id	sid	tid	token	lemma	pos
ejpam-4343	1	1	european	european	PROPN
ejpam-4343	1	2	journal	journal	PROPN
ejpam-4343	1	3	of	of	ADP
ejpam-4343	1	4	pure	pure	ADJ
ejpam-4343	1	5	and	and	CCONJ
ejpam-4343	1	6	applied	apply	VERB
ejpam-4343	1	7	mathematics	mathematic	NOUN
ejpam-4343	1	8	vol	vol	NOUN
ejpam-4343	1	9	.	.	PROPN
ejpam-4343	2	1	15	15	NUM
ejpam-4343	2	2	,	,	PUNCT
ejpam-4343	2	3	no	no	INTJ
ejpam-4343	2	4	.	.	NOUN
ejpam-4343	2	5	3	3	NUM
ejpam-4343	2	6	,	,	PUNCT
ejpam-4343	2	7	2022	2022	NUM
ejpam-4343	2	8	,	,	PUNCT
ejpam-4343	2	9	1023	1023	NUM
ejpam-4343	2	10	-	-	SYM
ejpam-4343	2	11	1046	1046	NUM
ejpam-4343	2	12	issn	issn	PROPN
ejpam-4343	2	13	1307	1307	NUM
ejpam-4343	2	14	-	-	SYM
ejpam-4343	2	15	5543	5543	NUM
ejpam-4343	2	16	–	–	PUNCT
ejpam-4343	2	17	ejpam.com	ejpam.com	X
ejpam-4343	2	18	published	publish	VERB
ejpam-4343	2	19	by	by	ADP
ejpam-4343	2	20	new	new	PROPN
ejpam-4343	2	21	york	york	PROPN
ejpam-4343	2	22	business	business	PROPN
ejpam-4343	2	23	global	global	PROPN
ejpam-4343	2	24	on	on	ADP
ejpam-4343	2	25	some	some	DET
ejpam-4343	2	26	closed	closed	ADJ
ejpam-4343	2	27	sets	set	NOUN
ejpam-4343	2	28	and	and	CCONJ
ejpam-4343	2	29	low	low	ADJ
ejpam-4343	2	30	separation	separation	NOUN
ejpam-4343	2	31	axioms	axiom	NOUN
ejpam-4343	2	32	via	via	ADP
ejpam-4343	2	33	topological	topological	ADJ
ejpam-4343	2	34	ideals	ideal	NOUN
ejpam-4343	2	35	chawalit	chawalit	VERB
ejpam-4343	2	36	boonpok	boonpok	NOUN
ejpam-4343	2	37	1	1	NUM
ejpam-4343	2	38	mathematics	mathematic	NOUN
ejpam-4343	2	39	and	and	CCONJ
ejpam-4343	2	40	applied	apply	VERB
ejpam-4343	2	41	mathematics	mathematics	PROPN
ejpam-4343	2	42	research	research	NOUN
ejpam-4343	2	43	unit	unit	NOUN
ejpam-4343	2	44	,	,	PUNCT
ejpam-4343	2	45	department	department	NOUN
ejpam-4343	2	46	of	of	ADP
ejpam-4343	2	47	mathematics	mathematic	NOUN
ejpam-4343	2	48	,	,	PUNCT
ejpam-4343	2	49	faculty	faculty	NOUN
ejpam-4343	2	50	of	of	ADP
ejpam-4343	2	51	science	science	NOUN
ejpam-4343	2	52	,	,	PUNCT
ejpam-4343	2	53	mahasarakham	mahasarakham	PROPN
ejpam-4343	2	54	university	university	PROPN
ejpam-4343	2	55	,	,	PUNCT
ejpam-4343	2	56	maha	maha	PROPN
ejpam-4343	2	57	sarakham	sarakham	PROPN
ejpam-4343	2	58	,	,	PUNCT
ejpam-4343	2	59	44150	44150	NUM
ejpam-4343	2	60	,	,	PUNCT
ejpam-4343	2	61	thailand	thailand	PROPN
ejpam-4343	2	62	abstract	abstract	PROPN
ejpam-4343	2	63	.	.	PUNCT
ejpam-4343	3	1	this	this	DET
ejpam-4343	3	2	paper	paper	NOUN
ejpam-4343	3	3	deals	deal	NOUN
ejpam-4343	3	4	with	with	ADP
ejpam-4343	3	5	the	the	DET
ejpam-4343	3	6	concepts	concept	NOUN
ejpam-4343	3	7	of	of	ADP
ejpam-4343	3	8	λp(⋆)-sets	λp(⋆)-sets	PRON
ejpam-4343	3	9	and	and	CCONJ
ejpam-4343	3	10	(	(	PUNCT
ejpam-4343	3	11	λ	λ	PROPN
ejpam-4343	3	12	,	,	PUNCT
ejpam-4343	3	13	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	3	14	sets	set	NOUN
ejpam-4343	3	15	which	which	PRON
ejpam-4343	3	16	are	be	AUX
ejpam-4343	3	17	defined	define	VERB
ejpam-4343	3	18	by	by	ADP
ejpam-4343	3	19	utilizing	utilize	VERB
ejpam-4343	3	20	the	the	DET
ejpam-4343	3	21	notions	notion	NOUN
ejpam-4343	3	22	of	of	ADP
ejpam-4343	3	23	pre	pre	ADJ
ejpam-4343	3	24	-	-	ADJ
ejpam-4343	3	25	i	i	PRON
ejpam-4343	3	26	-open	-open	NOUN
ejpam-4343	3	27	sets	set	NOUN
ejpam-4343	3	28	and	and	CCONJ
ejpam-4343	3	29	pre	pre	ADJ
ejpam-4343	3	30	-	-	ADJ
ejpam-4343	3	31	i	i	PRON
ejpam-4343	3	32	-closed	-close	VERB
ejpam-4343	3	33	sets	set	NOUN
ejpam-4343	3	34	.	.	PUNCT
ejpam-4343	4	1	moreover	moreover	ADV
ejpam-4343	4	2	,	,	PUNCT
ejpam-4343	4	3	we	we	PRON
ejpam-4343	4	4	investigate	investigate	VERB
ejpam-4343	4	5	some	some	DET
ejpam-4343	4	6	properties	property	NOUN
ejpam-4343	4	7	of	of	ADP
ejpam-4343	4	8	(	(	PUNCT
ejpam-4343	4	9	λ	λ	NOUN
ejpam-4343	4	10	,	,	PUNCT
ejpam-4343	4	11	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	4	12	disconnected	disconnected	ADJ
ejpam-4343	4	13	ideal	ideal	ADJ
ejpam-4343	4	14	topological	topological	ADJ
ejpam-4343	4	15	spaces	space	NOUN
ejpam-4343	4	16	.	.	PUNCT
ejpam-4343	5	1	several	several	ADJ
ejpam-4343	5	2	characterizations	characterization	NOUN
ejpam-4343	5	3	of	of	ADP
ejpam-4343	5	4	(	(	PUNCT
ejpam-4343	5	5	λ	λ	PROPN
ejpam-4343	5	6	,	,	PUNCT
ejpam-4343	5	7	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	5	8	functions	function	NOUN
ejpam-4343	5	9	are	be	AUX
ejpam-4343	5	10	discussed	discuss	VERB
ejpam-4343	5	11	.	.	PUNCT
ejpam-4343	6	1	especially	especially	ADV
ejpam-4343	6	2	,	,	PUNCT
ejpam-4343	6	3	we	we	PRON
ejpam-4343	6	4	introduce	introduce	VERB
ejpam-4343	6	5	and	and	CCONJ
ejpam-4343	6	6	characterize	characterize	VERB
ejpam-4343	6	7	some	some	DET
ejpam-4343	6	8	low	low	ADJ
ejpam-4343	6	9	separation	separation	NOUN
ejpam-4343	6	10	axioms	axiom	NOUN
ejpam-4343	6	11	of	of	ADP
ejpam-4343	6	12	ideal	ideal	ADJ
ejpam-4343	6	13	topologies	topology	NOUN
ejpam-4343	6	14	constructed	construct	VERB
ejpam-4343	6	15	by	by	ADP
ejpam-4343	6	16	the	the	DET
ejpam-4343	6	17	concepts	concept	NOUN
ejpam-4343	6	18	of	of	ADP
ejpam-4343	6	19	pre	pre	ADJ
ejpam-4343	6	20	-	-	ADJ
ejpam-4343	6	21	i	i	PRON
ejpam-4343	6	22	-open	-open	NOUN
ejpam-4343	6	23	sets	set	NOUN
ejpam-4343	6	24	and	and	CCONJ
ejpam-4343	6	25	the	the	DET
ejpam-4343	6	26	pre	pre	NOUN
ejpam-4343	6	27	-	-	ADJ
ejpam-4343	6	28	i	i	ADJ
ejpam-4343	6	29	-closure	-closure	NOUN
ejpam-4343	6	30	operator	operator	NOUN
ejpam-4343	6	31	.	.	PUNCT
ejpam-4343	7	1	2020	2020	NUM
ejpam-4343	7	2	mathematics	mathematics	PROPN
ejpam-4343	7	3	subject	subject	NOUN
ejpam-4343	7	4	classifications	classification	NOUN
ejpam-4343	7	5	:	:	PUNCT
ejpam-4343	7	6	54a05	54a05	NUM
ejpam-4343	7	7	,	,	PUNCT
ejpam-4343	7	8	54c05	54c05	NUM
ejpam-4343	7	9	,	,	PUNCT
ejpam-4343	7	10	54d10	54d10	NUM
ejpam-4343	7	11	,	,	PUNCT
ejpam-4343	7	12	54g05	54g05	NUM
ejpam-4343	7	13	key	key	ADJ
ejpam-4343	7	14	words	word	NOUN
ejpam-4343	7	15	and	and	CCONJ
ejpam-4343	7	16	phrases	phrase	NOUN
ejpam-4343	7	17	:	:	PUNCT
ejpam-4343	7	18	pre	pre	ADJ
ejpam-4343	7	19	-	-	ADJ
ejpam-4343	7	20	i	i	PRON
ejpam-4343	7	21	-open	-open	NOUN
ejpam-4343	7	22	set	set	NOUN
ejpam-4343	7	23	,	,	PUNCT
ejpam-4343	7	24	λp(⋆)-set	λp(⋆)-set	PROPN
ejpam-4343	7	25	,	,	PUNCT
ejpam-4343	7	26	(	(	PUNCT
ejpam-4343	7	27	λ	λ	PROPN
ejpam-4343	7	28	,	,	PUNCT
ejpam-4343	7	29	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	7	30	set	set	NOUN
ejpam-4343	7	31	,	,	PUNCT
ejpam-4343	7	32	(	(	PUNCT
ejpam-4343	7	33	λ	λ	NOUN
ejpam-4343	7	34	,	,	PUNCT
ejpam-4343	7	35	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	7	36	disconnected	disconnected	ADJ
ejpam-4343	7	37	,	,	PUNCT
ejpam-4343	7	38	pre	pre	ADJ
ejpam-4343	7	39	-	-	VERB
ejpam-4343	7	40	i	i	PRON
ejpam-4343	7	41	-t0	-t0	VERB
ejpam-4343	7	42	,	,	PUNCT
ejpam-4343	7	43	pre	pre	ADJ
ejpam-4343	7	44	-	-	PROPN
ejpam-4343	7	45	i	i	PRON
ejpam-4343	7	46	-t1	-t1	VERB
ejpam-4343	7	47	,	,	PUNCT
ejpam-4343	7	48	pre	pre	ADJ
ejpam-4343	7	49	-	-	VERB
ejpam-4343	7	50	i	i	PRON
ejpam-4343	7	51	-r0	-r0	NOUN
ejpam-4343	7	52	,	,	PUNCT
ejpam-4343	7	53	(	(	PUNCT
ejpam-4343	7	54	λ	λ	INTJ
ejpam-4343	7	55	,	,	PUNCT
ejpam-4343	7	56	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	7	57	function	function	NOUN
ejpam-4343	7	58	1	1	NUM
ejpam-4343	7	59	.	.	PUNCT
ejpam-4343	7	60	introduction	introduction	NOUN
ejpam-4343	7	61	openness	openness	NOUN
ejpam-4343	7	62	and	and	CCONJ
ejpam-4343	7	63	closedness	closedness	NOUN
ejpam-4343	7	64	are	be	AUX
ejpam-4343	7	65	fundamental	fundamental	ADJ
ejpam-4343	7	66	concept	concept	NOUN
ejpam-4343	7	67	for	for	ADP
ejpam-4343	7	68	the	the	DET
ejpam-4343	7	69	study	study	NOUN
ejpam-4343	7	70	and	and	CCONJ
ejpam-4343	7	71	investigation	investigation	NOUN
ejpam-4343	7	72	in	in	ADP
ejpam-4343	7	73	topological	topological	ADJ
ejpam-4343	7	74	spaces	space	NOUN
ejpam-4343	7	75	.	.	PUNCT
ejpam-4343	8	1	many	many	ADJ
ejpam-4343	8	2	mathematicians	mathematician	NOUN
ejpam-4343	8	3	introduced	introduce	VERB
ejpam-4343	8	4	and	and	CCONJ
ejpam-4343	8	5	studied	study	VERB
ejpam-4343	8	6	the	the	DET
ejpam-4343	8	7	various	various	ADJ
ejpam-4343	8	8	types	type	NOUN
ejpam-4343	8	9	of	of	ADP
ejpam-4343	8	10	generalizations	generalization	NOUN
ejpam-4343	8	11	of	of	ADP
ejpam-4343	8	12	open	open	ADJ
ejpam-4343	8	13	sets	set	NOUN
ejpam-4343	8	14	.	.	PUNCT
ejpam-4343	9	1	in	in	ADP
ejpam-4343	9	2	1982	1982	NUM
ejpam-4343	9	3	,	,	PUNCT
ejpam-4343	9	4	mashhour	mashhour	PROPN
ejpam-4343	9	5	et	et	PROPN
ejpam-4343	9	6	al	al	PROPN
ejpam-4343	9	7	.	.	PUNCT
ejpam-4343	10	1	[	[	X
ejpam-4343	10	2	28	28	NUM
ejpam-4343	10	3	]	]	PUNCT
ejpam-4343	10	4	introduced	introduce	VERB
ejpam-4343	10	5	the	the	DET
ejpam-4343	10	6	notion	notion	NOUN
ejpam-4343	10	7	of	of	ADP
ejpam-4343	10	8	preopen	preopen	ADJ
ejpam-4343	10	9	sets	set	NOUN
ejpam-4343	10	10	which	which	PRON
ejpam-4343	10	11	is	be	AUX
ejpam-4343	10	12	also	also	ADV
ejpam-4343	10	13	known	know	VERB
ejpam-4343	10	14	under	under	ADP
ejpam-4343	10	15	the	the	DET
ejpam-4343	10	16	name	name	NOUN
ejpam-4343	10	17	of	of	ADP
ejpam-4343	10	18	locally	locally	ADV
ejpam-4343	10	19	dense	dense	ADJ
ejpam-4343	10	20	sets	set	NOUN
ejpam-4343	10	21	[	[	X
ejpam-4343	10	22	12	12	NUM
ejpam-4343	10	23	]	]	PUNCT
ejpam-4343	10	24	in	in	ADP
ejpam-4343	10	25	the	the	DET
ejpam-4343	10	26	literature	literature	NOUN
ejpam-4343	10	27	.	.	PUNCT
ejpam-4343	11	1	kar	kar	PROPN
ejpam-4343	11	2	and	and	CCONJ
ejpam-4343	11	3	bhattacharya	bhattacharya	PROPN
ejpam-4343	11	4	[	[	X
ejpam-4343	11	5	25	25	NUM
ejpam-4343	11	6	]	]	PUNCT
ejpam-4343	11	7	introduced	introduce	VERB
ejpam-4343	11	8	new	new	ADJ
ejpam-4343	11	9	separation	separation	NOUN
ejpam-4343	11	10	axioms	axiom	VERB
ejpam-4343	11	11	pre	pre	ADJ
ejpam-4343	11	12	-	-	NOUN
ejpam-4343	11	13	t0	t0	ADJ
ejpam-4343	11	14	,	,	PUNCT
ejpam-4343	11	15	pre	pre	ADJ
ejpam-4343	11	16	-	-	NOUN
ejpam-4343	11	17	t1	t1	NOUN
ejpam-4343	11	18	and	and	CCONJ
ejpam-4343	11	19	pre	pre	NOUN
ejpam-4343	11	20	-	-	NOUN
ejpam-4343	11	21	t2	t2	NOUN
ejpam-4343	11	22	by	by	ADP
ejpam-4343	11	23	using	use	VERB
ejpam-4343	11	24	preopen	preopen	ADJ
ejpam-4343	11	25	sets	set	NOUN
ejpam-4343	11	26	due	due	ADP
ejpam-4343	11	27	to	to	ADP
ejpam-4343	11	28	mashhour	mashhour	PROPN
ejpam-4343	11	29	et	et	PROPN
ejpam-4343	11	30	al	al	PROPN
ejpam-4343	11	31	.	.	PUNCT
ejpam-4343	12	1	[	[	X
ejpam-4343	12	2	28	28	NUM
ejpam-4343	12	3	]	]	PUNCT
ejpam-4343	12	4	.	.	PUNCT
ejpam-4343	13	1	caldas	caldas	PROPN
ejpam-4343	13	2	[	[	X
ejpam-4343	13	3	5	5	NUM
ejpam-4343	13	4	]	]	PUNCT
ejpam-4343	13	5	and	and	CCONJ
ejpam-4343	13	6	jafari	jafari	PROPN
ejpam-4343	13	7	[	[	X
ejpam-4343	13	8	22	22	NUM
ejpam-4343	13	9	]	]	PUNCT
ejpam-4343	13	10	introduced	introduce	VERB
ejpam-4343	13	11	independently	independently	ADV
ejpam-4343	13	12	the	the	DET
ejpam-4343	13	13	notions	notion	NOUN
ejpam-4343	13	14	of	of	ADP
ejpam-4343	13	15	p	p	PROPN
ejpam-4343	13	16	-	-	PUNCT
ejpam-4343	13	17	d	d	NOUN
ejpam-4343	13	18	-	-	PUNCT
ejpam-4343	13	19	sets	set	NOUN
ejpam-4343	13	20	and	and	CCONJ
ejpam-4343	13	21	a	a	DET
ejpam-4343	13	22	separation	separation	NOUN
ejpam-4343	13	23	axiom	axiom	NOUN
ejpam-4343	13	24	p	p	NOUN
ejpam-4343	13	25	-	-	PUNCT
ejpam-4343	13	26	d1	d1	NOUN
ejpam-4343	13	27	which	which	PRON
ejpam-4343	13	28	is	be	AUX
ejpam-4343	13	29	strictly	strictly	ADV
ejpam-4343	13	30	between	between	ADP
ejpam-4343	13	31	pre	pre	NOUN
ejpam-4343	13	32	-	-	NOUN
ejpam-4343	13	33	t0	t0	NOUN
ejpam-4343	13	34	and	and	CCONJ
ejpam-4343	13	35	pre	pre	NOUN
ejpam-4343	13	36	-	-	NOUN
ejpam-4343	13	37	t1	t1	NOUN
ejpam-4343	13	38	.	.	PUNCT
ejpam-4343	14	1	in	in	ADP
ejpam-4343	14	2	[	[	X
ejpam-4343	14	3	7	7	NUM
ejpam-4343	14	4	]	]	PUNCT
ejpam-4343	14	5	,	,	PUNCT
ejpam-4343	14	6	the	the	DET
ejpam-4343	14	7	present	present	ADJ
ejpam-4343	14	8	authors	author	NOUN
ejpam-4343	14	9	introduced	introduce	VERB
ejpam-4343	14	10	two	two	NUM
ejpam-4343	14	11	new	new	ADJ
ejpam-4343	14	12	classes	class	NOUN
ejpam-4343	14	13	of	of	ADP
ejpam-4343	14	14	topological	topological	ADJ
ejpam-4343	14	15	spaces	space	NOUN
ejpam-4343	14	16	called	call	VERB
ejpam-4343	14	17	pre	pre	NOUN
ejpam-4343	14	18	-	-	ADJ
ejpam-4343	14	19	r0	r0	ADJ
ejpam-4343	14	20	and	and	CCONJ
ejpam-4343	14	21	pre	pre	ADJ
ejpam-4343	14	22	-	-	ADJ
ejpam-4343	14	23	r1	r1	ADJ
ejpam-4343	14	24	spaces	space	NOUN
ejpam-4343	14	25	in	in	ADP
ejpam-4343	14	26	terms	term	NOUN
ejpam-4343	14	27	of	of	ADP
ejpam-4343	14	28	concept	concept	NOUN
ejpam-4343	14	29	of	of	ADP
ejpam-4343	14	30	preopen	preopen	ADJ
ejpam-4343	14	31	sets	set	NOUN
ejpam-4343	14	32	and	and	CCONJ
ejpam-4343	14	33	investigated	investigate	VERB
ejpam-4343	14	34	some	some	PRON
ejpam-4343	14	35	of	of	ADP
ejpam-4343	14	36	their	their	PRON
ejpam-4343	14	37	fundamental	fundamental	ADJ
ejpam-4343	14	38	properties	property	NOUN
ejpam-4343	14	39	.	.	PUNCT
ejpam-4343	15	1	in	in	ADP
ejpam-4343	15	2	1986	1986	NUM
ejpam-4343	15	3	,	,	PUNCT
ejpam-4343	15	4	maki	maki	NOUN
ejpam-4343	15	5	[	[	X
ejpam-4343	15	6	27	27	NUM
ejpam-4343	15	7	]	]	PUNCT
ejpam-4343	15	8	introduced	introduce	VERB
ejpam-4343	15	9	the	the	DET
ejpam-4343	15	10	concept	concept	NOUN
ejpam-4343	15	11	of	of	ADP
ejpam-4343	15	12	λ	λ	NOUN
ejpam-4343	15	13	-	-	NOUN
ejpam-4343	15	14	sets	set	NOUN
ejpam-4343	15	15	in	in	ADP
ejpam-4343	15	16	topological	topological	ADJ
ejpam-4343	15	17	spaces	space	NOUN
ejpam-4343	15	18	as	as	ADP
ejpam-4343	15	19	the	the	DET
ejpam-4343	15	20	sets	set	NOUN
ejpam-4343	15	21	that	that	PRON
ejpam-4343	15	22	coincide	coincide	VERB
ejpam-4343	15	23	with	with	ADP
ejpam-4343	15	24	their	their	PRON
ejpam-4343	15	25	kernel	kernel	NOUN
ejpam-4343	15	26	.	.	PUNCT
ejpam-4343	16	1	the	the	DET
ejpam-4343	16	2	kernel	kernel	NOUN
ejpam-4343	16	3	of	of	ADP
ejpam-4343	16	4	a	a	DET
ejpam-4343	16	5	set	set	NOUN
ejpam-4343	16	6	a	a	PRON
ejpam-4343	16	7	is	be	AUX
ejpam-4343	16	8	the	the	DET
ejpam-4343	16	9	intersection	intersection	NOUN
ejpam-4343	16	10	of	of	ADP
ejpam-4343	16	11	all	all	DET
ejpam-4343	16	12	open	open	ADJ
ejpam-4343	16	13	superset	superset	NOUN
ejpam-4343	16	14	a.	a.	NOUN
ejpam-4343	16	15	arenas	arenas	PROPN
ejpam-4343	16	16	et	et	PROPN
ejpam-4343	16	17	al	al	PROPN
ejpam-4343	16	18	.	.	PUNCT
ejpam-4343	17	1	[	[	X
ejpam-4343	17	2	4	4	X
ejpam-4343	17	3	]	]	PUNCT
ejpam-4343	17	4	introduced	introduce	VERB
ejpam-4343	17	5	and	and	CCONJ
ejpam-4343	17	6	investigated	investigate	VERB
ejpam-4343	17	7	the	the	DET
ejpam-4343	17	8	concept	concept	NOUN
ejpam-4343	17	9	of	of	ADP
ejpam-4343	17	10	λ	λ	NOUN
ejpam-4343	17	11	-	-	ADJ
ejpam-4343	17	12	closed	closed	ADJ
ejpam-4343	17	13	sets	set	NOUN
ejpam-4343	17	14	by	by	ADP
ejpam-4343	17	15	involving	involve	VERB
ejpam-4343	17	16	λ	λ	NOUN
ejpam-4343	17	17	-	-	NOUN
ejpam-4343	17	18	sets	set	NOUN
ejpam-4343	17	19	and	and	CCONJ
ejpam-4343	17	20	closed	closed	ADJ
ejpam-4343	17	21	sets	set	NOUN
ejpam-4343	17	22	.	.	PUNCT
ejpam-4343	18	1	caldas	caldas	PROPN
ejpam-4343	18	2	et	et	PROPN
ejpam-4343	18	3	al	al	PROPN
ejpam-4343	18	4	.	.	PUNCT
ejpam-4343	19	1	[	[	X
ejpam-4343	19	2	10	10	NUM
ejpam-4343	19	3	]	]	PUNCT
ejpam-4343	19	4	introduced	introduce	VERB
ejpam-4343	19	5	the	the	DET
ejpam-4343	19	6	concept	concept	NOUN
ejpam-4343	19	7	of	of	ADP
ejpam-4343	19	8	λ	λ	NOUN
ejpam-4343	19	9	-	-	NOUN
ejpam-4343	19	10	closure	closure	NOUN
ejpam-4343	19	11	of	of	ADP
ejpam-4343	19	12	a	a	DET
ejpam-4343	19	13	set	set	NOUN
ejpam-4343	19	14	by	by	ADP
ejpam-4343	19	15	utilizing	utilize	VERB
ejpam-4343	19	16	the	the	DET
ejpam-4343	19	17	notion	notion	NOUN
ejpam-4343	19	18	of	of	ADP
ejpam-4343	19	19	λopen	λopen	ADJ
ejpam-4343	19	20	sets	set	NOUN
ejpam-4343	19	21	defined	define	VERB
ejpam-4343	19	22	in	in	ADP
ejpam-4343	19	23	[	[	X
ejpam-4343	19	24	4	4	NUM
ejpam-4343	19	25	]	]	PUNCT
ejpam-4343	19	26	.	.	PUNCT
ejpam-4343	20	1	in	in	ADP
ejpam-4343	20	2	[	[	X
ejpam-4343	20	3	9	9	NUM
ejpam-4343	20	4	]	]	PUNCT
ejpam-4343	20	5	,	,	PUNCT
ejpam-4343	20	6	the	the	DET
ejpam-4343	20	7	present	present	ADJ
ejpam-4343	20	8	authors	author	NOUN
ejpam-4343	20	9	introduced	introduce	VERB
ejpam-4343	20	10	and	and	CCONJ
ejpam-4343	20	11	studied	study	VERB
ejpam-4343	20	12	two	two	NUM
ejpam-4343	20	13	new	new	ADJ
ejpam-4343	20	14	low	low	ADJ
ejpam-4343	20	15	doi	doi	NOUN
ejpam-4343	20	16	:	:	PUNCT
ejpam-4343	20	17	https://doi.org/10.29020/nybg.ejpam.v15i3.4343	https://doi.org/10.29020/nybg.ejpam.v15i3.4343	NOUN
ejpam-4343	20	18	email	email	NOUN
ejpam-4343	20	19	address	address	NOUN
ejpam-4343	20	20	:	:	PUNCT
ejpam-4343	21	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4343	21	2	(	(	PUNCT
ejpam-4343	21	3	c.	c.	PROPN
ejpam-4343	21	4	boonpok	boonpok	PROPN
ejpam-4343	21	5	)	)	PUNCT
ejpam-4343	21	6	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4343	21	7	1023	1023	NUM
ejpam-4343	22	1	©	©	PROPN
ejpam-4343	22	2	2022	2022	NUM
ejpam-4343	22	3	ejpam	ejpam	VERB
ejpam-4343	22	4	all	all	DET
ejpam-4343	22	5	rights	right	NOUN
ejpam-4343	22	6	reserved	reserve	VERB
ejpam-4343	22	7	.	.	PUNCT
ejpam-4343	23	1	c.	c.	PROPN
ejpam-4343	23	2	boonpok	boonpok	PROPN
ejpam-4343	23	3	/	/	SYM
ejpam-4343	23	4	eur	eur	PROPN
ejpam-4343	23	5	.	.	PUNCT
ejpam-4343	24	1	j.	j.	PROPN
ejpam-4343	24	2	pure	pure	PROPN
ejpam-4343	24	3	appl	appl	PROPN
ejpam-4343	24	4	.	.	PROPN
ejpam-4343	24	5	math	math	PROPN
ejpam-4343	24	6	,	,	PUNCT
ejpam-4343	24	7	15	15	NUM
ejpam-4343	24	8	(	(	PUNCT
ejpam-4343	24	9	3	3	NUM
ejpam-4343	24	10	)	)	PUNCT
ejpam-4343	24	11	(	(	PUNCT
ejpam-4343	24	12	2022	2022	NUM
ejpam-4343	24	13	)	)	PUNCT
ejpam-4343	24	14	,	,	PUNCT
ejpam-4343	24	15	1023	1023	NUM
ejpam-4343	24	16	-	-	SYM
ejpam-4343	24	17	1046	1046	NUM
ejpam-4343	24	18	1024	1024	NUM
ejpam-4343	24	19	separation	separation	NOUN
ejpam-4343	24	20	axioms	axiom	NOUN
ejpam-4343	24	21	called	call	VERB
ejpam-4343	24	22	λ	λ	NOUN
ejpam-4343	24	23	-	-	NOUN
ejpam-4343	24	24	r0	r0	NOUN
ejpam-4343	24	25	and	and	CCONJ
ejpam-4343	24	26	λ	λ	NOUN
ejpam-4343	24	27	-	-	NOUN
ejpam-4343	24	28	r1	r1	NOUN
ejpam-4343	24	29	by	by	ADP
ejpam-4343	24	30	utilizing	utilize	VERB
ejpam-4343	24	31	the	the	DET
ejpam-4343	24	32	notions	notion	NOUN
ejpam-4343	24	33	of	of	ADP
ejpam-4343	24	34	λ	λ	NOUN
ejpam-4343	24	35	-	-	ADJ
ejpam-4343	24	36	open	open	ADJ
ejpam-4343	24	37	sets	set	NOUN
ejpam-4343	24	38	and	and	CCONJ
ejpam-4343	24	39	the	the	DET
ejpam-4343	24	40	λclosure	λclosure	NOUN
ejpam-4343	24	41	operator	operator	NOUN
ejpam-4343	24	42	.	.	PUNCT
ejpam-4343	25	1	jafari	jafari	PROPN
ejpam-4343	25	2	et	et	PROPN
ejpam-4343	25	3	al	al	PROPN
ejpam-4343	25	4	.	.	PUNCT
ejpam-4343	26	1	[	[	X
ejpam-4343	26	2	23	23	NUM
ejpam-4343	26	3	]	]	PUNCT
ejpam-4343	26	4	introduced	introduce	VERB
ejpam-4343	26	5	a	a	DET
ejpam-4343	26	6	new	new	ADJ
ejpam-4343	26	7	class	class	NOUN
ejpam-4343	26	8	of	of	ADP
ejpam-4343	26	9	functions	function	NOUN
ejpam-4343	26	10	between	between	ADP
ejpam-4343	26	11	topological	topological	ADJ
ejpam-4343	26	12	spaces	space	NOUN
ejpam-4343	26	13	,	,	PUNCT
ejpam-4343	26	14	namely	namely	ADV
ejpam-4343	26	15	almost	almost	ADV
ejpam-4343	26	16	λ	λ	ADJ
ejpam-4343	26	17	-	-	ADJ
ejpam-4343	26	18	continuous	continuous	ADJ
ejpam-4343	26	19	functions	function	NOUN
ejpam-4343	26	20	and	and	CCONJ
ejpam-4343	26	21	investigated	investigate	VERB
ejpam-4343	26	22	several	several	ADJ
ejpam-4343	26	23	characterizations	characterization	NOUN
ejpam-4343	26	24	of	of	ADP
ejpam-4343	26	25	almost	almost	ADV
ejpam-4343	26	26	λ	λ	ADJ
ejpam-4343	26	27	-	-	ADJ
ejpam-4343	26	28	continuous	continuous	ADJ
ejpam-4343	26	29	functions	function	NOUN
ejpam-4343	26	30	.	.	PUNCT
ejpam-4343	27	1	ekici	ekici	NOUN
ejpam-4343	27	2	et	et	PROPN
ejpam-4343	27	3	al	al	PROPN
ejpam-4343	27	4	.	.	PUNCT
ejpam-4343	28	1	[	[	X
ejpam-4343	28	2	16	16	NUM
ejpam-4343	28	3	]	]	PUNCT
ejpam-4343	28	4	introduced	introduce	VERB
ejpam-4343	28	5	a	a	DET
ejpam-4343	28	6	new	new	ADJ
ejpam-4343	28	7	class	class	NOUN
ejpam-4343	28	8	of	of	ADP
ejpam-4343	28	9	generalization	generalization	NOUN
ejpam-4343	28	10	of	of	ADP
ejpam-4343	28	11	continuous	continuous	ADJ
ejpam-4343	28	12	functions	function	NOUN
ejpam-4343	28	13	via	via	ADP
ejpam-4343	28	14	λ	λ	NOUN
ejpam-4343	28	15	-	-	ADJ
ejpam-4343	28	16	open	open	ADJ
ejpam-4343	28	17	sets	set	NOUN
ejpam-4343	28	18	called	call	VERB
ejpam-4343	28	19	weakly	weakly	ADJ
ejpam-4343	28	20	λ	λ	ADJ
ejpam-4343	28	21	-	-	ADJ
ejpam-4343	28	22	continuous	continuous	ADJ
ejpam-4343	28	23	functions	function	NOUN
ejpam-4343	28	24	and	and	CCONJ
ejpam-4343	28	25	investigated	investigate	VERB
ejpam-4343	28	26	some	some	DET
ejpam-4343	28	27	fundamental	fundamental	ADJ
ejpam-4343	28	28	properties	property	NOUN
ejpam-4343	28	29	of	of	ADP
ejpam-4343	28	30	such	such	ADJ
ejpam-4343	28	31	functions	function	NOUN
ejpam-4343	28	32	.	.	PUNCT
ejpam-4343	29	1	ganster	ganster	NOUN
ejpam-4343	29	2	et	et	PROPN
ejpam-4343	29	3	al	al	PROPN
ejpam-4343	29	4	.	.	PUNCT
ejpam-4343	30	1	[	[	X
ejpam-4343	30	2	19	19	NUM
ejpam-4343	30	3	]	]	PUNCT
ejpam-4343	30	4	introduced	introduce	VERB
ejpam-4343	30	5	the	the	DET
ejpam-4343	30	6	notions	notion	NOUN
ejpam-4343	30	7	of	of	ADP
ejpam-4343	30	8	a	a	DET
ejpam-4343	30	9	pre	pre	ADJ
ejpam-4343	30	10	-	-	ADJ
ejpam-4343	30	11	λ	λ	NOUN
ejpam-4343	30	12	-	-	NOUN
ejpam-4343	30	13	set	set	VERB
ejpam-4343	30	14	and	and	CCONJ
ejpam-4343	30	15	a	a	DET
ejpam-4343	30	16	pre	pre	ADJ
ejpam-4343	30	17	-	-	NOUN
ejpam-4343	30	18	v	v	ADJ
ejpam-4343	30	19	-set	-set	PUNCT
ejpam-4343	30	20	in	in	ADP
ejpam-4343	30	21	a	a	DET
ejpam-4343	30	22	topological	topological	ADJ
ejpam-4343	30	23	space	space	NOUN
ejpam-4343	30	24	and	and	CCONJ
ejpam-4343	30	25	investigated	investigate	VERB
ejpam-4343	30	26	the	the	DET
ejpam-4343	30	27	topologies	topology	NOUN
ejpam-4343	30	28	defined	define	VERB
ejpam-4343	30	29	by	by	ADP
ejpam-4343	30	30	these	these	DET
ejpam-4343	30	31	families	family	NOUN
ejpam-4343	30	32	of	of	ADP
ejpam-4343	30	33	sets	set	NOUN
ejpam-4343	30	34	.	.	PUNCT
ejpam-4343	31	1	veličko	veličko	PROPN
ejpam-4343	32	1	[	[	X
ejpam-4343	32	2	31	31	NUM
ejpam-4343	32	3	]	]	PUNCT
ejpam-4343	32	4	introduced	introduce	VERB
ejpam-4343	32	5	and	and	CCONJ
ejpam-4343	32	6	studied	study	VERB
ejpam-4343	32	7	the	the	DET
ejpam-4343	32	8	concepts	concept	NOUN
ejpam-4343	32	9	of	of	ADP
ejpam-4343	32	10	δ	δ	PROPN
ejpam-4343	32	11	-	-	ADJ
ejpam-4343	32	12	open	open	ADJ
ejpam-4343	32	13	sets	set	NOUN
ejpam-4343	32	14	,	,	PUNCT
ejpam-4343	32	15	δ	δ	NOUN
ejpam-4343	32	16	-	-	PUNCT
ejpam-4343	32	17	closure	closure	NOUN
ejpam-4343	32	18	operator	operator	NOUN
ejpam-4343	32	19	and	and	CCONJ
ejpam-4343	32	20	δ	δ	NOUN
ejpam-4343	32	21	-	-	PUNCT
ejpam-4343	32	22	closed	close	VERB
ejpam-4343	32	23	sets	set	NOUN
ejpam-4343	32	24	.	.	PUNCT
ejpam-4343	33	1	georgiou	georgiou	PROPN
ejpam-4343	33	2	et	et	PROPN
ejpam-4343	33	3	al	al	PROPN
ejpam-4343	33	4	.	.	PUNCT
ejpam-4343	34	1	[	[	X
ejpam-4343	34	2	20	20	NUM
ejpam-4343	34	3	]	]	PUNCT
ejpam-4343	34	4	by	by	ADP
ejpam-4343	34	5	considering	consider	VERB
ejpam-4343	34	6	the	the	DET
ejpam-4343	34	7	notion	notion	NOUN
ejpam-4343	34	8	of	of	ADP
ejpam-4343	34	9	δ	δ	PROPN
ejpam-4343	34	10	-	-	PUNCT
ejpam-4343	34	11	closed	close	VERB
ejpam-4343	34	12	sets	set	NOUN
ejpam-4343	34	13	,	,	PUNCT
ejpam-4343	34	14	introduced	introduce	VERB
ejpam-4343	34	15	and	and	CCONJ
ejpam-4343	34	16	investigated	investigate	VERB
ejpam-4343	34	17	λδ	λδ	PRON
ejpam-4343	34	18	-	-	PUNCT
ejpam-4343	34	19	sets	set	NOUN
ejpam-4343	34	20	,	,	PUNCT
ejpam-4343	34	21	(	(	PUNCT
ejpam-4343	34	22	λ	λ	X
ejpam-4343	34	23	,	,	PUNCT
ejpam-4343	34	24	δ)-closed	δ)-close	VERB
ejpam-4343	34	25	,	,	PUNCT
ejpam-4343	34	26	(	(	PUNCT
ejpam-4343	34	27	λ	λ	X
ejpam-4343	34	28	,	,	PUNCT
ejpam-4343	34	29	δ)-open	δ)-open	PUNCT
ejpam-4343	34	30	sets	set	NOUN
ejpam-4343	34	31	and	and	CCONJ
ejpam-4343	34	32	the	the	DET
ejpam-4343	34	33	(	(	PUNCT
ejpam-4343	34	34	λ	λ	NOUN
ejpam-4343	34	35	,	,	PUNCT
ejpam-4343	34	36	δ)-closure	δ)-closure	NOUN
ejpam-4343	34	37	operator	operator	NOUN
ejpam-4343	34	38	.	.	PUNCT
ejpam-4343	35	1	caldas	caldas	PROPN
ejpam-4343	35	2	and	and	CCONJ
ejpam-4343	35	3	jafari	jafari	PROPN
ejpam-4343	36	1	[	[	X
ejpam-4343	36	2	6	6	NUM
ejpam-4343	36	3	]	]	PUNCT
ejpam-4343	36	4	introduced	introduce	VERB
ejpam-4343	36	5	and	and	CCONJ
ejpam-4343	36	6	investigated	investigate	VERB
ejpam-4343	36	7	some	some	DET
ejpam-4343	36	8	new	new	ADJ
ejpam-4343	36	9	low	low	ADJ
ejpam-4343	36	10	separation	separation	NOUN
ejpam-4343	36	11	axioms	axiom	NOUN
ejpam-4343	36	12	by	by	ADP
ejpam-4343	36	13	using	use	VERB
ejpam-4343	36	14	the	the	DET
ejpam-4343	36	15	notions	notion	NOUN
ejpam-4343	36	16	of	of	ADP
ejpam-4343	36	17	(	(	PUNCT
ejpam-4343	36	18	λ	λ	PROPN
ejpam-4343	36	19	,	,	PUNCT
ejpam-4343	36	20	δ)-open	δ)-open	PUNCT
ejpam-4343	36	21	sets	set	NOUN
ejpam-4343	36	22	and	and	CCONJ
ejpam-4343	36	23	the	the	DET
ejpam-4343	36	24	(	(	PUNCT
ejpam-4343	36	25	λ	λ	NOUN
ejpam-4343	36	26	,	,	PUNCT
ejpam-4343	36	27	δ)-closure	δ)-closure	NOUN
ejpam-4343	36	28	operator	operator	NOUN
ejpam-4343	36	29	.	.	PUNCT
ejpam-4343	37	1	cammaroto	cammaroto	NOUN
ejpam-4343	37	2	and	and	CCONJ
ejpam-4343	37	3	noiri	noiri	ADV
ejpam-4343	38	1	[	[	X
ejpam-4343	38	2	11	11	NUM
ejpam-4343	38	3	]	]	PUNCT
ejpam-4343	38	4	introduced	introduce	VERB
ejpam-4343	38	5	and	and	CCONJ
ejpam-4343	38	6	investigated	investigate	VERB
ejpam-4343	38	7	three	three	NUM
ejpam-4343	38	8	topological	topological	ADJ
ejpam-4343	38	9	spaces	space	NOUN
ejpam-4343	38	10	(	(	PUNCT
ejpam-4343	38	11	x	x	X
ejpam-4343	38	12	,	,	PUNCT
ejpam-4343	38	13	λm	λm	NOUN
ejpam-4343	38	14	)	)	PUNCT
ejpam-4343	38	15	,	,	PUNCT
ejpam-4343	38	16	(	(	PUNCT
ejpam-4343	38	17	x	x	X
ejpam-4343	38	18	,	,	PUNCT
ejpam-4343	38	19	λ∗	λ∗	PROPN
ejpam-4343	38	20	mc	mc	PROPN
ejpam-4343	38	21	)	)	PUNCT
ejpam-4343	38	22	and	and	CCONJ
ejpam-4343	38	23	(	(	PUNCT
ejpam-4343	38	24	x	x	NOUN
ejpam-4343	38	25	,	,	PUNCT
ejpam-4343	38	26	λgλm	λgλm	ADJ
ejpam-4343	38	27	)	)	PUNCT
ejpam-4343	38	28	by	by	ADP
ejpam-4343	38	29	using	use	VERB
ejpam-4343	38	30	λm	λm	NOUN
ejpam-4343	38	31	-	-	PUNCT
ejpam-4343	38	32	sets	set	NOUN
ejpam-4343	38	33	,	,	PUNCT
ejpam-4343	38	34	(	(	PUNCT
ejpam-4343	38	35	λ	λ	X
ejpam-4343	38	36	,	,	PUNCT
ejpam-4343	38	37	m)-closed	m)-closed	ADJ
ejpam-4343	38	38	sets	set	NOUN
ejpam-4343	38	39	and	and	CCONJ
ejpam-4343	38	40	generalized	generalize	VERB
ejpam-4343	38	41	λm	λm	NOUN
ejpam-4343	38	42	-	-	PUNCT
ejpam-4343	38	43	sets	set	NOUN
ejpam-4343	38	44	,	,	PUNCT
ejpam-4343	38	45	respectively	respectively	ADV
ejpam-4343	38	46	.	.	PUNCT
ejpam-4343	39	1	caldas	caldas	PROPN
ejpam-4343	39	2	et	et	PROPN
ejpam-4343	39	3	al	al	PROPN
ejpam-4343	39	4	.	.	PUNCT
ejpam-4343	40	1	[	[	X
ejpam-4343	40	2	8	8	NUM
ejpam-4343	40	3	]	]	PUNCT
ejpam-4343	40	4	introduced	introduce	VERB
ejpam-4343	40	5	and	and	CCONJ
ejpam-4343	40	6	studied	study	VERB
ejpam-4343	40	7	two	two	NUM
ejpam-4343	40	8	new	new	ADJ
ejpam-4343	40	9	weak	weak	ADJ
ejpam-4343	40	10	separation	separation	NOUN
ejpam-4343	40	11	axioms	axiom	NOUN
ejpam-4343	40	12	called	call	VERB
ejpam-4343	40	13	λθ	λθ	NOUN
ejpam-4343	40	14	-	-	PUNCT
ejpam-4343	40	15	r0	r0	NOUN
ejpam-4343	40	16	and	and	CCONJ
ejpam-4343	40	17	λθ	λθ	NOUN
ejpam-4343	40	18	-	-	PUNCT
ejpam-4343	40	19	r1	r1	NOUN
ejpam-4343	40	20	spaces	space	NOUN
ejpam-4343	40	21	by	by	ADP
ejpam-4343	40	22	using	use	VERB
ejpam-4343	40	23	the	the	DET
ejpam-4343	40	24	notions	notion	NOUN
ejpam-4343	40	25	of	of	ADP
ejpam-4343	40	26	(	(	PUNCT
ejpam-4343	40	27	λ	λ	PROPN
ejpam-4343	40	28	,	,	PUNCT
ejpam-4343	40	29	θ)-open	θ)-open	VERB
ejpam-4343	40	30	sets	set	NOUN
ejpam-4343	40	31	and	and	CCONJ
ejpam-4343	40	32	the	the	DET
ejpam-4343	40	33	(	(	PUNCT
ejpam-4343	40	34	λ	λ	NOUN
ejpam-4343	40	35	,	,	PUNCT
ejpam-4343	40	36	θ)closure	θ)closure	NOUN
ejpam-4343	40	37	operator	operator	NOUN
ejpam-4343	40	38	.	.	PUNCT
ejpam-4343	41	1	the	the	DET
ejpam-4343	41	2	concept	concept	NOUN
ejpam-4343	41	3	of	of	ADP
ejpam-4343	41	4	ideals	ideal	NOUN
ejpam-4343	41	5	in	in	ADP
ejpam-4343	41	6	topological	topological	ADJ
ejpam-4343	41	7	spaces	space	NOUN
ejpam-4343	41	8	has	have	AUX
ejpam-4343	41	9	been	be	AUX
ejpam-4343	41	10	introduced	introduce	VERB
ejpam-4343	41	11	and	and	CCONJ
ejpam-4343	41	12	studied	study	VERB
ejpam-4343	41	13	by	by	ADP
ejpam-4343	41	14	kuratowski	kuratowski	ADJ
ejpam-4343	41	15	[	[	X
ejpam-4343	41	16	26	26	NUM
ejpam-4343	41	17	]	]	PUNCT
ejpam-4343	41	18	and	and	CCONJ
ejpam-4343	41	19	vaidyanathswamy	vaidyanathswamy	NOUN
ejpam-4343	41	20	[	[	X
ejpam-4343	41	21	30	30	NUM
ejpam-4343	41	22	]	]	PUNCT
ejpam-4343	41	23	.	.	PUNCT
ejpam-4343	42	1	the	the	DET
ejpam-4343	42	2	topology	topology	NOUN
ejpam-4343	42	3	τ	τ	PROPN
ejpam-4343	42	4	of	of	ADP
ejpam-4343	42	5	a	a	DET
ejpam-4343	42	6	space	space	NOUN
ejpam-4343	42	7	is	be	AUX
ejpam-4343	42	8	enlarged	enlarge	VERB
ejpam-4343	42	9	to	to	ADP
ejpam-4343	42	10	a	a	DET
ejpam-4343	42	11	topology	topology	NOUN
ejpam-4343	42	12	τ⋆	τ⋆	NOUN
ejpam-4343	42	13	using	use	VERB
ejpam-4343	42	14	an	an	DET
ejpam-4343	42	15	ideal	ideal	NOUN
ejpam-4343	42	16	i	i	PRON
ejpam-4343	42	17	whose	whose	DET
ejpam-4343	42	18	members	member	NOUN
ejpam-4343	42	19	are	be	AUX
ejpam-4343	42	20	disjoint	disjoint	ADJ
ejpam-4343	42	21	with	with	ADP
ejpam-4343	42	22	the	the	DET
ejpam-4343	42	23	members	member	NOUN
ejpam-4343	42	24	of	of	ADP
ejpam-4343	42	25	τ	τ	PROPN
ejpam-4343	42	26	.	.	PUNCT
ejpam-4343	43	1	every	every	DET
ejpam-4343	43	2	topological	topological	ADJ
ejpam-4343	43	3	space	space	NOUN
ejpam-4343	43	4	is	be	AUX
ejpam-4343	43	5	an	an	DET
ejpam-4343	43	6	ideal	ideal	ADJ
ejpam-4343	43	7	topological	topological	ADJ
ejpam-4343	43	8	space	space	NOUN
ejpam-4343	43	9	and	and	CCONJ
ejpam-4343	43	10	all	all	DET
ejpam-4343	43	11	the	the	DET
ejpam-4343	43	12	results	result	NOUN
ejpam-4343	43	13	of	of	ADP
ejpam-4343	43	14	ideal	ideal	ADJ
ejpam-4343	43	15	topological	topological	ADJ
ejpam-4343	43	16	spaces	space	NOUN
ejpam-4343	43	17	are	be	AUX
ejpam-4343	43	18	generalizations	generalization	NOUN
ejpam-4343	43	19	of	of	ADP
ejpam-4343	43	20	the	the	DET
ejpam-4343	43	21	results	result	NOUN
ejpam-4343	43	22	established	establish	VERB
ejpam-4343	43	23	in	in	ADP
ejpam-4343	43	24	topological	topological	ADJ
ejpam-4343	43	25	spaces	space	NOUN
ejpam-4343	43	26	.	.	PUNCT
ejpam-4343	44	1	in	in	ADP
ejpam-4343	44	2	1990	1990	NUM
ejpam-4343	44	3	,	,	PUNCT
ejpam-4343	44	4	janković	janković	ADJ
ejpam-4343	44	5	and	and	CCONJ
ejpam-4343	44	6	hamlett	hamlett	PROPN
ejpam-4343	45	1	[	[	X
ejpam-4343	45	2	24	24	NUM
ejpam-4343	45	3	]	]	PUNCT
ejpam-4343	45	4	introduced	introduce	VERB
ejpam-4343	45	5	the	the	DET
ejpam-4343	45	6	notion	notion	NOUN
ejpam-4343	45	7	of	of	ADP
ejpam-4343	45	8	i	i	PRON
ejpam-4343	45	9	-open	-open	PROPN
ejpam-4343	45	10	sets	set	NOUN
ejpam-4343	45	11	in	in	ADP
ejpam-4343	45	12	ideal	ideal	ADJ
ejpam-4343	45	13	topologial	topologial	ADJ
ejpam-4343	45	14	spaces	space	NOUN
ejpam-4343	45	15	.	.	PUNCT
ejpam-4343	46	1	abd	abd	PROPN
ejpam-4343	46	2	el	el	PROPN
ejpam-4343	46	3	-	-	PROPN
ejpam-4343	46	4	monsef	monsef	PROPN
ejpam-4343	46	5	et	et	PROPN
ejpam-4343	46	6	al	al	PROPN
ejpam-4343	46	7	.	.	PUNCT
ejpam-4343	47	1	[	[	X
ejpam-4343	47	2	18	18	NUM
ejpam-4343	47	3	]	]	PUNCT
ejpam-4343	47	4	further	far	ADV
ejpam-4343	47	5	investigated	investigate	VERB
ejpam-4343	47	6	i	i	PRON
ejpam-4343	47	7	-open	-open	PROPN
ejpam-4343	47	8	sets	set	NOUN
ejpam-4343	47	9	and	and	CCONJ
ejpam-4343	47	10	i	i	PRON
ejpam-4343	47	11	-continuous	-continuous	ADJ
ejpam-4343	47	12	functions	function	NOUN
ejpam-4343	47	13	.	.	PUNCT
ejpam-4343	48	1	later	later	ADV
ejpam-4343	48	2	,	,	PUNCT
ejpam-4343	48	3	several	several	ADJ
ejpam-4343	48	4	authors	author	NOUN
ejpam-4343	48	5	studied	study	VERB
ejpam-4343	48	6	ideal	ideal	ADJ
ejpam-4343	48	7	topological	topological	ADJ
ejpam-4343	48	8	spaces	space	NOUN
ejpam-4343	48	9	giving	give	VERB
ejpam-4343	48	10	several	several	ADJ
ejpam-4343	48	11	convenient	convenient	ADJ
ejpam-4343	48	12	definitions	definition	NOUN
ejpam-4343	48	13	.	.	PUNCT
ejpam-4343	49	1	some	some	DET
ejpam-4343	49	2	authors	author	NOUN
ejpam-4343	49	3	obtained	obtain	VERB
ejpam-4343	49	4	decompositions	decomposition	NOUN
ejpam-4343	49	5	of	of	ADP
ejpam-4343	49	6	continuity	continuity	NOUN
ejpam-4343	49	7	.	.	PUNCT
ejpam-4343	50	1	for	for	ADP
ejpam-4343	50	2	instance	instance	NOUN
ejpam-4343	50	3	,	,	PUNCT
ejpam-4343	50	4	açıkgöz	açıkgöz	PROPN
ejpam-4343	50	5	et	et	NOUN
ejpam-4343	50	6	al	al	PROPN
ejpam-4343	50	7	.	.	PUNCT
ejpam-4343	51	1	[	[	X
ejpam-4343	51	2	2	2	X
ejpam-4343	51	3	]	]	PUNCT
ejpam-4343	51	4	introduced	introduce	VERB
ejpam-4343	51	5	and	and	CCONJ
ejpam-4343	51	6	investigated	investigate	VERB
ejpam-4343	51	7	the	the	DET
ejpam-4343	51	8	notions	notion	NOUN
ejpam-4343	51	9	of	of	ADP
ejpam-4343	51	10	weakly	weakly	ADJ
ejpam-4343	51	11	-	-	PUNCT
ejpam-4343	51	12	i	i	PRON
ejpam-4343	51	13	-continuous	-continuous	ADJ
ejpam-4343	51	14	and	and	CCONJ
ejpam-4343	51	15	weak⋆-i	weak⋆-i	ADP
ejpam-4343	51	16	-continuous	-continuous	ADJ
ejpam-4343	51	17	functions	function	NOUN
ejpam-4343	51	18	in	in	ADP
ejpam-4343	51	19	ideal	ideal	ADJ
ejpam-4343	51	20	topological	topological	ADJ
ejpam-4343	51	21	spaces	space	NOUN
ejpam-4343	51	22	.	.	PUNCT
ejpam-4343	52	1	donthev	donthev	PROPN
ejpam-4343	53	1	[	[	X
ejpam-4343	53	2	15	15	NUM
ejpam-4343	53	3	]	]	PUNCT
ejpam-4343	53	4	introduced	introduce	VERB
ejpam-4343	53	5	the	the	DET
ejpam-4343	53	6	notion	notion	NOUN
ejpam-4343	53	7	of	of	ADP
ejpam-4343	53	8	pre	pre	ADJ
ejpam-4343	53	9	-	-	ADJ
ejpam-4343	53	10	i	i	PRON
ejpam-4343	53	11	-open	-open	NOUN
ejpam-4343	53	12	sets	set	NOUN
ejpam-4343	53	13	and	and	CCONJ
ejpam-4343	53	14	obtained	obtain	VERB
ejpam-4343	53	15	a	a	DET
ejpam-4343	53	16	decomposition	decomposition	NOUN
ejpam-4343	53	17	of	of	ADP
ejpam-4343	53	18	i	i	PROPN
ejpam-4343	53	19	-continuity	-continuity	PROPN
ejpam-4343	53	20	.	.	PUNCT
ejpam-4343	53	21	hatır	hatır	NOUN
ejpam-4343	53	22	and	and	CCONJ
ejpam-4343	53	23	noiri	noiri	ADV
ejpam-4343	54	1	[	[	X
ejpam-4343	54	2	21	21	NUM
ejpam-4343	54	3	]	]	PUNCT
ejpam-4343	54	4	introduced	introduce	VERB
ejpam-4343	54	5	α	α	X
ejpam-4343	54	6	-	-	PUNCT
ejpam-4343	54	7	i	i	PRON
ejpam-4343	54	8	-open	-open	NOUN
ejpam-4343	54	9	,	,	PUNCT
ejpam-4343	54	10	semi	semi	ADJ
ejpam-4343	54	11	-	-	ADJ
ejpam-4343	54	12	i	i	PRON
ejpam-4343	54	13	-open	-open	ADJ
ejpam-4343	54	14	and	and	CCONJ
ejpam-4343	54	15	β	β	NOUN
ejpam-4343	54	16	-	-	ADJ
ejpam-4343	54	17	i	i	PRON
ejpam-4343	54	18	-open	-open	NOUN
ejpam-4343	54	19	sets	set	NOUN
ejpam-4343	54	20	via	via	ADP
ejpam-4343	54	21	idealization	idealization	NOUN
ejpam-4343	54	22	and	and	CCONJ
ejpam-4343	54	23	using	use	VERB
ejpam-4343	54	24	these	these	DET
ejpam-4343	54	25	sets	set	NOUN
ejpam-4343	54	26	obtained	obtain	VERB
ejpam-4343	54	27	new	new	ADJ
ejpam-4343	54	28	decompositions	decomposition	NOUN
ejpam-4343	54	29	of	of	ADP
ejpam-4343	54	30	continuity	continuity	NOUN
ejpam-4343	54	31	.	.	PUNCT
ejpam-4343	55	1	in	in	ADP
ejpam-4343	55	2	[	[	X
ejpam-4343	55	3	3	3	NUM
ejpam-4343	55	4	]	]	PUNCT
ejpam-4343	55	5	,	,	PUNCT
ejpam-4343	55	6	the	the	DET
ejpam-4343	55	7	present	present	ADJ
ejpam-4343	55	8	authors	author	NOUN
ejpam-4343	55	9	studied	study	VERB
ejpam-4343	55	10	the	the	DET
ejpam-4343	55	11	concepts	concept	NOUN
ejpam-4343	55	12	of	of	ADP
ejpam-4343	55	13	α	α	PROPN
ejpam-4343	55	14	-	-	PUNCT
ejpam-4343	55	15	i	i	PRON
ejpam-4343	55	16	-continuity	-continuity	PROPN
ejpam-4343	55	17	and	and	CCONJ
ejpam-4343	55	18	α	α	NOUN
ejpam-4343	55	19	-	-	PUNCT
ejpam-4343	55	20	i	i	PRON
ejpam-4343	55	21	openness	openness	NOUN
ejpam-4343	55	22	in	in	ADP
ejpam-4343	55	23	ideal	ideal	ADJ
ejpam-4343	55	24	topological	topological	ADJ
ejpam-4343	55	25	spaces	space	NOUN
ejpam-4343	55	26	and	and	CCONJ
ejpam-4343	55	27	obtained	obtain	VERB
ejpam-4343	55	28	several	several	ADJ
ejpam-4343	55	29	characterizations	characterization	NOUN
ejpam-4343	55	30	of	of	ADP
ejpam-4343	55	31	these	these	DET
ejpam-4343	55	32	functions	function	NOUN
ejpam-4343	55	33	.	.	PUNCT
ejpam-4343	56	1	açıkgöz	açıkgöz	NOUN
ejpam-4343	56	2	et	et	NOUN
ejpam-4343	56	3	al	al	PROPN
ejpam-4343	56	4	.	.	PUNCT
ejpam-4343	57	1	[	[	X
ejpam-4343	57	2	1	1	X
ejpam-4343	57	3	]	]	PUNCT
ejpam-4343	57	4	introduced	introduce	VERB
ejpam-4343	57	5	two	two	NUM
ejpam-4343	57	6	new	new	ADJ
ejpam-4343	57	7	classes	class	NOUN
ejpam-4343	57	8	of	of	ADP
ejpam-4343	57	9	functions	function	NOUN
ejpam-4343	57	10	called	call	VERB
ejpam-4343	57	11	α	α	NOUN
ejpam-4343	57	12	-	-	PUNCT
ejpam-4343	57	13	i	i	NOUN
ejpam-4343	57	14	-preirresolute	-preirresolute	NOUN
ejpam-4343	57	15	functions	function	NOUN
ejpam-4343	57	16	and	and	CCONJ
ejpam-4343	57	17	β	β	NOUN
ejpam-4343	57	18	-	-	ADJ
ejpam-4343	57	19	i	i	NOUN
ejpam-4343	57	20	-preirresolute	-preirresolute	NOUN
ejpam-4343	57	21	functions	function	NOUN
ejpam-4343	57	22	in	in	ADP
ejpam-4343	57	23	ideal	ideal	ADJ
ejpam-4343	57	24	topologial	topologial	ADJ
ejpam-4343	57	25	spaces	space	NOUN
ejpam-4343	57	26	and	and	CCONJ
ejpam-4343	57	27	investigated	investigate	VERB
ejpam-4343	57	28	the	the	DET
ejpam-4343	57	29	relationships	relationship	NOUN
ejpam-4343	57	30	between	between	ADP
ejpam-4343	57	31	these	these	DET
ejpam-4343	57	32	classes	class	NOUN
ejpam-4343	57	33	of	of	ADP
ejpam-4343	57	34	functions	function	NOUN
ejpam-4343	57	35	and	and	CCONJ
ejpam-4343	57	36	other	other	ADJ
ejpam-4343	57	37	classes	class	NOUN
ejpam-4343	57	38	of	of	ADP
ejpam-4343	57	39	non	non	ADJ
ejpam-4343	57	40	-	-	ADJ
ejpam-4343	57	41	continuous	continuous	ADJ
ejpam-4343	57	42	functions	function	NOUN
ejpam-4343	57	43	.	.	PUNCT
ejpam-4343	58	1	in	in	ADP
ejpam-4343	58	2	2009	2009	NUM
ejpam-4343	58	3	,	,	PUNCT
ejpam-4343	58	4	ekici	ekici	NOUN
ejpam-4343	58	5	and	and	CCONJ
ejpam-4343	58	6	t.	t.	NOUN
ejpam-4343	58	7	noiri	noiri	PROPN
ejpam-4343	58	8	[	[	X
ejpam-4343	58	9	17	17	NUM
ejpam-4343	58	10	]	]	PUNCT
ejpam-4343	58	11	introduced	introduce	VERB
ejpam-4343	58	12	the	the	DET
ejpam-4343	58	13	concept	concept	NOUN
ejpam-4343	58	14	of	of	ADP
ejpam-4343	58	15	⋆-extremally	⋆-extremally	ADV
ejpam-4343	58	16	disconnected	disconnected	ADJ
ejpam-4343	58	17	ideal	ideal	ADJ
ejpam-4343	58	18	topological	topological	ADJ
ejpam-4343	58	19	spaces	space	NOUN
ejpam-4343	58	20	and	and	CCONJ
ejpam-4343	58	21	investigated	investigate	VERB
ejpam-4343	58	22	several	several	ADJ
ejpam-4343	58	23	characterizations	characterization	NOUN
ejpam-4343	58	24	of	of	ADP
ejpam-4343	58	25	⋆-extremally	⋆-extremally	ADV
ejpam-4343	58	26	disconnected	disconnected	ADJ
ejpam-4343	58	27	ideal	ideal	ADJ
ejpam-4343	58	28	topological	topological	ADJ
ejpam-4343	58	29	spaces	space	NOUN
ejpam-4343	58	30	.	.	PUNCT
ejpam-4343	59	1	the	the	DET
ejpam-4343	59	2	paper	paper	NOUN
ejpam-4343	59	3	is	be	AUX
ejpam-4343	59	4	organized	organize	VERB
ejpam-4343	59	5	as	as	SCONJ
ejpam-4343	59	6	follows	follow	VERB
ejpam-4343	59	7	.	.	PUNCT
ejpam-4343	60	1	in	in	ADP
ejpam-4343	60	2	section	section	NOUN
ejpam-4343	60	3	3	3	NUM
ejpam-4343	60	4	,	,	PUNCT
ejpam-4343	60	5	we	we	PRON
ejpam-4343	60	6	introduce	introduce	VERB
ejpam-4343	60	7	the	the	DET
ejpam-4343	60	8	notions	notion	NOUN
ejpam-4343	60	9	of	of	ADP
ejpam-4343	60	10	λp(⋆)-sets	λp(⋆)-set	NOUN
ejpam-4343	60	11	and	and	CCONJ
ejpam-4343	60	12	λp(⋆)-sets	λp(⋆)-sets	PROPN
ejpam-4343	60	13	.	.	PUNCT
ejpam-4343	61	1	moreover	moreover	ADV
ejpam-4343	61	2	,	,	PUNCT
ejpam-4343	61	3	some	some	DET
ejpam-4343	61	4	properties	property	NOUN
ejpam-4343	61	5	of	of	ADP
ejpam-4343	61	6	λp(⋆)-sets	λp(⋆)-set	NOUN
ejpam-4343	61	7	and	and	CCONJ
ejpam-4343	61	8	λp(⋆)-sets	λp(⋆)-set	NOUN
ejpam-4343	61	9	are	be	AUX
ejpam-4343	61	10	investigated	investigate	VERB
ejpam-4343	61	11	.	.	PUNCT
ejpam-4343	62	1	in	in	ADP
ejpam-4343	62	2	section	section	NOUN
ejpam-4343	62	3	4	4	NUM
ejpam-4343	62	4	,	,	PUNCT
ejpam-4343	62	5	we	we	PRON
ejpam-4343	62	6	define	define	VERB
ejpam-4343	62	7	(	(	PUNCT
ejpam-4343	62	8	λ	λ	PROPN
ejpam-4343	62	9	,	,	PUNCT
ejpam-4343	62	10	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	62	11	sets	set	NOUN
ejpam-4343	62	12	and	and	CCONJ
ejpam-4343	62	13	investigate	investigate	VERB
ejpam-4343	62	14	several	several	ADJ
ejpam-4343	62	15	characterizations	characterization	NOUN
ejpam-4343	62	16	of	of	ADP
ejpam-4343	62	17	(	(	PUNCT
ejpam-4343	62	18	λ	λ	NOUN
ejpam-4343	62	19	,	,	PUNCT
ejpam-4343	62	20	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	62	21	disconnected	disconnected	ADJ
ejpam-4343	62	22	.	.	PUNCT
ejpam-4343	63	1	in	in	ADP
ejpam-4343	63	2	section	section	NOUN
ejpam-4343	63	3	5	5	NUM
ejpam-4343	63	4	,	,	PUNCT
ejpam-4343	63	5	we	we	PRON
ejpam-4343	63	6	introduce	introduce	VERB
ejpam-4343	63	7	the	the	DET
ejpam-4343	63	8	concept	concept	NOUN
ejpam-4343	63	9	of	of	ADP
ejpam-4343	63	10	(	(	PUNCT
ejpam-4343	63	11	λ	λ	PROPN
ejpam-4343	63	12	,	,	PUNCT
ejpam-4343	63	13	p(⋆))continuous	p(⋆))continuous	ADJ
ejpam-4343	63	14	functions	function	NOUN
ejpam-4343	63	15	and	and	CCONJ
ejpam-4343	63	16	investigate	investigate	VERB
ejpam-4343	63	17	some	some	DET
ejpam-4343	63	18	characterizations	characterization	NOUN
ejpam-4343	63	19	of	of	ADP
ejpam-4343	63	20	such	such	ADJ
ejpam-4343	63	21	functions	function	NOUN
ejpam-4343	63	22	.	.	PUNCT
ejpam-4343	64	1	in	in	ADP
ejpam-4343	64	2	the	the	DET
ejpam-4343	64	3	last	last	ADJ
ejpam-4343	64	4	c.	c.	NOUN
ejpam-4343	64	5	boonpok	boonpok	PROPN
ejpam-4343	64	6	/	/	SYM
ejpam-4343	64	7	eur	eur	PROPN
ejpam-4343	64	8	.	.	PUNCT
ejpam-4343	65	1	j.	j.	PROPN
ejpam-4343	65	2	pure	pure	PROPN
ejpam-4343	65	3	appl	appl	PROPN
ejpam-4343	65	4	.	.	PROPN
ejpam-4343	65	5	math	math	PROPN
ejpam-4343	65	6	,	,	PUNCT
ejpam-4343	65	7	15	15	NUM
ejpam-4343	65	8	(	(	PUNCT
ejpam-4343	65	9	3	3	NUM
ejpam-4343	65	10	)	)	PUNCT
ejpam-4343	65	11	(	(	PUNCT
ejpam-4343	65	12	2022	2022	NUM
ejpam-4343	65	13	)	)	PUNCT
ejpam-4343	65	14	,	,	PUNCT
ejpam-4343	65	15	1023	1023	NUM
ejpam-4343	65	16	-	-	SYM
ejpam-4343	65	17	1046	1046	NUM
ejpam-4343	65	18	1025	1025	NUM
ejpam-4343	65	19	section	section	NOUN
ejpam-4343	65	20	,	,	PUNCT
ejpam-4343	65	21	we	we	PRON
ejpam-4343	65	22	introduce	introduce	VERB
ejpam-4343	65	23	and	and	CCONJ
ejpam-4343	65	24	study	study	VERB
ejpam-4343	65	25	some	some	DET
ejpam-4343	65	26	new	new	ADJ
ejpam-4343	65	27	low	low	ADJ
ejpam-4343	65	28	separation	separation	NOUN
ejpam-4343	65	29	axioms	axiom	NOUN
ejpam-4343	65	30	by	by	ADP
ejpam-4343	65	31	using	use	VERB
ejpam-4343	65	32	the	the	DET
ejpam-4343	65	33	concepts	concept	NOUN
ejpam-4343	65	34	of	of	ADP
ejpam-4343	65	35	pre	pre	ADJ
ejpam-4343	65	36	-	-	ADJ
ejpam-4343	65	37	i	i	PRON
ejpam-4343	65	38	-open	-open	NOUN
ejpam-4343	65	39	sets	set	NOUN
ejpam-4343	65	40	and	and	CCONJ
ejpam-4343	65	41	the	the	DET
ejpam-4343	65	42	pre	pre	NOUN
ejpam-4343	65	43	-	-	ADJ
ejpam-4343	65	44	i	i	ADJ
ejpam-4343	65	45	-closure	-closure	NOUN
ejpam-4343	65	46	operator	operator	NOUN
ejpam-4343	65	47	.	.	PUNCT
ejpam-4343	66	1	2	2	X
ejpam-4343	66	2	.	.	X
ejpam-4343	66	3	preliminaries	preliminary	NOUN
ejpam-4343	66	4	we	we	PRON
ejpam-4343	66	5	begin	begin	VERB
ejpam-4343	66	6	with	with	ADP
ejpam-4343	66	7	some	some	DET
ejpam-4343	66	8	definitions	definition	NOUN
ejpam-4343	66	9	and	and	CCONJ
ejpam-4343	66	10	known	know	VERB
ejpam-4343	66	11	results	result	NOUN
ejpam-4343	66	12	which	which	PRON
ejpam-4343	66	13	will	will	AUX
ejpam-4343	66	14	be	be	AUX
ejpam-4343	66	15	used	use	VERB
ejpam-4343	66	16	throughout	throughout	ADP
ejpam-4343	66	17	this	this	DET
ejpam-4343	66	18	paper	paper	NOUN
ejpam-4343	66	19	.	.	PUNCT
ejpam-4343	67	1	in	in	ADP
ejpam-4343	67	2	the	the	DET
ejpam-4343	67	3	present	present	ADJ
ejpam-4343	67	4	paper	paper	NOUN
ejpam-4343	67	5	,	,	PUNCT
ejpam-4343	67	6	spaces	space	NOUN
ejpam-4343	67	7	(	(	PUNCT
ejpam-4343	67	8	x	x	X
ejpam-4343	67	9	,	,	PUNCT
ejpam-4343	67	10	τ	τ	X
ejpam-4343	67	11	)	)	PUNCT
ejpam-4343	67	12	and	and	CCONJ
ejpam-4343	67	13	(	(	PUNCT
ejpam-4343	67	14	y	y	PROPN
ejpam-4343	67	15	,	,	PUNCT
ejpam-4343	67	16	σ	σ	PROPN
ejpam-4343	67	17	)	)	PUNCT
ejpam-4343	67	18	(	(	PUNCT
ejpam-4343	67	19	or	or	CCONJ
ejpam-4343	67	20	simply	simply	ADV
ejpam-4343	67	21	x	x	X
ejpam-4343	67	22	and	and	CCONJ
ejpam-4343	67	23	y	y	PROPN
ejpam-4343	67	24	)	)	PUNCT
ejpam-4343	67	25	always	always	ADV
ejpam-4343	67	26	mean	mean	VERB
ejpam-4343	67	27	topological	topological	ADJ
ejpam-4343	67	28	spaces	space	NOUN
ejpam-4343	67	29	on	on	ADP
ejpam-4343	67	30	which	which	PRON
ejpam-4343	67	31	no	no	DET
ejpam-4343	67	32	separation	separation	NOUN
ejpam-4343	67	33	axioms	axiom	NOUN
ejpam-4343	67	34	are	be	AUX
ejpam-4343	67	35	assumed	assume	VERB
ejpam-4343	67	36	unless	unless	SCONJ
ejpam-4343	67	37	explicitly	explicitly	ADV
ejpam-4343	67	38	stated	state	VERB
ejpam-4343	67	39	.	.	PUNCT
ejpam-4343	68	1	for	for	ADP
ejpam-4343	68	2	a	a	DET
ejpam-4343	68	3	subset	subset	NOUN
ejpam-4343	68	4	a	a	PRON
ejpam-4343	68	5	of	of	ADP
ejpam-4343	68	6	a	a	DET
ejpam-4343	68	7	topological	topological	ADJ
ejpam-4343	68	8	space	space	NOUN
ejpam-4343	68	9	(	(	PUNCT
ejpam-4343	68	10	x	x	X
ejpam-4343	68	11	,	,	PUNCT
ejpam-4343	68	12	τ	τ	PROPN
ejpam-4343	68	13	)	)	PUNCT
ejpam-4343	68	14	,	,	PUNCT
ejpam-4343	68	15	cl(a	cl(a	NUM
ejpam-4343	68	16	)	)	PUNCT
ejpam-4343	68	17	and	and	CCONJ
ejpam-4343	68	18	int(a	int(a	PROPN
ejpam-4343	68	19	)	)	PUNCT
ejpam-4343	68	20	represent	represent	VERB
ejpam-4343	68	21	the	the	DET
ejpam-4343	68	22	closure	closure	NOUN
ejpam-4343	68	23	and	and	CCONJ
ejpam-4343	68	24	the	the	DET
ejpam-4343	68	25	interior	interior	NOUN
ejpam-4343	68	26	of	of	ADP
ejpam-4343	68	27	a	a	PRON
ejpam-4343	68	28	,	,	PUNCT
ejpam-4343	68	29	respectively	respectively	ADV
ejpam-4343	68	30	.	.	PUNCT
ejpam-4343	69	1	a	a	DET
ejpam-4343	69	2	nonempty	nonempty	ADJ
ejpam-4343	69	3	collection	collection	NOUN
ejpam-4343	69	4	i	i	PRON
ejpam-4343	69	5	of	of	ADP
ejpam-4343	69	6	subsets	subset	NOUN
ejpam-4343	69	7	of	of	ADP
ejpam-4343	69	8	a	a	DET
ejpam-4343	69	9	set	set	NOUN
ejpam-4343	69	10	x	x	SYM
ejpam-4343	69	11	is	be	AUX
ejpam-4343	69	12	said	say	VERB
ejpam-4343	69	13	to	to	PART
ejpam-4343	69	14	be	be	AUX
ejpam-4343	69	15	an	an	DET
ejpam-4343	69	16	ideal	ideal	NOUN
ejpam-4343	69	17	on	on	ADP
ejpam-4343	69	18	x	x	SYM
ejpam-4343	69	19	if	if	SCONJ
ejpam-4343	69	20	i	i	PRON
ejpam-4343	69	21	satisfies	satisfy	VERB
ejpam-4343	69	22	the	the	DET
ejpam-4343	69	23	following	follow	VERB
ejpam-4343	69	24	two	two	NUM
ejpam-4343	69	25	properties	property	NOUN
ejpam-4343	69	26	:	:	PUNCT
ejpam-4343	69	27	(	(	PUNCT
ejpam-4343	69	28	i	i	NOUN
ejpam-4343	69	29	)	)	PUNCT
ejpam-4343	70	1	a	a	PRON
ejpam-4343	70	2	∈	∈	NOUN
ejpam-4343	71	1	i	i	PRON
ejpam-4343	71	2	and	and	CCONJ
ejpam-4343	71	3	b	b	X
ejpam-4343	71	4	⊆	⊆	NUM
ejpam-4343	71	5	a	a	DET
ejpam-4343	71	6	⇒	⇒	NOUN
ejpam-4343	71	7	b	b	X
ejpam-4343	71	8	∈	∈	PROPN
ejpam-4343	72	1	i	i	PRON
ejpam-4343	72	2	;	;	PUNCT
ejpam-4343	72	3	(	(	PUNCT
ejpam-4343	72	4	ii	ii	NOUN
ejpam-4343	72	5	)	)	PUNCT
ejpam-4343	72	6	a	a	PRON
ejpam-4343	72	7	∈	∈	PROPN
ejpam-4343	73	1	i	i	PRON
ejpam-4343	73	2	and	and	CCONJ
ejpam-4343	73	3	b	b	X
ejpam-4343	73	4	∈	∈	PROPN
ejpam-4343	73	5	i	i	PRON
ejpam-4343	73	6	⇒	⇒	VERB
ejpam-4343	73	7	a	a	DET
ejpam-4343	73	8	∪	∪	X
ejpam-4343	73	9	b	b	NOUN
ejpam-4343	73	10	∈	∈	NOUN
ejpam-4343	73	11	i	i	PRON
ejpam-4343	73	12	.	.	PUNCT
ejpam-4343	74	1	for	for	ADP
ejpam-4343	74	2	a	a	DET
ejpam-4343	74	3	topological	topological	ADJ
ejpam-4343	74	4	space	space	NOUN
ejpam-4343	74	5	(	(	PUNCT
ejpam-4343	74	6	x	x	X
ejpam-4343	74	7	,	,	PUNCT
ejpam-4343	74	8	τ	τ	X
ejpam-4343	74	9	)	)	PUNCT
ejpam-4343	74	10	with	with	ADP
ejpam-4343	74	11	an	an	DET
ejpam-4343	74	12	ideal	ideal	ADJ
ejpam-4343	74	13	i	i	PRON
ejpam-4343	74	14	on	on	ADP
ejpam-4343	74	15	x	x	NOUN
ejpam-4343	74	16	,	,	PUNCT
ejpam-4343	74	17	a	a	DET
ejpam-4343	74	18	set	set	NOUN
ejpam-4343	74	19	operator	operator	NOUN
ejpam-4343	74	20	(	(	PUNCT
ejpam-4343	74	21	.)⋆	.)⋆	NOUN
ejpam-4343	74	22	:	:	PUNCT
ejpam-4343	74	23	p(x	p(x	PROPN
ejpam-4343	74	24	)	)	PUNCT
ejpam-4343	74	25	→	→	SYM
ejpam-4343	74	26	p(x	p(x	PROPN
ejpam-4343	74	27	)	)	PUNCT
ejpam-4343	74	28	where	where	SCONJ
ejpam-4343	74	29	p(x	p(x	NOUN
ejpam-4343	74	30	)	)	PUNCT
ejpam-4343	74	31	is	be	AUX
ejpam-4343	74	32	the	the	DET
ejpam-4343	74	33	set	set	NOUN
ejpam-4343	74	34	of	of	ADP
ejpam-4343	74	35	all	all	DET
ejpam-4343	74	36	subsets	subset	NOUN
ejpam-4343	74	37	of	of	ADP
ejpam-4343	74	38	x	x	PRON
ejpam-4343	74	39	,	,	PUNCT
ejpam-4343	74	40	called	call	VERB
ejpam-4343	74	41	a	a	DET
ejpam-4343	74	42	local	local	ADJ
ejpam-4343	74	43	function	function	NOUN
ejpam-4343	74	44	[	[	X
ejpam-4343	74	45	26	26	NUM
ejpam-4343	74	46	]	]	PUNCT
ejpam-4343	74	47	of	of	ADP
ejpam-4343	74	48	a	a	PRON
ejpam-4343	74	49	with	with	ADP
ejpam-4343	74	50	respect	respect	NOUN
ejpam-4343	74	51	to	to	ADP
ejpam-4343	74	52	i	i	PRON
ejpam-4343	74	53	and	and	CCONJ
ejpam-4343	74	54	τ	τ	PROPN
ejpam-4343	74	55	is	be	AUX
ejpam-4343	74	56	defined	define	VERB
ejpam-4343	74	57	as	as	SCONJ
ejpam-4343	74	58	follows	follow	VERB
ejpam-4343	74	59	:	:	PUNCT
ejpam-4343	74	60	for	for	ADP
ejpam-4343	74	61	a	a	DET
ejpam-4343	74	62	⊆	⊆	NUM
ejpam-4343	74	63	x	x	SYM
ejpam-4343	74	64	,	,	PUNCT
ejpam-4343	74	65	a⋆(i	a⋆(i	NOUN
ejpam-4343	74	66	,	,	PUNCT
ejpam-4343	74	67	τ	τ	X
ejpam-4343	74	68	)	)	PUNCT
ejpam-4343	74	69	=	=	PRON
ejpam-4343	74	70	{	{	PUNCT
ejpam-4343	74	71	x	x	PUNCT
ejpam-4343	74	72	∈	∈	NOUN
ejpam-4343	74	73	x	x	INTJ
ejpam-4343	74	74	|	|	ADV
ejpam-4343	74	75	g	g	PROPN
ejpam-4343	74	76	∩	∩	VERB
ejpam-4343	74	77	a	a	DET
ejpam-4343	74	78	̸∈	̸∈	PROPN
ejpam-4343	74	79	i	i	PROPN
ejpam-4343	74	80	for	for	ADP
ejpam-4343	74	81	every	every	DET
ejpam-4343	74	82	g	g	PROPN
ejpam-4343	74	83	∈	∈	PROPN
ejpam-4343	74	84	τ(x	τ(x	NOUN
ejpam-4343	74	85	)	)	PUNCT
ejpam-4343	74	86	}	}	PUNCT
ejpam-4343	74	87	where	where	SCONJ
ejpam-4343	74	88	τ(x	τ(x	NOUN
ejpam-4343	74	89	)	)	PUNCT
ejpam-4343	74	90	=	=	PRON
ejpam-4343	74	91	{	{	PUNCT
ejpam-4343	74	92	g	g	PROPN
ejpam-4343	74	93	∈	∈	PROPN
ejpam-4343	74	94	τ	τ	X
ejpam-4343	74	95	|	|	ADV
ejpam-4343	74	96	x	x	X
ejpam-4343	74	97	∈	∈	PROPN
ejpam-4343	74	98	g	g	NOUN
ejpam-4343	74	99	}	}	PUNCT
ejpam-4343	74	100	.	.	PUNCT
ejpam-4343	75	1	a	a	DET
ejpam-4343	75	2	kuratowski	kuratowski	ADJ
ejpam-4343	75	3	closure	closure	NOUN
ejpam-4343	75	4	operator	operator	NOUN
ejpam-4343	75	5	cl⋆	cl⋆	PROPN
ejpam-4343	75	6	(	(	PUNCT
ejpam-4343	75	7	.	.	PUNCT
ejpam-4343	75	8	)	)	PUNCT
ejpam-4343	75	9	for	for	ADP
ejpam-4343	75	10	a	a	DET
ejpam-4343	75	11	topology	topology	NOUN
ejpam-4343	75	12	τ⋆(i	τ⋆(i	NOUN
ejpam-4343	75	13	,	,	PUNCT
ejpam-4343	75	14	τ	τ	PROPN
ejpam-4343	75	15	)	)	PUNCT
ejpam-4343	75	16	,	,	PUNCT
ejpam-4343	75	17	called	call	VERB
ejpam-4343	75	18	the	the	DET
ejpam-4343	75	19	⋆-topology	⋆-topology	NOUN
ejpam-4343	75	20	and	and	CCONJ
ejpam-4343	75	21	finer	fine	ADJ
ejpam-4343	75	22	than	than	ADP
ejpam-4343	75	23	τ	τ	PROPN
ejpam-4343	75	24	,	,	PUNCT
ejpam-4343	75	25	is	be	AUX
ejpam-4343	75	26	defined	define	VERB
ejpam-4343	75	27	by	by	ADP
ejpam-4343	75	28	cl⋆(a	cl⋆(a	NOUN
ejpam-4343	75	29	)	)	PUNCT
ejpam-4343	75	30	=	=	NOUN
ejpam-4343	75	31	a	a	DET
ejpam-4343	75	32	∪	∪	NOUN
ejpam-4343	75	33	a⋆	a⋆	NOUN
ejpam-4343	76	1	[	[	X
ejpam-4343	76	2	24	24	NUM
ejpam-4343	76	3	]	]	PUNCT
ejpam-4343	76	4	.	.	PUNCT
ejpam-4343	77	1	we	we	PRON
ejpam-4343	77	2	shall	shall	AUX
ejpam-4343	77	3	simply	simply	ADV
ejpam-4343	77	4	write	write	VERB
ejpam-4343	77	5	a⋆	a⋆	ADV
ejpam-4343	77	6	for	for	ADP
ejpam-4343	77	7	a⋆(i	a⋆(i	PROPN
ejpam-4343	77	8	,	,	PUNCT
ejpam-4343	77	9	τ	τ	PROPN
ejpam-4343	77	10	)	)	PUNCT
ejpam-4343	77	11	and	and	CCONJ
ejpam-4343	77	12	τ⋆	τ⋆	X
ejpam-4343	77	13	for	for	ADP
ejpam-4343	77	14	τ⋆(i	τ⋆(i	NOUN
ejpam-4343	77	15	,	,	PUNCT
ejpam-4343	77	16	τ	τ	PROPN
ejpam-4343	77	17	)	)	PUNCT
ejpam-4343	77	18	.	.	PUNCT
ejpam-4343	78	1	a	a	DET
ejpam-4343	78	2	basis	basis	NOUN
ejpam-4343	78	3	b(i	b(i	ADJ
ejpam-4343	78	4	,	,	PUNCT
ejpam-4343	78	5	τ	τ	X
ejpam-4343	78	6	)	)	PUNCT
ejpam-4343	78	7	for	for	ADP
ejpam-4343	78	8	τ⋆	τ⋆	PRON
ejpam-4343	78	9	can	can	AUX
ejpam-4343	78	10	be	be	AUX
ejpam-4343	78	11	described	describe	VERB
ejpam-4343	78	12	as	as	ADP
ejpam-4343	78	13	follows	follow	VERB
ejpam-4343	78	14	:	:	PUNCT
ejpam-4343	78	15	b(i	b(i	NUM
ejpam-4343	78	16	,	,	PUNCT
ejpam-4343	78	17	τ	τ	X
ejpam-4343	78	18	)	)	PUNCT
ejpam-4343	78	19	=	=	PRON
ejpam-4343	79	1	{	{	PUNCT
ejpam-4343	79	2	v	v	NOUN
ejpam-4343	79	3	−	−	NOUN
ejpam-4343	80	1	i	i	PRON
ejpam-4343	80	2	′	′	VERB
ejpam-4343	81	1	|	|	ADV
ejpam-4343	81	2	v	v	X
ejpam-4343	81	3	∈	∈	NOUN
ejpam-4343	81	4	τ	τ	X
ejpam-4343	82	1	and	and	CCONJ
ejpam-4343	82	2	i	i	PRON
ejpam-4343	82	3	′	′	VERB
ejpam-4343	83	1	∈	∈	INTJ
ejpam-4343	84	1	i	i	PRON
ejpam-4343	84	2	}	}	PUNCT
ejpam-4343	84	3	.	.	PUNCT
ejpam-4343	85	1	however	however	ADV
ejpam-4343	85	2	,	,	PUNCT
ejpam-4343	85	3	b(i	b(i	PROPN
ejpam-4343	85	4	,	,	PUNCT
ejpam-4343	85	5	τ	τ	X
ejpam-4343	85	6	)	)	PUNCT
ejpam-4343	85	7	is	be	AUX
ejpam-4343	85	8	not	not	PART
ejpam-4343	85	9	always	always	ADV
ejpam-4343	85	10	a	a	DET
ejpam-4343	85	11	topology	topology	NOUN
ejpam-4343	85	12	[	[	X
ejpam-4343	85	13	24	24	NUM
ejpam-4343	85	14	]	]	PUNCT
ejpam-4343	85	15	.	.	PUNCT
ejpam-4343	86	1	a	a	DET
ejpam-4343	86	2	subset	subset	NOUN
ejpam-4343	86	3	a	a	PRON
ejpam-4343	86	4	of	of	ADP
ejpam-4343	86	5	an	an	DET
ejpam-4343	86	6	ideal	ideal	ADJ
ejpam-4343	86	7	topological	topological	ADJ
ejpam-4343	86	8	space	space	NOUN
ejpam-4343	86	9	(	(	PUNCT
ejpam-4343	86	10	x	x	X
ejpam-4343	86	11	,	,	PUNCT
ejpam-4343	86	12	τ	τ	PROPN
ejpam-4343	86	13	,	,	PUNCT
ejpam-4343	86	14	i	i	PROPN
ejpam-4343	86	15	)	)	PUNCT
ejpam-4343	86	16	is	be	AUX
ejpam-4343	86	17	called	call	VERB
ejpam-4343	86	18	⋆-closed	⋆-closed	ADJ
ejpam-4343	86	19	(	(	PUNCT
ejpam-4343	86	20	τ⋆-closed	τ⋆-close	VERB
ejpam-4343	86	21	)	)	PUNCT
ejpam-4343	87	1	[	[	X
ejpam-4343	87	2	24	24	NUM
ejpam-4343	87	3	]	]	X
ejpam-4343	87	4	if	if	SCONJ
ejpam-4343	87	5	a⋆	a⋆	ADJ
ejpam-4343	87	6	⊆	⊆	NUM
ejpam-4343	87	7	a.	a.	NOUN
ejpam-4343	87	8	the	the	DET
ejpam-4343	87	9	interior	interior	NOUN
ejpam-4343	87	10	of	of	ADP
ejpam-4343	87	11	a	a	DET
ejpam-4343	87	12	subset	subset	NOUN
ejpam-4343	87	13	a	a	DET
ejpam-4343	87	14	in	in	ADP
ejpam-4343	87	15	(	(	PUNCT
ejpam-4343	87	16	x	x	X
ejpam-4343	87	17	,	,	PUNCT
ejpam-4343	87	18	τ⋆(i	τ⋆(i	NOUN
ejpam-4343	87	19	,	,	PUNCT
ejpam-4343	87	20	τ	τ	PROPN
ejpam-4343	87	21	)	)	PUNCT
ejpam-4343	87	22	)	)	PUNCT
ejpam-4343	87	23	is	be	AUX
ejpam-4343	87	24	denoted	denote	VERB
ejpam-4343	87	25	by	by	ADP
ejpam-4343	87	26	int⋆(a	int⋆(a	NOUN
ejpam-4343	87	27	)	)	PUNCT
ejpam-4343	87	28	.	.	PUNCT
ejpam-4343	88	1	a	a	DET
ejpam-4343	88	2	subset	subset	NOUN
ejpam-4343	88	3	a	a	PRON
ejpam-4343	88	4	of	of	ADP
ejpam-4343	88	5	an	an	DET
ejpam-4343	88	6	ideal	ideal	ADJ
ejpam-4343	88	7	topological	topological	ADJ
ejpam-4343	88	8	space	space	NOUN
ejpam-4343	88	9	(	(	PUNCT
ejpam-4343	88	10	x	x	X
ejpam-4343	88	11	,	,	PUNCT
ejpam-4343	88	12	τ	τ	PROPN
ejpam-4343	88	13	,	,	PUNCT
ejpam-4343	88	14	i	i	PROPN
ejpam-4343	88	15	)	)	PUNCT
ejpam-4343	88	16	is	be	AUX
ejpam-4343	88	17	said	say	VERB
ejpam-4343	88	18	to	to	PART
ejpam-4343	88	19	be	be	AUX
ejpam-4343	88	20	pre	pre	ADJ
ejpam-4343	88	21	-	-	ADJ
ejpam-4343	88	22	i	i	PRON
ejpam-4343	88	23	-open	-open	NOUN
ejpam-4343	89	1	[	[	X
ejpam-4343	89	2	15	15	NUM
ejpam-4343	89	3	]	]	X
ejpam-4343	89	4	if	if	SCONJ
ejpam-4343	89	5	a	a	DET
ejpam-4343	89	6	⊆	⊆	NUM
ejpam-4343	89	7	int(cl⋆(a	int(cl⋆(a	NUM
ejpam-4343	89	8	)	)	PUNCT
ejpam-4343	89	9	)	)	PUNCT
ejpam-4343	89	10	.	.	PUNCT
ejpam-4343	90	1	the	the	DET
ejpam-4343	90	2	complement	complement	NOUN
ejpam-4343	90	3	of	of	ADP
ejpam-4343	90	4	a	a	DET
ejpam-4343	90	5	pre	pre	ADJ
ejpam-4343	90	6	-	-	ADJ
ejpam-4343	90	7	i	i	PRON
ejpam-4343	90	8	-open	-open	ADJ
ejpam-4343	90	9	set	set	NOUN
ejpam-4343	90	10	is	be	AUX
ejpam-4343	90	11	called	call	VERB
ejpam-4343	90	12	pre	pre	ADJ
ejpam-4343	90	13	-	-	ADJ
ejpam-4343	90	14	i	i	PRON
ejpam-4343	90	15	-closed	-closed	ADJ
ejpam-4343	90	16	.	.	PUNCT
ejpam-4343	91	1	the	the	DET
ejpam-4343	91	2	family	family	NOUN
ejpam-4343	91	3	of	of	ADP
ejpam-4343	91	4	all	all	DET
ejpam-4343	91	5	pre	pre	ADJ
ejpam-4343	91	6	-	-	ADJ
ejpam-4343	91	7	i	i	ADJ
ejpam-4343	91	8	-open	-open	ADJ
ejpam-4343	91	9	sets	set	NOUN
ejpam-4343	91	10	of	of	ADP
ejpam-4343	91	11	an	an	DET
ejpam-4343	91	12	ideal	ideal	ADJ
ejpam-4343	91	13	topological	topological	ADJ
ejpam-4343	91	14	space	space	NOUN
ejpam-4343	91	15	(	(	PUNCT
ejpam-4343	91	16	x	x	X
ejpam-4343	91	17	,	,	PUNCT
ejpam-4343	91	18	τ	τ	PROPN
ejpam-4343	91	19	,	,	PUNCT
ejpam-4343	91	20	i	i	PROPN
ejpam-4343	91	21	)	)	PUNCT
ejpam-4343	91	22	is	be	AUX
ejpam-4343	91	23	denoted	denote	VERB
ejpam-4343	91	24	by	by	ADP
ejpam-4343	91	25	pio(x	pio(x	PROPN
ejpam-4343	91	26	,	,	PUNCT
ejpam-4343	91	27	τ	τ	PROPN
ejpam-4343	91	28	)	)	PUNCT
ejpam-4343	91	29	.	.	PUNCT
ejpam-4343	92	1	for	for	ADP
ejpam-4343	92	2	a	a	DET
ejpam-4343	92	3	subset	subset	NOUN
ejpam-4343	92	4	a	a	PRON
ejpam-4343	92	5	of	of	ADP
ejpam-4343	92	6	an	an	DET
ejpam-4343	92	7	ideal	ideal	ADJ
ejpam-4343	92	8	topological	topological	ADJ
ejpam-4343	92	9	space	space	NOUN
ejpam-4343	92	10	(	(	PUNCT
ejpam-4343	92	11	x	x	X
ejpam-4343	92	12	,	,	PUNCT
ejpam-4343	92	13	τ	τ	PROPN
ejpam-4343	92	14	,	,	PUNCT
ejpam-4343	92	15	i	i	NOUN
ejpam-4343	92	16	)	)	PUNCT
ejpam-4343	92	17	,	,	PUNCT
ejpam-4343	92	18	the	the	DET
ejpam-4343	92	19	intersection	intersection	NOUN
ejpam-4343	92	20	of	of	ADP
ejpam-4343	92	21	all	all	DET
ejpam-4343	92	22	pre	pre	ADJ
ejpam-4343	92	23	-	-	ADJ
ejpam-4343	92	24	i	i	PRON
ejpam-4343	92	25	-closed	-closed	ADJ
ejpam-4343	92	26	sets	set	NOUN
ejpam-4343	92	27	of	of	ADP
ejpam-4343	92	28	x	x	PUNCT
ejpam-4343	92	29	containing	contain	VERB
ejpam-4343	92	30	a	a	PRON
ejpam-4343	92	31	is	be	AUX
ejpam-4343	92	32	called	call	VERB
ejpam-4343	92	33	the	the	DET
ejpam-4343	92	34	pre	pre	NOUN
ejpam-4343	92	35	-	-	ADJ
ejpam-4343	92	36	i	i	ADJ
ejpam-4343	92	37	-closure	-closure	NOUN
ejpam-4343	92	38	[	[	X
ejpam-4343	92	39	13	13	NUM
ejpam-4343	92	40	]	]	PUNCT
ejpam-4343	92	41	of	of	ADP
ejpam-4343	92	42	a	a	PRON
ejpam-4343	92	43	and	and	CCONJ
ejpam-4343	92	44	is	be	AUX
ejpam-4343	92	45	denoted	denote	VERB
ejpam-4343	92	46	by	by	ADP
ejpam-4343	92	47	pıcl(a	pıcl(a	NOUN
ejpam-4343	92	48	)	)	PUNCT
ejpam-4343	92	49	.	.	PUNCT
ejpam-4343	93	1	the	the	DET
ejpam-4343	93	2	union	union	NOUN
ejpam-4343	93	3	of	of	ADP
ejpam-4343	93	4	all	all	DET
ejpam-4343	93	5	pre	pre	ADJ
ejpam-4343	93	6	-	-	ADJ
ejpam-4343	93	7	i	i	PRON
ejpam-4343	93	8	-open	-open	NOUN
ejpam-4343	93	9	sets	set	NOUN
ejpam-4343	93	10	contained	contain	VERB
ejpam-4343	93	11	in	in	ADP
ejpam-4343	93	12	a	a	PRON
ejpam-4343	93	13	is	be	AUX
ejpam-4343	93	14	called	call	VERB
ejpam-4343	93	15	the	the	DET
ejpam-4343	93	16	pre	pre	NOUN
ejpam-4343	93	17	-	-	ADJ
ejpam-4343	93	18	i	i	PRON
ejpam-4343	93	19	-interior	-interior	NOUN
ejpam-4343	93	20	of	of	ADP
ejpam-4343	93	21	a	a	PRON
ejpam-4343	93	22	and	and	CCONJ
ejpam-4343	93	23	is	be	AUX
ejpam-4343	93	24	denoted	denote	VERB
ejpam-4343	93	25	by	by	ADP
ejpam-4343	93	26	pıint(a	pıint(a	PROPN
ejpam-4343	93	27	)	)	PUNCT
ejpam-4343	93	28	.	.	PUNCT
ejpam-4343	94	1	lemma	lemma	PROPN
ejpam-4343	94	2	1	1	NUM
ejpam-4343	94	3	.	.	PUNCT
ejpam-4343	95	1	[	[	X
ejpam-4343	95	2	13	13	NUM
ejpam-4343	95	3	]	]	PUNCT
ejpam-4343	95	4	let	let	VERB
ejpam-4343	95	5	a	a	PRON
ejpam-4343	95	6	be	be	AUX
ejpam-4343	95	7	a	a	DET
ejpam-4343	95	8	subset	subset	NOUN
ejpam-4343	95	9	of	of	ADP
ejpam-4343	95	10	an	an	DET
ejpam-4343	95	11	ideal	ideal	ADJ
ejpam-4343	95	12	topological	topological	ADJ
ejpam-4343	95	13	space	space	NOUN
ejpam-4343	95	14	(	(	PUNCT
ejpam-4343	95	15	x	x	X
ejpam-4343	95	16	,	,	PUNCT
ejpam-4343	95	17	τ	τ	PROPN
ejpam-4343	95	18	,	,	PUNCT
ejpam-4343	95	19	i	i	PROPN
ejpam-4343	95	20	)	)	PUNCT
ejpam-4343	95	21	and	and	CCONJ
ejpam-4343	96	1	x	x	PUNCT
ejpam-4343	96	2	∈	∈	PROPN
ejpam-4343	96	3	x.	x.	NOUN
ejpam-4343	97	1	then	then	ADV
ejpam-4343	97	2	,	,	PUNCT
ejpam-4343	97	3	the	the	DET
ejpam-4343	97	4	following	follow	VERB
ejpam-4343	97	5	properties	property	NOUN
ejpam-4343	97	6	hold	hold	VERB
ejpam-4343	97	7	:	:	PUNCT
ejpam-4343	97	8	(	(	PUNCT
ejpam-4343	97	9	1	1	X
ejpam-4343	97	10	)	)	PUNCT
ejpam-4343	97	11	x	x	SYM
ejpam-4343	97	12	∈	∈	PROPN
ejpam-4343	97	13	pıcl(a	pıcl(a	NOUN
ejpam-4343	97	14	)	)	PUNCT
ejpam-4343	98	1	if	if	SCONJ
ejpam-4343	98	2	and	and	CCONJ
ejpam-4343	98	3	only	only	ADV
ejpam-4343	98	4	if	if	SCONJ
ejpam-4343	98	5	u	u	PROPN
ejpam-4343	98	6	∩a	∩a	PROPN
ejpam-4343	98	7	̸=	̸=	PROPN
ejpam-4343	98	8	∅	∅	NOUN
ejpam-4343	98	9	for	for	ADP
ejpam-4343	98	10	every	every	DET
ejpam-4343	98	11	pre	pre	ADJ
ejpam-4343	98	12	-	-	ADJ
ejpam-4343	98	13	i	i	PRON
ejpam-4343	98	14	-open	-open	NOUN
ejpam-4343	98	15	set	set	VERB
ejpam-4343	98	16	u	u	NOUN
ejpam-4343	98	17	of	of	ADP
ejpam-4343	98	18	x	x	SYM
ejpam-4343	98	19	containing	contain	VERB
ejpam-4343	98	20	x.	x.	NOUN
ejpam-4343	98	21	(	(	PUNCT
ejpam-4343	98	22	2	2	NUM
ejpam-4343	98	23	)	)	PUNCT
ejpam-4343	98	24	a	a	PRON
ejpam-4343	98	25	is	be	AUX
ejpam-4343	98	26	pre	pre	ADJ
ejpam-4343	98	27	-	-	ADJ
ejpam-4343	98	28	i	i	PRON
ejpam-4343	98	29	-closed	-close	VERB
ejpam-4343	98	30	if	if	SCONJ
ejpam-4343	98	31	and	and	CCONJ
ejpam-4343	98	32	only	only	ADV
ejpam-4343	98	33	if	if	SCONJ
ejpam-4343	98	34	a	a	DET
