	SYM
ejpam-4733	125	13	{	{	PUNCT
ejpam-4733	125	14	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	125	15	,	,	PUNCT
ejpam-4733	125	16	p	p	NOUN
ejpam-4733	125	17	)	)	PUNCT
ejpam-4733	125	18	or	or	CCONJ
ejpam-4733	125	19	{	{	PUNCT
ejpam-4733	125	20	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	125	21	,	,	PUNCT
ejpam-4733	125	22	p)∩{y}δp(λ	p)∩{y}δp(λ	PROPN
ejpam-4733	125	23	,	,	PUNCT
ejpam-4733	125	24	p	p	NOUN
ejpam-4733	125	25	)	)	PUNCT
ejpam-4733	125	26	=	=	PUNCT
ejpam-4733	125	27	∅.	∅.	NOUN
ejpam-4733	125	28	proof	proof	NOUN
ejpam-4733	125	29	.	.	PUNCT
ejpam-4733	126	1	(	(	PUNCT
ejpam-4733	126	2	1	1	X
ejpam-4733	126	3	)	)	PUNCT
ejpam-4733	126	4	⇒	⇒	NOUN
ejpam-4733	126	5	(	(	PUNCT
ejpam-4733	126	6	2	2	NUM
ejpam-4733	126	7	):	):	PUNCT
ejpam-4733	126	8	let	let	VERB
ejpam-4733	126	9	f	f	PRON
ejpam-4733	126	10	be	be	AUX
ejpam-4733	126	11	a	a	DET
ejpam-4733	126	12	δp(λ	δp(λ	NOUN
ejpam-4733	126	13	,	,	PUNCT
ejpam-4733	126	14	p)-closed	p)-close	VERB
ejpam-4733	126	15	set	set	NOUN
ejpam-4733	126	16	and	and	CCONJ
ejpam-4733	126	17	x	x	SYM
ejpam-4733	126	18	∈	∈	PROPN
ejpam-4733	126	19	x	x	X
ejpam-4733	126	20	−	−	PROPN
ejpam-4733	126	21	f	f	X
ejpam-4733	126	22	.	.	PUNCT
ejpam-4733	127	1	since	since	SCONJ
ejpam-4733	127	2	(	(	PUNCT
ejpam-4733	127	3	x	x	X
ejpam-4733	127	4	,	,	PUNCT
ejpam-4733	127	5	τ	τ	X
ejpam-4733	127	6	)	)	PUNCT
ejpam-4733	127	7	is	be	AUX
ejpam-4733	127	8	δp(λ	δp(λ	NOUN
ejpam-4733	127	9	,	,	PUNCT
ejpam-4733	127	10	p)-r0	p)-r0	NOUN
ejpam-4733	127	11	,	,	PUNCT
ejpam-4733	127	12	we	we	PRON
ejpam-4733	127	13	have	have	VERB
ejpam-4733	127	14	{	{	PUNCT
ejpam-4733	127	15	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	127	16	,	,	PUNCT
ejpam-4733	127	17	p	p	NOUN
ejpam-4733	127	18	)	)	PUNCT
ejpam-4733	127	19	⊆	⊆	NUM
ejpam-4733	127	20	x	x	SYM
ejpam-4733	128	1	−	−	PROPN
ejpam-4733	129	1	f	f	X
ejpam-4733	129	2	.	.	PUNCT
ejpam-4733	130	1	put	put	VERB
ejpam-4733	130	2	u	u	NOUN
ejpam-4733	130	3	=	=	NOUN
ejpam-4733	130	4	x	x	SYM
ejpam-4733	130	5	−	−	PROPN
ejpam-4733	130	6	{	{	PUNCT
ejpam-4733	130	7	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	130	8	,	,	PUNCT
ejpam-4733	130	9	p	p	NOUN
ejpam-4733	130	10	)	)	PUNCT
ejpam-4733	130	11	.	.	PUNCT
ejpam-4733	131	1	thus	thus	ADV
ejpam-4733	131	2	,	,	PUNCT
ejpam-4733	131	3	by	by	ADP
ejpam-4733	131	4	lemma	lemma	PROPN
ejpam-4733	131	5	1	1	NUM
ejpam-4733	131	6	,	,	PUNCT
ejpam-4733	131	7	u	u	PROPN
ejpam-4733	131	8	∈	∈	PROPN
ejpam-4733	131	9	δp(λ	δp(λ	NOUN
ejpam-4733	131	10	,	,	PUNCT
ejpam-4733	131	11	p)o(x	p)o(x	ADJ
ejpam-4733	131	12	,	,	PUNCT
ejpam-4733	131	13	τ	τ	PROPN
ejpam-4733	131	14	)	)	PUNCT
ejpam-4733	131	15	,	,	PUNCT
ejpam-4733	131	16	f	f	PROPN
ejpam-4733	131	17	⊆	⊆	NUM
ejpam-4733	131	18	u	u	NOUN
ejpam-4733	131	19	and	and	CCONJ
ejpam-4733	131	20	x	x	PUNCT
ejpam-4733	131	21	̸∈	̸∈	PROPN
ejpam-4733	131	22	u	u	PROPN
ejpam-4733	131	23	.	.	PUNCT
ejpam-4733	132	1	(	(	PUNCT
ejpam-4733	132	2	2	2	X
ejpam-4733	132	3	)	)	PUNCT
ejpam-4733	132	4	⇒	⇒	NOUN
ejpam-4733	132	5	(	(	PUNCT
ejpam-4733	132	6	3	3	NUM
ejpam-4733	132	7	):	):	PUNCT
ejpam-4733	132	8	let	let	VERB
ejpam-4733	132	9	f	f	PRON
ejpam-4733	132	10	be	be	AUX
ejpam-4733	132	11	a	a	DET
ejpam-4733	132	12	δp(λ	δp(λ	NOUN
ejpam-4733	132	13	,	,	PUNCT
ejpam-4733	132	14	p)-closed	p)-close	VERB
ejpam-4733	132	15	set	set	NOUN
ejpam-4733	132	16	and	and	CCONJ
ejpam-4733	132	17	x	x	SYM
ejpam-4733	132	18	∈	∈	PROPN
ejpam-4733	132	19	x	x	X
ejpam-4733	133	1	−	−	PROPN
ejpam-4733	133	2	f	f	X
ejpam-4733	133	3	.	.	PUNCT
ejpam-4733	134	1	by	by	ADP
ejpam-4733	134	2	(	(	PUNCT
ejpam-4733	134	3	2	2	NUM
ejpam-4733	134	4	)	)	PUNCT
ejpam-4733	134	5	,	,	PUNCT
ejpam-4733	134	6	there	there	PRON
ejpam-4733	134	7	exists	exist	VERB
ejpam-4733	134	8	u	u	PROPN
ejpam-4733	134	9	∈	∈	PROPN
ejpam-4733	134	10	δp(λ	δp(λ	NOUN
ejpam-4733	134	11	,	,	PUNCT
ejpam-4733	134	12	p)o(x	p)o(x	ADJ
ejpam-4733	134	13	,	,	PUNCT
ejpam-4733	134	14	τ	τ	PROPN
ejpam-4733	134	15	)	)	PUNCT
ejpam-4733	134	16	such	such	ADJ
ejpam-4733	134	17	that	that	SCONJ
ejpam-4733	134	18	f	f	PROPN
ejpam-4733	134	19	⊆	⊆	NUM
ejpam-4733	134	20	u	u	NOUN
ejpam-4733	134	21	and	and	CCONJ
ejpam-4733	134	22	x	x	PUNCT
ejpam-4733	134	23	̸∈	̸∈	PROPN
ejpam-4733	134	24	u	u	PROPN
ejpam-4733	134	25	.	.	PUNCT
ejpam-4733	135	1	since	since	SCONJ
ejpam-4733	135	2	u	u	PROPN
ejpam-4733	135	3	∈	∈	PROPN
ejpam-4733	135	4	δp(λ	δp(λ	NOUN
ejpam-4733	135	5	,	,	PUNCT
ejpam-4733	135	6	p)o(x	p)o(x	ADJ
ejpam-4733	135	7	,	,	PUNCT
ejpam-4733	135	8	τ	τ	PROPN
ejpam-4733	135	9	)	)	PUNCT
ejpam-4733	135	10	,	,	PUNCT
ejpam-4733	135	11	u	u	PROPN
ejpam-4733	135	12	∩	∩	NOUN
ejpam-4733	135	13	{	{	PUNCT
ejpam-4733	135	14	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	135	15	,	,	PUNCT
ejpam-4733	135	16	p	p	NOUN
ejpam-4733	135	17	)	)	PUNCT
ejpam-4733	135	18	=	=	NOUN
ejpam-4733	135	19	∅	∅	NOUN
ejpam-4733	135	20	and	and	CCONJ
ejpam-4733	135	21	hence	hence	ADV
ejpam-4733	135	22	f	f	PROPN
ejpam-4733	135	23	∩	∩	PROPN
ejpam-4733	135	24	{	{	PUNCT
ejpam-4733	135	25	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	135	26	,	,	PUNCT
ejpam-4733	135	27	p	p	NOUN
ejpam-4733	135	28	)	)	PUNCT
ejpam-4733	135	29	=	=	PUNCT
ejpam-4733	135	30	∅.	∅.	PROPN
ejpam-4733	135	31	c.	c.	PROPN
ejpam-4733	135	32	boonpok	boonpok	PROPN
ejpam-4733	135	33	,	,	PUNCT
ejpam-4733	135	34	m.	m.	NOUN
ejpam-4733	135	35	thongmoon	thongmoon	PROPN
ejpam-4733	135	36	/	/	SYM
ejpam-4733	135	37	eur	eur	PROPN
ejpam-4733	135	38	.	.	PUNCT
ejpam-4733	136	1	j.	j.	PROPN
ejpam-4733	136	2	pure	pure	PROPN
ejpam-4733	136	3	appl	appl	PROPN
ejpam-4733	136	4	.	.	PROPN
ejpam-4733	136	5	math	math	PROPN
ejpam-4733	136	6	,	,	PUNCT
ejpam-4733	136	7	16	16	NUM
ejpam-4733	136	8	(	(	PUNCT
ejpam-4733	136	9	3	3	NUM
ejpam-4733	136	10	)	)	PUNCT
ejpam-4733	136	11	(	(	PUNCT
ejpam-4733	136	12	2023	2023	NUM
ejpam-4733	136	13	)	)	PUNCT
ejpam-4733	136	14	,	,	PUNCT
ejpam-4733	136	15	1533	1533	NUM
ejpam-4733	136	16	-	-	SYM
ejpam-4733	136	17	1542	1542	NUM
ejpam-4733	136	18	1537	1537	NUM
ejpam-4733	136	19	(	(	PUNCT
ejpam-4733	136	20	3	3	NUM
ejpam-4733	136	21	)	)	PUNCT
ejpam-4733	136	22	⇒	⇒	NOUN
ejpam-4733	136	23	(	(	PUNCT
ejpam-4733	136	24	4	4	NUM
ejpam-4733	136	25	):	):	PUNCT
ejpam-4733	136	26	let	let	VERB
ejpam-4733	136	27	x	x	PRON
ejpam-4733	136	28	and	and	CCONJ
ejpam-4733	136	29	y	y	PROPN
ejpam-4733	136	30	be	be	AUX
ejpam-4733	136	31	distinct	distinct	ADJ
ejpam-4733	136	32	points	point	NOUN
ejpam-4733	136	33	of	of	ADP
ejpam-4733	136	34	x.	x.	NOUN
ejpam-4733	136	35	suppose	suppose	VERB
ejpam-4733	136	36	that	that	SCONJ
ejpam-4733	136	37	{	{	PUNCT
ejpam-4733	136	38	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	136	39	,	,	PUNCT
ejpam-4733	136	40	p)∩{y}δp(λ	p)∩{y}δp(λ	PROPN
ejpam-4733	136	41	,	,	PUNCT
ejpam-4733	136	42	p	p	NOUN
ejpam-4733	136	43	)	)	PUNCT
ejpam-4733	136	44	̸=	̸=	PROPN
ejpam-4733	136	45	∅.	∅.	VERB
ejpam-4733	136	46	by	by	ADP
ejpam-4733	136	47	(	(	PUNCT
ejpam-4733	136	48	3	3	NUM
ejpam-4733	136	49	)	)	PUNCT
ejpam-4733	136	50	,	,	PUNCT
ejpam-4733	136	51	we	we	PRON
ejpam-4733	136	52	have	have	VERB
ejpam-4733	136	53	x	x	PART
ejpam-4733	136	54	∈	∈	PROPN
ejpam-4733	136	55	{	{	PUNCT
ejpam-4733	136	56	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	136	57	,	,	PUNCT
ejpam-4733	136	58	p	p	NOUN
ejpam-4733	136	59	)	)	PUNCT
ejpam-4733	136	60	and	and	CCONJ
ejpam-4733	136	61	y	y	PROPN
ejpam-4733	136	62	∈	∈	PROPN
ejpam-4733	136	63	{	{	PUNCT
ejpam-4733	136	64	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	136	65	,	,	PUNCT
ejpam-4733	136	66	p	p	NOUN
ejpam-4733	136	67	)	)	PUNCT
ejpam-4733	136	68	.	.	PUNCT
ejpam-4733	137	1	by	by	ADP
ejpam-4733	137	2	lemma	lemma	PROPN
ejpam-4733	137	3	1	1	NUM
ejpam-4733	137	4	,	,	PUNCT
ejpam-4733	137	5	{	{	PUNCT
ejpam-4733	137	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	137	7	,	,	PUNCT
ejpam-4733	137	8	p	p	NOUN
ejpam-4733	137	9	)	)	PUNCT
ejpam-4733	137	10	⊆	⊆	NUM
ejpam-4733	137	11	{	{	PUNCT
ejpam-4733	137	12	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	137	13	,	,	PUNCT
ejpam-4733	137	14	p	p	NOUN
ejpam-4733	137	15	)	)	PUNCT
ejpam-4733	137	16	⊆	⊆	NUM
ejpam-4733	137	17	{	{	PUNCT
ejpam-4733	137	18	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	137	19	,	,	PUNCT
ejpam-4733	137	20	p	p	NOUN
ejpam-4733	137	21	)	)	PUNCT
ejpam-4733	137	22	and	and	CCONJ
ejpam-4733	137	23	hence	hence	ADV
ejpam-4733	137	24	{	{	PUNCT
ejpam-4733	137	25	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	137	26	,	,	PUNCT
ejpam-4733	137	27	p	p	NOUN
ejpam-4733	137	28	)	)	PUNCT
ejpam-4733	137	29	=	=	SYM
ejpam-4733	137	30	{	{	PUNCT
ejpam-4733	137	31	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	137	32	,	,	PUNCT
ejpam-4733	137	33	p	p	NOUN
ejpam-4733	137	34	)	)	PUNCT
ejpam-4733	137	35	.	.	PUNCT
ejpam-4733	138	1	(	(	PUNCT
ejpam-4733	138	2	4	4	X
ejpam-4733	138	3	)	)	PUNCT
ejpam-4733	138	4	⇒	⇒	NOUN
ejpam-4733	138	5	(	(	PUNCT
ejpam-4733	138	6	1	1	NUM
ejpam-4733	138	7	):	):	PUNCT
ejpam-4733	138	8	let	let	VERB
ejpam-4733	138	9	v	v	NUM
ejpam-4733	138	10	∈	∈	PROPN
ejpam-4733	138	11	δp(λ	δp(λ	NOUN
ejpam-4733	138	12	,	,	PUNCT
ejpam-4733	138	13	p)o(x	p)o(x	ADJ
ejpam-4733	138	14	,	,	PUNCT
ejpam-4733	138	15	τ	τ	PROPN
ejpam-4733	138	16	)	)	PUNCT
ejpam-4733	138	17	and	and	CCONJ
ejpam-4733	138	18	x	x	PUNCT
ejpam-4733	138	19	∈	∈	NOUN
ejpam-4733	138	20	v	v	NOUN
ejpam-4733	138	21	.	.	PUNCT
ejpam-4733	139	1	for	for	ADP
ejpam-4733	139	2	each	each	DET
ejpam-4733	139	3	y	y	PROPN
ejpam-4733	139	4	̸∈	̸∈	PROPN
ejpam-4733	139	5	v	v	PROPN
ejpam-4733	139	6	,	,	PUNCT
ejpam-4733	139	7	v	v	NOUN
ejpam-4733	139	8	∩	∩	NOUN
ejpam-4733	139	9	{	{	PUNCT
ejpam-4733	139	10	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	139	11	,	,	PUNCT
ejpam-4733	139	12	p	p	NOUN
ejpam-4733	139	13	)	)	PUNCT
ejpam-4733	139	14	=	=	NOUN
ejpam-4733	139	15	∅	∅	NOUN
ejpam-4733	139	16	and	and	CCONJ
ejpam-4733	139	17	hence	hence	ADV
ejpam-4733	139	18	x	x	X
ejpam-4733	139	19	̸∈	̸∈	PROPN
ejpam-4733	139	20	{	{	PUNCT
ejpam-4733	139	21	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	139	22	,	,	PUNCT
ejpam-4733	139	23	p	p	NOUN
ejpam-4733	139	24	)	)	PUNCT
ejpam-4733	139	25	.	.	PUNCT
ejpam-4733	140	1	thus	thus	ADV
ejpam-4733	140	2	,	,	PUNCT
ejpam-4733	140	3	{	{	PUNCT
ejpam-4733	140	4	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	140	5	,	,	PUNCT
ejpam-4733	140	6	p	p	NOUN
ejpam-4733	140	7	)	)	PUNCT
ejpam-4733	140	8	̸=	̸=	PROPN
ejpam-4733	140	9	{	{	PUNCT
ejpam-4733	140	10	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	140	11	,	,	PUNCT
ejpam-4733	140	12	p	p	NOUN
ejpam-4733	140	13	)	)	PUNCT
ejpam-4733	140	14	.	.	PUNCT
ejpam-4733	141	1	by	by	ADP
ejpam-4733	141	2	(	(	PUNCT
ejpam-4733	141	3	4	4	NUM
ejpam-4733	141	4	)	)	PUNCT
ejpam-4733	141	5	,	,	PUNCT
ejpam-4733	141	6	for	for	SCONJ
ejpam-4733	141	7	each	each	DET
ejpam-4733	141	8	y	y	PROPN
ejpam-4733	141	9	̸∈	̸∈	PROPN
ejpam-4733	141	10	v	v	PROPN
ejpam-4733	141	11	,	,	PUNCT
ejpam-4733	141	12	{	{	PUNCT
ejpam-4733	141	13	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	141	14	,	,	PUNCT
ejpam-4733	141	15	p	p	NOUN
ejpam-4733	141	16	)	)	PUNCT
ejpam-4733	141	17	∩	∩	NOUN
ejpam-4733	141	18	{	{	PUNCT
ejpam-4733	141	19	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	141	20	,	,	PUNCT
ejpam-4733	141	21	p	p	NOUN
ejpam-4733	141	22	)	)	PUNCT
ejpam-4733	141	23	=	=	PUNCT
ejpam-4733	141	24	∅.	∅.	NOUN
ejpam-4733	141	25	since	since	SCONJ
ejpam-4733	141	26	x	x	PROPN
ejpam-4733	141	27	−	−	PROPN
ejpam-4733	141	28	v	v	NOUN
ejpam-4733	141	29	is	be	AUX
ejpam-4733	141	30	δp(λ	δp(λ	NOUN
ejpam-4733	141	31	,	,	PUNCT
ejpam-4733	141	32	p)-closed	p)-close	VERB
ejpam-4733	141	33	,	,	PUNCT
ejpam-4733	141	34	y	y	PROPN
ejpam-4733	141	35	∈	∈	PROPN
ejpam-4733	141	36	{	{	PUNCT
ejpam-4733	141	37	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	141	38	,	,	PUNCT
ejpam-4733	141	39	p	p	NOUN
ejpam-4733	141	40	)	)	PUNCT
ejpam-4733	141	41	⊆	⊆	NUM
ejpam-4733	141	42	x	x	SYM
ejpam-4733	141	43	−	−	PROPN
ejpam-4733	141	44	v	v	NOUN
ejpam-4733	141	45	and	and	CCONJ
ejpam-4733	141	46	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4733	141	47	{	{	PUNCT
ejpam-4733	141	48	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	141	49	,	,	PUNCT
ejpam-4733	141	50	p	p	NOUN
ejpam-4733	141	51	)	)	PUNCT
ejpam-4733	141	52	=	=	PUNCT
ejpam-4733	141	53	x	x	PUNCT
ejpam-4733	141	54	−	−	NOUN
ejpam-4733	141	55	v	v	NOUN
ejpam-4733	141	56	.	.	PUNCT
ejpam-4733	142	1	thus	thus	ADV
ejpam-4733	142	2	,	,	PUNCT
ejpam-4733	142	3	{	{	PUNCT
ejpam-4733	142	4	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	142	5	,	,	PUNCT
ejpam-4733	142	6	p	p	NOUN
ejpam-4733	142	7	)	)	PUNCT
ejpam-4733	142	8	∩	∩	NOUN
ejpam-4733	142	9	(	(	PUNCT
ejpam-4733	142	10	x	x	SYM
ejpam-4733	142	11	−	−	PROPN
ejpam-4733	142	12	v	v	NOUN
ejpam-4733	142	13	)	)	PUNCT
ejpam-4733	142	14	=	=	SYM
ejpam-4733	142	15	{	{	PUNCT
ejpam-4733	142	16	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	142	17	,	,	PUNCT
ejpam-4733	142	18	p	p	NOUN
ejpam-4733	142	19	)	)	PUNCT
ejpam-4733	142	20	∩	∩	NOUN
ejpam-4733	142	21	[	[	X
ejpam-4733	142	22	∪y∈x−v	∪y∈x−v	ADJ
ejpam-4733	142	23	{	{	PUNCT
ejpam-4733	142	24	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	142	25	,	,	PUNCT
ejpam-4733	142	26	p	p	NOUN
ejpam-4733	142	27	)	)	PUNCT
ejpam-4733	142	28	]	]	PUNCT
ejpam-4733	143	1	=	=	PUNCT
ejpam-4733	143	2	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4733	144	1	[	[	X
ejpam-4733	144	2	{	{	PUNCT
ejpam-4733	144	3	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	144	4	,	,	PUNCT
ejpam-4733	144	5	p	p	NOUN
ejpam-4733	144	6	)	)	PUNCT
