id	sid	tid	token	lemma	pos
ejpam-4340	1	1	european	european	PROPN
ejpam-4340	1	2	journal	journal	PROPN
ejpam-4340	1	3	of	of	ADP
ejpam-4340	1	4	pure	pure	ADJ
ejpam-4340	1	5	and	and	CCONJ
ejpam-4340	1	6	applied	apply	VERB
ejpam-4340	1	7	mathematics	mathematic	NOUN
ejpam-4340	1	8	vol	vol	NOUN
ejpam-4340	1	9	.	.	PROPN
ejpam-4340	2	1	15	15	NUM
ejpam-4340	2	2	,	,	PUNCT
ejpam-4340	2	3	no	no	INTJ
ejpam-4340	2	4	.	.	NOUN
ejpam-4340	2	5	2	2	NUM
ejpam-4340	2	6	,	,	PUNCT
ejpam-4340	2	7	2022	2022	NUM
ejpam-4340	2	8	,	,	PUNCT
ejpam-4340	2	9	354	354	NUM
ejpam-4340	2	10	-	-	SYM
ejpam-4340	2	11	374	374	NUM
ejpam-4340	2	12	issn	issn	PROPN
ejpam-4340	2	13	1307	1307	NUM
ejpam-4340	2	14	-	-	SYM
ejpam-4340	2	15	5543	5543	NUM
ejpam-4340	2	16	–	–	PUNCT
ejpam-4340	2	17	ejpam.com	ejpam.com	X
ejpam-4340	2	18	published	publish	VERB
ejpam-4340	2	19	by	by	ADP
ejpam-4340	2	20	new	new	PROPN
ejpam-4340	2	21	york	york	PROPN
ejpam-4340	2	22	business	business	PROPN
ejpam-4340	2	23	global	global	ADJ
ejpam-4340	2	24	on	on	ADP
ejpam-4340	2	25	generalized	generalized	ADJ
ejpam-4340	2	26	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	2	27	sets	set	NOUN
ejpam-4340	2	28	pınar	pınar	PROPN
ejpam-4340	2	29	şaşmaz1	şaşmaz1	NOUN
ejpam-4340	2	30	,	,	PUNCT
ejpam-4340	2	31	murad	murad	NOUN
ejpam-4340	2	32	özkoç2,∗	özkoç2,∗	NOUN
ejpam-4340	2	33	1	1	NUM
ejpam-4340	2	34	muğla	muğla	NOUN
ejpam-4340	2	35	sıtkı	sıtkı	PROPN
ejpam-4340	2	36	koçman	koçman	PROPN
ejpam-4340	2	37	university	university	NOUN
ejpam-4340	2	38	,	,	PUNCT
ejpam-4340	2	39	graduate	graduate	NOUN
ejpam-4340	2	40	school	school	NOUN
ejpam-4340	2	41	of	of	ADP
ejpam-4340	2	42	applied	applied	ADJ
ejpam-4340	2	43	and	and	CCONJ
ejpam-4340	2	44	natural	natural	ADJ
ejpam-4340	2	45	sciences	science	NOUN
ejpam-4340	2	46	,	,	PUNCT
ejpam-4340	2	47	mathematics	mathematic	NOUN
ejpam-4340	2	48	,	,	PUNCT
ejpam-4340	2	49	48000	48000	NUM
ejpam-4340	2	50	menteşe	menteşe	NOUN
ejpam-4340	2	51	-	-	PUNCT
ejpam-4340	2	52	muğla	muğla	NOUN
ejpam-4340	2	53	,	,	PUNCT
ejpam-4340	2	54	turkey	turkey	NOUN
ejpam-4340	2	55	2	2	NUM
ejpam-4340	2	56	muğla	muğla	NOUN
ejpam-4340	2	57	sıtkı	sıtkı	ADJ
ejpam-4340	2	58	koçman	koçman	PROPN
ejpam-4340	2	59	university	university	NOUN
ejpam-4340	2	60	,	,	PUNCT
ejpam-4340	2	61	faculty	faculty	NOUN
ejpam-4340	2	62	of	of	ADP
ejpam-4340	2	63	science	science	NOUN
ejpam-4340	2	64	,	,	PUNCT
ejpam-4340	2	65	department	department	NOUN
ejpam-4340	2	66	of	of	ADP
ejpam-4340	2	67	mathematics	mathematic	NOUN
ejpam-4340	2	68	,	,	PUNCT
ejpam-4340	2	69	48000	48000	NUM
ejpam-4340	2	70	menteşe	menteşe	NOUN
ejpam-4340	2	71	-	-	PUNCT
ejpam-4340	2	72	muğla	muğla	NOUN
ejpam-4340	2	73	,	,	PUNCT
ejpam-4340	2	74	turkey	turkey	NOUN
ejpam-4340	2	75	abstract	abstract	NOUN
ejpam-4340	2	76	.	.	PUNCT
ejpam-4340	3	1	the	the	DET
ejpam-4340	3	2	aim	aim	NOUN
ejpam-4340	3	3	of	of	ADP
ejpam-4340	3	4	this	this	DET
ejpam-4340	3	5	paper	paper	NOUN
ejpam-4340	3	6	is	be	AUX
ejpam-4340	3	7	to	to	PART
ejpam-4340	3	8	introduce	introduce	VERB
ejpam-4340	3	9	and	and	CCONJ
ejpam-4340	3	10	study	study	VERB
ejpam-4340	3	11	a	a	DET
ejpam-4340	3	12	new	new	ADJ
ejpam-4340	3	13	type	type	NOUN
ejpam-4340	3	14	of	of	ADP
ejpam-4340	3	15	generalized	generalized	ADJ
ejpam-4340	3	16	closed	close	VERB
ejpam-4340	3	17	sets	set	NOUN
ejpam-4340	3	18	,	,	PUNCT
ejpam-4340	3	19	called	call	VERB
ejpam-4340	3	20	generalized	generalized	ADJ
ejpam-4340	3	21	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	3	22	(	(	PUNCT
ejpam-4340	3	23	briefly	briefly	ADV
ejpam-4340	3	24	,	,	PUNCT
ejpam-4340	3	25	gωe∗-closed	gωe∗-closed	ADJ
ejpam-4340	3	26	)	)	PUNCT
ejpam-4340	3	27	sets	set	NOUN
ejpam-4340	3	28	,	,	PUNCT
ejpam-4340	3	29	via	via	ADP
ejpam-4340	3	30	ωe∗-closure	ωe∗-closure	NOUN
ejpam-4340	3	31	operator	operator	NOUN
ejpam-4340	3	32	.	.	PUNCT
ejpam-4340	4	1	we	we	PRON
ejpam-4340	4	2	examine	examine	VERB
ejpam-4340	4	3	the	the	DET
ejpam-4340	4	4	fundamental	fundamental	ADJ
ejpam-4340	4	5	properties	property	NOUN
ejpam-4340	4	6	of	of	ADP
ejpam-4340	4	7	the	the	DET
ejpam-4340	4	8	class	class	NOUN
ejpam-4340	4	9	of	of	ADP
ejpam-4340	4	10	these	these	DET
ejpam-4340	4	11	sets	set	NOUN
ejpam-4340	4	12	.	.	PUNCT
ejpam-4340	5	1	the	the	DET
ejpam-4340	5	2	notion	notion	NOUN
ejpam-4340	5	3	of	of	ADP
ejpam-4340	5	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	5	5	set	set	NOUN
ejpam-4340	5	6	is	be	AUX
ejpam-4340	5	7	weaker	weak	ADJ
ejpam-4340	5	8	than	than	ADP
ejpam-4340	5	9	the	the	DET
ejpam-4340	5	10	notions	notion	NOUN
ejpam-4340	5	11	of	of	ADP
ejpam-4340	5	12	gωβ	gωβ	NOUN
ejpam-4340	5	13	-	-	PUNCT
ejpam-4340	5	14	closed	closed	ADJ
ejpam-4340	5	15	set	set	NOUN
ejpam-4340	5	16	and	and	CCONJ
ejpam-4340	5	17	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	5	18	set	set	NOUN
ejpam-4340	5	19	in	in	ADP
ejpam-4340	5	20	the	the	DET
ejpam-4340	5	21	literature	literature	NOUN
ejpam-4340	5	22	.	.	PUNCT
ejpam-4340	6	1	also	also	ADV
ejpam-4340	6	2	,	,	PUNCT
ejpam-4340	6	3	we	we	PRON
ejpam-4340	6	4	define	define	VERB
ejpam-4340	6	5	and	and	CCONJ
ejpam-4340	6	6	discuss	discuss	VERB
ejpam-4340	6	7	the	the	DET
ejpam-4340	6	8	notions	notion	NOUN
ejpam-4340	6	9	of	of	ADP
ejpam-4340	6	10	generalized	generalized	ADJ
ejpam-4340	6	11	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	6	12	and	and	CCONJ
ejpam-4340	6	13	generalized	generalized	ADJ
ejpam-4340	6	14	ωe∗-irresolute	ωe∗-irresolute	NOUN
ejpam-4340	6	15	functions	function	NOUN
ejpam-4340	6	16	.	.	PUNCT
ejpam-4340	7	1	2020	2020	NUM
ejpam-4340	7	2	mathematics	mathematic	NOUN
ejpam-4340	7	3	subject	subject	NOUN
ejpam-4340	7	4	classifications	classification	NOUN
ejpam-4340	7	5	:	:	PUNCT
ejpam-4340	7	6	54c05	54c05	NUM
ejpam-4340	7	7	,	,	PUNCT
ejpam-4340	7	8	54c08	54c08	NUM
ejpam-4340	7	9	key	key	ADJ
ejpam-4340	7	10	words	word	NOUN
ejpam-4340	7	11	and	and	CCONJ
ejpam-4340	7	12	phrases	phrase	NOUN
ejpam-4340	7	13	:	:	PUNCT
ejpam-4340	7	14	generalized	generalized	ADJ
ejpam-4340	7	15	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	7	16	set	set	NOUN
ejpam-4340	7	17	,	,	PUNCT
ejpam-4340	7	18	generalized	generalize	VERB
ejpam-4340	7	19	ωe∗-open	ωe∗-open	NOUN
ejpam-4340	7	20	set	set	NOUN
ejpam-4340	7	21	,	,	PUNCT
ejpam-4340	7	22	generalized	generalized	ADJ
ejpam-4340	7	23	ωe∗-neighborhood	ωe∗-neighborhood	NOUN
ejpam-4340	7	24	,	,	PUNCT
ejpam-4340	7	25	gωe∗-continuity	gωe∗-continuity	NOUN
ejpam-4340	7	26	,	,	PUNCT
ejpam-4340	7	27	gωe∗-irresoluteness	gωe∗-irresoluteness	PROPN
ejpam-4340	7	28	1	1	NUM
ejpam-4340	7	29	.	.	PUNCT
ejpam-4340	7	30	introduction	introduction	NOUN
ejpam-4340	7	31	the	the	DET
ejpam-4340	7	32	notion	notion	NOUN
ejpam-4340	7	33	of	of	ADP
ejpam-4340	7	34	the	the	DET
ejpam-4340	7	35	generalized	generalize	VERB
ejpam-4340	7	36	closed	close	VERB
ejpam-4340	7	37	set	set	NOUN
ejpam-4340	7	38	is	be	AUX
ejpam-4340	7	39	an	an	DET
ejpam-4340	7	40	important	important	ADJ
ejpam-4340	7	41	concept	concept	NOUN
ejpam-4340	7	42	in	in	ADP
ejpam-4340	7	43	the	the	DET
ejpam-4340	7	44	area	area	NOUN
ejpam-4340	7	45	of	of	ADP
ejpam-4340	7	46	general	general	ADJ
ejpam-4340	7	47	topology	topology	NOUN
ejpam-4340	7	48	.	.	PUNCT
ejpam-4340	8	1	it	it	PRON
ejpam-4340	8	2	was	be	AUX
ejpam-4340	8	3	first	first	ADV
ejpam-4340	8	4	introduced	introduce	VERB
ejpam-4340	8	5	by	by	ADP
ejpam-4340	8	6	levine	levine	PROPN
ejpam-4340	8	7	[	[	X
ejpam-4340	8	8	14	14	NUM
ejpam-4340	8	9	]	]	PUNCT
ejpam-4340	8	10	in	in	ADP
ejpam-4340	8	11	1970	1970	NUM
ejpam-4340	8	12	.	.	PUNCT
ejpam-4340	9	1	since	since	SCONJ
ejpam-4340	9	2	then	then	ADV
ejpam-4340	9	3	,	,	PUNCT
ejpam-4340	9	4	many	many	ADJ
ejpam-4340	9	5	forms	form	NOUN
ejpam-4340	9	6	of	of	ADP
ejpam-4340	9	7	this	this	DET
ejpam-4340	9	8	notion	notion	NOUN
ejpam-4340	9	9	such	such	ADJ
ejpam-4340	9	10	as	as	ADP
ejpam-4340	9	11	gα	gα	NOUN
ejpam-4340	9	12	-	-	PUNCT
ejpam-4340	9	13	closed	closed	ADJ
ejpam-4340	9	14	[	[	X
ejpam-4340	9	15	15	15	NUM
ejpam-4340	9	16	]	]	PUNCT
ejpam-4340	9	17	,	,	PUNCT
ejpam-4340	9	18	gs	gs	NOUN
ejpam-4340	9	19	-	-	PUNCT
ejpam-4340	9	20	closed	close	VERB
ejpam-4340	9	21	[	[	X
ejpam-4340	9	22	7	7	NUM
ejpam-4340	9	23	]	]	PUNCT
ejpam-4340	9	24	,	,	PUNCT
ejpam-4340	9	25	gp	gp	NOUN
ejpam-4340	9	26	-	-	ADJ
ejpam-4340	9	27	closed	closed	ADJ
ejpam-4340	9	28	[	[	X
ejpam-4340	9	29	16	16	NUM
ejpam-4340	9	30	]	]	X
ejpam-4340	9	31	,	,	PUNCT
ejpam-4340	9	32	gb	gb	ADV
ejpam-4340	9	33	-	-	PUNCT
ejpam-4340	9	34	closed	close	VERB
ejpam-4340	9	35	[	[	X
ejpam-4340	9	36	18	18	NUM
ejpam-4340	9	37	]	]	PUNCT
ejpam-4340	9	38	,	,	PUNCT
ejpam-4340	9	39	gβ	gβ	NOUN
ejpam-4340	9	40	-	-	PUNCT
ejpam-4340	9	41	closed	closed	ADJ
ejpam-4340	9	42	[	[	X
ejpam-4340	9	43	21	21	NUM
ejpam-4340	9	44	]	]	PUNCT
ejpam-4340	9	45	,	,	PUNCT
ejpam-4340	9	46	ge	ge	PROPN
ejpam-4340	9	47	-	-	PUNCT
ejpam-4340	9	48	closed	close	VERB
ejpam-4340	9	49	[	[	X
ejpam-4340	9	50	8	8	NUM
ejpam-4340	9	51	]	]	PUNCT
ejpam-4340	9	52	,	,	PUNCT
ejpam-4340	9	53	πge	πge	NOUN
ejpam-4340	9	54	-	-	PUNCT
ejpam-4340	9	55	closed	close	VERB
ejpam-4340	9	56	[	[	X
ejpam-4340	9	57	9	9	NUM
ejpam-4340	9	58	]	]	PUNCT
ejpam-4340	9	59	,	,	PUNCT
ejpam-4340	9	60	gω	gω	PROPN
ejpam-4340	9	61	-	-	PUNCT
ejpam-4340	9	62	closed	close	VERB
ejpam-4340	9	63	[	[	X
ejpam-4340	9	64	5	5	NUM
ejpam-4340	9	65	]	]	PUNCT
ejpam-4340	9	66	,	,	PUNCT
ejpam-4340	9	67	and	and	CCONJ
ejpam-4340	9	68	generalized	generalized	ADJ
ejpam-4340	9	69	ωβ	ωβ	ADJ
ejpam-4340	9	70	-	-	ADJ
ejpam-4340	9	71	closed	closed	ADJ
ejpam-4340	9	72	[	[	X
ejpam-4340	9	73	4	4	NUM
ejpam-4340	9	74	]	]	PUNCT
ejpam-4340	9	75	have	have	AUX
ejpam-4340	9	76	been	be	AUX
ejpam-4340	9	77	defined	define	VERB
ejpam-4340	9	78	and	and	CCONJ
ejpam-4340	9	79	studied	study	VERB
ejpam-4340	9	80	by	by	ADP
ejpam-4340	9	81	many	many	ADJ
ejpam-4340	9	82	mathematicians	mathematician	NOUN
ejpam-4340	9	83	.	.	PUNCT
ejpam-4340	10	1	moreover	moreover	ADV
ejpam-4340	10	2	,	,	PUNCT
ejpam-4340	10	3	the	the	DET
ejpam-4340	10	4	authors	author	NOUN
ejpam-4340	10	5	have	have	AUX
ejpam-4340	10	6	introduced	introduce	VERB
ejpam-4340	10	7	many	many	ADJ
ejpam-4340	10	8	new	new	ADJ
ejpam-4340	10	9	concepts	concept	NOUN
ejpam-4340	10	10	via	via	ADP
ejpam-4340	10	11	these	these	DET
ejpam-4340	10	12	new	new	ADJ
ejpam-4340	10	13	types	type	NOUN
ejpam-4340	10	14	of	of	ADP
ejpam-4340	10	15	sets	set	NOUN
ejpam-4340	10	16	.	.	PUNCT
ejpam-4340	11	1	they	they	PRON
ejpam-4340	11	2	have	have	AUX
ejpam-4340	11	3	also	also	ADV
ejpam-4340	11	4	investigated	investigate	VERB
ejpam-4340	11	5	some	some	PRON
ejpam-4340	11	6	of	of	ADP
ejpam-4340	11	7	their	their	PRON
ejpam-4340	11	8	fundamental	fundamental	ADJ
ejpam-4340	11	9	properties	property	NOUN
ejpam-4340	11	10	and	and	CCONJ
ejpam-4340	11	11	characterizations	characterization	NOUN
ejpam-4340	11	12	of	of	ADP
ejpam-4340	11	13	these	these	DET
ejpam-4340	11	14	concepts	concept	NOUN
ejpam-4340	11	15	.	.	PUNCT
ejpam-4340	12	1	furthermore	furthermore	ADV
ejpam-4340	12	2	,	,	PUNCT
ejpam-4340	12	3	they	they	PRON
ejpam-4340	12	4	have	have	AUX
ejpam-4340	12	5	not	not	PART
ejpam-4340	12	6	only	only	ADV
ejpam-4340	12	7	discussed	discuss	VERB
ejpam-4340	12	8	their	their	PRON
ejpam-4340	12	9	fundamental	fundamental	ADJ
ejpam-4340	12	10	properties	property	NOUN
ejpam-4340	12	11	but	but	CCONJ
ejpam-4340	12	12	also	also	ADV
ejpam-4340	12	13	put	put	VERB
ejpam-4340	12	14	forth	forth	ADP
ejpam-4340	12	15	the	the	DET
ejpam-4340	12	16	relationships	relationship	NOUN
ejpam-4340	12	17	between	between	ADP
ejpam-4340	12	18	them	they	PRON
ejpam-4340	12	19	and	and	CCONJ
ejpam-4340	12	20	the	the	DET
ejpam-4340	12	21	notions	notion	NOUN
ejpam-4340	12	22	in	in	ADP
ejpam-4340	12	23	the	the	DET
ejpam-4340	12	24	literature	literature	NOUN
ejpam-4340	12	25	.	.	PUNCT
ejpam-4340	13	1	in	in	ADP
ejpam-4340	13	2	this	this	DET
ejpam-4340	13	3	study	study	NOUN
ejpam-4340	13	4	,	,	PUNCT
ejpam-4340	13	5	we	we	PRON
ejpam-4340	13	6	define	define	VERB
ejpam-4340	13	7	a	a	DET
ejpam-4340	13	8	new	new	ADJ
ejpam-4340	13	9	concept	concept	NOUN
ejpam-4340	13	10	called	call	VERB
ejpam-4340	13	11	generalized	generalized	ADJ
ejpam-4340	13	12	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	13	13	sets	set	NOUN
ejpam-4340	13	14	via	via	ADP
ejpam-4340	13	15	the	the	DET
ejpam-4340	13	16	ωe∗closure	ωe∗closure	NOUN
ejpam-4340	13	17	operator	operator	NOUN
ejpam-4340	13	18	.	.	PUNCT
ejpam-4340	14	1	we	we	PRON
ejpam-4340	14	2	examine	examine	VERB
ejpam-4340	14	3	the	the	DET
ejpam-4340	14	4	relationships	relationship	NOUN
ejpam-4340	14	5	among	among	ADP
ejpam-4340	14	6	this	this	DET
ejpam-4340	14	7	new	new	ADJ
ejpam-4340	14	8	concept	concept	NOUN
ejpam-4340	14	9	and	and	CCONJ
ejpam-4340	14	10	some	some	DET
ejpam-4340	14	11	other	other	ADJ
ejpam-4340	14	12	concepts	concept	NOUN
ejpam-4340	14	13	existing	exist	VERB
ejpam-4340	14	14	in	in	ADP
ejpam-4340	14	15	the	the	DET
ejpam-4340	14	16	literature	literature	NOUN
ejpam-4340	14	17	such	such	ADJ
ejpam-4340	14	18	as	as	ADP
ejpam-4340	14	19	generalized	generalized	ADJ
ejpam-4340	14	20	β	β	NOUN
ejpam-4340	14	21	-	-	VERB
ejpam-4340	14	22	closed	closed	ADJ
ejpam-4340	14	23	,	,	PUNCT
ejpam-4340	14	24	generalized	generalize	VERB
ejpam-4340	14	25	e∗-closed	e∗-close	VERB
ejpam-4340	14	26	,	,	PUNCT
ejpam-4340	14	27	generalized	generalized	ADJ
ejpam-4340	14	28	ω	ω	NOUN
ejpam-4340	14	29	-	-	ADJ
ejpam-4340	14	30	closed	closed	ADJ
ejpam-4340	14	31	,	,	PUNCT
ejpam-4340	14	32	and	and	CCONJ
ejpam-4340	14	33	generalized	generalized	ADJ
ejpam-4340	14	34	ωβ	ωβ	ADJ
ejpam-4340	14	35	-	-	PUNCT
ejpam-4340	14	36	closed	closed	ADJ
ejpam-4340	14	37	.	.	PUNCT
ejpam-4340	15	1	in	in	ADP
ejpam-4340	15	2	addition	addition	NOUN
ejpam-4340	15	3	,	,	PUNCT
ejpam-4340	15	4	by	by	ADP
ejpam-4340	15	5	giving	give	VERB
ejpam-4340	15	6	the	the	DET
ejpam-4340	15	7	notion	notion	NOUN
ejpam-4340	15	8	of	of	ADP
ejpam-4340	15	9	ωe∗-limit	ωe∗-limit	ADJ
ejpam-4340	15	10	point	point	NOUN
ejpam-4340	15	11	,	,	PUNCT
ejpam-4340	15	12	we	we	PRON
ejpam-4340	15	13	prove	prove	VERB
ejpam-4340	15	14	that	that	SCONJ
ejpam-4340	15	15	the	the	DET
ejpam-4340	15	16	union	union	NOUN
ejpam-4340	15	17	of	of	ADP
ejpam-4340	15	18	two	two	NUM
ejpam-4340	15	19	generalized	generalized	ADJ
ejpam-4340	15	20	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	15	21	sets	set	NOUN
ejpam-4340	15	22	is	be	AUX
ejpam-4340	15	23	a	a	DET
ejpam-4340	15	24	generalized	generalized	ADJ
ejpam-4340	15	25	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	15	26	set	set	VERB
ejpam-4340	15	27	under	under	ADP
ejpam-4340	15	28	a	a	DET
ejpam-4340	15	29	special	special	ADJ
ejpam-4340	15	30	condition	condition	NOUN
ejpam-4340	15	31	.	.	PUNCT
ejpam-4340	16	1	furthermore	furthermore	ADV
ejpam-4340	16	2	,	,	PUNCT
ejpam-4340	16	3	the	the	DET
ejpam-4340	16	4	notions	notion	NOUN
ejpam-4340	16	5	of	of	ADP
ejpam-4340	16	6	generalized	generalized	ADJ
ejpam-4340	16	7	ωe∗-continuity	ωe∗-continuity	NOUN
ejpam-4340	16	8	and	and	CCONJ
ejpam-4340	16	9	generalized	generalized	ADJ
ejpam-4340	16	10	ωe∗-irresoluteness	ωe∗-irresoluteness	NOUN
ejpam-4340	16	11	have	have	AUX
ejpam-4340	16	12	been	be	AUX
ejpam-4340	16	13	introduced	introduce	VERB
ejpam-4340	16	14	and	and	CCONJ
ejpam-4340	16	15	finally	finally	ADV
ejpam-4340	16	16	many	many	ADJ
ejpam-4340	16	17	basic	basic	ADJ
ejpam-4340	16	18	properties	property	NOUN
ejpam-4340	16	19	of	of	ADP
ejpam-4340	16	20	such	such	ADJ
ejpam-4340	16	21	functions	function	NOUN
ejpam-4340	16	22	are	be	AUX
ejpam-4340	16	23	obtained	obtain	VERB
ejpam-4340	16	24	.	.	PUNCT
ejpam-4340	17	1	∗corresponding	∗corresponde	VERB
ejpam-4340	17	2	author	author	NOUN
ejpam-4340	17	3	.	.	PUNCT
ejpam-4340	18	1	doi	doi	NOUN
ejpam-4340	18	2	:	:	PUNCT
ejpam-4340	18	3	https://doi.org/10.29020/nybg.ejpam.v15i2.4340	https://doi.org/10.29020/nybg.ejpam.v15i2.4340	PROPN
ejpam-4340	18	4	email	email	NOUN
ejpam-4340	18	5	addresses	address	NOUN
ejpam-4340	18	6	:	:	PUNCT
ejpam-4340	18	7	pinarsasmaz@posta.mu.edu.tr	pinarsasmaz@posta.mu.edu.tr	PROPN
ejpam-4340	18	8	(	(	PUNCT
ejpam-4340	18	9	p.	p.	NOUN
ejpam-4340	18	10	şaşmaz	şaşmaz	PUNCT
ejpam-4340	18	11	)	)	PUNCT
ejpam-4340	18	12	,	,	PUNCT
ejpam-4340	18	13	murad.ozkoc@mu.edu.tr	murad.ozkoc@mu.edu.tr	PROPN
ejpam-4340	18	14	(	(	PUNCT
ejpam-4340	18	15	m.	m.	NOUN
ejpam-4340	18	16	özkoç	özkoç	PROPN
ejpam-4340	18	17	)	)	PUNCT
ejpam-4340	18	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4340	19	1	354	354	NUM
ejpam-4340	19	2	©	©	PROPN
ejpam-4340	19	3	2022	2022	NUM
ejpam-4340	19	4	ejpam	ejpam	VERB
ejpam-4340	19	5	all	all	DET
ejpam-4340	19	6	rights	right	NOUN
ejpam-4340	19	7	reserved	reserve	VERB
ejpam-4340	19	8	.	.	PUNCT
ejpam-4340	20	1	p.	p.	NOUN
ejpam-4340	20	2	şaşmaz	şaşmaz	NUM
ejpam-4340	20	3	,	,	PUNCT
ejpam-4340	20	4	m.	m.	NOUN
ejpam-4340	20	5	özkoç	özkoç	PROPN
ejpam-4340	20	6	/	/	SYM
ejpam-4340	20	7	eur	eur	PROPN
ejpam-4340	20	8	.	.	PUNCT
ejpam-4340	21	1	j.	j.	PROPN
ejpam-4340	21	2	pure	pure	PROPN
ejpam-4340	21	3	appl	appl	PROPN
ejpam-4340	21	4	.	.	PROPN
ejpam-4340	21	5	math	math	PROPN
ejpam-4340	21	6	,	,	PUNCT
ejpam-4340	21	7	15	15	NUM
ejpam-4340	21	8	(	(	PUNCT
ejpam-4340	21	9	2	2	NUM
ejpam-4340	21	10	)	)	PUNCT
ejpam-4340	21	11	(	(	PUNCT
ejpam-4340	21	12	2022	2022	NUM
ejpam-4340	21	13	)	)	PUNCT
ejpam-4340	21	14	,	,	PUNCT
ejpam-4340	21	15	354	354	NUM
ejpam-4340	21	16	-	-	SYM
ejpam-4340	21	17	374	374	NUM
ejpam-4340	21	18	355	355	NUM
ejpam-4340	21	19	2	2	NUM
ejpam-4340	21	20	.	.	PUNCT
ejpam-4340	21	21	preliminaries	preliminary	NOUN
ejpam-4340	21	22	throughout	throughout	ADP
ejpam-4340	21	23	this	this	DET
ejpam-4340	21	24	present	present	ADJ
ejpam-4340	21	25	paper	paper	NOUN
ejpam-4340	21	26	,	,	PUNCT
ejpam-4340	21	27	x	x	PUNCT
ejpam-4340	21	28	and	and	CCONJ
ejpam-4340	21	29	y	y	PROPN
ejpam-4340	21	30	represent	represent	VERB
ejpam-4340	21	31	topological	topological	ADJ
ejpam-4340	21	32	spaces	space	NOUN
ejpam-4340	21	33	.	.	PUNCT
ejpam-4340	22	1	for	for	ADP
ejpam-4340	22	2	a	a	DET
ejpam-4340	22	3	subset	subset	NOUN
ejpam-4340	22	4	a	a	PRON
ejpam-4340	22	5	of	of	ADP
ejpam-4340	22	6	a	a	DET
ejpam-4340	22	7	space	space	NOUN
ejpam-4340	22	8	x	x	NOUN
ejpam-4340	22	9	,	,	PUNCT
ejpam-4340	22	10	cl(a	cl(a	NUM
ejpam-4340	22	11	)	)	PUNCT
ejpam-4340	22	12	and	and	CCONJ
ejpam-4340	22	13	int(a	int(a	PROPN
ejpam-4340	22	14	)	)	PUNCT
ejpam-4340	22	15	denote	denote	VERB
ejpam-4340	22	16	the	the	DET
ejpam-4340	22	17	closure	closure	NOUN
ejpam-4340	22	18	of	of	ADP
ejpam-4340	22	19	a	a	PRON
ejpam-4340	22	20	and	and	CCONJ
ejpam-4340	22	21	the	the	DET
ejpam-4340	22	22	interior	interior	NOUN
ejpam-4340	22	23	of	of	ADP
ejpam-4340	22	24	a	a	PRON
ejpam-4340	22	25	,	,	PUNCT
ejpam-4340	22	26	respectively	respectively	ADV
ejpam-4340	22	27	.	.	PUNCT
ejpam-4340	23	1	the	the	DET
ejpam-4340	23	2	family	family	NOUN
ejpam-4340	23	3	of	of	ADP
ejpam-4340	23	4	all	all	DET
ejpam-4340	23	5	closed	closed	ADJ
ejpam-4340	23	6	(	(	PUNCT
ejpam-4340	23	7	resp	resp	NOUN
ejpam-4340	23	8	.	.	PUNCT
ejpam-4340	24	1	open	open	ADJ
ejpam-4340	24	2	)	)	PUNCT
ejpam-4340	24	3	sets	set	NOUN
ejpam-4340	24	4	of	of	ADP
ejpam-4340	24	5	x	x	SYM
ejpam-4340	24	6	is	be	AUX
ejpam-4340	24	7	denoted	denote	VERB
ejpam-4340	24	8	c(x	c(x	NOUN
ejpam-4340	24	9	)	)	PUNCT
ejpam-4340	24	10	(	(	PUNCT
ejpam-4340	24	11	resp	resp	NOUN
ejpam-4340	24	12	.	.	PUNCT
ejpam-4340	25	1	o(x	o(x	ADJ
ejpam-4340	25	2	)	)	PUNCT
ejpam-4340	25	3	or	or	CCONJ
ejpam-4340	25	4	τ	τ	PROPN
ejpam-4340	25	5	)	)	PUNCT
ejpam-4340	25	6	and	and	CCONJ
ejpam-4340	25	7	the	the	DET
ejpam-4340	25	8	family	family	NOUN
ejpam-4340	25	9	of	of	ADP
ejpam-4340	25	10	all	all	DET
ejpam-4340	25	11	closed	closed	ADJ
ejpam-4340	25	12	(	(	PUNCT
ejpam-4340	25	13	resp	resp	NOUN
ejpam-4340	25	14	.	.	PUNCT
ejpam-4340	26	1	open	open	ADJ
ejpam-4340	26	2	)	)	PUNCT
ejpam-4340	26	3	sets	set	NOUN
ejpam-4340	26	4	of	of	ADP
ejpam-4340	26	5	x	x	PUNCT
ejpam-4340	26	6	containing	contain	VERB
ejpam-4340	26	7	a	a	DET
ejpam-4340	26	8	point	point	NOUN
ejpam-4340	26	9	x	x	PUNCT
ejpam-4340	26	10	of	of	ADP
ejpam-4340	26	11	x	x	PRON
ejpam-4340	26	12	is	be	AUX
ejpam-4340	26	13	denoted	denote	VERB
ejpam-4340	26	14	by	by	ADP
ejpam-4340	26	15	c(x	c(x	NOUN
ejpam-4340	26	16	,	,	PUNCT
ejpam-4340	26	17	x	x	NOUN
ejpam-4340	26	18	)	)	PUNCT
ejpam-4340	26	19	(	(	PUNCT
ejpam-4340	26	20	resp	resp	NOUN
ejpam-4340	26	21	.	.	PUNCT
ejpam-4340	27	1	o(x	o(x	ADJ
ejpam-4340	27	2	,	,	PUNCT
ejpam-4340	27	3	x	x	NOUN
ejpam-4340	27	4	)	)	PUNCT
ejpam-4340	27	5	)	)	PUNCT
ejpam-4340	27	6	.	.	PUNCT
ejpam-4340	28	1	the	the	DET
ejpam-4340	28	2	family	family	NOUN
ejpam-4340	28	3	of	of	ADP
ejpam-4340	28	4	all	all	DET
ejpam-4340	28	5	neighborhood	neighborhood	NOUN
ejpam-4340	28	6	of	of	ADP
ejpam-4340	28	7	a	a	DET
ejpam-4340	28	8	point	point	NOUN
ejpam-4340	28	9	x	x	X
ejpam-4340	28	10	∈	∈	NOUN
ejpam-4340	28	11	x	x	AUX
ejpam-4340	28	12	is	be	AUX
ejpam-4340	28	13	denoted	denote	VERB
ejpam-4340	28	14	by	by	ADP
ejpam-4340	28	15	n	n	PROPN
ejpam-4340	28	16	(	(	PUNCT
ejpam-4340	28	17	x	x	NOUN
ejpam-4340	28	18	)	)	PUNCT
ejpam-4340	28	19	.	.	PUNCT
ejpam-4340	29	1	definition	definition	NOUN
ejpam-4340	29	2	1	1	NUM
ejpam-4340	29	3	.	.	PUNCT
ejpam-4340	30	1	a	a	DET
ejpam-4340	30	2	subset	subset	NOUN
ejpam-4340	30	3	a	a	PRON
ejpam-4340	30	4	of	of	ADP
ejpam-4340	30	5	a	a	DET
ejpam-4340	30	6	space	space	NOUN
ejpam-4340	30	7	x	x	PUNCT
ejpam-4340	30	8	is	be	AUX
ejpam-4340	30	9	called	call	VERB
ejpam-4340	30	10	:	:	PUNCT
ejpam-4340	30	11	(	(	PUNCT
ejpam-4340	30	12	a	a	X
ejpam-4340	30	13	)	)	PUNCT
ejpam-4340	30	14	regular	regular	ADJ
ejpam-4340	30	15	open	open	ADJ
ejpam-4340	30	16	[	[	X
ejpam-4340	30	17	20	20	NUM
ejpam-4340	30	18	]	]	PUNCT
ejpam-4340	30	19	if	if	SCONJ
ejpam-4340	30	20	a	a	PRON
ejpam-4340	30	21	=	=	X
ejpam-4340	30	22	int(cl(a	int(cl(a	PROPN
ejpam-4340	30	23	)	)	PUNCT
ejpam-4340	30	24	)	)	PUNCT
ejpam-4340	30	25	.	.	PUNCT
ejpam-4340	31	1	the	the	DET
ejpam-4340	31	2	complement	complement	NOUN
ejpam-4340	31	3	of	of	ADP
ejpam-4340	31	4	a	a	DET
ejpam-4340	31	5	regular	regular	ADJ
ejpam-4340	31	6	open	open	ADJ
ejpam-4340	31	7	set	set	NOUN
ejpam-4340	31	8	is	be	AUX
ejpam-4340	31	9	called	call	VERB
ejpam-4340	31	10	regular	regular	ADV
ejpam-4340	31	11	closed	closed	ADJ
ejpam-4340	31	12	.	.	PUNCT
ejpam-4340	32	1	a	a	DET
ejpam-4340	32	2	point	point	NOUN
ejpam-4340	32	3	x	x	X
ejpam-4340	32	4	∈	∈	NOUN
ejpam-4340	32	5	x	x	PUNCT
ejpam-4340	32	6	is	be	AUX
ejpam-4340	32	7	said	say	VERB
ejpam-4340	32	8	to	to	PART
ejpam-4340	32	9	be	be	AUX
ejpam-4340	32	10	the	the	DET
ejpam-4340	32	11	δ	δ	NOUN
ejpam-4340	32	12	-	-	PUNCT
ejpam-4340	32	13	cluster	cluster	NOUN
ejpam-4340	32	14	point	point	NOUN
ejpam-4340	32	15	[	[	X
ejpam-4340	32	16	22	22	NUM
ejpam-4340	32	17	]	]	PUNCT
ejpam-4340	32	18	of	of	ADP
ejpam-4340	32	19	a	a	DET
ejpam-4340	32	20	if	if	SCONJ
ejpam-4340	32	21	int(cl(u))∩a	int(cl(u))∩a	NOUN
ejpam-4340	32	22	̸=	̸=	PROPN
ejpam-4340	32	23	∅	∅	NOUN
ejpam-4340	32	24	for	for	ADP
ejpam-4340	32	25	each	each	DET
ejpam-4340	32	26	open	open	ADJ
ejpam-4340	32	27	neighborhood	neighborhood	NOUN
ejpam-4340	32	28	u	u	NOUN
ejpam-4340	32	29	of	of	ADP
ejpam-4340	32	30	x.	x.	NOUN
ejpam-4340	32	31	the	the	DET
ejpam-4340	32	32	set	set	NOUN
ejpam-4340	32	33	of	of	ADP
ejpam-4340	32	34	all	all	DET
ejpam-4340	32	35	δ	δ	NOUN
ejpam-4340	32	36	-	-	PUNCT
ejpam-4340	32	37	cluster	cluster	NOUN
ejpam-4340	32	38	points	point	NOUN
ejpam-4340	32	39	of	of	ADP
ejpam-4340	32	40	a	a	PRON
ejpam-4340	32	41	is	be	AUX
ejpam-4340	32	42	called	call	VERB
ejpam-4340	32	43	the	the	DET
ejpam-4340	32	44	δ	δ	NOUN
ejpam-4340	32	45	-	-	NOUN
ejpam-4340	32	46	closure	closure	NOUN
ejpam-4340	32	47	of	of	ADP
ejpam-4340	32	48	a	a	PRON
ejpam-4340	32	49	and	and	CCONJ
ejpam-4340	32	50	is	be	AUX
ejpam-4340	32	51	denoted	denote	VERB
ejpam-4340	32	52	by	by	ADP
ejpam-4340	32	53	δ	δ	PROPN
ejpam-4340	32	54	-	-	PUNCT
ejpam-4340	32	55	cl(a	cl(a	NUM
ejpam-4340	32	56	)	)	PUNCT
ejpam-4340	32	57	.	.	PUNCT
ejpam-4340	33	1	if	if	SCONJ
ejpam-4340	33	2	a	a	DET
ejpam-4340	33	3	=	=	X
ejpam-4340	33	4	δ	δ	PROPN
ejpam-4340	33	5	-	-	PUNCT
ejpam-4340	33	6	cl(a	cl(a	NUM
ejpam-4340	33	7	)	)	PUNCT
ejpam-4340	33	8	,	,	PUNCT
ejpam-4340	33	9	then	then	ADV
ejpam-4340	33	10	a	a	PRON
ejpam-4340	33	11	is	be	AUX
ejpam-4340	33	12	called	call	VERB
ejpam-4340	33	13	δ	δ	NOUN
ejpam-4340	33	14	-	-	PUNCT
ejpam-4340	33	15	closed	close	VERB
ejpam-4340	33	16	[	[	X
ejpam-4340	33	17	22	22	NUM
ejpam-4340	33	18	]	]	PUNCT
ejpam-4340	33	19	,	,	PUNCT
ejpam-4340	33	20	and	and	CCONJ
ejpam-4340	33	21	the	the	DET
ejpam-4340	33	22	complement	complement	NOUN
ejpam-4340	33	23	of	of	ADP
ejpam-4340	33	24	a	a	DET
ejpam-4340	33	25	δ	δ	NOUN
ejpam-4340	33	26	-	-	PUNCT
ejpam-4340	33	27	closed	close	VERB
ejpam-4340	33	28	set	set	NOUN
ejpam-4340	33	29	is	be	AUX
ejpam-4340	33	30	called	call	VERB
ejpam-4340	34	1	δ	δ	NOUN
ejpam-4340	34	2	-	-	NOUN
ejpam-4340	34	3	open	open	ADJ
ejpam-4340	34	4	.	.	PUNCT
ejpam-4340	35	1	the	the	DET
ejpam-4340	35	2	set	set	NOUN
ejpam-4340	35	3	{	{	PUNCT
ejpam-4340	35	4	x|(∃u	x|(∃u	PROPN
ejpam-4340	35	5	∈	∈	PROPN
ejpam-4340	35	6	o(x	o(x	PROPN
ejpam-4340	35	7	,	,	PUNCT
ejpam-4340	35	8	x))(int(cl(u	x))(int(cl(u	PROPN
ejpam-4340	35	9	)	)	PUNCT
ejpam-4340	35	10	)	)	PUNCT
ejpam-4340	36	1	⊆	⊆	NUM
ejpam-4340	36	2	a	a	PRON
ejpam-4340	36	3	)	)	PUNCT
ejpam-4340	36	4	}	}	PUNCT
ejpam-4340	36	5	is	be	AUX
ejpam-4340	36	6	called	call	VERB
ejpam-4340	36	7	the	the	DET
ejpam-4340	36	8	δ	δ	NOUN
ejpam-4340	36	9	-	-	NOUN
ejpam-4340	36	10	interior	interior	NOUN
ejpam-4340	36	11	of	of	ADP
ejpam-4340	36	12	a	a	PRON
ejpam-4340	36	13	and	and	CCONJ
ejpam-4340	36	14	is	be	AUX
ejpam-4340	36	15	denoted	denote	VERB
ejpam-4340	36	16	by	by	ADP
ejpam-4340	36	17	δ	δ	PROPN
ejpam-4340	36	18	-	-	PUNCT
ejpam-4340	36	19	int(a	int(a	PROPN
ejpam-4340	36	20	)	)	PUNCT
ejpam-4340	36	21	.	.	PUNCT
ejpam-4340	37	1	(	(	PUNCT
ejpam-4340	37	2	b	b	X
ejpam-4340	37	3	)	)	PUNCT
ejpam-4340	37	4	β	β	X
ejpam-4340	37	5	-	-	VERB
ejpam-4340	37	6	open	open	ADJ
ejpam-4340	38	1	[	[	X
ejpam-4340	38	2	1	1	NUM
ejpam-4340	38	3	]	]	X
ejpam-4340	38	4	if	if	SCONJ
ejpam-4340	38	5	a	a	DET
ejpam-4340	38	6	⊆	⊆	NUM
ejpam-4340	38	7	cl(int(cl(a	cl(int(cl(a	NOUN
ejpam-4340	38	8	)	)	PUNCT
ejpam-4340	38	9	)	)	PUNCT
ejpam-4340	38	10	)	)	PUNCT
ejpam-4340	38	11	.	.	PUNCT
ejpam-4340	39	1	the	the	DET
ejpam-4340	39	2	complement	complement	NOUN
ejpam-4340	39	3	of	of	ADP
ejpam-4340	39	4	a	a	DET
ejpam-4340	39	5	β	β	X
ejpam-4340	39	6	-	-	ADJ
ejpam-4340	39	7	open	open	ADJ
ejpam-4340	39	8	set	set	NOUN
ejpam-4340	39	9	is	be	AUX
ejpam-4340	39	10	called	call	VERB
ejpam-4340	39	11	β	β	NOUN
ejpam-4340	39	12	-	-	VERB
ejpam-4340	39	13	closed	closed	ADJ
ejpam-4340	39	14	.	.	PUNCT
ejpam-4340	40	1	the	the	DET
ejpam-4340	40	2	intersection	intersection	NOUN
ejpam-4340	40	3	of	of	ADP
ejpam-4340	40	4	all	all	DET
ejpam-4340	40	5	β	β	ADJ
ejpam-4340	40	6	-	-	ADJ
ejpam-4340	40	7	closed	closed	ADJ
ejpam-4340	40	8	sets	set	NOUN
ejpam-4340	40	9	containing	contain	VERB
ejpam-4340	40	10	a	a	PRON
ejpam-4340	40	11	is	be	AUX
ejpam-4340	40	12	called	call	VERB
ejpam-4340	40	13	the	the	DET
ejpam-4340	40	14	β	β	NOUN
ejpam-4340	40	15	-	-	NOUN
ejpam-4340	40	16	closure	closure	NOUN
ejpam-4340	40	17	of	of	ADP
ejpam-4340	40	18	a	a	PRON
ejpam-4340	40	19	and	and	CCONJ
ejpam-4340	40	20	is	be	AUX
ejpam-4340	40	21	denoted	denote	VERB
ejpam-4340	40	22	by	by	ADP
ejpam-4340	40	23	β	β	NOUN
ejpam-4340	40	24	-	-	PUNCT
ejpam-4340	40	25	cl(a	cl(a	NUM
ejpam-4340	40	26	)	)	PUNCT
ejpam-4340	40	27	.	.	PUNCT
ejpam-4340	41	1	the	the	DET
ejpam-4340	41	2	union	union	NOUN
ejpam-4340	41	3	of	of	ADP
ejpam-4340	41	4	all	all	DET
ejpam-4340	41	5	β	β	ADJ
ejpam-4340	41	6	-	-	ADJ
ejpam-4340	41	7	open	open	ADJ
ejpam-4340	41	8	sets	set	NOUN
ejpam-4340	41	9	of	of	ADP
ejpam-4340	41	10	x	x	PUNCT
ejpam-4340	41	11	contained	contain	VERB
ejpam-4340	41	12	in	in	ADP
ejpam-4340	41	13	a	a	PRON
ejpam-4340	41	14	is	be	AUX
ejpam-4340	41	15	called	call	VERB
ejpam-4340	41	16	the	the	DET
ejpam-4340	41	17	β	β	NOUN
ejpam-4340	41	18	-	-	NOUN
ejpam-4340	41	19	interior	interior	ADJ
ejpam-4340	41	20	of	of	ADP
ejpam-4340	41	21	a	a	PRON
ejpam-4340	41	22	and	and	CCONJ
ejpam-4340	41	23	is	be	AUX
ejpam-4340	41	24	denoted	denote	VERB
ejpam-4340	41	25	by	by	ADP
ejpam-4340	41	26	β	β	NOUN
ejpam-4340	41	27	-	-	PUNCT
ejpam-4340	41	28	int(a	int(a	NOUN
ejpam-4340	41	29	)	)	PUNCT
ejpam-4340	41	30	.	.	PUNCT
ejpam-4340	42	1	(	(	PUNCT
ejpam-4340	42	2	c	c	X
ejpam-4340	42	3	)	)	PUNCT
ejpam-4340	42	4	a	a	DET
ejpam-4340	42	5	-	-	PUNCT
ejpam-4340	42	6	open	open	ADJ
ejpam-4340	42	7	[	[	X
ejpam-4340	42	8	10	10	NUM
ejpam-4340	42	9	]	]	X
ejpam-4340	42	10	if	if	SCONJ
ejpam-4340	42	11	a	a	DET
ejpam-4340	42	12	⊆	⊆	NUM
ejpam-4340	42	13	int(cl(δ	int(cl(δ	PROPN
ejpam-4340	42	14	-	-	PUNCT
ejpam-4340	42	15	int(a	int(a	NOUN
ejpam-4340	42	16	)	)	PUNCT
ejpam-4340	42	17	)	)	PUNCT
ejpam-4340	42	18	)	)	PUNCT
ejpam-4340	42	19	.	.	PUNCT
ejpam-4340	43	1	the	the	DET
ejpam-4340	43	2	complement	complement	NOUN
ejpam-4340	43	3	of	of	ADP
ejpam-4340	43	4	an	an	DET
ejpam-4340	43	5	a	a	PRON
ejpam-4340	43	6	-	-	PUNCT
ejpam-4340	43	7	open	open	ADJ
ejpam-4340	43	8	set	set	NOUN
ejpam-4340	43	9	is	be	AUX
ejpam-4340	43	10	called	call	VERB
ejpam-4340	43	11	a	a	PRON
ejpam-4340	43	12	-	-	PUNCT
ejpam-4340	43	13	closed	closed	ADJ
ejpam-4340	43	14	[	[	X
ejpam-4340	43	15	10	10	NUM
ejpam-4340	43	16	]	]	PUNCT
ejpam-4340	43	17	.	.	PUNCT
ejpam-4340	44	1	the	the	DET
ejpam-4340	44	2	intersection	intersection	NOUN
ejpam-4340	44	3	of	of	ADP
ejpam-4340	44	4	all	all	DET
ejpam-4340	44	5	a	a	PRON
ejpam-4340	44	6	-	-	PUNCT
ejpam-4340	44	7	closed	closed	ADJ
ejpam-4340	44	8	sets	set	NOUN
ejpam-4340	44	9	containing	contain	VERB
ejpam-4340	44	10	a	a	PRON
ejpam-4340	44	11	is	be	AUX
ejpam-4340	44	12	called	call	VERB
ejpam-4340	44	13	the	the	DET
ejpam-4340	44	14	a	a	DET
ejpam-4340	44	15	-	-	PUNCT
ejpam-4340	44	16	closure	closure	NOUN
ejpam-4340	44	17	[	[	X
ejpam-4340	44	18	10	10	NUM
ejpam-4340	44	19	]	]	PUNCT
ejpam-4340	44	20	of	of	ADP
ejpam-4340	44	21	a	a	PRON
ejpam-4340	44	22	and	and	CCONJ
ejpam-4340	44	23	is	be	AUX
ejpam-4340	44	24	denoted	denote	VERB
ejpam-4340	44	25	by	by	ADP
ejpam-4340	44	26	a	a	DET
ejpam-4340	44	27	-	-	PUNCT
ejpam-4340	44	28	cl(a	cl(a	NUM
ejpam-4340	44	29	)	)	PUNCT
ejpam-4340	44	30	.	.	PUNCT
ejpam-4340	45	1	the	the	DET
ejpam-4340	45	2	union	union	NOUN
ejpam-4340	45	3	of	of	ADP
ejpam-4340	45	4	all	all	DET
ejpam-4340	45	5	a	a	PRON
ejpam-4340	45	6	-	-	PUNCT
ejpam-4340	45	7	open	open	ADJ
ejpam-4340	45	8	sets	set	NOUN
ejpam-4340	45	9	of	of	ADP
ejpam-4340	45	10	x	x	PUNCT
ejpam-4340	45	11	contained	contain	VERB
ejpam-4340	45	12	in	in	ADP
ejpam-4340	45	13	a	a	PRON
ejpam-4340	45	14	is	be	AUX
ejpam-4340	45	15	called	call	VERB
ejpam-4340	45	16	the	the	DET
ejpam-4340	45	17	a	a	DET
ejpam-4340	45	18	-	-	PUNCT
ejpam-4340	45	19	interior	interior	NOUN
ejpam-4340	45	20	[	[	X
ejpam-4340	45	21	10	10	NUM
ejpam-4340	45	22	]	]	PUNCT
ejpam-4340	45	23	of	of	ADP
ejpam-4340	45	24	a	a	PRON
ejpam-4340	45	25	and	and	CCONJ
ejpam-4340	45	26	is	be	AUX
ejpam-4340	45	27	denoted	denote	VERB
ejpam-4340	45	28	by	by	ADP
ejpam-4340	45	29	a	a	DET
ejpam-4340	45	30	-	-	PUNCT
ejpam-4340	45	31	int(a	int(a	NOUN
ejpam-4340	45	32	)	)	PUNCT
ejpam-4340	45	33	.	.	PUNCT
ejpam-4340	46	1	(	(	PUNCT
ejpam-4340	46	2	d	d	X
ejpam-4340	46	3	)	)	PUNCT
ejpam-4340	46	4	e∗-open	e∗-open	NOUN
ejpam-4340	47	1	[	[	PUNCT
ejpam-4340	47	2	11	11	NUM
ejpam-4340	47	3	]	]	X
ejpam-4340	47	4	if	if	SCONJ
ejpam-4340	47	5	a	a	DET
ejpam-4340	47	6	⊆	⊆	NUM
ejpam-4340	47	7	cl(int(δ	cl(int(δ	NOUN
ejpam-4340	47	8	-	-	PUNCT
ejpam-4340	47	9	cl(a	cl(a	NUM
ejpam-4340	47	10	)	)	PUNCT
ejpam-4340	47	11	)	)	PUNCT
ejpam-4340	47	12	)	)	PUNCT
ejpam-4340	47	13	.	.	PUNCT
ejpam-4340	48	1	the	the	DET
ejpam-4340	48	2	complement	complement	NOUN
ejpam-4340	48	3	of	of	ADP
ejpam-4340	48	4	an	an	DET
ejpam-4340	48	5	e∗-open	e∗-open	ADJ
ejpam-4340	48	6	set	set	NOUN
ejpam-4340	48	7	is	be	AUX
ejpam-4340	48	8	called	call	VERB
ejpam-4340	48	9	e∗-closed	e∗-close	VERB
ejpam-4340	48	10	[	[	PUNCT
ejpam-4340	48	11	11	11	NUM
ejpam-4340	48	12	]	]	PUNCT
ejpam-4340	48	13	.	.	PUNCT
ejpam-4340	49	1	the	the	DET
ejpam-4340	49	2	intersection	intersection	NOUN
ejpam-4340	49	3	of	of	ADP
ejpam-4340	49	4	all	all	DET
ejpam-4340	49	5	e∗-closed	e∗-close	VERB
ejpam-4340	49	6	sets	set	NOUN
ejpam-4340	49	7	containing	contain	VERB
ejpam-4340	49	8	a	a	PRON
ejpam-4340	49	9	is	be	AUX
ejpam-4340	49	10	called	call	VERB
ejpam-4340	49	11	the	the	DET
ejpam-4340	49	12	e∗-closure	e∗-closure	NOUN
ejpam-4340	49	13	[	[	X
ejpam-4340	49	14	11	11	NUM
ejpam-4340	49	15	]	]	PUNCT
ejpam-4340	49	16	of	of	ADP
ejpam-4340	49	17	a	a	PRON
ejpam-4340	49	18	and	and	CCONJ
ejpam-4340	49	19	is	be	AUX
ejpam-4340	49	20	denoted	denote	VERB
ejpam-4340	49	21	by	by	ADP
ejpam-4340	49	22	e∗-cl(a	e∗-cl(a	PROPN
ejpam-4340	49	23	)	)	PUNCT
ejpam-4340	49	24	.	.	PUNCT
ejpam-4340	50	1	the	the	DET
ejpam-4340	50	2	union	union	NOUN
ejpam-4340	50	3	of	of	ADP
ejpam-4340	50	4	all	all	DET
ejpam-4340	50	5	e∗-open	e∗-open	ADJ
ejpam-4340	50	6	sets	set	NOUN
ejpam-4340	50	7	of	of	ADP
ejpam-4340	50	8	x	x	PUNCT
ejpam-4340	50	9	contained	contain	VERB
ejpam-4340	50	10	in	in	ADP
ejpam-4340	50	11	a	a	PRON
ejpam-4340	50	12	is	be	AUX
ejpam-4340	50	13	called	call	VERB
ejpam-4340	50	14	the	the	DET
ejpam-4340	50	15	e∗-interior	e∗-interior	PROPN
ejpam-4340	50	16	[	[	NOUN
ejpam-4340	50	17	11	11	NUM
ejpam-4340	50	18	]	]	PUNCT
ejpam-4340	50	19	of	of	ADP
ejpam-4340	50	20	a	a	PRON
ejpam-4340	50	21	and	and	CCONJ
ejpam-4340	50	22	is	be	AUX
ejpam-4340	50	23	denoted	denote	VERB
ejpam-4340	50	24	by	by	ADP
ejpam-4340	50	25	e∗-int(a	e∗-int(a	PROPN
ejpam-4340	50	26	)	)	PUNCT
ejpam-4340	50	27	.	.	PUNCT
ejpam-4340	51	1	(	(	PUNCT
ejpam-4340	51	2	e	e	X
ejpam-4340	51	3	)	)	PUNCT
ejpam-4340	51	4	ω	ω	NOUN
ejpam-4340	51	5	-	-	NOUN
ejpam-4340	51	6	open	open	ADJ
ejpam-4340	51	7	[	[	X
ejpam-4340	51	8	6	6	NUM
ejpam-4340	51	9	]	]	PUNCT
ejpam-4340	51	10	(	(	PUNCT
ejpam-4340	51	11	resp	resp	NOUN
ejpam-4340	51	12	.	.	PUNCT
ejpam-4340	52	1	ωβ	ωβ	ADJ
ejpam-4340	52	2	-	-	ADJ
ejpam-4340	52	3	open	open	ADJ
ejpam-4340	52	4	[	[	X
ejpam-4340	52	5	2	2	NUM
ejpam-4340	52	6	]	]	PUNCT
ejpam-4340	52	7	)	)	PUNCT
ejpam-4340	52	8	if	if	SCONJ
ejpam-4340	52	9	for	for	ADP
ejpam-4340	52	10	every	every	DET
ejpam-4340	52	11	x	x	SYM
ejpam-4340	52	12	∈	∈	PROPN
ejpam-4340	52	13	a	a	DET
ejpam-4340	52	14	there	there	PRON
ejpam-4340	52	15	exists	exist	VERB
ejpam-4340	52	16	an	an	DET
ejpam-4340	52	17	open	open	ADJ
ejpam-4340	52	18	(	(	PUNCT
ejpam-4340	52	19	resp	resp	NOUN
ejpam-4340	52	20	.	.	PUNCT
ejpam-4340	53	1	βopen	βopen	ADJ
ejpam-4340	53	2	)	)	PUNCT
ejpam-4340	53	3	set	set	VERB
ejpam-4340	53	4	u	u	NOUN
ejpam-4340	53	5	containing	contain	VERB
ejpam-4340	53	6	x	x	PUNCT
ejpam-4340	53	7	such	such	ADJ
ejpam-4340	53	8	that	that	SCONJ
ejpam-4340	53	9	u	u	NOUN
ejpam-4340	53	10	\a	\a	ADJ
ejpam-4340	53	11	is	be	AUX
ejpam-4340	53	12	countable	countable	ADJ
ejpam-4340	53	13	.	.	PUNCT
ejpam-4340	54	1	the	the	DET
ejpam-4340	54	2	complement	complement	NOUN
ejpam-4340	54	3	of	of	ADP
ejpam-4340	54	4	an	an	DET
ejpam-4340	54	5	ω	ω	ADV
ejpam-4340	54	6	-	-	ADJ
ejpam-4340	54	7	open	open	ADJ
ejpam-4340	54	8	set	set	NOUN
ejpam-4340	54	9	(	(	PUNCT
ejpam-4340	54	10	resp	resp	NOUN
ejpam-4340	54	11	.	.	PUNCT
ejpam-4340	55	1	wβ	wβ	ADP
ejpam-4340	55	2	-	-	PUNCT
ejpam-4340	55	3	open	open	ADJ
ejpam-4340	55	4	set	set	NOUN
ejpam-4340	55	5	)	)	PUNCT
ejpam-4340	55	6	is	be	AUX
ejpam-4340	55	7	said	say	VERB
ejpam-4340	55	8	to	to	PART
ejpam-4340	55	9	be	be	AUX
ejpam-4340	55	10	ω	ω	NOUN
ejpam-4340	55	11	-	-	ADJ
ejpam-4340	55	12	closed	closed	ADJ
ejpam-4340	55	13	(	(	PUNCT
ejpam-4340	55	14	resp	resp	NOUN
ejpam-4340	55	15	.	.	PUNCT
ejpam-4340	56	1	ωβ	ωβ	ADJ
ejpam-4340	56	2	-	-	PUNCT
ejpam-4340	56	3	closed	closed	ADJ
ejpam-4340	56	4	)	)	PUNCT
ejpam-4340	56	5	.	.	PUNCT
ejpam-4340	57	1	the	the	DET
ejpam-4340	57	2	family	family	NOUN
ejpam-4340	57	3	of	of	ADP
ejpam-4340	57	4	all	all	DET
ejpam-4340	57	5	regular	regular	ADJ
ejpam-4340	57	6	open	open	ADJ
ejpam-4340	57	7	(	(	PUNCT
ejpam-4340	57	8	resp	resp	NOUN
ejpam-4340	57	9	.	.	PUNCT
ejpam-4340	58	1	regular	regular	ADJ
ejpam-4340	58	2	closed	closed	ADJ
ejpam-4340	58	3	,	,	PUNCT
ejpam-4340	58	4	β	β	X
ejpam-4340	58	5	-	-	ADJ
ejpam-4340	58	6	open	open	ADJ
ejpam-4340	58	7	,	,	PUNCT
ejpam-4340	58	8	β	β	NOUN
ejpam-4340	58	9	-	-	VERB
ejpam-4340	58	10	closed	closed	ADJ
ejpam-4340	58	11	,	,	PUNCT
ejpam-4340	58	12	a	a	DET
ejpam-4340	58	13	-	-	PUNCT
ejpam-4340	58	14	open	open	ADJ
ejpam-4340	58	15	,	,	PUNCT
ejpam-4340	58	16	aclosed	aclose	VERB
ejpam-4340	58	17	,	,	PUNCT
ejpam-4340	58	18	e∗-open	e∗-open	ADJ
ejpam-4340	58	19	,	,	PUNCT
ejpam-4340	58	20	e∗-closed	e∗-close	VERB
ejpam-4340	58	21	,	,	PUNCT
ejpam-4340	58	22	ω	ω	NOUN
ejpam-4340	58	23	-	-	ADJ
ejpam-4340	58	24	open	open	ADJ
ejpam-4340	58	25	,	,	PUNCT
ejpam-4340	58	26	ω	ω	NOUN
ejpam-4340	58	27	-	-	ADJ
ejpam-4340	58	28	closed	closed	ADJ
ejpam-4340	58	29	,	,	PUNCT
ejpam-4340	58	30	ωβ	ωβ	ADJ
ejpam-4340	58	31	-	-	ADJ
ejpam-4340	58	32	open	open	ADJ
ejpam-4340	58	33	,	,	PUNCT
ejpam-4340	58	34	ωβ	ωβ	ADJ
ejpam-4340	58	35	-	-	PUNCT
ejpam-4340	58	36	closed	closed	ADJ
ejpam-4340	58	37	)	)	PUNCT
ejpam-4340	58	38	subsets	subset	NOUN
ejpam-4340	58	39	of	of	ADP
ejpam-4340	58	40	x	x	PROPN
ejpam-4340	58	41	is	be	AUX
ejpam-4340	58	42	denoted	denote	VERB
ejpam-4340	58	43	by	by	ADP
ejpam-4340	58	44	ro(x	ro(x	ADJ
ejpam-4340	58	45	)	)	PUNCT
ejpam-4340	58	46	(	(	PUNCT
ejpam-4340	58	47	resp	resp	NOUN
ejpam-4340	58	48	.	.	PUNCT
ejpam-4340	58	49	rc(x	rc(x	PROPN
ejpam-4340	58	50	)	)	PUNCT
ejpam-4340	58	51	,	,	PUNCT
ejpam-4340	58	52	βo(x	βo(x	NUM
ejpam-4340	58	53	)	)	PUNCT
ejpam-4340	58	54	,	,	PUNCT
ejpam-4340	58	55	βc(x	βc(x	NUM
ejpam-4340	58	56	)	)	PUNCT
ejpam-4340	58	57	,	,	PUNCT
ejpam-4340	58	58	ao(x	ao(x	NUM
ejpam-4340	58	59	)	)	PUNCT
ejpam-4340	58	60	,	,	PUNCT
ejpam-4340	58	61	ac(x	ac(x	NOUN
ejpam-4340	58	62	)	)	PUNCT
ejpam-4340	58	63	,	,	PUNCT
ejpam-4340	58	64	e∗o(x	e∗o(x	PROPN
ejpam-4340	58	65	)	)	PUNCT
ejpam-4340	58	66	,	,	PUNCT
ejpam-4340	58	67	e∗c(x	e∗c(x	PROPN
ejpam-4340	58	68	)	)	PUNCT
ejpam-4340	58	69	,	,	PUNCT
ejpam-4340	58	70	ωo(x	ωo(x	NUM
ejpam-4340	58	71	)	)	PUNCT
ejpam-4340	58	72	,	,	PUNCT
ejpam-4340	58	73	ωc(x	ωc(x	NOUN
ejpam-4340	58	74	)	)	PUNCT
ejpam-4340	58	75	,	,	PUNCT
ejpam-4340	58	76	ωβo(x	ωβo(x	NUM
ejpam-4340	58	77	)	)	PUNCT
ejpam-4340	58	78	,	,	PUNCT
ejpam-4340	58	79	ωβc(x	ωβc(x	PROPN
ejpam-4340	58	80	)	)	PUNCT
ejpam-4340	58	81	)	)	PUNCT
ejpam-4340	58	82	.	.	PUNCT
ejpam-4340	59	1	the	the	DET
ejpam-4340	59	2	family	family	NOUN
ejpam-4340	59	3	of	of	ADP
ejpam-4340	59	4	all	all	DET
ejpam-4340	59	5	regular	regular	ADJ
ejpam-4340	59	6	open	open	ADJ
ejpam-4340	59	7	(	(	PUNCT
ejpam-4340	59	8	resp	resp	NOUN
ejpam-4340	59	9	.	.	PUNCT
ejpam-4340	60	1	regular	regular	ADJ
ejpam-4340	60	2	closed	closed	ADJ
ejpam-4340	60	3	,	,	PUNCT
ejpam-4340	60	4	β	β	X
ejpam-4340	60	5	-	-	ADJ
ejpam-4340	60	6	open	open	ADJ
ejpam-4340	60	7	,	,	PUNCT
ejpam-4340	60	8	β	β	NOUN
ejpam-4340	60	9	-	-	VERB
ejpam-4340	60	10	closed	closed	ADJ
ejpam-4340	60	11	,	,	PUNCT
ejpam-4340	60	12	a	a	DET
ejpam-4340	60	13	-	-	PUNCT
ejpam-4340	60	14	open	open	ADJ
ejpam-4340	60	15	,	,	PUNCT
ejpam-4340	60	16	a	a	PRON
ejpam-4340	60	17	-	-	PUNCT
ejpam-4340	60	18	closed	closed	ADJ
ejpam-4340	60	19	,	,	PUNCT
ejpam-4340	60	20	e∗-open	e∗-open	ADJ
ejpam-4340	60	21	,	,	PUNCT
ejpam-4340	60	22	e∗-closed	e∗-close	VERB
ejpam-4340	60	23	,	,	PUNCT
ejpam-4340	60	24	ω	ω	NOUN
ejpam-4340	60	25	-	-	ADJ
ejpam-4340	60	26	open	open	ADJ
ejpam-4340	60	27	,	,	PUNCT
ejpam-4340	60	28	ω	ω	NOUN
ejpam-4340	60	29	-	-	ADJ
ejpam-4340	60	30	closed	closed	ADJ
ejpam-4340	60	31	,	,	PUNCT
ejpam-4340	60	32	ωβ	ωβ	ADJ
ejpam-4340	60	33	-	-	ADJ
ejpam-4340	60	34	open	open	ADJ
ejpam-4340	60	35	,	,	PUNCT
ejpam-4340	60	36	ωβ	ωβ	ADJ
ejpam-4340	60	37	-	-	PUNCT
ejpam-4340	60	38	closed	closed	ADJ
ejpam-4340	60	39	)	)	PUNCT
ejpam-4340	60	40	sets	set	NOUN
ejpam-4340	60	41	of	of	ADP
ejpam-4340	60	42	x	x	PUNCT
ejpam-4340	60	43	containing	contain	VERB
ejpam-4340	60	44	a	a	DET
ejpam-4340	60	45	point	point	NOUN
ejpam-4340	60	46	x	x	PUNCT
ejpam-4340	60	47	of	of	ADP
ejpam-4340	60	48	x	x	PRON
ejpam-4340	60	49	is	be	AUX
ejpam-4340	60	50	denoted	denote	VERB
ejpam-4340	60	51	by	by	ADP
ejpam-4340	60	52	ro(x	ro(x	ADJ
ejpam-4340	60	53	,	,	PUNCT
ejpam-4340	60	54	x	x	X
ejpam-4340	60	55	)	)	PUNCT
ejpam-4340	60	56	(	(	PUNCT
ejpam-4340	60	57	resp	resp	NOUN
ejpam-4340	60	58	.	.	PUNCT
ejpam-4340	61	1	rc(x	rc(x	PROPN
ejpam-4340	61	2	,	,	PUNCT
ejpam-4340	61	3	x	x	NOUN
ejpam-4340	61	4	)	)	PUNCT
ejpam-4340	61	5	,	,	PUNCT
ejpam-4340	61	6	βo(x	βo(x	PUNCT
ejpam-4340	61	7	,	,	PUNCT
ejpam-4340	61	8	x	x	X
ejpam-4340	61	9	)	)	PUNCT
ejpam-4340	61	10	,	,	PUNCT
ejpam-4340	61	11	βc(x	βc(x	NUM
ejpam-4340	61	12	,	,	PUNCT
ejpam-4340	61	13	x	x	NOUN
ejpam-4340	61	14	)	)	PUNCT
ejpam-4340	61	15	,	,	PUNCT
ejpam-4340	61	16	ao(x	ao(x	ADJ
ejpam-4340	61	17	,	,	PUNCT
ejpam-4340	61	18	x	x	X
ejpam-4340	61	19	)	)	PUNCT
ejpam-4340	61	20	,	,	PUNCT
ejpam-4340	61	21	ac(x	ac(x	NOUN
ejpam-4340	61	22	,	,	PUNCT
ejpam-4340	61	23	x	x	NOUN
ejpam-4340	61	24	)	)	PUNCT
ejpam-4340	61	25	,	,	PUNCT
ejpam-4340	61	26	e∗o(x	e∗o(x	PROPN
ejpam-4340	61	27	,	,	PUNCT
ejpam-4340	61	28	x	x	NOUN
ejpam-4340	61	29	)	)	PUNCT
ejpam-4340	61	30	,	,	PUNCT
ejpam-4340	61	31	e∗c(x	e∗c(x	PROPN
ejpam-4340	61	32	,	,	PUNCT
ejpam-4340	61	33	x	x	NOUN
ejpam-4340	61	34	)	)	PUNCT
ejpam-4340	61	35	,	,	PUNCT
ejpam-4340	61	36	ωo(x	ωo(x	PROPN
ejpam-4340	61	37	,	,	PUNCT
ejpam-4340	61	38	x	x	NOUN
ejpam-4340	61	39	)	)	PUNCT
ejpam-4340	61	40	,	,	PUNCT
ejpam-4340	61	41	ωc(x	ωc(x	NOUN
ejpam-4340	61	42	,	,	PUNCT
ejpam-4340	61	43	x	x	NOUN
ejpam-4340	61	44	)	)	PUNCT
ejpam-4340	61	45	,	,	PUNCT
ejpam-4340	61	46	ωβo(x	ωβo(x	PROPN
ejpam-4340	61	47	,	,	PUNCT
ejpam-4340	61	48	x	x	NOUN
ejpam-4340	61	49	)	)	PUNCT
ejpam-4340	61	50	,	,	PUNCT
ejpam-4340	61	51	ωβc(x	ωβc(x	PROPN
ejpam-4340	61	52	,	,	PUNCT
ejpam-4340	61	53	x	x	NOUN
ejpam-4340	61	54	)	)	PUNCT
ejpam-4340	61	55	)	)	PUNCT
ejpam-4340	61	56	.	.	PUNCT
ejpam-4340	62	1	definition	definition	NOUN
ejpam-4340	62	2	2	2	NUM
ejpam-4340	62	3	.	.	PUNCT
ejpam-4340	62	4	let	let	VERB
ejpam-4340	62	5	a	a	DET
ejpam-4340	62	6	be	be	AUX
ejpam-4340	62	7	a	a	DET
ejpam-4340	62	8	subset	subset	NOUN
ejpam-4340	62	9	of	of	ADP
ejpam-4340	62	10	a	a	DET
ejpam-4340	62	11	space	space	NOUN
ejpam-4340	62	12	x.	x.	NOUN
ejpam-4340	63	1	a	a	PRON
ejpam-4340	63	2	is	be	AUX
ejpam-4340	63	3	said	say	VERB
ejpam-4340	63	4	to	to	PART
ejpam-4340	63	5	be	be	AUX
ejpam-4340	63	6	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	63	7	[	[	X
ejpam-4340	63	8	19	19	NUM
ejpam-4340	63	9	]	]	X
ejpam-4340	63	10	(	(	PUNCT
ejpam-4340	63	11	resp	resp	NOUN
ejpam-4340	63	12	.	.	PUNCT
ejpam-4340	64	1	ωaopen	ωaopen	VERB
ejpam-4340	65	1	[	[	X
ejpam-4340	65	2	19	19	NUM
ejpam-4340	65	3	]	]	SYM
ejpam-4340	65	4	)	)	PUNCT
ejpam-4340	65	5	if	if	SCONJ
ejpam-4340	65	6	for	for	ADP
ejpam-4340	65	7	every	every	DET
ejpam-4340	65	8	x	x	PROPN
ejpam-4340	65	9	∈	∈	PROPN
ejpam-4340	65	10	a	a	PRON
ejpam-4340	65	11	,	,	PUNCT
ejpam-4340	65	12	there	there	PRON
ejpam-4340	65	13	exists	exist	VERB
ejpam-4340	65	14	an	an	DET
ejpam-4340	65	15	e∗-open	e∗-open	ADJ
ejpam-4340	65	16	(	(	PUNCT
ejpam-4340	65	17	resp	resp	NOUN
ejpam-4340	65	18	.	.	PUNCT
ejpam-4340	66	1	a	a	X
ejpam-4340	66	2	-	-	PUNCT
ejpam-4340	66	3	open	open	ADJ