ejpam-4343	98	35	=	=	SYM
ejpam-4343	98	36	pıcl(a	pıcl(a	NOUN
ejpam-4343	98	37	)	)	PUNCT
ejpam-4343	98	38	.	.	PUNCT
ejpam-4343	99	1	(	(	PUNCT
ejpam-4343	99	2	3	3	X
ejpam-4343	99	3	)	)	PUNCT
ejpam-4343	99	4	x	x	NOUN
ejpam-4343	100	1	−	−	PROPN
ejpam-4343	100	2	pıcl(a	pıcl(a	NOUN
ejpam-4343	100	3	)	)	PUNCT
ejpam-4343	100	4	=	=	SYM
ejpam-4343	100	5	pıint(x	pıint(x	NOUN
ejpam-4343	100	6	−a	−a	NOUN
ejpam-4343	100	7	)	)	PUNCT
ejpam-4343	100	8	.	.	PUNCT
ejpam-4343	101	1	(	(	PUNCT
ejpam-4343	101	2	4	4	X
ejpam-4343	101	3	)	)	PUNCT
ejpam-4343	101	4	x	x	SYM
ejpam-4343	101	5	−	−	NOUN
ejpam-4343	101	6	pıint(a	pıint(a	PROPN
ejpam-4343	101	7	)	)	PUNCT
ejpam-4343	102	1	=	=	SYM
ejpam-4343	102	2	pıcl(x	pıcl(x	PROPN
ejpam-4343	102	3	−a	−a	NOUN
ejpam-4343	102	4	)	)	PUNCT
ejpam-4343	102	5	.	.	PUNCT
ejpam-4343	103	1	lemma	lemma	PROPN
ejpam-4343	103	2	2	2	NUM
ejpam-4343	103	3	.	.	PUNCT
ejpam-4343	104	1	[	[	X
ejpam-4343	104	2	14	14	NUM
ejpam-4343	104	3	,	,	PUNCT
ejpam-4343	104	4	29	29	NUM
ejpam-4343	104	5	]	]	PUNCT
ejpam-4343	104	6	let	let	VERB
ejpam-4343	104	7	a	a	PRON
ejpam-4343	104	8	be	be	AUX
ejpam-4343	104	9	a	a	DET
ejpam-4343	104	10	subset	subset	NOUN
ejpam-4343	104	11	of	of	ADP
ejpam-4343	104	12	an	an	DET
ejpam-4343	104	13	ideal	ideal	ADJ
ejpam-4343	104	14	topological	topological	ADJ
ejpam-4343	104	15	space	space	NOUN
ejpam-4343	104	16	(	(	PUNCT
ejpam-4343	104	17	x	x	X
ejpam-4343	104	18	,	,	PUNCT
ejpam-4343	104	19	τ	τ	PROPN
ejpam-4343	104	20	,	,	PUNCT
ejpam-4343	104	21	i	i	NOUN
ejpam-4343	104	22	)	)	PUNCT
ejpam-4343	104	23	.	.	PUNCT
ejpam-4343	105	1	then	then	ADV
ejpam-4343	105	2	,	,	PUNCT
ejpam-4343	105	3	pıcl(a	pıcl(a	NOUN
ejpam-4343	105	4	)	)	PUNCT
ejpam-4343	105	5	=	=	NOUN
ejpam-4343	105	6	a	a	DET
ejpam-4343	105	7	∪	∪	ADJ
ejpam-4343	105	8	cl(int⋆(a	cl(int⋆(a	NOUN
ejpam-4343	105	9	)	)	PUNCT
ejpam-4343	105	10	)	)	PUNCT
ejpam-4343	105	11	.	.	PUNCT
ejpam-4343	106	1	c.	c.	PROPN
ejpam-4343	106	2	boonpok	boonpok	PROPN
ejpam-4343	106	3	/	/	SYM
ejpam-4343	106	4	eur	eur	PROPN
ejpam-4343	106	5	.	.	PUNCT
ejpam-4343	107	1	j.	j.	PROPN
ejpam-4343	107	2	pure	pure	PROPN
ejpam-4343	107	3	appl	appl	PROPN
ejpam-4343	107	4	.	.	PROPN
ejpam-4343	107	5	math	math	PROPN
ejpam-4343	107	6	,	,	PUNCT
ejpam-4343	107	7	15	15	NUM
ejpam-4343	107	8	(	(	PUNCT
ejpam-4343	107	9	3	3	NUM
ejpam-4343	107	10	)	)	PUNCT
ejpam-4343	107	11	(	(	PUNCT
ejpam-4343	107	12	2022	2022	NUM
ejpam-4343	107	13	)	)	PUNCT
ejpam-4343	107	14	,	,	PUNCT
ejpam-4343	107	15	1023	1023	NUM
ejpam-4343	107	16	-	-	SYM
ejpam-4343	107	17	1046	1046	NUM
ejpam-4343	107	18	1026	1026	NUM
ejpam-4343	107	19	3	3	NUM
ejpam-4343	107	20	.	.	PUNCT
ejpam-4343	108	1	some	some	DET
ejpam-4343	108	2	properties	property	NOUN
ejpam-4343	108	3	of	of	ADP
ejpam-4343	108	4	λp(⋆)-sets	λp(⋆)-set	NOUN
ejpam-4343	108	5	and	and	CCONJ
ejpam-4343	108	6	δp(⋆)-sets	δp(⋆)-set	NOUN
ejpam-4343	108	7	in	in	ADP
ejpam-4343	108	8	this	this	DET
ejpam-4343	108	9	section	section	NOUN
ejpam-4343	108	10	,	,	PUNCT
ejpam-4343	108	11	we	we	PRON
ejpam-4343	108	12	introduce	introduce	VERB
ejpam-4343	108	13	the	the	DET
ejpam-4343	108	14	notions	notion	NOUN
ejpam-4343	108	15	of	of	ADP
ejpam-4343	108	16	λp(⋆)-sets	λp(⋆)-set	NOUN
ejpam-4343	108	17	and	and	CCONJ
ejpam-4343	108	18	δp(⋆)-sets	δp(⋆)-set	NOUN
ejpam-4343	108	19	.	.	PUNCT
ejpam-4343	109	1	moreover	moreover	ADV
ejpam-4343	109	2	,	,	PUNCT
ejpam-4343	109	3	several	several	ADJ
ejpam-4343	109	4	properties	property	NOUN
ejpam-4343	109	5	of	of	ADP
ejpam-4343	109	6	λp(⋆)-sets	λp(⋆)-set	NOUN
ejpam-4343	109	7	and	and	CCONJ
ejpam-4343	109	8	δp(⋆)-sets	δp(⋆)-set	NOUN
ejpam-4343	109	9	are	be	AUX
ejpam-4343	109	10	discussed	discuss	VERB
ejpam-4343	109	11	.	.	PUNCT
ejpam-4343	110	1	definition	definition	NOUN
ejpam-4343	110	2	1	1	NUM
ejpam-4343	110	3	.	.	PUNCT
ejpam-4343	111	1	let	let	VERB
ejpam-4343	111	2	a	a	DET
ejpam-4343	111	3	be	be	AUX
ejpam-4343	111	4	a	a	DET
ejpam-4343	111	5	subset	subset	NOUN
ejpam-4343	111	6	of	of	ADP
ejpam-4343	111	7	an	an	DET
ejpam-4343	111	8	ideal	ideal	ADJ
ejpam-4343	111	9	topological	topological	ADJ
ejpam-4343	111	10	space	space	NOUN
ejpam-4343	111	11	(	(	PUNCT
ejpam-4343	111	12	x	x	X
ejpam-4343	111	13	,	,	PUNCT
ejpam-4343	111	14	τ	τ	PROPN
ejpam-4343	111	15	,	,	PUNCT
ejpam-4343	111	16	i	i	NOUN
ejpam-4343	111	17	)	)	PUNCT
ejpam-4343	111	18	.	.	PUNCT
ejpam-4343	112	1	a	a	DET
ejpam-4343	112	2	subset	subset	ADJ
ejpam-4343	112	3	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	112	4	)	)	PUNCT
ejpam-4343	112	5	is	be	AUX
ejpam-4343	112	6	defined	define	VERB
ejpam-4343	112	7	as	as	SCONJ
ejpam-4343	112	8	follows	follow	VERB
ejpam-4343	112	9	:	:	PUNCT
ejpam-4343	112	10	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	112	11	)	)	PUNCT
ejpam-4343	113	1	=	=	PUNCT
ejpam-4343	114	1	∩{u	∩{u	PROPN
ejpam-4343	114	2	|	|	ADV
ejpam-4343	114	3	a	a	DET
ejpam-4343	114	4	⊆	⊆	NUM
ejpam-4343	114	5	u	u	NOUN
ejpam-4343	114	6	;	;	PUNCT
ejpam-4343	114	7	u	u	NOUN
ejpam-4343	114	8	is	be	AUX
ejpam-4343	114	9	pre	pre	ADJ
ejpam-4343	114	10	-	-	ADJ
ejpam-4343	114	11	i	i	PRON
ejpam-4343	114	12	-open	-open	NOUN
ejpam-4343	114	13	}	}	PUNCT
ejpam-4343	114	14	.	.	PUNCT
ejpam-4343	115	1	proposition	proposition	NOUN
ejpam-4343	115	2	1	1	NUM
ejpam-4343	115	3	.	.	PUNCT
ejpam-4343	116	1	for	for	ADP
ejpam-4343	116	2	subsets	subset	NOUN
ejpam-4343	116	3	a	a	PRON
ejpam-4343	116	4	,	,	PUNCT
ejpam-4343	116	5	b	b	NOUN
ejpam-4343	116	6	and	and	CCONJ
ejpam-4343	116	7	cγ(γ	cγ(γ	CCONJ
ejpam-4343	116	8	∈	∈	PROPN
ejpam-4343	116	9	γ	γ	PROPN
ejpam-4343	116	10	)	)	PUNCT
ejpam-4343	116	11	of	of	ADP
ejpam-4343	116	12	an	an	DET
ejpam-4343	116	13	ideal	ideal	ADJ
ejpam-4343	116	14	topological	topological	ADJ
ejpam-4343	116	15	space	space	NOUN
ejpam-4343	116	16	(	(	PUNCT
ejpam-4343	116	17	x	x	X
ejpam-4343	116	18	,	,	PUNCT
ejpam-4343	116	19	τ	τ	PROPN
ejpam-4343	116	20	,	,	PUNCT
ejpam-4343	116	21	i	i	NOUN
ejpam-4343	116	22	)	)	PUNCT
ejpam-4343	116	23	,	,	PUNCT
ejpam-4343	116	24	the	the	DET
ejpam-4343	116	25	following	follow	VERB
ejpam-4343	116	26	properties	property	NOUN
ejpam-4343	116	27	hold	hold	VERB
ejpam-4343	116	28	:	:	PUNCT
ejpam-4343	116	29	(	(	PUNCT
ejpam-4343	116	30	1	1	X
ejpam-4343	116	31	)	)	PUNCT
ejpam-4343	116	32	a	a	DET
ejpam-4343	116	33	⊆	⊆	NUM
ejpam-4343	116	34	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	116	35	)	)	PUNCT
ejpam-4343	116	36	.	.	PUNCT
ejpam-4343	117	1	(	(	PUNCT
ejpam-4343	117	2	2	2	X
ejpam-4343	117	3	)	)	PUNCT
ejpam-4343	117	4	if	if	SCONJ
ejpam-4343	117	5	a	a	DET
ejpam-4343	117	6	⊆	⊆	NUM
ejpam-4343	117	7	b	b	NOUN
ejpam-4343	117	8	,	,	PUNCT
ejpam-4343	117	9	then	then	ADV
ejpam-4343	117	10	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	117	11	)	)	PUNCT
ejpam-4343	117	12	⊆	⊆	NUM
ejpam-4343	117	13	λp(⋆)(b	λp(⋆)(b	PROPN
ejpam-4343	117	14	)	)	PUNCT
ejpam-4343	117	15	.	.	PUNCT
ejpam-4343	118	1	(	(	PUNCT
ejpam-4343	118	2	3	3	X
ejpam-4343	118	3	)	)	PUNCT
ejpam-4343	118	4	λp(⋆)(λp(⋆)(a	λp(⋆)(λp(⋆)(a	NOUN
ejpam-4343	118	5	)	)	PUNCT
ejpam-4343	118	6	)	)	PUNCT
ejpam-4343	119	1	=	=	SYM
ejpam-4343	119	2	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	119	3	)	)	PUNCT
ejpam-4343	119	4	.	.	PUNCT
ejpam-4343	120	1	(	(	PUNCT
ejpam-4343	120	2	4	4	X
ejpam-4343	120	3	)	)	PUNCT
ejpam-4343	120	4	if	if	SCONJ
ejpam-4343	120	5	a	a	PRON
ejpam-4343	120	6	is	be	AUX
ejpam-4343	120	7	a	a	DET
ejpam-4343	120	8	pre	pre	ADJ
ejpam-4343	120	9	-	-	ADJ
ejpam-4343	120	10	i	i	PRON
ejpam-4343	120	11	-open	-open	NOUN
ejpam-4343	120	12	set	set	NOUN
ejpam-4343	120	13	,	,	PUNCT
ejpam-4343	120	14	then	then	ADV
ejpam-4343	120	15	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	120	16	)	)	PUNCT
ejpam-4343	120	17	=	=	SYM
ejpam-4343	120	18	a.	a.	NOUN
ejpam-4343	120	19	(	(	PUNCT
ejpam-4343	120	20	5	5	NUM
ejpam-4343	120	21	)	)	PUNCT
ejpam-4343	120	22	λp(⋆)(∪{cγ	λp(⋆)(∪{cγ	NOUN
ejpam-4343	120	23	|γ	|γ	PROPN
ejpam-4343	120	24	∈	∈	PROPN
ejpam-4343	120	25	γ	γ	PROPN
ejpam-4343	120	26	}	}	PUNCT
ejpam-4343	120	27	)	)	PUNCT
ejpam-4343	120	28	=	=	SYM
ejpam-4343	120	29	∪{λp(⋆)(cγ)|γ	∪{λp(⋆)(cγ)|γ	PROPN
ejpam-4343	120	30	∈	∈	PROPN
ejpam-4343	120	31	γ	γ	X
ejpam-4343	120	32	}	}	PUNCT
ejpam-4343	120	33	.	.	PUNCT
ejpam-4343	121	1	(	(	PUNCT
ejpam-4343	121	2	6	6	X
ejpam-4343	121	3	)	)	PUNCT
ejpam-4343	121	4	λp(⋆)(∩{cγ	λp(⋆)(∩{cγ	NOUN
ejpam-4343	121	5	|γ	|γ	ADP
ejpam-4343	121	6	∈	∈	PROPN
ejpam-4343	121	7	γ	γ	PROPN
ejpam-4343	121	8	}	}	PUNCT
ejpam-4343	121	9	)	)	PUNCT
ejpam-4343	122	1	⊆	⊆	NUM
ejpam-4343	122	2	∩{λp(⋆)(cγ)|γ	∩{λp(⋆)(cγ)|γ	PUNCT
ejpam-4343	122	3	∈	∈	PROPN
ejpam-4343	122	4	γ	γ	X
ejpam-4343	122	5	}	}	PUNCT
ejpam-4343	122	6	.	.	PUNCT
ejpam-4343	123	1	proof	proof	NOUN
ejpam-4343	123	2	.	.	PUNCT
ejpam-4343	124	1	we	we	PRON
ejpam-4343	124	2	prove	prove	VERB
ejpam-4343	124	3	only	only	ADV
ejpam-4343	124	4	properties	property	NOUN
ejpam-4343	124	5	(	(	PUNCT
ejpam-4343	124	6	5	5	NUM
ejpam-4343	124	7	)	)	PUNCT
ejpam-4343	124	8	and	and	CCONJ
ejpam-4343	124	9	(	(	PUNCT
ejpam-4343	124	10	6	6	NUM
ejpam-4343	124	11	)	)	PUNCT
ejpam-4343	124	12	since	since	SCONJ
ejpam-4343	124	13	the	the	DET
ejpam-4343	124	14	others	other	NOUN
ejpam-4343	124	15	are	be	AUX
ejpam-4343	124	16	immediate	immediate	ADJ
ejpam-4343	124	17	consequences	consequence	NOUN
ejpam-4343	124	18	of	of	ADP
ejpam-4343	124	19	definition	definition	NOUN
ejpam-4343	124	20	1	1	NUM
ejpam-4343	124	21	.	.	PUNCT
ejpam-4343	125	1	(	(	PUNCT
ejpam-4343	125	2	5	5	NUM
ejpam-4343	125	3	)	)	PUNCT
ejpam-4343	125	4	first	first	ADV
ejpam-4343	125	5	for	for	ADP
ejpam-4343	125	6	each	each	DET
ejpam-4343	125	7	γ	γ	PROPN
ejpam-4343	125	8	∈	∈	PROPN
ejpam-4343	125	9	γ	γ	X
ejpam-4343	125	10	,	,	PUNCT
ejpam-4343	125	11	λp(⋆)(cγ	λp(⋆)(cγ	PROPN
ejpam-4343	125	12	)	)	PUNCT
ejpam-4343	125	13	⊆	⊆	NUM
ejpam-4343	125	14	λp(⋆)(∪γ∈γcγ	λp(⋆)(∪γ∈γcγ	NUM
ejpam-4343	125	15	)	)	PUNCT
ejpam-4343	125	16	.	.	PUNCT
ejpam-4343	126	1	thus	thus	ADV
ejpam-4343	126	2	,	,	PUNCT
ejpam-4343	126	3	∪γ∈γλp(⋆)(cγ	∪γ∈γλp(⋆)(cγ	VERB
ejpam-4343	126	4	)	)	PUNCT
ejpam-4343	126	5	⊆	⊆	NUM
ejpam-4343	126	6	λp(⋆)(∪γ∈γcγ	λp(⋆)(∪γ∈γcγ	NUM
ejpam-4343	126	7	)	)	PUNCT
ejpam-4343	126	8	.	.	PUNCT
ejpam-4343	127	1	on	on	ADP
ejpam-4343	127	2	the	the	DET
ejpam-4343	127	3	other	other	ADJ
ejpam-4343	127	4	hand	hand	NOUN
ejpam-4343	127	5	,	,	PUNCT
ejpam-4343	127	6	let	let	VERB
ejpam-4343	127	7	x	x	SYM
ejpam-4343	127	8	̸∈	̸∈	PROPN
ejpam-4343	127	9	∪γ∈γλp(⋆)(cγ	∪γ∈γλp(⋆)(cγ	ADJ
ejpam-4343	127	10	)	)	PUNCT
ejpam-4343	127	11	.	.	PUNCT
ejpam-4343	128	1	then	then	ADV
ejpam-4343	128	2	,	,	PUNCT
ejpam-4343	128	3	x	x	PROPN
ejpam-4343	128	4	̸∈	̸∈	PROPN
ejpam-4343	128	5	λp(⋆)(cγ	λp(⋆)(cγ	PROPN
ejpam-4343	128	6	)	)	PUNCT
ejpam-4343	128	7	for	for	ADP
ejpam-4343	128	8	each	each	DET
ejpam-4343	128	9	γ	γ	PROPN
ejpam-4343	128	10	∈	∈	PROPN
ejpam-4343	128	11	γ	γ	NOUN
ejpam-4343	128	12	and	and	CCONJ
ejpam-4343	128	13	so	so	ADV
ejpam-4343	128	14	there	there	PRON
ejpam-4343	128	15	exists	exist	VERB
ejpam-4343	128	16	a	a	DET
ejpam-4343	128	17	pre	pre	ADJ
ejpam-4343	128	18	-	-	ADJ
ejpam-4343	128	19	i	i	PRON
ejpam-4343	128	20	-open	-open	NOUN
ejpam-4343	128	21	set	set	VERB
ejpam-4343	128	22	vγ	vγ	NOUN
ejpam-4343	128	23	such	such	ADJ
ejpam-4343	128	24	that	that	DET
ejpam-4343	128	25	cγ	cγ	NOUN
ejpam-4343	128	26	⊆	⊆	NUM
ejpam-4343	128	27	vγ	vγ	NOUN
ejpam-4343	128	28	and	and	CCONJ
ejpam-4343	128	29	x	x	PART
ejpam-4343	128	30	̸∈	̸∈	PROPN
ejpam-4343	128	31	vγ	vγ	NOUN
ejpam-4343	128	32	for	for	ADP
ejpam-4343	128	33	each	each	DET
ejpam-4343	128	34	γ	γ	PROPN
ejpam-4343	128	35	∈	∈	PROPN
ejpam-4343	128	36	γ	γ	NOUN
ejpam-4343	128	37	.	.	PUNCT
ejpam-4343	129	1	thus	thus	ADV
ejpam-4343	129	2	,	,	PUNCT
ejpam-4343	129	3	∪γ∈γcγ	∪γ∈γcγ	PROPN
ejpam-4343	129	4	⊆	⊆	NUM
ejpam-4343	129	5	∪γ∈γvγ	∪γ∈γvγ	ADJ
ejpam-4343	129	6	and	and	CCONJ
ejpam-4343	129	7	hence	hence	ADV
ejpam-4343	129	8	∪γ∈γvγ	∪γ∈γvγ	ADJ
ejpam-4343	129	9	is	be	AUX
ejpam-4343	129	10	a	a	DET
ejpam-4343	129	11	pre	pre	ADJ
ejpam-4343	129	12	-	-	ADJ
ejpam-4343	129	13	i	i	PRON
ejpam-4343	129	14	-open	-open	NOUN
ejpam-4343	129	15	set	set	VERB
ejpam-4343	129	16	which	which	PRON
ejpam-4343	129	17	does	do	AUX
ejpam-4343	129	18	not	not	PART
ejpam-4343	129	19	contain	contain	VERB
ejpam-4343	129	20	x.	x.	NOUN
ejpam-4343	130	1	this	this	PRON
ejpam-4343	130	2	implies	imply	VERB
ejpam-4343	130	3	that	that	SCONJ
ejpam-4343	130	4	x	x	PROPN
ejpam-4343	130	5	̸∈	̸∈	PROPN
ejpam-4343	130	6	λp(⋆)(∪γ∈γcγ	λp(⋆)(∪γ∈γcγ	PROPN
ejpam-4343	130	7	)	)	PUNCT
ejpam-4343	130	8	.	.	PUNCT
ejpam-4343	131	1	therefore	therefore	ADV
ejpam-4343	131	2	,	,	PUNCT
ejpam-4343	131	3	λp(⋆)(∪γ∈γcγ	λp(⋆)(∪γ∈γcγ	PROPN
ejpam-4343	131	4	)	)	PUNCT
ejpam-4343	131	5	⊆	⊆	NUM
ejpam-4343	131	6	∪γ∈γλp(⋆)(cγ	∪γ∈γλp(⋆)(cγ	NUM
ejpam-4343	131	7	)	)	PUNCT
ejpam-4343	131	8	.	.	PUNCT
ejpam-4343	132	1	consequently	consequently	ADV
ejpam-4343	132	2	,	,	PUNCT
ejpam-4343	132	3	we	we	PRON
ejpam-4343	132	4	obtain	obtain	VERB
ejpam-4343	132	5	λp(⋆)(∪γ∈γcγ	λp(⋆)(∪γ∈γcγ	NOUN
ejpam-4343	132	6	)	)	PUNCT
ejpam-4343	132	7	=	=	PUNCT
ejpam-4343	132	8	∪γ∈γλp(⋆)(cγ	∪γ∈γλp(⋆)(cγ	ADJ
ejpam-4343	132	9	)	)	PUNCT
ejpam-4343	132	10	.	.	PUNCT
ejpam-4343	133	1	(	(	PUNCT
ejpam-4343	133	2	6	6	X
ejpam-4343	133	3	)	)	PUNCT
ejpam-4343	133	4	suppose	suppose	VERB
ejpam-4343	133	5	that	that	SCONJ
ejpam-4343	133	6	x	x	PROPN
ejpam-4343	133	7	̸∈	̸∈	PROPN
ejpam-4343	133	8	∩γ∈γλp(⋆)(cγ	∩γ∈γλp(⋆)(cγ	PROPN
ejpam-4343	133	9	)	)	PUNCT
ejpam-4343	133	10	.	.	PUNCT
ejpam-4343	134	1	there	there	PRON
ejpam-4343	134	2	exists	exist	VERB
ejpam-4343	134	3	γ0	γ0	NOUN
ejpam-4343	134	4	∈	∈	PROPN
ejpam-4343	134	5	γ	γ	NOUN
ejpam-4343	134	6	such	such	ADJ
ejpam-4343	134	7	that	that	SCONJ
ejpam-4343	134	8	x	x	X
ejpam-4343	134	9	̸∈	̸∈	PROPN
ejpam-4343	134	10	λp(⋆)(cγ0	λp(⋆)(cγ0	PROPN
ejpam-4343	134	11	)	)	PUNCT
ejpam-4343	134	12	and	and	CCONJ
ejpam-4343	134	13	there	there	PRON
ejpam-4343	134	14	exists	exist	VERB
ejpam-4343	134	15	a	a	DET
ejpam-4343	134	16	pre	pre	ADJ
ejpam-4343	134	17	-	-	ADJ
ejpam-4343	134	18	i	i	PRON
ejpam-4343	134	19	-open	-open	NOUN
ejpam-4343	134	20	set	set	VERB
ejpam-4343	134	21	v	v	ADP
ejpam-4343	134	22	such	such	ADJ
ejpam-4343	134	23	that	that	SCONJ
ejpam-4343	134	24	x	x	PART
ejpam-4343	134	25	̸∈	̸∈	PROPN
ejpam-4343	134	26	v	v	NOUN
ejpam-4343	134	27	and	and	CCONJ
ejpam-4343	134	28	cγ0	cγ0	VERB
ejpam-4343	134	29	⊆	⊆	NUM
ejpam-4343	134	30	v	v	NOUN
ejpam-4343	134	31	.	.	PUNCT
ejpam-4343	135	1	therefore	therefore	ADV
ejpam-4343	135	2	,	,	PUNCT
ejpam-4343	135	3	∩γ∈γcγ	∩γ∈γcγ	PROPN
ejpam-4343	135	4	⊆	⊆	NUM
ejpam-4343	135	5	cγ0	cγ0	NOUN
ejpam-4343	135	6	⊆	⊆	NUM
ejpam-4343	135	7	v.	v.	ADP
ejpam-4343	135	8	thus	thus	ADV
ejpam-4343	135	9	,	,	PUNCT
ejpam-4343	135	10	x	x	PROPN
ejpam-4343	135	11	̸∈	̸∈	PROPN
ejpam-4343	135	12	λp(⋆)(∩γ∈γcγ	λp(⋆)(∩γ∈γcγ	PROPN
ejpam-4343	135	13	)	)	PUNCT
ejpam-4343	135	14	and	and	CCONJ
ejpam-4343	135	15	hence	hence	ADV
ejpam-4343	135	16	λp(⋆)(∩γ∈γcγ	λp(⋆)(∩γ∈γcγ	PROPN
ejpam-4343	135	17	)	)	PUNCT
ejpam-4343	135	18	⊆	⊆	NUM
ejpam-4343	135	19	∩γ∈γλp(⋆)(cγ	∩γ∈γλp(⋆)(cγ	NOUN
ejpam-4343	135	20	)	)	PUNCT
ejpam-4343	135	21	.	.	PUNCT
ejpam-4343	136	1	remark	remark	PROPN
ejpam-4343	136	2	1	1	NUM
ejpam-4343	136	3	.	.	PUNCT
ejpam-4343	137	1	in	in	ADP
ejpam-4343	137	2	proposition	proposition	NOUN
ejpam-4343	137	3	1(6	1(6	NUM
ejpam-4343	137	4	)	)	PUNCT
ejpam-4343	137	5	,	,	PUNCT
ejpam-4343	137	6	the	the	DET
ejpam-4343	137	7	converse	converse	NOUN
ejpam-4343	137	8	is	be	AUX
ejpam-4343	137	9	not	not	PART
ejpam-4343	137	10	always	always	ADV
ejpam-4343	137	11	true	true	ADJ
ejpam-4343	137	12	as	as	SCONJ
ejpam-4343	137	13	the	the	DET
ejpam-4343	137	14	following	follow	VERB
ejpam-4343	137	15	example	example	NOUN
ejpam-4343	137	16	shows	show	NOUN
ejpam-4343	137	17	.	.	PUNCT
ejpam-4343	138	1	example	example	NOUN
ejpam-4343	139	1	1	1	NUM
ejpam-4343	139	2	.	.	PUNCT
ejpam-4343	139	3	let	let	VERB
ejpam-4343	139	4	x	x	PUNCT
ejpam-4343	139	5	=	=	PRON
ejpam-4343	139	6	{	{	PUNCT
ejpam-4343	139	7	−1	−1	NOUN
ejpam-4343	139	8	,	,	PUNCT
ejpam-4343	139	9	1	1	NUM
ejpam-4343	139	10	}	}	PUNCT
ejpam-4343	139	11	with	with	ADP
ejpam-4343	139	12	a	a	DET
ejpam-4343	139	13	topology	topology	NOUN
ejpam-4343	139	14	τ	τ	X
ejpam-4343	139	15	=	=	SYM
ejpam-4343	139	16	{	{	PUNCT
ejpam-4343	139	17	∅	∅	NOUN
ejpam-4343	139	18	,	,	PUNCT
ejpam-4343	139	19	{	{	PUNCT
ejpam-4343	139	20	−1	−1	NOUN
ejpam-4343	139	21	}	}	PUNCT
ejpam-4343	139	22	,	,	PUNCT
ejpam-4343	139	23	x	x	X
ejpam-4343	139	24	}	}	PUNCT
ejpam-4343	139	25	and	and	CCONJ
ejpam-4343	139	26	an	an	DET
ejpam-4343	139	27	ideal	ideal	NOUN
ejpam-4343	140	1	i	i	X
ejpam-4343	140	2	=	=	SYM
ejpam-4343	140	3	{	{	PUNCT
ejpam-4343	140	4	∅	∅	NOUN
ejpam-4343	140	5	,	,	PUNCT
ejpam-4343	140	6	{	{	PUNCT
ejpam-4343	140	7	1	1	NUM
ejpam-4343	140	8	}	}	PUNCT
ejpam-4343	140	9	}	}	PUNCT
ejpam-4343	140	10	.	.	PUNCT
ejpam-4343	141	1	let	let	VERB
ejpam-4343	141	2	a	a	PRON
ejpam-4343	141	3	=	=	X
ejpam-4343	141	4	{	{	PUNCT
ejpam-4343	141	5	−1	−1	NOUN
ejpam-4343	141	6	}	}	PUNCT
ejpam-4343	141	7	and	and	CCONJ
ejpam-4343	141	8	b	b	X
ejpam-4343	141	9	=	=	PUNCT
ejpam-4343	141	10	{	{	PUNCT
ejpam-4343	141	11	1	1	NUM
ejpam-4343	141	12	}	}	PUNCT
ejpam-4343	141	13	.	.	PUNCT
ejpam-4343	142	1	then	then	ADV
ejpam-4343	142	2	,	,	PUNCT
ejpam-4343	142	3	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	142	4	∩b	∩b	PROPN
ejpam-4343	142	5	)	)	PUNCT
ejpam-4343	142	6	=	=	SYM
ejpam-4343	142	7	λp(⋆)(∅	λp(⋆)(∅	NOUN
ejpam-4343	142	8	)	)	PUNCT
ejpam-4343	142	9	=	=	NOUN
ejpam-4343	142	10	∅	∅	NOUN
ejpam-4343	142	11	and	and	CCONJ
ejpam-4343	142	12	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	142	13	)	)	PUNCT
ejpam-4343	142	14	∩	∩	NOUN
ejpam-4343	142	15	λp(⋆)(b	λp(⋆)(b	PROPN
ejpam-4343	142	16	)	)	PUNCT
ejpam-4343	142	17	=	=	PRON
ejpam-4343	142	18	{	{	PUNCT
ejpam-4343	142	19	−1	−1	NOUN
ejpam-4343	142	20	}	}	PUNCT
ejpam-4343	142	21	.	.	PUNCT
ejpam-4343	143	1	c.	c.	PROPN
ejpam-4343	143	2	boonpok	boonpok	PROPN
ejpam-4343	143	3	/	/	SYM
ejpam-4343	143	4	eur	eur	PROPN
ejpam-4343	143	5	.	.	PUNCT
ejpam-4343	144	1	j.	j.	PROPN
ejpam-4343	144	2	pure	pure	PROPN
ejpam-4343	144	3	appl	appl	PROPN
ejpam-4343	144	4	.	.	PROPN
ejpam-4343	144	5	math	math	PROPN
ejpam-4343	144	6	,	,	PUNCT
ejpam-4343	144	7	15	15	NUM
ejpam-4343	144	8	(	(	PUNCT
ejpam-4343	144	9	3	3	NUM
ejpam-4343	144	10	)	)	PUNCT
ejpam-4343	144	11	(	(	PUNCT
ejpam-4343	144	12	2022	2022	NUM
ejpam-4343	144	13	)	)	PUNCT
ejpam-4343	144	14	,	,	PUNCT
ejpam-4343	144	15	1023	1023	NUM
ejpam-4343	144	16	-	-	SYM
ejpam-4343	144	17	1046	1046	NUM
ejpam-4343	144	18	1027	1027	NUM
ejpam-4343	144	19	definition	definition	NOUN
ejpam-4343	144	20	2	2	NUM
ejpam-4343	144	21	.	.	PUNCT
ejpam-4343	145	1	a	a	DET
ejpam-4343	145	2	subset	subset	NOUN
ejpam-4343	145	3	a	a	PRON
ejpam-4343	145	4	of	of	ADP
ejpam-4343	145	5	an	an	DET
ejpam-4343	145	6	ideal	ideal	ADJ
ejpam-4343	145	7	topological	topological	ADJ
ejpam-4343	145	8	space	space	NOUN
ejpam-4343	145	9	(	(	PUNCT
ejpam-4343	145	10	x	x	X
ejpam-4343	145	11	,	,	PUNCT
ejpam-4343	145	12	τ	τ	PROPN
ejpam-4343	145	13	,	,	PUNCT
ejpam-4343	145	14	i	i	PROPN
ejpam-4343	145	15	)	)	PUNCT
ejpam-4343	145	16	is	be	AUX
ejpam-4343	145	17	called	call	VERB
ejpam-4343	145	18	a	a	DET
ejpam-4343	145	19	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	145	20	if	if	SCONJ
ejpam-4343	145	21	a	a	DET
ejpam-4343	145	22	=	=	PUNCT
ejpam-4343	145	23	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	145	24	)	)	PUNCT
ejpam-4343	145	25	.	.	PUNCT
ejpam-4343	146	1	the	the	DET
ejpam-4343	146	2	family	family	NOUN
ejpam-4343	146	3	of	of	ADP
ejpam-4343	146	4	all	all	DET
ejpam-4343	146	5	λp(⋆)-sets	λp(⋆)-sets	PRON
ejpam-4343	146	6	of	of	ADP
ejpam-4343	146	7	an	an	DET
ejpam-4343	146	8	ideal	ideal	ADJ
ejpam-4343	146	9	topological	topological	ADJ
ejpam-4343	146	10	space	space	NOUN
ejpam-4343	146	11	(	(	PUNCT
ejpam-4343	146	12	x	x	X
ejpam-4343	146	13	,	,	PUNCT
ejpam-4343	146	14	τ	τ	PROPN
ejpam-4343	146	15	,	,	PUNCT
ejpam-4343	146	16	i	i	PROPN
ejpam-4343	146	17	)	)	PUNCT
ejpam-4343	146	18	is	be	AUX
ejpam-4343	146	19	denoted	denote	VERB
ejpam-4343	146	20	by	by	ADP
ejpam-4343	146	21	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	146	22	)	)	PUNCT
ejpam-4343	146	23	.	.	PUNCT
ejpam-4343	147	1	proposition	proposition	NOUN
ejpam-4343	147	2	2	2	NUM
ejpam-4343	147	3	.	.	X
ejpam-4343	147	4	for	for	ADP
ejpam-4343	147	5	subsets	subset	NOUN
ejpam-4343	147	6	a	a	PRON
ejpam-4343	147	7	and	and	CCONJ
ejpam-4343	147	8	bγ(γ	bγ(γ	NOUN
ejpam-4343	147	9	∈	∈	PROPN
ejpam-4343	147	10	γ	γ	PROPN
ejpam-4343	147	11	)	)	PUNCT
ejpam-4343	147	12	of	of	ADP
ejpam-4343	147	13	an	an	DET
ejpam-4343	147	14	ideal	ideal	ADJ
ejpam-4343	147	15	topological	topological	ADJ
ejpam-4343	147	16	space	space	NOUN
ejpam-4343	147	17	(	(	PUNCT
ejpam-4343	147	18	x	x	X
ejpam-4343	147	19	,	,	PUNCT
ejpam-4343	147	20	τ	τ	PROPN
ejpam-4343	147	21	,	,	PUNCT
ejpam-4343	147	22	i	i	NOUN
ejpam-4343	147	23	)	)	PUNCT
ejpam-4343	147	24	,	,	PUNCT
ejpam-4343	147	25	the	the	DET
ejpam-4343	147	26	following	follow	VERB
ejpam-4343	147	27	properties	property	NOUN
ejpam-4343	147	28	hold	hold	VERB
ejpam-4343	147	29	:	:	PUNCT
ejpam-4343	147	30	(	(	PUNCT
ejpam-4343	147	31	1	1	X
ejpam-4343	147	32	)	)	PUNCT
ejpam-4343	147	33	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	147	34	)	)	PUNCT
ejpam-4343	147	35	is	be	AUX
ejpam-4343	147	36	a	a	DET
ejpam-4343	147	37	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	147	38	.	.	PUNCT
ejpam-4343	148	1	(	(	PUNCT
ejpam-4343	148	2	2	2	X
ejpam-4343	148	3	)	)	PUNCT
ejpam-4343	148	4	if	if	SCONJ
ejpam-4343	148	5	a	a	PRON
ejpam-4343	148	6	is	be	AUX
ejpam-4343	148	7	a	a	DET
ejpam-4343	148	8	pre	pre	ADJ
ejpam-4343	148	9	-	-	ADJ
ejpam-4343	148	10	i	i	PRON
ejpam-4343	148	11	-open	-open	NOUN
ejpam-4343	148	12	set	set	NOUN
ejpam-4343	148	13	,	,	PUNCT
ejpam-4343	148	14	then	then	ADV
ejpam-4343	148	15	a	a	PRON
ejpam-4343	148	16	is	be	AUX
ejpam-4343	148	17	a	a	DET
ejpam-4343	148	18	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	148	19	.	.	PUNCT
ejpam-4343	149	1	(	(	PUNCT
ejpam-4343	149	2	3	3	X
ejpam-4343	149	3	)	)	PUNCT
ejpam-4343	149	4	if	if	SCONJ
ejpam-4343	149	5	bγ	bγ	PRON
ejpam-4343	149	6	is	be	AUX
ejpam-4343	149	7	a	a	DET
ejpam-4343	149	8	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	149	9	for	for	ADP
ejpam-4343	149	10	each	each	DET
ejpam-4343	149	11	γ	γ	PROPN
ejpam-4343	149	12	∈	∈	PROPN
ejpam-4343	149	13	γ	γ	X
ejpam-4343	149	14	,	,	PUNCT
ejpam-4343	149	15	then	then	ADV
ejpam-4343	149	16	∪γ∈γbγ	∪γ∈γbγ	PROPN
ejpam-4343	149	17	is	be	AUX
ejpam-4343	149	18	a	a	DET
ejpam-4343	149	19	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	149	20	.	.	PUNCT
ejpam-4343	150	1	(	(	PUNCT
ejpam-4343	150	2	4	4	X
ejpam-4343	150	3	)	)	PUNCT
ejpam-4343	150	4	if	if	SCONJ
ejpam-4343	150	5	bγ	bγ	PRON
ejpam-4343	150	6	is	be	AUX
ejpam-4343	150	7	a	a	DET
ejpam-4343	150	8	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	150	9	for	for	ADP
ejpam-4343	150	10	each	each	DET
ejpam-4343	150	11	γ	γ	PROPN
ejpam-4343	150	12	∈	∈	PROPN
ejpam-4343	150	13	γ	γ	X
ejpam-4343	150	14	,	,	PUNCT
ejpam-4343	150	15	then	then	ADV
ejpam-4343	150	16	∩γ∈γbγ	∩γ∈γbγ	X
ejpam-4343	150	17	is	be	AUX
ejpam-4343	150	18	a	a	DET
ejpam-4343	150	19	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	150	20	.	.	PUNCT
ejpam-4343	151	1	proof	proof	NOUN
ejpam-4343	151	2	.	.	PUNCT
ejpam-4343	152	1	(	(	PUNCT
ejpam-4343	152	2	1	1	X
ejpam-4343	152	3	)	)	PUNCT
ejpam-4343	152	4	and	and	CCONJ
ejpam-4343	152	5	(	(	PUNCT
ejpam-4343	152	6	2	2	X
ejpam-4343	152	7	)	)	PUNCT
ejpam-4343	152	8	are	be	AUX
ejpam-4343	152	9	obvious	obvious	ADJ
ejpam-4343	152	10	.	.	PUNCT
ejpam-4343	153	1	(	(	PUNCT
ejpam-4343	153	2	3	3	X
ejpam-4343	153	3	)	)	PUNCT
ejpam-4343	153	4	let	let	VERB
ejpam-4343	153	5	bγ	bγ	PRON
ejpam-4343	153	6	be	be	AUX
ejpam-4343	153	7	a	a	DET
ejpam-4343	153	8	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	153	9	for	for	ADP
ejpam-4343	153	10	each	each	DET
ejpam-4343	153	11	γ	γ	PROPN
ejpam-4343	153	12	∈	∈	PROPN
ejpam-4343	153	13	γ	γ	X
ejpam-4343	153	14	.	.	PUNCT
ejpam-4343	154	1	then	then	ADV
ejpam-4343	154	2	,	,	PUNCT
ejpam-4343	154	3	by	by	ADP
ejpam-4343	154	4	proposition	proposition	NOUN
ejpam-4343	154	5	1(5	1(5	NUM
ejpam-4343	154	6	)	)	PUNCT
ejpam-4343	154	7	,	,	PUNCT
ejpam-4343	154	8	we	we	PRON
ejpam-4343	154	9	have	have	VERB
ejpam-4343	154	10	∪γ∈γbγ	∪γ∈γbγ	PROPN
ejpam-4343	154	11	=	=	SYM
ejpam-4343	154	12	∪γ∈γλp(⋆)(bγ	∪γ∈γλp(⋆)(bγ	X
ejpam-4343	154	13	)	)	PUNCT
ejpam-4343	154	14	=	=	SYM
ejpam-4343	154	15	λp(⋆)(∪γ∈γbγ	λp(⋆)(∪γ∈γbγ	PRON
ejpam-4343	154	16	)	)	PUNCT
ejpam-4343	154	17	⊇	⊇	PROPN
ejpam-4343	154	18	∪γ∈γbγ	∪γ∈γbγ	PROPN
ejpam-4343	154	19	.	.	PUNCT
ejpam-4343	155	1	thus	thus	ADV
ejpam-4343	155	2	,	,	PUNCT
ejpam-4343	155	3	∪γ∈γbγ	∪γ∈γbγ	PROPN
ejpam-4343	155	4	=	=	PUNCT
ejpam-4343	155	5	λp(⋆)(∪γ∈γbγ	λp(⋆)(∪γ∈γbγ	PROPN
ejpam-4343	155	6	)	)	PUNCT
ejpam-4343	155	7	and	and	CCONJ
ejpam-4343	155	8	hence	hence	ADV
ejpam-4343	155	9	∪γ∈γbγ	∪γ∈γbγ	PROPN
ejpam-4343	155	10	is	be	AUX
ejpam-4343	155	11	a	a	DET
ejpam-4343	155	12	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	155	13	.	.	PUNCT
ejpam-4343	156	1	(	(	PUNCT
ejpam-4343	156	2	4	4	X
ejpam-4343	156	3	)	)	PUNCT
ejpam-4343	156	4	let	let	VERB
ejpam-4343	156	5	bγ	bγ	PRON
ejpam-4343	156	6	be	be	AUX
ejpam-4343	156	7	a	a	DET
ejpam-4343	156	8	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	156	9	for	for	ADP
ejpam-4343	156	10	each	each	DET
ejpam-4343	156	11	γ	γ	PROPN
ejpam-4343	156	12	∈	∈	PROPN
ejpam-4343	156	13	γ	γ	NOUN
ejpam-4343	156	14	.	.	PUNCT
ejpam-4343	156	15	thus	thus	ADV
ejpam-4343	156	16	,	,	PUNCT
ejpam-4343	156	17	by	by	ADP
ejpam-4343	156	18	proposition	proposition	NOUN
ejpam-4343	156	19	1(6	1(6	NUM
ejpam-4343	156	20	)	)	PUNCT
ejpam-4343	156	21	,	,	PUNCT
ejpam-4343	156	22	∩γ∈γbγ	∩γ∈γbγ	X
ejpam-4343	156	23	=	=	SYM
ejpam-4343	156	24	∩γ∈γλp(⋆)(bγ	∩γ∈γλp(⋆)(bγ	PROPN
ejpam-4343	156	25	)	)	PUNCT
ejpam-4343	156	26	⊇	⊇	PROPN
ejpam-4343	156	27	λp(⋆)(∩γ∈γbγ	λp(⋆)(∩γ∈γbγ	PROPN
ejpam-4343	156	28	)	)	PUNCT
ejpam-4343	156	29	⊇	⊇	PROPN
ejpam-4343	156	30	∩γ∈γbγ	∩γ∈γbγ	X
ejpam-4343	156	31	and	and	CCONJ
ejpam-4343	156	32	hence	hence	ADV
ejpam-4343	156	33	∩γ∈γbγ	∩γ∈γbγ	VERB
ejpam-4343	156	34	=	=	PUNCT
ejpam-4343	156	35	λp(⋆)(∩γ∈γbγ	λp(⋆)(∩γ∈γbγ	PROPN
ejpam-4343	156	36	)	)	PUNCT
ejpam-4343	156	37	.	.	PUNCT
ejpam-4343	157	1	this	this	PRON
ejpam-4343	157	2	shows	show	VERB
ejpam-4343	157	3	that	that	SCONJ
ejpam-4343	157	4	∩γ∈γbγ	∩γ∈γbγ	PRON
ejpam-4343	157	5	is	be	AUX
ejpam-4343	157	6	a	a	DET
ejpam-4343	157	7	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	157	8	.	.	PUNCT
ejpam-4343	158	1	proposition	proposition	NOUN
ejpam-4343	158	2	3	3	NUM
ejpam-4343	158	3	.	.	X
ejpam-4343	159	1	for	for	ADP
ejpam-4343	159	2	an	an	DET
ejpam-4343	159	3	ideal	ideal	ADJ
ejpam-4343	159	4	topological	topological	ADJ
ejpam-4343	159	5	space	space	NOUN
ejpam-4343	159	6	(	(	PUNCT
ejpam-4343	159	7	x	x	X
ejpam-4343	159	8	,	,	PUNCT
ejpam-4343	159	9	τ	τ	PROPN
ejpam-4343	159	10	,	,	PUNCT
ejpam-4343	159	11	i	i	NOUN
ejpam-4343	159	12	)	)	PUNCT
ejpam-4343	159	13	,	,	PUNCT
ejpam-4343	159	14	the	the	DET
ejpam-4343	159	15	pair	pair	NOUN
ejpam-4343	159	16	(	(	PUNCT
ejpam-4343	159	17	x	x	NOUN
ejpam-4343	159	18	,	,	PUNCT
ejpam-4343	159	19	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	159	20	)	)	PUNCT
ejpam-4343	159	21	)	)	PUNCT
ejpam-4343	159	22	is	be	AUX
ejpam-4343	159	23	an	an	DET
ejpam-4343	159	24	alexandroff	alexandroff	ADJ
ejpam-4343	159	25	space	space	NOUN
ejpam-4343	159	26	.	.	PUNCT
ejpam-4343	160	1	proof	proof	NOUN
ejpam-4343	160	2	.	.	PUNCT
ejpam-4343	161	1	(	(	PUNCT
ejpam-4343	161	2	1	1	X
ejpam-4343	161	3	)	)	PUNCT
ejpam-4343	161	4	∅	∅	NOUN
ejpam-4343	161	5	,	,	PUNCT
ejpam-4343	161	6	x	x	PROPN
ejpam-4343	161	7	∈	∈	PROPN
ejpam-4343	161	8	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	161	9	)	)	PUNCT
ejpam-4343	161	10	since	since	SCONJ
ejpam-4343	161	11	∅	∅	NOUN
ejpam-4343	161	12	,	,	PUNCT
ejpam-4343	161	13	x	x	PROPN
ejpam-4343	161	14	∈	∈	PROPN
ejpam-4343	161	15	pio(x	pio(x	PROPN
ejpam-4343	161	16	,	,	PUNCT
ejpam-4343	161	17	τ	τ	PROPN
ejpam-4343	161	18	)	)	PUNCT
ejpam-4343	161	19	and	and	CCONJ
ejpam-4343	161	20	pio(x	pio(x	PROPN
ejpam-4343	161	21	,	,	PUNCT
ejpam-4343	161	22	τ	τ	PROPN
ejpam-4343	161	23	)	)	PUNCT
ejpam-4343	161	24	⊆	⊆	NUM
ejpam-4343	161	25	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	161	26	)	)	PUNCT
ejpam-4343	161	27	.	.	PUNCT
ejpam-4343	162	1	(	(	PUNCT
ejpam-4343	162	2	2	2	X
ejpam-4343	162	3	)	)	PUNCT
ejpam-4343	162	4	let	let	VERB
ejpam-4343	162	5	vγ	vγ	NOUN
ejpam-4343	162	6	∈	∈	PROPN
ejpam-4343	162	7	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	162	8	)	)	PUNCT
ejpam-4343	162	9	for	for	ADP
ejpam-4343	162	10	each	each	DET
ejpam-4343	162	11	γ	γ	PROPN
ejpam-4343	162	12	∈	∈	PROPN
ejpam-4343	162	13	γ	γ	X
ejpam-4343	162	14	.	.	PUNCT
ejpam-4343	163	1	then	then	ADV
ejpam-4343	163	2	,	,	PUNCT
ejpam-4343	163	3	we	we	PRON
ejpam-4343	163	4	have	have	VERB
ejpam-4343	163	5	∪γ∈γ	∪γ∈γ	PROPN
ejpam-4343	163	6	vγ	vγ	NOUN
ejpam-4343	163	7	∈	∈	PROPN
ejpam-4343	163	8	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	163	9	)	)	PUNCT
ejpam-4343	163	10	by	by	ADP
ejpam-4343	163	11	proposition	proposition	NOUN
ejpam-4343	163	12	2(3	2(3	NUM
ejpam-4343	163	13	)	)	PUNCT
ejpam-4343	163	14	.	.	PUNCT
ejpam-4343	164	1	(	(	PUNCT
ejpam-4343	164	2	3	3	X
ejpam-4343	164	3	)	)	PUNCT
ejpam-4343	164	4	let	let	VERB
ejpam-4343	164	5	vγ	vγ	NOUN
ejpam-4343	164	6	∈	∈	PROPN
ejpam-4343	164	7	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	164	8	)	)	PUNCT
ejpam-4343	164	9	for	for	ADP
ejpam-4343	164	10	each	each	DET
ejpam-4343	164	11	γ	γ	PROPN
ejpam-4343	164	12	∈	∈	PROPN
ejpam-4343	164	13	γ	γ	X
ejpam-4343	164	14	.	.	PUNCT
ejpam-4343	165	1	then	then	ADV
ejpam-4343	165	2	,	,	PUNCT
ejpam-4343	165	3	we	we	PRON
ejpam-4343	165	4	have	have	VERB
ejpam-4343	165	5	∩γ∈γ	∩γ∈γ	ADJ
ejpam-4343	165	6	vγ	vγ	PROPN
ejpam-4343	165	7	∈	∈	PROPN
ejpam-4343	165	8	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	165	9	)	)	PUNCT
ejpam-4343	165	10	by	by	ADP
ejpam-4343	165	11	proposition	proposition	NOUN
ejpam-4343	165	12	2(4	2(4	NUM
ejpam-4343	165	13	)	)	PUNCT
ejpam-4343	165	14	.	.	PUNCT
ejpam-4343	166	1	proposition	proposition	NOUN
ejpam-4343	166	2	4	4	NUM
ejpam-4343	166	3	.	.	PUNCT
ejpam-4343	167	1	let	let	AUX
ejpam-4343	167	2	(	(	PUNCT
ejpam-4343	167	3	x	x	X
ejpam-4343	167	4	,	,	PUNCT
ejpam-4343	167	5	τ	τ	PROPN
ejpam-4343	167	6	,	,	PUNCT
ejpam-4343	167	7	i	i	PRON
ejpam-4343	167	8	)	)	PUNCT
ejpam-4343	167	9	be	be	AUX
ejpam-4343	167	10	an	an	DET
ejpam-4343	167	11	ideal	ideal	ADJ
ejpam-4343	167	12	topological	topological	ADJ
ejpam-4343	167	13	space	space	NOUN
ejpam-4343	167	14	.	.	PUNCT
ejpam-4343	168	1	then	then	ADV
ejpam-4343	168	2	,	,	PUNCT
ejpam-4343	168	3	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	168	4	)	)	PUNCT
ejpam-4343	168	5	=	=	SYM
ejpam-4343	168	6	λλp(⋆	λλp(⋆	X
ejpam-4343	168	7	)	)	PUNCT
ejpam-4343	168	8	(	(	PUNCT
ejpam-4343	168	9	x	x	NOUN
ejpam-4343	168	10	)	)	PUNCT
ejpam-4343	168	11	.	.	PUNCT
ejpam-4343	169	1	proof	proof	NOUN
ejpam-4343	169	2	.	.	PUNCT
ejpam-4343	170	1	by	by	ADP
ejpam-4343	170	2	proposition	proposition	NOUN
ejpam-4343	170	3	2(2	2(2	NUM
ejpam-4343	170	4	)	)	PUNCT
ejpam-4343	170	5	,	,	PUNCT
ejpam-4343	170	6	pi	pi	NOUN
ejpam-4343	170	7	o(x	o(x	PROPN
ejpam-4343	170	8	)	)	PUNCT
ejpam-4343	170	9	⊆	⊆	NUM
ejpam-4343	170	10	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	170	11	)	)	PUNCT
ejpam-4343	170	12	.	.	PUNCT
ejpam-4343	171	1	for	for	ADP
ejpam-4343	171	2	any	any	DET
ejpam-4343	171	3	subset	subset	NOUN
ejpam-4343	171	4	a	a	PRON
ejpam-4343	171	5	of	of	ADP
ejpam-4343	171	6	x	x	PRON
ejpam-4343	171	7	,	,	PUNCT
ejpam-4343	171	8	we	we	PRON
ejpam-4343	171	9	have	have	VERB
ejpam-4343	171	10	λλp(⋆	λλp(⋆	NOUN
ejpam-4343	171	11	)	)	PUNCT
ejpam-4343	172	1	(	(	PUNCT
ejpam-4343	172	2	a	a	X
ejpam-4343	172	3	)	)	PUNCT
ejpam-4343	172	4	=	=	SYM
ejpam-4343	173	1	∩{u	∩{u	PROPN
ejpam-4343	173	2	|	|	ADV
ejpam-4343	173	3	a	a	DET
ejpam-4343	173	4	⊆	⊆	NUM
ejpam-4343	173	5	u	u	NOUN
ejpam-4343	173	6	;	;	PUNCT
ejpam-4343	173	7	u	u	PROPN
ejpam-4343	173	8	∈	∈	PROPN
ejpam-4343	173	9	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	173	10	)	)	PUNCT
ejpam-4343	173	11	}	}	PUNCT
ejpam-4343	174	1	⊆	⊆	NUM
ejpam-4343	174	2	∩{u	∩{u	NUM
ejpam-4343	174	3	|	|	ADV
ejpam-4343	174	4	a	a	DET
ejpam-4343	174	5	⊆	⊆	NUM
ejpam-4343	174	6	u	u	NOUN
ejpam-4343	174	7	;	;	PUNCT
ejpam-4343	174	8	u	u	PROPN
ejpam-4343	174	9	∈	∈	PROPN
ejpam-4343	174	10	pi	pi	NOUN
ejpam-4343	174	11	o(x	o(x	PROPN
ejpam-4343	174	12	)	)	PUNCT
ejpam-4343	174	13	}	}	PUNCT
ejpam-4343	174	14	=	=	SYM
ejpam-4343	174	15	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	174	16	)	)	PUNCT
ejpam-4343	174	17	and	and	CCONJ
ejpam-4343	174	18	hence	hence	ADV
ejpam-4343	174	19	λλp(⋆	λλp(⋆	NUM
ejpam-4343	174	20	)	)	PUNCT
ejpam-4343	174	21	(	(	PUNCT
ejpam-4343	174	22	a	a	X
ejpam-4343	174	23	)	)	PUNCT
ejpam-4343	174	24	⊆	⊆	NUM
ejpam-4343	174	25	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	174	26	)	)	PUNCT
ejpam-4343	174	27	.	.	PUNCT
ejpam-4343	175	1	on	on	ADP
ejpam-4343	175	2	the	the	DET
ejpam-4343	175	3	other	other	ADJ
ejpam-4343	175	4	hand	hand	NOUN
ejpam-4343	175	5	,	,	PUNCT
ejpam-4343	175	6	suppose	suppose	VERB
ejpam-4343	175	7	that	that	SCONJ
ejpam-4343	175	8	x	x	PROPN
ejpam-4343	175	9	̸∈	̸∈	PROPN
ejpam-4343	175	10	λλp(⋆	λλp(⋆	PROPN
ejpam-4343	175	11	)	)	PUNCT
ejpam-4343	175	12	(	(	PUNCT
ejpam-4343	175	13	a	a	NOUN
ejpam-4343	175	14	)	)	PUNCT
ejpam-4343	175	15	.	.	PUNCT
ejpam-4343	176	1	then	then	ADV
ejpam-4343	176	2	,	,	PUNCT
ejpam-4343	176	3	there	there	PRON
ejpam-4343	176	4	exists	exist	VERB
ejpam-4343	176	5	u	u	PROPN
ejpam-4343	176	6	∈	∈	PROPN
ejpam-4343	176	7	λp(⋆)(x	λp(⋆)(x	PROPN
ejpam-4343	176	8	)	)	PUNCT
ejpam-4343	176	9	such	such	ADJ
ejpam-4343	176	10	that	that	SCONJ
ejpam-4343	176	11	a	a	DET
ejpam-4343	176	12	⊆	⊆	NUM
ejpam-4343	176	13	u	u	NOUN
ejpam-4343	176	14	and	and	CCONJ
ejpam-4343	176	15	x	x	PUNCT
ejpam-4343	176	16	̸∈	̸∈	PROPN
ejpam-4343	176	17	u	u	PROPN
ejpam-4343	176	18	.	.	PUNCT
ejpam-4343	177	1	since	since	SCONJ
ejpam-4343	177	2	x	x	PROPN
ejpam-4343	177	3	̸∈	̸∈	PROPN
ejpam-4343	177	4	u	u	PROPN
ejpam-4343	177	5	,	,	PUNCT
ejpam-4343	177	6	there	there	PRON
ejpam-4343	177	7	exists	exist	VERB
ejpam-4343	177	8	a	a	DET
ejpam-4343	177	9	pre	pre	ADJ
ejpam-4343	177	10	-	-	ADJ
ejpam-4343	177	11	i	i	PRON
ejpam-4343	177	12	-open	-open	NOUN
ejpam-4343	177	13	set	set	VERB
ejpam-4343	177	14	v	v	ADP
ejpam-4343	177	15	such	such	ADJ
ejpam-4343	177	16	that	that	DET
ejpam-4343	177	17	u	u	PROPN
ejpam-4343	177	18	⊆	⊆	NUM
ejpam-4343	177	19	v	v	NOUN
ejpam-4343	177	20	and	and	CCONJ
ejpam-4343	177	21	x	x	PART
ejpam-4343	177	22	̸∈	̸∈	PROPN
ejpam-4343	177	23	v	v	NUM
ejpam-4343	177	24	.	.	PUNCT
ejpam-4343	178	1	thus	thus	ADV
ejpam-4343	178	2	,	,	PUNCT
ejpam-4343	178	3	x	x	PROPN
ejpam-4343	178	4	̸∈	̸∈	PROPN
ejpam-4343	178	5	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	178	6	)	)	PUNCT
ejpam-4343	178	7	and	and	CCONJ
ejpam-4343	178	8	hence	hence	ADV
ejpam-4343	178	9	λλp(⋆	λλp(⋆	NUM
ejpam-4343	178	10	)	)	PUNCT
ejpam-4343	178	11	(	(	PUNCT
ejpam-4343	178	12	a	a	X
ejpam-4343	178	13	)	)	PUNCT
ejpam-4343	178	14	⊇	⊇	NOUN
ejpam-4343	178	15	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	178	16	)	)	PUNCT
ejpam-4343	178	17	.	.	PUNCT
ejpam-4343	179	1	consequently	consequently	ADV
ejpam-4343	179	2	,	,	PUNCT
ejpam-4343	179	3	we	we	PRON
ejpam-4343	179	4	obtain	obtain	VERB
ejpam-4343	179	5	λλp(⋆	λλp(⋆	NOUN
ejpam-4343	179	6	)	)	PUNCT
ejpam-4343	179	7	(	(	PUNCT
ejpam-4343	179	8	a	a	X
ejpam-4343	179	9	)	)	PUNCT
ejpam-4343	179	10	=	=	SYM
ejpam-4343	179	11	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	179	12	)	)	PUNCT
ejpam-4343	179	13	.	.	PUNCT
ejpam-4343	180	1	definition	definition	NOUN
ejpam-4343	180	2	3	3	X
ejpam-4343	180	3	.	.	PUNCT
ejpam-4343	181	1	let	let	VERB
ejpam-4343	181	2	a	a	DET
ejpam-4343	181	3	be	be	AUX
ejpam-4343	181	4	a	a	DET
ejpam-4343	181	5	subset	subset	NOUN
ejpam-4343	181	6	of	of	ADP
ejpam-4343	181	7	an	an	DET
ejpam-4343	181	8	ideal	ideal	ADJ
ejpam-4343	181	9	topological	topological	ADJ
ejpam-4343	181	10	space	space	NOUN
ejpam-4343	181	11	(	(	PUNCT
ejpam-4343	181	12	x	x	X
ejpam-4343	181	13	,	,	PUNCT
ejpam-4343	181	14	τ	τ	PROPN
ejpam-4343	181	15	,	,	PUNCT
ejpam-4343	181	16	i	i	NOUN
ejpam-4343	181	17	)	)	PUNCT
ejpam-4343	181	18	.	.	PUNCT
ejpam-4343	182	1	a	a	DET
ejpam-4343	182	2	subset	subset	ADJ
ejpam-4343	182	3	δp(⋆)(a	δp(⋆)(a	NOUN
ejpam-4343	182	4	)	)	PUNCT
ejpam-4343	182	5	is	be	AUX
ejpam-4343	182	6	defined	define	VERB
ejpam-4343	182	7	as	as	SCONJ
ejpam-4343	182	8	follows	follow	VERB
ejpam-4343	182	9	:	:	PUNCT
ejpam-4343	182	10	δp(⋆)(a	δp(⋆)(a	NOUN
ejpam-4343	182	11	)	)	PUNCT
ejpam-4343	183	1	=	=	SYM
ejpam-4343	183	2	∪{f	∪{f	PROPN
ejpam-4343	183	3	|	|	ADV
ejpam-4343	183	4	f	f	NOUN
ejpam-4343	183	5	⊆	⊆	NUM
ejpam-4343	183	6	a;f	a;f	NOUN
ejpam-4343	183	7	is	be	AUX
ejpam-4343	183	8	pre	pre	ADJ
ejpam-4343	183	9	-	-	ADJ
ejpam-4343	183	10	i	i	PRON
ejpam-4343	183	11	-closed	-closed	ADJ
ejpam-4343	183	12	}	}	PUNCT
ejpam-4343	183	13	.	.	PUNCT
ejpam-4343	184	1	c.	c.	PROPN
ejpam-4343	184	2	boonpok	boonpok	PROPN
ejpam-4343	184	3	/	/	SYM
ejpam-4343	184	4	eur	eur	PROPN
ejpam-4343	184	5	.	.	PUNCT
ejpam-4343	185	1	j.	j.	PROPN
ejpam-4343	185	2	pure	pure	PROPN
ejpam-4343	185	3	appl	appl	PROPN
ejpam-4343	185	4	.	.	PROPN
ejpam-4343	185	5	math	math	PROPN
ejpam-4343	185	6	,	,	PUNCT
ejpam-4343	185	7	15	15	NUM
ejpam-4343	185	8	(	(	PUNCT
ejpam-4343	185	9	3	3	NUM
ejpam-4343	185	10	)	)	PUNCT
ejpam-4343	185	11	(	(	PUNCT
ejpam-4343	185	12	2022	2022	NUM
ejpam-4343	185	13	)	)	PUNCT
ejpam-4343	185	14	,	,	PUNCT
ejpam-4343	185	15	1023	1023	NUM
ejpam-4343	185	16	-	-	SYM
ejpam-4343	185	17	1046	1046	NUM
ejpam-4343	185	18	1028	1028	NUM
ejpam-4343	185	19	definition	definition	NOUN
ejpam-4343	185	20	4	4	NUM
ejpam-4343	185	21	.	.	PUNCT
ejpam-4343	186	1	a	a	DET
ejpam-4343	186	2	subset	subset	NOUN
ejpam-4343	186	3	a	a	PRON
ejpam-4343	186	4	of	of	ADP
ejpam-4343	186	5	an	an	DET
ejpam-4343	186	6	ideal	ideal	ADJ
ejpam-4343	186	7	topological	topological	ADJ
ejpam-4343	186	8	space	space	NOUN
ejpam-4343	186	9	(	(	PUNCT
ejpam-4343	186	10	x	x	X
ejpam-4343	186	11	,	,	PUNCT
ejpam-4343	186	12	τ	τ	PROPN
ejpam-4343	186	13	,	,	PUNCT
ejpam-4343	186	14	i	i	PROPN
ejpam-4343	186	15	)	)	PUNCT
ejpam-4343	186	16	is	be	AUX
ejpam-4343	186	17	called	call	VERB
ejpam-4343	186	18	a	a	DET
ejpam-4343	186	19	δp(⋆)-set	δp(⋆)-set	NOUN
ejpam-4343	186	20	if	if	SCONJ
ejpam-4343	186	21	a	a	DET
ejpam-4343	186	22	=	=	NOUN
ejpam-4343	186	23	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	186	24	)	)	PUNCT
ejpam-4343	186	25	.	.	PUNCT
ejpam-4343	187	1	the	the	DET
ejpam-4343	187	2	family	family	NOUN
ejpam-4343	187	3	of	of	ADP
ejpam-4343	187	4	all	all	DET
ejpam-4343	187	5	δp(⋆)-sets	δp(⋆)-set	NOUN
ejpam-4343	187	6	of	of	ADP
ejpam-4343	187	7	an	an	DET
ejpam-4343	187	8	ideal	ideal	ADJ
ejpam-4343	187	9	topological	topological	ADJ
ejpam-4343	187	10	space	space	NOUN
ejpam-4343	187	11	(	(	PUNCT
ejpam-4343	187	12	x	x	X
ejpam-4343	187	13	,	,	PUNCT
ejpam-4343	187	14	τ	τ	PROPN
ejpam-4343	187	15	,	,	PUNCT
ejpam-4343	187	16	i	i	PROPN
ejpam-4343	187	17	)	)	PUNCT
ejpam-4343	187	18	is	be	AUX
ejpam-4343	187	19	denoted	denote	VERB
ejpam-4343	187	20	by	by	ADP
ejpam-4343	187	21	δp(⋆)(x	δp(⋆)(x	PROPN
ejpam-4343	187	22	)	)	PUNCT
ejpam-4343	187	23	.	.	PUNCT
ejpam-4343	188	1	proposition	proposition	NOUN
ejpam-4343	188	2	5	5	NUM
ejpam-4343	188	3	.	.	PUNCT
ejpam-4343	188	4	for	for	ADP
ejpam-4343	188	5	subsets	subset	NOUN
ejpam-4343	188	6	a	a	PRON
ejpam-4343	188	7	,	,	PUNCT
ejpam-4343	188	8	b	b	NOUN
ejpam-4343	188	9	and	and	CCONJ
ejpam-4343	188	10	cγ(γ	cγ(γ	CCONJ
ejpam-4343	188	11	∈	∈	PROPN
ejpam-4343	188	12	γ	γ	PROPN
ejpam-4343	188	13	)	)	PUNCT
ejpam-4343	188	14	of	of	ADP
ejpam-4343	188	15	an	an	DET
ejpam-4343	188	16	ideal	ideal	ADJ
ejpam-4343	188	17	topological	topological	ADJ
ejpam-4343	188	18	space	space	NOUN
ejpam-4343	188	19	(	(	PUNCT
ejpam-4343	188	20	x	x	X
ejpam-4343	188	21	,	,	PUNCT
ejpam-4343	188	22	τ	τ	PROPN
ejpam-4343	188	23	,	,	PUNCT
ejpam-4343	188	24	i	i	NOUN
ejpam-4343	188	25	)	)	PUNCT
ejpam-4343	188	26	,	,	PUNCT
ejpam-4343	188	27	the	the	DET
ejpam-4343	188	28	following	follow	VERB
ejpam-4343	188	29	properties	property	NOUN
ejpam-4343	188	30	hold	hold	VERB
ejpam-4343	188	31	:	:	PUNCT
ejpam-4343	188	32	(	(	PUNCT
ejpam-4343	188	33	1	1	X
ejpam-4343	188	34	)	)	PUNCT
ejpam-4343	188	35	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	188	36	)	)	PUNCT
ejpam-4343	189	1	⊆	⊆	NUM
ejpam-4343	189	2	a.	a.	NOUN
ejpam-4343	189	3	(	(	PUNCT
ejpam-4343	189	4	2	2	NUM
ejpam-4343	189	5	)	)	PUNCT
ejpam-4343	189	6	if	if	SCONJ
ejpam-4343	189	7	a	a	DET
ejpam-4343	189	8	⊆	⊆	NUM
ejpam-4343	189	9	b	b	NOUN
ejpam-4343	189	10	,	,	PUNCT
ejpam-4343	189	11	then	then	ADV
ejpam-4343	189	12	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	189	13	)	)	PUNCT
ejpam-4343	189	14	⊆	⊆	NUM
ejpam-4343	189	15	δp(⋆)(b	δp(⋆)(b	PROPN
ejpam-4343	189	16	)	)	PUNCT
ejpam-4343	189	17	.	.	PUNCT
ejpam-4343	190	1	(	(	PUNCT
ejpam-4343	190	2	3	3	X
ejpam-4343	190	3	)	)	PUNCT
ejpam-4343	190	4	δp(⋆)(δp(⋆)(a	δp(⋆)(δp(⋆)(a	NUM
ejpam-4343	190	5	)	)	PUNCT
ejpam-4343	190	6	)	)	PUNCT
ejpam-4343	191	1	=	=	SYM
ejpam-4343	191	2	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	191	3	)	)	PUNCT
ejpam-4343	191	4	.	.	PUNCT
ejpam-4343	192	1	(	(	PUNCT
ejpam-4343	192	2	4	4	X
ejpam-4343	192	3	)	)	PUNCT
ejpam-4343	192	4	if	if	SCONJ
ejpam-4343	192	5	a	a	PRON
ejpam-4343	192	6	is	be	AUX
ejpam-4343	192	7	a	a	DET
ejpam-4343	192	8	pre	pre	ADJ
ejpam-4343	192	9	-	-	ADJ
ejpam-4343	192	10	i	i	PRON
ejpam-4343	192	11	-closed	-close	VERB
ejpam-4343	192	12	set	set	NOUN
ejpam-4343	192	13	,	,	PUNCT
ejpam-4343	192	14	then	then	ADV
ejpam-4343	192	15	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	192	16	)	)	PUNCT
ejpam-4343	192	17	=	=	SYM
ejpam-4343	192	18	a.	a.	NOUN
ejpam-4343	192	19	(	(	PUNCT
ejpam-4343	192	20	5	5	NUM
ejpam-4343	192	21	)	)	PUNCT
ejpam-4343	192	22	δp(⋆)(∩{cγ	δp(⋆)(∩{cγ	NOUN
ejpam-4343	192	23	|γ	|γ	ADP
ejpam-4343	192	24	∈	∈	PROPN
ejpam-4343	192	25	γ	γ	X
ejpam-4343	192	26	}	}	PUNCT
ejpam-4343	192	27	)	)	PUNCT
ejpam-4343	192	28	=	=	PUNCT
ejpam-4343	192	29	∩{δp(⋆)(cγ)|γ	∩{δp(⋆)(cγ)|γ	PUNCT
ejpam-4343	192	30	∈	∈	PROPN
ejpam-4343	192	31	γ	γ	X
ejpam-4343	192	32	}	}	PUNCT
ejpam-4343	192	33	.	.	PUNCT
ejpam-4343	193	1	(	(	PUNCT
ejpam-4343	193	2	6	6	X
ejpam-4343	193	3	)	)	PUNCT
ejpam-4343	193	4	δp(⋆)(∪{cγ	δp(⋆)(∪{cγ	NOUN
ejpam-4343	193	5	|γ	|γ	ADP
ejpam-4343	193	6	∈	∈	PROPN
ejpam-4343	193	7	γ	γ	PROPN
ejpam-4343	193	8	}	}	PUNCT
ejpam-4343	193	9	)	)	PUNCT
ejpam-4343	193	10	⊇	⊇	PROPN
ejpam-4343	193	11	∪{δp(⋆)(cγ)|γ	∪{δp(⋆)(cγ)|γ	PROPN
ejpam-4343	193	12	∈	∈	PROPN
ejpam-4343	193	13	γ	γ	X
ejpam-4343	193	14	}	}	PUNCT
ejpam-4343	193	15	.	.	PUNCT
ejpam-4343	194	1	(	(	PUNCT
ejpam-4343	194	2	7	7	X
ejpam-4343	194	3	)	)	PUNCT
ejpam-4343	194	4	λp(⋆)(x	λp(⋆)(x	NOUN
ejpam-4343	194	5	−a	−a	NOUN
ejpam-4343	194	6	)	)	PUNCT
ejpam-4343	195	1	=	=	PUNCT
ejpam-4343	195	2	x	x	X
ejpam-4343	196	1	−	−	PROPN
ejpam-4343	196	2	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	196	3	)	)	PUNCT
ejpam-4343	196	4	and	and	CCONJ
ejpam-4343	196	5	δp(⋆)(x	δp(⋆)(x	PROPN
ejpam-4343	196	6	−a	−a	NOUN
ejpam-4343	196	7	)	)	PUNCT
ejpam-4343	196	8	=	=	PUNCT
ejpam-4343	197	1	x	x	X
ejpam-4343	198	1	−	−	ADP
ejpam-4343	198	2	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	198	3	)	)	PUNCT
ejpam-4343	198	4	.	.	PUNCT
ejpam-4343	199	1	proposition	proposition	NOUN
ejpam-4343	199	2	6	6	NUM
ejpam-4343	199	3	.	.	PUNCT
ejpam-4343	200	1	for	for	ADP
ejpam-4343	200	2	subsets	subset	NOUN
ejpam-4343	200	3	a	a	PRON
ejpam-4343	200	4	and	and	CCONJ
ejpam-4343	200	5	bγ(γ	bγ(γ	NOUN
ejpam-4343	200	6	∈	∈	PROPN
ejpam-4343	200	7	γ	γ	PROPN
ejpam-4343	200	8	)	)	PUNCT
ejpam-4343	200	9	of	of	ADP
ejpam-4343	200	10	an	an	DET
ejpam-4343	200	11	ideal	ideal	ADJ
ejpam-4343	200	12	topological	topological	ADJ
ejpam-4343	200	13	space	space	NOUN
ejpam-4343	200	14	(	(	PUNCT
ejpam-4343	200	15	x	x	X
ejpam-4343	200	16	,	,	PUNCT
ejpam-4343	200	17	τ	τ	PROPN
ejpam-4343	200	18	,	,	PUNCT
ejpam-4343	200	19	i	i	NOUN
ejpam-4343	200	20	)	)	PUNCT
ejpam-4343	200	21	,	,	PUNCT
ejpam-4343	200	22	the	the	DET
ejpam-4343	200	23	following	follow	VERB
ejpam-4343	200	24	properties	property	NOUN
ejpam-4343	200	25	hold	hold	VERB
ejpam-4343	200	26	:	:	PUNCT
ejpam-4343	200	27	(	(	PUNCT
ejpam-4343	200	28	1	1	X
ejpam-4343	200	29	)	)	PUNCT
ejpam-4343	200	30	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	200	31	)	)	PUNCT
ejpam-4343	200	32	is	be	AUX
ejpam-4343	200	33	a	a	DET
ejpam-4343	200	34	δp(⋆)-set	δp(⋆)-set	PROPN
ejpam-4343	200	35	.	.	PUNCT
ejpam-4343	201	1	(	(	PUNCT
ejpam-4343	201	2	2	2	X
ejpam-4343	201	3	)	)	PUNCT
ejpam-4343	201	4	if	if	SCONJ
ejpam-4343	201	5	a	a	PRON
ejpam-4343	201	6	is	be	AUX
ejpam-4343	201	7	a	a	DET
ejpam-4343	201	8	pre	pre	ADJ
ejpam-4343	201	9	-	-	ADJ
ejpam-4343	201	10	i	i	PRON
ejpam-4343	201	11	-closed	-close	VERB
ejpam-4343	201	12	set	set	NOUN
ejpam-4343	201	13	,	,	PUNCT
ejpam-4343	201	14	then	then	ADV
ejpam-4343	201	15	a	a	PRON
ejpam-4343	201	16	is	be	AUX
ejpam-4343	201	17	a	a	DET
ejpam-4343	201	18	δp(⋆)-set	δp(⋆)-set	PROPN
ejpam-4343	201	19	.	.	PUNCT
ejpam-4343	202	1	(	(	PUNCT
ejpam-4343	202	2	3	3	X
ejpam-4343	202	3	)	)	PUNCT
ejpam-4343	202	4	if	if	SCONJ
ejpam-4343	202	5	bγ	bγ	PRON
ejpam-4343	202	6	is	be	AUX
ejpam-4343	202	7	a	a	DET
ejpam-4343	202	8	δp(⋆)-set	δp(⋆)-set	NOUN
ejpam-4343	202	9	for	for	ADP
ejpam-4343	202	10	each	each	DET
ejpam-4343	202	11	γ	γ	PROPN
ejpam-4343	202	12	∈	∈	PROPN
ejpam-4343	202	13	γ	γ	X
ejpam-4343	202	14	,	,	PUNCT
ejpam-4343	202	15	then	then	ADV
ejpam-4343	202	16	∩γ∈γbγ	∩γ∈γbγ	X
ejpam-4343	202	17	is	be	AUX
ejpam-4343	202	18	a	a	DET
ejpam-4343	202	19	δp(⋆)-set	δp(⋆)-set	PROPN
ejpam-4343	202	20	.	.	PUNCT
ejpam-4343	203	1	(	(	PUNCT
ejpam-4343	203	2	4	4	X
ejpam-4343	203	3	)	)	PUNCT
ejpam-4343	203	4	if	if	SCONJ
ejpam-4343	203	5	bγ	bγ	PRON
ejpam-4343	203	6	is	be	AUX
ejpam-4343	203	7	a	a	DET
ejpam-4343	203	8	δp(⋆)-set	δp(⋆)-set	NOUN
ejpam-4343	203	9	for	for	ADP
ejpam-4343	203	10	each	each	DET
ejpam-4343	203	11	γ	γ	PROPN
ejpam-4343	203	12	∈	∈	PROPN
ejpam-4343	203	13	γ	γ	X
ejpam-4343	203	14	,	,	PUNCT
ejpam-4343	203	15	then	then	ADV
ejpam-4343	203	16	∪γ∈γbγ	∪γ∈γbγ	PROPN
ejpam-4343	203	17	is	be	AUX
ejpam-4343	203	18	a	a	DET
ejpam-4343	203	19	δp(⋆)-set	δp(⋆)-set	PROPN
ejpam-4343	203	20	.	.	PUNCT
ejpam-4343	203	21	proposition	proposition	NOUN
ejpam-4343	203	22	7	7	NUM
ejpam-4343	203	23	.	.	PUNCT
ejpam-4343	204	1	let	let	VERB
ejpam-4343	204	2	a	a	DET
ejpam-4343	204	3	be	be	AUX
ejpam-4343	204	4	a	a	DET
ejpam-4343	204	5	subset	subset	NOUN
ejpam-4343	204	6	of	of	ADP
ejpam-4343	204	7	an	an	DET
ejpam-4343	204	8	ideal	ideal	ADJ
ejpam-4343	204	9	topological	topological	ADJ
ejpam-4343	204	10	space	space	NOUN
ejpam-4343	204	11	(	(	PUNCT
ejpam-4343	204	12	x	x	X
ejpam-4343	204	13	,	,	PUNCT
ejpam-4343	204	14	τ	τ	PROPN
ejpam-4343	204	15	,	,	PUNCT
ejpam-4343	204	16	i	i	NOUN
ejpam-4343	204	17	)	)	PUNCT
ejpam-4343	204	18	.	.	PUNCT
ejpam-4343	205	1	then	then	ADV
ejpam-4343	205	2	,	,	PUNCT
ejpam-4343	205	3	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	205	4	)	)	PUNCT
ejpam-4343	205	5	=	=	PRON
ejpam-4343	206	1	{	{	PUNCT
ejpam-4343	206	2	x	x	PUNCT
ejpam-4343	206	3	∈	∈	PROPN
ejpam-4343	206	4	x	x	PUNCT
ejpam-4343	206	5	|	|	ADV
ejpam-4343	206	6	pıcl({x	pıcl({x	PROPN
ejpam-4343	206	7	}	}	PUNCT
ejpam-4343	206	8	)	)	PUNCT
ejpam-4343	206	9	∩a	∩a	PROPN
ejpam-4343	206	10	̸=	̸=	PROPN
ejpam-4343	206	11	∅	∅	NOUN
ejpam-4343	206	12	}	}	PUNCT
ejpam-4343	206	13	.	.	PUNCT
ejpam-4343	207	1	proof	proof	NOUN
ejpam-4343	207	2	.	.	PUNCT
ejpam-4343	208	1	let	let	VERB
ejpam-4343	208	2	x	x	X
ejpam-4343	208	3	∈	∈	PROPN
ejpam-4343	208	4	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	208	5	)	)	PUNCT
ejpam-4343	208	6	.	.	PUNCT
ejpam-4343	209	1	suppose	suppose	VERB
ejpam-4343	209	2	that	that	SCONJ
ejpam-4343	209	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	209	4	}	}	PUNCT
ejpam-4343	209	5	)	)	PUNCT
ejpam-4343	209	6	∩	∩	NOUN
ejpam-4343	209	7	a	a	DET
ejpam-4343	209	8	=	=	SYM
ejpam-4343	209	9	∅.	∅.	NOUN
ejpam-4343	209	10	then	then	ADV
ejpam-4343	209	11	,	,	PUNCT
ejpam-4343	209	12	x	x	PROPN
ejpam-4343	209	13	̸∈	̸∈	PROPN
ejpam-4343	209	14	x	x	X
ejpam-4343	209	15	−	−	PROPN
ejpam-4343	209	16	pıcl({x	pıcl({x	PROPN
ejpam-4343	209	17	}	}	PUNCT
ejpam-4343	209	18	)	)	PUNCT
ejpam-4343	209	19	which	which	PRON
ejpam-4343	209	20	is	be	AUX
ejpam-4343	209	21	a	a	DET
ejpam-4343	209	22	pre	pre	ADJ
ejpam-4343	209	23	-	-	ADJ
ejpam-4343	209	24	i	i	PRON
ejpam-4343	209	25	-open	-open	NOUN
ejpam-4343	209	26	set	set	VERB
ejpam-4343	209	27	containing	contain	VERB
ejpam-4343	209	28	a.	a.	NOUN
ejpam-4343	209	29	this	this	PRON
ejpam-4343	209	30	is	be	AUX
ejpam-4343	209	31	a	a	DET
ejpam-4343	209	32	contradiction	contradiction	NOUN
ejpam-4343	209	33	.	.	PUNCT
ejpam-4343	210	1	thus	thus	ADV
ejpam-4343	210	2	,	,	PUNCT
ejpam-4343	210	3	pıcl({x})∩a	pıcl({x})∩a	PROPN
ejpam-4343	210	4	̸=	̸=	PROPN
ejpam-4343	210	5	∅.	∅.	ADV
ejpam-4343	210	6	on	on	ADP
ejpam-4343	210	7	the	the	DET
ejpam-4343	210	8	other	other	ADJ
ejpam-4343	210	9	hand	hand	NOUN
ejpam-4343	210	10	,	,	PUNCT
ejpam-4343	210	11	let	let	VERB
ejpam-4343	210	12	x	x	PUNCT
ejpam-4343	210	13	∈	∈	PROPN
ejpam-4343	210	14	x	x	X
ejpam-4343	210	15	such	such	ADJ
ejpam-4343	210	16	that	that	SCONJ
ejpam-4343	210	17	pıcl({x})∩a	pıcl({x})∩a	PROPN
ejpam-4343	210	18	̸=	̸=	PROPN
ejpam-4343	210	19	∅	∅	NOUN
ejpam-4343	210	20	and	and	CCONJ
ejpam-4343	210	21	suppose	suppose	VERB
ejpam-4343	210	22	that	that	SCONJ
ejpam-4343	210	23	x	x	PROPN
ejpam-4343	210	24	̸∈	̸∈	PROPN
ejpam-4343	210	25	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	210	26	)	)	PUNCT
ejpam-4343	210	27	.	.	PUNCT
ejpam-4343	211	1	then	then	ADV
ejpam-4343	211	2	,	,	PUNCT
ejpam-4343	211	3	there	there	PRON
ejpam-4343	211	4	exists	exist	VERB
ejpam-4343	211	5	a	a	DET
ejpam-4343	211	6	pre	pre	ADJ
ejpam-4343	211	7	-	-	ADJ
ejpam-4343	211	8	i	i	PRON
ejpam-4343	211	9	-open	-open	NOUN
ejpam-4343	211	10	set	set	VERB
ejpam-4343	211	11	u	u	NOUN
ejpam-4343	211	12	containing	contain	VERB
ejpam-4343	211	13	a	a	PRON
ejpam-4343	211	14	and	and	CCONJ
ejpam-4343	211	15	x	x	X
ejpam-4343	211	16	̸∈	̸∈	PROPN
ejpam-4343	211	17	u	u	PROPN
ejpam-4343	211	18	.	.	PUNCT
ejpam-4343	212	1	let	let	VERB
ejpam-4343	212	2	y	y	PROPN
ejpam-4343	212	3	∈	∈	PROPN
ejpam-4343	212	4	pıcl({x	pıcl({x	PROPN
ejpam-4343	212	5	}	}	PUNCT
ejpam-4343	212	6	)	)	PUNCT
ejpam-4343	212	7	∩	∩	ADJ
ejpam-4343	212	8	a.	a.	NOUN
ejpam-4343	212	9	then	then	ADV
ejpam-4343	212	10	,	,	PUNCT
ejpam-4343	212	11	we	we	PRON
ejpam-4343	212	12	have	have	VERB
ejpam-4343	212	13	y	y	PROPN
ejpam-4343	212	14	∈	∈	PROPN
ejpam-4343	212	15	pıcl({x	pıcl({x	PROPN
ejpam-4343	212	16	}	}	PUNCT
ejpam-4343	212	17	)	)	PUNCT
ejpam-4343	212	18	and	and	CCONJ
ejpam-4343	212	19	y	y	PROPN
ejpam-4343	212	20	∈	∈	PROPN
ejpam-4343	212	21	a	a	DET
ejpam-4343	212	22	⊆	⊆	NUM
ejpam-4343	212	23	u	u	NOUN
ejpam-4343	212	24	.	.	PUNCT
ejpam-4343	213	1	therefore	therefore	ADV
ejpam-4343	213	2	,	,	PUNCT
ejpam-4343	213	3	u	u	PROPN
ejpam-4343	213	4	∩	∩	NOUN
ejpam-4343	213	5	{	{	PUNCT
ejpam-4343	213	6	x	x	NOUN
ejpam-4343	213	7	}	}	PUNCT
ejpam-4343	213	8	̸=	̸=	PROPN
ejpam-4343	213	9	∅.	∅.	ADP
ejpam-4343	213	10	this	this	PRON
ejpam-4343	213	11	is	be	AUX
ejpam-4343	213	12	a	a	DET
ejpam-4343	213	13	contradiction	contradiction	NOUN
ejpam-4343	213	14	.	.	PUNCT
ejpam-4343	214	1	consequently	consequently	ADV
ejpam-4343	214	2	,	,	PUNCT
ejpam-4343	214	3	we	we	PRON
ejpam-4343	214	4	obtain	obtain	VERB
ejpam-4343	214	5	x	x	SYM
ejpam-4343	214	6	∈	∈	NOUN
ejpam-4343	214	7	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	214	8	)	)	PUNCT
ejpam-4343	214	9	.	.	PUNCT
ejpam-4343	215	1	definition	definition	NOUN
ejpam-4343	215	2	5	5	NUM
ejpam-4343	215	3	.	.	PUNCT
ejpam-4343	216	1	let	let	VERB
ejpam-4343	216	2	a	a	DET
ejpam-4343	216	3	be	be	AUX
ejpam-4343	216	4	a	a	DET
ejpam-4343	216	5	subset	subset	NOUN
ejpam-4343	216	6	of	of	ADP
ejpam-4343	216	7	an	an	DET
ejpam-4343	216	8	ideal	ideal	ADJ
ejpam-4343	216	9	topological	topological	ADJ
ejpam-4343	216	10	space	space	NOUN
ejpam-4343	216	11	(	(	PUNCT
ejpam-4343	216	12	x	x	X
ejpam-4343	216	13	,	,	PUNCT
ejpam-4343	216	14	τ	τ	PROPN
ejpam-4343	216	15	,	,	PUNCT
ejpam-4343	216	16	i	i	PROPN
ejpam-4343	216	17	)	)	PUNCT
ejpam-4343	217	1	and	and	CCONJ
ejpam-4343	217	2	x	x	PUNCT
ejpam-4343	217	3	∈	∈	PROPN
ejpam-4343	217	4	x.	x.	NOUN
ejpam-4343	217	5	then	then	ADV
ejpam-4343	217	6	≺	≺	VERB
ejpam-4343	217	7	x	x	SYM
ejpam-4343	217	8	≻p(⋆	≻p(⋆	NOUN
ejpam-4343	217	9	)	)	PUNCT
ejpam-4343	217	10	is	be	AUX
ejpam-4343	217	11	defined	define	VERB
ejpam-4343	217	12	by	by	ADP
ejpam-4343	217	13	≺	≺	NOUN
ejpam-4343	217	14	x	x	SYM
ejpam-4343	217	15	≻p(⋆)=	≻p(⋆)=	PROPN
ejpam-4343	217	16	pıcl({x	pıcl({x	X
ejpam-4343	217	17	}	}	PUNCT
ejpam-4343	217	18	)	)	PUNCT
ejpam-4343	217	19	∩	∩	NOUN
ejpam-4343	217	20	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	217	21	}	}	PUNCT
ejpam-4343	217	22	)	)	PUNCT
ejpam-4343	217	23	.	.	PUNCT
ejpam-4343	218	1	proposition	proposition	NOUN
ejpam-4343	218	2	8	8	NUM
ejpam-4343	218	3	.	.	PUNCT
ejpam-4343	219	1	for	for	ADP
ejpam-4343	219	2	an	an	DET
ejpam-4343	219	3	ideal	ideal	ADJ
ejpam-4343	219	4	topological	topological	ADJ
ejpam-4343	219	5	space	space	NOUN
ejpam-4343	219	6	(	(	PUNCT
ejpam-4343	219	7	x	x	X
ejpam-4343	219	8	,	,	PUNCT
ejpam-4343	219	9	τ	τ	PROPN
ejpam-4343	219	10	,	,	PUNCT
ejpam-4343	219	11	i	i	NOUN
ejpam-4343	219	12	)	)	PUNCT
ejpam-4343	219	13	,	,	PUNCT
ejpam-4343	219	14	the	the	DET
ejpam-4343	219	15	following	follow	VERB
ejpam-4343	219	16	properties	property	NOUN
ejpam-4343	219	17	hold	hold	VERB
ejpam-4343	219	18	:	:	PUNCT
ejpam-4343	219	19	(	(	PUNCT
ejpam-4343	219	20	1	1	X
ejpam-4343	219	21	)	)	PUNCT
ejpam-4343	219	22	for	for	ADP
ejpam-4343	219	23	each	each	DET
ejpam-4343	219	24	x	x	SYM
ejpam-4343	219	25	∈	∈	PROPN
ejpam-4343	219	26	x	x	NOUN
ejpam-4343	219	27	,	,	PUNCT
ejpam-4343	219	28	λp(⋆)(≺	λp(⋆)(≺	NOUN
ejpam-4343	219	29	x	x	X
ejpam-4343	219	30	≻p(⋆	≻p(⋆	NUM
ejpam-4343	219	31	)	)	PUNCT
ejpam-4343	219	32	)	)	PUNCT
ejpam-4343	220	1	=	=	SYM
ejpam-4343	220	2	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	220	3	}	}	PUNCT
ejpam-4343	220	4	)	)	PUNCT
ejpam-4343	220	5	.	.	PUNCT
ejpam-4343	221	1	c.	c.	PROPN
ejpam-4343	221	2	boonpok	boonpok	PROPN
ejpam-4343	221	3	/	/	SYM
ejpam-4343	221	4	eur	eur	PROPN
ejpam-4343	221	5	.	.	PUNCT
ejpam-4343	222	1	j.	j.	PROPN
ejpam-4343	222	2	pure	pure	PROPN
ejpam-4343	222	3	appl	appl	PROPN
ejpam-4343	222	4	.	.	PROPN
ejpam-4343	222	5	math	math	PROPN
ejpam-4343	222	6	,	,	PUNCT
ejpam-4343	222	7	15	15	NUM
ejpam-4343	222	8	(	(	PUNCT
ejpam-4343	222	9	3	3	NUM
ejpam-4343	222	10	)	)	PUNCT
ejpam-4343	222	11	(	(	PUNCT
ejpam-4343	222	12	2022	2022	NUM
ejpam-4343	222	13	)	)	PUNCT
ejpam-4343	222	14	,	,	PUNCT
ejpam-4343	222	15	1023	1023	NUM
ejpam-4343	222	16	-	-	SYM
ejpam-4343	222	17	1046	1046	NUM
ejpam-4343	222	18	1029	1029	NUM
ejpam-4343	222	19	(	(	PUNCT
ejpam-4343	222	20	2	2	NUM
ejpam-4343	222	21	)	)	PUNCT
ejpam-4343	222	22	for	for	ADP
ejpam-4343	222	23	each	each	DET
ejpam-4343	222	24	x	x	SYM
ejpam-4343	222	25	∈	∈	PROPN
ejpam-4343	222	26	x	x	NOUN
ejpam-4343	222	27	,	,	PUNCT
ejpam-4343	222	28	pıcl(≺	pıcl(≺	NOUN
ejpam-4343	222	29	x	x	X
ejpam-4343	222	30	≻p(⋆	≻p(⋆	NUM
ejpam-4343	222	31	)	)	PUNCT
ejpam-4343	222	32	)	)	PUNCT
ejpam-4343	223	1	=	=	PUNCT
ejpam-4343	223	2	pıcl({x	pıcl({x	PROPN
ejpam-4343	223	3	}	}	PUNCT
ejpam-4343	223	4	)	)	PUNCT
ejpam-4343	223	5	.	.	PUNCT
ejpam-4343	224	1	(	(	PUNCT
ejpam-4343	224	2	3	3	X
ejpam-4343	224	3	)	)	PUNCT
ejpam-4343	224	4	for	for	ADP
ejpam-4343	224	5	each	each	DET
ejpam-4343	224	6	pre	pre	ADJ
ejpam-4343	224	7	-	-	ADJ
ejpam-4343	224	8	i	i	PRON
ejpam-4343	224	9	-open	-open	NOUN
ejpam-4343	224	10	set	set	VERB
ejpam-4343	224	11	v	v	ADP
ejpam-4343	224	12	and	and	CCONJ
ejpam-4343	224	13	each	each	DET
ejpam-4343	224	14	x	x	SYM
ejpam-4343	224	15	∈	∈	PROPN
ejpam-4343	224	16	v	v	NOUN
ejpam-4343	224	17	,	,	PUNCT
ejpam-4343	224	18	≺	≺	NOUN
ejpam-4343	224	19	x	x	SYM
ejpam-4343	224	20	≻p(⋆)⊆	≻p(⋆)⊆	NOUN
ejpam-4343	224	21	v	v	NOUN
ejpam-4343	224	22	.	.	PUNCT
ejpam-4343	225	1	(	(	PUNCT
ejpam-4343	225	2	4	4	NUM
ejpam-4343	225	3	)	)	PUNCT
ejpam-4343	225	4	for	for	ADP
ejpam-4343	225	5	each	each	DET
ejpam-4343	225	6	pre	pre	NOUN
ejpam-4343	225	7	-	-	ADJ
ejpam-4343	225	8	i	i	PRON
ejpam-4343	225	9	-closed	-close	VERB
ejpam-4343	226	1	set	set	VERB
ejpam-4343	226	2	f	f	NOUN
ejpam-4343	227	1	and	and	CCONJ
ejpam-4343	227	2	each	each	DET
ejpam-4343	227	3	x	x	SYM
ejpam-4343	227	4	∈	∈	PROPN
ejpam-4343	227	5	f	f	PROPN
ejpam-4343	227	6	,	,	PUNCT
ejpam-4343	227	7	≺	≺	NOUN
ejpam-4343	227	8	x	x	SYM
ejpam-4343	227	9	≻p(⋆)⊆	≻p(⋆)⊆	X
ejpam-4343	227	10	f	f	X
ejpam-4343	227	11	.	.	PUNCT
ejpam-4343	228	1	proof	proof	NOUN
ejpam-4343	228	2	.	.	PUNCT
ejpam-4343	229	1	(	(	PUNCT
ejpam-4343	229	2	1	1	X
ejpam-4343	229	3	)	)	PUNCT
ejpam-4343	229	4	let	let	VERB
ejpam-4343	229	5	x	x	PUNCT
ejpam-4343	229	6	∈	∈	PROPN
ejpam-4343	229	7	x.	x.	NOUN
ejpam-4343	229	8	then	then	ADV
ejpam-4343	229	9	,	,	PUNCT
ejpam-4343	229	10	{	{	PUNCT
ejpam-4343	229	11	x	x	NOUN
ejpam-4343	229	12	}	}	PUNCT
ejpam-4343	229	13	⊆	⊆	NUM
ejpam-4343	229	14	pıcl({x})∩λp(⋆)({x	pıcl({x})∩λp(⋆)({x	NOUN
ejpam-4343	229	15	}	}	PUNCT
ejpam-4343	229	16	)	)	PUNCT
ejpam-4343	230	1	=	=	NOUN
ejpam-4343	230	2	≺	≺	NOUN
ejpam-4343	230	3	x	x	SYM
ejpam-4343	230	4	≻p(⋆	≻p(⋆	NOUN
ejpam-4343	230	5	)	)	PUNCT
ejpam-4343	230	6	,	,	PUNCT
ejpam-4343	230	7	by	by	ADP
ejpam-4343	230	8	proposition	proposition	NOUN
ejpam-4343	230	9	1(2	1(2	NUM
ejpam-4343	230	10	)	)	PUNCT
ejpam-4343	230	11	,	,	PUNCT
ejpam-4343	230	12	λp(⋆)(≺	λp(⋆)(≺	NOUN
ejpam-4343	230	13	x	x	X
ejpam-4343	230	14	≻p(⋆	≻p(⋆	NUM
ejpam-4343	230	15	)	)	PUNCT
ejpam-4343	230	16	)	)	PUNCT
ejpam-4343	230	17	⊇	⊇	NOUN
ejpam-4343	230	18	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	230	19	}	}	PUNCT
ejpam-4343	230	20	)	)	PUNCT
ejpam-4343	230	21	.	.	PUNCT
ejpam-4343	231	1	on	on	ADP
ejpam-4343	231	2	the	the	DET
ejpam-4343	231	3	other	other	ADJ
ejpam-4343	231	4	hand	hand	NOUN
ejpam-4343	231	5	,	,	PUNCT
ejpam-4343	231	6	suppose	suppose	VERB
ejpam-4343	231	7	that	that	SCONJ
ejpam-4343	231	8	y	y	PROPN
ejpam-4343	231	9	̸∈	̸∈	PROPN
ejpam-4343	231	10	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	231	11	}	}	PUNCT
ejpam-4343	231	12	)	)	PUNCT
ejpam-4343	231	13	.	.	PUNCT
ejpam-4343	232	1	then	then	ADV
ejpam-4343	232	2	,	,	PUNCT
ejpam-4343	232	3	there	there	PRON
ejpam-4343	232	4	exists	exist	VERB
ejpam-4343	232	5	a	a	DET
ejpam-4343	232	6	pre	pre	NOUN
ejpam-4343	232	7	-	-	ADJ
ejpam-4343	232	8	i	i	PRON
ejpam-4343	232	9	open	open	ADJ
ejpam-4343	232	10	set	set	VERB
ejpam-4343	232	11	u	u	PRON
ejpam-4343	232	12	such	such	ADJ
ejpam-4343	232	13	that	that	SCONJ
ejpam-4343	232	14	x	x	SYM
ejpam-4343	232	15	∈	∈	PROPN
ejpam-4343	232	16	u	u	NOUN
ejpam-4343	232	17	and	and	CCONJ
ejpam-4343	232	18	y	y	PROPN
ejpam-4343	232	19	̸∈	̸∈	PROPN
ejpam-4343	232	20	u	u	PROPN
ejpam-4343	232	21	.	.	PUNCT
ejpam-4343	233	1	since	since	SCONJ
ejpam-4343	233	2	≺	≺	NOUN
ejpam-4343	233	3	x	x	SYM
ejpam-4343	233	4	≻p(⋆)⊆	≻p(⋆)⊆	X
ejpam-4343	233	5	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	233	6	}	}	PUNCT
ejpam-4343	233	7	)	)	PUNCT
ejpam-4343	233	8	⊆	⊆	NUM
ejpam-4343	233	9	λp(⋆)(u	λp(⋆)(u	PROPN
ejpam-4343	233	10	)	)	PUNCT
ejpam-4343	233	11	=	=	SYM
ejpam-4343	233	12	u	u	NOUN
ejpam-4343	233	13	,	,	PUNCT
ejpam-4343	233	14	we	we	PRON
ejpam-4343	233	15	have	have	VERB
ejpam-4343	233	16	λp(⋆)(≺	λp(⋆)(≺	NOUN
ejpam-4343	233	17	x	x	X
ejpam-4343	233	18	≻p(⋆	≻p(⋆	NOUN
ejpam-4343	233	19	)	)	PUNCT
ejpam-4343	233	20	)	)	PUNCT
ejpam-4343	234	1	⊆	⊆	NUM
ejpam-4343	234	2	u	u	NOUN
ejpam-4343	234	3	and	and	CCONJ
ejpam-4343	234	4	hence	hence	ADV
ejpam-4343	234	5	y	y	PROPN
ejpam-4343	234	6	̸∈	̸∈	PROPN
ejpam-4343	234	7	λp(⋆)(≺	λp(⋆)(≺	PROPN
ejpam-4343	234	8	x	x	X
ejpam-4343	234	9	≻p(⋆	≻p(⋆	NUM
ejpam-4343	234	10	)	)	PUNCT
ejpam-4343	234	11	)	)	PUNCT
ejpam-4343	234	12	.	.	PUNCT
ejpam-4343	235	1	thus	thus	ADV
ejpam-4343	235	2	,	,	PUNCT
ejpam-4343	235	3	λp(⋆)(≺	λp(⋆)(≺	NOUN
ejpam-4343	235	4	x	x	X
ejpam-4343	235	5	≻p(⋆	≻p(⋆	NUM
ejpam-4343	235	6	)	)	PUNCT
ejpam-4343	235	7	)	)	PUNCT
ejpam-4343	235	8	⊆	⊆	NUM
ejpam-4343	235	9	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	235	10	}	}	PUNCT
ejpam-4343	235	11	)	)	PUNCT
ejpam-4343	235	12	.	.	PUNCT
ejpam-4343	236	1	consequently	consequently	ADV
ejpam-4343	236	2	,	,	PUNCT
ejpam-4343	236	3	we	we	PRON
ejpam-4343	236	4	obtain	obtain	VERB
ejpam-4343	236	5	λp(⋆)(≺	λp(⋆)(≺	NOUN
ejpam-4343	236	6	x	x	X
ejpam-4343	236	7	≻p(⋆	≻p(⋆	NUM
ejpam-4343	236	8	)	)	PUNCT
ejpam-4343	236	9	)	)	PUNCT
ejpam-4343	237	1	=	=	SYM
ejpam-4343	237	2	λp(⋆)({x	λp(⋆)({x	ADJ
ejpam-4343	237	3	}	}	PUNCT
ejpam-4343	237	4	)	)	PUNCT
ejpam-4343	237	5	.	.	PUNCT
ejpam-4343	238	1	(	(	PUNCT
ejpam-4343	238	2	2	2	X
ejpam-4343	238	3	)	)	PUNCT
ejpam-4343	238	4	let	let	VERB
ejpam-4343	238	5	x	x	PUNCT
ejpam-4343	238	6	∈	∈	PROPN
ejpam-4343	238	7	x.	x.	NOUN
ejpam-4343	238	8	since	since	SCONJ
ejpam-4343	238	9	{	{	PUNCT
ejpam-4343	238	10	x	x	X
ejpam-4343	238	11	}	}	PUNCT