ejpam-4733	144	7	∩	∩	NOUN
ejpam-4733	144	8	{	{	PUNCT
ejpam-4733	144	9	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	144	10	,	,	PUNCT
ejpam-4733	144	11	p	p	NOUN
ejpam-4733	144	12	)	)	PUNCT
ejpam-4733	144	13	]	]	PUNCT
ejpam-4733	144	14	=	=	PUNCT
ejpam-4733	144	15	∅	∅	NOUN
ejpam-4733	144	16	and	and	CCONJ
ejpam-4733	144	17	hence	hence	ADV
ejpam-4733	144	18	{	{	PUNCT
ejpam-4733	144	19	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	144	20	,	,	PUNCT
ejpam-4733	144	21	p	p	NOUN
ejpam-4733	144	22	)	)	PUNCT
ejpam-4733	144	23	⊆	⊆	NUM
ejpam-4733	144	24	v	v	NOUN
ejpam-4733	144	25	.	.	PUNCT
ejpam-4733	145	1	this	this	PRON
ejpam-4733	145	2	shows	show	VERB
ejpam-4733	145	3	that	that	SCONJ
ejpam-4733	145	4	(	(	PUNCT
ejpam-4733	145	5	x	x	X
ejpam-4733	145	6	,	,	PUNCT
ejpam-4733	145	7	τ	τ	X
ejpam-4733	145	8	)	)	PUNCT
ejpam-4733	145	9	is	be	AUX
ejpam-4733	145	10	δp(λ	δp(λ	NOUN
ejpam-4733	145	11	,	,	PUNCT
ejpam-4733	145	12	p)-r0	p)-r0	X
ejpam-4733	145	13	.	.	PUNCT
ejpam-4733	146	1	corollary	corollary	ADJ
ejpam-4733	146	2	1	1	NUM
ejpam-4733	146	3	.	.	PUNCT
ejpam-4733	147	1	a	a	DET
ejpam-4733	147	2	topological	topological	ADJ
ejpam-4733	147	3	space	space	NOUN
ejpam-4733	147	4	(	(	PUNCT
ejpam-4733	147	5	x	x	X
ejpam-4733	147	6	,	,	PUNCT
ejpam-4733	147	7	τ	τ	X
ejpam-4733	147	8	)	)	PUNCT
ejpam-4733	147	9	is	be	AUX
ejpam-4733	147	10	δp(λ	δp(λ	NOUN
ejpam-4733	147	11	,	,	PUNCT
ejpam-4733	147	12	p)-r0	p)-r0	NOUN
ejpam-4733	147	13	if	if	SCONJ
ejpam-4733	147	14	and	and	CCONJ
ejpam-4733	147	15	only	only	ADV
ejpam-4733	147	16	if	if	SCONJ
ejpam-4733	147	17	,	,	PUNCT
ejpam-4733	147	18	for	for	ADP
ejpam-4733	147	19	any	any	DET
ejpam-4733	147	20	points	point	NOUN
ejpam-4733	147	21	x	x	PUNCT
ejpam-4733	147	22	and	and	CCONJ
ejpam-4733	147	23	y	y	PROPN
ejpam-4733	147	24	in	in	ADP
ejpam-4733	147	25	x	x	SYM
ejpam-4733	147	26	,	,	PUNCT
ejpam-4733	147	27	{	{	PUNCT
ejpam-4733	147	28	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	147	29	,	,	PUNCT
ejpam-4733	147	30	p	p	NOUN
ejpam-4733	147	31	)	)	PUNCT
ejpam-4733	147	32	̸=	̸=	PROPN
ejpam-4733	147	33	{	{	PUNCT
ejpam-4733	147	34	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	147	35	,	,	PUNCT
ejpam-4733	147	36	p	p	NOUN
ejpam-4733	147	37	)	)	PUNCT
ejpam-4733	147	38	implies	imply	VERB
ejpam-4733	147	39	{	{	PUNCT
ejpam-4733	147	40	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	147	41	,	,	PUNCT
ejpam-4733	147	42	p	p	NOUN
ejpam-4733	147	43	)	)	PUNCT
ejpam-4733	147	44	∩	∩	NOUN
ejpam-4733	147	45	{	{	PUNCT
ejpam-4733	147	46	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	147	47	,	,	PUNCT
ejpam-4733	147	48	p	p	NOUN
ejpam-4733	147	49	)	)	PUNCT
ejpam-4733	147	50	=	=	PUNCT
ejpam-4733	147	51	∅.	∅.	NOUN
ejpam-4733	147	52	proof	proof	NOUN
ejpam-4733	147	53	.	.	PUNCT
ejpam-4733	148	1	this	this	PRON
ejpam-4733	148	2	is	be	AUX
ejpam-4733	148	3	obvious	obvious	ADJ
ejpam-4733	148	4	by	by	ADP
ejpam-4733	148	5	theorem	theorem	NOUN
ejpam-4733	148	6	4	4	NUM
ejpam-4733	148	7	.	.	PUNCT
ejpam-4733	148	8	conversely	conversely	ADV
ejpam-4733	148	9	,	,	PUNCT
ejpam-4733	148	10	let	let	VERB
ejpam-4733	148	11	u	u	PRON
ejpam-4733	148	12	∈	∈	PROPN
ejpam-4733	148	13	δp(λ	δp(λ	NOUN
ejpam-4733	148	14	,	,	PUNCT
ejpam-4733	148	15	p)o(x	p)o(x	ADJ
ejpam-4733	148	16	,	,	PUNCT
ejpam-4733	148	17	τ	τ	PROPN
ejpam-4733	148	18	)	)	PUNCT
ejpam-4733	148	19	and	and	CCONJ
ejpam-4733	148	20	x	x	PUNCT
ejpam-4733	148	21	∈	∈	PROPN
ejpam-4733	148	22	u	u	NOUN
ejpam-4733	148	23	.	.	PUNCT
ejpam-4733	149	1	if	if	SCONJ
ejpam-4733	149	2	y	y	PROPN
ejpam-4733	149	3	̸∈	̸∈	PROPN
ejpam-4733	149	4	u	u	PROPN
ejpam-4733	149	5	,	,	PUNCT
ejpam-4733	149	6	then	then	ADV
ejpam-4733	149	7	u∩{y}δp(λ	u∩{y}δp(λ	PROPN
ejpam-4733	149	8	,	,	PUNCT
ejpam-4733	149	9	p	p	NOUN
ejpam-4733	149	10	)	)	PUNCT
ejpam-4733	149	11	=	=	PUNCT
ejpam-4733	149	12	∅.	∅.	ADP
ejpam-4733	149	13	thus	thus	ADV
ejpam-4733	149	14	,	,	PUNCT
ejpam-4733	149	15	x	x	PROPN
ejpam-4733	149	16	̸∈	̸∈	PROPN
ejpam-4733	149	17	{	{	PUNCT
ejpam-4733	149	18	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	149	19	,	,	PUNCT
ejpam-4733	149	20	p	p	NOUN
ejpam-4733	149	21	)	)	PUNCT
ejpam-4733	149	22	and	and	CCONJ
ejpam-4733	149	23	{	{	PUNCT
ejpam-4733	149	24	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	149	25	,	,	PUNCT
ejpam-4733	149	26	p	p	NOUN
ejpam-4733	149	27	)	)	PUNCT
ejpam-4733	149	28	̸=	̸=	PROPN
ejpam-4733	149	29	{	{	PUNCT
ejpam-4733	149	30	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	149	31	,	,	PUNCT
ejpam-4733	149	32	p	p	NOUN
ejpam-4733	149	33	)	)	PUNCT
ejpam-4733	149	34	.	.	PUNCT
ejpam-4733	150	1	by	by	ADP
ejpam-4733	150	2	the	the	DET
ejpam-4733	150	3	hypothesis	hypothesis	NOUN
ejpam-4733	150	4	,	,	PUNCT
ejpam-4733	150	5	{	{	PUNCT
ejpam-4733	150	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	150	7	,	,	PUNCT
ejpam-4733	150	8	p	p	NOUN
ejpam-4733	150	9	)	)	PUNCT
ejpam-4733	150	10	∩	∩	NOUN
ejpam-4733	150	11	{	{	PUNCT
ejpam-4733	150	12	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	150	13	,	,	PUNCT
ejpam-4733	150	14	p	p	NOUN
ejpam-4733	150	15	)	)	PUNCT
ejpam-4733	150	16	=	=	NOUN
ejpam-4733	150	17	∅	∅	NOUN
ejpam-4733	150	18	and	and	CCONJ
ejpam-4733	150	19	hence	hence	ADV
ejpam-4733	150	20	y	y	PROPN
ejpam-4733	150	21	̸∈	̸∈	PROPN
ejpam-4733	150	22	{	{	PUNCT
ejpam-4733	150	23	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	150	24	,	,	PUNCT
ejpam-4733	150	25	p	p	NOUN
ejpam-4733	150	26	)	)	PUNCT
ejpam-4733	150	27	.	.	PUNCT
ejpam-4733	151	1	this	this	PRON
ejpam-4733	151	2	shows	show	VERB
ejpam-4733	151	3	that	that	SCONJ
ejpam-4733	151	4	{	{	PUNCT
ejpam-4733	151	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	151	6	,	,	PUNCT
ejpam-4733	151	7	p	p	NOUN
ejpam-4733	151	8	)	)	PUNCT
ejpam-4733	151	9	⊆	⊆	NUM
ejpam-4733	151	10	u	u	NOUN
ejpam-4733	151	11	.	.	PUNCT
ejpam-4733	152	1	thus	thus	ADV
ejpam-4733	152	2	,	,	PUNCT
ejpam-4733	152	3	(	(	PUNCT
ejpam-4733	152	4	x	x	X
ejpam-4733	152	5	,	,	PUNCT
ejpam-4733	152	6	τ	τ	X
ejpam-4733	152	7	)	)	PUNCT
ejpam-4733	152	8	is	be	AUX
ejpam-4733	152	9	δp(λ	δp(λ	NOUN
ejpam-4733	152	10	,	,	PUNCT
ejpam-4733	152	11	p)-r0	p)-r0	X
ejpam-4733	152	12	.	.	PUNCT
ejpam-4733	153	1	definition	definition	NOUN
ejpam-4733	153	2	9	9	NUM
ejpam-4733	153	3	.	.	PUNCT
ejpam-4733	154	1	let	let	VERB
ejpam-4733	154	2	a	a	DET
ejpam-4733	154	3	be	be	AUX
ejpam-4733	154	4	a	a	DET
ejpam-4733	154	5	subset	subset	NOUN
ejpam-4733	154	6	of	of	ADP
ejpam-4733	154	7	a	a	DET
ejpam-4733	154	8	topological	topological	ADJ
ejpam-4733	154	9	space	space	NOUN
ejpam-4733	154	10	(	(	PUNCT
ejpam-4733	154	11	x	x	X
ejpam-4733	154	12	,	,	PUNCT
ejpam-4733	154	13	τ	τ	PROPN
ejpam-4733	154	14	)	)	PUNCT
ejpam-4733	154	15	.	.	PUNCT
ejpam-4733	155	1	the	the	DET
ejpam-4733	155	2	δp(λ	δp(λ	NOUN
ejpam-4733	155	3	,	,	PUNCT
ejpam-4733	155	4	p)-kernel	p)-kernel	NOUN
ejpam-4733	155	5	of	of	ADP
ejpam-4733	155	6	a	a	PRON
ejpam-4733	155	7	,	,	PUNCT
ejpam-4733	155	8	denoted	denote	VERB
ejpam-4733	155	9	by	by	ADP
ejpam-4733	155	10	δp(λ	δp(λ	NOUN
ejpam-4733	155	11	,	,	PUNCT
ejpam-4733	155	12	p)ker(a	p)ker(a	NUM
ejpam-4733	155	13	)	)	PUNCT
ejpam-4733	155	14	,	,	PUNCT
ejpam-4733	155	15	is	be	AUX
ejpam-4733	155	16	defined	define	VERB
ejpam-4733	155	17	to	to	PART
ejpam-4733	155	18	be	be	AUX
ejpam-4733	155	19	the	the	DET
ejpam-4733	155	20	set	set	NOUN
ejpam-4733	155	21	δp(λ	δp(λ	NOUN
ejpam-4733	155	22	,	,	PUNCT
ejpam-4733	155	23	p)ker(a	p)ker(a	NUM
ejpam-4733	155	24	)	)	PUNCT
ejpam-4733	155	25	=	=	SYM
ejpam-4733	155	26	∩{u	∩{u	PROPN
ejpam-4733	155	27	∈	∈	PROPN
ejpam-4733	155	28	δp(λ	δp(λ	NOUN
ejpam-4733	155	29	,	,	PUNCT
ejpam-4733	155	30	p)o(x	p)o(x	ADJ
ejpam-4733	155	31	,	,	PUNCT
ejpam-4733	155	32	τ	τ	PROPN
ejpam-4733	155	33	)	)	PUNCT
ejpam-4733	155	34	|	|	ADV
ejpam-4733	155	35	a	a	DET
ejpam-4733	155	36	⊆	⊆	NUM
ejpam-4733	155	37	u	u	NOUN
ejpam-4733	155	38	}	}	PUNCT
ejpam-4733	155	39	.	.	PUNCT
ejpam-4733	156	1	lemma	lemma	PROPN
ejpam-4733	156	2	5	5	NUM
ejpam-4733	156	3	.	.	PUNCT
ejpam-4733	157	1	for	for	ADP
ejpam-4733	157	2	subsets	subset	NOUN
ejpam-4733	157	3	a	a	DET
ejpam-4733	157	4	,	,	PUNCT
ejpam-4733	157	5	b	b	PROPN
ejpam-4733	157	6	of	of	ADP
ejpam-4733	157	7	a	a	DET
ejpam-4733	157	8	topological	topological	ADJ
ejpam-4733	157	9	space	space	NOUN
ejpam-4733	157	10	(	(	PUNCT
ejpam-4733	157	11	x	x	X
ejpam-4733	157	12	,	,	PUNCT
ejpam-4733	157	13	τ	τ	PROPN
ejpam-4733	157	14	)	)	PUNCT
ejpam-4733	157	15	,	,	PUNCT
ejpam-4733	157	16	the	the	DET
ejpam-4733	157	17	following	follow	VERB
ejpam-4733	157	18	properties	property	NOUN
ejpam-4733	157	19	hold	hold	VERB
ejpam-4733	157	20	:	:	PUNCT
ejpam-4733	157	21	(	(	PUNCT
ejpam-4733	157	22	1	1	X
ejpam-4733	157	23	)	)	PUNCT
ejpam-4733	157	24	a	a	DET
ejpam-4733	157	25	⊆	⊆	NUM
ejpam-4733	157	26	δp(λ	δp(λ	NOUN
ejpam-4733	157	27	,	,	PUNCT
ejpam-4733	157	28	p)ker(a	p)ker(a	NUM
ejpam-4733	157	29	)	)	PUNCT
ejpam-4733	157	30	.	.	PUNCT
ejpam-4733	158	1	(	(	PUNCT
ejpam-4733	158	2	2	2	X
ejpam-4733	158	3	)	)	PUNCT
ejpam-4733	158	4	if	if	SCONJ
ejpam-4733	158	5	a	a	DET
ejpam-4733	158	6	⊆	⊆	NUM
ejpam-4733	158	7	b	b	NOUN
ejpam-4733	158	8	,	,	PUNCT
ejpam-4733	158	9	then	then	ADV
ejpam-4733	158	10	δp(λ	δp(λ	NOUN
ejpam-4733	158	11	,	,	PUNCT
ejpam-4733	158	12	p)ker(a	p)ker(a	NUM
ejpam-4733	158	13	)	)	PUNCT
ejpam-4733	158	14	⊆	⊆	NUM
ejpam-4733	158	15	δp(λ	δp(λ	NOUN
ejpam-4733	158	16	,	,	PUNCT
ejpam-4733	158	17	p)ker(b	p)ker(b	NUM
ejpam-4733	158	18	)	)	PUNCT
ejpam-4733	158	19	.	.	PUNCT
ejpam-4733	159	1	(	(	PUNCT
ejpam-4733	159	2	3	3	NUM
ejpam-4733	159	3	)	)	PUNCT
ejpam-4733	159	4	δp(λ	δp(λ	NOUN
ejpam-4733	159	5	,	,	PUNCT
ejpam-4733	159	6	p)ker(δp(λ	p)ker(δp(λ	PROPN
ejpam-4733	159	7	,	,	PUNCT
ejpam-4733	159	8	sp)ker(a	sp)ker(a	NOUN
ejpam-4733	159	9	)	)	PUNCT
ejpam-4733	159	10	)	)	PUNCT
ejpam-4733	160	1	=	=	SYM
ejpam-4733	160	2	δp(λ	δp(λ	NOUN
ejpam-4733	160	3	,	,	PUNCT
ejpam-4733	160	4	p)ker(a	p)ker(a	NUM
ejpam-4733	160	5	)	)	PUNCT
ejpam-4733	160	6	.	.	PUNCT
ejpam-4733	161	1	(	(	PUNCT
ejpam-4733	161	2	4	4	X
ejpam-4733	161	3	)	)	PUNCT
ejpam-4733	161	4	if	if	SCONJ
ejpam-4733	161	5	a	a	PRON
ejpam-4733	161	6	is	be	AUX
ejpam-4733	161	7	δp(λ	δp(λ	NOUN
ejpam-4733	161	8	,	,	PUNCT
ejpam-4733	161	9	p)-open	p)-open	ADJ
ejpam-4733	161	10	,	,	PUNCT
ejpam-4733	161	11	δp(λ	δp(λ	NOUN
ejpam-4733	161	12	,	,	PUNCT
ejpam-4733	161	13	p)ker(a	p)ker(a	NUM
ejpam-4733	161	14	)	)	PUNCT
ejpam-4733	161	15	=	=	NOUN
ejpam-4733	161	16	a.	a.	NOUN
ejpam-4733	161	17	theorem	theorem	NOUN
ejpam-4733	161	18	5	5	NUM
ejpam-4733	161	19	.	.	X
ejpam-4733	162	1	for	for	ADP
ejpam-4733	162	2	any	any	DET
ejpam-4733	162	3	points	point	NOUN
ejpam-4733	162	4	x	x	PUNCT
ejpam-4733	162	5	and	and	CCONJ
ejpam-4733	162	6	y	y	PROPN
ejpam-4733	162	7	in	in	ADP
ejpam-4733	162	8	a	a	DET
ejpam-4733	162	9	topological	topological	ADJ
ejpam-4733	162	10	space	space	NOUN
ejpam-4733	162	11	(	(	PUNCT
ejpam-4733	162	12	x	x	X
ejpam-4733	162	13	,	,	PUNCT
ejpam-4733	162	14	τ	τ	PROPN
ejpam-4733	162	15	)	)	PUNCT
ejpam-4733	162	16	,	,	PUNCT
ejpam-4733	162	17	the	the	DET
ejpam-4733	162	18	following	follow	VERB
ejpam-4733	162	19	properties	property	NOUN
ejpam-4733	162	20	are	be	AUX
ejpam-4733	162	21	equivalent	equivalent	ADJ
ejpam-4733	162	22	:	:	PUNCT
ejpam-4733	162	23	c.	c.	PROPN
ejpam-4733	162	24	boonpok	boonpok	PROPN
ejpam-4733	162	25	,	,	PUNCT
ejpam-4733	162	26	m.	m.	NOUN
ejpam-4733	162	27	thongmoon	thongmoon	PROPN
ejpam-4733	162	28	/	/	SYM
ejpam-4733	162	29	eur	eur	PROPN
ejpam-4733	162	30	.	.	PUNCT
ejpam-4733	163	1	j.	j.	PROPN
ejpam-4733	163	2	pure	pure	PROPN
ejpam-4733	163	3	appl	appl	PROPN
ejpam-4733	163	4	.	.	PROPN
ejpam-4733	163	5	math	math	PROPN
ejpam-4733	163	6	,	,	PUNCT
ejpam-4733	163	7	16	16	NUM
ejpam-4733	163	8	(	(	PUNCT
ejpam-4733	163	9	3	3	NUM
ejpam-4733	163	10	)	)	PUNCT
ejpam-4733	163	11	(	(	PUNCT
ejpam-4733	163	12	2023	2023	NUM
ejpam-4733	163	13	)	)	PUNCT
ejpam-4733	163	14	,	,	PUNCT
ejpam-4733	163	15	1533	1533	NUM
ejpam-4733	163	16	-	-	SYM
ejpam-4733	163	17	1542	1542	NUM
ejpam-4733	163	18	1538	1538	NUM
ejpam-4733	163	19	(	(	PUNCT
ejpam-4733	163	20	1	1	NUM
ejpam-4733	163	21	)	)	PUNCT
ejpam-4733	163	22	δp(λ	δp(λ	NOUN
ejpam-4733	163	23	,	,	PUNCT
ejpam-4733	163	24	p)ker({x	p)ker({x	NOUN
ejpam-4733	163	25	}	}	PUNCT
ejpam-4733	163	26	)	)	PUNCT
ejpam-4733	163	27	̸=	̸=	PROPN
ejpam-4733	163	28	δp(λ	δp(λ	NOUN
ejpam-4733	163	29	,	,	PUNCT
ejpam-4733	163	30	p)ker({y	p)ker({y	NOUN
ejpam-4733	163	31	}	}	PUNCT
ejpam-4733	163	32	)	)	PUNCT
ejpam-4733	163	33	.	.	PUNCT
ejpam-4733	164	1	(	(	PUNCT
ejpam-4733	164	2	2	2	X
ejpam-4733	164	3	)	)	PUNCT
ejpam-4733	164	4	{	{	PUNCT
ejpam-4733	164	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	164	6	,	,	PUNCT
ejpam-4733	164	7	p	p	NOUN
ejpam-4733	164	8	)	)	PUNCT
ejpam-4733	164	9	̸=	̸=	PROPN
ejpam-4733	164	10	{	{	PUNCT
ejpam-4733	164	11	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	164	12	,	,	PUNCT
ejpam-4733	164	13	p	p	NOUN
ejpam-4733	164	14	)	)	PUNCT
ejpam-4733	164	15	.	.	PUNCT
ejpam-4733	165	1	proof	proof	NOUN
ejpam-4733	165	2	.	.	PUNCT
ejpam-4733	166	1	(	(	PUNCT
ejpam-4733	166	2	1	1	X
ejpam-4733	166	3	)	)	PUNCT
ejpam-4733	166	4	⇒	⇒	NOUN
ejpam-4733	166	5	(	(	PUNCT
ejpam-4733	166	6	2	2	NUM
ejpam-4733	166	7	):	):	PUNCT
ejpam-4733	166	8	suppose	suppose	VERB
ejpam-4733	166	9	that	that	SCONJ
ejpam-4733	166	10	δp(λ	δp(λ	NOUN
ejpam-4733	166	11	,	,	PUNCT
ejpam-4733	166	12	p)ker({x	p)ker({x	ADJ
ejpam-4733	166	13	}	}	PUNCT
ejpam-4733	166	14	)	)	PUNCT
ejpam-4733	166	15	̸=	̸=	PROPN
ejpam-4733	166	16	δp(λ	δp(λ	NOUN
ejpam-4733	166	17	,	,	PUNCT
ejpam-4733	166	18	p)ker({y	p)ker({y	NOUN
ejpam-4733	166	19	}	}	PUNCT
