id	sid	tid	token	lemma	pos
ejpam-6375	1	1	european	european	PROPN
ejpam-6375	1	2	journal	journal	PROPN
ejpam-6375	1	3	of	of	ADP
ejpam-6375	1	4	pure	pure	ADJ
ejpam-6375	1	5	and	and	CCONJ
ejpam-6375	1	6	applied	applied	ADJ
ejpam-6375	1	7	mathematics	mathematic	NOUN
ejpam-6375	1	8	2025	2025	NUM
ejpam-6375	1	9	,	,	PUNCT
ejpam-6375	1	10	vol	vol	NOUN
ejpam-6375	1	11	.	.	PROPN
ejpam-6375	1	12	18	18	NUM
ejpam-6375	1	13	,	,	PUNCT
ejpam-6375	1	14	issue	issue	NOUN
ejpam-6375	1	15	4	4	NUM
ejpam-6375	1	16	,	,	PUNCT
ejpam-6375	1	17	article	article	NOUN
ejpam-6375	1	18	number	number	NOUN
ejpam-6375	1	19	6375	6375	NUM
ejpam-6375	1	20	issn	issn	PROPN
ejpam-6375	1	21	1307	1307	NUM
ejpam-6375	1	22	-	-	SYM
ejpam-6375	1	23	5543	5543	NUM
ejpam-6375	1	24	–	–	PUNCT
ejpam-6375	1	25	ejpam.com	ejpam.com	X
ejpam-6375	1	26	published	publish	VERB
ejpam-6375	1	27	by	by	ADP
ejpam-6375	1	28	new	new	PROPN
ejpam-6375	1	29	york	york	PROPN
ejpam-6375	1	30	business	business	PROPN
ejpam-6375	1	31	global	global	PROPN
ejpam-6375	2	1	an	an	DET
ejpam-6375	2	2	exploration	exploration	NOUN
ejpam-6375	2	3	of	of	ADP
ejpam-6375	2	4	compactness	compactness	NOUN
ejpam-6375	2	5	and	and	CCONJ
ejpam-6375	2	6	separation	separation	NOUN
ejpam-6375	2	7	axioms	axiom	NOUN
ejpam-6375	2	8	in	in	ADP
ejpam-6375	2	9	generalized	generalized	ADJ
ejpam-6375	2	10	primal	primal	ADJ
ejpam-6375	2	11	topological	topological	ADJ
ejpam-6375	2	12	spaces	space	NOUN
ejpam-6375	2	13	muhammad	muhammad	PROPN
ejpam-6375	2	14	shahbaz1	shahbaz1	PROPN
ejpam-6375	2	15	,	,	PUNCT
ejpam-6375	2	16	tayyab	tayyab	PROPN
ejpam-6375	2	17	kamran1	kamran1	PROPN
ejpam-6375	2	18	,	,	PUNCT
ejpam-6375	2	19	mariam	mariam	PROPN
ejpam-6375	2	20	imtiaz2	imtiaz2	PROPN
ejpam-6375	2	21	,	,	PUNCT
ejpam-6375	2	22	umar	umar	PROPN
ejpam-6375	2	23	ishtiaq3	ishtiaq3	PROPN
ejpam-6375	2	24	,	,	PUNCT
ejpam-6375	2	25	mohammad	mohammad	PROPN
ejpam-6375	2	26	akram4,∗	akram4,∗	PROPN
ejpam-6375	2	27	,	,	PUNCT
ejpam-6375	2	28	ioan	ioan	NOUN
ejpam-6375	2	29	-	-	PUNCT
ejpam-6375	2	30	lucian	lucian	PROPN
ejpam-6375	2	31	popa5,6	popa5,6	PROPN
ejpam-6375	2	32	1	1	NUM
ejpam-6375	2	33	department	department	NOUN
ejpam-6375	2	34	of	of	ADP
ejpam-6375	2	35	mathematics	mathematic	NOUN
ejpam-6375	2	36	,	,	PUNCT
ejpam-6375	2	37	quaid	quaid	PROPN
ejpam-6375	2	38	-	-	PUNCT
ejpam-6375	2	39	i	i	PROPN
ejpam-6375	2	40	-	-	PUNCT
ejpam-6375	2	41	azam	azam	PROPN
ejpam-6375	2	42	university	university	PROPN
ejpam-6375	2	43	islamabad	islamabad	PROPN
ejpam-6375	2	44	,	,	PUNCT
ejpam-6375	2	45	pakistan	pakistan	PROPN
ejpam-6375	2	46	2	2	NUM
ejpam-6375	2	47	department	department	NOUN
ejpam-6375	2	48	of	of	ADP
ejpam-6375	2	49	mathematics	mathematic	NOUN
ejpam-6375	2	50	,	,	PUNCT
ejpam-6375	2	51	the	the	DET
ejpam-6375	2	52	islamia	islamia	PROPN
ejpam-6375	2	53	university	university	PROPN
ejpam-6375	2	54	of	of	ADP
ejpam-6375	2	55	bahawalpur	bahawalpur	PROPN
ejpam-6375	2	56	,	,	PUNCT
ejpam-6375	2	57	bahawalnagar	bahawalnagar	NOUN
ejpam-6375	2	58	campus	campus	PROPN
ejpam-6375	2	59	,	,	PUNCT
ejpam-6375	2	60	pakistan	pakistan	PROPN
ejpam-6375	2	61	3	3	NUM
ejpam-6375	2	62	office	office	NOUN
ejpam-6375	2	63	of	of	ADP
ejpam-6375	2	64	research	research	NOUN
ejpam-6375	2	65	,	,	PUNCT
ejpam-6375	2	66	innovation	innovation	NOUN
ejpam-6375	2	67	and	and	CCONJ
ejpam-6375	2	68	commercialization	commercialization	NOUN
ejpam-6375	2	69	,	,	PUNCT
ejpam-6375	2	70	university	university	NOUN
ejpam-6375	2	71	of	of	ADP
ejpam-6375	2	72	management	management	NOUN
ejpam-6375	2	73	and	and	CCONJ
ejpam-6375	2	74	technology	technology	NOUN
ejpam-6375	2	75	,	,	PUNCT
ejpam-6375	2	76	lahore	lahore	NOUN
ejpam-6375	2	77	54770	54770	NUM
ejpam-6375	2	78	,	,	PUNCT
ejpam-6375	2	79	pakistan	pakistan	PROPN
ejpam-6375	2	80	4	4	NUM
ejpam-6375	2	81	department	department	NOUN
ejpam-6375	2	82	of	of	ADP
ejpam-6375	2	83	mathematics	mathematic	NOUN
ejpam-6375	2	84	,	,	PUNCT
ejpam-6375	2	85	faculty	faculty	NOUN
ejpam-6375	2	86	of	of	ADP
ejpam-6375	2	87	science	science	NOUN
ejpam-6375	2	88	,	,	PUNCT
ejpam-6375	2	89	islamic	islamic	PROPN
ejpam-6375	2	90	university	university	PROPN
ejpam-6375	2	91	of	of	ADP
ejpam-6375	2	92	madinah	madinah	PROPN
ejpam-6375	2	93	,	,	PUNCT
ejpam-6375	2	94	madinah	madinah	PROPN
ejpam-6375	2	95	42351	42351	NUM
ejpam-6375	2	96	,	,	PUNCT
ejpam-6375	2	97	saudi	saudi	PROPN
ejpam-6375	2	98	arabia	arabia	PROPN
ejpam-6375	2	99	5	5	NUM
ejpam-6375	2	100	department	department	NOUN
ejpam-6375	2	101	of	of	ADP
ejpam-6375	2	102	computing	computing	NOUN
ejpam-6375	2	103	,	,	PUNCT
ejpam-6375	2	104	mathematics	mathematic	NOUN
ejpam-6375	2	105	and	and	CCONJ
ejpam-6375	2	106	electronics	electronic	NOUN
ejpam-6375	2	107	,	,	PUNCT
ejpam-6375	2	108	“	"	PUNCT
ejpam-6375	2	109	1	1	NUM
ejpam-6375	2	110	decembrie	decembrie	NOUN
ejpam-6375	2	111	1918	1918	NUM
ejpam-6375	2	112	”	"	PUNCT
ejpam-6375	2	113	university	university	PROPN
ejpam-6375	2	114	of	of	ADP
ejpam-6375	2	115	alba	alba	PROPN
ejpam-6375	2	116	iulia	iulia	PROPN
ejpam-6375	2	117	,	,	PUNCT
ejpam-6375	2	118	510009	510009	NUM
ejpam-6375	2	119	alba	alba	PROPN
ejpam-6375	2	120	iulia	iulia	PROPN
ejpam-6375	2	121	,	,	PUNCT
ejpam-6375	2	122	romania	romania	PROPN
ejpam-6375	2	123	6	6	NUM
ejpam-6375	2	124	faculty	faculty	NOUN
ejpam-6375	2	125	of	of	ADP
ejpam-6375	2	126	mathematics	mathematic	NOUN
ejpam-6375	2	127	and	and	CCONJ
ejpam-6375	2	128	computer	computer	NOUN
ejpam-6375	2	129	science	science	NOUN
ejpam-6375	2	130	,	,	PUNCT
ejpam-6375	2	131	transilvania	transilvania	PROPN
ejpam-6375	2	132	university	university	PROPN
ejpam-6375	2	133	of	of	ADP
ejpam-6375	2	134	brasov	brasov	NOUN
ejpam-6375	2	135	,	,	PUNCT
ejpam-6375	2	136	iuliu	iuliu	PROPN
ejpam-6375	2	137	maniu	maniu	PROPN
ejpam-6375	2	138	street	street	PROPN
ejpam-6375	2	139	50	50	NUM
ejpam-6375	2	140	,	,	PUNCT
ejpam-6375	2	141	500091	500091	NUM
ejpam-6375	2	142	brasov	brasov	NOUN
ejpam-6375	2	143	,	,	PUNCT
ejpam-6375	2	144	romania	romania	PROPN
ejpam-6375	2	145	abstract	abstract	NOUN
ejpam-6375	2	146	.	.	PUNCT
ejpam-6375	3	1	the	the	DET
ejpam-6375	3	2	research	research	NOUN
ejpam-6375	3	3	explores	explore	NOUN
ejpam-6375	3	4	s∗	s∗	VERB
ejpam-6375	3	5	g	g	PROPN
ejpam-6375	3	6	-compactness	-compactness	NOUN
ejpam-6375	3	7	together	together	ADV
ejpam-6375	3	8	with	with	ADP
ejpam-6375	3	9	s∗	s∗	PROPN
ejpam-6375	3	10	g	g	PROPN
ejpam-6375	3	11	-connectedness	-connectedness	NOUN
ejpam-6375	3	12	in	in	ADP
ejpam-6375	3	13	generalized	generalized	ADJ
ejpam-6375	3	14	primal	primal	ADJ
ejpam-6375	3	15	topological	topological	ADJ
ejpam-6375	3	16	spaces	space	NOUN
ejpam-6375	3	17	to	to	PART
ejpam-6375	3	18	enhance	enhance	VERB
ejpam-6375	3	19	theoretical	theoretical	ADJ
ejpam-6375	3	20	knowledge	knowledge	NOUN
ejpam-6375	3	21	of	of	ADP
ejpam-6375	3	22	non	non	ADJ
ejpam-6375	3	23	-	-	ADJ
ejpam-6375	3	24	classical	classical	ADJ
ejpam-6375	3	25	topological	topological	ADJ
ejpam-6375	3	26	systems	system	NOUN
ejpam-6375	3	27	.	.	PUNCT
ejpam-6375	4	1	this	this	DET
ejpam-6375	4	2	paper	paper	NOUN
ejpam-6375	4	3	provides	provide	VERB
ejpam-6375	4	4	an	an	DET
ejpam-6375	4	5	extensive	extensive	ADJ
ejpam-6375	4	6	analysis	analysis	NOUN
ejpam-6375	4	7	of	of	ADP
ejpam-6375	4	8	these	these	DET
ejpam-6375	4	9	two	two	NUM
ejpam-6375	4	10	concepts	concept	NOUN
ejpam-6375	4	11	to	to	PART
ejpam-6375	4	12	show	show	VERB
ejpam-6375	4	13	their	their	PRON
ejpam-6375	4	14	characteristics	characteristic	NOUN
ejpam-6375	4	15	and	and	CCONJ
ejpam-6375	4	16	potential	potential	ADJ
ejpam-6375	4	17	applications	application	NOUN
ejpam-6375	4	18	.	.	PUNCT
ejpam-6375	5	1	the	the	DET
ejpam-6375	5	2	research	research	NOUN
ejpam-6375	5	3	examines	examine	VERB
ejpam-6375	5	4	the	the	DET
ejpam-6375	5	5	relationship	relationship	NOUN
ejpam-6375	5	6	dynamics	dynamic	NOUN
ejpam-6375	5	7	between	between	ADP
ejpam-6375	5	8	t0	t0	PROPN
ejpam-6375	5	9	,	,	PUNCT
ejpam-6375	5	10	t1	t1	NOUN
ejpam-6375	5	11	,	,	PUNCT
ejpam-6375	5	12	and	and	CCONJ
ejpam-6375	5	13	t2	t2	NOUN
ejpam-6375	5	14	separation	separation	NOUN
ejpam-6375	5	15	axioms	axiom	NOUN
ejpam-6375	5	16	and	and	CCONJ
ejpam-6375	5	17	these	these	DET
ejpam-6375	5	18	concepts	concept	NOUN
ejpam-6375	5	19	throughout	throughout	ADP
ejpam-6375	5	20	their	their	PRON
ejpam-6375	5	21	expanded	expand	VERB
ejpam-6375	5	22	theoretical	theoretical	ADJ
ejpam-6375	5	23	framework	framework	NOUN
ejpam-6375	5	24	.	.	PUNCT
ejpam-6375	6	1	examining	examine	VERB
ejpam-6375	6	2	s∗	s∗	PROPN
ejpam-6375	6	3	g	g	PROPN
ejpam-6375	6	4	-compactness	-compactness	NOUN
ejpam-6375	6	5	and	and	CCONJ
ejpam-6375	6	6	s∗	s∗	PROPN
ejpam-6375	6	7	g	g	PROPN
ejpam-6375	6	8	-connectedness	-connectedness	NOUN
ejpam-6375	6	9	independently	independently	ADV
ejpam-6375	6	10	provides	provide	VERB
ejpam-6375	6	11	an	an	DET
ejpam-6375	6	12	advanced	advanced	ADJ
ejpam-6375	6	13	understanding	understanding	NOUN
ejpam-6375	6	14	of	of	ADP
ejpam-6375	6	15	generalized	generalized	ADJ
ejpam-6375	6	16	primal	primal	ADJ
ejpam-6375	6	17	spaces	space	NOUN
ejpam-6375	6	18	,	,	PUNCT
ejpam-6375	6	19	although	although	SCONJ
ejpam-6375	6	20	they	they	PRON
ejpam-6375	6	21	diverge	diverge	VERB
ejpam-6375	6	22	from	from	ADP
ejpam-6375	6	23	standard	standard	ADJ
ejpam-6375	6	24	separation	separation	NOUN
ejpam-6375	6	25	properties	property	NOUN
ejpam-6375	6	26	.	.	PUNCT
ejpam-6375	7	1	this	this	DET
ejpam-6375	7	2	study	study	NOUN
ejpam-6375	7	3	simultaneously	simultaneously	ADV
ejpam-6375	7	4	supports	support	VERB
ejpam-6375	7	5	theoretical	theoretical	ADJ
ejpam-6375	7	6	research	research	NOUN
ejpam-6375	7	7	of	of	ADP
ejpam-6375	7	8	these	these	DET
ejpam-6375	7	9	domains	domain	NOUN
ejpam-6375	7	10	while	while	SCONJ
ejpam-6375	7	11	building	build	VERB
ejpam-6375	7	12	essential	essential	ADJ
ejpam-6375	7	13	foundations	foundation	NOUN
ejpam-6375	7	14	for	for	ADP
ejpam-6375	7	15	math	math	NOUN
ejpam-6375	7	16	investigations	investigation	NOUN
ejpam-6375	7	17	in	in	ADP
ejpam-6375	7	18	this	this	DET
ejpam-6375	7	19	field	field	NOUN
ejpam-6375	7	20	.	.	PUNCT
ejpam-6375	8	1	2020	2020	NUM
ejpam-6375	8	2	mathematics	mathematic	NOUN
ejpam-6375	8	3	subject	subject	NOUN
ejpam-6375	8	4	classifications	classification	NOUN
ejpam-6375	8	5	:	:	PUNCT
ejpam-6375	8	6	54a05	54a05	NUM
ejpam-6375	8	7	,	,	PUNCT
ejpam-6375	8	8	54a10	54a10	NUM
ejpam-6375	8	9	key	key	ADJ
ejpam-6375	8	10	words	word	NOUN
ejpam-6375	8	11	and	and	CCONJ
ejpam-6375	8	12	phrases	phrase	NOUN
ejpam-6375	8	13	:	:	PUNCT
ejpam-6375	8	14	generalized	generalized	ADJ
ejpam-6375	8	15	topological	topological	ADJ
ejpam-6375	8	16	space	space	NOUN
ejpam-6375	8	17	,	,	PUNCT
ejpam-6375	8	18	hausdorff	hausdorff	NOUN
ejpam-6375	8	19	space	space	NOUN
ejpam-6375	8	20	,	,	PUNCT
ejpam-6375	8	21	compactness	compactness	NOUN
ejpam-6375	8	22	,	,	PUNCT
ejpam-6375	8	23	primal	primal	ADJ
ejpam-6375	8	24	topological	topological	ADJ
ejpam-6375	8	25	space	space	NOUN
ejpam-6375	8	26	1	1	NUM
ejpam-6375	8	27	.	.	PUNCT
ejpam-6375	8	28	introduction	introduction	NOUN
ejpam-6375	8	29	topology	topology	NOUN
ejpam-6375	8	30	,	,	PUNCT
ejpam-6375	8	31	as	as	ADP
ejpam-6375	8	32	a	a	DET
ejpam-6375	8	33	core	core	ADJ
ejpam-6375	8	34	branch	branch	NOUN
ejpam-6375	8	35	of	of	ADP
ejpam-6375	8	36	modern	modern	ADJ
ejpam-6375	8	37	mathematics	mathematic	NOUN
ejpam-6375	8	38	,	,	PUNCT
ejpam-6375	8	39	offers	offer	VERB
ejpam-6375	8	40	powerful	powerful	ADJ
ejpam-6375	8	41	tools	tool	NOUN
ejpam-6375	8	42	for	for	ADP
ejpam-6375	8	43	understanding	understand	VERB
ejpam-6375	8	44	ideas	idea	NOUN
ejpam-6375	8	45	such	such	ADJ
ejpam-6375	8	46	as	as	ADP
ejpam-6375	8	47	convergence	convergence	NOUN
ejpam-6375	8	48	,	,	PUNCT
ejpam-6375	8	49	continuity	continuity	NOUN
ejpam-6375	8	50	,	,	PUNCT
ejpam-6375	8	51	and	and	CCONJ
ejpam-6375	8	52	separation	separation	NOUN
ejpam-6375	8	53	axioms	axiom	NOUN
ejpam-6375	8	54	in	in	ADP
ejpam-6375	8	55	different	different	ADJ
ejpam-6375	8	56	types	type	NOUN
ejpam-6375	8	57	of	of	ADP
ejpam-6375	8	58	∗corresponding	∗corresponde	VERB
ejpam-6375	8	59	author	author	NOUN
ejpam-6375	8	60	.	.	PUNCT
ejpam-6375	9	1	doi	doi	NOUN
ejpam-6375	9	2	:	:	PUNCT
ejpam-6375	9	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6375	https://doi.org/10.29020/nybg.ejpam.v18i4.6375	PROPN
ejpam-6375	9	4	email	email	NOUN
ejpam-6375	9	5	addresses	address	NOUN
ejpam-6375	9	6	:	:	PUNCT
ejpam-6375	9	7	mshahbaz@math.qau.edu.pk	mshahbaz@math.qau.edu.pk	PROPN
ejpam-6375	9	8	(	(	PUNCT
ejpam-6375	9	9	m.	m.	PROPN
ejpam-6375	9	10	shahbaz	shahbaz	PROPN
ejpam-6375	9	11	)	)	PUNCT
ejpam-6375	9	12	,	,	PUNCT
ejpam-6375	9	13	tkamran@qau.edu.pk	tkamran@qau.edu.pk	PROPN
ejpam-6375	9	14	(	(	PUNCT
ejpam-6375	9	15	t.	t.	PROPN
ejpam-6375	9	16	kamran	kamran	PROPN
ejpam-6375	9	17	)	)	PUNCT
ejpam-6375	9	18	,	,	PUNCT
ejpam-6375	9	19	mariamimtiaz122@gmail.com	mariamimtiaz122@gmail.com	X
ejpam-6375	9	20	(	(	PUNCT
ejpam-6375	9	21	m.	m.	PROPN
ejpam-6375	9	22	imtiaz	imtiaz	PROPN
ejpam-6375	9	23	)	)	PUNCT
ejpam-6375	9	24	,	,	PUNCT
ejpam-6375	9	25	umarishtiaq000@gmail.com	umarishtiaq000@gmail.com	PROPN
ejpam-6375	10	1	(	(	PUNCT
ejpam-6375	10	2	u.	u.	PROPN
ejpam-6375	10	3	ishtiaq	ishtiaq	PROPN
ejpam-6375	10	4	)	)	PUNCT
ejpam-6375	10	5	,	,	PUNCT
ejpam-6375	10	6	akramkhan_20@rediffmail.com	akramkhan_20@rediffmail.com	X
ejpam-6375	10	7	(	(	PUNCT
ejpam-6375	10	8	m.	m.	NOUN
ejpam-6375	10	9	akram	akram	PROPN
ejpam-6375	10	10	)	)	PUNCT
ejpam-6375	10	11	,	,	PUNCT
ejpam-6375	10	12	lucian.popa@uab.ro	lucian.popa@uab.ro	NOUN
ejpam-6375	10	13	(	(	PUNCT
ejpam-6375	10	14	i.	i.	PROPN
ejpam-6375	10	15	l.	l.	PROPN
ejpam-6375	10	16	popa	popa	PROPN
ejpam-6375	10	17	)	)	PUNCT
ejpam-6375	10	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6375	11	1	1	1	NUM
ejpam-6375	11	2	copyright	copyright	NOUN
ejpam-6375	11	3	:	:	PUNCT
ejpam-6375	11	4	©	©	PROPN
ejpam-6375	11	5	2025	2025	NUM
ejpam-6375	11	6	the	the	DET
ejpam-6375	11	7	author(s	author(s	NOUN
ejpam-6375	11	8	)	)	PUNCT
ejpam-6375	11	9	.	.	PUNCT
ejpam-6375	12	1	(	(	PUNCT
ejpam-6375	12	2	cc	cc	NOUN
ejpam-6375	12	3	by	by	ADP
ejpam-6375	12	4	-	-	PUNCT
ejpam-6375	12	5	nc	nc	PROPN
ejpam-6375	12	6	4.0	4.0	NUM
ejpam-6375	12	7	)	)	PUNCT
ejpam-6375	12	8	m.	m.	NOUN
ejpam-6375	12	9	shahbaz	shahbaz	PROPN
ejpam-6375	12	10	et	et	PROPN
ejpam-6375	12	11	al	al	PROPN
ejpam-6375	12	12	.	.	PUNCT
ejpam-6375	12	13	/	/	SYM
ejpam-6375	12	14	eur	eur	PROPN
ejpam-6375	12	15	.	.	PUNCT
ejpam-6375	13	1	j.	j.	PROPN
ejpam-6375	13	2	pure	pure	PROPN
ejpam-6375	13	3	appl	appl	PROPN
ejpam-6375	13	4	.	.	PROPN
ejpam-6375	13	5	math	math	PROPN
ejpam-6375	13	6	,	,	PUNCT
ejpam-6375	13	7	18	18	NUM
ejpam-6375	13	8	(	(	PUNCT
ejpam-6375	13	9	4	4	NUM
ejpam-6375	13	10	)	)	PUNCT
ejpam-6375	13	11	(	(	PUNCT
ejpam-6375	13	12	2025	2025	NUM
ejpam-6375	13	13	)	)	PUNCT
ejpam-6375	13	14	,	,	PUNCT
ejpam-6375	13	15	6375	6375	NUM
ejpam-6375	13	16	2	2	NUM
ejpam-6375	13	17	of	of	ADP
ejpam-6375	13	18	22	22	NUM
ejpam-6375	13	19	spaces	space	NOUN
ejpam-6375	13	20	.	.	PUNCT
ejpam-6375	14	1	the	the	DET
ejpam-6375	14	2	theoretical	theoretical	ADJ
ejpam-6375	14	3	and	and	CCONJ
ejpam-6375	14	4	practical	practical	ADJ
ejpam-6375	14	5	world	world	NOUN
ejpam-6375	14	6	bases	base	VERB
ejpam-6375	14	7	its	its	PRON
ejpam-6375	14	8	work	work	NOUN
ejpam-6375	14	9	on	on	ADP
ejpam-6375	14	10	compactness	compactness	NOUN
ejpam-6375	14	11	and	and	CCONJ
ejpam-6375	14	12	connectedness	connectedness	NOUN
ejpam-6375	14	13	—	—	PUNCT
ejpam-6375	14	14	influential	influential	ADJ
ejpam-6375	14	15	concepts	concept	NOUN
ejpam-6375	14	16	which	which	PRON
ejpam-6375	14	17	lead	lead	VERB
ejpam-6375	14	18	topology	topology	NOUN
ejpam-6375	14	19	to	to	ADP
ejpam-6375	14	20	its	its	PRON
ejpam-6375	14	21	most	most	ADV
ejpam-6375	14	22	important	important	ADJ
ejpam-6375	14	23	advancements	advancement	NOUN
ejpam-6375	14	24	.	.	PUNCT
ejpam-6375	15	1	the	the	DET
ejpam-6375	15	2	recent	recent	ADJ
ejpam-6375	15	3	study	study	NOUN
ejpam-6375	15	4	conducted	conduct	VERB
ejpam-6375	15	5	by	by	ADP
ejpam-6375	15	6	shahbaz	shahbaz	PROPN
ejpam-6375	15	7	et	et	PROPN
ejpam-6375	15	8	al	al	PROPN
ejpam-6375	15	9	.	.	PUNCT
ejpam-6375	16	1	[	[	X
ejpam-6375	16	2	1	1	NUM
ejpam-6375	16	3	,	,	PUNCT
ejpam-6375	16	4	2	2	NUM
ejpam-6375	16	5	]	]	PUNCT
ejpam-6375	16	6	investigates	investigate	VERB
ejpam-6375	16	7	various	various	ADJ
ejpam-6375	16	8	continuity	continuity	NOUN
ejpam-6375	16	9	,	,	PUNCT
ejpam-6375	16	10	compactness	compactness	NOUN
ejpam-6375	16	11	,	,	PUNCT
ejpam-6375	16	12	and	and	CCONJ
ejpam-6375	16	13	connectedness	connectedness	NOUN
ejpam-6375	16	14	frameworks	framework	NOUN
ejpam-6375	16	15	that	that	PRON
ejpam-6375	16	16	generate	generate	VERB
ejpam-6375	16	17	novel	novel	ADJ
ejpam-6375	16	18	insights	insight	NOUN
ejpam-6375	16	19	which	which	PRON
ejpam-6375	16	20	define	define	VERB
ejpam-6375	16	21	modern	modern	ADJ
ejpam-6375	16	22	topological	topological	ADJ
ejpam-6375	16	23	research	research	NOUN
ejpam-6375	16	24	.	.	PUNCT
ejpam-6375	17	1	their	their	PRON
ejpam-6375	17	2	research	research	NOUN
ejpam-6375	17	3	on	on	ADP
ejpam-6375	17	4	generalized	generalized	ADJ
ejpam-6375	17	5	topological	topological	ADJ
ejpam-6375	17	6	spaces	space	NOUN
ejpam-6375	17	7	leads	lead	VERB
ejpam-6375	17	8	the	the	DET
ejpam-6375	17	9	way	way	NOUN
ejpam-6375	17	10	toward	toward	ADP
ejpam-6375	17	11	investigating	investigate	VERB
ejpam-6375	17	12	particular	particular	ADJ
ejpam-6375	17	13	cases	case	NOUN
ejpam-6375	17	14	of	of	ADP
ejpam-6375	17	15	spaces	space	NOUN
ejpam-6375	17	16	especially	especially	ADV
ejpam-6375	17	17	within	within	ADP
ejpam-6375	17	18	generalized	generalized	ADJ
ejpam-6375	17	19	primal	primal	ADJ
ejpam-6375	17	20	topological	topological	ADJ
ejpam-6375	17	21	spaces	space	NOUN
ejpam-6375	17	22	.	.	PUNCT
ejpam-6375	18	1	topological	topological	ADJ
ejpam-6375	18	2	ideas	idea	NOUN
ejpam-6375	18	3	face	face	VERB
ejpam-6375	18	4	increasing	increase	VERB
ejpam-6375	18	5	interest	interest	NOUN
ejpam-6375	18	6	from	from	ADP
ejpam-6375	18	7	scholars	scholar	NOUN
ejpam-6375	18	8	to	to	PART
ejpam-6375	18	9	rethink	rethink	VERB
ejpam-6375	18	10	traditional	traditional	ADJ
ejpam-6375	18	11	interpretations	interpretation	NOUN
ejpam-6375	18	12	through	through	ADP
ejpam-6375	18	13	abstract	abstract	ADJ
ejpam-6375	18	14	flexible	flexible	ADJ
ejpam-6375	18	15	frameworks	framework	NOUN
ejpam-6375	18	16	within	within	ADP
ejpam-6375	18	17	the	the	DET
ejpam-6375	18	18	last	last	ADJ
ejpam-6375	18	19	few	few	ADJ
ejpam-6375	18	20	years	year	NOUN
ejpam-6375	18	21	.	.	PUNCT
ejpam-6375	19	1	generalized	generalize	VERB
ejpam-6375	19	2	primal	primal	ADJ
ejpam-6375	19	3	topological	topological	ADJ
ejpam-6375	19	4	spaces	space	NOUN
ejpam-6375	19	5	emerged	emerge	VERB
ejpam-6375	19	6	because	because	SCONJ
ejpam-6375	19	7	of	of	ADP
ejpam-6375	19	8	heightened	heighten	VERB
ejpam-6375	19	9	interest	interest	NOUN
ejpam-6375	19	10	and	and	CCONJ
ejpam-6375	19	11	created	create	VERB
ejpam-6375	19	12	new	new	ADJ
ejpam-6375	19	13	ways	way	NOUN
ejpam-6375	19	14	to	to	PART
ejpam-6375	19	15	understand	understand	VERB
ejpam-6375	19	16	familiar	familiar	ADJ
ejpam-6375	19	17	concepts	concept	NOUN
ejpam-6375	19	18	.	.	PUNCT
ejpam-6375	20	1	historical	historical	ADJ
ejpam-6375	20	2	fundamental	fundamental	ADJ
ejpam-6375	20	3	principles	principle	NOUN
ejpam-6375	20	4	about	about	ADP
ejpam-6375	20	5	open	open	ADJ
ejpam-6375	20	6	and	and	CCONJ
ejpam-6375	20	7	closed	closed	ADJ
ejpam-6375	20	8	sets	set	NOUN
ejpam-6375	20	9	form	form	VERB
ejpam-6375	20	10	the	the	DET
ejpam-6375	20	11	basis	basis	NOUN
ejpam-6375	20	12	of	of	ADP
ejpam-6375	20	13	developing	develop	VERB
ejpam-6375	20	14	these	these	DET
ejpam-6375	20	15	spaces	space	NOUN
ejpam-6375	20	16	.	.	PUNCT
ejpam-6375	21	1	the	the	DET
ejpam-6375	21	2	generalized	generalize	VERB
ejpam-6375	21	3	closed	close	VERB
ejpam-6375	21	4	sets	set	NOUN
ejpam-6375	21	5	theory	theory	NOUN
ejpam-6375	21	6	of	of	ADP
ejpam-6375	21	7	levine	levine	PROPN
ejpam-6375	21	8	from	from	ADP
ejpam-6375	21	9	1970	1970	NUM
ejpam-6375	21	10	provided	provide	VERB
ejpam-6375	21	11	innovative	innovative	ADJ
ejpam-6375	21	12	methods	method	NOUN
ejpam-6375	21	13	for	for	ADP
ejpam-6375	21	14	examining	examine	VERB
ejpam-6375	21	15	covering	cover	VERB
ejpam-6375	21	16	properties	property	NOUN
ejpam-6375	21	17	and	and	CCONJ
ejpam-6375	21	18	separation	separation	NOUN
ejpam-6375	21	19	even	even	ADV
ejpam-6375	21	20	though	though	SCONJ
ejpam-6375	21	21	his	his	PRON
ejpam-6375	21	22	original	original	ADJ
ejpam-6375	21	23	research	research	NOUN
ejpam-6375	21	24	investigated	investigate	VERB
ejpam-6375	21	25	generalized	generalized	ADJ
ejpam-6375	21	26	topology	topology	NOUN
ejpam-6375	21	27	but	but	CCONJ
ejpam-6375	21	28	these	these	DET
ejpam-6375	21	29	principles	principle	NOUN
ejpam-6375	21	30	align	align	VERB
ejpam-6375	21	31	with	with	ADP
ejpam-6375	21	32	contemporary	contemporary	ADJ
ejpam-6375	21	33	primal	primal	ADJ
ejpam-6375	21	34	environment	environment	NOUN
ejpam-6375	21	35	research	research	NOUN
ejpam-6375	21	36	.	.	PUNCT
ejpam-6375	22	1	császár	császár	PROPN
ejpam-6375	22	2	developed	develop	VERB
ejpam-6375	22	3	a	a	DET
ejpam-6375	22	4	set	set	NOUN
ejpam-6375	22	5	of	of	ADP
ejpam-6375	22	6	theoretical	theoretical	ADJ
ejpam-6375	22	7	generalized	generalized	ADJ
ejpam-6375	22	8	open	open	ADJ
ejpam-6375	22	9	sets	set	NOUN
ejpam-6375	22	10	[	[	X
ejpam-6375	22	11	3	3	X
ejpam-6375	22	12	]	]	PUNCT
ejpam-6375	22	13	that	that	PRON
ejpam-6375	22	14	built	build	VERB
ejpam-6375	22	15	the	the	DET
ejpam-6375	22	16	essential	essential	ADJ
ejpam-6375	22	17	framework	framework	NOUN
ejpam-6375	22	18	for	for	ADP
ejpam-6375	22	19	open	open	ADJ
ejpam-6375	22	20	set	set	VERB
ejpam-6375	22	21	analysis	analysis	NOUN
ejpam-6375	22	22	for	for	ADP
ejpam-6375	22	23	non	non	ADJ
ejpam-6375	22	24	-	-	ADJ
ejpam-6375	22	25	traditional	traditional	ADJ
ejpam-6375	22	26	mathematical	mathematical	ADJ
ejpam-6375	22	27	spaces	space	NOUN
ejpam-6375	22	28	.	.	PUNCT
ejpam-6375	23	1	maki	maki	NOUN
ejpam-6375	23	2	,	,	PUNCT
ejpam-6375	23	3	balachandran	balachandran	NOUN
ejpam-6375	23	4	and	and	CCONJ
ejpam-6375	23	5	devi	devi	VERB
ejpam-6375	23	6	[	[	X
ejpam-6375	23	7	4	4	NUM
ejpam-6375	23	8	]	]	PUNCT
ejpam-6375	23	9	,	,	PUNCT
ejpam-6375	23	10	maki	maki	NOUN
ejpam-6375	23	11	,	,	PUNCT
ejpam-6375	23	12	rao	rao	NOUN
ejpam-6375	23	13	,	,	PUNCT
ejpam-6375	23	14	and	and	CCONJ
ejpam-6375	23	15	gani	gani	X
ejpam-6375	24	1	[	[	X
ejpam-6375	24	2	5	5	NUM
ejpam-6375	24	3	]	]	PUNCT
ejpam-6375	24	4	,	,	PUNCT
ejpam-6375	24	5	and	and	CCONJ
ejpam-6375	24	6	navalagi	navalagi	ADJ
ejpam-6375	24	7	along	along	ADP
ejpam-6375	24	8	with	with	ADP
ejpam-6375	24	9	page	page	NOUN
ejpam-6375	25	1	[	[	X
ejpam-6375	25	2	6	6	NUM
ejpam-6375	25	3	]	]	PUNCT
ejpam-6375	25	4	,	,	PUNCT
ejpam-6375	25	5	have	have	AUX
ejpam-6375	25	6	made	make	VERB
ejpam-6375	25	7	essential	essential	ADJ
ejpam-6375	25	8	scholarly	scholarly	ADJ
ejpam-6375	25	9	contributions	contribution	NOUN
ejpam-6375	25	10	to	to	ADP
ejpam-6375	25	11	the	the	DET
ejpam-6375	25	12	field	field	NOUN
ejpam-6375	25	13	through	through	ADP
ejpam-6375	25	14	their	their	PRON
ejpam-6375	25	15	semigeneralized	semigeneralize	VERB
ejpam-6375	25	16	closed	closed	ADJ
ejpam-6375	25	17	sets	set	NOUN
ejpam-6375	25	18	and	and	CCONJ
ejpam-6375	25	19	their	their	PRON
ejpam-6375	25	20	connected	connected	ADJ
ejpam-6375	25	21	structures	structure	NOUN
ejpam-6375	25	22	of	of	ADP
ejpam-6375	25	23	semi	semi	ADJ
ejpam-6375	25	24	-	-	ADJ
ejpam-6375	25	25	open	open	ADJ
ejpam-6375	25	26	and	and	CCONJ
ejpam-6375	25	27	pre	pre	ADJ
ejpam-6375	25	28	-	-	ADJ
ejpam-6375	25	29	open	open	ADJ
ejpam-6375	25	30	sets	set	NOUN
ejpam-6375	25	31	.	.	PUNCT
ejpam-6375	26	1	these	these	DET
ejpam-6375	26	2	fundamental	fundamental	ADJ
ejpam-6375	26	3	theoretical	theoretical	ADJ
ejpam-6375	26	4	foundations	foundation	NOUN
ejpam-6375	26	5	have	have	AUX
ejpam-6375	26	6	set	set	VERB
ejpam-6375	26	7	the	the	DET
ejpam-6375	26	8	base	base	NOUN
ejpam-6375	26	9	for	for	ADP
ejpam-6375	26	10	analyzing	analyze	VERB
ejpam-6375	26	11	analogous	analogous	ADJ
ejpam-6375	26	12	features	feature	NOUN
ejpam-6375	26	13	within	within	ADP
ejpam-6375	26	14	primal	primal	ADJ
ejpam-6375	26	15	spaces	space	NOUN
ejpam-6375	26	16	with	with	ADP
ejpam-6375	26	17	generalization	generalization	NOUN
ejpam-6375	26	18	.	.	PUNCT
ejpam-6375	27	1	the	the	DET
ejpam-6375	27	2	field	field	NOUN
ejpam-6375	27	3	of	of	ADP
ejpam-6375	27	4	generalized	generalized	ADJ
ejpam-6375	27	5	primal	primal	ADJ
ejpam-6375	27	6	topology	topology	NOUN
ejpam-6375	27	7	continues	continue	VERB
ejpam-6375	27	8	to	to	PART
ejpam-6375	27	9	evolve	evolve	VERB
ejpam-6375	27	10	through	through	ADP
ejpam-6375	27	11	choquet	choquet	NOUN
ejpam-6375	27	12	’s	’s	PART
ejpam-6375	27	13	initial	initial	ADJ
ejpam-6375	27	14	research	research	NOUN
ejpam-6375	27	15	on	on	ADP
ejpam-6375	27	16	grills	grill	NOUN
ejpam-6375	27	17	and	and	CCONJ
ejpam-6375	27	18	filters	filter	NOUN
ejpam-6375	27	19	from	from	ADP
ejpam-6375	27	20	1961	1961	NUM
ejpam-6375	27	21	which	which	PRON
ejpam-6375	27	22	remains	remain	VERB
ejpam-6375	27	23	influential	influential	ADJ
ejpam-6375	27	24	in	in	ADP
ejpam-6375	27	25	current	current	ADJ
ejpam-6375	27	26	investigations	investigation	NOUN
ejpam-6375	27	27	.	.	PUNCT
ejpam-6375	28	1	structural	structural	ADJ
ejpam-6375	28	2	analysis	analysis	NOUN
ejpam-6375	28	3	of	of	ADP
ejpam-6375	28	4	primal	primal	ADJ
ejpam-6375	28	5	spaces	space	NOUN
ejpam-6375	28	6	received	receive	VERB
ejpam-6375	28	7	clearer	clear	ADJ
ejpam-6375	28	8	explanations	explanation	NOUN
ejpam-6375	28	9	through	through	ADP
ejpam-6375	28	10	research	research	NOUN
ejpam-6375	28	11	conducted	conduct	VERB
ejpam-6375	28	12	by	by	ADP
ejpam-6375	28	13	acharjee	acharjee	NOUN
ejpam-6375	28	14	,	,	PUNCT
ejpam-6375	28	15	özkoç	özkoç	NOUN
ejpam-6375	28	16	,	,	PUNCT
ejpam-6375	28	17	and	and	CCONJ
ejpam-6375	28	18	issaka	issaka	NOUN
ejpam-6375	28	19	[	[	X
ejpam-6375	28	20	7	7	NUM
ejpam-6375	28	21	]	]	PUNCT
ejpam-6375	28	22	and	and	CCONJ
ejpam-6375	28	23	özkoç	özkoç	VERB
ejpam-6375	28	24	and	and	CCONJ
ejpam-6375	28	25	köstel	köstel	NOUN
ejpam-6375	28	26	[	[	X
ejpam-6375	28	27	8	8	NUM
ejpam-6375	28	28	]	]	PUNCT
ejpam-6375	28	29	.	.	PUNCT
ejpam-6375	29	1	saadi	saadi	NOUN
ejpam-6375	29	2	and	and	CCONJ
ejpam-6375	29	3	malki	malki	VERB
ejpam-6375	29	4	[	[	X
ejpam-6375	29	5	9	9	NUM
ejpam-6375	29	6	,	,	PUNCT
ejpam-6375	29	7	10	10	NUM
ejpam-6375	29	8	]	]	PUNCT
ejpam-6375	29	9	developed	develop	VERB
ejpam-6375	29	10	our	our	PRON
ejpam-6375	29	11	knowledge	knowledge	NOUN
ejpam-6375	29	12	of	of	ADP
ejpam-6375	29	13	these	these	DET
ejpam-6375	29	14	spaces	space	NOUN
ejpam-6375	29	15	through	through	ADP
ejpam-6375	29	16	their	their	PRON
ejpam-6375	29	17	examinations	examination	NOUN
ejpam-6375	29	18	of	of	ADP
ejpam-6375	29	19	different	different	ADJ
ejpam-6375	29	20	open	open	ADJ
ejpam-6375	29	21	set	set	VERB
ejpam-6375	29	22	categories	category	NOUN
ejpam-6375	29	23	.	.	PUNCT
ejpam-6375	30	1	a	a	DET
ejpam-6375	30	2	combination	combination	NOUN
ejpam-6375	30	3	of	of	ADP
ejpam-6375	30	4	above	above	ADV
ejpam-6375	30	5	-	-	PUNCT
ejpam-6375	30	6	mentioned	mention	VERB
ejpam-6375	30	7	aspects	aspect	NOUN
ejpam-6375	30	8	results	result	NOUN
ejpam-6375	30	9	in	in	ADP
ejpam-6375	30	10	the	the	DET
ejpam-6375	30	11	adaptability	adaptability	NOUN
ejpam-6375	30	12	of	of	ADP
ejpam-6375	30	13	primal	primal	ADJ
ejpam-6375	30	14	spaces	space	NOUN
ejpam-6375	30	15	being	be	AUX
ejpam-6375	30	16	demonstrated	demonstrate	VERB
ejpam-6375	30	17	also	also	ADV
ejpam-6375	30	18	through	through	ADP
ejpam-6375	30	19	extension	extension	NOUN
ejpam-6375	30	20	to	to	ADP
ejpam-6375	30	21	the	the	DET
ejpam-6375	30	22	soft	soft	ADJ
ejpam-6375	30	23	structures	structure	NOUN
ejpam-6375	30	24	[	[	X
ejpam-6375	30	25	11	11	NUM
ejpam-6375	30	26	]	]	PUNCT
ejpam-6375	30	27	.	.	PUNCT
ejpam-6375	31	1	besides	besides	SCONJ
ejpam-6375	31	2	these	these	PRON
ejpam-6375	31	3	,	,	PUNCT
ejpam-6375	31	4	new	new	ADJ
ejpam-6375	31	5	operator	operator	NOUN
ejpam-6375	31	6	-	-	PUNCT
ejpam-6375	31	7	based	base	VERB
ejpam-6375	31	8	solutions	solution	NOUN
ejpam-6375	31	9	have	have	AUX
ejpam-6375	31	10	been	be	AUX
ejpam-6375	31	11	suggested	suggest	VERB
ejpam-6375	31	12	to	to	PART
ejpam-6375	31	13	reinforce	reinforce	VERB
ejpam-6375	31	14	the	the	DET
ejpam-6375	31	15	primal	primal	ADJ
ejpam-6375	31	16	topology	topology	NOUN
ejpam-6375	31	17	framework	framework	NOUN
ejpam-6375	31	18	[	[	X
ejpam-6375	31	19	12	12	NUM
ejpam-6375	31	20	]	]	PUNCT
ejpam-6375	31	21	.	.	PUNCT
ejpam-6375	32	1	the	the	DET
ejpam-6375	32	2	most	most	ADV
ejpam-6375	32	3	significant	significant	ADJ
ejpam-6375	32	4	advancement	advancement	NOUN
ejpam-6375	32	5	emerged	emerge	VERB
ejpam-6375	32	6	through	through	ADP
ejpam-6375	32	7	missier	missier	NOUN
ejpam-6375	32	8	and	and	CCONJ
ejpam-6375	32	9	jesti	jesti	PROPN
ejpam-6375	32	10	’s	’s	PART
ejpam-6375	32	11	research	research	NOUN
ejpam-6375	32	12	which	which	PRON
ejpam-6375	32	13	proposed	propose	VERB
ejpam-6375	32	14	s∗	s∗	PROPN
ejpam-6375	32	15	g	g	PROPN
ejpam-6375	32	16	-open	-open	PROPN
ejpam-6375	32	17	sets	set	NOUN
ejpam-6375	32	18	[	[	X
ejpam-6375	32	19	13	13	NUM
ejpam-6375	32	20	]	]	PUNCT
ejpam-6375	32	21	.	.	PUNCT
ejpam-6375	33	1	the	the	DET
ejpam-6375	33	2	research	research	NOUN
ejpam-6375	33	3	initiated	initiate	VERB
ejpam-6375	33	4	by	by	ADP
ejpam-6375	33	5	missier	missier	NOUN
ejpam-6375	33	6	and	and	CCONJ
ejpam-6375	33	7	jesti	jesti	VERB
ejpam-6375	34	1	[	[	X
ejpam-6375	34	2	13	13	NUM
ejpam-6375	34	3	]	]	PUNCT
ejpam-6375	34	4	created	create	VERB
ejpam-6375	34	5	a	a	DET
ejpam-6375	34	6	whole	whole	ADJ
ejpam-6375	34	7	family	family	NOUN
ejpam-6375	34	8	of	of	ADP
ejpam-6375	34	9	connected	connect	VERB
ejpam-6375	34	10	functions	function	NOUN
ejpam-6375	34	11	for	for	ADP
ejpam-6375	34	12	improved	improved	ADJ
ejpam-6375	34	13	set	set	VERB
ejpam-6375	34	14	operation	operation	NOUN
ejpam-6375	34	15	analysis	analysis	NOUN
ejpam-6375	34	16	in	in	ADP
ejpam-6375	34	17	generalized	generalized	ADJ
ejpam-6375	34	18	primal	primal	ADJ
ejpam-6375	34	19	spaces	space	NOUN
ejpam-6375	34	20	.	.	PUNCT
ejpam-6375	35	1	the	the	DET
ejpam-6375	35	2	initial	initial	ADJ
ejpam-6375	35	3	concept	concept	NOUN
ejpam-6375	35	4	of	of	ADP
ejpam-6375	35	5	µ-compactness	µ-compactness	NOUN
ejpam-6375	35	6	created	create	VERB
ejpam-6375	35	7	by	by	ADP
ejpam-6375	35	8	thomas	thomas	PROPN
ejpam-6375	35	9	and	and	CCONJ
ejpam-6375	35	10	john	john	PROPN
ejpam-6375	35	11	[	[	X
ejpam-6375	35	12	14	14	NUM
ejpam-6375	35	13	]	]	PUNCT
ejpam-6375	35	14	started	start	VERB
ejpam-6375	35	15	in	in	ADP
ejpam-6375	35	16	general	general	ADJ
ejpam-6375	35	17	settings	setting	NOUN
ejpam-6375	35	18	before	before	SCONJ
ejpam-6375	35	19	it	it	PRON
ejpam-6375	35	20	helped	help	VERB
ejpam-6375	35	21	researchers	researcher	NOUN
ejpam-6375	35	22	understand	understand	VERB
ejpam-6375	35	23	compactness	compactness	NOUN
ejpam-6375	35	24	definitions	definition	NOUN
ejpam-6375	35	25	in	in	ADP
ejpam-6375	35	26	specific	specific	ADJ
ejpam-6375	35	27	spaces	space	NOUN
ejpam-6375	35	28	.	.	PUNCT
ejpam-6375	36	1	the	the	DET
ejpam-6375	36	2	investigation	investigation	NOUN
ejpam-6375	36	3	examines	examine	VERB
ejpam-6375	36	4	basic	basic	ADJ
ejpam-6375	36	5	space	space	NOUN
ejpam-6375	36	6	differentiation	differentiation	NOUN
ejpam-6375	36	7	criteria	criterion	NOUN
ejpam-6375	36	8	known	know	VERB
ejpam-6375	36	9	as	as	ADP
ejpam-6375	36	10	t0	t0	PROPN
ejpam-6375	36	11	,	,	PUNCT
ejpam-6375	36	12	t1	t1	NOUN
ejpam-6375	36	13	and	and	CCONJ
ejpam-6375	36	14	t2	t2	NOUN
ejpam-6375	36	15	separation	separation	NOUN
ejpam-6375	36	16	axioms	axiom	NOUN
ejpam-6375	36	17	because	because	SCONJ
ejpam-6375	36	18	they	they	PRON
ejpam-6375	36	19	serve	serve	VERB
ejpam-6375	36	20	as	as	ADP
ejpam-6375	36	21	fundamental	fundamental	ADJ
ejpam-6375	36	22	tools	tool	NOUN
ejpam-6375	36	23	to	to	PART
ejpam-6375	36	24	distinguish	distinguish	VERB
ejpam-6375	36	25	space	space	NOUN
ejpam-6375	36	26	types	type	NOUN
ejpam-6375	36	27	.	.	PUNCT
ejpam-6375	37	1	m.	m.	NOUN
ejpam-6375	37	2	shahbaz	shahbaz	PROPN
ejpam-6375	37	3	et	et	PROPN
ejpam-6375	37	4	al	al	PROPN
ejpam-6375	37	5	.	.	PUNCT
ejpam-6375	37	6	/	/	SYM
ejpam-6375	37	7	eur	eur	PROPN
ejpam-6375	37	8	.	.	PUNCT
ejpam-6375	38	1	j.	j.	PROPN
ejpam-6375	38	2	pure	pure	PROPN
ejpam-6375	38	3	appl	appl	PROPN
ejpam-6375	38	4	.	.	PROPN
ejpam-6375	38	5	math	math	PROPN
ejpam-6375	38	6	,	,	PUNCT
ejpam-6375	38	7	18	18	NUM
ejpam-6375	38	8	(	(	PUNCT
ejpam-6375	38	9	4	4	NUM
ejpam-6375	38	10	)	)	PUNCT
ejpam-6375	38	11	(	(	PUNCT
ejpam-6375	38	12	2025	2025	NUM
ejpam-6375	38	13	)	)	PUNCT
ejpam-6375	38	14	,	,	PUNCT
ejpam-6375	38	15	6375	6375	NUM
ejpam-6375	38	16	3	3	NUM
ejpam-6375	38	17	of	of	ADP
ejpam-6375	38	18	22	22	NUM
ejpam-6375	38	19	these	these	DET
ejpam-6375	38	20	axioms	axiom	NOUN
ejpam-6375	38	21	introduced	introduce	VERB
ejpam-6375	38	22	by	by	ADP
ejpam-6375	38	23	urysohn	urysohn	PROPN
ejpam-6375	39	1	[	[	X
ejpam-6375	39	2	15	15	NUM
ejpam-6375	39	3	]	]	PUNCT
ejpam-6375	39	4	then	then	ADV
ejpam-6375	39	5	improved	improve	VERB
ejpam-6375	39	6	by	by	ADP
ejpam-6375	39	7	freudenthal	freudenthal	NOUN
ejpam-6375	39	8	and	and	CCONJ
ejpam-6375	39	9	est	est	X
ejpam-6375	39	10	[	[	X
ejpam-6375	39	11	16	16	NUM
ejpam-6375	39	12	]	]	PUNCT
ejpam-6375	39	13	continue	continue	VERB
ejpam-6375	39	14	to	to	PART
ejpam-6375	39	15	play	play	VERB
ejpam-6375	39	16	an	an	DET
ejpam-6375	39	17	important	important	ADJ
ejpam-6375	39	18	role	role	NOUN
ejpam-6375	39	19	in	in	ADP
ejpam-6375	39	20	contemporary	contemporary	ADJ
ejpam-6375	39	21	topological	topological	ADJ
ejpam-6375	39	22	classifying	classify	VERB
ejpam-6375	39	23	systems	system	NOUN
ejpam-6375	39	24	.	.	PUNCT
ejpam-6375	40	1	stone	stone	NOUN
ejpam-6375	41	1	[	[	X
ejpam-6375	41	2	17	17	NUM
ejpam-6375	41	3	]	]	PUNCT
ejpam-6375	41	4	proved	prove	VERB
ejpam-6375	41	5	that	that	SCONJ
ejpam-6375	41	6	any	any	DET
ejpam-6375	41	7	topological	topological	ADJ
ejpam-6375	41	8	space	space	NOUN
ejpam-6375	41	9	can	can	AUX
ejpam-6375	41	10	become	become	VERB
ejpam-6375	41	11	a	a	DET
ejpam-6375	41	12	t0	t0	PROPN
ejpam-6375	41	13	space	space	NOUN
ejpam-6375	41	14	through	through	ADP
ejpam-6375	41	15	the	the	DET
ejpam-6375	41	16	process	process	NOUN
ejpam-6375	41	17	of	of	ADP
ejpam-6375	41	18	point	point	NOUN
ejpam-6375	41	19	merging	merge	VERB
ejpam-6375	41	20	to	to	PART
ejpam-6375	41	21	merge	merge	VERB
ejpam-6375	41	22	indistinguishable	indistinguishable	ADJ
ejpam-6375	41	23	points	point	NOUN
ejpam-6375	41	24	.	.	PUNCT
ejpam-6375	42	1	meanwhile	meanwhile	ADV
ejpam-6375	42	2	youngs	young	NOUN
ejpam-6375	42	3	[	[	X
ejpam-6375	42	4	18	18	NUM
ejpam-6375	42	5	]	]	PUNCT
ejpam-6375	42	6	examined	examine	VERB
ejpam-6375	42	7	separation	separation	NOUN
ejpam-6375	42	8	conditions	condition	NOUN
ejpam-6375	42	9	between	between	ADP
ejpam-6375	42	10	t0	t0	PROPN
ejpam-6375	42	11	and	and	CCONJ
ejpam-6375	42	12	t1	t1	NOUN
ejpam-6375	42	13	.	.	PUNCT
ejpam-6375	43	1	the	the	DET
ejpam-6375	43	2	current	current	ADJ
ejpam-6375	43	3	research	research	NOUN
ejpam-6375	43	4	established	establish	VERB
ejpam-6375	43	5	s∗	s∗	PROPN
ejpam-6375	43	6	g	g	PROPN
ejpam-6375	43	7	-compact	-compact	PROPN
ejpam-6375	43	8	and	and	CCONJ
ejpam-6375	43	9	s∗	s∗	VERB
ejpam-6375	43	10	g	g	PROPN
ejpam-6375	43	11	-connected	-connected	ADJ
ejpam-6375	43	12	spaces	space	NOUN
ejpam-6375	43	13	as	as	ADP
ejpam-6375	43	14	its	its	PRON
ejpam-6375	43	15	main	main	ADJ
ejpam-6375	43	16	focus	focus	NOUN
ejpam-6375	43	17	inside	inside	ADP
ejpam-6375	43	18	generalized	generalized	ADJ
ejpam-6375	43	19	primal	primal	ADJ
ejpam-6375	43	20	topological	topological	ADJ
ejpam-6375	43	21	worlds	world	NOUN
ejpam-6375	43	22	.	.	PUNCT
ejpam-6375	44	1	the	the	DET
ejpam-6375	44	2	paper	paper	NOUN
ejpam-6375	44	3	studies	study	NOUN
ejpam-6375	44	4	how	how	SCONJ
ejpam-6375	44	5	spaces	space	NOUN
ejpam-6375	44	6	change	change	VERB
ejpam-6375	44	7	under	under	ADP
ejpam-6375	44	8	the	the	DET
ejpam-6375	44	9	influence	influence	NOUN
ejpam-6375	44	10	of	of	ADP
ejpam-6375	44	11	t0	t0	PROPN
ejpam-6375	44	12	,	,	PUNCT
ejpam-6375	44	13	t1	t1	NOUN
ejpam-6375	44	14	and	and	CCONJ
ejpam-6375	44	15	t2	t2	NOUN
ejpam-6375	44	16	separation	separation	NOUN
ejpam-6375	44	17	axioms	axiom	NOUN
ejpam-6375	44	18	in	in	ADP
ejpam-6375	44	19	their	their	PRON
ejpam-6375	44	20	internal	internal	ADJ
ejpam-6375	44	21	structure	structure	NOUN
ejpam-6375	44	22	.	.	PUNCT
ejpam-6375	45	1	thus	thus	ADV
ejpam-6375	45	2	this	this	DET
ejpam-6375	45	3	piece	piece	NOUN
ejpam-6375	45	4	develops	develop	VERB
ejpam-6375	45	5	the	the	DET
ejpam-6375	45	6	concepts	concept	NOUN
ejpam-6375	45	7	of	of	ADP
ejpam-6375	45	8	s∗	s∗	PROPN
ejpam-6375	45	9	g	g	PROPN
ejpam-6375	45	10	-t0	-t0	PROPN
ejpam-6375	45	11	,	,	PUNCT
ejpam-6375	45	12	s∗	s∗	PROPN
ejpam-6375	45	13	g	g	PROPN
ejpam-6375	45	14	-t1	-t1	PROPN
ejpam-6375	45	15	and	and	CCONJ
ejpam-6375	45	16	s∗	s∗	AUX
ejpam-6375	45	17	g	g	PROPN
ejpam-6375	45	18	-t2	-t2	PROPN
ejpam-6375	45	19	spaces	space	VERB
ejpam-6375	45	20	to	to	PART
ejpam-6375	45	21	demonstrate	demonstrate	VERB
ejpam-6375	45	22	their	their	PRON
ejpam-6375	45	23	alignment	alignment	NOUN
ejpam-6375	45	24	with	with	ADP
ejpam-6375	45	25	s∗	s∗	PROPN
ejpam-6375	45	26	g	g	PROPN
ejpam-6375	45	27	-set	-set	PUNCT
ejpam-6375	45	28	characteristics	characteristic	NOUN
ejpam-6375	45	29	.	.	PUNCT
ejpam-6375	46	1	the	the	DET
ejpam-6375	46	2	study	study	NOUN
ejpam-6375	46	3	explores	explore	VERB
ejpam-6375	46	4	theoretical	theoretical	ADJ
ejpam-6375	46	5	growth	growth	NOUN
ejpam-6375	46	6	through	through	ADP
ejpam-6375	46	7	its	its	PRON
ejpam-6375	46	8	results	result	NOUN
ejpam-6375	46	9	which	which	PRON
ejpam-6375	46	10	establish	establish	VERB
ejpam-6375	46	11	foundations	foundation	NOUN
ejpam-6375	46	12	for	for	ADP
ejpam-6375	46	13	future	future	ADJ
ejpam-6375	46	14	research	research	NOUN
ejpam-6375	46	15	in	in	ADP
ejpam-6375	46	16	generalized	generalized	ADJ
ejpam-6375	46	17	primal	primal	ADJ
ejpam-6375	46	18	topology	topology	NOUN
ejpam-6375	46	19	.	.	PUNCT
ejpam-6375	47	1	2	2	X
ejpam-6375	47	2	.	.	X
ejpam-6375	47	3	preliminaries	preliminary	NOUN
ejpam-6375	47	4	this	this	DET
ejpam-6375	47	5	section	section	NOUN
ejpam-6375	47	6	provides	provide	VERB
ejpam-6375	47	7	some	some	DET
ejpam-6375	47	8	findings	finding	NOUN
ejpam-6375	47	9	and	and	CCONJ
ejpam-6375	47	10	definitions	definition	NOUN
ejpam-6375	47	11	from	from	ADP
ejpam-6375	47	12	the	the	DET
ejpam-6375	47	13	literature	literature	NOUN
ejpam-6375	47	14	in	in	ADP
ejpam-6375	47	15	order	order	NOUN
ejpam-6375	47	16	to	to	PART
ejpam-6375	47	17	clarify	clarify	VERB
ejpam-6375	47	18	the	the	DET
ejpam-6375	47	19	main	main	ADJ
ejpam-6375	47	20	part	part	NOUN
ejpam-6375	47	21	.	.	PUNCT
ejpam-6375	48	1	definition	definition	NOUN
ejpam-6375	48	2	2.1	2.1	NUM
ejpam-6375	48	3	.	.	PUNCT
ejpam-6375	49	1	[	[	X
ejpam-6375	49	2	19	19	NUM
ejpam-6375	49	3	]	]	X
ejpam-6375	49	4	an	an	DET
ejpam-6375	49	5	empty	empty	ADJ
ejpam-6375	49	6	set	set	NOUN
ejpam-6375	49	7	does	do	AUX
ejpam-6375	49	8	not	not	PART
ejpam-6375	49	9	exist	exist	VERB
ejpam-6375	49	10	yet	yet	ADV
ejpam-6375	49	11	the	the	DET
ejpam-6375	49	12	set	set	NOUN
ejpam-6375	49	13	v	v	ADP
ejpam-6375	49	14	̸=	̸=	PROPN
ejpam-6375	49	15	∅.	∅.	ADP
ejpam-6375	49	16	a	a	DET
ejpam-6375	49	17	collection	collection	NOUN
ejpam-6375	49	18	gτ	gτ	PROPN
ejpam-6375	49	19	⊆	⊆	NUM
ejpam-6375	49	20	2v	2v	PROPN
ejpam-6375	49	21	satisfies	satisfy	VERB
ejpam-6375	49	22	the	the	DET
ejpam-6375	49	23	criteria	criterion	NOUN
ejpam-6375	49	24	to	to	PART
ejpam-6375	49	25	be	be	AUX
ejpam-6375	49	26	considered	consider	VERB
ejpam-6375	49	27	a	a	DET
ejpam-6375	49	28	generalized	generalized	ADJ
ejpam-6375	49	29	topology	topology	NOUN
ejpam-6375	49	30	(	(	PUNCT
ejpam-6375	49	31	gt	gt	PROPN
ejpam-6375	49	32	)	)	PUNCT
ejpam-6375	49	33	on	on	ADP
ejpam-6375	49	34	v	v	PRON
ejpam-6375	49	35	if	if	SCONJ
ejpam-6375	49	36	it	it	PRON
ejpam-6375	49	37	contains	contain	VERB
ejpam-6375	49	38	the	the	DET
ejpam-6375	49	39	empty	empty	ADJ
ejpam-6375	49	40	set	set	NOUN
ejpam-6375	49	41	and	and	CCONJ
ejpam-6375	49	42	all	all	DET
ejpam-6375	49	43	possible	possible	ADJ
ejpam-6375	49	44	unions	union	NOUN
ejpam-6375	49	45	of	of	ADP
ejpam-6375	49	46	non	non	ADJ
ejpam-6375	49	47	-	-	ADJ
ejpam-6375	49	48	empty	empty	ADJ
ejpam-6375	49	49	subclasses	subclass	NOUN
ejpam-6375	49	50	within	within	ADP
ejpam-6375	49	51	gτ	gτ	PROPN
ejpam-6375	49	52	are	be	AUX
ejpam-6375	49	53	also	also	ADV
ejpam-6375	49	54	contained	contain	VERB
ejpam-6375	49	55	in	in	ADP
ejpam-6375	49	56	gτ	gτ	PROPN
ejpam-6375	49	57	.	.	PUNCT
ejpam-6375	50	1	the	the	DET
ejpam-6375	50	2	pair	pair	NOUN
ejpam-6375	50	3	made	make	VERB
ejpam-6375	50	4	up	up	ADP
ejpam-6375	50	5	of	of	ADP
ejpam-6375	50	6	(	(	PUNCT
ejpam-6375	50	7	v	v	NOUN
ejpam-6375	50	8	,	,	PUNCT
ejpam-6375	50	9	gτ	gτ	PROPN
ejpam-6375	50	10	)	)	PUNCT
ejpam-6375	50	11	represents	represent	VERB
ejpam-6375	50	12	a	a	DET
ejpam-6375	50	13	gts	gts	NOUN
ejpam-6375	50	14	(	(	PUNCT
ejpam-6375	50	15	generalized	generalize	VERB
ejpam-6375	50	16	topological	topological	ADJ
ejpam-6375	50	17	space	space	NOUN
ejpam-6375	50	18	)	)	PUNCT
ejpam-6375	50	19	.	.	PUNCT
ejpam-6375	51	1	remark	remark	VERB
ejpam-6375	51	2	2.1	2.1	NUM
ejpam-6375	51	3	.	.	PUNCT
ejpam-6375	52	1	[	[	X
ejpam-6375	52	2	3	3	X
ejpam-6375	52	3	]	]	PUNCT
ejpam-6375	52	4	each	each	DET
ejpam-6375	52	5	member	member	NOUN
ejpam-6375	52	6	of	of	ADP
ejpam-6375	52	7	the	the	DET
ejpam-6375	52	8	set	set	NOUN
ejpam-6375	52	9	gτ	gτ	NOUN
ejpam-6375	52	10	is	be	AUX
ejpam-6375	52	11	identified	identify	VERB
ejpam-6375	52	12	as	as	ADP
ejpam-6375	52	13	open	open	ADJ
ejpam-6375	52	14	in	in	ADP
ejpam-6375	52	15	the	the	DET
ejpam-6375	52	16	space	space	NOUN
ejpam-6375	52	17	.	.	PUNCT
ejpam-6375	53	1	any	any	DET
ejpam-6375	53	2	set	set	NOUN
ejpam-6375	53	3	e	e	NOUN
ejpam-6375	53	4	present	present	NOUN
ejpam-6375	53	5	in	in	ADP
ejpam-6375	53	6	the	the	DET
ejpam-6375	53	7	context	context	NOUN
ejpam-6375	53	8	of	of	ADP
ejpam-6375	53	9	(	(	PUNCT
ejpam-6375	53	10	v	v	NOUN
ejpam-6375	53	11	,	,	PUNCT
ejpam-6375	53	12	gτ	gτ	NOUN
ejpam-6375	53	13	)	)	PUNCT
ejpam-6375	53	14	forms	form	VERB
ejpam-6375	53	15	an	an	DET
ejpam-6375	53	16	essential	essential	ADJ
ejpam-6375	53	17	part	part	NOUN
ejpam-6375	53	18	of	of	ADP
ejpam-6375	53	19	our	our	PRON
ejpam-6375	53	20	consideration	consideration	NOUN
ejpam-6375	53	21	.	.	PUNCT
ejpam-6375	54	1	it	it	PRON
ejpam-6375	54	2	becomes	become	VERB
ejpam-6375	54	3	a	a	DET
ejpam-6375	54	4	closed	closed	ADJ
ejpam-6375	54	5	set	set	NOUN
ejpam-6375	54	6	whenever	whenever	SCONJ
ejpam-6375	54	7	the	the	DET
ejpam-6375	54	8	complement	complement	NOUN
ejpam-6375	54	9	of	of	ADP
ejpam-6375	54	10	e	e	PROPN
ejpam-6375	54	11	relative	relative	ADJ
ejpam-6375	54	12	to	to	ADP
ejpam-6375	54	13	the	the	DET
ejpam-6375	54	14	set	set	NOUN
ejpam-6375	54	15	v	v	NOUN
ejpam-6375	54	16	is	be	AUX
ejpam-6375	54	17	open	open	ADJ
ejpam-6375	54	18	.	.	PUNCT
ejpam-6375	55	1	the	the	DET
ejpam-6375	55	2	closure	closure	NOUN
ejpam-6375	55	3	of	of	ADP
ejpam-6375	55	4	a	a	DET
ejpam-6375	55	5	set	set	NOUN
ejpam-6375	55	6	e	e	NOUN
ejpam-6375	55	7	receives	receive	VERB
ejpam-6375	55	8	the	the	DET
ejpam-6375	55	9	notation	notation	NOUN
ejpam-6375	55	10	clg(e	clg(e	PROPN
ejpam-6375	55	11	)	)	PUNCT
ejpam-6375	55	12	through	through	ADP
ejpam-6375	55	13	intersection	intersection	NOUN
ejpam-6375	55	14	of	of	ADP
ejpam-6375	55	15	every	every	DET
ejpam-6375	55	16	closed	close	VERB
ejpam-6375	55	17	set	set	NOUN
ejpam-6375	55	18	which	which	PRON
ejpam-6375	55	19	contains	contain	VERB
ejpam-6375	55	20	e.	e.	PROPN
ejpam-6375	55	21	interior	interior	PROPN
ejpam-6375	55	22	of	of	ADP
ejpam-6375	55	23	a	a	DET
ejpam-6375	55	24	set	set	NOUN
ejpam-6375	55	25	known	know	VERB
ejpam-6375	55	26	as	as	ADP
ejpam-6375	55	27	intg(e	intg(e	NOUN
ejpam-6375	55	28	)	)	PUNCT
ejpam-6375	55	29	consist	consist	NOUN
ejpam-6375	55	30	of	of	ADP
ejpam-6375	55	31	every	every	DET
ejpam-6375	55	32	open	open	ADJ
ejpam-6375	55	33	subset	subset	NOUN
ejpam-6375	55	34	found	find	VERB
ejpam-6375	55	35	within	within	ADP
ejpam-6375	55	36	e.	e.	PROPN
ejpam-6375	55	37	definition	definition	NOUN
ejpam-6375	55	38	2.2	2.2	NUM
ejpam-6375	55	39	.	.	PUNCT
ejpam-6375	56	1	[	[	X
ejpam-6375	56	2	3	3	X
ejpam-6375	56	3	]	]	PUNCT
ejpam-6375	56	4	according	accord	VERB
ejpam-6375	56	5	to	to	ADP
ejpam-6375	56	6	gts	gts	NOUN
ejpam-6375	56	7	a	a	DET
ejpam-6375	56	8	ψ	ψ	NOUN
ejpam-6375	56	9	operator	operator	NOUN
ejpam-6375	56	10	maps	map	NOUN
ejpam-6375	56	11	elements	element	NOUN
ejpam-6375	56	12	x	x	PUNCT
ejpam-6375	56	13	from	from	ADP
ejpam-6375	56	14	v	v	NUM
ejpam-6375	56	15	to	to	ADP
ejpam-6375	56	16	sets	set	NOUN
ejpam-6375	56	17	in	in	ADP
ejpam-6375	56	18	22	22	NUM
ejpam-6375	56	19	v	v	NOUN
ejpam-6375	57	1	and	and	CCONJ
ejpam-6375	57	2	it	it	PRON
ejpam-6375	57	3	fulfills	fulfill	VERB
ejpam-6375	57	4	the	the	DET
ejpam-6375	57	5	condition	condition	NOUN
ejpam-6375	57	6	where	where	SCONJ
ejpam-6375	57	7	x	x	SYM
ejpam-6375	57	8	∈	∈	PROPN
ejpam-6375	57	9	f	f	X
ejpam-6375	57	10	whenever	whenever	SCONJ
ejpam-6375	57	11	f	f	PROPN
ejpam-6375	57	12	belongs	belong	VERB