ejpam-4343	238	12	⊆≺	⊆≺	NUM
ejpam-4343	238	13	x	x	SYM
ejpam-4343	238	14	≻p(⋆	≻p(⋆	NUM
ejpam-4343	238	15	)	)	PUNCT
ejpam-4343	238	16	,	,	PUNCT
ejpam-4343	238	17	we	we	PRON
ejpam-4343	238	18	have	have	VERB
ejpam-4343	238	19	pıcl({x	pıcl({x	NUM
ejpam-4343	238	20	}	}	PUNCT
ejpam-4343	238	21	)	)	PUNCT
ejpam-4343	239	1	⊆	⊆	NUM
ejpam-4343	239	2	pıcl(≺	pıcl(≺	NOUN
ejpam-4343	239	3	x	x	X
ejpam-4343	239	4	≻p(⋆	≻p(⋆	NUM
ejpam-4343	239	5	)	)	PUNCT
ejpam-4343	239	6	)	)	PUNCT
ejpam-4343	239	7	.	.	PUNCT
ejpam-4343	240	1	on	on	ADP
ejpam-4343	240	2	the	the	DET
ejpam-4343	240	3	other	other	ADJ
ejpam-4343	240	4	hand	hand	NOUN
ejpam-4343	240	5	,	,	PUNCT
ejpam-4343	240	6	we	we	PRON
ejpam-4343	240	7	have	have	VERB
ejpam-4343	240	8	≺	≺	NOUN
ejpam-4343	240	9	x	x	SYM
ejpam-4343	240	10	≻p(⋆)⊆	≻p(⋆)⊆	X
ejpam-4343	240	11	pıcl({x	pıcl({x	X
ejpam-4343	240	12	}	}	PUNCT
ejpam-4343	240	13	)	)	PUNCT
ejpam-4343	241	1	and	and	CCONJ
ejpam-4343	241	2	so	so	ADV
ejpam-4343	241	3	pıcl(≺	pıcl(≺	ADV
ejpam-4343	241	4	x	x	X
ejpam-4343	241	5	≻p(⋆	≻p(⋆	NUM
ejpam-4343	241	6	)	)	PUNCT
ejpam-4343	241	7	)	)	PUNCT
ejpam-4343	242	1	⊆	⊆	NUM
ejpam-4343	242	2	pıcl(pıcl({x	pıcl(pıcl({x	NOUN
ejpam-4343	242	3	}	}	PUNCT
ejpam-4343	242	4	)	)	PUNCT
ejpam-4343	242	5	)	)	PUNCT
ejpam-4343	243	1	=	=	PUNCT
ejpam-4343	243	2	pıcl({x	pıcl({x	PROPN
ejpam-4343	243	3	}	}	PUNCT
ejpam-4343	243	4	)	)	PUNCT
ejpam-4343	243	5	.	.	PUNCT
ejpam-4343	244	1	thus	thus	ADV
ejpam-4343	244	2	,	,	PUNCT
ejpam-4343	244	3	pıcl(≺	pıcl(≺	NOUN
ejpam-4343	244	4	x	x	PUNCT
ejpam-4343	244	5	≻p(⋆	≻p(⋆	NUM
ejpam-4343	244	6	)	)	PUNCT
ejpam-4343	244	7	)	)	PUNCT
ejpam-4343	244	8	=	=	PUNCT
ejpam-4343	244	9	pıcl({x	pıcl({x	PROPN
ejpam-4343	244	10	}	}	PUNCT
ejpam-4343	244	11	)	)	PUNCT
ejpam-4343	244	12	.	.	PUNCT
ejpam-4343	245	1	(	(	PUNCT
ejpam-4343	245	2	3	3	X
ejpam-4343	245	3	)	)	PUNCT
ejpam-4343	245	4	let	let	VERB
ejpam-4343	245	5	v	v	PART
ejpam-4343	245	6	be	be	AUX
ejpam-4343	245	7	any	any	DET
ejpam-4343	245	8	pre	pre	ADJ
ejpam-4343	245	9	-	-	ADJ
ejpam-4343	245	10	i	i	PRON
ejpam-4343	245	11	-open	-open	NOUN
ejpam-4343	245	12	set	set	VERB
ejpam-4343	245	13	and	and	CCONJ
ejpam-4343	245	14	x	x	PART
ejpam-4343	245	15	∈	∈	PROPN
ejpam-4343	245	16	v	v	NOUN
ejpam-4343	245	17	.	.	PUNCT
ejpam-4343	246	1	then	then	ADV
ejpam-4343	246	2	,	,	PUNCT
ejpam-4343	246	3	λp(⋆)(≺	λp(⋆)(≺	NOUN
ejpam-4343	246	4	x	x	X
ejpam-4343	246	5	≻p(⋆	≻p(⋆	NUM
ejpam-4343	246	6	)	)	PUNCT
ejpam-4343	246	7	)	)	PUNCT
ejpam-4343	247	1	⊆	⊆	NUM
ejpam-4343	247	2	v	v	NOUN
ejpam-4343	247	3	and	and	CCONJ
ejpam-4343	247	4	hence	hence	ADV
ejpam-4343	247	5	≺	≺	NOUN
ejpam-4343	247	6	x	x	SYM
ejpam-4343	247	7	≻p(⋆)⊆	≻p(⋆)⊆	NUM
ejpam-4343	247	8	v	v	NOUN
ejpam-4343	247	9	.	.	PUNCT
ejpam-4343	248	1	(	(	PUNCT
ejpam-4343	248	2	4	4	X
ejpam-4343	248	3	)	)	PUNCT
ejpam-4343	248	4	let	let	VERB
ejpam-4343	248	5	f	f	PRON
ejpam-4343	248	6	be	be	AUX
ejpam-4343	248	7	any	any	DET
ejpam-4343	248	8	pre	pre	NOUN
ejpam-4343	248	9	-	-	ADJ
ejpam-4343	248	10	i	i	PRON
ejpam-4343	248	11	-closed	-close	VERB
ejpam-4343	248	12	set	set	NOUN
ejpam-4343	248	13	and	and	CCONJ
ejpam-4343	249	1	x	x	SYM
ejpam-4343	249	2	∈	∈	PROPN
ejpam-4343	249	3	f	f	X
ejpam-4343	249	4	.	.	PUNCT
ejpam-4343	250	1	therefore	therefore	ADV
ejpam-4343	250	2	,	,	PUNCT
ejpam-4343	250	3	we	we	PRON
ejpam-4343	250	4	have	have	VERB
ejpam-4343	250	5	≺	≺	NOUN
ejpam-4343	250	6	x	x	SYM
ejpam-4343	250	7	≻p(⋆)=	≻p(⋆)=	PROPN
ejpam-4343	250	8	pıcl({x	pıcl({x	PROPN
ejpam-4343	250	9	}	}	PUNCT
ejpam-4343	250	10	)	)	PUNCT
ejpam-4343	250	11	∩	∩	NOUN
ejpam-4343	250	12	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	250	13	}	}	PUNCT
ejpam-4343	250	14	)	)	PUNCT
ejpam-4343	251	1	⊆	⊆	NUM
ejpam-4343	251	2	pıcl({x	pıcl({x	NUM
ejpam-4343	251	3	}	}	PUNCT
ejpam-4343	251	4	)	)	PUNCT
ejpam-4343	251	5	⊆	⊆	NUM
ejpam-4343	251	6	pıcl(f	pıcl(f	NOUN
ejpam-4343	251	7	)	)	PUNCT
ejpam-4343	252	1	=	=	PUNCT
ejpam-4343	252	2	f.	f.	PROPN
ejpam-4343	252	3	4	4	X
ejpam-4343	252	4	.	.	PUNCT
ejpam-4343	253	1	(	(	PUNCT
ejpam-4343	253	2	λ	λ	X
ejpam-4343	253	3	,	,	PUNCT
ejpam-4343	253	4	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	253	5	sets	set	NOUN
ejpam-4343	253	6	in	in	ADP
ejpam-4343	253	7	this	this	DET
ejpam-4343	253	8	section	section	NOUN
ejpam-4343	253	9	,	,	PUNCT
ejpam-4343	253	10	we	we	PRON
ejpam-4343	253	11	introduce	introduce	VERB
ejpam-4343	253	12	the	the	DET
ejpam-4343	253	13	concept	concept	NOUN
ejpam-4343	253	14	of	of	ADP
ejpam-4343	253	15	(	(	PUNCT
ejpam-4343	253	16	λ	λ	PROPN
ejpam-4343	253	17	,	,	PUNCT
ejpam-4343	253	18	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	253	19	sets	set	NOUN
ejpam-4343	253	20	and	and	CCONJ
ejpam-4343	253	21	investigate	investigate	VERB
ejpam-4343	253	22	some	some	DET
ejpam-4343	253	23	properties	property	NOUN
ejpam-4343	253	24	of	of	ADP
ejpam-4343	253	25	(	(	PUNCT
ejpam-4343	253	26	λ	λ	PROPN
ejpam-4343	253	27	,	,	PUNCT
ejpam-4343	253	28	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	253	29	sets	set	NOUN
ejpam-4343	253	30	.	.	PUNCT
ejpam-4343	254	1	definition	definition	NOUN
ejpam-4343	254	2	6	6	NUM
ejpam-4343	254	3	.	.	PUNCT
ejpam-4343	255	1	a	a	DET
ejpam-4343	255	2	subset	subset	NOUN
ejpam-4343	255	3	a	a	PRON
ejpam-4343	255	4	of	of	ADP
ejpam-4343	255	5	an	an	DET
ejpam-4343	255	6	ideal	ideal	ADJ
ejpam-4343	255	7	topological	topological	ADJ
ejpam-4343	255	8	space	space	NOUN
ejpam-4343	255	9	(	(	PUNCT
ejpam-4343	255	10	x	x	X
ejpam-4343	255	11	,	,	PUNCT
ejpam-4343	255	12	τ	τ	PROPN
ejpam-4343	255	13	,	,	PUNCT
ejpam-4343	255	14	i	i	PROPN
ejpam-4343	255	15	)	)	PUNCT
ejpam-4343	255	16	is	be	AUX
ejpam-4343	255	17	said	say	VERB
ejpam-4343	255	18	to	to	PART
ejpam-4343	255	19	be	be	AUX
ejpam-4343	255	20	(	(	PUNCT
ejpam-4343	255	21	λ	λ	X
ejpam-4343	255	22	,	,	PUNCT
ejpam-4343	255	23	p(⋆))closed	p(⋆))close	VERB
ejpam-4343	255	24	if	if	SCONJ
ejpam-4343	255	25	a	a	DET
ejpam-4343	255	26	=	=	X
ejpam-4343	255	27	t	t	NOUN
ejpam-4343	255	28	∩c	∩c	NOUN
ejpam-4343	255	29	,	,	PUNCT
ejpam-4343	255	30	where	where	SCONJ
ejpam-4343	255	31	t	t	PROPN
ejpam-4343	255	32	is	be	AUX
ejpam-4343	255	33	a	a	DET
ejpam-4343	255	34	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	255	35	and	and	CCONJ
ejpam-4343	255	36	c	c	NOUN
ejpam-4343	255	37	is	be	AUX
ejpam-4343	255	38	a	a	DET
ejpam-4343	255	39	pre	pre	ADJ
ejpam-4343	255	40	-	-	ADJ
ejpam-4343	255	41	i	i	PRON
ejpam-4343	255	42	-closed	-close	VERB
ejpam-4343	255	43	set	set	NOUN
ejpam-4343	255	44	.	.	PUNCT
ejpam-4343	256	1	the	the	DET
ejpam-4343	256	2	collection	collection	NOUN
ejpam-4343	256	3	of	of	ADP
ejpam-4343	256	4	all	all	DET
ejpam-4343	256	5	(	(	PUNCT
ejpam-4343	256	6	λ	λ	PROPN
ejpam-4343	256	7	,	,	PUNCT
ejpam-4343	256	8	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	256	9	sets	set	NOUN
ejpam-4343	256	10	in	in	ADP
ejpam-4343	256	11	an	an	DET
ejpam-4343	256	12	ideal	ideal	ADJ
ejpam-4343	256	13	topological	topological	ADJ
ejpam-4343	256	14	space	space	NOUN
ejpam-4343	256	15	(	(	PUNCT
ejpam-4343	256	16	x	x	X
ejpam-4343	256	17	,	,	PUNCT
ejpam-4343	256	18	τ	τ	PROPN
ejpam-4343	256	19	,	,	PUNCT
ejpam-4343	256	20	i	i	PROPN
ejpam-4343	256	21	)	)	PUNCT
ejpam-4343	256	22	is	be	AUX
ejpam-4343	256	23	denoted	denote	VERB
ejpam-4343	256	24	by	by	ADP
ejpam-4343	256	25	(	(	PUNCT
ejpam-4343	256	26	λ	λ	PROPN
ejpam-4343	256	27	,	,	PUNCT
ejpam-4343	256	28	p(⋆))c(x	p(⋆))c(x	NOUN
ejpam-4343	256	29	)	)	PUNCT
ejpam-4343	256	30	.	.	PUNCT
ejpam-4343	257	1	theorem	theorem	NOUN
ejpam-4343	257	2	1	1	NUM
ejpam-4343	257	3	.	.	X
ejpam-4343	257	4	for	for	ADP
ejpam-4343	257	5	a	a	DET
ejpam-4343	257	6	subset	subset	NOUN
ejpam-4343	257	7	a	a	PRON
ejpam-4343	257	8	of	of	ADP
ejpam-4343	257	9	an	an	DET
ejpam-4343	257	10	ideal	ideal	ADJ
ejpam-4343	257	11	topological	topological	ADJ
ejpam-4343	257	12	space	space	NOUN
ejpam-4343	257	13	(	(	PUNCT
ejpam-4343	257	14	x	x	X
ejpam-4343	257	15	,	,	PUNCT
ejpam-4343	257	16	τ	τ	PROPN
ejpam-4343	257	17	,	,	PUNCT
ejpam-4343	257	18	i	i	NOUN
ejpam-4343	257	19	)	)	PUNCT
ejpam-4343	257	20	,	,	PUNCT
ejpam-4343	257	21	the	the	DET
ejpam-4343	257	22	following	follow	VERB
ejpam-4343	257	23	properties	property	NOUN
ejpam-4343	257	24	are	be	AUX
ejpam-4343	257	25	equivalent	equivalent	ADJ
ejpam-4343	257	26	:	:	PUNCT
ejpam-4343	257	27	(	(	PUNCT
ejpam-4343	257	28	1	1	X
ejpam-4343	257	29	)	)	PUNCT
ejpam-4343	257	30	a	a	PRON
ejpam-4343	257	31	is	be	AUX
ejpam-4343	257	32	(	(	PUNCT
ejpam-4343	257	33	λ	λ	X
ejpam-4343	257	34	,	,	PUNCT
ejpam-4343	257	35	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	257	36	.	.	PUNCT
ejpam-4343	258	1	(	(	PUNCT
ejpam-4343	258	2	2	2	X
ejpam-4343	258	3	)	)	PUNCT
ejpam-4343	258	4	a	a	DET
ejpam-4343	258	5	=	=	SYM
ejpam-4343	258	6	t	t	PROPN
ejpam-4343	258	7	∩	∩	X
ejpam-4343	258	8	pıcl(a	pıcl(a	NOUN
ejpam-4343	258	9	)	)	PUNCT
ejpam-4343	258	10	,	,	PUNCT
ejpam-4343	258	11	where	where	SCONJ
ejpam-4343	258	12	t	t	PROPN
ejpam-4343	258	13	is	be	AUX
ejpam-4343	258	14	a	a	DET
ejpam-4343	258	15	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	258	16	.	.	PUNCT
ejpam-4343	259	1	(	(	PUNCT
ejpam-4343	259	2	3	3	X
ejpam-4343	259	3	)	)	PUNCT
ejpam-4343	259	4	a	a	DET
ejpam-4343	259	5	=	=	SYM
ejpam-4343	259	6	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	259	7	)	)	PUNCT
ejpam-4343	259	8	∩	∩	NOUN
ejpam-4343	259	9	pıcl(a	pıcl(a	NOUN
ejpam-4343	259	10	)	)	PUNCT
ejpam-4343	259	11	.	.	PUNCT
ejpam-4343	260	1	c.	c.	PROPN
ejpam-4343	260	2	boonpok	boonpok	PROPN
ejpam-4343	260	3	/	/	SYM
ejpam-4343	260	4	eur	eur	PROPN
ejpam-4343	260	5	.	.	PUNCT
ejpam-4343	261	1	j.	j.	PROPN
ejpam-4343	261	2	pure	pure	PROPN
ejpam-4343	261	3	appl	appl	PROPN
ejpam-4343	261	4	.	.	PROPN
ejpam-4343	261	5	math	math	PROPN
ejpam-4343	261	6	,	,	PUNCT
ejpam-4343	261	7	15	15	NUM
ejpam-4343	261	8	(	(	PUNCT
ejpam-4343	261	9	3	3	NUM
ejpam-4343	261	10	)	)	PUNCT
ejpam-4343	261	11	(	(	PUNCT
ejpam-4343	261	12	2022	2022	NUM
ejpam-4343	261	13	)	)	PUNCT
ejpam-4343	261	14	,	,	PUNCT
ejpam-4343	261	15	1023	1023	NUM
ejpam-4343	261	16	-	-	SYM
ejpam-4343	261	17	1046	1046	NUM
ejpam-4343	261	18	1030	1030	NUM
ejpam-4343	261	19	proof	proof	NOUN
ejpam-4343	261	20	.	.	PUNCT
ejpam-4343	262	1	(	(	PUNCT
ejpam-4343	262	2	1	1	X
ejpam-4343	262	3	)	)	PUNCT
ejpam-4343	262	4	⇒	⇒	NOUN
ejpam-4343	262	5	(	(	PUNCT
ejpam-4343	262	6	2	2	NUM
ejpam-4343	262	7	):	):	PUNCT
ejpam-4343	262	8	suppose	suppose	VERB
ejpam-4343	262	9	that	that	SCONJ
ejpam-4343	262	10	a	a	DET
ejpam-4343	262	11	=	=	SYM
ejpam-4343	262	12	t	t	PROPN
ejpam-4343	262	13	∩	∩	ADJ
ejpam-4343	262	14	c	c	NOUN
ejpam-4343	262	15	,	,	PUNCT
ejpam-4343	262	16	where	where	SCONJ
ejpam-4343	262	17	t	t	PROPN
ejpam-4343	262	18	is	be	AUX
ejpam-4343	262	19	a	a	DET
ejpam-4343	262	20	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	262	21	and	and	CCONJ
ejpam-4343	262	22	c	c	NOUN
ejpam-4343	262	23	is	be	AUX
ejpam-4343	262	24	a	a	DET
ejpam-4343	262	25	prei	prei	NOUN
ejpam-4343	262	26	-closed	-close	VERB
ejpam-4343	262	27	set	set	NOUN
ejpam-4343	262	28	.	.	PUNCT
ejpam-4343	263	1	since	since	SCONJ
ejpam-4343	263	2	a	a	DET
ejpam-4343	263	3	⊆	⊆	NUM
ejpam-4343	263	4	c	c	NOUN
ejpam-4343	263	5	,	,	PUNCT
ejpam-4343	263	6	we	we	PRON
ejpam-4343	263	7	have	have	VERB
ejpam-4343	263	8	pıcl(a	pıcl(a	NOUN
ejpam-4343	263	9	)	)	PUNCT
ejpam-4343	263	10	⊆	⊆	NUM
ejpam-4343	263	11	c	c	NOUN
ejpam-4343	263	12	and	and	CCONJ
ejpam-4343	263	13	a	a	DET
ejpam-4343	263	14	=	=	SYM
ejpam-4343	263	15	t	t	PROPN
ejpam-4343	263	16	∩	∩	PROPN
ejpam-4343	263	17	c	c	PROPN
ejpam-4343	263	18	⊇	⊇	PROPN
ejpam-4343	263	19	t	t	PROPN
ejpam-4343	263	20	∩	∩	ADJ
ejpam-4343	263	21	pıcl(a	pıcl(a	NOUN
ejpam-4343	263	22	)	)	PUNCT
ejpam-4343	263	23	⊇	⊇	PROPN
ejpam-4343	263	24	a.	a.	NOUN
ejpam-4343	263	25	consequently	consequently	ADV
ejpam-4343	263	26	,	,	PUNCT
ejpam-4343	263	27	we	we	PRON
ejpam-4343	263	28	obtain	obtain	VERB
ejpam-4343	263	29	a	a	DET
ejpam-4343	263	30	=	=	SYM
ejpam-4343	263	31	t	t	NOUN
ejpam-4343	263	32	∩	∩	X
ejpam-4343	263	33	pıcl(a	pıcl(a	NOUN
ejpam-4343	263	34	)	)	PUNCT
ejpam-4343	263	35	.	.	PUNCT
ejpam-4343	264	1	(	(	PUNCT
ejpam-4343	264	2	2	2	X
ejpam-4343	264	3	)	)	PUNCT
ejpam-4343	264	4	⇒	⇒	NOUN
ejpam-4343	264	5	(	(	PUNCT
ejpam-4343	264	6	3	3	NUM
ejpam-4343	264	7	):	):	PUNCT
ejpam-4343	264	8	suppose	suppose	VERB
ejpam-4343	264	9	that	that	SCONJ
ejpam-4343	264	10	a	a	DET
ejpam-4343	264	11	=	=	SYM
ejpam-4343	264	12	t	t	NOUN
ejpam-4343	264	13	∩	∩	X
ejpam-4343	264	14	pıcl(a	pıcl(a	NOUN
ejpam-4343	264	15	)	)	PUNCT
ejpam-4343	264	16	,	,	PUNCT
ejpam-4343	264	17	where	where	SCONJ
ejpam-4343	264	18	t	t	PROPN
ejpam-4343	264	19	is	be	AUX
ejpam-4343	264	20	a	a	DET
ejpam-4343	264	21	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	264	22	.	.	PUNCT
ejpam-4343	265	1	since	since	SCONJ
ejpam-4343	265	2	a	a	DET
ejpam-4343	265	3	⊆	⊆	NUM
ejpam-4343	265	4	t	t	NOUN
ejpam-4343	265	5	,	,	PUNCT
ejpam-4343	265	6	we	we	PRON
ejpam-4343	265	7	have	have	VERB
ejpam-4343	265	8	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	265	9	)	)	PUNCT
ejpam-4343	266	1	⊆	⊆	NUM
ejpam-4343	266	2	λp(⋆)(t	λp(⋆)(t	PROPN
ejpam-4343	266	3	)	)	PUNCT
ejpam-4343	266	4	=	=	SYM
ejpam-4343	266	5	t	t	NOUN
ejpam-4343	266	6	and	and	CCONJ
ejpam-4343	266	7	hence	hence	ADV
ejpam-4343	266	8	a	a	DET
ejpam-4343	266	9	⊆	⊆	NUM
ejpam-4343	266	10	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	266	11	)	)	PUNCT
ejpam-4343	266	12	∩	∩	ADJ
ejpam-4343	266	13	pıcl(a	pıcl(a	NOUN
ejpam-4343	266	14	)	)	PUNCT
ejpam-4343	266	15	⊆	⊆	NUM
ejpam-4343	266	16	t	t	NOUN
ejpam-4343	266	17	∩	∩	ADJ
ejpam-4343	266	18	pıcl(a	pıcl(a	NOUN
ejpam-4343	266	19	)	)	PUNCT
ejpam-4343	266	20	=	=	SYM
ejpam-4343	266	21	a.	a.	NOUN
ejpam-4343	266	22	thus	thus	ADV
ejpam-4343	266	23	,	,	PUNCT
ejpam-4343	266	24	a	a	DET
ejpam-4343	266	25	=	=	SYM
ejpam-4343	266	26	λp(⋆)(a	λp(⋆)(a	NOUN
ejpam-4343	266	27	)	)	PUNCT
ejpam-4343	266	28	∩	∩	NOUN
ejpam-4343	266	29	pıcl(a	pıcl(a	NOUN
ejpam-4343	266	30	)	)	PUNCT
ejpam-4343	266	31	.	.	PUNCT
ejpam-4343	267	1	(	(	PUNCT
ejpam-4343	267	2	3	3	X
ejpam-4343	267	3	)	)	PUNCT
ejpam-4343	267	4	⇒	⇒	NOUN
ejpam-4343	267	5	(	(	PUNCT
ejpam-4343	267	6	1	1	NUM
ejpam-4343	267	7	):	):	PUNCT
ejpam-4343	267	8	since	since	SCONJ
ejpam-4343	267	9	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	267	10	)	)	PUNCT
ejpam-4343	267	11	is	be	AUX
ejpam-4343	267	12	λp(⋆)-set	λp(⋆)-set	VERB
ejpam-4343	267	13	,	,	PUNCT
ejpam-4343	267	14	pıcl(a	pıcl(a	NOUN
ejpam-4343	267	15	)	)	PUNCT
ejpam-4343	267	16	is	be	AUX
ejpam-4343	267	17	a	a	DET
ejpam-4343	267	18	pre	pre	NOUN
ejpam-4343	267	19	-	-	ADJ
ejpam-4343	267	20	i	i	PRON
ejpam-4343	267	21	-closed	-close	VERB
ejpam-4343	267	22	set	set	NOUN
ejpam-4343	267	23	and	and	CCONJ
ejpam-4343	267	24	a	a	DET
ejpam-4343	267	25	=	=	PUNCT
ejpam-4343	267	26	λp(⋆)(a	λp(⋆)(a	PROPN
ejpam-4343	267	27	)	)	PUNCT
ejpam-4343	267	28	∩	∩	NOUN
ejpam-4343	267	29	pıcl(a	pıcl(a	NOUN
ejpam-4343	267	30	)	)	PUNCT
ejpam-4343	267	31	.	.	PUNCT
ejpam-4343	268	1	this	this	PRON
ejpam-4343	268	2	shows	show	VERB
ejpam-4343	268	3	that	that	SCONJ
ejpam-4343	268	4	a	a	PRON
ejpam-4343	268	5	is	be	AUX
ejpam-4343	268	6	(	(	PUNCT
ejpam-4343	268	7	λ	λ	PROPN
ejpam-4343	268	8	,	,	PUNCT
ejpam-4343	268	9	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	268	10	.	.	PUNCT
ejpam-4343	269	1	remark	remark	PROPN
ejpam-4343	269	2	2	2	NUM
ejpam-4343	269	3	.	.	PUNCT
ejpam-4343	270	1	every	every	DET
ejpam-4343	270	2	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	270	3	(	(	PUNCT
ejpam-4343	270	4	resp	resp	NOUN
ejpam-4343	270	5	.	.	PUNCT
ejpam-4343	271	1	pre	pre	VERB
ejpam-4343	271	2	-	-	ADJ
ejpam-4343	271	3	i	i	PRON
ejpam-4343	271	4	-closed	-close	VERB
ejpam-4343	271	5	set	set	NOUN
ejpam-4343	271	6	)	)	PUNCT
ejpam-4343	271	7	is	be	AUX
ejpam-4343	271	8	(	(	PUNCT
ejpam-4343	271	9	λ	λ	X
ejpam-4343	271	10	,	,	PUNCT
ejpam-4343	271	11	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	271	12	.	.	PUNCT
ejpam-4343	272	1	the	the	DET
ejpam-4343	272	2	converse	converse	NOUN
ejpam-4343	272	3	of	of	ADP
ejpam-4343	272	4	remark	remark	NOUN
ejpam-4343	272	5	2	2	NUM
ejpam-4343	272	6	is	be	AUX
ejpam-4343	272	7	not	not	PART
ejpam-4343	272	8	true	true	ADJ
ejpam-4343	272	9	in	in	ADP
ejpam-4343	272	10	general	general	ADJ
ejpam-4343	272	11	as	as	SCONJ
ejpam-4343	272	12	shown	show	VERB
ejpam-4343	272	13	by	by	ADP
ejpam-4343	272	14	the	the	DET
ejpam-4343	272	15	following	follow	VERB
ejpam-4343	272	16	example	example	NOUN
ejpam-4343	272	17	.	.	PUNCT
ejpam-4343	273	1	example	example	NOUN
ejpam-4343	274	1	2	2	NUM
ejpam-4343	274	2	.	.	PUNCT
ejpam-4343	274	3	let	let	VERB
ejpam-4343	274	4	x	x	PUNCT
ejpam-4343	274	5	=	=	PRON
ejpam-4343	274	6	{	{	PUNCT
ejpam-4343	274	7	1	1	NUM
ejpam-4343	274	8	,	,	PUNCT
ejpam-4343	274	9	2	2	NUM
ejpam-4343	274	10	}	}	PUNCT
ejpam-4343	274	11	with	with	ADP
ejpam-4343	274	12	a	a	DET
ejpam-4343	274	13	topology	topology	NOUN
ejpam-4343	274	14	τ	τ	X
ejpam-4343	274	15	=	=	SYM
ejpam-4343	274	16	{	{	PUNCT
ejpam-4343	274	17	∅	∅	NOUN
ejpam-4343	274	18	,	,	PUNCT
ejpam-4343	274	19	{	{	PUNCT
ejpam-4343	274	20	1	1	NUM
ejpam-4343	274	21	}	}	PUNCT
ejpam-4343	274	22	,	,	PUNCT
ejpam-4343	274	23	x	x	NOUN
ejpam-4343	274	24	}	}	PUNCT
ejpam-4343	274	25	and	and	CCONJ
ejpam-4343	274	26	an	an	DET
ejpam-4343	274	27	ideal	ideal	NOUN
ejpam-4343	274	28	i	i	X
ejpam-4343	274	29	=	=	SYM
ejpam-4343	274	30	{	{	PUNCT
ejpam-4343	274	31	∅	∅	NOUN
ejpam-4343	274	32	,	,	PUNCT
ejpam-4343	274	33	{	{	PUNCT
ejpam-4343	274	34	1	1	NUM
ejpam-4343	274	35	}	}	PUNCT
ejpam-4343	274	36	}	}	PUNCT
ejpam-4343	274	37	.	.	PUNCT
ejpam-4343	275	1	let	let	VERB
ejpam-4343	275	2	a	a	PRON
ejpam-4343	275	3	=	=	X
ejpam-4343	275	4	{	{	PUNCT
ejpam-4343	275	5	1	1	NUM
ejpam-4343	275	6	}	}	PUNCT
ejpam-4343	275	7	.	.	PUNCT
ejpam-4343	276	1	then	then	ADV
ejpam-4343	276	2	,	,	PUNCT
ejpam-4343	276	3	a	a	DET
ejpam-4343	276	4	is	be	AUX
ejpam-4343	276	5	(	(	PUNCT
ejpam-4343	276	6	λ	λ	X
ejpam-4343	276	7	,	,	PUNCT
ejpam-4343	276	8	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	276	9	but	but	CCONJ
ejpam-4343	276	10	it	it	PRON
ejpam-4343	276	11	is	be	AUX
ejpam-4343	276	12	not	not	PART
ejpam-4343	276	13	pre	pre	ADJ
ejpam-4343	276	14	-	-	ADJ
ejpam-4343	276	15	i	i	PRON
ejpam-4343	276	16	-closed	-closed	ADJ
ejpam-4343	276	17	.	.	PUNCT
ejpam-4343	277	1	moreover	moreover	ADV
ejpam-4343	277	2	,	,	PUNCT
ejpam-4343	277	3	let	let	VERB
ejpam-4343	277	4	b	b	NOUN
ejpam-4343	277	5	=	=	PRON
ejpam-4343	277	6	{	{	PUNCT
ejpam-4343	277	7	2	2	NUM
ejpam-4343	277	8	}	}	PUNCT
ejpam-4343	277	9	,	,	PUNCT
ejpam-4343	277	10	then	then	ADV
ejpam-4343	277	11	b	b	X
ejpam-4343	277	12	is	be	AUX
ejpam-4343	277	13	(	(	PUNCT
ejpam-4343	277	14	λ	λ	X
ejpam-4343	277	15	,	,	PUNCT
ejpam-4343	277	16	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	277	17	but	but	CCONJ
ejpam-4343	277	18	it	it	PRON
ejpam-4343	277	19	is	be	AUX
ejpam-4343	277	20	not	not	PART
ejpam-4343	277	21	a	a	DET
ejpam-4343	277	22	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	277	23	.	.	PUNCT
ejpam-4343	278	1	definition	definition	NOUN
ejpam-4343	278	2	7	7	NUM
ejpam-4343	278	3	.	.	PUNCT
ejpam-4343	279	1	a	a	DET
ejpam-4343	279	2	subset	subset	NOUN
ejpam-4343	279	3	a	a	PRON
ejpam-4343	279	4	of	of	ADP
ejpam-4343	279	5	an	an	DET
ejpam-4343	279	6	ideal	ideal	ADJ
ejpam-4343	279	7	topological	topological	ADJ
ejpam-4343	279	8	space	space	NOUN
ejpam-4343	279	9	(	(	PUNCT
ejpam-4343	279	10	x	x	X
ejpam-4343	279	11	,	,	PUNCT
ejpam-4343	279	12	τ	τ	PROPN
ejpam-4343	279	13	,	,	PUNCT
ejpam-4343	279	14	i	i	PROPN
ejpam-4343	279	15	)	)	PUNCT
ejpam-4343	279	16	is	be	AUX
ejpam-4343	279	17	said	say	VERB
ejpam-4343	279	18	to	to	PART
ejpam-4343	279	19	be	be	AUX
ejpam-4343	279	20	(	(	PUNCT
ejpam-4343	279	21	λ	λ	X
ejpam-4343	279	22	,	,	PUNCT
ejpam-4343	279	23	p(⋆))open	p(⋆))open	VERB
ejpam-4343	279	24	if	if	SCONJ
ejpam-4343	279	25	the	the	DET
ejpam-4343	279	26	complement	complement	NOUN
ejpam-4343	279	27	of	of	ADP
ejpam-4343	279	28	a	a	DET
ejpam-4343	279	29	is	is	NOUN
ejpam-4343	279	30	(	(	PUNCT
ejpam-4343	279	31	λ	λ	X
ejpam-4343	279	32	,	,	PUNCT
ejpam-4343	279	33	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	279	34	.	.	PUNCT
ejpam-4343	280	1	the	the	DET
ejpam-4343	280	2	collection	collection	NOUN
ejpam-4343	280	3	of	of	ADP
ejpam-4343	280	4	all	all	DET
ejpam-4343	280	5	(	(	PUNCT
ejpam-4343	280	6	λ	λ	PROPN
ejpam-4343	280	7	,	,	PUNCT
ejpam-4343	280	8	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	280	9	sets	set	NOUN
ejpam-4343	280	10	in	in	ADP
ejpam-4343	280	11	an	an	DET
ejpam-4343	280	12	ideal	ideal	ADJ
ejpam-4343	280	13	topological	topological	ADJ
ejpam-4343	280	14	space	space	NOUN
ejpam-4343	280	15	(	(	PUNCT
ejpam-4343	280	16	x	x	X
ejpam-4343	280	17	,	,	PUNCT
ejpam-4343	280	18	τ	τ	PROPN
ejpam-4343	280	19	,	,	PUNCT
ejpam-4343	280	20	i	i	PROPN
ejpam-4343	280	21	)	)	PUNCT
ejpam-4343	280	22	is	be	AUX
ejpam-4343	280	23	denoted	denote	VERB
ejpam-4343	280	24	by	by	ADP
ejpam-4343	280	25	λp(⋆)o(x	λp(⋆)o(x	PROPN
ejpam-4343	280	26	)	)	PUNCT
ejpam-4343	280	27	.	.	PUNCT
ejpam-4343	281	1	theorem	theorem	NOUN
ejpam-4343	281	2	2	2	NUM
ejpam-4343	281	3	.	.	PUNCT
ejpam-4343	282	1	let	let	VERB
ejpam-4343	282	2	aγ(γ	aγ(γ	NUM
ejpam-4343	282	3	∈	∈	PROPN
ejpam-4343	282	4	γ	γ	X
ejpam-4343	282	5	)	)	PUNCT
ejpam-4343	282	6	be	be	VERB
ejpam-4343	282	7	a	a	DET
ejpam-4343	282	8	subset	subset	NOUN
ejpam-4343	282	9	of	of	ADP
ejpam-4343	282	10	an	an	DET
ejpam-4343	282	11	ideal	ideal	ADJ
ejpam-4343	282	12	topological	topological	ADJ
ejpam-4343	282	13	space	space	NOUN
ejpam-4343	282	14	(	(	PUNCT
ejpam-4343	282	15	x	x	X
ejpam-4343	282	16	,	,	PUNCT
ejpam-4343	282	17	τ	τ	PROPN
ejpam-4343	282	18	,	,	PUNCT
ejpam-4343	282	19	i	i	NOUN
ejpam-4343	282	20	)	)	PUNCT
ejpam-4343	282	21	.	.	PUNCT
ejpam-4343	283	1	then	then	ADV
ejpam-4343	283	2	,	,	PUNCT
ejpam-4343	283	3	the	the	DET
ejpam-4343	283	4	following	follow	VERB
ejpam-4343	283	5	properties	property	NOUN
ejpam-4343	283	6	hold	hold	VERB
ejpam-4343	283	7	:	:	PUNCT
ejpam-4343	283	8	(	(	PUNCT
ejpam-4343	283	9	1	1	X
ejpam-4343	283	10	)	)	PUNCT
ejpam-4343	283	11	if	if	SCONJ
ejpam-4343	283	12	aγ	aγ	PRON
ejpam-4343	283	13	is	be	AUX
ejpam-4343	283	14	(	(	PUNCT
ejpam-4343	283	15	λ	λ	X
ejpam-4343	283	16	,	,	PUNCT
ejpam-4343	283	17	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	283	18	for	for	ADP
ejpam-4343	283	19	each	each	DET
ejpam-4343	283	20	γ	γ	PROPN
ejpam-4343	283	21	∈	∈	PROPN
ejpam-4343	283	22	γ	γ	X
ejpam-4343	283	23	,	,	PUNCT
ejpam-4343	283	24	then	then	ADV
ejpam-4343	283	25	∩{aγ	∩{aγ	VERB
ejpam-4343	283	26	|	|	ADV
ejpam-4343	283	27	γ	γ	PROPN
ejpam-4343	283	28	∈	∈	PROPN
ejpam-4343	283	29	γ	γ	X
ejpam-4343	283	30	}	}	PUNCT
ejpam-4343	283	31	is	be	AUX
ejpam-4343	283	32	(	(	PUNCT
ejpam-4343	283	33	λ	λ	X
ejpam-4343	283	34	,	,	PUNCT
ejpam-4343	283	35	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	283	36	.	.	PUNCT
ejpam-4343	284	1	(	(	PUNCT
ejpam-4343	284	2	2	2	X
ejpam-4343	284	3	)	)	PUNCT
ejpam-4343	284	4	if	if	SCONJ
ejpam-4343	284	5	aγ	aγ	PRON
ejpam-4343	284	6	is	be	AUX
ejpam-4343	284	7	(	(	PUNCT
ejpam-4343	284	8	λ	λ	X
ejpam-4343	284	9	,	,	PUNCT
ejpam-4343	284	10	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	284	11	for	for	ADP
ejpam-4343	284	12	each	each	DET
ejpam-4343	284	13	γ	γ	PROPN
ejpam-4343	284	14	∈	∈	PROPN
ejpam-4343	284	15	γ	γ	X
ejpam-4343	284	16	,	,	PUNCT
ejpam-4343	284	17	then	then	ADV
ejpam-4343	284	18	∪{aγ	∪{aγ	PROPN
ejpam-4343	284	19	|	|	ADV
ejpam-4343	284	20	γ	γ	PROPN
ejpam-4343	284	21	∈	∈	PROPN
ejpam-4343	284	22	γ	γ	X
ejpam-4343	284	23	}	}	PUNCT
ejpam-4343	284	24	is	be	AUX
ejpam-4343	284	25	(	(	PUNCT
ejpam-4343	284	26	λ	λ	X
ejpam-4343	284	27	,	,	PUNCT
ejpam-4343	284	28	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	284	29	.	.	PUNCT
ejpam-4343	285	1	proof	proof	NOUN
ejpam-4343	285	2	.	.	PUNCT
ejpam-4343	286	1	(	(	PUNCT
ejpam-4343	286	2	1	1	X
ejpam-4343	286	3	)	)	PUNCT
ejpam-4343	286	4	suppose	suppose	VERB
ejpam-4343	286	5	that	that	SCONJ
ejpam-4343	286	6	aγ	aγ	PRON
ejpam-4343	286	7	is	be	AUX
ejpam-4343	286	8	(	(	PUNCT
ejpam-4343	286	9	λ	λ	X
ejpam-4343	286	10	,	,	PUNCT
ejpam-4343	286	11	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	286	12	for	for	ADP
ejpam-4343	286	13	each	each	DET
ejpam-4343	286	14	γ	γ	PROPN
ejpam-4343	286	15	∈	∈	PROPN
ejpam-4343	286	16	γ	γ	X
ejpam-4343	286	17	.	.	PROPN
ejpam-4343	287	1	then	then	ADV
ejpam-4343	287	2	,	,	PUNCT
ejpam-4343	287	3	for	for	ADP
ejpam-4343	287	4	each	each	DET
ejpam-4343	287	5	γ	γ	NOUN
ejpam-4343	287	6	,	,	PUNCT
ejpam-4343	287	7	there	there	PRON
ejpam-4343	287	8	exist	exist	VERB
ejpam-4343	287	9	a	a	DET
ejpam-4343	287	10	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	287	11	tγ	tγ	NOUN
ejpam-4343	287	12	and	and	CCONJ
ejpam-4343	287	13	a	a	DET
ejpam-4343	287	14	pre	pre	NOUN
ejpam-4343	287	15	-	-	ADJ
ejpam-4343	287	16	i	i	PRON
ejpam-4343	287	17	-closed	-close	VERB
ejpam-4343	287	18	set	set	NOUN
ejpam-4343	287	19	cγ	cγ	ADP
ejpam-4343	287	20	such	such	ADJ
ejpam-4343	287	21	that	that	SCONJ
ejpam-4343	287	22	aγ	aγ	PRON
ejpam-4343	287	23	=	=	SYM
ejpam-4343	287	24	tγ	tγ	PROPN
ejpam-4343	287	25	∩	∩	ADJ
ejpam-4343	287	26	cγ	cγ	NOUN
ejpam-4343	287	27	.	.	PUNCT
ejpam-4343	288	1	thus	thus	ADV
ejpam-4343	288	2	,	,	PUNCT
ejpam-4343	288	3	∩γ∈γaγ	∩γ∈γaγ	PUNCT
ejpam-4343	288	4	=	=	SYM
ejpam-4343	288	5	∩γ∈γ(tγ	∩γ∈γ(tγ	NOUN
ejpam-4343	288	6	∩	∩	NOUN
ejpam-4343	288	7	cγ	cγ	NOUN
ejpam-4343	288	8	)	)	PUNCT
ejpam-4343	288	9	=	=	SYM
ejpam-4343	288	10	(	(	PUNCT
ejpam-4343	288	11	∩γ∈γtγ	∩γ∈γtγ	NOUN
ejpam-4343	288	12	)	)	PUNCT
ejpam-4343	288	13	∩	∩	NOUN
ejpam-4343	288	14	(	(	PUNCT
ejpam-4343	288	15	∩γ∈γcγ	∩γ∈γcγ	NOUN
ejpam-4343	288	16	)	)	PUNCT
ejpam-4343	288	17	.	.	PUNCT
ejpam-4343	289	1	since	since	SCONJ
ejpam-4343	289	2	∩γ∈γcγ	∩γ∈γcγ	PROPN
ejpam-4343	289	3	is	be	AUX
ejpam-4343	289	4	pre	pre	ADJ
ejpam-4343	289	5	-	-	ADJ
ejpam-4343	289	6	i	i	PRON
ejpam-4343	289	7	-closed	-close	VERB
ejpam-4343	289	8	and	and	CCONJ
ejpam-4343	289	9	∩γ∈γtγ	∩γ∈γtγ	PROPN
ejpam-4343	289	10	is	be	AUX
ejpam-4343	289	11	a	a	DET
ejpam-4343	289	12	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	289	13	by	by	ADP
ejpam-4343	289	14	proposition	proposition	NOUN
ejpam-4343	289	15	2(4	2(4	NUM
ejpam-4343	289	16	)	)	PUNCT
ejpam-4343	289	17	,	,	PUNCT
ejpam-4343	289	18	we	we	PRON
ejpam-4343	289	19	have	have	VERB
ejpam-4343	289	20	∩γ∈γaγ	∩γ∈γaγ	VERB
ejpam-4343	289	21	is	be	AUX
ejpam-4343	289	22	(	(	PUNCT
ejpam-4343	289	23	λ	λ	X
ejpam-4343	289	24	,	,	PUNCT
ejpam-4343	289	25	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	289	26	.	.	PUNCT
ejpam-4343	290	1	(	(	PUNCT
ejpam-4343	290	2	2	2	X
ejpam-4343	290	3	)	)	PUNCT
ejpam-4343	290	4	let	let	VERB
ejpam-4343	290	5	aγ	aγ	PRON
ejpam-4343	290	6	is	be	AUX
ejpam-4343	290	7	(	(	PUNCT
ejpam-4343	290	8	λ	λ	X
ejpam-4343	290	9	,	,	PUNCT
ejpam-4343	290	10	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	290	11	for	for	ADP
ejpam-4343	290	12	each	each	DET
ejpam-4343	290	13	γ	γ	PROPN
ejpam-4343	290	14	∈	∈	PROPN
ejpam-4343	290	15	γ	γ	X
ejpam-4343	290	16	.	.	PUNCT
ejpam-4343	291	1	then	then	ADV
ejpam-4343	291	2	,	,	PUNCT
ejpam-4343	291	3	x	x	PUNCT
ejpam-4343	291	4	−	−	NOUN
ejpam-4343	292	1	aγ	aγ	PRON
ejpam-4343	292	2	is	be	AUX
ejpam-4343	292	3	(	(	PUNCT
ejpam-4343	292	4	λ	λ	X
ejpam-4343	292	5	,	,	PUNCT
ejpam-4343	292	6	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	292	7	.	.	PUNCT
ejpam-4343	293	1	since	since	SCONJ
ejpam-4343	293	2	x	x	X
ejpam-4343	293	3	−	−	PROPN
ejpam-4343	293	4	∪γ∈γaγ	∪γ∈γaγ	NOUN
ejpam-4343	293	5	=	=	SYM
ejpam-4343	293	6	∩γ∈γ(x	∩γ∈γ(x	NOUN
ejpam-4343	293	7	−aγ	−aγ	PROPN
ejpam-4343	293	8	)	)	PUNCT
ejpam-4343	293	9	and	and	CCONJ
ejpam-4343	293	10	by	by	ADP
ejpam-4343	293	11	(	(	PUNCT
ejpam-4343	293	12	1	1	NUM
ejpam-4343	293	13	)	)	PUNCT
ejpam-4343	293	14	,	,	PUNCT
ejpam-4343	293	15	∪γ∈γaγ	∪γ∈γaγ	PROPN
ejpam-4343	293	16	is	be	AUX
ejpam-4343	293	17	(	(	PUNCT
ejpam-4343	293	18	λ	λ	INTJ
ejpam-4343	293	19	,	,	PUNCT
ejpam-4343	293	20	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	293	21	.	.	PUNCT
ejpam-4343	294	1	theorem	theorem	NOUN
ejpam-4343	294	2	3	3	NUM
ejpam-4343	294	3	.	.	X
ejpam-4343	294	4	for	for	ADP
ejpam-4343	294	5	a	a	DET
ejpam-4343	294	6	subset	subset	NOUN
ejpam-4343	294	7	a	a	PRON
ejpam-4343	294	8	of	of	ADP
ejpam-4343	294	9	an	an	DET
ejpam-4343	294	10	ideal	ideal	ADJ
ejpam-4343	294	11	topological	topological	ADJ
ejpam-4343	294	12	space	space	NOUN
ejpam-4343	294	13	(	(	PUNCT
ejpam-4343	294	14	x	x	X
ejpam-4343	294	15	,	,	PUNCT
ejpam-4343	294	16	τ	τ	PROPN
ejpam-4343	294	17	,	,	PUNCT
ejpam-4343	294	18	i	i	NOUN
ejpam-4343	294	19	)	)	PUNCT
ejpam-4343	294	20	,	,	PUNCT
ejpam-4343	294	21	the	the	DET
ejpam-4343	294	22	following	follow	VERB
ejpam-4343	294	23	properties	property	NOUN
ejpam-4343	294	24	are	be	AUX
ejpam-4343	294	25	equivalent	equivalent	ADJ
ejpam-4343	294	26	:	:	PUNCT
ejpam-4343	294	27	(	(	PUNCT
ejpam-4343	294	28	1	1	X
ejpam-4343	294	29	)	)	PUNCT
ejpam-4343	294	30	a	a	PRON
ejpam-4343	294	31	is	be	AUX
ejpam-4343	294	32	(	(	PUNCT
ejpam-4343	294	33	λ	λ	INTJ
ejpam-4343	294	34	,	,	PUNCT
ejpam-4343	294	35	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	294	36	.	.	PUNCT
ejpam-4343	295	1	(	(	PUNCT
ejpam-4343	295	2	2	2	X
ejpam-4343	295	3	)	)	PUNCT
ejpam-4343	295	4	a	a	DET
ejpam-4343	295	5	=	=	X
ejpam-4343	295	6	s	s	X
ejpam-4343	295	7	∪	∪	ADJ
ejpam-4343	295	8	v	v	NOUN
ejpam-4343	295	9	,	,	PUNCT
ejpam-4343	295	10	where	where	SCONJ
ejpam-4343	295	11	s	s	NOUN
ejpam-4343	295	12	is	be	AUX
ejpam-4343	295	13	a	a	DET
ejpam-4343	295	14	δp(⋆)-set	δp(⋆)-set	NOUN
ejpam-4343	295	15	and	and	CCONJ
ejpam-4343	295	16	v	v	NOUN
ejpam-4343	295	17	is	be	AUX
ejpam-4343	295	18	a	a	DET
ejpam-4343	295	19	pre	pre	ADJ
ejpam-4343	295	20	-	-	ADJ
ejpam-4343	295	21	i	i	PRON
ejpam-4343	295	22	-open	-open	NOUN
ejpam-4343	295	23	set	set	NOUN
ejpam-4343	295	24	.	.	PUNCT
ejpam-4343	296	1	(	(	PUNCT
ejpam-4343	296	2	3	3	X
ejpam-4343	296	3	)	)	PUNCT
ejpam-4343	296	4	a	a	DET
ejpam-4343	296	5	=	=	X
ejpam-4343	296	6	s	s	X
ejpam-4343	296	7	∪	∪	ADJ
ejpam-4343	296	8	pıint(a	pıint(a	PROPN
ejpam-4343	296	9	)	)	PUNCT
ejpam-4343	296	10	,	,	PUNCT
ejpam-4343	296	11	where	where	SCONJ
ejpam-4343	296	12	s	s	NOUN
ejpam-4343	296	13	is	be	AUX
ejpam-4343	296	14	a	a	DET
ejpam-4343	296	15	δp(⋆)-set	δp(⋆)-set	PROPN
ejpam-4343	296	16	.	.	PUNCT
ejpam-4343	297	1	c.	c.	PROPN
ejpam-4343	297	2	boonpok	boonpok	PROPN
ejpam-4343	297	3	/	/	SYM
ejpam-4343	297	4	eur	eur	PROPN
ejpam-4343	297	5	.	.	PUNCT
ejpam-4343	298	1	j.	j.	PROPN
ejpam-4343	298	2	pure	pure	PROPN
ejpam-4343	298	3	appl	appl	PROPN
ejpam-4343	298	4	.	.	PROPN
ejpam-4343	298	5	math	math	PROPN
ejpam-4343	298	6	,	,	PUNCT
ejpam-4343	298	7	15	15	NUM
ejpam-4343	298	8	(	(	PUNCT
ejpam-4343	298	9	3	3	NUM
ejpam-4343	298	10	)	)	PUNCT
ejpam-4343	298	11	(	(	PUNCT
ejpam-4343	298	12	2022	2022	NUM
ejpam-4343	298	13	)	)	PUNCT
ejpam-4343	298	14	,	,	PUNCT
ejpam-4343	298	15	1023	1023	NUM
ejpam-4343	298	16	-	-	SYM
ejpam-4343	298	17	1046	1046	NUM
ejpam-4343	298	18	1031	1031	NUM
ejpam-4343	298	19	(	(	PUNCT
ejpam-4343	298	20	4	4	NUM
ejpam-4343	298	21	)	)	PUNCT
ejpam-4343	298	22	a	a	DET
ejpam-4343	298	23	=	=	SYM
ejpam-4343	298	24	δp(⋆)(a	δp(⋆)(a	PROPN
ejpam-4343	298	25	)	)	PUNCT
ejpam-4343	298	26	∪	∪	ADP
ejpam-4343	298	27	pıint(a	pıint(a	PROPN
ejpam-4343	298	28	)	)	PUNCT
ejpam-4343	298	29	.	.	PUNCT
ejpam-4343	299	1	proof	proof	NOUN
ejpam-4343	299	2	.	.	PUNCT
ejpam-4343	300	1	the	the	DET
ejpam-4343	300	2	proof	proof	NOUN
ejpam-4343	300	3	follows	follow	VERB
ejpam-4343	300	4	from	from	ADP
ejpam-4343	300	5	theorem	theorem	ADJ
ejpam-4343	300	6	1	1	NUM
ejpam-4343	300	7	.	.	PUNCT
ejpam-4343	300	8	definition	definition	NOUN
ejpam-4343	300	9	8	8	NUM
ejpam-4343	300	10	.	.	PUNCT
ejpam-4343	301	1	let	let	VERB
ejpam-4343	301	2	a	a	DET
ejpam-4343	301	3	be	be	AUX
ejpam-4343	301	4	a	a	DET
ejpam-4343	301	5	subset	subset	NOUN
ejpam-4343	301	6	of	of	ADP
ejpam-4343	301	7	an	an	DET
ejpam-4343	301	8	ideal	ideal	ADJ
ejpam-4343	301	9	topological	topological	ADJ
ejpam-4343	301	10	space	space	NOUN
ejpam-4343	301	11	(	(	PUNCT
ejpam-4343	301	12	x	x	X
ejpam-4343	301	13	,	,	PUNCT
ejpam-4343	301	14	τ	τ	PROPN
ejpam-4343	301	15	,	,	PUNCT
ejpam-4343	301	16	i	i	NOUN
ejpam-4343	301	17	)	)	PUNCT
ejpam-4343	301	18	.	.	PUNCT
ejpam-4343	302	1	a	a	DET
ejpam-4343	302	2	point	point	NOUN
ejpam-4343	302	3	x	x	X
ejpam-4343	302	4	∈	∈	NOUN
ejpam-4343	302	5	x	x	PUNCT
ejpam-4343	302	6	is	be	AUX
ejpam-4343	302	7	called	call	VERB
ejpam-4343	302	8	a	a	DET
ejpam-4343	302	9	(	(	PUNCT
ejpam-4343	302	10	λ	λ	PROPN
ejpam-4343	302	11	,	,	PUNCT
ejpam-4343	302	12	p(⋆))-cluster	p(⋆))-cluster	NOUN
ejpam-4343	302	13	point	point	NOUN
ejpam-4343	302	14	of	of	ADP
ejpam-4343	302	15	a	a	DET
ejpam-4343	302	16	if	if	SCONJ
ejpam-4343	302	17	a∩u	a∩u	VERB
ejpam-4343	302	18	̸=	̸=	NOUN
ejpam-4343	302	19	∅	∅	NOUN
ejpam-4343	302	20	for	for	ADP
ejpam-4343	302	21	every	every	DET
ejpam-4343	302	22	(	(	PUNCT
ejpam-4343	302	23	λ	λ	PROPN
ejpam-4343	302	24	,	,	PUNCT
ejpam-4343	302	25	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	302	26	set	set	VERB
ejpam-4343	302	27	u	u	NOUN
ejpam-4343	302	28	containing	contain	VERB
ejpam-4343	302	29	x.	x.	NOUN
ejpam-4343	302	30	the	the	DET
ejpam-4343	302	31	set	set	NOUN
ejpam-4343	302	32	of	of	ADP
ejpam-4343	302	33	all	all	DET
ejpam-4343	302	34	(	(	PUNCT
ejpam-4343	302	35	λ	λ	PROPN
ejpam-4343	302	36	,	,	PUNCT
ejpam-4343	302	37	p(⋆))-cluster	p(⋆))-cluster	NOUN
ejpam-4343	302	38	points	point	NOUN
ejpam-4343	302	39	of	of	ADP
ejpam-4343	302	40	a	a	PRON
ejpam-4343	302	41	is	be	AUX
ejpam-4343	302	42	called	call	VERB
ejpam-4343	302	43	the	the	DET
ejpam-4343	302	44	(	(	PUNCT
ejpam-4343	302	45	λ	λ	PROPN
ejpam-4343	302	46	,	,	PUNCT
ejpam-4343	302	47	p(⋆))-closure	p(⋆))-closure	NOUN
ejpam-4343	302	48	of	of	ADP
ejpam-4343	302	49	a	a	PRON
ejpam-4343	302	50	and	and	CCONJ
ejpam-4343	302	51	is	be	AUX
ejpam-4343	302	52	denoted	denote	VERB
ejpam-4343	302	53	by	by	ADP
ejpam-4343	302	54	a(λ	a(λ	PROPN
ejpam-4343	302	55	,	,	PUNCT
ejpam-4343	302	56	p(⋆	p(⋆	PROPN
ejpam-4343	302	57	)	)	PUNCT
ejpam-4343	302	58	)	)	PUNCT
ejpam-4343	302	59	.	.	PUNCT
ejpam-4343	303	1	lemma	lemma	PROPN
ejpam-4343	303	2	3	3	X
ejpam-4343	303	3	.	.	PUNCT
ejpam-4343	304	1	let	let	VERB
ejpam-4343	304	2	a	a	PRON
ejpam-4343	304	3	and	and	CCONJ
ejpam-4343	304	4	b	b	NOUN
ejpam-4343	304	5	be	be	AUX
ejpam-4343	304	6	subsets	subset	NOUN
ejpam-4343	304	7	of	of	ADP
ejpam-4343	304	8	an	an	DET
ejpam-4343	304	9	ideal	ideal	ADJ
ejpam-4343	304	10	topological	topological	ADJ
ejpam-4343	304	11	space	space	NOUN
ejpam-4343	304	12	(	(	PUNCT
ejpam-4343	304	13	x	x	X
ejpam-4343	304	14	,	,	PUNCT
ejpam-4343	304	15	τ	τ	PROPN
ejpam-4343	304	16	,	,	PUNCT
ejpam-4343	304	17	i	i	NOUN
ejpam-4343	304	18	)	)	PUNCT
ejpam-4343	304	19	.	.	PUNCT
ejpam-4343	305	1	for	for	ADP
ejpam-4343	305	2	the	the	DET
ejpam-4343	305	3	(	(	PUNCT
ejpam-4343	305	4	λ	λ	PROPN
ejpam-4343	305	5	,	,	PUNCT
ejpam-4343	305	6	p(⋆))-closure	p(⋆))-closure	NOUN
ejpam-4343	305	7	,	,	PUNCT
ejpam-4343	305	8	the	the	DET
ejpam-4343	305	9	following	follow	VERB
ejpam-4343	305	10	properties	property	NOUN
ejpam-4343	305	11	hold	hold	VERB
ejpam-4343	305	12	:	:	PUNCT
ejpam-4343	305	13	(	(	PUNCT
ejpam-4343	305	14	1	1	X
ejpam-4343	305	15	)	)	PUNCT
ejpam-4343	305	16	a	a	DET
ejpam-4343	305	17	⊆	⊆	NUM
ejpam-4343	305	18	a(λ	a(λ	ADJ
ejpam-4343	305	19	,	,	PUNCT
ejpam-4343	305	20	p(⋆	p(⋆	PROPN
ejpam-4343	305	21	)	)	PUNCT
ejpam-4343	305	22	)	)	PUNCT
ejpam-4343	306	1	and	and	CCONJ
ejpam-4343	306	2	[	[	X
ejpam-4343	306	3	a(λ	a(λ	ADV
ejpam-4343	306	4	,	,	PUNCT
ejpam-4343	306	5	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	306	6	,	,	PUNCT
ejpam-4343	306	7	p(⋆	p(⋆	PROPN
ejpam-4343	306	8	)	)	PUNCT
ejpam-4343	306	9	)	)	PUNCT
ejpam-4343	307	1	=	=	PUNCT
ejpam-4343	307	2	a(λ	a(λ	ADV
ejpam-4343	307	3	,	,	PUNCT
ejpam-4343	307	4	p(⋆	p(⋆	PROPN
ejpam-4343	307	5	)	)	PUNCT
ejpam-4343	307	6	)	)	PUNCT
ejpam-4343	307	7	.	.	PUNCT
ejpam-4343	308	1	(	(	PUNCT
ejpam-4343	308	2	2	2	NUM
ejpam-4343	308	3	)	)	PUNCT
ejpam-4343	308	4	a(λ	a(λ	ADV
ejpam-4343	308	5	,	,	PUNCT
ejpam-4343	308	6	p(⋆	p(⋆	PROPN
ejpam-4343	308	7	)	)	PUNCT
ejpam-4343	308	8	)	)	PUNCT
ejpam-4343	309	1	=	=	PRON
ejpam-4343	309	2	∩{f	∩{f	NOUN
ejpam-4343	309	3	|	|	ADV
ejpam-4343	309	4	a	a	DET
ejpam-4343	309	5	⊆	⊆	NUM
ejpam-4343	309	6	f	f	NOUN
ejpam-4343	309	7	and	and	CCONJ
ejpam-4343	309	8	f	f	PROPN
ejpam-4343	309	9	is	be	AUX
ejpam-4343	309	10	(	(	PUNCT
ejpam-4343	309	11	λ	λ	X
ejpam-4343	309	12	,	,	PUNCT
ejpam-4343	309	13	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	309	14	}	}	PUNCT
ejpam-4343	309	15	.	.	PUNCT
ejpam-4343	310	1	(	(	PUNCT
ejpam-4343	310	2	3	3	X
ejpam-4343	310	3	)	)	PUNCT
ejpam-4343	310	4	if	if	SCONJ
ejpam-4343	310	5	a	a	DET
ejpam-4343	310	6	⊆	⊆	NUM
ejpam-4343	310	7	b	b	NOUN
ejpam-4343	310	8	,	,	PUNCT
ejpam-4343	310	9	then	then	ADV
ejpam-4343	310	10	a(λ	a(λ	ADV
ejpam-4343	310	11	,	,	PUNCT
ejpam-4343	310	12	p(⋆	p(⋆	PROPN
ejpam-4343	310	13	)	)	PUNCT
ejpam-4343	310	14	)	)	PUNCT
ejpam-4343	310	15	⊆	⊆	NUM
ejpam-4343	310	16	b(λ	b(λ	NOUN
ejpam-4343	310	17	,	,	PUNCT
ejpam-4343	310	18	p(⋆	p(⋆	PROPN
ejpam-4343	310	19	)	)	PUNCT
ejpam-4343	310	20	)	)	PUNCT
ejpam-4343	310	21	.	.	PUNCT
ejpam-4343	311	1	(	(	PUNCT
ejpam-4343	311	2	4	4	X
ejpam-4343	311	3	)	)	PUNCT
ejpam-4343	311	4	a	a	PRON
ejpam-4343	311	5	is	be	AUX
ejpam-4343	311	6	(	(	PUNCT
ejpam-4343	311	7	λ	λ	X
ejpam-4343	311	8	,	,	PUNCT
ejpam-4343	311	9	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	311	10	if	if	SCONJ
ejpam-4343	311	11	and	and	CCONJ
ejpam-4343	311	12	only	only	ADV
ejpam-4343	311	13	if	if	SCONJ
ejpam-4343	311	14	a(λ	a(λ	ADV
ejpam-4343	311	15	,	,	PUNCT
ejpam-4343	311	16	p(⋆	p(⋆	PROPN
ejpam-4343	311	17	)	)	PUNCT
ejpam-4343	311	18	)	)	PUNCT
ejpam-4343	312	1	=	=	PUNCT
ejpam-4343	312	2	a.	a.	NOUN
ejpam-4343	312	3	(	(	PUNCT
ejpam-4343	312	4	5	5	NUM
ejpam-4343	312	5	)	)	PUNCT
ejpam-4343	312	6	a(λ	a(λ	ADV
ejpam-4343	312	7	,	,	PUNCT
ejpam-4343	312	8	p(⋆	p(⋆	PROPN
ejpam-4343	312	9	)	)	PUNCT
ejpam-4343	312	10	)	)	PUNCT
ejpam-4343	312	11	is	be	AUX
ejpam-4343	312	12	(	(	PUNCT
ejpam-4343	312	13	λ	λ	X
ejpam-4343	312	14	,	,	PUNCT
ejpam-4343	312	15	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	312	16	.	.	PUNCT
ejpam-4343	313	1	definition	definition	NOUN
ejpam-4343	313	2	9	9	NUM
ejpam-4343	313	3	.	.	PUNCT
ejpam-4343	314	1	let	let	VERB
ejpam-4343	314	2	a	a	DET
ejpam-4343	314	3	be	be	AUX
ejpam-4343	314	4	a	a	DET
ejpam-4343	314	5	subset	subset	NOUN
ejpam-4343	314	6	of	of	ADP
ejpam-4343	314	7	an	an	DET
ejpam-4343	314	8	ideal	ideal	ADJ
ejpam-4343	314	9	topological	topological	ADJ
ejpam-4343	314	10	space	space	NOUN
ejpam-4343	314	11	(	(	PUNCT
ejpam-4343	314	12	x	x	X
ejpam-4343	314	13	,	,	PUNCT
ejpam-4343	314	14	τ	τ	PROPN
ejpam-4343	314	15	,	,	PUNCT
ejpam-4343	314	16	i	i	NOUN
ejpam-4343	314	17	)	)	PUNCT
ejpam-4343	314	18	.	.	PUNCT
ejpam-4343	315	1	the	the	DET
ejpam-4343	315	2	union	union	NOUN
ejpam-4343	315	3	of	of	ADP
ejpam-4343	315	4	all	all	DET
ejpam-4343	315	5	(	(	PUNCT
ejpam-4343	315	6	λ	λ	PROPN
ejpam-4343	315	7	,	,	PUNCT
ejpam-4343	315	8	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	315	9	sets	set	NOUN
ejpam-4343	315	10	contained	contain	VERB
ejpam-4343	315	11	in	in	ADP
ejpam-4343	315	12	a	a	PRON
ejpam-4343	315	13	is	be	AUX
ejpam-4343	315	14	called	call	VERB
ejpam-4343	315	15	the	the	DET
ejpam-4343	315	16	(	(	PUNCT
ejpam-4343	315	17	λ	λ	PROPN
ejpam-4343	315	18	,	,	PUNCT
ejpam-4343	315	19	p(⋆))-interior	p(⋆))-interior	ADJ
ejpam-4343	315	20	of	of	ADP
ejpam-4343	315	21	a	a	PRON
ejpam-4343	315	22	and	and	CCONJ
ejpam-4343	315	23	is	be	AUX
ejpam-4343	315	24	denoted	denote	VERB
ejpam-4343	315	25	by	by	ADP
ejpam-4343	315	26	a(λ	a(λ	PROPN
ejpam-4343	315	27	,	,	PUNCT
ejpam-4343	315	28	p(⋆	p(⋆	PROPN
ejpam-4343	315	29	)	)	PUNCT
ejpam-4343	315	30	)	)	PUNCT
ejpam-4343	315	31	.	.	PUNCT
ejpam-4343	316	1	lemma	lemma	PROPN
ejpam-4343	316	2	4	4	X
ejpam-4343	316	3	.	.	PUNCT
ejpam-4343	317	1	let	let	VERB
ejpam-4343	317	2	a	a	PRON
ejpam-4343	317	3	and	and	CCONJ
ejpam-4343	317	4	b	b	NOUN
ejpam-4343	317	5	be	be	AUX
ejpam-4343	317	6	subsets	subset	NOUN
ejpam-4343	317	7	of	of	ADP
ejpam-4343	317	8	an	an	DET
ejpam-4343	317	9	ideal	ideal	ADJ
ejpam-4343	317	10	topological	topological	ADJ
ejpam-4343	317	11	space	space	NOUN
ejpam-4343	317	12	(	(	PUNCT
ejpam-4343	317	13	x	x	X
ejpam-4343	317	14	,	,	PUNCT
ejpam-4343	317	15	τ	τ	PROPN
ejpam-4343	317	16	,	,	PUNCT
ejpam-4343	317	17	i	i	NOUN
ejpam-4343	317	18	)	)	PUNCT
ejpam-4343	317	19	.	.	PUNCT
ejpam-4343	318	1	for	for	ADP
ejpam-4343	318	2	the	the	DET
ejpam-4343	318	3	(	(	PUNCT
ejpam-4343	318	4	λ	λ	PROPN
ejpam-4343	318	5	,	,	PUNCT
ejpam-4343	318	6	p(⋆))-interior	p(⋆))-interior	PROPN
ejpam-4343	318	7	,	,	PUNCT
ejpam-4343	318	8	the	the	DET
ejpam-4343	318	9	following	follow	VERB
ejpam-4343	318	10	properties	property	NOUN
ejpam-4343	318	11	hold	hold	VERB
ejpam-4343	318	12	:	:	PUNCT
ejpam-4343	318	13	(	(	PUNCT
ejpam-4343	318	14	1	1	X
ejpam-4343	318	15	)	)	PUNCT
ejpam-4343	318	16	[	[	X
ejpam-4343	318	17	a(λ	a(λ	ADV
ejpam-4343	318	18	,	,	PUNCT
ejpam-4343	318	19	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	318	20	,	,	PUNCT
ejpam-4343	318	21	p(⋆	p(⋆	PROPN
ejpam-4343	318	22	)	)	PUNCT
ejpam-4343	318	23	)	)	PUNCT
ejpam-4343	319	1	=	=	PUNCT
ejpam-4343	319	2	a(λ	a(λ	ADV
ejpam-4343	319	3	,	,	PUNCT
ejpam-4343	319	4	p(⋆	p(⋆	PROPN
ejpam-4343	319	5	)	)	PUNCT
ejpam-4343	319	6	)	)	PUNCT
ejpam-4343	319	7	.	.	PUNCT
ejpam-4343	320	1	(	(	PUNCT
ejpam-4343	320	2	2	2	X
ejpam-4343	320	3	)	)	PUNCT
ejpam-4343	320	4	if	if	SCONJ
ejpam-4343	320	5	a	a	DET
ejpam-4343	320	6	⊆	⊆	NUM
ejpam-4343	320	7	b	b	NOUN
ejpam-4343	320	8	,	,	PUNCT
ejpam-4343	320	9	then	then	ADV
ejpam-4343	320	10	a(λ	a(λ	ADV
ejpam-4343	320	11	,	,	PUNCT
ejpam-4343	320	12	p(⋆	p(⋆	PROPN
ejpam-4343	320	13	)	)	PUNCT
ejpam-4343	320	14	)	)	PUNCT
ejpam-4343	320	15	⊆	⊆	NUM
ejpam-4343	320	16	b(λ	b(λ	NOUN
ejpam-4343	320	17	,	,	PUNCT
ejpam-4343	320	18	p(⋆	p(⋆	PROPN
ejpam-4343	320	19	)	)	PUNCT
ejpam-4343	320	20	)	)	PUNCT
ejpam-4343	320	21	.	.	PUNCT
ejpam-4343	321	1	(	(	PUNCT
ejpam-4343	321	2	4	4	X
ejpam-4343	321	3	)	)	PUNCT
ejpam-4343	321	4	a	a	PRON
ejpam-4343	321	5	is	be	AUX
ejpam-4343	321	6	(	(	PUNCT
ejpam-4343	321	7	λ	λ	X
ejpam-4343	321	8	,	,	PUNCT
ejpam-4343	321	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	321	10	if	if	SCONJ
ejpam-4343	321	11	and	and	CCONJ
ejpam-4343	321	12	only	only	ADV
ejpam-4343	321	13	if	if	SCONJ
ejpam-4343	321	14	a(λ	a(λ	ADV
ejpam-4343	321	15	,	,	PUNCT
ejpam-4343	321	16	p(⋆	p(⋆	PROPN
ejpam-4343	321	17	)	)	PUNCT
ejpam-4343	321	18	)	)	PUNCT
ejpam-4343	322	1	=	=	PUNCT
ejpam-4343	322	2	a.	a.	NOUN
ejpam-4343	322	3	(	(	PUNCT
ejpam-4343	322	4	5	5	NUM
ejpam-4343	322	5	)	)	PUNCT
ejpam-4343	322	6	a(λ	a(λ	ADV
ejpam-4343	322	7	,	,	PUNCT
ejpam-4343	322	8	p(⋆	p(⋆	PROPN
ejpam-4343	322	9	)	)	PUNCT
ejpam-4343	322	10	)	)	PUNCT
ejpam-4343	322	11	is	be	AUX
ejpam-4343	322	12	(	(	PUNCT
ejpam-4343	322	13	λ	λ	INTJ
ejpam-4343	322	14	,	,	PUNCT
ejpam-4343	322	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	322	16	.	.	PUNCT
ejpam-4343	323	1	definition	definition	NOUN
ejpam-4343	323	2	10	10	NUM
ejpam-4343	323	3	.	.	PUNCT
ejpam-4343	324	1	a	a	DET
ejpam-4343	324	2	subset	subset	NOUN
ejpam-4343	324	3	a	a	PRON
ejpam-4343	324	4	of	of	ADP
ejpam-4343	324	5	an	an	DET
ejpam-4343	324	6	ideal	ideal	ADJ
ejpam-4343	324	7	topological	topological	ADJ
ejpam-4343	324	8	space	space	NOUN
ejpam-4343	324	9	(	(	PUNCT
ejpam-4343	324	10	x	x	X
ejpam-4343	324	11	,	,	PUNCT
ejpam-4343	324	12	τ	τ	PROPN
ejpam-4343	324	13	,	,	PUNCT
ejpam-4343	324	14	i	i	PROPN
ejpam-4343	324	15	)	)	PUNCT
ejpam-4343	324	16	is	be	AUX
ejpam-4343	324	17	called	call	VERB
ejpam-4343	324	18	semi-(λ	semi-(λ	PROPN
ejpam-4343	324	19	,	,	PUNCT
ejpam-4343	324	20	p(⋆))open	p(⋆))open	PROPN
ejpam-4343	324	21	(	(	PUNCT
ejpam-4343	324	22	resp	resp	NOUN
ejpam-4343	324	23	.	.	PUNCT
ejpam-4343	325	1	pre-(λ	pre-(λ	ADJ
ejpam-4343	325	2	,	,	PUNCT
ejpam-4343	325	3	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	325	4	,	,	PUNCT
ejpam-4343	325	5	α-(λ	α-(λ	PROPN
ejpam-4343	325	6	,	,	PUNCT
ejpam-4343	325	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	325	8	,	,	PUNCT
ejpam-4343	325	9	β-(λ	β-(λ	PRON
ejpam-4343	325	10	,	,	PUNCT
ejpam-4343	325	11	p(⋆))-open	p(⋆))-open	NOUN
ejpam-4343	325	12	)	)	PUNCT
ejpam-4343	325	13	if	if	SCONJ
ejpam-4343	325	14	a	a	DET
ejpam-4343	325	15	⊆	⊆	NUM
ejpam-4343	325	16	[	[	X
ejpam-4343	325	17	a(λ	a(λ	ADV
ejpam-4343	325	18	,	,	PUNCT
ejpam-4343	325	19	p(⋆	p(⋆	PROPN
ejpam-4343	325	20	)	)	PUNCT
ejpam-4343	325	21	)	)	PUNCT
ejpam-4343	325	22	]	]	PUNCT
ejpam-4343	326	1	(	(	PUNCT
ejpam-4343	326	2	λ	λ	X
ejpam-4343	326	3	,	,	PUNCT
ejpam-4343	326	4	p(⋆	p(⋆	PROPN
ejpam-4343	326	5	)	)	PUNCT
ejpam-4343	326	6	)	)	PUNCT
ejpam-4343	326	7	(	(	PUNCT
ejpam-4343	326	8	resp	resp	NOUN
ejpam-4343	326	9	.	.	PUNCT
ejpam-4343	327	1	a	a	DET
ejpam-4343	327	2	⊆	⊆	NUM
ejpam-4343	327	3	[	[	X
ejpam-4343	327	4	a(λ	a(λ	ADV
ejpam-4343	327	5	,	,	PUNCT
ejpam-4343	327	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	327	7	,	,	PUNCT
ejpam-4343	327	8	p(⋆	p(⋆	PROPN
ejpam-4343	327	9	)	)	PUNCT
ejpam-4343	327	10	)	)	PUNCT
ejpam-4343	327	11	,	,	PUNCT
ejpam-4343	327	12	a	a	DET
ejpam-4343	327	13	⊆	⊆	NUM
ejpam-4343	327	14	[	[	X
ejpam-4343	327	15	[	[	X
ejpam-4343	327	16	a(λ	a(λ	ADJ
ejpam-4343	327	17	,	,	PUNCT
ejpam-4343	327	18	p(⋆	p(⋆	PROPN
ejpam-4343	327	19	)	)	PUNCT
ejpam-4343	327	20	)	)	PUNCT
ejpam-4343	327	21	]	]	PUNCT
ejpam-4343	327	22	(	(	PUNCT
ejpam-4343	327	23	λ	λ	X
ejpam-4343	327	24	,	,	PUNCT
ejpam-4343	327	25	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	327	26	,	,	PUNCT
ejpam-4343	327	27	p(⋆	p(⋆	PROPN
ejpam-4343	327	28	)	)	PUNCT
ejpam-4343	327	29	)	)	PUNCT
ejpam-4343	327	30	,	,	PUNCT
ejpam-4343	327	31	a	a	DET
ejpam-4343	327	32	⊆	⊆	NUM
ejpam-4343	327	33	[	[	X
ejpam-4343	327	34	[	[	X
ejpam-4343	327	35	a(λ	a(λ	ADV
ejpam-4343	327	36	,	,	PUNCT
ejpam-4343	327	37	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	327	38	,	,	PUNCT
ejpam-4343	327	39	p(⋆	p(⋆	PROPN
ejpam-4343	327	40	)	)	PUNCT
ejpam-4343	327	41	)	)	PUNCT
ejpam-4343	327	42	]	]	PUNCT
ejpam-4343	327	43	(	(	PUNCT
ejpam-4343	327	44	λ	λ	X
ejpam-4343	327	45	,	,	PUNCT
ejpam-4343	327	46	p(⋆	p(⋆	PROPN
ejpam-4343	327	47	)	)	PUNCT
ejpam-4343	327	48	)	)	PUNCT
ejpam-4343	327	49	)	)	PUNCT
ejpam-4343	327	50	.	.	PUNCT
ejpam-4343	328	1	the	the	DET
ejpam-4343	328	2	complement	complement	NOUN
ejpam-4343	328	3	of	of	ADP
ejpam-4343	328	4	a	a	DET
ejpam-4343	328	5	semi-(λ	semi-(λ	PROPN
ejpam-4343	328	6	,	,	PUNCT
ejpam-4343	328	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	328	8	(	(	PUNCT
ejpam-4343	328	9	resp	resp	NOUN
ejpam-4343	328	10	.	.	PUNCT
ejpam-4343	329	1	pre-(λ	pre-(λ	ADJ
ejpam-4343	329	2	,	,	PUNCT
ejpam-4343	329	3	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	329	4	,	,	PUNCT
ejpam-4343	329	5	α-(λ	α-(λ	PROPN
ejpam-4343	329	6	,	,	PUNCT
ejpam-4343	329	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	329	8	,	,	PUNCT
ejpam-4343	329	9	β(λ	β(λ	X
ejpam-4343	329	10	,	,	PUNCT
ejpam-4343	329	11	p(⋆))-open	p(⋆))-open	NOUN
ejpam-4343	329	12	)	)	PUNCT
ejpam-4343	329	13	set	set	NOUN
ejpam-4343	329	14	is	be	AUX
ejpam-4343	329	15	called	call	VERB
ejpam-4343	329	16	semi-(λ	semi-(λ	PROPN
ejpam-4343	329	17	,	,	PUNCT
ejpam-4343	329	18	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	329	19	(	(	PUNCT
ejpam-4343	329	20	resp	resp	NOUN
ejpam-4343	329	21	.	.	PUNCT
ejpam-4343	330	1	pre-(λ	pre-(λ	PROPN
ejpam-4343	330	2	,	,	PUNCT
ejpam-4343	330	3	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	330	4	,	,	PUNCT
ejpam-4343	330	5	α-(λ	α-(λ	NOUN
ejpam-4343	330	6	,	,	PUNCT
ejpam-4343	330	7	p(⋆))closed	p(⋆))close	VERB
ejpam-4343	330	8	,	,	PUNCT
ejpam-4343	330	9	β-(λ	β-(λ	PRON
ejpam-4343	330	10	,	,	PUNCT
ejpam-4343	330	11	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	330	12	)	)	PUNCT
ejpam-4343	330	13	.	.	PUNCT
ejpam-4343	331	1	definition	definition	NOUN
ejpam-4343	331	2	11	11	NUM
ejpam-4343	331	3	.	.	PUNCT
ejpam-4343	332	1	an	an	DET
ejpam-4343	332	2	ideal	ideal	ADJ
ejpam-4343	332	3	topological	topological	ADJ
ejpam-4343	332	4	space	space	NOUN
ejpam-4343	332	5	(	(	PUNCT
ejpam-4343	332	6	x	x	X
ejpam-4343	332	7	,	,	PUNCT
ejpam-4343	332	8	τ	τ	PROPN
ejpam-4343	332	9	,	,	PUNCT
ejpam-4343	332	10	i	i	PROPN
ejpam-4343	332	11	)	)	PUNCT
ejpam-4343	332	12	is	be	AUX
ejpam-4343	332	13	called	call	VERB
ejpam-4343	332	14	(	(	PUNCT
ejpam-4343	332	15	λ	λ	NOUN
ejpam-4343	332	16	,	,	PUNCT
ejpam-4343	332	17	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	332	18	disconnected	disconnected	ADJ
ejpam-4343	332	19	if	if	SCONJ
ejpam-4343	332	20	the	the	DET
ejpam-4343	332	21	(	(	PUNCT
ejpam-4343	332	22	λ	λ	PROPN
ejpam-4343	332	23	,	,	PUNCT
ejpam-4343	332	24	p(⋆))-closure	p(⋆))-closure	NOUN
ejpam-4343	332	25	of	of	ADP
ejpam-4343	332	26	every	every	DET
ejpam-4343	332	27	(	(	PUNCT
ejpam-4343	332	28	λ	λ	PROPN
ejpam-4343	332	29	,	,	PUNCT
ejpam-4343	332	30	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	332	31	set	set	NOUN
ejpam-4343	332	32	of	of	ADP
ejpam-4343	332	33	x	x	PUNCT
ejpam-4343	332	34	is	be	AUX
ejpam-4343	332	35	(	(	PUNCT
ejpam-4343	332	36	λ	λ	INTJ
ejpam-4343	332	37	,	,	PUNCT
ejpam-4343	332	38	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	332	39	.	.	PUNCT
ejpam-4343	332	40	example	example	NOUN
ejpam-4343	333	1	3	3	X
ejpam-4343	333	2	.	.	PUNCT
ejpam-4343	333	3	let	let	VERB
ejpam-4343	333	4	x	x	PUNCT
ejpam-4343	333	5	=	=	PRON
ejpam-4343	333	6	{	{	PUNCT
ejpam-4343	333	7	a	a	PRON
ejpam-4343	333	8	,	,	PUNCT
ejpam-4343	333	9	b	b	NOUN
ejpam-4343	333	10	,	,	PUNCT
ejpam-4343	333	11	c	c	NOUN
ejpam-4343	333	12	}	}	PUNCT
ejpam-4343	333	13	with	with	ADP
ejpam-4343	333	14	a	a	DET
ejpam-4343	333	15	topology	topology	NOUN
ejpam-4343	333	16	τ	τ	X
ejpam-4343	333	17	=	=	SYM
ejpam-4343	333	18	{	{	PUNCT
ejpam-4343	333	19	∅	∅	NOUN
ejpam-4343	333	20	,	,	PUNCT
ejpam-4343	333	21	{	{	PUNCT
ejpam-4343	333	22	a	a	X
ejpam-4343	333	23	}	}	PUNCT
ejpam-4343	333	24	,	,	PUNCT
ejpam-4343	333	25	{	{	PUNCT
ejpam-4343	333	26	a	a	DET
ejpam-4343	333	27	,	,	PUNCT
ejpam-4343	333	28	b	b	NOUN
ejpam-4343	333	29	}	}	PUNCT
ejpam-4343	333	30	,	,	PUNCT
ejpam-4343	333	31	{	{	PUNCT
ejpam-4343	333	32	a	a	X
ejpam-4343	333	33	,	,	PUNCT
ejpam-4343	333	34	c	c	NOUN
ejpam-4343	333	35	}	}	PUNCT
ejpam-4343	333	36	,	,	PUNCT
ejpam-4343	333	37	x	x	NOUN
ejpam-4343	333	38	}	}	PUNCT
ejpam-4343	333	39	and	and	CCONJ
ejpam-4343	333	40	an	an	DET
ejpam-4343	333	41	ideal	ideal	NOUN
ejpam-4343	333	42	i	i	X
ejpam-4343	333	43	=	=	SYM
ejpam-4343	333	44	{	{	PUNCT
ejpam-4343	333	45	∅	∅	NOUN
ejpam-4343	333	46	,	,	PUNCT
ejpam-4343	333	47	{	{	PUNCT
ejpam-4343	333	48	c	c	NOUN
ejpam-4343	333	49	}	}	PUNCT
ejpam-4343	333	50	}	}	PUNCT
ejpam-4343	333	51	.	.	PUNCT
ejpam-4343	334	1	then	then	ADV
ejpam-4343	334	2	(	(	PUNCT
ejpam-4343	334	3	x	x	X
ejpam-4343	334	4	,	,	PUNCT
ejpam-4343	334	5	τ	τ	PROPN
ejpam-4343	334	6	,	,	PUNCT
ejpam-4343	334	7	i	i	PROPN
ejpam-4343	334	8	)	)	PUNCT
ejpam-4343	334	9	is	be	AUX
ejpam-4343	334	10	a	a	DET
ejpam-4343	334	11	(	(	PUNCT
ejpam-4343	334	12	λ	λ	NOUN
ejpam-4343	334	13	,	,	PUNCT
ejpam-4343	334	14	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	334	15	disconnected	disconnected	ADJ
ejpam-4343	334	16	space	space	NOUN
ejpam-4343	334	17	.	.	PUNCT
ejpam-4343	335	1	c.	c.	PROPN
ejpam-4343	335	2	boonpok	boonpok	PROPN
ejpam-4343	335	3	/	/	SYM
ejpam-4343	335	4	eur	eur	PROPN
ejpam-4343	335	5	.	.	PUNCT
ejpam-4343	336	1	j.	j.	PROPN
ejpam-4343	336	2	pure	pure	PROPN
ejpam-4343	336	3	appl	appl	PROPN
ejpam-4343	336	4	.	.	PROPN
ejpam-4343	336	5	math	math	PROPN
ejpam-4343	336	6	,	,	PUNCT
ejpam-4343	336	7	15	15	NUM
ejpam-4343	336	8	(	(	PUNCT
ejpam-4343	336	9	3	3	NUM
ejpam-4343	336	10	)	)	PUNCT
ejpam-4343	336	11	(	(	PUNCT
ejpam-4343	336	12	2022	2022	NUM
ejpam-4343	336	13	)	)	PUNCT
ejpam-4343	336	14	,	,	PUNCT
ejpam-4343	336	15	1023	1023	NUM
ejpam-4343	336	16	-	-	SYM
ejpam-4343	336	17	1046	1046	NUM
ejpam-4343	336	18	1032	1032	NUM
ejpam-4343	336	19	theorem	theorem	VERB
ejpam-4343	336	20	4	4	NUM
ejpam-4343	336	21	.	.	X
ejpam-4343	336	22	for	for	ADP
ejpam-4343	336	23	an	an	DET
ejpam-4343	336	24	ideal	ideal	ADJ
ejpam-4343	336	25	topological	topological	ADJ
ejpam-4343	336	26	space	space	NOUN
ejpam-4343	336	27	(	(	PUNCT
ejpam-4343	336	28	x	x	X
ejpam-4343	336	29	,	,	PUNCT
ejpam-4343	336	30	τ	τ	PROPN
ejpam-4343	336	31	,	,	PUNCT
ejpam-4343	336	32	i	i	NOUN
ejpam-4343	336	33	)	)	PUNCT
ejpam-4343	336	34	,	,	PUNCT
ejpam-4343	336	35	the	the	DET
ejpam-4343	336	36	following	follow	VERB
ejpam-4343	336	37	properties	property	NOUN
ejpam-4343	336	38	are	be	AUX
ejpam-4343	336	39	equivalent	equivalent	ADJ
ejpam-4343	336	40	:	:	PUNCT
ejpam-4343	336	41	(	(	PUNCT
ejpam-4343	336	42	1	1	X
ejpam-4343	336	43	)	)	PUNCT
ejpam-4343	336	44	(	(	PUNCT
ejpam-4343	336	45	x	x	X
ejpam-4343	336	46	,	,	PUNCT
ejpam-4343	336	47	τ	τ	PROPN
ejpam-4343	336	48	,	,	PUNCT
ejpam-4343	336	49	i	i	PROPN
ejpam-4343	336	50	)	)	PUNCT
ejpam-4343	336	51	is	be	AUX
ejpam-4343	336	52	(	(	PUNCT
ejpam-4343	336	53	λ	λ	INTJ
ejpam-4343	336	54	,	,	PUNCT
ejpam-4343	336	55	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	336	56	disconnected	disconnected	ADJ
ejpam-4343	336	57	.	.	PUNCT
ejpam-4343	337	1	(	(	PUNCT
ejpam-4343	337	2	2	2	X
ejpam-4343	337	3	)	)	PUNCT
ejpam-4343	337	4	f(λ	f(λ	NOUN
ejpam-4343	337	5	,	,	PUNCT
ejpam-4343	337	6	p(⋆	p(⋆	PROPN
ejpam-4343	337	7	)	)	PUNCT
ejpam-4343	337	8	)	)	PUNCT
ejpam-4343	337	9	is	be	AUX
ejpam-4343	337	10	(	(	PUNCT
ejpam-4343	337	11	λ	λ	X
ejpam-4343	337	12	,	,	PUNCT
ejpam-4343	337	13	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	337	14	for	for	ADP
ejpam-4343	337	15	every	every	DET
ejpam-4343	337	16	(	(	PUNCT
ejpam-4343	337	17	λ	λ	PROPN
ejpam-4343	337	18	,	,	PUNCT
ejpam-4343	337	19	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	337	20	set	set	VERB
ejpam-4343	337	21	f	f	PROPN
ejpam-4343	337	22	of	of	ADP
ejpam-4343	337	23	x.	x.	PROPN
ejpam-4343	337	24	(	(	PUNCT
ejpam-4343	337	25	3	3	NUM
ejpam-4343	337	26	)	)	PUNCT
ejpam-4343	337	27	[	[	X
ejpam-4343	337	28	a(λ	a(λ	ADV
ejpam-4343	337	29	,	,	PUNCT
ejpam-4343	337	30	p(⋆	p(⋆	PROPN
ejpam-4343	337	31	)	)	PUNCT
ejpam-4343	337	32	)	)	PUNCT
ejpam-4343	337	33	]	]	PUNCT
ejpam-4343	338	1	(	(	PUNCT
ejpam-4343	338	2	λ	λ	X
ejpam-4343	338	3	,	,	PUNCT
ejpam-4343	338	4	p(⋆	p(⋆	PROPN
ejpam-4343	338	5	)	)	PUNCT
ejpam-4343	338	6	)	)	PUNCT
ejpam-4343	339	1	⊆	⊆	NUM
ejpam-4343	339	2	[	[	X
ejpam-4343	339	3	a(λ	a(λ	ADV
ejpam-4343	339	4	,	,	PUNCT
ejpam-4343	339	5	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	339	6	,	,	PUNCT
ejpam-4343	339	7	p(⋆	p(⋆	PROPN
ejpam-4343	339	8	)	)	PUNCT
ejpam-4343	339	9	)	)	PUNCT
ejpam-4343	339	10	for	for	ADP
ejpam-4343	339	11	every	every	DET
ejpam-4343	339	12	subset	subset	NOUN
ejpam-4343	339	13	a	a	PRON
ejpam-4343	339	14	of	of	ADP
ejpam-4343	339	15	x.	x.	NOUN
ejpam-4343	339	16	(	(	PUNCT
ejpam-4343	339	17	4	4	NUM
ejpam-4343	339	18	)	)	PUNCT
ejpam-4343	339	19	every	every	DET
ejpam-4343	339	20	semi-(λ	semi-(λ	PROPN
ejpam-4343	339	21	,	,	PUNCT
ejpam-4343	339	22	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	339	23	set	set	NOUN
ejpam-4343	339	24	is	be	AUX
ejpam-4343	339	25	pre-(λ	pre-(λ	PROPN
ejpam-4343	339	26	,	,	PUNCT
ejpam-4343	339	27	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	339	28	.	.	PUNCT
ejpam-4343	340	1	(	(	PUNCT
ejpam-4343	340	2	5	5	X
ejpam-4343	340	3	)	)	PUNCT
ejpam-4343	340	4	the	the	DET
ejpam-4343	340	5	(	(	PUNCT
ejpam-4343	340	6	λ	λ	PROPN
ejpam-4343	340	7	,	,	PUNCT
ejpam-4343	340	8	p(⋆))-closure	p(⋆))-closure	NOUN
ejpam-4343	340	9	of	of	ADP
ejpam-4343	340	10	every	every	DET
ejpam-4343	340	11	β-(λ	β-(λ	NOUN
ejpam-4343	340	12	,	,	PUNCT
ejpam-4343	340	13	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	340	14	set	set	NOUN
ejpam-4343	340	15	of	of	ADP
ejpam-4343	340	16	x	x	PUNCT
ejpam-4343	340	17	is	be	AUX
ejpam-4343	340	18	(	(	PUNCT
ejpam-4343	340	19	λ	λ	INTJ
ejpam-4343	340	20	,	,	PUNCT
ejpam-4343	340	21	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	340	22	.	.	PUNCT
ejpam-4343	341	1	(	(	PUNCT
ejpam-4343	341	2	6	6	NUM
ejpam-4343	341	3	)	)	PUNCT
ejpam-4343	341	4	every	every	DET
ejpam-4343	341	5	β-(λ	β-(λ	NOUN
ejpam-4343	341	6	,	,	PUNCT
ejpam-4343	341	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	341	8	set	set	NOUN
ejpam-4343	341	9	is	be	AUX
ejpam-4343	341	10	pre-(λ	pre-(λ	PROPN
ejpam-4343	341	11	,	,	PUNCT
ejpam-4343	341	12	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	341	13	.	.	PUNCT
ejpam-4343	342	1	(	(	PUNCT
ejpam-4343	342	2	7	7	X
ejpam-4343	342	3	)	)	PUNCT
ejpam-4343	342	4	for	for	ADP
ejpam-4343	342	5	every	every	DET
ejpam-4343	342	6	subset	subset	NOUN
ejpam-4343	342	7	a	a	PRON
ejpam-4343	342	8	of	of	ADP
ejpam-4343	342	9	x	x	PRON
ejpam-4343	342	10	,	,	PUNCT
ejpam-4343	342	11	a	a	PRON
ejpam-4343	342	12	is	be	AUX
ejpam-4343	342	13	α-(λ	α-(λ	NOUN
ejpam-4343	342	14	,	,	PUNCT
ejpam-4343	342	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	342	16	if	if	SCONJ
ejpam-4343	342	17	and	and	CCONJ
ejpam-4343	342	18	only	only	ADV
ejpam-4343	342	19	if	if	SCONJ
ejpam-4343	342	20	it	it	PRON
ejpam-4343	342	21	is	be	AUX
ejpam-4343	342	22	semi-(λ	semi-(λ	PROPN
ejpam-4343	342	23	,	,	PUNCT
ejpam-4343	342	24	p(⋆))open	p(⋆))open	PROPN
ejpam-4343	342	25	.	.	PUNCT
ejpam-4343	343	1	proof	proof	NOUN
ejpam-4343	343	2	.	.	PUNCT
ejpam-4343	344	1	(	(	PUNCT
ejpam-4343	344	2	1	1	X
ejpam-4343	344	3	)	)	PUNCT
ejpam-4343	344	4	⇒	⇒	NOUN
ejpam-4343	344	5	(	(	PUNCT
ejpam-4343	344	6	2	2	NUM
ejpam-4343	344	7	):	):	PUNCT
ejpam-4343	344	8	let	let	VERB
ejpam-4343	344	9	a	a	DET
ejpam-4343	344	10	be	be	AUX
ejpam-4343	344	11	any	any	DET
ejpam-4343	344	12	(	(	PUNCT
ejpam-4343	344	13	λ	λ	PROPN
ejpam-4343	344	14	,	,	PUNCT
ejpam-4343	344	15	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	344	16	set	set	NOUN
ejpam-4343	344	17	.	.	PUNCT
ejpam-4343	345	1	then	then	ADV
ejpam-4343	345	2	,	,	PUNCT
ejpam-4343	345	3	x	x	PUNCT
ejpam-4343	345	4	−	−	NOUN
ejpam-4343	345	5	a	a	PRON
ejpam-4343	345	6	is	be	AUX
ejpam-4343	345	7	(	(	PUNCT
ejpam-4343	345	8	λ	λ	PROPN
ejpam-4343	345	9	,	,	PUNCT
ejpam-4343	345	10	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	345	11	and	and	CCONJ
ejpam-4343	345	12	by	by	ADP
ejpam-4343	345	13	(	(	PUNCT
ejpam-4343	345	14	1	1	NUM
ejpam-4343	345	15	)	)	PUNCT
ejpam-4343	345	16	,	,	PUNCT
ejpam-4343	345	17	we	we	PRON
ejpam-4343	345	18	have	have	VERB
ejpam-4343	345	19	(	(	PUNCT
ejpam-4343	345	20	x	x	NOUN
ejpam-4343	345	21	−	−	NOUN
ejpam-4343	345	22	a)(λ	a)(λ	PROPN
ejpam-4343	345	23	,	,	PUNCT
ejpam-4343	345	24	p(⋆	p(⋆	PROPN
ejpam-4343	345	25	)	)	PUNCT
ejpam-4343	345	26	)	)	PUNCT
ejpam-4343	346	1	=	=	PUNCT
ejpam-4343	346	2	x	x	X
ejpam-4343	346	3	−	−	NOUN
ejpam-4343	346	4	a(λ	a(λ	ADV
ejpam-4343	346	5	,	,	PUNCT
ejpam-4343	346	6	p(⋆	p(⋆	PROPN
ejpam-4343	346	7	)	)	PUNCT
ejpam-4343	346	8	)	)	PUNCT
ejpam-4343	346	9	is	be	AUX
ejpam-4343	346	10	(	(	PUNCT
ejpam-4343	346	11	λ	λ	INTJ
ejpam-4343	346	12	,	,	PUNCT
ejpam-4343	346	13	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	346	14	.	.	PUNCT
ejpam-4343	347	1	thus	thus	ADV
ejpam-4343	347	2	,	,	PUNCT
ejpam-4343	347	3	a(λ	a(λ	ADV
ejpam-4343	347	4	,	,	PUNCT
ejpam-4343	347	5	p(⋆	p(⋆	PROPN
ejpam-4343	347	6	)	)	PUNCT
ejpam-4343	347	7	)	)	PUNCT
ejpam-4343	348	1	is	be	AUX
ejpam-4343	348	2	(	(	PUNCT
ejpam-4343	348	3	λ	λ	X
ejpam-4343	348	4	,	,	PUNCT
ejpam-4343	348	5	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	348	6	.	.	PUNCT
ejpam-4343	349	1	(	(	PUNCT
ejpam-4343	349	2	2	2	X
ejpam-4343	349	3	)	)	PUNCT
ejpam-4343	349	4	⇒	⇒	NOUN
ejpam-4343	349	5	(	(	PUNCT
ejpam-4343	349	6	3	3	NUM
ejpam-4343	349	7	):	):	PUNCT
ejpam-4343	349	8	let	let	VERB
ejpam-4343	349	9	a	a	DET
ejpam-4343	349	10	be	be	AUX
ejpam-4343	349	11	any	any	DET
ejpam-4343	349	12	subset	subset	NOUN
ejpam-4343	349	13	of	of	ADP
ejpam-4343	349	14	x.	x.	NOUN
ejpam-4343	349	15	then	then	ADV
ejpam-4343	349	16	,	,	PUNCT
ejpam-4343	349	17	x	x	NOUN
ejpam-4343	349	18	−	−	NOUN
ejpam-4343	349	19	a(λ	a(λ	ADV
ejpam-4343	349	20	,	,	PUNCT
ejpam-4343	349	21	p(⋆	p(⋆	PROPN
ejpam-4343	349	22	)	)	PUNCT
ejpam-4343	349	23	)	)	PUNCT
ejpam-4343	350	1	is	be	AUX
ejpam-4343	350	2	(	(	PUNCT
ejpam-4343	350	3	λ	λ	X
ejpam-4343	350	4	,	,	PUNCT
ejpam-4343	350	5	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	350	6	and	and	CCONJ
ejpam-4343	350	7	by	by	ADP
ejpam-4343	350	8	(	(	PUNCT
ejpam-4343	350	9	2	2	NUM
ejpam-4343	350	10	)	)	PUNCT
ejpam-4343	350	11	,	,	PUNCT
ejpam-4343	351	1	[	[	X
ejpam-4343	351	2	x	x	X
ejpam-4343	351	3	−	−	NOUN
ejpam-4343	351	4	a(λ	a(λ	PROPN
ejpam-4343	351	5	,	,	PUNCT
ejpam-4343	351	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	351	7	,	,	PUNCT
ejpam-4343	351	8	p(⋆	p(⋆	PROPN
ejpam-4343	351	9	)	)	PUNCT
ejpam-4343	351	10	)	)	PUNCT
ejpam-4343	351	11	is	be	AUX
ejpam-4343	351	12	(	(	PUNCT
ejpam-4343	351	13	λ	λ	X
ejpam-4343	351	14	,	,	PUNCT
ejpam-4343	351	15	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	351	16	.	.	PUNCT
ejpam-4343	352	1	therefore	therefore	ADV
ejpam-4343	352	2	,	,	PUNCT
ejpam-4343	352	3	we	we	PRON
ejpam-4343	352	4	have	have	VERB
ejpam-4343	352	5	[	[	X
ejpam-4343	352	6	a(λ	a(λ	ADV
ejpam-4343	352	7	,	,	PUNCT
ejpam-4343	352	8	p(⋆	p(⋆	PROPN
ejpam-4343	352	9	)	)	PUNCT
ejpam-4343	352	10	)	)	PUNCT
ejpam-4343	352	11	]	]	PUNCT
ejpam-4343	353	1	(	(	PUNCT
ejpam-4343	353	2	λ	λ	X
ejpam-4343	353	3	,	,	PUNCT
