id	sid	tid	token	lemma	pos
ejpam-5256	1	1	european	european	PROPN
ejpam-5256	1	2	journal	journal	PROPN
ejpam-5256	1	3	of	of	ADP
ejpam-5256	1	4	pure	pure	ADJ
ejpam-5256	1	5	and	and	CCONJ
ejpam-5256	1	6	applied	apply	VERB
ejpam-5256	1	7	mathematics	mathematic	NOUN
ejpam-5256	1	8	vol	vol	NOUN
ejpam-5256	1	9	.	.	PROPN
ejpam-5256	2	1	17	17	NUM
ejpam-5256	2	2	,	,	PUNCT
ejpam-5256	2	3	no	no	INTJ
ejpam-5256	2	4	.	.	NOUN
ejpam-5256	2	5	3	3	NUM
ejpam-5256	2	6	,	,	PUNCT
ejpam-5256	2	7	2024	2024	NUM
ejpam-5256	2	8	,	,	PUNCT
ejpam-5256	2	9	1869	1869	NUM
ejpam-5256	2	10	-	-	SYM
ejpam-5256	2	11	1876	1876	NUM
ejpam-5256	2	12	issn	issn	PROPN
ejpam-5256	2	13	1307	1307	NUM
ejpam-5256	2	14	-	-	SYM
ejpam-5256	2	15	5543	5543	NUM
ejpam-5256	2	16	–	–	PUNCT
ejpam-5256	2	17	ejpam.com	ejpam.com	X
ejpam-5256	2	18	published	publish	VERB
ejpam-5256	2	19	by	by	ADP
ejpam-5256	2	20	new	new	PROPN
ejpam-5256	2	21	york	york	PROPN
ejpam-5256	2	22	business	business	PROPN
ejpam-5256	2	23	global	global	ADJ
ejpam-5256	2	24	properties	property	NOUN
ejpam-5256	2	25	of	of	ADP
ejpam-5256	2	26	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	2	27	sets	set	NOUN
ejpam-5256	2	28	in	in	ADP
ejpam-5256	2	29	generalized	generalized	ADJ
ejpam-5256	2	30	topological	topological	ADJ
ejpam-5256	2	31	spaces	space	NOUN
ejpam-5256	2	32	jeeranunt	jeeranunt	NOUN
ejpam-5256	2	33	khampakdee	khampakdee	NOUN
ejpam-5256	2	34	mathematics	mathematic	NOUN
ejpam-5256	2	35	and	and	CCONJ
ejpam-5256	2	36	applied	apply	VERB
ejpam-5256	2	37	mathematics	mathematics	PROPN
ejpam-5256	2	38	research	research	NOUN
ejpam-5256	2	39	unit	unit	NOUN
ejpam-5256	2	40	,	,	PUNCT
ejpam-5256	2	41	department	department	NOUN
ejpam-5256	2	42	of	of	ADP
ejpam-5256	2	43	mathematics	mathematic	NOUN
ejpam-5256	2	44	,	,	PUNCT
ejpam-5256	2	45	faculty	faculty	NOUN
ejpam-5256	2	46	of	of	ADP
ejpam-5256	2	47	science	science	NOUN
ejpam-5256	2	48	,	,	PUNCT
ejpam-5256	2	49	mahasarakham	mahasarakham	PROPN
ejpam-5256	2	50	university	university	PROPN
ejpam-5256	2	51	,	,	PUNCT
ejpam-5256	2	52	maha	maha	PROPN
ejpam-5256	2	53	sarakham	sarakham	PROPN
ejpam-5256	2	54	,	,	PUNCT
ejpam-5256	2	55	44150	44150	NUM
ejpam-5256	2	56	,	,	PUNCT
ejpam-5256	2	57	thailand	thailand	PROPN
ejpam-5256	2	58	abstract	abstract	PROPN
ejpam-5256	2	59	.	.	PUNCT
ejpam-5256	3	1	a	a	DET
ejpam-5256	3	2	study	study	NOUN
ejpam-5256	3	3	of	of	ADP
ejpam-5256	3	4	θ̃-open	θ̃-open	PROPN
ejpam-5256	3	5	sets	set	NOUN
ejpam-5256	3	6	in	in	ADP
ejpam-5256	3	7	generalized	generalized	ADJ
ejpam-5256	3	8	topological	topological	ADJ
ejpam-5256	3	9	spaces	space	NOUN
ejpam-5256	3	10	started	start	VERB
ejpam-5256	3	11	in	in	ADP
ejpam-5256	3	12	2011	2011	NUM
ejpam-5256	3	13	by	by	ADP
ejpam-5256	3	14	min	min	NOUN
ejpam-5256	4	1	[	[	X
ejpam-5256	4	2	4	4	NUM
ejpam-5256	4	3	]	]	PUNCT
ejpam-5256	4	4	.	.	PUNCT
ejpam-5256	5	1	in	in	ADP
ejpam-5256	5	2	this	this	DET
ejpam-5256	5	3	paper	paper	NOUN
ejpam-5256	5	4	,	,	PUNCT
ejpam-5256	5	5	we	we	PRON
ejpam-5256	5	6	introduce	introduce	VERB
ejpam-5256	5	7	the	the	DET
ejpam-5256	5	8	concepts	concept	NOUN
ejpam-5256	5	9	of	of	ADP
ejpam-5256	5	10	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	5	11	sets	set	NOUN
ejpam-5256	5	12	,	,	PUNCT
ejpam-5256	5	13	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	5	14	sets	set	NOUN
ejpam-5256	5	15	,	,	PUNCT
ejpam-5256	5	16	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	5	17	functions	function	NOUN
ejpam-5256	5	18	and	and	CCONJ
ejpam-5256	5	19	sθ̃-irresolute	sθ̃-irresolute	NOUN
ejpam-5256	5	20	functions	function	NOUN
ejpam-5256	5	21	on	on	ADP
ejpam-5256	5	22	generalized	generalized	ADJ
ejpam-5256	5	23	topological	topological	ADJ
ejpam-5256	5	24	spaces	space	NOUN
ejpam-5256	5	25	.	.	PUNCT
ejpam-5256	6	1	we	we	PRON
ejpam-5256	6	2	also	also	ADV
ejpam-5256	6	3	study	study	VERB
ejpam-5256	6	4	some	some	DET
ejpam-5256	6	5	basic	basic	ADJ
ejpam-5256	6	6	properties	property	NOUN
ejpam-5256	6	7	of	of	ADP
ejpam-5256	6	8	such	such	ADJ
ejpam-5256	6	9	sets	set	NOUN
ejpam-5256	6	10	and	and	CCONJ
ejpam-5256	6	11	functions	function	NOUN
ejpam-5256	6	12	.	.	PUNCT
ejpam-5256	7	1	2020	2020	NUM
ejpam-5256	7	2	mathematics	mathematic	NOUN
ejpam-5256	7	3	subject	subject	NOUN
ejpam-5256	7	4	classifications	classification	NOUN
ejpam-5256	7	5	:	:	PUNCT
ejpam-5256	7	6	54a05	54a05	NUM
ejpam-5256	7	7	key	key	ADJ
ejpam-5256	7	8	words	word	NOUN
ejpam-5256	7	9	and	and	CCONJ
ejpam-5256	7	10	phrases	phrase	NOUN
ejpam-5256	7	11	:	:	PUNCT
ejpam-5256	7	12	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	7	13	set	set	NOUN
ejpam-5256	7	14	,	,	PUNCT
ejpam-5256	7	15	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	7	16	function	function	NOUN
ejpam-5256	7	17	,	,	PUNCT
ejpam-5256	7	18	sθ̃-irresolute	sθ̃-irresolute	NOUN
ejpam-5256	7	19	function	function	NOUN
ejpam-5256	7	20	.	.	PUNCT
ejpam-5256	8	1	1	1	X
ejpam-5256	8	2	.	.	X
ejpam-5256	8	3	introduction	introduction	NOUN
ejpam-5256	8	4	in	in	ADP
ejpam-5256	8	5	2002	2002	NUM
ejpam-5256	8	6	,	,	PUNCT
ejpam-5256	8	7	császár	császár	NOUN
ejpam-5256	9	1	[	[	X
ejpam-5256	9	2	1	1	X
ejpam-5256	9	3	]	]	PUNCT
ejpam-5256	9	4	defined	define	VERB
ejpam-5256	9	5	the	the	DET
ejpam-5256	9	6	concept	concept	NOUN
ejpam-5256	9	7	of	of	ADP
ejpam-5256	9	8	a	a	DET
ejpam-5256	9	9	generalized	generalized	ADJ
ejpam-5256	9	10	topological	topological	ADJ
ejpam-5256	9	11	space	space	NOUN
ejpam-5256	9	12	which	which	PRON
ejpam-5256	9	13	is	be	AUX
ejpam-5256	9	14	an	an	DET
ejpam-5256	9	15	extension	extension	NOUN
ejpam-5256	9	16	of	of	ADP
ejpam-5256	9	17	the	the	DET
ejpam-5256	9	18	idea	idea	NOUN
ejpam-5256	9	19	from	from	ADP
ejpam-5256	9	20	a	a	DET
ejpam-5256	9	21	topological	topological	ADJ
ejpam-5256	9	22	space	space	NOUN
ejpam-5256	9	23	.	.	PUNCT
ejpam-5256	10	1	in	in	ADP
ejpam-5256	10	2	addition	addition	NOUN
ejpam-5256	10	3	,	,	PUNCT
ejpam-5256	10	4	the	the	DET
ejpam-5256	10	5	concepts	concept	NOUN
ejpam-5256	10	6	of	of	ADP
ejpam-5256	10	7	the	the	DET
ejpam-5256	10	8	interior	interior	NOUN
ejpam-5256	10	9	of	of	ADP
ejpam-5256	10	10	sets	set	NOUN
ejpam-5256	10	11	and	and	CCONJ
ejpam-5256	10	12	the	the	DET
ejpam-5256	10	13	closure	closure	NOUN
ejpam-5256	10	14	of	of	ADP
ejpam-5256	10	15	sets	set	NOUN
ejpam-5256	10	16	in	in	ADP
ejpam-5256	10	17	topological	topological	ADJ
ejpam-5256	10	18	spaces	space	NOUN
ejpam-5256	10	19	were	be	AUX
ejpam-5256	10	20	introduced	introduce	VERB
ejpam-5256	10	21	.	.	PUNCT
ejpam-5256	11	1	if	if	SCONJ
ejpam-5256	11	2	a	a	PRON
ejpam-5256	11	3	is	be	AUX
ejpam-5256	11	4	a	a	DET
ejpam-5256	11	5	subset	subset	NOUN
ejpam-5256	11	6	of	of	ADP
ejpam-5256	11	7	x	x	PRON
ejpam-5256	11	8	,	,	PUNCT
ejpam-5256	11	9	then	then	ADV
ejpam-5256	11	10	the	the	DET
ejpam-5256	11	11	symbols	symbol	NOUN
ejpam-5256	11	12	iµ(a	iµ(a	ADJ
ejpam-5256	11	13	)	)	PUNCT
ejpam-5256	11	14	and	and	CCONJ
ejpam-5256	11	15	cµ(a	cµ(a	NUM
ejpam-5256	11	16	)	)	PUNCT
ejpam-5256	11	17	are	be	AUX
ejpam-5256	11	18	used	use	VERB
ejpam-5256	11	19	to	to	PART
ejpam-5256	11	20	represent	represent	VERB
ejpam-5256	11	21	the	the	DET
ejpam-5256	11	22	interior	interior	NOUN
ejpam-5256	11	23	of	of	ADP
ejpam-5256	11	24	a	a	PRON
ejpam-5256	11	25	and	and	CCONJ
ejpam-5256	11	26	the	the	DET
ejpam-5256	11	27	closure	closure	NOUN
ejpam-5256	11	28	of	of	ADP
ejpam-5256	11	29	a	a	PRON
ejpam-5256	11	30	,	,	PUNCT
ejpam-5256	11	31	respectively	respectively	ADV
ejpam-5256	11	32	,	,	PUNCT
ejpam-5256	11	33	in	in	ADP
ejpam-5256	11	34	a	a	DET
ejpam-5256	11	35	generalized	generalized	ADJ
ejpam-5256	11	36	topological	topological	ADJ
ejpam-5256	11	37	space	space	NOUN
ejpam-5256	11	38	(	(	PUNCT
ejpam-5256	11	39	x,µ	x,µ	NOUN
ejpam-5256	11	40	)	)	PUNCT
ejpam-5256	11	41	.	.	PUNCT
ejpam-5256	12	1	in	in	ADP
ejpam-5256	12	2	[	[	X
ejpam-5256	12	3	1	1	NUM
ejpam-5256	12	4	]	]	PUNCT
ejpam-5256	12	5	,	,	PUNCT
ejpam-5256	12	6	the	the	DET
ejpam-5256	12	7	concept	concept	NOUN
ejpam-5256	12	8	of	of	ADP
ejpam-5256	12	9	(	(	PUNCT
ejpam-5256	12	10	µ	µ	NUM
ejpam-5256	12	11	,	,	PUNCT
ejpam-5256	12	12	µ′)-continuous	µ′)-continuous	ADJ
ejpam-5256	12	13	functions	function	NOUN
ejpam-5256	12	14	on	on	ADP
ejpam-5256	12	15	generalized	generalized	ADJ
ejpam-5256	12	16	topological	topological	ADJ
ejpam-5256	12	17	spaces	space	NOUN
ejpam-5256	12	18	was	be	AUX
ejpam-5256	12	19	also	also	ADV
ejpam-5256	12	20	introduced	introduce	VERB
ejpam-5256	12	21	.	.	PUNCT
ejpam-5256	13	1	in	in	ADP
ejpam-5256	13	2	2005	2005	NUM
ejpam-5256	13	3	,	,	PUNCT
ejpam-5256	13	4	császár	császár	NOUN
ejpam-5256	14	1	[	[	X
ejpam-5256	14	2	2	2	X
ejpam-5256	14	3	]	]	PUNCT
ejpam-5256	14	4	used	use	VERB
ejpam-5256	14	5	the	the	DET
ejpam-5256	14	6	concepts	concept	NOUN
ejpam-5256	14	7	of	of	ADP
ejpam-5256	14	8	semi	semi	ADJ
ejpam-5256	14	9	-	-	ADJ
ejpam-5256	14	10	open	open	ADJ
ejpam-5256	14	11	sets	set	NOUN
ejpam-5256	14	12	and	and	CCONJ
ejpam-5256	14	13	semi	semi	ADJ
ejpam-5256	14	14	-	-	ADJ
ejpam-5256	14	15	closed	closed	ADJ
ejpam-5256	14	16	sets	set	NOUN
ejpam-5256	14	17	in	in	ADP
ejpam-5256	14	18	topological	topological	ADJ
ejpam-5256	14	19	spaces	space	NOUN
ejpam-5256	14	20	to	to	PART
ejpam-5256	14	21	define	define	VERB
ejpam-5256	14	22	such	such	ADJ
ejpam-5256	14	23	sets	set	NOUN
ejpam-5256	14	24	in	in	ADP
ejpam-5256	14	25	generalized	generalized	ADJ
ejpam-5256	14	26	topological	topological	ADJ
ejpam-5256	14	27	spaces	space	NOUN
ejpam-5256	14	28	,	,	PUNCT
ejpam-5256	14	29	and	and	CCONJ
ejpam-5256	14	30	proved	prove	VERB
ejpam-5256	14	31	that	that	SCONJ
ejpam-5256	14	32	if	if	SCONJ
ejpam-5256	14	33	a	a	DET
ejpam-5256	14	34	⊆	⊆	NUM
ejpam-5256	14	35	x	x	NOUN
ejpam-5256	14	36	,	,	PUNCT
ejpam-5256	14	37	then	then	ADV
ejpam-5256	14	38	iµ(iµ(a	iµ(iµ(a	ADJ
ejpam-5256	14	39	)	)	PUNCT
ejpam-5256	14	40	)	)	PUNCT
ejpam-5256	14	41	=	=	SYM
ejpam-5256	14	42	iµ(a	iµ(a	X
ejpam-5256	14	43	)	)	PUNCT
ejpam-5256	14	44	and	and	CCONJ
ejpam-5256	14	45	cµ(cµ(a	cµ(cµ(a	NOUN
ejpam-5256	14	46	)	)	PUNCT
ejpam-5256	14	47	)	)	PUNCT
ejpam-5256	15	1	=	=	SYM
ejpam-5256	15	2	cµ(a	cµ(a	PROPN
ejpam-5256	15	3	)	)	PUNCT
ejpam-5256	15	4	.	.	PUNCT
ejpam-5256	16	1	moreover	moreover	ADV
ejpam-5256	16	2	,	,	PUNCT
ejpam-5256	16	3	if	if	SCONJ
ejpam-5256	16	4	a	a	DET
ejpam-5256	16	5	⊆	⊆	NUM
ejpam-5256	16	6	b	b	NOUN
ejpam-5256	16	7	⊆	⊆	NUM
ejpam-5256	16	8	x	x	NUM
ejpam-5256	16	9	,	,	PUNCT
ejpam-5256	16	10	then	then	ADV
ejpam-5256	16	11	iµ(a	iµ(a	ADJ
ejpam-5256	16	12	)	)	PUNCT
ejpam-5256	16	13	⊆	⊆	NUM
ejpam-5256	16	14	iµ(b	iµ(b	NOUN
ejpam-5256	16	15	)	)	PUNCT
ejpam-5256	16	16	and	and	CCONJ
ejpam-5256	16	17	cµ(a	cµ(a	ADJ
ejpam-5256	16	18	)	)	PUNCT
ejpam-5256	16	19	⊆	⊆	NUM
ejpam-5256	16	20	cµ(b	cµ(b	NOUN
ejpam-5256	16	21	)	)	PUNCT
ejpam-5256	16	22	.	.	PUNCT
ejpam-5256	17	1	in	in	ADP
ejpam-5256	17	2	2008	2008	NUM
ejpam-5256	17	3	,	,	PUNCT
ejpam-5256	17	4	császár	császár	NOUN
ejpam-5256	18	1	[	[	X
ejpam-5256	18	2	3	3	X
ejpam-5256	18	3	]	]	PUNCT
ejpam-5256	18	4	defined	define	VERB
ejpam-5256	18	5	the	the	DET
ejpam-5256	18	6	family	family	NOUN
ejpam-5256	18	7	θ(µ	θ(µ	NOUN
ejpam-5256	18	8	)	)	PUNCT
ejpam-5256	19	1	=	=	SYM
ejpam-5256	19	2	θ	θ	PROPN
ejpam-5256	19	3	⊆	⊆	NUM
ejpam-5256	19	4	p	p	X
ejpam-5256	19	5	(	(	PUNCT
ejpam-5256	19	6	x	x	X
ejpam-5256	19	7	)	)	PUNCT
ejpam-5256	19	8	in	in	ADP
ejpam-5256	19	9	a	a	DET
ejpam-5256	19	10	generalized	generalized	ADJ
ejpam-5256	19	11	topological	topological	ADJ
ejpam-5256	19	12	space	space	NOUN
ejpam-5256	19	13	(	(	PUNCT
ejpam-5256	19	14	x,µ	x,µ	NOUN
ejpam-5256	19	15	)	)	PUNCT
ejpam-5256	19	16	.	.	PUNCT
ejpam-5256	20	1	a	a	DET
ejpam-5256	20	2	subset	subset	NOUN
ejpam-5256	20	3	a	a	DET
ejpam-5256	20	4	of	of	ADP
ejpam-5256	20	5	(	(	PUNCT
ejpam-5256	20	6	x,µ	x,µ	NOUN
ejpam-5256	20	7	)	)	PUNCT
ejpam-5256	20	8	is	be	AUX
ejpam-5256	20	9	an	an	DET
ejpam-5256	20	10	element	element	NOUN
ejpam-5256	20	11	of	of	ADP
ejpam-5256	20	12	θ(µ	θ(µ	NOUN
ejpam-5256	20	13	)	)	PUNCT
ejpam-5256	20	14	if	if	SCONJ
ejpam-5256	20	15	and	and	CCONJ
ejpam-5256	20	16	only	only	ADV
ejpam-5256	20	17	if	if	SCONJ
ejpam-5256	20	18	there	there	PRON
ejpam-5256	20	19	exists	exist	VERB
ejpam-5256	20	20	m	m	VERB
ejpam-5256	20	21	∈	∈	PROPN
ejpam-5256	20	22	µ	µ	PRON
ejpam-5256	20	23	such	such	ADJ
ejpam-5256	20	24	that	that	SCONJ
ejpam-5256	20	25	x	x	SYM
ejpam-5256	20	26	∈	∈	PROPN
ejpam-5256	20	27	m	m	NOUN
ejpam-5256	20	28	and	and	CCONJ
ejpam-5256	20	29	m	m	PROPN
ejpam-5256	20	30	⊆	⊆	NUM
ejpam-5256	20	31	cµ(m	cµ(m	NOUN
ejpam-5256	20	32	)	)	PUNCT
ejpam-5256	20	33	⊆	⊆	NUM
ejpam-5256	20	34	a	a	PRON
ejpam-5256	20	35	for	for	ADP
ejpam-5256	20	36	all	all	DET
ejpam-5256	20	37	x	x	SYM
ejpam-5256	20	38	∈	∈	NOUN
ejpam-5256	20	39	a.	a.	NOUN
ejpam-5256	20	40	the	the	DET
ejpam-5256	20	41	elements	element	NOUN
ejpam-5256	20	42	of	of	ADP
ejpam-5256	20	43	θ(µ	θ(µ	NOUN
ejpam-5256	20	44	)	)	PUNCT
ejpam-5256	20	45	are	be	AUX
ejpam-5256	20	46	called	call	VERB
ejpam-5256	20	47	θ	θ	ADJ
ejpam-5256	20	48	-	-	ADJ
ejpam-5256	20	49	open	open	ADJ
ejpam-5256	20	50	sets	set	NOUN
ejpam-5256	20	51	in	in	ADP
ejpam-5256	20	52	(	(	PUNCT
ejpam-5256	20	53	x,µ	x,µ	NOUN
ejpam-5256	20	54	)	)	PUNCT
ejpam-5256	20	55	.	.	PUNCT
ejpam-5256	21	1	the	the	DET
ejpam-5256	21	2	complements	complement	NOUN
ejpam-5256	21	3	of	of	ADP
ejpam-5256	21	4	θ	θ	ADJ
ejpam-5256	21	5	-	-	ADJ
ejpam-5256	21	6	open	open	ADJ
ejpam-5256	21	7	sets	set	NOUN
ejpam-5256	21	8	are	be	AUX
ejpam-5256	21	9	called	call	VERB
ejpam-5256	21	10	θ	θ	ADJ
ejpam-5256	21	11	-	-	PUNCT
ejpam-5256	21	12	closed	closed	ADJ
ejpam-5256	21	13	sets	set	NOUN
ejpam-5256	21	14	.	.	PUNCT
ejpam-5256	22	1	in	in	ADP
ejpam-5256	22	2	2011	2011	NUM
ejpam-5256	22	3	,	,	PUNCT
ejpam-5256	22	4	the	the	DET
ejpam-5256	22	5	concept	concept	NOUN
ejpam-5256	22	6	of	of	ADP
ejpam-5256	22	7	the	the	DET
ejpam-5256	22	8	collection	collection	NOUN
ejpam-5256	22	9	θ(µ	θ(µ	PROPN
ejpam-5256	22	10	)	)	PUNCT
ejpam-5256	22	11	was	be	AUX
ejpam-5256	22	12	developed	develop	VERB
ejpam-5256	22	13	into	into	ADP
ejpam-5256	22	14	the	the	DET
ejpam-5256	22	15	collection	collection	NOUN
ejpam-5256	22	16	θ̃(µ	θ̃(µ	NOUN
ejpam-5256	22	17	)	)	PUNCT
ejpam-5256	22	18	=	=	PUNCT
ejpam-5256	23	1	θ̃	θ̃	NOUN
ejpam-5256	23	2	⊆	⊆	NUM
ejpam-5256	23	3	p	p	NOUN
ejpam-5256	23	4	(	(	PUNCT
ejpam-5256	23	5	x	x	NOUN
ejpam-5256	23	6	)	)	PUNCT
ejpam-5256	23	7	by	by	ADP
ejpam-5256	23	8	min	min	NOUN
ejpam-5256	24	1	[	[	X
ejpam-5256	24	2	4	4	NUM
ejpam-5256	24	3	]	]	PUNCT
ejpam-5256	24	4	.	.	PUNCT
ejpam-5256	25	1	additionally	additionally	ADV
ejpam-5256	25	2	,	,	PUNCT
ejpam-5256	25	3	he	he	PRON
ejpam-5256	25	4	also	also	ADV
ejpam-5256	25	5	concluded	conclude	VERB
ejpam-5256	25	6	that	that	SCONJ
ejpam-5256	25	7	θ	θ	PROPN
ejpam-5256	25	8	⊆	⊆	NUM
ejpam-5256	25	9	θ̃	θ̃	PROPN
ejpam-5256	25	10	⊆	⊆	NUM
ejpam-5256	25	11	µ.	µ.	NOUN
ejpam-5256	25	12	in	in	ADP
ejpam-5256	25	13	2011	2011	NUM
ejpam-5256	25	14	,	,	PUNCT
ejpam-5256	25	15	roy	roy	PROPN
ejpam-5256	25	16	[	[	X
ejpam-5256	25	17	5	5	NUM
ejpam-5256	25	18	]	]	PUNCT
ejpam-5256	25	19	studied	study	VERB
ejpam-5256	25	20	some	some	DET
ejpam-5256	25	21	properties	property	NOUN
ejpam-5256	25	22	of	of	ADP
ejpam-5256	25	23	(	(	PUNCT
ejpam-5256	25	24	µ	µ	NUM
ejpam-5256	25	25	,	,	PUNCT
ejpam-5256	25	26	µ′)-continuous	µ′)-continuous	ADJ
ejpam-5256	25	27	functions	function	NOUN
ejpam-5256	25	28	on	on	ADP
ejpam-5256	25	29	generalized	generalized	ADJ
ejpam-5256	25	30	topological	topological	ADJ
ejpam-5256	25	31	spaces	space	NOUN
ejpam-5256	25	32	.	.	PUNCT
ejpam-5256	26	1	in	in	ADP
ejpam-5256	26	2	this	this	DET
ejpam-5256	26	3	paper	paper	NOUN
ejpam-5256	26	4	,	,	PUNCT
ejpam-5256	26	5	we	we	PRON
ejpam-5256	26	6	use	use	VERB
ejpam-5256	26	7	the	the	DET
ejpam-5256	26	8	concept	concept	NOUN
ejpam-5256	26	9	of	of	ADP
ejpam-5256	26	10	the	the	DET
ejpam-5256	26	11	above	above	ADJ
ejpam-5256	26	12	researches	research	NOUN
ejpam-5256	26	13	to	to	PART
ejpam-5256	26	14	define	define	VERB
ejpam-5256	26	15	new	new	ADJ
ejpam-5256	26	16	types	type	NOUN
ejpam-5256	26	17	of	of	ADP
ejpam-5256	26	18	sets	set	NOUN
ejpam-5256	26	19	in	in	ADP
ejpam-5256	26	20	generalized	generalized	ADJ
ejpam-5256	26	21	topological	topological	ADJ
ejpam-5256	26	22	spaces	space	NOUN
ejpam-5256	26	23	,	,	PUNCT
ejpam-5256	26	24	along	along	ADP
ejpam-5256	26	25	with	with	ADP
ejpam-5256	26	26	studying	study	VERB
ejpam-5256	26	27	some	some	DET
ejpam-5256	26	28	basic	basic	ADJ
ejpam-5256	26	29	properties	property	NOUN
ejpam-5256	26	30	of	of	ADP
ejpam-5256	26	31	such	such	ADJ
ejpam-5256	26	32	sets	set	NOUN
ejpam-5256	26	33	.	.	PUNCT
ejpam-5256	27	1	doi	doi	NOUN
ejpam-5256	27	2	:	:	PUNCT
ejpam-5256	27	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5256	https://doi.org/10.29020/nybg.ejpam.v17i3.5256	DET
ejpam-5256	27	4	email	email	NOUN
ejpam-5256	27	5	address	address	NOUN
ejpam-5256	27	6	:	:	PUNCT
ejpam-5256	28	1	jeeranunt.k@msu.ac.th	jeeranunt.k@msu.ac.th	INTJ
ejpam-5256	28	2	(	(	PUNCT
ejpam-5256	28	3	j.	j.	PROPN
ejpam-5256	28	4	khampakdee	khampakdee	PROPN
ejpam-5256	28	5	)	)	PUNCT
ejpam-5256	28	6	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5256	28	7	1869	1869	NUM
ejpam-5256	29	1	©	©	ADP
ejpam-5256	29	2	2024	2024	NUM
ejpam-5256	29	3	ejpam	ejpam	NOUN
ejpam-5256	29	4	all	all	DET
ejpam-5256	29	5	rights	right	NOUN
ejpam-5256	29	6	reserved	reserve	VERB
ejpam-5256	29	7	.	.	PUNCT
ejpam-5256	30	1	j.	j.	PROPN
ejpam-5256	30	2	khampakdee	khampakdee	PROPN
ejpam-5256	30	3	/	/	PUNCT
ejpam-5256	30	4	eur	eur	PROPN
ejpam-5256	30	5	.	.	PUNCT
ejpam-5256	31	1	j.	j.	PROPN
ejpam-5256	31	2	pure	pure	PROPN
ejpam-5256	31	3	appl	appl	PROPN
ejpam-5256	31	4	.	.	PROPN
ejpam-5256	31	5	math	math	PROPN
ejpam-5256	31	6	,	,	PUNCT
ejpam-5256	31	7	17	17	NUM
ejpam-5256	31	8	(	(	PUNCT
ejpam-5256	31	9	3	3	NUM
ejpam-5256	31	10	)	)	PUNCT
ejpam-5256	31	11	(	(	PUNCT
ejpam-5256	31	12	2024	2024	NUM
ejpam-5256	31	13	)	)	PUNCT
ejpam-5256	31	14	,	,	PUNCT
ejpam-5256	31	15	1869	1869	NUM
ejpam-5256	31	16	-	-	SYM
ejpam-5256	31	17	1876	1876	NUM
ejpam-5256	31	18	1870	1870	NUM
ejpam-5256	31	19	2	2	NUM
ejpam-5256	31	20	.	.	PUNCT
ejpam-5256	31	21	preliminaries	preliminary	NOUN
ejpam-5256	31	22	definition	definition	NOUN
ejpam-5256	31	23	1	1	NUM
ejpam-5256	31	24	.	.	PUNCT
ejpam-5256	32	1	[	[	X
ejpam-5256	32	2	1	1	X
ejpam-5256	32	3	]	]	AUX
ejpam-5256	32	4	let	let	VERB
ejpam-5256	32	5	x	x	PRON
ejpam-5256	32	6	be	be	AUX
ejpam-5256	32	7	a	a	DET
ejpam-5256	32	8	non	non	ADJ
ejpam-5256	32	9	-	-	ADJ
ejpam-5256	32	10	empty	empty	ADJ
ejpam-5256	32	11	set	set	NOUN
ejpam-5256	32	12	,	,	PUNCT
ejpam-5256	32	13	p	p	X
ejpam-5256	32	14	(	(	PUNCT
ejpam-5256	32	15	x	x	NOUN
ejpam-5256	32	16	)	)	PUNCT
ejpam-5256	32	17	denotes	denote	VERB
ejpam-5256	32	18	the	the	DET
ejpam-5256	32	19	power	power	NOUN
ejpam-5256	32	20	set	set	NOUN
ejpam-5256	32	21	of	of	ADP
ejpam-5256	32	22	x	x	PRON
ejpam-5256	32	23	,	,	PUNCT
ejpam-5256	32	24	we	we	PRON
ejpam-5256	32	25	call	call	VERB
ejpam-5256	32	26	a	a	DET
ejpam-5256	32	27	collection	collection	NOUN
ejpam-5256	32	28	µ	µ	PRON
ejpam-5256	32	29	⊆	⊆	NUM
ejpam-5256	32	30	p	p	NOUN
ejpam-5256	32	31	(	(	PUNCT
ejpam-5256	32	32	x	x	X
ejpam-5256	32	33	)	)	PUNCT
ejpam-5256	32	34	a	a	DET
ejpam-5256	32	35	generalized	generalized	ADJ
ejpam-5256	32	36	topology	topology	NOUN
ejpam-5256	32	37	on	on	ADP
ejpam-5256	32	38	x	x	SYM
ejpam-5256	32	39	if	if	SCONJ
ejpam-5256	32	40	∅	∅	NOUN
ejpam-5256	32	41	∈	∈	PROPN
ejpam-5256	32	42	µ	µ	NOUN
ejpam-5256	32	43	and	and	CCONJ
ejpam-5256	32	44	⋃	⋃	ADP
ejpam-5256	32	45	α∈j	α∈j	NOUN
ejpam-5256	32	46	gα	gα	ADP
ejpam-5256	32	47	∈	∈	PROPN
ejpam-5256	32	48	µ	µ	NOUN
ejpam-5256	32	49	for	for	ADP
ejpam-5256	32	50	each	each	DET
ejpam-5256	32	51	gα	gα	ADP
ejpam-5256	32	52	∈	∈	PROPN
ejpam-5256	32	53	µ	µ	NOUN
ejpam-5256	32	54	and	and	CCONJ
ejpam-5256	32	55	α	α	PRON
ejpam-5256	32	56	∈	∈	PROPN
ejpam-5256	32	57	j	j	PROPN
ejpam-5256	32	58	.	.	PUNCT
ejpam-5256	33	1	the	the	DET
ejpam-5256	33	2	elements	element	NOUN
ejpam-5256	33	3	in	in	ADP
ejpam-5256	33	4	µ	µ	PROPN
ejpam-5256	33	5	is	be	AUX
ejpam-5256	33	6	called	call	VERB
ejpam-5256	33	7	µ-open	µ-open	NOUN
ejpam-5256	33	8	sets	set	NOUN
ejpam-5256	33	9	in	in	ADP
ejpam-5256	33	10	x.	x.	NOUN
ejpam-5256	33	11	the	the	DET
ejpam-5256	33	12	complement	complement	NOUN
ejpam-5256	33	13	of	of	ADP
ejpam-5256	33	14	each	each	DET
ejpam-5256	33	15	µ-open	µ-open	NOUN
ejpam-5256	33	16	set	set	VERB
ejpam-5256	33	17	is	be	AUX
ejpam-5256	33	18	called	call	VERB
ejpam-5256	33	19	µ-closed	µ-close	VERB
ejpam-5256	33	20	in	in	ADP