ejpam-4340	66	4	)	)	PUNCT
ejpam-4340	66	5	set	set	VERB
ejpam-4340	66	6	u	u	NOUN
ejpam-4340	66	7	containing	contain	VERB
ejpam-4340	66	8	x	x	PUNCT
ejpam-4340	66	9	such	such	ADJ
ejpam-4340	66	10	that	that	SCONJ
ejpam-4340	66	11	u	u	NOUN
ejpam-4340	66	12	\a	\a	ADJ
ejpam-4340	67	1	is	be	AUX
ejpam-4340	67	2	countable	countable	ADJ
ejpam-4340	67	3	.	.	PUNCT
ejpam-4340	68	1	the	the	DET
ejpam-4340	68	2	complement	complement	NOUN
ejpam-4340	68	3	of	of	ADP
ejpam-4340	68	4	an	an	DET
ejpam-4340	68	5	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	68	6	(	(	PUNCT
ejpam-4340	68	7	resp	resp	NOUN
ejpam-4340	68	8	.	.	PUNCT
ejpam-4340	69	1	ωa	ωa	ADJ
ejpam-4340	69	2	-	-	PUNCT
ejpam-4340	69	3	open	open	ADJ
ejpam-4340	69	4	)	)	PUNCT
ejpam-4340	69	5	set	set	NOUN
ejpam-4340	69	6	is	be	AUX
ejpam-4340	69	7	called	call	VERB
ejpam-4340	69	8	p.	p.	PROPN
ejpam-4340	70	1	şaşmaz	şaşmaz	PUNCT
ejpam-4340	70	2	,	,	PUNCT
ejpam-4340	70	3	m.	m.	NOUN
ejpam-4340	70	4	özkoç	özkoç	PROPN
ejpam-4340	70	5	/	/	SYM
ejpam-4340	70	6	eur	eur	PROPN
ejpam-4340	70	7	.	.	PUNCT
ejpam-4340	71	1	j.	j.	PROPN
ejpam-4340	71	2	pure	pure	PROPN
ejpam-4340	71	3	appl	appl	PROPN
ejpam-4340	71	4	.	.	PROPN
ejpam-4340	71	5	math	math	PROPN
ejpam-4340	71	6	,	,	PUNCT
ejpam-4340	71	7	15	15	NUM
ejpam-4340	71	8	(	(	PUNCT
ejpam-4340	71	9	2	2	NUM
ejpam-4340	71	10	)	)	PUNCT
ejpam-4340	71	11	(	(	PUNCT
ejpam-4340	71	12	2022	2022	NUM
ejpam-4340	71	13	)	)	PUNCT
ejpam-4340	71	14	,	,	PUNCT
ejpam-4340	71	15	354	354	NUM
ejpam-4340	71	16	-	-	SYM
ejpam-4340	71	17	374	374	NUM
ejpam-4340	71	18	356	356	NUM
ejpam-4340	71	19	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	71	20	(	(	PUNCT
ejpam-4340	71	21	resp	resp	NOUN
ejpam-4340	71	22	.	.	PUNCT
ejpam-4340	72	1	ωa	ωa	ADJ
ejpam-4340	72	2	-	-	PUNCT
ejpam-4340	72	3	closed	closed	ADJ
ejpam-4340	72	4	)	)	PUNCT
ejpam-4340	72	5	.	.	PUNCT
ejpam-4340	73	1	the	the	DET
ejpam-4340	73	2	family	family	NOUN
ejpam-4340	73	3	of	of	ADP
ejpam-4340	73	4	all	all	DET
ejpam-4340	73	5	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	73	6	(	(	PUNCT
ejpam-4340	73	7	resp	resp	NOUN
ejpam-4340	73	8	.	.	PUNCT
ejpam-4340	74	1	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	74	2	,	,	PUNCT
ejpam-4340	74	3	ωa	ωa	ADJ
ejpam-4340	74	4	-	-	PUNCT
ejpam-4340	74	5	open	open	ADJ
ejpam-4340	74	6	,	,	PUNCT
ejpam-4340	74	7	ωaclosed	ωaclosed	ADJ
ejpam-4340	74	8	)	)	PUNCT
ejpam-4340	74	9	sets	set	NOUN
ejpam-4340	74	10	of	of	ADP
ejpam-4340	74	11	x	x	PUNCT
ejpam-4340	74	12	will	will	AUX
ejpam-4340	74	13	be	be	AUX
ejpam-4340	74	14	denoted	denote	VERB
ejpam-4340	74	15	by	by	ADP
ejpam-4340	74	16	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	74	17	)	)	PUNCT
ejpam-4340	74	18	(	(	PUNCT
ejpam-4340	74	19	resp	resp	NOUN
ejpam-4340	74	20	.	.	PUNCT
ejpam-4340	75	1	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	75	2	)	)	PUNCT
ejpam-4340	75	3	,	,	PUNCT
ejpam-4340	75	4	ωao(x	ωao(x	PROPN
ejpam-4340	75	5	)	)	PUNCT
ejpam-4340	75	6	,	,	PUNCT
ejpam-4340	75	7	ωac(x	ωac(x	NOUN
ejpam-4340	75	8	)	)	PUNCT
ejpam-4340	75	9	)	)	PUNCT
ejpam-4340	75	10	.	.	PUNCT
ejpam-4340	76	1	the	the	DET
ejpam-4340	76	2	family	family	NOUN
ejpam-4340	76	3	of	of	ADP
ejpam-4340	76	4	all	all	DET
ejpam-4340	76	5	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	76	6	(	(	PUNCT
ejpam-4340	76	7	resp	resp	NOUN
ejpam-4340	76	8	.	.	PUNCT
ejpam-4340	77	1	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	77	2	,	,	PUNCT
ejpam-4340	77	3	ωa	ωa	ADJ
ejpam-4340	77	4	-	-	PUNCT
ejpam-4340	77	5	open	open	ADJ
ejpam-4340	77	6	,	,	PUNCT
ejpam-4340	77	7	ωa	ωa	ADJ
ejpam-4340	77	8	-	-	PUNCT
ejpam-4340	77	9	closed	closed	ADJ
ejpam-4340	77	10	)	)	PUNCT
ejpam-4340	77	11	sets	set	NOUN
ejpam-4340	77	12	of	of	ADP
ejpam-4340	77	13	x	x	PUNCT
ejpam-4340	77	14	containing	contain	VERB
ejpam-4340	77	15	a	a	DET
ejpam-4340	77	16	point	point	NOUN
ejpam-4340	77	17	x	x	PUNCT
ejpam-4340	77	18	of	of	ADP
ejpam-4340	77	19	x	x	PRON
ejpam-4340	77	20	will	will	AUX
ejpam-4340	77	21	be	be	AUX
ejpam-4340	77	22	denoted	denote	VERB
ejpam-4340	77	23	by	by	ADP
ejpam-4340	77	24	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	77	25	,	,	PUNCT
ejpam-4340	77	26	x	x	X
ejpam-4340	77	27	)	)	PUNCT
ejpam-4340	77	28	(	(	PUNCT
ejpam-4340	77	29	resp	resp	NOUN
ejpam-4340	77	30	.	.	PUNCT
ejpam-4340	78	1	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	78	2	,	,	PUNCT
ejpam-4340	78	3	x	x	NOUN
ejpam-4340	78	4	)	)	PUNCT
ejpam-4340	78	5	,	,	PUNCT
ejpam-4340	78	6	ωao(x	ωao(x	PROPN
ejpam-4340	78	7	,	,	PUNCT
ejpam-4340	78	8	x	x	NOUN
ejpam-4340	78	9	)	)	PUNCT
ejpam-4340	78	10	,	,	PUNCT
ejpam-4340	78	11	ωac(x	ωac(x	NOUN
ejpam-4340	78	12	,	,	PUNCT
ejpam-4340	78	13	x	x	NOUN
ejpam-4340	78	14	)	)	PUNCT
ejpam-4340	78	15	)	)	PUNCT
ejpam-4340	78	16	.	.	PUNCT
ejpam-4340	79	1	open	open	ADJ
ejpam-4340	79	2	→	→	SYM
ejpam-4340	79	3	β	β	X
ejpam-4340	79	4	-	-	ADJ
ejpam-4340	79	5	open	open	ADJ
ejpam-4340	79	6	→	→	SYM
ejpam-4340	79	7	e∗-open	e∗-open	ADJ
ejpam-4340	79	8	↓	↓	PROPN
ejpam-4340	79	9	↓	↓	PROPN
ejpam-4340	79	10	↓	↓	PROPN
ejpam-4340	79	11	ω	ω	PROPN
ejpam-4340	79	12	-	-	NOUN
ejpam-4340	79	13	open	open	ADJ
ejpam-4340	79	14	→	→	SYM
ejpam-4340	79	15	ωβ	ωβ	ADJ
ejpam-4340	79	16	-	-	ADJ
ejpam-4340	79	17	open	open	ADJ
ejpam-4340	79	18	→	→	SYM
ejpam-4340	79	19	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	79	20	←	←	PROPN
ejpam-4340	79	21	ωa	ωa	ADJ
ejpam-4340	79	22	-	-	PUNCT
ejpam-4340	79	23	open	open	ADJ
ejpam-4340	79	24	definition	definition	NOUN
ejpam-4340	79	25	3	3	NUM
ejpam-4340	79	26	.	.	PUNCT
ejpam-4340	80	1	[	[	X
ejpam-4340	80	2	19	19	NUM
ejpam-4340	80	3	]	]	PUNCT
ejpam-4340	80	4	let	let	VERB
ejpam-4340	80	5	a	a	PRON
ejpam-4340	80	6	be	be	AUX
ejpam-4340	80	7	a	a	DET
ejpam-4340	80	8	subset	subset	NOUN
ejpam-4340	80	9	of	of	ADP
ejpam-4340	80	10	a	a	DET
ejpam-4340	80	11	space	space	NOUN
ejpam-4340	80	12	x.	x.	NOUN
ejpam-4340	81	1	the	the	DET
ejpam-4340	81	2	union	union	NOUN
ejpam-4340	81	3	of	of	ADP
ejpam-4340	81	4	all	all	DET
ejpam-4340	81	5	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	81	6	subsets	subset	NOUN
ejpam-4340	81	7	of	of	ADP
ejpam-4340	81	8	x	x	PUNCT
ejpam-4340	81	9	contained	contain	VERB
ejpam-4340	81	10	in	in	ADP
ejpam-4340	81	11	a	a	PRON
ejpam-4340	81	12	is	be	AUX
ejpam-4340	81	13	called	call	VERB
ejpam-4340	81	14	the	the	DET
ejpam-4340	81	15	ωe∗-interior	ωe∗-interior	NOUN
ejpam-4340	81	16	of	of	ADP
ejpam-4340	81	17	a	a	PRON
ejpam-4340	81	18	and	and	CCONJ
ejpam-4340	81	19	is	be	AUX
ejpam-4340	81	20	denoted	denote	VERB
ejpam-4340	81	21	by	by	ADP
ejpam-4340	81	22	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	81	23	)	)	PUNCT
ejpam-4340	81	24	.	.	PUNCT
ejpam-4340	82	1	theorem	theorem	NOUN
ejpam-4340	82	2	1	1	NUM
ejpam-4340	82	3	.	.	PUNCT
ejpam-4340	83	1	[	[	X
ejpam-4340	83	2	19	19	NUM
ejpam-4340	83	3	]	]	PUNCT
ejpam-4340	83	4	let	let	VERB
ejpam-4340	83	5	a	a	PRON
ejpam-4340	83	6	be	be	AUX
ejpam-4340	83	7	a	a	DET
ejpam-4340	83	8	subset	subset	NOUN
ejpam-4340	83	9	of	of	ADP
ejpam-4340	83	10	a	a	DET
ejpam-4340	83	11	space	space	NOUN
ejpam-4340	83	12	x.	x.	NOUN
ejpam-4340	83	13	then	then	ADV
ejpam-4340	83	14	the	the	DET
ejpam-4340	83	15	following	follow	VERB
ejpam-4340	83	16	properties	property	NOUN
ejpam-4340	83	17	hold	hold	VERB
ejpam-4340	83	18	:	:	PUNCT
ejpam-4340	83	19	(	(	PUNCT
ejpam-4340	83	20	a	a	X
ejpam-4340	83	21	)	)	PUNCT
ejpam-4340	83	22	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	83	23	)	)	PUNCT
ejpam-4340	83	24	⊆	⊆	NUM
ejpam-4340	83	25	a	a	PRON
ejpam-4340	83	26	,	,	PUNCT
ejpam-4340	83	27	(	(	PUNCT
ejpam-4340	83	28	b	b	NOUN
ejpam-4340	83	29	)	)	PUNCT
ejpam-4340	83	30	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	83	31	)	)	PUNCT
ejpam-4340	83	32	∈	∈	PROPN
ejpam-4340	83	33	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	83	34	)	)	PUNCT
ejpam-4340	83	35	,	,	PUNCT
ejpam-4340	83	36	(	(	PUNCT
ejpam-4340	83	37	c	c	X
ejpam-4340	83	38	)	)	PUNCT
ejpam-4340	83	39	x	x	SYM
ejpam-4340	83	40	∈	∈	NOUN
ejpam-4340	83	41	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	83	42	)	)	PUNCT
ejpam-4340	84	1	if	if	SCONJ
ejpam-4340	84	2	and	and	CCONJ
ejpam-4340	84	3	only	only	ADV
ejpam-4340	84	4	if	if	SCONJ
ejpam-4340	84	5	there	there	PRON
ejpam-4340	84	6	exists	exist	VERB
ejpam-4340	84	7	u	u	PROPN
ejpam-4340	84	8	∈	∈	PROPN
ejpam-4340	84	9	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	84	10	,	,	PUNCT
ejpam-4340	84	11	x	x	NOUN
ejpam-4340	84	12	)	)	PUNCT
ejpam-4340	84	13	such	such	ADJ
ejpam-4340	84	14	that	that	SCONJ
ejpam-4340	84	15	u	u	PROPN
ejpam-4340	84	16	⊆	⊆	SYM
ejpam-4340	84	17	a	a	PRON
ejpam-4340	84	18	,	,	PUNCT
ejpam-4340	84	19	(	(	PUNCT
ejpam-4340	84	20	d	d	X
ejpam-4340	84	21	)	)	PUNCT
ejpam-4340	84	22	a	a	DET
ejpam-4340	84	23	⊆	⊆	NUM
ejpam-4340	84	24	b	b	NOUN
ejpam-4340	84	25	⇒	⇒	NOUN
ejpam-4340	84	26	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	84	27	)	)	PUNCT
ejpam-4340	84	28	⊆	⊆	NUM
ejpam-4340	84	29	ωe∗-int(b	ωe∗-int(b	NOUN
ejpam-4340	84	30	)	)	PUNCT
ejpam-4340	84	31	,	,	PUNCT
ejpam-4340	84	32	(	(	PUNCT
ejpam-4340	84	33	e	e	NOUN
ejpam-4340	84	34	)	)	PUNCT
ejpam-4340	84	35	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	84	36	)	)	PUNCT
ejpam-4340	84	37	∪	∪	NOUN
ejpam-4340	84	38	ωe∗-int(b	ωe∗-int(b	NOUN
ejpam-4340	84	39	)	)	PUNCT
ejpam-4340	84	40	⊆	⊆	NUM
ejpam-4340	84	41	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	84	42	∪b	∪b	NUM
ejpam-4340	84	43	)	)	PUNCT
ejpam-4340	84	44	,	,	PUNCT
ejpam-4340	84	45	(	(	PUNCT
ejpam-4340	84	46	f	f	X
ejpam-4340	84	47	)	)	PUNCT
ejpam-4340	84	48	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	84	49	∩b	∩b	NOUN
ejpam-4340	84	50	)	)	PUNCT
ejpam-4340	84	51	⊆	⊆	NUM
ejpam-4340	84	52	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	84	53	)	)	PUNCT
ejpam-4340	84	54	∩	∩	ADJ
ejpam-4340	84	55	ωe∗-int(b	ωe∗-int(b	NOUN
ejpam-4340	84	56	)	)	PUNCT
ejpam-4340	84	57	,	,	PUNCT
ejpam-4340	84	58	(	(	PUNCT
ejpam-4340	84	59	g	g	NOUN
ejpam-4340	84	60	)	)	PUNCT
ejpam-4340	84	61	a	a	DET
ejpam-4340	84	62	∈	∈	PROPN
ejpam-4340	84	63	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	84	64	)	)	PUNCT
ejpam-4340	84	65	if	if	SCONJ
ejpam-4340	85	1	and	and	CCONJ
ejpam-4340	85	2	only	only	ADV
ejpam-4340	85	3	if	if	SCONJ
ejpam-4340	85	4	a	a	DET
ejpam-4340	85	5	=	=	NOUN
ejpam-4340	85	6	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	85	7	)	)	PUNCT
ejpam-4340	85	8	,	,	PUNCT
ejpam-4340	85	9	(	(	PUNCT
ejpam-4340	85	10	h	h	NOUN
ejpam-4340	85	11	)	)	PUNCT
ejpam-4340	85	12	ωe∗-int(ωe∗-int(a	ωe∗-int(ωe∗-int(a	NOUN
ejpam-4340	85	13	)	)	PUNCT
ejpam-4340	85	14	)	)	PUNCT
ejpam-4340	86	1	=	=	SYM
ejpam-4340	86	2	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	86	3	)	)	PUNCT
ejpam-4340	86	4	.	.	PUNCT
ejpam-4340	87	1	definition	definition	NOUN
ejpam-4340	87	2	4	4	NUM
ejpam-4340	87	3	.	.	PUNCT
ejpam-4340	88	1	[	[	X
ejpam-4340	88	2	19	19	NUM
ejpam-4340	88	3	]	]	PUNCT
ejpam-4340	88	4	let	let	VERB
ejpam-4340	88	5	a	a	PRON
ejpam-4340	88	6	be	be	AUX
ejpam-4340	88	7	a	a	DET
ejpam-4340	88	8	subset	subset	NOUN
ejpam-4340	88	9	of	of	ADP
ejpam-4340	88	10	a	a	DET
ejpam-4340	88	11	space	space	NOUN
ejpam-4340	88	12	x.	x.	NOUN
ejpam-4340	89	1	the	the	DET
ejpam-4340	89	2	intersection	intersection	NOUN
ejpam-4340	89	3	of	of	ADP
ejpam-4340	89	4	all	all	DET
ejpam-4340	89	5	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	89	6	subsets	subset	NOUN
ejpam-4340	89	7	of	of	ADP
ejpam-4340	89	8	x	x	PUNCT
ejpam-4340	89	9	containing	contain	VERB
ejpam-4340	89	10	a	a	PRON
ejpam-4340	89	11	is	be	AUX
ejpam-4340	89	12	called	call	VERB
ejpam-4340	89	13	the	the	DET
ejpam-4340	89	14	ωe∗-closure	ωe∗-closure	NOUN
ejpam-4340	89	15	of	of	ADP
ejpam-4340	89	16	a	a	PRON
ejpam-4340	89	17	and	and	CCONJ
ejpam-4340	89	18	is	be	AUX
ejpam-4340	89	19	denoted	denote	VERB
ejpam-4340	89	20	by	by	ADP
ejpam-4340	89	21	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	89	22	)	)	PUNCT
ejpam-4340	89	23	.	.	PUNCT
ejpam-4340	90	1	theorem	theorem	NOUN
ejpam-4340	90	2	2	2	NUM
ejpam-4340	90	3	.	.	PUNCT
ejpam-4340	91	1	[	[	X
ejpam-4340	91	2	19	19	NUM
ejpam-4340	91	3	]	]	PUNCT
ejpam-4340	91	4	let	let	VERB
ejpam-4340	91	5	a	a	PRON
ejpam-4340	91	6	and	and	CCONJ
ejpam-4340	91	7	b	b	NOUN
ejpam-4340	91	8	be	be	AUX
ejpam-4340	91	9	two	two	NUM
ejpam-4340	91	10	subsets	subset	NOUN
ejpam-4340	91	11	of	of	ADP
ejpam-4340	91	12	a	a	DET
ejpam-4340	91	13	space	space	NOUN
ejpam-4340	91	14	x.	x.	NOUN
ejpam-4340	91	15	then	then	ADV
ejpam-4340	91	16	the	the	DET
ejpam-4340	91	17	following	follow	VERB
ejpam-4340	91	18	properties	property	NOUN
ejpam-4340	91	19	hold	hold	VERB
ejpam-4340	91	20	:	:	PUNCT
ejpam-4340	91	21	(	(	PUNCT
ejpam-4340	91	22	a	a	X
ejpam-4340	91	23	)	)	PUNCT
ejpam-4340	91	24	a	a	DET
ejpam-4340	91	25	⊆	⊆	NUM
ejpam-4340	91	26	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	27	)	)	PUNCT
ejpam-4340	91	28	,	,	PUNCT
ejpam-4340	91	29	(	(	PUNCT
ejpam-4340	91	30	b	b	X
ejpam-4340	91	31	)	)	PUNCT
ejpam-4340	91	32	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	33	)	)	PUNCT
ejpam-4340	91	34	∈	∈	PROPN
ejpam-4340	91	35	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	91	36	)	)	PUNCT
ejpam-4340	91	37	,	,	PUNCT
ejpam-4340	91	38	(	(	PUNCT
ejpam-4340	91	39	c	c	X
ejpam-4340	91	40	)	)	PUNCT
ejpam-4340	91	41	x	x	SYM
ejpam-4340	91	42	∈	∈	PROPN
ejpam-4340	91	43	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	44	)	)	PUNCT
ejpam-4340	91	45	if	if	SCONJ
ejpam-4340	91	46	and	and	CCONJ
ejpam-4340	91	47	only	only	ADV
ejpam-4340	91	48	if	if	SCONJ
ejpam-4340	91	49	a	a	DET
ejpam-4340	91	50	∩	∩	ADJ
ejpam-4340	91	51	u	u	ADJ
ejpam-4340	91	52	̸=	̸=	PROPN
ejpam-4340	91	53	∅	∅	NOUN
ejpam-4340	91	54	for	for	ADP
ejpam-4340	91	55	every	every	DET
ejpam-4340	91	56	u	u	PROPN
ejpam-4340	91	57	∈	∈	PROPN
ejpam-4340	91	58	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	91	59	,	,	PUNCT
ejpam-4340	91	60	x	x	NOUN
ejpam-4340	91	61	)	)	PUNCT
ejpam-4340	91	62	,	,	PUNCT
ejpam-4340	91	63	(	(	PUNCT
ejpam-4340	91	64	d	d	X
ejpam-4340	91	65	)	)	PUNCT
ejpam-4340	91	66	a	a	DET
ejpam-4340	91	67	⊆	⊆	NUM
ejpam-4340	91	68	b	b	NOUN
ejpam-4340	91	69	⇒	⇒	NOUN
ejpam-4340	91	70	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	71	)	)	PUNCT
ejpam-4340	91	72	⊆	⊆	NUM
ejpam-4340	91	73	ωe∗-cl(b	ωe∗-cl(b	NOUN
ejpam-4340	91	74	)	)	PUNCT
ejpam-4340	91	75	,	,	PUNCT
ejpam-4340	91	76	(	(	PUNCT
ejpam-4340	91	77	e	e	NOUN
ejpam-4340	91	78	)	)	PUNCT
ejpam-4340	91	79	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	80	)	)	PUNCT
ejpam-4340	91	81	∪	∪	ADP
ejpam-4340	91	82	ωe∗-cl(b	ωe∗-cl(b	NOUN
ejpam-4340	91	83	)	)	PUNCT
ejpam-4340	91	84	⊆	⊆	NUM
ejpam-4340	91	85	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	86	∪b	∪b	NOUN
ejpam-4340	91	87	)	)	PUNCT
ejpam-4340	91	88	,	,	PUNCT
ejpam-4340	91	89	(	(	PUNCT
ejpam-4340	91	90	f	f	X
ejpam-4340	91	91	)	)	PUNCT
ejpam-4340	91	92	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	93	∩b	∩b	NOUN
ejpam-4340	91	94	)	)	PUNCT
ejpam-4340	91	95	⊆	⊆	NUM
ejpam-4340	91	96	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	91	97	)	)	PUNCT
ejpam-4340	91	98	∩	∩	NOUN
ejpam-4340	91	99	ωe∗-cl(b	ωe∗-cl(b	NOUN
ejpam-4340	91	100	)	)	PUNCT
ejpam-4340	91	101	,	,	PUNCT
ejpam-4340	91	102	(	(	PUNCT
ejpam-4340	91	103	g	g	NOUN
ejpam-4340	91	104	)	)	PUNCT
ejpam-4340	91	105	a	a	DET
ejpam-4340	91	106	∈	∈	PROPN
ejpam-4340	91	107	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	91	108	)	)	PUNCT
ejpam-4340	92	1	if	if	SCONJ
ejpam-4340	92	2	and	and	CCONJ
ejpam-4340	92	3	only	only	ADV
ejpam-4340	92	4	if	if	SCONJ
ejpam-4340	92	5	a	a	DET
ejpam-4340	92	6	=	=	SYM
ejpam-4340	92	7	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	92	8	)	)	PUNCT
ejpam-4340	92	9	,	,	PUNCT
ejpam-4340	92	10	(	(	PUNCT
ejpam-4340	92	11	h	h	NOUN
ejpam-4340	92	12	)	)	PUNCT
ejpam-4340	92	13	ωe∗-cl(ωe∗-cl(a	ωe∗-cl(ωe∗-cl(a	NUM
ejpam-4340	92	14	)	)	PUNCT
ejpam-4340	92	15	)	)	PUNCT
ejpam-4340	93	1	=	=	PUNCT
ejpam-4340	93	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	93	3	)	)	PUNCT
ejpam-4340	93	4	,	,	PUNCT
ejpam-4340	93	5	(	(	PUNCT
ejpam-4340	93	6	i	i	NOUN
ejpam-4340	93	7	)	)	PUNCT
ejpam-4340	93	8	ωe∗-cl(x	ωe∗-cl(x	NUM
ejpam-4340	93	9	\a	\a	ADJ
ejpam-4340	93	10	)	)	PUNCT
ejpam-4340	93	11	=	=	SYM
ejpam-4340	93	12	x	x	SYM
ejpam-4340	93	13	\	\	NOUN
ejpam-4340	93	14	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	93	15	)	)	PUNCT
ejpam-4340	93	16	.	.	PUNCT
ejpam-4340	94	1	lemma	lemma	PROPN
ejpam-4340	94	2	1	1	NUM
ejpam-4340	94	3	.	.	PUNCT
ejpam-4340	95	1	[	[	X
ejpam-4340	95	2	19	19	NUM
ejpam-4340	95	3	]	]	PUNCT
ejpam-4340	95	4	let	let	VERB
ejpam-4340	95	5	x	x	PRON
ejpam-4340	95	6	be	be	AUX
ejpam-4340	95	7	a	a	DET
ejpam-4340	95	8	topological	topological	ADJ
ejpam-4340	95	9	space	space	NOUN
ejpam-4340	95	10	.	.	PUNCT
ejpam-4340	96	1	then	then	ADV
ejpam-4340	96	2	the	the	DET
ejpam-4340	96	3	following	follow	VERB
ejpam-4340	96	4	properties	property	NOUN
ejpam-4340	96	5	hold	hold	VERB
ejpam-4340	96	6	:	:	PUNCT
ejpam-4340	96	7	(	(	PUNCT
ejpam-4340	96	8	a	a	X
ejpam-4340	96	9	)	)	PUNCT
ejpam-4340	96	10	the	the	DET
ejpam-4340	96	11	union	union	NOUN
ejpam-4340	96	12	of	of	ADP
ejpam-4340	96	13	any	any	DET
ejpam-4340	96	14	family	family	NOUN
ejpam-4340	96	15	of	of	ADP
ejpam-4340	96	16	we∗-open	we∗-open	PROPN
ejpam-4340	96	17	sets	set	NOUN
ejpam-4340	96	18	is	be	AUX
ejpam-4340	96	19	we∗-open	we∗-open	NOUN
ejpam-4340	96	20	,	,	PUNCT
ejpam-4340	96	21	(	(	PUNCT
ejpam-4340	96	22	b	b	X
ejpam-4340	96	23	)	)	PUNCT
ejpam-4340	96	24	the	the	DET
ejpam-4340	96	25	intersection	intersection	NOUN
ejpam-4340	96	26	of	of	ADP
ejpam-4340	96	27	an	an	DET
ejpam-4340	96	28	wa	wa	ADJ
ejpam-4340	96	29	-	-	ADJ
ejpam-4340	96	30	open	open	ADJ
ejpam-4340	96	31	set	set	NOUN
ejpam-4340	96	32	and	and	CCONJ
ejpam-4340	96	33	an	an	DET
ejpam-4340	96	34	we∗-open	we∗-open	NOUN
ejpam-4340	96	35	set	set	NOUN
ejpam-4340	96	36	is	be	AUX
ejpam-4340	96	37	we∗-open	we∗-open	NOUN
ejpam-4340	96	38	.	.	PUNCT
ejpam-4340	97	1	definition	definition	NOUN
ejpam-4340	97	2	5	5	NUM
ejpam-4340	97	3	.	.	PUNCT
ejpam-4340	98	1	let	let	VERB
ejpam-4340	98	2	a	a	DET
ejpam-4340	98	3	be	be	AUX
ejpam-4340	98	4	a	a	DET
ejpam-4340	98	5	subset	subset	NOUN
ejpam-4340	98	6	of	of	ADP
ejpam-4340	98	7	a	a	DET
ejpam-4340	98	8	space	space	NOUN
ejpam-4340	98	9	x.	x.	NOUN
ejpam-4340	99	1	the	the	DET
ejpam-4340	99	2	intersection	intersection	NOUN
ejpam-4340	99	3	of	of	ADP
ejpam-4340	99	4	all	all	DET
ejpam-4340	99	5	open	open	ADJ
ejpam-4340	99	6	sets	set	NOUN
ejpam-4340	99	7	in	in	ADP
ejpam-4340	99	8	x	x	PUNCT
ejpam-4340	99	9	containing	contain	VERB
ejpam-4340	99	10	a	a	PRON
ejpam-4340	99	11	is	be	AUX
ejpam-4340	99	12	called	call	VERB
ejpam-4340	99	13	the	the	DET
ejpam-4340	99	14	kernel	kernel	NOUN
ejpam-4340	99	15	[	[	X
ejpam-4340	99	16	17	17	NUM
ejpam-4340	99	17	]	]	PUNCT
ejpam-4340	99	18	of	of	ADP
ejpam-4340	99	19	a	a	PRON
ejpam-4340	99	20	and	and	CCONJ
ejpam-4340	99	21	is	be	AUX
ejpam-4340	99	22	denoted	denote	VERB
ejpam-4340	99	23	by	by	ADP
ejpam-4340	99	24	ker(a	ker(a	PROPN
ejpam-4340	99	25	)	)	PUNCT
ejpam-4340	99	26	.	.	PUNCT
ejpam-4340	100	1	lemma	lemma	PROPN
ejpam-4340	100	2	2	2	NUM
ejpam-4340	100	3	.	.	PUNCT
ejpam-4340	101	1	[	[	X
ejpam-4340	101	2	17	17	NUM
ejpam-4340	101	3	]	]	PUNCT
ejpam-4340	101	4	the	the	DET
ejpam-4340	101	5	followings	following	NOUN
ejpam-4340	101	6	hold	hold	VERB
ejpam-4340	101	7	for	for	ADP
ejpam-4340	101	8	subsets	subset	NOUN
ejpam-4340	101	9	a	a	PRON
ejpam-4340	101	10	and	and	CCONJ
ejpam-4340	101	11	b	b	NOUN
ejpam-4340	101	12	of	of	ADP
ejpam-4340	101	13	a	a	DET
ejpam-4340	101	14	space	space	NOUN
ejpam-4340	101	15	x.	x.	NOUN
ejpam-4340	101	16	(	(	PUNCT
ejpam-4340	101	17	a	a	X
ejpam-4340	101	18	)	)	PUNCT
ejpam-4340	101	19	x	x	SYM
ejpam-4340	101	20	∈	∈	PROPN
ejpam-4340	101	21	ker(a	ker(a	PROPN
ejpam-4340	101	22	)	)	PUNCT
ejpam-4340	102	1	if	if	SCONJ
ejpam-4340	102	2	and	and	CCONJ
ejpam-4340	102	3	only	only	ADV
ejpam-4340	102	4	if	if	SCONJ
ejpam-4340	102	5	a	a	DET
ejpam-4340	102	6	∩	∩	NOUN
ejpam-4340	102	7	f	f	X
ejpam-4340	102	8	̸=	̸=	PROPN
ejpam-4340	102	9	∅	∅	NOUN
ejpam-4340	102	10	for	for	ADP
ejpam-4340	102	11	any	any	DET
ejpam-4340	102	12	f	f	PROPN
ejpam-4340	102	13	∈	∈	PROPN
ejpam-4340	102	14	c(x	c(x	NOUN
ejpam-4340	102	15	,	,	PUNCT
ejpam-4340	102	16	x	x	NOUN
ejpam-4340	102	17	)	)	PUNCT
ejpam-4340	102	18	,	,	PUNCT
ejpam-4340	102	19	p.	p.	NOUN
ejpam-4340	102	20	şaşmaz	şaşmaz	NUM
ejpam-4340	102	21	,	,	PUNCT
ejpam-4340	102	22	m.	m.	NOUN
ejpam-4340	102	23	özkoç	özkoç	PROPN
ejpam-4340	102	24	/	/	SYM
ejpam-4340	102	25	eur	eur	PROPN
ejpam-4340	102	26	.	.	PUNCT
ejpam-4340	103	1	j.	j.	PROPN
ejpam-4340	103	2	pure	pure	PROPN
ejpam-4340	103	3	appl	appl	PROPN
ejpam-4340	103	4	.	.	PROPN
ejpam-4340	103	5	math	math	PROPN
ejpam-4340	103	6	,	,	PUNCT
ejpam-4340	103	7	15	15	NUM
ejpam-4340	103	8	(	(	PUNCT
ejpam-4340	103	9	2	2	NUM
ejpam-4340	103	10	)	)	PUNCT
ejpam-4340	103	11	(	(	PUNCT
ejpam-4340	103	12	2022	2022	NUM
ejpam-4340	103	13	)	)	PUNCT
ejpam-4340	103	14	,	,	PUNCT
ejpam-4340	103	15	354	354	NUM
ejpam-4340	103	16	-	-	SYM
ejpam-4340	103	17	374	374	NUM
ejpam-4340	103	18	357	357	NUM
ejpam-4340	103	19	(	(	PUNCT
ejpam-4340	103	20	b	b	NOUN
ejpam-4340	103	21	)	)	PUNCT
ejpam-4340	103	22	a	a	DET
ejpam-4340	103	23	⊆	⊆	NUM
ejpam-4340	103	24	ker(a	ker(a	NOUN
ejpam-4340	103	25	)	)	PUNCT
ejpam-4340	103	26	,	,	PUNCT
ejpam-4340	103	27	(	(	PUNCT
ejpam-4340	103	28	c	c	X
ejpam-4340	103	29	)	)	PUNCT
ejpam-4340	103	30	if	if	SCONJ
ejpam-4340	103	31	a	a	PRON
ejpam-4340	103	32	is	be	AUX
ejpam-4340	103	33	open	open	ADJ
ejpam-4340	103	34	in	in	ADP
ejpam-4340	103	35	x	x	NOUN
ejpam-4340	103	36	,	,	PUNCT
ejpam-4340	103	37	then	then	ADV
ejpam-4340	103	38	a	a	DET
ejpam-4340	103	39	=	=	SYM
ejpam-4340	103	40	ker(a	ker(a	PROPN
ejpam-4340	103	41	)	)	PUNCT
ejpam-4340	103	42	,	,	PUNCT
ejpam-4340	103	43	(	(	PUNCT
ejpam-4340	103	44	d	d	X
ejpam-4340	103	45	)	)	PUNCT
ejpam-4340	103	46	if	if	SCONJ
ejpam-4340	103	47	a	a	DET
ejpam-4340	103	48	⊆	⊆	NUM
ejpam-4340	103	49	b	b	NOUN
ejpam-4340	103	50	,	,	PUNCT
ejpam-4340	103	51	then	then	ADV
ejpam-4340	103	52	ker(a	ker(a	PROPN
ejpam-4340	103	53	)	)	PUNCT
ejpam-4340	103	54	⊆	⊆	NUM
ejpam-4340	103	55	ker(b	ker(b	PROPN
ejpam-4340	103	56	)	)	PUNCT
ejpam-4340	103	57	.	.	PUNCT
ejpam-4340	104	1	definition	definition	NOUN
ejpam-4340	104	2	6	6	NUM
ejpam-4340	104	3	.	.	PUNCT
ejpam-4340	105	1	a	a	DET
ejpam-4340	105	2	function	function	NOUN
ejpam-4340	105	3	f	f	NOUN
ejpam-4340	105	4	:	:	PUNCT
ejpam-4340	105	5	x	x	X
ejpam-4340	105	6	→	→	SYM
ejpam-4340	105	7	y	y	PROPN
ejpam-4340	105	8	is	be	AUX
ejpam-4340	105	9	called	call	VERB
ejpam-4340	105	10	:	:	PUNCT
ejpam-4340	105	11	(	(	PUNCT
ejpam-4340	105	12	a	a	X
ejpam-4340	105	13	)	)	PUNCT
ejpam-4340	105	14	e∗-continuous	e∗-continuous	ADJ
ejpam-4340	105	15	[	[	X
ejpam-4340	105	16	11	11	NUM
ejpam-4340	105	17	]	]	PUNCT
ejpam-4340	105	18	if	if	SCONJ
ejpam-4340	105	19	f−1[v	f−1[v	NOUN
ejpam-4340	105	20	]	]	PUNCT
ejpam-4340	105	21	∈	∈	PROPN
ejpam-4340	105	22	e∗o(x	e∗o(x	PROPN
ejpam-4340	105	23	)	)	PUNCT
ejpam-4340	105	24	for	for	SCONJ
ejpam-4340	105	25	each	each	DET
ejpam-4340	105	26	open	open	ADJ
ejpam-4340	105	27	set	set	VERB
ejpam-4340	105	28	v	v	NOUN
ejpam-4340	105	29	of	of	ADP
ejpam-4340	105	30	y	y	PROPN
ejpam-4340	105	31	,	,	PUNCT
ejpam-4340	105	32	(	(	PUNCT
ejpam-4340	105	33	b	b	NOUN
ejpam-4340	105	34	)	)	PUNCT
ejpam-4340	105	35	β	β	NOUN
ejpam-4340	105	36	-	-	ADJ
ejpam-4340	105	37	continuous	continuous	ADJ
ejpam-4340	105	38	[	[	X
ejpam-4340	105	39	1	1	NUM
ejpam-4340	105	40	]	]	PUNCT
ejpam-4340	105	41	if	if	SCONJ
ejpam-4340	105	42	f−1[v	f−1[v	NOUN
ejpam-4340	105	43	]	]	PUNCT
ejpam-4340	105	44	∈	∈	PROPN
ejpam-4340	105	45	ωβo(x	ωβo(x	NUM
ejpam-4340	105	46	)	)	PUNCT
ejpam-4340	105	47	for	for	ADP
ejpam-4340	105	48	each	each	DET
ejpam-4340	105	49	open	open	ADJ
ejpam-4340	105	50	set	set	VERB
ejpam-4340	105	51	v	v	NOUN
ejpam-4340	105	52	of	of	ADP
ejpam-4340	105	53	y	y	PROPN
ejpam-4340	105	54	,	,	PUNCT
ejpam-4340	105	55	(	(	PUNCT
ejpam-4340	105	56	c	c	X
ejpam-4340	105	57	)	)	PUNCT
ejpam-4340	105	58	ω	ω	NOUN
ejpam-4340	105	59	-	-	NOUN
ejpam-4340	105	60	continuous	continuous	ADJ
ejpam-4340	105	61	[	[	X
ejpam-4340	105	62	13	13	NUM
ejpam-4340	105	63	]	]	PUNCT
ejpam-4340	105	64	if	if	SCONJ
ejpam-4340	105	65	f−1[v	f−1[v	NOUN
ejpam-4340	105	66	]	]	X
ejpam-4340	105	67	∈	∈	PROPN
ejpam-4340	105	68	ωo(x	ωo(x	NUM
ejpam-4340	105	69	)	)	PUNCT
ejpam-4340	105	70	for	for	SCONJ
ejpam-4340	105	71	each	each	DET
ejpam-4340	105	72	open	open	ADJ
ejpam-4340	105	73	set	set	VERB
ejpam-4340	105	74	v	v	NOUN
ejpam-4340	105	75	of	of	ADP
ejpam-4340	105	76	y	y	PROPN
ejpam-4340	105	77	,	,	PUNCT
ejpam-4340	105	78	(	(	PUNCT
ejpam-4340	105	79	d	d	X
ejpam-4340	105	80	)	)	PUNCT
ejpam-4340	105	81	ωβ	ωβ	ADJ
ejpam-4340	105	82	-	-	ADJ
ejpam-4340	105	83	continuous	continuous	ADJ
ejpam-4340	105	84	[	[	X
ejpam-4340	105	85	3	3	NUM
ejpam-4340	105	86	]	]	X
ejpam-4340	105	87	if	if	SCONJ
ejpam-4340	105	88	for	for	ADP
ejpam-4340	105	89	each	each	DET
ejpam-4340	105	90	x	x	SYM
ejpam-4340	105	91	∈	∈	PROPN
ejpam-4340	105	92	x	x	X
ejpam-4340	105	93	and	and	CCONJ
ejpam-4340	105	94	each	each	DET
ejpam-4340	105	95	open	open	ADJ
ejpam-4340	105	96	set	set	VERB
ejpam-4340	105	97	v	v	NOUN
ejpam-4340	105	98	in	in	ADP
ejpam-4340	105	99	y	y	NOUN
ejpam-4340	105	100	containing	contain	VERB
ejpam-4340	105	101	f(x	f(x	PROPN
ejpam-4340	105	102	)	)	PUNCT
ejpam-4340	105	103	,	,	PUNCT
ejpam-4340	105	104	there	there	PRON
ejpam-4340	105	105	exists	exist	VERB
ejpam-4340	105	106	an	an	DET
ejpam-4340	105	107	ωβ	ωβ	ADJ
ejpam-4340	105	108	-	-	PUNCT
ejpam-4340	105	109	open	open	ADJ
ejpam-4340	105	110	u	u	NOUN
ejpam-4340	105	111	in	in	ADP
ejpam-4340	105	112	x	x	PUNCT
ejpam-4340	105	113	containing	contain	VERB
ejpam-4340	105	114	x	x	PUNCT
ejpam-4340	105	115	such	such	ADJ
ejpam-4340	105	116	that	that	SCONJ
ejpam-4340	105	117	f	f	PROPN
ejpam-4340	106	1	[	[	X
ejpam-4340	106	2	u	u	X
ejpam-4340	106	3	]	]	PUNCT
ejpam-4340	106	4	⊆	⊆	NUM
ejpam-4340	106	5	v	v	NOUN
ejpam-4340	106	6	,	,	PUNCT
ejpam-4340	106	7	(	(	PUNCT
ejpam-4340	106	8	e	e	NOUN
ejpam-4340	106	9	)	)	PUNCT
ejpam-4340	106	10	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	106	11	[	[	X
ejpam-4340	106	12	19	19	NUM
ejpam-4340	106	13	]	]	PUNCT
ejpam-4340	106	14	at	at	ADP
ejpam-4340	106	15	a	a	DET
ejpam-4340	106	16	point	point	NOUN
ejpam-4340	106	17	x	x	SYM
ejpam-4340	106	18	∈	∈	NOUN
ejpam-4340	106	19	x	x	INTJ
ejpam-4340	106	20	if	if	SCONJ
ejpam-4340	106	21	for	for	ADP
ejpam-4340	106	22	every	every	DET
ejpam-4340	106	23	open	open	NOUN
ejpam-4340	106	24	set	set	VERB
ejpam-4340	106	25	v	v	NOUN
ejpam-4340	106	26	in	in	ADP
ejpam-4340	106	27	y	y	NOUN
ejpam-4340	106	28	containing	contain	VERB
ejpam-4340	106	29	f(x	f(x	PROPN
ejpam-4340	106	30	)	)	PUNCT
ejpam-4340	106	31	,	,	PUNCT
ejpam-4340	106	32	there	there	PRON
ejpam-4340	106	33	exists	exist	VERB
ejpam-4340	106	34	an	an	DET
ejpam-4340	106	35	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	106	36	set	set	VERB
ejpam-4340	106	37	u	u	NOUN
ejpam-4340	106	38	in	in	ADP
ejpam-4340	106	39	x	x	SYM
ejpam-4340	106	40	containing	contain	VERB
ejpam-4340	106	41	x	x	PUNCT
ejpam-4340	106	42	such	such	ADJ
ejpam-4340	106	43	that	that	SCONJ
ejpam-4340	106	44	f	f	PROPN
ejpam-4340	107	1	[	[	X
ejpam-4340	107	2	u	u	X
ejpam-4340	107	3	]	]	PUNCT
ejpam-4340	107	4	⊆	⊆	NUM
ejpam-4340	107	5	v	v	NOUN
ejpam-4340	107	6	.	.	PUNCT
ejpam-4340	108	1	definition	definition	NOUN
ejpam-4340	108	2	7	7	NUM
ejpam-4340	108	3	.	.	PUNCT
ejpam-4340	109	1	let	let	VERB
ejpam-4340	109	2	a	a	DET
ejpam-4340	109	3	be	be	AUX
ejpam-4340	109	4	a	a	DET
ejpam-4340	109	5	subset	subset	NOUN
ejpam-4340	109	6	of	of	ADP
ejpam-4340	109	7	a	a	DET
ejpam-4340	109	8	space	space	NOUN
ejpam-4340	109	9	x.	x.	NOUN
ejpam-4340	109	10	a	a	PRON
ejpam-4340	109	11	is	be	AUX
ejpam-4340	109	12	said	say	VERB
ejpam-4340	109	13	to	to	PART
ejpam-4340	109	14	be	be	AUX
ejpam-4340	109	15	generalized	generalize	VERB
ejpam-4340	109	16	closed	close	VERB
ejpam-4340	109	17	[	[	X
ejpam-4340	109	18	14](briefly	14](briefly	NUM
ejpam-4340	109	19	,	,	PUNCT
ejpam-4340	109	20	g	g	NOUN
ejpam-4340	109	21	-	-	PUNCT
ejpam-4340	109	22	closed	closed	ADJ
ejpam-4340	109	23	)	)	PUNCT
ejpam-4340	109	24	(	(	PUNCT
ejpam-4340	109	25	resp	resp	NOUN
ejpam-4340	109	26	.	.	PUNCT
ejpam-4340	110	1	generalized	generalize	VERB
ejpam-4340	110	2	ω	ω	NOUN
ejpam-4340	110	3	-	-	PUNCT
ejpam-4340	110	4	closed	closed	ADJ
ejpam-4340	110	5	[	[	X
ejpam-4340	110	6	5](briefly	5](briefly	NUM
ejpam-4340	110	7	,	,	PUNCT
ejpam-4340	110	8	gω	gω	PROPN
ejpam-4340	110	9	-	-	PUNCT
ejpam-4340	110	10	closed	closed	ADJ
ejpam-4340	110	11	)	)	PUNCT
ejpam-4340	111	1	,	,	PUNCT
ejpam-4340	111	2	generalized	generalize	VERB
ejpam-4340	111	3	β	β	X
ejpam-4340	111	4	-	-	ADJ
ejpam-4340	111	5	closed	closed	ADJ
ejpam-4340	111	6	[	[	X
ejpam-4340	111	7	21](briefly	21](briefly	NUM
ejpam-4340	111	8	,	,	PUNCT
ejpam-4340	111	9	gβ	gβ	NOUN
ejpam-4340	111	10	-	-	PUNCT
ejpam-4340	111	11	closed	closed	ADJ
ejpam-4340	111	12	)	)	PUNCT
ejpam-4340	111	13	,	,	PUNCT
ejpam-4340	111	14	generalized	generalize	VERB
ejpam-4340	111	15	e∗-closed	e∗-close	VERB
ejpam-4340	111	16	[	[	X
ejpam-4340	111	17	12](briefly	12](briefly	NUM
ejpam-4340	111	18	,	,	PUNCT
ejpam-4340	111	19	ge∗-closed	ge∗-close	VERB
ejpam-4340	111	20	)	)	PUNCT
ejpam-4340	111	21	,	,	PUNCT
ejpam-4340	111	22	generalized	generalized	ADJ
ejpam-4340	111	23	ωβ	ωβ	ADJ
ejpam-4340	111	24	-	-	ADJ
ejpam-4340	111	25	closed	closed	ADJ
ejpam-4340	111	26	[	[	X
ejpam-4340	111	27	4](briefly	4](briefly	ADV
ejpam-4340	111	28	,	,	PUNCT
ejpam-4340	111	29	gωβ	gωβ	NOUN
ejpam-4340	111	30	-	-	PUNCT
ejpam-4340	111	31	closed	closed	ADJ
ejpam-4340	111	32	)	)	PUNCT
ejpam-4340	111	33	)	)	PUNCT
ejpam-4340	111	34	if	if	SCONJ
ejpam-4340	111	35	cl(a	cl(a	NUM
ejpam-4340	111	36	)	)	PUNCT
ejpam-4340	111	37	⊆	⊆	NUM
ejpam-4340	111	38	u	u	NOUN
ejpam-4340	111	39	(	(	PUNCT
ejpam-4340	111	40	resp	resp	NOUN
ejpam-4340	111	41	.	.	PUNCT
ejpam-4340	111	42	ω	ω	PROPN
ejpam-4340	111	43	-	-	PUNCT
ejpam-4340	111	44	cl(a	cl(a	NUM
ejpam-4340	111	45	)	)	PUNCT
ejpam-4340	111	46	⊆	⊆	NUM
ejpam-4340	111	47	u	u	NOUN
ejpam-4340	111	48	,	,	PUNCT
ejpam-4340	111	49	β	β	NOUN
ejpam-4340	111	50	-	-	PUNCT
ejpam-4340	111	51	cl(a	cl(a	NUM
ejpam-4340	111	52	)	)	PUNCT
ejpam-4340	111	53	⊆	⊆	NUM
ejpam-4340	111	54	u	u	NOUN
ejpam-4340	111	55	,	,	PUNCT
ejpam-4340	111	56	e∗-cl(a	e∗-cl(a	PROPN
ejpam-4340	111	57	)	)	PUNCT
ejpam-4340	111	58	⊆	⊆	NUM
ejpam-4340	111	59	u	u	NOUN
ejpam-4340	111	60	,	,	PUNCT
ejpam-4340	111	61	ωβ	ωβ	NOUN
ejpam-4340	111	62	-	-	NUM
ejpam-4340	111	63	cl(a	cl(a	NUM
ejpam-4340	111	64	)	)	PUNCT
ejpam-4340	111	65	⊆	⊆	NUM
ejpam-4340	111	66	u	u	NOUN
ejpam-4340	111	67	)	)	PUNCT
ejpam-4340	111	68	whenever	whenever	SCONJ
ejpam-4340	111	69	u	u	PROPN
ejpam-4340	111	70	∈	∈	PROPN
ejpam-4340	111	71	o(x	o(x	PROPN
ejpam-4340	111	72	)	)	PUNCT
ejpam-4340	111	73	and	and	CCONJ
ejpam-4340	111	74	a	a	DET
ejpam-4340	111	75	⊆	⊆	NUM
ejpam-4340	111	76	u.	u.	NOUN
ejpam-4340	111	77	the	the	DET
ejpam-4340	111	78	complement	complement	NOUN
ejpam-4340	111	79	of	of	ADP
ejpam-4340	111	80	a	a	DET
ejpam-4340	111	81	g	g	NOUN
ejpam-4340	111	82	-	-	PUNCT
ejpam-4340	111	83	closed	closed	ADJ
ejpam-4340	111	84	(	(	PUNCT
ejpam-4340	111	85	resp	resp	NOUN
ejpam-4340	111	86	.	.	PUNCT
ejpam-4340	112	1	gω	gω	PROPN
ejpam-4340	112	2	-	-	PUNCT
ejpam-4340	112	3	closed	close	VERB
ejpam-4340	112	4	[	[	X
ejpam-4340	112	5	5	5	NUM
ejpam-4340	112	6	]	]	PUNCT
ejpam-4340	112	7	,	,	PUNCT
ejpam-4340	112	8	gβ	gβ	NOUN
ejpam-4340	112	9	-	-	PUNCT
ejpam-4340	112	10	closed	closed	ADJ
ejpam-4340	112	11	[	[	X
ejpam-4340	112	12	21	21	NUM
ejpam-4340	112	13	]	]	PUNCT
ejpam-4340	112	14	,	,	PUNCT
ejpam-4340	112	15	ge∗-closed	ge∗-close	VERB
ejpam-4340	112	16	[	[	X
ejpam-4340	112	17	12	12	NUM
ejpam-4340	112	18	]	]	PUNCT
ejpam-4340	112	19	,	,	PUNCT
ejpam-4340	112	20	gωβ	gωβ	NOUN
ejpam-4340	112	21	-	-	PUNCT
ejpam-4340	112	22	closed	closed	ADJ
ejpam-4340	112	23	[	[	X
ejpam-4340	112	24	4	4	NUM
ejpam-4340	112	25	]	]	PUNCT
ejpam-4340	112	26	)	)	PUNCT
ejpam-4340	112	27	set	set	NOUN
ejpam-4340	112	28	is	be	AUX
ejpam-4340	112	29	called	call	VERB
ejpam-4340	112	30	a	a	DET
ejpam-4340	112	31	generalized	generalized	ADJ
ejpam-4340	112	32	open	open	NOUN
ejpam-4340	112	33	(	(	PUNCT
ejpam-4340	112	34	briefly	briefly	ADV
ejpam-4340	112	35	,	,	PUNCT
ejpam-4340	112	36	gopen)(resp	gopen)(resp	PROPN
ejpam-4340	112	37	.	.	PUNCT
ejpam-4340	113	1	generalized	generalize	VERB
ejpam-4340	113	2	ω	ω	NOUN
ejpam-4340	113	3	-	-	ADJ
ejpam-4340	113	4	open	open	ADJ
ejpam-4340	113	5	[	[	X
ejpam-4340	113	6	5](briefly	5](briefly	NUM
ejpam-4340	113	7	,	,	PUNCT
ejpam-4340	113	8	gω	gω	NOUN
ejpam-4340	113	9	-	-	PUNCT
ejpam-4340	113	10	open	open	ADJ
ejpam-4340	113	11	)	)	PUNCT
ejpam-4340	113	12	,	,	PUNCT
ejpam-4340	113	13	generalized	generalize	VERB
ejpam-4340	113	14	β	β	X
ejpam-4340	113	15	-	-	ADJ
ejpam-4340	113	16	open	open	ADJ
ejpam-4340	113	17	[	[	X
ejpam-4340	113	18	21](briefly	21](briefly	NUM
ejpam-4340	113	19	,	,	PUNCT
ejpam-4340	113	20	gβopen	gβopen	NOUN
ejpam-4340	113	21	)	)	PUNCT
ejpam-4340	113	22	,	,	PUNCT
ejpam-4340	113	23	generalized	generalize	VERB
ejpam-4340	113	24	e∗-open	e∗-open	PROPN
ejpam-4340	113	25	[	[	X
ejpam-4340	113	26	12](briefly	12](briefly	NUM
ejpam-4340	113	27	,	,	PUNCT
ejpam-4340	113	28	ge∗-open	ge∗-open	NOUN
ejpam-4340	113	29	)	)	PUNCT
ejpam-4340	113	30	,	,	PUNCT
ejpam-4340	113	31	generalized	generalized	ADJ
ejpam-4340	113	32	ωβ	ωβ	ADJ
ejpam-4340	113	33	-	-	ADJ
ejpam-4340	113	34	open	open	ADJ
ejpam-4340	113	35	[	[	X
ejpam-4340	113	36	4](briefly	4](briefly	ADV
ejpam-4340	113	37	,	,	PUNCT
ejpam-4340	113	38	gωβopen	gωβopen	NOUN
ejpam-4340	113	39	)	)	PUNCT
ejpam-4340	113	40	)	)	PUNCT
ejpam-4340	113	41	.	.	PUNCT
ejpam-4340	114	1	the	the	DET
ejpam-4340	114	2	family	family	NOUN
ejpam-4340	114	3	of	of	ADP
ejpam-4340	114	4	all	all	DET
ejpam-4340	114	5	g	g	NOUN
ejpam-4340	114	6	-	-	PUNCT
ejpam-4340	114	7	closed	closed	ADJ
ejpam-4340	114	8	(	(	PUNCT
ejpam-4340	114	9	resp	resp	NOUN
ejpam-4340	114	10	.	.	PUNCT
ejpam-4340	115	1	gω	gω	PROPN
ejpam-4340	115	2	-	-	PUNCT
ejpam-4340	115	3	closed	close	VERB
ejpam-4340	115	4	[	[	X
ejpam-4340	115	5	5	5	NUM
ejpam-4340	115	6	]	]	PUNCT
ejpam-4340	115	7	,	,	PUNCT
ejpam-4340	115	8	gβ	gβ	NOUN
ejpam-4340	115	9	-	-	PUNCT
ejpam-4340	115	10	closed	closed	ADJ
ejpam-4340	115	11	[	[	X
ejpam-4340	115	12	21	21	NUM
ejpam-4340	115	13	]	]	PUNCT
ejpam-4340	115	14	,	,	PUNCT
ejpam-4340	115	15	ge∗-closed	ge∗-close	VERB
ejpam-4340	115	16	[	[	X
ejpam-4340	115	17	12	12	NUM
ejpam-4340	115	18	]	]	PUNCT
ejpam-4340	115	19	,	,	PUNCT
ejpam-4340	115	20	gωβ	gωβ	NOUN
ejpam-4340	115	21	-	-	PUNCT
ejpam-4340	115	22	closed	closed	ADJ
ejpam-4340	115	23	[	[	X
ejpam-4340	115	24	4	4	NUM
ejpam-4340	115	25	]	]	SYM
ejpam-4340	115	26	)	)	PUNCT
ejpam-4340	115	27	sets	set	NOUN
ejpam-4340	115	28	of	of	ADP
ejpam-4340	115	29	x	x	PUNCT
ejpam-4340	115	30	will	will	AUX
ejpam-4340	115	31	be	be	AUX
ejpam-4340	115	32	denoted	denote	VERB
ejpam-4340	115	33	by	by	ADP
ejpam-4340	115	34	gc(x	gc(x	NOUN
ejpam-4340	115	35	)	)	PUNCT
ejpam-4340	115	36	(	(	PUNCT
ejpam-4340	115	37	resp	resp	NOUN
ejpam-4340	115	38	.	.	PUNCT
ejpam-4340	116	1	gωc(x	gωc(x	NOUN
ejpam-4340	116	2	)	)	PUNCT
ejpam-4340	116	3	,	,	PUNCT
ejpam-4340	116	4	gβc(x	gβc(x	PROPN
ejpam-4340	116	5	)	)	PUNCT
ejpam-4340	116	6	,	,	PUNCT
ejpam-4340	116	7	ge∗c(x	ge∗c(x	NOUN
ejpam-4340	116	8	)	)	PUNCT
ejpam-4340	116	9	,	,	PUNCT
ejpam-4340	116	10	gωβc(x	gωβc(x	NOUN
ejpam-4340	116	11	)	)	PUNCT
ejpam-4340	116	12	)	)	PUNCT
ejpam-4340	116	13	.	.	PUNCT
ejpam-4340	117	1	the	the	DET
ejpam-4340	117	2	family	family	NOUN
ejpam-4340	117	3	of	of	ADP
ejpam-4340	117	4	all	all	DET
ejpam-4340	117	5	g	g	NOUN
ejpam-4340	117	6	-	-	PUNCT
ejpam-4340	117	7	open	open	ADJ
ejpam-4340	117	8	(	(	PUNCT
ejpam-4340	117	9	resp	resp	NOUN
ejpam-4340	117	10	.	.	PUNCT
ejpam-4340	118	1	gω	gω	PROPN
ejpam-4340	118	2	-	-	PUNCT
ejpam-4340	118	3	open	open	NOUN
ejpam-4340	119	1	[	[	X
ejpam-4340	119	2	5	5	NUM
ejpam-4340	119	3	]	]	PUNCT
ejpam-4340	119	4	,	,	PUNCT
ejpam-4340	119	5	gβ	gβ	NOUN
ejpam-4340	119	6	-	-	PUNCT
ejpam-4340	119	7	open	open	ADJ
ejpam-4340	120	1	[	[	X
ejpam-4340	120	2	21	21	NUM
ejpam-4340	120	3	]	]	PUNCT
ejpam-4340	120	4	,	,	PUNCT
ejpam-4340	120	5	ge∗-open[12	ge∗-open[12	PROPN
ejpam-4340	120	6	]	]	PUNCT
ejpam-4340	120	7	,	,	PUNCT
ejpam-4340	120	8	gωβ	gωβ	NOUN
ejpam-4340	120	9	-	-	ADJ
ejpam-4340	120	10	open	open	ADJ
ejpam-4340	120	11	[	[	X
ejpam-4340	120	12	4	4	NUM
ejpam-4340	120	13	]	]	SYM
ejpam-4340	120	14	)	)	PUNCT
ejpam-4340	120	15	sets	set	NOUN
ejpam-4340	120	16	of	of	ADP
ejpam-4340	120	17	x	x	PUNCT
ejpam-4340	120	18	will	will	AUX
ejpam-4340	120	19	be	be	AUX
ejpam-4340	120	20	denoted	denote	VERB
ejpam-4340	120	21	by	by	ADP
ejpam-4340	120	22	go(x	go(x	ADJ
ejpam-4340	120	23	)	)	PUNCT
ejpam-4340	120	24	(	(	PUNCT
ejpam-4340	120	25	resp	resp	NOUN
ejpam-4340	120	26	.	.	PUNCT
ejpam-4340	121	1	gωo(x	gωo(x	PROPN
ejpam-4340	121	2	)	)	PUNCT
ejpam-4340	121	3	,	,	PUNCT
ejpam-4340	121	4	gβo(x	gβo(x	PROPN
ejpam-4340	121	5	)	)	PUNCT
ejpam-4340	121	6	,	,	PUNCT
ejpam-4340	121	7	ge∗o(x	ge∗o(x	PROPN
ejpam-4340	121	8	)	)	PUNCT
ejpam-4340	121	9	,	,	PUNCT
ejpam-4340	121	10	gωβo(x	gωβo(x	NOUN
ejpam-4340	121	11	)	)	PUNCT
ejpam-4340	121	12	)	)	PUNCT
ejpam-4340	121	13	.	.	PUNCT
ejpam-4340	122	1	3	3	X
ejpam-4340	122	2	.	.	NUM
ejpam-4340	122	3	generalized	generalize	VERB
ejpam-4340	122	4	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	122	5	sets	set	NOUN
ejpam-4340	122	6	definition	definition	NOUN
ejpam-4340	122	7	8	8	NUM
ejpam-4340	122	8	.	.	PUNCT
ejpam-4340	123	1	a	a	DET
ejpam-4340	123	2	subset	subset	NOUN
ejpam-4340	123	3	a	a	PRON
ejpam-4340	123	4	of	of	ADP
ejpam-4340	123	5	a	a	DET
ejpam-4340	123	6	space	space	NOUN
ejpam-4340	123	7	x	x	PUNCT
ejpam-4340	123	8	is	be	AUX
ejpam-4340	123	9	called	call	VERB
ejpam-4340	123	10	generalized	generalized	ADJ
ejpam-4340	123	11	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	123	12	set	set	NOUN
ejpam-4340	123	13	(	(	PUNCT
ejpam-4340	123	14	briefly	briefly	ADV
ejpam-4340	123	15	,	,	PUNCT
ejpam-4340	123	16	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	123	17	set	set	NOUN
ejpam-4340	123	18	)	)	PUNCT
ejpam-4340	123	19	if	if	SCONJ
ejpam-4340	123	20	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	123	21	)	)	PUNCT
ejpam-4340	123	22	⊆	⊆	NUM
ejpam-4340	123	23	u	u	NOUN
ejpam-4340	123	24	whenever	whenever	SCONJ
ejpam-4340	123	25	u	u	PROPN
ejpam-4340	123	26	∈	∈	PROPN
ejpam-4340	123	27	o(x	o(x	PROPN
ejpam-4340	123	28	)	)	PUNCT
ejpam-4340	123	29	and	and	CCONJ
ejpam-4340	123	30	a	a	DET
ejpam-4340	123	31	⊆	⊆	NUM
ejpam-4340	123	32	u.	u.	NOUN
ejpam-4340	123	33	we	we	PRON
ejpam-4340	123	34	denote	denote	VERB
ejpam-4340	123	35	the	the	DET
ejpam-4340	123	36	family	family	NOUN
ejpam-4340	123	37	of	of	ADP
ejpam-4340	123	38	all	all	DET
ejpam-4340	123	39	generalized	generalize	VERB
ejpam-4340	123	40	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	123	41	subsets	subset	NOUN
ejpam-4340	123	42	of	of	ADP
ejpam-4340	123	43	a	a	DET
ejpam-4340	123	44	space	space	NOUN
ejpam-4340	123	45	x	x	PUNCT
ejpam-4340	123	46	by	by	ADP
ejpam-4340	123	47	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	123	48	)	)	PUNCT
ejpam-4340	123	49	.	.	PUNCT
ejpam-4340	124	1	proposition	proposition	NOUN
ejpam-4340	124	2	1	1	X
ejpam-4340	124	3	.	.	PUNCT
ejpam-4340	125	1	let	let	VERB
ejpam-4340	125	2	x	x	PRON
ejpam-4340	125	3	be	be	AUX
ejpam-4340	125	4	a	a	DET
ejpam-4340	125	5	topological	topological	ADJ
ejpam-4340	125	6	space	space	NOUN
ejpam-4340	125	7	.	.	PUNCT
ejpam-4340	126	1	then	then	ADV
ejpam-4340	126	2	the	the	DET
ejpam-4340	126	3	followings	following	NOUN
ejpam-4340	126	4	hold	hold	VERB
ejpam-4340	126	5	:	:	PUNCT
ejpam-4340	126	6	(	(	PUNCT
ejpam-4340	126	7	a	a	X
ejpam-4340	126	8	)	)	PUNCT
ejpam-4340	126	9	if	if	SCONJ
ejpam-4340	126	10	x	x	PRON
ejpam-4340	126	11	is	be	AUX
ejpam-4340	126	12	a	a	DET
ejpam-4340	126	13	countable	countable	ADJ
ejpam-4340	126	14	space	space	NOUN
ejpam-4340	126	15	,	,	PUNCT
ejpam-4340	126	16	then	then	ADV
ejpam-4340	126	17	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	126	18	)	)	PUNCT
ejpam-4340	127	1	=	=	SYM
ejpam-4340	127	2	2x	2x	NOUN
ejpam-4340	127	3	,	,	PUNCT
ejpam-4340	127	4	(	(	PUNCT
ejpam-4340	127	5	b	b	X
ejpam-4340	127	6	)	)	PUNCT
ejpam-4340	127	7	if	if	SCONJ
ejpam-4340	127	8	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	127	9	)	)	PUNCT
ejpam-4340	127	10	=	=	SYM
ejpam-4340	127	11	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	127	12	)	)	PUNCT
ejpam-4340	127	13	,	,	PUNCT
ejpam-4340	127	14	then	then	ADV
ejpam-4340	127	15	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	127	16	)	)	PUNCT
ejpam-4340	127	17	=	=	SYM
ejpam-4340	128	1	2x	2x	NOUN
ejpam-4340	128	2	.	.	PUNCT
ejpam-4340	129	1	proof	proof	NOUN
ejpam-4340	129	2	.	.	PUNCT
ejpam-4340	130	1	(	(	PUNCT
ejpam-4340	130	2	a	a	X
ejpam-4340	130	3	)	)	PUNCT
ejpam-4340	130	4	let	let	VERB
ejpam-4340	130	5	a	a	DET
ejpam-4340	130	6	∈	∈	NOUN
ejpam-4340	130	7	2x	2x	NOUN
ejpam-4340	130	8	and	and	CCONJ
ejpam-4340	130	9	a	a	DET
ejpam-4340	130	10	⊆	⊆	NUM
ejpam-4340	130	11	u	u	NOUN
ejpam-4340	130	12	∈	∈	PROPN
ejpam-4340	130	13	o(x	o(x	PROPN
ejpam-4340	130	14	)	)	PUNCT
ejpam-4340	130	15	.	.	PUNCT
ejpam-4340	131	1	|x|	|x|	PROPN
ejpam-4340	131	2	≤	≤	NUM
ejpam-4340	131	3	ℵ0	ℵ0	PROPN
ejpam-4340	131	4	⇒	⇒	ADJ
ejpam-4340	131	5	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	131	6	)	)	PUNCT
ejpam-4340	131	7	=	=	PUNCT
ejpam-4340	132	1	2x	2x	NOUN
ejpam-4340	132	2	a	a	DET
ejpam-4340	132	3	∈	∈	PROPN
ejpam-4340	132	4	2x	2x	NUM
ejpam-4340	132	5	}	}	PUNCT
ejpam-4340	132	6	⇒	⇒	VERB
ejpam-4340	132	7	a	a	DET
ejpam-4340	132	8	∈	∈	PROPN
ejpam-4340	132	9	ωe∗c(x)⇒	ωe∗c(x)⇒	X
ejpam-4340	132	10	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	132	11	)	)	PUNCT
ejpam-4340	132	12	=	=	PUNCT
ejpam-4340	132	13	a	a	DET
ejpam-4340	132	14	a	a	DET
ejpam-4340	132	15	⊆	⊆	NUM
ejpam-4340	132	16	u	u	NOUN
ejpam-4340	132	17	∈	∈	PROPN
ejpam-4340	132	18	o(x	o(x	PROPN
ejpam-4340	132	19	)	)	PUNCT
ejpam-4340	132	20	}	}	PUNCT
ejpam-4340	132	21	⇒	⇒	VERB
ejpam-4340	132	22	⇒	⇒	NOUN
ejpam-4340	132	23	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	132	24	)	)	PUNCT
ejpam-4340	132	25	⊆	⊆	NUM
ejpam-4340	132	26	u	u	NOUN
ejpam-4340	132	27	this	this	PRON
ejpam-4340	132	28	means	mean	VERB
ejpam-4340	132	29	that	that	SCONJ
ejpam-4340	132	30	a	a	DET
ejpam-4340	132	31	∈	∈	PROPN
ejpam-4340	132	32	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	132	33	)	)	PUNCT
ejpam-4340	132	34	.	.	PUNCT
ejpam-4340	133	1	then	then	ADV
ejpam-4340	133	2	we	we	PRON
ejpam-4340	133	3	have	have	VERB
ejpam-4340	133	4	2x	2x	NUM
ejpam-4340	133	5	⊆	⊆	NUM
ejpam-4340	133	6	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	133	7	)	)	PUNCT
ejpam-4340	133	8	.	.	PUNCT
ejpam-4340	134	1	on	on	ADP
ejpam-4340	134	2	the	the	DET
ejpam-4340	134	3	other	other	ADJ
ejpam-4340	134	4	hand	hand	NOUN
ejpam-4340	134	5	,	,	PUNCT
ejpam-4340	134	6	we	we	PRON
ejpam-4340	134	7	have	have	VERB
ejpam-4340	134	8	always	always	ADV
ejpam-4340	134	9	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	134	10	)	)	PUNCT
ejpam-4340	134	11	⊆	⊆	NUM
ejpam-4340	134	12	2x	2x	NUM
ejpam-4340	134	13	.	.	PUNCT
ejpam-4340	135	1	therefore	therefore	ADV
ejpam-4340	135	2	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	135	3	)	)	PUNCT
