id	sid	tid	token	lemma	pos
ejpam-4367	1	1	european	european	PROPN
ejpam-4367	1	2	journal	journal	PROPN
ejpam-4367	1	3	of	of	ADP
ejpam-4367	1	4	pure	pure	ADJ
ejpam-4367	1	5	and	and	CCONJ
ejpam-4367	1	6	applied	apply	VERB
ejpam-4367	1	7	mathematics	mathematic	NOUN
ejpam-4367	1	8	vol	vol	NOUN
ejpam-4367	1	9	.	.	PROPN
ejpam-4367	2	1	15	15	NUM
ejpam-4367	2	2	,	,	PUNCT
ejpam-4367	2	3	no	no	INTJ
ejpam-4367	2	4	.	.	NOUN
ejpam-4367	2	5	3	3	NUM
ejpam-4367	2	6	,	,	PUNCT
ejpam-4367	2	7	2022	2022	NUM
ejpam-4367	2	8	,	,	PUNCT
ejpam-4367	2	9	878	878	NUM
ejpam-4367	2	10	-	-	SYM
ejpam-4367	2	11	886	886	NUM
ejpam-4367	2	12	issn	issn	PROPN
ejpam-4367	2	13	1307	1307	NUM
ejpam-4367	2	14	-	-	SYM
ejpam-4367	2	15	5543	5543	NUM
ejpam-4367	2	16	–	–	PUNCT
ejpam-4367	2	17	ejpam.com	ejpam.com	X
ejpam-4367	2	18	published	publish	VERB
ejpam-4367	2	19	by	by	ADP
ejpam-4367	2	20	new	new	PROPN
ejpam-4367	2	21	york	york	PROPN
ejpam-4367	2	22	business	business	PROPN
ejpam-4367	2	23	global	global	VERB
ejpam-4367	2	24	some	some	DET
ejpam-4367	2	25	applications	application	NOUN
ejpam-4367	2	26	of	of	ADP
ejpam-4367	2	27	(	(	PUNCT
ejpam-4367	2	28	λ	λ	PROPN
ejpam-4367	2	29	,	,	PUNCT
ejpam-4367	2	30	sp)-open	sp)-open	ADJ
ejpam-4367	2	31	sets	set	NOUN
ejpam-4367	2	32	in	in	ADP
ejpam-4367	2	33	topological	topological	ADJ
ejpam-4367	2	34	spaces	space	NOUN
ejpam-4367	2	35	chawalit	chawalit	VERB
ejpam-4367	2	36	boonpok1	boonpok1	PROPN
ejpam-4367	2	37	,	,	PUNCT
ejpam-4367	2	38	chokchai	chokchai	ADJ
ejpam-4367	2	39	viriyapong1,∗	viriyapong1,∗	NOUN
ejpam-4367	2	40	1	1	NUM
ejpam-4367	2	41	mathematics	mathematic	NOUN
ejpam-4367	2	42	and	and	CCONJ
ejpam-4367	2	43	applied	apply	VERB
ejpam-4367	2	44	mathematics	mathematics	PROPN
ejpam-4367	2	45	research	research	NOUN
ejpam-4367	2	46	unit	unit	NOUN
ejpam-4367	2	47	,	,	PUNCT
ejpam-4367	2	48	department	department	NOUN
ejpam-4367	2	49	of	of	ADP
ejpam-4367	2	50	mathematics	mathematic	NOUN
ejpam-4367	2	51	,	,	PUNCT
ejpam-4367	2	52	faculty	faculty	NOUN
ejpam-4367	2	53	of	of	ADP
ejpam-4367	2	54	science	science	NOUN
ejpam-4367	2	55	,	,	PUNCT
ejpam-4367	2	56	mahasarakham	mahasarakham	PROPN
ejpam-4367	2	57	university	university	PROPN
ejpam-4367	2	58	,	,	PUNCT
ejpam-4367	2	59	maha	maha	PROPN
ejpam-4367	2	60	sarakham	sarakham	PROPN
ejpam-4367	2	61	,	,	PUNCT
ejpam-4367	2	62	44150	44150	NUM
ejpam-4367	2	63	,	,	PUNCT
ejpam-4367	2	64	thailand	thailand	PROPN
ejpam-4367	2	65	abstract	abstract	PROPN
ejpam-4367	2	66	.	.	PUNCT
ejpam-4367	3	1	our	our	PRON
ejpam-4367	3	2	main	main	ADJ
ejpam-4367	3	3	purpose	purpose	NOUN
ejpam-4367	3	4	is	be	AUX
ejpam-4367	3	5	to	to	PART
ejpam-4367	3	6	introduce	introduce	VERB
ejpam-4367	3	7	some	some	DET
ejpam-4367	3	8	weak	weak	ADJ
ejpam-4367	3	9	separation	separation	NOUN
ejpam-4367	3	10	axioms	axiom	NOUN
ejpam-4367	3	11	by	by	ADP
ejpam-4367	3	12	utilizing	utilize	VERB
ejpam-4367	3	13	the	the	DET
ejpam-4367	3	14	concepts	concept	NOUN
ejpam-4367	3	15	of	of	ADP
ejpam-4367	3	16	(	(	PUNCT
ejpam-4367	3	17	λ	λ	PROPN
ejpam-4367	3	18	,	,	PUNCT
ejpam-4367	3	19	sp)-open	sp)-open	ADJ
ejpam-4367	3	20	sets	set	NOUN
ejpam-4367	3	21	and	and	CCONJ
ejpam-4367	3	22	the	the	DET
ejpam-4367	3	23	(	(	PUNCT
ejpam-4367	3	24	λ	λ	PROPN
ejpam-4367	3	25	,	,	PUNCT
ejpam-4367	3	26	sp)-closure	sp)-closure	NOUN
ejpam-4367	3	27	operator	operator	NOUN
ejpam-4367	3	28	.	.	PUNCT
ejpam-4367	4	1	in	in	ADP
ejpam-4367	4	2	particular	particular	ADJ
ejpam-4367	4	3	,	,	PUNCT
ejpam-4367	4	4	some	some	DET
ejpam-4367	4	5	characterizations	characterization	NOUN
ejpam-4367	4	6	of	of	ADP
ejpam-4367	4	7	(	(	PUNCT
ejpam-4367	4	8	λ	λ	PROPN
ejpam-4367	4	9	,	,	PUNCT
ejpam-4367	4	10	sp)-r0	sp)-r0	PROPN
ejpam-4367	4	11	and	and	CCONJ
ejpam-4367	4	12	(	(	PUNCT
ejpam-4367	4	13	λ	λ	PROPN
ejpam-4367	4	14	,	,	PUNCT
ejpam-4367	4	15	sp)-r1	sp)-r1	NOUN
ejpam-4367	4	16	topological	topological	ADJ
ejpam-4367	4	17	spaces	space	NOUN
ejpam-4367	4	18	are	be	AUX
ejpam-4367	4	19	investigated	investigate	VERB
ejpam-4367	4	20	.	.	PUNCT
ejpam-4367	5	1	2020	2020	NUM
ejpam-4367	5	2	mathematics	mathematic	NOUN
ejpam-4367	5	3	subject	subject	NOUN
ejpam-4367	5	4	classifications	classification	NOUN
ejpam-4367	5	5	:	:	PUNCT
ejpam-4367	5	6	54a05	54a05	NUM
ejpam-4367	5	7	,	,	PUNCT
ejpam-4367	5	8	54d10	54d10	NUM
ejpam-4367	5	9	key	key	ADJ
ejpam-4367	5	10	words	word	NOUN
ejpam-4367	5	11	and	and	CCONJ
ejpam-4367	5	12	phrases	phrase	NOUN
ejpam-4367	5	13	:	:	PUNCT
ejpam-4367	5	14	(	(	PUNCT
ejpam-4367	5	15	λ	λ	NOUN
ejpam-4367	5	16	,	,	PUNCT
ejpam-4367	5	17	sp)-open	sp)-open	ADJ
ejpam-4367	5	18	set	set	NOUN
ejpam-4367	5	19	,	,	PUNCT
ejpam-4367	5	20	(	(	PUNCT
ejpam-4367	5	21	λ	λ	INTJ
ejpam-4367	5	22	,	,	PUNCT
ejpam-4367	5	23	sp)-r0	sp)-r0	PROPN
ejpam-4367	5	24	space	space	NOUN
ejpam-4367	5	25	,	,	PUNCT
ejpam-4367	5	26	(	(	PUNCT
ejpam-4367	5	27	λ	λ	INTJ
ejpam-4367	5	28	,	,	PUNCT
ejpam-4367	5	29	sp)-r1	sp)-r1	NOUN
ejpam-4367	5	30	space	space	NOUN
ejpam-4367	5	31	1	1	NUM
ejpam-4367	5	32	.	.	PUNCT
ejpam-4367	5	33	introduction	introduction	NOUN
ejpam-4367	5	34	the	the	DET
ejpam-4367	5	35	concept	concept	NOUN
ejpam-4367	5	36	of	of	ADP
ejpam-4367	5	37	r0	r0	NOUN
ejpam-4367	5	38	topological	topological	ADJ
ejpam-4367	5	39	spaces	space	NOUN
ejpam-4367	5	40	was	be	AUX
ejpam-4367	5	41	first	first	ADV
ejpam-4367	5	42	introduced	introduce	VERB
ejpam-4367	5	43	by	by	ADP
ejpam-4367	5	44	shanin	shanin	PROPN
ejpam-4367	6	1	[	[	X
ejpam-4367	6	2	18	18	NUM
ejpam-4367	6	3	]	]	PUNCT
ejpam-4367	6	4	in	in	ADP
ejpam-4367	6	5	1961	1961	NUM
ejpam-4367	6	6	,	,	PUNCT
ejpam-4367	6	7	davis	davis	PROPN
ejpam-4367	6	8	[	[	X
ejpam-4367	6	9	7	7	X
ejpam-4367	6	10	]	]	PUNCT
ejpam-4367	6	11	introduced	introduce	VERB
ejpam-4367	6	12	the	the	DET
ejpam-4367	6	13	concept	concept	NOUN
ejpam-4367	6	14	of	of	ADP
ejpam-4367	6	15	a	a	DET
ejpam-4367	6	16	separation	separation	NOUN
ejpam-4367	6	17	axiom	axiom	NOUN
ejpam-4367	6	18	called	call	VERB
ejpam-4367	6	19	r1	r1	PROPN
ejpam-4367	6	20	.	.	PUNCT
ejpam-4367	7	1	dube	dube	PROPN
ejpam-4367	8	1	[	[	X
ejpam-4367	8	2	9	9	NUM
ejpam-4367	8	3	]	]	PUNCT
ejpam-4367	8	4	and	and	CCONJ
ejpam-4367	8	5	naimpally	naimpally	ADV
ejpam-4367	8	6	[	[	X
ejpam-4367	8	7	15	15	NUM
ejpam-4367	8	8	]	]	X
ejpam-4367	8	9	further	far	ADV
ejpam-4367	8	10	investigated	investigate	VERB
ejpam-4367	8	11	characterizations	characterization	NOUN
ejpam-4367	8	12	of	of	ADP
ejpam-4367	8	13	r0	r0	PROPN
ejpam-4367	8	14	topological	topological	ADJ
ejpam-4367	8	15	spaces	space	NOUN
ejpam-4367	8	16	and	and	CCONJ
ejpam-4367	8	17	several	several	ADJ
ejpam-4367	8	18	interesting	interesting	ADJ
ejpam-4367	8	19	results	result	NOUN
ejpam-4367	8	20	have	have	AUX
ejpam-4367	8	21	been	be	AUX
ejpam-4367	8	22	obtained	obtain	VERB
ejpam-4367	8	23	in	in	ADP
ejpam-4367	8	24	various	various	ADJ
ejpam-4367	8	25	contexts	contexts	NOUN
ejpam-4367	8	26	.	.	PUNCT
ejpam-4367	9	1	murdeshwar	murdeshwar	NOUN
ejpam-4367	9	2	and	and	CCONJ
ejpam-4367	9	3	naimpally	naimpally	ADV
ejpam-4367	9	4	[	[	X
ejpam-4367	9	5	14	14	NUM
ejpam-4367	9	6	]	]	PUNCT
ejpam-4367	9	7	and	and	CCONJ
ejpam-4367	9	8	dube	dube	PROPN
ejpam-4367	10	1	[	[	X
ejpam-4367	10	2	10	10	NUM
ejpam-4367	10	3	]	]	PUNCT
ejpam-4367	10	4	studied	study	VERB
ejpam-4367	10	5	some	some	PRON
ejpam-4367	10	6	of	of	ADP
ejpam-4367	10	7	the	the	DET
ejpam-4367	10	8	fundamental	fundamental	ADJ
ejpam-4367	10	9	properties	property	NOUN
ejpam-4367	10	10	of	of	ADP
ejpam-4367	10	11	r1	r1	PROPN
ejpam-4367	10	12	topological	topological	ADJ
ejpam-4367	10	13	spaces	space	NOUN
ejpam-4367	10	14	.	.	PUNCT
ejpam-4367	11	1	as	as	SCONJ
ejpam-4367	11	2	natural	natural	ADJ
ejpam-4367	11	3	generalizations	generalization	NOUN
ejpam-4367	11	4	of	of	ADP
ejpam-4367	11	5	the	the	DET
ejpam-4367	11	6	separation	separation	NOUN
ejpam-4367	11	7	axioms	axiom	VERB
ejpam-4367	11	8	r0	r0	NOUN
ejpam-4367	11	9	and	and	CCONJ
ejpam-4367	11	10	r1	r1	PROPN
ejpam-4367	11	11	,	,	PUNCT
ejpam-4367	11	12	the	the	DET
ejpam-4367	11	13	concepts	concept	NOUN
ejpam-4367	11	14	of	of	ADP
ejpam-4367	11	15	semi	semi	ADJ
ejpam-4367	11	16	-	-	ADJ
ejpam-4367	11	17	r0	r0	ADJ
ejpam-4367	11	18	and	and	CCONJ
ejpam-4367	11	19	semir1	semir1	NOUN
ejpam-4367	11	20	were	be	AUX
ejpam-4367	11	21	introduced	introduce	VERB
ejpam-4367	11	22	and	and	CCONJ
ejpam-4367	11	23	investigated	investigate	VERB
ejpam-4367	11	24	by	by	ADP
ejpam-4367	11	25	maheshwari	maheshwari	NOUN
ejpam-4367	11	26	and	and	CCONJ
ejpam-4367	11	27	prasad	prasad	PROPN
ejpam-4367	12	1	[	[	X
ejpam-4367	12	2	13	13	NUM
ejpam-4367	12	3	]	]	PUNCT
ejpam-4367	12	4	and	and	CCONJ
ejpam-4367	12	5	dorsett	dorsett	PROPN
ejpam-4367	13	1	[	[	X
ejpam-4367	13	2	8	8	NUM
ejpam-4367	13	3	]	]	PUNCT
ejpam-4367	13	4	.	.	PUNCT
ejpam-4367	14	1	in	in	ADP
ejpam-4367	14	2	[	[	X
ejpam-4367	14	3	4	4	NUM
ejpam-4367	14	4	]	]	PUNCT
ejpam-4367	14	5	,	,	PUNCT
ejpam-4367	14	6	the	the	DET
ejpam-4367	14	7	concepts	concept	NOUN
ejpam-4367	14	8	of	of	ADP
ejpam-4367	14	9	the	the	DET
ejpam-4367	14	10	(	(	PUNCT
ejpam-4367	14	11	λ	λ	NOUN
ejpam-4367	14	12	,	,	PUNCT
ejpam-4367	14	13	θ)-closure	θ)-closure	PUNCT
ejpam-4367	14	14	and	and	CCONJ
ejpam-4367	14	15	(	(	PUNCT
ejpam-4367	14	16	λ	λ	NOUN
ejpam-4367	14	17	,	,	PUNCT
ejpam-4367	14	18	θ)-open	θ)-open	VERB
ejpam-4367	14	19	sets	set	NOUN
ejpam-4367	14	20	were	be	AUX
ejpam-4367	14	21	introduced	introduce	VERB
ejpam-4367	14	22	by	by	ADP
ejpam-4367	14	23	using	use	VERB
ejpam-4367	14	24	the	the	DET
ejpam-4367	14	25	θ	θ	ADJ
ejpam-4367	14	26	-	-	PUNCT
ejpam-4367	14	27	closure	closure	NOUN
ejpam-4367	14	28	operator	operator	NOUN
ejpam-4367	14	29	and	and	CCONJ
ejpam-4367	14	30	θ	θ	ADJ
ejpam-4367	14	31	-	-	ADJ
ejpam-4367	14	32	open	open	ADJ
ejpam-4367	14	33	sets	set	NOUN
ejpam-4367	14	34	due	due	ADP
ejpam-4367	14	35	to	to	ADP
ejpam-4367	14	36	velčko	velčko	NOUN
ejpam-4367	14	37	[	[	X
ejpam-4367	14	38	19	19	NUM
ejpam-4367	14	39	]	]	PUNCT
ejpam-4367	14	40	.	.	PUNCT
ejpam-4367	15	1	caldas	caldas	PROPN
ejpam-4367	15	2	et	et	PROPN
ejpam-4367	15	3	al	al	PROPN
ejpam-4367	15	4	.	.	PUNCT
ejpam-4367	16	1	[	[	X
ejpam-4367	16	2	5	5	NUM
ejpam-4367	16	3	]	]	PUNCT
ejpam-4367	16	4	introduced	introduce	VERB
ejpam-4367	16	5	and	and	CCONJ
ejpam-4367	16	6	studied	study	VERB
ejpam-4367	16	7	two	two	NUM
ejpam-4367	16	8	new	new	ADJ
ejpam-4367	16	9	weak	weak	ADJ
ejpam-4367	16	10	separation	separation	NOUN
ejpam-4367	16	11	axioms	axiom	NOUN
ejpam-4367	16	12	called	call	VERB
ejpam-4367	16	13	λθ	λθ	NOUN
ejpam-4367	16	14	-	-	PUNCT
ejpam-4367	16	15	r0	r0	NOUN
ejpam-4367	16	16	and	and	CCONJ
ejpam-4367	16	17	λθ	λθ	NOUN
ejpam-4367	16	18	-	-	PUNCT
ejpam-4367	16	19	r1	r1	NOUN
ejpam-4367	16	20	by	by	ADP
ejpam-4367	16	21	using	use	VERB
ejpam-4367	16	22	the	the	DET
ejpam-4367	16	23	notions	notion	NOUN
ejpam-4367	16	24	of	of	ADP
ejpam-4367	16	25	(	(	PUNCT
ejpam-4367	16	26	λ	λ	PROPN
ejpam-4367	16	27	,	,	PUNCT
ejpam-4367	16	28	θ)-open	θ)-open	VERB
ejpam-4367	16	29	sets	set	NOUN
ejpam-4367	16	30	and	and	CCONJ
ejpam-4367	16	31	the	the	DET
ejpam-4367	16	32	(	(	PUNCT
ejpam-4367	16	33	λ	λ	NOUN
ejpam-4367	16	34	,	,	PUNCT
ejpam-4367	16	35	θ)-closure	θ)-closure	NOUN
ejpam-4367	16	36	operator	operator	NOUN
ejpam-4367	16	37	.	.	PUNCT
ejpam-4367	17	1	in	in	ADP
ejpam-4367	17	2	2005	2005	NUM
ejpam-4367	17	3	,	,	PUNCT
ejpam-4367	17	4	cammaroto	cammaroto	NOUN
ejpam-4367	17	5	and	and	CCONJ
ejpam-4367	17	6	noiri	noiri	ADV
ejpam-4367	17	7	[	[	X
ejpam-4367	17	8	6	6	NUM
ejpam-4367	17	9	]	]	PUNCT
ejpam-4367	17	10	introduce	introduce	VERB
ejpam-4367	17	11	a	a	DET
ejpam-4367	17	12	weak	weak	ADJ
ejpam-4367	17	13	separation	separation	NOUN
ejpam-4367	17	14	axiom	axiom	NOUN
ejpam-4367	17	15	m	m	NOUN
ejpam-4367	17	16	-	-	PUNCT
ejpam-4367	17	17	r0	r0	NOUN
ejpam-4367	17	18	in	in	ADP
ejpam-4367	17	19	m	m	NOUN
ejpam-4367	17	20	-	-	NOUN
ejpam-4367	17	21	spaces	space	NOUN
ejpam-4367	17	22	which	which	PRON
ejpam-4367	17	23	are	be	AUX
ejpam-4367	17	24	equivalent	equivalent	ADJ
ejpam-4367	17	25	to	to	ADP
ejpam-4367	17	26	generalized	generalize	VERB
ejpam-4367	17	27	topological	topological	ADJ
ejpam-4367	17	28	spaces	space	NOUN
ejpam-4367	17	29	due	due	ADP
ejpam-4367	17	30	to	to	ADP
ejpam-4367	17	31	lugojan	lugojan	NOUN
ejpam-4367	17	32	[	[	X
ejpam-4367	17	33	12	12	NUM
ejpam-4367	17	34	]	]	PUNCT
ejpam-4367	17	35	.	.	PUNCT
ejpam-4367	18	1	in	in	ADP
ejpam-4367	18	2	2006	2006	NUM
ejpam-4367	18	3	,	,	PUNCT
ejpam-4367	18	4	noiri	noiri	ADV
ejpam-4367	18	5	[	[	X
ejpam-4367	18	6	16	16	NUM
ejpam-4367	18	7	]	]	PUNCT
ejpam-4367	18	8	introduced	introduce	VERB
ejpam-4367	18	9	the	the	DET
ejpam-4367	18	10	notion	notion	NOUN
ejpam-4367	18	11	of	of	ADP
ejpam-4367	18	12	m	m	NOUN
ejpam-4367	18	13	-	-	PUNCT
ejpam-4367	18	14	r1	r1	ADJ
ejpam-4367	18	15	spaces	space	NOUN
ejpam-4367	18	16	and	and	CCONJ
ejpam-4367	18	17	investigated	investigate	VERB
ejpam-4367	18	18	several	several	ADJ
ejpam-4367	18	19	characterizations	characterization	NOUN
ejpam-4367	18	20	of	of	ADP
ejpam-4367	18	21	m	m	NOUN
ejpam-4367	18	22	-	-	PUNCT
ejpam-4367	18	23	r0	r0	NOUN
ejpam-4367	18	24	spaces	space	NOUN
ejpam-4367	18	25	and	and	CCONJ
ejpam-4367	18	26	m	m	NOUN
ejpam-4367	18	27	-	-	PUNCT
ejpam-4367	18	28	r1	r1	PROPN
ejpam-4367	18	29	spaces	space	NOUN
ejpam-4367	18	30	.	.	PUNCT
ejpam-4367	19	1	abd	abd	PROPN
ejpam-4367	19	2	el	el	PROPN
ejpam-4367	19	3	-	-	PROPN
ejpam-4367	19	4	monsef	monsef	PROPN
ejpam-4367	19	5	et	et	PROPN
ejpam-4367	19	6	al	al	PROPN
ejpam-4367	19	7	.	.	PUNCT
ejpam-4367	20	1	[	[	X
ejpam-4367	20	2	11	11	NUM
ejpam-4367	20	3	]	]	PUNCT
ejpam-4367	20	4	introduced	introduce	VERB
ejpam-4367	20	5	a	a	DET
ejpam-4367	20	6	weak	weak	ADJ
ejpam-4367	20	7	form	form	NOUN
ejpam-4367	20	8	of	of	ADP
ejpam-4367	20	9	open	open	ADJ
ejpam-4367	20	10	sets	set	NOUN
ejpam-4367	20	11	called	call	VERB
ejpam-4367	20	12	β	β	NOUN
ejpam-4367	20	13	-	-	ADJ
ejpam-4367	20	14	open	open	ADJ
ejpam-4367	20	15	sets	set	NOUN
ejpam-4367	20	16	.	.	PUNCT
ejpam-4367	21	1	this	this	DET
ejpam-4367	21	2	notion	notion	NOUN
ejpam-4367	21	3	was	be	AUX
ejpam-4367	21	4	also	also	ADV
ejpam-4367	21	5	called	call	VERB
ejpam-4367	21	6	semi	semi	ADJ
ejpam-4367	21	7	-	-	ADJ
ejpam-4367	21	8	preopen	preopen	ADJ
ejpam-4367	21	9	sets	set	NOUN
ejpam-4367	21	10	in	in	ADP
ejpam-4367	21	11	the	the	DET
ejpam-4367	21	12	sense	sense	NOUN
ejpam-4367	21	13	of	of	ADP
ejpam-4367	21	14	andrijević	andrijević	NOUN
ejpam-4367	21	15	[	[	X
ejpam-4367	21	16	1	1	NUM
ejpam-4367	21	17	]	]	PUNCT
ejpam-4367	21	18	.	.	PUNCT
ejpam-4367	22	1	noiri	noiri	PROPN
ejpam-4367	22	2	and	and	CCONJ
ejpam-4367	22	3	hatir	hatir	PROPN
ejpam-4367	23	1	[	[	X
ejpam-4367	23	2	17	17	NUM
ejpam-4367	23	3	]	]	PUNCT
ejpam-4367	23	4	introduced	introduce	VERB
ejpam-4367	23	5	the	the	DET
ejpam-4367	23	6	notion	notion	NOUN
ejpam-4367	23	7	of	of	ADP
ejpam-4367	23	8	λsp	λsp	NOUN
ejpam-4367	23	9	-	-	PUNCT
ejpam-4367	23	10	sets	set	NOUN
ejpam-4367	23	11	in	in	ADP
ejpam-4367	23	12	terms	term	NOUN
ejpam-4367	23	13	of	of	ADP
ejpam-4367	23	14	the	the	DET
ejpam-4367	23	15	concept	concept	NOUN
ejpam-4367	23	16	of	of	ADP
ejpam-4367	23	17	β	β	ADJ
ejpam-4367	23	18	-	-	ADJ
ejpam-4367	23	19	open	open	ADJ
ejpam-4367	23	20	sets	set	NOUN
ejpam-4367	23	21	and	and	CCONJ
ejpam-4367	23	22	investigated	investigate	VERB
ejpam-4367	23	23	the	the	DET
ejpam-4367	23	24	notion	notion	NOUN
ejpam-4367	23	25	of	of	ADP
ejpam-4367	23	26	λsp	λsp	NOUN
ejpam-4367	23	27	-	-	PUNCT
ejpam-4367	23	28	closed	close	VERB
ejpam-4367	23	29	sets	set	NOUN
ejpam-4367	23	30	by	by	ADP
ejpam-4367	23	31	using	use	VERB
ejpam-4367	23	32	λsp	λsp	NOUN
ejpam-4367	23	33	-	-	PUNCT
ejpam-4367	23	34	sets	set	NOUN
ejpam-4367	23	35	.	.	PUNCT
ejpam-4367	24	1	in	in	ADP
ejpam-4367	24	2	[	[	X
ejpam-4367	24	3	2	2	NUM
ejpam-4367	24	4	]	]	PUNCT
ejpam-4367	24	5	,	,	PUNCT
ejpam-4367	24	6	the	the	DET
ejpam-4367	24	7	author	author	NOUN
ejpam-4367	24	8	introduced	introduce	VERB
ejpam-4367	24	9	the	the	DET
ejpam-4367	24	10	concepts	concept	NOUN
ejpam-4367	24	11	∗corresponding	∗corresponde	VERB
ejpam-4367	24	12	author	author	NOUN
ejpam-4367	24	13	.	.	PUNCT
ejpam-4367	25	1	doi	doi	NOUN
ejpam-4367	25	2	:	:	PUNCT
ejpam-4367	25	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4367	https://doi.org/10.29020/nybg.ejpam.v15i3.4367	NOUN
ejpam-4367	25	4	email	email	NOUN
ejpam-4367	25	5	addresses	address	NOUN
ejpam-4367	25	6	:	:	PUNCT
ejpam-4367	26	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	INTJ
ejpam-4367	26	2	(	(	PUNCT
ejpam-4367	26	3	c.	c.	PROPN
ejpam-4367	26	4	boonpok	boonpok	PROPN
ejpam-4367	26	5	)	)	PUNCT
ejpam-4367	26	6	,	,	PUNCT
ejpam-4367	26	7	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	INTJ
ejpam-4367	26	8	(	(	PUNCT
ejpam-4367	26	9	c.	c.	PROPN
ejpam-4367	26	10	viriyapong	viriyapong	PROPN
ejpam-4367	26	11	)	)	PUNCT
ejpam-4367	26	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4367	27	1	878	878	NUM
ejpam-4367	27	2	©	©	PROPN
ejpam-4367	27	3	2022	2022	NUM
ejpam-4367	27	4	ejpam	ejpam	VERB
ejpam-4367	27	5	all	all	DET
ejpam-4367	27	6	rights	right	NOUN
ejpam-4367	27	7	reserved	reserve	VERB
ejpam-4367	27	8	.	.	PUNCT
ejpam-4367	28	1	c.	c.	PROPN
ejpam-4367	28	2	boonpok	boonpok	PROPN
ejpam-4367	28	3	,	,	PUNCT
ejpam-4367	28	4	c.	c.	PROPN
ejpam-4367	28	5	viriyapong	viriyapong	PROPN
ejpam-4367	28	6	/	/	SYM
ejpam-4367	28	7	eur	eur	PROPN
ejpam-4367	28	8	.	.	PUNCT
ejpam-4367	29	1	j.	j.	PROPN
ejpam-4367	29	2	pure	pure	PROPN
ejpam-4367	29	3	appl	appl	PROPN
ejpam-4367	29	4	.	.	PROPN
ejpam-4367	29	5	math	math	PROPN
ejpam-4367	29	6	,	,	PUNCT
ejpam-4367	29	7	15	15	NUM
ejpam-4367	29	8	(	(	PUNCT
ejpam-4367	29	9	3	3	NUM
ejpam-4367	29	10	)	)	PUNCT
ejpam-4367	29	11	(	(	PUNCT
ejpam-4367	29	12	2022	2022	NUM
ejpam-4367	29	13	)	)	PUNCT
ejpam-4367	29	14	,	,	PUNCT
ejpam-4367	29	15	878	878	NUM
ejpam-4367	29	16	-	-	SYM
ejpam-4367	29	17	886	886	NUM
ejpam-4367	29	18	879	879	NUM
ejpam-4367	29	19	of	of	ADP
ejpam-4367	29	20	(	(	PUNCT
ejpam-4367	29	21	λ	λ	PROPN
ejpam-4367	29	22	,	,	PUNCT
ejpam-4367	29	23	sp)-open	sp)-open	ADJ
ejpam-4367	29	24	sets	set	NOUN
ejpam-4367	29	25	and	and	CCONJ
ejpam-4367	29	26	(	(	PUNCT
ejpam-4367	29	27	λ	λ	PROPN
ejpam-4367	29	28	,	,	PUNCT
ejpam-4367	29	29	sp)-closed	sp)-close	VERB
ejpam-4367	29	30	sets	set	NOUN
ejpam-4367	29	31	which	which	PRON
ejpam-4367	29	32	are	be	AUX
ejpam-4367	29	33	defined	define	VERB
ejpam-4367	29	34	by	by	ADP
ejpam-4367	29	35	utilizing	utilize	VERB
ejpam-4367	29	36	the	the	DET
ejpam-4367	29	37	notions	notion	NOUN
ejpam-4367	29	38	of	of	ADP
ejpam-4367	29	39	λsp	λsp	NOUN
ejpam-4367	29	40	-	-	PUNCT
ejpam-4367	29	41	sets	set	NOUN
ejpam-4367	29	42	and	and	CCONJ
ejpam-4367	29	43	β	β	NOUN
ejpam-4367	29	44	-	-	ADJ
ejpam-4367	29	45	closed	closed	ADJ
ejpam-4367	29	46	sets	set	NOUN
ejpam-4367	29	47	.	.	PUNCT
ejpam-4367	30	1	in	in	ADP
ejpam-4367	30	2	this	this	DET
ejpam-4367	30	3	paper	paper	NOUN
ejpam-4367	30	4	,	,	PUNCT
ejpam-4367	30	5	introduce	introduce	VERB
ejpam-4367	30	6	some	some	DET
ejpam-4367	30	7	weak	weak	ADJ
ejpam-4367	30	8	separation	separation	NOUN
ejpam-4367	30	9	axioms	axiom	NOUN
ejpam-4367	30	10	by	by	ADP
ejpam-4367	30	11	utilizing	utilize	VERB
ejpam-4367	30	12	the	the	DET
ejpam-4367	30	13	concepts	concept	NOUN
ejpam-4367	30	14	of	of	ADP
ejpam-4367	30	15	(	(	PUNCT
ejpam-4367	30	16	λ	λ	PROPN
ejpam-4367	30	17	,	,	PUNCT
ejpam-4367	30	18	sp)-open	sp)-open	ADJ
ejpam-4367	30	19	sets	set	NOUN
ejpam-4367	30	20	and	and	CCONJ
ejpam-4367	30	21	the	the	DET
ejpam-4367	30	22	(	(	PUNCT
ejpam-4367	30	23	λ	λ	PROPN
ejpam-4367	30	24	,	,	PUNCT
ejpam-4367	30	25	sp)-closure	sp)-closure	NOUN
ejpam-4367	30	26	operator	operator	NOUN
ejpam-4367	30	27	.	.	PUNCT
ejpam-4367	31	1	furthermore	furthermore	ADV
ejpam-4367	31	2	,	,	PUNCT
ejpam-4367	31	3	several	several	ADJ
ejpam-4367	31	4	characterizations	characterization	NOUN
ejpam-4367	31	5	of	of	ADP
ejpam-4367	31	6	(	(	PUNCT
ejpam-4367	31	7	λ	λ	PROPN
ejpam-4367	31	8	,	,	PUNCT
ejpam-4367	31	9	sp)-r0	sp)-r0	PROPN
ejpam-4367	31	10	and	and	CCONJ
ejpam-4367	31	11	(	(	PUNCT
ejpam-4367	31	12	λ	λ	PROPN
ejpam-4367	31	13	,	,	PUNCT
ejpam-4367	31	14	sp)-r1	sp)-r1	NOUN
ejpam-4367	31	15	topological	topological	ADJ
ejpam-4367	31	16	spaces	space	NOUN
ejpam-4367	31	17	are	be	AUX
ejpam-4367	31	18	discussed	discuss	VERB
ejpam-4367	31	19	.	.	PUNCT
ejpam-4367	32	1	2	2	X
ejpam-4367	32	2	.	.	X
ejpam-4367	32	3	preliminaries	preliminary	NOUN
ejpam-4367	32	4	we	we	PRON
ejpam-4367	32	5	begin	begin	VERB
ejpam-4367	32	6	with	with	ADP
ejpam-4367	32	7	some	some	DET
ejpam-4367	32	8	definitions	definition	NOUN
ejpam-4367	32	9	and	and	CCONJ
ejpam-4367	32	10	known	know	VERB
ejpam-4367	32	11	results	result	NOUN
ejpam-4367	32	12	which	which	PRON
ejpam-4367	32	13	will	will	AUX
ejpam-4367	32	14	be	be	AUX
ejpam-4367	32	15	used	use	VERB
ejpam-4367	32	16	throughout	throughout	ADP
ejpam-4367	32	17	this	this	DET
ejpam-4367	32	18	paper	paper	NOUN
ejpam-4367	32	19	.	.	PUNCT
ejpam-4367	33	1	in	in	ADP
ejpam-4367	33	2	the	the	DET
ejpam-4367	33	3	present	present	ADJ
ejpam-4367	33	4	paper	paper	NOUN
ejpam-4367	33	5	,	,	PUNCT
ejpam-4367	33	6	spaces	space	NOUN
ejpam-4367	33	7	(	(	PUNCT
ejpam-4367	33	8	x	x	X
ejpam-4367	33	9	,	,	PUNCT
ejpam-4367	33	10	τ	τ	X
ejpam-4367	33	11	)	)	PUNCT
ejpam-4367	33	12	and	and	CCONJ
ejpam-4367	33	13	(	(	PUNCT
ejpam-4367	33	14	y	y	PROPN
ejpam-4367	33	15	,	,	PUNCT
ejpam-4367	33	16	σ	σ	PROPN
ejpam-4367	33	17	)	)	PUNCT
ejpam-4367	33	18	(	(	PUNCT
ejpam-4367	33	19	or	or	CCONJ
ejpam-4367	33	20	simply	simply	ADV
ejpam-4367	33	21	x	x	X
ejpam-4367	33	22	and	and	CCONJ
ejpam-4367	33	23	y	y	PROPN
ejpam-4367	33	24	)	)	PUNCT
ejpam-4367	33	25	always	always	ADV
ejpam-4367	33	26	mean	mean	VERB
ejpam-4367	33	27	topological	topological	ADJ
ejpam-4367	33	28	spaces	space	NOUN
ejpam-4367	33	29	on	on	ADP
ejpam-4367	33	30	which	which	PRON
ejpam-4367	33	31	no	no	DET
ejpam-4367	33	32	separation	separation	NOUN
ejpam-4367	33	33	axioms	axiom	NOUN
ejpam-4367	33	34	are	be	AUX
ejpam-4367	33	35	assumed	assume	VERB
ejpam-4367	33	36	unless	unless	SCONJ
ejpam-4367	33	37	explicitly	explicitly	ADV
ejpam-4367	33	38	stated	state	VERB
ejpam-4367	33	39	.	.	PUNCT
ejpam-4367	34	1	for	for	ADP
ejpam-4367	34	2	a	a	DET
ejpam-4367	34	3	subset	subset	NOUN
ejpam-4367	34	4	a	a	PRON
ejpam-4367	34	5	of	of	ADP
ejpam-4367	34	6	a	a	DET
ejpam-4367	34	7	topological	topological	ADJ
ejpam-4367	34	8	space	space	NOUN
ejpam-4367	34	9	(	(	PUNCT
ejpam-4367	34	10	x	x	X
ejpam-4367	34	11	,	,	PUNCT
ejpam-4367	34	12	τ	τ	PROPN
ejpam-4367	34	13	)	)	PUNCT
ejpam-4367	34	14	,	,	PUNCT
ejpam-4367	34	15	cl(a	cl(a	NUM
ejpam-4367	34	16	)	)	PUNCT
ejpam-4367	34	17	and	and	CCONJ
ejpam-4367	34	18	int(a	int(a	PROPN
ejpam-4367	34	19	)	)	PUNCT
ejpam-4367	34	20	represent	represent	VERB
ejpam-4367	34	21	the	the	DET
ejpam-4367	34	22	closure	closure	NOUN
ejpam-4367	34	23	and	and	CCONJ
ejpam-4367	34	24	the	the	DET
ejpam-4367	34	25	interior	interior	NOUN
ejpam-4367	34	26	of	of	ADP
ejpam-4367	34	27	a	a	PRON
ejpam-4367	34	28	,	,	PUNCT
ejpam-4367	34	29	respectively	respectively	ADV