ejpam-4343	353	4	p(⋆	p(⋆	PROPN
ejpam-4343	353	5	)	)	PUNCT
ejpam-4343	353	6	)	)	PUNCT
ejpam-4343	353	7	is	be	AUX
ejpam-4343	353	8	(	(	PUNCT
ejpam-4343	353	9	λ	λ	X
ejpam-4343	353	10	,	,	PUNCT
ejpam-4343	353	11	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	353	12	and	and	CCONJ
ejpam-4343	353	13	hence	hence	ADV
ejpam-4343	353	14	[	[	X
ejpam-4343	353	15	a(λ	a(λ	ADV
ejpam-4343	353	16	,	,	PUNCT
ejpam-4343	353	17	p(⋆	p(⋆	PROPN
ejpam-4343	353	18	)	)	PUNCT
ejpam-4343	353	19	)	)	PUNCT
ejpam-4343	353	20	]	]	PUNCT
ejpam-4343	354	1	(	(	PUNCT
ejpam-4343	354	2	λ	λ	X
ejpam-4343	354	3	,	,	PUNCT
ejpam-4343	354	4	p(⋆	p(⋆	PROPN
ejpam-4343	354	5	)	)	PUNCT
ejpam-4343	354	6	)	)	PUNCT
ejpam-4343	355	1	⊆	⊆	NUM
ejpam-4343	355	2	[	[	X
ejpam-4343	355	3	a(λ	a(λ	ADV
ejpam-4343	355	4	,	,	PUNCT
ejpam-4343	355	5	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	355	6	,	,	PUNCT
ejpam-4343	355	7	p(⋆	p(⋆	PROPN
ejpam-4343	355	8	)	)	PUNCT
ejpam-4343	355	9	)	)	PUNCT
ejpam-4343	355	10	.	.	PUNCT
ejpam-4343	356	1	(	(	PUNCT
ejpam-4343	356	2	3	3	X
ejpam-4343	356	3	)	)	PUNCT
ejpam-4343	356	4	⇒	⇒	NOUN
ejpam-4343	356	5	(	(	PUNCT
ejpam-4343	356	6	4	4	NUM
ejpam-4343	356	7	):	):	PUNCT
ejpam-4343	356	8	let	let	VERB
ejpam-4343	356	9	v	v	PART
ejpam-4343	356	10	be	be	AUX
ejpam-4343	356	11	any	any	DET
ejpam-4343	356	12	semi-(λ	semi-(λ	PROPN
ejpam-4343	356	13	,	,	PUNCT
ejpam-4343	356	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	356	15	set	set	NOUN
ejpam-4343	356	16	.	.	PUNCT
ejpam-4343	357	1	by	by	ADP
ejpam-4343	357	2	(	(	PUNCT
ejpam-4343	357	3	3	3	NUM
ejpam-4343	357	4	)	)	PUNCT
ejpam-4343	357	5	,	,	PUNCT
ejpam-4343	357	6	we	we	PRON
ejpam-4343	357	7	have	have	VERB
ejpam-4343	357	8	v	v	ADP
ejpam-4343	357	9	⊆	⊆	NUM
ejpam-4343	357	10	[	[	X
ejpam-4343	357	11	v(λ	v(λ	PROPN
ejpam-4343	357	12	,	,	PUNCT
ejpam-4343	357	13	p(⋆	p(⋆	PROPN
ejpam-4343	357	14	)	)	PUNCT
ejpam-4343	357	15	)	)	PUNCT
ejpam-4343	357	16	]	]	PUNCT
ejpam-4343	358	1	(	(	PUNCT
ejpam-4343	358	2	λ	λ	X
ejpam-4343	358	3	,	,	PUNCT
ejpam-4343	358	4	p(⋆	p(⋆	PROPN
ejpam-4343	358	5	)	)	PUNCT
ejpam-4343	358	6	)	)	PUNCT
ejpam-4343	359	1	⊆	⊆	NUM
ejpam-4343	359	2	[	[	X
ejpam-4343	359	3	v	v	X
ejpam-4343	359	4	(	(	PUNCT
ejpam-4343	359	5	λ	λ	PROPN
ejpam-4343	359	6	,	,	PUNCT
ejpam-4343	359	7	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	359	8	,	,	PUNCT
ejpam-4343	359	9	p(⋆	p(⋆	PROPN
ejpam-4343	359	10	)	)	PUNCT
ejpam-4343	359	11	)	)	PUNCT
ejpam-4343	359	12	and	and	CCONJ
ejpam-4343	359	13	hence	hence	ADV
ejpam-4343	359	14	v	v	NOUN
ejpam-4343	359	15	is	be	AUX
ejpam-4343	359	16	pre-(λ	pre-(λ	ADJ
ejpam-4343	359	17	,	,	PUNCT
ejpam-4343	359	18	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	359	19	.	.	PUNCT
ejpam-4343	360	1	(	(	PUNCT
ejpam-4343	360	2	4	4	X
ejpam-4343	360	3	)	)	PUNCT
ejpam-4343	360	4	⇒	⇒	NOUN
ejpam-4343	360	5	(	(	PUNCT
ejpam-4343	360	6	5	5	NUM
ejpam-4343	360	7	):	):	PUNCT
ejpam-4343	360	8	let	let	VERB
ejpam-4343	360	9	v	v	PART
ejpam-4343	360	10	be	be	AUX
ejpam-4343	360	11	any	any	DET
ejpam-4343	360	12	β-(λ	β-(λ	NOUN
ejpam-4343	360	13	,	,	PUNCT
ejpam-4343	360	14	p(⋆))-open	p(⋆))-open	PROPN
ejpam-4343	360	15	set	set	NOUN
ejpam-4343	360	16	.	.	PUNCT
ejpam-4343	361	1	then	then	ADV
ejpam-4343	361	2	,	,	PUNCT
ejpam-4343	361	3	v	v	INTJ
ejpam-4343	361	4	(	(	PUNCT
ejpam-4343	361	5	λ	λ	PROPN
ejpam-4343	361	6	,	,	PUNCT
ejpam-4343	361	7	p(⋆	p(⋆	PROPN
ejpam-4343	361	8	)	)	PUNCT
ejpam-4343	361	9	)	)	PUNCT
ejpam-4343	361	10	is	be	AUX
ejpam-4343	361	11	semi-(λ	semi-(λ	PROPN
ejpam-4343	361	12	,	,	PUNCT
ejpam-4343	361	13	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	361	14	and	and	CCONJ
ejpam-4343	361	15	by	by	ADP
ejpam-4343	361	16	(	(	PUNCT
ejpam-4343	361	17	4	4	NUM
ejpam-4343	361	18	)	)	PUNCT
ejpam-4343	361	19	,	,	PUNCT
ejpam-4343	361	20	we	we	PRON
ejpam-4343	361	21	have	have	VERB
ejpam-4343	361	22	v	v	NUM
ejpam-4343	361	23	(	(	PUNCT
ejpam-4343	361	24	λ	λ	PROPN
ejpam-4343	361	25	,	,	PUNCT
ejpam-4343	361	26	p(⋆	p(⋆	PROPN
ejpam-4343	361	27	)	)	PUNCT
ejpam-4343	361	28	)	)	PUNCT
ejpam-4343	361	29	is	be	AUX
ejpam-4343	361	30	pre-(λ	pre-(λ	PROPN
ejpam-4343	361	31	,	,	PUNCT
ejpam-4343	361	32	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	361	33	.	.	PUNCT
ejpam-4343	362	1	thus	thus	ADV
ejpam-4343	362	2	,	,	PUNCT
ejpam-4343	362	3	v	v	INTJ
ejpam-4343	362	4	(	(	PUNCT
ejpam-4343	362	5	λ	λ	PROPN
ejpam-4343	362	6	,	,	PUNCT
ejpam-4343	362	7	p(⋆	p(⋆	PROPN
ejpam-4343	362	8	)	)	PUNCT
ejpam-4343	362	9	)	)	PUNCT
ejpam-4343	363	1	⊆	⊆	NUM
ejpam-4343	363	2	[	[	X
ejpam-4343	363	3	v	v	X
ejpam-4343	363	4	(	(	PUNCT
ejpam-4343	363	5	λ	λ	PROPN
ejpam-4343	363	6	,	,	PUNCT
ejpam-4343	363	7	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	363	8	,	,	PUNCT
ejpam-4343	363	9	p(⋆	p(⋆	PROPN
ejpam-4343	363	10	)	)	PUNCT
ejpam-4343	363	11	)	)	PUNCT
ejpam-4343	363	12	and	and	CCONJ
ejpam-4343	363	13	hence	hence	ADV
ejpam-4343	363	14	v	v	NOUN
ejpam-4343	363	15	(	(	PUNCT
ejpam-4343	363	16	λ	λ	PROPN
ejpam-4343	363	17	,	,	PUNCT
ejpam-4343	363	18	p(⋆	p(⋆	PROPN
ejpam-4343	363	19	)	)	PUNCT
ejpam-4343	363	20	)	)	PUNCT
ejpam-4343	363	21	is	be	AUX
ejpam-4343	363	22	(	(	PUNCT
ejpam-4343	363	23	λ	λ	INTJ
ejpam-4343	363	24	,	,	PUNCT
ejpam-4343	363	25	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	363	26	.	.	PUNCT
ejpam-4343	364	1	(	(	PUNCT
ejpam-4343	364	2	5	5	X
ejpam-4343	364	3	)	)	PUNCT
ejpam-4343	364	4	⇒	⇒	NOUN
ejpam-4343	364	5	(	(	PUNCT
ejpam-4343	364	6	6	6	NUM
ejpam-4343	364	7	):	):	PUNCT
ejpam-4343	364	8	let	let	VERB
ejpam-4343	364	9	v	v	PART
ejpam-4343	364	10	be	be	AUX
ejpam-4343	364	11	any	any	DET
ejpam-4343	364	12	β-(λ	β-(λ	NOUN
ejpam-4343	364	13	,	,	PUNCT
ejpam-4343	364	14	p(⋆))-open	p(⋆))-open	PROPN
ejpam-4343	364	15	set	set	NOUN
ejpam-4343	364	16	.	.	PUNCT
ejpam-4343	365	1	by	by	ADP
ejpam-4343	365	2	(	(	PUNCT
ejpam-4343	365	3	5	5	NUM
ejpam-4343	365	4	)	)	PUNCT
ejpam-4343	365	5	,	,	PUNCT
ejpam-4343	365	6	v	v	X
ejpam-4343	365	7	(	(	PUNCT
ejpam-4343	365	8	λ	λ	PROPN
ejpam-4343	365	9	,	,	PUNCT
ejpam-4343	365	10	p(⋆	p(⋆	PROPN
ejpam-4343	365	11	)	)	PUNCT
ejpam-4343	365	12	)	)	PUNCT
ejpam-4343	366	1	=	=	PUNCT
ejpam-4343	367	1	[	[	X
ejpam-4343	367	2	v	v	X
ejpam-4343	367	3	(	(	PUNCT
ejpam-4343	367	4	λ	λ	PROPN
ejpam-4343	367	5	,	,	PUNCT
ejpam-4343	367	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	367	7	,	,	PUNCT
ejpam-4343	367	8	p(⋆	p(⋆	PROPN
ejpam-4343	367	9	)	)	PUNCT
ejpam-4343	367	10	)	)	PUNCT
ejpam-4343	367	11	.	.	PUNCT
ejpam-4343	368	1	therefore	therefore	ADV
ejpam-4343	368	2	,	,	PUNCT
ejpam-4343	368	3	v	v	ADP
ejpam-4343	368	4	⊆	⊆	NUM
ejpam-4343	368	5	v	v	NOUN
ejpam-4343	368	6	(	(	PUNCT
ejpam-4343	368	7	λ	λ	PROPN
ejpam-4343	368	8	,	,	PUNCT
ejpam-4343	368	9	p(⋆	p(⋆	PROPN
ejpam-4343	368	10	)	)	PUNCT
ejpam-4343	368	11	)	)	PUNCT
ejpam-4343	369	1	=	=	PUNCT
ejpam-4343	370	1	[	[	X
ejpam-4343	370	2	v	v	X
ejpam-4343	370	3	(	(	PUNCT
ejpam-4343	370	4	λ	λ	PROPN
ejpam-4343	370	5	,	,	PUNCT
ejpam-4343	370	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	370	7	,	,	PUNCT
ejpam-4343	370	8	p(⋆	p(⋆	PROPN
ejpam-4343	370	9	)	)	PUNCT
ejpam-4343	370	10	)	)	PUNCT
ejpam-4343	370	11	and	and	CCONJ
ejpam-4343	370	12	hence	hence	ADV
ejpam-4343	370	13	v	v	NOUN
ejpam-4343	370	14	is	be	AUX
ejpam-4343	370	15	pre-(λ	pre-(λ	ADJ
ejpam-4343	370	16	,	,	PUNCT
ejpam-4343	370	17	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	370	18	.	.	PUNCT
ejpam-4343	371	1	(	(	PUNCT
ejpam-4343	371	2	6	6	NUM
ejpam-4343	371	3	)	)	PUNCT
ejpam-4343	371	4	⇒	⇒	NOUN
ejpam-4343	371	5	(	(	PUNCT
ejpam-4343	371	6	7	7	NUM
ejpam-4343	371	7	):	):	PUNCT
ejpam-4343	371	8	let	let	VERB
ejpam-4343	371	9	v	v	PART
ejpam-4343	371	10	be	be	AUX
ejpam-4343	371	11	any	any	DET
ejpam-4343	371	12	semi-(λ	semi-(λ	PROPN
ejpam-4343	371	13	,	,	PUNCT
ejpam-4343	371	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	371	15	set	set	NOUN
ejpam-4343	371	16	.	.	PUNCT
ejpam-4343	372	1	then	then	ADV
ejpam-4343	372	2	,	,	PUNCT
ejpam-4343	372	3	v	v	NOUN
ejpam-4343	372	4	is	be	AUX
ejpam-4343	372	5	β-(λ	β-(λ	PRON
ejpam-4343	372	6	,	,	PUNCT
ejpam-4343	372	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	372	8	and	and	CCONJ
ejpam-4343	372	9	by	by	ADP
ejpam-4343	372	10	(	(	PUNCT
ejpam-4343	372	11	6	6	NUM
ejpam-4343	372	12	)	)	PUNCT
ejpam-4343	372	13	,	,	PUNCT
ejpam-4343	372	14	v	v	NOUN
ejpam-4343	372	15	is	be	AUX
ejpam-4343	372	16	pre-(λ	pre-(λ	PROPN
ejpam-4343	372	17	,	,	PUNCT
ejpam-4343	372	18	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	372	19	.	.	PUNCT
ejpam-4343	373	1	since	since	SCONJ
ejpam-4343	373	2	v	v	NOUN
ejpam-4343	373	3	is	be	AUX
ejpam-4343	373	4	semi-(λ	semi-(λ	PROPN
ejpam-4343	373	5	,	,	PUNCT
ejpam-4343	373	6	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	373	7	and	and	CCONJ
ejpam-4343	373	8	pre-(λ	pre-(λ	PROPN
ejpam-4343	373	9	,	,	PUNCT
ejpam-4343	373	10	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	373	11	,	,	PUNCT
ejpam-4343	373	12	v	v	NOUN
ejpam-4343	373	13	is	be	AUX
ejpam-4343	373	14	α-(λ	α-(λ	NOUN
ejpam-4343	373	15	,	,	PUNCT
ejpam-4343	373	16	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	373	17	.	.	PUNCT
ejpam-4343	374	1	(	(	PUNCT
ejpam-4343	374	2	7	7	X
ejpam-4343	374	3	)	)	PUNCT
ejpam-4343	374	4	⇒	⇒	NOUN
ejpam-4343	374	5	(	(	PUNCT
ejpam-4343	374	6	1	1	NUM
ejpam-4343	374	7	):	):	PUNCT
ejpam-4343	374	8	let	let	VERB
ejpam-4343	374	9	v	v	PART
ejpam-4343	374	10	be	be	AUX
ejpam-4343	374	11	any	any	DET
ejpam-4343	374	12	(	(	PUNCT
ejpam-4343	374	13	λ	λ	PROPN
ejpam-4343	374	14	,	,	PUNCT
ejpam-4343	374	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	374	16	set	set	NOUN
ejpam-4343	374	17	.	.	PUNCT
ejpam-4343	375	1	then	then	ADV
ejpam-4343	375	2	,	,	PUNCT
ejpam-4343	375	3	v	v	INTJ
ejpam-4343	375	4	(	(	PUNCT
ejpam-4343	375	5	λ	λ	PROPN
ejpam-4343	375	6	,	,	PUNCT
ejpam-4343	375	7	p(⋆	p(⋆	PROPN
ejpam-4343	375	8	)	)	PUNCT
ejpam-4343	375	9	)	)	PUNCT
ejpam-4343	375	10	is	be	AUX
ejpam-4343	375	11	semi-(λ	semi-(λ	PROPN
ejpam-4343	375	12	,	,	PUNCT
ejpam-4343	375	13	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	375	14	and	and	CCONJ
ejpam-4343	375	15	by	by	ADP
ejpam-4343	375	16	(	(	PUNCT
ejpam-4343	375	17	7	7	NUM
ejpam-4343	375	18	)	)	PUNCT
ejpam-4343	375	19	,	,	PUNCT
ejpam-4343	375	20	we	we	PRON
ejpam-4343	375	21	have	have	VERB
ejpam-4343	375	22	v	v	NUM
ejpam-4343	375	23	(	(	PUNCT
ejpam-4343	375	24	λ	λ	PROPN
ejpam-4343	375	25	,	,	PUNCT
ejpam-4343	375	26	p(⋆	p(⋆	PROPN
ejpam-4343	375	27	)	)	PUNCT
ejpam-4343	375	28	)	)	PUNCT
ejpam-4343	375	29	is	be	AUX
ejpam-4343	375	30	α-(λ	α-(λ	NOUN
ejpam-4343	375	31	,	,	PUNCT
ejpam-4343	375	32	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	375	33	.	.	PUNCT
ejpam-4343	376	1	thus	thus	ADV
ejpam-4343	376	2	,	,	PUNCT
ejpam-4343	376	3	v	v	INTJ
ejpam-4343	376	4	(	(	PUNCT
ejpam-4343	376	5	λ	λ	PROPN
ejpam-4343	376	6	,	,	PUNCT
ejpam-4343	376	7	p(⋆	p(⋆	PROPN
ejpam-4343	376	8	)	)	PUNCT
ejpam-4343	376	9	)	)	PUNCT
ejpam-4343	377	1	⊆	⊆	NUM
ejpam-4343	378	1	[	[	X
ejpam-4343	378	2	[	[	X
ejpam-4343	378	3	[	[	X
ejpam-4343	378	4	v	v	X
ejpam-4343	378	5	(	(	PUNCT
ejpam-4343	378	6	λ	λ	PROPN
ejpam-4343	378	7	,	,	PUNCT
ejpam-4343	378	8	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	378	9	,	,	PUNCT
ejpam-4343	378	10	p(⋆	p(⋆	PROPN
ejpam-4343	378	11	)	)	PUNCT
ejpam-4343	378	12	)	)	PUNCT
ejpam-4343	378	13	]	]	PUNCT
ejpam-4343	378	14	(	(	PUNCT
ejpam-4343	378	15	λ	λ	X
ejpam-4343	378	16	,	,	PUNCT
ejpam-4343	378	17	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	378	18	,	,	PUNCT
ejpam-4343	378	19	p(⋆	p(⋆	PROPN
ejpam-4343	378	20	)	)	PUNCT
ejpam-4343	378	21	)	)	PUNCT
ejpam-4343	379	1	=	=	PUNCT
ejpam-4343	380	1	[	[	X
ejpam-4343	380	2	v	v	X
ejpam-4343	380	3	(	(	PUNCT
ejpam-4343	380	4	λ	λ	PROPN
ejpam-4343	380	5	,	,	PUNCT
ejpam-4343	380	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	380	7	,	,	PUNCT
ejpam-4343	380	8	p(⋆	p(⋆	PROPN
ejpam-4343	380	9	)	)	PUNCT
ejpam-4343	380	10	)	)	PUNCT
ejpam-4343	380	11	and	and	CCONJ
ejpam-4343	380	12	hence	hence	ADV
ejpam-4343	380	13	v	v	NOUN
ejpam-4343	380	14	(	(	PUNCT
ejpam-4343	380	15	λ	λ	PROPN
ejpam-4343	380	16	,	,	PUNCT
ejpam-4343	380	17	p(⋆	p(⋆	PROPN
ejpam-4343	380	18	)	)	PUNCT
ejpam-4343	380	19	)	)	PUNCT
ejpam-4343	381	1	=	=	PUNCT
ejpam-4343	382	1	[	[	X
ejpam-4343	382	2	v	v	X
ejpam-4343	382	3	(	(	PUNCT
ejpam-4343	382	4	λ	λ	PROPN
ejpam-4343	382	5	,	,	PUNCT
ejpam-4343	382	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	382	7	,	,	PUNCT
ejpam-4343	382	8	p(⋆	p(⋆	PROPN
ejpam-4343	382	9	)	)	PUNCT
ejpam-4343	382	10	)	)	PUNCT
ejpam-4343	382	11	.	.	PUNCT
ejpam-4343	383	1	therefore	therefore	ADV
ejpam-4343	383	2	,	,	PUNCT
ejpam-4343	383	3	v	v	INTJ
ejpam-4343	383	4	(	(	PUNCT
ejpam-4343	383	5	λ	λ	PROPN
ejpam-4343	383	6	,	,	PUNCT
ejpam-4343	383	7	p(⋆	p(⋆	PROPN
ejpam-4343	383	8	)	)	PUNCT
ejpam-4343	383	9	)	)	PUNCT
ejpam-4343	383	10	is	be	AUX
ejpam-4343	383	11	(	(	PUNCT
ejpam-4343	383	12	λ	λ	INTJ
ejpam-4343	383	13	,	,	PUNCT
ejpam-4343	383	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	383	15	.	.	PUNCT
ejpam-4343	384	1	this	this	PRON
ejpam-4343	384	2	shows	show	VERB
ejpam-4343	384	3	that	that	SCONJ
ejpam-4343	384	4	(	(	PUNCT
ejpam-4343	384	5	x	x	X
ejpam-4343	384	6	,	,	PUNCT
ejpam-4343	384	7	τ	τ	PROPN
ejpam-4343	384	8	,	,	PUNCT
ejpam-4343	384	9	i	i	PROPN
ejpam-4343	384	10	)	)	PUNCT
ejpam-4343	384	11	is	be	AUX
ejpam-4343	384	12	(	(	PUNCT
ejpam-4343	384	13	λ	λ	INTJ
ejpam-4343	384	14	,	,	PUNCT
ejpam-4343	384	15	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	384	16	disconnected	disconnected	ADJ
ejpam-4343	384	17	.	.	PUNCT
ejpam-4343	385	1	c.	c.	PROPN
ejpam-4343	385	2	boonpok	boonpok	PROPN
ejpam-4343	385	3	/	/	SYM
ejpam-4343	385	4	eur	eur	PROPN
ejpam-4343	385	5	.	.	PUNCT
ejpam-4343	386	1	j.	j.	PROPN
ejpam-4343	386	2	pure	pure	PROPN
ejpam-4343	386	3	appl	appl	PROPN
ejpam-4343	386	4	.	.	PROPN
ejpam-4343	386	5	math	math	PROPN
ejpam-4343	386	6	,	,	PUNCT
ejpam-4343	386	7	15	15	NUM
ejpam-4343	386	8	(	(	PUNCT
ejpam-4343	386	9	3	3	NUM
ejpam-4343	386	10	)	)	PUNCT
ejpam-4343	386	11	(	(	PUNCT
ejpam-4343	386	12	2022	2022	NUM
ejpam-4343	386	13	)	)	PUNCT
ejpam-4343	386	14	,	,	PUNCT
ejpam-4343	386	15	1023	1023	NUM
ejpam-4343	386	16	-	-	SYM
ejpam-4343	386	17	1046	1046	NUM
ejpam-4343	386	18	1033	1033	NUM
ejpam-4343	386	19	definition	definition	NOUN
ejpam-4343	386	20	12	12	NUM
ejpam-4343	386	21	.	.	PUNCT
ejpam-4343	387	1	an	an	DET
ejpam-4343	387	2	ideal	ideal	ADJ
ejpam-4343	387	3	topological	topological	ADJ
ejpam-4343	387	4	space	space	NOUN
ejpam-4343	387	5	(	(	PUNCT
ejpam-4343	387	6	x	x	X
ejpam-4343	387	7	,	,	PUNCT
ejpam-4343	387	8	τ	τ	PROPN
ejpam-4343	387	9	,	,	PUNCT
ejpam-4343	387	10	i	i	PROPN
ejpam-4343	387	11	)	)	PUNCT
ejpam-4343	387	12	is	be	AUX
ejpam-4343	387	13	said	say	VERB
ejpam-4343	387	14	to	to	PART
ejpam-4343	387	15	be	be	AUX
ejpam-4343	387	16	(	(	PUNCT
ejpam-4343	387	17	λ	λ	X
ejpam-4343	387	18	,	,	PUNCT
ejpam-4343	387	19	p(⋆))-normal	p(⋆))-normal	ADJ
ejpam-4343	387	20	if	if	SCONJ
ejpam-4343	387	21	,	,	PUNCT
ejpam-4343	387	22	for	for	ADP
ejpam-4343	387	23	any	any	DET
ejpam-4343	387	24	pair	pair	NOUN
ejpam-4343	387	25	disjoint	disjoint	NOUN
ejpam-4343	387	26	(	(	PUNCT
ejpam-4343	387	27	λ	λ	NOUN
ejpam-4343	387	28	,	,	PUNCT
ejpam-4343	387	29	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	387	30	sets	set	VERB
ejpam-4343	387	31	u	u	NOUN
ejpam-4343	387	32	and	and	CCONJ
ejpam-4343	387	33	v	v	NOUN
ejpam-4343	387	34	,	,	PUNCT
ejpam-4343	387	35	there	there	PRON
ejpam-4343	387	36	exist	exist	VERB
ejpam-4343	387	37	disjoint	disjoint	NOUN
ejpam-4343	387	38	(	(	PUNCT
ejpam-4343	387	39	λ	λ	PROPN
ejpam-4343	387	40	,	,	PUNCT
ejpam-4343	387	41	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	387	42	sets	set	NOUN
ejpam-4343	387	43	f	f	PROPN
ejpam-4343	387	44	and	and	CCONJ
ejpam-4343	387	45	h	h	NOUN
ejpam-4343	387	46	such	such	ADJ
ejpam-4343	388	1	that	that	SCONJ
ejpam-4343	388	2	u	u	PROPN
ejpam-4343	388	3	⊆	⊆	NUM
ejpam-4343	388	4	f	f	PROPN
ejpam-4343	388	5	and	and	CCONJ
ejpam-4343	388	6	v	v	ADP
ejpam-4343	388	7	⊆	⊆	NUM
ejpam-4343	388	8	h.	h.	NOUN
ejpam-4343	388	9	theorem	theorem	NOUN
ejpam-4343	388	10	5	5	NUM
ejpam-4343	388	11	.	.	X
ejpam-4343	388	12	for	for	ADP
ejpam-4343	388	13	an	an	DET
ejpam-4343	388	14	ideal	ideal	ADJ
ejpam-4343	388	15	topological	topological	ADJ
ejpam-4343	388	16	space	space	NOUN
ejpam-4343	388	17	(	(	PUNCT
ejpam-4343	388	18	x	x	X
ejpam-4343	388	19	,	,	PUNCT
ejpam-4343	388	20	τ	τ	PROPN
ejpam-4343	388	21	,	,	PUNCT
ejpam-4343	388	22	i	i	NOUN
ejpam-4343	388	23	)	)	PUNCT
ejpam-4343	388	24	,	,	PUNCT
ejpam-4343	388	25	the	the	DET
ejpam-4343	388	26	following	follow	VERB
ejpam-4343	388	27	properties	property	NOUN
ejpam-4343	388	28	are	be	AUX
ejpam-4343	388	29	equivalent	equivalent	ADJ
ejpam-4343	388	30	:	:	PUNCT
ejpam-4343	388	31	(	(	PUNCT
ejpam-4343	388	32	1	1	X
ejpam-4343	388	33	)	)	PUNCT
ejpam-4343	388	34	(	(	PUNCT
ejpam-4343	388	35	x	x	X
ejpam-4343	388	36	,	,	PUNCT
ejpam-4343	388	37	τ	τ	PROPN
ejpam-4343	388	38	,	,	PUNCT
ejpam-4343	388	39	i	i	PROPN
ejpam-4343	388	40	)	)	PUNCT
ejpam-4343	388	41	is	be	AUX
ejpam-4343	388	42	(	(	PUNCT
ejpam-4343	388	43	λ	λ	X
ejpam-4343	388	44	,	,	PUNCT
ejpam-4343	388	45	p(⋆))-normal	p(⋆))-normal	ADJ
ejpam-4343	388	46	.	.	PUNCT
ejpam-4343	389	1	(	(	PUNCT
ejpam-4343	389	2	2	2	NUM
ejpam-4343	389	3	)	)	PUNCT
ejpam-4343	389	4	(	(	PUNCT
ejpam-4343	389	5	x	x	X
ejpam-4343	389	6	,	,	PUNCT
ejpam-4343	389	7	τ	τ	PROPN
ejpam-4343	389	8	,	,	PUNCT
ejpam-4343	389	9	i	i	PROPN
ejpam-4343	389	10	)	)	PUNCT
ejpam-4343	389	11	is	be	AUX
ejpam-4343	389	12	(	(	PUNCT
ejpam-4343	389	13	λ	λ	INTJ
ejpam-4343	389	14	,	,	PUNCT
ejpam-4343	389	15	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	389	16	disconnected	disconnected	ADJ
ejpam-4343	389	17	.	.	PUNCT
ejpam-4343	390	1	proof	proof	NOUN
ejpam-4343	390	2	.	.	PUNCT
ejpam-4343	391	1	(	(	PUNCT
ejpam-4343	391	2	1	1	X
ejpam-4343	391	3	)	)	PUNCT
ejpam-4343	391	4	⇒	⇒	NOUN
ejpam-4343	391	5	(	(	PUNCT
ejpam-4343	391	6	2	2	NUM
ejpam-4343	391	7	):	):	PUNCT
ejpam-4343	391	8	let	let	VERB
ejpam-4343	391	9	u	u	PRON
ejpam-4343	391	10	be	be	AUX
ejpam-4343	391	11	any	any	DET
ejpam-4343	391	12	(	(	PUNCT
ejpam-4343	391	13	λ	λ	PROPN
ejpam-4343	391	14	,	,	PUNCT
ejpam-4343	391	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	391	16	set	set	NOUN
ejpam-4343	391	17	.	.	PUNCT
ejpam-4343	392	1	then	then	ADV
ejpam-4343	392	2	,	,	PUNCT
ejpam-4343	392	3	we	we	PRON
ejpam-4343	392	4	have	have	VERB
ejpam-4343	392	5	u	u	NOUN
ejpam-4343	392	6	and	and	CCONJ
ejpam-4343	392	7	v	v	NOUN
ejpam-4343	392	8	=	=	NOUN
ejpam-4343	392	9	x	x	SYM
ejpam-4343	392	10	−	−	PROPN
ejpam-4343	392	11	u	u	NOUN
ejpam-4343	392	12	(	(	PUNCT
ejpam-4343	392	13	λ	λ	PROPN
ejpam-4343	392	14	,	,	PUNCT
ejpam-4343	392	15	p(⋆	p(⋆	PROPN
ejpam-4343	392	16	)	)	PUNCT
ejpam-4343	392	17	)	)	PUNCT
ejpam-4343	392	18	are	be	AUX
ejpam-4343	392	19	disjoint	disjoint	NOUN
ejpam-4343	392	20	(	(	PUNCT
ejpam-4343	392	21	λ	λ	NOUN
ejpam-4343	392	22	,	,	PUNCT
ejpam-4343	392	23	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	392	24	sets	set	NOUN
ejpam-4343	392	25	.	.	PUNCT
ejpam-4343	393	1	there	there	PRON
ejpam-4343	393	2	exist	exist	VERB
ejpam-4343	393	3	disjoint	disjoint	NOUN
ejpam-4343	393	4	(	(	PUNCT
ejpam-4343	393	5	λ	λ	PROPN
ejpam-4343	393	6	,	,	PUNCT
ejpam-4343	393	7	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	393	8	sets	set	NOUN
ejpam-4343	393	9	f	f	PROPN
ejpam-4343	393	10	and	and	CCONJ
ejpam-4343	393	11	h	h	NOUN
ejpam-4343	393	12	such	such	ADJ
ejpam-4343	394	1	that	that	SCONJ
ejpam-4343	394	2	u	u	PROPN
ejpam-4343	394	3	⊆	⊆	NUM
ejpam-4343	394	4	f	f	PROPN
ejpam-4343	394	5	and	and	CCONJ
ejpam-4343	394	6	v	v	ADP
ejpam-4343	394	7	⊆	⊆	NUM
ejpam-4343	394	8	h.	h.	NOUN
ejpam-4343	394	9	since	since	SCONJ
ejpam-4343	394	10	u	u	PROPN
ejpam-4343	394	11	(	(	PUNCT
ejpam-4343	394	12	λ	λ	PROPN
ejpam-4343	394	13	,	,	PUNCT
ejpam-4343	394	14	p(⋆	p(⋆	PROPN
ejpam-4343	394	15	)	)	PUNCT
ejpam-4343	394	16	)	)	PUNCT
ejpam-4343	395	1	⊆	⊆	NUM
ejpam-4343	395	2	f	f	X
ejpam-4343	395	3	(	(	PUNCT
ejpam-4343	395	4	λ	λ	PROPN
ejpam-4343	395	5	,	,	PUNCT
ejpam-4343	395	6	p(⋆	p(⋆	PROPN
ejpam-4343	395	7	)	)	PUNCT
ejpam-4343	395	8	)	)	PUNCT
ejpam-4343	396	1	=	=	PUNCT
ejpam-4343	396	2	f	f	PROPN
ejpam-4343	397	1	⊆	⊆	NUM
ejpam-4343	397	2	x	x	PUNCT
ejpam-4343	397	3	−h	−h	VERB
ejpam-4343	397	4	⊆	⊆	NUM
ejpam-4343	397	5	x	x	SYM
ejpam-4343	397	6	−	−	NOUN
ejpam-4343	397	7	v	v	NOUN
ejpam-4343	397	8	=	=	SYM
ejpam-4343	397	9	u	u	PROPN
ejpam-4343	397	10	(	(	PUNCT
ejpam-4343	397	11	λ	λ	PROPN
ejpam-4343	397	12	,	,	PUNCT
ejpam-4343	397	13	p(⋆	p(⋆	PROPN
ejpam-4343	397	14	)	)	PUNCT
ejpam-4343	397	15	)	)	PUNCT
ejpam-4343	397	16	,	,	PUNCT
ejpam-4343	397	17	we	we	PRON
ejpam-4343	397	18	have	have	VERB
ejpam-4343	397	19	u	u	NOUN
ejpam-4343	397	20	(	(	PUNCT
ejpam-4343	397	21	λ	λ	PROPN
ejpam-4343	397	22	,	,	PUNCT
ejpam-4343	397	23	p(⋆	p(⋆	PROPN
ejpam-4343	397	24	)	)	PUNCT
ejpam-4343	397	25	)	)	PUNCT
ejpam-4343	398	1	=	=	SYM
ejpam-4343	398	2	f	f	X
ejpam-4343	398	3	.	.	PUNCT
ejpam-4343	399	1	since	since	SCONJ
ejpam-4343	399	2	v	v	NUM
ejpam-4343	399	3	⊆	⊆	NUM
ejpam-4343	399	4	h	h	NOUN
ejpam-4343	399	5	⊆	⊆	NUM
ejpam-4343	399	6	x	x	SYM
ejpam-4343	399	7	−	−	NUM
ejpam-4343	399	8	f	f	NOUN
ejpam-4343	399	9	=	=	SYM
ejpam-4343	399	10	v	v	PROPN
ejpam-4343	399	11	,	,	PUNCT
ejpam-4343	399	12	v	v	NOUN
ejpam-4343	399	13	=	=	PUNCT
ejpam-4343	399	14	h.	h.	PROPN
ejpam-4343	400	1	thus	thus	ADV
ejpam-4343	400	2	,	,	PUNCT
ejpam-4343	400	3	u	u	PROPN
ejpam-4343	400	4	(	(	PUNCT
ejpam-4343	400	5	λ	λ	PROPN
ejpam-4343	400	6	,	,	PUNCT
ejpam-4343	400	7	p(⋆	p(⋆	PROPN
ejpam-4343	400	8	)	)	PUNCT
ejpam-4343	400	9	)	)	PUNCT
ejpam-4343	400	10	=	=	PUNCT
ejpam-4343	401	1	x	x	PUNCT
ejpam-4343	401	2	−h	−h	ADV
ejpam-4343	401	3	is	be	AUX
ejpam-4343	401	4	(	(	PUNCT
ejpam-4343	401	5	λ	λ	INTJ
ejpam-4343	401	6	,	,	PUNCT
ejpam-4343	401	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	401	8	.	.	PUNCT
ejpam-4343	402	1	this	this	PRON
ejpam-4343	402	2	shows	show	VERB
ejpam-4343	402	3	that	that	SCONJ
ejpam-4343	402	4	(	(	PUNCT
ejpam-4343	402	5	x	x	X
ejpam-4343	402	6	,	,	PUNCT
ejpam-4343	402	7	τ	τ	PROPN
ejpam-4343	402	8	,	,	PUNCT
ejpam-4343	402	9	i	i	PROPN
ejpam-4343	402	10	)	)	PUNCT
ejpam-4343	402	11	is	be	AUX
ejpam-4343	402	12	(	(	PUNCT
ejpam-4343	402	13	λ	λ	INTJ
ejpam-4343	402	14	,	,	PUNCT
ejpam-4343	402	15	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	402	16	disconnected	disconnected	ADJ
ejpam-4343	402	17	.	.	PUNCT
ejpam-4343	403	1	(	(	PUNCT
ejpam-4343	403	2	2	2	X
ejpam-4343	403	3	)	)	PUNCT
ejpam-4343	403	4	⇒	⇒	NOUN
ejpam-4343	403	5	(	(	PUNCT
ejpam-4343	403	6	1	1	NUM
ejpam-4343	403	7	):	):	PUNCT
ejpam-4343	403	8	let	let	VERB
ejpam-4343	403	9	u	u	PRON
ejpam-4343	403	10	and	and	CCONJ
ejpam-4343	403	11	v	v	NOUN
ejpam-4343	403	12	be	be	AUX
ejpam-4343	403	13	any	any	DET
ejpam-4343	403	14	two	two	NUM
ejpam-4343	403	15	disjoint	disjoint	NOUN
ejpam-4343	403	16	(	(	PUNCT
ejpam-4343	403	17	λ	λ	NOUN
ejpam-4343	403	18	,	,	PUNCT
ejpam-4343	403	19	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	403	20	sets	set	NOUN
ejpam-4343	403	21	.	.	PUNCT
ejpam-4343	404	1	then	then	ADV
ejpam-4343	404	2	,	,	PUNCT
ejpam-4343	404	3	u	u	PROPN
ejpam-4343	404	4	(	(	PUNCT
ejpam-4343	404	5	λ	λ	PROPN
ejpam-4343	404	6	,	,	PUNCT
ejpam-4343	404	7	p(⋆	p(⋆	PROPN
ejpam-4343	404	8	)	)	PUNCT
ejpam-4343	404	9	)	)	PUNCT
ejpam-4343	405	1	and	and	CCONJ
ejpam-4343	406	1	x	x	X
ejpam-4343	406	2	−	−	PROPN
ejpam-4343	406	3	u	u	NOUN
ejpam-4343	406	4	(	(	PUNCT
ejpam-4343	406	5	λ	λ	PROPN
ejpam-4343	406	6	,	,	PUNCT
ejpam-4343	406	7	p(⋆	p(⋆	PROPN
ejpam-4343	406	8	)	)	PUNCT
ejpam-4343	406	9	)	)	PUNCT
ejpam-4343	406	10	are	be	AUX
ejpam-4343	406	11	disjoint	disjoint	NOUN
ejpam-4343	406	12	(	(	PUNCT
ejpam-4343	406	13	λ	λ	PROPN
ejpam-4343	406	14	,	,	PUNCT
ejpam-4343	406	15	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	406	16	sets	set	NOUN
ejpam-4343	406	17	containing	contain	VERB
ejpam-4343	406	18	u	u	NOUN
ejpam-4343	406	19	and	and	CCONJ
ejpam-4343	406	20	v	v	NOUN
ejpam-4343	406	21	,	,	PUNCT
ejpam-4343	406	22	respectively	respectively	ADV
ejpam-4343	406	23	.	.	PUNCT
ejpam-4343	407	1	thus	thus	ADV
ejpam-4343	407	2	,	,	PUNCT
ejpam-4343	407	3	(	(	PUNCT
ejpam-4343	407	4	x	x	X
ejpam-4343	407	5	,	,	PUNCT
ejpam-4343	407	6	τ	τ	PROPN
ejpam-4343	407	7	,	,	PUNCT
ejpam-4343	407	8	i	i	PROPN
ejpam-4343	407	9	)	)	PUNCT
ejpam-4343	407	10	is	be	AUX
ejpam-4343	407	11	(	(	PUNCT
ejpam-4343	407	12	λ	λ	INTJ
ejpam-4343	407	13	,	,	PUNCT
ejpam-4343	407	14	p(⋆))-normal	p(⋆))-normal	ADJ
ejpam-4343	407	15	.	.	PUNCT
ejpam-4343	408	1	definition	definition	NOUN
ejpam-4343	408	2	13	13	NUM
ejpam-4343	408	3	.	.	PUNCT
ejpam-4343	409	1	a	a	DET
ejpam-4343	409	2	subset	subset	NOUN
ejpam-4343	409	3	a	a	PRON
ejpam-4343	409	4	of	of	ADP
ejpam-4343	409	5	an	an	DET
ejpam-4343	409	6	ideal	ideal	ADJ
ejpam-4343	409	7	topological	topological	ADJ
ejpam-4343	409	8	space	space	NOUN
ejpam-4343	409	9	(	(	PUNCT
ejpam-4343	409	10	x	x	X
ejpam-4343	409	11	,	,	PUNCT
ejpam-4343	409	12	τ	τ	PROPN
ejpam-4343	409	13	,	,	PUNCT
ejpam-4343	409	14	i	i	PROPN
ejpam-4343	409	15	)	)	PUNCT
ejpam-4343	409	16	is	be	AUX
ejpam-4343	409	17	said	say	VERB
ejpam-4343	409	18	to	to	PART
ejpam-4343	409	19	be	be	AUX
ejpam-4343	409	20	:	:	PUNCT
ejpam-4343	409	21	(	(	PUNCT
ejpam-4343	409	22	1	1	X
ejpam-4343	409	23	)	)	PUNCT
ejpam-4343	409	24	regular	regular	ADJ
ejpam-4343	409	25	(	(	PUNCT
ejpam-4343	409	26	λ	λ	PROPN
ejpam-4343	409	27	,	,	PUNCT
ejpam-4343	409	28	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	409	29	if	if	SCONJ
ejpam-4343	409	30	a	a	PRON
ejpam-4343	409	31	=	=	X
ejpam-4343	410	1	[	[	X
ejpam-4343	410	2	a(λ	a(λ	PROPN
ejpam-4343	410	3	,	,	PUNCT
ejpam-4343	410	4	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	410	5	,	,	PUNCT
ejpam-4343	410	6	p(⋆	p(⋆	PROPN
ejpam-4343	410	7	)	)	PUNCT
ejpam-4343	410	8	)	)	PUNCT
ejpam-4343	410	9	;	;	PUNCT
ejpam-4343	410	10	(	(	PUNCT
ejpam-4343	410	11	2	2	X
ejpam-4343	410	12	)	)	PUNCT
ejpam-4343	410	13	regular	regular	ADJ
ejpam-4343	410	14	(	(	PUNCT
ejpam-4343	410	15	λ	λ	PROPN
ejpam-4343	410	16	,	,	PUNCT
ejpam-4343	410	17	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	410	18	if	if	SCONJ
ejpam-4343	410	19	a	a	PRON
ejpam-4343	410	20	=	=	X
ejpam-4343	411	1	[	[	X
ejpam-4343	411	2	a(λ	a(λ	ADV
ejpam-4343	411	3	,	,	PUNCT
ejpam-4343	411	4	p(⋆	p(⋆	PROPN
ejpam-4343	411	5	)	)	PUNCT
ejpam-4343	411	6	)	)	PUNCT
ejpam-4343	411	7	]	]	PUNCT
ejpam-4343	412	1	(	(	PUNCT
ejpam-4343	412	2	λ	λ	X
ejpam-4343	412	3	,	,	PUNCT
ejpam-4343	412	4	p(⋆	p(⋆	PROPN
ejpam-4343	412	5	)	)	PUNCT
ejpam-4343	412	6	)	)	PUNCT
ejpam-4343	412	7	.	.	PUNCT
ejpam-4343	413	1	theorem	theorem	VERB
ejpam-4343	413	2	6	6	NUM
ejpam-4343	413	3	.	.	PUNCT
ejpam-4343	413	4	for	for	ADP
ejpam-4343	413	5	an	an	DET
ejpam-4343	413	6	ideal	ideal	ADJ
ejpam-4343	413	7	topological	topological	ADJ
ejpam-4343	413	8	space	space	NOUN
ejpam-4343	413	9	(	(	PUNCT
ejpam-4343	413	10	x	x	X
ejpam-4343	413	11	,	,	PUNCT
ejpam-4343	413	12	τ	τ	PROPN
ejpam-4343	413	13	,	,	PUNCT
ejpam-4343	413	14	i	i	NOUN
ejpam-4343	413	15	)	)	PUNCT
ejpam-4343	413	16	,	,	PUNCT
ejpam-4343	413	17	the	the	DET
ejpam-4343	413	18	following	follow	VERB
ejpam-4343	413	19	properties	property	NOUN
ejpam-4343	413	20	are	be	AUX
ejpam-4343	413	21	equivalent	equivalent	ADJ
ejpam-4343	413	22	:	:	PUNCT
ejpam-4343	413	23	(	(	PUNCT
ejpam-4343	413	24	1	1	X
ejpam-4343	413	25	)	)	PUNCT
ejpam-4343	413	26	(	(	PUNCT
ejpam-4343	413	27	x	x	X
ejpam-4343	413	28	,	,	PUNCT
ejpam-4343	413	29	τ	τ	PROPN
ejpam-4343	413	30	,	,	PUNCT
ejpam-4343	413	31	i	i	PROPN
ejpam-4343	413	32	)	)	PUNCT
ejpam-4343	413	33	is	be	AUX
ejpam-4343	413	34	(	(	PUNCT
ejpam-4343	413	35	λ	λ	INTJ
ejpam-4343	413	36	,	,	PUNCT
ejpam-4343	413	37	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	413	38	disconnected	disconnected	ADJ
ejpam-4343	413	39	.	.	PUNCT
ejpam-4343	414	1	(	(	PUNCT
ejpam-4343	414	2	2	2	X
ejpam-4343	414	3	)	)	PUNCT
ejpam-4343	414	4	every	every	ADV
ejpam-4343	414	5	regular	regular	ADJ
ejpam-4343	414	6	(	(	PUNCT
ejpam-4343	414	7	λ	λ	PROPN
ejpam-4343	414	8	,	,	PUNCT
ejpam-4343	414	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	414	10	set	set	VERB
ejpam-4343	414	11	is	be	AUX
ejpam-4343	414	12	(	(	PUNCT
ejpam-4343	414	13	λ	λ	X
ejpam-4343	414	14	,	,	PUNCT
ejpam-4343	414	15	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	414	16	.	.	PUNCT
ejpam-4343	415	1	(	(	PUNCT
ejpam-4343	415	2	3	3	X
ejpam-4343	415	3	)	)	PUNCT
ejpam-4343	415	4	every	every	ADV
ejpam-4343	415	5	regular	regular	ADJ
ejpam-4343	415	6	(	(	PUNCT
ejpam-4343	415	7	λ	λ	PROPN
ejpam-4343	415	8	,	,	PUNCT
ejpam-4343	415	9	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	415	10	set	set	NOUN
ejpam-4343	415	11	is	be	AUX
ejpam-4343	415	12	(	(	PUNCT
ejpam-4343	415	13	λ	λ	INTJ
ejpam-4343	415	14	,	,	PUNCT
ejpam-4343	415	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	415	16	.	.	PUNCT
ejpam-4343	416	1	proof	proof	NOUN
ejpam-4343	416	2	.	.	PUNCT
ejpam-4343	417	1	(	(	PUNCT
ejpam-4343	417	2	1	1	X
ejpam-4343	417	3	)	)	PUNCT
ejpam-4343	417	4	⇒	⇒	NOUN
ejpam-4343	417	5	(	(	PUNCT
ejpam-4343	417	6	2	2	NUM
ejpam-4343	417	7	):	):	PUNCT
ejpam-4343	417	8	suppose	suppose	VERB
ejpam-4343	417	9	that	that	SCONJ
ejpam-4343	417	10	(	(	PUNCT
ejpam-4343	417	11	x	x	X
ejpam-4343	417	12	,	,	PUNCT
ejpam-4343	417	13	τ	τ	PROPN
ejpam-4343	417	14	,	,	PUNCT
ejpam-4343	417	15	i	i	PROPN
ejpam-4343	417	16	)	)	PUNCT
ejpam-4343	417	17	is	be	AUX
ejpam-4343	417	18	(	(	PUNCT
ejpam-4343	417	19	λ	λ	INTJ
ejpam-4343	417	20	,	,	PUNCT
ejpam-4343	417	21	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	417	22	disconnected	disconnect	VERB
ejpam-4343	417	23	.	.	PUNCT
ejpam-4343	418	1	let	let	VERB
ejpam-4343	418	2	v	v	PART
ejpam-4343	418	3	be	be	AUX
ejpam-4343	418	4	any	any	PRON
ejpam-4343	418	5	regular	regular	ADJ
ejpam-4343	418	6	(	(	PUNCT
ejpam-4343	418	7	λ	λ	PROPN
ejpam-4343	418	8	,	,	PUNCT
ejpam-4343	418	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	418	10	set	set	NOUN
ejpam-4343	418	11	.	.	PUNCT
ejpam-4343	419	1	then	then	ADV
ejpam-4343	419	2	,	,	PUNCT
ejpam-4343	419	3	we	we	PRON
ejpam-4343	419	4	have	have	VERB
ejpam-4343	419	5	v	v	NOUN
ejpam-4343	419	6	=	=	PUNCT
ejpam-4343	420	1	[	[	X
ejpam-4343	420	2	v	v	X
ejpam-4343	420	3	(	(	PUNCT
ejpam-4343	420	4	λ	λ	PROPN
ejpam-4343	420	5	,	,	PUNCT
ejpam-4343	420	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	420	7	,	,	PUNCT
ejpam-4343	420	8	p(⋆	p(⋆	PROPN
ejpam-4343	420	9	)	)	PUNCT
ejpam-4343	420	10	)	)	PUNCT
ejpam-4343	420	11	.	.	PUNCT
ejpam-4343	421	1	since	since	SCONJ
ejpam-4343	421	2	v	v	NOUN
ejpam-4343	421	3	(	(	PUNCT
ejpam-4343	421	4	λ	λ	PROPN
ejpam-4343	421	5	,	,	PUNCT
ejpam-4343	421	6	p(⋆	p(⋆	PROPN
ejpam-4343	421	7	)	)	PUNCT
ejpam-4343	421	8	)	)	PUNCT
ejpam-4343	421	9	is	be	AUX
ejpam-4343	421	10	(	(	PUNCT
ejpam-4343	421	11	λ	λ	INTJ
ejpam-4343	421	12	,	,	PUNCT
ejpam-4343	421	13	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	421	14	,	,	PUNCT
ejpam-4343	421	15	v	v	NOUN
ejpam-4343	421	16	=	=	PUNCT
ejpam-4343	422	1	[	[	X
ejpam-4343	422	2	v	v	X
ejpam-4343	422	3	(	(	PUNCT
ejpam-4343	422	4	λ	λ	PROPN
ejpam-4343	422	5	,	,	PUNCT
ejpam-4343	422	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	422	7	,	,	PUNCT
ejpam-4343	422	8	p(⋆	p(⋆	PROPN
ejpam-4343	422	9	)	)	PUNCT
ejpam-4343	422	10	)	)	PUNCT
ejpam-4343	423	1	=	=	SYM
ejpam-4343	423	2	v	v	X
ejpam-4343	423	3	(	(	PUNCT
ejpam-4343	423	4	λ	λ	PROPN
ejpam-4343	423	5	,	,	PUNCT
ejpam-4343	423	6	p(⋆	p(⋆	PROPN
ejpam-4343	423	7	)	)	PUNCT
ejpam-4343	423	8	)	)	PUNCT
ejpam-4343	423	9	and	and	CCONJ
ejpam-4343	423	10	hence	hence	ADV
ejpam-4343	423	11	v	v	NOUN
ejpam-4343	423	12	is	be	AUX
ejpam-4343	423	13	(	(	PUNCT
ejpam-4343	423	14	λ	λ	PROPN
ejpam-4343	423	15	,	,	PUNCT
ejpam-4343	423	16	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	423	17	.	.	PUNCT
ejpam-4343	424	1	(	(	PUNCT
ejpam-4343	424	2	2	2	X
ejpam-4343	424	3	)	)	PUNCT
ejpam-4343	424	4	⇒	⇒	NOUN
ejpam-4343	424	5	(	(	PUNCT
ejpam-4343	424	6	1	1	NUM
ejpam-4343	424	7	):	):	PUNCT
ejpam-4343	424	8	suppose	suppose	VERB
ejpam-4343	424	9	that	that	SCONJ
ejpam-4343	424	10	every	every	DET
ejpam-4343	424	11	regular	regular	ADJ
ejpam-4343	424	12	(	(	PUNCT
ejpam-4343	424	13	λ	λ	PROPN
ejpam-4343	424	14	,	,	PUNCT
ejpam-4343	424	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	424	16	set	set	VERB
ejpam-4343	424	17	is	be	AUX
ejpam-4343	424	18	(	(	PUNCT
ejpam-4343	424	19	λ	λ	X
ejpam-4343	424	20	,	,	PUNCT
ejpam-4343	424	21	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	424	22	.	.	PUNCT
ejpam-4343	425	1	let	let	VERB
ejpam-4343	425	2	v	v	PART
ejpam-4343	425	3	be	be	AUX
ejpam-4343	425	4	any	any	DET
ejpam-4343	425	5	(	(	PUNCT
ejpam-4343	425	6	λ	λ	PROPN
ejpam-4343	425	7	,	,	PUNCT
ejpam-4343	425	8	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	425	9	set	set	NOUN
ejpam-4343	425	10	.	.	PUNCT
ejpam-4343	426	1	then	then	ADV
ejpam-4343	426	2	,	,	PUNCT
ejpam-4343	426	3	we	we	PRON
ejpam-4343	426	4	have	have	VERB
ejpam-4343	426	5	[	[	X
ejpam-4343	426	6	v	v	X
ejpam-4343	426	7	(	(	PUNCT
ejpam-4343	426	8	λ	λ	PROPN
ejpam-4343	426	9	,	,	PUNCT
ejpam-4343	426	10	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	426	11	,	,	PUNCT
ejpam-4343	426	12	p(⋆	p(⋆	PROPN
ejpam-4343	426	13	)	)	PUNCT
ejpam-4343	426	14	)	)	PUNCT
ejpam-4343	427	1	is	be	AUX
ejpam-4343	427	2	regular	regular	ADJ
ejpam-4343	427	3	(	(	PUNCT
ejpam-4343	427	4	λ	λ	PROPN
ejpam-4343	427	5	,	,	PUNCT
ejpam-4343	427	6	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	427	7	and	and	CCONJ
ejpam-4343	427	8	by	by	ADP
ejpam-4343	427	9	(	(	PUNCT
ejpam-4343	427	10	2	2	NUM
ejpam-4343	427	11	)	)	PUNCT
ejpam-4343	427	12	,	,	PUNCT
ejpam-4343	428	1	[	[	X
ejpam-4343	428	2	v	v	X
ejpam-4343	428	3	(	(	PUNCT
ejpam-4343	428	4	λ	λ	PROPN
ejpam-4343	428	5	,	,	PUNCT
ejpam-4343	428	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	428	7	,	,	PUNCT
ejpam-4343	428	8	p(⋆	p(⋆	PROPN
ejpam-4343	428	9	)	)	PUNCT
ejpam-4343	428	10	)	)	PUNCT
ejpam-4343	428	11	is	be	AUX
ejpam-4343	428	12	closed	close	VERB
ejpam-4343	428	13	.	.	PUNCT
ejpam-4343	429	1	thus	thus	ADV
ejpam-4343	429	2	,	,	PUNCT
ejpam-4343	429	3	v	v	INTJ
ejpam-4343	429	4	(	(	PUNCT
ejpam-4343	429	5	λ	λ	PROPN
ejpam-4343	429	6	,	,	PUNCT
ejpam-4343	429	7	p(⋆	p(⋆	PROPN
ejpam-4343	429	8	)	)	PUNCT
ejpam-4343	429	9	)	)	PUNCT
ejpam-4343	430	1	⊆	⊆	NUM
ejpam-4343	431	1	[	[	X
ejpam-4343	431	2	[	[	X
ejpam-4343	431	3	v	v	X
ejpam-4343	431	4	(	(	PUNCT
ejpam-4343	431	5	λ	λ	PROPN
ejpam-4343	431	6	,	,	PUNCT
ejpam-4343	431	7	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	431	8	,	,	PUNCT
ejpam-4343	431	9	p(⋆	p(⋆	PROPN
ejpam-4343	431	10	)	)	PUNCT
ejpam-4343	431	11	)	)	PUNCT
ejpam-4343	431	12	]	]	PUNCT
ejpam-4343	431	13	(	(	PUNCT
ejpam-4343	431	14	λ	λ	X
ejpam-4343	431	15	,	,	PUNCT
ejpam-4343	431	16	p(⋆	p(⋆	PROPN
ejpam-4343	431	17	)	)	PUNCT
ejpam-4343	431	18	)	)	PUNCT
ejpam-4343	432	1	=	=	PUNCT
ejpam-4343	433	1	[	[	X
ejpam-4343	433	2	v	v	X
ejpam-4343	433	3	(	(	PUNCT
ejpam-4343	433	4	λ	λ	PROPN
ejpam-4343	433	5	,	,	PUNCT
ejpam-4343	433	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	433	7	,	,	PUNCT
ejpam-4343	433	8	p(⋆	p(⋆	PROPN
ejpam-4343	433	9	)	)	PUNCT
ejpam-4343	433	10	)	)	PUNCT
ejpam-4343	433	11	and	and	CCONJ
ejpam-4343	433	12	hence	hence	ADV
ejpam-4343	433	13	v	v	NOUN
ejpam-4343	433	14	(	(	PUNCT
ejpam-4343	433	15	λ	λ	PROPN
ejpam-4343	433	16	,	,	PUNCT
ejpam-4343	433	17	p(⋆	p(⋆	PROPN
ejpam-4343	433	18	)	)	PUNCT
ejpam-4343	433	19	)	)	PUNCT
ejpam-4343	433	20	is	be	AUX
ejpam-4343	433	21	(	(	PUNCT
ejpam-4343	433	22	λ	λ	INTJ
ejpam-4343	433	23	,	,	PUNCT
ejpam-4343	433	24	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	433	25	.	.	PUNCT
ejpam-4343	434	1	therefore	therefore	ADV
ejpam-4343	434	2	,	,	PUNCT
ejpam-4343	434	3	(	(	PUNCT
ejpam-4343	434	4	x	x	X
ejpam-4343	434	5	,	,	PUNCT
ejpam-4343	434	6	τ	τ	PROPN
ejpam-4343	434	7	,	,	PUNCT
ejpam-4343	434	8	i	i	PROPN
ejpam-4343	434	9	)	)	PUNCT
ejpam-4343	434	10	is	be	AUX
ejpam-4343	434	11	(	(	PUNCT
ejpam-4343	434	12	λ	λ	INTJ
ejpam-4343	434	13	,	,	PUNCT
ejpam-4343	434	14	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	434	15	disconnected	disconnected	ADJ
ejpam-4343	434	16	.	.	PUNCT
ejpam-4343	435	1	(	(	PUNCT
ejpam-4343	435	2	2	2	X
ejpam-4343	435	3	)	)	PUNCT
ejpam-4343	435	4	⇔	⇔	X
ejpam-4343	435	5	(	(	PUNCT
ejpam-4343	435	6	3	3	NUM
ejpam-4343	435	7	):	):	PUNCT
ejpam-4343	435	8	this	this	PRON
ejpam-4343	435	9	is	be	AUX
ejpam-4343	435	10	obvious	obvious	ADJ
ejpam-4343	435	11	.	.	PUNCT
ejpam-4343	436	1	c.	c.	PROPN
ejpam-4343	436	2	boonpok	boonpok	PROPN
ejpam-4343	436	3	/	/	SYM
ejpam-4343	436	4	eur	eur	PROPN
ejpam-4343	436	5	.	.	PUNCT
ejpam-4343	437	1	j.	j.	PROPN
ejpam-4343	437	2	pure	pure	PROPN
ejpam-4343	437	3	appl	appl	PROPN
ejpam-4343	437	4	.	.	PROPN
ejpam-4343	437	5	math	math	PROPN
ejpam-4343	437	6	,	,	PUNCT
ejpam-4343	437	7	15	15	NUM
ejpam-4343	437	8	(	(	PUNCT
ejpam-4343	437	9	3	3	NUM
ejpam-4343	437	10	)	)	PUNCT
ejpam-4343	437	11	(	(	PUNCT
ejpam-4343	437	12	2022	2022	NUM
ejpam-4343	437	13	)	)	PUNCT
ejpam-4343	437	14	,	,	PUNCT
ejpam-4343	437	15	1023	1023	NUM
ejpam-4343	437	16	-	-	SYM
ejpam-4343	437	17	1046	1046	NUM
ejpam-4343	437	18	1034	1034	NUM
ejpam-4343	437	19	theorem	theorem	VERB
ejpam-4343	437	20	7	7	NUM
ejpam-4343	437	21	.	.	X
ejpam-4343	437	22	for	for	ADP
ejpam-4343	437	23	an	an	DET
ejpam-4343	437	24	ideal	ideal	ADJ
ejpam-4343	437	25	topological	topological	ADJ
ejpam-4343	437	26	space	space	NOUN
ejpam-4343	437	27	(	(	PUNCT
ejpam-4343	437	28	x	x	X
ejpam-4343	437	29	,	,	PUNCT
ejpam-4343	437	30	τ	τ	PROPN
ejpam-4343	437	31	,	,	PUNCT
ejpam-4343	437	32	i	i	NOUN
ejpam-4343	437	33	)	)	PUNCT
ejpam-4343	437	34	,	,	PUNCT
ejpam-4343	437	35	the	the	DET
ejpam-4343	437	36	following	follow	VERB
ejpam-4343	437	37	properties	property	NOUN
ejpam-4343	437	38	are	be	AUX
ejpam-4343	437	39	equivalent	equivalent	ADJ
ejpam-4343	437	40	:	:	PUNCT
ejpam-4343	437	41	(	(	PUNCT
ejpam-4343	437	42	1	1	X
ejpam-4343	437	43	)	)	PUNCT
ejpam-4343	437	44	(	(	PUNCT
ejpam-4343	437	45	x	x	X
ejpam-4343	437	46	,	,	PUNCT
ejpam-4343	437	47	τ	τ	PROPN
ejpam-4343	437	48	,	,	PUNCT
ejpam-4343	437	49	i	i	PROPN
ejpam-4343	437	50	)	)	PUNCT
ejpam-4343	437	51	is	be	AUX
ejpam-4343	437	52	(	(	PUNCT
ejpam-4343	437	53	λ	λ	INTJ
ejpam-4343	437	54	,	,	PUNCT
ejpam-4343	437	55	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	437	56	disconnected	disconnected	ADJ
ejpam-4343	437	57	.	.	PUNCT
ejpam-4343	438	1	(	(	PUNCT
ejpam-4343	438	2	2	2	X
ejpam-4343	438	3	)	)	PUNCT
ejpam-4343	438	4	the	the	DET
ejpam-4343	438	5	(	(	PUNCT
ejpam-4343	438	6	λ	λ	PROPN
ejpam-4343	438	7	,	,	PUNCT
ejpam-4343	438	8	p(⋆))-closure	p(⋆))-closure	NOUN
ejpam-4343	438	9	of	of	ADP
ejpam-4343	438	10	every	every	DET
ejpam-4343	438	11	semi-(λ	semi-(λ	PROPN
ejpam-4343	438	12	,	,	PUNCT
ejpam-4343	438	13	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	438	14	set	set	VERB
ejpam-4343	438	15	is	be	AUX
ejpam-4343	438	16	(	(	PUNCT
ejpam-4343	438	17	λ	λ	INTJ
ejpam-4343	438	18	,	,	PUNCT
ejpam-4343	438	19	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	438	20	.	.	PUNCT
ejpam-4343	439	1	(	(	PUNCT
ejpam-4343	439	2	3	3	X
ejpam-4343	439	3	)	)	PUNCT
ejpam-4343	439	4	the	the	DET
ejpam-4343	439	5	(	(	PUNCT
ejpam-4343	439	6	λ	λ	PROPN
ejpam-4343	439	7	,	,	PUNCT
ejpam-4343	439	8	p(⋆))-closure	p(⋆))-closure	NOUN
ejpam-4343	439	9	of	of	ADP
ejpam-4343	439	10	every	every	DET
ejpam-4343	439	11	pre-(λ	pre-(λ	PROPN
ejpam-4343	439	12	,	,	PUNCT
ejpam-4343	439	13	p(⋆))-open	p(⋆))-open	PROPN
ejpam-4343	439	14	set	set	VERB
ejpam-4343	439	15	is	be	AUX
ejpam-4343	439	16	(	(	PUNCT
ejpam-4343	439	17	λ	λ	INTJ
ejpam-4343	439	18	,	,	PUNCT
ejpam-4343	439	19	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	439	20	.	.	PUNCT
ejpam-4343	440	1	(	(	PUNCT
ejpam-4343	440	2	4	4	X
ejpam-4343	440	3	)	)	PUNCT
ejpam-4343	440	4	the	the	DET
ejpam-4343	440	5	(	(	PUNCT
ejpam-4343	440	6	λ	λ	PROPN
ejpam-4343	440	7	,	,	PUNCT
ejpam-4343	440	8	p(⋆))-closure	p(⋆))-closure	NOUN
ejpam-4343	440	9	of	of	ADP
ejpam-4343	440	10	every	every	DET
ejpam-4343	440	11	regular	regular	ADJ
ejpam-4343	440	12	(	(	PUNCT
ejpam-4343	440	13	λ	λ	PROPN
ejpam-4343	440	14	,	,	PUNCT
ejpam-4343	440	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	440	16	set	set	VERB
ejpam-4343	440	17	is	be	AUX
ejpam-4343	440	18	(	(	PUNCT
ejpam-4343	440	19	λ	λ	INTJ
ejpam-4343	440	20	,	,	PUNCT
ejpam-4343	440	21	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	440	22	.	.	PUNCT
ejpam-4343	441	1	proof	proof	NOUN
ejpam-4343	441	2	.	.	PUNCT
ejpam-4343	442	1	(	(	PUNCT
ejpam-4343	442	2	1	1	X
ejpam-4343	442	3	)	)	PUNCT
ejpam-4343	442	4	⇒	⇒	NOUN
ejpam-4343	442	5	(	(	PUNCT
ejpam-4343	442	6	2	2	NUM
ejpam-4343	442	7	):	):	PUNCT
ejpam-4343	442	8	let	let	VERB
ejpam-4343	442	9	v	v	PART
ejpam-4343	442	10	be	be	AUX
ejpam-4343	442	11	any	any	DET
ejpam-4343	442	12	semi-(λ	semi-(λ	PROPN
ejpam-4343	442	13	,	,	PUNCT
ejpam-4343	442	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	442	15	set	set	NOUN
ejpam-4343	442	16	.	.	PUNCT
ejpam-4343	443	1	then	then	ADV
ejpam-4343	443	2	,	,	PUNCT
ejpam-4343	443	3	v	v	NOUN
ejpam-4343	443	4	is	be	AUX
ejpam-4343	443	5	β-(λ	β-(λ	PRON
ejpam-4343	443	6	,	,	PUNCT
ejpam-4343	443	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	443	8	and	and	CCONJ
ejpam-4343	443	9	by	by	ADP
ejpam-4343	443	10	theorem	theorem	ADJ
ejpam-4343	443	11	4	4	NUM
ejpam-4343	443	12	,	,	PUNCT
ejpam-4343	443	13	v	v	NOUN
ejpam-4343	443	14	(	(	PUNCT
ejpam-4343	443	15	λ	λ	PROPN
ejpam-4343	443	16	,	,	PUNCT
ejpam-4343	443	17	p(⋆	p(⋆	PROPN
ejpam-4343	443	18	)	)	PUNCT
ejpam-4343	443	19	)	)	PUNCT
ejpam-4343	444	1	is	be	AUX
ejpam-4343	444	2	(	(	PUNCT
ejpam-4343	444	3	λ	λ	INTJ
ejpam-4343	444	4	,	,	PUNCT
ejpam-4343	444	5	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	444	6	.	.	PUNCT
ejpam-4343	445	1	(	(	PUNCT
ejpam-4343	445	2	2	2	X
ejpam-4343	445	3	)	)	PUNCT
ejpam-4343	445	4	⇒	⇒	NOUN
ejpam-4343	445	5	(	(	PUNCT
ejpam-4343	445	6	4	4	NUM
ejpam-4343	445	7	):	):	PUNCT
ejpam-4343	445	8	let	let	VERB
ejpam-4343	445	9	v	v	PART
ejpam-4343	445	10	be	be	AUX
ejpam-4343	445	11	any	any	PRON
ejpam-4343	445	12	regular	regular	ADJ
ejpam-4343	445	13	(	(	PUNCT
ejpam-4343	445	14	λ	λ	PROPN
ejpam-4343	445	15	,	,	PUNCT
ejpam-4343	445	16	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	445	17	set	set	NOUN
ejpam-4343	445	18	.	.	PUNCT
ejpam-4343	446	1	then	then	ADV
ejpam-4343	446	2	,	,	PUNCT
ejpam-4343	446	3	v	v	INTJ
ejpam-4343	446	4	(	(	PUNCT
ejpam-4343	446	5	λ	λ	PROPN
ejpam-4343	446	6	,	,	PUNCT
ejpam-4343	446	7	p(⋆	p(⋆	PROPN
ejpam-4343	446	8	)	)	PUNCT
ejpam-4343	446	9	)	)	PUNCT
ejpam-4343	446	10	is	be	AUX
ejpam-4343	446	11	semi-(λ	semi-(λ	PROPN
ejpam-4343	446	12	,	,	PUNCT
ejpam-4343	446	13	p(⋆))open	p(⋆))open	PROPN
ejpam-4343	446	14	and	and	CCONJ
ejpam-4343	446	15	by	by	ADP
ejpam-4343	446	16	(	(	PUNCT
ejpam-4343	446	17	2	2	NUM
ejpam-4343	446	18	)	)	PUNCT
ejpam-4343	446	19	,	,	PUNCT
ejpam-4343	446	20	we	we	PRON
ejpam-4343	446	21	have	have	VERB
ejpam-4343	446	22	[	[	X
ejpam-4343	446	23	v	v	X
ejpam-4343	446	24	(	(	PUNCT
ejpam-4343	446	25	λ	λ	PROPN
ejpam-4343	446	26	,	,	PUNCT
ejpam-4343	446	27	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	446	28	,	,	PUNCT
ejpam-4343	446	29	p(⋆	p(⋆	PROPN
ejpam-4343	446	30	)	)	PUNCT
ejpam-4343	446	31	)	)	PUNCT
ejpam-4343	447	1	=	=	SYM
ejpam-4343	447	2	v	v	X
ejpam-4343	447	3	(	(	PUNCT
ejpam-4343	447	4	λ	λ	PROPN
ejpam-4343	447	5	,	,	PUNCT
ejpam-4343	447	6	p(⋆	p(⋆	PROPN
ejpam-4343	447	7	)	)	PUNCT
ejpam-4343	447	8	)	)	PUNCT
ejpam-4343	447	9	is	be	AUX
ejpam-4343	447	10	(	(	PUNCT
ejpam-4343	447	11	λ	λ	INTJ
ejpam-4343	447	12	,	,	PUNCT
ejpam-4343	447	13	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	447	14	.	.	PUNCT
ejpam-4343	448	1	(	(	PUNCT
ejpam-4343	448	2	4	4	X
ejpam-4343	448	3	)	)	PUNCT
ejpam-4343	448	4	⇒	⇒	NOUN
ejpam-4343	448	5	(	(	PUNCT
ejpam-4343	448	6	1	1	NUM
ejpam-4343	448	7	):	):	PUNCT
ejpam-4343	448	8	let	let	VERB
ejpam-4343	448	9	v	v	PART
ejpam-4343	448	10	be	be	AUX
ejpam-4343	448	11	any	any	DET
ejpam-4343	448	12	(	(	PUNCT
ejpam-4343	448	13	λ	λ	PROPN
ejpam-4343	448	14	,	,	PUNCT
ejpam-4343	448	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	448	16	set	set	NOUN
ejpam-4343	448	17	.	.	PUNCT
ejpam-4343	449	1	then	then	ADV
ejpam-4343	449	2	,	,	PUNCT
ejpam-4343	449	3	we	we	PRON
ejpam-4343	449	4	have	have	VERB
ejpam-4343	449	5	[	[	X
ejpam-4343	449	6	v	v	X
ejpam-4343	449	7	(	(	PUNCT
ejpam-4343	449	8	λ	λ	PROPN
ejpam-4343	449	9	,	,	PUNCT
ejpam-4343	449	10	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	449	11	,	,	PUNCT
ejpam-4343	449	12	p(⋆	p(⋆	PROPN
ejpam-4343	449	13	)	)	PUNCT
ejpam-4343	449	14	)	)	PUNCT
ejpam-4343	450	1	is	be	AUX
ejpam-4343	450	2	regular	regular	ADJ
ejpam-4343	450	3	(	(	PUNCT
ejpam-4343	450	4	λ	λ	PROPN
ejpam-4343	450	5	,	,	PUNCT
ejpam-4343	450	6	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	450	7	and	and	CCONJ
ejpam-4343	450	8	by	by	ADP
ejpam-4343	450	9	(	(	PUNCT
ejpam-4343	450	10	4	4	NUM
ejpam-4343	450	11	)	)	PUNCT
ejpam-4343	450	12	,	,	PUNCT
ejpam-4343	451	1	[	[	X
ejpam-4343	451	2	[	[	X
ejpam-4343	451	3	v	v	X
ejpam-4343	451	4	(	(	PUNCT
ejpam-4343	451	5	λ	λ	PROPN
ejpam-4343	451	6	,	,	PUNCT
ejpam-4343	451	7	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	451	8	,	,	PUNCT
ejpam-4343	451	9	p(⋆	p(⋆	PROPN
ejpam-4343	451	10	)	)	PUNCT
ejpam-4343	451	11	)	)	PUNCT
ejpam-4343	451	12	]	]	PUNCT
ejpam-4343	451	13	(	(	PUNCT
ejpam-4343	451	14	λ	λ	X
ejpam-4343	451	15	,	,	PUNCT
ejpam-4343	451	16	p(⋆	p(⋆	PROPN
ejpam-4343	451	17	)	)	PUNCT
ejpam-4343	451	18	)	)	PUNCT
ejpam-4343	451	19	is	be	AUX
ejpam-4343	451	20	(	(	PUNCT
ejpam-4343	451	21	λ	λ	INTJ
ejpam-4343	451	22	,	,	PUNCT
ejpam-4343	451	23	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	451	24	.	.	PUNCT
ejpam-4343	452	1	thus	thus	ADV
ejpam-4343	452	2	,	,	PUNCT
ejpam-4343	452	3	v	v	INTJ
ejpam-4343	452	4	(	(	PUNCT
ejpam-4343	452	5	λ	λ	PROPN
ejpam-4343	452	6	,	,	PUNCT
ejpam-4343	452	7	p(⋆	p(⋆	PROPN
ejpam-4343	452	8	)	)	PUNCT
ejpam-4343	452	9	)	)	PUNCT
ejpam-4343	453	1	⊆	⊆	NUM
ejpam-4343	454	1	[	[	X
ejpam-4343	454	2	[	[	X
ejpam-4343	454	3	v	v	X
ejpam-4343	454	4	(	(	PUNCT
ejpam-4343	454	5	λ	λ	PROPN
ejpam-4343	454	6	,	,	PUNCT
ejpam-4343	454	7	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	454	8	,	,	PUNCT
ejpam-4343	454	9	p(⋆	p(⋆	PROPN
ejpam-4343	454	10	)	)	PUNCT
ejpam-4343	454	11	)	)	PUNCT
ejpam-4343	454	12	]	]	PUNCT
ejpam-4343	454	13	(	(	PUNCT
ejpam-4343	454	14	λ	λ	X
ejpam-4343	454	15	,	,	PUNCT
ejpam-4343	454	16	p(⋆	p(⋆	PROPN
ejpam-4343	454	17	)	)	PUNCT
ejpam-4343	454	18	)	)	PUNCT
ejpam-4343	455	1	=	=	PUNCT
ejpam-4343	456	1	[	[	X
ejpam-4343	456	2	[	[	X
ejpam-4343	456	3	[	[	X
ejpam-4343	456	4	v	v	X
ejpam-4343	456	5	(	(	PUNCT
ejpam-4343	456	6	λ	λ	PROPN
ejpam-4343	456	7	,	,	PUNCT
ejpam-4343	456	8	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	456	9	,	,	PUNCT
ejpam-4343	456	10	p(⋆	p(⋆	PROPN
ejpam-4343	456	11	)	)	PUNCT
ejpam-4343	456	12	)	)	PUNCT
ejpam-4343	456	13	]	]	PUNCT
ejpam-4343	456	14	(	(	PUNCT
ejpam-4343	456	15	λ	λ	X
ejpam-4343	456	16	,	,	PUNCT
ejpam-4343	456	17	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	456	18	,	,	PUNCT
ejpam-4343	456	19	p(⋆	p(⋆	PROPN
ejpam-4343	456	20	)	)	PUNCT
ejpam-4343	456	21	)	)	PUNCT
ejpam-4343	457	1	=	=	PUNCT
ejpam-4343	458	1	[	[	X
ejpam-4343	458	2	v	v	X
ejpam-4343	458	3	(	(	PUNCT
ejpam-4343	458	4	λ	λ	PROPN
ejpam-4343	458	5	,	,	PUNCT
ejpam-4343	458	6	p(⋆))](λ	p(⋆))](λ	PROPN
ejpam-4343	458	7	,	,	PUNCT
ejpam-4343	458	8	p(⋆	p(⋆	PROPN
ejpam-4343	458	9	)	)	PUNCT
ejpam-4343	458	10	)	)	PUNCT
ejpam-4343	458	11	and	and	CCONJ
ejpam-4343	458	12	hence	hence	ADV
ejpam-4343	458	13	v	v	NOUN
ejpam-4343	458	14	(	(	PUNCT
ejpam-4343	458	15	λ	λ	PROPN
ejpam-4343	458	16	,	,	PUNCT
ejpam-4343	458	17	p(⋆	p(⋆	PROPN
ejpam-4343	458	18	)	)	PUNCT
ejpam-4343	458	19	)	)	PUNCT
ejpam-4343	458	20	is	be	AUX
ejpam-4343	458	21	(	(	PUNCT
ejpam-4343	458	22	λ	λ	INTJ
ejpam-4343	458	23	,	,	PUNCT
ejpam-4343	458	24	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	458	25	.	.	PUNCT
ejpam-4343	459	1	therefore	therefore	ADV
ejpam-4343	459	2	,	,	PUNCT
ejpam-4343	459	3	(	(	PUNCT
ejpam-4343	459	4	x	x	X
ejpam-4343	459	5	,	,	PUNCT
ejpam-4343	459	6	τ	τ	PROPN
ejpam-4343	459	7	,	,	PUNCT
ejpam-4343	459	8	i	i	PROPN
ejpam-4343	459	9	)	)	PUNCT
ejpam-4343	459	10	is	be	AUX
ejpam-4343	459	11	(	(	PUNCT
ejpam-4343	459	12	λ	λ	INTJ
ejpam-4343	459	13	,	,	PUNCT
ejpam-4343	459	14	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	459	15	disconnected	disconnected	ADJ
ejpam-4343	459	16	.	.	PUNCT
ejpam-4343	460	1	(	(	PUNCT
ejpam-4343	460	2	1	1	X
ejpam-4343	460	3	)	)	PUNCT
ejpam-4343	460	4	⇒	⇒	NOUN
ejpam-4343	460	5	(	(	PUNCT
ejpam-4343	460	6	3	3	NUM
ejpam-4343	460	7	):	):	PUNCT
ejpam-4343	460	8	let	let	VERB
ejpam-4343	460	9	v	v	PART
ejpam-4343	460	10	be	be	AUX
ejpam-4343	460	11	any	any	DET
ejpam-4343	460	12	pre-(λ	pre-(λ	PROPN
ejpam-4343	460	13	,	,	PUNCT
ejpam-4343	460	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	460	15	set	set	NOUN
ejpam-4343	460	16	.	.	PUNCT
ejpam-4343	461	1	then	then	ADV
ejpam-4343	461	2	,	,	PUNCT
ejpam-4343	461	3	v	v	NOUN
ejpam-4343	461	4	is	be	AUX
ejpam-4343	461	5	β-(λ	β-(λ	PRON
ejpam-4343	461	6	,	,	PUNCT
ejpam-4343	461	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	461	8	and	and	CCONJ
ejpam-4343	461	9	by	by	ADP
ejpam-4343	461	10	theorem	theorem	ADJ
ejpam-4343	461	11	4	4	NUM
ejpam-4343	461	12	,	,	PUNCT
ejpam-4343	461	13	v	v	NOUN
ejpam-4343	461	14	(	(	PUNCT
ejpam-4343	461	15	λ	λ	PROPN
ejpam-4343	461	16	,	,	PUNCT
ejpam-4343	461	17	p(⋆	p(⋆	PROPN
ejpam-4343	461	18	)	)	PUNCT
ejpam-4343	461	19	)	)	PUNCT
ejpam-4343	462	1	is	be	AUX
ejpam-4343	462	2	(	(	PUNCT
ejpam-4343	462	3	λ	λ	INTJ
ejpam-4343	462	4	,	,	PUNCT
ejpam-4343	462	5	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	462	6	.	.	PUNCT
ejpam-4343	463	1	(	(	PUNCT
ejpam-4343	463	2	3	3	X
ejpam-4343	463	3	)	)	PUNCT
ejpam-4343	463	4	⇒	⇒	NOUN
ejpam-4343	463	5	(	(	PUNCT
ejpam-4343	463	6	4	4	NUM
ejpam-4343	463	7	):	):	PUNCT
ejpam-4343	463	8	let	let	VERB
ejpam-4343	463	9	v	v	PART
ejpam-4343	463	10	be	be	AUX
ejpam-4343	463	11	any	any	PRON
ejpam-4343	463	12	regular	regular	ADJ
ejpam-4343	463	13	(	(	PUNCT
ejpam-4343	463	14	λ	λ	PROPN
ejpam-4343	463	15	,	,	PUNCT
ejpam-4343	463	16	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	463	17	set	set	NOUN
ejpam-4343	463	18	.	.	PUNCT
ejpam-4343	464	1	then	then	ADV
ejpam-4343	464	2	,	,	PUNCT
ejpam-4343	464	3	we	we	PRON
ejpam-4343	464	4	have	have	VERB
ejpam-4343	464	5	v	v	NOUN
ejpam-4343	464	6	is	be	AUX
ejpam-4343	464	7	pre-(λ	pre-(λ	ADV
ejpam-4343	464	8	,	,	PUNCT
ejpam-4343	464	9	p(⋆))open	p(⋆))open	PROPN
ejpam-4343	464	10	,	,	PUNCT
ejpam-4343	464	11	by	by	ADP
ejpam-4343	464	12	(	(	PUNCT
ejpam-4343	464	13	3	3	NUM
ejpam-4343	464	14	)	)	PUNCT
ejpam-4343	464	15	,	,	PUNCT
ejpam-4343	464	16	v	v	X
ejpam-4343	464	17	(	(	PUNCT
ejpam-4343	464	18	λ	λ	PROPN
ejpam-4343	464	19	,	,	PUNCT
ejpam-4343	464	20	p(⋆	p(⋆	PROPN
ejpam-4343	464	21	)	)	PUNCT
ejpam-4343	464	22	)	)	PUNCT
ejpam-4343	465	1	is	be	AUX
ejpam-4343	465	2	(	(	PUNCT
ejpam-4343	465	3	λ	λ	INTJ
ejpam-4343	465	4	,	,	PUNCT
ejpam-4343	465	5	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	465	6	.	.	PUNCT
ejpam-4343	466	1	5	5	X
ejpam-4343	466	2	.	.	PUNCT
ejpam-4343	466	3	characterizations	characterization	NOUN
ejpam-4343	466	4	of	of	ADP
ejpam-4343	466	5	(	(	PUNCT
ejpam-4343	466	6	λ	λ	PROPN
ejpam-4343	466	7	,	,	PUNCT
ejpam-4343	466	8	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	466	9	functions	function	NOUN
ejpam-4343	466	10	in	in	ADP
ejpam-4343	466	11	this	this	DET
ejpam-4343	466	12	section	section	NOUN
ejpam-4343	466	13	,	,	PUNCT
ejpam-4343	466	14	we	we	PRON
ejpam-4343	466	15	introduce	introduce	VERB
ejpam-4343	466	16	the	the	DET
ejpam-4343	466	17	notion	notion	NOUN
ejpam-4343	466	18	of	of	ADP
ejpam-4343	466	19	(	(	PUNCT
ejpam-4343	466	20	λ	λ	PROPN
ejpam-4343	466	21	,	,	PUNCT
ejpam-4343	466	22	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	466	23	functions	function	NOUN
ejpam-4343	466	24	.	.	PUNCT
ejpam-4343	467	1	in	in	ADP
ejpam-4343	467	2	particular	particular	ADJ
ejpam-4343	467	3	,	,	PUNCT
ejpam-4343	467	4	several	several	ADJ
ejpam-4343	467	5	characterizations	characterization	NOUN
ejpam-4343	467	6	of	of	ADP
ejpam-4343	467	7	(	(	PUNCT
ejpam-4343	467	8	λ	λ	PROPN
ejpam-4343	467	9	,	,	PUNCT
ejpam-4343	467	10	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	467	11	functions	function	NOUN
ejpam-4343	467	12	are	be	AUX
ejpam-4343	467	13	investigated	investigate	VERB
ejpam-4343	467	14	.	.	PUNCT
ejpam-4343	468	1	definition	definition	NOUN
ejpam-4343	468	2	14	14	NUM
ejpam-4343	468	3	.	.	PUNCT
ejpam-4343	469	1	a	a	DET
ejpam-4343	469	2	function	function	NOUN
ejpam-4343	469	3	f	f	NOUN
ejpam-4343	469	4	:	:	PUNCT
ejpam-4343	469	5	(	(	PUNCT
ejpam-4343	469	6	x	x	X
ejpam-4343	469	7	,	,	PUNCT
ejpam-4343	469	8	τ	τ	PROPN
ejpam-4343	469	9	,	,	PUNCT
ejpam-4343	469	10	i	i	NOUN
ejpam-4343	469	11	)	)	PUNCT
ejpam-4343	469	12	→	→	PUNCT
ejpam-4343	469	13	(	(	PUNCT
ejpam-4343	469	14	y	y	PROPN
ejpam-4343	469	15	,	,	PUNCT
ejpam-4343	469	16	σ	σ	PROPN
ejpam-4343	469	17	,	,	PUNCT
ejpam-4343	469	18	j	j	PROPN
ejpam-4343	469	19	)	)	PUNCT
ejpam-4343	469	20	is	be	AUX
ejpam-4343	469	21	called	call	VERB
ejpam-4343	469	22	(	(	PUNCT
ejpam-4343	469	23	λ	λ	X
ejpam-4343	469	24	,	,	PUNCT
ejpam-4343	469	25	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	469	26	at	at	ADP
ejpam-4343	469	27	a	a	DET
ejpam-4343	469	28	point	point	NOUN
ejpam-4343	469	29	x	x	SYM
ejpam-4343	469	30	∈	∈	NOUN
ejpam-4343	469	31	x	x	INTJ
ejpam-4343	469	32	if	if	SCONJ
ejpam-4343	469	33	,	,	PUNCT
ejpam-4343	469	34	for	for	SCONJ
ejpam-4343	469	35	each	each	DET
ejpam-4343	469	36	(	(	PUNCT
ejpam-4343	469	37	λ	λ	PROPN
ejpam-4343	469	38	,	,	PUNCT
ejpam-4343	469	39	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	469	40	set	set	VERB
ejpam-4343	469	41	v	v	NOUN
ejpam-4343	469	42	of	of	ADP
ejpam-4343	469	43	y	y	NOUN
ejpam-4343	469	44	containing	contain	VERB
ejpam-4343	469	45	f(x	f(x	PROPN
ejpam-4343	469	46	)	)	PUNCT
ejpam-4343	469	47	,	,	PUNCT
ejpam-4343	469	48	there	there	PRON
ejpam-4343	469	49	exists	exist	VERB
ejpam-4343	469	50	a	a	PRON
ejpam-4343	469	51	(	(	PUNCT
ejpam-4343	469	52	λ	λ	PROPN
ejpam-4343	469	53	,	,	PUNCT
ejpam-4343	469	54	p(⋆))open	p(⋆))open	PROPN
ejpam-4343	469	55	set	set	VERB
ejpam-4343	469	56	u	u	NOUN
ejpam-4343	469	57	of	of	ADP
ejpam-4343	469	58	x	x	PUNCT
ejpam-4343	469	59	containing	contain	VERB
ejpam-4343	469	60	x	x	PUNCT
ejpam-4343	469	61	such	such	ADJ
ejpam-4343	469	62	that	that	DET
ejpam-4343	469	63	f(u	f(u	PROPN
ejpam-4343	469	64	)	)	PUNCT
ejpam-4343	469	65	⊆	⊆	NUM
ejpam-4343	469	66	v	v	NOUN
ejpam-4343	469	67	.	.	PUNCT
ejpam-4343	470	1	a	a	DET
ejpam-4343	470	2	function	function	NOUN
ejpam-4343	470	3	f	f	NOUN
ejpam-4343	470	4	:	:	PUNCT
ejpam-4343	470	5	(	(	PUNCT
ejpam-4343	470	6	x	x	X
ejpam-4343	470	7	,	,	PUNCT
ejpam-4343	470	8	τ	τ	PROPN
ejpam-4343	470	9	,	,	PUNCT
ejpam-4343	470	10	i	i	NOUN
ejpam-4343	470	11	)	)	PUNCT
ejpam-4343	470	12	→	→	PUNCT
ejpam-4343	470	13	(	(	PUNCT
ejpam-4343	470	14	y	y	PROPN
ejpam-4343	470	15	,	,	PUNCT
ejpam-4343	470	16	σ	σ	PROPN
ejpam-4343	470	17	,	,	PUNCT
ejpam-4343	470	18	j	j	PROPN
ejpam-4343	470	19	)	)	PUNCT
ejpam-4343	470	20	is	be	AUX
ejpam-4343	470	21	called	call	VERB
ejpam-4343	470	22	(	(	PUNCT
ejpam-4343	470	23	λ	λ	X
ejpam-4343	470	24	,	,	PUNCT
ejpam-4343	470	25	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	470	26	if	if	SCONJ
ejpam-4343	470	27	f	f	PROPN
ejpam-4343	470	28	has	have	VERB
ejpam-4343	470	29	this	this	DET
ejpam-4343	470	30	property	property	NOUN
ejpam-4343	470	31	at	at	ADP
ejpam-4343	470	32	each	each	DET
ejpam-4343	470	33	point	point	NOUN
ejpam-4343	470	34	x	x	X
ejpam-4343	470	35	∈	∈	PROPN
ejpam-4343	470	36	x.	x.	NOUN
ejpam-4343	470	37	example	example	NOUN
ejpam-4343	470	38	4	4	X
ejpam-4343	470	39	.	.	PUNCT
ejpam-4343	471	1	let	let	VERB
ejpam-4343	471	2	x	x	PUNCT
ejpam-4343	471	3	=	=	PRON
ejpam-4343	471	4	{	{	PUNCT
ejpam-4343	471	5	a	a	PRON
ejpam-4343	471	6	,	,	PUNCT
ejpam-4343	471	7	b	b	NOUN
ejpam-4343	471	8	,	,	PUNCT
ejpam-4343	471	9	c	c	NOUN
ejpam-4343	471	10	}	}	PUNCT
ejpam-4343	471	11	with	with	ADP
ejpam-4343	471	12	a	a	DET
ejpam-4343	471	13	topology	topology	NOUN
ejpam-4343	471	14	τ	τ	X
ejpam-4343	471	15	=	=	SYM
ejpam-4343	471	16	{	{	PUNCT
ejpam-4343	471	17	∅	∅	NOUN
ejpam-4343	471	18	,	,	PUNCT
ejpam-4343	471	19	{	{	PUNCT
ejpam-4343	471	20	a	a	X
ejpam-4343	471	21	}	}	PUNCT
ejpam-4343	471	22	,	,	PUNCT
ejpam-4343	471	23	x	x	NOUN
ejpam-4343	471	24	}	}	PUNCT
ejpam-4343	471	25	and	and	CCONJ
ejpam-4343	471	26	an	an	DET
ejpam-4343	471	27	ideal	ideal	NOUN
ejpam-4343	471	28	i	i	X
ejpam-4343	471	29	=	=	SYM
ejpam-4343	471	30	{	{	PUNCT
ejpam-4343	471	31	∅	∅	NOUN
ejpam-4343	471	32	,	,	PUNCT
ejpam-4343	471	33	{	{	PUNCT
ejpam-4343	471	34	a	a	X
ejpam-4343	471	35	}	}	PUNCT
ejpam-4343	471	36	}	}	PUNCT
ejpam-4343	471	37	.	.	PUNCT
ejpam-4343	472	1	let	let	VERB
ejpam-4343	472	2	y	y	PROPN
ejpam-4343	472	3	=	=	PUNCT
ejpam-4343	472	4	{	{	PUNCT
ejpam-4343	472	5	1	1	NUM
ejpam-4343	472	6	,	,	PUNCT
ejpam-4343	472	7	2	2	NUM
ejpam-4343	472	8	,	,	PUNCT
ejpam-4343	472	9	3	3	NUM
ejpam-4343	472	10	}	}	PUNCT
ejpam-4343	472	11	with	with	ADP
ejpam-4343	472	12	a	a	DET
ejpam-4343	472	13	topology	topology	NOUN
ejpam-4343	472	14	σ	σ	NOUN
ejpam-4343	472	15	=	=	SYM
ejpam-4343	472	16	{	{	PUNCT
ejpam-4343	472	17	∅	∅	NOUN
ejpam-4343	472	18	,	,	PUNCT
ejpam-4343	472	19	{	{	PUNCT
ejpam-4343	472	20	1	1	NUM
ejpam-4343	472	21	}	}	PUNCT
ejpam-4343	472	22	,	,	PUNCT
ejpam-4343	472	23	{	{	PUNCT
ejpam-4343	472	24	2	2	NUM
ejpam-4343	472	25	}	}	PUNCT
ejpam-4343	472	26	,	,	PUNCT
ejpam-4343	472	27	{	{	PUNCT
ejpam-4343	472	28	1	1	NUM
ejpam-4343	472	29	,	,	PUNCT
ejpam-4343	472	30	2	2	NUM
ejpam-4343	472	31	}	}	PUNCT
ejpam-4343	472	32	,	,	PUNCT
ejpam-4343	472	33	y	y	PROPN
ejpam-4343	472	34	}	}	PUNCT
ejpam-4343	472	35	and	and	CCONJ
ejpam-4343	472	36	an	an	DET
ejpam-4343	472	37	ideal	ideal	NOUN
ejpam-4343	472	38	j	j	PROPN
ejpam-4343	472	39	=	=	PUNCT
ejpam-4343	472	40	{	{	PUNCT
ejpam-4343	472	41	∅	∅	NOUN
ejpam-4343	472	42	,	,	PUNCT
ejpam-4343	472	43	{	{	PUNCT
ejpam-4343	472	44	2	2	NUM
ejpam-4343	472	45	}	}	PUNCT
ejpam-4343	472	46	}	}	PUNCT
ejpam-4343	472	47	.	.	PUNCT
ejpam-4343	473	1	a	a	DET
ejpam-4343	473	2	function	function	NOUN
ejpam-4343	473	3	f	f	NOUN
ejpam-4343	473	4	:	:	PUNCT
ejpam-4343	473	5	(	(	PUNCT
ejpam-4343	473	6	x	x	X
ejpam-4343	473	7	,	,	PUNCT
ejpam-4343	473	8	τ	τ	PROPN
ejpam-4343	473	9	,	,	PUNCT
ejpam-4343	473	10	i	i	NOUN
ejpam-4343	473	11	)	)	PUNCT
ejpam-4343	473	12	→	→	PUNCT
ejpam-4343	473	13	(	(	PUNCT
ejpam-4343	473	14	y	y	PROPN
ejpam-4343	473	15	,	,	PUNCT
ejpam-4343	473	16	σ	σ	PROPN
ejpam-4343	473	17	,	,	PUNCT
ejpam-4343	473	18	j	j	PROPN
ejpam-4343	473	19	)	)	PUNCT
ejpam-4343	473	20	is	be	AUX
ejpam-4343	473	21	defined	define	VERB
ejpam-4343	473	22	as	as	SCONJ
ejpam-4343	473	23	follows	follow	VERB
ejpam-4343	473	24	:	:	PUNCT
ejpam-4343	473	25	f(a	f(a	NOUN
ejpam-4343	473	26	)	)	PUNCT
ejpam-4343	474	1	=	=	SYM
ejpam-4343	474	2	1	1	NUM
ejpam-4343	474	3	,	,	PUNCT
ejpam-4343	474	4	f(b	f(b	PROPN
ejpam-4343	474	5	)	)	PUNCT
ejpam-4343	474	6	=	=	SYM
ejpam-4343	474	7	2	2	NUM
ejpam-4343	474	8	and	and	CCONJ
ejpam-4343	474	9	f(c	f(c	PROPN
ejpam-4343	474	10	)	)	PUNCT
ejpam-4343	475	1	=	=	SYM
ejpam-4343	476	1	3	3	X
ejpam-4343	476	2	.	.	PUNCT
ejpam-4343	476	3	then	then	ADV
ejpam-4343	476	4	,	,	PUNCT
ejpam-4343	476	5	f	f	PROPN
ejpam-4343	476	6	is	be	AUX
ejpam-4343	476	7	(	(	PUNCT
ejpam-4343	476	8	λ	λ	INTJ
ejpam-4343	476	9	,	,	PUNCT
ejpam-4343	476	10	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	476	11	.	.	PUNCT
ejpam-4343	477	1	lemma	lemma	PROPN
ejpam-4343	477	2	5	5	X
ejpam-4343	477	3	.	.	PUNCT
ejpam-4343	477	4	let	let	VERB
ejpam-4343	477	5	a	a	DET
ejpam-4343	477	6	be	be	AUX
ejpam-4343	477	7	a	a	DET
ejpam-4343	477	8	subset	subset	NOUN
ejpam-4343	477	9	of	of	ADP
ejpam-4343	477	10	an	an	DET
ejpam-4343	477	11	ideal	ideal	ADJ
ejpam-4343	477	12	topological	topological	ADJ
ejpam-4343	477	13	space	space	NOUN
ejpam-4343	477	14	(	(	PUNCT
ejpam-4343	477	15	x	x	X
ejpam-4343	477	16	,	,	PUNCT
ejpam-4343	477	17	τ	τ	PROPN
ejpam-4343	477	18	,	,	PUNCT
ejpam-4343	477	19	i	i	NOUN
ejpam-4343	477	20	)	)	PUNCT
ejpam-4343	477	21	.	.	PUNCT
ejpam-4343	478	1	then	then	ADV
ejpam-4343	478	2	,	,	PUNCT
ejpam-4343	478	3	x	x	PUNCT
ejpam-4343	478	4	∈	∈	PROPN
ejpam-4343	478	5	a(λ	a(λ	ADV
ejpam-4343	478	6	,	,	PUNCT
ejpam-4343	478	7	p(⋆	p(⋆	PROPN
ejpam-4343	478	8	)	)	PUNCT
ejpam-4343	478	9	)	)	PUNCT
ejpam-4343	479	1	if	if	SCONJ
ejpam-4343	479	2	and	and	CCONJ
ejpam-4343	479	3	only	only	ADV
ejpam-4343	479	4	if	if	SCONJ
ejpam-4343	479	5	a	a	DET
ejpam-4343	479	6	∩	∩	ADJ
ejpam-4343	479	7	u	u	ADJ
ejpam-4343	479	8	̸=	̸=	PROPN
ejpam-4343	479	9	∅	∅	NOUN
ejpam-4343	479	10	for	for	ADP
ejpam-4343	479	11	every	every	DET
ejpam-4343	479	12	(	(	PUNCT
ejpam-4343	479	13	λ	λ	PROPN
ejpam-4343	479	14	,	,	PUNCT
ejpam-4343	479	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	479	16	set	set	VERB
ejpam-4343	479	17	u	u	NOUN
ejpam-4343	479	18	of	of	ADP
ejpam-4343	479	19	x	x	SYM
ejpam-4343	479	20	containing	contain	VERB
ejpam-4343	479	21	x.	x.	NOUN
ejpam-4343	479	22	theorem	theorem	VERB
ejpam-4343	479	23	8	8	NUM
ejpam-4343	479	24	.	.	PUNCT