ejpam-4340	135	4	=	=	SYM
ejpam-4340	136	1	2x	2x	NOUN
ejpam-4340	136	2	.	.	PUNCT
ejpam-4340	137	1	p.	p.	NOUN
ejpam-4340	137	2	şaşmaz	şaşmaz	NUM
ejpam-4340	137	3	,	,	PUNCT
ejpam-4340	137	4	m.	m.	NOUN
ejpam-4340	137	5	özkoç	özkoç	PROPN
ejpam-4340	137	6	/	/	SYM
ejpam-4340	137	7	eur	eur	PROPN
ejpam-4340	137	8	.	.	PUNCT
ejpam-4340	138	1	j.	j.	PROPN
ejpam-4340	138	2	pure	pure	PROPN
ejpam-4340	138	3	appl	appl	PROPN
ejpam-4340	138	4	.	.	PROPN
ejpam-4340	138	5	math	math	PROPN
ejpam-4340	138	6	,	,	PUNCT
ejpam-4340	138	7	15	15	NUM
ejpam-4340	138	8	(	(	PUNCT
ejpam-4340	138	9	2	2	NUM
ejpam-4340	138	10	)	)	PUNCT
ejpam-4340	138	11	(	(	PUNCT
ejpam-4340	138	12	2022	2022	NUM
ejpam-4340	138	13	)	)	PUNCT
ejpam-4340	138	14	,	,	PUNCT
ejpam-4340	138	15	354	354	NUM
ejpam-4340	138	16	-	-	SYM
ejpam-4340	138	17	374	374	NUM
ejpam-4340	138	18	358	358	NUM
ejpam-4340	138	19	(	(	PUNCT
ejpam-4340	138	20	b	b	X
ejpam-4340	138	21	)	)	PUNCT
ejpam-4340	138	22	let	let	VERB
ejpam-4340	138	23	a	a	DET
ejpam-4340	138	24	∈	∈	NOUN
ejpam-4340	138	25	2x	2x	NOUN
ejpam-4340	138	26	and	and	CCONJ
ejpam-4340	138	27	a	a	DET
ejpam-4340	138	28	⊆	⊆	NUM
ejpam-4340	138	29	u	u	NOUN
ejpam-4340	138	30	∈	∈	PROPN
ejpam-4340	138	31	o(x	o(x	PROPN
ejpam-4340	138	32	)	)	PUNCT
ejpam-4340	138	33	.	.	PUNCT
ejpam-4340	139	1	a	a	DET
ejpam-4340	139	2	⊆	⊆	NUM
ejpam-4340	139	3	u	u	NOUN
ejpam-4340	139	4	∈	∈	PROPN
ejpam-4340	139	5	o(x	o(x	PROPN
ejpam-4340	139	6	)	)	PUNCT
ejpam-4340	139	7	o(x	o(x	PROPN
ejpam-4340	139	8	)	)	PUNCT
ejpam-4340	139	9	⊆	⊆	NUM
ejpam-4340	139	10	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	139	11	)	)	PUNCT
ejpam-4340	139	12	=	=	SYM
ejpam-4340	139	13	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	139	14	)	)	PUNCT
ejpam-4340	139	15	}	}	PUNCT
ejpam-4340	139	16	⇒	⇒	VERB
ejpam-4340	139	17	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	139	18	)	)	PUNCT
ejpam-4340	139	19	⊆	⊆	NUM
ejpam-4340	139	20	ωe∗-cl(u	ωe∗-cl(u	NUM
ejpam-4340	139	21	)	)	PUNCT
ejpam-4340	139	22	=	=	SYM
ejpam-4340	139	23	u	u	NOUN
ejpam-4340	139	24	this	this	PRON
ejpam-4340	139	25	is	be	AUX
ejpam-4340	139	26	means	mean	VERB
ejpam-4340	139	27	that	that	SCONJ
ejpam-4340	139	28	a	a	DET
ejpam-4340	139	29	∈	∈	PROPN
ejpam-4340	139	30	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	139	31	)	)	PUNCT
ejpam-4340	139	32	.	.	PUNCT
ejpam-4340	140	1	then	then	ADV
ejpam-4340	140	2	we	we	PRON
ejpam-4340	140	3	have	have	VERB
ejpam-4340	140	4	2x	2x	NUM
ejpam-4340	140	5	⊆	⊆	NUM
ejpam-4340	140	6	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	140	7	)	)	PUNCT
ejpam-4340	140	8	.	.	PUNCT
ejpam-4340	141	1	on	on	ADP
ejpam-4340	141	2	the	the	DET
ejpam-4340	141	3	other	other	ADJ
ejpam-4340	141	4	hand	hand	NOUN
ejpam-4340	141	5	,	,	PUNCT
ejpam-4340	141	6	we	we	PRON
ejpam-4340	141	7	have	have	VERB
ejpam-4340	141	8	always	always	ADV
ejpam-4340	141	9	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	141	10	)	)	PUNCT
ejpam-4340	141	11	⊆	⊆	NUM
ejpam-4340	141	12	2x	2x	NUM
ejpam-4340	141	13	.	.	PUNCT
ejpam-4340	142	1	therefore	therefore	ADV
ejpam-4340	142	2	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	142	3	)	)	PUNCT
ejpam-4340	142	4	=	=	SYM
ejpam-4340	143	1	2x	2x	X
ejpam-4340	143	2	.	.	PUNCT
ejpam-4340	144	1	remark	remark	NOUN
ejpam-4340	144	2	1	1	NUM
ejpam-4340	144	3	.	.	PUNCT
ejpam-4340	145	1	the	the	DET
ejpam-4340	145	2	following	follow	VERB
ejpam-4340	145	3	diagram	diagram	NOUN
ejpam-4340	145	4	follows	follow	VERB
ejpam-4340	145	5	immediately	immediately	ADV
ejpam-4340	145	6	from	from	ADP
ejpam-4340	145	7	the	the	DET
ejpam-4340	145	8	definitions	definition	NOUN
ejpam-4340	145	9	in	in	ADP
ejpam-4340	145	10	which	which	PRON
ejpam-4340	145	11	none	none	NOUN
ejpam-4340	145	12	of	of	ADP
ejpam-4340	145	13	the	the	DET
ejpam-4340	145	14	implications	implication	NOUN
ejpam-4340	145	15	is	be	AUX
ejpam-4340	145	16	reversible	reversible	ADJ
ejpam-4340	145	17	.	.	PUNCT
ejpam-4340	146	1	also	also	ADV
ejpam-4340	146	2	,	,	PUNCT
ejpam-4340	146	3	examples	example	NOUN
ejpam-4340	146	4	for	for	ADP
ejpam-4340	146	5	the	the	DET
ejpam-4340	146	6	other	other	ADJ
ejpam-4340	146	7	implications	implication	NOUN
ejpam-4340	146	8	are	be	AUX
ejpam-4340	146	9	shown	show	VERB
ejpam-4340	146	10	in	in	ADP
ejpam-4340	146	11	the	the	DET
ejpam-4340	146	12	related	related	ADJ
ejpam-4340	146	13	papers	paper	NOUN
ejpam-4340	146	14	.	.	PUNCT
ejpam-4340	147	1	ω	ω	X
ejpam-4340	147	2	-	-	PUNCT
ejpam-4340	147	3	closed	close	VERB
ejpam-4340	147	4	gω	gω	PROPN
ejpam-4340	147	5	-	-	PUNCT
ejpam-4340	147	6	closed	close	VERB
ejpam-4340	147	7	gωβ	gωβ	NOUN
ejpam-4340	147	8	-	-	PUNCT
ejpam-4340	147	9	closed	closed	ADJ
ejpam-4340	147	10	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	147	11	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	147	12	closed	close	VERB
ejpam-4340	147	13	g	g	NOUN
ejpam-4340	147	14	-	-	PUNCT
ejpam-4340	147	15	closed	close	VERB
ejpam-4340	147	16	gβ	gβ	NOUN
ejpam-4340	147	17	-	-	PUNCT
ejpam-4340	147	18	closed	closed	ADJ
ejpam-4340	147	19	ge∗-closed	ge∗-close	VERB
ejpam-4340	147	20	e∗-closed	e∗-close	VERB
ejpam-4340	147	21	figure	figure	NOUN
ejpam-4340	147	22	1	1	NUM
ejpam-4340	147	23	:	:	PUNCT
ejpam-4340	147	24	relationships	relationship	NOUN
ejpam-4340	147	25	between	between	ADP
ejpam-4340	147	26	some	some	DET
ejpam-4340	147	27	types	type	NOUN
ejpam-4340	147	28	of	of	ADP
ejpam-4340	147	29	closed	closed	ADJ
ejpam-4340	147	30	sets	set	NOUN
ejpam-4340	147	31	example	example	NOUN
ejpam-4340	147	32	1	1	X
ejpam-4340	147	33	.	.	PUNCT
ejpam-4340	148	1	let	let	VERB
ejpam-4340	148	2	x	x	PUNCT
ejpam-4340	148	3	=	=	PRON
ejpam-4340	148	4	{	{	PUNCT
ejpam-4340	148	5	a	a	PRON
ejpam-4340	148	6	,	,	PUNCT
ejpam-4340	148	7	b	b	NOUN
ejpam-4340	148	8	,	,	PUNCT
ejpam-4340	148	9	c	c	NOUN
ejpam-4340	148	10	}	}	PUNCT
ejpam-4340	148	11	with	with	ADP
ejpam-4340	148	12	the	the	DET
ejpam-4340	148	13	topology	topology	NOUN
ejpam-4340	148	14	τ	τ	X
ejpam-4340	148	15	=	=	PUNCT
ejpam-4340	148	16	{	{	PUNCT
ejpam-4340	148	17	∅	∅	NOUN
ejpam-4340	148	18	,	,	PUNCT
ejpam-4340	148	19	x	x	X
ejpam-4340	148	20	,	,	PUNCT
ejpam-4340	148	21	{	{	PUNCT
ejpam-4340	148	22	a	a	X
ejpam-4340	148	23	}	}	PUNCT
ejpam-4340	148	24	,	,	PUNCT
ejpam-4340	148	25	{	{	PUNCT
ejpam-4340	148	26	b	b	NOUN
ejpam-4340	148	27	}	}	PUNCT
ejpam-4340	148	28	,	,	PUNCT
ejpam-4340	148	29	{	{	PUNCT
ejpam-4340	148	30	a	a	PRON
ejpam-4340	148	31	,	,	PUNCT
ejpam-4340	148	32	b	b	NOUN
ejpam-4340	148	33	}	}	PUNCT
ejpam-4340	148	34	}	}	PUNCT
ejpam-4340	148	35	and	and	CCONJ
ejpam-4340	148	36	a	a	PRON
ejpam-4340	148	37	=	=	X
ejpam-4340	148	38	{	{	PUNCT
ejpam-4340	148	39	a	a	PROPN
ejpam-4340	148	40	,	,	PUNCT
ejpam-4340	148	41	b	b	NOUN
ejpam-4340	148	42	}	}	PUNCT
ejpam-4340	148	43	.	.	PUNCT
ejpam-4340	149	1	then	then	ADV
ejpam-4340	149	2	a	a	PRON
ejpam-4340	149	3	is	be	AUX
ejpam-4340	149	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	149	5	since	since	SCONJ
ejpam-4340	149	6	x	x	PRON
ejpam-4340	149	7	is	be	AUX
ejpam-4340	149	8	countable	countable	ADJ
ejpam-4340	149	9	.	.	PUNCT
ejpam-4340	150	1	but	but	CCONJ
ejpam-4340	150	2	a	a	PRON
ejpam-4340	150	3	is	be	AUX
ejpam-4340	150	4	not	not	PART
ejpam-4340	150	5	ge∗-closed	ge∗-close	VERB
ejpam-4340	150	6	since	since	SCONJ
ejpam-4340	150	7	a	a	DET
ejpam-4340	150	8	⊆	⊆	NUM
ejpam-4340	150	9	{	{	PUNCT
ejpam-4340	150	10	a	a	PRON
ejpam-4340	150	11	,	,	PUNCT
ejpam-4340	150	12	b	b	NOUN
ejpam-4340	150	13	}	}	PUNCT
ejpam-4340	150	14	∈	∈	PROPN
ejpam-4340	150	15	o(x	o(x	PROPN
ejpam-4340	150	16	)	)	PUNCT
ejpam-4340	150	17	but	but	CCONJ
ejpam-4340	150	18	e∗-cl(a	e∗-cl(a	ADJ
ejpam-4340	150	19	)	)	PUNCT
ejpam-4340	150	20	=	=	SYM
ejpam-4340	151	1	x	x	SYM
ejpam-4340	151	2	⊈	⊈	X
ejpam-4340	151	3	{	{	PUNCT
ejpam-4340	151	4	a	a	NOUN
ejpam-4340	151	5	,	,	PUNCT
ejpam-4340	151	6	b	b	NOUN
ejpam-4340	151	7	}	}	PUNCT
ejpam-4340	151	8	.	.	PUNCT
ejpam-4340	152	1	question	question	NOUN
ejpam-4340	152	2	:	:	PUNCT
ejpam-4340	152	3	is	be	AUX
ejpam-4340	152	4	there	there	PRON
ejpam-4340	152	5	an	an	DET
ejpam-4340	152	6	example	example	NOUN
ejpam-4340	152	7	of	of	ADP
ejpam-4340	152	8	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	152	9	set	set	NOUN
ejpam-4340	152	10	which	which	PRON
ejpam-4340	152	11	is	be	AUX
ejpam-4340	152	12	not	not	PART
ejpam-4340	152	13	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	152	14	?	?	PUNCT
ejpam-4340	153	1	theorem	theorem	NOUN
ejpam-4340	153	2	3	3	X
ejpam-4340	153	3	.	.	PUNCT
ejpam-4340	153	4	let	let	VERB
ejpam-4340	153	5	a	a	DET
ejpam-4340	153	6	be	be	AUX
ejpam-4340	153	7	a	a	DET
ejpam-4340	153	8	subset	subset	NOUN
ejpam-4340	153	9	of	of	ADP
ejpam-4340	153	10	a	a	DET
ejpam-4340	153	11	space	space	NOUN
ejpam-4340	153	12	x.	x.	NOUN
ejpam-4340	154	1	if	if	SCONJ
ejpam-4340	154	2	a	a	PRON
ejpam-4340	154	3	is	be	AUX
ejpam-4340	154	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	154	5	,	,	PUNCT
ejpam-4340	154	6	then	then	ADV
ejpam-4340	154	7	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	154	8	)	)	PUNCT
ejpam-4340	154	9	\a	\a	VERB
ejpam-4340	154	10	does	do	AUX
ejpam-4340	154	11	not	not	PART
ejpam-4340	154	12	contain	contain	VERB
ejpam-4340	154	13	any	any	DET
ejpam-4340	154	14	non	non	ADJ
ejpam-4340	154	15	-	-	ADJ
ejpam-4340	154	16	empty	empty	ADJ
ejpam-4340	154	17	closed	closed	ADJ
ejpam-4340	154	18	sets	set	NOUN
ejpam-4340	154	19	.	.	PUNCT
ejpam-4340	155	1	proof	proof	NOUN
ejpam-4340	155	2	.	.	PUNCT
ejpam-4340	156	1	suppose	suppose	VERB
ejpam-4340	156	2	that	that	SCONJ
ejpam-4340	156	3	f	f	PROPN
ejpam-4340	156	4	∈	∈	PROPN
ejpam-4340	156	5	c(x	c(x	NOUN
ejpam-4340	156	6	)	)	PUNCT
ejpam-4340	156	7	\	\	NOUN
ejpam-4340	156	8	{	{	PUNCT
ejpam-4340	156	9	∅	∅	NOUN
ejpam-4340	156	10	}	}	PUNCT
ejpam-4340	156	11	and	and	CCONJ
ejpam-4340	156	12	f	f	PROPN
ejpam-4340	156	13	⊆	⊆	NUM
ejpam-4340	156	14	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	156	15	)	)	PUNCT
ejpam-4340	156	16	\a	\a	ADJ
ejpam-4340	156	17	.	.	PUNCT
ejpam-4340	157	1	(	(	PUNCT
ejpam-4340	157	2	f	f	PROPN
ejpam-4340	157	3	∈	∈	PROPN
ejpam-4340	157	4	c(x	c(x	NOUN
ejpam-4340	157	5	)	)	PUNCT
ejpam-4340	157	6	\	\	NOUN
ejpam-4340	157	7	{	{	PUNCT
ejpam-4340	157	8	∅})(f	∅})(f	PROPN
ejpam-4340	157	9	⊆	⊆	NUM
ejpam-4340	157	10	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	157	11	)	)	PUNCT
ejpam-4340	157	12	\a)⇒	\a)⇒	NOUN
ejpam-4340	157	13	a	a	DET
ejpam-4340	157	14	⊆	⊆	NUM
ejpam-4340	157	15	x	x	SYM
ejpam-4340	157	16	\	\	PROPN
ejpam-4340	157	17	f	f	PROPN
ejpam-4340	157	18	∈	∈	PROPN
ejpam-4340	157	19	o(x	o(x	PROPN
ejpam-4340	157	20	)	)	PUNCT
ejpam-4340	157	21	a	a	DET
ejpam-4340	157	22	∈	∈	PROPN
ejpam-4340	157	23	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	157	24	)	)	PUNCT
ejpam-4340	157	25	}	}	PUNCT
ejpam-4340	157	26	⇒	⇒	VERB
ejpam-4340	157	27	⇒	⇒	NOUN
ejpam-4340	157	28	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	157	29	)	)	PUNCT
ejpam-4340	157	30	⊆	⊆	NUM
ejpam-4340	157	31	x	x	SYM
ejpam-4340	157	32	\	\	PROPN
ejpam-4340	157	33	f	f	PROPN
ejpam-4340	157	34	⇒	⇒	NOUN
ejpam-4340	157	35	f	f	PROPN
ejpam-4340	157	36	⊆	⊆	NUM
ejpam-4340	157	37	x	x	SYM
ejpam-4340	157	38	\	\	PROPN
ejpam-4340	157	39	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	157	40	)	)	PUNCT
ejpam-4340	157	41	f	f	PROPN
ejpam-4340	157	42	⊆	⊆	NUM
ejpam-4340	157	43	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	157	44	)	)	PUNCT
ejpam-4340	157	45	\a	\a	ADJ
ejpam-4340	157	46	}	}	PUNCT
ejpam-4340	157	47	⇒	⇒	NOUN
ejpam-4340	157	48	f	f	X
ejpam-4340	157	49	=	=	NOUN
ejpam-4340	157	50	∅	∅	NOUN
ejpam-4340	157	51	this	this	PRON
ejpam-4340	157	52	contradicts	contradict	VERB
ejpam-4340	157	53	with	with	ADP
ejpam-4340	157	54	f	f	PROPN
ejpam-4340	157	55	̸=	̸=	PROPN
ejpam-4340	157	56	∅.	∅.	ADV
ejpam-4340	157	57	theorem	theorem	VERB
ejpam-4340	157	58	4	4	NUM
ejpam-4340	157	59	.	.	PUNCT
ejpam-4340	157	60	let	let	VERB
ejpam-4340	157	61	a	a	DET
ejpam-4340	157	62	be	be	AUX
ejpam-4340	157	63	a	a	DET
ejpam-4340	157	64	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	157	65	subset	subset	NOUN
ejpam-4340	157	66	of	of	ADP
ejpam-4340	157	67	a	a	DET
ejpam-4340	157	68	space	space	NOUN
ejpam-4340	157	69	x.	x.	NOUN
ejpam-4340	157	70	then	then	ADV
ejpam-4340	157	71	a	a	PRON
ejpam-4340	157	72	is	be	AUX
ejpam-4340	157	73	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	157	74	if	if	SCONJ
ejpam-4340	157	75	and	and	CCONJ
ejpam-4340	157	76	only	only	ADV
ejpam-4340	157	77	if	if	SCONJ
ejpam-4340	157	78	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	157	79	)	)	PUNCT
ejpam-4340	157	80	\a	\a	VERB
ejpam-4340	157	81	is	be	AUX
ejpam-4340	157	82	closed	closed	ADJ
ejpam-4340	157	83	.	.	PUNCT
ejpam-4340	158	1	proof	proof	NOUN
ejpam-4340	158	2	.	.	PUNCT
ejpam-4340	159	1	(	(	PUNCT
ejpam-4340	159	2	⇒	⇒	PROPN
ejpam-4340	159	3	)	)	PUNCT
ejpam-4340	159	4	:	:	PUNCT
ejpam-4340	159	5	it	it	PRON
ejpam-4340	159	6	is	be	AUX
ejpam-4340	159	7	obvious	obvious	ADJ
ejpam-4340	159	8	.	.	PUNCT
ejpam-4340	160	1	(	(	PUNCT
ejpam-4340	160	2	⇐	⇐	NOUN
ejpam-4340	160	3	)	)	PUNCT
ejpam-4340	160	4	:	:	PUNCT
ejpam-4340	160	5	let	let	VERB
ejpam-4340	160	6	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	160	7	)	)	PUNCT
ejpam-4340	160	8	\a	\a	VERB
ejpam-4340	160	9	∈	∈	PROPN
ejpam-4340	160	10	c(x	c(x	NOUN
ejpam-4340	160	11	)	)	PUNCT
ejpam-4340	160	12	.	.	PUNCT
ejpam-4340	161	1	p.	p.	NOUN
ejpam-4340	161	2	şaşmaz	şaşmaz	NUM
ejpam-4340	161	3	,	,	PUNCT
ejpam-4340	161	4	m.	m.	NOUN
ejpam-4340	161	5	özkoç	özkoç	PROPN
ejpam-4340	161	6	/	/	SYM
ejpam-4340	161	7	eur	eur	PROPN
ejpam-4340	161	8	.	.	PUNCT
ejpam-4340	162	1	j.	j.	PROPN
ejpam-4340	162	2	pure	pure	PROPN
ejpam-4340	162	3	appl	appl	PROPN
ejpam-4340	162	4	.	.	PROPN
ejpam-4340	162	5	math	math	PROPN
ejpam-4340	162	6	,	,	PUNCT
ejpam-4340	162	7	15	15	NUM
ejpam-4340	162	8	(	(	PUNCT
ejpam-4340	162	9	2	2	NUM
ejpam-4340	162	10	)	)	PUNCT
ejpam-4340	162	11	(	(	PUNCT
ejpam-4340	162	12	2022	2022	NUM
ejpam-4340	162	13	)	)	PUNCT
ejpam-4340	162	14	,	,	PUNCT
ejpam-4340	162	15	354	354	NUM
ejpam-4340	162	16	-	-	SYM
ejpam-4340	162	17	374	374	NUM
ejpam-4340	162	18	359	359	NUM
ejpam-4340	162	19	a	a	DET
ejpam-4340	162	20	∈	∈	PROPN
ejpam-4340	162	21	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	162	22	)	)	PUNCT
ejpam-4340	162	23	theorem	theorem	VERB
ejpam-4340	162	24	3⇒	3⇒	NUM
ejpam-4340	162	25	(	(	PUNCT
ejpam-4340	162	26	∀f	∀f	PROPN
ejpam-4340	162	27	∈	∈	PROPN
ejpam-4340	162	28	c(x))[f	c(x))[f	PUNCT
ejpam-4340	162	29	̸=	̸=	PROPN
ejpam-4340	162	30	∅	∅	NOUN
ejpam-4340	162	31	⇒	⇒	NOUN
ejpam-4340	162	32	f	f	PROPN
ejpam-4340	162	33	⊈	⊈	PROPN
ejpam-4340	162	34	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	162	35	)	)	PUNCT
ejpam-4340	162	36	\a	\a	NUM
ejpam-4340	162	37	]	]	X
ejpam-4340	162	38	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	162	39	)	)	PUNCT
ejpam-4340	162	40	\a	\a	VERB
ejpam-4340	162	41	∈	∈	PROPN
ejpam-4340	162	42	c(x	c(x	NOUN
ejpam-4340	162	43	)	)	PUNCT
ejpam-4340	162	44	}	}	PUNCT
ejpam-4340	162	45	⇒	⇒	VERB
ejpam-4340	162	46	⇒	⇒	NOUN
ejpam-4340	162	47	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	162	48	)	)	PUNCT
ejpam-4340	162	49	\a	\a	VERB
ejpam-4340	163	1	=	=	PUNCT
ejpam-4340	163	2	∅	∅	NOUN
ejpam-4340	163	3	⇒	⇒	NOUN
ejpam-4340	163	4	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	163	5	)	)	PUNCT
ejpam-4340	163	6	⊆	⊆	SYM
ejpam-4340	163	7	a	a	DET
ejpam-4340	163	8	a	a	DET
ejpam-4340	163	9	⊆	⊆	NUM
ejpam-4340	163	10	x	x	SYM
ejpam-4340	163	11	⇒	⇒	VERB
ejpam-4340	163	12	a	a	DET
ejpam-4340	163	13	⊆	⊆	NUM
ejpam-4340	163	14	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	163	15	)	)	PUNCT
ejpam-4340	163	16	}	}	PUNCT
ejpam-4340	163	17	⇒	⇒	VERB
ejpam-4340	163	18	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	163	19	)	)	PUNCT
ejpam-4340	163	20	=	=	SYM
ejpam-4340	164	1	a⇒	a⇒	X
ejpam-4340	164	2	a	a	DET
ejpam-4340	164	3	∈	∈	PROPN
ejpam-4340	164	4	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	164	5	)	)	PUNCT
ejpam-4340	164	6	.	.	PUNCT
ejpam-4340	165	1	definition	definition	NOUN
ejpam-4340	165	2	9	9	NUM
ejpam-4340	165	3	.	.	PUNCT
ejpam-4340	166	1	a	a	DET
ejpam-4340	166	2	space	space	NOUN
ejpam-4340	166	3	x	x	PUNCT
ejpam-4340	166	4	is	be	AUX
ejpam-4340	166	5	called	call	VERB
ejpam-4340	166	6	an	an	DET
ejpam-4340	166	7	ωe∗-locally	ωe∗-locally	ADV
ejpam-4340	166	8	indiscrete	indiscrete	ADJ
ejpam-4340	166	9	space	space	NOUN
ejpam-4340	166	10	if	if	SCONJ
ejpam-4340	166	11	every	every	DET
ejpam-4340	166	12	open	open	ADJ
ejpam-4340	166	13	set	set	NOUN
ejpam-4340	166	14	is	be	AUX
ejpam-4340	166	15	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	166	16	set	set	NOUN
ejpam-4340	166	17	.	.	PUNCT
ejpam-4340	167	1	proposition	proposition	NOUN
ejpam-4340	167	2	2	2	NUM
ejpam-4340	167	3	.	.	PUNCT
ejpam-4340	168	1	let	let	VERB
ejpam-4340	168	2	x	x	PRON
ejpam-4340	168	3	be	be	AUX
ejpam-4340	168	4	a	a	DET
ejpam-4340	168	5	topological	topological	ADJ
ejpam-4340	168	6	space	space	NOUN
ejpam-4340	168	7	.	.	PUNCT
ejpam-4340	169	1	then	then	ADV
ejpam-4340	169	2	the	the	DET
ejpam-4340	169	3	following	following	NOUN
ejpam-4340	169	4	are	be	AUX
ejpam-4340	169	5	equivalent	equivalent	ADJ
ejpam-4340	169	6	.	.	PUNCT
ejpam-4340	170	1	(	(	PUNCT
ejpam-4340	170	2	a	a	X
ejpam-4340	170	3	)	)	PUNCT
ejpam-4340	170	4	x	x	X
ejpam-4340	170	5	is	be	AUX
ejpam-4340	170	6	ωe∗-locally	ωe∗-locally	ADV
ejpam-4340	170	7	indiscrete	indiscrete	ADJ
ejpam-4340	170	8	;	;	PUNCT
ejpam-4340	170	9	(	(	PUNCT
ejpam-4340	170	10	b	b	X
ejpam-4340	170	11	)	)	PUNCT
ejpam-4340	170	12	every	every	DET
ejpam-4340	170	13	subset	subset	NOUN
ejpam-4340	170	14	of	of	ADP
ejpam-4340	170	15	x	x	PRON
ejpam-4340	170	16	is	be	AUX
ejpam-4340	170	17	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	170	18	.	.	PUNCT
ejpam-4340	171	1	proof	proof	NOUN
ejpam-4340	171	2	.	.	PUNCT
ejpam-4340	172	1	(	(	PUNCT
ejpam-4340	172	2	a)⇒	a)⇒	PROPN
ejpam-4340	172	3	(	(	PUNCT
ejpam-4340	172	4	b	b	NOUN
ejpam-4340	172	5	)	)	PUNCT
ejpam-4340	172	6	:	:	PUNCT
ejpam-4340	172	7	let	let	VERB
ejpam-4340	172	8	a	a	DET
ejpam-4340	172	9	⊆	⊆	NUM
ejpam-4340	172	10	u	u	NOUN
ejpam-4340	172	11	∈	∈	PROPN
ejpam-4340	172	12	o(x	o(x	PROPN
ejpam-4340	172	13	)	)	PUNCT
ejpam-4340	172	14	.	.	PUNCT
ejpam-4340	173	1	a	a	DET
ejpam-4340	173	2	⊆	⊆	NUM
ejpam-4340	173	3	u	u	NOUN
ejpam-4340	173	4	∈	∈	PROPN
ejpam-4340	173	5	o(x	o(x	PROPN
ejpam-4340	173	6	)	)	PUNCT
ejpam-4340	173	7	hypothesis	hypothesis	NOUN
ejpam-4340	173	8	}	}	PUNCT
ejpam-4340	173	9	⇒	⇒	VERB
ejpam-4340	173	10	a	a	DET
ejpam-4340	173	11	⊆	⊆	NUM
ejpam-4340	173	12	u	u	NOUN
ejpam-4340	173	13	∈	∈	PROPN
ejpam-4340	173	14	ωe∗c(x)⇒	ωe∗c(x)⇒	X
ejpam-4340	173	15	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	173	16	)	)	PUNCT
ejpam-4340	173	17	⊆	⊆	NUM
ejpam-4340	173	18	ωe∗-cl(u	ωe∗-cl(u	NUM
ejpam-4340	173	19	)	)	PUNCT
ejpam-4340	173	20	=	=	PUNCT
ejpam-4340	174	1	u.	u.	NOUN
ejpam-4340	174	2	(	(	PUNCT
ejpam-4340	174	3	b)⇒	b)⇒	PROPN
ejpam-4340	174	4	(	(	PUNCT
ejpam-4340	174	5	a	a	NOUN
ejpam-4340	174	6	)	)	PUNCT
ejpam-4340	174	7	:	:	PUNCT
ejpam-4340	174	8	let	let	VERB
ejpam-4340	174	9	u	u	PRON
ejpam-4340	174	10	∈	∈	PROPN
ejpam-4340	174	11	o(x	o(x	PROPN
ejpam-4340	174	12	)	)	PUNCT
ejpam-4340	174	13	.	.	PUNCT
ejpam-4340	175	1	u	u	PROPN
ejpam-4340	175	2	∈	∈	PROPN
ejpam-4340	175	3	o(x	o(x	PROPN
ejpam-4340	175	4	)	)	PUNCT
ejpam-4340	175	5	hypothesis	hypothesis	NOUN
ejpam-4340	175	6	}	}	PUNCT
ejpam-4340	175	7	⇒	⇒	VERB
ejpam-4340	175	8	u	u	PROPN
ejpam-4340	175	9	∈	∈	PROPN
ejpam-4340	175	10	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	175	11	)	)	PUNCT
ejpam-4340	175	12	u	u	PROPN
ejpam-4340	175	13	∈	∈	PROPN
ejpam-4340	175	14	o(x	o(x	PROPN
ejpam-4340	175	15	)	)	PUNCT
ejpam-4340	175	16	}	}	PUNCT
ejpam-4340	175	17	⇒	⇒	VERB
ejpam-4340	175	18	ωe∗-cl(u	ωe∗-cl(u	NUM
ejpam-4340	175	19	)	)	PUNCT
ejpam-4340	175	20	⊆	⊆	NUM
ejpam-4340	175	21	u	u	NOUN
ejpam-4340	175	22	⇒	⇒	VERB
ejpam-4340	175	23	u	u	PROPN
ejpam-4340	175	24	∈	∈	PROPN
ejpam-4340	175	25	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	175	26	)	)	PUNCT
ejpam-4340	175	27	.	.	PUNCT
ejpam-4340	176	1	theorem	theorem	NOUN
ejpam-4340	176	2	5	5	NUM
ejpam-4340	176	3	.	.	PUNCT
ejpam-4340	177	1	let	let	VERB
ejpam-4340	177	2	a	a	DET
ejpam-4340	177	3	be	be	AUX
ejpam-4340	177	4	a	a	DET
ejpam-4340	177	5	subset	subset	NOUN
ejpam-4340	177	6	of	of	ADP
ejpam-4340	177	7	a	a	DET
ejpam-4340	177	8	space	space	NOUN
ejpam-4340	177	9	x.	x.	NOUN
ejpam-4340	177	10	if	if	SCONJ
ejpam-4340	177	11	a	a	PRON
ejpam-4340	177	12	is	be	AUX
ejpam-4340	177	13	both	both	PRON
ejpam-4340	177	14	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	177	15	and	and	CCONJ
ejpam-4340	177	16	open	open	ADJ
ejpam-4340	177	17	,	,	PUNCT
ejpam-4340	177	18	then	then	ADV
ejpam-4340	177	19	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	177	20	)	)	PUNCT
ejpam-4340	177	21	\a	\a	VERB
ejpam-4340	177	22	=	=	PUNCT
ejpam-4340	177	23	∅.	∅.	NOUN
ejpam-4340	177	24	proof	proof	NOUN
ejpam-4340	177	25	.	.	PUNCT
ejpam-4340	178	1	let	let	VERB
ejpam-4340	178	2	a	a	DET
ejpam-4340	178	3	∈	∈	PROPN
ejpam-4340	178	4	o(x	o(x	PROPN
ejpam-4340	178	5	)	)	PUNCT
ejpam-4340	178	6	∩	∩	ADJ
ejpam-4340	178	7	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	178	8	)	)	PUNCT
ejpam-4340	178	9	.	.	PUNCT
ejpam-4340	179	1	a	a	DET
ejpam-4340	179	2	∈	∈	PROPN
ejpam-4340	179	3	o(x	o(x	PROPN
ejpam-4340	179	4	)	)	PUNCT
ejpam-4340	179	5	∩	∩	NOUN
ejpam-4340	179	6	gωe∗c(x)⇒	gωe∗c(x)⇒	PROPN
ejpam-4340	179	7	(	(	PUNCT
ejpam-4340	179	8	a	a	DET
ejpam-4340	179	9	∈	∈	PROPN
ejpam-4340	179	10	o(x))(a	o(x))(a	PROPN
ejpam-4340	179	11	∈	∈	PROPN
ejpam-4340	179	12	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	179	13	)	)	PUNCT
ejpam-4340	179	14	)	)	PUNCT
ejpam-4340	179	15	⇒	⇒	NOUN
ejpam-4340	179	16	(	(	PUNCT
ejpam-4340	179	17	a	a	DET
ejpam-4340	179	18	∈	∈	PROPN
ejpam-4340	179	19	o(x))(∀u	o(x))(∀u	NOUN
ejpam-4340	179	20	∈	∈	NOUN
ejpam-4340	179	21	o(x))(a	o(x))(a	PROPN
ejpam-4340	179	22	⊆	⊆	NUM
ejpam-4340	179	23	u	u	NOUN
ejpam-4340	179	24	⇒	⇒	NOUN
ejpam-4340	179	25	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	179	26	)	)	PUNCT
ejpam-4340	179	27	⊆	⊆	NUM
ejpam-4340	179	28	u	u	NOUN
ejpam-4340	179	29	)	)	PUNCT
ejpam-4340	179	30	⇒	⇒	NOUN
ejpam-4340	179	31	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	179	32	)	)	PUNCT
ejpam-4340	179	33	⊆	⊆	NUM
ejpam-4340	179	34	a	a	DET
ejpam-4340	179	35	⇒	⇒	NOUN
ejpam-4340	179	36	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	179	37	)	)	PUNCT
ejpam-4340	179	38	\a	\a	VERB
ejpam-4340	179	39	=	=	PUNCT
ejpam-4340	179	40	∅.	∅.	NOUN
ejpam-4340	179	41	theorem	theorem	VERB
ejpam-4340	179	42	6	6	NUM
ejpam-4340	179	43	.	.	PUNCT
ejpam-4340	179	44	let	let	VERB
ejpam-4340	179	45	a	a	PRON
ejpam-4340	179	46	and	and	CCONJ
ejpam-4340	179	47	b	b	NOUN
ejpam-4340	179	48	be	be	AUX
ejpam-4340	179	49	subsets	subset	NOUN
ejpam-4340	179	50	of	of	ADP
ejpam-4340	179	51	a	a	DET
ejpam-4340	179	52	space	space	NOUN
ejpam-4340	179	53	x.	x.	NOUN
ejpam-4340	180	1	if	if	SCONJ
ejpam-4340	180	2	a	a	PRON
ejpam-4340	180	3	is	be	AUX
ejpam-4340	180	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	180	5	and	and	CCONJ
ejpam-4340	180	6	b	b	NOUN
ejpam-4340	180	7	is	be	AUX
ejpam-4340	180	8	any	any	DET
ejpam-4340	180	9	set	set	NOUN
ejpam-4340	180	10	such	such	ADJ
ejpam-4340	180	11	that	that	SCONJ
ejpam-4340	180	12	a	a	DET
ejpam-4340	180	13	⊆	⊆	NUM
ejpam-4340	180	14	b	b	SYM
ejpam-4340	180	15	⊆	⊆	NUM
ejpam-4340	180	16	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	180	17	)	)	PUNCT
ejpam-4340	180	18	,	,	PUNCT
ejpam-4340	180	19	then	then	ADV
ejpam-4340	180	20	b	b	PROPN
ejpam-4340	180	21	is	be	AUX
ejpam-4340	180	22	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	180	23	.	.	PUNCT
ejpam-4340	181	1	proof	proof	NOUN
ejpam-4340	181	2	.	.	PUNCT
ejpam-4340	182	1	let	let	VERB
ejpam-4340	182	2	b	b	NOUN
ejpam-4340	182	3	⊆	⊆	NUM
ejpam-4340	182	4	u	u	NOUN
ejpam-4340	182	5	∈	∈	PROPN
ejpam-4340	182	6	o(x	o(x	PROPN
ejpam-4340	182	7	)	)	PUNCT
ejpam-4340	182	8	.	.	PUNCT
ejpam-4340	183	1	b	b	X
ejpam-4340	183	2	⊆	⊆	NUM
ejpam-4340	183	3	u	u	NOUN
ejpam-4340	183	4	∈	∈	PROPN
ejpam-4340	183	5	o(x	o(x	PROPN
ejpam-4340	183	6	)	)	PUNCT
ejpam-4340	183	7	hypothesis	hypothesis	NOUN
ejpam-4340	183	8	}	}	PUNCT
ejpam-4340	183	9	⇒	⇒	NOUN
ejpam-4340	183	10	(	(	PUNCT
ejpam-4340	183	11	a	a	DET
ejpam-4340	183	12	⊆	⊆	NUM
ejpam-4340	183	13	b	b	NOUN
ejpam-4340	183	14	⊆	⊆	NUM
ejpam-4340	183	15	u)(a	u)(a	NUM
ejpam-4340	183	16	⊆	⊆	NUM
ejpam-4340	183	17	b	b	NOUN
ejpam-4340	183	18	⊆	⊆	NUM
ejpam-4340	183	19	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	183	20	)	)	PUNCT
ejpam-4340	183	21	⊆	⊆	NUM
ejpam-4340	183	22	u	u	NOUN
ejpam-4340	183	23	)	)	PUNCT
ejpam-4340	183	24	⇒	⇒	NOUN
ejpam-4340	183	25	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	183	26	)	)	PUNCT
ejpam-4340	183	27	⊆	⊆	NUM
ejpam-4340	183	28	ωe∗-cl(b	ωe∗-cl(b	NUM
ejpam-4340	183	29	)	)	PUNCT
ejpam-4340	183	30	⊆	⊆	NUM
ejpam-4340	183	31	ωe∗-cl(ωe∗-cl(a	ωe∗-cl(ωe∗-cl(a	NUM
ejpam-4340	183	32	)	)	PUNCT
ejpam-4340	183	33	)	)	PUNCT
ejpam-4340	184	1	=	=	PUNCT
ejpam-4340	184	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	184	3	)	)	PUNCT
ejpam-4340	184	4	⊆	⊆	NUM
ejpam-4340	184	5	u.	u.	NOUN
ejpam-4340	184	6	definition	definition	NOUN
ejpam-4340	184	7	10	10	NUM
ejpam-4340	184	8	.	.	PUNCT
ejpam-4340	185	1	let	let	VERB
ejpam-4340	185	2	a	a	DET
ejpam-4340	185	3	be	be	AUX
ejpam-4340	185	4	a	a	DET
ejpam-4340	185	5	subset	subset	NOUN
ejpam-4340	185	6	of	of	ADP
ejpam-4340	185	7	a	a	DET
ejpam-4340	185	8	space	space	NOUN
ejpam-4340	185	9	x.	x.	NOUN
ejpam-4340	185	10	a	a	DET
ejpam-4340	185	11	point	point	NOUN
ejpam-4340	185	12	x	x	X
ejpam-4340	185	13	∈	∈	NOUN
ejpam-4340	185	14	x	x	PUNCT
ejpam-4340	185	15	is	be	AUX
ejpam-4340	185	16	said	say	VERB
ejpam-4340	185	17	to	to	PART
ejpam-4340	185	18	be	be	AUX
ejpam-4340	185	19	an	an	DET
ejpam-4340	185	20	ωe∗-limit	ωe∗-limit	ADJ
ejpam-4340	185	21	point	point	NOUN
ejpam-4340	185	22	of	of	ADP
ejpam-4340	185	23	a	a	DET
ejpam-4340	185	24	if	if	NOUN
ejpam-4340	185	25	for	for	SCONJ
ejpam-4340	185	26	each	each	DET
ejpam-4340	185	27	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	185	28	set	set	VERB
ejpam-4340	185	29	u	u	NOUN
ejpam-4340	185	30	containing	contain	VERB
ejpam-4340	185	31	x	x	PRON
ejpam-4340	185	32	,	,	PUNCT
ejpam-4340	185	33	we	we	PRON
ejpam-4340	185	34	have	have	VERB
ejpam-4340	185	35	u	u	NOUN
ejpam-4340	185	36	∩	∩	NOUN
ejpam-4340	185	37	(	(	PUNCT
ejpam-4340	185	38	a	a	DET
ejpam-4340	185	39	\	\	PROPN
ejpam-4340	185	40	{	{	PUNCT
ejpam-4340	185	41	x	x	NOUN
ejpam-4340	185	42	}	}	PUNCT
ejpam-4340	185	43	)	)	PUNCT
ejpam-4340	185	44	̸=	̸=	PROPN
ejpam-4340	185	45	∅.	∅.	ADP
ejpam-4340	185	46	the	the	DET
ejpam-4340	185	47	set	set	NOUN
ejpam-4340	185	48	of	of	ADP
ejpam-4340	185	49	all	all	DET
ejpam-4340	185	50	ωe∗-limit	ωe∗-limit	ADJ
ejpam-4340	185	51	points	point	NOUN
ejpam-4340	185	52	of	of	ADP
ejpam-4340	185	53	a	a	PRON
ejpam-4340	185	54	is	be	AUX
ejpam-4340	185	55	called	call	VERB
ejpam-4340	185	56	the	the	DET
ejpam-4340	185	57	ωe∗-derived	ωe∗-derive	VERB
ejpam-4340	185	58	set	set	NOUN
ejpam-4340	185	59	of	of	ADP
ejpam-4340	185	60	a	a	PRON
ejpam-4340	185	61	and	and	CCONJ
ejpam-4340	185	62	is	be	AUX
ejpam-4340	185	63	denoted	denote	VERB
ejpam-4340	185	64	by	by	ADP
ejpam-4340	185	65	dωe∗(a	dωe∗(a	NOUN
ejpam-4340	185	66	)	)	PUNCT
ejpam-4340	185	67	.	.	PUNCT
ejpam-4340	186	1	lemma	lemma	PROPN
ejpam-4340	186	2	3	3	X
ejpam-4340	186	3	.	.	PUNCT
ejpam-4340	187	1	let	let	VERB
ejpam-4340	187	2	a	a	DET
ejpam-4340	187	3	be	be	AUX
ejpam-4340	187	4	a	a	DET
ejpam-4340	187	5	subset	subset	NOUN
ejpam-4340	187	6	of	of	ADP
ejpam-4340	187	7	a	a	DET
ejpam-4340	187	8	space	space	NOUN
ejpam-4340	187	9	x.	x.	NOUN
ejpam-4340	188	1	if	if	SCONJ
ejpam-4340	188	2	d(a	d(a	PROPN
ejpam-4340	188	3	)	)	PUNCT
ejpam-4340	188	4	=	=	SYM
ejpam-4340	188	5	dωe∗(a	dωe∗(a	PROPN
ejpam-4340	188	6	)	)	PUNCT
ejpam-4340	188	7	,	,	PUNCT
ejpam-4340	188	8	then	then	ADV
ejpam-4340	188	9	cl(a	cl(a	PUNCT
ejpam-4340	188	10	)	)	PUNCT
ejpam-4340	188	11	=	=	SYM
ejpam-4340	188	12	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	188	13	)	)	PUNCT
ejpam-4340	189	1	.	.	PUNCT
ejpam-4340	190	1	proof	proof	NOUN
ejpam-4340	190	2	.	.	PUNCT
ejpam-4340	191	1	it	it	PRON
ejpam-4340	191	2	is	be	AUX
ejpam-4340	191	3	clear	clear	ADJ
ejpam-4340	191	4	.	.	PUNCT
ejpam-4340	192	1	p.	p.	NOUN
ejpam-4340	192	2	şaşmaz	şaşmaz	NUM
ejpam-4340	192	3	,	,	PUNCT
ejpam-4340	192	4	m.	m.	NOUN
ejpam-4340	192	5	özkoç	özkoç	PROPN
ejpam-4340	192	6	/	/	SYM
ejpam-4340	192	7	eur	eur	PROPN
ejpam-4340	192	8	.	.	PUNCT
ejpam-4340	193	1	j.	j.	PROPN
ejpam-4340	193	2	pure	pure	PROPN
ejpam-4340	193	3	appl	appl	PROPN
ejpam-4340	193	4	.	.	PROPN
ejpam-4340	193	5	math	math	PROPN
ejpam-4340	193	6	,	,	PUNCT
ejpam-4340	193	7	15	15	NUM
ejpam-4340	193	8	(	(	PUNCT
ejpam-4340	193	9	2	2	NUM
ejpam-4340	193	10	)	)	PUNCT
ejpam-4340	193	11	(	(	PUNCT
ejpam-4340	193	12	2022	2022	NUM
ejpam-4340	193	13	)	)	PUNCT
ejpam-4340	193	14	,	,	PUNCT
ejpam-4340	193	15	354	354	NUM
ejpam-4340	193	16	-	-	SYM
ejpam-4340	193	17	374	374	NUM
ejpam-4340	193	18	360	360	NUM
ejpam-4340	193	19	proposition	proposition	NOUN
ejpam-4340	193	20	3	3	X
ejpam-4340	193	21	.	.	PUNCT
ejpam-4340	194	1	let	let	VERB
ejpam-4340	194	2	a	a	PRON
ejpam-4340	194	3	and	and	CCONJ
ejpam-4340	194	4	b	b	NOUN
ejpam-4340	194	5	be	be	AUX
ejpam-4340	194	6	two	two	NUM
ejpam-4340	194	7	subsets	subset	NOUN
ejpam-4340	194	8	of	of	ADP
ejpam-4340	194	9	a	a	DET
ejpam-4340	194	10	space	space	NOUN
ejpam-4340	194	11	x.	x.	NOUN
ejpam-4340	194	12	then	then	ADV
ejpam-4340	194	13	the	the	DET
ejpam-4340	194	14	following	follow	VERB
ejpam-4340	194	15	properties	property	NOUN
ejpam-4340	194	16	hold	hold	VERB
ejpam-4340	194	17	:	:	PUNCT
ejpam-4340	194	18	(	(	PUNCT
ejpam-4340	194	19	a	a	X
ejpam-4340	194	20	)	)	PUNCT
ejpam-4340	194	21	dωe∗(a	dωe∗(a	NOUN
ejpam-4340	194	22	)	)	PUNCT
ejpam-4340	195	1	⊆	⊆	NUM
ejpam-4340	195	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	195	3	)	)	PUNCT
ejpam-4340	195	4	,	,	PUNCT
ejpam-4340	195	5	(	(	PUNCT
ejpam-4340	195	6	b	b	X
ejpam-4340	195	7	)	)	PUNCT
ejpam-4340	195	8	a	a	DET
ejpam-4340	195	9	⊆	⊆	NUM
ejpam-4340	195	10	b	b	NOUN
ejpam-4340	195	11	⇒	⇒	NOUN
ejpam-4340	195	12	dωe∗(a	dωe∗(a	NUM
ejpam-4340	195	13	)	)	PUNCT
ejpam-4340	195	14	⊆	⊆	NUM
ejpam-4340	195	15	dωe∗(b	dωe∗(b	NUM
ejpam-4340	195	16	)	)	PUNCT
ejpam-4340	195	17	,	,	PUNCT
ejpam-4340	195	18	(	(	PUNCT
ejpam-4340	195	19	c	c	X
ejpam-4340	195	20	)	)	PUNCT
ejpam-4340	195	21	dωe∗(a	dωe∗(a	NOUN
ejpam-4340	195	22	)	)	PUNCT
ejpam-4340	195	23	∪dωe∗(b	∪dωe∗(b	PROPN
ejpam-4340	195	24	)	)	PUNCT
ejpam-4340	195	25	⊆	⊆	NUM
ejpam-4340	195	26	dωe∗-cl(a	dωe∗-cl(a	ADJ
ejpam-4340	195	27	∪b	∪b	NOUN
ejpam-4340	195	28	)	)	PUNCT
ejpam-4340	195	29	,	,	PUNCT
ejpam-4340	195	30	(	(	PUNCT
ejpam-4340	195	31	d	d	X
ejpam-4340	195	32	)	)	PUNCT
ejpam-4340	195	33	dωe∗-cl(a	dωe∗-cl(a	ADJ
ejpam-4340	195	34	∩b	∩b	NOUN
ejpam-4340	195	35	)	)	PUNCT
ejpam-4340	195	36	⊆	⊆	NUM
ejpam-4340	195	37	dωe∗-cl(a	dωe∗-cl(a	ADJ
ejpam-4340	195	38	)	)	PUNCT
ejpam-4340	195	39	∩dωe∗-cl(b	∩dωe∗-cl(b	NOUN
ejpam-4340	195	40	)	)	PUNCT
ejpam-4340	195	41	,	,	PUNCT
ejpam-4340	195	42	(	(	PUNCT
ejpam-4340	195	43	e	e	NOUN
ejpam-4340	195	44	)	)	PUNCT
ejpam-4340	195	45	dωe∗(a	dωe∗(a	NOUN
ejpam-4340	195	46	)	)	PUNCT
ejpam-4340	195	47	⊆	⊆	NUM
ejpam-4340	195	48	d(a	d(a	PROPN
ejpam-4340	195	49	)	)	PUNCT
ejpam-4340	195	50	,	,	PUNCT
ejpam-4340	195	51	(	(	PUNCT
ejpam-4340	195	52	f	f	X
ejpam-4340	195	53	)	)	PUNCT
ejpam-4340	195	54	a	a	DET
ejpam-4340	195	55	∈	∈	PROPN
ejpam-4340	195	56	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	195	57	)	)	PUNCT
ejpam-4340	195	58	if	if	SCONJ
ejpam-4340	195	59	and	and	CCONJ
ejpam-4340	195	60	only	only	ADV
ejpam-4340	195	61	if	if	SCONJ
ejpam-4340	195	62	dωe∗(a	dωe∗(a	NOUN
ejpam-4340	195	63	)	)	PUNCT
ejpam-4340	195	64	⊆	⊆	NUM
ejpam-4340	195	65	a	a	PRON
ejpam-4340	195	66	,	,	PUNCT
ejpam-4340	195	67	(	(	PUNCT
ejpam-4340	195	68	g	g	NOUN
ejpam-4340	195	69	)	)	PUNCT
ejpam-4340	195	70	a	a	DET
ejpam-4340	195	71	∪dωe∗(a	∪dωe∗(a	NOUN
ejpam-4340	195	72	)	)	PUNCT
ejpam-4340	195	73	∈	∈	PROPN
ejpam-4340	195	74	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	195	75	)	)	PUNCT
ejpam-4340	195	76	,	,	PUNCT
ejpam-4340	195	77	(	(	PUNCT
ejpam-4340	195	78	h	h	NOUN
ejpam-4340	195	79	)	)	PUNCT
ejpam-4340	195	80	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	195	81	)	)	PUNCT
ejpam-4340	195	82	=	=	PUNCT
ejpam-4340	195	83	a	a	DET
ejpam-4340	195	84	∪dωe∗(a	∪dωe∗(a	NOUN
ejpam-4340	195	85	)	)	PUNCT
ejpam-4340	195	86	.	.	PUNCT
ejpam-4340	196	1	proof	proof	NOUN
ejpam-4340	196	2	.	.	PUNCT
ejpam-4340	197	1	the	the	DET
ejpam-4340	197	2	proofs	proof	NOUN
ejpam-4340	197	3	of	of	ADP
ejpam-4340	197	4	above	above	ADJ
ejpam-4340	197	5	results	result	NOUN
ejpam-4340	197	6	are	be	AUX
ejpam-4340	197	7	standard	standard	ADJ
ejpam-4340	197	8	.	.	PUNCT
ejpam-4340	198	1	hence	hence	ADV
ejpam-4340	198	2	,	,	PUNCT
ejpam-4340	198	3	they	they	PRON
ejpam-4340	198	4	are	be	AUX
ejpam-4340	198	5	omitted	omit	VERB
ejpam-4340	198	6	.	.	PUNCT
ejpam-4340	199	1	corollary	corollary	ADJ
ejpam-4340	199	2	1	1	NUM
ejpam-4340	199	3	.	.	PUNCT
ejpam-4340	200	1	let	let	VERB
ejpam-4340	200	2	a	a	DET
ejpam-4340	200	3	be	be	AUX
ejpam-4340	200	4	a	a	DET
ejpam-4340	200	5	subset	subset	NOUN
ejpam-4340	200	6	of	of	ADP
ejpam-4340	200	7	a	a	DET
ejpam-4340	200	8	space	space	NOUN
ejpam-4340	200	9	x.	x.	NOUN
ejpam-4340	200	10	if	if	SCONJ
ejpam-4340	200	11	d(a	d(a	PROPN
ejpam-4340	200	12	)	)	PUNCT
ejpam-4340	200	13	⊆	⊆	NUM
ejpam-4340	200	14	dωe∗(a	dωe∗(a	NUM
ejpam-4340	200	15	)	)	PUNCT
ejpam-4340	200	16	,	,	PUNCT
ejpam-4340	200	17	then	then	ADV
ejpam-4340	200	18	for	for	ADP
ejpam-4340	200	19	any	any	DET
ejpam-4340	200	20	subsets	subset	NOUN
ejpam-4340	200	21	f	f	PROPN
ejpam-4340	200	22	and	and	CCONJ
ejpam-4340	200	23	b	b	PROPN
ejpam-4340	200	24	of	of	ADP
ejpam-4340	200	25	x	x	PRON
ejpam-4340	200	26	,	,	PUNCT
ejpam-4340	200	27	we	we	PRON
ejpam-4340	200	28	have	have	VERB
ejpam-4340	200	29	ωe∗-cl(f	ωe∗-cl(f	NOUN
ejpam-4340	200	30	∪b	∪b	NOUN
ejpam-4340	200	31	)	)	PUNCT
ejpam-4340	200	32	=	=	SYM
ejpam-4340	200	33	ωe∗-cl(f	ωe∗-cl(f	NOUN
ejpam-4340	200	34	)	)	PUNCT
ejpam-4340	200	35	∪	∪	ADP
ejpam-4340	200	36	ωe∗-cl(b	ωe∗-cl(b	NOUN
ejpam-4340	200	37	)	)	PUNCT
ejpam-4340	200	38	.	.	PUNCT
ejpam-4340	201	1	proof	proof	NOUN
ejpam-4340	201	2	.	.	PUNCT
ejpam-4340	202	1	it	it	PRON
ejpam-4340	202	2	is	be	AUX
ejpam-4340	202	3	obvious	obvious	ADJ
ejpam-4340	202	4	.	.	PUNCT
ejpam-4340	203	1	proposition	proposition	NOUN
ejpam-4340	203	2	4	4	NUM
ejpam-4340	203	3	.	.	PUNCT
ejpam-4340	204	1	let	let	VERB
ejpam-4340	204	2	a	a	PRON
ejpam-4340	204	3	and	and	CCONJ
ejpam-4340	204	4	b	b	NOUN
ejpam-4340	204	5	be	be	AUX
ejpam-4340	204	6	subsets	subset	NOUN
ejpam-4340	204	7	of	of	ADP
ejpam-4340	204	8	a	a	DET
ejpam-4340	204	9	space	space	NOUN
ejpam-4340	204	10	x.	x.	NOUN
ejpam-4340	204	11	if	if	SCONJ
ejpam-4340	204	12	a	a	PRON
ejpam-4340	204	13	and	and	CCONJ
ejpam-4340	204	14	b	b	NOUN
ejpam-4340	204	15	are	be	AUX
ejpam-4340	204	16	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	204	17	sets	set	NOUN
ejpam-4340	204	18	such	such	ADJ
ejpam-4340	204	19	that	that	SCONJ
ejpam-4340	204	20	d(a	d(a	PROPN
ejpam-4340	204	21	)	)	PUNCT
ejpam-4340	204	22	⊆	⊆	NUM
ejpam-4340	204	23	dωe∗(a	dωe∗(a	NUM
ejpam-4340	204	24	)	)	PUNCT
ejpam-4340	204	25	and	and	CCONJ
ejpam-4340	204	26	d(b	d(b	NOUN
ejpam-4340	204	27	)	)	PUNCT
ejpam-4340	204	28	⊆	⊆	NUM
ejpam-4340	204	29	dωe∗(b	dωe∗(b	NUM
ejpam-4340	204	30	)	)	PUNCT
ejpam-4340	204	31	,	,	PUNCT
ejpam-4340	204	32	then	then	ADV
ejpam-4340	204	33	a	a	DET
ejpam-4340	204	34	∪b	∪b	VERB
ejpam-4340	204	35	is	be	AUX
ejpam-4340	204	36	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	204	37	.	.	PUNCT
ejpam-4340	205	1	proof	proof	NOUN
ejpam-4340	205	2	.	.	PUNCT
ejpam-4340	206	1	let	let	VERB
ejpam-4340	206	2	a	a	DET
ejpam-4340	206	3	∪b	∪b	NUM
ejpam-4340	206	4	⊆	⊆	NUM
ejpam-4340	206	5	u	u	NOUN
ejpam-4340	206	6	∈	∈	PROPN
ejpam-4340	206	7	o(x	o(x	PROPN
ejpam-4340	206	8	)	)	PUNCT
ejpam-4340	206	9	.	.	PUNCT
ejpam-4340	207	1	a	a	DET
ejpam-4340	207	2	∪b	∪b	NUM
ejpam-4340	207	3	⊆	⊆	NUM
ejpam-4340	207	4	u	u	NOUN
ejpam-4340	207	5	∈	∈	PROPN
ejpam-4340	207	6	o(x)⇒	o(x)⇒	PROPN
ejpam-4340	207	7	(	(	PUNCT
ejpam-4340	207	8	a	a	DET
ejpam-4340	207	9	⊆	⊆	NUM
ejpam-4340	207	10	u	u	NOUN
ejpam-4340	207	11	∈	∈	NOUN
ejpam-4340	207	12	o(x))(b	o(x))(b	ADJ
ejpam-4340	207	13	⊆	⊆	NUM
ejpam-4340	207	14	u	u	NOUN
ejpam-4340	207	15	∈	∈	PROPN
ejpam-4340	207	16	o(x	o(x	PROPN
ejpam-4340	207	17	)	)	PUNCT
ejpam-4340	207	18	)	)	PUNCT
ejpam-4340	208	1	a	a	DET
ejpam-4340	208	2	,	,	PUNCT
ejpam-4340	208	3	b	b	PROPN
ejpam-4340	208	4	∈	∈	PROPN
ejpam-4340	208	5	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	208	6	)	)	PUNCT
ejpam-4340	208	7	}	}	PUNCT
ejpam-4340	208	8	⇒	⇒	VERB
ejpam-4340	208	9	⇒	⇒	NOUN
ejpam-4340	208	10	(	(	PUNCT
ejpam-4340	208	11	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	208	12	)	)	PUNCT
ejpam-4340	208	13	⊆	⊆	NUM
ejpam-4340	208	14	u)(ωe∗-cl(b	u)(ωe∗-cl(b	NOUN
ejpam-4340	208	15	)	)	PUNCT
ejpam-4340	208	16	⊆	⊆	NUM
ejpam-4340	208	17	u	u	NOUN
ejpam-4340	208	18	)	)	PUNCT
ejpam-4340	208	19	(	(	PUNCT
ejpam-4340	208	20	d(a	d(a	PROPN
ejpam-4340	208	21	)	)	PUNCT
ejpam-4340	208	22	⊆	⊆	NUM
ejpam-4340	208	23	dωe∗(a))(d(b	dωe∗(a))(d(b	NOUN
ejpam-4340	208	24	)	)	PUNCT
ejpam-4340	208	25	⊆	⊆	NUM
ejpam-4340	208	26	dωe∗(b	dωe∗(b	NUM
ejpam-4340	208	27	)	)	PUNCT
ejpam-4340	208	28	)	)	PUNCT
ejpam-4340	208	29	}	}	PUNCT
ejpam-4340	208	30	⇒	⇒	VERB
ejpam-4340	208	31	⇒	⇒	NOUN
ejpam-4340	208	32	cl(a	cl(a	NUM
ejpam-4340	208	33	∪b	∪b	NOUN
ejpam-4340	208	34	)	)	PUNCT
ejpam-4340	208	35	=	=	SYM
ejpam-4340	208	36	cl(a	cl(a	X
ejpam-4340	208	37	)	)	PUNCT
ejpam-4340	208	38	∪	∪	ADJ
ejpam-4340	208	39	cl(b	cl(b	NOUN
ejpam-4340	208	40	)	)	PUNCT
ejpam-4340	208	41	=	=	SYM
ejpam-4340	208	42	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	208	43	)	)	PUNCT
ejpam-4340	208	44	∪	∪	ADP
ejpam-4340	208	45	ωe∗-cl(b	ωe∗-cl(b	NOUN
ejpam-4340	208	46	)	)	PUNCT
ejpam-4340	208	47	=	=	PUNCT
ejpam-4340	208	48	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	208	49	∪b	∪b	NOUN
ejpam-4340	208	50	)	)	PUNCT
ejpam-4340	208	51	⊆	⊆	NUM
ejpam-4340	208	52	u.	u.	NOUN
ejpam-4340	208	53	proposition	proposition	NOUN
ejpam-4340	208	54	5	5	NUM
ejpam-4340	208	55	.	.	PUNCT
ejpam-4340	208	56	let	let	VERB
ejpam-4340	208	57	a	a	PRON
ejpam-4340	208	58	and	and	CCONJ
ejpam-4340	208	59	b	b	NOUN
ejpam-4340	208	60	be	be	AUX
ejpam-4340	208	61	subsets	subset	NOUN
ejpam-4340	208	62	of	of	ADP
ejpam-4340	208	63	a	a	DET
ejpam-4340	208	64	space	space	NOUN
ejpam-4340	208	65	x.	x.	NOUN
ejpam-4340	209	1	then	then	ADV
ejpam-4340	209	2	the	the	DET
ejpam-4340	209	3	following	follow	VERB
ejpam-4340	209	4	properties	property	NOUN
ejpam-4340	209	5	hold	hold	VERB
ejpam-4340	209	6	:	:	PUNCT
ejpam-4340	209	7	(	(	PUNCT
ejpam-4340	209	8	a	a	X
ejpam-4340	209	9	)	)	PUNCT
ejpam-4340	209	10	if	if	SCONJ
ejpam-4340	209	11	a	a	PRON
ejpam-4340	209	12	is	be	AUX
ejpam-4340	209	13	open	open	ADJ
ejpam-4340	209	14	and	and	CCONJ
ejpam-4340	209	15	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	209	16	and	and	CCONJ
ejpam-4340	209	17	b	b	NOUN
ejpam-4340	209	18	is	be	AUX
ejpam-4340	209	19	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	209	20	,	,	PUNCT
ejpam-4340	209	21	then	then	ADV
ejpam-4340	209	22	a	a	DET
ejpam-4340	209	23	∩b	∩b	NOUN
ejpam-4340	209	24	is	be	AUX
ejpam-4340	209	25	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	209	26	,	,	PUNCT
ejpam-4340	209	27	(	(	PUNCT
ejpam-4340	209	28	b	b	X
ejpam-4340	209	29	)	)	PUNCT
ejpam-4340	209	30	if	if	SCONJ
ejpam-4340	209	31	a	a	PRON
ejpam-4340	209	32	is	be	AUX
ejpam-4340	209	33	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	209	34	and	and	CCONJ
ejpam-4340	209	35	b	b	NOUN
ejpam-4340	209	36	is	be	AUX
ejpam-4340	209	37	closed	closed	ADJ
ejpam-4340	209	38	,	,	PUNCT
ejpam-4340	209	39	then	then	ADV
ejpam-4340	209	40	a	a	DET
ejpam-4340	209	41	∩b	∩b	NOUN
ejpam-4340	209	42	is	be	AUX
ejpam-4340	209	43	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	209	44	.	.	PUNCT
ejpam-4340	210	1	proof	proof	NOUN
ejpam-4340	210	2	.	.	PUNCT
ejpam-4340	211	1	(	(	PUNCT
ejpam-4340	211	2	a	a	X
ejpam-4340	211	3	)	)	PUNCT
ejpam-4340	211	4	let	let	VERB
ejpam-4340	211	5	a	a	DET
ejpam-4340	211	6	∈	∈	PROPN
ejpam-4340	211	7	o(x	o(x	PROPN
ejpam-4340	211	8	)	)	PUNCT
ejpam-4340	211	9	∩	∩	ADJ
ejpam-4340	211	10	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	211	11	)	)	PUNCT
ejpam-4340	211	12	.	.	PUNCT
ejpam-4340	212	1	a	a	DET
ejpam-4340	212	2	∈	∈	PROPN
ejpam-4340	212	3	o(x	o(x	PROPN
ejpam-4340	212	4	)	)	PUNCT
ejpam-4340	212	5	∩	∩	ADJ
ejpam-4340	212	6	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	212	7	)	)	PUNCT
ejpam-4340	212	8	theorem	theorem	VERB
ejpam-4340	212	9	5⇒	5⇒	NUM
ejpam-4340	212	10	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	212	11	)	)	PUNCT
ejpam-4340	212	12	\a	\a	VERB
ejpam-4340	213	1	=	=	PUNCT
ejpam-4340	213	2	∅	∅	NOUN
ejpam-4340	213	3	⇒	⇒	NOUN
ejpam-4340	213	4	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	213	5	)	)	PUNCT
ejpam-4340	213	6	=	=	SYM
ejpam-4340	213	7	a	a	DET
ejpam-4340	213	8	⇒	⇒	NOUN
ejpam-4340	213	9	a	a	DET
ejpam-4340	213	10	∈	∈	PROPN
ejpam-4340	213	11	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	213	12	)	)	PUNCT
ejpam-4340	213	13	b	b	NOUN
ejpam-4340	213	14	∈	∈	PROPN
ejpam-4340	213	15	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	213	16	)	)	PUNCT
ejpam-4340	213	17	}	}	PUNCT
ejpam-4340	213	18	⇒	⇒	VERB
ejpam-4340	213	19	a	a	DET
ejpam-4340	213	20	∩b	∩b	NOUN
ejpam-4340	213	21	∈	∈	NOUN
ejpam-4340	213	22	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	213	23	)	)	PUNCT
ejpam-4340	213	24	⊆	⊆	NUM
ejpam-4340	213	25	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	213	26	)	)	PUNCT
ejpam-4340	213	27	.	.	PUNCT
ejpam-4340	214	1	(	(	PUNCT
ejpam-4340	214	2	b	b	X
ejpam-4340	214	3	)	)	PUNCT
ejpam-4340	214	4	let	let	VERB
ejpam-4340	214	5	a	a	DET
ejpam-4340	214	6	∩b	∩b	NOUN
ejpam-4340	214	7	⊆	⊆	NUM
ejpam-4340	214	8	u	u	NOUN
ejpam-4340	214	9	∈	∈	PROPN
ejpam-4340	214	10	o(x	o(x	PROPN
ejpam-4340	214	11	)	)	PUNCT
ejpam-4340	214	12	.	.	PUNCT
ejpam-4340	215	1	a	a	DET
ejpam-4340	215	2	∩b	∩b	NOUN
ejpam-4340	215	3	⊆	⊆	NUM
ejpam-4340	215	4	u	u	NOUN
ejpam-4340	215	5	∈	∈	PROPN
ejpam-4340	215	6	o(x	o(x	PROPN
ejpam-4340	215	7	)	)	PUNCT
ejpam-4340	215	8	b	b	PROPN
ejpam-4340	215	9	∈	∈	PROPN
ejpam-4340	215	10	c(x)⇒	c(x)⇒	PROPN
ejpam-4340	215	11	x	x	SYM
ejpam-4340	215	12	\b	\b	PROPN
ejpam-4340	215	13	∈	∈	PROPN
ejpam-4340	215	14	o(x	o(x	PROPN
ejpam-4340	215	15	)	)	PUNCT
ejpam-4340	215	16	}	}	PUNCT
ejpam-4340	215	17	⇒	⇒	VERB
ejpam-4340	215	18	⇒	⇒	NOUN
ejpam-4340	215	19	a	a	DET
ejpam-4340	215	20	⊆	⊆	NUM
ejpam-4340	215	21	(	(	PUNCT
ejpam-4340	215	22	a	a	DET
ejpam-4340	215	23	∩b	∩b	NOUN
ejpam-4340	215	24	)	)	PUNCT
ejpam-4340	215	25	∪	∪	NOUN
ejpam-4340	215	26	(	(	PUNCT
ejpam-4340	215	27	x	x	NOUN
ejpam-4340	215	28	\b	\b	ADJ
ejpam-4340	215	29	)	)	PUNCT
ejpam-4340	215	30	⊆	⊆	NUM
ejpam-4340	215	31	u	u	NOUN
ejpam-4340	215	32	∪	∪	X
ejpam-4340	215	33	(	(	PUNCT
ejpam-4340	215	34	x	x	NOUN
ejpam-4340	215	35	\b	\b	ADJ
ejpam-4340	215	36	)	)	PUNCT
ejpam-4340	215	37	∈	∈	PROPN
ejpam-4340	215	38	o(x	o(x	PROPN
ejpam-4340	215	39	)	)	PUNCT
ejpam-4340	215	40	a	a	DET
ejpam-4340	215	41	∈	∈	PROPN
