id	sid	tid	token	lemma	pos
ejpam-4386	1	1	european	european	PROPN
ejpam-4386	1	2	journal	journal	PROPN
ejpam-4386	1	3	of	of	ADP
ejpam-4386	1	4	pure	pure	ADJ
ejpam-4386	1	5	and	and	CCONJ
ejpam-4386	1	6	applied	apply	VERB
ejpam-4386	1	7	mathematics	mathematic	NOUN
ejpam-4386	1	8	vol	vol	NOUN
ejpam-4386	1	9	.	.	PROPN
ejpam-4386	2	1	15	15	NUM
ejpam-4386	2	2	,	,	PUNCT
ejpam-4386	2	3	no	no	INTJ
ejpam-4386	2	4	.	.	NOUN
ejpam-4386	2	5	3	3	NUM
ejpam-4386	2	6	,	,	PUNCT
ejpam-4386	2	7	2022	2022	NUM
ejpam-4386	2	8	,	,	PUNCT
ejpam-4386	2	9	821	821	NUM
ejpam-4386	2	10	-	-	SYM
ejpam-4386	2	11	829	829	NUM
ejpam-4386	2	12	issn	issn	PROPN
ejpam-4386	2	13	1307	1307	NUM
ejpam-4386	2	14	-	-	SYM
ejpam-4386	2	15	5543	5543	NUM
ejpam-4386	2	16	–	–	PUNCT
ejpam-4386	3	1	ejpam.com	ejpam.com	X
ejpam-4386	3	2	published	publish	VERB
ejpam-4386	3	3	by	by	ADP
ejpam-4386	3	4	new	new	PROPN
ejpam-4386	3	5	york	york	PROPN
ejpam-4386	3	6	business	business	PROPN
ejpam-4386	3	7	global	global	ADJ
ejpam-4386	3	8	semi	semi	ADJ
ejpam-4386	3	9	-	-	ADJ
ejpam-4386	3	10	regularization	regularization	ADJ
ejpam-4386	3	11	topological	topological	ADJ
ejpam-4386	3	12	spaces	space	NOUN
ejpam-4386	3	13	dina	dina	PROPN
ejpam-4386	3	14	abuzaid1	abuzaid1	PROPN
ejpam-4386	3	15	,	,	PUNCT
ejpam-4386	3	16	nouf	nouf	NOUN
ejpam-4386	3	17	alfarsi1,∗	alfarsi1,∗	NOUN
ejpam-4386	3	18	,	,	PUNCT
ejpam-4386	3	19	lutfi	lutfi	PROPN
ejpam-4386	3	20	kalantan1	kalantan1	PROPN
ejpam-4386	3	21	1	1	NUM
ejpam-4386	3	22	king	king	PROPN
ejpam-4386	3	23	abdulaziz	abdulaziz	PROPN
ejpam-4386	3	24	university	university	PROPN
ejpam-4386	3	25	,	,	PUNCT
ejpam-4386	3	26	department	department	NOUN
ejpam-4386	3	27	of	of	ADP
ejpam-4386	3	28	mathematics	mathematic	NOUN
ejpam-4386	3	29	,	,	PUNCT
ejpam-4386	3	30	p.o.box	p.o.box	PROPN
ejpam-4386	3	31	80203	80203	NUM
ejpam-4386	3	32	,	,	PUNCT
ejpam-4386	3	33	jeddah	jeddah	PROPN
ejpam-4386	3	34	21589	21589	NUM
ejpam-4386	3	35	,	,	PUNCT
ejpam-4386	3	36	saudi	saudi	PROPN
ejpam-4386	3	37	arabia	arabia	PROPN
ejpam-4386	3	38	abstract	abstract	NOUN
ejpam-4386	3	39	.	.	PUNCT
ejpam-4386	4	1	if	if	SCONJ
ejpam-4386	4	2	(	(	PUNCT
ejpam-4386	4	3	x	x	X
ejpam-4386	4	4	,	,	PUNCT
ejpam-4386	4	5	τ	τ	PROPN
ejpam-4386	4	6	)	)	PUNCT
ejpam-4386	4	7	is	be	AUX
ejpam-4386	4	8	a	a	DET
ejpam-4386	4	9	topological	topological	ADJ
ejpam-4386	4	10	space	space	NOUN
ejpam-4386	4	11	,	,	PUNCT
ejpam-4386	4	12	then	then	ADV
ejpam-4386	4	13	the	the	DET
ejpam-4386	4	14	semi	semi	ADJ
ejpam-4386	4	15	-	-	ADJ
ejpam-4386	4	16	regularization	regularization	ADJ
ejpam-4386	4	17	topology	topology	NOUN
ejpam-4386	4	18	τ	τ	X
ejpam-4386	4	19	s	s	VERB
ejpam-4386	4	20	on	on	ADP
ejpam-4386	4	21	x	x	PUNCT
ejpam-4386	4	22	of	of	ADP
ejpam-4386	4	23	τ	τ	PROPN
ejpam-4386	4	24	is	be	AUX
ejpam-4386	4	25	the	the	DET
ejpam-4386	4	26	coarser	coarse	ADJ
ejpam-4386	4	27	topology	topology	NOUN
ejpam-4386	4	28	on	on	ADP
ejpam-4386	4	29	x	x	PUNCT
ejpam-4386	4	30	generated	generate	VERB
ejpam-4386	4	31	by	by	ADP
ejpam-4386	4	32	the	the	DET
ejpam-4386	4	33	family	family	NOUN
ejpam-4386	4	34	of	of	ADP
ejpam-4386	4	35	all	all	DET
ejpam-4386	4	36	open	open	ADJ
ejpam-4386	4	37	domains	domain	NOUN
ejpam-4386	4	38	of	of	ADP
ejpam-4386	4	39	(	(	PUNCT
ejpam-4386	4	40	x	x	INTJ
ejpam-4386	4	41	,	,	PUNCT
ejpam-4386	4	42	τ	τ	PROPN
ejpam-4386	4	43	)	)	PUNCT
ejpam-4386	4	44	where	where	SCONJ
ejpam-4386	4	45	a	a	DET
ejpam-4386	4	46	subset	subset	NOUN
ejpam-4386	4	47	u	u	NOUN
ejpam-4386	4	48	is	be	AUX
ejpam-4386	4	49	called	call	VERB
ejpam-4386	4	50	an	an	DET
ejpam-4386	4	51	open	open	ADJ
ejpam-4386	4	52	domain	domain	NOUN
ejpam-4386	4	53	if	if	SCONJ
ejpam-4386	4	54	u	u	NOUN
ejpam-4386	4	55	=	=	SYM
ejpam-4386	4	56	int(u	int(u	PROPN
ejpam-4386	4	57	)	)	PUNCT
ejpam-4386	4	58	.	.	PUNCT
ejpam-4386	5	1	in	in	ADP
ejpam-4386	5	2	this	this	DET
ejpam-4386	5	3	paper	paper	NOUN
ejpam-4386	5	4	,	,	PUNCT
ejpam-4386	5	5	we	we	PRON
ejpam-4386	5	6	study	study	VERB
ejpam-4386	5	7	the	the	DET
ejpam-4386	5	8	semi	semi	NOUN
ejpam-4386	5	9	-	-	NOUN
ejpam-4386	5	10	regularity	regularity	NOUN
ejpam-4386	5	11	of	of	ADP
ejpam-4386	5	12	some	some	DET
ejpam-4386	5	13	generated	generate	VERB
ejpam-4386	5	14	spaces	space	NOUN
ejpam-4386	5	15	and	and	CCONJ
ejpam-4386	5	16	some	some	DET
ejpam-4386	5	17	properties	property	NOUN
ejpam-4386	5	18	of	of	ADP
ejpam-4386	5	19	weaker	weak	ADJ
ejpam-4386	5	20	version	version	NOUN
ejpam-4386	5	21	of	of	ADP
ejpam-4386	5	22	normality	normality	NOUN
ejpam-4386	5	23	of	of	ADP
ejpam-4386	5	24	the	the	DET
ejpam-4386	5	25	semi	semi	ADJ
ejpam-4386	5	26	-	-	ADJ
ejpam-4386	5	27	regularization	regularization	ADJ
ejpam-4386	5	28	space	space	NOUN
ejpam-4386	5	29	(	(	PUNCT
ejpam-4386	5	30	x	x	X
ejpam-4386	5	31	,	,	PUNCT
ejpam-4386	5	32	τ	τ	PROPN
ejpam-4386	5	33	s	s	PART
ejpam-4386	5	34	)	)	PUNCT
ejpam-4386	5	35	of	of	ADP
ejpam-4386	5	36	a	a	DET
ejpam-4386	5	37	space	space	NOUN
ejpam-4386	5	38	(	(	PUNCT
ejpam-4386	5	39	x	x	X
ejpam-4386	5	40	,	,	PUNCT
ejpam-4386	5	41	τ	τ	PROPN
ejpam-4386	5	42	)	)	PUNCT
ejpam-4386	5	43	.	.	PUNCT
ejpam-4386	6	1	2020	2020	NUM
ejpam-4386	6	2	mathematics	mathematic	NOUN
ejpam-4386	6	3	subject	subject	NOUN
ejpam-4386	6	4	classifications	classification	NOUN
ejpam-4386	6	5	:	:	PUNCT
ejpam-4386	6	6	54a10	54a10	NUM
ejpam-4386	6	7	,	,	PUNCT
ejpam-4386	6	8	54d10	54d10	NUM
ejpam-4386	6	9	.	.	PUNCT
ejpam-4386	7	1	key	key	ADJ
ejpam-4386	7	2	words	word	NOUN
ejpam-4386	7	3	and	and	CCONJ
ejpam-4386	7	4	phrases	phrase	NOUN
ejpam-4386	7	5	:	:	PUNCT
ejpam-4386	7	6	semi	semi	ADJ
ejpam-4386	7	7	-	-	ADJ
ejpam-4386	7	8	regular	regular	ADJ
ejpam-4386	7	9	,	,	PUNCT
ejpam-4386	7	10	semi	semi	ADJ
ejpam-4386	7	11	-	-	NOUN
ejpam-4386	7	12	regularization	regularization	ADJ
ejpam-4386	7	13	,	,	PUNCT
ejpam-4386	7	14	alexandroff	alexandroff	NOUN
ejpam-4386	7	15	duplicate	duplicate	NOUN
ejpam-4386	7	16	,	,	PUNCT
ejpam-4386	7	17	closed	closed	ADJ
ejpam-4386	7	18	extension	extension	NOUN
ejpam-4386	7	19	,	,	PUNCT
ejpam-4386	7	20	discrete	discrete	ADJ
ejpam-4386	7	21	extension	extension	NOUN
ejpam-4386	7	22	,	,	PUNCT
ejpam-4386	7	23	open	open	ADJ
ejpam-4386	7	24	extension	extension	NOUN
ejpam-4386	7	25	,	,	PUNCT
ejpam-4386	7	26	c	c	NOUN
ejpam-4386	7	27	-	-	PUNCT
ejpam-4386	7	28	normality	normality	ADJ
ejpam-4386	7	29	,	,	PUNCT
ejpam-4386	7	30	epi	epi	NOUN
ejpam-4386	7	31	-	-	NOUN
ejpam-4386	7	32	normality	normality	ADJ
ejpam-4386	7	33	,	,	PUNCT
ejpam-4386	7	34	submetrizability	submetrizability	NOUN
ejpam-4386	7	35	,	,	PUNCT
ejpam-4386	7	36	scattered	scatter	VERB
ejpam-4386	7	37	.	.	PUNCT
ejpam-4386	8	1	if	if	SCONJ
ejpam-4386	8	2	(	(	PUNCT
ejpam-4386	8	3	x	x	X
ejpam-4386	8	4	,	,	PUNCT
ejpam-4386	8	5	τ	τ	PROPN
ejpam-4386	8	6	)	)	PUNCT
ejpam-4386	8	7	is	be	AUX
ejpam-4386	8	8	a	a	DET
ejpam-4386	8	9	topological	topological	ADJ
ejpam-4386	8	10	space	space	NOUN
ejpam-4386	8	11	,	,	PUNCT
ejpam-4386	8	12	then	then	ADV
ejpam-4386	8	13	the	the	DET
ejpam-4386	8	14	semi	semi	ADJ
ejpam-4386	8	15	-	-	ADJ
ejpam-4386	8	16	regularization	regularization	ADJ
ejpam-4386	8	17	topology	topology	NOUN
ejpam-4386	8	18	τ	τ	X
ejpam-4386	8	19	s	s	VERB
ejpam-4386	8	20	on	on	ADP
ejpam-4386	8	21	x	x	PUNCT
ejpam-4386	8	22	of	of	ADP
ejpam-4386	8	23	τ	τ	PROPN
ejpam-4386	8	24	is	be	AUX
ejpam-4386	8	25	the	the	DET
ejpam-4386	8	26	coarser	coarse	ADJ
ejpam-4386	8	27	topology	topology	NOUN
ejpam-4386	8	28	on	on	ADP
ejpam-4386	8	29	x	x	PUNCT
ejpam-4386	8	30	generated	generate	VERB
ejpam-4386	8	31	by	by	ADP
ejpam-4386	8	32	the	the	DET
ejpam-4386	8	33	family	family	NOUN
ejpam-4386	8	34	of	of	ADP
ejpam-4386	8	35	all	all	DET
ejpam-4386	8	36	open	open	ADJ
ejpam-4386	8	37	domains	domain	NOUN
ejpam-4386	8	38	of	of	ADP
ejpam-4386	8	39	(	(	PUNCT
ejpam-4386	8	40	x	x	INTJ
ejpam-4386	8	41	,	,	PUNCT
ejpam-4386	8	42	τ	τ	PROPN
ejpam-4386	8	43	)	)	PUNCT
ejpam-4386	8	44	where	where	SCONJ
ejpam-4386	8	45	a	a	DET
ejpam-4386	8	46	subset	subset	NOUN
ejpam-4386	8	47	u	u	NOUN
ejpam-4386	8	48	is	be	AUX
ejpam-4386	8	49	called	call	VERB
ejpam-4386	8	50	an	an	DET
ejpam-4386	8	51	open	open	ADJ
ejpam-4386	8	52	domain	domain	NOUN
ejpam-4386	8	53	if	if	SCONJ
ejpam-4386	8	54	u	u	NOUN
ejpam-4386	8	55	=	=	SYM
ejpam-4386	8	56	int(u	int(u	PROPN
ejpam-4386	8	57	)	)	PUNCT
ejpam-4386	8	58	.	.	PUNCT
ejpam-4386	9	1	in	in	ADP
ejpam-4386	9	2	this	this	DET
ejpam-4386	9	3	paper	paper	NOUN
ejpam-4386	9	4	,	,	PUNCT
ejpam-4386	9	5	we	we	PRON
ejpam-4386	9	6	study	study	VERB
ejpam-4386	9	7	some	some	DET
ejpam-4386	9	8	properties	property	NOUN
ejpam-4386	9	9	of	of	ADP
ejpam-4386	9	10	weaker	weak	ADJ
ejpam-4386	9	11	version	version	NOUN
ejpam-4386	9	12	of	of	ADP
ejpam-4386	9	13	normality	normality	NOUN
ejpam-4386	9	14	of	of	ADP
ejpam-4386	9	15	the	the	DET
ejpam-4386	9	16	semi	semi	ADJ
ejpam-4386	9	17	-	-	ADJ
ejpam-4386	9	18	regularization	regularization	ADJ
ejpam-4386	9	19	space	space	NOUN
ejpam-4386	9	20	(	(	PUNCT
ejpam-4386	9	21	x	x	X
ejpam-4386	9	22	,	,	PUNCT
ejpam-4386	9	23	τ	τ	PROPN
ejpam-4386	9	24	s	s	PART
ejpam-4386	9	25	)	)	PUNCT
ejpam-4386	9	26	of	of	ADP
ejpam-4386	9	27	a	a	DET
ejpam-4386	9	28	space	space	NOUN
ejpam-4386	9	29	(	(	PUNCT
ejpam-4386	9	30	x	x	X
ejpam-4386	9	31	,	,	PUNCT
ejpam-4386	9	32	τ	τ	PROPN
ejpam-4386	9	33	)	)	PUNCT
ejpam-4386	9	34	.	.	PUNCT
ejpam-4386	10	1	also	also	ADV
ejpam-4386	10	2	,	,	PUNCT
ejpam-4386	10	3	we	we	PRON
ejpam-4386	10	4	study	study	VERB
ejpam-4386	10	5	the	the	DET
ejpam-4386	10	6	semi	semi	NOUN
ejpam-4386	10	7	-	-	NOUN
ejpam-4386	10	8	regularity	regularity	NOUN
ejpam-4386	10	9	of	of	ADP
ejpam-4386	10	10	some	some	DET
ejpam-4386	10	11	generated	generate	VERB
ejpam-4386	10	12	spaces	space	NOUN
ejpam-4386	10	13	.	.	PUNCT
ejpam-4386	11	1	this	this	DET
ejpam-4386	11	2	paper	paper	NOUN
ejpam-4386	11	3	may	may	AUX
ejpam-4386	11	4	considered	consider	VERB
ejpam-4386	11	5	as	as	ADP
ejpam-4386	11	6	a	a	DET
ejpam-4386	11	7	continuation	continuation	NOUN
ejpam-4386	11	8	of	of	ADP
ejpam-4386	11	9	the	the	DET
ejpam-4386	11	10	study	study	NOUN
ejpam-4386	11	11	of	of	ADP
ejpam-4386	11	12	mrs̆ević	mrs̆ević	PROPN
ejpam-4386	11	13	,	,	PUNCT
ejpam-4386	11	14	reilly	reilly	ADV
ejpam-4386	11	15	and	and	CCONJ
ejpam-4386	11	16	vamanamurthy	vamanamurthy	ADJ
ejpam-4386	11	17	in	in	ADP
ejpam-4386	11	18	[	[	X
ejpam-4386	11	19	10	10	NUM
ejpam-4386	11	20	]	]	PUNCT
ejpam-4386	11	21	.	.	PUNCT
ejpam-4386	12	1	throughout	throughout	ADP
ejpam-4386	12	2	this	this	DET
ejpam-4386	12	3	paper	paper	NOUN
ejpam-4386	12	4	,	,	PUNCT
ejpam-4386	12	5	we	we	PRON
ejpam-4386	12	6	denote	denote	VERB
ejpam-4386	12	7	the	the	DET
ejpam-4386	12	8	set	set	NOUN
ejpam-4386	12	9	of	of	ADP
ejpam-4386	12	10	positive	positive	ADJ
ejpam-4386	12	11	integers	integer	NOUN
ejpam-4386	12	12	by	by	ADP
ejpam-4386	12	13	n	n	CCONJ
ejpam-4386	12	14	,	,	PUNCT
ejpam-4386	12	15	the	the	DET
ejpam-4386	12	16	rationals	rational	NOUN
ejpam-4386	12	17	by	by	ADP
ejpam-4386	12	18	q	q	NOUN
ejpam-4386	12	19	,	,	PUNCT
ejpam-4386	12	20	the	the	DET
ejpam-4386	12	21	irrationals	irrational	NOUN
ejpam-4386	12	22	by	by	ADP
ejpam-4386	12	23	p	p	NOUN
ejpam-4386	12	24	,	,	PUNCT
ejpam-4386	12	25	and	and	CCONJ
ejpam-4386	12	26	the	the	DET
ejpam-4386	12	27	set	set	NOUN
ejpam-4386	12	28	of	of	ADP
ejpam-4386	12	29	real	real	ADJ
ejpam-4386	12	30	numbers	number	NOUN
ejpam-4386	12	31	by	by	ADP
ejpam-4386	12	32	r.	r.	PROPN
ejpam-4386	12	33	a	a	DET
ejpam-4386	12	34	t4	t4	PROPN
ejpam-4386	12	35	space	space	NOUN
ejpam-4386	12	36	is	be	AUX
ejpam-4386	12	37	a	a	DET
ejpam-4386	12	38	t1	t1	NOUN
ejpam-4386	12	39	normal	normal	ADJ
ejpam-4386	12	40	space	space	NOUN
ejpam-4386	12	41	and	and	CCONJ
ejpam-4386	12	42	a	a	DET
ejpam-4386	12	43	tychonoff	tychonoff	NOUN
ejpam-4386	12	44	space	space	NOUN
ejpam-4386	12	45	(	(	PUNCT
ejpam-4386	12	46	t3	t3	NOUN
ejpam-4386	12	47	1	1	NUM
ejpam-4386	12	48	2	2	NUM
ejpam-4386	12	49	)	)	PUNCT
ejpam-4386	12	50	is	be	AUX
ejpam-4386	12	51	a	a	DET
ejpam-4386	12	52	t1	t1	NOUN
ejpam-4386	12	53	completely	completely	ADV
ejpam-4386	12	54	regular	regular	ADJ
ejpam-4386	12	55	space	space	NOUN
ejpam-4386	12	56	.	.	PUNCT
ejpam-4386	13	1	we	we	PRON
ejpam-4386	13	2	do	do	AUX
ejpam-4386	13	3	not	not	PART
ejpam-4386	13	4	assume	assume	VERB
ejpam-4386	13	5	t2	t2	NOUN
ejpam-4386	13	6	in	in	ADP
ejpam-4386	13	7	the	the	DET
ejpam-4386	13	8	definition	definition	NOUN
ejpam-4386	13	9	of	of	ADP
ejpam-4386	13	10	compactness	compactness	NOUN
ejpam-4386	13	11	and	and	CCONJ
ejpam-4386	13	12	countable	countable	ADJ
ejpam-4386	13	13	compactness	compactness	NOUN
ejpam-4386	13	14	.	.	PUNCT
ejpam-4386	14	1	for	for	ADP
ejpam-4386	14	2	a	a	DET
ejpam-4386	14	3	subset	subset	NOUN
ejpam-4386	14	4	a	a	PRON
ejpam-4386	14	5	of	of	ADP
ejpam-4386	14	6	a	a	DET
ejpam-4386	14	7	space	space	NOUN
ejpam-4386	14	8	x	x	NOUN
ejpam-4386	14	9	,	,	PUNCT
ejpam-4386	14	10	inta	inta	PROPN
ejpam-4386	14	11	and	and	CCONJ
ejpam-4386	14	12	a	a	DET
ejpam-4386	14	13	denote	denote	NOUN
ejpam-4386	14	14	the	the	DET
ejpam-4386	14	15	interior	interior	NOUN
ejpam-4386	14	16	and	and	CCONJ
ejpam-4386	14	17	the	the	DET
ejpam-4386	14	18	closure	closure	NOUN
ejpam-4386	14	19	of	of	ADP
ejpam-4386	14	20	a	a	PRON
ejpam-4386	14	21	,	,	PUNCT
ejpam-4386	14	22	respectively	respectively	ADV
ejpam-4386	14	23	.	.	PUNCT
ejpam-4386	15	1	if	if	SCONJ
ejpam-4386	15	2	two	two	NUM
ejpam-4386	15	3	topologies	topology	NOUN
ejpam-4386	15	4	τ	τ	X
ejpam-4386	15	5	and	and	CCONJ
ejpam-4386	15	6	τ	τ	PROPN
ejpam-4386	15	7	′	′	NOUN
ejpam-4386	15	8	on	on	ADP
ejpam-4386	15	9	a	a	DET
ejpam-4386	15	10	set	set	NOUN
ejpam-4386	15	11	x	x	SYM
ejpam-4386	15	12	are	be	AUX
ejpam-4386	15	13	considered	consider	VERB
ejpam-4386	15	14	,	,	PUNCT
ejpam-4386	15	15	we	we	PRON
ejpam-4386	15	16	denote	denote	VERB
ejpam-4386	15	17	the	the	DET
ejpam-4386	15	18	interior	interior	NOUN
ejpam-4386	15	19	of	of	ADP
ejpam-4386	15	20	a	a	DET
ejpam-4386	15	21	in	in	ADP
ejpam-4386	15	22	(	(	PUNCT
ejpam-4386	15	23	x	x	INTJ
ejpam-4386	15	24	,	,	PUNCT
ejpam-4386	15	25	τ	τ	PROPN
ejpam-4386	15	26	)	)	PUNCT
ejpam-4386	15	27	by	by	ADP
ejpam-4386	15	28	int	int	NOUN
ejpam-4386	15	29	τa	τa	ADJ
ejpam-4386	15	30	and	and	CCONJ
ejpam-4386	15	31	int	int	NOUN
ejpam-4386	15	32	τ	τ	X
ejpam-4386	15	33	′a	′a	NOUN
ejpam-4386	15	34	for	for	ADP
ejpam-4386	15	35	the	the	DET
ejpam-4386	15	36	interior	interior	NOUN
ejpam-4386	15	37	of	of	ADP
ejpam-4386	15	38	a	a	DET
ejpam-4386	15	39	in	in	ADP
ejpam-4386	15	40	(	(	PUNCT
ejpam-4386	15	41	x	x	INTJ
ejpam-4386	15	42	,	,	PUNCT
ejpam-4386	15	43	τ	τ	PROPN
ejpam-4386	15	44	′	′	NUM
ejpam-4386	15	45	)	)	PUNCT
ejpam-4386	15	46	.	.	PUNCT
ejpam-4386	16	1	we	we	PRON
ejpam-4386	16	2	denote	denote	VERB
ejpam-4386	16	3	the	the	DET
ejpam-4386	16	4	closure	closure	NOUN
ejpam-4386	16	5	of	of	ADP
ejpam-4386	16	6	a	a	DET
ejpam-4386	16	7	in	in	ADP
ejpam-4386	16	8	(	(	PUNCT
ejpam-4386	16	9	x	x	INTJ
ejpam-4386	16	10	,	,	PUNCT
ejpam-4386	16	11	τ	τ	PROPN
ejpam-4386	16	12	′	′	NUM
ejpam-4386	16	13	)	)	PUNCT
ejpam-4386	16	14	by	by	ADP
ejpam-4386	16	15	a	a	DET
ejpam-4386	16	16	τ	τ	NOUN
ejpam-4386	16	17	′	′	NUM
ejpam-4386	17	1	and	and	CCONJ
ejpam-4386	17	2	,	,	PUNCT
ejpam-4386	17	3	similarly	similarly	ADV
ejpam-4386	17	4	,	,	PUNCT
ejpam-4386	17	5	a	a	DET
ejpam-4386	17	6	τ	τ	PROPN
ejpam-4386	17	7	denotes	denote	VERB
ejpam-4386	17	8	the	the	DET
ejpam-4386	17	9	closure	closure	NOUN
ejpam-4386	17	10	of	of	ADP
ejpam-4386	17	11	a	a	DET
ejpam-4386	17	12	in	in	ADP
ejpam-4386	17	13	(	(	PUNCT
ejpam-4386	17	14	x	x	INTJ
ejpam-4386	17	15	,	,	PUNCT
ejpam-4386	17	16	τ	τ	PROPN
ejpam-4386	17	17	)	)	PUNCT
ejpam-4386	17	18	.	.	PUNCT
ejpam-4386	18	1	we	we	PRON
ejpam-4386	18	2	denote	denote	VERB
ejpam-4386	18	3	an	an	DET
ejpam-4386	18	4	order	order	NOUN
ejpam-4386	18	5	pair	pair	NOUN
ejpam-4386	18	6	by	by	ADP
ejpam-4386	18	7	⟨x	⟨x	NUM
ejpam-4386	18	8	,	,	PUNCT
ejpam-4386	18	9	y⟩.	y⟩.	NUM
ejpam-4386	18	10	∗corresponding	∗corresponde	VERB
ejpam-4386	18	11	author	author	NOUN
ejpam-4386	18	12	.	.	PUNCT
ejpam-4386	19	1	doi	doi	NOUN
ejpam-4386	19	2	:	:	PUNCT
ejpam-4386	19	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4386	https://doi.org/10.29020/nybg.ejpam.v15i3.4386	PROPN
ejpam-4386	19	4	email	email	NOUN
ejpam-4386	19	5	addresses	address	NOUN
ejpam-4386	19	6	:	:	PUNCT
ejpam-4386	19	7	dabuzaid@kau.edu.sa	dabuzaid@kau.edu.sa	NOUN
ejpam-4386	19	8	,	,	PUNCT
ejpam-4386	19	9	dina.abuzaid@gmail.com	dina.abuzaid@gmail.com	X
ejpam-4386	19	10	(	(	PUNCT
ejpam-4386	19	11	d.	d.	PROPN
ejpam-4386	19	12	abuzaid	abuzaid	PROPN
ejpam-4386	19	13	)	)	PUNCT
ejpam-4386	19	14	,	,	PUNCT
ejpam-4386	19	15	nhamdanalfarsi@stu.kau.edu.sa	nhamdanalfarsi@stu.kau.edu.sa	PROPN
ejpam-4386	19	16	,	,	PUNCT
ejpam-4386	19	17	noufalfarsi98@gmail.com	noufalfarsi98@gmail.com	X
ejpam-4386	19	18	(	(	PUNCT
ejpam-4386	19	19	n.	n.	NOUN
ejpam-4386	19	20	alfarsi	alfarsi	PROPN
ejpam-4386	19	21	)	)	PUNCT
ejpam-4386	19	22	,	,	PUNCT
ejpam-4386	19	23	lnkalantan@hotmail.com	lnkalantan@hotmail.com	PROPN
ejpam-4386	19	24	,	,	PUNCT
ejpam-4386	19	25	lkalantan@kau.edu.sa	lkalantan@kau.edu.sa	PROPN
ejpam-4386	19	26	(	(	PUNCT
ejpam-4386	19	27	l.	l.	PROPN
ejpam-4386	19	28	kalantan	kalantan	PROPN
ejpam-4386	19	29	)	)	PUNCT
ejpam-4386	19	30	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4386	20	1	821	821	NUM
ejpam-4386	20	2	©	©	PROPN
ejpam-4386	20	3	2022	2022	NUM
ejpam-4386	20	4	ejpam	ejpam	VERB
ejpam-4386	20	5	all	all	DET
ejpam-4386	20	6	rights	right	NOUN
ejpam-4386	20	7	reserved	reserve	VERB
ejpam-4386	20	8	.	.	PUNCT
ejpam-4386	21	1	d.	d.	PROPN
ejpam-4386	21	2	abuzaid	abuzaid	PROPN
ejpam-4386	21	3	,	,	PUNCT
ejpam-4386	21	4	n.	n.	PROPN
ejpam-4386	21	5	alfarsi	alfarsi	NOUN
ejpam-4386	21	6	,	,	PUNCT
ejpam-4386	21	7	l.	l.	PROPN
ejpam-4386	21	8	kalantan	kalantan	PROPN
ejpam-4386	21	9	/	/	SYM
ejpam-4386	21	10	eur	eur	PROPN
ejpam-4386	21	11	.	.	PUNCT
ejpam-4386	22	1	j.	j.	PROPN
ejpam-4386	22	2	pure	pure	PROPN
ejpam-4386	22	3	appl	appl	PROPN
ejpam-4386	22	4	.	.	PROPN
ejpam-4386	22	5	math	math	PROPN
ejpam-4386	22	6	,	,	PUNCT
ejpam-4386	22	7	15	15	NUM
ejpam-4386	22	8	(	(	PUNCT
ejpam-4386	22	9	3	3	NUM
ejpam-4386	22	10	)	)	PUNCT
ejpam-4386	22	11	(	(	PUNCT
ejpam-4386	22	12	2022	2022	NUM
ejpam-4386	22	13	)	)	PUNCT
ejpam-4386	22	14	,	,	PUNCT
ejpam-4386	22	15	821	821	NUM
ejpam-4386	22	16	-	-	SYM
ejpam-4386	22	17	829	829	NUM
ejpam-4386	22	18	822	822	NUM
ejpam-4386	22	19	1	1	NUM
ejpam-4386	22	20	.	.	PUNCT
ejpam-4386	23	1	semi	semi	ADJ
ejpam-4386	23	2	-	-	NOUN
ejpam-4386	23	3	regularity	regularity	ADJ
ejpam-4386	23	4	we	we	PRON
ejpam-4386	23	5	have	have	VERB
ejpam-4386	23	6	to	to	PART
ejpam-4386	23	7	start	start	VERB
ejpam-4386	23	8	by	by	ADP
ejpam-4386	23	9	recalling	recall	VERB
ejpam-4386	23	10	some	some	DET
ejpam-4386	23	11	basic	basic	ADJ
ejpam-4386	23	12	definitions	definition	NOUN
ejpam-4386	23	13	.	.	PUNCT
ejpam-4386	24	1	definition	definition	NOUN
ejpam-4386	24	2	1	1	NUM
ejpam-4386	24	3	.	.	PUNCT
ejpam-4386	25	1	a	a	DET
ejpam-4386	25	2	subset	subset	NOUN
ejpam-4386	25	3	a	a	PRON
ejpam-4386	25	4	of	of	ADP
ejpam-4386	25	5	a	a	DET
ejpam-4386	25	6	space	space	NOUN
ejpam-4386	25	7	x	x	PUNCT
ejpam-4386	25	8	is	be	AUX
ejpam-4386	25	9	called	call	VERB
ejpam-4386	25	10	closed	closed	ADJ
ejpam-4386	25	11	domain	domain	NOUN
ejpam-4386	25	12	[	[	X
ejpam-4386	25	13	6	6	NUM
ejpam-4386	25	14	]	]	PUNCT
ejpam-4386	25	15	,	,	PUNCT
ejpam-4386	25	16	called	call	VERB
ejpam-4386	25	17	also	also	ADV
ejpam-4386	25	18	regular	regular	ADJ
ejpam-4386	25	19	closed	closed	ADJ
ejpam-4386	25	20	,	,	PUNCT
ejpam-4386	25	21	κ	κ	NOUN
ejpam-4386	25	22	-	-	PUNCT
ejpam-4386	25	23	closed	closed	ADJ
ejpam-4386	25	24	[	[	X
ejpam-4386	25	25	9	9	NUM
ejpam-4386	25	26	]	]	PUNCT
ejpam-4386	25	27	,	,	PUNCT
ejpam-4386	25	28	if	if	SCONJ
ejpam-4386	25	29	a	a	DET
ejpam-4386	25	30	=	=	X
ejpam-4386	25	31	inta	inta	PROPN
ejpam-4386	25	32	.	.	PUNCT
ejpam-4386	26	1	a	a	DET
ejpam-4386	26	2	subset	subset	NOUN
ejpam-4386	26	3	a	a	PRON
ejpam-4386	26	4	of	of	ADP
ejpam-4386	26	5	a	a	DET
ejpam-4386	26	6	space	space	NOUN
ejpam-4386	26	7	x	x	PUNCT
ejpam-4386	26	8	is	be	AUX
ejpam-4386	26	9	called	call	VERB
ejpam-4386	26	10	open	open	ADJ
ejpam-4386	26	11	domain	domain	NOUN
ejpam-4386	26	12	[	[	X
ejpam-4386	26	13	6	6	NUM
ejpam-4386	26	14	]	]	PUNCT
ejpam-4386	26	15	,	,	PUNCT
ejpam-4386	26	16	called	call	VERB
ejpam-4386	26	17	also	also	ADV
ejpam-4386	26	18	regular	regular	ADJ
ejpam-4386	26	19	open	open	ADJ
ejpam-4386	26	20	,	,	PUNCT
ejpam-4386	26	21	κ	κ	NOUN
ejpam-4386	26	22	-	-	ADJ
ejpam-4386	26	23	open	open	ADJ
ejpam-4386	26	24	,	,	PUNCT
ejpam-4386	26	25	if	if	SCONJ
ejpam-4386	26	26	a	a	PRON
ejpam-4386	26	27	=	=	X
ejpam-4386	26	28	int(a	int(a	NOUN
ejpam-4386	26	29	)	)	PUNCT
ejpam-4386	26	30	.	.	PUNCT
ejpam-4386	27	1	it	it	PRON
ejpam-4386	27	2	is	be	AUX
ejpam-4386	27	3	easy	easy	ADJ
ejpam-4386	27	4	to	to	PART
ejpam-4386	27	5	see	see	VERB
ejpam-4386	27	6	that	that	SCONJ
ejpam-4386	27	7	a	a	DET
ejpam-4386	27	8	subset	subset	NOUN
ejpam-4386	27	9	is	be	AUX
ejpam-4386	27	10	an	an	DET
ejpam-4386	27	11	open	open	ADJ
ejpam-4386	27	12	domain	domain	NOUN
ejpam-4386	27	13	if	if	SCONJ
ejpam-4386	27	14	and	and	CCONJ
ejpam-4386	27	15	only	only	ADV
ejpam-4386	27	16	if	if	SCONJ
ejpam-4386	27	17	it	it	PRON
ejpam-4386	27	18	is	be	AUX
ejpam-4386	27	19	the	the	DET
ejpam-4386	27	20	interior	interior	NOUN
ejpam-4386	27	21	of	of	ADP
ejpam-4386	27	22	a	a	DET
ejpam-4386	27	23	closed	closed	ADJ
ejpam-4386	27	24	set	set	NOUN
ejpam-4386	27	25	,	,	PUNCT
ejpam-4386	27	26	a	a	DET
ejpam-4386	27	27	subset	subset	NOUN
ejpam-4386	27	28	is	be	AUX
ejpam-4386	27	29	a	a	DET
ejpam-4386	27	30	closed	closed	ADJ
ejpam-4386	27	31	domain	domain	NOUN
ejpam-4386	27	32	if	if	SCONJ
ejpam-4386	28	1	and	and	CCONJ
ejpam-4386	28	2	only	only	ADV
ejpam-4386	28	3	if	if	SCONJ
ejpam-4386	28	4	it	it	PRON
ejpam-4386	28	5	is	be	AUX
ejpam-4386	28	6	the	the	DET
ejpam-4386	28	7	closure	closure	NOUN
ejpam-4386	28	8	of	of	ADP
ejpam-4386	28	9	an	an	DET
ejpam-4386	28	10	open	open	ADJ
ejpam-4386	28	11	set	set	NOUN
ejpam-4386	28	12	,	,	PUNCT
ejpam-4386	28	13	the	the	DET
ejpam-4386	28	14	complement	complement	NOUN
ejpam-4386	28	15	of	of	ADP
ejpam-4386	28	16	a	a	DET
ejpam-4386	28	17	closed	closed	ADJ
ejpam-4386	28	18	domain	domain	NOUN
ejpam-4386	28	19	is	be	AUX
ejpam-4386	28	20	an	an	DET
ejpam-4386	28	21	open	open	ADJ
ejpam-4386	28	22	domain	domain	NOUN
ejpam-4386	28	23	and	and	CCONJ
ejpam-4386	28	24	the	the	DET
ejpam-4386	28	25	complement	complement	NOUN
ejpam-4386	28	26	of	of	ADP
ejpam-4386	28	27	an	an	DET
ejpam-4386	28	28	open	open	ADJ
ejpam-4386	28	29	domain	domain	NOUN
ejpam-4386	28	30	is	be	AUX
ejpam-4386	28	31	a	a	DET
ejpam-4386	28	32	closed	closed	ADJ
ejpam-4386	28	33	domain	domain	NOUN
ejpam-4386	28	34	[	[	X
ejpam-4386	28	35	6	6	NUM
ejpam-4386	28	36	]	]	PUNCT
ejpam-4386	28	37	.	.	PUNCT
ejpam-4386	29	1	now	now	ADV
ejpam-4386	29	2	,	,	PUNCT
ejpam-4386	29	3	let	let	VERB
ejpam-4386	29	4	(	(	PUNCT
ejpam-4386	29	5	x	x	X
ejpam-4386	29	6	,	,	PUNCT
ejpam-4386	29	7	τ	τ	PROPN
ejpam-4386	29	8	)	)	PUNCT
ejpam-4386	29	9	be	be	AUX
ejpam-4386	29	10	a	a	DET
ejpam-4386	29	11	topological	topological	ADJ
ejpam-4386	29	12	space	space	NOUN
ejpam-4386	29	13	and	and	CCONJ
ejpam-4386	29	14	let	let	VERB
ejpam-4386	29	15	od	od	PROPN
ejpam-4386	29	16	denotes	denote	VERB
ejpam-4386	29	17	the	the	DET
ejpam-4386	29	18	family	family	NOUN
ejpam-4386	29	19	of	of	ADP
ejpam-4386	29	20	all	all	DET
ejpam-4386	29	21	open	open	ADJ
ejpam-4386	29	22	domains	domain	NOUN
ejpam-4386	29	23	in	in	ADP
ejpam-4386	29	24	(	(	PUNCT
ejpam-4386	29	25	x	x	INTJ
ejpam-4386	29	26	,	,	PUNCT
ejpam-4386	29	27	τ	τ	PROPN
ejpam-4386	29	28	)	)	PUNCT
ejpam-4386	29	29	.	.	PUNCT
ejpam-4386	30	1	since	since	SCONJ
ejpam-4386	30	2	x	x	PRON
ejpam-4386	30	3	is	be	AUX
ejpam-4386	30	4	an	an	DET
ejpam-4386	30	5	open	open	ADJ
ejpam-4386	30	6	domain	domain	NOUN
ejpam-4386	30	7	and	and	CCONJ
ejpam-4386	30	8	an	an	DET
ejpam-4386	30	9	intersection	intersection	NOUN
ejpam-4386	30	10	of	of	ADP
ejpam-4386	30	11	two	two	NUM
ejpam-4386	30	12	open	open	ADJ
ejpam-4386	30	13	domains	domain	NOUN
ejpam-4386	30	14	is	be	AUX
ejpam-4386	30	15	an	an	DET
ejpam-4386	30	16	open	open	ADJ
ejpam-4386	30	17	domain	domain	NOUN
ejpam-4386	30	18	[	[	X
ejpam-4386	30	19	6	6	NUM
ejpam-4386	30	20	,	,	PUNCT
ejpam-4386	30	21	1.1.c	1.1.c	NUM
ejpam-4386	30	22	]	]	PUNCT
ejpam-4386	30	23	,	,	PUNCT
ejpam-4386	30	24	then	then	ADV
ejpam-4386	30	25	we	we	PRON
ejpam-4386	30	26	have	have	VERB
ejpam-4386	30	27	the	the	DET
ejpam-4386	30	28	following	follow	VERB
ejpam-4386	30	29	definition	definition	NOUN
ejpam-4386	30	30	[	[	X
ejpam-4386	30	31	6	6	NUM
ejpam-4386	30	32	]	]	PUNCT
ejpam-4386	30	33	.	.	PUNCT
ejpam-4386	31	1	definition	definition	NOUN
ejpam-4386	31	2	2	2	NUM
ejpam-4386	31	3	.	.	PUNCT
ejpam-4386	32	1	if	if	SCONJ
ejpam-4386	32	2	(	(	PUNCT
ejpam-4386	32	3	x	x	X
ejpam-4386	32	4	,	,	PUNCT
ejpam-4386	32	5	τ	τ	PROPN
ejpam-4386	32	6	)	)	PUNCT
ejpam-4386	32	7	is	be	AUX
ejpam-4386	32	8	a	a	DET
ejpam-4386	32	9	topological	topological	ADJ
ejpam-4386	32	10	space	space	NOUN
ejpam-4386	32	11	,	,	PUNCT
ejpam-4386	32	12	then	then	ADV
ejpam-4386	32	13	the	the	DET
ejpam-4386	32	14	semi	semi	ADJ
ejpam-4386	32	15	-	-	ADJ
ejpam-4386	32	16	regularization	regularization	ADJ
ejpam-4386	32	17	topology	topology	NOUN
ejpam-4386	32	18	τ	τ	X
ejpam-4386	32	19	s	s	VERB
ejpam-4386	32	20	on	on	ADP
ejpam-4386	32	21	x	x	PUNCT
ejpam-4386	32	22	of	of	ADP
ejpam-4386	32	23	τ	τ	PROPN
ejpam-4386	32	24	is	be	AUX
ejpam-4386	32	25	the	the	DET
ejpam-4386	32	26	coarser	coarse	ADJ
ejpam-4386	32	27	topology	topology	NOUN
ejpam-4386	32	28	on	on	ADP
ejpam-4386	32	29	x	x	PUNCT
ejpam-4386	32	30	generated	generate	VERB
ejpam-4386	32	31	by	by	ADP
ejpam-4386	32	32	the	the	DET
ejpam-4386	32	33	family	family	NOUN
ejpam-4386	32	34	of	of	ADP
ejpam-4386	32	35	all	all	DET