ejpam-4367	34	30	.	.	PUNCT
ejpam-4367	35	1	a	a	DET
ejpam-4367	35	2	subset	subset	NOUN
ejpam-4367	35	3	a	a	PRON
ejpam-4367	35	4	of	of	ADP
ejpam-4367	35	5	a	a	DET
ejpam-4367	35	6	topological	topological	ADJ
ejpam-4367	35	7	space	space	NOUN
ejpam-4367	35	8	(	(	PUNCT
ejpam-4367	35	9	x	x	X
ejpam-4367	35	10	,	,	PUNCT
ejpam-4367	35	11	τ	τ	X
ejpam-4367	35	12	)	)	PUNCT
ejpam-4367	35	13	is	be	AUX
ejpam-4367	35	14	said	say	VERB
ejpam-4367	35	15	to	to	PART
ejpam-4367	35	16	be	be	AUX
ejpam-4367	35	17	β	β	X
ejpam-4367	35	18	-	-	ADJ
ejpam-4367	35	19	open	open	ADJ
ejpam-4367	36	1	[	[	X
ejpam-4367	36	2	11	11	NUM
ejpam-4367	36	3	]	]	X
ejpam-4367	36	4	if	if	SCONJ
ejpam-4367	36	5	a	a	DET
ejpam-4367	36	6	⊆	⊆	NUM
ejpam-4367	36	7	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-4367	36	8	)	)	PUNCT
ejpam-4367	36	9	)	)	PUNCT
ejpam-4367	36	10	)	)	PUNCT
ejpam-4367	36	11	.	.	PUNCT
ejpam-4367	37	1	the	the	DET
ejpam-4367	37	2	complement	complement	NOUN
ejpam-4367	37	3	of	of	ADP
ejpam-4367	37	4	a	a	DET
ejpam-4367	37	5	β	β	X
ejpam-4367	37	6	-	-	ADJ
ejpam-4367	37	7	open	open	ADJ
ejpam-4367	37	8	set	set	NOUN
ejpam-4367	37	9	is	be	AUX
ejpam-4367	37	10	called	call	VERB
ejpam-4367	37	11	β	β	NOUN
ejpam-4367	37	12	-	-	VERB
ejpam-4367	37	13	closed	closed	ADJ
ejpam-4367	37	14	.	.	PUNCT
ejpam-4367	38	1	the	the	DET
ejpam-4367	38	2	family	family	NOUN
ejpam-4367	38	3	of	of	ADP
ejpam-4367	38	4	all	all	DET
ejpam-4367	38	5	β	β	ADJ
ejpam-4367	38	6	-	-	ADJ
ejpam-4367	38	7	open	open	ADJ
ejpam-4367	38	8	sets	set	NOUN
ejpam-4367	38	9	of	of	ADP
ejpam-4367	38	10	a	a	DET
ejpam-4367	38	11	topological	topological	ADJ
ejpam-4367	38	12	space	space	NOUN
ejpam-4367	38	13	(	(	PUNCT
ejpam-4367	38	14	x	x	X
ejpam-4367	38	15	,	,	PUNCT
ejpam-4367	38	16	τ	τ	X
ejpam-4367	38	17	)	)	PUNCT
ejpam-4367	38	18	is	be	AUX
ejpam-4367	38	19	denoted	denote	VERB
ejpam-4367	38	20	by	by	ADP
ejpam-4367	38	21	β(x	β(x	PROPN
ejpam-4367	38	22	,	,	PUNCT
ejpam-4367	38	23	τ	τ	PROPN
ejpam-4367	38	24	)	)	PUNCT
ejpam-4367	38	25	.	.	PUNCT
ejpam-4367	39	1	a	a	DET
ejpam-4367	39	2	subset	subset	NOUN
ejpam-4367	39	3	λsp(a	λsp(a	NOUN
ejpam-4367	39	4	)	)	PUNCT
ejpam-4367	40	1	[	[	X
ejpam-4367	40	2	17	17	NUM
ejpam-4367	40	3	]	]	PUNCT
ejpam-4367	40	4	is	be	AUX
ejpam-4367	40	5	defined	define	VERB
ejpam-4367	40	6	as	as	SCONJ
ejpam-4367	40	7	follows	follow	VERB
ejpam-4367	40	8	:	:	PUNCT
ejpam-4367	40	9	λsp(a	λsp(a	NUM
ejpam-4367	40	10	)	)	PUNCT
ejpam-4367	40	11	=	=	PUNCT
ejpam-4367	41	1	∩{u	∩{u	PROPN
ejpam-4367	41	2	|	|	ADV
ejpam-4367	41	3	a	a	DET
ejpam-4367	41	4	⊆	⊆	NUM
ejpam-4367	41	5	u	u	NOUN
ejpam-4367	41	6	,	,	PUNCT
ejpam-4367	41	7	u	u	NOUN
ejpam-4367	41	8	∈	∈	PROPN
ejpam-4367	41	9	β(x	β(x	PROPN
ejpam-4367	41	10	,	,	PUNCT
ejpam-4367	41	11	τ	τ	X
ejpam-4367	41	12	)	)	PUNCT
ejpam-4367	41	13	}	}	PUNCT
ejpam-4367	41	14	.	.	PUNCT
ejpam-4367	42	1	a	a	DET
ejpam-4367	42	2	subset	subset	NOUN
ejpam-4367	42	3	b	b	NOUN
ejpam-4367	42	4	of	of	ADP
ejpam-4367	42	5	a	a	DET
ejpam-4367	42	6	topological	topological	ADJ
ejpam-4367	42	7	space	space	NOUN
ejpam-4367	42	8	(	(	PUNCT
ejpam-4367	42	9	x	x	X
ejpam-4367	42	10	,	,	PUNCT
ejpam-4367	42	11	τ	τ	X
ejpam-4367	42	12	)	)	PUNCT
ejpam-4367	42	13	is	be	AUX
ejpam-4367	42	14	called	call	VERB
ejpam-4367	42	15	a	a	DET
ejpam-4367	42	16	λsp	λsp	NOUN
ejpam-4367	42	17	-	-	PUNCT
ejpam-4367	42	18	set	set	NOUN
ejpam-4367	42	19	[	[	X
ejpam-4367	42	20	17	17	NUM
ejpam-4367	42	21	]	]	PUNCT
ejpam-4367	42	22	if	if	SCONJ
ejpam-4367	42	23	b	b	PROPN
ejpam-4367	42	24	=	=	SYM
ejpam-4367	42	25	λsp(b	λsp(b	PROPN
ejpam-4367	42	26	)	)	PUNCT
ejpam-4367	42	27	.	.	PUNCT
ejpam-4367	43	1	a	a	DET
ejpam-4367	43	2	subset	subset	NOUN
ejpam-4367	43	3	a	a	PRON
ejpam-4367	43	4	of	of	ADP
ejpam-4367	43	5	a	a	DET
ejpam-4367	43	6	topological	topological	ADJ
ejpam-4367	43	7	space	space	NOUN
ejpam-4367	43	8	(	(	PUNCT
ejpam-4367	43	9	x	x	X
ejpam-4367	43	10	,	,	PUNCT
ejpam-4367	43	11	τ	τ	X
ejpam-4367	43	12	)	)	PUNCT
ejpam-4367	43	13	is	be	AUX
ejpam-4367	43	14	called	call	VERB
ejpam-4367	43	15	(	(	PUNCT
ejpam-4367	43	16	λ	λ	X
ejpam-4367	43	17	,	,	PUNCT
ejpam-4367	43	18	sp)-closed	sp)-close	VERB
ejpam-4367	43	19	[	[	PUNCT
ejpam-4367	43	20	2	2	X
ejpam-4367	43	21	]	]	X
ejpam-4367	43	22	if	if	SCONJ
ejpam-4367	43	23	a	a	DET
ejpam-4367	43	24	=	=	X
ejpam-4367	43	25	t	t	NOUN
ejpam-4367	43	26	∩c	∩c	NOUN
ejpam-4367	43	27	,	,	PUNCT
ejpam-4367	43	28	where	where	SCONJ
ejpam-4367	43	29	t	t	PROPN
ejpam-4367	43	30	is	be	AUX
ejpam-4367	43	31	a	a	DET
ejpam-4367	43	32	λsp	λsp	NOUN
ejpam-4367	43	33	-	-	PUNCT
ejpam-4367	43	34	set	set	VERB
ejpam-4367	43	35	and	and	CCONJ
ejpam-4367	43	36	c	c	NOUN
ejpam-4367	43	37	is	be	AUX
ejpam-4367	43	38	a	a	DET
ejpam-4367	43	39	β	β	NOUN
ejpam-4367	43	40	-	-	ADJ
ejpam-4367	43	41	closed	closed	ADJ
ejpam-4367	43	42	set	set	NOUN
ejpam-4367	43	43	.	.	PUNCT
ejpam-4367	44	1	the	the	DET
ejpam-4367	44	2	complement	complement	NOUN
ejpam-4367	44	3	of	of	ADP
ejpam-4367	44	4	a	a	DET
ejpam-4367	44	5	(	(	PUNCT
ejpam-4367	44	6	λ	λ	PROPN
ejpam-4367	44	7	,	,	PUNCT
ejpam-4367	44	8	sp)-closed	sp)-close	VERB
ejpam-4367	44	9	set	set	VERB
ejpam-4367	44	10	is	be	AUX
ejpam-4367	44	11	called	call	VERB
ejpam-4367	44	12	(	(	PUNCT
ejpam-4367	44	13	λ	λ	NOUN
ejpam-4367	44	14	,	,	PUNCT
ejpam-4367	44	15	sp)-open	sp)-open	NOUN
ejpam-4367	44	16	.	.	PUNCT
ejpam-4367	45	1	the	the	DET
ejpam-4367	45	2	family	family	NOUN
ejpam-4367	45	3	of	of	ADP
ejpam-4367	45	4	all	all	DET
ejpam-4367	45	5	(	(	PUNCT
ejpam-4367	45	6	λ	λ	NOUN
ejpam-4367	45	7	,	,	PUNCT
ejpam-4367	45	8	sp)-open	sp)-open	ADJ
ejpam-4367	45	9	(	(	PUNCT
ejpam-4367	45	10	resp	resp	NOUN
ejpam-4367	45	11	.	.	PUNCT
ejpam-4367	46	1	(	(	PUNCT
ejpam-4367	46	2	λ	λ	X
ejpam-4367	46	3	,	,	PUNCT
ejpam-4367	46	4	sp)-closed	sp)-closed	ADJ
ejpam-4367	46	5	)	)	PUNCT
ejpam-4367	46	6	sets	set	NOUN
ejpam-4367	46	7	in	in	ADP
ejpam-4367	46	8	a	a	DET
ejpam-4367	46	9	topological	topological	ADJ
ejpam-4367	46	10	space	space	NOUN
ejpam-4367	46	11	(	(	PUNCT
ejpam-4367	46	12	x	x	X
ejpam-4367	46	13	,	,	PUNCT
ejpam-4367	46	14	τ	τ	X
ejpam-4367	46	15	)	)	PUNCT
ejpam-4367	46	16	is	be	AUX
ejpam-4367	46	17	denoted	denote	VERB
ejpam-4367	46	18	by	by	ADP
ejpam-4367	46	19	λspo(x	λspo(x	PROPN
ejpam-4367	46	20	,	,	PUNCT
ejpam-4367	46	21	τ	τ	PROPN
ejpam-4367	46	22	)	)	PUNCT
ejpam-4367	46	23	(	(	PUNCT
ejpam-4367	46	24	resp	resp	NOUN
ejpam-4367	46	25	.	.	PUNCT
ejpam-4367	47	1	λspc(x	λspc(x	NOUN
ejpam-4367	47	2	,	,	PUNCT
ejpam-4367	47	3	τ	τ	PROPN
ejpam-4367	47	4	)	)	PUNCT
ejpam-4367	47	5	)	)	PUNCT
ejpam-4367	47	6	.	.	PUNCT
ejpam-4367	48	1	let	let	VERB
ejpam-4367	48	2	a	a	DET
ejpam-4367	48	3	be	be	AUX
ejpam-4367	48	4	a	a	DET
ejpam-4367	48	5	subset	subset	NOUN
ejpam-4367	48	6	of	of	ADP
ejpam-4367	48	7	a	a	DET
ejpam-4367	48	8	topological	topological	ADJ
ejpam-4367	48	9	space	space	NOUN
ejpam-4367	48	10	(	(	PUNCT
ejpam-4367	48	11	x	x	X
ejpam-4367	48	12	,	,	PUNCT
ejpam-4367	48	13	τ	τ	PROPN
ejpam-4367	48	14	)	)	PUNCT
ejpam-4367	48	15	.	.	PUNCT
ejpam-4367	49	1	a	a	DET
ejpam-4367	49	2	point	point	NOUN
ejpam-4367	49	3	x	x	X
ejpam-4367	49	4	∈	∈	NOUN
ejpam-4367	49	5	x	x	PUNCT
ejpam-4367	49	6	is	be	AUX
ejpam-4367	49	7	called	call	VERB
ejpam-4367	49	8	a	a	DET
ejpam-4367	49	9	(	(	PUNCT
ejpam-4367	49	10	λ	λ	NOUN
ejpam-4367	49	11	,	,	PUNCT
ejpam-4367	49	12	sp)-cluster	sp)-cluster	NOUN
ejpam-4367	49	13	point	point	NOUN
ejpam-4367	49	14	[	[	X
ejpam-4367	49	15	2	2	X
ejpam-4367	49	16	]	]	PUNCT
ejpam-4367	49	17	of	of	ADP
ejpam-4367	49	18	a	a	PRON
ejpam-4367	49	19	if	if	SCONJ
ejpam-4367	49	20	a	a	DET
ejpam-4367	49	21	∩	∩	ADJ
ejpam-4367	49	22	u	u	ADJ
ejpam-4367	49	23	̸=	̸=	PROPN
ejpam-4367	49	24	∅	∅	NOUN
ejpam-4367	49	25	for	for	ADP
ejpam-4367	49	26	every	every	DET
ejpam-4367	49	27	(	(	PUNCT
ejpam-4367	49	28	λ	λ	NOUN
ejpam-4367	49	29	,	,	PUNCT
ejpam-4367	49	30	sp)-open	sp)-open	NOUN
ejpam-4367	49	31	set	set	VERB
ejpam-4367	49	32	u	u	NOUN
ejpam-4367	49	33	of	of	ADP
ejpam-4367	49	34	x	x	SYM
ejpam-4367	49	35	containing	contain	VERB
ejpam-4367	49	36	x.	x.	NOUN
ejpam-4367	49	37	the	the	DET
ejpam-4367	49	38	set	set	NOUN
ejpam-4367	49	39	of	of	ADP
ejpam-4367	49	40	all	all	DET
ejpam-4367	49	41	(	(	PUNCT
ejpam-4367	49	42	λ	λ	PROPN
ejpam-4367	49	43	,	,	PUNCT
ejpam-4367	49	44	sp)-cluster	sp)-cluster	NOUN
ejpam-4367	49	45	points	point	NOUN
ejpam-4367	49	46	of	of	ADP
ejpam-4367	49	47	a	a	PRON
ejpam-4367	49	48	is	be	AUX
ejpam-4367	49	49	called	call	VERB
ejpam-4367	49	50	the	the	DET
ejpam-4367	49	51	(	(	PUNCT
ejpam-4367	49	52	λ	λ	PROPN
ejpam-4367	49	53	,	,	PUNCT
ejpam-4367	49	54	sp)-closure	sp)-closure	NOUN
ejpam-4367	49	55	[	[	X
ejpam-4367	49	56	2	2	NUM
ejpam-4367	49	57	]	]	PUNCT
ejpam-4367	49	58	of	of	ADP
ejpam-4367	49	59	a	a	PRON
ejpam-4367	49	60	and	and	CCONJ
ejpam-4367	49	61	is	be	AUX
ejpam-4367	49	62	denoted	denote	VERB
ejpam-4367	49	63	by	by	ADP
ejpam-4367	49	64	a(λ	a(λ	ADV
ejpam-4367	49	65	,	,	PUNCT
ejpam-4367	49	66	sp	sp	NOUN
ejpam-4367	49	67	)	)	PUNCT
ejpam-4367	49	68	.	.	PUNCT
ejpam-4367	50	1	the	the	DET
ejpam-4367	50	2	union	union	NOUN
ejpam-4367	50	3	of	of	ADP
ejpam-4367	50	4	all	all	DET
ejpam-4367	50	5	(	(	PUNCT
ejpam-4367	50	6	λ	λ	NOUN
ejpam-4367	50	7	,	,	PUNCT
ejpam-4367	50	8	sp)-open	sp)-open	ADJ
ejpam-4367	50	9	sets	set	NOUN
ejpam-4367	50	10	contained	contain	VERB
ejpam-4367	50	11	in	in	ADP
ejpam-4367	50	12	a	a	PRON
ejpam-4367	50	13	is	be	AUX
ejpam-4367	50	14	called	call	VERB
ejpam-4367	50	15	the	the	DET
ejpam-4367	50	16	(	(	PUNCT
ejpam-4367	50	17	λ	λ	PROPN
ejpam-4367	50	18	,	,	PUNCT
ejpam-4367	50	19	sp)-interior	sp)-interior	NOUN
ejpam-4367	50	20	[	[	X
ejpam-4367	50	21	2	2	NUM
ejpam-4367	50	22	]	]	PUNCT
ejpam-4367	50	23	of	of	ADP
ejpam-4367	50	24	a	a	PRON
ejpam-4367	50	25	and	and	CCONJ
ejpam-4367	50	26	is	be	AUX
ejpam-4367	50	27	denoted	denote	VERB
ejpam-4367	50	28	by	by	ADP
ejpam-4367	50	29	a(λ	a(λ	ADV
ejpam-4367	50	30	,	,	PUNCT
ejpam-4367	50	31	sp	sp	NOUN
ejpam-4367	50	32	)	)	PUNCT
ejpam-4367	50	33	.	.	PUNCT
ejpam-4367	51	1	lemma	lemma	PROPN
ejpam-4367	51	2	1	1	NUM
ejpam-4367	51	3	.	.	PUNCT
ejpam-4367	52	1	[	[	X
ejpam-4367	52	2	2	2	X
ejpam-4367	52	3	]	]	PUNCT
ejpam-4367	52	4	let	let	VERB
ejpam-4367	52	5	a	a	PRON
ejpam-4367	52	6	and	and	CCONJ
ejpam-4367	52	7	b	b	NOUN
ejpam-4367	52	8	be	be	AUX
ejpam-4367	52	9	subsets	subset	NOUN
ejpam-4367	52	10	of	of	ADP
ejpam-4367	52	11	a	a	DET
ejpam-4367	52	12	topological	topological	ADJ
ejpam-4367	52	13	space	space	NOUN
ejpam-4367	52	14	(	(	PUNCT
ejpam-4367	52	15	x	x	X
ejpam-4367	52	16	,	,	PUNCT
ejpam-4367	52	17	τ	τ	PROPN
ejpam-4367	52	18	)	)	PUNCT
ejpam-4367	52	19	.	.	PUNCT
ejpam-4367	53	1	for	for	ADP
ejpam-4367	53	2	the	the	DET
ejpam-4367	53	3	(	(	PUNCT
ejpam-4367	53	4	λ	λ	PROPN
ejpam-4367	53	5	,	,	PUNCT
ejpam-4367	53	6	sp)-closure	sp)-closure	NOUN
ejpam-4367	53	7	,	,	PUNCT
ejpam-4367	53	8	the	the	DET
ejpam-4367	53	9	following	follow	VERB
ejpam-4367	53	10	properties	property	NOUN
ejpam-4367	53	11	hold	hold	VERB
ejpam-4367	53	12	:	:	PUNCT
ejpam-4367	53	13	(	(	PUNCT
ejpam-4367	53	14	1	1	X
ejpam-4367	53	15	)	)	PUNCT
ejpam-4367	53	16	a	a	DET
ejpam-4367	53	17	⊆	⊆	NUM
ejpam-4367	53	18	a(λ	a(λ	ADJ
ejpam-4367	53	19	,	,	PUNCT
ejpam-4367	53	20	sp	sp	NOUN
ejpam-4367	53	21	)	)	PUNCT
ejpam-4367	53	22	and	and	CCONJ
ejpam-4367	53	23	[	[	X
ejpam-4367	53	24	a(λ	a(λ	ADV
ejpam-4367	53	25	,	,	PUNCT
ejpam-4367	53	26	sp)](λ	sp)](λ	PROPN
ejpam-4367	53	27	,	,	PUNCT
ejpam-4367	53	28	sp	sp	NOUN
ejpam-4367	53	29	)	)	PUNCT
ejpam-4367	53	30	=	=	PUNCT
ejpam-4367	53	31	a(λ	a(λ	ADV
ejpam-4367	53	32	,	,	PUNCT
ejpam-4367	53	33	sp	sp	NOUN
ejpam-4367	53	34	)	)	PUNCT
ejpam-4367	53	35	.	.	PUNCT
ejpam-4367	54	1	(	(	PUNCT
ejpam-4367	54	2	2	2	X
ejpam-4367	54	3	)	)	PUNCT
ejpam-4367	54	4	if	if	SCONJ
ejpam-4367	54	5	a	a	DET
ejpam-4367	54	6	⊆	⊆	NUM
ejpam-4367	54	7	b	b	NOUN
ejpam-4367	54	8	,	,	PUNCT
ejpam-4367	54	9	then	then	ADV
ejpam-4367	54	10	a(λ	a(λ	ADV
ejpam-4367	54	11	,	,	PUNCT
ejpam-4367	54	12	sp	sp	NOUN
ejpam-4367	54	13	)	)	PUNCT
ejpam-4367	54	14	⊆	⊆	NUM
ejpam-4367	54	15	b(λ	b(λ	NOUN
ejpam-4367	54	16	,	,	PUNCT
ejpam-4367	54	17	sp	sp	NOUN
ejpam-4367	54	18	)	)	PUNCT
ejpam-4367	54	19	.	.	PUNCT
ejpam-4367	55	1	(	(	PUNCT
ejpam-4367	55	2	3	3	X
ejpam-4367	55	3	)	)	PUNCT
ejpam-4367	55	4	a(λ	a(λ	ADV
ejpam-4367	55	5	,	,	PUNCT
ejpam-4367	55	6	sp	sp	NOUN
ejpam-4367	55	7	)	)	PUNCT
ejpam-4367	55	8	is	be	AUX
ejpam-4367	55	9	(	(	PUNCT
ejpam-4367	55	10	λ	λ	X
ejpam-4367	55	11	,	,	PUNCT
ejpam-4367	55	12	sp)-closed	sp)-close	VERB
ejpam-4367	55	13	.	.	PUNCT
ejpam-4367	56	1	(	(	PUNCT
ejpam-4367	56	2	4	4	X
ejpam-4367	56	3	)	)	PUNCT
ejpam-4367	56	4	a	a	PRON
ejpam-4367	56	5	is	be	AUX
ejpam-4367	56	6	(	(	PUNCT
ejpam-4367	56	7	λ	λ	X
ejpam-4367	56	8	,	,	PUNCT
ejpam-4367	56	9	sp)-closed	sp)-close	VERB
ejpam-4367	56	10	if	if	SCONJ
ejpam-4367	56	11	and	and	CCONJ
ejpam-4367	56	12	only	only	ADV
ejpam-4367	56	13	if	if	SCONJ
ejpam-4367	56	14	a(λ	a(λ	ADV
ejpam-4367	56	15	,	,	PUNCT
ejpam-4367	56	16	sp	sp	NOUN
ejpam-4367	56	17	)	)	PUNCT
ejpam-4367	56	18	=	=	PUNCT
ejpam-4367	56	19	a.	a.	NOUN
ejpam-4367	56	20	lemma	lemma	PROPN
ejpam-4367	56	21	2	2	X
ejpam-4367	56	22	.	.	PUNCT
ejpam-4367	57	1	[	[	X
ejpam-4367	57	2	2	2	NUM
ejpam-4367	57	3	]	]	PUNCT
ejpam-4367	57	4	for	for	ADP
ejpam-4367	57	5	subsets	subset	NOUN
ejpam-4367	57	6	a	a	PRON
ejpam-4367	57	7	and	and	CCONJ
ejpam-4367	57	8	b	b	NOUN
ejpam-4367	57	9	of	of	ADP
ejpam-4367	57	10	a	a	DET
ejpam-4367	57	11	topological	topological	ADJ
ejpam-4367	57	12	space	space	NOUN
ejpam-4367	57	13	(	(	PUNCT
ejpam-4367	57	14	x	x	X
ejpam-4367	57	15	,	,	PUNCT
ejpam-4367	57	16	τ	τ	PROPN
ejpam-4367	57	17	)	)	PUNCT
ejpam-4367	57	18	,	,	PUNCT
ejpam-4367	57	19	the	the	DET
ejpam-4367	57	20	following	follow	VERB
ejpam-4367	57	21	properties	property	NOUN
ejpam-4367	57	22	hold	hold	VERB
ejpam-4367	57	23	:	:	PUNCT
ejpam-4367	57	24	(	(	PUNCT
ejpam-4367	57	25	1	1	X
ejpam-4367	57	26	)	)	PUNCT
ejpam-4367	57	27	a(λ	a(λ	ADV
ejpam-4367	57	28	,	,	PUNCT
ejpam-4367	57	29	sp	sp	NOUN
ejpam-4367	57	30	)	)	PUNCT
ejpam-4367	57	31	⊆	⊆	NUM
ejpam-4367	57	32	a	a	DET
ejpam-4367	57	33	and	and	CCONJ
ejpam-4367	57	34	[	[	X
ejpam-4367	57	35	a(λ	a(λ	ADV
ejpam-4367	57	36	,	,	PUNCT
ejpam-4367	57	37	sp)](λ	sp)](λ	PROPN
ejpam-4367	57	38	,	,	PUNCT
ejpam-4367	57	39	sp	sp	NOUN
ejpam-4367	57	40	)	)	PUNCT
ejpam-4367	57	41	=	=	PUNCT
ejpam-4367	57	42	a(λ	a(λ	ADV
ejpam-4367	57	43	,	,	PUNCT
ejpam-4367	57	44	sp	sp	NOUN
ejpam-4367	57	45	)	)	PUNCT
ejpam-4367	57	46	.	.	PUNCT
ejpam-4367	58	1	(	(	PUNCT
ejpam-4367	58	2	2	2	X
ejpam-4367	58	3	)	)	PUNCT
ejpam-4367	58	4	if	if	SCONJ
ejpam-4367	58	5	a	a	DET
ejpam-4367	58	6	⊆	⊆	NUM
ejpam-4367	58	7	b	b	NOUN
ejpam-4367	58	8	,	,	PUNCT
ejpam-4367	58	9	then	then	ADV
ejpam-4367	58	10	a(λ	a(λ	ADV
ejpam-4367	58	11	,	,	PUNCT
ejpam-4367	58	12	sp	sp	NOUN
ejpam-4367	58	13	)	)	PUNCT
ejpam-4367	58	14	⊆	⊆	NUM
ejpam-4367	58	15	b(λ	b(λ	NOUN
ejpam-4367	58	16	,	,	PUNCT
ejpam-4367	58	17	sp	sp	NOUN
ejpam-4367	58	18	)	)	PUNCT
ejpam-4367	58	19	.	.	PUNCT
ejpam-4367	59	1	(	(	PUNCT
ejpam-4367	59	2	3	3	X
ejpam-4367	59	3	)	)	PUNCT
ejpam-4367	59	4	a(λ	a(λ	ADV
ejpam-4367	59	5	,	,	PUNCT
ejpam-4367	59	6	sp	sp	NOUN
ejpam-4367	59	7	)	)	PUNCT
ejpam-4367	59	8	is	be	AUX
ejpam-4367	59	9	(	(	PUNCT
ejpam-4367	59	10	λ	λ	INTJ
ejpam-4367	59	11	,	,	PUNCT
ejpam-4367	59	12	sp)-open	sp)-open	NOUN
ejpam-4367	59	13	.	.	PUNCT
ejpam-4367	60	1	(	(	PUNCT
ejpam-4367	60	2	4	4	X
ejpam-4367	60	3	)	)	PUNCT
ejpam-4367	60	4	a	a	DET
ejpam-4367	60	5	is	be	AUX
ejpam-4367	60	6	(	(	PUNCT
ejpam-4367	60	7	λ	λ	NOUN
ejpam-4367	60	8	,	,	PUNCT
ejpam-4367	60	9	sp)-open	sp)-open	ADJ
ejpam-4367	60	10	if	if	SCONJ
ejpam-4367	60	11	and	and	CCONJ
ejpam-4367	60	12	only	only	ADV
ejpam-4367	60	13	if	if	SCONJ
ejpam-4367	60	14	a(λ	a(λ	ADV
ejpam-4367	60	15	,	,	PUNCT
ejpam-4367	60	16	sp	sp	NOUN
ejpam-4367	60	17	)	)	PUNCT
ejpam-4367	60	18	=	=	SYM
ejpam-4367	60	19	a.	a.	NOUN
ejpam-4367	60	20	c.	c.	PROPN
ejpam-4367	60	21	boonpok	boonpok	PROPN
ejpam-4367	60	22	,	,	PUNCT
ejpam-4367	60	23	c.	c.	PROPN
ejpam-4367	60	24	viriyapong	viriyapong	PROPN
ejpam-4367	60	25	/	/	SYM
ejpam-4367	60	26	eur	eur	PROPN
ejpam-4367	60	27	.	.	PUNCT
ejpam-4367	61	1	j.	j.	PROPN
ejpam-4367	61	2	pure	pure	PROPN
ejpam-4367	61	3	appl	appl	PROPN
ejpam-4367	61	4	.	.	PROPN
ejpam-4367	61	5	math	math	PROPN
ejpam-4367	61	6	,	,	PUNCT
ejpam-4367	61	7	15	15	NUM
ejpam-4367	61	8	(	(	PUNCT
ejpam-4367	61	9	3	3	NUM
ejpam-4367	61	10	)	)	PUNCT
ejpam-4367	61	11	(	(	PUNCT
ejpam-4367	61	12	2022	2022	NUM
ejpam-4367	61	13	)	)	PUNCT
ejpam-4367	61	14	,	,	PUNCT
ejpam-4367	61	15	878	878	NUM
ejpam-4367	61	16	-	-	SYM
ejpam-4367	61	17	886	886	NUM
ejpam-4367	61	18	880	880	NUM
ejpam-4367	61	19	(	(	PUNCT
ejpam-4367	61	20	5	5	NUM
ejpam-4367	61	21	)	)	PUNCT
ejpam-4367	62	1	[	[	X
ejpam-4367	62	2	x	x	X
ejpam-4367	62	3	−a](λ	−a](λ	PROPN
ejpam-4367	62	4	,	,	PUNCT
ejpam-4367	62	5	sp	sp	NOUN
ejpam-4367	62	6	)	)	PUNCT
ejpam-4367	62	7	=	=	SYM
ejpam-4367	62	8	x	x	SYM
ejpam-4367	62	9	−a(λ	−a(λ	NOUN
ejpam-4367	62	10	,	,	PUNCT
ejpam-4367	62	11	sp	sp	NOUN
ejpam-4367	62	12	)	)	PUNCT
ejpam-4367	62	13	.	.	PUNCT
ejpam-4367	63	1	(	(	PUNCT
ejpam-4367	63	2	6	6	NUM
ejpam-4367	63	3	)	)	PUNCT
ejpam-4367	64	1	[	[	X
ejpam-4367	64	2	x	x	X
ejpam-4367	64	3	−a](λ	−a](λ	PROPN
ejpam-4367	64	4	,	,	PUNCT
ejpam-4367	64	5	sp	sp	NOUN
ejpam-4367	64	6	)	)	PUNCT
ejpam-4367	64	7	=	=	SYM
ejpam-4367	64	8	x	x	SYM
ejpam-4367	64	9	−a(λ	−a(λ	NOUN
ejpam-4367	64	10	,	,	PUNCT
ejpam-4367	64	11	sp	sp	NOUN
ejpam-4367	64	12	)	)	PUNCT
ejpam-4367	64	13	.	.	PUNCT
ejpam-4367	65	1	3	3	X
ejpam-4367	65	2	.	.	X
ejpam-4367	65	3	characterizations	characterization	NOUN
ejpam-4367	65	4	of	of	ADP
ejpam-4367	65	5	(	(	PUNCT
ejpam-4367	65	6	λ	λ	PROPN
ejpam-4367	65	7	,	,	PUNCT
ejpam-4367	65	8	sp)-r0	sp)-r0	PROPN
ejpam-4367	65	9	topological	topological	ADJ
ejpam-4367	65	10	spaces	space	NOUN
ejpam-4367	65	11	in	in	ADP
ejpam-4367	65	12	this	this	DET
ejpam-4367	65	13	section	section	NOUN
ejpam-4367	65	14	,	,	PUNCT
ejpam-4367	65	15	we	we	PRON
ejpam-4367	65	16	introduce	introduce	VERB
ejpam-4367	65	17	the	the	DET
ejpam-4367	65	18	notion	notion	NOUN
ejpam-4367	65	19	of	of	ADP
ejpam-4367	65	20	(	(	PUNCT
ejpam-4367	65	21	λ	λ	PROPN
ejpam-4367	65	22	,	,	PUNCT
ejpam-4367	65	23	sp)-r0	sp)-r0	PROPN
ejpam-4367	65	24	topological	topological	ADJ
ejpam-4367	65	25	spaces	space	NOUN
ejpam-4367	65	26	.	.	PUNCT
ejpam-4367	66	1	moreover	moreover	ADV
ejpam-4367	66	2	,	,	PUNCT
ejpam-4367	66	3	several	several	ADJ
ejpam-4367	66	4	characterizations	characterization	NOUN
ejpam-4367	66	5	of	of	ADP
ejpam-4367	66	6	(	(	PUNCT
ejpam-4367	66	7	λ	λ	PROPN
ejpam-4367	66	8	,	,	PUNCT
ejpam-4367	66	9	sp)-r0	sp)-r0	PROPN
ejpam-4367	66	10	topological	topological	ADJ
ejpam-4367	66	11	spaces	space	NOUN
ejpam-4367	66	12	are	be	AUX
ejpam-4367	66	13	discussed	discuss	VERB
ejpam-4367	66	14	.	.	PUNCT
ejpam-4367	67	1	definition	definition	NOUN
ejpam-4367	67	2	1	1	NUM
ejpam-4367	67	3	.	.	PUNCT
ejpam-4367	68	1	a	a	DET
ejpam-4367	68	2	topological	topological	ADJ
ejpam-4367	68	3	space	space	NOUN
ejpam-4367	68	4	(	(	PUNCT
ejpam-4367	68	5	x	x	X
ejpam-4367	68	6	,	,	PUNCT
ejpam-4367	68	7	τ	τ	X
ejpam-4367	68	8	)	)	PUNCT
ejpam-4367	68	9	is	be	AUX
ejpam-4367	68	10	called	call	VERB
ejpam-4367	68	11	(	(	PUNCT
ejpam-4367	68	12	λ	λ	PROPN
ejpam-4367	68	13	,	,	PUNCT
ejpam-4367	68	14	sp)-r0	sp)-r0	VERB
ejpam-4367	68	15	if	if	SCONJ
ejpam-4367	68	16	,	,	PUNCT
ejpam-4367	68	17	for	for	ADP
ejpam-4367	68	18	each	each	DET
ejpam-4367	68	19	(	(	PUNCT
ejpam-4367	68	20	λ	λ	PROPN
ejpam-4367	68	21	,	,	PUNCT
ejpam-4367	68	22	sp)-open	sp)-open	ADJ
ejpam-4367	68	23	set	set	VERB
ejpam-4367	68	24	u	u	NOUN
ejpam-4367	68	25	and	and	CCONJ
ejpam-4367	68	26	each	each	DET
ejpam-4367	68	27	x	x	SYM
ejpam-4367	68	28	∈	∈	PROPN
ejpam-4367	68	29	u	u	NOUN
ejpam-4367	68	30	,	,	PUNCT
ejpam-4367	68	31	{	{	PUNCT
ejpam-4367	68	32	x}(λ	x}(λ	PROPN
ejpam-4367	68	33	,	,	PUNCT
ejpam-4367	68	34	sp	sp	NOUN
ejpam-4367	68	35	)	)	PUNCT
ejpam-4367	68	36	⊆	⊆	NUM
ejpam-4367	68	37	u	u	NOUN
ejpam-4367	68	38	.	.	PUNCT
ejpam-4367	69	1	theorem	theorem	NOUN
ejpam-4367	69	2	1	1	NUM
ejpam-4367	69	3	.	.	X
ejpam-4367	70	1	for	for	ADP
ejpam-4367	70	2	a	a	DET
ejpam-4367	70	3	topological	topological	ADJ
ejpam-4367	70	4	space	space	NOUN
ejpam-4367	70	5	(	(	PUNCT
ejpam-4367	70	6	x	x	X
ejpam-4367	70	7	,	,	PUNCT
ejpam-4367	70	8	τ	τ	PROPN
ejpam-4367	70	9	)	)	PUNCT
ejpam-4367	70	10	,	,	PUNCT
ejpam-4367	70	11	the	the	DET
ejpam-4367	70	12	following	follow	VERB
ejpam-4367	70	13	properties	property	NOUN
ejpam-4367	70	14	are	be	AUX
ejpam-4367	70	15	equivalent	equivalent	ADJ
ejpam-4367	70	16	:	:	PUNCT
ejpam-4367	70	17	(	(	PUNCT
ejpam-4367	70	18	1	1	X
ejpam-4367	70	19	)	)	PUNCT
ejpam-4367	70	20	(	(	PUNCT
ejpam-4367	70	21	x	x	X
ejpam-4367	70	22	,	,	PUNCT
ejpam-4367	70	23	τ	τ	X
ejpam-4367	70	24	)	)	PUNCT
ejpam-4367	70	25	is	be	AUX
ejpam-4367	70	26	(	(	PUNCT
ejpam-4367	70	27	λ	λ	PROPN
ejpam-4367	70	28	,	,	PUNCT
ejpam-4367	70	29	sp)-r0	sp)-r0	PROPN
ejpam-4367	70	30	.	.	PUNCT
ejpam-4367	71	1	(	(	PUNCT
ejpam-4367	71	2	2	2	NUM
ejpam-4367	71	3	)	)	PUNCT
ejpam-4367	71	4	for	for	SCONJ
ejpam-4367	71	5	each	each	DET
ejpam-4367	71	6	(	(	PUNCT
ejpam-4367	71	7	λ	λ	PROPN
ejpam-4367	71	8	,	,	PUNCT
ejpam-4367	71	9	sp)-closed	sp)-close	VERB
ejpam-4367	72	1	set	set	VERB
ejpam-4367	72	2	f	f	PROPN
ejpam-4367	72	3	and	and	CCONJ
ejpam-4367	72	4	each	each	DET
ejpam-4367	72	5	x	x	SYM
ejpam-4367	72	6	∈	∈	PROPN
ejpam-4367	72	7	x−f	x−f	PROPN
ejpam-4367	72	8	,	,	PUNCT
ejpam-4367	72	9	there	there	PRON
ejpam-4367	72	10	exists	exist	VERB
ejpam-4367	72	11	u	u	PROPN
ejpam-4367	72	12	∈	∈	PROPN
ejpam-4367	72	13	λspo(x	λspo(x	PROPN
ejpam-4367	72	14	,	,	PUNCT
ejpam-4367	72	15	τ	τ	PROPN
ejpam-4367	72	16	)	)	PUNCT
ejpam-4367	72	17	such	such	ADJ
ejpam-4367	72	18	that	that	SCONJ
ejpam-4367	72	19	f	f	PROPN
ejpam-4367	72	20	⊆	⊆	NUM
ejpam-4367	72	21	u	u	NOUN
ejpam-4367	72	22	and	and	CCONJ
ejpam-4367	72	23	x	x	PUNCT
ejpam-4367	72	24	̸∈	̸∈	PROPN
ejpam-4367	72	25	u	u	PROPN
ejpam-4367	72	26	.	.	PUNCT
ejpam-4367	73	1	(	(	PUNCT
ejpam-4367	73	2	3	3	X
ejpam-4367	73	3	)	)	PUNCT
ejpam-4367	73	4	for	for	ADP
ejpam-4367	73	5	each	each	DET
ejpam-4367	73	6	(	(	PUNCT
ejpam-4367	73	7	λ	λ	PROPN
ejpam-4367	73	8	,	,	PUNCT
ejpam-4367	73	9	sp)-closed	sp)-close	VERB
ejpam-4367	74	1	set	set	VERB
ejpam-4367	74	2	f	f	PROPN
ejpam-4367	74	3	and	and	CCONJ
ejpam-4367	74	4	each	each	DET
ejpam-4367	74	5	x	x	SYM
ejpam-4367	74	6	∈	∈	PROPN
ejpam-4367	74	7	x	x	X
ejpam-4367	75	1	−	−	PROPN
ejpam-4367	75	2	f	f	PROPN
ejpam-4367	75	3	,	,	PUNCT
ejpam-4367	75	4	f	f	PROPN
ejpam-4367	75	5	∩	∩	PROPN
ejpam-4367	75	6	{	{	PUNCT
ejpam-4367	75	7	x}(λ	x}(λ	PROPN
ejpam-4367	75	8	,	,	PUNCT
ejpam-4367	75	9	sp	sp	NOUN
ejpam-4367	75	10	)	)	PUNCT
ejpam-4367	75	11	=	=	SYM
ejpam-4367	75	12	∅.	∅.	X
ejpam-4367	75	13	(	(	PUNCT
ejpam-4367	75	14	4	4	NUM
ejpam-4367	75	15	)	)	PUNCT
ejpam-4367	75	16	for	for	ADP
ejpam-4367	75	17	any	any	DET
ejpam-4367	75	18	distinct	distinct	ADJ
ejpam-4367	75	19	points	point	NOUN
ejpam-4367	75	20	x	x	NOUN
ejpam-4367	75	21	,	,	PUNCT