ejpam-4340	215	42	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	215	43	)	)	PUNCT
ejpam-4340	215	44	}	}	PUNCT
ejpam-4340	215	45	⇒	⇒	VERB
ejpam-4340	215	46	p.	p.	NOUN
ejpam-4340	215	47	şaşmaz	şaşmaz	NUM
ejpam-4340	215	48	,	,	PUNCT
ejpam-4340	215	49	m.	m.	NOUN
ejpam-4340	215	50	özkoç	özkoç	PROPN
ejpam-4340	215	51	/	/	SYM
ejpam-4340	215	52	eur	eur	PROPN
ejpam-4340	215	53	.	.	PUNCT
ejpam-4340	216	1	j.	j.	PROPN
ejpam-4340	216	2	pure	pure	PROPN
ejpam-4340	216	3	appl	appl	PROPN
ejpam-4340	216	4	.	.	PROPN
ejpam-4340	216	5	math	math	PROPN
ejpam-4340	216	6	,	,	PUNCT
ejpam-4340	216	7	15	15	NUM
ejpam-4340	216	8	(	(	PUNCT
ejpam-4340	216	9	2	2	NUM
ejpam-4340	216	10	)	)	PUNCT
ejpam-4340	216	11	(	(	PUNCT
ejpam-4340	216	12	2022	2022	NUM
ejpam-4340	216	13	)	)	PUNCT
ejpam-4340	216	14	,	,	PUNCT
ejpam-4340	216	15	354	354	NUM
ejpam-4340	216	16	-	-	SYM
ejpam-4340	216	17	374	374	NUM
ejpam-4340	216	18	361	361	NUM
ejpam-4340	216	19	⇒	⇒	NOUN
ejpam-4340	216	20	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	216	21	)	)	PUNCT
ejpam-4340	216	22	⊆	⊆	NUM
ejpam-4340	216	23	u	u	NOUN
ejpam-4340	216	24	∪	∪	ADV
ejpam-4340	216	25	(	(	PUNCT
ejpam-4340	216	26	x	x	X
ejpam-4340	216	27	\b)⇒	\b)⇒	VERB
ejpam-4340	216	28	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	216	29	∩b	∩b	NOUN
ejpam-4340	216	30	)	)	PUNCT
ejpam-4340	216	31	⊆	⊆	NUM
ejpam-4340	216	32	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	216	33	)	)	PUNCT
ejpam-4340	216	34	∩	∩	NOUN
ejpam-4340	216	35	ωe∗-cl(b	ωe∗-cl(b	NOUN
ejpam-4340	216	36	)	)	PUNCT
ejpam-4340	216	37	⊆	⊆	NUM
ejpam-4340	216	38	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	216	39	)	)	PUNCT
ejpam-4340	216	40	∩	∩	NOUN
ejpam-4340	216	41	cl(b	cl(b	NOUN
ejpam-4340	216	42	)	)	PUNCT
ejpam-4340	216	43	=	=	SYM
ejpam-4340	217	1	ωe∗-cl(a	ωe∗-cl(a	X
ejpam-4340	217	2	)	)	PUNCT
ejpam-4340	217	3	∩b	∩b	NOUN
ejpam-4340	217	4	⊆	⊆	NUM
ejpam-4340	217	5	(	(	PUNCT
ejpam-4340	217	6	u	u	NOUN
ejpam-4340	217	7	∪	∪	X
ejpam-4340	217	8	(	(	PUNCT
ejpam-4340	217	9	x	x	NOUN
ejpam-4340	217	10	\b	\b	ADJ
ejpam-4340	217	11	)	)	PUNCT
ejpam-4340	217	12	)	)	PUNCT
ejpam-4340	217	13	∩b	∩b	NOUN
ejpam-4340	218	1	=	=	NOUN
ejpam-4340	218	2	u	u	NOUN
ejpam-4340	218	3	∩b	∩b	NOUN
ejpam-4340	218	4	⊆	⊆	NUM
ejpam-4340	218	5	u.	u.	NOUN
ejpam-4340	218	6	theorem	theorem	VERB
ejpam-4340	218	7	7	7	NUM
ejpam-4340	218	8	.	.	PUNCT
ejpam-4340	219	1	let	let	VERB
ejpam-4340	219	2	a	a	DET
ejpam-4340	219	3	be	be	AUX
ejpam-4340	219	4	a	a	DET
ejpam-4340	219	5	subset	subset	NOUN
ejpam-4340	219	6	of	of	ADP
ejpam-4340	219	7	a	a	DET
ejpam-4340	219	8	space	space	NOUN
ejpam-4340	219	9	x.	x.	NOUN
ejpam-4340	219	10	a	a	PRON
ejpam-4340	219	11	is	be	AUX
ejpam-4340	219	12	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	219	13	if	if	SCONJ
ejpam-4340	219	14	and	and	CCONJ
ejpam-4340	219	15	only	only	ADV
ejpam-4340	219	16	if	if	SCONJ
ejpam-4340	219	17	cl({x})∩a	cl({x})∩a	PRON
ejpam-4340	219	18	̸=	̸=	PROPN
ejpam-4340	219	19	∅	∅	NOUN
ejpam-4340	219	20	for	for	ADP
ejpam-4340	219	21	every	every	DET
ejpam-4340	219	22	x	x	PROPN
ejpam-4340	219	23	∈	∈	PROPN
ejpam-4340	219	24	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	219	25	)	)	PUNCT
ejpam-4340	219	26	.	.	PUNCT
ejpam-4340	220	1	proof	proof	NOUN
ejpam-4340	220	2	.	.	PUNCT
ejpam-4340	221	1	(	(	PUNCT
ejpam-4340	221	2	⇒	⇒	PROPN
ejpam-4340	221	3	)	)	PUNCT
ejpam-4340	221	4	:	:	PUNCT
ejpam-4340	221	5	suppose	suppose	VERB
ejpam-4340	221	6	that	that	SCONJ
ejpam-4340	221	7	x	x	PUNCT
ejpam-4340	221	8	∈	∈	PROPN
ejpam-4340	221	9	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	221	10	)	)	PUNCT
ejpam-4340	221	11	and	and	CCONJ
ejpam-4340	221	12	cl({x	cl({x	NOUN
ejpam-4340	221	13	}	}	PUNCT
ejpam-4340	221	14	)	)	PUNCT
ejpam-4340	222	1	∩a	∩a	PROPN
ejpam-4340	222	2	=	=	PUNCT
ejpam-4340	222	3	∅.	∅.	NOUN
ejpam-4340	222	4	cl({x	cl({x	NUM
ejpam-4340	222	5	}	}	PUNCT
ejpam-4340	222	6	)	)	PUNCT
ejpam-4340	223	1	∩a	∩a	NOUN
ejpam-4340	223	2	=	=	PUNCT
ejpam-4340	223	3	∅	∅	NOUN
ejpam-4340	223	4	⇒	⇒	NOUN
ejpam-4340	223	5	a	a	DET
ejpam-4340	223	6	⊆	⊆	NUM
ejpam-4340	223	7	x	x	SYM
ejpam-4340	223	8	\	\	NOUN
ejpam-4340	223	9	cl({x	cl({x	NUM
ejpam-4340	223	10	}	}	PUNCT
ejpam-4340	223	11	)	)	PUNCT
ejpam-4340	223	12	∈	∈	PROPN
ejpam-4340	223	13	o(x	o(x	PROPN
ejpam-4340	223	14	)	)	PUNCT
ejpam-4340	223	15	a	a	DET
ejpam-4340	223	16	∈	∈	PROPN
ejpam-4340	223	17	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	223	18	)	)	PUNCT
ejpam-4340	223	19	}	}	PUNCT
ejpam-4340	223	20	⇒	⇒	VERB
ejpam-4340	223	21	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	223	22	)	)	PUNCT
ejpam-4340	223	23	⊆	⊆	NUM
ejpam-4340	223	24	x	x	SYM
ejpam-4340	223	25	\	\	NOUN
ejpam-4340	223	26	cl({x	cl({x	NUM
ejpam-4340	223	27	}	}	PUNCT
ejpam-4340	223	28	)	)	PUNCT
ejpam-4340	223	29	⇒	⇒	NOUN
ejpam-4340	223	30	x	x	X
ejpam-4340	223	31	/∈	/∈	PUNCT
ejpam-4340	223	32	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	223	33	)	)	PUNCT
ejpam-4340	223	34	this	this	PRON
ejpam-4340	223	35	contradicts	contradict	VERB
ejpam-4340	223	36	with	with	ADP
ejpam-4340	223	37	x	x	PROPN
ejpam-4340	223	38	∈	∈	PROPN
ejpam-4340	223	39	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	223	40	)	)	PUNCT
ejpam-4340	223	41	.	.	PUNCT
ejpam-4340	224	1	(	(	PUNCT
ejpam-4340	224	2	⇐	⇐	NOUN
ejpam-4340	224	3	)	)	PUNCT
ejpam-4340	224	4	:	:	PUNCT
ejpam-4340	224	5	let	let	VERB
ejpam-4340	224	6	a	a	DET
ejpam-4340	224	7	⊆	⊆	NUM
ejpam-4340	224	8	u	u	NOUN
ejpam-4340	224	9	∈	∈	PROPN
ejpam-4340	224	10	o(x	o(x	PROPN
ejpam-4340	224	11	)	)	PUNCT
ejpam-4340	224	12	and	and	CCONJ
ejpam-4340	224	13	x	x	PUNCT
ejpam-4340	224	14	∈	∈	PROPN
ejpam-4340	224	15	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	224	16	)	)	PUNCT
ejpam-4340	224	17	.	.	PUNCT
ejpam-4340	225	1	x	x	PUNCT
ejpam-4340	225	2	∈	∈	PROPN
ejpam-4340	225	3	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	225	4	)	)	PUNCT
ejpam-4340	225	5	hypothesis	hypothesis	NOUN
ejpam-4340	225	6	}	}	PUNCT
ejpam-4340	225	7	⇒	⇒	VERB
ejpam-4340	225	8	a	a	DET
ejpam-4340	225	9	∩	∩	NOUN
ejpam-4340	225	10	cl({x	cl({x	NOUN
ejpam-4340	225	11	}	}	PUNCT
ejpam-4340	225	12	)	)	PUNCT
ejpam-4340	225	13	̸=	̸=	PROPN
ejpam-4340	225	14	∅	∅	NOUN
ejpam-4340	225	15	⇒	⇒	NOUN
ejpam-4340	225	16	(	(	PUNCT
ejpam-4340	225	17	∃y	∃y	PROPN
ejpam-4340	225	18	∈	∈	PROPN
ejpam-4340	225	19	x)(y	x)(y	PUNCT
ejpam-4340	225	20	∈	∈	PROPN
ejpam-4340	225	21	a	a	DET
ejpam-4340	225	22	∩	∩	NOUN
ejpam-4340	225	23	cl({x	cl({x	NOUN
ejpam-4340	225	24	}	}	PUNCT
ejpam-4340	225	25	)	)	PUNCT
ejpam-4340	225	26	)	)	PUNCT
ejpam-4340	226	1	⇒	⇒	NOUN
ejpam-4340	226	2	(	(	PUNCT
ejpam-4340	226	3	y	y	PROPN
ejpam-4340	226	4	∈	∈	PROPN
ejpam-4340	226	5	a)(y	a)(y	PROPN
ejpam-4340	226	6	∈	∈	PROPN
ejpam-4340	226	7	cl({x	cl({x	NOUN
ejpam-4340	226	8	}	}	PUNCT
ejpam-4340	226	9	)	)	PUNCT
ejpam-4340	226	10	)	)	PUNCT
ejpam-4340	226	11	a	a	DET
ejpam-4340	226	12	⊆	⊆	NUM
ejpam-4340	226	13	u	u	NOUN
ejpam-4340	226	14	∈	∈	PROPN
ejpam-4340	226	15	o(x	o(x	PROPN
ejpam-4340	226	16	)	)	PUNCT
ejpam-4340	226	17	}	}	PUNCT
ejpam-4340	226	18	⇒	⇒	NOUN
ejpam-4340	226	19	(	(	PUNCT
ejpam-4340	226	20	y	y	PROPN
ejpam-4340	226	21	∈	∈	PROPN
ejpam-4340	226	22	a	a	DET
ejpam-4340	226	23	⊆	⊆	NUM
ejpam-4340	226	24	u	u	NOUN
ejpam-4340	226	25	∈	∈	PROPN
ejpam-4340	226	26	o(x))(y	o(x))(y	PROPN
ejpam-4340	226	27	∈	∈	PROPN
ejpam-4340	226	28	cl({x	cl({x	NOUN
ejpam-4340	226	29	}	}	PUNCT
ejpam-4340	226	30	)	)	PUNCT
ejpam-4340	226	31	)	)	PUNCT
ejpam-4340	226	32	⇒	⇒	NOUN
ejpam-4340	226	33	(	(	PUNCT
ejpam-4340	226	34	u	u	NOUN
ejpam-4340	226	35	∈	∈	PROPN
ejpam-4340	226	36	o(x	o(x	PROPN
ejpam-4340	226	37	,	,	PUNCT
ejpam-4340	226	38	y))(y	y))(y	PROPN
ejpam-4340	226	39	∈	∈	PROPN
ejpam-4340	226	40	cl({x}))⇒	cl({x}))⇒	PROPN
ejpam-4340	226	41	u	u	PROPN
ejpam-4340	226	42	∩	∩	NOUN
ejpam-4340	226	43	{	{	PUNCT
ejpam-4340	226	44	x	x	NOUN
ejpam-4340	226	45	}	}	PUNCT
ejpam-4340	226	46	=	=	NOUN
ejpam-4340	226	47	̸	̸	NOUN
ejpam-4340	226	48	∅	∅	NOUN
ejpam-4340	226	49	⇒	⇒	NOUN
ejpam-4340	226	50	x	x	X
ejpam-4340	226	51	∈	∈	PROPN
ejpam-4340	226	52	u.	u.	NOUN
ejpam-4340	226	53	theorem	theorem	VERB
ejpam-4340	226	54	8	8	NUM
ejpam-4340	226	55	.	.	PUNCT
ejpam-4340	227	1	let	let	VERB
ejpam-4340	227	2	x	x	PRON
ejpam-4340	227	3	be	be	AUX
ejpam-4340	227	4	a	a	DET
ejpam-4340	227	5	space	space	NOUN
ejpam-4340	227	6	.	.	PUNCT
ejpam-4340	228	1	for	for	ADP
ejpam-4340	228	2	an	an	DET
ejpam-4340	228	3	element	element	NOUN
ejpam-4340	228	4	x	x	SYM
ejpam-4340	228	5	∈	∈	PROPN
ejpam-4340	228	6	x	x	NOUN
ejpam-4340	228	7	,	,	PUNCT
ejpam-4340	228	8	either	either	CCONJ
ejpam-4340	228	9	{	{	PUNCT
ejpam-4340	228	10	x	x	X
ejpam-4340	228	11	}	}	PUNCT
ejpam-4340	228	12	is	be	AUX
ejpam-4340	228	13	closed	closed	ADJ
ejpam-4340	228	14	or	or	CCONJ
ejpam-4340	228	15	x	x	SYM
ejpam-4340	228	16	\	\	X
ejpam-4340	228	17	{	{	PUNCT
ejpam-4340	228	18	x	x	NOUN
ejpam-4340	228	19	}	}	PUNCT
ejpam-4340	228	20	is	be	AUX
ejpam-4340	228	21	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	228	22	.	.	PUNCT
ejpam-4340	229	1	proof	proof	NOUN
ejpam-4340	229	2	.	.	PUNCT
ejpam-4340	230	1	suppose	suppose	VERB
ejpam-4340	230	2	that	that	SCONJ
ejpam-4340	230	3	{	{	PUNCT
ejpam-4340	230	4	x	x	NOUN
ejpam-4340	230	5	}	}	PUNCT
ejpam-4340	230	6	/∈	/∈	PUNCT
ejpam-4340	230	7	c(x	c(x	NOUN
ejpam-4340	230	8	)	)	PUNCT
ejpam-4340	230	9	.	.	PUNCT
ejpam-4340	231	1	{	{	PUNCT
ejpam-4340	231	2	x	x	X
ejpam-4340	231	3	}	}	PUNCT
ejpam-4340	231	4	/∈	/∈	PUNCT
ejpam-4340	231	5	c(x)⇒	c(x)⇒	NOUN
ejpam-4340	231	6	x	x	SYM
ejpam-4340	231	7	\	\	X
ejpam-4340	231	8	{	{	PUNCT
ejpam-4340	231	9	x	x	NOUN
ejpam-4340	231	10	}	}	PUNCT
ejpam-4340	231	11	/∈	/∈	PUNCT
ejpam-4340	232	1	o(x	o(x	PROPN
ejpam-4340	232	2	)	)	PUNCT
ejpam-4340	232	3	x	x	SYM
ejpam-4340	232	4	\	\	X
ejpam-4340	232	5	{	{	PUNCT
ejpam-4340	232	6	x	x	NOUN
ejpam-4340	232	7	}	}	PUNCT
ejpam-4340	232	8	⊆	⊆	NUM
ejpam-4340	232	9	x	x	SYM
ejpam-4340	232	10	∈	∈	PROPN
ejpam-4340	232	11	o(x	o(x	PROPN
ejpam-4340	232	12	)	)	PUNCT
ejpam-4340	232	13	}	}	PUNCT
ejpam-4340	232	14	⇒	⇒	VERB
ejpam-4340	232	15	ωe∗-cl(x	ωe∗-cl(x	NUM
ejpam-4340	232	16	\	\	NOUN
ejpam-4340	232	17	{	{	PUNCT
ejpam-4340	232	18	x	x	NOUN
ejpam-4340	232	19	}	}	PUNCT
ejpam-4340	232	20	)	)	PUNCT
ejpam-4340	232	21	⊆	⊆	NUM
ejpam-4340	232	22	ωe∗-cl(x	ωe∗-cl(x	NUM
ejpam-4340	232	23	)	)	PUNCT
ejpam-4340	232	24	=	=	PUNCT
ejpam-4340	232	25	x.	x.	NOUN
ejpam-4340	232	26	definition	definition	NOUN
ejpam-4340	232	27	11	11	NUM
ejpam-4340	232	28	.	.	PUNCT
ejpam-4340	233	1	a	a	DET
ejpam-4340	233	2	space	space	NOUN
ejpam-4340	233	3	x	x	PUNCT
ejpam-4340	233	4	is	be	AUX
ejpam-4340	233	5	said	say	VERB
ejpam-4340	233	6	to	to	PART
ejpam-4340	233	7	be	be	AUX
ejpam-4340	233	8	an	an	DET
ejpam-4340	233	9	ωe∗-t	ωe∗-t	NUM
ejpam-4340	233	10	1	1	NUM
ejpam-4340	233	11	2	2	NUM
ejpam-4340	233	12	space	space	NOUN
ejpam-4340	233	13	if	if	SCONJ
ejpam-4340	233	14	for	for	ADP
ejpam-4340	233	15	every	every	DET
ejpam-4340	233	16	generalized	generalize	VERB
ejpam-4340	233	17	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	233	18	set	set	NOUN
ejpam-4340	233	19	is	be	AUX
ejpam-4340	233	20	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	233	21	.	.	PUNCT
ejpam-4340	234	1	example	example	NOUN
ejpam-4340	235	1	2	2	NUM
ejpam-4340	235	2	.	.	X
ejpam-4340	235	3	any	any	DET
ejpam-4340	235	4	set	set	NOUN
ejpam-4340	235	5	with	with	ADP
ejpam-4340	235	6	indiscrete	indiscrete	ADJ
ejpam-4340	235	7	topology	topology	NOUN
ejpam-4340	235	8	is	be	AUX
ejpam-4340	235	9	an	an	DET
ejpam-4340	235	10	example	example	NOUN
ejpam-4340	235	11	for	for	ADP
ejpam-4340	235	12	an	an	DET
ejpam-4340	235	13	ωe∗-t	ωe∗-t	PROPN
ejpam-4340	235	14	1	1	NUM
ejpam-4340	235	15	2	2	NUM
ejpam-4340	235	16	space	space	NOUN
ejpam-4340	235	17	.	.	PUNCT
ejpam-4340	236	1	theorem	theorem	VERB
ejpam-4340	236	2	9	9	NUM
ejpam-4340	236	3	.	.	PUNCT
ejpam-4340	237	1	let	let	VERB
ejpam-4340	237	2	x	x	PRON
ejpam-4340	237	3	be	be	AUX
ejpam-4340	237	4	a	a	DET
ejpam-4340	237	5	space	space	NOUN
ejpam-4340	237	6	.	.	PUNCT
ejpam-4340	238	1	x	x	PUNCT
ejpam-4340	238	2	is	be	AUX
ejpam-4340	238	3	an	an	DET
ejpam-4340	238	4	ωe∗-t	ωe∗-t	NUM
ejpam-4340	238	5	1	1	NUM
ejpam-4340	238	6	2	2	NUM
ejpam-4340	238	7	space	space	NOUN
ejpam-4340	238	8	if	if	SCONJ
ejpam-4340	238	9	and	and	CCONJ
ejpam-4340	238	10	only	only	ADV
ejpam-4340	238	11	if	if	SCONJ
ejpam-4340	238	12	every	every	DET
ejpam-4340	238	13	singleton	singleton	NOUN
ejpam-4340	238	14	is	be	AUX
ejpam-4340	238	15	either	either	CCONJ
ejpam-4340	238	16	closed	closed	ADJ
ejpam-4340	238	17	or	or	CCONJ
ejpam-4340	238	18	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	238	19	.	.	PUNCT
ejpam-4340	239	1	proof	proof	NOUN
ejpam-4340	239	2	.	.	PUNCT
ejpam-4340	240	1	(	(	PUNCT
ejpam-4340	240	2	⇒	⇒	PROPN
ejpam-4340	240	3	)	)	PUNCT
ejpam-4340	240	4	:	:	PUNCT
ejpam-4340	240	5	suppose	suppose	VERB
ejpam-4340	240	6	that	that	SCONJ
ejpam-4340	240	7	{	{	PUNCT
ejpam-4340	240	8	x	x	NOUN
ejpam-4340	240	9	}	}	PUNCT
ejpam-4340	240	10	/∈	/∈	PUNCT
ejpam-4340	240	11	c(x	c(x	NOUN
ejpam-4340	240	12	)	)	PUNCT
ejpam-4340	240	13	.	.	PUNCT
ejpam-4340	241	1	{	{	PUNCT
ejpam-4340	241	2	x	x	X
ejpam-4340	241	3	}	}	PUNCT
ejpam-4340	241	4	/∈	/∈	PUNCT
ejpam-4340	241	5	c(x	c(x	NOUN
ejpam-4340	241	6	)	)	PUNCT
ejpam-4340	241	7	theorem	theorem	VERB
ejpam-4340	241	8	8⇒	8⇒	PROPN
ejpam-4340	241	9	x	x	SYM
ejpam-4340	241	10	\	\	PROPN
ejpam-4340	241	11	{	{	PUNCT
ejpam-4340	241	12	x	x	NOUN
ejpam-4340	241	13	}	}	PUNCT
ejpam-4340	241	14	∈	∈	PROPN
ejpam-4340	241	15	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	241	16	)	)	PUNCT
ejpam-4340	241	17	x	x	PUNCT
ejpam-4340	242	1	is	be	AUX
ejpam-4340	242	2	ωe∗-t	ωe∗-t	NUM
ejpam-4340	242	3	1	1	NUM
ejpam-4340	242	4	2	2	NUM
ejpam-4340	242	5	space	space	NOUN
ejpam-4340	242	6	}	}	PUNCT
ejpam-4340	242	7	⇒	⇒	VERB
ejpam-4340	242	8	x	x	X
ejpam-4340	242	9	\	\	X
ejpam-4340	242	10	{	{	PUNCT
ejpam-4340	242	11	x	x	NOUN
ejpam-4340	242	12	}	}	PUNCT
ejpam-4340	242	13	∈	∈	PROPN
ejpam-4340	242	14	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	242	15	)	)	PUNCT
ejpam-4340	242	16	⇒	⇒	NOUN
ejpam-4340	242	17	{	{	PUNCT
ejpam-4340	242	18	x	x	NOUN
ejpam-4340	242	19	}	}	PUNCT
ejpam-4340	242	20	∈	∈	PROPN
ejpam-4340	242	21	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	242	22	)	)	PUNCT
ejpam-4340	242	23	.	.	PUNCT
ejpam-4340	243	1	(	(	PUNCT
ejpam-4340	243	2	⇐	⇐	NOUN
ejpam-4340	243	3	)	)	PUNCT
ejpam-4340	243	4	:	:	PUNCT
ejpam-4340	243	5	let	let	VERB
ejpam-4340	243	6	a	a	DET
ejpam-4340	243	7	∈	∈	PROPN
ejpam-4340	243	8	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	243	9	)	)	PUNCT
ejpam-4340	243	10	and	and	CCONJ
ejpam-4340	243	11	x	x	PUNCT
ejpam-4340	243	12	∈	∈	PROPN
ejpam-4340	243	13	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	243	14	)	)	PUNCT
ejpam-4340	243	15	.	.	PUNCT
ejpam-4340	244	1	p.	p.	NOUN
ejpam-4340	244	2	şaşmaz	şaşmaz	NUM
ejpam-4340	244	3	,	,	PUNCT
ejpam-4340	244	4	m.	m.	NOUN
ejpam-4340	244	5	özkoç	özkoç	PROPN
ejpam-4340	244	6	/	/	SYM
ejpam-4340	244	7	eur	eur	PROPN
ejpam-4340	244	8	.	.	PUNCT
ejpam-4340	245	1	j.	j.	PROPN
ejpam-4340	245	2	pure	pure	PROPN
ejpam-4340	245	3	appl	appl	PROPN
ejpam-4340	245	4	.	.	PROPN
ejpam-4340	245	5	math	math	PROPN
ejpam-4340	245	6	,	,	PUNCT
ejpam-4340	245	7	15	15	NUM
ejpam-4340	245	8	(	(	PUNCT
ejpam-4340	245	9	2	2	NUM
ejpam-4340	245	10	)	)	PUNCT
ejpam-4340	245	11	(	(	PUNCT
ejpam-4340	245	12	2022	2022	NUM
ejpam-4340	245	13	)	)	PUNCT
ejpam-4340	245	14	,	,	PUNCT
ejpam-4340	245	15	354	354	NUM
ejpam-4340	245	16	-	-	SYM
ejpam-4340	245	17	374	374	NUM
ejpam-4340	245	18	362	362	NUM
ejpam-4340	245	19	1st	1st	ADJ
ejpam-4340	245	20	case	case	NOUN
ejpam-4340	245	21	:	:	PUNCT
ejpam-4340	245	22	let	let	VERB
ejpam-4340	245	23	{	{	PUNCT
ejpam-4340	245	24	x	x	NOUN
ejpam-4340	245	25	}	}	PUNCT
ejpam-4340	245	26	∈	∈	PROPN
ejpam-4340	245	27	c(x	c(x	NOUN
ejpam-4340	245	28	)	)	PUNCT
ejpam-4340	245	29	and	and	CCONJ
ejpam-4340	245	30	suppose	suppose	VERB
ejpam-4340	245	31	that	that	SCONJ
ejpam-4340	245	32	x	x	SYM
ejpam-4340	245	33	/∈	/∈	PUNCT
ejpam-4340	245	34	a.	a.	NOUN
ejpam-4340	245	35	(	(	PUNCT
ejpam-4340	245	36	{	{	PUNCT
ejpam-4340	245	37	x	x	NOUN
ejpam-4340	245	38	}	}	PUNCT
ejpam-4340	245	39	∈	∈	PROPN
ejpam-4340	245	40	c(x))(x	c(x))(x	PROPN
ejpam-4340	245	41	/∈	/∈	PUNCT
ejpam-4340	246	1	a	a	PRON
ejpam-4340	246	2	)	)	PUNCT
ejpam-4340	246	3	x	x	SYM
ejpam-4340	246	4	∈	∈	PROPN
ejpam-4340	246	5	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	246	6	)	)	PUNCT
ejpam-4340	246	7	}	}	PUNCT
ejpam-4340	246	8	⇒	⇒	VERB
ejpam-4340	246	9	x	x	PUNCT
ejpam-4340	246	10	∈	∈	PROPN
ejpam-4340	246	11	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	246	12	)	)	PUNCT
ejpam-4340	246	13	\a⇒	\a⇒	PROPN
ejpam-4340	246	14	{	{	PUNCT
ejpam-4340	246	15	x	x	NOUN
ejpam-4340	246	16	}	}	PUNCT
ejpam-4340	246	17	⊆	⊆	NUM
ejpam-4340	246	18	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	246	19	)	)	PUNCT
ejpam-4340	246	20	\a	\a	VERB
ejpam-4340	247	1	this	this	DET
ejpam-4340	247	2	result	result	NOUN
ejpam-4340	247	3	contradicts	contradict	VERB
ejpam-4340	247	4	with	with	ADP
ejpam-4340	247	5	theorem	theorem	ADJ
ejpam-4340	247	6	3	3	NUM
ejpam-4340	247	7	.	.	PUNCT
ejpam-4340	248	1	hence	hence	ADV
ejpam-4340	248	2	,	,	PUNCT
ejpam-4340	248	3	x	x	PUNCT
ejpam-4340	248	4	∈	∈	NOUN
ejpam-4340	248	5	a.	a.	NOUN
ejpam-4340	248	6	this	this	PRON
ejpam-4340	248	7	means	mean	VERB
ejpam-4340	248	8	that	that	SCONJ
ejpam-4340	248	9	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	248	10	)	)	PUNCT
ejpam-4340	248	11	⊆	⊆	NUM
ejpam-4340	248	12	a.	a.	NOUN
ejpam-4340	248	13	then	then	ADV
ejpam-4340	248	14	,	,	PUNCT
ejpam-4340	248	15	we	we	PRON
ejpam-4340	248	16	have	have	VERB
ejpam-4340	248	17	a	a	DET
ejpam-4340	248	18	∈	∈	NOUN
ejpam-4340	248	19	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	248	20	.	.	PUNCT
ejpam-4340	249	1	2nd	2nd	ADJ
ejpam-4340	249	2	case	case	NOUN
ejpam-4340	249	3	:	:	PUNCT
ejpam-4340	249	4	let	let	VERB
ejpam-4340	249	5	{	{	PUNCT
ejpam-4340	249	6	x	x	NOUN
ejpam-4340	249	7	}	}	PUNCT
ejpam-4340	249	8	∈	∈	PROPN
ejpam-4340	249	9	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	249	10	)	)	PUNCT
ejpam-4340	249	11	.	.	PUNCT
ejpam-4340	250	1	x	x	X
ejpam-4340	250	2	∈	∈	NOUN
ejpam-4340	250	3	ωe∗-cl(a)⇒	ωe∗-cl(a)⇒	PROPN
ejpam-4340	250	4	(	(	PUNCT
ejpam-4340	250	5	∀u	∀u	NOUN
ejpam-4340	250	6	∈	∈	NOUN
ejpam-4340	250	7	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	250	8	,	,	PUNCT
ejpam-4340	250	9	x))(u	x))(u	PUNCT
ejpam-4340	250	10	∩a	∩a	PROPN
ejpam-4340	250	11	̸=	̸=	PROPN
ejpam-4340	250	12	∅	∅	NOUN
ejpam-4340	250	13	)	)	PUNCT
ejpam-4340	250	14	{	{	PUNCT
ejpam-4340	250	15	x	x	NOUN
ejpam-4340	250	16	}	}	PUNCT
ejpam-4340	250	17	∈	∈	PROPN
ejpam-4340	250	18	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	250	19	)	)	PUNCT
ejpam-4340	250	20	}	}	PUNCT
ejpam-4340	250	21	⇒	⇒	VERB
ejpam-4340	250	22	⇒	⇒	NOUN
ejpam-4340	250	23	(	(	PUNCT
ejpam-4340	250	24	{	{	PUNCT
ejpam-4340	250	25	x	x	ADJ
ejpam-4340	250	26	}	}	PUNCT
ejpam-4340	250	27	∈	∈	PROPN
ejpam-4340	250	28	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	250	29	,	,	PUNCT
ejpam-4340	250	30	x))({x	x))({x	NOUN
ejpam-4340	250	31	}	}	PUNCT
ejpam-4340	250	32	∩a	∩a	PROPN
ejpam-4340	250	33	̸=	̸=	PROPN
ejpam-4340	250	34	∅)⇒	∅)⇒	VERB
ejpam-4340	251	1	x	x	X
ejpam-4340	251	2	∈	∈	PROPN
ejpam-4340	251	3	a	a	DET
ejpam-4340	251	4	this	this	PRON
ejpam-4340	251	5	means	mean	VERB
ejpam-4340	251	6	that	that	SCONJ
ejpam-4340	251	7	a	a	PRON
ejpam-4340	251	8	is	be	AUX
ejpam-4340	251	9	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	251	10	.	.	PUNCT
ejpam-4340	252	1	definition	definition	NOUN
ejpam-4340	252	2	12	12	NUM
ejpam-4340	252	3	.	.	PUNCT
ejpam-4340	253	1	a	a	DET
ejpam-4340	253	2	space	space	NOUN
ejpam-4340	253	3	x	x	PUNCT
ejpam-4340	253	4	is	be	AUX
ejpam-4340	253	5	said	say	VERB
ejpam-4340	253	6	to	to	PART
ejpam-4340	253	7	be	be	AUX
ejpam-4340	253	8	an	an	DET
ejpam-4340	253	9	e∗-anti	e∗-anti	ADJ
ejpam-4340	253	10	-	-	ADJ
ejpam-4340	253	11	locally	locally	ADV
ejpam-4340	253	12	countable	countable	ADJ
ejpam-4340	253	13	if	if	SCONJ
ejpam-4340	253	14	each	each	DET
ejpam-4340	253	15	u	u	PROPN
ejpam-4340	253	16	∈	∈	PROPN
ejpam-4340	253	17	e∗o(x	e∗o(x	PROPN
ejpam-4340	253	18	)	)	PUNCT
ejpam-4340	253	19	\	\	NOUN
ejpam-4340	253	20	{	{	PUNCT
ejpam-4340	253	21	∅	∅	NOUN
ejpam-4340	253	22	}	}	PUNCT
ejpam-4340	253	23	is	be	AUX
ejpam-4340	253	24	uncountable	uncountable	ADJ
ejpam-4340	253	25	.	.	PUNCT
ejpam-4340	254	1	theorem	theorem	ADJ
ejpam-4340	254	2	10	10	NUM
ejpam-4340	254	3	.	.	PUNCT
ejpam-4340	255	1	let	let	VERB
ejpam-4340	255	2	x	x	PRON
ejpam-4340	255	3	be	be	AUX
ejpam-4340	255	4	a	a	DET
ejpam-4340	255	5	space	space	NOUN
ejpam-4340	255	6	.	.	PUNCT
ejpam-4340	256	1	if	if	SCONJ
ejpam-4340	256	2	x	x	PRON
ejpam-4340	256	3	is	be	AUX
ejpam-4340	256	4	e∗-anti	e∗-anti	NOUN
ejpam-4340	256	5	-	-	ADJ
ejpam-4340	256	6	locally	locally	ADV
ejpam-4340	256	7	countable	countable	ADJ
ejpam-4340	256	8	and	and	CCONJ
ejpam-4340	256	9	ωe∗-t	ωe∗-t	NUM
ejpam-4340	256	10	1	1	NUM
ejpam-4340	256	11	2	2	NUM
ejpam-4340	256	12	space	space	NOUN
ejpam-4340	256	13	,	,	PUNCT
ejpam-4340	256	14	then	then	ADV
ejpam-4340	256	15	x	x	PUNCT
ejpam-4340	256	16	is	be	AUX
ejpam-4340	256	17	t1	t1	NOUN
ejpam-4340	256	18	space	space	NOUN
ejpam-4340	256	19	.	.	PUNCT
ejpam-4340	257	1	proof	proof	NOUN
ejpam-4340	257	2	.	.	PUNCT
ejpam-4340	258	1	let	let	VERB
ejpam-4340	258	2	x	x	PUNCT
ejpam-4340	258	3	∈	∈	PROPN
ejpam-4340	258	4	x	x	PUNCT
ejpam-4340	258	5	and	and	CCONJ
ejpam-4340	258	6	suppose	suppose	VERB
ejpam-4340	258	7	that	that	SCONJ
ejpam-4340	258	8	{	{	PUNCT
ejpam-4340	258	9	x	x	NOUN
ejpam-4340	258	10	}	}	PUNCT
ejpam-4340	258	11	/∈	/∈	PUNCT
ejpam-4340	258	12	c(x	c(x	NOUN
ejpam-4340	258	13	)	)	PUNCT
ejpam-4340	258	14	.	.	PUNCT
ejpam-4340	259	1	{	{	PUNCT
ejpam-4340	259	2	x	x	X
ejpam-4340	259	3	}	}	PUNCT
ejpam-4340	259	4	/∈	/∈	PUNCT
ejpam-4340	259	5	c(x	c(x	NOUN
ejpam-4340	259	6	)	)	PUNCT
ejpam-4340	259	7	theorem	theorem	VERB
ejpam-4340	259	8	8⇒	8⇒	PROPN
ejpam-4340	259	9	x	x	SYM
ejpam-4340	259	10	\	\	PROPN
ejpam-4340	259	11	{	{	PUNCT
ejpam-4340	259	12	x	x	NOUN
ejpam-4340	259	13	}	}	PUNCT
ejpam-4340	259	14	∈	∈	PROPN
ejpam-4340	259	15	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	259	16	)	)	PUNCT
ejpam-4340	259	17	x	x	PUNCT
ejpam-4340	260	1	is	be	AUX
ejpam-4340	260	2	ωe∗-t	ωe∗-t	NUM
ejpam-4340	260	3	1	1	NUM
ejpam-4340	260	4	2	2	NUM
ejpam-4340	260	5	}	}	PUNCT
ejpam-4340	260	6	⇒	⇒	NOUN
ejpam-4340	260	7	x	x	X
ejpam-4340	260	8	\	\	X
ejpam-4340	260	9	{	{	PUNCT
ejpam-4340	260	10	x	x	NOUN
ejpam-4340	260	11	}	}	PUNCT
ejpam-4340	260	12	∈	∈	PROPN
ejpam-4340	260	13	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	260	14	)	)	PUNCT
ejpam-4340	260	15	⇒	⇒	NOUN
ejpam-4340	260	16	x	x	PUNCT
ejpam-4340	260	17	∈	∈	PROPN
ejpam-4340	260	18	{	{	PUNCT
ejpam-4340	260	19	x	x	NOUN
ejpam-4340	260	20	}	}	PUNCT
ejpam-4340	260	21	∈	∈	X
ejpam-4340	260	22	ωe∗o(x)⇒	ωe∗o(x)⇒	NOUN
ejpam-4340	260	23	(	(	PUNCT
ejpam-4340	260	24	∃u	∃u	PROPN
ejpam-4340	260	25	∈	∈	PROPN
ejpam-4340	260	26	e∗o(x	e∗o(x	PROPN
ejpam-4340	260	27	,	,	PUNCT
ejpam-4340	260	28	x))(|u	x))(|u	PROPN
ejpam-4340	260	29	\	\	PROPN
ejpam-4340	260	30	{	{	PUNCT
ejpam-4340	260	31	x}|	x}|	PROPN
ejpam-4340	260	32	≤	≤	PUNCT
ejpam-4340	260	33	ℵ0	ℵ0	PROPN
ejpam-4340	260	34	)	)	PUNCT
ejpam-4340	260	35	this	this	PRON
ejpam-4340	260	36	contradicts	contradict	VERB
ejpam-4340	260	37	the	the	DET
ejpam-4340	260	38	fact	fact	NOUN
ejpam-4340	260	39	that	that	SCONJ
ejpam-4340	260	40	x	x	PRON
ejpam-4340	260	41	is	be	AUX
ejpam-4340	260	42	e∗-anti	e∗-anti	NOUN
ejpam-4340	260	43	-	-	ADJ
ejpam-4340	260	44	locally	locally	ADV
ejpam-4340	260	45	countable	countable	ADJ
ejpam-4340	260	46	.	.	PUNCT
ejpam-4340	261	1	then	then	ADV
ejpam-4340	261	2	,	,	PUNCT
ejpam-4340	261	3	{	{	PUNCT
ejpam-4340	261	4	x	x	NOUN
ejpam-4340	261	5	}	}	PUNCT
ejpam-4340	261	6	∈	∈	PROPN
ejpam-4340	261	7	c(x	c(x	NOUN
ejpam-4340	261	8	)	)	PUNCT
ejpam-4340	261	9	for	for	ADP
ejpam-4340	261	10	all	all	PRON
ejpam-4340	261	11	x	x	SYM
ejpam-4340	261	12	∈	∈	PROPN
ejpam-4340	261	13	x.	x.	NOUN
ejpam-4340	261	14	namely	namely	ADV
ejpam-4340	261	15	,	,	PUNCT
ejpam-4340	261	16	x	x	X
ejpam-4340	261	17	is	be	AUX
ejpam-4340	261	18	t1	t1	NOUN
ejpam-4340	261	19	space	space	NOUN
ejpam-4340	261	20	.	.	PUNCT
ejpam-4340	262	1	proposition	proposition	NOUN
ejpam-4340	262	2	6	6	NUM
ejpam-4340	262	3	.	.	PUNCT
ejpam-4340	263	1	let	let	VERB
ejpam-4340	263	2	a	a	DET
ejpam-4340	263	3	be	be	AUX
ejpam-4340	263	4	a	a	DET
ejpam-4340	263	5	subset	subset	NOUN
ejpam-4340	263	6	of	of	ADP
ejpam-4340	263	7	a	a	DET
ejpam-4340	263	8	space	space	NOUN
ejpam-4340	263	9	x.	x.	NOUN
ejpam-4340	263	10	a	a	PRON
ejpam-4340	263	11	is	be	AUX
ejpam-4340	263	12	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	263	13	set	set	VERB
ejpam-4340	263	14	if	if	SCONJ
ejpam-4340	263	15	and	and	CCONJ
ejpam-4340	263	16	only	only	ADV
ejpam-4340	263	17	if	if	SCONJ
ejpam-4340	263	18	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	263	19	)	)	PUNCT
ejpam-4340	263	20	⊆	⊆	NUM
ejpam-4340	263	21	ker(a	ker(a	NOUN
ejpam-4340	263	22	)	)	PUNCT
ejpam-4340	263	23	.	.	PUNCT
ejpam-4340	264	1	proof	proof	NOUN
ejpam-4340	264	2	.	.	PUNCT
ejpam-4340	265	1	(	(	PUNCT
ejpam-4340	265	2	⇒	⇒	PROPN
ejpam-4340	265	3	)	)	PUNCT
ejpam-4340	265	4	:	:	PUNCT
ejpam-4340	265	5	let	let	VERB
ejpam-4340	265	6	a	a	DET
ejpam-4340	265	7	∈	∈	PROPN
ejpam-4340	265	8	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	265	9	)	)	PUNCT
ejpam-4340	265	10	.	.	PUNCT
ejpam-4340	266	1	a	a	DET
ejpam-4340	266	2	∈	∈	PROPN
ejpam-4340	266	3	gωe∗c(x)⇒	gωe∗c(x)⇒	PROPN
ejpam-4340	266	4	(	(	PUNCT
ejpam-4340	266	5	∀u	∀u	NOUN
ejpam-4340	266	6	∈	∈	NOUN
ejpam-4340	266	7	o(x))(a	o(x))(a	NOUN
ejpam-4340	266	8	⊆	⊆	NUM
ejpam-4340	266	9	u	u	NOUN
ejpam-4340	266	10	⇒	⇒	NOUN
ejpam-4340	266	11	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	266	12	)	)	PUNCT
ejpam-4340	266	13	⊆	⊆	NUM
ejpam-4340	266	14	u	u	NOUN
ejpam-4340	266	15	)	)	PUNCT
ejpam-4340	266	16	ker(a	ker(a	PROPN
ejpam-4340	266	17	)	)	PUNCT
ejpam-4340	266	18	:	:	PUNCT
ejpam-4340	267	1	=	=	PUNCT
ejpam-4340	267	2	∩{u	∩{u	PUNCT
ejpam-4340	267	3	|(a	|(a	PROPN
ejpam-4340	267	4	⊆	⊆	NUM
ejpam-4340	267	5	u)(u	u)(u	ADJ
ejpam-4340	267	6	∈	∈	PROPN
ejpam-4340	267	7	o(x	o(x	PROPN
ejpam-4340	267	8	)	)	PUNCT
ejpam-4340	267	9	)	)	PUNCT
ejpam-4340	267	10	}	}	PUNCT
ejpam-4340	267	11	}	}	PUNCT
ejpam-4340	267	12	⇒	⇒	VERB
ejpam-4340	267	13	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	267	14	)	)	PUNCT
ejpam-4340	267	15	⊆	⊆	NUM
ejpam-4340	267	16	ker(a	ker(a	NOUN
ejpam-4340	267	17	)	)	PUNCT
ejpam-4340	267	18	.	.	PUNCT
ejpam-4340	268	1	(	(	PUNCT
ejpam-4340	268	2	⇐	⇐	NOUN
ejpam-4340	268	3	)	)	PUNCT
ejpam-4340	268	4	:	:	PUNCT
ejpam-4340	268	5	let	let	VERB
ejpam-4340	268	6	a	a	DET
ejpam-4340	268	7	⊆	⊆	NUM
ejpam-4340	268	8	u	u	NOUN
ejpam-4340	268	9	∈	∈	PROPN
ejpam-4340	268	10	o(x	o(x	PROPN
ejpam-4340	268	11	)	)	PUNCT
ejpam-4340	268	12	.	.	PUNCT
ejpam-4340	269	1	a	a	DET
ejpam-4340	269	2	⊆	⊆	NUM
ejpam-4340	269	3	u	u	NOUN
ejpam-4340	269	4	⇒	⇒	X
ejpam-4340	269	5	ker(a	ker(a	PROPN
ejpam-4340	269	6	)	)	PUNCT
ejpam-4340	269	7	⊆	⊆	NUM
ejpam-4340	269	8	ker(u	ker(u	PROPN
ejpam-4340	269	9	)	)	PUNCT
ejpam-4340	269	10	hypothesis	hypothesis	NOUN
ejpam-4340	269	11	}	}	PUNCT
ejpam-4340	269	12	⇒	⇒	VERB
ejpam-4340	269	13	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	269	14	)	)	PUNCT
ejpam-4340	269	15	⊆	⊆	NUM
ejpam-4340	269	16	ker(a	ker(a	X
ejpam-4340	269	17	)	)	PUNCT
ejpam-4340	269	18	⊆	⊆	NUM
ejpam-4340	269	19	ker(u	ker(u	PROPN
ejpam-4340	269	20	)	)	PUNCT
ejpam-4340	269	21	u	u	NOUN
ejpam-4340	269	22	∈	∈	PROPN
ejpam-4340	269	23	o(x)⇒	o(x)⇒	PROPN
ejpam-4340	269	24	u	u	NOUN
ejpam-4340	269	25	=	=	PROPN
ejpam-4340	269	26	ker(u	ker(u	PROPN
ejpam-4340	269	27	)	)	PUNCT
ejpam-4340	269	28	}	}	PUNCT
ejpam-4340	269	29	⇒	⇒	VERB
ejpam-4340	269	30	⇒	⇒	NOUN
ejpam-4340	269	31	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	269	32	)	)	PUNCT
ejpam-4340	269	33	⊆	⊆	NUM
ejpam-4340	269	34	u.	u.	NOUN
ejpam-4340	269	35	4	4	NUM
ejpam-4340	269	36	.	.	PUNCT
ejpam-4340	269	37	generalized	generalize	VERB
ejpam-4340	269	38	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	269	39	sets	set	NOUN
ejpam-4340	269	40	and	and	CCONJ
ejpam-4340	269	41	generalized	generalized	ADJ
ejpam-4340	269	42	ωe∗-neighborhoods	ωe∗-neighborhood	NOUN
ejpam-4340	269	43	definition	definition	NOUN
ejpam-4340	269	44	13	13	NUM
ejpam-4340	269	45	.	.	PUNCT
ejpam-4340	270	1	a	a	DET
ejpam-4340	270	2	subset	subset	NOUN
ejpam-4340	270	3	a	a	PRON
ejpam-4340	270	4	of	of	ADP
ejpam-4340	270	5	a	a	DET
ejpam-4340	270	6	space	space	NOUN
ejpam-4340	270	7	x	x	PUNCT
ejpam-4340	270	8	is	be	AUX
ejpam-4340	270	9	called	call	VERB
ejpam-4340	270	10	generalized	generalize	VERB
ejpam-4340	270	11	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	270	12	if	if	SCONJ
ejpam-4340	270	13	its	its	PRON
ejpam-4340	270	14	complement	complement	NOUN
ejpam-4340	270	15	is	be	AUX
ejpam-4340	270	16	generalized	generalize	VERB
ejpam-4340	270	17	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	270	18	.	.	PUNCT
ejpam-4340	271	1	we	we	PRON
ejpam-4340	271	2	denote	denote	VERB
ejpam-4340	271	3	the	the	DET
ejpam-4340	271	4	family	family	NOUN
ejpam-4340	271	5	of	of	ADP
ejpam-4340	271	6	all	all	DET
ejpam-4340	271	7	generalized	generalize	VERB
ejpam-4340	271	8	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	271	9	subsets	subset	NOUN
ejpam-4340	271	10	of	of	ADP
ejpam-4340	271	11	a	a	DET
ejpam-4340	271	12	space	space	NOUN
ejpam-4340	271	13	x	x	PUNCT
ejpam-4340	271	14	by	by	ADP
ejpam-4340	271	15	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	271	16	)	)	PUNCT
ejpam-4340	271	17	.	.	PUNCT
ejpam-4340	272	1	p.	p.	NOUN
ejpam-4340	272	2	şaşmaz	şaşmaz	NUM
ejpam-4340	272	3	,	,	PUNCT
ejpam-4340	272	4	m.	m.	NOUN
ejpam-4340	272	5	özkoç	özkoç	PROPN
ejpam-4340	272	6	/	/	SYM
ejpam-4340	272	7	eur	eur	PROPN
ejpam-4340	272	8	.	.	PUNCT
ejpam-4340	273	1	j.	j.	PROPN
ejpam-4340	273	2	pure	pure	PROPN
ejpam-4340	273	3	appl	appl	PROPN
ejpam-4340	273	4	.	.	PROPN
ejpam-4340	273	5	math	math	PROPN
ejpam-4340	273	6	,	,	PUNCT
ejpam-4340	273	7	15	15	NUM
ejpam-4340	273	8	(	(	PUNCT
ejpam-4340	273	9	2	2	NUM
ejpam-4340	273	10	)	)	PUNCT
ejpam-4340	273	11	(	(	PUNCT
ejpam-4340	273	12	2022	2022	NUM
ejpam-4340	273	13	)	)	PUNCT
ejpam-4340	273	14	,	,	PUNCT
ejpam-4340	273	15	354	354	NUM
ejpam-4340	273	16	-	-	SYM
ejpam-4340	273	17	374	374	NUM
ejpam-4340	273	18	363	363	NUM
ejpam-4340	273	19	corollary	corollary	ADJ
ejpam-4340	273	20	2	2	NUM
ejpam-4340	273	21	.	.	PUNCT
ejpam-4340	274	1	let	let	VERB
ejpam-4340	274	2	a	a	DET
ejpam-4340	274	3	be	be	AUX
ejpam-4340	274	4	a	a	DET
ejpam-4340	274	5	subset	subset	NOUN
ejpam-4340	274	6	of	of	ADP
ejpam-4340	274	7	a	a	DET
ejpam-4340	274	8	space	space	NOUN
ejpam-4340	274	9	x.	x.	NOUN
ejpam-4340	274	10	a	a	PRON
ejpam-4340	274	11	is	be	AUX
ejpam-4340	274	12	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	274	13	set	set	VERB
ejpam-4340	274	14	if	if	SCONJ
ejpam-4340	274	15	and	and	CCONJ
ejpam-4340	274	16	only	only	ADV
ejpam-4340	274	17	if	if	SCONJ
ejpam-4340	274	18	f	f	PROPN
ejpam-4340	274	19	⊆	⊆	NUM
ejpam-4340	274	20	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	274	21	)	)	PUNCT
ejpam-4340	274	22	,	,	PUNCT
ejpam-4340	274	23	where	where	SCONJ
ejpam-4340	274	24	f	f	PROPN
ejpam-4340	274	25	is	be	AUX
ejpam-4340	274	26	closed	close	VERB
ejpam-4340	274	27	set	set	VERB
ejpam-4340	274	28	and	and	CCONJ
ejpam-4340	274	29	f	f	NOUN
ejpam-4340	274	30	⊆	⊆	NUM
ejpam-4340	274	31	a.	a.	NOUN
ejpam-4340	274	32	proof	proof	NOUN
ejpam-4340	274	33	.	.	PUNCT
ejpam-4340	275	1	(	(	PUNCT
ejpam-4340	275	2	⇒	⇒	PROPN
ejpam-4340	275	3	)	)	PUNCT
ejpam-4340	275	4	:	:	PUNCT
ejpam-4340	275	5	let	let	VERB
ejpam-4340	275	6	a	a	DET
ejpam-4340	275	7	∈	∈	PROPN
ejpam-4340	275	8	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	275	9	)	)	PUNCT
ejpam-4340	275	10	and	and	CCONJ
ejpam-4340	275	11	f	f	PROPN
ejpam-4340	275	12	∈	∈	PROPN
ejpam-4340	275	13	c(x	c(x	NOUN
ejpam-4340	275	14	)	)	PUNCT
ejpam-4340	275	15	such	such	ADJ
ejpam-4340	275	16	that	that	SCONJ
ejpam-4340	275	17	f	f	PROPN
ejpam-4340	275	18	⊆	⊆	NUM
ejpam-4340	275	19	a.	a.	NOUN
ejpam-4340	275	20	a	a	DET
ejpam-4340	275	21	∈	∈	PROPN
ejpam-4340	275	22	gωe∗o(x)⇒	gωe∗o(x)⇒	NOUN
ejpam-4340	275	23	x	x	PUNCT
ejpam-4340	275	24	\a	\a	ADJ
ejpam-4340	275	25	∈	∈	PROPN
ejpam-4340	275	26	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	275	27	)	)	PUNCT
ejpam-4340	275	28	a	a	DET
ejpam-4340	275	29	⊇	⊇	PROPN
ejpam-4340	275	30	f	f	PROPN
ejpam-4340	275	31	∈	∈	PROPN
ejpam-4340	275	32	c(x)⇒	c(x)⇒	PROPN
ejpam-4340	275	33	x	x	PUNCT
ejpam-4340	275	34	\a	\a	VERB
ejpam-4340	275	35	⊆	⊆	NUM
ejpam-4340	275	36	x	x	SYM
ejpam-4340	275	37	\	\	PROPN
ejpam-4340	275	38	f	f	PROPN
ejpam-4340	275	39	∈	∈	PROPN
ejpam-4340	275	40	o(x	o(x	PROPN
ejpam-4340	275	41	)	)	PUNCT
ejpam-4340	275	42	}	}	PUNCT
ejpam-4340	275	43	⇒	⇒	VERB
ejpam-4340	275	44	⇒	⇒	NOUN
ejpam-4340	275	45	ωe∗-cl(x	ωe∗-cl(x	NUM
ejpam-4340	275	46	\a	\a	NUM
ejpam-4340	275	47	)	)	PUNCT
ejpam-4340	276	1	=	=	SYM
ejpam-4340	276	2	x	x	SYM
ejpam-4340	276	3	\	\	NOUN
ejpam-4340	276	4	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	276	5	)	)	PUNCT
ejpam-4340	276	6	⊆	⊆	NUM
ejpam-4340	276	7	x	x	SYM
ejpam-4340	276	8	\	\	PROPN
ejpam-4340	276	9	f	f	PROPN
ejpam-4340	276	10	⇒	⇒	NOUN
ejpam-4340	276	11	f	f	PROPN
ejpam-4340	276	12	⊆	⊆	NUM
ejpam-4340	276	13	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	276	14	)	)	PUNCT
ejpam-4340	276	15	.	.	PUNCT
ejpam-4340	277	1	(	(	PUNCT
ejpam-4340	277	2	⇐	⇐	NOUN
ejpam-4340	277	3	)	)	PUNCT
ejpam-4340	277	4	:	:	PUNCT
ejpam-4340	277	5	let	let	VERB
ejpam-4340	277	6	x	x	PUNCT
ejpam-4340	277	7	\a	\a	VERB
ejpam-4340	277	8	⊆	⊆	NUM
ejpam-4340	277	9	u	u	NOUN
ejpam-4340	277	10	∈	∈	PROPN
ejpam-4340	277	11	o(x	o(x	PROPN
ejpam-4340	277	12	)	)	PUNCT
ejpam-4340	277	13	.	.	PUNCT
ejpam-4340	278	1	x	x	PUNCT
ejpam-4340	278	2	\a	\a	VERB
ejpam-4340	278	3	⊆	⊆	NUM
ejpam-4340	278	4	u	u	PROPN
ejpam-4340	278	5	∈	∈	PROPN
ejpam-4340	278	6	o(x)⇒	o(x)⇒	PROPN
ejpam-4340	278	7	(	(	PUNCT
ejpam-4340	278	8	x	x	SYM
ejpam-4340	278	9	\	\	NOUN
ejpam-4340	278	10	u	u	PROPN
ejpam-4340	278	11	∈	∈	PROPN
ejpam-4340	278	12	c(x))(x	c(x))(x	PROPN
ejpam-4340	278	13	\	\	NOUN
ejpam-4340	278	14	u	u	PROPN
ejpam-4340	278	15	⊆	⊆	NUM
ejpam-4340	278	16	a	a	DET
ejpam-4340	278	17	)	)	PUNCT
ejpam-4340	278	18	hypothesis	hypothesis	NOUN
ejpam-4340	278	19	}	}	PUNCT
ejpam-4340	278	20	⇒	⇒	VERB
ejpam-4340	278	21	x	x	PUNCT
ejpam-4340	278	22	\	\	X
ejpam-4340	278	23	u	u	PROPN
ejpam-4340	278	24	⊆	⊆	NUM
ejpam-4340	278	25	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	278	26	)	)	PUNCT
ejpam-4340	278	27	⇒	⇒	VERB
ejpam-4340	278	28	ωe∗-cl(x	ωe∗-cl(x	NUM
ejpam-4340	278	29	\a	\a	NUM
ejpam-4340	278	30	)	)	PUNCT
ejpam-4340	279	1	=	=	SYM
ejpam-4340	279	2	x	x	SYM
ejpam-4340	279	3	\	\	NOUN
ejpam-4340	279	4	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	279	5	)	)	PUNCT
ejpam-4340	279	6	⊆	⊆	NUM
ejpam-4340	279	7	u.	u.	NOUN
ejpam-4340	279	8	proposition	proposition	NOUN
ejpam-4340	279	9	7	7	NUM
ejpam-4340	279	10	.	.	PUNCT
ejpam-4340	280	1	let	let	VERB
ejpam-4340	280	2	a	a	PRON
ejpam-4340	280	3	and	and	CCONJ
ejpam-4340	280	4	b	b	NOUN
ejpam-4340	280	5	be	be	AUX
ejpam-4340	280	6	subsets	subset	NOUN
ejpam-4340	280	7	of	of	ADP
ejpam-4340	280	8	a	a	DET
ejpam-4340	280	9	space	space	NOUN
ejpam-4340	280	10	x.	x.	NOUN
ejpam-4340	281	1	if	if	SCONJ
ejpam-4340	281	2	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	281	3	)	)	PUNCT
ejpam-4340	281	4	⊆	⊆	NUM
ejpam-4340	281	5	b	b	X
ejpam-4340	281	6	⊆	⊆	NUM
ejpam-4340	281	7	a	a	PRON
ejpam-4340	281	8	and	and	CCONJ
ejpam-4340	281	9	a	a	PRON
ejpam-4340	281	10	is	be	AUX
ejpam-4340	281	11	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	281	12	,	,	PUNCT
ejpam-4340	281	13	then	then	ADV
ejpam-4340	281	14	b	b	PROPN
ejpam-4340	281	15	is	be	AUX
ejpam-4340	281	16	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	281	17	.	.	PUNCT
ejpam-4340	282	1	proof	proof	NOUN
ejpam-4340	282	2	.	.	PUNCT
ejpam-4340	283	1	it	it	PRON
ejpam-4340	283	2	is	be	AUX
ejpam-4340	283	3	clear	clear	ADJ
ejpam-4340	283	4	from	from	ADP
ejpam-4340	283	5	theorem	theorem	ADJ
ejpam-4340	283	6	6	6	NUM
ejpam-4340	283	7	.	.	PUNCT
ejpam-4340	283	8	proposition	proposition	NOUN
ejpam-4340	283	9	8	8	NUM
ejpam-4340	283	10	.	.	PUNCT
ejpam-4340	284	1	let	let	VERB
ejpam-4340	284	2	a	a	DET
ejpam-4340	284	3	be	be	AUX
ejpam-4340	284	4	a	a	DET
ejpam-4340	284	5	subset	subset	NOUN
ejpam-4340	284	6	of	of	ADP
ejpam-4340	284	7	a	a	DET
ejpam-4340	284	8	space	space	NOUN
ejpam-4340	284	9	x.	x.	NOUN
ejpam-4340	284	10	if	if	SCONJ
ejpam-4340	284	11	a	a	PRON
ejpam-4340	284	12	is	be	AUX
ejpam-4340	284	13	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	284	14	,	,	PUNCT
ejpam-4340	284	15	then	then	ADV
ejpam-4340	284	16	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	284	17	)	)	PUNCT
ejpam-4340	284	18	\	\	NOUN
ejpam-4340	285	1	a	a	PRON
ejpam-4340	285	2	is	be	AUX
ejpam-4340	285	3	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	285	4	.	.	PUNCT
ejpam-4340	286	1	proof	proof	NOUN
ejpam-4340	286	2	.	.	PUNCT
ejpam-4340	287	1	it	it	PRON
ejpam-4340	287	2	is	be	AUX
ejpam-4340	287	3	clear	clear	ADJ
ejpam-4340	287	4	from	from	ADP
ejpam-4340	287	5	theorem	theorem	ADJ
ejpam-4340	287	6	4	4	NUM
ejpam-4340	287	7	.	.	NOUN
ejpam-4340	287	8	remark	remark	NOUN
ejpam-4340	287	9	2	2	NUM
ejpam-4340	287	10	.	.	PUNCT
ejpam-4340	288	1	let	let	VERB
ejpam-4340	288	2	a	a	DET
ejpam-4340	288	3	be	be	AUX
ejpam-4340	288	4	a	a	DET
ejpam-4340	288	5	subset	subset	NOUN
ejpam-4340	288	6	of	of	ADP
ejpam-4340	288	7	a	a	DET
ejpam-4340	288	8	space	space	NOUN
ejpam-4340	288	9	x.	x.	NOUN
ejpam-4340	288	10	then	then	ADV
ejpam-4340	288	11	ωe∗-int(ωe∗-cl(a	ωe∗-int(ωe∗-cl(a	NUM
ejpam-4340	288	12	)	)	PUNCT
ejpam-4340	288	13	\a	\a	NUM
ejpam-4340	288	14	)	)	PUNCT
ejpam-4340	289	1	=	=	PUNCT
ejpam-4340	289	2	∅.	∅.	NOUN
ejpam-4340	289	3	proposition	proposition	NOUN
ejpam-4340	289	4	9	9	NUM
ejpam-4340	289	5	.	.	PUNCT
ejpam-4340	290	1	let	let	VERB
ejpam-4340	290	2	a	a	PRON
ejpam-4340	290	3	and	and	CCONJ
ejpam-4340	290	4	b	b	NOUN
ejpam-4340	290	5	be	be	AUX
ejpam-4340	290	6	two	two	NUM
ejpam-4340	290	7	subsets	subset	NOUN
ejpam-4340	290	8	of	of	ADP
ejpam-4340	290	9	a	a	DET
ejpam-4340	290	10	space	space	NOUN
ejpam-4340	290	11	x.	x.	NOUN
ejpam-4340	291	1	if	if	SCONJ
ejpam-4340	291	2	a	a	DET
ejpam-4340	291	3	⊆	⊆	NUM
ejpam-4340	291	4	b	b	SYM
ejpam-4340	291	5	⊆	⊆	NUM
ejpam-4340	291	6	x	x	PUNCT
ejpam-4340	291	7	and	and	CCONJ
ejpam-4340	291	8	ωe∗-cl(a)\a	ωe∗-cl(a)\a	PROPN
ejpam-4340	291	9	is	be	AUX
ejpam-4340	291	10	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	291	11	,	,	PUNCT
ejpam-4340	291	12	then	then	ADV
ejpam-4340	291	13	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	291	14	)	)	PUNCT
ejpam-4340	291	15	\b	\b	NOUN
ejpam-4340	291	16	is	be	AUX
ejpam-4340	291	17	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	291	18	.	.	PUNCT
ejpam-4340	292	1	proof	proof	NOUN
ejpam-4340	292	2	.	.	PUNCT
ejpam-4340	293	1	let	let	VERB
ejpam-4340	293	2	f	f	PROPN
ejpam-4340	293	3	∈	∈	PROPN
ejpam-4340	293	4	c(x	c(x	NOUN
ejpam-4340	293	5	)	)	PUNCT
ejpam-4340	293	6	such	such	ADJ
ejpam-4340	293	7	that	that	SCONJ
ejpam-4340	293	8	f	f	PROPN
ejpam-4340	293	9	⊆	⊆	NUM
ejpam-4340	293	10	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	293	11	)	)	PUNCT
ejpam-4340	293	12	\b	\b	NOUN
ejpam-4340	293	13	.	.	PUNCT
ejpam-4340	294	1	a	a	DET
ejpam-4340	294	2	⊆	⊆	NUM
ejpam-4340	294	3	b	b	NOUN
ejpam-4340	294	4	⇒	⇒	X
ejpam-4340	294	5	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	294	6	)	)	PUNCT
ejpam-4340	294	7	\a	\a	VERB
ejpam-4340	294	8	⊆	⊆	NUM
ejpam-4340	294	9	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	294	10	)	)	PUNCT
ejpam-4340	294	11	\b	\b	NOUN
ejpam-4340	294	12	)	)	PUNCT
ejpam-4340	294	13	(	(	PUNCT
ejpam-4340	294	14	f	f	PROPN
ejpam-4340	294	15	∈	∈	PROPN
ejpam-4340	294	16	c(x))(f	c(x))(f	VERB
ejpam-4340	294	17	⊆	⊆	NUM
ejpam-4340	294	18	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	294	19	)	)	PUNCT
ejpam-4340	294	20	\b	\b	ADJ
ejpam-4340	294	21	)	)	PUNCT
ejpam-4340	294	22	}	}	PUNCT
ejpam-4340	294	23	⇒	⇒	VERB
ejpam-4340	294	24	⇒	⇒	NOUN
ejpam-4340	294	25	(	(	PUNCT
ejpam-4340	294	26	f	f	PROPN
ejpam-4340	294	27	∈	∈	PROPN
ejpam-4340	294	28	c(x))(f	c(x))(f	VERB
ejpam-4340	294	29	⊆	⊆	NUM
ejpam-4340	294	30	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	294	31	)	)	PUNCT
ejpam-4340	294	32	\a	\a	VERB
ejpam-4340	295	1	⊆	⊆	NUM
ejpam-4340	295	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	295	3	)	)	PUNCT
ejpam-4340	295	4	\b	\b	ADJ
ejpam-4340	295	5	)	)	PUNCT
ejpam-4340	295	6	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	295	7	)	)	PUNCT
ejpam-4340	295	8	\a	\a	VERB
ejpam-4340	295	9	∈	∈	PROPN
ejpam-4340	295	10	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	295	11	)	)	PUNCT
ejpam-4340	295	12	}	}	PUNCT
ejpam-4340	295	13	corollary	corollary	ADJ
ejpam-4340	295	14	2⇒	2⇒	PROPN
ejpam-4340	295	15	⇒	⇒	NOUN
ejpam-4340	295	16	f	f	PROPN
ejpam-4340	295	17	⊆	⊆	NUM
ejpam-4340	295	18	ωe∗-int(ωe∗-cl(a	ωe∗-int(ωe∗-cl(a	NUM
ejpam-4340	295	19	)	)	PUNCT
ejpam-4340	295	20	\a	\a	NUM
ejpam-4340	295	21	)	)	PUNCT
ejpam-4340	296	1	⊆	⊆	NUM
ejpam-4340	296	2	ωe∗-int(ωe∗-cl(a	ωe∗-int(ωe∗-cl(a	NUM
ejpam-4340	296	3	)	)	PUNCT
ejpam-4340	296	4	\b	\b	NUM
ejpam-4340	296	5	)	)	PUNCT
ejpam-4340	296	6	.	.	PUNCT
ejpam-4340	297	1	proposition	proposition	NOUN
ejpam-4340	297	2	10	10	NUM
ejpam-4340	297	3	.	.	PUNCT
ejpam-4340	298	1	let	let	VERB
ejpam-4340	298	2	a	a	DET
ejpam-4340	298	3	be	be	AUX
ejpam-4340	298	4	a	a	DET
ejpam-4340	298	5	subset	subset	NOUN
ejpam-4340	298	6	of	of	ADP
ejpam-4340	298	7	a	a	DET
ejpam-4340	298	8	space	space	NOUN
ejpam-4340	298	9	x.	x.	NOUN
ejpam-4340	299	1	if	if	SCONJ
ejpam-4340	299	2	a	a	PRON
ejpam-4340	299	3	is	be	AUX
ejpam-4340	299	4	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	299	5	,	,	PUNCT
ejpam-4340	299	6	then	then	ADV
ejpam-4340	299	7	u	u	NOUN
ejpam-4340	299	8	=	=	NOUN
ejpam-4340	299	9	x	x	INTJ
ejpam-4340	299	10	whenever	whenever	SCONJ
ejpam-4340	299	11	u	u	NOUN
ejpam-4340	299	12	is	be	AUX
ejpam-4340	299	13	open	open	ADJ
ejpam-4340	299	14	in	in	ADP
ejpam-4340	299	15	x	x	PUNCT
ejpam-4340	299	16	and	and	CCONJ
ejpam-4340	299	17	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	299	18	)	)	PUNCT
ejpam-4340	299	19	∪	∪	NOUN
ejpam-4340	299	20	(	(	PUNCT
ejpam-4340	299	21	x	x	NOUN
ejpam-4340	299	22	\a	\a	NUM
ejpam-4340	299	23	)	)	PUNCT
ejpam-4340	299	24	⊆	⊆	NUM
ejpam-4340	299	25	u.	u.	NOUN
ejpam-4340	299	26	proof	proof	NOUN
ejpam-4340	299	27	.	.	PUNCT
ejpam-4340	300	1	let	let	VERB
ejpam-4340	300	2	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	300	3	)	)	PUNCT
ejpam-4340	300	4	∪	∪	NOUN
ejpam-4340	300	5	(	(	PUNCT
ejpam-4340	300	6	x	x	NOUN
ejpam-4340	300	7	\a	\a	NUM
ejpam-4340	300	8	)	)	PUNCT
ejpam-4340	300	9	⊆	⊆	NUM
ejpam-4340	300	10	u	u	NOUN
ejpam-4340	300	11	∈	∈	PROPN
ejpam-4340	300	12	o(x	o(x	PROPN
ejpam-4340	300	13	)	)	PUNCT
ejpam-4340	300	14	.	.	PUNCT
ejpam-4340	301	1	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	301	2	)	)	PUNCT
ejpam-4340	302	1	∪	∪	NOUN
ejpam-4340	302	2	(	(	PUNCT
ejpam-4340	302	3	x	x	NOUN
ejpam-4340	302	4	\a	\a	NUM
ejpam-4340	302	5	)	)	PUNCT
ejpam-4340	302	6	⊆	⊆	NUM
ejpam-4340	302	7	u	u	NOUN