ejpam-4343	480	1	for	for	ADP
ejpam-4343	480	2	a	a	DET
ejpam-4343	480	3	function	function	NOUN
ejpam-4343	480	4	f	f	NOUN
ejpam-4343	480	5	:	:	PUNCT
ejpam-4343	480	6	(	(	PUNCT
ejpam-4343	480	7	x	x	X
ejpam-4343	480	8	,	,	PUNCT
ejpam-4343	480	9	τ	τ	PROPN
ejpam-4343	480	10	,	,	PUNCT
ejpam-4343	480	11	i	i	NOUN
ejpam-4343	480	12	)	)	PUNCT
ejpam-4343	480	13	→	→	PUNCT
ejpam-4343	480	14	(	(	PUNCT
ejpam-4343	480	15	y	y	PROPN
ejpam-4343	480	16	,	,	PUNCT
ejpam-4343	480	17	σ	σ	PROPN
ejpam-4343	480	18	,	,	PUNCT
ejpam-4343	480	19	j	j	PROPN
ejpam-4343	480	20	)	)	PUNCT
ejpam-4343	480	21	,	,	PUNCT
ejpam-4343	480	22	the	the	DET
ejpam-4343	480	23	following	follow	VERB
ejpam-4343	480	24	properties	property	NOUN
ejpam-4343	480	25	are	be	AUX
ejpam-4343	480	26	equivalent	equivalent	ADJ
ejpam-4343	480	27	:	:	PUNCT
ejpam-4343	480	28	c.	c.	PROPN
ejpam-4343	480	29	boonpok	boonpok	PROPN
ejpam-4343	480	30	/	/	SYM
ejpam-4343	480	31	eur	eur	PROPN
ejpam-4343	480	32	.	.	PUNCT
ejpam-4343	481	1	j.	j.	PROPN
ejpam-4343	481	2	pure	pure	PROPN
ejpam-4343	481	3	appl	appl	PROPN
ejpam-4343	481	4	.	.	PROPN
ejpam-4343	481	5	math	math	PROPN
ejpam-4343	481	6	,	,	PUNCT
ejpam-4343	481	7	15	15	NUM
ejpam-4343	481	8	(	(	PUNCT
ejpam-4343	481	9	3	3	NUM
ejpam-4343	481	10	)	)	PUNCT
ejpam-4343	481	11	(	(	PUNCT
ejpam-4343	481	12	2022	2022	NUM
ejpam-4343	481	13	)	)	PUNCT
ejpam-4343	481	14	,	,	PUNCT
ejpam-4343	481	15	1023	1023	NUM
ejpam-4343	481	16	-	-	SYM
ejpam-4343	481	17	1046	1046	NUM
ejpam-4343	481	18	1035	1035	NUM
ejpam-4343	481	19	(	(	PUNCT
ejpam-4343	481	20	1	1	NUM
ejpam-4343	481	21	)	)	PUNCT
ejpam-4343	481	22	f	f	PROPN
ejpam-4343	481	23	is	be	AUX
ejpam-4343	481	24	(	(	PUNCT
ejpam-4343	481	25	λ	λ	X
ejpam-4343	481	26	,	,	PUNCT
ejpam-4343	481	27	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	481	28	at	at	ADP
ejpam-4343	481	29	x	x	PROPN
ejpam-4343	481	30	∈	∈	PROPN
ejpam-4343	481	31	x.	x.	NOUN
ejpam-4343	481	32	(	(	PUNCT
ejpam-4343	481	33	2	2	NUM
ejpam-4343	481	34	)	)	PUNCT
ejpam-4343	481	35	x	x	SYM
ejpam-4343	481	36	∈	∈	PROPN
ejpam-4343	482	1	[	[	X
ejpam-4343	482	2	f−1(v	f−1(v	NOUN
ejpam-4343	482	3	)	)	PUNCT
ejpam-4343	482	4	]	]	PUNCT
ejpam-4343	482	5	(	(	PUNCT
ejpam-4343	482	6	λ	λ	X
ejpam-4343	482	7	,	,	PUNCT
ejpam-4343	482	8	p(⋆	p(⋆	PROPN
ejpam-4343	482	9	)	)	PUNCT
ejpam-4343	482	10	)	)	PUNCT
ejpam-4343	482	11	for	for	SCONJ
ejpam-4343	482	12	every	every	DET
ejpam-4343	482	13	(	(	PUNCT
ejpam-4343	482	14	λ	λ	PROPN
ejpam-4343	482	15	,	,	PUNCT
ejpam-4343	482	16	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	482	17	set	set	VERB
ejpam-4343	482	18	v	v	NOUN
ejpam-4343	482	19	of	of	ADP
ejpam-4343	482	20	y	y	NOUN
ejpam-4343	482	21	containing	contain	VERB
ejpam-4343	482	22	f(x	f(x	PROPN
ejpam-4343	482	23	)	)	PUNCT
ejpam-4343	482	24	.	.	PUNCT
ejpam-4343	483	1	(	(	PUNCT
ejpam-4343	483	2	3	3	X
ejpam-4343	483	3	)	)	PUNCT
ejpam-4343	483	4	x	x	SYM
ejpam-4343	483	5	∈	∈	PROPN
ejpam-4343	483	6	f−1([f(a)](λ	f−1([f(a)](λ	X
ejpam-4343	483	7	,	,	PUNCT
ejpam-4343	483	8	p(⋆	p(⋆	PROPN
ejpam-4343	483	9	)	)	PUNCT
ejpam-4343	483	10	)	)	PUNCT
ejpam-4343	483	11	)	)	PUNCT
ejpam-4343	484	1	for	for	ADP
ejpam-4343	484	2	every	every	DET
ejpam-4343	484	3	subset	subset	NOUN
ejpam-4343	484	4	a	a	PRON
ejpam-4343	484	5	of	of	ADP
ejpam-4343	484	6	x	x	SYM
ejpam-4343	484	7	such	such	ADJ
ejpam-4343	484	8	that	that	SCONJ
ejpam-4343	484	9	x	x	SYM
ejpam-4343	484	10	∈	∈	PROPN
ejpam-4343	484	11	a(λ	a(λ	ADV
ejpam-4343	484	12	,	,	PUNCT
ejpam-4343	484	13	p(⋆	p(⋆	PROPN
ejpam-4343	484	14	)	)	PUNCT
ejpam-4343	484	15	)	)	PUNCT
ejpam-4343	484	16	.	.	PUNCT
ejpam-4343	485	1	(	(	PUNCT
ejpam-4343	485	2	4	4	X
ejpam-4343	485	3	)	)	PUNCT
ejpam-4343	485	4	x	x	SYM
ejpam-4343	485	5	∈	∈	PROPN
ejpam-4343	485	6	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	485	7	,	,	PUNCT
ejpam-4343	485	8	p(⋆	p(⋆	PROPN
ejpam-4343	485	9	)	)	PUNCT
ejpam-4343	485	10	)	)	PUNCT
ejpam-4343	485	11	)	)	PUNCT
ejpam-4343	486	1	for	for	ADP
ejpam-4343	486	2	every	every	DET
ejpam-4343	486	3	subset	subset	NOUN
ejpam-4343	486	4	b	b	PROPN
ejpam-4343	486	5	of	of	ADP
ejpam-4343	486	6	y	y	PRON
ejpam-4343	486	7	such	such	ADJ
ejpam-4343	486	8	that	that	SCONJ
ejpam-4343	486	9	x	x	SYM
ejpam-4343	486	10	∈	∈	PROPN
ejpam-4343	486	11	[	[	X
ejpam-4343	486	12	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	486	13	,	,	PUNCT
ejpam-4343	486	14	p(⋆	p(⋆	PROPN
ejpam-4343	486	15	)	)	PUNCT
ejpam-4343	486	16	)	)	PUNCT
ejpam-4343	486	17	.	.	PUNCT
ejpam-4343	487	1	(	(	PUNCT
ejpam-4343	487	2	5	5	X
ejpam-4343	487	3	)	)	PUNCT
ejpam-4343	487	4	x	x	SYM
ejpam-4343	488	1	∈	∈	PROPN
ejpam-4343	488	2	[	[	X
ejpam-4343	488	3	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	488	4	,	,	PUNCT
ejpam-4343	488	5	p(⋆	p(⋆	PROPN
ejpam-4343	488	6	)	)	PUNCT
ejpam-4343	488	7	)	)	PUNCT
ejpam-4343	488	8	for	for	ADP
ejpam-4343	488	9	every	every	DET
ejpam-4343	488	10	subset	subset	NOUN
ejpam-4343	488	11	b	b	PROPN
ejpam-4343	488	12	of	of	ADP
ejpam-4343	488	13	y	y	PRON
ejpam-4343	488	14	such	such	ADJ
ejpam-4343	488	15	that	that	SCONJ
ejpam-4343	488	16	x	x	SYM
ejpam-4343	488	17	∈	∈	PROPN
ejpam-4343	488	18	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	488	19	,	,	PUNCT
ejpam-4343	488	20	p(⋆	p(⋆	PROPN
ejpam-4343	488	21	)	)	PUNCT
ejpam-4343	488	22	)	)	PUNCT
ejpam-4343	488	23	)	)	PUNCT
ejpam-4343	488	24	.	.	PUNCT
ejpam-4343	489	1	(	(	PUNCT
ejpam-4343	489	2	6	6	NUM
ejpam-4343	489	3	)	)	PUNCT
ejpam-4343	489	4	x	x	SYM
ejpam-4343	489	5	∈	∈	PROPN
ejpam-4343	489	6	f−1(k	f−1(k	PROPN
ejpam-4343	489	7	)	)	PUNCT
ejpam-4343	489	8	for	for	ADP
ejpam-4343	489	9	every	every	DET
ejpam-4343	489	10	(	(	PUNCT
ejpam-4343	489	11	λ	λ	PROPN
ejpam-4343	489	12	,	,	PUNCT
ejpam-4343	489	13	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	489	14	set	set	VERB
ejpam-4343	489	15	k	k	PROPN
ejpam-4343	489	16	of	of	ADP
ejpam-4343	489	17	y	y	PROPN
ejpam-4343	489	18	such	such	ADJ
ejpam-4343	489	19	that	that	SCONJ
ejpam-4343	489	20	x	x	SYM
ejpam-4343	489	21	∈	∈	PROPN
ejpam-4343	489	22	[	[	X
ejpam-4343	489	23	f−1(k)](λ	f−1(k)](λ	PROPN
ejpam-4343	489	24	,	,	PUNCT
ejpam-4343	489	25	p(⋆	p(⋆	PROPN
ejpam-4343	489	26	)	)	PUNCT
ejpam-4343	489	27	)	)	PUNCT
ejpam-4343	489	28	.	.	PUNCT
ejpam-4343	490	1	proof	proof	NOUN
ejpam-4343	490	2	.	.	PUNCT
ejpam-4343	491	1	(	(	PUNCT
ejpam-4343	491	2	1	1	X
ejpam-4343	491	3	)	)	PUNCT
ejpam-4343	491	4	⇒	⇒	NOUN
ejpam-4343	491	5	(	(	PUNCT
ejpam-4343	491	6	2	2	NUM
ejpam-4343	491	7	):	):	PUNCT
ejpam-4343	491	8	let	let	VERB
ejpam-4343	491	9	v	v	PART
ejpam-4343	491	10	be	be	AUX
ejpam-4343	491	11	any	any	DET
ejpam-4343	491	12	(	(	PUNCT
ejpam-4343	491	13	λ	λ	NOUN
ejpam-4343	491	14	,	,	PUNCT
ejpam-4343	491	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	491	16	set	set	NOUN
ejpam-4343	491	17	of	of	ADP
ejpam-4343	491	18	y	y	PROPN
ejpam-4343	491	19	containing	contain	VERB
ejpam-4343	491	20	f(x	f(x	PROPN
ejpam-4343	491	21	)	)	PUNCT
ejpam-4343	491	22	.	.	PUNCT
ejpam-4343	492	1	then	then	ADV
ejpam-4343	492	2	,	,	PUNCT
ejpam-4343	492	3	there	there	PRON
ejpam-4343	492	4	exists	exist	VERB
ejpam-4343	492	5	a	a	DET
ejpam-4343	492	6	(	(	PUNCT
ejpam-4343	492	7	λ	λ	NOUN
ejpam-4343	492	8	,	,	PUNCT
ejpam-4343	492	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	492	10	set	set	VERB
ejpam-4343	492	11	u	u	NOUN
ejpam-4343	492	12	of	of	ADP
ejpam-4343	492	13	x	x	PUNCT
ejpam-4343	492	14	containing	contain	VERB
ejpam-4343	492	15	x	x	PUNCT
ejpam-4343	492	16	such	such	ADJ
ejpam-4343	492	17	that	that	DET
ejpam-4343	492	18	f(u	f(u	PROPN
ejpam-4343	492	19	)	)	PUNCT
ejpam-4343	492	20	⊆	⊆	NUM
ejpam-4343	492	21	v	v	NOUN
ejpam-4343	492	22	.	.	PUNCT
ejpam-4343	493	1	thus	thus	ADV
ejpam-4343	493	2	,	,	PUNCT
ejpam-4343	493	3	u	u	PROPN
ejpam-4343	493	4	⊆	⊆	NUM
ejpam-4343	493	5	f−1(v	f−1(v	NOUN
ejpam-4343	493	6	)	)	PUNCT
ejpam-4343	493	7	.	.	PUNCT
ejpam-4343	494	1	since	since	SCONJ
ejpam-4343	494	2	u	u	NOUN
ejpam-4343	494	3	is	be	AUX
ejpam-4343	494	4	(	(	PUNCT
ejpam-4343	494	5	λ	λ	X
ejpam-4343	494	6	,	,	PUNCT
ejpam-4343	494	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	494	8	in	in	ADP
ejpam-4343	494	9	x	x	PRON
ejpam-4343	494	10	,	,	PUNCT
ejpam-4343	494	11	we	we	PRON
ejpam-4343	494	12	have	have	VERB
ejpam-4343	494	13	x	x	X
ejpam-4343	494	14	∈	∈	PROPN
ejpam-4343	494	15	[	[	X
ejpam-4343	494	16	f−1(v	f−1(v	NOUN
ejpam-4343	494	17	)	)	PUNCT
ejpam-4343	494	18	]	]	PUNCT
ejpam-4343	494	19	(	(	PUNCT
ejpam-4343	494	20	λ	λ	X
ejpam-4343	494	21	,	,	PUNCT
ejpam-4343	494	22	p(⋆	p(⋆	PROPN
ejpam-4343	494	23	)	)	PUNCT
ejpam-4343	494	24	)	)	PUNCT
ejpam-4343	494	25	.	.	PUNCT
ejpam-4343	495	1	(	(	PUNCT
ejpam-4343	495	2	2	2	X
ejpam-4343	495	3	)	)	PUNCT
ejpam-4343	495	4	⇒	⇒	NOUN
ejpam-4343	495	5	(	(	PUNCT
ejpam-4343	495	6	3	3	NUM
ejpam-4343	495	7	):	):	PUNCT
ejpam-4343	495	8	let	let	VERB
ejpam-4343	495	9	a	a	PRON
ejpam-4343	495	10	be	be	AUX
ejpam-4343	495	11	any	any	DET
ejpam-4343	495	12	subset	subset	NOUN
ejpam-4343	495	13	of	of	ADP
ejpam-4343	495	14	x	x	PUNCT
ejpam-4343	495	15	and	and	CCONJ
ejpam-4343	495	16	x	x	PROPN
ejpam-4343	495	17	∈	∈	PROPN
ejpam-4343	495	18	a(λ	a(λ	ADV
ejpam-4343	495	19	,	,	PUNCT
ejpam-4343	495	20	p(⋆	p(⋆	PROPN
ejpam-4343	495	21	)	)	PUNCT
ejpam-4343	495	22	)	)	PUNCT
ejpam-4343	495	23	.	.	PUNCT
ejpam-4343	496	1	let	let	VERB
ejpam-4343	496	2	v	v	PART
ejpam-4343	496	3	be	be	AUX
ejpam-4343	496	4	any	any	DET
ejpam-4343	496	5	(	(	PUNCT
ejpam-4343	496	6	λ	λ	NOUN
ejpam-4343	496	7	,	,	PUNCT
ejpam-4343	496	8	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	496	9	set	set	NOUN
ejpam-4343	496	10	of	of	ADP
ejpam-4343	496	11	y	y	PROPN
ejpam-4343	496	12	containing	contain	VERB
ejpam-4343	496	13	f(x	f(x	PROPN
ejpam-4343	496	14	)	)	PUNCT
ejpam-4343	496	15	.	.	PUNCT
ejpam-4343	497	1	by	by	ADP
ejpam-4343	497	2	(	(	PUNCT
ejpam-4343	497	3	2	2	NUM
ejpam-4343	497	4	)	)	PUNCT
ejpam-4343	497	5	,	,	PUNCT
ejpam-4343	497	6	x	x	PUNCT
ejpam-4343	497	7	∈	∈	PROPN
ejpam-4343	497	8	[	[	X
ejpam-4343	497	9	f−1(v	f−1(v	NOUN
ejpam-4343	497	10	)	)	PUNCT
ejpam-4343	497	11	]	]	PUNCT
ejpam-4343	497	12	(	(	PUNCT
ejpam-4343	497	13	λ	λ	X
ejpam-4343	497	14	,	,	PUNCT
ejpam-4343	497	15	p(⋆	p(⋆	PROPN
ejpam-4343	497	16	)	)	PUNCT
ejpam-4343	497	17	)	)	PUNCT
ejpam-4343	497	18	and	and	CCONJ
ejpam-4343	497	19	there	there	PRON
ejpam-4343	497	20	exists	exist	VERB
ejpam-4343	497	21	a	a	DET
ejpam-4343	497	22	(	(	PUNCT
ejpam-4343	497	23	λ	λ	NOUN
ejpam-4343	497	24	,	,	PUNCT
ejpam-4343	497	25	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	497	26	set	set	VERB
ejpam-4343	497	27	u	u	NOUN
ejpam-4343	497	28	of	of	ADP
ejpam-4343	497	29	x	x	SYM
ejpam-4343	497	30	such	such	ADJ
ejpam-4343	497	31	that	that	SCONJ
ejpam-4343	497	32	x	x	SYM
ejpam-4343	497	33	∈	∈	PROPN
ejpam-4343	497	34	u	u	NOUN
ejpam-4343	497	35	⊆	⊆	NUM
ejpam-4343	497	36	f−1(v	f−1(v	NOUN
ejpam-4343	497	37	)	)	PUNCT
ejpam-4343	497	38	.	.	PUNCT
ejpam-4343	498	1	since	since	SCONJ
ejpam-4343	498	2	x	x	PROPN
ejpam-4343	498	3	∈	∈	PROPN
ejpam-4343	498	4	a(λ	a(λ	ADV
ejpam-4343	498	5	,	,	PUNCT
ejpam-4343	498	6	p(⋆	p(⋆	PROPN
ejpam-4343	498	7	)	)	PUNCT
ejpam-4343	498	8	)	)	PUNCT
ejpam-4343	498	9	,	,	PUNCT
ejpam-4343	498	10	by	by	ADP
ejpam-4343	498	11	lemma	lemma	PROPN
ejpam-4343	498	12	5	5	NUM
ejpam-4343	498	13	,	,	PUNCT
ejpam-4343	498	14	u	u	NOUN
ejpam-4343	498	15	∩a	∩a	PROPN
ejpam-4343	498	16	̸=	̸=	PROPN
ejpam-4343	498	17	∅	∅	NOUN
ejpam-4343	498	18	and	and	CCONJ
ejpam-4343	498	19	∅	∅	NOUN
ejpam-4343	498	20	=	=	NOUN
ejpam-4343	498	21	̸	̸	ADV
ejpam-4343	498	22	f(u	f(u	PROPN
ejpam-4343	498	23	∩a	∩a	PROPN
ejpam-4343	498	24	)	)	PUNCT
ejpam-4343	498	25	⊆	⊆	NUM
ejpam-4343	498	26	f(u	f(u	PROPN
ejpam-4343	498	27	)	)	PUNCT
ejpam-4343	498	28	∩	∩	ADJ
ejpam-4343	498	29	f(a	f(a	NOUN
ejpam-4343	498	30	)	)	PUNCT
ejpam-4343	498	31	⊆	⊆	NUM
ejpam-4343	498	32	v	v	ADP
ejpam-4343	498	33	∩	∩	ADJ
ejpam-4343	498	34	f(a	f(a	NOUN
ejpam-4343	498	35	)	)	PUNCT
ejpam-4343	498	36	.	.	PUNCT
ejpam-4343	499	1	thus	thus	ADV
ejpam-4343	499	2	,	,	PUNCT
ejpam-4343	499	3	f(x	f(x	PROPN
ejpam-4343	499	4	)	)	PUNCT
ejpam-4343	499	5	∈	∈	PROPN
ejpam-4343	500	1	[	[	X
ejpam-4343	500	2	f(a)](λ	f(a)](λ	NUM
ejpam-4343	500	3	,	,	PUNCT
ejpam-4343	500	4	p(⋆	p(⋆	PROPN
ejpam-4343	500	5	)	)	PUNCT
ejpam-4343	500	6	)	)	PUNCT
ejpam-4343	501	1	and	and	CCONJ
ejpam-4343	501	2	hence	hence	ADV
ejpam-4343	501	3	x	x	X
ejpam-4343	501	4	∈	∈	PROPN
ejpam-4343	501	5	f−1([f(a)](λ	f−1([f(a)](λ	X
ejpam-4343	501	6	,	,	PUNCT
ejpam-4343	501	7	p(⋆	p(⋆	PROPN
ejpam-4343	501	8	)	)	PUNCT
ejpam-4343	501	9	)	)	PUNCT
ejpam-4343	501	10	)	)	PUNCT
ejpam-4343	501	11	.	.	PUNCT
ejpam-4343	502	1	(	(	PUNCT
ejpam-4343	502	2	3	3	X
ejpam-4343	502	3	)	)	PUNCT
ejpam-4343	502	4	⇒	⇒	NOUN
ejpam-4343	502	5	(	(	PUNCT
ejpam-4343	502	6	4	4	NUM
ejpam-4343	502	7	):	):	PUNCT
ejpam-4343	502	8	let	let	VERB
ejpam-4343	502	9	b	b	X
ejpam-4343	502	10	be	be	AUX
ejpam-4343	502	11	any	any	DET
ejpam-4343	502	12	subset	subset	NOUN
ejpam-4343	502	13	of	of	ADP
ejpam-4343	502	14	y	y	PROPN
ejpam-4343	502	15	and	and	CCONJ
ejpam-4343	502	16	x	x	PROPN
ejpam-4343	502	17	∈	∈	PROPN
ejpam-4343	503	1	[	[	X
ejpam-4343	503	2	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	503	3	,	,	PUNCT
ejpam-4343	503	4	p(⋆	p(⋆	PROPN
ejpam-4343	503	5	)	)	PUNCT
ejpam-4343	503	6	)	)	PUNCT
ejpam-4343	503	7	.	.	PUNCT
ejpam-4343	504	1	by	by	ADP
ejpam-4343	504	2	(	(	PUNCT
ejpam-4343	504	3	3	3	NUM
ejpam-4343	504	4	)	)	PUNCT
ejpam-4343	504	5	,	,	PUNCT
ejpam-4343	504	6	x	x	PROPN
ejpam-4343	504	7	∈	∈	PROPN
ejpam-4343	504	8	f−1([f(f−1(b))](λ	f−1([f(f−1(b))](λ	PROPN
ejpam-4343	504	9	,	,	PUNCT
ejpam-4343	504	10	p(⋆	p(⋆	PROPN
ejpam-4343	504	11	)	)	PUNCT
ejpam-4343	504	12	)	)	PUNCT
ejpam-4343	504	13	)	)	PUNCT
ejpam-4343	505	1	⊆	⊆	NUM
ejpam-4343	505	2	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	505	3	,	,	PUNCT
ejpam-4343	505	4	p(⋆	p(⋆	PROPN
ejpam-4343	505	5	)	)	PUNCT
ejpam-4343	505	6	)	)	PUNCT
ejpam-4343	505	7	)	)	PUNCT
ejpam-4343	505	8	and	and	CCONJ
ejpam-4343	505	9	hence	hence	ADV
ejpam-4343	505	10	x	x	PART
ejpam-4343	505	11	∈	∈	PROPN
ejpam-4343	505	12	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	505	13	,	,	PUNCT
ejpam-4343	505	14	p(⋆	p(⋆	PROPN
ejpam-4343	505	15	)	)	PUNCT
ejpam-4343	505	16	)	)	PUNCT
ejpam-4343	505	17	)	)	PUNCT
ejpam-4343	505	18	.	.	PUNCT
ejpam-4343	506	1	(	(	PUNCT
ejpam-4343	506	2	4	4	X
ejpam-4343	506	3	)	)	PUNCT
ejpam-4343	506	4	⇒	⇒	NOUN
ejpam-4343	506	5	(	(	PUNCT
ejpam-4343	506	6	5	5	NUM
ejpam-4343	506	7	):	):	PUNCT
ejpam-4343	506	8	let	let	VERB
ejpam-4343	506	9	b	b	X
ejpam-4343	506	10	be	be	AUX
ejpam-4343	506	11	any	any	DET
ejpam-4343	506	12	subset	subset	NOUN
ejpam-4343	506	13	of	of	ADP
ejpam-4343	506	14	y	y	PRON
ejpam-4343	506	15	such	such	ADJ
ejpam-4343	506	16	that	that	SCONJ
ejpam-4343	506	17	x	x	SYM
ejpam-4343	506	18	̸∈	̸∈	PROPN
ejpam-4343	506	19	[	[	X
ejpam-4343	506	20	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	506	21	,	,	PUNCT
ejpam-4343	506	22	p(⋆	p(⋆	PROPN
ejpam-4343	506	23	)	)	PUNCT
ejpam-4343	506	24	)	)	PUNCT
ejpam-4343	506	25	.	.	PUNCT
ejpam-4343	507	1	then	then	ADV
ejpam-4343	507	2	,	,	PUNCT
ejpam-4343	507	3	x	x	PUNCT
ejpam-4343	507	4	∈	∈	NOUN
ejpam-4343	507	5	x	x	X
ejpam-4343	507	6	−	−	PROPN
ejpam-4343	508	1	[	[	X
ejpam-4343	508	2	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	508	3	,	,	PUNCT
ejpam-4343	508	4	p(⋆	p(⋆	PROPN
ejpam-4343	508	5	)	)	PUNCT
ejpam-4343	508	6	)	)	PUNCT
ejpam-4343	509	1	=	=	PUNCT
ejpam-4343	510	1	[	[	X
ejpam-4343	510	2	x	x	X
ejpam-4343	510	3	−	−	NOUN
ejpam-4343	510	4	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	510	5	,	,	PUNCT
ejpam-4343	510	6	p(⋆	p(⋆	PROPN
ejpam-4343	510	7	)	)	PUNCT
ejpam-4343	510	8	)	)	PUNCT
ejpam-4343	511	1	=	=	PUNCT
ejpam-4343	512	1	[	[	X
ejpam-4343	512	2	f−1(y	f−1(y	PROPN
ejpam-4343	512	3	−b)](λ	−b)](λ	PROPN
ejpam-4343	512	4	,	,	PUNCT
ejpam-4343	512	5	p(⋆	p(⋆	PROPN
ejpam-4343	512	6	)	)	PUNCT
ejpam-4343	512	7	)	)	PUNCT
ejpam-4343	512	8	and	and	CCONJ
ejpam-4343	512	9	by	by	ADP
ejpam-4343	512	10	(	(	PUNCT
ejpam-4343	512	11	4	4	NUM
ejpam-4343	512	12	)	)	PUNCT
ejpam-4343	512	13	,	,	PUNCT
ejpam-4343	512	14	x	x	PUNCT
ejpam-4343	512	15	∈	∈	NOUN
ejpam-4343	512	16	f−1([y	f−1([y	NOUN
ejpam-4343	512	17	−	−	NOUN
ejpam-4343	512	18	b](λ	b](λ	NOUN
ejpam-4343	512	19	,	,	PUNCT
ejpam-4343	512	20	p(⋆	p(⋆	PROPN
ejpam-4343	512	21	)	)	PUNCT
ejpam-4343	512	22	)	)	PUNCT
ejpam-4343	512	23	)	)	PUNCT
ejpam-4343	513	1	=	=	SYM
ejpam-4343	513	2	f−1(y	f−1(y	PROPN
ejpam-4343	513	3	−	−	PROPN
ejpam-4343	514	1	b(λ	b(λ	PROPN
ejpam-4343	514	2	,	,	PUNCT
ejpam-4343	514	3	p(⋆	p(⋆	PROPN
ejpam-4343	514	4	)	)	PUNCT
ejpam-4343	514	5	)	)	PUNCT
ejpam-4343	514	6	)	)	PUNCT
ejpam-4343	515	1	=	=	PUNCT
ejpam-4343	515	2	x	x	SYM
ejpam-4343	515	3	−	−	PROPN
ejpam-4343	515	4	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	515	5	,	,	PUNCT
ejpam-4343	515	6	p(⋆	p(⋆	PROPN
ejpam-4343	515	7	)	)	PUNCT
ejpam-4343	515	8	)	)	PUNCT
ejpam-4343	515	9	)	)	PUNCT
ejpam-4343	515	10	.	.	PUNCT
ejpam-4343	516	1	thus	thus	ADV
ejpam-4343	516	2	,	,	PUNCT
ejpam-4343	516	3	x	x	PROPN
ejpam-4343	516	4	̸∈	̸∈	PROPN
ejpam-4343	516	5	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	516	6	,	,	PUNCT
ejpam-4343	516	7	p(⋆	p(⋆	PROPN
ejpam-4343	516	8	)	)	PUNCT
ejpam-4343	516	9	)	)	PUNCT
ejpam-4343	516	10	)	)	PUNCT
ejpam-4343	516	11	.	.	PUNCT
ejpam-4343	517	1	(	(	PUNCT
ejpam-4343	517	2	5	5	X
ejpam-4343	517	3	)	)	PUNCT
ejpam-4343	517	4	⇒	⇒	NOUN
ejpam-4343	517	5	(	(	PUNCT
ejpam-4343	517	6	6	6	NUM
ejpam-4343	517	7	):	):	PUNCT
ejpam-4343	517	8	let	let	VERB
ejpam-4343	517	9	k	k	PRON
ejpam-4343	517	10	be	be	AUX
ejpam-4343	517	11	any	any	DET
ejpam-4343	517	12	(	(	PUNCT
ejpam-4343	517	13	λ	λ	PROPN
ejpam-4343	517	14	,	,	PUNCT
ejpam-4343	517	15	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	517	16	set	set	NOUN
ejpam-4343	517	17	of	of	ADP
ejpam-4343	517	18	y	y	PRON
ejpam-4343	517	19	such	such	ADJ
ejpam-4343	517	20	that	that	SCONJ
ejpam-4343	517	21	x	x	PROPN
ejpam-4343	517	22	̸∈	̸∈	PROPN
ejpam-4343	517	23	f−1(k	f−1(k	PROPN
ejpam-4343	517	24	)	)	PUNCT
ejpam-4343	517	25	.	.	PUNCT
ejpam-4343	518	1	then	then	ADV
ejpam-4343	518	2	,	,	PUNCT
ejpam-4343	518	3	we	we	PRON
ejpam-4343	518	4	have	have	VERB
ejpam-4343	518	5	x	x	X
ejpam-4343	518	6	∈	∈	PROPN
ejpam-4343	518	7	x	x	SYM
ejpam-4343	518	8	−	−	PROPN
ejpam-4343	518	9	f−1(k	f−1(k	PROPN
ejpam-4343	518	10	)	)	PUNCT
ejpam-4343	519	1	=	=	SYM
ejpam-4343	519	2	f−1(y	f−1(y	PROPN
ejpam-4343	519	3	−k	−k	NOUN
ejpam-4343	519	4	)	)	PUNCT
ejpam-4343	519	5	=	=	SYM
ejpam-4343	520	1	f−1((y	f−1((y	NOUN
ejpam-4343	520	2	−k)(λ	−k)(λ	NOUN
ejpam-4343	520	3	,	,	PUNCT
ejpam-4343	520	4	p(⋆	p(⋆	PROPN
ejpam-4343	520	5	)	)	PUNCT
ejpam-4343	520	6	)	)	PUNCT
ejpam-4343	520	7	)	)	PUNCT
ejpam-4343	520	8	,	,	PUNCT
ejpam-4343	520	9	by	by	ADP
ejpam-4343	520	10	(	(	PUNCT
ejpam-4343	520	11	5	5	NUM
ejpam-4343	520	12	)	)	PUNCT
ejpam-4343	520	13	,	,	PUNCT
ejpam-4343	520	14	x	x	PUNCT
ejpam-4343	520	15	∈	∈	PROPN
ejpam-4343	520	16	[	[	X
ejpam-4343	520	17	f−1(y	f−1(y	PROPN
ejpam-4343	520	18	−k)](λ	−k)](λ	PROPN
ejpam-4343	520	19	,	,	PUNCT
ejpam-4343	520	20	p(⋆	p(⋆	PROPN
ejpam-4343	520	21	)	)	PUNCT
ejpam-4343	520	22	)	)	PUNCT
ejpam-4343	521	1	=	=	PUNCT
ejpam-4343	522	1	[	[	X
ejpam-4343	522	2	x	x	X
ejpam-4343	522	3	−	−	NOUN
ejpam-4343	522	4	f−1(k)](λ	f−1(k)](λ	PROPN
ejpam-4343	522	5	,	,	PUNCT
ejpam-4343	522	6	p(⋆	p(⋆	PROPN
ejpam-4343	522	7	)	)	PUNCT
ejpam-4343	522	8	)	)	PUNCT
ejpam-4343	523	1	=	=	PUNCT
ejpam-4343	523	2	x	x	X
ejpam-4343	524	1	−	−	PROPN
ejpam-4343	524	2	[	[	X
ejpam-4343	524	3	f−1(k)](λ	f−1(k)](λ	PROPN
ejpam-4343	524	4	,	,	PUNCT
ejpam-4343	524	5	p(⋆	p(⋆	PROPN
ejpam-4343	524	6	)	)	PUNCT
ejpam-4343	524	7	)	)	PUNCT
ejpam-4343	524	8	and	and	CCONJ
ejpam-4343	524	9	hence	hence	ADV
ejpam-4343	524	10	x	x	X
ejpam-4343	524	11	̸∈	̸∈	PROPN
ejpam-4343	524	12	[	[	X
ejpam-4343	524	13	f−1(k)](λ	f−1(k)](λ	PROPN
ejpam-4343	524	14	,	,	PUNCT
ejpam-4343	524	15	p(⋆	p(⋆	PROPN
ejpam-4343	524	16	)	)	PUNCT
ejpam-4343	524	17	)	)	PUNCT
ejpam-4343	524	18	.	.	PUNCT
ejpam-4343	525	1	(	(	PUNCT
ejpam-4343	525	2	6	6	X
ejpam-4343	525	3	)	)	PUNCT
ejpam-4343	525	4	⇒	⇒	NOUN
ejpam-4343	525	5	(	(	PUNCT
ejpam-4343	525	6	2	2	NUM
ejpam-4343	525	7	):	):	PUNCT
ejpam-4343	525	8	let	let	VERB
ejpam-4343	525	9	x	x	PUNCT
ejpam-4343	525	10	∈	∈	PROPN
ejpam-4343	525	11	x	x	PUNCT
ejpam-4343	525	12	and	and	CCONJ
ejpam-4343	525	13	let	let	VERB
ejpam-4343	525	14	v	v	PART
ejpam-4343	525	15	be	be	AUX
ejpam-4343	525	16	any	any	DET
ejpam-4343	525	17	(	(	PUNCT
ejpam-4343	525	18	λ	λ	NOUN
ejpam-4343	525	19	,	,	PUNCT
ejpam-4343	525	20	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	525	21	set	set	NOUN
ejpam-4343	525	22	of	of	ADP
ejpam-4343	525	23	y	y	PROPN
ejpam-4343	525	24	containing	contain	VERB
ejpam-4343	525	25	f(x	f(x	PROPN
ejpam-4343	525	26	)	)	PUNCT
ejpam-4343	525	27	.	.	PUNCT
ejpam-4343	526	1	suppose	suppose	VERB
ejpam-4343	526	2	that	that	SCONJ
ejpam-4343	526	3	x	x	PROPN
ejpam-4343	526	4	̸∈	̸∈	PROPN
ejpam-4343	526	5	[	[	X
ejpam-4343	526	6	f−1(v	f−1(v	PROPN
ejpam-4343	526	7	)	)	PUNCT
ejpam-4343	526	8	]	]	PUNCT
ejpam-4343	526	9	(	(	PUNCT
ejpam-4343	526	10	λ	λ	X
ejpam-4343	526	11	,	,	PUNCT
ejpam-4343	526	12	p(⋆	p(⋆	PROPN
ejpam-4343	526	13	)	)	PUNCT
ejpam-4343	526	14	)	)	PUNCT
ejpam-4343	526	15	.	.	PUNCT
ejpam-4343	527	1	then	then	ADV
ejpam-4343	527	2	,	,	PUNCT
ejpam-4343	527	3	x	x	PUNCT
ejpam-4343	527	4	∈	∈	NOUN
ejpam-4343	527	5	x	x	X
ejpam-4343	527	6	−	−	PROPN
ejpam-4343	528	1	[	[	X
ejpam-4343	528	2	f−1(v	f−1(v	NOUN
ejpam-4343	528	3	)	)	PUNCT
ejpam-4343	528	4	]	]	PUNCT
ejpam-4343	528	5	(	(	PUNCT
ejpam-4343	528	6	λ	λ	X
ejpam-4343	528	7	,	,	PUNCT
ejpam-4343	528	8	p(⋆	p(⋆	PROPN
ejpam-4343	528	9	)	)	PUNCT
ejpam-4343	528	10	)	)	PUNCT
ejpam-4343	529	1	=	=	PUNCT
ejpam-4343	530	1	[	[	X
ejpam-4343	530	2	x	x	X
ejpam-4343	530	3	−	−	PROPN
ejpam-4343	530	4	f−1(v	f−1(v	NOUN
ejpam-4343	530	5	)	)	PUNCT
ejpam-4343	530	6	]	]	PUNCT
ejpam-4343	530	7	(	(	PUNCT
ejpam-4343	530	8	λ	λ	X
ejpam-4343	530	9	,	,	PUNCT
ejpam-4343	530	10	p(⋆	p(⋆	PROPN
ejpam-4343	530	11	)	)	PUNCT
ejpam-4343	530	12	)	)	PUNCT
ejpam-4343	531	1	=	=	PUNCT
ejpam-4343	532	1	[	[	X
ejpam-4343	532	2	f−1(y	f−1(y	NOUN
ejpam-4343	532	3	−	−	PROPN
ejpam-4343	532	4	v	v	NOUN
ejpam-4343	532	5	)	)	PUNCT
ejpam-4343	532	6	]	]	PUNCT
ejpam-4343	532	7	(	(	PUNCT
ejpam-4343	532	8	λ	λ	X
ejpam-4343	532	9	,	,	PUNCT
ejpam-4343	532	10	p(⋆	p(⋆	PROPN
ejpam-4343	532	11	)	)	PUNCT
ejpam-4343	532	12	)	)	PUNCT
ejpam-4343	532	13	.	.	PUNCT
ejpam-4343	533	1	c.	c.	PROPN
ejpam-4343	533	2	boonpok	boonpok	PROPN
ejpam-4343	533	3	/	/	SYM
ejpam-4343	533	4	eur	eur	PROPN
ejpam-4343	533	5	.	.	PUNCT
ejpam-4343	534	1	j.	j.	PROPN
ejpam-4343	534	2	pure	pure	PROPN
ejpam-4343	534	3	appl	appl	PROPN
ejpam-4343	534	4	.	.	PROPN
ejpam-4343	534	5	math	math	PROPN
ejpam-4343	534	6	,	,	PUNCT
ejpam-4343	534	7	15	15	NUM
ejpam-4343	534	8	(	(	PUNCT
ejpam-4343	534	9	3	3	NUM
ejpam-4343	534	10	)	)	PUNCT
ejpam-4343	534	11	(	(	PUNCT
ejpam-4343	534	12	2022	2022	NUM
ejpam-4343	534	13	)	)	PUNCT
ejpam-4343	534	14	,	,	PUNCT
ejpam-4343	534	15	1023	1023	NUM
ejpam-4343	534	16	-	-	SYM
ejpam-4343	534	17	1046	1046	NUM
ejpam-4343	534	18	1036	1036	NUM
ejpam-4343	534	19	by	by	ADP
ejpam-4343	534	20	(	(	PUNCT
ejpam-4343	534	21	6	6	NUM
ejpam-4343	534	22	)	)	PUNCT
ejpam-4343	535	1	,	,	PUNCT
ejpam-4343	535	2	we	we	PRON
ejpam-4343	535	3	have	have	VERB
ejpam-4343	535	4	x	x	X
ejpam-4343	535	5	∈	∈	PROPN
ejpam-4343	535	6	f−1(y	f−1(y	NOUN
ejpam-4343	535	7	−	−	PROPN
ejpam-4343	535	8	v	v	NOUN
ejpam-4343	535	9	)	)	PUNCT
ejpam-4343	535	10	=	=	PUNCT
ejpam-4343	535	11	x	x	PUNCT
ejpam-4343	535	12	−	−	PROPN
ejpam-4343	535	13	f−1(v	f−1(v	PROPN
ejpam-4343	535	14	)	)	PUNCT
ejpam-4343	535	15	and	and	CCONJ
ejpam-4343	535	16	hence	hence	ADV
ejpam-4343	535	17	x	x	X
ejpam-4343	535	18	̸∈	̸∈	PROPN
ejpam-4343	535	19	f−1(v	f−1(v	PROPN
ejpam-4343	535	20	)	)	PUNCT
ejpam-4343	535	21	.	.	PUNCT
ejpam-4343	536	1	this	this	DET
ejpam-4343	536	2	contraries	contrary	NOUN
ejpam-4343	536	3	to	to	ADP
ejpam-4343	536	4	the	the	DET
ejpam-4343	536	5	hypothesis	hypothesis	NOUN
ejpam-4343	536	6	.	.	PUNCT
ejpam-4343	537	1	(	(	PUNCT
ejpam-4343	537	2	2	2	X
ejpam-4343	537	3	)	)	PUNCT
ejpam-4343	537	4	⇒	⇒	NOUN
ejpam-4343	537	5	(	(	PUNCT
ejpam-4343	537	6	1	1	NUM
ejpam-4343	537	7	):	):	PUNCT
ejpam-4343	537	8	let	let	VERB
ejpam-4343	537	9	v	v	PART
ejpam-4343	537	10	be	be	AUX
ejpam-4343	537	11	any	any	DET
ejpam-4343	537	12	(	(	PUNCT
ejpam-4343	537	13	λ	λ	NOUN
ejpam-4343	537	14	,	,	PUNCT
ejpam-4343	537	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	537	16	set	set	NOUN
ejpam-4343	537	17	of	of	ADP
ejpam-4343	537	18	y	y	PROPN
ejpam-4343	537	19	containing	contain	VERB
ejpam-4343	537	20	f(x	f(x	PROPN
ejpam-4343	537	21	)	)	PUNCT
ejpam-4343	537	22	.	.	PUNCT
ejpam-4343	538	1	by	by	ADP
ejpam-4343	538	2	(	(	PUNCT
ejpam-4343	538	3	2	2	NUM
ejpam-4343	538	4	)	)	PUNCT
ejpam-4343	538	5	,	,	PUNCT
ejpam-4343	538	6	we	we	PRON
ejpam-4343	538	7	have	have	VERB
ejpam-4343	538	8	x	x	X
ejpam-4343	538	9	∈	∈	PROPN
ejpam-4343	538	10	[	[	X
ejpam-4343	538	11	f−1(v	f−1(v	NOUN
ejpam-4343	538	12	)	)	PUNCT
ejpam-4343	538	13	]	]	PUNCT
ejpam-4343	538	14	(	(	PUNCT
ejpam-4343	538	15	λ	λ	X
ejpam-4343	538	16	,	,	PUNCT
ejpam-4343	538	17	p(⋆	p(⋆	PROPN
ejpam-4343	538	18	)	)	PUNCT
ejpam-4343	538	19	)	)	PUNCT
ejpam-4343	539	1	and	and	CCONJ
ejpam-4343	539	2	so	so	ADV
ejpam-4343	539	3	there	there	PRON
ejpam-4343	539	4	exists	exist	VERB
ejpam-4343	539	5	a	a	DET
ejpam-4343	539	6	(	(	PUNCT
ejpam-4343	539	7	λ	λ	NOUN
ejpam-4343	539	8	,	,	PUNCT
ejpam-4343	539	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	539	10	set	set	VERB
ejpam-4343	539	11	u	u	NOUN
ejpam-4343	539	12	of	of	ADP
ejpam-4343	539	13	x	x	PUNCT
ejpam-4343	539	14	containing	contain	VERB
ejpam-4343	539	15	x	x	PUNCT
ejpam-4343	539	16	such	such	ADJ
ejpam-4343	539	17	that	that	SCONJ
ejpam-4343	539	18	x	x	SYM
ejpam-4343	539	19	∈	∈	PROPN
ejpam-4343	539	20	u	u	NOUN
ejpam-4343	539	21	⊆	⊆	NUM
ejpam-4343	539	22	f−1(v	f−1(v	NOUN
ejpam-4343	539	23	)	)	PUNCT
ejpam-4343	539	24	;	;	PUNCT
ejpam-4343	539	25	hence	hence	ADV
ejpam-4343	539	26	f(u	f(u	PROPN
ejpam-4343	539	27	)	)	PUNCT
ejpam-4343	539	28	⊆	⊆	NUM
ejpam-4343	539	29	v	v	NOUN
ejpam-4343	539	30	.	.	PUNCT
ejpam-4343	540	1	this	this	PRON
ejpam-4343	540	2	shows	show	VERB
ejpam-4343	540	3	that	that	SCONJ
ejpam-4343	540	4	f	f	PROPN
ejpam-4343	540	5	is	be	AUX
ejpam-4343	540	6	(	(	PUNCT
ejpam-4343	540	7	λ	λ	X
ejpam-4343	540	8	,	,	PUNCT
ejpam-4343	540	9	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	540	10	at	at	ADP
ejpam-4343	540	11	x.	x.	NOUN
ejpam-4343	540	12	theorem	theorem	VERB
ejpam-4343	540	13	9	9	NUM
ejpam-4343	540	14	.	.	PUNCT
ejpam-4343	540	15	for	for	ADP
ejpam-4343	540	16	a	a	DET
ejpam-4343	540	17	function	function	NOUN
ejpam-4343	540	18	f	f	NOUN
ejpam-4343	540	19	:	:	PUNCT
ejpam-4343	540	20	(	(	PUNCT
ejpam-4343	540	21	x	x	X
ejpam-4343	540	22	,	,	PUNCT
ejpam-4343	540	23	τ	τ	PROPN
ejpam-4343	540	24	,	,	PUNCT
ejpam-4343	540	25	i	i	NOUN
ejpam-4343	540	26	)	)	PUNCT
ejpam-4343	540	27	→	→	PUNCT
ejpam-4343	540	28	(	(	PUNCT
ejpam-4343	540	29	y	y	PROPN
ejpam-4343	540	30	,	,	PUNCT
ejpam-4343	540	31	σ	σ	PROPN
ejpam-4343	540	32	,	,	PUNCT
ejpam-4343	540	33	j	j	PROPN
ejpam-4343	540	34	)	)	PUNCT
ejpam-4343	540	35	,	,	PUNCT
ejpam-4343	540	36	the	the	DET
ejpam-4343	540	37	following	follow	VERB
ejpam-4343	540	38	properties	property	NOUN
ejpam-4343	540	39	are	be	AUX
ejpam-4343	540	40	equivalent	equivalent	ADJ
ejpam-4343	540	41	:	:	PUNCT
ejpam-4343	540	42	(	(	PUNCT
ejpam-4343	540	43	1	1	X
ejpam-4343	540	44	)	)	PUNCT
ejpam-4343	540	45	f	f	PROPN
ejpam-4343	540	46	is	be	AUX
ejpam-4343	540	47	(	(	PUNCT
ejpam-4343	540	48	λ	λ	INTJ
ejpam-4343	540	49	,	,	PUNCT
ejpam-4343	540	50	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	540	51	.	.	PUNCT
ejpam-4343	541	1	(	(	PUNCT
ejpam-4343	541	2	2	2	X
ejpam-4343	541	3	)	)	PUNCT
ejpam-4343	541	4	f−1(v	f−1(v	NOUN
ejpam-4343	541	5	)	)	PUNCT
ejpam-4343	541	6	is	be	AUX
ejpam-4343	541	7	(	(	PUNCT
ejpam-4343	541	8	λ	λ	X
ejpam-4343	541	9	,	,	PUNCT
ejpam-4343	541	10	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	541	11	in	in	ADP
ejpam-4343	541	12	x	x	PUNCT
ejpam-4343	541	13	for	for	SCONJ
ejpam-4343	541	14	every	every	DET
ejpam-4343	541	15	(	(	PUNCT
ejpam-4343	541	16	λ	λ	PROPN
ejpam-4343	541	17	,	,	PUNCT
ejpam-4343	541	18	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	541	19	set	set	VERB
ejpam-4343	541	20	v	v	NOUN
ejpam-4343	541	21	of	of	ADP
ejpam-4343	541	22	y	y	PROPN
ejpam-4343	541	23	.	.	PUNCT
ejpam-4343	542	1	(	(	PUNCT
ejpam-4343	542	2	3	3	X
ejpam-4343	542	3	)	)	PUNCT
ejpam-4343	542	4	f(a(λ	f(a(λ	NOUN
ejpam-4343	542	5	,	,	PUNCT
ejpam-4343	542	6	p(⋆	p(⋆	PROPN
ejpam-4343	542	7	)	)	PUNCT
ejpam-4343	542	8	)	)	PUNCT
ejpam-4343	542	9	)	)	PUNCT
ejpam-4343	543	1	⊆	⊆	NUM
ejpam-4343	543	2	[	[	X
ejpam-4343	543	3	f(a)](λ	f(a)](λ	NUM
ejpam-4343	543	4	,	,	PUNCT
ejpam-4343	543	5	p(⋆	p(⋆	PROPN
ejpam-4343	543	6	)	)	PUNCT
ejpam-4343	543	7	)	)	PUNCT
ejpam-4343	543	8	for	for	ADP
ejpam-4343	543	9	every	every	DET
ejpam-4343	543	10	subset	subset	NOUN
ejpam-4343	543	11	a	a	PRON
ejpam-4343	543	12	of	of	ADP
ejpam-4343	543	13	x.	x.	NOUN
ejpam-4343	543	14	(	(	PUNCT
ejpam-4343	543	15	4	4	NUM
ejpam-4343	543	16	)	)	PUNCT
ejpam-4343	543	17	[	[	X
ejpam-4343	543	18	f−1(b)](λ	f−1(b)](λ	X
ejpam-4343	543	19	,	,	PUNCT
ejpam-4343	543	20	p(⋆	p(⋆	PROPN
ejpam-4343	543	21	)	)	PUNCT
ejpam-4343	543	22	)	)	PUNCT
ejpam-4343	544	1	⊆	⊆	NUM
ejpam-4343	544	2	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	544	3	,	,	PUNCT
ejpam-4343	544	4	p(⋆	p(⋆	PROPN
ejpam-4343	544	5	)	)	PUNCT
ejpam-4343	544	6	)	)	PUNCT
ejpam-4343	544	7	)	)	PUNCT
ejpam-4343	544	8	for	for	ADP
ejpam-4343	544	9	every	every	DET
ejpam-4343	544	10	subset	subset	NOUN
ejpam-4343	544	11	b	b	PROPN
ejpam-4343	544	12	of	of	ADP
ejpam-4343	544	13	y	y	PROPN
ejpam-4343	544	14	.	.	PUNCT
ejpam-4343	545	1	(	(	PUNCT
ejpam-4343	545	2	5	5	NUM
ejpam-4343	545	3	)	)	PUNCT
ejpam-4343	545	4	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	545	5	,	,	PUNCT
ejpam-4343	545	6	p(⋆	p(⋆	PROPN
ejpam-4343	545	7	)	)	PUNCT
ejpam-4343	545	8	)	)	PUNCT
ejpam-4343	545	9	)	)	PUNCT
ejpam-4343	546	1	⊆	⊆	NUM
ejpam-4343	546	2	[	[	X
ejpam-4343	546	3	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	546	4	,	,	PUNCT
ejpam-4343	546	5	p(⋆	p(⋆	PROPN
ejpam-4343	546	6	)	)	PUNCT
ejpam-4343	546	7	)	)	PUNCT
ejpam-4343	546	8	for	for	ADP
ejpam-4343	546	9	every	every	DET
ejpam-4343	546	10	subset	subset	NOUN
ejpam-4343	546	11	b	b	PROPN
ejpam-4343	546	12	of	of	ADP
ejpam-4343	546	13	y	y	PROPN
ejpam-4343	546	14	.	.	PUNCT
ejpam-4343	547	1	(	(	PUNCT
ejpam-4343	547	2	6	6	X
ejpam-4343	547	3	)	)	PUNCT
ejpam-4343	547	4	f−1(k	f−1(k	PROPN
ejpam-4343	547	5	)	)	PUNCT
ejpam-4343	547	6	is	be	AUX
ejpam-4343	547	7	(	(	PUNCT
ejpam-4343	547	8	λ	λ	X
ejpam-4343	547	9	,	,	PUNCT
ejpam-4343	547	10	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	547	11	in	in	ADP
ejpam-4343	547	12	x	x	PUNCT
ejpam-4343	547	13	for	for	SCONJ
ejpam-4343	547	14	every	every	DET
ejpam-4343	547	15	(	(	PUNCT
ejpam-4343	547	16	λ	λ	PROPN
ejpam-4343	547	17	,	,	PUNCT
ejpam-4343	547	18	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	547	19	set	set	VERB
ejpam-4343	547	20	k	k	PROPN
ejpam-4343	547	21	of	of	ADP
ejpam-4343	547	22	y	y	PROPN
ejpam-4343	547	23	.	.	PUNCT
ejpam-4343	548	1	proof	proof	NOUN
ejpam-4343	548	2	.	.	PUNCT
ejpam-4343	549	1	(	(	PUNCT
ejpam-4343	549	2	1	1	X
ejpam-4343	549	3	)	)	PUNCT
ejpam-4343	549	4	⇒	⇒	NOUN
ejpam-4343	549	5	(	(	PUNCT
ejpam-4343	549	6	2	2	NUM
ejpam-4343	549	7	):	):	PUNCT
ejpam-4343	549	8	let	let	VERB
ejpam-4343	549	9	v	v	PART
ejpam-4343	549	10	be	be	AUX
ejpam-4343	549	11	any	any	DET
ejpam-4343	549	12	(	(	PUNCT
ejpam-4343	549	13	λ	λ	NOUN
ejpam-4343	549	14	,	,	PUNCT
ejpam-4343	549	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	549	16	set	set	NOUN
ejpam-4343	549	17	of	of	ADP
ejpam-4343	549	18	y	y	PRON
ejpam-4343	549	19	such	such	ADJ
ejpam-4343	549	20	that	that	SCONJ
ejpam-4343	549	21	x	x	SYM
ejpam-4343	549	22	∈	∈	PROPN
ejpam-4343	549	23	f−1(v	f−1(v	NOUN
ejpam-4343	549	24	)	)	PUNCT
ejpam-4343	549	25	.	.	PUNCT
ejpam-4343	550	1	then	then	ADV
ejpam-4343	550	2	,	,	PUNCT
ejpam-4343	550	3	f(x	f(x	PROPN
ejpam-4343	550	4	)	)	PUNCT
ejpam-4343	550	5	∈	∈	PROPN
ejpam-4343	550	6	v	v	NOUN
ejpam-4343	550	7	and	and	CCONJ
ejpam-4343	550	8	there	there	PRON
ejpam-4343	550	9	exists	exist	VERB
ejpam-4343	550	10	a	a	DET
ejpam-4343	550	11	(	(	PUNCT
ejpam-4343	550	12	λ	λ	NOUN
ejpam-4343	550	13	,	,	PUNCT
ejpam-4343	550	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	550	15	set	set	VERB
ejpam-4343	550	16	u	u	NOUN
ejpam-4343	550	17	of	of	ADP
ejpam-4343	550	18	x	x	PUNCT
ejpam-4343	550	19	containing	contain	VERB
ejpam-4343	550	20	x	x	PUNCT
ejpam-4343	550	21	such	such	ADJ
ejpam-4343	550	22	that	that	DET
ejpam-4343	550	23	f(u	f(u	PROPN
ejpam-4343	550	24	)	)	PUNCT
ejpam-4343	550	25	⊆	⊆	NUM
ejpam-4343	550	26	v	v	NOUN
ejpam-4343	550	27	.	.	PUNCT
ejpam-4343	551	1	since	since	SCONJ
ejpam-4343	551	2	u	u	NOUN
ejpam-4343	551	3	is	be	AUX
ejpam-4343	551	4	(	(	PUNCT
ejpam-4343	551	5	λ	λ	X
ejpam-4343	551	6	,	,	PUNCT
ejpam-4343	551	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	551	8	in	in	ADP
ejpam-4343	551	9	x	x	PRON
ejpam-4343	551	10	,	,	PUNCT
ejpam-4343	551	11	x	x	SYM
ejpam-4343	551	12	∈	∈	PROPN
ejpam-4343	552	1	[	[	X
ejpam-4343	552	2	f−1(v	f−1(v	NOUN
ejpam-4343	552	3	)	)	PUNCT
ejpam-4343	552	4	]	]	PUNCT
ejpam-4343	552	5	(	(	PUNCT
ejpam-4343	552	6	λ	λ	X
ejpam-4343	552	7	,	,	PUNCT
ejpam-4343	552	8	p(⋆	p(⋆	PROPN
ejpam-4343	552	9	)	)	PUNCT
ejpam-4343	552	10	)	)	PUNCT
ejpam-4343	552	11	and	and	CCONJ
ejpam-4343	552	12	hence	hence	ADV
ejpam-4343	552	13	f−1(v	f−1(v	NOUN
ejpam-4343	552	14	)	)	PUNCT
ejpam-4343	552	15	⊆	⊆	NUM
ejpam-4343	553	1	[	[	X
ejpam-4343	553	2	f−1(v	f−1(v	NOUN
ejpam-4343	553	3	)	)	PUNCT
ejpam-4343	553	4	]	]	PUNCT
ejpam-4343	553	5	(	(	PUNCT
ejpam-4343	553	6	λ	λ	X
ejpam-4343	553	7	,	,	PUNCT
ejpam-4343	553	8	p(⋆	p(⋆	PROPN
ejpam-4343	553	9	)	)	PUNCT
ejpam-4343	553	10	)	)	PUNCT
ejpam-4343	553	11	.	.	PUNCT
ejpam-4343	554	1	thus	thus	ADV
ejpam-4343	554	2	,	,	PUNCT
ejpam-4343	554	3	f−1(v	f−1(v	PROPN
ejpam-4343	554	4	)	)	PUNCT
ejpam-4343	554	5	is	be	AUX
ejpam-4343	554	6	(	(	PUNCT
ejpam-4343	554	7	λ	λ	INTJ
ejpam-4343	554	8	,	,	PUNCT
ejpam-4343	554	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	554	10	.	.	PUNCT
ejpam-4343	555	1	(	(	PUNCT
ejpam-4343	555	2	2	2	X
ejpam-4343	555	3	)	)	PUNCT
ejpam-4343	555	4	⇒	⇒	NOUN
ejpam-4343	555	5	(	(	PUNCT
ejpam-4343	555	6	3	3	NUM
ejpam-4343	555	7	):	):	PUNCT
ejpam-4343	555	8	let	let	VERB
ejpam-4343	555	9	a	a	DET
ejpam-4343	555	10	be	be	AUX
ejpam-4343	555	11	any	any	DET
ejpam-4343	555	12	subset	subset	NOUN
ejpam-4343	555	13	of	of	ADP
ejpam-4343	555	14	x.	x.	NOUN
ejpam-4343	555	15	let	let	VERB
ejpam-4343	555	16	x	x	X
ejpam-4343	555	17	∈	∈	PROPN
ejpam-4343	555	18	a(λ	a(λ	ADV
ejpam-4343	555	19	,	,	PUNCT
ejpam-4343	555	20	p(⋆	p(⋆	PROPN
ejpam-4343	555	21	)	)	PUNCT
ejpam-4343	555	22	)	)	PUNCT
ejpam-4343	555	23	and	and	CCONJ
ejpam-4343	555	24	let	let	VERB
ejpam-4343	555	25	v	v	PART
ejpam-4343	555	26	be	be	AUX
ejpam-4343	555	27	any	any	DET
ejpam-4343	555	28	(	(	PUNCT
ejpam-4343	555	29	λ	λ	PROPN
ejpam-4343	555	30	,	,	PUNCT
ejpam-4343	555	31	p(⋆))open	p(⋆))open	PROPN
ejpam-4343	555	32	set	set	NOUN
ejpam-4343	555	33	of	of	ADP
ejpam-4343	555	34	y	y	PROPN
ejpam-4343	555	35	containing	contain	VERB
ejpam-4343	555	36	f(x	f(x	PROPN
ejpam-4343	555	37	)	)	PUNCT
ejpam-4343	555	38	.	.	PUNCT
ejpam-4343	556	1	by	by	ADP
ejpam-4343	556	2	(	(	PUNCT
ejpam-4343	556	3	2	2	NUM
ejpam-4343	556	4	)	)	PUNCT
ejpam-4343	556	5	,	,	PUNCT
ejpam-4343	556	6	we	we	PRON
ejpam-4343	556	7	have	have	VERB
ejpam-4343	556	8	x	x	X
ejpam-4343	556	9	∈	∈	PROPN
ejpam-4343	556	10	[	[	X
ejpam-4343	556	11	f−1(v	f−1(v	NOUN
ejpam-4343	556	12	)	)	PUNCT
ejpam-4343	556	13	]	]	PUNCT
ejpam-4343	556	14	(	(	PUNCT
ejpam-4343	556	15	λ	λ	X
ejpam-4343	556	16	,	,	PUNCT
ejpam-4343	556	17	p(⋆	p(⋆	PROPN
ejpam-4343	556	18	)	)	PUNCT
ejpam-4343	556	19	)	)	PUNCT
ejpam-4343	556	20	and	and	CCONJ
ejpam-4343	556	21	there	there	PRON
ejpam-4343	556	22	exists	exist	VERB
ejpam-4343	556	23	a	a	DET
ejpam-4343	556	24	(	(	PUNCT
ejpam-4343	556	25	λ	λ	NOUN
ejpam-4343	556	26	,	,	PUNCT
ejpam-4343	556	27	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	556	28	set	set	VERB
ejpam-4343	556	29	u	u	NOUN
ejpam-4343	556	30	of	of	ADP
ejpam-4343	556	31	x	x	SYM
ejpam-4343	556	32	such	such	ADJ
ejpam-4343	556	33	that	that	SCONJ
ejpam-4343	556	34	x	x	SYM
ejpam-4343	556	35	∈	∈	PROPN
ejpam-4343	556	36	u	u	NOUN
ejpam-4343	556	37	⊆	⊆	NUM
ejpam-4343	556	38	f−1(v	f−1(v	NOUN
ejpam-4343	556	39	)	)	PUNCT
ejpam-4343	556	40	.	.	PUNCT
ejpam-4343	557	1	since	since	SCONJ
ejpam-4343	557	2	x	x	PROPN
ejpam-4343	557	3	∈	∈	PROPN
ejpam-4343	557	4	a(λ	a(λ	ADV
ejpam-4343	557	5	,	,	PUNCT
ejpam-4343	557	6	p(⋆	p(⋆	PROPN
ejpam-4343	557	7	)	)	PUNCT
ejpam-4343	557	8	)	)	PUNCT
ejpam-4343	557	9	,	,	PUNCT
ejpam-4343	557	10	by	by	ADP
ejpam-4343	557	11	lemma	lemma	PROPN
ejpam-4343	557	12	5	5	NUM
ejpam-4343	557	13	,	,	PUNCT
ejpam-4343	557	14	u	u	NOUN
ejpam-4343	557	15	∩	∩	NOUN
ejpam-4343	557	16	a	a	DET
ejpam-4343	557	17	̸=	̸=	PROPN
ejpam-4343	557	18	∅	∅	NOUN
ejpam-4343	557	19	and	and	CCONJ
ejpam-4343	557	20	∅	∅	NOUN
ejpam-4343	557	21	̸=	̸=	PROPN
ejpam-4343	557	22	f(u	f(u	PROPN
ejpam-4343	557	23	∩	∩	NOUN
ejpam-4343	557	24	a	a	X
ejpam-4343	557	25	)	)	PUNCT
ejpam-4343	557	26	⊆	⊆	NUM
ejpam-4343	557	27	f(u	f(u	PROPN
ejpam-4343	557	28	)	)	PUNCT
ejpam-4343	557	29	∩	∩	ADJ
ejpam-4343	557	30	f(a	f(a	NOUN
ejpam-4343	557	31	)	)	PUNCT
ejpam-4343	557	32	⊆	⊆	NUM
ejpam-4343	557	33	v	v	ADP
ejpam-4343	557	34	∩	∩	ADJ
ejpam-4343	557	35	f(a	f(a	NOUN
ejpam-4343	557	36	)	)	PUNCT
ejpam-4343	557	37	.	.	PUNCT
ejpam-4343	558	1	thus	thus	ADV
ejpam-4343	558	2	,	,	PUNCT
ejpam-4343	558	3	f(x	f(x	PROPN
ejpam-4343	558	4	)	)	PUNCT
ejpam-4343	558	5	∈	∈	PROPN
ejpam-4343	559	1	[	[	X
ejpam-4343	559	2	f(a)](λ	f(a)](λ	NUM
ejpam-4343	559	3	,	,	PUNCT
ejpam-4343	559	4	p(⋆	p(⋆	PROPN
ejpam-4343	559	5	)	)	PUNCT
ejpam-4343	559	6	)	)	PUNCT
ejpam-4343	560	1	and	and	CCONJ
ejpam-4343	560	2	hence	hence	ADV
ejpam-4343	560	3	f(a(λ	f(a(λ	PROPN
ejpam-4343	560	4	,	,	PUNCT
ejpam-4343	560	5	p(⋆	p(⋆	PROPN
ejpam-4343	560	6	)	)	PUNCT
ejpam-4343	560	7	)	)	PUNCT
ejpam-4343	560	8	)	)	PUNCT
ejpam-4343	561	1	⊆	⊆	NUM
ejpam-4343	561	2	[	[	X
ejpam-4343	561	3	f(a)](λ	f(a)](λ	NUM
ejpam-4343	561	4	,	,	PUNCT
ejpam-4343	561	5	p(⋆	p(⋆	PROPN
ejpam-4343	561	6	)	)	PUNCT
ejpam-4343	561	7	)	)	PUNCT
ejpam-4343	561	8	.	.	PUNCT
ejpam-4343	562	1	(	(	PUNCT
ejpam-4343	562	2	3	3	X
ejpam-4343	562	3	)	)	PUNCT
ejpam-4343	562	4	⇒	⇒	NOUN
ejpam-4343	562	5	(	(	PUNCT
ejpam-4343	562	6	4	4	NUM
ejpam-4343	562	7	):	):	PUNCT
ejpam-4343	562	8	let	let	VERB
ejpam-4343	562	9	b	b	X
ejpam-4343	562	10	be	be	AUX
ejpam-4343	562	11	any	any	DET
ejpam-4343	562	12	subset	subset	NOUN
ejpam-4343	562	13	of	of	ADP
ejpam-4343	562	14	y	y	PROPN
ejpam-4343	562	15	.	.	PUNCT
ejpam-4343	563	1	by	by	ADP
ejpam-4343	563	2	(	(	PUNCT
ejpam-4343	563	3	3	3	NUM
ejpam-4343	563	4	)	)	PUNCT
ejpam-4343	563	5	,	,	PUNCT
ejpam-4343	563	6	f([f−1(b)](λ	f([f−1(b)](λ	PROPN
ejpam-4343	563	7	,	,	PUNCT
ejpam-4343	563	8	p(⋆	p(⋆	PROPN
ejpam-4343	563	9	)	)	PUNCT
ejpam-4343	563	10	)	)	PUNCT
ejpam-4343	563	11	)	)	PUNCT
ejpam-4343	564	1	⊆	⊆	NUM
ejpam-4343	564	2	[	[	X
ejpam-4343	564	3	f(f−1(b))](λ	f(f−1(b))](λ	X
ejpam-4343	564	4	,	,	PUNCT
ejpam-4343	564	5	p(⋆	p(⋆	PROPN
ejpam-4343	564	6	)	)	PUNCT
ejpam-4343	564	7	)	)	PUNCT
ejpam-4343	565	1	⊆	⊆	NUM
ejpam-4343	565	2	b(λ	b(λ	NOUN
ejpam-4343	565	3	,	,	PUNCT
ejpam-4343	565	4	p(⋆	p(⋆	PROPN
ejpam-4343	565	5	)	)	PUNCT
ejpam-4343	565	6	)	)	PUNCT
ejpam-4343	565	7	.	.	PUNCT
ejpam-4343	566	1	therefore	therefore	ADV
ejpam-4343	566	2	,	,	PUNCT
ejpam-4343	566	3	[	[	X
ejpam-4343	566	4	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	566	5	,	,	PUNCT
ejpam-4343	566	6	p(⋆	p(⋆	PROPN
ejpam-4343	566	7	)	)	PUNCT
ejpam-4343	566	8	)	)	PUNCT
ejpam-4343	567	1	⊆	⊆	NUM
ejpam-4343	567	2	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	567	3	,	,	PUNCT
ejpam-4343	567	4	p(⋆	p(⋆	PROPN
ejpam-4343	567	5	)	)	PUNCT
ejpam-4343	567	6	)	)	PUNCT
ejpam-4343	567	7	)	)	PUNCT
ejpam-4343	567	8	.	.	PUNCT
ejpam-4343	568	1	(	(	PUNCT
ejpam-4343	568	2	4	4	X
ejpam-4343	568	3	)	)	PUNCT
ejpam-4343	568	4	⇒	⇒	NOUN
ejpam-4343	568	5	(	(	PUNCT
ejpam-4343	568	6	5	5	NUM
ejpam-4343	568	7	):	):	PUNCT
ejpam-4343	568	8	let	let	VERB
ejpam-4343	568	9	b	b	X
ejpam-4343	568	10	be	be	AUX
ejpam-4343	568	11	any	any	DET
ejpam-4343	568	12	subset	subset	NOUN
ejpam-4343	568	13	of	of	ADP
ejpam-4343	568	14	y	y	PROPN
ejpam-4343	568	15	.	.	PUNCT
ejpam-4343	569	1	by	by	ADP
ejpam-4343	569	2	(	(	PUNCT
ejpam-4343	569	3	4	4	NUM
ejpam-4343	569	4	)	)	PUNCT
ejpam-4343	569	5	,	,	PUNCT
ejpam-4343	569	6	we	we	PRON
ejpam-4343	569	7	have	have	VERB
ejpam-4343	569	8	x	x	PART
ejpam-4343	569	9	−	−	PROPN
ejpam-4343	570	1	[	[	X
ejpam-4343	570	2	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	570	3	,	,	PUNCT
ejpam-4343	570	4	p(⋆	p(⋆	PROPN
ejpam-4343	570	5	)	)	PUNCT
ejpam-4343	570	6	)	)	PUNCT
ejpam-4343	571	1	=	=	PUNCT
ejpam-4343	572	1	[	[	X
ejpam-4343	572	2	x	x	X
ejpam-4343	572	3	−	−	NOUN
ejpam-4343	572	4	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	572	5	,	,	PUNCT
ejpam-4343	572	6	p(⋆	p(⋆	PROPN
ejpam-4343	572	7	)	)	PUNCT
ejpam-4343	572	8	)	)	PUNCT
ejpam-4343	573	1	=	=	PUNCT
ejpam-4343	574	1	[	[	X
ejpam-4343	574	2	f−1(y	f−1(y	PROPN
ejpam-4343	574	3	−b)](λ	−b)](λ	PROPN
ejpam-4343	574	4	,	,	PUNCT
ejpam-4343	574	5	p(⋆	p(⋆	PROPN
ejpam-4343	574	6	)	)	PUNCT
ejpam-4343	574	7	)	)	PUNCT
ejpam-4343	575	1	⊆	⊆	NUM
ejpam-4343	575	2	f−1([y	f−1([y	NOUN
ejpam-4343	575	3	−b](λ	−b](λ	NOUN
ejpam-4343	575	4	,	,	PUNCT
ejpam-4343	575	5	p(⋆	p(⋆	PROPN
ejpam-4343	575	6	)	)	PUNCT
ejpam-4343	575	7	)	)	PUNCT
ejpam-4343	575	8	)	)	PUNCT
ejpam-4343	576	1	=	=	SYM
ejpam-4343	576	2	f−1(y	f−1(y	PROPN
ejpam-4343	576	3	−b(λ	−b(λ	PROPN
ejpam-4343	576	4	,	,	PUNCT
ejpam-4343	576	5	p(⋆	p(⋆	PROPN
ejpam-4343	576	6	)	)	PUNCT
ejpam-4343	576	7	)	)	PUNCT
ejpam-4343	576	8	)	)	PUNCT
ejpam-4343	577	1	=	=	PUNCT
ejpam-4343	577	2	x	x	SYM
ejpam-4343	577	3	−	−	PROPN
ejpam-4343	577	4	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	577	5	,	,	PUNCT
ejpam-4343	577	6	p(⋆	p(⋆	PROPN
ejpam-4343	577	7	)	)	PUNCT
ejpam-4343	577	8	)	)	PUNCT
ejpam-4343	577	9	)	)	PUNCT
ejpam-4343	577	10	and	and	CCONJ
ejpam-4343	577	11	hence	hence	ADV
ejpam-4343	577	12	f−1(b(λ	f−1(b(λ	PROPN
ejpam-4343	577	13	,	,	PUNCT
ejpam-4343	577	14	p(⋆	p(⋆	PROPN
ejpam-4343	577	15	)	)	PUNCT
ejpam-4343	577	16	)	)	PUNCT
ejpam-4343	577	17	)	)	PUNCT
ejpam-4343	578	1	⊆	⊆	NUM
ejpam-4343	578	2	[	[	X
ejpam-4343	578	3	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	578	4	,	,	PUNCT
ejpam-4343	578	5	p(⋆	p(⋆	PROPN
ejpam-4343	578	6	)	)	PUNCT
ejpam-4343	578	7	)	)	PUNCT
ejpam-4343	578	8	.	.	PUNCT
ejpam-4343	579	1	c.	c.	PROPN
ejpam-4343	579	2	boonpok	boonpok	PROPN
ejpam-4343	579	3	/	/	SYM
ejpam-4343	579	4	eur	eur	PROPN
ejpam-4343	579	5	.	.	PUNCT
ejpam-4343	580	1	j.	j.	PROPN
ejpam-4343	580	2	pure	pure	PROPN
ejpam-4343	580	3	appl	appl	PROPN
ejpam-4343	580	4	.	.	PROPN
ejpam-4343	580	5	math	math	PROPN
ejpam-4343	580	6	,	,	PUNCT
ejpam-4343	580	7	15	15	NUM
ejpam-4343	580	8	(	(	PUNCT
ejpam-4343	580	9	3	3	NUM
ejpam-4343	580	10	)	)	PUNCT
ejpam-4343	580	11	(	(	PUNCT
ejpam-4343	580	12	2022	2022	NUM
ejpam-4343	580	13	)	)	PUNCT
ejpam-4343	580	14	,	,	PUNCT
ejpam-4343	580	15	1023	1023	NUM
ejpam-4343	580	16	-	-	SYM
ejpam-4343	580	17	1046	1046	NUM
ejpam-4343	580	18	1037	1037	NUM
ejpam-4343	580	19	(	(	PUNCT
ejpam-4343	580	20	5	5	NUM
ejpam-4343	580	21	)	)	PUNCT
ejpam-4343	580	22	⇒	⇒	NOUN
ejpam-4343	580	23	(	(	PUNCT
ejpam-4343	580	24	6	6	NUM
ejpam-4343	580	25	):	):	PUNCT
ejpam-4343	580	26	let	let	VERB
ejpam-4343	580	27	k	k	PRON
ejpam-4343	580	28	be	be	AUX
ejpam-4343	580	29	any	any	DET
ejpam-4343	580	30	(	(	PUNCT
ejpam-4343	580	31	λ	λ	PROPN
ejpam-4343	580	32	,	,	PUNCT
ejpam-4343	580	33	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	580	34	set	set	NOUN
ejpam-4343	580	35	of	of	ADP
ejpam-4343	580	36	y	y	PROPN
ejpam-4343	580	37	.	.	PUNCT
ejpam-4343	581	1	then	then	ADV
ejpam-4343	581	2	,	,	PUNCT
ejpam-4343	581	3	y	y	PROPN
ejpam-4343	581	4	−k	−k	NOUN
ejpam-4343	581	5	=	=	PUNCT
ejpam-4343	582	1	[	[	X
ejpam-4343	582	2	y	y	PROPN
ejpam-4343	582	3	−k](λ	−k](λ	NUM
ejpam-4343	582	4	,	,	PUNCT
ejpam-4343	582	5	p(⋆	p(⋆	PROPN
ejpam-4343	582	6	)	)	PUNCT
ejpam-4343	582	7	)	)	PUNCT
ejpam-4343	582	8	and	and	CCONJ
ejpam-4343	582	9	by	by	ADP
ejpam-4343	582	10	(	(	PUNCT
ejpam-4343	582	11	5	5	NUM
ejpam-4343	582	12	)	)	PUNCT
ejpam-4343	582	13	,	,	PUNCT
ejpam-4343	582	14	x	x	PUNCT
ejpam-4343	582	15	−	−	PROPN
ejpam-4343	582	16	f−1(k	f−1(k	PROPN
ejpam-4343	582	17	)	)	PUNCT
ejpam-4343	582	18	=	=	SYM
ejpam-4343	582	19	f−1(y	f−1(y	PROPN
ejpam-4343	582	20	−k	−k	NOUN
ejpam-4343	582	21	)	)	PUNCT
ejpam-4343	582	22	=	=	SYM
ejpam-4343	583	1	f−1([y	f−1([y	NOUN
ejpam-4343	583	2	−k](λ	−k](λ	NUM
ejpam-4343	583	3	,	,	PUNCT
ejpam-4343	583	4	p(⋆	p(⋆	PROPN
ejpam-4343	583	5	)	)	PUNCT
ejpam-4343	583	6	)	)	PUNCT
ejpam-4343	583	7	)	)	PUNCT
ejpam-4343	584	1	⊆	⊆	NUM
ejpam-4343	585	1	[	[	X
ejpam-4343	585	2	f−1(y	f−1(y	PROPN
ejpam-4343	585	3	−k)](λ	−k)](λ	PROPN
ejpam-4343	585	4	,	,	PUNCT
ejpam-4343	585	5	p(⋆	p(⋆	PROPN
ejpam-4343	585	6	)	)	PUNCT
ejpam-4343	585	7	)	)	PUNCT
ejpam-4343	586	1	=	=	PUNCT
ejpam-4343	587	1	[	[	X
ejpam-4343	587	2	x	x	X
ejpam-4343	587	3	−	−	NOUN
ejpam-4343	587	4	f−1(k)](λ	f−1(k)](λ	PROPN
ejpam-4343	587	5	,	,	PUNCT
ejpam-4343	587	6	p(⋆	p(⋆	PROPN
ejpam-4343	587	7	)	)	PUNCT
ejpam-4343	587	8	)	)	PUNCT
ejpam-4343	588	1	=	=	PUNCT
ejpam-4343	588	2	x	x	X
ejpam-4343	589	1	−	−	PROPN
ejpam-4343	589	2	[	[	X
ejpam-4343	589	3	f−1(k)](λ	f−1(k)](λ	PROPN
ejpam-4343	589	4	,	,	PUNCT
ejpam-4343	589	5	p(⋆	p(⋆	PROPN
ejpam-4343	589	6	)	)	PUNCT
ejpam-4343	589	7	)	)	PUNCT
ejpam-4343	589	8	.	.	PUNCT
ejpam-4343	590	1	thus	thus	ADV
ejpam-4343	590	2	,	,	PUNCT
ejpam-4343	590	3	[	[	X
ejpam-4343	590	4	f−1(k)](λ	f−1(k)](λ	PROPN
ejpam-4343	590	5	,	,	PUNCT
ejpam-4343	590	6	p(⋆	p(⋆	PROPN
ejpam-4343	590	7	)	)	PUNCT
ejpam-4343	590	8	)	)	PUNCT
ejpam-4343	591	1	⊆	⊆	NUM
ejpam-4343	591	2	f−1(k	f−1(k	PROPN
ejpam-4343	591	3	)	)	PUNCT
ejpam-4343	591	4	and	and	CCONJ
ejpam-4343	591	5	hence	hence	ADV
ejpam-4343	591	6	f−1(k	f−1(k	PROPN
ejpam-4343	591	7	)	)	PUNCT
ejpam-4343	591	8	is	be	AUX
ejpam-4343	591	9	(	(	PUNCT
ejpam-4343	591	10	λ	λ	X
ejpam-4343	591	11	,	,	PUNCT
ejpam-4343	591	12	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	591	13	.	.	PUNCT
ejpam-4343	592	1	(	(	PUNCT
ejpam-4343	592	2	6	6	NUM
ejpam-4343	592	3	)	)	PUNCT
ejpam-4343	592	4	⇒	⇒	NOUN
ejpam-4343	592	5	(	(	PUNCT
ejpam-4343	592	6	2	2	NUM
ejpam-4343	592	7	):	):	PUNCT
ejpam-4343	592	8	the	the	DET
ejpam-4343	592	9	proof	proof	NOUN
ejpam-4343	592	10	is	be	AUX
ejpam-4343	592	11	obvious	obvious	ADJ
ejpam-4343	592	12	.	.	PUNCT
ejpam-4343	593	1	(	(	PUNCT
ejpam-4343	593	2	2	2	X
ejpam-4343	593	3	)	)	PUNCT
ejpam-4343	593	4	⇒	⇒	NOUN
ejpam-4343	593	5	(	(	PUNCT
ejpam-4343	593	6	1	1	NUM
ejpam-4343	593	7	):	):	PUNCT
ejpam-4343	593	8	let	let	VERB
ejpam-4343	593	9	x	x	PUNCT
ejpam-4343	593	10	∈	∈	PROPN
ejpam-4343	593	11	x	x	PUNCT
ejpam-4343	593	12	and	and	CCONJ
ejpam-4343	593	13	let	let	VERB
ejpam-4343	593	14	v	v	PART
ejpam-4343	593	15	be	be	AUX
ejpam-4343	593	16	any	any	DET
ejpam-4343	593	17	(	(	PUNCT
ejpam-4343	593	18	λ	λ	NOUN
ejpam-4343	593	19	,	,	PUNCT
ejpam-4343	593	20	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	593	21	set	set	NOUN
ejpam-4343	593	22	of	of	ADP
ejpam-4343	593	23	y	y	PROPN
ejpam-4343	593	24	containing	contain	VERB
ejpam-4343	593	25	f(x	f(x	PROPN
ejpam-4343	593	26	)	)	PUNCT
ejpam-4343	593	27	.	.	PUNCT
ejpam-4343	594	1	by	by	ADP
ejpam-4343	594	2	(	(	PUNCT
ejpam-4343	594	3	2	2	NUM
ejpam-4343	594	4	)	)	PUNCT
ejpam-4343	594	5	,	,	PUNCT
ejpam-4343	594	6	x	x	PUNCT
ejpam-4343	594	7	∈	∈	PROPN
ejpam-4343	594	8	[	[	X
ejpam-4343	594	9	f−1(v	f−1(v	NOUN
ejpam-4343	594	10	)	)	PUNCT
ejpam-4343	594	11	]	]	PUNCT
ejpam-4343	594	12	(	(	PUNCT
ejpam-4343	594	13	λ	λ	X
ejpam-4343	594	14	,	,	PUNCT
ejpam-4343	594	15	p(⋆	p(⋆	PROPN
ejpam-4343	594	16	)	)	PUNCT
ejpam-4343	594	17	)	)	PUNCT
ejpam-4343	595	1	and	and	CCONJ
ejpam-4343	595	2	so	so	ADV
ejpam-4343	595	3	there	there	PRON
ejpam-4343	595	4	exists	exist	VERB
ejpam-4343	595	5	a	a	DET
ejpam-4343	595	6	(	(	PUNCT
ejpam-4343	595	7	λ	λ	NOUN
ejpam-4343	595	8	,	,	PUNCT
ejpam-4343	595	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	595	10	set	set	VERB
ejpam-4343	595	11	u	u	NOUN
ejpam-4343	595	12	of	of	ADP
ejpam-4343	595	13	x	x	PUNCT
ejpam-4343	595	14	containing	contain	VERB
ejpam-4343	595	15	x	x	PUNCT
ejpam-4343	595	16	such	such	ADJ
ejpam-4343	595	17	that	that	SCONJ
ejpam-4343	595	18	x	x	SYM
ejpam-4343	595	19	∈	∈	PROPN
ejpam-4343	595	20	u	u	NOUN
ejpam-4343	595	21	⊆	⊆	NUM
ejpam-4343	595	22	f−1(v	f−1(v	NOUN
ejpam-4343	595	23	)	)	PUNCT
ejpam-4343	595	24	;	;	PUNCT
ejpam-4343	595	25	hence	hence	ADV
ejpam-4343	595	26	f(u	f(u	PROPN
ejpam-4343	595	27	)	)	PUNCT
ejpam-4343	595	28	⊆	⊆	NUM
ejpam-4343	595	29	v	v	NOUN
ejpam-4343	595	30	.	.	PUNCT
ejpam-4343	596	1	thus	thus	ADV
ejpam-4343	596	2	,	,	PUNCT
ejpam-4343	596	3	f	f	PROPN
ejpam-4343	596	4	is	be	AUX
ejpam-4343	596	5	(	(	PUNCT
ejpam-4343	596	6	λ	λ	X
ejpam-4343	596	7	,	,	PUNCT
ejpam-4343	596	8	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	596	9	at	at	ADP
ejpam-4343	596	10	x.	x.	NOUN
ejpam-4343	596	11	this	this	PRON
ejpam-4343	596	12	shows	show	VERB
ejpam-4343	596	13	that	that	SCONJ
ejpam-4343	596	14	f	f	PROPN
ejpam-4343	596	15	is	be	AUX
ejpam-4343	596	16	(	(	PUNCT
ejpam-4343	596	17	λ	λ	INTJ
ejpam-4343	596	18	,	,	PUNCT
ejpam-4343	596	19	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	596	20	.	.	PUNCT
ejpam-4343	597	1	definition	definition	NOUN
ejpam-4343	597	2	15	15	NUM
ejpam-4343	597	3	.	.	PUNCT
ejpam-4343	598	1	an	an	DET
ejpam-4343	598	2	ideal	ideal	ADJ
ejpam-4343	598	3	topological	topological	ADJ
ejpam-4343	598	4	space	space	NOUN
ejpam-4343	598	5	(	(	PUNCT
ejpam-4343	598	6	x	x	X
ejpam-4343	598	7	,	,	PUNCT
ejpam-4343	598	8	τ	τ	PROPN
ejpam-4343	598	9	,	,	PUNCT
ejpam-4343	598	10	i	i	PROPN
ejpam-4343	598	11	)	)	PUNCT
ejpam-4343	598	12	is	be	AUX
ejpam-4343	598	13	said	say	VERB
ejpam-4343	598	14	to	to	PART
ejpam-4343	598	15	be	be	AUX
ejpam-4343	598	16	(	(	PUNCT
ejpam-4343	598	17	λ	λ	X
ejpam-4343	598	18	,	,	PUNCT
ejpam-4343	598	19	p(⋆))-connected	p(⋆))-connecte	VERB
ejpam-4343	598	20	if	if	SCONJ
ejpam-4343	598	21	x	x	PRON
ejpam-4343	598	22	can	can	AUX
ejpam-4343	598	23	not	not	PART
ejpam-4343	598	24	be	be	AUX
ejpam-4343	598	25	written	write	VERB
ejpam-4343	598	26	as	as	ADP
ejpam-4343	598	27	a	a	DET
ejpam-4343	598	28	disjoint	disjoint	NOUN
ejpam-4343	598	29	union	union	NOUN
ejpam-4343	598	30	of	of	ADP
ejpam-4343	598	31	two	two	NUM
ejpam-4343	598	32	nonempty	nonempty	ADJ
ejpam-4343	598	33	(	(	PUNCT
ejpam-4343	598	34	λ	λ	NOUN
ejpam-4343	598	35	,	,	PUNCT
ejpam-4343	598	36	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	598	37	sets	set	NOUN
ejpam-4343	598	38	of	of	ADP
ejpam-4343	598	39	x.	x.	NOUN
ejpam-4343	598	40	example	example	NOUN
ejpam-4343	599	1	5	5	NUM
ejpam-4343	599	2	.	.	PUNCT
ejpam-4343	600	1	let	let	VERB
ejpam-4343	600	2	x	x	PUNCT
ejpam-4343	600	3	=	=	PRON
ejpam-4343	600	4	{	{	PUNCT
ejpam-4343	600	5	a	a	PRON
ejpam-4343	600	6	,	,	PUNCT
ejpam-4343	600	7	b	b	NOUN
ejpam-4343	600	8	,	,	PUNCT
ejpam-4343	600	9	c	c	NOUN
ejpam-4343	600	10	}	}	PUNCT
ejpam-4343	600	11	with	with	ADP
ejpam-4343	600	12	a	a	DET
ejpam-4343	600	13	topology	topology	NOUN
ejpam-4343	600	14	τ	τ	X
ejpam-4343	600	15	=	=	SYM
ejpam-4343	600	16	{	{	PUNCT
ejpam-4343	600	17	∅	∅	NOUN
ejpam-4343	600	18	,	,	PUNCT
ejpam-4343	600	19	{	{	PUNCT
ejpam-4343	600	20	a	a	DET
ejpam-4343	600	21	,	,	PUNCT
ejpam-4343	600	22	b	b	NOUN
ejpam-4343	600	23	}	}	PUNCT
ejpam-4343	600	24	,	,	PUNCT
ejpam-4343	600	25	x	x	NOUN
ejpam-4343	600	26	}	}	PUNCT
ejpam-4343	600	27	and	and	CCONJ
ejpam-4343	600	28	an	an	DET
ejpam-4343	600	29	ideal	ideal	NOUN
ejpam-4343	600	30	i	i	X
ejpam-4343	600	31	=	=	SYM
ejpam-4343	600	32	{	{	PUNCT
ejpam-4343	600	33	∅	∅	NOUN
ejpam-4343	600	34	,	,	PUNCT
ejpam-4343	600	35	{	{	PUNCT
ejpam-4343	600	36	a	a	X
ejpam-4343	600	37	}	}	PUNCT
ejpam-4343	600	38	,	,	PUNCT
ejpam-4343	600	39	{	{	PUNCT
ejpam-4343	600	40	b	b	NOUN
ejpam-4343	600	41	}	}	PUNCT
ejpam-4343	600	42	,	,	PUNCT
ejpam-4343	600	43	{	{	PUNCT
ejpam-4343	600	44	a	a	PRON
ejpam-4343	600	45	,	,	PUNCT
ejpam-4343	600	46	b	b	NOUN
ejpam-4343	600	47	}	}	PUNCT
ejpam-4343	600	48	}	}	PUNCT
ejpam-4343	600	49	.	.	PUNCT
ejpam-4343	601	1	then	then	ADV
ejpam-4343	601	2	,	,	PUNCT
ejpam-4343	601	3	(	(	PUNCT
ejpam-4343	601	4	x	x	X
ejpam-4343	601	5	,	,	PUNCT
ejpam-4343	601	6	τ	τ	PROPN
ejpam-4343	601	7	,	,	PUNCT
ejpam-4343	601	8	i	i	PROPN
ejpam-4343	601	9	)	)	PUNCT
ejpam-4343	601	10	is	be	AUX
ejpam-4343	601	11	(	(	PUNCT
ejpam-4343	601	12	λ	λ	X
ejpam-4343	601	13	,	,	PUNCT
ejpam-4343	601	14	p(⋆))-connected	p(⋆))-connecte	VERB
ejpam-4343	601	15	.	.	PUNCT
ejpam-4343	602	1	proposition	proposition	NOUN
ejpam-4343	602	2	9	9	NUM
ejpam-4343	602	3	.	.	PUNCT
ejpam-4343	603	1	if	if	SCONJ
ejpam-4343	603	2	f	f	PROPN
ejpam-4343	603	3	:	:	PUNCT
ejpam-4343	603	4	(	(	PUNCT
ejpam-4343	603	5	x	x	X
ejpam-4343	603	6	,	,	PUNCT
ejpam-4343	603	7	τ	τ	PROPN
ejpam-4343	603	8	,	,	PUNCT
ejpam-4343	603	9	i	i	NOUN
ejpam-4343	603	10	)	)	PUNCT
ejpam-4343	603	11	→	→	PUNCT
ejpam-4343	603	12	(	(	PUNCT
ejpam-4343	603	13	y	y	PROPN
ejpam-4343	603	14	,	,	PUNCT
ejpam-4343	603	15	σ	σ	PROPN
ejpam-4343	603	16	,	,	PUNCT
ejpam-4343	603	17	j	j	PROPN
ejpam-4343	603	18	)	)	PUNCT
ejpam-4343	603	19	is	be	AUX
ejpam-4343	603	20	a	a	DET
ejpam-4343	603	21	(	(	PUNCT
ejpam-4343	603	22	λ	λ	NOUN
ejpam-4343	603	23	,	,	PUNCT
ejpam-4343	603	24	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	603	25	surjection	surjection	NOUN
ejpam-4343	603	26	and	and	CCONJ
ejpam-4343	603	27	(	(	PUNCT
ejpam-4343	603	28	x	x	X
ejpam-4343	603	29	,	,	PUNCT
ejpam-4343	603	30	τ	τ	PROPN
ejpam-4343	603	31	,	,	PUNCT
ejpam-4343	603	32	i	i	PROPN
ejpam-4343	603	33	)	)	PUNCT
ejpam-4343	603	34	is	be	AUX
ejpam-4343	603	35	(	(	PUNCT
ejpam-4343	603	36	λ	λ	X
ejpam-4343	603	37	,	,	PUNCT
ejpam-4343	603	38	p(⋆))-connected	p(⋆))-connecte	VERB
ejpam-4343	603	39	,	,	PUNCT
ejpam-4343	603	40	then	then	ADV
ejpam-4343	603	41	(	(	PUNCT
ejpam-4343	603	42	y	y	PROPN
ejpam-4343	603	43	,	,	PUNCT
ejpam-4343	603	44	σ	σ	PROPN
ejpam-4343	603	45	,	,	PUNCT
ejpam-4343	603	46	j	j	PROPN
ejpam-4343	603	47	)	)	PUNCT
ejpam-4343	603	48	is	be	AUX
ejpam-4343	603	49	(	(	PUNCT
ejpam-4343	603	50	λ	λ	X
ejpam-4343	603	51	,	,	PUNCT
ejpam-4343	603	52	p(⋆))-connected	p(⋆))-connecte	VERB
ejpam-4343	603	53	.	.	PUNCT
ejpam-4343	604	1	proof	proof	NOUN
ejpam-4343	604	2	.	.	PUNCT
ejpam-4343	605	1	suppose	suppose	VERB
ejpam-4343	605	2	that	that	SCONJ
ejpam-4343	605	3	(	(	PUNCT
ejpam-4343	605	4	y	y	PROPN
ejpam-4343	605	5	,	,	PUNCT
ejpam-4343	605	6	σ	σ	PROPN
ejpam-4343	605	7	,	,	PUNCT
ejpam-4343	605	8	j	j	PROPN
ejpam-4343	605	9	)	)	PUNCT
ejpam-4343	605	10	is	be	AUX
ejpam-4343	605	11	not	not	PART
ejpam-4343	605	12	(	(	PUNCT
ejpam-4343	605	13	λ	λ	X
ejpam-4343	605	14	,	,	PUNCT
ejpam-4343	605	15	p(⋆))-connected	p(⋆))-connecte	VERB
ejpam-4343	605	16	.	.	PUNCT
ejpam-4343	606	1	there	there	PRON
ejpam-4343	606	2	exist	exist	VERB
ejpam-4343	606	3	nonempty	nonempty	ADJ
ejpam-4343	606	4	(	(	PUNCT
ejpam-4343	606	5	λ	λ	NOUN
ejpam-4343	606	6	,	,	PUNCT
ejpam-4343	606	7	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	606	8	sets	set	VERB
ejpam-4343	606	9	u	u	NOUN
ejpam-4343	606	10	and	and	CCONJ
ejpam-4343	606	11	v	v	NOUN
ejpam-4343	606	12	of	of	ADP
ejpam-4343	606	13	y	y	PRON
ejpam-4343	606	14	such	such	ADJ
ejpam-4343	606	15	that	that	SCONJ
ejpam-4343	606	16	u	u	PROPN
ejpam-4343	606	17	∩	∩	NOUN
ejpam-4343	606	18	v	v	NOUN
ejpam-4343	606	19	=	=	NOUN
ejpam-4343	606	20	∅	∅	NOUN
ejpam-4343	606	21	and	and	CCONJ
ejpam-4343	606	22	u	u	NOUN
ejpam-4343	606	23	∪	∪	NOUN
ejpam-4343	606	24	v	v	ADP
ejpam-4343	606	25	=	=	SYM
ejpam-4343	606	26	y	y	PROPN
ejpam-4343	606	27	.	.	PUNCT
ejpam-4343	607	1	then	then	ADV
ejpam-4343	607	2	,	,	PUNCT
ejpam-4343	607	3	we	we	PRON
ejpam-4343	607	4	have	have	VERB
ejpam-4343	607	5	f−1(u)∩	f−1(u)∩	PROPN
ejpam-4343	607	6	f−1(v	f−1(v	PROPN
ejpam-4343	607	7	)	)	PUNCT
ejpam-4343	608	1	=	=	NOUN
ejpam-4343	608	2	∅	∅	NOUN
ejpam-4343	608	3	and	and	CCONJ
ejpam-4343	608	4	f−1(u)∪	f−1(u)∪	PROPN
ejpam-4343	608	5	f−1(v	f−1(v	PROPN
ejpam-4343	608	6	)	)	PUNCT
ejpam-4343	609	1	=	=	PUNCT
ejpam-4343	610	1	x.	x.	NOUN
ejpam-4343	610	2	moreover	moreover	ADV
ejpam-4343	610	3	,	,	PUNCT
ejpam-4343	610	4	f−1(u	f−1(u	PROPN
ejpam-4343	610	5	)	)	PUNCT
ejpam-4343	610	6	and	and	CCONJ
ejpam-4343	610	7	f−1(v	f−1(v	PROPN
ejpam-4343	610	8	)	)	PUNCT
ejpam-4343	610	9	are	be	AUX
ejpam-4343	610	10	nonempty	nonempty	ADJ
ejpam-4343	610	11	(	(	PUNCT
ejpam-4343	610	12	λ	λ	NOUN
ejpam-4343	610	13	,	,	PUNCT
ejpam-4343	610	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	610	15	sets	set	NOUN
ejpam-4343	610	16	of	of	ADP
ejpam-4343	610	17	x.	x.	NOUN
ejpam-4343	611	1	this	this	PRON
ejpam-4343	611	2	shows	show	VERB
ejpam-4343	611	3	that	that	SCONJ
ejpam-4343	611	4	(	(	PUNCT
ejpam-4343	611	5	x	x	X
ejpam-4343	611	6	,	,	PUNCT
ejpam-4343	611	7	τ	τ	PROPN
ejpam-4343	611	8	,	,	PUNCT
ejpam-4343	611	9	i	i	PROPN
ejpam-4343	611	10	)	)	PUNCT
ejpam-4343	611	11	is	be	AUX
ejpam-4343	611	12	not	not	PART
ejpam-4343	611	13	(	(	PUNCT
ejpam-4343	611	14	λ	λ	X
ejpam-4343	611	15	,	,	PUNCT
ejpam-4343	611	16	p(⋆))-connected	p(⋆))-connecte	VERB
ejpam-4343	611	17	.	.	PUNCT
ejpam-4343	612	1	definition	definition	NOUN
ejpam-4343	612	2	16	16	NUM
ejpam-4343	612	3	.	.	PUNCT
ejpam-4343	613	1	an	an	DET
ejpam-4343	613	2	ideal	ideal	ADJ
ejpam-4343	613	3	topological	topological	ADJ
ejpam-4343	613	4	space	space	NOUN
ejpam-4343	613	5	(	(	PUNCT
ejpam-4343	613	6	x	x	X
ejpam-4343	613	7	,	,	PUNCT
ejpam-4343	613	8	τ	τ	PROPN
ejpam-4343	613	9	,	,	PUNCT
ejpam-4343	613	10	i	i	PROPN
ejpam-4343	613	11	)	)	PUNCT
ejpam-4343	613	12	is	be	AUX
ejpam-4343	613	13	said	say	VERB
ejpam-4343	613	14	to	to	PART
ejpam-4343	613	15	be	be	AUX
ejpam-4343	613	16	(	(	PUNCT
ejpam-4343	613	17	λ	λ	X
ejpam-4343	613	18	,	,	PUNCT
ejpam-4343	613	19	p(⋆))-compact	p(⋆))-compact	VERB
ejpam-4343	613	20	if	if	SCONJ
ejpam-4343	613	21	every	every	DET
ejpam-4343	613	22	cover	cover	NOUN
ejpam-4343	613	23	of	of	ADP
ejpam-4343	613	24	x	x	PUNCT
ejpam-4343	613	25	by	by	ADP
ejpam-4343	613	26	(	(	PUNCT
ejpam-4343	613	27	λ	λ	INTJ
ejpam-4343	613	28	,	,	PUNCT
ejpam-4343	613	29	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	613	30	sets	set	NOUN
ejpam-4343	613	31	of	of	ADP
ejpam-4343	613	32	x	x	PUNCT
ejpam-4343	613	33	has	have	VERB
ejpam-4343	613	34	a	a	DET
ejpam-4343	613	35	finite	finite	ADJ
ejpam-4343	613	36	subcover	subcover	PROPN
ejpam-4343	613	37	.	.	PUNCT
ejpam-4343	614	1	proposition	proposition	NOUN
ejpam-4343	614	2	10	10	NUM
ejpam-4343	614	3	.	.	PUNCT
ejpam-4343	615	1	if	if	SCONJ
ejpam-4343	615	2	f	f	PROPN
ejpam-4343	615	3	:	:	PUNCT
ejpam-4343	615	4	(	(	PUNCT
ejpam-4343	615	5	x	x	X
ejpam-4343	615	6	,	,	PUNCT
ejpam-4343	615	7	τ	τ	PROPN
ejpam-4343	615	8	,	,	PUNCT
ejpam-4343	615	9	i	i	NOUN
ejpam-4343	615	10	)	)	PUNCT
ejpam-4343	615	11	→	→	PUNCT
ejpam-4343	615	12	(	(	PUNCT
ejpam-4343	615	13	y	y	PROPN
ejpam-4343	615	14	,	,	PUNCT
ejpam-4343	615	15	σ	σ	PROPN
ejpam-4343	615	16	,	,	PUNCT
ejpam-4343	615	17	j	j	PROPN
ejpam-4343	615	18	)	)	PUNCT
ejpam-4343	615	19	is	be	AUX
ejpam-4343	615	20	a	a	DET
ejpam-4343	615	21	(	(	PUNCT
ejpam-4343	615	22	λ	λ	NOUN
ejpam-4343	615	23	,	,	PUNCT
ejpam-4343	615	24	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	615	25	surjection	surjection	NOUN
ejpam-4343	615	26	and	and	CCONJ
ejpam-4343	615	27	(	(	PUNCT
ejpam-4343	615	28	x	x	X
ejpam-4343	615	29	,	,	PUNCT
ejpam-4343	615	30	τ	τ	PROPN
ejpam-4343	615	31	,	,	PUNCT
ejpam-4343	615	32	i	i	PROPN
ejpam-4343	615	33	)	)	PUNCT
ejpam-4343	615	34	is	be	AUX
ejpam-4343	615	35	(	(	PUNCT
ejpam-4343	615	36	λ	λ	INTJ
ejpam-4343	615	37	,	,	PUNCT
ejpam-4343	615	38	p(⋆))-compact	p(⋆))-compact	VERB
ejpam-4343	615	39	,	,	PUNCT
ejpam-4343	615	40	then	then	ADV
ejpam-4343	615	41	(	(	PUNCT
ejpam-4343	615	42	y	y	PROPN
ejpam-4343	615	43	,	,	PUNCT
ejpam-4343	615	44	σ	σ	PROPN
ejpam-4343	615	45	,	,	PUNCT
ejpam-4343	615	46	j	j	PROPN
ejpam-4343	615	47	)	)	PUNCT
ejpam-4343	615	48	is	be	AUX
ejpam-4343	615	49	(	(	PUNCT
ejpam-4343	615	50	λ	λ	INTJ
ejpam-4343	615	51	,	,	PUNCT
ejpam-4343	615	52	p(⋆))-compact	p(⋆))-compact	ADJ
ejpam-4343	615	53	.	.	PUNCT
ejpam-4343	616	1	proof	proof	NOUN
ejpam-4343	616	2	.	.	PUNCT
ejpam-4343	617	1	let	let	VERB
ejpam-4343	617	2	{	{	PUNCT
ejpam-4343	617	3	vγ	vγ	VERB
ejpam-4343	617	4	|	|	ADV
ejpam-4343	617	5	γ	γ	PROPN
ejpam-4343	617	6	∈	∈	PROPN
ejpam-4343	617	7	γ	γ	AUX
ejpam-4343	617	8	}	}	PUNCT
ejpam-4343	617	9	be	be	VERB
ejpam-4343	617	10	any	any	DET
ejpam-4343	617	11	cover	cover	NOUN
ejpam-4343	617	12	of	of	ADP
ejpam-4343	617	13	y	y	PROPN
ejpam-4343	617	14	by	by	ADP
ejpam-4343	617	15	(	(	PUNCT
ejpam-4343	617	16	λ	λ	INTJ
ejpam-4343	617	17	,	,	PUNCT
ejpam-4343	617	18	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	617	19	sets	set	NOUN
ejpam-4343	617	20	of	of	ADP
ejpam-4343	617	21	y	y	PROPN
ejpam-4343	617	22	.	.	PUNCT
ejpam-4343	618	1	since	since	SCONJ