ejpam-4367	75	22	y	y	PROPN
ejpam-4367	75	23	in	in	ADP
ejpam-4367	75	24	x	x	X
ejpam-4367	75	25	,	,	PUNCT
ejpam-4367	75	26	{	{	PUNCT
ejpam-4367	75	27	x}(λ	x}(λ	PROPN
ejpam-4367	75	28	,	,	PUNCT
ejpam-4367	75	29	sp	sp	NOUN
ejpam-4367	75	30	)	)	PUNCT
ejpam-4367	75	31	=	=	SYM
ejpam-4367	75	32	{	{	PUNCT
ejpam-4367	75	33	y}(λ	y}(λ	PROPN
ejpam-4367	75	34	,	,	PUNCT
ejpam-4367	75	35	sp	sp	NOUN
ejpam-4367	75	36	)	)	PUNCT
ejpam-4367	75	37	or	or	CCONJ
ejpam-4367	75	38	{	{	PUNCT
ejpam-4367	75	39	x}(λ	x}(λ	PROPN
ejpam-4367	75	40	,	,	PUNCT
ejpam-4367	75	41	sp	sp	NOUN
ejpam-4367	75	42	)	)	PUNCT
ejpam-4367	75	43	∩	∩	NOUN
ejpam-4367	75	44	{	{	PUNCT
ejpam-4367	75	45	y}(λ	y}(λ	PROPN
ejpam-4367	75	46	,	,	PUNCT
ejpam-4367	75	47	sp	sp	NOUN
ejpam-4367	75	48	)	)	PUNCT
ejpam-4367	75	49	=	=	PUNCT
ejpam-4367	75	50	∅.	∅.	NOUN
ejpam-4367	75	51	proof	proof	NOUN
ejpam-4367	75	52	.	.	PUNCT
ejpam-4367	76	1	(	(	PUNCT
ejpam-4367	76	2	1	1	X
ejpam-4367	76	3	)	)	PUNCT
ejpam-4367	76	4	⇒	⇒	NOUN
ejpam-4367	76	5	(	(	PUNCT
ejpam-4367	76	6	2	2	NUM
ejpam-4367	76	7	):	):	PUNCT
ejpam-4367	76	8	let	let	VERB
ejpam-4367	76	9	f	f	PRON
ejpam-4367	76	10	be	be	AUX
ejpam-4367	76	11	a	a	DET
ejpam-4367	76	12	(	(	PUNCT
ejpam-4367	76	13	λ	λ	NOUN
ejpam-4367	76	14	,	,	PUNCT
ejpam-4367	76	15	sp)-closed	sp)-close	VERB
ejpam-4367	76	16	set	set	ADJ
ejpam-4367	76	17	and	and	CCONJ
ejpam-4367	76	18	let	let	VERB
ejpam-4367	76	19	x	x	SYM
ejpam-4367	76	20	∈	∈	PROPN
ejpam-4367	76	21	x	x	X
ejpam-4367	76	22	−	−	PROPN
ejpam-4367	76	23	f	f	X
ejpam-4367	76	24	.	.	PUNCT
ejpam-4367	77	1	since	since	SCONJ
ejpam-4367	77	2	(	(	PUNCT
ejpam-4367	77	3	x	x	X
ejpam-4367	77	4	,	,	PUNCT
ejpam-4367	77	5	τ	τ	X
ejpam-4367	77	6	)	)	PUNCT
ejpam-4367	77	7	is	be	AUX
ejpam-4367	77	8	(	(	PUNCT
ejpam-4367	77	9	λ	λ	PROPN
ejpam-4367	77	10	,	,	PUNCT
ejpam-4367	77	11	sp)-r0	sp)-r0	PROPN
ejpam-4367	77	12	,	,	PUNCT
ejpam-4367	77	13	we	we	PRON
ejpam-4367	77	14	have	have	VERB
ejpam-4367	77	15	{	{	PUNCT
ejpam-4367	77	16	x}(λ	x}(λ	PROPN
ejpam-4367	77	17	,	,	PUNCT
ejpam-4367	77	18	sp	sp	NOUN
ejpam-4367	77	19	)	)	PUNCT
ejpam-4367	77	20	⊆	⊆	NUM
ejpam-4367	77	21	x	x	SYM
ejpam-4367	77	22	−	−	PROPN
ejpam-4367	77	23	f	f	X
ejpam-4367	77	24	.	.	PUNCT
ejpam-4367	78	1	put	put	VERB
ejpam-4367	78	2	u	u	NOUN
ejpam-4367	78	3	=	=	NOUN
ejpam-4367	78	4	x	x	SYM
ejpam-4367	78	5	−	−	PROPN
ejpam-4367	78	6	{	{	PUNCT
ejpam-4367	78	7	x}(λ	x}(λ	PROPN
ejpam-4367	78	8	,	,	PUNCT
ejpam-4367	78	9	sp	sp	NOUN
ejpam-4367	78	10	)	)	PUNCT
ejpam-4367	78	11	.	.	PUNCT
ejpam-4367	79	1	thus	thus	ADV
ejpam-4367	79	2	,	,	PUNCT
ejpam-4367	79	3	by	by	ADP
ejpam-4367	79	4	lemma	lemma	PROPN
ejpam-4367	79	5	1	1	NUM
ejpam-4367	79	6	,	,	PUNCT
ejpam-4367	79	7	u	u	PROPN
ejpam-4367	79	8	∈	∈	PROPN
ejpam-4367	79	9	λspo(x	λspo(x	PROPN
ejpam-4367	79	10	,	,	PUNCT
ejpam-4367	79	11	τ	τ	PROPN
ejpam-4367	79	12	)	)	PUNCT
ejpam-4367	79	13	,	,	PUNCT
ejpam-4367	79	14	f	f	PROPN
ejpam-4367	79	15	⊆	⊆	NUM
ejpam-4367	79	16	u	u	NOUN
ejpam-4367	79	17	and	and	CCONJ
ejpam-4367	79	18	x	x	PUNCT
ejpam-4367	79	19	̸∈	̸∈	PROPN
ejpam-4367	79	20	u	u	PROPN
ejpam-4367	79	21	.	.	PUNCT
ejpam-4367	80	1	(	(	PUNCT
ejpam-4367	80	2	2	2	X
ejpam-4367	80	3	)	)	PUNCT
ejpam-4367	80	4	⇒	⇒	NOUN
ejpam-4367	80	5	(	(	PUNCT
ejpam-4367	80	6	3	3	NUM
ejpam-4367	80	7	):	):	PUNCT
ejpam-4367	80	8	let	let	VERB
ejpam-4367	80	9	f	f	PRON
ejpam-4367	80	10	be	be	AUX
ejpam-4367	80	11	a	a	DET
ejpam-4367	80	12	(	(	PUNCT
ejpam-4367	80	13	λ	λ	NOUN
ejpam-4367	80	14	,	,	PUNCT
ejpam-4367	80	15	sp)-closed	sp)-close	VERB
ejpam-4367	80	16	set	set	ADJ
ejpam-4367	80	17	and	and	CCONJ
ejpam-4367	80	18	let	let	VERB
ejpam-4367	80	19	x	x	SYM
ejpam-4367	80	20	∈	∈	PROPN
ejpam-4367	80	21	x	x	X
ejpam-4367	80	22	−	−	PROPN
ejpam-4367	80	23	f	f	X
ejpam-4367	80	24	.	.	PUNCT
ejpam-4367	81	1	by	by	ADP
ejpam-4367	81	2	(	(	PUNCT
ejpam-4367	81	3	2	2	NUM
ejpam-4367	81	4	)	)	PUNCT
ejpam-4367	81	5	,	,	PUNCT
ejpam-4367	81	6	there	there	PRON
ejpam-4367	81	7	exists	exist	VERB
ejpam-4367	81	8	u	u	PROPN
ejpam-4367	81	9	∈	∈	PROPN
ejpam-4367	81	10	λspo(x	λspo(x	PROPN
ejpam-4367	81	11	,	,	PUNCT
ejpam-4367	81	12	τ	τ	PROPN
ejpam-4367	81	13	)	)	PUNCT
ejpam-4367	81	14	such	such	ADJ
ejpam-4367	81	15	that	that	SCONJ
ejpam-4367	81	16	f	f	PROPN
ejpam-4367	81	17	⊆	⊆	NUM
ejpam-4367	81	18	u	u	NOUN
ejpam-4367	81	19	and	and	CCONJ
ejpam-4367	81	20	x	x	PUNCT
ejpam-4367	81	21	̸∈	̸∈	PROPN
ejpam-4367	81	22	u	u	PROPN
ejpam-4367	81	23	.	.	PUNCT
ejpam-4367	82	1	since	since	SCONJ
ejpam-4367	82	2	u	u	PROPN
ejpam-4367	82	3	∈	∈	PROPN
ejpam-4367	82	4	λspo(x	λspo(x	PROPN
ejpam-4367	82	5	,	,	PUNCT
ejpam-4367	82	6	τ	τ	PROPN
ejpam-4367	82	7	)	)	PUNCT
ejpam-4367	82	8	,	,	PUNCT
ejpam-4367	82	9	u	u	PROPN
ejpam-4367	82	10	∩	∩	NOUN
ejpam-4367	82	11	{	{	PUNCT
ejpam-4367	82	12	x}(λ	x}(λ	PROPN
ejpam-4367	82	13	,	,	PUNCT
ejpam-4367	82	14	sp	sp	NOUN
ejpam-4367	82	15	)	)	PUNCT
ejpam-4367	82	16	=	=	NOUN
ejpam-4367	82	17	∅	∅	NOUN
ejpam-4367	82	18	and	and	CCONJ
ejpam-4367	82	19	hence	hence	ADV
ejpam-4367	82	20	f	f	PROPN
ejpam-4367	82	21	∩	∩	PROPN
ejpam-4367	82	22	{	{	PUNCT
ejpam-4367	82	23	x}(λ	x}(λ	PROPN
ejpam-4367	82	24	,	,	PUNCT
ejpam-4367	82	25	sp	sp	NOUN
ejpam-4367	82	26	)	)	PUNCT
ejpam-4367	82	27	=	=	SYM
ejpam-4367	82	28	∅.	∅.	X
ejpam-4367	82	29	(	(	PUNCT
ejpam-4367	82	30	3	3	NUM
ejpam-4367	82	31	)	)	PUNCT
ejpam-4367	82	32	⇒	⇒	NOUN
ejpam-4367	82	33	(	(	PUNCT
ejpam-4367	82	34	4	4	NUM
ejpam-4367	82	35	):	):	PUNCT
ejpam-4367	82	36	let	let	VERB
ejpam-4367	82	37	x	x	PRON
ejpam-4367	82	38	and	and	CCONJ
ejpam-4367	82	39	y	y	PROPN
ejpam-4367	82	40	be	be	AUX
ejpam-4367	82	41	distinct	distinct	ADJ
ejpam-4367	82	42	points	point	NOUN
ejpam-4367	82	43	of	of	ADP
ejpam-4367	82	44	x.	x.	NOUN
ejpam-4367	82	45	suppose	suppose	VERB
ejpam-4367	82	46	that	that	SCONJ
ejpam-4367	82	47	{	{	PUNCT
ejpam-4367	82	48	x}(λ	x}(λ	PROPN
ejpam-4367	82	49	,	,	PUNCT
ejpam-4367	82	50	sp	sp	NOUN
ejpam-4367	82	51	)	)	PUNCT
ejpam-4367	82	52	∩{y}(λ	∩{y}(λ	NOUN
ejpam-4367	82	53	,	,	PUNCT
ejpam-4367	82	54	sp	sp	NOUN
ejpam-4367	82	55	)	)	PUNCT
ejpam-4367	82	56	̸=	̸=	PROPN
ejpam-4367	82	57	∅.	∅.	VERB
ejpam-4367	82	58	by	by	ADP
ejpam-4367	82	59	(	(	PUNCT
ejpam-4367	82	60	3	3	NUM
ejpam-4367	82	61	)	)	PUNCT
ejpam-4367	82	62	,	,	PUNCT
ejpam-4367	82	63	x	x	PUNCT
ejpam-4367	82	64	∈	∈	PROPN
ejpam-4367	82	65	{	{	PUNCT
ejpam-4367	82	66	y}(λ	y}(λ	PROPN
ejpam-4367	82	67	,	,	PUNCT
ejpam-4367	82	68	sp	sp	NOUN
ejpam-4367	82	69	)	)	PUNCT
ejpam-4367	82	70	and	and	CCONJ
ejpam-4367	82	71	y	y	PROPN
ejpam-4367	82	72	∈	∈	PROPN
ejpam-4367	82	73	{	{	PUNCT
ejpam-4367	82	74	x}(λ	x}(λ	PROPN
ejpam-4367	82	75	,	,	PUNCT
ejpam-4367	82	76	sp	sp	NOUN
ejpam-4367	82	77	)	)	PUNCT
ejpam-4367	82	78	.	.	PUNCT
ejpam-4367	83	1	by	by	ADP
ejpam-4367	83	2	lemma	lemma	PROPN
ejpam-4367	83	3	1	1	NUM
ejpam-4367	83	4	,	,	PUNCT
ejpam-4367	83	5	{	{	PUNCT
ejpam-4367	83	6	x}(λ	x}(λ	PROPN
ejpam-4367	83	7	,	,	PUNCT
ejpam-4367	83	8	sp	sp	NOUN
ejpam-4367	83	9	)	)	PUNCT
ejpam-4367	83	10	⊆	⊆	NUM
ejpam-4367	83	11	{	{	PUNCT
ejpam-4367	83	12	y}(λ	y}(λ	PROPN
ejpam-4367	83	13	,	,	PUNCT
ejpam-4367	83	14	sp	sp	NOUN
ejpam-4367	83	15	)	)	PUNCT
ejpam-4367	83	16	⊆	⊆	NUM
ejpam-4367	83	17	{	{	PUNCT
ejpam-4367	83	18	x}(λ	x}(λ	PROPN
ejpam-4367	83	19	,	,	PUNCT
ejpam-4367	83	20	sp	sp	NOUN
ejpam-4367	83	21	)	)	PUNCT
ejpam-4367	83	22	and	and	CCONJ
ejpam-4367	83	23	hence	hence	ADV
ejpam-4367	83	24	{	{	PUNCT
ejpam-4367	83	25	x}(λ	x}(λ	PROPN
ejpam-4367	83	26	,	,	PUNCT
ejpam-4367	83	27	sp	sp	NOUN
ejpam-4367	83	28	)	)	PUNCT
ejpam-4367	83	29	=	=	SYM
ejpam-4367	83	30	{	{	PUNCT
ejpam-4367	83	31	y}(λ	y}(λ	PROPN
ejpam-4367	83	32	,	,	PUNCT
ejpam-4367	83	33	sp	sp	NOUN
ejpam-4367	83	34	)	)	PUNCT
ejpam-4367	83	35	.	.	PUNCT
ejpam-4367	84	1	(	(	PUNCT
ejpam-4367	84	2	4	4	X
ejpam-4367	84	3	)	)	PUNCT
ejpam-4367	84	4	⇒	⇒	NOUN
ejpam-4367	84	5	(	(	PUNCT
ejpam-4367	84	6	1	1	NUM
ejpam-4367	84	7	):	):	PUNCT
ejpam-4367	84	8	let	let	VERB
ejpam-4367	84	9	v	v	NUM
ejpam-4367	84	10	∈	∈	PROPN
ejpam-4367	84	11	λspo(x	λspo(x	NOUN
ejpam-4367	84	12	,	,	PUNCT
ejpam-4367	84	13	τ	τ	X
ejpam-4367	84	14	)	)	PUNCT
ejpam-4367	84	15	and	and	CCONJ
ejpam-4367	84	16	let	let	VERB
ejpam-4367	84	17	x	x	SYM
ejpam-4367	84	18	∈	∈	PROPN
ejpam-4367	84	19	v	v	NOUN
ejpam-4367	84	20	.	.	PUNCT
ejpam-4367	85	1	for	for	ADP
ejpam-4367	85	2	each	each	DET
ejpam-4367	85	3	y	y	PROPN
ejpam-4367	85	4	̸∈	̸∈	PROPN
ejpam-4367	85	5	v	v	PROPN
ejpam-4367	85	6	,	,	PUNCT
ejpam-4367	85	7	v	v	NOUN
ejpam-4367	85	8	∩	∩	NOUN
ejpam-4367	85	9	{	{	PUNCT
ejpam-4367	85	10	y}(λ	y}(λ	PROPN
ejpam-4367	85	11	,	,	PUNCT
ejpam-4367	85	12	sp	sp	NOUN
ejpam-4367	85	13	)	)	PUNCT
ejpam-4367	85	14	=	=	NOUN
ejpam-4367	85	15	∅	∅	NOUN
ejpam-4367	85	16	and	and	CCONJ
ejpam-4367	85	17	hence	hence	ADV
ejpam-4367	85	18	x	x	X
ejpam-4367	85	19	̸∈	̸∈	PROPN
ejpam-4367	85	20	{	{	PUNCT
ejpam-4367	85	21	y}(λ	y}(λ	PROPN
ejpam-4367	85	22	,	,	PUNCT
ejpam-4367	85	23	sp	sp	NOUN
ejpam-4367	85	24	)	)	PUNCT
ejpam-4367	85	25	.	.	PUNCT
ejpam-4367	86	1	thus	thus	ADV
ejpam-4367	86	2	,	,	PUNCT
ejpam-4367	86	3	{	{	PUNCT
ejpam-4367	86	4	x}(λ	x}(λ	PROPN
ejpam-4367	86	5	,	,	PUNCT
ejpam-4367	86	6	sp	sp	NOUN
ejpam-4367	86	7	)	)	PUNCT
ejpam-4367	86	8	̸=	̸=	PROPN
ejpam-4367	86	9	{	{	PUNCT
ejpam-4367	86	10	y}(λ	y}(λ	PROPN
ejpam-4367	86	11	,	,	PUNCT
ejpam-4367	86	12	sp	sp	NOUN
ejpam-4367	86	13	)	)	PUNCT
ejpam-4367	86	14	.	.	PUNCT
ejpam-4367	87	1	by	by	ADP
ejpam-4367	87	2	(	(	PUNCT
ejpam-4367	87	3	4	4	NUM
ejpam-4367	87	4	)	)	PUNCT
ejpam-4367	87	5	,	,	PUNCT
ejpam-4367	87	6	for	for	ADP
ejpam-4367	87	7	each	each	DET
ejpam-4367	87	8	y	y	PROPN
ejpam-4367	87	9	̸∈	̸∈	PROPN
ejpam-4367	87	10	v	v	PROPN
ejpam-4367	87	11	,	,	PUNCT
ejpam-4367	87	12	{	{	PUNCT
ejpam-4367	87	13	x}(λ	x}(λ	PROPN
ejpam-4367	87	14	,	,	PUNCT
ejpam-4367	87	15	sp	sp	NOUN
ejpam-4367	87	16	)	)	PUNCT
ejpam-4367	87	17	∩	∩	NOUN
ejpam-4367	87	18	{	{	PUNCT
ejpam-4367	87	19	y}(λ	y}(λ	PROPN
ejpam-4367	87	20	,	,	PUNCT
ejpam-4367	87	21	sp	sp	NOUN
ejpam-4367	87	22	)	)	PUNCT
ejpam-4367	87	23	=	=	PUNCT
ejpam-4367	87	24	∅.	∅.	NOUN
ejpam-4367	87	25	since	since	SCONJ
ejpam-4367	87	26	x−v	x−v	PROPN
ejpam-4367	87	27	is	be	AUX
ejpam-4367	87	28	(	(	PUNCT
ejpam-4367	87	29	λ	λ	X
ejpam-4367	87	30	,	,	PUNCT
ejpam-4367	87	31	sp)-closed	sp)-close	VERB
ejpam-4367	87	32	,	,	PUNCT
ejpam-4367	87	33	y	y	PROPN
ejpam-4367	87	34	∈	∈	PROPN
ejpam-4367	87	35	{	{	PUNCT
ejpam-4367	87	36	y}(λ	y}(λ	PROPN
ejpam-4367	87	37	,	,	PUNCT
ejpam-4367	87	38	sp	sp	NOUN
ejpam-4367	87	39	)	)	PUNCT
ejpam-4367	87	40	⊆	⊆	NUM
ejpam-4367	87	41	x−v	x−v	PROPN
ejpam-4367	87	42	and	and	CCONJ
ejpam-4367	87	43	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4367	87	44	{	{	PUNCT
ejpam-4367	87	45	y}(λ	y}(λ	PROPN
ejpam-4367	87	46	,	,	PUNCT
ejpam-4367	87	47	sp	sp	NOUN
ejpam-4367	87	48	)	)	PUNCT
ejpam-4367	87	49	=	=	SYM
ejpam-4367	87	50	x−v	x−v	PROPN
ejpam-4367	87	51	.	.	PUNCT
ejpam-4367	88	1	thus	thus	ADV
ejpam-4367	88	2	,	,	PUNCT
ejpam-4367	88	3	{	{	PUNCT
ejpam-4367	88	4	x}(λ	x}(λ	PROPN
ejpam-4367	88	5	,	,	PUNCT
ejpam-4367	88	6	sp	sp	NOUN
ejpam-4367	88	7	)	)	PUNCT
ejpam-4367	88	8	∩	∩	NOUN
ejpam-4367	88	9	(	(	PUNCT
ejpam-4367	88	10	x	x	SYM
ejpam-4367	88	11	−	−	PROPN
ejpam-4367	88	12	v	v	NOUN
ejpam-4367	88	13	)	)	PUNCT
ejpam-4367	88	14	=	=	SYM
ejpam-4367	88	15	{	{	PUNCT
ejpam-4367	88	16	x}(λ	x}(λ	PROPN
ejpam-4367	88	17	,	,	PUNCT
ejpam-4367	88	18	sp	sp	NOUN
ejpam-4367	88	19	)	)	PUNCT
ejpam-4367	88	20	∩	∩	NOUN
ejpam-4367	88	21	[	[	X
ejpam-4367	88	22	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4367	88	23	{	{	PUNCT
ejpam-4367	88	24	y}(λ	y}(λ	PROPN
ejpam-4367	88	25	,	,	PUNCT
ejpam-4367	88	26	sp	sp	NOUN
ejpam-4367	88	27	)	)	PUNCT
ejpam-4367	88	28	]	]	PUNCT
ejpam-4367	89	1	=	=	PUNCT
ejpam-4367	89	2	∪y∈x−v	∪y∈x−v	PROPN
ejpam-4367	90	1	[	[	X
ejpam-4367	90	2	{	{	PUNCT
ejpam-4367	90	3	x}(λ	x}(λ	PROPN
ejpam-4367	90	4	,	,	PUNCT
ejpam-4367	90	5	sp	sp	NOUN
ejpam-4367	90	6	)	)	PUNCT
ejpam-4367	90	7	∩	∩	NOUN
ejpam-4367	90	8	{	{	PUNCT
ejpam-4367	90	9	y}(λ	y}(λ	PROPN
ejpam-4367	90	10	,	,	PUNCT
ejpam-4367	90	11	sp	sp	NOUN
ejpam-4367	90	12	)	)	PUNCT
ejpam-4367	90	13	]	]	PUNCT
ejpam-4367	90	14	=	=	PUNCT
ejpam-4367	90	15	∅	∅	NOUN
ejpam-4367	90	16	and	and	CCONJ
ejpam-4367	90	17	hence	hence	ADV
ejpam-4367	90	18	{	{	PUNCT
ejpam-4367	90	19	x}(λ	x}(λ	PROPN
ejpam-4367	90	20	,	,	PUNCT
ejpam-4367	90	21	sp	sp	NOUN
ejpam-4367	90	22	)	)	PUNCT
ejpam-4367	90	23	⊆	⊆	NUM
ejpam-4367	90	24	v	v	NOUN
ejpam-4367	90	25	.	.	PUNCT
ejpam-4367	91	1	this	this	PRON
ejpam-4367	91	2	shows	show	VERB
ejpam-4367	91	3	that	that	SCONJ
ejpam-4367	91	4	(	(	PUNCT
ejpam-4367	91	5	x	x	X
ejpam-4367	91	6	,	,	PUNCT
ejpam-4367	91	7	τ	τ	X
ejpam-4367	91	8	)	)	PUNCT
ejpam-4367	91	9	is	be	AUX
ejpam-4367	91	10	(	(	PUNCT
ejpam-4367	91	11	λ	λ	PROPN
ejpam-4367	91	12	,	,	PUNCT
ejpam-4367	91	13	sp)-r0	sp)-r0	PROPN
ejpam-4367	91	14	.	.	PUNCT
ejpam-4367	91	15	corollary	corollary	ADJ
ejpam-4367	91	16	1	1	NUM
ejpam-4367	91	17	.	.	PUNCT
ejpam-4367	92	1	a	a	DET
ejpam-4367	92	2	topological	topological	ADJ
ejpam-4367	92	3	space	space	NOUN
ejpam-4367	92	4	(	(	PUNCT
ejpam-4367	92	5	x	x	X
ejpam-4367	92	6	,	,	PUNCT
ejpam-4367	92	7	τ	τ	X
ejpam-4367	92	8	)	)	PUNCT
ejpam-4367	92	9	is	be	AUX
ejpam-4367	92	10	(	(	PUNCT
ejpam-4367	92	11	λ	λ	X
ejpam-4367	92	12	,	,	PUNCT
ejpam-4367	92	13	sp)-r0	sp)-r0	NOUN
ejpam-4367	93	1	if	if	SCONJ
ejpam-4367	93	2	and	and	CCONJ
ejpam-4367	93	3	only	only	ADV
ejpam-4367	93	4	if	if	SCONJ
ejpam-4367	93	5	,	,	PUNCT
ejpam-4367	93	6	for	for	ADP
ejpam-4367	93	7	any	any	DET
ejpam-4367	93	8	points	point	NOUN
ejpam-4367	93	9	x	x	PUNCT
ejpam-4367	93	10	and	and	CCONJ
ejpam-4367	93	11	y	y	PROPN
ejpam-4367	93	12	in	in	ADP
ejpam-4367	93	13	x	x	PRON
ejpam-4367	93	14	,	,	PUNCT
ejpam-4367	93	15	{	{	PUNCT
ejpam-4367	93	16	x}(λ	x}(λ	PROPN
ejpam-4367	93	17	,	,	PUNCT
ejpam-4367	93	18	sp	sp	NOUN
ejpam-4367	93	19	)	)	PUNCT
ejpam-4367	93	20	̸=	̸=	PROPN
ejpam-4367	93	21	{	{	PUNCT
ejpam-4367	93	22	y}(λ	y}(λ	PROPN
ejpam-4367	93	23	,	,	PUNCT
ejpam-4367	93	24	sp	sp	NOUN
ejpam-4367	93	25	)	)	PUNCT
ejpam-4367	93	26	implies	imply	VERB
ejpam-4367	93	27	{	{	PUNCT
ejpam-4367	93	28	x}(λ	x}(λ	PROPN
ejpam-4367	93	29	,	,	PUNCT
ejpam-4367	93	30	sp	sp	NOUN
ejpam-4367	93	31	)	)	PUNCT
ejpam-4367	93	32	∩	∩	NOUN
ejpam-4367	93	33	{	{	PUNCT
ejpam-4367	93	34	y}(λ	y}(λ	PROPN
ejpam-4367	93	35	,	,	PUNCT
ejpam-4367	93	36	sp	sp	NOUN
ejpam-4367	93	37	)	)	PUNCT
ejpam-4367	93	38	=	=	PUNCT
ejpam-4367	93	39	∅.	∅.	NOUN
ejpam-4367	93	40	proof	proof	NOUN
ejpam-4367	93	41	.	.	PUNCT
ejpam-4367	94	1	this	this	PRON
ejpam-4367	94	2	is	be	AUX
ejpam-4367	94	3	obvious	obvious	ADJ
ejpam-4367	94	4	by	by	ADP
ejpam-4367	94	5	theorem	theorem	NOUN
ejpam-4367	94	6	1	1	NUM
ejpam-4367	94	7	.	.	PUNCT
ejpam-4367	95	1	conversely	conversely	ADV
ejpam-4367	95	2	,	,	PUNCT
ejpam-4367	95	3	let	let	VERB
ejpam-4367	95	4	u	u	PRON
ejpam-4367	95	5	∈	∈	PROPN
ejpam-4367	95	6	λspo(x	λspo(x	PROPN
ejpam-4367	95	7	,	,	PUNCT
ejpam-4367	95	8	τ	τ	X
ejpam-4367	95	9	)	)	PUNCT
ejpam-4367	95	10	and	and	CCONJ
ejpam-4367	95	11	let	let	VERB
ejpam-4367	95	12	x	x	PUNCT
ejpam-4367	95	13	∈	∈	PROPN
ejpam-4367	95	14	u	u	NOUN
ejpam-4367	95	15	.	.	PUNCT
ejpam-4367	96	1	if	if	SCONJ
ejpam-4367	96	2	y	y	PROPN
ejpam-4367	96	3	̸∈	̸∈	PROPN
ejpam-4367	96	4	u	u	PROPN
ejpam-4367	96	5	,	,	PUNCT
ejpam-4367	96	6	then	then	ADV
ejpam-4367	96	7	u	u	NOUN
ejpam-4367	96	8	∩	∩	PROPN
ejpam-4367	96	9	{	{	PUNCT
ejpam-4367	96	10	y}(λ	y}(λ	PROPN
ejpam-4367	96	11	,	,	PUNCT
ejpam-4367	96	12	sp	sp	NOUN
ejpam-4367	96	13	)	)	PUNCT
ejpam-4367	96	14	=	=	NOUN
ejpam-4367	96	15	∅.	∅.	ADP
ejpam-4367	96	16	thus	thus	ADV
ejpam-4367	96	17	,	,	PUNCT
ejpam-4367	96	18	x	x	PROPN
ejpam-4367	96	19	̸∈	̸∈	PROPN
ejpam-4367	96	20	{	{	PUNCT
ejpam-4367	96	21	y}(λ	y}(λ	PROPN
ejpam-4367	96	22	,	,	PUNCT
ejpam-4367	96	23	sp	sp	NOUN
ejpam-4367	96	24	)	)	PUNCT
ejpam-4367	96	25	and	and	CCONJ
ejpam-4367	96	26	{	{	PUNCT
ejpam-4367	96	27	x}(λ	x}(λ	PROPN
ejpam-4367	96	28	,	,	PUNCT
ejpam-4367	96	29	sp	sp	NOUN
ejpam-4367	96	30	)	)	PUNCT
ejpam-4367	96	31	̸=	̸=	PROPN
ejpam-4367	96	32	{	{	PUNCT
ejpam-4367	96	33	y}(λ	y}(λ	PROPN
ejpam-4367	96	34	,	,	PUNCT
ejpam-4367	96	35	sp	sp	NOUN
ejpam-4367	96	36	)	)	PUNCT
ejpam-4367	96	37	.	.	PUNCT
ejpam-4367	97	1	by	by	ADP
ejpam-4367	97	2	the	the	DET
ejpam-4367	97	3	hypothesis	hypothesis	NOUN
ejpam-4367	97	4	,	,	PUNCT
ejpam-4367	97	5	{	{	PUNCT
ejpam-4367	97	6	x}(λ	x}(λ	PROPN
ejpam-4367	97	7	,	,	PUNCT
ejpam-4367	97	8	sp	sp	NOUN
ejpam-4367	97	9	)	)	PUNCT
ejpam-4367	97	10	∩	∩	NOUN
ejpam-4367	97	11	{	{	PUNCT
ejpam-4367	97	12	y}(λ	y}(λ	PROPN
ejpam-4367	97	13	,	,	PUNCT
ejpam-4367	97	14	sp	sp	NOUN
ejpam-4367	97	15	)	)	PUNCT
ejpam-4367	97	16	=	=	NOUN
ejpam-4367	97	17	∅	∅	NOUN
ejpam-4367	97	18	and	and	CCONJ
ejpam-4367	97	19	hence	hence	ADV
ejpam-4367	97	20	y	y	PROPN
ejpam-4367	97	21	̸∈	̸∈	PROPN
ejpam-4367	97	22	{	{	PUNCT
ejpam-4367	97	23	x}(λ	x}(λ	PROPN
ejpam-4367	97	24	,	,	PUNCT
ejpam-4367	97	25	sp	sp	NOUN
ejpam-4367	97	26	)	)	PUNCT
ejpam-4367	97	27	.	.	PUNCT
ejpam-4367	98	1	this	this	PRON
ejpam-4367	98	2	shows	show	VERB
ejpam-4367	98	3	that	that	SCONJ
ejpam-4367	98	4	{	{	PUNCT
ejpam-4367	98	5	x}(λ	x}(λ	PROPN
ejpam-4367	98	6	,	,	PUNCT
ejpam-4367	98	7	sp	sp	NOUN
ejpam-4367	98	8	)	)	PUNCT
ejpam-4367	98	9	⊆	⊆	NUM
ejpam-4367	98	10	u	u	NOUN
ejpam-4367	98	11	.	.	PUNCT
ejpam-4367	99	1	thus	thus	ADV
ejpam-4367	99	2	,	,	PUNCT
ejpam-4367	99	3	(	(	PUNCT
ejpam-4367	99	4	x	x	X
ejpam-4367	99	5	,	,	PUNCT
ejpam-4367	99	6	τ	τ	X
ejpam-4367	99	7	)	)	PUNCT
ejpam-4367	99	8	is	be	AUX
ejpam-4367	99	9	(	(	PUNCT
ejpam-4367	99	10	λ	λ	PROPN
ejpam-4367	99	11	,	,	PUNCT
ejpam-4367	99	12	sp)-r0	sp)-r0	PROPN
ejpam-4367	99	13	.	.	PUNCT
ejpam-4367	100	1	c.	c.	PROPN
ejpam-4367	100	2	boonpok	boonpok	PROPN
ejpam-4367	100	3	,	,	PUNCT
ejpam-4367	100	4	c.	c.	PROPN
ejpam-4367	100	5	viriyapong	viriyapong	PROPN
ejpam-4367	100	6	/	/	SYM
ejpam-4367	100	7	eur	eur	PROPN
ejpam-4367	100	8	.	.	PUNCT
ejpam-4367	101	1	j.	j.	PROPN
ejpam-4367	101	2	pure	pure	PROPN
ejpam-4367	101	3	appl	appl	PROPN
ejpam-4367	101	4	.	.	PROPN
ejpam-4367	101	5	math	math	PROPN
ejpam-4367	101	6	,	,	PUNCT
ejpam-4367	101	7	15	15	NUM
ejpam-4367	101	8	(	(	PUNCT
ejpam-4367	101	9	3	3	NUM
ejpam-4367	101	10	)	)	PUNCT
ejpam-4367	101	11	(	(	PUNCT
ejpam-4367	101	12	2022	2022	NUM
ejpam-4367	101	13	)	)	PUNCT
ejpam-4367	101	14	,	,	PUNCT
ejpam-4367	101	15	878	878	NUM
ejpam-4367	101	16	-	-	SYM
ejpam-4367	101	17	886	886	NUM
ejpam-4367	101	18	881	881	NUM
ejpam-4367	101	19	definition	definition	NOUN
ejpam-4367	101	20	2	2	NUM
ejpam-4367	101	21	.	.	PUNCT
ejpam-4367	102	1	[	[	X
ejpam-4367	102	2	3	3	X
ejpam-4367	102	3	]	]	PUNCT
ejpam-4367	102	4	let	let	VERB
ejpam-4367	102	5	a	a	PRON
ejpam-4367	102	6	be	be	AUX
ejpam-4367	102	7	a	a	DET
ejpam-4367	102	8	subset	subset	NOUN
ejpam-4367	102	9	of	of	ADP
ejpam-4367	102	10	a	a	DET
ejpam-4367	102	11	topological	topological	ADJ
ejpam-4367	102	12	space	space	NOUN
ejpam-4367	102	13	(	(	PUNCT
ejpam-4367	102	14	x	x	X
ejpam-4367	102	15	,	,	PUNCT
ejpam-4367	102	16	τ	τ	PROPN
ejpam-4367	102	17	)	)	PUNCT
ejpam-4367	102	18	.	.	PUNCT
ejpam-4367	103	1	a	a	DET
ejpam-4367	103	2	subset	subset	NOUN
ejpam-4367	103	3	λ(λ	λ(λ	ADP
ejpam-4367	103	4	,	,	PUNCT
ejpam-4367	103	5	sp	sp	NOUN
ejpam-4367	103	6	)	)	PUNCT
ejpam-4367	103	7	is	be	AUX
ejpam-4367	103	8	defined	define	VERB
ejpam-4367	103	9	as	as	SCONJ
ejpam-4367	103	10	follows	follow	VERB
ejpam-4367	103	11	:	:	PUNCT
ejpam-4367	104	1	λ(λ	λ(λ	ADV
ejpam-4367	104	2	,	,	PUNCT
ejpam-4367	104	3	sp)(a	sp)(a	PROPN
ejpam-4367	104	4	)	)	PUNCT
ejpam-4367	105	1	=	=	PUNCT
ejpam-4367	106	1	∩{u	∩{u	PROPN
ejpam-4367	106	2	|	|	ADV
ejpam-4367	106	3	a	a	DET
ejpam-4367	106	4	⊆	⊆	NUM
ejpam-4367	106	5	u	u	NOUN
ejpam-4367	106	6	,	,	PUNCT
ejpam-4367	106	7	u	u	PROPN
ejpam-4367	106	8	∈	∈	PROPN
ejpam-4367	106	9	λspo(x	λspo(x	PROPN
ejpam-4367	106	10	,	,	PUNCT
ejpam-4367	106	11	τ	τ	PROPN
ejpam-4367	106	12	)	)	PUNCT
ejpam-4367	106	13	}	}	PUNCT
ejpam-4367	106	14	.	.	PUNCT
ejpam-4367	107	1	lemma	lemma	PROPN
ejpam-4367	107	2	3	3	X
ejpam-4367	107	3	.	.	PUNCT
ejpam-4367	108	1	[	[	X
ejpam-4367	108	2	3	3	X
ejpam-4367	108	3	]	]	PUNCT
ejpam-4367	108	4	for	for	ADP
ejpam-4367	108	5	subsets	subset	NOUN
ejpam-4367	108	6	a	a	DET
ejpam-4367	108	7	,	,	PUNCT
ejpam-4367	108	8	b	b	PROPN
ejpam-4367	108	9	of	of	ADP
ejpam-4367	108	10	a	a	DET
ejpam-4367	108	11	topological	topological	ADJ
ejpam-4367	108	12	space	space	NOUN
ejpam-4367	108	13	(	(	PUNCT
ejpam-4367	108	14	x	x	X
ejpam-4367	108	15	,	,	PUNCT
ejpam-4367	108	16	τ	τ	PROPN
ejpam-4367	108	17	)	)	PUNCT
ejpam-4367	108	18	,	,	PUNCT
ejpam-4367	108	19	the	the	DET
ejpam-4367	108	20	following	follow	VERB
ejpam-4367	108	21	properties	property	NOUN
ejpam-4367	108	22	hold	hold	VERB
ejpam-4367	108	23	:	:	PUNCT
ejpam-4367	108	24	(	(	PUNCT
ejpam-4367	108	25	1	1	X
ejpam-4367	108	26	)	)	PUNCT
ejpam-4367	108	27	a	a	DET
ejpam-4367	108	28	⊆	⊆	NUM
ejpam-4367	108	29	λ(λ	λ(λ	NOUN
ejpam-4367	108	30	,	,	PUNCT
ejpam-4367	108	31	sp)(a	sp)(a	PROPN
ejpam-4367	108	32	)	)	PUNCT
ejpam-4367	108	33	.	.	PUNCT
ejpam-4367	109	1	(	(	PUNCT
ejpam-4367	109	2	2	2	X
ejpam-4367	109	3	)	)	PUNCT
ejpam-4367	109	4	if	if	SCONJ
ejpam-4367	109	5	a	a	DET
ejpam-4367	109	6	⊆	⊆	NUM
ejpam-4367	109	7	b	b	NOUN
ejpam-4367	109	8	,	,	PUNCT
ejpam-4367	109	9	then	then	ADV
ejpam-4367	109	10	λ(λ	λ(λ	PROPN
ejpam-4367	109	11	,	,	PUNCT
ejpam-4367	109	12	sp)(a	sp)(a	PROPN
ejpam-4367	109	13	)	)	PUNCT
ejpam-4367	109	14	⊆	⊆	NUM
ejpam-4367	109	15	λ(λ	λ(λ	NOUN
ejpam-4367	109	16	,	,	PUNCT
ejpam-4367	109	17	sp)(b	sp)(b	PROPN
ejpam-4367	109	18	)	)	PUNCT
ejpam-4367	109	19	.	.	PUNCT
ejpam-4367	110	1	(	(	PUNCT
ejpam-4367	110	2	3	3	X
ejpam-4367	110	3	)	)	PUNCT
ejpam-4367	110	4	λ(λ	λ(λ	ADV
ejpam-4367	110	5	,	,	PUNCT
ejpam-4367	110	6	sp)[λ(λ	sp)[λ(λ	NOUN
ejpam-4367	110	7	,	,	PUNCT
ejpam-4367	110	8	sp)(a	sp)(a	PROPN
ejpam-4367	110	9	)	)	PUNCT
ejpam-4367	110	10	]	]	PUNCT
ejpam-4367	111	1	=	=	PUNCT
ejpam-4367	111	2	λ(λ	λ(λ	PROPN
ejpam-4367	111	3	,	,	PUNCT
ejpam-4367	111	4	sp)(a	sp)(a	PROPN
ejpam-4367	111	5	)	)	PUNCT
ejpam-4367	111	6	.	.	PUNCT
ejpam-4367	112	1	(	(	PUNCT
ejpam-4367	112	2	4	4	X
ejpam-4367	112	3	)	)	PUNCT
ejpam-4367	112	4	if	if	SCONJ
ejpam-4367	112	5	a	a	PRON
ejpam-4367	112	6	is	be	AUX
ejpam-4367	112	7	(	(	PUNCT
ejpam-4367	112	8	λ	λ	NOUN
ejpam-4367	112	9	,	,	PUNCT
ejpam-4367	112	10	sp)-open	sp)-open	ADJ
ejpam-4367	112	11	,	,	PUNCT
ejpam-4367	112	12	λ(λ	λ(λ	ADV
ejpam-4367	112	13	,	,	PUNCT
ejpam-4367	112	14	sp)(a	sp)(a	PROPN
ejpam-4367	112	15	)	)	PUNCT
ejpam-4367	113	1	=	=	PUNCT
ejpam-4367	113	2	a.	a.	NOUN
ejpam-4367	113	3	lemma	lemma	PROPN
ejpam-4367	113	4	4	4	X
ejpam-4367	113	5	.	.	PUNCT
ejpam-4367	114	1	[	[	X
ejpam-4367	114	2	3	3	X
ejpam-4367	114	3	]	]	X
ejpam-4367	114	4	let	let	VERB
ejpam-4367	114	5	(	(	PUNCT
ejpam-4367	114	6	x	x	NOUN
ejpam-4367	114	7	,	,	PUNCT
ejpam-4367	114	8	τ	τ	X
ejpam-4367	114	9	)	)	PUNCT
ejpam-4367	114	10	be	be	VERB
ejpam-4367	114	11	a	a	DET
ejpam-4367	114	12	topological	topological	ADJ
ejpam-4367	114	13	space	space	NOUN
ejpam-4367	114	14	and	and	CCONJ