ejpam-4386	32	36	open	open	ADJ
ejpam-4386	32	37	domains	domain	NOUN
ejpam-4386	32	38	of	of	ADP
ejpam-4386	32	39	(	(	PUNCT
ejpam-4386	32	40	x	x	INTJ
ejpam-4386	32	41	,	,	PUNCT
ejpam-4386	32	42	τ	τ	PROPN
ejpam-4386	32	43	)	)	PUNCT
ejpam-4386	32	44	.	.	PUNCT
ejpam-4386	33	1	(	(	PUNCT
ejpam-4386	33	2	x	x	X
ejpam-4386	33	3	,	,	PUNCT
ejpam-4386	33	4	τ	τ	PROPN
ejpam-4386	33	5	)	)	PUNCT
ejpam-4386	33	6	is	be	AUX
ejpam-4386	33	7	called	call	VERB
ejpam-4386	33	8	semi	semi	ADJ
ejpam-4386	33	9	-	-	ADJ
ejpam-4386	33	10	regular	regular	ADJ
ejpam-4386	33	11	if	if	SCONJ
ejpam-4386	33	12	τ=	τ=	PROPN
ejpam-4386	33	13	τ	τ	X
ejpam-4386	33	14	s.	s.	PROPN
ejpam-4386	33	15	(	(	PUNCT
ejpam-4386	33	16	x	x	X
ejpam-4386	33	17	,	,	PUNCT
ejpam-4386	33	18	τ	τ	PROPN
ejpam-4386	33	19	s	s	PART
ejpam-4386	33	20	)	)	PUNCT
ejpam-4386	33	21	is	be	AUX
ejpam-4386	33	22	called	call	VERB
ejpam-4386	33	23	the	the	DET
ejpam-4386	33	24	semi	semi	ADJ
ejpam-4386	33	25	-	-	ADJ
ejpam-4386	33	26	regularization	regularization	ADJ
ejpam-4386	33	27	topological	topological	ADJ
ejpam-4386	33	28	space	space	NOUN
ejpam-4386	33	29	of	of	ADP
ejpam-4386	33	30	(	(	PUNCT
ejpam-4386	33	31	x	x	INTJ
ejpam-4386	33	32	,	,	PUNCT
ejpam-4386	33	33	τ	τ	PROPN
ejpam-4386	33	34	)	)	PUNCT
ejpam-4386	33	35	,	,	PUNCT
ejpam-4386	33	36	see	see	VERB
ejpam-4386	33	37	[	[	X
ejpam-4386	33	38	10	10	NUM
ejpam-4386	33	39	]	]	PUNCT
ejpam-4386	33	40	.	.	PUNCT
ejpam-4386	34	1	since	since	SCONJ
ejpam-4386	34	2	any	any	DET
ejpam-4386	34	3	open	open	ADJ
ejpam-4386	34	4	domain	domain	NOUN
ejpam-4386	34	5	is	be	AUX
ejpam-4386	34	6	an	an	DET
ejpam-4386	34	7	open	open	ADJ
ejpam-4386	34	8	set	set	NOUN
ejpam-4386	34	9	,	,	PUNCT
ejpam-4386	34	10	then	then	ADV
ejpam-4386	34	11	for	for	ADP
ejpam-4386	34	12	any	any	DET
ejpam-4386	34	13	space	space	NOUN
ejpam-4386	34	14	(	(	PUNCT
ejpam-4386	34	15	x	x	X
ejpam-4386	34	16	,	,	PUNCT
ejpam-4386	34	17	τ	τ	PROPN
ejpam-4386	34	18	)	)	PUNCT
ejpam-4386	34	19	,	,	PUNCT
ejpam-4386	34	20	we	we	PRON
ejpam-4386	34	21	have	have	VERB
ejpam-4386	34	22	that	that	SCONJ
ejpam-4386	34	23	τ	τ	PROPN
ejpam-4386	34	24	s	s	PART
ejpam-4386	34	25	is	be	AUX
ejpam-4386	34	26	coarser	coarse	ADJ
ejpam-4386	34	27	than	than	ADP
ejpam-4386	34	28	τ	τ	PROPN
ejpam-4386	34	29	,	,	PUNCT
ejpam-4386	34	30	that	that	ADV
ejpam-4386	34	31	is	is	ADV
ejpam-4386	34	32	,	,	PUNCT
ejpam-4386	34	33	τ	τ	PROPN
ejpam-4386	34	34	s	s	PROPN
ejpam-4386	34	35	⊆	⊆	NUM
ejpam-4386	34	36	τ	τ	X
ejpam-4386	34	37	.	.	PUNCT
ejpam-4386	35	1	note	note	VERB
ejpam-4386	35	2	that	that	SCONJ
ejpam-4386	35	3	if	if	SCONJ
ejpam-4386	35	4	∅	∅	NOUN
ejpam-4386	35	5	̸=	̸=	PROPN
ejpam-4386	35	6	u	u	NOUN
ejpam-4386	35	7	⊆	⊆	NUM
ejpam-4386	35	8	x	x	NOUN
ejpam-4386	35	9	,	,	PUNCT
ejpam-4386	35	10	then	then	ADV
ejpam-4386	35	11	u	u	PROPN
ejpam-4386	35	12	∈	∈	PROPN
ejpam-4386	35	13	τ	τ	X
ejpam-4386	35	14	s	s	X
ejpam-4386	35	15	if	if	SCONJ
ejpam-4386	35	16	and	and	CCONJ
ejpam-4386	35	17	only	only	ADV
ejpam-4386	35	18	if	if	SCONJ
ejpam-4386	35	19	u	u	PROPN
ejpam-4386	35	20	=	=	PUNCT
ejpam-4386	35	21	⋃	⋃	NOUN
ejpam-4386	35	22	α∈λ	α∈λ	NOUN
ejpam-4386	35	23	vα	vα	NOUN
ejpam-4386	35	24	with	with	ADP
ejpam-4386	35	25	vα	vα	PROPN
ejpam-4386	35	26	is	be	AUX
ejpam-4386	35	27	an	an	DET
ejpam-4386	35	28	open	open	ADJ
ejpam-4386	35	29	domain	domain	NOUN
ejpam-4386	35	30	in	in	ADP
ejpam-4386	35	31	(	(	PUNCT
ejpam-4386	35	32	x	x	INTJ
ejpam-4386	35	33	,	,	PUNCT
ejpam-4386	35	34	τ	τ	PROPN
ejpam-4386	35	35	)	)	PUNCT
ejpam-4386	35	36	for	for	ADP
ejpam-4386	35	37	each	each	DET
ejpam-4386	35	38	α	α	PRON
ejpam-4386	35	39	∈	∈	PROPN
ejpam-4386	35	40	λ	λ	PROPN
ejpam-4386	35	41	.	.	PUNCT
ejpam-4386	36	1	equivalently	equivalently	ADV
ejpam-4386	36	2	u	u	X
ejpam-4386	36	3	∈	∈	PROPN
ejpam-4386	36	4	τ	τ	X
ejpam-4386	36	5	s	s	X
ejpam-4386	36	6	if	if	SCONJ
ejpam-4386	36	7	and	and	CCONJ
ejpam-4386	36	8	only	only	ADV
ejpam-4386	36	9	if	if	SCONJ
ejpam-4386	36	10	for	for	ADP
ejpam-4386	36	11	each	each	DET
ejpam-4386	36	12	x	x	SYM
ejpam-4386	36	13	∈	∈	PROPN
ejpam-4386	36	14	u	u	NOUN
ejpam-4386	36	15	there	there	PRON
ejpam-4386	36	16	exists	exist	VERB
ejpam-4386	36	17	an	an	DET
ejpam-4386	36	18	open	open	ADJ
ejpam-4386	36	19	domain	domain	NOUN
ejpam-4386	36	20	g	g	NOUN
ejpam-4386	36	21	in	in	ADP
ejpam-4386	36	22	(	(	PUNCT
ejpam-4386	36	23	x	x	INTJ
ejpam-4386	36	24	,	,	PUNCT
ejpam-4386	36	25	τ	τ	PROPN
ejpam-4386	36	26	)	)	PUNCT
ejpam-4386	37	1	such	such	ADJ
ejpam-4386	37	2	that	that	SCONJ
ejpam-4386	37	3	x	x	SYM
ejpam-4386	37	4	∈	∈	NOUN
ejpam-4386	37	5	g	g	NOUN
ejpam-4386	37	6	⊆	⊆	NUM
ejpam-4386	37	7	u	u	NOUN
ejpam-4386	37	8	,	,	PUNCT
ejpam-4386	37	9	[	[	X
ejpam-4386	37	10	11	11	NUM
ejpam-4386	37	11	]	]	PUNCT
ejpam-4386	37	12	.	.	PUNCT
ejpam-4386	38	1	if	if	SCONJ
ejpam-4386	38	2	cf	cf	NOUN
ejpam-4386	38	3	is	be	AUX
ejpam-4386	38	4	the	the	DET
ejpam-4386	38	5	finite	finite	ADJ
ejpam-4386	38	6	complement	complement	NOUN
ejpam-4386	38	7	topology	topology	NOUN
ejpam-4386	38	8	on	on	ADP
ejpam-4386	38	9	an	an	DET
ejpam-4386	38	10	infinite	infinite	ADJ
ejpam-4386	38	11	set	set	NOUN
ejpam-4386	38	12	,	,	PUNCT
ejpam-4386	38	13	then	then	ADV
ejpam-4386	38	14	cfs	cfs	PROPN
ejpam-4386	39	1	=	=	PUNCT
ejpam-4386	39	2	i	i	PROPN
ejpam-4386	39	3	,	,	PUNCT
ejpam-4386	39	4	where	where	SCONJ
ejpam-4386	39	5	i	i	PRON
ejpam-4386	39	6	is	be	AUX
ejpam-4386	39	7	the	the	DET
ejpam-4386	39	8	indiscrete	indiscrete	ADJ
ejpam-4386	39	9	topology	topology	NOUN
ejpam-4386	39	10	.	.	PUNCT
ejpam-4386	40	1	if	if	SCONJ
ejpam-4386	40	2	cc	cc	PROPN
ejpam-4386	40	3	is	be	AUX
ejpam-4386	40	4	the	the	DET
ejpam-4386	40	5	countable	countable	ADJ
ejpam-4386	40	6	complement	complement	NOUN
ejpam-4386	40	7	topology	topology	NOUN
ejpam-4386	40	8	on	on	ADP
ejpam-4386	40	9	an	an	DET
ejpam-4386	40	10	uncountable	uncountable	ADJ
ejpam-4386	40	11	set	set	NOUN
ejpam-4386	40	12	,	,	PUNCT
ejpam-4386	40	13	then	then	ADV
ejpam-4386	40	14	ccs	ccs	PROPN
ejpam-4386	40	15	=	=	PROPN
ejpam-4386	40	16	i.	i.	PROPN
ejpam-4386	40	17	if	if	SCONJ
ejpam-4386	40	18	x	x	PRON
ejpam-4386	40	19	=	=	NOUN
ejpam-4386	40	20	{	{	PUNCT
ejpam-4386	40	21	⟨x	⟨x	NUM
ejpam-4386	40	22	,	,	PUNCT
ejpam-4386	40	23	y⟩	y⟩	NOUN
ejpam-4386	40	24	:	:	PUNCT
ejpam-4386	40	25	y	y	NOUN
ejpam-4386	40	26	≥	≥	NOUN
ejpam-4386	40	27	0	0	NUM
ejpam-4386	40	28	}	}	PUNCT
ejpam-4386	40	29	,	,	PUNCT
ejpam-4386	40	30	the	the	DET
ejpam-4386	40	31	closed	closed	ADJ
ejpam-4386	40	32	upper	upper	ADJ
ejpam-4386	40	33	half	half	NOUN
ejpam-4386	40	34	plan	plan	NOUN
ejpam-4386	40	35	,	,	PUNCT
ejpam-4386	40	36	then	then	ADV
ejpam-4386	40	37	the	the	DET
ejpam-4386	40	38	semiregularization	semiregularization	NOUN
ejpam-4386	40	39	topology	topology	NOUN
ejpam-4386	40	40	of	of	ADP
ejpam-4386	40	41	the	the	DET
ejpam-4386	40	42	half	half	ADJ
ejpam-4386	40	43	-	-	PUNCT
ejpam-4386	40	44	disc	disc	NOUN
ejpam-4386	40	45	topology	topology	NOUN
ejpam-4386	40	46	on	on	ADP
ejpam-4386	40	47	x	x	PUNCT
ejpam-4386	41	1	[	[	X
ejpam-4386	41	2	12	12	NUM
ejpam-4386	41	3	,	,	PUNCT
ejpam-4386	41	4	example	example	NOUN
ejpam-4386	41	5	78	78	NUM
ejpam-4386	41	6	]	]	PUNCT
ejpam-4386	41	7	,	,	PUNCT
ejpam-4386	41	8	is	be	AUX
ejpam-4386	41	9	the	the	DET
ejpam-4386	41	10	usual	usual	ADJ
ejpam-4386	41	11	metric	metric	ADJ
ejpam-4386	41	12	topology	topology	NOUN
ejpam-4386	41	13	u	u	NOUN
ejpam-4386	41	14	on	on	ADP
ejpam-4386	41	15	x.	x.	NOUN
ejpam-4386	41	16	if	if	SCONJ
ejpam-4386	41	17	x	x	PRON
ejpam-4386	41	18	is	be	AUX
ejpam-4386	41	19	regular	regular	ADJ
ejpam-4386	41	20	,	,	PUNCT
ejpam-4386	41	21	then	then	ADV
ejpam-4386	41	22	it	it	PRON
ejpam-4386	41	23	is	be	AUX
ejpam-4386	41	24	semi	semi	ADJ
ejpam-4386	41	25	-	-	ADJ
ejpam-4386	41	26	regular	regular	ADJ
ejpam-4386	41	27	[	[	X
ejpam-4386	41	28	6	6	NUM
ejpam-4386	41	29	,	,	PUNCT
ejpam-4386	41	30	1.1.8	1.1.8	NUM
ejpam-4386	41	31	]	]	PUNCT
ejpam-4386	41	32	.	.	PUNCT
ejpam-4386	42	1	the	the	DET
ejpam-4386	42	2	converse	converse	NOUN
ejpam-4386	42	3	is	be	AUX
ejpam-4386	42	4	not	not	PART
ejpam-4386	42	5	always	always	ADV
ejpam-4386	42	6	true	true	ADJ
ejpam-4386	42	7	.	.	PUNCT
ejpam-4386	43	1	as	as	ADP
ejpam-4386	43	2	an	an	DET
ejpam-4386	43	3	example	example	NOUN
ejpam-4386	43	4	the	the	DET
ejpam-4386	43	5	simplified	simplified	ADJ
ejpam-4386	43	6	arens	arens	PROPN
ejpam-4386	43	7	square	square	NOUN
ejpam-4386	43	8	[	[	X
ejpam-4386	43	9	12	12	NUM
ejpam-4386	43	10	,	,	PUNCT
ejpam-4386	43	11	example	example	NOUN
ejpam-4386	43	12	81	81	NUM
ejpam-4386	43	13	]	]	PUNCT
ejpam-4386	43	14	.	.	PUNCT
ejpam-4386	44	1	2	2	X
ejpam-4386	44	2	.	.	X
ejpam-4386	44	3	semi	semi	NOUN
ejpam-4386	44	4	-	-	NOUN
ejpam-4386	44	5	regularity	regularity	NOUN
ejpam-4386	44	6	of	of	ADP
ejpam-4386	44	7	generated	generate	VERB
ejpam-4386	44	8	spaces	space	NOUN
ejpam-4386	44	9	there	there	PRON
ejpam-4386	44	10	are	be	VERB
ejpam-4386	44	11	many	many	ADJ
ejpam-4386	44	12	ways	way	NOUN
ejpam-4386	44	13	of	of	ADP
ejpam-4386	44	14	generating	generate	VERB
ejpam-4386	44	15	new	new	ADJ
ejpam-4386	44	16	spaces	space	NOUN
ejpam-4386	44	17	from	from	ADP
ejpam-4386	44	18	old	old	ADJ
ejpam-4386	44	19	ones	one	NOUN
ejpam-4386	44	20	.	.	PUNCT
ejpam-4386	45	1	in	in	ADP
ejpam-4386	45	2	this	this	DET
ejpam-4386	45	3	section	section	NOUN
ejpam-4386	45	4	,	,	PUNCT
ejpam-4386	45	5	we	we	PRON
ejpam-4386	45	6	study	study	VERB
ejpam-4386	45	7	the	the	DET
ejpam-4386	45	8	semi	semi	NOUN
ejpam-4386	45	9	-	-	NOUN
ejpam-4386	45	10	regularity	regularity	NOUN
ejpam-4386	45	11	of	of	ADP
ejpam-4386	45	12	the	the	DET
ejpam-4386	45	13	alexandroff	alexandroff	NOUN
ejpam-4386	45	14	duplicate	duplicate	NOUN
ejpam-4386	45	15	,	,	PUNCT
ejpam-4386	45	16	the	the	DET
ejpam-4386	45	17	closed	closed	ADJ
ejpam-4386	45	18	extension	extension	NOUN
ejpam-4386	45	19	,	,	PUNCT
ejpam-4386	45	20	the	the	DET
ejpam-4386	45	21	discrete	discrete	ADJ
ejpam-4386	45	22	extension	extension	NOUN
ejpam-4386	45	23	,	,	PUNCT
ejpam-4386	45	24	and	and	CCONJ
ejpam-4386	45	25	the	the	DET
ejpam-4386	45	26	open	open	ADJ
ejpam-4386	45	27	extension	extension	NOUN
ejpam-4386	45	28	of	of	ADP
ejpam-4386	45	29	a	a	DET
ejpam-4386	45	30	given	give	VERB
ejpam-4386	45	31	space	space	NOUN
ejpam-4386	45	32	x.	x.	NOUN
ejpam-4386	45	33	recall	recall	VERB
ejpam-4386	45	34	that	that	SCONJ
ejpam-4386	45	35	the	the	DET
ejpam-4386	45	36	alexandroff	alexandroff	NOUN
ejpam-4386	45	37	duplicate	duplicate	VERB
ejpam-4386	45	38	space	space	NOUN
ejpam-4386	45	39	a(x	a(x	NOUN
ejpam-4386	45	40	)	)	PUNCT
ejpam-4386	45	41	of	of	ADP
ejpam-4386	45	42	a	a	DET
ejpam-4386	45	43	space	space	NOUN
ejpam-4386	45	44	x	x	PUNCT
ejpam-4386	45	45	is	be	AUX
ejpam-4386	45	46	defined	define	VERB
ejpam-4386	45	47	as	as	SCONJ
ejpam-4386	45	48	follows	follow	VERB
ejpam-4386	45	49	:	:	PUNCT
ejpam-4386	45	50	let	let	VERB
ejpam-4386	45	51	x	x	PRON
ejpam-4386	45	52	be	be	AUX
ejpam-4386	45	53	any	any	DET
ejpam-4386	45	54	topological	topological	ADJ
ejpam-4386	45	55	space	space	NOUN
ejpam-4386	45	56	.	.	PUNCT
ejpam-4386	46	1	let	let	VERB
ejpam-4386	46	2	x	x	X
ejpam-4386	46	3	′	′	NUM
ejpam-4386	47	1	=	=	PUNCT
ejpam-4386	47	2	x	x	SYM
ejpam-4386	47	3	×	×	NOUN
ejpam-4386	47	4	{	{	PUNCT
ejpam-4386	47	5	1	1	NUM
ejpam-4386	47	6	}	}	PUNCT
ejpam-4386	47	7	,	,	PUNCT
ejpam-4386	47	8	so	so	ADV
ejpam-4386	47	9	x	x	SYM
ejpam-4386	47	10	′	′	NOUN
ejpam-4386	47	11	is	be	AUX
ejpam-4386	47	12	just	just	ADV
ejpam-4386	47	13	a	a	DET
ejpam-4386	47	14	copy	copy	NOUN
ejpam-4386	47	15	of	of	ADP
ejpam-4386	47	16	x.	x.	NOUN
ejpam-4386	47	17	note	note	VERB
ejpam-4386	47	18	that	that	SCONJ
ejpam-4386	47	19	x	x	X
ejpam-4386	47	20	∩x	∩x	NOUN
ejpam-4386	48	1	′	′	NUM
ejpam-4386	48	2	=	=	PUNCT
ejpam-4386	48	3	∅.	∅.	AUX
ejpam-4386	48	4	let	let	VERB
ejpam-4386	48	5	a(x	a(x	NOUN
ejpam-4386	48	6	)	)	PUNCT
ejpam-4386	48	7	=	=	PUNCT
ejpam-4386	48	8	x	x	SYM
ejpam-4386	48	9	∪x	∪x	X
ejpam-4386	48	10	′.	′.	NOUN
ejpam-4386	48	11	for	for	ADP
ejpam-4386	48	12	simplicity	simplicity	NOUN
ejpam-4386	48	13	,	,	PUNCT
ejpam-4386	48	14	for	for	ADP
ejpam-4386	48	15	an	an	DET
ejpam-4386	48	16	element	element	NOUN
ejpam-4386	48	17	x	x	SYM
ejpam-4386	48	18	∈	∈	PROPN
ejpam-4386	48	19	x	x	NOUN
ejpam-4386	48	20	,	,	PUNCT
ejpam-4386	48	21	we	we	PRON
ejpam-4386	48	22	will	will	AUX
ejpam-4386	48	23	denote	denote	VERB
ejpam-4386	48	24	the	the	DET
ejpam-4386	48	25	element	element	NOUN
ejpam-4386	48	26	⟨x	⟨x	VERB
ejpam-4386	48	27	,	,	PUNCT
ejpam-4386	48	28	1⟩	1⟩	NUM
ejpam-4386	48	29	in	in	ADP
ejpam-4386	48	30	x	x	X
ejpam-4386	48	31	′	′	NUM
ejpam-4386	48	32	by	by	ADP
ejpam-4386	48	33	x′	x′	PROPN
ejpam-4386	48	34	and	and	CCONJ
ejpam-4386	48	35	for	for	ADP
ejpam-4386	48	36	a	a	DET
ejpam-4386	48	37	subset	subset	NOUN
ejpam-4386	48	38	b	b	NOUN
ejpam-4386	48	39	⊆	⊆	NUM
ejpam-4386	48	40	x	x	NUM
ejpam-4386	48	41	,	,	PUNCT
ejpam-4386	48	42	let	let	VERB
ejpam-4386	48	43	b′	b′	NOUN
ejpam-4386	48	44	=	=	PUNCT
ejpam-4386	48	45	{	{	PUNCT
ejpam-4386	48	46	x′	x′	PROPN
ejpam-4386	48	47	:	:	PUNCT
ejpam-4386	48	48	x	x	SYM
ejpam-4386	48	49	∈	∈	PROPN
ejpam-4386	48	50	b	b	AUX
ejpam-4386	48	51	}	}	PUNCT
ejpam-4386	48	52	=	=	SYM
ejpam-4386	48	53	b×{1	b×{1	NOUN
ejpam-4386	48	54	}	}	PUNCT
ejpam-4386	48	55	⊆	⊆	NUM
ejpam-4386	48	56	x	x	SYM
ejpam-4386	48	57	′.	′.	NOUN
ejpam-4386	48	58	for	for	ADP
ejpam-4386	48	59	each	each	DET
ejpam-4386	48	60	x′	x′	PROPN
ejpam-4386	48	61	∈	∈	PROPN
ejpam-4386	48	62	x	x	SYM
ejpam-4386	48	63	′	′	NOUN
ejpam-4386	48	64	,	,	PUNCT
ejpam-4386	48	65	let	let	VERB
ejpam-4386	48	66	b(x′	b(x′	NUM
ejpam-4386	48	67	)	)	PUNCT
ejpam-4386	48	68	=	=	PRON
ejpam-4386	48	69	{	{	PUNCT
ejpam-4386	48	70	{	{	PUNCT
ejpam-4386	48	71	x′	x′	NUM
ejpam-4386	48	72	}	}	PUNCT
ejpam-4386	48	73	}	}	PUNCT
ejpam-4386	48	74	.	.	PUNCT
ejpam-4386	49	1	for	for	ADP
ejpam-4386	49	2	each	each	DET
ejpam-4386	49	3	x	x	SYM
ejpam-4386	49	4	∈	∈	PROPN
ejpam-4386	49	5	x	x	NOUN
ejpam-4386	49	6	,	,	PUNCT
ejpam-4386	49	7	let	let	VERB
ejpam-4386	49	8	b(x	b(x	NOUN
ejpam-4386	49	9	)	)	PUNCT
ejpam-4386	49	10	=	=	SYM
ejpam-4386	50	1	{	{	PUNCT
ejpam-4386	50	2	u∪(u	u∪(u	NOUN
ejpam-4386	50	3	′\e	′\e	VERB
ejpam-4386	50	4	)	)	PUNCT
ejpam-4386	50	5	:	:	PUNCT
ejpam-4386	50	6	u	u	NOUN
ejpam-4386	50	7	is	be	AUX
ejpam-4386	50	8	open	open	ADJ
ejpam-4386	50	9	in	in	ADP
ejpam-4386	50	10	x	x	PUNCT
ejpam-4386	50	11	with	with	ADP
ejpam-4386	50	12	x	x	PROPN
ejpam-4386	50	13	∈	∈	PROPN
ejpam-4386	50	14	u	u	NOUN
ejpam-4386	50	15	and	and	CCONJ
ejpam-4386	50	16	e	e	NOUN
ejpam-4386	50	17	is	be	AUX
ejpam-4386	50	18	a	a	DET
ejpam-4386	50	19	finite	finite	NOUN
ejpam-4386	50	20	subset	subset	VERB
ejpam-4386	50	21	ofx	ofx	PROPN
ejpam-4386	50	22	′	′	NUM
ejpam-4386	50	23	}	}	PUNCT
ejpam-4386	50	24	.	.	PUNCT
ejpam-4386	51	1	then	then	ADV
ejpam-4386	51	2	b	b	X
ejpam-4386	51	3	=	=	PRON
ejpam-4386	51	4	{	{	PUNCT
ejpam-4386	51	5	b(x	b(x	NOUN
ejpam-4386	51	6	)	)	PUNCT
ejpam-4386	51	7	:	:	PUNCT
ejpam-4386	51	8	x	x	X
ejpam-4386	51	9	∈	∈	NOUN
ejpam-4386	51	10	x}∪{b(x′	x}∪{b(x′	NUM
ejpam-4386	51	11	)	)	PUNCT
ejpam-4386	51	12	:	:	PUNCT
ejpam-4386	52	1	x′	x′	X
ejpam-4386	52	2	∈	∈	NOUN
ejpam-4386	52	3	x	x	SYM
ejpam-4386	52	4	′	′	NUM
ejpam-4386	52	5	}	}	PUNCT
ejpam-4386	52	6	d.	d.	PROPN
ejpam-4386	52	7	abuzaid	abuzaid	PROPN
ejpam-4386	52	8	,	,	PUNCT
ejpam-4386	52	9	n.	n.	PROPN
ejpam-4386	52	10	alfarsi	alfarsi	NOUN
ejpam-4386	52	11	,	,	PUNCT
ejpam-4386	52	12	l.	l.	PROPN
ejpam-4386	52	13	kalantan	kalantan	PROPN
ejpam-4386	52	14	/	/	SYM
ejpam-4386	52	15	eur	eur	PROPN
ejpam-4386	52	16	.	.	PUNCT
ejpam-4386	53	1	j.	j.	PROPN
ejpam-4386	53	2	pure	pure	PROPN
ejpam-4386	53	3	appl	appl	PROPN
ejpam-4386	53	4	.	.	PROPN
ejpam-4386	53	5	math	math	PROPN
ejpam-4386	53	6	,	,	PUNCT
ejpam-4386	53	7	15	15	NUM
ejpam-4386	53	8	(	(	PUNCT
ejpam-4386	53	9	3	3	NUM
ejpam-4386	53	10	)	)	PUNCT
ejpam-4386	53	11	(	(	PUNCT
ejpam-4386	53	12	2022	2022	NUM
ejpam-4386	53	13	)	)	PUNCT
ejpam-4386	53	14	,	,	PUNCT
ejpam-4386	53	15	821	821	NUM
ejpam-4386	53	16	-	-	SYM
ejpam-4386	53	17	829	829	NUM
ejpam-4386	53	18	823	823	NUM
ejpam-4386	53	19	will	will	AUX
ejpam-4386	53	20	generate	generate	VERB
ejpam-4386	53	21	a	a	DET
ejpam-4386	53	22	unique	unique	ADJ
ejpam-4386	53	23	topology	topology	NOUN
ejpam-4386	53	24	on	on	ADP
ejpam-4386	53	25	a(x	a(x	NOUN
ejpam-4386	53	26	)	)	PUNCT
ejpam-4386	53	27	such	such	ADJ
ejpam-4386	53	28	that	that	SCONJ
ejpam-4386	53	29	b	b	PROPN
ejpam-4386	53	30	is	be	AUX
ejpam-4386	53	31	its	its	PRON
ejpam-4386	53	32	neighborhood	neighborhood	NOUN
ejpam-4386	53	33	system	system	NOUN
ejpam-4386	53	34	.	.	PUNCT
ejpam-4386	54	1	a(x	a(x	NOUN
ejpam-4386	54	2	)	)	PUNCT
ejpam-4386	54	3	with	with	ADP
ejpam-4386	54	4	this	this	DET
ejpam-4386	54	5	topology	topology	NOUN
ejpam-4386	54	6	is	be	AUX
ejpam-4386	54	7	called	call	VERB
ejpam-4386	54	8	the	the	DET
ejpam-4386	54	9	alexandroff	alexandroff	ADJ
ejpam-4386	54	10	duplicate	duplicate	NOUN
ejpam-4386	54	11	of	of	ADP
ejpam-4386	54	12	x	x	PUNCT
ejpam-4386	54	13	[	[	X
ejpam-4386	54	14	2	2	NUM
ejpam-4386	54	15	,	,	PUNCT
ejpam-4386	54	16	5	5	NUM
ejpam-4386	54	17	]	]	PUNCT
ejpam-4386	54	18	.	.	PUNCT
ejpam-4386	55	1	our	our	PRON
ejpam-4386	55	2	goal	goal	NOUN
ejpam-4386	55	3	is	be	AUX
ejpam-4386	55	4	to	to	PART
ejpam-4386	55	5	show	show	VERB
ejpam-4386	55	6	that	that	SCONJ
ejpam-4386	55	7	“	"	PUNCT
ejpam-4386	55	8	if	if	SCONJ
ejpam-4386	55	9	x	x	PRON
ejpam-4386	55	10	is	be	AUX
ejpam-4386	55	11	semi	semi	ADJ
ejpam-4386	55	12	-	-	ADJ
ejpam-4386	55	13	regular	regular	ADJ
ejpam-4386	55	14	,	,	PUNCT
ejpam-4386	55	15	then	then	ADV
ejpam-4386	55	16	so	so	ADV
ejpam-4386	55	17	is	be	AUX
ejpam-4386	55	18	its	its	PRON
ejpam-4386	55	19	alexandroff	alexandroff	NOUN
ejpam-4386	55	20	duplicate	duplicate	VERB
ejpam-4386	55	21	a(x	a(x	NOUN
ejpam-4386	55	22	)	)	PUNCT
ejpam-4386	55	23	”	"	PUNCT
ejpam-4386	55	24	.	.	PUNCT
ejpam-4386	56	1	in	in	ADP
ejpam-4386	56	2	order	order	NOUN
ejpam-4386	56	3	to	to	PART
ejpam-4386	56	4	show	show	VERB
ejpam-4386	56	5	this	this	PRON
ejpam-4386	56	6	we	we	PRON
ejpam-4386	56	7	will	will	AUX
ejpam-4386	56	8	follow	follow	VERB
ejpam-4386	56	9	five	five	NUM
ejpam-4386	56	10	steps	step	NOUN
ejpam-4386	56	11	expressed	express	VERB
ejpam-4386	56	12	in	in	ADP
ejpam-4386	56	13	the	the	DET
ejpam-4386	56	14	following	follow	VERB
ejpam-4386	56	15	lemmas	lemma	NOUN
ejpam-4386	56	16	and	and	CCONJ
ejpam-4386	56	17	theorem	theorem	VERB
ejpam-4386	56	18	1	1	NUM
ejpam-4386	56	19	.	.	PUNCT
ejpam-4386	56	20	as	as	ADP
ejpam-4386	56	21	a	a	DET
ejpam-4386	56	22	notation	notation	NOUN
ejpam-4386	56	23	we	we	PRON
ejpam-4386	56	24	will	will	AUX
ejpam-4386	56	25	call	call	VERB
ejpam-4386	56	26	a	a	DET
ejpam-4386	56	27	subset	subset	NOUN
ejpam-4386	56	28	which	which	PRON
ejpam-4386	56	29	is	be	AUX
ejpam-4386	56	30	closed	close	VERB
ejpam-4386	56	31	and	and	CCONJ
ejpam-4386	56	32	open	open	ADJ
ejpam-4386	56	33	by	by	ADP
ejpam-4386	56	34	clopen	clopen	ADJ
ejpam-4386	56	35	.	.	PUNCT
ejpam-4386	57	1	lemma	lemma	PROPN
ejpam-4386	57	2	1	1	NUM
ejpam-4386	57	3	.	.	PUNCT
ejpam-4386	58	1	for	for	ADP
ejpam-4386	58	2	each	each	DET
ejpam-4386	58	3	x′	x′	PROPN
ejpam-4386	58	4	∈	∈	PROPN
ejpam-4386	58	5	x	x	SYM
ejpam-4386	58	6	′	′	NUM
ejpam-4386	58	7	,	,	PUNCT
ejpam-4386	58	8	{	{	PUNCT
ejpam-4386	58	9	x′	x′	X
ejpam-4386	58	10	}	}	PUNCT
ejpam-4386	58	11	is	be	AUX
ejpam-4386	58	12	clopen	clopen	ADJ
ejpam-4386	58	13	in	in	ADP
ejpam-4386	58	14	a(x	a(x	NOUN
ejpam-4386	58	15	)	)	PUNCT
ejpam-4386	58	16	.	.	PUNCT
ejpam-4386	59	1	proof	proof	NOUN
ejpam-4386	59	2	.	.	PUNCT
ejpam-4386	60	1	let	let	VERB
ejpam-4386	60	2	x′	x′	PROPN
ejpam-4386	60	3	∈	∈	PROPN
ejpam-4386	60	4	x	x	INTJ
ejpam-4386	60	5	′	′	NUM
ejpam-4386	60	6	be	be	AUX
ejpam-4386	60	7	arbitrary	arbitrary	ADJ
ejpam-4386	60	8	.	.	PUNCT
ejpam-4386	61	1	we	we	PRON
ejpam-4386	61	2	need	need	VERB
ejpam-4386	61	3	only	only	ADV
ejpam-4386	61	4	to	to	PART
ejpam-4386	61	5	show	show	VERB
ejpam-4386	61	6	that	that	SCONJ
ejpam-4386	61	7	{	{	PUNCT
ejpam-4386	61	8	x′	x′	NUM
ejpam-4386	61	9	}	}	PUNCT
ejpam-4386	61	10	is	be	AUX
ejpam-4386	61	11	closed	closed	ADJ
ejpam-4386	61	12	.	.	PUNCT
ejpam-4386	62	1	so	so	ADV
ejpam-4386	62	2	,	,	PUNCT
ejpam-4386	62	3	let	let	VERB
ejpam-4386	62	4	a	a	DET
ejpam-4386	62	5	∈	∈	PROPN
ejpam-4386	62	6	a(x	a(x	PROPN
ejpam-4386	62	7	)	)	PUNCT
ejpam-4386	62	8	\	\	NOUN
ejpam-4386	62	9	{	{	PUNCT
ejpam-4386	62	10	x′	x′	PROPN
ejpam-4386	62	11	}	}	PUNCT
ejpam-4386	62	12	be	be	AUX
ejpam-4386	62	13	arbitrary	arbitrary	ADJ
ejpam-4386	62	14	.	.	PUNCT
ejpam-4386	63	1	if	if	SCONJ
ejpam-4386	63	2	a	a	DET
ejpam-4386	63	3	∈	∈	PROPN
ejpam-4386	63	4	x	x	SYM
ejpam-4386	63	5	′	′	NOUN
ejpam-4386	63	6	,	,	PUNCT
ejpam-4386	63	7	then	then	ADV
ejpam-4386	63	8	{	{	PUNCT
ejpam-4386	63	9	a	a	PRON
ejpam-4386	63	10	}	}	PUNCT
ejpam-4386	63	11	is	be	AUX
ejpam-4386	63	12	an	an	DET
ejpam-4386	63	13	open	open	ADJ
ejpam-4386	63	14	neighborhood	neighborhood	NOUN
ejpam-4386	63	15	of	of	ADP
ejpam-4386	63	16	a	a	PRON
ejpam-4386	63	17	in	in	ADP
ejpam-4386	63	18	a(x	a(x	NOUN
ejpam-4386	63	19	)	)	PUNCT
ejpam-4386	63	20	with	with	ADP
ejpam-4386	63	21	{	{	PUNCT
ejpam-4386	63	22	a	a	PRON
ejpam-4386	63	23	}	}	PUNCT
ejpam-4386	63	24	⊂	⊂	PROPN
ejpam-4386	63	25	a(x	a(x	PROPN
ejpam-4386	63	26	)	)	PUNCT
ejpam-4386	63	27	\	\	NOUN
ejpam-4386	63	28	{	{	PUNCT
ejpam-4386	63	29	x′	x′	NUM
ejpam-4386	63	30	}	}	PUNCT
ejpam-4386	63	31	.	.	PUNCT
ejpam-4386	64	1	if	if	SCONJ
ejpam-4386	64	2	a	a	DET
ejpam-4386	64	3	∈	∈	PROPN
ejpam-4386	64	4	x	x	NOUN
ejpam-4386	64	5	,	,	PUNCT
ejpam-4386	64	6	pick	pick	VERB
ejpam-4386	64	7	any	any	DET
ejpam-4386	64	8	open	open	ADJ
ejpam-4386	64	9	neighborhood	neighborhood	NOUN
ejpam-4386	64	10	u	u	NOUN
ejpam-4386	64	11	⊆	⊆	NUM
ejpam-4386	64	12	x	x	X
ejpam-4386	64	13	of	of	ADP
ejpam-4386	64	14	a.	a.	NOUN
ejpam-4386	64	15	then	then	ADV
ejpam-4386	64	16	u	u	NOUN
ejpam-4386	64	17	∪	∪	ADJ
ejpam-4386	64	18	(	(	PUNCT
ejpam-4386	64	19	u	u	NOUN
ejpam-4386	64	20	′	′	NOUN
ejpam-4386	64	21	\	\	NOUN
ejpam-4386	64	22	{	{	PUNCT
ejpam-4386	64	23	x′	x′	NUM
ejpam-4386	64	24	}	}	PUNCT
ejpam-4386	64	25	)	)	PUNCT
ejpam-4386	64	26	is	be	AUX
ejpam-4386	64	27	an	an	DET
ejpam-4386	64	28	open	open	ADJ
ejpam-4386	64	29	neighborhood	neighborhood	NOUN
ejpam-4386	64	30	of	of	ADP
ejpam-4386	64	31	a	a	PRON
ejpam-4386	64	32	in	in	ADP
ejpam-4386	64	33	a(x	a(x	NOUN
ejpam-4386	64	34	)	)	PUNCT
ejpam-4386	64	35	with	with	ADP
ejpam-4386	64	36	u	u	NOUN
ejpam-4386	64	37	∪	∪	X
ejpam-4386	64	38	(	(	PUNCT
ejpam-4386	64	39	u	u	NOUN
ejpam-4386	64	40	′	′	NOUN
ejpam-4386	64	41	\	\	NOUN
ejpam-4386	64	42	{	{	PUNCT
ejpam-4386	64	43	x′	x′	NUM
ejpam-4386	64	44	}	}	PUNCT
ejpam-4386	64	45	)	)	PUNCT
ejpam-4386	64	46	⊆	⊆	X
ejpam-4386	64	47	a(x	a(x	NOUN
ejpam-4386	64	48	)	)	PUNCT
ejpam-4386	64	49	\	\	NOUN
ejpam-4386	64	50	{	{	PUNCT
ejpam-4386	64	51	x′	x′	NUM
ejpam-4386	64	52	}	}	PUNCT
ejpam-4386	64	53	.	.	PUNCT
ejpam-4386	65	1	thus	thus	ADV
ejpam-4386	65	2	a(x	a(x	NOUN
ejpam-4386	65	3	)	)	PUNCT
ejpam-4386	65	4	\	\	NOUN
ejpam-4386	65	5	{	{	PUNCT
ejpam-4386	65	6	x′	x′	PROPN
ejpam-4386	65	7	}	}	PUNCT
ejpam-4386	65	8	is	be	AUX
ejpam-4386	65	9	open	open	ADJ
ejpam-4386	65	10	in	in	ADP
ejpam-4386	65	11	a(x	a(x	NOUN
ejpam-4386	65	12	)	)	PUNCT
ejpam-4386	65	13	.	.	PUNCT
ejpam-4386	66	1	therefore	therefore	ADV
ejpam-4386	66	2	,	,	PUNCT
ejpam-4386	66	3	{	{	PUNCT
ejpam-4386	66	4	x′	x′	PROPN
ejpam-4386	66	5	}	}	PUNCT
ejpam-4386	66	6	is	be	AUX
ejpam-4386	66	7	closed	closed	ADJ
ejpam-4386	66	8	.	.	PUNCT
ejpam-4386	67	1	lemma	lemma	PROPN
ejpam-4386	67	2	2	2	X
ejpam-4386	67	3	.	.	PUNCT
ejpam-4386	68	1	let	let	AUX
ejpam-4386	68	2	(	(	PUNCT
ejpam-4386	68	3	x	x	X
ejpam-4386	68	4	,	,	PUNCT
ejpam-4386	68	5	τ	τ	PROPN
ejpam-4386	68	6	)	)	PUNCT
ejpam-4386	68	7	be	be	AUX
ejpam-4386	68	8	any	any	DET
ejpam-4386	68	9	topological	topological	ADJ
ejpam-4386	68	10	space	space	NOUN
ejpam-4386	68	11	.	.	PUNCT
ejpam-4386	69	1	if	if	SCONJ
ejpam-4386	69	2	c	c	PROPN
ejpam-4386	69	3	is	be	AUX
ejpam-4386	69	4	clopen	clopen	ADJ
ejpam-4386	69	5	in	in	ADP
ejpam-4386	69	6	x	x	PUNCT
ejpam-4386	69	7	and	and	CCONJ
ejpam-4386	69	8	b	b	PROPN
ejpam-4386	69	9	is	be	AUX
ejpam-4386	69	10	an	an	DET
ejpam-4386	69	11	open	open	ADJ
ejpam-4386	69	12	domain	domain	NOUN
ejpam-4386	69	13	in	in	ADP
ejpam-4386	69	14	x	x	SYM
ejpam-4386	69	15	,	,	PUNCT
ejpam-4386	69	16	then	then	ADV
ejpam-4386	69	17	b	b	X
ejpam-4386	69	18	\	\	PROPN
ejpam-4386	69	19	c	c	PROPN
ejpam-4386	69	20	is	be	AUX
ejpam-4386	69	21	an	an	DET
ejpam-4386	69	22	open	open	ADJ
ejpam-4386	69	23	domain	domain	NOUN
ejpam-4386	69	24	in	in	ADP
ejpam-4386	69	25	x.	x.	NOUN
ejpam-4386	69	26	proof	proof	NOUN
ejpam-4386	69	27	.	.	PUNCT
ejpam-4386	70	1	we	we	PRON
ejpam-4386	70	2	want	want	VERB
ejpam-4386	70	3	to	to	PART
ejpam-4386	70	4	show	show	VERB
ejpam-4386	70	5	that	that	SCONJ
ejpam-4386	70	6	,	,	PUNCT
ejpam-4386	70	7	int(b	int(b	PROPN
ejpam-4386	70	8	\	\	NOUN
ejpam-4386	70	9	c	c	NOUN
ejpam-4386	70	10	)	)	PUNCT
ejpam-4386	70	11	=	=	SYM
ejpam-4386	70	12	b	b	X
ejpam-4386	70	13	\c	\c	NOUN
ejpam-4386	70	14	.	.	PUNCT
ejpam-4386	71	1	we	we	PRON
ejpam-4386	71	2	always	always	ADV
ejpam-4386	71	3	have	have	VERB
ejpam-4386	71	4	b	b	NOUN
ejpam-4386	71	5	\c	\c	NOUN
ejpam-4386	71	6	⊆	⊆	NUM
ejpam-4386	71	7	b	b	NOUN
ejpam-4386	71	8	\	\	PROPN
ejpam-4386	71	9	c	c	PROPN
ejpam-4386	71	10	,	,	PUNCT
ejpam-4386	71	11	by	by	ADP
ejpam-4386	71	12	taking	take	VERB
ejpam-4386	71	13	the	the	DET
ejpam-4386	71	14	interior	interior	NOUN
ejpam-4386	71	15	in	in	ADP
ejpam-4386	71	16	both	both	DET
ejpam-4386	71	17	sides	side	NOUN
ejpam-4386	71	18	we	we	PRON
ejpam-4386	71	19	get	get	VERB
ejpam-4386	71	20	int(b	int(b	PRON
ejpam-4386	71	21	\	\	NOUN
ejpam-4386	71	22	c	c	NOUN
ejpam-4386	71	23	)	)	PUNCT
ejpam-4386	71	24	⊆	⊆	NUM
ejpam-4386	71	25	int(b	int(b	NOUN
ejpam-4386	71	26	\	\	NOUN
ejpam-4386	71	27	c	c	NOUN
ejpam-4386	71	28	)	)	PUNCT