ejpam-6375	57	13	to	to	ADP
ejpam-6375	57	14	the	the	DET
ejpam-6375	57	15	image	image	NOUN
ejpam-6375	57	16	of	of	ADP
ejpam-6375	57	17	x.	x.	NOUN
ejpam-6375	57	18	according	accord	VERB
ejpam-6375	57	19	to	to	ADP
ejpam-6375	57	20	definition	definition	NOUN
ejpam-6375	57	21	a	a	DET
ejpam-6375	57	22	generalized	generalized	ADJ
ejpam-6375	57	23	neighbourhood	neighbourhood	NOUN
ejpam-6375	57	24	of	of	ADP
ejpam-6375	57	25	point	point	NOUN
ejpam-6375	57	26	x	x	PUNCT
ejpam-6375	57	27	in	in	ADP
ejpam-6375	57	28	set	set	NOUN
ejpam-6375	57	29	v	v	NUM
ejpam-6375	57	30	refers	refer	VERB
ejpam-6375	57	31	to	to	ADP
ejpam-6375	57	32	the	the	DET
ejpam-6375	57	33	element	element	NOUN
ejpam-6375	57	34	f	f	PROPN
ejpam-6375	57	35	∈	∈	PROPN
ejpam-6375	57	36	ψ(x	ψ(x	PROPN
ejpam-6375	57	37	)	)	PUNCT
ejpam-6375	57	38	.	.	PUNCT
ejpam-6375	58	1	the	the	DET
ejpam-6375	58	2	collection	collection	NOUN
ejpam-6375	58	3	of	of	ADP
ejpam-6375	58	4	every	every	DET
ejpam-6375	58	5	generalized	generalized	ADJ
ejpam-6375	58	6	neighbourhood	neighbourhood	NOUN
ejpam-6375	58	7	that	that	PRON
ejpam-6375	58	8	exists	exist	VERB
ejpam-6375	58	9	within	within	ADP
ejpam-6375	58	10	v	v	NOUN
ejpam-6375	58	11	gets	get	AUX
ejpam-6375	58	12	symbolized	symbolize	VERB
ejpam-6375	58	13	by	by	ADP
ejpam-6375	58	14	ψ(v	ψ(v	PROPN
ejpam-6375	58	15	)	)	PUNCT
ejpam-6375	58	16	.	.	PUNCT
ejpam-6375	59	1	definition	definition	NOUN
ejpam-6375	59	2	2.3	2.3	NUM
ejpam-6375	59	3	.	.	PUNCT
ejpam-6375	60	1	[	[	X
ejpam-6375	60	2	4–6	4–6	X
ejpam-6375	60	3	,	,	PUNCT
ejpam-6375	60	4	20	20	NUM
ejpam-6375	60	5	]	]	PUNCT
ejpam-6375	60	6	(	(	PUNCT
ejpam-6375	60	7	i	i	NOUN
ejpam-6375	60	8	)	)	PUNCT
ejpam-6375	60	9	consider	consider	VERB
ejpam-6375	60	10	a	a	DET
ejpam-6375	60	11	gts	gts	NOUN
ejpam-6375	60	12	(	(	PUNCT
ejpam-6375	60	13	v	v	NOUN
ejpam-6375	60	14	,	,	PUNCT
ejpam-6375	60	15	gτ	gτ	PROPN
ejpam-6375	60	16	)	)	PUNCT
ejpam-6375	60	17	.	.	PUNCT
ejpam-6375	61	1	if	if	SCONJ
ejpam-6375	61	2	a	a	DET
ejpam-6375	61	3	set	set	NOUN
ejpam-6375	61	4	e	e	NOUN
ejpam-6375	61	5	in	in	ADP
ejpam-6375	61	6	a	a	DET
ejpam-6375	61	7	gts	gts	NOUN
ejpam-6375	61	8	has	have	VERB
ejpam-6375	61	9	an	an	DET
ejpam-6375	61	10	open	open	ADJ
ejpam-6375	61	11	container	container	NOUN
ejpam-6375	61	12	set	set	NOUN
ejpam-6375	61	13	f	f	PROPN
ejpam-6375	61	14	that	that	PRON
ejpam-6375	61	15	satisfies	satisfy	VERB
ejpam-6375	61	16	the	the	DET
ejpam-6375	61	17	condition	condition	NOUN
ejpam-6375	61	18	f	f	PROPN
ejpam-6375	61	19	⊆	⊆	NUM
ejpam-6375	61	20	e	e	PROPN
ejpam-6375	61	21	⊆	⊆	NUM
ejpam-6375	61	22	cl(f	cl(f	NUM
ejpam-6375	61	23	)	)	PUNCT
ejpam-6375	61	24	or	or	CCONJ
ejpam-6375	61	25	if	if	SCONJ
ejpam-6375	61	26	e	e	PROPN
ejpam-6375	61	27	⊆	⊆	NUM
ejpam-6375	61	28	cl(int(e	cl(int(e	PROPN
ejpam-6375	61	29	)	)	PUNCT
ejpam-6375	61	30	)	)	PUNCT
ejpam-6375	61	31	then	then	ADV
ejpam-6375	61	32	it	it	PRON
ejpam-6375	61	33	is	be	AUX
ejpam-6375	61	34	known	know	VERB
ejpam-6375	61	35	as	as	ADP
ejpam-6375	61	36	generalized	generalized	ADJ
ejpam-6375	61	37	gτ	gτ	PROPN
ejpam-6375	61	38	-semi	-semi	NOUN
ejpam-6375	61	39	-	-	PUNCT
ejpam-6375	61	40	open	open	ADJ
ejpam-6375	61	41	.	.	PUNCT
ejpam-6375	62	1	m.	m.	NOUN
ejpam-6375	62	2	shahbaz	shahbaz	PROPN
ejpam-6375	62	3	et	et	PROPN
ejpam-6375	62	4	al	al	PROPN
ejpam-6375	62	5	.	.	PUNCT
ejpam-6375	62	6	/	/	SYM
ejpam-6375	62	7	eur	eur	PROPN
ejpam-6375	62	8	.	.	PUNCT
ejpam-6375	63	1	j.	j.	PROPN
ejpam-6375	63	2	pure	pure	PROPN
ejpam-6375	63	3	appl	appl	PROPN
ejpam-6375	63	4	.	.	PROPN
ejpam-6375	63	5	math	math	PROPN
ejpam-6375	63	6	,	,	PUNCT
ejpam-6375	63	7	18	18	NUM
ejpam-6375	63	8	(	(	PUNCT
ejpam-6375	63	9	4	4	NUM
ejpam-6375	63	10	)	)	PUNCT
ejpam-6375	63	11	(	(	PUNCT
ejpam-6375	63	12	2025	2025	NUM
ejpam-6375	63	13	)	)	PUNCT
ejpam-6375	63	14	,	,	PUNCT
ejpam-6375	63	15	6375	6375	NUM
ejpam-6375	63	16	4	4	NUM
ejpam-6375	63	17	of	of	ADP
ejpam-6375	63	18	22	22	NUM
ejpam-6375	63	19	(	(	PUNCT
ejpam-6375	63	20	ii	ii	NOUN
ejpam-6375	63	21	)	)	PUNCT
ejpam-6375	63	22	the	the	DET
ejpam-6375	63	23	complement	complement	NOUN
ejpam-6375	63	24	of	of	ADP
ejpam-6375	63	25	a	a	DET
ejpam-6375	63	26	generalized	generalize	VERB
ejpam-6375	63	27	gτ	gτ	NOUN
ejpam-6375	63	28	-semi	-semi	NOUN
ejpam-6375	63	29	-	-	PUNCT
ejpam-6375	63	30	open	open	ADJ
ejpam-6375	63	31	set	set	NOUN
ejpam-6375	63	32	results	result	NOUN
ejpam-6375	63	33	in	in	ADP
ejpam-6375	63	34	a	a	DET
ejpam-6375	63	35	generalized	generalize	VERB
ejpam-6375	63	36	gτ	gτ	NOUN
ejpam-6375	63	37	-semiclosed	-semiclose	VERB
ejpam-6375	63	38	set	set	NOUN
ejpam-6375	63	39	.	.	PUNCT
ejpam-6375	64	1	all	all	DET
ejpam-6375	64	2	generalized	generalize	VERB
ejpam-6375	64	3	gτ	gτ	PROPN
ejpam-6375	64	4	-semi	-semi	NOUN
ejpam-6375	64	5	-	-	PUNCT
ejpam-6375	64	6	open	open	ADJ
ejpam-6375	64	7	sets	set	NOUN
ejpam-6375	64	8	within	within	ADP
ejpam-6375	64	9	(	(	PUNCT
ejpam-6375	64	10	v	v	NOUN
ejpam-6375	64	11	,	,	PUNCT
ejpam-6375	64	12	gτ	gτ	NOUN
ejpam-6375	64	13	)	)	PUNCT
ejpam-6375	64	14	form	form	VERB
ejpam-6375	64	15	the	the	DET
ejpam-6375	64	16	set	set	NOUN
ejpam-6375	64	17	known	know	VERB
ejpam-6375	64	18	as	as	ADP
ejpam-6375	64	19	so(v	so(v	NOUN
ejpam-6375	64	20	,	,	PUNCT
ejpam-6375	64	21	gτ	gτ	PROPN
ejpam-6375	64	22	)	)	PUNCT
ejpam-6375	64	23	.	.	PUNCT
ejpam-6375	65	1	(	(	PUNCT
ejpam-6375	65	2	iii	iii	X
ejpam-6375	65	3	)	)	PUNCT
ejpam-6375	65	4	in	in	ADP
ejpam-6375	65	5	a	a	DET
ejpam-6375	65	6	generalized	generalize	VERB
ejpam-6375	65	7	gτ	gτ	NOUN
ejpam-6375	65	8	-semi	-semi	NOUN
ejpam-6375	65	9	-	-	PUNCT
ejpam-6375	65	10	open	open	ADJ
ejpam-6375	65	11	setting	set	VERB
ejpam-6375	65	12	the	the	DET
ejpam-6375	65	13	overall	overall	ADJ
ejpam-6375	65	14	collection	collection	NOUN
ejpam-6375	65	15	of	of	ADP
ejpam-6375	65	16	such	such	ADJ
ejpam-6375	65	17	sets	set	NOUN
ejpam-6375	65	18	contained	contain	VERB
ejpam-6375	65	19	within	within	ADP
ejpam-6375	65	20	e	e	NOUN
ejpam-6375	65	21	forms	form	VERB
ejpam-6375	65	22	the	the	DET
ejpam-6375	65	23	generalized	generalized	ADJ
ejpam-6375	65	24	semi	semi	ADJ
ejpam-6375	65	25	-	-	ADJ
ejpam-6375	65	26	interior	interior	ADJ
ejpam-6375	65	27	which	which	PRON
ejpam-6375	65	28	is	be	AUX
ejpam-6375	65	29	denoted	denote	VERB
ejpam-6375	65	30	as	as	ADP
ejpam-6375	65	31	sgint(e	sgint(e	PROPN
ejpam-6375	65	32	)	)	PUNCT
ejpam-6375	65	33	.	.	PUNCT
ejpam-6375	66	1	(	(	PUNCT
ejpam-6375	66	2	iv	iv	X
ejpam-6375	66	3	)	)	PUNCT
ejpam-6375	66	4	a	a	DET
ejpam-6375	66	5	generalized	generalized	ADJ
ejpam-6375	66	6	semi	semi	ADJ
ejpam-6375	66	7	-	-	ADJ
ejpam-6375	66	8	closure	closure	ADJ
ejpam-6375	66	9	consists	consist	VERB
ejpam-6375	66	10	of	of	ADP
ejpam-6375	66	11	the	the	DET
ejpam-6375	66	12	intersection	intersection	NOUN
ejpam-6375	66	13	between	between	ADP
ejpam-6375	66	14	all	all	DET
ejpam-6375	66	15	generalized	generalized	ADJ
ejpam-6375	66	16	semiclosed	semiclose	VERB
ejpam-6375	66	17	sets	set	NOUN
ejpam-6375	66	18	of	of	ADP
ejpam-6375	66	19	v	v	NOUN
ejpam-6375	66	20	that	that	PRON
ejpam-6375	66	21	contain	contain	VERB
ejpam-6375	66	22	e.	e.	PROPN
ejpam-6375	66	23	this	this	DET
ejpam-6375	66	24	set	set	NOUN
ejpam-6375	66	25	bears	bear	VERB
ejpam-6375	66	26	the	the	DET
ejpam-6375	66	27	notation	notation	NOUN
ejpam-6375	66	28	sgcl(e	sgcl(e	PROPN
ejpam-6375	66	29	)	)	PUNCT
ejpam-6375	66	30	.	.	PUNCT
ejpam-6375	67	1	definition	definition	NOUN
ejpam-6375	67	2	2.4	2.4	NUM
ejpam-6375	67	3	.	.	PUNCT
ejpam-6375	68	1	[	[	X
ejpam-6375	68	2	14	14	NUM
ejpam-6375	68	3	]	]	X
ejpam-6375	68	4	if	if	SCONJ
ejpam-6375	68	5	(	(	PUNCT
ejpam-6375	68	6	v	v	NOUN
ejpam-6375	68	7	,	,	PUNCT
ejpam-6375	68	8	gτ	gτ	PROPN
ejpam-6375	68	9	)	)	PUNCT
ejpam-6375	68	10	has	have	VERB
ejpam-6375	68	11	a	a	DET
ejpam-6375	68	12	finite	finite	ADJ
ejpam-6375	68	13	subcover	subcover	NOUN
ejpam-6375	68	14	for	for	ADP
ejpam-6375	68	15	every	every	DET
ejpam-6375	68	16	open	open	ADJ
ejpam-6375	68	17	cover	cover	NOUN
ejpam-6375	68	18	,	,	PUNCT
ejpam-6375	68	19	then	then	ADV
ejpam-6375	68	20	a	a	DET
ejpam-6375	68	21	generalized	generalized	ADJ
ejpam-6375	68	22	topological	topological	ADJ
ejpam-6375	68	23	space	space	NOUN
ejpam-6375	68	24	(	(	PUNCT
ejpam-6375	68	25	v	v	NOUN
ejpam-6375	68	26	,	,	PUNCT
ejpam-6375	68	27	gτ	gτ	PROPN
ejpam-6375	68	28	)	)	PUNCT
ejpam-6375	68	29	is	be	AUX
ejpam-6375	68	30	called	call	VERB
ejpam-6375	68	31	gτ	gτ	PROPN
ejpam-6375	68	32	-compact	-compact	PROPN
ejpam-6375	68	33	.	.	PUNCT
ejpam-6375	69	1	definition	definition	NOUN
ejpam-6375	69	2	2.5	2.5	NUM
ejpam-6375	69	3	.	.	PUNCT
ejpam-6375	70	1	[	[	X
ejpam-6375	70	2	1	1	X
ejpam-6375	70	3	]	]	X
ejpam-6375	70	4	assume	assume	VERB
ejpam-6375	70	5	(	(	PUNCT
ejpam-6375	70	6	v	v	NOUN
ejpam-6375	70	7	gτ	gτ	PROPN
ejpam-6375	70	8	1	1	NUM
ejpam-6375	70	9	)	)	PUNCT
ejpam-6375	70	10	and	and	CCONJ
ejpam-6375	70	11	(	(	PUNCT
ejpam-6375	70	12	z	z	NOUN
ejpam-6375	70	13	,	,	PUNCT
ejpam-6375	70	14	gτ	gτ	PROPN
ejpam-6375	70	15	2	2	NUM
ejpam-6375	70	16	)	)	PUNCT
ejpam-6375	70	17	as	as	ADP
ejpam-6375	70	18	gts	gts	NOUN
ejpam-6375	70	19	.	.	PUNCT
ejpam-6375	71	1	a	a	DET
ejpam-6375	71	2	mapping	mapping	NOUN
ejpam-6375	71	3	j	j	NOUN
ejpam-6375	71	4	:	:	PUNCT
ejpam-6375	71	5	(	(	PUNCT
ejpam-6375	71	6	v	v	NOUN
ejpam-6375	71	7	,	,	PUNCT
ejpam-6375	71	8	gτ	gτ	PROPN
ejpam-6375	71	9	1	1	NUM
ejpam-6375	71	10	)	)	PUNCT
ejpam-6375	71	11	→	→	PUNCT
ejpam-6375	71	12	(	(	PUNCT
ejpam-6375	71	13	z	z	NOUN
ejpam-6375	71	14	,	,	PUNCT
ejpam-6375	71	15	gτ	gτ	PROPN
ejpam-6375	71	16	2	2	NUM
ejpam-6375	71	17	)	)	PUNCT
ejpam-6375	71	18	is	be	AUX
ejpam-6375	71	19	classified	classify	VERB
ejpam-6375	71	20	as	as	ADP
ejpam-6375	71	21	gτ	gτ	PROPN
ejpam-6375	71	22	-s∗	-s∗	PROPN
ejpam-6375	71	23	g	g	PROPN
ejpam-6375	71	24	-irresolute	-irresolute	PROPN
ejpam-6375	71	25	when	when	SCONJ
ejpam-6375	71	26	the	the	DET
ejpam-6375	71	27	preimage	preimage	NOUN
ejpam-6375	71	28	of	of	ADP
ejpam-6375	71	29	every	every	DET
ejpam-6375	71	30	gτ	gτ	PROPN
ejpam-6375	71	31	-s∗	-s∗	PROPN
ejpam-6375	71	32	g	g	PROPN
ejpam-6375	71	33	-open	-open	NOUN
ejpam-6375	71	34	set	set	VERB
ejpam-6375	71	35	in	in	ADP
ejpam-6375	71	36	(	(	PUNCT
ejpam-6375	71	37	z	z	NOUN
ejpam-6375	71	38	,	,	PUNCT
ejpam-6375	71	39	gτ	gτ	PROPN
ejpam-6375	71	40	2	2	NUM
ejpam-6375	71	41	)	)	PUNCT
ejpam-6375	71	42	is	be	AUX
ejpam-6375	71	43	a	a	DET
ejpam-6375	71	44	gτ	gτ	NOUN
ejpam-6375	71	45	-s∗	-s∗	NOUN
ejpam-6375	71	46	g	g	PROPN
ejpam-6375	71	47	-open	-open	NOUN
ejpam-6375	71	48	set	set	VERB
ejpam-6375	71	49	in	in	ADP
ejpam-6375	71	50	(	(	PUNCT
ejpam-6375	71	51	v	v	NOUN
ejpam-6375	71	52	,	,	PUNCT
ejpam-6375	71	53	gτ	gτ	PROPN
ejpam-6375	71	54	1	1	NUM
ejpam-6375	71	55	)	)	PUNCT
ejpam-6375	71	56	.	.	PUNCT
ejpam-6375	72	1	main	main	ADJ
ejpam-6375	72	2	results	result	NOUN
ejpam-6375	72	3	3	3	NUM
ejpam-6375	72	4	.	.	X
ejpam-6375	72	5	gpt	gpt	NOUN
ejpam-6375	72	6	-	-	PUNCT
ejpam-6375	72	7	s∗	s∗	PROPN
ejpam-6375	72	8	g	g	PROPN
ejpam-6375	72	9	-compact	-compact	PROPN
ejpam-6375	72	10	space	space	NOUN
ejpam-6375	72	11	in	in	ADP
ejpam-6375	72	12	gpts	gpt	NOUN
ejpam-6375	72	13	this	this	DET
ejpam-6375	72	14	particular	particular	ADJ
ejpam-6375	72	15	section	section	NOUN
ejpam-6375	72	16	evolves	evolve	VERB
ejpam-6375	72	17	to	to	ADP
ejpam-6375	72	18	generalized	generalize	VERB
ejpam-6375	72	19	primal	primal	ADJ
ejpam-6375	72	20	topological	topological	ADJ
ejpam-6375	72	21	spaces	space	NOUN
ejpam-6375	72	22	.	.	PUNCT
ejpam-6375	73	1	the	the	DET
ejpam-6375	73	2	extension	extension	NOUN
ejpam-6375	73	3	of	of	ADP
ejpam-6375	73	4	the	the	DET
ejpam-6375	73	5	previous	previous	ADJ
ejpam-6375	73	6	ideas	idea	NOUN
ejpam-6375	73	7	is	be	AUX
ejpam-6375	73	8	emphasized	emphasize	VERB
ejpam-6375	73	9	in	in	ADP
ejpam-6375	73	10	this	this	DET
ejpam-6375	73	11	part	part	NOUN
ejpam-6375	73	12	,	,	PUNCT
ejpam-6375	73	13	which	which	PRON
ejpam-6375	73	14	also	also	ADV
ejpam-6375	73	15	uses	use	VERB
ejpam-6375	73	16	the	the	DET
ejpam-6375	73	17	kuratowski	kuratowski	ADJ
ejpam-6375	73	18	closure	closure	NOUN
ejpam-6375	73	19	operator	operator	NOUN
ejpam-6375	73	20	to	to	PART
ejpam-6375	73	21	examine	examine	VERB
ejpam-6375	73	22	closure	closure	NOUN
ejpam-6375	73	23	features	feature	NOUN
ejpam-6375	73	24	in	in	ADP
ejpam-6375	73	25	primal	primal	ADJ
ejpam-6375	73	26	topological	topological	ADJ
ejpam-6375	73	27	spaces	space	NOUN
ejpam-6375	73	28	.	.	PUNCT
ejpam-6375	74	1	definition	definition	NOUN
ejpam-6375	74	2	3.1	3.1	NUM
ejpam-6375	74	3	.	.	PUNCT
ejpam-6375	75	1	[	[	X
ejpam-6375	75	2	21	21	NUM
ejpam-6375	75	3	]	]	PUNCT
ejpam-6375	75	4	assume	assume	VERB
ejpam-6375	75	5	that	that	SCONJ
ejpam-6375	75	6	v	v	ADP
ejpam-6375	75	7	̸=	̸=	PROPN
ejpam-6375	75	8	∅.	∅.	ADP
ejpam-6375	75	9	a	a	DET
ejpam-6375	75	10	grill	grill	NOUN
ejpam-6375	75	11	on	on	ADP
ejpam-6375	75	12	v	v	NUM
ejpam-6375	75	13	is	be	AUX
ejpam-6375	75	14	a	a	DET
ejpam-6375	75	15	family	family	NOUN
ejpam-6375	75	16	g	g	PROPN
ejpam-6375	75	17	⊆	⊆	NUM
ejpam-6375	75	18	2v	2v	NUM
ejpam-6375	75	19	if	if	SCONJ
ejpam-6375	75	20	the	the	DET
ejpam-6375	75	21	following	follow	VERB
ejpam-6375	75	22	criteria	criterion	NOUN
ejpam-6375	75	23	are	be	AUX
ejpam-6375	75	24	met	meet	VERB
ejpam-6375	75	25	:	:	PUNCT
ejpam-6375	75	26	(	(	PUNCT
ejpam-6375	75	27	i	i	NOUN
ejpam-6375	75	28	)	)	PUNCT
ejpam-6375	75	29	∅	∅	NOUN
ejpam-6375	75	30	is	be	AUX
ejpam-6375	75	31	not	not	PART
ejpam-6375	75	32	a	a	DET
ejpam-6375	75	33	member	member	NOUN
ejpam-6375	75	34	of	of	ADP
ejpam-6375	75	35	g.	g.	PROPN
ejpam-6375	75	36	(	(	PUNCT
ejpam-6375	75	37	ii	ii	PROPN
ejpam-6375	75	38	)	)	PUNCT
ejpam-6375	75	39	for	for	ADP
ejpam-6375	75	40	d	d	PROPN
ejpam-6375	75	41	,	,	PUNCT
ejpam-6375	75	42	e	e	PROPN
ejpam-6375	75	43	⊆	⊆	NUM
ejpam-6375	75	44	v	v	ADP
ejpam-6375	75	45	having	have	VERB
ejpam-6375	75	46	d	d	NOUN
ejpam-6375	75	47	⊆	⊆	NUM
ejpam-6375	75	48	e	e	NOUN
ejpam-6375	75	49	implies	imply	VERB
ejpam-6375	75	50	e	e	PROPN
ejpam-6375	75	51	∈	∈	PROPN
ejpam-6375	75	52	g	g	PROPN
ejpam-6375	75	53	if	if	SCONJ
ejpam-6375	75	54	d	d	PROPN
ejpam-6375	75	55	∈	∈	PROPN
ejpam-6375	75	56	g.	g.	PROPN
ejpam-6375	75	57	(	(	PUNCT
ejpam-6375	75	58	iii	iii	NOUN
ejpam-6375	75	59	)	)	PUNCT
ejpam-6375	75	60	for	for	ADP
ejpam-6375	75	61	d	d	PROPN
ejpam-6375	75	62	,	,	PUNCT
ejpam-6375	75	63	e	e	PROPN
ejpam-6375	75	64	⊆	⊆	NUM
ejpam-6375	75	65	v	v	NOUN
ejpam-6375	75	66	,	,	PUNCT
ejpam-6375	75	67	then	then	ADV
ejpam-6375	75	68	d	d	X
ejpam-6375	75	69	∪	∪	X
ejpam-6375	75	70	e	e	PROPN
ejpam-6375	75	71	∈	∈	PROPN
ejpam-6375	75	72	g	g	PROPN
ejpam-6375	75	73	,	,	PUNCT
ejpam-6375	75	74	whenever	whenever	SCONJ
ejpam-6375	75	75	d	d	PROPN
ejpam-6375	75	76	∈	∈	PROPN
ejpam-6375	75	77	g	g	NOUN
ejpam-6375	75	78	or	or	CCONJ
ejpam-6375	75	79	e	e	PROPN
ejpam-6375	75	80	∈	∈	PROPN
ejpam-6375	75	81	g.	g.	NOUN
ejpam-6375	75	82	definition	definition	NOUN
ejpam-6375	75	83	3.2	3.2	NUM
ejpam-6375	75	84	.	.	PUNCT
ejpam-6375	76	1	[	[	X
ejpam-6375	76	2	8	8	NUM
ejpam-6375	76	3	]	]	PUNCT
ejpam-6375	76	4	assume	assume	VERB
ejpam-6375	76	5	that	that	SCONJ
ejpam-6375	76	6	v	v	ADP
ejpam-6375	76	7	̸=	̸=	PROPN
ejpam-6375	76	8	∅.	∅.	ADP
ejpam-6375	76	9	a	a	DET
ejpam-6375	76	10	primal	primal	NOUN
ejpam-6375	76	11	on	on	ADP
ejpam-6375	76	12	v	v	NUM
ejpam-6375	76	13	is	be	AUX
ejpam-6375	76	14	a	a	DET
ejpam-6375	76	15	collection	collection	NOUN
ejpam-6375	76	16	p	p	NOUN
ejpam-6375	76	17	of	of	ADP
ejpam-6375	76	18	2v	2v	PROPN
ejpam-6375	76	19	if	if	SCONJ
ejpam-6375	76	20	the	the	DET
ejpam-6375	76	21	following	follow	VERB
ejpam-6375	76	22	criteria	criterion	NOUN
ejpam-6375	76	23	are	be	AUX
ejpam-6375	76	24	met	meet	VERB
ejpam-6375	76	25	:	:	PUNCT
ejpam-6375	76	26	(	(	PUNCT
ejpam-6375	76	27	i	i	NOUN
ejpam-6375	76	28	)	)	PUNCT
ejpam-6375	76	29	v	v	ADP
ejpam-6375	76	30	̸∈	̸∈	PROPN
ejpam-6375	76	31	p.	p.	PROPN
ejpam-6375	76	32	(	(	PUNCT
ejpam-6375	76	33	ii	ii	PROPN
ejpam-6375	76	34	)	)	PUNCT
ejpam-6375	77	1	if	if	SCONJ
ejpam-6375	77	2	d	d	PROPN
ejpam-6375	77	3	∈	∈	PROPN
ejpam-6375	77	4	p	p	NOUN
ejpam-6375	77	5	and	and	CCONJ
ejpam-6375	77	6	e	e	NOUN
ejpam-6375	77	7	⊆	⊆	NUM
ejpam-6375	77	8	d	d	NOUN
ejpam-6375	77	9	then	then	ADV
ejpam-6375	77	10	e	e	PROPN
ejpam-6375	77	11	∈	∈	PROPN
ejpam-6375	77	12	p.	p.	NOUN
ejpam-6375	77	13	(	(	PUNCT
ejpam-6375	77	14	iii	iii	X
ejpam-6375	77	15	)	)	PUNCT
ejpam-6375	77	16	if	if	SCONJ
ejpam-6375	77	17	d	d	PROPN
ejpam-6375	77	18	∩	∩	NOUN
ejpam-6375	77	19	e	e	NOUN
ejpam-6375	77	20	∈	∈	PROPN
ejpam-6375	77	21	p	p	X
ejpam-6375	77	22	,	,	PUNCT
ejpam-6375	77	23	then	then	ADV
ejpam-6375	77	24	d	d	PROPN
ejpam-6375	77	25	∈	∈	PROPN
ejpam-6375	77	26	p	p	NOUN
ejpam-6375	77	27	or	or	CCONJ
ejpam-6375	77	28	e	e	NOUN
ejpam-6375	77	29	∈	∈	PROPN
ejpam-6375	77	30	p.	p.	NOUN
ejpam-6375	77	31	a	a	DET
ejpam-6375	77	32	pair	pair	NOUN
ejpam-6375	77	33	(	(	PUNCT
ejpam-6375	77	34	v	v	NOUN
ejpam-6375	77	35	,	,	PUNCT
ejpam-6375	77	36	gτ	gτ	PROPN
ejpam-6375	77	37	)	)	PUNCT
ejpam-6375	77	38	with	with	ADP
ejpam-6375	77	39	a	a	DET
ejpam-6375	77	40	primal	primal	ADJ
ejpam-6375	77	41	p	p	NOUN
ejpam-6375	77	42	on	on	ADP
ejpam-6375	77	43	v	v	NUM
ejpam-6375	77	44	is	be	AUX
ejpam-6375	77	45	termed	term	VERB
ejpam-6375	77	46	generalized	generalized	ADJ
ejpam-6375	77	47	primal	primal	ADJ
ejpam-6375	77	48	topological	topological	ADJ
ejpam-6375	77	49	space	space	NOUN
ejpam-6375	77	50	(	(	PUNCT
ejpam-6375	77	51	gpts	gpt	NOUN
ejpam-6375	77	52	)	)	PUNCT
ejpam-6375	77	53	symbolized	symbolize	VERB
ejpam-6375	77	54	as	as	ADP
ejpam-6375	77	55	(	(	PUNCT
ejpam-6375	77	56	v	v	NOUN
ejpam-6375	77	57	,	,	PUNCT
ejpam-6375	77	58	gτ	gτ	INTJ
ejpam-6375	77	59	,	,	PUNCT
ejpam-6375	77	60	p	p	NOUN
ejpam-6375	77	61	)	)	PUNCT
ejpam-6375	77	62	.	.	PUNCT
ejpam-6375	78	1	the	the	DET
ejpam-6375	78	2	members	member	NOUN
ejpam-6375	78	3	of	of	ADP
ejpam-6375	78	4	(	(	PUNCT
ejpam-6375	78	5	v	v	NOUN
ejpam-6375	78	6	,	,	PUNCT
ejpam-6375	78	7	gτ	gτ	INTJ
ejpam-6375	78	8	,	,	PUNCT
ejpam-6375	78	9	p	p	X
ejpam-6375	78	10	)	)	PUNCT
ejpam-6375	78	11	are	be	AUX
ejpam-6375	78	12	known	know	VERB
ejpam-6375	78	13	as	as	ADP
ejpam-6375	78	14	gpt	gpt	NOUN
ejpam-6375	78	15	-	-	PUNCT
ejpam-6375	78	16	open	open	ADJ
ejpam-6375	78	17	sets	set	NOUN
ejpam-6375	78	18	,	,	PUNCT
ejpam-6375	78	19	and	and	CCONJ
ejpam-6375	78	20	their	their	PRON
ejpam-6375	78	21	complements	complement	NOUN
ejpam-6375	78	22	are	be	AUX
ejpam-6375	78	23	considered	consider	VERB
ejpam-6375	78	24	gpt	gpt	NOUN
ejpam-6375	78	25	-	-	PUNCT
ejpam-6375	78	26	closed	close	VERB
ejpam-6375	78	27	sets	set	NOUN
ejpam-6375	78	28	.	.	PUNCT
ejpam-6375	79	1	definition	definition	NOUN
ejpam-6375	79	2	3.3	3.3	NUM
ejpam-6375	79	3	.	.	PUNCT
ejpam-6375	80	1	[	[	X
ejpam-6375	80	2	9	9	NUM
ejpam-6375	80	3	]	]	PUNCT
ejpam-6375	80	4	assume	assume	VERB
ejpam-6375	80	5	a	a	DET
ejpam-6375	80	6	⊆	⊆	NUM
ejpam-6375	80	7	v.	v.	SCONJ
ejpam-6375	80	8	let	let	VERB
ejpam-6375	80	9	an	an	DET
ejpam-6375	80	10	operator	operator	NOUN
ejpam-6375	80	11	(	(	PUNCT
ejpam-6375	80	12	.	.	PUNCT
ejpam-6375	80	13	)	)	PUNCT
ejpam-6375	81	1	◦	◦	NOUN
ejpam-6375	81	2	:	:	PUNCT
ejpam-6375	81	3	2v	2v	PROPN
ejpam-6375	81	4	→	→	SYM
ejpam-6375	81	5	2v	2v	PROPN
ejpam-6375	81	6	in	in	ADP
ejpam-6375	81	7	gpts	gpt	NOUN
ejpam-6375	81	8	is	be	AUX
ejpam-6375	81	9	defined	define	VERB
ejpam-6375	81	10	as	as	ADP
ejpam-6375	81	11	a	a	DET
ejpam-6375	81	12	◦	◦	NOUN
ejpam-6375	81	13	(	(	PUNCT
ejpam-6375	81	14	v	v	NOUN
ejpam-6375	81	15	,	,	PUNCT
ejpam-6375	81	16	gτ	gτ	INTJ
ejpam-6375	81	17	,	,	PUNCT
ejpam-6375	81	18	p	p	X
ejpam-6375	81	19	)	)	PUNCT
ejpam-6375	81	20	=	=	SYM
ejpam-6375	82	1	{	{	PUNCT
ejpam-6375	82	2	x	x	PUNCT
ejpam-6375	82	3	∈	∈	PROPN
ejpam-6375	82	4	v	v	NOUN
ejpam-6375	82	5	:	:	PUNCT
ejpam-6375	82	6	a	a	DET
ejpam-6375	82	7	c	c	NOUN
ejpam-6375	82	8	∪	∪	NOUN
ejpam-6375	82	9	oc	oc	ADP
ejpam-6375	82	10	∈	∈	PROPN
ejpam-6375	82	11	p	p	NOUN
ejpam-6375	82	12	,	,	PUNCT
ejpam-6375	82	13	∀	∀	NOUN
ejpam-6375	82	14	o	o	NOUN
ejpam-6375	82	15	∈	∈	PROPN
ejpam-6375	82	16	ψ(x	ψ(x	NOUN
ejpam-6375	82	17	)	)	PUNCT
ejpam-6375	82	18	}	}	PUNCT
ejpam-6375	82	19	where	where	SCONJ
ejpam-6375	82	20	o	o	NOUN
ejpam-6375	82	21	is	be	AUX
ejpam-6375	82	22	generalized	generalize	VERB
ejpam-6375	82	23	primal	primal	ADJ
ejpam-6375	82	24	neighbourhood	neighbourhood	NOUN
ejpam-6375	82	25	of	of	ADP
ejpam-6375	82	26	x	x	PUNCT
ejpam-6375	82	27	in	in	ADP
ejpam-6375	82	28	v	v	NOUN
ejpam-6375	82	29	and	and	CCONJ
ejpam-6375	82	30	the	the	DET
ejpam-6375	82	31	collection	collection	NOUN
ejpam-6375	82	32	of	of	ADP
ejpam-6375	82	33	all	all	DET
ejpam-6375	82	34	generalized	generalized	ADJ
ejpam-6375	82	35	neighbourhood	neighbourhood	NOUN
ejpam-6375	82	36	of	of	ADP
ejpam-6375	82	37	v	v	NOUN
ejpam-6375	82	38	is	be	AUX
ejpam-6375	82	39	termed	term	VERB
ejpam-6375	82	40	as	as	ADP
ejpam-6375	82	41	ψ(v	ψ(v	PROPN
ejpam-6375	82	42	)	)	PUNCT
ejpam-6375	82	43	.	.	PUNCT
ejpam-6375	83	1	m.	m.	PROPN
ejpam-6375	83	2	shahbaz	shahbaz	PROPN
ejpam-6375	83	3	et	et	PROPN
ejpam-6375	83	4	al	al	PROPN
ejpam-6375	83	5	.	.	PUNCT
ejpam-6375	83	6	/	/	SYM
ejpam-6375	83	7	eur	eur	PROPN
ejpam-6375	83	8	.	.	PUNCT
ejpam-6375	84	1	j.	j.	PROPN
ejpam-6375	84	2	pure	pure	PROPN
ejpam-6375	84	3	appl	appl	PROPN
ejpam-6375	84	4	.	.	PROPN
ejpam-6375	84	5	math	math	PROPN
ejpam-6375	84	6	,	,	PUNCT
ejpam-6375	84	7	18	18	NUM
ejpam-6375	84	8	(	(	PUNCT
ejpam-6375	84	9	4	4	NUM
ejpam-6375	84	10	)	)	PUNCT
ejpam-6375	84	11	(	(	PUNCT
ejpam-6375	84	12	2025	2025	NUM
ejpam-6375	84	13	)	)	PUNCT
ejpam-6375	84	14	,	,	PUNCT
ejpam-6375	84	15	6375	6375	NUM
ejpam-6375	84	16	5	5	NUM
ejpam-6375	84	17	of	of	ADP
ejpam-6375	84	18	22	22	NUM
ejpam-6375	84	19	definition	definition	NOUN
ejpam-6375	84	20	3.4	3.4	NUM
ejpam-6375	84	21	.	.	PUNCT
ejpam-6375	85	1	let	let	VERB
ejpam-6375	85	2	a	a	DET
ejpam-6375	85	3	⊆	⊆	NUM
ejpam-6375	85	4	v	v	NOUN
ejpam-6375	85	5	in	in	ADP
ejpam-6375	85	6	gpts	gpt	NOUN
ejpam-6375	85	7	.	.	PUNCT
ejpam-6375	86	1	the	the	DET
ejpam-6375	86	2	generalized	generalized	ADJ
ejpam-6375	86	3	kuratowski	kuratowski	ADJ
ejpam-6375	86	4	closure	closure	NOUN
ejpam-6375	86	5	operator	operator	NOUN
ejpam-6375	86	6	cl	cl	NOUN
ejpam-6375	86	7	◦	◦	NOUN
ejpam-6375	86	8	is	be	AUX
ejpam-6375	86	9	defined	define	VERB
ejpam-6375	86	10	as	as	ADP
ejpam-6375	86	11	cl	cl	NOUN
ejpam-6375	86	12	◦	◦	NOUN
ejpam-6375	86	13	(a	(a	NOUN
ejpam-6375	86	14	)	)	PUNCT
ejpam-6375	86	15	=	=	PUNCT
ejpam-6375	87	1	a∪	a∪	ADP
ejpam-6375	87	2	a	a	DET
ejpam-6375	87	3	◦	◦	NOUN
ejpam-6375	87	4	,	,	PUNCT
ejpam-6375	87	5	with	with	ADP
ejpam-6375	87	6	the	the	DET
ejpam-6375	87	7	condition	condition	NOUN
ejpam-6375	87	8	:	:	PUNCT
ejpam-6375	87	9	cl	cl	NOUN
ejpam-6375	87	10	◦	◦	NOUN
ejpam-6375	87	11	(a	(a	NOUN
ejpam-6375	87	12	∪b	∪b	NOUN
ejpam-6375	87	13	)	)	PUNCT
ejpam-6375	87	14	⊇	⊇	NOUN
ejpam-6375	87	15	cl	cl	NOUN
ejpam-6375	87	16	◦	◦	NOUN
ejpam-6375	87	17	(a	(a	NOUN
ejpam-6375	87	18	)	)	PUNCT
ejpam-6375	87	19	∪	∪	ADP
ejpam-6375	87	20	cl	cl	NOUN
ejpam-6375	87	21	◦	◦	NOUN
ejpam-6375	87	22	(b	(b	NOUN
ejpam-6375	87	23	)	)	PUNCT
ejpam-6375	87	24	.	.	PUNCT
ejpam-6375	88	1	remark	remark	PROPN
ejpam-6375	88	2	3.1	3.1	NUM
ejpam-6375	88	3	.	.	PUNCT
ejpam-6375	89	1	the	the	DET
ejpam-6375	89	2	operator	operator	NOUN
ejpam-6375	89	3	cl	cl	NOUN
ejpam-6375	89	4	◦	◦	NOUN
ejpam-6375	89	5	satisfies	satisfie	NOUN
ejpam-6375	89	6	the	the	DET
ejpam-6375	89	7	following	follow	VERB
ejpam-6375	89	8	properties	property	NOUN
ejpam-6375	89	9	:	:	PUNCT
ejpam-6375	89	10	(	(	PUNCT
ejpam-6375	89	11	i	i	NOUN
ejpam-6375	89	12	)	)	PUNCT
ejpam-6375	89	13	extensivity	extensivity	NOUN
ejpam-6375	89	14	:	:	PUNCT
ejpam-6375	89	15	a	a	DET
ejpam-6375	89	16	⊆	⊆	NUM
ejpam-6375	89	17	cl	cl	NOUN
ejpam-6375	89	18	◦	◦	NOUN
ejpam-6375	89	19	(a	(a	NUM
ejpam-6375	89	20	)	)	PUNCT
ejpam-6375	89	21	.	.	PUNCT
ejpam-6375	90	1	(	(	PUNCT
ejpam-6375	90	2	ii	ii	NOUN
ejpam-6375	90	3	)	)	PUNCT
ejpam-6375	90	4	monotonicity	monotonicity	NOUN
ejpam-6375	90	5	:	:	PUNCT
ejpam-6375	90	6	if	if	SCONJ
ejpam-6375	90	7	a	a	DET
ejpam-6375	90	8	⊆	⊆	NUM
ejpam-6375	90	9	b	b	NOUN
ejpam-6375	90	10	,	,	PUNCT
ejpam-6375	90	11	then	then	ADV
ejpam-6375	90	12	cl	cl	VERB
ejpam-6375	90	13	◦	◦	NOUN
ejpam-6375	90	14	(a	(a	NOUN
ejpam-6375	90	15	)	)	PUNCT
ejpam-6375	90	16	⊆	⊆	NUM
ejpam-6375	90	17	cl	cl	NOUN
ejpam-6375	90	18	◦	◦	NOUN
ejpam-6375	90	19	(b	(b	NOUN
ejpam-6375	90	20	)	)	PUNCT
ejpam-6375	90	21	.	.	PUNCT
ejpam-6375	91	1	(	(	PUNCT
ejpam-6375	91	2	iii	iii	X
ejpam-6375	91	3	)	)	PUNCT
ejpam-6375	91	4	idempotency	idempotency	NOUN
ejpam-6375	91	5	:	:	PUNCT
ejpam-6375	91	6	cl	cl	NOUN
ejpam-6375	91	7	◦	◦	NOUN
ejpam-6375	91	8	(cl	(cl	NOUN
ejpam-6375	91	9	◦	◦	NOUN
ejpam-6375	91	10	(a	(a	NUM
ejpam-6375	91	11	)	)	PUNCT
ejpam-6375	91	12	)	)	PUNCT
ejpam-6375	92	1	=	=	PUNCT
ejpam-6375	92	2	cl	cl	NOUN
ejpam-6375	92	3	◦	◦	NOUN
ejpam-6375	92	4	(a	(a	NUM
ejpam-6375	92	5	)	)	PUNCT
ejpam-6375	92	6	.	.	PUNCT
ejpam-6375	93	1	(	(	PUNCT
ejpam-6375	93	2	iv	iv	X
ejpam-6375	93	3	)	)	PUNCT
ejpam-6375	93	4	generalized	generalize	VERB
ejpam-6375	93	5	subset	subset	NOUN
ejpam-6375	93	6	union	union	NOUN
ejpam-6375	93	7	property	property	NOUN
ejpam-6375	93	8	:	:	PUNCT
ejpam-6375	93	9	cl	cl	VERB
ejpam-6375	93	10	◦	◦	NOUN
ejpam-6375	93	11	(a	(a	NOUN
ejpam-6375	93	12	∪b	∪b	NOUN
ejpam-6375	93	13	)	)	PUNCT
ejpam-6375	93	14	⊇	⊇	NOUN
ejpam-6375	93	15	cl	cl	NOUN
ejpam-6375	93	16	◦	◦	NOUN
ejpam-6375	93	17	(a	(a	NOUN
ejpam-6375	93	18	)	)	PUNCT
ejpam-6375	93	19	∪	∪	ADP
ejpam-6375	93	20	cl	cl	NOUN
ejpam-6375	93	21	◦	◦	NOUN
ejpam-6375	93	22	(b	(b	NOUN
ejpam-6375	93	23	)	)	PUNCT
ejpam-6375	93	24	.	.	PUNCT
ejpam-6375	94	1	in	in	ADP
ejpam-6375	94	2	standard	standard	ADJ
ejpam-6375	94	3	topological	topological	ADJ
ejpam-6375	94	4	spaces	space	NOUN
ejpam-6375	94	5	,	,	PUNCT
ejpam-6375	94	6	cl	cl	NOUN
ejpam-6375	94	7	◦	◦	NOUN
ejpam-6375	94	8	reduces	reduce	VERB
ejpam-6375	94	9	to	to	ADP
ejpam-6375	94	10	the	the	DET
ejpam-6375	94	11	classical	classical	ADJ
ejpam-6375	94	12	kuratowski	kuratowski	ADJ
ejpam-6375	94	13	closure	closure	NOUN
ejpam-6375	94	14	operator	operator	NOUN
ejpam-6375	94	15	.	.	PUNCT
ejpam-6375	95	1	proof	proof	NOUN
ejpam-6375	95	2	.	.	PUNCT
ejpam-6375	96	1	1	1	X
ejpam-6375	96	2	.	.	X
ejpam-6375	96	3	extensivity	extensivity	NOUN
ejpam-6375	96	4	by	by	ADP
ejpam-6375	96	5	definition	definition	NOUN
ejpam-6375	96	6	,	,	PUNCT
ejpam-6375	96	7	the	the	DET
ejpam-6375	96	8	generalized	generalized	ADJ
ejpam-6375	96	9	kuratowski	kuratowski	ADJ
ejpam-6375	96	10	closure	closure	NOUN
ejpam-6375	96	11	operator	operator	NOUN
ejpam-6375	96	12	cl	cl	NOUN
ejpam-6375	96	13	◦	◦	NOUN
ejpam-6375	96	14	(a	(a	NOUN
ejpam-6375	96	15	)	)	PUNCT
ejpam-6375	96	16	consists	consist	VERB
ejpam-6375	96	17	of	of	ADP
ejpam-6375	96	18	all	all	DET
ejpam-6375	96	19	points	point	NOUN
ejpam-6375	96	20	in	in	ADP
ejpam-6375	96	21	a.	a.	NOUN
ejpam-6375	96	22	all	all	PRON
ejpam-6375	96	23	generalized	generalize	VERB
ejpam-6375	96	24	limit	limit	NOUN
ejpam-6375	96	25	points	point	NOUN
ejpam-6375	96	26	of	of	ADP
ejpam-6375	96	27	a	a	DET
ejpam-6375	96	28	,	,	PUNCT
ejpam-6375	96	29	i.e.	i.e.	X
ejpam-6375	96	30	,	,	PUNCT
ejpam-6375	96	31	points	point	VERB
ejpam-6375	96	32	where	where	SCONJ
ejpam-6375	96	33	every	every	DET
ejpam-6375	96	34	generalized	generalize	VERB
ejpam-6375	96	35	primal	primal	ADJ
ejpam-6375	96	36	neighbourhood	neighbourhood	NOUN
ejpam-6375	96	37	intersects	intersect	NOUN
ejpam-6375	96	38	a.	a.	NOUN
ejpam-6375	96	39	since	since	SCONJ
ejpam-6375	96	40	every	every	DET
ejpam-6375	96	41	point	point	NOUN
ejpam-6375	96	42	in	in	ADP
ejpam-6375	96	43	a	a	PRON
ejpam-6375	96	44	is	be	AUX
ejpam-6375	96	45	trivially	trivially	ADV
ejpam-6375	96	46	in	in	ADP
ejpam-6375	96	47	its	its	PRON
ejpam-6375	96	48	closure	closure	NOUN
ejpam-6375	96	49	,	,	PUNCT
ejpam-6375	96	50	thus	thus	ADV
ejpam-6375	96	51	a	a	DET
ejpam-6375	96	52	⊆	⊆	NUM
ejpam-6375	96	53	cl	cl	NOUN
ejpam-6375	96	54	◦	◦	NOUN
ejpam-6375	96	55	(a	(a	NUM
ejpam-6375	96	56	)	)	PUNCT
ejpam-6375	96	57	.	.	PUNCT
ejpam-6375	97	1	2	2	X
ejpam-6375	97	2	.	.	X
ejpam-6375	97	3	monotonicity	monotonicity	NOUN
ejpam-6375	97	4	suppose	suppose	VERB
ejpam-6375	97	5	a	a	DET
ejpam-6375	97	6	⊆	⊆	NUM
ejpam-6375	97	7	b.	b.	NOUN
ejpam-6375	97	8	any	any	DET
ejpam-6375	97	9	generalized	generalize	VERB
ejpam-6375	97	10	primal	primal	ADJ
ejpam-6375	97	11	neighbourhood	neighbourhood	NOUN
ejpam-6375	97	12	of	of	ADP
ejpam-6375	97	13	a	a	DET
ejpam-6375	97	14	point	point	NOUN
ejpam-6375	97	15	x	x	PUNCT
ejpam-6375	97	16	that	that	PRON
ejpam-6375	97	17	intersects	intersect	VERB
ejpam-6375	97	18	a	a	DET
ejpam-6375	97	19	also	also	ADV
ejpam-6375	97	20	intersects	intersect	NOUN
ejpam-6375	97	21	b.	b.	PROPN
ejpam-6375	97	22	hence	hence	ADV
ejpam-6375	97	23	,	,	PUNCT
ejpam-6375	97	24	any	any	DET
ejpam-6375	97	25	generalized	generalize	VERB
ejpam-6375	97	26	limit	limit	NOUN
ejpam-6375	97	27	point	point	NOUN
ejpam-6375	97	28	of	of	ADP
ejpam-6375	97	29	a	a	PRON
ejpam-6375	97	30	must	must	AUX
ejpam-6375	97	31	also	also	ADV
ejpam-6375	97	32	be	be	AUX
ejpam-6375	97	33	a	a	DET
ejpam-6375	97	34	generalized	generalize	VERB
ejpam-6375	97	35	limit	limit	NOUN
ejpam-6375	97	36	point	point	NOUN
ejpam-6375	97	37	of	of	ADP
ejpam-6375	97	38	b	b	NOUN
ejpam-6375	97	39	,	,	PUNCT
ejpam-6375	97	40	implying	implying	ADJ
ejpam-6375	97	41	cl	cl	NOUN
ejpam-6375	97	42	◦	◦	NOUN
ejpam-6375	97	43	(a	(a	NOUN
ejpam-6375	97	44	)	)	PUNCT
ejpam-6375	97	45	⊆	⊆	NUM
ejpam-6375	97	46	cl	cl	NOUN
ejpam-6375	97	47	◦	◦	NOUN
ejpam-6375	97	48	(b	(b	NOUN
ejpam-6375	97	49	)	)	PUNCT
ejpam-6375	97	50	.	.	PUNCT
ejpam-6375	98	1	3	3	X
ejpam-6375	98	2	.	.	X
ejpam-6375	98	3	idempotency	idempotency	NOUN
ejpam-6375	98	4	expanding	expand	VERB
ejpam-6375	98	5	cl	cl	NOUN
ejpam-6375	98	6	◦	◦	NOUN
ejpam-6375	98	7	(cl	(cl	NOUN
ejpam-6375	98	8	◦	◦	NOUN
ejpam-6375	98	9	(a	(a	NUM
ejpam-6375	98	10	)	)	PUNCT
ejpam-6375	98	11	)	)	PUNCT
ejpam-6375	98	12	.	.	PUNCT
ejpam-6375	99	1	applying	apply	VERB
ejpam-6375	99	2	the	the	DET
ejpam-6375	99	3	closure	closure	NOUN
ejpam-6375	99	4	operator	operator	NOUN
ejpam-6375	99	5	twice	twice	ADV
ejpam-6375	99	6	,	,	PUNCT
ejpam-6375	99	7	cl	cl	NOUN
ejpam-6375	99	8	◦	◦	NOUN
ejpam-6375	99	9	(cl	(cl	NOUN
ejpam-6375	99	10	◦	◦	NOUN
ejpam-6375	99	11	(a	(a	NUM
ejpam-6375	99	12	)	)	PUNCT
ejpam-6375	99	13	)	)	PUNCT
ejpam-6375	100	1	=	=	PUNCT
ejpam-6375	100	2	cl	cl	NOUN
ejpam-6375	100	3	◦	◦	NOUN
ejpam-6375	100	4	(a∪a	(a∪a	NOUN
ejpam-6375	100	5	◦	◦	NOUN
ejpam-6375	100	6	)	)	PUNCT
ejpam-6375	100	7	.	.	PUNCT
ejpam-6375	101	1	using	use	VERB
ejpam-6375	101	2	the	the	DET
ejpam-6375	101	3	definition	definition	NOUN
ejpam-6375	101	4	of	of	ADP
ejpam-6375	101	5	closure	closure	NOUN
ejpam-6375	101	6	again	again	ADV
ejpam-6375	101	7	.	.	PUNCT
ejpam-6375	102	1	by	by	ADP
ejpam-6375	102	2	the	the	DET
ejpam-6375	102	3	definition	definition	NOUN
ejpam-6375	102	4	of	of	ADP
ejpam-6375	102	5	the	the	DET
ejpam-6375	102	6	closure	closure	NOUN
ejpam-6375	102	7	operator	operator	NOUN
ejpam-6375	102	8	cl	cl	NOUN
ejpam-6375	102	9	◦	◦	NOUN
ejpam-6375	102	10	(a∪a	(a∪a	NOUN
ejpam-6375	102	11	◦	◦	NOUN
ejpam-6375	102	12	)	)	PUNCT
ejpam-6375	102	13	=	=	SYM
ejpam-6375	102	14	(	(	PUNCT
ejpam-6375	102	15	a∪a	a∪a	ADV
ejpam-6375	102	16	◦	◦	NOUN
ejpam-6375	102	17	)∪(a∪a	)∪(a∪a	NOUN
ejpam-6375	102	18	◦	◦	NOUN
ejpam-6375	102	19	)	)	PUNCT
ejpam-6375	102	20	◦	◦	NOUN
ejpam-6375	102	21	.	.	NOUN
ejpam-6375	103	1	since	since	SCONJ
ejpam-6375	103	2	closure	closure	NOUN
ejpam-6375	103	3	always	always	ADV
ejpam-6375	103	4	includes	include	VERB
ejpam-6375	103	5	the	the	DET
ejpam-6375	103	6	interior	interior	NOUN
ejpam-6375	103	7	,	,	PUNCT
ejpam-6375	103	8	(	(	PUNCT
ejpam-6375	103	9	a	a	DET
ejpam-6375	103	10	∪	∪	NOUN
ejpam-6375	103	11	a	a	DET
ejpam-6375	103	12	◦	◦	NOUN
ejpam-6375	103	13	)	)	PUNCT
ejpam-6375	103	14	◦	◦	NOUN
ejpam-6375	103	15	=	=	PUNCT
ejpam-6375	103	16	a	a	DET
ejpam-6375	103	17	◦	◦	NOUN
ejpam-6375	103	18	.	.	PUNCT
ejpam-6375	104	1	by	by	ADP
ejpam-6375	104	2	substituting	substitute	VERB
ejpam-6375	104	3	this	this	PRON
ejpam-6375	104	4	,	,	PUNCT
ejpam-6375	104	5	we	we	PRON
ejpam-6375	104	6	get	get	VERB
ejpam-6375	104	7	cl	cl	NOUN
ejpam-6375	104	8	◦	◦	NOUN
ejpam-6375	104	9	(cl	(cl	NOUN
ejpam-6375	104	10	◦	◦	NOUN
ejpam-6375	104	11	(a	(a	NUM
ejpam-6375	104	12	)	)	PUNCT
ejpam-6375	104	13	)	)	PUNCT
ejpam-6375	105	1	=	=	PRON
ejpam-6375	105	2	(	(	PUNCT
ejpam-6375	105	3	a	a	DET
ejpam-6375	105	4	∪	∪	NOUN
ejpam-6375	105	5	a	a	DET
ejpam-6375	105	6	◦	◦	NOUN
ejpam-6375	105	7	)	)	PUNCT
ejpam-6375	105	8	∪	∪	ADP
ejpam-6375	105	9	a	a	DET
ejpam-6375	105	10	◦	◦	NOUN
ejpam-6375	105	11	.	.	PUNCT
ejpam-6375	106	1	since	since	SCONJ
ejpam-6375	106	2	a	a	DET
ejpam-6375	106	3	◦	◦	NOUN
ejpam-6375	106	4	is	be	AUX
ejpam-6375	106	5	already	already	ADV
ejpam-6375	106	6	included	include	VERB
ejpam-6375	106	7	in	in	ADP
ejpam-6375	106	8	cl	cl	NOUN
ejpam-6375	106	9	◦	◦	NOUN
ejpam-6375	106	10	(a	(a	NUM
ejpam-6375	106	11	)	)	PUNCT
ejpam-6375	106	12	,	,	PUNCT
ejpam-6375	106	13	implies	imply	VERB
ejpam-6375	106	14	cl	cl	NOUN
ejpam-6375	106	15	◦	◦	NOUN
ejpam-6375	106	16	(cl	(cl	NOUN
ejpam-6375	106	17	◦	◦	NOUN
ejpam-6375	106	18	(a	(a	NUM
ejpam-6375	106	19	)	)	PUNCT
ejpam-6375	106	20	)	)	PUNCT
ejpam-6375	107	1	=	=	PUNCT
ejpam-6375	107	2	cl	cl	NOUN
ejpam-6375	107	3	◦	◦	NOUN
ejpam-6375	107	4	(a	(a	NUM
ejpam-6375	107	5	)	)	PUNCT
ejpam-6375	107	6	.	.	PUNCT
ejpam-6375	108	1	4	4	X
ejpam-6375	108	2	.	.	X
ejpam-6375	108	3	generalized	generalize	VERB
ejpam-6375	108	4	finite	finite	PROPN
ejpam-6375	108	5	union	union	NOUN
ejpam-6375	108	6	property	property	NOUN
ejpam-6375	108	7	consider	consider	VERB
ejpam-6375	108	8	x	x	X
ejpam-6375	108	9	∈	∈	PROPN
ejpam-6375	108	10	cl	cl	NOUN
ejpam-6375	108	11	◦	◦	NOUN
ejpam-6375	108	12	(a)∪cl	(a)∪cl	NOUN
ejpam-6375	108	13	◦	◦	NOUN
ejpam-6375	108	14	(b	(b	NOUN
ejpam-6375	108	15	)	)	PUNCT
ejpam-6375	108	16	.	.	PUNCT
ejpam-6375	109	1	this	this	PRON
ejpam-6375	109	2	means	mean	VERB
ejpam-6375	109	3	x	x	PRON
ejpam-6375	109	4	is	be	AUX
ejpam-6375	109	5	either	either	CCONJ
ejpam-6375	109	6	in	in	ADP
ejpam-6375	109	7	cl	cl	NOUN
ejpam-6375	109	8	◦	◦	NOUN
ejpam-6375	109	9	(a	(a	NOUN
ejpam-6375	109	10	)	)	PUNCT
ejpam-6375	109	11	or	or	CCONJ
ejpam-6375	109	12	cl	cl	NOUN
ejpam-6375	109	13	◦	◦	NOUN
ejpam-6375	109	14	(b	(b	NOUN
ejpam-6375	109	15	)	)	PUNCT
ejpam-6375	109	16	,	,	PUNCT
ejpam-6375	109	17	so	so	CCONJ
ejpam-6375	109	18	every	every	DET
ejpam-6375	109	19	generalized	generalized	ADJ
ejpam-6375	109	20	primal	primal	ADJ
ejpam-6375	109	21	neighbourhood	neighbourhood	NOUN
ejpam-6375	109	22	of	of	ADP
ejpam-6375	109	23	x	x	NOUN
ejpam-6375	109	24	intersects	intersect	NOUN
ejpam-6375	109	25	either	either	CCONJ
ejpam-6375	109	26	a	a	PRON
ejpam-6375	109	27	or	or	CCONJ
ejpam-6375	109	28	b.	b.	PROPN
ejpam-6375	109	29	therefore	therefore	ADV
ejpam-6375	109	30	,	,	PUNCT
ejpam-6375	109	31	every	every	DET
ejpam-6375	109	32	generalized	generalize	VERB
ejpam-6375	109	33	primal	primal	ADJ
ejpam-6375	109	34	neighbourhood	neighbourhood	NOUN
ejpam-6375	109	35	of	of	ADP
ejpam-6375	109	36	x	x	PART
ejpam-6375	109	37	intersects	intersect	NOUN
ejpam-6375	109	38	a∪b	a∪b	NOUN
ejpam-6375	109	39	,	,	PUNCT
ejpam-6375	109	40	implying	implying	ADJ
ejpam-6375	109	41	cl	cl	NOUN
ejpam-6375	109	42	◦	◦	NOUN
ejpam-6375	109	43	(a∪b	(a∪b	NOUN
ejpam-6375	109	44	)	)	PUNCT
ejpam-6375	109	45	⊇	⊇	PROPN
ejpam-6375	109	46	cl	cl	NOUN
ejpam-6375	109	47	◦	◦	NOUN
ejpam-6375	109	48	(a)∪cl	(a)∪cl	NOUN
ejpam-6375	109	49	◦	◦	NOUN
ejpam-6375	109	50	(b	(b	NOUN
ejpam-6375	109	51	)	)	PUNCT
ejpam-6375	109	52	.	.	PUNCT
ejpam-6375	110	1	definition	definition	NOUN
ejpam-6375	110	2	3.5	3.5	NUM
ejpam-6375	110	3	.	.	PUNCT
ejpam-6375	111	1	(	(	PUNCT
ejpam-6375	111	2	i	i	NOUN
ejpam-6375	111	3	)	)	PUNCT
ejpam-6375	111	4	assume	assume	VERB
ejpam-6375	111	5	(	(	PUNCT
ejpam-6375	111	6	v	v	NOUN
ejpam-6375	111	7	,	,	PUNCT
ejpam-6375	111	8	gτ	gτ	INTJ
ejpam-6375	111	9	,	,	PUNCT
ejpam-6375	111	10	p	p	NOUN
ejpam-6375	111	11	)	)	PUNCT
ejpam-6375	111	12	as	as	ADP
ejpam-6375	111	13	a	a	DET
ejpam-6375	111	14	gpts	gpt	NOUN
ejpam-6375	111	15	.	.	PUNCT
ejpam-6375	112	1	if	if	SCONJ
ejpam-6375	112	2	there	there	PRON
ejpam-6375	112	3	exists	exist	VERB
ejpam-6375	112	4	an	an	DET
ejpam-6375	112	5	open	open	ADJ
ejpam-6375	112	6	set	set	NOUN
ejpam-6375	112	7	f	f	PROPN
ejpam-6375	112	8	in	in	ADP
ejpam-6375	112	9	v	v	NUM
ejpam-6375	112	10	such	such	ADJ
ejpam-6375	112	11	that	that	SCONJ
ejpam-6375	112	12	f	f	PROPN
ejpam-6375	112	13	⊆	⊆	NUM
ejpam-6375	112	14	e	e	PROPN
ejpam-6375	112	15	⊆	⊆	NUM
ejpam-6375	112	16	cl	cl	NOUN
ejpam-6375	112	17	◦	◦	NOUN
ejpam-6375	112	18	(	(	PUNCT
ejpam-6375	112	19	f	f	X
ejpam-6375	112	20	)	)	PUNCT
ejpam-6375	112	21	or	or	CCONJ
ejpam-6375	112	22	equivalently	equivalently	ADV
ejpam-6375	112	23	if	if	SCONJ
ejpam-6375	112	24	e	e	PROPN
ejpam-6375	112	25	⊆	⊆	NUM
ejpam-6375	112	26	cl	cl	NOUN
ejpam-6375	112	27	◦	◦	NOUN
ejpam-6375	112	28	(int(e	(int(e	PROPN
ejpam-6375	112	29	)	)	PUNCT
ejpam-6375	112	30	)	)	PUNCT
ejpam-6375	112	31	,	,	PUNCT
ejpam-6375	112	32	then	then	ADV
ejpam-6375	112	33	the	the	DET
ejpam-6375	112	34	subset	subset	ADJ
ejpam-6375	112	35	e	e	PROPN
ejpam-6375	112	36	of	of	ADP
ejpam-6375	112	37	(	(	PUNCT
ejpam-6375	112	38	v	v	NOUN
ejpam-6375	112	39	,	,	PUNCT
ejpam-6375	112	40	gτ	gτ	INTJ
ejpam-6375	112	41	,	,	PUNCT
ejpam-6375	112	42	p	p	X
ejpam-6375	112	43	)	)	PUNCT
ejpam-6375	112	44	is	be	AUX
ejpam-6375	112	45	known	know	VERB
ejpam-6375	112	46	as	as	ADP
ejpam-6375	112	47	generalized	generalize	VERB
ejpam-6375	112	48	primal	primal	ADJ
ejpam-6375	112	49	semi	semi	ADJ
ejpam-6375	112	50	-	-	ADJ
ejpam-6375	112	51	open	open	ADJ
ejpam-6375	112	52	[	[	X
ejpam-6375	112	53	10	10	NUM
ejpam-6375	112	54	]	]	PUNCT
ejpam-6375	112	55	.	.	PUNCT
ejpam-6375	113	1	(	(	PUNCT
ejpam-6375	113	2	ii	ii	X
ejpam-6375	113	3	)	)	PUNCT
ejpam-6375	113	4	the	the	DET
ejpam-6375	113	5	generalized	generalize	VERB
ejpam-6375	113	6	primal	primal	ADJ
ejpam-6375	113	7	semi	semi	ADJ
ejpam-6375	113	8	-	-	ADJ
ejpam-6375	113	9	closed	closed	ADJ
ejpam-6375	113	10	set	set	NOUN
ejpam-6375	113	11	exists	exist	VERB
ejpam-6375	113	12	as	as	ADP
ejpam-6375	113	13	the	the	DET
ejpam-6375	113	14	complement	complement	NOUN
ejpam-6375	113	15	of	of	ADP
ejpam-6375	113	16	generalized	generalized	ADJ
ejpam-6375	113	17	primal	primal	ADJ
ejpam-6375	113	18	semi	semi	ADJ
ejpam-6375	113	19	-	-	ADJ
ejpam-6375	113	20	open	open	ADJ
ejpam-6375	113	21	sets	set	NOUN
ejpam-6375	113	22	.	.	PUNCT
ejpam-6375	114	1	all	all	DET
ejpam-6375	114	2	generalized	generalize	VERB
ejpam-6375	114	3	primal	primal	ADJ
ejpam-6375	114	4	semi	semi	ADJ
ejpam-6375	114	5	-	-	ADJ
ejpam-6375	114	6	open	open	ADJ
ejpam-6375	114	7	sets	set	NOUN
ejpam-6375	114	8	make	make	VERB
ejpam-6375	114	9	up	up	ADP
ejpam-6375	114	10	the	the	DET
ejpam-6375	114	11	collection	collection	NOUN
ejpam-6375	114	12	known	know	VERB
ejpam-6375	114	13	as	as	ADP
ejpam-6375	114	14	gpt	gpt	NOUN
ejpam-6375	114	15	-	-	PUNCT
ejpam-6375	114	16	so	so	ADV
ejpam-6375	114	17	in	in	ADP
ejpam-6375	114	18	(	(	PUNCT
ejpam-6375	114	19	v	v	NOUN
ejpam-6375	114	20	,	,	PUNCT
ejpam-6375	114	21	gτ	gτ	INTJ
ejpam-6375	114	22	,	,	PUNCT
ejpam-6375	114	23	p	p	NOUN
ejpam-6375	114	24	)	)	PUNCT
ejpam-6375	114	25	[	[	X
ejpam-6375	114	26	10	10	NUM
ejpam-6375	114	27	]	]	PUNCT
ejpam-6375	114	28	.	.	PUNCT
ejpam-6375	115	1	m.	m.	PROPN
ejpam-6375	115	2	shahbaz	shahbaz	PROPN
ejpam-6375	115	3	et	et	PROPN
ejpam-6375	115	4	al	al	PROPN
ejpam-6375	115	5	.	.	PUNCT
ejpam-6375	115	6	/	/	SYM
ejpam-6375	115	7	eur	eur	PROPN
ejpam-6375	115	8	.	.	PUNCT
ejpam-6375	116	1	j.	j.	PROPN
ejpam-6375	116	2	pure	pure	PROPN
ejpam-6375	116	3	appl	appl	PROPN
ejpam-6375	116	4	.	.	PROPN
ejpam-6375	116	5	math	math	PROPN
ejpam-6375	116	6	,	,	PUNCT
ejpam-6375	116	7	18	18	NUM
ejpam-6375	116	8	(	(	PUNCT
ejpam-6375	116	9	4	4	NUM
ejpam-6375	116	10	)	)	PUNCT
ejpam-6375	116	11	(	(	PUNCT
ejpam-6375	116	12	2025	2025	NUM
ejpam-6375	116	13	)	)	PUNCT
ejpam-6375	116	14	,	,	PUNCT
ejpam-6375	116	15	6375	6375	NUM
ejpam-6375	116	16	6	6	NUM
ejpam-6375	116	17	of	of	ADP
ejpam-6375	116	18	22	22	NUM
ejpam-6375	116	19	(	(	PUNCT
ejpam-6375	116	20	iii	iii	NOUN
ejpam-6375	116	21	)	)	PUNCT
ejpam-6375	116	22	the	the	DET
ejpam-6375	116	23	union	union	NOUN
ejpam-6375	116	24	of	of	ADP
ejpam-6375	116	25	all	all	DET
ejpam-6375	116	26	gpt	gpt	NOUN
ejpam-6375	116	27	-	-	PUNCT
ejpam-6375	116	28	so	so	NOUN
ejpam-6375	116	29	sets	set	NOUN
ejpam-6375	116	30	of	of	ADP
ejpam-6375	116	31	v	v	NOUN
ejpam-6375	116	32	contained	contain	VERB
ejpam-6375	116	33	in	in	ADP
ejpam-6375	116	34	e	e	PROPN
ejpam-6375	116	35	is	be	AUX
ejpam-6375	116	36	called	call	VERB
ejpam-6375	116	37	gpt	gpt	NOUN
ejpam-6375	116	38	-	-	PUNCT
ejpam-6375	116	39	semi−interior	semi−interior	NOUN
ejpam-6375	116	40	of	of	ADP
ejpam-6375	116	41	e	e	NOUN
ejpam-6375	116	42	(	(	PUNCT
ejpam-6375	116	43	briefly	briefly	NOUN
ejpam-6375	116	44	gpt	gpt	NOUN
ejpam-6375	116	45	-	-	PUNCT
ejpam-6375	116	46	sint(e	sint(e	NOUN
ejpam-6375	116	47	)	)	PUNCT
ejpam-6375	116	48	.	.	PUNCT
ejpam-6375	117	1	(	(	PUNCT
ejpam-6375	117	2	iv	iv	X
ejpam-6375	117	3	)	)	PUNCT
ejpam-6375	117	4	the	the	DET
ejpam-6375	117	5	intersection	intersection	NOUN
ejpam-6375	117	6	of	of	ADP
ejpam-6375	117	7	all	all	DET
ejpam-6375	117	8	gpt	gpt	NOUN
ejpam-6375	117	9	-	-	PUNCT
ejpam-6375	117	10	sc	sc	NOUN
ejpam-6375	117	11	sets	set	NOUN
ejpam-6375	117	12	of	of	ADP
ejpam-6375	117	13	v	v	NOUN
ejpam-6375	117	14	containing	contain	VERB
ejpam-6375	117	15	e	e	NOUN
ejpam-6375	117	16	is	be	AUX
ejpam-6375	117	17	gpt	gpt	NOUN
ejpam-6375	117	18	-	-	PUNCT
ejpam-6375	117	19	semi	semi	NOUN
ejpam-6375	117	20	-	-	NOUN
ejpam-6375	117	21	closure	closure	NOUN
ejpam-6375	117	22	of	of	ADP
ejpam-6375	117	23	e	e	NOUN
ejpam-6375	117	24	(	(	PUNCT
ejpam-6375	117	25	briefly	briefly	NOUN
ejpam-6375	117	26	gpt	gpt	NOUN
ejpam-6375	117	27	-	-	PUNCT
ejpam-6375	117	28	scl(e	scl(e	PROPN
ejpam-6375	117	29	)	)	PUNCT
ejpam-6375	117	30	)	)	PUNCT
ejpam-6375	117	31	.	.	PUNCT
ejpam-6375	118	1	definition	definition	NOUN
ejpam-6375	118	2	3.6	3.6	NUM
ejpam-6375	118	3	.	.	PUNCT
ejpam-6375	119	1	[	[	X
ejpam-6375	119	2	1	1	NUM
ejpam-6375	119	3	]	]	PUNCT
ejpam-6375	119	4	(	(	PUNCT
ejpam-6375	119	5	i	i	NOUN
ejpam-6375	119	6	)	)	PUNCT
ejpam-6375	119	7	consider	consider	VERB
ejpam-6375	119	8	(	(	PUNCT
ejpam-6375	119	9	v	v	NOUN
ejpam-6375	119	10	,	,	PUNCT
ejpam-6375	119	11	gτ	gτ	INTJ
ejpam-6375	119	12	,	,	PUNCT
ejpam-6375	119	13	p	p	X
ejpam-6375	119	14	)	)	PUNCT
ejpam-6375	119	15	be	be	AUX
ejpam-6375	119	16	a	a	DET
ejpam-6375	119	17	gpts	gpt	NOUN
ejpam-6375	119	18	.	.	PUNCT
ejpam-6375	120	1	if	if	SCONJ
ejpam-6375	120	2	gpt	gpt	NOUN
ejpam-6375	120	3	-	-	PUNCT
ejpam-6375	120	4	scl(e	scl(e	PROPN
ejpam-6375	120	5	)	)	PUNCT
ejpam-6375	120	6	⊆	⊆	NUM
ejpam-6375	120	7	f	f	NOUN
ejpam-6375	120	8	whenever	whenever	SCONJ
ejpam-6375	120	9	e	e	PROPN
ejpam-6375	120	10	⊆	⊆	NUM
ejpam-6375	120	11	f	f	PROPN
ejpam-6375	120	12	and	and	CCONJ
ejpam-6375	120	13	f	f	PROPN
ejpam-6375	120	14	is	be	AUX
ejpam-6375	120	15	gptsemi−open	gptsemi−open	VERB
ejpam-6375	120	16	in	in	ADP
ejpam-6375	120	17	v	v	NOUN
ejpam-6375	120	18	,	,	PUNCT
ejpam-6375	120	19	then	then	ADV
ejpam-6375	120	20	the	the	DET
ejpam-6375	120	21	subset	subset	ADJ
ejpam-6375	120	22	e	e	PROPN
ejpam-6375	120	23	of	of	ADP
ejpam-6375	120	24	v	v	NUM
ejpam-6375	120	25	is	be	AUX
ejpam-6375	120	26	known	know	VERB
ejpam-6375	120	27	as	as	ADP
ejpam-6375	120	28	generalized	generalized	ADJ
ejpam-6375	120	29	primal	primal	ADJ
ejpam-6375	120	30	semigeneralized	semigeneralize	VERB
ejpam-6375	120	31	closed	close	VERB
ejpam-6375	120	32	(	(	PUNCT
ejpam-6375	120	33	briefly	briefly	ADV
ejpam-6375	120	34	gptsg	gptsg	NOUN
ejpam-6375	120	35	-	-	PUNCT
ejpam-6375	120	36	closed	close	VERB
ejpam-6375	120	37	)	)	PUNCT
ejpam-6375	120	38	.	.	PUNCT
ejpam-6375	121	1	(	(	PUNCT
ejpam-6375	121	2	ii	ii	NOUN
ejpam-6375	121	3	)	)	PUNCT
ejpam-6375	121	4	if	if	SCONJ
ejpam-6375	121	5	cl(e	cl(e	NUM
ejpam-6375	121	6	)	)	PUNCT
ejpam-6375	121	7	⊆	⊆	NUM
ejpam-6375	121	8	f	f	NOUN
ejpam-6375	121	9	whenever	whenever	SCONJ
ejpam-6375	121	10	e	e	PROPN
ejpam-6375	121	11	⊆	⊆	NUM
ejpam-6375	121	12	f	f	PROPN
ejpam-6375	121	13	and	and	CCONJ
ejpam-6375	121	14	f	f	PROPN
ejpam-6375	121	15	is	be	AUX
ejpam-6375	121	16	gptsemi−open	gptsemi−open	VERB
ejpam-6375	121	17	in	in	ADP
ejpam-6375	121	18	v	v	NOUN
ejpam-6375	121	19	,	,	PUNCT
ejpam-6375	121	20	then	then	ADV
ejpam-6375	121	21	e	e	X
ejpam-6375	121	22	as	as	ADP
ejpam-6375	121	23	a	a	DET
ejpam-6375	121	24	subset	subset	NOUN
ejpam-6375	121	25	of	of	ADP
ejpam-6375	121	26	a	a	DET
ejpam-6375	121	27	space	space	NOUN
ejpam-6375	121	28	v	v	NOUN
ejpam-6375	121	29	is	be	AUX
ejpam-6375	121	30	generalized	generalize	VERB
ejpam-6375	121	31	primal	primal	ADJ
ejpam-6375	121	32	semi−star	semi−star	ADV
ejpam-6375	121	33	generalized	generalize	VERB
ejpam-6375	121	34	closed	close	VERB
ejpam-6375	121	35	(	(	PUNCT
ejpam-6375	121	36	briefly	briefly	NOUN
ejpam-6375	121	37	gpt	gpt	NOUN
ejpam-6375	121	38	-	-	PUNCT
ejpam-6375	121	39	s∗-closed	s∗-close	VERB
ejpam-6375	121	40	)	)	PUNCT
ejpam-6375	121	41	.	.	PUNCT
ejpam-6375	122	1	(	(	PUNCT
ejpam-6375	122	2	iii	iii	X
ejpam-6375	122	3	)	)	PUNCT
ejpam-6375	122	4	the	the	DET
ejpam-6375	122	5	complement	complement	NOUN
ejpam-6375	122	6	of	of	ADP
ejpam-6375	122	7	gptsg	gptsg	NOUN
ejpam-6375	122	8	-	-	PUNCT
ejpam-6375	122	9	closed	close	VERB
ejpam-6375	122	10	set	set	NOUN
ejpam-6375	122	11	(	(	PUNCT
ejpam-6375	122	12	gpt	gpt	NOUN