ejpam-4733	166	20	)	)	PUNCT
ejpam-4733	166	21	.	.	PUNCT
ejpam-4733	167	1	then	then	ADV
ejpam-4733	167	2	,	,	PUNCT
ejpam-4733	167	3	there	there	PRON
ejpam-4733	167	4	exists	exist	VERB
ejpam-4733	167	5	a	a	DET
ejpam-4733	167	6	point	point	NOUN
ejpam-4733	167	7	z	z	NOUN
ejpam-4733	167	8	∈	∈	PROPN
ejpam-4733	167	9	x	x	PUNCT
ejpam-4733	167	10	such	such	ADJ
ejpam-4733	167	11	that	that	SCONJ
ejpam-4733	167	12	z	z	PROPN
ejpam-4733	167	13	∈	∈	PROPN
ejpam-4733	167	14	δp(λ	δp(λ	NOUN
ejpam-4733	167	15	,	,	PUNCT
ejpam-4733	167	16	p)ker({x	p)ker({x	NOUN
ejpam-4733	167	17	}	}	PUNCT
ejpam-4733	167	18	)	)	PUNCT
ejpam-4733	167	19	and	and	CCONJ
ejpam-4733	167	20	z	z	PROPN
ejpam-4733	167	21	̸∈	̸∈	PROPN
ejpam-4733	167	22	δp(λ	δp(λ	NOUN
ejpam-4733	167	23	,	,	PUNCT
ejpam-4733	167	24	p)ker({y	p)ker({y	NOUN
ejpam-4733	167	25	}	}	PUNCT
ejpam-4733	167	26	)	)	PUNCT
ejpam-4733	167	27	or	or	CCONJ
ejpam-4733	167	28	z	z	NOUN
ejpam-4733	167	29	∈	∈	PROPN
ejpam-4733	167	30	δp(λ	δp(λ	NOUN
ejpam-4733	167	31	,	,	PUNCT
ejpam-4733	167	32	p)ker({y	p)ker({y	NOUN
ejpam-4733	167	33	}	}	PUNCT
ejpam-4733	167	34	)	)	PUNCT
ejpam-4733	167	35	and	and	CCONJ
ejpam-4733	167	36	z	z	PROPN
ejpam-4733	167	37	̸∈	̸∈	PROPN
ejpam-4733	167	38	δp(λ	δp(λ	NOUN
ejpam-4733	167	39	,	,	PUNCT
ejpam-4733	167	40	p)ker({x	p)ker({x	NOUN
ejpam-4733	167	41	}	}	PUNCT
ejpam-4733	167	42	)	)	PUNCT
ejpam-4733	167	43	.	.	PUNCT
ejpam-4733	168	1	we	we	PRON
ejpam-4733	168	2	prove	prove	VERB
ejpam-4733	168	3	only	only	ADV
ejpam-4733	168	4	the	the	DET
ejpam-4733	168	5	first	first	ADJ
ejpam-4733	168	6	case	case	NOUN
ejpam-4733	168	7	being	be	AUX
ejpam-4733	168	8	the	the	DET
ejpam-4733	168	9	second	second	ADJ
ejpam-4733	168	10	analogous	analogous	NOUN
ejpam-4733	168	11	.	.	PUNCT
ejpam-4733	169	1	from	from	ADP
ejpam-4733	169	2	z	z	PROPN
ejpam-4733	169	3	∈	∈	PROPN
ejpam-4733	169	4	δp(λ	δp(λ	NOUN
ejpam-4733	169	5	,	,	PUNCT
ejpam-4733	169	6	p)ker({x	p)ker({x	ADJ
ejpam-4733	169	7	}	}	PUNCT
ejpam-4733	169	8	)	)	PUNCT
ejpam-4733	170	1	it	it	PRON
ejpam-4733	170	2	follows	follow	VERB
ejpam-4733	170	3	that	that	SCONJ
ejpam-4733	170	4	{	{	PUNCT
ejpam-4733	170	5	x	x	NOUN
ejpam-4733	170	6	}	}	PUNCT
ejpam-4733	170	7	∩	∩	NOUN
ejpam-4733	170	8	{	{	PUNCT
ejpam-4733	170	9	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	170	10	,	,	PUNCT
ejpam-4733	170	11	p	p	NOUN
ejpam-4733	170	12	)	)	PUNCT
ejpam-4733	170	13	̸=	̸=	PROPN
ejpam-4733	170	14	∅	∅	NOUN
ejpam-4733	170	15	which	which	PRON
ejpam-4733	170	16	implies	imply	VERB
ejpam-4733	170	17	x	x	X
ejpam-4733	170	18	∈	∈	PROPN
ejpam-4733	170	19	{	{	PUNCT
ejpam-4733	170	20	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	170	21	,	,	PUNCT
ejpam-4733	170	22	p	p	NOUN
ejpam-4733	170	23	)	)	PUNCT
ejpam-4733	170	24	.	.	PUNCT
ejpam-4733	171	1	by	by	ADP
ejpam-4733	171	2	z	z	PROPN
ejpam-4733	171	3	̸∈	̸∈	PROPN
ejpam-4733	171	4	δp(λ	δp(λ	NOUN
ejpam-4733	171	5	,	,	PUNCT
ejpam-4733	171	6	p)ker({y	p)ker({y	NOUN
ejpam-4733	171	7	}	}	PUNCT
ejpam-4733	171	8	)	)	PUNCT
ejpam-4733	171	9	,	,	PUNCT
ejpam-4733	171	10	we	we	PRON
ejpam-4733	171	11	have	have	VERB
ejpam-4733	171	12	{	{	PUNCT
ejpam-4733	171	13	y	y	NOUN
ejpam-4733	171	14	}	}	PUNCT
ejpam-4733	171	15	∩	∩	NOUN
ejpam-4733	171	16	{	{	PUNCT
ejpam-4733	171	17	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	171	18	,	,	PUNCT
ejpam-4733	171	19	p	p	NOUN
ejpam-4733	171	20	)	)	PUNCT
ejpam-4733	171	21	=	=	PUNCT
ejpam-4733	171	22	∅.	∅.	NOUN
ejpam-4733	171	23	since	since	SCONJ
ejpam-4733	171	24	x	x	PROPN
ejpam-4733	171	25	∈	∈	PROPN
ejpam-4733	171	26	{	{	PUNCT
ejpam-4733	171	27	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	171	28	,	,	PUNCT
ejpam-4733	171	29	p	p	NOUN
ejpam-4733	171	30	)	)	PUNCT
ejpam-4733	171	31	,	,	PUNCT
ejpam-4733	171	32	{	{	PUNCT
ejpam-4733	171	33	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	171	34	,	,	PUNCT
ejpam-4733	171	35	p	p	NOUN
ejpam-4733	171	36	)	)	PUNCT
ejpam-4733	171	37	⊆	⊆	NUM
ejpam-4733	171	38	{	{	PUNCT
ejpam-4733	171	39	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	171	40	,	,	PUNCT
ejpam-4733	171	41	p	p	NOUN
ejpam-4733	171	42	)	)	PUNCT
ejpam-4733	171	43	and	and	CCONJ
ejpam-4733	171	44	{	{	PUNCT
ejpam-4733	171	45	y	y	NOUN
ejpam-4733	171	46	}	}	PUNCT
ejpam-4733	171	47	∩	∩	NOUN
ejpam-4733	171	48	{	{	PUNCT
ejpam-4733	171	49	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	171	50	,	,	PUNCT
ejpam-4733	171	51	p	p	NOUN
ejpam-4733	171	52	)	)	PUNCT
ejpam-4733	171	53	=	=	PUNCT
ejpam-4733	171	54	∅.	∅.	VERB
ejpam-4733	171	55	therefore	therefore	ADV
ejpam-4733	171	56	,	,	PUNCT
ejpam-4733	171	57	{	{	PUNCT
ejpam-4733	171	58	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	171	59	,	,	PUNCT
ejpam-4733	171	60	p	p	NOUN
ejpam-4733	171	61	)	)	PUNCT
ejpam-4733	171	62	̸=	̸=	PROPN
ejpam-4733	171	63	{	{	PUNCT
ejpam-4733	171	64	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	171	65	,	,	PUNCT
ejpam-4733	171	66	p	p	NOUN
ejpam-4733	171	67	)	)	PUNCT
ejpam-4733	171	68	.	.	PUNCT
ejpam-4733	172	1	thus	thus	ADV
ejpam-4733	172	2	,	,	PUNCT
ejpam-4733	172	3	δp(λ	δp(λ	NOUN
ejpam-4733	172	4	,	,	PUNCT
ejpam-4733	172	5	p)ker({x	p)ker({x	ADJ
ejpam-4733	172	6	}	}	PUNCT
ejpam-4733	172	7	)	)	PUNCT
ejpam-4733	172	8	̸=	̸=	PROPN
ejpam-4733	172	9	δp(λ	δp(λ	NOUN
ejpam-4733	172	10	,	,	PUNCT
ejpam-4733	172	11	p)ker({y	p)ker({y	NOUN
ejpam-4733	172	12	}	}	PUNCT
ejpam-4733	172	13	)	)	PUNCT
ejpam-4733	172	14	implies	imply	VERB
ejpam-4733	172	15	that	that	SCONJ
ejpam-4733	172	16	{	{	PUNCT
ejpam-4733	172	17	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	172	18	,	,	PUNCT
ejpam-4733	172	19	p	p	NOUN
ejpam-4733	172	20	)	)	PUNCT
ejpam-4733	172	21	̸=	̸=	PROPN
ejpam-4733	172	22	{	{	PUNCT
ejpam-4733	172	23	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	172	24	,	,	PUNCT
ejpam-4733	172	25	p	p	NOUN
ejpam-4733	172	26	)	)	PUNCT
ejpam-4733	172	27	.	.	PUNCT
ejpam-4733	173	1	(	(	PUNCT
ejpam-4733	173	2	2	2	X
ejpam-4733	173	3	)	)	PUNCT
ejpam-4733	173	4	⇒	⇒	NOUN
ejpam-4733	173	5	(	(	PUNCT
ejpam-4733	173	6	1	1	NUM
ejpam-4733	173	7	):	):	PUNCT
ejpam-4733	173	8	suppose	suppose	VERB
ejpam-4733	173	9	that	that	SCONJ
ejpam-4733	173	10	{	{	PUNCT
ejpam-4733	173	11	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	173	12	,	,	PUNCT
ejpam-4733	173	13	p	p	NOUN
ejpam-4733	173	14	)	)	PUNCT
ejpam-4733	173	15	̸=	̸=	PROPN
ejpam-4733	173	16	{	{	PUNCT
ejpam-4733	173	17	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	173	18	,	,	PUNCT
ejpam-4733	173	19	p	p	NOUN
ejpam-4733	173	20	)	)	PUNCT
ejpam-4733	173	21	.	.	PUNCT
ejpam-4733	174	1	there	there	PRON
ejpam-4733	174	2	exists	exist	VERB
ejpam-4733	174	3	a	a	DET
ejpam-4733	174	4	point	point	NOUN
ejpam-4733	174	5	z	z	NOUN
ejpam-4733	174	6	∈	∈	PROPN
ejpam-4733	174	7	x	x	PUNCT
ejpam-4733	174	8	such	such	ADJ
ejpam-4733	174	9	that	that	SCONJ
ejpam-4733	174	10	z	z	PROPN
ejpam-4733	174	11	∈	∈	PROPN
ejpam-4733	174	12	{	{	PUNCT
ejpam-4733	174	13	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	174	14	,	,	PUNCT
ejpam-4733	174	15	p	p	NOUN
ejpam-4733	174	16	)	)	PUNCT
ejpam-4733	174	17	and	and	CCONJ
ejpam-4733	174	18	z	z	PROPN
ejpam-4733	174	19	̸∈	̸∈	PROPN
ejpam-4733	174	20	{	{	PUNCT
ejpam-4733	174	21	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	174	22	,	,	PUNCT
ejpam-4733	174	23	p	p	NOUN
ejpam-4733	174	24	)	)	PUNCT
ejpam-4733	174	25	or	or	CCONJ
ejpam-4733	174	26	z	z	NOUN
ejpam-4733	174	27	∈	∈	PROPN
ejpam-4733	174	28	{	{	PUNCT
ejpam-4733	174	29	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	174	30	,	,	PUNCT
ejpam-4733	174	31	p	p	NOUN
ejpam-4733	174	32	)	)	PUNCT
ejpam-4733	174	33	and	and	CCONJ
ejpam-4733	174	34	z	z	PROPN
ejpam-4733	174	35	̸∈	̸∈	PROPN
ejpam-4733	174	36	{	{	PUNCT
ejpam-4733	174	37	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	174	38	,	,	PUNCT
ejpam-4733	174	39	p	p	NOUN
ejpam-4733	174	40	)	)	PUNCT
ejpam-4733	174	41	.	.	PUNCT
ejpam-4733	175	1	we	we	PRON
ejpam-4733	175	2	prove	prove	VERB
ejpam-4733	175	3	only	only	ADV
ejpam-4733	175	4	the	the	DET
ejpam-4733	175	5	first	first	ADJ
ejpam-4733	175	6	case	case	NOUN
ejpam-4733	175	7	being	be	AUX
ejpam-4733	175	8	the	the	DET
ejpam-4733	175	9	second	second	ADJ
ejpam-4733	175	10	analogous	analogous	NOUN
ejpam-4733	175	11	.	.	PUNCT
ejpam-4733	176	1	it	it	PRON
ejpam-4733	176	2	follows	follow	VERB
ejpam-4733	176	3	that	that	SCONJ
ejpam-4733	176	4	there	there	PRON
ejpam-4733	176	5	exists	exist	VERB
ejpam-4733	176	6	a	a	DET
ejpam-4733	176	7	δp(λ	δp(λ	NOUN
ejpam-4733	176	8	,	,	PUNCT
ejpam-4733	176	9	p)open	p)open	PROPN
ejpam-4733	176	10	set	set	NOUN
ejpam-4733	176	11	containing	contain	VERB
ejpam-4733	176	12	z	z	PROPN
ejpam-4733	176	13	and	and	CCONJ
ejpam-4733	176	14	therefore	therefore	ADV
ejpam-4733	176	15	x	x	X
ejpam-4733	176	16	but	but	CCONJ
ejpam-4733	176	17	not	not	PART
ejpam-4733	176	18	y	y	NOUN
ejpam-4733	176	19	,	,	PUNCT
ejpam-4733	176	20	namely	namely	ADV
ejpam-4733	176	21	,	,	PUNCT
ejpam-4733	176	22	y	y	PROPN
ejpam-4733	176	23	̸∈	̸∈	PROPN
ejpam-4733	176	24	δp(λ	δp(λ	NOUN
ejpam-4733	176	25	,	,	PUNCT
ejpam-4733	176	26	p)ker({x	p)ker({x	NOUN
ejpam-4733	176	27	}	}	PUNCT
ejpam-4733	176	28	)	)	PUNCT
ejpam-4733	176	29	and	and	CCONJ
ejpam-4733	176	30	thus	thus	ADV
ejpam-4733	176	31	δp(λ	δp(λ	NOUN
ejpam-4733	176	32	,	,	PUNCT
ejpam-4733	176	33	p)ker({x	p)ker({x	NOUN
ejpam-4733	176	34	}	}	PUNCT
ejpam-4733	176	35	)	)	PUNCT
ejpam-4733	176	36	̸=	̸=	PROPN
ejpam-4733	176	37	δp(λ	δp(λ	NOUN
ejpam-4733	176	38	,	,	PUNCT
ejpam-4733	176	39	p)ker({y	p)ker({y	NOUN
ejpam-4733	176	40	}	}	PUNCT
ejpam-4733	176	41	)	)	PUNCT
ejpam-4733	176	42	.	.	PUNCT
ejpam-4733	177	1	lemma	lemma	PROPN
ejpam-4733	177	2	6	6	NUM
ejpam-4733	177	3	.	.	PUNCT
ejpam-4733	178	1	let	let	VERB
ejpam-4733	178	2	(	(	PUNCT
ejpam-4733	178	3	x	x	NOUN
ejpam-4733	178	4	,	,	PUNCT
ejpam-4733	178	5	τ	τ	X
ejpam-4733	178	6	)	)	PUNCT
ejpam-4733	178	7	be	be	VERB
ejpam-4733	178	8	a	a	DET
ejpam-4733	178	9	topological	topological	ADJ
ejpam-4733	178	10	space	space	NOUN
ejpam-4733	178	11	and	and	CCONJ
ejpam-4733	178	12	x	x	NOUN
ejpam-4733	178	13	,	,	PUNCT
ejpam-4733	178	14	y	y	PROPN
ejpam-4733	178	15	∈	∈	PROPN
ejpam-4733	178	16	x.	x.	NOUN
ejpam-4733	179	1	then	then	ADV
ejpam-4733	179	2	,	,	PUNCT
ejpam-4733	179	3	the	the	DET
ejpam-4733	179	4	following	follow	VERB
ejpam-4733	179	5	properties	property	NOUN
ejpam-4733	179	6	hold	hold	VERB
ejpam-4733	179	7	:	:	PUNCT
ejpam-4733	179	8	(	(	PUNCT
ejpam-4733	179	9	1	1	X
ejpam-4733	179	10	)	)	PUNCT
ejpam-4733	179	11	y	y	PROPN
ejpam-4733	179	12	∈	∈	PROPN
ejpam-4733	179	13	δp(λ	δp(λ	NOUN
ejpam-4733	179	14	,	,	PUNCT
ejpam-4733	179	15	p)ker({x	p)ker({x	ADJ
ejpam-4733	179	16	}	}	PUNCT
ejpam-4733	179	17	)	)	PUNCT
ejpam-4733	180	1	if	if	SCONJ
ejpam-4733	180	2	and	and	CCONJ
ejpam-4733	180	3	only	only	ADV
ejpam-4733	180	4	if	if	SCONJ
ejpam-4733	180	5	x	x	SYM
ejpam-4733	180	6	∈	∈	NOUN
ejpam-4733	180	7	{	{	PUNCT
ejpam-4733	180	8	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	180	9	,	,	PUNCT
ejpam-4733	180	10	p	p	NOUN
ejpam-4733	180	11	)	)	PUNCT
ejpam-4733	180	12	.	.	PUNCT
ejpam-4733	181	1	(	(	PUNCT
ejpam-4733	181	2	2	2	NUM
ejpam-4733	181	3	)	)	PUNCT
ejpam-4733	181	4	δp(λ	δp(λ	NOUN
ejpam-4733	181	5	,	,	PUNCT
ejpam-4733	181	6	p)ker({x	p)ker({x	NOUN
ejpam-4733	181	7	}	}	PUNCT
ejpam-4733	181	8	)	)	PUNCT
ejpam-4733	182	1	=	=	SYM
ejpam-4733	182	2	δp(λ	δp(λ	NOUN
ejpam-4733	182	3	,	,	PUNCT
ejpam-4733	182	4	p)ker({y	p)ker({y	NOUN
ejpam-4733	182	5	}	}	PUNCT
ejpam-4733	182	6	)	)	PUNCT
ejpam-4733	182	7	if	if	SCONJ
ejpam-4733	182	8	and	and	CCONJ
ejpam-4733	182	9	only	only	ADV
ejpam-4733	182	10	if	if	SCONJ
ejpam-4733	182	11	{	{	PUNCT
ejpam-4733	182	12	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	182	13	,	,	PUNCT
ejpam-4733	182	14	p	p	NOUN
ejpam-4733	182	15	)	)	PUNCT
ejpam-4733	182	16	=	=	SYM
ejpam-4733	182	17	{	{	PUNCT
ejpam-4733	182	18	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	182	19	,	,	PUNCT
ejpam-4733	182	20	p	p	NOUN
ejpam-4733	182	21	)	)	PUNCT
ejpam-4733	182	22	.	.	PUNCT
ejpam-4733	183	1	proof	proof	NOUN
ejpam-4733	183	2	.	.	PUNCT
ejpam-4733	184	1	(	(	PUNCT
ejpam-4733	184	2	1	1	X
ejpam-4733	184	3	)	)	PUNCT
ejpam-4733	184	4	let	let	VERB
ejpam-4733	184	5	x	x	SYM
ejpam-4733	184	6	̸∈	̸∈	PROPN
ejpam-4733	184	7	{	{	PUNCT
ejpam-4733	184	8	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	184	9	,	,	PUNCT
ejpam-4733	184	10	p	p	NOUN
ejpam-4733	184	11	)	)	PUNCT
ejpam-4733	184	12	.	.	PUNCT
ejpam-4733	185	1	then	then	ADV
ejpam-4733	185	2	,	,	PUNCT
ejpam-4733	185	3	there	there	PRON
ejpam-4733	185	4	exists	exist	VERB
ejpam-4733	185	5	u	u	PROPN
ejpam-4733	185	6	∈	∈	PROPN
ejpam-4733	185	7	δp(λ	δp(λ	NOUN
ejpam-4733	185	8	,	,	PUNCT
ejpam-4733	185	9	p)o(x	p)o(x	ADJ
ejpam-4733	185	10	,	,	PUNCT
ejpam-4733	185	11	τ	τ	PROPN
ejpam-4733	185	12	)	)	PUNCT
ejpam-4733	185	13	such	such	ADJ
ejpam-4733	185	14	that	that	SCONJ
ejpam-4733	185	15	x	x	SYM
ejpam-4733	185	16	∈	∈	PROPN
ejpam-4733	185	17	u	u	NOUN
ejpam-4733	185	18	and	and	CCONJ
ejpam-4733	185	19	y	y	PROPN
ejpam-4733	185	20	̸∈	̸∈	PROPN
ejpam-4733	185	21	u	u	PROPN
ejpam-4733	185	22	.	.	PUNCT
ejpam-4733	186	1	thus	thus	ADV
ejpam-4733	186	2	,	,	PUNCT
ejpam-4733	186	3	y	y	PROPN
ejpam-4733	186	4	̸∈	̸∈	PROPN
ejpam-4733	186	5	δp(λ	δp(λ	NOUN
ejpam-4733	186	6	,	,	PUNCT
ejpam-4733	186	7	p)ker({x	p)ker({x	NOUN
ejpam-4733	186	8	}	}	PUNCT
ejpam-4733	186	9	)	)	PUNCT
ejpam-4733	186	10	.	.	PUNCT
ejpam-4733	187	1	the	the	DET
ejpam-4733	187	2	converse	converse	NOUN
ejpam-4733	187	3	is	be	AUX
ejpam-4733	187	4	similarly	similarly	ADV
ejpam-4733	187	5	shown	show	VERB
ejpam-4733	187	6	.	.	PUNCT