ejpam-4386	71	29	.	.	PUNCT
ejpam-4386	72	1	but	but	CCONJ
ejpam-4386	72	2	int(b	int(b	NOUN
ejpam-4386	72	3	\	\	NOUN
ejpam-4386	72	4	c	c	X
ejpam-4386	72	5	)	)	PUNCT
ejpam-4386	72	6	=	=	SYM
ejpam-4386	72	7	int(b	int(b	PROPN
ejpam-4386	72	8	∩	∩	NOUN
ejpam-4386	72	9	(	(	PUNCT
ejpam-4386	72	10	x	x	SYM
ejpam-4386	72	11	\c	\c	NOUN
ejpam-4386	72	12	)	)	PUNCT
ejpam-4386	72	13	)	)	PUNCT
ejpam-4386	73	1	=	=	PRON
ejpam-4386	73	2	int(b)∩	int(b)∩	VERB
ejpam-4386	73	3	int(x	int(x	NOUN
ejpam-4386	73	4	\c	\c	NOUN
ejpam-4386	73	5	)	)	PUNCT
ejpam-4386	74	1	=	=	SYM
ejpam-4386	75	1	b∩	b∩	NOUN
ejpam-4386	75	2	(	(	PUNCT
ejpam-4386	75	3	x	x	SYM
ejpam-4386	75	4	\c	\c	NOUN
ejpam-4386	75	5	)	)	PUNCT
ejpam-4386	76	1	=	=	SYM
ejpam-4386	76	2	b	b	X
ejpam-4386	76	3	\c	\c	NOUN
ejpam-4386	76	4	,	,	PUNCT
ejpam-4386	76	5	because	because	SCONJ
ejpam-4386	76	6	c	c	PROPN
ejpam-4386	76	7	is	be	AUX
ejpam-4386	76	8	clopen	clopen	ADJ
ejpam-4386	76	9	and	and	CCONJ
ejpam-4386	76	10	b	b	NOUN
ejpam-4386	76	11	is	be	AUX
ejpam-4386	76	12	an	an	DET
ejpam-4386	76	13	open	open	ADJ
ejpam-4386	76	14	domain	domain	NOUN
ejpam-4386	76	15	so	so	SCONJ
ejpam-4386	76	16	b	b	PROPN
ejpam-4386	76	17	is	be	AUX
ejpam-4386	76	18	an	an	DET
ejpam-4386	76	19	open	open	ADJ
ejpam-4386	76	20	set	set	NOUN
ejpam-4386	76	21	,	,	PUNCT
ejpam-4386	76	22	thus	thus	ADV
ejpam-4386	76	23	b	b	X
ejpam-4386	76	24	\c	\c	NOUN
ejpam-4386	76	25	is	be	AUX
ejpam-4386	76	26	an	an	DET
ejpam-4386	76	27	open	open	ADJ
ejpam-4386	76	28	set	set	NOUN
ejpam-4386	76	29	.	.	PUNCT
ejpam-4386	77	1	thus	thus	ADV
ejpam-4386	77	2	,	,	PUNCT
ejpam-4386	77	3	b	b	X
ejpam-4386	77	4	\c	\c	NOUN
ejpam-4386	77	5	⊆	⊆	NUM
ejpam-4386	77	6	int(b	int(b	VERB
ejpam-4386	77	7	\	\	NOUN
ejpam-4386	77	8	c	c	NOUN
ejpam-4386	77	9	)	)	PUNCT
ejpam-4386	77	10	.	.	PUNCT
ejpam-4386	78	1	now	now	ADV
ejpam-4386	78	2	,	,	PUNCT
ejpam-4386	78	3	we	we	PRON
ejpam-4386	78	4	always	always	ADV
ejpam-4386	78	5	have	have	VERB
ejpam-4386	78	6	int(b	int(b	NOUN
ejpam-4386	78	7	\	\	NOUN
ejpam-4386	78	8	c	c	NOUN
ejpam-4386	78	9	)	)	PUNCT
ejpam-4386	79	1	⊆	⊆	NUM
ejpam-4386	79	2	b	b	PROPN
ejpam-4386	79	3	\	\	PROPN
ejpam-4386	79	4	c.	c.	NOUN
ejpam-4386	79	5	thus	thus	ADV
ejpam-4386	79	6	int(b	int(b	PROPN
ejpam-4386	79	7	\	\	NOUN
ejpam-4386	79	8	c	c	NOUN
ejpam-4386	79	9	)	)	PUNCT
ejpam-4386	79	10	⊆	⊆	NUM
ejpam-4386	79	11	b	b	NOUN
ejpam-4386	79	12	\	\	PROPN
ejpam-4386	79	13	c	c	PROPN
ejpam-4386	79	14	=	=	SYM
ejpam-4386	79	15	b	b	PROPN
ejpam-4386	79	16	∩	∩	NOUN
ejpam-4386	79	17	(	(	PUNCT
ejpam-4386	79	18	x	x	SYM
ejpam-4386	79	19	\	\	PROPN
ejpam-4386	79	20	c	c	X
ejpam-4386	79	21	)	)	PUNCT
ejpam-4386	79	22	⊆	⊆	NUM
ejpam-4386	79	23	b∩	b∩	NOUN
ejpam-4386	79	24	(	(	PUNCT
ejpam-4386	79	25	x	x	SYM
ejpam-4386	79	26	\	\	PROPN
ejpam-4386	79	27	c	c	X
ejpam-4386	79	28	)	)	PUNCT
ejpam-4386	79	29	=	=	SYM
ejpam-4386	79	30	b	b	NOUN
ejpam-4386	79	31	∩	∩	NOUN
ejpam-4386	79	32	(	(	PUNCT
ejpam-4386	79	33	x	x	SYM
ejpam-4386	79	34	\	\	PROPN
ejpam-4386	79	35	c	c	X
ejpam-4386	79	36	)	)	PUNCT
ejpam-4386	79	37	,	,	PUNCT
ejpam-4386	79	38	because	because	SCONJ
ejpam-4386	79	39	c	c	PROPN
ejpam-4386	79	40	is	be	AUX
ejpam-4386	79	41	clopen	clopen	ADJ
ejpam-4386	79	42	in	in	ADP
ejpam-4386	79	43	x.	x.	NOUN
ejpam-4386	79	44	int(b	int(b	PROPN
ejpam-4386	79	45	\	\	PROPN
ejpam-4386	79	46	c)⊆	c)⊆	PROPN
ejpam-4386	79	47	b	b	X
ejpam-4386	79	48	∩	∩	NOUN
ejpam-4386	79	49	(	(	PUNCT
ejpam-4386	79	50	x	x	SYM
ejpam-4386	79	51	\	\	PROPN
ejpam-4386	79	52	c	c	X
ejpam-4386	79	53	)	)	PUNCT
ejpam-4386	79	54	then	then	ADV
ejpam-4386	79	55	,	,	PUNCT
ejpam-4386	79	56	int(int(b	int(int(b	VERB
ejpam-4386	79	57	\	\	PROPN
ejpam-4386	79	58	c	c	X
ejpam-4386	79	59	)	)	PUNCT
ejpam-4386	79	60	)	)	PUNCT
ejpam-4386	80	1	⊆	⊆	NUM
ejpam-4386	80	2	int(b∩	int(b∩	PUNCT
ejpam-4386	80	3	(	(	PUNCT
ejpam-4386	80	4	x	x	NOUN
ejpam-4386	80	5	\c	\c	NOUN
ejpam-4386	80	6	)	)	PUNCT
ejpam-4386	80	7	)	)	PUNCT
ejpam-4386	80	8	thus	thus	ADV
ejpam-4386	80	9	,	,	PUNCT
ejpam-4386	80	10	int(b	int(b	PROPN
ejpam-4386	80	11	\	\	ADJ
ejpam-4386	80	12	c	c	NOUN
ejpam-4386	80	13	)	)	PUNCT
ejpam-4386	80	14	⊆	⊆	NUM
ejpam-4386	80	15	int(b)∩	int(b)∩	ADP
ejpam-4386	80	16	int(x	int(x	NOUN
ejpam-4386	80	17	\c	\c	NOUN
ejpam-4386	80	18	)	)	PUNCT
ejpam-4386	80	19	hence	hence	ADV
ejpam-4386	80	20	,	,	PUNCT
ejpam-4386	80	21	int(b	int(b	PROPN
ejpam-4386	80	22	\	\	NOUN
ejpam-4386	80	23	c	c	NOUN
ejpam-4386	80	24	)	)	PUNCT
ejpam-4386	80	25	⊆	⊆	NUM
ejpam-4386	80	26	b∩	b∩	NOUN
ejpam-4386	80	27	(	(	PUNCT
ejpam-4386	80	28	x	x	SYM
ejpam-4386	80	29	\c	\c	NOUN
ejpam-4386	80	30	)	)	PUNCT
ejpam-4386	80	31	=	=	SYM
ejpam-4386	80	32	b	b	X
ejpam-4386	80	33	\	\	PROPN
ejpam-4386	80	34	c	c	NOUN
ejpam-4386	80	35	,	,	PUNCT
ejpam-4386	80	36	because	because	SCONJ
ejpam-4386	80	37	b	b	NOUN
ejpam-4386	80	38	is	be	AUX
ejpam-4386	80	39	an	an	DET
ejpam-4386	80	40	open	open	ADJ
ejpam-4386	80	41	domain	domain	NOUN
ejpam-4386	80	42	and	and	CCONJ
ejpam-4386	80	43	c	c	NOUN
ejpam-4386	80	44	is	be	AUX
ejpam-4386	80	45	clopen	clopen	ADJ
ejpam-4386	80	46	in	in	ADP
ejpam-4386	80	47	x.	x.	NOUN
ejpam-4386	80	48	thus	thus	ADV
ejpam-4386	80	49	,	,	PUNCT
ejpam-4386	80	50	int(b	int(b	PROPN
ejpam-4386	80	51	\	\	ADJ
ejpam-4386	80	52	c	c	NOUN
ejpam-4386	80	53	)	)	PUNCT
ejpam-4386	80	54	⊆	⊆	NUM
ejpam-4386	80	55	b	b	PROPN
ejpam-4386	80	56	\	\	PROPN
ejpam-4386	80	57	c.	c.	PROPN
ejpam-4386	80	58	therefore	therefore	ADV
ejpam-4386	80	59	,	,	PUNCT
ejpam-4386	80	60	int(b	int(b	PROPN
ejpam-4386	80	61	\	\	ADJ
ejpam-4386	80	62	c	c	NOUN
ejpam-4386	80	63	)	)	PUNCT
ejpam-4386	80	64	=	=	SYM
ejpam-4386	80	65	b	b	PROPN
ejpam-4386	80	66	\	\	PROPN
ejpam-4386	80	67	c.	c.	PROPN
ejpam-4386	80	68	thus	thus	ADV
ejpam-4386	80	69	,	,	PUNCT
ejpam-4386	80	70	b	b	X
ejpam-4386	80	71	\	\	PROPN
ejpam-4386	80	72	c	c	PROPN
ejpam-4386	80	73	is	be	AUX
ejpam-4386	80	74	an	an	DET
ejpam-4386	80	75	open	open	ADJ
ejpam-4386	80	76	domain	domain	NOUN
ejpam-4386	80	77	.	.	PUNCT
ejpam-4386	81	1	notice	notice	VERB
ejpam-4386	81	2	that	that	SCONJ
ejpam-4386	81	3	,	,	PUNCT
ejpam-4386	81	4	if	if	SCONJ
ejpam-4386	81	5	u	u	NOUN
ejpam-4386	81	6	is	be	AUX
ejpam-4386	81	7	any	any	DET
ejpam-4386	81	8	non	non	ADJ
ejpam-4386	81	9	-	-	ADJ
ejpam-4386	81	10	empty	empty	ADJ
ejpam-4386	81	11	open	open	ADJ
ejpam-4386	81	12	set	set	NOUN
ejpam-4386	81	13	in	in	ADP
ejpam-4386	81	14	x	x	NOUN
ejpam-4386	81	15	,	,	PUNCT
ejpam-4386	81	16	then	then	ADV
ejpam-4386	81	17	u	u	NOUN
ejpam-4386	81	18	∪	∪	X
ejpam-4386	81	19	(	(	PUNCT
ejpam-4386	81	20	u	u	NOUN
ejpam-4386	81	21	′	′	ADJ
ejpam-4386	81	22	\	\	NOUN
ejpam-4386	81	23	∅	∅	NOUN
ejpam-4386	81	24	)	)	PUNCT
ejpam-4386	81	25	=	=	SYM
ejpam-4386	81	26	u	u	NOUN
ejpam-4386	81	27	∪u	∪u	NUM
ejpam-4386	81	28	′	′	NUM
ejpam-4386	81	29	is	be	AUX
ejpam-4386	81	30	a	a	DET
ejpam-4386	81	31	basic	basic	ADJ
ejpam-4386	81	32	open	open	ADJ
ejpam-4386	81	33	neighborhood	neighborhood	NOUN
ejpam-4386	81	34	in	in	ADP
ejpam-4386	81	35	a(x	a(x	NOUN
ejpam-4386	81	36	)	)	PUNCT
ejpam-4386	81	37	of	of	ADP
ejpam-4386	81	38	any	any	DET
ejpam-4386	81	39	x	x	SYM
ejpam-4386	81	40	∈	∈	PROPN
ejpam-4386	81	41	u	u	NOUN
ejpam-4386	81	42	.	.	PUNCT
ejpam-4386	82	1	so	so	ADV
ejpam-4386	82	2	,	,	PUNCT
ejpam-4386	82	3	we	we	PRON
ejpam-4386	82	4	establish	establish	VERB
ejpam-4386	82	5	that	that	SCONJ
ejpam-4386	82	6	the	the	DET
ejpam-4386	82	7	following	follow	VERB
ejpam-4386	82	8	lemma	lemma	PROPN
ejpam-4386	82	9	.	.	PUNCT
ejpam-4386	82	10	lemma	lemma	PROPN
ejpam-4386	82	11	3	3	X
ejpam-4386	82	12	.	.	PUNCT
ejpam-4386	83	1	if	if	SCONJ
ejpam-4386	83	2	u	u	NOUN
ejpam-4386	83	3	is	be	AUX
ejpam-4386	83	4	an	an	DET
ejpam-4386	83	5	open	open	ADJ
ejpam-4386	83	6	set	set	NOUN
ejpam-4386	83	7	in	in	ADP
ejpam-4386	83	8	x	x	NOUN
ejpam-4386	83	9	,	,	PUNCT
ejpam-4386	83	10	then	then	ADV
ejpam-4386	83	11	u	u	NOUN
ejpam-4386	83	12	∪u	∪u	NUM
ejpam-4386	83	13	′	′	NUM
ejpam-4386	83	14	is	be	AUX
ejpam-4386	83	15	an	an	DET
ejpam-4386	83	16	open	open	ADJ
ejpam-4386	83	17	set	set	NOUN
ejpam-4386	83	18	in	in	ADP
ejpam-4386	83	19	a(x	a(x	NOUN
ejpam-4386	83	20	)	)	PUNCT
ejpam-4386	83	21	theorem	theorem	NOUN
ejpam-4386	83	22	1	1	NUM
ejpam-4386	83	23	.	.	PUNCT
ejpam-4386	84	1	if	if	SCONJ
ejpam-4386	84	2	u	u	NOUN
ejpam-4386	84	3	is	be	AUX
ejpam-4386	84	4	an	an	DET
ejpam-4386	84	5	open	open	ADJ
ejpam-4386	84	6	domain	domain	NOUN
ejpam-4386	84	7	in	in	ADP
ejpam-4386	84	8	x	x	NOUN
ejpam-4386	84	9	,	,	PUNCT
ejpam-4386	84	10	then	then	ADV
ejpam-4386	84	11	u	u	NOUN
ejpam-4386	84	12	∪u	∪u	NUM
ejpam-4386	84	13	′	′	NUM
ejpam-4386	84	14	is	be	AUX
ejpam-4386	84	15	an	an	DET
ejpam-4386	84	16	open	open	ADJ
ejpam-4386	84	17	domain	domain	NOUN
ejpam-4386	84	18	in	in	ADP
ejpam-4386	84	19	a(x	a(x	NOUN
ejpam-4386	84	20	)	)	PUNCT
ejpam-4386	84	21	.	.	PUNCT
ejpam-4386	85	1	proof	proof	NOUN
ejpam-4386	85	2	.	.	PUNCT
ejpam-4386	86	1	let	let	VERB
ejpam-4386	86	2	u	u	PRON
ejpam-4386	86	3	be	be	AUX
ejpam-4386	86	4	an	an	DET
ejpam-4386	86	5	open	open	ADJ
ejpam-4386	86	6	domain	domain	NOUN
ejpam-4386	86	7	in	in	ADP
ejpam-4386	86	8	x.	x.	NOUN
ejpam-4386	86	9	we	we	PRON
ejpam-4386	86	10	show	show	VERB
ejpam-4386	86	11	that	that	SCONJ
ejpam-4386	86	12	u	u	PROPN
ejpam-4386	86	13	∪	∪	ADP
ejpam-4386	86	14	u	u	NOUN
ejpam-4386	86	15	′	′	NOUN
ejpam-4386	86	16	=	=	PUNCT
ejpam-4386	87	1	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	87	2	∪	∪	ADJ
ejpam-4386	87	3	u	u	NOUN
ejpam-4386	87	4	′	′	NOUN
ejpam-4386	87	5	a(x	a(x	NOUN
ejpam-4386	87	6	)	)	PUNCT
ejpam-4386	87	7	)	)	PUNCT
ejpam-4386	88	1	first	first	ADV
ejpam-4386	88	2	,	,	PUNCT
ejpam-4386	88	3	we	we	PRON
ejpam-4386	88	4	show	show	VERB
ejpam-4386	88	5	that	that	SCONJ
ejpam-4386	88	6	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	88	7	∪	∪	ADJ
ejpam-4386	88	8	u	u	NOUN
ejpam-4386	88	9	′	′	NOUN
ejpam-4386	88	10	a(x	a(x	NOUN
ejpam-4386	88	11	)	)	PUNCT
ejpam-4386	88	12	)	)	PUNCT
ejpam-4386	89	1	⊆	⊆	NUM
ejpam-4386	89	2	u	u	NOUN
ejpam-4386	89	3	∪	∪	VERB
ejpam-4386	89	4	u	u	NOUN
ejpam-4386	89	5	′.	′.	NOUN
ejpam-4386	89	6	let	let	VERB
ejpam-4386	89	7	x	x	X
ejpam-4386	89	8	∈	∈	PROPN
ejpam-4386	89	9	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	89	10	∪	∪	ADJ
ejpam-4386	89	11	u	u	NOUN
ejpam-4386	89	12	′	′	NOUN
ejpam-4386	89	13	a(x	a(x	NOUN
ejpam-4386	89	14	)	)	PUNCT
ejpam-4386	89	15	)	)	PUNCT
ejpam-4386	89	16	be	be	AUX
ejpam-4386	89	17	arbitrary	arbitrary	ADJ
ejpam-4386	89	18	,	,	PUNCT
ejpam-4386	89	19	then	then	ADV
ejpam-4386	89	20	x	x	PART
ejpam-4386	89	21	∈	∈	PROPN
ejpam-4386	89	22	u	u	NOUN
ejpam-4386	89	23	∪	∪	VERB
ejpam-4386	89	24	u	u	NOUN
ejpam-4386	89	25	′	′	NOUN
ejpam-4386	89	26	a(x	a(x	NOUN
ejpam-4386	89	27	)	)	PUNCT
ejpam-4386	89	28	.	.	PUNCT
ejpam-4386	90	1	there	there	PRON
ejpam-4386	90	2	are	be	VERB
ejpam-4386	90	3	only	only	ADV
ejpam-4386	90	4	two	two	NUM
ejpam-4386	90	5	cases	case	NOUN
ejpam-4386	90	6	.	.	PUNCT
ejpam-4386	91	1	case	case	NOUN
ejpam-4386	91	2	1	1	NUM
ejpam-4386	91	3	:	:	PUNCT
ejpam-4386	91	4	x	x	SYM
ejpam-4386	91	5	∈	∈	NOUN
ejpam-4386	91	6	x	x	SYM
ejpam-4386	91	7	′.	′.	NOUN
ejpam-4386	91	8	so	so	ADV
ejpam-4386	91	9	,	,	PUNCT
ejpam-4386	91	10	{	{	PUNCT
ejpam-4386	91	11	x	x	X
ejpam-4386	91	12	}	}	PUNCT
ejpam-4386	91	13	is	be	AUX
ejpam-4386	91	14	an	an	DET
ejpam-4386	91	15	open	open	ADJ
ejpam-4386	91	16	set	set	NOUN
ejpam-4386	91	17	in	in	ADP
ejpam-4386	91	18	a(x	a(x	NOUN
ejpam-4386	91	19	)	)	PUNCT
ejpam-4386	91	20	with	with	ADP
ejpam-4386	91	21	x	x	PROPN
ejpam-4386	91	22	∈	∈	PROPN
ejpam-4386	91	23	{	{	PUNCT
ejpam-4386	91	24	x	x	NOUN
ejpam-4386	91	25	}	}	PUNCT
ejpam-4386	91	26	and	and	CCONJ
ejpam-4386	91	27	{	{	PUNCT
ejpam-4386	91	28	x	x	NOUN
ejpam-4386	91	29	}	}	PUNCT
ejpam-4386	91	30	∩	∩	NOUN
ejpam-4386	91	31	(	(	PUNCT
ejpam-4386	91	32	u	u	PROPN
ejpam-4386	91	33	∪	∪	VERB
ejpam-4386	91	34	u	u	NOUN
ejpam-4386	91	35	′	′	NOUN
ejpam-4386	91	36	)	)	PUNCT
ejpam-4386	91	37	̸=	̸=	NOUN
ejpam-4386	91	38	∅	∅	NOUN
ejpam-4386	91	39	,	,	PUNCT
ejpam-4386	91	40	then	then	ADV
ejpam-4386	91	41	x	x	PART
ejpam-4386	91	42	∈	∈	PROPN
ejpam-4386	91	43	u	u	NOUN
ejpam-4386	91	44	′	′	NUM
ejpam-4386	91	45	⊆	⊆	NUM
ejpam-4386	91	46	u	u	NOUN
ejpam-4386	91	47	∪	∪	VERB
ejpam-4386	91	48	u	u	NOUN
ejpam-4386	91	49	′.	′.	NOUN
ejpam-4386	91	50	thus	thus	ADV
ejpam-4386	91	51	,	,	PUNCT
ejpam-4386	91	52	x	x	PUNCT
ejpam-4386	91	53	∈	∈	PROPN
ejpam-4386	91	54	u	u	NOUN
ejpam-4386	91	55	∪	∪	VERB
ejpam-4386	91	56	u	u	NOUN
ejpam-4386	91	57	′.	′.	NOUN
ejpam-4386	91	58	case	case	NOUN
ejpam-4386	91	59	2	2	NUM
ejpam-4386	91	60	:	:	PUNCT
ejpam-4386	91	61	x	x	SYM
ejpam-4386	91	62	∈	∈	NOUN
ejpam-4386	91	63	x.	x.	NOUN
ejpam-4386	91	64	since	since	SCONJ
ejpam-4386	91	65	x	x	PROPN
ejpam-4386	91	66	∈	∈	PROPN
ejpam-4386	91	67	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	91	68	∪	∪	ADJ
ejpam-4386	91	69	u	u	NOUN
ejpam-4386	91	70	′	′	NOUN
ejpam-4386	91	71	a(x	a(x	NOUN
ejpam-4386	91	72	)	)	PUNCT
ejpam-4386	91	73	)	)	PUNCT
ejpam-4386	91	74	,	,	PUNCT
ejpam-4386	91	75	then	then	ADV
ejpam-4386	91	76	there	there	PRON
ejpam-4386	91	77	exist	exist	VERB
ejpam-4386	91	78	an	an	DET
ejpam-4386	91	79	open	open	ADJ
ejpam-4386	91	80	set	set	VERB
ejpam-4386	91	81	v	v	NOUN
ejpam-4386	91	82	in	in	ADP
ejpam-4386	91	83	x	x	PUNCT
ejpam-4386	91	84	with	with	ADP
ejpam-4386	91	85	x	x	PROPN
ejpam-4386	91	86	∈	∈	NOUN
ejpam-4386	91	87	v	v	ADP
ejpam-4386	91	88	such	such	ADJ
ejpam-4386	91	89	that	that	SCONJ
ejpam-4386	91	90	x	x	SYM
ejpam-4386	91	91	∈	∈	NOUN
ejpam-4386	91	92	v	v	NOUN
ejpam-4386	91	93	∪	∪	X
ejpam-4386	91	94	(	(	PUNCT
ejpam-4386	91	95	v	v	NOUN
ejpam-4386	91	96	′	′	NUM
ejpam-4386	91	97	\	\	NOUN
ejpam-4386	92	1	e	e	X
ejpam-4386	92	2	)	)	PUNCT
ejpam-4386	92	3	⊆	⊆	NUM
ejpam-4386	92	4	u	u	NOUN
ejpam-4386	92	5	∪	∪	VERB
ejpam-4386	92	6	u	u	NOUN
ejpam-4386	92	7	′	′	NOUN
ejpam-4386	92	8	a(x	a(x	NOUN
ejpam-4386	92	9	)	)	PUNCT
ejpam-4386	92	10	=	=	SYM
ejpam-4386	92	11	u	u	NOUN
ejpam-4386	92	12	a(x	a(x	PROPN
ejpam-4386	92	13	)	)	PUNCT
ejpam-4386	92	14	∪	∪	ADP
ejpam-4386	92	15	u	u	NOUN
ejpam-4386	92	16	′	′	NOUN
ejpam-4386	92	17	a(x	a(x	NOUN
ejpam-4386	92	18	)	)	PUNCT
ejpam-4386	92	19	where	where	SCONJ
ejpam-4386	92	20	e	e	NOUN
ejpam-4386	92	21	is	be	AUX
ejpam-4386	92	22	a	a	DET
ejpam-4386	92	23	finite	finite	NOUN
ejpam-4386	92	24	subset	subset	NOUN
ejpam-4386	92	25	of	of	ADP
ejpam-4386	92	26	x	x	PROPN
ejpam-4386	92	27	′.	′.	PROPN
ejpam-4386	92	28	thus	thus	ADV
ejpam-4386	92	29	,	,	PUNCT
ejpam-4386	92	30	x	x	SYM
ejpam-4386	92	31	∈	∈	NOUN
ejpam-4386	92	32	v	v	ADP
ejpam-4386	92	33	⊆	⊆	NUM
ejpam-4386	92	34	u	u	NOUN
ejpam-4386	92	35	a(x	a(x	NOUN
ejpam-4386	92	36	)	)	PUNCT
ejpam-4386	92	37	,	,	PUNCT
ejpam-4386	92	38	but	but	CCONJ
ejpam-4386	92	39	u	u	NOUN
ejpam-4386	92	40	a(x	a(x	PROPN
ejpam-4386	92	41	)	)	PUNCT
ejpam-4386	92	42	=	=	SYM
ejpam-4386	92	43	u	u	NOUN
ejpam-4386	92	44	x	x	X
ejpam-4386	92	45	.	.	PUNCT
ejpam-4386	93	1	so	so	ADV
ejpam-4386	93	2	,	,	PUNCT
ejpam-4386	93	3	x	x	SYM
ejpam-4386	93	4	∈	∈	NOUN
ejpam-4386	93	5	v	v	ADP
ejpam-4386	93	6	⊆	⊆	NUM
ejpam-4386	93	7	u	u	NOUN
ejpam-4386	93	8	x	x	X
ejpam-4386	93	9	,	,	PUNCT
ejpam-4386	93	10	by	by	ADP
ejpam-4386	93	11	taking	take	VERB
ejpam-4386	93	12	the	the	DET
ejpam-4386	93	13	interior	interior	NOUN
ejpam-4386	93	14	of	of	ADP
ejpam-4386	93	15	both	both	DET
ejpam-4386	93	16	sides	side	NOUN
ejpam-4386	93	17	with	with	ADP
ejpam-4386	93	18	respect	respect	NOUN
ejpam-4386	93	19	to	to	ADP
ejpam-4386	93	20	x	x	NOUN
ejpam-4386	93	21	we	we	PRON
ejpam-4386	93	22	get	get	VERB
ejpam-4386	93	23	,	,	PUNCT
ejpam-4386	93	24	intx(v	intx(v	NOUN
ejpam-4386	93	25	)	)	PUNCT
ejpam-4386	93	26	⊆	⊆	NUM
ejpam-4386	93	27	intx(u	intx(u	NOUN
ejpam-4386	93	28	x	x	PUNCT
ejpam-4386	93	29	)	)	PUNCT
ejpam-4386	93	30	,	,	PUNCT
ejpam-4386	93	31	then	then	ADV
ejpam-4386	93	32	v	v	VERB
ejpam-4386	93	33	⊆	⊆	NUM
ejpam-4386	93	34	u	u	NOUN
ejpam-4386	93	35	as	as	ADP
ejpam-4386	93	36	v	v	NUM
ejpam-4386	93	37	is	be	AUX
ejpam-4386	93	38	an	an	DET
ejpam-4386	93	39	open	open	ADJ
ejpam-4386	93	40	set	set	NOUN
ejpam-4386	93	41	in	in	ADP
ejpam-4386	93	42	x	x	PUNCT
ejpam-4386	93	43	and	and	CCONJ
ejpam-4386	93	44	u	u	NOUN
ejpam-4386	93	45	is	be	AUX
ejpam-4386	93	46	an	an	DET
ejpam-4386	93	47	open	open	ADJ
ejpam-4386	93	48	domain	domain	NOUN
ejpam-4386	93	49	in	in	ADP
ejpam-4386	93	50	x.	x.	NOUN
ejpam-4386	93	51	thus	thus	ADV
ejpam-4386	93	52	,	,	PUNCT
ejpam-4386	93	53	x	x	PUNCT
ejpam-4386	93	54	∈	∈	PROPN
ejpam-4386	93	55	u	u	NOUN
ejpam-4386	93	56	⊆	⊆	NUM
ejpam-4386	93	57	u	u	NOUN
ejpam-4386	93	58	∪	∪	VERB
ejpam-4386	93	59	u	u	NOUN
ejpam-4386	93	60	′.	′.	NOUN
ejpam-4386	93	61	hence	hence	ADV
ejpam-4386	93	62	d.	d.	PROPN
ejpam-4386	93	63	abuzaid	abuzaid	PROPN
ejpam-4386	93	64	,	,	PUNCT
ejpam-4386	93	65	n.	n.	PROPN
ejpam-4386	93	66	alfarsi	alfarsi	NOUN
ejpam-4386	93	67	,	,	PUNCT
ejpam-4386	93	68	l.	l.	PROPN
ejpam-4386	93	69	kalantan	kalantan	PROPN
ejpam-4386	93	70	/	/	SYM
ejpam-4386	93	71	eur	eur	PROPN
ejpam-4386	93	72	.	.	PUNCT
ejpam-4386	94	1	j.	j.	PROPN
ejpam-4386	94	2	pure	pure	PROPN
ejpam-4386	94	3	appl	appl	PROPN
ejpam-4386	94	4	.	.	PROPN
ejpam-4386	94	5	math	math	PROPN
ejpam-4386	94	6	,	,	PUNCT
ejpam-4386	94	7	15	15	NUM
ejpam-4386	94	8	(	(	PUNCT
ejpam-4386	94	9	3	3	NUM
ejpam-4386	94	10	)	)	PUNCT
ejpam-4386	94	11	(	(	PUNCT
ejpam-4386	94	12	2022	2022	NUM
ejpam-4386	94	13	)	)	PUNCT
ejpam-4386	94	14	,	,	PUNCT
ejpam-4386	94	15	821	821	NUM
ejpam-4386	94	16	-	-	SYM
ejpam-4386	94	17	829	829	NUM
ejpam-4386	94	18	824	824	NUM
ejpam-4386	94	19	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	94	20	∪	∪	ADJ
ejpam-4386	94	21	u	u	NOUN
ejpam-4386	94	22	′	′	NOUN
ejpam-4386	94	23	a(x	a(x	NOUN
ejpam-4386	94	24	)	)	PUNCT
ejpam-4386	94	25	)	)	PUNCT
ejpam-4386	95	1	⊆	⊆	NUM
ejpam-4386	95	2	u	u	NOUN
ejpam-4386	95	3	∪	∪	VERB
ejpam-4386	95	4	u	u	NOUN
ejpam-4386	95	5	′.	′.	NOUN
ejpam-4386	95	6	now	now	ADV
ejpam-4386	95	7	,	,	PUNCT
ejpam-4386	95	8	we	we	PRON
ejpam-4386	95	9	show	show	VERB
ejpam-4386	95	10	that	that	SCONJ
ejpam-4386	95	11	u	u	PROPN
ejpam-4386	95	12	∪	∪	ADP
ejpam-4386	95	13	u	u	NOUN
ejpam-4386	95	14	′	′	NUM
ejpam-4386	95	15	⊆	⊆	NUM
ejpam-4386	95	16	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	95	17	∪	∪	ADJ
ejpam-4386	95	18	u	u	NOUN
ejpam-4386	95	19	′	′	NOUN
ejpam-4386	95	20	a(x	a(x	NOUN
ejpam-4386	95	21	)	)	PUNCT
ejpam-4386	95	22	)	)	PUNCT
ejpam-4386	95	23	.	.	PUNCT
ejpam-4386	96	1	note	note	VERB
ejpam-4386	96	2	that	that	SCONJ
ejpam-4386	96	3	we	we	PRON
ejpam-4386	96	4	always	always	ADV
ejpam-4386	96	5	have	have	VERB
ejpam-4386	96	6	u	u	NOUN
ejpam-4386	96	7	∪	∪	ADJ
ejpam-4386	96	8	u	u	NOUN
ejpam-4386	96	9	′	′	NUM
ejpam-4386	96	10	⊆	⊆	NUM
ejpam-4386	96	11	u	u	NOUN
ejpam-4386	96	12	∪	∪	VERB
ejpam-4386	96	13	u	u	NOUN
ejpam-4386	96	14	′	′	NUM
ejpam-4386	96	15	a(x	a(x	NOUN
ejpam-4386	96	16	)	)	PUNCT
ejpam-4386	96	17	.	.	PUNCT
ejpam-4386	97	1	since	since	SCONJ
ejpam-4386	97	2	u	u	NOUN
ejpam-4386	97	3	is	be	AUX
ejpam-4386	97	4	an	an	DET
ejpam-4386	97	5	open	open	ADJ
ejpam-4386	97	6	domain	domain	NOUN
ejpam-4386	97	7	in	in	ADP
ejpam-4386	97	8	x	x	NOUN
ejpam-4386	97	9	,	,	PUNCT
ejpam-4386	97	10	then	then	ADV
ejpam-4386	97	11	u	u	NOUN
ejpam-4386	97	12	is	be	AUX
ejpam-4386	97	13	an	an	DET
ejpam-4386	97	14	open	open	ADJ
ejpam-4386	97	15	set	set	NOUN
ejpam-4386	97	16	in	in	ADP
ejpam-4386	97	17	x	x	SYM
ejpam-4386	97	18	,	,	PUNCT
ejpam-4386	97	19	so	so	ADV
ejpam-4386	97	20	by	by	ADP
ejpam-4386	97	21	lemma	lemma	PROPN
ejpam-4386	97	22	3	3	NUM
ejpam-4386	97	23	,	,	PUNCT
ejpam-4386	97	24	u	u	NOUN
ejpam-4386	97	25	∪	∪	NOUN
ejpam-4386	97	26	u	u	NOUN
ejpam-4386	97	27	′	′	NOUN
ejpam-4386	97	28	is	be	AUX
ejpam-4386	97	29	an	an	DET
ejpam-4386	97	30	open	open	ADJ
ejpam-4386	97	31	set	set	NOUN
ejpam-4386	97	32	in	in	ADP
ejpam-4386	97	33	a(x	a(x	NOUN
ejpam-4386	97	34	)	)	PUNCT
ejpam-4386	97	35	.	.	PUNCT
ejpam-4386	98	1	then	then	ADV
ejpam-4386	98	2	by	by	ADP
ejpam-4386	98	3	taking	take	VERB
ejpam-4386	98	4	the	the	DET
ejpam-4386	98	5	interior	interior	NOUN
ejpam-4386	98	6	of	of	ADP
ejpam-4386	98	7	both	both	DET
ejpam-4386	98	8	sides	side	NOUN
ejpam-4386	98	9	with	with	ADP
ejpam-4386	98	10	respect	respect	NOUN
ejpam-4386	98	11	to	to	ADP
ejpam-4386	98	12	a(x	a(x	NOUN
ejpam-4386	98	13	)	)	PUNCT
ejpam-4386	98	14	we	we	PRON
ejpam-4386	98	15	get	get	VERB
ejpam-4386	98	16	,	,	PUNCT
ejpam-4386	98	17	u	u	NOUN
ejpam-4386	98	18	∪	∪	NOUN
ejpam-4386	98	19	u	u	NOUN
ejpam-4386	98	20	′	′	NOUN
ejpam-4386	98	21	=	=	PUNCT
ejpam-4386	99	1	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	99	2	∪	∪	ADJ
ejpam-4386	99	3	u	u	NOUN
ejpam-4386	99	4	′	′	NUM
ejpam-4386	99	5	)	)	PUNCT
ejpam-4386	99	6	⊆	⊆	NUM
ejpam-4386	99	7	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	99	8	∪	∪	ADJ
ejpam-4386	99	9	u	u	NOUN
ejpam-4386	99	10	′	′	NOUN
ejpam-4386	99	11	a(x	a(x	NOUN
ejpam-4386	99	12	)	)	PUNCT
ejpam-4386	99	13	)	)	PUNCT
ejpam-4386	99	14	.	.	PUNCT
ejpam-4386	100	1	hence	hence	ADV
ejpam-4386	100	2	,	,	PUNCT
ejpam-4386	100	3	u	u	PROPN
ejpam-4386	100	4	∪	∪	VERB
ejpam-4386	100	5	u	u	NOUN
ejpam-4386	100	6	′	′	NOUN
ejpam-4386	100	7	=	=	PUNCT
ejpam-4386	101	1	inta(x)(u	inta(x)(u	PROPN
ejpam-4386	101	2	∪	∪	ADJ
ejpam-4386	101	3	u	u	NOUN
ejpam-4386	101	4	′	′	NOUN
ejpam-4386	101	5	a(x	a(x	NOUN
ejpam-4386	101	6	)	)	PUNCT
ejpam-4386	101	7	)	)	PUNCT
ejpam-4386	101	8	.	.	PUNCT
ejpam-4386	102	1	therefore	therefore	ADV
ejpam-4386	102	2	,	,	PUNCT
ejpam-4386	102	3	u	u	NOUN
ejpam-4386	102	4	∪	∪	VERB
ejpam-4386	102	5	u	u	NOUN
ejpam-4386	102	6	′	′	NOUN
ejpam-4386	102	7	is	be	AUX
ejpam-4386	102	8	an	an	DET
ejpam-4386	102	9	open	open	ADJ
ejpam-4386	102	10	domain	domain	NOUN
ejpam-4386	102	11	in	in	ADP
ejpam-4386	102	12	a(x	a(x	NOUN
ejpam-4386	102	13	)	)	PUNCT
ejpam-4386	102	14	.	.	PUNCT
ejpam-4386	103	1	as	as	ADP
ejpam-4386	103	2	an	an	DET
ejpam-4386	103	3	immediate	immediate	ADJ
ejpam-4386	103	4	consequence	consequence	NOUN
ejpam-4386	103	5	of	of	ADP
ejpam-4386	103	6	the	the	DET
ejpam-4386	103	7	theorem	theorem	NOUN
ejpam-4386	103	8	1	1	NUM
ejpam-4386	103	9	is	be	AUX
ejpam-4386	103	10	the	the	DET
ejpam-4386	103	11	following	follow	VERB
ejpam-4386	103	12	corollary	corollary	ADJ
ejpam-4386	103	13	1	1	NUM
ejpam-4386	103	14	.	.	PUNCT
ejpam-4386	103	15	corollary	corollary	ADJ
ejpam-4386	103	16	1	1	NUM
ejpam-4386	103	17	.	.	PUNCT
ejpam-4386	104	1	if	if	SCONJ
ejpam-4386	104	2	u	u	NOUN
ejpam-4386	104	3	is	be	AUX
ejpam-4386	104	4	an	an	DET
ejpam-4386	104	5	open	open	ADJ
ejpam-4386	104	6	domain	domain	NOUN
ejpam-4386	104	7	in	in	ADP
ejpam-4386	104	8	x	x	NOUN
ejpam-4386	104	9	,	,	PUNCT
ejpam-4386	104	10	then	then	ADV
ejpam-4386	104	11	u	u	PROPN
ejpam-4386	104	12	∪(u	∪(u	NOUN
ejpam-4386	104	13	′	′	NUM
ejpam-4386	104	14	\e	\e	PROPN
ejpam-4386	104	15	)	)	PUNCT
ejpam-4386	104	16	is	be	AUX
ejpam-4386	104	17	an	an	DET
ejpam-4386	104	18	open	open	ADJ
ejpam-4386	104	19	domain	domain	NOUN
ejpam-4386	104	20	in	in	ADP
ejpam-4386	104	21	a(x	a(x	NOUN
ejpam-4386	104	22	)	)	PUNCT
ejpam-4386	104	23	where	where	SCONJ
ejpam-4386	104	24	e	e	NOUN
ejpam-4386	104	25	is	be	AUX
ejpam-4386	104	26	a	a	DET
ejpam-4386	104	27	finite	finite	NOUN
ejpam-4386	104	28	subset	subset	NOUN
ejpam-4386	104	29	of	of	ADP
ejpam-4386	104	30	x	x	SYM
ejpam-4386	104	31	′	′	NOUN
ejpam-4386	104	32	.	.	PUNCT
ejpam-4386	105	1	proof	proof	NOUN
ejpam-4386	105	2	.	.	PUNCT
ejpam-4386	106	1	let	let	VERB
ejpam-4386	106	2	u	u	PRON
ejpam-4386	106	3	be	be	AUX
ejpam-4386	106	4	any	any	DET
ejpam-4386	106	5	open	open	ADJ
ejpam-4386	106	6	domain	domain	NOUN
ejpam-4386	106	7	in	in	ADP
ejpam-4386	106	8	x.	x.	NOUN
ejpam-4386	106	9	by	by	ADP
ejpam-4386	106	10	lemma	lemma	PROPN
ejpam-4386	106	11	1	1	NUM
ejpam-4386	106	12	,	,	PUNCT
ejpam-4386	106	13	we	we	PRON
ejpam-4386	106	14	have	have	VERB
ejpam-4386	106	15	any	any	DET
ejpam-4386	106	16	singleton	singleton	NOUN
ejpam-4386	106	17	in	in	ADP
ejpam-4386	106	18	x	x	PROPN
ejpam-4386	106	19	′	′	NUM
ejpam-4386	106	20	is	be	AUX
ejpam-4386	106	21	clopen	clopen	ADJ
ejpam-4386	106	22	in	in	ADP
ejpam-4386	106	23	a(x	a(x	NOUN
ejpam-4386	106	24	)	)	PUNCT
ejpam-4386	106	25	,	,	PUNCT
ejpam-4386	106	26	then	then	ADV
ejpam-4386	106	27	e	e	PROPN
ejpam-4386	106	28	is	be	AUX
ejpam-4386	106	29	clopen	clopen	ADJ
ejpam-4386	106	30	in	in	ADP
ejpam-4386	106	31	a(x	a(x	NOUN
ejpam-4386	106	32	)	)	PUNCT
ejpam-4386	106	33	because	because	SCONJ
ejpam-4386	106	34	finite	finite	PROPN
ejpam-4386	106	35	union	union	NOUN
ejpam-4386	106	36	of	of	ADP
ejpam-4386	106	37	closed	closed	ADJ
ejpam-4386	106	38	sets	set	NOUN
ejpam-4386	106	39	is	be	AUX
ejpam-4386	106	40	closed	closed	ADJ
ejpam-4386	106	41	and	and	CCONJ
ejpam-4386	106	42	arbitrary	arbitrary	ADJ
ejpam-4386	106	43	union	union	NOUN
ejpam-4386	106	44	of	of	ADP
ejpam-4386	106	45	open	open	ADJ
ejpam-4386	106	46	sets	set	NOUN
ejpam-4386	106	47	is	be	AUX
ejpam-4386	106	48	open	open	ADJ
ejpam-4386	106	49	.	.	PUNCT
ejpam-4386	107	1	also	also	ADV
ejpam-4386	107	2	,	,	PUNCT
ejpam-4386	107	3	by	by	ADP
ejpam-4386	107	4	theorem	theorem	NOUN
ejpam-4386	107	5	1	1	NUM
ejpam-4386	107	6	,	,	PUNCT
ejpam-4386	107	7	we	we	PRON
ejpam-4386	107	8	have	have	VERB
ejpam-4386	107	9	u	u	NOUN
ejpam-4386	107	10	∪	∪	NOUN
ejpam-4386	107	11	u	u	NOUN
ejpam-4386	107	12	′	′	NOUN
ejpam-4386	107	13	is	be	AUX
ejpam-4386	107	14	an	an	DET
ejpam-4386	107	15	open	open	ADJ
ejpam-4386	107	16	domain	domain	NOUN
ejpam-4386	107	17	in	in	ADP
ejpam-4386	107	18	a(x	a(x	NOUN
ejpam-4386	107	19	)	)	PUNCT
ejpam-4386	107	20	.	.	PUNCT
ejpam-4386	108	1	but	but	CCONJ
ejpam-4386	108	2	,	,	PUNCT
ejpam-4386	108	3	u	u	PROPN
ejpam-4386	108	4	∪(u	∪(u	NOUN
ejpam-4386	108	5	′	′	NUM
ejpam-4386	108	6	\e	\e	PROPN
ejpam-4386	108	7	)	)	PUNCT
ejpam-4386	109	1	=	=	PRON
ejpam-4386	109	2	(	(	PUNCT
ejpam-4386	109	3	u	u	NOUN
ejpam-4386	109	4	∪u	∪u	NOUN
ejpam-4386	109	5	′	′	NUM
ejpam-4386	109	6	)	)	PUNCT
ejpam-4386	109	7	\e	\e	PROPN
ejpam-4386	109	8	is	be	AUX