ejpam-6375	122	13	-	-	PUNCT
ejpam-6375	122	14	s∗g	s∗g	NOUN
ejpam-6375	122	15	-	-	PUNCT
ejpam-6375	122	16	closed	close	VERB
ejpam-6375	122	17	set	set	NOUN
ejpam-6375	122	18	)	)	PUNCT
ejpam-6375	122	19	is	be	AUX
ejpam-6375	122	20	generalized	generalize	VERB
ejpam-6375	122	21	primal	primal	ADJ
ejpam-6375	122	22	semigeneralized	semigeneralize	VERB
ejpam-6375	122	23	open	open	ADJ
ejpam-6375	122	24	(	(	PUNCT
ejpam-6375	122	25	generalized	generalize	VERB
ejpam-6375	122	26	primal	primal	ADJ
ejpam-6375	122	27	semi	semi	ADJ
ejpam-6375	122	28	-	-	ADJ
ejpam-6375	122	29	star	star	ADJ
ejpam-6375	122	30	generalized	generalize	VERB
ejpam-6375	122	31	open	open	NOUN
ejpam-6375	122	32	)	)	PUNCT
ejpam-6375	122	33	.	.	PUNCT
ejpam-6375	123	1	it	it	PRON
ejpam-6375	123	2	is	be	AUX
ejpam-6375	123	3	represented	represent	VERB
ejpam-6375	123	4	by	by	ADP
ejpam-6375	123	5	gpt	gpt	NOUN
ejpam-6375	123	6	-	-	PUNCT
ejpam-6375	123	7	sg	sg	NOUN
ejpam-6375	123	8	-	-	PUNCT
ejpam-6375	123	9	open	open	ADJ
ejpam-6375	123	10	(	(	PUNCT
ejpam-6375	123	11	gpt	gpt	NOUN
ejpam-6375	123	12	-	-	PUNCT
ejpam-6375	123	13	s∗g	s∗g	NOUN
ejpam-6375	123	14	-	-	NOUN
ejpam-6375	123	15	open	open	ADJ
ejpam-6375	123	16	)	)	PUNCT
ejpam-6375	123	17	appropriately	appropriately	ADV
ejpam-6375	123	18	.	.	PUNCT
ejpam-6375	124	1	(	(	PUNCT
ejpam-6375	124	2	iv	iv	X
ejpam-6375	124	3	)	)	PUNCT
ejpam-6375	124	4	the	the	DET
ejpam-6375	124	5	generalized	generalize	VERB
ejpam-6375	124	6	primal	primal	ADJ
ejpam-6375	124	7	semi	semi	ADJ
ejpam-6375	124	8	-	-	ADJ
ejpam-6375	124	9	generalized	generalized	ADJ
ejpam-6375	124	10	interior	interior	NOUN
ejpam-6375	124	11	(	(	PUNCT
ejpam-6375	124	12	briefly	briefly	NOUN
ejpam-6375	124	13	gpt	gpt	NOUN
ejpam-6375	124	14	-	-	PUNCT
ejpam-6375	124	15	sint∗(e	sint∗(e	NOUN
ejpam-6375	124	16	)	)	PUNCT
ejpam-6375	124	17	)	)	PUNCT
ejpam-6375	124	18	of	of	ADP
ejpam-6375	124	19	e	e	PROPN
ejpam-6375	124	20	is	be	AUX
ejpam-6375	124	21	indicated	indicate	VERB
ejpam-6375	124	22	as	as	ADP
ejpam-6375	124	23	the	the	DET
ejpam-6375	124	24	union	union	NOUN
ejpam-6375	124	25	of	of	ADP
ejpam-6375	124	26	all	all	DET
ejpam-6375	124	27	gpt	gpt	NOUN
ejpam-6375	124	28	-	-	PUNCT
ejpam-6375	124	29	sg	sg	ADP
ejpam-6375	124	30	-	-	PUNCT
ejpam-6375	124	31	open	open	ADJ
ejpam-6375	124	32	sets	set	NOUN
ejpam-6375	124	33	of	of	ADP
ejpam-6375	124	34	v	v	NOUN
ejpam-6375	124	35	contained	contain	VERB
ejpam-6375	124	36	in	in	ADP
ejpam-6375	124	37	e.	e.	PROPN
ejpam-6375	124	38	(	(	PUNCT
ejpam-6375	124	39	v	v	NOUN
ejpam-6375	124	40	)	)	PUNCT
ejpam-6375	124	41	the	the	DET
ejpam-6375	124	42	intersection	intersection	NOUN
ejpam-6375	124	43	of	of	ADP
ejpam-6375	124	44	all	all	DET
ejpam-6375	124	45	gptsg−closed	gptsg−close	VERB
ejpam-6375	124	46	sets	set	NOUN
ejpam-6375	124	47	in	in	ADP
ejpam-6375	124	48	v	v	NOUN
ejpam-6375	124	49	containing	contain	VERB
ejpam-6375	124	50	e	e	NOUN
ejpam-6375	124	51	when	when	SCONJ
ejpam-6375	124	52	e	e	PROPN
ejpam-6375	124	53	is	be	AUX
ejpam-6375	124	54	a	a	DET
ejpam-6375	124	55	subset	subset	NOUN
ejpam-6375	124	56	of	of	ADP
ejpam-6375	124	57	v	v	NOUN
ejpam-6375	124	58	is	be	AUX
ejpam-6375	124	59	called	call	VERB
ejpam-6375	124	60	generalized	generalized	ADJ
ejpam-6375	124	61	primal	primal	ADJ
ejpam-6375	124	62	semi−generalized	semi−generalize	VERB
ejpam-6375	124	63	closure	closure	NOUN
ejpam-6375	124	64	(	(	PUNCT
ejpam-6375	124	65	briefly	briefly	NOUN
ejpam-6375	124	66	gpt	gpt	NOUN
ejpam-6375	124	67	-	-	PUNCT
ejpam-6375	124	68	scl∗(e	scl∗(e	NOUN
ejpam-6375	124	69	)	)	PUNCT
ejpam-6375	124	70	)	)	PUNCT
ejpam-6375	124	71	of	of	ADP
ejpam-6375	124	72	e.	e.	PROPN
ejpam-6375	124	73	definition	definition	PROPN
ejpam-6375	124	74	3.7	3.7	NUM
ejpam-6375	124	75	.	.	PUNCT
ejpam-6375	125	1	assume	assume	VERB
ejpam-6375	125	2	a	a	DET
ejpam-6375	125	3	gpts	gpt	NOUN
ejpam-6375	125	4	(	(	PUNCT
ejpam-6375	125	5	v	v	NOUN
ejpam-6375	125	6	,	,	PUNCT
ejpam-6375	125	7	gτ	gτ	INTJ
ejpam-6375	125	8	,	,	PUNCT
ejpam-6375	125	9	p	p	X
ejpam-6375	125	10	)	)	PUNCT
ejpam-6375	125	11	and	and	CCONJ
ejpam-6375	125	12	a	a	DET
ejpam-6375	125	13	subset	subset	NOUN
ejpam-6375	125	14	e	e	X
ejpam-6375	125	15	of	of	ADP
ejpam-6375	125	16	(	(	PUNCT
ejpam-6375	125	17	v	v	NOUN
ejpam-6375	125	18	,	,	PUNCT
ejpam-6375	125	19	gτ	gτ	INTJ
ejpam-6375	125	20	,	,	PUNCT
ejpam-6375	125	21	p	p	NOUN
ejpam-6375	125	22	)	)	PUNCT
ejpam-6375	125	23	,	,	PUNCT
ejpam-6375	125	24	a	a	DET
ejpam-6375	125	25	collection	collection	NOUN
ejpam-6375	125	26	{	{	PUNCT
ejpam-6375	125	27	eαi	eαi	NOUN
ejpam-6375	125	28	:	:	PUNCT
ejpam-6375	125	29	i	i	PROPN
ejpam-6375	125	30	∈	∈	PROPN
ejpam-6375	125	31	⋏	⋏	PROPN
ejpam-6375	125	32	}	}	PUNCT
ejpam-6375	125	33	of	of	ADP
ejpam-6375	125	34	gpt	gpt	NOUN
ejpam-6375	125	35	-	-	PUNCT
ejpam-6375	125	36	s∗	s∗	PROPN
ejpam-6375	125	37	g	g	PROPN
ejpam-6375	125	38	-open	-open	NOUN
ejpam-6375	125	39	set	set	VERB
ejpam-6375	125	40	in	in	ADP
ejpam-6375	125	41	gpt	gpt	NOUN
ejpam-6375	125	42	is	be	AUX
ejpam-6375	125	43	referred	refer	VERB
ejpam-6375	125	44	to	to	ADP
ejpam-6375	125	45	as	as	ADP
ejpam-6375	125	46	gpt	gpt	NOUN
ejpam-6375	125	47	-	-	PUNCT
ejpam-6375	125	48	s∗	s∗	PROPN
ejpam-6375	125	49	g	g	PROPN
ejpam-6375	125	50	-open	-open	ADJ
ejpam-6375	125	51	cover	cover	NOUN
ejpam-6375	125	52	of	of	ADP
ejpam-6375	125	53	e	e	NOUN
ejpam-6375	125	54	if	if	SCONJ
ejpam-6375	125	55	e	e	PROPN
ejpam-6375	125	56	⊂	⊂	PROPN
ejpam-6375	125	57	∪	∪	PROPN
ejpam-6375	125	58	i	i	PRON
ejpam-6375	125	59	∈	∈	PROPN
ejpam-6375	125	60	⋏	⋏	PROPN
ejpam-6375	125	61	eαi	eαi	NOUN
ejpam-6375	125	62	.	.	PUNCT
ejpam-6375	126	1	definition	definition	NOUN
ejpam-6375	126	2	3.8	3.8	NUM
ejpam-6375	126	3	.	.	PUNCT
ejpam-6375	127	1	if	if	SCONJ
ejpam-6375	127	2	every	every	DET
ejpam-6375	127	3	gpt	gpt	NOUN
ejpam-6375	127	4	-	-	PUNCT
ejpam-6375	127	5	s∗	s∗	PROPN
ejpam-6375	127	6	g	g	PROPN
ejpam-6375	127	7	-open	-open	ADJ
ejpam-6375	127	8	cover	cover	NOUN
ejpam-6375	127	9	of	of	ADP
ejpam-6375	127	10	(	(	PUNCT
ejpam-6375	127	11	v	v	NOUN
ejpam-6375	127	12	,	,	PUNCT
ejpam-6375	127	13	gτ	gτ	INTJ
ejpam-6375	127	14	,	,	PUNCT
ejpam-6375	127	15	p	p	X
ejpam-6375	127	16	)	)	PUNCT
ejpam-6375	127	17	has	have	VERB
ejpam-6375	127	18	a	a	DET
ejpam-6375	127	19	finite	finite	ADJ
ejpam-6375	127	20	subcover	subcover	NOUN
ejpam-6375	127	21	,	,	PUNCT
ejpam-6375	127	22	then	then	ADV
ejpam-6375	127	23	the	the	DET
ejpam-6375	127	24	gpts	gpt	NOUN
ejpam-6375	127	25	is	be	AUX
ejpam-6375	127	26	called	call	VERB
ejpam-6375	127	27	gpt	gpt	NOUN
ejpam-6375	127	28	-	-	PUNCT
ejpam-6375	127	29	s∗	s∗	PROPN
ejpam-6375	127	30	g	g	PROPN
ejpam-6375	127	31	-compact	-compact	PROPN
ejpam-6375	127	32	.	.	PUNCT
ejpam-6375	128	1	definition	definition	NOUN
ejpam-6375	128	2	3.9	3.9	NUM
ejpam-6375	128	3	.	.	PUNCT
ejpam-6375	129	1	the	the	DET
ejpam-6375	129	2	subset	subset	NOUN
ejpam-6375	129	3	e	e	PROPN
ejpam-6375	129	4	of	of	ADP
ejpam-6375	129	5	gpts	gpt	NOUN
ejpam-6375	129	6	is	be	AUX
ejpam-6375	129	7	named	name	VERB
ejpam-6375	129	8	as	as	ADP
ejpam-6375	129	9	gpt	gpt	NOUN
ejpam-6375	129	10	-	-	PUNCT
ejpam-6375	129	11	s∗	s∗	PROPN
ejpam-6375	129	12	g	g	PROPN
ejpam-6375	129	13	-compact	-compact	PROPN
ejpam-6375	129	14	relative	relative	ADJ
ejpam-6375	129	15	of	of	ADP
ejpam-6375	129	16	v	v	NOUN
ejpam-6375	129	17	if	if	SCONJ
ejpam-6375	129	18	there	there	PRON
ejpam-6375	129	19	exists	exist	VERB
ejpam-6375	129	20	⋏o	⋏o	PROPN
ejpam-6375	129	21	of	of	ADP
ejpam-6375	129	22	⋏	⋏	PROPN
ejpam-6375	129	23	as	as	ADP
ejpam-6375	129	24	a	a	DET
ejpam-6375	129	25	finite	finite	NOUN
ejpam-6375	129	26	subset	subset	NOUN
ejpam-6375	129	27	which	which	PRON
ejpam-6375	129	28	satisfies	satisfy	VERB
ejpam-6375	129	29	e	e	PROPN
ejpam-6375	129	30	⊂	⊂	PROPN
ejpam-6375	129	31	∪	∪	X
ejpam-6375	129	32	{	{	PUNCT
ejpam-6375	129	33	zi	zi	NOUN
ejpam-6375	129	34	:	:	PUNCT
ejpam-6375	129	35	i	i	PRON
ejpam-6375	129	36	∈	∈	VERB
ejpam-6375	129	37	⋏o	⋏o	PROPN
ejpam-6375	129	38	}	}	PUNCT
ejpam-6375	129	39	for	for	ADP
ejpam-6375	129	40	each	each	DET
ejpam-6375	129	41	{	{	PUNCT
ejpam-6375	129	42	zi	zi	NOUN
ejpam-6375	129	43	:	:	PUNCT
ejpam-6375	129	44	i	i	PRON
ejpam-6375	129	45	∈	∈	PROPN
ejpam-6375	129	46	⋏	⋏	PROPN
ejpam-6375	129	47	}	}	PUNCT
ejpam-6375	129	48	consisting	consist	VERB
ejpam-6375	129	49	of	of	ADP
ejpam-6375	129	50	gpt	gpt	NOUN
ejpam-6375	129	51	-	-	PUNCT
ejpam-6375	129	52	s∗	s∗	PROPN
ejpam-6375	129	53	g	g	PROPN
ejpam-6375	129	54	-open	-open	NOUN
ejpam-6375	129	55	subset	subset	NOUN
ejpam-6375	129	56	of	of	ADP
ejpam-6375	129	57	v	v	NOUN
ejpam-6375	129	58	such	such	ADJ
ejpam-6375	129	59	that	that	SCONJ
ejpam-6375	129	60	e	e	PROPN
ejpam-6375	129	61	⊂	⊂	PROPN
ejpam-6375	129	62	∪	∪	X
ejpam-6375	129	63	{	{	PUNCT
ejpam-6375	129	64	zi	zi	NOUN
ejpam-6375	129	65	:	:	PUNCT
ejpam-6375	129	66	i	i	PRON
ejpam-6375	129	67	∈	∈	PROPN
ejpam-6375	129	68	⋏	⋏	PROPN
ejpam-6375	129	69	}	}	PUNCT
ejpam-6375	129	70	.	.	PUNCT
ejpam-6375	130	1	definition	definition	NOUN
ejpam-6375	130	2	3.10	3.10	NUM
ejpam-6375	130	3	.	.	PUNCT
ejpam-6375	131	1	assume	assume	VERB
ejpam-6375	131	2	e	e	NOUN
ejpam-6375	131	3	as	as	ADP
ejpam-6375	131	4	a	a	DET
ejpam-6375	131	5	subset	subset	NOUN
ejpam-6375	131	6	of	of	ADP
ejpam-6375	131	7	gpts	gpt	NOUN
ejpam-6375	131	8	.	.	PUNCT
ejpam-6375	132	1	a	a	DET
ejpam-6375	132	2	subset	subset	NOUN
ejpam-6375	132	3	e	e	NOUN
ejpam-6375	132	4	of	of	ADP
ejpam-6375	132	5	v	v	NOUN
ejpam-6375	132	6	is	be	AUX
ejpam-6375	132	7	named	name	VERB
ejpam-6375	132	8	gpt	gpt	NOUN
ejpam-6375	132	9	-	-	PUNCT
ejpam-6375	132	10	s∗	s∗	PROPN
ejpam-6375	132	11	g	g	PROPN
ejpam-6375	132	12	compact	compact	ADJ
ejpam-6375	132	13	when	when	SCONJ
ejpam-6375	132	14	it	it	PRON
ejpam-6375	132	15	maintains	maintain	VERB
ejpam-6375	132	16	this	this	DET
ejpam-6375	132	17	property	property	NOUN
ejpam-6375	132	18	as	as	ADP
ejpam-6375	132	19	a	a	DET
ejpam-6375	132	20	subspace	subspace	NOUN
ejpam-6375	132	21	of	of	ADP
ejpam-6375	132	22	v.	v.	CCONJ
ejpam-6375	132	23	theorem	theorem	ADJ
ejpam-6375	132	24	3.1	3.1	NUM
ejpam-6375	132	25	.	.	PUNCT
ejpam-6375	133	1	(	(	PUNCT
ejpam-6375	133	2	i	i	NOUN
ejpam-6375	133	3	)	)	PUNCT
ejpam-6375	133	4	every	every	DET
ejpam-6375	133	5	gpt	gpt	NOUN
ejpam-6375	133	6	-	-	PUNCT
ejpam-6375	133	7	s∗	s∗	PROPN
ejpam-6375	133	8	g	g	PROPN
ejpam-6375	133	9	-compact	-compact	PROPN
ejpam-6375	133	10	space	space	NOUN
ejpam-6375	133	11	is	be	AUX
ejpam-6375	133	12	gpt	gpt	NOUN
ejpam-6375	133	13	-	-	PUNCT
ejpam-6375	133	14	compact	compact	ADJ
ejpam-6375	133	15	.	.	PUNCT
ejpam-6375	134	1	(	(	PUNCT
ejpam-6375	134	2	ii	ii	NOUN
ejpam-6375	134	3	)	)	PUNCT
ejpam-6375	134	4	the	the	DET
ejpam-6375	134	5	property	property	NOUN
ejpam-6375	134	6	of	of	ADP
ejpam-6375	134	7	being	be	AUX
ejpam-6375	134	8	gpt	gpt	NOUN
ejpam-6375	134	9	-	-	PUNCT
ejpam-6375	134	10	semi	semi	ADJ
ejpam-6375	134	11	-	-	ADJ
ejpam-6375	134	12	compact	compact	ADJ
ejpam-6375	134	13	implies	imply	VERB
ejpam-6375	134	14	gpt	gpt	NOUN
ejpam-6375	134	15	-	-	PUNCT
ejpam-6375	134	16	s∗	s∗	PROPN
ejpam-6375	134	17	g	g	PROPN
ejpam-6375	134	18	-compactness	-compactness	NOUN
ejpam-6375	134	19	.	.	PUNCT
ejpam-6375	135	1	proof	proof	NOUN
ejpam-6375	135	2	.	.	PUNCT
ejpam-6375	136	1	(	(	PUNCT
ejpam-6375	136	2	i	i	NOUN
ejpam-6375	136	3	)	)	PUNCT
ejpam-6375	136	4	let	let	VERB
ejpam-6375	136	5	u	u	PRON
ejpam-6375	136	6	=	=	PUNCT
ejpam-6375	136	7	{	{	PUNCT
ejpam-6375	136	8	ui	ui	NOUN
ejpam-6375	136	9	:	:	PUNCT
ejpam-6375	136	10	i	i	PROPN
ejpam-6375	136	11	∈	∈	PROPN
ejpam-6375	136	12	λ	λ	PROPN
ejpam-6375	136	13	,	,	PUNCT
ejpam-6375	136	14	ui	ui	PROPN
ejpam-6375	136	15	∈	∈	PROPN
ejpam-6375	136	16	gpt	gpt	NOUN
ejpam-6375	136	17	}	}	PUNCT
ejpam-6375	136	18	be	be	AUX
ejpam-6375	136	19	an	an	DET
ejpam-6375	136	20	gpt	gpt	NOUN
ejpam-6375	136	21	-	-	PUNCT
ejpam-6375	136	22	open	open	ADJ
ejpam-6375	136	23	cover	cover	NOUN
ejpam-6375	136	24	of	of	ADP
ejpam-6375	136	25	x	x	PRON
ejpam-6375	136	26	,	,	PUNCT
ejpam-6375	136	27	so	so	ADV
ejpam-6375	136	28	x	x	NOUN
ejpam-6375	136	29	=	=	PUNCT
ejpam-6375	136	30	⋃	⋃	NOUN
ejpam-6375	136	31	i∈λ	i∈λ	NOUN
ejpam-6375	136	32	ui	ui	NOUN
ejpam-6375	136	33	.	.	PUNCT
ejpam-6375	137	1	because	because	SCONJ
ejpam-6375	137	2	every	every	DET
ejpam-6375	137	3	gpt	gpt	NOUN
ejpam-6375	137	4	-	-	PUNCT
ejpam-6375	137	5	open	open	ADJ
ejpam-6375	137	6	set	set	NOUN
ejpam-6375	137	7	is	be	AUX
ejpam-6375	137	8	gpt	gpt	NOUN
ejpam-6375	137	9	-	-	PUNCT
ejpam-6375	137	10	s∗	s∗	PROPN
ejpam-6375	137	11	g	g	PROPN
ejpam-6375	137	12	,	,	PUNCT
ejpam-6375	137	13	u	u	PROPN
ejpam-6375	137	14	is	be	AUX
ejpam-6375	137	15	also	also	ADV
ejpam-6375	137	16	a	a	DET
ejpam-6375	137	17	gpt	gpt	NOUN
ejpam-6375	137	18	-	-	PUNCT
ejpam-6375	137	19	s∗	s∗	NOUN
ejpam-6375	137	20	g	g	PROPN
ejpam-6375	137	21	-open	-open	ADJ
ejpam-6375	137	22	cover	cover	NOUN
ejpam-6375	137	23	of	of	ADP
ejpam-6375	137	24	x.	x.	NOUN
ejpam-6375	137	25	if	if	SCONJ
ejpam-6375	137	26	x	x	PRON
ejpam-6375	137	27	is	be	AUX
ejpam-6375	137	28	gpts∗	gpts∗	PROPN
ejpam-6375	137	29	g	g	PROPN
ejpam-6375	137	30	-compact	-compact	NOUN
ejpam-6375	137	31	,	,	PUNCT
ejpam-6375	137	32	then	then	ADV
ejpam-6375	137	33	by	by	ADP
ejpam-6375	137	34	definition	definition	NOUN
ejpam-6375	137	35	every	every	DET
ejpam-6375	137	36	gpt	gpt	NOUN
ejpam-6375	137	37	-	-	PUNCT
ejpam-6375	137	38	s∗	s∗	PROPN
ejpam-6375	137	39	g	g	PROPN
ejpam-6375	137	40	-open	-open	ADJ
ejpam-6375	137	41	cover	cover	NOUN
ejpam-6375	137	42	of	of	ADP
ejpam-6375	137	43	x	x	PUNCT
ejpam-6375	137	44	has	have	VERB
ejpam-6375	137	45	a	a	DET
ejpam-6375	137	46	finite	finite	ADJ
ejpam-6375	137	47	subcover	subcover	NOUN
ejpam-6375	137	48	,	,	PUNCT
ejpam-6375	137	49	hence	hence	ADV
ejpam-6375	137	50	there	there	PRON
ejpam-6375	137	51	exist	exist	VERB
ejpam-6375	137	52	a	a	DET
ejpam-6375	137	53	finite	finite	NOUN
ejpam-6375	137	54	subset	subset	NOUN
ejpam-6375	137	55	λ0	λ0	NOUN
ejpam-6375	137	56	=	=	SYM
ejpam-6375	137	57	{	{	PUNCT
ejpam-6375	137	58	i1	i1	NOUN
ejpam-6375	137	59	,	,	PUNCT
ejpam-6375	137	60	.	.	PUNCT
ejpam-6375	137	61	.	.	PUNCT
ejpam-6375	138	1	.	.	PUNCT
ejpam-6375	139	1	,	,	PUNCT
ejpam-6375	139	2	in	in	ADP
ejpam-6375	139	3	}	}	PUNCT
ejpam-6375	139	4	⊆	⊆	NUM
ejpam-6375	139	5	λ	λ	NOUN
ejpam-6375	139	6	such	such	ADJ
ejpam-6375	139	7	that	that	SCONJ
ejpam-6375	139	8	x	x	SYM
ejpam-6375	139	9	=	=	SYM
ejpam-6375	139	10	⋃n	⋃n	PROPN
ejpam-6375	139	11	k=1	k=1	PROPN
ejpam-6375	139	12	uik	uik	NOUN
ejpam-6375	139	13	.	.	PUNCT
ejpam-6375	140	1	therefore	therefore	ADV
ejpam-6375	140	2	x	x	PRON
ejpam-6375	140	3	admits	admit	VERB
ejpam-6375	140	4	a	a	DET
ejpam-6375	140	5	finite	finite	ADJ
ejpam-6375	140	6	subcover	subcover	NOUN
ejpam-6375	140	7	of	of	ADP
ejpam-6375	140	8	the	the	DET
ejpam-6375	140	9	gpt	gpt	NOUN
ejpam-6375	140	10	-	-	PUNCT
ejpam-6375	140	11	open	open	ADJ
ejpam-6375	140	12	cover	cover	NOUN
ejpam-6375	140	13	.	.	PUNCT
ejpam-6375	141	1	m.	m.	NOUN
ejpam-6375	141	2	shahbaz	shahbaz	PROPN
ejpam-6375	141	3	et	et	PROPN
ejpam-6375	141	4	al	al	PROPN
ejpam-6375	141	5	.	.	PUNCT
ejpam-6375	141	6	/	/	SYM
ejpam-6375	141	7	eur	eur	PROPN
ejpam-6375	141	8	.	.	PUNCT
ejpam-6375	142	1	j.	j.	PROPN
ejpam-6375	142	2	pure	pure	PROPN
ejpam-6375	142	3	appl	appl	PROPN
ejpam-6375	142	4	.	.	PROPN
ejpam-6375	142	5	math	math	PROPN
ejpam-6375	142	6	,	,	PUNCT
ejpam-6375	142	7	18	18	NUM
ejpam-6375	142	8	(	(	PUNCT
ejpam-6375	142	9	4	4	NUM
ejpam-6375	142	10	)	)	PUNCT
ejpam-6375	142	11	(	(	PUNCT
ejpam-6375	142	12	2025	2025	NUM
ejpam-6375	142	13	)	)	PUNCT
ejpam-6375	142	14	,	,	PUNCT
ejpam-6375	142	15	6375	6375	NUM
ejpam-6375	142	16	7	7	NUM
ejpam-6375	142	17	of	of	ADP
ejpam-6375	142	18	22	22	NUM
ejpam-6375	142	19	(	(	PUNCT
ejpam-6375	142	20	ii	ii	NOUN
ejpam-6375	142	21	)	)	PUNCT
ejpam-6375	142	22	since	since	SCONJ
ejpam-6375	142	23	every	every	DET
ejpam-6375	142	24	gpt	gpt	NOUN
ejpam-6375	142	25	-	-	PUNCT
ejpam-6375	142	26	s∗	s∗	PROPN
ejpam-6375	142	27	g	g	PROPN
ejpam-6375	142	28	-open	-open	ADJ
ejpam-6375	142	29	set	set	NOUN
ejpam-6375	142	30	is	be	AUX
ejpam-6375	142	31	a	a	DET
ejpam-6375	142	32	gpt	gpt	NOUN
ejpam-6375	142	33	-	-	PUNCT
ejpam-6375	142	34	semi	semi	ADJ
ejpam-6375	142	35	-	-	ADJ
ejpam-6375	142	36	open	open	ADJ
ejpam-6375	142	37	set	set	NOUN
ejpam-6375	142	38	,	,	PUNCT
ejpam-6375	142	39	the	the	DET
ejpam-6375	142	40	result	result	NOUN
ejpam-6375	142	41	follows	follow	VERB
ejpam-6375	142	42	similarly	similarly	ADV
ejpam-6375	142	43	to	to	ADP
ejpam-6375	142	44	part	part	NOUN
ejpam-6375	142	45	(	(	PUNCT
ejpam-6375	142	46	i	i	NOUN
ejpam-6375	142	47	)	)	PUNCT
ejpam-6375	142	48	.	.	PUNCT
ejpam-6375	143	1	example	example	NOUN
ejpam-6375	143	2	3.1	3.1	NUM
ejpam-6375	143	3	.	.	PUNCT
ejpam-6375	144	1	let	let	VERB
ejpam-6375	144	2	v	v	PART
ejpam-6375	144	3	be	be	AUX
ejpam-6375	144	4	the	the	DET
ejpam-6375	144	5	set	set	NOUN
ejpam-6375	144	6	of	of	ADP
ejpam-6375	144	7	all	all	DET
ejpam-6375	144	8	bounded	bounded	ADJ
ejpam-6375	144	9	spherical	spherical	ADJ
ejpam-6375	144	10	regions	region	NOUN
ejpam-6375	144	11	in	in	ADP
ejpam-6375	144	12	3	3	NUM
ejpam-6375	144	13	-	-	PUNCT
ejpam-6375	144	14	dimensional	dimensional	ADJ
ejpam-6375	144	15	euclidean	euclidean	ADJ
ejpam-6375	144	16	space	space	NOUN
ejpam-6375	144	17	:	:	PUNCT
ejpam-6375	145	1	v	v	X
ejpam-6375	145	2	=	=	PUNCT
ejpam-6375	145	3	{	{	PUNCT
ejpam-6375	145	4	s	s	NOUN
ejpam-6375	145	5	⊆	⊆	NUM
ejpam-6375	145	6	r3	r3	NOUN
ejpam-6375	146	1	|	|	ADV
ejpam-6375	146	2	s	s	VERB
ejpam-6375	146	3	is	be	AUX
ejpam-6375	146	4	a	a	DET
ejpam-6375	146	5	bounded	bounded	ADJ
ejpam-6375	146	6	spherical	spherical	ADJ
ejpam-6375	146	7	region	region	NOUN
ejpam-6375	146	8	}	}	PUNCT
ejpam-6375	146	9	.	.	PUNCT
ejpam-6375	147	1	where	where	SCONJ
ejpam-6375	147	2	s	s	NOUN
ejpam-6375	147	3	is	be	AUX
ejpam-6375	147	4	defined	define	VERB
ejpam-6375	147	5	as	as	ADP
ejpam-6375	147	6	:	:	PUNCT
ejpam-6375	147	7	s	s	PART
ejpam-6375	147	8	=	=	PUNCT
ejpam-6375	147	9	{	{	PUNCT
ejpam-6375	147	10	(	(	PUNCT
ejpam-6375	147	11	l	l	NOUN
ejpam-6375	147	12	,	,	PUNCT
ejpam-6375	147	13	m	m	PROPN
ejpam-6375	147	14	,	,	PUNCT
ejpam-6375	147	15	n	n	CCONJ
ejpam-6375	147	16	)	)	PUNCT
ejpam-6375	147	17	∈	∈	PROPN
ejpam-6375	147	18	r3	r3	PROPN
ejpam-6375	147	19	|	|	ADV
ejpam-6375	147	20	√	√	PROPN
ejpam-6375	147	21	(	(	PUNCT
ejpam-6375	147	22	l	l	NOUN
ejpam-6375	147	23	−	−	PROPN
ejpam-6375	147	24	a)2	a)2	NOUN
ejpam-6375	147	25	+	+	CCONJ
ejpam-6375	147	26	(	(	PUNCT
ejpam-6375	147	27	m−	m−	PROPN
ejpam-6375	147	28	b)2	b)2	PROPN
ejpam-6375	148	1	+	+	CCONJ
ejpam-6375	148	2	(	(	PUNCT
ejpam-6375	148	3	n−	n−	NOUN
ejpam-6375	148	4	c)2	c)2	VERB
ejpam-6375	148	5	≤	≤	ADJ
ejpam-6375	148	6	r	r	NOUN
ejpam-6375	148	7	}	}	PUNCT
ejpam-6375	148	8	,	,	PUNCT
ejpam-6375	148	9	with	with	ADP
ejpam-6375	148	10	r	r	NOUN
ejpam-6375	148	11	>	>	X
ejpam-6375	148	12	0	0	PUNCT
ejpam-6375	149	1	as	as	ADP
ejpam-6375	149	2	the	the	DET
ejpam-6375	149	3	radius	radius	NOUN
ejpam-6375	149	4	of	of	ADP
ejpam-6375	149	5	the	the	DET
ejpam-6375	149	6	spherical	spherical	ADJ
ejpam-6375	149	7	region	region	NOUN
ejpam-6375	149	8	and	and	CCONJ
ejpam-6375	149	9	a	a	DET
ejpam-6375	149	10	,	,	PUNCT
ejpam-6375	149	11	b	b	NOUN
ejpam-6375	149	12	,	,	PUNCT
ejpam-6375	149	13	c	c	PROPN
ejpam-6375	149	14	∈	∈	PROPN
ejpam-6375	149	15	r.	r.	PROPN
ejpam-6375	149	16	a	a	DET
ejpam-6375	149	17	bounded	bound	VERB
ejpam-6375	149	18	spherical	spherical	ADJ
ejpam-6375	149	19	region	region	NOUN
ejpam-6375	149	20	is	be	AUX
ejpam-6375	149	21	a	a	DET
ejpam-6375	149	22	subset	subset	NOUN
ejpam-6375	149	23	of	of	ADP
ejpam-6375	149	24	r3	r3	PROPN
ejpam-6375	149	25	consisting	consist	VERB
ejpam-6375	149	26	of	of	ADP
ejpam-6375	149	27	points	point	NOUN
ejpam-6375	149	28	within	within	ADP
ejpam-6375	149	29	a	a	DET
ejpam-6375	149	30	sphere	sphere	NOUN
ejpam-6375	149	31	of	of	ADP
ejpam-6375	149	32	finite	finite	ADJ
ejpam-6375	149	33	radius	radius	NOUN
ejpam-6375	149	34	.	.	PUNCT
ejpam-6375	150	1	gτ	gτ	PROPN
ejpam-6375	150	2	=	=	PUNCT
ejpam-6375	150	3	{	{	PUNCT
ejpam-6375	150	4	e	e	PROPN
ejpam-6375	150	5	⊆	⊆	NUM
ejpam-6375	150	6	v	v	ADP
ejpam-6375	150	7	\	\	PROPN
ejpam-6375	150	8	sr1	sr1	PROPN
ejpam-6375	150	9	|	|	PROPN
ejpam-6375	150	10	sr1	sr1	PROPN
ejpam-6375	150	11	=	=	PRON
ejpam-6375	150	12	{	{	PUNCT
ejpam-6375	150	13	s	s	NOUN
ejpam-6375	150	14	∈	∈	NOUN
ejpam-6375	150	15	v	v	ADP
ejpam-6375	150	16	|	|	ADV
ejpam-6375	150	17	radius(s	radius(s	NOUN
ejpam-6375	150	18	)	)	PUNCT
ejpam-6375	150	19	=	=	SYM
ejpam-6375	150	20	r1	r1	PROPN
ejpam-6375	150	21	}	}	PUNCT
ejpam-6375	150	22	}	}	PUNCT
ejpam-6375	150	23	,	,	PUNCT
ejpam-6375	150	24	where	where	SCONJ
ejpam-6375	150	25	r1	r1	PROPN
ejpam-6375	150	26	>	>	X
ejpam-6375	150	27	0	0	PUNCT
ejpam-6375	150	28	is	be	AUX
ejpam-6375	150	29	a	a	DET
ejpam-6375	150	30	fixed	fix	VERB
ejpam-6375	150	31	radius	radius	NOUN
ejpam-6375	150	32	,	,	PUNCT
ejpam-6375	150	33	and	and	CCONJ
ejpam-6375	150	34	sr1	sr1	PROPN
ejpam-6375	150	35	is	be	AUX
ejpam-6375	150	36	the	the	DET
ejpam-6375	150	37	set	set	NOUN
ejpam-6375	150	38	of	of	ADP
ejpam-6375	150	39	all	all	DET
ejpam-6375	150	40	spherical	spherical	ADJ
ejpam-6375	150	41	regions	region	NOUN
ejpam-6375	150	42	in	in	ADP
ejpam-6375	150	43	v	v	NOUN
ejpam-6375	150	44	with	with	ADP
ejpam-6375	150	45	radius	radius	NOUN
ejpam-6375	150	46	exactly	exactly	ADV
ejpam-6375	150	47	r1	r1	NOUN
ejpam-6375	150	48	,	,	PUNCT
ejpam-6375	150	49	forms	form	VERB
ejpam-6375	150	50	a	a	DET
ejpam-6375	150	51	generalized	generalized	ADJ
ejpam-6375	150	52	topology	topology	NOUN
ejpam-6375	150	53	on	on	ADP
ejpam-6375	150	54	v.	v.	INTJ
ejpam-6375	150	55	we	we	PRON
ejpam-6375	150	56	now	now	ADV
ejpam-6375	150	57	prove	prove	VERB
ejpam-6375	150	58	that	that	SCONJ
ejpam-6375	150	59	gτ	gτ	PROPN
ejpam-6375	150	60	satisfies	satisfy	VERB
ejpam-6375	150	61	the	the	DET
ejpam-6375	150	62	axioms	axiom	NOUN
ejpam-6375	150	63	of	of	ADP
ejpam-6375	150	64	a	a	DET
ejpam-6375	150	65	generalized	generalized	ADJ
ejpam-6375	150	66	topology	topology	NOUN
ejpam-6375	150	67	.	.	PUNCT
ejpam-6375	151	1	a	a	DET
ejpam-6375	151	2	collection	collection	NOUN
ejpam-6375	151	3	gτ	gτ	NOUN
ejpam-6375	151	4	forms	form	VERB
ejpam-6375	151	5	a	a	DET
ejpam-6375	151	6	generalized	generalized	ADJ
ejpam-6375	151	7	topology	topology	NOUN
ejpam-6375	151	8	if	if	SCONJ
ejpam-6375	151	9	it	it	PRON
ejpam-6375	151	10	satisfies	satisfy	VERB
ejpam-6375	151	11	the	the	DET
ejpam-6375	151	12	following	follow	VERB
ejpam-6375	151	13	conditions	condition	NOUN
ejpam-6375	151	14	:	:	PUNCT
ejpam-6375	151	15	condition	condition	NOUN
ejpam-6375	151	16	1	1	NUM
ejpam-6375	151	17	:	:	PUNCT
ejpam-6375	151	18	∅	∅	NOUN
ejpam-6375	151	19	∈	∈	PROPN
ejpam-6375	151	20	gτ	gτ	VERB
ejpam-6375	151	21	the	the	DET
ejpam-6375	151	22	empty	empty	ADJ
ejpam-6375	151	23	set	set	NOUN
ejpam-6375	151	24	trivially	trivially	ADV
ejpam-6375	151	25	belongs	belong	VERB
ejpam-6375	151	26	to	to	ADP
ejpam-6375	151	27	gτ	gτ	PROPN
ejpam-6375	151	28	since	since	SCONJ
ejpam-6375	151	29	there	there	PRON
ejpam-6375	151	30	is	be	VERB
ejpam-6375	151	31	no	no	DET
ejpam-6375	151	32	restriction	restriction	NOUN
ejpam-6375	151	33	preventing	prevent	VERB
ejpam-6375	151	34	∅	∅	NOUN
ejpam-6375	151	35	from	from	ADP
ejpam-6375	151	36	being	be	AUX
ejpam-6375	151	37	included	include	VERB
ejpam-6375	151	38	.	.	PUNCT
ejpam-6375	152	1	condition	condition	NOUN
ejpam-6375	152	2	2	2	NUM
ejpam-6375	152	3	:	:	PUNCT
ejpam-6375	152	4	arbitrary	arbitrary	ADJ
ejpam-6375	152	5	unions	union	NOUN
ejpam-6375	152	6	of	of	ADP
ejpam-6375	152	7	elements	element	NOUN
ejpam-6375	152	8	of	of	ADP
ejpam-6375	152	9	gτ	gτ	PROPN
ejpam-6375	152	10	remain	remain	VERB
ejpam-6375	152	11	in	in	ADP
ejpam-6375	152	12	gτ	gτ	PROPN
ejpam-6375	152	13	let	let	VERB
ejpam-6375	152	14	{	{	PUNCT
ejpam-6375	152	15	eαi}i∈i	eαi}i∈i	PART
ejpam-6375	152	16	be	be	AUX
ejpam-6375	152	17	a	a	DET
ejpam-6375	152	18	family	family	NOUN
ejpam-6375	152	19	of	of	ADP
ejpam-6375	152	20	sets	set	NOUN
ejpam-6375	152	21	in	in	ADP
ejpam-6375	152	22	gτ	gτ	PROPN
ejpam-6375	152	23	,	,	PUNCT
ejpam-6375	152	24	meaning	mean	VERB
ejpam-6375	152	25	each	each	DET
ejpam-6375	152	26	eαi	eαi	NOUN
ejpam-6375	152	27	⊆	⊆	NUM
ejpam-6375	152	28	v	v	ADP
ejpam-6375	152	29	\	\	PROPN
ejpam-6375	152	30	sr	sr	PROPN
ejpam-6375	152	31	.	.	PUNCT
ejpam-6375	153	1	consider	consider	VERB
ejpam-6375	153	2	their	their	PRON
ejpam-6375	153	3	union	union	NOUN
ejpam-6375	153	4	:	:	PUNCT
ejpam-6375	153	5	e	e	X
ejpam-6375	153	6	=	=	PUNCT
ejpam-6375	153	7	⋃	⋃	PROPN
ejpam-6375	153	8	i∈i	i∈i	ADJ
ejpam-6375	153	9	vi	vi	PROPN
ejpam-6375	153	10	.	.	PUNCT
ejpam-6375	154	1	since	since	SCONJ
ejpam-6375	154	2	each	each	DET
ejpam-6375	154	3	eαi	eαi	NOUN
ejpam-6375	154	4	excludes	exclude	VERB
ejpam-6375	154	5	all	all	DET
ejpam-6375	154	6	spherical	spherical	ADJ
ejpam-6375	154	7	regions	region	NOUN
ejpam-6375	154	8	of	of	ADP
ejpam-6375	154	9	radius	radius	NOUN
ejpam-6375	154	10	exactly	exactly	ADV
ejpam-6375	154	11	r	r	NOUN
ejpam-6375	154	12	,	,	PUNCT
ejpam-6375	154	13	their	their	PRON
ejpam-6375	154	14	union	union	NOUN
ejpam-6375	154	15	e	e	NOUN
ejpam-6375	154	16	must	must	AUX
ejpam-6375	154	17	also	also	ADV
ejpam-6375	154	18	exclude	exclude	VERB
ejpam-6375	154	19	all	all	DET
ejpam-6375	154	20	such	such	ADJ
ejpam-6375	154	21	regions	region	NOUN
ejpam-6375	154	22	.	.	PUNCT
ejpam-6375	155	1	that	that	PRON
ejpam-6375	155	2	is	be	AUX
ejpam-6375	155	3	,	,	PUNCT
ejpam-6375	155	4	e	e	PROPN
ejpam-6375	155	5	⊆	⊆	NUM
ejpam-6375	155	6	v	v	ADP
ejpam-6375	155	7	\	\	PROPN
ejpam-6375	155	8	sr	sr	PROPN
ejpam-6375	155	9	.	.	PUNCT
ejpam-6375	156	1	thus	thus	ADV
ejpam-6375	156	2	,	,	PUNCT
ejpam-6375	156	3	sα	sα	ADV
ejpam-6375	156	4	satisfies	satisfy	VERB
ejpam-6375	156	5	the	the	DET
ejpam-6375	156	6	definition	definition	NOUN
ejpam-6375	156	7	of	of	ADP
ejpam-6375	156	8	gτ	gτ	PROPN
ejpam-6375	156	9	,	,	PUNCT
ejpam-6375	156	10	ensuring	ensure	VERB
ejpam-6375	156	11	that	that	SCONJ
ejpam-6375	156	12	arbitrary	arbitrary	ADJ
ejpam-6375	156	13	unions	union	NOUN
ejpam-6375	156	14	remain	remain	VERB
ejpam-6375	156	15	in	in	ADP
ejpam-6375	156	16	gτ	gτ	PROPN
ejpam-6375	156	17	.	.	PUNCT
ejpam-6375	157	1	since	since	SCONJ
ejpam-6375	157	2	gτ	gτ	PROPN
ejpam-6375	157	3	satisfies	satisfy	VERB
ejpam-6375	157	4	both	both	DET
ejpam-6375	157	5	required	required	ADJ
ejpam-6375	157	6	conditions	condition	NOUN
ejpam-6375	157	7	,	,	PUNCT
ejpam-6375	157	8	so	so	SCONJ
ejpam-6375	157	9	it	it	PRON
ejpam-6375	157	10	forms	form	VERB
ejpam-6375	157	11	a	a	DET
ejpam-6375	157	12	generalized	generalized	ADJ
ejpam-6375	157	13	topology	topology	NOUN
ejpam-6375	157	14	on	on	ADP
ejpam-6375	157	15	v	v	NOUN
ejpam-6375	157	16	and	and	CCONJ
ejpam-6375	157	17	define	define	VERB
ejpam-6375	157	18	a	a	DET
ejpam-6375	157	19	collection	collection	NOUN
ejpam-6375	157	20	gpt	gpt	NOUN
ejpam-6375	157	21	⊆	⊆	NUM
ejpam-6375	157	22	2v	2v	NUM
ejpam-6375	157	23	as	as	ADP
ejpam-6375	157	24	:	:	PUNCT
ejpam-6375	157	25	gpt	gpt	NOUN
ejpam-6375	157	26	=	=	SYM
ejpam-6375	157	27	{	{	PUNCT
ejpam-6375	157	28	e	e	PROPN
ejpam-6375	157	29	⊆	⊆	NUM
ejpam-6375	157	30	v	v	ADP
ejpam-6375	157	31	|	|	ADV
ejpam-6375	157	32	all	all	DET
ejpam-6375	157	33	spherical	spherical	ADJ
ejpam-6375	157	34	region	region	NOUN
ejpam-6375	157	35	in	in	ADP
ejpam-6375	157	36	e	e	PROPN
ejpam-6375	157	37	have	have	AUX
ejpam-6375	157	38	radius	radiu	VERB
ejpam-6375	157	39	strictly	strictly	ADV
ejpam-6375	157	40	less	less	ADJ
ejpam-6375	157	41	than	than	ADP
ejpam-6375	157	42	r	r	NOUN
ejpam-6375	157	43	,	,	PUNCT
ejpam-6375	157	44	where	where	SCONJ
ejpam-6375	157	45	r	r	NOUN
ejpam-6375	157	46	>	>	X
ejpam-6375	157	47	0	0	NUM
ejpam-6375	157	48	}	}	PUNCT
ejpam-6375	157	49	.	.	PUNCT
ejpam-6375	158	1	to	to	PART
ejpam-6375	158	2	determine	determine	VERB
ejpam-6375	158	3	whether	whether	SCONJ
ejpam-6375	158	4	e	e	NOUN
ejpam-6375	158	5	is	be	AUX
ejpam-6375	158	6	gpt	gpt	NOUN
ejpam-6375	158	7	-	-	PUNCT
ejpam-6375	158	8	semi	semi	NOUN
ejpam-6375	158	9	-	-	ADJ
ejpam-6375	158	10	compact	compact	ADJ
ejpam-6375	158	11	,	,	PUNCT
ejpam-6375	158	12	consider	consider	VERB
ejpam-6375	158	13	a	a	DET
ejpam-6375	158	14	family	family	NOUN
ejpam-6375	158	15	{	{	PUNCT
ejpam-6375	158	16	zi	zi	NOUN
ejpam-6375	158	17	:	:	PUNCT
ejpam-6375	158	18	i	i	PRON
ejpam-6375	158	19	∈	∈	PROPN
ejpam-6375	158	20	⋏	⋏	PROPN
ejpam-6375	158	21	}	}	PUNCT
ejpam-6375	158	22	of	of	ADP
ejpam-6375	158	23	gpt	gpt	NOUN
ejpam-6375	158	24	-	-	PUNCT
ejpam-6375	158	25	s∗	s∗	PROPN
ejpam-6375	158	26	g	g	PROPN
ejpam-6375	158	27	open	open	ADJ
ejpam-6375	158	28	sets	set	NOUN
ejpam-6375	158	29	satisfying	satisfy	VERB
ejpam-6375	158	30	e	e	NOUN
ejpam-6375	158	31	⊆	⊆	X
ejpam-6375	158	32	⋃	⋃	NOUN
ejpam-6375	158	33	{	{	PUNCT
ejpam-6375	158	34	zi	zi	NOUN
ejpam-6375	158	35	:	:	PUNCT
ejpam-6375	158	36	i	i	PRON
ejpam-6375	158	37	∈	∈	PROPN
ejpam-6375	158	38	⋏	⋏	PROPN
ejpam-6375	158	39	}	}	PUNCT
ejpam-6375	158	40	.	.	PUNCT
ejpam-6375	159	1	within	within	ADP
ejpam-6375	159	2	this	this	DET
ejpam-6375	159	3	covering	covering	NOUN
ejpam-6375	159	4	,	,	PUNCT
ejpam-6375	159	5	a	a	DET
ejpam-6375	159	6	finite	finite	NOUN
ejpam-6375	159	7	subcover	subcover	PROPN
ejpam-6375	159	8	can	can	AUX
ejpam-6375	159	9	always	always	ADV
ejpam-6375	159	10	be	be	AUX
ejpam-6375	159	11	extracted	extract	VERB
ejpam-6375	159	12	.	.	PUNCT
ejpam-6375	160	1	hence	hence	ADV
ejpam-6375	160	2	,	,	PUNCT
ejpam-6375	160	3	e	e	PROPN
ejpam-6375	160	4	is	be	AUX
ejpam-6375	160	5	gpt	gpt	NOUN
ejpam-6375	160	6	-	-	PUNCT
ejpam-6375	160	7	s∗	s∗	PROPN
ejpam-6375	160	8	g	g	PROPN
ejpam-6375	160	9	-semi	-semi	NOUN
ejpam-6375	160	10	-	-	PUNCT
ejpam-6375	160	11	compact	compact	ADJ
ejpam-6375	160	12	.	.	PUNCT
ejpam-6375	161	1	next	next	ADV
ejpam-6375	161	2	,	,	PUNCT
ejpam-6375	161	3	let	let	VERB
ejpam-6375	161	4	{	{	PUNCT
ejpam-6375	161	5	zi	zi	NOUN
ejpam-6375	161	6	:	:	PUNCT
ejpam-6375	162	1	i	i	PRON
ejpam-6375	162	2	∈	∈	PROPN
ejpam-6375	162	3	⋏	⋏	PROPN
ejpam-6375	162	4	}	}	PUNCT
ejpam-6375	162	5	be	be	AUX
ejpam-6375	162	6	a	a	DET
ejpam-6375	162	7	collection	collection	NOUN
ejpam-6375	162	8	of	of	ADP
ejpam-6375	162	9	gpt	gpt	NOUN
ejpam-6375	162	10	-	-	PUNCT
ejpam-6375	162	11	s∗	s∗	PROPN
ejpam-6375	162	12	g	g	PROPN
ejpam-6375	162	13	-open	-open	NOUN
ejpam-6375	162	14	sets	set	NOUN
ejpam-6375	162	15	such	such	ADJ
ejpam-6375	162	16	that	that	SCONJ
ejpam-6375	162	17	e	e	PROPN
ejpam-6375	162	18	⊆	⊆	NUM
ejpam-6375	162	19	⋃	⋃	NOUN
ejpam-6375	162	20	{	{	PUNCT
ejpam-6375	162	21	zi	zi	NOUN
ejpam-6375	162	22	:	:	PUNCT
ejpam-6375	162	23	i	i	PRON
ejpam-6375	162	24	∈	∈	PROPN
ejpam-6375	162	25	⋏	⋏	PROPN
ejpam-6375	162	26	}	}	PUNCT
ejpam-6375	162	27	.	.	PUNCT
ejpam-6375	163	1	each	each	DET
ejpam-6375	163	2	point	point	NOUN
ejpam-6375	163	3	in	in	ADP
ejpam-6375	163	4	e	e	NOUN
ejpam-6375	163	5	must	must	AUX
ejpam-6375	163	6	be	be	AUX
ejpam-6375	163	7	contained	contain	VERB
ejpam-6375	163	8	within	within	ADP
ejpam-6375	163	9	at	at	ADV
ejpam-6375	163	10	least	least	ADV
ejpam-6375	163	11	one	one	NUM
ejpam-6375	163	12	zi	zi	NOUN
ejpam-6375	163	13	,	,	PUNCT
ejpam-6375	163	14	and	and	CCONJ
ejpam-6375	163	15	thus	thus	ADV
ejpam-6375	163	16	a	a	DET
ejpam-6375	163	17	finite	finite	NOUN
ejpam-6375	163	18	subset	subset	VERB
ejpam-6375	163	19	⋏o	⋏o	PROPN
ejpam-6375	163	20	⊆	⊆	PROPN
ejpam-6375	163	21	⋏	⋏	PROPN
ejpam-6375	163	22	is	be	AUX
ejpam-6375	163	23	sufficient	sufficient	ADJ
ejpam-6375	163	24	to	to	PART
ejpam-6375	163	25	cover	cover	VERB
ejpam-6375	163	26	e.	e.	PROPN
ejpam-6375	163	27	consequently	consequently	ADV
ejpam-6375	163	28	,	,	PUNCT
ejpam-6375	163	29	e	e	PROPN
ejpam-6375	163	30	is	be	AUX
ejpam-6375	163	31	gpt	gpt	NOUN
ejpam-6375	163	32	-	-	PUNCT
ejpam-6375	163	33	s∗	s∗	PROPN
ejpam-6375	163	34	g	g	PROPN
ejpam-6375	163	35	-compact	-compact	PROPN
ejpam-6375	163	36	.	.	PUNCT
ejpam-6375	164	1	m.	m.	NOUN
ejpam-6375	164	2	shahbaz	shahbaz	PROPN
ejpam-6375	164	3	et	et	PROPN
ejpam-6375	164	4	al	al	PROPN
ejpam-6375	164	5	.	.	PUNCT
ejpam-6375	164	6	/	/	SYM
ejpam-6375	164	7	eur	eur	PROPN
ejpam-6375	164	8	.	.	PUNCT
ejpam-6375	165	1	j.	j.	PROPN
ejpam-6375	165	2	pure	pure	PROPN
ejpam-6375	165	3	appl	appl	PROPN
ejpam-6375	165	4	.	.	PROPN
ejpam-6375	165	5	math	math	PROPN
ejpam-6375	165	6	,	,	PUNCT
ejpam-6375	165	7	18	18	NUM
ejpam-6375	165	8	(	(	PUNCT
ejpam-6375	165	9	4	4	NUM
ejpam-6375	165	10	)	)	PUNCT
ejpam-6375	165	11	(	(	PUNCT
ejpam-6375	165	12	2025	2025	NUM
ejpam-6375	165	13	)	)	PUNCT
ejpam-6375	165	14	,	,	PUNCT
ejpam-6375	165	15	6375	6375	NUM
ejpam-6375	165	16	8	8	NUM
ejpam-6375	165	17	of	of	ADP
ejpam-6375	165	18	22	22	NUM
ejpam-6375	165	19	all	all	PRON
ejpam-6375	165	20	open	open	ADJ
ejpam-6375	165	21	covers	cover	NOUN
ejpam-6375	165	22	from	from	ADP
ejpam-6375	165	23	gpt	gpt	NOUN
ejpam-6375	165	24	naturally	naturally	ADV
ejpam-6375	165	25	have	have	VERB
ejpam-6375	165	26	finite	finite	NOUN
ejpam-6375	165	27	subcovers	subcover	NOUN
ejpam-6375	165	28	in	in	ADP
ejpam-6375	165	29	the	the	DET
ejpam-6375	165	30	context	context	NOUN
ejpam-6375	165	31	of	of	ADP
ejpam-6375	165	32	e.	e.	PROPN
ejpam-6375	165	33	thus	thus	ADV
ejpam-6375	165	34	,	,	PUNCT
ejpam-6375	165	35	e	e	PROPN
ejpam-6375	165	36	is	be	AUX
ejpam-6375	165	37	gpt	gpt	NOUN
ejpam-6375	165	38	-	-	PUNCT
ejpam-6375	165	39	compact	compact	ADJ
ejpam-6375	165	40	.	.	PUNCT
ejpam-6375	166	1	theorem	theorem	VERB
ejpam-6375	166	2	3.2	3.2	NUM
ejpam-6375	166	3	.	.	PUNCT
ejpam-6375	167	1	a	a	DET
ejpam-6375	167	2	gpt	gpt	NOUN
ejpam-6375	167	3	-	-	PUNCT
ejpam-6375	167	4	s∗	s∗	PROPN
ejpam-6375	167	5	g	g	PROPN
ejpam-6375	167	6	-compact	-compact	PROPN
ejpam-6375	167	7	space	space	NOUN
ejpam-6375	167	8	makes	make	VERB
ejpam-6375	167	9	every	every	PRON
ejpam-6375	167	10	of	of	ADP
ejpam-6375	167	11	its	its	PRON
ejpam-6375	167	12	gpt	gpt	NOUN
ejpam-6375	167	13	-	-	PUNCT
ejpam-6375	167	14	s∗	s∗	PROPN
ejpam-6375	167	15	g	g	PROPN
ejpam-6375	167	16	-closed	-close	VERB
ejpam-6375	167	17	subsets	subset	NOUN
ejpam-6375	167	18	compact	compact	ADJ
ejpam-6375	167	19	relative	relative	ADJ
ejpam-6375	167	20	to	to	ADP
ejpam-6375	167	21	(	(	PUNCT
ejpam-6375	167	22	v	v	NOUN
ejpam-6375	167	23	,	,	PUNCT
ejpam-6375	167	24	gτ	gτ	INTJ
ejpam-6375	167	25	,	,	PUNCT
ejpam-6375	167	26	p	p	NOUN
ejpam-6375	167	27	)	)	PUNCT
ejpam-6375	167	28	.	.	PUNCT
ejpam-6375	168	1	proof	proof	NOUN
ejpam-6375	168	2	.	.	PUNCT
ejpam-6375	169	1	consider	consider	VERB
ejpam-6375	169	2	e	e	NOUN
ejpam-6375	169	3	as	as	ADP
ejpam-6375	169	4	a	a	DET
ejpam-6375	169	5	gpt	gpt	NOUN
ejpam-6375	169	6	-	-	PUNCT
ejpam-6375	169	7	s∗	s∗	PROPN
ejpam-6375	169	8	g	g	PROPN
ejpam-6375	169	9	-closed	-close	VERB
ejpam-6375	169	10	subset	subset	NOUN
ejpam-6375	169	11	of	of	ADP
ejpam-6375	169	12	gpt	gpt	NOUN
ejpam-6375	169	13	-	-	PUNCT
ejpam-6375	169	14	s∗	s∗	PROPN
ejpam-6375	169	15	g	g	PROPN
ejpam-6375	169	16	-compact	-compact	PROPN
ejpam-6375	169	17	space	space	NOUN
ejpam-6375	169	18	(	(	PUNCT
ejpam-6375	169	19	v	v	NOUN
ejpam-6375	169	20	,	,	PUNCT
ejpam-6375	169	21	gτ	gτ	INTJ
ejpam-6375	169	22	,	,	PUNCT
ejpam-6375	169	23	p	p	NOUN
ejpam-6375	169	24	)	)	PUNCT
ejpam-6375	169	25	implies	imply	VERB
ejpam-6375	169	26	ec	ec	PROPN
ejpam-6375	169	27	is	be	AUX
ejpam-6375	169	28	gpt	gpt	NOUN
ejpam-6375	169	29	-	-	PUNCT
ejpam-6375	169	30	s∗	s∗	NOUN
ejpam-6375	169	31	g	g	PROPN
ejpam-6375	169	32	-open	-open	NOUN
ejpam-6375	169	33	in	in	ADP
ejpam-6375	169	34	(	(	PUNCT
ejpam-6375	169	35	v	v	NOUN
ejpam-6375	169	36	,	,	PUNCT
ejpam-6375	170	1	gτ	gτ	INTJ
ejpam-6375	170	2	,	,	PUNCT
ejpam-6375	170	3	p	p	NOUN
ejpam-6375	170	4	)	)	PUNCT
ejpam-6375	170	5	.	.	PUNCT
ejpam-6375	171	1	the	the	DET
ejpam-6375	171	2	collection	collection	NOUN
ejpam-6375	171	3	{	{	PUNCT
ejpam-6375	171	4	zi	zi	NOUN
ejpam-6375	171	5	:	:	PUNCT
ejpam-6375	172	1	i	i	PRON
ejpam-6375	172	2	∈	∈	PROPN
ejpam-6375	172	3	⋏	⋏	PROPN
ejpam-6375	172	4	}	}	PUNCT
ejpam-6375	172	5	serves	serve	VERB
ejpam-6375	172	6	as	as	ADP
ejpam-6375	172	7	a	a	DET
ejpam-6375	172	8	cover	cover	NOUN
ejpam-6375	172	9	of	of	ADP
ejpam-6375	172	10	the	the	DET
ejpam-6375	172	11	set	set	NOUN
ejpam-6375	172	12	e	e	NOUN
ejpam-6375	172	13	since	since	SCONJ
ejpam-6375	172	14	each	each	DET
ejpam-6375	172	15	member	member	NOUN
ejpam-6375	172	16	of	of	ADP
ejpam-6375	172	17	the	the	DET
ejpam-6375	172	18	collection	collection	NOUN
ejpam-6375	172	19	is	be	AUX
ejpam-6375	172	20	an	an	DET
ejpam-6375	172	21	element	element	NOUN
ejpam-6375	172	22	from	from	ADP
ejpam-6375	172	23	the	the	DET
ejpam-6375	172	24	family	family	NOUN
ejpam-6375	172	25	of	of	ADP
ejpam-6375	172	26	gpt	gpt	PROPN
ejpam-6375	172	27	-	-	PUNCT
ejpam-6375	172	28	s∗	s∗	PROPN
ejpam-6375	172	29	g	g	PROPN
ejpam-6375	172	30	-open	-open	ADJ
ejpam-6375	172	31	subsets	subset	NOUN
ejpam-6375	172	32	of	of	ADP
ejpam-6375	172	33	v	v	NOUN
ejpam-6375	172	34	so	so	SCONJ
ejpam-6375	172	35	that	that	SCONJ
ejpam-6375	172	36	e	e	PROPN
ejpam-6375	172	37	⊂	⊂	PROPN
ejpam-6375	172	38	∪	∪	X
ejpam-6375	172	39	{	{	PUNCT
ejpam-6375	172	40	zi	zi	NOUN
ejpam-6375	172	41	:	:	PUNCT
ejpam-6375	172	42	i	i	PRON
ejpam-6375	172	43	∈	∈	PROPN
ejpam-6375	172	44	⋏	⋏	PROPN
ejpam-6375	172	45	}	}	PUNCT
ejpam-6375	172	46	implies	imply	VERB
ejpam-6375	172	47	ec	ec	PROPN
ejpam-6375	172	48	⊂	⊂	PROPN
ejpam-6375	172	49	∪	∪	X
ejpam-6375	172	50	{	{	PUNCT
ejpam-6375	172	51	zi	zi	NOUN
ejpam-6375	172	52	:	:	PUNCT
ejpam-6375	172	53	i	i	PRON
ejpam-6375	172	54	∈	∈	PROPN
ejpam-6375	172	55	⋏	⋏	PROPN
ejpam-6375	172	56	}	}	PUNCT
ejpam-6375	172	57	=	=	PUNCT
ejpam-6375	173	1	v.	v.	CCONJ
ejpam-6375	173	2	therefore	therefore	ADV
ejpam-6375	173	3	,	,	PUNCT
ejpam-6375	173	4	(	(	PUNCT
ejpam-6375	173	5	v	v	NOUN
ejpam-6375	173	6	,	,	PUNCT
ejpam-6375	173	7	gτ	gτ	INTJ
ejpam-6375	173	8	,	,	PUNCT
ejpam-6375	173	9	p	p	X
ejpam-6375	173	10	)	)	PUNCT
ejpam-6375	173	11	is	be	AUX
ejpam-6375	173	12	gpt	gpt	NOUN
ejpam-6375	173	13	-	-	PUNCT
ejpam-6375	173	14	s∗	s∗	PROPN
ejpam-6375	173	15	g	g	PROPN
ejpam-6375	173	16	-compact	-compact	NOUN
ejpam-6375	173	17	then	then	ADV
ejpam-6375	173	18	there	there	PRON
ejpam-6375	173	19	exists	exist	VERB
ejpam-6375	173	20	a	a	DET
ejpam-6375	173	21	finite	finite	NOUN
ejpam-6375	173	22	subset	subset	NOUN
ejpam-6375	173	23	eo	eo	PROPN
ejpam-6375	173	24	of	of	ADP
ejpam-6375	173	25	e	e	PROPN
ejpam-6375	173	26	so	so	SCONJ
ejpam-6375	173	27	that	that	SCONJ
ejpam-6375	173	28	e	e	PROPN
ejpam-6375	173	29	⊂	⊂	PROPN
ejpam-6375	173	30	ec	ec	PROPN
ejpam-6375	173	31	∪	∪	X
ejpam-6375	173	32	{	{	PUNCT
ejpam-6375	173	33	zi	zi	NOUN
ejpam-6375	173	34	:	:	PUNCT
ejpam-6375	173	35	i	i	PRON
ejpam-6375	173	36	∈	∈	PROPN
ejpam-6375	173	37	⋏	⋏	PROPN
ejpam-6375	173	38	}	}	PUNCT
ejpam-6375	173	39	=	=	PUNCT
ejpam-6375	174	1	v.	v.	CCONJ
ejpam-6375	174	2	then	then	ADV
ejpam-6375	174	3	,	,	PUNCT
ejpam-6375	174	4	e	e	PROPN
ejpam-6375	174	5	⊂	⊂	PROPN
ejpam-6375	174	6	∪	∪	X
ejpam-6375	174	7	{	{	PUNCT
ejpam-6375	174	8	zi	zi	NOUN
ejpam-6375	174	9	:	:	PUNCT
ejpam-6375	174	10	i	i	PRON
ejpam-6375	174	11	∈	∈	PROPN
ejpam-6375	174	12	⋏	⋏	PROPN
ejpam-6375	174	13	}	}	PUNCT
ejpam-6375	174	14	and	and	CCONJ
ejpam-6375	174	15	therefore	therefore	ADV
ejpam-6375	174	16	e	e	PROPN
ejpam-6375	174	17	is	be	AUX
ejpam-6375	174	18	gpt	gpt	NOUN
ejpam-6375	174	19	-	-	PUNCT
ejpam-6375	174	20	s∗	s∗	PROPN
ejpam-6375	174	21	g	g	PROPN
ejpam-6375	174	22	-compact	-compact	NOUN
ejpam-6375	174	23	relative	relative	ADJ
ejpam-6375	174	24	to	to	ADP
ejpam-6375	174	25	v.	v.	ADP
ejpam-6375	174	26	example	example	NOUN
ejpam-6375	174	27	3.2	3.2	NUM
ejpam-6375	174	28	.	.	PUNCT
ejpam-6375	175	1	assume	assume	VERB
ejpam-6375	175	2	v	v	ADP
ejpam-6375	175	3	=	=	SYM
ejpam-6375	175	4	r	r	NOUN
ejpam-6375	175	5	and	and	CCONJ
ejpam-6375	175	6	(	(	PUNCT
ejpam-6375	175	7	v	v	NOUN
ejpam-6375	175	8	,	,	PUNCT
ejpam-6375	175	9	gτ	gτ	INTJ
ejpam-6375	175	10	,	,	PUNCT
ejpam-6375	175	11	p	p	AUX
ejpam-6375	175	12	)	)	PUNCT
ejpam-6375	175	13	be	be	AUX
ejpam-6375	175	14	defined	define	VERB
ejpam-6375	175	15	as	as	SCONJ
ejpam-6375	175	16	follows	follow	VERB
ejpam-6375	175	17	:	:	PUNCT
ejpam-6375	175	18	u	u	PROPN
ejpam-6375	175	19	∈	∈	NOUN
ejpam-6375	176	1	gτ	gτ	INTJ
ejpam-6375	176	2	if	if	SCONJ
ejpam-6375	176	3	and	and	CCONJ
ejpam-6375	176	4	only	only	ADV
ejpam-6375	176	5	if	if	SCONJ
ejpam-6375	176	6	either	either	CCONJ
ejpam-6375	176	7	u	u	NOUN
ejpam-6375	176	8	=	=	NOUN
ejpam-6375	176	9	∅	∅	NOUN
ejpam-6375	176	10	or	or	CCONJ
ejpam-6375	176	11	1	1	NUM
ejpam-6375	176	12	∈	∈	NOUN
ejpam-6375	176	13	u	u	NOUN
ejpam-6375	176	14	,	,	PUNCT
ejpam-6375	176	15	see	see	VERB
ejpam-6375	176	16	example	example	NOUN
ejpam-6375	176	17	10	10	NUM
ejpam-6375	176	18	in	in	ADP
ejpam-6375	176	19	[	[	X
ejpam-6375	176	20	22	22	NUM
ejpam-6375	176	21	]	]	PUNCT
ejpam-6375	176	22	.	.	PUNCT
ejpam-6375	177	1	let	let	VERB
ejpam-6375	177	2	gpt	gpt	NOUN
ejpam-6375	177	3	be	be	AUX
ejpam-6375	177	4	defined	define	VERB
ejpam-6375	177	5	on	on	ADP
ejpam-6375	177	6	r	r	NOUN
ejpam-6375	177	7	as	as	SCONJ
ejpam-6375	177	8	follows	follow	VERB
ejpam-6375	177	9	:	:	PUNCT
ejpam-6375	177	10	u	u	PROPN
ejpam-6375	177	11	∈	∈	PROPN
ejpam-6375	177	12	gpt	gpt	NOUN
ejpam-6375	178	1	if	if	SCONJ
ejpam-6375	178	2	and	and	CCONJ
ejpam-6375	178	3	only	only	ADV
ejpam-6375	178	4	if	if	SCONJ
ejpam-6375	178	5	1	1	NUM
ejpam-6375	178	6	/∈	/∈	SYM
ejpam-6375	178	7	u	u	NOUN
ejpam-6375	178	8	.	.	PUNCT
ejpam-6375	179	1	then	then	ADV
ejpam-6375	179	2	,	,	PUNCT
ejpam-6375	179	3	(	(	PUNCT
ejpam-6375	179	4	v	v	NOUN
ejpam-6375	179	5	,	,	PUNCT
ejpam-6375	179	6	gτ	gτ	INTJ
ejpam-6375	179	7	,	,	PUNCT
ejpam-6375	179	8	p	p	X
ejpam-6375	179	9	)	)	PUNCT
ejpam-6375	179	10	is	be	AUX
ejpam-6375	179	11	a	a	DET
ejpam-6375	179	12	generalized	generalized	ADJ
ejpam-6375	179	13	primal	primal	ADJ
ejpam-6375	179	14	topology	topology	NOUN
ejpam-6375	179	15	.	.	PUNCT
ejpam-6375	180	1	now	now	ADV
ejpam-6375	180	2	,	,	PUNCT
ejpam-6375	180	3	consider	consider	VERB
ejpam-6375	180	4	the	the	DET
ejpam-6375	180	5	subset	subset	NOUN
ejpam-6375	180	6	n	n	PROPN
ejpam-6375	180	7	⊂	⊂	PROPN
ejpam-6375	180	8	r.	r.	PROPN
ejpam-6375	180	9	let	let	VERB
ejpam-6375	180	10	s	s	PRON
ejpam-6375	180	11	is	be	AUX
ejpam-6375	180	12	index	index	NOUN
ejpam-6375	180	13	set	set	VERB
ejpam-6375	180	14	and	and	CCONJ
ejpam-6375	180	15	{	{	PUNCT
ejpam-6375	180	16	vη}η∈s	vη}η∈s	PROPN
ejpam-6375	180	17	be	be	AUX
ejpam-6375	180	18	a	a	DET
ejpam-6375	180	19	gpt	gpt	NOUN
ejpam-6375	180	20	-	-	PUNCT
ejpam-6375	180	21	s∗	s∗	NOUN
ejpam-6375	180	22	g	g	PROPN
ejpam-6375	180	23	-open	-open	ADJ
ejpam-6375	180	24	cover	cover	NOUN
ejpam-6375	180	25	of	of	ADP
ejpam-6375	180	26	n	n	CCONJ
ejpam-6375	180	27	such	such	ADJ
ejpam-6375	180	28	that	that	SCONJ
ejpam-6375	180	29	vη	vη	NUM
ejpam-6375	180	30	̸=	̸=	PROPN
ejpam-6375	180	31	∅	∅	NOUN
ejpam-6375	180	32	for	for	ADP
ejpam-6375	180	33	every	every	DET
ejpam-6375	180	34	η	η	PROPN
ejpam-6375	180	35	∈	∈	PROPN
ejpam-6375	180	36	s	s	PART
ejpam-6375	180	37	.	.	PUNCT
ejpam-6375	181	1	this	this	PRON
ejpam-6375	181	2	implies	imply	VERB
ejpam-6375	181	3	that	that	SCONJ
ejpam-6375	181	4	:	:	PUNCT
ejpam-6375	181	5	n	n	PRON
ejpam-6375	181	6	⊆	⊆	NUM
ejpam-6375	181	7	⋃	⋃	NOUN
ejpam-6375	181	8	η∈s	η∈	NOUN
ejpam-6375	181	9	vη	vη	NOUN
ejpam-6375	181	10	.	.	PUNCT
ejpam-6375	181	11	let	let	VERB
ejpam-6375	181	12	s0	s0	PROPN
ejpam-6375	181	13	=	=	PUNCT
ejpam-6375	181	14	{	{	PUNCT
ejpam-6375	181	15	vi}ni=1	vi}ni=1	PROPN
ejpam-6375	181	16	⊆	⊆	NUM
ejpam-6375	181	17	{	{	PUNCT
ejpam-6375	181	18	vη}η∈s	vη}η∈s	PROPN
ejpam-6375	181	19	.	.	PUNCT
ejpam-6375	182	1	then	then	ADV
ejpam-6375	182	2	,	,	PUNCT
ejpam-6375	182	3	for	for	ADP
ejpam-6375	182	4	any	any	DET
ejpam-6375	182	5	x	x	SYM
ejpam-6375	182	6	∈	∈	PROPN
ejpam-6375	182	7	n	n	CCONJ
ejpam-6375	182	8	\	\	PROPN
ejpam-6375	182	9	⋃n	⋃n	PROPN
ejpam-6375	182	10	i=1	i=1	PROPN
ejpam-6375	182	11	vi	vi	PROPN
ejpam-6375	182	12	,	,	PUNCT
ejpam-6375	182	13	it	it	PRON
ejpam-6375	182	14	must	must	AUX
ejpam-6375	182	15	follow	follow	VERB
ejpam-6375	182	16	that	that	PRON
ejpam-6375	182	17	r	r	NOUN
ejpam-6375	182	18	\	\	PUNCT
ejpam-6375	182	19	[	[	PUNCT
ejpam-6375	182	20	n	n	CCONJ
ejpam-6375	182	21	\	\	PROPN
ejpam-6375	182	22	⋃n	⋃n	PROPN
ejpam-6375	182	23	i=1	i=1	PROPN
ejpam-6375	182	24	vi	vi	X
ejpam-6375	182	25	]	]	PUNCT
ejpam-6375	182	26	/∈	/∈	PUNCT
ejpam-6375	182	27	gpt	gpt	NOUN
ejpam-6375	182	28	.	.	PUNCT
ejpam-6375	183	1	thus	thus	ADV
ejpam-6375	183	2	,	,	PUNCT
ejpam-6375	183	3	there	there	PRON
ejpam-6375	183	4	exists	exist	VERB
ejpam-6375	183	5	a	a	DET
ejpam-6375	183	6	finite	finite	ADJ
ejpam-6375	183	7	subcover	subcover	PROPN
ejpam-6375	183	8	s0	s0	PROPN
ejpam-6375	183	9	that	that	PRON
ejpam-6375	183	10	covers	cover	VERB
ejpam-6375	183	11	n	n	CCONJ
ejpam-6375	183	12	,	,	PUNCT
ejpam-6375	183	13	proving	prove	VERB
ejpam-6375	183	14	that	that	SCONJ
ejpam-6375	183	15	n	n	PRON
ejpam-6375	183	16	is	be	AUX
ejpam-6375	183	17	gpt	gpt	NOUN
ejpam-6375	183	18	-	-	PUNCT
ejpam-6375	183	19	s∗	s∗	PROPN
ejpam-6375	183	20	g	g	PROPN
ejpam-6375	183	21	-compact	-compact	PROPN
ejpam-6375	183	22	relative	relative	ADJ
ejpam-6375	183	23	to	to	ADP
ejpam-6375	183	24	(	(	PUNCT
ejpam-6375	183	25	v	v	NOUN
ejpam-6375	183	26	,	,	PUNCT
ejpam-6375	183	27	gτ	gτ	INTJ
ejpam-6375	183	28	,	,	PUNCT
ejpam-6375	183	29	p	p	NOUN
ejpam-6375	183	30	)	)	PUNCT
ejpam-6375	183	31	.	.	PUNCT
ejpam-6375	184	1	theorem	theorem	VERB
ejpam-6375	184	2	3.3	3.3	NUM
ejpam-6375	184	3	.	.	PUNCT
ejpam-6375	185	1	consider	consider	VERB
ejpam-6375	185	2	a	a	DET
ejpam-6375	185	3	gpt	gpt	NOUN
ejpam-6375	185	4	-	-	PUNCT
ejpam-6375	185	5	s∗	s∗	PROPN