ejpam-5256	33	21	x.	x.	NOUN
ejpam-5256	33	22	the	the	DET
ejpam-5256	33	23	pair	pair	NOUN
ejpam-5256	33	24	(	(	PUNCT
ejpam-5256	33	25	x,µ	x,µ	NOUN
ejpam-5256	33	26	)	)	PUNCT
ejpam-5256	33	27	is	be	AUX
ejpam-5256	33	28	called	call	VERB
ejpam-5256	33	29	a	a	DET
ejpam-5256	33	30	generalized	generalized	ADJ
ejpam-5256	33	31	topological	topological	ADJ
ejpam-5256	33	32	space	space	NOUN
ejpam-5256	33	33	.	.	PUNCT
ejpam-5256	34	1	for	for	ADP
ejpam-5256	34	2	a	a	DET
ejpam-5256	34	3	⊆	⊆	NUM
ejpam-5256	34	4	x	x	NOUN
ejpam-5256	34	5	,	,	PUNCT
ejpam-5256	34	6	the	the	DET
ejpam-5256	34	7	symbol	symbol	NOUN
ejpam-5256	34	8	iµ(a	iµ(a	ADJ
ejpam-5256	34	9	)	)	PUNCT
ejpam-5256	34	10	represents	represent	VERB
ejpam-5256	34	11	the	the	DET
ejpam-5256	34	12	interior	interior	NOUN
ejpam-5256	34	13	of	of	ADP
ejpam-5256	34	14	a	a	PRON
ejpam-5256	34	15	which	which	PRON
ejpam-5256	34	16	is	be	AUX
ejpam-5256	34	17	the	the	DET
ejpam-5256	34	18	union	union	NOUN
ejpam-5256	34	19	of	of	ADP
ejpam-5256	34	20	all	all	DET
ejpam-5256	34	21	µ-open	µ-open	NOUN
ejpam-5256	34	22	sets	set	NOUN
ejpam-5256	34	23	contained	contain	VERB
ejpam-5256	34	24	in	in	ADP
ejpam-5256	34	25	a	a	PRON
ejpam-5256	34	26	,	,	PUNCT
ejpam-5256	34	27	and	and	CCONJ
ejpam-5256	34	28	the	the	DET
ejpam-5256	34	29	symbol	symbol	NOUN
ejpam-5256	34	30	cµ(a	cµ(a	PUNCT
ejpam-5256	34	31	)	)	PUNCT
ejpam-5256	34	32	represents	represent	VERB
ejpam-5256	34	33	the	the	DET
ejpam-5256	34	34	closure	closure	NOUN
ejpam-5256	34	35	of	of	ADP
ejpam-5256	34	36	a	a	PRON
ejpam-5256	34	37	,	,	PUNCT
ejpam-5256	34	38	which	which	PRON
ejpam-5256	34	39	means	mean	VERB
ejpam-5256	34	40	the	the	DET
ejpam-5256	34	41	intersection	intersection	NOUN
ejpam-5256	34	42	of	of	ADP
ejpam-5256	34	43	all	all	DET
ejpam-5256	34	44	µ-closed	µ-close	VERB
ejpam-5256	34	45	sets	set	NOUN
ejpam-5256	34	46	containing	contain	VERB
ejpam-5256	34	47	a.	a.	PROPN
ejpam-5256	34	48	lemma	lemma	PROPN
ejpam-5256	34	49	1	1	NUM
ejpam-5256	34	50	.	.	PUNCT
ejpam-5256	35	1	[	[	X
ejpam-5256	35	2	2	2	X
ejpam-5256	35	3	]	]	PUNCT
ejpam-5256	35	4	let	let	VERB
ejpam-5256	35	5	a	a	PRON
ejpam-5256	35	6	be	be	AUX
ejpam-5256	35	7	a	a	DET
ejpam-5256	35	8	subset	subset	NOUN
ejpam-5256	35	9	of	of	ADP
ejpam-5256	35	10	a	a	DET
ejpam-5256	35	11	generalized	generalized	ADJ
ejpam-5256	35	12	topological	topological	ADJ
ejpam-5256	35	13	space	space	NOUN
ejpam-5256	35	14	(	(	PUNCT
ejpam-5256	35	15	x,µ	x,µ	NOUN
ejpam-5256	35	16	)	)	PUNCT
ejpam-5256	35	17	.	.	PUNCT
ejpam-5256	36	1	then	then	ADV
ejpam-5256	36	2	:	:	PUNCT
ejpam-5256	36	3	1	1	X
ejpam-5256	36	4	.	.	X
ejpam-5256	36	5	iµ(a	iµ(a	ADJ
ejpam-5256	36	6	)	)	PUNCT
ejpam-5256	36	7	is	be	AUX
ejpam-5256	37	1	the	the	DET
ejpam-5256	37	2	largest	large	ADJ
ejpam-5256	37	3	µ-open	µ-open	NOUN
ejpam-5256	37	4	set	set	NOUN
ejpam-5256	37	5	contained	contain	VERB
ejpam-5256	37	6	in	in	ADP
ejpam-5256	37	7	a	a	DET
ejpam-5256	37	8	,	,	PUNCT
ejpam-5256	37	9	2	2	NUM
ejpam-5256	37	10	.	.	NUM
ejpam-5256	37	11	cµ(a	cµ(a	NUM
ejpam-5256	37	12	)	)	PUNCT
ejpam-5256	37	13	is	be	AUX
ejpam-5256	37	14	the	the	DET
ejpam-5256	37	15	smallest	small	ADJ
ejpam-5256	37	16	µ-closed	µ-close	VERB
ejpam-5256	37	17	set	set	VERB
ejpam-5256	37	18	containing	contain	VERB
ejpam-5256	37	19	a	a	DET
ejpam-5256	37	20	,	,	PUNCT
ejpam-5256	37	21	3	3	NUM
ejpam-5256	37	22	.	.	PUNCT
ejpam-5256	37	23	cµ(x	cµ(x	VERB
ejpam-5256	37	24	−a	−a	ADJ
ejpam-5256	37	25	)	)	PUNCT
ejpam-5256	38	1	=	=	PUNCT
ejpam-5256	38	2	x	x	X
ejpam-5256	38	3	−	−	NOUN
ejpam-5256	38	4	iµ(a	iµ(a	PROPN
ejpam-5256	38	5	)	)	PUNCT
ejpam-5256	38	6	.	.	PUNCT
ejpam-5256	39	1	definition	definition	NOUN
ejpam-5256	39	2	2	2	NUM
ejpam-5256	39	3	.	.	PUNCT
ejpam-5256	40	1	[	[	X
ejpam-5256	40	2	2	2	NUM
ejpam-5256	40	3	]	]	X
ejpam-5256	40	4	let	let	VERB
ejpam-5256	40	5	(	(	PUNCT
ejpam-5256	40	6	x,µ	x,µ	NOUN
ejpam-5256	40	7	)	)	PUNCT
ejpam-5256	40	8	be	be	VERB
ejpam-5256	40	9	a	a	DET
ejpam-5256	40	10	subset	subset	NOUN
ejpam-5256	40	11	of	of	ADP
ejpam-5256	40	12	a	a	DET
ejpam-5256	40	13	generalized	generalized	ADJ
ejpam-5256	40	14	topological	topological	ADJ
ejpam-5256	40	15	space	space	NOUN
ejpam-5256	40	16	.	.	PUNCT
ejpam-5256	41	1	then	then	ADV
ejpam-5256	41	2	,	,	PUNCT
ejpam-5256	41	3	a	a	DET
ejpam-5256	41	4	subset	subset	NOUN
ejpam-5256	41	5	a	a	PRON
ejpam-5256	41	6	of	of	ADP
ejpam-5256	41	7	x	x	PRON
ejpam-5256	41	8	is	be	AUX
ejpam-5256	41	9	called	call	VERB
ejpam-5256	41	10	a	a	DET
ejpam-5256	41	11	µ-semi	µ-semi	ADJ
ejpam-5256	41	12	-	-	ADJ
ejpam-5256	41	13	open	open	ADJ
ejpam-5256	41	14	set	set	NOUN
ejpam-5256	41	15	if	if	SCONJ
ejpam-5256	41	16	a	a	DET
ejpam-5256	41	17	⊆	⊆	NUM
ejpam-5256	41	18	cµ(iµ(a	cµ(iµ(a	NOUN
ejpam-5256	41	19	)	)	PUNCT
ejpam-5256	41	20	)	)	PUNCT
ejpam-5256	41	21	.	.	PUNCT
ejpam-5256	42	1	the	the	DET
ejpam-5256	42	2	complement	complement	NOUN
ejpam-5256	42	3	of	of	ADP
ejpam-5256	42	4	µ-semi	µ-semi	ADJ
ejpam-5256	42	5	-	-	ADJ
ejpam-5256	42	6	open	open	ADJ
ejpam-5256	42	7	sets	set	NOUN
ejpam-5256	42	8	is	be	AUX
ejpam-5256	42	9	called	call	VERB
ejpam-5256	42	10	µ-semi	µ-semi	ADJ
ejpam-5256	42	11	-	-	ADJ
ejpam-5256	42	12	closed	closed	ADJ
ejpam-5256	42	13	sets	set	NOUN
ejpam-5256	42	14	.	.	PUNCT
ejpam-5256	43	1	the	the	DET
ejpam-5256	43	2	intersection	intersection	NOUN
ejpam-5256	43	3	of	of	ADP
ejpam-5256	43	4	all	all	DET
ejpam-5256	43	5	µ-semi	µ-semi	NOUN
ejpam-5256	43	6	-	-	ADJ
ejpam-5256	43	7	closed	closed	ADJ
ejpam-5256	43	8	sets	set	NOUN
ejpam-5256	43	9	containing	contain	VERB
ejpam-5256	43	10	a	a	PRON
ejpam-5256	43	11	is	be	AUX
ejpam-5256	43	12	denoted	denote	VERB
ejpam-5256	43	13	by	by	ADP
ejpam-5256	43	14	cσ(a	cσ(a	NOUN
ejpam-5256	43	15	)	)	PUNCT
ejpam-5256	43	16	.	.	PUNCT
ejpam-5256	44	1	definition	definition	NOUN
ejpam-5256	44	2	3	3	NUM
ejpam-5256	44	3	.	.	PUNCT
ejpam-5256	45	1	[	[	X
ejpam-5256	45	2	3	3	X
ejpam-5256	45	3	]	]	PUNCT
ejpam-5256	45	4	let	let	VERB
ejpam-5256	45	5	a	a	PRON
ejpam-5256	45	6	be	be	AUX
ejpam-5256	45	7	a	a	DET
ejpam-5256	45	8	subset	subset	NOUN
ejpam-5256	45	9	of	of	ADP
ejpam-5256	45	10	a	a	DET
ejpam-5256	45	11	generalized	generalized	ADJ
ejpam-5256	45	12	topological	topological	ADJ
ejpam-5256	45	13	space	space	NOUN
ejpam-5256	45	14	(	(	PUNCT
ejpam-5256	45	15	x,µ	x,µ	NOUN
ejpam-5256	45	16	)	)	PUNCT
ejpam-5256	45	17	.	.	PUNCT
ejpam-5256	46	1	then	then	ADV
ejpam-5256	46	2	,	,	PUNCT
ejpam-5256	46	3	a	a	PRON
ejpam-5256	46	4	is	be	AUX
ejpam-5256	46	5	an	an	DET
ejpam-5256	46	6	element	element	NOUN
ejpam-5256	46	7	of	of	ADP
ejpam-5256	46	8	a	a	DET
ejpam-5256	46	9	collection	collection	NOUN
ejpam-5256	46	10	θ(µ	θ(µ	NOUN
ejpam-5256	46	11	)	)	PUNCT
ejpam-5256	46	12	=	=	SYM
ejpam-5256	46	13	θ	θ	PROPN
ejpam-5256	46	14	⊆	⊆	NUM
ejpam-5256	46	15	p	p	X
ejpam-5256	46	16	(	(	PUNCT
ejpam-5256	46	17	x	x	NOUN
ejpam-5256	46	18	)	)	PUNCT
ejpam-5256	46	19	if	if	SCONJ
ejpam-5256	46	20	and	and	CCONJ
ejpam-5256	46	21	only	only	ADV
ejpam-5256	46	22	if	if	SCONJ
ejpam-5256	46	23	there	there	PRON
ejpam-5256	46	24	exists	exist	VERB
ejpam-5256	46	25	m	m	VERB
ejpam-5256	46	26	∈	∈	PROPN
ejpam-5256	46	27	µ	µ	PRON
ejpam-5256	46	28	such	such	ADJ
ejpam-5256	46	29	that	that	SCONJ
ejpam-5256	46	30	x	x	SYM
ejpam-5256	46	31	∈m	∈m	NOUN
ejpam-5256	46	32	and	and	CCONJ
ejpam-5256	46	33	m	m	NOUN
ejpam-5256	46	34	⊆	⊆	NUM
ejpam-5256	46	35	cµ(m	cµ(m	NOUN
ejpam-5256	46	36	)	)	PUNCT
ejpam-5256	46	37	⊆	⊆	NUM
ejpam-5256	46	38	a	a	PRON
ejpam-5256	46	39	for	for	ADP
ejpam-5256	46	40	all	all	DET
ejpam-5256	46	41	x	x	SYM
ejpam-5256	46	42	∈	∈	NOUN
ejpam-5256	46	43	a.	a.	NOUN
ejpam-5256	46	44	the	the	DET
ejpam-5256	46	45	elements	element	NOUN
ejpam-5256	46	46	of	of	ADP
ejpam-5256	46	47	θ(µ	θ(µ	NOUN
ejpam-5256	46	48	)	)	PUNCT
ejpam-5256	46	49	are	be	AUX
ejpam-5256	46	50	called	call	VERB
ejpam-5256	46	51	θ	θ	ADJ
ejpam-5256	46	52	-	-	ADJ
ejpam-5256	46	53	open	open	ADJ
ejpam-5256	46	54	sets	set	NOUN
ejpam-5256	46	55	in	in	ADP
ejpam-5256	46	56	x.	x.	NOUN
ejpam-5256	47	1	the	the	DET
ejpam-5256	47	2	complements	complement	NOUN
ejpam-5256	47	3	of	of	ADP
ejpam-5256	47	4	θ	θ	ADJ
ejpam-5256	47	5	-	-	ADJ
ejpam-5256	47	6	open	open	ADJ
ejpam-5256	47	7	sets	set	NOUN
ejpam-5256	47	8	are	be	AUX
ejpam-5256	47	9	called	call	VERB
ejpam-5256	47	10	θ	θ	ADJ
ejpam-5256	47	11	-	-	PUNCT
ejpam-5256	47	12	closed	closed	ADJ
ejpam-5256	47	13	sets	set	NOUN
ejpam-5256	47	14	.	.	PUNCT
ejpam-5256	48	1	definition	definition	NOUN
ejpam-5256	48	2	4	4	NUM
ejpam-5256	48	3	.	.	PUNCT
ejpam-5256	49	1	[	[	X
ejpam-5256	49	2	4	4	X
ejpam-5256	49	3	]	]	X
ejpam-5256	49	4	let	let	VERB
ejpam-5256	49	5	(	(	PUNCT
ejpam-5256	49	6	x,µ	x,µ	NOUN
ejpam-5256	49	7	)	)	PUNCT
ejpam-5256	49	8	be	be	VERB
ejpam-5256	49	9	a	a	DET
ejpam-5256	49	10	generalized	generalized	ADJ
ejpam-5256	49	11	topological	topological	ADJ
ejpam-5256	49	12	space	space	NOUN
ejpam-5256	49	13	and	and	CCONJ
ejpam-5256	49	14	θ̃(µ	θ̃(µ	NOUN
ejpam-5256	49	15	)	)	PUNCT
ejpam-5256	49	16	=	=	PUNCT
ejpam-5256	50	1	θ̃	θ̃	NOUN
ejpam-5256	50	2	⊆	⊆	NUM
ejpam-5256	50	3	p	p	NOUN
ejpam-5256	50	4	(	(	PUNCT
ejpam-5256	50	5	x	x	NOUN
ejpam-5256	50	6	)	)	PUNCT
ejpam-5256	50	7	.	.	PUNCT
ejpam-5256	51	1	a	a	DET
ejpam-5256	51	2	subset	subset	NOUN
ejpam-5256	51	3	a	a	PRON
ejpam-5256	51	4	of	of	ADP
ejpam-5256	51	5	x	x	PUNCT
ejpam-5256	51	6	is	be	AUX
ejpam-5256	51	7	an	an	DET
ejpam-5256	51	8	element	element	NOUN
ejpam-5256	51	9	of	of	ADP
ejpam-5256	51	10	θ̃	θ̃	NOUN
ejpam-5256	51	11	if	if	SCONJ
ejpam-5256	51	12	and	and	CCONJ
ejpam-5256	51	13	only	only	ADV
ejpam-5256	51	14	if	if	SCONJ
ejpam-5256	51	15	there	there	PRON
ejpam-5256	51	16	exists	exist	VERB
ejpam-5256	51	17	m	m	VERB
ejpam-5256	51	18	∈	∈	PROPN
ejpam-5256	51	19	µ	µ	PRON
ejpam-5256	51	20	such	such	ADJ
ejpam-5256	51	21	that	that	SCONJ
ejpam-5256	51	22	x	x	SYM
ejpam-5256	51	23	∈	∈	PROPN
ejpam-5256	51	24	m	m	NOUN
ejpam-5256	51	25	⊆	⊆	NUM
ejpam-5256	51	26	cµ(m	cµ(m	NOUN
ejpam-5256	51	27	)	)	PUNCT
ejpam-5256	51	28	∩mµ	∩mµ	VERB
ejpam-5256	51	29	⊆	⊆	NUM
ejpam-5256	51	30	a	a	PRON
ejpam-5256	51	31	for	for	ADP
ejpam-5256	51	32	all	all	DET
ejpam-5256	51	33	x	x	SYM
ejpam-5256	51	34	∈	∈	PROPN
ejpam-5256	51	35	a	a	PRON
ejpam-5256	51	36	,	,	PUNCT
ejpam-5256	51	37	where	where	SCONJ
ejpam-5256	51	38	mµ	mµ	ADP
ejpam-5256	51	39	=	=	PUNCT
ejpam-5256	51	40	∪{m	∪{m	PROPN
ejpam-5256	51	41	⊆	⊆	NUM
ejpam-5256	51	42	x	x	SYM
ejpam-5256	51	43	:	:	PUNCT
ejpam-5256	51	44	m	m	VERB
ejpam-5256	51	45	∈	∈	NOUN
ejpam-5256	51	46	µ	µ	X
ejpam-5256	51	47	}	}	PUNCT
ejpam-5256	51	48	.	.	PUNCT
ejpam-5256	52	1	the	the	DET
ejpam-5256	52	2	elements	element	NOUN
ejpam-5256	52	3	of	of	ADP
ejpam-5256	52	4	θ̃	θ̃	PROPN
ejpam-5256	52	5	are	be	AUX
ejpam-5256	52	6	called	call	VERB
ejpam-5256	52	7	θ̃-open	θ̃-open	PROPN
ejpam-5256	52	8	sets	set	NOUN
ejpam-5256	52	9	in	in	ADP
ejpam-5256	52	10	x.	x.	NOUN
ejpam-5256	52	11	the	the	DET
ejpam-5256	52	12	complements	complement	NOUN
ejpam-5256	52	13	of	of	ADP
ejpam-5256	52	14	θ̃-open	θ̃-open	NOUN
ejpam-5256	52	15	sets	set	NOUN
ejpam-5256	52	16	are	be	AUX
ejpam-5256	52	17	called	call	VERB
ejpam-5256	52	18	θ̃-closed	θ̃-close	VERB
ejpam-5256	52	19	sets	set	NOUN
ejpam-5256	52	20	.	.	PUNCT
ejpam-5256	53	1	additionally	additionally	ADV
ejpam-5256	53	2	,	,	PUNCT
ejpam-5256	53	3	in	in	ADP
ejpam-5256	53	4	[	[	PUNCT
ejpam-5256	53	5	4	4	NUM
ejpam-5256	53	6	]	]	PUNCT
ejpam-5256	53	7	,	,	PUNCT
ejpam-5256	53	8	the	the	DET
ejpam-5256	53	9	relationship	relationship	NOUN
ejpam-5256	53	10	between	between	ADP
ejpam-5256	53	11	θ	θ	PROPN
ejpam-5256	53	12	,	,	PUNCT
ejpam-5256	53	13	θ̃	θ̃	NOUN
ejpam-5256	53	14	and	and	CCONJ
ejpam-5256	53	15	µ	µ	NOUN
ejpam-5256	53	16	was	be	AUX
ejpam-5256	53	17	concluded	conclude	VERB
ejpam-5256	53	18	as	as	SCONJ
ejpam-5256	53	19	follows	follow	VERB
ejpam-5256	53	20	:	:	PUNCT
ejpam-5256	53	21	θ	θ	PROPN
ejpam-5256	53	22	⊆	⊆	NUM
ejpam-5256	53	23	θ̃	θ̃	PROPN
ejpam-5256	53	24	⊆	⊆	NUM
ejpam-5256	53	25	µ	µ	NOUN
ejpam-5256	53	26	.	.	PUNCT
ejpam-5256	54	1	definition	definition	NOUN
ejpam-5256	54	2	5	5	NUM
ejpam-5256	54	3	.	.	PUNCT
ejpam-5256	55	1	[	[	X
ejpam-5256	55	2	1	1	X
ejpam-5256	55	3	]	]	X
ejpam-5256	55	4	let	let	VERB
ejpam-5256	55	5	(	(	PUNCT
ejpam-5256	55	6	x,µ	x,µ	NOUN
ejpam-5256	55	7	)	)	PUNCT
ejpam-5256	55	8	and	and	CCONJ
ejpam-5256	55	9	(	(	PUNCT
ejpam-5256	55	10	y	y	NOUN
ejpam-5256	55	11	,	,	PUNCT
ejpam-5256	55	12	µ′	µ′	PUNCT
ejpam-5256	55	13	)	)	PUNCT
ejpam-5256	55	14	be	be	AUX
ejpam-5256	55	15	generalized	generalize	VERB
ejpam-5256	55	16	topological	topological	ADJ
ejpam-5256	55	17	spaces	space	NOUN
ejpam-5256	55	18	.	.	PUNCT
ejpam-5256	56	1	then	then	ADV
ejpam-5256	56	2	a	a	DET
ejpam-5256	56	3	function	function	NOUN
ejpam-5256	56	4	f	f	X
ejpam-5256	56	5	from	from	ADP
ejpam-5256	56	6	(	(	PUNCT
ejpam-5256	56	7	x,µ	x,µ	NOUN
ejpam-5256	56	8	)	)	PUNCT
ejpam-5256	56	9	into	into	ADP
ejpam-5256	56	10	(	(	PUNCT
ejpam-5256	56	11	y	y	NOUN
ejpam-5256	56	12	,	,	PUNCT
ejpam-5256	56	13	µ′	µ′	NUM
ejpam-5256	56	14	)	)	PUNCT
ejpam-5256	56	15	is	be	AUX
ejpam-5256	56	16	called	call	VERB
ejpam-5256	56	17	(	(	PUNCT
ejpam-5256	56	18	µ	µ	NUM
ejpam-5256	56	19	,	,	PUNCT
ejpam-5256	56	20	µ′)-continuous	µ′)-continuous	ADJ
ejpam-5256	56	21	if	if	SCONJ
ejpam-5256	56	22	f−1(g	f−1(g	PROPN
ejpam-5256	56	23	)	)	PUNCT
ejpam-5256	56	24	is	be	AUX
ejpam-5256	56	25	µ-open	µ-open	NOUN
ejpam-5256	56	26	in	in	ADP
ejpam-5256	56	27	x	x	PUNCT
ejpam-5256	56	28	for	for	SCONJ
ejpam-5256	56	29	each	each	DET
ejpam-5256	56	30	µ′-open	µ′-open	NUM
ejpam-5256	56	31	set	set	VERB
ejpam-5256	56	32	g	g	NOUN
ejpam-5256	56	33	in	in	ADP
ejpam-5256	56	34	y	y	PROPN
ejpam-5256	56	35	.	.	PUNCT
ejpam-5256	57	1	3	3	X
ejpam-5256	57	2	.	.	X
ejpam-5256	57	3	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	57	4	sets	set	NOUN
ejpam-5256	57	5	in	in	ADP
ejpam-5256	57	6	this	this	DET
ejpam-5256	57	7	section	section	NOUN
ejpam-5256	57	8	,	,	PUNCT
ejpam-5256	57	9	we	we	PRON
ejpam-5256	57	10	introduce	introduce	VERB
ejpam-5256	57	11	the	the	DET
ejpam-5256	57	12	concept	concept	NOUN
ejpam-5256	57	13	of	of	ADP
ejpam-5256	57	14	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	57	15	sets	set	NOUN
ejpam-5256	57	16	.	.	PUNCT
ejpam-5256	58	1	furthermore	furthermore	ADV
ejpam-5256	58	2	,	,	PUNCT
ejpam-5256	58	3	some	some	DET
ejpam-5256	58	4	properties	property	NOUN
ejpam-5256	58	5	of	of	ADP
ejpam-5256	58	6	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	58	7	sets	set	NOUN
ejpam-5256	58	8	are	be	AUX
ejpam-5256	58	9	studied	study	VERB
ejpam-5256	58	10	.	.	PUNCT
ejpam-5256	59	1	definition	definition	NOUN
ejpam-5256	59	2	6	6	NUM
ejpam-5256	59	3	.	.	PUNCT
ejpam-5256	60	1	let	let	VERB
ejpam-5256	60	2	(	(	PUNCT
ejpam-5256	60	3	x,µ	x,µ	NOUN
ejpam-5256	60	4	)	)	PUNCT
ejpam-5256	60	5	be	be	VERB
ejpam-5256	60	6	a	a	DET
ejpam-5256	60	7	generalized	generalized	ADJ
ejpam-5256	60	8	topological	topological	ADJ
ejpam-5256	60	9	space	space	NOUN
ejpam-5256	60	10	and	and	CCONJ
ejpam-5256	60	11	a	a	DET
ejpam-5256	60	12	⊆	⊆	NUM
ejpam-5256	60	13	x.	x.	NOUN
ejpam-5256	60	14	define	define	VERB
ejpam-5256	60	15	the	the	DET
ejpam-5256	60	16	collection	collection	NOUN
ejpam-5256	60	17	sθ̃(µ	sθ̃(µ	NOUN
ejpam-5256	60	18	)	)	PUNCT
ejpam-5256	60	19	=	=	PUNCT
ejpam-5256	60	20	sθ̃	sθ̃	X
ejpam-5256	60	21	⊆	⊆	NUM
ejpam-5256	60	22	p	p	X
ejpam-5256	60	23	(	(	PUNCT
ejpam-5256	60	24	x	x	NOUN
ejpam-5256	60	25	)	)	PUNCT
ejpam-5256	60	26	by	by	ADP
ejpam-5256	60	27	a	a	DET
ejpam-5256	60	28	∈	∈	PROPN
ejpam-5256	60	29	sθ̃	sθ̃	NOUN
ejpam-5256	60	30	if	if	SCONJ
ejpam-5256	60	31	for	for	ADP
ejpam-5256	60	32	each	each	DET
ejpam-5256	60	33	x	x	SYM
ejpam-5256	60	34	∈	∈	PROPN
ejpam-5256	60	35	a	a	DET
ejpam-5256	60	36	there	there	PRON
ejpam-5256	60	37	exists	exist	VERB
ejpam-5256	60	38	m	m	VERB
ejpam-5256	60	39	∈	∈	PROPN
ejpam-5256	60	40	µ	µ	PRON
ejpam-5256	60	41	such	such	ADJ
ejpam-5256	60	42	that	that	SCONJ
ejpam-5256	60	43	x	x	SYM
ejpam-5256	60	44	∈	∈	PROPN
ejpam-5256	60	45	m	m	NOUN
ejpam-5256	60	46	⊆	⊆	NUM
ejpam-5256	60	47	cσ(m	cσ(m	NUM
ejpam-5256	60	48	)	)	PUNCT
ejpam-5256	60	49	∩mµ	∩mµ	VERB
ejpam-5256	60	50	⊆	⊆	NUM
ejpam-5256	60	51	a	a	NOUN
ejpam-5256	60	52	,	,	PUNCT
ejpam-5256	60	53	where	where	SCONJ
ejpam-5256	60	54	mµ	mµ	ADP
ejpam-5256	60	55	=	=	PUNCT
ejpam-5256	60	56	∪{m	∪{m	PROPN
ejpam-5256	60	57	⊆	⊆	NUM
ejpam-5256	60	58	x	x	SYM
ejpam-5256	60	59	:	:	PUNCT
ejpam-5256	60	60	m	m	VERB
ejpam-5256	60	61	∈	∈	NOUN
ejpam-5256	60	62	µ	µ	X
ejpam-5256	60	63	}	}	PUNCT
ejpam-5256	60	64	.	.	PUNCT
ejpam-5256	61	1	the	the	DET
ejpam-5256	61	2	elements	element	NOUN
ejpam-5256	61	3	in	in	ADP
ejpam-5256	61	4	sθ̃	sθ̃	PROPN
ejpam-5256	61	5	are	be	AUX
ejpam-5256	61	6	called	call	VERB
ejpam-5256	61	7	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	61	8	sets	set	NOUN
ejpam-5256	61	9	and	and	CCONJ
ejpam-5256	61	10	the	the	DET
ejpam-5256	61	11	complements	complement	NOUN
ejpam-5256	61	12	are	be	AUX
ejpam-5256	61	13	called	call	VERB
ejpam-5256	61	14	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	61	15	sets	set	NOUN
ejpam-5256	61	16	in	in	ADP
ejpam-5256	61	17	x.	x.	PROPN
ejpam-5256	61	18	j.	j.	PROPN
ejpam-5256	61	19	khampakdee	khampakdee	PROPN
ejpam-5256	61	20	/	/	PUNCT
ejpam-5256	61	21	eur	eur	PROPN
ejpam-5256	61	22	.	.	PUNCT
ejpam-5256	62	1	j.	j.	PROPN
ejpam-5256	62	2	pure	pure	PROPN
ejpam-5256	62	3	appl	appl	PROPN
ejpam-5256	62	4	.	.	PROPN
ejpam-5256	62	5	math	math	PROPN
ejpam-5256	62	6	,	,	PUNCT
ejpam-5256	62	7	17	17	NUM
ejpam-5256	62	8	(	(	PUNCT
ejpam-5256	62	9	3	3	NUM
ejpam-5256	62	10	)	)	PUNCT
ejpam-5256	62	11	(	(	PUNCT
ejpam-5256	62	12	2024	2024	NUM
ejpam-5256	62	13	)	)	PUNCT
ejpam-5256	62	14	,	,	PUNCT
ejpam-5256	62	15	1869	1869	NUM
ejpam-5256	62	16	-	-	SYM
ejpam-5256	62	17	1876	1876	NUM
ejpam-5256	62	18	1871	1871	NUM
ejpam-5256	62	19	theorem	theorem	NOUN
ejpam-5256	62	20	1	1	NUM
ejpam-5256	62	21	.	.	PUNCT
ejpam-5256	63	1	let	let	VERB
ejpam-5256	63	2	(	(	PUNCT
ejpam-5256	63	3	x,µ	x,µ	NOUN
ejpam-5256	63	4	)	)	PUNCT
ejpam-5256	63	5	be	be	VERB
ejpam-5256	63	6	a	a	DET
ejpam-5256	63	7	generalized	generalized	ADJ
ejpam-5256	63	8	topological	topological	ADJ
ejpam-5256	63	9	space	space	NOUN
ejpam-5256	63	10	.	.	PUNCT
ejpam-5256	64	1	then	then	ADV
ejpam-5256	64	2	,	,	PUNCT
ejpam-5256	64	3	sθ̃	sθ̃	PRON
ejpam-5256	64	4	is	be	AUX
ejpam-5256	64	5	a	a	DET
ejpam-5256	64	6	generalized	generalized	ADJ
ejpam-5256	64	7	topology	topology	NOUN
ejpam-5256	64	8	on	on	ADP
ejpam-5256	64	9	x.	x.	NOUN
ejpam-5256	64	10	proof	proof	NOUN
ejpam-5256	64	11	.	.	PUNCT
ejpam-5256	65	1	by	by	ADP
ejpam-5256	65	2	definition	definition	NOUN
ejpam-5256	65	3	6	6	NUM
ejpam-5256	65	4	,	,	PUNCT
ejpam-5256	65	5	we	we	PRON
ejpam-5256	65	6	can	can	AUX
ejpam-5256	65	7	clearly	clearly	ADV
ejpam-5256	65	8	see	see	VERB
ejpam-5256	65	9	that	that	SCONJ
ejpam-5256	65	10	∅	∅	NOUN
ejpam-5256	65	11	∈	∈	PROPN
ejpam-5256	65	12	sθ̃.	sθ̃.	PROPN
ejpam-5256	65	13	assume	assume	VERB
ejpam-5256	65	14	that	that	SCONJ
ejpam-5256	65	15	ai	ai	VERB
ejpam-5256	65	16	∈	∈	PRON
ejpam-5256	65	17	sθ̃	sθ̃	PRON
ejpam-5256	65	18	for	for	ADP
ejpam-5256	65	19	all	all	PRON
ejpam-5256	66	1	i	i	PRON
ejpam-5256	66	2	∈	∈	PROPN
ejpam-5256	66	3	j	j	X
ejpam-5256	66	4	.	.	PUNCT
ejpam-5256	67	1	let	let	VERB
ejpam-5256	67	2	x	x	PUNCT
ejpam-5256	67	3	∈	∈	PROPN
ejpam-5256	67	4	⋃	⋃	NOUN
ejpam-5256	67	5	i∈j	i∈j	NOUN
ejpam-5256	67	6	ai	ai	VERB
ejpam-5256	67	7	,	,	PUNCT
ejpam-5256	67	8	we	we	PRON
ejpam-5256	67	9	get	get	VERB
ejpam-5256	67	10	x	x	PUNCT
ejpam-5256	67	11	∈	∈	NOUN
ejpam-5256	67	12	ai	ai	VERB
ejpam-5256	67	13	for	for	ADP
ejpam-5256	67	14	some	some	PRON
ejpam-5256	67	15	i	i	PRON
ejpam-5256	67	16	∈	∈	PROPN
ejpam-5256	67	17	j	j	PROPN
ejpam-5256	67	18	.	.	PUNCT
ejpam-5256	68	1	by	by	ADP
ejpam-5256	68	2	the	the	DET
ejpam-5256	68	3	assumption	assumption	NOUN
ejpam-5256	68	4	,	,	PUNCT
ejpam-5256	68	5	there	there	PRON
ejpam-5256	68	6	exists	exist	VERB
ejpam-5256	68	7	m	m	PROPN
ejpam-5256	68	8	∈	∈	PROPN
ejpam-5256	68	9	µ	µ	PRON
ejpam-5256	68	10	such	such	ADJ
ejpam-5256	68	11	that	that	SCONJ
ejpam-5256	68	12	x	x	PUNCT
ejpam-5256	68	13	∈m	∈m	ADP
ejpam-5256	68	14	⊆	⊆	NUM
ejpam-5256	68	15	cσ(m	cσ(m	NOUN
ejpam-5256	68	16	)	)	PUNCT
ejpam-5256	68	17	∩mµ	∩mµ	VERB
ejpam-5256	69	1	⊆	⊆	NUM
ejpam-5256	69	2	ai	ai	VERB
ejpam-5256	69	3	⊆	⊆	NUM
ejpam-5256	69	4	⋃	⋃	NOUN
ejpam-5256	69	5	i∈j	i∈j	NOUN
ejpam-5256	69	6	ai	ai	VERB
ejpam-5256	69	7	,	,	PUNCT
ejpam-5256	69	8	where	where	SCONJ
ejpam-5256	69	9	mµ	mµ	ADP
ejpam-5256	69	10	=	=	PUNCT
ejpam-5256	69	11	∪{m	∪{m	PROPN
ejpam-5256	69	12	⊆	⊆	NUM