ejpam-4343	618	2	f	f	PROPN
ejpam-4343	618	3	is	be	AUX
ejpam-4343	618	4	(	(	PUNCT
ejpam-4343	618	5	λ	λ	INTJ
ejpam-4343	618	6	,	,	PUNCT
ejpam-4343	618	7	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	618	8	,	,	PUNCT
ejpam-4343	618	9	by	by	ADP
ejpam-4343	618	10	theorem	theorem	NOUN
ejpam-4343	618	11	9	9	NUM
ejpam-4343	618	12	,	,	PUNCT
ejpam-4343	618	13	{	{	PUNCT
ejpam-4343	618	14	f−1(vγ	f−1(vγ	PROPN
ejpam-4343	618	15	)	)	PUNCT
ejpam-4343	618	16	|	|	ADV
ejpam-4343	618	17	γ	γ	PROPN
ejpam-4343	618	18	∈	∈	PROPN
ejpam-4343	618	19	γ	γ	X
ejpam-4343	618	20	}	}	PUNCT
ejpam-4343	618	21	is	be	AUX
ejpam-4343	618	22	a	a	DET
ejpam-4343	618	23	cover	cover	NOUN
ejpam-4343	618	24	of	of	ADP
ejpam-4343	618	25	x	x	PUNCT
ejpam-4343	618	26	by	by	ADP
ejpam-4343	618	27	(	(	PUNCT
ejpam-4343	618	28	λ	λ	INTJ
ejpam-4343	618	29	,	,	PUNCT
ejpam-4343	618	30	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	618	31	sets	set	NOUN
ejpam-4343	618	32	of	of	ADP
ejpam-4343	618	33	x.	x.	NOUN
ejpam-4343	618	34	thus	thus	ADV
ejpam-4343	618	35	,	,	PUNCT
ejpam-4343	618	36	there	there	PRON
ejpam-4343	618	37	exists	exist	VERB
ejpam-4343	618	38	a	a	DET
ejpam-4343	618	39	finite	finite	NOUN
ejpam-4343	618	40	subset	subset	NOUN
ejpam-4343	618	41	γ0	γ0	NOUN
ejpam-4343	618	42	of	of	ADP
ejpam-4343	618	43	γ	γ	NOUN
ejpam-4343	618	44	such	such	ADJ
ejpam-4343	618	45	that	that	SCONJ
ejpam-4343	618	46	x	x	SYM
ejpam-4343	618	47	=	=	SYM
ejpam-4343	618	48	∪{f−1(vγ	∪{f−1(vγ	PROPN
ejpam-4343	618	49	)	)	PUNCT
ejpam-4343	619	1	|	|	ADV
ejpam-4343	619	2	γ	γ	PROPN
ejpam-4343	619	3	∈	∈	PROPN
ejpam-4343	619	4	γ0	γ0	PROPN
ejpam-4343	619	5	}	}	PUNCT
ejpam-4343	619	6	.	.	PUNCT
ejpam-4343	620	1	since	since	SCONJ
ejpam-4343	620	2	f	f	PROPN
ejpam-4343	620	3	is	be	AUX
ejpam-4343	620	4	surjective	surjective	ADJ
ejpam-4343	620	5	,	,	PUNCT
ejpam-4343	620	6	y	y	PROPN
ejpam-4343	620	7	=	=	SYM
ejpam-4343	620	8	f(x	f(x	PROPN
ejpam-4343	620	9	)	)	PUNCT
ejpam-4343	620	10	=	=	PRON
ejpam-4343	621	1	∪{vγ	∪{vγ	PROPN
ejpam-4343	621	2	|	|	ADV
ejpam-4343	621	3	γ	γ	PROPN
ejpam-4343	621	4	∈	∈	PROPN
ejpam-4343	621	5	γ0	γ0	PROPN
ejpam-4343	621	6	}	}	PUNCT
ejpam-4343	621	7	.	.	PUNCT
ejpam-4343	622	1	this	this	PRON
ejpam-4343	622	2	shows	show	VERB
ejpam-4343	622	3	that	that	SCONJ
ejpam-4343	622	4	(	(	PUNCT
ejpam-4343	622	5	y	y	PROPN
ejpam-4343	622	6	,	,	PUNCT
ejpam-4343	622	7	σ	σ	PROPN
ejpam-4343	622	8	,	,	PUNCT
ejpam-4343	622	9	j	j	PROPN
ejpam-4343	622	10	)	)	PUNCT
ejpam-4343	622	11	is	be	AUX
ejpam-4343	622	12	(	(	PUNCT
ejpam-4343	622	13	λ	λ	PROPN
ejpam-4343	622	14	,	,	PUNCT
ejpam-4343	622	15	p(⋆))compact	p(⋆))compact	PROPN
ejpam-4343	622	16	.	.	PUNCT
ejpam-4343	623	1	definition	definition	NOUN
ejpam-4343	623	2	17	17	NUM
ejpam-4343	623	3	.	.	PUNCT
ejpam-4343	624	1	a	a	DET
ejpam-4343	624	2	subset	subset	NOUN
ejpam-4343	624	3	a	a	PRON
ejpam-4343	624	4	of	of	ADP
ejpam-4343	624	5	an	an	DET
ejpam-4343	624	6	ideal	ideal	ADJ
ejpam-4343	624	7	topological	topological	ADJ
ejpam-4343	624	8	space	space	NOUN
ejpam-4343	624	9	(	(	PUNCT
ejpam-4343	624	10	x	x	X
ejpam-4343	624	11	,	,	PUNCT
ejpam-4343	624	12	τ	τ	PROPN
ejpam-4343	624	13	,	,	PUNCT
ejpam-4343	624	14	i	i	PROPN
ejpam-4343	624	15	)	)	PUNCT
ejpam-4343	624	16	is	be	AUX
ejpam-4343	624	17	said	say	VERB
ejpam-4343	624	18	to	to	PART
ejpam-4343	624	19	be	be	AUX
ejpam-4343	624	20	a	a	DET
ejpam-4343	624	21	(	(	PUNCT
ejpam-4343	624	22	λ	λ	PROPN
ejpam-4343	624	23	,	,	PUNCT
ejpam-4343	624	24	p(⋆))neighbourhood	p(⋆))neighbourhood	PROPN
ejpam-4343	624	25	of	of	ADP
ejpam-4343	624	26	x	x	PRON
ejpam-4343	624	27	if	if	SCONJ
ejpam-4343	624	28	there	there	PRON
ejpam-4343	624	29	exists	exist	VERB
ejpam-4343	624	30	a	a	DET
ejpam-4343	624	31	(	(	PUNCT
ejpam-4343	624	32	λ	λ	NOUN
ejpam-4343	624	33	,	,	PUNCT
ejpam-4343	624	34	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	624	35	set	set	VERB
ejpam-4343	624	36	u	u	NOUN
ejpam-4343	624	37	of	of	ADP
ejpam-4343	624	38	x	x	SYM
ejpam-4343	624	39	such	such	ADJ
ejpam-4343	624	40	that	that	SCONJ
ejpam-4343	624	41	x	x	SYM
ejpam-4343	624	42	∈	∈	NUM
ejpam-4343	624	43	u	u	NOUN
ejpam-4343	624	44	⊆	⊆	NUM
ejpam-4343	624	45	a.	a.	NOUN
ejpam-4343	624	46	c.	c.	NOUN
ejpam-4343	624	47	boonpok	boonpok	PROPN
ejpam-4343	624	48	/	/	SYM
ejpam-4343	624	49	eur	eur	PROPN
ejpam-4343	624	50	.	.	PUNCT
ejpam-4343	625	1	j.	j.	PROPN
ejpam-4343	625	2	pure	pure	PROPN
ejpam-4343	625	3	appl	appl	PROPN
ejpam-4343	625	4	.	.	PROPN
ejpam-4343	625	5	math	math	PROPN
ejpam-4343	625	6	,	,	PUNCT
ejpam-4343	625	7	15	15	NUM
ejpam-4343	625	8	(	(	PUNCT
ejpam-4343	625	9	3	3	NUM
ejpam-4343	625	10	)	)	PUNCT
ejpam-4343	625	11	(	(	PUNCT
ejpam-4343	625	12	2022	2022	NUM
ejpam-4343	625	13	)	)	PUNCT
ejpam-4343	625	14	,	,	PUNCT
ejpam-4343	625	15	1023	1023	NUM
ejpam-4343	625	16	-	-	SYM
ejpam-4343	625	17	1046	1046	NUM
ejpam-4343	625	18	1038	1038	NUM
ejpam-4343	625	19	definition	definition	NOUN
ejpam-4343	625	20	18	18	NUM
ejpam-4343	625	21	.	.	PUNCT
ejpam-4343	626	1	let	let	VERB
ejpam-4343	626	2	a	a	DET
ejpam-4343	626	3	be	be	AUX
ejpam-4343	626	4	a	a	DET
ejpam-4343	626	5	subset	subset	NOUN
ejpam-4343	626	6	of	of	ADP
ejpam-4343	626	7	an	an	DET
ejpam-4343	626	8	ideal	ideal	ADJ
ejpam-4343	626	9	topological	topological	ADJ
ejpam-4343	626	10	space	space	NOUN
ejpam-4343	626	11	(	(	PUNCT
ejpam-4343	626	12	x	x	X
ejpam-4343	626	13	,	,	PUNCT
ejpam-4343	626	14	τ	τ	PROPN
ejpam-4343	626	15	,	,	PUNCT
ejpam-4343	626	16	i	i	NOUN
ejpam-4343	626	17	)	)	PUNCT
ejpam-4343	626	18	.	.	PUNCT
ejpam-4343	627	1	a	a	DET
ejpam-4343	627	2	subset	subset	NOUN
ejpam-4343	627	3	λ(λ	λ(λ	ADP
ejpam-4343	627	4	,	,	PUNCT
ejpam-4343	627	5	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	627	6	)	)	PUNCT
ejpam-4343	627	7	is	be	AUX
ejpam-4343	627	8	defined	define	VERB
ejpam-4343	627	9	as	as	SCONJ
ejpam-4343	627	10	follows	follow	VERB
ejpam-4343	627	11	:	:	PUNCT
ejpam-4343	627	12	λ(λ	λ(λ	ADV
ejpam-4343	627	13	,	,	PUNCT
ejpam-4343	627	14	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	627	15	)	)	PUNCT
ejpam-4343	628	1	=	=	PUNCT
ejpam-4343	629	1	∩{u	∩{u	PROPN
ejpam-4343	629	2	|	|	ADV
ejpam-4343	629	3	a	a	DET
ejpam-4343	629	4	⊆	⊆	NUM
ejpam-4343	629	5	u	u	NOUN
ejpam-4343	629	6	;	;	PUNCT
ejpam-4343	629	7	u	u	NOUN
ejpam-4343	629	8	is	be	AUX
ejpam-4343	629	9	(	(	PUNCT
ejpam-4343	629	10	λ	λ	X
ejpam-4343	629	11	,	,	PUNCT
ejpam-4343	629	12	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	629	13	}	}	PUNCT
ejpam-4343	629	14	.	.	PUNCT
ejpam-4343	630	1	proposition	proposition	NOUN
ejpam-4343	630	2	11	11	NUM
ejpam-4343	630	3	.	.	PUNCT
ejpam-4343	631	1	for	for	ADP
ejpam-4343	631	2	subsets	subset	NOUN
ejpam-4343	631	3	a	a	PRON
ejpam-4343	631	4	,	,	PUNCT
ejpam-4343	631	5	b	b	NOUN
ejpam-4343	631	6	and	and	CCONJ
ejpam-4343	631	7	cγ(γ	cγ(γ	CCONJ
ejpam-4343	631	8	∈	∈	PROPN
ejpam-4343	631	9	γ	γ	PROPN
ejpam-4343	631	10	)	)	PUNCT
ejpam-4343	631	11	of	of	ADP
ejpam-4343	631	12	an	an	DET
ejpam-4343	631	13	ideal	ideal	ADJ
ejpam-4343	631	14	topological	topological	ADJ
ejpam-4343	631	15	space	space	NOUN
ejpam-4343	631	16	(	(	PUNCT
ejpam-4343	631	17	x	x	X
ejpam-4343	631	18	,	,	PUNCT
ejpam-4343	631	19	τ	τ	PROPN
ejpam-4343	631	20	,	,	PUNCT
ejpam-4343	631	21	i	i	NOUN
ejpam-4343	631	22	)	)	PUNCT
ejpam-4343	631	23	,	,	PUNCT
ejpam-4343	631	24	the	the	DET
ejpam-4343	631	25	following	follow	VERB
ejpam-4343	631	26	properties	property	NOUN
ejpam-4343	631	27	hold	hold	VERB
ejpam-4343	631	28	:	:	PUNCT
ejpam-4343	631	29	(	(	PUNCT
ejpam-4343	631	30	1	1	X
ejpam-4343	631	31	)	)	PUNCT
ejpam-4343	631	32	a	a	DET
ejpam-4343	631	33	⊆	⊆	NUM
ejpam-4343	631	34	λ(λ	λ(λ	NOUN
ejpam-4343	631	35	,	,	PUNCT
ejpam-4343	631	36	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	631	37	)	)	PUNCT
ejpam-4343	631	38	.	.	PUNCT
ejpam-4343	632	1	(	(	PUNCT
ejpam-4343	632	2	2	2	X
ejpam-4343	632	3	)	)	PUNCT
ejpam-4343	632	4	if	if	SCONJ
ejpam-4343	632	5	a	a	DET
ejpam-4343	632	6	⊆	⊆	NUM
ejpam-4343	632	7	b	b	NOUN
ejpam-4343	632	8	,	,	PUNCT
ejpam-4343	632	9	then	then	ADV
ejpam-4343	632	10	λ(λ	λ(λ	PROPN
ejpam-4343	632	11	,	,	PUNCT
ejpam-4343	632	12	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	632	13	)	)	PUNCT
ejpam-4343	632	14	⊆	⊆	NUM
ejpam-4343	632	15	λ(λ	λ(λ	PROPN
ejpam-4343	632	16	,	,	PUNCT
ejpam-4343	632	17	p(⋆))(b	p(⋆))(b	NUM
ejpam-4343	632	18	)	)	PUNCT
ejpam-4343	632	19	.	.	PUNCT
ejpam-4343	633	1	(	(	PUNCT
ejpam-4343	633	2	3	3	X
ejpam-4343	633	3	)	)	PUNCT
ejpam-4343	633	4	λ(λ	λ(λ	ADV
ejpam-4343	633	5	,	,	PUNCT
ejpam-4343	633	6	p(⋆))(λ(λ	p(⋆))(λ(λ	ADV
ejpam-4343	633	7	,	,	PUNCT
ejpam-4343	633	8	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	633	9	)	)	PUNCT
ejpam-4343	633	10	)	)	PUNCT
ejpam-4343	634	1	=	=	PUNCT
ejpam-4343	634	2	λ(λ	λ(λ	ADV
ejpam-4343	634	3	,	,	PUNCT
ejpam-4343	634	4	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	634	5	)	)	PUNCT
ejpam-4343	634	6	.	.	PUNCT
ejpam-4343	635	1	(	(	PUNCT
ejpam-4343	635	2	4	4	X
ejpam-4343	635	3	)	)	PUNCT
ejpam-4343	635	4	if	if	SCONJ
ejpam-4343	635	5	a	a	PRON
ejpam-4343	635	6	is	be	AUX
ejpam-4343	635	7	a	a	DET
ejpam-4343	635	8	(	(	PUNCT
ejpam-4343	635	9	λ	λ	PROPN
ejpam-4343	635	10	,	,	PUNCT
ejpam-4343	635	11	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	635	12	set	set	NOUN
ejpam-4343	635	13	,	,	PUNCT
ejpam-4343	635	14	then	then	ADV
ejpam-4343	635	15	λ(λ	λ(λ	PROPN
ejpam-4343	635	16	,	,	PUNCT
ejpam-4343	635	17	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	635	18	)	)	PUNCT
ejpam-4343	635	19	=	=	SYM
ejpam-4343	635	20	a.	a.	NOUN
ejpam-4343	635	21	(	(	PUNCT
ejpam-4343	635	22	5	5	NUM
ejpam-4343	635	23	)	)	PUNCT
ejpam-4343	635	24	λ(λ	λ(λ	ADV
ejpam-4343	635	25	,	,	PUNCT
ejpam-4343	635	26	p(⋆))(∩{cγ	p(⋆))(∩{cγ	NOUN
ejpam-4343	635	27	|γ	|γ	ADP
ejpam-4343	635	28	∈	∈	PROPN
ejpam-4343	635	29	γ	γ	PROPN
ejpam-4343	635	30	}	}	PUNCT
ejpam-4343	635	31	)	)	PUNCT
ejpam-4343	636	1	⊆	⊆	NUM
ejpam-4343	636	2	∩{λ(λ	∩{λ(λ	NOUN
ejpam-4343	636	3	,	,	PUNCT
ejpam-4343	636	4	p(⋆))(cγ)|γ	p(⋆))(cγ)|γ	PROPN
ejpam-4343	636	5	∈	∈	PROPN
ejpam-4343	636	6	γ	γ	X
ejpam-4343	636	7	}	}	PUNCT
ejpam-4343	636	8	.	.	PUNCT
ejpam-4343	637	1	(	(	PUNCT
ejpam-4343	637	2	6	6	NUM
ejpam-4343	637	3	)	)	PUNCT
ejpam-4343	637	4	λ(λ	λ(λ	ADV
ejpam-4343	637	5	,	,	PUNCT
ejpam-4343	637	6	p(⋆))(∪{cγ	p(⋆))(∪{cγ	PROPN
ejpam-4343	637	7	|γ	|γ	ADP
ejpam-4343	637	8	∈	∈	PROPN
ejpam-4343	637	9	γ	γ	X
ejpam-4343	637	10	}	}	PUNCT
ejpam-4343	637	11	)	)	PUNCT
ejpam-4343	637	12	=	=	SYM
ejpam-4343	637	13	∪{λ(λ	∪{λ(λ	NUM
ejpam-4343	637	14	,	,	PUNCT
ejpam-4343	637	15	p(⋆))(cγ)|γ	p(⋆))(cγ)|γ	PROPN
ejpam-4343	637	16	∈	∈	PROPN
ejpam-4343	637	17	γ	γ	X
ejpam-4343	637	18	}	}	PUNCT
ejpam-4343	637	19	.	.	PUNCT
ejpam-4343	638	1	lemma	lemma	PROPN
ejpam-4343	638	2	6	6	NUM
ejpam-4343	638	3	.	.	PUNCT
ejpam-4343	639	1	let	let	VERB
ejpam-4343	639	2	a	a	DET
ejpam-4343	639	3	be	be	AUX
ejpam-4343	639	4	a	a	DET
ejpam-4343	639	5	subset	subset	NOUN
ejpam-4343	639	6	of	of	ADP
ejpam-4343	639	7	an	an	DET
ejpam-4343	639	8	ideal	ideal	ADJ
ejpam-4343	639	9	topological	topological	ADJ
ejpam-4343	639	10	space	space	NOUN
ejpam-4343	639	11	(	(	PUNCT
ejpam-4343	639	12	x	x	X
ejpam-4343	639	13	,	,	PUNCT
ejpam-4343	639	14	τ	τ	PROPN
ejpam-4343	639	15	,	,	PUNCT
ejpam-4343	639	16	i	i	PROPN
ejpam-4343	639	17	)	)	PUNCT
ejpam-4343	640	1	and	and	CCONJ
ejpam-4343	640	2	x	x	PUNCT
ejpam-4343	640	3	∈	∈	PROPN
ejpam-4343	640	4	x.	x.	NOUN
ejpam-4343	640	5	then	then	ADV
ejpam-4343	640	6	,	,	PUNCT
ejpam-4343	640	7	x	x	PUNCT
ejpam-4343	640	8	∈	∈	PROPN
ejpam-4343	640	9	λ(λ	λ(λ	PROPN
ejpam-4343	640	10	,	,	PUNCT
ejpam-4343	640	11	p(⋆))(a	p(⋆))(a	NOUN
ejpam-4343	640	12	)	)	PUNCT
ejpam-4343	641	1	if	if	SCONJ
ejpam-4343	641	2	and	and	CCONJ
ejpam-4343	641	3	only	only	ADV
ejpam-4343	641	4	if	if	SCONJ
ejpam-4343	641	5	a∩f	a∩f	PROPN
ejpam-4343	641	6	̸=	̸=	PROPN
ejpam-4343	641	7	∅	∅	NOUN
ejpam-4343	641	8	for	for	ADP
ejpam-4343	641	9	every	every	DET
ejpam-4343	641	10	(	(	PUNCT
ejpam-4343	641	11	λ	λ	PROPN
ejpam-4343	641	12	,	,	PUNCT
ejpam-4343	641	13	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	641	14	set	set	VERB
ejpam-4343	641	15	f	f	PROPN
ejpam-4343	641	16	of	of	ADP
ejpam-4343	641	17	x	x	PUNCT
ejpam-4343	641	18	with	with	ADP
ejpam-4343	641	19	x	x	PROPN
ejpam-4343	641	20	∈	∈	PROPN
ejpam-4343	641	21	f	f	PROPN
ejpam-4343	641	22	.	.	PUNCT
ejpam-4343	642	1	theorem	theorem	ADJ
ejpam-4343	642	2	10	10	NUM
ejpam-4343	642	3	.	.	PUNCT
ejpam-4343	643	1	for	for	ADP
ejpam-4343	643	2	a	a	DET
ejpam-4343	643	3	function	function	NOUN
ejpam-4343	643	4	f	f	NOUN
ejpam-4343	643	5	:	:	PUNCT
ejpam-4343	643	6	(	(	PUNCT
ejpam-4343	643	7	x	x	X
ejpam-4343	643	8	,	,	PUNCT
ejpam-4343	643	9	τ	τ	PROPN
ejpam-4343	643	10	,	,	PUNCT
ejpam-4343	643	11	i	i	NOUN
ejpam-4343	643	12	)	)	PUNCT
ejpam-4343	643	13	→	→	PUNCT
ejpam-4343	643	14	(	(	PUNCT
ejpam-4343	643	15	y	y	PROPN
ejpam-4343	643	16	,	,	PUNCT
ejpam-4343	643	17	σ	σ	PROPN
ejpam-4343	643	18	,	,	PUNCT
ejpam-4343	643	19	j	j	PROPN
ejpam-4343	643	20	)	)	PUNCT
ejpam-4343	643	21	,	,	PUNCT
ejpam-4343	643	22	the	the	DET
ejpam-4343	643	23	following	follow	VERB
ejpam-4343	643	24	properties	property	NOUN
ejpam-4343	643	25	are	be	AUX
ejpam-4343	643	26	equivalent	equivalent	ADJ
ejpam-4343	643	27	:	:	PUNCT
ejpam-4343	643	28	(	(	PUNCT
ejpam-4343	643	29	1	1	X
ejpam-4343	643	30	)	)	PUNCT
ejpam-4343	643	31	f	f	PROPN
ejpam-4343	643	32	is	be	AUX
ejpam-4343	643	33	(	(	PUNCT
ejpam-4343	643	34	λ	λ	INTJ
ejpam-4343	643	35	,	,	PUNCT
ejpam-4343	643	36	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	643	37	.	.	PUNCT
ejpam-4343	644	1	(	(	PUNCT
ejpam-4343	644	2	2	2	NUM
ejpam-4343	644	3	)	)	PUNCT
ejpam-4343	644	4	for	for	ADP
ejpam-4343	644	5	each	each	DET
ejpam-4343	644	6	x	x	SYM
ejpam-4343	644	7	∈	∈	PROPN
ejpam-4343	644	8	x	x	X
ejpam-4343	644	9	and	and	CCONJ
ejpam-4343	644	10	each	each	DET
ejpam-4343	644	11	(	(	PUNCT
ejpam-4343	644	12	λ	λ	PROPN
ejpam-4343	644	13	,	,	PUNCT
ejpam-4343	644	14	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	644	15	set	set	VERB
ejpam-4343	644	16	v	v	NOUN
ejpam-4343	644	17	of	of	ADP
ejpam-4343	644	18	y	y	PRON
ejpam-4343	644	19	such	such	ADJ
ejpam-4343	644	20	that	that	SCONJ
ejpam-4343	644	21	f(x	f(x	PROPN
ejpam-4343	644	22	)	)	PUNCT
ejpam-4343	644	23	∈	∈	PROPN
ejpam-4343	644	24	v	v	NOUN
ejpam-4343	644	25	,	,	PUNCT
ejpam-4343	644	26	f−1(v	f−1(v	PROPN
ejpam-4343	644	27	)	)	PUNCT
ejpam-4343	644	28	is	be	AUX
ejpam-4343	644	29	a	a	DET
ejpam-4343	644	30	(	(	PUNCT
ejpam-4343	644	31	λ	λ	NOUN
ejpam-4343	644	32	,	,	PUNCT
ejpam-4343	644	33	p(⋆))-neighbourhood	p(⋆))-neighbourhood	NOUN
ejpam-4343	644	34	of	of	ADP
ejpam-4343	644	35	x.	x.	PROPN
ejpam-4343	644	36	(	(	PUNCT
ejpam-4343	644	37	3	3	NUM
ejpam-4343	644	38	)	)	PUNCT
ejpam-4343	644	39	f(a(λ	f(a(λ	NOUN
ejpam-4343	644	40	,	,	PUNCT
ejpam-4343	644	41	p(⋆	p(⋆	PROPN
ejpam-4343	644	42	)	)	PUNCT
ejpam-4343	644	43	)	)	PUNCT
ejpam-4343	644	44	)	)	PUNCT
ejpam-4343	645	1	⊆	⊆	X
ejpam-4343	645	2	λ(λ	λ(λ	ADP
ejpam-4343	645	3	,	,	PUNCT
ejpam-4343	645	4	p(⋆))(f(a	p(⋆))(f(a	NOUN
ejpam-4343	645	5	)	)	PUNCT
ejpam-4343	645	6	)	)	PUNCT
ejpam-4343	645	7	for	for	ADP
ejpam-4343	645	8	every	every	DET
ejpam-4343	645	9	subset	subset	NOUN
ejpam-4343	645	10	a	a	PRON
ejpam-4343	645	11	of	of	ADP
ejpam-4343	645	12	x.	x.	NOUN
ejpam-4343	645	13	(	(	PUNCT
ejpam-4343	645	14	4	4	NUM
ejpam-4343	645	15	)	)	PUNCT
ejpam-4343	646	1	[	[	X
ejpam-4343	646	2	f−1(b)](λ	f−1(b)](λ	X
ejpam-4343	646	3	,	,	PUNCT
ejpam-4343	646	4	p(⋆	p(⋆	PROPN
ejpam-4343	646	5	)	)	PUNCT
ejpam-4343	646	6	)	)	PUNCT
ejpam-4343	647	1	⊆	⊆	NUM
ejpam-4343	647	2	f−1(λ(λ	f−1(λ(λ	NOUN
ejpam-4343	647	3	,	,	PUNCT
ejpam-4343	647	4	p(⋆))(b	p(⋆))(b	NUM
ejpam-4343	647	5	)	)	PUNCT
ejpam-4343	647	6	)	)	PUNCT
ejpam-4343	647	7	for	for	ADP
ejpam-4343	647	8	every	every	DET
ejpam-4343	647	9	subset	subset	NOUN
ejpam-4343	647	10	b	b	PROPN
ejpam-4343	647	11	of	of	ADP
ejpam-4343	647	12	y	y	PROPN
ejpam-4343	647	13	.	.	PUNCT
ejpam-4343	648	1	proof	proof	NOUN
ejpam-4343	648	2	.	.	PUNCT
ejpam-4343	649	1	(	(	PUNCT
ejpam-4343	649	2	1	1	X
ejpam-4343	649	3	)	)	PUNCT
ejpam-4343	649	4	⇒	⇒	NOUN
ejpam-4343	649	5	(	(	PUNCT
ejpam-4343	649	6	2	2	NUM
ejpam-4343	649	7	):	):	PUNCT
ejpam-4343	649	8	let	let	VERB
ejpam-4343	649	9	x	x	PUNCT
ejpam-4343	649	10	∈	∈	PROPN
ejpam-4343	649	11	x	x	PUNCT
ejpam-4343	649	12	and	and	CCONJ
ejpam-4343	649	13	let	let	VERB
ejpam-4343	649	14	v	v	PART
ejpam-4343	649	15	be	be	AUX
ejpam-4343	649	16	any	any	DET
ejpam-4343	649	17	(	(	PUNCT
ejpam-4343	649	18	λ	λ	NOUN
ejpam-4343	649	19	,	,	PUNCT
ejpam-4343	649	20	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	649	21	set	set	NOUN
ejpam-4343	649	22	of	of	ADP
ejpam-4343	649	23	y	y	PRON
ejpam-4343	649	24	such	such	ADJ
ejpam-4343	649	25	that	that	SCONJ
ejpam-4343	649	26	f(x	f(x	PROPN
ejpam-4343	649	27	)	)	PUNCT
ejpam-4343	649	28	∈	∈	PROPN
ejpam-4343	649	29	v	v	NOUN
ejpam-4343	649	30	.	.	PUNCT
ejpam-4343	650	1	since	since	SCONJ
ejpam-4343	650	2	f	f	PROPN
ejpam-4343	650	3	is	be	AUX
ejpam-4343	650	4	(	(	PUNCT
ejpam-4343	650	5	λ	λ	INTJ
ejpam-4343	650	6	,	,	PUNCT
ejpam-4343	650	7	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	650	8	,	,	PUNCT
ejpam-4343	650	9	there	there	PRON
ejpam-4343	650	10	exists	exist	VERB
ejpam-4343	650	11	a	a	DET
ejpam-4343	650	12	(	(	PUNCT
ejpam-4343	650	13	λ	λ	NOUN
ejpam-4343	650	14	,	,	PUNCT
ejpam-4343	650	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	650	16	set	set	VERB
ejpam-4343	650	17	u	u	NOUN
ejpam-4343	650	18	of	of	ADP
ejpam-4343	650	19	x	x	PUNCT
ejpam-4343	650	20	containing	contain	VERB
ejpam-4343	650	21	x	x	PUNCT
ejpam-4343	650	22	such	such	ADJ
ejpam-4343	650	23	that	that	DET
ejpam-4343	650	24	f(u	f(u	PROPN
ejpam-4343	650	25	)	)	PUNCT
ejpam-4343	650	26	⊆	⊆	NUM
ejpam-4343	650	27	v	v	NOUN
ejpam-4343	650	28	.	.	PUNCT
ejpam-4343	651	1	thus	thus	ADV
ejpam-4343	651	2	,	,	PUNCT
ejpam-4343	651	3	x	x	PUNCT
ejpam-4343	651	4	∈	∈	PROPN
ejpam-4343	651	5	u	u	NOUN
ejpam-4343	651	6	⊆	⊆	NUM
ejpam-4343	651	7	f−1(v	f−1(v	NOUN
ejpam-4343	651	8	)	)	PUNCT
ejpam-4343	651	9	and	and	CCONJ
ejpam-4343	651	10	hence	hence	ADV
ejpam-4343	651	11	f−1(v	f−1(v	PROPN
ejpam-4343	651	12	)	)	PUNCT
ejpam-4343	651	13	is	be	AUX
ejpam-4343	651	14	a	a	DET
ejpam-4343	651	15	(	(	PUNCT
ejpam-4343	651	16	λ	λ	PROPN
ejpam-4343	651	17	,	,	PUNCT
ejpam-4343	651	18	p(⋆))neighbourhood	p(⋆))neighbourhood	PROPN
ejpam-4343	651	19	of	of	ADP
ejpam-4343	651	20	x.	x.	PROPN
ejpam-4343	651	21	(	(	PUNCT
ejpam-4343	651	22	2	2	NUM
ejpam-4343	651	23	)	)	PUNCT
ejpam-4343	651	24	⇒	⇒	NOUN
ejpam-4343	651	25	(	(	PUNCT
ejpam-4343	651	26	1	1	NUM
ejpam-4343	651	27	):	):	PUNCT
ejpam-4343	651	28	let	let	VERB
ejpam-4343	651	29	x	x	PUNCT
ejpam-4343	651	30	∈	∈	PROPN
ejpam-4343	651	31	x	x	PUNCT
ejpam-4343	651	32	and	and	CCONJ
ejpam-4343	651	33	let	let	VERB
ejpam-4343	651	34	v	v	PART
ejpam-4343	651	35	be	be	AUX
ejpam-4343	651	36	any	any	DET
ejpam-4343	651	37	(	(	PUNCT
ejpam-4343	651	38	λ	λ	NOUN
ejpam-4343	651	39	,	,	PUNCT
ejpam-4343	651	40	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	651	41	set	set	NOUN
ejpam-4343	651	42	of	of	ADP
ejpam-4343	651	43	y	y	PROPN
ejpam-4343	651	44	containing	contain	VERB
ejpam-4343	651	45	f(x	f(x	PROPN
ejpam-4343	651	46	)	)	PUNCT
ejpam-4343	651	47	.	.	PUNCT
ejpam-4343	652	1	by	by	ADP
ejpam-4343	652	2	(	(	PUNCT
ejpam-4343	652	3	2	2	NUM
ejpam-4343	652	4	)	)	PUNCT
ejpam-4343	652	5	,	,	PUNCT
ejpam-4343	652	6	f−1(v	f−1(v	PROPN
ejpam-4343	652	7	)	)	PUNCT
ejpam-4343	652	8	is	be	AUX
ejpam-4343	652	9	a	a	DET
ejpam-4343	652	10	(	(	PUNCT
ejpam-4343	652	11	λ	λ	NOUN
ejpam-4343	652	12	,	,	PUNCT
ejpam-4343	652	13	p(⋆))-neighbourhood	p(⋆))-neighbourhood	NOUN
ejpam-4343	652	14	of	of	ADP
ejpam-4343	652	15	x	x	PUNCT
ejpam-4343	652	16	and	and	CCONJ
ejpam-4343	652	17	there	there	PRON
ejpam-4343	652	18	exists	exist	VERB
ejpam-4343	652	19	a	a	DET
ejpam-4343	652	20	(	(	PUNCT
ejpam-4343	652	21	λ	λ	NOUN
ejpam-4343	652	22	,	,	PUNCT
ejpam-4343	652	23	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	652	24	set	set	VERB
ejpam-4343	652	25	u	u	NOUN
ejpam-4343	652	26	of	of	ADP
ejpam-4343	652	27	x	x	SYM
ejpam-4343	652	28	such	such	ADJ
ejpam-4343	652	29	that	that	SCONJ
ejpam-4343	652	30	x	x	SYM
ejpam-4343	652	31	∈	∈	PROPN
ejpam-4343	652	32	u	u	NOUN
ejpam-4343	652	33	⊆	⊆	NUM
ejpam-4343	652	34	f−1(v	f−1(v	NOUN
ejpam-4343	652	35	)	)	PUNCT
ejpam-4343	652	36	.	.	PUNCT
ejpam-4343	653	1	thus	thus	ADV
ejpam-4343	653	2	,	,	PUNCT
ejpam-4343	653	3	f(u	f(u	PROPN
ejpam-4343	653	4	)	)	PUNCT
ejpam-4343	653	5	⊆	⊆	NUM
ejpam-4343	653	6	v	v	NOUN
ejpam-4343	653	7	and	and	CCONJ
ejpam-4343	653	8	hence	hence	ADV
ejpam-4343	653	9	f	f	PROPN
ejpam-4343	653	10	is	be	AUX
ejpam-4343	653	11	(	(	PUNCT
ejpam-4343	653	12	λ	λ	INTJ
ejpam-4343	653	13	,	,	PUNCT
ejpam-4343	653	14	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	653	15	.	.	PUNCT
ejpam-4343	654	1	(	(	PUNCT
ejpam-4343	654	2	1	1	X
ejpam-4343	654	3	)	)	PUNCT
ejpam-4343	654	4	⇒	⇒	NOUN
ejpam-4343	654	5	(	(	PUNCT
ejpam-4343	654	6	3	3	NUM
ejpam-4343	654	7	):	):	PUNCT
ejpam-4343	654	8	let	let	VERB
ejpam-4343	654	9	a	a	PRON
ejpam-4343	654	10	be	be	AUX
ejpam-4343	654	11	any	any	DET
ejpam-4343	654	12	subset	subset	NOUN
ejpam-4343	654	13	of	of	ADP
ejpam-4343	654	14	x	x	PUNCT
ejpam-4343	654	15	and	and	CCONJ
ejpam-4343	654	16	let	let	VERB
ejpam-4343	654	17	y	y	PROPN
ejpam-4343	654	18	̸∈	̸∈	PROPN
ejpam-4343	654	19	λ(λ	λ(λ	PROPN
ejpam-4343	654	20	,	,	PUNCT
ejpam-4343	654	21	p(⋆))(f(a	p(⋆))(f(a	NOUN
ejpam-4343	654	22	)	)	PUNCT
ejpam-4343	654	23	)	)	PUNCT
ejpam-4343	654	24	.	.	PUNCT
ejpam-4343	655	1	by	by	ADP
ejpam-4343	655	2	lemma	lemma	PROPN
ejpam-4343	655	3	6	6	NUM
ejpam-4343	655	4	,	,	PUNCT
ejpam-4343	655	5	there	there	PRON
ejpam-4343	655	6	exists	exist	VERB
ejpam-4343	655	7	a	a	DET
ejpam-4343	655	8	(	(	PUNCT
ejpam-4343	655	9	λ	λ	PROPN
ejpam-4343	655	10	,	,	PUNCT
ejpam-4343	655	11	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	655	12	set	set	VERB
ejpam-4343	655	13	f	f	PROPN
ejpam-4343	655	14	of	of	ADP
ejpam-4343	655	15	y	y	PRON
ejpam-4343	655	16	such	such	ADJ
ejpam-4343	655	17	that	that	SCONJ
ejpam-4343	655	18	y	y	PROPN
ejpam-4343	655	19	∈	∈	PROPN
ejpam-4343	655	20	f	f	PROPN
ejpam-4343	655	21	and	and	CCONJ
ejpam-4343	655	22	f(a	f(a	NOUN
ejpam-4343	655	23	)	)	PUNCT
ejpam-4343	655	24	∩	∩	NOUN
ejpam-4343	655	25	f	f	X
ejpam-4343	655	26	=	=	PUNCT
ejpam-4343	655	27	∅.	∅.	NOUN
ejpam-4343	655	28	thus	thus	ADV
ejpam-4343	655	29	,	,	PUNCT
ejpam-4343	655	30	a	a	DET
ejpam-4343	655	31	∩	∩	ADJ
ejpam-4343	655	32	f−1(f	f−1(f	NOUN
ejpam-4343	655	33	)	)	PUNCT
ejpam-4343	656	1	=	=	NOUN
ejpam-4343	656	2	∅	∅	NOUN
ejpam-4343	656	3	and	and	CCONJ
ejpam-4343	656	4	hence	hence	ADV
ejpam-4343	656	5	f−1(f	f−1(f	PROPN
ejpam-4343	656	6	)	)	PUNCT
ejpam-4343	656	7	∩	∩	PROPN
ejpam-4343	656	8	a(λ	a(λ	ADV
ejpam-4343	656	9	,	,	PUNCT
ejpam-4343	656	10	p(⋆	p(⋆	PROPN
ejpam-4343	656	11	)	)	PUNCT
ejpam-4343	656	12	)	)	PUNCT
ejpam-4343	657	1	=	=	PUNCT
ejpam-4343	657	2	∅.	∅.	VERB
ejpam-4343	657	3	therefore	therefore	ADV
ejpam-4343	657	4	,	,	PUNCT
ejpam-4343	657	5	f(a(λ	f(a(λ	PROPN
ejpam-4343	657	6	,	,	PUNCT
ejpam-4343	657	7	p(⋆	p(⋆	PROPN
ejpam-4343	657	8	)	)	PUNCT
ejpam-4343	657	9	)	)	PUNCT
ejpam-4343	657	10	)	)	PUNCT
ejpam-4343	657	11	∩	∩	NOUN
ejpam-4343	657	12	f	f	X
ejpam-4343	658	1	=	=	PUNCT
ejpam-4343	658	2	∅.	∅.	PROPN
ejpam-4343	658	3	this	this	PRON
ejpam-4343	658	4	shows	show	VERB
ejpam-4343	658	5	that	that	SCONJ
ejpam-4343	658	6	y	y	PROPN
ejpam-4343	658	7	̸∈	̸∈	PROPN
ejpam-4343	658	8	f(a(λ	f(a(λ	PROPN
ejpam-4343	658	9	,	,	PUNCT
ejpam-4343	658	10	p(⋆	p(⋆	PROPN
ejpam-4343	658	11	)	)	PUNCT
ejpam-4343	658	12	)	)	PUNCT
ejpam-4343	658	13	)	)	PUNCT
ejpam-4343	658	14	.	.	PUNCT
ejpam-4343	659	1	consequently	consequently	ADV
ejpam-4343	659	2	,	,	PUNCT
ejpam-4343	659	3	we	we	PRON
ejpam-4343	659	4	obtain	obtain	VERB
ejpam-4343	659	5	f(a(λ	f(a(λ	NOUN
ejpam-4343	659	6	,	,	PUNCT
ejpam-4343	659	7	p(⋆	p(⋆	PROPN
ejpam-4343	659	8	)	)	PUNCT
ejpam-4343	659	9	)	)	PUNCT
ejpam-4343	659	10	)	)	PUNCT
ejpam-4343	660	1	⊆	⊆	X
ejpam-4343	660	2	λ(λ	λ(λ	ADP
ejpam-4343	660	3	,	,	PUNCT
ejpam-4343	660	4	p(⋆))(f(a	p(⋆))(f(a	NOUN
ejpam-4343	660	5	)	)	PUNCT
ejpam-4343	660	6	)	)	PUNCT
ejpam-4343	660	7	.	.	PUNCT
ejpam-4343	661	1	(	(	PUNCT
ejpam-4343	661	2	3	3	X
ejpam-4343	661	3	)	)	PUNCT
ejpam-4343	661	4	⇒	⇒	NOUN
ejpam-4343	661	5	(	(	PUNCT
ejpam-4343	661	6	4	4	NUM
ejpam-4343	661	7	):	):	PUNCT
ejpam-4343	661	8	let	let	VERB
ejpam-4343	661	9	b	b	X
ejpam-4343	661	10	be	be	AUX
ejpam-4343	661	11	any	any	DET
ejpam-4343	661	12	subset	subset	NOUN
ejpam-4343	661	13	of	of	ADP
ejpam-4343	661	14	y	y	PROPN
ejpam-4343	661	15	.	.	PUNCT
ejpam-4343	662	1	by	by	ADP
ejpam-4343	662	2	(	(	PUNCT
ejpam-4343	662	3	3	3	NUM
ejpam-4343	662	4	)	)	PUNCT
ejpam-4343	662	5	and	and	CCONJ
ejpam-4343	662	6	proposition	proposition	NOUN
ejpam-4343	662	7	11(2	11(2	NUM
ejpam-4343	662	8	)	)	PUNCT
ejpam-4343	662	9	,	,	PUNCT
ejpam-4343	662	10	we	we	PRON
ejpam-4343	662	11	have	have	VERB
ejpam-4343	662	12	f([f−1(b)](λ	f([f−1(b)](λ	PROPN
ejpam-4343	662	13	,	,	PUNCT
ejpam-4343	662	14	p(⋆	p(⋆	PROPN
ejpam-4343	662	15	)	)	PUNCT
ejpam-4343	662	16	)	)	PUNCT
ejpam-4343	662	17	)	)	PUNCT
ejpam-4343	663	1	⊆	⊆	NUM
ejpam-4343	663	2	λ(λ	λ(λ	NOUN
ejpam-4343	663	3	,	,	PUNCT
ejpam-4343	663	4	p(⋆))(f(f	p(⋆))(f(f	PROPN
ejpam-4343	663	5	−1(b	−1(b	NOUN
ejpam-4343	663	6	)	)	PUNCT
ejpam-4343	663	7	)	)	PUNCT
ejpam-4343	663	8	)	)	PUNCT
ejpam-4343	664	1	⊆	⊆	X
ejpam-4343	664	2	λ(λ	λ(λ	ADP
ejpam-4343	664	3	,	,	PUNCT
ejpam-4343	664	4	p(⋆))(b	p(⋆))(b	NUM
ejpam-4343	664	5	)	)	PUNCT
ejpam-4343	664	6	and	and	CCONJ
ejpam-4343	664	7	hence	hence	ADV
ejpam-4343	664	8	[	[	X
ejpam-4343	664	9	f−1(b)](λ	f−1(b)](λ	PROPN
ejpam-4343	664	10	,	,	PUNCT
ejpam-4343	664	11	p(⋆	p(⋆	PROPN
ejpam-4343	664	12	)	)	PUNCT
ejpam-4343	664	13	)	)	PUNCT
ejpam-4343	664	14	⊆	⊆	NUM
ejpam-4343	664	15	f−1(λ(λ	f−1(λ(λ	NOUN
ejpam-4343	664	16	,	,	PUNCT
ejpam-4343	664	17	p(⋆))(b	p(⋆))(b	NUM
ejpam-4343	664	18	)	)	PUNCT
ejpam-4343	664	19	)	)	PUNCT
ejpam-4343	664	20	.	.	PUNCT
ejpam-4343	665	1	(	(	PUNCT
ejpam-4343	665	2	4	4	X
ejpam-4343	665	3	)	)	PUNCT
ejpam-4343	665	4	⇒	⇒	NOUN
ejpam-4343	665	5	(	(	PUNCT
ejpam-4343	665	6	1	1	NUM
ejpam-4343	665	7	):	):	PUNCT
ejpam-4343	665	8	let	let	VERB
ejpam-4343	665	9	v	v	PART
ejpam-4343	665	10	be	be	AUX
ejpam-4343	665	11	any	any	DET
ejpam-4343	665	12	(	(	PUNCT
ejpam-4343	665	13	λ	λ	NOUN
ejpam-4343	665	14	,	,	PUNCT
ejpam-4343	665	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	665	16	set	set	NOUN
ejpam-4343	665	17	of	of	ADP
ejpam-4343	665	18	y	y	PROPN
ejpam-4343	665	19	.	.	PUNCT
ejpam-4343	666	1	by	by	ADP
ejpam-4343	666	2	(	(	PUNCT
ejpam-4343	666	3	4	4	NUM
ejpam-4343	666	4	)	)	PUNCT
ejpam-4343	666	5	and	and	CCONJ
ejpam-4343	666	6	proposition	proposition	NOUN
ejpam-4343	666	7	11(4	11(4	NUM
ejpam-4343	666	8	)	)	PUNCT
ejpam-4343	666	9	,	,	PUNCT
ejpam-4343	666	10	[	[	X
ejpam-4343	666	11	f−1(v	f−1(v	NOUN
ejpam-4343	666	12	)	)	PUNCT
ejpam-4343	666	13	]	]	PUNCT
ejpam-4343	666	14	(	(	PUNCT
ejpam-4343	666	15	λ	λ	X
ejpam-4343	666	16	,	,	PUNCT
ejpam-4343	666	17	p(⋆	p(⋆	PROPN
ejpam-4343	666	18	)	)	PUNCT
ejpam-4343	666	19	)	)	PUNCT
ejpam-4343	667	1	⊆	⊆	NUM
ejpam-4343	667	2	f−1(λ(λ	f−1(λ(λ	NOUN
ejpam-4343	667	3	,	,	PUNCT
ejpam-4343	667	4	p(⋆))(v	p(⋆))(v	X
ejpam-4343	667	5	)	)	PUNCT
ejpam-4343	667	6	)	)	PUNCT
ejpam-4343	668	1	=	=	SYM
ejpam-4343	668	2	f−1(v	f−1(v	PROPN
ejpam-4343	668	3	)	)	PUNCT
ejpam-4343	668	4	and	and	CCONJ
ejpam-4343	668	5	hence	hence	ADV
ejpam-4343	668	6	[	[	X
ejpam-4343	668	7	f−1(v	f−1(v	NOUN
ejpam-4343	668	8	)	)	PUNCT
ejpam-4343	668	9	]	]	PUNCT
ejpam-4343	668	10	(	(	PUNCT
ejpam-4343	668	11	λ	λ	X
ejpam-4343	668	12	,	,	PUNCT
ejpam-4343	668	13	p(⋆	p(⋆	PROPN
ejpam-4343	668	14	)	)	PUNCT
ejpam-4343	668	15	)	)	PUNCT
ejpam-4343	669	1	=	=	SYM
ejpam-4343	669	2	f−1(v	f−1(v	PROPN
ejpam-4343	669	3	)	)	PUNCT
ejpam-4343	669	4	.	.	PUNCT
ejpam-4343	670	1	thus	thus	ADV
ejpam-4343	670	2	,	,	PUNCT
ejpam-4343	670	3	f−1(v	f−1(v	PROPN
ejpam-4343	670	4	)	)	PUNCT
ejpam-4343	670	5	is	be	AUX
ejpam-4343	670	6	(	(	PUNCT
ejpam-4343	670	7	λ	λ	INTJ
ejpam-4343	670	8	,	,	PUNCT
ejpam-4343	670	9	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	670	10	,	,	PUNCT
ejpam-4343	670	11	by	by	ADP
ejpam-4343	670	12	theorem	theorem	NOUN
ejpam-4343	670	13	9	9	NUM
ejpam-4343	670	14	,	,	PUNCT
ejpam-4343	670	15	f	f	X
ejpam-4343	670	16	is	be	AUX
ejpam-4343	670	17	(	(	PUNCT
ejpam-4343	670	18	λ	λ	INTJ
ejpam-4343	670	19	,	,	PUNCT
ejpam-4343	670	20	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	670	21	.	.	PUNCT
ejpam-4343	671	1	c.	c.	PROPN
ejpam-4343	671	2	boonpok	boonpok	PROPN
ejpam-4343	671	3	/	/	SYM
ejpam-4343	671	4	eur	eur	PROPN
ejpam-4343	671	5	.	.	PUNCT
ejpam-4343	672	1	j.	j.	PROPN
ejpam-4343	672	2	pure	pure	PROPN
ejpam-4343	672	3	appl	appl	PROPN
ejpam-4343	672	4	.	.	PROPN
ejpam-4343	672	5	math	math	PROPN
ejpam-4343	672	6	,	,	PUNCT
ejpam-4343	672	7	15	15	NUM
ejpam-4343	672	8	(	(	PUNCT
ejpam-4343	672	9	3	3	NUM
ejpam-4343	672	10	)	)	PUNCT
ejpam-4343	672	11	(	(	PUNCT
ejpam-4343	672	12	2022	2022	NUM
ejpam-4343	672	13	)	)	PUNCT
ejpam-4343	672	14	,	,	PUNCT
ejpam-4343	672	15	1023	1023	NUM
ejpam-4343	672	16	-	-	SYM
ejpam-4343	672	17	1046	1046	NUM
ejpam-4343	672	18	1039	1039	NUM
ejpam-4343	672	19	6	6	NUM
ejpam-4343	672	20	.	.	PUNCT
ejpam-4343	673	1	some	some	DET
ejpam-4343	673	2	low	low	ADJ
ejpam-4343	673	3	separation	separation	NOUN
ejpam-4343	673	4	axioms	axiom	NOUN
ejpam-4343	673	5	we	we	PRON
ejpam-4343	673	6	begin	begin	VERB
ejpam-4343	673	7	this	this	DET
ejpam-4343	673	8	section	section	NOUN
ejpam-4343	673	9	by	by	ADP
ejpam-4343	673	10	introducing	introduce	VERB
ejpam-4343	673	11	some	some	DET
ejpam-4343	673	12	low	low	ADJ
ejpam-4343	673	13	separation	separation	NOUN
ejpam-4343	673	14	axioms	axiom	NOUN
ejpam-4343	673	15	.	.	PUNCT
ejpam-4343	674	1	definition	definition	NOUN
ejpam-4343	674	2	19	19	NUM
ejpam-4343	674	3	.	.	PUNCT
ejpam-4343	675	1	an	an	DET
ejpam-4343	675	2	ideal	ideal	ADJ
ejpam-4343	675	3	topological	topological	ADJ
ejpam-4343	675	4	space	space	NOUN
ejpam-4343	675	5	(	(	PUNCT
ejpam-4343	675	6	x	x	X
ejpam-4343	675	7	,	,	PUNCT
ejpam-4343	675	8	τ	τ	PROPN
ejpam-4343	675	9	,	,	PUNCT
ejpam-4343	675	10	i	i	PROPN
ejpam-4343	675	11	)	)	PUNCT
ejpam-4343	675	12	is	be	AUX
ejpam-4343	675	13	said	say	VERB
ejpam-4343	675	14	to	to	PART
ejpam-4343	675	15	be	be	AUX
ejpam-4343	675	16	:	:	PUNCT
ejpam-4343	675	17	(	(	PUNCT
ejpam-4343	675	18	i	i	NOUN
ejpam-4343	675	19	)	)	PUNCT
ejpam-4343	675	20	pre	pre	VERB
ejpam-4343	675	21	-	-	VERB
ejpam-4343	675	22	i	i	PRON
ejpam-4343	675	23	-t0	-t0	VERB
ejpam-4343	675	24	if	if	SCONJ
ejpam-4343	675	25	,	,	PUNCT
ejpam-4343	675	26	for	for	ADP
ejpam-4343	675	27	each	each	DET
ejpam-4343	675	28	pair	pair	NOUN
ejpam-4343	675	29	of	of	ADP
ejpam-4343	675	30	distinct	distinct	ADJ
ejpam-4343	675	31	points	point	NOUN
ejpam-4343	675	32	of	of	ADP
ejpam-4343	675	33	x	x	NOUN
ejpam-4343	675	34	,	,	PUNCT
ejpam-4343	675	35	there	there	PRON
ejpam-4343	675	36	exists	exist	VERB
ejpam-4343	675	37	a	a	DET
ejpam-4343	675	38	pre	pre	ADJ
ejpam-4343	675	39	-	-	ADJ
ejpam-4343	675	40	i	i	PRON
ejpam-4343	675	41	-open	-open	NOUN
ejpam-4343	675	42	set	set	VERB
ejpam-4343	675	43	containing	contain	VERB
ejpam-4343	675	44	one	one	NUM
ejpam-4343	675	45	of	of	ADP
ejpam-4343	675	46	the	the	DET
ejpam-4343	675	47	points	point	NOUN
ejpam-4343	675	48	but	but	CCONJ
ejpam-4343	675	49	not	not	PART
ejpam-4343	675	50	the	the	DET
ejpam-4343	675	51	other	other	ADJ
ejpam-4343	675	52	;	;	PUNCT
ejpam-4343	675	53	(	(	PUNCT
ejpam-4343	675	54	ii	ii	NOUN
ejpam-4343	675	55	)	)	PUNCT
ejpam-4343	675	56	pre	pre	NOUN
ejpam-4343	675	57	-	-	PROPN
ejpam-4343	675	58	i	i	PRON
ejpam-4343	675	59	-t1	-t1	VERB
ejpam-4343	675	60	if	if	SCONJ
ejpam-4343	675	61	,	,	PUNCT
ejpam-4343	675	62	for	for	ADP
ejpam-4343	675	63	each	each	DET
ejpam-4343	675	64	pair	pair	NOUN
ejpam-4343	675	65	of	of	ADP
ejpam-4343	675	66	distinct	distinct	ADJ
ejpam-4343	675	67	points	point	NOUN
ejpam-4343	675	68	x	x	PUNCT
ejpam-4343	675	69	and	and	CCONJ
ejpam-4343	675	70	y	y	PROPN
ejpam-4343	675	71	of	of	ADP
ejpam-4343	675	72	x	x	PRON
ejpam-4343	675	73	,	,	PUNCT
ejpam-4343	675	74	there	there	PRON
ejpam-4343	675	75	exists	exist	VERB
ejpam-4343	675	76	a	a	DET
ejpam-4343	675	77	pair	pair	NOUN
ejpam-4343	675	78	of	of	ADP
ejpam-4343	675	79	pre	pre	ADJ
ejpam-4343	675	80	-	-	ADJ
ejpam-4343	675	81	i	i	PRON
ejpam-4343	675	82	-open	-open	NOUN
ejpam-4343	675	83	sets	set	VERB
ejpam-4343	675	84	one	one	NUM
ejpam-4343	675	85	containing	contain	VERB
ejpam-4343	675	86	x	x	PUNCT
ejpam-4343	675	87	but	but	CCONJ
ejpam-4343	675	88	not	not	PART
ejpam-4343	675	89	y	y	PROPN
ejpam-4343	675	90	and	and	CCONJ
ejpam-4343	675	91	the	the	DET
ejpam-4343	675	92	other	other	ADJ
ejpam-4343	675	93	containing	contain	VERB
ejpam-4343	675	94	y	y	PROPN
ejpam-4343	675	95	but	but	CCONJ
ejpam-4343	675	96	not	not	PART
ejpam-4343	675	97	x	x	ADP
ejpam-4343	675	98	;	;	PUNCT
ejpam-4343	675	99	(	(	PUNCT
ejpam-4343	675	100	iii	iii	NOUN
ejpam-4343	675	101	)	)	PUNCT
ejpam-4343	675	102	pre	pre	NOUN
ejpam-4343	675	103	-	-	PROPN
ejpam-4343	675	104	i	i	PRON
ejpam-4343	675	105	-r0	-r0	PROPN
ejpam-4343	675	106	if	if	SCONJ
ejpam-4343	675	107	every	every	DET
ejpam-4343	675	108	pre	pre	ADJ
ejpam-4343	675	109	-	-	ADJ
ejpam-4343	675	110	i	i	PRON
ejpam-4343	675	111	-open	-open	ADJ
ejpam-4343	675	112	set	set	NOUN
ejpam-4343	675	113	contains	contain	VERB
ejpam-4343	675	114	the	the	DET
ejpam-4343	675	115	pre	pre	NOUN
ejpam-4343	675	116	-	-	ADJ
ejpam-4343	675	117	i	i	ADJ
ejpam-4343	675	118	-closure	-closure	NOUN
ejpam-4343	675	119	of	of	ADP
ejpam-4343	675	120	each	each	PRON
ejpam-4343	675	121	of	of	ADP
ejpam-4343	675	122	its	its	PRON
ejpam-4343	675	123	singletons	singleton	NOUN
ejpam-4343	675	124	.	.	PUNCT
ejpam-4343	676	1	example	example	NOUN
ejpam-4343	676	2	6	6	NUM
ejpam-4343	676	3	.	.	PUNCT
ejpam-4343	677	1	let	let	VERB
ejpam-4343	677	2	x	x	PUNCT
ejpam-4343	677	3	=	=	PRON
ejpam-4343	677	4	{	{	PUNCT
ejpam-4343	677	5	a	a	PRON
ejpam-4343	677	6	,	,	PUNCT
ejpam-4343	677	7	b	b	NOUN
ejpam-4343	677	8	,	,	PUNCT
ejpam-4343	677	9	c	c	NOUN
ejpam-4343	677	10	}	}	PUNCT
ejpam-4343	677	11	with	with	ADP
ejpam-4343	677	12	a	a	DET
ejpam-4343	677	13	topology	topology	NOUN
ejpam-4343	677	14	τ	τ	X
ejpam-4343	677	15	=	=	SYM
ejpam-4343	677	16	{	{	PUNCT
ejpam-4343	677	17	∅	∅	NOUN
ejpam-4343	677	18	,	,	PUNCT
ejpam-4343	677	19	{	{	PUNCT
ejpam-4343	677	20	b	b	NOUN
ejpam-4343	677	21	}	}	PUNCT
ejpam-4343	677	22	,	,	PUNCT
ejpam-4343	677	23	{	{	PUNCT
ejpam-4343	677	24	b	b	X
ejpam-4343	677	25	,	,	PUNCT
ejpam-4343	677	26	c	c	NOUN
ejpam-4343	677	27	}	}	PUNCT
ejpam-4343	677	28	,	,	PUNCT
ejpam-4343	677	29	x	x	NOUN
ejpam-4343	677	30	}	}	PUNCT
ejpam-4343	677	31	and	and	CCONJ
ejpam-4343	677	32	an	an	DET
ejpam-4343	677	33	ideal	ideal	NOUN
ejpam-4343	677	34	i	i	X
ejpam-4343	677	35	=	=	SYM
ejpam-4343	677	36	{	{	PUNCT
ejpam-4343	677	37	∅	∅	NOUN
ejpam-4343	677	38	,	,	PUNCT
ejpam-4343	677	39	{	{	PUNCT
ejpam-4343	677	40	b	b	NOUN
ejpam-4343	677	41	}	}	PUNCT
ejpam-4343	677	42	}	}	PUNCT
ejpam-4343	677	43	.	.	PUNCT
ejpam-4343	678	1	then	then	ADV
ejpam-4343	678	2	,	,	PUNCT
ejpam-4343	678	3	(	(	PUNCT
ejpam-4343	678	4	x	x	X
ejpam-4343	678	5	,	,	PUNCT
ejpam-4343	678	6	τ	τ	PROPN
ejpam-4343	678	7	,	,	PUNCT
ejpam-4343	678	8	i	i	PROPN
ejpam-4343	678	9	)	)	PUNCT
ejpam-4343	678	10	is	be	AUX
ejpam-4343	678	11	a	a	DET
ejpam-4343	678	12	pre	pre	ADJ
ejpam-4343	678	13	-	-	ADJ
ejpam-4343	678	14	t1	t1	ADJ
ejpam-4343	678	15	space	space	NOUN
ejpam-4343	678	16	.	.	PUNCT
ejpam-4343	679	1	remark	remark	PROPN
ejpam-4343	679	2	3	3	NUM
ejpam-4343	679	3	.	.	PUNCT
ejpam-4343	680	1	for	for	ADP
ejpam-4343	680	2	an	an	DET
ejpam-4343	680	3	ideal	ideal	ADJ
ejpam-4343	680	4	topological	topological	ADJ
ejpam-4343	680	5	space	space	NOUN
ejpam-4343	680	6	(	(	PUNCT
ejpam-4343	680	7	x	x	X
ejpam-4343	680	8	,	,	PUNCT
ejpam-4343	680	9	τ	τ	PROPN
ejpam-4343	680	10	,	,	PUNCT
ejpam-4343	680	11	i	i	NOUN
ejpam-4343	680	12	)	)	PUNCT
ejpam-4343	680	13	,	,	PUNCT
ejpam-4343	680	14	the	the	DET
ejpam-4343	680	15	following	follow	VERB
ejpam-4343	680	16	implications	implication	NOUN
ejpam-4343	680	17	hold	hold	VERB
ejpam-4343	680	18	:	:	PUNCT
ejpam-4343	680	19	pre	pre	ADJ
ejpam-4343	680	20	-	-	VERB
ejpam-4343	680	21	i	i	PRON
ejpam-4343	680	22	-r0	-r0	NOUN
ejpam-4343	681	1	⇐	⇐	VERB
ejpam-4343	681	2	pre	pre	ADJ
ejpam-4343	681	3	-	-	VERB
ejpam-4343	681	4	i	i	PRON
ejpam-4343	681	5	-t1	-t1	VERB
ejpam-4343	681	6	⇒	⇒	NOUN
ejpam-4343	682	1	pre	pre	VERB
ejpam-4343	682	2	-	-	VERB
ejpam-4343	682	3	i	i	PRON
ejpam-4343	682	4	-t0	-t0	VERB
ejpam-4343	682	5	.	.	PUNCT
ejpam-4343	683	1	the	the	DET
ejpam-4343	683	2	following	follow	VERB
ejpam-4343	683	3	examples	example	NOUN
ejpam-4343	683	4	show	show	VERB
ejpam-4343	683	5	that	that	SCONJ
ejpam-4343	683	6	these	these	DET
ejpam-4343	683	7	implications	implication	NOUN
ejpam-4343	683	8	are	be	AUX
ejpam-4343	683	9	not	not	PART
ejpam-4343	683	10	reversible	reversible	ADJ
ejpam-4343	683	11	.	.	PUNCT
ejpam-4343	684	1	example	example	NOUN
ejpam-4343	685	1	7	7	NUM
ejpam-4343	685	2	.	.	PUNCT
ejpam-4343	686	1	let	let	AUX
ejpam-4343	686	2	(	(	PUNCT
ejpam-4343	686	3	x	x	X
ejpam-4343	686	4	,	,	PUNCT
ejpam-4343	686	5	τ	τ	PROPN
ejpam-4343	686	6	,	,	PUNCT
ejpam-4343	686	7	i	i	PRON
ejpam-4343	686	8	)	)	PUNCT
ejpam-4343	686	9	be	be	AUX
ejpam-4343	686	10	the	the	DET
ejpam-4343	686	11	same	same	ADJ
ejpam-4343	686	12	ideal	ideal	ADJ
ejpam-4343	686	13	topological	topological	ADJ
ejpam-4343	686	14	space	space	NOUN
ejpam-4343	686	15	as	as	ADP
ejpam-4343	686	16	in	in	ADP
ejpam-4343	686	17	example	example	NOUN
ejpam-4343	686	18	2	2	NUM
ejpam-4343	686	19	.	.	PUNCT
ejpam-4343	687	1	then	then	ADV
ejpam-4343	687	2	,	,	PUNCT
ejpam-4343	687	3	(	(	PUNCT
ejpam-4343	687	4	x	x	X
ejpam-4343	687	5	,	,	PUNCT
ejpam-4343	687	6	τ	τ	PROPN
ejpam-4343	687	7	,	,	PUNCT
ejpam-4343	687	8	i	i	PROPN
ejpam-4343	687	9	)	)	PUNCT
ejpam-4343	687	10	is	be	AUX
ejpam-4343	687	11	a	a	DET
ejpam-4343	687	12	pre	pre	NOUN
ejpam-4343	687	13	-	-	ADJ
ejpam-4343	687	14	i	i	PRON
ejpam-4343	687	15	-t0	-t0	VERB
ejpam-4343	687	16	space	space	NOUN
ejpam-4343	687	17	but	but	CCONJ
ejpam-4343	687	18	(	(	PUNCT
ejpam-4343	687	19	x	x	X
ejpam-4343	687	20	,	,	PUNCT
ejpam-4343	687	21	τ	τ	PROPN
ejpam-4343	687	22	,	,	PUNCT
ejpam-4343	687	23	i	i	PROPN
ejpam-4343	687	24	)	)	PUNCT
ejpam-4343	687	25	is	be	AUX
ejpam-4343	687	26	not	not	PART
ejpam-4343	687	27	pre	pre	ADJ
ejpam-4343	687	28	-	-	ADJ
ejpam-4343	687	29	i	i	PRON
ejpam-4343	687	30	-t1	-t1	VERB
ejpam-4343	687	31	.	.	PUNCT
ejpam-4343	687	32	example	example	NOUN
ejpam-4343	688	1	8	8	NUM
ejpam-4343	688	2	.	.	PUNCT
ejpam-4343	689	1	let	let	VERB
ejpam-4343	689	2	x	x	PUNCT
ejpam-4343	689	3	=	=	PRON
ejpam-4343	689	4	{	{	PUNCT
ejpam-4343	689	5	a	a	DET
ejpam-4343	689	6	,	,	PUNCT
ejpam-4343	689	7	b	b	NOUN
ejpam-4343	689	8	}	}	PUNCT
ejpam-4343	689	9	with	with	ADP
ejpam-4343	689	10	a	a	DET
ejpam-4343	689	11	topology	topology	NOUN
ejpam-4343	689	12	τ	τ	X
ejpam-4343	689	13	=	=	SYM
ejpam-4343	689	14	{	{	PUNCT
ejpam-4343	689	15	∅	∅	NOUN
ejpam-4343	689	16	,	,	PUNCT
ejpam-4343	689	17	x	x	NOUN
ejpam-4343	689	18	}	}	PUNCT
ejpam-4343	689	19	and	and	CCONJ
ejpam-4343	689	20	an	an	DET
ejpam-4343	689	21	ideal	ideal	NOUN
ejpam-4343	690	1	i	i	X
ejpam-4343	690	2	=	=	SYM
ejpam-4343	690	3	{	{	PUNCT
ejpam-4343	690	4	∅	∅	NOUN
ejpam-4343	690	5	,	,	PUNCT
ejpam-4343	690	6	{	{	PUNCT
ejpam-4343	690	7	a	a	X
ejpam-4343	690	8	}	}	PUNCT
ejpam-4343	690	9	,	,	PUNCT
ejpam-4343	690	10	{	{	PUNCT
ejpam-4343	690	11	b	b	NOUN
ejpam-4343	690	12	}	}	PUNCT
ejpam-4343	690	13	,	,	PUNCT
ejpam-4343	690	14	x	x	NOUN
ejpam-4343	690	15	}	}	PUNCT
ejpam-4343	690	16	.	.	PUNCT
ejpam-4343	691	1	then	then	ADV
ejpam-4343	691	2	,	,	PUNCT
ejpam-4343	691	3	(	(	PUNCT
ejpam-4343	691	4	x	x	X
ejpam-4343	691	5	,	,	PUNCT
ejpam-4343	691	6	τ	τ	PROPN
ejpam-4343	691	7	,	,	PUNCT
ejpam-4343	691	8	i	i	PROPN
ejpam-4343	691	9	)	)	PUNCT
ejpam-4343	691	10	is	be	AUX
ejpam-4343	691	11	a	a	DET
ejpam-4343	691	12	pre	pre	NOUN
ejpam-4343	691	13	-	-	ADJ
ejpam-4343	691	14	i	i	PRON
ejpam-4343	691	15	-r0	-r0	NOUN
ejpam-4343	691	16	space	space	NOUN
ejpam-4343	692	1	but	but	CCONJ
ejpam-4343	692	2	(	(	PUNCT
ejpam-4343	692	3	x	x	X
ejpam-4343	692	4	,	,	PUNCT
ejpam-4343	692	5	τ	τ	PROPN
ejpam-4343	692	6	,	,	PUNCT
ejpam-4343	692	7	i	i	PROPN
ejpam-4343	692	8	)	)	PUNCT
ejpam-4343	692	9	is	be	AUX
ejpam-4343	692	10	not	not	PART
ejpam-4343	692	11	pre	pre	ADJ
ejpam-4343	692	12	-	-	ADJ
ejpam-4343	692	13	i	i	PRON
ejpam-4343	692	14	-t1	-t1	VERB
ejpam-4343	692	15	.	.	PUNCT
ejpam-4343	693	1	theorem	theorem	VERB
ejpam-4343	693	2	11	11	NUM
ejpam-4343	693	3	.	.	PUNCT
ejpam-4343	694	1	an	an	DET
ejpam-4343	694	2	ideal	ideal	ADJ
ejpam-4343	694	3	topological	topological	ADJ
ejpam-4343	694	4	space	space	NOUN
ejpam-4343	694	5	(	(	PUNCT
ejpam-4343	694	6	x	x	X
ejpam-4343	694	7	,	,	PUNCT
ejpam-4343	694	8	τ	τ	PROPN
ejpam-4343	694	9	,	,	PUNCT
ejpam-4343	694	10	i	i	PROPN
ejpam-4343	694	11	)	)	PUNCT
ejpam-4343	694	12	is	be	AUX
ejpam-4343	694	13	pre	pre	VERB
ejpam-4343	694	14	-	-	ADJ
ejpam-4343	694	15	i	i	PRON
ejpam-4343	694	16	-t0	-t0	VERB
ejpam-4343	694	17	if	if	SCONJ
ejpam-4343	695	1	and	and	CCONJ
ejpam-4343	695	2	only	only	ADV
ejpam-4343	695	3	if	if	SCONJ
ejpam-4343	695	4	,	,	PUNCT
ejpam-4343	695	5	for	for	ADP
ejpam-4343	695	6	each	each	DET
ejpam-4343	695	7	x	x	SYM
ejpam-4343	695	8	∈	∈	PROPN
ejpam-4343	695	9	x	x	NOUN
ejpam-4343	695	10	,	,	PUNCT
ejpam-4343	695	11	the	the	DET
ejpam-4343	695	12	singleton	singleton	NOUN
ejpam-4343	695	13	{	{	PUNCT
ejpam-4343	695	14	x	x	NOUN
ejpam-4343	695	15	}	}	PUNCT
ejpam-4343	695	16	is	be	AUX
ejpam-4343	695	17	(	(	PUNCT
ejpam-4343	695	18	λ	λ	X
ejpam-4343	695	19	,	,	PUNCT
ejpam-4343	695	20	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	695	21	.	.	PUNCT
ejpam-4343	696	1	proof	proof	NOUN
ejpam-4343	696	2	.	.	PUNCT
ejpam-4343	697	1	suppose	suppose	VERB
ejpam-4343	697	2	that	that	SCONJ
ejpam-4343	697	3	(	(	PUNCT
ejpam-4343	697	4	x	x	X
ejpam-4343	697	5	,	,	PUNCT
ejpam-4343	697	6	τ	τ	PROPN
ejpam-4343	697	7	,	,	PUNCT
ejpam-4343	697	8	i	i	PROPN
ejpam-4343	697	9	)	)	PUNCT
ejpam-4343	697	10	is	be	AUX
ejpam-4343	697	11	a	a	DET
ejpam-4343	697	12	pre	pre	NOUN
ejpam-4343	697	13	-	-	ADJ
ejpam-4343	697	14	i	i	PRON
ejpam-4343	697	15	-t0	-t0	PROPN
ejpam-4343	697	16	space	space	NOUN
ejpam-4343	697	17	.	.	PUNCT
ejpam-4343	698	1	for	for	ADP
ejpam-4343	698	2	each	each	DET
ejpam-4343	698	3	x	x	SYM
ejpam-4343	698	4	∈	∈	PROPN
ejpam-4343	698	5	x	x	X
ejpam-4343	698	6	,	,	PUNCT
ejpam-4343	698	7	we	we	PRON
ejpam-4343	698	8	have	have	VERB
ejpam-4343	698	9	{	{	PUNCT
ejpam-4343	698	10	x	x	NOUN
ejpam-4343	698	11	}	}	PUNCT
ejpam-4343	698	12	⊆	⊆	NUM
ejpam-4343	698	13	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	698	14	}	}	PUNCT
ejpam-4343	698	15	)	)	PUNCT
ejpam-4343	698	16	∩	∩	PROPN
ejpam-4343	698	17	pıcl({x	pıcl({x	PROPN
ejpam-4343	698	18	}	}	PUNCT
ejpam-4343	698	19	)	)	PUNCT
ejpam-4343	698	20	.	.	PUNCT
ejpam-4343	699	1	if	if	SCONJ
ejpam-4343	699	2	y	y	PROPN
ejpam-4343	699	3	̸=	̸=	PROPN
ejpam-4343	699	4	x	x	NUM
ejpam-4343	699	5	,	,	PUNCT
ejpam-4343	699	6	(	(	PUNCT
ejpam-4343	699	7	i	i	NOUN
ejpam-4343	699	8	)	)	PUNCT
ejpam-4343	699	9	there	there	PRON
ejpam-4343	699	10	exists	exist	VERB
ejpam-4343	699	11	a	a	DET
ejpam-4343	699	12	pre	pre	ADJ
ejpam-4343	699	13	-	-	ADJ
ejpam-4343	699	14	i	i	PRON
ejpam-4343	699	15	-open	-open	NOUN
ejpam-4343	699	16	set	set	VERB
ejpam-4343	699	17	u	u	NOUN
ejpam-4343	699	18	such	such	ADJ
ejpam-4343	699	19	that	that	SCONJ
ejpam-4343	699	20	y	y	PROPN
ejpam-4343	699	21	̸∈	̸∈	PROPN
ejpam-4343	699	22	u	u	PROPN
ejpam-4343	699	23	and	and	CCONJ
ejpam-4343	699	24	x	x	SYM
ejpam-4343	699	25	∈	∈	PROPN
ejpam-4343	699	26	u	u	NOUN
ejpam-4343	699	27	or	or	CCONJ
ejpam-4343	699	28	(	(	PUNCT
ejpam-4343	699	29	ii	ii	NOUN
ejpam-4343	699	30	)	)	PUNCT
ejpam-4343	699	31	there	there	PRON
ejpam-4343	699	32	exists	exist	VERB
ejpam-4343	699	33	a	a	DET
ejpam-4343	699	34	pre	pre	ADJ
ejpam-4343	699	35	-	-	ADJ
ejpam-4343	699	36	i	i	PRON
ejpam-4343	699	37	-open	-open	NOUN
ejpam-4343	699	38	set	set	VERB
ejpam-4343	699	39	v	v	ADP
ejpam-4343	699	40	such	such	ADJ
ejpam-4343	699	41	that	that	SCONJ
ejpam-4343	699	42	x	x	PART
ejpam-4343	699	43	̸∈	̸∈	PROPN
ejpam-4343	699	44	v	v	PROPN
ejpam-4343	699	45	and	and	CCONJ
ejpam-4343	699	46	y	y	PROPN
ejpam-4343	699	47	∈	∈	PROPN
ejpam-4343	699	48	v	v	NOUN
ejpam-4343	699	49	.	.	PUNCT
ejpam-4343	700	1	in	in	ADP
ejpam-4343	700	2	case	case	NOUN
ejpam-4343	700	3	of	of	ADP
ejpam-4343	700	4	(	(	PUNCT
ejpam-4343	700	5	i	i	PROPN
ejpam-4343	700	6	)	)	PUNCT
ejpam-4343	700	7	,	,	PUNCT
ejpam-4343	700	8	y	y	PROPN
ejpam-4343	700	9	̸∈	̸∈	PROPN
ejpam-4343	700	10	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	700	11	}	}	PUNCT
ejpam-4343	700	12	)	)	PUNCT
ejpam-4343	700	13	and	and	CCONJ
ejpam-4343	700	14	y	y	PROPN
ejpam-4343	700	15	̸∈	̸∈	PROPN
ejpam-4343	700	16	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	700	17	}	}	PUNCT
ejpam-4343	700	18	)	)	PUNCT
ejpam-4343	700	19	∩	∩	PROPN
ejpam-4343	700	20	pıcl({x	pıcl({x	PROPN
ejpam-4343	700	21	}	}	PUNCT
ejpam-4343	700	22	)	)	PUNCT
ejpam-4343	700	23	.	.	PUNCT
ejpam-4343	701	1	this	this	PRON
ejpam-4343	701	2	shows	show	VERB
ejpam-4343	701	3	that	that	SCONJ
ejpam-4343	701	4	{	{	PUNCT
ejpam-4343	701	5	x	x	NOUN
ejpam-4343	701	6	}	}	PUNCT
ejpam-4343	701	7	⊇	⊇	NOUN
ejpam-4343	701	8	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	701	9	}	}	PUNCT
ejpam-4343	701	10	)	)	PUNCT
ejpam-4343	701	11	∩	∩	PROPN
ejpam-4343	701	12	pıcl({x	pıcl({x	PROPN
ejpam-4343	701	13	}	}	PUNCT
ejpam-4343	701	14	)	)	PUNCT
ejpam-4343	701	15	.	.	PUNCT
ejpam-4343	702	1	in	in	ADP
ejpam-4343	702	2	case	case	NOUN
ejpam-4343	702	3	(	(	PUNCT
ejpam-4343	702	4	ii	ii	NOUN
ejpam-4343	702	5	)	)	PUNCT
ejpam-4343	702	6	,	,	PUNCT
ejpam-4343	702	7	y	y	PROPN
ejpam-4343	702	8	̸∈	̸∈	PROPN
ejpam-4343	702	9	pıcl({x	pıcl({x	PROPN
ejpam-4343	702	10	}	}	PUNCT
ejpam-4343	702	11	)	)	PUNCT
ejpam-4343	702	12	and	and	CCONJ
ejpam-4343	702	13	y	y	PROPN
ejpam-4343	702	14	̸∈	̸∈	PROPN
ejpam-4343	702	15	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	702	16	}	}	PUNCT
ejpam-4343	702	17	)	)	PUNCT
ejpam-4343	702	18	∩	∩	PROPN
ejpam-4343	702	19	pıcl({x	pıcl({x	PROPN
ejpam-4343	702	20	}	}	PUNCT
ejpam-4343	702	21	)	)	PUNCT
ejpam-4343	702	22	.	.	PUNCT
ejpam-4343	703	1	thus	thus	ADV
ejpam-4343	703	2	,	,	PUNCT
ejpam-4343	703	3	{	{	PUNCT
ejpam-4343	703	4	x	x	NOUN
ejpam-4343	703	5	}	}	PUNCT
ejpam-4343	703	6	⊇	⊇	NOUN
ejpam-4343	703	7	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	703	8	}	}	PUNCT
ejpam-4343	703	9	)	)	PUNCT
ejpam-4343	703	10	∩	∩	PROPN
ejpam-4343	703	11	pıcl({x	pıcl({x	X
ejpam-4343	703	12	}	}	PUNCT
ejpam-4343	703	13	)	)	PUNCT
ejpam-4343	703	14	and	and	CCONJ
ejpam-4343	703	15	hence	hence	ADV
ejpam-4343	703	16	{	{	PUNCT
ejpam-4343	703	17	x	x	NOUN
ejpam-4343	703	18	}	}	PUNCT
ejpam-4343	703	19	=	=	SYM
ejpam-4343	703	20	λp(⋆)({x	λp(⋆)({x	ADJ
ejpam-4343	703	21	}	}	PUNCT
ejpam-4343	703	22	)	)	PUNCT
ejpam-4343	703	23	∩	∩	PROPN
ejpam-4343	703	24	pıcl({x	pıcl({x	PROPN
ejpam-4343	703	25	}	}	PUNCT
ejpam-4343	703	26	)	)	PUNCT
ejpam-4343	703	27	.	.	PUNCT
ejpam-4343	704	1	conversely	conversely	ADV
ejpam-4343	704	2	,	,	PUNCT
ejpam-4343	704	3	suppose	suppose	VERB
ejpam-4343	704	4	that	that	SCONJ
ejpam-4343	704	5	(	(	PUNCT
ejpam-4343	704	6	x	x	X
ejpam-4343	704	7	,	,	PUNCT
ejpam-4343	704	8	τ	τ	PROPN
ejpam-4343	704	9	,	,	PUNCT
ejpam-4343	704	10	i	i	PROPN
ejpam-4343	704	11	)	)	PUNCT
ejpam-4343	704	12	is	be	AUX
ejpam-4343	704	13	not	not	PART
ejpam-4343	704	14	pre	pre	ADJ
ejpam-4343	704	15	-	-	ADJ
ejpam-4343	704	16	i	i	PRON
ejpam-4343	704	17	-t0	-t0	VERB
ejpam-4343	704	18	.	.	PUNCT
ejpam-4343	704	19	there	there	PRON
ejpam-4343	704	20	exist	exist	VERB
ejpam-4343	704	21	two	two	NUM
ejpam-4343	704	22	distinct	distinct	ADJ
ejpam-4343	704	23	points	point	NOUN
ejpam-4343	705	1	x	x	X
ejpam-4343	705	2	,	,	PUNCT
ejpam-4343	705	3	y	y	PROPN
ejpam-4343	705	4	of	of	ADP
ejpam-4343	705	5	x	x	INTJ
ejpam-4343	705	6	such	such	ADJ
ejpam-4343	705	7	that	that	SCONJ
ejpam-4343	705	8	(	(	PUNCT
ejpam-4343	705	9	i	i	NOUN
ejpam-4343	705	10	)	)	PUNCT
ejpam-4343	705	11	y	y	PROPN
ejpam-4343	705	12	∈	∈	PROPN
ejpam-4343	705	13	u	u	NOUN
ejpam-4343	705	14	for	for	ADP
ejpam-4343	705	15	every	every	DET
ejpam-4343	705	16	pre	pre	ADJ
ejpam-4343	705	17	-	-	ADJ
ejpam-4343	705	18	i	i	PRON
ejpam-4343	705	19	-open	-open	VERB
ejpam-4343	705	20	set	set	VERB
ejpam-4343	705	21	u	u	NOUN
ejpam-4343	705	22	containing	contain	VERB
ejpam-4343	705	23	x	x	PUNCT
ejpam-4343	705	24	and	and	CCONJ
ejpam-4343	705	25	(	(	PUNCT
ejpam-4343	705	26	ii	ii	NOUN
ejpam-4343	705	27	)	)	PUNCT
ejpam-4343	705	28	x	x	SYM
ejpam-4343	705	29	∈	∈	NOUN
ejpam-4343	705	30	v	v	NOUN
ejpam-4343	705	31	for	for	ADP
ejpam-4343	705	32	every	every	DET
ejpam-4343	705	33	pre	pre	ADJ
ejpam-4343	705	34	-	-	ADJ
ejpam-4343	705	35	i	i	PRON
ejpam-4343	705	36	-open	-open	NOUN
ejpam-4343	705	37	set	set	VERB
ejpam-4343	705	38	v	v	NOUN
ejpam-4343	705	39	containing	contain	VERB
ejpam-4343	705	40	y.	y.	NOUN
ejpam-4343	705	41	from	from	ADP
ejpam-4343	705	42	(	(	PUNCT
ejpam-4343	705	43	i	i	NOUN
ejpam-4343	705	44	)	)	PUNCT
ejpam-4343	705	45	and	and	CCONJ
ejpam-4343	705	46	(	(	PUNCT
ejpam-4343	705	47	ii	ii	NOUN
ejpam-4343	705	48	)	)	PUNCT
ejpam-4343	705	49	,	,	PUNCT
ejpam-4343	705	50	we	we	PRON
ejpam-4343	705	51	obtain	obtain	VERB
ejpam-4343	705	52	y	y	PROPN
ejpam-4343	705	53	∈	∈	PROPN
ejpam-4343	705	54	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	705	55	}	}	PUNCT
ejpam-4343	705	56	)	)	PUNCT
ejpam-4343	705	57	and	and	CCONJ
ejpam-4343	705	58	y	y	PROPN
ejpam-4343	705	59	∈	∈	PROPN
ejpam-4343	705	60	pıcl({x	pıcl({x	PROPN
ejpam-4343	705	61	}	}	PUNCT
ejpam-4343	705	62	)	)	PUNCT
ejpam-4343	705	63	,	,	PUNCT
ejpam-4343	705	64	respectively	respectively	ADV
ejpam-4343	705	65	.	.	PUNCT
ejpam-4343	706	1	thus	thus	ADV
ejpam-4343	706	2	,	,	PUNCT
ejpam-4343	706	3	y	y	PROPN
ejpam-4343	706	4	∈	∈	PROPN
ejpam-4343	706	5	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	706	6	}	}	PUNCT
ejpam-4343	706	7	)	)	PUNCT
ejpam-4343	706	8	∩	∩	PROPN
ejpam-4343	706	9	pıcl({x	pıcl({x	PROPN
ejpam-4343	706	10	}	}	PUNCT
ejpam-4343	706	11	)	)	PUNCT
ejpam-4343	706	12	.	.	PUNCT
ejpam-4343	707	1	by	by	ADP
ejpam-4343	707	2	theorem	theorem	NOUN
ejpam-4343	707	3	1	1	NUM
ejpam-4343	707	4	,	,	PUNCT
ejpam-4343	707	5	{	{	PUNCT
ejpam-4343	707	6	x	x	NOUN
ejpam-4343	707	7	}	}	PUNCT
ejpam-4343	707	8	=	=	SYM
ejpam-4343	707	9	λp(⋆)({x	λp(⋆)({x	ADJ
ejpam-4343	707	10	}	}	PUNCT
ejpam-4343	707	11	)	)	PUNCT
ejpam-4343	707	12	∩	∩	PROPN
ejpam-4343	707	13	pıcl({x	pıcl({x	X
ejpam-4343	707	14	}	}	PUNCT
ejpam-4343	707	15	)	)	PUNCT
ejpam-4343	707	16	since	since	SCONJ
ejpam-4343	707	17	{	{	PUNCT
ejpam-4343	707	18	x	x	X
ejpam-4343	707	19	}	}	PUNCT
ejpam-4343	707	20	is	be	AUX
ejpam-4343	707	21	(	(	PUNCT
ejpam-4343	707	22	λ	λ	X
ejpam-4343	707	23	,	,	PUNCT
ejpam-4343	707	24	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	707	25	.	.	PUNCT
ejpam-4343	708	1	this	this	PRON
ejpam-4343	708	2	is	be	AUX
ejpam-4343	708	3	contrary	contrary	ADJ
ejpam-4343	708	4	to	to	ADP
ejpam-4343	708	5	x	x	SYM
ejpam-4343	708	6	̸=	̸=	PROPN
ejpam-4343	708	7	y.	y.	PROPN
ejpam-4343	708	8	c.	c.	PROPN
ejpam-4343	708	9	boonpok	boonpok	PROPN
ejpam-4343	708	10	/	/	SYM
ejpam-4343	708	11	eur	eur	PROPN
ejpam-4343	708	12	.	.	PUNCT
ejpam-4343	709	1	j.	j.	PROPN
ejpam-4343	709	2	pure	pure	PROPN
ejpam-4343	709	3	appl	appl	PROPN
ejpam-4343	709	4	.	.	PROPN
ejpam-4343	709	5	math	math	PROPN
ejpam-4343	709	6	,	,	PUNCT
ejpam-4343	709	7	15	15	NUM
ejpam-4343	709	8	(	(	PUNCT
ejpam-4343	709	9	3	3	NUM
ejpam-4343	709	10	)	)	PUNCT
ejpam-4343	709	11	(	(	PUNCT
ejpam-4343	709	12	2022	2022	NUM
ejpam-4343	709	13	)	)	PUNCT
ejpam-4343	709	14	,	,	PUNCT
ejpam-4343	709	15	1023	1023	NUM
ejpam-4343	709	16	-	-	SYM
ejpam-4343	709	17	1046	1046	NUM
ejpam-4343	709	18	1040	1040	NUM
ejpam-4343	709	19	theorem	theorem	NOUN
ejpam-4343	709	20	12	12	NUM
ejpam-4343	709	21	.	.	PUNCT
ejpam-4343	710	1	for	for	ADP
ejpam-4343	710	2	an	an	DET
ejpam-4343	710	3	ideal	ideal	ADJ
ejpam-4343	710	4	topological	topological	ADJ
ejpam-4343	710	5	space	space	NOUN
ejpam-4343	710	6	(	(	PUNCT
ejpam-4343	710	7	x	x	X
ejpam-4343	710	8	,	,	PUNCT
ejpam-4343	710	9	τ	τ	PROPN
ejpam-4343	710	10	,	,	PUNCT
ejpam-4343	710	11	i	i	NOUN
ejpam-4343	710	12	)	)	PUNCT
ejpam-4343	710	13	,	,	PUNCT
ejpam-4343	710	14	the	the	DET
ejpam-4343	710	15	following	follow	VERB
ejpam-4343	710	16	properties	property	NOUN
ejpam-4343	710	17	are	be	AUX
ejpam-4343	710	18	equivalent	equivalent	ADJ
ejpam-4343	710	19	:	:	PUNCT
ejpam-4343	710	20	(	(	PUNCT
ejpam-4343	710	21	1	1	X
ejpam-4343	710	22	)	)	PUNCT
ejpam-4343	710	23	(	(	PUNCT
ejpam-4343	710	24	x	x	X
ejpam-4343	710	25	,	,	PUNCT
ejpam-4343	710	26	τ	τ	PROPN
ejpam-4343	710	27	,	,	PUNCT
ejpam-4343	710	28	i	i	PROPN
ejpam-4343	710	29	)	)	PUNCT
ejpam-4343	710	30	is	be	AUX
ejpam-4343	710	31	pre	pre	ADJ
ejpam-4343	710	32	-	-	ADJ
ejpam-4343	710	33	i	i	PRON
ejpam-4343	710	34	-t1	-t1	VERB
ejpam-4343	710	35	.	.	PUNCT
ejpam-4343	711	1	(	(	PUNCT
ejpam-4343	711	2	2	2	X
ejpam-4343	711	3	)	)	PUNCT
ejpam-4343	711	4	for	for	ADP
ejpam-4343	711	5	each	each	DET
ejpam-4343	711	6	x	x	SYM
ejpam-4343	711	7	∈	∈	PROPN
ejpam-4343	711	8	x	x	NOUN
ejpam-4343	711	9	,	,	PUNCT
ejpam-4343	711	10	the	the	DET
ejpam-4343	711	11	singleton	singleton	NOUN
ejpam-4343	711	12	{	{	PUNCT
ejpam-4343	711	13	x	x	NOUN
ejpam-4343	711	14	}	}	PUNCT
ejpam-4343	711	15	is	be	AUX
ejpam-4343	711	16	a	a	DET
ejpam-4343	711	17	pre	pre	ADJ
ejpam-4343	711	18	-	-	ADJ
ejpam-4343	711	19	i	i	PRON
ejpam-4343	711	20	-closed	-close	VERB
ejpam-4343	711	21	set	set	NOUN
ejpam-4343	711	22	.	.	PUNCT
ejpam-4343	712	1	(	(	PUNCT
ejpam-4343	712	2	3	3	X
ejpam-4343	712	3	)	)	PUNCT
ejpam-4343	712	4	for	for	ADP
ejpam-4343	712	5	each	each	DET
ejpam-4343	712	6	x	x	SYM
ejpam-4343	712	7	∈	∈	PROPN
ejpam-4343	712	8	x	x	NOUN
ejpam-4343	712	9	,	,	PUNCT
ejpam-4343	712	10	the	the	DET
ejpam-4343	712	11	singleton	singleton	NOUN
ejpam-4343	712	12	{	{	PUNCT
ejpam-4343	712	13	x	x	NOUN
ejpam-4343	712	14	}	}	PUNCT
ejpam-4343	712	15	is	be	AUX
ejpam-4343	712	16	a	a	DET
ejpam-4343	712	17	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	712	18	.	.	PUNCT
ejpam-4343	713	1	proof	proof	NOUN
ejpam-4343	713	2	.	.	PUNCT
ejpam-4343	714	1	(	(	PUNCT
ejpam-4343	714	2	1	1	X
ejpam-4343	714	3	)	)	PUNCT
ejpam-4343	714	4	⇒	⇒	NOUN
ejpam-4343	714	5	(	(	PUNCT
ejpam-4343	714	6	2	2	NUM
ejpam-4343	714	7	):	):	PUNCT
ejpam-4343	714	8	let	let	VERB
ejpam-4343	714	9	y	y	PROPN
ejpam-4343	714	10	∈	∈	PROPN
ejpam-4343	714	11	x	x	X
ejpam-4343	714	12	and	and	CCONJ
ejpam-4343	714	13	x	x	SYM
ejpam-4343	714	14	∈	∈	NOUN
ejpam-4343	714	15	x	x	PUNCT
ejpam-4343	714	16	−{y	−{y	NOUN
ejpam-4343	714	17	}	}	PUNCT
ejpam-4343	714	18	.	.	PUNCT
ejpam-4343	715	1	there	there	PRON
ejpam-4343	715	2	exists	exist	VERB
ejpam-4343	715	3	a	a	DET
ejpam-4343	715	4	pre	pre	ADJ
ejpam-4343	715	5	-	-	ADJ
ejpam-4343	715	6	i	i	PRON
ejpam-4343	715	7	-open	-open	NOUN
ejpam-4343	715	8	set	set	VERB
ejpam-4343	715	9	gx	gx	PROPN
ejpam-4343	715	10	such	such	ADJ
ejpam-4343	715	11	that	that	SCONJ
ejpam-4343	715	12	x	x	SYM
ejpam-4343	715	13	∈	∈	PROPN
ejpam-4343	715	14	gx	gx	PROPN
ejpam-4343	715	15	and	and	CCONJ
ejpam-4343	715	16	y	y	PROPN
ejpam-4343	715	17	̸∈	̸∈	PROPN
ejpam-4343	715	18	gx	gx	PROPN
ejpam-4343	715	19	.	.	PUNCT
ejpam-4343	716	1	therefore	therefore	ADV
ejpam-4343	716	2	,	,	PUNCT
ejpam-4343	716	3	we	we	PRON
ejpam-4343	716	4	have	have	VERB
ejpam-4343	716	5	x	x	PART
ejpam-4343	716	6	−	−	PROPN
ejpam-4343	716	7	{	{	PUNCT
ejpam-4343	716	8	y	y	NOUN
ejpam-4343	716	9	}	}	PUNCT
ejpam-4343	716	10	=	=	SYM
ejpam-4343	716	11	∪x∈x−{y}gx	∪x∈x−{y}gx	PROPN
ejpam-4343	716	12	and	and	CCONJ
ejpam-4343	716	13	hence	hence	ADV
ejpam-4343	716	14	{	{	PUNCT
ejpam-4343	716	15	y	y	NOUN
ejpam-4343	716	16	}	}	PUNCT
ejpam-4343	716	17	is	be	AUX
ejpam-4343	716	18	a	a	DET
ejpam-4343	716	19	pre	pre	ADJ
ejpam-4343	716	20	-	-	ADJ
ejpam-4343	716	21	i	i	PRON
ejpam-4343	716	22	-closed	-close	VERB
ejpam-4343	716	23	set	set	NOUN
ejpam-4343	716	24	.	.	PUNCT
ejpam-4343	717	1	(	(	PUNCT
ejpam-4343	717	2	2	2	X
ejpam-4343	717	3	)	)	PUNCT
ejpam-4343	717	4	⇒	⇒	NOUN
ejpam-4343	717	5	(	(	PUNCT
ejpam-4343	717	6	3	3	NUM
ejpam-4343	717	7	):	):	PUNCT
ejpam-4343	717	8	let	let	VERB
ejpam-4343	717	9	x	x	PUNCT
ejpam-4343	717	10	∈	∈	PROPN
ejpam-4343	717	11	x	x	X
ejpam-4343	717	12	and	and	CCONJ
ejpam-4343	717	13	y	y	PROPN
ejpam-4343	717	14	∈	∈	PROPN
ejpam-4343	717	15	x	x	PUNCT
ejpam-4343	717	16	−	−	PROPN
ejpam-4343	717	17	{	{	PUNCT
ejpam-4343	717	18	x	x	NOUN
ejpam-4343	717	19	}	}	PUNCT
ejpam-4343	717	20	.	.	PUNCT
ejpam-4343	718	1	then	then	ADV
ejpam-4343	718	2	,	,	PUNCT
ejpam-4343	718	3	we	we	PRON
ejpam-4343	718	4	have	have	VERB
ejpam-4343	718	5	x	x	X
ejpam-4343	718	6	∈	∈	PROPN
ejpam-4343	718	7	x	x	X
ejpam-4343	718	8	−	−	PROPN
ejpam-4343	718	9	{	{	PUNCT
ejpam-4343	718	10	y	y	NOUN
ejpam-4343	718	11	}	}	PUNCT
ejpam-4343	718	12	and	and	CCONJ
ejpam-4343	718	13	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	718	14	}	}	PUNCT
ejpam-4343	718	15	)	)	PUNCT
ejpam-4343	719	1	⊆	⊆	NUM
ejpam-4343	719	2	x	x	SYM
ejpam-4343	719	3	−	−	PROPN
ejpam-4343	719	4	{	{	PUNCT
ejpam-4343	719	5	y	y	NOUN
ejpam-4343	719	6	}	}	PUNCT
ejpam-4343	719	7	.	.	PUNCT
ejpam-4343	720	1	thus	thus	ADV
ejpam-4343	720	2	,	,	PUNCT
ejpam-4343	720	3	y	y	PROPN
ejpam-4343	720	4	̸∈	̸∈	PROPN
ejpam-4343	720	5	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	720	6	}	}	PUNCT
ejpam-4343	720	7	)	)	PUNCT
ejpam-4343	720	8	and	and	CCONJ
ejpam-4343	720	9	hence	hence	ADV
ejpam-4343	720	10	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	720	11	}	}	PUNCT
ejpam-4343	720	12	)	)	PUNCT
ejpam-4343	721	1	⊆	⊆	NUM
ejpam-4343	721	2	{	{	PUNCT
ejpam-4343	721	3	x	x	NOUN
ejpam-4343	721	4	}	}	PUNCT
ejpam-4343	721	5	.	.	PUNCT
ejpam-4343	722	1	this	this	PRON
ejpam-4343	722	2	implies	imply	VERB
ejpam-4343	722	3	that	that	SCONJ
ejpam-4343	722	4	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	722	5	}	}	PUNCT
ejpam-4343	722	6	)	)	PUNCT
ejpam-4343	723	1	=	=	PRON
ejpam-4343	723	2	{	{	PUNCT
ejpam-4343	723	3	x	x	NOUN
ejpam-4343	723	4	}	}	PUNCT
ejpam-4343	723	5	.	.	PUNCT
ejpam-4343	724	1	consequently	consequently	ADV
ejpam-4343	724	2	,	,	PUNCT
ejpam-4343	724	3	we	we	PRON
ejpam-4343	724	4	obtain	obtain	VERB
ejpam-4343	724	5	{	{	PUNCT
ejpam-4343	724	6	x	x	NOUN
ejpam-4343	724	7	}	}	PUNCT
ejpam-4343	724	8	is	be	AUX
ejpam-4343	724	9	a	a	DET
ejpam-4343	724	10	λp(⋆)-set	λp(⋆)-set	NOUN
ejpam-4343	724	11	.	.	PUNCT
ejpam-4343	725	1	(	(	PUNCT
ejpam-4343	725	2	3	3	X
ejpam-4343	725	3	)	)	PUNCT
ejpam-4343	725	4	⇒	⇒	NOUN
ejpam-4343	725	5	(	(	PUNCT
ejpam-4343	725	6	1	1	NUM
ejpam-4343	725	7	):	):	PUNCT
ejpam-4343	725	8	let	let	VERB
ejpam-4343	725	9	x	x	PRON
ejpam-4343	725	10	and	and	CCONJ
ejpam-4343	725	11	y	y	PROPN
ejpam-4343	725	12	be	be	AUX
ejpam-4343	725	13	any	any	DET
ejpam-4343	725	14	distinct	distinct	ADJ
ejpam-4343	725	15	points	point	NOUN
ejpam-4343	725	16	of	of	ADP
ejpam-4343	725	17	x.	x.	NOUN
ejpam-4343	725	18	then	then	ADV
ejpam-4343	725	19	,	,	PUNCT
ejpam-4343	725	20	we	we	PRON
ejpam-4343	725	21	have	have	VERB
ejpam-4343	725	22	y	y	PROPN
ejpam-4343	725	23	̸∈	̸∈	PROPN
ejpam-4343	725	24	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	725	25	}	}	PUNCT
ejpam-4343	725	26	)	)	PUNCT
ejpam-4343	725	27	and	and	CCONJ
ejpam-4343	725	28	so	so	ADV
ejpam-4343	725	29	there	there	PRON
ejpam-4343	725	30	exists	exist	VERB
ejpam-4343	725	31	a	a	DET
ejpam-4343	725	32	pre	pre	ADJ
ejpam-4343	725	33	-	-	ADJ
ejpam-4343	725	34	i	i	PRON
ejpam-4343	725	35	-open	-open	NOUN
ejpam-4343	725	36	set	set	VERB
ejpam-4343	725	37	u	u	PRON
ejpam-4343	725	38	such	such	ADJ
ejpam-4343	725	39	that	that	SCONJ
ejpam-4343	725	40	x	x	SYM
ejpam-4343	725	41	∈	∈	PROPN
ejpam-4343	725	42	u	u	NOUN
ejpam-4343	725	43	and	and	CCONJ
ejpam-4343	725	44	y	y	PROPN
ejpam-4343	725	45	̸∈	̸∈	PROPN
ejpam-4343	725	46	u	u	PROPN
ejpam-4343	725	47	.	.	PUNCT
ejpam-4343	726	1	similarly	similarly	ADV
ejpam-4343	726	2	,	,	PUNCT
ejpam-4343	726	3	x	x	PROPN
ejpam-4343	726	4	̸∈	̸∈	PROPN
ejpam-4343	726	5	λp(⋆)({y	λp(⋆)({y	PROPN
ejpam-4343	726	6	}	}	PUNCT
ejpam-4343	726	7	)	)	PUNCT
ejpam-4343	726	8	and	and	CCONJ
ejpam-4343	726	9	there	there	PRON
ejpam-4343	726	10	exists	exist	VERB
ejpam-4343	726	11	a	a	DET
ejpam-4343	726	12	pre	pre	ADJ
ejpam-4343	726	13	-	-	ADJ
ejpam-4343	726	14	i	i	PRON
ejpam-4343	726	15	-open	-open	NOUN
ejpam-4343	726	16	set	set	VERB
ejpam-4343	726	17	v	v	ADP
ejpam-4343	726	18	such	such	ADJ
ejpam-4343	726	19	that	that	SCONJ
ejpam-4343	726	20	y	y	PROPN
ejpam-4343	726	21	∈	∈	PROPN
ejpam-4343	726	22	v	v	NOUN
ejpam-4343	726	23	and	and	CCONJ
ejpam-4343	726	24	x	x	PUNCT
ejpam-4343	726	25	̸∈	̸∈	PROPN
ejpam-4343	726	26	v	v	PROPN
ejpam-4343	726	27	.	.	PUNCT
ejpam-4343	727	1	this	this	PRON
ejpam-4343	727	2	shows	show	VERB
ejpam-4343	727	3	that	that	SCONJ
ejpam-4343	727	4	(	(	PUNCT
ejpam-4343	727	5	x	x	X
ejpam-4343	727	6	,	,	PUNCT
ejpam-4343	727	7	τ	τ	PROPN
ejpam-4343	727	8	,	,	PUNCT
ejpam-4343	727	9	i	i	PROPN
ejpam-4343	727	10	)	)	PUNCT
ejpam-4343	727	11	is	be	AUX
ejpam-4343	727	12	a	a	DET
ejpam-4343	727	13	pre	pre	NOUN
ejpam-4343	727	14	-	-	ADJ
ejpam-4343	727	15	i	i	PRON
ejpam-4343	727	16	-t1	-t1	NOUN
ejpam-4343	727	17	space	space	NOUN
ejpam-4343	727	18	.	.	PUNCT