ejpam-4733	188	1	(	(	PUNCT
ejpam-4733	188	2	2	2	X
ejpam-4733	188	3	)	)	PUNCT
ejpam-4733	188	4	suppose	suppose	VERB
ejpam-4733	188	5	that	that	SCONJ
ejpam-4733	188	6	δp(λ	δp(λ	NOUN
ejpam-4733	188	7	,	,	PUNCT
ejpam-4733	188	8	p)ker({x	p)ker({x	ADJ
ejpam-4733	188	9	}	}	PUNCT
ejpam-4733	188	10	)	)	PUNCT
ejpam-4733	188	11	=	=	SYM
ejpam-4733	189	1	δp(λ	δp(λ	NOUN
ejpam-4733	189	2	,	,	PUNCT
ejpam-4733	189	3	p)ker({y	p)ker({y	NOUN
ejpam-4733	189	4	}	}	PUNCT
ejpam-4733	189	5	)	)	PUNCT
ejpam-4733	189	6	for	for	ADP
ejpam-4733	189	7	any	any	DET
ejpam-4733	189	8	x	x	NOUN
ejpam-4733	189	9	,	,	PUNCT
ejpam-4733	189	10	y	y	PROPN
ejpam-4733	189	11	∈	∈	PROPN
ejpam-4733	189	12	x.	x.	VERB
ejpam-4733	190	1	since	since	SCONJ
ejpam-4733	190	2	x	x	PROPN
ejpam-4733	190	3	∈	∈	PROPN
ejpam-4733	190	4	δp(λ	δp(λ	NOUN
ejpam-4733	190	5	,	,	PUNCT
ejpam-4733	190	6	p)ker({x	p)ker({x	ADJ
ejpam-4733	190	7	}	}	PUNCT
ejpam-4733	190	8	)	)	PUNCT
ejpam-4733	190	9	,	,	PUNCT
ejpam-4733	190	10	x	x	PUNCT
ejpam-4733	190	11	∈	∈	PROPN
ejpam-4733	190	12	δp(λ	δp(λ	NOUN
ejpam-4733	190	13	,	,	PUNCT
ejpam-4733	190	14	p)ker({y	p)ker({y	NOUN
ejpam-4733	190	15	}	}	PUNCT
ejpam-4733	190	16	)	)	PUNCT
ejpam-4733	190	17	,	,	PUNCT
ejpam-4733	190	18	by	by	ADP
ejpam-4733	190	19	(	(	PUNCT
ejpam-4733	190	20	1	1	NUM
ejpam-4733	190	21	)	)	PUNCT
ejpam-4733	190	22	,	,	PUNCT
ejpam-4733	190	23	y	y	PROPN
ejpam-4733	190	24	∈	∈	PROPN
ejpam-4733	190	25	{	{	PUNCT
ejpam-4733	190	26	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	190	27	,	,	PUNCT
ejpam-4733	190	28	p	p	NOUN
ejpam-4733	190	29	)	)	PUNCT
ejpam-4733	190	30	.	.	PUNCT
ejpam-4733	191	1	by	by	ADP
ejpam-4733	191	2	lemma	lemma	PROPN
ejpam-4733	191	3	1	1	NUM
ejpam-4733	191	4	,	,	PUNCT
ejpam-4733	191	5	{	{	PUNCT
ejpam-4733	191	6	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	191	7	,	,	PUNCT
ejpam-4733	191	8	p	p	NOUN
ejpam-4733	191	9	)	)	PUNCT
ejpam-4733	191	10	⊆	⊆	NUM
ejpam-4733	191	11	{	{	PUNCT
ejpam-4733	191	12	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	191	13	,	,	PUNCT
ejpam-4733	191	14	p	p	NOUN
ejpam-4733	191	15	)	)	PUNCT
ejpam-4733	191	16	.	.	PUNCT
ejpam-4733	192	1	similarly	similarly	ADV
ejpam-4733	192	2	,	,	PUNCT
ejpam-4733	192	3	we	we	PRON
ejpam-4733	192	4	have	have	VERB
ejpam-4733	192	5	{	{	PUNCT
ejpam-4733	192	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	192	7	,	,	PUNCT
ejpam-4733	192	8	p	p	NOUN
ejpam-4733	192	9	)	)	PUNCT
ejpam-4733	192	10	⊆	⊆	NUM
ejpam-4733	192	11	{	{	PUNCT
ejpam-4733	192	12	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	192	13	,	,	PUNCT
ejpam-4733	192	14	p	p	NOUN
ejpam-4733	192	15	)	)	PUNCT
ejpam-4733	192	16	and	and	CCONJ
ejpam-4733	192	17	hence	hence	ADV
ejpam-4733	192	18	{	{	PUNCT
ejpam-4733	192	19	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	192	20	,	,	PUNCT
ejpam-4733	192	21	p	p	NOUN
ejpam-4733	192	22	)	)	PUNCT
ejpam-4733	192	23	=	=	SYM
ejpam-4733	192	24	{	{	PUNCT
ejpam-4733	192	25	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	192	26	,	,	PUNCT
ejpam-4733	192	27	p	p	NOUN
ejpam-4733	192	28	)	)	PUNCT
ejpam-4733	192	29	.	.	PUNCT
ejpam-4733	193	1	conversely	conversely	ADV
ejpam-4733	193	2	,	,	PUNCT
ejpam-4733	193	3	suppose	suppose	VERB
ejpam-4733	193	4	that	that	SCONJ
ejpam-4733	193	5	{	{	PUNCT
ejpam-4733	193	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	193	7	,	,	PUNCT
ejpam-4733	193	8	p	p	NOUN
ejpam-4733	193	9	)	)	PUNCT
ejpam-4733	193	10	=	=	SYM
ejpam-4733	193	11	{	{	PUNCT
ejpam-4733	193	12	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	193	13	,	,	PUNCT
ejpam-4733	193	14	p	p	NOUN
ejpam-4733	193	15	)	)	PUNCT
ejpam-4733	193	16	.	.	PUNCT
ejpam-4733	194	1	since	since	SCONJ
ejpam-4733	194	2	x	x	PROPN
ejpam-4733	194	3	∈	∈	PROPN
ejpam-4733	194	4	{	{	PUNCT
ejpam-4733	194	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	194	6	,	,	PUNCT
ejpam-4733	194	7	p	p	NOUN
ejpam-4733	194	8	)	)	PUNCT
ejpam-4733	194	9	,	,	PUNCT
ejpam-4733	194	10	we	we	PRON
ejpam-4733	194	11	have	have	VERB
ejpam-4733	194	12	x	x	PART
ejpam-4733	194	13	∈	∈	PROPN
ejpam-4733	194	14	{	{	PUNCT
ejpam-4733	194	15	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	194	16	,	,	PUNCT
ejpam-4733	194	17	p	p	NOUN
ejpam-4733	194	18	)	)	PUNCT
ejpam-4733	194	19	and	and	CCONJ
ejpam-4733	194	20	by	by	ADP
ejpam-4733	194	21	(	(	PUNCT
ejpam-4733	194	22	1	1	NUM
ejpam-4733	194	23	)	)	PUNCT
ejpam-4733	194	24	,	,	PUNCT
ejpam-4733	194	25	y	y	PROPN
ejpam-4733	194	26	∈	∈	PROPN
ejpam-4733	194	27	δp(λ	δp(λ	NOUN
ejpam-4733	194	28	,	,	PUNCT
ejpam-4733	194	29	p)ker({x	p)ker({x	ADJ
ejpam-4733	194	30	}	}	PUNCT
ejpam-4733	194	31	)	)	PUNCT
ejpam-4733	194	32	.	.	PUNCT
ejpam-4733	195	1	by	by	ADP
ejpam-4733	195	2	lemma	lemma	PROPN
ejpam-4733	195	3	5	5	NUM
ejpam-4733	195	4	,	,	PUNCT
ejpam-4733	195	5	δp(λ	δp(λ	NOUN
ejpam-4733	195	6	,	,	PUNCT
ejpam-4733	195	7	p)ker({y	p)ker({y	NOUN
ejpam-4733	195	8	}	}	PUNCT
ejpam-4733	195	9	)	)	PUNCT
ejpam-4733	195	10	⊆	⊆	NUM
ejpam-4733	195	11	δp(λ	δp(λ	NOUN
ejpam-4733	195	12	,	,	PUNCT
ejpam-4733	195	13	p)ker(δp(λ	p)ker(δp(λ	PROPN
ejpam-4733	195	14	,	,	PUNCT
ejpam-4733	195	15	p)ker({x	p)ker({x	NOUN
ejpam-4733	195	16	}	}	PUNCT
ejpam-4733	195	17	)	)	PUNCT
ejpam-4733	195	18	)	)	PUNCT
ejpam-4733	196	1	=	=	SYM
ejpam-4733	196	2	δp(λ	δp(λ	NOUN
ejpam-4733	196	3	,	,	PUNCT
ejpam-4733	196	4	p)ker({x	p)ker({x	NOUN
ejpam-4733	196	5	}	}	PUNCT
ejpam-4733	196	6	)	)	PUNCT
ejpam-4733	196	7	.	.	PUNCT
ejpam-4733	197	1	similarly	similarly	ADV
ejpam-4733	197	2	,	,	PUNCT
ejpam-4733	197	3	we	we	PRON
ejpam-4733	197	4	have	have	VERB
ejpam-4733	197	5	δp(λ	δp(λ	NOUN
ejpam-4733	197	6	,	,	PUNCT
ejpam-4733	197	7	p)ker({x	p)ker({x	ADJ
ejpam-4733	197	8	}	}	PUNCT
ejpam-4733	197	9	)	)	PUNCT
ejpam-4733	198	1	⊆	⊆	NUM
ejpam-4733	198	2	δp(λ	δp(λ	NOUN
ejpam-4733	198	3	,	,	PUNCT
ejpam-4733	198	4	p)ker({y	p)ker({y	NOUN
ejpam-4733	198	5	}	}	PUNCT
ejpam-4733	198	6	)	)	PUNCT
ejpam-4733	198	7	and	and	CCONJ
ejpam-4733	198	8	hence	hence	ADV
ejpam-4733	198	9	δp(λ	δp(λ	NOUN
ejpam-4733	198	10	,	,	PUNCT
ejpam-4733	198	11	p)ker({x	p)ker({x	NOUN
ejpam-4733	198	12	}	}	PUNCT
ejpam-4733	198	13	)	)	PUNCT
ejpam-4733	199	1	=	=	SYM
ejpam-4733	199	2	δp(λ	δp(λ	NOUN
ejpam-4733	199	3	,	,	PUNCT
ejpam-4733	199	4	p)ker({y	p)ker({y	NOUN
ejpam-4733	199	5	}	}	PUNCT
ejpam-4733	199	6	)	)	PUNCT
ejpam-4733	199	7	.	.	PUNCT
ejpam-4733	200	1	c.	c.	PROPN
ejpam-4733	200	2	boonpok	boonpok	PROPN
ejpam-4733	200	3	,	,	PUNCT
ejpam-4733	200	4	m.	m.	NOUN
ejpam-4733	200	5	thongmoon	thongmoon	PROPN
ejpam-4733	200	6	/	/	SYM
ejpam-4733	200	7	eur	eur	PROPN
ejpam-4733	200	8	.	.	PUNCT
ejpam-4733	201	1	j.	j.	PROPN
ejpam-4733	201	2	pure	pure	PROPN
ejpam-4733	201	3	appl	appl	PROPN
ejpam-4733	201	4	.	.	PROPN
ejpam-4733	201	5	math	math	PROPN
ejpam-4733	201	6	,	,	PUNCT
ejpam-4733	201	7	16	16	NUM
ejpam-4733	201	8	(	(	PUNCT
ejpam-4733	201	9	3	3	NUM
ejpam-4733	201	10	)	)	PUNCT
ejpam-4733	201	11	(	(	PUNCT
ejpam-4733	201	12	2023	2023	NUM
ejpam-4733	201	13	)	)	PUNCT
ejpam-4733	201	14	,	,	PUNCT
ejpam-4733	201	15	1533	1533	NUM
ejpam-4733	201	16	-	-	SYM
ejpam-4733	201	17	1542	1542	NUM
ejpam-4733	201	18	1539	1539	NUM
ejpam-4733	201	19	theorem	theorem	VERB
ejpam-4733	201	20	6	6	NUM
ejpam-4733	201	21	.	.	PUNCT
ejpam-4733	202	1	a	a	DET
ejpam-4733	202	2	topological	topological	ADJ
ejpam-4733	202	3	space	space	NOUN
ejpam-4733	202	4	(	(	PUNCT
ejpam-4733	202	5	x	x	X
ejpam-4733	202	6	,	,	PUNCT
ejpam-4733	202	7	τ	τ	X
ejpam-4733	202	8	)	)	PUNCT
ejpam-4733	202	9	is	be	AUX
ejpam-4733	202	10	δp(λ	δp(λ	NOUN
ejpam-4733	202	11	,	,	PUNCT
ejpam-4733	202	12	p)-r0	p)-r0	NOUN
ejpam-4733	202	13	if	if	SCONJ
ejpam-4733	202	14	and	and	CCONJ
ejpam-4733	202	15	only	only	ADV
ejpam-4733	202	16	if	if	SCONJ
ejpam-4733	202	17	for	for	ADP
ejpam-4733	202	18	each	each	DET
ejpam-4733	202	19	points	point	NOUN
ejpam-4733	202	20	x	x	PUNCT
ejpam-4733	202	21	and	and	CCONJ
ejpam-4733	202	22	y	y	PROPN
ejpam-4733	202	23	in	in	ADP
ejpam-4733	202	24	x	x	PROPN
ejpam-4733	202	25	,	,	PUNCT
ejpam-4733	202	26	δp(λ	δp(λ	NOUN
ejpam-4733	202	27	,	,	PUNCT
ejpam-4733	202	28	p)ker({x	p)ker({x	ADJ
ejpam-4733	202	29	}	}	PUNCT
ejpam-4733	202	30	)	)	PUNCT
ejpam-4733	202	31	̸=	̸=	PROPN
ejpam-4733	202	32	δp(λ	δp(λ	NOUN
ejpam-4733	202	33	,	,	PUNCT
ejpam-4733	202	34	p)ker({y	p)ker({y	NOUN
ejpam-4733	202	35	}	}	PUNCT
ejpam-4733	202	36	)	)	PUNCT
ejpam-4733	202	37	implies	imply	VERB
ejpam-4733	202	38	δp(λ	δp(λ	NOUN
ejpam-4733	202	39	,	,	PUNCT
ejpam-4733	202	40	p)ker({x	p)ker({x	ADJ
ejpam-4733	202	41	}	}	PUNCT
ejpam-4733	202	42	)	)	PUNCT
ejpam-4733	202	43	∩	∩	NOUN
ejpam-4733	202	44	δp(λ	δp(λ	NOUN
ejpam-4733	202	45	,	,	PUNCT
ejpam-4733	202	46	p)ker({y	p)ker({y	NOUN
ejpam-4733	202	47	}	}	PUNCT
ejpam-4733	202	48	)	)	PUNCT
ejpam-4733	203	1	=	=	PUNCT
ejpam-4733	203	2	∅.	∅.	NOUN
ejpam-4733	203	3	proof	proof	NOUN
ejpam-4733	203	4	.	.	PUNCT
ejpam-4733	204	1	let	let	VERB
ejpam-4733	204	2	(	(	PUNCT
ejpam-4733	204	3	x	x	NOUN
ejpam-4733	204	4	,	,	PUNCT
ejpam-4733	204	5	τ	τ	X
ejpam-4733	204	6	)	)	PUNCT
ejpam-4733	204	7	be	be	AUX
ejpam-4733	204	8	δp(λ	δp(λ	NOUN
ejpam-4733	204	9	,	,	PUNCT
ejpam-4733	204	10	p)-r0	p)-r0	NOUN
ejpam-4733	204	11	.	.	PUNCT
ejpam-4733	205	1	suppose	suppose	VERB
ejpam-4733	205	2	that	that	SCONJ
ejpam-4733	205	3	δp(λ	δp(λ	NOUN
ejpam-4733	205	4	,	,	PUNCT
ejpam-4733	205	5	p)ker({x	p)ker({x	NOUN
ejpam-4733	205	6	}	}	PUNCT
ejpam-4733	205	7	)	)	PUNCT
ejpam-4733	205	8	∩	∩	NOUN
ejpam-4733	205	9	δp(λ	δp(λ	NOUN
ejpam-4733	205	10	,	,	PUNCT
ejpam-4733	205	11	p)ker({y	p)ker({y	NOUN
ejpam-4733	205	12	}	}	PUNCT
ejpam-4733	205	13	)	)	PUNCT
ejpam-4733	205	14	̸=	̸=	PROPN
ejpam-4733	205	15	∅.	∅.	ADV
ejpam-4733	205	16	let	let	VERB
ejpam-4733	205	17	z	z	PROPN
ejpam-4733	205	18	∈	∈	PROPN
ejpam-4733	205	19	δp(λ	δp(λ	NOUN
ejpam-4733	205	20	,	,	PUNCT
ejpam-4733	205	21	p)ker({x})∩δp(λ	p)ker({x})∩δp(λ	NOUN
ejpam-4733	205	22	,	,	PUNCT
ejpam-4733	205	23	p)ker({y	p)ker({y	NOUN
ejpam-4733	205	24	}	}	PUNCT
ejpam-4733	205	25	)	)	PUNCT
ejpam-4733	205	26	.	.	PUNCT
ejpam-4733	206	1	then	then	ADV
ejpam-4733	206	2	,	,	PUNCT
ejpam-4733	206	3	z	z	PROPN
ejpam-4733	206	4	∈	∈	PROPN
ejpam-4733	206	5	δp(λ	δp(λ	NOUN
ejpam-4733	206	6	,	,	PUNCT
ejpam-4733	206	7	p)ker({x	p)ker({x	NOUN
ejpam-4733	206	8	}	}	PUNCT
ejpam-4733	206	9	)	)	PUNCT
ejpam-4733	206	10	and	and	CCONJ
ejpam-4733	206	11	by	by	ADP
ejpam-4733	206	12	lemma	lemma	PROPN
ejpam-4733	206	13	6	6	NUM
ejpam-4733	206	14	,	,	PUNCT
ejpam-4733	206	15	x	x	SYM
ejpam-4733	206	16	∈	∈	PROPN
ejpam-4733	206	17	{	{	PUNCT
ejpam-4733	206	18	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	206	19	,	,	PUNCT
ejpam-4733	206	20	p	p	NOUN
ejpam-4733	206	21	)	)	PUNCT
ejpam-4733	206	22	.	.	PUNCT
ejpam-4733	207	1	thus	thus	ADV
ejpam-4733	207	2	,	,	PUNCT
ejpam-4733	207	3	x	x	SYM
ejpam-4733	207	4	∈	∈	PROPN
ejpam-4733	207	5	{	{	PUNCT
ejpam-4733	207	6	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	207	7	,	,	PUNCT
ejpam-4733	207	8	p	p	NOUN
ejpam-4733	207	9	)	)	PUNCT
ejpam-4733	207	10	∩	∩	NOUN
ejpam-4733	207	11	{	{	PUNCT
ejpam-4733	207	12	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	207	13	,	,	PUNCT
ejpam-4733	207	14	p	p	NOUN
ejpam-4733	207	15	)	)	PUNCT
ejpam-4733	207	16	and	and	CCONJ
ejpam-4733	207	17	by	by	ADP
ejpam-4733	207	18	corollary	corollary	ADJ
ejpam-4733	207	19	1	1	NUM
ejpam-4733	207	20	,	,	PUNCT
ejpam-4733	207	21	{	{	PUNCT
ejpam-4733	207	22	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	207	23	,	,	PUNCT
ejpam-4733	207	24	p	p	NOUN
ejpam-4733	207	25	)	)	PUNCT
ejpam-4733	207	26	=	=	SYM
ejpam-4733	207	27	{	{	PUNCT
ejpam-4733	207	28	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	207	29	,	,	PUNCT
ejpam-4733	207	30	p	p	NOUN
ejpam-4733	207	31	)	)	PUNCT
ejpam-4733	207	32	.	.	PUNCT
ejpam-4733	208	1	similarly	similarly	ADV
ejpam-4733	208	2	,	,	PUNCT
ejpam-4733	208	3	we	we	PRON
ejpam-4733	208	4	have	have	VERB
ejpam-4733	208	5	{	{	PUNCT
ejpam-4733	208	6	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	208	7	,	,	PUNCT
ejpam-4733	208	8	p	p	NOUN
ejpam-4733	208	9	)	)	PUNCT
ejpam-4733	208	10	=	=	SYM
ejpam-4733	208	11	{	{	PUNCT
ejpam-4733	208	12	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	208	13	,	,	PUNCT
ejpam-4733	208	14	p	p	NOUN
ejpam-4733	208	15	)	)	PUNCT
ejpam-4733	208	16	and	and	CCONJ
ejpam-4733	208	17	hence	hence	ADV
ejpam-4733	208	18	{	{	PUNCT
ejpam-4733	208	19	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	208	20	,	,	PUNCT
ejpam-4733	208	21	p	p	NOUN
ejpam-4733	208	22	)	)	PUNCT
ejpam-4733	208	23	=	=	SYM
ejpam-4733	208	24	{	{	PUNCT
ejpam-4733	208	25	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	208	26	,	,	PUNCT
ejpam-4733	208	27	p	p	NOUN
ejpam-4733	208	28	)	)	PUNCT
ejpam-4733	208	29	,	,	PUNCT
ejpam-4733	208	30	by	by	ADP
ejpam-4733	208	31	lemma	lemma	PROPN
ejpam-4733	208	32	6	6	NUM
ejpam-4733	208	33	,	,	PUNCT
ejpam-4733	208	34	δp(λ	δp(λ	NOUN
ejpam-4733	208	35	,	,	PUNCT
ejpam-4733	208	36	p)ker({x	p)ker({x	NOUN
ejpam-4733	208	37	}	}	PUNCT
ejpam-4733	208	38	)	)	PUNCT
ejpam-4733	208	39	=	=	SYM
ejpam-4733	209	1	δp(λ	δp(λ	NOUN
ejpam-4733	209	2	,	,	PUNCT
ejpam-4733	209	3	p)ker({y	p)ker({y	NOUN
ejpam-4733	209	4	}	}	PUNCT
ejpam-4733	209	5	)	)	PUNCT
ejpam-4733	209	6	.	.	PUNCT
ejpam-4733	210	1	conversely	conversely	ADV
ejpam-4733	210	2	,	,	PUNCT
ejpam-4733	210	3	we	we	PRON
ejpam-4733	210	4	show	show	VERB
ejpam-4733	210	5	the	the	DET
ejpam-4733	210	6	sufficiency	sufficiency	NOUN
ejpam-4733	210	7	by	by	ADP
ejpam-4733	210	8	using	use	VERB
ejpam-4733	210	9	corollary	corollary	ADJ
ejpam-4733	210	10	1	1	NUM
ejpam-4733	210	11	.	.	PUNCT
ejpam-4733	210	12	suppose	suppose	VERB
ejpam-4733	210	13	that	that	SCONJ