ejpam-5256	69	13	x	x	SYM
ejpam-5256	69	14	:	:	PUNCT
ejpam-5256	69	15	m	m	VERB
ejpam-5256	69	16	∈	∈	NOUN
ejpam-5256	69	17	µ	µ	X
ejpam-5256	69	18	}	}	PUNCT
ejpam-5256	69	19	.	.	PUNCT
ejpam-5256	70	1	thus	thus	ADV
ejpam-5256	70	2	,	,	PUNCT
ejpam-5256	70	3	⋃	⋃	PUNCT
ejpam-5256	70	4	i∈j	i∈j	NOUN
ejpam-5256	70	5	ai	ai	VERB
ejpam-5256	70	6	∈	∈	PROPN
ejpam-5256	70	7	sθ̃.	sθ̃.	NOUN
ejpam-5256	70	8	therefore	therefore	ADV
ejpam-5256	70	9	,	,	PUNCT
ejpam-5256	70	10	sθ̃	sθ̃	PRON
ejpam-5256	70	11	is	be	AUX
ejpam-5256	70	12	a	a	DET
ejpam-5256	70	13	generalized	generalized	ADJ
ejpam-5256	70	14	topology	topology	NOUN
ejpam-5256	70	15	on	on	ADP
ejpam-5256	70	16	x.	x.	NOUN
ejpam-5256	70	17	theorem	theorem	VERB
ejpam-5256	70	18	2	2	X
ejpam-5256	70	19	.	.	PUNCT
ejpam-5256	71	1	let	let	VERB
ejpam-5256	71	2	(	(	PUNCT
ejpam-5256	71	3	x,µ	x,µ	NOUN
ejpam-5256	71	4	)	)	PUNCT
ejpam-5256	71	5	be	be	VERB
ejpam-5256	71	6	a	a	DET
ejpam-5256	71	7	generalized	generalized	ADJ
ejpam-5256	71	8	topological	topological	ADJ
ejpam-5256	71	9	space	space	NOUN
ejpam-5256	71	10	.	.	PUNCT
ejpam-5256	72	1	then	then	ADV
ejpam-5256	72	2	sθ̃	sθ̃	PROPN
ejpam-5256	72	3	⊆	⊆	NUM
ejpam-5256	72	4	µ.	µ.	NOUN
ejpam-5256	72	5	proof	proof	NOUN
ejpam-5256	72	6	.	.	PUNCT
ejpam-5256	73	1	let	let	VERB
ejpam-5256	73	2	a	a	DET
ejpam-5256	73	3	⊆	⊆	NUM
ejpam-5256	73	4	x	x	NOUN
ejpam-5256	73	5	and	and	CCONJ
ejpam-5256	73	6	a	a	DET
ejpam-5256	73	7	∈	∈	PROPN
ejpam-5256	73	8	sθ̃.	sθ̃.	NOUN
ejpam-5256	73	9	if	if	SCONJ
ejpam-5256	73	10	a	a	DET
ejpam-5256	73	11	=	=	NOUN
ejpam-5256	73	12	∅	∅	NOUN
ejpam-5256	73	13	,	,	PUNCT
ejpam-5256	73	14	then	then	ADV
ejpam-5256	73	15	a	a	DET
ejpam-5256	73	16	∈	∈	PROPN
ejpam-5256	73	17	µ.	µ.	NOUN
ejpam-5256	73	18	if	if	SCONJ
ejpam-5256	73	19	a	a	DET
ejpam-5256	73	20	̸=	̸=	PROPN
ejpam-5256	73	21	∅	∅	NOUN
ejpam-5256	73	22	,	,	PUNCT
ejpam-5256	73	23	let	let	VERB
ejpam-5256	73	24	x	x	PUNCT
ejpam-5256	73	25	∈	∈	VERB
ejpam-5256	73	26	a.	a.	NOUN
ejpam-5256	73	27	since	since	SCONJ
ejpam-5256	73	28	a	a	DET
ejpam-5256	73	29	∈	∈	NOUN
ejpam-5256	73	30	sθ̃	sθ̃	NOUN
ejpam-5256	73	31	,	,	PUNCT
ejpam-5256	73	32	there	there	PRON
ejpam-5256	73	33	exists	exist	VERB
ejpam-5256	73	34	mx	mx	PROPN
ejpam-5256	73	35	∈	∈	PROPN
ejpam-5256	73	36	µ	µ	PRON
ejpam-5256	73	37	such	such	ADJ
ejpam-5256	73	38	that	that	SCONJ
ejpam-5256	73	39	x	x	SYM
ejpam-5256	73	40	∈	∈	PROPN
ejpam-5256	73	41	mx	mx	PROPN
ejpam-5256	73	42	⊆	⊆	NUM
ejpam-5256	73	43	cσ(mx	cσ(mx	PROPN
ejpam-5256	73	44	)	)	PUNCT
ejpam-5256	73	45	∩	∩	NOUN
ejpam-5256	73	46	mµ	mµ	VERB
ejpam-5256	73	47	⊆	⊆	NUM
ejpam-5256	73	48	a	a	PRON
ejpam-5256	73	49	for	for	ADP
ejpam-5256	73	50	each	each	DET
ejpam-5256	73	51	x	x	SYM
ejpam-5256	73	52	∈	∈	PROPN
ejpam-5256	73	53	a.	a.	NOUN
ejpam-5256	73	54	hence	hence	ADV
ejpam-5256	73	55	,	,	PUNCT
ejpam-5256	73	56	⋃	⋃	ADV
ejpam-5256	73	57	x∈a	x∈a	ADJ
ejpam-5256	73	58	{	{	PUNCT
ejpam-5256	73	59	x	x	NOUN
ejpam-5256	73	60	}	}	PUNCT
ejpam-5256	73	61	⊆	⊆	NUM
ejpam-5256	73	62	⋃	⋃	NOUN
ejpam-5256	73	63	x∈a	x∈a	ADJ
ejpam-5256	73	64	mx	mx	NOUN
ejpam-5256	73	65	⊆	⊆	NUM
ejpam-5256	73	66	a.	a.	NOUN
ejpam-5256	73	67	as	as	ADP
ejpam-5256	73	68	a	a	DET
ejpam-5256	73	69	=	=	SYM
ejpam-5256	73	70	⋃	⋃	NOUN
ejpam-5256	73	71	x∈a	x∈a	NOUN
ejpam-5256	73	72	{	{	PUNCT
ejpam-5256	73	73	x	x	NOUN
ejpam-5256	73	74	}	}	PUNCT
ejpam-5256	73	75	,	,	PUNCT
ejpam-5256	73	76	a	a	DET
ejpam-5256	73	77	⊆	⊆	NUM
ejpam-5256	73	78	⋃	⋃	NOUN
ejpam-5256	73	79	x∈a	x∈a	ADJ
ejpam-5256	73	80	mx	mx	NOUN
ejpam-5256	73	81	⊆	⊆	NUM
ejpam-5256	73	82	a.	a.	NOUN
ejpam-5256	73	83	consequently	consequently	ADV
ejpam-5256	73	84	,	,	PUNCT
ejpam-5256	73	85	a	a	DET
ejpam-5256	73	86	=	=	PUNCT
ejpam-5256	73	87	⋃	⋃	NOUN
ejpam-5256	73	88	x∈a	x∈a	ADJ
ejpam-5256	73	89	mx	mx	PROPN
ejpam-5256	73	90	∈	∈	PROPN
ejpam-5256	73	91	µ.	µ.	NOUN
ejpam-5256	73	92	therefore	therefore	ADV
ejpam-5256	73	93	,	,	PUNCT
ejpam-5256	73	94	sθ̃	sθ̃	PROPN
ejpam-5256	73	95	⊆	⊆	NUM
ejpam-5256	73	96	µ.	µ.	NOUN
ejpam-5256	73	97	corollary	corollary	ADJ
ejpam-5256	73	98	1	1	NUM
ejpam-5256	73	99	.	.	PUNCT
ejpam-5256	74	1	let	let	VERB
ejpam-5256	74	2	a	a	DET
ejpam-5256	74	3	be	be	AUX
ejpam-5256	74	4	a	a	DET
ejpam-5256	74	5	subset	subset	NOUN
ejpam-5256	74	6	of	of	ADP
ejpam-5256	74	7	a	a	DET
ejpam-5256	74	8	generalized	generalized	ADJ
ejpam-5256	74	9	topological	topological	ADJ
ejpam-5256	74	10	space	space	NOUN
ejpam-5256	74	11	(	(	PUNCT
ejpam-5256	74	12	x,µ	x,µ	NOUN
ejpam-5256	74	13	)	)	PUNCT
ejpam-5256	74	14	.	.	PUNCT
ejpam-5256	75	1	if	if	SCONJ
ejpam-5256	75	2	a	a	PRON
ejpam-5256	75	3	is	be	AUX
ejpam-5256	75	4	an	an	DET
ejpam-5256	75	5	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	75	6	set	set	NOUN
ejpam-5256	75	7	,	,	PUNCT
ejpam-5256	75	8	then	then	ADV
ejpam-5256	75	9	a	a	PRON
ejpam-5256	75	10	is	be	AUX
ejpam-5256	75	11	µ-closed	µ-close	VERB
ejpam-5256	75	12	.	.	PUNCT
ejpam-5256	76	1	theorem	theorem	NOUN
ejpam-5256	76	2	3	3	X
ejpam-5256	76	3	.	.	PUNCT
ejpam-5256	77	1	let	let	VERB
ejpam-5256	77	2	(	(	PUNCT
ejpam-5256	77	3	x,µ	x,µ	NOUN
ejpam-5256	77	4	)	)	PUNCT
ejpam-5256	77	5	be	be	VERB
ejpam-5256	77	6	a	a	DET
ejpam-5256	77	7	generalized	generalized	ADJ
ejpam-5256	77	8	topological	topological	ADJ
ejpam-5256	77	9	space	space	NOUN
ejpam-5256	77	10	.	.	PUNCT
ejpam-5256	78	1	then	then	ADV
ejpam-5256	78	2	,	,	PUNCT
ejpam-5256	78	3	θ̃	θ̃	PROPN
ejpam-5256	78	4	⊆	⊆	NUM
ejpam-5256	78	5	sθ̃.	sθ̃.	NOUN
ejpam-5256	78	6	proof	proof	NOUN
ejpam-5256	78	7	.	.	PUNCT
ejpam-5256	79	1	let	let	VERB
ejpam-5256	79	2	a	a	PRON
ejpam-5256	79	3	be	be	AUX
ejpam-5256	79	4	an	an	DET
ejpam-5256	79	5	arbitrary	arbitrary	ADJ
ejpam-5256	79	6	element	element	NOUN
ejpam-5256	79	7	in	in	ADP
ejpam-5256	79	8	θ̃.	θ̃.	NOUN
ejpam-5256	79	9	assume	assume	VERB
ejpam-5256	79	10	that	that	SCONJ
ejpam-5256	79	11	x	x	PUNCT
ejpam-5256	79	12	∈	∈	NOUN
ejpam-5256	79	13	a.	a.	NOUN
ejpam-5256	79	14	since	since	SCONJ
ejpam-5256	79	15	a	a	DET
ejpam-5256	79	16	∈	∈	PROPN
ejpam-5256	79	17	θ̃	θ̃	NOUN
ejpam-5256	79	18	,	,	PUNCT
ejpam-5256	79	19	there	there	PRON
ejpam-5256	79	20	exists	exist	VERB
ejpam-5256	79	21	m	m	PROPN
ejpam-5256	79	22	∈	∈	PROPN
ejpam-5256	79	23	µ	µ	PRON
ejpam-5256	79	24	such	such	ADJ
ejpam-5256	79	25	that	that	SCONJ
ejpam-5256	79	26	x	x	SYM
ejpam-5256	79	27	∈	∈	PROPN
ejpam-5256	79	28	m	m	NOUN
ejpam-5256	79	29	⊆	⊆	NUM
ejpam-5256	79	30	cµ(m	cµ(m	NOUN
ejpam-5256	79	31	)	)	PUNCT
ejpam-5256	79	32	∩mµ	∩mµ	VERB
ejpam-5256	79	33	⊆	⊆	NUM
ejpam-5256	79	34	a.	a.	NOUN
ejpam-5256	79	35	as	as	ADP
ejpam-5256	79	36	m	m	PROPN
ejpam-5256	79	37	⊆	⊆	NUM
ejpam-5256	79	38	cσ(m	cσ(m	NUM
ejpam-5256	79	39	)	)	PUNCT
ejpam-5256	79	40	⊆	⊆	NUM
ejpam-5256	79	41	cµ(m	cµ(m	NOUN
ejpam-5256	79	42	)	)	PUNCT
ejpam-5256	79	43	for	for	ADP
ejpam-5256	79	44	all	all	DET
ejpam-5256	79	45	x	x	SYM
ejpam-5256	79	46	∈	∈	NOUN
ejpam-5256	79	47	a.	a.	NOUN
ejpam-5256	79	48	accordingly	accordingly	ADV
ejpam-5256	79	49	,	,	PUNCT
ejpam-5256	79	50	a	a	DET
ejpam-5256	79	51	∈	∈	PROPN
ejpam-5256	79	52	sθ̃.	sθ̃.	NOUN
ejpam-5256	79	53	therefore	therefore	ADV
ejpam-5256	79	54	,	,	PUNCT
ejpam-5256	79	55	θ̃	θ̃	PROPN
ejpam-5256	79	56	⊆	⊆	NUM
ejpam-5256	79	57	sθ̃.	sθ̃.	NOUN
ejpam-5256	79	58	in	in	ADP
ejpam-5256	79	59	a	a	DET
ejpam-5256	79	60	generalized	generalized	ADJ
ejpam-5256	79	61	topological	topological	ADJ
ejpam-5256	79	62	space	space	NOUN
ejpam-5256	79	63	(	(	PUNCT
ejpam-5256	79	64	x,µ	x,µ	NOUN
ejpam-5256	79	65	)	)	PUNCT
ejpam-5256	79	66	,	,	PUNCT
ejpam-5256	79	67	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	79	68	sets	set	NOUN
ejpam-5256	79	69	may	may	AUX
ejpam-5256	79	70	not	not	PART
ejpam-5256	79	71	be	be	AUX
ejpam-5256	79	72	θ̃-open	θ̃-open	ADJ
ejpam-5256	79	73	sets	set	NOUN
ejpam-5256	79	74	as	as	ADP
ejpam-5256	79	75	the	the	DET
ejpam-5256	79	76	following	follow	VERB
ejpam-5256	79	77	example	example	NOUN
ejpam-5256	79	78	.	.	PUNCT
ejpam-5256	80	1	example	example	NOUN
ejpam-5256	81	1	1	1	NUM
ejpam-5256	81	2	.	.	PUNCT
ejpam-5256	81	3	let	let	VERB
ejpam-5256	81	4	x	x	PUNCT
ejpam-5256	81	5	=	=	PRON
ejpam-5256	81	6	{	{	PUNCT
ejpam-5256	81	7	a	a	PRON
ejpam-5256	81	8	,	,	PUNCT
ejpam-5256	81	9	b	b	NOUN
ejpam-5256	81	10	,	,	PUNCT
ejpam-5256	81	11	c	c	NOUN
ejpam-5256	81	12	,	,	PUNCT
ejpam-5256	81	13	d	d	NOUN
ejpam-5256	81	14	}	}	PUNCT
ejpam-5256	81	15	,	,	PUNCT
ejpam-5256	81	16	µ	µ	X
ejpam-5256	81	17	=	=	SYM
ejpam-5256	81	18	{	{	PUNCT
ejpam-5256	81	19	∅	∅	NOUN
ejpam-5256	81	20	,	,	PUNCT
ejpam-5256	81	21	{	{	PUNCT
ejpam-5256	81	22	a	a	X
ejpam-5256	81	23	}	}	PUNCT
ejpam-5256	81	24	,	,	PUNCT
ejpam-5256	81	25	{	{	PUNCT
ejpam-5256	81	26	b	b	NOUN
ejpam-5256	81	27	}	}	PUNCT
ejpam-5256	81	28	,	,	PUNCT
ejpam-5256	81	29	{	{	PUNCT
ejpam-5256	81	30	a	a	DET
ejpam-5256	81	31	,	,	PUNCT
ejpam-5256	81	32	b	b	NOUN
ejpam-5256	81	33	}	}	PUNCT
ejpam-5256	81	34	,	,	PUNCT
ejpam-5256	81	35	{	{	PUNCT
ejpam-5256	81	36	a	a	PRON
ejpam-5256	81	37	,	,	PUNCT
ejpam-5256	81	38	b	b	NOUN
ejpam-5256	81	39	,	,	PUNCT
ejpam-5256	81	40	c	c	NOUN
ejpam-5256	81	41	}	}	PUNCT
ejpam-5256	81	42	}	}	PUNCT
ejpam-5256	81	43	and	and	CCONJ
ejpam-5256	81	44	a	a	DET
ejpam-5256	81	45	=	=	X
ejpam-5256	81	46	{	{	PUNCT
ejpam-5256	81	47	b	b	NOUN
ejpam-5256	81	48	}	}	PUNCT
ejpam-5256	81	49	.	.	PUNCT
ejpam-5256	82	1	then	then	ADV
ejpam-5256	82	2	,	,	PUNCT
ejpam-5256	82	3	x	x	X
ejpam-5256	82	4	,	,	PUNCT
ejpam-5256	82	5	{	{	PUNCT
ejpam-5256	82	6	b	b	NOUN
ejpam-5256	82	7	,	,	PUNCT
ejpam-5256	82	8	c	c	NOUN
ejpam-5256	82	9	,	,	PUNCT
ejpam-5256	82	10	d	d	NOUN
ejpam-5256	82	11	}	}	PUNCT
ejpam-5256	82	12	,	,	PUNCT
ejpam-5256	82	13	{	{	PUNCT
ejpam-5256	82	14	a	a	PRON
ejpam-5256	82	15	,	,	PUNCT
ejpam-5256	82	16	c	c	NOUN
ejpam-5256	82	17	,	,	PUNCT
ejpam-5256	82	18	d	d	NOUN
ejpam-5256	82	19	}	}	PUNCT
ejpam-5256	82	20	,	,	PUNCT
ejpam-5256	82	21	{	{	PUNCT
ejpam-5256	82	22	c	c	X
ejpam-5256	82	23	,	,	PUNCT
ejpam-5256	82	24	d	d	NOUN
ejpam-5256	82	25	}	}	PUNCT
ejpam-5256	82	26	,	,	PUNCT
ejpam-5256	82	27	{	{	PUNCT
ejpam-5256	82	28	d	d	X
ejpam-5256	82	29	}	}	PUNCT
ejpam-5256	82	30	are	be	AUX
ejpam-5256	82	31	µ-closed	µ-close	VERB
ejpam-5256	82	32	and	and	CCONJ
ejpam-5256	82	33	mµ	mµ	VERB
ejpam-5256	82	34	=	=	PUNCT
ejpam-5256	82	35	{	{	PUNCT
ejpam-5256	82	36	a	a	DET
ejpam-5256	82	37	,	,	PUNCT
ejpam-5256	82	38	b	b	NOUN
ejpam-5256	82	39	,	,	PUNCT
ejpam-5256	82	40	c	c	NOUN
ejpam-5256	82	41	}	}	PUNCT
ejpam-5256	82	42	.	.	PUNCT
ejpam-5256	83	1	moreover	moreover	ADV
ejpam-5256	83	2	,	,	PUNCT
ejpam-5256	83	3	only	only	ADV
ejpam-5256	83	4	∅	∅	NOUN
ejpam-5256	83	5	,	,	PUNCT
ejpam-5256	83	6	{	{	PUNCT
ejpam-5256	83	7	a	a	X
ejpam-5256	83	8	}	}	PUNCT
ejpam-5256	83	9	,	,	PUNCT
ejpam-5256	83	10	{	{	PUNCT
ejpam-5256	83	11	b	b	NOUN
ejpam-5256	83	12	}	}	PUNCT
ejpam-5256	83	13	,	,	PUNCT
ejpam-5256	83	14	{	{	PUNCT
ejpam-5256	83	15	c	c	X
ejpam-5256	83	16	}	}	PUNCT
ejpam-5256	83	17	,	,	PUNCT
ejpam-5256	83	18	{	{	PUNCT
ejpam-5256	83	19	d	d	X
ejpam-5256	83	20	}	}	PUNCT
ejpam-5256	83	21	,	,	PUNCT
ejpam-5256	83	22	{	{	PUNCT
ejpam-5256	83	23	a	a	X
ejpam-5256	83	24	,	,	PUNCT
ejpam-5256	83	25	c	c	NOUN
ejpam-5256	83	26	}	}	PUNCT
ejpam-5256	83	27	,	,	PUNCT
ejpam-5256	83	28	{	{	PUNCT
ejpam-5256	83	29	a	a	PRON
ejpam-5256	83	30	,	,	PUNCT
ejpam-5256	83	31	d	d	NOUN
ejpam-5256	83	32	}	}	PUNCT
ejpam-5256	83	33	,	,	PUNCT
ejpam-5256	83	34	{	{	PUNCT
ejpam-5256	83	35	b	b	X
ejpam-5256	83	36	,	,	PUNCT
ejpam-5256	83	37	c	c	NOUN
ejpam-5256	83	38	}	}	PUNCT
ejpam-5256	83	39	,	,	PUNCT
ejpam-5256	83	40	{	{	PUNCT
ejpam-5256	83	41	b	b	X
ejpam-5256	83	42	,	,	PUNCT
ejpam-5256	83	43	d	d	NOUN
ejpam-5256	83	44	}	}	PUNCT
ejpam-5256	83	45	,	,	PUNCT
ejpam-5256	83	46	{	{	PUNCT
ejpam-5256	83	47	c	c	X
ejpam-5256	83	48	,	,	PUNCT
ejpam-5256	83	49	d	d	NOUN
ejpam-5256	83	50	}	}	PUNCT
ejpam-5256	83	51	,	,	PUNCT
ejpam-5256	83	52	{	{	PUNCT
ejpam-5256	83	53	a	a	DET
ejpam-5256	83	54	,	,	PUNCT
ejpam-5256	83	55	b	b	NOUN
ejpam-5256	83	56	,	,	PUNCT
ejpam-5256	83	57	c	c	NOUN
ejpam-5256	83	58	}	}	PUNCT
ejpam-5256	83	59	,	,	PUNCT
ejpam-5256	83	60	{	{	PUNCT
ejpam-5256	83	61	a	a	PRON
ejpam-5256	83	62	,	,	PUNCT
ejpam-5256	83	63	c	c	NOUN
ejpam-5256	83	64	,	,	PUNCT
ejpam-5256	83	65	d	d	NOUN
ejpam-5256	83	66	}	}	PUNCT
ejpam-5256	83	67	,	,	PUNCT
ejpam-5256	83	68	{	{	PUNCT
ejpam-5256	83	69	b	b	X
ejpam-5256	83	70	,	,	PUNCT
ejpam-5256	83	71	c	c	NOUN
ejpam-5256	83	72	,	,	PUNCT
ejpam-5256	83	73	d	d	NOUN
ejpam-5256	83	74	}	}	PUNCT
ejpam-5256	83	75	and	and	CCONJ
ejpam-5256	83	76	x	x	X
ejpam-5256	83	77	are	be	AUX
ejpam-5256	83	78	µ-semi	µ-semi	NOUN
ejpam-5256	83	79	-	-	ADJ
ejpam-5256	83	80	closed	closed	ADJ
ejpam-5256	83	81	sets	set	NOUN
ejpam-5256	83	82	.	.	PUNCT
ejpam-5256	84	1	consider	consider	VERB
ejpam-5256	84	2	b	b	NOUN
ejpam-5256	84	3	∈	∈	PROPN
ejpam-5256	84	4	a	a	PRON
ejpam-5256	84	5	,	,	PUNCT
ejpam-5256	84	6	then	then	ADV
ejpam-5256	84	7	there	there	PRON
ejpam-5256	84	8	exists	exist	VERB
ejpam-5256	84	9	{	{	PUNCT
ejpam-5256	84	10	b	b	X
ejpam-5256	84	11	}	}	PUNCT
ejpam-5256	84	12	∈	∈	PROPN
ejpam-5256	84	13	µ	µ	NOUN
ejpam-5256	84	14	such	such	ADJ
ejpam-5256	84	15	that	that	PRON
ejpam-5256	84	16	b	b	X
ejpam-5256	84	17	∈	∈	PROPN
ejpam-5256	84	18	{	{	PUNCT
ejpam-5256	84	19	b	b	NOUN
ejpam-5256	84	20	}	}	PUNCT
ejpam-5256	84	21	⊆	⊆	NUM
ejpam-5256	84	22	cσ({b})∩mµ	cσ({b})∩mµ	NOUN
ejpam-5256	84	23	=	=	SYM
ejpam-5256	84	24	{	{	PUNCT
ejpam-5256	84	25	b	b	NOUN
ejpam-5256	84	26	}	}	PUNCT
ejpam-5256	84	27	.	.	PUNCT
ejpam-5256	85	1	hence	hence	ADV
ejpam-5256	85	2	,	,	PUNCT
ejpam-5256	85	3	a	a	DET
ejpam-5256	85	4	∈	∈	PROPN
ejpam-5256	85	5	sθ̃.	sθ̃.	NOUN
ejpam-5256	85	6	consider	consider	VERB
ejpam-5256	85	7	each	each	DET
ejpam-5256	85	8	µ-open	µ-open	NOUN
ejpam-5256	85	9	set	set	VERB
ejpam-5256	85	10	m	m	VERB
ejpam-5256	85	11	containing	contain	VERB
ejpam-5256	85	12	b.	b.	PROPN
ejpam-5256	86	1	if	if	SCONJ
ejpam-5256	86	2	m	m	VERB
ejpam-5256	86	3	=	=	SYM
ejpam-5256	86	4	{	{	PUNCT
ejpam-5256	86	5	b	b	X
ejpam-5256	86	6	}	}	PUNCT
ejpam-5256	86	7	∈	∈	PROPN
ejpam-5256	86	8	µ	µ	NOUN
ejpam-5256	86	9	,	,	PUNCT
ejpam-5256	86	10	then	then	ADV
ejpam-5256	86	11	b	b	PROPN
ejpam-5256	86	12	∈	∈	PROPN
ejpam-5256	86	13	{	{	PUNCT
ejpam-5256	86	14	b	b	NOUN
ejpam-5256	86	15	}	}	PUNCT
ejpam-5256	86	16	⊆	⊆	NUM
ejpam-5256	86	17	cµ({b	cµ({b	NUM
ejpam-5256	86	18	}	}	PUNCT
ejpam-5256	86	19	)	)	PUNCT
ejpam-5256	86	20	∩mµ	∩mµ	PROPN
ejpam-5256	86	21	=	=	SYM
ejpam-5256	86	22	{	{	PUNCT
ejpam-5256	86	23	b	b	PROPN
ejpam-5256	86	24	,	,	PUNCT
ejpam-5256	86	25	c	c	NOUN
ejpam-5256	86	26	}	}	PUNCT
ejpam-5256	86	27	̸⊆	̸⊆	NOUN
ejpam-5256	86	28	{	{	PUNCT
ejpam-5256	86	29	b	b	NOUN
ejpam-5256	86	30	}	}	PUNCT
ejpam-5256	86	31	.	.	PUNCT
ejpam-5256	87	1	if	if	SCONJ
ejpam-5256	87	2	m	m	VERB
ejpam-5256	87	3	=	=	X
ejpam-5256	87	4	{	{	PUNCT
ejpam-5256	87	5	a	a	PRON
ejpam-5256	87	6	,	,	PUNCT
ejpam-5256	87	7	b	b	NOUN
ejpam-5256	87	8	}	}	PUNCT
ejpam-5256	87	9	∈	∈	PROPN
ejpam-5256	87	10	µ	µ	NOUN
ejpam-5256	87	11	,	,	PUNCT
ejpam-5256	87	12	then	then	ADV
ejpam-5256	87	13	b	b	PROPN
ejpam-5256	87	14	∈	∈	PROPN
ejpam-5256	87	15	{	{	PUNCT
ejpam-5256	87	16	a	a	PROPN
ejpam-5256	87	17	,	,	PUNCT
ejpam-5256	87	18	b	b	NOUN
ejpam-5256	87	19	}	}	PUNCT
ejpam-5256	87	20	⊆	⊆	NUM
ejpam-5256	87	21	cµ({a	cµ({a	NOUN
ejpam-5256	87	22	,	,	PUNCT
ejpam-5256	87	23	b	b	NOUN
ejpam-5256	87	24	}	}	PUNCT
ejpam-5256	87	25	)	)	PUNCT
ejpam-5256	87	26	∩mµ	∩mµ	PROPN
ejpam-5256	88	1	=	=	PUNCT
ejpam-5256	89	1	{	{	PUNCT
ejpam-5256	89	2	a	a	DET
ejpam-5256	89	3	,	,	PUNCT
ejpam-5256	89	4	b	b	NOUN
ejpam-5256	89	5	,	,	PUNCT
ejpam-5256	89	6	c	c	NOUN
ejpam-5256	89	7	}	}	PUNCT
ejpam-5256	89	8	̸⊆	̸⊆	NOUN
ejpam-5256	89	9	{	{	PUNCT
ejpam-5256	89	10	b	b	NOUN
ejpam-5256	89	11	}	}	PUNCT
ejpam-5256	89	12	.	.	PUNCT
ejpam-5256	90	1	if	if	SCONJ
ejpam-5256	90	2	m	m	VERB
ejpam-5256	90	3	=	=	X
ejpam-5256	90	4	{	{	PUNCT
ejpam-5256	90	5	a	a	PRON
ejpam-5256	90	6	,	,	PUNCT
ejpam-5256	90	7	b	b	NOUN
ejpam-5256	90	8	,	,	PUNCT
ejpam-5256	90	9	c	c	NOUN
ejpam-5256	90	10	}	}	PUNCT
ejpam-5256	90	11	∈	∈	PROPN
ejpam-5256	90	12	µ	µ	NOUN
ejpam-5256	90	13	,	,	PUNCT
ejpam-5256	90	14	then	then	ADV
ejpam-5256	90	15	b	b	PROPN
ejpam-5256	90	16	∈	∈	PROPN
ejpam-5256	90	17	{	{	PUNCT
ejpam-5256	90	18	a	a	PROPN
ejpam-5256	90	19	,	,	PUNCT
ejpam-5256	90	20	b	b	NOUN
ejpam-5256	90	21	,	,	PUNCT
ejpam-5256	90	22	c	c	NOUN
ejpam-5256	90	23	}	}	PUNCT
ejpam-5256	90	24	⊆	⊆	NUM
ejpam-5256	90	25	cµ({a	cµ({a	ADJ
ejpam-5256	90	26	,	,	PUNCT
ejpam-5256	90	27	b	b	NOUN
ejpam-5256	90	28	,	,	PUNCT
ejpam-5256	90	29	c	c	NOUN
ejpam-5256	90	30	}	}	PUNCT
ejpam-5256	90	31	)	)	PUNCT
ejpam-5256	90	32	∩mµ	∩mµ	PROPN
ejpam-5256	90	33	=	=	PUNCT
ejpam-5256	90	34	{	{	PUNCT
ejpam-5256	90	35	a	a	PRON
ejpam-5256	90	36	,	,	PUNCT
ejpam-5256	90	37	b	b	NOUN
ejpam-5256	90	38	,	,	PUNCT
ejpam-5256	90	39	c	c	NOUN
ejpam-5256	90	40	}	}	PUNCT
ejpam-5256	90	41	̸⊆	̸⊆	NOUN
ejpam-5256	90	42	{	{	PUNCT
ejpam-5256	90	43	b	b	NOUN
ejpam-5256	90	44	}	}	PUNCT
ejpam-5256	90	45	.	.	PUNCT
ejpam-5256	91	1	from	from	ADP
ejpam-5256	91	2	the	the	DET
ejpam-5256	91	3	above	above	ADJ
ejpam-5256	91	4	three	three	NUM
ejpam-5256	91	5	cases	case	NOUN
ejpam-5256	91	6	,	,	PUNCT
ejpam-5256	91	7	a	a	DET
ejpam-5256	91	8	̸∈	̸∈	PROPN
ejpam-5256	91	9	θ̃.	θ̃.	PROPN
ejpam-5256	91	10	therefore	therefore	ADV
ejpam-5256	91	11	,	,	PUNCT
ejpam-5256	91	12	sθ̃	sθ̃	PROPN
ejpam-5256	91	13	̸⊆	̸⊆	X
ejpam-5256	91	14	θ̃.	θ̃.	NOUN
ejpam-5256	91	15	by	by	ADP
ejpam-5256	91	16	the	the	DET
ejpam-5256	91	17	generalized	generalized	ADJ
ejpam-5256	91	18	topological	topological	ADJ
ejpam-5256	91	19	space	space	NOUN
ejpam-5256	91	20	(	(	PUNCT
ejpam-5256	91	21	x,µ	x,µ	NOUN
ejpam-5256	91	22	)	)	PUNCT
ejpam-5256	91	23	in	in	ADP
ejpam-5256	91	24	example	example	NOUN
ejpam-5256	91	25	1	1	NUM
ejpam-5256	91	26	,	,	PUNCT
ejpam-5256	91	27	we	we	PRON
ejpam-5256	91	28	can	can	AUX
ejpam-5256	91	29	a	a	PRON
ejpam-5256	91	30	is	be	AUX
ejpam-5256	91	31	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	91	32	but	but	CCONJ
ejpam-5256	91	33	not	not	PART
ejpam-5256	91	34	θ̃-open	θ̃-open	VERB
ejpam-5256	91	35	.	.	PUNCT
ejpam-5256	92	1	even	even	ADV
ejpam-5256	92	2	though	though	SCONJ
ejpam-5256	92	3	µ	µ	NOUN
ejpam-5256	92	4	is	be	AUX
ejpam-5256	92	5	a	a	DET
ejpam-5256	92	6	topology	topology	NOUN
ejpam-5256	92	7	on	on	ADP
ejpam-5256	92	8	x	x	SYM
ejpam-5256	92	9	,	,	PUNCT
ejpam-5256	92	10	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	92	11	sets	set	NOUN
ejpam-5256	92	12	may	may	AUX
ejpam-5256	92	13	not	not	PART
ejpam-5256	92	14	be	be	AUX
ejpam-5256	92	15	θ̃-open	θ̃-open	ADJ
ejpam-5256	92	16	as	as	ADP
ejpam-5256	92	17	in	in	ADP
ejpam-5256	92	18	the	the	DET
ejpam-5256	92	19	following	follow	VERB
ejpam-5256	92	20	example	example	NOUN
ejpam-5256	92	21	.	.	PUNCT
ejpam-5256	93	1	j.	j.	PROPN
ejpam-5256	93	2	khampakdee	khampakdee	PROPN
ejpam-5256	93	3	/	/	PUNCT
ejpam-5256	93	4	eur	eur	PROPN
ejpam-5256	93	5	.	.	PUNCT
ejpam-5256	94	1	j.	j.	PROPN
ejpam-5256	94	2	pure	pure	PROPN
ejpam-5256	94	3	appl	appl	PROPN
ejpam-5256	94	4	.	.	PROPN
ejpam-5256	94	5	math	math	PROPN
ejpam-5256	94	6	,	,	PUNCT
ejpam-5256	94	7	17	17	NUM
ejpam-5256	94	8	(	(	PUNCT
ejpam-5256	94	9	3	3	NUM
ejpam-5256	94	10	)	)	PUNCT
ejpam-5256	94	11	(	(	PUNCT
ejpam-5256	94	12	2024	2024	NUM
ejpam-5256	94	13	)	)	PUNCT
ejpam-5256	94	14	,	,	PUNCT
ejpam-5256	94	15	1869	1869	NUM
ejpam-5256	94	16	-	-	SYM
ejpam-5256	94	17	1876	1876	NUM
ejpam-5256	94	18	1872	1872	NUM
ejpam-5256	94	19	example	example	NOUN
ejpam-5256	95	1	2	2	NUM
ejpam-5256	95	2	.	.	PUNCT
ejpam-5256	96	1	let	let	VERB
ejpam-5256	96	2	x	x	PUNCT
ejpam-5256	96	3	=	=	PRON
ejpam-5256	96	4	{	{	PUNCT
ejpam-5256	96	5	a	a	PRON
ejpam-5256	96	6	,	,	PUNCT
ejpam-5256	96	7	b	b	NOUN
ejpam-5256	96	8	,	,	PUNCT
ejpam-5256	96	9	c	c	NOUN
ejpam-5256	96	10	,	,	PUNCT
ejpam-5256	96	11	d	d	NOUN
ejpam-5256	96	12	}	}	PUNCT
ejpam-5256	96	13	,	,	PUNCT
ejpam-5256	96	14	µ	µ	X
ejpam-5256	96	15	=	=	SYM
ejpam-5256	96	16	{	{	PUNCT
ejpam-5256	96	17	∅	∅	NOUN
ejpam-5256	96	18	,	,	PUNCT
ejpam-5256	96	19	{	{	PUNCT
ejpam-5256	96	20	a	a	X
ejpam-5256	96	21	}	}	PUNCT
ejpam-5256	96	22	,	,	PUNCT
ejpam-5256	96	23	{	{	PUNCT
ejpam-5256	96	24	b	b	NOUN
ejpam-5256	96	25	}	}	PUNCT
ejpam-5256	96	26	,	,	PUNCT
ejpam-5256	96	27	{	{	PUNCT
ejpam-5256	96	28	a	a	DET
ejpam-5256	96	29	,	,	PUNCT
ejpam-5256	96	30	b	b	NOUN
ejpam-5256	96	31	}	}	PUNCT
ejpam-5256	96	32	,	,	PUNCT
ejpam-5256	96	33	{	{	PUNCT
ejpam-5256	96	34	a	a	PRON
ejpam-5256	96	35	,	,	PUNCT
ejpam-5256	96	36	b	b	NOUN
ejpam-5256	96	37	,	,	PUNCT
ejpam-5256	96	38	c	c	NOUN
ejpam-5256	96	39	}	}	PUNCT
ejpam-5256	96	40	,	,	PUNCT