ejpam-4340	302	8	∈	∈	PROPN
ejpam-4340	302	9	o(x)⇒	o(x)⇒	PROPN
ejpam-4340	302	10	(	(	PUNCT
ejpam-4340	302	11	x	x	SYM
ejpam-4340	302	12	\	\	NOUN
ejpam-4340	302	13	u	u	PROPN
ejpam-4340	302	14	∈	∈	PROPN
ejpam-4340	302	15	c(x))(x	c(x))(x	PROPN
ejpam-4340	302	16	\	\	NOUN
ejpam-4340	302	17	u	u	NOUN
ejpam-4340	302	18	⊆	⊆	NUM
ejpam-4340	302	19	x	x	SYM
ejpam-4340	302	20	\	\	X
ejpam-4340	303	1	[	[	X
ejpam-4340	303	2	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	303	3	)	)	PUNCT
ejpam-4340	303	4	∪	∪	NOUN
ejpam-4340	303	5	(	(	PUNCT
ejpam-4340	303	6	x	x	NOUN
ejpam-4340	303	7	\a	\a	NUM
ejpam-4340	303	8	)	)	PUNCT
ejpam-4340	303	9	]	]	PUNCT
ejpam-4340	303	10	⇒	⇒	NOUN
ejpam-4340	303	11	(	(	PUNCT
ejpam-4340	303	12	x	x	SYM
ejpam-4340	303	13	\	\	NOUN
ejpam-4340	303	14	u	u	PROPN
ejpam-4340	303	15	∈	∈	PROPN
ejpam-4340	303	16	c(x))(x	c(x))(x	PROPN
ejpam-4340	303	17	\	\	NOUN
ejpam-4340	303	18	u	u	NOUN
ejpam-4340	303	19	⊆	⊆	NUM
ejpam-4340	303	20	x	x	SYM
ejpam-4340	303	21	\	\	X
ejpam-4340	304	1	[	[	X
ejpam-4340	304	2	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	304	3	)	)	PUNCT
ejpam-4340	304	4	∪	∪	NOUN
ejpam-4340	304	5	(	(	PUNCT
ejpam-4340	304	6	x	x	NOUN
ejpam-4340	304	7	\a	\a	NUM
ejpam-4340	304	8	)	)	PUNCT
ejpam-4340	304	9	]	]	PUNCT
ejpam-4340	304	10	=	=	PUNCT
ejpam-4340	304	11	ωe∗-cl(x	ωe∗-cl(x	NUM
ejpam-4340	304	12	\a	\a	NUM
ejpam-4340	304	13	)	)	PUNCT
ejpam-4340	304	14	\	\	PUNCT
ejpam-4340	305	1	(	(	PUNCT
ejpam-4340	305	2	x	x	SYM
ejpam-4340	305	3	\a	\a	NUM
ejpam-4340	305	4	)	)	PUNCT
ejpam-4340	305	5	a	a	DET
ejpam-4340	305	6	∈	∈	PROPN
ejpam-4340	305	7	gωe∗o(x)⇒	gωe∗o(x)⇒	NOUN
ejpam-4340	305	8	x	x	PUNCT
ejpam-4340	305	9	\a	\a	ADJ
ejpam-4340	305	10	∈	∈	PROPN
ejpam-4340	305	11	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	305	12	)	)	PUNCT
ejpam-4340	305	13	}	}	PUNCT
ejpam-4340	305	14	⇒	⇒	NOUN
ejpam-4340	305	15	theorem	theorem	VERB
ejpam-4340	305	16	3⇒	3⇒	NUM
ejpam-4340	305	17	x	x	SYM
ejpam-4340	305	18	\	\	NOUN
ejpam-4340	305	19	u	u	NOUN
ejpam-4340	305	20	=	=	NOUN
ejpam-4340	305	21	∅	∅	NOUN
ejpam-4340	305	22	⇒	⇒	NOUN
ejpam-4340	305	23	x	x	X
ejpam-4340	305	24	⊆	⊆	NUM
ejpam-4340	305	25	u	u	NOUN
ejpam-4340	305	26	u	u	NOUN
ejpam-4340	305	27	⊆	⊆	NUM
ejpam-4340	305	28	x	x	SYM
ejpam-4340	305	29	}	}	PUNCT
ejpam-4340	305	30	⇒	⇒	VERB
ejpam-4340	305	31	u	u	NOUN
ejpam-4340	305	32	=	=	PROPN
ejpam-4340	305	33	x.	x.	PROPN
ejpam-4340	305	34	p.	p.	NOUN
ejpam-4340	305	35	şaşmaz	şaşmaz	NUM
ejpam-4340	305	36	,	,	PUNCT
ejpam-4340	305	37	m.	m.	NOUN
ejpam-4340	305	38	özkoç	özkoç	PROPN
ejpam-4340	305	39	/	/	SYM
ejpam-4340	305	40	eur	eur	PROPN
ejpam-4340	305	41	.	.	PUNCT
ejpam-4340	306	1	j.	j.	PROPN
ejpam-4340	306	2	pure	pure	PROPN
ejpam-4340	306	3	appl	appl	PROPN
ejpam-4340	306	4	.	.	PROPN
ejpam-4340	306	5	math	math	PROPN
ejpam-4340	306	6	,	,	PUNCT
ejpam-4340	306	7	15	15	NUM
ejpam-4340	306	8	(	(	PUNCT
ejpam-4340	306	9	2	2	NUM
ejpam-4340	306	10	)	)	PUNCT
ejpam-4340	306	11	(	(	PUNCT
ejpam-4340	306	12	2022	2022	NUM
ejpam-4340	306	13	)	)	PUNCT
ejpam-4340	306	14	,	,	PUNCT
ejpam-4340	306	15	354	354	NUM
ejpam-4340	306	16	-	-	SYM
ejpam-4340	306	17	374	374	NUM
ejpam-4340	306	18	364	364	NUM
ejpam-4340	306	19	theorem	theorem	NOUN
ejpam-4340	306	20	11	11	NUM
ejpam-4340	306	21	.	.	PUNCT
ejpam-4340	307	1	let	let	VERB
ejpam-4340	307	2	a	a	PRON
ejpam-4340	307	3	and	and	CCONJ
ejpam-4340	307	4	b	b	NOUN
ejpam-4340	307	5	be	be	AUX
ejpam-4340	307	6	two	two	NUM
ejpam-4340	307	7	subsets	subset	NOUN
ejpam-4340	307	8	of	of	ADP
ejpam-4340	307	9	a	a	DET
ejpam-4340	307	10	space	space	NOUN
ejpam-4340	307	11	x.	x.	NOUN
ejpam-4340	307	12	then	then	ADV
ejpam-4340	307	13	the	the	DET
ejpam-4340	307	14	following	follow	VERB
ejpam-4340	307	15	properties	property	NOUN
ejpam-4340	307	16	hold	hold	VERB
ejpam-4340	307	17	:	:	PUNCT
ejpam-4340	307	18	(	(	PUNCT
ejpam-4340	307	19	a	a	X
ejpam-4340	307	20	)	)	PUNCT
ejpam-4340	307	21	if	if	SCONJ
ejpam-4340	307	22	a	a	PRON
ejpam-4340	307	23	is	be	AUX
ejpam-4340	307	24	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	307	25	and	and	CCONJ
ejpam-4340	307	26	b	b	NOUN
ejpam-4340	307	27	is	be	AUX
ejpam-4340	307	28	ωa	ωa	ADV
ejpam-4340	307	29	-	-	PUNCT
ejpam-4340	307	30	open	open	ADJ
ejpam-4340	307	31	,	,	PUNCT
ejpam-4340	307	32	then	then	ADV
ejpam-4340	307	33	a	a	DET
ejpam-4340	307	34	∩b	∩b	NOUN
ejpam-4340	307	35	is	be	AUX
ejpam-4340	307	36	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	307	37	,	,	PUNCT
ejpam-4340	307	38	(	(	PUNCT
ejpam-4340	307	39	b	b	X
ejpam-4340	307	40	)	)	PUNCT
ejpam-4340	307	41	if	if	SCONJ
ejpam-4340	307	42	b	b	PROPN
ejpam-4340	307	43	is	be	AUX
ejpam-4340	307	44	gωe∗-open	gωe∗-open	NOUN
ejpam-4340	307	45	and	and	CCONJ
ejpam-4340	307	46	ωe∗-int(b	ωe∗-int(b	NOUN
ejpam-4340	307	47	)	)	PUNCT
ejpam-4340	307	48	⊆	⊆	NUM
ejpam-4340	307	49	a	a	PRON
ejpam-4340	307	50	,	,	PUNCT
ejpam-4340	307	51	then	then	ADV
ejpam-4340	307	52	a	a	DET
ejpam-4340	307	53	∩b	∩b	NOUN
ejpam-4340	307	54	is	be	AUX
ejpam-4340	307	55	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	307	56	.	.	PUNCT
ejpam-4340	308	1	proof	proof	NOUN
ejpam-4340	308	2	.	.	PUNCT
ejpam-4340	309	1	(	(	PUNCT
ejpam-4340	309	2	a	a	X
ejpam-4340	309	3	)	)	PUNCT
ejpam-4340	309	4	let	let	VERB
ejpam-4340	309	5	f	f	PROPN
ejpam-4340	309	6	∈	∈	PROPN
ejpam-4340	309	7	c(x	c(x	NOUN
ejpam-4340	309	8	)	)	PUNCT
ejpam-4340	309	9	such	such	ADJ
ejpam-4340	309	10	that	that	SCONJ
ejpam-4340	309	11	f	f	PROPN
ejpam-4340	309	12	⊆	⊆	NUM
ejpam-4340	309	13	a	a	DET
ejpam-4340	309	14	∩b	∩b	NOUN
ejpam-4340	309	15	.	.	PUNCT
ejpam-4340	310	1	(	(	PUNCT
ejpam-4340	310	2	f	f	PROPN
ejpam-4340	310	3	∈	∈	PROPN
ejpam-4340	310	4	c(x))(f	c(x))(f	VERB
ejpam-4340	310	5	⊆	⊆	NUM
ejpam-4340	310	6	a	a	PRON
ejpam-4340	310	7	∩b)⇒	∩b)⇒	NOUN
ejpam-4340	311	1	(	(	PUNCT
ejpam-4340	312	1	f	f	PROPN
ejpam-4340	312	2	∈	∈	PROPN
ejpam-4340	312	3	c(x))(f	c(x))(f	VERB
ejpam-4340	312	4	⊆	⊆	NUM
ejpam-4340	312	5	a	a	DET
ejpam-4340	312	6	∩b	∩b	NOUN
ejpam-4340	312	7	⊆	⊆	NUM
ejpam-4340	312	8	a	a	NOUN
ejpam-4340	312	9	)	)	PUNCT
ejpam-4340	312	10	a	a	DET
ejpam-4340	312	11	∈	∈	NOUN
ejpam-4340	312	12	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	312	13	)	)	PUNCT
ejpam-4340	312	14	}	}	PUNCT
ejpam-4340	312	15	corollary	corollary	ADJ
ejpam-4340	312	16	2⇒	2⇒	PROPN
ejpam-4340	312	17	⇒	⇒	NOUN
ejpam-4340	312	18	f	f	PROPN
ejpam-4340	312	19	⊆	⊆	NUM
ejpam-4340	312	20	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	312	21	)	)	PUNCT
ejpam-4340	313	1	b	b	NOUN
ejpam-4340	313	2	∈	∈	PROPN
ejpam-4340	313	3	ωao(x	ωao(x	NUM
ejpam-4340	313	4	)	)	PUNCT
ejpam-4340	313	5	}	}	PUNCT
ejpam-4340	313	6	⇒	⇒	VERB
ejpam-4340	313	7	f	f	NOUN
ejpam-4340	313	8	=	=	SYM
ejpam-4340	313	9	f	f	PROPN
ejpam-4340	313	10	∩b	∩b	NOUN
ejpam-4340	313	11	⊆	⊆	NUM
ejpam-4340	313	12	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	313	13	)	)	PUNCT
ejpam-4340	313	14	∩b	∩b	NOUN
ejpam-4340	313	15	=	=	NOUN
ejpam-4340	313	16	ωe∗-int(a	ωe∗-int(a	NOUN
ejpam-4340	313	17	∩b	∩b	NOUN
ejpam-4340	313	18	)	)	PUNCT
ejpam-4340	313	19	.	.	PUNCT
ejpam-4340	314	1	(	(	PUNCT
ejpam-4340	314	2	b	b	X
ejpam-4340	314	3	)	)	PUNCT
ejpam-4340	314	4	let	let	VERB
ejpam-4340	314	5	b	b	NOUN
ejpam-4340	314	6	∈	∈	PROPN
ejpam-4340	314	7	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	314	8	)	)	PUNCT
ejpam-4340	314	9	and	and	CCONJ
ejpam-4340	314	10	ωe∗-int(b	ωe∗-int(b	X
ejpam-4340	314	11	)	)	PUNCT
ejpam-4340	314	12	⊆	⊆	NUM
ejpam-4340	314	13	a.	a.	NOUN
ejpam-4340	314	14	ωe∗-int(b	ωe∗-int(b	NOUN
ejpam-4340	314	15	)	)	PUNCT
ejpam-4340	314	16	⊆	⊆	NUM
ejpam-4340	314	17	a⇒	a⇒	PROPN
ejpam-4340	314	18	b	b	PROPN
ejpam-4340	314	19	∩	∩	ADJ
ejpam-4340	314	20	ωe∗-int(b	ωe∗-int(b	NOUN
ejpam-4340	314	21	)	)	PUNCT
ejpam-4340	314	22	⊆	⊆	NUM
ejpam-4340	314	23	a	a	DET
ejpam-4340	314	24	∩b	∩b	NOUN
ejpam-4340	314	25	⊆	⊆	NUM
ejpam-4340	314	26	b	b	SYM
ejpam-4340	314	27	b	b	PROPN
ejpam-4340	314	28	∈	∈	PROPN
ejpam-4340	314	29	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	314	30	)	)	PUNCT
ejpam-4340	314	31	}	}	PUNCT
ejpam-4340	314	32	proposition	proposition	NOUN
ejpam-4340	314	33	7⇒	7⇒	NOUN
ejpam-4340	314	34	a	a	DET
ejpam-4340	314	35	∩b	∩b	NOUN
ejpam-4340	314	36	∈	∈	NOUN
ejpam-4340	314	37	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	314	38	)	)	PUNCT
ejpam-4340	314	39	.	.	PUNCT
ejpam-4340	315	1	definition	definition	NOUN
ejpam-4340	315	2	14	14	NUM
ejpam-4340	315	3	.	.	PUNCT
ejpam-4340	316	1	let	let	VERB
ejpam-4340	316	2	x	x	PRON
ejpam-4340	316	3	be	be	AUX
ejpam-4340	316	4	a	a	DET
ejpam-4340	316	5	space	space	NOUN
ejpam-4340	316	6	and	and	CCONJ
ejpam-4340	316	7	x	x	PUNCT
ejpam-4340	316	8	∈	∈	PROPN
ejpam-4340	316	9	x.	x.	NOUN
ejpam-4340	316	10	a	a	DET
ejpam-4340	316	11	subset	subset	NOUN
ejpam-4340	316	12	n	n	NOUN
ejpam-4340	316	13	of	of	ADP
ejpam-4340	316	14	x	x	AUX
ejpam-4340	316	15	is	be	AUX
ejpam-4340	316	16	called	call	VERB
ejpam-4340	316	17	a	a	DET
ejpam-4340	316	18	gωe∗neighborhood	gωe∗neighborhood	NOUN
ejpam-4340	316	19	of	of	ADP
ejpam-4340	316	20	x	x	PRON
ejpam-4340	316	21	if	if	SCONJ
ejpam-4340	316	22	there	there	PRON
ejpam-4340	316	23	exists	exist	VERB
ejpam-4340	316	24	a	a	DET
ejpam-4340	316	25	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	316	26	set	set	VERB
ejpam-4340	316	27	u	u	PRON
ejpam-4340	316	28	such	such	ADJ
ejpam-4340	316	29	that	that	SCONJ
ejpam-4340	316	30	x	x	SYM
ejpam-4340	316	31	∈	∈	PROPN
ejpam-4340	316	32	u	u	NOUN
ejpam-4340	316	33	⊆	⊆	NUM
ejpam-4340	316	34	n.	n.	NOUN
ejpam-4340	316	35	the	the	DET
ejpam-4340	316	36	set	set	NOUN
ejpam-4340	316	37	of	of	ADP
ejpam-4340	316	38	all	all	DET
ejpam-4340	316	39	gωe∗-neighborhoods	gωe∗-neighborhood	NOUN
ejpam-4340	316	40	of	of	ADP
ejpam-4340	316	41	x	x	SYM
ejpam-4340	316	42	is	be	AUX
ejpam-4340	316	43	called	call	VERB
ejpam-4340	316	44	the	the	DET
ejpam-4340	316	45	gωe∗-neighborhood	gωe∗-neighborhood	PROPN
ejpam-4340	316	46	system	system	NOUN
ejpam-4340	316	47	at	at	ADP
ejpam-4340	316	48	x	x	X
ejpam-4340	316	49	,	,	PUNCT
ejpam-4340	316	50	and	and	CCONJ
ejpam-4340	316	51	is	be	AUX
ejpam-4340	316	52	denoted	denote	VERB
ejpam-4340	316	53	by	by	ADP
ejpam-4340	316	54	ngωe∗(x	ngωe∗(x	NOUN
ejpam-4340	316	55	)	)	PUNCT
ejpam-4340	316	56	.	.	PUNCT
ejpam-4340	317	1	definition	definition	NOUN
ejpam-4340	317	2	15	15	NUM
ejpam-4340	317	3	.	.	PUNCT
ejpam-4340	318	1	let	let	VERB
ejpam-4340	318	2	x	x	PRON
ejpam-4340	318	3	be	be	AUX
ejpam-4340	318	4	a	a	DET
ejpam-4340	318	5	space	space	NOUN
ejpam-4340	318	6	and	and	CCONJ
ejpam-4340	318	7	a	a	DET
ejpam-4340	318	8	⊆	⊆	NUM
ejpam-4340	318	9	x.	x.	NOUN
ejpam-4340	318	10	a	a	DET
ejpam-4340	318	11	subset	subset	NOUN
ejpam-4340	318	12	n	n	NOUN
ejpam-4340	318	13	of	of	ADP
ejpam-4340	318	14	x	x	PROPN
ejpam-4340	318	15	is	be	AUX
ejpam-4340	318	16	called	call	VERB
ejpam-4340	318	17	a	a	DET
ejpam-4340	318	18	gωe∗neighborhood	gωe∗neighborhood	NOUN
ejpam-4340	318	19	of	of	ADP
ejpam-4340	318	20	a	a	PRON
ejpam-4340	318	21	if	if	SCONJ
ejpam-4340	318	22	there	there	PRON
ejpam-4340	318	23	exists	exist	VERB
ejpam-4340	318	24	a	a	DET
ejpam-4340	318	25	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	318	26	set	set	VERB
ejpam-4340	318	27	u	u	PRON
ejpam-4340	318	28	such	such	ADJ
ejpam-4340	318	29	that	that	SCONJ
ejpam-4340	318	30	a	a	DET
ejpam-4340	318	31	⊆	⊆	NUM
ejpam-4340	318	32	u	u	NOUN
ejpam-4340	318	33	⊆	⊆	NUM
ejpam-4340	318	34	n.	n.	NOUN
ejpam-4340	318	35	corollary	corollary	NOUN
ejpam-4340	318	36	3	3	X
ejpam-4340	318	37	.	.	PUNCT
ejpam-4340	319	1	let	let	VERB
ejpam-4340	319	2	x	x	PRON
ejpam-4340	319	3	be	be	AUX
ejpam-4340	319	4	a	a	DET
ejpam-4340	319	5	space	space	NOUN
ejpam-4340	319	6	and	and	CCONJ
ejpam-4340	319	7	x	x	PUNCT
ejpam-4340	319	8	∈	∈	NOUN
ejpam-4340	319	9	x.	x.	NOUN
ejpam-4340	320	1	every	every	DET
ejpam-4340	320	2	neighborhood	neighborhood	NOUN
ejpam-4340	320	3	n	n	NOUN
ejpam-4340	320	4	of	of	ADP
ejpam-4340	320	5	x	x	PUNCT
ejpam-4340	320	6	is	be	AUX
ejpam-4340	320	7	a	a	DET
ejpam-4340	320	8	gωe∗neighborhood	gωe∗neighborhood	NOUN
ejpam-4340	320	9	of	of	ADP
ejpam-4340	320	10	x.	x.	NOUN
ejpam-4340	320	11	remark	remark	PROPN
ejpam-4340	320	12	3	3	NUM
ejpam-4340	320	13	.	.	PUNCT
ejpam-4340	321	1	a	a	DET
ejpam-4340	321	2	gωe∗-neighborhood	gωe∗-neighborhood	PROPN
ejpam-4340	321	3	n	n	PROPN
ejpam-4340	321	4	of	of	ADP
ejpam-4340	321	5	x	x	PUNCT
ejpam-4340	321	6	in	in	ADP
ejpam-4340	321	7	a	a	DET
ejpam-4340	321	8	space	space	NOUN
ejpam-4340	321	9	x	x	AUX
ejpam-4340	321	10	need	need	AUX
ejpam-4340	321	11	not	not	PART
ejpam-4340	321	12	be	be	AUX
ejpam-4340	321	13	a	a	DET
ejpam-4340	321	14	neighborhood	neighborhood	NOUN
ejpam-4340	321	15	of	of	ADP
ejpam-4340	321	16	x	x	PART
ejpam-4340	321	17	as	as	SCONJ
ejpam-4340	321	18	shown	show	VERB
ejpam-4340	321	19	by	by	ADP
ejpam-4340	321	20	the	the	DET
ejpam-4340	321	21	following	follow	VERB
ejpam-4340	321	22	example	example	NOUN
ejpam-4340	321	23	.	.	PUNCT
ejpam-4340	322	1	example	example	NOUN
ejpam-4340	323	1	3	3	X
ejpam-4340	323	2	.	.	PUNCT
ejpam-4340	323	3	let	let	VERB
ejpam-4340	323	4	x	x	PUNCT
ejpam-4340	323	5	=	=	PRON
ejpam-4340	323	6	{	{	PUNCT
ejpam-4340	323	7	a	a	PRON
ejpam-4340	323	8	,	,	PUNCT
ejpam-4340	323	9	b	b	NOUN
ejpam-4340	323	10	,	,	PUNCT
ejpam-4340	323	11	c	c	NOUN
ejpam-4340	323	12	,	,	PUNCT
ejpam-4340	323	13	d	d	NOUN
ejpam-4340	323	14	}	}	PUNCT
ejpam-4340	323	15	with	with	ADP
ejpam-4340	323	16	a	a	DET
ejpam-4340	323	17	topology	topology	NOUN
ejpam-4340	323	18	τ	τ	X
ejpam-4340	323	19	=	=	SYM
ejpam-4340	323	20	{	{	PUNCT
ejpam-4340	323	21	∅	∅	NOUN
ejpam-4340	323	22	,	,	PUNCT
ejpam-4340	323	23	x	x	X
ejpam-4340	323	24	,	,	PUNCT
ejpam-4340	323	25	{	{	PUNCT
ejpam-4340	323	26	a	a	X
ejpam-4340	323	27	}	}	PUNCT
ejpam-4340	323	28	,	,	PUNCT
ejpam-4340	323	29	{	{	PUNCT
ejpam-4340	323	30	b	b	NOUN
ejpam-4340	323	31	}	}	PUNCT
ejpam-4340	323	32	,	,	PUNCT
ejpam-4340	323	33	{	{	PUNCT
ejpam-4340	323	34	a	a	DET
ejpam-4340	323	35	,	,	PUNCT
ejpam-4340	323	36	b	b	NOUN
ejpam-4340	323	37	}	}	PUNCT
ejpam-4340	323	38	,	,	PUNCT
ejpam-4340	323	39	{	{	PUNCT
ejpam-4340	323	40	a	a	PRON
ejpam-4340	323	41	,	,	PUNCT
ejpam-4340	323	42	b	b	NOUN
ejpam-4340	323	43	,	,	PUNCT
ejpam-4340	323	44	c	c	NOUN
ejpam-4340	323	45	}	}	PUNCT
ejpam-4340	323	46	}	}	PUNCT
ejpam-4340	323	47	and	and	CCONJ
ejpam-4340	323	48	a	a	DET
ejpam-4340	323	49	=	=	X
ejpam-4340	323	50	{	{	PUNCT
ejpam-4340	323	51	a	a	X
ejpam-4340	323	52	,	,	PUNCT
ejpam-4340	323	53	c	c	NOUN
ejpam-4340	323	54	}	}	PUNCT
ejpam-4340	323	55	.	.	PUNCT
ejpam-4340	324	1	since	since	SCONJ
ejpam-4340	324	2	x	x	PRON
ejpam-4340	324	3	is	be	AUX
ejpam-4340	324	4	countable	countable	ADJ
ejpam-4340	324	5	,	,	PUNCT
ejpam-4340	324	6	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	324	7	)	)	PUNCT
ejpam-4340	324	8	=	=	SYM
ejpam-4340	324	9	2x	2x	NOUN
ejpam-4340	324	10	.	.	PUNCT
ejpam-4340	325	1	then	then	ADV
ejpam-4340	325	2	,	,	PUNCT
ejpam-4340	325	3	a	a	PRON
ejpam-4340	325	4	is	be	AUX
ejpam-4340	325	5	a	a	DET
ejpam-4340	325	6	gωe∗-neighborhood	gωe∗-neighborhood	NOUN
ejpam-4340	325	7	of	of	ADP
ejpam-4340	325	8	the	the	DET
ejpam-4340	325	9	point	point	NOUN
ejpam-4340	325	10	c	c	NOUN
ejpam-4340	325	11	,	,	PUNCT
ejpam-4340	325	12	since	since	SCONJ
ejpam-4340	325	13	{	{	PUNCT
ejpam-4340	325	14	c	c	X
ejpam-4340	325	15	}	}	PUNCT
ejpam-4340	325	16	is	be	AUX
ejpam-4340	325	17	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	325	18	set	set	VERB
ejpam-4340	325	19	such	such	ADJ
ejpam-4340	325	20	that	that	SCONJ
ejpam-4340	325	21	c	c	PROPN
ejpam-4340	325	22	∈	∈	PROPN
ejpam-4340	325	23	{	{	PUNCT
ejpam-4340	325	24	c	c	NOUN
ejpam-4340	325	25	}	}	PUNCT
ejpam-4340	325	26	⊆	⊆	NUM
ejpam-4340	325	27	{	{	PUNCT
ejpam-4340	325	28	a	a	NOUN
ejpam-4340	325	29	,	,	PUNCT
ejpam-4340	325	30	c	c	NOUN
ejpam-4340	325	31	}	}	PUNCT
ejpam-4340	325	32	.	.	PUNCT
ejpam-4340	326	1	however	however	ADV
ejpam-4340	326	2	,	,	PUNCT
ejpam-4340	326	3	the	the	DET
ejpam-4340	326	4	set	set	NOUN
ejpam-4340	326	5	{	{	PUNCT
ejpam-4340	326	6	a	a	NOUN
ejpam-4340	326	7	,	,	PUNCT
ejpam-4340	326	8	c	c	NOUN
ejpam-4340	326	9	}	}	PUNCT
ejpam-4340	326	10	is	be	AUX
ejpam-4340	326	11	not	not	PART
ejpam-4340	326	12	a	a	DET
ejpam-4340	326	13	neighborhood	neighborhood	NOUN
ejpam-4340	326	14	of	of	ADP
ejpam-4340	326	15	the	the	DET
ejpam-4340	326	16	point	point	NOUN
ejpam-4340	326	17	c	c	NOUN
ejpam-4340	326	18	,	,	PUNCT
ejpam-4340	326	19	since	since	SCONJ
ejpam-4340	326	20	there	there	PRON
ejpam-4340	326	21	exists	exist	VERB
ejpam-4340	326	22	no	no	DET
ejpam-4340	326	23	open	open	ADJ
ejpam-4340	326	24	set	set	NOUN
ejpam-4340	326	25	u	u	PRON
ejpam-4340	326	26	such	such	ADJ
ejpam-4340	326	27	that	that	SCONJ
ejpam-4340	326	28	c	c	PROPN
ejpam-4340	326	29	∈	∈	PROPN
ejpam-4340	326	30	u	u	NOUN
ejpam-4340	326	31	⊆	⊆	NUM
ejpam-4340	326	32	{	{	PUNCT
ejpam-4340	326	33	a	a	NOUN
ejpam-4340	326	34	,	,	PUNCT
ejpam-4340	326	35	c	c	NOUN
ejpam-4340	326	36	}	}	PUNCT
ejpam-4340	326	37	.	.	PUNCT
ejpam-4340	327	1	theorem	theorem	NOUN
ejpam-4340	327	2	12	12	NUM
ejpam-4340	327	3	.	.	PUNCT
ejpam-4340	328	1	let	let	VERB
ejpam-4340	328	2	n	n	PRON
ejpam-4340	328	3	be	be	AUX
ejpam-4340	328	4	a	a	DET
ejpam-4340	328	5	subset	subset	NOUN
ejpam-4340	328	6	of	of	ADP
ejpam-4340	328	7	a	a	DET
ejpam-4340	328	8	space	space	NOUN
ejpam-4340	328	9	x	x	PUNCT
ejpam-4340	328	10	and	and	CCONJ
ejpam-4340	328	11	x	x	SYM
ejpam-4340	328	12	∈	∈	PROPN
ejpam-4340	328	13	x.	x.	NOUN
ejpam-4340	329	1	if	if	SCONJ
ejpam-4340	329	2	n	n	PROPN
ejpam-4340	329	3	is	be	AUX
ejpam-4340	329	4	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	329	5	,	,	PUNCT
ejpam-4340	329	6	then	then	ADV
ejpam-4340	329	7	n	n	PRON
ejpam-4340	329	8	is	be	AUX
ejpam-4340	329	9	a	a	DET
ejpam-4340	329	10	gωe∗-neighborhood	gωe∗-neighborhood	PROPN
ejpam-4340	329	11	of	of	ADP
ejpam-4340	329	12	x.	x.	NOUN
ejpam-4340	329	13	proof	proof	NOUN
ejpam-4340	329	14	.	.	PUNCT
ejpam-4340	330	1	it	it	PRON
ejpam-4340	330	2	is	be	AUX
ejpam-4340	330	3	clear	clear	ADJ
ejpam-4340	330	4	.	.	PUNCT
ejpam-4340	331	1	theorem	theorem	ADJ
ejpam-4340	331	2	13	13	NUM
ejpam-4340	331	3	.	.	PUNCT
ejpam-4340	332	1	let	let	VERB
ejpam-4340	332	2	n	n	PRON
ejpam-4340	332	3	and	and	CCONJ
ejpam-4340	332	4	f	f	PROPN
ejpam-4340	332	5	be	be	AUX
ejpam-4340	332	6	two	two	NUM
ejpam-4340	332	7	subsets	subset	NOUN
ejpam-4340	332	8	of	of	ADP
ejpam-4340	332	9	a	a	DET
ejpam-4340	332	10	space	space	NOUN
ejpam-4340	332	11	x	x	PUNCT
ejpam-4340	332	12	and	and	CCONJ
ejpam-4340	332	13	x	x	SYM
ejpam-4340	332	14	∈	∈	PROPN
ejpam-4340	332	15	x.	x.	NOUN
ejpam-4340	333	1	if	if	SCONJ
ejpam-4340	333	2	f	f	PROPN
ejpam-4340	333	3	is	be	AUX
ejpam-4340	333	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	333	5	and	and	CCONJ
ejpam-4340	333	6	x	x	PART
ejpam-4340	333	7	∈	∈	NOUN
ejpam-4340	333	8	x	x	PUNCT
ejpam-4340	333	9	\	\	PROPN
ejpam-4340	333	10	f	f	PROPN
ejpam-4340	333	11	,	,	PUNCT
ejpam-4340	333	12	then	then	ADV
ejpam-4340	333	13	there	there	PRON
ejpam-4340	333	14	exists	exist	VERB
ejpam-4340	333	15	a	a	DET
ejpam-4340	333	16	gωe∗-neighborhood	gωe∗-neighborhood	PROPN
ejpam-4340	333	17	n	n	PROPN
ejpam-4340	333	18	of	of	ADP
ejpam-4340	333	19	x	x	PUNCT
ejpam-4340	333	20	such	such	ADJ
ejpam-4340	333	21	that	that	SCONJ
ejpam-4340	333	22	n	n	NOUN
ejpam-4340	333	23	∩	∩	ADJ
ejpam-4340	333	24	f	f	X
ejpam-4340	333	25	=	=	PUNCT
ejpam-4340	333	26	∅.	∅.	NOUN
ejpam-4340	333	27	proof	proof	NOUN
ejpam-4340	333	28	.	.	PUNCT
ejpam-4340	334	1	it	it	PRON
ejpam-4340	334	2	is	be	AUX
ejpam-4340	334	3	clear	clear	ADJ
ejpam-4340	334	4	.	.	PUNCT
ejpam-4340	335	1	p.	p.	NOUN
ejpam-4340	335	2	şaşmaz	şaşmaz	NUM
ejpam-4340	335	3	,	,	PUNCT
ejpam-4340	335	4	m.	m.	NOUN
ejpam-4340	335	5	özkoç	özkoç	PROPN
ejpam-4340	335	6	/	/	SYM
ejpam-4340	335	7	eur	eur	PROPN
ejpam-4340	335	8	.	.	PUNCT
ejpam-4340	336	1	j.	j.	PROPN
ejpam-4340	336	2	pure	pure	PROPN
ejpam-4340	336	3	appl	appl	PROPN
ejpam-4340	336	4	.	.	PROPN
ejpam-4340	336	5	math	math	PROPN
ejpam-4340	336	6	,	,	PUNCT
ejpam-4340	336	7	15	15	NUM
ejpam-4340	336	8	(	(	PUNCT
ejpam-4340	336	9	2	2	NUM
ejpam-4340	336	10	)	)	PUNCT
ejpam-4340	336	11	(	(	PUNCT
ejpam-4340	336	12	2022	2022	NUM
ejpam-4340	336	13	)	)	PUNCT
ejpam-4340	336	14	,	,	PUNCT
ejpam-4340	336	15	354	354	NUM
ejpam-4340	336	16	-	-	SYM
ejpam-4340	336	17	374	374	NUM
ejpam-4340	336	18	365	365	NUM
ejpam-4340	336	19	theorem	theorem	NOUN
ejpam-4340	336	20	14	14	NUM
ejpam-4340	336	21	.	.	PUNCT
ejpam-4340	337	1	let	let	VERB
ejpam-4340	337	2	n	n	PRON
ejpam-4340	337	3	be	be	AUX
ejpam-4340	337	4	a	a	DET
ejpam-4340	337	5	subset	subset	NOUN
ejpam-4340	337	6	of	of	ADP
ejpam-4340	337	7	a	a	DET
ejpam-4340	337	8	space	space	NOUN
ejpam-4340	337	9	x	x	PUNCT
ejpam-4340	337	10	and	and	CCONJ
ejpam-4340	337	11	x	x	SYM
ejpam-4340	337	12	∈	∈	PROPN
ejpam-4340	337	13	x.	x.	NOUN
ejpam-4340	337	14	then	then	ADV
ejpam-4340	337	15	the	the	DET
ejpam-4340	337	16	following	follow	VERB
ejpam-4340	337	17	properties	property	NOUN
ejpam-4340	337	18	hold	hold	VERB
ejpam-4340	337	19	:	:	PUNCT
ejpam-4340	337	20	(	(	PUNCT
ejpam-4340	337	21	a	a	X
ejpam-4340	337	22	)	)	PUNCT
ejpam-4340	337	23	for	for	ADP
ejpam-4340	337	24	all	all	DET
ejpam-4340	337	25	x	x	SYM
ejpam-4340	337	26	∈	∈	PROPN
ejpam-4340	337	27	x	x	NOUN
ejpam-4340	337	28	,	,	PUNCT
ejpam-4340	337	29	ngωe∗(x	ngωe∗(x	ADJ
ejpam-4340	337	30	)	)	PUNCT
ejpam-4340	337	31	̸=	̸=	PROPN
ejpam-4340	337	32	∅	∅	NOUN
ejpam-4340	337	33	,	,	PUNCT
ejpam-4340	337	34	(	(	PUNCT
ejpam-4340	337	35	b	b	X
ejpam-4340	337	36	)	)	PUNCT
ejpam-4340	337	37	if	if	SCONJ
ejpam-4340	337	38	n	n	PRON
ejpam-4340	337	39	∈	∈	PROPN
ejpam-4340	337	40	ngωe∗(x	ngωe∗(x	NOUN
ejpam-4340	337	41	)	)	PUNCT
ejpam-4340	337	42	,	,	PUNCT
ejpam-4340	337	43	then	then	ADV
ejpam-4340	337	44	x	x	SYM
ejpam-4340	337	45	∈	∈	PROPN
ejpam-4340	337	46	n	n	CCONJ
ejpam-4340	337	47	,	,	PUNCT
ejpam-4340	337	48	(	(	PUNCT
ejpam-4340	337	49	c	c	X
ejpam-4340	337	50	)	)	PUNCT
ejpam-4340	337	51	if	if	SCONJ
ejpam-4340	337	52	n	n	PRON
ejpam-4340	337	53	∈	∈	PROPN
ejpam-4340	337	54	ngωe∗(x	ngωe∗(x	NOUN
ejpam-4340	337	55	)	)	PUNCT
ejpam-4340	337	56	and	and	CCONJ
ejpam-4340	337	57	n	n	CCONJ
ejpam-4340	337	58	⊆m	⊆m	VERB
ejpam-4340	337	59	⊆	⊆	NUM
ejpam-4340	337	60	x	x	NOUN
ejpam-4340	337	61	,	,	PUNCT
ejpam-4340	337	62	then	then	ADV
ejpam-4340	337	63	m	m	NOUN
ejpam-4340	337	64	∈	∈	NOUN
ejpam-4340	337	65	ngωe∗(x	ngωe∗(x	NOUN
ejpam-4340	337	66	)	)	PUNCT
ejpam-4340	337	67	,	,	PUNCT
ejpam-4340	337	68	(	(	PUNCT
ejpam-4340	337	69	d	d	X
ejpam-4340	337	70	)	)	PUNCT
ejpam-4340	337	71	if	if	SCONJ
ejpam-4340	337	72	n	n	PRON
ejpam-4340	337	73	∈	∈	PROPN
ejpam-4340	337	74	ngωe∗(x	ngωe∗(x	NOUN
ejpam-4340	337	75	)	)	PUNCT
ejpam-4340	337	76	,	,	PUNCT
ejpam-4340	337	77	then	then	ADV
ejpam-4340	337	78	there	there	PRON
ejpam-4340	337	79	exists	exist	VERB
ejpam-4340	337	80	m	m	VERB
ejpam-4340	337	81	∈	∈	NOUN
ejpam-4340	337	82	ngωe∗(x	ngωe∗(x	NOUN
ejpam-4340	337	83	)	)	PUNCT
ejpam-4340	337	84	such	such	ADJ
ejpam-4340	337	85	that	that	SCONJ
ejpam-4340	337	86	m	m	PROPN
ejpam-4340	337	87	⊆	⊆	NUM
ejpam-4340	337	88	n	n	NOUN
ejpam-4340	337	89	and	and	CCONJ
ejpam-4340	337	90	n	n	PRON
ejpam-4340	337	91	∈	∈	PROPN
ejpam-4340	337	92	ngωe∗(y	ngωe∗(y	PROPN
ejpam-4340	337	93	)	)	PUNCT
ejpam-4340	337	94	for	for	ADP
ejpam-4340	337	95	every	every	DET
ejpam-4340	337	96	y	y	PROPN
ejpam-4340	337	97	∈m	∈m	NOUN
ejpam-4340	337	98	.	.	PUNCT
ejpam-4340	338	1	proof	proof	NOUN
ejpam-4340	338	2	.	.	PUNCT
ejpam-4340	339	1	straightforward	straightforward	ADJ
ejpam-4340	339	2	.	.	PUNCT
ejpam-4340	340	1	definition	definition	NOUN
ejpam-4340	340	2	16	16	NUM
ejpam-4340	340	3	.	.	PUNCT
ejpam-4340	341	1	let	let	VERB
ejpam-4340	341	2	a	a	DET
ejpam-4340	341	3	be	be	AUX
ejpam-4340	341	4	a	a	DET
ejpam-4340	341	5	subset	subset	NOUN
ejpam-4340	341	6	of	of	ADP
ejpam-4340	341	7	a	a	DET
ejpam-4340	341	8	space	space	NOUN
ejpam-4340	341	9	x.	x.	NOUN
ejpam-4340	342	1	the	the	DET
ejpam-4340	342	2	intersection	intersection	NOUN
ejpam-4340	342	3	of	of	ADP
ejpam-4340	342	4	all	all	DET
ejpam-4340	342	5	generalized	generalize	VERB
ejpam-4340	342	6	ωe∗closed	ωe∗close	VERB
ejpam-4340	342	7	(	(	PUNCT
ejpam-4340	342	8	resp	resp	NOUN
ejpam-4340	342	9	.	.	PUNCT
ejpam-4340	343	1	generalized	generalize	VERB
ejpam-4340	343	2	closed	close	VERB
ejpam-4340	343	3	[	[	X
ejpam-4340	343	4	14	14	NUM
ejpam-4340	343	5	]	]	SYM
ejpam-4340	343	6	)	)	PUNCT
ejpam-4340	343	7	subsets	subset	NOUN
ejpam-4340	343	8	of	of	ADP
ejpam-4340	343	9	x	x	PUNCT
ejpam-4340	343	10	containing	contain	VERB
ejpam-4340	343	11	a	a	PRON
ejpam-4340	343	12	is	be	AUX
ejpam-4340	343	13	called	call	VERB
ejpam-4340	343	14	the	the	DET
ejpam-4340	343	15	generalized	generalized	ADJ
ejpam-4340	343	16	ωe∗-closure	ωe∗-closure	NOUN
ejpam-4340	343	17	(	(	PUNCT
ejpam-4340	343	18	resp	resp	NOUN
ejpam-4340	343	19	.	.	PUNCT
ejpam-4340	344	1	generalized	generalize	VERB
ejpam-4340	344	2	closure	closure	NOUN
ejpam-4340	344	3	[	[	X
ejpam-4340	344	4	14	14	NUM
ejpam-4340	344	5	]	]	SYM
ejpam-4340	344	6	)	)	PUNCT
ejpam-4340	344	7	of	of	ADP
ejpam-4340	344	8	a	a	PRON
ejpam-4340	344	9	and	and	CCONJ
ejpam-4340	344	10	is	be	AUX
ejpam-4340	344	11	denoted	denote	VERB
ejpam-4340	344	12	by	by	ADP
ejpam-4340	344	13	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	344	14	)	)	PUNCT
ejpam-4340	344	15	(	(	PUNCT
ejpam-4340	344	16	resp	resp	NOUN
ejpam-4340	344	17	.	.	PUNCT
ejpam-4340	345	1	g	g	NOUN
ejpam-4340	345	2	-	-	PUNCT
ejpam-4340	345	3	cl(a	cl(a	NUM
ejpam-4340	345	4	)	)	PUNCT
ejpam-4340	345	5	)	)	PUNCT
ejpam-4340	345	6	.	.	PUNCT
ejpam-4340	346	1	the	the	DET
ejpam-4340	346	2	proofs	proof	NOUN
ejpam-4340	346	3	of	of	ADP
ejpam-4340	346	4	the	the	DET
ejpam-4340	346	5	following	following	ADJ
ejpam-4340	346	6	results	result	NOUN
ejpam-4340	346	7	are	be	AUX
ejpam-4340	346	8	standard	standard	ADJ
ejpam-4340	346	9	,	,	PUNCT
ejpam-4340	346	10	hence	hence	ADV
ejpam-4340	346	11	they	they	PRON
ejpam-4340	346	12	are	be	AUX
ejpam-4340	346	13	omitted	omit	VERB
ejpam-4340	346	14	.	.	PUNCT
ejpam-4340	347	1	theorem	theorem	VERB
ejpam-4340	347	2	15	15	NUM
ejpam-4340	347	3	.	.	PUNCT
ejpam-4340	348	1	let	let	VERB
ejpam-4340	348	2	a	a	PRON
ejpam-4340	348	3	and	and	CCONJ
ejpam-4340	348	4	b	b	NOUN
ejpam-4340	348	5	be	be	AUX
ejpam-4340	348	6	subsets	subset	NOUN
ejpam-4340	348	7	of	of	ADP
ejpam-4340	348	8	a	a	DET
ejpam-4340	348	9	space	space	NOUN
ejpam-4340	348	10	x	x	PUNCT
ejpam-4340	348	11	and	and	CCONJ
ejpam-4340	348	12	x	x	SYM
ejpam-4340	348	13	∈	∈	PROPN
ejpam-4340	348	14	x.	x.	NOUN
ejpam-4340	348	15	then	then	ADV
ejpam-4340	348	16	the	the	DET
ejpam-4340	348	17	following	follow	VERB
ejpam-4340	348	18	properties	property	NOUN
ejpam-4340	348	19	hold	hold	VERB
ejpam-4340	348	20	:	:	PUNCT
ejpam-4340	348	21	(	(	PUNCT
ejpam-4340	348	22	a	a	X
ejpam-4340	348	23	)	)	PUNCT
ejpam-4340	348	24	x	x	SYM
ejpam-4340	348	25	∈	∈	PROPN
ejpam-4340	348	26	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	348	27	)	)	PUNCT
ejpam-4340	348	28	iff	iff	PROPN
ejpam-4340	348	29	v	v	ADP
ejpam-4340	348	30	∩a	∩a	PROPN
ejpam-4340	348	31	̸=	̸=	PROPN
ejpam-4340	348	32	∅	∅	NOUN
ejpam-4340	348	33	for	for	ADP
ejpam-4340	348	34	every	every	DET
ejpam-4340	348	35	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	348	36	set	set	VERB
ejpam-4340	348	37	v	v	NOUN
ejpam-4340	348	38	containing	contain	VERB
ejpam-4340	348	39	x	x	X
ejpam-4340	348	40	,	,	PUNCT
ejpam-4340	348	41	(	(	PUNCT
ejpam-4340	348	42	b	b	NOUN
ejpam-4340	348	43	)	)	PUNCT
ejpam-4340	348	44	gωe∗-cl(∅	gωe∗-cl(∅	NOUN
ejpam-4340	348	45	)	)	PUNCT
ejpam-4340	348	46	=	=	SYM
ejpam-4340	348	47	∅	∅	NOUN
ejpam-4340	348	48	and	and	CCONJ
ejpam-4340	348	49	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	348	50	)	)	PUNCT
ejpam-4340	349	1	=	=	SYM
ejpam-4340	350	1	x	x	X
ejpam-4340	350	2	,	,	PUNCT
ejpam-4340	350	3	(	(	PUNCT
ejpam-4340	350	4	c	c	X
ejpam-4340	350	5	)	)	PUNCT
ejpam-4340	350	6	if	if	SCONJ
ejpam-4340	350	7	a	a	DET
ejpam-4340	350	8	⊆	⊆	NUM
ejpam-4340	350	9	b	b	NOUN
ejpam-4340	350	10	,	,	PUNCT
ejpam-4340	350	11	then	then	ADV
ejpam-4340	350	12	gωe∗-cl(a	gωe∗-cl(a	ADJ
ejpam-4340	350	13	)	)	PUNCT
ejpam-4340	350	14	⊆	⊆	NUM
ejpam-4340	350	15	gωe∗-cl(b	gωe∗-cl(b	NOUN
ejpam-4340	350	16	)	)	PUNCT
ejpam-4340	350	17	,	,	PUNCT
ejpam-4340	350	18	(	(	PUNCT
ejpam-4340	350	19	d	d	X
ejpam-4340	350	20	)	)	PUNCT
ejpam-4340	350	21	a	a	DET
ejpam-4340	350	22	⊆	⊆	NUM
ejpam-4340	350	23	gωe∗-cl(a	gωe∗-cl(a	NOUN
ejpam-4340	350	24	)	)	PUNCT
ejpam-4340	350	25	⊆	⊆	NUM
ejpam-4340	350	26	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	350	27	)	)	PUNCT
ejpam-4340	350	28	⊆	⊆	NUM
ejpam-4340	350	29	cl(a	cl(a	NUM
ejpam-4340	350	30	)	)	PUNCT
ejpam-4340	350	31	,	,	PUNCT
ejpam-4340	350	32	(	(	PUNCT
ejpam-4340	350	33	e	e	X
ejpam-4340	350	34	)	)	PUNCT
ejpam-4340	350	35	a	a	DET
ejpam-4340	350	36	⊆	⊆	NUM
ejpam-4340	350	37	gωe∗-cl(a	gωe∗-cl(a	NOUN
ejpam-4340	350	38	)	)	PUNCT
ejpam-4340	350	39	⊆	⊆	NUM
ejpam-4340	350	40	g	g	NOUN
ejpam-4340	350	41	-	-	PUNCT
ejpam-4340	350	42	cl(a	cl(a	NUM
ejpam-4340	350	43	)	)	PUNCT
ejpam-4340	350	44	⊆	⊆	NUM
ejpam-4340	350	45	cl(a	cl(a	NUM
ejpam-4340	350	46	)	)	PUNCT
ejpam-4340	350	47	,	,	PUNCT
ejpam-4340	350	48	(	(	PUNCT
ejpam-4340	350	49	f	f	X
ejpam-4340	350	50	)	)	PUNCT
ejpam-4340	350	51	gωe∗-cl(a	gωe∗-cl(a	ADJ
ejpam-4340	350	52	)	)	PUNCT
ejpam-4340	350	53	∪	∪	PROPN
ejpam-4340	350	54	gωe∗-cl(b	gωe∗-cl(b	PROPN
ejpam-4340	350	55	)	)	PUNCT
ejpam-4340	350	56	⊆	⊆	NUM
ejpam-4340	350	57	gωe∗-cl(a	gωe∗-cl(a	NOUN
ejpam-4340	350	58	∪b	∪b	NOUN
ejpam-4340	350	59	)	)	PUNCT
ejpam-4340	350	60	,	,	PUNCT
ejpam-4340	350	61	(	(	PUNCT
ejpam-4340	350	62	g	g	NOUN
ejpam-4340	350	63	)	)	PUNCT
ejpam-4340	350	64	gωe∗-cl(a	gωe∗-cl(a	ADJ
ejpam-4340	350	65	∩b	∩b	NOUN
ejpam-4340	350	66	)	)	PUNCT
ejpam-4340	350	67	⊆	⊆	NUM
ejpam-4340	350	68	gωe∗-cl(a	gωe∗-cl(a	ADJ
ejpam-4340	350	69	)	)	PUNCT
ejpam-4340	350	70	∩	∩	ADJ
ejpam-4340	350	71	gωe∗-cl(b	gωe∗-cl(b	NOUN
ejpam-4340	350	72	)	)	PUNCT
ejpam-4340	350	73	,	,	PUNCT
ejpam-4340	350	74	(	(	PUNCT
ejpam-4340	350	75	h	h	X
ejpam-4340	350	76	)	)	PUNCT
ejpam-4340	350	77	a	a	DET
ejpam-4340	350	78	∈	∈	PROPN
ejpam-4340	350	79	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	350	80	)	)	PUNCT
ejpam-4340	350	81	if	if	SCONJ
ejpam-4340	350	82	and	and	CCONJ
ejpam-4340	350	83	only	only	ADV
ejpam-4340	350	84	if	if	SCONJ
ejpam-4340	350	85	a	a	DET
ejpam-4340	350	86	=	=	X
ejpam-4340	350	87	gωe∗-cl(a	gωe∗-cl(a	NOUN
ejpam-4340	350	88	)	)	PUNCT
ejpam-4340	350	89	,	,	PUNCT
ejpam-4340	350	90	(	(	PUNCT
ejpam-4340	350	91	i	i	NOUN
ejpam-4340	350	92	)	)	PUNCT
ejpam-4340	350	93	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	350	94	)	)	PUNCT
ejpam-4340	350	95	=	=	SYM
ejpam-4340	350	96	gωe∗-cl(gωe∗-cl(a	gωe∗-cl(gωe∗-cl(a	PROPN
ejpam-4340	350	97	)	)	PUNCT
ejpam-4340	350	98	)	)	PUNCT
ejpam-4340	350	99	,	,	PUNCT
ejpam-4340	350	100	(	(	PUNCT
ejpam-4340	350	101	j	j	NOUN
ejpam-4340	350	102	)	)	PUNCT
ejpam-4340	350	103	gωe∗-cl(a	gωe∗-cl(a	ADJ
ejpam-4340	350	104	)	)	PUNCT
ejpam-4340	350	105	∈	∈	PROPN
ejpam-4340	350	106	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	350	107	)	)	PUNCT
ejpam-4340	350	108	.	.	PUNCT
ejpam-4340	351	1	definition	definition	NOUN
ejpam-4340	351	2	17	17	NUM
ejpam-4340	351	3	.	.	PUNCT
ejpam-4340	352	1	let	let	VERB
ejpam-4340	352	2	x	x	PRON
ejpam-4340	352	3	be	be	AUX
ejpam-4340	352	4	a	a	DET
ejpam-4340	352	5	topological	topological	ADJ
ejpam-4340	352	6	space	space	NOUN
ejpam-4340	352	7	.	.	PUNCT
ejpam-4340	353	1	(	(	PUNCT
ejpam-4340	353	2	a	a	X
ejpam-4340	353	3	)	)	PUNCT
ejpam-4340	354	1	[	[	X
ejpam-4340	354	2	14	14	NUM
ejpam-4340	354	3	]	]	X
ejpam-4340	354	4	τ∗	τ∗	X
ejpam-4340	354	5	=	=	SYM
ejpam-4340	354	6	{	{	PUNCT
ejpam-4340	354	7	u	u	NOUN
ejpam-4340	354	8	⊆	⊆	NUM
ejpam-4340	354	9	x|cl∗(x	x|cl∗(x	PROPN
ejpam-4340	354	10	\	\	PROPN
ejpam-4340	354	11	u	u	NOUN
ejpam-4340	354	12	)	)	PUNCT
ejpam-4340	354	13	=	=	SYM
ejpam-4340	354	14	x	x	SYM
ejpam-4340	354	15	\	\	PROPN
ejpam-4340	354	16	u	u	NOUN
ejpam-4340	354	17	}	}	PUNCT
ejpam-4340	354	18	,	,	PUNCT
ejpam-4340	354	19	(	(	PUNCT
ejpam-4340	354	20	b	b	NOUN
ejpam-4340	354	21	)	)	PUNCT
ejpam-4340	354	22	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	354	23	=	=	PUNCT
ejpam-4340	354	24	{	{	PUNCT
ejpam-4340	354	25	v	v	ADP
ejpam-4340	354	26	⊆	⊆	NUM
ejpam-4340	354	27	x|gωe∗-cl(x	x|gωe∗-cl(x	PUNCT
ejpam-4340	354	28	\	\	PROPN
ejpam-4340	355	1	v	v	X
ejpam-4340	355	2	)	)	PUNCT
ejpam-4340	355	3	=	=	PUNCT
ejpam-4340	356	1	x	x	SYM
ejpam-4340	356	2	\	\	PROPN
ejpam-4340	356	3	v	v	X
ejpam-4340	356	4	}	}	PUNCT
ejpam-4340	356	5	.	.	PUNCT
ejpam-4340	357	1	proposition	proposition	NOUN
ejpam-4340	357	2	11	11	NUM
ejpam-4340	357	3	.	.	PUNCT
ejpam-4340	358	1	for	for	ADP
ejpam-4340	358	2	a	a	DET
ejpam-4340	358	3	subset	subset	NOUN
ejpam-4340	358	4	a	a	PRON
ejpam-4340	358	5	of	of	ADP
ejpam-4340	358	6	x	x	PRON
ejpam-4340	358	7	,	,	PUNCT
ejpam-4340	358	8	the	the	DET
ejpam-4340	358	9	following	follow	VERB
ejpam-4340	358	10	properties	property	NOUN
ejpam-4340	358	11	hold	hold	VERB
ejpam-4340	358	12	:	:	PUNCT
ejpam-4340	358	13	(	(	PUNCT
ejpam-4340	358	14	a	a	X
ejpam-4340	358	15	)	)	PUNCT
ejpam-4340	358	16	τ	τ	PROPN
ejpam-4340	358	17	⊆	⊆	NUM
ejpam-4340	358	18	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	358	19	)	)	PUNCT
ejpam-4340	358	20	⊆	⊆	NUM
ejpam-4340	358	21	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	358	22	,	,	PUNCT
ejpam-4340	358	23	(	(	PUNCT
ejpam-4340	358	24	b	b	X
ejpam-4340	358	25	)	)	PUNCT
ejpam-4340	358	26	τ	τ	PROPN
ejpam-4340	358	27	⊆	⊆	NUM
ejpam-4340	358	28	go(x	go(x	NUM
ejpam-4340	358	29	)	)	PUNCT
ejpam-4340	358	30	⊆	⊆	NUM
ejpam-4340	358	31	τ∗	τ∗	NOUN
ejpam-4340	358	32	⊆	⊆	NUM
ejpam-4340	358	33	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	358	34	.	.	PUNCT
ejpam-4340	359	1	theorem	theorem	VERB
ejpam-4340	359	2	16	16	NUM
ejpam-4340	359	3	.	.	PUNCT
ejpam-4340	360	1	let	let	VERB
ejpam-4340	360	2	x	x	PRON
ejpam-4340	360	3	be	be	AUX
ejpam-4340	360	4	a	a	DET
ejpam-4340	360	5	topological	topological	ADJ
ejpam-4340	360	6	space	space	NOUN
ejpam-4340	360	7	.	.	PUNCT
ejpam-4340	361	1	if	if	SCONJ
ejpam-4340	361	2	the	the	DET
ejpam-4340	361	3	family	family	NOUN
ejpam-4340	361	4	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	361	5	)	)	PUNCT
ejpam-4340	361	6	is	be	AUX
ejpam-4340	361	7	a	a	DET
ejpam-4340	361	8	topology	topology	NOUN
ejpam-4340	361	9	on	on	ADP
ejpam-4340	361	10	x	x	NOUN
ejpam-4340	361	11	,	,	PUNCT
ejpam-4340	361	12	then	then	ADV
ejpam-4340	361	13	the	the	DET
ejpam-4340	361	14	family	family	NOUN
ejpam-4340	361	15	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	361	16	is	be	AUX
ejpam-4340	361	17	a	a	DET
ejpam-4340	361	18	topology	topology	NOUN
ejpam-4340	361	19	on	on	ADP
ejpam-4340	361	20	x.	x.	NOUN
ejpam-4340	361	21	proof	proof	NOUN
ejpam-4340	361	22	.	.	PUNCT
ejpam-4340	362	1	it	it	PRON
ejpam-4340	362	2	is	be	AUX
ejpam-4340	362	3	obvious	obvious	ADJ
ejpam-4340	362	4	that	that	SCONJ
ejpam-4340	362	5	∅	∅	NOUN
ejpam-4340	362	6	,	,	PUNCT
ejpam-4340	362	7	x	x	SYM
ejpam-4340	362	8	∈	∈	NOUN
ejpam-4340	362	9	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	362	10	.	.	PUNCT
ejpam-4340	363	1	let	let	VERB
ejpam-4340	363	2	a	a	DET
ejpam-4340	363	3	,	,	PUNCT
ejpam-4340	363	4	b	b	PROPN
ejpam-4340	363	5	∈	∈	PROPN
ejpam-4340	363	6	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	363	7	.	.	PUNCT
ejpam-4340	364	1	a	a	DET
ejpam-4340	364	2	,	,	PUNCT
ejpam-4340	364	3	b	b	PROPN
ejpam-4340	364	4	∈	∈	PROPN
ejpam-4340	364	5	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	364	6	⇒	⇒	PROPN
ejpam-4340	364	7	(	(	PUNCT
ejpam-4340	364	8	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	364	9	\a	\a	ADV
ejpam-4340	364	10	)	)	PUNCT
ejpam-4340	365	1	=	=	PUNCT
ejpam-4340	365	2	x	x	X
ejpam-4340	365	3	\a)(gωe∗-cl(x	\a)(gωe∗-cl(x	PROPN
ejpam-4340	365	4	\b	\b	NOUN
ejpam-4340	365	5	)	)	PUNCT
ejpam-4340	365	6	=	=	SYM
ejpam-4340	365	7	x	x	SYM
ejpam-4340	365	8	\b	\b	ADJ
ejpam-4340	365	9	)	)	PUNCT
ejpam-4340	365	10	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	365	11	)	)	PUNCT
ejpam-4340	365	12	is	be	AUX
ejpam-4340	365	13	a	a	DET
ejpam-4340	365	14	topology	topology	NOUN
ejpam-4340	365	15	on	on	ADP
ejpam-4340	365	16	x	x	SYM
ejpam-4340	365	17	}	}	PUNCT
ejpam-4340	365	18	⇒	⇒	VERB
ejpam-4340	365	19	⇒	⇒	NOUN
ejpam-4340	365	20	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	365	21	\a	\a	PRON
ejpam-4340	365	22	)	)	PUNCT
ejpam-4340	365	23	∪	∪	VERB
ejpam-4340	365	24	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	365	25	\b	\b	NOUN
ejpam-4340	365	26	)	)	PUNCT
ejpam-4340	366	1	=	=	PRON
ejpam-4340	366	2	(	(	PUNCT
ejpam-4340	366	3	x	x	NOUN
ejpam-4340	366	4	\a	\a	ADJ
ejpam-4340	366	5	)	)	PUNCT
ejpam-4340	366	6	∪	∪	ADP
ejpam-4340	366	7	(	(	PUNCT
ejpam-4340	366	8	x	x	NOUN
ejpam-4340	366	9	\b	\b	ADJ
ejpam-4340	366	10	)	)	PUNCT
ejpam-4340	366	11	⇒	⇒	NOUN
ejpam-4340	366	12	gωe∗-cl((x	gωe∗-cl((x	VERB
ejpam-4340	366	13	\a	\a	PRON
ejpam-4340	366	14	)	)	PUNCT
ejpam-4340	366	15	∪	∪	NOUN
ejpam-4340	366	16	(	(	PUNCT
ejpam-4340	366	17	x	x	NOUN
ejpam-4340	366	18	\b	\b	ADJ
ejpam-4340	366	19	)	)	PUNCT
ejpam-4340	366	20	)	)	PUNCT
ejpam-4340	367	1	=	=	SYM
ejpam-4340	367	2	gωe∗-cl(x	gωe∗-cl(x	X
ejpam-4340	367	3	\	\	PROPN
ejpam-4340	368	1	(	(	PUNCT
ejpam-4340	368	2	a	a	DET
ejpam-4340	368	3	∩b	∩b	NOUN
ejpam-4340	368	4	)	)	PUNCT
ejpam-4340	368	5	)	)	PUNCT
ejpam-4340	369	1	=	=	PUNCT
ejpam-4340	370	1	x	x	SYM
ejpam-4340	370	2	\	\	PROPN
ejpam-4340	370	3	(	(	PUNCT
ejpam-4340	370	4	a	a	DET
ejpam-4340	370	5	∩b	∩b	NOUN
ejpam-4340	370	6	)	)	PUNCT
ejpam-4340	370	7	p.	p.	NOUN
ejpam-4340	370	8	şaşmaz	şaşmaz	NUM
ejpam-4340	370	9	,	,	PUNCT
ejpam-4340	370	10	m.	m.	NOUN
ejpam-4340	370	11	özkoç	özkoç	PROPN
ejpam-4340	370	12	/	/	SYM
ejpam-4340	370	13	eur	eur	PROPN
ejpam-4340	370	14	.	.	PUNCT
ejpam-4340	371	1	j.	j.	PROPN
ejpam-4340	371	2	pure	pure	PROPN
ejpam-4340	371	3	appl	appl	PROPN
ejpam-4340	371	4	.	.	PROPN
ejpam-4340	371	5	math	math	PROPN
ejpam-4340	371	6	,	,	PUNCT
ejpam-4340	371	7	15	15	NUM
ejpam-4340	371	8	(	(	PUNCT
ejpam-4340	371	9	2	2	NUM
ejpam-4340	371	10	)	)	PUNCT
ejpam-4340	371	11	(	(	PUNCT
ejpam-4340	371	12	2022	2022	NUM
ejpam-4340	371	13	)	)	PUNCT
ejpam-4340	371	14	,	,	PUNCT
ejpam-4340	371	15	354	354	NUM
ejpam-4340	371	16	-	-	SYM
ejpam-4340	371	17	374	374	NUM
ejpam-4340	371	18	366	366	NUM
ejpam-4340	371	19	⇒	⇒	NOUN
ejpam-4340	371	20	a	a	DET
ejpam-4340	371	21	∩b	∩b	NOUN
ejpam-4340	371	22	∈	∈	PROPN
ejpam-4340	371	23	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	371	24	.	.	PUNCT
ejpam-4340	372	1	now	now	ADV
ejpam-4340	372	2	,	,	PUNCT
ejpam-4340	372	3	let	let	VERB
ejpam-4340	372	4	a	a	DET
ejpam-4340	372	5	⊆	⊆	NUM
ejpam-4340	372	6	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	372	7	.	.	PUNCT
ejpam-4340	373	1	a	a	DET
ejpam-4340	373	2	∈	∈	PROPN
ejpam-4340	373	3	a	a	DET
ejpam-4340	373	4	⊆	⊆	NUM
ejpam-4340	373	5	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	373	6	⇒	⇒	VERB
ejpam-4340	373	7	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	373	8	\a	\a	PUNCT
ejpam-4340	373	9	)	)	PUNCT
ejpam-4340	374	1	=	=	PUNCT
ejpam-4340	375	1	x	x	PUNCT
ejpam-4340	375	2	\a⇒	\a⇒	PROPN
ejpam-4340	375	3	x	x	PUNCT
ejpam-4340	375	4	\a	\a	ADJ
ejpam-4340	375	5	∈	∈	PROPN
ejpam-4340	375	6	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	375	7	)	)	PUNCT
ejpam-4340	375	8	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	375	9	)	)	PUNCT
ejpam-4340	375	10	is	be	AUX
ejpam-4340	375	11	a	a	DET
ejpam-4340	375	12	topology	topology	NOUN
ejpam-4340	375	13	on	on	ADP
ejpam-4340	375	14	x	x	SYM
ejpam-4340	375	15	}	}	PUNCT
ejpam-4340	375	16	⇒	⇒	VERB
ejpam-4340	375	17	⇒	⇒	NOUN
ejpam-4340	375	18	x	x	PUNCT
ejpam-4340	375	19	\	\	PROPN
ejpam-4340	375	20	(	(	PUNCT
ejpam-4340	375	21	∪a	∪a	NUM
ejpam-4340	375	22	)	)	PUNCT
ejpam-4340	375	23	=	=	PRON
ejpam-4340	375	24	∩a∈a(x	∩a∈a(x	NOUN
ejpam-4340	375	25	\a	\a	ADJ
ejpam-4340	375	26	)	)	PUNCT
ejpam-4340	375	27	∈	∈	PROPN
ejpam-4340	375	28	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	375	29	)	)	PUNCT
ejpam-4340	375	30	⇒	⇒	VERB
ejpam-4340	375	31	gωe∗-cl(∩a∈a(x	gωe∗-cl(∩a∈a(x	NOUN
ejpam-4340	375	32	\a	\a	ADJ
ejpam-4340	375	33	)	)	PUNCT
ejpam-4340	375	34	)	)	PUNCT
ejpam-4340	376	1	=	=	PRON
ejpam-4340	376	2	∩a∈a(x	∩a∈a(x	NOUN
ejpam-4340	376	3	\a	\a	ADJ
ejpam-4340	376	4	)	)	PUNCT
ejpam-4340	377	1	=	=	PUNCT
ejpam-4340	377	2	x	x	SYM
ejpam-4340	377	3	\	\	PROPN
ejpam-4340	377	4	(	(	PUNCT
ejpam-4340	377	5	∪a	∪a	NUM
ejpam-4340	377	6	)	)	PUNCT
ejpam-4340	377	7	⇒	⇒	NOUN
ejpam-4340	377	8	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	377	9	\	\	PROPN
ejpam-4340	377	10	(	(	PUNCT
ejpam-4340	377	11	∪a	∪a	NUM
ejpam-4340	377	12	)	)	PUNCT
ejpam-4340	377	13	)	)	PUNCT
ejpam-4340	378	1	=	=	PUNCT
ejpam-4340	378	2	x	x	SYM
ejpam-4340	378	3	\	\	PROPN
ejpam-4340	378	4	(	(	PUNCT
ejpam-4340	378	5	∪a	∪a	NUM
ejpam-4340	378	6	)	)	PUNCT
ejpam-4340	378	7	⇒	⇒	VERB
ejpam-4340	378	8	∪a	∪a	NUM
ejpam-4340	378	9	∈	∈	PROPN
ejpam-4340	378	10	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	378	11	.	.	PUNCT
ejpam-4340	379	1	theorem	theorem	VERB
ejpam-4340	379	2	17	17	NUM
ejpam-4340	379	3	.	.	PUNCT
ejpam-4340	380	1	let	let	VERB
ejpam-4340	380	2	x	x	PRON
ejpam-4340	380	3	be	be	AUX
ejpam-4340	380	4	a	a	DET
ejpam-4340	380	5	topological	topological	ADJ
ejpam-4340	380	6	space	space	NOUN
ejpam-4340	380	7	.	.	PUNCT
ejpam-4340	381	1	then	then	ADV
ejpam-4340	381	2	the	the	DET
ejpam-4340	381	3	following	follow	VERB
ejpam-4340	381	4	properties	property	NOUN
ejpam-4340	381	5	hold	hold	VERB
ejpam-4340	381	6	:	:	PUNCT
ejpam-4340	381	7	(	(	PUNCT
ejpam-4340	381	8	a	a	X
ejpam-4340	381	9	)	)	PUNCT
ejpam-4340	381	10	a	a	DET
ejpam-4340	381	11	space	space	NOUN
ejpam-4340	381	12	x	x	PUNCT
ejpam-4340	381	13	is	be	AUX
ejpam-4340	381	14	ωe∗-t	ωe∗-t	NUM
ejpam-4340	381	15	1	1	NUM
ejpam-4340	381	16	2	2	NUM
ejpam-4340	381	17	if	if	SCONJ
ejpam-4340	381	18	and	and	CCONJ
ejpam-4340	381	19	only	only	ADV
ejpam-4340	381	20	if	if	SCONJ
ejpam-4340	381	21	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	381	22	=	=	PUNCT
ejpam-4340	381	23	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	381	24	)	)	PUNCT
ejpam-4340	381	25	,	,	PUNCT
ejpam-4340	381	26	(	(	PUNCT
ejpam-4340	381	27	b	b	X
ejpam-4340	381	28	)	)	PUNCT
ejpam-4340	381	29	every	every	PRON
ejpam-4340	381	30	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	381	31	is	be	AUX
ejpam-4340	381	32	closed	close	VERB
ejpam-4340	381	33	if	if	SCONJ
ejpam-4340	381	34	and	and	CCONJ
ejpam-4340	382	1	only	only	ADV
ejpam-4340	382	2	if	if	SCONJ
ejpam-4340	382	3	τ∗ωe∗	τ∗ωe∗	PROPN
ejpam-4340	382	4	=	=	SYM
ejpam-4340	382	5	τ	τ	PROPN
ejpam-4340	382	6	.	.	PUNCT
ejpam-4340	382	7	proof	proof	NOUN
ejpam-4340	382	8	.	.	PUNCT
ejpam-4340	383	1	(	(	PUNCT
ejpam-4340	383	2	a	a	X
ejpam-4340	383	3	)	)	PUNCT
ejpam-4340	383	4	(	(	PUNCT
ejpam-4340	383	5	⇒	⇒	PROPN
ejpam-4340	383	6	)	)	PUNCT
ejpam-4340	383	7	:	:	PUNCT
ejpam-4340	383	8	let	let	VERB
ejpam-4340	383	9	a	a	DET
ejpam-4340	383	10	∈	∈	PROPN
ejpam-4340	383	11	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	383	12	.	.	PUNCT
ejpam-4340	384	1	a	a	DET
ejpam-4340	384	2	∈	∈	PROPN
ejpam-4340	384	3	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	384	4	⇒	⇒	NOUN
ejpam-4340	384	5	x	x	PUNCT
ejpam-4340	384	6	\a	\a	ADJ
ejpam-4340	384	7	=	=	PUNCT
ejpam-4340	384	8	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	384	9	\a)⇒	\a)⇒	PROPN
ejpam-4340	384	10	x	x	SYM
ejpam-4340	384	11	\a	\a	ADJ
ejpam-4340	384	12	∈	∈	PROPN