ejpam-4367	114	15	x	x	NOUN
ejpam-4367	114	16	,	,	PUNCT
ejpam-4367	114	17	y	y	PROPN
ejpam-4367	114	18	∈	∈	PROPN
ejpam-4367	114	19	x.	x.	NOUN
ejpam-4367	114	20	then	then	ADV
ejpam-4367	114	21	,	,	PUNCT
ejpam-4367	114	22	the	the	DET
ejpam-4367	114	23	following	follow	VERB
ejpam-4367	114	24	properties	property	NOUN
ejpam-4367	114	25	hold	hold	VERB
ejpam-4367	114	26	:	:	PUNCT
ejpam-4367	114	27	(	(	PUNCT
ejpam-4367	114	28	1	1	X
ejpam-4367	114	29	)	)	PUNCT
ejpam-4367	114	30	y	y	PROPN
ejpam-4367	114	31	∈	∈	PROPN
ejpam-4367	114	32	λ(λ	λ(λ	PROPN
ejpam-4367	114	33	,	,	PUNCT
ejpam-4367	114	34	sp)({x	sp)({x	PROPN
ejpam-4367	114	35	}	}	PUNCT
ejpam-4367	114	36	)	)	PUNCT
ejpam-4367	115	1	if	if	SCONJ
ejpam-4367	115	2	and	and	CCONJ
ejpam-4367	115	3	only	only	ADV
ejpam-4367	115	4	if	if	SCONJ
ejpam-4367	115	5	x	x	SYM
ejpam-4367	115	6	∈	∈	PROPN
ejpam-4367	115	7	{	{	PUNCT
ejpam-4367	115	8	y}(λ	y}(λ	PROPN
ejpam-4367	115	9	,	,	PUNCT
ejpam-4367	115	10	sp	sp	NOUN
ejpam-4367	115	11	)	)	PUNCT
ejpam-4367	115	12	.	.	PUNCT
ejpam-4367	116	1	(	(	PUNCT
ejpam-4367	116	2	2	2	X
ejpam-4367	116	3	)	)	PUNCT
ejpam-4367	116	4	λ(λ	λ(λ	ADV
ejpam-4367	116	5	,	,	PUNCT
ejpam-4367	116	6	sp)({x	sp)({x	NOUN
ejpam-4367	116	7	}	}	PUNCT
ejpam-4367	116	8	)	)	PUNCT
ejpam-4367	117	1	=	=	SYM
ejpam-4367	117	2	λ(λ	λ(λ	PROPN
ejpam-4367	117	3	,	,	PUNCT
ejpam-4367	117	4	sp)({y	sp)({y	NOUN
ejpam-4367	117	5	}	}	PUNCT
ejpam-4367	117	6	)	)	PUNCT
ejpam-4367	117	7	if	if	SCONJ
ejpam-4367	117	8	and	and	CCONJ
ejpam-4367	117	9	only	only	ADV
ejpam-4367	117	10	if	if	SCONJ
ejpam-4367	117	11	{	{	PUNCT
ejpam-4367	117	12	x}(λ	x}(λ	PROPN
ejpam-4367	117	13	,	,	PUNCT
ejpam-4367	117	14	sp	sp	NOUN
ejpam-4367	117	15	)	)	PUNCT
ejpam-4367	117	16	=	=	SYM
ejpam-4367	117	17	{	{	PUNCT
ejpam-4367	117	18	y}(λ	y}(λ	PROPN
ejpam-4367	117	19	,	,	PUNCT
ejpam-4367	117	20	sp	sp	NOUN
ejpam-4367	117	21	)	)	PUNCT
ejpam-4367	117	22	.	.	PUNCT
ejpam-4367	118	1	theorem	theorem	NOUN
ejpam-4367	118	2	2	2	NUM
ejpam-4367	118	3	.	.	PUNCT
ejpam-4367	118	4	a	a	DET
ejpam-4367	118	5	topological	topological	ADJ
ejpam-4367	118	6	space	space	NOUN
ejpam-4367	118	7	(	(	PUNCT
ejpam-4367	118	8	x	x	X
ejpam-4367	118	9	,	,	PUNCT
ejpam-4367	118	10	τ	τ	X
ejpam-4367	118	11	)	)	PUNCT
ejpam-4367	118	12	is	be	AUX
ejpam-4367	118	13	(	(	PUNCT
ejpam-4367	118	14	λ	λ	X
ejpam-4367	118	15	,	,	PUNCT
ejpam-4367	118	16	sp)-r0	sp)-r0	NOUN
ejpam-4367	119	1	if	if	SCONJ
ejpam-4367	119	2	and	and	CCONJ
ejpam-4367	119	3	only	only	ADV
ejpam-4367	119	4	if	if	SCONJ
ejpam-4367	119	5	,	,	PUNCT
ejpam-4367	119	6	for	for	ADP
ejpam-4367	119	7	each	each	DET
ejpam-4367	119	8	points	point	NOUN
ejpam-4367	119	9	x	x	PUNCT
ejpam-4367	119	10	and	and	CCONJ
ejpam-4367	119	11	y	y	PROPN
ejpam-4367	119	12	in	in	ADP
ejpam-4367	119	13	x	x	PRON
ejpam-4367	119	14	,	,	PUNCT
ejpam-4367	119	15	λ(λ	λ(λ	PROPN
ejpam-4367	119	16	,	,	PUNCT
ejpam-4367	119	17	sp)({x	sp)({x	PROPN
ejpam-4367	119	18	}	}	PUNCT
ejpam-4367	119	19	)	)	PUNCT
ejpam-4367	119	20	̸=	̸=	PROPN
ejpam-4367	119	21	λ(λ	λ(λ	PROPN
ejpam-4367	119	22	,	,	PUNCT
ejpam-4367	119	23	sp)({y	sp)({y	NOUN
ejpam-4367	119	24	}	}	PUNCT
ejpam-4367	119	25	)	)	PUNCT
ejpam-4367	119	26	implies	imply	VERB
ejpam-4367	119	27	λ(λ	λ(λ	PROPN
ejpam-4367	119	28	,	,	PUNCT
ejpam-4367	119	29	sp)({x	sp)({x	NOUN
ejpam-4367	119	30	}	}	PUNCT
ejpam-4367	119	31	)	)	PUNCT
ejpam-4367	119	32	∩	∩	NOUN
ejpam-4367	119	33	λ(λ	λ(λ	ADP
ejpam-4367	119	34	,	,	PUNCT
ejpam-4367	119	35	sp)({y	sp)({y	NOUN
ejpam-4367	119	36	}	}	PUNCT
ejpam-4367	119	37	)	)	PUNCT
ejpam-4367	120	1	=	=	PUNCT
ejpam-4367	120	2	∅.	∅.	NOUN
ejpam-4367	120	3	proof	proof	NOUN
ejpam-4367	120	4	.	.	PUNCT
ejpam-4367	121	1	let	let	AUX
ejpam-4367	121	2	(	(	PUNCT
ejpam-4367	121	3	x	x	NOUN
ejpam-4367	121	4	,	,	PUNCT
ejpam-4367	121	5	τ	τ	X
ejpam-4367	121	6	)	)	PUNCT
ejpam-4367	121	7	be	be	AUX
ejpam-4367	121	8	(	(	PUNCT
ejpam-4367	121	9	λ	λ	PROPN
ejpam-4367	121	10	,	,	PUNCT
ejpam-4367	121	11	sp)-r0	sp)-r0	PROPN
ejpam-4367	121	12	.	.	PUNCT
ejpam-4367	121	13	suppose	suppose	VERB
ejpam-4367	121	14	that	that	SCONJ
ejpam-4367	121	15	λ(λ	λ(λ	PROPN
ejpam-4367	121	16	,	,	PUNCT
ejpam-4367	121	17	sp)({x	sp)({x	PROPN
ejpam-4367	121	18	}	}	PUNCT
ejpam-4367	121	19	)	)	PUNCT
ejpam-4367	121	20	∩	∩	NOUN
ejpam-4367	121	21	λ(λ	λ(λ	ADP
ejpam-4367	121	22	,	,	PUNCT
ejpam-4367	121	23	sp)({y	sp)({y	NOUN
ejpam-4367	121	24	}	}	PUNCT
ejpam-4367	121	25	)	)	PUNCT
ejpam-4367	121	26	̸=	̸=	PROPN
ejpam-4367	121	27	∅.	∅.	ADV
ejpam-4367	121	28	let	let	VERB
ejpam-4367	121	29	z	z	PROPN
ejpam-4367	121	30	∈	∈	PROPN
ejpam-4367	121	31	λ(λ	λ(λ	PROPN
ejpam-4367	121	32	,	,	PUNCT
ejpam-4367	121	33	sp)({x	sp)({x	PROPN
ejpam-4367	121	34	}	}	PUNCT
ejpam-4367	121	35	)	)	PUNCT
ejpam-4367	121	36	∩	∩	NOUN
ejpam-4367	121	37	λ(λ	λ(λ	ADP
ejpam-4367	121	38	,	,	PUNCT
ejpam-4367	121	39	sp)({y	sp)({y	NOUN
ejpam-4367	121	40	}	}	PUNCT
ejpam-4367	121	41	)	)	PUNCT
ejpam-4367	121	42	.	.	PUNCT
ejpam-4367	122	1	then	then	ADV
ejpam-4367	122	2	,	,	PUNCT
ejpam-4367	122	3	z	z	PROPN
ejpam-4367	122	4	∈	∈	PROPN
ejpam-4367	122	5	λ(λ	λ(λ	PROPN
ejpam-4367	122	6	,	,	PUNCT
ejpam-4367	122	7	sp)({x	sp)({x	PROPN
ejpam-4367	122	8	}	}	PUNCT
ejpam-4367	122	9	)	)	PUNCT
ejpam-4367	122	10	and	and	CCONJ
ejpam-4367	122	11	by	by	ADP
ejpam-4367	122	12	lemma	lemma	PROPN
ejpam-4367	122	13	4	4	NUM
ejpam-4367	122	14	,	,	PUNCT
ejpam-4367	122	15	x	x	SYM
ejpam-4367	122	16	∈	∈	NOUN
ejpam-4367	122	17	{	{	PUNCT
ejpam-4367	122	18	z}(λ	z}(λ	PROPN
ejpam-4367	122	19	,	,	PUNCT
ejpam-4367	122	20	sp	sp	NOUN
ejpam-4367	122	21	)	)	PUNCT
ejpam-4367	122	22	.	.	PUNCT
ejpam-4367	123	1	thus	thus	ADV
ejpam-4367	123	2	,	,	PUNCT
ejpam-4367	123	3	x	x	SYM
ejpam-4367	123	4	∈	∈	PROPN
ejpam-4367	123	5	{	{	PUNCT
ejpam-4367	123	6	z}(λ	z}(λ	PROPN
ejpam-4367	123	7	,	,	PUNCT
ejpam-4367	123	8	sp)∩{x}(λ	sp)∩{x}(λ	NOUN
ejpam-4367	123	9	,	,	PUNCT
ejpam-4367	123	10	sp	sp	NOUN
ejpam-4367	123	11	)	)	PUNCT
ejpam-4367	123	12	and	and	CCONJ
ejpam-4367	123	13	by	by	ADP
ejpam-4367	123	14	corollary	corollary	ADJ
ejpam-4367	123	15	1	1	NUM
ejpam-4367	123	16	,	,	PUNCT
ejpam-4367	123	17	{	{	PUNCT
ejpam-4367	123	18	z}(λ	z}(λ	PROPN
ejpam-4367	123	19	,	,	PUNCT
ejpam-4367	123	20	sp	sp	NOUN
ejpam-4367	123	21	)	)	PUNCT
ejpam-4367	123	22	=	=	SYM
ejpam-4367	123	23	{	{	PUNCT
ejpam-4367	123	24	x}(λ	x}(λ	PROPN
ejpam-4367	123	25	,	,	PUNCT
ejpam-4367	123	26	sp	sp	NOUN
ejpam-4367	123	27	)	)	PUNCT
ejpam-4367	123	28	.	.	PUNCT
ejpam-4367	124	1	similarly	similarly	ADV
ejpam-4367	124	2	,	,	PUNCT
ejpam-4367	124	3	we	we	PRON
ejpam-4367	124	4	have	have	AUX
ejpam-4367	124	5	{	{	PUNCT
ejpam-4367	124	6	z}(λ	z}(λ	PROPN
ejpam-4367	124	7	,	,	PUNCT
ejpam-4367	124	8	sp	sp	NOUN
ejpam-4367	124	9	)	)	PUNCT
ejpam-4367	124	10	=	=	SYM
ejpam-4367	124	11	{	{	PUNCT
ejpam-4367	124	12	y}(λ	y}(λ	PROPN
ejpam-4367	124	13	,	,	PUNCT
ejpam-4367	124	14	sp	sp	NOUN
ejpam-4367	124	15	)	)	PUNCT
ejpam-4367	124	16	and	and	CCONJ
ejpam-4367	124	17	hence	hence	ADV
ejpam-4367	124	18	{	{	PUNCT
ejpam-4367	124	19	x}(λ	x}(λ	PROPN
ejpam-4367	124	20	,	,	PUNCT
ejpam-4367	124	21	sp	sp	NOUN
ejpam-4367	124	22	)	)	PUNCT
ejpam-4367	124	23	=	=	SYM
ejpam-4367	124	24	{	{	PUNCT
ejpam-4367	124	25	y}(λ	y}(λ	PROPN
ejpam-4367	124	26	,	,	PUNCT
ejpam-4367	124	27	sp	sp	NOUN
ejpam-4367	124	28	)	)	PUNCT
ejpam-4367	124	29	,	,	PUNCT
ejpam-4367	124	30	by	by	ADP
ejpam-4367	124	31	lemma	lemma	PROPN
ejpam-4367	124	32	4	4	NUM
ejpam-4367	124	33	,	,	PUNCT
ejpam-4367	124	34	λ(λ	λ(λ	ADV
ejpam-4367	124	35	,	,	PUNCT
ejpam-4367	124	36	sp)({x	sp)({x	NOUN
ejpam-4367	124	37	}	}	PUNCT
ejpam-4367	124	38	)	)	PUNCT
ejpam-4367	125	1	=	=	SYM
ejpam-4367	125	2	λ(λ	λ(λ	PROPN
ejpam-4367	125	3	,	,	PUNCT
ejpam-4367	125	4	sp)({y	sp)({y	NOUN
ejpam-4367	125	5	}	}	PUNCT
ejpam-4367	125	6	)	)	PUNCT
ejpam-4367	125	7	.	.	PUNCT
ejpam-4367	126	1	conversely	conversely	ADV
ejpam-4367	126	2	,	,	PUNCT
ejpam-4367	126	3	we	we	PRON
ejpam-4367	126	4	show	show	VERB
ejpam-4367	126	5	the	the	DET
ejpam-4367	126	6	sufficiency	sufficiency	NOUN
ejpam-4367	126	7	by	by	ADP
ejpam-4367	126	8	using	use	VERB
ejpam-4367	126	9	corollary	corollary	ADJ
ejpam-4367	126	10	1	1	NUM
ejpam-4367	126	11	.	.	PUNCT
ejpam-4367	126	12	suppose	suppose	VERB
ejpam-4367	126	13	that	that	SCONJ
ejpam-4367	126	14	{	{	PUNCT
ejpam-4367	126	15	x}(λ	x}(λ	PROPN
ejpam-4367	126	16	,	,	PUNCT
ejpam-4367	126	17	sp	sp	NOUN
ejpam-4367	126	18	)	)	PUNCT
ejpam-4367	126	19	̸=	̸=	PROPN
ejpam-4367	126	20	{	{	PUNCT
ejpam-4367	126	21	y}(λ	y}(λ	PROPN
ejpam-4367	126	22	,	,	PUNCT
ejpam-4367	126	23	sp	sp	NOUN
ejpam-4367	126	24	)	)	PUNCT
ejpam-4367	126	25	.	.	PUNCT
ejpam-4367	127	1	by	by	ADP
ejpam-4367	127	2	lemma	lemma	PROPN
ejpam-4367	127	3	4	4	NUM
ejpam-4367	127	4	,	,	PUNCT
ejpam-4367	127	5	λ(λ	λ(λ	ADV
ejpam-4367	127	6	,	,	PUNCT
ejpam-4367	127	7	sp)({x	sp)({x	PROPN
ejpam-4367	127	8	}	}	PUNCT
ejpam-4367	127	9	)	)	PUNCT
ejpam-4367	128	1	̸=	̸=	PROPN
ejpam-4367	128	2	λ(λ	λ(λ	PROPN
ejpam-4367	128	3	,	,	PUNCT
ejpam-4367	128	4	sp)({y	sp)({y	NOUN
ejpam-4367	128	5	}	}	PUNCT
ejpam-4367	128	6	)	)	PUNCT
ejpam-4367	128	7	and	and	CCONJ
ejpam-4367	128	8	hence	hence	ADV
ejpam-4367	128	9	λ(λ	λ(λ	ADV
ejpam-4367	128	10	,	,	PUNCT
ejpam-4367	128	11	sp)({x})∩λ(λ	sp)({x})∩λ(λ	PROPN
ejpam-4367	128	12	,	,	PUNCT
ejpam-4367	128	13	sp)({y	sp)({y	NOUN
ejpam-4367	128	14	}	}	PUNCT
ejpam-4367	128	15	)	)	PUNCT
ejpam-4367	128	16	=	=	PUNCT
ejpam-4367	128	17	∅.	∅.	ADP
ejpam-4367	128	18	thus	thus	ADV
ejpam-4367	128	19	,	,	PUNCT
ejpam-4367	128	20	{	{	PUNCT
ejpam-4367	128	21	x}(λ	x}(λ	PROPN
ejpam-4367	128	22	,	,	PUNCT
ejpam-4367	128	23	sp)∩{y}(λ	sp)∩{y}(λ	NOUN
ejpam-4367	128	24	,	,	PUNCT
ejpam-4367	128	25	sp	sp	NOUN
ejpam-4367	128	26	)	)	PUNCT
ejpam-4367	128	27	=	=	PUNCT
ejpam-4367	128	28	∅.	∅.	NOUN
ejpam-4367	128	29	in	in	ADP
ejpam-4367	128	30	fact	fact	NOUN
ejpam-4367	128	31	,	,	PUNCT
ejpam-4367	128	32	assume	assume	VERB
ejpam-4367	128	33	that	that	SCONJ
ejpam-4367	128	34	z	z	PROPN
ejpam-4367	128	35	∈	∈	PROPN
ejpam-4367	128	36	{	{	PUNCT
ejpam-4367	128	37	x}(λ	x}(λ	PROPN
ejpam-4367	128	38	,	,	PUNCT
ejpam-4367	128	39	sp)∩{y}(λ	sp)∩{y}(λ	NOUN
ejpam-4367	128	40	,	,	PUNCT
ejpam-4367	128	41	sp	sp	NOUN
ejpam-4367	128	42	)	)	PUNCT
ejpam-4367	128	43	.	.	PUNCT
ejpam-4367	129	1	then	then	ADV
ejpam-4367	129	2	,	,	PUNCT
ejpam-4367	129	3	z	z	PROPN
ejpam-4367	129	4	∈	∈	PROPN
ejpam-4367	129	5	{	{	PUNCT
ejpam-4367	129	6	x}(λ	x}(λ	PROPN
ejpam-4367	129	7	,	,	PUNCT
ejpam-4367	129	8	sp	sp	NOUN
ejpam-4367	129	9	)	)	PUNCT
ejpam-4367	129	10	implies	imply	VERB
ejpam-4367	129	11	x	x	X
ejpam-4367	129	12	∈	∈	PROPN
ejpam-4367	129	13	λ(λ	λ(λ	PROPN
ejpam-4367	129	14	,	,	PUNCT
ejpam-4367	129	15	sp)({z	sp)({z	PROPN
ejpam-4367	129	16	}	}	PUNCT
ejpam-4367	129	17	)	)	PUNCT
ejpam-4367	129	18	and	and	CCONJ
ejpam-4367	129	19	hence	hence	ADV
ejpam-4367	129	20	x	x	X
ejpam-4367	129	21	∈	∈	PROPN
ejpam-4367	129	22	λ(λ	λ(λ	PROPN
ejpam-4367	129	23	,	,	PUNCT
ejpam-4367	129	24	sp)({z	sp)({z	PROPN
ejpam-4367	129	25	}	}	PUNCT
ejpam-4367	129	26	)	)	PUNCT
ejpam-4367	129	27	∩	∩	NOUN
ejpam-4367	129	28	λ(λ	λ(λ	ADP
ejpam-4367	129	29	,	,	PUNCT
ejpam-4367	129	30	sp)({x	sp)({x	PROPN
ejpam-4367	129	31	}	}	PUNCT
ejpam-4367	129	32	)	)	PUNCT
ejpam-4367	129	33	.	.	PUNCT
ejpam-4367	130	1	by	by	ADP
ejpam-4367	130	2	the	the	DET
ejpam-4367	130	3	hypothesis	hypothesis	NOUN
ejpam-4367	130	4	,	,	PUNCT
ejpam-4367	130	5	λ(λ	λ(λ	ADV
ejpam-4367	130	6	,	,	PUNCT
ejpam-4367	130	7	sp)({z	sp)({z	PROPN
ejpam-4367	130	8	}	}	PUNCT
ejpam-4367	130	9	)	)	PUNCT
ejpam-4367	130	10	=	=	SYM
ejpam-4367	130	11	λ(λ	λ(λ	PROPN
ejpam-4367	130	12	,	,	PUNCT
ejpam-4367	130	13	sp)({x	sp)({x	NOUN
ejpam-4367	130	14	}	}	PUNCT
ejpam-4367	130	15	)	)	PUNCT
ejpam-4367	130	16	and	and	CCONJ
ejpam-4367	130	17	by	by	ADP
ejpam-4367	130	18	lemma	lemma	PROPN
ejpam-4367	130	19	4	4	NUM
ejpam-4367	130	20	,	,	PUNCT
ejpam-4367	130	21	{	{	PUNCT
ejpam-4367	130	22	z}(λ	z}(λ	PROPN
ejpam-4367	130	23	,	,	PUNCT
ejpam-4367	130	24	sp	sp	NOUN
ejpam-4367	130	25	)	)	PUNCT
ejpam-4367	130	26	=	=	SYM
ejpam-4367	130	27	{	{	PUNCT
ejpam-4367	130	28	x}(λ	x}(λ	PROPN
ejpam-4367	130	29	,	,	PUNCT
ejpam-4367	130	30	sp	sp	NOUN
ejpam-4367	130	31	)	)	PUNCT
ejpam-4367	130	32	.	.	PUNCT
ejpam-4367	131	1	similarly	similarly	ADV
ejpam-4367	131	2	,	,	PUNCT
ejpam-4367	131	3	we	we	PRON
ejpam-4367	131	4	have	have	AUX
ejpam-4367	131	5	{	{	PUNCT
ejpam-4367	131	6	z}(λ	z}(λ	PROPN
ejpam-4367	131	7	,	,	PUNCT
ejpam-4367	131	8	sp	sp	NOUN
ejpam-4367	131	9	)	)	PUNCT
ejpam-4367	131	10	=	=	SYM
ejpam-4367	131	11	{	{	PUNCT
ejpam-4367	131	12	y}(λ	y}(λ	PROPN
ejpam-4367	131	13	,	,	PUNCT
ejpam-4367	131	14	sp	sp	NOUN
ejpam-4367	131	15	)	)	PUNCT
ejpam-4367	131	16	and	and	CCONJ
ejpam-4367	131	17	hence	hence	ADV
ejpam-4367	131	18	{	{	PUNCT
ejpam-4367	131	19	x}(λ	x}(λ	PROPN
ejpam-4367	131	20	,	,	PUNCT
ejpam-4367	131	21	sp	sp	NOUN
ejpam-4367	131	22	)	)	PUNCT
ejpam-4367	131	23	=	=	SYM
ejpam-4367	131	24	{	{	PUNCT
ejpam-4367	131	25	y}(λ	y}(λ	PROPN
ejpam-4367	131	26	,	,	PUNCT
ejpam-4367	131	27	sp	sp	NOUN
ejpam-4367	131	28	)	)	PUNCT
ejpam-4367	131	29	.	.	PUNCT
ejpam-4367	132	1	this	this	PRON
ejpam-4367	132	2	contradicts	contradict	VERB
ejpam-4367	132	3	that	that	SCONJ
ejpam-4367	132	4	{	{	PUNCT
ejpam-4367	132	5	x}(λ	x}(λ	PROPN
ejpam-4367	132	6	,	,	PUNCT
ejpam-4367	132	7	sp	sp	NOUN
ejpam-4367	132	8	)	)	PUNCT
ejpam-4367	132	9	̸=	̸=	PROPN
ejpam-4367	132	10	{	{	PUNCT
ejpam-4367	132	11	y}(λ	y}(λ	PROPN
ejpam-4367	132	12	,	,	PUNCT
ejpam-4367	132	13	sp	sp	NOUN
ejpam-4367	132	14	)	)	PUNCT
ejpam-4367	132	15	.	.	PUNCT
ejpam-4367	133	1	thus	thus	ADV
ejpam-4367	133	2	,	,	PUNCT
ejpam-4367	133	3	{	{	PUNCT
ejpam-4367	133	4	x}(λ	x}(λ	PROPN
ejpam-4367	133	5	,	,	PUNCT
ejpam-4367	133	6	sp	sp	NOUN
ejpam-4367	133	7	)	)	PUNCT
ejpam-4367	133	8	∩	∩	NOUN
ejpam-4367	133	9	{	{	PUNCT
ejpam-4367	133	10	y}(λ	y}(λ	PROPN
ejpam-4367	133	11	,	,	PUNCT
ejpam-4367	133	12	sp	sp	NOUN
ejpam-4367	133	13	)	)	PUNCT
ejpam-4367	133	14	=	=	PUNCT
ejpam-4367	133	15	∅.	∅.	ADP
ejpam-4367	133	16	this	this	PRON
ejpam-4367	133	17	shows	show	VERB
ejpam-4367	133	18	that	that	SCONJ
ejpam-4367	133	19	(	(	PUNCT
ejpam-4367	133	20	x	x	X
ejpam-4367	133	21	,	,	PUNCT
ejpam-4367	133	22	τ	τ	X
ejpam-4367	133	23	)	)	PUNCT
ejpam-4367	133	24	is	be	AUX
ejpam-4367	133	25	(	(	PUNCT
ejpam-4367	133	26	λ	λ	PROPN
ejpam-4367	133	27	,	,	PUNCT
ejpam-4367	133	28	sp)-r0	sp)-r0	PROPN
ejpam-4367	133	29	.	.	PUNCT
ejpam-4367	133	30	theorem	theorem	VERB
ejpam-4367	133	31	3	3	NUM
ejpam-4367	133	32	.	.	X
ejpam-4367	133	33	for	for	ADP
ejpam-4367	133	34	a	a	DET
ejpam-4367	133	35	topological	topological	ADJ
ejpam-4367	133	36	space	space	NOUN
ejpam-4367	133	37	(	(	PUNCT
ejpam-4367	133	38	x	x	X
ejpam-4367	133	39	,	,	PUNCT
ejpam-4367	133	40	τ	τ	PROPN
ejpam-4367	133	41	)	)	PUNCT
ejpam-4367	133	42	,	,	PUNCT
ejpam-4367	133	43	the	the	DET
ejpam-4367	133	44	following	follow	VERB
ejpam-4367	133	45	properties	property	NOUN
ejpam-4367	133	46	are	be	AUX
ejpam-4367	133	47	equivalent	equivalent	ADJ
ejpam-4367	133	48	:	:	PUNCT
ejpam-4367	133	49	(	(	PUNCT
ejpam-4367	133	50	1	1	X
ejpam-4367	133	51	)	)	PUNCT
ejpam-4367	133	52	(	(	PUNCT
ejpam-4367	133	53	x	x	X
ejpam-4367	133	54	,	,	PUNCT
ejpam-4367	133	55	τ	τ	X
ejpam-4367	133	56	)	)	PUNCT
ejpam-4367	133	57	is	be	AUX
ejpam-4367	133	58	(	(	PUNCT
ejpam-4367	133	59	λ	λ	PROPN
ejpam-4367	133	60	,	,	PUNCT
ejpam-4367	133	61	sp)-r0	sp)-r0	PROPN
ejpam-4367	133	62	.	.	PUNCT
ejpam-4367	134	1	c.	c.	PROPN
ejpam-4367	134	2	boonpok	boonpok	PROPN
ejpam-4367	134	3	,	,	PUNCT
ejpam-4367	134	4	c.	c.	PROPN
ejpam-4367	134	5	viriyapong	viriyapong	PROPN
ejpam-4367	134	6	/	/	SYM
ejpam-4367	134	7	eur	eur	PROPN
ejpam-4367	134	8	.	.	PUNCT
ejpam-4367	135	1	j.	j.	PROPN
ejpam-4367	135	2	pure	pure	PROPN
ejpam-4367	135	3	appl	appl	PROPN
ejpam-4367	135	4	.	.	PROPN
ejpam-4367	135	5	math	math	PROPN
ejpam-4367	135	6	,	,	PUNCT
ejpam-4367	135	7	15	15	NUM
ejpam-4367	135	8	(	(	PUNCT
ejpam-4367	135	9	3	3	NUM
ejpam-4367	135	10	)	)	PUNCT
ejpam-4367	135	11	(	(	PUNCT
ejpam-4367	135	12	2022	2022	NUM
ejpam-4367	135	13	)	)	PUNCT
ejpam-4367	135	14	,	,	PUNCT
ejpam-4367	135	15	878	878	NUM
ejpam-4367	135	16	-	-	SYM
ejpam-4367	135	17	886	886	NUM
ejpam-4367	135	18	882	882	NUM
ejpam-4367	135	19	(	(	PUNCT
ejpam-4367	135	20	2	2	NUM
ejpam-4367	135	21	)	)	PUNCT
ejpam-4367	135	22	x	x	SYM
ejpam-4367	135	23	∈	∈	PROPN
ejpam-4367	135	24	{	{	PUNCT
ejpam-4367	135	25	y}(λ	y}(λ	PROPN
ejpam-4367	135	26	,	,	PUNCT
ejpam-4367	135	27	sp	sp	NOUN
ejpam-4367	135	28	)	)	PUNCT
ejpam-4367	135	29	if	if	SCONJ
ejpam-4367	136	1	and	and	CCONJ
ejpam-4367	136	2	only	only	ADV
ejpam-4367	136	3	if	if	SCONJ
ejpam-4367	136	4	y	y	PROPN
ejpam-4367	136	5	∈	∈	PROPN
ejpam-4367	136	6	{	{	PUNCT
ejpam-4367	136	7	x}(λ	x}(λ	PROPN
ejpam-4367	136	8	,	,	PUNCT
ejpam-4367	136	9	sp	sp	NOUN
ejpam-4367	136	10	)	)	PUNCT
ejpam-4367	136	11	.	.	PUNCT
ejpam-4367	137	1	proof	proof	NOUN
ejpam-4367	137	2	.	.	PUNCT
ejpam-4367	138	1	(	(	PUNCT
ejpam-4367	138	2	1	1	X
ejpam-4367	138	3	)	)	PUNCT
ejpam-4367	138	4	⇒	⇒	NOUN
ejpam-4367	138	5	(	(	PUNCT
ejpam-4367	138	6	2	2	NUM
ejpam-4367	138	7	):	):	PUNCT
ejpam-4367	138	8	suppose	suppose	VERB
ejpam-4367	138	9	that	that	SCONJ
ejpam-4367	138	10	x	x	SYM
ejpam-4367	138	11	∈	∈	PROPN
ejpam-4367	138	12	{	{	PUNCT
ejpam-4367	138	13	y}(λ	y}(λ	PROPN
ejpam-4367	138	14	,	,	PUNCT
ejpam-4367	138	15	sp	sp	NOUN
ejpam-4367	138	16	)	)	PUNCT
ejpam-4367	138	17	.	.	PUNCT
ejpam-4367	139	1	by	by	ADP
ejpam-4367	139	2	lemma	lemma	PROPN
ejpam-4367	139	3	4	4	NUM
ejpam-4367	139	4	,	,	PUNCT
ejpam-4367	139	5	y	y	PROPN
ejpam-4367	139	6	∈	∈	PROPN
ejpam-4367	139	7	λ(λ	λ(λ	PROPN
ejpam-4367	139	8	,	,	PUNCT
ejpam-4367	139	9	sp)({x	sp)({x	PROPN
ejpam-4367	139	10	}	}	PUNCT
ejpam-4367	139	11	)	)	PUNCT
ejpam-4367	139	12	and	and	CCONJ
ejpam-4367	139	13	hence	hence	ADV
ejpam-4367	139	14	λ(λ	λ(λ	PROPN
ejpam-4367	139	15	,	,	PUNCT
ejpam-4367	139	16	sp)({x	sp)({x	PROPN
ejpam-4367	139	17	}	}	PUNCT
ejpam-4367	139	18	)	)	PUNCT
ejpam-4367	139	19	∩	∩	NOUN
ejpam-4367	139	20	λ(λ	λ(λ	ADP
ejpam-4367	139	21	,	,	PUNCT
ejpam-4367	139	22	sp)({y	sp)({y	NOUN
ejpam-4367	139	23	}	}	PUNCT
ejpam-4367	139	24	)	)	PUNCT
ejpam-4367	140	1	̸=	̸=	PROPN
ejpam-4367	140	2	∅.	∅.	VERB
ejpam-4367	140	3	by	by	ADP
ejpam-4367	140	4	theorem	theorem	NOUN
ejpam-4367	140	5	2	2	NUM
ejpam-4367	140	6	,	,	PUNCT
ejpam-4367	140	7	λ(λ	λ(λ	ADV
ejpam-4367	140	8	,	,	PUNCT
ejpam-4367	140	9	sp)({x	sp)({x	NOUN
ejpam-4367	140	10	}	}	PUNCT
ejpam-4367	140	11	)	)	PUNCT
ejpam-4367	140	12	=	=	SYM
ejpam-4367	140	13	λ(λ	λ(λ	PROPN
ejpam-4367	140	14	,	,	PUNCT
ejpam-4367	140	15	sp)({y	sp)({y	NOUN
ejpam-4367	140	16	}	}	PUNCT
ejpam-4367	140	17	)	)	PUNCT
ejpam-4367	140	18	and	and	CCONJ
ejpam-4367	140	19	hence	hence	ADV
ejpam-4367	140	20	x	x	X
ejpam-4367	140	21	∈	∈	PROPN
ejpam-4367	140	22	λ(λ	λ(λ	PROPN
ejpam-4367	140	23	,	,	PUNCT
ejpam-4367	140	24	sp)({y	sp)({y	NOUN
ejpam-4367	140	25	}	}	PUNCT
ejpam-4367	140	26	)	)	PUNCT
ejpam-4367	140	27	.	.	PUNCT
ejpam-4367	141	1	thus	thus	ADV
ejpam-4367	141	2	,	,	PUNCT
ejpam-4367	141	3	by	by	ADP
ejpam-4367	141	4	lemma	lemma	PROPN
ejpam-4367	141	5	4	4	NUM
ejpam-4367	141	6	,	,	PUNCT
ejpam-4367	141	7	y	y	PROPN
ejpam-4367	141	8	∈	∈	PROPN
ejpam-4367	141	9	{	{	PUNCT
ejpam-4367	141	10	x}(λ	x}(λ	PROPN
ejpam-4367	141	11	,	,	PUNCT
ejpam-4367	141	12	sp	sp	NOUN
ejpam-4367	141	13	)	)	PUNCT
ejpam-4367	141	14	.	.	PUNCT
ejpam-4367	142	1	the	the	DET
ejpam-4367	142	2	converse	converse	NOUN
ejpam-4367	142	3	is	be	AUX
ejpam-4367	142	4	similarly	similarly	ADV
ejpam-4367	142	5	shown	show	VERB
ejpam-4367	142	6	.	.	PUNCT
ejpam-4367	143	1	(	(	PUNCT
ejpam-4367	143	2	2	2	X
ejpam-4367	143	3	)	)	PUNCT
ejpam-4367	143	4	⇒	⇒	NOUN
ejpam-4367	143	5	(	(	PUNCT
ejpam-4367	143	6	1	1	NUM
ejpam-4367	143	7	):	):	PUNCT
ejpam-4367	143	8	let	let	VERB
ejpam-4367	143	9	u	u	PRON
ejpam-4367	143	10	∈	∈	PROPN
ejpam-4367	143	11	λspo(x	λspo(x	PROPN
ejpam-4367	143	12	,	,	PUNCT
ejpam-4367	143	13	τ	τ	X
ejpam-4367	143	14	)	)	PUNCT
ejpam-4367	143	15	and	and	CCONJ
ejpam-4367	143	16	let	let	VERB
ejpam-4367	143	17	x	x	PUNCT
ejpam-4367	143	18	∈	∈	PROPN
ejpam-4367	143	19	u	u	NOUN
ejpam-4367	143	20	.	.	PUNCT
ejpam-4367	144	1	if	if	SCONJ
ejpam-4367	144	2	y	y	PROPN
ejpam-4367	144	3	̸∈	̸∈	PROPN
ejpam-4367	144	4	u	u	PROPN
ejpam-4367	144	5	,	,	PUNCT
ejpam-4367	144	6	then	then	ADV
ejpam-4367	144	7	u	u	NOUN
ejpam-4367	144	8	∩	∩	PROPN
ejpam-4367	144	9	{	{	PUNCT
ejpam-4367	144	10	y}(λ	y}(λ	PROPN
ejpam-4367	144	11	,	,	PUNCT
ejpam-4367	144	12	sp	sp	NOUN
ejpam-4367	144	13	)	)	PUNCT
ejpam-4367	144	14	=	=	NOUN
ejpam-4367	144	15	∅.	∅.	ADP
ejpam-4367	144	16	thus	thus	ADV
ejpam-4367	144	17	,	,	PUNCT
ejpam-4367	144	18	x	x	PROPN
ejpam-4367	144	19	̸∈	̸∈	PROPN
ejpam-4367	144	20	{	{	PUNCT
ejpam-4367	144	21	y}(λ	y}(λ	PROPN
ejpam-4367	144	22	,	,	PUNCT
ejpam-4367	144	23	sp	sp	NOUN
ejpam-4367	144	24	)	)	PUNCT
ejpam-4367	144	25	and	and	CCONJ
ejpam-4367	144	26	y	y	PROPN
ejpam-4367	144	27	̸∈	̸∈	PROPN
ejpam-4367	144	28	{	{	PUNCT
ejpam-4367	144	29	x}(λ	x}(λ	PROPN
ejpam-4367	144	30	,	,	PUNCT
ejpam-4367	144	31	sp	sp	NOUN
ejpam-4367	144	32	)	)	PUNCT
ejpam-4367	144	33	.	.	PUNCT
ejpam-4367	145	1	this	this	PRON
ejpam-4367	145	2	implies	imply	VERB
ejpam-4367	145	3	that	that	SCONJ
ejpam-4367	145	4	{	{	PUNCT
ejpam-4367	145	5	x}(λ	x}(λ	PROPN
ejpam-4367	145	6	,	,	PUNCT
ejpam-4367	145	7	sp	sp	NOUN
ejpam-4367	145	8	)	)	PUNCT
ejpam-4367	145	9	⊆	⊆	NUM
ejpam-4367	145	10	u	u	NOUN
ejpam-4367	145	11	.	.	PUNCT
ejpam-4367	146	1	therefore	therefore	ADV
ejpam-4367	146	2	,	,	PUNCT
ejpam-4367	146	3	(	(	PUNCT
ejpam-4367	146	4	x	x	X
ejpam-4367	146	5	,	,	PUNCT
ejpam-4367	146	6	τ	τ	X
ejpam-4367	146	7	)	)	PUNCT
ejpam-4367	146	8	is	be	AUX
ejpam-4367	146	9	(	(	PUNCT
ejpam-4367	146	10	λ	λ	PROPN
ejpam-4367	146	11	,	,	PUNCT
ejpam-4367	146	12	sp)-r0	sp)-r0	PROPN
ejpam-4367	146	13	.	.	PUNCT
ejpam-4367	146	14	theorem	theorem	VERB
ejpam-4367	146	15	4	4	NUM
ejpam-4367	146	16	.	.	X
ejpam-4367	146	17	for	for	ADP
ejpam-4367	146	18	a	a	DET
ejpam-4367	146	19	topological	topological	ADJ
ejpam-4367	146	20	space	space	NOUN
ejpam-4367	146	21	(	(	PUNCT
ejpam-4367	146	22	x	x	X
ejpam-4367	146	23	,	,	PUNCT
ejpam-4367	146	24	τ	τ	PROPN
ejpam-4367	146	25	)	)	PUNCT
ejpam-4367	146	26	,	,	PUNCT
ejpam-4367	146	27	the	the	DET
ejpam-4367	146	28	following	follow	VERB
ejpam-4367	146	29	properties	property	NOUN
ejpam-4367	146	30	are	be	AUX
ejpam-4367	146	31	equivalent	equivalent	ADJ
ejpam-4367	146	32	:	:	PUNCT
ejpam-4367	146	33	(	(	PUNCT
ejpam-4367	146	34	1	1	X
ejpam-4367	146	35	)	)	PUNCT
ejpam-4367	146	36	(	(	PUNCT
ejpam-4367	146	37	x	x	X
ejpam-4367	146	38	,	,	PUNCT
ejpam-4367	146	39	τ	τ	X
ejpam-4367	146	40	)	)	PUNCT
ejpam-4367	146	41	is	be	AUX
ejpam-4367	146	42	(	(	PUNCT
ejpam-4367	146	43	λ	λ	PROPN
ejpam-4367	146	44	,	,	PUNCT
ejpam-4367	146	45	sp)-r0	sp)-r0	PROPN
ejpam-4367	146	46	.	.	PUNCT
ejpam-4367	147	1	(	(	PUNCT
ejpam-4367	147	2	2	2	NUM
ejpam-4367	147	3	)	)	PUNCT
ejpam-4367	147	4	for	for	ADP
ejpam-4367	147	5	each	each	DET
ejpam-4367	147	6	nonempty	nonempty	NOUN
ejpam-4367	147	7	subset	subset	VERB
ejpam-4367	147	8	a	a	PRON