ejpam-4733	210	14	{	{	PUNCT
ejpam-4733	210	15	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	210	16	,	,	PUNCT
ejpam-4733	210	17	p	p	NOUN
ejpam-4733	210	18	)	)	PUNCT
ejpam-4733	210	19	̸=	̸=	PROPN
ejpam-4733	210	20	{	{	PUNCT
ejpam-4733	210	21	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	210	22	,	,	PUNCT
ejpam-4733	210	23	p	p	NOUN
ejpam-4733	210	24	)	)	PUNCT
ejpam-4733	210	25	.	.	PUNCT
ejpam-4733	211	1	by	by	ADP
ejpam-4733	211	2	lemma	lemma	PROPN
ejpam-4733	211	3	6	6	NUM
ejpam-4733	211	4	,	,	PUNCT
ejpam-4733	211	5	δp(λ	δp(λ	NOUN
ejpam-4733	211	6	,	,	PUNCT
ejpam-4733	211	7	p)ker({x	p)ker({x	ADJ
ejpam-4733	211	8	}	}	PUNCT
ejpam-4733	211	9	)	)	PUNCT
ejpam-4733	211	10	̸=	̸=	PROPN
ejpam-4733	211	11	δp(λ	δp(λ	NOUN
ejpam-4733	211	12	,	,	PUNCT
ejpam-4733	211	13	p)ker({y	p)ker({y	NOUN
ejpam-4733	211	14	}	}	PUNCT
ejpam-4733	211	15	)	)	PUNCT
ejpam-4733	211	16	and	and	CCONJ
ejpam-4733	211	17	hence	hence	ADV
ejpam-4733	211	18	δp(λ	δp(λ	NOUN
ejpam-4733	211	19	,	,	PUNCT
ejpam-4733	211	20	p)ker({x	p)ker({x	NOUN
ejpam-4733	211	21	}	}	PUNCT
ejpam-4733	211	22	)	)	PUNCT
ejpam-4733	211	23	∩	∩	NOUN
ejpam-4733	211	24	δp(λ	δp(λ	NOUN
ejpam-4733	211	25	,	,	PUNCT
ejpam-4733	211	26	p)ker({y	p)ker({y	NOUN
ejpam-4733	211	27	}	}	PUNCT
ejpam-4733	211	28	)	)	PUNCT
ejpam-4733	211	29	=	=	PUNCT
ejpam-4733	211	30	∅.	∅.	ADP
ejpam-4733	211	31	thus	thus	ADV
ejpam-4733	211	32	,	,	PUNCT
ejpam-4733	211	33	{	{	PUNCT
ejpam-4733	211	34	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	211	35	,	,	PUNCT
ejpam-4733	211	36	p	p	NOUN
ejpam-4733	211	37	)	)	PUNCT
ejpam-4733	211	38	∩	∩	NOUN
ejpam-4733	211	39	{	{	PUNCT
ejpam-4733	211	40	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	211	41	,	,	PUNCT
ejpam-4733	211	42	p	p	NOUN
ejpam-4733	211	43	)	)	PUNCT
ejpam-4733	211	44	=	=	PUNCT
ejpam-4733	211	45	∅.	∅.	NOUN
ejpam-4733	211	46	in	in	ADP
ejpam-4733	211	47	fact	fact	NOUN
ejpam-4733	211	48	,	,	PUNCT
ejpam-4733	211	49	assume	assume	VERB
ejpam-4733	211	50	that	that	SCONJ
ejpam-4733	211	51	z	z	PROPN
ejpam-4733	211	52	∈	∈	PROPN
ejpam-4733	211	53	{	{	PUNCT
ejpam-4733	211	54	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	211	55	,	,	PUNCT
ejpam-4733	211	56	p	p	NOUN
ejpam-4733	211	57	)	)	PUNCT
ejpam-4733	211	58	∩	∩	NOUN
ejpam-4733	211	59	{	{	PUNCT
ejpam-4733	211	60	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	211	61	,	,	PUNCT
ejpam-4733	211	62	p	p	NOUN
ejpam-4733	211	63	)	)	PUNCT
ejpam-4733	211	64	.	.	PUNCT
ejpam-4733	212	1	then	then	ADV
ejpam-4733	212	2	,	,	PUNCT
ejpam-4733	212	3	z	z	PROPN
ejpam-4733	212	4	∈	∈	PROPN
ejpam-4733	212	5	{	{	PUNCT
ejpam-4733	212	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	212	7	,	,	PUNCT
ejpam-4733	212	8	p	p	NOUN
ejpam-4733	212	9	)	)	PUNCT
ejpam-4733	212	10	implies	imply	VERB
ejpam-4733	212	11	x	x	PART
ejpam-4733	212	12	∈	∈	PROPN
ejpam-4733	212	13	δp(λ	δp(λ	NOUN
ejpam-4733	212	14	,	,	PUNCT
ejpam-4733	212	15	p)ker({z	p)ker({z	NOUN
ejpam-4733	212	16	}	}	PUNCT
ejpam-4733	212	17	)	)	PUNCT
ejpam-4733	212	18	and	and	CCONJ
ejpam-4733	212	19	hence	hence	ADV
ejpam-4733	212	20	x	x	PART
ejpam-4733	212	21	∈	∈	NOUN
ejpam-4733	212	22	δp(λ	δp(λ	NOUN
ejpam-4733	212	23	,	,	PUNCT
ejpam-4733	212	24	p)ker({z	p)ker({z	NOUN
ejpam-4733	212	25	}	}	PUNCT
ejpam-4733	212	26	)	)	PUNCT
ejpam-4733	212	27	∩	∩	NOUN
ejpam-4733	212	28	δp(λ	δp(λ	NOUN
ejpam-4733	212	29	,	,	PUNCT
ejpam-4733	212	30	p)ker({x	p)ker({x	NOUN
ejpam-4733	212	31	}	}	PUNCT
ejpam-4733	212	32	)	)	PUNCT
ejpam-4733	212	33	.	.	PUNCT
ejpam-4733	213	1	by	by	ADP
ejpam-4733	213	2	the	the	DET
ejpam-4733	213	3	hypothesis	hypothesis	NOUN
ejpam-4733	213	4	,	,	PUNCT
ejpam-4733	213	5	δp(λ	δp(λ	NOUN
ejpam-4733	213	6	,	,	PUNCT
ejpam-4733	213	7	p)ker({z	p)ker({z	NOUN
ejpam-4733	213	8	}	}	PUNCT
ejpam-4733	213	9	)	)	PUNCT
ejpam-4733	213	10	=	=	SYM
ejpam-4733	213	11	δp(λ	δp(λ	NOUN
ejpam-4733	213	12	,	,	PUNCT
ejpam-4733	213	13	p)ker({x	p)ker({x	NOUN
ejpam-4733	213	14	}	}	PUNCT
ejpam-4733	213	15	)	)	PUNCT
ejpam-4733	213	16	and	and	CCONJ
ejpam-4733	213	17	by	by	ADP
ejpam-4733	213	18	lemma	lemma	PROPN
ejpam-4733	213	19	6	6	NUM
ejpam-4733	213	20	,	,	PUNCT
ejpam-4733	213	21	{	{	PUNCT
ejpam-4733	213	22	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	213	23	,	,	PUNCT
ejpam-4733	213	24	p	p	NOUN
ejpam-4733	213	25	)	)	PUNCT
ejpam-4733	213	26	=	=	SYM
ejpam-4733	213	27	{	{	PUNCT
ejpam-4733	213	28	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	213	29	,	,	PUNCT
ejpam-4733	213	30	p	p	NOUN
ejpam-4733	213	31	)	)	PUNCT
ejpam-4733	213	32	.	.	PUNCT
ejpam-4733	214	1	similarly	similarly	ADV
ejpam-4733	214	2	,	,	PUNCT
ejpam-4733	214	3	we	we	PRON
ejpam-4733	214	4	have	have	VERB
ejpam-4733	214	5	{	{	PUNCT
ejpam-4733	214	6	z}δp(λ	z}δp(λ	PROPN
ejpam-4733	214	7	,	,	PUNCT
ejpam-4733	214	8	p	p	NOUN
ejpam-4733	214	9	)	)	PUNCT
ejpam-4733	214	10	=	=	SYM
ejpam-4733	214	11	{	{	PUNCT
ejpam-4733	214	12	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	214	13	,	,	PUNCT
ejpam-4733	214	14	p	p	NOUN
ejpam-4733	214	15	)	)	PUNCT
ejpam-4733	214	16	and	and	CCONJ
ejpam-4733	214	17	hence	hence	ADV
ejpam-4733	214	18	{	{	PUNCT
ejpam-4733	214	19	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	214	20	,	,	PUNCT
ejpam-4733	214	21	p	p	NOUN
ejpam-4733	214	22	)	)	PUNCT
ejpam-4733	214	23	=	=	SYM
ejpam-4733	214	24	{	{	PUNCT
ejpam-4733	214	25	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	214	26	,	,	PUNCT
ejpam-4733	214	27	p	p	NOUN
ejpam-4733	214	28	)	)	PUNCT
ejpam-4733	214	29	.	.	PUNCT
ejpam-4733	215	1	this	this	PRON
ejpam-4733	215	2	contradicts	contradict	VERB
ejpam-4733	215	3	that	that	SCONJ
ejpam-4733	215	4	{	{	PUNCT
ejpam-4733	215	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	215	6	,	,	PUNCT
ejpam-4733	215	7	p	p	NOUN
ejpam-4733	215	8	)	)	PUNCT
ejpam-4733	215	9	̸=	̸=	PROPN
ejpam-4733	215	10	{	{	PUNCT
ejpam-4733	215	11	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	215	12	,	,	PUNCT
ejpam-4733	215	13	p	p	NOUN
ejpam-4733	215	14	)	)	PUNCT
ejpam-4733	215	15	.	.	PUNCT
ejpam-4733	216	1	thus	thus	ADV
ejpam-4733	216	2	,	,	PUNCT
ejpam-4733	216	3	{	{	PUNCT
ejpam-4733	216	4	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	216	5	,	,	PUNCT
ejpam-4733	216	6	p)∩{y}δp(λ	p)∩{y}δp(λ	PROPN
ejpam-4733	216	7	,	,	PUNCT
ejpam-4733	216	8	p	p	NOUN
ejpam-4733	216	9	)	)	PUNCT
ejpam-4733	216	10	=	=	PUNCT
ejpam-4733	216	11	∅.	∅.	ADP
ejpam-4733	216	12	this	this	PRON
ejpam-4733	216	13	shows	show	VERB
ejpam-4733	216	14	that	that	SCONJ
ejpam-4733	216	15	(	(	PUNCT
ejpam-4733	216	16	x	x	X
ejpam-4733	216	17	,	,	PUNCT
ejpam-4733	216	18	τ	τ	X
ejpam-4733	216	19	)	)	PUNCT
ejpam-4733	216	20	is	be	AUX
ejpam-4733	216	21	δp(λ	δp(λ	NOUN
ejpam-4733	216	22	,	,	PUNCT
ejpam-4733	216	23	p)-r0	p)-r0	X
ejpam-4733	216	24	.	.	PUNCT
ejpam-4733	217	1	theorem	theorem	VERB
ejpam-4733	217	2	7	7	NUM
ejpam-4733	217	3	.	.	X
ejpam-4733	217	4	for	for	ADP
ejpam-4733	217	5	a	a	DET
ejpam-4733	217	6	topological	topological	ADJ
ejpam-4733	217	7	space	space	NOUN
ejpam-4733	217	8	(	(	PUNCT
ejpam-4733	217	9	x	x	X
ejpam-4733	217	10	,	,	PUNCT
ejpam-4733	217	11	τ	τ	PROPN
ejpam-4733	217	12	)	)	PUNCT
ejpam-4733	217	13	,	,	PUNCT
ejpam-4733	217	14	the	the	DET
ejpam-4733	217	15	following	follow	VERB
ejpam-4733	217	16	properties	property	NOUN
ejpam-4733	217	17	are	be	AUX
ejpam-4733	217	18	equivalent	equivalent	ADJ
ejpam-4733	217	19	:	:	PUNCT
ejpam-4733	217	20	(	(	PUNCT
ejpam-4733	217	21	1	1	X
ejpam-4733	217	22	)	)	PUNCT
ejpam-4733	217	23	(	(	PUNCT
ejpam-4733	217	24	x	x	X
ejpam-4733	217	25	,	,	PUNCT
ejpam-4733	217	26	τ	τ	X
ejpam-4733	217	27	)	)	PUNCT
ejpam-4733	217	28	is	be	AUX
ejpam-4733	217	29	δp(λ	δp(λ	NOUN
ejpam-4733	217	30	,	,	PUNCT
ejpam-4733	217	31	p)-r0	p)-r0	X
ejpam-4733	217	32	.	.	PUNCT
ejpam-4733	218	1	(	(	PUNCT
ejpam-4733	218	2	2	2	X
ejpam-4733	218	3	)	)	PUNCT
ejpam-4733	218	4	x	x	SYM
ejpam-4733	218	5	∈	∈	PROPN
ejpam-4733	218	6	{	{	PUNCT
ejpam-4733	218	7	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	218	8	,	,	PUNCT
ejpam-4733	218	9	p	p	NOUN
ejpam-4733	218	10	)	)	PUNCT
ejpam-4733	219	1	if	if	SCONJ
ejpam-4733	219	2	and	and	CCONJ
ejpam-4733	219	3	only	only	ADV
ejpam-4733	219	4	if	if	SCONJ
ejpam-4733	219	5	y	y	PROPN
ejpam-4733	219	6	∈	∈	PROPN
ejpam-4733	219	7	{	{	PUNCT
ejpam-4733	219	8	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	219	9	,	,	PUNCT
ejpam-4733	219	10	p	p	NOUN
ejpam-4733	219	11	)	)	PUNCT
ejpam-4733	219	12	.	.	PUNCT
ejpam-4733	220	1	proof	proof	NOUN
ejpam-4733	220	2	.	.	PUNCT
ejpam-4733	221	1	(	(	PUNCT
ejpam-4733	221	2	1	1	X
ejpam-4733	221	3	)	)	PUNCT
ejpam-4733	221	4	⇒	⇒	NOUN
ejpam-4733	221	5	(	(	PUNCT
ejpam-4733	221	6	2	2	NUM
ejpam-4733	221	7	):	):	PUNCT
ejpam-4733	221	8	suppose	suppose	VERB
ejpam-4733	221	9	that	that	SCONJ
ejpam-4733	221	10	x	x	SYM
ejpam-4733	221	11	∈	∈	PROPN
ejpam-4733	221	12	{	{	PUNCT
ejpam-4733	221	13	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	221	14	,	,	PUNCT
ejpam-4733	221	15	p	p	NOUN
ejpam-4733	221	16	)	)	PUNCT
ejpam-4733	221	17	.	.	PUNCT
ejpam-4733	222	1	by	by	ADP
ejpam-4733	222	2	lemma	lemma	PROPN
ejpam-4733	222	3	6	6	NUM
ejpam-4733	222	4	,	,	PUNCT
ejpam-4733	222	5	y	y	PROPN
ejpam-4733	222	6	∈	∈	PROPN
ejpam-4733	222	7	δp(λ	δp(λ	NOUN
ejpam-4733	222	8	,	,	PUNCT
ejpam-4733	222	9	p)ker({x	p)ker({x	NOUN
ejpam-4733	222	10	}	}	PUNCT
ejpam-4733	222	11	)	)	PUNCT
ejpam-4733	222	12	and	and	CCONJ
ejpam-4733	222	13	hence	hence	ADV
ejpam-4733	222	14	δp(λ	δp(λ	NOUN
ejpam-4733	222	15	,	,	PUNCT
ejpam-4733	222	16	p)ker({x	p)ker({x	NOUN
ejpam-4733	222	17	}	}	PUNCT
ejpam-4733	222	18	)	)	PUNCT
ejpam-4733	222	19	∩	∩	NOUN
ejpam-4733	222	20	δp(λ	δp(λ	NOUN
ejpam-4733	222	21	,	,	PUNCT
ejpam-4733	222	22	p)ker({y	p)ker({y	NOUN
ejpam-4733	222	23	}	}	PUNCT
ejpam-4733	222	24	)	)	PUNCT
ejpam-4733	223	1	̸=	̸=	PROPN
ejpam-4733	223	2	∅.	∅.	VERB
ejpam-4733	223	3	by	by	ADP
ejpam-4733	223	4	theorem	theorem	ADJ
ejpam-4733	223	5	6	6	NUM
ejpam-4733	223	6	,	,	PUNCT
ejpam-4733	223	7	δp(λ	δp(λ	NOUN
ejpam-4733	223	8	,	,	PUNCT
ejpam-4733	223	9	p)ker({x	p)ker({x	NOUN
ejpam-4733	223	10	}	}	PUNCT
ejpam-4733	223	11	)	)	PUNCT
ejpam-4733	223	12	=	=	SYM
ejpam-4733	223	13	δp(λ	δp(λ	NOUN
ejpam-4733	223	14	,	,	PUNCT
ejpam-4733	223	15	p)ker({y	p)ker({y	NOUN
ejpam-4733	223	16	}	}	PUNCT
ejpam-4733	223	17	)	)	PUNCT
ejpam-4733	223	18	and	and	CCONJ
ejpam-4733	223	19	hence	hence	ADV
ejpam-4733	223	20	x	x	PART
ejpam-4733	223	21	∈	∈	PROPN
ejpam-4733	223	22	δp(λ	δp(λ	NOUN
ejpam-4733	223	23	,	,	PUNCT
ejpam-4733	223	24	p)ker({y	p)ker({y	NOUN
ejpam-4733	223	25	}	}	PUNCT
ejpam-4733	223	26	)	)	PUNCT
ejpam-4733	223	27	.	.	PUNCT
ejpam-4733	224	1	thus	thus	ADV
ejpam-4733	224	2	,	,	PUNCT
ejpam-4733	224	3	by	by	ADP
ejpam-4733	224	4	lemma	lemma	PROPN
ejpam-4733	224	5	6	6	NUM
ejpam-4733	224	6	,	,	PUNCT
ejpam-4733	224	7	y	y	PROPN
ejpam-4733	224	8	∈	∈	PROPN
ejpam-4733	224	9	{	{	PUNCT
ejpam-4733	224	10	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	224	11	,	,	PUNCT
ejpam-4733	224	12	p	p	NOUN
ejpam-4733	224	13	)	)	PUNCT
ejpam-4733	224	14	.	.	PUNCT
ejpam-4733	225	1	the	the	DET
ejpam-4733	225	2	converse	converse	NOUN
ejpam-4733	225	3	is	be	AUX
ejpam-4733	225	4	similarly	similarly	ADV
ejpam-4733	225	5	shown	show	VERB
ejpam-4733	225	6	.	.	PUNCT
ejpam-4733	226	1	(	(	PUNCT
ejpam-4733	226	2	2	2	X
ejpam-4733	226	3	)	)	PUNCT
ejpam-4733	226	4	⇒	⇒	NOUN
ejpam-4733	226	5	(	(	PUNCT
ejpam-4733	226	6	1	1	NUM
ejpam-4733	226	7	):	):	PUNCT
ejpam-4733	226	8	let	let	VERB
ejpam-4733	226	9	u	u	PRON
ejpam-4733	226	10	∈	∈	PROPN
ejpam-4733	226	11	δp(λ	δp(λ	NOUN
ejpam-4733	226	12	,	,	PUNCT
ejpam-4733	226	13	p)o(x	p)o(x	ADJ
ejpam-4733	226	14	,	,	PUNCT
ejpam-4733	226	15	τ	τ	PROPN
ejpam-4733	226	16	)	)	PUNCT
ejpam-4733	226	17	and	and	CCONJ
ejpam-4733	226	18	x	x	PUNCT
ejpam-4733	226	19	∈	∈	PROPN
ejpam-4733	226	20	u	u	NOUN
ejpam-4733	226	21	.	.	PUNCT
ejpam-4733	227	1	if	if	SCONJ
ejpam-4733	227	2	y	y	PROPN
ejpam-4733	227	3	̸∈	̸∈	PROPN
ejpam-4733	227	4	u	u	PROPN
ejpam-4733	227	5	,	,	PUNCT
ejpam-4733	227	6	then	then	ADV
ejpam-4733	227	7	u	u	NOUN
ejpam-4733	227	8	∩	∩	NOUN
ejpam-4733	227	9	{	{	PUNCT
ejpam-4733	227	10	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	227	11	,	,	PUNCT
ejpam-4733	227	12	p	p	NOUN
ejpam-4733	227	13	)	)	PUNCT
ejpam-4733	227	14	=	=	PUNCT
ejpam-4733	227	15	∅.	∅.	ADP
ejpam-4733	227	16	thus	thus	ADV
ejpam-4733	227	17	,	,	PUNCT
ejpam-4733	227	18	x	x	PROPN
ejpam-4733	227	19	̸∈	̸∈	PROPN
ejpam-4733	227	20	{	{	PUNCT
ejpam-4733	227	21	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	227	22	,	,	PUNCT
ejpam-4733	227	23	p	p	NOUN
ejpam-4733	227	24	)	)	PUNCT
ejpam-4733	227	25	and	and	CCONJ
ejpam-4733	227	26	y	y	PROPN
ejpam-4733	227	27	̸∈	̸∈	PROPN
ejpam-4733	227	28	{	{	PUNCT
ejpam-4733	227	29	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	227	30	,	,	PUNCT
ejpam-4733	227	31	p	p	NOUN
ejpam-4733	227	32	)	)	PUNCT
ejpam-4733	227	33	.	.	PUNCT
ejpam-4733	228	1	this	this	PRON
ejpam-4733	228	2	implies	imply	VERB
ejpam-4733	228	3	that	that	SCONJ
ejpam-4733	228	4	{	{	PUNCT
ejpam-4733	228	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	228	6	,	,	PUNCT
ejpam-4733	228	7	p	p	NOUN
ejpam-4733	228	8	)	)	PUNCT
ejpam-4733	228	9	⊆	⊆	NUM
ejpam-4733	228	10	u	u	NOUN
ejpam-4733	228	11	.	.	PUNCT
ejpam-4733	229	1	therefore	therefore	ADV
ejpam-4733	229	2	,	,	PUNCT
ejpam-4733	229	3	(	(	PUNCT
ejpam-4733	229	4	x	x	X
ejpam-4733	229	5	,	,	PUNCT
ejpam-4733	229	6	τ	τ	X
ejpam-4733	229	7	)	)	PUNCT
ejpam-4733	229	8	is	be	AUX
ejpam-4733	229	9	δp(λ	δp(λ	NOUN
ejpam-4733	229	10	,	,	PUNCT
ejpam-4733	229	11	p)-r0	p)-r0	NOUN
ejpam-4733	229	12	.	.	PUNCT
ejpam-4733	230	1	c.	c.	PROPN