ejpam-4340	384	13	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	384	14	)	)	PUNCT
ejpam-4340	384	15	x	x	PUNCT
ejpam-4340	384	16	is	be	AUX
ejpam-4340	384	17	ωe∗-t	ωe∗-t	NUM
ejpam-4340	384	18	1	1	NUM
ejpam-4340	384	19	2	2	NUM
ejpam-4340	384	20	}	}	PUNCT
ejpam-4340	384	21	⇒	⇒	VERB
ejpam-4340	384	22	⇒	⇒	NOUN
ejpam-4340	384	23	x	x	PUNCT
ejpam-4340	384	24	\a	\a	ADJ
ejpam-4340	384	25	∈	∈	NOUN
ejpam-4340	384	26	ωe∗c(x)⇒	ωe∗c(x)⇒	ADP
ejpam-4340	384	27	a	a	DET
ejpam-4340	384	28	∈	∈	PROPN
ejpam-4340	384	29	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	384	30	)	)	PUNCT
ejpam-4340	384	31	.	.	PUNCT
ejpam-4340	385	1	(	(	PUNCT
ejpam-4340	385	2	⇐	⇐	NOUN
ejpam-4340	385	3	)	)	PUNCT
ejpam-4340	385	4	:	:	PUNCT
ejpam-4340	385	5	let	let	VERB
ejpam-4340	385	6	a	a	DET
ejpam-4340	385	7	∈	∈	PROPN
ejpam-4340	385	8	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	385	9	)	)	PUNCT
ejpam-4340	385	10	.	.	PUNCT
ejpam-4340	386	1	a	a	DET
ejpam-4340	386	2	∈	∈	PROPN
ejpam-4340	386	3	gωe∗c(x)⇒	gωe∗c(x)⇒	PROPN
ejpam-4340	386	4	a	a	DET
ejpam-4340	386	5	=	=	NOUN
ejpam-4340	386	6	gωe∗-cl(a)⇒	gωe∗-cl(a)⇒	NOUN
ejpam-4340	386	7	x	x	SYM
ejpam-4340	386	8	\a	\a	ADJ
ejpam-4340	386	9	∈	∈	PROPN
ejpam-4340	386	10	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	386	11	hypothesis	hypothesis	NOUN
ejpam-4340	386	12	}	}	PUNCT
ejpam-4340	386	13	⇒	⇒	VERB
ejpam-4340	386	14	x	x	PUNCT
ejpam-4340	386	15	\a	\a	ADJ
ejpam-4340	386	16	∈	∈	PROPN
ejpam-4340	386	17	ωe∗o(x	ωe∗o(x	NUM
ejpam-4340	386	18	)	)	PUNCT
ejpam-4340	386	19	⇒	⇒	VERB
ejpam-4340	386	20	a	a	DET
ejpam-4340	386	21	∈	∈	PROPN
ejpam-4340	386	22	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	386	23	)	)	PUNCT
ejpam-4340	386	24	.	.	PUNCT
ejpam-4340	387	1	(	(	PUNCT
ejpam-4340	387	2	b	b	X
ejpam-4340	387	3	)	)	PUNCT
ejpam-4340	387	4	(	(	PUNCT
ejpam-4340	387	5	⇒	⇒	PROPN
ejpam-4340	387	6	)	)	PUNCT
ejpam-4340	387	7	:	:	PUNCT
ejpam-4340	387	8	let	let	VERB
ejpam-4340	387	9	a	a	DET
ejpam-4340	387	10	∈	∈	PROPN
ejpam-4340	387	11	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	387	12	.	.	PUNCT
ejpam-4340	388	1	a	a	DET
ejpam-4340	388	2	∈	∈	PROPN
ejpam-4340	388	3	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	388	4	⇒	⇒	NOUN
ejpam-4340	388	5	x	x	PUNCT
ejpam-4340	388	6	\a	\a	ADJ
ejpam-4340	388	7	=	=	PUNCT
ejpam-4340	388	8	gωe∗-cl(x	gωe∗-cl(x	PROPN
ejpam-4340	388	9	\a)⇒	\a)⇒	PROPN
ejpam-4340	388	10	x	x	SYM
ejpam-4340	388	11	\a	\a	ADJ
ejpam-4340	388	12	∈	∈	PROPN
ejpam-4340	388	13	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	388	14	)	)	PUNCT
ejpam-4340	388	15	hypothesis	hypothesis	NOUN
ejpam-4340	388	16	}	}	PUNCT
ejpam-4340	388	17	⇒	⇒	VERB
ejpam-4340	388	18	⇒	⇒	NOUN
ejpam-4340	388	19	x	x	PUNCT
ejpam-4340	388	20	\a	\a	PROPN
ejpam-4340	388	21	∈	∈	PROPN
ejpam-4340	388	22	c(x)⇒	c(x)⇒	VERB
ejpam-4340	388	23	a	a	DET
ejpam-4340	388	24	∈	∈	PROPN
ejpam-4340	388	25	τ	τ	X
ejpam-4340	388	26	.	.	PUNCT
ejpam-4340	389	1	(	(	PUNCT
ejpam-4340	389	2	⇐	⇐	NOUN
ejpam-4340	389	3	)	)	PUNCT
ejpam-4340	389	4	:	:	PUNCT
ejpam-4340	389	5	let	let	VERB
ejpam-4340	389	6	a	a	DET
ejpam-4340	389	7	∈	∈	PROPN
ejpam-4340	389	8	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	389	9	)	)	PUNCT
ejpam-4340	389	10	.	.	PUNCT
ejpam-4340	390	1	a	a	DET
ejpam-4340	390	2	∈	∈	PROPN
ejpam-4340	390	3	gωe∗c(x)⇒	gωe∗c(x)⇒	PROPN
ejpam-4340	390	4	a	a	DET
ejpam-4340	390	5	=	=	NOUN
ejpam-4340	390	6	gωe∗-cl(a)⇒	gωe∗-cl(a)⇒	NOUN
ejpam-4340	390	7	x	x	SYM
ejpam-4340	390	8	\a	\a	ADJ
ejpam-4340	390	9	∈	∈	PROPN
ejpam-4340	390	10	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	390	11	hypothesis	hypothesis	NOUN
ejpam-4340	390	12	}	}	PUNCT
ejpam-4340	390	13	⇒	⇒	VERB
ejpam-4340	390	14	x	x	PUNCT
ejpam-4340	390	15	\a	\a	VERB
ejpam-4340	390	16	∈	∈	PROPN
ejpam-4340	390	17	τ	τ	PROPN
ejpam-4340	390	18	⇒	⇒	VERB
ejpam-4340	390	19	a	a	DET
ejpam-4340	390	20	∈	∈	PROPN
ejpam-4340	390	21	c(x	c(x	NOUN
ejpam-4340	390	22	)	)	PUNCT
ejpam-4340	390	23	.	.	PUNCT
ejpam-4340	391	1	5	5	X
ejpam-4340	391	2	.	.	X
ejpam-4340	391	3	gωe∗-continuity	gωe∗-continuity	NOUN
ejpam-4340	391	4	,	,	PUNCT
ejpam-4340	391	5	gωe∗-irresoluteness	gωe∗-irresoluteness	PROPN
ejpam-4340	391	6	and	and	CCONJ
ejpam-4340	391	7	gωe∗-closedness	gωe∗-closedness	PROPN
ejpam-4340	391	8	definition	definition	NOUN
ejpam-4340	391	9	18	18	NUM
ejpam-4340	391	10	.	.	PUNCT
ejpam-4340	392	1	a	a	DET
ejpam-4340	392	2	function	function	NOUN
ejpam-4340	392	3	f	f	NOUN
ejpam-4340	392	4	:	:	PUNCT
ejpam-4340	392	5	x	x	X
ejpam-4340	392	6	→	→	SYM
ejpam-4340	392	7	y	y	PROPN
ejpam-4340	392	8	is	be	AUX
ejpam-4340	392	9	said	say	VERB
ejpam-4340	392	10	to	to	PART
ejpam-4340	392	11	be	be	AUX
ejpam-4340	392	12	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	392	13	(	(	PUNCT
ejpam-4340	392	14	resp	resp	NOUN
ejpam-4340	392	15	.	.	PUNCT
ejpam-4340	393	1	gωβcontinuous	gωβcontinuous	ADJ
ejpam-4340	393	2	[	[	X
ejpam-4340	393	3	4	4	NUM
ejpam-4340	393	4	]	]	PUNCT
ejpam-4340	393	5	)	)	PUNCT
ejpam-4340	393	6	if	if	SCONJ
ejpam-4340	393	7	f−1[v	f−1[v	NOUN
ejpam-4340	393	8	]	]	PUNCT
ejpam-4340	393	9	is	be	AUX
ejpam-4340	393	10	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	393	11	(	(	PUNCT
ejpam-4340	393	12	resp	resp	NOUN
ejpam-4340	393	13	.	.	PUNCT
ejpam-4340	394	1	gωβ	gωβ	ADJ
ejpam-4340	394	2	-	-	PUNCT
ejpam-4340	394	3	closed	closed	ADJ
ejpam-4340	394	4	[	[	X
ejpam-4340	394	5	4	4	NUM
ejpam-4340	394	6	]	]	PUNCT
ejpam-4340	394	7	)	)	PUNCT
ejpam-4340	394	8	in	in	ADP
ejpam-4340	394	9	x	x	PUNCT
ejpam-4340	394	10	for	for	ADP
ejpam-4340	394	11	every	every	DET
ejpam-4340	394	12	closed	close	VERB
ejpam-4340	394	13	set	set	VERB
ejpam-4340	394	14	v	v	NOUN
ejpam-4340	394	15	of	of	ADP
ejpam-4340	394	16	y.	y.	PROPN
ejpam-4340	394	17	corollary	corollary	NOUN
ejpam-4340	394	18	4	4	NUM
ejpam-4340	394	19	.	.	PUNCT
ejpam-4340	395	1	let	let	VERB
ejpam-4340	395	2	f	f	NOUN
ejpam-4340	395	3	:	:	PUNCT
ejpam-4340	395	4	x	x	X
ejpam-4340	395	5	→	→	SYM
ejpam-4340	395	6	y	y	X
ejpam-4340	395	7	be	be	AUX
ejpam-4340	395	8	a	a	DET
ejpam-4340	395	9	function	function	NOUN
ejpam-4340	395	10	.	.	PUNCT
ejpam-4340	396	1	f	f	PROPN
ejpam-4340	396	2	is	be	AUX
ejpam-4340	396	3	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	396	4	if	if	SCONJ
ejpam-4340	396	5	and	and	CCONJ
ejpam-4340	396	6	only	only	ADV
ejpam-4340	396	7	if	if	SCONJ
ejpam-4340	396	8	the	the	DET
ejpam-4340	396	9	inverse	inverse	ADJ
ejpam-4340	396	10	image	image	NOUN
ejpam-4340	396	11	of	of	ADP
ejpam-4340	396	12	every	every	DET
ejpam-4340	396	13	open	open	ADJ
ejpam-4340	396	14	set	set	NOUN
ejpam-4340	396	15	in	in	ADP
ejpam-4340	396	16	y	y	PROPN
ejpam-4340	396	17	is	be	AUX
ejpam-4340	396	18	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	396	19	in	in	ADP
ejpam-4340	396	20	x.	x.	PROPN
ejpam-4340	396	21	p.	p.	PROPN
ejpam-4340	396	22	şaşmaz	şaşmaz	NUM
ejpam-4340	396	23	,	,	PUNCT
ejpam-4340	396	24	m.	m.	NOUN
ejpam-4340	396	25	özkoç	özkoç	PROPN
ejpam-4340	396	26	/	/	SYM
ejpam-4340	396	27	eur	eur	PROPN
ejpam-4340	396	28	.	.	PUNCT
ejpam-4340	397	1	j.	j.	PROPN
ejpam-4340	397	2	pure	pure	PROPN
ejpam-4340	397	3	appl	appl	PROPN
ejpam-4340	397	4	.	.	PROPN
ejpam-4340	397	5	math	math	PROPN
ejpam-4340	397	6	,	,	PUNCT
ejpam-4340	397	7	15	15	NUM
ejpam-4340	397	8	(	(	PUNCT
ejpam-4340	397	9	2	2	NUM
ejpam-4340	397	10	)	)	PUNCT
ejpam-4340	397	11	(	(	PUNCT
ejpam-4340	397	12	2022	2022	NUM
ejpam-4340	397	13	)	)	PUNCT
ejpam-4340	397	14	,	,	PUNCT
ejpam-4340	397	15	354	354	NUM
ejpam-4340	397	16	-	-	SYM
ejpam-4340	397	17	374	374	NUM
ejpam-4340	397	18	367	367	NUM
ejpam-4340	397	19	remark	remark	NOUN
ejpam-4340	397	20	4	4	NUM
ejpam-4340	397	21	.	.	PUNCT
ejpam-4340	398	1	every	every	DET
ejpam-4340	398	2	continuous	continuous	ADJ
ejpam-4340	398	3	function	function	NOUN
ejpam-4340	398	4	is	be	AUX
ejpam-4340	398	5	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	398	6	but	but	CCONJ
ejpam-4340	398	7	the	the	DET
ejpam-4340	398	8	converse	converse	NOUN
ejpam-4340	398	9	need	need	VERB
ejpam-4340	398	10	not	not	PART
ejpam-4340	398	11	to	to	PART
ejpam-4340	398	12	be	be	AUX
ejpam-4340	398	13	true	true	ADJ
ejpam-4340	398	14	as	as	SCONJ
ejpam-4340	398	15	shown	show	VERB
ejpam-4340	398	16	by	by	ADP
ejpam-4340	398	17	the	the	DET
ejpam-4340	398	18	following	follow	VERB
ejpam-4340	398	19	example	example	NOUN
ejpam-4340	398	20	.	.	PUNCT
ejpam-4340	399	1	example	example	NOUN
ejpam-4340	400	1	4	4	NUM
ejpam-4340	400	2	.	.	PUNCT
ejpam-4340	400	3	consider	consider	VERB
ejpam-4340	400	4	the	the	DET
ejpam-4340	400	5	real	real	ADJ
ejpam-4340	400	6	numbers	number	NOUN
ejpam-4340	400	7	r	r	NOUN
ejpam-4340	400	8	with	with	ADP
ejpam-4340	400	9	usual	usual	ADJ
ejpam-4340	400	10	topology	topology	NOUN
ejpam-4340	400	11	and	and	CCONJ
ejpam-4340	400	12	let	let	VERB
ejpam-4340	400	13	y	y	PROPN
ejpam-4340	400	14	=	=	PUNCT
ejpam-4340	400	15	{	{	PUNCT
ejpam-4340	400	16	1	1	NUM
ejpam-4340	400	17	,	,	PUNCT
ejpam-4340	400	18	2	2	NUM
ejpam-4340	400	19	}	}	PUNCT
ejpam-4340	400	20	with	with	ADP
ejpam-4340	400	21	the	the	DET
ejpam-4340	400	22	topology	topology	NOUN
ejpam-4340	400	23	τ	τ	X
ejpam-4340	400	24	=	=	SYM
ejpam-4340	400	25	{	{	PUNCT
ejpam-4340	400	26	∅	∅	NOUN
ejpam-4340	400	27	,	,	PUNCT
ejpam-4340	400	28	y	y	PROPN
ejpam-4340	400	29	,	,	PUNCT
ejpam-4340	400	30	{	{	PUNCT
ejpam-4340	400	31	1	1	NUM
ejpam-4340	400	32	}	}	PUNCT
ejpam-4340	400	33	}	}	PUNCT
ejpam-4340	400	34	.	.	PUNCT
ejpam-4340	401	1	define	define	VERB
ejpam-4340	401	2	the	the	DET
ejpam-4340	401	3	function	function	NOUN
ejpam-4340	401	4	f	f	NOUN
ejpam-4340	401	5	:	:	PUNCT
ejpam-4340	401	6	r→	r→	PROPN
ejpam-4340	401	7	y	y	PROPN
ejpam-4340	401	8	by	by	ADP
ejpam-4340	401	9	f(x	f(x	PROPN
ejpam-4340	401	10	)	)	PUNCT
ejpam-4340	402	1	=	=	PRON
ejpam-4340	402	2	{	{	PUNCT
ejpam-4340	402	3	1	1	NUM
ejpam-4340	402	4	,	,	PUNCT
ejpam-4340	402	5	x	x	PUNCT
ejpam-4340	402	6	∈	∈	PROPN
ejpam-4340	402	7	q	q	PROPN
ejpam-4340	402	8	2	2	NUM
ejpam-4340	402	9	,	,	PUNCT
ejpam-4340	402	10	x	x	SYM
ejpam-4340	402	11	∈	∈	NOUN
ejpam-4340	402	12	r	r	NOUN
ejpam-4340	402	13	\q	\q	NOUN
ejpam-4340	402	14	.	.	PUNCT
ejpam-4340	403	1	then	then	ADV
ejpam-4340	403	2	the	the	DET
ejpam-4340	403	3	function	function	NOUN
ejpam-4340	403	4	f	f	PROPN
ejpam-4340	403	5	is	be	AUX
ejpam-4340	403	6	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	403	7	but	but	CCONJ
ejpam-4340	403	8	not	not	PART
ejpam-4340	403	9	continuous	continuous	ADJ
ejpam-4340	403	10	since	since	SCONJ
ejpam-4340	403	11	f−1[{2	f−1[{2	NOUN
ejpam-4340	403	12	}	}	PUNCT
ejpam-4340	403	13	]	]	PUNCT
ejpam-4340	403	14	=	=	SYM
ejpam-4340	403	15	r	r	NOUN
ejpam-4340	403	16	\q	\q	NOUN
ejpam-4340	403	17	is	be	AUX
ejpam-4340	403	18	not	not	PART
ejpam-4340	403	19	closed	close	VERB
ejpam-4340	403	20	in	in	ADP
ejpam-4340	403	21	r.	r.	PROPN
ejpam-4340	403	22	remark	remark	PROPN
ejpam-4340	403	23	5	5	NUM
ejpam-4340	403	24	.	.	PUNCT
ejpam-4340	404	1	let	let	VERB
ejpam-4340	404	2	f	f	NOUN
ejpam-4340	404	3	:	:	PUNCT
ejpam-4340	404	4	x	x	X
ejpam-4340	404	5	→	→	SYM
ejpam-4340	404	6	y	y	X
ejpam-4340	404	7	be	be	AUX
ejpam-4340	404	8	a	a	DET
ejpam-4340	404	9	function	function	NOUN
ejpam-4340	404	10	.	.	PUNCT
ejpam-4340	405	1	then	then	ADV
ejpam-4340	405	2	the	the	DET
ejpam-4340	405	3	following	follow	VERB
ejpam-4340	405	4	properties	property	NOUN
ejpam-4340	405	5	hold	hold	VERB
ejpam-4340	405	6	:	:	PUNCT
ejpam-4340	405	7	(	(	PUNCT
ejpam-4340	405	8	a	a	X
ejpam-4340	405	9	)	)	PUNCT
ejpam-4340	405	10	if	if	SCONJ
ejpam-4340	405	11	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	405	12	=	=	PUNCT
ejpam-4340	405	13	τ	τ	PROPN
ejpam-4340	405	14	in	in	ADP
ejpam-4340	405	15	x	x	NOUN
ejpam-4340	405	16	,	,	PUNCT
ejpam-4340	405	17	then	then	ADV
ejpam-4340	405	18	the	the	DET
ejpam-4340	405	19	notion	notion	NOUN
ejpam-4340	405	20	of	of	ADP
ejpam-4340	405	21	continuity	continuity	NOUN
ejpam-4340	405	22	and	and	CCONJ
ejpam-4340	405	23	the	the	DET
ejpam-4340	405	24	notion	notion	NOUN
ejpam-4340	405	25	of	of	ADP
ejpam-4340	405	26	gωe∗-continuity	gωe∗-continuity	NOUN
ejpam-4340	405	27	coincide	coincide	NOUN
ejpam-4340	405	28	.	.	PUNCT
ejpam-4340	406	1	(	(	PUNCT
ejpam-4340	406	2	b	b	X
ejpam-4340	406	3	)	)	PUNCT
ejpam-4340	406	4	every	every	DET
ejpam-4340	406	5	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	406	6	function	function	NOUN
ejpam-4340	406	7	defined	define	VERB
ejpam-4340	406	8	on	on	ADP
ejpam-4340	406	9	ωe∗-t	ωe∗-t	PROPN
ejpam-4340	406	10	1	1	NUM
ejpam-4340	406	11	2	2	NUM
ejpam-4340	406	12	space	space	NOUN
ejpam-4340	406	13	is	be	AUX
ejpam-4340	406	14	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	406	15	.	.	PUNCT
ejpam-4340	407	1	remark	remark	NOUN
ejpam-4340	407	2	6	6	NUM
ejpam-4340	407	3	.	.	PUNCT
ejpam-4340	408	1	the	the	DET
ejpam-4340	408	2	following	follow	VERB
ejpam-4340	408	3	diagram	diagram	NOUN
ejpam-4340	408	4	follows	follow	VERB
ejpam-4340	408	5	immediately	immediately	ADV
ejpam-4340	408	6	from	from	ADP
ejpam-4340	408	7	the	the	DET
ejpam-4340	408	8	definitions	definition	NOUN
ejpam-4340	408	9	in	in	ADP
ejpam-4340	408	10	which	which	PRON
ejpam-4340	408	11	none	none	NOUN
ejpam-4340	408	12	of	of	ADP
ejpam-4340	408	13	the	the	DET
ejpam-4340	408	14	implications	implication	NOUN
ejpam-4340	408	15	is	be	AUX
ejpam-4340	408	16	reversible	reversible	ADJ
ejpam-4340	408	17	.	.	PUNCT
ejpam-4340	409	1	continuous	continuous	ADJ
ejpam-4340	409	2	→	→	SYM
ejpam-4340	409	3	ωβ	ωβ	ADJ
ejpam-4340	409	4	-	-	ADJ
ejpam-4340	409	5	continuous	continuous	ADJ
ejpam-4340	409	6	→	→	NOUN
ejpam-4340	409	7	gωβ	gωβ	ADJ
ejpam-4340	409	8	-	-	ADJ
ejpam-4340	409	9	continuous	continuous	ADJ
ejpam-4340	409	10	↓	↓	PROPN
ejpam-4340	409	11	↓	↓	PROPN
ejpam-4340	409	12	↓	↓	PROPN
ejpam-4340	409	13	e∗-continuous	e∗-continuous	PROPN
ejpam-4340	409	14	→	→	SYM
ejpam-4340	409	15	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	409	16	→	→	NOUN
ejpam-4340	409	17	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	409	18	theorem	theorem	NOUN
ejpam-4340	409	19	18	18	NUM
ejpam-4340	409	20	.	.	PUNCT
ejpam-4340	410	1	let	let	VERB
ejpam-4340	410	2	f	f	NOUN
ejpam-4340	410	3	:	:	PUNCT
ejpam-4340	410	4	x	x	X
ejpam-4340	410	5	→	→	SYM
ejpam-4340	410	6	y	y	X
ejpam-4340	410	7	be	be	AUX
ejpam-4340	410	8	a	a	DET
ejpam-4340	410	9	function	function	NOUN
ejpam-4340	410	10	.	.	PUNCT
ejpam-4340	411	1	if	if	SCONJ
ejpam-4340	411	2	f	f	PROPN
ejpam-4340	411	3	is	be	AUX
ejpam-4340	411	4	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	411	5	,	,	PUNCT
ejpam-4340	411	6	then	then	ADV
ejpam-4340	411	7	f	f	PROPN
ejpam-4340	412	1	[	[	X
ejpam-4340	412	2	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	412	3	)	)	PUNCT
ejpam-4340	412	4	]	]	PUNCT
ejpam-4340	413	1	⊆	⊆	NUM
ejpam-4340	413	2	cl(f	cl(f	NOUN
ejpam-4340	414	1	[	[	X
ejpam-4340	414	2	a	a	X
ejpam-4340	414	3	]	]	X
ejpam-4340	414	4	)	)	PUNCT
ejpam-4340	414	5	for	for	ADP
ejpam-4340	414	6	every	every	DET
ejpam-4340	414	7	subset	subset	NOUN
ejpam-4340	414	8	a	a	PRON
ejpam-4340	414	9	of	of	ADP
ejpam-4340	414	10	x.	x.	NOUN
ejpam-4340	414	11	proof	proof	NOUN
ejpam-4340	414	12	.	.	PUNCT
ejpam-4340	415	1	let	let	VERB
ejpam-4340	415	2	a	a	DET
ejpam-4340	415	3	⊆	⊆	NUM
ejpam-4340	415	4	x.	x.	NOUN
ejpam-4340	415	5	a	a	DET
ejpam-4340	415	6	⊆	⊆	NUM
ejpam-4340	415	7	x	x	SYM
ejpam-4340	415	8	⇒	⇒	NOUN
ejpam-4340	415	9	cl(f	cl(f	PROPN
ejpam-4340	416	1	[	[	X
ejpam-4340	416	2	a	a	X
ejpam-4340	416	3	]	]	X
ejpam-4340	416	4	)	)	PUNCT
ejpam-4340	416	5	∈	∈	PROPN
ejpam-4340	416	6	c(y	c(y	PROPN
ejpam-4340	416	7	)	)	PUNCT
ejpam-4340	417	1	f	f	PROPN
ejpam-4340	417	2	is	be	AUX
ejpam-4340	417	3	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	417	4	}	}	PUNCT
ejpam-4340	417	5	⇒	⇒	NOUN
ejpam-4340	417	6	f−1[cl(f	f−1[cl(f	PROPN
ejpam-4340	418	1	[	[	X
ejpam-4340	418	2	a	a	X
ejpam-4340	418	3	]	]	X
ejpam-4340	418	4	)	)	PUNCT
ejpam-4340	418	5	]	]	PUNCT
ejpam-4340	418	6	∈	∈	PROPN
ejpam-4340	418	7	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	418	8	)	)	PUNCT
ejpam-4340	418	9	⇒	⇒	PROPN
ejpam-4340	418	10	gωe∗-cl(f−1[cl(f	gωe∗-cl(f−1[cl(f	NOUN
ejpam-4340	419	1	[	[	X
ejpam-4340	419	2	a	a	X
ejpam-4340	419	3	]	]	X
ejpam-4340	419	4	)	)	PUNCT
ejpam-4340	419	5	]	]	PUNCT
ejpam-4340	419	6	)	)	PUNCT
ejpam-4340	420	1	=	=	PUNCT
ejpam-4340	420	2	f−1[cl(f	f−1[cl(f	PROPN
ejpam-4340	421	1	[	[	X
ejpam-4340	421	2	a	a	X
ejpam-4340	421	3	]	]	X
ejpam-4340	421	4	)	)	PUNCT
ejpam-4340	421	5	]	]	PUNCT
ejpam-4340	421	6	a	a	DET
ejpam-4340	421	7	⊆	⊆	NUM
ejpam-4340	421	8	f−1[f	f−1[f	NOUN
ejpam-4340	421	9	[	[	X
ejpam-4340	421	10	a	a	X
ejpam-4340	421	11	]	]	X
ejpam-4340	421	12	]	]	X
ejpam-4340	421	13	⊆	⊆	NUM
ejpam-4340	421	14	f−1[cl(f	f−1[cl(f	PROPN
ejpam-4340	421	15	[	[	PUNCT
ejpam-4340	421	16	a])]⇒	a])]⇒	PROPN
ejpam-4340	421	17	gωe∗-cl(a	gωe∗-cl(a	NOUN
ejpam-4340	421	18	)	)	PUNCT
ejpam-4340	421	19	⊆	⊆	NUM
ejpam-4340	421	20	gωe∗-cl(f−1[cl(f	gωe∗-cl(f−1[cl(f	NOUN
ejpam-4340	422	1	[	[	X
ejpam-4340	422	2	a	a	X
ejpam-4340	422	3	]	]	X
ejpam-4340	422	4	)	)	PUNCT
ejpam-4340	422	5	]	]	PUNCT
ejpam-4340	422	6	)	)	PUNCT
ejpam-4340	422	7	}	}	PUNCT
ejpam-4340	422	8	⇒	⇒	VERB
ejpam-4340	422	9	⇒	⇒	PROPN
ejpam-4340	422	10	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	422	11	)	)	PUNCT
ejpam-4340	423	1	⊆	⊆	NUM
ejpam-4340	423	2	f−1[cl(f	f−1[cl(f	PROPN
ejpam-4340	424	1	[	[	X
ejpam-4340	424	2	a	a	X
ejpam-4340	424	3	]	]	X
ejpam-4340	424	4	)	)	PUNCT
ejpam-4340	424	5	]	]	PUNCT
ejpam-4340	424	6	⇒	⇒	X
ejpam-4340	424	7	f	f	PROPN
ejpam-4340	425	1	[	[	X
ejpam-4340	425	2	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	425	3	)	)	PUNCT
ejpam-4340	425	4	]	]	PUNCT
ejpam-4340	426	1	⊆	⊆	NUM
ejpam-4340	426	2	cl(f	cl(f	NOUN
ejpam-4340	427	1	[	[	X
ejpam-4340	427	2	a	a	X
ejpam-4340	427	3	]	]	X
ejpam-4340	427	4	)	)	PUNCT
ejpam-4340	427	5	.	.	PUNCT
ejpam-4340	428	1	theorem	theorem	NOUN
ejpam-4340	428	2	19	19	NUM
ejpam-4340	428	3	.	.	PUNCT
ejpam-4340	429	1	let	let	VERB
ejpam-4340	429	2	f	f	NOUN
ejpam-4340	429	3	:	:	PUNCT
ejpam-4340	429	4	x	x	X
ejpam-4340	429	5	→	→	SYM
ejpam-4340	429	6	y	y	X
ejpam-4340	429	7	be	be	AUX
ejpam-4340	429	8	a	a	DET
ejpam-4340	429	9	function	function	NOUN
ejpam-4340	429	10	.	.	PUNCT
ejpam-4340	430	1	if	if	SCONJ
ejpam-4340	430	2	for	for	ADP
ejpam-4340	430	3	each	each	DET
ejpam-4340	430	4	point	point	NOUN
ejpam-4340	430	5	x	x	X
ejpam-4340	430	6	∈	∈	NOUN
ejpam-4340	430	7	x	x	X
ejpam-4340	430	8	and	and	CCONJ
ejpam-4340	430	9	each	each	DET
ejpam-4340	430	10	open	open	ADJ
ejpam-4340	430	11	set	set	VERB
ejpam-4340	430	12	v	v	NOUN
ejpam-4340	430	13	containing	contain	VERB
ejpam-4340	430	14	f(x	f(x	PROPN
ejpam-4340	430	15	)	)	PUNCT
ejpam-4340	430	16	there	there	PRON
ejpam-4340	430	17	exists	exist	VERB
ejpam-4340	430	18	a	a	DET
ejpam-4340	430	19	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	430	20	set	set	VERB
ejpam-4340	430	21	u	u	NOUN
ejpam-4340	430	22	containing	contain	VERB
ejpam-4340	430	23	x	x	PUNCT
ejpam-4340	430	24	such	such	ADJ
ejpam-4340	430	25	that	that	SCONJ
ejpam-4340	430	26	f	f	PROPN
ejpam-4340	431	1	[	[	X
ejpam-4340	431	2	u	u	X
ejpam-4340	431	3	]	]	PUNCT
ejpam-4340	431	4	⊆	⊆	NUM
ejpam-4340	431	5	v	v	NOUN
ejpam-4340	431	6	,	,	PUNCT
ejpam-4340	431	7	then	then	ADV
ejpam-4340	431	8	f	f	PROPN
ejpam-4340	432	1	[	[	X
ejpam-4340	432	2	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	432	3	)	)	PUNCT
ejpam-4340	432	4	]	]	PUNCT
ejpam-4340	433	1	⊆	⊆	NUM
ejpam-4340	433	2	cl(f	cl(f	NOUN
ejpam-4340	434	1	[	[	X
ejpam-4340	434	2	a	a	X
ejpam-4340	434	3	]	]	X
ejpam-4340	434	4	)	)	PUNCT
ejpam-4340	434	5	for	for	ADP
ejpam-4340	434	6	every	every	DET
ejpam-4340	434	7	subset	subset	NOUN
ejpam-4340	434	8	a	a	PRON
ejpam-4340	434	9	of	of	ADP
ejpam-4340	434	10	x.	x.	NOUN
ejpam-4340	434	11	proof	proof	NOUN
ejpam-4340	434	12	.	.	PUNCT
ejpam-4340	435	1	let	let	VERB
ejpam-4340	435	2	y	y	PROPN
ejpam-4340	435	3	∈	∈	PROPN
ejpam-4340	435	4	f	f	PROPN
ejpam-4340	436	1	[	[	X
ejpam-4340	436	2	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	436	3	)	)	PUNCT
ejpam-4340	436	4	]	]	PUNCT
ejpam-4340	436	5	.	.	PUNCT
ejpam-4340	437	1	y	y	PROPN
ejpam-4340	437	2	∈	∈	PROPN
ejpam-4340	437	3	f	f	X
ejpam-4340	438	1	[	[	X
ejpam-4340	438	2	gωe∗-cl(a)]⇒	gωe∗-cl(a)]⇒	X
ejpam-4340	438	3	(	(	PUNCT
ejpam-4340	438	4	∃x	∃x	PROPN
ejpam-4340	438	5	∈	∈	PROPN
ejpam-4340	438	6	gωe∗-cl(a))(f(x	gωe∗-cl(a))(f(x	NOUN
ejpam-4340	438	7	)	)	PUNCT
ejpam-4340	438	8	=	=	SYM
ejpam-4340	438	9	y	y	X
ejpam-4340	438	10	)	)	PUNCT
ejpam-4340	438	11	⇒	⇒	NOUN
ejpam-4340	438	12	(	(	PUNCT
ejpam-4340	438	13	∀u	∀u	NOUN
ejpam-4340	438	14	∈	∈	NOUN
ejpam-4340	438	15	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	438	16	,	,	PUNCT
ejpam-4340	438	17	x))(u	x))(u	PROPN
ejpam-4340	438	18	∩a	∩a	PROPN
ejpam-4340	438	19	̸=	̸=	PROPN
ejpam-4340	438	20	∅)(f(x	∅)(f(x	NOUN
ejpam-4340	438	21	)	)	PUNCT
ejpam-4340	438	22	=	=	SYM
ejpam-4340	438	23	y	y	X
ejpam-4340	438	24	)	)	PUNCT
ejpam-4340	438	25	hypothesis	hypothesis	NOUN
ejpam-4340	438	26	}	}	PUNCT
ejpam-4340	438	27	⇒	⇒	VERB
ejpam-4340	438	28	⇒	⇒	NOUN
ejpam-4340	438	29	(	(	PUNCT
ejpam-4340	438	30	∀v	∀v	PROPN
ejpam-4340	438	31	∈	∈	PROPN
ejpam-4340	438	32	o(y	o(y	PROPN
ejpam-4340	438	33	,	,	PUNCT
ejpam-4340	438	34	f(x)))(u	f(x)))(u	PROPN
ejpam-4340	438	35	∈	∈	PROPN
ejpam-4340	438	36	gωe∗o(x	gωe∗o(x	NOUN
ejpam-4340	438	37	,	,	PUNCT
ejpam-4340	438	38	x))(∅	x))(∅	ADJ
ejpam-4340	439	1	=	=	SYM
ejpam-4340	439	2	̸	̸	NUM
ejpam-4340	439	3	f	f	NOUN
ejpam-4340	440	1	[	[	X
ejpam-4340	440	2	u	u	X
ejpam-4340	440	3	∩a	∩a	X
ejpam-4340	440	4	]	]	PUNCT
ejpam-4340	440	5	⊆	⊆	NUM
ejpam-4340	440	6	f	f	X
ejpam-4340	441	1	[	[	X
ejpam-4340	441	2	u	u	X
ejpam-4340	441	3	]	]	PUNCT
ejpam-4340	441	4	∩	∩	PROPN
ejpam-4340	441	5	f	f	X
ejpam-4340	441	6	[	[	X
ejpam-4340	441	7	a	a	X
ejpam-4340	441	8	]	]	X
ejpam-4340	441	9	⊆	⊆	NUM
ejpam-4340	441	10	v	v	NOUN
ejpam-4340	441	11	∩	∩	ADJ
ejpam-4340	441	12	f	f	X
ejpam-4340	442	1	[	[	X
ejpam-4340	442	2	a	a	X
ejpam-4340	442	3	]	]	X
ejpam-4340	442	4	)	)	PUNCT
ejpam-4340	442	5	⇒	⇒	NOUN
ejpam-4340	442	6	y	y	PROPN
ejpam-4340	442	7	=	=	SYM
ejpam-4340	442	8	f(x	f(x	PROPN
ejpam-4340	442	9	)	)	PUNCT
ejpam-4340	442	10	∈	∈	PROPN
ejpam-4340	442	11	cl(f	cl(f	NOUN
ejpam-4340	443	1	[	[	X
ejpam-4340	443	2	a	a	X
ejpam-4340	443	3	]	]	X
ejpam-4340	443	4	)	)	PUNCT
ejpam-4340	443	5	.	.	PUNCT
ejpam-4340	444	1	p.	p.	NOUN
ejpam-4340	444	2	şaşmaz	şaşmaz	NUM
ejpam-4340	444	3	,	,	PUNCT
ejpam-4340	444	4	m.	m.	NOUN
ejpam-4340	444	5	özkoç	özkoç	PROPN
ejpam-4340	444	6	/	/	SYM
ejpam-4340	444	7	eur	eur	PROPN
ejpam-4340	444	8	.	.	PUNCT
ejpam-4340	445	1	j.	j.	PROPN
ejpam-4340	445	2	pure	pure	PROPN
ejpam-4340	445	3	appl	appl	PROPN
ejpam-4340	445	4	.	.	PROPN
ejpam-4340	445	5	math	math	PROPN
ejpam-4340	445	6	,	,	PUNCT
ejpam-4340	445	7	15	15	NUM
ejpam-4340	445	8	(	(	PUNCT
ejpam-4340	445	9	2	2	NUM
ejpam-4340	445	10	)	)	PUNCT
ejpam-4340	445	11	(	(	PUNCT
ejpam-4340	445	12	2022	2022	NUM
ejpam-4340	445	13	)	)	PUNCT
ejpam-4340	445	14	,	,	PUNCT
ejpam-4340	445	15	354	354	NUM
ejpam-4340	445	16	-	-	SYM
ejpam-4340	445	17	374	374	NUM
ejpam-4340	445	18	368	368	NUM
ejpam-4340	445	19	theorem	theorem	NOUN
ejpam-4340	445	20	20	20	NUM
ejpam-4340	445	21	.	.	PUNCT
ejpam-4340	446	1	let	let	VERB
ejpam-4340	446	2	f	f	NOUN
ejpam-4340	446	3	:	:	PUNCT
ejpam-4340	446	4	x	x	X
ejpam-4340	446	5	→	→	SYM
ejpam-4340	446	6	y	y	X
ejpam-4340	446	7	be	be	AUX
ejpam-4340	446	8	a	a	DET
ejpam-4340	446	9	function	function	NOUN
ejpam-4340	446	10	.	.	PUNCT
ejpam-4340	447	1	then	then	ADV
ejpam-4340	447	2	the	the	DET
ejpam-4340	447	3	following	follow	VERB
ejpam-4340	447	4	statements	statement	NOUN
ejpam-4340	447	5	are	be	AUX
ejpam-4340	447	6	equivalent	equivalent	ADJ
ejpam-4340	447	7	:	:	PUNCT
ejpam-4340	447	8	(	(	PUNCT
ejpam-4340	447	9	a	a	X
ejpam-4340	447	10	)	)	PUNCT
ejpam-4340	447	11	f	f	NOUN
ejpam-4340	448	1	[	[	X
ejpam-4340	448	2	gωe∗-cl(a	gωe∗-cl(a	NOUN
ejpam-4340	448	3	)	)	PUNCT
ejpam-4340	448	4	]	]	PUNCT
ejpam-4340	449	1	⊆	⊆	NUM
ejpam-4340	449	2	cl(f	cl(f	NOUN
ejpam-4340	450	1	[	[	X
ejpam-4340	450	2	a	a	X
ejpam-4340	450	3	]	]	X
ejpam-4340	450	4	)	)	PUNCT
ejpam-4340	450	5	for	for	ADP
ejpam-4340	450	6	every	every	DET
ejpam-4340	450	7	subset	subset	NOUN
ejpam-4340	450	8	a	a	PRON
ejpam-4340	450	9	of	of	ADP
ejpam-4340	450	10	x	x	PRON
ejpam-4340	450	11	;	;	PUNCT
ejpam-4340	450	12	(	(	PUNCT
ejpam-4340	450	13	b	b	X
ejpam-4340	450	14	)	)	PUNCT
ejpam-4340	450	15	if	if	SCONJ
ejpam-4340	450	16	τ∗ωe∗	τ∗ωe∗	ADJ
ejpam-4340	450	17	is	be	AUX
ejpam-4340	450	18	a	a	DET
ejpam-4340	450	19	topology	topology	NOUN
ejpam-4340	450	20	on	on	ADP
ejpam-4340	450	21	x	x	NOUN
ejpam-4340	450	22	,	,	PUNCT
ejpam-4340	450	23	then	then	ADV
ejpam-4340	450	24	f	f	X
ejpam-4340	450	25	:	:	PUNCT
ejpam-4340	450	26	(	(	PUNCT
ejpam-4340	450	27	x	x	NOUN
ejpam-4340	450	28	,	,	PUNCT
ejpam-4340	450	29	τ∗ωe∗)→	τ∗ωe∗)→	PROPN
ejpam-4340	450	30	(	(	PUNCT
ejpam-4340	450	31	y	y	PROPN
ejpam-4340	450	32	,	,	PUNCT
ejpam-4340	450	33	σ	σ	PROPN
ejpam-4340	450	34	)	)	PUNCT
ejpam-4340	450	35	is	be	AUX
ejpam-4340	450	36	continuous	continuous	ADJ
ejpam-4340	450	37	.	.	PUNCT
ejpam-4340	451	1	proof	proof	NOUN
ejpam-4340	451	2	.	.	PUNCT
ejpam-4340	452	1	(	(	PUNCT
ejpam-4340	452	2	a)⇒	a)⇒	PROPN
ejpam-4340	452	3	(	(	PUNCT
ejpam-4340	452	4	b	b	NOUN
ejpam-4340	452	5	)	)	PUNCT
ejpam-4340	452	6	:	:	PUNCT
ejpam-4340	452	7	let	let	VERB
ejpam-4340	452	8	a	a	DET
ejpam-4340	452	9	∈	∈	PROPN
ejpam-4340	452	10	c(y	c(y	PROPN
ejpam-4340	452	11	)	)	PUNCT
ejpam-4340	452	12	.	.	PUNCT
ejpam-4340	453	1	a	a	DET
ejpam-4340	453	2	∈	∈	PROPN
ejpam-4340	453	3	c(y	c(y	PROPN
ejpam-4340	453	4	)	)	PUNCT
ejpam-4340	453	5	⇒	⇒	PROPN
ejpam-4340	453	6	f−1[a	f−1[a	PROPN
ejpam-4340	453	7	]	]	PUNCT
ejpam-4340	453	8	⊆	⊆	NUM
ejpam-4340	453	9	x	x	SYM
ejpam-4340	453	10	hypothesis	hypothesis	NOUN
ejpam-4340	453	11	}	}	PUNCT
ejpam-4340	453	12	⇒	⇒	VERB
ejpam-4340	453	13	f	f	PROPN
ejpam-4340	454	1	[	[	X
ejpam-4340	454	2	gωe∗-cl(f−1[a	gωe∗-cl(f−1[a	NOUN
ejpam-4340	454	3	]	]	X
ejpam-4340	454	4	)	)	PUNCT
ejpam-4340	454	5	]	]	PUNCT
ejpam-4340	455	1	⊆	⊆	NUM
ejpam-4340	455	2	cl(f	cl(f	NOUN
ejpam-4340	455	3	[	[	X
ejpam-4340	455	4	f−1[a	f−1[a	X
ejpam-4340	455	5	]	]	X
ejpam-4340	455	6	]	]	X
ejpam-4340	455	7	)	)	PUNCT
ejpam-4340	455	8	⊆	⊆	NUM
ejpam-4340	455	9	cl(a	cl(a	NUM
ejpam-4340	455	10	)	)	PUNCT
ejpam-4340	455	11	=	=	SYM
ejpam-4340	455	12	a	a	DET
ejpam-4340	455	13	⇒	⇒	X
ejpam-4340	455	14	gωe∗-cl(f−1[a	gωe∗-cl(f−1[a	NOUN
ejpam-4340	455	15	]	]	PUNCT
ejpam-4340	455	16	)	)	PUNCT
ejpam-4340	455	17	⊆	⊆	NUM
ejpam-4340	455	18	f−1[a	f−1[a	NUM
ejpam-4340	455	19	]	]	X
ejpam-4340	455	20	f−1[a	f−1[a	X
ejpam-4340	455	21	]	]	X
ejpam-4340	455	22	⊆	⊆	NUM
ejpam-4340	455	23	gωe∗-cl(f−1[a	gωe∗-cl(f−1[a	NOUN
ejpam-4340	455	24	]	]	X
ejpam-4340	455	25	)	)	PUNCT
ejpam-4340	455	26	}	}	PUNCT
ejpam-4340	455	27	⇒	⇒	VERB
ejpam-4340	455	28	f−1[a	f−1[a	PROPN
ejpam-4340	455	29	]	]	PUNCT
ejpam-4340	455	30	=	=	SYM
ejpam-4340	455	31	gωe∗-cl(f−1[a	gωe∗-cl(f−1[a	NOUN
ejpam-4340	455	32	]	]	PUNCT
ejpam-4340	455	33	)	)	PUNCT
ejpam-4340	455	34	⇒	⇒	NOUN
ejpam-4340	455	35	x	x	PUNCT
ejpam-4340	455	36	\	\	X
ejpam-4340	455	37	f−1[a	f−1[a	PROPN
ejpam-4340	455	38	]	]	PUNCT
ejpam-4340	455	39	∈	∈	PROPN
ejpam-4340	455	40	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	455	41	⇒	⇒	PROPN
ejpam-4340	455	42	f−1[a	f−1[a	PROPN
ejpam-4340	455	43	]	]	X
ejpam-4340	455	44	∈	∈	PROPN
ejpam-4340	455	45	c(x	c(x	NOUN
ejpam-4340	455	46	,	,	PUNCT
ejpam-4340	455	47	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	455	48	)	)	PUNCT
ejpam-4340	455	49	.	.	PUNCT
ejpam-4340	456	1	(	(	PUNCT
ejpam-4340	456	2	b)⇒	b)⇒	NOUN
ejpam-4340	456	3	(	(	PUNCT
ejpam-4340	456	4	a	a	NOUN
ejpam-4340	456	5	)	)	PUNCT
ejpam-4340	456	6	:	:	PUNCT
ejpam-4340	456	7	let	let	VERB
ejpam-4340	456	8	a	a	DET
ejpam-4340	456	9	⊆	⊆	NUM
ejpam-4340	456	10	x.	x.	NOUN
ejpam-4340	456	11	a	a	DET
ejpam-4340	456	12	⊆	⊆	NUM
ejpam-4340	456	13	x	x	SYM
ejpam-4340	456	14	⇒	⇒	NOUN
ejpam-4340	456	15	cl(f	cl(f	PROPN
ejpam-4340	457	1	[	[	X
ejpam-4340	457	2	a	a	X
ejpam-4340	457	3	]	]	X
ejpam-4340	457	4	)	)	PUNCT
ejpam-4340	457	5	∈	∈	PROPN
ejpam-4340	457	6	c(y	c(y	PROPN
ejpam-4340	457	7	)	)	PUNCT
ejpam-4340	457	8	hypothesis	hypothesis	NOUN
ejpam-4340	457	9	}	}	PUNCT
ejpam-4340	457	10	⇒	⇒	VERB
ejpam-4340	457	11	x	x	PUNCT
ejpam-4340	457	12	\	\	X
ejpam-4340	458	1	f−1[cl(f	f−1[cl(f	PROPN
ejpam-4340	459	1	[	[	X
ejpam-4340	459	2	a	a	X
ejpam-4340	459	3	]	]	X
ejpam-4340	459	4	)	)	PUNCT
ejpam-4340	459	5	]	]	PUNCT
ejpam-4340	459	6	∈	∈	PROPN
ejpam-4340	459	7	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	459	8	⇒	⇒	PROPN
ejpam-4340	459	9	f−1[cl(f	f−1[cl(f	PROPN
ejpam-4340	460	1	[	[	X
ejpam-4340	460	2	a	a	X
ejpam-4340	460	3	]	]	X
ejpam-4340	460	4	)	)	PUNCT
ejpam-4340	460	5	]	]	PUNCT
ejpam-4340	460	6	∈	∈	PROPN
ejpam-4340	460	7	c(x	c(x	NOUN
ejpam-4340	460	8	,	,	PUNCT
ejpam-4340	460	9	τ∗ωe∗	τ∗ωe∗	NOUN
ejpam-4340	460	10	)	)	PUNCT
ejpam-4340	460	11	⇒	⇒	PROPN
ejpam-4340	460	12	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	460	13	)	)	PUNCT
ejpam-4340	460	14	⊆	⊆	NUM
ejpam-4340	460	15	gωe∗-cl(f−1[cl(f	gωe∗-cl(f−1[cl(f	NOUN
ejpam-4340	461	1	[	[	X
ejpam-4340	461	2	a	a	X
ejpam-4340	461	3	]	]	X
ejpam-4340	461	4	)	)	PUNCT
ejpam-4340	461	5	]	]	PUNCT
ejpam-4340	461	6	)	)	PUNCT
ejpam-4340	462	1	=	=	PUNCT
ejpam-4340	462	2	f−1[cl(f	f−1[cl(f	PROPN
ejpam-4340	463	1	[	[	X
ejpam-4340	463	2	a	a	X
ejpam-4340	463	3	]	]	X
ejpam-4340	463	4	)	)	PUNCT
ejpam-4340	463	5	]	]	PUNCT
ejpam-4340	463	6	⇒	⇒	X
ejpam-4340	463	7	f	f	PROPN
ejpam-4340	464	1	[	[	X
ejpam-4340	464	2	gωe∗-cl(a	gωe∗-cl(a	PROPN
ejpam-4340	464	3	)	)	PUNCT
ejpam-4340	464	4	]	]	PUNCT
ejpam-4340	465	1	⊆	⊆	NUM
ejpam-4340	465	2	cl(f	cl(f	NOUN
ejpam-4340	466	1	[	[	X
ejpam-4340	466	2	a	a	X
ejpam-4340	466	3	]	]	X
ejpam-4340	466	4	)	)	PUNCT
ejpam-4340	466	5	.	.	PUNCT
ejpam-4340	467	1	definition	definition	NOUN
ejpam-4340	467	2	19	19	NUM
ejpam-4340	467	3	.	.	PUNCT
ejpam-4340	468	1	a	a	DET
ejpam-4340	468	2	function	function	NOUN
ejpam-4340	468	3	f	f	NOUN
ejpam-4340	468	4	:	:	PUNCT
ejpam-4340	468	5	x	x	X
ejpam-4340	468	6	→	→	SYM
ejpam-4340	468	7	y	y	PROPN
ejpam-4340	468	8	is	be	AUX
ejpam-4340	468	9	said	say	VERB
ejpam-4340	468	10	to	to	PART
ejpam-4340	468	11	be	be	AUX
ejpam-4340	468	12	pre	pre	ADJ
ejpam-4340	468	13	-	-	ADJ
ejpam-4340	468	14	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	468	15	if	if	SCONJ
ejpam-4340	468	16	f	f	PROPN
ejpam-4340	468	17	[	[	X
ejpam-4340	468	18	f	f	X
ejpam-4340	468	19	]	]	X
ejpam-4340	468	20	is	be	AUX
ejpam-4340	468	21	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	468	22	in	in	ADP
ejpam-4340	468	23	y	y	PROPN
ejpam-4340	468	24	for	for	ADP
ejpam-4340	468	25	every	every	DET
ejpam-4340	468	26	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	468	27	set	set	VERB
ejpam-4340	468	28	f	f	PROPN
ejpam-4340	468	29	of	of	ADP
ejpam-4340	468	30	x.	x.	PROPN
ejpam-4340	468	31	definition	definition	NOUN
ejpam-4340	468	32	20	20	NUM
ejpam-4340	468	33	.	.	PUNCT
ejpam-4340	469	1	a	a	DET
ejpam-4340	469	2	function	function	NOUN
ejpam-4340	469	3	f	f	NOUN
ejpam-4340	469	4	:	:	PUNCT
ejpam-4340	469	5	x	x	X
ejpam-4340	469	6	→	→	SYM
ejpam-4340	469	7	y	y	PROPN
ejpam-4340	469	8	is	be	AUX
ejpam-4340	469	9	said	say	VERB
ejpam-4340	469	10	to	to	PART
ejpam-4340	469	11	be	be	AUX
ejpam-4340	469	12	pre	pre	VERB
ejpam-4340	469	13	-	-	ADJ
ejpam-4340	469	14	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	469	15	if	if	SCONJ
ejpam-4340	469	16	f	f	PROPN
ejpam-4340	469	17	[	[	X
ejpam-4340	469	18	u	u	X
ejpam-4340	469	19	]	]	PUNCT
ejpam-4340	469	20	is	be	AUX
ejpam-4340	469	21	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	469	22	in	in	ADP
ejpam-4340	469	23	y	y	PROPN
ejpam-4340	469	24	for	for	ADP
ejpam-4340	469	25	every	every	DET
ejpam-4340	469	26	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	469	27	set	set	NOUN
ejpam-4340	469	28	u	u	NOUN
ejpam-4340	469	29	of	of	ADP
ejpam-4340	469	30	x.	x.	PROPN
ejpam-4340	469	31	theorem	theorem	VERB
ejpam-4340	469	32	21	21	NUM
ejpam-4340	469	33	.	.	PUNCT
ejpam-4340	470	1	let	let	VERB
ejpam-4340	470	2	f	f	NOUN
ejpam-4340	470	3	:	:	PUNCT
ejpam-4340	470	4	x	x	X
ejpam-4340	470	5	→	→	SYM
ejpam-4340	470	6	y	y	X
ejpam-4340	470	7	be	be	AUX
ejpam-4340	470	8	a	a	DET
ejpam-4340	470	9	function	function	NOUN
ejpam-4340	470	10	.	.	PUNCT
ejpam-4340	471	1	if	if	SCONJ
ejpam-4340	471	2	f	f	PROPN
ejpam-4340	471	3	is	be	AUX
ejpam-4340	471	4	continuous	continuous	ADJ
ejpam-4340	471	5	and	and	CCONJ
ejpam-4340	471	6	pre	pre	ADJ
ejpam-4340	471	7	-	-	ADJ
ejpam-4340	471	8	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	471	9	,	,	PUNCT
ejpam-4340	471	10	then	then	ADV
ejpam-4340	471	11	f	f	PROPN
ejpam-4340	471	12	is	be	AUX
ejpam-4340	471	13	pre	pre	ADJ
ejpam-4340	471	14	-	-	ADJ
ejpam-4340	471	15	gωe∗-open	gωe∗-open	ADJ
ejpam-4340	471	16	.	.	PUNCT
ejpam-4340	472	1	proof	proof	NOUN
ejpam-4340	472	2	.	.	PUNCT
ejpam-4340	473	1	let	let	VERB
ejpam-4340	473	2	a	a	DET
ejpam-4340	473	3	∈	∈	PROPN
ejpam-4340	473	4	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	473	5	)	)	PUNCT
ejpam-4340	473	6	and	and	CCONJ
ejpam-4340	473	7	f	f	X
ejpam-4340	474	1	[	[	X
ejpam-4340	474	2	a	a	X
ejpam-4340	474	3	]	]	X
ejpam-4340	474	4	⊆	⊆	NUM
ejpam-4340	474	5	u	u	NOUN
ejpam-4340	474	6	∈	∈	NOUN
ejpam-4340	474	7	o(y	o(y	PROPN
ejpam-4340	474	8	)	)	PUNCT
ejpam-4340	474	9	.	.	PUNCT
ejpam-4340	475	1	(	(	PUNCT
ejpam-4340	475	2	a	a	DET
ejpam-4340	475	3	∈	∈	NOUN
ejpam-4340	475	4	gωe∗c(x))(f	gωe∗c(x))(f	NOUN
ejpam-4340	476	1	[	[	X
ejpam-4340	476	2	a	a	X
ejpam-4340	476	3	]	]	X
ejpam-4340	476	4	⊆	⊆	NUM
ejpam-4340	476	5	u	u	NOUN
ejpam-4340	476	6	∈	∈	NOUN
ejpam-4340	476	7	o(y	o(y	PROPN
ejpam-4340	476	8	)	)	PUNCT
ejpam-4340	476	9	)	)	PUNCT
ejpam-4340	477	1	f	f	PROPN
ejpam-4340	477	2	is	be	AUX
ejpam-4340	477	3	continuous	continuous	ADJ
ejpam-4340	477	4	}	}	PUNCT
ejpam-4340	477	5	⇒	⇒	NOUN
ejpam-4340	477	6	⇒	⇒	NOUN
ejpam-4340	477	7	(	(	PUNCT
ejpam-4340	477	8	a	a	DET
ejpam-4340	477	9	⊆	⊆	NUM
ejpam-4340	477	10	f−1[u	f−1[u	NOUN
ejpam-4340	477	11	]	]	PUNCT
ejpam-4340	477	12	∈	∈	NOUN
ejpam-4340	477	13	o(x))(ωe∗-cl(a	o(x))(ωe∗-cl(a	NOUN
ejpam-4340	477	14	)	)	PUNCT
ejpam-4340	477	15	⊆	⊆	NUM
ejpam-4340	477	16	f−1[u	f−1[u	NOUN
ejpam-4340	477	17	]	]	NOUN
ejpam-4340	477	18	)	)	PUNCT
ejpam-4340	477	19	⇒	⇒	NOUN
ejpam-4340	477	20	f	f	X
ejpam-4340	478	1	[	[	X
ejpam-4340	478	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	478	3	)	)	PUNCT
ejpam-4340	478	4	]	]	PUNCT
ejpam-4340	478	5	⊆	⊆	NUM
ejpam-4340	478	6	u	u	NOUN
ejpam-4340	478	7	f	f	PROPN
ejpam-4340	478	8	is	be	AUX
ejpam-4340	478	9	pre	pre	ADJ
ejpam-4340	478	10	-	-	ADJ
ejpam-4340	478	11	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	478	12	function	function	NOUN
ejpam-4340	478	13	}	}	PUNCT
ejpam-4340	478	14	⇒	⇒	VERB
ejpam-4340	478	15	⇒	⇒	NOUN
ejpam-4340	478	16	ωe∗-cl(f	ωe∗-cl(f	PROPN
ejpam-4340	479	1	[	[	X
ejpam-4340	479	2	a	a	X
ejpam-4340	479	3	]	]	X
ejpam-4340	479	4	)	)	PUNCT
ejpam-4340	479	5	⊆	⊆	NUM
ejpam-4340	479	6	ωe∗-cl(f	ωe∗-cl(f	NOUN
ejpam-4340	479	7	[	[	X
ejpam-4340	479	8	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	479	9	)	)	PUNCT
ejpam-4340	479	10	]	]	PUNCT
ejpam-4340	479	11	)	)	PUNCT
ejpam-4340	480	1	=	=	PUNCT
ejpam-4340	480	2	f	f	X
ejpam-4340	481	1	[	[	X
ejpam-4340	481	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	481	3	)	)	PUNCT
ejpam-4340	481	4	]	]	PUNCT
ejpam-4340	481	5	⊆	⊆	NUM
ejpam-4340	481	6	u.	u.	NOUN
ejpam-4340	481	7	definition	definition	NOUN
ejpam-4340	481	8	21	21	NUM
ejpam-4340	481	9	.	.	PUNCT
ejpam-4340	482	1	a	a	DET
ejpam-4340	482	2	function	function	NOUN
ejpam-4340	482	3	f	f	NOUN
ejpam-4340	482	4	:	:	PUNCT
ejpam-4340	482	5	x	x	X
ejpam-4340	482	6	→	→	SYM
ejpam-4340	482	7	y	y	PROPN
ejpam-4340	482	8	is	be	AUX
ejpam-4340	482	9	said	say	VERB
ejpam-4340	482	10	to	to	PART
ejpam-4340	482	11	be	be	AUX
ejpam-4340	482	12	gωe∗-irresolute	gωe∗-irresolute	NOUN
ejpam-4340	482	13	if	if	SCONJ
ejpam-4340	482	14	f−1[v	f−1[v	NOUN
ejpam-4340	482	15	]	]	PUNCT
ejpam-4340	482	16	is	be	AUX
ejpam-4340	482	17	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	482	18	in	in	ADP
ejpam-4340	482	19	x	x	PUNCT
ejpam-4340	482	20	for	for	ADP
ejpam-4340	482	21	every	every	DET
ejpam-4340	482	22	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	482	23	set	set	VERB
ejpam-4340	482	24	v	v	NOUN
ejpam-4340	482	25	of	of	ADP
ejpam-4340	482	26	y.	y.	PROPN
ejpam-4340	482	27	corollary	corollary	PROPN
ejpam-4340	482	28	5	5	NUM
ejpam-4340	482	29	.	.	PUNCT
ejpam-4340	483	1	let	let	VERB
ejpam-4340	483	2	f	f	NOUN
ejpam-4340	483	3	:	:	PUNCT
ejpam-4340	483	4	x	x	X
ejpam-4340	483	5	→	→	SYM
ejpam-4340	483	6	y	y	X
ejpam-4340	483	7	be	be	AUX
ejpam-4340	483	8	a	a	DET
ejpam-4340	483	9	function	function	NOUN
ejpam-4340	483	10	.	.	PUNCT
ejpam-4340	484	1	f	f	PROPN
ejpam-4340	484	2	is	be	AUX
ejpam-4340	484	3	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	484	4	if	if	SCONJ
ejpam-4340	484	5	f−1[v	f−1[v	NOUN
ejpam-4340	484	6	]	]	PUNCT
ejpam-4340	484	7	is	be	AUX
ejpam-4340	484	8	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	484	9	in	in	ADP
ejpam-4340	484	10	x	x	PUNCT
ejpam-4340	484	11	for	for	SCONJ
ejpam-4340	484	12	every	every	DET
ejpam-4340	484	13	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	484	14	set	set	VERB
ejpam-4340	484	15	v	v	NOUN
ejpam-4340	484	16	of	of	ADP
ejpam-4340	484	17	y.	y.	PROPN
ejpam-4340	484	18	proposition	proposition	NOUN
ejpam-4340	484	19	12	12	NUM
ejpam-4340	484	20	.	.	PUNCT
ejpam-4340	485	1	let	let	VERB
ejpam-4340	485	2	f	f	NOUN
ejpam-4340	485	3	:	:	PUNCT
ejpam-4340	485	4	(	(	PUNCT
ejpam-4340	485	5	x	x	X
ejpam-4340	485	6	,	,	PUNCT
ejpam-4340	485	7	τ	τ	X
ejpam-4340	485	8	)	)	PUNCT
ejpam-4340	485	9	→	→	SYM
ejpam-4340	485	10	(	(	PUNCT
ejpam-4340	485	11	y	y	PROPN
ejpam-4340	485	12	,	,	PUNCT
ejpam-4340	485	13	σ	σ	PROPN
ejpam-4340	485	14	)	)	PUNCT
ejpam-4340	485	15	be	be	AUX
ejpam-4340	485	16	a	a	DET
ejpam-4340	485	17	function	function	NOUN
ejpam-4340	485	18	.	.	PUNCT
ejpam-4340	486	1	if	if	SCONJ
ejpam-4340	486	2	f	f	PROPN
ejpam-4340	486	3	is	be	AUX
ejpam-4340	486	4	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	486	5	and	and	CCONJ
ejpam-4340	486	6	σ∗	σ∗	ADJ
ejpam-4340	486	7	ωe∗	ωe∗	NOUN
ejpam-4340	486	8	=	=	PROPN
ejpam-4340	486	9	σ	σ	PROPN
ejpam-4340	486	10	holds	hold	VERB
ejpam-4340	486	11	,	,	PUNCT
ejpam-4340	486	12	then	then	ADV
ejpam-4340	486	13	f	f	PROPN
ejpam-4340	486	14	is	be	AUX
ejpam-4340	486	15	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	486	16	.	.	PUNCT
ejpam-4340	487	1	the	the	DET
ejpam-4340	487	2	proof	proof	NOUN
ejpam-4340	487	3	follows	follow	VERB
ejpam-4340	487	4	from	from	ADP
ejpam-4340	487	5	remark	remark	NOUN
ejpam-4340	487	6	5	5	NUM
ejpam-4340	487	7	.	.	PUNCT
ejpam-4340	488	1	p.	p.	NOUN
ejpam-4340	488	2	şaşmaz	şaşmaz	NUM
ejpam-4340	488	3	,	,	PUNCT
ejpam-4340	488	4	m.	m.	NOUN
ejpam-4340	488	5	özkoç	özkoç	PROPN
ejpam-4340	488	6	/	/	SYM
ejpam-4340	488	7	eur	eur	PROPN
ejpam-4340	488	8	.	.	PUNCT
ejpam-4340	489	1	j.	j.	PROPN
ejpam-4340	489	2	pure	pure	PROPN
ejpam-4340	489	3	appl	appl	PROPN
ejpam-4340	489	4	.	.	PROPN
ejpam-4340	489	5	math	math	PROPN
ejpam-4340	489	6	,	,	PUNCT
ejpam-4340	489	7	15	15	NUM
ejpam-4340	489	8	(	(	PUNCT
ejpam-4340	489	9	2	2	NUM
ejpam-4340	489	10	)	)	PUNCT
ejpam-4340	489	11	(	(	PUNCT
ejpam-4340	489	12	2022	2022	NUM
ejpam-4340	489	13	)	)	PUNCT
ejpam-4340	489	14	,	,	PUNCT
ejpam-4340	489	15	354	354	NUM
ejpam-4340	489	16	-	-	SYM
ejpam-4340	489	17	374	374	NUM
ejpam-4340	489	18	369	369	NUM
ejpam-4340	489	19	theorem	theorem	NOUN
ejpam-4340	489	20	22	22	NUM
ejpam-4340	489	21	.	.	PUNCT
ejpam-4340	490	1	let	let	VERB
ejpam-4340	490	2	f	f	NOUN
ejpam-4340	490	3	:	:	PUNCT
ejpam-4340	490	4	x	x	X
ejpam-4340	490	5	→	→	SYM
ejpam-4340	490	6	y	y	X
ejpam-4340	490	7	be	be	AUX
ejpam-4340	490	8	a	a	DET
ejpam-4340	490	9	function	function	NOUN
ejpam-4340	490	10	.	.	PUNCT
ejpam-4340	491	1	if	if	SCONJ
ejpam-4340	491	2	f	f	PROPN
ejpam-4340	491	3	is	be	AUX
ejpam-4340	491	4	an	an	DET
ejpam-4340	491	5	ωe∗-irresolute	ωe∗-irresolute	NOUN
ejpam-4340	491	6	open	open	ADJ
ejpam-4340	491	7	bijection	bijection	NOUN
ejpam-4340	491	8	,	,	PUNCT
ejpam-4340	491	9	then	then	ADV
ejpam-4340	491	10	f	f	PROPN
ejpam-4340	491	11	is	be	AUX
ejpam-4340	491	12	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	491	13	.	.	PUNCT
ejpam-4340	492	1	proof	proof	NOUN
ejpam-4340	492	2	.	.	PUNCT
ejpam-4340	493	1	let	let	VERB
ejpam-4340	493	2	f	f	PROPN
ejpam-4340	493	3	∈	∈	PROPN
ejpam-4340	493	4	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	493	5	)	)	PUNCT
ejpam-4340	493	6	and	and	CCONJ
ejpam-4340	493	7	f−1[f	f−1[f	VERB
ejpam-4340	493	8	]	]	PUNCT
ejpam-4340	494	1	⊆	⊆	NUM
ejpam-4340	494	2	u	u	NOUN
ejpam-4340	494	3	∈	∈	PROPN
ejpam-4340	494	4	o(x	o(x	PROPN
ejpam-4340	494	5	)	)	PUNCT
ejpam-4340	494	6	.	.	PUNCT
ejpam-4340	495	1	f−1[f	f−1[f	VERB
ejpam-4340	495	2	]	]	PUNCT
ejpam-4340	496	1	⊆	⊆	NUM
ejpam-4340	496	2	u	u	NOUN
ejpam-4340	496	3	∈	∈	PROPN
ejpam-4340	496	4	o(x	o(x	PROPN
ejpam-4340	496	5	)	)	PUNCT
ejpam-4340	496	6	f	f	PROPN
ejpam-4340	496	7	is	be	AUX
ejpam-4340	496	8	open	open	ADJ
ejpam-4340	496	9	bijection	bijection	NOUN
ejpam-4340	496	10	}	}	PUNCT
ejpam-4340	496	11	⇒	⇒	VERB
ejpam-4340	496	12	f	f	PROPN
ejpam-4340	496	13	⊆	⊆	NUM
ejpam-4340	496	14	f	f	X
ejpam-4340	497	1	[	[	X
ejpam-4340	497	2	u	u	X
ejpam-4340	497	3	]	]	X
ejpam-4340	497	4	∈	∈	PROPN
ejpam-4340	497	5	o(y	o(y	PROPN
ejpam-4340	497	6	)	)	PUNCT
ejpam-4340	498	1	f	f	PROPN
ejpam-4340	498	2	∈	∈	PROPN
ejpam-4340	498	3	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	498	4	)	)	PUNCT
ejpam-4340	498	5	}	}	PUNCT
ejpam-4340	498	6	⇒	⇒	VERB
ejpam-4340	498	7	ωe∗-cl(f	ωe∗-cl(f	NUM
ejpam-4340	498	8	)	)	PUNCT
ejpam-4340	499	1	⊆	⊆	NUM
ejpam-4340	499	2	f	f	X
ejpam-4340	499	3	[	[	X
ejpam-4340	499	4	u	u	X
ejpam-4340	499	5	]	]	PUNCT
ejpam-4340	499	6	⇒	⇒	PROPN
ejpam-4340	499	7	f−1[ωe∗-cl(f	f−1[ωe∗-cl(f	PROPN
ejpam-4340	499	8	)	)	PUNCT
ejpam-4340	499	9	]	]	PUNCT
ejpam-4340	500	1	⊆	⊆	NUM
ejpam-4340	500	2	u	u	NOUN
ejpam-4340	500	3	f	f	PROPN
ejpam-4340	500	4	is	be	AUX
ejpam-4340	500	5	ωe∗-irresolute	ωe∗-irresolute	NOUN
ejpam-4340	500	6	}	}	PUNCT
ejpam-4340	500	7	⇒	⇒	VERB
ejpam-4340	500	8	⇒	⇒	NOUN
ejpam-4340	500	9	ωe∗-cl(f−1[f	ωe∗-cl(f−1[f	ADV
ejpam-4340	500	10	]	]	X
ejpam-4340	500	11	)	)	PUNCT
ejpam-4340	501	1	⊆	⊆	NUM
ejpam-4340	501	2	ωe∗-cl(f−1[ωe∗-cl(f	ωe∗-cl(f−1[ωe∗-cl(f	PROPN
ejpam-4340	501	3	)	)	PUNCT
ejpam-4340	501	4	]	]	X
ejpam-4340	501	5	)	)	PUNCT
ejpam-4340	501	6	=	=	SYM
ejpam-4340	501	7	f−1[ωe∗-cl(f	f−1[ωe∗-cl(f	NOUN
ejpam-4340	501	8	)	)	PUNCT
ejpam-4340	501	9	]	]	PUNCT
ejpam-4340	502	1	⊆	⊆	NUM
ejpam-4340	502	2	u.	u.	NOUN
ejpam-4340	502	3	theorem	theorem	VERB
ejpam-4340	502	4	23	23	NUM
ejpam-4340	502	5	.	.	PUNCT
ejpam-4340	503	1	let	let	VERB
ejpam-4340	503	2	f	f	NOUN
ejpam-4340	503	3	:	:	PUNCT
ejpam-4340	503	4	x	x	X
ejpam-4340	503	5	→	→	SYM
ejpam-4340	503	6	y	y	PROPN
ejpam-4340	503	7	and	and	CCONJ
ejpam-4340	503	8	g	g	PROPN
ejpam-4340	503	9	:	:	PUNCT
ejpam-4340	503	10	y	y	PROPN
ejpam-4340	503	11	→	→	SYM
ejpam-4340	503	12	z	z	X
ejpam-4340	503	13	be	be	AUX
ejpam-4340	503	14	any	any	DET
ejpam-4340	503	15	two	two	NUM
ejpam-4340	503	16	functions	function	NOUN
ejpam-4340	503	17	.	.	PUNCT
ejpam-4340	504	1	then	then	ADV
ejpam-4340	504	2	the	the	DET
ejpam-4340	504	3	following	follow	VERB
ejpam-4340	504	4	properties	property	NOUN
ejpam-4340	504	5	hold	hold	VERB
ejpam-4340	504	6	:	:	PUNCT
ejpam-4340	504	7	(	(	PUNCT
ejpam-4340	504	8	a	a	X
ejpam-4340	504	9	)	)	PUNCT
ejpam-4340	504	10	if	if	SCONJ