ejpam-4386	109	9	an	an	DET
ejpam-4386	109	10	open	open	ADJ
ejpam-4386	109	11	domain	domain	NOUN
ejpam-4386	109	12	in	in	ADP
ejpam-4386	109	13	a(x	a(x	NOUN
ejpam-4386	109	14	)	)	PUNCT
ejpam-4386	109	15	by	by	ADP
ejpam-4386	109	16	lemma	lemma	PROPN
ejpam-4386	109	17	2	2	NUM
ejpam-4386	109	18	.	.	PUNCT
ejpam-4386	109	19	theorem	theorem	NOUN
ejpam-4386	109	20	2	2	NUM
ejpam-4386	109	21	.	.	PUNCT
ejpam-4386	110	1	if	if	SCONJ
ejpam-4386	110	2	(	(	PUNCT
ejpam-4386	110	3	x	x	X
ejpam-4386	110	4	,	,	PUNCT
ejpam-4386	110	5	τ	τ	PROPN
ejpam-4386	110	6	)	)	PUNCT
ejpam-4386	110	7	is	be	AUX
ejpam-4386	110	8	semi	semi	ADJ
ejpam-4386	110	9	-	-	ADJ
ejpam-4386	110	10	regular	regular	ADJ
ejpam-4386	110	11	,	,	PUNCT
ejpam-4386	110	12	then	then	ADV
ejpam-4386	110	13	so	so	ADV
ejpam-4386	110	14	is	be	AUX
ejpam-4386	110	15	its	its	PRON
ejpam-4386	110	16	alexandroff	alexandroff	NOUN
ejpam-4386	110	17	duplicate	duplicate	VERB
ejpam-4386	110	18	a(x	a(x	NOUN
ejpam-4386	110	19	)	)	PUNCT
ejpam-4386	110	20	.	.	PUNCT
ejpam-4386	111	1	proof	proof	NOUN
ejpam-4386	111	2	.	.	PUNCT
ejpam-4386	112	1	let	let	VERB
ejpam-4386	112	2	(	(	PUNCT
ejpam-4386	112	3	x	x	X
ejpam-4386	112	4	,	,	PUNCT
ejpam-4386	112	5	τ	τ	PROPN
ejpam-4386	112	6	)	)	PUNCT
ejpam-4386	112	7	be	be	AUX
ejpam-4386	112	8	any	any	DET
ejpam-4386	112	9	semi	semi	ADJ
ejpam-4386	112	10	-	-	ADJ
ejpam-4386	112	11	regular	regular	ADJ
ejpam-4386	112	12	topological	topological	ADJ
ejpam-4386	112	13	space	space	NOUN
ejpam-4386	112	14	.	.	PUNCT
ejpam-4386	113	1	let	let	VERB
ejpam-4386	113	2	∅	∅	NOUN
ejpam-4386	113	3	=	=	NOUN
ejpam-4386	113	4	̸	̸	ADJ
ejpam-4386	113	5	w	w	AUX
ejpam-4386	113	6	be	be	AUX
ejpam-4386	113	7	any	any	DET
ejpam-4386	113	8	open	open	ADJ
ejpam-4386	113	9	set	set	NOUN
ejpam-4386	113	10	in	in	ADP
ejpam-4386	113	11	a(x	a(x	NOUN
ejpam-4386	113	12	)	)	PUNCT
ejpam-4386	113	13	and	and	CCONJ
ejpam-4386	113	14	let	let	VERB
ejpam-4386	113	15	x	x	X
ejpam-4386	113	16	∈	∈	PROPN
ejpam-4386	113	17	w	w	AUX
ejpam-4386	113	18	be	be	AUX
ejpam-4386	113	19	arbitrary	arbitrary	ADJ
ejpam-4386	113	20	.	.	PUNCT
ejpam-4386	114	1	to	to	PART
ejpam-4386	114	2	show	show	VERB
ejpam-4386	114	3	that	that	SCONJ
ejpam-4386	114	4	a(x	a(x	NOUN
ejpam-4386	114	5	)	)	PUNCT
ejpam-4386	114	6	is	be	AUX
ejpam-4386	114	7	semi	semi	ADJ
ejpam-4386	114	8	-	-	ADJ
ejpam-4386	114	9	regular	regular	ADJ
ejpam-4386	114	10	,	,	PUNCT
ejpam-4386	114	11	it	it	PRON
ejpam-4386	114	12	is	be	AUX
ejpam-4386	114	13	enough	enough	ADJ
ejpam-4386	114	14	to	to	PART
ejpam-4386	114	15	exhibit	exhibit	VERB
ejpam-4386	114	16	an	an	DET
ejpam-4386	114	17	open	open	ADJ
ejpam-4386	114	18	domain	domain	NOUN
ejpam-4386	114	19	subset	subset	VERB
ejpam-4386	114	20	g	g	NOUN
ejpam-4386	114	21	in	in	ADP
ejpam-4386	114	22	a(x	a(x	NOUN
ejpam-4386	114	23	)	)	PUNCT
ejpam-4386	115	1	such	such	ADJ
ejpam-4386	115	2	that	that	SCONJ
ejpam-4386	115	3	x	x	SYM
ejpam-4386	115	4	∈	∈	NOUN
ejpam-4386	115	5	g	g	NOUN
ejpam-4386	115	6	⊆	⊆	NUM
ejpam-4386	115	7	w	w	NOUN
ejpam-4386	115	8	.	.	PUNCT
ejpam-4386	116	1	for	for	ADP
ejpam-4386	116	2	such	such	DET
ejpam-4386	116	3	an	an	DET
ejpam-4386	116	4	x	x	NOUN
ejpam-4386	116	5	,	,	PUNCT
ejpam-4386	116	6	we	we	PRON
ejpam-4386	116	7	have	have	VERB
ejpam-4386	116	8	only	only	ADV
ejpam-4386	116	9	two	two	NUM
ejpam-4386	116	10	cases	case	NOUN
ejpam-4386	116	11	.	.	PUNCT
ejpam-4386	117	1	case	case	NOUN
ejpam-4386	117	2	1	1	NUM
ejpam-4386	117	3	:	:	PUNCT
ejpam-4386	117	4	x	x	SYM
ejpam-4386	117	5	∈	∈	NOUN
ejpam-4386	117	6	x	x	SYM
ejpam-4386	117	7	′.	′.	NOUN
ejpam-4386	117	8	since	since	SCONJ
ejpam-4386	117	9	any	any	DET
ejpam-4386	117	10	clopen	clopen	ADJ
ejpam-4386	117	11	subset	subset	NOUN
ejpam-4386	117	12	is	be	AUX
ejpam-4386	117	13	an	an	DET
ejpam-4386	117	14	open	open	ADJ
ejpam-4386	117	15	domain	domain	NOUN
ejpam-4386	117	16	,	,	PUNCT
ejpam-4386	117	17	then	then	ADV
ejpam-4386	117	18	by	by	ADP
ejpam-4386	117	19	lemma	lemma	PROPN
ejpam-4386	117	20	1	1	NUM
ejpam-4386	117	21	,	,	PUNCT
ejpam-4386	117	22	there	there	PRON
ejpam-4386	117	23	exist	exist	VERB
ejpam-4386	117	24	an	an	DET
ejpam-4386	117	25	open	open	ADJ
ejpam-4386	117	26	domain	domain	NOUN
ejpam-4386	117	27	g	g	NOUN
ejpam-4386	117	28	=	=	SYM
ejpam-4386	117	29	{	{	PUNCT
ejpam-4386	117	30	x	x	NOUN
ejpam-4386	117	31	}	}	PUNCT
ejpam-4386	117	32	in	in	ADP
ejpam-4386	117	33	a(x	a(x	NOUN
ejpam-4386	117	34	)	)	PUNCT
ejpam-4386	117	35	such	such	ADJ
ejpam-4386	117	36	that	that	SCONJ
ejpam-4386	117	37	x	x	SYM
ejpam-4386	117	38	∈	∈	NOUN
ejpam-4386	117	39	g	g	NOUN
ejpam-4386	117	40	⊆	⊆	NUM
ejpam-4386	117	41	w	w	NOUN
ejpam-4386	117	42	.	.	PUNCT
ejpam-4386	118	1	case	case	NOUN
ejpam-4386	118	2	2	2	NUM
ejpam-4386	118	3	:	:	PUNCT
ejpam-4386	118	4	x	x	SYM
ejpam-4386	118	5	∈	∈	PROPN
ejpam-4386	118	6	x.	x.	NOUN
ejpam-4386	118	7	since	since	SCONJ
ejpam-4386	118	8	w	w	PROPN
ejpam-4386	118	9	is	be	AUX
ejpam-4386	118	10	an	an	DET
ejpam-4386	118	11	open	open	ADJ
ejpam-4386	118	12	set	set	NOUN
ejpam-4386	118	13	in	in	ADP
ejpam-4386	118	14	a(x	a(x	NOUN
ejpam-4386	118	15	)	)	PUNCT
ejpam-4386	118	16	with	with	ADP
ejpam-4386	118	17	x	x	PROPN
ejpam-4386	118	18	∈	∈	PROPN
ejpam-4386	118	19	w	w	NOUN
ejpam-4386	118	20	,	,	PUNCT
ejpam-4386	118	21	then	then	ADV
ejpam-4386	118	22	there	there	PRON
ejpam-4386	118	23	exists	exist	VERB
ejpam-4386	118	24	an	an	DET
ejpam-4386	118	25	open	open	ADJ
ejpam-4386	118	26	set	set	NOUN
ejpam-4386	118	27	u	u	NOUN
ejpam-4386	118	28	in	in	ADP
ejpam-4386	118	29	x	x	PUNCT
ejpam-4386	118	30	with	with	ADP
ejpam-4386	118	31	x	x	PROPN
ejpam-4386	118	32	∈	∈	PROPN
ejpam-4386	118	33	u	u	NOUN
ejpam-4386	118	34	and	and	CCONJ
ejpam-4386	118	35	u	u	NOUN
ejpam-4386	118	36	∪	∪	NOUN
ejpam-4386	118	37	(	(	PUNCT
ejpam-4386	118	38	u	u	NOUN
ejpam-4386	118	39	′	′	NOUN
ejpam-4386	119	1	\	\	NOUN
ejpam-4386	119	2	e	e	X
ejpam-4386	119	3	)	)	PUNCT
ejpam-4386	119	4	⊆	⊆	NUM
ejpam-4386	119	5	w	w	NOUN
ejpam-4386	119	6	,	,	PUNCT
ejpam-4386	119	7	where	where	SCONJ
ejpam-4386	119	8	e	e	NOUN
ejpam-4386	119	9	is	be	AUX
ejpam-4386	119	10	a	a	DET
ejpam-4386	119	11	finite	finite	NOUN
ejpam-4386	119	12	subset	subset	NOUN
ejpam-4386	119	13	of	of	ADP
ejpam-4386	119	14	x	x	PROPN
ejpam-4386	119	15	′.	′.	NOUN
ejpam-4386	119	16	now	now	ADV
ejpam-4386	119	17	,	,	PUNCT
ejpam-4386	119	18	since	since	SCONJ
ejpam-4386	119	19	(	(	PUNCT
ejpam-4386	119	20	x	x	X
ejpam-4386	119	21	,	,	PUNCT
ejpam-4386	119	22	τ	τ	PROPN
ejpam-4386	119	23	)	)	PUNCT
ejpam-4386	119	24	is	be	AUX
ejpam-4386	119	25	semi	semi	ADJ
ejpam-4386	119	26	-	-	ADJ
ejpam-4386	119	27	regular	regular	ADJ
ejpam-4386	119	28	,	,	PUNCT
ejpam-4386	119	29	then	then	ADV
ejpam-4386	119	30	there	there	PRON
ejpam-4386	119	31	exists	exist	VERB
ejpam-4386	119	32	an	an	DET
ejpam-4386	119	33	open	open	ADJ
ejpam-4386	119	34	domain	domain	NOUN
ejpam-4386	119	35	v	v	NOUN
ejpam-4386	119	36	in	in	ADP
ejpam-4386	119	37	x	x	PUNCT
ejpam-4386	119	38	such	such	ADJ
ejpam-4386	119	39	that	that	SCONJ
ejpam-4386	119	40	x	x	SYM
ejpam-4386	119	41	∈	∈	NOUN
ejpam-4386	119	42	v	v	ADP
ejpam-4386	119	43	⊆	⊆	NUM
ejpam-4386	119	44	u	u	NOUN
ejpam-4386	119	45	.	.	PUNCT
ejpam-4386	120	1	thus	thus	ADV
ejpam-4386	120	2	,	,	PUNCT
ejpam-4386	120	3	x	x	SYM
ejpam-4386	120	4	∈	∈	PROPN
ejpam-4386	120	5	(	(	PUNCT
ejpam-4386	120	6	v	v	NOUN
ejpam-4386	120	7	∪	∪	X
ejpam-4386	120	8	(	(	PUNCT
ejpam-4386	120	9	v	v	NOUN
ejpam-4386	120	10	\e	\e	PROPN
ejpam-4386	120	11	)	)	PUNCT
ejpam-4386	120	12	)	)	PUNCT
ejpam-4386	120	13	⊆	⊆	X
ejpam-4386	120	14	(	(	PUNCT
ejpam-4386	120	15	u	u	NOUN
ejpam-4386	120	16	∪	∪	X
ejpam-4386	120	17	(	(	PUNCT
ejpam-4386	120	18	u	u	NOUN
ejpam-4386	120	19	\e	\e	PROPN
ejpam-4386	120	20	)	)	PUNCT
ejpam-4386	120	21	)	)	PUNCT
ejpam-4386	121	1	⊆	⊆	X
ejpam-4386	121	2	w	w	NOUN
ejpam-4386	121	3	by	by	ADP
ejpam-4386	121	4	corollary	corollary	ADJ
ejpam-4386	121	5	1	1	NUM
ejpam-4386	121	6	,	,	PUNCT
ejpam-4386	121	7	we	we	PRON
ejpam-4386	121	8	get	get	VERB
ejpam-4386	121	9	v	v	ADP
ejpam-4386	121	10	∪	∪	ADJ
ejpam-4386	121	11	(	(	PUNCT
ejpam-4386	121	12	v	v	NOUN
ejpam-4386	121	13	′	′	NUM
ejpam-4386	121	14	\e	\e	PROPN
ejpam-4386	121	15	)	)	PUNCT
ejpam-4386	122	1	=	=	SYM
ejpam-4386	122	2	g	g	NOUN
ejpam-4386	122	3	is	be	AUX
ejpam-4386	122	4	an	an	DET
ejpam-4386	122	5	open	open	ADJ
ejpam-4386	122	6	domain	domain	NOUN
ejpam-4386	122	7	in	in	ADP
ejpam-4386	122	8	a(x	a(x	NOUN
ejpam-4386	122	9	)	)	PUNCT
ejpam-4386	122	10	such	such	ADJ
ejpam-4386	122	11	that	that	SCONJ
ejpam-4386	122	12	x	x	SYM
ejpam-4386	122	13	∈	∈	NOUN
ejpam-4386	122	14	g	g	NOUN
ejpam-4386	122	15	⊆	⊆	NUM
ejpam-4386	122	16	w	w	NOUN
ejpam-4386	122	17	.	.	PUNCT
ejpam-4386	123	1	therefore	therefore	ADV
ejpam-4386	123	2	,	,	PUNCT
ejpam-4386	123	3	a(x	a(x	PROPN
ejpam-4386	123	4	)	)	PUNCT
ejpam-4386	123	5	is	be	AUX
ejpam-4386	123	6	semi	semi	ADJ
ejpam-4386	123	7	-	-	ADJ
ejpam-4386	123	8	regular	regular	ADJ
ejpam-4386	123	9	.	.	PUNCT
ejpam-4386	124	1	definition	definition	NOUN
ejpam-4386	124	2	3	3	X
ejpam-4386	124	3	.	.	PUNCT
ejpam-4386	125	1	let	let	AUX
ejpam-4386	125	2	(	(	PUNCT
ejpam-4386	125	3	x	x	X
ejpam-4386	125	4	,	,	PUNCT
ejpam-4386	125	5	τ	τ	PROPN
ejpam-4386	125	6	)	)	PUNCT
ejpam-4386	125	7	be	be	AUX
ejpam-4386	125	8	a	a	DET
ejpam-4386	125	9	topological	topological	ADJ
ejpam-4386	125	10	space	space	NOUN
ejpam-4386	125	11	and	and	CCONJ
ejpam-4386	125	12	let	let	VERB
ejpam-4386	125	13	p	p	PRON
ejpam-4386	125	14	be	be	AUX
ejpam-4386	125	15	an	an	DET
ejpam-4386	125	16	object	object	NOUN
ejpam-4386	125	17	not	not	PART
ejpam-4386	125	18	in	in	ADP
ejpam-4386	125	19	x	x	NOUN
ejpam-4386	125	20	,	,	PUNCT
ejpam-4386	125	21	that	that	ADV
ejpam-4386	125	22	is	is	ADV
ejpam-4386	125	23	,	,	PUNCT
ejpam-4386	125	24	p	p	PROPN
ejpam-4386	125	25	̸∈	̸∈	PROPN
ejpam-4386	125	26	x.	x.	PROPN
ejpam-4386	125	27	put	put	VERB
ejpam-4386	125	28	xp	xp	NOUN
ejpam-4386	126	1	=	=	NOUN
ejpam-4386	126	2	x	x	SYM
ejpam-4386	126	3	∪	∪	X
ejpam-4386	126	4	{	{	PUNCT
ejpam-4386	126	5	p	p	NOUN
ejpam-4386	126	6	}	}	PUNCT
ejpam-4386	126	7	.	.	PUNCT
ejpam-4386	127	1	define	define	VERB
ejpam-4386	127	2	a	a	DET
ejpam-4386	127	3	topology	topology	NOUN
ejpam-4386	127	4	τ	τ	X
ejpam-4386	127	5	⋆	⋆	VERB
ejpam-4386	127	6	on	on	ADP
ejpam-4386	127	7	xp	xp	INTJ
ejpam-4386	127	8	by	by	ADP
ejpam-4386	127	9	τ	τ	PROPN
ejpam-4386	127	10	⋆	⋆	X
ejpam-4386	127	11	=	=	NOUN
ejpam-4386	127	12	{	{	PUNCT
ejpam-4386	127	13	∅	∅	NOUN
ejpam-4386	127	14	}	}	PUNCT
ejpam-4386	127	15	∪	∪	VERB
ejpam-4386	127	16	{	{	PUNCT
ejpam-4386	127	17	u	u	NOUN
ejpam-4386	127	18	∪	∪	NOUN
ejpam-4386	127	19	{	{	PUNCT
ejpam-4386	127	20	p	p	NOUN
ejpam-4386	127	21	}	}	PUNCT
ejpam-4386	127	22	:	:	PUNCT
ejpam-4386	127	23	u	u	PROPN
ejpam-4386	127	24	∈	∈	PROPN
ejpam-4386	127	25	τ	τ	X
ejpam-4386	127	26	}	}	PUNCT
ejpam-4386	127	27	.	.	PUNCT
ejpam-4386	128	1	the	the	DET
ejpam-4386	128	2	space	space	NOUN
ejpam-4386	128	3	(	(	PUNCT
ejpam-4386	128	4	xp	xp	INTJ
ejpam-4386	128	5	,	,	PUNCT
ejpam-4386	128	6	τ	τ	PROPN
ejpam-4386	128	7	⋆	⋆	VERB
ejpam-4386	128	8	)	)	PUNCT
ejpam-4386	128	9	is	be	AUX
ejpam-4386	128	10	called	call	VERB
ejpam-4386	128	11	the	the	DET
ejpam-4386	128	12	closed	closed	ADJ
ejpam-4386	128	13	extension	extension	NOUN
ejpam-4386	128	14	space	space	NOUN
ejpam-4386	128	15	of	of	ADP
ejpam-4386	128	16	(	(	PUNCT
ejpam-4386	128	17	x	x	INTJ
ejpam-4386	128	18	,	,	PUNCT
ejpam-4386	128	19	τ	τ	PROPN
ejpam-4386	128	20	)	)	PUNCT
ejpam-4386	128	21	,	,	PUNCT
ejpam-4386	128	22	see	see	VERB
ejpam-4386	128	23	[	[	X
ejpam-4386	128	24	12	12	NUM
ejpam-4386	128	25	,	,	PUNCT
ejpam-4386	128	26	example	example	NOUN
ejpam-4386	128	27	12	12	NUM
ejpam-4386	128	28	]	]	PUNCT
ejpam-4386	128	29	.	.	PUNCT
ejpam-4386	129	1	consider	consider	VERB
ejpam-4386	129	2	the	the	DET
ejpam-4386	129	3	particular	particular	ADJ
ejpam-4386	129	4	point	point	NOUN
ejpam-4386	129	5	topology	topology	NOUN
ejpam-4386	129	6	τ	τ	X
ejpam-4386	129	7	p	p	X
ejpam-4386	129	8	=	=	X
ejpam-4386	129	9	{	{	PUNCT
ejpam-4386	129	10	w	w	PROPN
ejpam-4386	129	11	⊆	⊆	NUM
ejpam-4386	129	12	xp	xp	NOUN
ejpam-4386	129	13	:	:	PUNCT
ejpam-4386	129	14	p	p	X
ejpam-4386	129	15	∈	∈	PROPN
ejpam-4386	129	16	w	w	X
ejpam-4386	129	17	}	}	PUNCT
ejpam-4386	129	18	on	on	ADP
ejpam-4386	129	19	xp	xp	PROPN
ejpam-4386	129	20	,	,	PUNCT
ejpam-4386	129	21	[	[	X
ejpam-4386	129	22	12	12	NUM
ejpam-4386	129	23	,	,	PUNCT
ejpam-4386	129	24	example	example	NOUN
ejpam-4386	129	25	10	10	NUM
ejpam-4386	129	26	]	]	PUNCT
ejpam-4386	129	27	.	.	PUNCT
ejpam-4386	130	1	it	it	PRON
ejpam-4386	130	2	is	be	AUX
ejpam-4386	130	3	easy	easy	ADJ
ejpam-4386	130	4	to	to	PART
ejpam-4386	130	5	see	see	VERB
ejpam-4386	130	6	that	that	SCONJ
ejpam-4386	130	7	τ	τ	PROPN
ejpam-4386	130	8	⋆	⋆	VERB
ejpam-4386	130	9	is	be	AUX
ejpam-4386	130	10	coarser	coarse	ADJ
ejpam-4386	130	11	than	than	SCONJ
ejpam-4386	130	12	τ	τ	PROPN
ejpam-4386	130	13	p	p	NOUN
ejpam-4386	130	14	,	,	PUNCT
ejpam-4386	130	15	that	that	ADV
ejpam-4386	130	16	is	is	ADV
ejpam-4386	130	17	,	,	PUNCT
ejpam-4386	130	18	τ	τ	PROPN
ejpam-4386	130	19	⋆	⋆	VERB
ejpam-4386	130	20	⊆	⊆	NUM
ejpam-4386	130	21	τ	τ	PROPN
ejpam-4386	130	22	p.	p.	NOUN
ejpam-4386	130	23	notice	notice	VERB
ejpam-4386	130	24	that	that	SCONJ
ejpam-4386	130	25	the	the	DET
ejpam-4386	130	26	closed	closed	ADJ
ejpam-4386	130	27	extension	extension	NOUN
ejpam-4386	130	28	(	(	PUNCT
ejpam-4386	130	29	xp	xp	INTJ
ejpam-4386	130	30	,	,	PUNCT
ejpam-4386	130	31	τ	τ	PROPN
ejpam-4386	130	32	⋆	⋆	NOUN
ejpam-4386	130	33	)	)	PUNCT
ejpam-4386	130	34	of	of	ADP
ejpam-4386	130	35	a	a	DET
ejpam-4386	130	36	space	space	NOUN
ejpam-4386	130	37	(	(	PUNCT
ejpam-4386	130	38	x	x	X
ejpam-4386	130	39	,	,	PUNCT
ejpam-4386	130	40	τ	τ	PROPN
ejpam-4386	130	41	)	)	PUNCT
ejpam-4386	130	42	is	be	AUX
ejpam-4386	130	43	not	not	PART
ejpam-4386	130	44	semi	semi	ADJ
ejpam-4386	130	45	-	-	ADJ
ejpam-4386	130	46	regular	regular	ADJ
ejpam-4386	130	47	regardless	regardless	ADJ
ejpam-4386	130	48	wither	wither	NOUN
ejpam-4386	130	49	(	(	PUNCT
ejpam-4386	130	50	x	x	X
ejpam-4386	130	51	,	,	PUNCT
ejpam-4386	130	52	τ	τ	PROPN
ejpam-4386	130	53	)	)	PUNCT
ejpam-4386	130	54	is	be	AUX
ejpam-4386	130	55	semi	semi	ADJ
ejpam-4386	130	56	-	-	ADJ
ejpam-4386	130	57	regular	regular	ADJ
ejpam-4386	130	58	or	or	CCONJ
ejpam-4386	130	59	not	not	PART
ejpam-4386	130	60	.	.	PUNCT
ejpam-4386	131	1	d.	d.	PROPN
ejpam-4386	131	2	abuzaid	abuzaid	PROPN
ejpam-4386	131	3	,	,	PUNCT
ejpam-4386	131	4	n.	n.	PROPN
ejpam-4386	131	5	alfarsi	alfarsi	NOUN
ejpam-4386	131	6	,	,	PUNCT
ejpam-4386	131	7	l.	l.	PROPN
ejpam-4386	131	8	kalantan	kalantan	PROPN
ejpam-4386	131	9	/	/	SYM
ejpam-4386	131	10	eur	eur	PROPN
ejpam-4386	131	11	.	.	PUNCT
ejpam-4386	132	1	j.	j.	PROPN
ejpam-4386	132	2	pure	pure	PROPN
ejpam-4386	132	3	appl	appl	PROPN
ejpam-4386	132	4	.	.	PROPN
ejpam-4386	132	5	math	math	PROPN
ejpam-4386	132	6	,	,	PUNCT
ejpam-4386	132	7	15	15	NUM
ejpam-4386	132	8	(	(	PUNCT
ejpam-4386	132	9	3	3	NUM
ejpam-4386	132	10	)	)	PUNCT
ejpam-4386	132	11	(	(	PUNCT
ejpam-4386	132	12	2022	2022	NUM
ejpam-4386	132	13	)	)	PUNCT
ejpam-4386	132	14	,	,	PUNCT
ejpam-4386	132	15	821	821	NUM
ejpam-4386	132	16	-	-	SYM
ejpam-4386	132	17	829	829	NUM
ejpam-4386	132	18	825	825	NUM
ejpam-4386	132	19	example	example	NOUN
ejpam-4386	132	20	1	1	NUM
ejpam-4386	132	21	.	.	PUNCT
ejpam-4386	133	1	let	let	AUX
ejpam-4386	133	2	(	(	PUNCT
ejpam-4386	133	3	x	x	X
ejpam-4386	133	4	,	,	PUNCT
ejpam-4386	133	5	τ	τ	PROPN
ejpam-4386	133	6	)	)	PUNCT
ejpam-4386	133	7	be	be	VERB
ejpam-4386	133	8	the	the	DET
ejpam-4386	133	9	simplified	simplified	ADJ
ejpam-4386	133	10	arens	arens	PROPN
ejpam-4386	133	11	square	square	ADJ
ejpam-4386	133	12	topological	topological	ADJ
ejpam-4386	133	13	space	space	NOUN
ejpam-4386	133	14	,	,	PUNCT
ejpam-4386	133	15	[	[	X
ejpam-4386	133	16	12	12	NUM
ejpam-4386	133	17	,	,	PUNCT
ejpam-4386	133	18	example	example	NOUN
ejpam-4386	133	19	81	81	NUM
ejpam-4386	133	20	]	]	PUNCT
ejpam-4386	133	21	.	.	PUNCT
ejpam-4386	134	1	so	so	ADV
ejpam-4386	134	2	,	,	PUNCT
ejpam-4386	134	3	x	x	SYM
ejpam-4386	134	4	=	=	PRON
ejpam-4386	134	5	{	{	PUNCT
ejpam-4386	134	6	⟨0	⟨0	PROPN
ejpam-4386	134	7	,	,	PUNCT
ejpam-4386	134	8	0⟩	0⟩	PROPN
ejpam-4386	134	9	,	,	PUNCT
ejpam-4386	134	10	⟨1	⟨1	PROPN
ejpam-4386	134	11	,	,	PUNCT
ejpam-4386	134	12	0⟩	0⟩	NUM
ejpam-4386	134	13	}	}	PUNCT
ejpam-4386	134	14	∪	∪	X
ejpam-4386	134	15	{	{	PUNCT
ejpam-4386	134	16	⟨x	⟨x	NUM
ejpam-4386	134	17	,	,	PUNCT
ejpam-4386	134	18	y⟩	y⟩	NOUN
ejpam-4386	134	19	:	:	PUNCT
ejpam-4386	135	1	0	0	PUNCT
ejpam-4386	135	2	<	<	X
ejpam-4386	135	3	x	x	X
ejpam-4386	135	4	,	,	PUNCT
ejpam-4386	135	5	y	y	PROPN
ejpam-4386	135	6	<	<	X
ejpam-4386	135	7	1	1	NUM
ejpam-4386	135	8	}	}	PUNCT
ejpam-4386	135	9	.	.	PUNCT
ejpam-4386	136	1	the	the	DET
ejpam-4386	136	2	topology	topology	NOUN
ejpam-4386	136	3	τ	τ	PROPN
ejpam-4386	136	4	on	on	ADP
ejpam-4386	136	5	x	x	PROPN
ejpam-4386	136	6	is	be	AUX
ejpam-4386	136	7	generated	generate	VERB
ejpam-4386	136	8	by	by	ADP
ejpam-4386	136	9	the	the	DET
ejpam-4386	136	10	following	follow	VERB
ejpam-4386	136	11	neighborhood	neighborhood	NOUN
ejpam-4386	136	12	system	system	NOUN
ejpam-4386	136	13	:	:	PUNCT
ejpam-4386	136	14	for	for	ADP
ejpam-4386	136	15	each	each	DET
ejpam-4386	136	16	⟨x	⟨x	VERB
ejpam-4386	136	17	,	,	PUNCT
ejpam-4386	136	18	y⟩	y⟩	NOUN
ejpam-4386	136	19	∈	∈	PROPN
ejpam-4386	136	20	{	{	PUNCT
ejpam-4386	136	21	⟨x	⟨x	NUM
ejpam-4386	136	22	,	,	PUNCT
ejpam-4386	136	23	y⟩	y⟩	NOUN
ejpam-4386	136	24	:	:	PUNCT
ejpam-4386	136	25	0	0	PUNCT
ejpam-4386	136	26	<	<	X
ejpam-4386	136	27	x	x	X
ejpam-4386	136	28	,	,	PUNCT
ejpam-4386	136	29	y	y	PROPN
ejpam-4386	136	30	<	<	X
ejpam-4386	136	31	1	1	NUM
ejpam-4386	136	32	}	}	PUNCT
ejpam-4386	136	33	,	,	PUNCT
ejpam-4386	136	34	let	let	VERB
ejpam-4386	136	35	b(⟨x	b(⟨x	NOUN
ejpam-4386	136	36	,	,	PUNCT
ejpam-4386	136	37	y⟩	y⟩	NOUN
ejpam-4386	136	38	)	)	PUNCT
ejpam-4386	136	39	=	=	PRON
ejpam-4386	136	40	{	{	PUNCT
ejpam-4386	136	41	bd(⟨x	bd(⟨x	PROPN
ejpam-4386	136	42	,	,	PUNCT
ejpam-4386	136	43	y⟩	y⟩	NOUN
ejpam-4386	136	44	;	;	PUNCT
ejpam-4386	137	1	ϵ	ϵ	X
ejpam-4386	137	2	)	)	PUNCT
ejpam-4386	137	3	⊂	⊂	PROPN
ejpam-4386	137	4	s	s	VERB
ejpam-4386	137	5	:	:	PUNCT
ejpam-4386	137	6	ϵ	ϵ	X
ejpam-4386	137	7	>	>	X
ejpam-4386	137	8	0	0	PUNCT
ejpam-4386	137	9	}	}	PUNCT
ejpam-4386	137	10	where	where	SCONJ
ejpam-4386	137	11	d	d	NOUN
ejpam-4386	137	12	is	be	AUX
ejpam-4386	137	13	the	the	DET
ejpam-4386	137	14	usual	usual	ADJ
ejpam-4386	137	15	metric	metric	NOUN
ejpam-4386	137	16	on	on	ADP
ejpam-4386	137	17	r2	r2	PROPN
ejpam-4386	137	18	and	and	CCONJ
ejpam-4386	137	19	bd(⟨x	bd(⟨x	NOUN
ejpam-4386	137	20	,	,	PUNCT
ejpam-4386	137	21	y⟩	y⟩	NOUN
ejpam-4386	137	22	;	;	PUNCT
ejpam-4386	137	23	ϵ	ϵ	X
ejpam-4386	137	24	)	)	PUNCT
ejpam-4386	137	25	is	be	AUX
ejpam-4386	137	26	the	the	DET
ejpam-4386	137	27	open	open	ADJ
ejpam-4386	137	28	ball	ball	NOUN
ejpam-4386	137	29	centered	center	VERB
ejpam-4386	137	30	at	at	ADP
ejpam-4386	137	31	⟨x	⟨x	NUM
ejpam-4386	137	32	,	,	PUNCT
ejpam-4386	137	33	y⟩	y⟩	NOUN
ejpam-4386	137	34	of	of	ADP
ejpam-4386	137	35	radius	radius	NOUN
ejpam-4386	137	36	ϵ	ϵ	X
ejpam-4386	137	37	>	>	X
ejpam-4386	137	38	0	0	PUNCT
ejpam-4386	138	1	so	so	SCONJ
ejpam-4386	138	2	that	that	SCONJ
ejpam-4386	138	3	ϵ	ϵ	PROPN
ejpam-4386	138	4	is	be	AUX
ejpam-4386	138	5	small	small	ADJ
ejpam-4386	138	6	enough	enough	ADV
ejpam-4386	138	7	to	to	PART
ejpam-4386	138	8	make	make	VERB
ejpam-4386	138	9	the	the	DET
ejpam-4386	138	10	open	open	ADJ
ejpam-4386	138	11	ball	ball	NOUN
ejpam-4386	138	12	bd(⟨x	bd(⟨x	PROPN
ejpam-4386	138	13	,	,	PUNCT
ejpam-4386	138	14	y⟩	y⟩	NOUN
ejpam-4386	138	15	;	;	PUNCT
ejpam-4386	138	16	ϵ	ϵ	X
ejpam-4386	138	17	)	)	PUNCT
ejpam-4386	138	18	is	be	AUX
ejpam-4386	138	19	contained	contain	VERB
ejpam-4386	138	20	in	in	ADP
ejpam-4386	138	21	{	{	PUNCT
ejpam-4386	138	22	⟨x	⟨x	NUM
ejpam-4386	138	23	,	,	PUNCT
ejpam-4386	138	24	y⟩	y⟩	NOUN
ejpam-4386	138	25	:	:	PUNCT
ejpam-4386	138	26	0	0	PUNCT
ejpam-4386	138	27	<	<	X
ejpam-4386	138	28	x	x	X
ejpam-4386	138	29	,	,	PUNCT
ejpam-4386	138	30	y	y	PROPN
ejpam-4386	138	31	<	<	X
ejpam-4386	138	32	1	1	NUM
ejpam-4386	138	33	}	}	PUNCT
ejpam-4386	138	34	.	.	PUNCT
ejpam-4386	139	1	let	let	VERB
ejpam-4386	139	2	b(⟨0	b(⟨0	NOUN
ejpam-4386	139	3	,	,	PUNCT
ejpam-4386	139	4	0⟩	0⟩	PROPN
ejpam-4386	139	5	)	)	PUNCT
ejpam-4386	140	1	=	=	SYM
ejpam-4386	140	2	{	{	PUNCT
ejpam-4386	140	3	un(⟨0	un(⟨0	NOUN
ejpam-4386	140	4	,	,	PUNCT
ejpam-4386	140	5	0⟩	0⟩	PROPN
ejpam-4386	140	6	)	)	PUNCT
ejpam-4386	140	7	:	:	PUNCT
ejpam-4386	141	1	n	n	X
ejpam-4386	141	2	∈	∈	PROPN
ejpam-4386	141	3	n	n	CCONJ
ejpam-4386	141	4	}	}	PUNCT
ejpam-4386	141	5	,	,	PUNCT
ejpam-4386	141	6	where	where	SCONJ
ejpam-4386	141	7	for	for	ADP
ejpam-4386	141	8	each	each	DET
ejpam-4386	141	9	n	n	PRON
ejpam-4386	141	10	∈	∈	PROPN
ejpam-4386	141	11	n	n	CCONJ
ejpam-4386	141	12	,	,	PUNCT
ejpam-4386	141	13	we	we	PRON
ejpam-4386	141	14	have	have	VERB
ejpam-4386	141	15	un(⟨0	un(⟨0	NOUN
ejpam-4386	141	16	,	,	PUNCT
ejpam-4386	141	17	0⟩	0⟩	PROPN
ejpam-4386	141	18	)	)	PUNCT
ejpam-4386	142	1	=	=	PRON
ejpam-4386	142	2	{	{	PUNCT
ejpam-4386	142	3	{	{	PUNCT
ejpam-4386	142	4	⟨0	⟨0	NOUN
ejpam-4386	142	5	,	,	PUNCT
ejpam-4386	142	6	0⟩	0⟩	PROPN
ejpam-4386	142	7	}	}	PUNCT
ejpam-4386	142	8	⋃	⋃	ADV
ejpam-4386	142	9	{	{	PUNCT
ejpam-4386	142	10	⟨x	⟨x	VERB
ejpam-4386	142	11	,	,	PUNCT
ejpam-4386	142	12	y⟩	y⟩	NOUN
ejpam-4386	142	13	∈	∈	PROPN
ejpam-4386	142	14	s	s	PART
ejpam-4386	142	15	:	:	PUNCT
ejpam-4386	142	16	0	0	NUM
ejpam-4386	142	17	<	<	X
ejpam-4386	142	18	x	x	X
ejpam-4386	142	19	<	<	X
ejpam-4386	142	20	1	1	NUM
ejpam-4386	142	21	2	2	NUM
ejpam-4386	142	22	and	and	CCONJ
ejpam-4386	142	23	0	0	NUM
ejpam-4386	142	24	<	<	X
ejpam-4386	142	25	y	y	X
ejpam-4386	142	26	<	<	X
ejpam-4386	142	27	1	1	NUM
ejpam-4386	142	28	n	n	PROPN
ejpam-4386	142	29	}	}	PUNCT
ejpam-4386	142	30	.	.	PUNCT
ejpam-4386	143	1	let	let	VERB
ejpam-4386	143	2	b(⟨1	b(⟨1	ADJ
ejpam-4386	143	3	,	,	PUNCT
ejpam-4386	143	4	0⟩	0⟩	PROPN
ejpam-4386	143	5	)	)	PUNCT
ejpam-4386	144	1	=	=	SYM
ejpam-4386	144	2	{	{	PUNCT
ejpam-4386	144	3	un(⟨1	un(⟨1	NOUN
ejpam-4386	144	4	,	,	PUNCT
ejpam-4386	144	5	0⟩	0⟩	PROPN
ejpam-4386	144	6	)	)	PUNCT
ejpam-4386	144	7	:	:	PUNCT
ejpam-4386	144	8	n	n	X
ejpam-4386	144	9	∈	∈	PROPN
ejpam-4386	144	10	n	n	CCONJ
ejpam-4386	144	11	}	}	PUNCT
ejpam-4386	144	12	,	,	PUNCT
ejpam-4386	144	13	where	where	SCONJ
ejpam-4386	144	14	for	for	ADP
ejpam-4386	144	15	each	each	DET
ejpam-4386	144	16	n	n	PRON
ejpam-4386	144	17	∈	∈	PROPN
ejpam-4386	144	18	n	n	CCONJ
ejpam-4386	144	19	,	,	PUNCT
ejpam-4386	144	20	we	we	PRON
ejpam-4386	144	21	have	have	VERB
ejpam-4386	144	22	un(⟨1	un(⟨1	NOUN
ejpam-4386	144	23	,	,	PUNCT
ejpam-4386	144	24	0⟩	0⟩	PROPN
ejpam-4386	144	25	)	)	PUNCT
ejpam-4386	145	1	=	=	PRON
ejpam-4386	145	2	{	{	PUNCT
ejpam-4386	145	3	{	{	PUNCT
ejpam-4386	145	4	⟨1	⟨1	PROPN
ejpam-4386	145	5	,	,	PUNCT
ejpam-4386	145	6	0⟩	0⟩	PROPN
ejpam-4386	145	7	}	}	PUNCT
ejpam-4386	145	8	⋃	⋃	ADV
ejpam-4386	145	9	{	{	PUNCT
ejpam-4386	145	10	⟨x	⟨x	VERB
ejpam-4386	145	11	,	,	PUNCT
ejpam-4386	145	12	y⟩	y⟩	NOUN
ejpam-4386	145	13	∈	∈	PROPN
ejpam-4386	145	14	s	s	PART
ejpam-4386	145	15	:	:	PUNCT
ejpam-4386	145	16	1	1	NUM
ejpam-4386	145	17	2	2	NUM
ejpam-4386	145	18	<	<	X
ejpam-4386	145	19	x	x	X
ejpam-4386	145	20	<	<	X
ejpam-4386	145	21	1	1	NUM
ejpam-4386	145	22	and	and	CCONJ
ejpam-4386	145	23	0	0	NUM
ejpam-4386	145	24	<	<	X
ejpam-4386	145	25	y	y	X
ejpam-4386	145	26	<	<	X
ejpam-4386	145	27	1	1	NUM
ejpam-4386	145	28	n	n	PROPN
ejpam-4386	145	29	}	}	PUNCT
ejpam-4386	145	30	.	.	PUNCT
ejpam-4386	146	1	in	in	ADP
ejpam-4386	146	2	[	[	X
ejpam-4386	146	3	12	12	NUM
ejpam-4386	146	4	,	,	PUNCT
ejpam-4386	146	5	example	example	NOUN
ejpam-4386	146	6	81	81	NUM
ejpam-4386	146	7	]	]	PUNCT
ejpam-4386	146	8	,	,	PUNCT
ejpam-4386	146	9	it	it	PRON
ejpam-4386	146	10	was	be	AUX
ejpam-4386	146	11	shown	show	VERB
ejpam-4386	146	12	that	that	SCONJ
ejpam-4386	146	13	the	the	DET
ejpam-4386	146	14	simplified	simplified	ADJ
ejpam-4386	146	15	arens	arens	PROPN
ejpam-4386	146	16	square	square	ADJ
ejpam-4386	146	17	space	space	NOUN
ejpam-4386	146	18	(	(	PUNCT
ejpam-4386	146	19	x	x	X
ejpam-4386	146	20	,	,	PUNCT
ejpam-4386	146	21	τ	τ	PROPN
ejpam-4386	146	22	)	)	PUNCT
ejpam-4386	146	23	is	be	AUX
ejpam-4386	146	24	semi	semi	ADJ
ejpam-4386	146	25	-	-	ADJ
ejpam-4386	146	26	regular	regular	ADJ
ejpam-4386	146	27	.	.	PUNCT
ejpam-4386	147	1	let	let	VERB
ejpam-4386	147	2	u	u	PRON
ejpam-4386	147	3	be	be	AUX
ejpam-4386	147	4	any	any	DET
ejpam-4386	147	5	non	non	ADJ
ejpam-4386	147	6	-	-	ADJ
ejpam-4386	147	7	empty	empty	ADJ
ejpam-4386	147	8	proper	proper	ADJ
ejpam-4386	147	9	open	open	ADJ
ejpam-4386	147	10	subset	subset	NOUN
ejpam-4386	147	11	of	of	ADP
ejpam-4386	147	12	x	x	PRON
ejpam-4386	147	13	,	,	PUNCT
ejpam-4386	147	14	then	then	ADV
ejpam-4386	147	15	u	u	NOUN
ejpam-4386	147	16	∪	∪	X
ejpam-4386	147	17	{	{	PUNCT
ejpam-4386	147	18	p	p	NOUN
ejpam-4386	147	19	}	}	PUNCT
ejpam-4386	147	20	is	be	AUX
ejpam-4386	147	21	an	an	DET
ejpam-4386	147	22	open	open	ADJ
ejpam-4386	147	23	set	set	NOUN
ejpam-4386	147	24	in	in	ADP
ejpam-4386	147	25	xp	xp	INTJ
ejpam-4386	147	26	such	such	ADJ
ejpam-4386	147	27	that	that	SCONJ
ejpam-4386	147	28	u	u	PROPN
ejpam-4386	147	29	̸=	̸=	PROPN
ejpam-4386	147	30	xp	xp	INTJ
ejpam-4386	147	31	.	.	PUNCT
ejpam-4386	148	1	now	now	ADV
ejpam-4386	148	2	,	,	PUNCT
ejpam-4386	148	3	u	u	NOUN
ejpam-4386	148	4	∪	∪	VERB
ejpam-4386	148	5	{	{	PUNCT
ejpam-4386	148	6	p	p	NOUN
ejpam-4386	148	7	}	}	PUNCT
ejpam-4386	148	8	τ⋆	τ⋆	PUNCT
ejpam-4386	148	9	=	=	PUNCT
ejpam-4386	148	10	u	u	SYM
ejpam-4386	148	11	τ⋆	τ⋆	NOUN
ejpam-4386	148	12	∪{p	∪{p	PROPN
ejpam-4386	148	13	}	}	PUNCT
ejpam-4386	148	14	τ⋆	τ⋆	PUNCT
ejpam-4386	148	15	=	=	PUNCT
ejpam-4386	148	16	u	u	SYM
ejpam-4386	148	17	τ⋆	τ⋆	NOUN
ejpam-4386	148	18	∪xp	∪xp	VERB
ejpam-4386	148	19	=	=	SYM