ejpam-4367	147	9	of	of	ADP
ejpam-4367	147	10	x	x	PUNCT
ejpam-4367	147	11	and	and	CCONJ
ejpam-4367	147	12	each	each	DET
ejpam-4367	147	13	u	u	PROPN
ejpam-4367	147	14	∈	∈	PROPN
ejpam-4367	147	15	λspo(x	λspo(x	PROPN
ejpam-4367	147	16	,	,	PUNCT
ejpam-4367	147	17	τ	τ	PROPN
ejpam-4367	147	18	)	)	PUNCT
ejpam-4367	147	19	such	such	ADJ
ejpam-4367	147	20	that	that	SCONJ
ejpam-4367	147	21	a	a	DET
ejpam-4367	147	22	∩	∩	ADJ
ejpam-4367	147	23	u	u	NOUN
ejpam-4367	147	24	̸=	̸=	PROPN
ejpam-4367	147	25	∅	∅	NOUN
ejpam-4367	147	26	,	,	PUNCT
ejpam-4367	147	27	there	there	PRON
ejpam-4367	147	28	exists	exist	VERB
ejpam-4367	147	29	a	a	DET
ejpam-4367	147	30	(	(	PUNCT
ejpam-4367	147	31	λ	λ	PROPN
ejpam-4367	147	32	,	,	PUNCT
ejpam-4367	147	33	sp)-closed	sp)-close	VERB
ejpam-4367	147	34	set	set	VERB
ejpam-4367	147	35	f	f	PROPN
ejpam-4367	147	36	such	such	ADJ
ejpam-4367	147	37	that	that	SCONJ
ejpam-4367	147	38	a	a	DET
ejpam-4367	147	39	∩	∩	ADJ
ejpam-4367	147	40	f	f	PROPN
ejpam-4367	147	41	̸=	̸=	PROPN
ejpam-4367	147	42	∅	∅	NOUN
ejpam-4367	147	43	and	and	CCONJ
ejpam-4367	147	44	f	f	PROPN
ejpam-4367	147	45	⊆	⊆	NUM
ejpam-4367	147	46	u	u	NOUN
ejpam-4367	147	47	.	.	PUNCT
ejpam-4367	148	1	(	(	PUNCT
ejpam-4367	148	2	3	3	X
ejpam-4367	148	3	)	)	PUNCT
ejpam-4367	148	4	f	f	NOUN
ejpam-4367	149	1	=	=	SYM
ejpam-4367	149	2	λ(λ	λ(λ	PROPN
ejpam-4367	149	3	,	,	PUNCT
ejpam-4367	149	4	sp)(f	sp)(f	PROPN
ejpam-4367	149	5	)	)	PUNCT
ejpam-4367	150	1	for	for	SCONJ
ejpam-4367	150	2	each	each	PRON
ejpam-4367	150	3	(	(	PUNCT
ejpam-4367	150	4	λ	λ	PROPN
ejpam-4367	150	5	,	,	PUNCT
ejpam-4367	150	6	sp)-closed	sp)-close	VERB
ejpam-4367	150	7	set	set	VERB
ejpam-4367	150	8	f	f	PROPN
ejpam-4367	150	9	.	.	PUNCT
ejpam-4367	151	1	(	(	PUNCT
ejpam-4367	151	2	4	4	NUM
ejpam-4367	151	3	)	)	PUNCT
ejpam-4367	151	4	{	{	PUNCT
ejpam-4367	151	5	x}(λ	x}(λ	PROPN
ejpam-4367	151	6	,	,	PUNCT
ejpam-4367	151	7	sp	sp	NOUN
ejpam-4367	151	8	)	)	PUNCT
ejpam-4367	151	9	=	=	SYM
ejpam-4367	152	1	λ(λ	λ(λ	PROPN
ejpam-4367	152	2	,	,	PUNCT
ejpam-4367	152	3	sp)({x	sp)({x	PROPN
ejpam-4367	152	4	}	}	PUNCT
ejpam-4367	152	5	)	)	PUNCT
ejpam-4367	152	6	for	for	ADP
ejpam-4367	152	7	each	each	DET
ejpam-4367	152	8	x	x	SYM
ejpam-4367	152	9	∈	∈	PROPN
ejpam-4367	152	10	x.	x.	NOUN
ejpam-4367	152	11	(	(	PUNCT
ejpam-4367	152	12	5	5	NUM
ejpam-4367	152	13	)	)	PUNCT
ejpam-4367	152	14	{	{	PUNCT
ejpam-4367	152	15	x}(λ	x}(λ	PROPN
ejpam-4367	152	16	,	,	PUNCT
ejpam-4367	152	17	sp	sp	NOUN
ejpam-4367	152	18	)	)	PUNCT
ejpam-4367	152	19	⊆	⊆	NUM
ejpam-4367	152	20	λ(λ	λ(λ	PROPN
ejpam-4367	152	21	,	,	PUNCT
ejpam-4367	152	22	sp)({x	sp)({x	PROPN
ejpam-4367	152	23	}	}	PUNCT
ejpam-4367	152	24	)	)	PUNCT
ejpam-4367	152	25	for	for	ADP
ejpam-4367	152	26	each	each	DET
ejpam-4367	152	27	x	x	SYM
ejpam-4367	152	28	∈	∈	PROPN
ejpam-4367	152	29	x.	x.	NOUN
ejpam-4367	152	30	proof	proof	NOUN
ejpam-4367	152	31	.	.	PUNCT
ejpam-4367	153	1	(	(	PUNCT
ejpam-4367	153	2	1	1	X
ejpam-4367	153	3	)	)	PUNCT
ejpam-4367	153	4	⇒	⇒	NOUN
ejpam-4367	153	5	(	(	PUNCT
ejpam-4367	153	6	2	2	NUM
ejpam-4367	153	7	):	):	PUNCT
ejpam-4367	153	8	let	let	VERB
ejpam-4367	153	9	a	a	PRON
ejpam-4367	153	10	be	be	AUX
ejpam-4367	153	11	a	a	DET
ejpam-4367	153	12	nonempty	nonempty	ADJ
ejpam-4367	153	13	subset	subset	NOUN
ejpam-4367	153	14	of	of	ADP
ejpam-4367	153	15	x	x	PUNCT
ejpam-4367	153	16	and	and	CCONJ
ejpam-4367	153	17	let	let	VERB
ejpam-4367	153	18	u	u	PRON
ejpam-4367	153	19	∈	∈	PROPN
ejpam-4367	153	20	λspo(x	λspo(x	PROPN
ejpam-4367	153	21	,	,	PUNCT
ejpam-4367	153	22	τ	τ	PROPN
ejpam-4367	153	23	)	)	PUNCT
ejpam-4367	153	24	such	such	ADJ
ejpam-4367	153	25	that	that	SCONJ
ejpam-4367	153	26	a	a	DET
ejpam-4367	153	27	∩	∩	ADJ
ejpam-4367	153	28	u	u	NOUN
ejpam-4367	153	29	̸=	̸=	PROPN
ejpam-4367	153	30	∅.	∅.	VERB
ejpam-4367	153	31	then	then	ADV
ejpam-4367	153	32	,	,	PUNCT
ejpam-4367	153	33	there	there	PRON
ejpam-4367	153	34	exists	exist	VERB
ejpam-4367	153	35	x	x	X
ejpam-4367	153	36	∈	∈	PROPN
ejpam-4367	153	37	a	a	DET
ejpam-4367	153	38	∩	∩	ADJ
ejpam-4367	153	39	u	u	NOUN
ejpam-4367	153	40	and	and	CCONJ
ejpam-4367	153	41	hence	hence	ADV
ejpam-4367	153	42	{	{	PUNCT
ejpam-4367	153	43	x}(λ	x}(λ	PROPN
ejpam-4367	153	44	,	,	PUNCT
ejpam-4367	153	45	sp	sp	NOUN
ejpam-4367	153	46	)	)	PUNCT
ejpam-4367	153	47	⊆	⊆	NUM
ejpam-4367	153	48	u	u	NOUN
ejpam-4367	153	49	.	.	PUNCT
ejpam-4367	154	1	put	put	VERB
ejpam-4367	154	2	f	f	X
ejpam-4367	154	3	=	=	SYM
ejpam-4367	154	4	{	{	PUNCT
ejpam-4367	154	5	x}(λ	x}(λ	PROPN
ejpam-4367	154	6	,	,	PUNCT
ejpam-4367	154	7	sp	sp	NOUN
ejpam-4367	154	8	)	)	PUNCT
ejpam-4367	154	9	,	,	PUNCT
ejpam-4367	154	10	by	by	ADP
ejpam-4367	154	11	lemma	lemma	PROPN
ejpam-4367	154	12	1	1	NUM
ejpam-4367	154	13	,	,	PUNCT
ejpam-4367	154	14	f	f	PROPN
ejpam-4367	154	15	is	be	AUX
ejpam-4367	154	16	(	(	PUNCT
ejpam-4367	154	17	λ	λ	X
ejpam-4367	154	18	,	,	PUNCT
ejpam-4367	154	19	sp)-closed	sp)-close	VERB
ejpam-4367	154	20	,	,	PUNCT
ejpam-4367	154	21	a	a	DET
ejpam-4367	154	22	∩	∩	ADJ
ejpam-4367	154	23	f	f	PROPN
ejpam-4367	154	24	̸=	̸=	PROPN
ejpam-4367	154	25	∅	∅	NOUN
ejpam-4367	154	26	and	and	CCONJ
ejpam-4367	154	27	f	f	PROPN
ejpam-4367	154	28	⊆	⊆	NUM
ejpam-4367	154	29	u	u	NOUN
ejpam-4367	154	30	.	.	PUNCT
ejpam-4367	155	1	(	(	PUNCT
ejpam-4367	155	2	2	2	X
ejpam-4367	155	3	)	)	PUNCT
ejpam-4367	155	4	⇒	⇒	NOUN
ejpam-4367	155	5	(	(	PUNCT
ejpam-4367	155	6	3	3	NUM
ejpam-4367	155	7	):	):	PUNCT
ejpam-4367	155	8	let	let	VERB
ejpam-4367	155	9	f	f	PRON
ejpam-4367	155	10	be	be	AUX
ejpam-4367	155	11	any	any	DET
ejpam-4367	155	12	(	(	PUNCT
ejpam-4367	155	13	λ	λ	PROPN
ejpam-4367	155	14	,	,	PUNCT
ejpam-4367	155	15	sp)-closed	sp)-close	VERB
ejpam-4367	155	16	set	set	NOUN
ejpam-4367	155	17	of	of	ADP
ejpam-4367	155	18	x.	x.	NOUN
ejpam-4367	155	19	by	by	ADP
ejpam-4367	155	20	lemma	lemma	PROPN
ejpam-4367	155	21	3	3	NUM
ejpam-4367	155	22	,	,	PUNCT
ejpam-4367	155	23	we	we	PRON
ejpam-4367	155	24	have	have	VERB
ejpam-4367	155	25	f	f	PROPN
ejpam-4367	155	26	⊆	⊆	NUM
ejpam-4367	155	27	λ(λ	λ(λ	PROPN
ejpam-4367	155	28	,	,	PUNCT
ejpam-4367	155	29	sp)(f	sp)(f	PROPN
ejpam-4367	155	30	)	)	PUNCT
ejpam-4367	155	31	.	.	PUNCT
ejpam-4367	156	1	next	next	ADV
ejpam-4367	156	2	,	,	PUNCT
ejpam-4367	156	3	we	we	PRON
ejpam-4367	156	4	show	show	VERB
ejpam-4367	156	5	f	f	PROPN
ejpam-4367	156	6	⊇	⊇	PROPN
ejpam-4367	156	7	λ(λ	λ(λ	PROPN
ejpam-4367	156	8	,	,	PUNCT
ejpam-4367	156	9	sp)(f	sp)(f	PROPN
ejpam-4367	156	10	)	)	PUNCT
ejpam-4367	156	11	.	.	PUNCT
ejpam-4367	157	1	let	let	VERB
ejpam-4367	157	2	x	x	SYM
ejpam-4367	157	3	̸∈	̸∈	PROPN
ejpam-4367	157	4	f	f	PROPN
ejpam-4367	157	5	.	.	PUNCT
ejpam-4367	158	1	then	then	ADV
ejpam-4367	158	2	,	,	PUNCT
ejpam-4367	158	3	x	x	PUNCT
ejpam-4367	158	4	∈	∈	NOUN
ejpam-4367	158	5	x	x	X
ejpam-4367	158	6	−	−	PROPN
ejpam-4367	158	7	f	f	PROPN
ejpam-4367	158	8	∈	∈	PROPN
ejpam-4367	158	9	λspo(x	λspo(x	PROPN
ejpam-4367	158	10	,	,	PUNCT
ejpam-4367	158	11	τ	τ	PROPN
ejpam-4367	158	12	)	)	PUNCT
ejpam-4367	158	13	and	and	CCONJ
ejpam-4367	158	14	by	by	ADP
ejpam-4367	158	15	(	(	PUNCT
ejpam-4367	158	16	2	2	NUM
ejpam-4367	158	17	)	)	PUNCT
ejpam-4367	158	18	,	,	PUNCT
ejpam-4367	158	19	there	there	PRON
ejpam-4367	158	20	exists	exist	VERB
ejpam-4367	158	21	a	a	DET
ejpam-4367	158	22	(	(	PUNCT
ejpam-4367	158	23	λ	λ	PROPN
ejpam-4367	158	24	,	,	PUNCT
ejpam-4367	158	25	sp)-closed	sp)-close	VERB
ejpam-4367	158	26	set	set	ADJ
ejpam-4367	158	27	k	k	ADP
ejpam-4367	158	28	such	such	ADJ
ejpam-4367	158	29	that	that	SCONJ
ejpam-4367	158	30	x	x	SYM
ejpam-4367	158	31	∈	∈	PROPN
ejpam-4367	158	32	k	k	PROPN
ejpam-4367	158	33	and	and	CCONJ
ejpam-4367	158	34	k	k	PROPN
ejpam-4367	158	35	⊆	⊆	NUM
ejpam-4367	158	36	x	x	SYM
ejpam-4367	158	37	−	−	PROPN
ejpam-4367	158	38	f	f	NOUN
ejpam-4367	158	39	.	.	PUNCT
ejpam-4367	159	1	now	now	ADV
ejpam-4367	159	2	,	,	PUNCT
ejpam-4367	159	3	put	put	VERB
ejpam-4367	159	4	u	u	NOUN
ejpam-4367	159	5	=	=	NOUN
ejpam-4367	159	6	x	x	SYM
ejpam-4367	159	7	−k	−k	VERB
ejpam-4367	159	8	.	.	PUNCT
ejpam-4367	160	1	then	then	ADV
ejpam-4367	160	2	,	,	PUNCT
ejpam-4367	160	3	f	f	PROPN
ejpam-4367	160	4	⊆	⊆	NUM
ejpam-4367	160	5	u	u	X
ejpam-4367	160	6	∈	∈	PROPN
ejpam-4367	160	7	λspo(x	λspo(x	PROPN
ejpam-4367	160	8	,	,	PUNCT
ejpam-4367	160	9	τ	τ	PROPN
ejpam-4367	160	10	)	)	PUNCT
ejpam-4367	160	11	and	and	CCONJ
ejpam-4367	160	12	x	x	PUNCT
ejpam-4367	160	13	̸∈	̸∈	PROPN
ejpam-4367	160	14	u	u	PROPN
ejpam-4367	160	15	.	.	PUNCT
ejpam-4367	161	1	thus	thus	ADV
ejpam-4367	161	2	,	,	PUNCT
ejpam-4367	161	3	x	x	PROPN
ejpam-4367	161	4	̸∈	̸∈	PROPN
ejpam-4367	161	5	λ(λ	λ(λ	PROPN
ejpam-4367	161	6	,	,	PUNCT
ejpam-4367	161	7	sp)(f	sp)(f	PROPN
ejpam-4367	161	8	)	)	PUNCT
ejpam-4367	161	9	.	.	PUNCT
ejpam-4367	162	1	this	this	PRON
ejpam-4367	162	2	shows	show	VERB
ejpam-4367	162	3	that	that	SCONJ
ejpam-4367	162	4	f	f	PROPN
ejpam-4367	162	5	⊇	⊇	PROPN
ejpam-4367	162	6	λ(λ	λ(λ	PROPN
ejpam-4367	162	7	,	,	PUNCT
ejpam-4367	162	8	sp)(f	sp)(f	PROPN
ejpam-4367	162	9	)	)	PUNCT
ejpam-4367	162	10	.	.	PUNCT
ejpam-4367	163	1	(	(	PUNCT
ejpam-4367	163	2	3	3	X
ejpam-4367	163	3	)	)	PUNCT
ejpam-4367	163	4	⇒	⇒	NOUN
ejpam-4367	163	5	(	(	PUNCT
ejpam-4367	163	6	4	4	NUM
ejpam-4367	163	7	):	):	PUNCT
ejpam-4367	163	8	let	let	VERB
ejpam-4367	163	9	x	x	PUNCT
ejpam-4367	163	10	∈	∈	PROPN
ejpam-4367	163	11	x	x	PUNCT
ejpam-4367	163	12	and	and	CCONJ
ejpam-4367	163	13	let	let	VERB
ejpam-4367	163	14	y	y	PROPN
ejpam-4367	163	15	̸∈	̸∈	PROPN
ejpam-4367	163	16	λ(λ	λ(λ	PROPN
ejpam-4367	163	17	,	,	PUNCT
ejpam-4367	163	18	sp)({x	sp)({x	PROPN
ejpam-4367	163	19	}	}	PUNCT
ejpam-4367	163	20	)	)	PUNCT
ejpam-4367	163	21	.	.	PUNCT
ejpam-4367	164	1	then	then	ADV
ejpam-4367	164	2	,	,	PUNCT
ejpam-4367	164	3	there	there	PRON
ejpam-4367	164	4	exists	exist	VERB
ejpam-4367	164	5	u	u	PROPN
ejpam-4367	164	6	∈	∈	PROPN
ejpam-4367	164	7	λspo(x	λspo(x	PROPN
ejpam-4367	164	8	,	,	PUNCT
ejpam-4367	164	9	τ	τ	PROPN
ejpam-4367	164	10	)	)	PUNCT
ejpam-4367	164	11	such	such	ADJ
ejpam-4367	164	12	that	that	SCONJ
ejpam-4367	164	13	x	x	SYM
ejpam-4367	164	14	∈	∈	PROPN
ejpam-4367	164	15	u	u	NOUN
ejpam-4367	164	16	and	and	CCONJ
ejpam-4367	164	17	y	y	PROPN
ejpam-4367	164	18	̸∈	̸∈	PROPN
ejpam-4367	164	19	u	u	PROPN
ejpam-4367	164	20	.	.	PUNCT
ejpam-4367	165	1	thus	thus	ADV
ejpam-4367	165	2	,	,	PUNCT
ejpam-4367	165	3	u	u	PROPN
ejpam-4367	165	4	∩	∩	NOUN
ejpam-4367	165	5	{	{	PUNCT
ejpam-4367	165	6	y}(λ	y}(λ	PROPN
ejpam-4367	165	7	,	,	PUNCT
ejpam-4367	165	8	sp	sp	NOUN
ejpam-4367	165	9	)	)	PUNCT
ejpam-4367	165	10	=	=	PUNCT
ejpam-4367	165	11	∅.	∅.	X
ejpam-4367	165	12	by	by	ADP
ejpam-4367	165	13	(	(	PUNCT
ejpam-4367	165	14	3	3	NUM
ejpam-4367	165	15	)	)	PUNCT
ejpam-4367	165	16	,	,	PUNCT
ejpam-4367	165	17	u	u	PROPN
ejpam-4367	165	18	∩	∩	X
ejpam-4367	165	19	λ(λ	λ(λ	PROPN
ejpam-4367	165	20	,	,	PUNCT
ejpam-4367	165	21	sp)({y}(λ	sp)({y}(λ	PROPN
ejpam-4367	165	22	,	,	PUNCT
ejpam-4367	165	23	sp	sp	NOUN
ejpam-4367	165	24	)	)	PUNCT
ejpam-4367	165	25	)	)	PUNCT
ejpam-4367	166	1	=	=	PUNCT
ejpam-4367	166	2	∅.	∅.	NOUN
ejpam-4367	166	3	since	since	SCONJ
ejpam-4367	166	4	x	x	PROPN
ejpam-4367	166	5	̸∈	̸∈	PROPN
ejpam-4367	166	6	λ(λ	λ(λ	PROPN
ejpam-4367	166	7	,	,	PUNCT
ejpam-4367	166	8	sp)({y}(λ	sp)({y}(λ	PROPN
ejpam-4367	166	9	,	,	PUNCT
ejpam-4367	166	10	sp	sp	NOUN
ejpam-4367	166	11	)	)	PUNCT
ejpam-4367	166	12	)	)	PUNCT
ejpam-4367	166	13	,	,	PUNCT
ejpam-4367	166	14	there	there	PRON
ejpam-4367	166	15	exists	exist	VERB
ejpam-4367	166	16	v	v	ADP
ejpam-4367	166	17	∈	∈	PROPN
ejpam-4367	166	18	λspo(x	λspo(x	NOUN
ejpam-4367	166	19	,	,	PUNCT
ejpam-4367	166	20	τ	τ	PROPN
ejpam-4367	166	21	)	)	PUNCT
ejpam-4367	166	22	such	such	ADJ
ejpam-4367	166	23	that	that	SCONJ
ejpam-4367	166	24	{	{	PUNCT
ejpam-4367	166	25	y}(λ	y}(λ	PROPN
ejpam-4367	166	26	,	,	PUNCT
ejpam-4367	166	27	sp	sp	NOUN
ejpam-4367	166	28	)	)	PUNCT
ejpam-4367	166	29	⊆	⊆	NUM
ejpam-4367	166	30	v	v	NOUN
ejpam-4367	166	31	and	and	CCONJ
ejpam-4367	166	32	x	x	PART
ejpam-4367	166	33	̸∈	̸∈	PROPN
ejpam-4367	166	34	v	v	NUM
ejpam-4367	166	35	.	.	PUNCT
ejpam-4367	167	1	thus	thus	ADV
ejpam-4367	167	2	,	,	PUNCT
ejpam-4367	167	3	v	v	ADP
ejpam-4367	167	4	∩	∩	NOUN
ejpam-4367	167	5	{	{	PUNCT
ejpam-4367	167	6	x}(λ	x}(λ	PROPN
ejpam-4367	167	7	,	,	PUNCT
ejpam-4367	167	8	sp	sp	NOUN
ejpam-4367	167	9	)	)	PUNCT
ejpam-4367	167	10	=	=	PUNCT
ejpam-4367	167	11	∅.	∅.	NOUN
ejpam-4367	167	12	since	since	SCONJ
ejpam-4367	167	13	y	y	PROPN
ejpam-4367	167	14	∈	∈	PROPN
ejpam-4367	167	15	v	v	NOUN
ejpam-4367	167	16	,	,	PUNCT
ejpam-4367	167	17	y	y	PROPN
ejpam-4367	167	18	̸∈	̸∈	PROPN
ejpam-4367	167	19	{	{	PUNCT
ejpam-4367	167	20	x}(λ	x}(λ	PROPN
ejpam-4367	167	21	,	,	PUNCT
ejpam-4367	167	22	sp	sp	NOUN
ejpam-4367	167	23	)	)	PUNCT
ejpam-4367	167	24	and	and	CCONJ
ejpam-4367	167	25	hence	hence	ADV
ejpam-4367	167	26	{	{	PUNCT
ejpam-4367	167	27	x}(λ	x}(λ	PROPN
ejpam-4367	167	28	,	,	PUNCT
ejpam-4367	167	29	sp	sp	NOUN
ejpam-4367	167	30	)	)	PUNCT
ejpam-4367	167	31	⊆	⊆	NUM
ejpam-4367	167	32	λ(λ	λ(λ	PROPN
ejpam-4367	167	33	,	,	PUNCT
ejpam-4367	167	34	sp)({x	sp)({x	PROPN
ejpam-4367	167	35	}	}	PUNCT
ejpam-4367	167	36	)	)	PUNCT
ejpam-4367	167	37	.	.	PUNCT
ejpam-4367	168	1	moreover	moreover	ADV
ejpam-4367	168	2	,	,	PUNCT
ejpam-4367	168	3	{	{	PUNCT
ejpam-4367	168	4	x}(λ	x}(λ	PROPN
ejpam-4367	168	5	,	,	PUNCT
ejpam-4367	168	6	sp	sp	NOUN
ejpam-4367	168	7	)	)	PUNCT
ejpam-4367	168	8	⊆	⊆	NUM
ejpam-4367	168	9	λ(λ	λ(λ	PROPN
ejpam-4367	168	10	,	,	PUNCT
ejpam-4367	168	11	sp)({x	sp)({x	PROPN
ejpam-4367	168	12	}	}	PUNCT
ejpam-4367	168	13	)	)	PUNCT
ejpam-4367	169	1	⊆	⊆	NUM
ejpam-4367	169	2	λ(λ	λ(λ	ADP
ejpam-4367	169	3	,	,	PUNCT
ejpam-4367	169	4	sp)({x}(λ	sp)({x}(λ	NOUN
ejpam-4367	169	5	,	,	PUNCT
ejpam-4367	169	6	sp	sp	NOUN
ejpam-4367	169	7	)	)	PUNCT
ejpam-4367	169	8	)	)	PUNCT
ejpam-4367	169	9	=	=	PRON
ejpam-4367	169	10	{	{	PUNCT
ejpam-4367	169	11	x}(λ	x}(λ	PROPN
ejpam-4367	169	12	,	,	PUNCT
ejpam-4367	169	13	sp	sp	NOUN
ejpam-4367	169	14	)	)	PUNCT
ejpam-4367	169	15	.	.	PUNCT
ejpam-4367	170	1	this	this	PRON
ejpam-4367	170	2	shows	show	VERB
ejpam-4367	170	3	that	that	SCONJ
ejpam-4367	170	4	{	{	PUNCT
ejpam-4367	170	5	x}(λ	x}(λ	PROPN
ejpam-4367	170	6	,	,	PUNCT
ejpam-4367	170	7	sp	sp	NOUN
ejpam-4367	170	8	)	)	PUNCT
ejpam-4367	170	9	=	=	SYM
ejpam-4367	170	10	λ(λ	λ(λ	PROPN
ejpam-4367	170	11	,	,	PUNCT
ejpam-4367	170	12	sp)({x	sp)({x	PROPN
ejpam-4367	170	13	}	}	PUNCT
ejpam-4367	170	14	)	)	PUNCT
ejpam-4367	170	15	.	.	PUNCT
ejpam-4367	171	1	(	(	PUNCT
ejpam-4367	171	2	4	4	X
ejpam-4367	171	3	)	)	PUNCT
ejpam-4367	171	4	⇒	⇒	NOUN
ejpam-4367	171	5	(	(	PUNCT
ejpam-4367	171	6	5	5	NUM
ejpam-4367	171	7	):	):	PUNCT
ejpam-4367	171	8	the	the	DET
ejpam-4367	171	9	proof	proof	NOUN
ejpam-4367	171	10	is	be	AUX
ejpam-4367	171	11	obvious	obvious	ADJ
ejpam-4367	171	12	.	.	PUNCT
ejpam-4367	172	1	(	(	PUNCT
ejpam-4367	172	2	5	5	X
ejpam-4367	172	3	)	)	PUNCT
ejpam-4367	172	4	⇒	⇒	NOUN
ejpam-4367	172	5	(	(	PUNCT
ejpam-4367	172	6	1	1	NUM
ejpam-4367	172	7	):	):	PUNCT
ejpam-4367	172	8	let	let	VERB
ejpam-4367	172	9	u	u	PRON
ejpam-4367	172	10	∈	∈	PROPN
ejpam-4367	172	11	λspo(x	λspo(x	PROPN
ejpam-4367	172	12	,	,	PUNCT
ejpam-4367	172	13	τ	τ	X
ejpam-4367	172	14	)	)	PUNCT
ejpam-4367	172	15	and	and	CCONJ
ejpam-4367	172	16	let	let	VERB
ejpam-4367	172	17	x	x	PUNCT
ejpam-4367	172	18	∈	∈	PROPN
ejpam-4367	172	19	u	u	NOUN
ejpam-4367	172	20	.	.	PUNCT
ejpam-4367	173	1	if	if	SCONJ
ejpam-4367	173	2	y	y	PROPN
ejpam-4367	173	3	̸∈	̸∈	PROPN
ejpam-4367	173	4	u	u	PROPN
ejpam-4367	173	5	,	,	PUNCT
ejpam-4367	173	6	then	then	ADV
ejpam-4367	173	7	u	u	NOUN
ejpam-4367	173	8	∩	∩	PROPN
ejpam-4367	173	9	{	{	PUNCT
ejpam-4367	173	10	y}(λ	y}(λ	PROPN
ejpam-4367	173	11	,	,	PUNCT
ejpam-4367	173	12	sp	sp	NOUN
ejpam-4367	173	13	)	)	PUNCT
ejpam-4367	173	14	=	=	SYM
ejpam-4367	173	15	∅	∅	NOUN
ejpam-4367	173	16	and	and	CCONJ
ejpam-4367	173	17	x	x	PART
ejpam-4367	173	18	̸∈	̸∈	PROPN
ejpam-4367	173	19	{	{	PUNCT
ejpam-4367	173	20	y}(λ	y}(λ	PROPN
ejpam-4367	173	21	,	,	PUNCT
ejpam-4367	173	22	sp	sp	NOUN
ejpam-4367	173	23	)	)	PUNCT
ejpam-4367	173	24	.	.	PUNCT
ejpam-4367	174	1	by	by	ADP
ejpam-4367	174	2	lemma	lemma	PROPN
ejpam-4367	174	3	4	4	NUM
ejpam-4367	174	4	,	,	PUNCT
ejpam-4367	174	5	y	y	PROPN
ejpam-4367	174	6	̸∈	̸∈	PROPN
ejpam-4367	174	7	λ(λ	λ(λ	PROPN
ejpam-4367	174	8	,	,	PUNCT
ejpam-4367	174	9	sp)({x	sp)({x	PROPN
ejpam-4367	174	10	}	}	PUNCT
ejpam-4367	174	11	)	)	PUNCT
ejpam-4367	174	12	and	and	CCONJ
ejpam-4367	174	13	by	by	ADP
ejpam-4367	174	14	(	(	PUNCT
ejpam-4367	174	15	5	5	NUM
ejpam-4367	174	16	)	)	PUNCT
ejpam-4367	174	17	,	,	PUNCT
ejpam-4367	174	18	y	y	PROPN
ejpam-4367	174	19	̸∈	̸∈	PROPN
ejpam-4367	174	20	{	{	PUNCT
ejpam-4367	174	21	x}(λ	x}(λ	PROPN
ejpam-4367	174	22	,	,	PUNCT
ejpam-4367	174	23	sp	sp	NOUN
ejpam-4367	174	24	)	)	PUNCT
ejpam-4367	174	25	.	.	PUNCT
ejpam-4367	175	1	thus	thus	ADV
ejpam-4367	175	2	,	,	PUNCT
ejpam-4367	175	3	{	{	PUNCT
ejpam-4367	175	4	x}(λ	x}(λ	PROPN
ejpam-4367	175	5	,	,	PUNCT
ejpam-4367	175	6	sp	sp	NOUN
ejpam-4367	175	7	)	)	PUNCT
ejpam-4367	175	8	⊆	⊆	NUM
ejpam-4367	175	9	u	u	NOUN
ejpam-4367	175	10	and	and	CCONJ
ejpam-4367	175	11	hence	hence	ADV
ejpam-4367	175	12	(	(	PUNCT
ejpam-4367	175	13	x	x	X
ejpam-4367	175	14	,	,	PUNCT
ejpam-4367	175	15	τ	τ	X
ejpam-4367	175	16	)	)	PUNCT
ejpam-4367	175	17	is	be	AUX
ejpam-4367	175	18	(	(	PUNCT
ejpam-4367	175	19	λ	λ	PROPN
ejpam-4367	175	20	,	,	PUNCT
ejpam-4367	175	21	sp)-r0	sp)-r0	PROPN
ejpam-4367	175	22	.	.	PUNCT
ejpam-4367	175	23	corollary	corollary	ADJ
ejpam-4367	175	24	2	2	NUM
ejpam-4367	175	25	.	.	PUNCT
ejpam-4367	176	1	a	a	DET
ejpam-4367	176	2	topological	topological	ADJ
ejpam-4367	176	3	space	space	NOUN
ejpam-4367	176	4	(	(	PUNCT
ejpam-4367	176	5	x	x	X
ejpam-4367	176	6	,	,	PUNCT
ejpam-4367	176	7	τ	τ	X
ejpam-4367	176	8	)	)	PUNCT
ejpam-4367	176	9	is	be	AUX
ejpam-4367	176	10	(	(	PUNCT
ejpam-4367	176	11	λ	λ	X
ejpam-4367	176	12	,	,	PUNCT
ejpam-4367	176	13	sp)-r0	sp)-r0	NOUN
ejpam-4367	177	1	if	if	SCONJ
ejpam-4367	177	2	and	and	CCONJ
ejpam-4367	177	3	only	only	ADV
ejpam-4367	177	4	if	if	SCONJ
ejpam-4367	177	5	λ(λ	λ(λ	PROPN
ejpam-4367	177	6	,	,	PUNCT
ejpam-4367	177	7	sp)({x	sp)({x	PROPN
ejpam-4367	177	8	}	}	PUNCT
ejpam-4367	177	9	)	)	PUNCT
ejpam-4367	177	10	⊆	⊆	X
ejpam-4367	177	11	{	{	PUNCT
ejpam-4367	177	12	x}(λ	x}(λ	PROPN
ejpam-4367	177	13	,	,	PUNCT
ejpam-4367	177	14	sp	sp	NOUN
ejpam-4367	177	15	)	)	PUNCT
ejpam-4367	177	16	for	for	ADP
ejpam-4367	177	17	each	each	DET
ejpam-4367	177	18	x	x	SYM
ejpam-4367	177	19	∈	∈	PROPN
ejpam-4367	177	20	x.	x.	NOUN
ejpam-4367	177	21	c.	c.	PROPN
ejpam-4367	177	22	boonpok	boonpok	PROPN
ejpam-4367	177	23	,	,	PUNCT
ejpam-4367	177	24	c.	c.	PROPN
ejpam-4367	177	25	viriyapong	viriyapong	PROPN
ejpam-4367	177	26	/	/	SYM
ejpam-4367	177	27	eur	eur	PROPN
ejpam-4367	177	28	.	.	PUNCT
ejpam-4367	178	1	j.	j.	PROPN
ejpam-4367	178	2	pure	pure	PROPN
ejpam-4367	178	3	appl	appl	PROPN
ejpam-4367	178	4	.	.	PROPN
ejpam-4367	178	5	math	math	PROPN
ejpam-4367	178	6	,	,	PUNCT
ejpam-4367	178	7	15	15	NUM
ejpam-4367	178	8	(	(	PUNCT
ejpam-4367	178	9	3	3	NUM
ejpam-4367	178	10	)	)	PUNCT
ejpam-4367	178	11	(	(	PUNCT
ejpam-4367	178	12	2022	2022	NUM
ejpam-4367	178	13	)	)	PUNCT
ejpam-4367	178	14	,	,	PUNCT
ejpam-4367	178	15	878	878	NUM
ejpam-4367	178	16	-	-	SYM
ejpam-4367	178	17	886	886	NUM
ejpam-4367	178	18	883	883	NUM
ejpam-4367	178	19	proof	proof	NOUN
ejpam-4367	178	20	.	.	PUNCT
ejpam-4367	179	1	this	this	PRON
ejpam-4367	179	2	is	be	AUX
ejpam-4367	179	3	obvious	obvious	ADJ
ejpam-4367	179	4	by	by	ADP
ejpam-4367	179	5	theorem	theorem	NOUN
ejpam-4367	179	6	4	4	NUM
ejpam-4367	179	7	.	.	PUNCT
ejpam-4367	179	8	conversely	conversely	ADV
ejpam-4367	179	9	,	,	PUNCT
ejpam-4367	179	10	let	let	VERB
ejpam-4367	179	11	x	x	X
ejpam-4367	179	12	∈	∈	PROPN
ejpam-4367	179	13	{	{	PUNCT
ejpam-4367	179	14	y}(λ	y}(λ	PROPN
ejpam-4367	179	15	,	,	PUNCT
ejpam-4367	179	16	sp	sp	NOUN
ejpam-4367	179	17	)	)	PUNCT
ejpam-4367	179	18	.	.	PUNCT
ejpam-4367	180	1	thus	thus	ADV
ejpam-4367	180	2	,	,	PUNCT
ejpam-4367	180	3	by	by	ADP
ejpam-4367	180	4	lemma	lemma	PROPN
ejpam-4367	180	5	4	4	NUM
ejpam-4367	180	6	,	,	PUNCT
ejpam-4367	180	7	we	we	PRON
ejpam-4367	180	8	have	have	VERB
ejpam-4367	180	9	y	y	PROPN
ejpam-4367	180	10	∈	∈	PROPN
ejpam-4367	180	11	λ(λ	λ(λ	PROPN
ejpam-4367	180	12	,	,	PUNCT
ejpam-4367	180	13	sp)({x	sp)({x	PROPN
ejpam-4367	180	14	}	}	PUNCT
ejpam-4367	180	15	)	)	PUNCT
ejpam-4367	180	16	and	and	CCONJ
ejpam-4367	180	17	hence	hence	ADV
ejpam-4367	180	18	y	y	PROPN
ejpam-4367	180	19	∈	∈	PROPN
ejpam-4367	180	20	{	{	PUNCT
ejpam-4367	180	21	x}(λ	x}(λ	PROPN
ejpam-4367	180	22	,	,	PUNCT
ejpam-4367	180	23	sp	sp	NOUN
ejpam-4367	180	24	)	)	PUNCT
ejpam-4367	180	25	.	.	PUNCT
ejpam-4367	181	1	similarly	similarly	ADV
ejpam-4367	181	2	,	,	PUNCT
ejpam-4367	181	3	if	if	SCONJ
ejpam-4367	181	4	y	y	PROPN
ejpam-4367	181	5	∈	∈	PROPN
ejpam-4367	181	6	{	{	PUNCT
ejpam-4367	181	7	x}(λ	x}(λ	PROPN
ejpam-4367	181	8	,	,	PUNCT
ejpam-4367	181	9	sp	sp	NOUN
ejpam-4367	181	10	)	)	PUNCT
ejpam-4367	181	11	,	,	PUNCT
ejpam-4367	181	12	then	then	ADV
ejpam-4367	181	13	x	x	SYM
ejpam-4367	181	14	∈	∈	PROPN
ejpam-4367	181	15	{	{	PUNCT
ejpam-4367	181	16	y}(λ	y}(λ	PROPN
ejpam-4367	181	17	,	,	PUNCT
ejpam-4367	181	18	sp	sp	NOUN
ejpam-4367	181	19	)	)	PUNCT
ejpam-4367	181	20	.	.	PUNCT
ejpam-4367	182	1	it	it	PRON
ejpam-4367	182	2	follows	follow	VERB
ejpam-4367	182	3	from	from	ADP
ejpam-4367	182	4	theorem	theorem	ADJ
ejpam-4367	182	5	3	3	NUM
ejpam-4367	182	6	that	that	PRON
ejpam-4367	182	7	(	(	PUNCT
ejpam-4367	182	8	x	x	X
ejpam-4367	182	9	,	,	PUNCT
ejpam-4367	182	10	τ	τ	X
ejpam-4367	182	11	)	)	PUNCT
ejpam-4367	182	12	is	be	AUX
ejpam-4367	182	13	(	(	PUNCT
ejpam-4367	182	14	λ	λ	PROPN
ejpam-4367	182	15	,	,	PUNCT
ejpam-4367	182	16	sp)-r0	sp)-r0	PROPN
ejpam-4367	182	17	.	.	PUNCT
ejpam-4367	183	1	definition	definition	NOUN
ejpam-4367	183	2	3	3	NUM
ejpam-4367	183	3	.	.	PUNCT
ejpam-4367	184	1	[	[	X
ejpam-4367	184	2	3	3	X
ejpam-4367	184	3	]	]	X
ejpam-4367	184	4	let	let	VERB
ejpam-4367	184	5	(	(	PUNCT
ejpam-4367	184	6	x	x	NOUN
ejpam-4367	184	7	,	,	PUNCT
ejpam-4367	184	8	τ	τ	X
ejpam-4367	184	9	)	)	PUNCT
ejpam-4367	184	10	be	be	VERB
ejpam-4367	184	11	a	a	DET
ejpam-4367	184	12	topological	topological	ADJ
ejpam-4367	184	13	space	space	NOUN
ejpam-4367	184	14	and	and	CCONJ
ejpam-4367	184	15	x	x	PUNCT
ejpam-4367	184	16	∈	∈	PROPN
ejpam-4367	184	17	x.	x.	NOUN
ejpam-4367	184	18	a	a	DET
ejpam-4367	184	19	subset	subset	NOUN
ejpam-4367	184	20	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	184	21	is	be	AUX
ejpam-4367	184	22	defined	define	VERB
ejpam-4367	184	23	as	as	SCONJ
ejpam-4367	184	24	follows	follow	VERB
ejpam-4367	184	25	:	:	PUNCT
ejpam-4367	184	26	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	184	27	=	=	SYM
ejpam-4367	184	28	λ(λ	λ(λ	PROPN
ejpam-4367	184	29	,	,	PUNCT
ejpam-4367	184	30	sp)({x	sp)({x	NOUN
ejpam-4367	184	31	}	}	PUNCT
ejpam-4367	184	32	)	)	PUNCT
ejpam-4367	184	33	∩	∩	NOUN
ejpam-4367	184	34	{	{	PUNCT
ejpam-4367	184	35	x}(λ	x}(λ	PROPN
ejpam-4367	184	36	,	,	PUNCT
ejpam-4367	184	37	sp	sp	NOUN
ejpam-4367	184	38	)	)	PUNCT