ejpam-4340	504	11	g	g	PROPN
ejpam-4340	504	12	is	be	AUX
ejpam-4340	504	13	continuous	continuous	ADJ
ejpam-4340	504	14	and	and	CCONJ
ejpam-4340	504	15	f	f	PROPN
ejpam-4340	504	16	is	be	AUX
ejpam-4340	504	17	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	504	18	,	,	PUNCT
ejpam-4340	504	19	then	then	ADV
ejpam-4340	504	20	g	g	PROPN
ejpam-4340	504	21	◦	◦	PROPN
ejpam-4340	504	22	f	f	PROPN
ejpam-4340	504	23	is	be	AUX
ejpam-4340	504	24	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	504	25	,	,	PUNCT
ejpam-4340	504	26	(	(	PUNCT
ejpam-4340	504	27	b	b	X
ejpam-4340	504	28	)	)	PUNCT
ejpam-4340	504	29	if	if	SCONJ
ejpam-4340	504	30	g	g	PROPN
ejpam-4340	504	31	is	be	AUX
ejpam-4340	504	32	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	504	33	and	and	CCONJ
ejpam-4340	504	34	f	f	PROPN
ejpam-4340	504	35	is	be	AUX
ejpam-4340	504	36	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	504	37	,	,	PUNCT
ejpam-4340	504	38	then	then	ADV
ejpam-4340	504	39	g	g	PROPN
ejpam-4340	504	40	◦	◦	PROPN
ejpam-4340	504	41	f	f	PROPN
ejpam-4340	504	42	is	be	AUX
ejpam-4340	504	43	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	504	44	,	,	PUNCT
ejpam-4340	504	45	(	(	PUNCT
ejpam-4340	504	46	c	c	X
ejpam-4340	504	47	)	)	PUNCT
ejpam-4340	504	48	if	if	SCONJ
ejpam-4340	504	49	g	g	PROPN
ejpam-4340	504	50	is	be	AUX
ejpam-4340	504	51	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	504	52	and	and	CCONJ
ejpam-4340	504	53	f	f	PROPN
ejpam-4340	504	54	is	be	AUX
ejpam-4340	504	55	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	504	56	,	,	PUNCT
ejpam-4340	504	57	then	then	ADV
ejpam-4340	504	58	g	g	PROPN
ejpam-4340	504	59	◦	◦	PROPN
ejpam-4340	504	60	f	f	PROPN
ejpam-4340	504	61	is	be	AUX
ejpam-4340	504	62	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	504	63	,	,	PUNCT
ejpam-4340	504	64	(	(	PUNCT
ejpam-4340	504	65	d	d	X
ejpam-4340	504	66	)	)	PUNCT
ejpam-4340	504	67	if	if	SCONJ
ejpam-4340	504	68	g	g	PROPN
ejpam-4340	504	69	is	be	AUX
ejpam-4340	504	70	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	504	71	and	and	CCONJ
ejpam-4340	504	72	f	f	PROPN
ejpam-4340	504	73	is	be	AUX
ejpam-4340	504	74	ωe∗-irresolute	ωe∗-irresolute	NOUN
ejpam-4340	504	75	and	and	CCONJ
ejpam-4340	504	76	y	y	PROPN
ejpam-4340	504	77	is	be	AUX
ejpam-4340	504	78	ωe∗-t	ωe∗-t	NUM
ejpam-4340	504	79	1	1	NUM
ejpam-4340	504	80	2	2	NUM
ejpam-4340	504	81	space	space	NOUN
ejpam-4340	504	82	,	,	PUNCT
ejpam-4340	504	83	then	then	ADV
ejpam-4340	504	84	g	g	PROPN
ejpam-4340	504	85	◦	◦	PROPN
ejpam-4340	504	86	f	f	PROPN
ejpam-4340	504	87	is	be	AUX
ejpam-4340	504	88	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	504	89	,	,	PUNCT
ejpam-4340	504	90	(	(	PUNCT
ejpam-4340	504	91	e	e	NOUN
ejpam-4340	504	92	)	)	PUNCT
ejpam-4340	504	93	if	if	SCONJ
ejpam-4340	504	94	g	g	PROPN
ejpam-4340	504	95	and	and	CCONJ
ejpam-4340	504	96	f	f	PROPN
ejpam-4340	504	97	are	be	AUX
ejpam-4340	504	98	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	504	99	and	and	CCONJ
ejpam-4340	504	100	σ∗	σ∗	ADJ
ejpam-4340	504	101	ωe∗	ωe∗	NOUN
ejpam-4340	505	1	=	=	PROPN
ejpam-4340	505	2	σ	σ	PROPN
ejpam-4340	505	3	,	,	PUNCT
ejpam-4340	505	4	then	then	ADV
ejpam-4340	505	5	g	g	PROPN
ejpam-4340	505	6	◦	◦	PROPN
ejpam-4340	505	7	f	f	PROPN
ejpam-4340	505	8	is	be	AUX
ejpam-4340	505	9	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	505	10	.	.	PUNCT
ejpam-4340	506	1	proof	proof	NOUN
ejpam-4340	506	2	.	.	PUNCT
ejpam-4340	507	1	straightforward	straightforward	ADJ
ejpam-4340	507	2	.	.	PUNCT
ejpam-4340	508	1	theorem	theorem	VERB
ejpam-4340	508	2	24	24	NUM
ejpam-4340	508	3	.	.	PUNCT
ejpam-4340	509	1	let	let	VERB
ejpam-4340	509	2	f	f	NOUN
ejpam-4340	509	3	:	:	PUNCT
ejpam-4340	509	4	x	x	X
ejpam-4340	509	5	→	→	SYM
ejpam-4340	509	6	y	y	X
ejpam-4340	509	7	be	be	AUX
ejpam-4340	509	8	a	a	DET
ejpam-4340	509	9	function	function	NOUN
ejpam-4340	509	10	.	.	PUNCT
ejpam-4340	510	1	then	then	ADV
ejpam-4340	510	2	the	the	DET
ejpam-4340	510	3	following	follow	VERB
ejpam-4340	510	4	properties	property	NOUN
ejpam-4340	510	5	hold	hold	VERB
ejpam-4340	510	6	:	:	PUNCT
ejpam-4340	510	7	(	(	PUNCT
ejpam-4340	510	8	a	a	X
ejpam-4340	510	9	)	)	PUNCT
ejpam-4340	510	10	if	if	SCONJ
ejpam-4340	510	11	f	f	PROPN
ejpam-4340	510	12	is	be	AUX
ejpam-4340	510	13	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	510	14	and	and	CCONJ
ejpam-4340	510	15	x	x	X
ejpam-4340	510	16	is	be	AUX
ejpam-4340	510	17	ωe∗-t	ωe∗-t	NUM
ejpam-4340	510	18	1	1	NUM
ejpam-4340	510	19	2	2	NUM
ejpam-4340	510	20	space	space	NOUN
ejpam-4340	510	21	,	,	PUNCT
ejpam-4340	510	22	then	then	ADV
ejpam-4340	510	23	f	f	PROPN
ejpam-4340	510	24	is	be	AUX
ejpam-4340	510	25	ωe∗-irresolute	ωe∗-irresolute	NOUN
ejpam-4340	510	26	,	,	PUNCT
ejpam-4340	510	27	(	(	PUNCT
ejpam-4340	510	28	b	b	X
ejpam-4340	510	29	)	)	PUNCT
ejpam-4340	510	30	if	if	SCONJ
ejpam-4340	510	31	f	f	PROPN
ejpam-4340	510	32	is	be	AUX
ejpam-4340	510	33	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	510	34	and	and	CCONJ
ejpam-4340	510	35	x	x	PRON
ejpam-4340	510	36	is	be	AUX
ejpam-4340	510	37	ωe∗-t	ωe∗-t	NUM
ejpam-4340	510	38	1	1	NUM
ejpam-4340	510	39	2	2	NUM
ejpam-4340	510	40	space	space	NOUN
ejpam-4340	510	41	,	,	PUNCT
ejpam-4340	510	42	then	then	ADV
ejpam-4340	510	43	f	f	PROPN
ejpam-4340	510	44	is	be	AUX
ejpam-4340	510	45	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	510	46	.	.	PUNCT
ejpam-4340	511	1	proof	proof	NOUN
ejpam-4340	511	2	.	.	PUNCT
ejpam-4340	512	1	straightforward	straightforward	ADJ
ejpam-4340	512	2	.	.	PUNCT
ejpam-4340	513	1	theorem	theorem	VERB
ejpam-4340	513	2	25	25	NUM
ejpam-4340	513	3	.	.	PUNCT
ejpam-4340	514	1	let	let	VERB
ejpam-4340	514	2	f	f	NOUN
ejpam-4340	514	3	:	:	PUNCT
ejpam-4340	514	4	x	x	X
ejpam-4340	514	5	→	→	SYM
ejpam-4340	514	6	y	y	X
ejpam-4340	514	7	be	be	AUX
ejpam-4340	514	8	a	a	DET
ejpam-4340	514	9	pre	pre	ADJ
ejpam-4340	514	10	-	-	ADJ
ejpam-4340	514	11	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	514	12	and	and	CCONJ
ejpam-4340	514	13	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	514	14	surjection	surjection	NOUN
ejpam-4340	514	15	.	.	PUNCT
ejpam-4340	515	1	if	if	SCONJ
ejpam-4340	515	2	x	x	PRON
ejpam-4340	515	3	is	be	AUX
ejpam-4340	515	4	ωe∗-t	ωe∗-t	NUM
ejpam-4340	515	5	1	1	NUM
ejpam-4340	515	6	2	2	NUM
ejpam-4340	515	7	space	space	NOUN
ejpam-4340	515	8	,	,	PUNCT
ejpam-4340	515	9	then	then	ADV
ejpam-4340	515	10	y	y	PROPN
ejpam-4340	515	11	is	be	AUX
ejpam-4340	515	12	ωe∗-t	ωe∗-t	NUM
ejpam-4340	515	13	1	1	NUM
ejpam-4340	515	14	2	2	NUM
ejpam-4340	515	15	space	space	NOUN
ejpam-4340	515	16	.	.	PUNCT
ejpam-4340	516	1	proof	proof	NOUN
ejpam-4340	516	2	.	.	PUNCT
ejpam-4340	517	1	let	let	VERB
ejpam-4340	517	2	f	f	PROPN
ejpam-4340	517	3	∈	∈	PROPN
ejpam-4340	517	4	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	517	5	)	)	PUNCT
ejpam-4340	517	6	.	.	PUNCT
ejpam-4340	518	1	f	f	PROPN
ejpam-4340	518	2	∈	∈	PROPN
ejpam-4340	518	3	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	518	4	)	)	PUNCT
ejpam-4340	519	1	f	f	PROPN
ejpam-4340	519	2	is	be	AUX
ejpam-4340	519	3	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	519	4	}	}	PUNCT
ejpam-4340	519	5	⇒	⇒	NOUN
ejpam-4340	519	6	f−1[f	f−1[f	VERB
ejpam-4340	519	7	]	]	PUNCT
ejpam-4340	519	8	∈	∈	PROPN
ejpam-4340	519	9	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	519	10	)	)	PUNCT
ejpam-4340	519	11	x	x	PUNCT
ejpam-4340	519	12	is	be	AUX
ejpam-4340	519	13	ωe∗-t	ωe∗-t	NUM
ejpam-4340	519	14	1	1	NUM
ejpam-4340	519	15	2	2	NUM
ejpam-4340	519	16	}	}	PUNCT
ejpam-4340	519	17	⇒	⇒	NOUN
ejpam-4340	519	18	f−1[f	f−1[f	VERB
ejpam-4340	519	19	]	]	PUNCT
ejpam-4340	519	20	∈	∈	PROPN
ejpam-4340	519	21	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	519	22	)	)	PUNCT
ejpam-4340	519	23	⇒	⇒	NOUN
ejpam-4340	519	24	f−1[f	f−1[f	VERB
ejpam-4340	519	25	]	]	PUNCT
ejpam-4340	519	26	∈	∈	PROPN
ejpam-4340	519	27	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	519	28	)	)	PUNCT
ejpam-4340	519	29	f	f	PROPN
ejpam-4340	519	30	is	be	AUX
ejpam-4340	519	31	pre	pre	ADJ
ejpam-4340	519	32	-	-	ADJ
ejpam-4340	519	33	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	519	34	surjection	surjection	NOUN
ejpam-4340	519	35	}	}	PUNCT
ejpam-4340	519	36	⇒	⇒	NOUN
ejpam-4340	519	37	f	f	PROPN
ejpam-4340	520	1	[	[	X
ejpam-4340	520	2	f−1[f	f−1[f	VERB
ejpam-4340	520	3	]	]	X
ejpam-4340	520	4	]	]	PUNCT
ejpam-4340	520	5	=	=	PUNCT
ejpam-4340	520	6	f	f	PROPN
ejpam-4340	520	7	∈	∈	PROPN
ejpam-4340	520	8	ωe∗c(y	ωe∗c(y	PROPN
ejpam-4340	520	9	)	)	PUNCT
ejpam-4340	520	10	.	.	PUNCT
ejpam-4340	521	1	definition	definition	NOUN
ejpam-4340	521	2	22	22	NUM
ejpam-4340	521	3	.	.	PUNCT
ejpam-4340	522	1	a	a	DET
ejpam-4340	522	2	function	function	NOUN
ejpam-4340	522	3	f	f	NOUN
ejpam-4340	522	4	:	:	PUNCT
ejpam-4340	522	5	x	x	X
ejpam-4340	522	6	→	→	SYM
ejpam-4340	522	7	y	y	PROPN
ejpam-4340	522	8	is	be	AUX
ejpam-4340	522	9	said	say	VERB
ejpam-4340	522	10	to	to	PART
ejpam-4340	522	11	be	be	AUX
ejpam-4340	522	12	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	522	13	if	if	SCONJ
ejpam-4340	522	14	f−1[v	f−1[v	NOUN
ejpam-4340	522	15	]	]	PUNCT
ejpam-4340	522	16	is	be	AUX
ejpam-4340	522	17	gωe∗closed	gωe∗close	VERB
ejpam-4340	522	18	in	in	ADP
ejpam-4340	522	19	x	x	PUNCT
ejpam-4340	522	20	for	for	ADP
ejpam-4340	522	21	every	every	DET
ejpam-4340	522	22	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	522	23	set	set	VERB
ejpam-4340	522	24	v	v	NOUN
ejpam-4340	522	25	of	of	ADP
ejpam-4340	522	26	y.	y.	PROPN
ejpam-4340	522	27	p.	p.	PROPN
ejpam-4340	522	28	şaşmaz	şaşmaz	PROPN
ejpam-4340	522	29	,	,	PUNCT
ejpam-4340	522	30	m.	m.	NOUN
ejpam-4340	522	31	özkoç	özkoç	PROPN
ejpam-4340	522	32	/	/	SYM
ejpam-4340	522	33	eur	eur	PROPN
ejpam-4340	522	34	.	.	PUNCT
ejpam-4340	523	1	j.	j.	PROPN
ejpam-4340	523	2	pure	pure	PROPN
ejpam-4340	523	3	appl	appl	PROPN
ejpam-4340	523	4	.	.	PROPN
ejpam-4340	523	5	math	math	PROPN
ejpam-4340	523	6	,	,	PUNCT
ejpam-4340	523	7	15	15	NUM
ejpam-4340	523	8	(	(	PUNCT
ejpam-4340	523	9	2	2	NUM
ejpam-4340	523	10	)	)	PUNCT
ejpam-4340	523	11	(	(	PUNCT
ejpam-4340	523	12	2022	2022	NUM
ejpam-4340	523	13	)	)	PUNCT
ejpam-4340	523	14	,	,	PUNCT
ejpam-4340	523	15	354	354	NUM
ejpam-4340	523	16	-	-	SYM
ejpam-4340	523	17	374	374	NUM
ejpam-4340	523	18	370	370	NUM
ejpam-4340	523	19	remark	remark	NOUN
ejpam-4340	523	20	7	7	NUM
ejpam-4340	523	21	.	.	PUNCT
ejpam-4340	523	22	recall	recall	VERB
ejpam-4340	523	23	that	that	SCONJ
ejpam-4340	523	24	every	every	DET
ejpam-4340	523	25	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	523	26	function	function	NOUN
ejpam-4340	523	27	is	be	AUX
ejpam-4340	523	28	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	523	29	function	function	NOUN
ejpam-4340	523	30	and	and	CCONJ
ejpam-4340	523	31	every	every	DET
ejpam-4340	523	32	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	523	33	function	function	NOUN
ejpam-4340	523	34	is	be	AUX
ejpam-4340	523	35	gωe∗-continuous	gωe∗-continuous	ADJ
ejpam-4340	523	36	function	function	NOUN
ejpam-4340	523	37	.	.	PUNCT
ejpam-4340	524	1	proposition	proposition	NOUN
ejpam-4340	524	2	13	13	NUM
ejpam-4340	524	3	.	.	PUNCT
ejpam-4340	525	1	let	let	VERB
ejpam-4340	525	2	f	f	NOUN
ejpam-4340	525	3	:	:	PUNCT
ejpam-4340	525	4	x	x	X
ejpam-4340	525	5	→	→	SYM
ejpam-4340	525	6	y	y	X
ejpam-4340	525	7	be	be	AUX
ejpam-4340	525	8	a	a	DET
ejpam-4340	525	9	function	function	NOUN
ejpam-4340	525	10	.	.	PUNCT
ejpam-4340	526	1	if	if	SCONJ
ejpam-4340	526	2	f	f	PROPN
ejpam-4340	526	3	is	be	AUX
ejpam-4340	526	4	an	an	DET
ejpam-4340	526	5	open	open	ADJ
ejpam-4340	526	6	bijection	bijection	NOUN
ejpam-4340	526	7	and	and	CCONJ
ejpam-4340	526	8	g∗ωe∗continuous	g∗ωe∗continuous	PROPN
ejpam-4340	526	9	,	,	PUNCT
ejpam-4340	526	10	then	then	ADV
ejpam-4340	526	11	f	f	PROPN
ejpam-4340	526	12	is	be	AUX
ejpam-4340	526	13	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	526	14	.	.	PUNCT
ejpam-4340	527	1	proof	proof	NOUN
ejpam-4340	527	2	.	.	PUNCT
ejpam-4340	528	1	let	let	VERB
ejpam-4340	528	2	a	a	DET
ejpam-4340	528	3	∈	∈	PROPN
ejpam-4340	528	4	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	528	5	)	)	PUNCT
ejpam-4340	528	6	and	and	CCONJ
ejpam-4340	528	7	f−1[a	f−1[a	PROPN
ejpam-4340	528	8	]	]	PUNCT
ejpam-4340	528	9	⊆	⊆	NUM
ejpam-4340	528	10	u	u	NOUN
ejpam-4340	528	11	∈	∈	PROPN
ejpam-4340	528	12	o(x	o(x	PROPN
ejpam-4340	528	13	)	)	PUNCT
ejpam-4340	528	14	.	.	PUNCT
ejpam-4340	529	1	f−1[a	f−1[a	PROPN
ejpam-4340	529	2	]	]	PUNCT
ejpam-4340	529	3	⊆	⊆	NUM
ejpam-4340	529	4	u	u	X
ejpam-4340	529	5	∈	∈	PROPN
ejpam-4340	529	6	o(x	o(x	PROPN
ejpam-4340	529	7	)	)	PUNCT
ejpam-4340	529	8	f	f	PROPN
ejpam-4340	529	9	is	be	AUX
ejpam-4340	529	10	open	open	ADJ
ejpam-4340	529	11	bijection	bijection	NOUN
ejpam-4340	529	12	}	}	PUNCT
ejpam-4340	529	13	⇒	⇒	NOUN
ejpam-4340	529	14	f	f	PROPN
ejpam-4340	530	1	[	[	X
ejpam-4340	530	2	f−1[a	f−1[a	X
ejpam-4340	530	3	]	]	X
ejpam-4340	530	4	]	]	X
ejpam-4340	530	5	=	=	PUNCT
ejpam-4340	530	6	a	a	PRON
ejpam-4340	530	7	⊆	⊆	NUM
ejpam-4340	530	8	f	f	X
ejpam-4340	530	9	[	[	X
ejpam-4340	530	10	u	u	X
ejpam-4340	530	11	]	]	X
ejpam-4340	530	12	∈	∈	PROPN
ejpam-4340	530	13	o(y	o(y	PROPN
ejpam-4340	530	14	)	)	PUNCT
ejpam-4340	530	15	a	a	DET
ejpam-4340	530	16	∈	∈	PROPN
ejpam-4340	530	17	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	530	18	)	)	PUNCT
ejpam-4340	530	19	}	}	PUNCT
ejpam-4340	530	20	⇒	⇒	VERB
ejpam-4340	530	21	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	530	22	)	)	PUNCT
ejpam-4340	530	23	⊆	⊆	NUM
ejpam-4340	530	24	f	f	X
ejpam-4340	531	1	[	[	X
ejpam-4340	531	2	u	u	X
ejpam-4340	531	3	]	]	PUNCT
ejpam-4340	531	4	⇒	⇒	PROPN
ejpam-4340	531	5	f−1[ωe∗-cl(a	f−1[ωe∗-cl(a	NOUN
ejpam-4340	531	6	)	)	PUNCT
ejpam-4340	531	7	]	]	PUNCT
ejpam-4340	532	1	⊆	⊆	NUM
ejpam-4340	532	2	u	u	NOUN
ejpam-4340	532	3	f	f	PROPN
ejpam-4340	532	4	is	be	AUX
ejpam-4340	532	5	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	532	6	}	}	PUNCT
ejpam-4340	532	7	⇒	⇒	NOUN
ejpam-4340	532	8	f−1[ωe∗-cl(a	f−1[ωe∗-cl(a	NOUN
ejpam-4340	532	9	)	)	PUNCT
ejpam-4340	532	10	]	]	PUNCT
ejpam-4340	532	11	∈	∈	PROPN
ejpam-4340	532	12	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	532	13	)	)	PUNCT
ejpam-4340	532	14	⇒	⇒	PROPN
ejpam-4340	532	15	ωe∗-cl(f−1[a	ωe∗-cl(f−1[a	PROPN
ejpam-4340	532	16	]	]	PUNCT
ejpam-4340	532	17	)	)	PUNCT
ejpam-4340	532	18	⊆	⊆	NUM
ejpam-4340	532	19	ωe∗-cl(f−1[ωe∗-cl(a	ωe∗-cl(f−1[ωe∗-cl(a	NUM
ejpam-4340	532	20	)	)	PUNCT
ejpam-4340	532	21	]	]	PUNCT
ejpam-4340	532	22	)	)	PUNCT
ejpam-4340	533	1	⊆	⊆	NUM
ejpam-4340	533	2	u.	u.	NOUN
ejpam-4340	533	3	proposition	proposition	NOUN
ejpam-4340	533	4	14	14	NUM
ejpam-4340	533	5	.	.	PUNCT
ejpam-4340	534	1	let	let	VERB
ejpam-4340	534	2	f	f	NOUN
ejpam-4340	534	3	:	:	PUNCT
ejpam-4340	534	4	x	x	X
ejpam-4340	534	5	→	→	SYM
ejpam-4340	534	6	y	y	X
ejpam-4340	534	7	be	be	AUX
ejpam-4340	534	8	a	a	DET
ejpam-4340	534	9	pre	pre	ADJ
ejpam-4340	534	10	-	-	ADJ
ejpam-4340	534	11	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	534	12	and	and	CCONJ
ejpam-4340	534	13	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	534	14	bijection	bijection	NOUN
ejpam-4340	534	15	open	open	ADJ
ejpam-4340	534	16	function	function	NOUN
ejpam-4340	534	17	.	.	PUNCT
ejpam-4340	535	1	if	if	SCONJ
ejpam-4340	535	2	x	x	PRON
ejpam-4340	535	3	is	be	AUX
ejpam-4340	535	4	ωe∗-t	ωe∗-t	NUM
ejpam-4340	535	5	1	1	NUM
ejpam-4340	535	6	2	2	NUM
ejpam-4340	535	7	space	space	NOUN
ejpam-4340	535	8	,	,	PUNCT
ejpam-4340	535	9	then	then	ADV
ejpam-4340	535	10	y	y	PROPN
ejpam-4340	535	11	is	be	AUX
ejpam-4340	535	12	ωe∗-t	ωe∗-t	NUM
ejpam-4340	535	13	1	1	NUM
ejpam-4340	535	14	2	2	NUM
ejpam-4340	535	15	space	space	NOUN
ejpam-4340	535	16	.	.	PUNCT
ejpam-4340	536	1	proof	proof	NOUN
ejpam-4340	536	2	.	.	PUNCT
ejpam-4340	537	1	let	let	VERB
ejpam-4340	537	2	a	a	DET
ejpam-4340	537	3	∈	∈	PROPN
ejpam-4340	537	4	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	537	5	)	)	PUNCT
ejpam-4340	537	6	.	.	PUNCT
ejpam-4340	538	1	a	a	DET
ejpam-4340	538	2	∈	∈	PROPN
ejpam-4340	538	3	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	538	4	)	)	PUNCT
ejpam-4340	539	1	f	f	PROPN
ejpam-4340	539	2	is	be	AUX
ejpam-4340	539	3	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	539	4	open	open	ADJ
ejpam-4340	539	5	bijection	bijection	NOUN
ejpam-4340	539	6	}	}	PUNCT
ejpam-4340	539	7	proposition	proposition	NOUN
ejpam-4340	539	8	13⇒	13⇒	NUM
ejpam-4340	539	9	f−1[a	f−1[a	PROPN
ejpam-4340	539	10	]	]	X
ejpam-4340	539	11	∈	∈	PROPN
ejpam-4340	539	12	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	539	13	)	)	PUNCT
ejpam-4340	539	14	x	x	PUNCT
ejpam-4340	539	15	is	be	AUX
ejpam-4340	539	16	ωe∗-t	ωe∗-t	NUM
ejpam-4340	539	17	1	1	NUM
ejpam-4340	539	18	2	2	NUM
ejpam-4340	539	19	space	space	NOUN
ejpam-4340	539	20	}	}	PUNCT
ejpam-4340	539	21	⇒	⇒	VERB
ejpam-4340	539	22	⇒	⇒	NOUN
ejpam-4340	539	23	f−1[a	f−1[a	PROPN
ejpam-4340	539	24	]	]	X
ejpam-4340	539	25	∈	∈	PROPN
ejpam-4340	539	26	ωe∗c(x	ωe∗c(x	NOUN
ejpam-4340	539	27	)	)	PUNCT
ejpam-4340	539	28	f	f	PROPN
ejpam-4340	539	29	is	be	AUX
ejpam-4340	539	30	pre	pre	ADJ
ejpam-4340	539	31	-	-	ADJ
ejpam-4340	539	32	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	539	33	bijection	bijection	NOUN
ejpam-4340	539	34	}	}	PUNCT
ejpam-4340	539	35	⇒	⇒	NOUN
ejpam-4340	539	36	f	f	PROPN
ejpam-4340	540	1	[	[	X
ejpam-4340	540	2	f−1[a	f−1[a	X
ejpam-4340	540	3	]	]	X
ejpam-4340	540	4	]	]	X
ejpam-4340	540	5	=	=	PUNCT
ejpam-4340	540	6	a	a	DET
ejpam-4340	540	7	∈	∈	PROPN
ejpam-4340	540	8	ωe∗c(y	ωe∗c(y	PROPN
ejpam-4340	540	9	)	)	PUNCT
ejpam-4340	540	10	.	.	PUNCT
ejpam-4340	541	1	definition	definition	NOUN
ejpam-4340	541	2	23	23	NUM
ejpam-4340	541	3	.	.	PUNCT
ejpam-4340	542	1	a	a	DET
ejpam-4340	542	2	function	function	NOUN
ejpam-4340	542	3	f	f	NOUN
ejpam-4340	542	4	:	:	PUNCT
ejpam-4340	542	5	x	x	X
ejpam-4340	542	6	→	→	SYM
ejpam-4340	542	7	y	y	PROPN
ejpam-4340	542	8	is	be	AUX
ejpam-4340	542	9	said	say	VERB
ejpam-4340	542	10	to	to	PART
ejpam-4340	542	11	be	be	AUX
ejpam-4340	542	12	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	542	13	if	if	SCONJ
ejpam-4340	542	14	f	f	PROPN
ejpam-4340	542	15	[	[	X
ejpam-4340	542	16	f	f	X
ejpam-4340	542	17	]	]	X
ejpam-4340	542	18	is	be	AUX
ejpam-4340	542	19	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	542	20	in	in	ADP
ejpam-4340	542	21	y	y	PROPN
ejpam-4340	542	22	for	for	ADP
ejpam-4340	542	23	every	every	DET
ejpam-4340	542	24	closed	close	VERB
ejpam-4340	542	25	set	set	VERB
ejpam-4340	542	26	f	f	PROPN
ejpam-4340	542	27	of	of	ADP
ejpam-4340	542	28	x.	x.	PROPN
ejpam-4340	542	29	remark	remark	PROPN
ejpam-4340	542	30	8	8	NUM
ejpam-4340	542	31	.	.	PUNCT
ejpam-4340	543	1	every	every	DET
ejpam-4340	543	2	closed	close	VERB
ejpam-4340	543	3	function	function	NOUN
ejpam-4340	543	4	is	be	AUX
ejpam-4340	543	5	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	543	6	function	function	NOUN
ejpam-4340	543	7	but	but	CCONJ
ejpam-4340	543	8	not	not	PART
ejpam-4340	543	9	conversely	conversely	ADV
ejpam-4340	543	10	.	.	PUNCT
ejpam-4340	544	1	example	example	NOUN
ejpam-4340	544	2	5	5	NUM
ejpam-4340	544	3	.	.	PUNCT
ejpam-4340	545	1	let	let	VERB
ejpam-4340	545	2	x	x	PUNCT
ejpam-4340	545	3	=	=	PRON
ejpam-4340	545	4	{	{	PUNCT
ejpam-4340	545	5	1	1	NUM
ejpam-4340	545	6	,	,	PUNCT
ejpam-4340	545	7	2	2	NUM
ejpam-4340	545	8	}	}	PUNCT
ejpam-4340	545	9	with	with	ADP
ejpam-4340	545	10	the	the	DET
ejpam-4340	545	11	topologies	topology	NOUN
ejpam-4340	545	12	τ	τ	X
ejpam-4340	545	13	=	=	SYM
ejpam-4340	545	14	{	{	PUNCT
ejpam-4340	545	15	x	x	NOUN
ejpam-4340	545	16	,	,	PUNCT
ejpam-4340	545	17	∅	∅	NOUN
ejpam-4340	545	18	,	,	PUNCT
ejpam-4340	545	19	{	{	PUNCT
ejpam-4340	545	20	1	1	NUM
ejpam-4340	545	21	}	}	PUNCT
ejpam-4340	545	22	}	}	PUNCT
ejpam-4340	545	23	and	and	CCONJ
ejpam-4340	545	24	σ	σ	X
ejpam-4340	545	25	=	=	SYM
ejpam-4340	545	26	{	{	PUNCT
ejpam-4340	545	27	x	x	NOUN
ejpam-4340	545	28	,	,	PUNCT
ejpam-4340	545	29	∅	∅	NOUN
ejpam-4340	545	30	,	,	PUNCT
ejpam-4340	545	31	{	{	PUNCT
ejpam-4340	545	32	2	2	NUM
ejpam-4340	545	33	}	}	PUNCT
ejpam-4340	545	34	}	}	PUNCT
ejpam-4340	545	35	.	.	PUNCT
ejpam-4340	546	1	let	let	VERB
ejpam-4340	546	2	f	f	NOUN
ejpam-4340	546	3	:	:	PUNCT
ejpam-4340	546	4	(	(	PUNCT
ejpam-4340	546	5	x	x	X
ejpam-4340	546	6	,	,	PUNCT
ejpam-4340	546	7	τ	τ	X
ejpam-4340	546	8	)	)	PUNCT
ejpam-4340	546	9	→	→	SYM
ejpam-4340	546	10	(	(	PUNCT
ejpam-4340	546	11	x	x	X
ejpam-4340	546	12	,	,	PUNCT
ejpam-4340	546	13	σ	σ	PROPN
ejpam-4340	546	14	)	)	PUNCT
ejpam-4340	546	15	be	be	VERB
ejpam-4340	546	16	the	the	DET
ejpam-4340	546	17	identity	identity	NOUN
ejpam-4340	546	18	function	function	NOUN
ejpam-4340	546	19	.	.	PUNCT
ejpam-4340	547	1	then	then	ADV
ejpam-4340	547	2	f	f	PROPN
ejpam-4340	547	3	is	be	AUX
ejpam-4340	547	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	547	5	but	but	CCONJ
ejpam-4340	547	6	not	not	PART
ejpam-4340	547	7	closed	close	VERB
ejpam-4340	547	8	since	since	SCONJ
ejpam-4340	547	9	f	f	PROPN
ejpam-4340	548	1	[	[	X
ejpam-4340	548	2	{	{	PUNCT
ejpam-4340	548	3	2	2	NUM
ejpam-4340	548	4	}	}	PUNCT
ejpam-4340	548	5	]	]	PUNCT
ejpam-4340	548	6	=	=	PUNCT
ejpam-4340	548	7	{	{	PUNCT
ejpam-4340	548	8	2	2	NUM
ejpam-4340	548	9	}	}	PUNCT
ejpam-4340	548	10	is	be	AUX
ejpam-4340	548	11	not	not	PART
ejpam-4340	548	12	closed	close	VERB
ejpam-4340	548	13	in	in	ADP
ejpam-4340	548	14	x.	x.	NOUN
ejpam-4340	548	15	theorem	theorem	AUX
ejpam-4340	548	16	26	26	NUM
ejpam-4340	548	17	.	.	PUNCT
ejpam-4340	549	1	let	let	VERB
ejpam-4340	549	2	f	f	NOUN
ejpam-4340	549	3	:	:	PUNCT
ejpam-4340	549	4	x	x	X
ejpam-4340	549	5	→	→	SYM
ejpam-4340	549	6	y	y	X
ejpam-4340	549	7	be	be	AUX
ejpam-4340	549	8	a	a	DET
ejpam-4340	549	9	function	function	NOUN
ejpam-4340	549	10	.	.	PUNCT
ejpam-4340	550	1	then	then	ADV
ejpam-4340	550	2	,	,	PUNCT
ejpam-4340	550	3	f	f	PROPN
ejpam-4340	550	4	is	be	AUX
ejpam-4340	550	5	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	550	6	if	if	SCONJ
ejpam-4340	550	7	and	and	CCONJ
ejpam-4340	550	8	only	only	ADV
ejpam-4340	550	9	if	if	SCONJ
ejpam-4340	550	10	for	for	ADP
ejpam-4340	550	11	each	each	DET
ejpam-4340	550	12	subset	subset	NOUN
ejpam-4340	550	13	s	s	PROPN
ejpam-4340	550	14	of	of	ADP
ejpam-4340	550	15	y	y	PROPN
ejpam-4340	550	16	and	and	CCONJ
ejpam-4340	550	17	for	for	ADP
ejpam-4340	550	18	each	each	DET
ejpam-4340	550	19	open	open	ADJ
ejpam-4340	550	20	set	set	VERB
ejpam-4340	550	21	u	u	NOUN
ejpam-4340	550	22	containing	contain	VERB
ejpam-4340	550	23	f−1[s	f−1[s	NOUN
ejpam-4340	550	24	]	]	PUNCT
ejpam-4340	550	25	,	,	PUNCT
ejpam-4340	550	26	there	there	PRON
ejpam-4340	550	27	exists	exist	VERB
ejpam-4340	550	28	a	a	DET
ejpam-4340	550	29	gωe∗-open	gωe∗-open	PROPN
ejpam-4340	550	30	set	set	VERB
ejpam-4340	550	31	v	v	NOUN
ejpam-4340	550	32	of	of	ADP
ejpam-4340	550	33	y	y	PRON
ejpam-4340	550	34	such	such	ADJ
ejpam-4340	550	35	that	that	PRON
ejpam-4340	550	36	s	s	VERB
ejpam-4340	550	37	⊆	⊆	NUM
ejpam-4340	550	38	v	v	NOUN
ejpam-4340	550	39	and	and	CCONJ
ejpam-4340	550	40	f−1[v	f−1[v	NOUN
ejpam-4340	550	41	]	]	PUNCT
ejpam-4340	550	42	⊆	⊆	NUM
ejpam-4340	550	43	u.	u.	NOUN
ejpam-4340	550	44	proof	proof	NOUN
ejpam-4340	550	45	.	.	PUNCT
ejpam-4340	551	1	(	(	PUNCT
ejpam-4340	551	2	⇒	⇒	PROPN
ejpam-4340	551	3	)	)	PUNCT
ejpam-4340	551	4	:	:	PUNCT
ejpam-4340	551	5	let	let	VERB
ejpam-4340	551	6	s	s	PRON
ejpam-4340	551	7	⊆	⊆	NUM
ejpam-4340	551	8	y	y	PROPN
ejpam-4340	551	9	and	and	CCONJ
ejpam-4340	551	10	f−1[s	f−1[s	PROPN
ejpam-4340	551	11	]	]	PUNCT
ejpam-4340	551	12	⊆	⊆	NUM
ejpam-4340	551	13	u	u	NOUN
ejpam-4340	551	14	∈	∈	PROPN
ejpam-4340	551	15	o(x	o(x	PROPN
ejpam-4340	551	16	)	)	PUNCT
ejpam-4340	551	17	.	.	PUNCT
ejpam-4340	552	1	f−1[s	f−1[s	NOUN
ejpam-4340	552	2	]	]	PUNCT
ejpam-4340	552	3	⊆	⊆	NUM
ejpam-4340	552	4	u	u	NOUN
ejpam-4340	552	5	∈	∈	PROPN
ejpam-4340	552	6	o(x)⇒	o(x)⇒	PROPN
ejpam-4340	552	7	(	(	PUNCT
ejpam-4340	552	8	x	x	SYM
ejpam-4340	552	9	\	\	NOUN
ejpam-4340	552	10	u	u	PROPN
ejpam-4340	552	11	∈	∈	PROPN
ejpam-4340	552	12	c(x))(x	c(x))(x	PROPN
ejpam-4340	552	13	\	\	NOUN
ejpam-4340	552	14	u	u	NOUN
ejpam-4340	552	15	⊆	⊆	NUM
ejpam-4340	552	16	x	x	SYM
ejpam-4340	552	17	\	\	PROPN
ejpam-4340	552	18	f−1[s	f−1[s	NOUN
ejpam-4340	552	19	]	]	PUNCT
ejpam-4340	552	20	)	)	PUNCT
ejpam-4340	553	1	f	f	PROPN
ejpam-4340	553	2	is	be	AUX
ejpam-4340	553	3	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	553	4	}	}	PUNCT
ejpam-4340	553	5	⇒	⇒	VERB
ejpam-4340	553	6	⇒	⇒	NOUN
ejpam-4340	553	7	(	(	PUNCT
ejpam-4340	553	8	f	f	X
ejpam-4340	554	1	[	[	X
ejpam-4340	554	2	x	x	X
ejpam-4340	554	3	\	\	PROPN
ejpam-4340	554	4	u	u	PROPN
ejpam-4340	554	5	]	]	PUNCT
ejpam-4340	554	6	∈	∈	PROPN
ejpam-4340	554	7	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	554	8	)	)	PUNCT
ejpam-4340	554	9	)	)	PUNCT
ejpam-4340	555	1	(	(	PUNCT
ejpam-4340	555	2	f	f	X
ejpam-4340	555	3	[	[	X
ejpam-4340	555	4	x	x	X
ejpam-4340	555	5	\	\	PROPN
ejpam-4340	555	6	u	u	NOUN
ejpam-4340	555	7	]	]	PUNCT
ejpam-4340	555	8	⊆	⊆	NUM
ejpam-4340	555	9	f	f	X
ejpam-4340	555	10	[	[	X
ejpam-4340	555	11	x	x	X
ejpam-4340	555	12	\	\	PROPN
ejpam-4340	555	13	f−1[s	f−1[s	NOUN
ejpam-4340	555	14	]	]	X
ejpam-4340	555	15	]	]	PUNCT
ejpam-4340	556	1	=	=	PUNCT
ejpam-4340	556	2	f	f	X
ejpam-4340	557	1	[	[	X
ejpam-4340	557	2	f−1[y	f−1[y	X
ejpam-4340	557	3	\	\	PROPN
ejpam-4340	557	4	s	s	X
ejpam-4340	557	5	]	]	X
ejpam-4340	557	6	]	]	X
ejpam-4340	557	7	⊆	⊆	NUM
ejpam-4340	557	8	y	y	PROPN
ejpam-4340	557	9	\	\	PROPN
ejpam-4340	557	10	s	s	PROPN
ejpam-4340	557	11	)	)	PUNCT
ejpam-4340	557	12	⇒	⇒	NOUN
ejpam-4340	557	13	(	(	PUNCT
ejpam-4340	557	14	y	y	PROPN
ejpam-4340	557	15	\	\	PROPN
ejpam-4340	557	16	f	f	PROPN
ejpam-4340	558	1	[	[	X
ejpam-4340	558	2	x	x	X
ejpam-4340	558	3	\	\	PROPN
ejpam-4340	558	4	u	u	PROPN
ejpam-4340	558	5	]	]	PUNCT
ejpam-4340	558	6	∈	∈	PROPN
ejpam-4340	558	7	gωe∗o(y	gωe∗o(y	NOUN
ejpam-4340	558	8	)	)	PUNCT
ejpam-4340	558	9	)	)	PUNCT
ejpam-4340	559	1	(	(	PUNCT
ejpam-4340	559	2	s	s	VERB
ejpam-4340	559	3	⊆	⊆	NUM
ejpam-4340	559	4	y	y	NOUN
ejpam-4340	559	5	\	\	PROPN
ejpam-4340	559	6	f	f	PROPN
ejpam-4340	560	1	[	[	X
ejpam-4340	560	2	x	x	X
ejpam-4340	560	3	\	\	PROPN
ejpam-4340	560	4	u	u	NOUN
ejpam-4340	560	5	]	]	X
ejpam-4340	560	6	)	)	PUNCT
ejpam-4340	560	7	v	v	ADP
ejpam-4340	560	8	:	:	PUNCT
ejpam-4340	560	9	=	=	SYM
ejpam-4340	560	10	y	y	NOUN
ejpam-4340	560	11	\	\	PROPN
ejpam-4340	560	12	f	f	PROPN
ejpam-4340	561	1	[	[	X
ejpam-4340	561	2	x	x	X
ejpam-4340	561	3	\	\	PROPN
ejpam-4340	561	4	u	u	NOUN
ejpam-4340	561	5	]	]	PUNCT
ejpam-4340	561	6	}	}	PUNCT
ejpam-4340	561	7	⇒	⇒	VERB
ejpam-4340	561	8	⇒	⇒	NOUN
ejpam-4340	561	9	(	(	PUNCT
ejpam-4340	561	10	v	v	NUM
ejpam-4340	561	11	∈	∈	PROPN
ejpam-4340	561	12	gωe∗o(y	gωe∗o(y	NOUN
ejpam-4340	561	13	)	)	PUNCT
ejpam-4340	561	14	)	)	PUNCT
ejpam-4340	562	1	(	(	PUNCT
ejpam-4340	562	2	s	s	VERB
ejpam-4340	562	3	⊆	⊆	NUM
ejpam-4340	562	4	v	v	NOUN
ejpam-4340	562	5	)	)	PUNCT
ejpam-4340	562	6	(	(	PUNCT
ejpam-4340	562	7	f−1[v	f−1[v	NOUN
ejpam-4340	562	8	]	]	PUNCT
ejpam-4340	562	9	⊆	⊆	NUM
ejpam-4340	562	10	u	u	NOUN
ejpam-4340	562	11	)	)	PUNCT
ejpam-4340	562	12	.	.	PUNCT
ejpam-4340	563	1	p.	p.	NOUN
ejpam-4340	563	2	şaşmaz	şaşmaz	NUM
ejpam-4340	563	3	,	,	PUNCT
ejpam-4340	563	4	m.	m.	NOUN
ejpam-4340	563	5	özkoç	özkoç	PROPN
ejpam-4340	563	6	/	/	SYM
ejpam-4340	563	7	eur	eur	PROPN
ejpam-4340	563	8	.	.	PUNCT
ejpam-4340	564	1	j.	j.	PROPN
ejpam-4340	564	2	pure	pure	PROPN
ejpam-4340	564	3	appl	appl	PROPN
ejpam-4340	564	4	.	.	PROPN
ejpam-4340	564	5	math	math	PROPN
ejpam-4340	564	6	,	,	PUNCT
ejpam-4340	564	7	15	15	NUM
ejpam-4340	564	8	(	(	PUNCT
ejpam-4340	564	9	2	2	NUM
ejpam-4340	564	10	)	)	PUNCT
ejpam-4340	564	11	(	(	PUNCT
ejpam-4340	564	12	2022	2022	NUM
ejpam-4340	564	13	)	)	PUNCT
ejpam-4340	564	14	,	,	PUNCT
ejpam-4340	564	15	354	354	NUM
ejpam-4340	564	16	-	-	SYM
ejpam-4340	564	17	374	374	NUM
ejpam-4340	564	18	371	371	NUM
ejpam-4340	564	19	(	(	PUNCT
ejpam-4340	564	20	⇐	⇐	PROPN
ejpam-4340	564	21	)	)	PUNCT
ejpam-4340	564	22	:	:	PUNCT
ejpam-4340	564	23	let	let	VERB
ejpam-4340	564	24	f	f	PROPN
ejpam-4340	564	25	∈	∈	PROPN
ejpam-4340	564	26	c(x	c(x	PROPN
ejpam-4340	564	27	)	)	PUNCT
ejpam-4340	564	28	.	.	PUNCT
ejpam-4340	565	1	f	f	PROPN
ejpam-4340	565	2	∈	∈	PROPN
ejpam-4340	565	3	c(x)⇒	c(x)⇒	PROPN
ejpam-4340	565	4	f−1[y	f−1[y	X
ejpam-4340	565	5	\	\	PROPN
ejpam-4340	565	6	f	f	PROPN
ejpam-4340	566	1	[	[	X
ejpam-4340	566	2	f	f	X
ejpam-4340	566	3	]	]	X
ejpam-4340	566	4	]	]	X
ejpam-4340	566	5	⊆	⊆	NUM
ejpam-4340	566	6	x	x	SYM
ejpam-4340	566	7	\	\	PROPN
ejpam-4340	566	8	f	f	PROPN
ejpam-4340	566	9	∈	∈	PROPN
ejpam-4340	566	10	o(x	o(x	PROPN
ejpam-4340	566	11	)	)	PUNCT
ejpam-4340	566	12	hypothesis	hypothesis	NOUN
ejpam-4340	566	13	}	}	PUNCT
ejpam-4340	566	14	⇒	⇒	VERB
ejpam-4340	566	15	⇒	⇒	NOUN
ejpam-4340	566	16	(	(	PUNCT
ejpam-4340	566	17	∃v	∃v	PROPN
ejpam-4340	566	18	∈	∈	PROPN
ejpam-4340	566	19	gωe∗o(y	gωe∗o(y	NOUN
ejpam-4340	566	20	)	)	PUNCT
ejpam-4340	566	21	)	)	PUNCT
ejpam-4340	567	1	(	(	PUNCT
ejpam-4340	567	2	y	y	PROPN
ejpam-4340	567	3	\	\	PROPN
ejpam-4340	567	4	f	f	PROPN
ejpam-4340	568	1	[	[	X
ejpam-4340	568	2	f	f	X
ejpam-4340	568	3	]	]	X
ejpam-4340	568	4	⊆	⊆	NUM
ejpam-4340	568	5	v	v	NOUN
ejpam-4340	568	6	)	)	PUNCT
ejpam-4340	568	7	(	(	PUNCT
ejpam-4340	568	8	f−1[v	f−1[v	NOUN
ejpam-4340	568	9	]	]	PUNCT
ejpam-4340	568	10	⊆	⊆	NUM
ejpam-4340	568	11	x	x	SYM
ejpam-4340	568	12	\	\	PROPN
ejpam-4340	568	13	f	f	PROPN
ejpam-4340	568	14	)	)	PUNCT
ejpam-4340	568	15	⇒	⇒	NOUN
ejpam-4340	568	16	(	(	PUNCT
ejpam-4340	568	17	∃v	∃v	PROPN
ejpam-4340	568	18	∈	∈	PROPN
ejpam-4340	568	19	gωe∗o(y	gωe∗o(y	NOUN
ejpam-4340	568	20	)	)	PUNCT
ejpam-4340	568	21	)	)	PUNCT
ejpam-4340	569	1	(	(	PUNCT
ejpam-4340	569	2	y	y	PROPN
ejpam-4340	569	3	\	\	PROPN
ejpam-4340	569	4	v	v	ADP
ejpam-4340	569	5	⊆	⊆	NUM
ejpam-4340	569	6	f	f	NOUN
ejpam-4340	570	1	[	[	X
ejpam-4340	570	2	f	f	X
ejpam-4340	570	3	]	]	X
ejpam-4340	570	4	⊆	⊆	NUM
ejpam-4340	570	5	f	f	X
ejpam-4340	570	6	[	[	X
ejpam-4340	570	7	x	x	X
ejpam-4340	570	8	\	\	ADJ
ejpam-4340	570	9	f−1[v	f−1[v	NOUN
ejpam-4340	570	10	]	]	PUNCT
ejpam-4340	570	11	]	]	PUNCT
ejpam-4340	570	12	⊆	⊆	NUM
ejpam-4340	570	13	y	y	PROPN
ejpam-4340	570	14	\	\	PROPN
ejpam-4340	570	15	v	v	NOUN
ejpam-4340	570	16	)	)	PUNCT
ejpam-4340	570	17	⇒	⇒	NOUN
ejpam-4340	570	18	(	(	PUNCT
ejpam-4340	570	19	y	y	PROPN
ejpam-4340	570	20	\	\	PROPN
ejpam-4340	570	21	v	v	PROPN
ejpam-4340	570	22	∈	∈	PROPN
ejpam-4340	570	23	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	570	24	)	)	PUNCT
ejpam-4340	570	25	)	)	PUNCT
ejpam-4340	570	26	(	(	PUNCT
ejpam-4340	570	27	y	y	PROPN
ejpam-4340	570	28	\	\	PROPN
ejpam-4340	570	29	v	v	PROPN
ejpam-4340	570	30	=	=	SYM
ejpam-4340	570	31	f	f	X
ejpam-4340	571	1	[	[	X
ejpam-4340	571	2	f	f	X
ejpam-4340	571	3	]	]	X
ejpam-4340	571	4	)	)	PUNCT
ejpam-4340	571	5	⇒	⇒	NOUN
ejpam-4340	571	6	f	f	PROPN
ejpam-4340	572	1	[	[	X
ejpam-4340	572	2	f	f	X
ejpam-4340	572	3	]	]	X
ejpam-4340	572	4	∈	∈	PROPN
ejpam-4340	572	5	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	572	6	)	)	PUNCT
ejpam-4340	572	7	.	.	PUNCT
ejpam-4340	573	1	theorem	theorem	NOUN
ejpam-4340	573	2	27	27	NUM
ejpam-4340	573	3	.	.	PUNCT
ejpam-4340	574	1	let	let	VERB
ejpam-4340	574	2	f	f	NOUN
ejpam-4340	574	3	:	:	PUNCT
ejpam-4340	574	4	x	x	X
ejpam-4340	574	5	→	→	SYM
ejpam-4340	574	6	y	y	X
ejpam-4340	574	7	be	be	AUX
ejpam-4340	574	8	a	a	DET
ejpam-4340	574	9	function	function	NOUN
ejpam-4340	574	10	.	.	PUNCT
ejpam-4340	575	1	if	if	SCONJ
ejpam-4340	575	2	f	f	PROPN
ejpam-4340	575	3	is	be	AUX
ejpam-4340	575	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	575	5	,	,	PUNCT
ejpam-4340	575	6	then	then	ADV
ejpam-4340	575	7	gωe∗-cl(f	gωe∗-cl(f	PROPN
ejpam-4340	576	1	[	[	X
ejpam-4340	576	2	a	a	X
ejpam-4340	576	3	]	]	X
ejpam-4340	576	4	)	)	PUNCT
ejpam-4340	576	5	⊆	⊆	NUM
ejpam-4340	576	6	f	f	X
ejpam-4340	576	7	[	[	X
ejpam-4340	576	8	cl(a	cl(a	X
ejpam-4340	576	9	)	)	PUNCT
ejpam-4340	576	10	]	]	PUNCT
ejpam-4340	576	11	for	for	ADP
ejpam-4340	576	12	every	every	DET
ejpam-4340	576	13	subset	subset	NOUN
ejpam-4340	576	14	a	a	DET
ejpam-4340	576	15	of	of	ADP
ejpam-4340	576	16	x.	x.	NOUN
ejpam-4340	576	17	proof	proof	NOUN
ejpam-4340	576	18	.	.	PUNCT
ejpam-4340	577	1	let	let	VERB
ejpam-4340	577	2	a	a	DET
ejpam-4340	577	3	⊆	⊆	NUM
ejpam-4340	577	4	x.	x.	NOUN
ejpam-4340	577	5	a	a	DET
ejpam-4340	577	6	⊆	⊆	NUM
ejpam-4340	577	7	x	x	SYM
ejpam-4340	577	8	⇒	⇒	NOUN
ejpam-4340	577	9	cl(a	cl(a	NUM
ejpam-4340	577	10	)	)	PUNCT
ejpam-4340	577	11	∈	∈	PROPN
ejpam-4340	577	12	c(x	c(x	NOUN
ejpam-4340	577	13	)	)	PUNCT
ejpam-4340	577	14	f	f	PROPN
ejpam-4340	577	15	is	be	AUX
ejpam-4340	577	16	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	577	17	}	}	PUNCT
ejpam-4340	577	18	⇒	⇒	NOUN
ejpam-4340	577	19	f	f	PROPN
ejpam-4340	578	1	[	[	X
ejpam-4340	578	2	cl(a	cl(a	X
ejpam-4340	578	3	)	)	PUNCT
ejpam-4340	578	4	]	]	PUNCT
ejpam-4340	578	5	∈	∈	PROPN
ejpam-4340	578	6	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	578	7	)	)	PUNCT
ejpam-4340	578	8	⇒	⇒	NOUN
ejpam-4340	578	9	gωe∗-cl(f	gωe∗-cl(f	NOUN
ejpam-4340	579	1	[	[	X
ejpam-4340	579	2	a	a	X
ejpam-4340	579	3	]	]	X
ejpam-4340	579	4	)	)	PUNCT
ejpam-4340	579	5	⊆	⊆	NUM
ejpam-4340	579	6	gωe∗-cl(f	gωe∗-cl(f	NOUN
ejpam-4340	579	7	[	[	X
ejpam-4340	579	8	cl(a	cl(a	X
ejpam-4340	579	9	)	)	PUNCT
ejpam-4340	579	10	]	]	PUNCT
ejpam-4340	579	11	)	)	PUNCT
ejpam-4340	580	1	=	=	PUNCT
ejpam-4340	580	2	f	f	PROPN
ejpam-4340	581	1	[	[	X
ejpam-4340	581	2	cl(a	cl(a	X
ejpam-4340	581	3	)	)	PUNCT
ejpam-4340	581	4	]	]	PUNCT
ejpam-4340	581	5	.	.	PUNCT
ejpam-4340	582	1	theorem	theorem	PROPN
ejpam-4340	582	2	28	28	NUM
ejpam-4340	582	3	.	.	PUNCT
ejpam-4340	583	1	let	let	VERB
ejpam-4340	583	2	f	f	NOUN
ejpam-4340	583	3	:	:	PUNCT
ejpam-4340	583	4	x	x	X
ejpam-4340	583	5	→	→	SYM
ejpam-4340	583	6	y	y	X
ejpam-4340	583	7	be	be	AUX
ejpam-4340	583	8	a	a	DET
ejpam-4340	583	9	function	function	NOUN
ejpam-4340	583	10	.	.	PUNCT
ejpam-4340	584	1	if	if	SCONJ
ejpam-4340	584	2	f	f	PROPN
ejpam-4340	584	3	is	be	AUX
ejpam-4340	584	4	continuous	continuous	ADJ
ejpam-4340	584	5	,	,	PUNCT
ejpam-4340	584	6	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	584	7	and	and	CCONJ
ejpam-4340	584	8	a	a	PRON
ejpam-4340	584	9	is	be	AUX
ejpam-4340	584	10	a	a	DET
ejpam-4340	584	11	g	g	NOUN
ejpam-4340	584	12	-	-	PUNCT
ejpam-4340	584	13	closed	close	VERB
ejpam-4340	584	14	subset	subset	NOUN
ejpam-4340	584	15	of	of	ADP
ejpam-4340	584	16	x	x	PRON
ejpam-4340	584	17	,	,	PUNCT
ejpam-4340	584	18	then	then	ADV
ejpam-4340	584	19	f	f	PROPN
ejpam-4340	585	1	[	[	X
ejpam-4340	585	2	a	a	X
ejpam-4340	585	3	]	]	X
ejpam-4340	585	4	is	be	AUX
ejpam-4340	585	5	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	585	6	.	.	PUNCT
ejpam-4340	586	1	proof	proof	NOUN
ejpam-4340	586	2	.	.	PUNCT
ejpam-4340	587	1	let	let	VERB
ejpam-4340	587	2	f	f	PROPN
ejpam-4340	587	3	[	[	X
ejpam-4340	587	4	a	a	X
ejpam-4340	587	5	]	]	X
ejpam-4340	587	6	⊆	⊆	NUM
ejpam-4340	587	7	u	u	NOUN
ejpam-4340	587	8	∈	∈	NOUN
ejpam-4340	587	9	o(y	o(y	PROPN
ejpam-4340	587	10	)	)	PUNCT
ejpam-4340	587	11	.	.	PUNCT
ejpam-4340	588	1	f	f	X
ejpam-4340	589	1	[	[	X
ejpam-4340	589	2	a	a	X
ejpam-4340	589	3	]	]	X
ejpam-4340	589	4	⊆	⊆	NUM
ejpam-4340	589	5	u	u	NOUN
ejpam-4340	589	6	∈	∈	NOUN
ejpam-4340	589	7	o(y	o(y	PROPN
ejpam-4340	589	8	)	)	PUNCT
ejpam-4340	590	1	f	f	PROPN
ejpam-4340	590	2	is	be	AUX
ejpam-4340	590	3	continuous	continuous	ADJ
ejpam-4340	590	4	}	}	PUNCT
ejpam-4340	590	5	⇒	⇒	NOUN
ejpam-4340	590	6	a	a	DET
ejpam-4340	590	7	⊆	⊆	NUM
ejpam-4340	590	8	f−1[f	f−1[f	NOUN
ejpam-4340	590	9	[	[	X
ejpam-4340	590	10	a	a	X
ejpam-4340	590	11	]	]	X
ejpam-4340	590	12	]	]	X
ejpam-4340	590	13	⊆	⊆	NUM
ejpam-4340	590	14	f−1[u	f−1[u	NOUN
ejpam-4340	590	15	]	]	PUNCT
ejpam-4340	590	16	∈	∈	PROPN
ejpam-4340	590	17	o(x	o(x	PROPN
ejpam-4340	590	18	)	)	PUNCT
ejpam-4340	590	19	a	a	DET
ejpam-4340	590	20	∈	∈	PROPN
ejpam-4340	590	21	gc(x	gc(x	NOUN
ejpam-4340	590	22	)	)	PUNCT
ejpam-4340	590	23	}	}	PUNCT
ejpam-4340	590	24	⇒	⇒	NOUN
ejpam-4340	590	25	cl(a	cl(a	NUM
ejpam-4340	590	26	)	)	PUNCT
ejpam-4340	590	27	⊆	⊆	NUM
ejpam-4340	590	28	f−1[u	f−1[u	NOUN
ejpam-4340	590	29	]	]	PUNCT
ejpam-4340	590	30	⇒	⇒	NOUN
ejpam-4340	590	31	(	(	PUNCT
ejpam-4340	590	32	cl(a	cl(a	X
ejpam-4340	590	33	)	)	PUNCT
ejpam-4340	590	34	∈	∈	NOUN
ejpam-4340	590	35	c(x))(f	c(x))(f	VERB
ejpam-4340	591	1	[	[	X
ejpam-4340	591	2	a	a	X
ejpam-4340	591	3	]	]	X
ejpam-4340	591	4	⊆	⊆	NUM
ejpam-4340	591	5	f	f	X
ejpam-4340	591	6	[	[	X
ejpam-4340	591	7	cl(a	cl(a	X
ejpam-4340	591	8	)	)	PUNCT
ejpam-4340	591	9	]	]	PUNCT
ejpam-4340	592	1	⊆	⊆	NUM
ejpam-4340	592	2	f	f	X
ejpam-4340	592	3	[	[	X
ejpam-4340	592	4	f−1[u	f−1[u	X
ejpam-4340	592	5	]	]	X
ejpam-4340	592	6	]	]	X
ejpam-4340	592	7	⊆	⊆	NUM
ejpam-4340	592	8	u	u	NOUN
ejpam-4340	592	9	)	)	PUNCT
ejpam-4340	592	10	f	f	PROPN
ejpam-4340	592	11	is	be	AUX
ejpam-4340	592	12	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	592	13	}	}	PUNCT
ejpam-4340	592	14	⇒	⇒	VERB
ejpam-4340	592	15	⇒	⇒	NOUN
ejpam-4340	592	16	(	(	PUNCT
ejpam-4340	592	17	f	f	X
ejpam-4340	592	18	[	[	X
ejpam-4340	592	19	cl(a	cl(a	X
ejpam-4340	592	20	)	)	PUNCT
ejpam-4340	592	21	]	]	PUNCT
ejpam-4340	593	1	∈	∈	PROPN
ejpam-4340	593	2	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	593	3	)	)	PUNCT
ejpam-4340	593	4	)	)	PUNCT
ejpam-4340	594	1	(	(	PUNCT
ejpam-4340	594	2	f	f	X
ejpam-4340	595	1	[	[	X
ejpam-4340	595	2	a	a	X
ejpam-4340	595	3	]	]	X
ejpam-4340	595	4	⊆	⊆	NUM
ejpam-4340	595	5	f	f	X
ejpam-4340	595	6	[	[	X
ejpam-4340	595	7	cl(a	cl(a	X
ejpam-4340	595	8	)	)	PUNCT
ejpam-4340	595	9	]	]	PUNCT
ejpam-4340	595	10	⊆	⊆	NUM
ejpam-4340	595	11	u	u	NOUN
ejpam-4340	595	12	)	)	PUNCT
ejpam-4340	595	13	⇒	⇒	NOUN
ejpam-4340	595	14	ωe∗-cl(f	ωe∗-cl(f	NOUN
ejpam-4340	596	1	[	[	X
ejpam-4340	596	2	a	a	X
ejpam-4340	596	3	]	]	X
ejpam-4340	596	4	)	)	PUNCT
ejpam-4340	596	5	⊆	⊆	NUM
ejpam-4340	596	6	ωe∗-cl(f	ωe∗-cl(f	NUM
ejpam-4340	596	7	[	[	X
ejpam-4340	596	8	cl(a	cl(a	X
ejpam-4340	596	9	)	)	PUNCT
ejpam-4340	596	10	]	]	PUNCT
ejpam-4340	596	11	)	)	PUNCT
ejpam-4340	597	1	⊆	⊆	NUM
ejpam-4340	597	2	u.	u.	NOUN
ejpam-4340	597	3	theorem	theorem	NOUN
ejpam-4340	597	4	29	29	NUM
ejpam-4340	597	5	.	.	PUNCT
ejpam-4340	598	1	let	let	VERB
ejpam-4340	598	2	f	f	NOUN
ejpam-4340	598	3	:	:	PUNCT
ejpam-4340	598	4	x	x	X
ejpam-4340	598	5	→	→	SYM
ejpam-4340	598	6	y	y	X
ejpam-4340	598	7	be	be	AUX
ejpam-4340	598	8	an	an	DET
ejpam-4340	598	9	open	open	ADJ
ejpam-4340	598	10	bijection	bijection	NOUN
ejpam-4340	598	11	.	.	PUNCT
ejpam-4340	599	1	if	if	SCONJ
ejpam-4340	599	2	f	f	PROPN
ejpam-4340	599	3	is	be	AUX
ejpam-4340	599	4	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	599	5	,	,	PUNCT
ejpam-4340	599	6	then	then	ADV
ejpam-4340	599	7	f	f	PROPN
ejpam-4340	599	8	is	be	AUX
ejpam-4340	599	9	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	599	10	.	.	PUNCT
ejpam-4340	600	1	proof	proof	NOUN
ejpam-4340	600	2	.	.	PUNCT
ejpam-4340	601	1	let	let	VERB
ejpam-4340	601	2	v	v	NUM
ejpam-4340	601	3	∈	∈	PROPN
ejpam-4340	601	4	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	601	5	)	)	PUNCT
ejpam-4340	601	6	and	and	CCONJ
ejpam-4340	601	7	f−1[v	f−1[v	NOUN
ejpam-4340	601	8	]	]	PUNCT
ejpam-4340	601	9	⊆	⊆	NUM
ejpam-4340	601	10	u	u	NOUN
ejpam-4340	601	11	∈	∈	PROPN
ejpam-4340	601	12	o(x	o(x	PROPN
ejpam-4340	601	13	)	)	PUNCT
ejpam-4340	601	14	.	.	PUNCT
ejpam-4340	602	1	(	(	PUNCT
ejpam-4340	602	2	v	v	NUM
ejpam-4340	602	3	∈	∈	PROPN
ejpam-4340	602	4	gωe∗c(y	gωe∗c(y	NOUN
ejpam-4340	602	5	)	)	PUNCT
ejpam-4340	602	6	)	)	PUNCT
ejpam-4340	603	1	(	(	PUNCT
ejpam-4340	603	2	f−1[v	f−1[v	NOUN
ejpam-4340	603	3	]	]	PUNCT
ejpam-4340	603	4	⊆	⊆	NUM
ejpam-4340	603	5	u	u	NOUN
ejpam-4340	603	6	∈	∈	PROPN
ejpam-4340	603	7	o(x	o(x	PROPN
ejpam-4340	603	8	)	)	PUNCT
ejpam-4340	603	9	)	)	PUNCT
ejpam-4340	604	1	f	f	PROPN
ejpam-4340	604	2	is	be	AUX
ejpam-4340	604	3	open	open	ADJ
ejpam-4340	604	4	bijection	bijection	NOUN
ejpam-4340	604	5	}	}	PUNCT
ejpam-4340	604	6	⇒	⇒	VERB
ejpam-4340	604	7	⇒	⇒	NOUN
ejpam-4340	604	8	(	(	PUNCT
ejpam-4340	604	9	f	f	X
ejpam-4340	605	1	[	[	X
ejpam-4340	605	2	f−1[v	f−1[v	X
ejpam-4340	605	3	]	]	X
ejpam-4340	605	4	]	]	X
ejpam-4340	605	5	=	=	SYM
ejpam-4340	605	6	v	v	ADP
ejpam-4340	605	7	⊆	⊆	NUM
ejpam-4340	605	8	f	f	X
ejpam-4340	606	1	[	[	X
ejpam-4340	606	2	u	u	X
ejpam-4340	606	3	]	]	X
ejpam-4340	606	4	∈	∈	PROPN
ejpam-4340	606	5	o(y	o(y	PROPN
ejpam-4340	606	6	)	)	PUNCT
ejpam-4340	606	7	)	)	PUNCT
ejpam-4340	606	8	(	(	PUNCT
ejpam-4340	606	9	ωe∗-cl(v	ωe∗-cl(v	NUM
ejpam-4340	606	10	)	)	PUNCT
ejpam-4340	606	11	⊆	⊆	NUM
ejpam-4340	606	12	f	f	X
ejpam-4340	607	1	[	[	X
ejpam-4340	607	2	u	u	X
ejpam-4340	607	3	]	]	X
ejpam-4340	607	4	)	)	PUNCT
ejpam-4340	607	5	f	f	PROPN
ejpam-4340	607	6	is	be	AUX
ejpam-4340	607	7	g∗ωe∗-continuous	g∗ωe∗-continuous	ADJ
ejpam-4340	607	8	}	}	PUNCT
ejpam-4340	607	9	⇒	⇒	NOUN
ejpam-4340	607	10	⇒	⇒	NOUN
ejpam-4340	607	11	(	(	PUNCT
ejpam-4340	607	12	f−1[ωe∗-cl(v	f−1[ωe∗-cl(v	NOUN
ejpam-4340	607	13	)	)	PUNCT
ejpam-4340	607	14	]	]	PUNCT
ejpam-4340	608	1	∈	∈	PROPN
ejpam-4340	608	2	gωe∗c(x))(f−1[ωe∗-cl(v	gωe∗c(x))(f−1[ωe∗-cl(v	NOUN
ejpam-4340	608	3	)	)	PUNCT
ejpam-4340	608	4	]	]	PUNCT
ejpam-4340	608	5	⊆	⊆	NUM
ejpam-4340	608	6	u	u	NOUN
ejpam-4340	608	7	)	)	PUNCT
ejpam-4340	608	8	⇒	⇒	NOUN
ejpam-4340	608	9	ωe∗-cl(f−1[v	ωe∗-cl(f−1[v	PUNCT
ejpam-4340	608	10	]	]	X
ejpam-4340	608	11	)	)	PUNCT
ejpam-4340	609	1	⊆	⊆	NUM
ejpam-4340	609	2	ωe∗-cl(f−1[ωe∗-cl(v	ωe∗-cl(f−1[ωe∗-cl(v	NUM
ejpam-4340	609	3	)	)	PUNCT
ejpam-4340	609	4	]	]	PUNCT
ejpam-4340	609	5	)	)	PUNCT
ejpam-4340	610	1	⊆	⊆	NUM
ejpam-4340	610	2	u.	u.	NOUN
ejpam-4340	610	3	theorem	theorem	NOUN
ejpam-4340	610	4	30	30	NUM
ejpam-4340	610	5	.	.	PUNCT
ejpam-4340	611	1	let	let	VERB
ejpam-4340	611	2	f	f	NOUN
ejpam-4340	611	3	:	:	PUNCT
ejpam-4340	611	4	x	x	X
ejpam-4340	611	5	→	→	SYM
ejpam-4340	611	6	y	y	X
ejpam-4340	611	7	be	be	AUX
ejpam-4340	611	8	a	a	DET
ejpam-4340	611	9	function	function	NOUN
ejpam-4340	611	10	.	.	PUNCT
ejpam-4340	612	1	if	if	SCONJ
ejpam-4340	612	2	f	f	PROPN
ejpam-4340	612	3	is	be	AUX
ejpam-4340	612	4	a	a	DET
ejpam-4340	612	5	continuous	continuous	ADJ
ejpam-4340	612	6	pre	pre	ADJ
ejpam-4340	612	7	-	-	ADJ
ejpam-4340	612	8	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	612	9	bijection	bijection	NOUN
ejpam-4340	612	10	,	,	PUNCT
ejpam-4340	612	11	then	then	ADV
ejpam-4340	612	12	the	the	DET
ejpam-4340	612	13	inverse	inverse	NOUN
ejpam-4340	612	14	function	function	NOUN
ejpam-4340	612	15	of	of	ADP
ejpam-4340	612	16	f	f	PROPN
ejpam-4340	612	17	is	be	AUX
ejpam-4340	612	18	gωe∗-irresolute	gωe∗-irresolute	PROPN
ejpam-4340	612	19	.	.	PUNCT
ejpam-4340	613	1	proof	proof	NOUN
ejpam-4340	613	2	.	.	PUNCT
ejpam-4340	614	1	let	let	VERB
ejpam-4340	614	2	a	a	DET
ejpam-4340	614	3	∈	∈	PROPN
ejpam-4340	614	4	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	614	5	)	)	PUNCT
ejpam-4340	614	6	and	and	CCONJ
ejpam-4340	614	7	(	(	PUNCT
ejpam-4340	614	8	f−1)−1[a	f−1)−1[a	X
ejpam-4340	614	9	]	]	PUNCT
ejpam-4340	615	1	=	=	SYM
ejpam-4340	615	2	f	f	X
ejpam-4340	616	1	[	[	X
ejpam-4340	616	2	a	a	X
ejpam-4340	616	3	]	]	X
ejpam-4340	616	4	⊆	⊆	NUM
ejpam-4340	616	5	u	u	NOUN
ejpam-4340	616	6	∈	∈	NOUN
ejpam-4340	616	7	o(y	o(y	PROPN
ejpam-4340	616	8	)	)	PUNCT
ejpam-4340	616	9	.	.	PUNCT
ejpam-4340	617	1	f	f	X
ejpam-4340	618	1	[	[	X
ejpam-4340	618	2	a	a	X
ejpam-4340	618	3	]	]	X
ejpam-4340	618	4	⊆	⊆	NUM
ejpam-4340	618	5	u	u	NOUN
ejpam-4340	618	6	∈	∈	NOUN
ejpam-4340	618	7	o(y	o(y	PROPN
ejpam-4340	618	8	)	)	PUNCT
ejpam-4340	619	1	f	f	PROPN
ejpam-4340	619	2	is	be	AUX