ejpam-4733	230	2	boonpok	boonpok	PROPN
ejpam-4733	230	3	,	,	PUNCT
ejpam-4733	230	4	m.	m.	NOUN
ejpam-4733	230	5	thongmoon	thongmoon	PROPN
ejpam-4733	230	6	/	/	SYM
ejpam-4733	230	7	eur	eur	PROPN
ejpam-4733	230	8	.	.	PUNCT
ejpam-4733	231	1	j.	j.	PROPN
ejpam-4733	231	2	pure	pure	PROPN
ejpam-4733	231	3	appl	appl	PROPN
ejpam-4733	231	4	.	.	PROPN
ejpam-4733	231	5	math	math	PROPN
ejpam-4733	231	6	,	,	PUNCT
ejpam-4733	231	7	16	16	NUM
ejpam-4733	231	8	(	(	PUNCT
ejpam-4733	231	9	3	3	NUM
ejpam-4733	231	10	)	)	PUNCT
ejpam-4733	231	11	(	(	PUNCT
ejpam-4733	231	12	2023	2023	NUM
ejpam-4733	231	13	)	)	PUNCT
ejpam-4733	231	14	,	,	PUNCT
ejpam-4733	231	15	1533	1533	NUM
ejpam-4733	231	16	-	-	SYM
ejpam-4733	231	17	1542	1542	NUM
ejpam-4733	231	18	1540	1540	NUM
ejpam-4733	231	19	theorem	theorem	VERB
ejpam-4733	231	20	8	8	NUM
ejpam-4733	231	21	.	.	PUNCT
ejpam-4733	232	1	for	for	ADP
ejpam-4733	232	2	a	a	DET
ejpam-4733	232	3	topological	topological	ADJ
ejpam-4733	232	4	space	space	NOUN
ejpam-4733	232	5	(	(	PUNCT
ejpam-4733	232	6	x	x	X
ejpam-4733	232	7	,	,	PUNCT
ejpam-4733	232	8	τ	τ	PROPN
ejpam-4733	232	9	)	)	PUNCT
ejpam-4733	232	10	,	,	PUNCT
ejpam-4733	232	11	the	the	DET
ejpam-4733	232	12	following	follow	VERB
ejpam-4733	232	13	properties	property	NOUN
ejpam-4733	232	14	are	be	AUX
ejpam-4733	232	15	equivalent	equivalent	ADJ
ejpam-4733	232	16	:	:	PUNCT
ejpam-4733	232	17	(	(	PUNCT
ejpam-4733	232	18	1	1	X
ejpam-4733	232	19	)	)	PUNCT
ejpam-4733	232	20	(	(	PUNCT
ejpam-4733	232	21	x	x	X
ejpam-4733	232	22	,	,	PUNCT
ejpam-4733	232	23	τ	τ	X
ejpam-4733	232	24	)	)	PUNCT
ejpam-4733	232	25	is	be	AUX
ejpam-4733	232	26	δp(λ	δp(λ	NOUN
ejpam-4733	232	27	,	,	PUNCT
ejpam-4733	232	28	p)-r0	p)-r0	X
ejpam-4733	232	29	.	.	PUNCT
ejpam-4733	233	1	(	(	PUNCT
ejpam-4733	233	2	2	2	X
ejpam-4733	233	3	)	)	PUNCT
ejpam-4733	233	4	for	for	ADP
ejpam-4733	233	5	each	each	DET
ejpam-4733	233	6	nonempty	nonempty	NOUN
ejpam-4733	233	7	subset	subset	VERB
ejpam-4733	233	8	a	a	PRON
ejpam-4733	233	9	of	of	ADP
ejpam-4733	233	10	x	x	PUNCT
ejpam-4733	233	11	and	and	CCONJ
ejpam-4733	233	12	each	each	DET
ejpam-4733	233	13	u	u	PROPN
ejpam-4733	233	14	∈	∈	PROPN
ejpam-4733	233	15	δp(λ	δp(λ	NOUN
ejpam-4733	233	16	,	,	PUNCT
ejpam-4733	233	17	p)o(x	p)o(x	ADJ
ejpam-4733	233	18	,	,	PUNCT
ejpam-4733	233	19	τ	τ	PROPN
ejpam-4733	233	20	)	)	PUNCT
ejpam-4733	233	21	such	such	ADJ
ejpam-4733	233	22	that	that	PRON
ejpam-4733	233	23	a∩u	a∩u	PROPN
ejpam-4733	233	24	̸=	̸=	NOUN
ejpam-4733	233	25	∅	∅	NOUN
ejpam-4733	233	26	,	,	PUNCT
ejpam-4733	233	27	there	there	PRON
ejpam-4733	233	28	exists	exist	VERB
ejpam-4733	233	29	a	a	DET
ejpam-4733	233	30	δp(λ	δp(λ	NOUN
ejpam-4733	233	31	,	,	PUNCT
ejpam-4733	233	32	p)-closed	p)-close	VERB
ejpam-4733	233	33	set	set	NOUN
ejpam-4733	233	34	f	f	PROPN
ejpam-4733	233	35	such	such	ADJ
ejpam-4733	233	36	that	that	SCONJ
ejpam-4733	233	37	a	a	DET
ejpam-4733	233	38	∩	∩	ADJ
ejpam-4733	233	39	f	f	PROPN
ejpam-4733	233	40	̸=	̸=	PROPN
ejpam-4733	233	41	∅	∅	NOUN
ejpam-4733	233	42	and	and	CCONJ
ejpam-4733	233	43	f	f	PROPN
ejpam-4733	233	44	⊆	⊆	NUM
ejpam-4733	233	45	u	u	NOUN
ejpam-4733	233	46	.	.	PUNCT
ejpam-4733	234	1	(	(	PUNCT
ejpam-4733	234	2	3	3	X
ejpam-4733	234	3	)	)	PUNCT
ejpam-4733	234	4	f	f	NOUN
ejpam-4733	234	5	=	=	SYM
ejpam-4733	234	6	δp(λ	δp(λ	PROPN
ejpam-4733	234	7	,	,	PUNCT
ejpam-4733	234	8	p)ker(f	p)ker(f	NOUN
ejpam-4733	234	9	)	)	PUNCT
ejpam-4733	234	10	for	for	ADP
ejpam-4733	234	11	each	each	DET
ejpam-4733	234	12	δp(λ	δp(λ	NOUN
ejpam-4733	234	13	,	,	PUNCT
ejpam-4733	234	14	p)-closed	p)-close	VERB
ejpam-4733	234	15	set	set	VERB
ejpam-4733	234	16	f	f	X
ejpam-4733	234	17	.	.	PUNCT
ejpam-4733	235	1	(	(	PUNCT
ejpam-4733	235	2	4	4	X
ejpam-4733	235	3	)	)	PUNCT
ejpam-4733	235	4	{	{	PUNCT
ejpam-4733	235	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	235	6	,	,	PUNCT
ejpam-4733	235	7	p	p	NOUN
ejpam-4733	235	8	)	)	PUNCT
ejpam-4733	235	9	=	=	SYM
ejpam-4733	235	10	δp(λ	δp(λ	NOUN
ejpam-4733	235	11	,	,	PUNCT
ejpam-4733	235	12	p)ker({x	p)ker({x	NOUN
ejpam-4733	235	13	}	}	PUNCT
ejpam-4733	235	14	)	)	PUNCT
ejpam-4733	235	15	for	for	ADP
ejpam-4733	235	16	each	each	DET
ejpam-4733	235	17	x	x	SYM
ejpam-4733	235	18	∈	∈	PROPN
ejpam-4733	235	19	x.	x.	NOUN
ejpam-4733	235	20	(	(	PUNCT
ejpam-4733	235	21	5	5	NUM
ejpam-4733	235	22	)	)	PUNCT
ejpam-4733	235	23	{	{	PUNCT
ejpam-4733	235	24	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	235	25	,	,	PUNCT
ejpam-4733	235	26	p	p	NOUN
ejpam-4733	235	27	)	)	PUNCT
ejpam-4733	235	28	⊆	⊆	NUM
ejpam-4733	235	29	δp(λ	δp(λ	NOUN
ejpam-4733	235	30	,	,	PUNCT
ejpam-4733	235	31	p)ker({x	p)ker({x	NOUN
ejpam-4733	235	32	}	}	PUNCT
ejpam-4733	235	33	)	)	PUNCT
ejpam-4733	235	34	for	for	ADP
ejpam-4733	235	35	each	each	DET
ejpam-4733	235	36	x	x	SYM
ejpam-4733	235	37	∈	∈	PROPN
ejpam-4733	235	38	x.	x.	NOUN
ejpam-4733	235	39	proof	proof	NOUN
ejpam-4733	235	40	.	.	PUNCT
ejpam-4733	236	1	(	(	PUNCT
ejpam-4733	236	2	1	1	X
ejpam-4733	236	3	)	)	PUNCT
ejpam-4733	236	4	⇒	⇒	NOUN
ejpam-4733	236	5	(	(	PUNCT
ejpam-4733	236	6	2	2	NUM
ejpam-4733	236	7	):	):	PUNCT
ejpam-4733	236	8	let	let	VERB
ejpam-4733	236	9	a	a	PRON
ejpam-4733	236	10	be	be	AUX
ejpam-4733	236	11	a	a	DET
ejpam-4733	236	12	nonempty	nonempty	ADJ
ejpam-4733	236	13	subset	subset	NOUN
ejpam-4733	236	14	of	of	ADP
ejpam-4733	236	15	x	x	PUNCT
ejpam-4733	236	16	and	and	CCONJ
ejpam-4733	236	17	u	u	PROPN
ejpam-4733	236	18	∈	∈	PROPN
ejpam-4733	236	19	δp(λ	δp(λ	NOUN
ejpam-4733	236	20	,	,	PUNCT
ejpam-4733	236	21	p)o(x	p)o(x	ADJ
ejpam-4733	236	22	,	,	PUNCT
ejpam-4733	236	23	τ	τ	PROPN
ejpam-4733	236	24	)	)	PUNCT
ejpam-4733	236	25	such	such	ADJ
ejpam-4733	236	26	that	that	SCONJ
ejpam-4733	236	27	a	a	DET
ejpam-4733	236	28	∩	∩	ADJ
ejpam-4733	236	29	u	u	NOUN
ejpam-4733	236	30	̸=	̸=	PROPN
ejpam-4733	236	31	∅.	∅.	VERB
ejpam-4733	236	32	then	then	ADV
ejpam-4733	236	33	,	,	PUNCT
ejpam-4733	236	34	there	there	PRON
ejpam-4733	236	35	exists	exist	VERB
ejpam-4733	236	36	x	x	X
ejpam-4733	236	37	∈	∈	PROPN
ejpam-4733	236	38	a	a	DET
ejpam-4733	236	39	∩	∩	ADJ
ejpam-4733	236	40	u	u	NOUN
ejpam-4733	236	41	and	and	CCONJ
ejpam-4733	236	42	hence	hence	ADV
ejpam-4733	236	43	{	{	PUNCT
ejpam-4733	236	44	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	236	45	,	,	PUNCT
ejpam-4733	236	46	p	p	NOUN
ejpam-4733	236	47	)	)	PUNCT
ejpam-4733	236	48	⊆	⊆	NUM
ejpam-4733	236	49	u	u	NOUN
ejpam-4733	236	50	.	.	PUNCT
ejpam-4733	237	1	put	put	VERB
ejpam-4733	237	2	f	f	X
ejpam-4733	237	3	=	=	PRON
ejpam-4733	237	4	{	{	PUNCT
ejpam-4733	237	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	237	6	,	,	PUNCT
ejpam-4733	237	7	p	p	NOUN
ejpam-4733	237	8	)	)	PUNCT
ejpam-4733	237	9	.	.	PUNCT
ejpam-4733	238	1	then	then	ADV
ejpam-4733	238	2	,	,	PUNCT
ejpam-4733	238	3	f	f	PROPN
ejpam-4733	238	4	is	be	AUX
ejpam-4733	238	5	δp(λ	δp(λ	NOUN
ejpam-4733	238	6	,	,	PUNCT
ejpam-4733	238	7	p)-closed	p)-close	VERB
ejpam-4733	238	8	,	,	PUNCT
ejpam-4733	238	9	a	a	DET
ejpam-4733	238	10	∩	∩	ADJ
ejpam-4733	238	11	f	f	PROPN
ejpam-4733	238	12	̸=	̸=	PROPN
ejpam-4733	238	13	∅	∅	NOUN
ejpam-4733	238	14	and	and	CCONJ
ejpam-4733	238	15	f	f	PROPN
ejpam-4733	238	16	⊆	⊆	NUM
ejpam-4733	238	17	u	u	NOUN
ejpam-4733	238	18	.	.	PUNCT
ejpam-4733	239	1	(	(	PUNCT
ejpam-4733	239	2	2	2	X
ejpam-4733	239	3	)	)	PUNCT
ejpam-4733	239	4	⇒	⇒	NOUN
ejpam-4733	239	5	(	(	PUNCT
ejpam-4733	239	6	3	3	NUM
ejpam-4733	239	7	):	):	PUNCT
ejpam-4733	239	8	let	let	VERB
ejpam-4733	239	9	f	f	PRON
ejpam-4733	239	10	be	be	AUX
ejpam-4733	239	11	any	any	DET
ejpam-4733	239	12	δp(λ	δp(λ	NOUN
ejpam-4733	239	13	,	,	PUNCT
ejpam-4733	239	14	p)-closed	p)-close	VERB
ejpam-4733	239	15	set	set	NOUN
ejpam-4733	239	16	of	of	ADP
ejpam-4733	239	17	x.	x.	NOUN
ejpam-4733	239	18	by	by	ADP
ejpam-4733	239	19	lemma	lemma	PROPN
ejpam-4733	239	20	5	5	NUM
ejpam-4733	239	21	,	,	PUNCT
ejpam-4733	239	22	we	we	PRON
ejpam-4733	239	23	have	have	VERB
ejpam-4733	239	24	f	f	PROPN
ejpam-4733	239	25	⊆	⊆	NUM
ejpam-4733	239	26	δp(λ	δp(λ	NOUN
ejpam-4733	239	27	,	,	PUNCT
ejpam-4733	239	28	p)ker(f	p)ker(f	NOUN
ejpam-4733	239	29	)	)	PUNCT
ejpam-4733	239	30	.	.	PUNCT
ejpam-4733	240	1	next	next	ADV
ejpam-4733	240	2	,	,	PUNCT
ejpam-4733	240	3	we	we	PRON
ejpam-4733	240	4	show	show	VERB
ejpam-4733	240	5	f	f	PROPN
ejpam-4733	240	6	⊇	⊇	PROPN
ejpam-4733	240	7	δp(λ	δp(λ	PROPN
ejpam-4733	240	8	,	,	PUNCT
ejpam-4733	240	9	p)ker(f	p)ker(f	NOUN
ejpam-4733	240	10	)	)	PUNCT
ejpam-4733	240	11	.	.	PUNCT
ejpam-4733	241	1	let	let	VERB
ejpam-4733	241	2	x	x	SYM
ejpam-4733	241	3	̸∈	̸∈	PROPN
ejpam-4733	241	4	f	f	PROPN
ejpam-4733	241	5	.	.	PUNCT
ejpam-4733	242	1	then	then	ADV
ejpam-4733	242	2	,	,	PUNCT
ejpam-4733	242	3	x	x	PUNCT
ejpam-4733	242	4	∈	∈	NOUN
ejpam-4733	242	5	x	x	X
ejpam-4733	242	6	−	−	PROPN
ejpam-4733	242	7	f	f	PROPN
ejpam-4733	242	8	∈	∈	PROPN
ejpam-4733	242	9	δp(λ	δp(λ	PROPN
ejpam-4733	242	10	,	,	PUNCT
ejpam-4733	242	11	p)o(x	p)o(x	ADJ
ejpam-4733	242	12	,	,	PUNCT
ejpam-4733	242	13	τ	τ	PROPN
ejpam-4733	242	14	)	)	PUNCT
ejpam-4733	242	15	and	and	CCONJ
ejpam-4733	242	16	by	by	ADP
ejpam-4733	242	17	(	(	PUNCT
ejpam-4733	242	18	2	2	NUM
ejpam-4733	242	19	)	)	PUNCT
ejpam-4733	242	20	,	,	PUNCT
ejpam-4733	242	21	there	there	PRON
ejpam-4733	242	22	exists	exist	VERB
ejpam-4733	242	23	a	a	DET
ejpam-4733	242	24	δp(λ	δp(λ	NOUN
ejpam-4733	242	25	,	,	PUNCT
ejpam-4733	242	26	p)-closed	p)-close	VERB
ejpam-4733	242	27	set	set	NOUN
ejpam-4733	242	28	k	k	ADP
ejpam-4733	242	29	such	such	ADJ
ejpam-4733	242	30	that	that	SCONJ
ejpam-4733	242	31	x	x	SYM
ejpam-4733	242	32	∈	∈	PROPN
ejpam-4733	242	33	k	k	PROPN
ejpam-4733	242	34	and	and	CCONJ
ejpam-4733	242	35	k	k	PROPN
ejpam-4733	242	36	⊆	⊆	NUM
ejpam-4733	242	37	x	x	SYM
ejpam-4733	242	38	−	−	PROPN
ejpam-4733	242	39	f	f	NOUN
ejpam-4733	242	40	.	.	PUNCT
ejpam-4733	243	1	now	now	ADV
ejpam-4733	243	2	,	,	PUNCT
ejpam-4733	243	3	put	put	VERB
ejpam-4733	243	4	u	u	NOUN
ejpam-4733	243	5	=	=	NOUN
ejpam-4733	243	6	x	x	PROPN
ejpam-4733	243	7	−	−	PROPN
ejpam-4733	244	1	k.	k.	NOUN
ejpam-4733	244	2	then	then	ADV
ejpam-4733	244	3	,	,	PUNCT
ejpam-4733	244	4	f	f	PROPN
ejpam-4733	244	5	⊆	⊆	NUM
ejpam-4733	244	6	u	u	PROPN
ejpam-4733	244	7	∈	∈	PROPN
ejpam-4733	244	8	δp(λ	δp(λ	NOUN
ejpam-4733	244	9	,	,	PUNCT
ejpam-4733	244	10	p)o(x	p)o(x	ADJ
ejpam-4733	244	11	,	,	PUNCT
ejpam-4733	244	12	τ	τ	PROPN
ejpam-4733	244	13	)	)	PUNCT
ejpam-4733	244	14	and	and	CCONJ
ejpam-4733	244	15	x	x	PUNCT
ejpam-4733	244	16	̸∈	̸∈	PROPN
ejpam-4733	244	17	u	u	PROPN
ejpam-4733	244	18	.	.	PUNCT
ejpam-4733	245	1	thus	thus	ADV
ejpam-4733	245	2	,	,	PUNCT
ejpam-4733	245	3	x	x	PROPN
ejpam-4733	245	4	̸∈	̸∈	PROPN
ejpam-4733	245	5	δp(λ	δp(λ	NOUN
ejpam-4733	245	6	,	,	PUNCT
ejpam-4733	245	7	p)ker(f	p)ker(f	NOUN
ejpam-4733	245	8	)	)	PUNCT
ejpam-4733	245	9	.	.	PUNCT
ejpam-4733	246	1	this	this	PRON
ejpam-4733	246	2	shows	show	VERB
ejpam-4733	246	3	that	that	SCONJ
ejpam-4733	246	4	f	f	PROPN
ejpam-4733	246	5	⊇	⊇	PROPN
ejpam-4733	246	6	δp(λ	δp(λ	PROPN
ejpam-4733	246	7	,	,	PUNCT
ejpam-4733	246	8	p)ker(f	p)ker(f	NOUN
ejpam-4733	246	9	)	)	PUNCT
ejpam-4733	246	10	.	.	PUNCT
ejpam-4733	247	1	(	(	PUNCT
ejpam-4733	247	2	3	3	X
ejpam-4733	247	3	)	)	PUNCT
ejpam-4733	247	4	⇒	⇒	NOUN
ejpam-4733	247	5	(	(	PUNCT
ejpam-4733	247	6	4	4	NUM
ejpam-4733	247	7	):	):	PUNCT
ejpam-4733	247	8	let	let	VERB
ejpam-4733	247	9	x	x	PUNCT
ejpam-4733	247	10	∈	∈	PROPN
ejpam-4733	247	11	x	x	X
ejpam-4733	247	12	and	and	CCONJ
ejpam-4733	247	13	y	y	PROPN
ejpam-4733	247	14	̸∈	̸∈	PROPN
ejpam-4733	247	15	δp(λ	δp(λ	NOUN
ejpam-4733	247	16	,	,	PUNCT
ejpam-4733	247	17	p)ker({x	p)ker({x	NOUN
ejpam-4733	247	18	}	}	PUNCT
ejpam-4733	247	19	)	)	PUNCT
ejpam-4733	247	20	.	.	PUNCT
ejpam-4733	248	1	there	there	PRON
ejpam-4733	248	2	exists	exist	VERB
ejpam-4733	248	3	u	u	PROPN
ejpam-4733	248	4	∈	∈	PROPN
ejpam-4733	248	5	δp(λ	δp(λ	NOUN
ejpam-4733	248	6	,	,	PUNCT
ejpam-4733	248	7	p)o(x	p)o(x	ADJ
ejpam-4733	248	8	,	,	PUNCT
ejpam-4733	248	9	τ	τ	PROPN
ejpam-4733	248	10	)	)	PUNCT
ejpam-4733	248	11	such	such	ADJ
ejpam-4733	248	12	that	that	SCONJ
ejpam-4733	248	13	x	x	SYM
ejpam-4733	248	14	∈	∈	PROPN
ejpam-4733	248	15	u	u	NOUN
ejpam-4733	248	16	and	and	CCONJ
ejpam-4733	248	17	y	y	PROPN
ejpam-4733	248	18	̸∈	̸∈	PROPN
ejpam-4733	248	19	u	u	PROPN
ejpam-4733	248	20	.	.	PUNCT
ejpam-4733	249	1	thus	thus	ADV
ejpam-4733	249	2	,	,	PUNCT
ejpam-4733	249	3	u	u	PROPN
ejpam-4733	249	4	∩	∩	NOUN
ejpam-4733	249	5	{	{	PUNCT
ejpam-4733	249	6	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	249	7	,	,	PUNCT
ejpam-4733	249	8	p	p	NOUN
ejpam-4733	249	9	)	)	PUNCT
ejpam-4733	249	10	=	=	PUNCT
ejpam-4733	249	11	∅.	∅.	X
ejpam-4733	249	12	by	by	ADP
ejpam-4733	249	13	(	(	PUNCT
ejpam-4733	249	14	3	3	NUM
ejpam-4733	249	15	)	)	PUNCT
ejpam-4733	249	16	,	,	PUNCT
ejpam-4733	249	17	u	u	PROPN
ejpam-4733	249	18	∩	∩	NOUN
ejpam-4733	249	19	δp(λ	δp(λ	NOUN
ejpam-4733	249	20	,	,	PUNCT
ejpam-4733	249	21	p)ker({y}δp(λ	p)ker({y}δp(λ	NOUN
ejpam-4733	249	22	,	,	PUNCT
ejpam-4733	249	23	p	p	NOUN
ejpam-4733	249	24	)	)	PUNCT
ejpam-4733	249	25	)	)	PUNCT
ejpam-4733	249	26	=	=	PUNCT
ejpam-4733	249	27	∅.	∅.	NOUN
ejpam-4733	249	28	since	since	SCONJ
ejpam-4733	249	29	x	x	PROPN
ejpam-4733	249	30	̸∈	̸∈	PROPN
ejpam-4733	249	31	δp(λ	δp(λ	NOUN
ejpam-4733	249	32	,	,	PUNCT
ejpam-4733	249	33	p)ker({y}δp(λ	p)ker({y}δp(λ	NUM
ejpam-4733	249	34	,	,	PUNCT
ejpam-4733	249	35	p	p	NOUN
ejpam-4733	249	36	)	)	PUNCT
ejpam-4733	249	37	)	)	PUNCT
ejpam-4733	249	38	,	,	PUNCT
ejpam-4733	249	39	there	there	PRON
ejpam-4733	249	40	exists	exist	VERB
ejpam-4733	249	41	v	v	ADP
ejpam-4733	249	42	∈	∈	PROPN
ejpam-4733	249	43	δp(λ	δp(λ	NOUN