ejpam-4367	184	39	.	.	PUNCT
ejpam-4367	185	1	corollary	corollary	ADJ
ejpam-4367	185	2	3	3	NUM
ejpam-4367	185	3	.	.	PUNCT
ejpam-4367	186	1	a	a	DET
ejpam-4367	186	2	topological	topological	ADJ
ejpam-4367	186	3	space	space	NOUN
ejpam-4367	186	4	(	(	PUNCT
ejpam-4367	186	5	x	x	X
ejpam-4367	186	6	,	,	PUNCT
ejpam-4367	186	7	τ	τ	X
ejpam-4367	186	8	)	)	PUNCT
ejpam-4367	186	9	is	be	AUX
ejpam-4367	186	10	(	(	PUNCT
ejpam-4367	186	11	λ	λ	X
ejpam-4367	186	12	,	,	PUNCT
ejpam-4367	186	13	sp)-r0	sp)-r0	NOUN
ejpam-4367	187	1	if	if	SCONJ
ejpam-4367	187	2	and	and	CCONJ
ejpam-4367	187	3	only	only	ADV
ejpam-4367	187	4	if	if	SCONJ
ejpam-4367	187	5	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	187	6	=	=	SYM
ejpam-4367	187	7	{	{	PUNCT
ejpam-4367	187	8	x}(λ	x}(λ	PROPN
ejpam-4367	187	9	,	,	PUNCT
ejpam-4367	187	10	sp	sp	NOUN
ejpam-4367	187	11	)	)	PUNCT
ejpam-4367	187	12	for	for	ADP
ejpam-4367	187	13	each	each	DET
ejpam-4367	187	14	x	x	SYM
ejpam-4367	187	15	∈	∈	PROPN
ejpam-4367	187	16	x.	x.	NOUN
ejpam-4367	187	17	proof	proof	NOUN
ejpam-4367	187	18	.	.	PUNCT
ejpam-4367	188	1	let	let	VERB
ejpam-4367	189	1	x	x	SYM
ejpam-4367	189	2	∈	∈	PROPN
ejpam-4367	189	3	x.	x.	NOUN
ejpam-4367	189	4	by	by	ADP
ejpam-4367	189	5	theorem	theorem	NOUN
ejpam-4367	189	6	4	4	NUM
ejpam-4367	189	7	,	,	PUNCT
ejpam-4367	189	8	λ(λ	λ(λ	ADV
ejpam-4367	189	9	,	,	PUNCT
ejpam-4367	189	10	sp)({x	sp)({x	NOUN
ejpam-4367	189	11	}	}	PUNCT
ejpam-4367	189	12	)	)	PUNCT
ejpam-4367	190	1	=	=	PRON
ejpam-4367	190	2	{	{	PUNCT
ejpam-4367	190	3	x}(λ	x}(λ	PROPN
ejpam-4367	190	4	,	,	PUNCT
ejpam-4367	190	5	sp	sp	NOUN
ejpam-4367	190	6	)	)	PUNCT
ejpam-4367	190	7	.	.	PUNCT
ejpam-4367	191	1	thus	thus	ADV
ejpam-4367	191	2	,	,	PUNCT
ejpam-4367	191	3	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	191	4	=	=	SYM
ejpam-4367	191	5	λ(λ	λ(λ	PROPN
ejpam-4367	191	6	,	,	PUNCT
ejpam-4367	191	7	sp)({x	sp)({x	NOUN
ejpam-4367	191	8	}	}	PUNCT
ejpam-4367	191	9	)	)	PUNCT
ejpam-4367	191	10	∩	∩	NOUN
ejpam-4367	191	11	{	{	PUNCT
ejpam-4367	191	12	x}(λ	x}(λ	PROPN
ejpam-4367	191	13	,	,	PUNCT
ejpam-4367	191	14	sp	sp	NOUN
ejpam-4367	191	15	)	)	PUNCT
ejpam-4367	191	16	=	=	SYM
ejpam-4367	191	17	{	{	PUNCT
ejpam-4367	191	18	x}(λ	x}(λ	PROPN
ejpam-4367	191	19	,	,	PUNCT
ejpam-4367	191	20	sp	sp	NOUN
ejpam-4367	191	21	)	)	PUNCT
ejpam-4367	191	22	.	.	PUNCT
ejpam-4367	192	1	conversely	conversely	ADV
ejpam-4367	192	2	,	,	PUNCT
ejpam-4367	192	3	let	let	VERB
ejpam-4367	192	4	x	x	X
ejpam-4367	192	5	∈	∈	PROPN
ejpam-4367	192	6	x.	x.	NOUN
ejpam-4367	192	7	by	by	ADP
ejpam-4367	192	8	the	the	DET
ejpam-4367	192	9	hypothesis	hypothesis	NOUN
ejpam-4367	192	10	,	,	PUNCT
ejpam-4367	192	11	{	{	PUNCT
ejpam-4367	192	12	x}(λ	x}(λ	PROPN
ejpam-4367	192	13	,	,	PUNCT
ejpam-4367	192	14	sp	sp	NOUN
ejpam-4367	192	15	)	)	PUNCT
ejpam-4367	192	16	=	=	SYM
ejpam-4367	192	17	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	192	18	=	=	SYM
ejpam-4367	192	19	λ(λ	λ(λ	PROPN
ejpam-4367	192	20	,	,	PUNCT
ejpam-4367	192	21	sp)({x	sp)({x	NOUN
ejpam-4367	192	22	}	}	PUNCT
ejpam-4367	192	23	)	)	PUNCT
ejpam-4367	192	24	∩	∩	NOUN
ejpam-4367	192	25	{	{	PUNCT
ejpam-4367	192	26	x}(λ	x}(λ	PROPN
ejpam-4367	192	27	,	,	PUNCT
ejpam-4367	192	28	sp	sp	NOUN
ejpam-4367	192	29	)	)	PUNCT
ejpam-4367	192	30	⊆	⊆	NUM
ejpam-4367	192	31	λ(λ	λ(λ	PROPN
ejpam-4367	192	32	,	,	PUNCT
ejpam-4367	192	33	sp)({x	sp)({x	PROPN
ejpam-4367	192	34	}	}	PUNCT
ejpam-4367	192	35	)	)	PUNCT
ejpam-4367	192	36	.	.	PUNCT
ejpam-4367	193	1	it	it	PRON
ejpam-4367	193	2	follows	follow	VERB
ejpam-4367	193	3	from	from	ADP
ejpam-4367	193	4	theorem	theorem	ADJ
ejpam-4367	193	5	4	4	NUM
ejpam-4367	193	6	that	that	PRON
ejpam-4367	193	7	(	(	PUNCT
ejpam-4367	193	8	x	x	X
ejpam-4367	193	9	,	,	PUNCT
ejpam-4367	193	10	τ	τ	X
ejpam-4367	193	11	)	)	PUNCT
ejpam-4367	193	12	is	be	AUX
ejpam-4367	193	13	(	(	PUNCT
ejpam-4367	193	14	λ	λ	PROPN
ejpam-4367	193	15	,	,	PUNCT
ejpam-4367	193	16	sp)-r0	sp)-r0	PROPN
ejpam-4367	193	17	.	.	PROPN
ejpam-4367	193	18	4	4	X
ejpam-4367	193	19	.	.	PUNCT
ejpam-4367	193	20	characterizations	characterization	NOUN
ejpam-4367	193	21	of	of	ADP
ejpam-4367	193	22	(	(	PUNCT
ejpam-4367	193	23	λ	λ	PROPN
ejpam-4367	193	24	,	,	PUNCT
ejpam-4367	193	25	sp)-r1	sp)-r1	NOUN
ejpam-4367	193	26	topological	topological	ADJ
ejpam-4367	193	27	spaces	space	NOUN
ejpam-4367	193	28	we	we	PRON
ejpam-4367	193	29	begin	begin	VERB
ejpam-4367	193	30	this	this	DET
ejpam-4367	193	31	section	section	NOUN
ejpam-4367	193	32	by	by	ADP
ejpam-4367	193	33	introducing	introduce	VERB
ejpam-4367	193	34	the	the	DET
ejpam-4367	193	35	notion	notion	NOUN
ejpam-4367	193	36	of	of	ADP
ejpam-4367	193	37	(	(	PUNCT
ejpam-4367	193	38	λ	λ	PROPN
ejpam-4367	193	39	,	,	PUNCT
ejpam-4367	193	40	sp)-r1	sp)-r1	NOUN
ejpam-4367	193	41	topological	topological	ADJ
ejpam-4367	193	42	spaces	space	NOUN
ejpam-4367	193	43	.	.	PUNCT
ejpam-4367	194	1	definition	definition	NOUN
ejpam-4367	194	2	4	4	NUM
ejpam-4367	194	3	.	.	PUNCT
ejpam-4367	195	1	a	a	DET
ejpam-4367	195	2	topological	topological	ADJ
ejpam-4367	195	3	space	space	NOUN
ejpam-4367	195	4	(	(	PUNCT
ejpam-4367	195	5	x	x	X
ejpam-4367	195	6	,	,	PUNCT
ejpam-4367	195	7	τ	τ	X
ejpam-4367	195	8	)	)	PUNCT
ejpam-4367	195	9	is	be	AUX
ejpam-4367	195	10	said	say	VERB
ejpam-4367	195	11	to	to	PART
ejpam-4367	195	12	be	be	AUX
ejpam-4367	195	13	(	(	PUNCT
ejpam-4367	195	14	λ	λ	X
ejpam-4367	195	15	,	,	PUNCT
ejpam-4367	195	16	sp)-r1	sp)-r1	NOUN
ejpam-4367	195	17	if	if	SCONJ
ejpam-4367	195	18	,	,	PUNCT
ejpam-4367	195	19	for	for	ADP
ejpam-4367	195	20	each	each	DET
ejpam-4367	195	21	points	point	NOUN
ejpam-4367	195	22	x	x	PRON
ejpam-4367	195	23	,	,	PUNCT
ejpam-4367	195	24	y	y	PROPN
ejpam-4367	195	25	in	in	ADP
ejpam-4367	195	26	x	x	PUNCT
ejpam-4367	195	27	with	with	ADP
ejpam-4367	195	28	{	{	PUNCT
ejpam-4367	195	29	x}(λ	x}(λ	PROPN
ejpam-4367	195	30	,	,	PUNCT
ejpam-4367	195	31	sp	sp	NOUN
ejpam-4367	195	32	)	)	PUNCT
ejpam-4367	195	33	̸=	̸=	PROPN
ejpam-4367	195	34	{	{	PUNCT
ejpam-4367	195	35	y}(λ	y}(λ	PROPN
ejpam-4367	195	36	,	,	PUNCT
ejpam-4367	195	37	sp	sp	NOUN
ejpam-4367	195	38	)	)	PUNCT
ejpam-4367	195	39	,	,	PUNCT
ejpam-4367	195	40	there	there	PRON
ejpam-4367	195	41	exist	exist	VERB
ejpam-4367	195	42	disjoint	disjoint	NOUN
ejpam-4367	195	43	(	(	PUNCT
ejpam-4367	195	44	λ	λ	NOUN
ejpam-4367	195	45	,	,	PUNCT
ejpam-4367	195	46	sp)-open	sp)-open	NOUN
ejpam-4367	195	47	sets	set	VERB
ejpam-4367	195	48	u	u	NOUN
ejpam-4367	195	49	and	and	CCONJ
ejpam-4367	195	50	v	v	ADP
ejpam-4367	195	51	such	such	ADJ
ejpam-4367	195	52	that	that	SCONJ
ejpam-4367	195	53	{	{	PUNCT
ejpam-4367	195	54	x}(λ	x}(λ	PROPN
ejpam-4367	195	55	,	,	PUNCT
ejpam-4367	195	56	sp	sp	NOUN
ejpam-4367	195	57	)	)	PUNCT
ejpam-4367	195	58	⊆	⊆	NUM
ejpam-4367	195	59	u	u	NOUN
ejpam-4367	195	60	and	and	CCONJ
ejpam-4367	195	61	{	{	PUNCT
ejpam-4367	195	62	y}(λ	y}(λ	PROPN
ejpam-4367	195	63	,	,	PUNCT
ejpam-4367	195	64	sp	sp	NOUN
ejpam-4367	195	65	)	)	PUNCT
ejpam-4367	195	66	⊆	⊆	NUM
ejpam-4367	195	67	v	v	NOUN
ejpam-4367	195	68	.	.	PUNCT
ejpam-4367	196	1	theorem	theorem	ADJ
ejpam-4367	196	2	5	5	NUM
ejpam-4367	196	3	.	.	PUNCT
ejpam-4367	197	1	a	a	DET
ejpam-4367	197	2	topological	topological	ADJ
ejpam-4367	197	3	space	space	NOUN
ejpam-4367	197	4	(	(	PUNCT
ejpam-4367	197	5	x	x	X
ejpam-4367	197	6	,	,	PUNCT
ejpam-4367	197	7	τ	τ	X
ejpam-4367	197	8	)	)	PUNCT
ejpam-4367	197	9	is	be	AUX
ejpam-4367	197	10	(	(	PUNCT
ejpam-4367	197	11	λ	λ	X
ejpam-4367	197	12	,	,	PUNCT
ejpam-4367	197	13	sp)-r1	sp)-r1	NOUN
ejpam-4367	197	14	if	if	SCONJ
ejpam-4367	197	15	and	and	CCONJ
ejpam-4367	197	16	only	only	ADV
ejpam-4367	197	17	if	if	SCONJ
ejpam-4367	197	18	,	,	PUNCT
ejpam-4367	197	19	for	for	ADP
ejpam-4367	197	20	any	any	DET
ejpam-4367	197	21	points	point	NOUN
ejpam-4367	197	22	x	x	X
ejpam-4367	197	23	,	,	PUNCT
ejpam-4367	197	24	y	y	PROPN
ejpam-4367	197	25	in	in	ADP
ejpam-4367	197	26	x	x	PUNCT
ejpam-4367	197	27	with	with	ADP
ejpam-4367	197	28	{	{	PUNCT
ejpam-4367	197	29	x}(λ	x}(λ	PROPN
ejpam-4367	197	30	,	,	PUNCT
ejpam-4367	197	31	sp	sp	NOUN
ejpam-4367	197	32	)	)	PUNCT
ejpam-4367	197	33	̸=	̸=	PROPN
ejpam-4367	197	34	{	{	PUNCT
ejpam-4367	197	35	y}(λ	y}(λ	PROPN
ejpam-4367	197	36	,	,	PUNCT
ejpam-4367	197	37	sp	sp	NOUN
ejpam-4367	197	38	)	)	PUNCT
ejpam-4367	197	39	,	,	PUNCT
ejpam-4367	197	40	there	there	PRON
ejpam-4367	197	41	exist	exist	VERB
ejpam-4367	197	42	(	(	PUNCT
ejpam-4367	197	43	λ	λ	X
ejpam-4367	197	44	,	,	PUNCT
ejpam-4367	197	45	sp)-closed	sp)-close	VERB
ejpam-4367	197	46	sets	set	VERB
ejpam-4367	197	47	f	f	PROPN
ejpam-4367	197	48	and	and	CCONJ
ejpam-4367	197	49	k	k	PROPN
ejpam-4367	198	1	such	such	ADJ
ejpam-4367	198	2	that	that	SCONJ
ejpam-4367	198	3	x	x	SYM
ejpam-4367	198	4	∈	∈	PROPN
ejpam-4367	198	5	f	f	PROPN
ejpam-4367	198	6	,	,	PUNCT
ejpam-4367	198	7	y	y	PROPN
ejpam-4367	198	8	̸∈	̸∈	PROPN
ejpam-4367	198	9	f	f	PROPN
ejpam-4367	198	10	,	,	PUNCT
ejpam-4367	198	11	y	y	PROPN
ejpam-4367	198	12	∈	∈	PROPN
ejpam-4367	198	13	k	k	PROPN
ejpam-4367	198	14	,	,	PUNCT
ejpam-4367	198	15	x	x	PROPN
ejpam-4367	198	16	̸∈	̸∈	PROPN
ejpam-4367	198	17	k	k	PROPN
ejpam-4367	198	18	and	and	CCONJ
ejpam-4367	198	19	x	x	X
ejpam-4367	198	20	=	=	SYM
ejpam-4367	198	21	f	f	PROPN
ejpam-4367	198	22	∪k	∪k	PROPN
ejpam-4367	198	23	.	.	PUNCT
ejpam-4367	198	24	proof	proof	NOUN
ejpam-4367	198	25	.	.	PUNCT
ejpam-4367	199	1	let	let	VERB
ejpam-4367	199	2	x	x	PRON
ejpam-4367	199	3	and	and	CCONJ
ejpam-4367	199	4	y	y	PROPN
ejpam-4367	199	5	be	be	AUX
ejpam-4367	199	6	any	any	DET
ejpam-4367	199	7	points	point	NOUN
ejpam-4367	199	8	in	in	ADP
ejpam-4367	199	9	x	x	PUNCT
ejpam-4367	199	10	with	with	ADP
ejpam-4367	199	11	{	{	PUNCT
ejpam-4367	199	12	x}(λ	x}(λ	PROPN
ejpam-4367	199	13	,	,	PUNCT
ejpam-4367	199	14	sp	sp	NOUN
ejpam-4367	199	15	)	)	PUNCT
ejpam-4367	199	16	̸=	̸=	PROPN
ejpam-4367	199	17	{	{	PUNCT
ejpam-4367	199	18	y}(λ	y}(λ	PROPN
ejpam-4367	199	19	,	,	PUNCT
ejpam-4367	199	20	sp	sp	NOUN
ejpam-4367	199	21	)	)	PUNCT
ejpam-4367	199	22	.	.	PUNCT
ejpam-4367	200	1	then	then	ADV
ejpam-4367	200	2	,	,	PUNCT
ejpam-4367	200	3	there	there	PRON
ejpam-4367	200	4	exist	exist	VERB
ejpam-4367	200	5	disjoint	disjoint	NOUN
ejpam-4367	200	6	u	u	NOUN
ejpam-4367	200	7	,	,	PUNCT
ejpam-4367	200	8	v	v	NOUN
ejpam-4367	200	9	∈	∈	PROPN
ejpam-4367	200	10	λspo(x	λspo(x	NOUN
ejpam-4367	200	11	,	,	PUNCT
ejpam-4367	200	12	τ	τ	PROPN
ejpam-4367	200	13	)	)	PUNCT
ejpam-4367	200	14	such	such	ADJ
ejpam-4367	200	15	that	that	SCONJ
ejpam-4367	200	16	{	{	PUNCT
ejpam-4367	200	17	x}(λ	x}(λ	PROPN
ejpam-4367	200	18	,	,	PUNCT
ejpam-4367	200	19	sp	sp	NOUN
ejpam-4367	200	20	)	)	PUNCT
ejpam-4367	200	21	⊆	⊆	NUM
ejpam-4367	200	22	u	u	NOUN
ejpam-4367	200	23	and	and	CCONJ
ejpam-4367	200	24	{	{	PUNCT
ejpam-4367	200	25	y}(λ	y}(λ	PROPN
ejpam-4367	200	26	,	,	PUNCT
ejpam-4367	200	27	sp	sp	NOUN
ejpam-4367	200	28	)	)	PUNCT
ejpam-4367	200	29	⊆	⊆	NUM
ejpam-4367	200	30	v	v	NOUN
ejpam-4367	200	31	.	.	PUNCT
ejpam-4367	201	1	now	now	ADV
ejpam-4367	201	2	,	,	PUNCT
ejpam-4367	201	3	put	put	VERB
ejpam-4367	201	4	f	f	PROPN
ejpam-4367	201	5	=	=	SYM
ejpam-4367	201	6	x−v	x−v	PROPN
ejpam-4367	201	7	and	and	CCONJ
ejpam-4367	201	8	k	k	X
ejpam-4367	202	1	=	=	PUNCT
ejpam-4367	202	2	x−u	x−u	PROPN
ejpam-4367	202	3	.	.	PUNCT
ejpam-4367	203	1	then	then	ADV
ejpam-4367	203	2	,	,	PUNCT
ejpam-4367	203	3	f	f	PROPN
ejpam-4367	203	4	and	and	CCONJ
ejpam-4367	203	5	k	k	PROPN
ejpam-4367	203	6	are	be	AUX
ejpam-4367	203	7	(	(	PUNCT
ejpam-4367	203	8	λ	λ	X
ejpam-4367	203	9	,	,	PUNCT
ejpam-4367	203	10	sp)-closed	sp)-close	VERB
ejpam-4367	203	11	sets	set	NOUN
ejpam-4367	203	12	of	of	ADP
ejpam-4367	203	13	x	x	SYM
ejpam-4367	203	14	such	such	ADJ
ejpam-4367	203	15	that	that	SCONJ
ejpam-4367	203	16	x	x	SYM
ejpam-4367	203	17	∈	∈	PROPN
ejpam-4367	203	18	f	f	PROPN
ejpam-4367	203	19	,	,	PUNCT
ejpam-4367	203	20	y	y	PROPN
ejpam-4367	203	21	̸∈	̸∈	PROPN
ejpam-4367	203	22	f	f	PROPN
ejpam-4367	203	23	,	,	PUNCT
ejpam-4367	203	24	y	y	PROPN
ejpam-4367	203	25	∈	∈	PROPN
ejpam-4367	203	26	k	k	PROPN
ejpam-4367	203	27	,	,	PUNCT
ejpam-4367	203	28	x	x	PROPN
ejpam-4367	203	29	̸∈	̸∈	PROPN
ejpam-4367	203	30	k	k	PROPN
ejpam-4367	203	31	and	and	CCONJ
ejpam-4367	203	32	x	x	X
ejpam-4367	203	33	=	=	SYM
ejpam-4367	203	34	f	f	PROPN
ejpam-4367	203	35	∪k	∪k	PROPN
ejpam-4367	203	36	.	.	PUNCT
ejpam-4367	204	1	conversely	conversely	ADV
ejpam-4367	204	2	,	,	PUNCT
ejpam-4367	204	3	let	let	VERB
ejpam-4367	204	4	x	x	PRON
ejpam-4367	204	5	and	and	CCONJ
ejpam-4367	204	6	y	y	PROPN
ejpam-4367	204	7	be	be	AUX
ejpam-4367	204	8	any	any	DET
ejpam-4367	204	9	points	point	NOUN
ejpam-4367	204	10	in	in	ADP
ejpam-4367	204	11	x	x	INTJ
ejpam-4367	204	12	such	such	ADJ
ejpam-4367	204	13	that	that	SCONJ
ejpam-4367	204	14	{	{	PUNCT
ejpam-4367	204	15	x}(λ	x}(λ	PROPN
ejpam-4367	204	16	,	,	PUNCT
ejpam-4367	204	17	sp	sp	NOUN
ejpam-4367	204	18	)	)	PUNCT
ejpam-4367	204	19	̸=	̸=	PROPN
ejpam-4367	204	20	{	{	PUNCT
ejpam-4367	204	21	y}(λ	y}(λ	PROPN
ejpam-4367	204	22	,	,	PUNCT
ejpam-4367	204	23	sp	sp	NOUN
ejpam-4367	204	24	)	)	PUNCT
ejpam-4367	204	25	.	.	PUNCT
ejpam-4367	205	1	then	then	ADV
ejpam-4367	205	2	,	,	PUNCT
ejpam-4367	205	3	{	{	PUNCT
ejpam-4367	205	4	x}(λ	x}(λ	PROPN
ejpam-4367	205	5	,	,	PUNCT
ejpam-4367	205	6	sp	sp	NOUN
ejpam-4367	205	7	)	)	PUNCT
ejpam-4367	205	8	∩	∩	NOUN
ejpam-4367	205	9	{	{	PUNCT
ejpam-4367	205	10	y}(λ	y}(λ	PROPN
ejpam-4367	205	11	,	,	PUNCT
ejpam-4367	205	12	sp	sp	NOUN
ejpam-4367	205	13	)	)	PUNCT
ejpam-4367	205	14	=	=	PUNCT
ejpam-4367	205	15	∅.	∅.	NOUN
ejpam-4367	205	16	in	in	ADP
ejpam-4367	205	17	fact	fact	NOUN
ejpam-4367	205	18	,	,	PUNCT
ejpam-4367	205	19	if	if	SCONJ
ejpam-4367	205	20	z	z	PROPN
ejpam-4367	205	21	∈	∈	PROPN
ejpam-4367	205	22	{	{	PUNCT
ejpam-4367	205	23	x}(λ	x}(λ	PROPN
ejpam-4367	205	24	,	,	PUNCT
ejpam-4367	205	25	sp	sp	NOUN
ejpam-4367	205	26	)	)	PUNCT
ejpam-4367	205	27	∩	∩	NOUN
ejpam-4367	205	28	{	{	PUNCT
ejpam-4367	205	29	y}(λ	y}(λ	PROPN
ejpam-4367	205	30	,	,	PUNCT
ejpam-4367	205	31	sp	sp	NOUN
ejpam-4367	205	32	)	)	PUNCT
ejpam-4367	205	33	,	,	PUNCT
ejpam-4367	205	34	then	then	ADV
ejpam-4367	205	35	{	{	PUNCT
ejpam-4367	205	36	z}(λ	z}(λ	PROPN
ejpam-4367	205	37	,	,	PUNCT
ejpam-4367	205	38	sp	sp	NOUN
ejpam-4367	205	39	)	)	PUNCT
ejpam-4367	205	40	̸=	̸=	PROPN
ejpam-4367	205	41	{	{	PUNCT
ejpam-4367	205	42	x}(λ	x}(λ	PROPN
ejpam-4367	205	43	,	,	PUNCT
ejpam-4367	205	44	sp	sp	NOUN
ejpam-4367	205	45	)	)	PUNCT
ejpam-4367	205	46	or	or	CCONJ
ejpam-4367	205	47	{	{	PUNCT
ejpam-4367	205	48	z}(λ	z}(λ	PROPN
ejpam-4367	205	49	,	,	PUNCT
ejpam-4367	205	50	sp	sp	NOUN
ejpam-4367	205	51	)	)	PUNCT
ejpam-4367	205	52	̸=	̸=	PROPN
ejpam-4367	205	53	{	{	PUNCT
ejpam-4367	205	54	y}(λ	y}(λ	PROPN
ejpam-4367	205	55	,	,	PUNCT
ejpam-4367	205	56	sp	sp	NOUN
ejpam-4367	205	57	)	)	PUNCT
ejpam-4367	205	58	.	.	PUNCT
ejpam-4367	206	1	in	in	ADP
ejpam-4367	206	2	case	case	NOUN
ejpam-4367	206	3	{	{	PUNCT
ejpam-4367	206	4	z}(λ	z}(λ	PROPN
ejpam-4367	206	5	,	,	PUNCT
ejpam-4367	206	6	sp	sp	NOUN
ejpam-4367	206	7	)	)	PUNCT
ejpam-4367	206	8	̸=	̸=	PROPN
ejpam-4367	206	9	{	{	PUNCT
ejpam-4367	206	10	x}(λ	x}(λ	PROPN
ejpam-4367	206	11	,	,	PUNCT
ejpam-4367	206	12	sp	sp	NOUN
ejpam-4367	206	13	)	)	PUNCT
ejpam-4367	206	14	,	,	PUNCT
ejpam-4367	206	15	by	by	ADP
ejpam-4367	206	16	the	the	DET
ejpam-4367	206	17	hypothesis	hypothesis	NOUN
ejpam-4367	206	18	,	,	PUNCT
ejpam-4367	206	19	there	there	PRON
ejpam-4367	206	20	exists	exist	VERB
ejpam-4367	206	21	a	a	DET
ejpam-4367	206	22	(	(	PUNCT
ejpam-4367	206	23	λ	λ	PROPN
ejpam-4367	206	24	,	,	PUNCT
ejpam-4367	206	25	sp)-closed	sp)-close	VERB
ejpam-4367	206	26	set	set	VERB
ejpam-4367	206	27	f	f	PROPN
ejpam-4367	206	28	such	such	ADJ
ejpam-4367	206	29	that	that	SCONJ
ejpam-4367	206	30	x	x	SYM
ejpam-4367	206	31	∈	∈	PROPN
ejpam-4367	206	32	f	f	PROPN
ejpam-4367	206	33	and	and	CCONJ
ejpam-4367	206	34	z	z	PROPN
ejpam-4367	206	35	̸∈	̸∈	PROPN
ejpam-4367	206	36	f	f	PROPN
ejpam-4367	206	37	.	.	PUNCT
ejpam-4367	207	1	then	then	ADV
ejpam-4367	207	2	,	,	PUNCT
ejpam-4367	207	3	z	z	PROPN
ejpam-4367	207	4	∈	∈	PROPN
ejpam-4367	207	5	{	{	PUNCT
ejpam-4367	207	6	x}(λ	x}(λ	PROPN
ejpam-4367	207	7	,	,	PUNCT
ejpam-4367	207	8	sp	sp	NOUN
ejpam-4367	207	9	)	)	PUNCT
ejpam-4367	207	10	⊆	⊆	NUM
ejpam-4367	207	11	f	f	NOUN
ejpam-4367	207	12	.	.	PUNCT
ejpam-4367	208	1	this	this	PRON
ejpam-4367	208	2	contradicts	contradict	VERB
ejpam-4367	208	3	that	that	SCONJ
ejpam-4367	208	4	z	z	PROPN
ejpam-4367	208	5	̸∈	̸∈	PROPN
ejpam-4367	208	6	f	f	PROPN
ejpam-4367	208	7	.	.	PUNCT
ejpam-4367	209	1	in	in	ADP
ejpam-4367	209	2	case	case	NOUN
ejpam-4367	209	3	{	{	PUNCT
ejpam-4367	209	4	z}(λ	z}(λ	PROPN
ejpam-4367	209	5	,	,	PUNCT
ejpam-4367	209	6	sp	sp	NOUN
ejpam-4367	209	7	)	)	PUNCT
ejpam-4367	209	8	̸=	̸=	PROPN
ejpam-4367	209	9	{	{	PUNCT
ejpam-4367	209	10	y}(λ	y}(λ	PROPN
ejpam-4367	209	11	,	,	PUNCT
ejpam-4367	209	12	sp	sp	NOUN
ejpam-4367	209	13	)	)	PUNCT
ejpam-4367	209	14	,	,	PUNCT
ejpam-4367	209	15	similarly	similarly	ADV
ejpam-4367	209	16	,	,	PUNCT
ejpam-4367	209	17	this	this	PRON
ejpam-4367	209	18	leads	lead	VERB
ejpam-4367	209	19	to	to	ADP
ejpam-4367	209	20	the	the	DET
ejpam-4367	209	21	contradiction	contradiction	NOUN
ejpam-4367	209	22	.	.	PUNCT
ejpam-4367	210	1	thus	thus	ADV
ejpam-4367	210	2	,	,	PUNCT
ejpam-4367	210	3	{	{	PUNCT
ejpam-4367	210	4	x}(λ	x}(λ	PROPN
ejpam-4367	210	5	,	,	PUNCT
ejpam-4367	210	6	sp	sp	NOUN
ejpam-4367	210	7	)	)	PUNCT
ejpam-4367	210	8	∩	∩	NOUN
ejpam-4367	210	9	{	{	PUNCT
ejpam-4367	210	10	y}(λ	y}(λ	PROPN
ejpam-4367	210	11	,	,	PUNCT
ejpam-4367	210	12	sp	sp	NOUN
ejpam-4367	210	13	)	)	PUNCT
ejpam-4367	210	14	=	=	NOUN
ejpam-4367	210	15	∅	∅	NOUN
ejpam-4367	210	16	,	,	PUNCT
ejpam-4367	210	17	by	by	ADP
ejpam-4367	210	18	corollary	corollary	ADJ
ejpam-4367	210	19	1	1	NUM
ejpam-4367	210	20	,	,	PUNCT
ejpam-4367	210	21	(	(	PUNCT
ejpam-4367	210	22	x	x	X
ejpam-4367	210	23	,	,	PUNCT
ejpam-4367	210	24	τ	τ	X
ejpam-4367	210	25	)	)	PUNCT
ejpam-4367	210	26	is	be	AUX
ejpam-4367	210	27	(	(	PUNCT
ejpam-4367	210	28	λ	λ	PROPN
ejpam-4367	210	29	,	,	PUNCT
ejpam-4367	210	30	sp)-r0	sp)-r0	PROPN
ejpam-4367	210	31	.	.	PUNCT
ejpam-4367	211	1	by	by	ADP
ejpam-4367	211	2	the	the	DET
ejpam-4367	211	3	hypothesis	hypothesis	NOUN
ejpam-4367	211	4	,	,	PUNCT
ejpam-4367	211	5	there	there	PRON
ejpam-4367	211	6	exist	exist	VERB
ejpam-4367	211	7	(	(	PUNCT
ejpam-4367	211	8	λ	λ	X
ejpam-4367	211	9	,	,	PUNCT
ejpam-4367	211	10	sp)-closed	sp)-close	VERB
ejpam-4367	211	11	sets	set	VERB
ejpam-4367	211	12	f	f	PROPN
ejpam-4367	211	13	and	and	CCONJ
ejpam-4367	211	14	k	k	PROPN
ejpam-4367	211	15	such	such	ADJ
ejpam-4367	211	16	that	that	SCONJ
ejpam-4367	211	17	x	x	SYM
ejpam-4367	211	18	∈	∈	PROPN
ejpam-4367	211	19	f	f	PROPN
ejpam-4367	211	20	,	,	PUNCT
ejpam-4367	211	21	y	y	PROPN
ejpam-4367	211	22	̸∈	̸∈	PROPN
ejpam-4367	211	23	f	f	PROPN
ejpam-4367	211	24	,	,	PUNCT
ejpam-4367	211	25	y	y	PROPN
ejpam-4367	211	26	∈	∈	PROPN
ejpam-4367	211	27	k	k	PROPN
ejpam-4367	211	28	,	,	PUNCT
ejpam-4367	211	29	x	x	PROPN
ejpam-4367	211	30	̸∈	̸∈	PROPN
ejpam-4367	211	31	k	k	PROPN
ejpam-4367	211	32	and	and	CCONJ
ejpam-4367	211	33	x	x	X
ejpam-4367	211	34	=	=	SYM
ejpam-4367	211	35	f	f	PROPN
ejpam-4367	211	36	∪k	∪k	PROPN
ejpam-4367	211	37	.	.	PUNCT
ejpam-4367	212	1	put	put	VERB
ejpam-4367	212	2	u	u	NOUN
ejpam-4367	212	3	=	=	NOUN
ejpam-4367	212	4	x	x	SYM
ejpam-4367	212	5	−k	−k	ADJ
ejpam-4367	212	6	and	and	CCONJ
ejpam-4367	212	7	v	v	NOUN
ejpam-4367	212	8	=	=	NOUN
ejpam-4367	212	9	x	x	X
ejpam-4367	212	10	−	−	PROPN
ejpam-4367	212	11	f	f	X
ejpam-4367	212	12	.	.	PUNCT
ejpam-4367	213	1	then	then	ADV
ejpam-4367	213	2	,	,	PUNCT
ejpam-4367	213	3	x	x	PUNCT
ejpam-4367	213	4	∈	∈	PROPN
ejpam-4367	213	5	u	u	NOUN
ejpam-4367	213	6	∈	∈	PROPN
ejpam-4367	213	7	λspo(x	λspo(x	PROPN
ejpam-4367	213	8	,	,	PUNCT
ejpam-4367	213	9	τ	τ	PROPN
ejpam-4367	213	10	)	)	PUNCT
ejpam-4367	213	11	and	and	CCONJ
ejpam-4367	213	12	y	y	PROPN
ejpam-4367	213	13	∈	∈	PROPN
ejpam-4367	213	14	v	v	ADP
ejpam-4367	213	15	∈	∈	PROPN
ejpam-4367	213	16	λspo(x	λspo(x	NOUN
ejpam-4367	213	17	,	,	PUNCT
ejpam-4367	213	18	τ	τ	PROPN
ejpam-4367	213	19	)	)	PUNCT
ejpam-4367	213	20	.	.	PUNCT
ejpam-4367	214	1	since	since	SCONJ
ejpam-4367	214	2	(	(	PUNCT
ejpam-4367	214	3	x	x	X
ejpam-4367	214	4	,	,	PUNCT
ejpam-4367	214	5	τ	τ	X
ejpam-4367	214	6	)	)	PUNCT
ejpam-4367	214	7	is	be	AUX
ejpam-4367	214	8	(	(	PUNCT
ejpam-4367	214	9	λ	λ	PROPN
ejpam-4367	214	10	,	,	PUNCT
ejpam-4367	214	11	sp)-r0	sp)-r0	PROPN
ejpam-4367	214	12	,	,	PUNCT
ejpam-4367	214	13	we	we	PRON
ejpam-4367	214	14	have	have	VERB
ejpam-4367	214	15	{	{	PUNCT
ejpam-4367	214	16	x}(λ	x}(λ	PROPN
ejpam-4367	214	17	,	,	PUNCT
ejpam-4367	214	18	sp	sp	NOUN
ejpam-4367	214	19	)	)	PUNCT
ejpam-4367	214	20	⊆	⊆	NUM
ejpam-4367	214	21	u	u	NOUN
ejpam-4367	214	22	,	,	PUNCT
ejpam-4367	214	23	{	{	PUNCT
ejpam-4367	214	24	y}(λ	y}(λ	PROPN
ejpam-4367	214	25	,	,	PUNCT
ejpam-4367	214	26	sp	sp	NOUN
ejpam-4367	214	27	)	)	PUNCT
ejpam-4367	214	28	⊆	⊆	NUM
ejpam-4367	214	29	v	v	NOUN
ejpam-4367	214	30	and	and	CCONJ
ejpam-4367	214	31	also	also	ADV
ejpam-4367	214	32	u	u	NOUN
ejpam-4367	214	33	∩	∩	NOUN
ejpam-4367	214	34	v	v	NOUN
ejpam-4367	214	35	=	=	PUNCT
ejpam-4367	214	36	∅.	∅.	ADP
ejpam-4367	215	1	this	this	PRON
ejpam-4367	215	2	shows	show	VERB
ejpam-4367	215	3	that	that	SCONJ
ejpam-4367	215	4	(	(	PUNCT
ejpam-4367	215	5	x	x	X
ejpam-4367	215	6	,	,	PUNCT
ejpam-4367	215	7	τ	τ	X
ejpam-4367	215	8	)	)	PUNCT
ejpam-4367	215	9	is	be	AUX
ejpam-4367	215	10	(	(	PUNCT
ejpam-4367	215	11	λ	λ	PROPN
ejpam-4367	215	12	,	,	PUNCT
ejpam-4367	215	13	sp)-r1	sp)-r1	NOUN
ejpam-4367	215	14	.	.	PUNCT
ejpam-4367	216	1	c.	c.	PROPN
ejpam-4367	216	2	boonpok	boonpok	PROPN
ejpam-4367	216	3	,	,	PUNCT
ejpam-4367	216	4	c.	c.	PROPN
ejpam-4367	216	5	viriyapong	viriyapong	PROPN
ejpam-4367	216	6	/	/	SYM
ejpam-4367	216	7	eur	eur	PROPN
ejpam-4367	216	8	.	.	PUNCT
ejpam-4367	217	1	j.	j.	PROPN
ejpam-4367	217	2	pure	pure	PROPN
ejpam-4367	217	3	appl	appl	PROPN
ejpam-4367	217	4	.	.	PROPN
ejpam-4367	217	5	math	math	PROPN
ejpam-4367	217	6	,	,	PUNCT
ejpam-4367	217	7	15	15	NUM
ejpam-4367	217	8	(	(	PUNCT
ejpam-4367	217	9	3	3	NUM
ejpam-4367	217	10	)	)	PUNCT
ejpam-4367	217	11	(	(	PUNCT
ejpam-4367	217	12	2022	2022	NUM
ejpam-4367	217	13	)	)	PUNCT
ejpam-4367	217	14	,	,	PUNCT
ejpam-4367	217	15	878	878	NUM
ejpam-4367	217	16	-	-	SYM
ejpam-4367	217	17	886	886	NUM
ejpam-4367	217	18	884	884	NUM
ejpam-4367	217	19	definition	definition	NOUN
ejpam-4367	217	20	5	5	NUM
ejpam-4367	217	21	.	.	PUNCT
ejpam-4367	218	1	[	[	X
ejpam-4367	218	2	2	2	X
ejpam-4367	218	3	]	]	PUNCT
ejpam-4367	218	4	let	let	VERB
ejpam-4367	218	5	a	a	PRON
ejpam-4367	218	6	be	be	AUX
ejpam-4367	218	7	a	a	DET
ejpam-4367	218	8	subset	subset	NOUN
ejpam-4367	218	9	of	of	ADP
ejpam-4367	218	10	a	a	DET
ejpam-4367	218	11	topological	topological	ADJ
ejpam-4367	218	12	space	space	NOUN
ejpam-4367	218	13	(	(	PUNCT
ejpam-4367	218	14	x	x	X
ejpam-4367	218	15	,	,	PUNCT
ejpam-4367	218	16	τ	τ	PROPN
ejpam-4367	218	17	)	)	PUNCT
ejpam-4367	218	18	.	.	PUNCT
ejpam-4367	219	1	the	the	DET