ejpam-6375	185	6	g	g	PROPN
ejpam-6375	185	7	-continuous	-continuous	ADJ
ejpam-6375	185	8	surjective	surjective	ADJ
ejpam-6375	185	9	map	map	NOUN
ejpam-6375	186	1	f	f	X
ejpam-6375	186	2	:	:	PUNCT
ejpam-6375	186	3	(	(	PUNCT
ejpam-6375	186	4	v	v	NOUN
ejpam-6375	186	5	,	,	PUNCT
ejpam-6375	186	6	gτ	gτ	PROPN
ejpam-6375	186	7	1	1	NUM
ejpam-6375	186	8	,	,	PUNCT
ejpam-6375	186	9	pα	pα	NOUN
ejpam-6375	186	10	)	)	PUNCT
ejpam-6375	186	11	→	→	SYM
ejpam-6375	186	12	(	(	PUNCT
ejpam-6375	186	13	z	z	NOUN
ejpam-6375	186	14	,	,	PUNCT
ejpam-6375	186	15	gτ	gτ	PROPN
ejpam-6375	186	16	2	2	NUM
ejpam-6375	186	17	,	,	PUNCT
ejpam-6375	186	18	pβ	pβ	NOUN
ejpam-6375	186	19	)	)	PUNCT
ejpam-6375	186	20	from	from	ADP
ejpam-6375	186	21	v	v	NUM
ejpam-6375	186	22	to	to	ADP
ejpam-6375	186	23	z.	z.	PROPN
ejpam-6375	186	24	if	if	SCONJ
ejpam-6375	186	25	(	(	PUNCT
ejpam-6375	186	26	v	v	NOUN
ejpam-6375	186	27	,	,	PUNCT
ejpam-6375	186	28	gτ	gτ	PROPN
ejpam-6375	186	29	1	1	NUM
ejpam-6375	186	30	,	,	PUNCT
ejpam-6375	186	31	pα	pα	NOUN
ejpam-6375	186	32	)	)	PUNCT
ejpam-6375	186	33	is	be	AUX
ejpam-6375	186	34	gpt	gpt	NOUN
ejpam-6375	186	35	-	-	PUNCT
ejpam-6375	186	36	s∗	s∗	PROPN
ejpam-6375	186	37	g	g	PROPN
ejpam-6375	186	38	-compact	-compact	NOUN
ejpam-6375	186	39	,	,	PUNCT
ejpam-6375	186	40	then	then	ADV
ejpam-6375	186	41	(	(	PUNCT
ejpam-6375	186	42	z	z	NOUN
ejpam-6375	186	43	,	,	PUNCT
ejpam-6375	186	44	gτ	gτ	PROPN
ejpam-6375	186	45	2	2	NUM
ejpam-6375	186	46	,	,	PUNCT
ejpam-6375	186	47	pβ	pβ	NOUN
ejpam-6375	186	48	)	)	PUNCT
ejpam-6375	186	49	is	be	AUX
ejpam-6375	186	50	gpt	gpt	NOUN
ejpam-6375	186	51	-	-	PUNCT
ejpam-6375	186	52	compact	compact	ADJ
ejpam-6375	186	53	.	.	PUNCT
ejpam-6375	187	1	proof	proof	NOUN
ejpam-6375	187	2	.	.	PUNCT
ejpam-6375	188	1	consider	consider	VERB
ejpam-6375	188	2	{	{	PUNCT
ejpam-6375	188	3	eαi	eαi	NOUN
ejpam-6375	188	4	:	:	PUNCT
ejpam-6375	189	1	i	i	PROPN
ejpam-6375	189	2	∈	∈	PROPN
ejpam-6375	189	3	⋏	⋏	PROPN
ejpam-6375	189	4	}	}	PUNCT
ejpam-6375	189	5	,	,	PUNCT
ejpam-6375	189	6	an	an	DET
ejpam-6375	189	7	open	open	ADJ
ejpam-6375	189	8	cover	cover	NOUN
ejpam-6375	189	9	of	of	ADP
ejpam-6375	189	10	z.	z.	PROPN
ejpam-6375	189	11	as	as	SCONJ
ejpam-6375	189	12	f	f	PROPN
ejpam-6375	189	13	is	be	AUX
ejpam-6375	189	14	gpt	gpt	NOUN
ejpam-6375	189	15	-	-	PUNCT
ejpam-6375	189	16	s∗	s∗	PROPN
ejpam-6375	189	17	g	g	PROPN
ejpam-6375	189	18	-continuous	-continuous	ADJ
ejpam-6375	189	19	implies	implie	NOUN
ejpam-6375	189	20	{	{	PUNCT
ejpam-6375	189	21	f−1	f−1	PROPN
ejpam-6375	189	22	(	(	PUNCT
ejpam-6375	189	23	eαi	eαi	PROPN
ejpam-6375	189	24	)	)	PUNCT
ejpam-6375	189	25	:	:	PUNCT
ejpam-6375	190	1	i	i	PRON
ejpam-6375	190	2	∈	∈	PROPN
ejpam-6375	190	3	⋏	⋏	PROPN
ejpam-6375	190	4	}	}	PUNCT
ejpam-6375	190	5	is	be	AUX
ejpam-6375	190	6	a	a	DET
ejpam-6375	190	7	gpt	gpt	NOUN
ejpam-6375	190	8	-	-	PUNCT
ejpam-6375	190	9	s∗	s∗	NOUN
ejpam-6375	190	10	g	g	PROPN
ejpam-6375	190	11	-open	-open	ADJ
ejpam-6375	190	12	cover	cover	NOUN
ejpam-6375	190	13	of	of	ADP
ejpam-6375	190	14	v.	v.	ADP
ejpam-6375	190	15	furthermore	furthermore	ADV
ejpam-6375	190	16	,	,	PUNCT
ejpam-6375	190	17	there	there	PRON
ejpam-6375	190	18	exists	exist	VERB
ejpam-6375	190	19	a	a	DET
ejpam-6375	190	20	finite	finite	ADJ
ejpam-6375	190	21	gpt	gpt	NOUN
ejpam-6375	190	22	-	-	PUNCT
ejpam-6375	190	23	subcover	subcover	PROPN
ejpam-6375	190	24	{	{	PUNCT
ejpam-6375	190	25	f−1(eα1	f−1(eα1	PROPN
ejpam-6375	190	26	)	)	PUNCT
ejpam-6375	190	27	,	,	PUNCT
ejpam-6375	190	28	f−1(eα2	f−1(eα2	NOUN
ejpam-6375	190	29	)	)	PUNCT
ejpam-6375	190	30	,	,	PUNCT
ejpam-6375	190	31	f−1(eα3	f−1(eα3	NOUN
ejpam-6375	190	32	)	)	PUNCT
ejpam-6375	190	33	,	,	PUNCT
ejpam-6375	190	34	...	...	PUNCT
ejpam-6375	190	35	,	,	PUNCT
ejpam-6375	190	36	f−1(eαn	f−1(eαn	X
ejpam-6375	190	37	)	)	PUNCT
ejpam-6375	190	38	}	}	PUNCT
ejpam-6375	190	39	as	as	SCONJ
ejpam-6375	190	40	v	v	NOUN
ejpam-6375	190	41	is	be	AUX
ejpam-6375	190	42	gpt	gpt	NOUN
ejpam-6375	190	43	-	-	PUNCT
ejpam-6375	190	44	s∗	s∗	PROPN
ejpam-6375	190	45	g	g	PROPN
ejpam-6375	190	46	-compact	-compact	PROPN
ejpam-6375	190	47	.	.	PUNCT
ejpam-6375	191	1	the	the	DET
ejpam-6375	191	2	surjectiveness	surjectiveness	NOUN
ejpam-6375	191	3	of	of	ADP
ejpam-6375	191	4	v	v	PROPN
ejpam-6375	191	5	implies	imply	VERB
ejpam-6375	191	6	{	{	PUNCT
ejpam-6375	191	7	eα1	eα1	X
ejpam-6375	191	8	,	,	PUNCT
ejpam-6375	191	9	eα2	eα2	PROPN
ejpam-6375	191	10	,	,	PUNCT
ejpam-6375	191	11	eα3	eα3	X
ejpam-6375	191	12	,	,	PUNCT
ejpam-6375	191	13	...	...	PUNCT
ejpam-6375	191	14	,	,	PUNCT
ejpam-6375	191	15	eαn	eαn	AUX
ejpam-6375	191	16	}	}	PUNCT
ejpam-6375	191	17	is	be	AUX
ejpam-6375	191	18	a	a	DET
ejpam-6375	191	19	finite	finite	ADJ
ejpam-6375	191	20	gpt	gpt	NOUN
ejpam-6375	191	21	-	-	PUNCT
ejpam-6375	191	22	subcover	subcover	NOUN
ejpam-6375	191	23	of	of	ADP
ejpam-6375	191	24	z	z	PROPN
ejpam-6375	191	25	implies	imply	VERB
ejpam-6375	191	26	z	z	PROPN
ejpam-6375	191	27	is	be	AUX
ejpam-6375	191	28	gpt	gpt	NOUN
ejpam-6375	191	29	-	-	PUNCT
ejpam-6375	191	30	compact	compact	ADJ
ejpam-6375	191	31	.	.	PUNCT
ejpam-6375	192	1	theorem	theorem	VERB
ejpam-6375	192	2	3.4	3.4	NUM
ejpam-6375	192	3	.	.	PUNCT
ejpam-6375	193	1	consider	consider	VERB
ejpam-6375	193	2	a	a	DET
ejpam-6375	193	3	gpt	gpt	NOUN
ejpam-6375	193	4	-	-	PUNCT
ejpam-6375	193	5	s∗	s∗	PROPN
ejpam-6375	193	6	g	g	PROPN
ejpam-6375	193	7	-irresolute	-irresolute	ADJ
ejpam-6375	193	8	surjective	surjective	ADJ
ejpam-6375	193	9	map	map	NOUN
ejpam-6375	194	1	f	f	X
ejpam-6375	194	2	:	:	PUNCT
ejpam-6375	194	3	(	(	PUNCT
ejpam-6375	194	4	v	v	NOUN
ejpam-6375	194	5	,	,	PUNCT
ejpam-6375	194	6	gτ	gτ	PROPN
ejpam-6375	194	7	1	1	NUM
ejpam-6375	194	8	,	,	PUNCT
ejpam-6375	194	9	pα	pα	NOUN
ejpam-6375	194	10	)	)	PUNCT
ejpam-6375	194	11	→	→	SYM
ejpam-6375	194	12	(	(	PUNCT
ejpam-6375	194	13	z	z	NOUN
ejpam-6375	194	14	,	,	PUNCT
ejpam-6375	194	15	gτ	gτ	PROPN
ejpam-6375	194	16	2	2	NUM
ejpam-6375	194	17	,	,	PUNCT
ejpam-6375	194	18	pβ	pβ	NOUN
ejpam-6375	194	19	)	)	PUNCT
ejpam-6375	194	20	from	from	ADP
ejpam-6375	194	21	gpts	gpt	NOUN
ejpam-6375	194	22	v	v	NOUN
ejpam-6375	194	23	into	into	ADP
ejpam-6375	194	24	a	a	DET
ejpam-6375	194	25	gpts	gpt	NOUN
ejpam-6375	194	26	z.	z.	NOUN
ejpam-6375	195	1	if	if	SCONJ
ejpam-6375	195	2	(	(	PUNCT
ejpam-6375	195	3	v	v	NOUN
ejpam-6375	195	4	,	,	PUNCT
ejpam-6375	195	5	gτ	gτ	PROPN
ejpam-6375	195	6	1	1	NUM
ejpam-6375	195	7	,	,	PUNCT
ejpam-6375	195	8	pα	pα	NOUN
ejpam-6375	195	9	)	)	PUNCT
ejpam-6375	195	10	is	be	AUX
ejpam-6375	195	11	gpt	gpt	NOUN
ejpam-6375	195	12	-	-	PUNCT
ejpam-6375	195	13	s∗	s∗	PROPN
ejpam-6375	195	14	g	g	PROPN
ejpam-6375	195	15	-compact	-compact	NOUN
ejpam-6375	195	16	,	,	PUNCT
ejpam-6375	195	17	then	then	ADV
ejpam-6375	195	18	(	(	PUNCT
ejpam-6375	195	19	z	z	NOUN
ejpam-6375	195	20	,	,	PUNCT
ejpam-6375	195	21	gτ	gτ	PROPN
ejpam-6375	195	22	2	2	NUM
ejpam-6375	195	23	,	,	PUNCT
ejpam-6375	195	24	pβ	pβ	NOUN
ejpam-6375	195	25	)	)	PUNCT
ejpam-6375	195	26	is	be	AUX
ejpam-6375	195	27	gpt	gpt	NOUN
ejpam-6375	195	28	-	-	PUNCT
ejpam-6375	195	29	s∗	s∗	PROPN
ejpam-6375	195	30	g	g	PROPN
ejpam-6375	195	31	-compact	-compact	NOUN
ejpam-6375	195	32	.	.	PUNCT
ejpam-6375	196	1	proof	proof	NOUN
ejpam-6375	196	2	.	.	PUNCT
ejpam-6375	197	1	let	let	VERB
ejpam-6375	197	2	{	{	PUNCT
ejpam-6375	197	3	eαi	eαi	NOUN
ejpam-6375	197	4	:	:	PUNCT
ejpam-6375	197	5	i	i	PROPN
ejpam-6375	197	6	∈	∈	PROPN
ejpam-6375	197	7	⋏	⋏	PROPN
ejpam-6375	197	8	}	}	PUNCT
ejpam-6375	197	9	,	,	PUNCT
ejpam-6375	197	10	a	a	DET
ejpam-6375	197	11	gpt	gpt	NOUN
ejpam-6375	197	12	-	-	PUNCT
ejpam-6375	197	13	s∗	s∗	PROPN
ejpam-6375	197	14	g	g	PROPN
ejpam-6375	197	15	-open	-open	ADJ
ejpam-6375	197	16	cover	cover	NOUN
ejpam-6375	197	17	of	of	ADP
ejpam-6375	197	18	z.	z.	PROPN
ejpam-6375	197	19	as	as	SCONJ
ejpam-6375	197	20	f	f	PROPN
ejpam-6375	197	21	is	be	AUX
ejpam-6375	197	22	gpt	gpt	NOUN
ejpam-6375	197	23	-	-	PUNCT
ejpam-6375	197	24	s∗	s∗	PROPN
ejpam-6375	197	25	g	g	PROPN
ejpam-6375	197	26	-irresolute	-irresolute	PROPN
ejpam-6375	197	27	implies	imply	VERB
ejpam-6375	197	28	{	{	PUNCT
ejpam-6375	197	29	f−1	f−1	PROPN
ejpam-6375	197	30	(	(	PUNCT
ejpam-6375	197	31	eαi	eαi	PROPN
ejpam-6375	197	32	)	)	PUNCT
ejpam-6375	197	33	:	:	PUNCT
ejpam-6375	198	1	i	i	PRON
ejpam-6375	198	2	∈	∈	PROPN
ejpam-6375	198	3	⋏	⋏	PROPN
ejpam-6375	198	4	}	}	PUNCT
ejpam-6375	198	5	is	be	AUX
ejpam-6375	198	6	a	a	DET
ejpam-6375	198	7	gpt	gpt	NOUN
ejpam-6375	198	8	-	-	PUNCT
ejpam-6375	198	9	s∗	s∗	NOUN
ejpam-6375	198	10	g	g	PROPN
ejpam-6375	198	11	-open	-open	ADJ
ejpam-6375	198	12	cover	cover	NOUN
ejpam-6375	198	13	of	of	ADP
ejpam-6375	198	14	v.	v.	ADP
ejpam-6375	198	15	furthermore	furthermore	ADV
ejpam-6375	198	16	,	,	PUNCT
ejpam-6375	198	17	there	there	PRON
ejpam-6375	198	18	exists	exist	VERB
ejpam-6375	198	19	a	a	DET
ejpam-6375	198	20	finite	finite	ADJ
ejpam-6375	198	21	gpt	gpt	NOUN
ejpam-6375	198	22	-	-	PUNCT
ejpam-6375	198	23	subcover	subcover	PROPN
ejpam-6375	198	24	{	{	PUNCT
ejpam-6375	198	25	f−1(eα1	f−1(eα1	PROPN
ejpam-6375	198	26	)	)	PUNCT
ejpam-6375	198	27	,	,	PUNCT
ejpam-6375	198	28	f−1(eα2	f−1(eα2	NOUN
ejpam-6375	198	29	)	)	PUNCT
ejpam-6375	198	30	,	,	PUNCT
ejpam-6375	198	31	f−1(eα3	f−1(eα3	NOUN
ejpam-6375	198	32	)	)	PUNCT
ejpam-6375	198	33	,	,	PUNCT
ejpam-6375	198	34	...	...	PUNCT
ejpam-6375	198	35	,	,	PUNCT
ejpam-6375	198	36	f−1(eαn	f−1(eαn	X
ejpam-6375	198	37	)	)	PUNCT
ejpam-6375	198	38	}	}	PUNCT
ejpam-6375	198	39	as	as	SCONJ
ejpam-6375	198	40	v	v	NOUN
ejpam-6375	198	41	is	be	AUX
ejpam-6375	198	42	gpt	gpt	NOUN
ejpam-6375	198	43	-	-	PUNCT
ejpam-6375	198	44	s∗	s∗	NOUN
ejpam-6375	198	45	g	g	PROPN
ejpam-6375	198	46	-compact	-compact	PROPN
ejpam-6375	198	47	.	.	PUNCT
ejpam-6375	199	1	now	now	ADV
ejpam-6375	199	2	,	,	PUNCT
ejpam-6375	199	3	f	f	PROPN
ejpam-6375	199	4	is	be	AUX
ejpam-6375	199	5	onto	onto	ADP
ejpam-6375	199	6	implies	implie	NOUN
ejpam-6375	199	7	{	{	PUNCT
ejpam-6375	199	8	eα1	eα1	X
ejpam-6375	199	9	,	,	PUNCT
ejpam-6375	199	10	eα2	eα2	PROPN
ejpam-6375	199	11	,	,	PUNCT
ejpam-6375	199	12	eα3	eα3	X
ejpam-6375	199	13	,	,	PUNCT
ejpam-6375	199	14	...	...	PUNCT
ejpam-6375	199	15	,	,	PUNCT
ejpam-6375	199	16	eαn	eαn	AUX
ejpam-6375	199	17	}	}	PUNCT
ejpam-6375	199	18	is	be	AUX
ejpam-6375	199	19	a	a	DET
ejpam-6375	199	20	finite	finite	ADJ
ejpam-6375	199	21	gpt	gpt	NOUN
ejpam-6375	199	22	-	-	PUNCT
ejpam-6375	199	23	subcover	subcover	NOUN
ejpam-6375	199	24	of	of	ADP
ejpam-6375	199	25	z	z	PROPN
ejpam-6375	199	26	implies	imply	VERB
ejpam-6375	199	27	z	z	PROPN
ejpam-6375	199	28	is	be	AUX
ejpam-6375	199	29	gpts∗	gpts∗	PROPN
ejpam-6375	199	30	g	g	PROPN
ejpam-6375	199	31	-compact	-compact	PROPN
ejpam-6375	199	32	.	.	PUNCT
ejpam-6375	200	1	m.	m.	NOUN
ejpam-6375	200	2	shahbaz	shahbaz	PROPN
ejpam-6375	200	3	et	et	PROPN
ejpam-6375	200	4	al	al	PROPN
ejpam-6375	200	5	.	.	PUNCT
ejpam-6375	200	6	/	/	SYM
ejpam-6375	200	7	eur	eur	PROPN
ejpam-6375	200	8	.	.	PUNCT
ejpam-6375	201	1	j.	j.	PROPN
ejpam-6375	201	2	pure	pure	PROPN
ejpam-6375	201	3	appl	appl	PROPN
ejpam-6375	201	4	.	.	PROPN
ejpam-6375	201	5	math	math	PROPN
ejpam-6375	201	6	,	,	PUNCT
ejpam-6375	201	7	18	18	NUM
ejpam-6375	201	8	(	(	PUNCT
ejpam-6375	201	9	4	4	NUM
ejpam-6375	201	10	)	)	PUNCT
ejpam-6375	201	11	(	(	PUNCT
ejpam-6375	201	12	2025	2025	NUM
ejpam-6375	201	13	)	)	PUNCT
ejpam-6375	201	14	,	,	PUNCT
ejpam-6375	201	15	6375	6375	NUM
ejpam-6375	201	16	9	9	NUM
ejpam-6375	201	17	of	of	ADP
ejpam-6375	201	18	22	22	NUM
ejpam-6375	201	19	theorem	theorem	VERB
ejpam-6375	201	20	3.5	3.5	NUM
ejpam-6375	201	21	.	.	PUNCT
ejpam-6375	202	1	let	let	VERB
ejpam-6375	202	2	f	f	X
ejpam-6375	202	3	:	:	PUNCT
ejpam-6375	202	4	(	(	PUNCT
ejpam-6375	202	5	v	v	NOUN
ejpam-6375	202	6	,	,	PUNCT
ejpam-6375	202	7	gτ	gτ	PROPN
ejpam-6375	202	8	1	1	NUM
ejpam-6375	202	9	,	,	PUNCT
ejpam-6375	202	10	pα	pα	NOUN
ejpam-6375	202	11	)	)	PUNCT
ejpam-6375	202	12	→	→	SYM
ejpam-6375	202	13	(	(	PUNCT
ejpam-6375	202	14	z	z	NOUN
ejpam-6375	202	15	,	,	PUNCT
ejpam-6375	202	16	gτ	gτ	PROPN
ejpam-6375	202	17	2	2	NUM
ejpam-6375	202	18	,	,	PUNCT
ejpam-6375	202	19	pβ	pβ	NOUN
ejpam-6375	202	20	)	)	PUNCT
ejpam-6375	202	21	a	a	DET
ejpam-6375	202	22	gpt	gpt	NOUN
ejpam-6375	202	23	-	-	PUNCT
ejpam-6375	202	24	s∗	s∗	PROPN
ejpam-6375	202	25	g	g	PROPN
ejpam-6375	202	26	-irresolute	-irresolute	ADJ
ejpam-6375	202	27	map	map	NOUN
ejpam-6375	202	28	and	and	CCONJ
ejpam-6375	202	29	a	a	DET
ejpam-6375	202	30	subset	subset	NOUN
ejpam-6375	202	31	d	d	NOUN
ejpam-6375	202	32	of	of	ADP
ejpam-6375	202	33	(	(	PUNCT
ejpam-6375	202	34	v	v	NOUN
ejpam-6375	202	35	,	,	PUNCT
ejpam-6375	202	36	gτ	gτ	PROPN
ejpam-6375	202	37	1	1	NUM
ejpam-6375	202	38	,	,	PUNCT
ejpam-6375	202	39	pα	pα	NOUN
ejpam-6375	202	40	)	)	PUNCT
ejpam-6375	202	41	is	be	AUX
ejpam-6375	202	42	gpt	gpt	NOUN
ejpam-6375	202	43	-	-	PUNCT
ejpam-6375	202	44	s∗	s∗	PROPN
ejpam-6375	202	45	g	g	PROPN
ejpam-6375	202	46	-compact	-compact	PROPN
ejpam-6375	202	47	relative	relative	ADJ
ejpam-6375	202	48	to	to	ADP
ejpam-6375	202	49	v	v	NOUN
ejpam-6375	202	50	,	,	PUNCT
ejpam-6375	202	51	then	then	ADV
ejpam-6375	202	52	the	the	DET
ejpam-6375	202	53	image	image	NOUN
ejpam-6375	202	54	f(d	f(d	PROPN
ejpam-6375	202	55	)	)	PUNCT
ejpam-6375	202	56	is	be	AUX
ejpam-6375	202	57	gpt	gpt	NOUN
ejpam-6375	202	58	-	-	PUNCT
ejpam-6375	202	59	s∗	s∗	PROPN
ejpam-6375	202	60	g	g	PROPN
ejpam-6375	202	61	-compact	-compact	PROPN
ejpam-6375	202	62	relative	relative	ADJ
ejpam-6375	202	63	to	to	ADP
ejpam-6375	202	64	z.	z.	PROPN
ejpam-6375	202	65	proof	proof	NOUN
ejpam-6375	202	66	.	.	PUNCT
ejpam-6375	203	1	any	any	DET
ejpam-6375	203	2	collection	collection	NOUN
ejpam-6375	203	3	of	of	ADP
ejpam-6375	203	4	gpt	gpt	NOUN
ejpam-6375	203	5	-	-	PUNCT
ejpam-6375	203	6	s∗	s∗	PROPN
ejpam-6375	203	7	g	g	PROPN
ejpam-6375	203	8	-open	-open	PROPN
ejpam-6375	203	9	subsets	subset	NOUN
ejpam-6375	203	10	{	{	PUNCT
ejpam-6375	203	11	eαi	eαi	NOUN
ejpam-6375	203	12	:	:	PUNCT
ejpam-6375	203	13	i	i	PRON
ejpam-6375	203	14	∈	∈	PROPN
ejpam-6375	203	15	⋏	⋏	PROPN
ejpam-6375	203	16	}	}	PUNCT
ejpam-6375	203	17	of	of	ADP
ejpam-6375	203	18	z	z	NOUN
ejpam-6375	203	19	containing	contain	VERB
ejpam-6375	203	20	the	the	DET
ejpam-6375	203	21	image	image	NOUN
ejpam-6375	203	22	of	of	ADP
ejpam-6375	203	23	d	d	PROPN
ejpam-6375	203	24	under	under	ADP
ejpam-6375	203	25	the	the	DET
ejpam-6375	203	26	mapping	mapping	NOUN
ejpam-6375	203	27	f	f	NOUN
ejpam-6375	203	28	becomes	become	VERB
ejpam-6375	203	29	part	part	NOUN
ejpam-6375	203	30	of	of	ADP
ejpam-6375	203	31	the	the	DET
ejpam-6375	203	32	union	union	NOUN
ejpam-6375	203	33	function	function	NOUN
ejpam-6375	203	34	.	.	PUNCT
ejpam-6375	204	1	then	then	ADV
ejpam-6375	204	2	,	,	PUNCT
ejpam-6375	204	3	d	d	PROPN
ejpam-6375	204	4	⊂	⊂	PROPN
ejpam-6375	204	5	∪	∪	X
ejpam-6375	204	6	{	{	PUNCT
ejpam-6375	204	7	f−1(eαi	f−1(eαi	PROPN
ejpam-6375	204	8	)	)	PUNCT
ejpam-6375	204	9	:	:	PUNCT
ejpam-6375	204	10	i	i	PRON
ejpam-6375	204	11	∈	∈	PROPN
ejpam-6375	204	12	⋏	⋏	PROPN
ejpam-6375	204	13	}	}	PUNCT
ejpam-6375	204	14	holds	hold	VERB
ejpam-6375	204	15	.	.	PUNCT
ejpam-6375	205	1	d	d	NOUN
ejpam-6375	205	2	is	be	AUX
ejpam-6375	205	3	gpt	gpt	NOUN
ejpam-6375	205	4	-	-	PUNCT
ejpam-6375	205	5	s∗	s∗	PROPN
ejpam-6375	205	6	g	g	PROPN
ejpam-6375	205	7	-compact	-compact	PROPN
ejpam-6375	205	8	relative	relative	ADJ
ejpam-6375	205	9	to	to	ADP
ejpam-6375	205	10	v	v	NOUN
ejpam-6375	205	11	by	by	ADP
ejpam-6375	205	12	hypothesis	hypothesis	NOUN
ejpam-6375	205	13	,	,	PUNCT
ejpam-6375	205	14	then	then	ADV
ejpam-6375	205	15	there	there	PRON
ejpam-6375	205	16	exists	exist	VERB
ejpam-6375	205	17	⋏o	⋏o	PROPN
ejpam-6375	205	18	a	a	DET
ejpam-6375	205	19	finite	finite	NOUN
ejpam-6375	205	20	subset	subset	NOUN
ejpam-6375	205	21	of	of	ADP
ejpam-6375	205	22	⋏	⋏	PROPN
ejpam-6375	206	1	such	such	ADJ
ejpam-6375	206	2	that	that	SCONJ
ejpam-6375	206	3	d	d	X
ejpam-6375	206	4	⊂	⊂	PROPN
ejpam-6375	206	5	∪	∪	X
ejpam-6375	206	6	{	{	PUNCT
ejpam-6375	206	7	f−1(eαi	f−1(eαi	PROPN
ejpam-6375	206	8	)	)	PUNCT
ejpam-6375	206	9	:	:	PUNCT
ejpam-6375	207	1	i	i	PRON
ejpam-6375	207	2	∈	∈	VERB
ejpam-6375	207	3	⋏o	⋏o	PROPN
ejpam-6375	207	4	}	}	PUNCT
ejpam-6375	207	5	implies	imply	VERB
ejpam-6375	207	6	f(d	f(d	NOUN
ejpam-6375	207	7	)	)	PUNCT
ejpam-6375	207	8	⊂	⊂	PROPN
ejpam-6375	207	9	∪	∪	X
ejpam-6375	207	10	{	{	PUNCT
ejpam-6375	207	11	eαi	eαi	NOUN
ejpam-6375	207	12	:	:	PUNCT
ejpam-6375	207	13	i	i	PRON
ejpam-6375	207	14	∈	∈	VERB
ejpam-6375	207	15	⋏o	⋏o	PROPN
ejpam-6375	207	16	}	}	PUNCT
ejpam-6375	207	17	.	.	PUNCT
ejpam-6375	208	1	thus	thus	ADV
ejpam-6375	208	2	,	,	PUNCT
ejpam-6375	208	3	f(d	f(d	PROPN
ejpam-6375	208	4	)	)	PUNCT
ejpam-6375	208	5	is	be	AUX
ejpam-6375	208	6	gpt	gpt	NOUN
ejpam-6375	208	7	-	-	PUNCT
ejpam-6375	208	8	s∗	s∗	PROPN
ejpam-6375	208	9	g	g	PROPN
ejpam-6375	208	10	-compact	-compact	PROPN
ejpam-6375	208	11	relative	relative	ADJ
ejpam-6375	208	12	to	to	ADP
ejpam-6375	208	13	z.	z.	PROPN
ejpam-6375	208	14	theorem	theorem	PROPN
ejpam-6375	208	15	3.6	3.6	NUM
ejpam-6375	208	16	.	.	PUNCT
ejpam-6375	209	1	if	if	SCONJ
ejpam-6375	209	2	a	a	DET
ejpam-6375	209	3	surjective	surjective	ADJ
ejpam-6375	209	4	map	map	NOUN
ejpam-6375	209	5	f	f	X
ejpam-6375	209	6	:	:	PUNCT
ejpam-6375	209	7	(	(	PUNCT
ejpam-6375	209	8	v	v	NOUN
ejpam-6375	209	9	,	,	PUNCT
ejpam-6375	209	10	gτ	gτ	PROPN
ejpam-6375	209	11	1	1	NUM
ejpam-6375	209	12	,	,	PUNCT
ejpam-6375	209	13	pα	pα	NOUN
ejpam-6375	209	14	)	)	PUNCT
ejpam-6375	209	15	→	→	SYM
ejpam-6375	209	16	(	(	PUNCT
ejpam-6375	209	17	z	z	NOUN
ejpam-6375	209	18	,	,	PUNCT
ejpam-6375	209	19	gτ	gτ	PROPN
ejpam-6375	209	20	2	2	NUM
ejpam-6375	209	21	,	,	PUNCT
ejpam-6375	209	22	pβ	pβ	NOUN
ejpam-6375	209	23	)	)	PUNCT
ejpam-6375	209	24	is	be	AUX
ejpam-6375	209	25	strongly	strongly	ADV
ejpam-6375	209	26	gpt	gpt	NOUN
ejpam-6375	209	27	-	-	PUNCT
ejpam-6375	209	28	s∗	s∗	PROPN
ejpam-6375	209	29	g	g	NOUN
ejpam-6375	209	30	continuous	continuous	ADJ
ejpam-6375	209	31	and	and	CCONJ
ejpam-6375	209	32	(	(	PUNCT
ejpam-6375	209	33	v	v	NOUN
ejpam-6375	209	34	,	,	PUNCT
ejpam-6375	209	35	gτ	gτ	PROPN
ejpam-6375	209	36	1	1	NUM
ejpam-6375	209	37	,	,	PUNCT
ejpam-6375	209	38	pα	pα	NOUN
ejpam-6375	209	39	)	)	PUNCT
ejpam-6375	209	40	is	be	AUX
ejpam-6375	209	41	gpt	gpt	NOUN
ejpam-6375	209	42	-	-	PUNCT
ejpam-6375	209	43	compact	compact	ADJ
ejpam-6375	209	44	,	,	PUNCT
ejpam-6375	209	45	then	then	ADV
ejpam-6375	209	46	(	(	PUNCT
ejpam-6375	209	47	z	z	X
ejpam-6375	209	48	,	,	PUNCT
ejpam-6375	209	49	gτ	gτ	PROPN
ejpam-6375	209	50	2	2	NUM
ejpam-6375	209	51	,	,	PUNCT
ejpam-6375	209	52	pβ	pβ	NOUN
ejpam-6375	209	53	)	)	PUNCT
ejpam-6375	209	54	is	be	AUX
ejpam-6375	209	55	gpt	gpt	NOUN
ejpam-6375	209	56	-	-	PUNCT
ejpam-6375	209	57	s∗	s∗	PROPN
ejpam-6375	209	58	g	g	PROPN
ejpam-6375	209	59	-compact	-compact	NOUN
ejpam-6375	209	60	.	.	PUNCT
ejpam-6375	210	1	proof	proof	NOUN
ejpam-6375	210	2	.	.	PUNCT
ejpam-6375	211	1	let	let	VERB
ejpam-6375	211	2	a	a	DET
ejpam-6375	211	3	gpt	gpt	NOUN
ejpam-6375	211	4	-	-	PUNCT
ejpam-6375	211	5	s∗	s∗	NOUN
ejpam-6375	211	6	g	g	PROPN
ejpam-6375	211	7	-open	-open	ADJ
ejpam-6375	211	8	cover	cover	NOUN
ejpam-6375	211	9	{	{	PUNCT
ejpam-6375	211	10	eαi	eαi	NOUN
ejpam-6375	211	11	:	:	PUNCT
ejpam-6375	211	12	i	i	PROPN
ejpam-6375	211	13	∈	∈	PROPN
ejpam-6375	211	14	⋏	⋏	PROPN
ejpam-6375	211	15	}	}	PUNCT
ejpam-6375	211	16	of	of	ADP
ejpam-6375	211	17	z.	z.	PROPN
ejpam-6375	211	18	as	as	SCONJ
ejpam-6375	211	19	f	f	PROPN
ejpam-6375	211	20	exhibits	exhibit	VERB
ejpam-6375	211	21	strong	strong	ADJ
ejpam-6375	211	22	gpt	gpt	NOUN
ejpam-6375	211	23	-	-	PUNCT
ejpam-6375	211	24	s∗	s∗	PROPN
ejpam-6375	211	25	g	g	NOUN
ejpam-6375	211	26	continuity	continuity	NOUN
ejpam-6375	211	27	it	it	PRON
ejpam-6375	211	28	implies	imply	VERB
ejpam-6375	211	29	that	that	SCONJ
ejpam-6375	211	30	{	{	PUNCT
ejpam-6375	211	31	f−1	f−1	PROPN
ejpam-6375	211	32	(	(	PUNCT
ejpam-6375	211	33	eαi	eαi	PROPN
ejpam-6375	211	34	)	)	PUNCT
ejpam-6375	211	35	:	:	PUNCT
ejpam-6375	212	1	i	i	PRON
ejpam-6375	212	2	∈	∈	PROPN
ejpam-6375	212	3	⋏	⋏	PROPN
ejpam-6375	212	4	}	}	PUNCT
ejpam-6375	212	5	creates	create	VERB
ejpam-6375	212	6	an	an	DET
ejpam-6375	212	7	open	open	ADJ
ejpam-6375	212	8	cover	cover	NOUN
ejpam-6375	212	9	of	of	ADP
ejpam-6375	212	10	v.	v.	ADV
ejpam-6375	212	11	since	since	SCONJ
ejpam-6375	212	12	v	v	NOUN
ejpam-6375	212	13	has	have	VERB
ejpam-6375	212	14	the	the	DET
ejpam-6375	212	15	gpt	gpt	NOUN
ejpam-6375	212	16	-	-	PUNCT
ejpam-6375	212	17	compactness	compactness	NOUN
ejpam-6375	212	18	property	property	NOUN
ejpam-6375	212	19	it	it	PRON
ejpam-6375	212	20	contains	contain	VERB
ejpam-6375	212	21	the	the	DET
ejpam-6375	212	22	finite	finite	PROPN
ejpam-6375	212	23	subcover	subcover	PROPN
ejpam-6375	212	24	{	{	PUNCT
ejpam-6375	212	25	f−1(eα1	f−1(eα1	PROPN
ejpam-6375	212	26	)	)	PUNCT
ejpam-6375	212	27	,	,	PUNCT
ejpam-6375	212	28	f−1(eα2	f−1(eα2	NOUN
ejpam-6375	212	29	)	)	PUNCT
ejpam-6375	212	30	,	,	PUNCT
ejpam-6375	212	31	f−1(eα3	f−1(eα3	NOUN
ejpam-6375	212	32	)	)	PUNCT
ejpam-6375	212	33	,	,	PUNCT
ejpam-6375	212	34	...	...	PUNCT
ejpam-6375	212	35	,	,	PUNCT
ejpam-6375	212	36	f−1(eαn	f−1(eαn	X
ejpam-6375	212	37	)	)	PUNCT
ejpam-6375	212	38	}	}	PUNCT
ejpam-6375	212	39	.	.	PUNCT
ejpam-6375	213	1	the	the	DET
ejpam-6375	213	2	surjectiveness	surjectiveness	NOUN
ejpam-6375	213	3	of	of	ADP
ejpam-6375	213	4	map	map	NOUN
ejpam-6375	213	5	f	f	PROPN
ejpam-6375	213	6	leads	lead	VERB
ejpam-6375	213	7	to	to	ADP
ejpam-6375	213	8	the	the	DET
ejpam-6375	213	9	finite	finite	PROPN
ejpam-6375	213	10	subcover	subcover	PROPN
ejpam-6375	213	11	{	{	PUNCT
ejpam-6375	213	12	eα1	eα1	X
ejpam-6375	213	13	,	,	PUNCT
ejpam-6375	213	14	eα2	eα2	PROPN
ejpam-6375	213	15	,	,	PUNCT
ejpam-6375	213	16	eα3	eα3	X
ejpam-6375	213	17	,	,	PUNCT
ejpam-6375	213	18	...	...	PUNCT
ejpam-6375	213	19	,	,	PUNCT
ejpam-6375	213	20	eαn	eαn	PROPN
ejpam-6375	213	21	}	}	PUNCT
ejpam-6375	213	22	of	of	ADP
ejpam-6375	213	23	collection	collection	NOUN
ejpam-6375	213	24	z	z	NOUN
ejpam-6375	213	25	which	which	PRON
ejpam-6375	213	26	makes	make	VERB
ejpam-6375	213	27	z	z	NOUN
ejpam-6375	213	28	gpt	gpt	VERB
ejpam-6375	213	29	-	-	PUNCT
ejpam-6375	213	30	s∗	s∗	PROPN
ejpam-6375	213	31	g	g	PROPN
ejpam-6375	213	32	-compact	-compact	PROPN
ejpam-6375	213	33	.	.	PUNCT
ejpam-6375	214	1	example	example	NOUN
ejpam-6375	214	2	3.3	3.3	NUM
ejpam-6375	214	3	.	.	PUNCT
ejpam-6375	215	1	let	let	VERB
ejpam-6375	215	2	v	v	PART
ejpam-6375	215	3	be	be	AUX
ejpam-6375	215	4	the	the	DET
ejpam-6375	215	5	set	set	NOUN
ejpam-6375	215	6	of	of	ADP
ejpam-6375	215	7	all	all	DET
ejpam-6375	215	8	bounded	bounded	ADJ
ejpam-6375	215	9	spherical	spherical	ADJ
ejpam-6375	215	10	regions	region	NOUN
ejpam-6375	215	11	in	in	ADP
ejpam-6375	215	12	r3	r3	PROPN
ejpam-6375	215	13	,	,	PUNCT
ejpam-6375	215	14	where	where	SCONJ
ejpam-6375	215	15	a	a	DET
ejpam-6375	215	16	spherical	spherical	ADJ
ejpam-6375	215	17	region	region	NOUN
ejpam-6375	215	18	s	s	VERB
ejpam-6375	215	19	is	be	AUX
ejpam-6375	215	20	defined	define	VERB
ejpam-6375	215	21	as	as	ADP
ejpam-6375	215	22	:	:	PUNCT
ejpam-6375	215	23	s	s	PART
ejpam-6375	215	24	=	=	PUNCT
ejpam-6375	215	25	{	{	PUNCT
ejpam-6375	215	26	(	(	PUNCT
ejpam-6375	215	27	l	l	NOUN
ejpam-6375	215	28	,	,	PUNCT
ejpam-6375	215	29	m	m	PROPN
ejpam-6375	215	30	,	,	PUNCT
ejpam-6375	215	31	n	n	CCONJ
ejpam-6375	215	32	)	)	PUNCT
ejpam-6375	215	33	∈	∈	PROPN
ejpam-6375	215	34	r3	r3	PROPN
ejpam-6375	215	35	|	|	ADV
ejpam-6375	215	36	√	√	PROPN
ejpam-6375	215	37	(	(	PUNCT
ejpam-6375	215	38	l	l	NOUN
ejpam-6375	215	39	−	−	PROPN
ejpam-6375	215	40	a)2	a)2	NOUN
ejpam-6375	215	41	+	+	CCONJ
ejpam-6375	215	42	(	(	PUNCT
ejpam-6375	215	43	m−	m−	PROPN
ejpam-6375	215	44	b)2	b)2	PROPN
ejpam-6375	216	1	+	+	CCONJ
ejpam-6375	216	2	(	(	PUNCT
ejpam-6375	216	3	n−	n−	NOUN
ejpam-6375	216	4	c)2	c)2	VERB
ejpam-6375	216	5	≤	≤	ADJ
ejpam-6375	216	6	r	r	NOUN
ejpam-6375	216	7	}	}	PUNCT
ejpam-6375	216	8	,	,	PUNCT
ejpam-6375	216	9	with	with	ADP
ejpam-6375	216	10	r	r	NOUN
ejpam-6375	216	11	>	>	X
ejpam-6375	216	12	0	0	PUNCT
ejpam-6375	217	1	as	as	ADP
ejpam-6375	217	2	the	the	DET
ejpam-6375	217	3	radius	radius	NOUN
ejpam-6375	217	4	of	of	ADP
ejpam-6375	217	5	the	the	DET
ejpam-6375	217	6	spherical	spherical	ADJ
ejpam-6375	217	7	region	region	NOUN
ejpam-6375	217	8	and	and	CCONJ
ejpam-6375	217	9	a	a	DET
ejpam-6375	217	10	,	,	PUNCT
ejpam-6375	217	11	b	b	NOUN
ejpam-6375	217	12	,	,	PUNCT
ejpam-6375	217	13	c	c	PROPN
ejpam-6375	217	14	∈	∈	PROPN
ejpam-6375	217	15	r	r	NOUN
ejpam-6375	217	16	,	,	PUNCT
ejpam-6375	217	17	τg1	τg1	NOUN
ejpam-6375	217	18	=	=	PRON
ejpam-6375	217	19	{	{	PUNCT
ejpam-6375	217	20	e	e	PROPN
ejpam-6375	217	21	⊆	⊆	NUM
ejpam-6375	217	22	v	v	ADP
ejpam-6375	217	23	|	|	NOUN
ejpam-6375	217	24	e	e	NOUN
ejpam-6375	217	25	⊆	⊆	NUM
ejpam-6375	217	26	v	v	ADP
ejpam-6375	217	27	\	\	NOUN
ejpam-6375	217	28	{	{	PUNCT
ejpam-6375	217	29	s	s	X
ejpam-6375	217	30	|	|	ADV
ejpam-6375	217	31	radius(s	radius(s	NOUN
ejpam-6375	217	32	)	)	PUNCT
ejpam-6375	218	1	=	=	SYM
ejpam-6375	218	2	r	r	X
ejpam-6375	218	3	}	}	PUNCT
ejpam-6375	218	4	for	for	ADP
ejpam-6375	218	5	some	some	DET
ejpam-6375	218	6	fixed	fix	VERB
ejpam-6375	218	7	r	r	NOUN
ejpam-6375	218	8	>	>	X
ejpam-6375	218	9	0	0	NUM
ejpam-6375	218	10	}	}	PUNCT
ejpam-6375	218	11	(	(	PUNCT
ejpam-6375	218	12	see	see	VERB
ejpam-6375	218	13	example	example	NOUN
ejpam-6375	218	14	3.1	3.1	NUM
ejpam-6375	218	15	)	)	PUNCT
ejpam-6375	218	16	and	and	CCONJ
ejpam-6375	218	17	define	define	VERB
ejpam-6375	218	18	a	a	DET
ejpam-6375	218	19	collection	collection	NOUN
ejpam-6375	218	20	pα	pα	NOUN
ejpam-6375	218	21	⊆	⊆	NUM
ejpam-6375	218	22	2v	2v	NUM
ejpam-6375	218	23	as	as	SCONJ
ejpam-6375	218	24	:	:	PUNCT
ejpam-6375	218	25	pα	pα	INTJ
ejpam-6375	218	26	=	=	PUNCT
ejpam-6375	218	27	{	{	PUNCT
ejpam-6375	218	28	e	e	PROPN
ejpam-6375	218	29	⊆	⊆	NUM
ejpam-6375	218	30	v	v	ADP
ejpam-6375	218	31	|	|	ADV
ejpam-6375	218	32	all	all	DET
ejpam-6375	218	33	spherical	spherical	ADJ
ejpam-6375	218	34	region	region	NOUN
ejpam-6375	218	35	in	in	ADP
ejpam-6375	218	36	e	e	PROPN
ejpam-6375	218	37	have	have	AUX
ejpam-6375	218	38	radius	radiu	VERB
ejpam-6375	218	39	strictly	strictly	ADV
ejpam-6375	218	40	less	less	ADJ
ejpam-6375	218	41	than	than	ADP
ejpam-6375	218	42	r	r	NOUN
ejpam-6375	218	43	,	,	PUNCT
ejpam-6375	218	44	where	where	SCONJ
ejpam-6375	218	45	r	r	NOUN
ejpam-6375	218	46	>	>	X
ejpam-6375	218	47	0	0	NUM
ejpam-6375	218	48	}	}	PUNCT
ejpam-6375	218	49	.	.	PUNCT
ejpam-6375	219	1	let	let	VERB
ejpam-6375	219	2	z	z	NOUN
ejpam-6375	219	3	be	be	AUX
ejpam-6375	219	4	the	the	DET
ejpam-6375	219	5	set	set	NOUN
ejpam-6375	219	6	of	of	ADP
ejpam-6375	219	7	all	all	DET
ejpam-6375	219	8	bounded	bounded	ADJ
ejpam-6375	219	9	circular	circular	ADJ
ejpam-6375	219	10	regions	region	NOUN
ejpam-6375	219	11	in	in	ADP
ejpam-6375	219	12	r2	r2	PROPN
ejpam-6375	219	13	,	,	PUNCT
ejpam-6375	219	14	where	where	SCONJ
ejpam-6375	219	15	a	a	DET
ejpam-6375	219	16	circular	circular	ADJ
ejpam-6375	219	17	region	region	NOUN
ejpam-6375	219	18	c	c	NOUN
ejpam-6375	219	19	is	be	AUX
ejpam-6375	219	20	defined	define	VERB
ejpam-6375	219	21	as	as	ADP
ejpam-6375	219	22	:	:	PUNCT
ejpam-6375	219	23	c	c	X
ejpam-6375	219	24	=	=	SYM
ejpam-6375	219	25	{	{	PUNCT
ejpam-6375	219	26	(	(	PUNCT
ejpam-6375	219	27	l	l	NOUN
ejpam-6375	219	28	,	,	PUNCT
ejpam-6375	219	29	m	m	NOUN
ejpam-6375	219	30	)	)	PUNCT
ejpam-6375	219	31	∈	∈	PROPN
ejpam-6375	219	32	r2	r2	NOUN
ejpam-6375	219	33	|	|	ADV
ejpam-6375	219	34	√	√	PROPN
ejpam-6375	220	1	(	(	PUNCT
ejpam-6375	220	2	l	l	NOUN
ejpam-6375	220	3	−	−	PROPN
ejpam-6375	220	4	a)2	a)2	NOUN
ejpam-6375	220	5	+	+	CCONJ
ejpam-6375	220	6	(	(	PUNCT
ejpam-6375	220	7	m−	m−	PROPN
ejpam-6375	220	8	b)2	b)2	PROPN
ejpam-6375	220	9	≤	≤	PROPN
ejpam-6375	220	10	r1	r1	PROPN
ejpam-6375	220	11	}	}	PUNCT
ejpam-6375	220	12	,	,	PUNCT
ejpam-6375	220	13	with	with	ADP
ejpam-6375	220	14	(	(	PUNCT
ejpam-6375	220	15	a	a	DET
ejpam-6375	220	16	,	,	PUNCT
ejpam-6375	220	17	b	b	NOUN
ejpam-6375	220	18	)	)	PUNCT
ejpam-6375	220	19	is	be	AUX
ejpam-6375	220	20	the	the	DET
ejpam-6375	220	21	center	center	NOUN
ejpam-6375	220	22	of	of	ADP
ejpam-6375	220	23	the	the	DET
ejpam-6375	220	24	circular	circular	ADJ
ejpam-6375	220	25	region	region	NOUN
ejpam-6375	220	26	r1	r1	PROPN
ejpam-6375	220	27	>	>	X
ejpam-6375	220	28	0	0	PUNCT
ejpam-6375	221	1	as	as	ADP
ejpam-6375	221	2	the	the	DET
ejpam-6375	221	3	radius	radius	NOUN
ejpam-6375	221	4	of	of	ADP
ejpam-6375	221	5	the	the	DET
ejpam-6375	221	6	circular	circular	ADJ
ejpam-6375	221	7	region	region	NOUN
ejpam-6375	221	8	and	and	CCONJ
ejpam-6375	221	9	a	a	DET
ejpam-6375	221	10	,	,	PUNCT
ejpam-6375	221	11	b	b	X
ejpam-6375	221	12	∈	∈	PROPN
ejpam-6375	221	13	r	r	NOUN
ejpam-6375	221	14	,	,	PUNCT
ejpam-6375	221	15	τg2	τg2	X
ejpam-6375	221	16	=	=	SYM
ejpam-6375	221	17	{	{	PUNCT
ejpam-6375	221	18	e	e	PROPN
ejpam-6375	221	19	⊆	⊆	NUM
ejpam-6375	221	20	v	v	ADP
ejpam-6375	221	21	|	|	NOUN
ejpam-6375	221	22	e	e	NOUN
ejpam-6375	221	23	⊆	⊆	NUM
ejpam-6375	221	24	v	v	ADP
ejpam-6375	221	25	\	\	NOUN
ejpam-6375	221	26	{	{	PUNCT
ejpam-6375	221	27	c	c	NOUN
ejpam-6375	221	28	|	|	ADV
ejpam-6375	221	29	radius(c	radius(c	ADJ
ejpam-6375	221	30	)	)	PUNCT
ejpam-6375	221	31	=	=	SYM
ejpam-6375	221	32	r1	r1	PROPN
ejpam-6375	221	33	}	}	PUNCT
ejpam-6375	221	34	for	for	ADP
ejpam-6375	221	35	some	some	DET
ejpam-6375	221	36	fixed	fix	VERB
ejpam-6375	221	37	r1	r1	NOUN
ejpam-6375	221	38	>	>	X
ejpam-6375	221	39	0	0	NUM
ejpam-6375	221	40	}	}	PUNCT
ejpam-6375	221	41	is	be	AUX
ejpam-6375	221	42	generalized	generalized	ADJ
ejpam-6375	221	43	topology	topology	NOUN
ejpam-6375	221	44	,	,	PUNCT
ejpam-6375	221	45	similarly	similarly	ADV
ejpam-6375	221	46	step	step	VERB
ejpam-6375	221	47	(	(	PUNCT
ejpam-6375	221	48	see	see	VERB
ejpam-6375	221	49	example	example	NOUN
ejpam-6375	221	50	3.1	3.1	NUM
ejpam-6375	221	51	)	)	PUNCT
ejpam-6375	221	52	and	and	CCONJ
ejpam-6375	221	53	pβ	pβ	ADV
ejpam-6375	221	54	be	be	AUX
ejpam-6375	221	55	a	a	DET
ejpam-6375	221	56	primal	primal	NOUN
ejpam-6375	221	57	on	on	ADP
ejpam-6375	221	58	z	z	NOUN
ejpam-6375	221	59	such	such	ADJ
ejpam-6375	221	60	that	that	SCONJ
ejpam-6375	221	61	pβ	pβ	ADV
ejpam-6375	221	62	=	=	PRON
ejpam-6375	221	63	{	{	PUNCT
ejpam-6375	221	64	e	e	PROPN
ejpam-6375	221	65	⊆	⊆	NUM
ejpam-6375	221	66	v	v	ADP
ejpam-6375	221	67	|	|	ADV
ejpam-6375	221	68	e	e	NOUN
ejpam-6375	221	69	does	do	AUX
ejpam-6375	221	70	not	not	PART
ejpam-6375	221	71	contain	contain	VERB
ejpam-6375	221	72	any	any	DET
ejpam-6375	221	73	point	point	NOUN
ejpam-6375	221	74	on	on	ADP
ejpam-6375	221	75	the	the	DET
ejpam-6375	221	76	boundary	boundary	NOUN
ejpam-6375	221	77	}	}	PUNCT
ejpam-6375	221	78	.	.	PUNCT
ejpam-6375	222	1	a	a	DET
ejpam-6375	222	2	surjective	surjective	ADJ
ejpam-6375	222	3	map	map	NOUN
ejpam-6375	222	4	f	f	NOUN
ejpam-6375	222	5	:	:	PUNCT
ejpam-6375	222	6	v	v	PROPN
ejpam-6375	222	7	→	→	SYM
ejpam-6375	222	8	z	z	NOUN
ejpam-6375	222	9	from	from	ADP
ejpam-6375	222	10	a	a	DET
ejpam-6375	222	11	spherical	spherical	ADJ
ejpam-6375	222	12	region	region	NOUN
ejpam-6375	222	13	to	to	ADP
ejpam-6375	222	14	a	a	DET
ejpam-6375	222	15	circular	circular	ADJ
ejpam-6375	222	16	region	region	NOUN
ejpam-6375	222	17	is	be	AUX
ejpam-6375	222	18	defined	define	VERB
ejpam-6375	222	19	as	as	ADP
ejpam-6375	222	20	:	:	PUNCT
ejpam-6375	222	21	h(l	h(l	NUM
ejpam-6375	222	22	,	,	PUNCT
ejpam-6375	222	23	m	m	NOUN
ejpam-6375	222	24	,	,	PUNCT
ejpam-6375	222	25	n	n	CCONJ
ejpam-6375	222	26	)	)	PUNCT
ejpam-6375	222	27	=	=	SYM
ejpam-6375	223	1	(	(	PUNCT
ejpam-6375	223	2	r	r	NOUN
ejpam-6375	223	3	r1	r1	PROPN
ejpam-6375	223	4	l	l	PROPN
ejpam-6375	223	5	,	,	PUNCT
ejpam-6375	223	6	r	r	PROPN
ejpam-6375	223	7	r1	r1	PROPN
ejpam-6375	223	8	m	m	PROPN
ejpam-6375	223	9	)	)	PUNCT
ejpam-6375	223	10	.	.	PUNCT
ejpam-6375	224	1	m.	m.	PROPN
ejpam-6375	224	2	shahbaz	shahbaz	PROPN
ejpam-6375	224	3	et	et	PROPN
ejpam-6375	224	4	al	al	PROPN
ejpam-6375	224	5	.	.	PUNCT
ejpam-6375	224	6	/	/	SYM
ejpam-6375	224	7	eur	eur	PROPN
ejpam-6375	224	8	.	.	PUNCT
ejpam-6375	225	1	j.	j.	PROPN
ejpam-6375	225	2	pure	pure	PROPN
ejpam-6375	225	3	appl	appl	PROPN
ejpam-6375	225	4	.	.	PROPN
ejpam-6375	225	5	math	math	PROPN
ejpam-6375	225	6	,	,	PUNCT
ejpam-6375	225	7	18	18	NUM
ejpam-6375	225	8	(	(	PUNCT
ejpam-6375	225	9	4	4	NUM
ejpam-6375	225	10	)	)	PUNCT
ejpam-6375	225	11	(	(	PUNCT
ejpam-6375	225	12	2025	2025	NUM
ejpam-6375	225	13	)	)	PUNCT
ejpam-6375	225	14	,	,	PUNCT
ejpam-6375	225	15	6375	6375	NUM
ejpam-6375	225	16	10	10	NUM
ejpam-6375	225	17	of	of	ADP
ejpam-6375	225	18	22	22	NUM
ejpam-6375	225	19	this	this	DET
ejpam-6375	225	20	map	map	NOUN
ejpam-6375	225	21	is	be	AUX
ejpam-6375	225	22	surjective	surjective	ADJ
ejpam-6375	225	23	,	,	PUNCT
ejpam-6375	225	24	as	as	SCONJ
ejpam-6375	225	25	every	every	DET
ejpam-6375	225	26	element	element	NOUN
ejpam-6375	225	27	of	of	ADP
ejpam-6375	225	28	z	z	PROPN
ejpam-6375	225	29	has	have	VERB
ejpam-6375	225	30	a	a	DET
ejpam-6375	225	31	pre	pre	NOUN
ejpam-6375	225	32	-	-	NOUN
ejpam-6375	225	33	image	image	NOUN
ejpam-6375	225	34	in	in	ADP
ejpam-6375	225	35	v.	v.	ADP
ejpam-6375	225	36	assume	assume	VERB
ejpam-6375	225	37	that	that	SCONJ
ejpam-6375	225	38	f	f	PROPN
ejpam-6375	225	39	is	be	AUX
ejpam-6375	225	40	strongly	strongly	ADV
ejpam-6375	225	41	gpt	gpt	NOUN
ejpam-6375	225	42	-	-	PUNCT
ejpam-6375	225	43	s∗	s∗	NOUN
ejpam-6375	225	44	g	g	PROPN
ejpam-6375	225	45	-continuous	-continuous	ADJ
ejpam-6375	225	46	,	,	PUNCT
ejpam-6375	225	47	meaning	mean	VERB
ejpam-6375	225	48	it	it	PRON
ejpam-6375	225	49	preserves	preserve	VERB
ejpam-6375	225	50	the	the	DET
ejpam-6375	225	51	primal	primal	ADJ
ejpam-6375	225	52	structure	structure	NOUN
ejpam-6375	225	53	and	and	CCONJ
ejpam-6375	225	54	satisfies	satisfy	VERB
ejpam-6375	225	55	the	the	DET
ejpam-6375	225	56	required	require	VERB
ejpam-6375	225	57	continuity	continuity	NOUN
ejpam-6375	225	58	.	.	PUNCT
ejpam-6375	226	1	since	since	SCONJ
ejpam-6375	226	2	v	v	NOUN
ejpam-6375	226	3	is	be	AUX
ejpam-6375	226	4	gpt	gpt	NOUN
ejpam-6375	226	5	-	-	PUNCT
ejpam-6375	226	6	compact	compact	ADJ
ejpam-6375	226	7	,	,	PUNCT
ejpam-6375	226	8	and	and	CCONJ
ejpam-6375	226	9	f	f	PROPN
ejpam-6375	226	10	is	be	AUX
ejpam-6375	226	11	a	a	DET
ejpam-6375	226	12	surjective	surjective	ADJ
ejpam-6375	226	13	map	map	NOUN
ejpam-6375	226	14	that	that	PRON
ejpam-6375	226	15	is	be	AUX
ejpam-6375	226	16	strongly	strongly	ADV
ejpam-6375	226	17	gpt	gpt	NOUN
ejpam-6375	226	18	-	-	PUNCT
ejpam-6375	226	19	s∗	s∗	NOUN
ejpam-6375	226	20	g	g	PROPN
ejpam-6375	226	21	-continuous	-continuous	ADJ
ejpam-6375	226	22	.	.	PUNCT
ejpam-6375	227	1	thus	thus	ADV
ejpam-6375	227	2	,	,	PUNCT
ejpam-6375	227	3	z	z	NOUN
ejpam-6375	227	4	inherits	inherit	VERB
ejpam-6375	227	5	the	the	DET
ejpam-6375	227	6	gpt	gpt	NOUN
ejpam-6375	227	7	-	-	PUNCT
ejpam-6375	227	8	s∗	s∗	PROPN
ejpam-6375	227	9	g	g	PROPN
ejpam-6375	227	10	-compactness	-compactness	NOUN
ejpam-6375	227	11	.	.	PUNCT
ejpam-6375	228	1	theorem	theorem	VERB
ejpam-6375	228	2	3.7	3.7	NUM
ejpam-6375	228	3	.	.	PUNCT
ejpam-6375	229	1	if	if	SCONJ
ejpam-6375	229	2	a	a	DET
ejpam-6375	229	3	surjective	surjective	ADJ
ejpam-6375	229	4	map	map	NOUN
ejpam-6375	229	5	f	f	X
ejpam-6375	229	6	:	:	PUNCT
ejpam-6375	229	7	(	(	PUNCT
ejpam-6375	229	8	v	v	NOUN
ejpam-6375	229	9	,	,	PUNCT
ejpam-6375	229	10	gτ	gτ	PROPN
ejpam-6375	229	11	1	1	NUM
ejpam-6375	229	12	,	,	PUNCT
ejpam-6375	229	13	pα	pα	NOUN
ejpam-6375	229	14	)	)	PUNCT
ejpam-6375	229	15	→	→	SYM
ejpam-6375	229	16	(	(	PUNCT
ejpam-6375	229	17	z	z	NOUN
ejpam-6375	229	18	,	,	PUNCT
ejpam-6375	229	19	gτ	gτ	PROPN
ejpam-6375	229	20	2	2	NUM
ejpam-6375	229	21	,	,	PUNCT
ejpam-6375	229	22	pβ	pβ	NOUN
ejpam-6375	229	23	)	)	PUNCT
ejpam-6375	229	24	is	be	AUX
ejpam-6375	229	25	perfectly	perfectly	ADV
ejpam-6375	229	26	gpt	gpt	NOUN
ejpam-6375	229	27	-	-	PUNCT
ejpam-6375	229	28	s∗	s∗	PROPN
ejpam-6375	229	29	g	g	NOUN
ejpam-6375	229	30	continuous	continuous	ADJ
ejpam-6375	229	31	and	and	CCONJ
ejpam-6375	229	32	(	(	PUNCT
ejpam-6375	229	33	v	v	NOUN
ejpam-6375	229	34	,	,	PUNCT
ejpam-6375	229	35	gτ	gτ	PROPN
ejpam-6375	229	36	1	1	NUM
ejpam-6375	229	37	,	,	PUNCT
ejpam-6375	229	38	pα	pα	NOUN
ejpam-6375	229	39	)	)	PUNCT
ejpam-6375	229	40	is	be	AUX
ejpam-6375	229	41	gpt	gpt	NOUN
ejpam-6375	229	42	-	-	PUNCT
ejpam-6375	229	43	compact	compact	ADJ
ejpam-6375	229	44	,	,	PUNCT
ejpam-6375	229	45	then	then	ADV
ejpam-6375	229	46	(	(	PUNCT
ejpam-6375	229	47	z	z	X
ejpam-6375	229	48	,	,	PUNCT
ejpam-6375	229	49	gτ	gτ	PROPN
ejpam-6375	229	50	2	2	NUM
ejpam-6375	229	51	,	,	PUNCT
ejpam-6375	229	52	pβ	pβ	NOUN
ejpam-6375	229	53	)	)	PUNCT
ejpam-6375	229	54	is	be	AUX
ejpam-6375	229	55	gpt	gpt	NOUN
ejpam-6375	229	56	-	-	PUNCT
ejpam-6375	229	57	s∗	s∗	PROPN
ejpam-6375	229	58	g	g	PROPN
ejpam-6375	229	59	-compact	-compact	NOUN
ejpam-6375	229	60	.	.	PUNCT
ejpam-6375	230	1	proof	proof	NOUN
ejpam-6375	230	2	.	.	PUNCT
ejpam-6375	231	1	as	as	SCONJ
ejpam-6375	231	2	every	every	DET
ejpam-6375	231	3	perfectly	perfectly	ADV
ejpam-6375	231	4	gpt	gpt	NOUN
ejpam-6375	231	5	-	-	PUNCT
ejpam-6375	231	6	s∗	s∗	PROPN
ejpam-6375	231	7	g	g	PROPN
ejpam-6375	231	8	-continuous	-continuous	ADJ
ejpam-6375	231	9	is	be	AUX
ejpam-6375	231	10	strongly	strongly	ADV
ejpam-6375	231	11	gpt	gpt	NOUN
ejpam-6375	231	12	-	-	PUNCT
ejpam-6375	231	13	s∗	s∗	NOUN
ejpam-6375	231	14	g	g	PROPN
ejpam-6375	231	15	-continuous	-continuous	PROPN
ejpam-6375	231	16	.	.	PUNCT
ejpam-6375	232	1	theorem	theorem	VERB
ejpam-6375	232	2	3.6	3.6	NUM
ejpam-6375	232	3	leads	lead	NOUN
ejpam-6375	232	4	to	to	ADP
ejpam-6375	232	5	the	the	DET
ejpam-6375	232	6	obtained	obtain	VERB
ejpam-6375	232	7	result	result	NOUN
ejpam-6375	232	8	.	.	PUNCT
ejpam-6375	233	1	3.1	3.1	NUM
ejpam-6375	233	2	.	.	PUNCT
ejpam-6375	233	3	gpt	gpt	NOUN
ejpam-6375	233	4	-	-	PUNCT
ejpam-6375	233	5	s∗	s∗	PROPN
ejpam-6375	233	6	g	g	PROPN
ejpam-6375	233	7	-connected	-connected	ADJ
ejpam-6375	233	8	space	space	NOUN
ejpam-6375	233	9	in	in	ADP
ejpam-6375	233	10	gpts	gpt	NOUN
ejpam-6375	233	11	definition	definition	NOUN
ejpam-6375	233	12	3.11	3.11	NUM
ejpam-6375	233	13	.	.	PUNCT
ejpam-6375	234	1	assume	assume	VERB
ejpam-6375	234	2	a	a	DET
ejpam-6375	234	3	gpts	gpt	NOUN
ejpam-6375	234	4	and	and	CCONJ
ejpam-6375	234	5	two	two	NUM
ejpam-6375	234	6	disjoint	disjoint	ADJ
ejpam-6375	234	7	non	non	ADJ
ejpam-6375	234	8	-	-	ADJ
ejpam-6375	234	9	empty	empty	ADJ
ejpam-6375	234	10	open	open	ADJ
ejpam-6375	234	11	sets	set	NOUN
ejpam-6375	234	12	d	d	NOUN
ejpam-6375	234	13	and	and	CCONJ
ejpam-6375	234	14	e	e	PROPN
ejpam-6375	234	15	in	in	ADP
ejpam-6375	234	16	v	v	NUM
ejpam-6375	234	17	,	,	PUNCT
ejpam-6375	234	18	then	then	ADV
ejpam-6375	234	19	(	(	PUNCT
ejpam-6375	234	20	v	v	NOUN
ejpam-6375	234	21	,	,	PUNCT
ejpam-6375	234	22	gτ	gτ	INTJ
ejpam-6375	234	23	,	,	PUNCT
ejpam-6375	234	24	p	p	X
ejpam-6375	234	25	)	)	PUNCT
ejpam-6375	234	26	is	be	AUX
ejpam-6375	234	27	known	know	VERB
ejpam-6375	234	28	as	as	ADP
ejpam-6375	234	29	gpt	gpt	NOUN
ejpam-6375	234	30	-	-	PUNCT
ejpam-6375	234	31	disconnection	disconnection	NOUN
ejpam-6375	234	32	if	if	SCONJ
ejpam-6375	234	33	d	d	PROPN
ejpam-6375	234	34	∪	∪	X
ejpam-6375	234	35	e	e	X
ejpam-6375	234	36	=	=	SYM
ejpam-6375	234	37	v.	v.	ADP
ejpam-6375	234	38	definition	definition	NOUN
ejpam-6375	234	39	3.12	3.12	NUM
ejpam-6375	234	40	.	.	PUNCT
ejpam-6375	235	1	a	a	DET
ejpam-6375	235	2	gpts	gpt	NOUN
ejpam-6375	235	3	,	,	PUNCT
ejpam-6375	235	4	(	(	PUNCT
ejpam-6375	235	5	v	v	NOUN
ejpam-6375	235	6	,	,	PUNCT
ejpam-6375	235	7	gτ	gτ	INTJ
ejpam-6375	235	8	,	,	PUNCT
ejpam-6375	235	9	p	p	X
ejpam-6375	235	10	)	)	PUNCT
ejpam-6375	235	11	is	be	AUX
ejpam-6375	235	12	known	know	VERB
ejpam-6375	235	13	as	as	ADP
ejpam-6375	235	14	gpt	gpt	NOUN
ejpam-6375	235	15	-	-	PUNCT
ejpam-6375	235	16	connected	connect	VERB
ejpam-6375	235	17	space	space	NOUN
ejpam-6375	235	18	if	if	SCONJ
ejpam-6375	235	19	(	(	PUNCT
ejpam-6375	235	20	v	v	NOUN
ejpam-6375	235	21	,	,	PUNCT
ejpam-6375	235	22	gτ	gτ	INTJ
ejpam-6375	235	23	,	,	PUNCT
ejpam-6375	235	24	p	p	X
ejpam-6375	235	25	)	)	PUNCT
ejpam-6375	235	26	has	have	VERB
ejpam-6375	235	27	no	no	DET
ejpam-6375	235	28	gpt	gpt	NOUN
ejpam-6375	235	29	-	-	PUNCT
ejpam-6375	235	30	disconnection	disconnection	NOUN
ejpam-6375	235	31	.	.	PUNCT
ejpam-6375	236	1	definition	definition	NOUN
ejpam-6375	236	2	3.13	3.13	NUM
ejpam-6375	236	3	.	.	PUNCT
ejpam-6375	237	1	the	the	DET
ejpam-6375	237	2	condition	condition	NOUN
ejpam-6375	237	3	for	for	ADP
ejpam-6375	237	4	a	a	DET
ejpam-6375	237	5	space	space	NOUN
ejpam-6375	237	6	to	to	PART
ejpam-6375	237	7	become	become	VERB
ejpam-6375	237	8	gpt	gpt	NOUN
ejpam-6375	237	9	-	-	PUNCT
ejpam-6375	237	10	s∗	s∗	NOUN
ejpam-6375	237	11	g	g	PROPN
ejpam-6375	237	12	-connected	-connected	ADJ
ejpam-6375	237	13	exists	exist	NOUN
ejpam-6375	237	14	when	when	SCONJ
ejpam-6375	237	15	two	two	NUM
ejpam-6375	237	16	non	non	ADJ
ejpam-6375	237	17	-	-	ADJ
ejpam-6375	237	18	empty	empty	ADJ
ejpam-6375	237	19	gpt	gpt	NOUN
ejpam-6375	237	20	-	-	PUNCT
ejpam-6375	237	21	s∗	s∗	PROPN
ejpam-6375	237	22	g	g	PROPN
ejpam-6375	237	23	-open	-open	NOUN
ejpam-6375	237	24	sets	set	NOUN
ejpam-6375	237	25	d	d	NOUN
ejpam-6375	237	26	and	and	CCONJ
ejpam-6375	237	27	e	e	X
ejpam-6375	237	28	in	in	ADP
ejpam-6375	237	29	(	(	PUNCT
ejpam-6375	237	30	v	v	NOUN
ejpam-6375	237	31	,	,	PUNCT
ejpam-6375	237	32	gτ	gτ	INTJ
ejpam-6375	237	33	,	,	PUNCT
ejpam-6375	237	34	p	p	X
ejpam-6375	237	35	)	)	PUNCT
ejpam-6375	237	36	can	can	AUX
ejpam-6375	237	37	not	not	PART
ejpam-6375	237	38	be	be	AUX
ejpam-6375	237	39	disjoint	disjoint	ADJ
ejpam-6375	237	40	as	as	ADP
ejpam-6375	237	41	d	d	X
ejpam-6375	237	42	∪	∪	X
ejpam-6375	237	43	e	e	X
ejpam-6375	237	44	=	=	PUNCT
ejpam-6375	237	45	v.	v.	CCONJ
ejpam-6375	237	46	otherwise	otherwise	ADV
ejpam-6375	237	47	called	call	VERB
ejpam-6375	237	48	gpt	gpt	NOUN
ejpam-6375	237	49	-	-	PUNCT
ejpam-6375	237	50	s∗	s∗	PROPN
ejpam-6375	237	51	g	g	PROPN
ejpam-6375	237	52	-disconnected	-disconnecte	VERB
ejpam-6375	237	53	space	space	NOUN
ejpam-6375	237	54	if	if	SCONJ
ejpam-6375	237	55	d	d	PROPN
ejpam-6375	237	56	∪	∪	X
ejpam-6375	237	57	e	e	X
ejpam-6375	237	58	=	=	PUNCT
ejpam-6375	237	59	v.	v.	ADP
ejpam-6375	237	60	example	example	NOUN
ejpam-6375	237	61	3.4	3.4	NUM
ejpam-6375	237	62	.	.	PUNCT
ejpam-6375	237	63	assuming	assume	VERB
ejpam-6375	237	64	v	v	NOUN
ejpam-6375	237	65	=	=	SYM
ejpam-6375	237	66	{	{	PUNCT
ejpam-6375	237	67	j1	j1	PROPN
ejpam-6375	237	68	,	,	PUNCT
ejpam-6375	237	69	k1	k1	PROPN
ejpam-6375	237	70	,	,	PUNCT
ejpam-6375	237	71	l1	l1	PROPN
ejpam-6375	237	72	}	}	PUNCT
ejpam-6375	237	73	,	,	PUNCT
ejpam-6375	237	74	gτ	gτ	PROPN
ejpam-6375	237	75	=	=	PUNCT
ejpam-6375	237	76	{	{	PUNCT
ejpam-6375	237	77	∅	∅	NOUN
ejpam-6375	237	78	,	,	PUNCT
ejpam-6375	237	79	{	{	PUNCT
ejpam-6375	237	80	j1	j1	PROPN
ejpam-6375	237	81	,	,	PUNCT
ejpam-6375	237	82	k1	k1	NOUN
ejpam-6375	237	83	}	}	PUNCT
ejpam-6375	237	84	}	}	PUNCT
ejpam-6375	237	85	and	and	CCONJ
ejpam-6375	237	86	p	p	NOUN
ejpam-6375	237	87	=	=	NOUN
ejpam-6375	237	88	{	{	PUNCT
ejpam-6375	237	89	∅	∅	NOUN
ejpam-6375	237	90	,	,	PUNCT
ejpam-6375	237	91	{	{	PUNCT
ejpam-6375	237	92	j1	j1	PROPN
ejpam-6375	237	93	}	}	PUNCT
ejpam-6375	237	94	,	,	PUNCT
ejpam-6375	237	95	{	{	PUNCT
ejpam-6375	237	96	l1	l1	PROPN
ejpam-6375	237	97	}	}	PUNCT
ejpam-6375	237	98	,	,	PUNCT
ejpam-6375	237	99	{	{	PUNCT
ejpam-6375	237	100	j1	j1	PROPN
ejpam-6375	237	101	,	,	PUNCT
ejpam-6375	237	102	l1	l1	PROPN
ejpam-6375	237	103	}	}	PUNCT
ejpam-6375	237	104	}	}	PUNCT
ejpam-6375	237	105	.	.	PUNCT
ejpam-6375	238	1	in	in	ADP
ejpam-6375	238	2	this	this	DET
ejpam-6375	238	3	gpts	gpt	NOUN
ejpam-6375	238	4	(	(	PUNCT
ejpam-6375	238	5	v	v	NOUN
ejpam-6375	238	6	,	,	PUNCT
ejpam-6375	238	7	gτ	gτ	INTJ
ejpam-6375	238	8	,	,	PUNCT
ejpam-6375	238	9	p)o	p)o	PUNCT
ejpam-6375	238	10	=	=	PUNCT
ejpam-6375	238	11	{	{	PUNCT
ejpam-6375	238	12	∅	∅	NOUN
ejpam-6375	238	13	,	,	PUNCT
ejpam-6375	238	14	{	{	PUNCT
ejpam-6375	238	15	j1	j1	PROPN
ejpam-6375	238	16	,	,	PUNCT
ejpam-6375	238	17	k1	k1	NOUN
ejpam-6375	238	18	}	}	PUNCT
ejpam-6375	238	19	}	}	PUNCT
ejpam-6375	238	20	and	and	CCONJ
ejpam-6375	238	21	(	(	PUNCT
ejpam-6375	238	22	gτ	gτ	INTJ
ejpam-6375	238	23	,	,	PUNCT
ejpam-6375	238	24	p)s∗	p)s∗	PROPN
ejpam-6375	238	25	go	go	VERB
ejpam-6375	238	26	=	=	PUNCT
ejpam-6375	238	27	{	{	PUNCT
ejpam-6375	238	28	∅	∅	NOUN
ejpam-6375	238	29	,	,	PUNCT
ejpam-6375	238	30	{	{	PUNCT
ejpam-6375	238	31	j1	j1	PROPN
ejpam-6375	238	32	,	,	PUNCT
ejpam-6375	238	33	k1	k1	NOUN
ejpam-6375	238	34	}	}	PUNCT
ejpam-6375	238	35	,	,	PUNCT
ejpam-6375	238	36	v	v	NOUN
ejpam-6375	238	37	}	}	PUNCT
ejpam-6375	238	38	.	.	PUNCT
ejpam-6375	239	1	there	there	PRON
ejpam-6375	239	2	does	do	AUX
ejpam-6375	239	3	not	not	PART
ejpam-6375	239	4	exists	exist	VERB
ejpam-6375	239	5	two	two	NUM
ejpam-6375	239	6	disjoint	disjoint	ADJ
ejpam-6375	239	7	non	non	ADJ
ejpam-6375	239	8	-	-	ADJ
ejpam-6375	239	9	empty	empty	ADJ
ejpam-6375	239	10	gpt	gpt	NOUN
ejpam-6375	239	11	-	-	PUNCT
ejpam-6375	239	12	s∗	s∗	PROPN
ejpam-6375	239	13	g	g	PROPN
ejpam-6375	239	14	-open	-open	NOUN
ejpam-6375	239	15	sets	set	NOUN
ejpam-6375	239	16	d	d	NOUN
ejpam-6375	239	17	and	and	CCONJ
ejpam-6375	239	18	e	e	X
ejpam-6375	239	19	in	in	ADP
ejpam-6375	239	20	(	(	PUNCT
ejpam-6375	239	21	v	v	NOUN
ejpam-6375	239	22	,	,	PUNCT
ejpam-6375	239	23	gτ	gτ	INTJ
ejpam-6375	239	24	,	,	PUNCT
ejpam-6375	239	25	p	p	NOUN
ejpam-6375	239	26	)	)	PUNCT
ejpam-6375	239	27	such	such	ADJ
ejpam-6375	239	28	that	that	SCONJ
ejpam-6375	239	29	d	d	PROPN