ejpam-4386	148	20	xp	xp	INTJ
ejpam-4386	148	21	because	because	SCONJ
ejpam-4386	148	22	{	{	PUNCT
ejpam-4386	148	23	p	p	X
ejpam-4386	148	24	}	}	PUNCT
ejpam-4386	148	25	is	be	AUX
ejpam-4386	148	26	dense	dense	ADJ
ejpam-4386	148	27	in	in	SCONJ
ejpam-4386	148	28	(	(	PUNCT
ejpam-4386	148	29	xp	xp	INTJ
ejpam-4386	148	30	,	,	PUNCT
ejpam-4386	148	31	τ	τ	PROPN
ejpam-4386	148	32	⋆	⋆	NOUN
ejpam-4386	148	33	)	)	PUNCT
ejpam-4386	148	34	.	.	PUNCT
ejpam-4386	149	1	hence	hence	ADV
ejpam-4386	149	2	,	,	PUNCT
ejpam-4386	149	3	int	int	ADJ
ejpam-4386	149	4	τ⋆(u	τ⋆(u	NOUN
ejpam-4386	149	5	∪	∪	X
ejpam-4386	149	6	{	{	PUNCT
ejpam-4386	149	7	p	p	NOUN
ejpam-4386	149	8	}	}	PUNCT
ejpam-4386	149	9	τ⋆	τ⋆	NUM
ejpam-4386	149	10	)	)	PUNCT
ejpam-4386	150	1	=	=	PUNCT
ejpam-4386	150	2	int	int	NOUN
ejpam-4386	150	3	τ⋆	τ⋆	PRON
ejpam-4386	150	4	(	(	PUNCT
ejpam-4386	150	5	xp	xp	INTJ
ejpam-4386	150	6	)	)	PUNCT
ejpam-4386	150	7	=	=	SYM
ejpam-4386	150	8	xp	xp	NOUN
ejpam-4386	150	9	̸=	̸=	PROPN
ejpam-4386	150	10	u	u	NOUN
ejpam-4386	150	11	∪	∪	VERB
ejpam-4386	150	12	{	{	PUNCT
ejpam-4386	150	13	p	p	NOUN
ejpam-4386	150	14	}	}	PUNCT
ejpam-4386	150	15	.	.	PUNCT
ejpam-4386	151	1	thus	thus	ADV
ejpam-4386	151	2	the	the	DET
ejpam-4386	151	3	only	only	ADJ
ejpam-4386	151	4	open	open	ADJ
ejpam-4386	151	5	domains	domain	NOUN
ejpam-4386	151	6	in	in	ADP
ejpam-4386	151	7	(	(	PUNCT
ejpam-4386	151	8	xp	xp	INTJ
ejpam-4386	151	9	,	,	PUNCT
ejpam-4386	151	10	τ	τ	PROPN
ejpam-4386	151	11	⋆	⋆	VERB
ejpam-4386	151	12	)	)	PUNCT
ejpam-4386	151	13	are	be	AUX
ejpam-4386	151	14	xp	xp	ADJ
ejpam-4386	151	15	and	and	CCONJ
ejpam-4386	151	16	∅	∅	NOUN
ejpam-4386	151	17	,	,	PUNCT
ejpam-4386	151	18	then	then	ADV
ejpam-4386	151	19	τ	τ	PROPN
ejpam-4386	151	20	⋆	⋆	X
ejpam-4386	151	21	s	s	X
ejpam-4386	151	22	=	=	X
ejpam-4386	151	23	i	i	PROPN
ejpam-4386	151	24	on	on	ADP
ejpam-4386	151	25	xp	xp	INTJ
ejpam-4386	151	26	,	,	PUNCT
ejpam-4386	151	27	where	where	SCONJ
ejpam-4386	151	28	i	i	PRON
ejpam-4386	151	29	is	be	AUX
ejpam-4386	151	30	the	the	DET
ejpam-4386	151	31	indiscrete	indiscrete	ADJ
ejpam-4386	151	32	topology	topology	NOUN
ejpam-4386	151	33	.	.	PUNCT
ejpam-4386	152	1	therefore	therefore	ADV
ejpam-4386	152	2	,	,	PUNCT
ejpam-4386	152	3	the	the	DET
ejpam-4386	152	4	closed	closed	ADJ
ejpam-4386	152	5	extension	extension	NOUN
ejpam-4386	152	6	topological	topological	ADJ
ejpam-4386	152	7	space	space	NOUN
ejpam-4386	152	8	(	(	PUNCT
ejpam-4386	152	9	xp	xp	INTJ
ejpam-4386	152	10	,	,	PUNCT
ejpam-4386	152	11	τ	τ	PROPN
ejpam-4386	152	12	⋆	⋆	NOUN
ejpam-4386	152	13	)	)	PUNCT
ejpam-4386	152	14	of	of	ADP
ejpam-4386	152	15	the	the	DET
ejpam-4386	152	16	simplified	simplified	ADJ
ejpam-4386	152	17	arens	arens	PROPN
ejpam-4386	152	18	square	square	ADJ
ejpam-4386	152	19	space	space	NOUN
ejpam-4386	152	20	(	(	PUNCT
ejpam-4386	152	21	x	x	X
ejpam-4386	152	22	,	,	PUNCT
ejpam-4386	152	23	τ	τ	PROPN
ejpam-4386	152	24	)	)	PUNCT
ejpam-4386	152	25	is	be	AUX
ejpam-4386	152	26	not	not	PART
ejpam-4386	152	27	semi	semi	ADJ
ejpam-4386	152	28	-	-	ADJ
ejpam-4386	152	29	regular	regular	ADJ
ejpam-4386	152	30	.	.	PUNCT
ejpam-4386	153	1	definition	definition	NOUN
ejpam-4386	153	2	4	4	NUM
ejpam-4386	153	3	.	.	PUNCT
ejpam-4386	154	1	let	let	VERB
ejpam-4386	154	2	m	m	PRON
ejpam-4386	154	3	be	be	AUX
ejpam-4386	154	4	a	a	DET
ejpam-4386	154	5	non	non	ADJ
ejpam-4386	154	6	-	-	ADJ
ejpam-4386	154	7	empty	empty	ADJ
ejpam-4386	154	8	proper	proper	ADJ
ejpam-4386	154	9	subset	subset	NOUN
ejpam-4386	154	10	of	of	ADP
ejpam-4386	154	11	a	a	DET
ejpam-4386	154	12	topological	topological	ADJ
ejpam-4386	154	13	space	space	NOUN
ejpam-4386	154	14	(	(	PUNCT
ejpam-4386	154	15	x	x	X
ejpam-4386	154	16	,	,	PUNCT
ejpam-4386	154	17	τ	τ	PROPN
ejpam-4386	154	18	)	)	PUNCT
ejpam-4386	154	19	.	.	PUNCT
ejpam-4386	155	1	define	define	VERB
ejpam-4386	155	2	a	a	DET
ejpam-4386	155	3	new	new	ADJ
ejpam-4386	155	4	topology	topology	NOUN
ejpam-4386	155	5	τ	τ	X
ejpam-4386	155	6	(	(	PUNCT
ejpam-4386	155	7	m	m	NOUN
ejpam-4386	155	8	)	)	PUNCT
ejpam-4386	155	9	on	on	ADP
ejpam-4386	155	10	x	x	PUNCT
ejpam-4386	155	11	as	as	SCONJ
ejpam-4386	155	12	follows	follow	VERB
ejpam-4386	155	13	:	:	PUNCT
ejpam-4386	155	14	τ	τ	PROPN
ejpam-4386	155	15	(	(	PUNCT
ejpam-4386	155	16	m	m	NOUN
ejpam-4386	155	17	)	)	PUNCT
ejpam-4386	155	18	=	=	PRON
ejpam-4386	155	19	{	{	PUNCT
ejpam-4386	155	20	u	u	NOUN
ejpam-4386	155	21	∪k	∪k	PROPN
ejpam-4386	155	22	:	:	PUNCT
ejpam-4386	155	23	u	u	PROPN
ejpam-4386	155	24	∈	∈	PROPN
ejpam-4386	155	25	τ	τ	X
ejpam-4386	155	26	and	and	CCONJ
ejpam-4386	155	27	k	k	PROPN
ejpam-4386	155	28	⊆	⊆	NUM
ejpam-4386	155	29	x	x	X
ejpam-4386	155	30	\m	\m	NOUN
ejpam-4386	155	31	}	}	PUNCT
ejpam-4386	155	32	.	.	PUNCT
ejpam-4386	156	1	(	(	PUNCT
ejpam-4386	156	2	x	x	X
ejpam-4386	156	3	,	,	PUNCT
ejpam-4386	156	4	τ	τ	PROPN
ejpam-4386	156	5	(	(	PUNCT
ejpam-4386	156	6	m	m	NOUN
ejpam-4386	156	7	)	)	PUNCT
ejpam-4386	156	8	)	)	PUNCT
ejpam-4386	156	9	is	be	AUX
ejpam-4386	156	10	called	call	VERB
ejpam-4386	156	11	a	a	DET
ejpam-4386	156	12	discrete	discrete	ADJ
ejpam-4386	156	13	extension	extension	NOUN
ejpam-4386	156	14	of	of	ADP
ejpam-4386	156	15	(	(	PUNCT
ejpam-4386	156	16	x	x	INTJ
ejpam-4386	156	17	,	,	PUNCT
ejpam-4386	156	18	τ	τ	PROPN
ejpam-4386	156	19	)	)	PUNCT
ejpam-4386	156	20	and	and	CCONJ
ejpam-4386	156	21	we	we	PRON
ejpam-4386	156	22	denote	denote	VERB
ejpam-4386	156	23	(	(	PUNCT
ejpam-4386	156	24	x	x	X
ejpam-4386	156	25	,	,	PUNCT
ejpam-4386	156	26	τ	τ	PROPN
ejpam-4386	156	27	(	(	PUNCT
ejpam-4386	156	28	m	m	PROPN
ejpam-4386	156	29	)	)	PUNCT
ejpam-4386	156	30	)	)	PUNCT
ejpam-4386	156	31	,	,	PUNCT
ejpam-4386	156	32	simply	simply	ADV
ejpam-4386	156	33	,	,	PUNCT
ejpam-4386	156	34	by	by	ADP
ejpam-4386	156	35	xm	xm	PROPN
ejpam-4386	157	1	[	[	X
ejpam-4386	157	2	1	1	NUM
ejpam-4386	157	3	]	]	PUNCT
ejpam-4386	157	4	,	,	PUNCT
ejpam-4386	157	5	see	see	VERB
ejpam-4386	157	6	also	also	ADV
ejpam-4386	157	7	[	[	X
ejpam-4386	157	8	6	6	NUM
ejpam-4386	157	9	,	,	PUNCT
ejpam-4386	157	10	example	example	NOUN
ejpam-4386	157	11	5.1.22	5.1.22	NUM
ejpam-4386	157	12	]	]	PUNCT
ejpam-4386	157	13	.	.	PUNCT
ejpam-4386	158	1	observe	observe	VERB
ejpam-4386	158	2	that	that	SCONJ
ejpam-4386	158	3	if	if	SCONJ
ejpam-4386	158	4	u	u	NOUN
ejpam-4386	158	5	is	be	AUX
ejpam-4386	158	6	an	an	DET
ejpam-4386	158	7	open	open	ADJ
ejpam-4386	158	8	set	set	NOUN
ejpam-4386	158	9	in	in	ADP
ejpam-4386	158	10	x	x	NOUN
ejpam-4386	158	11	,	,	PUNCT
ejpam-4386	158	12	then	then	ADV
ejpam-4386	158	13	u	u	NOUN
ejpam-4386	158	14	is	be	AUX
ejpam-4386	158	15	also	also	ADV
ejpam-4386	158	16	open	open	ADJ
ejpam-4386	158	17	in	in	ADP
ejpam-4386	158	18	xm	xm	PROPN
ejpam-4386	158	19	because	because	SCONJ
ejpam-4386	158	20	we	we	PRON
ejpam-4386	158	21	can	can	AUX
ejpam-4386	158	22	write	write	VERB
ejpam-4386	158	23	u	u	NOUN
ejpam-4386	158	24	=	=	PROPN
ejpam-4386	158	25	u	u	NOUN
ejpam-4386	158	26	∪	∪	VERB
ejpam-4386	158	27	∅.	∅.	VERB
ejpam-4386	158	28	the	the	DET
ejpam-4386	158	29	space	space	NOUN
ejpam-4386	158	30	xm	xm	PROPN
ejpam-4386	158	31	has	have	VERB
ejpam-4386	158	32	the	the	DET
ejpam-4386	158	33	following	follow	VERB
ejpam-4386	158	34	neighborhood	neighborhood	NOUN
ejpam-4386	158	35	system	system	NOUN
ejpam-4386	158	36	:	:	PUNCT
ejpam-4386	158	37	for	for	SCONJ
ejpam-4386	158	38	each	each	DET
ejpam-4386	158	39	x	x	SYM
ejpam-4386	158	40	∈	∈	PROPN
ejpam-4386	158	41	x	x	NOUN
ejpam-4386	158	42	\m	\m	NOUN
ejpam-4386	158	43	,	,	PUNCT
ejpam-4386	158	44	let	let	VERB
ejpam-4386	158	45	b(x	b(x	NOUN
ejpam-4386	158	46	)	)	PUNCT
ejpam-4386	158	47	=	=	PRON
ejpam-4386	158	48	{	{	PUNCT
ejpam-4386	158	49	{	{	PUNCT
ejpam-4386	158	50	x	x	NOUN
ejpam-4386	158	51	}	}	PUNCT
ejpam-4386	158	52	}	}	PUNCT
ejpam-4386	158	53	and	and	CCONJ
ejpam-4386	158	54	for	for	ADP
ejpam-4386	158	55	each	each	DET
ejpam-4386	158	56	x	x	SYM
ejpam-4386	158	57	∈	∈	PROPN
ejpam-4386	158	58	m	m	VERB
ejpam-4386	158	59	,	,	PUNCT
ejpam-4386	158	60	let	let	VERB
ejpam-4386	158	61	b(x	b(x	NOUN
ejpam-4386	158	62	)	)	PUNCT
ejpam-4386	158	63	=	=	PRON
ejpam-4386	158	64	{	{	PUNCT
ejpam-4386	158	65	u	u	X
ejpam-4386	158	66	∈	∈	PROPN
ejpam-4386	158	67	τ	τ	X
ejpam-4386	158	68	:	:	PUNCT
ejpam-4386	158	69	x	x	SYM
ejpam-4386	158	70	∈	∈	PROPN
ejpam-4386	158	71	u	u	NOUN
ejpam-4386	158	72	}	}	PUNCT
ejpam-4386	158	73	.	.	PUNCT
ejpam-4386	159	1	if	if	SCONJ
ejpam-4386	159	2	x	x	PRON
ejpam-4386	159	3	is	be	AUX
ejpam-4386	159	4	a	a	DET
ejpam-4386	159	5	semiregular	semiregular	ADJ
ejpam-4386	159	6	topological	topological	ADJ
ejpam-4386	159	7	space	space	NOUN
ejpam-4386	159	8	and	and	CCONJ
ejpam-4386	159	9	∅	∅	NOUN
ejpam-4386	159	10	̸=	̸=	PROPN
ejpam-4386	159	11	m	m	NUM
ejpam-4386	159	12	⊂	⊂	PROPN
ejpam-4386	159	13	x	x	NOUN
ejpam-4386	159	14	,	,	PUNCT
ejpam-4386	159	15	then	then	ADV
ejpam-4386	159	16	the	the	DET
ejpam-4386	159	17	discrete	discrete	ADJ
ejpam-4386	159	18	extension	extension	NOUN
ejpam-4386	159	19	xm	xm	PROPN
ejpam-4386	159	20	may	may	AUX
ejpam-4386	159	21	not	not	PART
ejpam-4386	159	22	be	be	AUX
ejpam-4386	159	23	semi	semi	ADJ
ejpam-4386	159	24	-	-	ADJ
ejpam-4386	159	25	regular	regular	ADJ
ejpam-4386	159	26	as	as	SCONJ
ejpam-4386	159	27	can	can	AUX
ejpam-4386	159	28	be	be	AUX
ejpam-4386	159	29	shown	show	VERB
ejpam-4386	159	30	in	in	ADP
ejpam-4386	159	31	the	the	DET
ejpam-4386	159	32	following	follow	VERB
ejpam-4386	159	33	example	example	NOUN
ejpam-4386	159	34	.	.	PUNCT
ejpam-4386	160	1	example	example	NOUN
ejpam-4386	160	2	2	2	NUM
ejpam-4386	160	3	.	.	X
ejpam-4386	160	4	consider	consider	VERB
ejpam-4386	160	5	,	,	PUNCT
ejpam-4386	160	6	(	(	PUNCT
ejpam-4386	160	7	r	r	NOUN
ejpam-4386	160	8	,	,	PUNCT
ejpam-4386	160	9	i	i	PROPN
ejpam-4386	160	10	)	)	PUNCT
ejpam-4386	160	11	where	where	SCONJ
ejpam-4386	160	12	i	i	PRON
ejpam-4386	160	13	is	be	AUX
ejpam-4386	160	14	the	the	DET
ejpam-4386	160	15	indiscrete	indiscrete	ADJ
ejpam-4386	160	16	topology	topology	NOUN
ejpam-4386	160	17	.	.	PUNCT
ejpam-4386	161	1	it	it	PRON
ejpam-4386	161	2	is	be	AUX
ejpam-4386	161	3	clear	clear	ADJ
ejpam-4386	161	4	that	that	SCONJ
ejpam-4386	161	5	(	(	PUNCT
ejpam-4386	161	6	r	r	NOUN
ejpam-4386	161	7	,	,	PUNCT
ejpam-4386	161	8	i	i	PROPN
ejpam-4386	161	9	)	)	PUNCT
ejpam-4386	161	10	is	be	AUX
ejpam-4386	161	11	semi	semi	ADJ
ejpam-4386	161	12	-	-	ADJ
ejpam-4386	161	13	regular	regular	ADJ
ejpam-4386	161	14	.	.	PUNCT
ejpam-4386	162	1	put	put	VERB
ejpam-4386	162	2	m	m	PROPN
ejpam-4386	162	3	=	=	SYM
ejpam-4386	162	4	r\{0	r\{0	PROPN
ejpam-4386	162	5	}	}	PUNCT
ejpam-4386	162	6	.	.	PUNCT
ejpam-4386	163	1	then	then	ADV
ejpam-4386	163	2	,	,	PUNCT
ejpam-4386	163	3	the	the	DET
ejpam-4386	163	4	discrete	discrete	ADJ
ejpam-4386	163	5	extension	extension	NOUN
ejpam-4386	163	6	xm	xm	PROPN
ejpam-4386	163	7	can	can	AUX
ejpam-4386	163	8	be	be	AUX
ejpam-4386	163	9	describe	describe	NOUN
ejpam-4386	163	10	as	as	SCONJ
ejpam-4386	163	11	follows	follow	VERB
ejpam-4386	163	12	:	:	PUNCT
ejpam-4386	163	13	b(0	b(0	NOUN
ejpam-4386	163	14	)	)	PUNCT
ejpam-4386	163	15	=	=	PRON
ejpam-4386	163	16	{	{	PUNCT
ejpam-4386	163	17	{	{	PUNCT
ejpam-4386	163	18	0	0	NUM
ejpam-4386	163	19	}	}	PUNCT
ejpam-4386	163	20	}	}	PUNCT
ejpam-4386	163	21	and	and	CCONJ
ejpam-4386	163	22	for	for	ADP
ejpam-4386	163	23	each	each	DET
ejpam-4386	163	24	x	x	PUNCT
ejpam-4386	163	25	̸=	̸=	PROPN
ejpam-4386	163	26	0	0	NUM
ejpam-4386	163	27	,	,	PUNCT
ejpam-4386	163	28	b(x	b(x	NOUN
ejpam-4386	163	29	)	)	PUNCT
ejpam-4386	163	30	=	=	PRON
ejpam-4386	163	31	{	{	PUNCT
ejpam-4386	163	32	r	r	NOUN
ejpam-4386	163	33	}	}	PUNCT
ejpam-4386	163	34	.	.	PUNCT
ejpam-4386	164	1	xm	xm	PROPN
ejpam-4386	164	2	is	be	AUX
ejpam-4386	164	3	not	not	PART
ejpam-4386	164	4	semi	semi	ADJ
ejpam-4386	164	5	-	-	ADJ
ejpam-4386	164	6	regular	regular	ADJ
ejpam-4386	164	7	because	because	SCONJ
ejpam-4386	164	8	{	{	PUNCT
ejpam-4386	164	9	0	0	X
ejpam-4386	164	10	}	}	PUNCT
ejpam-4386	164	11	is	be	AUX
ejpam-4386	164	12	an	an	DET
ejpam-4386	164	13	open	open	ADJ
ejpam-4386	164	14	set	set	NOUN
ejpam-4386	164	15	in	in	ADP
ejpam-4386	164	16	xm	xm	PROPN
ejpam-4386	164	17	,	,	PUNCT
ejpam-4386	164	18	but	but	CCONJ
ejpam-4386	164	19	intxm	intxm	NOUN
ejpam-4386	164	20	(	(	PUNCT
ejpam-4386	164	21	{	{	PUNCT
ejpam-4386	164	22	0}xm	0}xm	NOUN
ejpam-4386	164	23	)	)	PUNCT
ejpam-4386	164	24	=	=	NOUN
ejpam-4386	165	1	intxm	intxm	NOUN
ejpam-4386	165	2	(	(	PUNCT
ejpam-4386	165	3	r)=	r)=	NOUN
ejpam-4386	165	4	r	r	PROPN
ejpam-4386	165	5	̸=	̸=	PROPN
ejpam-4386	165	6	{	{	PUNCT
ejpam-4386	165	7	0	0	NUM
ejpam-4386	165	8	}	}	PUNCT
ejpam-4386	165	9	.	.	PUNCT
ejpam-4386	166	1	thus	thus	ADV
ejpam-4386	166	2	,	,	PUNCT
ejpam-4386	166	3	{	{	PUNCT
ejpam-4386	166	4	0	0	X
ejpam-4386	166	5	}	}	PUNCT
ejpam-4386	166	6	is	be	AUX
ejpam-4386	166	7	not	not	PART
ejpam-4386	166	8	an	an	DET
ejpam-4386	166	9	open	open	ADJ
ejpam-4386	166	10	domain	domain	NOUN
ejpam-4386	166	11	in	in	ADP
ejpam-4386	166	12	xm	xm	PROPN
ejpam-4386	166	13	.	.	PUNCT
ejpam-4386	167	1	hence	hence	ADV
ejpam-4386	167	2	,	,	PUNCT
ejpam-4386	167	3	0	0	NUM
ejpam-4386	167	4	∈	∈	PROPN
ejpam-4386	167	5	{	{	PUNCT
ejpam-4386	167	6	0	0	NUM
ejpam-4386	167	7	}	}	PUNCT
ejpam-4386	167	8	with	with	ADP
ejpam-4386	167	9	{	{	PUNCT
ejpam-4386	167	10	0	0	NUM
ejpam-4386	167	11	}	}	PUNCT
ejpam-4386	167	12	is	be	AUX
ejpam-4386	167	13	an	an	DET
ejpam-4386	167	14	open	open	ADJ
ejpam-4386	167	15	set	set	NOUN
ejpam-4386	167	16	and	and	CCONJ
ejpam-4386	167	17	there	there	PRON
ejpam-4386	167	18	is	be	VERB
ejpam-4386	167	19	no	no	DET
ejpam-4386	167	20	open	open	ADJ
ejpam-4386	167	21	domain	domain	NOUN
ejpam-4386	167	22	g	g	NOUN
ejpam-4386	167	23	in	in	ADP
ejpam-4386	167	24	xm	xm	PROPN
ejpam-4386	167	25	satisfies	satisfie	NOUN
ejpam-4386	167	26	0	0	NUM
ejpam-4386	167	27	∈	∈	PROPN
ejpam-4386	167	28	g	g	ADP
ejpam-4386	167	29	⊆	⊆	NUM
ejpam-4386	167	30	{	{	PUNCT
ejpam-4386	167	31	0	0	NUM
ejpam-4386	167	32	}	}	PUNCT
ejpam-4386	167	33	.	.	PUNCT
ejpam-4386	168	1	therefore	therefore	ADV
ejpam-4386	168	2	,	,	PUNCT
ejpam-4386	168	3	xm	xm	PROPN
ejpam-4386	168	4	is	be	AUX
ejpam-4386	168	5	not	not	PART
ejpam-4386	168	6	semi	semi	ADJ
ejpam-4386	168	7	-	-	ADJ
ejpam-4386	168	8	regular	regular	ADJ
ejpam-4386	168	9	.	.	PUNCT
ejpam-4386	169	1	lemma	lemma	PROPN
ejpam-4386	169	2	4	4	X
ejpam-4386	169	3	.	.	PUNCT
ejpam-4386	170	1	let	let	AUX
ejpam-4386	170	2	(	(	PUNCT
ejpam-4386	170	3	x	x	X
ejpam-4386	170	4	,	,	PUNCT
ejpam-4386	170	5	τ	τ	PROPN
ejpam-4386	170	6	)	)	PUNCT
ejpam-4386	170	7	be	be	AUX
ejpam-4386	170	8	a	a	DET
ejpam-4386	170	9	topological	topological	ADJ
ejpam-4386	170	10	space	space	NOUN
ejpam-4386	170	11	.	.	PUNCT
ejpam-4386	171	1	let	let	VERB
ejpam-4386	171	2	m	m	PRON
ejpam-4386	171	3	be	be	AUX
ejpam-4386	171	4	any	any	DET
ejpam-4386	171	5	non	non	ADJ
ejpam-4386	171	6	-	-	ADJ
ejpam-4386	171	7	empty	empty	ADJ
ejpam-4386	171	8	proper	proper	ADJ
ejpam-4386	171	9	subset	subset	NOUN
ejpam-4386	171	10	of	of	ADP
ejpam-4386	171	11	x.	x.	NOUN
ejpam-4386	171	12	then	then	ADV
ejpam-4386	171	13	,	,	PUNCT
ejpam-4386	171	14	for	for	ADP
ejpam-4386	171	15	any	any	DET
ejpam-4386	171	16	open	open	ADJ
ejpam-4386	171	17	domain	domain	NOUN
ejpam-4386	171	18	u	u	NOUN
ejpam-4386	171	19	in	in	ADP
ejpam-4386	171	20	x	x	PROPN
ejpam-4386	171	21	,	,	PUNCT
ejpam-4386	171	22	u	u	NOUN
ejpam-4386	171	23	is	be	AUX
ejpam-4386	171	24	an	an	DET
ejpam-4386	171	25	open	open	ADJ
ejpam-4386	171	26	domain	domain	NOUN
ejpam-4386	171	27	in	in	ADP
ejpam-4386	171	28	xm	xm	PROPN
ejpam-4386	171	29	.	.	PUNCT
ejpam-4386	172	1	proof	proof	NOUN
ejpam-4386	172	2	.	.	PUNCT
ejpam-4386	173	1	let	let	VERB
ejpam-4386	173	2	u	u	PRON
ejpam-4386	173	3	be	be	AUX
ejpam-4386	173	4	any	any	DET
ejpam-4386	173	5	open	open	ADJ
ejpam-4386	173	6	domain	domain	NOUN
ejpam-4386	173	7	in	in	ADP
ejpam-4386	173	8	x	x	NOUN
ejpam-4386	173	9	,	,	PUNCT
ejpam-4386	173	10	we	we	PRON
ejpam-4386	173	11	always	always	ADV
ejpam-4386	173	12	have	have	VERB
ejpam-4386	173	13	u	u	NOUN
ejpam-4386	173	14	⊆	⊆	NUM
ejpam-4386	173	15	u	u	NOUN
ejpam-4386	173	16	xm	xm	PROPN
ejpam-4386	173	17	.	.	PUNCT
ejpam-4386	174	1	by	by	ADP
ejpam-4386	174	2	taking	take	VERB
ejpam-4386	174	3	the	the	DET
ejpam-4386	174	4	interior	interior	NOUN
ejpam-4386	174	5	of	of	ADP
ejpam-4386	174	6	both	both	DET
ejpam-4386	174	7	sides	side	NOUN
ejpam-4386	174	8	with	with	ADP
ejpam-4386	174	9	respect	respect	NOUN
ejpam-4386	174	10	to	to	ADP
ejpam-4386	174	11	xm	xm	PROPN
ejpam-4386	174	12	we	we	PRON
ejpam-4386	174	13	get	get	VERB
ejpam-4386	174	14	,	,	PUNCT
ejpam-4386	174	15	intxm	intxm	NOUN
ejpam-4386	174	16	(	(	PUNCT
ejpam-4386	174	17	u	u	NOUN
ejpam-4386	174	18	)	)	PUNCT
ejpam-4386	174	19	⊆	⊆	NUM
ejpam-4386	174	20	intxm	intxm	NOUN
ejpam-4386	174	21	(	(	PUNCT
ejpam-4386	174	22	u	u	NOUN
ejpam-4386	174	23	xm	xm	PROPN
ejpam-4386	174	24	)	)	PUNCT
ejpam-4386	174	25	.	.	PUNCT
ejpam-4386	175	1	but	but	CCONJ
ejpam-4386	175	2	since	since	SCONJ
ejpam-4386	175	3	u	u	NOUN
ejpam-4386	175	4	is	be	AUX
ejpam-4386	175	5	an	an	DET
ejpam-4386	175	6	open	open	ADJ
ejpam-4386	175	7	domain	domain	NOUN
ejpam-4386	175	8	in	in	ADP
ejpam-4386	175	9	x	x	NOUN
ejpam-4386	175	10	,	,	PUNCT
ejpam-4386	175	11	then	then	ADV
ejpam-4386	175	12	u	u	NOUN
ejpam-4386	175	13	is	be	AUX
ejpam-4386	175	14	an	an	DET
ejpam-4386	175	15	open	open	ADJ
ejpam-4386	175	16	set	set	NOUN
ejpam-4386	175	17	in	in	ADP
ejpam-4386	175	18	x.	x.	NOUN
ejpam-4386	175	19	thus	thus	ADV
ejpam-4386	175	20	,	,	PUNCT
ejpam-4386	175	21	u	u	NOUN
ejpam-4386	175	22	is	be	AUX
ejpam-4386	175	23	an	an	DET
ejpam-4386	175	24	open	open	ADJ
ejpam-4386	175	25	set	set	NOUN
ejpam-4386	175	26	in	in	ADP
ejpam-4386	175	27	xm	xm	PROPN
ejpam-4386	175	28	.	.	PUNCT
ejpam-4386	176	1	hence	hence	ADV
ejpam-4386	176	2	,	,	PUNCT
ejpam-4386	176	3	intxm	intxm	NOUN
ejpam-4386	176	4	(	(	PUNCT
ejpam-4386	176	5	u	u	NOUN
ejpam-4386	176	6	)	)	PUNCT
ejpam-4386	176	7	=	=	SYM
ejpam-4386	176	8	u	u	NOUN
ejpam-4386	176	9	,	,	PUNCT
ejpam-4386	176	10	therefore	therefore	ADV
ejpam-4386	176	11	u	u	PROPN
ejpam-4386	176	12	⊆	⊆	NUM
ejpam-4386	176	13	intxm	intxm	NOUN
ejpam-4386	176	14	(	(	PUNCT
ejpam-4386	176	15	u	u	NOUN
ejpam-4386	176	16	xm	xm	PROPN
ejpam-4386	176	17	)	)	PUNCT
ejpam-4386	176	18	...	...	PUNCT
ejpam-4386	177	1	⋆.	⋆.	PUNCT
ejpam-4386	177	2	now	now	ADV
ejpam-4386	177	3	,	,	PUNCT
ejpam-4386	177	4	let	let	VERB
ejpam-4386	177	5	x	x	X
ejpam-4386	177	6	∈	∈	PROPN
ejpam-4386	177	7	intxm	intxm	NOUN
ejpam-4386	177	8	(	(	PUNCT
ejpam-4386	177	9	u	u	NOUN
ejpam-4386	177	10	xm	xm	PROPN
ejpam-4386	177	11	)	)	PUNCT
ejpam-4386	177	12	be	be	AUX
ejpam-4386	177	13	arbitrary	arbitrary	ADJ
ejpam-4386	177	14	,	,	PUNCT
ejpam-4386	177	15	then	then	ADV
ejpam-4386	177	16	x	x	SYM
ejpam-4386	177	17	∈	∈	PROPN
ejpam-4386	177	18	(	(	PUNCT
ejpam-4386	177	19	u	u	NOUN
ejpam-4386	177	20	xm	xm	PROPN
ejpam-4386	177	21	)	)	PUNCT
ejpam-4386	177	22	.	.	PUNCT
ejpam-4386	178	1	there	there	PRON
ejpam-4386	178	2	are	be	VERB
ejpam-4386	178	3	only	only	ADV
ejpam-4386	178	4	two	two	NUM
ejpam-4386	178	5	cases	case	NOUN
ejpam-4386	178	6	.	.	PUNCT
ejpam-4386	179	1	case	case	NOUN
ejpam-4386	179	2	1	1	NUM
ejpam-4386	179	3	:	:	PUNCT
ejpam-4386	179	4	x	x	SYM
ejpam-4386	179	5	∈	∈	PROPN
ejpam-4386	179	6	x	x	X
ejpam-4386	179	7	\m	\m	NOUN
ejpam-4386	179	8	.	.	PUNCT
ejpam-4386	180	1	since	since	SCONJ
ejpam-4386	180	2	{	{	PUNCT
ejpam-4386	180	3	x	x	X
ejpam-4386	180	4	}	}	PUNCT
ejpam-4386	180	5	is	be	AUX
ejpam-4386	180	6	an	an	DET
ejpam-4386	180	7	open	open	ADJ
ejpam-4386	180	8	neighborhood	neighborhood	NOUN
ejpam-4386	180	9	of	of	ADP
ejpam-4386	180	10	x	x	PUNCT
ejpam-4386	180	11	in	in	ADP
ejpam-4386	180	12	xm	xm	PROPN
ejpam-4386	180	13	satisfies	satisfie	NOUN
ejpam-4386	180	14	{	{	PUNCT
ejpam-4386	180	15	x	x	NOUN
ejpam-4386	180	16	}	}	PUNCT
ejpam-4386	180	17	∩u	∩u	ADJ
ejpam-4386	180	18	̸=	̸=	PROPN
ejpam-4386	180	19	∅	∅	NOUN
ejpam-4386	180	20	,	,	PUNCT
ejpam-4386	180	21	d.	d.	PROPN
ejpam-4386	180	22	abuzaid	abuzaid	PROPN
ejpam-4386	180	23	,	,	PUNCT
ejpam-4386	180	24	n.	n.	PROPN
ejpam-4386	180	25	alfarsi	alfarsi	NOUN
ejpam-4386	180	26	,	,	PUNCT
ejpam-4386	180	27	l.	l.	PROPN
ejpam-4386	180	28	kalantan	kalantan	PROPN
ejpam-4386	180	29	/	/	SYM
ejpam-4386	180	30	eur	eur	PROPN
ejpam-4386	180	31	.	.	PUNCT
ejpam-4386	181	1	j.	j.	PROPN
ejpam-4386	181	2	pure	pure	PROPN
ejpam-4386	181	3	appl	appl	PROPN
ejpam-4386	181	4	.	.	PROPN
ejpam-4386	181	5	math	math	PROPN
ejpam-4386	181	6	,	,	PUNCT
ejpam-4386	181	7	15	15	NUM
ejpam-4386	181	8	(	(	PUNCT
ejpam-4386	181	9	3	3	NUM
ejpam-4386	181	10	)	)	PUNCT
ejpam-4386	181	11	(	(	PUNCT
ejpam-4386	181	12	2022	2022	NUM
ejpam-4386	181	13	)	)	PUNCT
ejpam-4386	181	14	,	,	PUNCT
ejpam-4386	181	15	821	821	NUM
ejpam-4386	181	16	-	-	SYM
ejpam-4386	181	17	829	829	NUM
ejpam-4386	181	18	826	826	NUM
ejpam-4386	181	19	then	then	ADV
ejpam-4386	181	20	x	x	SYM
ejpam-4386	181	21	∈	∈	PROPN
ejpam-4386	181	22	u	u	NOUN
ejpam-4386	181	23	.	.	PUNCT
ejpam-4386	182	1	case	case	NOUN
ejpam-4386	182	2	2	2	NUM
ejpam-4386	182	3	:	:	PUNCT
ejpam-4386	182	4	x	x	SYM
ejpam-4386	182	5	∈	∈	NOUN
ejpam-4386	182	6	m	m	VERB
ejpam-4386	182	7	.	.	PUNCT
ejpam-4386	183	1	since	since	SCONJ
ejpam-4386	183	2	x	x	PROPN
ejpam-4386	183	3	∈	∈	PROPN
ejpam-4386	183	4	intxm	intxm	NOUN
ejpam-4386	183	5	(	(	PUNCT
ejpam-4386	183	6	u	u	NOUN
ejpam-4386	183	7	xm	xm	PROPN
ejpam-4386	183	8	)	)	PUNCT
ejpam-4386	183	9	,	,	PUNCT
ejpam-4386	183	10	then	then	ADV
ejpam-4386	183	11	there	there	PRON
ejpam-4386	183	12	exist	exist	VERB
ejpam-4386	183	13	an	an	DET
ejpam-4386	183	14	open	open	ADJ
ejpam-4386	183	15	set	set	VERB
ejpam-4386	183	16	v	v	NOUN
ejpam-4386	183	17	in	in	ADP
ejpam-4386	183	18	x	x	PUNCT
ejpam-4386	183	19	such	such	ADJ
ejpam-4386	183	20	that	that	SCONJ
ejpam-4386	183	21	x	x	SYM
ejpam-4386	183	22	∈	∈	NOUN
ejpam-4386	183	23	v	v	ADP
ejpam-4386	183	24	⊆	⊆	NUM
ejpam-4386	183	25	u	u	NOUN
ejpam-4386	183	26	xm	xm	PROPN
ejpam-4386	183	27	⊆	⊆	NUM
ejpam-4386	183	28	u	u	NOUN
ejpam-4386	183	29	x	x	X
ejpam-4386	183	30	and	and	CCONJ
ejpam-4386	183	31	the	the	DET
ejpam-4386	183	32	last	last	ADJ
ejpam-4386	183	33	inclusion	inclusion	NOUN
ejpam-4386	183	34	is	be	AUX
ejpam-4386	183	35	true	true	ADJ
ejpam-4386	183	36	because	because	SCONJ
ejpam-4386	183	37	the	the	DET
ejpam-4386	183	38	topology	topology	NOUN
ejpam-4386	183	39	on	on	ADP
ejpam-4386	183	40	x	x	SYM
ejpam-4386	183	41	is	be	AUX
ejpam-4386	183	42	coarser	coarse	ADJ
ejpam-4386	183	43	than	than	ADP
ejpam-4386	183	44	the	the	DET
ejpam-4386	183	45	topology	topology	NOUN
ejpam-4386	183	46	on	on	ADP
ejpam-4386	183	47	xm	xm	PROPN
ejpam-4386	183	48	.	.	PUNCT
ejpam-4386	184	1	therefore	therefore	ADV
ejpam-4386	184	2	,	,	PUNCT
ejpam-4386	184	3	we	we	PRON
ejpam-4386	184	4	have	have	VERB
ejpam-4386	184	5	x	x	X
ejpam-4386	184	6	∈	∈	NOUN
ejpam-4386	184	7	v	v	ADP
ejpam-4386	184	8	⊆	⊆	NUM
ejpam-4386	184	9	u	u	NOUN
ejpam-4386	184	10	x	x	X
ejpam-4386	184	11	,	,	PUNCT
ejpam-4386	184	12	then	then	ADV
ejpam-4386	184	13	by	by	ADP
ejpam-4386	184	14	taking	take	VERB
ejpam-4386	184	15	the	the	DET
ejpam-4386	184	16	interior	interior	NOUN
ejpam-4386	184	17	of	of	ADP
ejpam-4386	184	18	both	both	DET
ejpam-4386	184	19	sides	side	NOUN
ejpam-4386	184	20	with	with	ADP
ejpam-4386	184	21	respect	respect	NOUN
ejpam-4386	184	22	to	to	ADP
ejpam-4386	184	23	x	x	NOUN
ejpam-4386	184	24	we	we	PRON
ejpam-4386	184	25	have	have	VERB
ejpam-4386	184	26	,	,	PUNCT
ejpam-4386	184	27	x	x	SYM
ejpam-4386	184	28	∈	∈	NOUN
ejpam-4386	184	29	intxv	intxv	NOUN
ejpam-4386	184	30	=	=	SYM
ejpam-4386	184	31	v	v	ADP
ejpam-4386	184	32	⊆	⊆	NUM
ejpam-4386	184	33	intx(u	intx(u	NOUN
ejpam-4386	184	34	x	x	PUNCT
ejpam-4386	184	35	)	)	PUNCT
ejpam-4386	185	1	=	=	SYM
ejpam-4386	185	2	u	u	NOUN
ejpam-4386	185	3	because	because	SCONJ
ejpam-4386	185	4	u	u	NOUN
ejpam-4386	185	5	is	be	AUX
ejpam-4386	185	6	an	an	DET
ejpam-4386	185	7	open	open	ADJ
ejpam-4386	185	8	domain	domain	NOUN
ejpam-4386	185	9	in	in	ADP
ejpam-4386	185	10	x	x	PUNCT
ejpam-4386	185	11	and	and	CCONJ
ejpam-4386	185	12	v	v	NOUN
ejpam-4386	185	13	is	be	AUX
ejpam-4386	185	14	an	an	DET
ejpam-4386	185	15	open	open	ADJ
ejpam-4386	185	16	set	set	NOUN
ejpam-4386	185	17	in	in	ADP
ejpam-4386	185	18	x.	x.	NOUN
ejpam-4386	185	19	hence	hence	ADV
ejpam-4386	185	20	,	,	PUNCT
ejpam-4386	185	21	x	x	PUNCT
ejpam-4386	185	22	∈	∈	PROPN
ejpam-4386	185	23	u	u	NOUN
ejpam-4386	185	24	,	,	PUNCT
ejpam-4386	185	25	thus	thus	ADV
ejpam-4386	185	26	intxm	intxm	ADJ
ejpam-4386	185	27	(	(	PUNCT
ejpam-4386	185	28	u	u	NOUN
ejpam-4386	185	29	xm	xm	PROPN
ejpam-4386	185	30	)	)	PUNCT
ejpam-4386	186	1	⊆	⊆	NUM
ejpam-4386	186	2	u	u	NOUN
ejpam-4386	186	3	...	...	PUNCT
ejpam-4386	186	4	⋆⋆.	⋆⋆.	NOUN
ejpam-4386	186	5	by	by	ADP
ejpam-4386	186	6	⋆	⋆	NOUN
ejpam-4386	186	7	and	and	CCONJ
ejpam-4386	186	8	⋆⋆	⋆⋆	PROPN
ejpam-4386	186	9	we	we	PRON
ejpam-4386	186	10	get	get	VERB
ejpam-4386	186	11	u	u	NOUN
ejpam-4386	186	12	=	=	NOUN
ejpam-4386	186	13	intxm	intxm	NOUN
ejpam-4386	186	14	(	(	PUNCT
ejpam-4386	186	15	u	u	NOUN
ejpam-4386	186	16	xm	xm	PROPN
ejpam-4386	186	17	)	)	PUNCT
ejpam-4386	186	18	.	.	PUNCT
ejpam-4386	187	1	therefore	therefore	ADV
ejpam-4386	187	2	,	,	PUNCT
ejpam-4386	187	3	u	u	NOUN
ejpam-4386	187	4	is	be	AUX
ejpam-4386	187	5	an	an	DET
ejpam-4386	187	6	open	open	ADJ
ejpam-4386	187	7	domain	domain	NOUN
ejpam-4386	187	8	in	in	ADP
ejpam-4386	187	9	xm	xm	PROPN
ejpam-4386	187	10	.	.	PUNCT
ejpam-4386	188	1	in	in	ADP
ejpam-4386	188	2	the	the	DET
ejpam-4386	188	3	next	next	ADJ
ejpam-4386	188	4	theorem	theorem	NOUN
ejpam-4386	188	5	,	,	PUNCT
ejpam-4386	188	6	we	we	PRON
ejpam-4386	188	7	will	will	AUX
ejpam-4386	188	8	use	use	VERB
ejpam-4386	188	9	the	the	DET
ejpam-4386	188	10	following	follow	VERB
ejpam-4386	188	11	fact	fact	NOUN
ejpam-4386	188	12	which	which	PRON
ejpam-4386	188	13	was	be	AUX
ejpam-4386	188	14	proved	prove	VERB
ejpam-4386	188	15	in	in	ADP
ejpam-4386	188	16	[	[	X