ejpam-4343	728	1	corollary	corollary	ADJ
ejpam-4343	728	2	1	1	NUM
ejpam-4343	728	3	.	.	PUNCT
ejpam-4343	729	1	for	for	ADP
ejpam-4343	729	2	an	an	DET
ejpam-4343	729	3	ideal	ideal	ADJ
ejpam-4343	729	4	topological	topological	ADJ
ejpam-4343	729	5	space	space	NOUN
ejpam-4343	729	6	(	(	PUNCT
ejpam-4343	729	7	x	x	X
ejpam-4343	729	8	,	,	PUNCT
ejpam-4343	729	9	τ	τ	PROPN
ejpam-4343	729	10	,	,	PUNCT
ejpam-4343	729	11	i	i	NOUN
ejpam-4343	729	12	)	)	PUNCT
ejpam-4343	729	13	,	,	PUNCT
ejpam-4343	729	14	the	the	DET
ejpam-4343	729	15	following	follow	VERB
ejpam-4343	729	16	properties	property	NOUN
ejpam-4343	729	17	are	be	AUX
ejpam-4343	729	18	equivalent	equivalent	ADJ
ejpam-4343	729	19	:	:	PUNCT
ejpam-4343	729	20	(	(	PUNCT
ejpam-4343	729	21	1	1	X
ejpam-4343	729	22	)	)	PUNCT
ejpam-4343	729	23	(	(	PUNCT
ejpam-4343	729	24	x	x	X
ejpam-4343	729	25	,	,	PUNCT
ejpam-4343	729	26	τ	τ	PROPN
ejpam-4343	729	27	,	,	PUNCT
ejpam-4343	729	28	i	i	PROPN
ejpam-4343	729	29	)	)	PUNCT
ejpam-4343	729	30	is	be	AUX
ejpam-4343	729	31	pre	pre	ADJ
ejpam-4343	729	32	-	-	ADJ
ejpam-4343	729	33	i	i	PRON
ejpam-4343	729	34	-t1	-t1	VERB
ejpam-4343	729	35	;	;	PUNCT
ejpam-4343	729	36	(	(	PUNCT
ejpam-4343	729	37	2	2	X
ejpam-4343	729	38	)	)	PUNCT
ejpam-4343	729	39	(	(	PUNCT
ejpam-4343	729	40	x	x	X
ejpam-4343	729	41	,	,	PUNCT
ejpam-4343	729	42	τ	τ	PROPN
ejpam-4343	729	43	,	,	PUNCT
ejpam-4343	729	44	i	i	PROPN
ejpam-4343	729	45	)	)	PUNCT
ejpam-4343	729	46	is	be	AUX
ejpam-4343	729	47	pre	pre	VERB
ejpam-4343	729	48	-	-	ADJ
ejpam-4343	729	49	i	i	PRON
ejpam-4343	729	50	-t0	-t0	ADJ
ejpam-4343	729	51	and	and	CCONJ
ejpam-4343	729	52	pre	pre	ADJ
ejpam-4343	729	53	-	-	PROPN
ejpam-4343	729	54	i	i	PRON
ejpam-4343	729	55	-r0	-r0	NOUN
ejpam-4343	729	56	.	.	PUNCT
ejpam-4343	730	1	proof	proof	NOUN
ejpam-4343	730	2	.	.	PUNCT
ejpam-4343	731	1	(	(	PUNCT
ejpam-4343	731	2	1	1	X
ejpam-4343	731	3	)	)	PUNCT
ejpam-4343	731	4	⇒	⇒	NOUN
ejpam-4343	731	5	(	(	PUNCT
ejpam-4343	731	6	2	2	NUM
ejpam-4343	731	7	):	):	PUNCT
ejpam-4343	731	8	suppose	suppose	VERB
ejpam-4343	731	9	that	that	SCONJ
ejpam-4343	731	10	(	(	PUNCT
ejpam-4343	731	11	x	x	X
ejpam-4343	731	12	,	,	PUNCT
ejpam-4343	731	13	τ	τ	PROPN
ejpam-4343	731	14	,	,	PUNCT
ejpam-4343	731	15	i	i	PROPN
ejpam-4343	731	16	)	)	PUNCT
ejpam-4343	731	17	is	be	AUX
ejpam-4343	731	18	pre	pre	ADJ
ejpam-4343	731	19	-	-	ADJ
ejpam-4343	731	20	i	i	PRON
ejpam-4343	731	21	-t1	-t1	VERB
ejpam-4343	731	22	.	.	PUNCT
ejpam-4343	732	1	by	by	ADP
ejpam-4343	732	2	remark	remark	NOUN
ejpam-4343	732	3	3	3	NUM
ejpam-4343	732	4	and	and	CCONJ
ejpam-4343	732	5	theorem	theorem	VERB
ejpam-4343	732	6	12	12	NUM
ejpam-4343	732	7	,	,	PUNCT
ejpam-4343	732	8	every	every	DET
ejpam-4343	732	9	pre	pre	NOUN
ejpam-4343	732	10	-	-	PROPN
ejpam-4343	732	11	i	i	PRON
ejpam-4343	732	12	-t1	-t1	NOUN
ejpam-4343	732	13	space	space	NOUN
ejpam-4343	732	14	is	be	AUX
ejpam-4343	732	15	pre	pre	ADJ
ejpam-4343	732	16	-	-	ADJ
ejpam-4343	732	17	i	i	PRON
ejpam-4343	732	18	-t0	-t0	ADJ
ejpam-4343	732	19	and	and	CCONJ
ejpam-4343	732	20	pre	pre	ADJ
ejpam-4343	732	21	-	-	PROPN
ejpam-4343	732	22	i	i	PRON
ejpam-4343	732	23	-r0	-r0	NOUN
ejpam-4343	732	24	.	.	PUNCT
ejpam-4343	733	1	(	(	PUNCT
ejpam-4343	733	2	2	2	X
ejpam-4343	733	3	)	)	PUNCT
ejpam-4343	733	4	⇒	⇒	NOUN
ejpam-4343	733	5	(	(	PUNCT
ejpam-4343	733	6	1	1	NUM
ejpam-4343	733	7	):	):	PUNCT
ejpam-4343	733	8	suppose	suppose	VERB
ejpam-4343	733	9	that	that	SCONJ
ejpam-4343	733	10	(	(	PUNCT
ejpam-4343	733	11	x	x	X
ejpam-4343	733	12	,	,	PUNCT
ejpam-4343	733	13	τ	τ	PROPN
ejpam-4343	733	14	,	,	PUNCT
ejpam-4343	733	15	i	i	PROPN
ejpam-4343	733	16	)	)	PUNCT
ejpam-4343	733	17	is	be	AUX
ejpam-4343	733	18	pre	pre	VERB
ejpam-4343	733	19	-	-	ADJ
ejpam-4343	733	20	i	i	PRON
ejpam-4343	733	21	-t0	-t0	ADJ
ejpam-4343	733	22	and	and	CCONJ
ejpam-4343	733	23	pre	pre	ADJ
ejpam-4343	733	24	-	-	PROPN
ejpam-4343	733	25	i	i	PRON
ejpam-4343	733	26	-r0	-r0	NOUN
ejpam-4343	733	27	.	.	PUNCT
ejpam-4343	734	1	since	since	SCONJ
ejpam-4343	734	2	(	(	PUNCT
ejpam-4343	734	3	x	x	X
ejpam-4343	734	4	,	,	PUNCT
ejpam-4343	734	5	τ	τ	PROPN
ejpam-4343	734	6	,	,	PUNCT
ejpam-4343	734	7	i	i	PROPN
ejpam-4343	734	8	)	)	PUNCT
ejpam-4343	734	9	is	be	AUX
ejpam-4343	734	10	pre	pre	ADJ
ejpam-4343	734	11	-	-	ADJ
ejpam-4343	734	12	i	i	PRON
ejpam-4343	734	13	-t0	-t0	VERB
ejpam-4343	734	14	,	,	PUNCT
ejpam-4343	734	15	for	for	ADP
ejpam-4343	734	16	any	any	DET
ejpam-4343	734	17	distinct	distinct	ADJ
ejpam-4343	734	18	points	point	NOUN
ejpam-4343	734	19	x	x	PUNCT
ejpam-4343	734	20	and	and	CCONJ
ejpam-4343	734	21	y	y	PROPN
ejpam-4343	734	22	of	of	ADP
ejpam-4343	734	23	x	x	PRON
ejpam-4343	734	24	,	,	PUNCT
ejpam-4343	734	25	there	there	PRON
ejpam-4343	734	26	exists	exist	VERB
ejpam-4343	734	27	a	a	DET
ejpam-4343	734	28	pre	pre	ADJ
ejpam-4343	734	29	-	-	ADJ
ejpam-4343	734	30	i	i	PRON
ejpam-4343	734	31	-open	-open	NOUN
ejpam-4343	734	32	set	set	VERB
ejpam-4343	734	33	v	v	ADP
ejpam-4343	734	34	such	such	ADJ
ejpam-4343	734	35	that	that	SCONJ
ejpam-4343	734	36	x	x	SYM
ejpam-4343	734	37	∈	∈	PROPN
ejpam-4343	734	38	v	v	NOUN
ejpam-4343	734	39	and	and	CCONJ
ejpam-4343	734	40	y	y	PROPN
ejpam-4343	734	41	̸∈	̸∈	PROPN
ejpam-4343	734	42	v	v	PROPN
ejpam-4343	734	43	.	.	PUNCT
ejpam-4343	735	1	since	since	SCONJ
ejpam-4343	735	2	(	(	PUNCT
ejpam-4343	735	3	x	x	X
ejpam-4343	735	4	,	,	PUNCT
ejpam-4343	735	5	τ	τ	PROPN
ejpam-4343	735	6	,	,	PUNCT
ejpam-4343	735	7	i	i	PROPN
ejpam-4343	735	8	)	)	PUNCT
ejpam-4343	735	9	is	be	AUX
ejpam-4343	735	10	pre	pre	VERB
ejpam-4343	735	11	-	-	ADJ
ejpam-4343	735	12	i	i	PRON
ejpam-4343	735	13	-r0	-r0	NOUN
ejpam-4343	735	14	,	,	PUNCT
ejpam-4343	735	15	we	we	PRON
ejpam-4343	735	16	have	have	VERB
ejpam-4343	735	17	pıcl({x	pıcl({x	X
ejpam-4343	735	18	}	}	PUNCT
ejpam-4343	735	19	)	)	PUNCT
ejpam-4343	736	1	⊆	⊆	NUM
ejpam-4343	736	2	v	v	NOUN
ejpam-4343	736	3	.	.	PUNCT
ejpam-4343	737	1	thus	thus	ADV
ejpam-4343	737	2	,	,	PUNCT
ejpam-4343	737	3	x	x	PROPN
ejpam-4343	737	4	̸∈	̸∈	PROPN
ejpam-4343	737	5	x	x	X
ejpam-4343	737	6	−	−	PROPN
ejpam-4343	737	7	pıcl({x	pıcl({x	PROPN
ejpam-4343	737	8	}	}	PUNCT
ejpam-4343	737	9	)	)	PUNCT
ejpam-4343	737	10	and	and	CCONJ
ejpam-4343	737	11	hence	hence	ADV
ejpam-4343	737	12	y	y	PROPN
ejpam-4343	737	13	∈	∈	PROPN
ejpam-4343	737	14	x	x	PUNCT
ejpam-4343	737	15	−	−	NOUN
ejpam-4343	737	16	v	v	ADP
ejpam-4343	737	17	⊆	⊆	NUM
ejpam-4343	737	18	x	x	SYM
ejpam-4343	737	19	−	−	PROPN
ejpam-4343	737	20	pıcl({x	pıcl({x	PROPN
ejpam-4343	737	21	}	}	PUNCT
ejpam-4343	737	22	)	)	PUNCT
ejpam-4343	737	23	.	.	PUNCT
ejpam-4343	738	1	therefore	therefore	ADV
ejpam-4343	738	2	,	,	PUNCT
ejpam-4343	738	3	(	(	PUNCT
ejpam-4343	738	4	x	x	X
ejpam-4343	738	5	,	,	PUNCT
ejpam-4343	738	6	τ	τ	PROPN
ejpam-4343	738	7	,	,	PUNCT
ejpam-4343	738	8	i	i	PROPN
ejpam-4343	738	9	)	)	PUNCT
ejpam-4343	738	10	is	be	AUX
ejpam-4343	738	11	pre	pre	ADJ
ejpam-4343	738	12	-	-	ADJ
ejpam-4343	738	13	i	i	PRON
ejpam-4343	738	14	-t1	-t1	VERB
ejpam-4343	738	15	.	.	PUNCT
ejpam-4343	739	1	lemma	lemma	PROPN
ejpam-4343	739	2	7	7	NUM
ejpam-4343	739	3	.	.	PUNCT
ejpam-4343	740	1	an	an	DET
ejpam-4343	740	2	ideal	ideal	ADJ
ejpam-4343	740	3	topological	topological	ADJ
ejpam-4343	740	4	space	space	NOUN
ejpam-4343	740	5	(	(	PUNCT
ejpam-4343	740	6	x	x	X
ejpam-4343	740	7	,	,	PUNCT
ejpam-4343	740	8	τ	τ	PROPN
ejpam-4343	740	9	,	,	PUNCT
ejpam-4343	740	10	i	i	PROPN
ejpam-4343	740	11	)	)	PUNCT
ejpam-4343	740	12	is	be	AUX
ejpam-4343	740	13	pre	pre	VERB
ejpam-4343	740	14	-	-	ADJ
ejpam-4343	740	15	i	i	PRON
ejpam-4343	740	16	-r0	-r0	INTJ
ejpam-4343	741	1	if	if	SCONJ
ejpam-4343	741	2	and	and	CCONJ
ejpam-4343	741	3	only	only	ADV
ejpam-4343	741	4	if	if	SCONJ
ejpam-4343	741	5	,	,	PUNCT
ejpam-4343	741	6	for	for	ADP
ejpam-4343	741	7	each	each	DET
ejpam-4343	741	8	prei	prei	NOUN
ejpam-4343	741	9	-open	-open	NOUN
ejpam-4343	741	10	set	set	VERB
ejpam-4343	741	11	u	u	NOUN
ejpam-4343	741	12	,	,	PUNCT
ejpam-4343	741	13	x	x	PROPN
ejpam-4343	741	14	∈	∈	PROPN
ejpam-4343	741	15	u	u	NOUN
ejpam-4343	741	16	implies	imply	VERB
ejpam-4343	741	17	cl(int⋆({x	cl(int⋆({x	NOUN
ejpam-4343	741	18	}	}	PUNCT
ejpam-4343	741	19	)	)	PUNCT
ejpam-4343	741	20	)	)	PUNCT
ejpam-4343	742	1	⊆	⊆	NUM
ejpam-4343	742	2	u	u	NOUN
ejpam-4343	742	3	.	.	PUNCT
ejpam-4343	743	1	proof	proof	NOUN
ejpam-4343	743	2	.	.	PUNCT
ejpam-4343	744	1	let	let	VERB
ejpam-4343	744	2	u	u	PRON
ejpam-4343	744	3	be	be	AUX
ejpam-4343	744	4	any	any	DET
ejpam-4343	744	5	pre	pre	ADJ
ejpam-4343	744	6	-	-	ADJ
ejpam-4343	744	7	i	i	PRON
ejpam-4343	744	8	-open	-open	NOUN
ejpam-4343	744	9	set	set	VERB
ejpam-4343	744	10	and	and	CCONJ
ejpam-4343	744	11	x	x	SYM
ejpam-4343	744	12	∈	∈	PROPN
ejpam-4343	744	13	u	u	NOUN
ejpam-4343	744	14	.	.	PUNCT
ejpam-4343	745	1	then	then	ADV
ejpam-4343	745	2	,	,	PUNCT
ejpam-4343	745	3	we	we	PRON
ejpam-4343	745	4	have	have	VERB
ejpam-4343	745	5	pıcl({x	pıcl({x	X
ejpam-4343	745	6	}	}	PUNCT
ejpam-4343	745	7	)	)	PUNCT
ejpam-4343	746	1	⊆	⊆	NUM
ejpam-4343	746	2	u	u	NOUN
ejpam-4343	746	3	and	and	CCONJ
ejpam-4343	746	4	by	by	ADP
ejpam-4343	746	5	lemma	lemma	PROPN
ejpam-4343	746	6	2	2	NUM
ejpam-4343	746	7	,	,	PUNCT
ejpam-4343	746	8	cl(int⋆({x	cl(int⋆({x	NOUN
ejpam-4343	746	9	}	}	PUNCT
ejpam-4343	746	10	)	)	PUNCT
ejpam-4343	746	11	)	)	PUNCT
ejpam-4343	747	1	⊆	⊆	NUM
ejpam-4343	747	2	u	u	NOUN
ejpam-4343	747	3	.	.	PUNCT
ejpam-4343	748	1	conversely	conversely	ADV
ejpam-4343	748	2	,	,	PUNCT
ejpam-4343	748	3	let	let	VERB
ejpam-4343	748	4	u	u	PRON
ejpam-4343	748	5	be	be	AUX
ejpam-4343	748	6	any	any	DET
ejpam-4343	748	7	pre	pre	ADJ
ejpam-4343	748	8	-	-	ADJ
ejpam-4343	748	9	i	i	PRON
ejpam-4343	748	10	-open	-open	NOUN
ejpam-4343	748	11	set	set	VERB
ejpam-4343	748	12	and	and	CCONJ
ejpam-4343	748	13	x	x	SYM
ejpam-4343	748	14	∈	∈	PROPN
ejpam-4343	748	15	u	u	NOUN
ejpam-4343	748	16	.	.	PUNCT
ejpam-4343	749	1	by	by	ADP
ejpam-4343	749	2	the	the	DET
ejpam-4343	749	3	hypothesis	hypothesis	NOUN
ejpam-4343	749	4	,	,	PUNCT
ejpam-4343	749	5	we	we	PRON
ejpam-4343	749	6	have	have	VERB
ejpam-4343	749	7	cl(int⋆({x	cl(int⋆({x	VERB
ejpam-4343	749	8	}	}	PUNCT
ejpam-4343	749	9	)	)	PUNCT
ejpam-4343	749	10	)	)	PUNCT
ejpam-4343	750	1	⊆	⊆	NUM
ejpam-4343	750	2	u	u	NOUN
ejpam-4343	750	3	and	and	CCONJ
ejpam-4343	750	4	by	by	ADP
ejpam-4343	750	5	lemma	lemma	PROPN
ejpam-4343	750	6	2	2	NUM
ejpam-4343	750	7	,	,	PUNCT
ejpam-4343	750	8	pıcl({x	pıcl({x	PROPN
ejpam-4343	750	9	}	}	PUNCT
ejpam-4343	750	10	)	)	PUNCT
ejpam-4343	750	11	⊆	⊆	NUM
ejpam-4343	750	12	u	u	NOUN
ejpam-4343	750	13	.	.	PUNCT
ejpam-4343	751	1	this	this	PRON
ejpam-4343	751	2	shows	show	VERB
ejpam-4343	751	3	that	that	SCONJ
ejpam-4343	751	4	(	(	PUNCT
ejpam-4343	751	5	x	x	X
ejpam-4343	751	6	,	,	PUNCT
ejpam-4343	751	7	τ	τ	PROPN
ejpam-4343	751	8	,	,	PUNCT
ejpam-4343	751	9	i	i	PROPN
ejpam-4343	751	10	)	)	PUNCT
ejpam-4343	751	11	is	be	AUX
ejpam-4343	751	12	a	a	DET
ejpam-4343	751	13	pre	pre	NOUN
ejpam-4343	751	14	-	-	ADJ
ejpam-4343	751	15	i	i	PRON
ejpam-4343	751	16	-r0	-r0	PROPN
ejpam-4343	751	17	space	space	NOUN
ejpam-4343	751	18	.	.	PUNCT
ejpam-4343	752	1	c.	c.	PROPN
ejpam-4343	752	2	boonpok	boonpok	PROPN
ejpam-4343	752	3	/	/	SYM
ejpam-4343	752	4	eur	eur	PROPN
ejpam-4343	752	5	.	.	PUNCT
ejpam-4343	753	1	j.	j.	PROPN
ejpam-4343	753	2	pure	pure	PROPN
ejpam-4343	753	3	appl	appl	PROPN
ejpam-4343	753	4	.	.	PROPN
ejpam-4343	753	5	math	math	PROPN
ejpam-4343	753	6	,	,	PUNCT
ejpam-4343	753	7	15	15	NUM
ejpam-4343	753	8	(	(	PUNCT
ejpam-4343	753	9	3	3	NUM
ejpam-4343	753	10	)	)	PUNCT
ejpam-4343	753	11	(	(	PUNCT
ejpam-4343	753	12	2022	2022	NUM
ejpam-4343	753	13	)	)	PUNCT
ejpam-4343	753	14	,	,	PUNCT
ejpam-4343	753	15	1023	1023	NUM
ejpam-4343	753	16	-	-	SYM
ejpam-4343	753	17	1046	1046	NUM
ejpam-4343	753	18	1041	1041	NUM
ejpam-4343	753	19	theorem	theorem	VERB
ejpam-4343	753	20	13	13	NUM
ejpam-4343	753	21	.	.	PUNCT
ejpam-4343	754	1	for	for	ADP
ejpam-4343	754	2	an	an	DET
ejpam-4343	754	3	ideal	ideal	ADJ
ejpam-4343	754	4	topological	topological	ADJ
ejpam-4343	754	5	space	space	NOUN
ejpam-4343	754	6	(	(	PUNCT
ejpam-4343	754	7	x	x	X
ejpam-4343	754	8	,	,	PUNCT
ejpam-4343	754	9	τ	τ	PROPN
ejpam-4343	754	10	,	,	PUNCT
ejpam-4343	754	11	i	i	NOUN
ejpam-4343	754	12	)	)	PUNCT
ejpam-4343	754	13	,	,	PUNCT
ejpam-4343	754	14	the	the	DET
ejpam-4343	754	15	following	follow	VERB
ejpam-4343	754	16	properties	property	NOUN
ejpam-4343	754	17	are	be	AUX
ejpam-4343	754	18	equivalent	equivalent	ADJ
ejpam-4343	754	19	:	:	PUNCT
ejpam-4343	754	20	(	(	PUNCT
ejpam-4343	754	21	1	1	X
ejpam-4343	754	22	)	)	PUNCT
ejpam-4343	754	23	(	(	PUNCT
ejpam-4343	754	24	x	x	X
ejpam-4343	754	25	,	,	PUNCT
ejpam-4343	754	26	τ	τ	PROPN
ejpam-4343	754	27	,	,	PUNCT
ejpam-4343	754	28	i	i	PROPN
ejpam-4343	754	29	)	)	PUNCT
ejpam-4343	754	30	is	be	AUX
ejpam-4343	754	31	pre	pre	VERB
ejpam-4343	754	32	-	-	ADJ
ejpam-4343	754	33	i	i	PRON
ejpam-4343	754	34	-r0	-r0	NOUN
ejpam-4343	754	35	.	.	PUNCT
ejpam-4343	755	1	(	(	PUNCT
ejpam-4343	755	2	2	2	X
ejpam-4343	755	3	)	)	PUNCT
ejpam-4343	755	4	for	for	ADP
ejpam-4343	755	5	each	each	DET
ejpam-4343	755	6	pre	pre	NOUN
ejpam-4343	755	7	-	-	ADJ
ejpam-4343	755	8	i	i	PRON
ejpam-4343	755	9	-closed	-close	VERB
ejpam-4343	755	10	set	set	VERB
ejpam-4343	755	11	f	f	NOUN
ejpam-4343	755	12	and	and	CCONJ
ejpam-4343	755	13	each	each	DET
ejpam-4343	755	14	x	x	SYM
ejpam-4343	755	15	∈	∈	PROPN
ejpam-4343	755	16	x	x	X
ejpam-4343	755	17	−	−	PROPN
ejpam-4343	755	18	f	f	NOUN
ejpam-4343	755	19	,	,	PUNCT
ejpam-4343	755	20	there	there	PRON
ejpam-4343	755	21	exists	exist	VERB
ejpam-4343	755	22	a	a	DET
ejpam-4343	755	23	pre	pre	ADJ
ejpam-4343	755	24	-	-	ADJ
ejpam-4343	755	25	i	i	PRON
ejpam-4343	755	26	-open	-open	NOUN
ejpam-4343	755	27	set	set	VERB
ejpam-4343	755	28	u	u	NOUN
ejpam-4343	755	29	such	such	ADJ
ejpam-4343	755	30	that	that	SCONJ
ejpam-4343	755	31	f	f	PROPN
ejpam-4343	755	32	⊆	⊆	NUM
ejpam-4343	755	33	u	u	NOUN
ejpam-4343	755	34	and	and	CCONJ
ejpam-4343	755	35	x	x	PUNCT
ejpam-4343	755	36	̸∈	̸∈	PROPN
ejpam-4343	755	37	u	u	PROPN
ejpam-4343	755	38	.	.	PUNCT
ejpam-4343	756	1	(	(	PUNCT
ejpam-4343	756	2	3	3	X
ejpam-4343	756	3	)	)	PUNCT
ejpam-4343	756	4	for	for	ADP
ejpam-4343	756	5	each	each	DET
ejpam-4343	756	6	pre	pre	NOUN
ejpam-4343	756	7	-	-	ADJ
ejpam-4343	756	8	i	i	PRON
ejpam-4343	756	9	-closed	-close	VERB
ejpam-4343	756	10	set	set	VERB
ejpam-4343	756	11	f	f	NOUN
ejpam-4343	756	12	and	and	CCONJ
ejpam-4343	756	13	each	each	DET
ejpam-4343	756	14	x	x	SYM
ejpam-4343	756	15	∈	∈	PROPN
ejpam-4343	756	16	x	x	X
ejpam-4343	756	17	−	−	PROPN
ejpam-4343	756	18	f	f	PROPN
ejpam-4343	756	19	,	,	PUNCT
ejpam-4343	756	20	pıcl({x	pıcl({x	PROPN
ejpam-4343	756	21	}	}	PUNCT
ejpam-4343	756	22	)	)	PUNCT
ejpam-4343	756	23	∩	∩	NOUN
ejpam-4343	756	24	f	f	X
ejpam-4343	756	25	=	=	PUNCT
ejpam-4343	756	26	∅.	∅.	X
ejpam-4343	756	27	(	(	PUNCT
ejpam-4343	756	28	4	4	NUM
ejpam-4343	756	29	)	)	PUNCT
ejpam-4343	756	30	for	for	ADP
ejpam-4343	756	31	any	any	DET
ejpam-4343	756	32	distinct	distinct	ADJ
ejpam-4343	756	33	points	point	NOUN
ejpam-4343	756	34	x	x	PUNCT
ejpam-4343	756	35	and	and	CCONJ
ejpam-4343	756	36	y	y	PROPN
ejpam-4343	756	37	of	of	ADP
ejpam-4343	756	38	x	x	PROPN
ejpam-4343	756	39	,	,	PUNCT
ejpam-4343	756	40	either	either	CCONJ
ejpam-4343	756	41	pıcl({x	pıcl({x	PROPN
ejpam-4343	756	42	}	}	PUNCT
ejpam-4343	756	43	)	)	PUNCT
ejpam-4343	757	1	=	=	SYM
ejpam-4343	757	2	pıcl({y	pıcl({y	PROPN
ejpam-4343	757	3	}	}	PUNCT
ejpam-4343	757	4	)	)	PUNCT
ejpam-4343	757	5	or	or	CCONJ
ejpam-4343	757	6	pıcl({x	pıcl({x	NUM
ejpam-4343	757	7	}	}	PUNCT
ejpam-4343	757	8	)	)	PUNCT
ejpam-4343	757	9	∩	∩	NOUN
ejpam-4343	757	10	pıcl({y	pıcl({y	X
ejpam-4343	757	11	}	}	PUNCT
ejpam-4343	757	12	)	)	PUNCT
ejpam-4343	757	13	=	=	PUNCT
ejpam-4343	757	14	∅.	∅.	NOUN
ejpam-4343	757	15	proof	proof	NOUN
ejpam-4343	757	16	.	.	PUNCT
ejpam-4343	758	1	(	(	PUNCT
ejpam-4343	758	2	1	1	X
ejpam-4343	758	3	)	)	PUNCT
ejpam-4343	758	4	⇒	⇒	NOUN
ejpam-4343	758	5	(	(	PUNCT
ejpam-4343	758	6	2	2	NUM
ejpam-4343	758	7	):	):	PUNCT
ejpam-4343	758	8	let	let	VERB
ejpam-4343	758	9	f	f	PRON
ejpam-4343	758	10	be	be	AUX
ejpam-4343	758	11	any	any	DET
ejpam-4343	758	12	pre	pre	NOUN
ejpam-4343	758	13	-	-	ADJ
ejpam-4343	758	14	i	i	PRON
ejpam-4343	758	15	-closed	-close	VERB
ejpam-4343	758	16	set	set	NOUN
ejpam-4343	758	17	and	and	CCONJ
ejpam-4343	758	18	x	x	SYM
ejpam-4343	758	19	∈	∈	PROPN
ejpam-4343	758	20	x	x	X
ejpam-4343	758	21	−	−	PROPN
ejpam-4343	758	22	f	f	X
ejpam-4343	758	23	.	.	PUNCT
ejpam-4343	759	1	then	then	ADV
ejpam-4343	759	2	,	,	PUNCT
ejpam-4343	759	3	we	we	PRON
ejpam-4343	759	4	have	have	VERB
ejpam-4343	759	5	x	x	X
ejpam-4343	759	6	∈	∈	PROPN
ejpam-4343	759	7	x−f	x−f	PROPN
ejpam-4343	759	8	and	and	CCONJ
ejpam-4343	759	9	by	by	ADP
ejpam-4343	759	10	(	(	PUNCT
ejpam-4343	759	11	1	1	NUM
ejpam-4343	759	12	)	)	PUNCT
ejpam-4343	759	13	,	,	PUNCT
ejpam-4343	759	14	pıcl({x	pıcl({x	PROPN
ejpam-4343	759	15	}	}	PUNCT
ejpam-4343	759	16	)	)	PUNCT
ejpam-4343	759	17	⊆	⊆	NUM
ejpam-4343	759	18	x−f	x−f	X
ejpam-4343	759	19	.	.	PUNCT
ejpam-4343	760	1	put	put	VERB
ejpam-4343	760	2	u	u	NOUN
ejpam-4343	760	3	=	=	PROPN
ejpam-4343	760	4	x−pıcl({x	x−pıcl({x	PROPN
ejpam-4343	760	5	}	}	PUNCT
ejpam-4343	760	6	)	)	PUNCT
ejpam-4343	760	7	,	,	PUNCT
ejpam-4343	760	8	then	then	ADV
ejpam-4343	760	9	u	u	NOUN
ejpam-4343	760	10	is	be	AUX
ejpam-4343	760	11	a	a	DET
ejpam-4343	760	12	pre	pre	ADJ
ejpam-4343	760	13	-	-	ADJ
ejpam-4343	760	14	i	i	PRON
ejpam-4343	760	15	-open	-open	NOUN
ejpam-4343	760	16	set	set	VERB
ejpam-4343	760	17	such	such	ADJ
ejpam-4343	760	18	that	that	SCONJ
ejpam-4343	760	19	f	f	PROPN
ejpam-4343	760	20	⊆	⊆	NUM
ejpam-4343	760	21	u	u	NOUN
ejpam-4343	760	22	and	and	CCONJ
ejpam-4343	760	23	x	x	PUNCT
ejpam-4343	760	24	̸∈	̸∈	PROPN
ejpam-4343	760	25	u	u	PROPN
ejpam-4343	760	26	.	.	PUNCT
ejpam-4343	761	1	(	(	PUNCT
ejpam-4343	761	2	2	2	X
ejpam-4343	761	3	)	)	PUNCT
ejpam-4343	761	4	⇒	⇒	NOUN
ejpam-4343	761	5	(	(	PUNCT
ejpam-4343	761	6	3	3	NUM
ejpam-4343	761	7	):	):	PUNCT
ejpam-4343	761	8	let	let	VERB
ejpam-4343	761	9	f	f	PRON
ejpam-4343	761	10	be	be	AUX
ejpam-4343	761	11	any	any	DET
ejpam-4343	761	12	pre	pre	NOUN
ejpam-4343	761	13	-	-	ADJ
ejpam-4343	761	14	i	i	PRON
ejpam-4343	761	15	-closed	-close	VERB
ejpam-4343	761	16	set	set	NOUN
ejpam-4343	761	17	and	and	CCONJ
ejpam-4343	761	18	x	x	SYM
ejpam-4343	761	19	∈	∈	PROPN
ejpam-4343	761	20	x	x	X
ejpam-4343	761	21	−	−	PROPN
ejpam-4343	761	22	f	f	X
ejpam-4343	761	23	.	.	PUNCT
ejpam-4343	762	1	then	then	ADV
ejpam-4343	762	2	by	by	ADP
ejpam-4343	762	3	(	(	PUNCT
ejpam-4343	762	4	2	2	NUM
ejpam-4343	762	5	)	)	PUNCT
ejpam-4343	762	6	,	,	PUNCT
ejpam-4343	762	7	there	there	PRON
ejpam-4343	762	8	exists	exist	VERB
ejpam-4343	762	9	a	a	DET
ejpam-4343	762	10	pre	pre	ADJ
ejpam-4343	762	11	-	-	ADJ
ejpam-4343	762	12	i	i	PRON
ejpam-4343	762	13	-open	-open	NOUN
ejpam-4343	762	14	set	set	VERB
ejpam-4343	762	15	v	v	ADP
ejpam-4343	762	16	such	such	ADJ
ejpam-4343	762	17	that	that	SCONJ
ejpam-4343	762	18	f	f	PROPN
ejpam-4343	762	19	⊆	⊆	NUM
ejpam-4343	762	20	v	v	NOUN
ejpam-4343	762	21	and	and	CCONJ
ejpam-4343	762	22	x	x	PART
ejpam-4343	762	23	̸∈	̸∈	PROPN
ejpam-4343	762	24	v	v	PROPN
ejpam-4343	762	25	.	.	PUNCT
ejpam-4343	763	1	since	since	SCONJ
ejpam-4343	763	2	v	v	NOUN
ejpam-4343	763	3	is	be	AUX
ejpam-4343	763	4	a	a	DET
ejpam-4343	763	5	pre	pre	ADJ
ejpam-4343	763	6	-	-	ADJ
ejpam-4343	763	7	i	i	PRON
ejpam-4343	763	8	-open	-open	NOUN
ejpam-4343	763	9	set	set	NOUN
ejpam-4343	763	10	,	,	PUNCT
ejpam-4343	763	11	we	we	PRON
ejpam-4343	763	12	have	have	VERB
ejpam-4343	763	13	pıcl({x	pıcl({x	NUM
ejpam-4343	763	14	}	}	PUNCT
ejpam-4343	763	15	)	)	PUNCT
ejpam-4343	763	16	∩	∩	ADJ
ejpam-4343	763	17	v	v	NOUN
ejpam-4343	763	18	=	=	NOUN
ejpam-4343	763	19	∅	∅	NOUN
ejpam-4343	763	20	and	and	CCONJ
ejpam-4343	763	21	hence	hence	ADV
ejpam-4343	763	22	pıcl({x	pıcl({x	PROPN
ejpam-4343	763	23	}	}	PUNCT
ejpam-4343	763	24	)	)	PUNCT
ejpam-4343	763	25	∩	∩	NOUN
ejpam-4343	763	26	f	f	X
ejpam-4343	763	27	=	=	PUNCT
ejpam-4343	763	28	∅.	∅.	X
ejpam-4343	763	29	(	(	PUNCT
ejpam-4343	763	30	3	3	NUM
ejpam-4343	763	31	)	)	PUNCT
ejpam-4343	763	32	⇒	⇒	NOUN
ejpam-4343	763	33	(	(	PUNCT
ejpam-4343	763	34	4	4	NUM
ejpam-4343	763	35	):	):	PUNCT
ejpam-4343	763	36	let	let	VERB
ejpam-4343	763	37	x	x	PRON
ejpam-4343	763	38	,	,	PUNCT
ejpam-4343	763	39	y	y	PROPN
ejpam-4343	763	40	be	be	VERB
ejpam-4343	763	41	any	any	DET
ejpam-4343	763	42	points	point	NOUN
ejpam-4343	763	43	ofx	ofx	NOUN
ejpam-4343	763	44	.	.	PUNCT
ejpam-4343	764	1	suppose	suppose	VERB
ejpam-4343	764	2	that	that	SCONJ
ejpam-4343	764	3	pıcl({x})∩pıcl({y	pıcl({x})∩pıcl({y	PROPN
ejpam-4343	764	4	}	}	PUNCT
ejpam-4343	764	5	)	)	PUNCT
ejpam-4343	765	1	̸=	̸=	PROPN
ejpam-4343	765	2	∅.	∅.	PRON
ejpam-4343	765	3	by	by	ADP
ejpam-4343	765	4	(	(	PUNCT
ejpam-4343	765	5	3	3	NUM
ejpam-4343	765	6	)	)	PUNCT
ejpam-4343	765	7	,	,	PUNCT
ejpam-4343	765	8	x	x	PROPN
ejpam-4343	765	9	̸∈	̸∈	PROPN
ejpam-4343	765	10	pıcl({y	pıcl({y	PROPN
ejpam-4343	765	11	}	}	PUNCT
ejpam-4343	765	12	)	)	PUNCT
ejpam-4343	765	13	and	and	CCONJ
ejpam-4343	765	14	y	y	PROPN
ejpam-4343	765	15	̸∈	̸∈	PROPN
ejpam-4343	765	16	pıcl({x	pıcl({x	PROPN
ejpam-4343	765	17	}	}	PUNCT
ejpam-4343	765	18	)	)	PUNCT
ejpam-4343	765	19	.	.	PUNCT
ejpam-4343	766	1	this	this	PRON
ejpam-4343	766	2	implies	imply	VERB
ejpam-4343	766	3	that	that	SCONJ
ejpam-4343	766	4	pıcl({x	pıcl({x	PROPN
ejpam-4343	766	5	}	}	PUNCT
ejpam-4343	766	6	)	)	PUNCT
ejpam-4343	766	7	⊆	⊆	NUM
ejpam-4343	766	8	pıcl({y	pıcl({y	NUM
ejpam-4343	766	9	}	}	PUNCT
ejpam-4343	766	10	)	)	PUNCT
ejpam-4343	766	11	⊆	⊆	NUM
ejpam-4343	766	12	pıcl({x	pıcl({x	NUM
ejpam-4343	766	13	}	}	PUNCT
ejpam-4343	766	14	)	)	PUNCT
ejpam-4343	766	15	.	.	PUNCT
ejpam-4343	767	1	consequently	consequently	ADV
ejpam-4343	767	2	,	,	PUNCT
ejpam-4343	767	3	we	we	PRON
ejpam-4343	767	4	obtain	obtain	VERB
ejpam-4343	767	5	pıcl({x	pıcl({x	PRON
ejpam-4343	767	6	}	}	PUNCT
ejpam-4343	767	7	)	)	PUNCT
ejpam-4343	768	1	=	=	SYM
ejpam-4343	768	2	pıcl({y	pıcl({y	PROPN
ejpam-4343	768	3	}	}	PUNCT
ejpam-4343	768	4	)	)	PUNCT
ejpam-4343	768	5	.	.	PUNCT
ejpam-4343	769	1	(	(	PUNCT
ejpam-4343	769	2	4	4	X
ejpam-4343	769	3	)	)	PUNCT
ejpam-4343	769	4	⇒	⇒	NOUN
ejpam-4343	769	5	(	(	PUNCT
ejpam-4343	769	6	1	1	NUM
ejpam-4343	769	7	):	):	PUNCT
ejpam-4343	769	8	let	let	VERB
ejpam-4343	769	9	u	u	PRON
ejpam-4343	769	10	be	be	AUX
ejpam-4343	769	11	any	any	DET
ejpam-4343	769	12	pre	pre	ADJ
ejpam-4343	769	13	-	-	ADJ
ejpam-4343	769	14	i	i	PRON
ejpam-4343	769	15	-open	-open	NOUN
ejpam-4343	769	16	set	set	VERB
ejpam-4343	769	17	and	and	CCONJ
ejpam-4343	769	18	x	x	SYM
ejpam-4343	769	19	∈	∈	PROPN
ejpam-4343	769	20	u	u	NOUN
ejpam-4343	769	21	.	.	PUNCT
ejpam-4343	770	1	for	for	ADP
ejpam-4343	770	2	each	each	DET
ejpam-4343	770	3	y	y	PROPN
ejpam-4343	770	4	̸∈	̸∈	PROPN
ejpam-4343	770	5	u	u	PROPN
ejpam-4343	770	6	,	,	PUNCT
ejpam-4343	770	7	we	we	PRON
ejpam-4343	770	8	have	have	VERB
ejpam-4343	770	9	pıcl({y	pıcl({y	NUM
ejpam-4343	770	10	}	}	PUNCT
ejpam-4343	770	11	)	)	PUNCT
ejpam-4343	770	12	∩	∩	ADJ
ejpam-4343	770	13	u	u	NOUN
ejpam-4343	770	14	=	=	NOUN
ejpam-4343	770	15	∅	∅	NOUN
ejpam-4343	770	16	and	and	CCONJ
ejpam-4343	770	17	so	so	ADV
ejpam-4343	770	18	x	x	X
ejpam-4343	770	19	̸∈	̸∈	PROPN
ejpam-4343	770	20	pıcl({y	pıcl({y	PROPN
ejpam-4343	770	21	}	}	PUNCT
ejpam-4343	770	22	)	)	PUNCT
ejpam-4343	770	23	.	.	PUNCT
ejpam-4343	771	1	therefore	therefore	ADV
ejpam-4343	771	2	,	,	PUNCT
ejpam-4343	771	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	771	4	}	}	PUNCT
ejpam-4343	771	5	)	)	PUNCT
ejpam-4343	771	6	̸=	̸=	PROPN
ejpam-4343	771	7	pıcl({y	pıcl({y	NUM
ejpam-4343	771	8	}	}	PUNCT
ejpam-4343	771	9	)	)	PUNCT
ejpam-4343	771	10	.	.	PUNCT
ejpam-4343	772	1	by	by	ADP
ejpam-4343	772	2	(	(	PUNCT
ejpam-4343	772	3	4	4	NUM
ejpam-4343	772	4	)	)	PUNCT
ejpam-4343	772	5	,	,	PUNCT
ejpam-4343	772	6	for	for	ADP
ejpam-4343	772	7	each	each	DET
ejpam-4343	772	8	y	y	PROPN
ejpam-4343	772	9	̸∈	̸∈	PROPN
ejpam-4343	772	10	u	u	PROPN
ejpam-4343	772	11	,	,	PUNCT
ejpam-4343	772	12	pıcl({x})∩pıcl({y	pıcl({x})∩pıcl({y	PROPN
ejpam-4343	772	13	}	}	PUNCT
ejpam-4343	772	14	)	)	PUNCT
ejpam-4343	772	15	=	=	PUNCT
ejpam-4343	772	16	∅.	∅.	PROPN
ejpam-4343	772	17	sincex−u	sincex−u	NOUN
ejpam-4343	772	18	is	be	AUX
ejpam-4343	772	19	pre	pre	ADJ
ejpam-4343	772	20	-	-	ADJ
ejpam-4343	772	21	i	i	PRON
ejpam-4343	772	22	-closed	-close	VERB
ejpam-4343	772	23	,	,	PUNCT
ejpam-4343	772	24	y	y	PROPN
ejpam-4343	772	25	∈	∈	PROPN
ejpam-4343	772	26	pıcl({y	pıcl({y	PROPN
ejpam-4343	772	27	}	}	PUNCT
ejpam-4343	772	28	)	)	PUNCT
ejpam-4343	773	1	⊆	⊆	NUM
ejpam-4343	773	2	x−u	x−u	X
ejpam-4343	773	3	and	and	CCONJ
ejpam-4343	773	4	x	x	SYM
ejpam-4343	773	5	−	−	NOUN
ejpam-4343	773	6	u	u	NOUN
ejpam-4343	773	7	=	=	PUNCT
ejpam-4343	773	8	∪y∈x−u	∪y∈x−u	NOUN
ejpam-4343	773	9	pıcl({y	pıcl({y	NOUN
ejpam-4343	773	10	}	}	PUNCT
ejpam-4343	773	11	)	)	PUNCT
ejpam-4343	773	12	.	.	PUNCT
ejpam-4343	774	1	thus	thus	ADV
ejpam-4343	774	2	,	,	PUNCT
ejpam-4343	774	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	774	4	}	}	PUNCT
ejpam-4343	774	5	)	)	PUNCT
ejpam-4343	774	6	∩	∩	NOUN
ejpam-4343	774	7	(	(	PUNCT
ejpam-4343	774	8	x	x	SYM
ejpam-4343	774	9	−	−	PROPN
ejpam-4343	774	10	u	u	NOUN
ejpam-4343	774	11	)	)	PUNCT
ejpam-4343	774	12	=	=	SYM
ejpam-4343	774	13	pıcl({x	pıcl({x	PROPN
ejpam-4343	774	14	}	}	PUNCT
ejpam-4343	774	15	)	)	PUNCT
ejpam-4343	774	16	∩	∩	NOUN
ejpam-4343	775	1	[	[	X
ejpam-4343	775	2	∪y∈x−v	∪y∈x−v	NOUN
ejpam-4343	775	3	pıcl({y	pıcl({y	PROPN
ejpam-4343	775	4	}	}	PUNCT
ejpam-4343	775	5	)	)	PUNCT
ejpam-4343	775	6	]	]	PUNCT
ejpam-4343	776	1	=	=	PUNCT
ejpam-4343	776	2	∪y∈x−u	∪y∈x−u	NOUN
ejpam-4343	776	3	[	[	X
ejpam-4343	776	4	pıcl({x	pıcl({x	NUM
ejpam-4343	776	5	}	}	PUNCT
ejpam-4343	776	6	)	)	PUNCT
ejpam-4343	776	7	∩	∩	NOUN
ejpam-4343	776	8	pıcl({y	pıcl({y	PROPN
ejpam-4343	776	9	}	}	PUNCT
ejpam-4343	776	10	)	)	PUNCT
ejpam-4343	776	11	]	]	PUNCT
ejpam-4343	777	1	=	=	PUNCT
ejpam-4343	777	2	∅	∅	NOUN
ejpam-4343	777	3	and	and	CCONJ
ejpam-4343	777	4	hence	hence	ADV
ejpam-4343	777	5	pıcl({x	pıcl({x	NUM
ejpam-4343	777	6	}	}	PUNCT
ejpam-4343	777	7	)	)	PUNCT
ejpam-4343	777	8	⊆	⊆	NUM
ejpam-4343	777	9	u	u	NOUN
ejpam-4343	777	10	.	.	PUNCT
ejpam-4343	778	1	this	this	PRON
ejpam-4343	778	2	shows	show	VERB
ejpam-4343	778	3	that	that	SCONJ
ejpam-4343	778	4	(	(	PUNCT
ejpam-4343	778	5	x	x	X
ejpam-4343	778	6	,	,	PUNCT
ejpam-4343	778	7	τ	τ	PROPN
ejpam-4343	778	8	,	,	PUNCT
ejpam-4343	778	9	i	i	PROPN
ejpam-4343	778	10	)	)	PUNCT
ejpam-4343	778	11	is	be	AUX
ejpam-4343	778	12	a	a	DET
ejpam-4343	778	13	pre	pre	NOUN
ejpam-4343	778	14	-	-	ADJ
ejpam-4343	778	15	i	i	PRON
ejpam-4343	778	16	-r0	-r0	PROPN
ejpam-4343	778	17	space	space	NOUN
ejpam-4343	778	18	.	.	PUNCT
ejpam-4343	779	1	corollary	corollary	ADJ
ejpam-4343	779	2	2	2	NUM
ejpam-4343	779	3	.	.	PUNCT
ejpam-4343	780	1	an	an	DET
ejpam-4343	780	2	ideal	ideal	ADJ
ejpam-4343	780	3	topological	topological	ADJ
ejpam-4343	780	4	space	space	NOUN
ejpam-4343	780	5	(	(	PUNCT
ejpam-4343	780	6	x	x	X
ejpam-4343	780	7	,	,	PUNCT
ejpam-4343	780	8	τ	τ	PROPN
ejpam-4343	780	9	,	,	PUNCT
ejpam-4343	780	10	i	i	PROPN
ejpam-4343	780	11	)	)	PUNCT
ejpam-4343	780	12	is	be	AUX
ejpam-4343	780	13	pre	pre	VERB
ejpam-4343	780	14	-	-	ADJ
ejpam-4343	780	15	i	i	PRON
ejpam-4343	780	16	-r0	-r0	INTJ
ejpam-4343	781	1	if	if	SCONJ
ejpam-4343	781	2	and	and	CCONJ
ejpam-4343	781	3	only	only	ADV
ejpam-4343	781	4	if	if	SCONJ
ejpam-4343	781	5	,	,	PUNCT
ejpam-4343	781	6	for	for	ADP
ejpam-4343	781	7	each	each	DET
ejpam-4343	781	8	x	x	NOUN
ejpam-4343	781	9	,	,	PUNCT
ejpam-4343	781	10	y	y	PROPN
ejpam-4343	781	11	∈	∈	PROPN
ejpam-4343	781	12	x	x	PROPN
ejpam-4343	781	13	,	,	PUNCT
ejpam-4343	781	14	pıcl({x	pıcl({x	PROPN
ejpam-4343	781	15	}	}	PUNCT
ejpam-4343	781	16	)	)	PUNCT
ejpam-4343	781	17	̸=	̸=	PROPN
ejpam-4343	781	18	pıcl({y	pıcl({y	NUM
ejpam-4343	781	19	}	}	PUNCT
ejpam-4343	781	20	)	)	PUNCT
ejpam-4343	781	21	implies	imply	VERB
ejpam-4343	781	22	pıcl({x	pıcl({x	X
ejpam-4343	781	23	}	}	PUNCT
ejpam-4343	781	24	)	)	PUNCT
ejpam-4343	781	25	∩	∩	NOUN
ejpam-4343	781	26	pıcl({y	pıcl({y	X
ejpam-4343	781	27	}	}	PUNCT
ejpam-4343	781	28	)	)	PUNCT
ejpam-4343	782	1	=	=	PUNCT
ejpam-4343	782	2	∅.	∅.	NOUN
ejpam-4343	782	3	proof	proof	NOUN
ejpam-4343	782	4	.	.	PUNCT
ejpam-4343	783	1	this	this	PRON
ejpam-4343	783	2	is	be	AUX
ejpam-4343	783	3	obvious	obvious	ADJ
ejpam-4343	783	4	by	by	ADP
ejpam-4343	783	5	theorem	theorem	NOUN
ejpam-4343	783	6	13(4	13(4	NUM
ejpam-4343	783	7	)	)	PUNCT
ejpam-4343	783	8	.	.	PUNCT
ejpam-4343	784	1	conversely	conversely	ADV
ejpam-4343	784	2	,	,	PUNCT
ejpam-4343	784	3	let	let	VERB
ejpam-4343	784	4	u	u	PRON
ejpam-4343	784	5	be	be	AUX
ejpam-4343	784	6	any	any	DET
ejpam-4343	784	7	pre	pre	ADJ
ejpam-4343	784	8	-	-	ADJ
ejpam-4343	784	9	i	i	PRON
ejpam-4343	784	10	-open	-open	NOUN
ejpam-4343	784	11	set	set	VERB
ejpam-4343	784	12	and	and	CCONJ
ejpam-4343	784	13	x	x	SYM
ejpam-4343	784	14	∈	∈	PROPN
ejpam-4343	784	15	u	u	NOUN
ejpam-4343	784	16	.	.	PUNCT
ejpam-4343	785	1	for	for	ADP
ejpam-4343	785	2	each	each	DET
ejpam-4343	785	3	y	y	PROPN
ejpam-4343	785	4	̸∈	̸∈	PROPN
ejpam-4343	785	5	u	u	PROPN
ejpam-4343	785	6	,	,	PUNCT
ejpam-4343	785	7	we	we	PRON
ejpam-4343	785	8	have	have	VERB
ejpam-4343	785	9	pıcl({y	pıcl({y	NUM
ejpam-4343	785	10	}	}	PUNCT
ejpam-4343	785	11	)	)	PUNCT
ejpam-4343	785	12	∩	∩	ADJ
ejpam-4343	785	13	u	u	NOUN
ejpam-4343	785	14	=	=	PUNCT
ejpam-4343	785	15	∅.	∅.	VERB
ejpam-4343	785	16	thus	thus	ADV
ejpam-4343	785	17	,	,	PUNCT
ejpam-4343	785	18	x	x	PROPN
ejpam-4343	785	19	̸∈	̸∈	PROPN
ejpam-4343	785	20	pıcl({y	pıcl({y	PROPN
ejpam-4343	785	21	}	}	PUNCT
ejpam-4343	785	22	)	)	PUNCT
ejpam-4343	785	23	and	and	CCONJ
ejpam-4343	785	24	hence	hence	ADV
ejpam-4343	785	25	pıcl({x	pıcl({x	PROPN
ejpam-4343	785	26	}	}	PUNCT
ejpam-4343	785	27	)	)	PUNCT
ejpam-4343	785	28	̸=	̸=	PROPN
ejpam-4343	785	29	pıcl({y	pıcl({y	NUM
ejpam-4343	785	30	}	}	PUNCT
ejpam-4343	785	31	)	)	PUNCT
ejpam-4343	785	32	.	.	PUNCT
ejpam-4343	786	1	by	by	ADP
ejpam-4343	786	2	the	the	DET
ejpam-4343	786	3	hypothesis	hypothesis	NOUN
ejpam-4343	786	4	,	,	PUNCT
ejpam-4343	786	5	pıcl({x	pıcl({x	PROPN
ejpam-4343	786	6	}	}	PUNCT
ejpam-4343	786	7	)	)	PUNCT
ejpam-4343	786	8	∩	∩	NOUN
ejpam-4343	786	9	pıcl({y	pıcl({y	X
ejpam-4343	786	10	}	}	PUNCT
ejpam-4343	786	11	)	)	PUNCT
ejpam-4343	787	1	=	=	NOUN
ejpam-4343	787	2	∅	∅	NOUN
ejpam-4343	788	1	and	and	CCONJ
ejpam-4343	788	2	so	so	ADV
ejpam-4343	788	3	y	y	PROPN
ejpam-4343	788	4	̸∈	̸∈	PROPN
ejpam-4343	788	5	pıcl({x	pıcl({x	PROPN
ejpam-4343	788	6	}	}	PUNCT
ejpam-4343	788	7	)	)	PUNCT
ejpam-4343	788	8	.	.	PUNCT
ejpam-4343	789	1	therefore	therefore	ADV
ejpam-4343	789	2	,	,	PUNCT
ejpam-4343	789	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	789	4	}	}	PUNCT
ejpam-4343	789	5	)	)	PUNCT
ejpam-4343	789	6	⊆	⊆	NUM
ejpam-4343	789	7	u	u	NOUN
ejpam-4343	789	8	.	.	PUNCT
ejpam-4343	790	1	this	this	PRON
ejpam-4343	790	2	shows	show	VERB
ejpam-4343	790	3	that	that	SCONJ
ejpam-4343	790	4	(	(	PUNCT
ejpam-4343	790	5	x	x	X
ejpam-4343	790	6	,	,	PUNCT
ejpam-4343	790	7	τ	τ	PROPN
ejpam-4343	790	8	,	,	PUNCT
ejpam-4343	790	9	i	i	PROPN
ejpam-4343	790	10	)	)	PUNCT
ejpam-4343	790	11	is	be	AUX
ejpam-4343	790	12	a	a	DET
ejpam-4343	790	13	pre	pre	NOUN
ejpam-4343	790	14	-	-	ADJ
ejpam-4343	790	15	i	i	PRON
ejpam-4343	790	16	-r0	-r0	PROPN
ejpam-4343	790	17	space	space	NOUN
ejpam-4343	790	18	.	.	PUNCT
ejpam-4343	791	1	lemma	lemma	PROPN
ejpam-4343	791	2	8	8	NUM
ejpam-4343	791	3	.	.	PUNCT
ejpam-4343	792	1	let	let	AUX
ejpam-4343	792	2	(	(	PUNCT
ejpam-4343	792	3	x	x	X
ejpam-4343	792	4	,	,	PUNCT
ejpam-4343	792	5	τ	τ	PROPN
ejpam-4343	792	6	,	,	PUNCT
ejpam-4343	792	7	i	i	PRON
ejpam-4343	792	8	)	)	PUNCT
ejpam-4343	792	9	be	be	AUX
ejpam-4343	792	10	an	an	DET
ejpam-4343	792	11	ideal	ideal	ADJ
ejpam-4343	792	12	topological	topological	ADJ
ejpam-4343	792	13	space	space	NOUN
ejpam-4343	792	14	and	and	CCONJ
ejpam-4343	792	15	x	x	NOUN
ejpam-4343	792	16	,	,	PUNCT
ejpam-4343	792	17	y	y	PROPN
ejpam-4343	792	18	∈	∈	PROPN
ejpam-4343	792	19	x.	x.	NOUN
ejpam-4343	793	1	then	then	ADV
ejpam-4343	793	2	,	,	PUNCT
ejpam-4343	793	3	y	y	PROPN
ejpam-4343	793	4	∈	∈	PROPN
ejpam-4343	793	5	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	793	6	}	}	PUNCT
ejpam-4343	793	7	)	)	PUNCT
ejpam-4343	794	1	if	if	SCONJ
ejpam-4343	794	2	and	and	CCONJ
ejpam-4343	794	3	only	only	ADV
ejpam-4343	794	4	if	if	SCONJ
ejpam-4343	794	5	x	x	PROPN
ejpam-4343	794	6	∈	∈	NOUN
ejpam-4343	794	7	pıcl({y	pıcl({y	PROPN
ejpam-4343	794	8	}	}	PUNCT
ejpam-4343	794	9	)	)	PUNCT
ejpam-4343	794	10	.	.	PUNCT
ejpam-4343	795	1	c.	c.	PROPN
ejpam-4343	795	2	boonpok	boonpok	PROPN
ejpam-4343	795	3	/	/	SYM
ejpam-4343	795	4	eur	eur	PROPN
ejpam-4343	795	5	.	.	PUNCT
ejpam-4343	796	1	j.	j.	PROPN
ejpam-4343	796	2	pure	pure	PROPN
ejpam-4343	796	3	appl	appl	PROPN
ejpam-4343	796	4	.	.	PROPN
ejpam-4343	796	5	math	math	PROPN
ejpam-4343	796	6	,	,	PUNCT
ejpam-4343	796	7	15	15	NUM
ejpam-4343	796	8	(	(	PUNCT
ejpam-4343	796	9	3	3	NUM
ejpam-4343	796	10	)	)	PUNCT
ejpam-4343	796	11	(	(	PUNCT
ejpam-4343	796	12	2022	2022	NUM
ejpam-4343	796	13	)	)	PUNCT
ejpam-4343	796	14	,	,	PUNCT
ejpam-4343	796	15	1023	1023	NUM
ejpam-4343	796	16	-	-	SYM
ejpam-4343	796	17	1046	1046	NUM
ejpam-4343	796	18	1042	1042	NUM
ejpam-4343	796	19	proof	proof	NOUN
ejpam-4343	796	20	.	.	PUNCT
ejpam-4343	796	21	suppose	suppose	VERB
ejpam-4343	796	22	that	that	SCONJ
ejpam-4343	796	23	y	y	PROPN
ejpam-4343	796	24	̸∈	̸∈	PROPN
ejpam-4343	796	25	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	796	26	}	}	PUNCT
ejpam-4343	796	27	)	)	PUNCT
ejpam-4343	796	28	.	.	PUNCT
ejpam-4343	797	1	then	then	ADV
ejpam-4343	797	2	,	,	PUNCT
ejpam-4343	797	3	there	there	PRON
ejpam-4343	797	4	exists	exist	VERB
ejpam-4343	797	5	a	a	DET
ejpam-4343	797	6	pre	pre	ADJ
ejpam-4343	797	7	-	-	ADJ
ejpam-4343	797	8	i	i	PRON
ejpam-4343	797	9	-open	-open	NOUN
ejpam-4343	797	10	set	set	VERB
ejpam-4343	797	11	v	v	NOUN
ejpam-4343	797	12	containing	contain	VERB
ejpam-4343	797	13	x	x	PUNCT
ejpam-4343	797	14	such	such	ADJ
ejpam-4343	797	15	that	that	SCONJ
ejpam-4343	797	16	y	y	PROPN
ejpam-4343	797	17	̸∈	̸∈	PROPN
ejpam-4343	797	18	v	v	PROPN
ejpam-4343	797	19	.	.	PUNCT
ejpam-4343	798	1	therefore	therefore	ADV
ejpam-4343	798	2	,	,	PUNCT
ejpam-4343	798	3	we	we	PRON
ejpam-4343	798	4	have	have	VERB
ejpam-4343	798	5	x	x	PROPN
ejpam-4343	798	6	̸∈	̸∈	PROPN
ejpam-4343	798	7	pıcl({y	pıcl({y	PROPN
ejpam-4343	798	8	}	}	PUNCT
ejpam-4343	798	9	)	)	PUNCT
ejpam-4343	798	10	.	.	PUNCT
ejpam-4343	799	1	conversely	conversely	ADV
ejpam-4343	799	2	,	,	PUNCT
ejpam-4343	799	3	suppose	suppose	VERB
ejpam-4343	799	4	that	that	SCONJ
ejpam-4343	799	5	x	x	PROPN
ejpam-4343	799	6	̸∈	̸∈	PROPN
ejpam-4343	799	7	pıcl({y	pıcl({y	PROPN
ejpam-4343	799	8	}	}	PUNCT
ejpam-4343	799	9	)	)	PUNCT
ejpam-4343	799	10	.	.	PUNCT
ejpam-4343	800	1	then	then	ADV
ejpam-4343	800	2	by	by	ADP
ejpam-4343	800	3	lemma	lemma	PROPN
ejpam-4343	800	4	1	1	NUM
ejpam-4343	800	5	,	,	PUNCT
ejpam-4343	800	6	there	there	PRON
ejpam-4343	800	7	exists	exist	VERB
ejpam-4343	800	8	a	a	DET
ejpam-4343	800	9	pre	pre	NOUN
ejpam-4343	800	10	-	-	ADJ
ejpam-4343	800	11	i	i	PRON
ejpam-4343	800	12	open	open	VERB
ejpam-4343	800	13	set	set	VERB
ejpam-4343	800	14	v	v	NOUN
ejpam-4343	800	15	containing	contain	VERB
ejpam-4343	800	16	x	x	PUNCT
ejpam-4343	800	17	such	such	ADJ
ejpam-4343	800	18	that	that	DET
ejpam-4343	800	19	v	v	NOUN
ejpam-4343	800	20	∩	∩	NOUN
ejpam-4343	800	21	{	{	PUNCT
ejpam-4343	800	22	y	y	NOUN
ejpam-4343	800	23	}	}	PUNCT
ejpam-4343	800	24	=	=	PUNCT
ejpam-4343	800	25	∅.	∅.	VERB
ejpam-4343	800	26	therefore	therefore	ADV
ejpam-4343	800	27	,	,	PUNCT
ejpam-4343	800	28	we	we	PRON
ejpam-4343	800	29	have	have	VERB
ejpam-4343	800	30	y	y	PROPN
ejpam-4343	800	31	̸∈	̸∈	PROPN
ejpam-4343	800	32	v	v	PROPN
ejpam-4343	800	33	and	and	CCONJ
ejpam-4343	800	34	hence	hence	ADV
ejpam-4343	800	35	y	y	PROPN
ejpam-4343	800	36	̸∈	̸∈	PROPN
ejpam-4343	800	37	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	800	38	}	}	PUNCT
ejpam-4343	800	39	)	)	PUNCT
ejpam-4343	800	40	.	.	PUNCT
ejpam-4343	801	1	lemma	lemma	PROPN
ejpam-4343	801	2	9	9	X
ejpam-4343	801	3	.	.	PUNCT
ejpam-4343	802	1	let	let	AUX
ejpam-4343	802	2	(	(	PUNCT
ejpam-4343	802	3	x	x	X
ejpam-4343	802	4	,	,	PUNCT
ejpam-4343	802	5	τ	τ	PROPN
ejpam-4343	802	6	,	,	PUNCT
ejpam-4343	802	7	i	i	PRON
ejpam-4343	802	8	)	)	PUNCT
ejpam-4343	802	9	be	be	AUX
ejpam-4343	802	10	an	an	DET
ejpam-4343	802	11	ideal	ideal	ADJ
ejpam-4343	802	12	topological	topological	ADJ
ejpam-4343	802	13	space	space	NOUN
ejpam-4343	802	14	and	and	CCONJ
ejpam-4343	802	15	x	x	NOUN
ejpam-4343	802	16	,	,	PUNCT
ejpam-4343	802	17	y	y	PROPN
ejpam-4343	802	18	∈	∈	PROPN
ejpam-4343	802	19	x.	x.	NOUN
ejpam-4343	802	20	then	then	ADV
ejpam-4343	802	21	,	,	PUNCT
ejpam-4343	802	22	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	802	23	}	}	PUNCT
ejpam-4343	802	24	)	)	PUNCT
ejpam-4343	803	1	=	=	SYM
ejpam-4343	803	2	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	803	3	}	}	PUNCT
ejpam-4343	803	4	)	)	PUNCT
ejpam-4343	804	1	if	if	SCONJ
ejpam-4343	804	2	and	and	CCONJ
ejpam-4343	804	3	only	only	ADV
ejpam-4343	804	4	if	if	SCONJ
ejpam-4343	804	5	pıcl({x	pıcl({x	X
ejpam-4343	804	6	}	}	PUNCT
ejpam-4343	804	7	)	)	PUNCT
ejpam-4343	805	1	=	=	SYM
ejpam-4343	805	2	pıcl({y	pıcl({y	PROPN
ejpam-4343	805	3	}	}	PUNCT
ejpam-4343	805	4	)	)	PUNCT
ejpam-4343	805	5	.	.	PUNCT
ejpam-4343	806	1	proof	proof	NOUN
ejpam-4343	806	2	.	.	PUNCT
ejpam-4343	807	1	let	let	VERB
ejpam-4343	807	2	x	x	PRON
ejpam-4343	807	3	,	,	PUNCT
ejpam-4343	807	4	y	y	PROPN
ejpam-4343	807	5	be	be	VERB
ejpam-4343	807	6	any	any	DET
ejpam-4343	807	7	points	point	NOUN
ejpam-4343	807	8	of	of	ADP
ejpam-4343	807	9	x.	x.	NOUN
ejpam-4343	807	10	suppose	suppose	VERB
ejpam-4343	807	11	that	that	SCONJ
ejpam-4343	807	12	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	807	13	}	}	PUNCT
ejpam-4343	807	14	)	)	PUNCT
ejpam-4343	808	1	=	=	SYM
ejpam-4343	808	2	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	808	3	}	}	PUNCT
ejpam-4343	808	4	)	)	PUNCT
ejpam-4343	808	5	.	.	PUNCT
ejpam-4343	809	1	since	since	SCONJ
ejpam-4343	809	2	x	x	PROPN
ejpam-4343	809	3	∈	∈	PROPN
ejpam-4343	809	4	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	809	5	}	}	PUNCT
ejpam-4343	809	6	)	)	PUNCT
ejpam-4343	809	7	,	,	PUNCT
ejpam-4343	809	8	we	we	PRON
ejpam-4343	809	9	have	have	VERB
ejpam-4343	809	10	x	x	X
ejpam-4343	809	11	∈	∈	PROPN
ejpam-4343	809	12	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	809	13	}	}	PUNCT
ejpam-4343	809	14	)	)	PUNCT
ejpam-4343	809	15	and	and	CCONJ
ejpam-4343	809	16	by	by	ADP
ejpam-4343	809	17	lemma	lemma	PROPN
ejpam-4343	809	18	8	8	NUM
ejpam-4343	809	19	,	,	PUNCT
ejpam-4343	809	20	y	y	PROPN
ejpam-4343	809	21	∈	∈	PROPN
ejpam-4343	809	22	pıcl({x	pıcl({x	PROPN
ejpam-4343	809	23	}	}	PUNCT
ejpam-4343	809	24	)	)	PUNCT
ejpam-4343	809	25	.	.	PUNCT
ejpam-4343	810	1	therefore	therefore	ADV
ejpam-4343	810	2	,	,	PUNCT
ejpam-4343	810	3	pıcl({y	pıcl({y	PROPN
ejpam-4343	810	4	}	}	PUNCT
ejpam-4343	810	5	)	)	PUNCT
ejpam-4343	810	6	⊆	⊆	NUM
ejpam-4343	810	7	pıcl({x	pıcl({x	NUM
ejpam-4343	810	8	}	}	PUNCT
ejpam-4343	810	9	)	)	PUNCT
ejpam-4343	810	10	.	.	PUNCT
ejpam-4343	811	1	similarly	similarly	ADV
ejpam-4343	811	2	,	,	PUNCT
ejpam-4343	811	3	we	we	PRON
ejpam-4343	811	4	have	have	VERB
ejpam-4343	811	5	pıcl({x	pıcl({x	X
ejpam-4343	811	6	}	}	PUNCT
ejpam-4343	811	7	)	)	PUNCT
ejpam-4343	811	8	⊆	⊆	NUM
ejpam-4343	811	9	pıcl({y	pıcl({y	NUM
ejpam-4343	811	10	}	}	PUNCT
ejpam-4343	811	11	)	)	PUNCT
ejpam-4343	811	12	and	and	CCONJ
ejpam-4343	811	13	hence	hence	ADV
ejpam-4343	811	14	pıcl({y	pıcl({y	NUM
ejpam-4343	811	15	}	}	PUNCT
ejpam-4343	811	16	)	)	PUNCT
ejpam-4343	811	17	=	=	PUNCT
ejpam-4343	811	18	pıcl({x	pıcl({x	PROPN
ejpam-4343	811	19	}	}	PUNCT
ejpam-4343	811	20	)	)	PUNCT
ejpam-4343	811	21	.	.	PUNCT
ejpam-4343	812	1	conversely	conversely	ADV
ejpam-4343	812	2	,	,	PUNCT
ejpam-4343	812	3	suppose	suppose	VERB
ejpam-4343	812	4	that	that	SCONJ
ejpam-4343	812	5	pıcl({x	pıcl({x	PROPN
ejpam-4343	812	6	}	}	PUNCT
ejpam-4343	812	7	)	)	PUNCT
ejpam-4343	812	8	=	=	SYM
ejpam-4343	812	9	pıcl({y	pıcl({y	PROPN
ejpam-4343	812	10	}	}	PUNCT
ejpam-4343	812	11	)	)	PUNCT
ejpam-4343	812	12	.	.	PUNCT
ejpam-4343	813	1	since	since	SCONJ
ejpam-4343	813	2	x	x	PROPN
ejpam-4343	813	3	∈	∈	PROPN
ejpam-4343	813	4	pıcl({x	pıcl({x	PROPN
ejpam-4343	813	5	}	}	PUNCT
ejpam-4343	813	6	)	)	PUNCT
ejpam-4343	813	7	,	,	PUNCT
ejpam-4343	813	8	x	x	PUNCT
ejpam-4343	813	9	∈	∈	NOUN
ejpam-4343	813	10	pıcl({y	pıcl({y	PROPN
ejpam-4343	813	11	}	}	PUNCT
ejpam-4343	813	12	)	)	PUNCT
ejpam-4343	813	13	and	and	CCONJ
ejpam-4343	813	14	by	by	ADP
ejpam-4343	813	15	lemma	lemma	PROPN
ejpam-4343	813	16	8	8	NUM
ejpam-4343	813	17	,	,	PUNCT
ejpam-4343	813	18	y	y	PROPN
ejpam-4343	813	19	∈	∈	PROPN
ejpam-4343	813	20	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	813	21	}	}	PUNCT
ejpam-4343	813	22	)	)	PUNCT
ejpam-4343	813	23	.	.	PUNCT
ejpam-4343	814	1	thus	thus	ADV
ejpam-4343	814	2	,	,	PUNCT
ejpam-4343	814	3	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	814	4	}	}	PUNCT
ejpam-4343	814	5	)	)	PUNCT
ejpam-4343	815	1	⊆	⊆	NUM
ejpam-4343	815	2	λp(⋆)(λp(⋆)({x	λp(⋆)(λp(⋆)({x	NOUN
ejpam-4343	815	3	}	}	PUNCT
ejpam-4343	815	4	)	)	PUNCT
ejpam-4343	815	5	)	)	PUNCT
ejpam-4343	816	1	=	=	SYM
ejpam-4343	816	2	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	816	3	}	}	PUNCT
ejpam-4343	816	4	)	)	PUNCT
ejpam-4343	816	5	.	.	PUNCT
ejpam-4343	817	1	similarly	similarly	ADV
ejpam-4343	817	2	,	,	PUNCT
ejpam-4343	817	3	we	we	PRON
ejpam-4343	817	4	have	have	VERB
ejpam-4343	817	5	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	817	6	}	}	PUNCT
ejpam-4343	817	7	)	)	PUNCT
ejpam-4343	818	1	⊆	⊆	NUM
ejpam-4343	818	2	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	818	3	}	}	PUNCT
ejpam-4343	818	4	)	)	PUNCT
ejpam-4343	818	5	and	and	CCONJ
ejpam-4343	818	6	hence	hence	ADV
ejpam-4343	818	7	λp(⋆))({x	λp(⋆))({x	PROPN
ejpam-4343	818	8	}	}	PUNCT
ejpam-4343	818	9	)	)	PUNCT
ejpam-4343	818	10	=	=	SYM
ejpam-4343	818	11	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	818	12	}	}	PUNCT
ejpam-4343	818	13	)	)	PUNCT
ejpam-4343	818	14	.	.	PUNCT
ejpam-4343	819	1	theorem	theorem	VERB
ejpam-4343	819	2	14	14	NUM
ejpam-4343	819	3	.	.	PUNCT
ejpam-4343	820	1	an	an	DET
ejpam-4343	820	2	ideal	ideal	ADJ
ejpam-4343	820	3	topological	topological	ADJ
ejpam-4343	820	4	space	space	NOUN
ejpam-4343	820	5	(	(	PUNCT
ejpam-4343	820	6	x	x	X
ejpam-4343	820	7	,	,	PUNCT
ejpam-4343	820	8	τ	τ	PROPN
ejpam-4343	820	9	,	,	PUNCT
ejpam-4343	820	10	i	i	PROPN
ejpam-4343	820	11	)	)	PUNCT
ejpam-4343	820	12	is	be	AUX
ejpam-4343	820	13	pre	pre	VERB
ejpam-4343	820	14	-	-	ADJ
ejpam-4343	820	15	i	i	PRON
ejpam-4343	820	16	-r0	-r0	INTJ
ejpam-4343	821	1	if	if	SCONJ
ejpam-4343	821	2	and	and	CCONJ
ejpam-4343	821	3	only	only	ADV
ejpam-4343	821	4	if	if	SCONJ
ejpam-4343	821	5	,	,	PUNCT
ejpam-4343	821	6	for	for	ADP
ejpam-4343	821	7	each	each	DET
ejpam-4343	821	8	x	x	NOUN
ejpam-4343	821	9	,	,	PUNCT
ejpam-4343	821	10	y	y	PROPN
ejpam-4343	821	11	∈	∈	PROPN
ejpam-4343	821	12	x	x	NOUN
ejpam-4343	821	13	,	,	PUNCT
ejpam-4343	821	14	λp(⋆)({x	λp(⋆)({x	ADJ
ejpam-4343	821	15	}	}	PUNCT
ejpam-4343	821	16	)	)	PUNCT
ejpam-4343	821	17	̸=	̸=	PROPN
ejpam-4343	821	18	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	821	19	}	}	PUNCT
ejpam-4343	821	20	)	)	PUNCT
ejpam-4343	821	21	implies	imply	VERB
ejpam-4343	821	22	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	821	23	}	}	PUNCT
ejpam-4343	821	24	)	)	PUNCT
ejpam-4343	821	25	∩	∩	ADJ
ejpam-4343	821	26	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	821	27	}	}	PUNCT
ejpam-4343	821	28	)	)	PUNCT
ejpam-4343	822	1	=	=	PUNCT
ejpam-4343	822	2	∅.	∅.	NOUN
ejpam-4343	822	3	proof	proof	NOUN
ejpam-4343	822	4	.	.	PUNCT
ejpam-4343	823	1	let	let	VERB
ejpam-4343	823	2	x	x	PRON
ejpam-4343	823	3	,	,	PUNCT
ejpam-4343	823	4	y	y	PROPN
ejpam-4343	823	5	be	be	VERB
ejpam-4343	823	6	any	any	DET
ejpam-4343	823	7	points	point	NOUN
ejpam-4343	823	8	of	of	ADP
ejpam-4343	823	9	x.	x.	NOUN
ejpam-4343	823	10	suppose	suppose	VERB
ejpam-4343	823	11	that	that	SCONJ
ejpam-4343	823	12	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	823	13	}	}	PUNCT
ejpam-4343	823	14	)	)	PUNCT
ejpam-4343	823	15	∩	∩	ADJ
ejpam-4343	823	16	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	823	17	}	}	PUNCT
ejpam-4343	823	18	)	)	PUNCT
ejpam-4343	823	19	̸=	̸=	PROPN
ejpam-4343	823	20	∅.	∅.	ADV
ejpam-4343	823	21	let	let	VERB
ejpam-4343	823	22	z	z	NOUN
ejpam-4343	823	23	∈	∈	VERB
ejpam-4343	823	24	λp(⋆)({x})∩λp(⋆)({y	λp(⋆)({x})∩λp(⋆)({y	NOUN
ejpam-4343	823	25	}	}	PUNCT
ejpam-4343	823	26	)	)	PUNCT
ejpam-4343	823	27	.	.	PUNCT
ejpam-4343	824	1	then	then	ADV
ejpam-4343	824	2	,	,	PUNCT
ejpam-4343	824	3	z	z	NOUN
ejpam-4343	824	4	∈	∈	PROPN
ejpam-4343	824	5	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	824	6	}	}	PUNCT
ejpam-4343	824	7	)	)	PUNCT
ejpam-4343	824	8	and	and	CCONJ
ejpam-4343	824	9	by	by	ADP
ejpam-4343	824	10	lemma	lemma	PROPN
ejpam-4343	824	11	8	8	NUM
ejpam-4343	824	12	,	,	PUNCT
ejpam-4343	824	13	we	we	PRON
ejpam-4343	824	14	have	have	VERB
ejpam-4343	824	15	x	x	X
ejpam-4343	824	16	∈	∈	NOUN
ejpam-4343	824	17	pıcl({z	pıcl({z	PROPN
ejpam-4343	824	18	}	}	PUNCT
ejpam-4343	824	19	)	)	PUNCT
ejpam-4343	824	20	.	.	PUNCT
ejpam-4343	825	1	therefore	therefore	ADV
ejpam-4343	825	2	,	,	PUNCT
ejpam-4343	825	3	x	x	X
ejpam-4343	825	4	∈	∈	PROPN
ejpam-4343	825	5	pıcl({z	pıcl({z	PROPN
ejpam-4343	825	6	}	}	PUNCT
ejpam-4343	825	7	)	)	PUNCT
ejpam-4343	825	8	∩	∩	PROPN
ejpam-4343	825	9	pıcl({x	pıcl({x	X
ejpam-4343	825	10	}	}	PUNCT
ejpam-4343	825	11	)	)	PUNCT
ejpam-4343	825	12	and	and	CCONJ
ejpam-4343	825	13	by	by	ADP
ejpam-4343	825	14	corollary	corollary	ADJ
ejpam-4343	825	15	2	2	NUM
ejpam-4343	825	16	,	,	PUNCT
ejpam-4343	825	17	x	x	SYM
ejpam-4343	825	18	∈	∈	NOUN
ejpam-4343	825	19	pıcl({z	pıcl({z	PROPN
ejpam-4343	825	20	}	}	PUNCT
ejpam-4343	825	21	)	)	PUNCT
ejpam-4343	825	22	=	=	PUNCT
ejpam-4343	825	23	pıcl({x	pıcl({x	PROPN
ejpam-4343	825	24	}	}	PUNCT
ejpam-4343	825	25	)	)	PUNCT
ejpam-4343	825	26	.	.	PUNCT
ejpam-4343	826	1	similarly	similarly	ADV
ejpam-4343	826	2	,	,	PUNCT
ejpam-4343	826	3	we	we	PRON
ejpam-4343	826	4	have	have	VERB
ejpam-4343	826	5	pıcl({z	pıcl({z	NOUN
ejpam-4343	826	6	}	}	PUNCT
ejpam-4343	826	7	)	)	PUNCT
ejpam-4343	827	1	=	=	PUNCT
ejpam-4343	827	2	pıcl({y	pıcl({y	PROPN
ejpam-4343	827	3	}	}	PUNCT
ejpam-4343	827	4	)	)	PUNCT
ejpam-4343	827	5	and	and	CCONJ
ejpam-4343	827	6	hence	hence	ADV
ejpam-4343	827	7	pıcl({x	pıcl({x	NUM
ejpam-4343	827	8	}	}	PUNCT
ejpam-4343	827	9	)	)	PUNCT
ejpam-4343	827	10	=	=	SYM
ejpam-4343	827	11	pıcl({y	pıcl({y	PROPN
ejpam-4343	827	12	}	}	PUNCT
ejpam-4343	827	13	)	)	PUNCT
ejpam-4343	827	14	.	.	PUNCT
ejpam-4343	828	1	by	by	ADP
ejpam-4343	828	2	lemma	lemma	PROPN
ejpam-4343	828	3	9	9	NUM
ejpam-4343	828	4	,	,	PUNCT
ejpam-4343	828	5	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	828	6	}	}	PUNCT
ejpam-4343	828	7	)	)	PUNCT
ejpam-4343	829	1	=	=	SYM
ejpam-4343	829	2	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	829	3	}	}	PUNCT
ejpam-4343	829	4	)	)	PUNCT
ejpam-4343	829	5	.	.	PUNCT
ejpam-4343	830	1	conversely	conversely	ADV
ejpam-4343	830	2	,	,	PUNCT
ejpam-4343	830	3	let	let	VERB
ejpam-4343	830	4	x	x	PRON
ejpam-4343	830	5	,	,	PUNCT
ejpam-4343	830	6	y	y	PROPN
ejpam-4343	830	7	be	be	VERB
ejpam-4343	830	8	any	any	DET
ejpam-4343	830	9	points	point	NOUN
ejpam-4343	830	10	of	of	ADP
ejpam-4343	830	11	x.	x.	NOUN
ejpam-4343	830	12	suppose	suppose	VERB
ejpam-4343	830	13	that	that	SCONJ
ejpam-4343	830	14	pıcl({x	pıcl({x	PROPN
ejpam-4343	830	15	}	}	PUNCT
ejpam-4343	830	16	)	)	PUNCT
ejpam-4343	830	17	̸=	̸=	PROPN
ejpam-4343	830	18	pıcl({y	pıcl({y	NUM
ejpam-4343	830	19	}	}	PUNCT
ejpam-4343	830	20	)	)	PUNCT
ejpam-4343	830	21	.	.	PUNCT
ejpam-4343	831	1	by	by	ADP
ejpam-4343	831	2	lemma	lemma	PROPN
ejpam-4343	831	3	9	9	NUM
ejpam-4343	831	4	,	,	PUNCT
ejpam-4343	831	5	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	831	6	}	}	PUNCT
ejpam-4343	831	7	)	)	PUNCT
ejpam-4343	832	1	̸=	̸=	PROPN
ejpam-4343	832	2	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	832	3	}	}	PUNCT
ejpam-4343	832	4	)	)	PUNCT
ejpam-4343	832	5	and	and	CCONJ
ejpam-4343	832	6	hence	hence	ADV
ejpam-4343	832	7	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	832	8	}	}	PUNCT
ejpam-4343	832	9	)	)	PUNCT
ejpam-4343	832	10	∩	∩	ADJ
ejpam-4343	832	11	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	832	12	}	}	PUNCT
ejpam-4343	832	13	)	)	PUNCT
ejpam-4343	833	1	=	=	PUNCT
ejpam-4343	833	2	∅.	∅.	VERB
ejpam-4343	833	3	therefore	therefore	ADV
ejpam-4343	833	4	,	,	PUNCT
ejpam-4343	833	5	pıcl({x	pıcl({x	PROPN
ejpam-4343	833	6	}	}	PUNCT
ejpam-4343	833	7	)	)	PUNCT
ejpam-4343	833	8	∩	∩	NOUN
ejpam-4343	833	9	pıcl({y	pıcl({y	X
ejpam-4343	833	10	}	}	PUNCT
ejpam-4343	833	11	)	)	PUNCT
ejpam-4343	833	12	=	=	PUNCT
ejpam-4343	833	13	∅.	∅.	VERB
ejpam-4343	833	14	in	in	ADP
ejpam-4343	833	15	fact	fact	NOUN
ejpam-4343	833	16	,	,	PUNCT
ejpam-4343	833	17	assume	assume	VERB
ejpam-4343	833	18	that	that	SCONJ
ejpam-4343	833	19	z	z	PROPN
ejpam-4343	833	20	∈	∈	PROPN
ejpam-4343	833	21	pıcl({x	pıcl({x	PROPN
ejpam-4343	833	22	}	}	PUNCT
ejpam-4343	833	23	)	)	PUNCT
ejpam-4343	833	24	∩	∩	NOUN
ejpam-4343	833	25	pıcl({y	pıcl({y	PROPN
ejpam-4343	833	26	}	}	PUNCT
ejpam-4343	833	27	)	)	PUNCT
ejpam-4343	833	28	.	.	PUNCT
ejpam-4343	834	1	then	then	ADV
ejpam-4343	834	2	,	,	PUNCT
ejpam-4343	834	3	z	z	PROPN
ejpam-4343	834	4	∈	∈	PROPN
ejpam-4343	834	5	pıcl({x	pıcl({x	PROPN
ejpam-4343	834	6	}	}	PUNCT
ejpam-4343	834	7	)	)	PUNCT
ejpam-4343	834	8	implies	imply	VERB
ejpam-4343	834	9	x	x	PUNCT
ejpam-4343	834	10	∈	∈	PROPN
ejpam-4343	834	11	λp(⋆)({z	λp(⋆)({z	NOUN
ejpam-4343	834	12	}	}	PUNCT
ejpam-4343	834	13	)	)	PUNCT
ejpam-4343	834	14	and	and	CCONJ
ejpam-4343	834	15	hence	hence	ADV
ejpam-4343	834	16	x	x	X
ejpam-4343	834	17	∈	∈	NOUN
ejpam-4343	834	18	λp(⋆)({z	λp(⋆)({z	NOUN
ejpam-4343	834	19	}	}	PUNCT
ejpam-4343	834	20	)	)	PUNCT
ejpam-4343	834	21	∩	∩	NOUN
ejpam-4343	834	22	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	834	23	}	}	PUNCT
ejpam-4343	834	24	)	)	PUNCT
ejpam-4343	834	25	.	.	PUNCT
ejpam-4343	835	1	by	by	ADP
ejpam-4343	835	2	the	the	DET
ejpam-4343	835	3	hypothesis	hypothesis	NOUN
ejpam-4343	835	4	,	,	PUNCT
ejpam-4343	835	5	λp(⋆)({z	λp(⋆)({z	X
ejpam-4343	835	6	}	}	PUNCT
ejpam-4343	835	7	)	)	PUNCT
ejpam-4343	835	8	=	=	SYM
ejpam-4343	835	9	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	835	10	}	}	PUNCT
ejpam-4343	835	11	)	)	PUNCT
ejpam-4343	835	12	and	and	CCONJ
ejpam-4343	835	13	by	by	ADP
ejpam-4343	835	14	lemma	lemma	PROPN
ejpam-4343	835	15	9	9	NUM
ejpam-4343	835	16	,	,	PUNCT
ejpam-4343	835	17	pıcl({z	pıcl({z	PROPN
ejpam-4343	835	18	}	}	PUNCT
ejpam-4343	835	19	)	)	PUNCT
ejpam-4343	835	20	=	=	PUNCT
ejpam-4343	835	21	pıcl({x	pıcl({x	PROPN
ejpam-4343	835	22	}	}	PUNCT
ejpam-4343	835	23	)	)	PUNCT
ejpam-4343	835	24	.	.	PUNCT
ejpam-4343	836	1	similarly	similarly	ADV
ejpam-4343	836	2	,	,	PUNCT
ejpam-4343	836	3	we	we	PRON
ejpam-4343	836	4	have	have	VERB
ejpam-4343	836	5	pıcl({z	pıcl({z	NOUN
ejpam-4343	836	6	}	}	PUNCT
ejpam-4343	836	7	)	)	PUNCT
ejpam-4343	837	1	=	=	PUNCT
ejpam-4343	837	2	pıcl({y	pıcl({y	PROPN
ejpam-4343	837	3	}	}	PUNCT
ejpam-4343	837	4	)	)	PUNCT
ejpam-4343	837	5	and	and	CCONJ
ejpam-4343	837	6	hence	hence	ADV
ejpam-4343	837	7	pıcl({x	pıcl({x	NUM
ejpam-4343	837	8	}	}	PUNCT
ejpam-4343	837	9	)	)	PUNCT
ejpam-4343	837	10	=	=	SYM
ejpam-4343	837	11	pıcl({y	pıcl({y	PROPN
ejpam-4343	837	12	}	}	PUNCT
ejpam-4343	837	13	)	)	PUNCT
ejpam-4343	837	14	.	.	PUNCT
ejpam-4343	838	1	this	this	PRON
ejpam-4343	838	2	contradicts	contradict	VERB
ejpam-4343	838	3	that	that	SCONJ
ejpam-4343	838	4	pıcl({x	pıcl({x	PROPN
ejpam-4343	838	5	}	}	PUNCT
ejpam-4343	838	6	)	)	PUNCT
ejpam-4343	838	7	̸=	̸=	PROPN
ejpam-4343	838	8	pıcl({y	pıcl({y	NUM
ejpam-4343	838	9	}	}	PUNCT
ejpam-4343	838	10	)	)	PUNCT
ejpam-4343	838	11	.	.	PUNCT
ejpam-4343	839	1	thus	thus	ADV
ejpam-4343	839	2	,	,	PUNCT
ejpam-4343	839	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	839	4	}	}	PUNCT
ejpam-4343	839	5	)	)	PUNCT
ejpam-4343	839	6	∩	∩	NOUN
ejpam-4343	839	7	pıcl({y	pıcl({y	X
ejpam-4343	839	8	}	}	PUNCT
ejpam-4343	839	9	)	)	PUNCT
ejpam-4343	840	1	=	=	PUNCT
ejpam-4343	840	2	∅.	∅.	ADP
ejpam-4343	840	3	this	this	PRON
ejpam-4343	840	4	shows	show	VERB
ejpam-4343	840	5	that	that	SCONJ
ejpam-4343	840	6	(	(	PUNCT
ejpam-4343	840	7	x	x	X
ejpam-4343	840	8	,	,	PUNCT
ejpam-4343	840	9	τ	τ	PROPN
ejpam-4343	840	10	,	,	PUNCT
ejpam-4343	840	11	i	i	PROPN
ejpam-4343	840	12	)	)	PUNCT
ejpam-4343	840	13	is	be	AUX
ejpam-4343	840	14	pre	pre	VERB
ejpam-4343	840	15	-	-	ADJ
ejpam-4343	840	16	i	i	PRON
ejpam-4343	840	17	-r0	-r0	PROPN
ejpam-4343	840	18	.	.	PUNCT
ejpam-4343	841	1	theorem	theorem	PROPN
ejpam-4343	841	2	15	15	NUM
ejpam-4343	841	3	.	.	PUNCT
ejpam-4343	842	1	for	for	ADP
ejpam-4343	842	2	an	an	DET
ejpam-4343	842	3	ideal	ideal	ADJ
ejpam-4343	842	4	topological	topological	ADJ
ejpam-4343	842	5	space	space	NOUN
ejpam-4343	842	6	(	(	PUNCT
ejpam-4343	842	7	x	x	X
ejpam-4343	842	8	,	,	PUNCT
ejpam-4343	842	9	τ	τ	PROPN
ejpam-4343	842	10	,	,	PUNCT
ejpam-4343	842	11	i	i	NOUN
ejpam-4343	842	12	)	)	PUNCT
ejpam-4343	842	13	,	,	PUNCT
ejpam-4343	842	14	the	the	DET
ejpam-4343	842	15	following	follow	VERB
ejpam-4343	842	16	properties	property	NOUN
ejpam-4343	842	17	are	be	AUX
ejpam-4343	842	18	equivalent	equivalent	ADJ
ejpam-4343	842	19	:	:	PUNCT
ejpam-4343	842	20	(	(	PUNCT
ejpam-4343	842	21	1	1	X
ejpam-4343	842	22	)	)	PUNCT
ejpam-4343	842	23	(	(	PUNCT
ejpam-4343	842	24	x	x	X
ejpam-4343	842	25	,	,	PUNCT
ejpam-4343	842	26	τ	τ	PROPN
ejpam-4343	842	27	,	,	PUNCT
ejpam-4343	842	28	i	i	PROPN
ejpam-4343	842	29	)	)	PUNCT
ejpam-4343	842	30	is	be	AUX
ejpam-4343	842	31	pre	pre	VERB
ejpam-4343	842	32	-	-	ADJ
ejpam-4343	842	33	i	i	PRON
ejpam-4343	842	34	-r0	-r0	NOUN
ejpam-4343	842	35	;	;	PUNCT
ejpam-4343	842	36	(	(	PUNCT
ejpam-4343	842	37	2	2	X
ejpam-4343	842	38	)	)	PUNCT
ejpam-4343	842	39	x	x	SYM
ejpam-4343	842	40	∈	∈	PROPN
ejpam-4343	842	41	pi	pi	NOUN
ejpam-4343	842	42	cl({y	cl({y	NOUN
ejpam-4343	842	43	}	}	PUNCT
ejpam-4343	842	44	)	)	PUNCT
ejpam-4343	843	1	if	if	SCONJ
ejpam-4343	843	2	and	and	CCONJ
ejpam-4343	843	3	only	only	ADV
ejpam-4343	843	4	if	if	SCONJ
ejpam-4343	843	5	y	y	PROPN
ejpam-4343	843	6	∈	∈	PROPN
ejpam-4343	843	7	pıcl({x	pıcl({x	PROPN
ejpam-4343	843	8	}	}	PUNCT
ejpam-4343	843	9	)	)	PUNCT
ejpam-4343	843	10	.	.	PUNCT
ejpam-4343	844	1	proof	proof	NOUN
ejpam-4343	844	2	.	.	PUNCT
ejpam-4343	845	1	(	(	PUNCT
ejpam-4343	845	2	1	1	X
ejpam-4343	845	3	)	)	PUNCT
ejpam-4343	845	4	⇒	⇒	NOUN
ejpam-4343	845	5	(	(	PUNCT
ejpam-4343	845	6	2	2	NUM
ejpam-4343	845	7	):	):	PUNCT
ejpam-4343	845	8	suppose	suppose	VERB
ejpam-4343	845	9	that	that	SCONJ
ejpam-4343	845	10	(	(	PUNCT
ejpam-4343	845	11	x	x	X
ejpam-4343	845	12	,	,	PUNCT
ejpam-4343	845	13	τ	τ	PROPN
ejpam-4343	845	14	,	,	PUNCT
ejpam-4343	845	15	i	i	PROPN
ejpam-4343	845	16	)	)	PUNCT
ejpam-4343	845	17	is	be	AUX
ejpam-4343	845	18	pre	pre	VERB
ejpam-4343	845	19	-	-	ADJ
ejpam-4343	845	20	i	i	PRON
ejpam-4343	845	21	-r0	-r0	PROPN
ejpam-4343	845	22	and	and	CCONJ
ejpam-4343	845	23	x	x	PUNCT
ejpam-4343	845	24	∈	∈	PROPN
ejpam-4343	845	25	pıcl({y	pıcl({y	PROPN
ejpam-4343	845	26	}	}	PUNCT
ejpam-4343	845	27	)	)	PUNCT
ejpam-4343	845	28	.	.	PUNCT
ejpam-4343	846	1	by	by	ADP
ejpam-4343	846	2	lemma	lemma	PROPN
ejpam-4343	846	3	8	8	NUM
ejpam-4343	846	4	,	,	PUNCT
ejpam-4343	846	5	we	we	PRON
ejpam-4343	846	6	have	have	VERB
ejpam-4343	846	7	y	y	PROPN
ejpam-4343	846	8	∈	∈	PROPN
ejpam-4343	846	9	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	846	10	}	}	PUNCT
ejpam-4343	846	11	)	)	PUNCT
ejpam-4343	846	12	.	.	PUNCT
ejpam-4343	847	1	thus	thus	ADV
ejpam-4343	847	2	,	,	PUNCT
ejpam-4343	847	3	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	847	4	}	}	PUNCT
ejpam-4343	847	5	)	)	PUNCT
ejpam-4343	847	6	∩	∩	ADJ
ejpam-4343	847	7	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	847	8	}	}	PUNCT
ejpam-4343	847	9	)	)	PUNCT
ejpam-4343	847	10	̸=	̸=	NOUN
ejpam-4343	847	11	∅	∅	NOUN
ejpam-4343	847	12	and	and	CCONJ
ejpam-4343	847	13	by	by	ADP
ejpam-4343	847	14	theorem	theorem	NOUN
ejpam-4343	847	15	14	14	NUM
ejpam-4343	847	16	,	,	PUNCT
ejpam-4343	847	17	c.	c.	NOUN
ejpam-4343	847	18	boonpok	boonpok	PROPN
ejpam-4343	847	19	/	/	SYM
ejpam-4343	847	20	eur	eur	PROPN
ejpam-4343	847	21	.	.	PUNCT
ejpam-4343	848	1	j.	j.	PROPN
ejpam-4343	848	2	pure	pure	PROPN
ejpam-4343	848	3	appl	appl	PROPN
ejpam-4343	848	4	.	.	PROPN
ejpam-4343	848	5	math	math	PROPN
ejpam-4343	848	6	,	,	PUNCT
ejpam-4343	848	7	15	15	NUM
ejpam-4343	848	8	(	(	PUNCT
ejpam-4343	848	9	3	3	NUM
ejpam-4343	848	10	)	)	PUNCT
ejpam-4343	848	11	(	(	PUNCT
ejpam-4343	848	12	2022	2022	NUM
ejpam-4343	848	13	)	)	PUNCT
ejpam-4343	848	14	,	,	PUNCT
ejpam-4343	848	15	1023	1023	NUM
ejpam-4343	848	16	-	-	SYM
ejpam-4343	848	17	1046	1046	NUM
ejpam-4343	848	18	1043	1043	NUM
ejpam-4343	848	19	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	848	20	}	}	PUNCT
ejpam-4343	848	21	)	)	PUNCT
ejpam-4343	849	1	=	=	SYM
ejpam-4343	849	2	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	849	3	}	}	PUNCT
ejpam-4343	849	4	)	)	PUNCT
ejpam-4343	849	5	.	.	PUNCT
ejpam-4343	850	1	therefore	therefore	ADV
ejpam-4343	850	2	,	,	PUNCT
ejpam-4343	850	3	x	x	PUNCT
ejpam-4343	850	4	∈	∈	PROPN
ejpam-4343	850	5	λp(⋆)({y	λp(⋆)({y	NOUN
ejpam-4343	850	6	}	}	PUNCT
ejpam-4343	850	7	)	)	PUNCT
ejpam-4343	850	8	and	and	CCONJ
ejpam-4343	850	9	by	by	ADP
ejpam-4343	850	10	lemma	lemma	PROPN
ejpam-4343	850	11	8	8	NUM
ejpam-4343	850	12	,	,	PUNCT
ejpam-4343	850	13	y	y	PROPN
ejpam-4343	850	14	∈	∈	PROPN
ejpam-4343	850	15	pıcl({x	pıcl({x	PROPN
ejpam-4343	850	16	}	}	PUNCT
ejpam-4343	850	17	)	)	PUNCT
ejpam-4343	850	18	.	.	PUNCT
ejpam-4343	851	1	the	the	DET
ejpam-4343	851	2	converse	converse	NOUN
ejpam-4343	851	3	is	be	AUX
ejpam-4343	851	4	similarly	similarly	ADV
ejpam-4343	851	5	shown	show	VERB
ejpam-4343	851	6	.	.	PUNCT
ejpam-4343	852	1	(	(	PUNCT
ejpam-4343	852	2	2	2	X
ejpam-4343	852	3	)	)	PUNCT
ejpam-4343	852	4	⇒	⇒	NOUN
ejpam-4343	852	5	(	(	PUNCT
ejpam-4343	852	6	1	1	NUM
ejpam-4343	852	7	):	):	PUNCT
ejpam-4343	852	8	let	let	VERB
ejpam-4343	852	9	v	v	PART
ejpam-4343	852	10	be	be	AUX
ejpam-4343	852	11	any	any	DET
ejpam-4343	852	12	pre	pre	ADJ
ejpam-4343	852	13	-	-	ADJ
ejpam-4343	852	14	i	i	PRON
ejpam-4343	852	15	-open	-open	NOUN
ejpam-4343	852	16	set	set	VERB
ejpam-4343	852	17	and	and	CCONJ
ejpam-4343	852	18	x	x	PART
ejpam-4343	852	19	∈	∈	PROPN
ejpam-4343	852	20	v	v	NOUN
ejpam-4343	852	21	.	.	PUNCT
ejpam-4343	853	1	for	for	ADP
ejpam-4343	853	2	each	each	DET
ejpam-4343	853	3	y	y	PROPN
ejpam-4343	853	4	̸∈	̸∈	PROPN
ejpam-4343	853	5	v	v	PROPN
ejpam-4343	853	6	,	,	PUNCT
ejpam-4343	853	7	we	we	PRON
ejpam-4343	853	8	have	have	VERB
ejpam-4343	853	9	pıcl({x	pıcl({x	NUM
ejpam-4343	853	10	}	}	PUNCT
ejpam-4343	853	11	)	)	PUNCT
ejpam-4343	853	12	∩	∩	NOUN
ejpam-4343	853	13	v	v	NOUN
ejpam-4343	853	14	=	=	PUNCT
ejpam-4343	853	15	∅.	∅.	NOUN
ejpam-4343	853	16	this	this	PRON
ejpam-4343	853	17	implies	imply	VERB
ejpam-4343	853	18	that	that	SCONJ
ejpam-4343	853	19	x	x	PROPN
ejpam-4343	853	20	̸∈	̸∈	PROPN
ejpam-4343	853	21	pıcl({y	pıcl({y	PROPN
ejpam-4343	853	22	}	}	PUNCT
ejpam-4343	853	23	)	)	PUNCT
ejpam-4343	853	24	and	and	CCONJ
ejpam-4343	853	25	y	y	PROPN
ejpam-4343	853	26	̸∈	̸∈	PROPN
ejpam-4343	853	27	pıcl({x	pıcl({x	PROPN
ejpam-4343	853	28	}	}	PUNCT
ejpam-4343	853	29	)	)	PUNCT
ejpam-4343	853	30	.	.	PUNCT
ejpam-4343	854	1	thus	thus	ADV
ejpam-4343	854	2	,	,	PUNCT
ejpam-4343	854	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	854	4	}	}	PUNCT
ejpam-4343	854	5	)	)	PUNCT
ejpam-4343	854	6	⊆	⊆	NUM
ejpam-4343	854	7	v	v	NOUN
ejpam-4343	854	8	and	and	CCONJ
ejpam-4343	854	9	hence	hence	ADV
ejpam-4343	854	10	(	(	PUNCT
ejpam-4343	854	11	x	x	X
ejpam-4343	854	12	,	,	PUNCT
ejpam-4343	854	13	τ	τ	PROPN
ejpam-4343	854	14	,	,	PUNCT
ejpam-4343	854	15	i	i	PROPN
ejpam-4343	854	16	)	)	PUNCT
ejpam-4343	854	17	is	be	AUX
ejpam-4343	854	18	pre	pre	VERB
ejpam-4343	854	19	-	-	ADJ
ejpam-4343	854	20	i	i	PRON
ejpam-4343	854	21	-r0	-r0	PROPN
ejpam-4343	854	22	.	.	PUNCT
ejpam-4343	855	1	theorem	theorem	VERB
ejpam-4343	855	2	16	16	NUM
ejpam-4343	855	3	.	.	PUNCT
ejpam-4343	856	1	for	for	ADP
ejpam-4343	856	2	an	an	DET
ejpam-4343	856	3	ideal	ideal	ADJ
ejpam-4343	856	4	topological	topological	ADJ
ejpam-4343	856	5	space	space	NOUN
ejpam-4343	856	6	(	(	PUNCT
ejpam-4343	856	7	x	x	X
ejpam-4343	856	8	,	,	PUNCT
ejpam-4343	856	9	τ	τ	PROPN
ejpam-4343	856	10	,	,	PUNCT
ejpam-4343	856	11	i	i	NOUN
ejpam-4343	856	12	)	)	PUNCT
ejpam-4343	856	13	,	,	PUNCT
ejpam-4343	856	14	the	the	DET
ejpam-4343	856	15	following	follow	VERB
ejpam-4343	856	16	properties	property	NOUN
ejpam-4343	856	17	are	be	AUX
ejpam-4343	856	18	equivalent	equivalent	ADJ
ejpam-4343	856	19	:	:	PUNCT
ejpam-4343	856	20	(	(	PUNCT
ejpam-4343	856	21	1	1	X
ejpam-4343	856	22	)	)	PUNCT
ejpam-4343	856	23	(	(	PUNCT
ejpam-4343	856	24	x	x	X
ejpam-4343	856	25	,	,	PUNCT
ejpam-4343	856	26	τ	τ	PROPN
ejpam-4343	856	27	,	,	PUNCT
ejpam-4343	856	28	i	i	PROPN
ejpam-4343	856	29	)	)	PUNCT
ejpam-4343	856	30	is	be	AUX
ejpam-4343	856	31	pre	pre	VERB
ejpam-4343	856	32	-	-	ADJ
ejpam-4343	856	33	i	i	PRON
ejpam-4343	856	34	-r0	-r0	NOUN
ejpam-4343	856	35	.	.	PUNCT
ejpam-4343	857	1	(	(	PUNCT