ejpam-5256	96	41	x	x	NOUN
ejpam-5256	96	42	}	}	PUNCT
ejpam-5256	96	43	and	and	CCONJ
ejpam-5256	96	44	a	a	DET
ejpam-5256	96	45	=	=	X
ejpam-5256	96	46	{	{	PUNCT
ejpam-5256	96	47	b	b	NOUN
ejpam-5256	96	48	}	}	PUNCT
ejpam-5256	96	49	.	.	PUNCT
ejpam-5256	97	1	clearly	clearly	ADV
ejpam-5256	97	2	,	,	PUNCT
ejpam-5256	97	3	x	x	X
ejpam-5256	97	4	,	,	PUNCT
ejpam-5256	97	5	{	{	PUNCT
ejpam-5256	97	6	b	b	NOUN
ejpam-5256	97	7	,	,	PUNCT
ejpam-5256	97	8	c	c	NOUN
ejpam-5256	97	9	,	,	PUNCT
ejpam-5256	97	10	d	d	NOUN
ejpam-5256	97	11	}	}	PUNCT
ejpam-5256	97	12	,	,	PUNCT
ejpam-5256	97	13	{	{	PUNCT
ejpam-5256	97	14	a	a	PRON
ejpam-5256	97	15	,	,	PUNCT
ejpam-5256	97	16	c	c	NOUN
ejpam-5256	97	17	,	,	PUNCT
ejpam-5256	97	18	d	d	NOUN
ejpam-5256	97	19	}	}	PUNCT
ejpam-5256	97	20	,	,	PUNCT
ejpam-5256	97	21	{	{	PUNCT
ejpam-5256	97	22	c	c	X
ejpam-5256	97	23	,	,	PUNCT
ejpam-5256	97	24	d	d	NOUN
ejpam-5256	97	25	}	}	PUNCT
ejpam-5256	97	26	,	,	PUNCT
ejpam-5256	97	27	{	{	PUNCT
ejpam-5256	97	28	d	d	X
ejpam-5256	97	29	}	}	PUNCT
ejpam-5256	97	30	,	,	PUNCT
ejpam-5256	97	31	∅	∅	NOUN
ejpam-5256	97	32	are	be	AUX
ejpam-5256	97	33	µ-closed	µ-close	VERB
ejpam-5256	97	34	and	and	CCONJ
ejpam-5256	97	35	mµ	mµ	VERB
ejpam-5256	97	36	=	=	NOUN
ejpam-5256	97	37	x.	x.	NOUN
ejpam-5256	97	38	moreover	moreover	ADV
ejpam-5256	97	39	,	,	PUNCT
ejpam-5256	97	40	only	only	ADV
ejpam-5256	97	41	∅	∅	NOUN
ejpam-5256	97	42	,	,	PUNCT
ejpam-5256	97	43	{	{	PUNCT
ejpam-5256	97	44	a	a	X
ejpam-5256	97	45	}	}	PUNCT
ejpam-5256	97	46	,	,	PUNCT
ejpam-5256	97	47	{	{	PUNCT
ejpam-5256	97	48	b	b	NOUN
ejpam-5256	97	49	}	}	PUNCT
ejpam-5256	97	50	,	,	PUNCT
ejpam-5256	97	51	{	{	PUNCT
ejpam-5256	97	52	c	c	X
ejpam-5256	97	53	}	}	PUNCT
ejpam-5256	97	54	,	,	PUNCT
ejpam-5256	97	55	{	{	PUNCT
ejpam-5256	97	56	d	d	X
ejpam-5256	97	57	}	}	PUNCT
ejpam-5256	97	58	,	,	PUNCT
ejpam-5256	97	59	{	{	PUNCT
ejpam-5256	97	60	a	a	X
ejpam-5256	97	61	,	,	PUNCT
ejpam-5256	97	62	c	c	NOUN
ejpam-5256	97	63	}	}	PUNCT
ejpam-5256	97	64	,	,	PUNCT
ejpam-5256	97	65	{	{	PUNCT
ejpam-5256	97	66	a	a	PRON
ejpam-5256	97	67	,	,	PUNCT
ejpam-5256	97	68	d	d	NOUN
ejpam-5256	97	69	}	}	PUNCT
ejpam-5256	97	70	,	,	PUNCT
ejpam-5256	97	71	{	{	PUNCT
ejpam-5256	97	72	b	b	X
ejpam-5256	97	73	,	,	PUNCT
ejpam-5256	97	74	c	c	NOUN
ejpam-5256	97	75	}	}	PUNCT
ejpam-5256	97	76	,	,	PUNCT
ejpam-5256	97	77	{	{	PUNCT
ejpam-5256	97	78	b	b	NOUN
ejpam-5256	97	79	,	,	PUNCT
ejpam-5256	97	80	d},{c	d},{c	INTJ
ejpam-5256	97	81	,	,	PUNCT
ejpam-5256	97	82	d	d	NOUN
ejpam-5256	97	83	}	}	PUNCT
ejpam-5256	97	84	,	,	PUNCT
ejpam-5256	97	85	{	{	PUNCT
ejpam-5256	97	86	a	a	PRON
ejpam-5256	97	87	,	,	PUNCT
ejpam-5256	97	88	c	c	NOUN
ejpam-5256	97	89	,	,	PUNCT
ejpam-5256	97	90	d	d	NOUN
ejpam-5256	97	91	}	}	PUNCT
ejpam-5256	97	92	,	,	PUNCT
ejpam-5256	97	93	{	{	PUNCT
ejpam-5256	97	94	b	b	X
ejpam-5256	97	95	,	,	PUNCT
ejpam-5256	97	96	c	c	NOUN
ejpam-5256	97	97	,	,	PUNCT
ejpam-5256	97	98	d	d	NOUN
ejpam-5256	97	99	}	}	PUNCT
ejpam-5256	97	100	,	,	PUNCT
ejpam-5256	97	101	x	x	PRON
ejpam-5256	97	102	are	be	AUX
ejpam-5256	97	103	µ-semi	µ-semi	NOUN
ejpam-5256	97	104	-	-	ADJ
ejpam-5256	97	105	closed	closed	ADJ
ejpam-5256	97	106	sets	set	NOUN
ejpam-5256	97	107	.	.	PUNCT
ejpam-5256	98	1	consider	consider	VERB
ejpam-5256	98	2	b	b	NOUN
ejpam-5256	98	3	∈	∈	PROPN
ejpam-5256	98	4	a	a	PRON
ejpam-5256	98	5	,	,	PUNCT
ejpam-5256	98	6	then	then	ADV
ejpam-5256	98	7	there	there	PRON
ejpam-5256	98	8	exists	exist	VERB
ejpam-5256	98	9	{	{	PUNCT
ejpam-5256	98	10	b	b	X
ejpam-5256	98	11	}	}	PUNCT
ejpam-5256	98	12	∈	∈	PROPN
ejpam-5256	98	13	µ	µ	NOUN
ejpam-5256	98	14	such	such	ADJ
ejpam-5256	99	1	that	that	DET
ejpam-5256	99	2	b	b	X
ejpam-5256	99	3	∈	∈	PROPN
ejpam-5256	99	4	{	{	PUNCT
ejpam-5256	99	5	b	b	NOUN
ejpam-5256	99	6	}	}	PUNCT
ejpam-5256	99	7	⊆	⊆	NUM
ejpam-5256	99	8	cσ({b	cσ({b	NUM
ejpam-5256	99	9	}	}	PUNCT
ejpam-5256	99	10	)	)	PUNCT
ejpam-5256	99	11	∩mµ	∩mµ	PROPN
ejpam-5256	99	12	=	=	SYM
ejpam-5256	99	13	{	{	PUNCT
ejpam-5256	99	14	b	b	NOUN
ejpam-5256	99	15	}	}	PUNCT
ejpam-5256	99	16	.	.	PUNCT
ejpam-5256	100	1	thus	thus	ADV
ejpam-5256	100	2	,	,	PUNCT
ejpam-5256	100	3	a	a	DET
ejpam-5256	100	4	∈	∈	PROPN
ejpam-5256	100	5	sθ̃.	sθ̃.	NOUN
ejpam-5256	100	6	since	since	SCONJ
ejpam-5256	100	7	{	{	PUNCT
ejpam-5256	100	8	b	b	NOUN
ejpam-5256	100	9	}	}	PUNCT
ejpam-5256	100	10	,	,	PUNCT
ejpam-5256	100	11	{	{	PUNCT
ejpam-5256	100	12	a	a	DET
ejpam-5256	100	13	,	,	PUNCT
ejpam-5256	100	14	b	b	NOUN
ejpam-5256	100	15	}	}	PUNCT
ejpam-5256	100	16	,	,	PUNCT
ejpam-5256	100	17	{	{	PUNCT
ejpam-5256	100	18	a	a	DET
ejpam-5256	100	19	,	,	PUNCT
ejpam-5256	100	20	b	b	NOUN
ejpam-5256	100	21	,	,	PUNCT
ejpam-5256	100	22	c	c	NOUN
ejpam-5256	100	23	}	}	PUNCT
ejpam-5256	100	24	,	,	PUNCT
ejpam-5256	100	25	x	x	PRON
ejpam-5256	100	26	are	be	AUX
ejpam-5256	100	27	µ-open	µ-open	NOUN
ejpam-5256	100	28	sets	set	NOUN
ejpam-5256	100	29	containing	contain	VERB
ejpam-5256	100	30	b.	b.	PROPN
ejpam-5256	100	31	consider	consider	VERB
ejpam-5256	100	32	each	each	DET
ejpam-5256	100	33	µ-open	µ-open	NOUN
ejpam-5256	100	34	set	set	VERB
ejpam-5256	100	35	m	m	AUX
ejpam-5256	100	36	containing	contain	VERB
ejpam-5256	100	37	b.	b.	PROPN
ejpam-5256	101	1	if	if	SCONJ
ejpam-5256	101	2	m	m	VERB
ejpam-5256	101	3	=	=	SYM
ejpam-5256	101	4	{	{	PUNCT
ejpam-5256	101	5	b	b	X
ejpam-5256	101	6	}	}	PUNCT
ejpam-5256	101	7	∈	∈	PROPN
ejpam-5256	101	8	µ	µ	NOUN
ejpam-5256	101	9	,	,	PUNCT
ejpam-5256	101	10	then	then	ADV
ejpam-5256	101	11	b	b	PROPN
ejpam-5256	101	12	∈	∈	PROPN
ejpam-5256	101	13	{	{	PUNCT
ejpam-5256	101	14	b	b	NOUN
ejpam-5256	101	15	}	}	PUNCT
ejpam-5256	101	16	⊆	⊆	NUM
ejpam-5256	101	17	cµ({b	cµ({b	NUM
ejpam-5256	101	18	}	}	PUNCT
ejpam-5256	101	19	)	)	PUNCT
ejpam-5256	101	20	∩mµ	∩mµ	PROPN
ejpam-5256	101	21	=	=	SYM
ejpam-5256	101	22	{	{	PUNCT
ejpam-5256	101	23	b	b	PROPN
ejpam-5256	101	24	,	,	PUNCT
ejpam-5256	101	25	c	c	NOUN
ejpam-5256	101	26	,	,	PUNCT
ejpam-5256	101	27	d	d	NOUN
ejpam-5256	101	28	}	}	PUNCT
ejpam-5256	101	29	̸⊆	̸⊆	NOUN
ejpam-5256	101	30	{	{	PUNCT
ejpam-5256	101	31	b	b	NOUN
ejpam-5256	101	32	}	}	PUNCT
ejpam-5256	101	33	.	.	PUNCT
ejpam-5256	102	1	if	if	SCONJ
ejpam-5256	102	2	m	m	VERB
ejpam-5256	102	3	=	=	X
ejpam-5256	102	4	{	{	PUNCT
ejpam-5256	102	5	a	a	PRON
ejpam-5256	102	6	,	,	PUNCT
ejpam-5256	102	7	b	b	NOUN
ejpam-5256	102	8	}	}	PUNCT
ejpam-5256	102	9	∈	∈	PROPN
ejpam-5256	102	10	µ	µ	NOUN
ejpam-5256	102	11	,	,	PUNCT
ejpam-5256	102	12	then	then	ADV
ejpam-5256	102	13	b	b	PROPN
ejpam-5256	102	14	∈	∈	PROPN
ejpam-5256	102	15	{	{	PUNCT
ejpam-5256	102	16	a	a	PROPN
ejpam-5256	102	17	,	,	PUNCT
ejpam-5256	102	18	b	b	NOUN
ejpam-5256	102	19	}	}	PUNCT
ejpam-5256	102	20	⊆	⊆	NUM
ejpam-5256	102	21	cµ({a	cµ({a	NOUN
ejpam-5256	102	22	,	,	PUNCT
ejpam-5256	102	23	b	b	NOUN
ejpam-5256	102	24	}	}	PUNCT
ejpam-5256	102	25	)	)	PUNCT
ejpam-5256	102	26	∩mµ	∩mµ	PROPN
ejpam-5256	103	1	=	=	SYM
ejpam-5256	103	2	x	x	SYM
ejpam-5256	103	3	̸⊆	̸⊆	NOUN
ejpam-5256	103	4	{	{	PUNCT
ejpam-5256	103	5	b	b	NOUN
ejpam-5256	103	6	}	}	PUNCT
ejpam-5256	103	7	.	.	PUNCT
ejpam-5256	104	1	if	if	SCONJ
ejpam-5256	104	2	m	m	VERB
ejpam-5256	104	3	=	=	X
ejpam-5256	104	4	{	{	PUNCT
ejpam-5256	104	5	a	a	PRON
ejpam-5256	104	6	,	,	PUNCT
ejpam-5256	104	7	b	b	NOUN
ejpam-5256	104	8	,	,	PUNCT
ejpam-5256	104	9	c	c	NOUN
ejpam-5256	104	10	}	}	PUNCT
ejpam-5256	104	11	∈	∈	PROPN
ejpam-5256	104	12	µ	µ	NOUN
ejpam-5256	104	13	,	,	PUNCT
ejpam-5256	104	14	then	then	ADV
ejpam-5256	104	15	b	b	PROPN
ejpam-5256	104	16	∈	∈	PROPN
ejpam-5256	104	17	{	{	PUNCT
ejpam-5256	104	18	a	a	PROPN
ejpam-5256	104	19	,	,	PUNCT
ejpam-5256	104	20	b	b	NOUN
ejpam-5256	104	21	,	,	PUNCT
ejpam-5256	104	22	c	c	NOUN
ejpam-5256	104	23	}	}	PUNCT
ejpam-5256	104	24	⊆	⊆	NUM
ejpam-5256	104	25	cµ({a	cµ({a	ADJ
ejpam-5256	104	26	,	,	PUNCT
ejpam-5256	104	27	b	b	NOUN
ejpam-5256	104	28	,	,	PUNCT
ejpam-5256	104	29	c	c	NOUN
ejpam-5256	104	30	}	}	PUNCT
ejpam-5256	104	31	)	)	PUNCT
ejpam-5256	104	32	∩mµ	∩mµ	PROPN
ejpam-5256	105	1	=	=	SYM
ejpam-5256	105	2	x	x	SYM
ejpam-5256	105	3	̸⊆	̸⊆	NOUN
ejpam-5256	105	4	{	{	PUNCT
ejpam-5256	105	5	b	b	NOUN
ejpam-5256	105	6	}	}	PUNCT
ejpam-5256	105	7	.	.	PUNCT
ejpam-5256	106	1	if	if	SCONJ
ejpam-5256	106	2	m	m	ADV
ejpam-5256	106	3	=	=	PUNCT
ejpam-5256	106	4	x	x	SYM
ejpam-5256	106	5	∈	∈	PROPN
ejpam-5256	106	6	µ	µ	NOUN
ejpam-5256	106	7	,	,	PUNCT
ejpam-5256	106	8	then	then	ADV
ejpam-5256	106	9	b	b	PROPN
ejpam-5256	106	10	∈	∈	PROPN
ejpam-5256	106	11	x	x	PUNCT
ejpam-5256	106	12	⊆	⊆	NUM
ejpam-5256	106	13	cµ(x	cµ(x	ADJ
ejpam-5256	106	14	)	)	PUNCT
ejpam-5256	106	15	∩mµ	∩mµ	NOUN
ejpam-5256	106	16	=	=	SYM
ejpam-5256	106	17	x	x	SYM
ejpam-5256	106	18	̸⊆	̸⊆	NOUN
ejpam-5256	106	19	{	{	PUNCT
ejpam-5256	106	20	b	b	NOUN
ejpam-5256	106	21	}	}	PUNCT
ejpam-5256	106	22	.	.	PUNCT
ejpam-5256	107	1	from	from	ADP
ejpam-5256	107	2	the	the	DET
ejpam-5256	107	3	above	above	ADJ
ejpam-5256	107	4	four	four	NUM
ejpam-5256	107	5	cases	case	NOUN
ejpam-5256	107	6	,	,	PUNCT
ejpam-5256	107	7	a	a	DET
ejpam-5256	107	8	̸∈	̸∈	PROPN
ejpam-5256	107	9	θ̃.	θ̃.	PROPN
ejpam-5256	107	10	therefore	therefore	ADV
ejpam-5256	107	11	,	,	PUNCT
ejpam-5256	107	12	sθ̃	sθ̃	NOUN
ejpam-5256	107	13	̸⊆	̸⊆	PUNCT
ejpam-5256	107	14	θ̃	θ̃	NOUN
ejpam-5256	107	15	even	even	ADV
ejpam-5256	107	16	though	though	SCONJ
ejpam-5256	107	17	µ	µ	NOUN
ejpam-5256	107	18	is	be	AUX
ejpam-5256	107	19	a	a	DET
ejpam-5256	107	20	topology	topology	NOUN
ejpam-5256	107	21	on	on	ADP
ejpam-5256	107	22	x.	x.	PROPN
ejpam-5256	107	23	corollary	corollary	PROPN
ejpam-5256	107	24	2	2	X
ejpam-5256	107	25	.	.	PUNCT
ejpam-5256	108	1	let	let	VERB
ejpam-5256	108	2	(	(	PUNCT
ejpam-5256	108	3	x,µ	x,µ	NOUN
ejpam-5256	108	4	)	)	PUNCT
ejpam-5256	108	5	be	be	VERB
ejpam-5256	108	6	a	a	DET
ejpam-5256	108	7	generalized	generalized	ADJ
ejpam-5256	108	8	topological	topological	ADJ
ejpam-5256	108	9	space	space	NOUN
ejpam-5256	108	10	.	.	PUNCT
ejpam-5256	109	1	then	then	ADV
ejpam-5256	109	2	θ	θ	PROPN
ejpam-5256	109	3	⊆	⊆	X
ejpam-5256	109	4	θ̃	θ̃	PROPN
ejpam-5256	109	5	⊆	⊆	NUM
ejpam-5256	109	6	sθ̃	sθ̃	X
ejpam-5256	109	7	⊆	⊆	NUM
ejpam-5256	109	8	µ.	µ.	NOUN
ejpam-5256	109	9	in	in	ADP
ejpam-5256	109	10	generalized	generalized	ADJ
ejpam-5256	109	11	topological	topological	ADJ
ejpam-5256	109	12	spaces	space	NOUN
ejpam-5256	109	13	,	,	PUNCT
ejpam-5256	109	14	µ-open	µ-open	PROPN
ejpam-5256	109	15	sets	set	NOUN
ejpam-5256	109	16	need	need	AUX
ejpam-5256	109	17	not	not	PART
ejpam-5256	109	18	be	be	AUX
ejpam-5256	109	19	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	109	20	as	as	SCONJ
ejpam-5256	109	21	can	can	AUX
ejpam-5256	109	22	be	be	AUX
ejpam-5256	109	23	seen	see	VERB
ejpam-5256	109	24	from	from	ADP
ejpam-5256	109	25	the	the	DET
ejpam-5256	109	26	following	follow	VERB
ejpam-5256	109	27	example	example	NOUN
ejpam-5256	109	28	.	.	PUNCT
ejpam-5256	110	1	example	example	NOUN
ejpam-5256	111	1	3	3	NUM
ejpam-5256	111	2	.	.	PUNCT
ejpam-5256	111	3	by	by	ADP
ejpam-5256	111	4	the	the	DET
ejpam-5256	111	5	generalized	generalized	ADJ
ejpam-5256	111	6	topological	topological	ADJ
ejpam-5256	111	7	space	space	NOUN
ejpam-5256	111	8	(	(	PUNCT
ejpam-5256	111	9	x,µ	x,µ	NOUN
ejpam-5256	111	10	)	)	PUNCT
ejpam-5256	111	11	in	in	ADP
ejpam-5256	111	12	example	example	NOUN
ejpam-5256	111	13	2	2	X
ejpam-5256	111	14	.	.	PUNCT
ejpam-5256	111	15	let	let	VERB
ejpam-5256	111	16	a	a	DET
ejpam-5256	111	17	=	=	X
ejpam-5256	111	18	{	{	PUNCT
ejpam-5256	111	19	a	a	PROPN
ejpam-5256	111	20	,	,	PUNCT
ejpam-5256	111	21	b	b	NOUN
ejpam-5256	111	22	,	,	PUNCT
ejpam-5256	111	23	c	c	NOUN
ejpam-5256	111	24	}	}	PUNCT
ejpam-5256	111	25	,	,	PUNCT
ejpam-5256	111	26	then	then	ADV
ejpam-5256	111	27	a	a	DET
ejpam-5256	111	28	∈	∈	PROPN
ejpam-5256	111	29	µ.	µ.	NOUN
ejpam-5256	111	30	consider	consider	VERB
ejpam-5256	111	31	c	c	NOUN
ejpam-5256	111	32	∈	∈	PROPN
ejpam-5256	111	33	a.	a.	NOUN
ejpam-5256	111	34	since	since	SCONJ
ejpam-5256	111	35	only	only	ADV
ejpam-5256	111	36	{	{	PUNCT
ejpam-5256	111	37	a	a	PRON
ejpam-5256	111	38	,	,	PUNCT
ejpam-5256	111	39	b	b	NOUN
ejpam-5256	111	40	,	,	PUNCT
ejpam-5256	111	41	c	c	NOUN
ejpam-5256	111	42	}	}	PUNCT
ejpam-5256	111	43	and	and	CCONJ
ejpam-5256	111	44	x	x	PRON
ejpam-5256	111	45	are	be	AUX
ejpam-5256	111	46	µ-open	µ-open	NOUN
ejpam-5256	111	47	sets	set	NOUN
ejpam-5256	111	48	containing	contain	VERB
ejpam-5256	111	49	c.	c.	NOUN
ejpam-5256	111	50	consider	consider	VERB
ejpam-5256	111	51	each	each	DET
ejpam-5256	111	52	µ-open	µ-open	NOUN
ejpam-5256	111	53	set	set	VERB
ejpam-5256	111	54	m	m	VERB
ejpam-5256	111	55	containing	contain	VERB
ejpam-5256	111	56	c.	c.	NOUN
ejpam-5256	111	57	if	if	SCONJ
ejpam-5256	111	58	m	m	VERB
ejpam-5256	111	59	=	=	X
ejpam-5256	111	60	{	{	PUNCT
ejpam-5256	111	61	a	a	PRON
ejpam-5256	111	62	,	,	PUNCT
ejpam-5256	111	63	b	b	NOUN
ejpam-5256	111	64	,	,	PUNCT
ejpam-5256	111	65	c	c	NOUN
ejpam-5256	111	66	}	}	PUNCT
ejpam-5256	111	67	∈	∈	PROPN
ejpam-5256	111	68	µ	µ	NOUN
ejpam-5256	111	69	,	,	PUNCT
ejpam-5256	111	70	then	then	ADV
ejpam-5256	111	71	c	c	PROPN
ejpam-5256	111	72	∈	∈	PROPN
ejpam-5256	111	73	{	{	PUNCT
ejpam-5256	111	74	a	a	PROPN
ejpam-5256	111	75	,	,	PUNCT
ejpam-5256	111	76	b	b	NOUN
ejpam-5256	111	77	,	,	PUNCT
ejpam-5256	111	78	c	c	NOUN
ejpam-5256	111	79	}	}	PUNCT
ejpam-5256	111	80	⊆	⊆	NUM
ejpam-5256	111	81	cµ({a	cµ({a	ADJ
ejpam-5256	111	82	,	,	PUNCT
ejpam-5256	111	83	b	b	NOUN
ejpam-5256	111	84	,	,	PUNCT
ejpam-5256	111	85	c	c	NOUN
ejpam-5256	111	86	}	}	PUNCT
ejpam-5256	111	87	)	)	PUNCT
ejpam-5256	111	88	∩mµ	∩mµ	PROPN
ejpam-5256	112	1	=	=	SYM
ejpam-5256	112	2	x	x	SYM
ejpam-5256	112	3	̸⊆	̸⊆	X
ejpam-5256	112	4	{	{	PUNCT
ejpam-5256	112	5	a	a	DET
ejpam-5256	112	6	,	,	PUNCT
ejpam-5256	112	7	b	b	NOUN
ejpam-5256	112	8	,	,	PUNCT
ejpam-5256	112	9	c	c	NOUN
ejpam-5256	112	10	}	}	PUNCT
ejpam-5256	112	11	.	.	PUNCT
ejpam-5256	113	1	if	if	SCONJ
ejpam-5256	113	2	m	m	ADV
ejpam-5256	113	3	=	=	PUNCT
ejpam-5256	113	4	x	x	SYM
ejpam-5256	113	5	∈	∈	PROPN
ejpam-5256	113	6	µ	µ	NOUN
ejpam-5256	113	7	,	,	PUNCT
ejpam-5256	113	8	then	then	ADV
ejpam-5256	113	9	c	c	NOUN
ejpam-5256	113	10	∈	∈	PROPN
ejpam-5256	113	11	x	x	PUNCT
ejpam-5256	113	12	⊆	⊆	NUM
ejpam-5256	113	13	cµ(x	cµ(x	ADJ
ejpam-5256	113	14	)	)	PUNCT
ejpam-5256	113	15	∩mµ	∩mµ	NOUN
ejpam-5256	113	16	=	=	SYM
ejpam-5256	113	17	x	x	SYM
ejpam-5256	113	18	̸⊆	̸⊆	X
ejpam-5256	113	19	{	{	PUNCT
ejpam-5256	113	20	a	a	DET
ejpam-5256	113	21	,	,	PUNCT
ejpam-5256	113	22	b	b	NOUN
ejpam-5256	113	23	,	,	PUNCT
ejpam-5256	113	24	c	c	NOUN
ejpam-5256	113	25	}	}	PUNCT
ejpam-5256	113	26	.	.	PUNCT
ejpam-5256	114	1	from	from	ADP
ejpam-5256	114	2	the	the	DET
ejpam-5256	114	3	above	above	ADJ
ejpam-5256	114	4	two	two	NUM
ejpam-5256	114	5	cases	case	NOUN
ejpam-5256	114	6	,	,	PUNCT
ejpam-5256	114	7	a	a	DET
ejpam-5256	114	8	̸∈	̸∈	PROPN
ejpam-5256	114	9	sθ̃.	sθ̃.	PROPN
ejpam-5256	114	10	in	in	ADP
ejpam-5256	114	11	[	[	X
ejpam-5256	114	12	4	4	NUM
ejpam-5256	114	13	]	]	PUNCT
ejpam-5256	114	14	,	,	PUNCT
ejpam-5256	114	15	min	min	PROPN
ejpam-5256	114	16	defined	define	VERB
ejpam-5256	114	17	the	the	DET
ejpam-5256	114	18	set	set	NOUN
ejpam-5256	114	19	γθ̃(a	γθ̃(a	NOUN
ejpam-5256	114	20	)	)	PUNCT
ejpam-5256	114	21	,	,	PUNCT
ejpam-5256	114	22	where	where	SCONJ
ejpam-5256	114	23	a	a	PRON
ejpam-5256	114	24	is	be	AUX
ejpam-5256	114	25	a	a	DET
ejpam-5256	114	26	subset	subset	NOUN
ejpam-5256	114	27	of	of	ADP
ejpam-5256	114	28	a	a	DET
ejpam-5256	114	29	generalized	generalized	ADJ
ejpam-5256	114	30	topological	topological	ADJ
ejpam-5256	114	31	space	space	NOUN
ejpam-5256	114	32	(	(	PUNCT
ejpam-5256	114	33	x,µ	x,µ	NOUN
ejpam-5256	114	34	)	)	PUNCT
ejpam-5256	114	35	as	as	SCONJ
ejpam-5256	114	36	follows	follow	VERB
ejpam-5256	114	37	:	:	PUNCT
ejpam-5256	114	38	x	x	SYM
ejpam-5256	114	39	∈	∈	NOUN
ejpam-5256	114	40	γθ̃(a	γθ̃(a	VERB
ejpam-5256	114	41	)	)	PUNCT
ejpam-5256	114	42	if	if	SCONJ
ejpam-5256	114	43	cµ(g)∩mµ	cµ(g)∩mµ	PROPN
ejpam-5256	114	44	∩a	∩a	PROPN
ejpam-5256	114	45	̸=	̸=	PROPN
ejpam-5256	114	46	∅	∅	NOUN
ejpam-5256	114	47	for	for	ADP
ejpam-5256	114	48	each	each	DET
ejpam-5256	114	49	g	g	PROPN
ejpam-5256	114	50	∈	∈	PROPN
ejpam-5256	114	51	µ	µ	PRON
ejpam-5256	114	52	such	such	ADJ
ejpam-5256	114	53	that	that	SCONJ
ejpam-5256	114	54	x	x	SYM
ejpam-5256	114	55	∈	∈	PROPN
ejpam-5256	114	56	g	g	NOUN
ejpam-5256	114	57	,	,	PUNCT
ejpam-5256	114	58	where	where	SCONJ
ejpam-5256	114	59	mµ	mµ	ADP
ejpam-5256	114	60	=	=	PUNCT
ejpam-5256	114	61	∪{m	∪{m	PROPN
ejpam-5256	114	62	⊆	⊆	NUM
ejpam-5256	114	63	x	x	SYM
ejpam-5256	114	64	:	:	PUNCT
ejpam-5256	114	65	m	m	VERB
ejpam-5256	114	66	∈	∈	PROPN
ejpam-5256	114	67	µ	µ	X
ejpam-5256	114	68	}	}	PUNCT
ejpam-5256	114	69	.	.	PUNCT
ejpam-5256	115	1	in	in	ADP
ejpam-5256	115	2	this	this	DET
ejpam-5256	115	3	paper	paper	NOUN
ejpam-5256	115	4	,	,	PUNCT
ejpam-5256	115	5	the	the	DET
ejpam-5256	115	6	aforementioned	aforementione	VERB
ejpam-5256	115	7	concepts	concept	NOUN
ejpam-5256	115	8	are	be	AUX
ejpam-5256	115	9	used	use	VERB
ejpam-5256	115	10	to	to	PART
ejpam-5256	115	11	define	define	VERB
ejpam-5256	115	12	the	the	DET
ejpam-5256	115	13	set	set	NOUN
ejpam-5256	115	14	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	115	15	)	)	PUNCT
ejpam-5256	115	16	as	as	SCONJ
ejpam-5256	115	17	follows	follow	VERB
ejpam-5256	115	18	.	.	PUNCT
ejpam-5256	116	1	definition	definition	NOUN
ejpam-5256	116	2	7	7	NUM
ejpam-5256	116	3	.	.	PUNCT
ejpam-5256	117	1	let	let	VERB
ejpam-5256	117	2	a	a	DET
ejpam-5256	117	3	be	be	AUX
ejpam-5256	117	4	a	a	DET
ejpam-5256	117	5	subset	subset	NOUN
ejpam-5256	117	6	of	of	ADP
ejpam-5256	117	7	a	a	DET
ejpam-5256	117	8	generalized	generalized	ADJ
ejpam-5256	117	9	topological	topological	ADJ
ejpam-5256	117	10	space	space	NOUN
ejpam-5256	117	11	(	(	PUNCT
ejpam-5256	117	12	x,µ	x,µ	NOUN
ejpam-5256	117	13	)	)	PUNCT
ejpam-5256	117	14	.	.	PUNCT
ejpam-5256	118	1	the	the	DET
ejpam-5256	118	2	set	set	NOUN
ejpam-5256	118	3	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	118	4	)	)	PUNCT
ejpam-5256	118	5	is	be	AUX
ejpam-5256	118	6	a	a	DET
ejpam-5256	118	7	subset	subset	NOUN
ejpam-5256	118	8	of	of	ADP
ejpam-5256	118	9	x	x	PUNCT
ejpam-5256	118	10	defined	define	VERB
ejpam-5256	118	11	by	by	ADP
ejpam-5256	118	12	x	x	PROPN
ejpam-5256	118	13	∈	∈	PROPN
ejpam-5256	118	14	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	118	15	)	)	PUNCT
ejpam-5256	118	16	if	if	SCONJ
ejpam-5256	118	17	cσ(g	cσ(g	NOUN
ejpam-5256	118	18	)	)	PUNCT
ejpam-5256	118	19	∩mµ	∩mµ	PROPN
ejpam-5256	118	20	∩	∩	NOUN
ejpam-5256	118	21	a	a	DET
ejpam-5256	118	22	̸=	̸=	PROPN
ejpam-5256	118	23	∅	∅	NOUN
ejpam-5256	118	24	for	for	ADP
ejpam-5256	118	25	each	each	DET
ejpam-5256	118	26	g	g	PROPN
ejpam-5256	118	27	∈	∈	PROPN
ejpam-5256	118	28	µ	µ	PRON
ejpam-5256	118	29	such	such	ADJ
ejpam-5256	118	30	that	that	SCONJ
ejpam-5256	118	31	x	x	SYM
ejpam-5256	118	32	∈	∈	PROPN
ejpam-5256	118	33	g	g	NOUN
ejpam-5256	118	34	,	,	PUNCT
ejpam-5256	118	35	where	where	SCONJ
ejpam-5256	118	36	mµ	mµ	ADP
ejpam-5256	118	37	=	=	PUNCT
ejpam-5256	118	38	∪{m	∪{m	PROPN
ejpam-5256	118	39	⊆	⊆	NUM
ejpam-5256	118	40	x	x	SYM
ejpam-5256	118	41	:	:	PUNCT
ejpam-5256	118	42	m	m	VERB
ejpam-5256	118	43	∈	∈	NOUN
ejpam-5256	118	44	µ	µ	X
ejpam-5256	118	45	}	}	PUNCT
ejpam-5256	118	46	.	.	PUNCT
ejpam-5256	119	1	theorem	theorem	ADJ
ejpam-5256	119	2	4	4	NUM
ejpam-5256	119	3	.	.	PUNCT
ejpam-5256	120	1	let	let	VERB
ejpam-5256	120	2	a	a	PRON
ejpam-5256	120	3	and	and	CCONJ
ejpam-5256	120	4	b	b	NOUN
ejpam-5256	120	5	be	be	AUX
ejpam-5256	120	6	subsets	subset	NOUN
ejpam-5256	120	7	of	of	ADP
ejpam-5256	120	8	a	a	DET
ejpam-5256	120	9	generalized	generalized	ADJ
ejpam-5256	120	10	topological	topological	ADJ
ejpam-5256	120	11	space	space	NOUN
ejpam-5256	120	12	(	(	PUNCT
ejpam-5256	120	13	x,µ	x,µ	NOUN
ejpam-5256	120	14	)	)	PUNCT
ejpam-5256	120	15	.	.	PUNCT
ejpam-5256	121	1	if	if	SCONJ
ejpam-5256	121	2	a	a	DET
ejpam-5256	121	3	⊆	⊆	NUM
ejpam-5256	121	4	b	b	NOUN
ejpam-5256	121	5	⊆	⊆	NUM
ejpam-5256	121	6	x	x	SYM
ejpam-5256	121	7	,	,	PUNCT
ejpam-5256	121	8	then	then	ADV
ejpam-5256	121	9	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	121	10	)	)	PUNCT
ejpam-5256	121	11	⊆	⊆	NUM
ejpam-5256	121	12	γsθ̃(b	γsθ̃(b	PROPN
ejpam-5256	121	13	)	)	PUNCT
ejpam-5256	121	14	.	.	PUNCT
ejpam-5256	122	1	proof	proof	NOUN
ejpam-5256	122	2	.	.	PUNCT
ejpam-5256	123	1	let	let	VERB
ejpam-5256	123	2	a	a	DET
ejpam-5256	123	3	⊆	⊆	NUM
ejpam-5256	123	4	b	b	NOUN
ejpam-5256	123	5	⊆	⊆	NUM
ejpam-5256	123	6	x	x	PUNCT
ejpam-5256	123	7	and	and	CCONJ
ejpam-5256	123	8	x	x	SYM