ejpam-6375	239	30	∪	∪	SCONJ
ejpam-6375	239	31	e	e	NOUN
ejpam-6375	239	32	=	=	SYM
ejpam-6375	239	33	v	v	NOUN
ejpam-6375	239	34	implies	imply	VERB
ejpam-6375	239	35	(	(	PUNCT
ejpam-6375	239	36	v	v	NOUN
ejpam-6375	239	37	,	,	PUNCT
ejpam-6375	239	38	gτ	gτ	INTJ
ejpam-6375	239	39	,	,	PUNCT
ejpam-6375	239	40	p	p	X
ejpam-6375	239	41	)	)	PUNCT
ejpam-6375	239	42	is	be	AUX
ejpam-6375	239	43	gpt	gpt	NOUN
ejpam-6375	239	44	-	-	PUNCT
ejpam-6375	239	45	s∗	s∗	PROPN
ejpam-6375	239	46	g	g	PROPN
ejpam-6375	239	47	-connected	-connected	ADJ
ejpam-6375	239	48	space	space	NOUN
ejpam-6375	239	49	.	.	PUNCT
ejpam-6375	240	1	theorem	theorem	VERB
ejpam-6375	240	2	3.8	3.8	NUM
ejpam-6375	240	3	.	.	PUNCT
ejpam-6375	241	1	a	a	DET
ejpam-6375	241	2	space	space	NOUN
ejpam-6375	241	3	which	which	PRON
ejpam-6375	241	4	satisfies	satisfy	VERB
ejpam-6375	241	5	gpt	gpt	NOUN
ejpam-6375	241	6	-	-	PUNCT
ejpam-6375	241	7	s∗	s∗	PROPN
ejpam-6375	241	8	g	g	PROPN
ejpam-6375	241	9	-connectedness	-connectedness	NOUN
ejpam-6375	241	10	contains	contain	VERB
ejpam-6375	241	11	the	the	DET
ejpam-6375	241	12	condition	condition	NOUN
ejpam-6375	241	13	of	of	ADP
ejpam-6375	241	14	gpt	gpt	NOUN
ejpam-6375	241	15	-	-	PUNCT
ejpam-6375	241	16	connectedness	connectedness	NOUN
ejpam-6375	241	17	.	.	PUNCT
ejpam-6375	242	1	proof	proof	NOUN
ejpam-6375	242	2	.	.	PUNCT
ejpam-6375	243	1	assume	assume	VERB
ejpam-6375	243	2	a	a	DET
ejpam-6375	243	3	gpt	gpt	NOUN
ejpam-6375	243	4	-	-	PUNCT
ejpam-6375	243	5	s∗	s∗	PROPN
ejpam-6375	243	6	g	g	PROPN
ejpam-6375	243	7	-connected	-connected	ADJ
ejpam-6375	243	8	space	space	NOUN
ejpam-6375	243	9	(	(	PUNCT
ejpam-6375	243	10	v	v	NOUN
ejpam-6375	243	11	,	,	PUNCT
ejpam-6375	243	12	gτ	gτ	INTJ
ejpam-6375	243	13	,	,	PUNCT
ejpam-6375	243	14	p	p	NOUN
ejpam-6375	243	15	)	)	PUNCT
ejpam-6375	243	16	.	.	PUNCT
ejpam-6375	244	1	consider	consider	VERB
ejpam-6375	244	2	(	(	PUNCT
ejpam-6375	244	3	v	v	NOUN
ejpam-6375	244	4	,	,	PUNCT
ejpam-6375	244	5	gτ	gτ	INTJ
ejpam-6375	244	6	,	,	PUNCT
ejpam-6375	244	7	p	p	X
ejpam-6375	244	8	)	)	PUNCT
ejpam-6375	244	9	is	be	AUX
ejpam-6375	244	10	not	not	PART
ejpam-6375	244	11	a	a	DET
ejpam-6375	244	12	gptconnected	gptconnecte	VERB
ejpam-6375	244	13	space	space	NOUN
ejpam-6375	244	14	,	,	PUNCT
ejpam-6375	244	15	then	then	ADV
ejpam-6375	244	16	there	there	PRON
ejpam-6375	244	17	exist	exist	VERB
ejpam-6375	244	18	nonempty	nonempty	ADJ
ejpam-6375	244	19	open	open	ADJ
ejpam-6375	244	20	subsets	subset	NOUN
ejpam-6375	244	21	d	d	NOUN
ejpam-6375	244	22	and	and	CCONJ
ejpam-6375	244	23	e	e	NOUN
ejpam-6375	244	24	in	in	ADP
ejpam-6375	244	25	v	v	NUM
ejpam-6375	244	26	such	such	ADJ
ejpam-6375	244	27	that	that	SCONJ
ejpam-6375	244	28	their	their	PRON
ejpam-6375	244	29	union	union	NOUN
ejpam-6375	244	30	equals	equal	VERB
ejpam-6375	244	31	v	v	ADP
ejpam-6375	244	32	itself	itself	PRON
ejpam-6375	244	33	.	.	PUNCT
ejpam-6375	245	1	every	every	DET
ejpam-6375	245	2	open	open	ADJ
ejpam-6375	245	3	set	set	NOUN
ejpam-6375	245	4	holds	hold	VERB
ejpam-6375	245	5	the	the	DET
ejpam-6375	245	6	status	status	NOUN
ejpam-6375	245	7	of	of	ADP
ejpam-6375	245	8	being	be	AUX
ejpam-6375	245	9	both	both	CCONJ
ejpam-6375	245	10	gpt	gpt	NOUN
ejpam-6375	245	11	-	-	PUNCT
ejpam-6375	245	12	s∗	s∗	PROPN
ejpam-6375	245	13	g	g	PROPN
ejpam-6375	245	14	-open	-open	ADJ
ejpam-6375	245	15	and	and	CCONJ
ejpam-6375	245	16	gpt	gpt	NOUN
ejpam-6375	245	17	-	-	PUNCT
ejpam-6375	245	18	open	open	ADJ
ejpam-6375	245	19	in	in	ADP
ejpam-6375	245	20	the	the	DET
ejpam-6375	245	21	defined	define	VERB
ejpam-6375	245	22	topology	topology	NOUN
ejpam-6375	245	23	.	.	PUNCT
ejpam-6375	246	1	the	the	DET
ejpam-6375	246	2	pair	pair	NOUN
ejpam-6375	246	3	of	of	ADP
ejpam-6375	246	4	open	open	ADJ
ejpam-6375	246	5	sets	set	NOUN
ejpam-6375	246	6	d	d	NOUN
ejpam-6375	246	7	and	and	CCONJ
ejpam-6375	246	8	e	e	PROPN
ejpam-6375	246	9	belongs	belong	VERB
ejpam-6375	246	10	to	to	ADP
ejpam-6375	246	11	the	the	DET
ejpam-6375	246	12	class	class	NOUN
ejpam-6375	246	13	of	of	ADP
ejpam-6375	246	14	gpt	gpt	NOUN
ejpam-6375	246	15	-	-	PUNCT
ejpam-6375	246	16	s∗	s∗	PROPN
ejpam-6375	246	17	g	g	PROPN
ejpam-6375	246	18	-open	-open	ADJ
ejpam-6375	246	19	sets	set	NOUN
ejpam-6375	246	20	since	since	SCONJ
ejpam-6375	246	21	v	v	NOUN
ejpam-6375	246	22	=	=	SYM
ejpam-6375	246	23	d	d	X
ejpam-6375	246	24	∪	∪	X
ejpam-6375	246	25	e.	e.	PROPN
ejpam-6375	246	26	the	the	DET
ejpam-6375	246	27	finding	finding	NOUN
ejpam-6375	246	28	of	of	ADP
ejpam-6375	246	29	two	two	NUM
ejpam-6375	246	30	nonempty	nonempty	ADJ
ejpam-6375	246	31	open	open	ADJ
ejpam-6375	246	32	sets	set	NOUN
ejpam-6375	246	33	in	in	ADP
ejpam-6375	246	34	v	v	NUM
ejpam-6375	246	35	leads	lead	VERB
ejpam-6375	246	36	to	to	ADP
ejpam-6375	246	37	a	a	DET
ejpam-6375	246	38	contradiction	contradiction	NOUN
ejpam-6375	246	39	when	when	SCONJ
ejpam-6375	246	40	applying	apply	VERB
ejpam-6375	246	41	the	the	DET
ejpam-6375	246	42	definition	definition	NOUN
ejpam-6375	246	43	of	of	ADP
ejpam-6375	246	44	gpt	gpt	NOUN
ejpam-6375	246	45	-	-	PUNCT
ejpam-6375	246	46	s∗	s∗	PROPN
ejpam-6375	246	47	g	g	PROPN
ejpam-6375	246	48	-connected	-connected	ADJ
ejpam-6375	246	49	space	space	NOUN
ejpam-6375	246	50	.	.	PUNCT
ejpam-6375	247	1	a	a	DET
ejpam-6375	247	2	space	space	NOUN
ejpam-6375	247	3	represented	represent	VERB
ejpam-6375	247	4	by	by	ADP
ejpam-6375	247	5	(	(	PUNCT
ejpam-6375	247	6	v	v	NOUN
ejpam-6375	247	7	,	,	PUNCT
ejpam-6375	247	8	gτ	gτ	INTJ
ejpam-6375	247	9	,	,	PUNCT
ejpam-6375	247	10	p	p	NOUN
ejpam-6375	247	11	)	)	PUNCT
ejpam-6375	247	12	serves	serve	VERB
ejpam-6375	247	13	as	as	ADP
ejpam-6375	247	14	a	a	DET
ejpam-6375	247	15	connected	connected	ADJ
ejpam-6375	247	16	space	space	NOUN
ejpam-6375	247	17	under	under	ADP
ejpam-6375	247	18	the	the	DET
ejpam-6375	247	19	gpt	gpt	NOUN
ejpam-6375	247	20	structure	structure	NOUN
ejpam-6375	247	21	.	.	PUNCT
ejpam-6375	248	1	remark	remark	PROPN
ejpam-6375	248	2	3.2	3.2	NUM
ejpam-6375	248	3	.	.	PUNCT
ejpam-6375	249	1	an	an	DET
ejpam-6375	249	2	illustration	illustration	NOUN
ejpam-6375	249	3	disproves	disprove	VERB
ejpam-6375	249	4	the	the	DET
ejpam-6375	249	5	invalidity	invalidity	NOUN
ejpam-6375	249	6	of	of	ADP
ejpam-6375	249	7	the	the	DET
ejpam-6375	249	8	converse	converse	NOUN
ejpam-6375	249	9	statement	statement	NOUN
ejpam-6375	249	10	derived	derive	VERB
ejpam-6375	249	11	from	from	ADP
ejpam-6375	249	12	above	above	ADP
ejpam-6375	249	13	.	.	PUNCT
ejpam-6375	250	1	example	example	NOUN
ejpam-6375	250	2	3.5	3.5	NUM
ejpam-6375	250	3	.	.	PUNCT
ejpam-6375	251	1	assuming	assume	VERB
ejpam-6375	251	2	v	v	NOUN
ejpam-6375	251	3	=	=	SYM
ejpam-6375	251	4	{	{	PUNCT
ejpam-6375	251	5	j1	j1	PROPN
ejpam-6375	251	6	,	,	PUNCT
ejpam-6375	251	7	k1	k1	NOUN
ejpam-6375	251	8	,	,	PUNCT
ejpam-6375	251	9	l1	l1	PROPN
ejpam-6375	251	10	}	}	PUNCT
ejpam-6375	251	11	,	,	PUNCT
ejpam-6375	251	12	gτ	gτ	PROPN
ejpam-6375	251	13	=	=	PUNCT
ejpam-6375	251	14	{	{	PUNCT
ejpam-6375	251	15	∅	∅	NOUN
ejpam-6375	251	16	,	,	PUNCT
ejpam-6375	251	17	{	{	PUNCT
ejpam-6375	251	18	j1	j1	PROPN
ejpam-6375	251	19	}	}	PUNCT
ejpam-6375	251	20	,	,	PUNCT
ejpam-6375	251	21	{	{	PUNCT
ejpam-6375	251	22	k1	k1	NOUN
ejpam-6375	251	23	}	}	PUNCT
ejpam-6375	251	24	,	,	PUNCT
ejpam-6375	251	25	{	{	PUNCT
ejpam-6375	251	26	j1	j1	PROPN
ejpam-6375	251	27	,	,	PUNCT
ejpam-6375	251	28	k1	k1	NOUN
ejpam-6375	251	29	}	}	PUNCT
ejpam-6375	251	30	}	}	PUNCT
ejpam-6375	251	31	and	and	CCONJ
ejpam-6375	251	32	p	p	NOUN
ejpam-6375	251	33	=	=	NOUN
ejpam-6375	251	34	{	{	PUNCT
ejpam-6375	251	35	∅	∅	NOUN
ejpam-6375	251	36	,	,	PUNCT
ejpam-6375	251	37	{	{	PUNCT
ejpam-6375	251	38	j1	j1	PROPN
ejpam-6375	251	39	}	}	PUNCT
ejpam-6375	251	40	,	,	PUNCT
ejpam-6375	251	41	{	{	PUNCT
ejpam-6375	251	42	l1	l1	PROPN
ejpam-6375	251	43	}	}	PUNCT
ejpam-6375	251	44	,	,	PUNCT
ejpam-6375	251	45	{	{	PUNCT
ejpam-6375	251	46	j1	j1	PROPN
ejpam-6375	251	47	,	,	PUNCT
ejpam-6375	251	48	l1	l1	PROPN
ejpam-6375	251	49	}	}	PUNCT
ejpam-6375	251	50	}	}	PUNCT
ejpam-6375	251	51	.	.	PUNCT
ejpam-6375	252	1	in	in	ADP
ejpam-6375	252	2	this	this	DET
ejpam-6375	252	3	gpts	gpt	NOUN
ejpam-6375	252	4	,	,	PUNCT
ejpam-6375	252	5	(	(	PUNCT
ejpam-6375	252	6	v	v	NOUN
ejpam-6375	252	7	,	,	PUNCT
ejpam-6375	252	8	gτ	gτ	INTJ
ejpam-6375	252	9	,	,	PUNCT
ejpam-6375	252	10	p)o	p)o	PUNCT
ejpam-6375	252	11	=	=	PUNCT
ejpam-6375	252	12	{	{	PUNCT
ejpam-6375	252	13	∅	∅	NOUN
ejpam-6375	252	14	,	,	PUNCT
ejpam-6375	252	15	{	{	PUNCT
ejpam-6375	252	16	j1	j1	PROPN
ejpam-6375	252	17	}	}	PUNCT
ejpam-6375	252	18	,	,	PUNCT
ejpam-6375	252	19	{	{	PUNCT
ejpam-6375	252	20	k1	k1	NOUN
ejpam-6375	252	21	}	}	PUNCT
ejpam-6375	252	22	,	,	PUNCT
ejpam-6375	252	23	{	{	PUNCT
ejpam-6375	252	24	j1	j1	PROPN
ejpam-6375	252	25	,	,	PUNCT
ejpam-6375	252	26	k1	k1	NOUN
ejpam-6375	252	27	}	}	PUNCT
ejpam-6375	252	28	}	}	PUNCT
ejpam-6375	252	29	and	and	CCONJ
ejpam-6375	252	30	(	(	PUNCT
ejpam-6375	252	31	gτ	gτ	INTJ
ejpam-6375	252	32	,	,	PUNCT
ejpam-6375	252	33	p)s∗	p)s∗	PROPN
ejpam-6375	252	34	go	go	VERB
ejpam-6375	252	35	=	=	PUNCT
ejpam-6375	252	36	{	{	PUNCT
ejpam-6375	252	37	∅	∅	NOUN
ejpam-6375	252	38	,	,	PUNCT
ejpam-6375	252	39	{	{	PUNCT
ejpam-6375	252	40	j1	j1	PROPN
ejpam-6375	252	41	}	}	PUNCT
ejpam-6375	252	42	,	,	PUNCT
ejpam-6375	252	43	{	{	PUNCT
ejpam-6375	252	44	k1	k1	NOUN
ejpam-6375	252	45	}	}	PUNCT
ejpam-6375	252	46	,	,	PUNCT
ejpam-6375	252	47	{	{	PUNCT
ejpam-6375	252	48	j1	j1	PROPN
ejpam-6375	252	49	,	,	PUNCT
ejpam-6375	252	50	k1	k1	NOUN
ejpam-6375	252	51	}	}	PUNCT
ejpam-6375	252	52	,	,	PUNCT
ejpam-6375	252	53	{	{	PUNCT
ejpam-6375	252	54	j1	j1	PROPN
ejpam-6375	252	55	,	,	PUNCT
ejpam-6375	252	56	l1	l1	PROPN
ejpam-6375	252	57	}	}	PUNCT
ejpam-6375	252	58	,	,	PUNCT
ejpam-6375	252	59	{	{	PUNCT
ejpam-6375	252	60	k1	k1	NOUN
ejpam-6375	252	61	,	,	PUNCT
ejpam-6375	252	62	l1	l1	PROPN
ejpam-6375	252	63	}	}	PUNCT
ejpam-6375	252	64	}	}	PUNCT
ejpam-6375	252	65	.	.	PUNCT
ejpam-6375	253	1	there	there	PRON
ejpam-6375	253	2	does	do	AUX
ejpam-6375	253	3	not	not	PART
ejpam-6375	253	4	exists	exist	VERB
ejpam-6375	253	5	two	two	NUM
ejpam-6375	253	6	disjoint	disjoint	ADJ
ejpam-6375	253	7	non	non	ADJ
ejpam-6375	253	8	-	-	ADJ
ejpam-6375	253	9	empty	empty	ADJ
ejpam-6375	253	10	open	open	ADJ
ejpam-6375	253	11	sets	set	NOUN
ejpam-6375	253	12	d	d	NOUN
ejpam-6375	253	13	and	and	CCONJ
ejpam-6375	253	14	e	e	X
ejpam-6375	253	15	in	in	ADP
ejpam-6375	253	16	(	(	PUNCT
ejpam-6375	253	17	v	v	NOUN
ejpam-6375	253	18	,	,	PUNCT
ejpam-6375	253	19	gτ	gτ	INTJ
ejpam-6375	253	20	,	,	PUNCT
ejpam-6375	253	21	p	p	NOUN
ejpam-6375	253	22	)	)	PUNCT
ejpam-6375	253	23	such	such	ADJ
ejpam-6375	253	24	that	that	SCONJ
ejpam-6375	253	25	d	d	PROPN
ejpam-6375	253	26	∪	∪	SCONJ
ejpam-6375	253	27	e	e	NOUN
ejpam-6375	253	28	=	=	SYM
ejpam-6375	253	29	v	v	NOUN
ejpam-6375	253	30	implies	imply	VERB
ejpam-6375	253	31	(	(	PUNCT
ejpam-6375	253	32	v	v	NOUN
ejpam-6375	253	33	,	,	PUNCT
ejpam-6375	253	34	gτ	gτ	INTJ
ejpam-6375	253	35	,	,	PUNCT
ejpam-6375	253	36	p	p	NOUN
ejpam-6375	253	37	)	)	PUNCT
ejpam-6375	253	38	m.	m.	NOUN
ejpam-6375	253	39	shahbaz	shahbaz	PROPN
ejpam-6375	253	40	et	et	PROPN
ejpam-6375	253	41	al	al	PROPN
ejpam-6375	253	42	.	.	PUNCT
ejpam-6375	253	43	/	/	SYM
ejpam-6375	253	44	eur	eur	PROPN
ejpam-6375	253	45	.	.	PUNCT
ejpam-6375	254	1	j.	j.	PROPN
ejpam-6375	254	2	pure	pure	PROPN
ejpam-6375	254	3	appl	appl	PROPN
ejpam-6375	254	4	.	.	PROPN
ejpam-6375	254	5	math	math	PROPN
ejpam-6375	254	6	,	,	PUNCT
ejpam-6375	254	7	18	18	NUM
ejpam-6375	254	8	(	(	PUNCT
ejpam-6375	254	9	4	4	NUM
ejpam-6375	254	10	)	)	PUNCT
ejpam-6375	254	11	(	(	PUNCT
ejpam-6375	254	12	2025	2025	NUM
ejpam-6375	254	13	)	)	PUNCT
ejpam-6375	254	14	,	,	PUNCT
ejpam-6375	254	15	6375	6375	NUM
ejpam-6375	254	16	11	11	NUM
ejpam-6375	254	17	of	of	ADP
ejpam-6375	254	18	22	22	NUM
ejpam-6375	254	19	is	be	AUX
ejpam-6375	254	20	gpt	gpt	NOUN
ejpam-6375	254	21	-	-	PUNCT
ejpam-6375	254	22	s∗	s∗	PROPN
ejpam-6375	254	23	g	g	PROPN
ejpam-6375	254	24	-connected	-connected	ADJ
ejpam-6375	254	25	space	space	NOUN
ejpam-6375	254	26	.	.	PUNCT
ejpam-6375	255	1	but	but	CCONJ
ejpam-6375	255	2	two	two	NUM
ejpam-6375	255	3	disjoint	disjoint	ADJ
ejpam-6375	255	4	non	non	ADJ
ejpam-6375	255	5	-	-	ADJ
ejpam-6375	255	6	empty	empty	ADJ
ejpam-6375	255	7	gτ	gτ	NOUN
ejpam-6375	255	8	-s∗	-s∗	NOUN
ejpam-6375	255	9	g	g	NOUN
ejpam-6375	255	10	-open	-open	PROPN
ejpam-6375	255	11	sets	set	NOUN
ejpam-6375	255	12	{	{	PUNCT
ejpam-6375	255	13	j1	j1	NOUN
ejpam-6375	255	14	}	}	PUNCT
ejpam-6375	255	15	and	and	CCONJ
ejpam-6375	255	16	{	{	PUNCT
ejpam-6375	255	17	k1	k1	PROPN
ejpam-6375	255	18	,	,	PUNCT
ejpam-6375	255	19	l1	l1	PROPN
ejpam-6375	255	20	}	}	PUNCT
ejpam-6375	255	21	exists	exist	VERB
ejpam-6375	255	22	such	such	ADJ
ejpam-6375	255	23	that	that	SCONJ
ejpam-6375	255	24	{	{	PUNCT
ejpam-6375	255	25	j1	j1	PROPN
ejpam-6375	255	26	}	}	PUNCT
ejpam-6375	255	27	∪	∪	NOUN
ejpam-6375	255	28	{	{	PUNCT
ejpam-6375	255	29	k1	k1	NOUN
ejpam-6375	255	30	,	,	PUNCT
ejpam-6375	255	31	l1	l1	PROPN
ejpam-6375	255	32	}	}	PUNCT
ejpam-6375	255	33	=	=	SYM
ejpam-6375	255	34	v	v	NOUN
ejpam-6375	255	35	implies	imply	VERB
ejpam-6375	255	36	(	(	PUNCT
ejpam-6375	255	37	v	v	NOUN
ejpam-6375	255	38	,	,	PUNCT
ejpam-6375	255	39	gτ	gτ	INTJ
ejpam-6375	255	40	,	,	PUNCT
ejpam-6375	255	41	p	p	X
ejpam-6375	255	42	)	)	PUNCT
ejpam-6375	255	43	is	be	AUX
ejpam-6375	255	44	not	not	PART
ejpam-6375	255	45	gpt	gpt	NOUN
ejpam-6375	255	46	-	-	PUNCT
ejpam-6375	255	47	s∗	s∗	NOUN
ejpam-6375	255	48	g	g	PROPN
ejpam-6375	255	49	-connected	-connected	ADJ
ejpam-6375	255	50	space	space	NOUN
ejpam-6375	255	51	.	.	PUNCT
ejpam-6375	256	1	theorem	theorem	VERB
ejpam-6375	256	2	3.9	3.9	NUM
ejpam-6375	256	3	.	.	PUNCT
ejpam-6375	257	1	assume	assume	VERB
ejpam-6375	257	2	a	a	DET
ejpam-6375	257	3	gpts	gpt	NOUN
ejpam-6375	257	4	,	,	PUNCT
ejpam-6375	257	5	then	then	ADV
ejpam-6375	257	6	the	the	DET
ejpam-6375	257	7	subsequent	subsequent	ADJ
ejpam-6375	257	8	are	be	AUX
ejpam-6375	257	9	equivalent	equivalent	ADJ
ejpam-6375	257	10	.	.	PUNCT
ejpam-6375	258	1	i.	i.	PROPN
ejpam-6375	258	2	v	v	PROPN
ejpam-6375	258	3	is	be	AUX
ejpam-6375	258	4	gpt	gpt	NOUN
ejpam-6375	258	5	-	-	PUNCT
ejpam-6375	258	6	s∗	s∗	PROPN
ejpam-6375	258	7	g	g	PROPN
ejpam-6375	258	8	-connected	-connected	PROPN
ejpam-6375	258	9	.	.	PUNCT
ejpam-6375	259	1	ii	ii	PROPN
ejpam-6375	259	2	.	.	PROPN
ejpam-6375	259	3	∅	∅	NOUN
ejpam-6375	259	4	and	and	CCONJ
ejpam-6375	259	5	v	v	NOUN
ejpam-6375	259	6	are	be	AUX
ejpam-6375	259	7	only	only	ADV
ejpam-6375	259	8	gpt	gpt	NOUN
ejpam-6375	259	9	-	-	PUNCT
ejpam-6375	259	10	s∗	s∗	PROPN
ejpam-6375	259	11	g	g	PROPN
ejpam-6375	259	12	-open	-open	NOUN
ejpam-6375	259	13	set	set	NOUN
ejpam-6375	259	14	and	and	CCONJ
ejpam-6375	259	15	gpt	gpt	NOUN
ejpam-6375	259	16	-	-	PUNCT
ejpam-6375	259	17	s∗	s∗	PROPN
ejpam-6375	259	18	g	g	PROPN
ejpam-6375	259	19	-closed	-close	VERB
ejpam-6375	259	20	set	set	NOUN
ejpam-6375	259	21	in	in	ADP
ejpam-6375	259	22	v.	v.	PROPN
ejpam-6375	259	23	iii	iii	PROPN
ejpam-6375	259	24	.	.	PUNCT
ejpam-6375	260	1	any	any	DET
ejpam-6375	260	2	gpt	gpt	NOUN
ejpam-6375	260	3	-	-	PUNCT
ejpam-6375	260	4	s∗	s∗	NOUN
ejpam-6375	260	5	g	g	PROPN
ejpam-6375	260	6	-continuous	-continuous	ADJ
ejpam-6375	260	7	mapping	mapping	NOUN
ejpam-6375	261	1	f	f	NOUN
ejpam-6375	261	2	:	:	PUNCT
ejpam-6375	261	3	(	(	PUNCT
ejpam-6375	261	4	v	v	NOUN
ejpam-6375	261	5	,	,	PUNCT
ejpam-6375	261	6	gτ	gτ	PROPN
ejpam-6375	261	7	1	1	NUM
ejpam-6375	261	8	,	,	PUNCT
ejpam-6375	261	9	pα	pα	NOUN
ejpam-6375	261	10	)	)	PUNCT
ejpam-6375	261	11	→	→	SYM
ejpam-6375	261	12	(	(	PUNCT
ejpam-6375	261	13	z	z	NOUN
ejpam-6375	261	14	,	,	PUNCT
ejpam-6375	261	15	gτ	gτ	PROPN
ejpam-6375	261	16	2	2	NUM
ejpam-6375	261	17	,	,	PUNCT
ejpam-6375	261	18	pβ	pβ	ADV
ejpam-6375	261	19	)	)	PUNCT
ejpam-6375	261	20	is	be	AUX
ejpam-6375	261	21	a	a	DET
ejpam-6375	261	22	constant	constant	ADJ
ejpam-6375	261	23	map	map	NOUN
ejpam-6375	261	24	where	where	SCONJ
ejpam-6375	261	25	(	(	PUNCT
ejpam-6375	261	26	z	z	NOUN
ejpam-6375	261	27	,	,	PUNCT
ejpam-6375	261	28	gτ	gτ	PROPN
ejpam-6375	261	29	2	2	NUM
ejpam-6375	261	30	,	,	PUNCT
ejpam-6375	261	31	pβ	pβ	ADV
ejpam-6375	261	32	)	)	PUNCT
ejpam-6375	261	33	at	at	ADP
ejpam-6375	261	34	least	least	ADJ
ejpam-6375	261	35	two	two	NUM
ejpam-6375	261	36	-	-	PUNCT
ejpam-6375	261	37	point	point	NOUN
ejpam-6375	261	38	discrete	discrete	ADJ
ejpam-6375	261	39	space	space	NOUN
ejpam-6375	261	40	.	.	PUNCT
ejpam-6375	262	1	proof	proof	NOUN
ejpam-6375	262	2	.	.	PUNCT
ejpam-6375	263	1	(	(	PUNCT
ejpam-6375	263	2	i	i	NOUN
ejpam-6375	263	3	)	)	PUNCT
ejpam-6375	263	4	then	then	ADV
ejpam-6375	263	5	(	(	PUNCT
ejpam-6375	263	6	ii	ii	NOUN
ejpam-6375	263	7	):	):	PUNCT
ejpam-6375	263	8	let	let	VERB
ejpam-6375	263	9	a	a	DET
ejpam-6375	263	10	gpt	gpt	NOUN
ejpam-6375	263	11	-	-	PUNCT
ejpam-6375	263	12	s∗	s∗	PROPN
ejpam-6375	263	13	g	g	PROPN
ejpam-6375	263	14	-open	-open	NOUN
ejpam-6375	263	15	set	set	NOUN
ejpam-6375	263	16	and	and	CCONJ
ejpam-6375	263	17	gpt	gpt	NOUN
ejpam-6375	263	18	-	-	PUNCT
ejpam-6375	263	19	s∗	s∗	PROPN
ejpam-6375	263	20	g	g	PROPN
ejpam-6375	263	21	-closed	-close	VERB
ejpam-6375	263	22	set	set	VERB
ejpam-6375	263	23	e	e	NOUN
ejpam-6375	263	24	in	in	ADP
ejpam-6375	263	25	(	(	PUNCT
ejpam-6375	263	26	v	v	NOUN
ejpam-6375	263	27	,	,	PUNCT
ejpam-6375	263	28	gτ	gτ	INTJ
ejpam-6375	263	29	,	,	PUNCT
ejpam-6375	263	30	p	p	NOUN
ejpam-6375	263	31	)	)	PUNCT
ejpam-6375	263	32	implies	imply	VERB
ejpam-6375	263	33	ec	ec	PROPN
ejpam-6375	263	34	is	be	AUX
ejpam-6375	263	35	also	also	ADV
ejpam-6375	263	36	gpt	gpt	NOUN
ejpam-6375	263	37	-	-	PUNCT
ejpam-6375	263	38	s∗	s∗	PROPN
ejpam-6375	263	39	g	g	PROPN
ejpam-6375	263	40	-open	-open	NOUN
ejpam-6375	263	41	set	set	NOUN
ejpam-6375	263	42	and	and	CCONJ
ejpam-6375	263	43	gpt	gpt	NOUN
ejpam-6375	263	44	-	-	PUNCT
ejpam-6375	263	45	s∗	s∗	PROPN
ejpam-6375	263	46	g	g	PROPN
ejpam-6375	263	47	-closed	-close	VERB
ejpam-6375	263	48	set	set	VERB
ejpam-6375	263	49	in	in	ADP
ejpam-6375	263	50	(	(	PUNCT
ejpam-6375	263	51	v	v	NOUN
ejpam-6375	263	52	,	,	PUNCT
ejpam-6375	263	53	gτ	gτ	INTJ
ejpam-6375	263	54	,	,	PUNCT
ejpam-6375	263	55	p	p	NOUN
ejpam-6375	263	56	)	)	PUNCT
ejpam-6375	263	57	.	.	PUNCT
ejpam-6375	264	1	then	then	ADV
ejpam-6375	264	2	,	,	PUNCT
ejpam-6375	264	3	e	e	X
ejpam-6375	264	4	∪	∪	VERB
ejpam-6375	264	5	ec	ec	PROPN
ejpam-6375	264	6	=	=	PUNCT
ejpam-6375	264	7	v	v	PROPN
ejpam-6375	264	8	as	as	ADP
ejpam-6375	264	9	e	e	PROPN
ejpam-6375	264	10	and	and	CCONJ
ejpam-6375	264	11	ec	ec	PROPN
ejpam-6375	264	12	are	be	AUX
ejpam-6375	264	13	both	both	PRON
ejpam-6375	264	14	disjoint	disjoint	NOUN
ejpam-6375	264	15	gpt	gpt	NOUN
ejpam-6375	264	16	-	-	PUNCT
ejpam-6375	264	17	s∗	s∗	PROPN
ejpam-6375	264	18	g	g	PROPN
ejpam-6375	264	19	-open	-open	NOUN
ejpam-6375	264	20	sets	set	NOUN
ejpam-6375	264	21	which	which	PRON
ejpam-6375	264	22	implies	imply	VERB
ejpam-6375	264	23	contradiction	contradiction	NOUN
ejpam-6375	264	24	.	.	PUNCT
ejpam-6375	265	1	as	as	ADP
ejpam-6375	265	2	(	(	PUNCT
ejpam-6375	265	3	v	v	NOUN
ejpam-6375	265	4	,	,	PUNCT
ejpam-6375	265	5	gτ	gτ	INTJ
ejpam-6375	265	6	,	,	PUNCT
ejpam-6375	265	7	p	p	X
ejpam-6375	265	8	)	)	PUNCT
ejpam-6375	265	9	is	be	AUX
ejpam-6375	265	10	gpt	gpt	NOUN
ejpam-6375	265	11	-	-	PUNCT
ejpam-6375	265	12	s∗	s∗	PROPN
ejpam-6375	265	13	g	g	PROPN
ejpam-6375	265	14	-connected	-connected	ADJ
ejpam-6375	265	15	space	space	NOUN
ejpam-6375	265	16	.	.	PUNCT
ejpam-6375	266	1	hence	hence	ADV
ejpam-6375	266	2	,	,	PUNCT
ejpam-6375	266	3	v	v	NOUN
ejpam-6375	266	4	is	be	AUX
ejpam-6375	266	5	∅	∅	NOUN
ejpam-6375	266	6	or	or	CCONJ
ejpam-6375	266	7	v.	v.	ADJ
ejpam-6375	266	8	(	(	PUNCT
ejpam-6375	266	9	ii	ii	PROPN
ejpam-6375	266	10	)	)	PUNCT
ejpam-6375	266	11	then	then	ADV
ejpam-6375	266	12	(	(	PUNCT
ejpam-6375	266	13	i	i	NOUN
ejpam-6375	266	14	):	):	PUNCT
ejpam-6375	266	15	assume	assume	VERB
ejpam-6375	266	16	e	e	NOUN
ejpam-6375	266	17	and	and	CCONJ
ejpam-6375	266	18	d	d	PROPN
ejpam-6375	266	19	as	as	ADP
ejpam-6375	266	20	disjoint	disjoint	PROPN
ejpam-6375	266	21	gpt	gpt	NOUN
ejpam-6375	266	22	-	-	PUNCT
ejpam-6375	266	23	s∗	s∗	PROPN
ejpam-6375	266	24	g	g	PROPN
ejpam-6375	266	25	-open	-open	ADJ
ejpam-6375	266	26	sets	set	NOUN
ejpam-6375	266	27	in	in	ADP
ejpam-6375	266	28	(	(	PUNCT
ejpam-6375	266	29	v	v	NOUN
ejpam-6375	266	30	,	,	PUNCT
ejpam-6375	266	31	gτ	gτ	INTJ
ejpam-6375	266	32	,	,	PUNCT
ejpam-6375	266	33	p	p	X
ejpam-6375	266	34	)	)	PUNCT
ejpam-6375	266	35	and	and	CCONJ
ejpam-6375	266	36	e	e	X
ejpam-6375	266	37	∪	∪	X
ejpam-6375	266	38	d	d	X
ejpam-6375	266	39	=	=	SYM
ejpam-6375	266	40	v.	v.	PROPN
ejpam-6375	266	41	since	since	SCONJ
ejpam-6375	266	42	ec	ec	PROPN
ejpam-6375	266	43	=	=	SYM
ejpam-6375	266	44	d	d	PROPN
ejpam-6375	266	45	,	,	PUNCT
ejpam-6375	266	46	then	then	ADV
ejpam-6375	266	47	e	e	PROPN
ejpam-6375	266	48	is	be	AUX
ejpam-6375	266	49	gpt	gpt	NOUN
ejpam-6375	266	50	-	-	PUNCT
ejpam-6375	266	51	s∗	s∗	PROPN
ejpam-6375	266	52	g	g	PROPN
ejpam-6375	266	53	-open	-open	NOUN
ejpam-6375	266	54	set	set	NOUN
ejpam-6375	266	55	(	(	PUNCT
ejpam-6375	266	56	gpt	gpt	NOUN
ejpam-6375	266	57	-	-	PUNCT
ejpam-6375	266	58	s∗	s∗	PROPN
ejpam-6375	266	59	g	g	PROPN
ejpam-6375	266	60	-closed	-close	VERB
ejpam-6375	266	61	set	set	NOUN
ejpam-6375	266	62	)	)	PUNCT
ejpam-6375	266	63	.	.	PUNCT
ejpam-6375	267	1	then	then	ADV
ejpam-6375	267	2	,	,	PUNCT
ejpam-6375	267	3	by	by	ADP
ejpam-6375	267	4	hypothesis	hypothesis	NOUN
ejpam-6375	267	5	,	,	PUNCT
ejpam-6375	267	6	e	e	X
ejpam-6375	267	7	is	be	AUX
ejpam-6375	267	8	∅	∅	NOUN
ejpam-6375	267	9	or	or	CCONJ
ejpam-6375	267	10	v	v	NOUN
ejpam-6375	267	11	,	,	PUNCT
ejpam-6375	267	12	this	this	PRON
ejpam-6375	267	13	contradicts	contradict	VERB
ejpam-6375	267	14	.	.	PUNCT
ejpam-6375	268	1	hence	hence	ADV
ejpam-6375	268	2	,	,	PUNCT
ejpam-6375	268	3	(	(	PUNCT
ejpam-6375	268	4	v	v	NOUN
ejpam-6375	268	5	,	,	PUNCT
ejpam-6375	268	6	gτ	gτ	INTJ
ejpam-6375	268	7	,	,	PUNCT
ejpam-6375	268	8	p	p	X
ejpam-6375	268	9	)	)	PUNCT
ejpam-6375	268	10	is	be	AUX
ejpam-6375	268	11	gpt	gpt	NOUN
ejpam-6375	268	12	-	-	PUNCT
ejpam-6375	268	13	s∗	s∗	PROPN
ejpam-6375	268	14	g	g	PROPN
ejpam-6375	268	15	-connected	-connected	ADJ
ejpam-6375	268	16	space	space	NOUN
ejpam-6375	268	17	.	.	PUNCT
ejpam-6375	269	1	(	(	PUNCT
ejpam-6375	269	2	ii	ii	NOUN
ejpam-6375	269	3	)	)	PUNCT
ejpam-6375	269	4	then	then	ADV
ejpam-6375	269	5	(	(	PUNCT
ejpam-6375	269	6	iii	iii	X
ejpam-6375	269	7	):	):	PUNCT
ejpam-6375	269	8	a	a	DET
ejpam-6375	269	9	mapping	mapping	NOUN
ejpam-6375	269	10	f	f	NOUN
ejpam-6375	269	11	:	:	PUNCT
ejpam-6375	269	12	(	(	PUNCT
ejpam-6375	269	13	v	v	NOUN
ejpam-6375	269	14	,	,	PUNCT
ejpam-6375	269	15	gτ	gτ	PROPN
ejpam-6375	269	16	1	1	NUM
ejpam-6375	269	17	,	,	PUNCT
ejpam-6375	269	18	pα	pα	NOUN
ejpam-6375	269	19	)	)	PUNCT
ejpam-6375	269	20	→	→	SYM
ejpam-6375	269	21	(	(	PUNCT
ejpam-6375	269	22	z	z	NOUN
ejpam-6375	269	23	,	,	PUNCT
ejpam-6375	269	24	gτ	gτ	PROPN
ejpam-6375	269	25	2	2	NUM
ejpam-6375	269	26	,	,	PUNCT
ejpam-6375	269	27	pβ	pβ	ADP
ejpam-6375	269	28	)	)	PUNCT
ejpam-6375	269	29	that	that	PRON
ejpam-6375	269	30	is	be	AUX
ejpam-6375	269	31	both	both	DET
ejpam-6375	269	32	gpt	gpt	NOUN
ejpam-6375	269	33	-	-	PUNCT
ejpam-6375	269	34	s∗	s∗	PROPN
ejpam-6375	269	35	g	g	PROPN
ejpam-6375	269	36	-continuous	-continuous	ADJ
ejpam-6375	269	37	and	and	CCONJ
ejpam-6375	269	38	constant	constant	ADJ
ejpam-6375	269	39	functions	function	NOUN
ejpam-6375	269	40	to	to	ADP
ejpam-6375	269	41	at	at	ADV
ejpam-6375	269	42	least	least	ADV
ejpam-6375	269	43	two	two	NUM
ejpam-6375	269	44	-	-	PUNCT
ejpam-6375	269	45	point	point	NOUN
ejpam-6375	269	46	discrete	discrete	ADJ
ejpam-6375	269	47	space	space	NOUN
ejpam-6375	269	48	works	work	NOUN
ejpam-6375	269	49	as	as	ADP
ejpam-6375	269	50	an	an	DET
ejpam-6375	269	51	assumption	assumption	NOUN
ejpam-6375	269	52	.	.	PUNCT
ejpam-6375	270	1	the	the	DET
ejpam-6375	270	2	pre	pre	NOUN
ejpam-6375	270	3	-	-	NOUN
ejpam-6375	270	4	image	image	NOUN
ejpam-6375	270	5	of	of	ADP
ejpam-6375	270	6	any	any	DET
ejpam-6375	270	7	set	set	NOUN
ejpam-6375	270	8	point	point	NOUN
ejpam-6375	270	9	under	under	ADP
ejpam-6375	270	10	f	f	PROPN
ejpam-6375	270	11	satisfies	satisfie	NOUN
ejpam-6375	270	12	both	both	CCONJ
ejpam-6375	270	13	gpt	gpt	NOUN
ejpam-6375	270	14	-	-	PUNCT
ejpam-6375	270	15	s∗	s∗	PROPN
ejpam-6375	270	16	g	g	PROPN
ejpam-6375	270	17	-open	-open	ADJ
ejpam-6375	270	18	and	and	CCONJ
ejpam-6375	270	19	gpt	gpt	NOUN
ejpam-6375	270	20	-	-	PUNCT
ejpam-6375	270	21	s∗	s∗	PROPN
ejpam-6375	270	22	g	g	PROPN
ejpam-6375	270	23	-closed	-close	VERB
ejpam-6375	270	24	properties	property	NOUN
ejpam-6375	270	25	throughout	throughout	ADP
ejpam-6375	270	26	the	the	DET
ejpam-6375	270	27	elements	element	NOUN
ejpam-6375	270	28	of	of	ADP
ejpam-6375	270	29	z.	z.	PROPN
ejpam-6375	270	30	the	the	DET
ejpam-6375	270	31	domain	domain	NOUN
ejpam-6375	270	32	set	set	VERB
ejpam-6375	270	33	v	v	NOUN
ejpam-6375	270	34	equals	equal	VERB
ejpam-6375	270	35	the	the	DET
ejpam-6375	270	36	union	union	NOUN
ejpam-6375	270	37	of	of	ADP
ejpam-6375	270	38	these	these	DET
ejpam-6375	270	39	pre	pre	ADJ
ejpam-6375	270	40	-	-	ADJ
ejpam-6375	270	41	image	image	ADJ
ejpam-6375	270	42	sets	set	NOUN
ejpam-6375	270	43	f−1	f−1	PROPN
ejpam-6375	270	44	(	(	PUNCT
ejpam-6375	270	45	{	{	PUNCT
ejpam-6375	270	46	x	x	NOUN
ejpam-6375	270	47	}	}	PUNCT
ejpam-6375	270	48	)	)	PUNCT
ejpam-6375	270	49	,	,	PUNCT
ejpam-6375	270	50	while	while	SCONJ
ejpam-6375	270	51	each	each	DET
ejpam-6375	270	52	set	set	VERB
ejpam-6375	270	53	point	point	NOUN
ejpam-6375	270	54	x	x	PUNCT
ejpam-6375	270	55	belongs	belong	VERB
ejpam-6375	270	56	to	to	ADP
ejpam-6375	270	57	z	z	NOUN
ejpam-6375	270	58	making	make	VERB
ejpam-6375	270	59	the	the	DET
ejpam-6375	270	60	pre	pre	NOUN
ejpam-6375	270	61	-	-	NOUN
ejpam-6375	270	62	images	image	NOUN
ejpam-6375	270	63	both	both	DET
ejpam-6375	270	64	gpt	gpt	NOUN
ejpam-6375	270	65	-	-	PUNCT
ejpam-6375	270	66	s∗	s∗	PROPN
ejpam-6375	270	67	g	g	PROPN
ejpam-6375	270	68	-open	-open	ADJ
ejpam-6375	270	69	and	and	CCONJ
ejpam-6375	270	70	gpt	gpt	NOUN
ejpam-6375	270	71	-	-	PUNCT
ejpam-6375	270	72	s∗	s∗	PROPN
ejpam-6375	270	73	g	g	PROPN
ejpam-6375	270	74	-closed	-close	VERB
ejpam-6375	270	75	in	in	ADP
ejpam-6375	270	76	v.	v.	ADP
ejpam-6375	270	77	the	the	DET
ejpam-6375	270	78	failure	failure	NOUN
ejpam-6375	270	79	for	for	SCONJ
ejpam-6375	270	80	f	f	PROPN
ejpam-6375	270	81	to	to	PART
ejpam-6375	270	82	be	be	AUX
ejpam-6375	270	83	a	a	DET
ejpam-6375	270	84	proper	proper	ADJ
ejpam-6375	270	85	mapping	mapping	NOUN
ejpam-6375	270	86	occurs	occur	VERB
ejpam-6375	270	87	when	when	SCONJ
ejpam-6375	270	88	each	each	DET
ejpam-6375	270	89	inverse	inverse	NOUN
ejpam-6375	270	90	image	image	NOUN
ejpam-6375	270	91	set	set	VERB
ejpam-6375	270	92	f−1	f−1	PROPN
ejpam-6375	270	93	(	(	PUNCT
ejpam-6375	270	94	{	{	PUNCT
ejpam-6375	270	95	x	x	NOUN
ejpam-6375	270	96	}	}	PUNCT
ejpam-6375	270	97	)	)	PUNCT
ejpam-6375	270	98	equals	equal	VERB
ejpam-6375	270	99	either	either	CCONJ
ejpam-6375	270	100	empty	empty	ADJ
ejpam-6375	270	101	set	set	NOUN
ejpam-6375	270	102	or	or	CCONJ
ejpam-6375	270	103	the	the	DET
ejpam-6375	270	104	entire	entire	ADJ
ejpam-6375	270	105	space	space	NOUN
ejpam-6375	270	106	v	v	NOUN
ejpam-6375	270	107	for	for	ADP
ejpam-6375	270	108	all	all	DET
ejpam-6375	270	109	elements	element	NOUN
ejpam-6375	270	110	x	x	PUNCT
ejpam-6375	270	111	in	in	ADP
ejpam-6375	270	112	z.	z.	PROPN
ejpam-6375	270	113	among	among	ADP
ejpam-6375	270	114	all	all	DET
ejpam-6375	270	115	points	point	NOUN
ejpam-6375	270	116	x	x	VERB
ejpam-6375	270	117	in	in	ADP
ejpam-6375	270	118	z	z	NOUN
ejpam-6375	270	119	there	there	PRON
ejpam-6375	270	120	exists	exist	VERB
ejpam-6375	270	121	at	at	ADP
ejpam-6375	270	122	least	least	ADJ
ejpam-6375	270	123	one	one	NUM
ejpam-6375	270	124	where	where	SCONJ
ejpam-6375	270	125	the	the	DET
ejpam-6375	270	126	pre	pre	ADJ
ejpam-6375	270	127	-	-	ADJ
ejpam-6375	270	128	image	image	ADJ
ejpam-6375	270	129	f−1	f−1	PROPN
ejpam-6375	270	130	(	(	PUNCT
ejpam-6375	270	131	{	{	PUNCT
ejpam-6375	270	132	x	x	NOUN
ejpam-6375	270	133	}	}	PUNCT
ejpam-6375	270	134	)	)	PUNCT
ejpam-6375	270	135	consists	consist	VERB
ejpam-6375	270	136	of	of	ADP
ejpam-6375	270	137	v	v	NUM
ejpam-6375	270	138	so	so	CCONJ
ejpam-6375	270	139	f	f	PROPN
ejpam-6375	270	140	functions	function	NOUN
ejpam-6375	270	141	as	as	ADP
ejpam-6375	270	142	a	a	DET
ejpam-6375	270	143	constant	constant	ADJ
ejpam-6375	270	144	.	.	PUNCT
ejpam-6375	271	1	(	(	PUNCT
ejpam-6375	271	2	iii	iii	NOUN
ejpam-6375	271	3	)	)	PUNCT
ejpam-6375	271	4	then	then	ADV
ejpam-6375	271	5	(	(	PUNCT
ejpam-6375	271	6	ii	ii	NOUN
ejpam-6375	271	7	):	):	PUNCT
ejpam-6375	271	8	asume	asume	PROPN
ejpam-6375	271	9	e	e	PROPN
ejpam-6375	271	10	is	be	AUX
ejpam-6375	271	11	gpt	gpt	NOUN
ejpam-6375	271	12	-	-	PUNCT
ejpam-6375	271	13	s∗	s∗	NOUN
ejpam-6375	271	14	g	g	PROPN
ejpam-6375	271	15	-open	-open	ADJ
ejpam-6375	271	16	and	and	CCONJ
ejpam-6375	271	17	gpt	gpt	NOUN
ejpam-6375	271	18	-	-	PUNCT
ejpam-6375	271	19	s∗	s∗	PROPN
ejpam-6375	271	20	g	g	PROPN
ejpam-6375	271	21	-closed	-close	VERB
ejpam-6375	271	22	in	in	ADP
ejpam-6375	271	23	gpts	gpt	NOUN
ejpam-6375	271	24	where	where	SCONJ
ejpam-6375	271	25	e	e	NOUN
ejpam-6375	271	26	is	be	AUX
ejpam-6375	271	27	non	non	ADJ
ejpam-6375	271	28	-	-	ADJ
ejpam-6375	271	29	empty	empty	ADJ
ejpam-6375	271	30	set	set	NOUN
ejpam-6375	271	31	.	.	PUNCT
ejpam-6375	272	1	consider	consider	VERB
ejpam-6375	272	2	gpt	gpt	NOUN
ejpam-6375	272	3	-	-	PUNCT
ejpam-6375	272	4	s∗	s∗	PROPN
ejpam-6375	272	5	g	g	PROPN
ejpam-6375	272	6	-continuous	-continuous	ADJ
ejpam-6375	272	7	mapping	mapping	NOUN
ejpam-6375	273	1	f	f	NOUN
ejpam-6375	273	2	:	:	PUNCT
ejpam-6375	273	3	(	(	PUNCT
ejpam-6375	273	4	v	v	NOUN
ejpam-6375	273	5	,	,	PUNCT
ejpam-6375	273	6	gτ	gτ	PROPN
ejpam-6375	273	7	1	1	NUM
ejpam-6375	273	8	,	,	PUNCT
ejpam-6375	273	9	pα	pα	NOUN
ejpam-6375	273	10	)	)	PUNCT
ejpam-6375	273	11	→	→	SYM
ejpam-6375	273	12	(	(	PUNCT
ejpam-6375	273	13	z	z	NOUN
ejpam-6375	273	14	,	,	PUNCT
ejpam-6375	273	15	gτ	gτ	PROPN
ejpam-6375	273	16	2	2	NUM
ejpam-6375	273	17	,	,	PUNCT
ejpam-6375	273	18	pβ	pβ	NOUN
ejpam-6375	273	19	)	)	PUNCT
ejpam-6375	273	20	defined	define	VERB
ejpam-6375	273	21	as	as	ADP
ejpam-6375	273	22	f	f	PROPN
ejpam-6375	273	23	(	(	PUNCT
ejpam-6375	273	24	e	e	NOUN
ejpam-6375	273	25	)	)	PUNCT
ejpam-6375	273	26	=	=	SYM
ejpam-6375	273	27	{	{	PUNCT
ejpam-6375	273	28	x	x	NOUN
ejpam-6375	273	29	}	}	PUNCT
ejpam-6375	273	30	and	and	CCONJ
ejpam-6375	273	31	f	f	PROPN
ejpam-6375	273	32	(	(	PUNCT
ejpam-6375	273	33	ec	ec	PROPN
ejpam-6375	273	34	)	)	PUNCT
ejpam-6375	273	35	=	=	PRON
ejpam-6375	273	36	{	{	PUNCT
ejpam-6375	273	37	y	y	NOUN
ejpam-6375	273	38	}	}	PUNCT
ejpam-6375	273	39	,	,	PUNCT
ejpam-6375	273	40	where	where	SCONJ
ejpam-6375	273	41	x	x	X
ejpam-6375	273	42	,	,	PUNCT
ejpam-6375	273	43	y	y	PROPN
ejpam-6375	273	44	∈	∈	PROPN
ejpam-6375	273	45	z	z	PROPN
ejpam-6375	273	46	and	and	CCONJ
ejpam-6375	273	47	x	x	SYM
ejpam-6375	273	48	̸=	̸=	PROPN
ejpam-6375	273	49	y.	y.	NOUN
ejpam-6375	273	50	by	by	ADP
ejpam-6375	273	51	hypothesis	hypothesis	NOUN
ejpam-6375	273	52	,	,	PUNCT
ejpam-6375	273	53	f	f	PROPN
ejpam-6375	273	54	is	be	AUX
ejpam-6375	273	55	constant	constant	ADJ
ejpam-6375	273	56	map	map	NOUN
ejpam-6375	273	57	and	and	CCONJ
ejpam-6375	274	1	e	e	NOUN
ejpam-6375	274	2	=	=	PROPN
ejpam-6375	274	3	v.	v.	PROPN
ejpam-6375	274	4	theorem	theorem	VERB
ejpam-6375	274	5	3.10	3.10	NUM
ejpam-6375	274	6	.	.	PUNCT
ejpam-6375	275	1	let	let	VERB
ejpam-6375	275	2	f	f	X
ejpam-6375	275	3	:	:	PUNCT
ejpam-6375	275	4	(	(	PUNCT
ejpam-6375	275	5	v	v	NOUN
ejpam-6375	275	6	,	,	PUNCT
ejpam-6375	275	7	gτ	gτ	PROPN
ejpam-6375	275	8	1	1	NUM
ejpam-6375	275	9	,	,	PUNCT
ejpam-6375	275	10	pα	pα	NOUN
ejpam-6375	275	11	)	)	PUNCT
ejpam-6375	275	12	→	→	SYM
ejpam-6375	275	13	(	(	PUNCT
ejpam-6375	275	14	z	z	NOUN
ejpam-6375	275	15	,	,	PUNCT
ejpam-6375	275	16	gτ	gτ	PROPN
ejpam-6375	275	17	2	2	NUM
ejpam-6375	275	18	,	,	PUNCT
ejpam-6375	275	19	pβ	pβ	NOUN
ejpam-6375	275	20	)	)	PUNCT
ejpam-6375	275	21	is	be	AUX
ejpam-6375	275	22	a	a	DET
ejpam-6375	275	23	surjective	surjective	ADJ
ejpam-6375	275	24	,	,	PUNCT
ejpam-6375	275	25	gpt	gpt	NOUN
ejpam-6375	275	26	-	-	PUNCT
ejpam-6375	275	27	s∗	s∗	PROPN
ejpam-6375	275	28	g	g	PROPN
ejpam-6375	275	29	-continuous	-continuous	ADJ
ejpam-6375	275	30	mapping	mapping	NOUN
ejpam-6375	275	31	and	and	CCONJ
ejpam-6375	275	32	(	(	PUNCT
ejpam-6375	275	33	v	v	NOUN
ejpam-6375	275	34	,	,	PUNCT
ejpam-6375	275	35	gτ	gτ	PROPN
ejpam-6375	275	36	1	1	NUM
ejpam-6375	275	37	,	,	PUNCT
ejpam-6375	275	38	pα	pα	NOUN
ejpam-6375	275	39	)	)	PUNCT
ejpam-6375	275	40	is	be	AUX
ejpam-6375	275	41	gpt	gpt	NOUN
ejpam-6375	275	42	-	-	PUNCT
ejpam-6375	275	43	s∗	s∗	PROPN
ejpam-6375	275	44	g	g	PROPN
ejpam-6375	275	45	-connected	-connected	ADJ
ejpam-6375	275	46	space	space	NOUN
ejpam-6375	275	47	then	then	ADV
ejpam-6375	275	48	(	(	PUNCT
ejpam-6375	275	49	z	z	X
ejpam-6375	275	50	,	,	PUNCT
ejpam-6375	275	51	gτ	gτ	PROPN
ejpam-6375	275	52	2	2	NUM
ejpam-6375	275	53	,	,	PUNCT
ejpam-6375	275	54	pβ	pβ	NOUN
ejpam-6375	275	55	)	)	PUNCT
ejpam-6375	275	56	is	be	AUX
ejpam-6375	275	57	gpt	gpt	NOUN
ejpam-6375	275	58	-	-	PUNCT
ejpam-6375	275	59	connected	connect	VERB
ejpam-6375	275	60	space	space	NOUN
ejpam-6375	275	61	.	.	PUNCT
ejpam-6375	276	1	proof	proof	NOUN
ejpam-6375	276	2	.	.	PUNCT
ejpam-6375	277	1	the	the	DET
ejpam-6375	277	2	space	space	NOUN
ejpam-6375	277	3	(	(	PUNCT
ejpam-6375	277	4	z	z	NOUN
ejpam-6375	277	5	,	,	PUNCT
ejpam-6375	277	6	gτ	gτ	PROPN
ejpam-6375	277	7	2	2	NUM
ejpam-6375	277	8	,	,	PUNCT
ejpam-6375	277	9	pβ	pβ	ADV
ejpam-6375	277	10	)	)	PUNCT
ejpam-6375	277	11	fails	fail	VERB
ejpam-6375	277	12	to	to	PART
ejpam-6375	277	13	be	be	AUX
ejpam-6375	277	14	a	a	DET
ejpam-6375	277	15	gpt	gpt	NOUN
ejpam-6375	277	16	-	-	PUNCT
ejpam-6375	277	17	connected	connect	VERB
ejpam-6375	277	18	space	space	NOUN
ejpam-6375	277	19	.	.	PUNCT
ejpam-6375	278	1	the	the	DET
ejpam-6375	278	2	space	space	NOUN
ejpam-6375	278	3	contains	contain	VERB
ejpam-6375	278	4	two	two	NUM
ejpam-6375	278	5	non	non	ADJ
ejpam-6375	278	6	-	-	ADJ
ejpam-6375	278	7	empty	empty	ADJ
ejpam-6375	278	8	disjoint	disjoint	NOUN
ejpam-6375	278	9	open	open	ADJ
ejpam-6375	278	10	sets	set	NOUN
ejpam-6375	278	11	e	e	NOUN
ejpam-6375	278	12	and	and	CCONJ
ejpam-6375	278	13	d	d	X
ejpam-6375	278	14	which	which	PRON
ejpam-6375	278	15	exist	exist	VERB
ejpam-6375	278	16	within	within	ADP
ejpam-6375	278	17	z.	z.	PROPN
ejpam-6375	278	18	such	such	ADJ
ejpam-6375	278	19	as	as	ADP
ejpam-6375	278	20	e	e	NOUN
ejpam-6375	278	21	∪	∪	NOUN
ejpam-6375	278	22	d	d	X
ejpam-6375	278	23	=	=	PUNCT
ejpam-6375	278	24	z.	z.	PROPN
ejpam-6375	279	1	the	the	DET
ejpam-6375	279	2	fact	fact	NOUN
ejpam-6375	279	3	that	that	SCONJ
ejpam-6375	279	4	f	f	PROPN
ejpam-6375	279	5	demonstrates	demonstrate	VERB
ejpam-6375	279	6	gpt	gpt	NOUN
ejpam-6375	279	7	-	-	PUNCT
ejpam-6375	279	8	s∗	s∗	PROPN
ejpam-6375	279	9	g	g	PROPN
ejpam-6375	279	10	-continuity	-continuity	PROPN
ejpam-6375	279	11	creates	create	VERB
ejpam-6375	279	12	a	a	DET
ejpam-6375	279	13	relationship	relationship	NOUN
ejpam-6375	279	14	between	between	ADP
ejpam-6375	279	15	f−1(e	f−1(e	NOUN
ejpam-6375	279	16	)	)	PUNCT
ejpam-6375	279	17	∪	∪	ADP
ejpam-6375	279	18	f−1(d	f−1(d	PROPN
ejpam-6375	279	19	)	)	PUNCT
ejpam-6375	279	20	=	=	SYM
ejpam-6375	279	21	v	v	NOUN
ejpam-6375	279	22	and	and	CCONJ
ejpam-6375	279	23	f−1(e	f−1(e	NOUN
ejpam-6375	279	24	)	)	PUNCT
ejpam-6375	279	25	,	,	PUNCT
ejpam-6375	279	26	f−1(d	f−1(d	PROPN
ejpam-6375	279	27	)	)	PUNCT
ejpam-6375	279	28	which	which	PRON
ejpam-6375	279	29	implies	imply	VERB
ejpam-6375	279	30	a	a	DET
ejpam-6375	279	31	contradiction	contradiction	NOUN
ejpam-6375	279	32	.	.	PUNCT
ejpam-6375	280	1	thus	thus	ADV
ejpam-6375	280	2	,	,	PUNCT
ejpam-6375	280	3	v	v	NOUN
ejpam-6375	280	4	is	be	AUX
ejpam-6375	280	5	gpt	gpt	NOUN
ejpam-6375	280	6	-	-	PUNCT
ejpam-6375	280	7	s∗	s∗	PROPN
ejpam-6375	280	8	g	g	PROPN
ejpam-6375	280	9	-connected	-connected	PROPN
ejpam-6375	280	10	.	.	PUNCT
ejpam-6375	281	1	both	both	DET
ejpam-6375	281	2	conditions	condition	NOUN
ejpam-6375	281	3	demonstrate	demonstrate	VERB
ejpam-6375	281	4	that	that	SCONJ
ejpam-6375	281	5	the	the	DET
ejpam-6375	281	6	space	space	NOUN
ejpam-6375	281	7	z	z	PROPN
ejpam-6375	281	8	is	be	AUX
ejpam-6375	281	9	gpt	gpt	NOUN
ejpam-6375	281	10	-	-	PUNCT
ejpam-6375	281	11	connected	connect	VERB
ejpam-6375	281	12	.	.	PUNCT
ejpam-6375	282	1	m.	m.	NOUN
ejpam-6375	282	2	shahbaz	shahbaz	PROPN
ejpam-6375	282	3	et	et	PROPN
ejpam-6375	282	4	al	al	PROPN
ejpam-6375	282	5	.	.	PUNCT
ejpam-6375	282	6	/	/	SYM
ejpam-6375	282	7	eur	eur	PROPN
ejpam-6375	282	8	.	.	PUNCT
ejpam-6375	283	1	j.	j.	PROPN
ejpam-6375	283	2	pure	pure	PROPN
ejpam-6375	283	3	appl	appl	PROPN
ejpam-6375	283	4	.	.	PROPN
ejpam-6375	283	5	math	math	PROPN
ejpam-6375	283	6	,	,	PUNCT
ejpam-6375	283	7	18	18	NUM
ejpam-6375	283	8	(	(	PUNCT
ejpam-6375	283	9	4	4	NUM
ejpam-6375	283	10	)	)	PUNCT
ejpam-6375	283	11	(	(	PUNCT
ejpam-6375	283	12	2025	2025	NUM
ejpam-6375	283	13	)	)	PUNCT
ejpam-6375	283	14	,	,	PUNCT
ejpam-6375	283	15	6375	6375	NUM
ejpam-6375	283	16	12	12	NUM
ejpam-6375	283	17	of	of	ADP
ejpam-6375	283	18	22	22	NUM
ejpam-6375	283	19	theorem	theorem	NOUN
ejpam-6375	283	20	3.11	3.11	NUM
ejpam-6375	283	21	.	.	PUNCT
ejpam-6375	284	1	let	let	VERB
ejpam-6375	284	2	f	f	X
ejpam-6375	284	3	:	:	PUNCT
ejpam-6375	284	4	(	(	PUNCT
ejpam-6375	284	5	v	v	NOUN
ejpam-6375	284	6	,	,	PUNCT
ejpam-6375	284	7	gτ	gτ	PROPN
ejpam-6375	284	8	1	1	NUM
ejpam-6375	284	9	,	,	PUNCT
ejpam-6375	284	10	pα	pα	NOUN
ejpam-6375	284	11	)	)	PUNCT
ejpam-6375	284	12	→	→	SYM
ejpam-6375	284	13	(	(	PUNCT
ejpam-6375	284	14	z	z	NOUN
ejpam-6375	284	15	,	,	PUNCT
ejpam-6375	284	16	gτ	gτ	PROPN
ejpam-6375	284	17	2	2	NUM
ejpam-6375	284	18	,	,	PUNCT
ejpam-6375	284	19	pβ	pβ	NOUN
ejpam-6375	284	20	)	)	PUNCT
ejpam-6375	284	21	is	be	AUX
ejpam-6375	284	22	a	a	DET
ejpam-6375	284	23	surjective	surjective	ADJ
ejpam-6375	284	24	,	,	PUNCT
ejpam-6375	284	25	gpt	gpt	NOUN
ejpam-6375	284	26	-	-	PUNCT
ejpam-6375	284	27	s∗	s∗	PROPN
ejpam-6375	284	28	g	g	PROPN
ejpam-6375	284	29	-irresolute	-irresolute	ADJ
ejpam-6375	284	30	function	function	NOUN
ejpam-6375	284	31	and	and	CCONJ
ejpam-6375	284	32	(	(	PUNCT
ejpam-6375	284	33	v	v	NOUN
ejpam-6375	284	34	,	,	PUNCT
ejpam-6375	284	35	gτ	gτ	PROPN
ejpam-6375	284	36	1	1	NUM
ejpam-6375	284	37	,	,	PUNCT
ejpam-6375	284	38	pα	pα	NOUN
ejpam-6375	284	39	)	)	PUNCT
ejpam-6375	284	40	is	be	AUX
ejpam-6375	284	41	gpt	gpt	NOUN
ejpam-6375	284	42	-	-	PUNCT
ejpam-6375	284	43	s∗	s∗	PROPN
ejpam-6375	284	44	g	g	PROPN
ejpam-6375	284	45	-connected	-connected	ADJ
ejpam-6375	284	46	space	space	NOUN
ejpam-6375	284	47	,	,	PUNCT
ejpam-6375	284	48	then	then	ADV
ejpam-6375	284	49	(	(	PUNCT
ejpam-6375	284	50	z	z	X
ejpam-6375	284	51	,	,	PUNCT
ejpam-6375	284	52	gτ	gτ	PROPN
ejpam-6375	284	53	2	2	NUM
ejpam-6375	284	54	,	,	PUNCT
ejpam-6375	284	55	pβ	pβ	NOUN
ejpam-6375	284	56	)	)	PUNCT
ejpam-6375	284	57	is	be	AUX
ejpam-6375	284	58	gpt	gpt	NOUN
ejpam-6375	284	59	-	-	PUNCT
ejpam-6375	284	60	connected	connect	VERB
ejpam-6375	284	61	space	space	NOUN
ejpam-6375	284	62	.	.	PUNCT
ejpam-6375	285	1	proof	proof	NOUN
ejpam-6375	285	2	.	.	PUNCT
ejpam-6375	286	1	assume	assume	VERB
ejpam-6375	286	2	(	(	PUNCT
ejpam-6375	286	3	z	z	NOUN
ejpam-6375	286	4	,	,	PUNCT
ejpam-6375	286	5	gτ	gτ	PROPN
ejpam-6375	286	6	2	2	NUM
ejpam-6375	286	7	,	,	PUNCT
ejpam-6375	286	8	pβ	pβ	NOUN
ejpam-6375	286	9	)	)	PUNCT
ejpam-6375	286	10	is	be	AUX
ejpam-6375	286	11	gpt	gpt	NOUN
ejpam-6375	286	12	-	-	PUNCT
ejpam-6375	286	13	s∗	s∗	PROPN
ejpam-6375	286	14	g	g	PROPN
ejpam-6375	286	15	-connected	-connected	ADJ
ejpam-6375	286	16	space	space	NOUN
ejpam-6375	286	17	.	.	PUNCT
ejpam-6375	287	1	let	let	VERB
ejpam-6375	287	2	e	e	NOUN
ejpam-6375	287	3	and	and	CCONJ
ejpam-6375	287	4	d	d	AUX
ejpam-6375	287	5	be	be	AUX
ejpam-6375	287	6	two	two	NUM
ejpam-6375	287	7	non	non	ADJ
ejpam-6375	287	8	-	-	ADJ
ejpam-6375	287	9	empty	empty	ADJ
ejpam-6375	287	10	gpt	gpt	NOUN
ejpam-6375	287	11	-	-	PUNCT
ejpam-6375	287	12	s∗	s∗	PROPN
ejpam-6375	287	13	g	g	PROPN
ejpam-6375	287	14	-open	-open	ADJ
ejpam-6375	287	15	sets	set	NOUN
ejpam-6375	287	16	in	in	ADP
ejpam-6375	287	17	z.	z.	PROPN
ejpam-6375	287	18	such	such	ADJ
ejpam-6375	287	19	as	as	ADP
ejpam-6375	287	20	e	e	NOUN
ejpam-6375	287	21	∪	∪	NOUN
ejpam-6375	287	22	d	d	X
ejpam-6375	287	23	=	=	SYM
ejpam-6375	287	24	z.	z.	PROPN
ejpam-6375	287	25	as	as	SCONJ
ejpam-6375	287	26	f	f	PROPN
ejpam-6375	287	27	is	be	AUX
ejpam-6375	287	28	surjective	surjective	ADJ
ejpam-6375	287	29	,	,	PUNCT
ejpam-6375	287	30	gpt	gpt	NOUN
ejpam-6375	287	31	-	-	PUNCT
ejpam-6375	287	32	s∗	s∗	PROPN
ejpam-6375	287	33	g	g	PROPN
ejpam-6375	287	34	-continuous	-continuous	ADJ
ejpam-6375	287	35	so	so	ADV
ejpam-6375	287	36	f−1(e	f−1(e	NOUN
ejpam-6375	287	37	)	)	PUNCT
ejpam-6375	287	38	∪	∪	ADP
ejpam-6375	287	39	f−1(d	f−1(d	PROPN
ejpam-6375	287	40	)	)	PUNCT
ejpam-6375	287	41	=	=	SYM
ejpam-6375	287	42	v	v	NOUN
ejpam-6375	287	43	and	and	CCONJ
ejpam-6375	287	44	f−1(e	f−1(e	NOUN
ejpam-6375	287	45	)	)	PUNCT
ejpam-6375	287	46	,	,	PUNCT
ejpam-6375	287	47	f−1(d	f−1(d	PROPN
ejpam-6375	287	48	)	)	PUNCT
ejpam-6375	287	49	is	be	AUX
ejpam-6375	287	50	gpt	gpt	NOUN
ejpam-6375	287	51	-	-	PUNCT
ejpam-6375	287	52	s∗	s∗	PROPN
ejpam-6375	287	53	g	g	PROPN
ejpam-6375	287	54	-open	-open	NOUN
ejpam-6375	287	55	disjoint	disjoint	NOUN
ejpam-6375	287	56	subsets	subset	NOUN
ejpam-6375	287	57	in	in	ADP
ejpam-6375	287	58	v.	v.	ADP
ejpam-6375	287	59	this	this	PRON
ejpam-6375	287	60	implies	imply	VERB
ejpam-6375	287	61	a	a	DET
ejpam-6375	287	62	contradiction	contradiction	NOUN
ejpam-6375	287	63	.	.	PUNCT
ejpam-6375	288	1	so	so	ADV
ejpam-6375	288	2	,	,	PUNCT
ejpam-6375	288	3	v	v	NOUN
ejpam-6375	288	4	is	be	AUX
ejpam-6375	288	5	gpt	gpt	NOUN
ejpam-6375	288	6	-	-	PUNCT
ejpam-6375	288	7	s∗	s∗	PROPN
ejpam-6375	288	8	g	g	PROPN
ejpam-6375	288	9	-connected	-connected	PROPN
ejpam-6375	288	10	.	.	PUNCT
ejpam-6375	289	1	therefore	therefore	ADV
ejpam-6375	289	2	,	,	PUNCT
ejpam-6375	289	3	z	z	PROPN
ejpam-6375	289	4	is	be	AUX
ejpam-6375	289	5	also	also	ADV
ejpam-6375	289	6	gpt	gpt	NOUN
ejpam-6375	289	7	-	-	PUNCT
ejpam-6375	289	8	s∗	s∗	NOUN
ejpam-6375	289	9	g	g	PROPN
ejpam-6375	289	10	-connected	-connected	ADJ
ejpam-6375	289	11	space	space	NOUN
ejpam-6375	289	12	.	.	PUNCT
ejpam-6375	290	1	theorem	theorem	VERB
ejpam-6375	290	2	3.12	3.12	NUM
ejpam-6375	290	3	.	.	PUNCT
ejpam-6375	291	1	suppose	suppose	VERB
ejpam-6375	291	2	f	f	X
ejpam-6375	291	3	:	:	PUNCT
ejpam-6375	291	4	(	(	PUNCT
ejpam-6375	291	5	v	v	NOUN
ejpam-6375	291	6	,	,	PUNCT
ejpam-6375	291	7	gτ	gτ	PROPN
ejpam-6375	291	8	1	1	NUM
ejpam-6375	291	9	,	,	PUNCT
ejpam-6375	291	10	pα	pα	NOUN
ejpam-6375	291	11	)	)	PUNCT
ejpam-6375	291	12	→	→	SYM
ejpam-6375	291	13	(	(	PUNCT
ejpam-6375	291	14	z	z	NOUN
ejpam-6375	291	15	,	,	PUNCT
ejpam-6375	291	16	gτ	gτ	PROPN
ejpam-6375	291	17	2	2	NUM
ejpam-6375	291	18	,	,	PUNCT
ejpam-6375	291	19	pβ	pβ	ADV
ejpam-6375	291	20	)	)	PUNCT
ejpam-6375	291	21	is	be	AUX
ejpam-6375	291	22	a	a	DET
ejpam-6375	291	23	strongly	strongly	ADV
ejpam-6375	291	24	gpt	gpt	NOUN
ejpam-6375	291	25	-	-	PUNCT
ejpam-6375	291	26	s∗	s∗	NOUN
ejpam-6375	291	27	g	g	PROPN
ejpam-6375	291	28	-continuous	-continuous	ADJ
ejpam-6375	291	29	function	function	NOUN
ejpam-6375	291	30	and	and	CCONJ
ejpam-6375	291	31	(	(	PUNCT
ejpam-6375	291	32	v	v	NOUN
ejpam-6375	291	33	,	,	PUNCT
ejpam-6375	291	34	gτ	gτ	PROPN
ejpam-6375	291	35	1	1	NUM
ejpam-6375	291	36	,	,	PUNCT
ejpam-6375	291	37	pα	pα	NOUN
ejpam-6375	291	38	)	)	PUNCT
ejpam-6375	291	39	is	be	AUX
ejpam-6375	291	40	gpt	gpt	NOUN
ejpam-6375	291	41	-	-	PUNCT
ejpam-6375	291	42	connected	connect	VERB
ejpam-6375	291	43	space	space	NOUN
ejpam-6375	291	44	,	,	PUNCT
ejpam-6375	291	45	then	then	ADV
ejpam-6375	291	46	its	its	PRON
ejpam-6375	291	47	image	image	NOUN
ejpam-6375	291	48	is	be	AUX
ejpam-6375	291	49	gpt	gpt	NOUN
ejpam-6375	291	50	-	-	PUNCT
ejpam-6375	291	51	s∗	s∗	PROPN
ejpam-6375	291	52	g	g	PROPN
ejpam-6375	291	53	-connected	-connected	ADJ
ejpam-6375	291	54	space	space	NOUN
ejpam-6375	291	55	.	.	PUNCT
ejpam-6375	292	1	proof	proof	NOUN
ejpam-6375	292	2	.	.	PUNCT
ejpam-6375	293	1	consider	consider	VERB
ejpam-6375	293	2	f	f	NOUN
ejpam-6375	293	3	:	:	PUNCT
ejpam-6375	293	4	(	(	PUNCT
ejpam-6375	293	5	v	v	NOUN
ejpam-6375	293	6	,	,	PUNCT
ejpam-6375	293	7	gτ	gτ	PROPN
ejpam-6375	293	8	1	1	NUM
ejpam-6375	293	9	,	,	PUNCT
ejpam-6375	293	10	pα	pα	NOUN
ejpam-6375	293	11	)	)	PUNCT
ejpam-6375	293	12	→	→	SYM
ejpam-6375	293	13	(	(	PUNCT
ejpam-6375	293	14	z	z	NOUN
ejpam-6375	293	15	,	,	PUNCT
ejpam-6375	293	16	gτ	gτ	PROPN
ejpam-6375	293	17	2	2	NUM
ejpam-6375	293	18	,	,	PUNCT
ejpam-6375	293	19	pβ	pβ	ADV
ejpam-6375	293	20	)	)	PUNCT
ejpam-6375	293	21	is	be	AUX
ejpam-6375	293	22	a	a	DET
ejpam-6375	293	23	strongly	strongly	ADV
ejpam-6375	293	24	gpt	gpt	NOUN
ejpam-6375	293	25	-	-	PUNCT
ejpam-6375	293	26	s∗	s∗	PROPN
ejpam-6375	293	27	g	g	PROPN
ejpam-6375	293	28	-continuous	-continuous	ADJ
ejpam-6375	293	29	map	map	NOUN
ejpam-6375	293	30	and	and	CCONJ
ejpam-6375	293	31	(	(	PUNCT
ejpam-6375	293	32	v	v	NOUN
ejpam-6375	293	33	,	,	PUNCT
ejpam-6375	293	34	gτ	gτ	PROPN
ejpam-6375	293	35	1	1	NUM
ejpam-6375	293	36	,	,	PUNCT
ejpam-6375	293	37	pα	pα	NOUN
ejpam-6375	293	38	)	)	PUNCT
ejpam-6375	293	39	is	be	AUX
ejpam-6375	293	40	gpt	gpt	NOUN
ejpam-6375	293	41	-	-	PUNCT