ejpam-4733	249	44	,	,	PUNCT
ejpam-4733	249	45	p)o(x	p)o(x	ADJ
ejpam-4733	249	46	,	,	PUNCT
ejpam-4733	249	47	τ	τ	PROPN
ejpam-4733	249	48	)	)	PUNCT
ejpam-4733	249	49	such	such	ADJ
ejpam-4733	249	50	that	that	SCONJ
ejpam-4733	249	51	{	{	PUNCT
ejpam-4733	249	52	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	249	53	,	,	PUNCT
ejpam-4733	249	54	p	p	NOUN
ejpam-4733	249	55	)	)	PUNCT
ejpam-4733	249	56	⊆	⊆	NUM
ejpam-4733	249	57	v	v	NOUN
ejpam-4733	249	58	and	and	CCONJ
ejpam-4733	249	59	x	x	PART
ejpam-4733	249	60	̸∈	̸∈	PROPN
ejpam-4733	249	61	v	v	NUM
ejpam-4733	249	62	.	.	PUNCT
ejpam-4733	250	1	thus	thus	ADV
ejpam-4733	250	2	,	,	PUNCT
ejpam-4733	250	3	v	v	ADP
ejpam-4733	250	4	∩	∩	NOUN
ejpam-4733	250	5	{	{	PUNCT
ejpam-4733	250	6	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	250	7	,	,	PUNCT
ejpam-4733	250	8	p	p	NOUN
ejpam-4733	250	9	)	)	PUNCT
ejpam-4733	250	10	=	=	PUNCT
ejpam-4733	250	11	∅.	∅.	NOUN
ejpam-4733	250	12	since	since	SCONJ
ejpam-4733	250	13	y	y	PROPN
ejpam-4733	250	14	∈	∈	PROPN
ejpam-4733	250	15	v	v	NOUN
ejpam-4733	250	16	,	,	PUNCT
ejpam-4733	250	17	y	y	PROPN
ejpam-4733	250	18	̸∈	̸∈	PROPN
ejpam-4733	250	19	{	{	PUNCT
ejpam-4733	250	20	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	250	21	,	,	PUNCT
ejpam-4733	250	22	p	p	NOUN
ejpam-4733	250	23	)	)	PUNCT
ejpam-4733	250	24	and	and	CCONJ
ejpam-4733	250	25	hence	hence	ADV
ejpam-4733	250	26	{	{	PUNCT
ejpam-4733	250	27	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	250	28	,	,	PUNCT
ejpam-4733	250	29	p	p	NOUN
ejpam-4733	250	30	)	)	PUNCT
ejpam-4733	250	31	⊆	⊆	NUM
ejpam-4733	250	32	δp(λ	δp(λ	NOUN
ejpam-4733	250	33	,	,	PUNCT
ejpam-4733	250	34	p)ker({x	p)ker({x	NOUN
ejpam-4733	250	35	}	}	PUNCT
ejpam-4733	250	36	)	)	PUNCT
ejpam-4733	250	37	.	.	PUNCT
ejpam-4733	251	1	moreover	moreover	ADV
ejpam-4733	251	2	,	,	PUNCT
ejpam-4733	251	3	{	{	PUNCT
ejpam-4733	251	4	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	251	5	,	,	PUNCT
ejpam-4733	251	6	p	p	NOUN
ejpam-4733	251	7	)	)	PUNCT
ejpam-4733	251	8	⊆	⊆	NUM
ejpam-4733	251	9	δp(λ	δp(λ	NOUN
ejpam-4733	251	10	,	,	PUNCT
ejpam-4733	251	11	p)ker({x	p)ker({x	NOUN
ejpam-4733	251	12	}	}	PUNCT
ejpam-4733	251	13	)	)	PUNCT
ejpam-4733	251	14	⊆	⊆	NUM
ejpam-4733	251	15	δp(λ	δp(λ	NOUN
ejpam-4733	251	16	,	,	PUNCT
ejpam-4733	251	17	p)ker({x}δp(λ	p)ker({x}δp(λ	PRON
ejpam-4733	251	18	,	,	PUNCT
ejpam-4733	251	19	p	p	NOUN
ejpam-4733	251	20	)	)	PUNCT
ejpam-4733	251	21	)	)	PUNCT
ejpam-4733	252	1	=	=	PRON
ejpam-4733	252	2	{	{	PUNCT
ejpam-4733	252	3	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	252	4	,	,	PUNCT
ejpam-4733	252	5	p	p	NOUN
ejpam-4733	252	6	)	)	PUNCT
ejpam-4733	252	7	.	.	PUNCT
ejpam-4733	253	1	this	this	PRON
ejpam-4733	253	2	shows	show	VERB
ejpam-4733	253	3	that	that	SCONJ
ejpam-4733	253	4	{	{	PUNCT
ejpam-4733	253	5	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	253	6	,	,	PUNCT
ejpam-4733	253	7	p	p	NOUN
ejpam-4733	253	8	)	)	PUNCT
ejpam-4733	253	9	=	=	SYM
ejpam-4733	253	10	δp(λ	δp(λ	NOUN
ejpam-4733	253	11	,	,	PUNCT
ejpam-4733	253	12	p)ker({x	p)ker({x	NOUN
ejpam-4733	253	13	}	}	PUNCT
ejpam-4733	253	14	)	)	PUNCT
ejpam-4733	253	15	.	.	PUNCT
ejpam-4733	254	1	(	(	PUNCT
ejpam-4733	254	2	4	4	X
ejpam-4733	254	3	)	)	PUNCT
ejpam-4733	254	4	⇒	⇒	NOUN
ejpam-4733	254	5	(	(	PUNCT
ejpam-4733	254	6	5	5	NUM
ejpam-4733	254	7	):	):	PUNCT
ejpam-4733	254	8	the	the	DET
ejpam-4733	254	9	proof	proof	NOUN
ejpam-4733	254	10	is	be	AUX
ejpam-4733	254	11	obvious	obvious	ADJ
ejpam-4733	254	12	.	.	PUNCT
ejpam-4733	255	1	(	(	PUNCT
ejpam-4733	255	2	5	5	X
ejpam-4733	255	3	)	)	PUNCT
ejpam-4733	255	4	⇒	⇒	NOUN
ejpam-4733	255	5	(	(	PUNCT
ejpam-4733	255	6	1	1	NUM
ejpam-4733	255	7	):	):	PUNCT
ejpam-4733	255	8	let	let	VERB
ejpam-4733	255	9	u	u	PRON
ejpam-4733	255	10	∈	∈	PROPN
ejpam-4733	255	11	δp(λ	δp(λ	NOUN
ejpam-4733	255	12	,	,	PUNCT
ejpam-4733	255	13	p)o(x	p)o(x	ADJ
ejpam-4733	255	14	,	,	PUNCT
ejpam-4733	255	15	τ	τ	PROPN
ejpam-4733	255	16	)	)	PUNCT
ejpam-4733	255	17	and	and	CCONJ
ejpam-4733	255	18	x	x	PUNCT
ejpam-4733	255	19	∈	∈	PROPN
ejpam-4733	255	20	u	u	NOUN
ejpam-4733	255	21	.	.	PUNCT
ejpam-4733	256	1	if	if	SCONJ
ejpam-4733	256	2	y	y	PROPN
ejpam-4733	256	3	̸∈	̸∈	PROPN
ejpam-4733	256	4	u	u	PROPN
ejpam-4733	256	5	,	,	PUNCT
ejpam-4733	256	6	then	then	ADV
ejpam-4733	256	7	u	u	NOUN
ejpam-4733	256	8	∩	∩	NOUN
ejpam-4733	256	9	{	{	PUNCT
ejpam-4733	256	10	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	256	11	,	,	PUNCT
ejpam-4733	256	12	p	p	NOUN
ejpam-4733	256	13	)	)	PUNCT
ejpam-4733	256	14	=	=	NOUN
ejpam-4733	256	15	∅	∅	NOUN
ejpam-4733	256	16	and	and	CCONJ
ejpam-4733	256	17	x	x	PART
ejpam-4733	256	18	̸∈	̸∈	PROPN
ejpam-4733	256	19	{	{	PUNCT
ejpam-4733	256	20	y}δp(λ	y}δp(λ	PROPN
ejpam-4733	256	21	,	,	PUNCT
ejpam-4733	256	22	p	p	NOUN
ejpam-4733	256	23	)	)	PUNCT
ejpam-4733	256	24	.	.	PUNCT
ejpam-4733	257	1	by	by	ADP
ejpam-4733	257	2	lemma	lemma	PROPN
ejpam-4733	257	3	6	6	NUM
ejpam-4733	257	4	,	,	PUNCT
ejpam-4733	257	5	y	y	PROPN
ejpam-4733	257	6	̸∈	̸∈	PROPN
ejpam-4733	257	7	δp(λ	δp(λ	NOUN
ejpam-4733	257	8	,	,	PUNCT
ejpam-4733	257	9	p)ker({x	p)ker({x	NOUN
ejpam-4733	257	10	}	}	PUNCT
ejpam-4733	257	11	)	)	PUNCT
ejpam-4733	257	12	and	and	CCONJ
ejpam-4733	257	13	by	by	ADP
ejpam-4733	257	14	(	(	PUNCT
ejpam-4733	257	15	5	5	NUM
ejpam-4733	257	16	)	)	PUNCT
ejpam-4733	257	17	,	,	PUNCT
ejpam-4733	257	18	y	y	PROPN
ejpam-4733	257	19	̸∈	̸∈	PROPN
ejpam-4733	257	20	{	{	PUNCT
ejpam-4733	257	21	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	257	22	,	,	PUNCT
ejpam-4733	257	23	p	p	NOUN
ejpam-4733	257	24	)	)	PUNCT
ejpam-4733	257	25	.	.	PUNCT
ejpam-4733	258	1	thus	thus	ADV
ejpam-4733	258	2	,	,	PUNCT
ejpam-4733	258	3	{	{	PUNCT
ejpam-4733	258	4	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	258	5	,	,	PUNCT
ejpam-4733	258	6	p	p	NOUN
ejpam-4733	258	7	)	)	PUNCT
ejpam-4733	258	8	⊆	⊆	NUM
ejpam-4733	258	9	u	u	NOUN
ejpam-4733	258	10	and	and	CCONJ
ejpam-4733	258	11	hence	hence	ADV
ejpam-4733	258	12	(	(	PUNCT
ejpam-4733	258	13	x	x	X
ejpam-4733	258	14	,	,	PUNCT
ejpam-4733	258	15	τ	τ	X
ejpam-4733	258	16	)	)	PUNCT
ejpam-4733	258	17	is	be	AUX
ejpam-4733	258	18	δp(λ	δp(λ	NOUN
ejpam-4733	258	19	,	,	PUNCT
ejpam-4733	258	20	p)-r0	p)-r0	X
ejpam-4733	258	21	.	.	PUNCT
ejpam-4733	259	1	corollary	corollary	ADJ
ejpam-4733	259	2	2	2	NUM
ejpam-4733	259	3	.	.	PUNCT
ejpam-4733	260	1	a	a	DET
ejpam-4733	260	2	topological	topological	ADJ
ejpam-4733	260	3	space	space	NOUN
ejpam-4733	260	4	(	(	PUNCT
ejpam-4733	260	5	x	x	X
ejpam-4733	260	6	,	,	PUNCT
ejpam-4733	260	7	τ	τ	X
ejpam-4733	260	8	)	)	PUNCT
ejpam-4733	260	9	is	be	AUX
ejpam-4733	260	10	δp(λ	δp(λ	NOUN
ejpam-4733	260	11	,	,	PUNCT
ejpam-4733	260	12	p)-r0	p)-r0	NOUN
ejpam-4733	260	13	if	if	SCONJ
ejpam-4733	260	14	and	and	CCONJ
ejpam-4733	260	15	only	only	ADV
ejpam-4733	260	16	if	if	SCONJ
ejpam-4733	260	17	δp(λ	δp(λ	NOUN
ejpam-4733	260	18	,	,	PUNCT
ejpam-4733	260	19	p)ker({x	p)ker({x	NOUN
ejpam-4733	260	20	}	}	PUNCT
ejpam-4733	260	21	)	)	PUNCT
ejpam-4733	260	22	⊆	⊆	NUM
ejpam-4733	260	23	{	{	PUNCT
ejpam-4733	260	24	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	260	25	,	,	PUNCT
ejpam-4733	260	26	p	p	NOUN
ejpam-4733	260	27	)	)	PUNCT
ejpam-4733	260	28	for	for	ADP
ejpam-4733	260	29	each	each	DET
ejpam-4733	260	30	x	x	SYM
ejpam-4733	260	31	∈	∈	PROPN
ejpam-4733	260	32	x.	x.	NOUN
ejpam-4733	260	33	references	reference	VERB
ejpam-4733	260	34	1541	1541	NUM
ejpam-4733	260	35	proof	proof	NOUN
ejpam-4733	260	36	.	.	PUNCT
ejpam-4733	261	1	this	this	PRON
ejpam-4733	261	2	is	be	AUX
ejpam-4733	261	3	obvious	obvious	ADJ
ejpam-4733	261	4	by	by	ADP
ejpam-4733	261	5	theorem	theorem	NOUN
ejpam-4733	261	6	8	8	NUM
ejpam-4733	261	7	.	.	PUNCT
ejpam-4733	262	1	conversely	conversely	ADV
ejpam-4733	262	2	,	,	PUNCT
ejpam-4733	262	3	let	let	VERB
ejpam-4733	262	4	x	x	PRON
ejpam-4733	262	5	∈	∈	PROPN
ejpam-4733	262	6	{	{	PUNCT
ejpam-4733	262	7	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	262	8	,	,	PUNCT
ejpam-4733	262	9	p	p	NOUN
ejpam-4733	262	10	)	)	PUNCT
ejpam-4733	262	11	.	.	PUNCT
ejpam-4733	263	1	thus	thus	ADV
ejpam-4733	263	2	,	,	PUNCT
ejpam-4733	263	3	by	by	ADP
ejpam-4733	263	4	lemma	lemma	PROPN
ejpam-4733	263	5	6	6	NUM
ejpam-4733	263	6	,	,	PUNCT
ejpam-4733	263	7	y	y	PROPN
ejpam-4733	263	8	∈	∈	PROPN
ejpam-4733	263	9	δp(λ	δp(λ	NOUN
ejpam-4733	263	10	,	,	PUNCT
ejpam-4733	263	11	p)ker({x	p)ker({x	NOUN
ejpam-4733	263	12	}	}	PUNCT
ejpam-4733	263	13	)	)	PUNCT
ejpam-4733	263	14	and	and	CCONJ
ejpam-4733	263	15	hence	hence	ADV
ejpam-4733	263	16	y	y	PROPN
ejpam-4733	263	17	∈	∈	PROPN
ejpam-4733	263	18	{	{	PUNCT
ejpam-4733	263	19	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	263	20	,	,	PUNCT
ejpam-4733	263	21	p	p	NOUN
ejpam-4733	263	22	)	)	PUNCT
ejpam-4733	263	23	.	.	PUNCT
ejpam-4733	264	1	similarly	similarly	ADV
ejpam-4733	264	2	,	,	PUNCT
ejpam-4733	264	3	if	if	SCONJ
ejpam-4733	264	4	y	y	PROPN
ejpam-4733	264	5	∈	∈	PROPN
ejpam-4733	264	6	{	{	PUNCT
ejpam-4733	264	7	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	264	8	,	,	PUNCT
ejpam-4733	264	9	p	p	NOUN
ejpam-4733	264	10	)	)	PUNCT
ejpam-4733	264	11	,	,	PUNCT
ejpam-4733	264	12	then	then	ADV
ejpam-4733	264	13	x	x	X
ejpam-4733	264	14	∈	∈	PROPN
ejpam-4733	264	15	{	{	PUNCT
ejpam-4733	264	16	y}δp(λ	y}δp(λ	NOUN
ejpam-4733	264	17	,	,	PUNCT
ejpam-4733	264	18	p	p	NOUN
ejpam-4733	264	19	)	)	PUNCT
ejpam-4733	264	20	.	.	PUNCT
ejpam-4733	265	1	it	it	PRON
ejpam-4733	265	2	follows	follow	VERB
ejpam-4733	265	3	from	from	ADP
ejpam-4733	265	4	theorem	theorem	ADJ
ejpam-4733	265	5	7	7	NUM
ejpam-4733	265	6	that	that	SCONJ
ejpam-4733	265	7	(	(	PUNCT
ejpam-4733	265	8	x	x	X
ejpam-4733	265	9	,	,	PUNCT
ejpam-4733	265	10	τ	τ	X
ejpam-4733	265	11	)	)	PUNCT
ejpam-4733	265	12	is	be	AUX
ejpam-4733	265	13	δp(λ	δp(λ	NOUN
ejpam-4733	265	14	,	,	PUNCT
ejpam-4733	265	15	p)-r0	p)-r0	X
ejpam-4733	265	16	.	.	PUNCT
ejpam-4733	266	1	definition	definition	NOUN
ejpam-4733	266	2	10	10	NUM
ejpam-4733	266	3	.	.	PUNCT
ejpam-4733	267	1	let	let	VERB
ejpam-4733	267	2	(	(	PUNCT
ejpam-4733	267	3	x	x	NOUN
ejpam-4733	267	4	,	,	PUNCT
ejpam-4733	267	5	τ	τ	X
ejpam-4733	267	6	)	)	PUNCT
ejpam-4733	267	7	be	be	VERB
ejpam-4733	267	8	a	a	DET
ejpam-4733	267	9	topological	topological	ADJ
ejpam-4733	267	10	space	space	NOUN
ejpam-4733	267	11	and	and	CCONJ
ejpam-4733	267	12	x	x	PUNCT
ejpam-4733	267	13	∈	∈	PROPN
ejpam-4733	267	14	x.	x.	NOUN
ejpam-4733	267	15	a	a	DET
ejpam-4733	267	16	subset	subset	NOUN
ejpam-4733	267	17	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4733	267	18	,	,	PUNCT
ejpam-4733	267	19	p	p	NOUN
ejpam-4733	267	20	)	)	PUNCT
ejpam-4733	267	21	is	be	AUX
ejpam-4733	267	22	defined	define	VERB
ejpam-4733	267	23	as	as	SCONJ
ejpam-4733	267	24	follows	follow	VERB
ejpam-4733	267	25	:	:	PUNCT
ejpam-4733	267	26	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4733	267	27	,	,	PUNCT
ejpam-4733	267	28	p	p	NOUN
ejpam-4733	267	29	)	)	PUNCT
ejpam-4733	267	30	=	=	SYM
ejpam-4733	267	31	δp(λ	δp(λ	NOUN
ejpam-4733	267	32	,	,	PUNCT
ejpam-4733	267	33	p)ker({x	p)ker({x	NOUN
ejpam-4733	267	34	}	}	PUNCT
ejpam-4733	267	35	)	)	PUNCT
ejpam-4733	267	36	∩	∩	NOUN
ejpam-4733	267	37	{	{	PUNCT
ejpam-4733	267	38	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	267	39	,	,	PUNCT
ejpam-4733	267	40	p	p	NOUN
ejpam-4733	267	41	)	)	PUNCT
ejpam-4733	267	42	.	.	PUNCT
ejpam-4733	268	1	theorem	theorem	VERB
ejpam-4733	268	2	9	9	NUM
ejpam-4733	268	3	.	.	PUNCT
ejpam-4733	269	1	a	a	DET
ejpam-4733	269	2	topological	topological	ADJ
ejpam-4733	269	3	space	space	NOUN
ejpam-4733	269	4	(	(	PUNCT
ejpam-4733	269	5	x	x	X
ejpam-4733	269	6	,	,	PUNCT
ejpam-4733	269	7	τ	τ	X
ejpam-4733	269	8	)	)	PUNCT
ejpam-4733	269	9	is	be	AUX
ejpam-4733	269	10	δp(λ	δp(λ	NOUN
ejpam-4733	269	11	,	,	PUNCT
ejpam-4733	269	12	p)-r0	p)-r0	NOUN
ejpam-4733	269	13	if	if	SCONJ
ejpam-4733	269	14	and	and	CCONJ
ejpam-4733	269	15	only	only	ADV
ejpam-4733	269	16	if	if	SCONJ
ejpam-4733	269	17	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4733	269	18	,	,	PUNCT
ejpam-4733	269	19	p	p	NOUN
ejpam-4733	269	20	)	)	PUNCT
ejpam-4733	269	21	=	=	SYM
ejpam-4733	269	22	{	{	PUNCT
ejpam-4733	269	23	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	269	24	,	,	PUNCT
ejpam-4733	269	25	p	p	NOUN
ejpam-4733	269	26	)	)	PUNCT
ejpam-4733	269	27	for	for	ADP
ejpam-4733	269	28	each	each	DET
ejpam-4733	269	29	x	x	SYM
ejpam-4733	269	30	∈	∈	PROPN
ejpam-4733	269	31	x.	x.	NOUN
ejpam-4733	269	32	proof	proof	NOUN
ejpam-4733	269	33	.	.	PUNCT
ejpam-4733	270	1	let	let	VERB
ejpam-4733	271	1	x	x	SYM
ejpam-4733	271	2	∈	∈	PROPN
ejpam-4733	271	3	x.	x.	NOUN
ejpam-4733	271	4	by	by	ADP
ejpam-4733	271	5	theorem	theorem	ADJ
ejpam-4733	271	6	8	8	NUM
ejpam-4733	271	7	,	,	PUNCT
ejpam-4733	271	8	δp(λ	δp(λ	NOUN
ejpam-4733	271	9	,	,	PUNCT
ejpam-4733	271	10	p)ker({x	p)ker({x	NOUN
ejpam-4733	271	11	}	}	PUNCT
ejpam-4733	271	12	)	)	PUNCT
ejpam-4733	271	13	=	=	PRON
ejpam-4733	271	14	{	{	PUNCT
ejpam-4733	271	15	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	271	16	,	,	PUNCT
ejpam-4733	271	17	sp	sp	NOUN
ejpam-4733	271	18	)	)	PUNCT
ejpam-4733	271	19	.	.	PUNCT
ejpam-4733	272	1	thus	thus	ADV
ejpam-4733	272	2	,	,	PUNCT
ejpam-4733	272	3	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4733	272	4	,	,	PUNCT
ejpam-4733	272	5	p	p	NOUN
ejpam-4733	272	6	)	)	PUNCT
ejpam-4733	272	7	=	=	SYM
ejpam-4733	272	8	δp(λ	δp(λ	NOUN
ejpam-4733	272	9	,	,	PUNCT
ejpam-4733	272	10	p)ker({x	p)ker({x	NOUN
ejpam-4733	272	11	}	}	PUNCT
ejpam-4733	272	12	)	)	PUNCT
ejpam-4733	272	13	∩	∩	NOUN
ejpam-4733	272	14	{	{	PUNCT
ejpam-4733	272	15	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	272	16	,	,	PUNCT