ejpam-5256	123	9	∈	∈	PROPN
ejpam-5256	123	10	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	123	11	)	)	PUNCT
ejpam-5256	123	12	.	.	PUNCT
ejpam-5256	124	1	thus	thus	ADV
ejpam-5256	124	2	,	,	PUNCT
ejpam-5256	124	3	cσ(g	cσ(g	NOUN
ejpam-5256	124	4	)	)	PUNCT
ejpam-5256	124	5	∩mµ	∩mµ	PROPN
ejpam-5256	124	6	∩a	∩a	PROPN
ejpam-5256	124	7	̸=	̸=	PROPN
ejpam-5256	124	8	∅	∅	NOUN
ejpam-5256	124	9	for	for	ADP
ejpam-5256	124	10	all	all	PRON
ejpam-5256	124	11	g	g	PROPN
ejpam-5256	124	12	∈	∈	PROPN
ejpam-5256	124	13	µ	µ	PRON
ejpam-5256	124	14	such	such	ADJ
ejpam-5256	124	15	that	that	SCONJ
ejpam-5256	124	16	x	x	SYM
ejpam-5256	124	17	∈	∈	PROPN
ejpam-5256	124	18	g	g	NOUN
ejpam-5256	124	19	,	,	PUNCT
ejpam-5256	124	20	where	where	SCONJ
ejpam-5256	124	21	mµ	mµ	ADP
ejpam-5256	124	22	=	=	PUNCT
ejpam-5256	124	23	∪{m	∪{m	PROPN
ejpam-5256	124	24	⊆	⊆	NUM
ejpam-5256	124	25	x	x	SYM
ejpam-5256	124	26	:	:	PUNCT
ejpam-5256	124	27	m	m	VERB
ejpam-5256	124	28	∈	∈	NOUN
ejpam-5256	124	29	µ	µ	NOUN
ejpam-5256	124	30	}	}	PUNCT
ejpam-5256	124	31	.	.	PUNCT
ejpam-5256	125	1	let	let	VERB
ejpam-5256	125	2	h	h	PRON
ejpam-5256	125	3	be	be	AUX
ejpam-5256	125	4	an	an	DET
ejpam-5256	125	5	arbitrary	arbitrary	ADJ
ejpam-5256	125	6	µ-open	µ-open	NOUN
ejpam-5256	125	7	set	set	VERB
ejpam-5256	125	8	such	such	ADJ
ejpam-5256	125	9	that	that	SCONJ
ejpam-5256	125	10	x	x	SYM
ejpam-5256	125	11	∈	∈	PROPN
ejpam-5256	125	12	h.	h.	NOUN
ejpam-5256	125	13	by	by	ADP
ejpam-5256	125	14	the	the	DET
ejpam-5256	125	15	assumption	assumption	NOUN
ejpam-5256	125	16	,	,	PUNCT
ejpam-5256	125	17	cσ(h	cσ(h	NOUN
ejpam-5256	125	18	)	)	PUNCT
ejpam-5256	125	19	∩mµ	∩mµ	PROPN
ejpam-5256	125	20	∩	∩	NOUN
ejpam-5256	125	21	b	b	PROPN
ejpam-5256	125	22	̸=	̸=	PROPN
ejpam-5256	125	23	∅.	∅.	VERB
ejpam-5256	125	24	consequently	consequently	ADV
ejpam-5256	125	25	,	,	PUNCT
ejpam-5256	125	26	x	x	PROPN
ejpam-5256	125	27	∈	∈	PROPN
ejpam-5256	125	28	γsθ̃(b	γsθ̃(b	PROPN
ejpam-5256	125	29	)	)	PUNCT
ejpam-5256	125	30	.	.	PUNCT
ejpam-5256	126	1	therefore	therefore	ADV
ejpam-5256	126	2	,	,	PUNCT
ejpam-5256	126	3	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	126	4	)	)	PUNCT
ejpam-5256	126	5	⊆	⊆	NUM
ejpam-5256	126	6	γsθ̃(b	γsθ̃(b	PROPN
ejpam-5256	126	7	)	)	PUNCT
ejpam-5256	126	8	.	.	PUNCT
ejpam-5256	127	1	theorem	theorem	NOUN
ejpam-5256	127	2	5	5	NUM
ejpam-5256	127	3	.	.	PUNCT
ejpam-5256	128	1	let	let	VERB
ejpam-5256	128	2	(	(	PUNCT
ejpam-5256	128	3	x,µ	x,µ	NOUN
ejpam-5256	128	4	)	)	PUNCT
ejpam-5256	128	5	be	be	VERB
ejpam-5256	128	6	a	a	DET
ejpam-5256	128	7	generalized	generalized	ADJ
ejpam-5256	128	8	topological	topological	ADJ
ejpam-5256	128	9	space	space	NOUN
ejpam-5256	128	10	and	and	CCONJ
ejpam-5256	128	11	a	a	DET
ejpam-5256	128	12	⊆	⊆	NUM
ejpam-5256	128	13	x.	x.	NOUN
ejpam-5256	128	14	then	then	ADV
ejpam-5256	128	15	,	,	PUNCT
ejpam-5256	128	16	a	a	DET
ejpam-5256	128	17	⊆	⊆	NUM
ejpam-5256	128	18	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	128	19	)	)	PUNCT
ejpam-5256	128	20	⊆	⊆	NUM
ejpam-5256	128	21	γθ̃(a	γθ̃(a	NOUN
ejpam-5256	128	22	)	)	PUNCT
ejpam-5256	128	23	.	.	PUNCT
ejpam-5256	129	1	proof	proof	NOUN
ejpam-5256	129	2	.	.	PUNCT
ejpam-5256	130	1	assume	assume	VERB
ejpam-5256	130	2	that	that	SCONJ
ejpam-5256	130	3	x	x	PUNCT
ejpam-5256	130	4	∈	∈	NOUN
ejpam-5256	130	5	a.	a.	NOUN
ejpam-5256	130	6	if	if	SCONJ
ejpam-5256	130	7	x	x	X
ejpam-5256	130	8	/∈	/∈	PUNCT
ejpam-5256	131	1	mµ	mµ	INTJ
ejpam-5256	131	2	,	,	PUNCT
ejpam-5256	131	3	then	then	ADV
ejpam-5256	131	4	x	x	SYM
ejpam-5256	131	5	∈	∈	PROPN
ejpam-5256	131	6	x	x	X
ejpam-5256	131	7	−mµ.	−mµ.	PROPN
ejpam-5256	131	8	it	it	PRON
ejpam-5256	131	9	follows	follow	VERB
ejpam-5256	131	10	that	that	SCONJ
ejpam-5256	131	11	x	x	PUNCT
ejpam-5256	131	12	∈	∈	NOUN
ejpam-5256	131	13	x	x	X
ejpam-5256	131	14	and	and	CCONJ
ejpam-5256	131	15	x	x	NOUN
ejpam-5256	131	16	/∈	/∈	PUNCT
ejpam-5256	132	1	mµ	mµ	INTJ
ejpam-5256	132	2	,	,	PUNCT
ejpam-5256	132	3	where	where	SCONJ
ejpam-5256	132	4	mµ	mµ	ADP
ejpam-5256	132	5	=	=	PUNCT
ejpam-5256	132	6	∪{m	∪{m	PROPN
ejpam-5256	132	7	⊆	⊆	NUM
ejpam-5256	132	8	x	x	SYM
ejpam-5256	132	9	:	:	PUNCT
ejpam-5256	132	10	m	m	VERB
ejpam-5256	132	11	∈	∈	NOUN
ejpam-5256	132	12	µ	µ	X
ejpam-5256	132	13	}	}	PUNCT
ejpam-5256	132	14	.	.	PUNCT
ejpam-5256	133	1	thus	thus	ADV
ejpam-5256	133	2	,	,	PUNCT
ejpam-5256	133	3	there	there	PRON
ejpam-5256	133	4	is	be	VERB
ejpam-5256	133	5	no	no	DET
ejpam-5256	133	6	g	g	PROPN
ejpam-5256	133	7	∈	∈	PROPN
ejpam-5256	133	8	µ	µ	PRON
ejpam-5256	133	9	such	such	ADJ
ejpam-5256	133	10	that	that	SCONJ
ejpam-5256	133	11	x	x	SYM
ejpam-5256	133	12	∈	∈	PROPN
ejpam-5256	133	13	g.	g.	PROPN
ejpam-5256	133	14	j.	j.	PROPN
ejpam-5256	133	15	khampakdee	khampakdee	PROPN
ejpam-5256	133	16	/	/	PUNCT
ejpam-5256	133	17	eur	eur	PROPN
ejpam-5256	133	18	.	.	PUNCT
ejpam-5256	134	1	j.	j.	PROPN
ejpam-5256	134	2	pure	pure	PROPN
ejpam-5256	134	3	appl	appl	PROPN
ejpam-5256	134	4	.	.	PROPN
ejpam-5256	134	5	math	math	PROPN
ejpam-5256	134	6	,	,	PUNCT
ejpam-5256	134	7	17	17	NUM
ejpam-5256	134	8	(	(	PUNCT
ejpam-5256	134	9	3	3	NUM
ejpam-5256	134	10	)	)	PUNCT
ejpam-5256	134	11	(	(	PUNCT
ejpam-5256	134	12	2024	2024	NUM
ejpam-5256	134	13	)	)	PUNCT
ejpam-5256	134	14	,	,	PUNCT
ejpam-5256	134	15	1869	1869	NUM
ejpam-5256	134	16	-	-	SYM
ejpam-5256	134	17	1876	1876	NUM
ejpam-5256	134	18	1873	1873	NUM
ejpam-5256	134	19	by	by	ADP
ejpam-5256	134	20	definition	definition	NOUN
ejpam-5256	134	21	7	7	NUM
ejpam-5256	134	22	,	,	PUNCT
ejpam-5256	134	23	x	x	SYM
ejpam-5256	134	24	∈	∈	PROPN
ejpam-5256	134	25	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	134	26	)	)	PUNCT
ejpam-5256	134	27	.	.	PUNCT
ejpam-5256	135	1	if	if	SCONJ
ejpam-5256	135	2	x	x	SYM
ejpam-5256	135	3	∈	∈	PROPN
ejpam-5256	135	4	mµ	mµ	VERB
ejpam-5256	135	5	,	,	PUNCT
ejpam-5256	135	6	then	then	ADV
ejpam-5256	135	7	cσ(g	cσ(g	PUNCT
ejpam-5256	135	8	)	)	PUNCT
ejpam-5256	135	9	∩mµ	∩mµ	PROPN
ejpam-5256	135	10	∩	∩	NOUN
ejpam-5256	135	11	a	a	DET
ejpam-5256	135	12	̸=	̸=	PROPN
ejpam-5256	135	13	∅	∅	NOUN
ejpam-5256	135	14	for	for	ADP
ejpam-5256	135	15	each	each	DET
ejpam-5256	135	16	g	g	PROPN
ejpam-5256	135	17	∈	∈	PROPN
ejpam-5256	135	18	µ	µ	PRON
ejpam-5256	135	19	such	such	ADJ
ejpam-5256	135	20	that	that	SCONJ
ejpam-5256	135	21	x	x	SYM
ejpam-5256	135	22	∈	∈	PROPN
ejpam-5256	135	23	g.	g.	NOUN
ejpam-5256	135	24	hence	hence	ADV
ejpam-5256	135	25	,	,	PUNCT
ejpam-5256	135	26	x	x	PUNCT
ejpam-5256	135	27	∈	∈	PROPN
ejpam-5256	135	28	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	135	29	)	)	PUNCT
ejpam-5256	135	30	.	.	PUNCT
ejpam-5256	136	1	thus	thus	ADV
ejpam-5256	136	2	,	,	PUNCT
ejpam-5256	136	3	a	a	DET
ejpam-5256	136	4	⊆	⊆	NUM
ejpam-5256	136	5	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	136	6	)	)	PUNCT
ejpam-5256	136	7	.	.	PUNCT
ejpam-5256	137	1	by	by	ADP
ejpam-5256	137	2	definition	definition	NOUN
ejpam-5256	137	3	7	7	NUM
ejpam-5256	137	4	,	,	PUNCT
ejpam-5256	137	5	we	we	PRON
ejpam-5256	137	6	can	can	AUX
ejpam-5256	137	7	see	see	VERB
ejpam-5256	137	8	that	that	SCONJ
ejpam-5256	137	9	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	137	10	)	)	PUNCT
ejpam-5256	137	11	⊆	⊆	NUM
ejpam-5256	137	12	γθ̃(a	γθ̃(a	NOUN
ejpam-5256	137	13	)	)	PUNCT
ejpam-5256	137	14	,	,	PUNCT
ejpam-5256	137	15	where	where	SCONJ
ejpam-5256	137	16	a	a	PRON
ejpam-5256	137	17	is	be	AUX
ejpam-5256	137	18	a	a	DET
ejpam-5256	137	19	subset	subset	NOUN
ejpam-5256	137	20	of	of	ADP
ejpam-5256	137	21	a	a	DET
ejpam-5256	137	22	generalized	generalized	ADJ
ejpam-5256	137	23	topological	topological	ADJ
ejpam-5256	137	24	space	space	NOUN
ejpam-5256	137	25	(	(	PUNCT
ejpam-5256	137	26	x,µ	x,µ	NOUN
ejpam-5256	137	27	)	)	PUNCT
ejpam-5256	137	28	.	.	PUNCT
ejpam-5256	138	1	therefore	therefore	ADV
ejpam-5256	138	2	,	,	PUNCT
ejpam-5256	138	3	a	a	DET
ejpam-5256	138	4	⊆	⊆	NUM
ejpam-5256	138	5	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	138	6	)	)	PUNCT
ejpam-5256	138	7	⊆	⊆	NUM
ejpam-5256	138	8	γθ̃(a	γθ̃(a	NOUN
ejpam-5256	138	9	)	)	PUNCT
ejpam-5256	138	10	.	.	PUNCT
ejpam-5256	139	1	we	we	PRON
ejpam-5256	139	2	use	use	VERB
ejpam-5256	139	3	some	some	DET
ejpam-5256	139	4	properties	property	NOUN
ejpam-5256	139	5	of	of	ADP
ejpam-5256	139	6	the	the	DET
ejpam-5256	139	7	operation	operation	NOUN
ejpam-5256	139	8	γsθ̃	γsθ̃	NOUN
ejpam-5256	139	9	to	to	PART
ejpam-5256	139	10	characterize	characterize	VERB
ejpam-5256	139	11	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	139	12	sets	set	NOUN
ejpam-5256	139	13	as	as	ADP
ejpam-5256	139	14	the	the	DET
ejpam-5256	139	15	following	follow	VERB
ejpam-5256	139	16	theorem	theorem	NOUN
ejpam-5256	139	17	.	.	PUNCT
ejpam-5256	139	18	theorem	theorem	NOUN
ejpam-5256	139	19	6	6	NUM
ejpam-5256	139	20	.	.	PUNCT
ejpam-5256	140	1	let	let	VERB
ejpam-5256	140	2	(	(	PUNCT
ejpam-5256	140	3	x,µ	x,µ	NOUN
ejpam-5256	140	4	)	)	PUNCT
ejpam-5256	140	5	be	be	VERB
ejpam-5256	140	6	a	a	DET
ejpam-5256	140	7	generalized	generalized	ADJ
ejpam-5256	140	8	topological	topological	ADJ
ejpam-5256	140	9	space	space	NOUN
ejpam-5256	140	10	and	and	CCONJ
ejpam-5256	140	11	a	a	DET
ejpam-5256	140	12	⊆	⊆	NUM
ejpam-5256	140	13	x.	x.	NOUN
ejpam-5256	140	14	then	then	ADV
ejpam-5256	140	15	a	a	PRON
ejpam-5256	140	16	is	be	AUX
ejpam-5256	140	17	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	140	18	if	if	SCONJ
ejpam-5256	140	19	and	and	CCONJ
ejpam-5256	140	20	only	only	ADV
ejpam-5256	140	21	if	if	SCONJ
ejpam-5256	140	22	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	140	23	)	)	PUNCT
ejpam-5256	140	24	=	=	SYM
ejpam-5256	140	25	a.	a.	NOUN
ejpam-5256	140	26	proof	proof	NOUN
ejpam-5256	140	27	.	.	PUNCT
ejpam-5256	141	1	(	(	PUNCT
ejpam-5256	141	2	→	→	AUX
ejpam-5256	141	3	)	)	PUNCT
ejpam-5256	141	4	let	let	VERB
ejpam-5256	141	5	a	a	PRON
ejpam-5256	141	6	be	be	AUX
ejpam-5256	141	7	an	an	DET
ejpam-5256	141	8	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	141	9	.	.	PUNCT
ejpam-5256	142	1	then	then	ADV
ejpam-5256	142	2	,	,	PUNCT
ejpam-5256	142	3	x	x	PRON
ejpam-5256	142	4	−a	−a	NOUN
ejpam-5256	142	5	is	be	AUX
ejpam-5256	142	6	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	142	7	.	.	PUNCT
ejpam-5256	143	1	assume	assume	VERB
ejpam-5256	143	2	that	that	SCONJ
ejpam-5256	143	3	x	x	PUNCT
ejpam-5256	143	4	∈	∈	X
ejpam-5256	143	5	x	x	X
ejpam-5256	143	6	−a	−a	NOUN
ejpam-5256	143	7	.	.	PUNCT
ejpam-5256	144	1	hence	hence	ADV
ejpam-5256	144	2	,	,	PUNCT
ejpam-5256	144	3	there	there	PRON
ejpam-5256	144	4	exists	exist	VERB
ejpam-5256	144	5	m	m	PROPN
ejpam-5256	144	6	∈	∈	PROPN
ejpam-5256	144	7	µ	µ	PRON
ejpam-5256	144	8	such	such	ADJ
ejpam-5256	144	9	that	that	SCONJ
ejpam-5256	144	10	x	x	PUNCT
ejpam-5256	144	11	∈m	∈m	ADP
ejpam-5256	144	12	⊆	⊆	NUM
ejpam-5256	144	13	cσ(m)∩mµ	cσ(m)∩mµ	NUM
ejpam-5256	144	14	⊆	⊆	NUM
ejpam-5256	144	15	x−a	x−a	NOUN
ejpam-5256	144	16	,	,	PUNCT
ejpam-5256	144	17	where	where	SCONJ
ejpam-5256	144	18	mµ	mµ	ADP
ejpam-5256	144	19	=	=	PUNCT
ejpam-5256	144	20	∪{m	∪{m	PROPN
ejpam-5256	144	21	⊆	⊆	NUM
ejpam-5256	144	22	x	x	SYM
ejpam-5256	144	23	:	:	PUNCT
ejpam-5256	144	24	m	m	VERB
ejpam-5256	144	25	∈	∈	NOUN
ejpam-5256	144	26	µ	µ	X
ejpam-5256	144	27	}	}	PUNCT
ejpam-5256	144	28	.	.	PUNCT
ejpam-5256	145	1	thus	thus	ADV
ejpam-5256	145	2	,	,	PUNCT
ejpam-5256	145	3	cσ(m	cσ(m	ADJ
ejpam-5256	145	4	)	)	PUNCT
ejpam-5256	145	5	∩mµ	∩mµ	PROPN
ejpam-5256	145	6	∩	∩	NOUN
ejpam-5256	145	7	a	a	DET
ejpam-5256	145	8	=	=	NOUN
ejpam-5256	145	9	∅	∅	NOUN
ejpam-5256	145	10	,	,	PUNCT
ejpam-5256	145	11	for	for	ADP
ejpam-5256	145	12	some	some	DET
ejpam-5256	145	13	m	m	NOUN
ejpam-5256	145	14	∈	∈	PROPN
ejpam-5256	145	15	µ	µ	NOUN
ejpam-5256	145	16	and	and	CCONJ
ejpam-5256	145	17	x	x	PUNCT
ejpam-5256	145	18	∈	∈	PROPN
ejpam-5256	145	19	m	m	VERB
ejpam-5256	145	20	.	.	PUNCT
ejpam-5256	146	1	it	it	PRON
ejpam-5256	146	2	follows	follow	VERB
ejpam-5256	146	3	that	that	SCONJ
ejpam-5256	146	4	x	x	X
ejpam-5256	146	5	/∈	/∈	PUNCT
ejpam-5256	146	6	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	146	7	)	)	PUNCT
ejpam-5256	146	8	.	.	PUNCT
ejpam-5256	147	1	consequently	consequently	ADV
ejpam-5256	147	2	,	,	PUNCT
ejpam-5256	147	3	x	x	PUNCT
ejpam-5256	147	4	∈	∈	NOUN
ejpam-5256	147	5	x	x	X
ejpam-5256	147	6	−	−	PROPN
ejpam-5256	147	7	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	147	8	)	)	PUNCT
ejpam-5256	147	9	.	.	PUNCT
ejpam-5256	148	1	accordingly	accordingly	ADV
ejpam-5256	148	2	,	,	PUNCT
ejpam-5256	148	3	x	x	SYM
ejpam-5256	148	4	−a	−a	VERB
ejpam-5256	148	5	⊆	⊆	NUM
ejpam-5256	148	6	x	x	PUNCT
ejpam-5256	148	7	−	−	PROPN
ejpam-5256	148	8	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	148	9	)	)	PUNCT
ejpam-5256	148	10	.	.	PUNCT
ejpam-5256	149	1	therefore	therefore	ADV
ejpam-5256	149	2	,	,	PUNCT
ejpam-5256	149	3	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	149	4	)	)	PUNCT
ejpam-5256	149	5	⊆	⊆	NUM
ejpam-5256	149	6	a.	a.	NOUN
ejpam-5256	149	7	by	by	ADP
ejpam-5256	149	8	theorem	theorem	ADJ
ejpam-5256	149	9	5	5	NUM
ejpam-5256	149	10	,	,	PUNCT
ejpam-5256	149	11	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	149	12	)	)	PUNCT
ejpam-5256	149	13	=	=	SYM
ejpam-5256	149	14	a.	a.	NOUN
ejpam-5256	149	15	(	(	PUNCT
ejpam-5256	149	16	←	←	PROPN
ejpam-5256	149	17	)	)	PUNCT
ejpam-5256	149	18	suppose	suppose	VERB
ejpam-5256	149	19	that	that	SCONJ
ejpam-5256	149	20	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	149	21	)	)	PUNCT
ejpam-5256	149	22	=	=	SYM
ejpam-5256	149	23	a.	a.	NOUN
ejpam-5256	149	24	then	then	ADV
ejpam-5256	149	25	,	,	PUNCT
ejpam-5256	149	26	x	x	PUNCT
ejpam-5256	150	1	−	−	NOUN
ejpam-5256	150	2	a	a	PRON
ejpam-5256	150	3	=	=	NOUN
ejpam-5256	150	4	x	x	SYM
ejpam-5256	150	5	−	−	PROPN
ejpam-5256	150	6	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	150	7	)	)	PUNCT
ejpam-5256	150	8	.	.	PUNCT
ejpam-5256	151	1	assume	assume	VERB
ejpam-5256	151	2	that	that	SCONJ
ejpam-5256	151	3	x	x	PUNCT
ejpam-5256	151	4	∈	∈	NOUN
ejpam-5256	151	5	x	x	INTJ
ejpam-5256	151	6	−	−	NOUN
ejpam-5256	151	7	a.	a.	NOUN
ejpam-5256	151	8	it	it	PRON
ejpam-5256	151	9	follows	follow	VERB
ejpam-5256	151	10	that	that	SCONJ
ejpam-5256	151	11	x	x	X
ejpam-5256	151	12	/∈	/∈	PUNCT
ejpam-5256	151	13	γsθ̃(a	γsθ̃(a	PROPN
ejpam-5256	151	14	)	)	PUNCT
ejpam-5256	151	15	.	.	PUNCT
ejpam-5256	152	1	hence	hence	ADV
ejpam-5256	152	2	,	,	PUNCT
ejpam-5256	152	3	there	there	PRON
ejpam-5256	152	4	existsm	existsm	VERB
ejpam-5256	152	5	∈	∈	PROPN
ejpam-5256	152	6	µ	µ	PRON
ejpam-5256	152	7	such	such	ADJ
ejpam-5256	152	8	that	that	SCONJ
ejpam-5256	152	9	x	x	SYM
ejpam-5256	152	10	∈m	∈m	NOUN
ejpam-5256	152	11	and	and	CCONJ
ejpam-5256	152	12	cσ(m)∩mµ∩a	cσ(m)∩mµ∩a	NOUN
ejpam-5256	152	13	=	=	PUNCT
ejpam-5256	152	14	∅.	∅.	NOUN
ejpam-5256	152	15	thus	thus	ADV
ejpam-5256	152	16	,	,	PUNCT
ejpam-5256	152	17	x	x	PUNCT
ejpam-5256	152	18	∈m	∈m	ADP
ejpam-5256	152	19	⊆	⊆	NUM
ejpam-5256	152	20	cσ(m)∩mµ	cσ(m)∩mµ	NUM
ejpam-5256	152	21	⊆	⊆	NUM
ejpam-5256	152	22	x	x	SYM
ejpam-5256	152	23	−a	−a	ADV
ejpam-5256	152	24	.	.	PUNCT
ejpam-5256	153	1	accordingly	accordingly	ADV
ejpam-5256	153	2	,	,	PUNCT
ejpam-5256	153	3	x	x	PRON
ejpam-5256	153	4	−a	−a	NOUN
ejpam-5256	153	5	is	be	AUX
ejpam-5256	153	6	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	153	7	.	.	PUNCT
ejpam-5256	154	1	therefore	therefore	ADV
ejpam-5256	154	2	,	,	PUNCT
ejpam-5256	154	3	a	a	PRON
ejpam-5256	154	4	is	be	AUX
ejpam-5256	154	5	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	154	6	.	.	PUNCT
ejpam-5256	155	1	corollary	corollary	ADJ
ejpam-5256	155	2	3	3	X
ejpam-5256	155	3	.	.	PUNCT
ejpam-5256	156	1	γsθ̃(x	γsθ̃(x	NOUN
ejpam-5256	156	2	)	)	PUNCT
ejpam-5256	156	3	=	=	SYM
ejpam-5256	157	1	x	x	NOUN
ejpam-5256	157	2	,	,	PUNCT
ejpam-5256	157	3	where	where	SCONJ
ejpam-5256	157	4	(	(	PUNCT
ejpam-5256	157	5	x,µ	x,µ	NOUN
ejpam-5256	157	6	)	)	PUNCT
ejpam-5256	157	7	is	be	AUX
ejpam-5256	157	8	a	a	DET
ejpam-5256	157	9	generalized	generalized	ADJ
ejpam-5256	157	10	topological	topological	ADJ
ejpam-5256	157	11	space	space	NOUN
ejpam-5256	157	12	.	.	PUNCT
ejpam-5256	158	1	proof	proof	NOUN
ejpam-5256	158	2	.	.	PUNCT
ejpam-5256	159	1	by	by	ADP
ejpam-5256	159	2	theorem	theorem	NOUN
ejpam-5256	159	3	1	1	NUM
ejpam-5256	159	4	,	,	PUNCT
ejpam-5256	159	5	∅	∅	NOUN
ejpam-5256	159	6	is	be	AUX
ejpam-5256	159	7	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	159	8	.	.	PUNCT
ejpam-5256	160	1	it	it	PRON
ejpam-5256	160	2	follows	follow	VERB
ejpam-5256	160	3	that	that	SCONJ
ejpam-5256	160	4	x	x	PRON
ejpam-5256	160	5	is	be	AUX
ejpam-5256	160	6	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	160	7	.	.	PUNCT
ejpam-5256	161	1	by	by	ADP
ejpam-5256	161	2	theorem	theorem	NOUN
ejpam-5256	161	3	6	6	NUM
ejpam-5256	161	4	,	,	PUNCT
ejpam-5256	161	5	γsθ̃(x	γsθ̃(x	PROPN
ejpam-5256	161	6	)	)	PUNCT
ejpam-5256	161	7	=	=	PUNCT
ejpam-5256	161	8	x.	x.	NOUN
ejpam-5256	161	9	definition	definition	NOUN
ejpam-5256	161	10	8	8	NUM
ejpam-5256	161	11	.	.	PUNCT
ejpam-5256	162	1	let	let	VERB
ejpam-5256	162	2	(	(	PUNCT
ejpam-5256	162	3	x,µ	x,µ	NOUN
ejpam-5256	162	4	)	)	PUNCT
ejpam-5256	162	5	and	and	CCONJ
ejpam-5256	162	6	(	(	PUNCT
ejpam-5256	162	7	y	y	NOUN
ejpam-5256	162	8	,	,	PUNCT
ejpam-5256	162	9	µ′	µ′	PUNCT
ejpam-5256	162	10	)	)	PUNCT
ejpam-5256	163	1	be	be	AUX
ejpam-5256	163	2	generalized	generalize	VERB
ejpam-5256	163	3	topological	topological	ADJ
ejpam-5256	163	4	spaces	space	NOUN
ejpam-5256	163	5	.	.	PUNCT
ejpam-5256	164	1	then	then	ADV
ejpam-5256	164	2	,	,	PUNCT
ejpam-5256	164	3	a	a	DET
ejpam-5256	164	4	function	function	NOUN
ejpam-5256	164	5	f	f	NOUN
ejpam-5256	164	6	from	from	ADP
ejpam-5256	164	7	(	(	PUNCT
ejpam-5256	164	8	x,µ	x,µ	NOUN
ejpam-5256	164	9	)	)	PUNCT
ejpam-5256	164	10	into	into	ADP
ejpam-5256	164	11	(	(	PUNCT
ejpam-5256	164	12	y	y	NOUN
ejpam-5256	164	13	,	,	PUNCT
ejpam-5256	164	14	µ′	µ′	NUM
ejpam-5256	164	15	)	)	PUNCT
ejpam-5256	164	16	is	be	AUX
ejpam-5256	164	17	called	call	VERB
ejpam-5256	164	18	θ̃-continuous	θ̃-continuous	ADJ
ejpam-5256	164	19	if	if	SCONJ
ejpam-5256	164	20	f−1(g	f−1(g	PROPN
ejpam-5256	164	21	)	)	PUNCT
ejpam-5256	164	22	is	be	AUX
ejpam-5256	164	23	θ̃-open	θ̃-open	ADJ
ejpam-5256	164	24	in	in	ADP
ejpam-5256	164	25	x	x	PUNCT
ejpam-5256	164	26	for	for	SCONJ
ejpam-5256	164	27	each	each	DET
ejpam-5256	164	28	µ′-open	µ′-open	NUM
ejpam-5256	164	29	set	set	VERB
ejpam-5256	164	30	g	g	NOUN
ejpam-5256	164	31	in	in	ADP
ejpam-5256	164	32	y	y	PROPN
ejpam-5256	164	33	.	.	PUNCT
ejpam-5256	164	34	example	example	NOUN
ejpam-5256	165	1	4	4	NUM
ejpam-5256	165	2	.	.	PUNCT
ejpam-5256	165	3	let	let	VERB
ejpam-5256	165	4	x	x	PUNCT
ejpam-5256	165	5	=	=	PRON
ejpam-5256	165	6	{	{	PUNCT
ejpam-5256	165	7	a	a	PRON
ejpam-5256	165	8	,	,	PUNCT
ejpam-5256	165	9	b	b	NOUN
ejpam-5256	165	10	,	,	PUNCT
ejpam-5256	165	11	c	c	NOUN
ejpam-5256	165	12	,	,	PUNCT
ejpam-5256	165	13	d	d	NOUN
ejpam-5256	165	14	}	}	PUNCT
ejpam-5256	165	15	,	,	PUNCT
ejpam-5256	165	16	y	y	PROPN
ejpam-5256	165	17	=	=	SYM
ejpam-5256	165	18	{	{	PUNCT
ejpam-5256	165	19	t	t	PROPN
ejpam-5256	165	20	,	,	PUNCT
ejpam-5256	165	21	v	v	NOUN
ejpam-5256	165	22	,	,	PUNCT
ejpam-5256	165	23	w	w	NOUN
ejpam-5256	165	24	}	}	PUNCT
ejpam-5256	165	25	.	.	PUNCT
ejpam-5256	166	1	µ	µ	X
ejpam-5256	166	2	=	=	SYM
ejpam-5256	166	3	{	{	PUNCT
ejpam-5256	166	4	∅	∅	NOUN
ejpam-5256	166	5	,	,	PUNCT
ejpam-5256	166	6	{	{	PUNCT
ejpam-5256	166	7	a	a	X
ejpam-5256	166	8	}	}	PUNCT
ejpam-5256	166	9	,	,	PUNCT
ejpam-5256	166	10	{	{	PUNCT
ejpam-5256	166	11	b	b	NOUN
ejpam-5256	166	12	}	}	PUNCT
ejpam-5256	166	13	,	,	PUNCT
ejpam-5256	166	14	{	{	PUNCT
ejpam-5256	166	15	a	a	DET
ejpam-5256	166	16	,	,	PUNCT
ejpam-5256	166	17	b	b	NOUN
ejpam-5256	166	18	}	}	PUNCT
ejpam-5256	166	19	,	,	PUNCT
ejpam-5256	166	20	{	{	PUNCT
ejpam-5256	166	21	a	a	PRON
ejpam-5256	166	22	,	,	PUNCT
ejpam-5256	166	23	b	b	NOUN
ejpam-5256	166	24	,	,	PUNCT
ejpam-5256	166	25	c	c	NOUN
ejpam-5256	166	26	}	}	PUNCT
ejpam-5256	166	27	}	}	PUNCT
ejpam-5256	166	28	and	and	CCONJ
ejpam-5256	166	29	σ	σ	NUM
ejpam-5256	166	30	=	=	SYM
ejpam-5256	166	31	{	{	PUNCT
ejpam-5256	166	32	∅	∅	NOUN
ejpam-5256	166	33	,	,	PUNCT
ejpam-5256	166	34	{	{	PUNCT
ejpam-5256	166	35	t	t	NOUN
ejpam-5256	166	36	}	}	PUNCT
ejpam-5256	166	37	,	,	PUNCT
ejpam-5256	166	38	{	{	PUNCT
ejpam-5256	166	39	t	t	PROPN
ejpam-5256	166	40	,	,	PUNCT
ejpam-5256	166	41	v	v	NOUN
ejpam-5256	166	42	}	}	PUNCT
ejpam-5256	166	43	}	}	PUNCT
ejpam-5256	166	44	.	.	PUNCT
ejpam-5256	167	1	let	let	VERB
ejpam-5256	167	2	f	f	PRON
ejpam-5256	167	3	be	be	AUX
ejpam-5256	167	4	a	a	DET
ejpam-5256	167	5	function	function	NOUN