ejpam-6375	293	42	connected	connect	VERB
ejpam-6375	293	43	space	space	NOUN
ejpam-6375	293	44	.	.	PUNCT
ejpam-6375	294	1	let	let	VERB
ejpam-6375	294	2	(	(	PUNCT
ejpam-6375	294	3	z	z	NOUN
ejpam-6375	294	4	,	,	PUNCT
ejpam-6375	294	5	gτ	gτ	PROPN
ejpam-6375	294	6	2	2	NUM
ejpam-6375	294	7	,	,	PUNCT
ejpam-6375	294	8	pβ	pβ	ADV
ejpam-6375	294	9	)	)	PUNCT
ejpam-6375	294	10	is	be	AUX
ejpam-6375	294	11	not	not	PART
ejpam-6375	294	12	gpt	gpt	NOUN
ejpam-6375	294	13	-	-	PUNCT
ejpam-6375	294	14	s∗	s∗	NOUN
ejpam-6375	294	15	g	g	PROPN
ejpam-6375	294	16	-connected	-connected	ADJ
ejpam-6375	294	17	space	space	NOUN
ejpam-6375	294	18	for	for	ADP
ejpam-6375	294	19	gpt	gpt	NOUN
ejpam-6375	294	20	-	-	PUNCT
ejpam-6375	294	21	s∗	s∗	PROPN
ejpam-6375	294	22	g	g	PROPN
ejpam-6375	294	23	-open	-open	NOUN
ejpam-6375	294	24	sets	set	NOUN
ejpam-6375	294	25	e	e	NOUN
ejpam-6375	294	26	and	and	CCONJ
ejpam-6375	294	27	d	d	PROPN
ejpam-6375	294	28	in	in	ADP
ejpam-6375	294	29	z.	z.	PROPN
ejpam-6375	294	30	such	such	ADJ
ejpam-6375	294	31	as	as	ADP
ejpam-6375	294	32	e	e	NOUN
ejpam-6375	294	33	∪	∪	NOUN
ejpam-6375	294	34	d	d	X
ejpam-6375	294	35	=	=	SYM
ejpam-6375	294	36	z.	z.	PROPN
ejpam-6375	294	37	as	as	SCONJ
ejpam-6375	294	38	f	f	PROPN
ejpam-6375	294	39	is	be	AUX
ejpam-6375	294	40	strongly	strongly	ADV
ejpam-6375	294	41	gpt	gpt	NOUN
ejpam-6375	294	42	-	-	PUNCT
ejpam-6375	294	43	s∗	s∗	NOUN
ejpam-6375	294	44	g	g	PROPN
ejpam-6375	294	45	-continuous	-continuous	ADJ
ejpam-6375	294	46	so	so	ADV
ejpam-6375	294	47	f−1(e	f−1(e	NOUN
ejpam-6375	294	48	)	)	PUNCT
ejpam-6375	294	49	∪	∪	ADP
ejpam-6375	294	50	f−1(d	f−1(d	PROPN
ejpam-6375	294	51	)	)	PUNCT
ejpam-6375	294	52	=	=	SYM
ejpam-6375	294	53	v	v	NOUN
ejpam-6375	294	54	and	and	CCONJ
ejpam-6375	294	55	f−1(e	f−1(e	NOUN
ejpam-6375	294	56	)	)	PUNCT
ejpam-6375	294	57	,	,	PUNCT
ejpam-6375	294	58	f−1(d	f−1(d	PROPN
ejpam-6375	294	59	)	)	PUNCT
ejpam-6375	294	60	is	be	AUX
ejpam-6375	294	61	open	open	ADJ
ejpam-6375	294	62	disjoint	disjoint	NOUN
ejpam-6375	294	63	sets	set	NOUN
ejpam-6375	294	64	in	in	ADP
ejpam-6375	294	65	v.	v.	ADP
ejpam-6375	294	66	this	this	PRON
ejpam-6375	294	67	implies	imply	VERB
ejpam-6375	294	68	a	a	DET
ejpam-6375	294	69	contradiction	contradiction	NOUN
ejpam-6375	294	70	.	.	PUNCT
ejpam-6375	295	1	thus	thus	ADV
ejpam-6375	295	2	,	,	PUNCT
ejpam-6375	295	3	v	v	NOUN
ejpam-6375	295	4	is	be	AUX
ejpam-6375	295	5	gpt	gpt	NOUN
ejpam-6375	295	6	-	-	PUNCT
ejpam-6375	295	7	connected	connect	VERB
ejpam-6375	295	8	.	.	PUNCT
ejpam-6375	296	1	hence	hence	ADV
ejpam-6375	296	2	,	,	PUNCT
ejpam-6375	296	3	z	z	PROPN
ejpam-6375	296	4	is	be	AUX
ejpam-6375	296	5	gpt	gpt	NOUN
ejpam-6375	296	6	-	-	PUNCT
ejpam-6375	296	7	s∗	s∗	PROPN
ejpam-6375	296	8	g	g	PROPN
ejpam-6375	296	9	-connected	-connected	ADJ
ejpam-6375	296	10	space	space	NOUN
ejpam-6375	296	11	.	.	PUNCT
ejpam-6375	297	1	3.2	3.2	NUM
ejpam-6375	297	2	.	.	PUNCT
ejpam-6375	297	3	gpt	gpt	NOUN
ejpam-6375	297	4	-	-	PUNCT
ejpam-6375	297	5	s∗	s∗	PROPN
ejpam-6375	297	6	g	g	PROPN
ejpam-6375	297	7	-separation	-separation	PROPN
ejpam-6375	297	8	axioms	axioms	PROPN
ejpam-6375	297	9	definition	definition	NOUN
ejpam-6375	297	10	3.14	3.14	NUM
ejpam-6375	297	11	.	.	PUNCT
ejpam-6375	298	1	a	a	DET
ejpam-6375	298	2	gpts	gpt	NOUN
ejpam-6375	298	3	(	(	PUNCT
ejpam-6375	298	4	v	v	NOUN
ejpam-6375	298	5	,	,	PUNCT
ejpam-6375	298	6	gτ	gτ	INTJ
ejpam-6375	298	7	,	,	PUNCT
ejpam-6375	298	8	p	p	X
ejpam-6375	298	9	)	)	PUNCT
ejpam-6375	298	10	is	be	AUX
ejpam-6375	298	11	called	call	VERB
ejpam-6375	298	12	gpt	gpt	NOUN
ejpam-6375	298	13	-	-	PUNCT
ejpam-6375	298	14	s∗	s∗	PROPN
ejpam-6375	298	15	gtc	gtc	PROPN
ejpam-6375	298	16	space	space	NOUN
ejpam-6375	298	17	if	if	SCONJ
ejpam-6375	298	18	every	every	DET
ejpam-6375	298	19	gpt	gpt	NOUN
ejpam-6375	298	20	-	-	PUNCT
ejpam-6375	298	21	s∗	s∗	PROPN
ejpam-6375	298	22	g	g	PROPN
ejpam-6375	298	23	-closed	-close	VERB
ejpam-6375	298	24	is	be	AUX
ejpam-6375	298	25	closed	closed	ADJ
ejpam-6375	298	26	.	.	PUNCT
ejpam-6375	299	1	definition	definition	NOUN
ejpam-6375	299	2	3.15	3.15	NUM
ejpam-6375	299	3	.	.	PUNCT
ejpam-6375	300	1	let	let	VERB
ejpam-6375	300	2	e	e	PRON
ejpam-6375	300	3	be	be	AUX
ejpam-6375	300	4	a	a	DET
ejpam-6375	300	5	subset	subset	NOUN
ejpam-6375	300	6	of	of	ADP
ejpam-6375	300	7	v	v	NOUN
ejpam-6375	300	8	which	which	PRON
ejpam-6375	300	9	is	be	AUX
ejpam-6375	300	10	gpt	gpt	NOUN
ejpam-6375	300	11	-	-	PUNCT
ejpam-6375	300	12	t0	t0	NOUN
ejpam-6375	300	13	space	space	NOUN
ejpam-6375	300	14	when	when	SCONJ
ejpam-6375	300	15	every	every	DET
ejpam-6375	300	16	explicit	explicit	ADJ
ejpam-6375	300	17	point	point	NOUN
ejpam-6375	300	18	r	r	NOUN
ejpam-6375	300	19	,	,	PUNCT
ejpam-6375	300	20	s	s	PROPN
ejpam-6375	300	21	of	of	ADP
ejpam-6375	300	22	v	v	NOUN
ejpam-6375	300	23	satisfies	satisfie	NOUN
ejpam-6375	300	24	conditions	condition	NOUN
ejpam-6375	300	25	:	:	PUNCT
ejpam-6375	300	26	either	either	CCONJ
ejpam-6375	300	27	s	s	VERB
ejpam-6375	300	28	/∈	/∈	NOUN
ejpam-6375	300	29	mα	mα	PROPN
ejpam-6375	301	1	and	and	CCONJ
ejpam-6375	301	2	r	r	NOUN
ejpam-6375	301	3	∈	∈	PROPN
ejpam-6375	301	4	mα	mα	NOUN
ejpam-6375	301	5	or	or	CCONJ
ejpam-6375	301	6	r	r	NOUN
ejpam-6375	301	7	/∈	/∈	PUNCT
ejpam-6375	302	1	mα	mα	PROPN
ejpam-6375	302	2	,	,	PUNCT
ejpam-6375	302	3	s	s	PROPN
ejpam-6375	302	4	∈	∈	PROPN
ejpam-6375	302	5	mα	mα	PROPN
ejpam-6375	302	6	,	,	PUNCT
ejpam-6375	302	7	where	where	SCONJ
ejpam-6375	302	8	mα	mα	PROPN
ejpam-6375	302	9	is	be	AUX
ejpam-6375	302	10	an	an	DET
ejpam-6375	302	11	gpt	gpt	NOUN
ejpam-6375	302	12	-	-	PUNCT
ejpam-6375	302	13	open	open	ADJ
ejpam-6375	302	14	set	set	NOUN
ejpam-6375	302	15	of	of	ADP
ejpam-6375	302	16	v.	v.	ADP
ejpam-6375	302	17	definition	definition	NOUN
ejpam-6375	302	18	3.16	3.16	NUM
ejpam-6375	302	19	.	.	PUNCT
ejpam-6375	303	1	let	let	VERB
ejpam-6375	303	2	mα	mα	PROPN
ejpam-6375	303	3	⊆	⊆	NUM
ejpam-6375	303	4	v.	v.	ADP
ejpam-6375	303	5	e	e	PROPN
ejpam-6375	303	6	is	be	AUX
ejpam-6375	303	7	gpt	gpt	NOUN
ejpam-6375	303	8	-	-	PUNCT
ejpam-6375	303	9	t1	t1	NOUN
ejpam-6375	303	10	space	space	NOUN
ejpam-6375	303	11	if	if	SCONJ
ejpam-6375	303	12	for	for	ADP
ejpam-6375	303	13	every	every	DET
ejpam-6375	303	14	explicit	explicit	ADJ
ejpam-6375	303	15	point	point	NOUN
ejpam-6375	303	16	r	r	NOUN
ejpam-6375	303	17	and	and	CCONJ
ejpam-6375	303	18	s	s	PROPN
ejpam-6375	303	19	of	of	ADP
ejpam-6375	303	20	v	v	NOUN
ejpam-6375	303	21	,	,	PUNCT
ejpam-6375	303	22	s	s	PART
ejpam-6375	303	23	/∈	/∈	NOUN
ejpam-6375	304	1	mα	mα	PROPN
ejpam-6375	304	2	,	,	PUNCT
ejpam-6375	305	1	r	r	NOUN
ejpam-6375	305	2	∈	∈	PROPN
ejpam-6375	305	3	mα	mα	NOUN
ejpam-6375	305	4	and	and	CCONJ
ejpam-6375	305	5	r	r	NOUN
ejpam-6375	305	6	/∈	/∈	PUNCT
ejpam-6375	305	7	nα	nα	NOUN
ejpam-6375	305	8	,	,	PUNCT
ejpam-6375	305	9	s	s	PART
ejpam-6375	305	10	∈	∈	PROPN
ejpam-6375	305	11	nα	nα	NOUN
ejpam-6375	305	12	,	,	PUNCT
ejpam-6375	305	13	where	where	SCONJ
ejpam-6375	305	14	mα	mα	PROPN
ejpam-6375	305	15	and	and	CCONJ
ejpam-6375	305	16	nα	nα	PROPN
ejpam-6375	305	17	are	be	AUX
ejpam-6375	305	18	gpt	gpt	NOUN
ejpam-6375	305	19	-	-	PUNCT
ejpam-6375	305	20	open	open	ADJ
ejpam-6375	305	21	sets	set	NOUN
ejpam-6375	305	22	of	of	ADP
ejpam-6375	305	23	v.	v.	ADP
ejpam-6375	305	24	definition	definition	NOUN
ejpam-6375	305	25	3.17	3.17	NUM
ejpam-6375	305	26	.	.	PUNCT
ejpam-6375	306	1	let	let	VERB
ejpam-6375	306	2	e	e	NOUN
ejpam-6375	306	3	⊆	⊆	NUM
ejpam-6375	306	4	v.	v.	ADP
ejpam-6375	306	5	e	e	PROPN
ejpam-6375	306	6	is	be	AUX
ejpam-6375	306	7	known	know	VERB
ejpam-6375	306	8	as	as	ADP
ejpam-6375	306	9	gpt	gpt	NOUN
ejpam-6375	306	10	-	-	PUNCT
ejpam-6375	306	11	t2	t2	NOUN
ejpam-6375	306	12	space	space	NOUN
ejpam-6375	306	13	if	if	SCONJ
ejpam-6375	306	14	for	for	ADP
ejpam-6375	306	15	every	every	DET
ejpam-6375	306	16	explicit	explicit	ADJ
ejpam-6375	306	17	point	point	NOUN
ejpam-6375	306	18	r	r	NOUN
ejpam-6375	306	19	and	and	CCONJ
ejpam-6375	306	20	s	s	PROPN
ejpam-6375	306	21	of	of	ADP
ejpam-6375	306	22	v	v	NOUN
ejpam-6375	306	23	,	,	PUNCT
ejpam-6375	306	24	s	s	PART
ejpam-6375	306	25	/∈	/∈	NOUN
ejpam-6375	307	1	mα	mα	PROPN
ejpam-6375	307	2	,	,	PUNCT
ejpam-6375	308	1	r	r	NOUN
ejpam-6375	308	2	∈	∈	PROPN
ejpam-6375	308	3	mα	mα	NOUN
ejpam-6375	308	4	and	and	CCONJ
ejpam-6375	308	5	r	r	NOUN
ejpam-6375	308	6	/∈	/∈	PUNCT
ejpam-6375	308	7	nα	nα	NOUN
ejpam-6375	308	8	,	,	PUNCT
ejpam-6375	308	9	s	s	PART
ejpam-6375	308	10	∈	∈	PROPN
ejpam-6375	308	11	nα	nα	NOUN
ejpam-6375	308	12	,	,	PUNCT
ejpam-6375	308	13	where	where	SCONJ
ejpam-6375	308	14	mα	mα	PROPN
ejpam-6375	308	15	and	and	CCONJ
ejpam-6375	308	16	nα	nα	PROPN
ejpam-6375	308	17	are	be	AUX
ejpam-6375	308	18	disjoint	disjoint	NOUN
ejpam-6375	308	19	gpt	gpt	NOUN
ejpam-6375	308	20	-	-	PUNCT
ejpam-6375	308	21	open	open	ADJ
ejpam-6375	308	22	sets	set	NOUN
ejpam-6375	308	23	of	of	ADP
ejpam-6375	308	24	v.	v.	ADP
ejpam-6375	308	25	definition	definition	NOUN
ejpam-6375	308	26	3.18	3.18	NUM
ejpam-6375	308	27	.	.	PUNCT
ejpam-6375	309	1	the	the	DET
ejpam-6375	309	2	function	function	NOUN
ejpam-6375	309	3	j	j	PROPN
ejpam-6375	309	4	:	:	PUNCT
ejpam-6375	309	5	v	v	X
ejpam-6375	309	6	→	→	SYM
ejpam-6375	309	7	z	z	NOUN
ejpam-6375	309	8	performs	perform	VERB
ejpam-6375	309	9	as	as	ADP
ejpam-6375	309	10	an	an	DET
ejpam-6375	309	11	gpt	gpt	NOUN
ejpam-6375	309	12	-	-	PUNCT
ejpam-6375	309	13	s∗	s∗	NOUN
ejpam-6375	309	14	g	g	PROPN
ejpam-6375	309	15	-continuous	-continuous	ADJ
ejpam-6375	309	16	operator	operator	NOUN
ejpam-6375	309	17	.	.	PUNCT
ejpam-6375	310	1	a	a	DET
ejpam-6375	310	2	function	function	NOUN
ejpam-6375	310	3	is	be	AUX
ejpam-6375	310	4	gpt	gpt	NOUN
ejpam-6375	310	5	-	-	PUNCT
ejpam-6375	310	6	s∗	s∗	PROPN
ejpam-6375	310	7	g	g	PROPN
ejpam-6375	310	8	-continuous	-continuous	ADJ
ejpam-6375	310	9	when	when	SCONJ
ejpam-6375	310	10	its	its	PRON
ejpam-6375	310	11	inverse	inverse	NOUN
ejpam-6375	310	12	images	image	NOUN
ejpam-6375	310	13	is	be	AUX
ejpam-6375	310	14	gpt	gpt	NOUN
ejpam-6375	310	15	-	-	PUNCT
ejpam-6375	310	16	s∗	s∗	NOUN
ejpam-6375	310	17	g	g	PROPN
ejpam-6375	310	18	-open	-open	NOUN
ejpam-6375	310	19	in	in	ADP
ejpam-6375	310	20	(	(	PUNCT
ejpam-6375	310	21	v	v	NOUN
ejpam-6375	310	22	,	,	PUNCT
ejpam-6375	310	23	gτ	gτ	PROPN
ejpam-6375	310	24	1	1	NUM
ejpam-6375	310	25	,	,	PUNCT
ejpam-6375	310	26	pα	pα	NOUN
ejpam-6375	310	27	)	)	PUNCT
ejpam-6375	310	28	for	for	ADP
ejpam-6375	310	29	every	every	DET
ejpam-6375	310	30	open	open	ADJ
ejpam-6375	310	31	set	set	NOUN
ejpam-6375	310	32	in	in	ADP
ejpam-6375	310	33	(	(	PUNCT
ejpam-6375	310	34	z	z	NOUN
ejpam-6375	310	35	,	,	PUNCT
ejpam-6375	310	36	gτ	gτ	PROPN
ejpam-6375	310	37	2	2	NUM
ejpam-6375	310	38	,	,	PUNCT
ejpam-6375	310	39	pβ	pβ	NOUN
ejpam-6375	310	40	)	)	PUNCT
ejpam-6375	310	41	.	.	PUNCT
ejpam-6375	311	1	3.2.1	3.2.1	NUM
ejpam-6375	311	2	.	.	X
ejpam-6375	311	3	gpt	gpt	NOUN
ejpam-6375	311	4	-	-	PUNCT
ejpam-6375	311	5	s∗	s∗	PROPN
ejpam-6375	311	6	g	g	PROPN
ejpam-6375	311	7	-t0	-t0	PROPN
ejpam-6375	311	8	,	,	PUNCT
ejpam-6375	311	9	gpt	gpt	NOUN
ejpam-6375	311	10	-	-	PUNCT
ejpam-6375	311	11	s∗	s∗	PROPN
ejpam-6375	311	12	g	g	PROPN
ejpam-6375	311	13	-t1	-t1	PROPN
ejpam-6375	311	14	,	,	PUNCT
ejpam-6375	311	15	gpt	gpt	NOUN
ejpam-6375	311	16	-	-	PUNCT
ejpam-6375	311	17	s∗	s∗	PROPN
ejpam-6375	311	18	g	g	PROPN
ejpam-6375	311	19	-t2	-t2	PROPN
ejpam-6375	311	20	spaces	space	NOUN
ejpam-6375	311	21	definition	definition	NOUN
ejpam-6375	311	22	3.19	3.19	NUM
ejpam-6375	311	23	.	.	PUNCT
ejpam-6375	312	1	a	a	DET
ejpam-6375	312	2	subset	subset	NOUN
ejpam-6375	312	3	e	e	NOUN
ejpam-6375	312	4	of	of	ADP
ejpam-6375	312	5	v	v	PROPN
ejpam-6375	312	6	is	be	AUX
ejpam-6375	312	7	called	call	VERB
ejpam-6375	312	8	gpt	gpt	NOUN
ejpam-6375	312	9	-	-	PUNCT
ejpam-6375	312	10	s∗	s∗	PROPN
ejpam-6375	312	11	g	g	PROPN
ejpam-6375	312	12	-t0	-t0	NOUN
ejpam-6375	312	13	space	space	NOUN
ejpam-6375	312	14	if	if	SCONJ
ejpam-6375	312	15	for	for	ADP
ejpam-6375	312	16	any	any	DET
ejpam-6375	312	17	two	two	NUM
ejpam-6375	312	18	different	different	ADJ
ejpam-6375	312	19	points	point	NOUN
ejpam-6375	312	20	r	r	NOUN
ejpam-6375	312	21	and	and	CCONJ
ejpam-6375	312	22	s	s	PART
ejpam-6375	312	23	satisfies	satisfie	NOUN
ejpam-6375	312	24	either	either	CCONJ
ejpam-6375	312	25	s	s	VERB
ejpam-6375	312	26	/∈	/∈	NOUN
ejpam-6375	313	1	mα	mα	PROPN
ejpam-6375	314	1	and	and	CCONJ
ejpam-6375	314	2	r	r	NOUN
ejpam-6375	314	3	∈	∈	PROPN
ejpam-6375	314	4	mα	mα	NOUN
ejpam-6375	314	5	or	or	CCONJ
ejpam-6375	314	6	r	r	NOUN
ejpam-6375	314	7	/∈	/∈	PUNCT
ejpam-6375	315	1	mα	mα	PROPN
ejpam-6375	315	2	and	and	CCONJ
ejpam-6375	315	3	s∈	s∈	NOUN
ejpam-6375	315	4	mα	mα	PROPN
ejpam-6375	315	5	,	,	PUNCT
ejpam-6375	315	6	where	where	SCONJ
ejpam-6375	315	7	mα	mα	PROPN
ejpam-6375	315	8	is	be	AUX
ejpam-6375	315	9	gpt	gpt	NOUN
ejpam-6375	315	10	-	-	PUNCT
ejpam-6375	315	11	s∗	s∗	PROPN
ejpam-6375	315	12	g−open	g−open	NOUN
ejpam-6375	315	13	set	set	VERB
ejpam-6375	315	14	.	.	PUNCT
ejpam-6375	316	1	m.	m.	PROPN
ejpam-6375	316	2	shahbaz	shahbaz	PROPN
ejpam-6375	316	3	et	et	PROPN
ejpam-6375	316	4	al	al	PROPN
ejpam-6375	316	5	.	.	PUNCT
ejpam-6375	316	6	/	/	SYM
ejpam-6375	316	7	eur	eur	PROPN
ejpam-6375	316	8	.	.	PUNCT
ejpam-6375	317	1	j.	j.	PROPN
ejpam-6375	317	2	pure	pure	PROPN
ejpam-6375	317	3	appl	appl	PROPN
ejpam-6375	317	4	.	.	PROPN
ejpam-6375	317	5	math	math	PROPN
ejpam-6375	317	6	,	,	PUNCT
ejpam-6375	317	7	18	18	NUM
ejpam-6375	317	8	(	(	PUNCT
ejpam-6375	317	9	4	4	NUM
ejpam-6375	317	10	)	)	PUNCT
ejpam-6375	317	11	(	(	PUNCT
ejpam-6375	317	12	2025	2025	NUM
ejpam-6375	317	13	)	)	PUNCT
ejpam-6375	317	14	,	,	PUNCT
ejpam-6375	317	15	6375	6375	NUM
ejpam-6375	317	16	13	13	NUM
ejpam-6375	317	17	of	of	ADP
ejpam-6375	317	18	22	22	NUM
ejpam-6375	317	19	definition	definition	NOUN
ejpam-6375	317	20	3.20	3.20	NUM
ejpam-6375	317	21	.	.	PUNCT
ejpam-6375	318	1	a	a	DET
ejpam-6375	318	2	subset	subset	NOUN
ejpam-6375	318	3	e	e	NOUN
ejpam-6375	318	4	of	of	ADP
ejpam-6375	318	5	v	v	PROPN
ejpam-6375	318	6	is	be	AUX
ejpam-6375	318	7	called	call	VERB
ejpam-6375	318	8	gpt	gpt	NOUN
ejpam-6375	318	9	-	-	PUNCT
ejpam-6375	318	10	s∗	s∗	PROPN
ejpam-6375	318	11	g	g	PROPN
ejpam-6375	318	12	-t1	-t1	NOUN
ejpam-6375	318	13	space	space	NOUN
ejpam-6375	318	14	if	if	SCONJ
ejpam-6375	318	15	for	for	ADP
ejpam-6375	318	16	any	any	DET
ejpam-6375	318	17	two	two	NUM
ejpam-6375	318	18	distinct	distinct	ADJ
ejpam-6375	318	19	point	point	NOUN
ejpam-6375	318	20	r	r	NOUN
ejpam-6375	318	21	and	and	CCONJ
ejpam-6375	318	22	s	s	PROPN
ejpam-6375	318	23	of	of	ADP
ejpam-6375	318	24	v	v	NOUN
ejpam-6375	318	25	,	,	PUNCT
ejpam-6375	318	26	s	s	PART
ejpam-6375	318	27	/∈	/∈	NOUN
ejpam-6375	318	28	mα	mα	PROPN
ejpam-6375	318	29	,	,	PUNCT
ejpam-6375	318	30	r	r	NOUN
ejpam-6375	318	31	∈	∈	PROPN
ejpam-6375	318	32	mα	mα	NOUN
ejpam-6375	318	33	and	and	CCONJ
ejpam-6375	318	34	r	r	NOUN
ejpam-6375	318	35	/∈	/∈	PUNCT
ejpam-6375	318	36	nα	nα	NOUN
ejpam-6375	318	37	,	,	PUNCT
ejpam-6375	318	38	s	s	PART
ejpam-6375	318	39	∈	∈	PROPN
ejpam-6375	318	40	nα	nα	NOUN
ejpam-6375	318	41	,	,	PUNCT
ejpam-6375	318	42	where	where	SCONJ
ejpam-6375	318	43	mα	mα	PROPN
ejpam-6375	318	44	and	and	CCONJ
ejpam-6375	318	45	nα	nα	PROPN
ejpam-6375	318	46	are	be	AUX
ejpam-6375	318	47	gpt	gpt	NOUN
ejpam-6375	318	48	-	-	PUNCT
ejpam-6375	318	49	s∗	s∗	NOUN
ejpam-6375	318	50	g	g	PROPN
ejpam-6375	318	51	-open	-open	ADJ
ejpam-6375	318	52	sets	set	NOUN
ejpam-6375	318	53	of	of	ADP
ejpam-6375	318	54	v.	v.	ADP
ejpam-6375	318	55	definition	definition	NOUN
ejpam-6375	318	56	3.21	3.21	NUM
ejpam-6375	318	57	.	.	PUNCT
ejpam-6375	319	1	let	let	VERB
ejpam-6375	319	2	e	e	NOUN
ejpam-6375	319	3	⊆	⊆	NUM
ejpam-6375	319	4	v.	v.	ADP
ejpam-6375	319	5	e	e	PROPN
ejpam-6375	319	6	is	be	AUX
ejpam-6375	319	7	called	call	VERB
ejpam-6375	319	8	gpt	gpt	NOUN
ejpam-6375	319	9	-	-	PUNCT
ejpam-6375	319	10	s∗	s∗	PROPN
ejpam-6375	319	11	g	g	PROPN
ejpam-6375	319	12	-t2	-t2	PROPN
ejpam-6375	319	13	(	(	PUNCT
ejpam-6375	319	14	gpt	gpt	NOUN
ejpam-6375	319	15	-	-	PUNCT
ejpam-6375	319	16	s∗	s∗	PROPN
ejpam-6375	319	17	g	g	PROPN
ejpam-6375	319	18	-housdorff	-housdorff	NOUN
ejpam-6375	319	19	)	)	PUNCT
ejpam-6375	319	20	space	space	NOUN
ejpam-6375	319	21	if	if	SCONJ
ejpam-6375	319	22	there	there	PRON
ejpam-6375	319	23	exist	exist	VERB
ejpam-6375	319	24	two	two	NUM
ejpam-6375	319	25	disjoint	disjoint	NOUN
ejpam-6375	319	26	gpt	gpt	NOUN
ejpam-6375	319	27	-	-	PUNCT
ejpam-6375	319	28	s∗	s∗	PROPN
ejpam-6375	319	29	g	g	PROPN
ejpam-6375	319	30	-open	-open	PROPN
ejpam-6375	319	31	sets	set	NOUN
ejpam-6375	319	32	mα	mα	PROPN
ejpam-6375	319	33	and	and	CCONJ
ejpam-6375	319	34	nα	nα	VERB
ejpam-6375	319	35	for	for	ADP
ejpam-6375	319	36	any	any	DET
ejpam-6375	319	37	two	two	NUM
ejpam-6375	319	38	different	different	ADJ
ejpam-6375	319	39	points	point	NOUN
ejpam-6375	319	40	r	r	NOUN
ejpam-6375	319	41	and	and	CCONJ
ejpam-6375	319	42	s	s	PROPN
ejpam-6375	319	43	of	of	ADP
ejpam-6375	319	44	v	v	NOUN
ejpam-6375	319	45	,	,	PUNCT
ejpam-6375	319	46	s	s	PART
ejpam-6375	319	47	/∈	/∈	NOUN
ejpam-6375	320	1	mα	mα	PROPN
ejpam-6375	320	2	,	,	PUNCT
ejpam-6375	321	1	r	r	NOUN
ejpam-6375	321	2	∈	∈	PROPN
ejpam-6375	321	3	mα	mα	NOUN
ejpam-6375	321	4	and	and	CCONJ
ejpam-6375	321	5	r	r	NOUN
ejpam-6375	321	6	/∈	/∈	PUNCT
ejpam-6375	321	7	nα	nα	NOUN
ejpam-6375	321	8	,	,	PUNCT
ejpam-6375	321	9	s	s	PART
ejpam-6375	321	10	∈	∈	PROPN
ejpam-6375	321	11	nα	nα	NOUN
ejpam-6375	321	12	.	.	PUNCT
ejpam-6375	321	13	theorem	theorem	VERB
ejpam-6375	321	14	3.13	3.13	NUM
ejpam-6375	321	15	.	.	PUNCT
ejpam-6375	322	1	i.	i.	NOUN
ejpam-6375	322	2	if	if	SCONJ
ejpam-6375	322	3	v	v	NOUN
ejpam-6375	322	4	is	be	AUX
ejpam-6375	322	5	gpt	gpt	NOUN
ejpam-6375	322	6	-	-	PUNCT
ejpam-6375	322	7	t0	t0	NOUN
ejpam-6375	322	8	,	,	PUNCT
ejpam-6375	322	9	then	then	ADV
ejpam-6375	322	10	v	v	NOUN
ejpam-6375	322	11	is	be	AUX
ejpam-6375	322	12	gpt	gpt	NOUN
ejpam-6375	322	13	-	-	PUNCT
ejpam-6375	322	14	s∗	s∗	PROPN
ejpam-6375	322	15	g	g	PROPN
ejpam-6375	322	16	-t0	-t0	PROPN
ejpam-6375	322	17	.	.	PUNCT
ejpam-6375	322	18	ii	ii	PROPN
ejpam-6375	322	19	.	.	PUNCT
ejpam-6375	323	1	if	if	SCONJ
ejpam-6375	323	2	v	v	NOUN
ejpam-6375	323	3	is	be	AUX
ejpam-6375	323	4	gpt	gpt	NOUN
ejpam-6375	323	5	-	-	PUNCT
ejpam-6375	323	6	t1	t1	NOUN
ejpam-6375	323	7	,	,	PUNCT
ejpam-6375	323	8	then	then	ADV
ejpam-6375	323	9	v	v	NOUN
ejpam-6375	323	10	is	be	AUX
ejpam-6375	323	11	gpt	gpt	NOUN
ejpam-6375	323	12	-	-	PUNCT
ejpam-6375	323	13	s∗	s∗	PROPN
ejpam-6375	323	14	g	g	PROPN
ejpam-6375	323	15	-t0	-t0	PROPN
ejpam-6375	323	16	and	and	CCONJ
ejpam-6375	323	17	gpt	gpt	NOUN
ejpam-6375	323	18	-	-	PUNCT
ejpam-6375	323	19	s∗	s∗	PROPN
ejpam-6375	323	20	g	g	PROPN
ejpam-6375	323	21	-t1	-t1	X
ejpam-6375	323	22	.	.	PUNCT
ejpam-6375	324	1	iii	iii	X
ejpam-6375	324	2	.	.	PUNCT
ejpam-6375	325	1	if	if	SCONJ
ejpam-6375	325	2	v	v	NOUN
ejpam-6375	325	3	is	be	AUX
ejpam-6375	325	4	gpt	gpt	NOUN
ejpam-6375	325	5	-	-	PUNCT
ejpam-6375	325	6	t2	t2	NOUN
ejpam-6375	325	7	,	,	PUNCT
ejpam-6375	325	8	then	then	ADV
ejpam-6375	325	9	v	v	NOUN
ejpam-6375	325	10	is	be	AUX
ejpam-6375	325	11	gpt	gpt	NOUN
ejpam-6375	325	12	-	-	PUNCT
ejpam-6375	325	13	s∗	s∗	PROPN
ejpam-6375	325	14	g	g	PROPN
ejpam-6375	325	15	-t2	-t2	PROPN
ejpam-6375	325	16	.	.	PUNCT
ejpam-6375	326	1	iv	iv	X
ejpam-6375	326	2	.	.	PUNCT
ejpam-6375	327	1	if	if	SCONJ
ejpam-6375	327	2	v	v	NOUN
ejpam-6375	327	3	is	be	AUX
ejpam-6375	327	4	gpt	gpt	NOUN
ejpam-6375	327	5	-	-	PUNCT
ejpam-6375	327	6	s∗	s∗	PROPN
ejpam-6375	327	7	g	g	PROPN
ejpam-6375	327	8	-t2	-t2	PROPN
ejpam-6375	327	9	,	,	PUNCT
ejpam-6375	327	10	then	then	ADV
ejpam-6375	327	11	v	v	NOUN
ejpam-6375	327	12	is	be	AUX
ejpam-6375	327	13	gpt	gpt	NOUN
ejpam-6375	327	14	-	-	PUNCT
ejpam-6375	327	15	s∗	s∗	PROPN
ejpam-6375	327	16	g	g	PROPN
ejpam-6375	327	17	-t0	-t0	PROPN
ejpam-6375	327	18	.	.	PUNCT
ejpam-6375	328	1	v.	v.	INTJ
ejpam-6375	328	2	if	if	SCONJ
ejpam-6375	328	3	v	v	NOUN
ejpam-6375	328	4	is	be	AUX
ejpam-6375	328	5	gpt	gpt	NOUN
ejpam-6375	328	6	-	-	PUNCT
ejpam-6375	328	7	s∗	s∗	PROPN
ejpam-6375	328	8	g	g	PROPN
ejpam-6375	328	9	-t2	-t2	PROPN
ejpam-6375	328	10	,	,	PUNCT
ejpam-6375	328	11	then	then	ADV
ejpam-6375	328	12	v	v	NOUN
ejpam-6375	328	13	is	be	AUX
ejpam-6375	328	14	gpt	gpt	NOUN
ejpam-6375	328	15	-	-	PUNCT
ejpam-6375	328	16	s∗	s∗	NOUN
ejpam-6375	328	17	g	g	PROPN
ejpam-6375	328	18	-t1	-t1	X
ejpam-6375	328	19	.	.	PUNCT
ejpam-6375	329	1	proof	proof	NOUN
ejpam-6375	329	2	.	.	PUNCT
ejpam-6375	330	1	i	i	PRON
ejpam-6375	330	2	)	)	PUNCT
ejpam-6375	330	3	given	give	VERB
ejpam-6375	330	4	,	,	PUNCT
ejpam-6375	330	5	v	v	NOUN
ejpam-6375	330	6	is	be	AUX
ejpam-6375	330	7	gpt	gpt	NOUN
ejpam-6375	330	8	-	-	PUNCT
ejpam-6375	330	9	t0	t0	NOUN
ejpam-6375	330	10	space	space	NOUN
ejpam-6375	330	11	.	.	PUNCT
ejpam-6375	331	1	an	an	DET
ejpam-6375	331	2	open	open	ADJ
ejpam-6375	331	3	set	set	NOUN
ejpam-6375	331	4	f	f	PROPN
ejpam-6375	331	5	exists	exist	NOUN
ejpam-6375	331	6	which	which	PRON
ejpam-6375	331	7	covers	cover	VERB
ejpam-6375	331	8	every	every	DET
ejpam-6375	331	9	pair	pair	NOUN
ejpam-6375	331	10	of	of	ADP
ejpam-6375	331	11	points	point	NOUN
ejpam-6375	331	12	r	r	NOUN
ejpam-6375	331	13	,	,	PUNCT
ejpam-6375	331	14	s	s	AUX
ejpam-6375	331	15	belonging	belong	VERB
ejpam-6375	331	16	to	to	ADP
ejpam-6375	331	17	v	v	NOUN
ejpam-6375	331	18	such	such	ADJ
ejpam-6375	331	19	that	that	DET
ejpam-6375	331	20	s	s	PART
ejpam-6375	331	21	/∈	/∈	NOUN
ejpam-6375	331	22	f	f	PROPN
ejpam-6375	332	1	while	while	SCONJ
ejpam-6375	332	2	r	r	NOUN
ejpam-6375	332	3	∈	∈	PROPN
ejpam-6375	332	4	f	f	NOUN
ejpam-6375	332	5	or	or	CCONJ
ejpam-6375	332	6	r	r	NOUN
ejpam-6375	332	7	/∈	/∈	PUNCT
ejpam-6375	333	1	f	f	NOUN
ejpam-6375	334	1	but	but	CCONJ
ejpam-6375	334	2	s	s	PROPN
ejpam-6375	334	3	∈	∈	PROPN
ejpam-6375	334	4	f.	f.	NOUN
ejpam-6375	334	5	the	the	DET
ejpam-6375	334	6	family	family	NOUN
ejpam-6375	334	7	set	set	VERB
ejpam-6375	334	8	f	f	PROPN
ejpam-6375	334	9	belongs	belong	VERB
ejpam-6375	334	10	to	to	ADP
ejpam-6375	334	11	the	the	DET
ejpam-6375	334	12	collection	collection	NOUN
ejpam-6375	334	13	gpt	gpt	NOUN
ejpam-6375	334	14	-	-	PUNCT
ejpam-6375	334	15	s∗	s∗	PROPN
ejpam-6375	334	16	go(v	go(v	NOUN
ejpam-6375	334	17	)	)	PUNCT
ejpam-6375	334	18	while	while	SCONJ
ejpam-6375	334	19	satisfying	satisfy	VERB
ejpam-6375	334	20	two	two	NUM
ejpam-6375	334	21	conditions	condition	NOUN
ejpam-6375	334	22	:	:	PUNCT
ejpam-6375	334	23	s	s	X
ejpam-6375	334	24	/∈	/∈	PUNCT
ejpam-6375	335	1	f	f	PROPN
ejpam-6375	335	2	and	and	CCONJ
ejpam-6375	335	3	r	r	PROPN
ejpam-6375	335	4	∈	∈	PROPN
ejpam-6375	335	5	f	f	NOUN
ejpam-6375	335	6	or	or	CCONJ
ejpam-6375	335	7	r	r	NOUN
ejpam-6375	335	8	/∈	/∈	PUNCT
ejpam-6375	336	1	f	f	PROPN
ejpam-6375	336	2	and	and	CCONJ
ejpam-6375	336	3	s	s	PROPN
ejpam-6375	336	4	∈	∈	PROPN
ejpam-6375	336	5	f.	f.	PROPN
ejpam-6375	336	6	thus	thus	ADV
ejpam-6375	336	7	,	,	PUNCT
ejpam-6375	336	8	v	v	NOUN
ejpam-6375	336	9	is	be	AUX
ejpam-6375	336	10	gpt	gpt	NOUN
ejpam-6375	336	11	-	-	PUNCT
ejpam-6375	336	12	s∗	s∗	PROPN
ejpam-6375	336	13	g	g	PROPN
ejpam-6375	336	14	-t0	-t0	PROPN
ejpam-6375	336	15	space	space	NOUN
ejpam-6375	336	16	.	.	PUNCT
ejpam-6375	337	1	the	the	DET
ejpam-6375	337	2	demonstration	demonstration	NOUN
ejpam-6375	337	3	of	of	ADP
ejpam-6375	337	4	all	all	DET
ejpam-6375	337	5	proposed	propose	VERB
ejpam-6375	337	6	points	point	NOUN
ejpam-6375	337	7	ii	ii	PROPN
ejpam-6375	337	8	)	)	PUNCT
ejpam-6375	337	9	,	,	PUNCT
ejpam-6375	337	10	iii	iii	PROPN
ejpam-6375	337	11	)	)	PUNCT
ejpam-6375	337	12	,	,	PUNCT
ejpam-6375	337	13	iv	iv	X
ejpam-6375	337	14	)	)	PUNCT
ejpam-6375	337	15	,	,	PUNCT
ejpam-6375	337	16	and	and	CCONJ
ejpam-6375	337	17	v	v	NOUN
ejpam-6375	337	18	)	)	PUNCT
ejpam-6375	337	19	is	be	AUX
ejpam-6375	337	20	also	also	ADV
ejpam-6375	337	21	possible	possible	ADJ
ejpam-6375	337	22	in	in	ADP
ejpam-6375	337	23	the	the	DET
ejpam-6375	337	24	same	same	ADJ
ejpam-6375	337	25	way	way	NOUN
ejpam-6375	337	26	.	.	PUNCT
ejpam-6375	338	1	remark	remark	VERB
ejpam-6375	338	2	3.3	3.3	NUM
ejpam-6375	338	3	.	.	PUNCT
ejpam-6375	339	1	an	an	DET
ejpam-6375	339	2	illustration	illustration	NOUN
ejpam-6375	339	3	disproves	disprove	VERB
ejpam-6375	339	4	the	the	DET
ejpam-6375	339	5	invalidity	invalidity	NOUN
ejpam-6375	339	6	of	of	ADP
ejpam-6375	339	7	the	the	DET
ejpam-6375	339	8	converse	converse	NOUN
ejpam-6375	339	9	statement	statement	NOUN
ejpam-6375	339	10	derived	derive	VERB
ejpam-6375	339	11	from	from	ADP
ejpam-6375	339	12	above	above	ADP
ejpam-6375	339	13	.	.	PUNCT
ejpam-6375	339	14	example	example	NOUN
ejpam-6375	340	1	3.6	3.6	NUM
ejpam-6375	340	2	.	.	PUNCT
ejpam-6375	341	1	let	let	VERB
ejpam-6375	341	2	v	v	VERB
ejpam-6375	341	3	=	=	SYM
ejpam-6375	341	4	{	{	PUNCT
ejpam-6375	341	5	j1	j1	PROPN
ejpam-6375	341	6	,	,	PUNCT
ejpam-6375	341	7	k1	k1	NOUN
ejpam-6375	341	8	,	,	PUNCT
ejpam-6375	341	9	l1	l1	PROPN
ejpam-6375	341	10	}	}	PUNCT
ejpam-6375	341	11	and	and	CCONJ
ejpam-6375	341	12	gτ	gτ	PROPN
ejpam-6375	341	13	=	=	SYM
ejpam-6375	341	14	{	{	PUNCT
ejpam-6375	341	15	∅	∅	NOUN
ejpam-6375	341	16	,	,	PUNCT
ejpam-6375	341	17	v	v	NOUN
ejpam-6375	341	18	,	,	PUNCT
ejpam-6375	341	19	{	{	PUNCT
ejpam-6375	341	20	l1	l1	PROPN
ejpam-6375	341	21	}	}	PUNCT
ejpam-6375	341	22	,	,	PUNCT
ejpam-6375	341	23	{	{	PUNCT
ejpam-6375	341	24	j1	j1	PROPN
ejpam-6375	341	25	,	,	PUNCT
ejpam-6375	341	26	k1	k1	NOUN
ejpam-6375	341	27	}	}	PUNCT
ejpam-6375	341	28	}	}	PUNCT
ejpam-6375	341	29	,	,	PUNCT
ejpam-6375	341	30	p	p	NOUN
ejpam-6375	341	31	=	=	X
ejpam-6375	341	32	{	{	PUNCT
ejpam-6375	341	33	∅	∅	NOUN
ejpam-6375	341	34	,	,	PUNCT
ejpam-6375	341	35	{	{	PUNCT
ejpam-6375	341	36	j1	j1	PROPN
ejpam-6375	341	37	}	}	PUNCT
ejpam-6375	341	38	,	,	PUNCT
ejpam-6375	341	39	{	{	PUNCT
ejpam-6375	341	40	l1	l1	PROPN
ejpam-6375	341	41	}	}	PUNCT
ejpam-6375	341	42	,	,	PUNCT
ejpam-6375	341	43	{	{	PUNCT
ejpam-6375	341	44	j1	j1	PROPN
ejpam-6375	341	45	,	,	PUNCT
ejpam-6375	341	46	l1	l1	PROPN
ejpam-6375	341	47	}	}	PUNCT
ejpam-6375	341	48	}	}	PUNCT
ejpam-6375	341	49	.	.	PUNCT
ejpam-6375	342	1	in	in	ADP
ejpam-6375	342	2	(	(	PUNCT
ejpam-6375	342	3	v	v	NOUN
ejpam-6375	342	4	,	,	PUNCT
ejpam-6375	342	5	gτ	gτ	INTJ
ejpam-6375	342	6	,	,	PUNCT
ejpam-6375	342	7	p)o	p)o	PUNCT
ejpam-6375	342	8	=	=	PUNCT
ejpam-6375	342	9	{	{	PUNCT
ejpam-6375	342	10	∅	∅	NOUN
ejpam-6375	342	11	,	,	PUNCT
ejpam-6375	342	12	v	v	NOUN
ejpam-6375	342	13	,	,	PUNCT
ejpam-6375	342	14	{	{	PUNCT
ejpam-6375	342	15	l1	l1	PROPN
ejpam-6375	342	16	}	}	PUNCT
ejpam-6375	342	17	,	,	PUNCT
ejpam-6375	342	18	{	{	PUNCT
ejpam-6375	342	19	j1	j1	PROPN
ejpam-6375	342	20	,	,	PUNCT
ejpam-6375	342	21	k1	k1	NOUN
ejpam-6375	342	22	}	}	PUNCT
ejpam-6375	342	23	}	}	PUNCT
ejpam-6375	342	24	and	and	CCONJ
ejpam-6375	342	25	gpt	gpt	NOUN
ejpam-6375	342	26	-	-	PUNCT
ejpam-6375	342	27	s∗	s∗	NOUN
ejpam-6375	342	28	go	go	VERB
ejpam-6375	342	29	=	=	SYM
ejpam-6375	342	30	p(v	p(v	NOUN
ejpam-6375	342	31	)	)	PUNCT
ejpam-6375	342	32	.	.	PUNCT
ejpam-6375	343	1	hence	hence	ADV
ejpam-6375	343	2	,	,	PUNCT
ejpam-6375	343	3	(	(	PUNCT
ejpam-6375	343	4	v	v	NOUN
ejpam-6375	343	5	,	,	PUNCT
ejpam-6375	343	6	gτ	gτ	INTJ
ejpam-6375	343	7	,	,	PUNCT
ejpam-6375	343	8	p	p	X
ejpam-6375	343	9	)	)	PUNCT
ejpam-6375	343	10	is	be	AUX
ejpam-6375	343	11	◦	◦	NOUN
ejpam-6375	343	12	gpt	gpt	NOUN
ejpam-6375	343	13	-	-	PUNCT
ejpam-6375	343	14	s∗	s∗	PROPN
ejpam-6375	343	15	g	g	PROPN
ejpam-6375	343	16	-t0	-t0	PROPN
ejpam-6375	343	17	but	but	CCONJ
ejpam-6375	343	18	not	not	PART
ejpam-6375	343	19	gpt	gpt	NOUN
ejpam-6375	343	20	-	-	PUNCT
ejpam-6375	343	21	t0	t0	NOUN
ejpam-6375	343	22	.	.	PUNCT
ejpam-6375	344	1	no	no	DET
ejpam-6375	344	2	open	open	ADJ
ejpam-6375	344	3	set	set	NOUN
ejpam-6375	344	4	exists	exist	VERB
ejpam-6375	344	5	with	with	ADP
ejpam-6375	344	6	s	s	PRON
ejpam-6375	344	7	/∈	/∈	PUNCT
ejpam-6375	344	8	mα	mα	PROPN
ejpam-6375	344	9	,	,	PUNCT
ejpam-6375	344	10	r	r	NOUN
ejpam-6375	344	11	∈	∈	PROPN
ejpam-6375	344	12	mα	mα	NOUN
ejpam-6375	344	13	or	or	CCONJ
ejpam-6375	344	14	r	r	NOUN
ejpam-6375	344	15	/∈	/∈	PUNCT
ejpam-6375	345	1	mα	mα	PROPN
ejpam-6375	345	2	,	,	PUNCT
ejpam-6375	345	3	s	s	PROPN
ejpam-6375	345	4	∈	∈	PROPN
ejpam-6375	345	5	mα	mα	PROPN
ejpam-6375	345	6	,	,	PUNCT
ejpam-6375	345	7	where	where	SCONJ
ejpam-6375	345	8	mα	mα	PROPN
ejpam-6375	345	9	is	be	AUX
ejpam-6375	345	10	an	an	DET
ejpam-6375	345	11	open	open	ADJ
ejpam-6375	345	12	set	set	NOUN
ejpam-6375	345	13	of	of	ADP
ejpam-6375	345	14	v	v	NOUN
ejpam-6375	345	15	for	for	ADP
ejpam-6375	345	16	explicit	explicit	ADJ
ejpam-6375	345	17	points	point	NOUN
ejpam-6375	345	18	r	r	NOUN
ejpam-6375	345	19	and	and	CCONJ
ejpam-6375	345	20	s	s	PROPN
ejpam-6375	345	21	of	of	ADP
ejpam-6375	345	22	v.	v.	ADP
ejpam-6375	345	23	◦	◦	NOUN
ejpam-6375	346	1	gpt	gpt	NOUN
ejpam-6375	346	2	-	-	PUNCT
ejpam-6375	346	3	s∗	s∗	PROPN
ejpam-6375	346	4	g	g	PROPN
ejpam-6375	346	5	-t1	-t1	NOUN
ejpam-6375	346	6	space	space	NOUN
ejpam-6375	346	7	but	but	CCONJ
ejpam-6375	346	8	not	not	PART
ejpam-6375	346	9	gpt	gpt	NOUN
ejpam-6375	346	10	-	-	PUNCT
ejpam-6375	346	11	t1	t1	NOUN
ejpam-6375	346	12	space	space	NOUN
ejpam-6375	346	13	.	.	PUNCT
ejpam-6375	347	1	no	no	DET
ejpam-6375	347	2	two	two	NUM
ejpam-6375	347	3	open	open	ADJ
ejpam-6375	347	4	sets	set	NOUN
ejpam-6375	347	5	mα	mα	PROPN
ejpam-6375	347	6	and	and	CCONJ
ejpam-6375	347	7	nα	nα	AUX
ejpam-6375	347	8	exist	exist	VERB
ejpam-6375	347	9	with	with	ADP
ejpam-6375	347	10	s	s	PRON
ejpam-6375	347	11	/∈	/∈	NOUN
ejpam-6375	347	12	mα	mα	PROPN
ejpam-6375	347	13	,	,	PUNCT
ejpam-6375	347	14	r	r	NOUN
ejpam-6375	347	15	∈	∈	PROPN
ejpam-6375	347	16	mα	mα	NOUN
ejpam-6375	348	1	and	and	CCONJ
ejpam-6375	348	2	r	r	NOUN
ejpam-6375	348	3	/∈	/∈	PUNCT
ejpam-6375	348	4	nα	nα	NOUN
ejpam-6375	348	5	,	,	PUNCT
ejpam-6375	348	6	s	s	PART
ejpam-6375	348	7	∈	∈	PROPN
ejpam-6375	348	8	nα	nα	NOUN
ejpam-6375	348	9	for	for	ADP
ejpam-6375	348	10	any	any	DET
ejpam-6375	348	11	explicit	explicit	ADJ
ejpam-6375	348	12	points	point	NOUN
ejpam-6375	348	13	s	s	PART
ejpam-6375	348	14	and	and	CCONJ
ejpam-6375	348	15	r	r	NOUN
ejpam-6375	348	16	of	of	ADP
ejpam-6375	348	17	v.	v.	ADP
ejpam-6375	348	18	◦	◦	NOUN
ejpam-6375	348	19	gpt	gpt	NOUN
ejpam-6375	348	20	-	-	PUNCT
ejpam-6375	348	21	s∗	s∗	PROPN
ejpam-6375	349	1	g	g	PROPN
ejpam-6375	349	2	-t2	-t2	NOUN
ejpam-6375	349	3	space	space	NOUN
ejpam-6375	349	4	but	but	CCONJ
ejpam-6375	349	5	not	not	PART
ejpam-6375	349	6	gpt	gpt	NOUN
ejpam-6375	349	7	-	-	PUNCT
ejpam-6375	349	8	t2	t2	NOUN
ejpam-6375	349	9	space	space	NOUN
ejpam-6375	349	10	.	.	PUNCT
ejpam-6375	350	1	no	no	DET
ejpam-6375	350	2	two	two	NUM
ejpam-6375	350	3	distinct	distinct	ADJ
ejpam-6375	350	4	open	open	ADJ
ejpam-6375	350	5	sets	set	NOUN
ejpam-6375	350	6	mα	mα	PROPN
ejpam-6375	350	7	and	and	CCONJ
ejpam-6375	350	8	nα	nα	AUX
ejpam-6375	350	9	exist	exist	VERB
ejpam-6375	350	10	with	with	ADP
ejpam-6375	350	11	s	s	PRON
ejpam-6375	350	12	/∈	/∈	NOUN
ejpam-6375	350	13	mα	mα	PROPN
ejpam-6375	350	14	,	,	PUNCT
ejpam-6375	350	15	r	r	NOUN
ejpam-6375	350	16	∈	∈	PROPN
ejpam-6375	350	17	mα	mα	NOUN
ejpam-6375	351	1	and	and	CCONJ
ejpam-6375	351	2	r	r	NOUN
ejpam-6375	351	3	/∈	/∈	PUNCT
ejpam-6375	351	4	nα	nα	NOUN
ejpam-6375	351	5	,	,	PUNCT
ejpam-6375	351	6	s	s	PART
ejpam-6375	351	7	∈	∈	PROPN
ejpam-6375	351	8	nα	nα	NOUN
ejpam-6375	351	9	for	for	ADP
ejpam-6375	351	10	any	any	DET
ejpam-6375	351	11	explicit	explicit	ADJ
ejpam-6375	351	12	points	point	NOUN
ejpam-6375	351	13	s	s	PART
ejpam-6375	351	14	and	and	CCONJ
ejpam-6375	351	15	r	r	NOUN
ejpam-6375	351	16	of	of	ADP
ejpam-6375	351	17	v.	v.	CCONJ
ejpam-6375	351	18	theorem	theorem	ADJ
ejpam-6375	351	19	3.14	3.14	NUM
ejpam-6375	351	20	.	.	PUNCT
ejpam-6375	352	1	if	if	SCONJ
ejpam-6375	352	2	f	f	PROPN
ejpam-6375	352	3	is	be	AUX
ejpam-6375	352	4	bijective	bijective	ADJ
ejpam-6375	352	5	,	,	PUNCT
ejpam-6375	352	6	strongly	strongly	ADV
ejpam-6375	352	7	gpt	gpt	NOUN
ejpam-6375	352	8	-	-	PUNCT
ejpam-6375	352	9	s∗	s∗	NOUN
ejpam-6375	352	10	g	g	PROPN
ejpam-6375	352	11	-open	-open	ADJ
ejpam-6375	352	12	and	and	CCONJ
ejpam-6375	352	13	v	v	NOUN
ejpam-6375	352	14	is	be	AUX
ejpam-6375	352	15	gpt	gpt	NOUN
ejpam-6375	352	16	-	-	PUNCT
ejpam-6375	352	17	s∗	s∗	PROPN
ejpam-6375	352	18	g	g	PROPN
ejpam-6375	352	19	-t0	-t0	PROPN
ejpam-6375	352	20	,	,	PUNCT
ejpam-6375	352	21	then	then	ADV
ejpam-6375	352	22	z	z	PROPN
ejpam-6375	352	23	is	be	AUX
ejpam-6375	352	24	gpt	gpt	NOUN
ejpam-6375	352	25	-	-	PUNCT
ejpam-6375	352	26	s∗	s∗	PROPN
ejpam-6375	352	27	g	g	PROPN
ejpam-6375	352	28	-t0	-t0	PROPN
ejpam-6375	352	29	space	space	NOUN
ejpam-6375	352	30	.	.	PUNCT
ejpam-6375	353	1	proof	proof	NOUN
ejpam-6375	353	2	.	.	PUNCT
ejpam-6375	354	1	take	take	VERB
ejpam-6375	354	2	r2	r2	PROPN
ejpam-6375	354	3	and	and	CCONJ
ejpam-6375	354	4	s2	s2	PROPN
ejpam-6375	354	5	of	of	ADP
ejpam-6375	354	6	z	z	PROPN
ejpam-6375	354	7	with	with	ADP
ejpam-6375	354	8	r2	r2	PROPN
ejpam-6375	354	9	̸=	̸=	PROPN
ejpam-6375	354	10	s2	s2	PROPN
ejpam-6375	354	11	.	.	PUNCT
ejpam-6375	355	1	by	by	ADP
ejpam-6375	355	2	hypothesis	hypothesis	NOUN
ejpam-6375	355	3	,	,	PUNCT
ejpam-6375	355	4	r2	r2	PROPN
ejpam-6375	355	5	=	=	SYM
ejpam-6375	355	6	f(rα	f(rα	PROPN
ejpam-6375	355	7	)	)	PUNCT
ejpam-6375	355	8	and	and	CCONJ
ejpam-6375	355	9	s2	s2	NOUN
ejpam-6375	355	10	=	=	SYM
ejpam-6375	355	11	f(s1	f(s1	NOUN
ejpam-6375	355	12	)	)	PUNCT
ejpam-6375	355	13	where	where	SCONJ
ejpam-6375	355	14	rα	rα	ADJ
ejpam-6375	355	15	and	and	CCONJ
ejpam-6375	355	16	s1	s1	NOUN
ejpam-6375	355	17	are	be	AUX
ejpam-6375	355	18	the	the	DET
ejpam-6375	355	19	explicit	explicit	ADJ
ejpam-6375	355	20	points	point	NOUN
ejpam-6375	355	21	of	of	ADP
ejpam-6375	355	22	v.	v.	ADP
ejpam-6375	355	23	by	by	ADP
ejpam-6375	355	24	hypothesis	hypothesis	NOUN
ejpam-6375	355	25	,	,	PUNCT
ejpam-6375	355	26	mα	mα	PROPN
ejpam-6375	355	27	∈	∈	PROPN
ejpam-6375	355	28	gpt	gpt	NOUN
ejpam-6375	355	29	-	-	PUNCT
ejpam-6375	355	30	s∗	s∗	PROPN
ejpam-6375	355	31	go(v	go(v	ADV
ejpam-6375	355	32	)	)	PUNCT
ejpam-6375	355	33	with	with	ADP
ejpam-6375	355	34	rα	rα	PRON
ejpam-6375	355	35	∈	∈	PROPN
ejpam-6375	355	36	mα	mα	NOUN
ejpam-6375	355	37	and	and	CCONJ
ejpam-6375	355	38	s1	s1	PROPN
ejpam-6375	355	39	/∈	/∈	PUNCT
ejpam-6375	356	1	mα	mα	PROPN
ejpam-6375	356	2	.	.	PUNCT
ejpam-6375	356	3	therefore	therefore	ADV
ejpam-6375	356	4	,	,	PUNCT
ejpam-6375	356	5	f(rα	f(rα	NUM
ejpam-6375	356	6	)	)	PUNCT
ejpam-6375	356	7	∈	∈	PROPN
ejpam-6375	356	8	f(mα	f(mα	NUM
ejpam-6375	356	9	)	)	PUNCT
ejpam-6375	356	10	and	and	CCONJ
ejpam-6375	356	11	f(s1	f(s1	NOUN
ejpam-6375	356	12	)	)	PUNCT
ejpam-6375	356	13	/∈	/∈	PUNCT
ejpam-6375	357	1	f(mα	f(mα	NUM
ejpam-6375	357	2	)	)	PUNCT
ejpam-6375	357	3	.	.	PUNCT
ejpam-6375	358	1	f(mα	f(mα	NUM
ejpam-6375	358	2	)	)	PUNCT
ejpam-6375	358	3	∈	∈	PROPN
ejpam-6375	358	4	gpt	gpt	NOUN
ejpam-6375	358	5	-	-	PUNCT
ejpam-6375	358	6	s∗	s∗	NOUN
ejpam-6375	358	7	go(z	go(z	NOUN
ejpam-6375	358	8	)	)	PUNCT
ejpam-6375	358	9	as	as	ADP
ejpam-6375	358	10	v	v	NOUN
ejpam-6375	358	11	is	be	AUX
ejpam-6375	358	12	strongly	strongly	ADV
ejpam-6375	358	13	gpt	gpt	NOUN
ejpam-6375	358	14	-	-	PUNCT
ejpam-6375	358	15	s∗	s∗	PROPN
ejpam-6375	358	16	g	g	PROPN
ejpam-6375	358	17	-open	-open	NOUN
ejpam-6375	358	18	.	.	PUNCT
ejpam-6375	359	1	thus	thus	ADV
ejpam-6375	359	2	,	,	PUNCT
ejpam-6375	359	3	f(mα	f(mα	NOUN
ejpam-6375	359	4	)	)	PUNCT
ejpam-6375	359	5	is	be	AUX
ejpam-6375	359	6	gpt	gpt	NOUN
ejpam-6375	359	7	-	-	PUNCT
ejpam-6375	359	8	s∗	s∗	PROPN
ejpam-6375	359	9	g	g	PROPN
ejpam-6375	359	10	-open	-open	NOUN
ejpam-6375	359	11	set	set	VERB
ejpam-6375	359	12	in	in	ADP
ejpam-6375	359	13	z	z	NOUN
ejpam-6375	359	14	with	with	ADP
ejpam-6375	359	15	r2	r2	PROPN
ejpam-6375	359	16	∈	∈	PROPN
ejpam-6375	359	17	f(mα	f(mα	NOUN
ejpam-6375	359	18	)	)	PUNCT
ejpam-6375	359	19	and	and	CCONJ
ejpam-6375	359	20	s2	s2	NOUN
ejpam-6375	359	21	/∈	/∈	PUNCT
ejpam-6375	360	1	f(mα	f(mα	NOUN
ejpam-6375	360	2	)	)	PUNCT
ejpam-6375	360	3	.	.	PUNCT
ejpam-6375	361	1	so	so	ADV
ejpam-6375	361	2	,	,	PUNCT
ejpam-6375	361	3	z	z	PROPN
ejpam-6375	361	4	is	be	AUX
ejpam-6375	361	5	a	a	DET
ejpam-6375	361	6	gpt	gpt	NOUN
ejpam-6375	361	7	-	-	PUNCT
ejpam-6375	361	8	s∗	s∗	PROPN
ejpam-6375	361	9	g	g	PROPN
ejpam-6375	361	10	-t0	-t0	PROPN
ejpam-6375	361	11	space	space	NOUN
ejpam-6375	361	12	.	.	PUNCT
ejpam-6375	362	1	m.	m.	NOUN
ejpam-6375	362	2	shahbaz	shahbaz	PROPN
ejpam-6375	362	3	et	et	PROPN
ejpam-6375	362	4	al	al	PROPN
ejpam-6375	362	5	.	.	PUNCT
ejpam-6375	362	6	/	/	SYM
ejpam-6375	362	7	eur	eur	PROPN
ejpam-6375	362	8	.	.	PUNCT
ejpam-6375	363	1	j.	j.	PROPN
ejpam-6375	363	2	pure	pure	PROPN
ejpam-6375	363	3	appl	appl	PROPN
ejpam-6375	363	4	.	.	PROPN
ejpam-6375	363	5	math	math	PROPN
ejpam-6375	363	6	,	,	PUNCT
ejpam-6375	363	7	18	18	NUM
ejpam-6375	363	8	(	(	PUNCT
ejpam-6375	363	9	4	4	NUM
ejpam-6375	363	10	)	)	PUNCT
ejpam-6375	363	11	(	(	PUNCT
ejpam-6375	363	12	2025	2025	NUM
ejpam-6375	363	13	)	)	PUNCT
ejpam-6375	363	14	,	,	PUNCT
ejpam-6375	363	15	6375	6375	NUM
ejpam-6375	363	16	14	14	NUM
ejpam-6375	363	17	of	of	ADP
ejpam-6375	363	18	22	22	NUM
ejpam-6375	363	19	figure	figure	NOUN
ejpam-6375	363	20	1	1	NUM
ejpam-6375	363	21	:	:	PUNCT
ejpam-6375	363	22	illustrates	illustrate	VERB
ejpam-6375	363	23	the	the	DET
ejpam-6375	363	24	above	above	ADJ
ejpam-6375	363	25	theorems	theorem	NOUN
ejpam-6375	363	26	and	and	CCONJ
ejpam-6375	363	27	considerations	consideration	NOUN
ejpam-6375	363	28	.	.	PUNCT
ejpam-6375	364	1	theorem	theorem	VERB
ejpam-6375	364	2	3.15	3.15	NUM
ejpam-6375	364	3	.	.	PUNCT
ejpam-6375	365	1	if	if	SCONJ
ejpam-6375	365	2	f	f	PROPN
ejpam-6375	365	3	is	be	AUX
ejpam-6375	365	4	gpt	gpt	NOUN
ejpam-6375	365	5	-	-	PUNCT
ejpam-6375	365	6	s∗	s∗	PROPN
ejpam-6375	365	7	g	g	PROPN
ejpam-6375	365	8	-irresolute	-irresolute	PROPN
ejpam-6375	365	9	,	,	PUNCT
ejpam-6375	365	10	injective	injective	ADJ
ejpam-6375	365	11	,	,	PUNCT
ejpam-6375	365	12	and	and	CCONJ
ejpam-6375	365	13	z	z	NOUN
ejpam-6375	365	14	is	be	AUX
ejpam-6375	365	15	gpt	gpt	NOUN
ejpam-6375	365	16	-	-	PUNCT
ejpam-6375	365	17	s∗	s∗	PROPN
ejpam-6375	365	18	g	g	PROPN
ejpam-6375	365	19	-t0	-t0	PROPN
ejpam-6375	365	20	,	,	PUNCT
ejpam-6375	365	21	then	then	ADV
ejpam-6375	365	22	x	x	PUNCT
ejpam-6375	365	23	is	be	AUX
ejpam-6375	365	24	gpts∗	gpts∗	PROPN
ejpam-6375	365	25	g	g	PROPN
ejpam-6375	365	26	-t0	-t0	PROPN
ejpam-6375	365	27	space	space	NOUN
ejpam-6375	365	28	.	.	PUNCT
ejpam-6375	366	1	proof	proof	NOUN
ejpam-6375	366	2	.	.	PUNCT
ejpam-6375	367	1	let	let	VERB
ejpam-6375	367	2	rα	rα	INTJ
ejpam-6375	367	3	and	and	CCONJ
ejpam-6375	367	4	s1	s1	VERB
ejpam-6375	367	5	distinct	distinct	ADJ
ejpam-6375	367	6	points	point	NOUN
ejpam-6375	367	7	of	of	ADP
ejpam-6375	367	8	v.	v.	ADV
ejpam-6375	367	9	since	since	SCONJ
ejpam-6375	367	10	f	f	PROPN
ejpam-6375	367	11	is	be	AUX
ejpam-6375	367	12	injective	injective	ADJ
ejpam-6375	367	13	implies	implie	NOUN
ejpam-6375	367	14	f(s1	f(s1	NOUN
ejpam-6375	367	15	)	)	PUNCT
ejpam-6375	367	16	̸=	̸=	PROPN
ejpam-6375	367	17	f(rα	f(rα	NUM
ejpam-6375	367	18	)	)	PUNCT
ejpam-6375	367	19	.	.	PUNCT
ejpam-6375	368	1	as	as	SCONJ
ejpam-6375	368	2	z	z	PROPN
ejpam-6375	368	3	is	be	AUX
ejpam-6375	368	4	gpt	gpt	NOUN
ejpam-6375	368	5	-	-	PUNCT
ejpam-6375	368	6	s∗	s∗	PROPN
ejpam-6375	368	7	g	g	PROPN
ejpam-6375	368	8	-t0	-t0	PROPN
ejpam-6375	368	9	there	there	ADV
ejpam-6375	368	10	exists	exist	VERB
ejpam-6375	368	11	f	f	PROPN
ejpam-6375	368	12	∈	∈	PROPN
ejpam-6375	368	13	gpt	gpt	NOUN
ejpam-6375	368	14	-	-	PUNCT
ejpam-6375	368	15	s∗	s∗	NOUN
ejpam-6375	368	16	go(z	go(z	NOUN
ejpam-6375	368	17	)	)	PUNCT
ejpam-6375	368	18	such	such	ADJ
ejpam-6375	368	19	that	that	DET
ejpam-6375	368	20	f(rα	f(rα	PROPN
ejpam-6375	368	21	)	)	PUNCT
ejpam-6375	368	22	∈	∈	PROPN
ejpam-6375	368	23	f	f	X
ejpam-6375	368	24	,	,	PUNCT
ejpam-6375	368	25	f(s1	f(s1	NOUN
ejpam-6375	368	26	)	)	PUNCT
ejpam-6375	368	27	/∈	/∈	PUNCT
ejpam-6375	369	1	f	f	NOUN
ejpam-6375	370	1	or	or	CCONJ
ejpam-6375	370	2	exists	exist	VERB
ejpam-6375	370	3	fβ	fβ	ADP
ejpam-6375	370	4	∈	∈	PROPN
ejpam-6375	370	5	gpt	gpt	NOUN
ejpam-6375	370	6	-	-	PUNCT
ejpam-6375	370	7	s∗	s∗	NOUN
ejpam-6375	370	8	go(z	go(z	NOUN
ejpam-6375	370	9	)	)	PUNCT
ejpam-6375	370	10	such	such	ADJ
ejpam-6375	370	11	that	that	SCONJ
ejpam-6375	370	12	f(s1	f(s1	NOUN
ejpam-6375	370	13	)	)	PUNCT
ejpam-6375	370	14	∈	∈	PROPN
ejpam-6375	370	15	fβ	fβ	ADP
ejpam-6375	370	16	,	,	PUNCT
ejpam-6375	370	17	f(rα	f(rα	PROPN
ejpam-6375	370	18	)	)	PUNCT
ejpam-6375	370	19	/∈	/∈	PUNCT
ejpam-6375	371	1	fβ	fβ	ADV
ejpam-6375	371	2	with	with	ADP
ejpam-6375	371	3	f(s1	f(s1	NOUN
ejpam-6375	371	4	)	)	PUNCT
ejpam-6375	371	5	̸=	̸=	PROPN
ejpam-6375	371	6	f(rα	f(rα	NUM
ejpam-6375	371	7	)	)	PUNCT
ejpam-6375	371	8	.	.	PUNCT
ejpam-6375	372	1	as	as	SCONJ
ejpam-6375	372	2	f	f	PROPN
ejpam-6375	372	3	is	be	AUX
ejpam-6375	372	4	gpt	gpt	NOUN
ejpam-6375	372	5	-	-	PUNCT
ejpam-6375	372	6	s∗	s∗	PROPN
ejpam-6375	372	7	g	g	PROPN
ejpam-6375	372	8	-irresolute	-irresolute	PROPN
ejpam-6375	372	9	then	then	ADV
ejpam-6375	372	10	f−1	f−1	PROPN
ejpam-6375	372	11	(	(	PUNCT
ejpam-6375	372	12	f	f	X
ejpam-6375	372	13	)	)	PUNCT
ejpam-6375	372	14	∈	∈	PROPN
ejpam-6375	372	15	gpt	gpt	NOUN
ejpam-6375	372	16	-	-	PUNCT
ejpam-6375	372	17	s∗	s∗	PROPN
ejpam-6375	372	18	go(v	go(v	NOUN
ejpam-6375	372	19	)	)	PUNCT
ejpam-6375	372	20	,	,	PUNCT
ejpam-6375	372	21	there	there	PRON
ejpam-6375	372	22	exists	exist	VERB
ejpam-6375	372	23	f−1(rα	f−1(rα	PROPN
ejpam-6375	372	24	)	)	PUNCT
ejpam-6375	372	25	∈	∈	PROPN
ejpam-6375	372	26	f	f	PROPN
ejpam-6375	372	27	,	,	PUNCT
ejpam-6375	372	28	f−1(s1	f−1(s1	PROPN
ejpam-6375	372	29	)	)	PUNCT
ejpam-6375	372	30	/∈	/∈	PUNCT
ejpam-6375	373	1	f	f	NOUN
ejpam-6375	373	2	or	or	CCONJ
ejpam-6375	373	3	f−1(fβ	f−1(fβ	PROPN
ejpam-6375	373	4	)	)	PUNCT
ejpam-6375	373	5	∈	∈	PROPN
ejpam-6375	373	6	gpt	gpt	NOUN
ejpam-6375	373	7	-	-	PUNCT
ejpam-6375	373	8	s∗	s∗	PROPN
ejpam-6375	373	9	go(v	go(v	X
ejpam-6375	373	10	)	)	PUNCT
ejpam-6375	373	11	implies	imply	VERB
ejpam-6375	373	12	f−1(s1	f−1(s1	NOUN
ejpam-6375	373	13	)	)	PUNCT
ejpam-6375	373	14	∈	∈	PROPN
ejpam-6375	373	15	fβ	fβ	ADP
ejpam-6375	373	16	,	,	PUNCT
ejpam-6375	373	17	f−1(rα	f−1(rα	PROPN
ejpam-6375	373	18	)	)	PUNCT
ejpam-6375	373	19	/∈	/∈	PUNCT
ejpam-6375	374	1	fβ	fβ	INTJ
ejpam-6375	374	2	.	.	PUNCT
ejpam-6375	375	1	hence	hence	ADV
ejpam-6375	375	2	,	,	PUNCT
ejpam-6375	375	3	v	v	NOUN
ejpam-6375	375	4	is	be	AUX
ejpam-6375	375	5	gpt	gpt	NOUN
ejpam-6375	375	6	-	-	PUNCT
ejpam-6375	375	7	s∗	s∗	PROPN
ejpam-6375	375	8	g	g	PROPN
ejpam-6375	375	9	-t0	-t0	PROPN
ejpam-6375	375	10	space	space	NOUN
ejpam-6375	375	11	.	.	PUNCT
ejpam-6375	376	1	theorem	theorem	VERB
ejpam-6375	376	2	3.16	3.16	NUM
ejpam-6375	376	3	.	.	PUNCT
ejpam-6375	377	1	assume	assume	VERB
ejpam-6375	377	2	v	v	NUM
ejpam-6375	377	3	is	be	AUX
ejpam-6375	377	4	gpt	gpt	NOUN
ejpam-6375	377	5	-	-	PUNCT
ejpam-6375	377	6	s∗	s∗	PROPN
ejpam-6375	377	7	g	g	PROPN
ejpam-6375	377	8	-t1	-t1	PROPN
ejpam-6375	377	9	iff	iff	PROPN
ejpam-6375	377	10	s1	s1	PROPN
ejpam-6375	377	11	∈	∈	PROPN
ejpam-6375	377	12	v	v	ADP
ejpam-6375	377	13	singleton	singleton	PROPN
ejpam-6375	377	14	{	{	PUNCT
ejpam-6375	377	15	s1	s1	PROPN
ejpam-6375	377	16	}	}	PUNCT
ejpam-6375	377	17	∈	∈	PROPN
ejpam-6375	377	18	gpt	gpt	NOUN
ejpam-6375	377	19	-	-	PUNCT
ejpam-6375	377	20	s∗	s∗	NOUN
ejpam-6375	377	21	gc(v	gc(v	NOUN
ejpam-6375	377	22	)	)	PUNCT
ejpam-6375	377	23	.	.	PUNCT
ejpam-6375	378	1	proof	proof	NOUN
ejpam-6375	378	2	.	.	PUNCT
ejpam-6375	379	1	assume	assume	VERB
ejpam-6375	379	2	v	v	NUM
ejpam-6375	379	3	is	be	AUX
ejpam-6375	379	4	gpt	gpt	NOUN
ejpam-6375	379	5	-	-	PUNCT
ejpam-6375	379	6	s∗	s∗	NOUN
ejpam-6375	379	7	g	g	PROPN
ejpam-6375	379	8	-t1	-t1	PROPN
ejpam-6375	379	9	,	,	PUNCT
ejpam-6375	379	10	rα	rα	PROPN
ejpam-6375	379	11	∈	∈	PROPN
ejpam-6375	379	12	v.	v.	CCONJ
ejpam-6375	379	13	then	then	ADV
ejpam-6375	379	14	,	,	PUNCT
ejpam-6375	379	15	s1	s1	PROPN
ejpam-6375	379	16	∈	∈	PROPN
ejpam-6375	379	17	v	v	NOUN
ejpam-6375	379	18	{	{	PUNCT
ejpam-6375	379	19	rα	rα	PRON
ejpam-6375	379	20	}	}	PUNCT
ejpam-6375	379	21	implies	imply	VERB
ejpam-6375	379	22	rα	rα	VERB
ejpam-6375	379	23	̸=	̸=	PROPN
ejpam-6375	379	24	s1	s1	PROPN
ejpam-6375	379	25	∈	∈	PROPN
ejpam-6375	380	1	v.	v.	CCONJ
ejpam-6375	380	2	but	but	CCONJ
ejpam-6375	380	3	v	v	NOUN
ejpam-6375	380	4	is	be	AUX
ejpam-6375	380	5	gpt	gpt	NOUN
ejpam-6375	380	6	-	-	PUNCT
ejpam-6375	380	7	s∗	s∗	NOUN
ejpam-6375	380	8	g	g	PROPN
ejpam-6375	380	9	-t1	-t1	NOUN
ejpam-6375	380	10	space	space	NOUN
ejpam-6375	380	11	implies	imply	VERB
ejpam-6375	380	12	there	there	PRON
ejpam-6375	380	13	exists	exist	VERB
ejpam-6375	380	14	f	f	X
ejpam-6375	380	15	,	,	PUNCT
ejpam-6375	380	16	fβ	fβ	PROPN
ejpam-6375	380	17	∈	∈	PROPN
ejpam-6375	380	18	gpt	gpt	NOUN
ejpam-6375	380	19	-	-	PUNCT
ejpam-6375	380	20	s∗	s∗	PROPN
ejpam-6375	380	21	go(v	go(v	X
ejpam-6375	380	22	)	)	PUNCT
ejpam-6375	380	23	implies	imply	VERB
ejpam-6375	380	24	rα	rα	ADJ
ejpam-6375	380	25	/∈	/∈	PUNCT
ejpam-6375	381	1	f	f	X
ejpam-6375	381	2	,	,	PUNCT
ejpam-6375	381	3	s1	s1	PROPN
ejpam-6375	381	4	∈	∈	PROPN
ejpam-6375	381	5	fβ	fβ	ADP
ejpam-6375	381	6	⊆	⊆	NUM
ejpam-6375	381	7	(	(	PUNCT
ejpam-6375	381	8	v	v	NOUN
ejpam-6375	381	9	{	{	PUNCT
ejpam-6375	381	10	rα	rα	ADJ
ejpam-6375	381	11	}	}	PUNCT
ejpam-6375	381	12	)	)	PUNCT
ejpam-6375	381	13	.	.	PUNCT
ejpam-6375	382	1	also	also	ADV
ejpam-6375	382	2	s1	s1	PROPN
ejpam-6375	382	3	∈	∈	PROPN
ejpam-6375	382	4	fβ	fβ	ADP
ejpam-6375	382	5	⊆	⊆	NUM
ejpam-6375	382	6	(	(	PUNCT
ejpam-6375	382	7	v	v	NOUN
ejpam-6375	382	8	{	{	PUNCT
ejpam-6375	382	9	rα	rα	NOUN