ejpam-4386	188	17	1	1	NUM
ejpam-4386	188	18	]	]	PUNCT
ejpam-4386	188	19	:	:	PUNCT
ejpam-4386	188	20	“	"	PUNCT
ejpam-4386	188	21	if	if	SCONJ
ejpam-4386	188	22	x	x	PRON
ejpam-4386	188	23	is	be	AUX
ejpam-4386	188	24	t1	t1	NOUN
ejpam-4386	188	25	,	,	PUNCT
ejpam-4386	188	26	then	then	ADV
ejpam-4386	188	27	so	so	ADV
ejpam-4386	188	28	is	be	AUX
ejpam-4386	188	29	xm	xm	PROPN
ejpam-4386	188	30	for	for	ADP
ejpam-4386	188	31	any	any	DET
ejpam-4386	188	32	non	non	ADJ
ejpam-4386	188	33	-	-	ADJ
ejpam-4386	188	34	empty	empty	ADJ
ejpam-4386	188	35	proper	proper	ADJ
ejpam-4386	188	36	subset	subset	NOUN
ejpam-4386	188	37	m	m	NOUN
ejpam-4386	188	38	of	of	ADP
ejpam-4386	188	39	x	x	NOUN
ejpam-4386	188	40	”	"	PUNCT
ejpam-4386	188	41	.	.	PUNCT
ejpam-4386	189	1	theorem	theorem	NOUN
ejpam-4386	189	2	3	3	NUM
ejpam-4386	189	3	.	.	PUNCT
ejpam-4386	190	1	if	if	SCONJ
ejpam-4386	190	2	x	x	PRON
ejpam-4386	190	3	is	be	AUX
ejpam-4386	190	4	t1	t1	NOUN
ejpam-4386	190	5	and	and	CCONJ
ejpam-4386	190	6	semi	semi	ADJ
ejpam-4386	190	7	-	-	ADJ
ejpam-4386	190	8	regular	regular	ADJ
ejpam-4386	190	9	,	,	PUNCT
ejpam-4386	190	10	then	then	ADV
ejpam-4386	190	11	for	for	ADP
ejpam-4386	190	12	any	any	DET
ejpam-4386	190	13	non	non	ADJ
ejpam-4386	190	14	-	-	ADJ
ejpam-4386	190	15	empty	empty	ADJ
ejpam-4386	190	16	proper	proper	ADJ
ejpam-4386	190	17	subset	subset	NOUN
ejpam-4386	190	18	m	m	NOUN
ejpam-4386	190	19	of	of	ADP
ejpam-4386	190	20	x	x	PRON
ejpam-4386	190	21	,	,	PUNCT
ejpam-4386	190	22	we	we	PRON
ejpam-4386	190	23	have	have	VERB
ejpam-4386	190	24	that	that	SCONJ
ejpam-4386	190	25	the	the	DET
ejpam-4386	190	26	discrete	discrete	ADJ
ejpam-4386	190	27	extension	extension	NOUN
ejpam-4386	190	28	xm	xm	PROPN
ejpam-4386	190	29	of	of	ADP
ejpam-4386	190	30	x	x	PROPN
ejpam-4386	190	31	is	be	AUX
ejpam-4386	190	32	semi	semi	ADJ
ejpam-4386	190	33	-	-	ADJ
ejpam-4386	190	34	regular	regular	ADJ
ejpam-4386	190	35	.	.	PUNCT
ejpam-4386	191	1	proof	proof	NOUN
ejpam-4386	191	2	.	.	PUNCT
ejpam-4386	192	1	assume	assume	VERB
ejpam-4386	192	2	the	the	DET
ejpam-4386	192	3	hypotheses	hypothesis	NOUN
ejpam-4386	192	4	.	.	PUNCT
ejpam-4386	193	1	let	let	VERB
ejpam-4386	193	2	w	w	NOUN
ejpam-4386	193	3	be	be	AUX
ejpam-4386	193	4	an	an	DET
ejpam-4386	193	5	arbitrary	arbitrary	ADJ
ejpam-4386	193	6	non	non	ADJ
ejpam-4386	193	7	-	-	ADJ
ejpam-4386	193	8	empty	empty	ADJ
ejpam-4386	193	9	open	open	ADJ
ejpam-4386	193	10	set	set	NOUN
ejpam-4386	193	11	in	in	ADP
ejpam-4386	193	12	xm	xm	PROPN
ejpam-4386	193	13	.	.	PUNCT
ejpam-4386	194	1	let	let	VERB
ejpam-4386	194	2	x	x	SYM
ejpam-4386	194	3	∈	∈	PROPN
ejpam-4386	194	4	w	w	AUX
ejpam-4386	194	5	be	be	AUX
ejpam-4386	194	6	arbitrary	arbitrary	ADJ
ejpam-4386	194	7	.	.	PUNCT
ejpam-4386	195	1	there	there	PRON
ejpam-4386	195	2	are	be	VERB
ejpam-4386	195	3	only	only	ADV
ejpam-4386	195	4	two	two	NUM
ejpam-4386	195	5	cases	case	NOUN
ejpam-4386	195	6	.	.	PUNCT
ejpam-4386	196	1	case	case	NOUN
ejpam-4386	196	2	1	1	NUM
ejpam-4386	196	3	:	:	PUNCT
ejpam-4386	196	4	x	x	SYM
ejpam-4386	196	5	∈	∈	PROPN
ejpam-4386	196	6	x	x	X
ejpam-4386	196	7	\m	\m	NOUN
ejpam-4386	196	8	.	.	PUNCT
ejpam-4386	197	1	then	then	ADV
ejpam-4386	197	2	we	we	PRON
ejpam-4386	197	3	have	have	AUX
ejpam-4386	197	4	{	{	PUNCT
ejpam-4386	197	5	x	x	NOUN
ejpam-4386	197	6	}	}	PUNCT
ejpam-4386	197	7	is	be	AUX
ejpam-4386	197	8	an	an	DET
ejpam-4386	197	9	open	open	ADJ
ejpam-4386	197	10	neighborhood	neighborhood	NOUN
ejpam-4386	197	11	of	of	ADP
ejpam-4386	197	12	x	x	PUNCT
ejpam-4386	197	13	in	in	ADP
ejpam-4386	197	14	xm	xm	PROPN
ejpam-4386	197	15	.	.	PUNCT
ejpam-4386	198	1	since	since	SCONJ
ejpam-4386	198	2	x	x	PROPN
ejpam-4386	198	3	is	be	AUX
ejpam-4386	198	4	t1	t1	NOUN
ejpam-4386	198	5	,	,	PUNCT
ejpam-4386	198	6	then	then	ADV
ejpam-4386	198	7	xm	xm	PROPN
ejpam-4386	198	8	is	be	AUX
ejpam-4386	198	9	also	also	ADV
ejpam-4386	198	10	t1	t1	NOUN
ejpam-4386	198	11	.	.	PUNCT
ejpam-4386	199	1	thus	thus	ADV
ejpam-4386	199	2	{	{	PUNCT
ejpam-4386	199	3	x	x	X
ejpam-4386	199	4	}	}	PUNCT
ejpam-4386	199	5	is	be	AUX
ejpam-4386	199	6	also	also	ADV
ejpam-4386	199	7	closed	close	VERB
ejpam-4386	199	8	in	in	ADP
ejpam-4386	199	9	xm	xm	PROPN
ejpam-4386	199	10	.	.	PUNCT
ejpam-4386	200	1	hence	hence	ADV
ejpam-4386	200	2	{	{	PUNCT
ejpam-4386	200	3	x	x	X
ejpam-4386	200	4	}	}	PUNCT
ejpam-4386	200	5	is	be	AUX
ejpam-4386	200	6	clopen	clopen	ADJ
ejpam-4386	200	7	in	in	ADP
ejpam-4386	200	8	xm	xm	PROPN
ejpam-4386	200	9	,	,	PUNCT
ejpam-4386	200	10	thus	thus	ADV
ejpam-4386	200	11	{	{	PUNCT
ejpam-4386	200	12	x	x	X
ejpam-4386	200	13	}	}	PUNCT
ejpam-4386	200	14	is	be	AUX
ejpam-4386	200	15	an	an	DET
ejpam-4386	200	16	open	open	ADJ
ejpam-4386	200	17	domain	domain	NOUN
ejpam-4386	200	18	in	in	ADP
ejpam-4386	200	19	xm	xm	PROPN
ejpam-4386	201	1	such	such	ADJ
ejpam-4386	201	2	that	that	SCONJ
ejpam-4386	201	3	x	x	SYM
ejpam-4386	201	4	∈	∈	PROPN
ejpam-4386	201	5	{	{	PUNCT
ejpam-4386	201	6	x	x	NOUN
ejpam-4386	201	7	}	}	PUNCT
ejpam-4386	201	8	⊆	⊆	NUM
ejpam-4386	201	9	w	w	NOUN
ejpam-4386	201	10	.	.	PUNCT
ejpam-4386	201	11	case	case	NOUN
ejpam-4386	201	12	2	2	NUM
ejpam-4386	201	13	:	:	PUNCT
ejpam-4386	201	14	x	x	SYM
ejpam-4386	201	15	∈	∈	NOUN
ejpam-4386	201	16	m	m	VERB
ejpam-4386	201	17	.	.	PUNCT
ejpam-4386	202	1	since	since	SCONJ
ejpam-4386	202	2	x	x	PRON
ejpam-4386	202	3	is	be	AUX
ejpam-4386	202	4	semi	semi	ADJ
ejpam-4386	202	5	-	-	ADJ
ejpam-4386	202	6	regular	regular	ADJ
ejpam-4386	202	7	,	,	PUNCT
ejpam-4386	202	8	then	then	ADV
ejpam-4386	202	9	there	there	PRON
ejpam-4386	202	10	is	be	VERB
ejpam-4386	202	11	a	a	DET
ejpam-4386	202	12	base	base	NOUN
ejpam-4386	202	13	for	for	ADP
ejpam-4386	202	14	x	x	SYM
ejpam-4386	202	15	consisting	consist	VERB
ejpam-4386	202	16	of	of	ADP
ejpam-4386	202	17	open	open	ADJ
ejpam-4386	202	18	domains	domain	NOUN
ejpam-4386	202	19	.	.	PUNCT
ejpam-4386	203	1	thus	thus	ADV
ejpam-4386	203	2	,	,	PUNCT
ejpam-4386	203	3	there	there	PRON
ejpam-4386	203	4	exists	exist	VERB
ejpam-4386	203	5	an	an	DET
ejpam-4386	203	6	open	open	ADJ
ejpam-4386	203	7	domain	domain	NOUN
ejpam-4386	203	8	v	v	NOUN
ejpam-4386	203	9	in	in	ADP
ejpam-4386	203	10	x	x	PUNCT
ejpam-4386	203	11	such	such	ADJ
ejpam-4386	203	12	that	that	SCONJ
ejpam-4386	203	13	x	x	SYM
ejpam-4386	203	14	∈	∈	NOUN
ejpam-4386	203	15	v	v	ADP
ejpam-4386	203	16	⊆	⊆	NUM
ejpam-4386	203	17	w	w	NOUN
ejpam-4386	203	18	.	.	PUNCT
ejpam-4386	204	1	by	by	ADP
ejpam-4386	204	2	lemma	lemma	PROPN
ejpam-4386	204	3	4	4	NUM
ejpam-4386	204	4	,	,	PUNCT
ejpam-4386	204	5	we	we	PRON
ejpam-4386	204	6	get	get	VERB
ejpam-4386	204	7	v	v	NOUN
ejpam-4386	204	8	is	be	AUX
ejpam-4386	204	9	an	an	DET
ejpam-4386	204	10	open	open	ADJ
ejpam-4386	204	11	domain	domain	NOUN
ejpam-4386	204	12	in	in	ADP
ejpam-4386	204	13	xm	xm	PROPN
ejpam-4386	204	14	.	.	PUNCT
ejpam-4386	205	1	therefore	therefore	ADV
ejpam-4386	205	2	,	,	PUNCT
ejpam-4386	205	3	xm	xm	PROPN
ejpam-4386	205	4	is	be	AUX
ejpam-4386	205	5	semi	semi	ADJ
ejpam-4386	205	6	-	-	ADJ
ejpam-4386	205	7	regular	regular	ADJ
ejpam-4386	205	8	.	.	PUNCT
ejpam-4386	206	1	definition	definition	NOUN
ejpam-4386	206	2	5	5	NUM
ejpam-4386	206	3	.	.	PUNCT
ejpam-4386	207	1	let	let	AUX
ejpam-4386	207	2	(	(	PUNCT
ejpam-4386	207	3	x	x	X
ejpam-4386	207	4	,	,	PUNCT
ejpam-4386	207	5	τ	τ	PROPN
ejpam-4386	207	6	)	)	PUNCT
ejpam-4386	207	7	be	be	AUX
ejpam-4386	207	8	a	a	DET
ejpam-4386	207	9	topological	topological	ADJ
ejpam-4386	207	10	space	space	NOUN
ejpam-4386	207	11	and	and	CCONJ
ejpam-4386	207	12	let	let	VERB
ejpam-4386	207	13	p	p	PRON
ejpam-4386	207	14	be	be	AUX
ejpam-4386	207	15	an	an	DET
ejpam-4386	207	16	object	object	NOUN
ejpam-4386	207	17	not	not	PART
ejpam-4386	207	18	in	in	ADP
ejpam-4386	207	19	x	x	NOUN
ejpam-4386	207	20	,	,	PUNCT
ejpam-4386	207	21	that	that	ADV
ejpam-4386	207	22	is	is	ADV
ejpam-4386	207	23	,	,	PUNCT
ejpam-4386	207	24	p	p	PROPN
ejpam-4386	207	25	̸∈	̸∈	PROPN
ejpam-4386	207	26	x.	x.	PROPN
ejpam-4386	207	27	put	put	VERB
ejpam-4386	207	28	xp	xp	NOUN
ejpam-4386	208	1	=	=	NOUN
ejpam-4386	208	2	x	x	SYM
ejpam-4386	208	3	∪	∪	X
ejpam-4386	208	4	{	{	PUNCT
ejpam-4386	208	5	p	p	NOUN
ejpam-4386	208	6	}	}	PUNCT
ejpam-4386	208	7	.	.	PUNCT
ejpam-4386	209	1	define	define	VERB
ejpam-4386	209	2	a	a	DET
ejpam-4386	209	3	topology	topology	NOUN
ejpam-4386	209	4	τ	τ	NOUN
ejpam-4386	209	5	′	′	NUM
ejpam-4386	209	6	on	on	ADP
ejpam-4386	209	7	xp	xp	INTJ
ejpam-4386	209	8	by	by	ADP
ejpam-4386	209	9	τ	τ	PROPN
ejpam-4386	209	10	′	′	NUM
ejpam-4386	209	11	=	=	PUNCT
ejpam-4386	209	12	{	{	PUNCT
ejpam-4386	209	13	xp	xp	INTJ
ejpam-4386	209	14	}	}	PUNCT
ejpam-4386	209	15	∪	∪	NOUN
ejpam-4386	209	16	{	{	PUNCT
ejpam-4386	209	17	u	u	NOUN
ejpam-4386	209	18	:	:	PUNCT
ejpam-4386	209	19	u	u	PROPN
ejpam-4386	209	20	∈	∈	PROPN
ejpam-4386	209	21	τ	τ	X
ejpam-4386	209	22	}	}	PUNCT
ejpam-4386	209	23	=	=	SYM
ejpam-4386	209	24	{	{	PUNCT
ejpam-4386	209	25	xp	xp	INTJ
ejpam-4386	209	26	}	}	PUNCT
ejpam-4386	209	27	∪	∪	X
ejpam-4386	209	28	τ	τ	X
ejpam-4386	209	29	.	.	PUNCT
ejpam-4386	210	1	the	the	DET
ejpam-4386	210	2	space	space	NOUN
ejpam-4386	210	3	(	(	PUNCT
ejpam-4386	210	4	xp	xp	INTJ
ejpam-4386	210	5	,	,	PUNCT
ejpam-4386	210	6	τ	τ	PROPN
ejpam-4386	210	7	′	′	NUM
ejpam-4386	210	8	)	)	PUNCT
ejpam-4386	210	9	is	be	AUX
ejpam-4386	210	10	called	call	VERB
ejpam-4386	210	11	the	the	DET
ejpam-4386	210	12	open	open	ADJ
ejpam-4386	210	13	extension	extension	NOUN
ejpam-4386	210	14	space	space	NOUN
ejpam-4386	210	15	of	of	ADP
ejpam-4386	210	16	(	(	PUNCT
ejpam-4386	210	17	x	x	INTJ
ejpam-4386	210	18	,	,	PUNCT
ejpam-4386	210	19	τ	τ	PROPN
ejpam-4386	210	20	)	)	PUNCT
ejpam-4386	210	21	,	,	PUNCT
ejpam-4386	210	22	see	see	VERB
ejpam-4386	210	23	[	[	X
ejpam-4386	210	24	12	12	NUM
ejpam-4386	210	25	,	,	PUNCT
ejpam-4386	210	26	example	example	NOUN
ejpam-4386	210	27	16	16	NUM
ejpam-4386	210	28	]	]	PUNCT
ejpam-4386	210	29	.	.	PUNCT
ejpam-4386	211	1	observe	observe	VERB
ejpam-4386	211	2	that	that	SCONJ
ejpam-4386	211	3	(	(	PUNCT
ejpam-4386	211	4	x	x	X
ejpam-4386	211	5	,	,	PUNCT
ejpam-4386	211	6	τ	τ	PROPN
ejpam-4386	211	7	)	)	PUNCT
ejpam-4386	211	8	and	and	CCONJ
ejpam-4386	211	9	(	(	PUNCT
ejpam-4386	211	10	xp	xp	INTJ
ejpam-4386	211	11	,	,	PUNCT
ejpam-4386	211	12	τ	τ	PROPN
ejpam-4386	211	13	′	′	NUM
ejpam-4386	211	14	)	)	PUNCT
ejpam-4386	211	15	have	have	VERB
ejpam-4386	211	16	the	the	DET
ejpam-4386	211	17	same	same	ADJ
ejpam-4386	211	18	open	open	ADJ
ejpam-4386	211	19	sets	set	NOUN
ejpam-4386	211	20	except	except	SCONJ
ejpam-4386	211	21	for	for	ADP
ejpam-4386	211	22	xp	xp	PROPN
ejpam-4386	211	23	.	.	PUNCT
ejpam-4386	212	1	also	also	ADV
ejpam-4386	212	2	,	,	PUNCT
ejpam-4386	212	3	if	if	SCONJ
ejpam-4386	212	4	u	u	NOUN
ejpam-4386	212	5	is	be	AUX
ejpam-4386	212	6	an	an	DET
ejpam-4386	212	7	open	open	ADJ
ejpam-4386	212	8	domain	domain	NOUN
ejpam-4386	212	9	in	in	ADP
ejpam-4386	212	10	(	(	PUNCT
ejpam-4386	212	11	x	x	INTJ
ejpam-4386	212	12	,	,	PUNCT
ejpam-4386	212	13	τ	τ	PROPN
ejpam-4386	212	14	)	)	PUNCT
ejpam-4386	212	15	,	,	PUNCT
ejpam-4386	212	16	then	then	ADV
ejpam-4386	212	17	u	u	NOUN
ejpam-4386	212	18	is	be	AUX
ejpam-4386	212	19	an	an	DET
ejpam-4386	212	20	open	open	ADJ
ejpam-4386	212	21	domain	domain	NOUN
ejpam-4386	212	22	in	in	ADP
ejpam-4386	212	23	(	(	PUNCT
ejpam-4386	212	24	xp	xp	INTJ
ejpam-4386	212	25	,	,	PUNCT
ejpam-4386	212	26	τ	τ	PROPN
ejpam-4386	212	27	′	′	NUM
ejpam-4386	212	28	)	)	PUNCT
ejpam-4386	212	29	because	because	SCONJ
ejpam-4386	212	30	u	u	PRON
ejpam-4386	212	31	xp	xp	NOUN
ejpam-4386	212	32	=	=	SYM
ejpam-4386	212	33	u	u	PROPN
ejpam-4386	212	34	x	x	X
ejpam-4386	212	35	∪	∪	X
ejpam-4386	212	36	{	{	PUNCT
ejpam-4386	212	37	p	p	NOUN
ejpam-4386	212	38	}	}	PUNCT
ejpam-4386	212	39	as	as	ADP
ejpam-4386	212	40	the	the	DET
ejpam-4386	212	41	only	only	ADJ
ejpam-4386	212	42	open	open	ADJ
ejpam-4386	212	43	neighborhood	neighborhood	NOUN
ejpam-4386	212	44	of	of	ADP
ejpam-4386	212	45	p	p	NOUN
ejpam-4386	212	46	in	in	ADP
ejpam-4386	212	47	(	(	PUNCT
ejpam-4386	212	48	xp	xp	INTJ
ejpam-4386	212	49	,	,	PUNCT
ejpam-4386	212	50	τ	τ	PROPN
ejpam-4386	212	51	′	′	NUM
ejpam-4386	212	52	)	)	PUNCT
ejpam-4386	212	53	is	be	AUX
ejpam-4386	212	54	xp	xp	INTJ
ejpam-4386	212	55	itself	itself	PRON
ejpam-4386	212	56	.	.	PUNCT
ejpam-4386	213	1	thus	thus	ADV
ejpam-4386	213	2	,	,	PUNCT
ejpam-4386	213	3	intxp	intxp	PROPN
ejpam-4386	213	4	(	(	PUNCT
ejpam-4386	213	5	u	u	NOUN
ejpam-4386	213	6	xp	xp	ADV
ejpam-4386	213	7	)	)	PUNCT
ejpam-4386	214	1	=	=	PUNCT
ejpam-4386	215	1	intxp(u	intxp(u	ADJ
ejpam-4386	215	2	x	x	SYM
ejpam-4386	215	3	∪	∪	X
ejpam-4386	215	4	{	{	PUNCT
ejpam-4386	215	5	p	p	NOUN
ejpam-4386	215	6	}	}	PUNCT
ejpam-4386	215	7	)	)	PUNCT
ejpam-4386	215	8	=	=	SYM
ejpam-4386	215	9	intx(u	intx(u	NOUN
ejpam-4386	215	10	x	x	PUNCT
ejpam-4386	215	11	)	)	PUNCT
ejpam-4386	215	12	=	=	SYM
ejpam-4386	215	13	u	u	NOUN
ejpam-4386	215	14	.	.	PUNCT
ejpam-4386	216	1	it	it	PRON
ejpam-4386	216	2	is	be	AUX
ejpam-4386	216	3	easy	easy	ADJ
ejpam-4386	216	4	to	to	PART
ejpam-4386	216	5	see	see	VERB
ejpam-4386	216	6	that	that	SCONJ
ejpam-4386	216	7	if	if	SCONJ
ejpam-4386	216	8	u	u	NOUN
ejpam-4386	216	9	is	be	AUX
ejpam-4386	216	10	an	an	DET
ejpam-4386	216	11	open	open	ADJ
ejpam-4386	216	12	domain	domain	NOUN
ejpam-4386	216	13	in	in	ADP
ejpam-4386	216	14	(	(	PUNCT
ejpam-4386	216	15	xp	xp	INTJ
ejpam-4386	216	16	,	,	PUNCT
ejpam-4386	216	17	τ	τ	PROPN
ejpam-4386	216	18	′	′	NOUN
ejpam-4386	216	19	)	)	PUNCT
ejpam-4386	216	20	such	such	ADJ
ejpam-4386	216	21	that	that	SCONJ
ejpam-4386	216	22	p	p	PROPN
ejpam-4386	216	23	̸∈	̸∈	PROPN
ejpam-4386	216	24	u	u	PROPN
ejpam-4386	216	25	,	,	PUNCT
ejpam-4386	216	26	then	then	ADV
ejpam-4386	216	27	u	u	NOUN
ejpam-4386	216	28	is	be	AUX
ejpam-4386	216	29	an	an	DET
ejpam-4386	216	30	open	open	ADJ
ejpam-4386	216	31	domain	domain	NOUN
ejpam-4386	216	32	in	in	ADP
ejpam-4386	216	33	(	(	PUNCT
ejpam-4386	216	34	x	x	INTJ
ejpam-4386	216	35	,	,	PUNCT
ejpam-4386	216	36	τ	τ	PROPN
ejpam-4386	216	37	)	)	PUNCT
ejpam-4386	216	38	.	.	PUNCT
ejpam-4386	217	1	theorem	theorem	ADJ
ejpam-4386	217	2	4	4	NUM
ejpam-4386	217	3	.	.	PUNCT
ejpam-4386	218	1	(	(	PUNCT
ejpam-4386	218	2	x	x	X
ejpam-4386	218	3	,	,	PUNCT
ejpam-4386	218	4	τ	τ	PROPN
ejpam-4386	218	5	)	)	PUNCT
ejpam-4386	218	6	is	be	AUX
ejpam-4386	218	7	semi	semi	ADJ
ejpam-4386	218	8	-	-	ADJ
ejpam-4386	218	9	regular	regular	ADJ
ejpam-4386	218	10	if	if	SCONJ
ejpam-4386	219	1	and	and	CCONJ
ejpam-4386	219	2	only	only	ADV
ejpam-4386	219	3	if	if	SCONJ
ejpam-4386	219	4	(	(	PUNCT
ejpam-4386	219	5	xp	xp	INTJ
ejpam-4386	219	6	,	,	PUNCT
ejpam-4386	219	7	τ	τ	PROPN
ejpam-4386	219	8	′	′	NUM
ejpam-4386	219	9	)	)	PUNCT
ejpam-4386	219	10	is	be	AUX
ejpam-4386	219	11	semi	semi	ADJ
ejpam-4386	219	12	-	-	ADJ
ejpam-4386	219	13	regular	regular	ADJ
ejpam-4386	219	14	.	.	PUNCT
ejpam-4386	220	1	proof	proof	NOUN
ejpam-4386	220	2	.	.	PUNCT
ejpam-4386	221	1	assume	assume	VERB
ejpam-4386	221	2	that	that	SCONJ
ejpam-4386	221	3	(	(	PUNCT
ejpam-4386	221	4	x	x	X
ejpam-4386	221	5	,	,	PUNCT
ejpam-4386	221	6	τ	τ	PROPN
ejpam-4386	221	7	)	)	PUNCT
ejpam-4386	221	8	is	be	AUX
ejpam-4386	221	9	semi	semi	ADJ
ejpam-4386	221	10	-	-	ADJ
ejpam-4386	221	11	regular	regular	ADJ
ejpam-4386	221	12	.	.	PUNCT
ejpam-4386	222	1	to	to	PART
ejpam-4386	222	2	show	show	VERB
ejpam-4386	222	3	that	that	SCONJ
ejpam-4386	222	4	(	(	PUNCT
ejpam-4386	222	5	xp	xp	INTJ
ejpam-4386	222	6	,	,	PUNCT
ejpam-4386	222	7	τ	τ	PROPN
ejpam-4386	222	8	′	′	NUM
ejpam-4386	222	9	)	)	PUNCT
ejpam-4386	222	10	is	be	AUX
ejpam-4386	222	11	semi	semi	ADJ
ejpam-4386	222	12	-	-	ADJ
ejpam-4386	222	13	regular	regular	ADJ
ejpam-4386	222	14	,	,	PUNCT
ejpam-4386	222	15	we	we	PRON
ejpam-4386	222	16	only	only	ADV
ejpam-4386	222	17	need	need	VERB
ejpam-4386	222	18	to	to	PART
ejpam-4386	222	19	prove	prove	VERB
ejpam-4386	222	20	that	that	SCONJ
ejpam-4386	222	21	τ	τ	PROPN
ejpam-4386	222	22	′	′	NUM
ejpam-4386	222	23	⊆	⊆	NUM
ejpam-4386	222	24	τ	τ	X
ejpam-4386	222	25	′	′	NUM
ejpam-4386	222	26	s.	s.	PROPN
ejpam-4386	222	27	let	let	VERB
ejpam-4386	222	28	w	w	PROPN
ejpam-4386	222	29	∈	∈	PROPN
ejpam-4386	222	30	τ	τ	X
ejpam-4386	222	31	′	′	NUM
ejpam-4386	222	32	be	be	AUX
ejpam-4386	222	33	an	an	DET
ejpam-4386	222	34	arbitrary	arbitrary	ADJ
ejpam-4386	222	35	such	such	ADJ
ejpam-4386	222	36	that	that	DET
ejpam-4386	222	37	∅	∅	NOUN
ejpam-4386	222	38	=	=	NOUN
ejpam-4386	222	39	̸	̸	NUM
ejpam-4386	222	40	w	w	NOUN
ejpam-4386	223	1	̸=	̸=	PROPN
ejpam-4386	223	2	xp	xp	ADV
ejpam-4386	223	3	,	,	PUNCT
ejpam-4386	223	4	then	then	ADV
ejpam-4386	223	5	w	w	PROPN
ejpam-4386	223	6	∈	∈	PROPN
ejpam-4386	223	7	τ	τ	X
ejpam-4386	223	8	.	.	PUNCT
ejpam-4386	224	1	since	since	SCONJ
ejpam-4386	224	2	(	(	PUNCT
ejpam-4386	224	3	x	x	X
ejpam-4386	224	4	,	,	PUNCT
ejpam-4386	224	5	τ	τ	PROPN
ejpam-4386	224	6	)	)	PUNCT
ejpam-4386	224	7	is	be	AUX
ejpam-4386	224	8	semi	semi	ADJ
ejpam-4386	224	9	-	-	ADJ
ejpam-4386	224	10	regular	regular	ADJ
ejpam-4386	224	11	,	,	PUNCT
ejpam-4386	224	12	then	then	ADV
ejpam-4386	224	13	τ	τ	PROPN
ejpam-4386	224	14	=	=	PUNCT
ejpam-4386	224	15	τ	τ	PROPN
ejpam-4386	224	16	s.	s.	PROPN
ejpam-4386	225	1	so	so	ADV
ejpam-4386	225	2	,	,	PUNCT
ejpam-4386	225	3	the	the	DET
ejpam-4386	225	4	family	family	NOUN
ejpam-4386	225	5	of	of	ADP
ejpam-4386	225	6	all	all	DET
ejpam-4386	225	7	open	open	ADJ
ejpam-4386	225	8	domains	domain	NOUN
ejpam-4386	225	9	in	in	ADP
ejpam-4386	225	10	(	(	PUNCT
ejpam-4386	225	11	x	x	INTJ
ejpam-4386	225	12	,	,	PUNCT
ejpam-4386	225	13	τ	τ	PROPN
ejpam-4386	225	14	)	)	PUNCT
ejpam-4386	225	15	is	be	AUX
ejpam-4386	225	16	a	a	DET
ejpam-4386	225	17	base	base	NOUN
ejpam-4386	225	18	for	for	ADP
ejpam-4386	225	19	(	(	PUNCT
ejpam-4386	225	20	x	x	INTJ
ejpam-4386	225	21	,	,	PUNCT
ejpam-4386	225	22	τ	τ	PROPN
ejpam-4386	225	23	)	)	PUNCT
ejpam-4386	225	24	.	.	PUNCT
ejpam-4386	226	1	thus	thus	ADV
ejpam-4386	226	2	w	w	X
ejpam-4386	226	3	can	can	AUX
ejpam-4386	226	4	be	be	AUX
ejpam-4386	226	5	written	write	VERB
ejpam-4386	226	6	as	as	ADP
ejpam-4386	226	7	a	a	DET
ejpam-4386	226	8	union	union	NOUN
ejpam-4386	226	9	of	of	ADP
ejpam-4386	226	10	open	open	ADJ
ejpam-4386	226	11	domains	domain	NOUN
ejpam-4386	226	12	in	in	ADP
ejpam-4386	226	13	(	(	PUNCT
ejpam-4386	226	14	x	x	INTJ
ejpam-4386	226	15	,	,	PUNCT
ejpam-4386	226	16	τ	τ	PROPN
ejpam-4386	226	17	)	)	PUNCT
ejpam-4386	226	18	.	.	PUNCT
ejpam-4386	227	1	so	so	ADV
ejpam-4386	227	2	,	,	PUNCT
ejpam-4386	227	3	w	w	PROPN
ejpam-4386	227	4	can	can	AUX
ejpam-4386	227	5	be	be	AUX
ejpam-4386	227	6	written	write	VERB
ejpam-4386	227	7	as	as	ADP
ejpam-4386	227	8	a	a	DET
ejpam-4386	227	9	union	union	NOUN
ejpam-4386	227	10	of	of	ADP
ejpam-4386	227	11	open	open	ADJ
ejpam-4386	227	12	domains	domain	NOUN
ejpam-4386	227	13	in	in	ADP
ejpam-4386	227	14	(	(	PUNCT
ejpam-4386	227	15	xp	xp	INTJ
ejpam-4386	227	16	,	,	PUNCT
ejpam-4386	227	17	τ	τ	PROPN
ejpam-4386	227	18	′	′	NUM
ejpam-4386	227	19	)	)	PUNCT
ejpam-4386	227	20	.	.	PUNCT
ejpam-4386	228	1	thus	thus	ADV
ejpam-4386	228	2	w	w	ADP
ejpam-4386	228	3	∈	∈	PROPN
ejpam-4386	228	4	τ	τ	X
ejpam-4386	228	5	′	′	NUM
ejpam-4386	228	6	s.	s.	PROPN
ejpam-4386	228	7	thus	thus	ADV
ejpam-4386	228	8	τ	τ	PROPN
ejpam-4386	228	9	′	′	NUM
ejpam-4386	228	10	⊆	⊆	NUM
ejpam-4386	228	11	τ	τ	X
ejpam-4386	228	12	′	′	NUM
ejpam-4386	228	13	s.	s.	PROPN
ejpam-4386	228	14	therefore	therefore	ADV
ejpam-4386	228	15	(	(	PUNCT
ejpam-4386	228	16	xp	xp	INTJ
ejpam-4386	228	17	,	,	PUNCT
ejpam-4386	228	18	τ	τ	PROPN
ejpam-4386	228	19	′	′	NUM
ejpam-4386	228	20	)	)	PUNCT
ejpam-4386	228	21	is	be	AUX
ejpam-4386	228	22	semi	semi	ADJ
ejpam-4386	228	23	-	-	ADJ
ejpam-4386	228	24	regular	regular	ADJ
ejpam-4386	228	25	.	.	PUNCT
ejpam-4386	229	1	conversely	conversely	ADV
ejpam-4386	229	2	,	,	PUNCT
ejpam-4386	229	3	assume	assume	VERB
ejpam-4386	229	4	that	that	SCONJ
ejpam-4386	229	5	(	(	PUNCT
ejpam-4386	229	6	xp	xp	INTJ
ejpam-4386	229	7	,	,	PUNCT
ejpam-4386	229	8	τ	τ	PROPN
ejpam-4386	229	9	′	′	NUM
ejpam-4386	229	10	)	)	PUNCT
ejpam-4386	229	11	is	be	AUX
ejpam-4386	229	12	semi	semi	ADJ
ejpam-4386	229	13	-	-	ADJ
ejpam-4386	229	14	regular	regular	ADJ
ejpam-4386	229	15	,	,	PUNCT
ejpam-4386	229	16	that	that	ADV
ejpam-4386	229	17	is	is	ADV
ejpam-4386	229	18	,	,	PUNCT
ejpam-4386	229	19	τ	τ	X
ejpam-4386	229	20	′	′	NUM
ejpam-4386	230	1	=	=	PUNCT
ejpam-4386	230	2	τ	τ	PROPN
ejpam-4386	230	3	′	′	NUM
ejpam-4386	230	4	s.	s.	PROPN
ejpam-4386	230	5	to	to	PART
ejpam-4386	230	6	show	show	VERB
ejpam-4386	230	7	that	that	SCONJ
ejpam-4386	230	8	(	(	PUNCT
ejpam-4386	230	9	x	x	X
ejpam-4386	230	10	,	,	PUNCT
ejpam-4386	230	11	τ	τ	PROPN
ejpam-4386	230	12	)	)	PUNCT
ejpam-4386	230	13	is	be	AUX
ejpam-4386	230	14	semi	semi	ADJ
ejpam-4386	230	15	-	-	ADJ
ejpam-4386	230	16	regular	regular	ADJ
ejpam-4386	230	17	,	,	PUNCT
ejpam-4386	230	18	we	we	PRON
ejpam-4386	230	19	only	only	ADV
ejpam-4386	230	20	need	need	VERB
ejpam-4386	230	21	to	to	PART
ejpam-4386	230	22	show	show	VERB
ejpam-4386	230	23	that	that	SCONJ
ejpam-4386	230	24	τ	τ	PROPN
ejpam-4386	230	25	⊆	⊆	NUM
ejpam-4386	230	26	τ	τ	PROPN
ejpam-4386	230	27	s.	s.	PROPN
ejpam-4386	230	28	let	let	VERB
ejpam-4386	230	29	w	w	PROPN
ejpam-4386	230	30	∈	∈	PROPN
ejpam-4386	230	31	τ	τ	X
ejpam-4386	230	32	be	be	AUX
ejpam-4386	230	33	arbitrary	arbitrary	ADJ
ejpam-4386	230	34	,	,	PUNCT
ejpam-4386	230	35	then	then	ADV
ejpam-4386	230	36	p	p	X
ejpam-4386	230	37	/∈	/∈	PROPN
ejpam-4386	231	1	w	w	INTJ
ejpam-4386	231	2	.	.	PUNCT
ejpam-4386	232	1	but	but	CCONJ
ejpam-4386	232	2	w	w	PROPN
ejpam-4386	232	3	∈	∈	PROPN
ejpam-4386	232	4	τ	τ	X
ejpam-4386	232	5	′	′	NUM
ejpam-4386	232	6	implies	imply	VERB
ejpam-4386	232	7	that	that	SCONJ
ejpam-4386	232	8	w	w	NOUN
ejpam-4386	232	9	can	can	AUX
ejpam-4386	232	10	be	be	AUX
ejpam-4386	232	11	written	write	VERB
ejpam-4386	232	12	as	as	ADP
ejpam-4386	232	13	a	a	DET
ejpam-4386	232	14	union	union	NOUN
ejpam-4386	232	15	of	of	ADP
ejpam-4386	232	16	open	open	ADJ
ejpam-4386	232	17	domains	domain	NOUN
ejpam-4386	232	18	in	in	ADP
ejpam-4386	232	19	d.	d.	PROPN
ejpam-4386	232	20	abuzaid	abuzaid	PROPN
ejpam-4386	232	21	,	,	PUNCT
ejpam-4386	232	22	n.	n.	PROPN
ejpam-4386	232	23	alfarsi	alfarsi	NOUN
ejpam-4386	232	24	,	,	PUNCT
ejpam-4386	232	25	l.	l.	PROPN
ejpam-4386	232	26	kalantan	kalantan	PROPN
ejpam-4386	232	27	/	/	SYM
ejpam-4386	232	28	eur	eur	PROPN
ejpam-4386	232	29	.	.	PUNCT
ejpam-4386	233	1	j.	j.	PROPN
ejpam-4386	233	2	pure	pure	PROPN
ejpam-4386	233	3	appl	appl	PROPN
ejpam-4386	233	4	.	.	PROPN
ejpam-4386	233	5	math	math	PROPN
ejpam-4386	233	6	,	,	PUNCT
ejpam-4386	233	7	15	15	NUM
ejpam-4386	233	8	(	(	PUNCT
ejpam-4386	233	9	3	3	NUM
ejpam-4386	233	10	)	)	PUNCT
ejpam-4386	233	11	(	(	PUNCT
ejpam-4386	233	12	2022	2022	NUM
ejpam-4386	233	13	)	)	PUNCT
ejpam-4386	233	14	,	,	PUNCT
ejpam-4386	233	15	821	821	NUM
ejpam-4386	233	16	-	-	SYM
ejpam-4386	233	17	829	829	NUM
ejpam-4386	233	18	827	827	NUM
ejpam-4386	233	19	(	(	PUNCT
ejpam-4386	233	20	xp	xp	INTJ
ejpam-4386	233	21	,	,	PUNCT
ejpam-4386	233	22	τ	τ	PROPN
ejpam-4386	233	23	′	′	NUM
ejpam-4386	233	24	)	)	PUNCT
ejpam-4386	233	25	.	.	PUNCT
ejpam-4386	234	1	since	since	SCONJ
ejpam-4386	234	2	any	any	DET
ejpam-4386	234	3	open	open	ADJ
ejpam-4386	234	4	domain	domain	NOUN
ejpam-4386	234	5	in	in	ADP
ejpam-4386	234	6	(	(	PUNCT
ejpam-4386	234	7	xp	xp	INTJ
ejpam-4386	234	8	,	,	PUNCT
ejpam-4386	234	9	τ	τ	PROPN
ejpam-4386	234	10	′	′	NUM
ejpam-4386	234	11	)	)	PUNCT
ejpam-4386	234	12	which	which	PRON
ejpam-4386	234	13	does	do	AUX
ejpam-4386	234	14	not	not	PART
ejpam-4386	234	15	contain	contain	VERB
ejpam-4386	234	16	the	the	DET
ejpam-4386	234	17	element	element	NOUN
ejpam-4386	234	18	p	p	NOUN
ejpam-4386	234	19	is	be	AUX
ejpam-4386	234	20	also	also	ADV
ejpam-4386	234	21	an	an	DET
ejpam-4386	234	22	open	open	ADJ
ejpam-4386	234	23	domain	domain	NOUN
ejpam-4386	234	24	in	in	ADP
ejpam-4386	234	25	(	(	PUNCT
ejpam-4386	234	26	x	x	INTJ
ejpam-4386	234	27	,	,	PUNCT
ejpam-4386	234	28	τ	τ	PROPN
ejpam-4386	234	29	)	)	PUNCT
ejpam-4386	234	30	,	,	PUNCT
ejpam-4386	234	31	then	then	ADV
ejpam-4386	234	32	w	w	PROPN
ejpam-4386	234	33	∈	∈	PROPN
ejpam-4386	234	34	τ	τ	X
ejpam-4386	234	35	s	s	PROPN
ejpam-4386	234	36	,	,	PUNCT
ejpam-4386	234	37	implies	imply	VERB
ejpam-4386	234	38	that	that	SCONJ
ejpam-4386	234	39	τ	τ	PROPN
ejpam-4386	234	40	⊆	⊆	NUM
ejpam-4386	234	41	τ	τ	X
ejpam-4386	234	42	s	s	PART
ejpam-4386	234	43	and	and	CCONJ
ejpam-4386	234	44	hence	hence	ADV
ejpam-4386	234	45	(	(	PUNCT
ejpam-4386	234	46	x	x	X
ejpam-4386	234	47	,	,	PUNCT
ejpam-4386	234	48	τ	τ	PROPN
ejpam-4386	234	49	)	)	PUNCT
ejpam-4386	234	50	is	be	AUX
ejpam-4386	234	51	semi	semi	ADJ
ejpam-4386	234	52	-	-	ADJ
ejpam-4386	234	53	regular	regular	ADJ
ejpam-4386	234	54	.	.	PUNCT
ejpam-4386	235	1	3	3	X
ejpam-4386	235	2	.	.	X
ejpam-4386	235	3	new	new	ADJ
ejpam-4386	235	4	results	result	NOUN
ejpam-4386	235	5	about	about	ADP
ejpam-4386	235	6	semi	semi	ADJ
ejpam-4386	235	7	-	-	ADJ
ejpam-4386	235	8	regularization	regularization	ADJ
ejpam-4386	235	9	spaces	space	NOUN
ejpam-4386	235	10	in	in	ADP
ejpam-4386	235	11	this	this	DET
ejpam-4386	235	12	section	section	NOUN
ejpam-4386	235	13	,	,	PUNCT
ejpam-4386	235	14	we	we	PRON
ejpam-4386	235	15	study	study	VERB
ejpam-4386	235	16	the	the	DET
ejpam-4386	235	17	relationship	relationship	NOUN
ejpam-4386	235	18	between	between	ADP
ejpam-4386	235	19	a	a	DET
ejpam-4386	235	20	topological	topological	ADJ
ejpam-4386	235	21	space	space	NOUN
ejpam-4386	235	22	(	(	PUNCT
ejpam-4386	235	23	x	x	X
ejpam-4386	235	24	,	,	PUNCT
ejpam-4386	235	25	τ	τ	PROPN
ejpam-4386	235	26	)	)	PUNCT
ejpam-4386	235	27	and	and	CCONJ
ejpam-4386	235	28	its	its	PRON
ejpam-4386	235	29	semi	semi	ADJ
ejpam-4386	235	30	-	-	ADJ
ejpam-4386	235	31	regularization	regularization	ADJ
ejpam-4386	235	32	space	space	NOUN
ejpam-4386	235	33	(	(	PUNCT
ejpam-4386	235	34	x	x	X
ejpam-4386	235	35	,	,	PUNCT
ejpam-4386	235	36	τ	τ	PROPN
ejpam-4386	235	37	s	s	PART