ejpam-4733	272	17	p	p	NOUN
ejpam-4733	272	18	)	)	PUNCT
ejpam-4733	272	19	=	=	SYM
ejpam-4733	272	20	{	{	PUNCT
ejpam-4733	272	21	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	272	22	,	,	PUNCT
ejpam-4733	272	23	p	p	NOUN
ejpam-4733	272	24	)	)	PUNCT
ejpam-4733	272	25	.	.	PUNCT
ejpam-4733	273	1	conversely	conversely	ADV
ejpam-4733	273	2	,	,	PUNCT
ejpam-4733	273	3	let	let	VERB
ejpam-4733	273	4	x	x	X
ejpam-4733	273	5	∈	∈	PROPN
ejpam-4733	273	6	x.	x.	NOUN
ejpam-4733	273	7	by	by	ADP
ejpam-4733	273	8	the	the	DET
ejpam-4733	273	9	hypothesis	hypothesis	NOUN
ejpam-4733	273	10	,	,	PUNCT
ejpam-4733	273	11	{	{	PUNCT
ejpam-4733	273	12	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	273	13	,	,	PUNCT
ejpam-4733	273	14	p	p	NOUN
ejpam-4733	273	15	)	)	PUNCT
ejpam-4733	273	16	=	=	SYM
ejpam-4733	273	17	⟨x⟩δp(λ	⟨x⟩δp(λ	NOUN
ejpam-4733	273	18	,	,	PUNCT
ejpam-4733	273	19	p	p	NOUN
ejpam-4733	273	20	)	)	PUNCT
ejpam-4733	273	21	=	=	SYM
ejpam-4733	274	1	δp(λ	δp(λ	NOUN
ejpam-4733	274	2	,	,	PUNCT
ejpam-4733	274	3	p)ker({x	p)ker({x	NOUN
ejpam-4733	274	4	}	}	PUNCT
ejpam-4733	274	5	)	)	PUNCT
ejpam-4733	274	6	∩	∩	NOUN
ejpam-4733	274	7	{	{	PUNCT
ejpam-4733	274	8	x}δp(λ	x}δp(λ	PROPN
ejpam-4733	274	9	,	,	PUNCT
ejpam-4733	274	10	p	p	NOUN
ejpam-4733	274	11	)	)	PUNCT
ejpam-4733	274	12	⊆	⊆	NUM
ejpam-4733	274	13	δp(λ	δp(λ	NOUN
ejpam-4733	274	14	,	,	PUNCT
ejpam-4733	274	15	p)ker({x	p)ker({x	NOUN
ejpam-4733	274	16	}	}	PUNCT
ejpam-4733	274	17	)	)	PUNCT
ejpam-4733	274	18	.	.	PUNCT
ejpam-4733	275	1	it	it	PRON
ejpam-4733	275	2	follows	follow	VERB
ejpam-4733	275	3	from	from	ADP
ejpam-4733	275	4	theorem	theorem	ADJ
ejpam-4733	275	5	8	8	NUM
ejpam-4733	275	6	that	that	PRON
ejpam-4733	275	7	(	(	PUNCT
ejpam-4733	275	8	x	x	X
ejpam-4733	275	9	,	,	PUNCT
ejpam-4733	275	10	τ	τ	X
ejpam-4733	275	11	)	)	PUNCT
ejpam-4733	275	12	is	be	AUX
ejpam-4733	275	13	δp(λ	δp(λ	NOUN
ejpam-4733	275	14	,	,	PUNCT
ejpam-4733	275	15	p)-r0	p)-r0	X
ejpam-4733	275	16	.	.	PUNCT
ejpam-4733	276	1	acknowledgements	acknowledgement	VERB
ejpam-4733	276	2	this	this	DET
ejpam-4733	276	3	research	research	NOUN
ejpam-4733	276	4	project	project	NOUN
ejpam-4733	276	5	was	be	AUX
ejpam-4733	276	6	financially	financially	ADV
ejpam-4733	276	7	supported	support	VERB
ejpam-4733	276	8	by	by	ADP
ejpam-4733	276	9	mahasarakham	mahasarakham	PROPN
ejpam-4733	276	10	university	university	PROPN
ejpam-4733	276	11	.	.	PUNCT
ejpam-4733	277	1	references	reference	NOUN
ejpam-4733	277	2	[	[	X
ejpam-4733	277	3	1	1	NUM
ejpam-4733	277	4	]	]	PUNCT
ejpam-4733	277	5	c.	c.	PROPN
ejpam-4733	277	6	boonpok	boonpok	PROPN
ejpam-4733	277	7	and	and	CCONJ
ejpam-4733	277	8	c.	c.	PROPN
ejpam-4733	277	9	viriyapong	viriyapong	PROPN
ejpam-4733	277	10	.	.	PUNCT
ejpam-4733	278	1	on	on	ADP
ejpam-4733	278	2	(	(	PUNCT
ejpam-4733	278	3	λ	λ	PROPN
ejpam-4733	278	4	,	,	PUNCT
ejpam-4733	278	5	p)-closed	p)-close	VERB
ejpam-4733	278	6	sets	set	NOUN
ejpam-4733	278	7	and	and	CCONJ
ejpam-4733	278	8	the	the	DET
ejpam-4733	278	9	related	related	ADJ
ejpam-4733	278	10	notions	notion	NOUN
ejpam-4733	278	11	in	in	ADP
ejpam-4733	278	12	topological	topological	ADJ
ejpam-4733	278	13	spaces	space	NOUN
ejpam-4733	278	14	.	.	PUNCT
ejpam-4733	279	1	european	european	ADJ
ejpam-4733	279	2	journal	journal	PROPN
ejpam-4733	279	3	of	of	ADP
ejpam-4733	279	4	pure	pure	ADJ
ejpam-4733	279	5	and	and	CCONJ
ejpam-4733	279	6	applied	applied	ADJ
ejpam-4733	279	7	mathematics	mathematic	NOUN
ejpam-4733	279	8	,	,	PUNCT
ejpam-4733	279	9	15(2):415	15(2):415	PROPN
ejpam-4733	279	10	–	–	PUNCT
ejpam-4733	279	11	436	436	NUM
ejpam-4733	279	12	,	,	PUNCT
ejpam-4733	279	13	2022	2022	NUM
ejpam-4733	279	14	.	.	PUNCT
ejpam-4733	280	1	[	[	X
ejpam-4733	280	2	2	2	NUM
ejpam-4733	280	3	]	]	PUNCT
ejpam-4733	280	4	c.	c.	PROPN
ejpam-4733	280	5	boonpok	boonpok	PROPN
ejpam-4733	280	6	and	and	CCONJ
ejpam-4733	280	7	c.	c.	PROPN
ejpam-4733	280	8	viriyapong	viriyapong	PROPN
ejpam-4733	280	9	.	.	PUNCT
ejpam-4733	281	1	on	on	ADP
ejpam-4733	281	2	some	some	DET
ejpam-4733	281	3	forms	form	NOUN
ejpam-4733	281	4	of	of	ADP
ejpam-4733	281	5	closed	closed	ADJ
ejpam-4733	281	6	sets	set	NOUN
ejpam-4733	281	7	and	and	CCONJ
ejpam-4733	281	8	related	related	ADJ
ejpam-4733	281	9	topics	topic	NOUN
ejpam-4733	281	10	.	.	PUNCT
ejpam-4733	282	1	european	european	ADJ
ejpam-4733	282	2	journal	journal	PROPN
ejpam-4733	282	3	of	of	ADP
ejpam-4733	282	4	pure	pure	ADJ
ejpam-4733	282	5	and	and	CCONJ
ejpam-4733	282	6	applied	applied	ADJ
ejpam-4733	282	7	mathematics	mathematic	NOUN
ejpam-4733	282	8	,	,	PUNCT
ejpam-4733	282	9	16(1):336–362	16(1):336–362	NUM
ejpam-4733	282	10	,	,	PUNCT
ejpam-4733	282	11	2023	2023	NUM
ejpam-4733	282	12	.	.	PUNCT
ejpam-4733	283	1	[	[	X
ejpam-4733	283	2	3	3	NUM
ejpam-4733	283	3	]	]	PUNCT
ejpam-4733	283	4	m.	m.	NOUN
ejpam-4733	283	5	caldas	caldas	PROPN
ejpam-4733	283	6	,	,	PUNCT
ejpam-4733	283	7	t.	t.	NOUN
ejpam-4733	283	8	fukutake	fukutake	NOUN
ejpam-4733	283	9	,	,	PUNCT
ejpam-4733	283	10	s.	s.	PROPN
ejpam-4733	283	11	jafari	jafari	PROPN
ejpam-4733	283	12	,	,	PUNCT
ejpam-4733	283	13	and	and	CCONJ
ejpam-4733	283	14	t.	t.	PROPN
ejpam-4733	283	15	noiri	noiri	PROPN
ejpam-4733	283	16	.	.	PUNCT
ejpam-4733	284	1	some	some	DET
ejpam-4733	284	2	applications	application	NOUN
ejpam-4733	284	3	of	of	ADP
ejpam-4733	284	4	δ	δ	NOUN
ejpam-4733	284	5	-	-	PUNCT
ejpam-4733	284	6	preopen	preopen	ADJ
ejpam-4733	284	7	sets	set	NOUN
ejpam-4733	284	8	in	in	ADP
ejpam-4733	284	9	topological	topological	ADJ
ejpam-4733	284	10	spaces	space	NOUN
ejpam-4733	284	11	.	.	PUNCT
ejpam-4733	285	1	bulletin	bulletin	NOUN
ejpam-4733	285	2	of	of	ADP
ejpam-4733	285	3	the	the	DET
ejpam-4733	285	4	institute	institute	NOUN
ejpam-4733	285	5	of	of	ADP
ejpam-4733	285	6	mathematics	mathematics	PROPN
ejpam-4733	285	7	,	,	PUNCT
ejpam-4733	285	8	academia	academia	PROPN
ejpam-4733	285	9	sinica	sinica	PROPN
ejpam-4733	285	10	,	,	PUNCT
ejpam-4733	285	11	33(3):261–276	33(3):261–276	NOUN
ejpam-4733	285	12	,	,	PUNCT
ejpam-4733	285	13	2005	2005	NUM
ejpam-4733	285	14	.	.	PUNCT
ejpam-4733	286	1	[	[	X
ejpam-4733	286	2	4	4	NUM
ejpam-4733	286	3	]	]	PUNCT
ejpam-4733	286	4	m.	m.	NOUN
ejpam-4733	286	5	caldas	caldas	PROPN
ejpam-4733	286	6	,	,	PUNCT
ejpam-4733	286	7	m.	m.	NOUN
ejpam-4733	286	8	ganster	ganster	NOUN
ejpam-4733	286	9	,	,	PUNCT
ejpam-4733	286	10	d.	d.	PROPN
ejpam-4733	286	11	n.	n.	PROPN
ejpam-4733	286	12	georgiou	georgiou	PROPN
ejpam-4733	286	13	,	,	PUNCT
ejpam-4733	286	14	s.	s.	PROPN
ejpam-4733	286	15	jafari	jafari	PROPN
ejpam-4733	286	16	,	,	PUNCT
ejpam-4733	286	17	and	and	CCONJ
ejpam-4733	286	18	t.	t.	PROPN
ejpam-4733	286	19	noiri	noiri	PROPN
ejpam-4733	286	20	.	.	PUNCT
ejpam-4733	287	1	δ	δ	PROPN
ejpam-4733	287	2	-	-	PUNCT
ejpam-4733	287	3	semiopen	semiopen	ADJ
ejpam-4733	287	4	sets	set	NOUN
ejpam-4733	287	5	in	in	ADP
ejpam-4733	287	6	topological	topological	ADJ
ejpam-4733	287	7	spaces	space	NOUN
ejpam-4733	287	8	.	.	PUNCT
ejpam-4733	288	1	topology	topology	NOUN
ejpam-4733	288	2	proceedings	proceeding	NOUN
ejpam-4733	288	3	,	,	PUNCT
ejpam-4733	288	4	29(2):369–383	29(2):369–383	NUM
ejpam-4733	288	5	,	,	PUNCT
ejpam-4733	288	6	2005	2005	NUM
ejpam-4733	288	7	.	.	PUNCT
ejpam-4733	289	1	[	[	X
ejpam-4733	289	2	5	5	NUM
ejpam-4733	289	3	]	]	PUNCT
ejpam-4733	289	4	m.	m.	NOUN
ejpam-4733	289	5	caldas	caldas	PROPN
ejpam-4733	289	6	,	,	PUNCT
ejpam-4733	289	7	d.	d.	PROPN
ejpam-4733	289	8	n.	n.	PROPN
ejpam-4733	289	9	georgiou	georgiou	PROPN
ejpam-4733	289	10	,	,	PUNCT
ejpam-4733	289	11	s.	s.	PROPN
ejpam-4733	289	12	jafari	jafari	PROPN
ejpam-4733	289	13	,	,	PUNCT
ejpam-4733	289	14	and	and	CCONJ
ejpam-4733	289	15	t.	t.	PROPN
ejpam-4733	289	16	noiri	noiri	PROPN
ejpam-4733	289	17	.	.	PUNCT
ejpam-4733	290	1	more	more	ADV
ejpam-4733	290	2	on	on	ADP
ejpam-4733	290	3	δ	δ	PROPN
ejpam-4733	290	4	-	-	PUNCT
ejpam-4733	290	5	semiopen	semiopen	ADJ
ejpam-4733	290	6	sets	set	NOUN
ejpam-4733	290	7	.	.	PUNCT
ejpam-4733	291	1	note	note	VERB
ejpam-4733	291	2	di	di	PROPN
ejpam-4733	291	3	matematica	matematica	PROPN
ejpam-4733	291	4	,	,	PUNCT
ejpam-4733	291	5	22(2):1–14	22(2):1–14	PROPN
ejpam-4733	291	6	,	,	PUNCT
ejpam-4733	291	7	2003	2003	NUM
ejpam-4733	291	8	.	.	PUNCT
ejpam-4733	292	1	[	[	X
ejpam-4733	292	2	6	6	NUM
ejpam-4733	292	3	]	]	PUNCT
ejpam-4733	292	4	m.	m.	NOUN
ejpam-4733	292	5	ganster	ganster	NOUN
ejpam-4733	292	6	,	,	PUNCT
ejpam-4733	292	7	s.	s.	PROPN
ejpam-4733	292	8	jafari	jafari	PROPN
ejpam-4733	292	9	,	,	PUNCT
ejpam-4733	292	10	and	and	CCONJ
ejpam-4733	292	11	t.	t.	PROPN
ejpam-4733	292	12	noiri	noiri	PROPN
ejpam-4733	292	13	.	.	PUNCT
ejpam-4733	293	1	on	on	ADP
ejpam-4733	293	2	pre	pre	ADJ
ejpam-4733	293	3	-	-	ADJ
ejpam-4733	293	4	λ	λ	NOUN
ejpam-4733	293	5	-	-	NOUN
ejpam-4733	293	6	sets	set	NOUN
ejpam-4733	293	7	and	and	CCONJ
ejpam-4733	293	8	pre	pre	ADJ
ejpam-4733	293	9	-	-	ADJ
ejpam-4733	293	10	v	v	ADJ
ejpam-4733	293	11	-sets	-set	NOUN
ejpam-4733	293	12	.	.	PUNCT
ejpam-4733	294	1	acta	acta	PROPN
ejpam-4733	294	2	mathematica	mathematica	PROPN
ejpam-4733	294	3	hungarica	hungarica	PROPN
ejpam-4733	294	4	,	,	PUNCT
ejpam-4733	294	5	95:337–343	95:337–343	PROPN
ejpam-4733	294	6	,	,	PUNCT
ejpam-4733	294	7	2002	2002	NUM
ejpam-4733	294	8	.	.	PUNCT
ejpam-4733	295	1	references	reference	NOUN
ejpam-4733	295	2	1542	1542	NUM
ejpam-4733	295	3	[	[	X
ejpam-4733	295	4	7	7	NUM
ejpam-4733	295	5	]	]	PUNCT
ejpam-4733	295	6	a.	a.	NOUN
ejpam-4733	295	7	s.	s.	PROPN
ejpam-4733	295	8	mashhour	mashhour	PROPN
ejpam-4733	295	9	,	,	PUNCT
ejpam-4733	295	10	m.	m.	PROPN
ejpam-4733	295	11	e.	e.	PROPN
ejpam-4733	295	12	abd	abd	PROPN
ejpam-4733	295	13	el	el	PROPN
ejpam-4733	295	14	-	-	PROPN
ejpam-4733	295	15	monsef	monsef	ADJ
ejpam-4733	295	16	,	,	PUNCT
ejpam-4733	295	17	and	and	CCONJ
ejpam-4733	295	18	s.	s.	PROPN
ejpam-4733	295	19	n.	n.	PROPN
ejpam-4733	295	20	el	el	PROPN
ejpam-4733	295	21	-	-	PROPN
ejpam-4733	295	22	deeb	deeb	PROPN
ejpam-4733	295	23	.	.	PUNCT
ejpam-4733	296	1	on	on	ADP
ejpam-4733	296	2	precontinuous	precontinuous	ADJ
ejpam-4733	296	3	and	and	CCONJ
ejpam-4733	296	4	weak	weak	ADJ
ejpam-4733	296	5	precontinuous	precontinuous	ADJ
ejpam-4733	296	6	mappings	mapping	NOUN
ejpam-4733	296	7	.	.	PUNCT
ejpam-4733	297	1	proceedings	proceeding	NOUN
ejpam-4733	297	2	of	of	ADP
ejpam-4733	297	3	the	the	DET
ejpam-4733	297	4	mathematical	mathematical	ADJ
ejpam-4733	297	5	and	and	CCONJ
ejpam-4733	297	6	physical	physical	ADJ
ejpam-4733	297	7	society	society	NOUN
ejpam-4733	297	8	of	of	ADP
ejpam-4733	297	9	egypt	egypt	PROPN
ejpam-4733	297	10	,	,	PUNCT
ejpam-4733	297	11	53:47–53	53:47–53	NUM
ejpam-4733	297	12	,	,	PUNCT
ejpam-4733	297	13	1982	1982	NUM
ejpam-4733	297	14	.	.	PUNCT
ejpam-4733	298	1	[	[	X
ejpam-4733	298	2	8	8	X
ejpam-4733	298	3	]	]	PUNCT
ejpam-4733	298	4	s.	s.	PROPN
ejpam-4733	298	5	raychaudhuri	raychaudhuri	PROPN
ejpam-4733	298	6	and	and	CCONJ
ejpam-4733	298	7	m.	m.	PROPN
ejpam-4733	298	8	n.	n.	PROPN
ejpam-4733	298	9	mukherjee	mukherjee	PROPN
ejpam-4733	298	10	.	.	PUNCT
ejpam-4733	299	1	on	on	ADP
ejpam-4733	299	2	δ	δ	PROPN
ejpam-4733	299	3	-	-	PUNCT
ejpam-4733	299	4	almost	almost	ADV
ejpam-4733	299	5	continuity	continuity	NOUN
ejpam-4733	299	6	and	and	CCONJ
ejpam-4733	299	7	δ	δ	NOUN
ejpam-4733	299	8	-	-	PUNCT
ejpam-4733	299	9	preopen	preopen	ADJ
ejpam-4733	299	10	sets	set	NOUN
ejpam-4733	299	11	.	.	PUNCT
ejpam-4733	300	1	bulletin	bulletin	NOUN
ejpam-4733	300	2	of	of	ADP
ejpam-4733	300	3	the	the	DET
ejpam-4733	300	4	institute	institute	NOUN
ejpam-4733	300	5	of	of	ADP
ejpam-4733	300	6	mathematics	mathematics	PROPN
ejpam-4733	300	7	,	,	PUNCT
ejpam-4733	300	8	academia	academia	PROPN
ejpam-4733	300	9	sinica	sinica	PROPN
ejpam-4733	300	10	,	,	PUNCT
ejpam-4733	300	11	21:357–366	21:357–366	PROPN
ejpam-4733	300	12	,	,	PUNCT
ejpam-4733	300	13	1993	1993	NUM
ejpam-4733	300	14	.	.	PUNCT
ejpam-4733	301	1	[	[	X
ejpam-4733	301	2	9	9	NUM
ejpam-4733	301	3	]	]	PUNCT
ejpam-4733	301	4	s.	s.	PROPN
ejpam-4733	301	5	raychaudhuri	raychaudhuri	PROPN
ejpam-4733	301	6	and	and	CCONJ
ejpam-4733	301	7	m.	m.	PROPN
ejpam-4733	301	8	n.	n.	PROPN
ejpam-4733	301	9	mukherjee	mukherjee	PROPN
ejpam-4733	301	10	.	.	PUNCT
ejpam-4733	302	1	δp	δp	PRON
ejpam-4733	302	2	-	-	PUNCT
ejpam-4733	302	3	closedness	closedness	NOUN
ejpam-4733	302	4	for	for	ADP
ejpam-4733	302	5	topological	topological	ADJ
ejpam-4733	302	6	spaces	space	NOUN
ejpam-4733	302	7	.	.	PUNCT
ejpam-4733	303	1	the	the	DET
ejpam-4733	303	2	journal	journal	NOUN
ejpam-4733	303	3	of	of	ADP
ejpam-4733	303	4	the	the	DET
ejpam-4733	303	5	indian	indian	PROPN
ejpam-4733	303	6	academy	academy	PROPN
ejpam-4733	303	7	of	of	ADP
ejpam-4733	303	8	mathematics	mathematic	NOUN
ejpam-4733	303	9	,	,	PUNCT
ejpam-4733	303	10	18:89–99	18:89–99	NUM
ejpam-4733	303	11	,	,	PUNCT
ejpam-4733	303	12	1996	1996	NUM
ejpam-4733	303	13	.	.	PUNCT
ejpam-4733	304	1	[	[	X
ejpam-4733	304	2	10	10	NUM
ejpam-4733	304	3	]	]	X
ejpam-4733	304	4	n.	n.	NOUN
ejpam-4733	304	5	v.	v.	ADP
ejpam-4733	304	6	veličko	veličko	PROPN
ejpam-4733	304	7	.	.	PUNCT
ejpam-4733	305	1	h	h	NOUN
ejpam-4733	305	2	-	-	PUNCT
ejpam-4733	305	3	closed	close	VERB
ejpam-4733	305	4	topological	topological	ADJ
ejpam-4733	305	5	spaces	space	NOUN
ejpam-4733	305	6	.	.	PUNCT
ejpam-4733	306	1	american	american	PROPN
ejpam-4733	306	2	mathematical	mathematical	ADJ
ejpam-4733	306	3	society	society	NOUN
ejpam-4733	306	4	translations	translation	NOUN
ejpam-4733	306	5	,	,	PUNCT
ejpam-4733	306	6	78(2):102–118	78(2):102–118	NUM
ejpam-4733	306	7	,	,	PUNCT
ejpam-4733	306	8	1968	1968	NUM
ejpam-4733	306	9	.	.	PUNCT