ejpam-5256	167	6	from	from	ADP
ejpam-5256	167	7	(	(	PUNCT
ejpam-5256	167	8	x,µ	x,µ	NOUN
ejpam-5256	167	9	)	)	PUNCT
ejpam-5256	167	10	into	into	ADP
ejpam-5256	167	11	(	(	PUNCT
ejpam-5256	167	12	y	y	NOUN
ejpam-5256	167	13	,	,	PUNCT
ejpam-5256	167	14	µ′	µ′	NOUN
ejpam-5256	167	15	)	)	PUNCT
ejpam-5256	167	16	defined	define	VERB
ejpam-5256	167	17	by	by	ADP
ejpam-5256	167	18	f(a	f(a	PROPN
ejpam-5256	167	19	)	)	PUNCT
ejpam-5256	167	20	=	=	SYM
ejpam-5256	167	21	t	t	PROPN
ejpam-5256	167	22	,	,	PUNCT
ejpam-5256	167	23	f(b	f(b	PROPN
ejpam-5256	167	24	)	)	PUNCT
ejpam-5256	167	25	=	=	SYM
ejpam-5256	167	26	t	t	PROPN
ejpam-5256	167	27	,	,	PUNCT
ejpam-5256	167	28	f(c	f(c	PROPN
ejpam-5256	167	29	)	)	PUNCT
ejpam-5256	167	30	=	=	SYM
ejpam-5256	167	31	t	t	PROPN
ejpam-5256	167	32	,	,	PUNCT
ejpam-5256	167	33	f(d	f(d	PROPN
ejpam-5256	167	34	)	)	PUNCT
ejpam-5256	167	35	=	=	SYM
ejpam-5256	167	36	w.	w.	PROPN
ejpam-5256	167	37	then	then	ADV
ejpam-5256	167	38	,	,	PUNCT
ejpam-5256	167	39	only	only	ADV
ejpam-5256	167	40	∅	∅	NOUN
ejpam-5256	167	41	and	and	CCONJ
ejpam-5256	167	42	{	{	PUNCT
ejpam-5256	167	43	a	a	PRON
ejpam-5256	167	44	,	,	PUNCT
ejpam-5256	167	45	b	b	NOUN
ejpam-5256	167	46	,	,	PUNCT
ejpam-5256	167	47	c	c	NOUN
ejpam-5256	167	48	}	}	PUNCT
ejpam-5256	167	49	are	be	AUX
ejpam-5256	167	50	θ̃-open	θ̃-open	ADJ
ejpam-5256	167	51	sets	set	NOUN
ejpam-5256	167	52	in	in	ADP
ejpam-5256	167	53	x.	x.	NOUN
ejpam-5256	167	54	consider	consider	VERB
ejpam-5256	167	55	∅	∅	NOUN
ejpam-5256	167	56	,	,	PUNCT
ejpam-5256	167	57	{	{	PUNCT
ejpam-5256	167	58	t	t	NOUN
ejpam-5256	167	59	}	}	PUNCT
ejpam-5256	167	60	and	and	CCONJ
ejpam-5256	167	61	{	{	PUNCT
ejpam-5256	167	62	t	t	PROPN
ejpam-5256	167	63	,	,	PUNCT
ejpam-5256	167	64	v	v	NOUN
ejpam-5256	167	65	}	}	PUNCT
ejpam-5256	167	66	are	be	AUX
ejpam-5256	167	67	µ′-open	µ′-open	NUM
ejpam-5256	167	68	sets	set	NOUN
ejpam-5256	167	69	in	in	ADP
ejpam-5256	167	70	y	y	PROPN
ejpam-5256	167	71	.	.	PUNCT
ejpam-5256	168	1	then	then	ADV
ejpam-5256	168	2	f−1(∅	f−1(∅	VERB
ejpam-5256	168	3	)	)	PUNCT
ejpam-5256	169	1	=	=	SYM
ejpam-5256	169	2	∅	∅	NOUN
ejpam-5256	169	3	,	,	PUNCT
ejpam-5256	169	4	f−1({t	f−1({t	PROPN
ejpam-5256	169	5	}	}	PUNCT
ejpam-5256	169	6	)	)	PUNCT
ejpam-5256	169	7	=	=	PRON
ejpam-5256	169	8	{	{	PUNCT
ejpam-5256	169	9	a	a	PRON
ejpam-5256	169	10	,	,	PUNCT
ejpam-5256	169	11	b	b	NOUN
ejpam-5256	169	12	,	,	PUNCT
ejpam-5256	169	13	c	c	NOUN
ejpam-5256	169	14	}	}	PUNCT
ejpam-5256	169	15	and	and	CCONJ
ejpam-5256	169	16	f−1({t	f−1({t	ADJ
ejpam-5256	169	17	,	,	PUNCT
ejpam-5256	169	18	v	v	NOUN
ejpam-5256	169	19	}	}	PUNCT
ejpam-5256	169	20	)	)	PUNCT
ejpam-5256	169	21	=	=	PRON
ejpam-5256	169	22	{	{	PUNCT
ejpam-5256	169	23	a	a	PRON
ejpam-5256	169	24	,	,	PUNCT
ejpam-5256	169	25	b	b	NOUN
ejpam-5256	169	26	,	,	PUNCT
ejpam-5256	169	27	c	c	NOUN
ejpam-5256	169	28	}	}	PUNCT
ejpam-5256	169	29	.	.	PUNCT
ejpam-5256	170	1	thus	thus	ADV
ejpam-5256	170	2	,	,	PUNCT
ejpam-5256	170	3	f−1(g	f−1(g	PROPN
ejpam-5256	170	4	)	)	PUNCT
ejpam-5256	170	5	is	be	AUX
ejpam-5256	170	6	θ̃-open	θ̃-open	ADJ
ejpam-5256	170	7	in	in	ADP
ejpam-5256	170	8	x	x	PUNCT
ejpam-5256	170	9	for	for	SCONJ
ejpam-5256	170	10	each	each	DET
ejpam-5256	170	11	µ′-open	µ′-open	NUM
ejpam-5256	170	12	set	set	VERB
ejpam-5256	170	13	g	g	NOUN
ejpam-5256	170	14	in	in	ADP
ejpam-5256	170	15	y	y	PROPN
ejpam-5256	170	16	.	.	PUNCT
ejpam-5256	171	1	therefore	therefore	ADV
ejpam-5256	171	2	,	,	PUNCT
ejpam-5256	171	3	f	f	PROPN
ejpam-5256	171	4	is	be	AUX
ejpam-5256	171	5	a	a	DET
ejpam-5256	171	6	θ̃-continuous	θ̃-continuous	ADJ
ejpam-5256	171	7	function	function	NOUN
ejpam-5256	171	8	.	.	PUNCT
ejpam-5256	172	1	theorem	theorem	VERB
ejpam-5256	172	2	7	7	NUM
ejpam-5256	172	3	.	.	PUNCT
ejpam-5256	173	1	let	let	VERB
ejpam-5256	173	2	(	(	PUNCT
ejpam-5256	173	3	x,µ	x,µ	NOUN
ejpam-5256	173	4	)	)	PUNCT
ejpam-5256	173	5	and	and	CCONJ
ejpam-5256	173	6	(	(	PUNCT
ejpam-5256	173	7	y	y	NOUN
ejpam-5256	173	8	,	,	PUNCT
ejpam-5256	173	9	µ′	µ′	PUNCT
ejpam-5256	173	10	)	)	PUNCT
ejpam-5256	173	11	be	be	AUX
ejpam-5256	173	12	generalized	generalize	VERB
ejpam-5256	173	13	topological	topological	ADJ
ejpam-5256	173	14	spaces	space	NOUN
ejpam-5256	173	15	.	.	PUNCT
ejpam-5256	174	1	then	then	ADV
ejpam-5256	174	2	,	,	PUNCT
ejpam-5256	174	3	f	f	X
ejpam-5256	174	4	:	:	PUNCT
ejpam-5256	174	5	(	(	PUNCT
ejpam-5256	174	6	x,µ)→	x,µ)→	X
ejpam-5256	174	7	(	(	PUNCT
ejpam-5256	174	8	y	y	NOUN
ejpam-5256	174	9	,	,	PUNCT
ejpam-5256	174	10	µ′	µ′	NUM
ejpam-5256	174	11	)	)	PUNCT
ejpam-5256	174	12	is	be	AUX
ejpam-5256	174	13	θ̃-continuous	θ̃-continuous	ADJ
ejpam-5256	174	14	if	if	SCONJ
ejpam-5256	174	15	and	and	CCONJ
ejpam-5256	174	16	only	only	ADV
ejpam-5256	174	17	if	if	SCONJ
ejpam-5256	174	18	f−1(v	f−1(v	PROPN
ejpam-5256	174	19	)	)	PUNCT
ejpam-5256	174	20	is	be	AUX
ejpam-5256	174	21	θ̃-closed	θ̃-close	VERB
ejpam-5256	174	22	in	in	ADP
ejpam-5256	174	23	x	x	PUNCT
ejpam-5256	174	24	for	for	ADP
ejpam-5256	174	25	each	each	DET
ejpam-5256	174	26	µ′-closed	µ′-close	VERB
ejpam-5256	174	27	set	set	VERB
ejpam-5256	174	28	v	v	NOUN
ejpam-5256	174	29	in	in	ADP
ejpam-5256	174	30	y	y	PROPN
ejpam-5256	174	31	.	.	PUNCT
ejpam-5256	175	1	proof	proof	NOUN
ejpam-5256	175	2	.	.	PUNCT
ejpam-5256	176	1	(	(	PUNCT
ejpam-5256	176	2	→	→	NOUN
ejpam-5256	176	3	)	)	PUNCT
ejpam-5256	176	4	assume	assume	VERB
ejpam-5256	176	5	that	that	SCONJ
ejpam-5256	176	6	f	f	PROPN
ejpam-5256	176	7	is	be	AUX
ejpam-5256	176	8	a	a	DET
ejpam-5256	176	9	θ̃-continuous	θ̃-continuous	ADJ
ejpam-5256	176	10	function	function	NOUN
ejpam-5256	176	11	.	.	PUNCT
ejpam-5256	177	1	let	let	VERB
ejpam-5256	177	2	v	v	PART
ejpam-5256	177	3	be	be	AUX
ejpam-5256	177	4	a	a	DET
ejpam-5256	177	5	µ′-closed	µ′-close	VERB
ejpam-5256	177	6	set	set	NOUN
ejpam-5256	177	7	in	in	ADP
ejpam-5256	177	8	y	y	PROPN
ejpam-5256	177	9	.	.	PUNCT
ejpam-5256	178	1	then	then	ADV
ejpam-5256	178	2	,	,	PUNCT
ejpam-5256	178	3	y	y	PROPN
ejpam-5256	178	4	−v	−v	NOUN
ejpam-5256	178	5	is	be	AUX
ejpam-5256	178	6	µ′-open	µ′-open	NUM
ejpam-5256	178	7	in	in	ADP
ejpam-5256	178	8	y	y	PROPN
ejpam-5256	178	9	.	.	PUNCT
ejpam-5256	179	1	hence	hence	ADV
ejpam-5256	179	2	,	,	PUNCT
ejpam-5256	179	3	f−1(y	f−1(y	PROPN
ejpam-5256	179	4	−v	−v	NOUN
ejpam-5256	179	5	)	)	PUNCT
ejpam-5256	179	6	=	=	PUNCT
ejpam-5256	179	7	x−f−1(v	x−f−1(v	PUNCT
ejpam-5256	179	8	)	)	PUNCT
ejpam-5256	179	9	is	be	AUX
ejpam-5256	179	10	θ̃-open	θ̃-open	ADJ
ejpam-5256	179	11	in	in	ADP
ejpam-5256	179	12	x.	x.	PROPN
ejpam-5256	179	13	therefore	therefore	ADV
ejpam-5256	179	14	,	,	PUNCT
ejpam-5256	179	15	f−1(v	f−1(v	PROPN
ejpam-5256	179	16	)	)	PUNCT
ejpam-5256	179	17	is	be	AUX
ejpam-5256	179	18	θ̃-closed	θ̃-close	VERB
ejpam-5256	179	19	in	in	ADP
ejpam-5256	179	20	x.	x.	PROPN
ejpam-5256	179	21	(	(	PUNCT
ejpam-5256	179	22	←	←	PROPN
ejpam-5256	179	23	)	)	PUNCT
ejpam-5256	179	24	assume	assume	VERB
ejpam-5256	179	25	that	that	SCONJ
ejpam-5256	179	26	f−1(v	f−1(v	PROPN
ejpam-5256	179	27	)	)	PUNCT
ejpam-5256	179	28	is	be	AUX
ejpam-5256	179	29	θ̃-closed	θ̃-close	VERB
ejpam-5256	179	30	in	in	ADP
ejpam-5256	179	31	x	x	PUNCT
ejpam-5256	179	32	for	for	ADP
ejpam-5256	179	33	each	each	DET
ejpam-5256	179	34	µ′-closed	µ′-close	VERB
ejpam-5256	179	35	set	set	VERB
ejpam-5256	179	36	v	v	NOUN
ejpam-5256	179	37	in	in	ADP
ejpam-5256	179	38	y	y	PROPN
ejpam-5256	179	39	.	.	PUNCT
ejpam-5256	180	1	let	let	VERB
ejpam-5256	180	2	h	h	PRON
ejpam-5256	180	3	be	be	AUX
ejpam-5256	180	4	µ′-open	µ′-open	NUM
ejpam-5256	180	5	in	in	ADP
ejpam-5256	180	6	y	y	PROPN
ejpam-5256	180	7	.	.	PUNCT
ejpam-5256	181	1	then	then	ADV
ejpam-5256	181	2	y	y	PROPN
ejpam-5256	181	3	−h	−h	VERB
ejpam-5256	181	4	is	be	AUX
ejpam-5256	181	5	µ′-closed	µ′-closed	PROPN
ejpam-5256	181	6	in	in	ADP
ejpam-5256	181	7	y	y	PROPN
ejpam-5256	181	8	.	.	PUNCT
ejpam-5256	182	1	by	by	ADP
ejpam-5256	182	2	the	the	DET
ejpam-5256	182	3	assumption	assumption	NOUN
ejpam-5256	182	4	,	,	PUNCT
ejpam-5256	182	5	f−1(y	f−1(y	PROPN
ejpam-5256	182	6	−h	−h	NOUN
ejpam-5256	182	7	)	)	PUNCT
ejpam-5256	182	8	is	be	AUX
ejpam-5256	182	9	θ̃-closed	θ̃-close	VERB
ejpam-5256	182	10	in	in	ADP
ejpam-5256	182	11	x.	x.	NOUN
ejpam-5256	182	12	it	it	PRON
ejpam-5256	182	13	follows	follow	VERB
ejpam-5256	182	14	that	that	SCONJ
ejpam-5256	182	15	f−1(h	f−1(h	PROPN
ejpam-5256	182	16	)	)	PUNCT
ejpam-5256	182	17	is	be	AUX
ejpam-5256	182	18	θ̃-open	θ̃-open	ADJ
ejpam-5256	182	19	in	in	ADP
ejpam-5256	182	20	x.	x.	PROPN
ejpam-5256	182	21	hence	hence	ADV
ejpam-5256	182	22	,	,	PUNCT
ejpam-5256	182	23	f	f	PROPN
ejpam-5256	182	24	is	be	AUX
ejpam-5256	182	25	θ̃-continuous	θ̃-continuous	PROPN
ejpam-5256	182	26	.	.	PUNCT
ejpam-5256	183	1	j.	j.	PROPN
ejpam-5256	183	2	khampakdee	khampakdee	PROPN
ejpam-5256	183	3	/	/	PUNCT
ejpam-5256	183	4	eur	eur	PROPN
ejpam-5256	183	5	.	.	PUNCT
ejpam-5256	184	1	j.	j.	PROPN
ejpam-5256	184	2	pure	pure	PROPN
ejpam-5256	184	3	appl	appl	PROPN
ejpam-5256	184	4	.	.	PROPN
ejpam-5256	184	5	math	math	PROPN
ejpam-5256	184	6	,	,	PUNCT
ejpam-5256	184	7	17	17	NUM
ejpam-5256	184	8	(	(	PUNCT
ejpam-5256	184	9	3	3	NUM
ejpam-5256	184	10	)	)	PUNCT
ejpam-5256	184	11	(	(	PUNCT
ejpam-5256	184	12	2024	2024	NUM
ejpam-5256	184	13	)	)	PUNCT
ejpam-5256	184	14	,	,	PUNCT
ejpam-5256	184	15	1869	1869	NUM
ejpam-5256	184	16	-	-	SYM
ejpam-5256	184	17	1876	1876	NUM
ejpam-5256	184	18	1874	1874	NUM
ejpam-5256	184	19	definition	definition	NOUN
ejpam-5256	184	20	9	9	NUM
ejpam-5256	184	21	.	.	PUNCT
ejpam-5256	185	1	let	let	VERB
ejpam-5256	185	2	(	(	PUNCT
ejpam-5256	185	3	x,µ	x,µ	NOUN
ejpam-5256	185	4	)	)	PUNCT
ejpam-5256	185	5	and	and	CCONJ
ejpam-5256	185	6	(	(	PUNCT
ejpam-5256	185	7	y	y	NOUN
ejpam-5256	185	8	,	,	PUNCT
ejpam-5256	185	9	µ′	µ′	PUNCT
ejpam-5256	185	10	)	)	PUNCT
ejpam-5256	186	1	be	be	AUX
ejpam-5256	186	2	generalized	generalize	VERB
ejpam-5256	186	3	topological	topological	ADJ
ejpam-5256	186	4	spaces	space	NOUN
ejpam-5256	186	5	.	.	PUNCT
ejpam-5256	187	1	then	then	ADV
ejpam-5256	187	2	,	,	PUNCT
ejpam-5256	187	3	a	a	DET
ejpam-5256	187	4	function	function	NOUN
ejpam-5256	187	5	f	f	NOUN
ejpam-5256	187	6	from	from	ADP
ejpam-5256	187	7	(	(	PUNCT
ejpam-5256	187	8	x,µ	x,µ	NOUN
ejpam-5256	187	9	)	)	PUNCT
ejpam-5256	187	10	into	into	ADP
ejpam-5256	187	11	(	(	PUNCT
ejpam-5256	187	12	y	y	NOUN
ejpam-5256	187	13	,	,	PUNCT
ejpam-5256	187	14	µ′	µ′	NUM
ejpam-5256	187	15	)	)	PUNCT
ejpam-5256	187	16	is	be	AUX
ejpam-5256	187	17	called	call	VERB
ejpam-5256	187	18	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	187	19	if	if	SCONJ
ejpam-5256	187	20	f−1(g	f−1(g	PROPN
ejpam-5256	187	21	)	)	PUNCT
ejpam-5256	187	22	is	be	AUX
ejpam-5256	187	23	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	187	24	in	in	ADP
ejpam-5256	187	25	x	x	PUNCT
ejpam-5256	187	26	for	for	SCONJ
ejpam-5256	187	27	each	each	DET
ejpam-5256	187	28	µ′-open	µ′-open	NUM
ejpam-5256	187	29	set	set	VERB
ejpam-5256	187	30	g	g	NOUN
ejpam-5256	187	31	in	in	ADP
ejpam-5256	187	32	y	y	PROPN
ejpam-5256	187	33	.	.	PUNCT
ejpam-5256	188	1	it	it	PRON
ejpam-5256	188	2	is	be	AUX
ejpam-5256	188	3	easy	easy	ADJ
ejpam-5256	188	4	to	to	PART
ejpam-5256	188	5	see	see	VERB
ejpam-5256	188	6	that	that	SCONJ
ejpam-5256	188	7	all	all	DET
ejpam-5256	188	8	θ̃-continuous	θ̃-continuous	ADJ
ejpam-5256	188	9	functions	function	NOUN
ejpam-5256	188	10	are	be	AUX
ejpam-5256	188	11	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	188	12	.	.	PUNCT
ejpam-5256	189	1	but	but	CCONJ
ejpam-5256	189	2	the	the	DET
ejpam-5256	189	3	converse	converse	NOUN
ejpam-5256	189	4	need	need	VERB
ejpam-5256	189	5	not	not	PART
ejpam-5256	189	6	to	to	PART
ejpam-5256	189	7	be	be	AUX
ejpam-5256	189	8	true	true	ADJ
ejpam-5256	189	9	as	as	ADP
ejpam-5256	189	10	the	the	DET
ejpam-5256	189	11	following	follow	VERB
ejpam-5256	189	12	example	example	NOUN
ejpam-5256	189	13	.	.	PUNCT
ejpam-5256	190	1	example	example	NOUN
ejpam-5256	190	2	5	5	NUM
ejpam-5256	190	3	.	.	PUNCT
ejpam-5256	191	1	by	by	ADP
ejpam-5256	191	2	the	the	DET
ejpam-5256	191	3	generalized	generalized	ADJ
ejpam-5256	191	4	topological	topological	ADJ
ejpam-5256	191	5	spaces	space	NOUN
ejpam-5256	191	6	(	(	PUNCT
ejpam-5256	191	7	x,µ	x,µ	NOUN
ejpam-5256	191	8	)	)	PUNCT
ejpam-5256	191	9	and	and	CCONJ
ejpam-5256	191	10	(	(	PUNCT
ejpam-5256	191	11	y	y	NOUN
ejpam-5256	191	12	,	,	PUNCT
ejpam-5256	191	13	µ′	µ′	NOUN
ejpam-5256	191	14	)	)	PUNCT
ejpam-5256	191	15	in	in	ADP
ejpam-5256	191	16	example	example	NOUN
ejpam-5256	191	17	4	4	NUM
ejpam-5256	191	18	,	,	PUNCT
ejpam-5256	191	19	if	if	SCONJ
ejpam-5256	191	20	f	f	PROPN
ejpam-5256	191	21	is	be	AUX
ejpam-5256	191	22	a	a	DET
ejpam-5256	191	23	function	function	NOUN
ejpam-5256	191	24	from	from	ADP
ejpam-5256	191	25	(	(	PUNCT
ejpam-5256	191	26	x,µ	x,µ	NOUN
ejpam-5256	191	27	)	)	PUNCT
ejpam-5256	191	28	into	into	ADP
ejpam-5256	191	29	(	(	PUNCT
ejpam-5256	191	30	y	y	NOUN
ejpam-5256	191	31	,	,	PUNCT
ejpam-5256	191	32	µ′	µ′	NOUN
ejpam-5256	191	33	)	)	PUNCT
ejpam-5256	191	34	defined	define	VERB
ejpam-5256	191	35	by	by	ADP
ejpam-5256	191	36	f(a	f(a	PROPN
ejpam-5256	191	37	)	)	PUNCT
ejpam-5256	191	38	=	=	SYM
ejpam-5256	191	39	t	t	PROPN
ejpam-5256	191	40	,	,	PUNCT
ejpam-5256	191	41	f(b	f(b	PROPN
ejpam-5256	191	42	)	)	PUNCT
ejpam-5256	191	43	=	=	SYM
ejpam-5256	191	44	t	t	PROPN
ejpam-5256	191	45	,	,	PUNCT
ejpam-5256	191	46	f(c	f(c	PROPN
ejpam-5256	191	47	)	)	PUNCT
ejpam-5256	191	48	=	=	SYM
ejpam-5256	191	49	v	v	NOUN
ejpam-5256	191	50	,	,	PUNCT
ejpam-5256	191	51	f(d	f(d	PROPN
ejpam-5256	191	52	)	)	PUNCT
ejpam-5256	191	53	=	=	SYM
ejpam-5256	191	54	w.	w.	PROPN
ejpam-5256	191	55	then	then	ADV
ejpam-5256	191	56	,	,	PUNCT
ejpam-5256	191	57	only	only	ADV
ejpam-5256	191	58	∅	∅	NOUN
ejpam-5256	191	59	,	,	PUNCT
ejpam-5256	191	60	{	{	PUNCT
ejpam-5256	191	61	a	a	X
ejpam-5256	191	62	}	}	PUNCT
ejpam-5256	191	63	,	,	PUNCT
ejpam-5256	191	64	{	{	PUNCT
ejpam-5256	191	65	b	b	NOUN
ejpam-5256	191	66	}	}	PUNCT
ejpam-5256	191	67	,	,	PUNCT
ejpam-5256	191	68	{	{	PUNCT
ejpam-5256	191	69	a	a	DET
ejpam-5256	191	70	,	,	PUNCT
ejpam-5256	191	71	b	b	NOUN
ejpam-5256	191	72	}	}	PUNCT
ejpam-5256	191	73	and	and	CCONJ
ejpam-5256	191	74	{	{	PUNCT
ejpam-5256	191	75	a	a	PRON
ejpam-5256	191	76	,	,	PUNCT
ejpam-5256	191	77	b	b	NOUN
ejpam-5256	191	78	,	,	PUNCT
ejpam-5256	191	79	c	c	NOUN
ejpam-5256	191	80	}	}	PUNCT
ejpam-5256	191	81	are	be	AUX
ejpam-5256	191	82	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	191	83	in	in	ADP
ejpam-5256	191	84	x.	x.	NOUN
ejpam-5256	191	85	it	it	PRON
ejpam-5256	191	86	can	can	AUX
ejpam-5256	191	87	be	be	AUX
ejpam-5256	191	88	checked	check	VERB
ejpam-5256	191	89	that	that	DET
ejpam-5256	191	90	f−1(∅	f−1(∅	NOUN
ejpam-5256	191	91	)	)	PUNCT
ejpam-5256	192	1	=	=	SYM
ejpam-5256	192	2	∅	∅	NOUN
ejpam-5256	192	3	,	,	PUNCT
ejpam-5256	192	4	f−1({t	f−1({t	PROPN
ejpam-5256	192	5	}	}	PUNCT
ejpam-5256	192	6	)	)	PUNCT
ejpam-5256	192	7	=	=	PRON
ejpam-5256	192	8	{	{	PUNCT
ejpam-5256	192	9	a	a	DET
ejpam-5256	192	10	,	,	PUNCT
ejpam-5256	192	11	b	b	NOUN
ejpam-5256	192	12	}	}	PUNCT
ejpam-5256	192	13	and	and	CCONJ
ejpam-5256	192	14	f−1({t	f−1({t	ADJ
ejpam-5256	192	15	,	,	PUNCT
ejpam-5256	192	16	v	v	NOUN
ejpam-5256	192	17	}	}	PUNCT
ejpam-5256	192	18	)	)	PUNCT
ejpam-5256	192	19	=	=	PRON
ejpam-5256	192	20	{	{	PUNCT
ejpam-5256	192	21	a	a	PRON
ejpam-5256	192	22	,	,	PUNCT
ejpam-5256	192	23	b	b	NOUN
ejpam-5256	192	24	,	,	PUNCT
ejpam-5256	192	25	c	c	NOUN
ejpam-5256	192	26	}	}	PUNCT
ejpam-5256	192	27	.	.	PUNCT
ejpam-5256	193	1	thus	thus	ADV
ejpam-5256	193	2	,	,	PUNCT
ejpam-5256	193	3	f−1(g	f−1(g	PROPN
ejpam-5256	193	4	)	)	PUNCT
ejpam-5256	193	5	is	be	AUX
ejpam-5256	193	6	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	193	7	in	in	ADP
ejpam-5256	193	8	x	x	PUNCT
ejpam-5256	193	9	for	for	SCONJ
ejpam-5256	193	10	each	each	DET
ejpam-5256	193	11	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	193	12	set	set	VERB
ejpam-5256	193	13	g	g	NOUN
ejpam-5256	193	14	in	in	ADP
ejpam-5256	193	15	y	y	PROPN
ejpam-5256	193	16	.	.	PUNCT
ejpam-5256	194	1	consequently	consequently	ADV
ejpam-5256	194	2	,	,	PUNCT
ejpam-5256	194	3	f	f	PROPN
ejpam-5256	194	4	is	be	AUX
ejpam-5256	194	5	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	194	6	.	.	PUNCT
ejpam-5256	195	1	since	since	SCONJ
ejpam-5256	195	2	there	there	PRON
ejpam-5256	195	3	exists	exist	VERB
ejpam-5256	195	4	a	a	DET
ejpam-5256	195	5	µ′-open	µ′-open	NUM
ejpam-5256	195	6	set	set	VERB
ejpam-5256	195	7	{	{	PUNCT
ejpam-5256	195	8	t	t	NOUN
ejpam-5256	195	9	}	}	PUNCT
ejpam-5256	195	10	in	in	ADP
ejpam-5256	195	11	y	y	PROPN
ejpam-5256	195	12	such	such	ADJ
ejpam-5256	195	13	that	that	PRON
ejpam-5256	195	14	f−1({t	f−1({t	PROPN
ejpam-5256	195	15	}	}	PUNCT
ejpam-5256	195	16	)	)	PUNCT
ejpam-5256	195	17	=	=	PRON
ejpam-5256	195	18	{	{	PUNCT
ejpam-5256	195	19	a	a	DET
ejpam-5256	195	20	,	,	PUNCT
ejpam-5256	195	21	b	b	NOUN
ejpam-5256	195	22	}	}	PUNCT
ejpam-5256	195	23	is	be	AUX
ejpam-5256	195	24	not	not	PART
ejpam-5256	195	25	θ̃-open	θ̃-open	ADJ
ejpam-5256	195	26	in	in	ADP
ejpam-5256	195	27	x.	x.	PROPN
ejpam-5256	195	28	therefore	therefore	ADV
ejpam-5256	195	29	,	,	PUNCT
ejpam-5256	195	30	f	f	PROPN
ejpam-5256	195	31	is	be	AUX
ejpam-5256	195	32	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	195	33	but	but	CCONJ
ejpam-5256	195	34	f	f	PROPN
ejpam-5256	195	35	is	be	AUX
ejpam-5256	195	36	not	not	PART
ejpam-5256	195	37	θ̃-continuous	θ̃-continuous	PROPN
ejpam-5256	195	38	.	.	PUNCT
ejpam-5256	196	1	theorem	theorem	ADJ
ejpam-5256	196	2	8	8	NUM
ejpam-5256	196	3	.	.	PUNCT
ejpam-5256	197	1	let	let	VERB
ejpam-5256	197	2	(	(	PUNCT
ejpam-5256	197	3	x,µ	x,µ	NOUN
ejpam-5256	197	4	)	)	PUNCT
ejpam-5256	197	5	and	and	CCONJ
ejpam-5256	197	6	(	(	PUNCT
ejpam-5256	197	7	y	y	NOUN
ejpam-5256	197	8	,	,	PUNCT
ejpam-5256	197	9	µ′	µ′	PUNCT
ejpam-5256	197	10	)	)	PUNCT
ejpam-5256	198	1	be	be	AUX
ejpam-5256	198	2	generalized	generalize	VERB
ejpam-5256	198	3	topological	topological	ADJ
ejpam-5256	198	4	spaces	space	NOUN
ejpam-5256	198	5	.	.	PUNCT
ejpam-5256	199	1	then	then	ADV
ejpam-5256	199	2	,	,	PUNCT
ejpam-5256	199	3	for	for	ADP
ejpam-5256	199	4	a	a	DET
ejpam-5256	199	5	function	function	NOUN
ejpam-5256	199	6	f	f	NOUN
ejpam-5256	199	7	from	from	ADP
ejpam-5256	199	8	(	(	PUNCT
ejpam-5256	199	9	x,µ	x,µ	NOUN
ejpam-5256	199	10	)	)	PUNCT
ejpam-5256	199	11	into	into	ADP
ejpam-5256	199	12	(	(	PUNCT
ejpam-5256	199	13	y	y	NOUN
ejpam-5256	199	14	,	,	PUNCT
ejpam-5256	199	15	µ′	µ′	NUM
ejpam-5256	199	16	)	)	PUNCT
ejpam-5256	199	17	the	the	DET
ejpam-5256	199	18	followings	following	NOUN
ejpam-5256	199	19	are	be	AUX
ejpam-5256	199	20	equivalent	equivalent	ADJ
ejpam-5256	199	21	:	:	PUNCT
ejpam-5256	199	22	a	a	X
ejpam-5256	199	23	)	)	PUNCT
ejpam-5256	199	24	f	f	NOUN
ejpam-5256	199	25	is	be	AUX
ejpam-5256	199	26	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	199	27	;	;	PUNCT
ejpam-5256	199	28	b	b	X
ejpam-5256	199	29	)	)	PUNCT
ejpam-5256	199	30	f−1(f	f−1(f	NOUN
ejpam-5256	199	31	)	)	PUNCT
ejpam-5256	199	32	is	be	AUX
ejpam-5256	199	33	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	199	34	in	in	ADP
ejpam-5256	199	35	x	x	PUNCT
ejpam-5256	199	36	for	for	ADP
ejpam-5256	199	37	each	each	DET
ejpam-5256	199	38	µ′-closed	µ′-close	VERB
ejpam-5256	199	39	set	set	VERB
ejpam-5256	199	40	f	f	PROPN
ejpam-5256	199	41	in	in	ADP
ejpam-5256	199	42	y	y	PROPN
ejpam-5256	199	43	;	;	PUNCT
ejpam-5256	200	1	c	c	X
ejpam-5256	200	2	)	)	PUNCT
ejpam-5256	200	3	for	for	ADP
ejpam-5256	200	4	all	all	DET
ejpam-5256	200	5	x	x	SYM
ejpam-5256	200	6	∈	∈	PROPN
ejpam-5256	200	7	x	x	NOUN
ejpam-5256	200	8	,	,	PUNCT
ejpam-5256	200	9	there	there	PRON
ejpam-5256	200	10	exists	exist	VERB
ejpam-5256	200	11	an	an	DET
ejpam-5256	200	12	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	200	13	set	set	NOUN
ejpam-5256	200	14	h	h	NOUN
ejpam-5256	200	15	in	in	ADP
ejpam-5256	200	16	x	x	INTJ
ejpam-5256	200	17	such	such	ADJ
ejpam-5256	200	18	that	that	SCONJ
ejpam-5256	200	19	x	x	SYM
ejpam-5256	200	20	∈	∈	PROPN
ejpam-5256	200	21	h	h	NOUN
ejpam-5256	200	22	and	and	CCONJ
ejpam-5256	200	23	f(h	f(h	PROPN
ejpam-5256	200	24	)	)	PUNCT
ejpam-5256	200	25	⊆	⊆	NUM
ejpam-5256	200	26	g	g	NOUN
ejpam-5256	200	27	,	,	PUNCT
ejpam-5256	200	28	for	for	ADP
ejpam-5256	200	29	all	all	DET
ejpam-5256	200	30	g	g	PROPN
ejpam-5256	200	31	∈	∈	PROPN
ejpam-5256	200	32	µ′	µ′	NOUN
ejpam-5256	200	33	such	such	ADJ
ejpam-5256	200	34	that	that	SCONJ
ejpam-5256	200	35	f(x	f(x	PROPN
ejpam-5256	200	36	)	)	PUNCT
ejpam-5256	200	37	∈	∈	PROPN