ejpam-4340	619	3	continuous	continuous	ADJ
ejpam-4340	619	4	}	}	PUNCT
ejpam-4340	619	5	⇒	⇒	NOUN
ejpam-4340	619	6	a	a	DET
ejpam-4340	619	7	⊆	⊆	NUM
ejpam-4340	619	8	f−1[u	f−1[u	NOUN
ejpam-4340	619	9	]	]	PUNCT
ejpam-4340	619	10	∈	∈	PROPN
ejpam-4340	619	11	o(x	o(x	PROPN
ejpam-4340	619	12	)	)	PUNCT
ejpam-4340	619	13	a	a	DET
ejpam-4340	619	14	∈	∈	PROPN
ejpam-4340	619	15	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	619	16	)	)	PUNCT
ejpam-4340	619	17	}	}	PUNCT
ejpam-4340	619	18	⇒	⇒	VERB
ejpam-4340	619	19	p.	p.	NOUN
ejpam-4340	619	20	şaşmaz	şaşmaz	NUM
ejpam-4340	619	21	,	,	PUNCT
ejpam-4340	619	22	m.	m.	NOUN
ejpam-4340	619	23	özkoç	özkoç	PROPN
ejpam-4340	619	24	/	/	SYM
ejpam-4340	619	25	eur	eur	PROPN
ejpam-4340	619	26	.	.	PUNCT
ejpam-4340	620	1	j.	j.	PROPN
ejpam-4340	620	2	pure	pure	PROPN
ejpam-4340	620	3	appl	appl	PROPN
ejpam-4340	620	4	.	.	PROPN
ejpam-4340	620	5	math	math	PROPN
ejpam-4340	620	6	,	,	PUNCT
ejpam-4340	620	7	15	15	NUM
ejpam-4340	620	8	(	(	PUNCT
ejpam-4340	620	9	2	2	NUM
ejpam-4340	620	10	)	)	PUNCT
ejpam-4340	620	11	(	(	PUNCT
ejpam-4340	620	12	2022	2022	NUM
ejpam-4340	620	13	)	)	PUNCT
ejpam-4340	620	14	,	,	PUNCT
ejpam-4340	620	15	354	354	NUM
ejpam-4340	620	16	-	-	SYM
ejpam-4340	620	17	374	374	NUM
ejpam-4340	620	18	372	372	NUM
ejpam-4340	620	19	⇒	⇒	NOUN
ejpam-4340	620	20	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	620	21	)	)	PUNCT
ejpam-4340	620	22	⊆	⊆	NUM
ejpam-4340	620	23	f−1[u	f−1[u	NOUN
ejpam-4340	620	24	]	]	X
ejpam-4340	620	25	f	f	X
ejpam-4340	620	26	is	be	AUX
ejpam-4340	620	27	a	a	DET
ejpam-4340	620	28	pre	pre	ADJ
ejpam-4340	620	29	-	-	ADJ
ejpam-4340	620	30	ωe∗-closed	ωe∗-closed	ADJ
ejpam-4340	620	31	bijection	bijection	NOUN
ejpam-4340	620	32	}	}	PUNCT
ejpam-4340	620	33	⇒	⇒	NOUN
ejpam-4340	620	34	⇒	⇒	NOUN
ejpam-4340	620	35	(	(	PUNCT
ejpam-4340	620	36	f	f	X
ejpam-4340	621	1	[	[	X
ejpam-4340	621	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	621	3	)	)	PUNCT
ejpam-4340	621	4	]	]	PUNCT
ejpam-4340	621	5	∈	∈	PROPN
ejpam-4340	621	6	ωe∗c(y	ωe∗c(y	PROPN
ejpam-4340	621	7	)	)	PUNCT
ejpam-4340	621	8	)	)	PUNCT
ejpam-4340	622	1	(	(	PUNCT
ejpam-4340	622	2	f	f	X
ejpam-4340	623	1	[	[	X
ejpam-4340	623	2	a	a	X
ejpam-4340	623	3	]	]	X
ejpam-4340	623	4	⊆	⊆	NUM
ejpam-4340	623	5	f	f	X
ejpam-4340	623	6	[	[	X
ejpam-4340	623	7	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	623	8	)	)	PUNCT
ejpam-4340	623	9	]	]	PUNCT
ejpam-4340	624	1	⊆	⊆	NUM
ejpam-4340	624	2	f	f	X
ejpam-4340	624	3	[	[	X
ejpam-4340	624	4	f−1[u	f−1[u	X
ejpam-4340	624	5	]	]	X
ejpam-4340	624	6	]	]	X
ejpam-4340	624	7	=	=	SYM
ejpam-4340	624	8	u	u	NOUN
ejpam-4340	624	9	)	)	PUNCT
ejpam-4340	624	10	⇒	⇒	VERB
ejpam-4340	624	11	ωe∗-cl(f	ωe∗-cl(f	NOUN
ejpam-4340	625	1	[	[	X
ejpam-4340	625	2	a	a	X
ejpam-4340	625	3	]	]	X
ejpam-4340	625	4	)	)	PUNCT
ejpam-4340	625	5	⊆	⊆	NUM
ejpam-4340	625	6	ωe∗-cl(f	ωe∗-cl(f	NOUN
ejpam-4340	625	7	[	[	X
ejpam-4340	625	8	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	625	9	)	)	PUNCT
ejpam-4340	625	10	]	]	PUNCT
ejpam-4340	625	11	)	)	PUNCT
ejpam-4340	626	1	=	=	PUNCT
ejpam-4340	626	2	f	f	X
ejpam-4340	627	1	[	[	X
ejpam-4340	627	2	ωe∗-cl(a	ωe∗-cl(a	NUM
ejpam-4340	627	3	)	)	PUNCT
ejpam-4340	627	4	]	]	PUNCT
ejpam-4340	627	5	⊆	⊆	NUM
ejpam-4340	627	6	u.	u.	NOUN
ejpam-4340	627	7	theorem	theorem	NOUN
ejpam-4340	627	8	31	31	NUM
ejpam-4340	627	9	.	.	PUNCT
ejpam-4340	628	1	let	let	VERB
ejpam-4340	628	2	f	f	NOUN
ejpam-4340	628	3	:	:	PUNCT
ejpam-4340	628	4	x	x	X
ejpam-4340	628	5	→	→	SYM
ejpam-4340	628	6	y	y	PROPN
ejpam-4340	628	7	and	and	CCONJ
ejpam-4340	628	8	g	g	PROPN
ejpam-4340	628	9	:	:	PUNCT
ejpam-4340	628	10	y	y	PROPN
ejpam-4340	628	11	→	→	SYM
ejpam-4340	628	12	z	z	X
ejpam-4340	628	13	be	be	AUX
ejpam-4340	628	14	two	two	NUM
ejpam-4340	628	15	functions	function	NOUN
ejpam-4340	628	16	.	.	PUNCT
ejpam-4340	629	1	if	if	SCONJ
ejpam-4340	629	2	f	f	PROPN
ejpam-4340	629	3	is	be	AUX
ejpam-4340	629	4	a	a	DET
ejpam-4340	629	5	continuous	continuous	ADJ
ejpam-4340	629	6	surjection	surjection	NOUN
ejpam-4340	629	7	and	and	CCONJ
ejpam-4340	629	8	g	g	PROPN
ejpam-4340	629	9	◦	◦	NOUN
ejpam-4340	629	10	f	f	PROPN
ejpam-4340	629	11	is	be	AUX
ejpam-4340	629	12	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	629	13	,	,	PUNCT
ejpam-4340	629	14	then	then	ADV
ejpam-4340	629	15	g	g	PROPN
ejpam-4340	629	16	is	be	AUX
ejpam-4340	629	17	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	629	18	.	.	PUNCT
ejpam-4340	630	1	proof	proof	NOUN
ejpam-4340	630	2	.	.	PUNCT
ejpam-4340	631	1	let	let	VERB
ejpam-4340	631	2	v	v	NUM
ejpam-4340	631	3	∈	∈	PROPN
ejpam-4340	631	4	c(y	c(y	PROPN
ejpam-4340	631	5	)	)	PUNCT
ejpam-4340	631	6	.	.	PUNCT
ejpam-4340	632	1	v	v	X
ejpam-4340	632	2	∈	∈	PROPN
ejpam-4340	632	3	c(y	c(y	PROPN
ejpam-4340	632	4	)	)	PUNCT
ejpam-4340	633	1	f	f	PROPN
ejpam-4340	633	2	is	be	AUX
ejpam-4340	633	3	continuous	continuous	ADJ
ejpam-4340	633	4	}	}	PUNCT
ejpam-4340	633	5	⇒	⇒	VERB
ejpam-4340	633	6	f−1[v	f−1[v	NOUN
ejpam-4340	633	7	]	]	PUNCT
ejpam-4340	633	8	∈	∈	PROPN
ejpam-4340	633	9	c(x	c(x	NOUN
ejpam-4340	633	10	)	)	PUNCT
ejpam-4340	633	11	f	f	PROPN
ejpam-4340	633	12	is	be	AUX
ejpam-4340	633	13	surjective	surjective	ADJ
ejpam-4340	633	14	}	}	PUNCT
ejpam-4340	633	15	⇒	⇒	VERB
ejpam-4340	633	16	⇒	⇒	NOUN
ejpam-4340	633	17	g[f	g[f	PROPN
ejpam-4340	634	1	[	[	X
ejpam-4340	634	2	f−1[v	f−1[v	NOUN
ejpam-4340	634	3	]	]	X
ejpam-4340	634	4	]	]	X
ejpam-4340	634	5	]	]	X
ejpam-4340	635	1	=	=	X
ejpam-4340	635	2	(	(	PUNCT
ejpam-4340	635	3	g	g	PROPN
ejpam-4340	635	4	◦	◦	NOUN
ejpam-4340	635	5	f)[f−1[v	f)[f−1[v	X
ejpam-4340	635	6	]	]	X
ejpam-4340	635	7	]	]	X
ejpam-4340	635	8	=	=	PUNCT
ejpam-4340	635	9	g[v	g[v	NOUN
ejpam-4340	635	10	]	]	PUNCT
ejpam-4340	635	11	g	g	NOUN
ejpam-4340	635	12	◦	◦	NOUN
ejpam-4340	635	13	f	f	PROPN
ejpam-4340	635	14	is	be	AUX
ejpam-4340	635	15	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	635	16	}	}	PUNCT
ejpam-4340	635	17	⇒	⇒	NOUN
ejpam-4340	635	18	g[v	g[v	NOUN
ejpam-4340	635	19	]	]	PUNCT
ejpam-4340	635	20	∈	∈	PROPN
ejpam-4340	635	21	gωe∗c(x	gωe∗c(x	NOUN
ejpam-4340	635	22	)	)	PUNCT
ejpam-4340	635	23	.	.	PUNCT
ejpam-4340	636	1	theorem	theorem	ADJ
ejpam-4340	636	2	32	32	NUM
ejpam-4340	636	3	.	.	PUNCT
ejpam-4340	637	1	let	let	VERB
ejpam-4340	637	2	f	f	NOUN
ejpam-4340	637	3	:	:	PUNCT
ejpam-4340	637	4	x	x	X
ejpam-4340	637	5	→	→	SYM
ejpam-4340	637	6	y	y	X
ejpam-4340	637	7	be	be	AUX
ejpam-4340	637	8	a	a	DET
ejpam-4340	637	9	function	function	NOUN
ejpam-4340	637	10	.	.	PUNCT
ejpam-4340	638	1	if	if	SCONJ
ejpam-4340	638	2	f	f	PROPN
ejpam-4340	638	3	is	be	AUX
ejpam-4340	638	4	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	638	5	continuous	continuous	ADJ
ejpam-4340	638	6	and	and	CCONJ
ejpam-4340	638	7	x	x	PRON
ejpam-4340	638	8	is	be	AUX
ejpam-4340	638	9	normal	normal	ADJ
ejpam-4340	638	10	,	,	PUNCT
ejpam-4340	638	11	then	then	ADV
ejpam-4340	638	12	y	y	PROPN
ejpam-4340	638	13	is	be	AUX
ejpam-4340	638	14	ωe∗-normal	ωe∗-normal	ADJ
ejpam-4340	638	15	.	.	PUNCT
ejpam-4340	639	1	proof	proof	NOUN
ejpam-4340	639	2	.	.	PUNCT
ejpam-4340	640	1	let	let	VERB
ejpam-4340	640	2	a	a	DET
ejpam-4340	640	3	,	,	PUNCT
ejpam-4340	640	4	b	b	PROPN
ejpam-4340	640	5	∈	∈	PROPN
ejpam-4340	640	6	c(y	c(y	PROPN
ejpam-4340	640	7	)	)	PUNCT
ejpam-4340	640	8	and	and	CCONJ
ejpam-4340	640	9	a	a	DET
ejpam-4340	640	10	∩b	∩b	NOUN
ejpam-4340	640	11	=	=	X
ejpam-4340	640	12	∅.	∅.	X
ejpam-4340	640	13	(	(	PUNCT
ejpam-4340	640	14	a	a	PRON
ejpam-4340	640	15	,	,	PUNCT
ejpam-4340	640	16	b	b	PROPN
ejpam-4340	640	17	∈	∈	PROPN
ejpam-4340	640	18	c(y	c(y	PROPN
ejpam-4340	640	19	)	)	PUNCT
ejpam-4340	640	20	)	)	PUNCT
ejpam-4340	640	21	(	(	PUNCT
ejpam-4340	640	22	a	a	DET
ejpam-4340	640	23	∩b	∩b	NOUN
ejpam-4340	640	24	=	=	SYM
ejpam-4340	640	25	∅	∅	NOUN
ejpam-4340	640	26	)	)	PUNCT
ejpam-4340	640	27	f	f	PROPN
ejpam-4340	640	28	is	be	AUX
ejpam-4340	640	29	continuous	continuous	ADJ
ejpam-4340	640	30	}	}	PUNCT
ejpam-4340	640	31	⇒	⇒	NOUN
ejpam-4340	640	32	⇒	⇒	NOUN
ejpam-4340	640	33	(	(	PUNCT
ejpam-4340	640	34	f−1[a	f−1[a	PROPN
ejpam-4340	640	35	]	]	X
ejpam-4340	640	36	,	,	PUNCT
ejpam-4340	640	37	f−1[b	f−1[b	X
ejpam-4340	640	38	]	]	PUNCT
ejpam-4340	640	39	∈	∈	PROPN
ejpam-4340	640	40	c(x))(f−1[a	c(x))(f−1[a	NOUN
ejpam-4340	640	41	∩b	∩b	NOUN
ejpam-4340	640	42	]	]	PUNCT
ejpam-4340	640	43	=	=	SYM
ejpam-4340	641	1	f−1[a	f−1[a	X
ejpam-4340	641	2	]	]	X
ejpam-4340	641	3	∩	∩	NOUN
ejpam-4340	641	4	f−1[b	f−1[b	X
ejpam-4340	641	5	]	]	PUNCT
ejpam-4340	641	6	=	=	SYM
ejpam-4340	641	7	f−1[∅	f−1[∅	PROPN
ejpam-4340	641	8	]	]	PUNCT
ejpam-4340	641	9	=	=	SYM
ejpam-4340	641	10	∅	∅	NOUN
ejpam-4340	641	11	)	)	PUNCT
ejpam-4340	641	12	x	x	X
ejpam-4340	641	13	is	be	AUX
ejpam-4340	641	14	normal	normal	ADJ
ejpam-4340	641	15	}	}	PUNCT
ejpam-4340	641	16	⇒	⇒	VERB
ejpam-4340	641	17	⇒	⇒	NOUN
ejpam-4340	641	18	(	(	PUNCT
ejpam-4340	641	19	∃u	∃u	PROPN
ejpam-4340	641	20	∈	∈	PROPN
ejpam-4340	641	21	o(x	o(x	PROPN
ejpam-4340	641	22	,	,	PUNCT
ejpam-4340	641	23	f−1[a]))(∃v	f−1[a]))(∃v	PROPN
ejpam-4340	641	24	∈	∈	PROPN
ejpam-4340	641	25	o(x	o(x	PROPN
ejpam-4340	641	26	,	,	PUNCT
ejpam-4340	641	27	f−1[b]))(u	f−1[b]))(u	ADJ
ejpam-4340	641	28	∩	∩	ADJ
ejpam-4340	641	29	v	v	NOUN
ejpam-4340	641	30	=	=	SYM
ejpam-4340	641	31	∅	∅	NOUN
ejpam-4340	641	32	)	)	PUNCT
ejpam-4340	641	33	f	f	PROPN
ejpam-4340	641	34	is	be	AUX
ejpam-4340	641	35	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	641	36	}	}	PUNCT
ejpam-4340	641	37	theorem	theorem	VERB
ejpam-4340	641	38	26⇒	26⇒	NUM
ejpam-4340	641	39	⇒	⇒	NOUN
ejpam-4340	641	40	(	(	PUNCT
ejpam-4340	641	41	∃g	∃g	PROPN
ejpam-4340	641	42	,	,	PUNCT
ejpam-4340	641	43	h	h	NOUN
ejpam-4340	641	44	∈	∈	PROPN
ejpam-4340	641	45	gωe∗o(y	gωe∗o(y	NOUN
ejpam-4340	641	46	)	)	PUNCT
ejpam-4340	641	47	)	)	PUNCT
ejpam-4340	641	48	(	(	PUNCT
ejpam-4340	641	49	a	a	DET
ejpam-4340	641	50	⊆	⊆	NUM
ejpam-4340	641	51	g)(b	g)(b	ADJ
ejpam-4340	641	52	⊆	⊆	NUM
ejpam-4340	641	53	h)(f−1[g	h)(f−1[g	NOUN
ejpam-4340	641	54	]	]	PUNCT
ejpam-4340	641	55	⊆	⊆	NUM
ejpam-4340	641	56	u)(f−1[h	u)(f−1[h	NOUN
ejpam-4340	641	57	]	]	X
ejpam-4340	641	58	⊆	⊆	NUM
ejpam-4340	641	59	v	v	NOUN
ejpam-4340	641	60	)	)	PUNCT
ejpam-4340	641	61	(	(	PUNCT
ejpam-4340	641	62	u	u	NOUN
ejpam-4340	641	63	∩	∩	X
ejpam-4340	641	64	v	v	NOUN
ejpam-4340	641	65	=	=	SYM
ejpam-4340	641	66	∅	∅	NOUN
ejpam-4340	641	67	)	)	PUNCT
ejpam-4340	641	68	⇒	⇒	NOUN
ejpam-4340	641	69	(	(	PUNCT
ejpam-4340	641	70	a	a	DET
ejpam-4340	641	71	⊆	⊆	NUM
ejpam-4340	641	72	ωe∗-int(g))(b	ωe∗-int(g))(b	ADJ
ejpam-4340	641	73	⊆	⊆	NUM
ejpam-4340	641	74	ωe∗-int(h))(a	ωe∗-int(h))(a	NUM
ejpam-4340	641	75	∩b	∩b	NOUN
ejpam-4340	641	76	⊆	⊆	NUM
ejpam-4340	641	77	g	g	ADP
ejpam-4340	641	78	∩h)(f−1[g	∩h)(f−1[g	PROPN
ejpam-4340	641	79	]	]	PUNCT
ejpam-4340	641	80	∩	∩	NOUN
ejpam-4340	641	81	f−1[h	f−1[h	NOUN
ejpam-4340	641	82	]	]	X
ejpam-4340	641	83	=	=	SYM
ejpam-4340	641	84	∅	∅	NOUN
ejpam-4340	641	85	)	)	PUNCT
ejpam-4340	641	86	(	(	PUNCT
ejpam-4340	641	87	u	u	NOUN
ejpam-4340	641	88	′	′	NUM
ejpam-4340	641	89	:	:	PUNCT
ejpam-4340	641	90	=	=	SYM
ejpam-4340	641	91	ωe∗-int(g))(v	ωe∗-int(g))(v	X
ejpam-4340	641	92	′	′	NUM
ejpam-4340	641	93	:	:	PUNCT
ejpam-4340	642	1	=	=	NOUN
ejpam-4340	642	2	ωe∗-int(h	ωe∗-int(h	NOUN
ejpam-4340	642	3	)	)	PUNCT
ejpam-4340	642	4	)	)	PUNCT
ejpam-4340	642	5	}	}	PUNCT
ejpam-4340	642	6	⇒	⇒	VERB
ejpam-4340	642	7	⇒	⇒	NOUN
ejpam-4340	642	8	(	(	PUNCT
ejpam-4340	642	9	u	u	NOUN
ejpam-4340	642	10	′	′	PROPN
ejpam-4340	642	11	∈	∈	PROPN
ejpam-4340	642	12	ωe∗o(y	ωe∗o(y	NOUN
ejpam-4340	642	13	,	,	PUNCT
ejpam-4340	642	14	a))(v	a))(v	VERB
ejpam-4340	642	15	′	′	NUM
ejpam-4340	642	16	∈	∈	PROPN
ejpam-4340	642	17	ωe∗o(y	ωe∗o(y	NOUN
ejpam-4340	642	18	,	,	PUNCT
ejpam-4340	642	19	b))(u	b))(u	NUM
ejpam-4340	642	20	′	′	NUM
ejpam-4340	642	21	∩	∩	NOUN
ejpam-4340	642	22	v	v	ADP
ejpam-4340	642	23	′	′	NOUN
ejpam-4340	642	24	=	=	NOUN
ejpam-4340	642	25	∅	∅	NOUN
ejpam-4340	642	26	)	)	PUNCT
ejpam-4340	642	27	.	.	PUNCT
ejpam-4340	643	1	theorem	theorem	NOUN
ejpam-4340	643	2	33	33	NUM
ejpam-4340	643	3	.	.	PUNCT
ejpam-4340	644	1	let	let	VERB
ejpam-4340	644	2	f	f	NOUN
ejpam-4340	644	3	:	:	PUNCT
ejpam-4340	644	4	x	x	X
ejpam-4340	644	5	→	→	SYM
ejpam-4340	644	6	y	y	X
ejpam-4340	644	7	be	be	AUX
ejpam-4340	644	8	a	a	DET
ejpam-4340	644	9	function	function	NOUN
ejpam-4340	644	10	.	.	PUNCT
ejpam-4340	645	1	if	if	SCONJ
ejpam-4340	645	2	f	f	PROPN
ejpam-4340	645	3	is	be	AUX
ejpam-4340	645	4	a	a	DET
ejpam-4340	645	5	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	645	6	continuous	continuous	ADJ
ejpam-4340	645	7	surjection	surjection	NOUN
ejpam-4340	645	8	,	,	PUNCT
ejpam-4340	645	9	ωe∗-open	ωe∗-open	VERB
ejpam-4340	645	10	and	and	CCONJ
ejpam-4340	645	11	x	x	SYM
ejpam-4340	645	12	is	be	AUX
ejpam-4340	645	13	regular	regular	ADJ
ejpam-4340	645	14	,	,	PUNCT
ejpam-4340	645	15	then	then	ADV
ejpam-4340	645	16	y	y	PROPN
ejpam-4340	645	17	is	be	AUX
ejpam-4340	645	18	ωe∗-regular	ωe∗-regular	ADJ
ejpam-4340	645	19	.	.	PUNCT
ejpam-4340	646	1	proof	proof	NOUN
ejpam-4340	646	2	.	.	PUNCT
ejpam-4340	647	1	let	let	VERB
ejpam-4340	647	2	y	y	PROPN
ejpam-4340	647	3	∈	∈	PROPN
ejpam-4340	647	4	y	y	PROPN
ejpam-4340	647	5	and	and	CCONJ
ejpam-4340	647	6	u	u	PROPN
ejpam-4340	647	7	∈	∈	PROPN
ejpam-4340	647	8	o(y	o(y	PROPN
ejpam-4340	647	9	,	,	PUNCT
ejpam-4340	647	10	y	y	NOUN
ejpam-4340	647	11	)	)	PUNCT
ejpam-4340	647	12	.	.	PUNCT
ejpam-4340	648	1	(	(	PUNCT
ejpam-4340	648	2	y	y	PROPN
ejpam-4340	648	3	∈	∈	PROPN
ejpam-4340	648	4	y	y	PROPN
ejpam-4340	648	5	)	)	PUNCT
ejpam-4340	648	6	(	(	PUNCT
ejpam-4340	648	7	u	u	NOUN
ejpam-4340	648	8	∈	∈	PROPN
ejpam-4340	648	9	o(y	o(y	PROPN
ejpam-4340	648	10	,	,	PUNCT
ejpam-4340	648	11	y	y	NOUN
ejpam-4340	648	12	)	)	PUNCT
ejpam-4340	648	13	)	)	PUNCT
ejpam-4340	649	1	f	f	PROPN
ejpam-4340	649	2	is	be	AUX
ejpam-4340	649	3	continuous	continuous	ADJ
ejpam-4340	649	4	}	}	PUNCT
ejpam-4340	649	5	⇒	⇒	NOUN
ejpam-4340	649	6	(	(	PUNCT
ejpam-4340	649	7	∃x	∃x	NOUN
ejpam-4340	649	8	∈	∈	NOUN
ejpam-4340	649	9	x)(y	x)(y	PUNCT
ejpam-4340	650	1	=	=	SYM
ejpam-4340	650	2	f(x))(f−1[u	f(x))(f−1[u	PROPN
ejpam-4340	650	3	]	]	PUNCT
ejpam-4340	650	4	∈	∈	PROPN
ejpam-4340	650	5	o(x	o(x	PROPN
ejpam-4340	650	6	,	,	PUNCT
ejpam-4340	650	7	x	x	NOUN
ejpam-4340	650	8	)	)	PUNCT
ejpam-4340	650	9	)	)	PUNCT
ejpam-4340	651	1	x	x	X
ejpam-4340	651	2	is	be	AUX
ejpam-4340	651	3	regular	regular	ADJ
ejpam-4340	651	4	}	}	PUNCT
ejpam-4340	651	5	⇒	⇒	NOUN
ejpam-4340	651	6	⇒	⇒	NOUN
ejpam-4340	651	7	(	(	PUNCT
ejpam-4340	651	8	∃v	∃v	PROPN
ejpam-4340	651	9	∈	∈	PROPN
ejpam-4340	651	10	o(x	o(x	PROPN
ejpam-4340	651	11	,	,	PUNCT
ejpam-4340	651	12	x))(v	x))(v	NOUN
ejpam-4340	651	13	⊆	⊆	NUM
ejpam-4340	651	14	cl(v	cl(v	NOUN
ejpam-4340	651	15	)	)	PUNCT
ejpam-4340	651	16	⊆	⊆	NUM
ejpam-4340	651	17	f−1[u	f−1[u	NOUN
ejpam-4340	651	18	]	]	PUNCT
ejpam-4340	651	19	)	)	PUNCT
ejpam-4340	651	20	f	f	PROPN
ejpam-4340	651	21	is	be	AUX
ejpam-4340	651	22	gωe∗-closed	gωe∗-close	VERB
ejpam-4340	651	23	surjection	surjection	NOUN
ejpam-4340	651	24	}	}	PUNCT
ejpam-4340	651	25	⇒	⇒	VERB
ejpam-4340	651	26	⇒	⇒	NOUN
ejpam-4340	651	27	(	(	PUNCT
ejpam-4340	651	28	∃v	∃v	PROPN
ejpam-4340	651	29	∈	∈	PROPN
ejpam-4340	651	30	o(x	o(x	PROPN
ejpam-4340	651	31	,	,	PUNCT
ejpam-4340	651	32	x))(f	x))(f	PROPN
ejpam-4340	652	1	[	[	X
ejpam-4340	652	2	cl(v	cl(v	X
ejpam-4340	652	3	)	)	PUNCT
ejpam-4340	652	4	]	]	PUNCT
ejpam-4340	653	1	∈	∈	PROPN
ejpam-4340	653	2	gωe∗c(x))(y	gωe∗c(x))(y	PROPN
ejpam-4340	653	3	∈	∈	PROPN
ejpam-4340	653	4	f	f	X
ejpam-4340	654	1	[	[	X
ejpam-4340	654	2	v	v	X
ejpam-4340	654	3	]	]	PUNCT
ejpam-4340	654	4	⊆	⊆	NUM
ejpam-4340	654	5	f	f	X
ejpam-4340	655	1	[	[	X
ejpam-4340	655	2	cl(v	cl(v	X
ejpam-4340	655	3	)	)	PUNCT
ejpam-4340	655	4	]	]	PUNCT
ejpam-4340	656	1	⊆	⊆	NUM
ejpam-4340	656	2	u	u	NOUN
ejpam-4340	656	3	)	)	PUNCT
ejpam-4340	656	4	f	f	PROPN
ejpam-4340	656	5	is	be	AUX
ejpam-4340	656	6	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	656	7	}	}	PUNCT
ejpam-4340	656	8	⇒	⇒	VERB
ejpam-4340	656	9	⇒	⇒	NOUN
ejpam-4340	656	10	(	(	PUNCT
ejpam-4340	656	11	f	f	X
ejpam-4340	656	12	[	[	X
ejpam-4340	656	13	v	v	X
ejpam-4340	656	14	]	]	PUNCT
ejpam-4340	656	15	∈	∈	PROPN
ejpam-4340	656	16	ωe∗o(y	ωe∗o(y	PROPN
ejpam-4340	656	17	,	,	PUNCT
ejpam-4340	656	18	y))(ωe∗-cl(f	y))(ωe∗-cl(f	PUNCT
ejpam-4340	657	1	[	[	X
ejpam-4340	657	2	v	v	X
ejpam-4340	657	3	]	]	PUNCT
ejpam-4340	657	4	)	)	PUNCT
ejpam-4340	657	5	⊆	⊆	NUM
ejpam-4340	657	6	ωe∗-cl(f	ωe∗-cl(f	NOUN
ejpam-4340	658	1	[	[	X
ejpam-4340	658	2	cl(v	cl(v	X
ejpam-4340	658	3	)	)	PUNCT
ejpam-4340	658	4	]	]	PUNCT
ejpam-4340	658	5	)	)	PUNCT
ejpam-4340	658	6	⊆	⊆	NUM
ejpam-4340	658	7	u	u	NOUN
ejpam-4340	658	8	)	)	PUNCT
ejpam-4340	658	9	.	.	PUNCT
ejpam-4340	659	1	references	reference	NOUN
ejpam-4340	659	2	373	373	NUM
ejpam-4340	659	3	conclusion	conclusion	NOUN
ejpam-4340	659	4	many	many	ADJ
ejpam-4340	659	5	forms	form	NOUN
ejpam-4340	659	6	of	of	ADP
ejpam-4340	659	7	generalized	generalized	ADJ
ejpam-4340	659	8	closed	close	VERB
ejpam-4340	659	9	sets	set	NOUN
ejpam-4340	659	10	which	which	PRON
ejpam-4340	659	11	are	be	AUX
ejpam-4340	659	12	first	first	ADV
ejpam-4340	659	13	defined	define	VERB
ejpam-4340	659	14	by	by	ADP
ejpam-4340	659	15	levine	levine	PROPN
ejpam-4340	659	16	[	[	X
ejpam-4340	659	17	14	14	NUM
ejpam-4340	659	18	]	]	PUNCT
ejpam-4340	659	19	have	have	AUX
ejpam-4340	659	20	been	be	AUX
ejpam-4340	659	21	studied	study	VERB
ejpam-4340	659	22	by	by	ADP
ejpam-4340	659	23	many	many	ADJ
ejpam-4340	659	24	authors	author	NOUN
ejpam-4340	659	25	in	in	ADP
ejpam-4340	659	26	recent	recent	ADJ
ejpam-4340	659	27	years	year	NOUN
ejpam-4340	659	28	.	.	PUNCT
ejpam-4340	660	1	this	this	DET
ejpam-4340	660	2	paper	paper	NOUN
ejpam-4340	660	3	is	be	AUX
ejpam-4340	660	4	concerned	concern	VERB
ejpam-4340	660	5	with	with	ADP
ejpam-4340	660	6	the	the	DET
ejpam-4340	660	7	notion	notion	NOUN
ejpam-4340	660	8	of	of	ADP
ejpam-4340	660	9	generalized	generalized	ADJ
ejpam-4340	660	10	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	660	11	sets	set	NOUN
ejpam-4340	660	12	which	which	PRON
ejpam-4340	660	13	are	be	AUX
ejpam-4340	660	14	defined	define	VERB
ejpam-4340	660	15	by	by	ADP
ejpam-4340	660	16	utilizing	utilize	VERB
ejpam-4340	660	17	the	the	DET
ejpam-4340	660	18	concept	concept	NOUN
ejpam-4340	660	19	of	of	ADP
ejpam-4340	660	20	ωe∗-open	ωe∗-open	ADJ
ejpam-4340	660	21	set	set	NOUN
ejpam-4340	660	22	.	.	PUNCT
ejpam-4340	661	1	we	we	PRON
ejpam-4340	661	2	have	have	AUX
ejpam-4340	661	3	seen	see	VERB
ejpam-4340	661	4	that	that	SCONJ
ejpam-4340	661	5	this	this	DET
ejpam-4340	661	6	concept	concept	NOUN
ejpam-4340	661	7	is	be	AUX
ejpam-4340	661	8	weaker	weak	ADJ
ejpam-4340	661	9	than	than	ADP
ejpam-4340	661	10	many	many	ADJ
ejpam-4340	661	11	generalized	generalize	VERB
ejpam-4340	661	12	closed	close	VERB
ejpam-4340	661	13	set	set	VERB
ejpam-4340	661	14	forms	form	NOUN
ejpam-4340	661	15	in	in	ADP
ejpam-4340	661	16	the	the	DET
ejpam-4340	661	17	literature	literature	NOUN
ejpam-4340	661	18	as	as	SCONJ
ejpam-4340	661	19	will	will	AUX
ejpam-4340	661	20	be	be	AUX
ejpam-4340	661	21	seen	see	VERB
ejpam-4340	661	22	in	in	ADP
ejpam-4340	661	23	figure	figure	NOUN
ejpam-4340	661	24	1	1	NUM
ejpam-4340	661	25	.	.	PUNCT
ejpam-4340	662	1	in	in	ADP
ejpam-4340	662	2	addition	addition	NOUN
ejpam-4340	662	3	,	,	PUNCT
ejpam-4340	662	4	we	we	PRON
ejpam-4340	662	5	gave	give	VERB
ejpam-4340	662	6	some	some	DET
ejpam-4340	662	7	examples	example	NOUN
ejpam-4340	662	8	related	relate	VERB
ejpam-4340	662	9	to	to	ADP
ejpam-4340	662	10	the	the	DET
ejpam-4340	662	11	concept	concept	NOUN
ejpam-4340	662	12	but	but	CCONJ
ejpam-4340	662	13	we	we	PRON
ejpam-4340	662	14	could	could	AUX
ejpam-4340	662	15	not	not	PART
ejpam-4340	662	16	find	find	VERB
ejpam-4340	662	17	an	an	DET
ejpam-4340	662	18	example	example	NOUN
ejpam-4340	662	19	generalized	generalize	VERB
ejpam-4340	662	20	ωe∗-closed	ωe∗-close	VERB
ejpam-4340	662	21	set	set	NOUN
ejpam-4340	662	22	which	which	PRON
ejpam-4340	662	23	is	be	AUX
ejpam-4340	662	24	not	not	PART
ejpam-4340	662	25	ωe∗closed	ωe∗closed	ADJ
ejpam-4340	662	26	.	.	PUNCT
ejpam-4340	663	1	we	we	PRON
ejpam-4340	663	2	believe	believe	VERB
ejpam-4340	663	3	that	that	SCONJ
ejpam-4340	663	4	this	this	DET
ejpam-4340	663	5	study	study	NOUN
ejpam-4340	663	6	will	will	AUX
ejpam-4340	663	7	help	help	VERB
ejpam-4340	663	8	researchers	researcher	NOUN
ejpam-4340	663	9	to	to	PART
ejpam-4340	663	10	upgrade	upgrade	VERB
ejpam-4340	663	11	and	and	CCONJ
ejpam-4340	663	12	support	support	VERB
ejpam-4340	663	13	further	further	ADJ
ejpam-4340	663	14	studies	study	NOUN
ejpam-4340	663	15	related	relate	VERB
ejpam-4340	663	16	to	to	ADP
ejpam-4340	663	17	compactness	compactness	NOUN
ejpam-4340	663	18	and	and	CCONJ
ejpam-4340	663	19	connectedness	connectedness	NOUN
ejpam-4340	663	20	etc	etc	X
ejpam-4340	663	21	.	.	X
ejpam-4340	664	1	also	also	ADV
ejpam-4340	664	2	,	,	PUNCT
ejpam-4340	664	3	the	the	DET
ejpam-4340	664	4	objects	object	NOUN
ejpam-4340	664	5	considered	consider	VERB
ejpam-4340	664	6	in	in	ADP
ejpam-4340	664	7	the	the	DET
ejpam-4340	664	8	article	article	NOUN
ejpam-4340	664	9	may	may	AUX
ejpam-4340	664	10	find	find	VERB
ejpam-4340	664	11	an	an	DET
ejpam-4340	664	12	application	application	NOUN
ejpam-4340	664	13	in	in	ADP
ejpam-4340	664	14	the	the	DET
ejpam-4340	664	15	area	area	NOUN
ejpam-4340	664	16	of	of	ADP
ejpam-4340	664	17	both	both	CCONJ
ejpam-4340	664	18	pure	pure	ADJ
ejpam-4340	664	19	and	and	CCONJ
ejpam-4340	664	20	applied	apply	VERB
ejpam-4340	664	21	sciences	science	NOUN
ejpam-4340	664	22	such	such	ADJ
ejpam-4340	664	23	as	as	ADP
ejpam-4340	664	24	computational	computational	ADJ
ejpam-4340	664	25	topology	topology	NOUN
ejpam-4340	664	26	and	and	CCONJ
ejpam-4340	664	27	digital	digital	ADJ
ejpam-4340	664	28	topology	topology	NOUN
ejpam-4340	664	29	.	.	PUNCT
ejpam-4340	665	1	acknowledgements	acknowledgement	NOUN
ejpam-4340	665	2	we	we	PRON
ejpam-4340	665	3	would	would	AUX
ejpam-4340	665	4	like	like	VERB
ejpam-4340	665	5	to	to	PART
ejpam-4340	665	6	thank	thank	VERB
ejpam-4340	665	7	the	the	DET
ejpam-4340	665	8	anonymous	anonymous	ADJ
ejpam-4340	665	9	reviewers	reviewer	NOUN
ejpam-4340	665	10	for	for	ADP
ejpam-4340	665	11	their	their	PRON
ejpam-4340	665	12	careful	careful	ADJ
ejpam-4340	665	13	reading	reading	NOUN
ejpam-4340	665	14	of	of	ADP
ejpam-4340	665	15	our	our	PRON
ejpam-4340	665	16	manuscript	manuscript	NOUN
ejpam-4340	665	17	and	and	CCONJ
ejpam-4340	665	18	their	their	PRON
ejpam-4340	665	19	insightful	insightful	ADJ
ejpam-4340	665	20	comments	comment	NOUN
ejpam-4340	665	21	and	and	CCONJ
ejpam-4340	665	22	suggestions	suggestion	NOUN
ejpam-4340	665	23	.	.	PUNCT
ejpam-4340	666	1	references	reference	NOUN
ejpam-4340	666	2	[	[	X
ejpam-4340	666	3	1	1	NUM
ejpam-4340	666	4	]	]	PUNCT
ejpam-4340	666	5	m.	m.	NOUN
ejpam-4340	666	6	e.	e.	PROPN
ejpam-4340	666	7	abd	abd	PROPN
ejpam-4340	666	8	el	el	PROPN
ejpam-4340	666	9	-	-	PROPN
ejpam-4340	666	10	monsef	monsef	PROPN
ejpam-4340	666	11	,	,	PUNCT
ejpam-4340	666	12	s.	s.	PROPN
ejpam-4340	666	13	n.	n.	PROPN
ejpam-4340	666	14	el	el	PROPN
ejpam-4340	666	15	-	-	PROPN
ejpam-4340	666	16	deeb	deeb	PROPN
ejpam-4340	666	17	,	,	PUNCT
ejpam-4340	666	18	and	and	CCONJ
ejpam-4340	666	19	r.	r.	PROPN
ejpam-4340	666	20	a.	a.	PROPN
ejpam-4340	666	21	mahmoud	mahmoud	PROPN
ejpam-4340	666	22	.	.	PUNCT
ejpam-4340	667	1	β	β	X
ejpam-4340	667	2	-	-	ADJ
ejpam-4340	667	3	open	open	ADJ
ejpam-4340	667	4	sets	set	NOUN
ejpam-4340	667	5	and	and	CCONJ
ejpam-4340	667	6	βcontinuous	βcontinuous	ADJ
ejpam-4340	667	7	mappings	mapping	NOUN
ejpam-4340	667	8	.	.	PUNCT
ejpam-4340	668	1	bull	bull	PROPN
ejpam-4340	668	2	fac	fac	PROPN
ejpam-4340	668	3	sci	sci	PROPN
ejpam-4340	668	4	assiut	assiut	PROPN
ejpam-4340	668	5	univ	univ	PROPN
ejpam-4340	668	6	a	a	PRON
ejpam-4340	668	7	,	,	PUNCT
ejpam-4340	668	8	12(1):77–90	12(1):77–90	NUM
ejpam-4340	668	9	,	,	PUNCT
ejpam-4340	668	10	1983	1983	NUM
ejpam-4340	668	11	.	.	PUNCT
ejpam-4340	669	1	[	[	X
ejpam-4340	669	2	2	2	X
ejpam-4340	669	3	]	]	PUNCT
ejpam-4340	669	4	h.	h.	PROPN
ejpam-4340	669	5	h.	h.	PROPN
ejpam-4340	669	6	aljarrah	aljarrah	PROPN
ejpam-4340	669	7	,	,	PUNCT
ejpam-4340	669	8	m.	m.	PROPN
ejpam-4340	669	9	s.	s.	PROPN
ejpam-4340	669	10	m.	m.	PROPN
ejpam-4340	669	11	noorani	noorani	PROPN
ejpam-4340	669	12	,	,	PUNCT
ejpam-4340	669	13	and	and	CCONJ
ejpam-4340	669	14	t.	t.	PROPN
ejpam-4340	669	15	noiri	noiri	PROPN
ejpam-4340	669	16	.	.	PUNCT
ejpam-4340	670	1	on	on	ADP
ejpam-4340	670	2	ωβ	ωβ	ADJ
ejpam-4340	670	3	-	-	ADJ
ejpam-4340	670	4	open	open	ADJ
ejpam-4340	670	5	sets	set	NOUN
ejpam-4340	670	6	.	.	PUNCT
ejpam-4340	671	1	submitted	submit	VERB
ejpam-4340	671	2	.	.	PUNCT
ejpam-4340	672	1	[	[	X
ejpam-4340	672	2	3	3	X
ejpam-4340	672	3	]	]	X
ejpam-4340	672	4	h.	h.	PROPN
ejpam-4340	672	5	h.	h.	PROPN
ejpam-4340	672	6	aljarrah	aljarrah	PROPN
ejpam-4340	672	7	,	,	PUNCT
ejpam-4340	672	8	m.	m.	PROPN
ejpam-4340	672	9	s.	s.	PROPN
ejpam-4340	672	10	m.	m.	PROPN
ejpam-4340	672	11	noorani	noorani	PROPN
ejpam-4340	672	12	,	,	PUNCT
ejpam-4340	672	13	and	and	CCONJ
ejpam-4340	672	14	t.	t.	PROPN
ejpam-4340	672	15	noiri	noiri	PROPN
ejpam-4340	672	16	.	.	PUNCT
ejpam-4340	673	1	on	on	ADP
ejpam-4340	673	2	ωβ	ωβ	ADJ
ejpam-4340	673	3	-	-	ADJ
ejpam-4340	673	4	continuous	continuous	ADJ
ejpam-4340	673	5	functions	function	NOUN
ejpam-4340	673	6	.	.	PUNCT
ejpam-4340	674	1	eur	eur	PROPN
ejpam-4340	674	2	.	.	PUNCT
ejpam-4340	675	1	j.	j.	PROPN
ejpam-4340	675	2	pure	pure	PROPN
ejpam-4340	675	3	appl	appl	PROPN
ejpam-4340	675	4	.	.	PUNCT
ejpam-4340	675	5	math	math	PROPN
ejpam-4340	675	6	.	.	PUNCT
ejpam-4340	675	7	,	,	PUNCT
ejpam-4340	676	1	5:129–140	5:129–140	NUM
ejpam-4340	676	2	,	,	PUNCT
ejpam-4340	676	3	2012	2012	NUM
ejpam-4340	676	4	.	.	PUNCT
ejpam-4340	677	1	[	[	X
ejpam-4340	677	2	4	4	X
ejpam-4340	677	3	]	]	PUNCT
ejpam-4340	677	4	h.	h.	PROPN
ejpam-4340	677	5	h.	h.	PROPN
ejpam-4340	677	6	aljarrah	aljarrah	PROPN
ejpam-4340	677	7	,	,	PUNCT
ejpam-4340	677	8	m.	m.	PROPN
ejpam-4340	677	9	s.	s.	PROPN
ejpam-4340	677	10	m.	m.	PROPN
ejpam-4340	677	11	noorani	noorani	PROPN
ejpam-4340	677	12	,	,	PUNCT
ejpam-4340	677	13	and	and	CCONJ
ejpam-4340	677	14	t.	t.	PROPN
ejpam-4340	677	15	noiri	noiri	PROPN
ejpam-4340	677	16	.	.	PUNCT
ejpam-4340	678	1	on	on	ADP
ejpam-4340	678	2	generalized	generalized	ADJ
ejpam-4340	678	3	ωβ	ωβ	ADJ
ejpam-4340	678	4	-	-	PUNCT
ejpam-4340	678	5	closed	closed	ADJ
ejpam-4340	678	6	sets	set	NOUN
ejpam-4340	678	7	.	.	PUNCT
ejpam-4340	679	1	missouri	missouri	PROPN
ejpam-4340	679	2	j.	j.	PROPN
ejpam-4340	679	3	of	of	ADP
ejpam-4340	679	4	math	math	PROPN
ejpam-4340	679	5	.	.	PUNCT
ejpam-4340	680	1	sci	sci	PROPN
ejpam-4340	680	2	.	.	PROPN
ejpam-4340	680	3	,	,	PUNCT
ejpam-4340	680	4	26(1):70–87	26(1):70–87	NUM
ejpam-4340	680	5	,	,	PUNCT
ejpam-4340	680	6	2014	2014	NUM
ejpam-4340	680	7	.	.	PUNCT
ejpam-4340	681	1	[	[	X
ejpam-4340	681	2	5	5	X
ejpam-4340	681	3	]	]	PUNCT
ejpam-4340	681	4	k.	k.	PROPN
ejpam-4340	682	1	y.	y.	PROPN
ejpam-4340	682	2	al	al	PROPN
ejpam-4340	682	3	-	-	PROPN
ejpam-4340	682	4	zoubi	zoubi	PROPN
ejpam-4340	682	5	.	.	PUNCT
ejpam-4340	683	1	on	on	ADP
ejpam-4340	683	2	generalized	generalized	ADJ
ejpam-4340	683	3	ω	ω	VERB
ejpam-4340	683	4	-	-	PUNCT
ejpam-4340	683	5	closed	closed	ADJ
ejpam-4340	683	6	sets	set	NOUN
ejpam-4340	683	7	.	.	PUNCT
ejpam-4340	684	1	int	int	NOUN
ejpam-4340	684	2	.	.	PUNCT
ejpam-4340	685	1	j.	j.	PROPN
ejpam-4340	685	2	math	math	PROPN
ejpam-4340	685	3	.	.	PUNCT
ejpam-4340	686	1	math	math	NOUN
ejpam-4340	686	2	.	.	PUNCT
ejpam-4340	687	1	sci	sci	PROPN
ejpam-4340	687	2	.	.	PROPN
ejpam-4340	687	3	,	,	PUNCT
ejpam-4340	687	4	13:2011–2021	13:2011–2021	NUM
ejpam-4340	687	5	,	,	PUNCT
ejpam-4340	687	6	2005	2005	NUM
ejpam-4340	687	7	.	.	PUNCT
ejpam-4340	688	1	[	[	X
ejpam-4340	688	2	6	6	NUM
ejpam-4340	688	3	]	]	PUNCT
ejpam-4340	688	4	k.	k.	PROPN
ejpam-4340	689	1	y.	y.	PROPN
ejpam-4340	689	2	al	al	PROPN
ejpam-4340	689	3	-	-	PROPN
ejpam-4340	689	4	zoubi	zoubi	PROPN
ejpam-4340	689	5	and	and	CCONJ
ejpam-4340	689	6	b.	b.	PROPN
ejpam-4340	689	7	al	al	PROPN
ejpam-4340	689	8	-	-	PUNCT
ejpam-4340	689	9	nashef	nashef	PROPN
ejpam-4340	689	10	.	.	PUNCT
ejpam-4340	690	1	the	the	DET
ejpam-4340	690	2	topology	topology	NOUN
ejpam-4340	690	3	of	of	ADP
ejpam-4340	690	4	ω	ω	VERB
ejpam-4340	690	5	-	-	ADJ
ejpam-4340	690	6	open	open	ADJ
ejpam-4340	690	7	subsets	subset	NOUN
ejpam-4340	690	8	.	.	PUNCT
ejpam-4340	691	1	al	al	PROPN
ejpam-4340	691	2	-	-	PUNCT
ejpam-4340	691	3	manarah	manarah	PROPN
ejpam-4340	691	4	journal	journal	NOUN
ejpam-4340	691	5	,	,	PUNCT
ejpam-4340	691	6	9:169–179	9:169–179	PROPN
ejpam-4340	691	7	,	,	PUNCT
ejpam-4340	691	8	2003	2003	NUM
ejpam-4340	691	9	.	.	PUNCT
ejpam-4340	692	1	[	[	X
ejpam-4340	692	2	7	7	X
ejpam-4340	692	3	]	]	PUNCT
ejpam-4340	692	4	s.	s.	PROPN
ejpam-4340	692	5	p.	p.	PROPN
ejpam-4340	692	6	arya	arya	PROPN
ejpam-4340	692	7	and	and	CCONJ
ejpam-4340	692	8	t.	t.	PROPN
ejpam-4340	692	9	nour	nour	PROPN
ejpam-4340	692	10	.	.	PUNCT
ejpam-4340	693	1	characterizations	characterization	NOUN
ejpam-4340	693	2	of	of	ADP
ejpam-4340	693	3	s	s	NOUN
ejpam-4340	693	4	-	-	ADJ
ejpam-4340	693	5	normal	normal	ADJ
ejpam-4340	693	6	spaces	space	NOUN
ejpam-4340	693	7	.	.	PUNCT
ejpam-4340	694	1	indian	indian	PROPN
ejpam-4340	694	2	j.	j.	PROPN
ejpam-4340	694	3	pure	pure	PROPN
ejpam-4340	694	4	appl	appl	PROPN
ejpam-4340	694	5	.	.	PUNCT
ejpam-4340	694	6	math	math	PROPN
ejpam-4340	694	7	.	.	PUNCT
ejpam-4340	694	8	,	,	PUNCT
ejpam-4340	694	9	21:717–719	21:717–719	NUM
ejpam-4340	694	10	,	,	PUNCT
ejpam-4340	694	11	1990	1990	NUM
ejpam-4340	694	12	.	.	PUNCT
ejpam-4340	695	1	[	[	X
ejpam-4340	695	2	8	8	NUM
ejpam-4340	695	3	]	]	X
ejpam-4340	695	4	b.	b.	PROPN
ejpam-4340	695	5	s.	s.	PROPN
ejpam-4340	695	6	ayhan	ayhan	PROPN
ejpam-4340	695	7	and	and	CCONJ
ejpam-4340	695	8	m.	m.	NOUN
ejpam-4340	695	9	özkoç.	özkoç.	NOUN
ejpam-4340	695	10	on	on	ADP
ejpam-4340	695	11	generalized	generalize	VERB
ejpam-4340	695	12	e	e	ADJ
ejpam-4340	695	13	-	-	ADJ
ejpam-4340	695	14	closed	closed	ADJ
ejpam-4340	695	15	set	set	NOUN
ejpam-4340	695	16	.	.	PUNCT
ejpam-4340	696	1	j.	j.	PROPN
ejpam-4340	696	2	adv	adv	PROPN
ejpam-4340	696	3	.	.	PUNCT
ejpam-4340	696	4	stud	stud	PROPN
ejpam-4340	696	5	.	.	PUNCT
ejpam-4340	697	1	topol	topol	PROPN
ejpam-4340	697	2	.	.	PUNCT
ejpam-4340	697	3	,	,	PUNCT
ejpam-4340	697	4	5(1):14	5(1):14	NUM
ejpam-4340	697	5	–	–	PUNCT
ejpam-4340	697	6	21	21	NUM
ejpam-4340	697	7	,	,	PUNCT
ejpam-4340	697	8	2014	2014	NUM
ejpam-4340	697	9	.	.	PUNCT
ejpam-4340	698	1	[	[	X
ejpam-4340	698	2	9	9	NUM
ejpam-4340	698	3	]	]	X
ejpam-4340	698	4	b.	b.	PROPN
ejpam-4340	698	5	s.	s.	PROPN
ejpam-4340	698	6	ayhan	ayhan	PROPN
ejpam-4340	698	7	and	and	CCONJ
ejpam-4340	698	8	m.	m.	NOUN
ejpam-4340	698	9	özkoç.	özkoç.	NOUN
ejpam-4340	698	10	on	on	ADP
ejpam-4340	698	11	πge	πge	NOUN
ejpam-4340	698	12	-	-	PUNCT
ejpam-4340	698	13	closed	close	VERB
ejpam-4340	698	14	sets	set	NOUN
ejpam-4340	698	15	and	and	CCONJ
ejpam-4340	698	16	related	related	ADJ
ejpam-4340	698	17	topics	topic	NOUN
ejpam-4340	698	18	.	.	PUNCT
ejpam-4340	699	1	j.	j.	PROPN
ejpam-4340	699	2	adv	adv	PROPN
ejpam-4340	699	3	.	.	PUNCT
ejpam-4340	699	4	stud	stud	PROPN
ejpam-4340	699	5	.	.	PUNCT
ejpam-4340	700	1	topol	topol	PROPN
ejpam-4340	700	2	.	.	PUNCT
ejpam-4340	700	3	,	,	PUNCT
ejpam-4340	700	4	7(2):93–100	7(2):93–100	NUM
ejpam-4340	700	5	,	,	PUNCT
ejpam-4340	700	6	2016	2016	NUM
ejpam-4340	700	7	.	.	PUNCT
ejpam-4340	701	1	references	reference	NOUN
ejpam-4340	701	2	374	374	NUM
ejpam-4340	701	3	[	[	X
ejpam-4340	701	4	10	10	NUM
ejpam-4340	701	5	]	]	X
ejpam-4340	701	6	e.	e.	PROPN
ejpam-4340	701	7	ekici	ekici	PROPN
ejpam-4340	701	8	.	.	PUNCT
ejpam-4340	702	1	on	on	ADP
ejpam-4340	702	2	a	a	DET
ejpam-4340	702	3	-	-	PUNCT
ejpam-4340	702	4	open	open	ADJ
ejpam-4340	702	5	sets	set	NOUN
ejpam-4340	702	6	,	,	PUNCT
ejpam-4340	702	7	a∗-sets	a∗-set	NOUN
ejpam-4340	702	8	and	and	CCONJ
ejpam-4340	702	9	decompositions	decomposition	NOUN
ejpam-4340	702	10	of	of	ADP
ejpam-4340	702	11	continuity	continuity	NOUN
ejpam-4340	702	12	and	and	CCONJ
ejpam-4340	702	13	supercontinuity	supercontinuity	NOUN
ejpam-4340	702	14	.	.	PUNCT
ejpam-4340	703	1	ann	ann	PROPN
ejpam-4340	703	2	univ	univ	PROPN
ejpam-4340	703	3	sci	sci	PROPN
ejpam-4340	703	4	budapest	budapest	PROPN
ejpam-4340	703	5	eötvös	eötvös	PROPN
ejpam-4340	703	6	sect	sect	NOUN
ejpam-4340	703	7	math	math	NOUN
ejpam-4340	703	8	.	.	PUNCT
ejpam-4340	703	9	,	,	PUNCT
ejpam-4340	703	10	51:39–51	51:39–51	PROPN
ejpam-4340	703	11	,	,	PUNCT
ejpam-4340	703	12	2008	2008	NUM
ejpam-4340	703	13	.	.	PUNCT
ejpam-4340	704	1	[	[	X
ejpam-4340	704	2	11	11	NUM
ejpam-4340	704	3	]	]	X
ejpam-4340	704	4	e.	e.	PROPN
ejpam-4340	704	5	ekici	ekici	PROPN
ejpam-4340	704	6	.	.	PUNCT
ejpam-4340	705	1	e∗-open	e∗-open	ADJ
ejpam-4340	705	2	sets	set	NOUN
ejpam-4340	705	3	and	and	CCONJ
ejpam-4340	705	4	(	(	PUNCT
ejpam-4340	705	5	d	d	PROPN
ejpam-4340	705	6	,	,	PUNCT
ejpam-4340	705	7	s)∗-set	s)∗-set	PROPN
ejpam-4340	705	8	.	.	PUNCT
ejpam-4340	705	9	math	math	PROPN
ejpam-4340	705	10	.	.	PUNCT
ejpam-4340	706	1	morav	morav	PROPN
ejpam-4340	706	2	.	.	PUNCT
ejpam-4340	706	3	,	,	PUNCT
ejpam-4340	706	4	13(1):29–36	13(1):29–36	NUM
ejpam-4340	706	5	,	,	PUNCT
ejpam-4340	706	6	2009	2009	NUM
ejpam-4340	706	7	.	.	PUNCT
ejpam-4340	707	1	[	[	X
ejpam-4340	707	2	12	12	NUM
ejpam-4340	707	3	]	]	X
ejpam-4340	707	4	s.	s.	PROPN
ejpam-4340	707	5	erdem	erdem	PROPN
ejpam-4340	707	6	,	,	PUNCT
ejpam-4340	707	7	m.	m.	NOUN
ejpam-4340	707	8	özkoç	özkoç	PROPN
ejpam-4340	707	9	,	,	PUNCT
ejpam-4340	707	10	and	and	CCONJ
ejpam-4340	707	11	t.	t.	PROPN
ejpam-4340	707	12	noiri	noiri	PROPN
ejpam-4340	707	13	.	.	PUNCT
ejpam-4340	708	1	on	on	ADP
ejpam-4340	708	2	e∗-normal	e∗-normal	ADJ
ejpam-4340	708	3	spaces	space	NOUN
ejpam-4340	708	4	.	.	PUNCT
ejpam-4340	709	1	submitted	submit	VERB
ejpam-4340	709	2	.	.	PUNCT
ejpam-4340	710	1	[	[	X
ejpam-4340	710	2	13	13	NUM
ejpam-4340	710	3	]	]	PUNCT
ejpam-4340	710	4	h.	h.	PROPN
ejpam-4340	710	5	z.	z.	PROPN
ejpam-4340	710	6	hdeib	hdeib	PROPN
ejpam-4340	710	7	.	.	PUNCT
ejpam-4340	711	1	ω	ω	X
ejpam-4340	711	2	-	-	ADJ
ejpam-4340	711	3	continuous	continuous	ADJ
ejpam-4340	711	4	functions	function	NOUN
ejpam-4340	711	5	.	.	PUNCT
ejpam-4340	712	1	dirasat	dirasat	PROPN
ejpam-4340	712	2	journal	journal	PROPN
ejpam-4340	712	3	,	,	PUNCT
ejpam-4340	712	4	16(2):136–153	16(2):136–153	NUM
ejpam-4340	712	5	,	,	PUNCT
ejpam-4340	712	6	1989	1989	NUM
ejpam-4340	712	7	.	.	PUNCT
ejpam-4340	713	1	[	[	X
ejpam-4340	713	2	14	14	NUM
ejpam-4340	713	3	]	]	X
ejpam-4340	713	4	n.	n.	PROPN
ejpam-4340	713	5	levine	levine	PROPN
ejpam-4340	713	6	.	.	PUNCT
ejpam-4340	714	1	generalized	generalize	VERB
ejpam-4340	714	2	closed	close	VERB
ejpam-4340	714	3	sets	set	NOUN
ejpam-4340	714	4	in	in	ADP
ejpam-4340	714	5	topological	topological	ADJ
ejpam-4340	714	6	spaces	space	NOUN
ejpam-4340	714	7	.	.	PUNCT
ejpam-4340	715	1	rend	rend	VERB
ejpam-4340	715	2	.	.	PUNCT
ejpam-4340	716	1	circ	circ	PROPN
ejpam-4340	716	2	.	.	PUNCT
ejpam-4340	717	1	mat	mat	NOUN
ejpam-4340	717	2	.	.	PUNCT
ejpam-4340	717	3	palermo	palermo	PROPN
ejpam-4340	717	4	,	,	PUNCT
ejpam-4340	717	5	19:89–96	19:89–96	NUM
ejpam-4340	717	6	,	,	PUNCT
ejpam-4340	717	7	1970	1970	NUM
ejpam-4340	717	8	.	.	PUNCT
ejpam-4340	718	1	[	[	X
ejpam-4340	718	2	15	15	NUM
ejpam-4340	718	3	]	]	X
ejpam-4340	718	4	h.	h.	NOUN
ejpam-4340	718	5	maki	maki	PROPN
ejpam-4340	718	6	,	,	PUNCT
ejpam-4340	718	7	k.	k.	PROPN
ejpam-4340	718	8	balachandran	balachandran	PROPN
ejpam-4340	718	9	,	,	PUNCT
ejpam-4340	718	10	and	and	CCONJ
ejpam-4340	718	11	r.	r.	PROPN
ejpam-4340	718	12	devi	devi	PROPN
ejpam-4340	718	13	.	.	PUNCT
ejpam-4340	719	1	associated	associated	ADJ
ejpam-4340	719	2	topologies	topology	NOUN
ejpam-4340	719	3	of	of	ADP
ejpam-4340	719	4	generalized	generalized	ADJ
ejpam-4340	719	5	αclosed	αclose	VERB
ejpam-4340	719	6	sets	set	NOUN
ejpam-4340	719	7	and	and	CCONJ
ejpam-4340	719	8	α	α	X
ejpam-4340	719	9	-	-	ADJ
ejpam-4340	719	10	generalized	generalize	VERB
ejpam-4340	719	11	closed	closed	ADJ
ejpam-4340	719	12	sets	set	NOUN
ejpam-4340	719	13	.	.	PUNCT
ejpam-4340	720	1	mem	mem	PROPN
ejpam-4340	720	2	.	.	PUNCT
ejpam-4340	720	3	fac	fac	PROPN
ejpam-4340	720	4	.	.	PUNCT
ejpam-4340	720	5	sci	sci	PROPN
ejpam-4340	720	6	.	.	PROPN
ejpam-4340	720	7	kochi	kochi	PROPN
ejpam-4340	720	8	univ	univ	PROPN
ejpam-4340	720	9	.	.	PUNCT
ejpam-4340	720	10	ser	ser	PROPN
ejpam-4340	720	11	.	.	PUNCT
ejpam-4340	721	1	a	a	DET
ejpam-4340	721	2	math	math	NOUN
ejpam-4340	721	3	.	.	PUNCT
ejpam-4340	721	4	,	,	PUNCT
ejpam-4340	721	5	15:51–63	15:51–63	NUM
ejpam-4340	721	6	,	,	PUNCT
ejpam-4340	721	7	1994	1994	NUM
ejpam-4340	721	8	.	.	PUNCT
ejpam-4340	722	1	[	[	X
ejpam-4340	722	2	16	16	NUM
ejpam-4340	722	3	]	]	X
ejpam-4340	722	4	h.	h.	PROPN
ejpam-4340	722	5	maki	maki	PROPN
ejpam-4340	722	6	,	,	PUNCT
ejpam-4340	722	7	j.	j.	PROPN
ejpam-4340	722	8	umehara	umehara	PROPN
ejpam-4340	722	9	,	,	PUNCT
ejpam-4340	722	10	and	and	CCONJ
ejpam-4340	722	11	t.	t.	PROPN
ejpam-4340	722	12	noiri	noiri	PROPN
ejpam-4340	722	13	.	.	PUNCT
ejpam-4340	723	1	every	every	DET
ejpam-4340	723	2	topological	topological	ADJ
ejpam-4340	723	3	space	space	NOUN
ejpam-4340	723	4	is	be	AUX
ejpam-4340	723	5	pre	pre	ADJ
ejpam-4340	723	6	-	-	ADJ
ejpam-4340	723	7	t	t	ADJ
ejpam-4340	723	8	1	1	NUM
ejpam-4340	723	9	2	2	NUM
ejpam-4340	723	10	.	.	PUNCT
ejpam-4340	724	1	mem	mem	PROPN
ejpam-4340	724	2	.	.	PUNCT
ejpam-4340	725	1	fac	fac	PROPN
ejpam-4340	725	2	.	.	PUNCT
ejpam-4340	726	1	sci	sci	PROPN
ejpam-4340	726	2	.	.	PROPN
ejpam-4340	726	3	kochi	kochi	PROPN
ejpam-4340	726	4	univ	univ	PROPN
ejpam-4340	726	5	.	.	PUNCT
ejpam-4340	727	1	ser	ser	PROPN
ejpam-4340	727	2	a	a	DET
ejpam-4340	727	3	math	math	NOUN
ejpam-4340	727	4	.	.	PUNCT
ejpam-4340	727	5	,	,	PUNCT
ejpam-4340	727	6	17:33–42	17:33–42	PROPN
ejpam-4340	727	7	,	,	PUNCT
ejpam-4340	727	8	1996	1996	NUM
ejpam-4340	727	9	.	.	PUNCT
ejpam-4340	728	1	[	[	X
ejpam-4340	728	2	17	17	NUM
ejpam-4340	728	3	]	]	PUNCT
ejpam-4340	728	4	m.	m.	NOUN
ejpam-4340	728	5	mrsevic	mrsevic	ADJ
ejpam-4340	728	6	.	.	PUNCT
ejpam-4340	729	1	on	on	ADP
ejpam-4340	729	2	pairwise	pairwise	NOUN
ejpam-4340	729	3	r0	r0	NOUN
ejpam-4340	729	4	and	and	CCONJ
ejpam-4340	729	5	pairwise	pairwise	NOUN
ejpam-4340	729	6	r1	r1	NOUN
ejpam-4340	729	7	bitopological	bitopological	ADJ
ejpam-4340	729	8	spaces	space	NOUN
ejpam-4340	729	9	.	.	PUNCT
ejpam-4340	730	1	bull	bull	NOUN
ejpam-4340	730	2	.	.	PUNCT
ejpam-4340	731	1	math	math	NOUN
ejpam-4340	731	2	.	.	PUNCT
ejpam-4340	732	1	soc	soc	PROPN
ejpam-4340	732	2	.	.	PUNCT
ejpam-4340	733	1	sci	sci	PROPN
ejpam-4340	733	2	.	.	PUNCT
ejpam-4340	733	3	math	math	PROPN
ejpam-4340	733	4	rs	rs	PROPN
ejpam-4340	733	5	roumano	roumano	PROPN
ejpam-4340	733	6	(	(	PUNCT
ejpam-4340	733	7	n.s	n.s	PROPN
ejpam-4340	733	8	.	.	PROPN
ejpam-4340	733	9	)	)	PUNCT
ejpam-4340	733	10	,	,	PUNCT
ejpam-4340	733	11	30(78):141–148	30(78):141–148	NUM
ejpam-4340	733	12	,	,	PUNCT
ejpam-4340	733	13	1986	1986	NUM
ejpam-4340	733	14	.	.	PUNCT
ejpam-4340	734	1	[	[	X
ejpam-4340	734	2	18	18	NUM
ejpam-4340	734	3	]	]	PUNCT
ejpam-4340	734	4	a.	a.	NOUN
ejpam-4340	734	5	a.	a.	NOUN
ejpam-4340	734	6	omari	omari	PROPN
ejpam-4340	734	7	and	and	CCONJ
ejpam-4340	734	8	m.	m.	PROPN
ejpam-4340	734	9	s.	s.	PROPN
ejpam-4340	734	10	m.	m.	PROPN
ejpam-4340	734	11	noorani	noorani	PROPN
ejpam-4340	734	12	.	.	PUNCT
ejpam-4340	735	1	on	on	ADP
ejpam-4340	735	2	generalized	generalized	ADJ
ejpam-4340	735	3	b	b	X
ejpam-4340	735	4	-	-	PUNCT
ejpam-4340	735	5	closed	closed	ADJ
ejpam-4340	735	6	sets	set	NOUN
ejpam-4340	735	7	.	.	PUNCT
ejpam-4340	736	1	bull	bull	NOUN
ejpam-4340	736	2	.	.	PUNCT
ejpam-4340	737	1	malays	malays	PROPN
ejpam-4340	737	2	.	.	PUNCT
ejpam-4340	738	1	math	math	NOUN
ejpam-4340	738	2	.	.	PUNCT
ejpam-4340	739	1	sci	sci	PROPN
ejpam-4340	739	2	.	.	PROPN
ejpam-4340	739	3	soc	soc	PROPN
ejpam-4340	739	4	.	.	PUNCT
ejpam-4340	739	5	,	,	PUNCT
ejpam-4340	739	6	32(1):19–30	32(1):19–30	NUM
ejpam-4340	739	7	,	,	PUNCT
ejpam-4340	739	8	2009	2009	NUM
ejpam-4340	739	9	.	.	PUNCT
ejpam-4340	740	1	[	[	X
ejpam-4340	740	2	19	19	NUM
ejpam-4340	740	3	]	]	PUNCT
ejpam-4340	740	4	m.	m.	NOUN
ejpam-4340	740	5	özkoç	özkoç	PROPN
ejpam-4340	740	6	and	and	CCONJ
ejpam-4340	740	7	p.	p.	NOUN
ejpam-4340	740	8	şaşmaz	şaşmaz	PUNCT
ejpam-4340	740	9	.	.	PUNCT
ejpam-4340	741	1	on	on	ADP
ejpam-4340	741	2	contra	contra	PROPN
ejpam-4340	741	3	ωe∗-continuous	ωe∗-continuous	ADJ
ejpam-4340	741	4	functions	function	NOUN
ejpam-4340	741	5	.	.	PUNCT
ejpam-4340	742	1	poincare	poincare	PROPN
ejpam-4340	742	2	j.	j.	PROPN
ejpam-4340	742	3	anal	anal	PROPN
ejpam-4340	742	4	.	.	PUNCT
ejpam-4340	743	1	appl	appl	PROPN
ejpam-4340	743	2	.	.	PROPN
ejpam-4340	743	3	,	,	PUNCT
ejpam-4340	743	4	8(1(i)):51–65	8(1(i)):51–65	NUM
ejpam-4340	743	5	,	,	PUNCT
ejpam-4340	743	6	2021	2021	NUM
ejpam-4340	743	7	.	.	PUNCT
ejpam-4340	744	1	[	[	X
ejpam-4340	744	2	20	20	NUM
ejpam-4340	744	3	]	]	PUNCT
ejpam-4340	744	4	m.	m.	NOUN
ejpam-4340	744	5	h.	h.	PROPN
ejpam-4340	744	6	stone	stone	PROPN
ejpam-4340	744	7	.	.	PUNCT
ejpam-4340	745	1	applications	application	NOUN
ejpam-4340	745	2	of	of	ADP
ejpam-4340	745	3	the	the	DET
ejpam-4340	745	4	theory	theory	NOUN
ejpam-4340	745	5	of	of	ADP
ejpam-4340	745	6	boolean	boolean	ADJ
ejpam-4340	745	7	rings	ring	NOUN
ejpam-4340	745	8	to	to	ADP
ejpam-4340	745	9	general	general	ADJ
ejpam-4340	745	10	topology	topology	NOUN
ejpam-4340	745	11	.	.	PUNCT
ejpam-4340	746	1	trans	trans	PROPN
ejpam-4340	746	2	.	.	PUNCT
ejpam-4340	747	1	amer	amer	PROPN
ejpam-4340	747	2	.	.	PUNCT
ejpam-4340	747	3	math	math	PROPN
ejpam-4340	747	4	.	.	PUNCT
ejpam-4340	748	1	soc	soc	PROPN
ejpam-4340	748	2	.	.	PUNCT
ejpam-4340	748	3	,	,	PUNCT
ejpam-4340	748	4	41:375–381	41:375–381	PROPN
ejpam-4340	748	5	,	,	PUNCT
ejpam-4340	748	6	1937	1937	NUM
ejpam-4340	748	7	.	.	PUNCT
ejpam-4340	749	1	[	[	X
ejpam-4340	749	2	21	21	NUM
ejpam-4340	749	3	]	]	X
ejpam-4340	749	4	s.	s.	PROPN
ejpam-4340	749	5	tahiliani	tahiliani	PROPN
ejpam-4340	749	6	.	.	PUNCT
ejpam-4340	750	1	generalized	generalize	VERB
ejpam-4340	750	2	β	β	X
ejpam-4340	750	3	-	-	ADJ
ejpam-4340	750	4	closed	closed	ADJ
ejpam-4340	750	5	functions	function	NOUN
ejpam-4340	750	6	.	.	PUNCT
ejpam-4340	751	1	bull	bull	NOUN
ejpam-4340	751	2	.	.	PUNCT
ejpam-4340	752	1	call	call	NOUN
ejpam-4340	752	2	.	.	PUNCT
ejpam-4340	753	1	math	math	NOUN
ejpam-4340	753	2	.	.	PUNCT
ejpam-4340	754	1	soc	soc	PROPN
ejpam-4340	754	2	.	.	PUNCT
ejpam-4340	754	3	,	,	PUNCT
ejpam-4340	754	4	98:307–376	98:307–376	PROPN
ejpam-4340	754	5	,	,	PUNCT
ejpam-4340	754	6	2006	2006	NUM
ejpam-4340	754	7	.	.	PUNCT
ejpam-4340	755	1	[	[	X
ejpam-4340	755	2	22	22	NUM
ejpam-4340	755	3	]	]	X
ejpam-4340	755	4	n.	n.	PROPN
ejpam-4340	755	5	v.	v.	PROPN
ejpam-4340	755	6	velic̆ko	velic̆ko	PROPN
ejpam-4340	755	7	.	.	PUNCT
ejpam-4340	756	1	h	h	PROPN
ejpam-4340	756	2	-closed	-close	VERB
ejpam-4340	756	3	topological	topological	ADJ
ejpam-4340	756	4	spaces	space	NOUN
ejpam-4340	756	5	.	.	PUNCT
ejpam-4340	757	1	amer	amer	PROPN
ejpam-4340	757	2	.	.	PUNCT
ejpam-4340	757	3	math	math	PROPN
ejpam-4340	757	4	.	.	PUNCT
ejpam-4340	758	1	soc	soc	PROPN
ejpam-4340	758	2	.	.	PUNCT
ejpam-4340	759	1	transl	transl	PROPN
ejpam-4340	759	2	.	.	PUNCT
ejpam-4340	759	3	,	,	PUNCT
ejpam-4340	760	1	78(2):103–118	78(2):103–118	PROPN
ejpam-4340	760	2	,	,	PUNCT
ejpam-4340	760	3	1968	1968	NUM
ejpam-4340	760	4	.	.	PUNCT