ejpam-4343	857	2	2	2	X
ejpam-4343	857	3	)	)	PUNCT
ejpam-4343	857	4	for	for	ADP
ejpam-4343	857	5	each	each	DET
ejpam-4343	857	6	nonempty	nonempty	NOUN
ejpam-4343	857	7	subset	subset	VERB
ejpam-4343	857	8	a	a	PRON
ejpam-4343	857	9	of	of	ADP
ejpam-4343	857	10	x	x	X
ejpam-4343	857	11	and	and	CCONJ
ejpam-4343	857	12	each	each	DET
ejpam-4343	857	13	pre	pre	ADJ
ejpam-4343	857	14	-	-	ADJ
ejpam-4343	857	15	i	i	PRON
ejpam-4343	857	16	-open	-open	NOUN
ejpam-4343	857	17	set	set	VERB
ejpam-4343	857	18	v	v	ADP
ejpam-4343	857	19	such	such	DET
ejpam-4343	857	20	that	that	SCONJ
ejpam-4343	857	21	a∩	a∩	PROPN
ejpam-4343	857	22	v	v	ADP
ejpam-4343	857	23	̸=	̸=	PROPN
ejpam-4343	857	24	∅	∅	NOUN
ejpam-4343	857	25	,	,	PUNCT
ejpam-4343	857	26	there	there	PRON
ejpam-4343	857	27	exists	exist	VERB
ejpam-4343	857	28	a	a	DET
ejpam-4343	857	29	pre	pre	NOUN
ejpam-4343	857	30	-	-	ADJ
ejpam-4343	857	31	i	i	PRON
ejpam-4343	857	32	-closed	-close	VERB
ejpam-4343	857	33	set	set	VERB
ejpam-4343	857	34	f	f	PROPN
ejpam-4343	857	35	such	such	ADJ
ejpam-4343	857	36	that	that	SCONJ
ejpam-4343	857	37	a	a	DET
ejpam-4343	857	38	∩	∩	ADJ
ejpam-4343	857	39	f	f	PROPN
ejpam-4343	857	40	̸=	̸=	PROPN
ejpam-4343	857	41	∅	∅	NOUN
ejpam-4343	857	42	and	and	CCONJ
ejpam-4343	857	43	f	f	PROPN
ejpam-4343	857	44	⊆	⊆	NUM
ejpam-4343	857	45	v	v	NOUN
ejpam-4343	857	46	.	.	PUNCT
ejpam-4343	858	1	(	(	PUNCT
ejpam-4343	858	2	3	3	X
ejpam-4343	858	3	)	)	PUNCT
ejpam-4343	858	4	f	f	NOUN
ejpam-4343	858	5	=	=	SYM
ejpam-4343	858	6	λp(⋆)(f	λp(⋆)(f	PROPN
ejpam-4343	858	7	)	)	PUNCT
ejpam-4343	858	8	for	for	ADP
ejpam-4343	858	9	every	every	DET
ejpam-4343	858	10	pre	pre	PROPN
ejpam-4343	858	11	-	-	ADJ
ejpam-4343	858	12	i	i	PRON
ejpam-4343	858	13	-closed	-close	VERB
ejpam-4343	858	14	set	set	VERB
ejpam-4343	858	15	f	f	NOUN
ejpam-4343	858	16	.	.	PUNCT
ejpam-4343	859	1	(	(	PUNCT
ejpam-4343	859	2	4	4	X
ejpam-4343	859	3	)	)	PUNCT
ejpam-4343	859	4	pıcl({x	pıcl({x	NUM
ejpam-4343	859	5	}	}	PUNCT
ejpam-4343	859	6	)	)	PUNCT
ejpam-4343	859	7	=	=	SYM
ejpam-4343	859	8	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	859	9	}	}	PUNCT
ejpam-4343	859	10	)	)	PUNCT
ejpam-4343	859	11	for	for	ADP
ejpam-4343	859	12	each	each	DET
ejpam-4343	859	13	x	x	SYM
ejpam-4343	859	14	∈	∈	PROPN
ejpam-4343	859	15	x.	x.	NOUN
ejpam-4343	859	16	(	(	PUNCT
ejpam-4343	859	17	5	5	NUM
ejpam-4343	859	18	)	)	PUNCT
ejpam-4343	859	19	pıcl({x	pıcl({x	PROPN
ejpam-4343	859	20	}	}	PUNCT
ejpam-4343	859	21	)	)	PUNCT
ejpam-4343	859	22	⊆	⊆	NUM
ejpam-4343	859	23	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	859	24	}	}	PUNCT
ejpam-4343	859	25	)	)	PUNCT
ejpam-4343	859	26	for	for	ADP
ejpam-4343	859	27	each	each	DET
ejpam-4343	859	28	x	x	SYM
ejpam-4343	859	29	∈	∈	PROPN
ejpam-4343	859	30	x.	x.	NOUN
ejpam-4343	859	31	proof	proof	NOUN
ejpam-4343	859	32	.	.	PUNCT
ejpam-4343	860	1	(	(	PUNCT
ejpam-4343	860	2	1	1	X
ejpam-4343	860	3	)	)	PUNCT
ejpam-4343	860	4	⇒	⇒	NOUN
ejpam-4343	860	5	(	(	PUNCT
ejpam-4343	860	6	2	2	NUM
ejpam-4343	860	7	):	):	PUNCT
ejpam-4343	860	8	let	let	VERB
ejpam-4343	860	9	a	a	DET
ejpam-4343	860	10	be	be	AUX
ejpam-4343	860	11	any	any	DET
ejpam-4343	860	12	nonempty	nonempty	NOUN
ejpam-4343	860	13	subset	subset	NOUN
ejpam-4343	860	14	of	of	ADP
ejpam-4343	860	15	x	x	PUNCT
ejpam-4343	860	16	and	and	CCONJ
ejpam-4343	860	17	let	let	VERB
ejpam-4343	860	18	v	v	PART
ejpam-4343	860	19	be	be	AUX
ejpam-4343	860	20	any	any	DET
ejpam-4343	860	21	pre	pre	ADJ
ejpam-4343	860	22	-	-	ADJ
ejpam-4343	860	23	i	i	PRON
ejpam-4343	860	24	-open	-open	NOUN
ejpam-4343	860	25	set	set	VERB
ejpam-4343	860	26	such	such	ADJ
ejpam-4343	860	27	that	that	SCONJ
ejpam-4343	860	28	a	a	DET
ejpam-4343	860	29	∩	∩	NOUN
ejpam-4343	860	30	v	v	ADP
ejpam-4343	860	31	̸=	̸=	PROPN
ejpam-4343	860	32	∅.	∅.	NOUN
ejpam-4343	860	33	then	then	ADV
ejpam-4343	860	34	,	,	PUNCT
ejpam-4343	860	35	there	there	PRON
ejpam-4343	860	36	exists	exist	VERB
ejpam-4343	860	37	x	x	X
ejpam-4343	860	38	∈	∈	PROPN
ejpam-4343	860	39	a	a	DET
ejpam-4343	860	40	∩	∩	ADJ
ejpam-4343	860	41	v	v	NOUN
ejpam-4343	860	42	and	and	CCONJ
ejpam-4343	860	43	hence	hence	ADV
ejpam-4343	860	44	pıcl({x	pıcl({x	PROPN
ejpam-4343	860	45	}	}	PUNCT
ejpam-4343	860	46	)	)	PUNCT
ejpam-4343	860	47	⊆	⊆	NUM
ejpam-4343	860	48	v	v	NOUN
ejpam-4343	860	49	.	.	PUNCT
ejpam-4343	861	1	put	put	VERB
ejpam-4343	861	2	f	f	PROPN
ejpam-4343	861	3	=	=	PUNCT
ejpam-4343	861	4	pıcl({x	pıcl({x	PROPN
ejpam-4343	861	5	}	}	PUNCT
ejpam-4343	861	6	)	)	PUNCT
ejpam-4343	861	7	,	,	PUNCT
ejpam-4343	861	8	then	then	ADV
ejpam-4343	861	9	f	f	PROPN
ejpam-4343	861	10	is	be	AUX
ejpam-4343	861	11	pre	pre	ADJ
ejpam-4343	861	12	-	-	ADJ
ejpam-4343	861	13	i	i	PRON
ejpam-4343	861	14	-closed	-close	VERB
ejpam-4343	861	15	,	,	PUNCT
ejpam-4343	861	16	a	a	DET
ejpam-4343	861	17	∩	∩	ADJ
ejpam-4343	861	18	f	f	PROPN
ejpam-4343	861	19	̸=	̸=	PROPN
ejpam-4343	861	20	∅	∅	NOUN
ejpam-4343	861	21	and	and	CCONJ
ejpam-4343	861	22	f	f	PROPN
ejpam-4343	861	23	⊆	⊆	NUM
ejpam-4343	861	24	v	v	NOUN
ejpam-4343	861	25	.	.	PUNCT
ejpam-4343	862	1	(	(	PUNCT
ejpam-4343	862	2	2	2	X
ejpam-4343	862	3	)	)	PUNCT
ejpam-4343	862	4	⇒	⇒	NOUN
ejpam-4343	862	5	(	(	PUNCT
ejpam-4343	862	6	3	3	NUM
ejpam-4343	862	7	):	):	PUNCT
ejpam-4343	862	8	let	let	VERB
ejpam-4343	862	9	f	f	PRON
ejpam-4343	862	10	be	be	AUX
ejpam-4343	862	11	any	any	DET
ejpam-4343	862	12	pre	pre	NOUN
ejpam-4343	862	13	-	-	ADJ
ejpam-4343	862	14	i	i	PRON
ejpam-4343	862	15	-closed	-close	VERB
ejpam-4343	862	16	set	set	NOUN
ejpam-4343	862	17	and	and	CCONJ
ejpam-4343	862	18	x	x	PART
ejpam-4343	862	19	̸∈	̸∈	PROPN
ejpam-4343	862	20	f	f	PROPN
ejpam-4343	862	21	.	.	PUNCT
ejpam-4343	863	1	then	then	ADV
ejpam-4343	863	2	,	,	PUNCT
ejpam-4343	863	3	x	x	PUNCT
ejpam-4343	863	4	∈	∈	NOUN
ejpam-4343	863	5	x	x	X
ejpam-4343	863	6	−	−	PROPN
ejpam-4343	863	7	f	f	PROPN
ejpam-4343	863	8	and	and	CCONJ
ejpam-4343	863	9	by	by	ADP
ejpam-4343	863	10	(	(	PUNCT
ejpam-4343	863	11	2	2	NUM
ejpam-4343	863	12	)	)	PUNCT
ejpam-4343	863	13	,	,	PUNCT
ejpam-4343	863	14	there	there	PRON
ejpam-4343	863	15	exists	exist	VERB
ejpam-4343	863	16	a	a	DET
ejpam-4343	863	17	pre	pre	NOUN
ejpam-4343	863	18	-	-	ADJ
ejpam-4343	863	19	i	i	PRON
ejpam-4343	863	20	-closed	-close	VERB
ejpam-4343	863	21	set	set	VERB
ejpam-4343	863	22	k	k	ADP
ejpam-4343	863	23	such	such	ADJ
ejpam-4343	863	24	that	that	SCONJ
ejpam-4343	863	25	x	x	SYM
ejpam-4343	863	26	∈	∈	PROPN
ejpam-4343	863	27	k	k	PROPN
ejpam-4343	863	28	and	and	CCONJ
ejpam-4343	863	29	k	k	PROPN
ejpam-4343	863	30	⊆	⊆	NUM
ejpam-4343	863	31	x−f	x−f	PROPN
ejpam-4343	863	32	.	.	PUNCT
ejpam-4343	864	1	now	now	ADV
ejpam-4343	864	2	,	,	PUNCT
ejpam-4343	864	3	put	put	VERB
ejpam-4343	864	4	v	v	NOUN
ejpam-4343	864	5	=	=	SYM
ejpam-4343	864	6	x−k	x−k	PROPN
ejpam-4343	864	7	.	.	PUNCT
ejpam-4343	865	1	then	then	ADV
ejpam-4343	865	2	,	,	PUNCT
ejpam-4343	865	3	v	v	NOUN
ejpam-4343	865	4	is	be	AUX
ejpam-4343	865	5	a	a	DET
ejpam-4343	865	6	pre	pre	ADJ
ejpam-4343	865	7	-	-	ADJ
ejpam-4343	865	8	i	i	PRON
ejpam-4343	865	9	-open	-open	NOUN
ejpam-4343	865	10	set	set	VERB
ejpam-4343	865	11	such	such	ADJ
ejpam-4343	865	12	that	that	SCONJ
ejpam-4343	865	13	f	f	PROPN
ejpam-4343	865	14	⊆	⊆	NUM
ejpam-4343	865	15	v	v	NOUN
ejpam-4343	865	16	and	and	CCONJ
ejpam-4343	865	17	x	x	PART
ejpam-4343	865	18	̸∈	̸∈	PROPN
ejpam-4343	865	19	v	v	NUM
ejpam-4343	865	20	.	.	PUNCT
ejpam-4343	866	1	thus	thus	ADV
ejpam-4343	866	2	,	,	PUNCT
ejpam-4343	866	3	x	x	PROPN
ejpam-4343	866	4	̸∈	̸∈	PROPN
ejpam-4343	866	5	λp(⋆)(f	λp(⋆)(f	PROPN
ejpam-4343	866	6	)	)	PUNCT
ejpam-4343	866	7	and	and	CCONJ
ejpam-4343	866	8	hence	hence	ADV
ejpam-4343	866	9	f	f	PROPN
ejpam-4343	866	10	⊇	⊇	PROPN
ejpam-4343	866	11	λp(⋆)(f	λp(⋆)(f	PROPN
ejpam-4343	866	12	)	)	PUNCT
ejpam-4343	866	13	.	.	PUNCT
ejpam-4343	867	1	on	on	ADP
ejpam-4343	867	2	the	the	DET
ejpam-4343	867	3	other	other	ADJ
ejpam-4343	867	4	hand	hand	NOUN
ejpam-4343	867	5	,	,	PUNCT
ejpam-4343	867	6	we	we	PRON
ejpam-4343	867	7	have	have	VERB
ejpam-4343	867	8	f	f	PROPN
ejpam-4343	867	9	⊆	⊆	NUM
ejpam-4343	867	10	λp(⋆)(f	λp(⋆)(f	PROPN
ejpam-4343	867	11	)	)	PUNCT
ejpam-4343	867	12	.	.	PUNCT
ejpam-4343	868	1	consequently	consequently	ADV
ejpam-4343	868	2	,	,	PUNCT
ejpam-4343	868	3	we	we	PRON
ejpam-4343	868	4	obtain	obtain	VERB
ejpam-4343	868	5	f	f	NOUN
ejpam-4343	868	6	=	=	PUNCT
ejpam-4343	868	7	λp(⋆)(f	λp(⋆)(f	PROPN
ejpam-4343	868	8	)	)	PUNCT
ejpam-4343	868	9	.	.	PUNCT
ejpam-4343	869	1	(	(	PUNCT
ejpam-4343	869	2	3	3	X
ejpam-4343	869	3	)	)	PUNCT
ejpam-4343	869	4	⇒	⇒	NOUN
ejpam-4343	869	5	(	(	PUNCT
ejpam-4343	869	6	4	4	NUM
ejpam-4343	869	7	):	):	PUNCT
ejpam-4343	869	8	let	let	VERB
ejpam-4343	869	9	x	x	PUNCT
ejpam-4343	869	10	∈	∈	PROPN
ejpam-4343	869	11	x	x	X
ejpam-4343	869	12	and	and	CCONJ
ejpam-4343	869	13	y	y	PROPN
ejpam-4343	869	14	̸∈	̸∈	PROPN
ejpam-4343	869	15	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	869	16	}	}	PUNCT
ejpam-4343	869	17	)	)	PUNCT
ejpam-4343	869	18	.	.	PUNCT
ejpam-4343	870	1	then	then	ADV
ejpam-4343	870	2	,	,	PUNCT
ejpam-4343	870	3	there	there	PRON
ejpam-4343	870	4	exists	exist	VERB
ejpam-4343	870	5	a	a	DET
ejpam-4343	870	6	pre	pre	ADJ
ejpam-4343	870	7	-	-	ADJ
ejpam-4343	870	8	i	i	PRON
ejpam-4343	870	9	-open	-open	NOUN
ejpam-4343	870	10	set	set	VERB
ejpam-4343	870	11	u	u	PRON
ejpam-4343	870	12	such	such	ADJ
ejpam-4343	870	13	that	that	SCONJ
ejpam-4343	870	14	x	x	SYM
ejpam-4343	870	15	∈	∈	PROPN
ejpam-4343	870	16	u	u	NOUN
ejpam-4343	870	17	and	and	CCONJ
ejpam-4343	870	18	y	y	PROPN
ejpam-4343	870	19	̸∈	̸∈	PROPN
ejpam-4343	870	20	u	u	PROPN
ejpam-4343	870	21	.	.	PUNCT
ejpam-4343	871	1	therefore	therefore	ADV
ejpam-4343	871	2	,	,	PUNCT
ejpam-4343	871	3	pıcl({y})∩u	pıcl({y})∩u	PROPN
ejpam-4343	871	4	=	=	SYM
ejpam-4343	871	5	∅	∅	NOUN
ejpam-4343	871	6	and	and	CCONJ
ejpam-4343	871	7	by	by	ADP
ejpam-4343	871	8	(	(	PUNCT
ejpam-4343	871	9	3	3	NUM
ejpam-4343	871	10	)	)	PUNCT
ejpam-4343	871	11	,	,	PUNCT
ejpam-4343	871	12	λp(⋆)(pıcl({y}))∩u	λp(⋆)(pıcl({y}))∩u	VERB
ejpam-4343	871	13	=	=	PUNCT
ejpam-4343	871	14	∅.	∅.	ADV
ejpam-4343	871	15	since	since	SCONJ
ejpam-4343	871	16	x	x	PROPN
ejpam-4343	871	17	̸∈	̸∈	PROPN
ejpam-4343	871	18	λp(⋆)(pıcl({y	λp(⋆)(pıcl({y	PROPN
ejpam-4343	871	19	}	}	PUNCT
ejpam-4343	871	20	)	)	PUNCT
ejpam-4343	871	21	)	)	PUNCT
ejpam-4343	871	22	,	,	PUNCT
ejpam-4343	871	23	there	there	PRON
ejpam-4343	871	24	exists	exist	VERB
ejpam-4343	871	25	a	a	DET
ejpam-4343	871	26	pre	pre	ADJ
ejpam-4343	871	27	-	-	ADJ
ejpam-4343	871	28	i	i	PRON
ejpam-4343	871	29	-open	-open	NOUN
ejpam-4343	871	30	set	set	VERB
ejpam-4343	871	31	v	v	ADP
ejpam-4343	871	32	such	such	ADJ
ejpam-4343	871	33	that	that	SCONJ
ejpam-4343	871	34	pıcl({y	pıcl({y	PROPN
ejpam-4343	871	35	}	}	PUNCT
ejpam-4343	871	36	)	)	PUNCT
ejpam-4343	871	37	⊆	⊆	NUM
ejpam-4343	871	38	v	v	NOUN
ejpam-4343	871	39	and	and	CCONJ
ejpam-4343	871	40	x	x	PART
ejpam-4343	871	41	̸∈	̸∈	PROPN
ejpam-4343	871	42	v	v	NUM
ejpam-4343	871	43	.	.	PUNCT
ejpam-4343	872	1	thus	thus	ADV
ejpam-4343	872	2	,	,	PUNCT
ejpam-4343	872	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	872	4	}	}	PUNCT
ejpam-4343	872	5	)	)	PUNCT
ejpam-4343	872	6	∩	∩	NOUN
ejpam-4343	872	7	v	v	X
ejpam-4343	872	8	=	=	PUNCT
ejpam-4343	872	9	∅.	∅.	NOUN
ejpam-4343	872	10	since	since	SCONJ
ejpam-4343	872	11	y	y	PROPN
ejpam-4343	872	12	∈	∈	PROPN
ejpam-4343	872	13	v	v	NOUN
ejpam-4343	872	14	,	,	PUNCT
ejpam-4343	872	15	y	y	PROPN
ejpam-4343	872	16	̸∈	̸∈	PROPN
ejpam-4343	872	17	pıcl({x	pıcl({x	PROPN
ejpam-4343	872	18	}	}	PUNCT
ejpam-4343	872	19	)	)	PUNCT
ejpam-4343	872	20	.	.	PUNCT
ejpam-4343	873	1	therefore	therefore	ADV
ejpam-4343	873	2	,	,	PUNCT
ejpam-4343	873	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	873	4	}	}	PUNCT
ejpam-4343	873	5	)	)	PUNCT
ejpam-4343	873	6	⊆	⊆	NUM
ejpam-4343	873	7	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	873	8	}	}	PUNCT
ejpam-4343	873	9	)	)	PUNCT
ejpam-4343	873	10	.	.	PUNCT
ejpam-4343	874	1	moreover	moreover	ADV
ejpam-4343	874	2	,	,	PUNCT
ejpam-4343	874	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	874	4	}	}	PUNCT
ejpam-4343	874	5	)	)	PUNCT
ejpam-4343	874	6	⊆	⊆	NUM
ejpam-4343	874	7	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	874	8	}	}	PUNCT
ejpam-4343	874	9	)	)	PUNCT
ejpam-4343	874	10	⊆	⊆	NUM
ejpam-4343	874	11	λp(⋆)(pıcl({x	λp(⋆)(pıcl({x	NUM
ejpam-4343	874	12	}	}	PUNCT
ejpam-4343	874	13	)	)	PUNCT
ejpam-4343	874	14	)	)	PUNCT
ejpam-4343	875	1	=	=	PUNCT
ejpam-4343	875	2	pıcl({x	pıcl({x	PROPN
ejpam-4343	875	3	}	}	PUNCT
ejpam-4343	875	4	)	)	PUNCT
ejpam-4343	875	5	.	.	PUNCT
ejpam-4343	876	1	this	this	PRON
ejpam-4343	876	2	shows	show	VERB
ejpam-4343	876	3	that	that	SCONJ
ejpam-4343	876	4	pıcl({x	pıcl({x	PRON
ejpam-4343	876	5	}	}	PUNCT
ejpam-4343	876	6	)	)	PUNCT
ejpam-4343	876	7	=	=	SYM
ejpam-4343	876	8	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	876	9	}	}	PUNCT
ejpam-4343	876	10	)	)	PUNCT
ejpam-4343	876	11	.	.	PUNCT
ejpam-4343	877	1	(	(	PUNCT
ejpam-4343	877	2	4	4	X
ejpam-4343	877	3	)	)	PUNCT
ejpam-4343	877	4	⇒	⇒	NOUN
ejpam-4343	877	5	(	(	PUNCT
ejpam-4343	877	6	5	5	NUM
ejpam-4343	877	7	):	):	PUNCT
ejpam-4343	877	8	the	the	DET
ejpam-4343	877	9	proof	proof	NOUN
ejpam-4343	877	10	is	be	AUX
ejpam-4343	877	11	obvious	obvious	ADJ
ejpam-4343	877	12	.	.	PUNCT
ejpam-4343	878	1	(	(	PUNCT
ejpam-4343	878	2	5	5	X
ejpam-4343	878	3	)	)	PUNCT
ejpam-4343	878	4	⇒	⇒	NOUN
ejpam-4343	878	5	(	(	PUNCT
ejpam-4343	878	6	1	1	NUM
ejpam-4343	878	7	):	):	PUNCT
ejpam-4343	878	8	let	let	VERB
ejpam-4343	878	9	v	v	PART
ejpam-4343	878	10	be	be	AUX
ejpam-4343	878	11	any	any	DET
ejpam-4343	878	12	pre	pre	ADJ
ejpam-4343	878	13	-	-	ADJ
ejpam-4343	878	14	i	i	PRON
ejpam-4343	878	15	-open	-open	NOUN
ejpam-4343	878	16	set	set	VERB
ejpam-4343	878	17	and	and	CCONJ
ejpam-4343	878	18	x	x	PART
ejpam-4343	878	19	∈	∈	NOUN
ejpam-4343	878	20	v	v	NOUN
ejpam-4343	878	21	.	.	PUNCT
ejpam-4343	878	22	suppose	suppose	VERB
ejpam-4343	878	23	that	that	SCONJ
ejpam-4343	878	24	y	y	PROPN
ejpam-4343	878	25	̸∈	̸∈	PROPN
ejpam-4343	878	26	v	v	PROPN
ejpam-4343	878	27	.	.	PUNCT
ejpam-4343	879	1	then	then	ADV
ejpam-4343	879	2	,	,	PUNCT
ejpam-4343	879	3	pıcl({y	pıcl({y	PROPN
ejpam-4343	879	4	}	}	PUNCT
ejpam-4343	879	5	)	)	PUNCT
ejpam-4343	879	6	∩	∩	ADJ
ejpam-4343	879	7	v	v	NOUN
ejpam-4343	879	8	=	=	NOUN
ejpam-4343	879	9	∅	∅	NOUN
ejpam-4343	879	10	and	and	CCONJ
ejpam-4343	879	11	x	x	PUNCT
ejpam-4343	879	12	̸∈	̸∈	PROPN
ejpam-4343	879	13	pıcl({y	pıcl({y	PROPN
ejpam-4343	879	14	}	}	PUNCT
ejpam-4343	879	15	)	)	PUNCT
ejpam-4343	879	16	.	.	PUNCT
ejpam-4343	880	1	by	by	ADP
ejpam-4343	880	2	lemma	lemma	PROPN
ejpam-4343	880	3	8	8	NUM
ejpam-4343	880	4	,	,	PUNCT
ejpam-4343	880	5	y	y	PROPN
ejpam-4343	880	6	̸∈	̸∈	PROPN
ejpam-4343	880	7	λp(⋆)({x	λp(⋆)({x	ADV
ejpam-4343	880	8	}	}	PUNCT
ejpam-4343	880	9	)	)	PUNCT
ejpam-4343	880	10	and	and	CCONJ
ejpam-4343	880	11	by	by	ADP
ejpam-4343	880	12	(	(	PUNCT
ejpam-4343	880	13	5	5	NUM
ejpam-4343	880	14	)	)	PUNCT
ejpam-4343	880	15	,	,	PUNCT
ejpam-4343	880	16	we	we	PRON
ejpam-4343	880	17	have	have	VERB
ejpam-4343	880	18	y	y	PROPN
ejpam-4343	880	19	̸∈	̸∈	PROPN
ejpam-4343	880	20	pıcl({x	pıcl({x	PROPN
ejpam-4343	880	21	}	}	PUNCT
ejpam-4343	880	22	)	)	PUNCT
ejpam-4343	880	23	.	.	PUNCT
ejpam-4343	881	1	thus	thus	ADV
ejpam-4343	881	2	,	,	PUNCT
ejpam-4343	881	3	pıcl({x	pıcl({x	PROPN
ejpam-4343	881	4	}	}	PUNCT
ejpam-4343	881	5	)	)	PUNCT
ejpam-4343	881	6	⊆	⊆	NUM
ejpam-4343	881	7	v	v	NOUN
ejpam-4343	881	8	and	and	CCONJ
ejpam-4343	881	9	hence	hence	ADV
ejpam-4343	881	10	(	(	PUNCT
ejpam-4343	881	11	x	x	X
ejpam-4343	881	12	,	,	PUNCT
ejpam-4343	881	13	τ	τ	PROPN
ejpam-4343	881	14	,	,	PUNCT
ejpam-4343	881	15	i	i	PROPN
ejpam-4343	881	16	)	)	PUNCT
ejpam-4343	881	17	is	be	AUX
ejpam-4343	881	18	pre	pre	VERB
ejpam-4343	881	19	-	-	ADJ
ejpam-4343	881	20	i	i	PRON
ejpam-4343	881	21	-r0	-r0	PROPN
ejpam-4343	881	22	.	.	PUNCT
ejpam-4343	882	1	corollary	corollary	ADJ
ejpam-4343	882	2	3	3	NUM
ejpam-4343	882	3	.	.	PUNCT
ejpam-4343	883	1	an	an	DET
ejpam-4343	883	2	ideal	ideal	ADJ
ejpam-4343	883	3	topological	topological	ADJ
ejpam-4343	883	4	space	space	NOUN
ejpam-4343	883	5	(	(	PUNCT
ejpam-4343	883	6	x	x	X
ejpam-4343	883	7	,	,	PUNCT
ejpam-4343	883	8	τ	τ	PROPN
ejpam-4343	883	9	,	,	PUNCT
ejpam-4343	883	10	i	i	PROPN
ejpam-4343	883	11	)	)	PUNCT
ejpam-4343	883	12	is	be	AUX
ejpam-4343	883	13	pre	pre	VERB
ejpam-4343	883	14	-	-	ADJ
ejpam-4343	883	15	i	i	PRON
ejpam-4343	883	16	-r0	-r0	INTJ
ejpam-4343	884	1	if	if	SCONJ
ejpam-4343	884	2	and	and	CCONJ
ejpam-4343	884	3	only	only	ADV
ejpam-4343	884	4	if	if	SCONJ
ejpam-4343	884	5	λp(⋆)({x	λp(⋆)({x	ADJ
ejpam-4343	884	6	}	}	PUNCT
ejpam-4343	884	7	)	)	PUNCT
ejpam-4343	885	1	⊆	⊆	NUM
ejpam-4343	885	2	pıcl({x	pıcl({x	NUM
ejpam-4343	885	3	}	}	PUNCT
ejpam-4343	885	4	)	)	PUNCT
ejpam-4343	885	5	for	for	SCONJ
ejpam-4343	885	6	each	each	DET
ejpam-4343	885	7	x	x	SYM
ejpam-4343	885	8	∈	∈	PROPN
ejpam-4343	885	9	x.	x.	NOUN
ejpam-4343	885	10	references	reference	VERB
ejpam-4343	885	11	1044	1044	NUM
ejpam-4343	885	12	proof	proof	NOUN
ejpam-4343	885	13	.	.	PUNCT
ejpam-4343	886	1	this	this	PRON
ejpam-4343	886	2	is	be	AUX
ejpam-4343	886	3	obvious	obvious	ADJ
ejpam-4343	886	4	by	by	ADP
ejpam-4343	886	5	theorem	theorem	NOUN
ejpam-4343	886	6	16	16	NUM
ejpam-4343	886	7	.	.	PUNCT
ejpam-4343	887	1	conversely	conversely	ADV
ejpam-4343	887	2	,	,	PUNCT
ejpam-4343	887	3	suppose	suppose	VERB
ejpam-4343	887	4	that	that	SCONJ
ejpam-4343	887	5	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	887	6	}	}	PUNCT
ejpam-4343	887	7	)	)	PUNCT
ejpam-4343	887	8	⊆	⊆	NUM
ejpam-4343	887	9	pıcl({x	pıcl({x	NUM
ejpam-4343	887	10	}	}	PUNCT
ejpam-4343	887	11	)	)	PUNCT
ejpam-4343	887	12	for	for	SCONJ
ejpam-4343	887	13	each	each	DET
ejpam-4343	887	14	x	x	SYM
ejpam-4343	887	15	∈	∈	PROPN
ejpam-4343	887	16	x.	x.	NOUN
ejpam-4343	887	17	let	let	VERB
ejpam-4343	887	18	x	x	X
ejpam-4343	887	19	∈	∈	PROPN
ejpam-4343	887	20	pıcl({y	pıcl({y	PROPN
ejpam-4343	887	21	}	}	PUNCT
ejpam-4343	887	22	)	)	PUNCT
ejpam-4343	887	23	.	.	PUNCT
ejpam-4343	888	1	by	by	ADP
ejpam-4343	888	2	lemma	lemma	PROPN
ejpam-4343	888	3	8	8	NUM
ejpam-4343	888	4	,	,	PUNCT
ejpam-4343	888	5	we	we	PRON
ejpam-4343	888	6	have	have	VERB
ejpam-4343	888	7	y	y	PROPN
ejpam-4343	888	8	∈	∈	PROPN
ejpam-4343	888	9	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	888	10	}	}	PUNCT
ejpam-4343	888	11	)	)	PUNCT
ejpam-4343	888	12	and	and	CCONJ
ejpam-4343	888	13	hence	hence	ADV
ejpam-4343	888	14	y	y	PROPN
ejpam-4343	888	15	∈	∈	PROPN
ejpam-4343	888	16	pıcl({x	pıcl({x	PROPN
ejpam-4343	888	17	}	}	PUNCT
ejpam-4343	888	18	)	)	PUNCT
ejpam-4343	888	19	.	.	PUNCT
ejpam-4343	889	1	similarly	similarly	ADV
ejpam-4343	889	2	,	,	PUNCT
ejpam-4343	889	3	if	if	SCONJ
ejpam-4343	889	4	y	y	PROPN
ejpam-4343	889	5	∈	∈	PROPN
ejpam-4343	889	6	pıcl({x	pıcl({x	PROPN
ejpam-4343	889	7	}	}	PUNCT
ejpam-4343	889	8	)	)	PUNCT
ejpam-4343	889	9	,	,	PUNCT
ejpam-4343	889	10	then	then	ADV
ejpam-4343	889	11	x	x	X
ejpam-4343	889	12	∈	∈	PROPN
ejpam-4343	889	13	pıcl({y	pıcl({y	PROPN
ejpam-4343	889	14	}	}	PUNCT
ejpam-4343	889	15	)	)	PUNCT
ejpam-4343	889	16	.	.	PUNCT
ejpam-4343	890	1	it	it	PRON
ejpam-4343	890	2	follows	follow	VERB
ejpam-4343	890	3	from	from	ADP
ejpam-4343	890	4	theorem	theorem	ADJ
ejpam-4343	890	5	15	15	NUM
ejpam-4343	890	6	that	that	SCONJ
ejpam-4343	890	7	(	(	PUNCT
ejpam-4343	890	8	x	x	X
ejpam-4343	890	9	,	,	PUNCT
ejpam-4343	890	10	τ	τ	PROPN
ejpam-4343	890	11	,	,	PUNCT
ejpam-4343	890	12	i	i	PROPN
ejpam-4343	890	13	)	)	PUNCT
ejpam-4343	890	14	is	be	AUX
ejpam-4343	890	15	pre	pre	VERB
ejpam-4343	890	16	-	-	ADJ
ejpam-4343	890	17	i	i	PRON
ejpam-4343	890	18	-r0	-r0	PROPN
ejpam-4343	890	19	.	.	PUNCT
ejpam-4343	891	1	corollary	corollary	ADJ
ejpam-4343	891	2	4	4	NUM
ejpam-4343	891	3	.	.	PUNCT
ejpam-4343	892	1	an	an	DET
ejpam-4343	892	2	ideal	ideal	ADJ
ejpam-4343	892	3	topological	topological	ADJ
ejpam-4343	892	4	space	space	NOUN
ejpam-4343	892	5	(	(	PUNCT
ejpam-4343	892	6	x	x	X
ejpam-4343	892	7	,	,	PUNCT
ejpam-4343	892	8	τ	τ	PROPN
ejpam-4343	892	9	,	,	PUNCT
ejpam-4343	892	10	i	i	PROPN
ejpam-4343	892	11	)	)	PUNCT
ejpam-4343	892	12	is	be	AUX
ejpam-4343	892	13	pre	pre	VERB
ejpam-4343	892	14	-	-	ADJ
ejpam-4343	892	15	i	i	PRON
ejpam-4343	892	16	-r0	-r0	INTJ
ejpam-4343	893	1	if	if	SCONJ
ejpam-4343	893	2	and	and	CCONJ
ejpam-4343	893	3	only	only	ADV
ejpam-4343	893	4	if	if	SCONJ
ejpam-4343	893	5	≺	≺	NOUN
ejpam-4343	893	6	x	x	VERB
ejpam-4343	893	7	≻p(⋆)=	≻p(⋆)=	PROPN
ejpam-4343	893	8	pıcl({x	pıcl({x	PROPN
ejpam-4343	893	9	}	}	PUNCT
ejpam-4343	893	10	)	)	PUNCT
ejpam-4343	893	11	for	for	ADP
ejpam-4343	893	12	each	each	DET
ejpam-4343	893	13	x	x	SYM
ejpam-4343	893	14	∈	∈	PROPN
ejpam-4343	893	15	x.	x.	NOUN
ejpam-4343	893	16	proof	proof	NOUN
ejpam-4343	893	17	.	.	PUNCT
ejpam-4343	894	1	let	let	VERB
ejpam-4343	894	2	x	x	SYM
ejpam-4343	894	3	∈	∈	PROPN
ejpam-4343	894	4	x.	x.	NOUN
ejpam-4343	894	5	by	by	ADP
ejpam-4343	894	6	theorem	theorem	NOUN
ejpam-4343	894	7	16	16	NUM
ejpam-4343	894	8	,	,	PUNCT
ejpam-4343	894	9	we	we	PRON
ejpam-4343	894	10	have	have	VERB
ejpam-4343	894	11	pıcl({x	pıcl({x	X
ejpam-4343	894	12	}	}	PUNCT
ejpam-4343	894	13	)	)	PUNCT
ejpam-4343	894	14	=	=	SYM
ejpam-4343	894	15	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	894	16	}	}	PUNCT
ejpam-4343	894	17	)	)	PUNCT
ejpam-4343	894	18	and	and	CCONJ
ejpam-4343	894	19	hence	hence	ADV
ejpam-4343	894	20	pıcl({x	pıcl({x	PROPN
ejpam-4343	894	21	}	}	PUNCT
ejpam-4343	894	22	)	)	PUNCT
ejpam-4343	894	23	=	=	SYM
ejpam-4343	894	24	λp(⋆)({x	λp(⋆)({x	ADJ
ejpam-4343	894	25	}	}	PUNCT
ejpam-4343	894	26	)	)	PUNCT
ejpam-4343	894	27	∩	∩	PROPN
ejpam-4343	894	28	pıcl({x	pıcl({x	X
ejpam-4343	894	29	}	}	PUNCT
ejpam-4343	894	30	)	)	PUNCT
ejpam-4343	895	1	=	=	NOUN
ejpam-4343	895	2	≺	≺	NOUN
ejpam-4343	895	3	x	x	SYM
ejpam-4343	895	4	≻p(⋆	≻p(⋆	NOUN
ejpam-4343	895	5	)	)	PUNCT
ejpam-4343	895	6	.	.	PUNCT
ejpam-4343	896	1	conversely	conversely	ADV
ejpam-4343	896	2	,	,	PUNCT
ejpam-4343	896	3	suppose	suppose	VERB
ejpam-4343	896	4	that	that	SCONJ
ejpam-4343	896	5	≺	≺	NOUN
ejpam-4343	896	6	x	x	VERB
ejpam-4343	896	7	≻p(⋆)=	≻p(⋆)=	PROPN
ejpam-4343	896	8	pıcl({x	pıcl({x	PROPN
ejpam-4343	896	9	}	}	PUNCT
ejpam-4343	896	10	)	)	PUNCT
ejpam-4343	896	11	for	for	ADP
ejpam-4343	896	12	each	each	DET
ejpam-4343	896	13	x	x	SYM
ejpam-4343	896	14	∈	∈	PROPN
ejpam-4343	896	15	x.	x.	NOUN
ejpam-4343	896	16	let	let	VERB
ejpam-4343	897	1	x	x	X
ejpam-4343	897	2	∈	∈	PROPN
ejpam-4343	897	3	x.	x.	NOUN
ejpam-4343	897	4	by	by	ADP
ejpam-4343	897	5	the	the	DET
ejpam-4343	897	6	hypothesis	hypothesis	NOUN
ejpam-4343	897	7	,	,	PUNCT
ejpam-4343	897	8	we	we	PRON
ejpam-4343	897	9	have	have	VERB
ejpam-4343	897	10	pıcl({x	pıcl({x	X
ejpam-4343	897	11	}	}	PUNCT
ejpam-4343	897	12	)	)	PUNCT
ejpam-4343	898	1	=	=	NOUN
ejpam-4343	898	2	≺	≺	NOUN
ejpam-4343	898	3	x	x	SYM
ejpam-4343	898	4	≻p(⋆	≻p(⋆	NOUN
ejpam-4343	898	5	)	)	PUNCT
ejpam-4343	898	6	(	(	PUNCT
ejpam-4343	898	7	{	{	PUNCT
ejpam-4343	898	8	x	x	NOUN
ejpam-4343	898	9	}	}	PUNCT
ejpam-4343	898	10	)	)	PUNCT
ejpam-4343	898	11	=	=	PUNCT
ejpam-4343	898	12	pıcl({x	pıcl({x	PROPN
ejpam-4343	898	13	}	}	PUNCT
ejpam-4343	898	14	)	)	PUNCT
ejpam-4343	898	15	∩	∩	NOUN
ejpam-4343	898	16	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	898	17	}	}	PUNCT
ejpam-4343	898	18	)	)	PUNCT
ejpam-4343	898	19	⊆	⊆	NUM
ejpam-4343	898	20	λp(⋆)({x	λp(⋆)({x	NOUN
ejpam-4343	898	21	}	}	PUNCT
ejpam-4343	898	22	)	)	PUNCT
ejpam-4343	898	23	.	.	PUNCT
ejpam-4343	899	1	it	it	PRON
ejpam-4343	899	2	follows	follow	VERB
ejpam-4343	899	3	from	from	ADP
ejpam-4343	899	4	theorem	theorem	ADJ
ejpam-4343	899	5	16	16	NUM
ejpam-4343	899	6	that	that	SCONJ
ejpam-4343	899	7	(	(	PUNCT
ejpam-4343	899	8	x	x	X
ejpam-4343	899	9	,	,	PUNCT
ejpam-4343	899	10	τ	τ	PROPN
ejpam-4343	899	11	,	,	PUNCT
ejpam-4343	899	12	i	i	PROPN
ejpam-4343	899	13	)	)	PUNCT
ejpam-4343	899	14	is	be	AUX
ejpam-4343	899	15	pre	pre	VERB
ejpam-4343	899	16	-	-	ADJ
ejpam-4343	899	17	i	i	PRON
ejpam-4343	899	18	-r0	-r0	NOUN
ejpam-4343	899	19	.	.	PUNCT
ejpam-4343	900	1	7	7	X
ejpam-4343	900	2	.	.	X
ejpam-4343	900	3	conclusion	conclusion	NOUN
ejpam-4343	900	4	topology	topology	NOUN
ejpam-4343	900	5	plays	play	VERB
ejpam-4343	900	6	an	an	DET
ejpam-4343	900	7	important	important	ADJ
ejpam-4343	900	8	role	role	NOUN
ejpam-4343	900	9	in	in	ADP
ejpam-4343	900	10	both	both	CCONJ
ejpam-4343	900	11	pure	pure	ADJ
ejpam-4343	900	12	and	and	CCONJ
ejpam-4343	900	13	applied	apply	VERB
ejpam-4343	900	14	sciences	science	NOUN
ejpam-4343	900	15	such	such	ADJ
ejpam-4343	900	16	as	as	ADP
ejpam-4343	900	17	quantum	quantum	NOUN
ejpam-4343	900	18	physics	physics	NOUN
ejpam-4343	900	19	,	,	PUNCT
ejpam-4343	900	20	high	high	ADJ
ejpam-4343	900	21	energy	energy	NOUN
ejpam-4343	900	22	physics	physics	NOUN
ejpam-4343	900	23	,	,	PUNCT
ejpam-4343	900	24	data	datum	NOUN
ejpam-4343	900	25	mining	mining	NOUN
ejpam-4343	900	26	,	,	PUNCT
ejpam-4343	900	27	computational	computational	ADJ
ejpam-4343	900	28	topology	topology	NOUN
ejpam-4343	900	29	,	,	PUNCT
ejpam-4343	900	30	digital	digital	ADJ
ejpam-4343	900	31	topology	topology	NOUN
ejpam-4343	900	32	and	and	CCONJ
ejpam-4343	900	33	mathematical	mathematical	ADJ
ejpam-4343	900	34	sciences	science	NOUN
ejpam-4343	900	35	.	.	PUNCT
ejpam-4343	901	1	the	the	DET
ejpam-4343	901	2	notions	notion	NOUN
ejpam-4343	901	3	of	of	ADP
ejpam-4343	901	4	closed	closed	ADJ
ejpam-4343	901	5	sets	set	NOUN
ejpam-4343	901	6	and	and	CCONJ
ejpam-4343	901	7	low	low	ADJ
ejpam-4343	901	8	separation	separation	NOUN
ejpam-4343	901	9	axioms	axiom	NOUN
ejpam-4343	901	10	are	be	AUX
ejpam-4343	901	11	fundamental	fundamental	ADJ
ejpam-4343	901	12	with	with	ADP
ejpam-4343	901	13	respect	respect	NOUN
ejpam-4343	901	14	to	to	ADP
ejpam-4343	901	15	the	the	DET
ejpam-4343	901	16	investigation	investigation	NOUN
ejpam-4343	901	17	of	of	ADP
ejpam-4343	901	18	topological	topological	ADJ
ejpam-4343	901	19	spaces	space	NOUN
ejpam-4343	901	20	.	.	PUNCT
ejpam-4343	902	1	various	various	ADJ
ejpam-4343	902	2	types	type	NOUN
ejpam-4343	902	3	of	of	ADP
ejpam-4343	902	4	generalizations	generalization	NOUN
ejpam-4343	902	5	of	of	ADP
ejpam-4343	902	6	closed	closed	ADJ
ejpam-4343	902	7	sets	set	NOUN
ejpam-4343	902	8	and	and	CCONJ
ejpam-4343	902	9	some	some	DET
ejpam-4343	902	10	new	new	ADJ
ejpam-4343	902	11	separation	separation	NOUN
ejpam-4343	902	12	axioms	axiom	NOUN
ejpam-4343	902	13	have	have	AUX
ejpam-4343	902	14	been	be	AUX
ejpam-4343	902	15	researched	research	VERB
ejpam-4343	902	16	by	by	ADP
ejpam-4343	902	17	many	many	ADJ
ejpam-4343	902	18	mathematicians	mathematician	NOUN
ejpam-4343	902	19	.	.	PUNCT
ejpam-4343	903	1	this	this	DET
ejpam-4343	903	2	paper	paper	NOUN
ejpam-4343	903	3	is	be	AUX
ejpam-4343	903	4	concerned	concern	VERB
ejpam-4343	903	5	with	with	ADP
ejpam-4343	903	6	the	the	DET
ejpam-4343	903	7	concepts	concept	NOUN
ejpam-4343	903	8	of	of	ADP
ejpam-4343	903	9	λp(⋆)-sets	λp(⋆)-sets	PRON
ejpam-4343	903	10	and	and	CCONJ
ejpam-4343	903	11	(	(	PUNCT
ejpam-4343	903	12	λ	λ	PROPN
ejpam-4343	903	13	,	,	PUNCT
ejpam-4343	903	14	p(⋆))closed	p(⋆))close	VERB
ejpam-4343	903	15	sets	set	NOUN
ejpam-4343	903	16	which	which	PRON
ejpam-4343	903	17	are	be	AUX
ejpam-4343	903	18	defined	define	VERB
ejpam-4343	903	19	by	by	ADP
ejpam-4343	903	20	utilizing	utilize	VERB
ejpam-4343	903	21	the	the	DET
ejpam-4343	903	22	notions	notion	NOUN
ejpam-4343	903	23	of	of	ADP
ejpam-4343	903	24	pre	pre	ADJ
ejpam-4343	903	25	-	-	ADJ
ejpam-4343	903	26	i	i	PRON
ejpam-4343	903	27	-open	-open	NOUN
ejpam-4343	903	28	sets	set	NOUN
ejpam-4343	903	29	and	and	CCONJ
ejpam-4343	903	30	pre	pre	ADJ
ejpam-4343	903	31	-	-	ADJ
ejpam-4343	903	32	i	i	ADJ
ejpam-4343	903	33	-open	-open	NOUN
ejpam-4343	903	34	sets	set	NOUN
ejpam-4343	903	35	.	.	PUNCT
ejpam-4343	904	1	furthermore	furthermore	ADV
ejpam-4343	904	2	,	,	PUNCT
ejpam-4343	904	3	some	some	DET
ejpam-4343	904	4	properties	property	NOUN
ejpam-4343	904	5	of	of	ADP
ejpam-4343	904	6	(	(	PUNCT
ejpam-4343	904	7	λ	λ	PROPN
ejpam-4343	904	8	,	,	PUNCT
ejpam-4343	904	9	p(⋆))-closed	p(⋆))-close	VERB
ejpam-4343	904	10	sets	set	NOUN
ejpam-4343	904	11	and	and	CCONJ
ejpam-4343	904	12	(	(	PUNCT
ejpam-4343	904	13	λ	λ	INTJ
ejpam-4343	904	14	,	,	PUNCT
ejpam-4343	904	15	p(⋆))-open	p(⋆))-open	ADJ
ejpam-4343	904	16	sets	set	NOUN
ejpam-4343	904	17	are	be	AUX
ejpam-4343	904	18	considered	consider	VERB
ejpam-4343	904	19	.	.	PUNCT
ejpam-4343	905	1	several	several	ADJ
ejpam-4343	905	2	characterizations	characterization	NOUN
ejpam-4343	905	3	of	of	ADP
ejpam-4343	905	4	(	(	PUNCT
ejpam-4343	905	5	λ	λ	PROPN
ejpam-4343	905	6	,	,	PUNCT
ejpam-4343	905	7	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-4343	905	8	functions	function	NOUN
ejpam-4343	905	9	are	be	AUX
ejpam-4343	905	10	obtained	obtain	VERB
ejpam-4343	905	11	.	.	PUNCT
ejpam-4343	906	1	additionally	additionally	ADV
ejpam-4343	906	2	,	,	PUNCT
ejpam-4343	906	3	some	some	DET
ejpam-4343	906	4	characterizations	characterization	NOUN
ejpam-4343	906	5	of	of	ADP
ejpam-4343	906	6	(	(	PUNCT
ejpam-4343	906	7	λ	λ	NOUN
ejpam-4343	906	8	,	,	PUNCT
ejpam-4343	906	9	p(⋆))-extremally	p(⋆))-extremally	ADV
ejpam-4343	906	10	disconnected	disconnected	ADJ
ejpam-4343	906	11	and	and	CCONJ
ejpam-4343	906	12	pre	pre	ADJ
ejpam-4343	906	13	-	-	PROPN
ejpam-4343	906	14	i	i	PRON
ejpam-4343	906	15	-r0	-r0	PROPN
ejpam-4343	906	16	ideal	ideal	ADJ
ejpam-4343	906	17	topological	topological	ADJ
ejpam-4343	906	18	spaces	space	NOUN
ejpam-4343	906	19	are	be	AUX
ejpam-4343	906	20	explored	explore	VERB
ejpam-4343	906	21	.	.	PUNCT
ejpam-4343	907	1	the	the	DET
ejpam-4343	907	2	ideas	idea	NOUN
ejpam-4343	907	3	and	and	CCONJ
ejpam-4343	907	4	results	result	NOUN
ejpam-4343	907	5	of	of	ADP
ejpam-4343	907	6	this	this	DET
ejpam-4343	907	7	paper	paper	NOUN
ejpam-4343	907	8	may	may	AUX
ejpam-4343	907	9	motivate	motivate	VERB
ejpam-4343	907	10	further	further	ADJ
ejpam-4343	907	11	research	research	NOUN
ejpam-4343	907	12	.	.	PUNCT
ejpam-4343	908	1	acknowledgements	acknowledgement	NOUN
ejpam-4343	908	2	this	this	DET
ejpam-4343	908	3	research	research	NOUN
ejpam-4343	908	4	project	project	NOUN
ejpam-4343	908	5	was	be	AUX
ejpam-4343	908	6	financially	financially	ADV
ejpam-4343	908	7	supported	support	VERB
ejpam-4343	908	8	by	by	ADP
ejpam-4343	908	9	mahasarakham	mahasarakham	PROPN
ejpam-4343	908	10	university	university	PROPN
ejpam-4343	908	11	.	.	PUNCT
ejpam-4343	909	1	references	reference	NOUN
ejpam-4343	909	2	[	[	X
ejpam-4343	909	3	1	1	NUM
ejpam-4343	909	4	]	]	PUNCT
ejpam-4343	909	5	a.	a.	NOUN
ejpam-4343	909	6	açıkgöz	açıkgöz	PROPN
ejpam-4343	909	7	,	,	PUNCT
ejpam-4343	909	8	ş.	ş.	PROPN
ejpam-4343	909	9	yüksel	yüksel	PROPN
ejpam-4343	909	10	,	,	PUNCT
ejpam-4343	909	11	and	and	CCONJ
ejpam-4343	909	12	t.	t.	PROPN
ejpam-4343	909	13	noiri	noiri	PROPN
ejpam-4343	909	14	.	.	PUNCT
ejpam-4343	910	1	α	α	X
ejpam-4343	910	2	-	-	PUNCT
ejpam-4343	910	3	i	i	NOUN
ejpam-4343	910	4	-preirresolute	-preirresolute	NOUN
ejpam-4343	910	5	functions	function	NOUN
ejpam-4343	910	6	and	and	CCONJ
ejpam-4343	910	7	β	β	X
ejpam-4343	910	8	-	-	ADJ
ejpam-4343	910	9	i	i	PRON
ejpam-4343	910	10	preirresolute	preirresolute	PROPN
ejpam-4343	910	11	functions	function	NOUN
ejpam-4343	910	12	.	.	PUNCT
ejpam-4343	911	1	bulletin	bulletin	PROPN
ejpam-4343	911	2	malaysian	malaysian	PROPN
ejpam-4343	911	3	mathematical	mathematical	PROPN
ejpam-4343	911	4	sciences	sciences	PROPN
ejpam-4343	911	5	society	society	NOUN
ejpam-4343	911	6	,	,	PUNCT
ejpam-4343	911	7	28:1–8	28:1–8	NUM
ejpam-4343	911	8	,	,	PUNCT
ejpam-4343	911	9	2005	2005	NUM
ejpam-4343	911	10	.	.	PUNCT
ejpam-4343	912	1	[	[	X
ejpam-4343	912	2	2	2	NUM
ejpam-4343	912	3	]	]	PUNCT
ejpam-4343	912	4	a.	a.	NOUN
ejpam-4343	912	5	açıkgöz	açıkgöz	NOUN
ejpam-4343	912	6	,	,	PUNCT
ejpam-4343	912	7	t.	t.	PROPN
ejpam-4343	912	8	noiri	noiri	PROPN
ejpam-4343	912	9	,	,	PUNCT
ejpam-4343	912	10	and	and	CCONJ
ejpam-4343	912	11	ş.	ş.	PROPN
ejpam-4343	912	12	yüksel	yüksel	PROPN
ejpam-4343	912	13	.	.	PUNCT
ejpam-4343	913	1	a	a	DET
ejpam-4343	913	2	decomposition	decomposition	NOUN
ejpam-4343	913	3	of	of	ADP
ejpam-4343	913	4	continuity	continuity	NOUN
ejpam-4343	913	5	in	in	ADP
ejpam-4343	913	6	ideal	ideal	ADJ
ejpam-4343	913	7	topological	topological	ADJ
ejpam-4343	913	8	spaces	space	NOUN
ejpam-4343	913	9	.	.	PUNCT
ejpam-4343	914	1	acta	acta	PROPN
ejpam-4343	914	2	mathematica	mathematica	PROPN
ejpam-4343	914	3	hungarica	hungarica	PROPN
ejpam-4343	914	4	,	,	PUNCT
ejpam-4343	914	5	105:285–289	105:285–289	NUM
ejpam-4343	914	6	,	,	PUNCT
ejpam-4343	914	7	2004	2004	NUM
ejpam-4343	914	8	.	.	PUNCT
ejpam-4343	915	1	[	[	X
ejpam-4343	915	2	3	3	NUM
ejpam-4343	915	3	]	]	PUNCT
ejpam-4343	915	4	a.	a.	NOUN
ejpam-4343	915	5	açıkgöz	açıkgöz	NOUN
ejpam-4343	915	6	,	,	PUNCT
ejpam-4343	915	7	t.	t.	PROPN
ejpam-4343	915	8	noiri	noiri	PROPN
ejpam-4343	915	9	,	,	PUNCT
ejpam-4343	915	10	and	and	CCONJ
ejpam-4343	915	11	ş.	ş.	PROPN
ejpam-4343	915	12	yüksel	yüksel	PROPN
ejpam-4343	915	13	.	.	PUNCT
ejpam-4343	916	1	on	on	ADP
ejpam-4343	916	2	α	α	PROPN
ejpam-4343	916	3	-	-	ADJ
ejpam-4343	916	4	i	i	PRON
ejpam-4343	916	5	-continuous	-continuous	ADJ
ejpam-4343	916	6	and	and	CCONJ
ejpam-4343	916	7	α	α	NOUN
ejpam-4343	916	8	-	-	PUNCT
ejpam-4343	916	9	i	i	NOUN
ejpam-4343	916	10	-open	-open	NOUN
ejpam-4343	916	11	functions	function	NOUN
ejpam-4343	916	12	.	.	PUNCT
ejpam-4343	917	1	acta	acta	PROPN
ejpam-4343	917	2	mathematica	mathematica	PROPN
ejpam-4343	917	3	hungarica	hungarica	PROPN
ejpam-4343	917	4	,	,	PUNCT
ejpam-4343	917	5	105:27–37	105:27–37	PROPN
ejpam-4343	917	6	,	,	PUNCT
ejpam-4343	917	7	2004	2004	NUM
ejpam-4343	917	8	.	.	PUNCT
ejpam-4343	918	1	references	reference	NOUN
ejpam-4343	918	2	1045	1045	NUM
ejpam-4343	918	3	[	[	X
ejpam-4343	918	4	4	4	NUM
ejpam-4343	918	5	]	]	PUNCT
ejpam-4343	918	6	f.	f.	PROPN
ejpam-4343	918	7	g.	g.	PROPN
ejpam-4343	918	8	arenas	arenas	PROPN
ejpam-4343	918	9	,	,	PUNCT
ejpam-4343	918	10	j.	j.	PROPN
ejpam-4343	918	11	dontchev	dontchev	PROPN
ejpam-4343	918	12	,	,	PUNCT
ejpam-4343	918	13	and	and	CCONJ
ejpam-4343	918	14	m.	m.	NOUN
ejpam-4343	918	15	ganster	ganster	NOUN
ejpam-4343	918	16	.	.	PUNCT
ejpam-4343	919	1	on	on	ADP
ejpam-4343	919	2	λ	λ	NOUN
ejpam-4343	919	3	-	-	ADJ
ejpam-4343	919	4	closed	closed	ADJ
ejpam-4343	919	5	sets	set	NOUN
ejpam-4343	919	6	and	and	CCONJ
ejpam-4343	919	7	dual	dual	ADJ
ejpam-4343	919	8	of	of	ADP
ejpam-4343	919	9	generalized	generalized	ADJ
ejpam-4343	919	10	continuity	continuity	NOUN
ejpam-4343	919	11	.	.	PUNCT
ejpam-4343	920	1	questions	question	NOUN
ejpam-4343	920	2	and	and	CCONJ
ejpam-4343	920	3	answers	answer	NOUN
ejpam-4343	920	4	in	in	ADP
ejpam-4343	920	5	general	general	ADJ
ejpam-4343	920	6	topology	topology	NOUN
ejpam-4343	920	7	,	,	PUNCT
ejpam-4343	920	8	15:3–13	15:3–13	NUM
ejpam-4343	920	9	,	,	PUNCT
ejpam-4343	920	10	1997	1997	NUM
ejpam-4343	920	11	.	.	PUNCT
ejpam-4343	921	1	[	[	X
ejpam-4343	921	2	5	5	NUM
ejpam-4343	921	3	]	]	PUNCT
ejpam-4343	921	4	m.	m.	NOUN
ejpam-4343	921	5	caldas	caldas	PROPN
ejpam-4343	921	6	.	.	PUNCT
ejpam-4343	922	1	a	a	DET
ejpam-4343	922	2	separation	separation	NOUN
ejpam-4343	922	3	axiom	axiom	NOUN
ejpam-4343	922	4	between	between	ADP
ejpam-4343	922	5	pre	pre	NOUN
ejpam-4343	922	6	-	-	NOUN
ejpam-4343	922	7	t0	t0	NOUN
ejpam-4343	922	8	and	and	CCONJ
ejpam-4343	922	9	pre	pre	ADJ
ejpam-4343	922	10	-	-	NOUN
ejpam-4343	922	11	t1	t1	NOUN
ejpam-4343	922	12	.	.	PUNCT
ejpam-4343	923	1	east	east	PROPN
ejpam-4343	923	2	-	-	PUNCT
ejpam-4343	923	3	west	west	PROPN
ejpam-4343	923	4	journal	journal	PROPN
ejpam-4343	923	5	of	of	ADP
ejpam-4343	923	6	mathematics	mathematic	NOUN
ejpam-4343	923	7	,	,	PUNCT
ejpam-4343	923	8	3:171–177	3:171–177	NUM
ejpam-4343	923	9	,	,	PUNCT
ejpam-4343	923	10	2001	2001	NUM
ejpam-4343	923	11	.	.	PUNCT
ejpam-4343	924	1	[	[	X
ejpam-4343	924	2	6	6	NUM
ejpam-4343	924	3	]	]	PUNCT
ejpam-4343	924	4	m.	m.	NOUN
ejpam-4343	924	5	caldas	caldas	PROPN
ejpam-4343	924	6	and	and	CCONJ
ejpam-4343	924	7	s.	s.	PROPN
ejpam-4343	924	8	jafari	jafari	PROPN
ejpam-4343	924	9	.	.	PUNCT
ejpam-4343	925	1	on	on	ADP
ejpam-4343	925	2	some	some	DET
ejpam-4343	925	3	low	low	ADJ
ejpam-4343	925	4	separation	separation	NOUN
ejpam-4343	925	5	axioms	axiom	NOUN
ejpam-4343	925	6	in	in	ADP
ejpam-4343	925	7	topological	topological	ADJ
ejpam-4343	925	8	spaces	space	NOUN
ejpam-4343	925	9	.	.	PUNCT
ejpam-4343	926	1	houston	houston	PROPN
ejpam-4343	926	2	journal	journal	PROPN
ejpam-4343	926	3	of	of	ADP
ejpam-4343	926	4	mathematics	mathematics	PROPN
ejpam-4343	926	5	,	,	PUNCT
ejpam-4343	926	6	29:94–104	29:94–104	PROPN
ejpam-4343	926	7	,	,	PUNCT
ejpam-4343	926	8	2003	2003	NUM
ejpam-4343	926	9	.	.	PUNCT
ejpam-4343	927	1	[	[	X
ejpam-4343	927	2	7	7	X
ejpam-4343	927	3	]	]	X
ejpam-4343	927	4	m.	m.	NOUN
ejpam-4343	927	5	caldas	caldas	PROPN
ejpam-4343	927	6	,	,	PUNCT
ejpam-4343	927	7	s.	s.	PROPN
ejpam-4343	927	8	jafari	jafari	PROPN
ejpam-4343	927	9	,	,	PUNCT
ejpam-4343	927	10	and	and	CCONJ
ejpam-4343	927	11	t.	t.	PROPN
ejpam-4343	927	12	noiri	noiri	PROPN
ejpam-4343	927	13	.	.	PUNCT
ejpam-4343	928	1	characterizations	characterization	NOUN
ejpam-4343	928	2	of	of	ADP
ejpam-4343	928	3	pre	pre	ADJ
ejpam-4343	928	4	-	-	ADJ
ejpam-4343	928	5	r0	r0	ADJ
ejpam-4343	928	6	and	and	CCONJ
ejpam-4343	928	7	pre	pre	ADJ
ejpam-4343	928	8	-	-	ADJ
ejpam-4343	928	9	r1	r1	ADJ
ejpam-4343	928	10	topological	topological	ADJ
ejpam-4343	928	11	spaces	space	NOUN
ejpam-4343	928	12	.	.	PUNCT
ejpam-4343	929	1	topology	topology	NOUN
ejpam-4343	929	2	proceedings	proceeding	NOUN
ejpam-4343	929	3	,	,	PUNCT
ejpam-4343	929	4	25:17–30	25:17–30	PROPN
ejpam-4343	929	5	,	,	PUNCT
ejpam-4343	929	6	2000	2000	NUM
ejpam-4343	929	7	.	.	PUNCT
ejpam-4343	930	1	[	[	X
ejpam-4343	930	2	8	8	NUM
ejpam-4343	930	3	]	]	PUNCT
ejpam-4343	930	4	m.	m.	NOUN
ejpam-4343	930	5	caldas	caldas	PROPN
ejpam-4343	930	6	,	,	PUNCT
ejpam-4343	930	7	s.	s.	PROPN
ejpam-4343	930	8	jafari	jafari	PROPN
ejpam-4343	930	9	,	,	PUNCT
ejpam-4343	930	10	and	and	CCONJ
ejpam-4343	930	11	t.	t.	PROPN
ejpam-4343	930	12	noiri	noiri	PROPN
ejpam-4343	930	13	.	.	PUNCT
ejpam-4343	931	1	characterizations	characterization	NOUN
ejpam-4343	931	2	of	of	ADP
ejpam-4343	931	3	λθ	λθ	NOUN
ejpam-4343	931	4	-	-	PUNCT
ejpam-4343	931	5	r0	r0	NOUN
ejpam-4343	931	6	and	and	CCONJ
ejpam-4343	931	7	λθ	λθ	NOUN
ejpam-4343	931	8	-	-	PUNCT
ejpam-4343	931	9	r1	r1	NOUN
ejpam-4343	931	10	topological	topological	ADJ
ejpam-4343	931	11	spaces	space	NOUN
ejpam-4343	931	12	.	.	PUNCT
ejpam-4343	932	1	acta	acta	PROPN
ejpam-4343	932	2	mathematica	mathematica	PROPN
ejpam-4343	932	3	hungarica	hungarica	PROPN
ejpam-4343	932	4	,	,	PUNCT
ejpam-4343	932	5	103:85–95	103:85–95	NUM
ejpam-4343	932	6	,	,	PUNCT
ejpam-4343	932	7	2004	2004	NUM
ejpam-4343	932	8	.	.	PUNCT
ejpam-4343	933	1	[	[	X
ejpam-4343	933	2	9	9	NUM
ejpam-4343	933	3	]	]	PUNCT
ejpam-4343	933	4	m.	m.	NOUN
ejpam-4343	933	5	caldas	caldas	PROPN
ejpam-4343	933	6	and	and	CCONJ
ejpam-4343	933	7	j.	j.	PROPN
ejpam-4343	933	8	safari	safari	PROPN
ejpam-4343	933	9	.	.	PUNCT
ejpam-4343	934	1	on	on	ADP
ejpam-4343	934	2	some	some	DET
ejpam-4343	934	3	low	low	ADJ
ejpam-4343	934	4	separation	separation	NOUN
ejpam-4343	934	5	axioms	axiom	NOUN
ejpam-4343	934	6	via	via	ADP
ejpam-4343	934	7	λ	λ	NOUN
ejpam-4343	934	8	-	-	ADJ
ejpam-4343	934	9	open	open	ADJ
ejpam-4343	934	10	and	and	CCONJ
ejpam-4343	934	11	λ	λ	NOUN
ejpam-4343	934	12	-	-	NOUN
ejpam-4343	934	13	closure	closure	NOUN
ejpam-4343	934	14	operator	operator	NOUN
ejpam-4343	934	15	.	.	PUNCT
ejpam-4343	935	1	rendiconti	rendiconti	PROPN
ejpam-4343	935	2	del	del	PROPN
ejpam-4343	935	3	circolo	circolo	PROPN
ejpam-4343	935	4	di	di	X
ejpam-4343	935	5	palermo	palermo	NOUN
ejpam-4343	935	6	,	,	PUNCT
ejpam-4343	935	7	54:195–208	54:195–208	NUM
ejpam-4343	935	8	,	,	PUNCT
ejpam-4343	935	9	2005	2005	NUM
ejpam-4343	935	10	.	.	PUNCT
ejpam-4343	936	1	[	[	X
ejpam-4343	936	2	10	10	NUM
ejpam-4343	936	3	]	]	PUNCT
ejpam-4343	936	4	m.	m.	NOUN
ejpam-4343	936	5	caldas	caldas	PROPN
ejpam-4343	936	6	,	,	PUNCT
ejpam-4343	936	7	j.	j.	PROPN
ejpam-4343	936	8	safari	safari	PROPN
ejpam-4343	936	9	,	,	PUNCT
ejpam-4343	936	10	and	and	CCONJ
ejpam-4343	936	11	g.	g.	PROPN
ejpam-4343	936	12	navalagi	navalagi	PROPN
ejpam-4343	936	13	.	.	PUNCT
ejpam-4343	937	1	more	more	ADJ
ejpam-4343	937	2	on	on	ADP
ejpam-4343	937	3	λ	λ	NOUN
ejpam-4343	937	4	-	-	ADJ
ejpam-4343	937	5	closed	closed	ADJ
ejpam-4343	937	6	sets	set	NOUN
ejpam-4343	937	7	in	in	ADP
ejpam-4343	937	8	topological	topological	ADJ
ejpam-4343	937	9	spaces	space	NOUN
ejpam-4343	937	10	.	.	PUNCT
ejpam-4343	938	1	revista	revista	PROPN
ejpam-4343	938	2	colombiana	colombiana	PROPN
ejpam-4343	938	3	de	de	X
ejpam-4343	938	4	matemáticas	matemáticas	PROPN
ejpam-4343	938	5	,	,	PUNCT
ejpam-4343	938	6	41:355–369	41:355–369	NUM
ejpam-4343	938	7	,	,	PUNCT
ejpam-4343	938	8	2007	2007	NUM
ejpam-4343	938	9	.	.	PUNCT
ejpam-4343	939	1	[	[	X
ejpam-4343	939	2	11	11	NUM
ejpam-4343	939	3	]	]	X
ejpam-4343	939	4	f.	f.	NOUN
ejpam-4343	939	5	cammarato	cammarato	PROPN
ejpam-4343	939	6	and	and	CCONJ
ejpam-4343	939	7	t.	t.	PROPN
ejpam-4343	939	8	noiri	noiri	PROPN
ejpam-4343	939	9	.	.	PUNCT
ejpam-4343	940	1	on	on	ADP
ejpam-4343	940	2	λm	λm	NOUN
ejpam-4343	940	3	-	-	PUNCT
ejpam-4343	940	4	sets	set	NOUN
ejpam-4343	940	5	and	and	CCONJ
ejpam-4343	940	6	related	relate	VERB
ejpam-4343	940	7	topological	topological	ADJ
ejpam-4343	940	8	spaces	space	NOUN
ejpam-4343	940	9	.	.	PUNCT
ejpam-4343	941	1	acta	acta	PROPN
ejpam-4343	941	2	mathematica	mathematica	PROPN
ejpam-4343	941	3	hungarica	hungarica	PROPN
ejpam-4343	941	4	,	,	PUNCT
ejpam-4343	941	5	109:261–279	109:261–279	NUM
ejpam-4343	941	6	,	,	PUNCT
ejpam-4343	941	7	2004	2004	NUM
ejpam-4343	941	8	.	.	PUNCT
ejpam-4343	942	1	[	[	X
ejpam-4343	942	2	12	12	NUM
ejpam-4343	942	3	]	]	X
ejpam-4343	942	4	h.	h.	NOUN
ejpam-4343	942	5	corson	corson	PROPN
ejpam-4343	942	6	and	and	CCONJ
ejpam-4343	942	7	e.	e.	PROPN
ejpam-4343	942	8	michael	michael	PROPN
ejpam-4343	942	9	.	.	PUNCT
ejpam-4343	943	1	metrizability	metrizability	NOUN
ejpam-4343	943	2	of	of	ADP
ejpam-4343	943	3	certain	certain	ADJ
ejpam-4343	943	4	countable	countable	ADJ
ejpam-4343	943	5	unions	union	NOUN
ejpam-4343	943	6	.	.	PUNCT
ejpam-4343	944	1	illinois	illinois	PROPN
ejpam-4343	944	2	journal	journal	PROPN
ejpam-4343	944	3	of	of	ADP
ejpam-4343	944	4	mathematics	mathematic	NOUN
ejpam-4343	944	5	,	,	PUNCT
ejpam-4343	944	6	8:351–360	8:351–360	NUM
ejpam-4343	944	7	,	,	PUNCT
ejpam-4343	944	8	1964	1964	NUM
ejpam-4343	944	9	.	.	PUNCT
ejpam-4343	945	1	[	[	X
ejpam-4343	945	2	13	13	NUM
ejpam-4343	945	3	]	]	PUNCT
ejpam-4343	945	4	ş.	ş.	PROPN
ejpam-4343	945	5	yüksel	yüksel	PROPN
ejpam-4343	945	6	,	,	PUNCT
ejpam-4343	945	7	a.	a.	NOUN
ejpam-4343	945	8	açikgöz	açikgöz	PROPN
ejpam-4343	945	9	,	,	PUNCT
ejpam-4343	945	10	and	and	CCONJ
ejpam-4343	945	11	e.	e.	PROPN
ejpam-4343	945	12	gursel	gursel	PROPN
ejpam-4343	945	13	.	.	PUNCT
ejpam-4343	946	1	on	on	ADP
ejpam-4343	946	2	a	a	DET
ejpam-4343	946	3	new	new	ADJ
ejpam-4343	946	4	type	type	NOUN
ejpam-4343	946	5	of	of	ADP
ejpam-4343	946	6	continuous	continuous	ADJ
ejpam-4343	946	7	functions	function	NOUN
ejpam-4343	946	8	in	in	ADP
ejpam-4343	946	9	ideal	ideal	ADJ
ejpam-4343	946	10	topological	topological	ADJ
ejpam-4343	946	11	spaces	space	NOUN
ejpam-4343	946	12	.	.	PUNCT
ejpam-4343	947	1	journal	journal	PROPN
ejpam-4343	947	2	of	of	ADP
ejpam-4343	947	3	indian	indian	PROPN
ejpam-4343	947	4	academy	academy	PROPN
ejpam-4343	947	5	of	of	ADP
ejpam-4343	947	6	mathematics	mathematics	PROPN
ejpam-4343	947	7	,	,	PUNCT
ejpam-4343	947	8	28:427–438	28:427–438	NUM
ejpam-4343	947	9	,	,	PUNCT
ejpam-4343	947	10	2007	2007	NUM
ejpam-4343	947	11	.	.	PUNCT
ejpam-4343	948	1	[	[	X
ejpam-4343	948	2	14	14	NUM
ejpam-4343	948	3	]	]	PUNCT
ejpam-4343	948	4	ş.	ş.	PROPN
ejpam-4343	948	5	yüksel	yüksel	PROPN
ejpam-4343	948	6	,	,	PUNCT
ejpam-4343	948	7	t.	t.	PROPN
ejpam-4343	948	8	h.	h.	PROPN
ejpam-4343	948	9	şimşekler	şimşekler	PROPN
ejpam-4343	948	10	,	,	PUNCT
ejpam-4343	948	11	z.	z.	PROPN
ejpam-4343	948	12	güzel	güzel	PROPN
ejpam-4343	948	13	,	,	PUNCT
ejpam-4343	948	14	and	and	CCONJ
ejpam-4343	948	15	t.	t.	PROPN
ejpam-4343	948	16	noiri	noiri	PROPN
ejpam-4343	948	17	.	.	PUNCT
ejpam-4343	949	1	strongly	strongly	ADV
ejpam-4343	949	2	θ	θ	VERB
ejpam-4343	949	3	-	-	PUNCT
ejpam-4343	949	4	pre	pre	ADJ
ejpam-4343	949	5	-	-	ADJ
ejpam-4343	949	6	i	i	ADJ
ejpam-4343	949	7	-continuous	-continuous	ADJ
ejpam-4343	949	8	functions	function	NOUN
ejpam-4343	949	9	.	.	PUNCT
ejpam-4343	950	1	scientific	scientific	ADJ
ejpam-4343	950	2	studies	study	NOUN
ejpam-4343	950	3	and	and	CCONJ
ejpam-4343	950	4	research	research	NOUN
ejpam-4343	950	5	.	.	PUNCT
ejpam-4343	951	1	series	series	PROPN
ejpam-4343	951	2	mathematics	mathematics	PROPN
ejpam-4343	951	3	and	and	CCONJ
ejpam-4343	951	4	informatics	informatic	NOUN
ejpam-4343	951	5	,	,	PUNCT
ejpam-4343	951	6	20:111	20:111	NUM
ejpam-4343	951	7	–	–	PUNCT
ejpam-4343	951	8	126	126	NUM
ejpam-4343	951	9	,	,	PUNCT
ejpam-4343	951	10	2010	2010	NUM
ejpam-4343	951	11	.	.	PUNCT
ejpam-4343	952	1	[	[	X
ejpam-4343	952	2	15	15	NUM
ejpam-4343	952	3	]	]	X
ejpam-4343	952	4	j.	j.	PROPN
ejpam-4343	952	5	dontchev	dontchev	PROPN
ejpam-4343	952	6	.	.	PUNCT
ejpam-4343	952	7	idealization	idealization	NOUN
ejpam-4343	952	8	of	of	ADP
ejpam-4343	952	9	ganster	ganster	NOUN
ejpam-4343	952	10	-	-	PUNCT
ejpam-4343	952	11	reilly	reilly	ADJ
ejpam-4343	952	12	decomposition	decomposition	NOUN
ejpam-4343	952	13	theorems	theorem	NOUN
ejpam-4343	952	14	.	.	PUNCT
ejpam-4343	953	1	arxiv	arxiv	PROPN
ejpam-4343	953	2	:	:	PUNCT
ejpam-4343	953	3	math	math	NOUN
ejpam-4343	953	4	.	.	PUNCT
ejpam-4343	954	1	gn/9901017v1	gn/9901017v1	NOUN
ejpam-4343	954	2	,	,	PUNCT
ejpam-4343	954	3	pages	page	NOUN
ejpam-4343	954	4	1–13	1–13	NOUN
ejpam-4343	954	5	,	,	PUNCT
ejpam-4343	954	6	1999	1999	NUM
ejpam-4343	954	7	.	.	PUNCT
ejpam-4343	955	1	[	[	X
ejpam-4343	955	2	16	16	NUM
ejpam-4343	955	3	]	]	X
ejpam-4343	955	4	e.	e.	PROPN
ejpam-4343	955	5	ekici	ekici	PROPN
ejpam-4343	955	6	,	,	PUNCT
ejpam-4343	955	7	s.	s.	PROPN
ejpam-4343	955	8	jafari	jafari	PROPN
ejpam-4343	955	9	,	,	PUNCT
ejpam-4343	955	10	m.	m.	PROPN
ejpam-4343	955	11	caldas	caldas	PROPN
ejpam-4343	955	12	,	,	PUNCT
ejpam-4343	955	13	and	and	CCONJ
ejpam-4343	955	14	t.	t.	PROPN
ejpam-4343	955	15	noiri	noiri	PROPN
ejpam-4343	955	16	.	.	PUNCT
ejpam-4343	956	1	weakly	weakly	ADJ
ejpam-4343	956	2	λ	λ	ADJ
ejpam-4343	956	3	-	-	ADJ
ejpam-4343	956	4	continuous	continuous	ADJ
ejpam-4343	956	5	functions	function	NOUN
ejpam-4343	956	6	.	.	PUNCT
ejpam-4343	957	1	novi	novi	PROPN
ejpam-4343	957	2	sad	sad	PROPN
ejpam-4343	957	3	journal	journal	PROPN
ejpam-4343	957	4	of	of	ADP
ejpam-4343	957	5	mathematics	mathematic	NOUN
ejpam-4343	957	6	,	,	PUNCT
ejpam-4343	957	7	38:47–56	38:47–56	NUM
ejpam-4343	957	8	,	,	PUNCT
ejpam-4343	957	9	2008	2008	NUM
ejpam-4343	957	10	.	.	PUNCT
ejpam-4343	958	1	[	[	X
ejpam-4343	958	2	17	17	NUM
ejpam-4343	958	3	]	]	X
ejpam-4343	958	4	e.	e.	PROPN
ejpam-4343	958	5	ekici	ekici	PROPN
ejpam-4343	958	6	and	and	CCONJ
ejpam-4343	958	7	t.	t.	PROPN
ejpam-4343	958	8	noiri	noiri	PROPN
ejpam-4343	958	9	.	.	PUNCT
ejpam-4343	959	1	⋆-extremally	⋆-extremally	ADV
ejpam-4343	959	2	disconnected	disconnect	VERB
ejpam-4343	959	3	ideal	ideal	ADJ
ejpam-4343	959	4	topological	topological	ADJ
ejpam-4343	959	5	spaces	space	NOUN
ejpam-4343	959	6	.	.	PUNCT
ejpam-4343	960	1	acta	acta	PROPN
ejpam-4343	960	2	mathematica	mathematica	PROPN
ejpam-4343	960	3	hungarica	hungarica	PROPN
ejpam-4343	960	4	,	,	PUNCT
ejpam-4343	960	5	122:81–90	122:81–90	NUM
ejpam-4343	960	6	,	,	PUNCT
ejpam-4343	960	7	2009	2009	NUM
ejpam-4343	960	8	.	.	PUNCT
ejpam-4343	961	1	[	[	X
ejpam-4343	961	2	18	18	NUM
ejpam-4343	961	3	]	]	PUNCT
ejpam-4343	961	4	m.	m.	NOUN
ejpam-4343	961	5	e.	e.	PROPN
ejpam-4343	961	6	abd	abd	PROPN
ejpam-4343	962	1	el	el	PROPN
ejpam-4343	962	2	-	-	PROPN
ejpam-4343	962	3	monsef	monsef	PROPN
ejpam-4343	962	4	,	,	PUNCT
ejpam-4343	962	5	e.	e.	PROPN
ejpam-4343	962	6	f.	f.	PROPN
ejpam-4343	962	7	lashien	lashien	PROPN
ejpam-4343	962	8	,	,	PUNCT
ejpam-4343	962	9	and	and	CCONJ
ejpam-4343	962	10	a.	a.	NOUN
ejpam-4343	962	11	a.	a.	NOUN
ejpam-4343	962	12	nasef	nasef	PROPN
ejpam-4343	962	13	.	.	PUNCT
ejpam-4343	963	1	on	on	ADP
ejpam-4343	963	2	i	i	PRON
ejpam-4343	963	3	-open	-open	PROPN
ejpam-4343	963	4	sets	set	NOUN
ejpam-4343	963	5	and	and	CCONJ
ejpam-4343	963	6	i	i	PRON
ejpam-4343	963	7	continuous	continuous	ADJ
ejpam-4343	963	8	functions	function	NOUN
ejpam-4343	963	9	.	.	PUNCT
ejpam-4343	964	1	kyungpook	kyungpook	PROPN
ejpam-4343	964	2	mathematical	mathematical	PROPN
ejpam-4343	964	3	journal	journal	PROPN
ejpam-4343	964	4	,	,	PUNCT
ejpam-4343	964	5	32:21–30	32:21–30	NUM
ejpam-4343	964	6	,	,	PUNCT
ejpam-4343	964	7	1992	1992	NUM
ejpam-4343	964	8	.	.	PUNCT
ejpam-4343	965	1	[	[	X
ejpam-4343	965	2	19	19	NUM
ejpam-4343	965	3	]	]	PUNCT
ejpam-4343	965	4	m.	m.	NOUN
ejpam-4343	965	5	ganster	ganster	NOUN
ejpam-4343	965	6	,	,	PUNCT
ejpam-4343	965	7	s.	s.	PROPN
ejpam-4343	965	8	jafari	jafari	PROPN
ejpam-4343	965	9	,	,	PUNCT
ejpam-4343	965	10	and	and	CCONJ
ejpam-4343	965	11	t.	t.	PROPN
ejpam-4343	965	12	noiri	noiri	PROPN
ejpam-4343	965	13	.	.	PUNCT
ejpam-4343	966	1	on	on	ADP
ejpam-4343	966	2	pre	pre	ADJ
ejpam-4343	966	3	-	-	ADJ
ejpam-4343	966	4	λ	λ	NOUN
ejpam-4343	966	5	-	-	NOUN
ejpam-4343	966	6	sets	set	NOUN
ejpam-4343	966	7	and	and	CCONJ
ejpam-4343	966	8	pre	pre	ADJ
ejpam-4343	966	9	-	-	ADJ
ejpam-4343	966	10	v	v	ADJ
ejpam-4343	966	11	-sets	-set	NOUN
ejpam-4343	966	12	.	.	PUNCT
ejpam-4343	967	1	acta	acta	PROPN
ejpam-4343	967	2	mathematica	mathematica	PROPN
ejpam-4343	967	3	hungarica	hungarica	PROPN
ejpam-4343	967	4	,	,	PUNCT
ejpam-4343	967	5	95:337–343	95:337–343	PROPN
ejpam-4343	967	6	,	,	PUNCT
ejpam-4343	967	7	2002	2002	NUM
ejpam-4343	967	8	.	.	PUNCT
ejpam-4343	968	1	references	reference	NOUN
ejpam-4343	968	2	1046	1046	NUM
ejpam-4343	968	3	[	[	X
ejpam-4343	968	4	20	20	NUM
ejpam-4343	968	5	]	]	X
ejpam-4343	968	6	d.	d.	PROPN
ejpam-4343	968	7	n.	n.	PROPN
ejpam-4343	968	8	georgiuo	georgiuo	PROPN
ejpam-4343	968	9	,	,	PUNCT
ejpam-4343	968	10	s.	s.	PROPN
ejpam-4343	968	11	jafari	jafari	PROPN
ejpam-4343	968	12	,	,	PUNCT
ejpam-4343	968	13	and	and	CCONJ
ejpam-4343	968	14	t.	t.	PROPN
ejpam-4343	968	15	noiri	noiri	PROPN
ejpam-4343	968	16	.	.	PUNCT
ejpam-4343	969	1	properties	property	NOUN
ejpam-4343	969	2	of	of	ADP
ejpam-4343	969	3	(	(	PUNCT
ejpam-4343	969	4	λ	λ	PROPN
ejpam-4343	969	5	,	,	PUNCT
ejpam-4343	969	6	δ)-closed	δ)-close	VERB
ejpam-4343	969	7	sets	set	NOUN
ejpam-4343	969	8	in	in	ADP
ejpam-4343	969	9	topological	topological	ADJ
ejpam-4343	969	10	spaces	space	NOUN
ejpam-4343	969	11	.	.	PUNCT
ejpam-4343	970	1	bollettino	bollettino	PROPN
ejpam-4343	970	2	dell	dell	PROPN
ejpam-4343	970	3	’	'	PUNCT
ejpam-4343	970	4	unione	unione	PROPN
ejpam-4343	970	5	mathematica	mathematica	PROPN
ejpam-4343	970	6	italiana	italiana	PROPN
ejpam-4343	970	7	,	,	PUNCT
ejpam-4343	970	8	7:745–756	7:745–756	PROPN
ejpam-4343	970	9	,	,	PUNCT
ejpam-4343	970	10	2004	2004	NUM
ejpam-4343	970	11	.	.	PUNCT
ejpam-4343	971	1	[	[	X
ejpam-4343	971	2	21	21	NUM
ejpam-4343	971	3	]	]	X
ejpam-4343	971	4	e.	e.	PROPN
ejpam-4343	971	5	hatır	hatır	PROPN
ejpam-4343	971	6	and	and	CCONJ
ejpam-4343	971	7	t.	t.	PROPN
ejpam-4343	971	8	noiri	noiri	PROPN
ejpam-4343	971	9	.	.	PUNCT
ejpam-4343	972	1	on	on	ADP
ejpam-4343	972	2	decompositions	decomposition	NOUN
ejpam-4343	972	3	of	of	ADP
ejpam-4343	972	4	continuity	continuity	NOUN
ejpam-4343	972	5	via	via	ADP
ejpam-4343	972	6	idealization	idealization	NOUN
ejpam-4343	972	7	.	.	PUNCT
ejpam-4343	973	1	acta	acta	PROPN
ejpam-4343	973	2	mathematica	mathematica	PROPN
ejpam-4343	973	3	hungarica	hungarica	PROPN
ejpam-4343	973	4	,	,	PUNCT
ejpam-4343	973	5	96:341–349	96:341–349	PROPN
ejpam-4343	973	6	,	,	PUNCT
ejpam-4343	973	7	2002	2002	NUM
ejpam-4343	973	8	.	.	PUNCT
ejpam-4343	974	1	[	[	X
ejpam-4343	974	2	22	22	NUM
ejpam-4343	974	3	]	]	PUNCT
ejpam-4343	974	4	s.	s.	PROPN
ejpam-4343	974	5	jafari	jafari	PROPN
ejpam-4343	974	6	.	.	PUNCT
ejpam-4343	975	1	on	on	ADP
ejpam-4343	975	2	a	a	DET
ejpam-4343	975	3	weak	weak	ADJ
ejpam-4343	975	4	separation	separation	NOUN
ejpam-4343	975	5	axiom	axiom	NOUN
ejpam-4343	975	6	.	.	PUNCT
ejpam-4343	976	1	far	far	PROPN
ejpam-4343	976	2	east	east	PROPN
ejpam-4343	976	3	journal	journal	PROPN
ejpam-4343	976	4	of	of	ADP
ejpam-4343	976	5	mathematical	mathematical	ADJ
ejpam-4343	976	6	sciences	sciences	PROPN
ejpam-4343	976	7	,	,	PUNCT
ejpam-4343	976	8	3:779–787	3:779–787	NUM
ejpam-4343	976	9	,	,	PUNCT
ejpam-4343	976	10	2001	2001	NUM
ejpam-4343	976	11	.	.	PUNCT
ejpam-4343	977	1	[	[	X
ejpam-4343	977	2	23	23	NUM
ejpam-4343	977	3	]	]	PUNCT
ejpam-4343	977	4	s.	s.	PROPN
ejpam-4343	977	5	jafari	jafari	PROPN
ejpam-4343	977	6	,	,	PUNCT
ejpam-4343	977	7	s.	s.	PROPN
ejpam-4343	977	8	p.	p.	PROPN
ejpam-4343	977	9	moshokoa	moshokoa	PROPN
ejpam-4343	977	10	,	,	PUNCT
ejpam-4343	977	11	k.	k.	PROPN
ejpam-4343	977	12	r.	r.	PROPN
ejpam-4343	977	13	nailana	nailana	PROPN
ejpam-4343	977	14	,	,	PUNCT
ejpam-4343	977	15	and	and	CCONJ
ejpam-4343	977	16	t.	t.	PROPN
ejpam-4343	977	17	noiri	noiri	PROPN
ejpam-4343	977	18	.	.	PUNCT
ejpam-4343	978	1	on	on	ADP
ejpam-4343	978	2	almost	almost	ADV
ejpam-4343	978	3	λ	λ	ADJ
ejpam-4343	978	4	-	-	ADJ
ejpam-4343	978	5	continuous	continuous	ADJ
ejpam-4343	978	6	functions	function	NOUN
ejpam-4343	978	7	.	.	PUNCT
ejpam-4343	979	1	scientific	scientific	ADJ
ejpam-4343	979	2	studies	study	NOUN
ejpam-4343	979	3	and	and	CCONJ
ejpam-4343	979	4	research	research	NOUN
ejpam-4343	979	5	.	.	PUNCT
ejpam-4343	980	1	series	series	PROPN
ejpam-4343	980	2	mathematics	mathematics	PROPN
ejpam-4343	980	3	and	and	CCONJ
ejpam-4343	980	4	informatics	informatic	NOUN
ejpam-4343	980	5	,	,	PUNCT
ejpam-4343	980	6	20:93	20:93	NUM
ejpam-4343	980	7	–	–	PUNCT
ejpam-4343	980	8	102	102	NUM
ejpam-4343	980	9	,	,	PUNCT
ejpam-4343	980	10	2010	2010	NUM
ejpam-4343	980	11	.	.	PUNCT
ejpam-4343	981	1	[	[	X
ejpam-4343	981	2	24	24	NUM
ejpam-4343	981	3	]	]	X
ejpam-4343	981	4	d.	d.	PROPN
ejpam-4343	981	5	s.	s.	PROPN
ejpam-4343	981	6	janković	janković	PROPN
ejpam-4343	981	7	and	and	CCONJ
ejpam-4343	981	8	t.	t.	PROPN
ejpam-4343	981	9	r.	r.	PROPN
ejpam-4343	981	10	hamlett	hamlett	PROPN
ejpam-4343	981	11	.	.	PUNCT
ejpam-4343	982	1	new	new	ADJ
ejpam-4343	982	2	topologies	topology	NOUN
ejpam-4343	982	3	from	from	ADP
ejpam-4343	982	4	old	old	ADJ
ejpam-4343	982	5	via	via	ADP
ejpam-4343	982	6	ideals	ideal	NOUN
ejpam-4343	982	7	.	.	PUNCT
ejpam-4343	983	1	the	the	DET
ejpam-4343	983	2	american	american	PROPN
ejpam-4343	983	3	mathematical	mathematical	PROPN
ejpam-4343	983	4	monthly	monthly	ADV
ejpam-4343	983	5	,	,	PUNCT
ejpam-4343	983	6	97:295–310	97:295–310	PROPN
ejpam-4343	983	7	,	,	PUNCT
ejpam-4343	983	8	1990	1990	NUM
ejpam-4343	983	9	.	.	PUNCT
ejpam-4343	984	1	[	[	X
ejpam-4343	984	2	25	25	NUM
ejpam-4343	984	3	]	]	PUNCT
ejpam-4343	984	4	a.	a.	NOUN
ejpam-4343	984	5	kar	kar	PROPN
ejpam-4343	984	6	and	and	CCONJ
ejpam-4343	984	7	p.	p.	PROPN
ejpam-4343	984	8	bhattacharya	bhattacharya	PROPN
ejpam-4343	984	9	.	.	PUNCT
ejpam-4343	985	1	some	some	DET
ejpam-4343	985	2	weak	weak	ADJ
ejpam-4343	985	3	separation	separation	NOUN
ejpam-4343	985	4	axioms	axiom	NOUN
ejpam-4343	985	5	.	.	PUNCT
ejpam-4343	986	1	bulletin	bulletin	NOUN
ejpam-4343	986	2	calcutta	calcutta	PROPN
ejpam-4343	986	3	mathematical	mathematical	ADJ
ejpam-4343	986	4	society	society	NOUN
ejpam-4343	986	5	,	,	PUNCT
ejpam-4343	986	6	82:415–422	82:415–422	NUM
ejpam-4343	986	7	,	,	PUNCT
ejpam-4343	986	8	1990	1990	NUM
ejpam-4343	986	9	.	.	PUNCT
ejpam-4343	987	1	[	[	X
ejpam-4343	987	2	26	26	NUM
ejpam-4343	987	3	]	]	PUNCT
ejpam-4343	987	4	k.	k.	PROPN
ejpam-4343	987	5	kuratowski	kuratowski	PROPN
ejpam-4343	987	6	.	.	PUNCT
ejpam-4343	988	1	topology	topology	PROPN
ejpam-4343	988	2	,	,	PUNCT
ejpam-4343	988	3	vol	vol	NOUN
ejpam-4343	988	4	.	.	PUNCT
ejpam-4343	988	5	i.	i.	PROPN
ejpam-4343	988	6	academic	academic	PROPN
ejpam-4343	988	7	press	press	PROPN
ejpam-4343	988	8	,	,	PUNCT
ejpam-4343	988	9	new	new	PROPN
ejpam-4343	988	10	york	york	PROPN
ejpam-4343	988	11	,	,	PUNCT
ejpam-4343	988	12	1966	1966	NUM
ejpam-4343	988	13	.	.	PUNCT
ejpam-4343	989	1	[	[	X
ejpam-4343	989	2	27	27	NUM
ejpam-4343	989	3	]	]	X
ejpam-4343	989	4	h.	h.	PROPN
ejpam-4343	989	5	maki	maki	PROPN
ejpam-4343	989	6	.	.	PUNCT
ejpam-4343	990	1	generalized	generalize	VERB
ejpam-4343	990	2	λ	λ	NOUN
ejpam-4343	990	3	-	-	ADJ
ejpam-4343	990	4	closed	closed	ADJ
ejpam-4343	990	5	sets	set	NOUN
ejpam-4343	990	6	and	and	CCONJ
ejpam-4343	990	7	the	the	DET
ejpam-4343	990	8	associated	associated	ADJ
ejpam-4343	990	9	closure	closure	NOUN
ejpam-4343	990	10	operator	operator	NOUN
ejpam-4343	990	11	.	.	PUNCT
ejpam-4343	991	1	the	the	DET
ejpam-4343	991	2	special	special	ADJ
ejpam-4343	991	3	issue	issue	NOUN
ejpam-4343	991	4	in	in	ADP
ejpam-4343	991	5	commemoration	commemoration	NOUN
ejpam-4343	991	6	of	of	ADP
ejpam-4343	991	7	prof	prof	PROPN
ejpam-4343	991	8	.	.	PUNCT
ejpam-4343	992	1	kazusada	kazusada	PROPN
ejpam-4343	992	2	ikeda	ikeda	PROPN
ejpam-4343	992	3	’	'	PUNCT
ejpam-4343	992	4	retirement	retirement	NOUN
ejpam-4343	992	5	,	,	PUNCT
ejpam-4343	992	6	1	1	NUM
ejpam-4343	992	7	.	.	PUNCT
ejpam-4343	992	8	oct	oct	PROPN
ejpam-4343	992	9	.	.	PROPN
ejpam-4343	992	10	,	,	PUNCT
ejpam-4343	992	11	pages	page	NOUN
ejpam-4343	992	12	139	139	NUM
ejpam-4343	992	13	–	–	PUNCT
ejpam-4343	992	14	146	146	NUM
ejpam-4343	992	15	,	,	PUNCT
ejpam-4343	992	16	1986	1986	NUM
ejpam-4343	992	17	.	.	PUNCT
ejpam-4343	993	1	[	[	X
ejpam-4343	993	2	28	28	NUM
ejpam-4343	993	3	]	]	X
ejpam-4343	993	4	a.	a.	NOUN
ejpam-4343	993	5	s.	s.	PROPN
ejpam-4343	993	6	mashhour	mashhour	PROPN
ejpam-4343	993	7	,	,	PUNCT
ejpam-4343	993	8	m.	m.	PROPN
ejpam-4343	993	9	e.	e.	PROPN
ejpam-4343	993	10	abd	abd	PROPN
ejpam-4343	993	11	el	el	PROPN
ejpam-4343	993	12	-	-	PROPN
ejpam-4343	993	13	monsef	monsef	ADJ
ejpam-4343	993	14	,	,	PUNCT
ejpam-4343	993	15	and	and	CCONJ
ejpam-4343	993	16	s.	s.	PROPN
ejpam-4343	993	17	n.	n.	PROPN
ejpam-4343	993	18	el	el	PROPN
ejpam-4343	993	19	-	-	PROPN
ejpam-4343	993	20	deeb	deeb	PROPN
ejpam-4343	993	21	.	.	PUNCT
ejpam-4343	994	1	on	on	ADP
ejpam-4343	994	2	precontinuous	precontinuous	ADJ
ejpam-4343	994	3	and	and	CCONJ
ejpam-4343	994	4	weak	weak	ADJ
ejpam-4343	994	5	precontinuous	precontinuous	ADJ
ejpam-4343	994	6	mappings	mapping	NOUN
ejpam-4343	994	7	.	.	PUNCT
ejpam-4343	995	1	proceedings	proceeding	NOUN
ejpam-4343	995	2	of	of	ADP
ejpam-4343	995	3	the	the	DET
ejpam-4343	995	4	mathematical	mathematical	ADJ
ejpam-4343	995	5	and	and	CCONJ
ejpam-4343	995	6	physical	physical	ADJ
ejpam-4343	995	7	society	society	NOUN
ejpam-4343	995	8	of	of	ADP
ejpam-4343	995	9	egypt	egypt	PROPN
ejpam-4343	995	10	,	,	PUNCT
ejpam-4343	995	11	53:47–53	53:47–53	NUM
ejpam-4343	995	12	,	,	PUNCT
ejpam-4343	995	13	1982	1982	NUM
ejpam-4343	995	14	.	.	PUNCT
ejpam-4343	996	1	[	[	X
ejpam-4343	996	2	29	29	NUM
ejpam-4343	996	3	]	]	X
ejpam-4343	996	4	v.	v.	X
ejpam-4343	996	5	renukadevi	renukadevi	NOUN
ejpam-4343	996	6	.	.	PUNCT
ejpam-4343	997	1	note	note	NOUN
ejpam-4343	997	2	on	on	ADP
ejpam-4343	997	3	ir	ir	NOUN
ejpam-4343	997	4	-	-	ADJ
ejpam-4343	997	5	closed	closed	ADJ
ejpam-4343	997	6	and	and	CCONJ
ejpam-4343	997	7	air	air	NOUN
ejpam-4343	997	8	-	-	PUNCT
ejpam-4343	997	9	sets	set	NOUN
ejpam-4343	997	10	.	.	PUNCT
ejpam-4343	998	1	acta	acta	PROPN
ejpam-4343	998	2	mathematica	mathematica	PROPN
ejpam-4343	998	3	hungarica	hungarica	PROPN
ejpam-4343	998	4	,	,	PUNCT
ejpam-4343	998	5	122(4):329–338	122(4):329–338	NUM
ejpam-4343	998	6	,	,	PUNCT
ejpam-4343	998	7	2009	2009	NUM
ejpam-4343	998	8	.	.	PUNCT
ejpam-4343	999	1	[	[	X
ejpam-4343	999	2	30	30	NUM
ejpam-4343	999	3	]	]	X
ejpam-4343	999	4	v.	v.	CCONJ
ejpam-4343	999	5	vaidyanathswamy	vaidyanathswamy	NOUN
ejpam-4343	999	6	.	.	PUNCT
ejpam-4343	1000	1	the	the	DET
ejpam-4343	1000	2	localization	localization	NOUN
ejpam-4343	1000	3	theory	theory	NOUN
ejpam-4343	1000	4	in	in	ADP
ejpam-4343	1000	5	set	set	NOUN
ejpam-4343	1000	6	topology	topology	NOUN
ejpam-4343	1000	7	.	.	PUNCT
ejpam-4343	1001	1	proceedings	proceeding	NOUN
ejpam-4343	1001	2	of	of	ADP
ejpam-4343	1001	3	the	the	DET
ejpam-4343	1001	4	indian	indian	PROPN
ejpam-4343	1001	5	academy	academy	PROPN
ejpam-4343	1001	6	of	of	ADP
ejpam-4343	1001	7	sciences	sciences	PROPN
ejpam-4343	1001	8	,	,	PUNCT
ejpam-4343	1001	9	20:51–61	20:51–61	NUM
ejpam-4343	1001	10	,	,	PUNCT
ejpam-4343	1001	11	1945	1945	NUM
ejpam-4343	1001	12	.	.	PUNCT
ejpam-4343	1002	1	[	[	X
ejpam-4343	1002	2	31	31	NUM
ejpam-4343	1002	3	]	]	X
ejpam-4343	1002	4	n.	n.	NOUN
ejpam-4343	1002	5	v.	v.	PROPN
ejpam-4343	1002	6	veličko	veličko	PROPN
ejpam-4343	1002	7	.	.	PUNCT
ejpam-4343	1003	1	h	h	NOUN
ejpam-4343	1003	2	-	-	PUNCT
ejpam-4343	1003	3	closed	close	VERB
ejpam-4343	1003	4	topological	topological	ADJ
ejpam-4343	1003	5	spaces	space	NOUN
ejpam-4343	1003	6	.	.	PUNCT
ejpam-4343	1004	1	american	american	PROPN
ejpam-4343	1004	2	mathematical	mathematical	ADJ
ejpam-4343	1004	3	society	society	NOUN
ejpam-4343	1004	4	translations	translation	NOUN
ejpam-4343	1004	5	,	,	PUNCT
ejpam-4343	1004	6	78:102–118	78:102–118	NUM
ejpam-4343	1004	7	,	,	PUNCT
ejpam-4343	1004	8	1968	1968	NUM
ejpam-4343	1004	9	.	.	PUNCT