ejpam-5256	200	38	g	g	NOUN
ejpam-5256	200	39	proof	proof	NOUN
ejpam-5256	200	40	.	.	PUNCT
ejpam-5256	201	1	a	a	X
ejpam-5256	201	2	)	)	PUNCT
ejpam-5256	201	3	→	→	SYM
ejpam-5256	201	4	b	b	X
ejpam-5256	201	5	)	)	PUNCT
ejpam-5256	201	6	let	let	VERB
ejpam-5256	201	7	f	f	PRON
ejpam-5256	201	8	be	be	AUX
ejpam-5256	201	9	an	an	DET
ejpam-5256	201	10	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	201	11	function	function	NOUN
ejpam-5256	201	12	.	.	PUNCT
ejpam-5256	202	1	then	then	ADV
ejpam-5256	202	2	,	,	PUNCT
ejpam-5256	202	3	f−1(g	f−1(g	PROPN
ejpam-5256	202	4	)	)	PUNCT
ejpam-5256	202	5	is	be	AUX
ejpam-5256	202	6	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	202	7	in	in	ADP
ejpam-5256	202	8	x	x	PUNCT
ejpam-5256	202	9	for	for	SCONJ
ejpam-5256	202	10	all	all	DET
ejpam-5256	202	11	µ′-open	µ′-open	NUM
ejpam-5256	202	12	set	set	VERB
ejpam-5256	202	13	g	g	NOUN
ejpam-5256	202	14	in	in	ADP
ejpam-5256	202	15	y	y	PROPN
ejpam-5256	202	16	.	.	PUNCT
ejpam-5256	203	1	assume	assume	VERB
ejpam-5256	203	2	that	that	SCONJ
ejpam-5256	203	3	f	f	PROPN
ejpam-5256	203	4	is	be	AUX
ejpam-5256	203	5	µ′-closed	µ′-closed	PROPN
ejpam-5256	203	6	in	in	ADP
ejpam-5256	203	7	y	y	PROPN
ejpam-5256	203	8	.	.	PUNCT
ejpam-5256	204	1	we	we	PRON
ejpam-5256	204	2	get	get	VERB
ejpam-5256	204	3	y	y	NOUN
ejpam-5256	204	4	−	−	PROPN
ejpam-5256	204	5	f	f	PROPN
ejpam-5256	204	6	is	be	AUX
ejpam-5256	204	7	µ′-open	µ′-open	NUM
ejpam-5256	204	8	in	in	ADP
ejpam-5256	204	9	y	y	PROPN
ejpam-5256	204	10	.	.	PUNCT
ejpam-5256	205	1	hence	hence	ADV
ejpam-5256	205	2	f−1(y	f−1(y	PROPN
ejpam-5256	205	3	−f	−f	PROPN
ejpam-5256	205	4	)	)	PUNCT
ejpam-5256	205	5	is	be	AUX
ejpam-5256	205	6	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	205	7	in	in	ADP
ejpam-5256	205	8	x.	x.	PROPN
ejpam-5256	205	9	thus	thus	ADV
ejpam-5256	205	10	,	,	PUNCT
ejpam-5256	205	11	x−f−1(f	x−f−1(f	NUM
ejpam-5256	205	12	)	)	PUNCT
ejpam-5256	205	13	is	be	AUX
ejpam-5256	205	14	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	205	15	in	in	ADP
ejpam-5256	205	16	x.	x.	NOUN
ejpam-5256	205	17	therefore	therefore	ADV
ejpam-5256	205	18	,	,	PUNCT
ejpam-5256	205	19	f−1(f	f−1(f	PROPN
ejpam-5256	205	20	)	)	PUNCT
ejpam-5256	205	21	is	be	AUX
ejpam-5256	205	22	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	205	23	in	in	ADP
ejpam-5256	205	24	x.	x.	PROPN
ejpam-5256	205	25	b	b	PROPN
ejpam-5256	205	26	)	)	PUNCT
ejpam-5256	205	27	→	→	SYM
ejpam-5256	205	28	c	c	X
ejpam-5256	205	29	)	)	PUNCT
ejpam-5256	205	30	assume	assume	VERB
ejpam-5256	205	31	that	that	SCONJ
ejpam-5256	205	32	f−1(f	f−1(f	PROPN
ejpam-5256	205	33	)	)	PUNCT
ejpam-5256	205	34	is	be	AUX
ejpam-5256	205	35	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	205	36	in	in	ADP
ejpam-5256	205	37	x	x	PUNCT
ejpam-5256	205	38	for	for	ADP
ejpam-5256	205	39	each	each	DET
ejpam-5256	205	40	µ′-closed	µ′-close	VERB
ejpam-5256	205	41	set	set	VERB
ejpam-5256	205	42	f	f	PROPN
ejpam-5256	205	43	in	in	ADP
ejpam-5256	205	44	y	y	PROPN
ejpam-5256	205	45	.	.	PUNCT
ejpam-5256	206	1	let	let	VERB
ejpam-5256	206	2	x	x	PUNCT
ejpam-5256	206	3	∈	∈	PROPN
ejpam-5256	206	4	x	x	X
ejpam-5256	206	5	and	and	CCONJ
ejpam-5256	206	6	g	g	PROPN
ejpam-5256	206	7	∈	∈	PROPN
ejpam-5256	206	8	µ′	µ′	NOUN
ejpam-5256	206	9	such	such	ADJ
ejpam-5256	206	10	that	that	SCONJ
ejpam-5256	206	11	f(x	f(x	PROPN
ejpam-5256	206	12	)	)	PUNCT
ejpam-5256	206	13	∈	∈	PROPN
ejpam-5256	206	14	g.	g.	NOUN
ejpam-5256	206	15	then	then	ADV
ejpam-5256	206	16	,	,	PUNCT
ejpam-5256	206	17	there	there	PRON
ejpam-5256	206	18	exists	exist	VERB
ejpam-5256	206	19	a	a	DET
ejpam-5256	206	20	µ′-closed	µ′-closed	NUM
ejpam-5256	206	21	set	set	VERB
ejpam-5256	206	22	f	f	PROPN
ejpam-5256	206	23	in	in	ADP
ejpam-5256	206	24	y	y	PRON
ejpam-5256	207	1	such	such	ADJ
ejpam-5256	207	2	that	that	SCONJ
ejpam-5256	207	3	g	g	NOUN
ejpam-5256	207	4	=	=	PROPN
ejpam-5256	207	5	x−f	x−f	PROPN
ejpam-5256	207	6	.	.	PUNCT
ejpam-5256	208	1	since	since	SCONJ
ejpam-5256	208	2	f(x	f(x	PROPN
ejpam-5256	208	3	)	)	PUNCT
ejpam-5256	208	4	∈	∈	PROPN
ejpam-5256	208	5	g	g	PROPN
ejpam-5256	208	6	,	,	PUNCT
ejpam-5256	208	7	x	x	PROPN
ejpam-5256	208	8	∈	∈	PROPN
ejpam-5256	208	9	f−1(g	f−1(g	PROPN
ejpam-5256	208	10	)	)	PUNCT
ejpam-5256	208	11	=	=	SYM
ejpam-5256	208	12	f−1(x−f	f−1(x−f	ADJ
ejpam-5256	208	13	)	)	PUNCT
ejpam-5256	208	14	=	=	SYM
ejpam-5256	208	15	y	y	PROPN
ejpam-5256	208	16	−f−1(f	−f−1(f	NOUN
ejpam-5256	208	17	)	)	PUNCT
ejpam-5256	208	18	.	.	PUNCT
ejpam-5256	209	1	by	by	ADP
ejpam-5256	209	2	the	the	DET
ejpam-5256	209	3	assumption	assumption	NOUN
ejpam-5256	209	4	,	,	PUNCT
ejpam-5256	209	5	f−1(f	f−1(f	PROPN
ejpam-5256	209	6	)	)	PUNCT
ejpam-5256	209	7	is	be	AUX
ejpam-5256	209	8	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	209	9	in	in	ADP
ejpam-5256	209	10	y	y	PROPN
ejpam-5256	209	11	.	.	PUNCT
ejpam-5256	210	1	it	it	PRON
ejpam-5256	210	2	follows	follow	VERB
ejpam-5256	210	3	that	that	PRON
ejpam-5256	210	4	f−1(g	f−1(g	PROPN
ejpam-5256	210	5	)	)	PUNCT
ejpam-5256	210	6	is	be	AUX
ejpam-5256	210	7	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	210	8	in	in	ADP
ejpam-5256	210	9	x	x	X
ejpam-5256	210	10	and	and	CCONJ
ejpam-5256	210	11	f(f−1(g	f(f−1(g	PROPN
ejpam-5256	210	12	)	)	PUNCT
ejpam-5256	210	13	)	)	PUNCT
ejpam-5256	211	1	⊆	⊆	NUM
ejpam-5256	211	2	g.	g.	NOUN
ejpam-5256	211	3	c	c	NOUN
ejpam-5256	211	4	)	)	PUNCT
ejpam-5256	211	5	→	→	SYM
ejpam-5256	211	6	a	a	X
ejpam-5256	211	7	)	)	PUNCT
ejpam-5256	211	8	assume	assume	VERB
ejpam-5256	211	9	that	that	SCONJ
ejpam-5256	211	10	for	for	ADP
ejpam-5256	211	11	all	all	DET
ejpam-5256	211	12	x	x	SYM
ejpam-5256	211	13	∈	∈	NOUN
ejpam-5256	211	14	x	x	NOUN
ejpam-5256	211	15	,	,	PUNCT
ejpam-5256	211	16	there	there	PRON
ejpam-5256	211	17	exists	exist	VERB
ejpam-5256	211	18	an	an	DET
ejpam-5256	211	19	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	211	20	set	set	NOUN
ejpam-5256	211	21	h	h	NOUN
ejpam-5256	211	22	in	in	ADP
ejpam-5256	211	23	x	x	INTJ
ejpam-5256	212	1	such	such	ADJ
ejpam-5256	212	2	that	that	SCONJ
ejpam-5256	212	3	x	x	SYM
ejpam-5256	212	4	∈	∈	PROPN
ejpam-5256	212	5	h	h	NOUN
ejpam-5256	212	6	and	and	CCONJ
ejpam-5256	212	7	f(h	f(h	PROPN
ejpam-5256	212	8	)	)	PUNCT
ejpam-5256	212	9	⊆	⊆	NUM
ejpam-5256	212	10	g	g	NOUN
ejpam-5256	212	11	,	,	PUNCT
ejpam-5256	212	12	for	for	ADP
ejpam-5256	212	13	all	all	DET
ejpam-5256	212	14	g	g	PROPN
ejpam-5256	212	15	∈	∈	PROPN
ejpam-5256	212	16	µ′	µ′	NOUN
ejpam-5256	212	17	such	such	ADJ
ejpam-5256	212	18	that	that	SCONJ
ejpam-5256	212	19	f(x	f(x	PROPN
ejpam-5256	212	20	)	)	PUNCT
ejpam-5256	212	21	∈	∈	PROPN
ejpam-5256	212	22	g.	g.	NOUN
ejpam-5256	212	23	let	let	VERB
ejpam-5256	212	24	k	k	PRON
ejpam-5256	212	25	be	be	AUX
ejpam-5256	212	26	µ′-open	µ′-open	NUM
ejpam-5256	212	27	in	in	ADP
ejpam-5256	212	28	y	y	PROPN
ejpam-5256	212	29	.	.	PUNCT
ejpam-5256	213	1	case	case	NOUN
ejpam-5256	213	2	1	1	NUM
ejpam-5256	213	3	.	.	PUNCT
ejpam-5256	214	1	f−1(k	f−1(k	PROPN
ejpam-5256	214	2	)	)	PUNCT
ejpam-5256	215	1	=	=	PUNCT
ejpam-5256	215	2	∅.	∅.	NOUN
ejpam-5256	215	3	since	since	SCONJ
ejpam-5256	215	4	sθ̃	sθ̃	PRON
ejpam-5256	215	5	is	be	AUX
ejpam-5256	215	6	a	a	DET
ejpam-5256	215	7	generalized	generalized	ADJ
ejpam-5256	215	8	topology	topology	NOUN
ejpam-5256	215	9	on	on	ADP
ejpam-5256	215	10	x	x	NOUN
ejpam-5256	215	11	,	,	PUNCT
ejpam-5256	215	12	∅	∅	NOUN
ejpam-5256	215	13	∈	∈	PROPN
ejpam-5256	215	14	sθ̃.	sθ̃.	PROPN
ejpam-5256	215	15	hence	hence	ADV
ejpam-5256	215	16	,	,	PUNCT
ejpam-5256	215	17	f−1(k	f−1(k	PROPN
ejpam-5256	215	18	)	)	PUNCT
ejpam-5256	215	19	∈	∈	PROPN
ejpam-5256	215	20	sθ̃.	sθ̃.	PROPN
ejpam-5256	215	21	thus	thus	ADV
ejpam-5256	215	22	,	,	PUNCT
ejpam-5256	215	23	f−1(k	f−1(k	PROPN
ejpam-5256	215	24	)	)	PUNCT
ejpam-5256	215	25	is	be	AUX
ejpam-5256	215	26	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	215	27	in	in	ADP
ejpam-5256	215	28	x.	x.	NOUN
ejpam-5256	215	29	case	case	NOUN
ejpam-5256	215	30	2	2	NUM
ejpam-5256	215	31	.	.	PUNCT
ejpam-5256	216	1	f−1(k	f−1(k	PROPN
ejpam-5256	216	2	)	)	PUNCT
ejpam-5256	217	1	̸=	̸=	PROPN
ejpam-5256	217	2	∅.	∅.	ADV
ejpam-5256	217	3	let	let	VERB
ejpam-5256	217	4	x	x	SYM
ejpam-5256	217	5	∈	∈	PROPN
ejpam-5256	217	6	f−1(k	f−1(k	PROPN
ejpam-5256	217	7	)	)	PUNCT
ejpam-5256	217	8	,	,	PUNCT
ejpam-5256	217	9	then	then	ADV
ejpam-5256	217	10	f(x	f(x	PROPN
ejpam-5256	217	11	)	)	PUNCT
ejpam-5256	217	12	∈	∈	PROPN
ejpam-5256	217	13	k.	k.	PROPN
ejpam-5256	218	1	hence	hence	ADV
ejpam-5256	218	2	,	,	PUNCT
ejpam-5256	218	3	x	x	PUNCT
ejpam-5256	218	4	∈	∈	PROPN
ejpam-5256	218	5	x	x	PUNCT
ejpam-5256	218	6	such	such	ADJ
ejpam-5256	218	7	that	that	SCONJ
ejpam-5256	218	8	f(x	f(x	PROPN
ejpam-5256	218	9	)	)	PUNCT
ejpam-5256	218	10	∈	∈	PROPN
ejpam-5256	218	11	k	k	PROPN
ejpam-5256	218	12	and	and	CCONJ
ejpam-5256	218	13	k	k	PROPN
ejpam-5256	218	14	is	be	AUX
ejpam-5256	218	15	µ′-open	µ′-open	NUM
ejpam-5256	218	16	in	in	ADP
ejpam-5256	218	17	y	y	PROPN
ejpam-5256	218	18	.	.	PUNCT
ejpam-5256	219	1	by	by	ADP
ejpam-5256	219	2	the	the	DET
ejpam-5256	219	3	assumption	assumption	NOUN
ejpam-5256	219	4	,	,	PUNCT
ejpam-5256	219	5	there	there	PRON
ejpam-5256	219	6	exists	exist	VERB
ejpam-5256	219	7	an	an	DET
ejpam-5256	219	8	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	219	9	set	set	NOUN
ejpam-5256	219	10	hx	hx	NOUN
ejpam-5256	219	11	in	in	ADP
ejpam-5256	219	12	x	x	PUNCT
ejpam-5256	219	13	such	such	ADJ
ejpam-5256	219	14	that	that	SCONJ
ejpam-5256	219	15	x	x	SYM
ejpam-5256	219	16	∈	∈	PROPN
ejpam-5256	219	17	hx	hx	PROPN
ejpam-5256	219	18	and	and	CCONJ
ejpam-5256	219	19	f(hx	f(hx	PROPN
ejpam-5256	219	20	)	)	PUNCT
ejpam-5256	219	21	⊆	⊆	NUM
ejpam-5256	219	22	k.	k.	NOUN
ejpam-5256	219	23	it	it	PRON
ejpam-5256	219	24	follows	follow	VERB
ejpam-5256	219	25	that	that	SCONJ
ejpam-5256	219	26	hx	hx	PROPN
ejpam-5256	219	27	⊆	⊆	NUM
ejpam-5256	219	28	f−1(f(hx	f−1(f(hx	NOUN
ejpam-5256	219	29	)	)	PUNCT
ejpam-5256	219	30	)	)	PUNCT
ejpam-5256	220	1	⊆	⊆	NUM
ejpam-5256	220	2	f−1(k	f−1(k	NOUN
ejpam-5256	220	3	)	)	PUNCT
ejpam-5256	220	4	.	.	PUNCT
ejpam-5256	221	1	thus	thus	ADV
ejpam-5256	221	2	,	,	PUNCT
ejpam-5256	221	3	x	x	SYM
ejpam-5256	221	4	∈	∈	PROPN
ejpam-5256	221	5	hx	hx	PROPN
ejpam-5256	221	6	⊆	⊆	NUM
ejpam-5256	221	7	f−1(k	f−1(k	PROPN
ejpam-5256	221	8	)	)	PUNCT
ejpam-5256	221	9	.	.	PUNCT
ejpam-5256	222	1	as	as	SCONJ
ejpam-5256	222	2	x	x	PRON
ejpam-5256	222	3	is	be	AUX
ejpam-5256	222	4	an	an	DET
ejpam-5256	222	5	arbitrary	arbitrary	ADJ
ejpam-5256	222	6	element	element	NOUN
ejpam-5256	222	7	in	in	ADP
ejpam-5256	222	8	f−1(k	f−1(k	PROPN
ejpam-5256	222	9	)	)	PUNCT
ejpam-5256	222	10	,	,	PUNCT
ejpam-5256	222	11	{	{	PUNCT
ejpam-5256	222	12	x	x	NOUN
ejpam-5256	222	13	}	}	PUNCT
ejpam-5256	222	14	⊆	⊆	NUM
ejpam-5256	222	15	hx	hx	PROPN
ejpam-5256	222	16	⊆	⊆	NUM
ejpam-5256	222	17	f−1(k	f−1(k	PROPN
ejpam-5256	222	18	)	)	PUNCT
ejpam-5256	222	19	for	for	ADP
ejpam-5256	222	20	all	all	DET
ejpam-5256	222	21	x	x	SYM
ejpam-5256	222	22	∈	∈	PROPN
ejpam-5256	222	23	f−1(k	f−1(k	PROPN
ejpam-5256	222	24	)	)	PUNCT
ejpam-5256	222	25	.	.	PUNCT
ejpam-5256	223	1	hence	hence	ADV
ejpam-5256	223	2	,	,	PUNCT
ejpam-5256	223	3	⋃	⋃	PROPN
ejpam-5256	223	4	x∈f−1(k	x∈f−1(k	NOUN
ejpam-5256	223	5	)	)	PUNCT
ejpam-5256	223	6	{	{	PUNCT
ejpam-5256	223	7	x	x	X
ejpam-5256	223	8	}	}	PUNCT
ejpam-5256	223	9	⊆	⊆	NUM
ejpam-5256	223	10	⋃	⋃	NOUN
ejpam-5256	223	11	x∈f−1(k	x∈f−1(k	NOUN
ejpam-5256	223	12	)	)	PUNCT
ejpam-5256	223	13	hx	hx	PROPN
ejpam-5256	223	14	⊆	⊆	NUM
ejpam-5256	223	15	f−1(k	f−1(k	PROPN
ejpam-5256	223	16	)	)	PUNCT
ejpam-5256	223	17	.	.	PUNCT
ejpam-5256	224	1	thus	thus	ADV
ejpam-5256	224	2	f−1(k	f−1(k	PROPN
ejpam-5256	224	3	)	)	PUNCT
ejpam-5256	225	1	=	=	SYM
ejpam-5256	225	2	⋃	⋃	NOUN
ejpam-5256	225	3	x∈f−1(k	x∈f−1(k	NOUN
ejpam-5256	225	4	)	)	PUNCT
ejpam-5256	225	5	hx	hx	PROPN
ejpam-5256	225	6	.	.	PUNCT
ejpam-5256	226	1	as	as	SCONJ
ejpam-5256	226	2	sθ̃	sθ̃	PRON
ejpam-5256	226	3	is	be	AUX
ejpam-5256	226	4	a	a	DET
ejpam-5256	226	5	generalized	generalized	ADJ
ejpam-5256	226	6	topology	topology	NOUN
ejpam-5256	226	7	on	on	ADP
ejpam-5256	226	8	x	x	PUNCT
ejpam-5256	226	9	and	and	CCONJ
ejpam-5256	226	10	hx	hx	PROPN
ejpam-5256	226	11	is	be	AUX
ejpam-5256	226	12	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	226	13	for	for	ADP
ejpam-5256	226	14	each	each	DET
ejpam-5256	226	15	x	x	PROPN
ejpam-5256	226	16	∈	∈	PROPN
ejpam-5256	226	17	f−1(k	f−1(k	PROPN
ejpam-5256	226	18	)	)	PUNCT
ejpam-5256	226	19	,	,	PUNCT
ejpam-5256	226	20	⋃	⋃	PUNCT
ejpam-5256	226	21	x∈f−1(k	x∈f−1(k	NOUN
ejpam-5256	226	22	)	)	PUNCT
ejpam-5256	226	23	hx	hx	PROPN
ejpam-5256	226	24	is	be	AUX
ejpam-5256	226	25	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	226	26	in	in	ADP
ejpam-5256	226	27	x.	x.	NOUN
ejpam-5256	226	28	consequently	consequently	ADV
ejpam-5256	226	29	,	,	PUNCT
ejpam-5256	226	30	f−1(k	f−1(k	PROPN
ejpam-5256	226	31	)	)	PUNCT
ejpam-5256	226	32	is	be	AUX
ejpam-5256	226	33	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	226	34	in	in	ADP
ejpam-5256	226	35	x.	x.	NOUN
ejpam-5256	226	36	therefore	therefore	ADV
ejpam-5256	226	37	,	,	PUNCT
ejpam-5256	226	38	f	f	PROPN
ejpam-5256	226	39	is	be	AUX
ejpam-5256	226	40	a	a	DET
ejpam-5256	226	41	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	226	42	function	function	NOUN
ejpam-5256	226	43	.	.	PUNCT
ejpam-5256	227	1	j.	j.	PROPN
ejpam-5256	227	2	khampakdee	khampakdee	PROPN
ejpam-5256	227	3	/	/	PUNCT
ejpam-5256	227	4	eur	eur	PROPN
ejpam-5256	227	5	.	.	PUNCT
ejpam-5256	228	1	j.	j.	PROPN
ejpam-5256	228	2	pure	pure	PROPN
ejpam-5256	228	3	appl	appl	PROPN
ejpam-5256	228	4	.	.	PROPN
ejpam-5256	228	5	math	math	PROPN
ejpam-5256	228	6	,	,	PUNCT
ejpam-5256	228	7	17	17	NUM
ejpam-5256	228	8	(	(	PUNCT
ejpam-5256	228	9	3	3	NUM
ejpam-5256	228	10	)	)	PUNCT
ejpam-5256	228	11	(	(	PUNCT
ejpam-5256	228	12	2024	2024	NUM
ejpam-5256	228	13	)	)	PUNCT
ejpam-5256	228	14	,	,	PUNCT
ejpam-5256	228	15	1869	1869	NUM
ejpam-5256	228	16	-	-	SYM
ejpam-5256	228	17	1876	1876	NUM
ejpam-5256	228	18	1875	1875	NUM
ejpam-5256	228	19	theorem	theorem	VERB
ejpam-5256	228	20	9	9	NUM
ejpam-5256	228	21	.	.	PUNCT
ejpam-5256	229	1	let	let	VERB
ejpam-5256	229	2	(	(	PUNCT
ejpam-5256	229	3	x,µ	x,µ	NOUN
ejpam-5256	229	4	)	)	PUNCT
ejpam-5256	229	5	,	,	PUNCT
ejpam-5256	229	6	(	(	PUNCT
ejpam-5256	229	7	y	y	NOUN
ejpam-5256	229	8	,	,	PUNCT
ejpam-5256	229	9	µ′	µ′	NUM
ejpam-5256	229	10	)	)	PUNCT
ejpam-5256	229	11	and	and	CCONJ
ejpam-5256	229	12	(	(	PUNCT
ejpam-5256	229	13	z	z	NOUN
ejpam-5256	229	14	,	,	PUNCT
ejpam-5256	229	15	µ′′	µ′′	PROPN
ejpam-5256	229	16	)	)	PUNCT
ejpam-5256	229	17	be	be	AUX
ejpam-5256	229	18	generalized	generalize	VERB
ejpam-5256	229	19	topological	topological	ADJ
ejpam-5256	229	20	spaces	space	NOUN
ejpam-5256	229	21	.	.	PUNCT
ejpam-5256	230	1	if	if	SCONJ
ejpam-5256	230	2	f	f	PROPN
ejpam-5256	230	3	:	:	PUNCT
ejpam-5256	230	4	x	x	X
ejpam-5256	230	5	→	→	SYM
ejpam-5256	230	6	y	y	PROPN
ejpam-5256	230	7	is	be	AUX
ejpam-5256	230	8	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	230	9	and	and	CCONJ
ejpam-5256	230	10	g	g	NOUN
ejpam-5256	230	11	:	:	PUNCT
ejpam-5256	230	12	y	y	PROPN
ejpam-5256	230	13	→	→	SYM
ejpam-5256	230	14	z	z	PROPN
ejpam-5256	230	15	is	be	AUX
ejpam-5256	230	16	(	(	PUNCT
ejpam-5256	230	17	µ	µ	NUM
ejpam-5256	230	18	,	,	PUNCT
ejpam-5256	230	19	µ′)-continuous	µ′)-continuous	ADJ
ejpam-5256	230	20	,	,	PUNCT
ejpam-5256	230	21	then	then	ADV
ejpam-5256	230	22	g	g	PROPN
ejpam-5256	230	23	◦	◦	NOUN
ejpam-5256	230	24	f	f	X
ejpam-5256	230	25	:	:	PUNCT
ejpam-5256	230	26	x	x	X
ejpam-5256	230	27	→	→	SYM
ejpam-5256	230	28	z	z	NOUN
ejpam-5256	230	29	is	be	AUX
ejpam-5256	230	30	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	230	31	.	.	PUNCT
ejpam-5256	231	1	proof	proof	NOUN
ejpam-5256	231	2	.	.	PUNCT
ejpam-5256	232	1	assume	assume	VERB
ejpam-5256	232	2	that	that	SCONJ
ejpam-5256	232	3	v	v	X
ejpam-5256	232	4	∈	∈	PROPN
ejpam-5256	232	5	µ′′.	µ′′.	PUNCT
ejpam-5256	232	6	since	since	SCONJ
ejpam-5256	232	7	g	g	PROPN
ejpam-5256	232	8	is	be	AUX
ejpam-5256	232	9	(	(	PUNCT
ejpam-5256	232	10	µ	µ	NUM
ejpam-5256	232	11	,	,	PUNCT
ejpam-5256	232	12	µ′)-continuous	µ′)-continuous	ADJ
ejpam-5256	232	13	,	,	PUNCT
ejpam-5256	232	14	g−1(v	g−1(v	PROPN
ejpam-5256	232	15	)	)	PUNCT
ejpam-5256	232	16	is	be	AUX
ejpam-5256	232	17	µ′-open	µ′-open	NUM
ejpam-5256	232	18	in	in	ADP
ejpam-5256	232	19	y	y	PROPN
ejpam-5256	232	20	.	.	PUNCT
ejpam-5256	233	1	as	as	SCONJ
ejpam-5256	233	2	f	f	PROPN
ejpam-5256	233	3	is	be	AUX
ejpam-5256	233	4	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	233	5	,	,	PUNCT
ejpam-5256	233	6	f−1(g−1(v	f−1(g−1(v	PROPN
ejpam-5256	233	7	)	)	PUNCT
ejpam-5256	233	8	)	)	PUNCT
ejpam-5256	233	9	is	be	AUX
ejpam-5256	233	10	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	233	11	in	in	ADP
ejpam-5256	233	12	x.	x.	NOUN
ejpam-5256	234	1	it	it	PRON
ejpam-5256	234	2	follows	follow	VERB
ejpam-5256	234	3	that	that	SCONJ
ejpam-5256	234	4	(	(	PUNCT
ejpam-5256	234	5	g	g	NOUN
ejpam-5256	234	6	◦	◦	NOUN
ejpam-5256	234	7	f)−1(v	f)−1(v	NOUN
ejpam-5256	234	8	)	)	PUNCT
ejpam-5256	234	9	is	be	AUX
ejpam-5256	234	10	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	234	11	in	in	ADP
ejpam-5256	234	12	x	x	PUNCT
ejpam-5256	234	13	for	for	ADP
ejpam-5256	234	14	all	all	PRON
ejpam-5256	234	15	v	v	NOUN
ejpam-5256	234	16	∈	∈	NOUN
ejpam-5256	234	17	µ′′.	µ′′.	PUNCT
ejpam-5256	234	18	therefore	therefore	ADV
ejpam-5256	234	19	,	,	PUNCT
ejpam-5256	235	1	g	g	PROPN
ejpam-5256	235	2	◦	◦	NOUN
ejpam-5256	235	3	f	f	X
ejpam-5256	235	4	:	:	PUNCT
ejpam-5256	235	5	x	x	X
ejpam-5256	235	6	→	→	SYM
ejpam-5256	235	7	z	z	NOUN
ejpam-5256	235	8	is	be	AUX
ejpam-5256	235	9	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	235	10	.	.	PUNCT
ejpam-5256	236	1	theorem	theorem	ADJ
ejpam-5256	236	2	10	10	NUM
ejpam-5256	236	3	.	.	PUNCT
ejpam-5256	237	1	let	let	VERB
ejpam-5256	237	2	(	(	PUNCT
ejpam-5256	237	3	x,µ	x,µ	NOUN
ejpam-5256	237	4	)	)	PUNCT
ejpam-5256	237	5	and	and	CCONJ
ejpam-5256	237	6	(	(	PUNCT
ejpam-5256	237	7	y	y	NOUN
ejpam-5256	237	8	,	,	PUNCT
ejpam-5256	237	9	µ′	µ′	PUNCT
ejpam-5256	237	10	)	)	PUNCT
ejpam-5256	237	11	be	be	AUX
ejpam-5256	237	12	generalized	generalize	VERB
ejpam-5256	237	13	topological	topological	ADJ
ejpam-5256	237	14	spaces	space	NOUN
ejpam-5256	237	15	.	.	PUNCT
ejpam-5256	238	1	if	if	SCONJ
ejpam-5256	238	2	f	f	PROPN
ejpam-5256	238	3	:	:	PUNCT
ejpam-5256	238	4	x	x	X
ejpam-5256	238	5	→	→	PUNCT
ejpam-5256	238	6	x	x	X
ejpam-5256	238	7	is	be	AUX
ejpam-5256	238	8	an	an	DET
ejpam-5256	238	9	identity	identity	NOUN
ejpam-5256	238	10	function	function	NOUN
ejpam-5256	238	11	and	and	CCONJ
ejpam-5256	238	12	g	g	NOUN
ejpam-5256	238	13	:	:	PUNCT
ejpam-5256	238	14	x	x	X
ejpam-5256	238	15	→	→	SYM
ejpam-5256	238	16	y	y	PROPN
ejpam-5256	238	17	is	be	AUX
ejpam-5256	238	18	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	238	19	,	,	PUNCT
ejpam-5256	238	20	then	then	ADV
ejpam-5256	238	21	g	g	PROPN
ejpam-5256	238	22	◦	◦	NOUN
ejpam-5256	239	1	f	f	X
ejpam-5256	239	2	:	:	PUNCT
ejpam-5256	239	3	x	x	X
ejpam-5256	239	4	→	→	SYM
ejpam-5256	239	5	y	y	PROPN
ejpam-5256	239	6	is	be	AUX
ejpam-5256	239	7	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	239	8	.	.	PUNCT
ejpam-5256	240	1	proof	proof	NOUN
ejpam-5256	240	2	.	.	PUNCT
ejpam-5256	241	1	assume	assume	VERB
ejpam-5256	241	2	that	that	SCONJ
ejpam-5256	241	3	v	v	NOUN
ejpam-5256	241	4	is	be	AUX
ejpam-5256	241	5	µ′-open	µ′-open	NUM
ejpam-5256	241	6	.	.	PUNCT
ejpam-5256	242	1	since	since	SCONJ
ejpam-5256	242	2	g	g	PROPN
ejpam-5256	242	3	is	be	AUX
ejpam-5256	242	4	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	242	5	,	,	PUNCT
ejpam-5256	242	6	g−1(v	g−1(v	PROPN
ejpam-5256	242	7	)	)	PUNCT
ejpam-5256	242	8	is	be	AUX
ejpam-5256	242	9	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	242	10	in	in	ADP
ejpam-5256	242	11	x.	x.	NOUN
ejpam-5256	242	12	as	as	SCONJ
ejpam-5256	242	13	f	f	PROPN
ejpam-5256	242	14	is	be	AUX
ejpam-5256	242	15	an	an	DET
ejpam-5256	242	16	identity	identity	NOUN
ejpam-5256	242	17	function	function	NOUN
ejpam-5256	242	18	,	,	PUNCT
ejpam-5256	242	19	g−1(v	g−1(v	NOUN
ejpam-5256	242	20	)	)	PUNCT
ejpam-5256	243	1	=	=	SYM
ejpam-5256	243	2	f−1(g−1(v	f−1(g−1(v	PROPN
ejpam-5256	243	3	)	)	PUNCT
ejpam-5256	243	4	)	)	PUNCT
ejpam-5256	243	5	.	.	PUNCT
ejpam-5256	244	1	consequently	consequently	ADV
ejpam-5256	244	2	,	,	PUNCT
ejpam-5256	244	3	(	(	PUNCT
ejpam-5256	244	4	g	g	NOUN
ejpam-5256	244	5	◦	◦	NOUN
ejpam-5256	244	6	f)−1(v	f)−1(v	NOUN