ejpam-6375	382	10	}	}	PUNCT
ejpam-6375	382	11	)	)	PUNCT
ejpam-6375	382	12	implies	imply	VERB
ejpam-6375	382	13	(	(	PUNCT
ejpam-6375	382	14	v	v	X
ejpam-6375	382	15	{	{	PUNCT
ejpam-6375	382	16	rα	rα	NOUN
ejpam-6375	382	17	}	}	PUNCT
ejpam-6375	382	18	)	)	PUNCT
ejpam-6375	382	19	∈	∈	PROPN
ejpam-6375	382	20	gpt	gpt	NOUN
ejpam-6375	382	21	-	-	PUNCT
ejpam-6375	382	22	s∗	s∗	PROPN
ejpam-6375	382	23	go(v	go(v	NOUN
ejpam-6375	382	24	)	)	PUNCT
ejpam-6375	382	25	.	.	PUNCT
ejpam-6375	383	1	thus	thus	ADV
ejpam-6375	383	2	,	,	PUNCT
ejpam-6375	383	3	{	{	PUNCT
ejpam-6375	383	4	rα	rα	X
ejpam-6375	383	5	}	}	PUNCT
ejpam-6375	383	6	is	be	AUX
ejpam-6375	383	7	gpt	gpt	NOUN
ejpam-6375	383	8	-	-	PUNCT
ejpam-6375	383	9	s∗	s∗	PROPN
ejpam-6375	383	10	g	g	PROPN
ejpam-6375	383	11	-closed	-close	VERB
ejpam-6375	383	12	.	.	PUNCT
ejpam-6375	384	1	conversely	conversely	ADV
ejpam-6375	384	2	,	,	PUNCT
ejpam-6375	384	3	consider	consider	VERB
ejpam-6375	384	4	the	the	DET
ejpam-6375	384	5	distinct	distinct	ADJ
ejpam-6375	384	6	elements	element	NOUN
ejpam-6375	384	7	rα	rα	PART
ejpam-6375	384	8	̸=	̸=	PROPN
ejpam-6375	384	9	s1	s1	PROPN
ejpam-6375	384	10	∈	∈	PROPN
ejpam-6375	384	11	v	v	ADP
ejpam-6375	384	12	where	where	SCONJ
ejpam-6375	384	13	sets	set	VERB
ejpam-6375	384	14	{	{	PUNCT
ejpam-6375	384	15	rα	rα	VERB
ejpam-6375	384	16	}	}	PUNCT
ejpam-6375	384	17	and	and	CCONJ
ejpam-6375	384	18	{	{	PUNCT
ejpam-6375	384	19	s1	s1	NOUN
ejpam-6375	384	20	}	}	PUNCT
ejpam-6375	384	21	form	form	NOUN
ejpam-6375	384	22	gpt	gpt	NOUN
ejpam-6375	384	23	-	-	PUNCT
ejpam-6375	384	24	s∗	s∗	PROPN
ejpam-6375	384	25	g	g	PROPN
ejpam-6375	384	26	-closed	-close	VERB
ejpam-6375	384	27	sets	set	NOUN
ejpam-6375	384	28	and	and	CCONJ
ejpam-6375	384	29	their	their	PRON
ejpam-6375	384	30	complement	complement	NOUN
ejpam-6375	384	31	{	{	PUNCT
ejpam-6375	384	32	rα}c	rα}c	NOUN
ejpam-6375	384	33	represents	represent	VERB
ejpam-6375	384	34	an	an	DET
ejpam-6375	384	35	gpt	gpt	NOUN
ejpam-6375	384	36	-	-	PUNCT
ejpam-6375	384	37	s∗	s∗	PROPN
ejpam-6375	384	38	g	g	PROPN
ejpam-6375	384	39	-open	-open	NOUN
ejpam-6375	384	40	subset	subset	NOUN
ejpam-6375	384	41	.	.	PUNCT
ejpam-6375	385	1	certainly	certainly	ADV
ejpam-6375	385	2	,	,	PUNCT
ejpam-6375	385	3	{	{	PUNCT
ejpam-6375	385	4	rα	rα	INTJ
ejpam-6375	385	5	}	}	PUNCT
ejpam-6375	385	6	/∈	/∈	PUNCT
ejpam-6375	385	7	{	{	PUNCT
ejpam-6375	385	8	rα}c	rα}c	NOUN
ejpam-6375	385	9	and	and	CCONJ
ejpam-6375	385	10	{	{	PUNCT
ejpam-6375	385	11	s1	s1	NOUN
ejpam-6375	385	12	}	}	PUNCT
ejpam-6375	385	13	∈	∈	PROPN
ejpam-6375	385	14	{	{	PUNCT
ejpam-6375	385	15	rα}c	rα}c	NOUN
ejpam-6375	385	16	.	.	PUNCT
ejpam-6375	386	1	similarly	similarly	ADV
ejpam-6375	386	2	{	{	PUNCT
ejpam-6375	386	3	s1}c	s1}c	PROPN
ejpam-6375	386	4	is	be	AUX
ejpam-6375	386	5	gpt	gpt	NOUN
ejpam-6375	386	6	-	-	PUNCT
ejpam-6375	386	7	s∗	s∗	PROPN
ejpam-6375	386	8	g	g	PROPN
ejpam-6375	386	9	-open	-open	PROPN
ejpam-6375	386	10	,	,	PUNCT
ejpam-6375	386	11	{	{	PUNCT
ejpam-6375	386	12	s1	s1	NOUN
ejpam-6375	386	13	}	}	PUNCT
ejpam-6375	386	14	/∈	/∈	PUNCT
ejpam-6375	387	1	{	{	PUNCT
ejpam-6375	387	2	s1}c	s1}c	NOUN
ejpam-6375	387	3	and	and	CCONJ
ejpam-6375	387	4	{	{	PUNCT
ejpam-6375	387	5	rα	rα	ADJ
ejpam-6375	387	6	}	}	PUNCT
ejpam-6375	387	7	∈	∈	PROPN
ejpam-6375	387	8	{	{	PUNCT
ejpam-6375	387	9	s1}c	s1}c	NOUN
ejpam-6375	387	10	.	.	PUNCT
ejpam-6375	388	1	thus	thus	ADV
ejpam-6375	388	2	,	,	PUNCT
ejpam-6375	388	3	v	v	NOUN
ejpam-6375	388	4	is	be	AUX
ejpam-6375	388	5	gpt	gpt	NOUN
ejpam-6375	388	6	-	-	PUNCT
ejpam-6375	388	7	s∗	s∗	NOUN
ejpam-6375	388	8	g	g	PROPN
ejpam-6375	388	9	-t1	-t1	NOUN
ejpam-6375	388	10	space	space	NOUN
ejpam-6375	388	11	.	.	PUNCT
ejpam-6375	389	1	m.	m.	NOUN
ejpam-6375	389	2	shahbaz	shahbaz	PROPN
ejpam-6375	389	3	et	et	PROPN
ejpam-6375	389	4	al	al	PROPN
ejpam-6375	389	5	.	.	PUNCT
ejpam-6375	389	6	/	/	SYM
ejpam-6375	389	7	eur	eur	PROPN
ejpam-6375	389	8	.	.	PUNCT
ejpam-6375	390	1	j.	j.	PROPN
ejpam-6375	390	2	pure	pure	PROPN
ejpam-6375	390	3	appl	appl	PROPN
ejpam-6375	390	4	.	.	PROPN
ejpam-6375	390	5	math	math	PROPN
ejpam-6375	390	6	,	,	PUNCT
ejpam-6375	390	7	18	18	NUM
ejpam-6375	390	8	(	(	PUNCT
ejpam-6375	390	9	4	4	NUM
ejpam-6375	390	10	)	)	PUNCT
ejpam-6375	390	11	(	(	PUNCT
ejpam-6375	390	12	2025	2025	NUM
ejpam-6375	390	13	)	)	PUNCT
ejpam-6375	390	14	,	,	PUNCT
ejpam-6375	390	15	6375	6375	NUM
ejpam-6375	390	16	15	15	NUM
ejpam-6375	390	17	of	of	ADP
ejpam-6375	390	18	22	22	NUM
ejpam-6375	390	19	theorem	theorem	VERB
ejpam-6375	390	20	3.17	3.17	NUM
ejpam-6375	390	21	.	.	PUNCT
ejpam-6375	391	1	assume	assume	VERB
ejpam-6375	391	2	f	f	X
ejpam-6375	391	3	:	:	PUNCT
ejpam-6375	391	4	v	v	X
ejpam-6375	391	5	→	→	SYM
ejpam-6375	391	6	z.	z.	PROPN
ejpam-6375	391	7	then	then	ADV
ejpam-6375	391	8	the	the	DET
ejpam-6375	391	9	subsequent	subsequent	ADJ
ejpam-6375	391	10	results	result	NOUN
ejpam-6375	391	11	hold	hold	VERB
ejpam-6375	391	12	:	:	PUNCT
ejpam-6375	391	13	i	i	NOUN
ejpam-6375	391	14	)	)	PUNCT
ejpam-6375	391	15	if	if	SCONJ
ejpam-6375	391	16	f	f	PROPN
ejpam-6375	391	17	is	be	AUX
ejpam-6375	391	18	injective	injective	ADJ
ejpam-6375	391	19	,	,	PUNCT
ejpam-6375	391	20	gpt	gpt	NOUN
ejpam-6375	391	21	-	-	PUNCT
ejpam-6375	391	22	s∗	s∗	PROPN
ejpam-6375	391	23	g	g	PROPN
ejpam-6375	391	24	-continuous	-continuous	ADJ
ejpam-6375	391	25	,	,	PUNCT
ejpam-6375	391	26	z	z	PROPN
ejpam-6375	391	27	is	be	AUX
ejpam-6375	391	28	gpt	gpt	NOUN
ejpam-6375	391	29	-	-	PUNCT
ejpam-6375	391	30	t1	t1	NOUN
ejpam-6375	391	31	,	,	PUNCT
ejpam-6375	391	32	then	then	ADV
ejpam-6375	391	33	v	v	NOUN
ejpam-6375	391	34	is	be	AUX
ejpam-6375	392	1	gpt	gpt	NOUN
ejpam-6375	392	2	-	-	PUNCT
ejpam-6375	392	3	s∗	s∗	NOUN
ejpam-6375	392	4	g	g	PROPN
ejpam-6375	392	5	-t1	-t1	X
ejpam-6375	392	6	.	.	PUNCT
ejpam-6375	393	1	ii	ii	X
ejpam-6375	393	2	)	)	PUNCT
ejpam-6375	393	3	if	if	SCONJ
ejpam-6375	393	4	f	f	PROPN
ejpam-6375	393	5	is	be	AUX
ejpam-6375	393	6	injective	injective	ADJ
ejpam-6375	393	7	,	,	PUNCT
ejpam-6375	393	8	gpt	gpt	NOUN
ejpam-6375	393	9	-	-	PUNCT
ejpam-6375	393	10	s∗	s∗	PROPN
ejpam-6375	393	11	g	g	PROPN
ejpam-6375	393	12	-continuous	-continuous	ADJ
ejpam-6375	393	13	,	,	PUNCT
ejpam-6375	393	14	z	z	PROPN
ejpam-6375	393	15	is	be	AUX
ejpam-6375	393	16	gpt	gpt	NOUN
ejpam-6375	393	17	-	-	PUNCT
ejpam-6375	393	18	t2	t2	NOUN
ejpam-6375	393	19	,	,	PUNCT
ejpam-6375	393	20	then	then	ADV
ejpam-6375	393	21	v	v	NOUN
ejpam-6375	393	22	is	be	AUX
ejpam-6375	393	23	gpt	gpt	NOUN
ejpam-6375	393	24	-	-	PUNCT
ejpam-6375	393	25	s∗	s∗	PROPN
ejpam-6375	393	26	g	g	PROPN
ejpam-6375	393	27	-t2	-t2	PROPN
ejpam-6375	393	28	.	.	PUNCT
ejpam-6375	394	1	iii	iii	X
ejpam-6375	394	2	)	)	PUNCT
ejpam-6375	394	3	if	if	SCONJ
ejpam-6375	394	4	f	f	PROPN
ejpam-6375	394	5	is	be	AUX
ejpam-6375	394	6	injective	injective	ADJ
ejpam-6375	394	7	,	,	PUNCT
ejpam-6375	394	8	gpt	gpt	NOUN
ejpam-6375	394	9	-	-	PUNCT
ejpam-6375	394	10	s∗	s∗	PROPN
ejpam-6375	394	11	g	g	PROPN
ejpam-6375	394	12	-irresolute	-irresolute	PROPN
ejpam-6375	394	13	,	,	PUNCT
ejpam-6375	394	14	z	z	PROPN
ejpam-6375	394	15	is	be	AUX
ejpam-6375	394	16	gpt	gpt	NOUN
ejpam-6375	394	17	-	-	PUNCT
ejpam-6375	394	18	s∗	s∗	PROPN
ejpam-6375	394	19	g	g	PROPN
ejpam-6375	394	20	-t2	-t2	PROPN
ejpam-6375	394	21	,	,	PUNCT
ejpam-6375	394	22	then	then	ADV
ejpam-6375	394	23	v	v	NOUN
ejpam-6375	394	24	is	be	AUX
ejpam-6375	394	25	gpt	gpt	NOUN
ejpam-6375	394	26	-	-	PUNCT
ejpam-6375	394	27	s∗	s∗	PROPN
ejpam-6375	394	28	g	g	PROPN
ejpam-6375	394	29	-t2	-t2	PROPN
ejpam-6375	394	30	.	.	PUNCT
ejpam-6375	395	1	proof	proof	NOUN
ejpam-6375	395	2	.	.	PUNCT
ejpam-6375	396	1	i	i	PRON
ejpam-6375	396	2	)	)	PUNCT
ejpam-6375	396	3	assume	assume	VERB
ejpam-6375	396	4	rα	rα	ADJ
ejpam-6375	396	5	̸=	̸=	PROPN
ejpam-6375	396	6	s1	s1	NOUN
ejpam-6375	396	7	,	,	PUNCT
ejpam-6375	396	8	where	where	SCONJ
ejpam-6375	396	9	rα	rα	VERB
ejpam-6375	396	10	,	,	PUNCT
ejpam-6375	396	11	s1	s1	PROPN
ejpam-6375	396	12	∈	∈	PROPN
ejpam-6375	396	13	v	v	NOUN
ejpam-6375	396	14	,	,	PUNCT
ejpam-6375	396	15	then	then	ADV
ejpam-6375	396	16	f(rα	f(rα	NUM
ejpam-6375	396	17	)	)	PUNCT
ejpam-6375	397	1	=	=	SYM
ejpam-6375	397	2	r2	r2	NOUN
ejpam-6375	397	3	and	and	CCONJ
ejpam-6375	397	4	f(s1	f(s1	NOUN
ejpam-6375	397	5	)	)	PUNCT
ejpam-6375	397	6	=	=	SYM
ejpam-6375	397	7	s2	s2	PROPN
ejpam-6375	397	8	.	.	PUNCT
ejpam-6375	398	1	additionally	additionally	ADV
ejpam-6375	398	2	,	,	PUNCT
ejpam-6375	398	3	f(rα	f(rα	NUM
ejpam-6375	398	4	)	)	PUNCT
ejpam-6375	399	1	̸=	̸=	PROPN
ejpam-6375	399	2	f(s1	f(s1	NOUN
ejpam-6375	399	3	)	)	PUNCT
ejpam-6375	399	4	.	.	PUNCT
ejpam-6375	400	1	as	as	SCONJ
ejpam-6375	400	2	(	(	PUNCT
ejpam-6375	400	3	z	z	NOUN
ejpam-6375	400	4	,	,	PUNCT
ejpam-6375	400	5	gτ	gτ	PROPN
ejpam-6375	400	6	2	2	NUM
ejpam-6375	400	7	,	,	PUNCT
ejpam-6375	400	8	pβ	pβ	NOUN
ejpam-6375	400	9	)	)	PUNCT
ejpam-6375	400	10	gpt	gpt	NOUN
ejpam-6375	400	11	-	-	PUNCT
ejpam-6375	400	12	t1	t1	NOUN
ejpam-6375	400	13	implies	imply	VERB
ejpam-6375	400	14	r2	r2	PROPN
ejpam-6375	400	15	∈	∈	PROPN
ejpam-6375	400	16	mα	mα	PROPN
ejpam-6375	400	17	,	,	PUNCT
ejpam-6375	400	18	s2	s2	PROPN
ejpam-6375	400	19	/∈	/∈	PUNCT
ejpam-6375	401	1	mα	mα	PROPN
ejpam-6375	402	1	and	and	CCONJ
ejpam-6375	402	2	s2	s2	PROPN
ejpam-6375	402	3	∈	∈	PROPN
ejpam-6375	402	4	nα	nα	NOUN
ejpam-6375	402	5	,	,	PUNCT
ejpam-6375	402	6	r2	r2	PROPN
ejpam-6375	402	7	/∈	/∈	PUNCT
ejpam-6375	403	1	nα	nα	PROPN
ejpam-6375	403	2	.	.	PUNCT
ejpam-6375	404	1	then	then	ADV
ejpam-6375	404	2	rα	rα	VERB
ejpam-6375	404	3	∈	∈	PROPN
ejpam-6375	404	4	f−1(mα	f−1(mα	PROPN
ejpam-6375	404	5	)	)	PUNCT
ejpam-6375	404	6	,	,	PUNCT
ejpam-6375	404	7	rα	rα	ADJ
ejpam-6375	404	8	/∈	/∈	PUNCT
ejpam-6375	405	1	f−1(nα	f−1(nα	PROPN
ejpam-6375	405	2	)	)	PUNCT
ejpam-6375	405	3	and	and	CCONJ
ejpam-6375	405	4	s1	s1	PROPN
ejpam-6375	405	5	∈	∈	PROPN
ejpam-6375	405	6	f−1(nα	f−1(nα	PROPN
ejpam-6375	405	7	)	)	PUNCT
ejpam-6375	405	8	,	,	PUNCT
ejpam-6375	405	9	s1	s1	NOUN
ejpam-6375	405	10	/∈	/∈	PUNCT
ejpam-6375	405	11	f−1(mα	f−1(mα	NOUN
ejpam-6375	405	12	)	)	PUNCT
ejpam-6375	405	13	.	.	PUNCT
ejpam-6375	406	1	according	accord	VERB
ejpam-6375	406	2	to	to	ADP
ejpam-6375	406	3	the	the	DET
ejpam-6375	406	4	definition	definition	NOUN
ejpam-6375	406	5	of	of	ADP
ejpam-6375	406	6	gpt	gpt	NOUN
ejpam-6375	406	7	-	-	PUNCT
ejpam-6375	406	8	s∗	s∗	PROPN
ejpam-6375	406	9	g	g	PROPN
ejpam-6375	406	10	-continuity	-continuity	PROPN
ejpam-6375	406	11	,	,	PUNCT
ejpam-6375	406	12	f−1(mα	f−1(mα	PROPN
ejpam-6375	406	13	)	)	PUNCT
ejpam-6375	406	14	and	and	CCONJ
ejpam-6375	406	15	f−1(nα	f−1(nα	PROPN
ejpam-6375	406	16	)	)	PUNCT
ejpam-6375	406	17	∈	∈	PROPN
ejpam-6375	406	18	gpt	gpt	NOUN
ejpam-6375	406	19	-	-	PUNCT
ejpam-6375	406	20	s∗	s∗	PROPN
ejpam-6375	406	21	go(v	go(v	NOUN
ejpam-6375	406	22	)	)	PUNCT
ejpam-6375	406	23	.	.	PUNCT
ejpam-6375	407	1	for	for	ADP
ejpam-6375	407	2	rα	rα	ADJ
ejpam-6375	407	3	̸=	̸=	PROPN
ejpam-6375	407	4	s1	s1	NOUN
ejpam-6375	407	5	,	,	PUNCT
ejpam-6375	407	6	rα	rα	INTJ
ejpam-6375	407	7	,	,	PUNCT
ejpam-6375	407	8	s1	s1	PROPN
ejpam-6375	407	9	∈	∈	PROPN
ejpam-6375	407	10	v	v	NOUN
ejpam-6375	407	11	implies	imply	VERB
ejpam-6375	407	12	rα	rα	PROPN
ejpam-6375	407	13	∈	∈	PROPN
ejpam-6375	407	14	f−1(mα	f−1(mα	PROPN
ejpam-6375	407	15	)	)	PUNCT
ejpam-6375	407	16	,	,	PUNCT
ejpam-6375	407	17	rα	rα	ADJ
ejpam-6375	407	18	/∈	/∈	PUNCT
ejpam-6375	408	1	f−1(nα	f−1(nα	PROPN
ejpam-6375	408	2	)	)	PUNCT
ejpam-6375	408	3	and	and	CCONJ
ejpam-6375	408	4	s1	s1	PROPN
ejpam-6375	408	5	∈	∈	PROPN
ejpam-6375	408	6	f−1(nα	f−1(nα	PROPN
ejpam-6375	408	7	)	)	PUNCT
ejpam-6375	408	8	,	,	PUNCT
ejpam-6375	408	9	s1	s1	NOUN
ejpam-6375	408	10	/∈	/∈	PUNCT
ejpam-6375	408	11	f−1(mα	f−1(mα	NOUN
ejpam-6375	408	12	)	)	PUNCT
ejpam-6375	408	13	.	.	PUNCT
ejpam-6375	409	1	thus	thus	ADV
ejpam-6375	409	2	,	,	PUNCT
ejpam-6375	409	3	(	(	PUNCT
ejpam-6375	409	4	v	v	NOUN
ejpam-6375	409	5	,	,	PUNCT
ejpam-6375	409	6	gτ	gτ	PROPN
ejpam-6375	409	7	1	1	NUM
ejpam-6375	409	8	,	,	PUNCT
ejpam-6375	409	9	pα	pα	NOUN
ejpam-6375	409	10	)	)	PUNCT
ejpam-6375	409	11	is	be	AUX
ejpam-6375	409	12	gpt	gpt	NOUN
ejpam-6375	409	13	-	-	PUNCT
ejpam-6375	409	14	s∗	s∗	PROPN
ejpam-6375	409	15	g	g	PROPN
ejpam-6375	409	16	-t1	-t1	NOUN
ejpam-6375	409	17	space	space	NOUN
ejpam-6375	409	18	.	.	PUNCT
ejpam-6375	410	1	in	in	ADP
ejpam-6375	410	2	the	the	DET
ejpam-6375	410	3	same	same	ADJ
ejpam-6375	410	4	way	way	NOUN
ejpam-6375	410	5	,	,	PUNCT
ejpam-6375	410	6	ii	ii	NOUN
ejpam-6375	410	7	)	)	PUNCT
ejpam-6375	410	8	and	and	CCONJ
ejpam-6375	410	9	iii	iii	X
ejpam-6375	410	10	)	)	PUNCT
ejpam-6375	410	11	can	can	AUX
ejpam-6375	410	12	be	be	AUX
ejpam-6375	410	13	proven	prove	VERB
ejpam-6375	410	14	.	.	PUNCT
ejpam-6375	411	1	theorem	theorem	VERB
ejpam-6375	411	2	3.18	3.18	NUM
ejpam-6375	411	3	.	.	PUNCT
ejpam-6375	412	1	the	the	DET
ejpam-6375	412	2	subsequent	subsequent	ADJ
ejpam-6375	412	3	statements	statement	NOUN
ejpam-6375	412	4	are	be	AUX
ejpam-6375	412	5	equivalent	equivalent	ADJ
ejpam-6375	412	6	.	.	PUNCT
ejpam-6375	413	1	(	(	PUNCT
ejpam-6375	413	2	i	i	NOUN
ejpam-6375	413	3	)	)	PUNCT
ejpam-6375	413	4	v	v	NOUN
ejpam-6375	413	5	is	be	AUX
ejpam-6375	413	6	gpt	gpt	NOUN
ejpam-6375	413	7	-	-	PUNCT
ejpam-6375	413	8	s∗	s∗	PROPN
ejpam-6375	413	9	g	g	PROPN
ejpam-6375	413	10	-t2	-t2	PROPN
ejpam-6375	413	11	.	.	PUNCT
ejpam-6375	414	1	(	(	PUNCT
ejpam-6375	414	2	ii	ii	NOUN
ejpam-6375	414	3	)	)	PUNCT
ejpam-6375	414	4	if	if	SCONJ
ejpam-6375	414	5	rα	rα	PRON
ejpam-6375	414	6	∈	∈	PROPN
ejpam-6375	414	7	v	v	NOUN
ejpam-6375	414	8	,	,	PUNCT
ejpam-6375	414	9	then	then	ADV
ejpam-6375	414	10	rα	rα	VERB
ejpam-6375	414	11	̸=	̸=	PROPN
ejpam-6375	414	12	s1	s1	NOUN
ejpam-6375	414	13	,	,	PUNCT
ejpam-6375	414	14	there	there	PRON
ejpam-6375	414	15	exists	exist	VERB
ejpam-6375	414	16	uα	uα	PROPN
ejpam-6375	414	17	containing	contain	VERB
ejpam-6375	414	18	rα	rα	ADJ
ejpam-6375	414	19	and	and	CCONJ
ejpam-6375	414	20	s1	s1	PROPN
ejpam-6375	414	21	/∈	/∈	PUNCT
ejpam-6375	414	22	gpt	gpt	NOUN
ejpam-6375	414	23	-	-	PUNCT
ejpam-6375	414	24	s∗	s∗	NOUN
ejpam-6375	414	25	gcl(uα	gcl(uα	NOUN
ejpam-6375	414	26	)	)	PUNCT
ejpam-6375	414	27	.	.	PUNCT
ejpam-6375	415	1	proof	proof	NOUN
ejpam-6375	415	2	.	.	PUNCT
ejpam-6375	416	1	(	(	PUNCT
ejpam-6375	416	2	1	1	X
ejpam-6375	416	3	)	)	PUNCT
ejpam-6375	416	4	implies	imply	VERB
ejpam-6375	416	5	(	(	PUNCT
ejpam-6375	416	6	2	2	X
ejpam-6375	416	7	)	)	PUNCT
ejpam-6375	416	8	take	take	VERB
ejpam-6375	416	9	rα	rα	ADJ
ejpam-6375	416	10	∈	∈	PROPN
ejpam-6375	416	11	v	v	NOUN
ejpam-6375	416	12	and	and	CCONJ
ejpam-6375	416	13	s1	s1	PROPN
ejpam-6375	416	14	∈	∈	PROPN
ejpam-6375	416	15	v	v	NOUN
ejpam-6375	416	16	with	with	ADP
ejpam-6375	416	17	rα	rα	ADJ
ejpam-6375	416	18	̸=	̸=	PROPN
ejpam-6375	416	19	s1	s1	NOUN
ejpam-6375	416	20	there	there	ADV
ejpam-6375	416	21	exists	exist	VERB
ejpam-6375	416	22	disjoint	disjoint	NOUN
ejpam-6375	416	23	set	set	VERB
ejpam-6375	416	24	uα	uα	NOUN
ejpam-6375	416	25	and	and	CCONJ
ejpam-6375	416	26	v	v	ADP
ejpam-6375	416	27	∈	∈	PROPN
ejpam-6375	416	28	gpt	gpt	NOUN
ejpam-6375	416	29	-	-	PUNCT
ejpam-6375	416	30	s∗	s∗	PROPN
ejpam-6375	416	31	go(v	go(v	NOUN
ejpam-6375	416	32	)	)	PUNCT
ejpam-6375	416	33	such	such	ADJ
ejpam-6375	416	34	that	that	SCONJ
ejpam-6375	416	35	rα	rα	ADV
ejpam-6375	416	36	∈	∈	PROPN
ejpam-6375	416	37	uα	uα	NOUN
ejpam-6375	416	38	and	and	CCONJ
ejpam-6375	416	39	s1	s1	PROPN
ejpam-6375	416	40	∈	∈	PROPN
ejpam-6375	416	41	v.	v.	CCONJ
ejpam-6375	416	42	then	then	ADV
ejpam-6375	416	43	,	,	PUNCT
ejpam-6375	416	44	rα	rα	ADV
ejpam-6375	416	45	∈	∈	PROPN
ejpam-6375	416	46	uα	uα	PROPN
ejpam-6375	416	47	⊆	⊆	NUM
ejpam-6375	416	48	vc	vc	NOUN
ejpam-6375	416	49	and	and	CCONJ
ejpam-6375	416	50	vc	vc	PROPN
ejpam-6375	416	51	∈	∈	PROPN
ejpam-6375	416	52	gpt	gpt	NOUN
ejpam-6375	416	53	-	-	PUNCT
ejpam-6375	416	54	s∗	s∗	NOUN
ejpam-6375	416	55	gc(v	gc(v	NOUN
ejpam-6375	416	56	)	)	PUNCT
ejpam-6375	416	57	and	and	CCONJ
ejpam-6375	416	58	s1	s1	PROPN
ejpam-6375	416	59	/∈	/∈	PUNCT
ejpam-6375	417	1	vc	vc	PROPN
ejpam-6375	417	2	implies	imply	VERB
ejpam-6375	417	3	s1	s1	PROPN
ejpam-6375	417	4	/∈	/∈	PUNCT
ejpam-6375	417	5	gpt	gpt	NOUN
ejpam-6375	417	6	-	-	PUNCT
ejpam-6375	417	7	s∗	s∗	NOUN
ejpam-6375	417	8	gcl(uα	gcl(uα	NOUN
ejpam-6375	417	9	)	)	PUNCT
ejpam-6375	417	10	.	.	PUNCT
ejpam-6375	418	1	(	(	PUNCT
ejpam-6375	418	2	2	2	X
ejpam-6375	418	3	)	)	PUNCT
ejpam-6375	418	4	implies	imply	VERB
ejpam-6375	418	5	(	(	PUNCT
ejpam-6375	418	6	1	1	X
ejpam-6375	418	7	)	)	PUNCT
ejpam-6375	418	8	consider	consider	VERB
ejpam-6375	418	9	rα	rα	PRON
ejpam-6375	418	10	∈	∈	PROPN
ejpam-6375	418	11	v	v	NOUN
ejpam-6375	418	12	and	and	CCONJ
ejpam-6375	418	13	s1	s1	PROPN
ejpam-6375	418	14	∈	∈	PROPN
ejpam-6375	418	15	v	v	NOUN
ejpam-6375	418	16	with	with	ADP
ejpam-6375	418	17	rα	rα	ADJ
ejpam-6375	418	18	̸=	̸=	PROPN
ejpam-6375	418	19	s1	s1	NOUN
ejpam-6375	418	20	implies	imply	VERB
ejpam-6375	418	21	there	there	PRON
ejpam-6375	418	22	exists	exist	VERB
ejpam-6375	418	23	gpt	gpt	NOUN
ejpam-6375	418	24	-	-	PUNCT
ejpam-6375	418	25	s∗	s∗	PROPN
ejpam-6375	418	26	g	g	PROPN
ejpam-6375	418	27	-open	-open	PROPN
ejpam-6375	418	28	uα	uα	NOUN
ejpam-6375	418	29	containing	contain	VERB
ejpam-6375	418	30	rα	rα	ADP
ejpam-6375	418	31	such	such	ADJ
ejpam-6375	418	32	that	that	SCONJ
ejpam-6375	418	33	s1	s1	PROPN
ejpam-6375	418	34	/∈	/∈	PUNCT
ejpam-6375	418	35	gpt	gpt	NOUN
ejpam-6375	418	36	-	-	PUNCT
ejpam-6375	418	37	s∗	s∗	PROPN
ejpam-6375	418	38	gcl(uα	gcl(uα	NOUN
ejpam-6375	418	39	)	)	PUNCT
ejpam-6375	418	40	implies	imply	VERB
ejpam-6375	418	41	s1	s1	PROPN
ejpam-6375	418	42	∈	∈	PROPN
ejpam-6375	418	43	(	(	PUNCT
ejpam-6375	418	44	v	v	NOUN
ejpam-6375	418	45	(	(	PUNCT
ejpam-6375	418	46	gpt	gpt	NOUN
ejpam-6375	418	47	-	-	PUNCT
ejpam-6375	418	48	s∗	s∗	NOUN
ejpam-6375	418	49	gcl(uα	gcl(uα	NOUN
ejpam-6375	418	50	)	)	PUNCT
ejpam-6375	418	51	)	)	PUNCT
ejpam-6375	418	52	)	)	PUNCT
ejpam-6375	418	53	.	.	PUNCT
ejpam-6375	419	1	(	(	PUNCT
ejpam-6375	419	2	v	v	X
ejpam-6375	419	3	(	(	PUNCT
ejpam-6375	419	4	gpt	gpt	NOUN
ejpam-6375	419	5	-	-	PUNCT
ejpam-6375	419	6	s∗	s∗	NOUN
ejpam-6375	419	7	gcl(uα	gcl(uα	NOUN
ejpam-6375	419	8	)	)	PUNCT
ejpam-6375	419	9	)	)	PUNCT
ejpam-6375	419	10	)	)	PUNCT
ejpam-6375	420	1	∈	∈	PROPN
ejpam-6375	420	2	gpt	gpt	NOUN
ejpam-6375	420	3	-	-	PUNCT
ejpam-6375	420	4	s∗	s∗	PROPN
ejpam-6375	420	5	go(v	go(v	NOUN
ejpam-6375	420	6	)	)	PUNCT
ejpam-6375	420	7	and	and	CCONJ
ejpam-6375	420	8	rα	rα	INTJ
ejpam-6375	420	9	/∈	/∈	PUNCT
ejpam-6375	421	1	(	(	PUNCT
ejpam-6375	421	2	v	v	NOUN
ejpam-6375	421	3	(	(	PUNCT
ejpam-6375	421	4	gpt	gpt	NOUN
ejpam-6375	421	5	-	-	PUNCT
ejpam-6375	421	6	s∗	s∗	NOUN
ejpam-6375	421	7	gcl(uα	gcl(uα	NOUN
ejpam-6375	421	8	)	)	PUNCT
ejpam-6375	421	9	)	)	PUNCT
ejpam-6375	421	10	)	)	PUNCT
ejpam-6375	421	11	.	.	PUNCT
ejpam-6375	422	1	furthermore	furthermore	ADV
ejpam-6375	422	2	,	,	PUNCT
ejpam-6375	422	3	uα	uα	PROPN
ejpam-6375	422	4	∩	∩	X
ejpam-6375	422	5	(	(	PUNCT
ejpam-6375	422	6	v	v	NOUN
ejpam-6375	422	7	(	(	PUNCT
ejpam-6375	422	8	gpt	gpt	NOUN
ejpam-6375	422	9	-	-	PUNCT
ejpam-6375	422	10	s∗	s∗	NOUN
ejpam-6375	422	11	gcl(uα	gcl(uα	NOUN
ejpam-6375	422	12	)	)	PUNCT
ejpam-6375	422	13	)	)	PUNCT
ejpam-6375	422	14	)	)	PUNCT
ejpam-6375	423	1	=	=	PUNCT
ejpam-6375	423	2	∅.	∅.	VERB
ejpam-6375	423	3	so	so	ADV
ejpam-6375	423	4	v	v	NOUN
ejpam-6375	423	5	is	be	AUX
ejpam-6375	423	6	gpt	gpt	NOUN
ejpam-6375	423	7	-	-	PUNCT
ejpam-6375	423	8	s∗	s∗	PROPN
ejpam-6375	423	9	g	g	PROPN
ejpam-6375	423	10	-t2	-t2	PROPN
ejpam-6375	423	11	.	.	PUNCT
ejpam-6375	424	1	3.2.2	3.2.2	NUM
ejpam-6375	424	2	.	.	PUNCT
ejpam-6375	424	3	gpt	gpt	NOUN
ejpam-6375	424	4	-	-	PUNCT
ejpam-6375	424	5	s∗	s∗	PROPN
ejpam-6375	424	6	g	g	ADP
ejpam-6375	424	7	-regular	-regular	ADJ
ejpam-6375	424	8	space	space	NOUN
ejpam-6375	424	9	definition	definition	NOUN
ejpam-6375	424	10	3.22	3.22	NUM
ejpam-6375	424	11	.	.	PUNCT
ejpam-6375	425	1	if	if	SCONJ
ejpam-6375	425	2	for	for	ADP
ejpam-6375	425	3	all	all	DET
ejpam-6375	425	4	f	f	PROPN
ejpam-6375	425	5	∈	∈	PROPN
ejpam-6375	425	6	gpt	gpt	NOUN
ejpam-6375	425	7	-	-	PUNCT
ejpam-6375	425	8	s∗	s∗	PROPN
ejpam-6375	425	9	gc(v	gc(v	NOUN
ejpam-6375	425	10	)	)	PUNCT
ejpam-6375	425	11	and	and	CCONJ
ejpam-6375	425	12	r	r	NOUN
ejpam-6375	425	13	/∈	/∈	PUNCT
ejpam-6375	426	1	f	f	X
ejpam-6375	426	2	,	,	PUNCT
ejpam-6375	426	3	there	there	PRON
ejpam-6375	426	4	exists	exist	VERB
ejpam-6375	426	5	disjoint	disjoint	ADJ
ejpam-6375	426	6	open	open	ADJ
ejpam-6375	426	7	sets	set	NOUN
ejpam-6375	426	8	e	e	NOUN
ejpam-6375	426	9	and	and	CCONJ
ejpam-6375	426	10	d	d	ADP
ejpam-6375	426	11	such	such	ADJ
ejpam-6375	426	12	that	that	SCONJ
ejpam-6375	426	13	f	f	PROPN
ejpam-6375	426	14	⊆	⊆	NUM
ejpam-6375	426	15	e	e	NOUN
ejpam-6375	426	16	,	,	PUNCT
ejpam-6375	426	17	r	r	PROPN
ejpam-6375	426	18	∈	∈	PROPN
ejpam-6375	426	19	d.	d.	PROPN
ejpam-6375	426	20	theorem	theorem	VERB
ejpam-6375	426	21	3.19	3.19	NUM
ejpam-6375	426	22	.	.	PUNCT
ejpam-6375	427	1	the	the	DET
ejpam-6375	427	2	condition	condition	NOUN
ejpam-6375	427	3	of	of	ADP
ejpam-6375	427	4	being	be	AUX
ejpam-6375	427	5	gpt	gpt	NOUN
ejpam-6375	427	6	-	-	PUNCT
ejpam-6375	427	7	s∗	s∗	PROPN
ejpam-6375	427	8	g	g	ADP
ejpam-6375	427	9	-regular	-regular	PROPN
ejpam-6375	427	10	implies	imply	VERB
ejpam-6375	427	11	gpt	gpt	NOUN
ejpam-6375	427	12	-	-	PUNCT
ejpam-6375	427	13	regularity	regularity	NOUN
ejpam-6375	427	14	.	.	PUNCT
ejpam-6375	428	1	proof	proof	NOUN
ejpam-6375	428	2	.	.	PUNCT
ejpam-6375	429	1	let	let	VERB
ejpam-6375	429	2	v	v	PART
ejpam-6375	429	3	be	be	AUX
ejpam-6375	429	4	a	a	DET
ejpam-6375	429	5	gpt	gpt	NOUN
ejpam-6375	429	6	-	-	PUNCT
ejpam-6375	429	7	s∗	s∗	PROPN
ejpam-6375	429	8	g	g	NOUN
ejpam-6375	429	9	-regular	-regular	ADJ
ejpam-6375	429	10	space	space	NOUN
ejpam-6375	429	11	.	.	PUNCT
ejpam-6375	430	1	take	take	VERB
ejpam-6375	430	2	f	f	PROPN
ejpam-6375	430	3	∈	∈	PROPN
ejpam-6375	430	4	gpt	gpt	NOUN
ejpam-6375	430	5	-	-	PUNCT
ejpam-6375	430	6	s∗	s∗	PROPN
ejpam-6375	430	7	gc(v	gc(v	NOUN
ejpam-6375	430	8	)	)	PUNCT
ejpam-6375	430	9	and	and	CCONJ
ejpam-6375	430	10	r	r	NOUN
ejpam-6375	430	11	/∈	/∈	PUNCT
ejpam-6375	431	1	f.	f.	NOUN
ejpam-6375	431	2	as	as	ADP
ejpam-6375	431	3	v	v	NOUN
ejpam-6375	431	4	is	be	AUX
ejpam-6375	431	5	gpt	gpt	NOUN
ejpam-6375	431	6	-	-	PUNCT
ejpam-6375	431	7	s∗	s∗	PROPN
ejpam-6375	431	8	g	g	PROPN
ejpam-6375	431	9	-regular	-regular	ADJ
ejpam-6375	431	10	,	,	PUNCT
ejpam-6375	431	11	there	there	PRON
ejpam-6375	431	12	exists	exist	VERB
ejpam-6375	431	13	a	a	DET
ejpam-6375	431	14	pair	pair	NOUN
ejpam-6375	431	15	of	of	ADP
ejpam-6375	431	16	disjoint	disjoint	ADJ
ejpam-6375	431	17	open	open	ADJ
ejpam-6375	431	18	sets	set	NOUN
ejpam-6375	431	19	e	e	NOUN
ejpam-6375	431	20	and	and	CCONJ
ejpam-6375	431	21	d	d	ADP
ejpam-6375	431	22	such	such	ADJ
ejpam-6375	431	23	that	that	SCONJ
ejpam-6375	431	24	f	f	PROPN
ejpam-6375	431	25	⊆	⊆	NUM
ejpam-6375	431	26	e	e	NOUN
ejpam-6375	431	27	,	,	PUNCT
ejpam-6375	431	28	r	r	PROPN
ejpam-6375	431	29	∈	∈	PROPN
ejpam-6375	431	30	d.	d.	NOUN
ejpam-6375	431	31	hence	hence	ADV
ejpam-6375	431	32	,	,	PUNCT
ejpam-6375	431	33	v	v	NOUN
ejpam-6375	431	34	is	be	AUX
ejpam-6375	431	35	a	a	DET
ejpam-6375	431	36	gpt	gpt	NOUN
ejpam-6375	431	37	-	-	PUNCT
ejpam-6375	431	38	regular	regular	NOUN
ejpam-6375	431	39	.	.	PUNCT
ejpam-6375	432	1	example	example	NOUN
ejpam-6375	432	2	3.7	3.7	NUM
ejpam-6375	432	3	.	.	PUNCT
ejpam-6375	433	1	let	let	VERB
ejpam-6375	433	2	v	v	PART
ejpam-6375	433	3	be	be	AUX
ejpam-6375	433	4	the	the	DET
ejpam-6375	433	5	set	set	NOUN
ejpam-6375	433	6	of	of	ADP
ejpam-6375	433	7	all	all	DET
ejpam-6375	433	8	bounded	bounded	ADJ
ejpam-6375	433	9	spherical	spherical	ADJ
ejpam-6375	433	10	regions	region	NOUN
ejpam-6375	433	11	in	in	ADP
ejpam-6375	433	12	r3	r3	PROPN
ejpam-6375	433	13	,	,	PUNCT
ejpam-6375	433	14	where	where	SCONJ
ejpam-6375	433	15	a	a	DET
ejpam-6375	433	16	spherical	spherical	ADJ
ejpam-6375	433	17	region	region	NOUN
ejpam-6375	433	18	s	s	VERB
ejpam-6375	433	19	is	be	AUX
ejpam-6375	433	20	defined	define	VERB
ejpam-6375	433	21	as	as	ADP
ejpam-6375	433	22	:	:	PUNCT
ejpam-6375	433	23	s	s	PART
ejpam-6375	433	24	=	=	PUNCT
ejpam-6375	433	25	{	{	PUNCT
ejpam-6375	433	26	(	(	PUNCT
ejpam-6375	433	27	x	x	NOUN
ejpam-6375	433	28	,	,	PUNCT
ejpam-6375	433	29	y	y	PROPN
ejpam-6375	433	30	,	,	PUNCT
ejpam-6375	433	31	z	z	NOUN
ejpam-6375	433	32	)	)	PUNCT
ejpam-6375	433	33	∈	∈	PROPN
ejpam-6375	433	34	r3	r3	PROPN
ejpam-6375	433	35	|	|	ADV
ejpam-6375	433	36	√	√	PROPN
ejpam-6375	433	37	(	(	PUNCT
ejpam-6375	433	38	x−	x−	PROPN
ejpam-6375	433	39	a)2	a)2	PROPN
ejpam-6375	433	40	+	+	CCONJ
ejpam-6375	434	1	(	(	PUNCT
ejpam-6375	434	2	y	y	PROPN
ejpam-6375	434	3	−	−	PROPN
ejpam-6375	434	4	b)2	b)2	PROPN
ejpam-6375	434	5	+	+	CCONJ
ejpam-6375	434	6	(	(	PUNCT
ejpam-6375	434	7	z	z	NOUN
ejpam-6375	434	8	−	−	PROPN
ejpam-6375	434	9	c)2	c)2	NOUN
ejpam-6375	434	10	≤	≤	ADJ
ejpam-6375	434	11	r	r	NOUN
ejpam-6375	434	12	}	}	PUNCT
ejpam-6375	434	13	,	,	PUNCT
ejpam-6375	434	14	m.	m.	NOUN
ejpam-6375	434	15	shahbaz	shahbaz	PROPN
ejpam-6375	434	16	et	et	PROPN
ejpam-6375	434	17	al	al	PROPN
ejpam-6375	434	18	.	.	PUNCT
ejpam-6375	434	19	/	/	SYM
ejpam-6375	434	20	eur	eur	PROPN
ejpam-6375	434	21	.	.	PUNCT
ejpam-6375	435	1	j.	j.	PROPN
ejpam-6375	435	2	pure	pure	PROPN
ejpam-6375	435	3	appl	appl	PROPN
ejpam-6375	435	4	.	.	PROPN
ejpam-6375	435	5	math	math	PROPN
ejpam-6375	435	6	,	,	PUNCT
ejpam-6375	435	7	18	18	NUM
ejpam-6375	435	8	(	(	PUNCT
ejpam-6375	435	9	4	4	NUM
ejpam-6375	435	10	)	)	PUNCT
ejpam-6375	435	11	(	(	PUNCT
ejpam-6375	435	12	2025	2025	NUM
ejpam-6375	435	13	)	)	PUNCT
ejpam-6375	435	14	,	,	PUNCT
ejpam-6375	435	15	6375	6375	NUM
ejpam-6375	435	16	16	16	NUM
ejpam-6375	435	17	of	of	ADP
ejpam-6375	435	18	22	22	NUM
ejpam-6375	435	19	with	with	ADP
ejpam-6375	435	20	r	r	NOUN
ejpam-6375	435	21	>	>	X
ejpam-6375	435	22	0	0	PUNCT
ejpam-6375	436	1	as	as	ADP
ejpam-6375	436	2	the	the	DET
ejpam-6375	436	3	radius	radius	NOUN
ejpam-6375	436	4	of	of	ADP
ejpam-6375	436	5	the	the	DET
ejpam-6375	436	6	spherical	spherical	ADJ
ejpam-6375	436	7	region	region	NOUN
ejpam-6375	436	8	and	and	CCONJ
ejpam-6375	436	9	a	a	DET
ejpam-6375	436	10	,	,	PUNCT
ejpam-6375	436	11	b	b	NOUN
ejpam-6375	436	12	,	,	PUNCT
ejpam-6375	436	13	c	c	PROPN
ejpam-6375	436	14	∈	∈	PROPN
ejpam-6375	436	15	r	r	NOUN
ejpam-6375	436	16	,	,	PUNCT
ejpam-6375	436	17	τg	τg	NOUN
ejpam-6375	436	18	=	=	SYM
ejpam-6375	436	19	{	{	PUNCT
ejpam-6375	436	20	e	e	NOUN
ejpam-6375	436	21	⊆	⊆	NUM
ejpam-6375	436	22	v	v	ADP
ejpam-6375	436	23	|	|	NOUN
ejpam-6375	436	24	e	e	NOUN
ejpam-6375	436	25	⊆	⊆	NUM
ejpam-6375	436	26	v	v	ADP
ejpam-6375	436	27	\	\	NOUN
ejpam-6375	436	28	{	{	PUNCT
ejpam-6375	436	29	s	s	X
ejpam-6375	436	30	|	|	ADV
ejpam-6375	436	31	radius(s	radius(s	NOUN
ejpam-6375	436	32	)	)	PUNCT
ejpam-6375	436	33	=	=	SYM
ejpam-6375	436	34	r	r	X
ejpam-6375	436	35	}	}	PUNCT
ejpam-6375	436	36	for	for	ADP
ejpam-6375	436	37	some	some	DET
ejpam-6375	436	38	fixed	fix	VERB
ejpam-6375	436	39	r	r	NOUN
ejpam-6375	436	40	>	>	X
ejpam-6375	436	41	0	0	NUM
ejpam-6375	436	42	}	}	PUNCT
ejpam-6375	436	43	(	(	PUNCT
ejpam-6375	436	44	see	see	VERB
ejpam-6375	436	45	example	example	NOUN
ejpam-6375	436	46	3.1	3.1	NUM
ejpam-6375	436	47	)	)	PUNCT
ejpam-6375	436	48	and	and	CCONJ
ejpam-6375	436	49	define	define	VERB
ejpam-6375	436	50	a	a	DET
ejpam-6375	436	51	collection	collection	NOUN
ejpam-6375	436	52	gpt	gpt	NOUN
ejpam-6375	436	53	⊆	⊆	NUM
ejpam-6375	436	54	2v	2v	NUM
ejpam-6375	436	55	as	as	ADP
ejpam-6375	436	56	:	:	PUNCT
ejpam-6375	436	57	gpt	gpt	NOUN
ejpam-6375	436	58	=	=	SYM
ejpam-6375	436	59	{	{	PUNCT
ejpam-6375	436	60	e	e	PROPN
ejpam-6375	436	61	⊆	⊆	NUM
ejpam-6375	436	62	v	v	ADP
ejpam-6375	436	63	|	|	ADV
ejpam-6375	436	64	all	all	DET
ejpam-6375	436	65	spherical	spherical	ADJ
ejpam-6375	436	66	region	region	NOUN
ejpam-6375	436	67	in	in	ADP
ejpam-6375	436	68	e	e	PROPN
ejpam-6375	436	69	have	have	AUX
ejpam-6375	436	70	radius	radiu	VERB
ejpam-6375	436	71	strictly	strictly	ADV
ejpam-6375	436	72	less	less	ADJ
ejpam-6375	436	73	than	than	ADP
ejpam-6375	436	74	r	r	NOUN
ejpam-6375	436	75	,	,	PUNCT
ejpam-6375	436	76	where	where	SCONJ
ejpam-6375	436	77	r	r	NOUN
ejpam-6375	436	78	>	>	X
ejpam-6375	436	79	0	0	NUM
ejpam-6375	436	80	}	}	PUNCT
ejpam-6375	436	81	.	.	PUNCT
ejpam-6375	437	1	let	let	VERB
ejpam-6375	437	2	x	x	SYM
ejpam-6375	437	3	∈	∈	PROPN
ejpam-6375	437	4	v	v	NOUN
ejpam-6375	437	5	and	and	CCONJ
ejpam-6375	437	6	c	c	NOUN
ejpam-6375	437	7	⊆	⊆	PROPN
ejpam-6375	437	8	v	v	NOUN
ejpam-6375	437	9	be	be	AUX
ejpam-6375	437	10	a	a	DET
ejpam-6375	437	11	gpt	gpt	NOUN
ejpam-6375	437	12	-	-	PUNCT
ejpam-6375	437	13	s∗	s∗	PROPN
ejpam-6375	437	14	g	g	PROPN
ejpam-6375	437	15	-closed	-close	VERB
ejpam-6375	437	16	set	set	VERB
ejpam-6375	437	17	such	such	ADJ
ejpam-6375	437	18	that	that	SCONJ
ejpam-6375	437	19	x	x	PROPN
ejpam-6375	437	20	/∈	/∈	PROPN
ejpam-6375	437	21	c.	c.	NOUN
ejpam-6375	437	22	since	since	SCONJ
ejpam-6375	437	23	c	c	PROPN
ejpam-6375	437	24	is	be	AUX
ejpam-6375	437	25	gpt	gpt	NOUN
ejpam-6375	437	26	-	-	PUNCT
ejpam-6375	437	27	s∗	s∗	PROPN
ejpam-6375	437	28	g	g	PROPN
ejpam-6375	437	29	-closed	-close	VERB
ejpam-6375	437	30	,	,	PUNCT
ejpam-6375	437	31	its	its	PRON
ejpam-6375	437	32	complement	complement	NOUN
ejpam-6375	437	33	v\c	v\c	ADV
ejpam-6375	437	34	is	be	AUX
ejpam-6375	437	35	gpt	gpt	NOUN
ejpam-6375	437	36	-	-	PUNCT
ejpam-6375	437	37	s∗	s∗	NOUN
ejpam-6375	437	38	g	g	PROPN
ejpam-6375	437	39	-open	-open	PROPN
ejpam-6375	437	40	in	in	ADP
ejpam-6375	437	41	v.	v.	CCONJ
ejpam-6375	437	42	let	let	VERB
ejpam-6375	437	43	an	an	DET
ejpam-6375	437	44	gpt	gpt	NOUN
ejpam-6375	437	45	-	-	PUNCT
ejpam-6375	437	46	s∗	s∗	PROPN
ejpam-6375	437	47	g	g	PROPN
ejpam-6375	437	48	-open	-open	PROPN
ejpam-6375	437	49	set	set	VERB
ejpam-6375	437	50	u	u	PRON
ejpam-6375	437	51	such	such	ADJ
ejpam-6375	437	52	that	that	SCONJ
ejpam-6375	437	53	x	x	SYM
ejpam-6375	437	54	∈	∈	PROPN
ejpam-6375	437	55	u	u	NOUN
ejpam-6375	437	56	and	and	CCONJ
ejpam-6375	437	57	u	u	NOUN
ejpam-6375	437	58	∩c	∩c	NOUN
ejpam-6375	437	59	=	=	PUNCT
ejpam-6375	437	60	∅.	∅.	VERB
ejpam-6375	437	61	therefore	therefore	ADV
ejpam-6375	437	62	,	,	PUNCT
ejpam-6375	437	63	(	(	PUNCT
ejpam-6375	437	64	v	v	NOUN
ejpam-6375	437	65	,	,	PUNCT
ejpam-6375	437	66	gτ	gτ	NOUN
ejpam-6375	437	67	,	,	PUNCT
ejpam-6375	437	68	gpt	gpt	NOUN
ejpam-6375	437	69	)	)	PUNCT
ejpam-6375	437	70	,	,	PUNCT
ejpam-6375	437	71	satisfies	satisfy	VERB
ejpam-6375	437	72	the	the	DET
ejpam-6375	437	73	condition	condition	NOUN
ejpam-6375	437	74	for	for	ADP
ejpam-6375	437	75	gpt	gpt	NOUN
ejpam-6375	437	76	-	-	PUNCT
ejpam-6375	437	77	s∗	s∗	PROPN
ejpam-6375	437	78	g	g	PROPN
ejpam-6375	437	79	-regularity	-regularity	NOUN
ejpam-6375	437	80	.	.	PUNCT
ejpam-6375	438	1	now	now	ADV
ejpam-6375	438	2	,	,	PUNCT
ejpam-6375	438	3	we	we	PRON
ejpam-6375	438	4	check	check	VERB
ejpam-6375	438	5	if	if	SCONJ
ejpam-6375	438	6	v	v	NOUN
ejpam-6375	438	7	is	be	AUX
ejpam-6375	438	8	gpt	gpt	NOUN
ejpam-6375	438	9	-	-	PUNCT
ejpam-6375	438	10	regular	regular	NOUN
ejpam-6375	438	11	:	:	PUNCT
ejpam-6375	438	12	let	let	VERB
ejpam-6375	438	13	x	x	PUNCT
ejpam-6375	438	14	∈	∈	PROPN
ejpam-6375	438	15	v	v	NOUN
ejpam-6375	438	16	and	and	CCONJ
ejpam-6375	438	17	c	c	PROPN
ejpam-6375	438	18	⊂	⊂	PROPN
ejpam-6375	438	19	v	v	AUX
ejpam-6375	438	20	be	be	AUX
ejpam-6375	438	21	a	a	DET
ejpam-6375	438	22	closed	closed	ADJ
ejpam-6375	438	23	set	set	NOUN
ejpam-6375	438	24	such	such	ADJ
ejpam-6375	438	25	that	that	SCONJ
ejpam-6375	438	26	x	x	PROPN
ejpam-6375	438	27	/∈	/∈	PROPN
ejpam-6375	438	28	c.	c.	NOUN
ejpam-6375	438	29	since	since	SCONJ
ejpam-6375	438	30	every	every	DET
ejpam-6375	438	31	closed	close	VERB
ejpam-6375	438	32	set	set	NOUN
ejpam-6375	438	33	is	be	AUX
ejpam-6375	438	34	gpt	gpt	NOUN
ejpam-6375	438	35	-	-	PUNCT
ejpam-6375	438	36	s∗	s∗	PROPN
ejpam-6375	438	37	g	g	PROPN
ejpam-6375	438	38	closed	close	VERB
ejpam-6375	438	39	set	set	NOUN
ejpam-6375	438	40	.	.	PUNCT
ejpam-6375	439	1	by	by	ADP
ejpam-6375	439	2	construction	construction	NOUN
ejpam-6375	439	3	,	,	PUNCT
ejpam-6375	439	4	such	such	DET
ejpam-6375	439	5	a	a	DET
ejpam-6375	439	6	set	set	NOUN
ejpam-6375	439	7	u	u	NOUN
ejpam-6375	439	8	exists	exist	VERB
ejpam-6375	439	9	and	and	CCONJ
ejpam-6375	439	10	satisfies	satisfy	VERB
ejpam-6375	439	11	the	the	DET
ejpam-6375	439	12	gpt	gpt	NOUN
ejpam-6375	439	13	-	-	PUNCT
ejpam-6375	439	14	regularity	regularity	NOUN
ejpam-6375	439	15	condition	condition	NOUN
ejpam-6375	439	16	which	which	PRON
ejpam-6375	439	17	implies	imply	VERB
ejpam-6375	439	18	(	(	PUNCT
ejpam-6375	439	19	v	v	NOUN
ejpam-6375	439	20	,	,	PUNCT
ejpam-6375	439	21	gτ	gτ	NOUN
ejpam-6375	439	22	,	,	PUNCT
ejpam-6375	439	23	gpt	gpt	NOUN
ejpam-6375	439	24	)	)	PUNCT
ejpam-6375	439	25	is	be	AUX
ejpam-6375	439	26	gpt	gpt	NOUN
ejpam-6375	439	27	-	-	PUNCT
ejpam-6375	439	28	regular	regular	NOUN
ejpam-6375	439	29	.	.	PUNCT
ejpam-6375	440	1	thus	thus	ADV
ejpam-6375	440	2	,	,	PUNCT
ejpam-6375	440	3	(	(	PUNCT
ejpam-6375	440	4	v	v	NOUN
ejpam-6375	440	5	,	,	PUNCT
ejpam-6375	440	6	gτ	gτ	PROPN
ejpam-6375	440	7	,	,	PUNCT
ejpam-6375	440	8	gpt	gpt	NOUN
ejpam-6375	440	9	)	)	PUNCT
ejpam-6375	440	10	is	be	AUX
ejpam-6375	440	11	both	both	DET
ejpam-6375	440	12	gpt	gpt	NOUN
ejpam-6375	440	13	-	-	PUNCT
ejpam-6375	440	14	s∗	s∗	PROPN
ejpam-6375	440	15	g	g	ADP
ejpam-6375	440	16	-regular	-regular	ADJ
ejpam-6375	440	17	and	and	CCONJ
ejpam-6375	440	18	gpt	gpt	NOUN
ejpam-6375	440	19	-	-	PUNCT
ejpam-6375	440	20	regular	regular	NOUN
ejpam-6375	440	21	.	.	PUNCT
ejpam-6375	441	1	remark	remark	PROPN
ejpam-6375	441	2	3.4	3.4	NUM
ejpam-6375	441	3	.	.	PUNCT
ejpam-6375	442	1	every	every	DET
ejpam-6375	442	2	gpt	gpt	NOUN
ejpam-6375	442	3	-	-	PUNCT
ejpam-6375	442	4	regular	regular	NOUN
ejpam-6375	442	5	is	be	AUX
ejpam-6375	442	6	a	a	DET
ejpam-6375	442	7	not	not	PART
ejpam-6375	442	8	gpt	gpt	NOUN
ejpam-6375	442	9	-	-	PUNCT
ejpam-6375	442	10	s∗	s∗	NOUN
ejpam-6375	442	11	g	g	ADP
ejpam-6375	442	12	-regular	-regular	ADJ
ejpam-6375	442	13	space	space	NOUN
ejpam-6375	442	14	.	.	PUNCT
ejpam-6375	443	1	example	example	NOUN
ejpam-6375	443	2	3.8	3.8	NUM
ejpam-6375	443	3	.	.	PUNCT
ejpam-6375	444	1	let	let	VERB
ejpam-6375	444	2	v	v	VERB
ejpam-6375	444	3	=	=	SYM
ejpam-6375	444	4	{	{	PUNCT
ejpam-6375	444	5	j1	j1	PROPN
ejpam-6375	444	6	,	,	PUNCT
ejpam-6375	444	7	k1	k1	NOUN
ejpam-6375	444	8	,	,	PUNCT
ejpam-6375	444	9	l1	l1	PROPN
ejpam-6375	444	10	}	}	PUNCT
ejpam-6375	444	11	and	and	CCONJ
ejpam-6375	444	12	gτ	gτ	PROPN
ejpam-6375	444	13	=	=	SYM
ejpam-6375	444	14	{	{	PUNCT
ejpam-6375	444	15	∅	∅	NOUN
ejpam-6375	444	16	,	,	PUNCT
ejpam-6375	444	17	v	v	NOUN
ejpam-6375	444	18	,	,	PUNCT
ejpam-6375	444	19	{	{	PUNCT
ejpam-6375	444	20	l1	l1	PROPN
ejpam-6375	444	21	}	}	PUNCT
ejpam-6375	444	22	,	,	PUNCT
ejpam-6375	444	23	{	{	PUNCT
ejpam-6375	444	24	j1	j1	PROPN
ejpam-6375	444	25	,	,	PUNCT
ejpam-6375	444	26	k1	k1	NOUN
ejpam-6375	444	27	}	}	PUNCT
ejpam-6375	444	28	}	}	PUNCT
ejpam-6375	444	29	,	,	PUNCT
ejpam-6375	444	30	p	p	NOUN
ejpam-6375	444	31	=	=	X
ejpam-6375	444	32	{	{	PUNCT
ejpam-6375	444	33	∅	∅	NOUN
ejpam-6375	444	34	,	,	PUNCT
ejpam-6375	444	35	{	{	PUNCT
ejpam-6375	444	36	j1	j1	PROPN
ejpam-6375	444	37	}	}	PUNCT
ejpam-6375	444	38	,	,	PUNCT
ejpam-6375	444	39	{	{	PUNCT
ejpam-6375	444	40	l1	l1	PROPN
ejpam-6375	444	41	}	}	PUNCT
ejpam-6375	444	42	,	,	PUNCT
ejpam-6375	444	43	{	{	PUNCT
ejpam-6375	444	44	j1	j1	PROPN
ejpam-6375	444	45	,	,	PUNCT
ejpam-6375	444	46	l1	l1	PROPN
ejpam-6375	444	47	}	}	PUNCT
ejpam-6375	444	48	}	}	PUNCT
ejpam-6375	444	49	.	.	PUNCT
ejpam-6375	445	1	hence	hence	ADV
ejpam-6375	445	2	,	,	PUNCT
ejpam-6375	445	3	(	(	PUNCT
ejpam-6375	445	4	v	v	NOUN
ejpam-6375	445	5	,	,	PUNCT
ejpam-6375	445	6	gτ	gτ	INTJ
ejpam-6375	445	7	,	,	PUNCT
ejpam-6375	445	8	p	p	X
ejpam-6375	445	9	)	)	PUNCT
ejpam-6375	445	10	is	be	AUX
ejpam-6375	445	11	gpt	gpt	NOUN
ejpam-6375	445	12	-	-	PUNCT
ejpam-6375	445	13	regular	regular	ADJ
ejpam-6375	445	14	but	but	CCONJ
ejpam-6375	445	15	not	not	PART
ejpam-6375	445	16	gpt	gpt	NOUN
ejpam-6375	445	17	-	-	PUNCT
ejpam-6375	445	18	s∗	s∗	PROPN
ejpam-6375	445	19	g	g	ADP
ejpam-6375	445	20	-regular	-regular	ADJ
ejpam-6375	445	21	space	space	NOUN
ejpam-6375	445	22	.	.	PUNCT
ejpam-6375	446	1	for	for	ADP
ejpam-6375	446	2	{	{	PUNCT
ejpam-6375	446	3	k1	k1	NOUN
ejpam-6375	446	4	}	}	PUNCT
ejpam-6375	446	5	∈	∈	PROPN
ejpam-6375	446	6	gpt	gpt	NOUN
ejpam-6375	446	7	-	-	PUNCT
ejpam-6375	446	8	s∗	s∗	NOUN
ejpam-6375	446	9	gc(v	gc(v	NOUN
ejpam-6375	446	10	)	)	PUNCT
ejpam-6375	446	11	and	and	CCONJ
ejpam-6375	446	12	r	r	NOUN
ejpam-6375	446	13	/∈	/∈	PUNCT
ejpam-6375	446	14	{	{	PUNCT
ejpam-6375	446	15	k1	k1	NOUN
ejpam-6375	446	16	}	}	PUNCT
ejpam-6375	446	17	,	,	PUNCT
ejpam-6375	446	18	there	there	PRON
ejpam-6375	446	19	does	do	AUX
ejpam-6375	446	20	not	not	PART
ejpam-6375	446	21	exists	exist	VERB
ejpam-6375	446	22	disjoint	disjoint	ADJ
ejpam-6375	446	23	open	open	ADJ
ejpam-6375	446	24	sets	set	NOUN
ejpam-6375	446	25	e	e	NOUN
ejpam-6375	446	26	and	and	CCONJ
ejpam-6375	446	27	d	d	ADP
ejpam-6375	446	28	such	such	ADJ
ejpam-6375	446	29	that	that	SCONJ
ejpam-6375	446	30	{	{	PUNCT
ejpam-6375	446	31	k1	k1	NOUN
ejpam-6375	446	32	}	}	PUNCT
ejpam-6375	446	33	⊆	⊆	NUM
ejpam-6375	446	34	e	e	NOUN
ejpam-6375	446	35	,	,	PUNCT
ejpam-6375	446	36	r	r	PROPN
ejpam-6375	446	37	∈	∈	PROPN
ejpam-6375	446	38	d.	d.	PROPN
ejpam-6375	446	39	theorem	theorem	VERB
ejpam-6375	446	40	3.20	3.20	NUM
ejpam-6375	446	41	.	.	PUNCT
ejpam-6375	447	1	every	every	DET
ejpam-6375	447	2	gpt	gpt	NOUN
ejpam-6375	447	3	-	-	PUNCT
ejpam-6375	447	4	regular	regular	NOUN
ejpam-6375	447	5	with	with	ADP
ejpam-6375	447	6	gpt	gpt	NOUN
ejpam-6375	447	7	-	-	PUNCT
ejpam-6375	447	8	s∗	s∗	PROPN
ejpam-6375	447	9	gtc	gtc	PROPN
ejpam-6375	447	10	space	space	NOUN
ejpam-6375	447	11	is	be	AUX
ejpam-6375	447	12	gpt	gpt	NOUN
ejpam-6375	447	13	-	-	PUNCT
ejpam-6375	447	14	s∗	s∗	NOUN
ejpam-6375	447	15	g	g	NOUN
ejpam-6375	447	16	-regular	-regular	ADJ
ejpam-6375	447	17	.	.	PUNCT
ejpam-6375	448	1	proof	proof	NOUN
ejpam-6375	448	2	.	.	PUNCT
ejpam-6375	449	1	under	under	ADP
ejpam-6375	449	2	the	the	DET
ejpam-6375	449	3	condition	condition	NOUN
ejpam-6375	449	4	that	that	SCONJ
ejpam-6375	449	5	v	v	NOUN
ejpam-6375	449	6	is	be	AUX
ejpam-6375	449	7	gpt	gpt	NOUN
ejpam-6375	449	8	-	-	PUNCT
ejpam-6375	449	9	regular	regular	ADJ
ejpam-6375	449	10	and	and	CCONJ
ejpam-6375	449	11	gpt	gpt	NOUN
ejpam-6375	449	12	-	-	PUNCT
ejpam-6375	449	13	s∗	s∗	PROPN
ejpam-6375	449	14	gtc	gtc	PROPN
ejpam-6375	449	15	.	.	PROPN
ejpam-6375	449	16	take	take	VERB
ejpam-6375	449	17	a	a	DET
ejpam-6375	449	18	set	set	NOUN
ejpam-6375	449	19	f	f	NOUN
ejpam-6375	449	20	which	which	PRON
ejpam-6375	449	21	belongs	belong	VERB
ejpam-6375	449	22	to	to	ADP
ejpam-6375	449	23	gpt	gpt	NOUN
ejpam-6375	449	24	-	-	PUNCT
ejpam-6375	449	25	s∗	s∗	NOUN
ejpam-6375	449	26	gc(v	gc(v	NOUN
ejpam-6375	449	27	)	)	PUNCT
ejpam-6375	449	28	along	along	ADP
ejpam-6375	449	29	with	with	ADP
ejpam-6375	449	30	an	an	DET
ejpam-6375	449	31	element	element	NOUN
ejpam-6375	449	32	r	r	NOUN
ejpam-6375	449	33	belonging	belong	VERB
ejpam-6375	449	34	to	to	ADP
ejpam-6375	449	35	both	both	PRON
ejpam-6375	449	36	v	v	NOUN
ejpam-6375	449	37	and	and	CCONJ
ejpam-6375	449	38	non	non	ADJ
ejpam-6375	449	39	-	-	ADJ
ejpam-6375	449	40	corresponding	corresponding	ADJ
ejpam-6375	449	41	to	to	ADP
ejpam-6375	449	42	f.	f.	PROPN
ejpam-6375	449	43	because	because	SCONJ
ejpam-6375	449	44	v	v	NUM
ejpam-6375	449	45	functions	function	NOUN
ejpam-6375	449	46	as	as	ADP
ejpam-6375	449	47	a	a	DET
ejpam-6375	449	48	gpt	gpt	NOUN
ejpam-6375	449	49	-	-	PUNCT
ejpam-6375	449	50	s∗	s∗	PROPN
ejpam-6375	449	51	gtc	gtc	PROPN
ejpam-6375	449	52	space	space	NOUN
ejpam-6375	449	53	,	,	PUNCT
ejpam-6375	449	54	its	its	PRON
ejpam-6375	449	55	concluding	concluding	NOUN
ejpam-6375	449	56	that	that	SCONJ
ejpam-6375	449	57	f	f	PROPN
ejpam-6375	449	58	is	be	AUX
ejpam-6375	449	59	closed	closed	ADJ
ejpam-6375	449	60	yet	yet	ADV
ejpam-6375	449	61	r	r	NOUN
ejpam-6375	449	62	belongs	belong	VERB
ejpam-6375	449	63	to	to	ADP
ejpam-6375	449	64	the	the	DET
ejpam-6375	449	65	exterior	exterior	NOUN
ejpam-6375	449	66	of	of	ADP
ejpam-6375	449	67	f.	f.	PROPN
ejpam-6375	449	68	since	since	SCONJ
ejpam-6375	449	69	v	v	NUM
ejpam-6375	449	70	possesses	possess	VERB
ejpam-6375	449	71	the	the	DET
ejpam-6375	449	72	property	property	NOUN
ejpam-6375	449	73	of	of	ADP
ejpam-6375	449	74	gpt	gpt	NOUN
ejpam-6375	449	75	-	-	PUNCT
ejpam-6375	449	76	regularity	regularity	NOUN
ejpam-6375	449	77	a	a	DET
ejpam-6375	449	78	pair	pair	NOUN
ejpam-6375	449	79	of	of	ADP
ejpam-6375	449	80	open	open	ADJ
ejpam-6375	449	81	sets	set	NOUN
ejpam-6375	449	82	named	name	VERB
ejpam-6375	449	83	e	e	NOUN
ejpam-6375	449	84	and	and	CCONJ
ejpam-6375	449	85	d	d	PROPN
ejpam-6375	449	86	exists	exist	VERB
ejpam-6375	449	87	with	with	ADP
ejpam-6375	449	88	the	the	DET
ejpam-6375	449	89	property	property	NOUN
ejpam-6375	449	90	that	that	PRON
ejpam-6375	449	91	f	f	PROPN
ejpam-6375	449	92	belongs	belong	VERB
ejpam-6375	449	93	to	to	ADP
ejpam-6375	449	94	e	e	NOUN
ejpam-6375	449	95	while	while	SCONJ
ejpam-6375	449	96	r	r	NOUN
ejpam-6375	449	97	belongs	belong	VERB
ejpam-6375	449	98	to	to	ADP
ejpam-6375	449	99	d	d	PROPN
ejpam-6375	449	100	and	and	CCONJ
ejpam-6375	449	101	these	these	DET
ejpam-6375	449	102	open	open	ADJ
ejpam-6375	449	103	sets	set	NOUN
ejpam-6375	449	104	are	be	AUX
ejpam-6375	449	105	disjoint	disjoint	ADJ
ejpam-6375	449	106	.	.	PUNCT
ejpam-6375	450	1	the	the	DET
ejpam-6375	450	2	space	space	NOUN
ejpam-6375	450	3	v	v	NOUN
ejpam-6375	450	4	meets	meet	VERB
ejpam-6375	450	5	the	the	DET
ejpam-6375	450	6	criteria	criterion	NOUN
ejpam-6375	450	7	for	for	ADP
ejpam-6375	450	8	being	be	AUX
ejpam-6375	450	9	a	a	DET
ejpam-6375	450	10	gpt	gpt	NOUN
ejpam-6375	450	11	-	-	PUNCT
ejpam-6375	450	12	s∗	s∗	NOUN
ejpam-6375	450	13	g	g	PROPN
ejpam-6375	450	14	-regular	-regular	NOUN
ejpam-6375	450	15	.	.	PUNCT
ejpam-6375	451	1	theorem	theorem	VERB
ejpam-6375	451	2	3.21	3.21	NUM
ejpam-6375	451	3	.	.	PUNCT
ejpam-6375	452	1	if	if	SCONJ
ejpam-6375	452	2	v	v	NOUN
ejpam-6375	452	3	is	be	AUX
ejpam-6375	452	4	gpt	gpt	NOUN
ejpam-6375	452	5	-	-	PUNCT
ejpam-6375	452	6	s∗	s∗	PROPN
ejpam-6375	452	7	g	g	PROPN
ejpam-6375	452	8	-regular	-regular	ADJ
ejpam-6375	452	9	,	,	PUNCT
ejpam-6375	452	10	then	then	ADV
ejpam-6375	452	11	it	it	PRON
ejpam-6375	452	12	is	be	AUX
ejpam-6375	452	13	gpt	gpt	NOUN
ejpam-6375	452	14	-	-	PUNCT
ejpam-6375	452	15	regular	regular	ADJ
ejpam-6375	452	16	space	space	NOUN
ejpam-6375	452	17	.	.	PUNCT
ejpam-6375	453	1	proof	proof	NOUN
ejpam-6375	453	2	.	.	PUNCT
ejpam-6375	454	1	according	accord	VERB
ejpam-6375	454	2	to	to	ADP
ejpam-6375	454	3	the	the	DET
ejpam-6375	454	4	fact	fact	NOUN
ejpam-6375	454	5	,	,	PUNCT
ejpam-6375	454	6	every	every	DET
ejpam-6375	454	7	closed	close	VERB
ejpam-6375	454	8	set	set	NOUN
ejpam-6375	454	9	belongs	belong	VERB
ejpam-6375	454	10	to	to	ADP
ejpam-6375	454	11	gpt	gpt	NOUN
ejpam-6375	454	12	-	-	PUNCT
ejpam-6375	454	13	s∗	s∗	NOUN
ejpam-6375	454	14	gc(v	gc(v	NOUN
ejpam-6375	454	15	)	)	PUNCT
ejpam-6375	454	16	.	.	PUNCT
ejpam-6375	455	1	theorem	theorem	VERB
ejpam-6375	455	2	3.22	3.22	NUM
ejpam-6375	455	3	.	.	PUNCT
ejpam-6375	456	1	the	the	DET
ejpam-6375	456	2	subsequent	subsequent	ADJ
ejpam-6375	456	3	statements	statement	NOUN
ejpam-6375	456	4	are	be	AUX
ejpam-6375	456	5	equivalent	equivalent	ADJ
ejpam-6375	456	6	:	:	PUNCT
ejpam-6375	456	7	(	(	PUNCT
ejpam-6375	456	8	i	i	NOUN
ejpam-6375	456	9	)	)	PUNCT
ejpam-6375	456	10	v	v	NOUN
ejpam-6375	456	11	is	be	AUX
ejpam-6375	456	12	gpt	gpt	NOUN
ejpam-6375	456	13	-	-	PUNCT
ejpam-6375	456	14	s∗	s∗	NOUN
ejpam-6375	456	15	g	g	NOUN
ejpam-6375	456	16	-regular	-regular	NOUN
ejpam-6375	456	17	.	.	PUNCT
ejpam-6375	457	1	(	(	PUNCT
ejpam-6375	457	2	ii	ii	NOUN
ejpam-6375	457	3	)	)	PUNCT
ejpam-6375	457	4	for	for	ADP
ejpam-6375	457	5	all	all	DET
ejpam-6375	457	6	r	r	NOUN
ejpam-6375	457	7	∈	∈	PROPN
ejpam-6375	457	8	v	v	NOUN
ejpam-6375	457	9	and	and	CCONJ
ejpam-6375	457	10	each	each	DET
ejpam-6375	457	11	gpt	gpt	NOUN
ejpam-6375	457	12	-	-	PUNCT
ejpam-6375	457	13	s∗	s∗	PROPN
ejpam-6375	457	14	g	g	PROPN
ejpam-6375	457	15	-open	-open	PROPN
ejpam-6375	457	16	neighbourhood	neighbourhood	NOUN
ejpam-6375	457	17	uα	uα	ADP
ejpam-6375	457	18	there	there	PRON
ejpam-6375	457	19	exists	exist	VERB
ejpam-6375	457	20	open	open	ADJ
ejpam-6375	457	21	neighbourhood	neighbourhood	NOUN
ejpam-6375	457	22	nα	nα	NOUN
ejpam-6375	457	23	of	of	ADP
ejpam-6375	457	24	v	v	NOUN
ejpam-6375	457	25	such	such	ADJ
ejpam-6375	457	26	that	that	DET
ejpam-6375	457	27	cl	cl	NOUN
ejpam-6375	457	28	◦	◦	NOUN
ejpam-6375	457	29	(	(	PUNCT
ejpam-6375	457	30	nα	nα	NOUN
ejpam-6375	457	31	)	)	PUNCT
ejpam-6375	457	32	⊆	⊆	NUM
ejpam-6375	457	33	uα	uα	NOUN
ejpam-6375	458	1	.	.	PUNCT
ejpam-6375	458	2	proof	proof	NOUN
ejpam-6375	458	3	.	.	PUNCT
ejpam-6375	459	1	(	(	PUNCT
ejpam-6375	459	2	1	1	X
ejpam-6375	459	3	)	)	PUNCT
ejpam-6375	459	4	implies	imply	VERB
ejpam-6375	459	5	(	(	PUNCT
ejpam-6375	459	6	2	2	X
ejpam-6375	459	7	)	)	PUNCT
ejpam-6375	459	8	assume	assume	VERB
ejpam-6375	459	9	uα	uα	PROPN
ejpam-6375	459	10	is	be	AUX
ejpam-6375	459	11	gpt	gpt	NOUN
ejpam-6375	459	12	-	-	PUNCT
ejpam-6375	459	13	s∗	s∗	PROPN
ejpam-6375	459	14	g	g	PROPN
ejpam-6375	459	15	-neighbourhood	-neighbourhood	NOUN
ejpam-6375	459	16	of	of	ADP
ejpam-6375	459	17	r	r	NOUN
ejpam-6375	459	18	,	,	PUNCT
ejpam-6375	459	19	there	there	PRON