ejpam-4367	219	2	θ(λ	θ(λ	PROPN
ejpam-4367	219	3	,	,	PUNCT
ejpam-4367	219	4	sp)-closure	sp)-closure	NOUN
ejpam-4367	219	5	of	of	ADP
ejpam-4367	219	6	a	a	DET
ejpam-4367	219	7	,	,	PUNCT
ejpam-4367	219	8	aθ(λ	aθ(λ	NOUN
ejpam-4367	219	9	,	,	PUNCT
ejpam-4367	219	10	sp	sp	NOUN
ejpam-4367	219	11	)	)	PUNCT
ejpam-4367	219	12	,	,	PUNCT
ejpam-4367	219	13	is	be	AUX
ejpam-4367	219	14	defined	define	VERB
ejpam-4367	219	15	as	as	SCONJ
ejpam-4367	219	16	follows	follow	VERB
ejpam-4367	219	17	:	:	PUNCT
ejpam-4367	219	18	aθ(λ	aθ(λ	NOUN
ejpam-4367	219	19	,	,	PUNCT
ejpam-4367	219	20	sp	sp	NOUN
ejpam-4367	219	21	)	)	PUNCT
ejpam-4367	219	22	=	=	PRON
ejpam-4367	220	1	{	{	PUNCT
ejpam-4367	220	2	x	x	PUNCT
ejpam-4367	220	3	∈	∈	NOUN
ejpam-4367	220	4	x	x	PUNCT
ejpam-4367	220	5	|	|	ADV
ejpam-4367	220	6	a	a	DET
ejpam-4367	220	7	∩	∩	ADJ
ejpam-4367	220	8	u	u	NOUN
ejpam-4367	220	9	(	(	PUNCT
ejpam-4367	220	10	λ	λ	PROPN
ejpam-4367	220	11	,	,	PUNCT
ejpam-4367	220	12	sp	sp	NOUN
ejpam-4367	220	13	)	)	PUNCT
ejpam-4367	220	14	̸=	̸=	NOUN
ejpam-4367	220	15	∅	∅	NOUN
ejpam-4367	220	16	for	for	ADP
ejpam-4367	220	17	each	each	DET
ejpam-4367	220	18	u	u	PROPN
ejpam-4367	220	19	∈	∈	PROPN
ejpam-4367	220	20	λspo(x	λspo(x	PROPN
ejpam-4367	220	21	,	,	PUNCT
ejpam-4367	220	22	τ	τ	X
ejpam-4367	220	23	)	)	PUNCT
ejpam-4367	220	24	containing	contain	VERB
ejpam-4367	220	25	x	x	X
ejpam-4367	220	26	}	}	PUNCT
ejpam-4367	220	27	.	.	PUNCT
ejpam-4367	221	1	lemma	lemma	PROPN
ejpam-4367	221	2	5	5	NUM
ejpam-4367	221	3	.	.	PUNCT
ejpam-4367	222	1	if	if	SCONJ
ejpam-4367	222	2	a	a	DET
ejpam-4367	222	3	topological	topological	ADJ
ejpam-4367	222	4	space	space	NOUN
ejpam-4367	222	5	(	(	PUNCT
ejpam-4367	222	6	x	x	X
ejpam-4367	222	7	,	,	PUNCT
ejpam-4367	222	8	τ	τ	X
ejpam-4367	222	9	)	)	PUNCT
ejpam-4367	222	10	is	be	AUX
ejpam-4367	222	11	(	(	PUNCT
ejpam-4367	222	12	λ	λ	PROPN
ejpam-4367	222	13	,	,	PUNCT
ejpam-4367	222	14	sp)-r1	sp)-r1	NOUN
ejpam-4367	222	15	,	,	PUNCT
ejpam-4367	222	16	then	then	ADV
ejpam-4367	222	17	(	(	PUNCT
ejpam-4367	222	18	x	x	X
ejpam-4367	222	19	,	,	PUNCT
ejpam-4367	222	20	τ	τ	X
ejpam-4367	222	21	)	)	PUNCT
ejpam-4367	222	22	is	be	AUX
ejpam-4367	222	23	(	(	PUNCT
ejpam-4367	222	24	λ	λ	PROPN
ejpam-4367	222	25	,	,	PUNCT
ejpam-4367	222	26	sp)-r0	sp)-r0	NOUN
ejpam-4367	222	27	.	.	PUNCT
ejpam-4367	223	1	proof	proof	NOUN
ejpam-4367	223	2	.	.	PUNCT
ejpam-4367	224	1	let	let	VERB
ejpam-4367	224	2	u	u	PRON
ejpam-4367	224	3	∈	∈	PROPN
ejpam-4367	224	4	λspo(x	λspo(x	PROPN
ejpam-4367	224	5	,	,	PUNCT
ejpam-4367	224	6	τ	τ	X
ejpam-4367	224	7	)	)	PUNCT
ejpam-4367	224	8	and	and	CCONJ
ejpam-4367	224	9	let	let	VERB
ejpam-4367	224	10	x	x	PUNCT
ejpam-4367	224	11	∈	∈	PROPN
ejpam-4367	224	12	u	u	NOUN
ejpam-4367	224	13	.	.	PUNCT
ejpam-4367	225	1	if	if	SCONJ
ejpam-4367	225	2	y	y	PROPN
ejpam-4367	225	3	̸∈	̸∈	PROPN
ejpam-4367	225	4	u	u	PROPN
ejpam-4367	225	5	,	,	PUNCT
ejpam-4367	225	6	then	then	ADV
ejpam-4367	225	7	u	u	NOUN
ejpam-4367	225	8	∩	∩	PROPN
ejpam-4367	225	9	{	{	PUNCT
ejpam-4367	225	10	y}(λ	y}(λ	PROPN
ejpam-4367	225	11	,	,	PUNCT
ejpam-4367	225	12	sp	sp	NOUN
ejpam-4367	225	13	)	)	PUNCT
ejpam-4367	225	14	=	=	SYM
ejpam-4367	225	15	∅	∅	NOUN
ejpam-4367	225	16	and	and	CCONJ
ejpam-4367	225	17	x	x	PART
ejpam-4367	225	18	̸∈	̸∈	PROPN
ejpam-4367	225	19	{	{	PUNCT
ejpam-4367	225	20	y}(λ	y}(λ	PROPN
ejpam-4367	225	21	,	,	PUNCT
ejpam-4367	225	22	sp	sp	NOUN
ejpam-4367	225	23	)	)	PUNCT
ejpam-4367	225	24	.	.	PUNCT
ejpam-4367	226	1	therefore	therefore	ADV
ejpam-4367	226	2	,	,	PUNCT
ejpam-4367	226	3	{	{	PUNCT
ejpam-4367	226	4	x}(λ	x}(λ	PROPN
ejpam-4367	226	5	,	,	PUNCT
ejpam-4367	226	6	sp	sp	NOUN
ejpam-4367	226	7	)	)	PUNCT
ejpam-4367	226	8	̸=	̸=	PROPN
ejpam-4367	226	9	{	{	PUNCT
ejpam-4367	226	10	y}(λ	y}(λ	PROPN
ejpam-4367	226	11	,	,	PUNCT
ejpam-4367	226	12	sp	sp	NOUN
ejpam-4367	226	13	)	)	PUNCT
ejpam-4367	226	14	.	.	PUNCT
ejpam-4367	227	1	since	since	SCONJ
ejpam-4367	227	2	(	(	PUNCT
ejpam-4367	227	3	x	x	X
ejpam-4367	227	4	,	,	PUNCT
ejpam-4367	227	5	τ	τ	X
ejpam-4367	227	6	)	)	PUNCT
ejpam-4367	227	7	is	be	AUX
ejpam-4367	227	8	(	(	PUNCT
ejpam-4367	227	9	λ	λ	INTJ
ejpam-4367	227	10	,	,	PUNCT
ejpam-4367	227	11	sp)-r1	sp)-r1	NOUN
ejpam-4367	227	12	,	,	PUNCT
ejpam-4367	227	13	there	there	PRON
ejpam-4367	227	14	exists	exist	VERB
ejpam-4367	227	15	v	v	ADP
ejpam-4367	227	16	∈	∈	PROPN
ejpam-4367	227	17	λspo(x	λspo(x	NOUN
ejpam-4367	227	18	,	,	PUNCT
ejpam-4367	227	19	τ	τ	PROPN
ejpam-4367	227	20	)	)	PUNCT
ejpam-4367	227	21	such	such	ADJ
ejpam-4367	227	22	that	that	SCONJ
ejpam-4367	227	23	{	{	PUNCT
ejpam-4367	227	24	y}(λ	y}(λ	PROPN
ejpam-4367	227	25	,	,	PUNCT
ejpam-4367	227	26	sp	sp	NOUN
ejpam-4367	227	27	)	)	PUNCT
ejpam-4367	227	28	⊆	⊆	NUM
ejpam-4367	227	29	v	v	NOUN
ejpam-4367	227	30	and	and	CCONJ
ejpam-4367	227	31	x	x	PART
ejpam-4367	227	32	̸∈	̸∈	PROPN
ejpam-4367	227	33	v	v	NUM
ejpam-4367	227	34	.	.	PUNCT
ejpam-4367	228	1	thus	thus	ADV
ejpam-4367	228	2	,	,	PUNCT
ejpam-4367	228	3	v	v	ADP
ejpam-4367	228	4	∩	∩	NOUN
ejpam-4367	228	5	{	{	PUNCT
ejpam-4367	228	6	x}(λ	x}(λ	PROPN
ejpam-4367	228	7	,	,	PUNCT
ejpam-4367	228	8	sp	sp	NOUN
ejpam-4367	228	9	)	)	PUNCT
ejpam-4367	228	10	=	=	NOUN
ejpam-4367	228	11	∅	∅	NOUN
ejpam-4367	228	12	and	and	CCONJ
ejpam-4367	228	13	hence	hence	ADV
ejpam-4367	228	14	y	y	PROPN
ejpam-4367	228	15	̸∈	̸∈	PROPN
ejpam-4367	228	16	{	{	PUNCT
ejpam-4367	228	17	x}(λ	x}(λ	PROPN
ejpam-4367	228	18	,	,	PUNCT
ejpam-4367	228	19	sp	sp	NOUN
ejpam-4367	228	20	)	)	PUNCT
ejpam-4367	228	21	.	.	PUNCT
ejpam-4367	229	1	therefore	therefore	ADV
ejpam-4367	229	2	,	,	PUNCT
ejpam-4367	229	3	{	{	PUNCT
ejpam-4367	229	4	x}(λ	x}(λ	PROPN
ejpam-4367	229	5	,	,	PUNCT
ejpam-4367	229	6	sp	sp	NOUN
ejpam-4367	229	7	)	)	PUNCT
ejpam-4367	229	8	⊆	⊆	NUM
ejpam-4367	229	9	u	u	NOUN
ejpam-4367	229	10	.	.	PUNCT
ejpam-4367	230	1	this	this	PRON
ejpam-4367	230	2	shows	show	VERB
ejpam-4367	230	3	that	that	SCONJ
ejpam-4367	230	4	(	(	PUNCT
ejpam-4367	230	5	x	x	X
ejpam-4367	230	6	,	,	PUNCT
ejpam-4367	230	7	τ	τ	X
ejpam-4367	230	8	)	)	PUNCT
ejpam-4367	230	9	is	be	AUX
ejpam-4367	230	10	(	(	PUNCT
ejpam-4367	230	11	λ	λ	PROPN
ejpam-4367	230	12	,	,	PUNCT
ejpam-4367	230	13	sp)-r0	sp)-r0	PROPN
ejpam-4367	230	14	.	.	PUNCT
ejpam-4367	230	15	theorem	theorem	VERB
ejpam-4367	230	16	6	6	NUM
ejpam-4367	230	17	.	.	PUNCT
ejpam-4367	231	1	a	a	DET
ejpam-4367	231	2	topological	topological	ADJ
ejpam-4367	231	3	space	space	NOUN
ejpam-4367	231	4	(	(	PUNCT
ejpam-4367	231	5	x	x	X
ejpam-4367	231	6	,	,	PUNCT
ejpam-4367	231	7	τ	τ	X
ejpam-4367	231	8	)	)	PUNCT
ejpam-4367	231	9	is	be	AUX
ejpam-4367	231	10	(	(	PUNCT
ejpam-4367	231	11	λ	λ	X
ejpam-4367	231	12	,	,	PUNCT
ejpam-4367	231	13	sp)-r1	sp)-r1	NOUN
ejpam-4367	231	14	if	if	SCONJ
ejpam-4367	231	15	and	and	CCONJ
ejpam-4367	231	16	only	only	ADV
ejpam-4367	231	17	if	if	SCONJ
ejpam-4367	231	18	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	231	19	=	=	SYM
ejpam-4367	231	20	{	{	PUNCT
ejpam-4367	231	21	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	231	22	,	,	PUNCT
ejpam-4367	231	23	sp	sp	NOUN
ejpam-4367	231	24	)	)	PUNCT
ejpam-4367	231	25	for	for	ADP
ejpam-4367	231	26	each	each	DET
ejpam-4367	231	27	x	x	SYM
ejpam-4367	231	28	∈	∈	PROPN
ejpam-4367	231	29	x.	x.	NOUN
ejpam-4367	231	30	proof	proof	NOUN
ejpam-4367	231	31	.	.	PUNCT
ejpam-4367	232	1	let	let	VERB
ejpam-4367	232	2	(	(	PUNCT
ejpam-4367	232	3	x	x	NOUN
ejpam-4367	232	4	,	,	PUNCT
ejpam-4367	232	5	τ	τ	X
ejpam-4367	232	6	)	)	PUNCT
ejpam-4367	232	7	be	be	AUX
ejpam-4367	232	8	(	(	PUNCT
ejpam-4367	232	9	λ	λ	NOUN
ejpam-4367	232	10	,	,	PUNCT
ejpam-4367	232	11	sp)-r1	sp)-r1	NOUN
ejpam-4367	232	12	.	.	PUNCT
ejpam-4367	233	1	by	by	ADP
ejpam-4367	233	2	lemma	lemma	PROPN
ejpam-4367	233	3	5	5	NUM
ejpam-4367	233	4	,	,	PUNCT
ejpam-4367	233	5	(	(	PUNCT
ejpam-4367	233	6	x	x	X
ejpam-4367	233	7	,	,	PUNCT
ejpam-4367	233	8	τ	τ	X
ejpam-4367	233	9	)	)	PUNCT
ejpam-4367	233	10	is	be	AUX
ejpam-4367	233	11	(	(	PUNCT
ejpam-4367	233	12	λ	λ	PROPN
ejpam-4367	233	13	,	,	PUNCT
ejpam-4367	233	14	sp)-r0	sp)-r0	NOUN
ejpam-4367	233	15	and	and	CCONJ
ejpam-4367	233	16	by	by	ADP
ejpam-4367	233	17	corollary	corollary	ADJ
ejpam-4367	233	18	3	3	NUM
ejpam-4367	233	19	,	,	PUNCT
ejpam-4367	233	20	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	233	21	=	=	SYM
ejpam-4367	233	22	{	{	PUNCT
ejpam-4367	233	23	x}(λ	x}(λ	PROPN
ejpam-4367	233	24	,	,	PUNCT
ejpam-4367	233	25	sp	sp	NOUN
ejpam-4367	233	26	)	)	PUNCT
ejpam-4367	233	27	⊆	⊆	NUM
ejpam-4367	233	28	{	{	PUNCT
ejpam-4367	233	29	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	233	30	,	,	PUNCT
ejpam-4367	233	31	sp	sp	NOUN
ejpam-4367	233	32	)	)	PUNCT
ejpam-4367	233	33	for	for	ADP
ejpam-4367	233	34	each	each	DET
ejpam-4367	233	35	x	x	SYM
ejpam-4367	233	36	∈	∈	PROPN
ejpam-4367	233	37	x.	x.	NOUN
ejpam-4367	233	38	thus	thus	ADV
ejpam-4367	233	39	,	,	PUNCT
ejpam-4367	233	40	⟨x⟩sp	⟨x⟩sp	VERB
ejpam-4367	233	41	⊆	⊆	NUM
ejpam-4367	233	42	{	{	PUNCT
ejpam-4367	233	43	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	233	44	,	,	PUNCT
ejpam-4367	233	45	sp	sp	NOUN
ejpam-4367	233	46	)	)	PUNCT
ejpam-4367	233	47	for	for	ADP
ejpam-4367	233	48	each	each	DET
ejpam-4367	233	49	x	x	SYM
ejpam-4367	233	50	∈	∈	PROPN
ejpam-4367	233	51	x.	x.	NOUN
ejpam-4367	233	52	in	in	ADP
ejpam-4367	233	53	order	order	NOUN
ejpam-4367	233	54	to	to	PART
ejpam-4367	233	55	show	show	VERB
ejpam-4367	233	56	the	the	DET
ejpam-4367	233	57	opposite	opposite	ADJ
ejpam-4367	233	58	inclusion	inclusion	NOUN
ejpam-4367	233	59	,	,	PUNCT
ejpam-4367	233	60	suppose	suppose	VERB
ejpam-4367	233	61	that	that	SCONJ
ejpam-4367	233	62	y	y	PROPN
ejpam-4367	233	63	̸∈	̸∈	PROPN
ejpam-4367	233	64	⟨x⟩sp	⟨x⟩sp	PROPN
ejpam-4367	233	65	.	.	PUNCT
ejpam-4367	234	1	then	then	ADV
ejpam-4367	234	2	,	,	PUNCT
ejpam-4367	234	3	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	234	4	̸=	̸=	PROPN
ejpam-4367	234	5	⟨y⟩sp	⟨y⟩sp	NOUN
ejpam-4367	234	6	.	.	PUNCT
ejpam-4367	235	1	since	since	SCONJ
ejpam-4367	235	2	(	(	PUNCT
ejpam-4367	235	3	x	x	X
ejpam-4367	235	4	,	,	PUNCT
ejpam-4367	235	5	τ	τ	X
ejpam-4367	235	6	)	)	PUNCT
ejpam-4367	235	7	is	be	AUX
ejpam-4367	235	8	(	(	PUNCT
ejpam-4367	235	9	λ	λ	PROPN
ejpam-4367	235	10	,	,	PUNCT
ejpam-4367	235	11	sp)-r0	sp)-r0	PROPN
ejpam-4367	235	12	,	,	PUNCT
ejpam-4367	235	13	by	by	ADP
ejpam-4367	235	14	corollary	corollary	ADJ
ejpam-4367	235	15	3	3	NUM
ejpam-4367	235	16	,	,	PUNCT
ejpam-4367	235	17	{	{	PUNCT
ejpam-4367	235	18	x}(λ	x}(λ	PROPN
ejpam-4367	235	19	,	,	PUNCT
ejpam-4367	235	20	sp	sp	NOUN
ejpam-4367	235	21	)	)	PUNCT
ejpam-4367	235	22	̸=	̸=	PROPN
ejpam-4367	235	23	{	{	PUNCT
ejpam-4367	235	24	y}(λ	y}(λ	PROPN
ejpam-4367	235	25	,	,	PUNCT
ejpam-4367	235	26	sp	sp	NOUN
ejpam-4367	235	27	)	)	PUNCT
ejpam-4367	235	28	.	.	PUNCT
ejpam-4367	236	1	since	since	SCONJ
ejpam-4367	236	2	(	(	PUNCT
ejpam-4367	236	3	x	x	X
ejpam-4367	236	4	,	,	PUNCT
ejpam-4367	236	5	τ	τ	X
ejpam-4367	236	6	)	)	PUNCT
ejpam-4367	236	7	is	be	AUX
ejpam-4367	236	8	(	(	PUNCT
ejpam-4367	236	9	λ	λ	INTJ
ejpam-4367	236	10	,	,	PUNCT
ejpam-4367	236	11	sp)-r1	sp)-r1	NOUN
ejpam-4367	236	12	,	,	PUNCT
ejpam-4367	236	13	there	there	PRON
ejpam-4367	236	14	exist	exist	VERB
ejpam-4367	236	15	disjoint	disjoint	NOUN
ejpam-4367	236	16	(	(	PUNCT
ejpam-4367	236	17	λ	λ	NOUN
ejpam-4367	236	18	,	,	PUNCT
ejpam-4367	236	19	sp)-open	sp)-open	NOUN
ejpam-4367	236	20	sets	set	VERB
ejpam-4367	236	21	u	u	NOUN
ejpam-4367	236	22	and	and	CCONJ
ejpam-4367	236	23	v	v	NOUN
ejpam-4367	236	24	of	of	ADP
ejpam-4367	236	25	x	x	PUNCT
ejpam-4367	236	26	such	such	ADJ
ejpam-4367	236	27	that	that	SCONJ
ejpam-4367	236	28	{	{	PUNCT
ejpam-4367	236	29	x}(λ	x}(λ	PROPN
ejpam-4367	236	30	,	,	PUNCT
ejpam-4367	236	31	sp	sp	NOUN
ejpam-4367	236	32	)	)	PUNCT
ejpam-4367	236	33	⊆	⊆	NUM
ejpam-4367	236	34	u	u	NOUN
ejpam-4367	236	35	and	and	CCONJ
ejpam-4367	236	36	{	{	PUNCT
ejpam-4367	236	37	y}(λ	y}(λ	PROPN
ejpam-4367	236	38	,	,	PUNCT
ejpam-4367	236	39	sp	sp	NOUN
ejpam-4367	236	40	)	)	PUNCT
ejpam-4367	236	41	⊆	⊆	NUM
ejpam-4367	236	42	v	v	NOUN
ejpam-4367	236	43	.	.	PUNCT
ejpam-4367	237	1	since	since	SCONJ
ejpam-4367	237	2	{	{	PUNCT
ejpam-4367	237	3	x	x	NOUN
ejpam-4367	237	4	}	}	PUNCT
ejpam-4367	237	5	∩	∩	ADJ
ejpam-4367	237	6	v	v	ADP
ejpam-4367	237	7	(	(	PUNCT
ejpam-4367	237	8	λ	λ	PROPN
ejpam-4367	237	9	,	,	PUNCT
ejpam-4367	237	10	sp	sp	NOUN
ejpam-4367	237	11	)	)	PUNCT
ejpam-4367	237	12	⊆	⊆	NUM
ejpam-4367	237	13	u	u	NOUN
ejpam-4367	237	14	∩	∩	X
ejpam-4367	237	15	v	v	X
ejpam-4367	237	16	(	(	PUNCT
ejpam-4367	237	17	λ	λ	PROPN
ejpam-4367	237	18	,	,	PUNCT
ejpam-4367	237	19	sp	sp	NOUN
ejpam-4367	237	20	)	)	PUNCT
ejpam-4367	237	21	=	=	NOUN
ejpam-4367	237	22	∅	∅	NOUN
ejpam-4367	237	23	,	,	PUNCT
ejpam-4367	237	24	y	y	PROPN
ejpam-4367	237	25	̸∈	̸∈	PROPN
ejpam-4367	237	26	{	{	PUNCT
ejpam-4367	237	27	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	237	28	,	,	PUNCT
ejpam-4367	237	29	sp	sp	NOUN
ejpam-4367	237	30	)	)	PUNCT
ejpam-4367	237	31	.	.	PUNCT
ejpam-4367	238	1	thus	thus	ADV
ejpam-4367	238	2	,	,	PUNCT
ejpam-4367	238	3	{	{	PUNCT
ejpam-4367	238	4	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	238	5	,	,	PUNCT
ejpam-4367	238	6	sp	sp	NOUN
ejpam-4367	238	7	)	)	PUNCT
ejpam-4367	238	8	⊆	⊆	NUM
ejpam-4367	238	9	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	238	10	and	and	CCONJ
ejpam-4367	238	11	hence	hence	ADV
ejpam-4367	238	12	{	{	PUNCT
ejpam-4367	238	13	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	238	14	,	,	PUNCT
ejpam-4367	238	15	sp	sp	NOUN
ejpam-4367	238	16	)	)	PUNCT
ejpam-4367	238	17	=	=	SYM
ejpam-4367	238	18	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	238	19	.	.	PUNCT
ejpam-4367	239	1	conversely	conversely	ADV
ejpam-4367	239	2	,	,	PUNCT
ejpam-4367	239	3	suppose	suppose	VERB
ejpam-4367	239	4	that	that	SCONJ
ejpam-4367	239	5	{	{	PUNCT
ejpam-4367	239	6	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	239	7	,	,	PUNCT
ejpam-4367	239	8	sp	sp	NOUN
ejpam-4367	239	9	)	)	PUNCT
ejpam-4367	239	10	=	=	SYM
ejpam-4367	239	11	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	239	12	for	for	ADP
ejpam-4367	239	13	each	each	DET
ejpam-4367	239	14	x	x	SYM
ejpam-4367	239	15	∈	∈	PROPN
ejpam-4367	239	16	x.	x.	NOUN
ejpam-4367	239	17	then	then	ADV
ejpam-4367	239	18	,	,	PUNCT
ejpam-4367	239	19	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	239	20	=	=	SYM
ejpam-4367	239	21	{	{	PUNCT
ejpam-4367	239	22	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	239	23	,	,	PUNCT
ejpam-4367	239	24	sp	sp	NOUN
ejpam-4367	239	25	)	)	PUNCT
ejpam-4367	239	26	⊇	⊇	NOUN
ejpam-4367	239	27	{	{	PUNCT
ejpam-4367	239	28	x}(λ	x}(λ	PROPN
ejpam-4367	239	29	,	,	PUNCT
ejpam-4367	239	30	sp	sp	NOUN
ejpam-4367	239	31	)	)	PUNCT
ejpam-4367	239	32	⊇	⊇	NOUN
ejpam-4367	239	33	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	239	34	and	and	CCONJ
ejpam-4367	239	35	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	239	36	=	=	SYM
ejpam-4367	239	37	{	{	PUNCT
ejpam-4367	239	38	x}(λ	x}(λ	PROPN
ejpam-4367	239	39	,	,	PUNCT
ejpam-4367	239	40	sp	sp	NOUN
ejpam-4367	239	41	)	)	PUNCT
ejpam-4367	239	42	for	for	ADP
ejpam-4367	239	43	each	each	DET
ejpam-4367	239	44	x	x	SYM
ejpam-4367	239	45	∈	∈	PROPN
ejpam-4367	239	46	x.	x.	NOUN
ejpam-4367	239	47	by	by	ADP
ejpam-4367	239	48	corollary	corollary	ADJ
ejpam-4367	239	49	3	3	NUM
ejpam-4367	239	50	,	,	PUNCT
ejpam-4367	239	51	(	(	PUNCT
ejpam-4367	239	52	x	x	X
ejpam-4367	239	53	,	,	PUNCT
ejpam-4367	239	54	τ	τ	X
ejpam-4367	239	55	)	)	PUNCT
ejpam-4367	239	56	is	be	AUX
ejpam-4367	239	57	(	(	PUNCT
ejpam-4367	239	58	λ	λ	PROPN
ejpam-4367	239	59	,	,	PUNCT
ejpam-4367	239	60	sp)-r0	sp)-r0	PROPN
ejpam-4367	239	61	.	.	PUNCT
ejpam-4367	239	62	suppose	suppose	VERB
ejpam-4367	239	63	that	that	SCONJ
ejpam-4367	239	64	{	{	PUNCT
ejpam-4367	239	65	x}(λ	x}(λ	PROPN
ejpam-4367	239	66	,	,	PUNCT
ejpam-4367	239	67	sp	sp	NOUN
ejpam-4367	239	68	)	)	PUNCT
ejpam-4367	239	69	̸=	̸=	PROPN
ejpam-4367	239	70	{	{	PUNCT
ejpam-4367	239	71	y}(λ	y}(λ	PROPN
ejpam-4367	239	72	,	,	PUNCT
ejpam-4367	239	73	sp	sp	NOUN
ejpam-4367	239	74	)	)	PUNCT
ejpam-4367	239	75	.	.	PUNCT
ejpam-4367	240	1	thus	thus	ADV
ejpam-4367	240	2	,	,	PUNCT
ejpam-4367	240	3	by	by	ADP
ejpam-4367	240	4	corollary	corollary	ADJ
ejpam-4367	240	5	1	1	NUM
ejpam-4367	240	6	,	,	PUNCT
ejpam-4367	240	7	{	{	PUNCT
ejpam-4367	240	8	x}(λ	x}(λ	PROPN
ejpam-4367	240	9	,	,	PUNCT
ejpam-4367	240	10	sp	sp	NOUN
ejpam-4367	240	11	)	)	PUNCT
ejpam-4367	240	12	∩	∩	NOUN
ejpam-4367	240	13	{	{	PUNCT
ejpam-4367	240	14	y}(λ	y}(λ	PROPN
ejpam-4367	240	15	,	,	PUNCT
ejpam-4367	240	16	sp	sp	NOUN
ejpam-4367	240	17	)	)	PUNCT
ejpam-4367	240	18	=	=	PUNCT
ejpam-4367	240	19	∅.	∅.	NOUN
ejpam-4367	240	20	by	by	ADP
ejpam-4367	240	21	corollary	corollary	ADJ
ejpam-4367	240	22	3	3	NUM
ejpam-4367	240	23	,	,	PUNCT
ejpam-4367	240	24	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	240	25	∩	∩	ADJ
ejpam-4367	240	26	⟨y⟩sp	⟨y⟩sp	NOUN
ejpam-4367	240	27	=	=	NOUN
ejpam-4367	240	28	∅	∅	NOUN
ejpam-4367	240	29	and	and	CCONJ
ejpam-4367	240	30	hence	hence	ADV
ejpam-4367	240	31	{	{	PUNCT
ejpam-4367	240	32	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	240	33	,	,	PUNCT
ejpam-4367	240	34	sp	sp	NOUN
ejpam-4367	240	35	)	)	PUNCT
ejpam-4367	240	36	∩	∩	NOUN
ejpam-4367	240	37	{	{	PUNCT
ejpam-4367	240	38	y}θ(λ	y}θ(λ	PROPN
ejpam-4367	240	39	,	,	PUNCT
ejpam-4367	240	40	sp	sp	NOUN
ejpam-4367	240	41	)	)	PUNCT
ejpam-4367	240	42	=	=	PUNCT
ejpam-4367	240	43	∅.	∅.	NOUN
ejpam-4367	240	44	since	since	SCONJ
ejpam-4367	240	45	y	y	PROPN
ejpam-4367	240	46	̸∈	̸∈	PROPN
ejpam-4367	240	47	{	{	PUNCT
ejpam-4367	240	48	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	240	49	,	,	PUNCT
ejpam-4367	240	50	sp	sp	NOUN
ejpam-4367	240	51	)	)	PUNCT
ejpam-4367	240	52	,	,	PUNCT
ejpam-4367	240	53	there	there	PRON
ejpam-4367	240	54	exists	exist	VERB
ejpam-4367	240	55	a	a	DET
ejpam-4367	240	56	(	(	PUNCT
ejpam-4367	240	57	λ	λ	NOUN
ejpam-4367	240	58	,	,	PUNCT
ejpam-4367	240	59	sp)-open	sp)-open	NOUN
ejpam-4367	240	60	set	set	VERB
ejpam-4367	240	61	u	u	NOUN
ejpam-4367	240	62	of	of	ADP
ejpam-4367	240	63	x	x	SYM
ejpam-4367	240	64	such	such	ADJ
ejpam-4367	240	65	that	that	SCONJ
ejpam-4367	240	66	y	y	PROPN
ejpam-4367	240	67	∈	∈	PROPN
ejpam-4367	240	68	u	u	PROPN
ejpam-4367	240	69	⊆	⊆	NUM
ejpam-4367	240	70	u	u	PROPN
ejpam-4367	240	71	(	(	PUNCT
ejpam-4367	240	72	λ	λ	PROPN
ejpam-4367	240	73	,	,	PUNCT
ejpam-4367	240	74	sp	sp	NOUN
ejpam-4367	240	75	)	)	PUNCT
ejpam-4367	240	76	⊆	⊆	NUM
ejpam-4367	240	77	x	x	SYM
ejpam-4367	240	78	−	−	PROPN
ejpam-4367	240	79	{	{	PUNCT
ejpam-4367	240	80	x	x	NOUN
ejpam-4367	240	81	}	}	PUNCT
ejpam-4367	240	82	.	.	PUNCT
ejpam-4367	241	1	let	let	VERB
ejpam-4367	241	2	v	v	VERB
ejpam-4367	241	3	=	=	SYM
ejpam-4367	241	4	x	x	SYM
ejpam-4367	241	5	−	−	PROPN
ejpam-4367	241	6	u	u	NOUN
ejpam-4367	241	7	(	(	PUNCT
ejpam-4367	241	8	λ	λ	PROPN
ejpam-4367	241	9	,	,	PUNCT
ejpam-4367	241	10	sp	sp	NOUN
ejpam-4367	241	11	)	)	PUNCT
ejpam-4367	241	12	,	,	PUNCT
ejpam-4367	241	13	then	then	ADV
ejpam-4367	241	14	x	x	SYM
ejpam-4367	241	15	∈	∈	PROPN
ejpam-4367	241	16	v	v	ADP
ejpam-4367	241	17	∈	∈	X
ejpam-4367	241	18	λspo(x	λspo(x	NOUN
ejpam-4367	241	19	,	,	PUNCT
ejpam-4367	241	20	τ	τ	PROPN
ejpam-4367	241	21	)	)	PUNCT
ejpam-4367	241	22	.	.	PUNCT
ejpam-4367	242	1	since	since	SCONJ
ejpam-4367	242	2	(	(	PUNCT
ejpam-4367	242	3	x	x	X
ejpam-4367	242	4	,	,	PUNCT
ejpam-4367	242	5	τ	τ	X
ejpam-4367	242	6	)	)	PUNCT
ejpam-4367	242	7	is	be	AUX
ejpam-4367	242	8	(	(	PUNCT
ejpam-4367	242	9	λ	λ	PROPN
ejpam-4367	242	10	,	,	PUNCT
ejpam-4367	242	11	sp)-r0	sp)-r0	PROPN
ejpam-4367	242	12	,	,	PUNCT
ejpam-4367	242	13	{	{	PUNCT
ejpam-4367	242	14	y}(λ	y}(λ	PROPN
ejpam-4367	242	15	,	,	PUNCT
ejpam-4367	242	16	sp	sp	NOUN
ejpam-4367	242	17	)	)	PUNCT
ejpam-4367	242	18	⊆	⊆	NUM
ejpam-4367	242	19	u	u	NOUN
ejpam-4367	242	20	,	,	PUNCT
ejpam-4367	242	21	{	{	PUNCT
ejpam-4367	242	22	x}(λ	x}(λ	PROPN
ejpam-4367	242	23	,	,	PUNCT
ejpam-4367	242	24	sp	sp	NOUN
ejpam-4367	242	25	)	)	PUNCT
ejpam-4367	242	26	⊆	⊆	NUM
ejpam-4367	242	27	v	v	NOUN
ejpam-4367	242	28	and	and	CCONJ
ejpam-4367	242	29	u	u	NOUN
ejpam-4367	242	30	∩	∩	NOUN
ejpam-4367	242	31	v	v	NOUN
ejpam-4367	242	32	=	=	PUNCT
ejpam-4367	242	33	∅.	∅.	ADP
ejpam-4367	242	34	this	this	DET
ejpam-4367	242	35	shows	show	VERB
ejpam-4367	242	36	that	that	SCONJ
ejpam-4367	242	37	(	(	PUNCT
ejpam-4367	242	38	x	x	X
ejpam-4367	242	39	,	,	PUNCT
ejpam-4367	242	40	τ	τ	X
ejpam-4367	242	41	)	)	PUNCT
ejpam-4367	242	42	is	be	AUX
ejpam-4367	242	43	(	(	PUNCT
ejpam-4367	242	44	λ	λ	PROPN
ejpam-4367	242	45	,	,	PUNCT
ejpam-4367	242	46	sp)-r1	sp)-r1	NOUN
ejpam-4367	242	47	.	.	PUNCT
ejpam-4367	242	48	corollary	corollary	ADJ
ejpam-4367	242	49	4	4	NUM
ejpam-4367	242	50	.	.	PUNCT
ejpam-4367	243	1	a	a	DET
ejpam-4367	243	2	topological	topological	ADJ
ejpam-4367	243	3	space	space	NOUN
ejpam-4367	243	4	(	(	PUNCT
ejpam-4367	243	5	x	x	X
ejpam-4367	243	6	,	,	PUNCT
ejpam-4367	243	7	τ	τ	X
ejpam-4367	243	8	)	)	PUNCT
ejpam-4367	243	9	is	be	AUX
ejpam-4367	243	10	(	(	PUNCT
ejpam-4367	243	11	λ	λ	X
ejpam-4367	243	12	,	,	PUNCT
ejpam-4367	243	13	sp)-r1	sp)-r1	NOUN
ejpam-4367	243	14	if	if	SCONJ
ejpam-4367	243	15	and	and	CCONJ
ejpam-4367	243	16	only	only	ADV
ejpam-4367	243	17	if	if	SCONJ
ejpam-4367	243	18	{	{	PUNCT
ejpam-4367	243	19	x}(λ	x}(λ	PROPN
ejpam-4367	243	20	,	,	PUNCT
ejpam-4367	243	21	sp	sp	NOUN
ejpam-4367	243	22	)	)	PUNCT
ejpam-4367	243	23	=	=	PRON
ejpam-4367	243	24	{	{	PUNCT
ejpam-4367	243	25	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	243	26	,	,	PUNCT
ejpam-4367	243	27	sp	sp	NOUN
ejpam-4367	243	28	)	)	PUNCT
ejpam-4367	243	29	for	for	ADP
ejpam-4367	243	30	each	each	DET
ejpam-4367	243	31	x	x	SYM
ejpam-4367	243	32	∈	∈	PROPN
ejpam-4367	243	33	x.	x.	NOUN
ejpam-4367	243	34	proof	proof	NOUN
ejpam-4367	243	35	.	.	PUNCT
ejpam-4367	244	1	let	let	VERB
ejpam-4367	244	2	(	(	PUNCT
ejpam-4367	244	3	x	x	NOUN
ejpam-4367	244	4	,	,	PUNCT
ejpam-4367	244	5	τ	τ	X
ejpam-4367	244	6	)	)	PUNCT
ejpam-4367	244	7	be	be	AUX
ejpam-4367	244	8	(	(	PUNCT
ejpam-4367	244	9	λ	λ	NOUN
ejpam-4367	244	10	,	,	PUNCT
ejpam-4367	244	11	sp)-r1	sp)-r1	NOUN
ejpam-4367	244	12	.	.	PUNCT
ejpam-4367	245	1	by	by	ADP
ejpam-4367	245	2	theorem	theorem	NOUN
ejpam-4367	245	3	6	6	NUM
ejpam-4367	245	4	,	,	PUNCT
ejpam-4367	245	5	we	we	PRON
ejpam-4367	245	6	have	have	AUX
ejpam-4367	245	7	{	{	PUNCT
ejpam-4367	245	8	x}(λ	x}(λ	PROPN
ejpam-4367	245	9	,	,	PUNCT
ejpam-4367	245	10	sp	sp	NOUN
ejpam-4367	245	11	)	)	PUNCT
ejpam-4367	245	12	⊇	⊇	NOUN
ejpam-4367	245	13	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	245	14	=	=	SYM
ejpam-4367	245	15	{	{	PUNCT
ejpam-4367	245	16	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	245	17	,	,	PUNCT
ejpam-4367	245	18	sp	sp	NOUN
ejpam-4367	245	19	)	)	PUNCT
ejpam-4367	245	20	⊇	⊇	NOUN
ejpam-4367	245	21	{	{	PUNCT
ejpam-4367	245	22	x}(λ	x}(λ	PROPN
ejpam-4367	245	23	,	,	PUNCT
ejpam-4367	245	24	sp	sp	NOUN
ejpam-4367	245	25	)	)	PUNCT
ejpam-4367	245	26	and	and	CCONJ
ejpam-4367	245	27	hence	hence	ADV
ejpam-4367	245	28	{	{	PUNCT
ejpam-4367	245	29	x}(λ	x}(λ	PROPN