ejpam-4386	235	38	)	)	PUNCT
ejpam-4386	235	39	regarding	regard	VERB
ejpam-4386	235	40	a	a	DET
ejpam-4386	235	41	topological	topological	ADJ
ejpam-4386	235	42	property	property	NOUN
ejpam-4386	235	43	.	.	PUNCT
ejpam-4386	236	1	we	we	PRON
ejpam-4386	236	2	start	start	VERB
ejpam-4386	236	3	with	with	ADP
ejpam-4386	236	4	the	the	DET
ejpam-4386	236	5	property	property	NOUN
ejpam-4386	236	6	of	of	ADP
ejpam-4386	236	7	scattered	scatter	VERB
ejpam-4386	236	8	.	.	PUNCT
ejpam-4386	237	1	recall	recall	VERB
ejpam-4386	237	2	that	that	SCONJ
ejpam-4386	237	3	a	a	DET
ejpam-4386	237	4	space	space	NOUN
ejpam-4386	237	5	x	x	PUNCT
ejpam-4386	237	6	is	be	AUX
ejpam-4386	237	7	scattered	scatter	VERB
ejpam-4386	237	8	if	if	SCONJ
ejpam-4386	237	9	any	any	DET
ejpam-4386	237	10	non	non	ADJ
ejpam-4386	237	11	-	-	ADJ
ejpam-4386	237	12	empty	empty	ADJ
ejpam-4386	237	13	subset	subset	NOUN
ejpam-4386	237	14	of	of	ADP
ejpam-4386	237	15	x	x	PUNCT
ejpam-4386	237	16	has	have	VERB
ejpam-4386	237	17	an	an	DET
ejpam-4386	237	18	isolated	isolated	ADJ
ejpam-4386	237	19	point	point	NOUN
ejpam-4386	237	20	,	,	PUNCT
ejpam-4386	237	21	that	that	ADV
ejpam-4386	237	22	is	is	ADV
ejpam-4386	237	23	,	,	PUNCT
ejpam-4386	237	24	if	if	SCONJ
ejpam-4386	237	25	∅	∅	NOUN
ejpam-4386	237	26	=	=	NOUN
ejpam-4386	237	27	̸	̸	NUM
ejpam-4386	237	28	a	a	DET
ejpam-4386	237	29	⊆	⊆	NUM
ejpam-4386	237	30	x	x	NOUN
ejpam-4386	237	31	,	,	PUNCT
ejpam-4386	237	32	then	then	ADV
ejpam-4386	237	33	there	there	PRON
ejpam-4386	237	34	exists	exist	VERB
ejpam-4386	237	35	an	an	DET
ejpam-4386	237	36	element	element	NOUN
ejpam-4386	237	37	a	a	DET
ejpam-4386	237	38	∈	∈	PROPN
ejpam-4386	237	39	a	a	PRON
ejpam-4386	238	1	and	and	CCONJ
ejpam-4386	238	2	there	there	PRON
ejpam-4386	238	3	exists	exist	VERB
ejpam-4386	238	4	an	an	DET
ejpam-4386	238	5	open	open	ADJ
ejpam-4386	238	6	set	set	NOUN
ejpam-4386	238	7	u	u	PRON
ejpam-4386	238	8	such	such	ADJ
ejpam-4386	238	9	that	that	SCONJ
ejpam-4386	238	10	a	a	DET
ejpam-4386	238	11	∈	∈	PROPN
ejpam-4386	238	12	u	u	NOUN
ejpam-4386	238	13	and	and	CCONJ
ejpam-4386	238	14	u	u	NOUN
ejpam-4386	238	15	∩	∩	NOUN
ejpam-4386	238	16	a	a	X
ejpam-4386	238	17	=	=	X
ejpam-4386	238	18	{	{	PUNCT
ejpam-4386	238	19	a	a	PRON
ejpam-4386	238	20	}	}	PUNCT
ejpam-4386	238	21	.	.	PUNCT
ejpam-4386	239	1	it	it	PRON
ejpam-4386	239	2	is	be	AUX
ejpam-4386	239	3	easy	easy	ADJ
ejpam-4386	239	4	to	to	PART
ejpam-4386	239	5	see	see	VERB
ejpam-4386	239	6	that	that	SCONJ
ejpam-4386	239	7	if	if	SCONJ
ejpam-4386	239	8	(	(	PUNCT
ejpam-4386	239	9	x	x	X
ejpam-4386	239	10	,	,	PUNCT
ejpam-4386	239	11	τ	τ	PROPN
ejpam-4386	239	12	s	s	PART
ejpam-4386	239	13	)	)	PUNCT
ejpam-4386	239	14	is	be	AUX
ejpam-4386	239	15	scattered	scatter	VERB
ejpam-4386	239	16	,	,	PUNCT
ejpam-4386	239	17	then	then	ADV
ejpam-4386	239	18	so	so	ADV
ejpam-4386	239	19	is	be	AUX
ejpam-4386	239	20	(	(	PUNCT
ejpam-4386	239	21	x	x	INTJ
ejpam-4386	239	22	,	,	PUNCT
ejpam-4386	239	23	τ	τ	PROPN
ejpam-4386	239	24	)	)	PUNCT
ejpam-4386	239	25	.	.	PUNCT
ejpam-4386	240	1	this	this	PRON
ejpam-4386	240	2	follows	follow	VERB
ejpam-4386	240	3	from	from	ADP
ejpam-4386	240	4	the	the	DET
ejpam-4386	240	5	containment	containment	NOUN
ejpam-4386	240	6	τ	τ	PROPN
ejpam-4386	240	7	s	s	PROPN
ejpam-4386	240	8	⊆	⊆	NUM
ejpam-4386	240	9	τ	τ	X
ejpam-4386	240	10	.	.	PUNCT
ejpam-4386	241	1	but	but	CCONJ
ejpam-4386	241	2	the	the	DET
ejpam-4386	241	3	converse	converse	NOUN
ejpam-4386	241	4	is	be	AUX
ejpam-4386	241	5	not	not	PART
ejpam-4386	241	6	always	always	ADV
ejpam-4386	241	7	true	true	ADJ
ejpam-4386	241	8	as	as	SCONJ
ejpam-4386	241	9	can	can	AUX
ejpam-4386	241	10	be	be	AUX
ejpam-4386	241	11	shown	show	VERB
ejpam-4386	241	12	in	in	ADP
ejpam-4386	241	13	the	the	DET
ejpam-4386	241	14	following	follow	VERB
ejpam-4386	241	15	example	example	NOUN
ejpam-4386	241	16	.	.	PUNCT
ejpam-4386	242	1	example	example	NOUN
ejpam-4386	243	1	3	3	X
ejpam-4386	243	2	.	.	X
ejpam-4386	243	3	consider	consider	VERB
ejpam-4386	243	4	r	r	NOUN
ejpam-4386	243	5	with	with	ADP
ejpam-4386	243	6	the	the	DET
ejpam-4386	243	7	particular	particular	ADJ
ejpam-4386	243	8	point	point	NOUN
ejpam-4386	243	9	topology	topology	NOUN
ejpam-4386	243	10	τ	τ	PROPN
ejpam-4386	243	11	0	0	NUM
ejpam-4386	243	12	which	which	PRON
ejpam-4386	243	13	is	be	AUX
ejpam-4386	243	14	scattered	scatter	VERB
ejpam-4386	243	15	,	,	PUNCT
ejpam-4386	243	16	see	see	VERB
ejpam-4386	243	17	[	[	X
ejpam-4386	243	18	12	12	NUM
ejpam-4386	243	19	,	,	PUNCT
ejpam-4386	243	20	example	example	NOUN
ejpam-4386	243	21	10	10	NUM
ejpam-4386	243	22	]	]	PUNCT
ejpam-4386	243	23	.	.	PUNCT
ejpam-4386	244	1	but	but	CCONJ
ejpam-4386	244	2	the	the	DET
ejpam-4386	244	3	semi	semi	ADJ
ejpam-4386	244	4	-	-	NOUN
ejpam-4386	244	5	regularization	regularization	NOUN
ejpam-4386	244	6	of	of	ADP
ejpam-4386	244	7	(	(	PUNCT
ejpam-4386	244	8	r	r	NOUN
ejpam-4386	244	9	,	,	PUNCT
ejpam-4386	244	10	τ	τ	X
ejpam-4386	244	11	0	0	NUM
ejpam-4386	244	12	)	)	PUNCT
ejpam-4386	244	13	is	be	AUX
ejpam-4386	244	14	(	(	PUNCT
ejpam-4386	244	15	r	r	NOUN
ejpam-4386	244	16	,	,	PUNCT
ejpam-4386	244	17	i	i	PROPN
ejpam-4386	244	18	)	)	PUNCT
ejpam-4386	244	19	where	where	SCONJ
ejpam-4386	244	20	i	i	PRON
ejpam-4386	244	21	is	be	AUX
ejpam-4386	244	22	the	the	DET
ejpam-4386	244	23	indiscrete	indiscrete	ADJ
ejpam-4386	244	24	topology	topology	NOUN
ejpam-4386	244	25	which	which	PRON
ejpam-4386	244	26	is	be	AUX
ejpam-4386	244	27	not	not	PART
ejpam-4386	244	28	scattered	scatter	VERB
ejpam-4386	244	29	.	.	PUNCT
ejpam-4386	245	1	definition	definition	NOUN
ejpam-4386	245	2	6	6	NUM
ejpam-4386	245	3	.	.	PUNCT
ejpam-4386	246	1	a	a	DET
ejpam-4386	246	2	topological	topological	ADJ
ejpam-4386	246	3	space	space	NOUN
ejpam-4386	246	4	(	(	PUNCT
ejpam-4386	246	5	x	x	X
ejpam-4386	246	6	,	,	PUNCT
ejpam-4386	246	7	τ	τ	PROPN
ejpam-4386	246	8	)	)	PUNCT
ejpam-4386	246	9	is	be	AUX
ejpam-4386	246	10	called	call	VERB
ejpam-4386	246	11	epi	epi	NOUN
ejpam-4386	246	12	-	-	NOUN
ejpam-4386	246	13	normal	normal	ADJ
ejpam-4386	246	14	if	if	SCONJ
ejpam-4386	246	15	there	there	PRON
ejpam-4386	246	16	exists	exist	VERB
ejpam-4386	246	17	a	a	DET
ejpam-4386	246	18	coarser	coarse	ADJ
ejpam-4386	246	19	topology	topology	NOUN
ejpam-4386	246	20	τ	τ	NOUN
ejpam-4386	246	21	′	′	NOUN
ejpam-4386	246	22	on	on	ADP
ejpam-4386	246	23	x	x	INTJ
ejpam-4386	246	24	such	such	ADJ
ejpam-4386	246	25	that	that	SCONJ
ejpam-4386	246	26	(	(	PUNCT
ejpam-4386	246	27	x	x	X
ejpam-4386	246	28	,	,	PUNCT
ejpam-4386	246	29	τ	τ	PROPN
ejpam-4386	246	30	′	′	NUM
ejpam-4386	246	31	)	)	PUNCT
ejpam-4386	246	32	is	be	AUX
ejpam-4386	246	33	t4	t4	PROPN
ejpam-4386	246	34	,	,	PUNCT
ejpam-4386	246	35	see	see	VERB
ejpam-4386	246	36	[	[	X
ejpam-4386	246	37	3	3	NUM
ejpam-4386	246	38	]	]	PUNCT
ejpam-4386	246	39	.	.	PUNCT
ejpam-4386	247	1	lemma	lemma	PROPN
ejpam-4386	247	2	5	5	X
ejpam-4386	247	3	.	.	PUNCT
ejpam-4386	248	1	let	let	VERB
ejpam-4386	248	2	(	(	PUNCT
ejpam-4386	248	3	y	y	NOUN
ejpam-4386	248	4	,	,	PUNCT
ejpam-4386	248	5	ν	ν	PROPN
ejpam-4386	248	6	)	)	PUNCT
ejpam-4386	248	7	be	be	AUX
ejpam-4386	248	8	a	a	DET
ejpam-4386	248	9	regular	regular	ADJ
ejpam-4386	248	10	space	space	NOUN
ejpam-4386	248	11	.	.	PUNCT
ejpam-4386	249	1	if	if	SCONJ
ejpam-4386	249	2	f	f	PROPN
ejpam-4386	249	3	:	:	PUNCT
ejpam-4386	249	4	(	(	PUNCT
ejpam-4386	249	5	x	x	X
ejpam-4386	249	6	,	,	PUNCT
ejpam-4386	249	7	τ	τ	PROPN
ejpam-4386	249	8	)	)	PUNCT
ejpam-4386	249	9	−→	−→	NOUN
ejpam-4386	249	10	(	(	PUNCT
ejpam-4386	249	11	y	y	PROPN
ejpam-4386	249	12	,	,	PUNCT
ejpam-4386	249	13	ν	ν	NOUN
ejpam-4386	249	14	)	)	PUNCT
ejpam-4386	249	15	is	be	AUX
ejpam-4386	249	16	continuous	continuous	ADJ
ejpam-4386	249	17	,	,	PUNCT
ejpam-4386	249	18	then	then	ADV
ejpam-4386	249	19	f	f	X
ejpam-4386	249	20	:	:	PUNCT
ejpam-4386	249	21	(	(	PUNCT
ejpam-4386	249	22	x	x	X
ejpam-4386	249	23	,	,	PUNCT
ejpam-4386	249	24	τ	τ	PROPN
ejpam-4386	249	25	s	s	PART
ejpam-4386	249	26	)	)	PUNCT
ejpam-4386	249	27	−→	−→	NOUN
ejpam-4386	249	28	(	(	PUNCT
ejpam-4386	249	29	y	y	PROPN
ejpam-4386	249	30	,	,	PUNCT
ejpam-4386	249	31	ν	ν	NOUN
ejpam-4386	249	32	)	)	PUNCT
ejpam-4386	249	33	is	be	AUX
ejpam-4386	249	34	continuous	continuous	ADJ
ejpam-4386	249	35	,	,	PUNCT
ejpam-4386	249	36	[	[	X
ejpam-4386	249	37	8	8	NUM
ejpam-4386	249	38	]	]	PUNCT
ejpam-4386	249	39	.	.	PUNCT
ejpam-4386	250	1	theorem	theorem	NOUN
ejpam-4386	250	2	5	5	NUM
ejpam-4386	250	3	.	.	PUNCT
ejpam-4386	251	1	(	(	PUNCT
ejpam-4386	251	2	x	x	X
ejpam-4386	251	3	,	,	PUNCT
ejpam-4386	251	4	τ	τ	PROPN
ejpam-4386	251	5	)	)	PUNCT
ejpam-4386	251	6	is	be	AUX
ejpam-4386	251	7	epi	epi	NOUN
ejpam-4386	251	8	-	-	ADJ
ejpam-4386	251	9	normal	normal	ADJ
ejpam-4386	251	10	if	if	SCONJ
ejpam-4386	251	11	and	and	CCONJ
ejpam-4386	251	12	only	only	ADV
ejpam-4386	251	13	if	if	SCONJ
ejpam-4386	251	14	(	(	PUNCT
ejpam-4386	251	15	x	x	X
ejpam-4386	251	16	,	,	PUNCT
ejpam-4386	251	17	τ	τ	PROPN
ejpam-4386	251	18	s	s	PART
ejpam-4386	251	19	)	)	PUNCT
ejpam-4386	251	20	is	be	AUX
ejpam-4386	251	21	epi	epi	NOUN
ejpam-4386	251	22	-	-	ADJ
ejpam-4386	251	23	normal	normal	ADJ
ejpam-4386	251	24	.	.	PUNCT
ejpam-4386	252	1	proof	proof	NOUN
ejpam-4386	252	2	.	.	PUNCT
ejpam-4386	253	1	assume	assume	VERB
ejpam-4386	253	2	that	that	SCONJ
ejpam-4386	253	3	(	(	PUNCT
ejpam-4386	253	4	x	x	X
ejpam-4386	253	5	,	,	PUNCT
ejpam-4386	253	6	τ	τ	PROPN
ejpam-4386	253	7	)	)	PUNCT
ejpam-4386	253	8	is	be	AUX
ejpam-4386	253	9	epi	epi	NOUN
ejpam-4386	253	10	-	-	ADJ
ejpam-4386	253	11	normal	normal	ADJ
ejpam-4386	253	12	.	.	PUNCT
ejpam-4386	254	1	pick	pick	VERB
ejpam-4386	254	2	a	a	DET
ejpam-4386	254	3	coarser	coarse	ADJ
ejpam-4386	254	4	topology	topology	NOUN
ejpam-4386	254	5	τ	τ	NOUN
ejpam-4386	254	6	′	′	NOUN
ejpam-4386	254	7	on	on	ADP
ejpam-4386	254	8	x	x	INTJ
ejpam-4386	254	9	such	such	ADJ
ejpam-4386	254	10	that	that	SCONJ
ejpam-4386	254	11	(	(	PUNCT
ejpam-4386	254	12	x	x	X
ejpam-4386	254	13	,	,	PUNCT
ejpam-4386	254	14	τ	τ	PROPN
ejpam-4386	254	15	′	′	NUM
ejpam-4386	254	16	)	)	PUNCT
ejpam-4386	254	17	is	be	AUX
ejpam-4386	254	18	t4	t4	PROPN
ejpam-4386	254	19	.	.	PUNCT
ejpam-4386	255	1	consider	consider	VERB
ejpam-4386	255	2	the	the	DET
ejpam-4386	255	3	identity	identity	NOUN
ejpam-4386	255	4	function	function	NOUN
ejpam-4386	255	5	idx	idx	NOUN
ejpam-4386	255	6	:	:	PUNCT
ejpam-4386	255	7	(	(	PUNCT
ejpam-4386	255	8	x	x	X
ejpam-4386	255	9	,	,	PUNCT
ejpam-4386	255	10	τ	τ	PROPN
ejpam-4386	255	11	)	)	PUNCT
ejpam-4386	256	1	−→	−→	NOUN
ejpam-4386	256	2	(	(	PUNCT
ejpam-4386	256	3	x	x	INTJ
ejpam-4386	256	4	,	,	PUNCT
ejpam-4386	256	5	τ	τ	PROPN
ejpam-4386	256	6	′	′	NUM
ejpam-4386	256	7	)	)	PUNCT
ejpam-4386	256	8	which	which	PRON
ejpam-4386	256	9	is	be	AUX
ejpam-4386	256	10	continuous	continuous	ADJ
ejpam-4386	256	11	since	since	SCONJ
ejpam-4386	256	12	τ	τ	PROPN
ejpam-4386	256	13	′	′	NUM
ejpam-4386	256	14	⊆	⊆	NUM
ejpam-4386	256	15	τ	τ	X
ejpam-4386	256	16	.	.	PUNCT
ejpam-4386	257	1	then	then	ADV
ejpam-4386	257	2	,	,	PUNCT
ejpam-4386	257	3	by	by	ADP
ejpam-4386	257	4	lemma	lemma	PROPN
ejpam-4386	257	5	5	5	NUM
ejpam-4386	257	6	,	,	PUNCT
ejpam-4386	257	7	we	we	PRON
ejpam-4386	257	8	have	have	VERB
ejpam-4386	257	9	idx	idx	NOUN
ejpam-4386	257	10	:	:	PUNCT
ejpam-4386	257	11	(	(	PUNCT
ejpam-4386	257	12	x	x	X
ejpam-4386	257	13	,	,	PUNCT
ejpam-4386	257	14	τ	τ	PROPN
ejpam-4386	257	15	s	s	PART
ejpam-4386	257	16	)	)	PUNCT
ejpam-4386	257	17	−→	−→	NOUN
ejpam-4386	257	18	(	(	PUNCT
ejpam-4386	257	19	x	x	INTJ
ejpam-4386	257	20	,	,	PUNCT
ejpam-4386	257	21	τ	τ	PROPN
ejpam-4386	257	22	′	′	NUM
ejpam-4386	257	23	)	)	PUNCT
ejpam-4386	257	24	is	be	AUX
ejpam-4386	257	25	continuous	continuous	ADJ
ejpam-4386	257	26	,	,	PUNCT
ejpam-4386	257	27	hence	hence	ADV
ejpam-4386	257	28	τ	τ	PROPN
ejpam-4386	257	29	′	′	NUM
ejpam-4386	258	1	⊆	⊆	NUM
ejpam-4386	258	2	τ	τ	X
ejpam-4386	258	3	s	s	PART
ejpam-4386	258	4	.	.	PUNCT
ejpam-4386	259	1	thus	thus	ADV
ejpam-4386	259	2	,	,	PUNCT
ejpam-4386	259	3	(	(	PUNCT
ejpam-4386	259	4	x	x	X
ejpam-4386	259	5	,	,	PUNCT
ejpam-4386	259	6	τ	τ	PROPN
ejpam-4386	259	7	s	s	PART
ejpam-4386	259	8	)	)	PUNCT
ejpam-4386	259	9	is	be	AUX
ejpam-4386	259	10	epi	epi	NOUN
ejpam-4386	259	11	-	-	ADJ
ejpam-4386	259	12	normal	normal	ADJ
ejpam-4386	259	13	.	.	PUNCT
ejpam-4386	260	1	conversely	conversely	ADV
ejpam-4386	260	2	,	,	PUNCT
ejpam-4386	260	3	assume	assume	VERB
ejpam-4386	260	4	that	that	SCONJ
ejpam-4386	260	5	(	(	PUNCT
ejpam-4386	260	6	x	x	X
ejpam-4386	260	7	,	,	PUNCT
ejpam-4386	260	8	τ	τ	PROPN
ejpam-4386	260	9	s	s	PART
ejpam-4386	260	10	)	)	PUNCT
ejpam-4386	260	11	is	be	AUX
ejpam-4386	260	12	epi	epi	NOUN
ejpam-4386	260	13	-	-	ADJ
ejpam-4386	260	14	normal	normal	ADJ
ejpam-4386	260	15	.	.	PUNCT
ejpam-4386	261	1	then	then	ADV
ejpam-4386	261	2	there	there	PRON
ejpam-4386	261	3	exist	exist	VERB
ejpam-4386	261	4	a	a	DET
ejpam-4386	261	5	coarser	coarse	ADJ
ejpam-4386	261	6	topology	topology	NOUN
ejpam-4386	261	7	τ	τ	NOUN
ejpam-4386	261	8	′	′	NOUN
ejpam-4386	261	9	on	on	ADP
ejpam-4386	261	10	x	x	INTJ
ejpam-4386	261	11	such	such	ADJ
ejpam-4386	261	12	that	that	SCONJ
ejpam-4386	261	13	(	(	PUNCT
ejpam-4386	261	14	x	x	X
ejpam-4386	261	15	,	,	PUNCT
ejpam-4386	261	16	τ	τ	PROPN
ejpam-4386	261	17	′	′	NUM
ejpam-4386	261	18	)	)	PUNCT
ejpam-4386	261	19	is	be	AUX
ejpam-4386	261	20	t4	t4	PROPN
ejpam-4386	261	21	.	.	PUNCT
ejpam-4386	262	1	since	since	SCONJ
ejpam-4386	262	2	τ	τ	PROPN
ejpam-4386	262	3	s	s	PROPN
ejpam-4386	262	4	⊆	⊆	NUM
ejpam-4386	262	5	τ	τ	X
ejpam-4386	262	6	,	,	PUNCT
ejpam-4386	262	7	then	then	ADV
ejpam-4386	262	8	result	result	NOUN
ejpam-4386	262	9	follows	follow	VERB
ejpam-4386	262	10	.	.	PUNCT
ejpam-4386	263	1	definition	definition	NOUN
ejpam-4386	263	2	7	7	NUM
ejpam-4386	263	3	.	.	PUNCT
ejpam-4386	264	1	a	a	DET
ejpam-4386	264	2	topological	topological	ADJ
ejpam-4386	264	3	space	space	NOUN
ejpam-4386	264	4	x	x	PUNCT
ejpam-4386	264	5	is	be	AUX
ejpam-4386	264	6	called	call	VERB
ejpam-4386	264	7	submetrizable	submetrizable	ADJ
ejpam-4386	264	8	if	if	SCONJ
ejpam-4386	264	9	there	there	PRON
ejpam-4386	264	10	exists	exist	VERB
ejpam-4386	264	11	a	a	DET
ejpam-4386	264	12	metric	metric	ADJ
ejpam-4386	264	13	d	d	NOUN
ejpam-4386	264	14	on	on	ADP
ejpam-4386	264	15	x	x	SYM
ejpam-4386	264	16	such	such	ADJ
ejpam-4386	264	17	that	that	SCONJ
ejpam-4386	264	18	τ	τ	PROPN
ejpam-4386	264	19	d	d	PROPN
ejpam-4386	264	20	⊆	⊆	NUM
ejpam-4386	264	21	τ	τ	X
ejpam-4386	264	22	,	,	PUNCT
ejpam-4386	264	23	[	[	X
ejpam-4386	264	24	7	7	NUM
ejpam-4386	264	25	]	]	PUNCT
ejpam-4386	264	26	.	.	PUNCT
ejpam-4386	265	1	similar	similar	ADJ
ejpam-4386	265	2	argument	argument	NOUN
ejpam-4386	265	3	of	of	ADP
ejpam-4386	265	4	the	the	DET
ejpam-4386	265	5	proof	proof	NOUN
ejpam-4386	265	6	of	of	ADP
ejpam-4386	265	7	theorem	theorem	ADJ
ejpam-4386	265	8	5	5	NUM
ejpam-4386	265	9	gives	give	VERB
ejpam-4386	265	10	the	the	DET
ejpam-4386	265	11	following	follow	VERB
ejpam-4386	265	12	theorem	theorem	VERB
ejpam-4386	265	13	.	.	PUNCT
ejpam-4386	266	1	theorem	theorem	NOUN
ejpam-4386	266	2	6	6	NUM
ejpam-4386	266	3	.	.	PUNCT
ejpam-4386	267	1	(	(	PUNCT
ejpam-4386	267	2	x	x	X
ejpam-4386	267	3	,	,	PUNCT
ejpam-4386	267	4	τ	τ	PROPN
ejpam-4386	267	5	)	)	PUNCT
ejpam-4386	267	6	is	be	AUX
ejpam-4386	267	7	submetrizable	submetrizable	ADJ
ejpam-4386	267	8	if	if	SCONJ
ejpam-4386	267	9	and	and	CCONJ
ejpam-4386	267	10	only	only	ADV
ejpam-4386	267	11	if	if	SCONJ
ejpam-4386	267	12	(	(	PUNCT
ejpam-4386	267	13	x	x	X
ejpam-4386	267	14	,	,	PUNCT
ejpam-4386	267	15	τ	τ	PROPN
ejpam-4386	267	16	s	s	PART
ejpam-4386	267	17	)	)	PUNCT
ejpam-4386	267	18	is	be	AUX
ejpam-4386	267	19	submetrizable	submetrizable	ADJ
ejpam-4386	267	20	.	.	PUNCT
ejpam-4386	268	1	definition	definition	NOUN
ejpam-4386	268	2	8	8	NUM
ejpam-4386	268	3	.	.	PUNCT
ejpam-4386	269	1	a	a	DET
ejpam-4386	269	2	topological	topological	ADJ
ejpam-4386	269	3	space	space	NOUN
ejpam-4386	269	4	x	x	PUNCT
ejpam-4386	269	5	is	be	AUX
ejpam-4386	269	6	called	call	VERB
ejpam-4386	269	7	c	c	NOUN
ejpam-4386	269	8	-	-	NOUN
ejpam-4386	269	9	normal	normal	ADJ
ejpam-4386	269	10	if	if	SCONJ
ejpam-4386	269	11	there	there	PRON
ejpam-4386	269	12	exist	exist	VERB
ejpam-4386	269	13	a	a	DET
ejpam-4386	269	14	normal	normal	ADJ
ejpam-4386	269	15	space	space	NOUN
ejpam-4386	269	16	y	y	PROPN
ejpam-4386	269	17	and	and	CCONJ
ejpam-4386	269	18	a	a	DET
ejpam-4386	269	19	bijective	bijective	ADJ
ejpam-4386	269	20	function	function	NOUN
ejpam-4386	270	1	f	f	NOUN
ejpam-4386	270	2	:	:	PUNCT
ejpam-4386	270	3	x	x	PUNCT
ejpam-4386	270	4	−→	−→	NOUN
ejpam-4386	270	5	y	y	PROPN
ejpam-4386	270	6	such	such	ADJ
ejpam-4386	270	7	that	that	SCONJ
ejpam-4386	270	8	the	the	DET
ejpam-4386	270	9	restriction	restriction	NOUN
ejpam-4386	270	10	f	f	PROPN
ejpam-4386	270	11	|a	|a	VERB
ejpam-4386	270	12	:	:	PUNCT
ejpam-4386	270	13	a	a	DET
ejpam-4386	270	14	−→	−→	NOUN
ejpam-4386	270	15	f(a	f(a	NOUN
ejpam-4386	270	16	)	)	PUNCT
ejpam-4386	270	17	is	be	AUX
ejpam-4386	270	18	a	a	DET
ejpam-4386	270	19	homeomorphism	homeomorphism	NOUN
ejpam-4386	270	20	for	for	ADP
ejpam-4386	270	21	each	each	DET
ejpam-4386	270	22	compact	compact	ADJ
ejpam-4386	270	23	subspace	subspace	NOUN
ejpam-4386	270	24	a	a	DET
ejpam-4386	270	25	⊆	⊆	NUM
ejpam-4386	270	26	x	x	SYM
ejpam-4386	270	27	,	,	PUNCT
ejpam-4386	270	28	[	[	X
ejpam-4386	270	29	4	4	NUM
ejpam-4386	270	30	]	]	PUNCT
ejpam-4386	270	31	.	.	PUNCT
ejpam-4386	271	1	the	the	DET
ejpam-4386	271	2	following	follow	VERB
ejpam-4386	271	3	example	example	NOUN
ejpam-4386	271	4	shows	show	VERB
ejpam-4386	271	5	that	that	SCONJ
ejpam-4386	271	6	,	,	PUNCT
ejpam-4386	271	7	if	if	SCONJ
ejpam-4386	271	8	(	(	PUNCT
ejpam-4386	271	9	x	x	X
ejpam-4386	271	10	,	,	PUNCT
ejpam-4386	271	11	τ	τ	PROPN
ejpam-4386	271	12	s	s	PART
ejpam-4386	271	13	)	)	PUNCT
ejpam-4386	271	14	is	be	AUX
ejpam-4386	271	15	c	c	NOUN
ejpam-4386	271	16	-	-	ADJ
ejpam-4386	271	17	normal	normal	ADJ
ejpam-4386	271	18	,	,	PUNCT
ejpam-4386	271	19	then	then	ADV
ejpam-4386	271	20	(	(	PUNCT
ejpam-4386	271	21	x	x	X
ejpam-4386	271	22	,	,	PUNCT
ejpam-4386	271	23	τ	τ	PROPN
ejpam-4386	271	24	)	)	PUNCT
ejpam-4386	271	25	may	may	AUX
ejpam-4386	271	26	not	not	PART
ejpam-4386	271	27	be	be	AUX
ejpam-4386	271	28	c	c	NOUN
ejpam-4386	271	29	-	-	ADJ
ejpam-4386	271	30	normal	normal	ADJ
ejpam-4386	271	31	.	.	PUNCT
ejpam-4386	272	1	references	reference	NOUN
ejpam-4386	272	2	828	828	NUM
ejpam-4386	272	3	example	example	NOUN
ejpam-4386	272	4	4	4	NUM
ejpam-4386	272	5	.	.	X
ejpam-4386	272	6	consider	consider	VERB
ejpam-4386	272	7	r	r	NOUN
ejpam-4386	272	8	with	with	ADP
ejpam-4386	272	9	the	the	DET
ejpam-4386	272	10	particular	particular	ADJ
ejpam-4386	272	11	point	point	NOUN
ejpam-4386	272	12	topology	topology	NOUN
ejpam-4386	272	13	τ	τ	PROPN
ejpam-4386	272	14	0	0	NUM
ejpam-4386	272	15	which	which	PRON
ejpam-4386	272	16	is	be	AUX
ejpam-4386	272	17	not	not	PART
ejpam-4386	272	18	c	c	NOUN
ejpam-4386	272	19	-	-	ADJ
ejpam-4386	272	20	normal	normal	ADJ
ejpam-4386	272	21	see	see	VERB
ejpam-4386	272	22	[	[	X
ejpam-4386	272	23	4	4	NUM
ejpam-4386	272	24	,	,	PUNCT
ejpam-4386	272	25	example	example	NOUN
ejpam-4386	272	26	1.5	1.5	NUM
ejpam-4386	272	27	]	]	PUNCT
ejpam-4386	272	28	.	.	PUNCT
ejpam-4386	273	1	but	but	CCONJ
ejpam-4386	273	2	the	the	DET
ejpam-4386	273	3	semi	semi	ADJ
ejpam-4386	273	4	-	-	ADJ
ejpam-4386	273	5	regularization	regularization	ADJ
ejpam-4386	273	6	topological	topological	ADJ
ejpam-4386	273	7	space	space	NOUN
ejpam-4386	273	8	of	of	ADP
ejpam-4386	273	9	(	(	PUNCT
ejpam-4386	273	10	r	r	NOUN
ejpam-4386	273	11	,	,	PUNCT
ejpam-4386	273	12	τ	τ	PROPN
ejpam-4386	273	13	0	0	NUM
ejpam-4386	273	14	)	)	PUNCT
ejpam-4386	273	15	is	be	AUX
ejpam-4386	273	16	(	(	PUNCT
ejpam-4386	273	17	r	r	NOUN
ejpam-4386	273	18	,	,	PUNCT
ejpam-4386	273	19	i	i	NOUN
ejpam-4386	273	20	)	)	PUNCT
ejpam-4386	273	21	,	,	PUNCT
ejpam-4386	273	22	where	where	SCONJ
ejpam-4386	273	23	i	i	PRON
ejpam-4386	273	24	is	be	AUX
ejpam-4386	273	25	the	the	DET
ejpam-4386	273	26	indiscrete	indiscrete	ADJ
ejpam-4386	273	27	topology	topology	NOUN
ejpam-4386	273	28	,	,	PUNCT
ejpam-4386	273	29	which	which	PRON
ejpam-4386	273	30	is	be	AUX
ejpam-4386	273	31	a	a	DET
ejpam-4386	273	32	normal	normal	ADJ
ejpam-4386	273	33	space	space	NOUN
ejpam-4386	273	34	,	,	PUNCT
ejpam-4386	273	35	thus	thus	ADV
ejpam-4386	273	36	c	c	NOUN
ejpam-4386	273	37	-	-	ADJ
ejpam-4386	273	38	normal	normal	ADJ
ejpam-4386	273	39	.	.	PUNCT
ejpam-4386	274	1	lemma	lemma	PROPN
ejpam-4386	274	2	6	6	NUM
ejpam-4386	274	3	.	.	PUNCT
ejpam-4386	275	1	if	if	SCONJ
ejpam-4386	275	2	x	x	PRON
ejpam-4386	275	3	is	be	AUX
ejpam-4386	275	4	t1	t1	NOUN
ejpam-4386	275	5	and	and	CCONJ
ejpam-4386	275	6	c	c	NOUN
ejpam-4386	275	7	-	-	ADJ
ejpam-4386	275	8	normal	normal	ADJ
ejpam-4386	275	9	,	,	PUNCT
ejpam-4386	275	10	then	then	ADV
ejpam-4386	275	11	any	any	DET
ejpam-4386	275	12	witness	witness	NOUN
ejpam-4386	275	13	y	y	PROPN
ejpam-4386	275	14	of	of	ADP
ejpam-4386	275	15	its	its	PRON
ejpam-4386	275	16	c	c	NOUN
ejpam-4386	275	17	-	-	PUNCT
ejpam-4386	275	18	normality	normality	NOUN
ejpam-4386	275	19	is	be	AUX
ejpam-4386	275	20	t4	t4	PROPN
ejpam-4386	275	21	.	.	PUNCT
ejpam-4386	275	22	proof	proof	NOUN
ejpam-4386	275	23	.	.	PUNCT
ejpam-4386	276	1	assume	assume	VERB
ejpam-4386	276	2	that	that	SCONJ
ejpam-4386	276	3	x	x	PRON
ejpam-4386	276	4	is	be	AUX
ejpam-4386	276	5	t1	t1	NOUN
ejpam-4386	276	6	and	and	CCONJ
ejpam-4386	276	7	c	c	NOUN
ejpam-4386	276	8	-	-	ADJ
ejpam-4386	276	9	normal	normal	ADJ
ejpam-4386	276	10	.	.	PUNCT
ejpam-4386	277	1	pick	pick	VERB
ejpam-4386	277	2	a	a	DET
ejpam-4386	277	3	normal	normal	ADJ
ejpam-4386	277	4	space	space	NOUN
ejpam-4386	277	5	y	y	PROPN
ejpam-4386	277	6	and	and	CCONJ
ejpam-4386	277	7	a	a	DET
ejpam-4386	277	8	bijective	bijective	ADJ
ejpam-4386	277	9	function	function	NOUN
ejpam-4386	278	1	f	f	NOUN
ejpam-4386	278	2	:	:	PUNCT
ejpam-4386	278	3	x	x	PUNCT
ejpam-4386	278	4	−→	−→	NOUN
ejpam-4386	278	5	y	y	PROPN
ejpam-4386	278	6	such	such	ADJ
ejpam-4386	278	7	that	that	DET
ejpam-4386	278	8	f|a	f|a	NOUN
ejpam-4386	278	9	:	:	PUNCT
ejpam-4386	278	10	a	a	DET
ejpam-4386	278	11	−→	−→	NOUN
ejpam-4386	278	12	f(a	f(a	NOUN
ejpam-4386	278	13	)	)	PUNCT
ejpam-4386	278	14	is	be	AUX
ejpam-4386	278	15	a	a	DET
ejpam-4386	278	16	homeomorphism	homeomorphism	NOUN
ejpam-4386	278	17	for	for	ADP
ejpam-4386	278	18	each	each	DET
ejpam-4386	278	19	compact	compact	ADJ
ejpam-4386	278	20	subspace	subspace	NOUN
ejpam-4386	278	21	a	a	DET
ejpam-4386	278	22	⊆	⊆	NUM
ejpam-4386	278	23	x.	x.	NOUN
ejpam-4386	278	24	let	let	VERB
ejpam-4386	278	25	x	x	PRON
ejpam-4386	278	26	,	,	PUNCT
ejpam-4386	278	27	y	y	PROPN
ejpam-4386	278	28	be	be	VERB
ejpam-4386	278	29	any	any	DET
ejpam-4386	278	30	two	two	NUM
ejpam-4386	278	31	distinct	distinct	ADJ
ejpam-4386	278	32	elements	element	NOUN
ejpam-4386	278	33	in	in	ADP
ejpam-4386	278	34	y	y	PROPN
ejpam-4386	278	35	.	.	PUNCT
ejpam-4386	279	1	since	since	SCONJ
ejpam-4386	279	2	f	f	PROPN
ejpam-4386	279	3	is	be	AUX
ejpam-4386	279	4	bijective	bijective	ADJ
ejpam-4386	279	5	,	,	PUNCT
ejpam-4386	279	6	there	there	PRON
ejpam-4386	279	7	are	be	VERB
ejpam-4386	279	8	unique	unique	ADJ
ejpam-4386	279	9	elements	element	NOUN
ejpam-4386	279	10	a	a	PRON
ejpam-4386	279	11	,	,	PUNCT
ejpam-4386	279	12	b	b	X
ejpam-4386	279	13	∈	∈	PROPN
ejpam-4386	279	14	x	x	PUNCT
ejpam-4386	279	15	such	such	ADJ
ejpam-4386	279	16	that	that	DET
ejpam-4386	279	17	f(a	f(a	NOUN
ejpam-4386	279	18	)	)	PUNCT
ejpam-4386	280	1	=	=	SYM
ejpam-4386	280	2	x	x	X
ejpam-4386	280	3	and	and	CCONJ
ejpam-4386	280	4	f(b	f(b	PROPN
ejpam-4386	280	5	)	)	PUNCT
ejpam-4386	281	1	=	=	PUNCT
ejpam-4386	282	1	y	y	PROPN
ejpam-4386	282	2	such	such	ADJ
ejpam-4386	282	3	that	that	SCONJ
ejpam-4386	282	4	a	a	DET
ejpam-4386	282	5	̸=	̸=	PROPN
ejpam-4386	282	6	b.	b.	PROPN
ejpam-4386	282	7	consider	consider	VERB
ejpam-4386	282	8	{	{	PUNCT
ejpam-4386	282	9	a	a	DET
ejpam-4386	282	10	,	,	PUNCT
ejpam-4386	282	11	b	b	NOUN
ejpam-4386	282	12	}	}	PUNCT
ejpam-4386	282	13	which	which	PRON
ejpam-4386	282	14	is	be	AUX
ejpam-4386	282	15	a	a	DET
ejpam-4386	282	16	compact	compact	ADJ
ejpam-4386	282	17	subset	subset	NOUN
ejpam-4386	282	18	ofx	ofx	NOUN
ejpam-4386	282	19	.	.	PUNCT
ejpam-4386	283	1	this	this	PRON
ejpam-4386	283	2	implies	imply	VERB
ejpam-4386	283	3	f|{a	f|{a	PROPN
ejpam-4386	283	4	,	,	PUNCT
ejpam-4386	283	5	b	b	NOUN
ejpam-4386	283	6	}	}	PUNCT
ejpam-4386	283	7	:	:	PUNCT
ejpam-4386	283	8	{	{	PUNCT
ejpam-4386	283	9	a	a	PRON
ejpam-4386	283	10	,	,	PUNCT
ejpam-4386	283	11	b	b	NOUN
ejpam-4386	283	12	}	}	PUNCT
ejpam-4386	283	13	−→	−→	NOUN
ejpam-4386	283	14	{	{	PUNCT
ejpam-4386	283	15	x	x	NOUN
ejpam-4386	283	16	,	,	PUNCT
ejpam-4386	283	17	y	y	PRON
ejpam-4386	283	18	}	}	PUNCT
ejpam-4386	283	19	is	be	AUX
ejpam-4386	283	20	a	a	DET
ejpam-4386	283	21	homeomorphism	homeomorphism	NOUN
ejpam-4386	283	22	.	.	PUNCT
ejpam-4386	284	1	but	but	CCONJ
ejpam-4386	284	2	x	x	X
ejpam-4386	284	3	is	be	AUX
ejpam-4386	284	4	t1	t1	NOUN
ejpam-4386	284	5	,	,	PUNCT
ejpam-4386	284	6	thus	thus	ADV
ejpam-4386	284	7	{	{	PUNCT
ejpam-4386	284	8	a	a	DET
ejpam-4386	284	9	,	,	PUNCT
ejpam-4386	284	10	b	b	NOUN
ejpam-4386	284	11	}	}	PUNCT
ejpam-4386	284	12	is	be	AUX
ejpam-4386	284	13	a	a	DET
ejpam-4386	284	14	discrete	discrete	ADJ
ejpam-4386	284	15	subspace	subspace	NOUN
ejpam-4386	284	16	of	of	ADP
ejpam-4386	284	17	x	x	PRON
ejpam-4386	284	18	,	,	PUNCT
ejpam-4386	284	19	hence	hence	ADV
ejpam-4386	284	20	{	{	PUNCT
ejpam-4386	284	21	x	x	NOUN
ejpam-4386	284	22	,	,	PUNCT
ejpam-4386	284	23	y	y	PRON
ejpam-4386	284	24	}	}	PUNCT
ejpam-4386	284	25	is	be	AUX
ejpam-4386	284	26	a	a	DET
ejpam-4386	284	27	discrete	discrete	ADJ
ejpam-4386	284	28	subspace	subspace	NOUN
ejpam-4386	284	29	of	of	ADP
ejpam-4386	284	30	y	y	PROPN
ejpam-4386	284	31	,	,	PUNCT
ejpam-4386	284	32	then	then	ADV
ejpam-4386	284	33	there	there	PRON