ejpam-5256	244	7	)	)	PUNCT
ejpam-5256	245	1	=	=	PUNCT
ejpam-5256	245	2	g−1(v	g−1(v	NOUN
ejpam-5256	245	3	)	)	PUNCT
ejpam-5256	245	4	is	be	AUX
ejpam-5256	245	5	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	245	6	in	in	ADP
ejpam-5256	245	7	x	x	PUNCT
ejpam-5256	245	8	for	for	ADP
ejpam-5256	245	9	all	all	DET
ejpam-5256	245	10	v	v	NOUN
ejpam-5256	245	11	∈	∈	NOUN
ejpam-5256	245	12	µ′.	µ′.	NOUN
ejpam-5256	245	13	thus	thus	ADV
ejpam-5256	245	14	,	,	PUNCT
ejpam-5256	245	15	g	g	PROPN
ejpam-5256	245	16	◦	◦	NOUN
ejpam-5256	245	17	f	f	X
ejpam-5256	245	18	:	:	PUNCT
ejpam-5256	245	19	x	x	X
ejpam-5256	245	20	→	→	SYM
ejpam-5256	245	21	y	y	PROPN
ejpam-5256	245	22	is	be	AUX
ejpam-5256	245	23	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	245	24	.	.	PUNCT
ejpam-5256	246	1	definition	definition	NOUN
ejpam-5256	246	2	10	10	NUM
ejpam-5256	246	3	.	.	PUNCT
ejpam-5256	247	1	let	let	VERB
ejpam-5256	247	2	(	(	PUNCT
ejpam-5256	247	3	x,µ	x,µ	NOUN
ejpam-5256	247	4	)	)	PUNCT
ejpam-5256	247	5	and	and	CCONJ
ejpam-5256	247	6	(	(	PUNCT
ejpam-5256	247	7	y	y	NOUN
ejpam-5256	247	8	,	,	PUNCT
ejpam-5256	247	9	µ′	µ′	PUNCT
ejpam-5256	247	10	)	)	PUNCT
ejpam-5256	247	11	be	be	AUX
ejpam-5256	247	12	generalized	generalize	VERB
ejpam-5256	247	13	topological	topological	ADJ
ejpam-5256	247	14	spaces	space	NOUN
ejpam-5256	247	15	.	.	PUNCT
ejpam-5256	248	1	then	then	ADV
ejpam-5256	248	2	,	,	PUNCT
ejpam-5256	248	3	a	a	DET
ejpam-5256	248	4	function	function	NOUN
ejpam-5256	248	5	f	f	NOUN
ejpam-5256	248	6	from	from	ADP
ejpam-5256	248	7	(	(	PUNCT
ejpam-5256	248	8	x,µ	x,µ	NOUN
ejpam-5256	248	9	)	)	PUNCT
ejpam-5256	248	10	into	into	ADP
ejpam-5256	248	11	(	(	PUNCT
ejpam-5256	248	12	y	y	NOUN
ejpam-5256	248	13	,	,	PUNCT
ejpam-5256	248	14	µ′	µ′	NUM
ejpam-5256	248	15	)	)	PUNCT
ejpam-5256	248	16	is	be	AUX
ejpam-5256	248	17	called	call	VERB
ejpam-5256	248	18	sθ̃-irresolute	sθ̃-irresolute	NOUN
ejpam-5256	248	19	if	if	SCONJ
ejpam-5256	248	20	f−1(g	f−1(g	PROPN
ejpam-5256	248	21	)	)	PUNCT
ejpam-5256	248	22	is	be	AUX
ejpam-5256	248	23	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	248	24	in	in	ADP
ejpam-5256	248	25	x	x	PUNCT
ejpam-5256	248	26	for	for	SCONJ
ejpam-5256	248	27	each	each	DET
ejpam-5256	248	28	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	248	29	set	set	VERB
ejpam-5256	248	30	g	g	NOUN
ejpam-5256	248	31	in	in	ADP
ejpam-5256	248	32	y	y	PROPN
ejpam-5256	248	33	.	.	PUNCT
ejpam-5256	249	1	theorem	theorem	ADJ
ejpam-5256	249	2	11	11	NUM
ejpam-5256	249	3	.	.	PUNCT
ejpam-5256	250	1	let	let	VERB
ejpam-5256	250	2	(	(	PUNCT
ejpam-5256	250	3	x,µ	x,µ	NOUN
ejpam-5256	250	4	)	)	PUNCT
ejpam-5256	250	5	and	and	CCONJ
ejpam-5256	250	6	(	(	PUNCT
ejpam-5256	250	7	y	y	NOUN
ejpam-5256	250	8	,	,	PUNCT
ejpam-5256	250	9	µ′	µ′	PUNCT
ejpam-5256	250	10	)	)	PUNCT
ejpam-5256	251	1	be	be	AUX
ejpam-5256	251	2	generalized	generalize	VERB
ejpam-5256	251	3	topological	topological	ADJ
ejpam-5256	251	4	spaces	space	NOUN
ejpam-5256	251	5	.	.	PUNCT
ejpam-5256	252	1	then	then	ADV
ejpam-5256	252	2	,	,	PUNCT
ejpam-5256	252	3	f	f	X
ejpam-5256	252	4	:	:	PUNCT
ejpam-5256	252	5	(	(	PUNCT
ejpam-5256	252	6	x,µ)→	x,µ)→	X
ejpam-5256	252	7	(	(	PUNCT
ejpam-5256	252	8	y	y	NOUN
ejpam-5256	252	9	,	,	PUNCT
ejpam-5256	252	10	µ′	µ′	NUM
ejpam-5256	252	11	)	)	PUNCT
ejpam-5256	252	12	is	be	AUX
ejpam-5256	252	13	sθ̃-irresolute	sθ̃-irresolute	ADJ
ejpam-5256	252	14	if	if	SCONJ
ejpam-5256	252	15	and	and	CCONJ
ejpam-5256	252	16	only	only	ADV
ejpam-5256	252	17	if	if	SCONJ
ejpam-5256	252	18	f−1(f	f−1(f	PROPN
ejpam-5256	252	19	)	)	PUNCT
ejpam-5256	252	20	is	be	AUX
ejpam-5256	252	21	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	252	22	in	in	ADP
ejpam-5256	252	23	x	x	PUNCT
ejpam-5256	252	24	for	for	ADP
ejpam-5256	252	25	each	each	DET
ejpam-5256	252	26	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	252	27	set	set	VERB
ejpam-5256	252	28	f	f	PROPN
ejpam-5256	252	29	in	in	ADP
ejpam-5256	252	30	y	y	PROPN
ejpam-5256	252	31	.	.	PUNCT
ejpam-5256	253	1	proof	proof	NOUN
ejpam-5256	253	2	.	.	PUNCT
ejpam-5256	254	1	(	(	PUNCT
ejpam-5256	254	2	→	→	NOUN
ejpam-5256	254	3	)	)	PUNCT
ejpam-5256	254	4	let	let	VERB
ejpam-5256	254	5	f	f	PRON
ejpam-5256	254	6	be	be	AUX
ejpam-5256	254	7	a	a	DET
ejpam-5256	254	8	sθ̃-irresolute	sθ̃-irresolute	NOUN
ejpam-5256	254	9	function	function	NOUN
ejpam-5256	254	10	.	.	PUNCT
ejpam-5256	254	11	suppose	suppose	VERB
ejpam-5256	254	12	that	that	SCONJ
ejpam-5256	254	13	f	f	PROPN
ejpam-5256	254	14	is	be	AUX
ejpam-5256	254	15	a	a	DET
ejpam-5256	254	16	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	254	17	set	set	NOUN
ejpam-5256	254	18	in	in	ADP
ejpam-5256	254	19	y	y	PROPN
ejpam-5256	254	20	.	.	PUNCT
ejpam-5256	255	1	then	then	ADV
ejpam-5256	255	2	,	,	PUNCT
ejpam-5256	255	3	y	y	PROPN
ejpam-5256	255	4	−	−	PROPN
ejpam-5256	255	5	f	f	PROPN
ejpam-5256	255	6	is	be	AUX
ejpam-5256	255	7	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	255	8	in	in	ADP
ejpam-5256	255	9	y	y	PROPN
ejpam-5256	255	10	.	.	PUNCT
ejpam-5256	256	1	consequently	consequently	ADV
ejpam-5256	256	2	,	,	PUNCT
ejpam-5256	256	3	x	x	PUNCT
ejpam-5256	256	4	−	−	PROPN
ejpam-5256	256	5	f−1(f	f−1(f	PROPN
ejpam-5256	256	6	)	)	PUNCT
ejpam-5256	256	7	is	be	AUX
ejpam-5256	256	8	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	256	9	in	in	ADP
ejpam-5256	256	10	x.	x.	NOUN
ejpam-5256	257	1	it	it	PRON
ejpam-5256	257	2	follows	follow	VERB
ejpam-5256	257	3	that	that	SCONJ
ejpam-5256	257	4	f−1(f	f−1(f	PROPN
ejpam-5256	257	5	)	)	PUNCT
ejpam-5256	257	6	is	be	AUX
ejpam-5256	257	7	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	257	8	in	in	ADP
ejpam-5256	257	9	x.	x.	PROPN
ejpam-5256	257	10	(	(	PUNCT
ejpam-5256	257	11	←	←	PROPN
ejpam-5256	257	12	)	)	PUNCT
ejpam-5256	257	13	assume	assume	VERB
ejpam-5256	257	14	that	that	SCONJ
ejpam-5256	257	15	f−1(f	f−1(f	PROPN
ejpam-5256	257	16	)	)	PUNCT
ejpam-5256	257	17	is	be	AUX
ejpam-5256	257	18	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	257	19	in	in	ADP
ejpam-5256	257	20	x	x	PUNCT
ejpam-5256	257	21	for	for	ADP
ejpam-5256	257	22	each	each	DET
ejpam-5256	257	23	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	257	24	set	set	VERB
ejpam-5256	257	25	f	f	PROPN
ejpam-5256	257	26	in	in	ADP
ejpam-5256	257	27	y	y	PROPN
ejpam-5256	257	28	.	.	PUNCT
ejpam-5256	258	1	let	let	VERB
ejpam-5256	258	2	k	k	PRON
ejpam-5256	258	3	be	be	AUX
ejpam-5256	258	4	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	258	5	in	in	ADP
ejpam-5256	258	6	y	y	PROPN
ejpam-5256	258	7	.	.	PUNCT
ejpam-5256	259	1	then	then	ADV
ejpam-5256	259	2	,	,	PUNCT
ejpam-5256	259	3	y	y	PROPN
ejpam-5256	259	4	−k	−k	PROPN
ejpam-5256	259	5	is	be	AUX
ejpam-5256	259	6	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	259	7	in	in	ADP
ejpam-5256	259	8	y	y	PROPN
ejpam-5256	259	9	.	.	PUNCT
ejpam-5256	260	1	by	by	ADP
ejpam-5256	260	2	the	the	DET
ejpam-5256	260	3	assumption	assumption	NOUN
ejpam-5256	260	4	,	,	PUNCT
ejpam-5256	260	5	f−1(y	f−1(y	PROPN
ejpam-5256	260	6	−k	−k	PROPN
ejpam-5256	260	7	)	)	PUNCT
ejpam-5256	260	8	is	be	AUX
ejpam-5256	260	9	sθ̃-closed	sθ̃-close	VERB
ejpam-5256	260	10	in	in	ADP
ejpam-5256	260	11	x.	x.	NOUN
ejpam-5256	260	12	it	it	PRON
ejpam-5256	260	13	follows	follow	VERB
ejpam-5256	260	14	that	that	SCONJ
ejpam-5256	260	15	f−1(k	f−1(k	PROPN
ejpam-5256	260	16	)	)	PUNCT
ejpam-5256	260	17	is	be	AUX
ejpam-5256	260	18	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	260	19	in	in	ADP
ejpam-5256	260	20	x.	x.	PROPN
ejpam-5256	260	21	hence	hence	ADV
ejpam-5256	260	22	,	,	PUNCT
ejpam-5256	260	23	f	f	PROPN
ejpam-5256	260	24	is	be	AUX
ejpam-5256	260	25	sθ̃-irresolute	sθ̃-irresolute	VERB
ejpam-5256	260	26	.	.	PUNCT
ejpam-5256	261	1	theorem	theorem	NOUN
ejpam-5256	261	2	12	12	NUM
ejpam-5256	261	3	.	.	PUNCT
ejpam-5256	262	1	each	each	DET
ejpam-5256	262	2	sθ̃-irresolute	sθ̃-irresolute	NOUN
ejpam-5256	262	3	function	function	NOUN
ejpam-5256	262	4	is	be	AUX
ejpam-5256	262	5	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	262	6	.	.	PUNCT
ejpam-5256	263	1	proof	proof	NOUN
ejpam-5256	263	2	.	.	PUNCT
ejpam-5256	264	1	assume	assume	VERB
ejpam-5256	264	2	that	that	SCONJ
ejpam-5256	264	3	v	v	NOUN
ejpam-5256	264	4	is	be	AUX
ejpam-5256	264	5	µ′-open	µ′-open	NUM
ejpam-5256	264	6	in	in	ADP
ejpam-5256	264	7	y	y	PROPN
ejpam-5256	264	8	.	.	PUNCT
ejpam-5256	265	1	by	by	ADP
ejpam-5256	265	2	theorem	theorem	NOUN
ejpam-5256	265	3	3	3	NUM
ejpam-5256	265	4	,	,	PUNCT
ejpam-5256	265	5	v	v	NOUN
ejpam-5256	265	6	is	be	AUX
ejpam-5256	265	7	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	265	8	.	.	PUNCT
ejpam-5256	266	1	since	since	SCONJ
ejpam-5256	266	2	f	f	PROPN
ejpam-5256	266	3	is	be	AUX
ejpam-5256	266	4	sθ̃-irresolute	sθ̃-irresolute	ADV
ejpam-5256	266	5	,	,	PUNCT
ejpam-5256	266	6	f−1(v	f−1(v	PROPN
ejpam-5256	266	7	)	)	PUNCT
ejpam-5256	266	8	is	be	AUX
ejpam-5256	266	9	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	266	10	in	in	ADP
ejpam-5256	266	11	x.	x.	PROPN
ejpam-5256	266	12	thus	thus	ADV
ejpam-5256	266	13	,	,	PUNCT
ejpam-5256	266	14	f	f	PROPN
ejpam-5256	266	15	is	be	AUX
ejpam-5256	266	16	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	266	17	.	.	PUNCT
ejpam-5256	267	1	theorem	theorem	NOUN
ejpam-5256	267	2	13	13	NUM
ejpam-5256	267	3	.	.	PUNCT
ejpam-5256	268	1	let	let	VERB
ejpam-5256	268	2	(	(	PUNCT
ejpam-5256	268	3	x,µ	x,µ	NOUN
ejpam-5256	268	4	)	)	PUNCT
ejpam-5256	268	5	,	,	PUNCT
ejpam-5256	268	6	(	(	PUNCT
ejpam-5256	268	7	y	y	NOUN
ejpam-5256	268	8	,	,	PUNCT
ejpam-5256	268	9	µ′	µ′	NUM
ejpam-5256	268	10	)	)	PUNCT
ejpam-5256	268	11	and	and	CCONJ
ejpam-5256	268	12	(	(	PUNCT
ejpam-5256	268	13	z	z	NOUN
ejpam-5256	268	14	,	,	PUNCT
ejpam-5256	268	15	µ′′	µ′′	PROPN
ejpam-5256	268	16	)	)	PUNCT
ejpam-5256	268	17	be	be	AUX
ejpam-5256	268	18	generalized	generalize	VERB
ejpam-5256	268	19	topological	topological	ADJ
ejpam-5256	268	20	spaces	space	NOUN
ejpam-5256	268	21	.	.	PUNCT
ejpam-5256	269	1	then	then	ADV
ejpam-5256	269	2	,	,	PUNCT
ejpam-5256	269	3	g	g	PROPN
ejpam-5256	269	4	◦	◦	NOUN
ejpam-5256	269	5	f	f	X
ejpam-5256	269	6	:	:	PUNCT
ejpam-5256	269	7	x	x	X
ejpam-5256	269	8	→	→	SYM
ejpam-5256	269	9	z	z	NOUN
ejpam-5256	269	10	is	be	AUX
ejpam-5256	269	11	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	269	12	if	if	SCONJ
ejpam-5256	269	13	f	f	X
ejpam-5256	269	14	:	:	PUNCT
ejpam-5256	269	15	x	x	X
ejpam-5256	269	16	→	→	SYM
ejpam-5256	269	17	y	y	PROPN
ejpam-5256	269	18	is	be	AUX
ejpam-5256	269	19	sθ̃-irresolute	sθ̃-irresolute	ADJ
ejpam-5256	269	20	and	and	CCONJ
ejpam-5256	269	21	g	g	NOUN
ejpam-5256	269	22	:	:	PUNCT
ejpam-5256	269	23	x	x	X
ejpam-5256	269	24	→	→	SYM
ejpam-5256	269	25	z	z	NOUN
ejpam-5256	269	26	is	be	AUX
ejpam-5256	269	27	sθ̃continuous	sθ̃continuous	ADJ
ejpam-5256	269	28	.	.	PUNCT
ejpam-5256	270	1	proof	proof	NOUN
ejpam-5256	270	2	.	.	PUNCT
ejpam-5256	271	1	assume	assume	VERB
ejpam-5256	271	2	that	that	SCONJ
ejpam-5256	271	3	h	h	NOUN
ejpam-5256	271	4	is	be	AUX
ejpam-5256	271	5	µ′′-open	µ′′-open	ADJ
ejpam-5256	271	6	.	.	PUNCT
ejpam-5256	272	1	since	since	SCONJ
ejpam-5256	272	2	g	g	PROPN
ejpam-5256	272	3	is	be	AUX
ejpam-5256	272	4	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	272	5	,	,	PUNCT
ejpam-5256	272	6	g−1(h	g−1(h	PROPN
ejpam-5256	272	7	)	)	PUNCT
ejpam-5256	272	8	is	be	AUX
ejpam-5256	272	9	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	272	10	in	in	ADP
ejpam-5256	272	11	y	y	PROPN
ejpam-5256	272	12	.	.	PUNCT
ejpam-5256	273	1	as	as	SCONJ
ejpam-5256	273	2	f	f	PROPN
ejpam-5256	273	3	is	be	AUX
ejpam-5256	273	4	sθ̃-irresolute	sθ̃-irresolute	ADV
ejpam-5256	273	5	,	,	PUNCT
ejpam-5256	273	6	f−1(g−1(h	f−1(g−1(h	PROPN
ejpam-5256	273	7	)	)	PUNCT
ejpam-5256	273	8	)	)	PUNCT
ejpam-5256	274	1	=	=	PRON
ejpam-5256	274	2	(	(	PUNCT
ejpam-5256	274	3	g	g	PROPN
ejpam-5256	274	4	◦	◦	NOUN
ejpam-5256	274	5	f)−1(h	f)−1(h	NOUN
ejpam-5256	274	6	)	)	PUNCT
ejpam-5256	274	7	is	be	AUX
ejpam-5256	274	8	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	274	9	in	in	ADP
ejpam-5256	274	10	x	x	PUNCT
ejpam-5256	274	11	for	for	ADP
ejpam-5256	274	12	all	all	DET
ejpam-5256	274	13	h	h	NOUN
ejpam-5256	274	14	∈	∈	PROPN
ejpam-5256	274	15	µ′′.	µ′′.	PRON
ejpam-5256	274	16	thus	thus	ADV
ejpam-5256	274	17	,	,	PUNCT
ejpam-5256	274	18	g	g	PROPN
ejpam-5256	274	19	◦	◦	NOUN
ejpam-5256	274	20	f	f	PROPN
ejpam-5256	274	21	is	be	AUX
ejpam-5256	274	22	sθ̃-continuous	sθ̃-continuous	ADJ
ejpam-5256	274	23	.	.	PUNCT
ejpam-5256	275	1	theorem	theorem	NOUN
ejpam-5256	275	2	14	14	NUM
ejpam-5256	275	3	.	.	PUNCT
ejpam-5256	276	1	let	let	VERB
ejpam-5256	276	2	(	(	PUNCT
ejpam-5256	276	3	x,µ	x,µ	NOUN
ejpam-5256	276	4	)	)	PUNCT
ejpam-5256	276	5	and	and	CCONJ
ejpam-5256	276	6	(	(	PUNCT
ejpam-5256	276	7	y	y	NOUN
ejpam-5256	276	8	,	,	PUNCT
ejpam-5256	276	9	µ′	µ′	PUNCT
ejpam-5256	276	10	)	)	PUNCT
ejpam-5256	277	1	be	be	AUX
ejpam-5256	277	2	generalized	generalize	VERB
ejpam-5256	277	3	topological	topological	ADJ
ejpam-5256	277	4	spaces	space	NOUN
ejpam-5256	277	5	.	.	PUNCT
ejpam-5256	278	1	then	then	ADV
ejpam-5256	278	2	,	,	PUNCT
ejpam-5256	278	3	g	g	PROPN
ejpam-5256	278	4	◦	◦	NOUN
ejpam-5256	278	5	f	f	X
ejpam-5256	278	6	:	:	PUNCT
ejpam-5256	278	7	x	x	X
ejpam-5256	278	8	→	→	SYM
ejpam-5256	278	9	y	y	PROPN
ejpam-5256	278	10	is	be	AUX
ejpam-5256	278	11	sθ̃-irresolute	sθ̃-irresolute	ADJ
ejpam-5256	278	12	if	if	SCONJ
ejpam-5256	278	13	f	f	PROPN
ejpam-5256	278	14	:	:	PUNCT
ejpam-5256	278	15	x	x	X
ejpam-5256	278	16	→	→	PUNCT
ejpam-5256	278	17	x	x	X
ejpam-5256	278	18	is	be	AUX
ejpam-5256	278	19	an	an	DET
ejpam-5256	278	20	identity	identity	NOUN
ejpam-5256	278	21	function	function	NOUN
ejpam-5256	278	22	and	and	CCONJ
ejpam-5256	278	23	g	g	NOUN
ejpam-5256	278	24	:	:	PUNCT
ejpam-5256	278	25	x	x	X
ejpam-5256	278	26	→	→	SYM
ejpam-5256	278	27	y	y	PROPN
ejpam-5256	278	28	is	be	AUX
ejpam-5256	278	29	sθ̃-irresolute	sθ̃-irresolute	NOUN
ejpam-5256	278	30	.	.	PUNCT
ejpam-5256	279	1	proof	proof	NOUN
ejpam-5256	279	2	.	.	PUNCT
ejpam-5256	280	1	suppose	suppose	VERB
ejpam-5256	280	2	that	that	SCONJ
ejpam-5256	280	3	m	m	NOUN
ejpam-5256	280	4	is	be	AUX
ejpam-5256	280	5	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	280	6	in	in	ADP
ejpam-5256	280	7	y	y	PROPN
ejpam-5256	280	8	.	.	PUNCT
ejpam-5256	281	1	as	as	SCONJ
ejpam-5256	281	2	g	g	PROPN
ejpam-5256	281	3	is	be	AUX
ejpam-5256	281	4	sθ̃-irresolute	sθ̃-irresolute	ADV
ejpam-5256	281	5	,	,	PUNCT
ejpam-5256	281	6	g−1(m	g−1(m	PROPN
ejpam-5256	281	7	)	)	PUNCT
ejpam-5256	281	8	is	be	AUX
ejpam-5256	281	9	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	281	10	in	in	ADP
ejpam-5256	281	11	x.	x.	NOUN
ejpam-5256	281	12	since	since	SCONJ
ejpam-5256	281	13	f	f	PROPN
ejpam-5256	281	14	is	be	AUX
ejpam-5256	281	15	an	an	DET
ejpam-5256	281	16	identity	identity	NOUN
ejpam-5256	281	17	function	function	NOUN
ejpam-5256	281	18	,	,	PUNCT
ejpam-5256	281	19	f−1(g−1(m	f−1(g−1(m	NOUN
ejpam-5256	281	20	)	)	PUNCT
ejpam-5256	281	21	)	)	PUNCT
ejpam-5256	282	1	=	=	SYM
ejpam-5256	282	2	g−1(m	g−1(m	PROPN
ejpam-5256	282	3	)	)	PUNCT
ejpam-5256	282	4	is	be	AUX
ejpam-5256	282	5	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	282	6	in	in	ADP
ejpam-5256	282	7	x	x	PUNCT
ejpam-5256	282	8	for	for	SCONJ
ejpam-5256	282	9	each	each	DET
ejpam-5256	282	10	m	m	NOUN
ejpam-5256	282	11	is	be	AUX
ejpam-5256	282	12	sθ̃-open	sθ̃-open	ADJ
ejpam-5256	282	13	in	in	ADP
ejpam-5256	282	14	y	y	PROPN
ejpam-5256	282	15	.	.	PUNCT
ejpam-5256	283	1	thus	thus	ADV
ejpam-5256	283	2	,	,	PUNCT
ejpam-5256	283	3	g	g	PROPN
ejpam-5256	283	4	◦	◦	NOUN
ejpam-5256	283	5	f	f	X
ejpam-5256	283	6	:	:	PUNCT
ejpam-5256	283	7	x	x	X
ejpam-5256	283	8	→	→	SYM
ejpam-5256	283	9	y	y	PROPN
ejpam-5256	283	10	is	be	AUX
ejpam-5256	283	11	sθ̃-irresolute	sθ̃-irresolute	NOUN
ejpam-5256	283	12	.	.	PUNCT
ejpam-5256	284	1	references	reference	NOUN
ejpam-5256	284	2	1876	1876	NUM
ejpam-5256	284	3	acknowledgements	acknowledgement	NOUN
ejpam-5256	284	4	the	the	DET
ejpam-5256	284	5	author	author	NOUN
ejpam-5256	284	6	would	would	AUX
ejpam-5256	284	7	like	like	VERB
ejpam-5256	284	8	to	to	PART
ejpam-5256	284	9	thank	thank	VERB
ejpam-5256	284	10	the	the	DET
ejpam-5256	284	11	referees	referee	NOUN
ejpam-5256	284	12	for	for	ADP
ejpam-5256	284	13	their	their	PRON
ejpam-5256	284	14	valuable	valuable	ADJ
ejpam-5256	284	15	suggestions	suggestion	NOUN
ejpam-5256	284	16	,	,	PUNCT
ejpam-5256	284	17	which	which	PRON
ejpam-5256	284	18	enhanced	enhance	VERB
ejpam-5256	284	19	the	the	DET
ejpam-5256	284	20	presentation	presentation	NOUN
ejpam-5256	284	21	of	of	ADP
ejpam-5256	284	22	the	the	DET
ejpam-5256	284	23	paper	paper	NOUN
ejpam-5256	284	24	.	.	PUNCT
ejpam-5256	285	1	references	reference	NOUN
ejpam-5256	285	2	[	[	X
ejpam-5256	285	3	1	1	NUM
ejpam-5256	285	4	]	]	PUNCT
ejpam-5256	285	5	á.	á.	PROPN
ejpam-5256	285	6	császár	császár	PROPN
ejpam-5256	285	7	.	.	PUNCT
ejpam-5256	286	1	generalized	generalize	VERB
ejpam-5256	286	2	topology	topology	NOUN
ejpam-5256	286	3	,	,	PUNCT
ejpam-5256	286	4	generalized	generalize	VERB
ejpam-5256	286	5	continuity	continuity	NOUN
ejpam-5256	286	6	.	.	PUNCT
ejpam-5256	287	1	acta	acta	PROPN
ejpam-5256	287	2	mathematica	mathematica	PROPN
ejpam-5256	287	3	hungarica	hungarica	PROPN
ejpam-5256	287	4	,	,	PUNCT
ejpam-5256	287	5	96(4):351–357	96(4):351–357	NOUN
ejpam-5256	287	6	,	,	PUNCT
ejpam-5256	287	7	2002	2002	NUM
ejpam-5256	287	8	.	.	PUNCT
ejpam-5256	288	1	[	[	X
ejpam-5256	288	2	2	2	NUM
ejpam-5256	288	3	]	]	PUNCT
ejpam-5256	288	4	á.	á.	PRON
ejpam-5256	288	5	császár	császár	PROPN
ejpam-5256	288	6	.	.	PUNCT
ejpam-5256	289	1	generalized	generalize	VERB
ejpam-5256	289	2	open	open	ADJ
ejpam-5256	289	3	sets	set	NOUN
ejpam-5256	289	4	in	in	ADP
ejpam-5256	289	5	generalized	generalized	ADJ
ejpam-5256	289	6	topologies	topology	NOUN
ejpam-5256	289	7	.	.	PUNCT
ejpam-5256	290	1	acta	acta	PROPN
ejpam-5256	290	2	mathematica	mathematica	PROPN
ejpam-5256	290	3	hungarica	hungarica	PROPN
ejpam-5256	290	4	,	,	PUNCT
ejpam-5256	290	5	106(1	106(1	NUM
ejpam-5256	290	6	-	-	SYM
ejpam-5256	290	7	2):53–66	2):53–66	NUM
ejpam-5256	290	8	,	,	PUNCT
ejpam-5256	290	9	2005	2005	NUM
ejpam-5256	290	10	.	.	PUNCT
ejpam-5256	291	1	[	[	X
ejpam-5256	291	2	3	3	NUM
ejpam-5256	291	3	]	]	X
ejpam-5256	291	4	á.	á.	NOUN
ejpam-5256	291	5	császár	császár	PROPN
ejpam-5256	291	6	.	.	PUNCT
ejpam-5256	292	1	δ	δ	PROPN
ejpam-5256	292	2	-	-	PUNCT
ejpam-5256	292	3	and	and	CCONJ
ejpam-5256	292	4	θmodifications	θmodification	NOUN
ejpam-5256	292	5	of	of	ADP
ejpam-5256	292	6	generalized	generalized	ADJ
ejpam-5256	292	7	topologies	topology	NOUN
ejpam-5256	292	8	.	.	PUNCT
ejpam-5256	293	1	acta	acta	PROPN
ejpam-5256	293	2	mathematica	mathematica	PROPN
ejpam-5256	293	3	hungarica	hungarica	PROPN
ejpam-5256	293	4	,	,	PUNCT
ejpam-5256	293	5	120(3):275–279	120(3):275–279	NUM
ejpam-5256	293	6	,	,	PUNCT
ejpam-5256	293	7	2008	2008	NUM
ejpam-5256	293	8	.	.	PUNCT
ejpam-5256	294	1	[	[	X
ejpam-5256	294	2	4	4	X
ejpam-5256	294	3	]	]	PUNCT
ejpam-5256	294	4	w.	w.	PROPN
ejpam-5256	294	5	k.	k.	PROPN
ejpam-5256	294	6	min	min	PROPN
ejpam-5256	294	7	.	.	PROPN
ejpam-5256	294	8	remarks	remark	NOUN
ejpam-5256	294	9	on	on	ADP
ejpam-5256	294	10	θ	θ	ADJ
ejpam-5256	294	11	-	-	ADJ
ejpam-5256	294	12	open	open	ADJ
ejpam-5256	294	13	sets	set	NOUN
ejpam-5256	294	14	in	in	ADP
ejpam-5256	294	15	generalized	generalized	ADJ
ejpam-5256	294	16	topological	topological	ADJ
ejpam-5256	294	17	spaces	space	NOUN
ejpam-5256	294	18	.	.	PUNCT
ejpam-5256	295	1	applied	apply	VERB
ejpam-5256	295	2	mathematics	mathematics	NOUN
ejpam-5256	295	3	letters	letter	NOUN
ejpam-5256	295	4	,	,	PUNCT
ejpam-5256	295	5	24(2):165–168	24(2):165–168	PROPN
ejpam-5256	295	6	,	,	PUNCT
ejpam-5256	295	7	2011	2011	NUM
ejpam-5256	295	8	.	.	PUNCT
ejpam-5256	296	1	[	[	X
ejpam-5256	296	2	5	5	NUM
ejpam-5256	296	3	]	]	PUNCT
ejpam-5256	296	4	b.	b.	PROPN
ejpam-5256	296	5	roy	roy	PROPN
ejpam-5256	296	6	.	.	PROPN
ejpam-5256	297	1	on	on	ADP
ejpam-5256	297	2	a	a	DET
ejpam-5256	297	3	type	type	NOUN
ejpam-5256	297	4	of	of	ADP
ejpam-5256	297	5	generalized	generalized	ADJ
ejpam-5256	297	6	open	open	ADJ
ejpam-5256	297	7	sets	set	NOUN
ejpam-5256	297	8	.	.	PUNCT
ejpam-5256	298	1	applied	apply	VERB
ejpam-5256	298	2	generalized	generalized	ADJ
ejpam-5256	298	3	topology	topology	NOUN
ejpam-5256	298	4	,	,	PUNCT
ejpam-5256	298	5	12(2):163	12(2):163	NUM
ejpam-5256	298	6	–	–	PUNCT
ejpam-5256	298	7	173	173	NUM
ejpam-5256	298	8	,	,	PUNCT
ejpam-5256	298	9	2011	2011	NUM
ejpam-5256	298	10	.	.	PUNCT