ejpam-4367	245	30	,	,	PUNCT
ejpam-4367	245	31	sp	sp	NOUN
ejpam-4367	245	32	)	)	PUNCT
ejpam-4367	245	33	=	=	PRON
ejpam-4367	245	34	{	{	PUNCT
ejpam-4367	245	35	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	245	36	,	,	PUNCT
ejpam-4367	245	37	sp	sp	NOUN
ejpam-4367	245	38	)	)	PUNCT
ejpam-4367	245	39	for	for	ADP
ejpam-4367	245	40	each	each	DET
ejpam-4367	245	41	x	x	SYM
ejpam-4367	245	42	∈	∈	PROPN
ejpam-4367	245	43	x.	x.	NOUN
ejpam-4367	245	44	conversely	conversely	ADV
ejpam-4367	245	45	,	,	PUNCT
ejpam-4367	245	46	suppose	suppose	VERB
ejpam-4367	245	47	that	that	SCONJ
ejpam-4367	245	48	{	{	PUNCT
ejpam-4367	245	49	x}(λ	x}(λ	PROPN
ejpam-4367	245	50	,	,	PUNCT
ejpam-4367	245	51	sp	sp	NOUN
ejpam-4367	245	52	)	)	PUNCT
ejpam-4367	245	53	=	=	PRON
ejpam-4367	245	54	{	{	PUNCT
ejpam-4367	245	55	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	245	56	,	,	PUNCT
ejpam-4367	245	57	sp	sp	NOUN
ejpam-4367	245	58	)	)	PUNCT
ejpam-4367	245	59	for	for	ADP
ejpam-4367	245	60	each	each	DET
ejpam-4367	245	61	x	x	SYM
ejpam-4367	245	62	∈	∈	PROPN
ejpam-4367	245	63	x.	x.	NOUN
ejpam-4367	245	64	first	first	ADV
ejpam-4367	245	65	,	,	PUNCT
ejpam-4367	245	66	we	we	PRON
ejpam-4367	245	67	show	show	VERB
ejpam-4367	245	68	that	that	SCONJ
ejpam-4367	245	69	(	(	PUNCT
ejpam-4367	245	70	x	x	X
ejpam-4367	245	71	,	,	PUNCT
ejpam-4367	245	72	τ	τ	X
ejpam-4367	245	73	)	)	PUNCT
ejpam-4367	245	74	is	be	AUX
ejpam-4367	245	75	(	(	PUNCT
ejpam-4367	245	76	λ	λ	PROPN
ejpam-4367	245	77	,	,	PUNCT
ejpam-4367	245	78	sp)-r0	sp)-r0	PROPN
ejpam-4367	245	79	.	.	PUNCT
ejpam-4367	246	1	let	let	VERB
ejpam-4367	246	2	u	u	PRON
ejpam-4367	246	3	∈	∈	PROPN
ejpam-4367	246	4	λspo(x	λspo(x	PROPN
ejpam-4367	246	5	,	,	PUNCT
ejpam-4367	246	6	τ	τ	PROPN
ejpam-4367	246	7	)	)	PUNCT
ejpam-4367	246	8	and	and	CCONJ
ejpam-4367	246	9	x	x	PUNCT
ejpam-4367	246	10	∈	∈	PROPN
ejpam-4367	246	11	u	u	NOUN
ejpam-4367	246	12	.	.	PUNCT
ejpam-4367	247	1	let	let	VERB
ejpam-4367	247	2	y	y	PROPN
ejpam-4367	247	3	̸∈	̸∈	PROPN
ejpam-4367	247	4	u	u	PROPN
ejpam-4367	247	5	.	.	PUNCT
ejpam-4367	248	1	then	then	ADV
ejpam-4367	248	2	,	,	PUNCT
ejpam-4367	248	3	u	u	PROPN
ejpam-4367	248	4	∩	∩	NOUN
ejpam-4367	248	5	{	{	PUNCT
ejpam-4367	248	6	y}(λ	y}(λ	PROPN
ejpam-4367	248	7	,	,	PUNCT
ejpam-4367	248	8	sp	sp	NOUN
ejpam-4367	248	9	)	)	PUNCT
ejpam-4367	248	10	=	=	SYM
ejpam-4367	248	11	u	u	PROPN
ejpam-4367	248	12	∩	∩	NOUN
ejpam-4367	248	13	{	{	PUNCT
ejpam-4367	248	14	y}θ(λ	y}θ(λ	PROPN
ejpam-4367	248	15	,	,	PUNCT
ejpam-4367	248	16	sp	sp	NOUN
ejpam-4367	248	17	)	)	PUNCT
ejpam-4367	248	18	=	=	NOUN
ejpam-4367	248	19	∅.	∅.	ADP
ejpam-4367	248	20	thus	thus	ADV
ejpam-4367	248	21	,	,	PUNCT
ejpam-4367	248	22	x	x	PROPN
ejpam-4367	248	23	̸∈	̸∈	PROPN
ejpam-4367	248	24	{	{	PUNCT
ejpam-4367	248	25	y}θ(λ	y}θ(λ	PROPN
ejpam-4367	248	26	,	,	PUNCT
ejpam-4367	248	27	sp	sp	NOUN
ejpam-4367	248	28	)	)	PUNCT
ejpam-4367	248	29	.	.	PUNCT
ejpam-4367	249	1	there	there	PRON
ejpam-4367	249	2	exists	exist	VERB
ejpam-4367	249	3	v	v	ADP
ejpam-4367	249	4	∈	∈	PROPN
ejpam-4367	249	5	λspo(x	λspo(x	NOUN
ejpam-4367	249	6	,	,	PUNCT
ejpam-4367	249	7	τ	τ	PROPN
ejpam-4367	249	8	)	)	PUNCT
ejpam-4367	249	9	such	such	ADJ
ejpam-4367	249	10	that	that	SCONJ
ejpam-4367	249	11	x	x	SYM
ejpam-4367	249	12	∈	∈	PROPN
ejpam-4367	249	13	v	v	NOUN
ejpam-4367	249	14	and	and	CCONJ
ejpam-4367	249	15	y	y	PROPN
ejpam-4367	249	16	̸∈	̸∈	PROPN
ejpam-4367	249	17	v	v	PROPN
ejpam-4367	249	18	(	(	PUNCT
ejpam-4367	249	19	λ	λ	PROPN
ejpam-4367	249	20	,	,	PUNCT
ejpam-4367	249	21	sp	sp	NOUN
ejpam-4367	249	22	)	)	PUNCT
ejpam-4367	249	23	.	.	PUNCT
ejpam-4367	250	1	since	since	SCONJ
ejpam-4367	250	2	{	{	PUNCT
ejpam-4367	250	3	x}(λ	x}(λ	PROPN
ejpam-4367	250	4	,	,	PUNCT
ejpam-4367	250	5	sp	sp	NOUN
ejpam-4367	250	6	)	)	PUNCT
ejpam-4367	250	7	⊆	⊆	NUM
ejpam-4367	250	8	v	v	NOUN
ejpam-4367	250	9	(	(	PUNCT
ejpam-4367	250	10	λ	λ	NOUN
ejpam-4367	250	11	,	,	PUNCT
ejpam-4367	250	12	sp	sp	NOUN
ejpam-4367	250	13	)	)	PUNCT
ejpam-4367	250	14	,	,	PUNCT
ejpam-4367	250	15	y	y	PROPN
ejpam-4367	250	16	̸∈	̸∈	PROPN
ejpam-4367	250	17	{	{	PUNCT
ejpam-4367	250	18	x}(λ	x}(λ	PROPN
ejpam-4367	250	19	,	,	PUNCT
ejpam-4367	250	20	sp	sp	NOUN
ejpam-4367	250	21	)	)	PUNCT
ejpam-4367	250	22	.	.	PUNCT
ejpam-4367	251	1	this	this	PRON
ejpam-4367	251	2	shows	show	VERB
ejpam-4367	251	3	that	that	SCONJ
ejpam-4367	251	4	{	{	PUNCT
ejpam-4367	251	5	x}(λ	x}(λ	PROPN
ejpam-4367	251	6	,	,	PUNCT
ejpam-4367	251	7	sp	sp	NOUN
ejpam-4367	251	8	)	)	PUNCT
ejpam-4367	251	9	⊆	⊆	NUM
ejpam-4367	251	10	u	u	NOUN
ejpam-4367	251	11	and	and	CCONJ
ejpam-4367	251	12	hence	hence	ADV
ejpam-4367	251	13	(	(	PUNCT
ejpam-4367	251	14	x	x	X
ejpam-4367	251	15	,	,	PUNCT
ejpam-4367	251	16	τ	τ	X
ejpam-4367	251	17	)	)	PUNCT
ejpam-4367	251	18	is	be	AUX
ejpam-4367	251	19	(	(	PUNCT
ejpam-4367	251	20	λ	λ	PROPN
ejpam-4367	251	21	,	,	PUNCT
ejpam-4367	251	22	sp)-r0	sp)-r0	PROPN
ejpam-4367	251	23	.	.	PUNCT
ejpam-4367	251	24	by	by	ADP
ejpam-4367	251	25	corollary	corollary	ADJ
ejpam-4367	251	26	3	3	NUM
ejpam-4367	251	27	,	,	PUNCT
ejpam-4367	251	28	⟨x⟩sp	⟨x⟩sp	NOUN
ejpam-4367	251	29	=	=	SYM
ejpam-4367	251	30	{	{	PUNCT
ejpam-4367	251	31	x}(λ	x}(λ	PROPN
ejpam-4367	251	32	,	,	PUNCT
ejpam-4367	251	33	sp	sp	NOUN
ejpam-4367	251	34	)	)	PUNCT
ejpam-4367	251	35	=	=	PRON
ejpam-4367	251	36	{	{	PUNCT
ejpam-4367	251	37	x}θ(λ	x}θ(λ	PROPN
ejpam-4367	251	38	,	,	PUNCT
ejpam-4367	251	39	sp	sp	NOUN
ejpam-4367	251	40	)	)	PUNCT
ejpam-4367	251	41	for	for	ADP
ejpam-4367	251	42	each	each	DET
ejpam-4367	251	43	x	x	SYM
ejpam-4367	251	44	∈	∈	PROPN
ejpam-4367	251	45	x.	x.	NOUN
ejpam-4367	251	46	thus	thus	ADV
ejpam-4367	251	47	,	,	PUNCT
ejpam-4367	251	48	by	by	SCONJ
ejpam-4367	251	49	theorem	theorem	NOUN
ejpam-4367	251	50	6	6	NUM
ejpam-4367	251	51	,	,	PUNCT
ejpam-4367	251	52	(	(	PUNCT
ejpam-4367	251	53	x	x	X
ejpam-4367	251	54	,	,	PUNCT
ejpam-4367	251	55	τ	τ	X
ejpam-4367	251	56	)	)	PUNCT
ejpam-4367	251	57	is	be	AUX
ejpam-4367	251	58	(	(	PUNCT
ejpam-4367	251	59	λ	λ	INTJ
ejpam-4367	251	60	,	,	PUNCT
ejpam-4367	251	61	sp)-r1	sp)-r1	NOUN
ejpam-4367	251	62	.	.	PUNCT
ejpam-4367	252	1	references	reference	NOUN
ejpam-4367	252	2	885	885	NUM
ejpam-4367	252	3	acknowledgements	acknowledgement	NOUN
ejpam-4367	252	4	this	this	DET
ejpam-4367	252	5	research	research	NOUN
ejpam-4367	252	6	project	project	NOUN
ejpam-4367	252	7	was	be	AUX
ejpam-4367	252	8	financially	financially	ADV
ejpam-4367	252	9	supported	support	VERB
ejpam-4367	252	10	by	by	ADP
ejpam-4367	252	11	mahasarakham	mahasarakham	PROPN
ejpam-4367	252	12	university	university	PROPN
ejpam-4367	252	13	.	.	PUNCT
ejpam-4367	253	1	references	reference	NOUN
ejpam-4367	253	2	[	[	X
ejpam-4367	253	3	1	1	NUM
ejpam-4367	253	4	]	]	PUNCT
ejpam-4367	253	5	d.	d.	PROPN
ejpam-4367	253	6	andrijević.	andrijević.	PROPN
ejpam-4367	253	7	on	on	ADP
ejpam-4367	253	8	b	b	X
ejpam-4367	253	9	-	-	PUNCT
ejpam-4367	253	10	open	open	ADJ
ejpam-4367	253	11	sets	set	NOUN
ejpam-4367	253	12	.	.	PUNCT
ejpam-4367	254	1	matematički	matematički	PROPN
ejpam-4367	254	2	vesnik	vesnik	PROPN
ejpam-4367	254	3	,	,	PUNCT
ejpam-4367	254	4	48:59–64	48:59–64	PROPN
ejpam-4367	254	5	,	,	PUNCT
ejpam-4367	254	6	1996	1996	NUM
ejpam-4367	254	7	.	.	PUNCT
ejpam-4367	255	1	[	[	X
ejpam-4367	255	2	2	2	NUM
ejpam-4367	255	3	]	]	PUNCT
ejpam-4367	255	4	c.	c.	PROPN
ejpam-4367	255	5	boonpok	boonpok	PROPN
ejpam-4367	255	6	.	.	PUNCT
ejpam-4367	256	1	(	(	PUNCT
ejpam-4367	256	2	λ	λ	NOUN
ejpam-4367	256	3	,	,	PUNCT
ejpam-4367	256	4	sp)-closed	sp)-close	VERB
ejpam-4367	256	5	sets	set	NOUN
ejpam-4367	256	6	and	and	CCONJ
ejpam-4367	256	7	related	related	ADJ
ejpam-4367	256	8	topics	topic	NOUN
ejpam-4367	256	9	in	in	ADP
ejpam-4367	256	10	topological	topological	ADJ
ejpam-4367	256	11	spaces	space	NOUN
ejpam-4367	256	12	.	.	PUNCT
ejpam-4367	257	1	wseas	wseas	VERB
ejpam-4367	257	2	transactions	transaction	NOUN
ejpam-4367	257	3	on	on	ADP
ejpam-4367	257	4	mathematics	mathematic	NOUN
ejpam-4367	257	5	,	,	PUNCT
ejpam-4367	257	6	19:312–322	19:312–322	PROPN
ejpam-4367	257	7	,	,	PUNCT
ejpam-4367	257	8	2020	2020	NUM
ejpam-4367	257	9	.	.	PUNCT
ejpam-4367	258	1	[	[	X
ejpam-4367	258	2	3	3	X
ejpam-4367	258	3	]	]	PUNCT
ejpam-4367	258	4	c.	c.	PROPN
ejpam-4367	258	5	boonpok	boonpok	PROPN
ejpam-4367	258	6	and	and	CCONJ
ejpam-4367	258	7	c.	c.	PROPN
ejpam-4367	258	8	viriyapong	viriyapong	PROPN
ejpam-4367	258	9	.	.	PUNCT
ejpam-4367	259	1	on	on	ADP
ejpam-4367	259	2	generalized	generalized	ADJ
ejpam-4367	259	3	(	(	PUNCT
ejpam-4367	259	4	λ	λ	PROPN
ejpam-4367	259	5	,	,	PUNCT
ejpam-4367	259	6	sp)-closed	sp)-close	VERB
ejpam-4367	259	7	sets	set	NOUN
ejpam-4367	259	8	.	.	PUNCT
ejpam-4367	260	1	european	european	ADJ
ejpam-4367	260	2	journal	journal	PROPN
ejpam-4367	260	3	of	of	ADP
ejpam-4367	260	4	pure	pure	ADJ
ejpam-4367	260	5	and	and	CCONJ
ejpam-4367	260	6	applied	applied	ADJ
ejpam-4367	260	7	mathematics	mathematic	NOUN
ejpam-4367	260	8	,	,	PUNCT
ejpam-4367	260	9	(	(	PUNCT
ejpam-4367	260	10	accepted	accept	VERB
ejpam-4367	260	11	)	)	PUNCT
ejpam-4367	260	12	.	.	PUNCT
ejpam-4367	261	1	[	[	X
ejpam-4367	261	2	4	4	NUM
ejpam-4367	261	3	]	]	PUNCT
ejpam-4367	261	4	m.	m.	NOUN
ejpam-4367	261	5	caldas	caldas	PROPN
ejpam-4367	261	6	,	,	PUNCT
ejpam-4367	261	7	d.	d.	PROPN
ejpam-4367	261	8	n.	n.	PROPN
ejpam-4367	261	9	georgiou	georgiou	PROPN
ejpam-4367	261	10	,	,	PUNCT
ejpam-4367	261	11	s.	s.	PROPN
ejpam-4367	261	12	jafari	jafari	PROPN
ejpam-4367	261	13	,	,	PUNCT
ejpam-4367	261	14	and	and	CCONJ
ejpam-4367	261	15	t.	t.	PROPN
ejpam-4367	261	16	noiri	noiri	PROPN
ejpam-4367	261	17	.	.	PUNCT
ejpam-4367	262	1	on	on	ADP
ejpam-4367	262	2	(	(	PUNCT
ejpam-4367	262	3	λ	λ	PROPN
ejpam-4367	262	4	,	,	PUNCT
ejpam-4367	262	5	θ)-closed	θ)-close	VERB
ejpam-4367	262	6	sets	set	NOUN
ejpam-4367	262	7	.	.	PUNCT
ejpam-4367	263	1	questions	question	NOUN
ejpam-4367	263	2	and	and	CCONJ
ejpam-4367	263	3	answers	answer	NOUN
ejpam-4367	263	4	in	in	ADP
ejpam-4367	263	5	general	general	ADJ
ejpam-4367	263	6	topology	topology	NOUN
ejpam-4367	263	7	,	,	PUNCT
ejpam-4367	263	8	23:69–87	23:69–87	NUM
ejpam-4367	263	9	,	,	PUNCT
ejpam-4367	263	10	2005	2005	NUM
ejpam-4367	263	11	.	.	PUNCT
ejpam-4367	264	1	[	[	X
ejpam-4367	264	2	5	5	NUM
ejpam-4367	264	3	]	]	PUNCT
ejpam-4367	264	4	m.	m.	NOUN
ejpam-4367	264	5	caldas	caldas	PROPN
ejpam-4367	264	6	,	,	PUNCT
ejpam-4367	264	7	s.	s.	PROPN
ejpam-4367	264	8	jafari	jafari	PROPN
ejpam-4367	264	9	,	,	PUNCT
ejpam-4367	264	10	and	and	CCONJ
ejpam-4367	264	11	t.	t.	PROPN
ejpam-4367	264	12	noiri	noiri	PROPN
ejpam-4367	264	13	.	.	PUNCT
ejpam-4367	265	1	characterizations	characterization	NOUN
ejpam-4367	265	2	of	of	ADP
ejpam-4367	265	3	λθ	λθ	NOUN
ejpam-4367	265	4	-	-	PUNCT
ejpam-4367	265	5	r0	r0	NOUN
ejpam-4367	265	6	and	and	CCONJ
ejpam-4367	265	7	λθ	λθ	NOUN
ejpam-4367	265	8	-	-	PUNCT
ejpam-4367	265	9	r1	r1	NOUN
ejpam-4367	265	10	topological	topological	ADJ
ejpam-4367	265	11	spaces	space	NOUN
ejpam-4367	265	12	.	.	PUNCT
ejpam-4367	266	1	acta	acta	PROPN
ejpam-4367	266	2	mathematica	mathematica	PROPN
ejpam-4367	266	3	hungarica	hungarica	PROPN
ejpam-4367	266	4	,	,	PUNCT
ejpam-4367	266	5	103(1	103(1	NUM
ejpam-4367	266	6	-	-	SYM
ejpam-4367	266	7	2):89–95	2):89–95	NUM
ejpam-4367	266	8	,	,	PUNCT
ejpam-4367	266	9	2004	2004	NUM
ejpam-4367	266	10	.	.	PUNCT
ejpam-4367	267	1	[	[	X
ejpam-4367	267	2	6	6	NUM
ejpam-4367	267	3	]	]	PUNCT
ejpam-4367	267	4	f.	f.	PROPN
ejpam-4367	267	5	cammaroto	cammaroto	NOUN
ejpam-4367	267	6	and	and	CCONJ
ejpam-4367	267	7	t.	t.	PROPN
ejpam-4367	267	8	noiri	noiri	PROPN
ejpam-4367	267	9	.	.	PUNCT
ejpam-4367	268	1	on	on	ADP
ejpam-4367	268	2	λm	λm	NOUN
ejpam-4367	268	3	-	-	PUNCT
ejpam-4367	268	4	sets	set	NOUN
ejpam-4367	268	5	and	and	CCONJ
ejpam-4367	268	6	related	relate	VERB
ejpam-4367	268	7	topological	topological	ADJ
ejpam-4367	268	8	spaces	space	NOUN
ejpam-4367	268	9	.	.	PUNCT
ejpam-4367	269	1	acta	acta	PROPN
ejpam-4367	269	2	mathematica	mathematica	PROPN
ejpam-4367	269	3	hungarica	hungarica	PROPN
ejpam-4367	269	4	,	,	PUNCT
ejpam-4367	269	5	109:261–279	109:261–279	NUM
ejpam-4367	269	6	,	,	PUNCT
ejpam-4367	269	7	2005	2005	NUM
ejpam-4367	269	8	.	.	PUNCT
ejpam-4367	270	1	[	[	X
ejpam-4367	270	2	7	7	X
ejpam-4367	270	3	]	]	PUNCT
ejpam-4367	270	4	a.	a.	NOUN
ejpam-4367	270	5	s.	s.	PROPN
ejpam-4367	270	6	davis	davis	PROPN
ejpam-4367	270	7	.	.	PUNCT
ejpam-4367	271	1	indexed	index	VERB
ejpam-4367	271	2	systems	system	NOUN
ejpam-4367	271	3	of	of	ADP
ejpam-4367	271	4	neighborhoods	neighborhood	NOUN
ejpam-4367	271	5	for	for	ADP
ejpam-4367	271	6	general	general	ADJ
ejpam-4367	271	7	topological	topological	ADJ
ejpam-4367	271	8	spaces	space	NOUN
ejpam-4367	271	9	.	.	PUNCT
ejpam-4367	272	1	the	the	DET
ejpam-4367	272	2	american	american	PROPN
ejpam-4367	272	3	mathematical	mathematical	PROPN
ejpam-4367	272	4	monthly	monthly	ADV
ejpam-4367	272	5	,	,	PUNCT
ejpam-4367	272	6	68:886–893	68:886–893	PROPN
ejpam-4367	272	7	,	,	PUNCT
ejpam-4367	272	8	1961	1961	NUM
ejpam-4367	272	9	.	.	PUNCT
ejpam-4367	273	1	[	[	X
ejpam-4367	273	2	8	8	NUM
ejpam-4367	273	3	]	]	X
ejpam-4367	273	4	c.	c.	PROPN
ejpam-4367	273	5	dorsett	dorsett	PROPN
ejpam-4367	273	6	.	.	PUNCT
ejpam-4367	274	1	r0	r0	NOUN
ejpam-4367	274	2	and	and	CCONJ
ejpam-4367	274	3	r1	r1	PROPN
ejpam-4367	274	4	topological	topological	ADJ
ejpam-4367	274	5	spaces	space	NOUN
ejpam-4367	274	6	.	.	PUNCT
ejpam-4367	275	1	matematički	matematički	PROPN
ejpam-4367	275	2	vesnik	vesnik	PROPN
ejpam-4367	275	3	,	,	PUNCT
ejpam-4367	275	4	2(15)(30):117–122	2(15)(30):117–122	NUM
ejpam-4367	275	5	,	,	PUNCT
ejpam-4367	275	6	1978	1978	NUM
ejpam-4367	275	7	.	.	PUNCT
ejpam-4367	276	1	[	[	X
ejpam-4367	276	2	9	9	NUM
ejpam-4367	276	3	]	]	PUNCT
ejpam-4367	276	4	k.	k.	PROPN
ejpam-4367	276	5	k.	k.	PROPN
ejpam-4367	276	6	dube	dube	PROPN
ejpam-4367	276	7	.	.	PUNCT
ejpam-4367	277	1	a	a	DET
ejpam-4367	277	2	note	note	NOUN
ejpam-4367	277	3	on	on	ADP
ejpam-4367	277	4	r0	r0	PROPN
ejpam-4367	277	5	topological	topological	ADJ
ejpam-4367	277	6	spaces	space	NOUN
ejpam-4367	277	7	.	.	PUNCT
ejpam-4367	278	1	matematički	matematički	PROPN
ejpam-4367	278	2	vesnik	vesnik	PROPN
ejpam-4367	278	3	,	,	PUNCT
ejpam-4367	278	4	11:203–208	11:203–208	NUM
ejpam-4367	278	5	,	,	PUNCT
ejpam-4367	278	6	1974	1974	NUM
ejpam-4367	278	7	.	.	PUNCT
ejpam-4367	279	1	[	[	X
ejpam-4367	279	2	10	10	NUM
ejpam-4367	279	3	]	]	PUNCT
ejpam-4367	279	4	k.	k.	PROPN
ejpam-4367	279	5	k.	k.	PROPN
ejpam-4367	279	6	dube	dube	PROPN
ejpam-4367	279	7	.	.	PUNCT
ejpam-4367	280	1	a	a	DET
ejpam-4367	280	2	note	note	NOUN
ejpam-4367	280	3	on	on	ADP
ejpam-4367	280	4	r1	r1	PROPN
ejpam-4367	280	5	topological	topological	ADJ
ejpam-4367	280	6	spaces	space	NOUN
ejpam-4367	280	7	.	.	PUNCT
ejpam-4367	281	1	periodica	periodica	PROPN
ejpam-4367	281	2	mathematica	mathematica	PROPN
ejpam-4367	281	3	hungarica	hungarica	PROPN
ejpam-4367	281	4	,	,	PUNCT
ejpam-4367	281	5	13:267–271	13:267–271	NUM
ejpam-4367	281	6	,	,	PUNCT
ejpam-4367	281	7	1982	1982	NUM
ejpam-4367	281	8	.	.	PUNCT
ejpam-4367	282	1	[	[	X
ejpam-4367	282	2	11	11	NUM
ejpam-4367	282	3	]	]	PUNCT
ejpam-4367	282	4	m.	m.	NOUN
ejpam-4367	282	5	e.	e.	PROPN
ejpam-4367	282	6	abd	abd	PROPN
ejpam-4367	282	7	el	el	PROPN
ejpam-4367	282	8	-	-	PROPN
ejpam-4367	282	9	monsef	monsef	PROPN
ejpam-4367	282	10	,	,	PUNCT
ejpam-4367	282	11	s.	s.	PROPN
ejpam-4367	282	12	n.	n.	PROPN
ejpam-4367	282	13	el	el	PROPN
ejpam-4367	282	14	-	-	PROPN
ejpam-4367	282	15	deeb	deeb	PROPN
ejpam-4367	282	16	,	,	PUNCT
ejpam-4367	282	17	and	and	CCONJ
ejpam-4367	282	18	r.	r.	PROPN
ejpam-4367	282	19	a.	a.	PROPN
ejpam-4367	282	20	mahmoud	mahmoud	PROPN
ejpam-4367	282	21	.	.	PUNCT
ejpam-4367	283	1	β	β	X
ejpam-4367	283	2	-	-	ADJ
ejpam-4367	283	3	open	open	ADJ
ejpam-4367	283	4	sets	set	NOUN
ejpam-4367	283	5	and	and	CCONJ
ejpam-4367	283	6	βcontinuous	βcontinuous	ADJ
ejpam-4367	283	7	mappings	mapping	NOUN
ejpam-4367	283	8	.	.	PUNCT
ejpam-4367	284	1	bulletin	bulletin	NOUN
ejpam-4367	284	2	of	of	ADP
ejpam-4367	284	3	the	the	DET
ejpam-4367	284	4	faculty	faculty	NOUN
ejpam-4367	284	5	of	of	ADP
ejpam-4367	284	6	science	science	NOUN
ejpam-4367	284	7	.	.	PUNCT
ejpam-4367	285	1	assiut	assiut	PROPN
ejpam-4367	285	2	university	university	PROPN
ejpam-4367	285	3	,	,	PUNCT
ejpam-4367	285	4	12:77–90	12:77–90	NUM
ejpam-4367	285	5	,	,	PUNCT
ejpam-4367	285	6	1983	1983	NUM
ejpam-4367	285	7	.	.	PUNCT
ejpam-4367	286	1	[	[	X
ejpam-4367	286	2	12	12	NUM
ejpam-4367	286	3	]	]	X
ejpam-4367	286	4	s.	s.	PROPN
ejpam-4367	286	5	lugojan	lugojan	PROPN
ejpam-4367	286	6	.	.	PUNCT
ejpam-4367	287	1	generalized	generalized	ADJ
ejpam-4367	287	2	topology	topology	NOUN
ejpam-4367	287	3	.	.	PUNCT
ejpam-4367	288	1	studii	studii	PROPN
ejpam-4367	288	2	şi	şi	PROPN
ejpam-4367	288	3	cercetări	cercetări	PROPN
ejpam-4367	288	4	de	de	X
ejpam-4367	288	5	matematică	matematică	NOUN
ejpam-4367	288	6	,	,	PUNCT
ejpam-4367	288	7	34:348–360	34:348–360	PROPN
ejpam-4367	288	8	,	,	PUNCT
ejpam-4367	288	9	1982	1982	NUM
ejpam-4367	288	10	.	.	PUNCT
ejpam-4367	289	1	[	[	X
ejpam-4367	289	2	13	13	NUM
ejpam-4367	289	3	]	]	PUNCT
ejpam-4367	289	4	s.	s.	PROPN
ejpam-4367	289	5	n.	n.	PROPN
ejpam-4367	289	6	maheshwari	maheshwari	PROPN
ejpam-4367	289	7	and	and	CCONJ
ejpam-4367	289	8	r.	r.	PROPN
ejpam-4367	289	9	prasad	prasad	PROPN
ejpam-4367	289	10	.	.	PUNCT
ejpam-4367	290	1	on	on	ADP
ejpam-4367	290	2	(	(	PUNCT
ejpam-4367	290	3	r0)s	r0)s	NOUN
ejpam-4367	290	4	-	-	PUNCT
ejpam-4367	290	5	spaces	space	NOUN
ejpam-4367	290	6	.	.	PUNCT
ejpam-4367	291	1	portugaliae	portugaliae	PROPN
ejpam-4367	291	2	mathematica	mathematica	PROPN
ejpam-4367	291	3	,	,	PUNCT
ejpam-4367	291	4	34:213	34:213	NUM
ejpam-4367	291	5	–	–	PUNCT
ejpam-4367	291	6	217	217	NUM
ejpam-4367	291	7	,	,	PUNCT
ejpam-4367	291	8	1975	1975	NUM
ejpam-4367	291	9	.	.	PUNCT
ejpam-4367	292	1	[	[	X
ejpam-4367	292	2	14	14	NUM
ejpam-4367	292	3	]	]	PUNCT
ejpam-4367	292	4	m.	m.	NOUN
ejpam-4367	292	5	g.	g.	PROPN
ejpam-4367	292	6	murdeshwar	murdeshwar	PROPN
ejpam-4367	292	7	and	and	CCONJ
ejpam-4367	292	8	s.	s.	PROPN
ejpam-4367	292	9	a.	a.	PROPN
ejpam-4367	292	10	naimpally	naimpally	ADV
ejpam-4367	292	11	.	.	PUNCT
ejpam-4367	293	1	r1	r1	PROPN
ejpam-4367	293	2	topological	topological	ADJ
ejpam-4367	293	3	spaces	space	NOUN
ejpam-4367	293	4	.	.	PUNCT
ejpam-4367	294	1	canadian	canadian	ADJ
ejpam-4367	294	2	mathematical	mathematical	ADJ
ejpam-4367	294	3	bulletin	bulletin	NOUN
ejpam-4367	294	4	,	,	PUNCT
ejpam-4367	294	5	9:521–523	9:521–523	NOUN
ejpam-4367	294	6	,	,	PUNCT
ejpam-4367	294	7	1966	1966	NUM
ejpam-4367	294	8	.	.	PUNCT
ejpam-4367	295	1	[	[	X
ejpam-4367	295	2	15	15	NUM
ejpam-4367	295	3	]	]	X
ejpam-4367	295	4	s.	s.	PROPN
ejpam-4367	295	5	a.	a.	PROPN
ejpam-4367	295	6	naimpally	naimpally	ADV
ejpam-4367	295	7	.	.	PUNCT
ejpam-4367	296	1	on	on	ADP
ejpam-4367	296	2	r0	r0	PROPN
ejpam-4367	296	3	topological	topological	ADJ
ejpam-4367	296	4	spaces	space	NOUN
ejpam-4367	296	5	.	.	PUNCT
ejpam-4367	297	1	annales	annales	PROPN
ejpam-4367	297	2	universitatis	universitatis	PROPN
ejpam-4367	297	3	scientiarum	scientiarum	PROPN
ejpam-4367	297	4	budapestinensis	budapestinensis	PROPN
ejpam-4367	297	5	de	de	PROPN
ejpam-4367	297	6	rolando	rolando	PROPN
ejpam-4367	297	7	eötvös	eötvös	PROPN
ejpam-4367	297	8	nominatae	nominatae	NOUN
ejpam-4367	297	9	sectio	sectio	NOUN
ejpam-4367	297	10	mathematica	mathematica	PROPN
ejpam-4367	297	11	,	,	PUNCT
ejpam-4367	297	12	10:53–54	10:53–54	NUM
ejpam-4367	297	13	,	,	PUNCT
ejpam-4367	297	14	1967	1967	NUM
ejpam-4367	297	15	.	.	PUNCT
ejpam-4367	298	1	references	reference	NOUN
ejpam-4367	298	2	886	886	NUM
ejpam-4367	299	1	[	[	X
ejpam-4367	299	2	16	16	NUM
ejpam-4367	299	3	]	]	PUNCT
ejpam-4367	299	4	t.	t.	PROPN
ejpam-4367	299	5	noiri	noiri	PROPN
ejpam-4367	299	6	.	.	PUNCT
ejpam-4367	300	1	unified	unify	VERB
ejpam-4367	300	2	characterizations	characterization	NOUN
ejpam-4367	300	3	for	for	ADP
ejpam-4367	300	4	modifications	modification	NOUN
ejpam-4367	300	5	of	of	ADP
ejpam-4367	300	6	r0	r0	NOUN
ejpam-4367	300	7	and	and	CCONJ
ejpam-4367	300	8	r1	r1	PROPN
ejpam-4367	300	9	topological	topological	ADJ
ejpam-4367	300	10	spaces	space	NOUN
ejpam-4367	300	11	.	.	PUNCT
ejpam-4367	301	1	rendiconti	rendiconti	ADJ
ejpam-4367	301	2	del	del	PROPN
ejpam-4367	301	3	circolo	circolo	PROPN
ejpam-4367	301	4	matematico	matematico	X
ejpam-4367	301	5	de	de	X
ejpam-4367	301	6	palermo	palermo	X
ejpam-4367	301	7	(	(	PUNCT
ejpam-4367	301	8	2	2	NUM
ejpam-4367	301	9	)	)	PUNCT
ejpam-4367	301	10	,	,	PUNCT
ejpam-4367	301	11	60:29–42	60:29–42	PROPN
ejpam-4367	301	12	,	,	PUNCT
ejpam-4367	301	13	2006	2006	NUM
ejpam-4367	301	14	.	.	PUNCT
ejpam-4367	302	1	[	[	X
ejpam-4367	302	2	17	17	NUM
ejpam-4367	302	3	]	]	PUNCT
ejpam-4367	302	4	t.	t.	PROPN
ejpam-4367	302	5	noiri	noiri	PROPN
ejpam-4367	302	6	and	and	CCONJ
ejpam-4367	302	7	e.	e.	PROPN
ejpam-4367	302	8	hatir	hatir	PROPN
ejpam-4367	302	9	.	.	PUNCT
ejpam-4367	303	1	λsp	λsp	NOUN
ejpam-4367	303	2	-	-	PUNCT
ejpam-4367	303	3	sets	set	NOUN
ejpam-4367	303	4	and	and	CCONJ
ejpam-4367	303	5	some	some	DET
ejpam-4367	303	6	weak	weak	ADJ
ejpam-4367	303	7	separation	separation	NOUN
ejpam-4367	303	8	axioms	axiom	NOUN
ejpam-4367	303	9	.	.	PUNCT
ejpam-4367	304	1	acta	acta	PROPN
ejpam-4367	304	2	mathematica	mathematica	PROPN
ejpam-4367	304	3	hungarica	hungarica	PROPN
ejpam-4367	304	4	,	,	PUNCT
ejpam-4367	304	5	103(3):225–232	103(3):225–232	NUM
ejpam-4367	304	6	,	,	PUNCT
ejpam-4367	304	7	2004	2004	NUM
ejpam-4367	304	8	.	.	PUNCT
ejpam-4367	305	1	[	[	X
ejpam-4367	305	2	18	18	NUM
ejpam-4367	305	3	]	]	X
ejpam-4367	305	4	n.	n.	NOUN
ejpam-4367	305	5	a.	a.	NOUN
ejpam-4367	305	6	shanin	shanin	PROPN
ejpam-4367	305	7	.	.	PUNCT
ejpam-4367	306	1	on	on	ADP
ejpam-4367	306	2	separation	separation	NOUN
ejpam-4367	306	3	in	in	ADP
ejpam-4367	306	4	topological	topological	ADJ
ejpam-4367	306	5	spaces	space	NOUN
ejpam-4367	306	6	.	.	PUNCT
ejpam-4367	307	1	doklady	doklady	PROPN
ejpam-4367	307	2	akademii	akademii	NOUN
ejpam-4367	307	3	nauk	nauk	NOUN
ejpam-4367	307	4	sssr	sssr	NOUN
ejpam-4367	307	5	,	,	PUNCT
ejpam-4367	307	6	38:110–113	38:110–113	NUM
ejpam-4367	307	7	,	,	PUNCT
ejpam-4367	307	8	1943	1943	NUM
ejpam-4367	307	9	.	.	PUNCT
ejpam-4367	308	1	[	[	X
ejpam-4367	308	2	19	19	NUM
ejpam-4367	308	3	]	]	X
ejpam-4367	308	4	n.	n.	NOUN
ejpam-4367	308	5	v.	v.	PROPN
ejpam-4367	308	6	veličko	veličko	PROPN
ejpam-4367	308	7	.	.	PUNCT
ejpam-4367	309	1	h	h	NOUN
ejpam-4367	309	2	-	-	PUNCT
ejpam-4367	309	3	closed	close	VERB
ejpam-4367	309	4	topological	topological	ADJ
ejpam-4367	309	5	spaces	space	NOUN
ejpam-4367	309	6	.	.	PUNCT
ejpam-4367	310	1	american	american	PROPN
ejpam-4367	310	2	mathematical	mathematical	ADJ
ejpam-4367	310	3	society	society	NOUN
ejpam-4367	310	4	translations	translation	NOUN
ejpam-4367	310	5	:	:	PUNCT
ejpam-4367	310	6	series	series	NOUN
ejpam-4367	310	7	2	2	NUM
ejpam-4367	310	8	,	,	PUNCT
ejpam-4367	310	9	78:102–118	78:102–118	NUM
ejpam-4367	310	10	,	,	PUNCT
ejpam-4367	310	11	1968	1968	NUM
ejpam-4367	310	12	.	.	PUNCT