ejpam-4386	284	34	are	be	VERB
ejpam-4386	284	35	two	two	NUM
ejpam-4386	284	36	open	open	ADJ
ejpam-4386	284	37	neighborhoods	neighborhood	NOUN
ejpam-4386	284	38	u	u	NOUN
ejpam-4386	284	39	and	and	CCONJ
ejpam-4386	284	40	v	v	NOUN
ejpam-4386	284	41	of	of	ADP
ejpam-4386	284	42	x	x	X
ejpam-4386	284	43	and	and	CCONJ
ejpam-4386	284	44	y	y	PROPN
ejpam-4386	284	45	respectively	respectively	ADV
ejpam-4386	284	46	in	in	ADP
ejpam-4386	284	47	y	y	PRON
ejpam-4386	284	48	such	such	ADJ
ejpam-4386	284	49	that	that	SCONJ
ejpam-4386	284	50	u	u	PROPN
ejpam-4386	284	51	∩	∩	NOUN
ejpam-4386	284	52	{	{	PUNCT
ejpam-4386	284	53	x	x	NOUN
ejpam-4386	284	54	,	,	PUNCT
ejpam-4386	284	55	y	y	PROPN
ejpam-4386	284	56	}	}	PUNCT
ejpam-4386	284	57	=	=	SYM
ejpam-4386	284	58	{	{	PUNCT
ejpam-4386	284	59	x	x	NOUN
ejpam-4386	284	60	}	}	PUNCT
ejpam-4386	284	61	and	and	CCONJ
ejpam-4386	284	62	v	v	ADP
ejpam-4386	284	63	∩	∩	NOUN
ejpam-4386	284	64	{	{	PUNCT
ejpam-4386	284	65	x	x	NOUN
ejpam-4386	284	66	,	,	PUNCT
ejpam-4386	284	67	y	y	PROPN
ejpam-4386	284	68	}	}	PUNCT
ejpam-4386	284	69	=	=	SYM
ejpam-4386	284	70	{	{	PUNCT
ejpam-4386	284	71	y	y	NOUN
ejpam-4386	284	72	}	}	PUNCT
ejpam-4386	284	73	where	where	SCONJ
ejpam-4386	284	74	y	y	PROPN
ejpam-4386	284	75	/∈	/∈	PUNCT
ejpam-4386	284	76	u	u	PROPN
ejpam-4386	284	77	and	and	CCONJ
ejpam-4386	284	78	x	x	NOUN
ejpam-4386	284	79	/∈	/∈	NOUN
ejpam-4386	285	1	v	v	NOUN
ejpam-4386	285	2	.	.	PUNCT
ejpam-4386	286	1	thus	thus	ADV
ejpam-4386	286	2	y	y	PROPN
ejpam-4386	286	3	is	be	AUX
ejpam-4386	286	4	t1	t1	NOUN
ejpam-4386	286	5	and	and	CCONJ
ejpam-4386	286	6	given	give	VERB
ejpam-4386	286	7	that	that	SCONJ
ejpam-4386	286	8	y	y	PROPN
ejpam-4386	286	9	is	be	AUX
ejpam-4386	286	10	normal	normal	ADJ
ejpam-4386	286	11	,	,	PUNCT
ejpam-4386	286	12	thus	thus	ADV
ejpam-4386	286	13	y	y	PROPN
ejpam-4386	286	14	is	be	AUX
ejpam-4386	286	15	t4	t4	PROPN
ejpam-4386	286	16	.	.	PROPN
ejpam-4386	286	17	recall	recall	VERB
ejpam-4386	286	18	that	that	SCONJ
ejpam-4386	286	19	a	a	DET
ejpam-4386	286	20	topological	topological	ADJ
ejpam-4386	286	21	space	space	NOUN
ejpam-4386	286	22	x	x	PUNCT
ejpam-4386	286	23	is	be	AUX
ejpam-4386	286	24	called	call	VERB
ejpam-4386	286	25	a	a	DET
ejpam-4386	286	26	fréchet	fréchet	NOUN
ejpam-4386	286	27	space	space	NOUN
ejpam-4386	286	28	if	if	SCONJ
ejpam-4386	286	29	for	for	ADP
ejpam-4386	286	30	every	every	DET
ejpam-4386	286	31	a	a	DET
ejpam-4386	286	32	⊆	⊆	NUM
ejpam-4386	286	33	x	x	SYM
ejpam-4386	286	34	and	and	CCONJ
ejpam-4386	286	35	every	every	DET
ejpam-4386	286	36	x	x	SYM
ejpam-4386	286	37	∈	∈	PROPN
ejpam-4386	286	38	a	a	DET
ejpam-4386	286	39	there	there	PRON
ejpam-4386	286	40	exists	exist	VERB
ejpam-4386	286	41	a	a	DET
ejpam-4386	286	42	sequence	sequence	NOUN
ejpam-4386	286	43	(	(	PUNCT
ejpam-4386	286	44	an)n∈n	an)n∈n	NUM
ejpam-4386	286	45	of	of	ADP
ejpam-4386	286	46	points	point	NOUN
ejpam-4386	286	47	of	of	ADP
ejpam-4386	286	48	a	a	DET
ejpam-4386	286	49	such	such	ADJ
ejpam-4386	286	50	that	that	SCONJ
ejpam-4386	286	51	an	an	DET
ejpam-4386	286	52	−→	−→	NOUN
ejpam-4386	286	53	x	x	NOUN
ejpam-4386	286	54	,	,	PUNCT
ejpam-4386	286	55	[	[	X
ejpam-4386	286	56	6	6	NUM
ejpam-4386	286	57	]	]	PUNCT
ejpam-4386	286	58	.	.	PUNCT
ejpam-4386	287	1	lemma	lemma	PROPN
ejpam-4386	287	2	7	7	X
ejpam-4386	287	3	.	.	PUNCT
ejpam-4386	288	1	if	if	SCONJ
ejpam-4386	288	2	x	x	PRON
ejpam-4386	288	3	is	be	AUX
ejpam-4386	288	4	fréchet	fréchet	ADJ
ejpam-4386	288	5	and	and	CCONJ
ejpam-4386	288	6	c	c	NOUN
ejpam-4386	288	7	-	-	ADJ
ejpam-4386	288	8	normal	normal	ADJ
ejpam-4386	288	9	,	,	PUNCT
ejpam-4386	288	10	then	then	ADV
ejpam-4386	288	11	any	any	DET
ejpam-4386	288	12	witness	witness	NOUN
ejpam-4386	288	13	function	function	NOUN
ejpam-4386	288	14	of	of	ADP
ejpam-4386	288	15	its	its	PRON
ejpam-4386	288	16	c	c	NOUN
ejpam-4386	288	17	-	-	PUNCT
ejpam-4386	288	18	normality	normality	NOUN
ejpam-4386	288	19	is	be	AUX
ejpam-4386	288	20	continuous	continuous	ADJ
ejpam-4386	288	21	.	.	PUNCT
ejpam-4386	289	1	proof	proof	NOUN
ejpam-4386	289	2	.	.	PUNCT
ejpam-4386	290	1	assume	assume	VERB
ejpam-4386	290	2	that	that	SCONJ
ejpam-4386	290	3	x	x	PRON
ejpam-4386	290	4	is	be	AUX
ejpam-4386	290	5	fréchet	fréchet	ADJ
ejpam-4386	290	6	and	and	CCONJ
ejpam-4386	290	7	c	c	NOUN
ejpam-4386	290	8	-	-	ADJ
ejpam-4386	290	9	normal	normal	ADJ
ejpam-4386	290	10	.	.	PUNCT
ejpam-4386	291	1	let	let	VERB
ejpam-4386	291	2	f	f	NOUN
ejpam-4386	291	3	:	:	PUNCT
ejpam-4386	291	4	x	x	PUNCT
ejpam-4386	291	5	−→	−→	NOUN
ejpam-4386	291	6	y	y	NOUN
ejpam-4386	291	7	be	be	AUX
ejpam-4386	291	8	a	a	DET
ejpam-4386	291	9	witness	witness	NOUN
ejpam-4386	291	10	function	function	NOUN
ejpam-4386	291	11	of	of	ADP
ejpam-4386	291	12	the	the	DET
ejpam-4386	291	13	c	c	NOUN
ejpam-4386	291	14	-	-	PUNCT
ejpam-4386	291	15	normality	normality	NOUN
ejpam-4386	291	16	of	of	ADP
ejpam-4386	291	17	x.	x.	NOUN
ejpam-4386	291	18	let	let	VERB
ejpam-4386	291	19	a	a	DET
ejpam-4386	291	20	⊂	⊂	NOUN
ejpam-4386	291	21	x	x	X
ejpam-4386	291	22	and	and	CCONJ
ejpam-4386	291	23	let	let	VERB
ejpam-4386	291	24	y	y	PROPN
ejpam-4386	291	25	∈	∈	PROPN
ejpam-4386	291	26	f(a	f(a	PROPN
ejpam-4386	291	27	)	)	PUNCT
ejpam-4386	291	28	be	be	AUX
ejpam-4386	291	29	arbitrary	arbitrary	ADJ
ejpam-4386	291	30	.	.	PUNCT
ejpam-4386	292	1	pick	pick	VERB
ejpam-4386	292	2	the	the	DET
ejpam-4386	292	3	unique	unique	ADJ
ejpam-4386	292	4	element	element	NOUN
ejpam-4386	292	5	x	x	SYM
ejpam-4386	292	6	∈	∈	NOUN
ejpam-4386	292	7	x	x	PUNCT
ejpam-4386	292	8	such	such	ADJ
ejpam-4386	292	9	that	that	SCONJ
ejpam-4386	292	10	f(x	f(x	NOUN
ejpam-4386	292	11	)	)	PUNCT
ejpam-4386	293	1	=	=	PUNCT
ejpam-4386	294	1	y.	y.	NOUN
ejpam-4386	294	2	thus	thus	ADV
ejpam-4386	294	3	x	x	X
ejpam-4386	294	4	∈	∈	NOUN
ejpam-4386	294	5	a.	a.	NOUN
ejpam-4386	294	6	since	since	SCONJ
ejpam-4386	294	7	x	x	PRON
ejpam-4386	294	8	is	be	AUX
ejpam-4386	294	9	a	a	DET
ejpam-4386	294	10	fréchet	fréchet	NOUN
ejpam-4386	294	11	space	space	NOUN
ejpam-4386	294	12	,	,	PUNCT
ejpam-4386	294	13	then	then	ADV
ejpam-4386	294	14	there	there	PRON
ejpam-4386	294	15	exist	exist	VERB
ejpam-4386	294	16	a	a	DET
ejpam-4386	294	17	sequence	sequence	NOUN
ejpam-4386	294	18	(	(	PUNCT
ejpam-4386	294	19	an	an	NOUN
ejpam-4386	294	20	)	)	PUNCT
ejpam-4386	294	21	⊆	⊆	PROPN
ejpam-4386	294	22	a	a	DET
ejpam-4386	294	23	such	such	ADJ
ejpam-4386	294	24	that	that	SCONJ
ejpam-4386	294	25	(	(	PUNCT
ejpam-4386	294	26	an	an	X
ejpam-4386	294	27	)	)	PUNCT
ejpam-4386	294	28	converges	converge	NOUN
ejpam-4386	294	29	to	to	PART
ejpam-4386	294	30	x.	x.	VERB
ejpam-4386	294	31	the	the	DET
ejpam-4386	294	32	subspace	subspace	PROPN
ejpam-4386	294	33	b	b	PROPN
ejpam-4386	294	34	=	=	PRON
ejpam-4386	294	35	{	{	PUNCT
ejpam-4386	294	36	x	x	NOUN
ejpam-4386	294	37	,	,	PUNCT
ejpam-4386	294	38	an	an	DET
ejpam-4386	294	39	:	:	PUNCT
ejpam-4386	294	40	n	n	CCONJ
ejpam-4386	294	41	∈	∈	PROPN
ejpam-4386	294	42	n	n	CCONJ
ejpam-4386	294	43	}	}	PUNCT
ejpam-4386	294	44	of	of	ADP
ejpam-4386	294	45	x	x	SYM
ejpam-4386	294	46	is	be	AUX
ejpam-4386	294	47	compact	compact	ADJ
ejpam-4386	294	48	and	and	CCONJ
ejpam-4386	294	49	thus	thus	ADV
ejpam-4386	294	50	f|b	f|b	VERB
ejpam-4386	294	51	:	:	PUNCT
ejpam-4386	294	52	b	b	X
ejpam-4386	294	53	−→	−→	ADJ
ejpam-4386	294	54	f(b	f(b	PROPN
ejpam-4386	294	55	)	)	PUNCT
ejpam-4386	294	56	is	be	AUX
ejpam-4386	294	57	a	a	DET
ejpam-4386	294	58	homeomorphism	homeomorphism	NOUN
ejpam-4386	294	59	.	.	PUNCT
ejpam-4386	295	1	now	now	ADV
ejpam-4386	295	2	,	,	PUNCT
ejpam-4386	295	3	let	let	VERB
ejpam-4386	295	4	w	w	PRON
ejpam-4386	295	5	⊆	⊆	NUM
ejpam-4386	295	6	y	y	NOUN
ejpam-4386	295	7	be	be	AUX
ejpam-4386	295	8	any	any	DET
ejpam-4386	295	9	open	open	ADJ
ejpam-4386	295	10	neighborhood	neighborhood	NOUN
ejpam-4386	295	11	of	of	ADP
ejpam-4386	295	12	y	y	PROPN
ejpam-4386	295	13	,	,	PUNCT
ejpam-4386	295	14	then	then	ADV
ejpam-4386	295	15	w	w	PROPN
ejpam-4386	295	16	∩	∩	ADJ
ejpam-4386	295	17	f(b	f(b	PROPN
ejpam-4386	295	18	)	)	PUNCT
ejpam-4386	295	19	is	be	AUX
ejpam-4386	295	20	open	open	ADJ
ejpam-4386	295	21	in	in	ADP
ejpam-4386	295	22	the	the	DET
ejpam-4386	295	23	subspace	subspace	NOUN
ejpam-4386	295	24	f(b	f(b	PROPN
ejpam-4386	295	25	)	)	PUNCT
ejpam-4386	295	26	containing	contain	VERB
ejpam-4386	295	27	y.	y.	NOUN
ejpam-4386	295	28	since	since	SCONJ
ejpam-4386	295	29	f({an	f({an	PROPN
ejpam-4386	295	30	:	:	PUNCT
ejpam-4386	295	31	n	n	CCONJ
ejpam-4386	295	32	∈	∈	PROPN
ejpam-4386	295	33	n	n	CCONJ
ejpam-4386	295	34	}	}	PUNCT
ejpam-4386	295	35	)	)	PUNCT
ejpam-4386	295	36	⊆	⊆	NUM
ejpam-4386	295	37	f(b	f(b	NOUN
ejpam-4386	295	38	)	)	PUNCT
ejpam-4386	295	39	∩	∩	ADJ
ejpam-4386	295	40	f(a	f(a	NOUN
ejpam-4386	295	41	)	)	PUNCT
ejpam-4386	295	42	and	and	CCONJ
ejpam-4386	295	43	w	w	ADP
ejpam-4386	295	44	∩	∩	ADJ
ejpam-4386	295	45	f(b	f(b	X
ejpam-4386	295	46	)	)	PUNCT
ejpam-4386	295	47	̸=	̸=	NOUN
ejpam-4386	295	48	∅	∅	NOUN
ejpam-4386	295	49	,	,	PUNCT
ejpam-4386	295	50	then	then	ADV
ejpam-4386	295	51	w	w	PROPN
ejpam-4386	295	52	∩	∩	ADJ
ejpam-4386	295	53	f(a	f(a	NOUN
ejpam-4386	295	54	)	)	PUNCT
ejpam-4386	295	55	̸=	̸=	PROPN
ejpam-4386	295	56	∅.	∅.	PRON
ejpam-4386	295	57	hence	hence	ADV
ejpam-4386	295	58	y	y	PROPN
ejpam-4386	295	59	∈	∈	PROPN
ejpam-4386	295	60	f(a	f(a	PROPN
ejpam-4386	295	61	)	)	PUNCT
ejpam-4386	295	62	and	and	CCONJ
ejpam-4386	295	63	thus	thus	ADV
ejpam-4386	295	64	f(a	f(a	NOUN
ejpam-4386	295	65	)	)	PUNCT
ejpam-4386	295	66	⊆	⊆	NUM
ejpam-4386	295	67	f(a	f(a	NOUN
ejpam-4386	295	68	)	)	PUNCT
ejpam-4386	295	69	.	.	PUNCT
ejpam-4386	296	1	therefore	therefore	ADV
ejpam-4386	296	2	,	,	PUNCT
ejpam-4386	296	3	f	f	PROPN
ejpam-4386	296	4	is	be	AUX
ejpam-4386	296	5	continuous	continuous	ADJ
ejpam-4386	296	6	.	.	PUNCT
ejpam-4386	297	1	theorem	theorem	ADJ
ejpam-4386	297	2	7	7	NUM
ejpam-4386	297	3	.	.	PUNCT
ejpam-4386	298	1	if	if	SCONJ
ejpam-4386	298	2	(	(	PUNCT
ejpam-4386	298	3	x	x	X
ejpam-4386	298	4	,	,	PUNCT
ejpam-4386	298	5	τ	τ	PROPN
ejpam-4386	298	6	)	)	PUNCT
ejpam-4386	298	7	is	be	AUX
ejpam-4386	298	8	fréchet	fréchet	NOUN
ejpam-4386	298	9	,	,	PUNCT
ejpam-4386	298	10	t1	t1	NOUN
ejpam-4386	298	11	and	and	CCONJ
ejpam-4386	298	12	c	c	NOUN
ejpam-4386	298	13	-	-	ADJ
ejpam-4386	298	14	normal	normal	ADJ
ejpam-4386	298	15	,	,	PUNCT
ejpam-4386	298	16	then	then	ADV
ejpam-4386	298	17	its	its	PRON
ejpam-4386	298	18	semi	semi	ADJ
ejpam-4386	298	19	-	-	ADJ
ejpam-4386	298	20	regularization	regularization	ADJ
ejpam-4386	298	21	topological	topological	ADJ
ejpam-4386	298	22	space	space	NOUN
ejpam-4386	298	23	(	(	PUNCT
ejpam-4386	298	24	x	x	X
ejpam-4386	298	25	,	,	PUNCT
ejpam-4386	298	26	τ	τ	PROPN
ejpam-4386	298	27	s	s	PART
ejpam-4386	298	28	)	)	PUNCT
ejpam-4386	298	29	is	be	AUX
ejpam-4386	298	30	c	c	NOUN
ejpam-4386	298	31	-	-	ADJ
ejpam-4386	298	32	normal	normal	ADJ
ejpam-4386	298	33	proof	proof	NOUN
ejpam-4386	298	34	.	.	PUNCT
ejpam-4386	299	1	assume	assume	VERB
ejpam-4386	299	2	the	the	DET
ejpam-4386	299	3	hypothesis	hypothesis	NOUN
ejpam-4386	299	4	.	.	PUNCT
ejpam-4386	300	1	pick	pick	VERB
ejpam-4386	300	2	a	a	DET
ejpam-4386	300	3	normal	normal	ADJ
ejpam-4386	300	4	topological	topological	ADJ
ejpam-4386	300	5	space	space	NOUN
ejpam-4386	300	6	(	(	PUNCT
ejpam-4386	300	7	y	y	PROPN
ejpam-4386	300	8	,	,	PUNCT
ejpam-4386	300	9	τ	τ	PROPN
ejpam-4386	300	10	′	′	NUM
ejpam-4386	300	11	)	)	PUNCT
ejpam-4386	300	12	and	and	CCONJ
ejpam-4386	300	13	a	a	DET
ejpam-4386	300	14	bijective	bijective	ADJ
ejpam-4386	300	15	function	function	NOUN
ejpam-4386	300	16	f	f	NOUN
ejpam-4386	300	17	:	:	PUNCT
ejpam-4386	300	18	(	(	PUNCT
ejpam-4386	300	19	x	x	X
ejpam-4386	300	20	,	,	PUNCT
ejpam-4386	300	21	τ	τ	PROPN
ejpam-4386	300	22	)	)	PUNCT
ejpam-4386	300	23	−→	−→	NOUN
ejpam-4386	300	24	(	(	PUNCT
ejpam-4386	300	25	y	y	PROPN
ejpam-4386	300	26	,	,	PUNCT
ejpam-4386	300	27	τ	τ	PROPN
ejpam-4386	300	28	′	′	NUM
ejpam-4386	300	29	)	)	PUNCT
ejpam-4386	300	30	such	such	ADJ
ejpam-4386	300	31	that	that	DET
ejpam-4386	300	32	f|a	f|a	NOUN
ejpam-4386	300	33	:	:	PUNCT
ejpam-4386	300	34	a	a	DET
ejpam-4386	300	35	−→	−→	NOUN
ejpam-4386	300	36	f(a	f(a	NOUN
ejpam-4386	300	37	)	)	PUNCT
ejpam-4386	300	38	is	be	AUX
ejpam-4386	300	39	a	a	DET
ejpam-4386	300	40	homeomorphism	homeomorphism	NOUN
ejpam-4386	300	41	for	for	ADP
ejpam-4386	300	42	any	any	DET
ejpam-4386	300	43	compact	compact	ADJ
ejpam-4386	300	44	subspace	subspace	NOUN
ejpam-4386	300	45	a	a	PRON
ejpam-4386	300	46	of	of	ADP
ejpam-4386	300	47	x.	x.	NOUN
ejpam-4386	300	48	as	as	SCONJ
ejpam-4386	300	49	x	x	PROPN
ejpam-4386	300	50	is	be	AUX
ejpam-4386	300	51	fréchet	fréchet	ADJ
ejpam-4386	300	52	,	,	PUNCT
ejpam-4386	300	53	then	then	ADV
ejpam-4386	300	54	by	by	ADP
ejpam-4386	300	55	lemma	lemma	PROPN
ejpam-4386	300	56	7	7	NUM
ejpam-4386	300	57	,	,	PUNCT
ejpam-4386	300	58	f	f	PROPN
ejpam-4386	300	59	is	be	AUX
ejpam-4386	300	60	continuous	continuous	ADJ
ejpam-4386	300	61	and	and	CCONJ
ejpam-4386	300	62	by	by	ADP
ejpam-4386	300	63	lemma	lemma	PROPN
ejpam-4386	300	64	6	6	NUM
ejpam-4386	300	65	,	,	PUNCT
ejpam-4386	300	66	we	we	PRON
ejpam-4386	300	67	get	get	VERB
ejpam-4386	300	68	(	(	PUNCT
ejpam-4386	300	69	y	y	PROPN
ejpam-4386	300	70	,	,	PUNCT
ejpam-4386	300	71	τ	τ	PROPN
ejpam-4386	300	72	′	′	NUM
ejpam-4386	300	73	)	)	PUNCT
ejpam-4386	300	74	is	be	AUX
ejpam-4386	300	75	t4	t4	PROPN
ejpam-4386	300	76	.	.	PROPN
ejpam-4386	300	77	pick	pick	VERB
ejpam-4386	300	78	the	the	DET
ejpam-4386	300	79	same	same	ADJ
ejpam-4386	300	80	bijection	bijection	NOUN
ejpam-4386	300	81	function	function	NOUN
ejpam-4386	300	82	f	f	NOUN
ejpam-4386	300	83	:	:	PUNCT
ejpam-4386	300	84	(	(	PUNCT
ejpam-4386	300	85	x	x	X
ejpam-4386	300	86	,	,	PUNCT
ejpam-4386	300	87	τ	τ	PROPN
ejpam-4386	300	88	s	s	PART
ejpam-4386	300	89	)	)	PUNCT
ejpam-4386	300	90	−→	−→	NOUN
ejpam-4386	300	91	(	(	PUNCT
ejpam-4386	300	92	y	y	PROPN
ejpam-4386	300	93	,	,	PUNCT
ejpam-4386	300	94	τ	τ	PROPN
ejpam-4386	300	95	′	′	NUM
ejpam-4386	300	96	)	)	PUNCT
ejpam-4386	300	97	which	which	PRON
ejpam-4386	300	98	is	be	AUX
ejpam-4386	300	99	continuous	continuous	ADJ
ejpam-4386	300	100	by	by	ADP
ejpam-4386	300	101	lemma	lemma	PROPN
ejpam-4386	300	102	5	5	NUM
ejpam-4386	300	103	.	.	PUNCT
ejpam-4386	301	1	let	let	VERB
ejpam-4386	301	2	b	b	X
ejpam-4386	301	3	be	be	AUX
ejpam-4386	301	4	any	any	DET
ejpam-4386	301	5	compact	compact	ADJ
ejpam-4386	301	6	subset	subset	NOUN
ejpam-4386	301	7	of	of	ADP
ejpam-4386	301	8	(	(	PUNCT
ejpam-4386	301	9	x	x	INTJ
ejpam-4386	301	10	,	,	PUNCT
ejpam-4386	301	11	τ	τ	PROPN
ejpam-4386	301	12	s	s	PART
ejpam-4386	301	13	)	)	PUNCT
ejpam-4386	301	14	,	,	PUNCT
ejpam-4386	301	15	then	then	ADV
ejpam-4386	301	16	f|b	f|b	NOUN
ejpam-4386	301	17	:	:	PUNCT
ejpam-4386	301	18	b	b	X
ejpam-4386	301	19	−→	−→	ADJ
ejpam-4386	301	20	f(b	f(b	PROPN
ejpam-4386	301	21	)	)	PUNCT
ejpam-4386	301	22	is	be	AUX
ejpam-4386	301	23	bijective	bijective	ADJ
ejpam-4386	301	24	and	and	CCONJ
ejpam-4386	301	25	continuous	continuous	ADJ
ejpam-4386	301	26	,	,	PUNCT
ejpam-4386	301	27	thus	thus	ADV
ejpam-4386	301	28	by	by	ADP
ejpam-4386	301	29	[	[	X
ejpam-4386	301	30	6	6	NUM
ejpam-4386	301	31	,	,	PUNCT
ejpam-4386	301	32	theorem	theorem	VERB
ejpam-4386	301	33	3.1.13	3.1.13	NUM
ejpam-4386	301	34	]	]	PUNCT
ejpam-4386	301	35	f|b	f|b	NOUN
ejpam-4386	301	36	is	be	AUX
ejpam-4386	301	37	a	a	DET
ejpam-4386	301	38	homeomorphism	homeomorphism	NOUN
ejpam-4386	301	39	.	.	PUNCT
ejpam-4386	302	1	therefore	therefore	ADV
ejpam-4386	302	2	,	,	PUNCT
ejpam-4386	302	3	(	(	PUNCT
ejpam-4386	302	4	x	x	X
ejpam-4386	302	5	,	,	PUNCT
ejpam-4386	302	6	τ	τ	PROPN
ejpam-4386	302	7	s	s	PART
ejpam-4386	302	8	)	)	PUNCT
ejpam-4386	302	9	is	be	AUX
ejpam-4386	302	10	c	c	NOUN
ejpam-4386	302	11	-	-	ADJ
ejpam-4386	302	12	normal	normal	ADJ
ejpam-4386	302	13	.	.	PUNCT
ejpam-4386	303	1	references	reference	NOUN
ejpam-4386	303	2	[	[	X
ejpam-4386	303	3	1	1	X
ejpam-4386	303	4	]	]	PUNCT
ejpam-4386	303	5	a	a	DET
ejpam-4386	303	6	alawadi	alawadi	NOUN
ejpam-4386	303	7	,	,	PUNCT
ejpam-4386	303	8	l	l	PROPN
ejpam-4386	303	9	kalantan	kalantan	PROPN
ejpam-4386	303	10	,	,	PUNCT
ejpam-4386	303	11	and	and	CCONJ
ejpam-4386	303	12	m	m	PROPN
ejpam-4386	303	13	m	m	PROPN
ejpam-4386	303	14	saeed	saeed	PROPN
ejpam-4386	303	15	.	.	PUNCT
ejpam-4386	304	1	on	on	ADP
ejpam-4386	304	2	the	the	DET
ejpam-4386	304	3	discrete	discrete	ADJ
ejpam-4386	304	4	extension	extension	NOUN
ejpam-4386	304	5	spaces	space	NOUN
ejpam-4386	304	6	.	.	PUNCT
ejpam-4386	305	1	journal	journal	PROPN
ejpam-4386	305	2	of	of	ADP
ejpam-4386	305	3	mathematical	mathematical	ADJ
ejpam-4386	305	4	analysis	analysis	NOUN
ejpam-4386	305	5	,	,	PUNCT
ejpam-4386	305	6	9(2):150–157	9(2):150–157	NOUN
ejpam-4386	305	7	.	.	PROPN
ejpam-4386	305	8	,	,	PUNCT
ejpam-4386	305	9	2018	2018	NUM
ejpam-4386	305	10	.	.	PUNCT
ejpam-4386	306	1	references	reference	NOUN
ejpam-4386	306	2	829	829	NUM
ejpam-4386	307	1	[	[	X
ejpam-4386	307	2	2	2	NUM
ejpam-4386	307	3	]	]	X
ejpam-4386	307	4	p	p	X
ejpam-4386	307	5	s	s	NOUN
ejpam-4386	307	6	alexandroff	alexandroff	NOUN
ejpam-4386	308	1	and	and	CCONJ
ejpam-4386	308	2	p	p	X
ejpam-4386	308	3	s	s	VERB
ejpam-4386	308	4	urysohn	urysohn	NOUN
ejpam-4386	308	5	.	.	PUNCT
ejpam-4386	309	1	mémoire	mémoire	PROPN
ejpam-4386	309	2	sur	sur	PROPN
ejpam-4386	309	3	les	les	X
ejpam-4386	309	4	espaces	espace	NOUN
ejpam-4386	309	5	topologiques	topologique	NOUN
ejpam-4386	309	6	compacts	compact	NOUN
ejpam-4386	309	7	.	.	PUNCT
ejpam-4386	310	1	verh	verh	ADJ
ejpam-4386	310	2	.	.	PUNCT
ejpam-4386	311	1	konink	konink	PROPN
ejpam-4386	311	2	.	.	PUNCT
ejpam-4386	312	1	acad	acad	PROPN
ejpam-4386	312	2	.	.	PUNCT
ejpam-4386	313	1	wetensch	wetensch	PROPN
ejpam-4386	313	2	.	.	PUNCT
ejpam-4386	314	1	amsterdam	amsterdam	PROPN
ejpam-4386	314	2	,	,	PUNCT
ejpam-4386	314	3	14:1–96	14:1–96	PROPN
ejpam-4386	314	4	.	.	PROPN
ejpam-4386	314	5	,	,	PUNCT
ejpam-4386	314	6	1929	1929	NUM
ejpam-4386	314	7	.	.	PUNCT
ejpam-4386	315	1	[	[	X
ejpam-4386	315	2	3	3	NUM
ejpam-4386	315	3	]	]	X
ejpam-4386	315	4	s	s	PART
ejpam-4386	315	5	alzahrani	alzahrani	NOUN
ejpam-4386	315	6	and	and	CCONJ
ejpam-4386	315	7	l	l	PROPN
ejpam-4386	315	8	kalantan	kalantan	PROPN
ejpam-4386	315	9	.	.	PUNCT
ejpam-4386	316	1	epinormality	epinormality	NOUN
ejpam-4386	316	2	.	.	PUNCT
ejpam-4386	317	1	journal	journal	PROPN
ejpam-4386	317	2	of	of	ADP
ejpam-4386	317	3	nonlinear	nonlinear	PROPN
ejpam-4386	317	4	sciences	sciences	PROPN
ejpam-4386	317	5	&	&	CCONJ
ejpam-4386	317	6	applications	application	NOUN
ejpam-4386	317	7	,	,	PUNCT
ejpam-4386	317	8	9(9):5398–5402	9(9):5398–5402	PROPN
ejpam-4386	317	9	.	.	PROPN
ejpam-4386	317	10	,	,	PUNCT
ejpam-4386	317	11	2016	2016	NUM
ejpam-4386	317	12	.	.	PUNCT
ejpam-4386	318	1	[	[	X
ejpam-4386	318	2	4	4	NUM
ejpam-4386	318	3	]	]	SYM
ejpam-4386	318	4	s	s	PART
ejpam-4386	318	5	alzahrani	alzahrani	NOUN
ejpam-4386	318	6	and	and	CCONJ
ejpam-4386	318	7	l	l	PROPN
ejpam-4386	318	8	kalantan	kalantan	PROPN
ejpam-4386	318	9	.	.	PUNCT
ejpam-4386	319	1	c	c	X
ejpam-4386	319	2	-	-	PUNCT
ejpam-4386	319	3	normal	normal	ADJ
ejpam-4386	319	4	topological	topological	ADJ
ejpam-4386	319	5	property	property	NOUN
ejpam-4386	319	6	.	.	PUNCT
ejpam-4386	320	1	filomat	filomat	NOUN
ejpam-4386	320	2	,	,	PUNCT
ejpam-4386	320	3	31(2):407–411	31(2):407–411	PROPN
ejpam-4386	320	4	.	.	PROPN
ejpam-4386	320	5	,	,	PUNCT
ejpam-4386	320	6	2017	2017	NUM
ejpam-4386	320	7	.	.	PUNCT
ejpam-4386	321	1	[	[	X
ejpam-4386	321	2	5	5	NUM
ejpam-4386	321	3	]	]	X
ejpam-4386	321	4	r	r	NOUN
ejpam-4386	321	5	engelking	engelking	NOUN
ejpam-4386	321	6	.	.	PUNCT
ejpam-4386	322	1	on	on	ADP
ejpam-4386	322	2	the	the	DET
ejpam-4386	322	3	double	double	ADJ
ejpam-4386	322	4	circumference	circumference	NOUN
ejpam-4386	322	5	of	of	ADP
ejpam-4386	322	6	alexandroff	alexandroff	NOUN
ejpam-4386	322	7	.	.	PUNCT
ejpam-4386	323	1	bull	bull	NOUN
ejpam-4386	323	2	.	.	PUNCT
ejpam-4386	324	1	acad	acad	PROPN
ejpam-4386	324	2	.	.	PUNCT
ejpam-4386	325	1	pol	pol	PROPN
ejpam-4386	325	2	.	.	PUNCT
ejpam-4386	326	1	sci	sci	PROPN
ejpam-4386	326	2	.	.	PUNCT
ejpam-4386	326	3	ser	ser	PROPN
ejpam-4386	326	4	.	.	PUNCT
ejpam-4386	327	1	astron	astron	PROPN
ejpam-4386	327	2	.	.	PUNCT
ejpam-4386	327	3	math	math	NOUN
ejpam-4386	327	4	.	.	PUNCT
ejpam-4386	328	1	phys	phy	NOUN
ejpam-4386	328	2	.	.	PUNCT
ejpam-4386	328	3	,	,	PUNCT
ejpam-4386	328	4	16(8):629–634	16(8):629–634	PROPN
ejpam-4386	328	5	.	.	PROPN
ejpam-4386	328	6	,	,	PUNCT
ejpam-4386	328	7	1968	1968	NUM
ejpam-4386	328	8	.	.	PUNCT
ejpam-4386	329	1	[	[	X
ejpam-4386	329	2	6	6	NUM
ejpam-4386	329	3	]	]	X
ejpam-4386	329	4	r	r	NOUN
ejpam-4386	329	5	engelking	engelking	NOUN
ejpam-4386	329	6	.	.	PUNCT
ejpam-4386	330	1	general	general	ADJ
ejpam-4386	330	2	topology	topology	PROPN
ejpam-4386	330	3	.	.	PUNCT
ejpam-4386	331	1	pwn	pwn	PROPN
ejpam-4386	331	2	,	,	PUNCT
ejpam-4386	331	3	warszawa	warszawa	PROPN
ejpam-4386	331	4	,	,	PUNCT
ejpam-4386	331	5	1977	1977	NUM
ejpam-4386	331	6	.	.	PUNCT
ejpam-4386	332	1	[	[	X
ejpam-4386	332	2	7	7	X
ejpam-4386	332	3	]	]	SYM
ejpam-4386	332	4	g	g	PROPN
ejpam-4386	332	5	gruenhage	gruenhage	NOUN
ejpam-4386	332	6	.	.	PUNCT
ejpam-4386	333	1	generalized	generalize	VERB
ejpam-4386	333	2	metric	metric	ADJ
ejpam-4386	333	3	spaces	space	NOUN
ejpam-4386	333	4	.	.	PUNCT
ejpam-4386	334	1	handbook	handbook	NOUN
ejpam-4386	334	2	of	of	ADP
ejpam-4386	334	3	set	set	NOUN
ejpam-4386	334	4	-	-	PUNCT
ejpam-4386	334	5	theoretic	theoretic	NOUN
ejpam-4386	334	6	topology	topology	NOUN
ejpam-4386	334	7	,	,	PUNCT
ejpam-4386	334	8	pages	page	VERB
ejpam-4386	334	9	423–501	423–501	NUM
ejpam-4386	334	10	.	.	PROPN
ejpam-4386	334	11	,	,	PUNCT
ejpam-4386	334	12	1984	1984	NUM
ejpam-4386	334	13	.	.	PUNCT
ejpam-4386	335	1	[	[	X
ejpam-4386	335	2	8	8	NUM
ejpam-4386	335	3	]	]	SYM
ejpam-4386	335	4	l	l	NOUN
ejpam-4386	335	5	l	l	PROPN
ejpam-4386	335	6	herrington	herrington	PROPN
ejpam-4386	335	7	.	.	PUNCT
ejpam-4386	336	1	characterizations	characterization	NOUN
ejpam-4386	336	2	of	of	ADP
ejpam-4386	336	3	urysohn	urysohn	NOUN
ejpam-4386	336	4	-	-	PUNCT
ejpam-4386	336	5	closed	close	VERB
ejpam-4386	336	6	spaces	space	NOUN
ejpam-4386	336	7	.	.	PUNCT
ejpam-4386	337	1	proceedings	proceeding	NOUN
ejpam-4386	337	2	of	of	ADP
ejpam-4386	337	3	the	the	DET
ejpam-4386	337	4	american	american	PROPN
ejpam-4386	337	5	mathematical	mathematical	PROPN
ejpam-4386	337	6	society	society	NOUN
ejpam-4386	337	7	,	,	PUNCT
ejpam-4386	337	8	pages	page	NOUN
ejpam-4386	337	9	435–439	435–439	NUM
ejpam-4386	337	10	.	.	PROPN
ejpam-4386	337	11	,	,	PUNCT
ejpam-4386	337	12	1976	1976	NUM
ejpam-4386	337	13	.	.	PUNCT
ejpam-4386	338	1	[	[	X
ejpam-4386	338	2	9	9	NUM
ejpam-4386	338	3	]	]	SYM
ejpam-4386	338	4	l	l	NOUN
ejpam-4386	338	5	kalantan	kalantan	PROPN
ejpam-4386	338	6	.	.	PUNCT
ejpam-4386	339	1	results	result	VERB
ejpam-4386	339	2	about	about	ADP
ejpam-4386	339	3	κ	κ	NOUN
ejpam-4386	339	4	-	-	NOUN
ejpam-4386	339	5	normality	normality	NOUN
ejpam-4386	339	6	.	.	PUNCT
ejpam-4386	340	1	topology	topology	NOUN
ejpam-4386	340	2	and	and	CCONJ
ejpam-4386	340	3	its	its	PRON
ejpam-4386	340	4	applications	application	NOUN
ejpam-4386	340	5	,	,	PUNCT
ejpam-4386	340	6	125(1):47–62	125(1):47–62	NUM
ejpam-4386	340	7	.	.	PROPN
ejpam-4386	340	8	,	,	PUNCT
ejpam-4386	340	9	2002	2002	NUM
ejpam-4386	340	10	.	.	PUNCT
ejpam-4386	341	1	[	[	X
ejpam-4386	341	2	10	10	NUM
ejpam-4386	341	3	]	]	X
ejpam-4386	341	4	m	m	VERB
ejpam-4386	341	5	mrs̆ević	mrs̆ević	PROPN
ejpam-4386	341	6	,	,	PUNCT
ejpam-4386	341	7	i	i	PRON
ejpam-4386	341	8	l	l	PROPN
ejpam-4386	341	9	reilly	reilly	ADV
ejpam-4386	341	10	,	,	PUNCT
ejpam-4386	341	11	and	and	CCONJ
ejpam-4386	341	12	m	m	PROPN
ejpam-4386	341	13	k	k	NOUN
ejpam-4386	341	14	vamanamurthy	vamanamurthy	NOUN
ejpam-4386	341	15	.	.	PUNCT
ejpam-4386	342	1	on	on	ADP
ejpam-4386	342	2	semi	semi	ADJ
ejpam-4386	342	3	-	-	ADJ
ejpam-4386	342	4	regularization	regularization	ADJ
ejpam-4386	342	5	topologies	topology	NOUN
ejpam-4386	342	6	.	.	PUNCT
ejpam-4386	343	1	journal	journal	NOUN
ejpam-4386	343	2	of	of	ADP
ejpam-4386	343	3	the	the	DET
ejpam-4386	343	4	australian	australian	ADJ
ejpam-4386	343	5	mathematical	mathematical	ADJ
ejpam-4386	343	6	society	society	NOUN
ejpam-4386	343	7	,	,	PUNCT
ejpam-4386	343	8	38(1):40–54	38(1):40–54	NUM
ejpam-4386	343	9	.	.	NUM
ejpam-4386	343	10	,	,	PUNCT
ejpam-4386	343	11	1985	1985	NUM
ejpam-4386	343	12	.	.	PUNCT
ejpam-4386	344	1	[	[	X
ejpam-4386	344	2	11	11	NUM
ejpam-4386	344	3	]	]	PUNCT
ejpam-4386	344	4	t.	t.	PROPN
ejpam-4386	344	5	noiri	noiri	PROPN
ejpam-4386	344	6	and	and	CCONJ
ejpam-4386	344	7	v.	v.	ADP
ejpam-4386	344	8	popa	popa	NOUN
ejpam-4386	344	9	.	.	PUNCT
ejpam-4386	345	1	on	on	ADP
ejpam-4386	345	2	almost	almost	ADV
ejpam-4386	345	3	b	b	NOUN
ejpam-4386	345	4	-	-	PUNCT
ejpam-4386	345	5	continuous	continuous	ADJ
ejpam-4386	345	6	functions	function	NOUN
ejpam-4386	345	7	.	.	PUNCT
ejpam-4386	346	1	acta	acta	PROPN
ejpam-4386	346	2	math	math	PROPN
ejpam-4386	346	3	hungar	hungar	NOUN
ejpam-4386	346	4	,	,	PUNCT
ejpam-4386	346	5	79(4):329–339	79(4):329–339	PROPN
ejpam-4386	346	6	.	.	PROPN
ejpam-4386	346	7	,	,	PUNCT
ejpam-4386	346	8	1998	1998	NUM
ejpam-4386	346	9	.	.	PUNCT
ejpam-4386	347	1	[	[	X
ejpam-4386	347	2	12	12	NUM
ejpam-4386	347	3	]	]	PUNCT
ejpam-4386	347	4	l	l	NOUN
ejpam-4386	347	5	steen	steen	PROPN
ejpam-4386	347	6	and	and	CCONJ
ejpam-4386	347	7	j	j	PROPN
ejpam-4386	347	8	a	a	DET
ejpam-4386	347	9	seebach	seebach	NOUN
ejpam-4386	347	10	.	.	PUNCT
ejpam-4386	348	1	counterexamples	counterexample	NOUN
ejpam-4386	348	2	in	in	ADP
ejpam-4386	348	3	topology	topology	NOUN
ejpam-4386	348	4	.	.	PUNCT
ejpam-4386	349	1	dover	dover	PROPN
ejpam-4386	349	2	publications	publications	PROPN
ejpam-4386	349	3	inc	inc	PROPN
ejpam-4386	349	4	,	,	PUNCT
ejpam-4386	349	5	usa	usa	PROPN
ejpam-4386	349	6	,	,	PUNCT
ejpam-4386	349	7	1995	1995	NUM
ejpam-4386	349	8	.	.	PUNCT