ejpam-6375	459	20	exists	exist	VERB
ejpam-6375	459	21	e	e	PROPN
ejpam-6375	459	22	∈	∈	PROPN
ejpam-6375	459	23	gpts∗	gpts∗	PROPN
ejpam-6375	459	24	go(v	go(v	NOUN
ejpam-6375	459	25	)	)	PUNCT
ejpam-6375	459	26	such	such	ADJ
ejpam-6375	459	27	that	that	SCONJ
ejpam-6375	459	28	r	r	NOUN
ejpam-6375	459	29	∈	∈	NOUN
ejpam-6375	459	30	e	e	NOUN
ejpam-6375	459	31	⊆	⊆	NUM
ejpam-6375	459	32	uα	uα	PROPN
ejpam-6375	459	33	.	.	PUNCT
ejpam-6375	460	1	now	now	ADV
ejpam-6375	460	2	,	,	PUNCT
ejpam-6375	460	3	ec	ec	PROPN
ejpam-6375	460	4	∈	∈	PROPN
ejpam-6375	460	5	gpt	gpt	NOUN
ejpam-6375	460	6	-	-	PUNCT
ejpam-6375	460	7	s∗	s∗	NOUN
ejpam-6375	460	8	gc(v	gc(v	NOUN
ejpam-6375	460	9	)	)	PUNCT
ejpam-6375	460	10	and	and	CCONJ
ejpam-6375	460	11	r	r	NOUN
ejpam-6375	460	12	/∈	/∈	PUNCT
ejpam-6375	460	13	ec	ec	PROPN
ejpam-6375	460	14	.	.	PUNCT
ejpam-6375	460	15	from	from	ADP
ejpam-6375	460	16	(	(	PUNCT
ejpam-6375	460	17	1	1	NUM
ejpam-6375	460	18	)	)	PUNCT
ejpam-6375	460	19	,	,	PUNCT
ejpam-6375	460	20	there	there	PRON
ejpam-6375	460	21	exists	exist	VERB
ejpam-6375	460	22	rα	rα	ADJ
ejpam-6375	460	23	,	,	PUNCT
ejpam-6375	460	24	sα	sα	ADV
ejpam-6375	460	25	such	such	ADJ
ejpam-6375	460	26	that	that	SCONJ
ejpam-6375	460	27	ec	ec	PROPN
ejpam-6375	460	28	⊆	⊆	NUM
ejpam-6375	460	29	rα	rα	ADJ
ejpam-6375	460	30	,	,	PUNCT
ejpam-6375	460	31	r	r	NOUN
ejpam-6375	460	32	∈	∈	PROPN
ejpam-6375	460	33	sα	sα	ADJ
ejpam-6375	460	34	,	,	PUNCT
ejpam-6375	460	35	rα	rα	ADJ
ejpam-6375	460	36	∩	∩	NOUN
ejpam-6375	460	37	sα	sα	ADJ
ejpam-6375	460	38	=	=	PUNCT
ejpam-6375	460	39	∅.	∅.	AUX
ejpam-6375	460	40	thus	thus	ADV
ejpam-6375	460	41	sα	sα	ADV
ejpam-6375	461	1	⊆	⊆	NUM
ejpam-6375	461	2	mα	mα	PROPN
ejpam-6375	461	3	c.	c.	PROPN
ejpam-6375	461	4	now	now	ADV
ejpam-6375	461	5	,	,	PUNCT
ejpam-6375	461	6	cl	cl	NOUN
ejpam-6375	461	7	◦	◦	NOUN
ejpam-6375	461	8	(sα	(sα	PUNCT
ejpam-6375	461	9	)	)	PUNCT
ejpam-6375	462	1	⊆	⊆	NUM
ejpam-6375	462	2	cl	cl	NOUN
ejpam-6375	462	3	◦	◦	NOUN
ejpam-6375	462	4	(rα	(rα	NOUN
ejpam-6375	462	5	c	c	NOUN
ejpam-6375	462	6	)	)	PUNCT
ejpam-6375	462	7	=	=	SYM
ejpam-6375	462	8	ec	ec	PROPN
ejpam-6375	462	9	and	and	CCONJ
ejpam-6375	462	10	ec	ec	PROPN
ejpam-6375	462	11	⊆	⊆	NUM
ejpam-6375	462	12	rα	rα	NOUN
ejpam-6375	462	13	implies	imply	VERB
ejpam-6375	462	14	rα	rα	INTJ
ejpam-6375	462	15	c	c	NOUN
ejpam-6375	462	16	⊆	⊆	NUM
ejpam-6375	462	17	e	e	X
ejpam-6375	462	18	⊆	⊆	NUM
ejpam-6375	462	19	uα	uα	PROPN
ejpam-6375	462	20	.	.	PUNCT
ejpam-6375	463	1	thus	thus	ADV
ejpam-6375	463	2	cl	cl	VERB
ejpam-6375	463	3	◦	◦	NOUN
ejpam-6375	463	4	(sα	(sα	PUNCT
ejpam-6375	463	5	)	)	PUNCT
ejpam-6375	463	6	⊆	⊆	NUM
ejpam-6375	463	7	uα	uα	NOUN
ejpam-6375	463	8	.	.	PUNCT
ejpam-6375	464	1	(	(	PUNCT
ejpam-6375	464	2	2	2	X
ejpam-6375	464	3	)	)	PUNCT
ejpam-6375	464	4	implies	imply	VERB
ejpam-6375	464	5	(	(	PUNCT
ejpam-6375	464	6	1	1	X
ejpam-6375	464	7	)	)	PUNCT
ejpam-6375	464	8	consider	consider	VERB
ejpam-6375	464	9	gpt	gpt	NOUN
ejpam-6375	464	10	-	-	PUNCT
ejpam-6375	464	11	s∗	s∗	PROPN
ejpam-6375	464	12	g	g	PROPN
ejpam-6375	464	13	-closed	-closed	PROPN
ejpam-6375	464	14	f	f	PROPN
ejpam-6375	464	15	in	in	ADP
ejpam-6375	464	16	v	v	NOUN
ejpam-6375	464	17	and	and	CCONJ
ejpam-6375	464	18	r	r	NOUN
ejpam-6375	464	19	/∈	/∈	PUNCT
ejpam-6375	465	1	f	f	NOUN
ejpam-6375	465	2	or	or	CCONJ
ejpam-6375	465	3	r	r	NOUN
ejpam-6375	465	4	∈	∈	PROPN
ejpam-6375	465	5	(	(	PUNCT
ejpam-6375	465	6	f)c	f)c	NOUN
ejpam-6375	465	7	and	and	CCONJ
ejpam-6375	465	8	v	v	NOUN
ejpam-6375	465	9	is	be	AUX
ejpam-6375	465	10	gpt	gpt	NOUN
ejpam-6375	465	11	-	-	PUNCT
ejpam-6375	465	12	s∗	s∗	PROPN
ejpam-6375	465	13	g	g	PROPN
ejpam-6375	465	14	-open	-open	NOUN
ejpam-6375	465	15	implies	imply	VERB
ejpam-6375	465	16	(	(	PUNCT
ejpam-6375	465	17	f)c	f)c	ADV
ejpam-6375	465	18	is	be	AUX
ejpam-6375	465	19	m.	m.	NOUN
ejpam-6375	465	20	shahbaz	shahbaz	PROPN
ejpam-6375	465	21	et	et	PROPN
ejpam-6375	465	22	al	al	PROPN
ejpam-6375	466	1	.	.	PUNCT
ejpam-6375	466	2	/	/	SYM
ejpam-6375	466	3	eur	eur	PROPN
ejpam-6375	466	4	.	.	PUNCT
ejpam-6375	467	1	j.	j.	PROPN
ejpam-6375	467	2	pure	pure	PROPN
ejpam-6375	467	3	appl	appl	PROPN
ejpam-6375	467	4	.	.	PROPN
ejpam-6375	467	5	math	math	PROPN
ejpam-6375	467	6	,	,	PUNCT
ejpam-6375	467	7	18	18	NUM
ejpam-6375	467	8	(	(	PUNCT
ejpam-6375	467	9	4	4	NUM
ejpam-6375	467	10	)	)	PUNCT
ejpam-6375	467	11	(	(	PUNCT
ejpam-6375	467	12	2025	2025	NUM
ejpam-6375	467	13	)	)	PUNCT
ejpam-6375	467	14	,	,	PUNCT
ejpam-6375	467	15	6375	6375	NUM
ejpam-6375	467	16	17	17	NUM
ejpam-6375	467	17	of	of	ADP
ejpam-6375	467	18	22	22	NUM
ejpam-6375	467	19	gpt	gpt	NOUN
ejpam-6375	467	20	-	-	PUNCT
ejpam-6375	467	21	s∗	s∗	PROPN
ejpam-6375	467	22	g	g	PROPN
ejpam-6375	467	23	-neighbourhood	-neighbourhood	PROPN
ejpam-6375	467	24	of	of	ADP
ejpam-6375	467	25	r.	r.	PROPN
ejpam-6375	467	26	by	by	ADP
ejpam-6375	467	27	hypothesis	hypothesis	NOUN
ejpam-6375	468	1	,	,	PUNCT
ejpam-6375	468	2	there	there	PRON
ejpam-6375	468	3	exists	exist	VERB
ejpam-6375	468	4	an	an	DET
ejpam-6375	468	5	open	open	ADJ
ejpam-6375	468	6	neighbourhood	neighbourhood	NOUN
ejpam-6375	468	7	nα	nα	ADP
ejpam-6375	468	8	such	such	ADJ
ejpam-6375	468	9	that	that	SCONJ
ejpam-6375	468	10	r	r	PROPN
ejpam-6375	468	11	∈	∈	PROPN
ejpam-6375	468	12	nα	nα	NOUN
ejpam-6375	468	13	,	,	PUNCT
ejpam-6375	468	14	cl	cl	NOUN
ejpam-6375	468	15	◦	◦	NOUN
ejpam-6375	468	16	(	(	PUNCT
ejpam-6375	468	17	nα	nα	NOUN
ejpam-6375	468	18	)	)	PUNCT
ejpam-6375	468	19	⊆	⊆	NUM
ejpam-6375	468	20	(	(	PUNCT
ejpam-6375	468	21	f)c	f)c	ADV
ejpam-6375	468	22	implies	imply	VERB
ejpam-6375	468	23	f	f	PROPN
ejpam-6375	468	24	⊆	⊆	NUM
ejpam-6375	468	25	{	{	PUNCT
ejpam-6375	468	26	v	v	NUM
ejpam-6375	468	27	cl	cl	NOUN
ejpam-6375	468	28	◦	◦	NOUN
ejpam-6375	468	29	(	(	PUNCT
ejpam-6375	468	30	nα	nα	NOUN
ejpam-6375	468	31	)	)	PUNCT
ejpam-6375	468	32	}	}	PUNCT
ejpam-6375	468	33	and	and	CCONJ
ejpam-6375	468	34	nα	nα	ADP
ejpam-6375	468	35	∩	∩	NOUN
ejpam-6375	468	36	{	{	PUNCT
ejpam-6375	468	37	v	v	NOUN
ejpam-6375	468	38	cl	cl	NOUN
ejpam-6375	468	39	◦	◦	NOUN
ejpam-6375	468	40	(nα	(nα	PUNCT
ejpam-6375	468	41	)	)	PUNCT
ejpam-6375	468	42	}	}	PUNCT
ejpam-6375	468	43	=	=	PUNCT
ejpam-6375	468	44	∅.	∅.	VERB
ejpam-6375	468	45	thus	thus	ADV
ejpam-6375	468	46	,	,	PUNCT
ejpam-6375	468	47	v	v	NOUN
ejpam-6375	468	48	is	be	AUX
ejpam-6375	468	49	gpt	gpt	NOUN
ejpam-6375	468	50	-	-	PUNCT
ejpam-6375	468	51	s∗	s∗	NOUN
ejpam-6375	468	52	g	g	PROPN
ejpam-6375	468	53	-regular	-regular	NOUN
ejpam-6375	468	54	.	.	PUNCT
ejpam-6375	469	1	theorem	theorem	VERB
ejpam-6375	469	2	3.23	3.23	NUM
ejpam-6375	469	3	.	.	PUNCT
ejpam-6375	470	1	assume	assume	VERB
ejpam-6375	470	2	v	v	NUM
ejpam-6375	470	3	is	be	AUX
ejpam-6375	470	4	gpt	gpt	NOUN
ejpam-6375	470	5	-	-	PUNCT
ejpam-6375	470	6	s∗	s∗	PROPN
ejpam-6375	470	7	g	g	PROPN
ejpam-6375	470	8	-regular	-regular	ADJ
ejpam-6375	470	9	iff	iff	NOUN
ejpam-6375	470	10	for	for	ADP
ejpam-6375	470	11	every	every	DET
ejpam-6375	470	12	e	e	PROPN
ejpam-6375	470	13	∈	∈	PROPN
ejpam-6375	470	14	gpt	gpt	NOUN
ejpam-6375	470	15	-	-	PUNCT
ejpam-6375	470	16	s∗	s∗	NOUN
ejpam-6375	470	17	gc(v	gc(v	NOUN
ejpam-6375	470	18	)	)	PUNCT
ejpam-6375	470	19	and	and	CCONJ
ejpam-6375	470	20	point	point	VERB
ejpam-6375	470	21	p	p	X
ejpam-6375	470	22	∈	∈	PROPN
ejpam-6375	470	23	(	(	PUNCT
ejpam-6375	470	24	v	v	NOUN
ejpam-6375	470	25	e	e	NOUN
ejpam-6375	470	26	)	)	PUNCT
ejpam-6375	470	27	then	then	ADV
ejpam-6375	470	28	r	r	PROPN
ejpam-6375	470	29	∈	∈	PROPN
ejpam-6375	470	30	uα	uα	PROPN
ejpam-6375	470	31	,	,	PUNCT
ejpam-6375	470	32	e	e	NOUN
ejpam-6375	470	33	⊆	⊆	NUM
ejpam-6375	470	34	nα	nα	NOUN
ejpam-6375	470	35	and	and	CCONJ
ejpam-6375	470	36	cl	cl	NOUN
ejpam-6375	470	37	◦	◦	NOUN
ejpam-6375	470	38	(nα	(nα	SYM
ejpam-6375	470	39	)	)	PUNCT
ejpam-6375	470	40	∩	∩	ADJ
ejpam-6375	470	41	cl	cl	NOUN
ejpam-6375	470	42	◦	◦	NOUN
ejpam-6375	470	43	(uα	(uα	NOUN
ejpam-6375	470	44	)	)	PUNCT
ejpam-6375	470	45	=	=	NOUN
ejpam-6375	470	46	∅	∅	NOUN
ejpam-6375	470	47	,	,	PUNCT
ejpam-6375	470	48	where	where	SCONJ
ejpam-6375	470	49	uα	uα	PROPN
ejpam-6375	470	50	and	and	CCONJ
ejpam-6375	470	51	nα	nα	NOUN
ejpam-6375	470	52	are	be	AUX
ejpam-6375	470	53	open	open	ADJ
ejpam-6375	470	54	sets	set	NOUN
ejpam-6375	470	55	.	.	PUNCT
ejpam-6375	471	1	proof	proof	NOUN
ejpam-6375	471	2	.	.	PUNCT
ejpam-6375	472	1	given	give	VERB
ejpam-6375	472	2	that	that	DET
ejpam-6375	472	3	v	v	NOUN
ejpam-6375	472	4	is	be	AUX
ejpam-6375	472	5	gpt	gpt	NOUN
ejpam-6375	472	6	-	-	PUNCT
ejpam-6375	472	7	s∗	s∗	NOUN
ejpam-6375	472	8	g	g	PROPN
ejpam-6375	472	9	-regular	-regular	ADJ
ejpam-6375	472	10	.	.	PUNCT
ejpam-6375	473	1	assume	assume	VERB
ejpam-6375	473	2	e	e	X
ejpam-6375	473	3	∈	∈	PROPN
ejpam-6375	473	4	gpt	gpt	NOUN
ejpam-6375	473	5	-	-	PUNCT
ejpam-6375	473	6	s∗	s∗	NOUN
ejpam-6375	473	7	gc(v	gc(v	NOUN
ejpam-6375	473	8	)	)	PUNCT
ejpam-6375	473	9	and	and	CCONJ
ejpam-6375	473	10	r	r	NOUN
ejpam-6375	473	11	/∈	/∈	PROPN
ejpam-6375	474	1	e.	e.	PROPN
ejpam-6375	474	2	then	then	ADV
ejpam-6375	474	3	,	,	PUNCT
ejpam-6375	474	4	p	p	PROPN
ejpam-6375	474	5	∈	∈	PROPN
ejpam-6375	474	6	mα	mα	NOUN
ejpam-6375	474	7	and	and	CCONJ
ejpam-6375	474	8	e	e	NOUN
ejpam-6375	474	9	⊆	⊆	NUM
ejpam-6375	474	10	nα	nα	ADP
ejpam-6375	474	11	and	and	CCONJ
ejpam-6375	474	12	mα	mα	ADP
ejpam-6375	474	13	∩	∩	NOUN
ejpam-6375	474	14	nα	nα	NOUN
ejpam-6375	474	15	=	=	SYM
ejpam-6375	474	16	∅	∅	NOUN
ejpam-6375	474	17	,	,	PUNCT
ejpam-6375	474	18	where	where	SCONJ
ejpam-6375	474	19	mα	mα	PROPN
ejpam-6375	474	20	and	and	CCONJ
ejpam-6375	474	21	nα	nα	PROPN
ejpam-6375	474	22	are	be	AUX
ejpam-6375	474	23	open	open	ADJ
ejpam-6375	474	24	sets	set	NOUN
ejpam-6375	474	25	implies	imply	VERB
ejpam-6375	474	26	mα	mα	ADP
ejpam-6375	474	27	∩	∩	ADJ
ejpam-6375	474	28	cl	cl	NOUN
ejpam-6375	474	29	◦	◦	NOUN
ejpam-6375	474	30	(nα	(nα	PUNCT
ejpam-6375	474	31	)	)	PUNCT
ejpam-6375	474	32	=	=	PUNCT
ejpam-6375	474	33	∅.	∅.	NOUN
ejpam-6375	474	34	as	as	ADP
ejpam-6375	474	35	v	v	NOUN
ejpam-6375	474	36	is	be	AUX
ejpam-6375	474	37	gpt	gpt	NOUN
ejpam-6375	474	38	-	-	PUNCT
ejpam-6375	474	39	s∗	s∗	PROPN
ejpam-6375	474	40	g	g	PROPN
ejpam-6375	474	41	-regular	-regular	ADJ
ejpam-6375	474	42	,	,	PUNCT
ejpam-6375	474	43	p	p	PROPN
ejpam-6375	474	44	∈	∈	PROPN
ejpam-6375	474	45	rα	rα	ADJ
ejpam-6375	474	46	and	and	CCONJ
ejpam-6375	474	47	cl	cl	NOUN
ejpam-6375	474	48	◦	◦	NOUN
ejpam-6375	474	49	(nα	(nα	PUNCT
ejpam-6375	474	50	)	)	PUNCT
ejpam-6375	474	51	⊆	⊆	NUM
ejpam-6375	474	52	sα	sα	ADJ
ejpam-6375	474	53	,	,	PUNCT
ejpam-6375	474	54	rα	rα	ADJ
ejpam-6375	474	55	∩	∩	NOUN
ejpam-6375	474	56	nα	nα	NOUN
ejpam-6375	474	57	=	=	SYM
ejpam-6375	474	58	∅	∅	NOUN
ejpam-6375	474	59	where	where	SCONJ
ejpam-6375	474	60	rα	rα	ADJ
ejpam-6375	474	61	and	and	CCONJ
ejpam-6375	474	62	sα	sα	ADV
ejpam-6375	474	63	are	be	AUX
ejpam-6375	474	64	open	open	ADJ
ejpam-6375	474	65	.	.	PUNCT
ejpam-6375	475	1	furthermore	furthermore	ADV
ejpam-6375	475	2	,	,	PUNCT
ejpam-6375	475	3	cl	cl	NOUN
ejpam-6375	475	4	◦	◦	NOUN
ejpam-6375	475	5	(rα	(rα	PUNCT
ejpam-6375	475	6	)	)	PUNCT
ejpam-6375	476	1	∩	∩	NOUN
ejpam-6375	476	2	sα	sα	ADJ
ejpam-6375	476	3	=	=	PUNCT
ejpam-6375	476	4	∅.	∅.	AUX
ejpam-6375	476	5	assume	assume	VERB
ejpam-6375	476	6	uα	uα	PROPN
ejpam-6375	476	7	=	=	SYM
ejpam-6375	476	8	mα	mα	PROPN
ejpam-6375	476	9	∩	∩	NOUN
ejpam-6375	476	10	rα	rα	PART
ejpam-6375	476	11	implies	imply	VERB
ejpam-6375	476	12	p	p	PROPN
ejpam-6375	476	13	∈	∈	PROPN
ejpam-6375	476	14	uα	uα	PROPN
ejpam-6375	476	15	,	,	PUNCT
ejpam-6375	476	16	e	e	NOUN
ejpam-6375	476	17	⊆	⊆	NUM
ejpam-6375	476	18	nα	nα	NOUN
ejpam-6375	476	19	and	and	CCONJ
ejpam-6375	476	20	cl	cl	NOUN
ejpam-6375	476	21	◦	◦	NOUN
ejpam-6375	476	22	(nα	(nα	SYM
ejpam-6375	476	23	)	)	PUNCT
ejpam-6375	476	24	∩	∩	ADJ
ejpam-6375	476	25	cl	cl	NOUN
ejpam-6375	476	26	◦	◦	NOUN
ejpam-6375	476	27	(uα	(uα	NOUN
ejpam-6375	476	28	)	)	PUNCT
ejpam-6375	476	29	=	=	NOUN
ejpam-6375	476	30	∅	∅	NOUN
ejpam-6375	476	31	where	where	SCONJ
ejpam-6375	476	32	nα	nα	VERB
ejpam-6375	476	33	and	and	CCONJ
ejpam-6375	476	34	uα	uα	PRON
ejpam-6375	476	35	are	be	AUX
ejpam-6375	476	36	open	open	ADJ
ejpam-6375	476	37	in	in	ADP
ejpam-6375	476	38	v.	v.	ADP
ejpam-6375	476	39	on	on	ADP
ejpam-6375	476	40	the	the	DET
ejpam-6375	476	41	other	other	ADJ
ejpam-6375	476	42	hand	hand	NOUN
ejpam-6375	476	43	,	,	PUNCT
ejpam-6375	476	44	consider	consider	VERB
ejpam-6375	476	45	nα	nα	PRON
ejpam-6375	476	46	and	and	CCONJ
ejpam-6375	476	47	uα	uα	NOUN
ejpam-6375	476	48	are	be	AUX
ejpam-6375	476	49	open	open	ADJ
ejpam-6375	476	50	sets	set	NOUN
ejpam-6375	476	51	.	.	PUNCT
ejpam-6375	477	1	p	p	X
ejpam-6375	477	2	∈	∈	PROPN
ejpam-6375	477	3	uα	uα	PROPN
ejpam-6375	477	4	,	,	PUNCT
ejpam-6375	477	5	e	e	NOUN
ejpam-6375	477	6	⊆	⊆	NUM
ejpam-6375	477	7	nα	nα	NOUN
ejpam-6375	477	8	and	and	CCONJ
ejpam-6375	477	9	cl	cl	NOUN
ejpam-6375	477	10	◦	◦	NOUN
ejpam-6375	477	11	(nα	(nα	SYM
ejpam-6375	477	12	)	)	PUNCT
ejpam-6375	477	13	∩	∩	ADJ
ejpam-6375	477	14	cl	cl	NOUN
ejpam-6375	477	15	◦	◦	NOUN
ejpam-6375	477	16	(uα	(uα	NOUN
ejpam-6375	477	17	)	)	PUNCT
ejpam-6375	477	18	=	=	NOUN
ejpam-6375	477	19	∅	∅	NOUN
ejpam-6375	477	20	for	for	ADP
ejpam-6375	477	21	all	all	DET
ejpam-6375	477	22	e	e	PROPN
ejpam-6375	477	23	∈	∈	PROPN
ejpam-6375	477	24	gpt	gpt	NOUN
ejpam-6375	477	25	-	-	PUNCT
ejpam-6375	477	26	s∗	s∗	NOUN
ejpam-6375	477	27	gc(v	gc(v	NOUN
ejpam-6375	477	28	)	)	PUNCT
ejpam-6375	477	29	and	and	CCONJ
ejpam-6375	477	30	p	p	NOUN
ejpam-6375	477	31	∈	∈	PROPN
ejpam-6375	477	32	(	(	PUNCT
ejpam-6375	477	33	v	v	NOUN
ejpam-6375	477	34	e	e	NOUN
ejpam-6375	477	35	)	)	PUNCT
ejpam-6375	477	36	implies	imply	VERB
ejpam-6375	477	37	p	p	PROPN
ejpam-6375	477	38	∈	∈	PROPN
ejpam-6375	477	39	uα	uα	PROPN
ejpam-6375	477	40	,	,	PUNCT
ejpam-6375	477	41	e	e	NOUN
ejpam-6375	477	42	⊆	⊆	NUM
ejpam-6375	477	43	nα	nα	ADP
ejpam-6375	477	44	and	and	CCONJ
ejpam-6375	477	45	uα	uα	PROPN
ejpam-6375	477	46	∩	∩	NOUN
ejpam-6375	477	47	nα	nα	NOUN
ejpam-6375	477	48	=	=	PUNCT
ejpam-6375	477	49	∅.	∅.	VERB
ejpam-6375	477	50	thus	thus	ADV
ejpam-6375	477	51	,	,	PUNCT
ejpam-6375	477	52	v	v	NOUN
ejpam-6375	477	53	is	be	AUX
ejpam-6375	477	54	gpt	gpt	NOUN
ejpam-6375	477	55	-	-	PUNCT
ejpam-6375	477	56	s∗	s∗	NOUN
ejpam-6375	477	57	g	g	PROPN
ejpam-6375	477	58	-regular	-regular	NOUN
ejpam-6375	477	59	.	.	PUNCT
ejpam-6375	478	1	theorem	theorem	VERB
ejpam-6375	478	2	3.24	3.24	NUM
ejpam-6375	478	3	.	.	PUNCT
ejpam-6375	479	1	a	a	DET
ejpam-6375	479	2	subspace	subspace	NOUN
ejpam-6375	479	3	z	z	PROPN
ejpam-6375	479	4	of	of	ADP
ejpam-6375	479	5	gpt	gpt	NOUN
ejpam-6375	479	6	-	-	PUNCT
ejpam-6375	479	7	s∗	s∗	PROPN
ejpam-6375	479	8	g	g	ADP
ejpam-6375	479	9	-regular	-regular	ADJ
ejpam-6375	479	10	(	(	PUNCT
ejpam-6375	479	11	z	z	NOUN
ejpam-6375	479	12	,	,	PUNCT
ejpam-6375	479	13	gτ	gτ	INTJ
ejpam-6375	479	14	,	,	PUNCT
ejpam-6375	479	15	p	p	X
ejpam-6375	479	16	)	)	PUNCT
ejpam-6375	479	17	is	be	AUX
ejpam-6375	479	18	gpt	gpt	NOUN
ejpam-6375	479	19	-	-	PUNCT
ejpam-6375	479	20	s∗	s∗	NOUN
ejpam-6375	479	21	g	g	NOUN
ejpam-6375	479	22	-regular	-regular	ADJ
ejpam-6375	479	23	.	.	PUNCT
ejpam-6375	480	1	proof	proof	NOUN
ejpam-6375	480	2	.	.	PUNCT
ejpam-6375	481	1	obvious	obvious	ADJ
ejpam-6375	481	2	.	.	PUNCT
ejpam-6375	482	1	theorem	theorem	VERB
ejpam-6375	482	2	3.25	3.25	NUM
ejpam-6375	482	3	.	.	PUNCT
ejpam-6375	483	1	assume	assume	VERB
ejpam-6375	483	2	f	f	PROPN
ejpam-6375	483	3	is	be	AUX
ejpam-6375	483	4	bijective	bijective	ADJ
ejpam-6375	483	5	,	,	PUNCT
ejpam-6375	483	6	gpt	gpt	NOUN
ejpam-6375	483	7	-	-	PUNCT
ejpam-6375	483	8	s∗	s∗	PROPN
ejpam-6375	483	9	g	g	PROPN
ejpam-6375	483	10	-irresolute	-irresolute	ADJ
ejpam-6375	483	11	and	and	CCONJ
ejpam-6375	483	12	open	open	ADJ
ejpam-6375	483	13	map	map	NOUN
ejpam-6375	483	14	from	from	ADP
ejpam-6375	483	15	gpt	gpt	NOUN
ejpam-6375	483	16	-	-	PUNCT
ejpam-6375	483	17	s∗	s∗	PROPN
ejpam-6375	483	18	g	g	PROPN
ejpam-6375	483	19	regular	regular	ADJ
ejpam-6375	483	20	v	v	NOUN
ejpam-6375	483	21	into	into	ADP
ejpam-6375	483	22	z	z	PROPN
ejpam-6375	483	23	,	,	PUNCT
ejpam-6375	483	24	then	then	ADV
ejpam-6375	483	25	z	z	PROPN
ejpam-6375	483	26	is	be	AUX
ejpam-6375	483	27	gpt	gpt	NOUN
ejpam-6375	483	28	-	-	PUNCT
ejpam-6375	483	29	s∗	s∗	NOUN
ejpam-6375	483	30	g	g	NOUN
ejpam-6375	483	31	-regular	-regular	ADJ
ejpam-6375	483	32	.	.	PUNCT
ejpam-6375	484	1	proof	proof	NOUN
ejpam-6375	484	2	.	.	PUNCT
ejpam-6375	485	1	let	let	VERB
ejpam-6375	485	2	rα	rα	PRON
ejpam-6375	485	3	∈	∈	PROPN
ejpam-6375	485	4	z	z	PROPN
ejpam-6375	485	5	and	and	CCONJ
ejpam-6375	485	6	f	f	PROPN
ejpam-6375	485	7	∈	∈	PROPN
ejpam-6375	485	8	gpt	gpt	NOUN
ejpam-6375	485	9	-	-	PUNCT
ejpam-6375	485	10	s∗	s∗	PROPN
ejpam-6375	485	11	gc(v	gc(v	NOUN
ejpam-6375	485	12	)	)	PUNCT
ejpam-6375	485	13	and	and	CCONJ
ejpam-6375	485	14	rα	rα	INTJ
ejpam-6375	485	15	/∈	/∈	PUNCT
ejpam-6375	486	1	f.	f.	PROPN
ejpam-6375	486	2	furthermore	furthermore	ADV
ejpam-6375	486	3	,	,	PUNCT
ejpam-6375	486	4	f	f	PROPN
ejpam-6375	486	5	is	be	AUX
ejpam-6375	486	6	gpt	gpt	NOUN
ejpam-6375	486	7	-	-	PUNCT
ejpam-6375	486	8	s∗	s∗	PROPN
ejpam-6375	486	9	g	g	PROPN
ejpam-6375	486	10	-irresolute	-irresolute	PROPN
ejpam-6375	486	11	,	,	PUNCT
ejpam-6375	486	12	then	then	ADV
ejpam-6375	486	13	f−1(f	f−1(f	PROPN
ejpam-6375	486	14	)	)	PUNCT
ejpam-6375	486	15	∈	∈	PROPN
ejpam-6375	486	16	gpt	gpt	NOUN
ejpam-6375	486	17	-	-	PUNCT
ejpam-6375	486	18	s∗	s∗	PROPN
ejpam-6375	486	19	gc(v	gc(v	NOUN
ejpam-6375	486	20	)	)	PUNCT
ejpam-6375	486	21	.	.	PUNCT
ejpam-6375	487	1	now	now	ADV
ejpam-6375	487	2	,	,	PUNCT
ejpam-6375	487	3	assume	assume	VERB
ejpam-6375	487	4	rα	rα	ADJ
ejpam-6375	487	5	=	=	PUNCT
ejpam-6375	487	6	f(r	f(r	X
ejpam-6375	487	7	)	)	PUNCT
ejpam-6375	487	8	then	then	ADV
ejpam-6375	487	9	f−1(rα	f−1(rα	PROPN
ejpam-6375	487	10	)	)	PUNCT
ejpam-6375	487	11	=	=	SYM
ejpam-6375	488	1	r	r	NOUN
ejpam-6375	488	2	and	and	CCONJ
ejpam-6375	488	3	r	r	NOUN
ejpam-6375	488	4	/∈	/∈	PUNCT
ejpam-6375	488	5	f−1(f	f−1(f	PROPN
ejpam-6375	488	6	)	)	PUNCT
ejpam-6375	488	7	.	.	PUNCT
ejpam-6375	489	1	as	as	SCONJ
ejpam-6375	489	2	v	v	NOUN
ejpam-6375	489	3	is	be	AUX
ejpam-6375	489	4	gpt	gpt	NOUN
ejpam-6375	489	5	-	-	PUNCT
ejpam-6375	489	6	s∗	s∗	PROPN
ejpam-6375	489	7	g	g	ADP
ejpam-6375	489	8	-regular	-regular	ADJ
ejpam-6375	489	9	then	then	ADV
ejpam-6375	489	10	there	there	PRON
ejpam-6375	489	11	exists	exist	VERB
ejpam-6375	489	12	rα	rα	ADJ
ejpam-6375	489	13	and	and	CCONJ
ejpam-6375	489	14	sα	sα	VERB
ejpam-6375	489	15	such	such	ADJ
ejpam-6375	489	16	that	that	SCONJ
ejpam-6375	489	17	r	r	NOUN
ejpam-6375	489	18	∈	∈	PROPN
ejpam-6375	489	19	rα	rα	NOUN
ejpam-6375	489	20	and	and	CCONJ
ejpam-6375	489	21	f−1(f	f−1(f	PROPN
ejpam-6375	489	22	)	)	PUNCT
ejpam-6375	490	1	⊆	⊆	NUM
ejpam-6375	490	2	sα	sα	ADJ
ejpam-6375	490	3	,	,	PUNCT
ejpam-6375	490	4	rα	rα	ADJ
ejpam-6375	490	5	∩	∩	NOUN
ejpam-6375	490	6	sα	sα	ADJ
ejpam-6375	490	7	=	=	PUNCT
ejpam-6375	490	8	∅.	∅.	NOUN
ejpam-6375	490	9	since	since	SCONJ
ejpam-6375	490	10	f	f	PROPN
ejpam-6375	490	11	is	be	AUX
ejpam-6375	490	12	open	open	ADJ
ejpam-6375	490	13	and	and	CCONJ
ejpam-6375	490	14	bijective	bijective	ADJ
ejpam-6375	490	15	implies	imply	VERB
ejpam-6375	490	16	rα	rα	PROPN
ejpam-6375	490	17	∈	∈	PROPN
ejpam-6375	490	18	f(rα	f(rα	PROPN
ejpam-6375	490	19	)	)	PUNCT
ejpam-6375	490	20	,	,	PUNCT
ejpam-6375	490	21	f	f	PROPN
ejpam-6375	490	22	⊆	⊆	NUM
ejpam-6375	490	23	f(sα	f(sα	NOUN
ejpam-6375	490	24	)	)	PUNCT
ejpam-6375	490	25	and	and	CCONJ
ejpam-6375	490	26	f(rα	f(rα	PROPN
ejpam-6375	490	27	∩	∩	NOUN
ejpam-6375	490	28	sα	sα	NOUN
ejpam-6375	490	29	)	)	PUNCT
ejpam-6375	490	30	=	=	SYM
ejpam-6375	490	31	f(∅	f(∅	PROPN
ejpam-6375	490	32	)	)	PUNCT
ejpam-6375	490	33	=	=	PUNCT
ejpam-6375	490	34	∅.	∅.	NOUN
ejpam-6375	490	35	then	then	ADV
ejpam-6375	490	36	,	,	PUNCT
ejpam-6375	490	37	z	z	PROPN
ejpam-6375	490	38	is	be	AUX
ejpam-6375	490	39	gpt	gpt	NOUN
ejpam-6375	490	40	-	-	PUNCT
ejpam-6375	490	41	s∗	s∗	NOUN
ejpam-6375	490	42	g	g	NOUN
ejpam-6375	490	43	-regular	-regular	NOUN
ejpam-6375	490	44	.	.	PUNCT
ejpam-6375	491	1	3.2.3	3.2.3	X
ejpam-6375	491	2	.	.	X
ejpam-6375	491	3	gpt	gpt	NOUN
ejpam-6375	491	4	-	-	PUNCT
ejpam-6375	491	5	s∗	s∗	PROPN
ejpam-6375	491	6	g	g	PROPN
ejpam-6375	491	7	-normal	-normal	ADJ
ejpam-6375	491	8	space	space	NOUN
ejpam-6375	491	9	definition	definition	NOUN
ejpam-6375	491	10	3.23	3.23	NUM
ejpam-6375	491	11	.	.	PUNCT
ejpam-6375	492	1	assume	assume	VERB
ejpam-6375	492	2	v	v	NUM
ejpam-6375	492	3	is	be	AUX
ejpam-6375	492	4	gpt	gpt	NOUN
ejpam-6375	492	5	-	-	PUNCT
ejpam-6375	492	6	s∗	s∗	NOUN
ejpam-6375	492	7	gnormal	gnormal	NOUN
ejpam-6375	492	8	if	if	SCONJ
ejpam-6375	492	9	for	for	ADP
ejpam-6375	492	10	each	each	DET
ejpam-6375	492	11	pair	pair	NOUN
ejpam-6375	492	12	e	e	NOUN
ejpam-6375	492	13	,	,	PUNCT
ejpam-6375	492	14	d	d	PROPN
ejpam-6375	492	15	∈	∈	PROPN
ejpam-6375	492	16	gpt	gpt	NOUN
ejpam-6375	492	17	-	-	PUNCT
ejpam-6375	492	18	s∗	s∗	NOUN
ejpam-6375	492	19	gc(v	gc(v	NOUN
ejpam-6375	492	20	)	)	PUNCT
ejpam-6375	492	21	,	,	PUNCT
ejpam-6375	492	22	there	there	PRON
ejpam-6375	492	23	exists	exist	VERB
ejpam-6375	492	24	open	open	ADJ
ejpam-6375	492	25	sets	set	NOUN
ejpam-6375	492	26	rα	rα	ADJ
ejpam-6375	493	1	and	and	CCONJ
ejpam-6375	493	2	sα	sα	ADV
ejpam-6375	493	3	in	in	ADP
ejpam-6375	493	4	v	v	ADP
ejpam-6375	493	5	such	such	ADJ
ejpam-6375	493	6	that	that	SCONJ
ejpam-6375	493	7	d	d	PROPN
ejpam-6375	493	8	⊆	⊆	NUM
ejpam-6375	493	9	rα	rα	ADJ
ejpam-6375	493	10	and	and	CCONJ
ejpam-6375	493	11	e	e	NOUN
ejpam-6375	493	12	⊆	⊆	NUM
ejpam-6375	493	13	sα	sα	PROPN
ejpam-6375	493	14	.	.	PROPN
ejpam-6375	493	15	theorem	theorem	VERB
ejpam-6375	493	16	3.26	3.26	NUM
ejpam-6375	493	17	.	.	PUNCT
ejpam-6375	494	1	every	every	DET
ejpam-6375	494	2	gpt	gpt	NOUN
ejpam-6375	494	3	-	-	PUNCT
ejpam-6375	494	4	s∗	s∗	NOUN
ejpam-6375	494	5	g	g	PROPN
ejpam-6375	494	6	-normal	-normal	ADJ
ejpam-6375	494	7	is	be	AUX
ejpam-6375	494	8	gpt	gpt	NOUN
ejpam-6375	494	9	-	-	PUNCT
ejpam-6375	494	10	normal	normal	ADJ
ejpam-6375	494	11	.	.	PUNCT
ejpam-6375	495	1	proof	proof	NOUN
ejpam-6375	495	2	.	.	PUNCT
ejpam-6375	496	1	as	as	SCONJ
ejpam-6375	496	2	v	v	NOUN
ejpam-6375	496	3	is	be	AUX
ejpam-6375	496	4	a	a	DET
ejpam-6375	496	5	gpt	gpt	NOUN
ejpam-6375	496	6	-	-	PUNCT
ejpam-6375	496	7	s∗	s∗	NOUN
ejpam-6375	496	8	g	g	PROPN
ejpam-6375	496	9	-normal	-normal	NOUN
ejpam-6375	496	10	.	.	PUNCT
ejpam-6375	497	1	assume	assume	VERB
ejpam-6375	497	2	disjoint	disjoint	NOUN
ejpam-6375	497	3	sets	set	NOUN
ejpam-6375	497	4	e	e	NOUN
ejpam-6375	497	5	and	and	CCONJ
ejpam-6375	497	6	d	d	X
ejpam-6375	497	7	in	in	ADP
ejpam-6375	497	8	v.	v.	ADP
ejpam-6375	497	9	so	so	ADV
ejpam-6375	497	10	e	e	PROPN
ejpam-6375	497	11	,	,	PUNCT
ejpam-6375	497	12	d	d	PROPN
ejpam-6375	497	13	∈	∈	PROPN
ejpam-6375	497	14	gpts∗	gpts∗	PROPN
ejpam-6375	497	15	gc(v	gc(v	NOUN
ejpam-6375	497	16	)	)	PUNCT
ejpam-6375	497	17	.	.	PUNCT
ejpam-6375	498	1	as	as	SCONJ
ejpam-6375	498	2	v	v	NOUN
ejpam-6375	498	3	is	be	AUX
ejpam-6375	498	4	gpt	gpt	NOUN
ejpam-6375	498	5	-	-	PUNCT
ejpam-6375	498	6	s∗	s∗	NOUN
ejpam-6375	498	7	g	g	PROPN
ejpam-6375	498	8	-normal	-normal	ADJ
ejpam-6375	498	9	implies	imply	VERB
ejpam-6375	498	10	there	there	PRON
ejpam-6375	498	11	exist	exist	VERB
ejpam-6375	498	12	a	a	DET
ejpam-6375	498	13	pair	pair	NOUN
ejpam-6375	499	1	f	f	NOUN
ejpam-6375	499	2	,	,	PUNCT
ejpam-6375	499	3	hα	hα	ADP
ejpam-6375	499	4	such	such	ADJ
ejpam-6375	499	5	that	that	SCONJ
ejpam-6375	499	6	d	d	PROPN
ejpam-6375	499	7	⊆	⊆	NUM
ejpam-6375	499	8	f	f	NUM
ejpam-6375	499	9	,	,	PUNCT
ejpam-6375	499	10	e	e	PROPN
ejpam-6375	499	11	⊆	⊆	NUM
ejpam-6375	499	12	hα	hα	NOUN
ejpam-6375	499	13	.	.	PUNCT
ejpam-6375	500	1	thus	thus	ADV
ejpam-6375	500	2	,	,	PUNCT
ejpam-6375	500	3	v	v	NOUN
ejpam-6375	500	4	is	be	AUX
ejpam-6375	500	5	gpt	gpt	NOUN
ejpam-6375	500	6	-	-	PUNCT
ejpam-6375	500	7	normal	normal	ADJ
ejpam-6375	500	8	.	.	PUNCT
ejpam-6375	501	1	example	example	NOUN
ejpam-6375	501	2	3.9	3.9	NUM
ejpam-6375	501	3	.	.	PUNCT
ejpam-6375	502	1	consider	consider	VERB
ejpam-6375	502	2	v	v	NOUN
ejpam-6375	502	3	=	=	SYM
ejpam-6375	502	4	{	{	PUNCT
ejpam-6375	502	5	j1	j1	PROPN
ejpam-6375	502	6	,	,	PUNCT
ejpam-6375	502	7	k1	k1	NOUN
ejpam-6375	502	8	,	,	PUNCT
ejpam-6375	502	9	l1	l1	PROPN
ejpam-6375	502	10	}	}	PUNCT
ejpam-6375	502	11	and	and	CCONJ
ejpam-6375	502	12	gτ	gτ	PROPN
ejpam-6375	502	13	=	=	SYM
ejpam-6375	502	14	{	{	PUNCT
ejpam-6375	502	15	∅	∅	NOUN
ejpam-6375	502	16	,	,	PUNCT
ejpam-6375	502	17	v	v	NOUN
ejpam-6375	502	18	,	,	PUNCT
ejpam-6375	502	19	{	{	PUNCT
ejpam-6375	502	20	k1	k1	NOUN
ejpam-6375	502	21	}	}	PUNCT
ejpam-6375	502	22	,	,	PUNCT
ejpam-6375	502	23	{	{	PUNCT
ejpam-6375	502	24	l1	l1	PROPN
ejpam-6375	502	25	}	}	PUNCT
ejpam-6375	502	26	,	,	PUNCT
ejpam-6375	502	27	{	{	PUNCT
ejpam-6375	502	28	k1	k1	NOUN
ejpam-6375	502	29	,	,	PUNCT
ejpam-6375	502	30	l1	l1	PROPN
ejpam-6375	502	31	}	}	PUNCT
ejpam-6375	502	32	,	,	PUNCT
ejpam-6375	502	33	{	{	PUNCT
ejpam-6375	502	34	j1	j1	PROPN
ejpam-6375	502	35	,	,	PUNCT
ejpam-6375	502	36	k1	k1	NOUN
ejpam-6375	502	37	}	}	PUNCT
ejpam-6375	502	38	}	}	PUNCT
ejpam-6375	502	39	,	,	PUNCT
ejpam-6375	502	40	p	p	NOUN
ejpam-6375	502	41	=	=	X
ejpam-6375	502	42	{	{	PUNCT
ejpam-6375	502	43	∅	∅	NOUN
ejpam-6375	502	44	,	,	PUNCT
ejpam-6375	502	45	{	{	PUNCT
ejpam-6375	502	46	j1	j1	PROPN
ejpam-6375	502	47	}	}	PUNCT
ejpam-6375	502	48	,	,	PUNCT
ejpam-6375	502	49	{	{	PUNCT
ejpam-6375	502	50	l1	l1	PROPN
ejpam-6375	502	51	}	}	PUNCT
ejpam-6375	502	52	,	,	PUNCT
ejpam-6375	502	53	{	{	PUNCT
ejpam-6375	502	54	j1	j1	PROPN
ejpam-6375	502	55	,	,	PUNCT
ejpam-6375	502	56	l1	l1	PROPN
ejpam-6375	502	57	}	}	PUNCT
ejpam-6375	502	58	}	}	PUNCT
ejpam-6375	502	59	.	.	PUNCT
ejpam-6375	503	1	here	here	ADV
ejpam-6375	503	2	,	,	PUNCT
ejpam-6375	503	3	(	(	PUNCT
ejpam-6375	503	4	v	v	NOUN
ejpam-6375	503	5	,	,	PUNCT
ejpam-6375	503	6	gτ	gτ	INTJ
ejpam-6375	503	7	,	,	PUNCT
ejpam-6375	503	8	p	p	X
ejpam-6375	503	9	)	)	PUNCT
ejpam-6375	503	10	is	be	AUX
ejpam-6375	503	11	gpt	gpt	NOUN
ejpam-6375	503	12	-	-	PUNCT
ejpam-6375	503	13	normal	normal	ADJ
ejpam-6375	503	14	but	but	CCONJ
ejpam-6375	503	15	not	not	PART
ejpam-6375	503	16	gpt	gpt	NOUN
ejpam-6375	503	17	-	-	PUNCT
ejpam-6375	503	18	s∗	s∗	NOUN
ejpam-6375	503	19	g	g	PROPN
ejpam-6375	503	20	-normal	-normal	ADJ
ejpam-6375	503	21	space	space	NOUN
ejpam-6375	503	22	.	.	PUNCT
ejpam-6375	504	1	for	for	ADP
ejpam-6375	504	2	disjoint	disjoint	NOUN
ejpam-6375	504	3	sets	set	NOUN
ejpam-6375	504	4	{	{	PUNCT
ejpam-6375	504	5	j1	j1	PROPN
ejpam-6375	504	6	}	}	PUNCT
ejpam-6375	504	7	,	,	PUNCT
ejpam-6375	504	8	{	{	PUNCT
ejpam-6375	504	9	k1	k1	NOUN
ejpam-6375	504	10	,	,	PUNCT
ejpam-6375	504	11	l1	l1	PROPN
ejpam-6375	504	12	}	}	PUNCT
ejpam-6375	504	13	∈	∈	PROPN
ejpam-6375	504	14	gpt	gpt	NOUN
ejpam-6375	504	15	-	-	PUNCT
ejpam-6375	504	16	s∗	s∗	NOUN
ejpam-6375	504	17	gc(v	gc(v	NOUN
ejpam-6375	504	18	)	)	PUNCT
ejpam-6375	504	19	,	,	PUNCT
ejpam-6375	504	20	there	there	PRON
ejpam-6375	504	21	does	do	AUX
ejpam-6375	504	22	not	not	PART
ejpam-6375	504	23	exist	exist	VERB
ejpam-6375	504	24	open	open	ADJ
ejpam-6375	504	25	sets	set	NOUN
ejpam-6375	504	26	rα	rα	ADJ
ejpam-6375	504	27	and	and	CCONJ
ejpam-6375	504	28	sα	sα	ADV
ejpam-6375	504	29	in	in	ADP
ejpam-6375	504	30	v.	v.	ADP
ejpam-6375	504	31	m.	m.	PROPN
ejpam-6375	504	32	shahbaz	shahbaz	PROPN
ejpam-6375	504	33	et	et	PROPN
ejpam-6375	504	34	al	al	PROPN
ejpam-6375	504	35	.	.	PUNCT
ejpam-6375	504	36	/	/	SYM
ejpam-6375	504	37	eur	eur	PROPN
ejpam-6375	504	38	.	.	PUNCT
ejpam-6375	505	1	j.	j.	PROPN
ejpam-6375	505	2	pure	pure	PROPN
ejpam-6375	505	3	appl	appl	PROPN
ejpam-6375	505	4	.	.	PROPN
ejpam-6375	505	5	math	math	PROPN
ejpam-6375	505	6	,	,	PUNCT
ejpam-6375	505	7	18	18	NUM
ejpam-6375	505	8	(	(	PUNCT
ejpam-6375	505	9	4	4	NUM
ejpam-6375	505	10	)	)	PUNCT
ejpam-6375	505	11	(	(	PUNCT
ejpam-6375	505	12	2025	2025	NUM
ejpam-6375	505	13	)	)	PUNCT
ejpam-6375	505	14	,	,	PUNCT
ejpam-6375	505	15	6375	6375	NUM
ejpam-6375	505	16	18	18	NUM
ejpam-6375	505	17	of	of	ADP
ejpam-6375	505	18	22	22	NUM
ejpam-6375	505	19	theorem	theorem	VERB
ejpam-6375	505	20	3.27	3.27	NUM
ejpam-6375	505	21	.	.	PUNCT
ejpam-6375	506	1	if	if	SCONJ
ejpam-6375	506	2	z	z	NOUN
ejpam-6375	506	3	is	be	AUX
ejpam-6375	506	4	gpt	gpt	NOUN
ejpam-6375	506	5	-	-	PUNCT
ejpam-6375	506	6	normal	normal	ADJ
ejpam-6375	506	7	,	,	PUNCT
ejpam-6375	506	8	gpt	gpt	NOUN
ejpam-6375	506	9	-	-	PUNCT
ejpam-6375	506	10	s∗	s∗	PROPN
ejpam-6375	506	11	gtc	gtc	PROPN
ejpam-6375	506	12	space	space	NOUN
ejpam-6375	506	13	,	,	PUNCT
ejpam-6375	506	14	then	then	ADV
ejpam-6375	506	15	z	z	PROPN
ejpam-6375	506	16	is	be	AUX
ejpam-6375	506	17	gpt	gpt	NOUN
ejpam-6375	506	18	-	-	PUNCT
ejpam-6375	506	19	s∗	s∗	NOUN
ejpam-6375	506	20	g	g	NOUN
ejpam-6375	506	21	-normal	-normal	NOUN
ejpam-6375	506	22	.	.	PUNCT
ejpam-6375	507	1	proof	proof	NOUN
ejpam-6375	507	2	.	.	PUNCT
ejpam-6375	508	1	since	since	SCONJ
ejpam-6375	508	2	z	z	PROPN
ejpam-6375	508	3	is	be	AUX
ejpam-6375	508	4	gpt	gpt	NOUN
ejpam-6375	508	5	-	-	PUNCT
ejpam-6375	508	6	normal	normal	ADJ
ejpam-6375	508	7	.	.	PUNCT
ejpam-6375	509	1	consider	consider	VERB
ejpam-6375	509	2	disjoint	disjoint	NOUN
ejpam-6375	509	3	set	set	NOUN
ejpam-6375	509	4	e	e	NOUN
ejpam-6375	509	5	,	,	PUNCT
ejpam-6375	509	6	d	d	PROPN
ejpam-6375	509	7	∈	∈	PROPN
ejpam-6375	509	8	gpt	gpt	NOUN
ejpam-6375	509	9	-	-	PUNCT
ejpam-6375	509	10	s∗	s∗	PROPN
ejpam-6375	509	11	gc(z	gc(z	NUM
ejpam-6375	509	12	)	)	PUNCT
ejpam-6375	509	13	.	.	PUNCT
ejpam-6375	510	1	as	as	ADP
ejpam-6375	510	2	gpt	gpt	NOUN
ejpam-6375	510	3	-	-	PUNCT
ejpam-6375	510	4	s∗	s∗	PROPN
ejpam-6375	510	5	gtc	gtc	PROPN
ejpam-6375	510	6	space	space	NOUN
ejpam-6375	510	7	,	,	PUNCT
ejpam-6375	510	8	then	then	ADV
ejpam-6375	510	9	e	e	PROPN
ejpam-6375	510	10	and	and	CCONJ
ejpam-6375	510	11	d	d	PROPN
ejpam-6375	510	12	are	be	AUX
ejpam-6375	510	13	closed	closed	ADJ
ejpam-6375	510	14	.	.	PUNCT
ejpam-6375	511	1	since	since	SCONJ
ejpam-6375	511	2	z	z	PROPN
ejpam-6375	511	3	is	be	AUX
ejpam-6375	511	4	gpt	gpt	NOUN
ejpam-6375	511	5	-	-	PUNCT
ejpam-6375	511	6	normal	normal	ADJ
ejpam-6375	511	7	,	,	PUNCT
ejpam-6375	511	8	then	then	ADV
ejpam-6375	511	9	there	there	PRON
ejpam-6375	511	10	exist	exist	VERB
ejpam-6375	511	11	disjoint	disjoint	ADJ
ejpam-6375	511	12	open	open	ADJ
ejpam-6375	511	13	sets	set	NOUN
ejpam-6375	511	14	rα	rα	ADJ
ejpam-6375	511	15	and	and	CCONJ
ejpam-6375	511	16	sα	sα	ADV
ejpam-6375	511	17	in	in	ADP
ejpam-6375	511	18	z	z	PROPN
ejpam-6375	511	19	such	such	ADJ
ejpam-6375	511	20	that	that	SCONJ
ejpam-6375	511	21	e	e	PROPN
ejpam-6375	511	22	⊆	⊆	NUM
ejpam-6375	511	23	rα	rα	ADJ
ejpam-6375	511	24	and	and	CCONJ
ejpam-6375	511	25	d	d	PROPN
ejpam-6375	511	26	⊆	⊆	NUM
ejpam-6375	511	27	sα	sα	NOUN
ejpam-6375	511	28	.	.	PUNCT
ejpam-6375	512	1	thus	thus	ADV
ejpam-6375	512	2	,	,	PUNCT
ejpam-6375	512	3	z	z	PROPN
ejpam-6375	512	4	is	be	AUX
ejpam-6375	512	5	gpt	gpt	NOUN
ejpam-6375	512	6	-	-	PUNCT
ejpam-6375	512	7	s∗	s∗	NOUN
ejpam-6375	512	8	g	g	PROPN
ejpam-6375	512	9	-normal	-normal	NOUN
ejpam-6375	512	10	.	.	PUNCT
ejpam-6375	513	1	theorem	theorem	VERB
ejpam-6375	513	2	3.28	3.28	NUM
ejpam-6375	513	3	.	.	PUNCT
ejpam-6375	514	1	every	every	DET
ejpam-6375	514	2	gpt	gpt	NOUN
ejpam-6375	514	3	-	-	PUNCT
ejpam-6375	514	4	s∗	s∗	NOUN
ejpam-6375	514	5	g	g	PROPN
ejpam-6375	514	6	-normal	-normal	ADJ
ejpam-6375	514	7	is	be	AUX
ejpam-6375	514	8	gpt	gpt	NOUN
ejpam-6375	514	9	-	-	PUNCT
ejpam-6375	514	10	g	g	NOUN
ejpam-6375	514	11	-	-	PUNCT
ejpam-6375	514	12	normal	normal	ADJ
ejpam-6375	514	13	.	.	PUNCT
ejpam-6375	515	1	proof	proof	NOUN
ejpam-6375	515	2	.	.	PUNCT
ejpam-6375	516	1	as	as	SCONJ
ejpam-6375	516	2	v	v	NOUN
ejpam-6375	516	3	is	be	AUX
ejpam-6375	516	4	gpt	gpt	NOUN
ejpam-6375	516	5	-	-	PUNCT
ejpam-6375	516	6	s∗	s∗	NOUN
ejpam-6375	516	7	g	g	PROPN
ejpam-6375	516	8	-normal	-normal	NOUN
ejpam-6375	516	9	.	.	PUNCT
ejpam-6375	517	1	assume	assume	VERB
ejpam-6375	517	2	disjoint	disjoint	NOUN
ejpam-6375	517	3	set	set	PROPN
ejpam-6375	517	4	e	e	NOUN
ejpam-6375	517	5	,	,	PUNCT
ejpam-6375	517	6	d	d	PROPN
ejpam-6375	517	7	∈	∈	PROPN
ejpam-6375	517	8	gpt	gpt	NOUN
ejpam-6375	517	9	-	-	PUNCT
ejpam-6375	517	10	s∗	s∗	PROPN
ejpam-6375	517	11	gc(z	gc(z	X
ejpam-6375	517	12	)	)	PUNCT
ejpam-6375	517	13	implies	imply	VERB
ejpam-6375	517	14	there	there	PRON
ejpam-6375	517	15	exist	exist	VERB
ejpam-6375	517	16	a	a	DET
ejpam-6375	517	17	disjoint	disjoint	NOUN
ejpam-6375	517	18	rα	rα	NOUN
ejpam-6375	517	19	,	,	PUNCT
ejpam-6375	517	20	sα	sα	ADV
ejpam-6375	517	21	such	such	ADJ
ejpam-6375	517	22	that	that	SCONJ
ejpam-6375	517	23	e	e	PROPN
ejpam-6375	518	1	⊆	⊆	NUM
ejpam-6375	518	2	rα	rα	ADJ
ejpam-6375	518	3	and	and	CCONJ
ejpam-6375	518	4	d	d	PROPN
ejpam-6375	518	5	⊆	⊆	NUM
ejpam-6375	518	6	sα	sα	NOUN
ejpam-6375	518	7	.	.	PUNCT
ejpam-6375	519	1	thus	thus	ADV
ejpam-6375	519	2	,	,	PUNCT
ejpam-6375	519	3	v	v	NOUN
ejpam-6375	519	4	is	be	AUX
ejpam-6375	519	5	gpt	gpt	NOUN
ejpam-6375	519	6	-	-	PUNCT
ejpam-6375	519	7	g	g	NOUN
ejpam-6375	519	8	-	-	PUNCT
ejpam-6375	519	9	normal	normal	ADJ
ejpam-6375	519	10	.	.	PUNCT
ejpam-6375	520	1	remark	remark	PROPN
ejpam-6375	520	2	3.5	3.5	NUM
ejpam-6375	520	3	.	.	PUNCT
ejpam-6375	521	1	every	every	DET
ejpam-6375	521	2	gpt	gpt	NOUN
ejpam-6375	521	3	-	-	PUNCT
ejpam-6375	521	4	g	g	NOUN
ejpam-6375	521	5	-	-	PUNCT
ejpam-6375	521	6	normal	normal	ADJ
ejpam-6375	521	7	is	be	AUX
ejpam-6375	521	8	not	not	PART
ejpam-6375	521	9	gpt	gpt	NOUN
ejpam-6375	521	10	-	-	PUNCT
ejpam-6375	521	11	s∗	s∗	NOUN
ejpam-6375	521	12	g	g	PROPN
ejpam-6375	521	13	-normal	-normal	NOUN
ejpam-6375	521	14	.	.	PUNCT
ejpam-6375	521	15	example	example	NOUN
ejpam-6375	521	16	3.10	3.10	NUM
ejpam-6375	521	17	.	.	PUNCT
ejpam-6375	522	1	consider	consider	VERB
ejpam-6375	522	2	v	v	NOUN
ejpam-6375	522	3	=	=	SYM
ejpam-6375	522	4	{	{	PUNCT
ejpam-6375	522	5	j1	j1	PROPN
ejpam-6375	522	6	,	,	PUNCT
ejpam-6375	522	7	k1	k1	NOUN
ejpam-6375	522	8	,	,	PUNCT
ejpam-6375	522	9	l1	l1	PROPN
ejpam-6375	522	10	}	}	PUNCT
ejpam-6375	522	11	and	and	CCONJ
ejpam-6375	522	12	gτ	gτ	PROPN
ejpam-6375	522	13	=	=	SYM
ejpam-6375	522	14	{	{	PUNCT
ejpam-6375	522	15	∅	∅	NOUN
ejpam-6375	522	16	,	,	PUNCT
ejpam-6375	522	17	v	v	NOUN
ejpam-6375	522	18	,	,	PUNCT
ejpam-6375	522	19	{	{	PUNCT
ejpam-6375	522	20	k1	k1	NOUN
ejpam-6375	522	21	}	}	PUNCT
ejpam-6375	522	22	,	,	PUNCT
ejpam-6375	522	23	{	{	PUNCT
ejpam-6375	522	24	l1	l1	PROPN
ejpam-6375	522	25	}	}	PUNCT
ejpam-6375	522	26	,	,	PUNCT
ejpam-6375	522	27	{	{	PUNCT
ejpam-6375	522	28	k1	k1	NOUN
ejpam-6375	522	29	,	,	PUNCT
ejpam-6375	522	30	l1	l1	PROPN
ejpam-6375	522	31	}	}	PUNCT
ejpam-6375	522	32	,	,	PUNCT
ejpam-6375	522	33	{	{	PUNCT
ejpam-6375	522	34	j1	j1	PROPN
ejpam-6375	522	35	,	,	PUNCT
ejpam-6375	522	36	l1	l1	PROPN
ejpam-6375	522	37	}	}	PUNCT
ejpam-6375	522	38	}	}	PUNCT
ejpam-6375	522	39	,	,	PUNCT
ejpam-6375	522	40	p	p	NOUN
ejpam-6375	522	41	=	=	X
ejpam-6375	522	42	{	{	PUNCT
ejpam-6375	522	43	∅	∅	NOUN
ejpam-6375	522	44	,	,	PUNCT
ejpam-6375	522	45	{	{	PUNCT
ejpam-6375	522	46	j1	j1	PROPN
ejpam-6375	522	47	}	}	PUNCT
ejpam-6375	522	48	,	,	PUNCT
ejpam-6375	522	49	{	{	PUNCT
ejpam-6375	522	50	l1	l1	PROPN
ejpam-6375	522	51	}	}	PUNCT
ejpam-6375	522	52	,	,	PUNCT
ejpam-6375	522	53	{	{	PUNCT
ejpam-6375	522	54	j1	j1	PROPN
ejpam-6375	522	55	,	,	PUNCT
ejpam-6375	522	56	l1	l1	PROPN
ejpam-6375	522	57	}	}	PUNCT
ejpam-6375	522	58	}	}	PUNCT
ejpam-6375	522	59	.	.	PUNCT
ejpam-6375	523	1	here	here	ADV
ejpam-6375	523	2	,	,	PUNCT
ejpam-6375	523	3	(	(	PUNCT
ejpam-6375	523	4	v	v	NOUN
ejpam-6375	523	5	,	,	PUNCT
ejpam-6375	523	6	gτ	gτ	INTJ
ejpam-6375	523	7	,	,	PUNCT
ejpam-6375	523	8	p	p	X
ejpam-6375	523	9	)	)	PUNCT
ejpam-6375	523	10	is	be	AUX
ejpam-6375	523	11	gpt	gpt	NOUN
ejpam-6375	523	12	-	-	PUNCT
ejpam-6375	523	13	g	g	NOUN
ejpam-6375	523	14	-	-	PUNCT
ejpam-6375	523	15	normal	normal	ADJ
ejpam-6375	523	16	but	but	CCONJ
ejpam-6375	523	17	not	not	PART
ejpam-6375	523	18	gpt	gpt	NOUN
ejpam-6375	523	19	-	-	PUNCT
ejpam-6375	523	20	s∗	s∗	NOUN
ejpam-6375	523	21	g	g	PROPN
ejpam-6375	523	22	-normal	-normal	ADJ
ejpam-6375	523	23	space	space	NOUN
ejpam-6375	523	24	.	.	PUNCT
ejpam-6375	524	1	for	for	ADP
ejpam-6375	524	2	disjoint	disjoint	NOUN
ejpam-6375	524	3	sets	set	NOUN
ejpam-6375	524	4	{	{	PUNCT
ejpam-6375	524	5	j1	j1	PROPN
ejpam-6375	524	6	}	}	PUNCT
ejpam-6375	524	7	,	,	PUNCT
ejpam-6375	524	8	{	{	PUNCT
ejpam-6375	524	9	k1	k1	NOUN
ejpam-6375	524	10	,	,	PUNCT
ejpam-6375	524	11	l1	l1	PROPN
ejpam-6375	524	12	}	}	PUNCT
ejpam-6375	524	13	∈	∈	PROPN
ejpam-6375	524	14	gpt	gpt	NOUN
ejpam-6375	524	15	-	-	PUNCT
ejpam-6375	524	16	s∗	s∗	NOUN
ejpam-6375	524	17	gc(v	gc(v	NOUN
ejpam-6375	524	18	)	)	PUNCT
ejpam-6375	524	19	,	,	PUNCT
ejpam-6375	524	20	there	there	PRON
ejpam-6375	524	21	does	do	AUX
ejpam-6375	524	22	not	not	PART
ejpam-6375	524	23	exist	exist	VERB
ejpam-6375	524	24	open	open	ADJ
ejpam-6375	524	25	sets	set	NOUN
ejpam-6375	524	26	rα	rα	ADJ
ejpam-6375	524	27	and	and	CCONJ
ejpam-6375	524	28	sα	sα	ADV
ejpam-6375	524	29	in	in	ADP
ejpam-6375	524	30	v.	v.	CCONJ
ejpam-6375	524	31	theorem	theorem	ADJ
ejpam-6375	524	32	3.29	3.29	NUM
ejpam-6375	524	33	.	.	PUNCT
ejpam-6375	525	1	every	every	DET
ejpam-6375	525	2	gpt	gpt	NOUN
ejpam-6375	525	3	-	-	PUNCT
ejpam-6375	525	4	s∗	s∗	NOUN
ejpam-6375	525	5	g	g	PROPN
ejpam-6375	525	6	-normal	-normal	ADJ
ejpam-6375	525	7	is	be	AUX
ejpam-6375	525	8	gpt	gpt	NOUN
ejpam-6375	525	9	-	-	PUNCT
ejpam-6375	525	10	w	w	NOUN
ejpam-6375	525	11	-	-	PUNCT
ejpam-6375	525	12	normal	normal	ADJ
ejpam-6375	525	13	.	.	PUNCT
ejpam-6375	526	1	proof	proof	NOUN
ejpam-6375	526	2	.	.	PUNCT
ejpam-6375	527	1	similar	similar	ADJ
ejpam-6375	527	2	to	to	ADP
ejpam-6375	527	3	theorem	theorem	NOUN
ejpam-6375	527	4	3.28	3.28	NUM
ejpam-6375	527	5	.	.	PUNCT
ejpam-6375	527	6	remark	remark	NOUN
ejpam-6375	527	7	3.6	3.6	NUM
ejpam-6375	527	8	.	.	PUNCT
ejpam-6375	528	1	an	an	DET
ejpam-6375	528	2	illustration	illustration	NOUN
ejpam-6375	528	3	disproves	disprove	VERB
ejpam-6375	528	4	the	the	DET
ejpam-6375	528	5	invalidity	invalidity	NOUN
ejpam-6375	528	6	of	of	ADP
ejpam-6375	528	7	the	the	DET
ejpam-6375	528	8	converse	converse	NOUN
ejpam-6375	528	9	statement	statement	NOUN
ejpam-6375	528	10	derived	derive	VERB
ejpam-6375	528	11	from	from	ADP
ejpam-6375	528	12	above	above	ADP
ejpam-6375	528	13	.	.	PUNCT
ejpam-6375	529	1	example	example	NOUN
ejpam-6375	529	2	3.11	3.11	NUM
ejpam-6375	529	3	.	.	PUNCT
ejpam-6375	530	1	consider	consider	VERB
ejpam-6375	530	2	v	v	NOUN
ejpam-6375	530	3	=	=	SYM
ejpam-6375	530	4	{	{	PUNCT
ejpam-6375	530	5	j1	j1	PROPN
ejpam-6375	530	6	,	,	PUNCT
ejpam-6375	530	7	k1	k1	NOUN
ejpam-6375	530	8	,	,	PUNCT
ejpam-6375	530	9	l1	l1	PROPN
ejpam-6375	530	10	}	}	PUNCT
ejpam-6375	530	11	and	and	CCONJ
ejpam-6375	530	12	gτ	gτ	PROPN
ejpam-6375	530	13	=	=	SYM
ejpam-6375	530	14	{	{	PUNCT
ejpam-6375	530	15	∅	∅	NOUN
ejpam-6375	530	16	,	,	PUNCT
ejpam-6375	530	17	v	v	NOUN
ejpam-6375	530	18	,	,	PUNCT
ejpam-6375	530	19	{	{	PUNCT
ejpam-6375	530	20	k1	k1	NOUN
ejpam-6375	530	21	}	}	PUNCT
ejpam-6375	530	22	,	,	PUNCT
ejpam-6375	530	23	{	{	PUNCT
ejpam-6375	530	24	l1	l1	PROPN
ejpam-6375	530	25	}	}	PUNCT
ejpam-6375	530	26	,	,	PUNCT
ejpam-6375	530	27	{	{	PUNCT
ejpam-6375	530	28	k1	k1	NOUN
ejpam-6375	530	29	,	,	PUNCT
ejpam-6375	530	30	l1	l1	PROPN
ejpam-6375	530	31	}	}	PUNCT
ejpam-6375	530	32	,	,	PUNCT
ejpam-6375	530	33	{	{	PUNCT
ejpam-6375	530	34	j1	j1	PROPN
ejpam-6375	530	35	,	,	PUNCT
ejpam-6375	530	36	l1	l1	PROPN
ejpam-6375	530	37	}	}	PUNCT
ejpam-6375	530	38	}	}	PUNCT
ejpam-6375	530	39	,	,	PUNCT
ejpam-6375	530	40	p	p	NOUN
ejpam-6375	530	41	=	=	X
ejpam-6375	530	42	{	{	PUNCT
ejpam-6375	530	43	∅	∅	NOUN
ejpam-6375	530	44	,	,	PUNCT
ejpam-6375	530	45	{	{	PUNCT
ejpam-6375	530	46	j1	j1	PROPN
ejpam-6375	530	47	}	}	PUNCT
ejpam-6375	530	48	,	,	PUNCT
ejpam-6375	530	49	{	{	PUNCT
ejpam-6375	530	50	l1	l1	PROPN
ejpam-6375	530	51	}	}	PUNCT
ejpam-6375	530	52	,	,	PUNCT
ejpam-6375	530	53	{	{	PUNCT
ejpam-6375	530	54	j1	j1	PROPN
ejpam-6375	530	55	,	,	PUNCT
ejpam-6375	530	56	l1	l1	PROPN
ejpam-6375	530	57	}	}	PUNCT
ejpam-6375	530	58	}	}	PUNCT
ejpam-6375	530	59	.	.	PUNCT
ejpam-6375	531	1	here	here	ADV
ejpam-6375	531	2	,	,	PUNCT
ejpam-6375	531	3	(	(	PUNCT
ejpam-6375	531	4	v	v	NOUN
ejpam-6375	531	5	,	,	PUNCT
ejpam-6375	531	6	gτ	gτ	INTJ
ejpam-6375	531	7	,	,	PUNCT
ejpam-6375	531	8	p	p	X
ejpam-6375	531	9	)	)	PUNCT
ejpam-6375	531	10	is	be	AUX
ejpam-6375	531	11	gpt	gpt	NOUN
ejpam-6375	531	12	-	-	PUNCT
ejpam-6375	531	13	w	w	NOUN
ejpam-6375	531	14	-	-	PUNCT
ejpam-6375	531	15	normal	normal	ADJ
ejpam-6375	531	16	but	but	CCONJ
ejpam-6375	531	17	not	not	PART
ejpam-6375	531	18	gpt	gpt	NOUN
ejpam-6375	531	19	-	-	PUNCT
ejpam-6375	531	20	s∗	s∗	NOUN
ejpam-6375	531	21	g	g	PROPN
ejpam-6375	531	22	-normal	-normal	ADJ
ejpam-6375	531	23	space	space	NOUN
ejpam-6375	531	24	.	.	PUNCT
ejpam-6375	532	1	for	for	ADP
ejpam-6375	532	2	disjoint	disjoint	NOUN
ejpam-6375	532	3	sets	set	NOUN
ejpam-6375	532	4	{	{	PUNCT
ejpam-6375	532	5	j1	j1	PROPN
ejpam-6375	532	6	}	}	PUNCT
ejpam-6375	532	7	,	,	PUNCT
ejpam-6375	532	8	{	{	PUNCT
ejpam-6375	532	9	k1	k1	NOUN
ejpam-6375	532	10	,	,	PUNCT
ejpam-6375	532	11	l1	l1	PROPN
ejpam-6375	532	12	}	}	PUNCT
ejpam-6375	532	13	∈	∈	PROPN
ejpam-6375	532	14	gpt	gpt	NOUN
ejpam-6375	532	15	-	-	PUNCT
ejpam-6375	532	16	s∗	s∗	NOUN
ejpam-6375	532	17	gc(v	gc(v	NOUN
ejpam-6375	532	18	)	)	PUNCT
ejpam-6375	532	19	,	,	PUNCT
ejpam-6375	532	20	there	there	PRON
ejpam-6375	532	21	does	do	AUX
ejpam-6375	532	22	not	not	PART
ejpam-6375	532	23	exists	exist	VERB
ejpam-6375	532	24	open	open	ADJ
ejpam-6375	532	25	sets	set	NOUN
ejpam-6375	532	26	rα	rα	ADJ
ejpam-6375	532	27	and	and	CCONJ
ejpam-6375	532	28	sα	sα	ADV
ejpam-6375	532	29	in	in	ADP
ejpam-6375	532	30	v.	v.	CCONJ
ejpam-6375	532	31	theorem	theorem	ADJ
ejpam-6375	532	32	3.30	3.30	NUM
ejpam-6375	532	33	.	.	PUNCT
ejpam-6375	533	1	if	if	SCONJ
ejpam-6375	533	2	z	z	NOUN
ejpam-6375	533	3	is	be	AUX
ejpam-6375	533	4	gpt	gpt	NOUN
ejpam-6375	533	5	-	-	PUNCT
ejpam-6375	533	6	s∗	s∗	PROPN
ejpam-6375	533	7	g	g	PROPN
ejpam-6375	533	8	-closed	-close	VERB
ejpam-6375	533	9	subspace	subspace	NOUN
ejpam-6375	533	10	of	of	ADP
ejpam-6375	533	11	gpt	gpt	NOUN
ejpam-6375	533	12	-	-	PUNCT
ejpam-6375	533	13	s∗	s∗	PROPN
ejpam-6375	533	14	g	g	PROPN
ejpam-6375	533	15	-normal	-normal	ADJ
ejpam-6375	533	16	v	v	NOUN
ejpam-6375	533	17	,	,	PUNCT
ejpam-6375	533	18	then	then	ADV
ejpam-6375	533	19	z	z	PROPN
ejpam-6375	533	20	is	be	AUX
ejpam-6375	533	21	gpt	gpt	NOUN
ejpam-6375	533	22	-	-	PUNCT
ejpam-6375	533	23	s∗	s∗	PROPN
ejpam-6375	533	24	g	g	PROPN
ejpam-6375	533	25	normal	normal	ADJ
ejpam-6375	533	26	.	.	PUNCT
ejpam-6375	534	1	proof	proof	NOUN
ejpam-6375	534	2	.	.	PUNCT
ejpam-6375	535	1	assume	assume	VERB
ejpam-6375	535	2	v	v	NUM
ejpam-6375	535	3	is	be	AUX
ejpam-6375	535	4	gpt	gpt	NOUN
ejpam-6375	535	5	-	-	PUNCT
ejpam-6375	535	6	s∗	s∗	NOUN
ejpam-6375	535	7	g	g	PROPN
ejpam-6375	535	8	-normal	-normal	ADJ
ejpam-6375	535	9	and	and	CCONJ
ejpam-6375	535	10	z	z	NOUN
ejpam-6375	535	11	is	be	AUX
ejpam-6375	535	12	gpt	gpt	NOUN
ejpam-6375	535	13	-	-	PUNCT
ejpam-6375	535	14	s∗	s∗	PROPN
ejpam-6375	535	15	g	g	PROPN
ejpam-6375	535	16	-closed	-close	VERB
ejpam-6375	535	17	subspace	subspace	NOUN
ejpam-6375	535	18	.	.	PUNCT
ejpam-6375	536	1	consider	consider	VERB
ejpam-6375	536	2	a	a	DET
ejpam-6375	536	3	pair	pair	NOUN
ejpam-6375	536	4	of	of	ADP
ejpam-6375	536	5	disjoint	disjoint	NOUN
ejpam-6375	536	6	sets	set	NOUN
ejpam-6375	536	7	e	e	NOUN
ejpam-6375	536	8	and	and	CCONJ
ejpam-6375	536	9	d	d	PROPN
ejpam-6375	536	10	∈	∈	PROPN
ejpam-6375	536	11	gpt	gpt	NOUN
ejpam-6375	536	12	-	-	PUNCT
ejpam-6375	536	13	s∗	s∗	PROPN
ejpam-6375	536	14	gc(z	gc(z	X
ejpam-6375	536	15	)	)	PUNCT
ejpam-6375	536	16	implies	imply	VERB
ejpam-6375	536	17	there	there	PRON
ejpam-6375	536	18	exists	exist	VERB
ejpam-6375	536	19	f	f	X
ejpam-6375	536	20	,	,	PUNCT
ejpam-6375	536	21	hα	hα	ADP
ejpam-6375	536	22	∈	∈	PROPN
ejpam-6375	536	23	v	v	ADP
ejpam-6375	536	24	such	such	ADJ
ejpam-6375	536	25	that	that	SCONJ
ejpam-6375	536	26	e	e	PROPN
ejpam-6375	536	27	⊆	⊆	NUM
ejpam-6375	536	28	f	f	PROPN
ejpam-6375	536	29	and	and	CCONJ
ejpam-6375	536	30	d	d	PROPN
ejpam-6375	536	31	⊆	⊆	NUM
ejpam-6375	536	32	hα	hα	ADP
ejpam-6375	536	33	implies	imply	VERB
ejpam-6375	536	34	f	f	PROPN
ejpam-6375	536	35	∩	∩	PROPN
ejpam-6375	536	36	z	z	PROPN
ejpam-6375	536	37	and	and	CCONJ
ejpam-6375	536	38	hα	hα	ADP
ejpam-6375	536	39	∩	∩	NOUN
ejpam-6375	536	40	z	z	PROPN
ejpam-6375	536	41	are	be	AUX
ejpam-6375	536	42	open	open	ADJ
ejpam-6375	536	43	in	in	ADP
ejpam-6375	536	44	z.	z.	PROPN
ejpam-6375	536	45	furthermore	furthermore	ADV
ejpam-6375	536	46	,	,	PUNCT
ejpam-6375	536	47	e	e	PROPN
ejpam-6375	536	48	⊆	⊆	NUM
ejpam-6375	536	49	f	f	PROPN
ejpam-6375	536	50	and	and	CCONJ
ejpam-6375	536	51	d	d	PROPN
ejpam-6375	536	52	⊆	⊆	NUM
ejpam-6375	536	53	hα	hα	ADP
ejpam-6375	536	54	implies	imply	VERB
ejpam-6375	536	55	e	e	NOUN
ejpam-6375	536	56	∩	∩	NOUN
ejpam-6375	536	57	z	z	PROPN
ejpam-6375	536	58	⊆	⊆	NUM
ejpam-6375	536	59	z	z	NOUN
ejpam-6375	536	60	∩	∩	ADJ
ejpam-6375	536	61	f	f	PROPN
ejpam-6375	536	62	and	and	CCONJ
ejpam-6375	536	63	z	z	PROPN
ejpam-6375	536	64	∩	∩	PROPN
ejpam-6375	536	65	d	d	ADP
ejpam-6375	536	66	⊆	⊆	NUM
ejpam-6375	536	67	z	z	NOUN
ejpam-6375	536	68	∩	∩	NOUN
ejpam-6375	536	69	hα	hα	X
ejpam-6375	537	1	and	and	CCONJ
ejpam-6375	537	2	(	(	PUNCT
ejpam-6375	537	3	f	f	PROPN
ejpam-6375	537	4	∩	∩	PROPN
ejpam-6375	537	5	z	z	NOUN
ejpam-6375	537	6	)	)	PUNCT
ejpam-6375	537	7	∩	∩	NOUN
ejpam-6375	537	8	(	(	PUNCT
ejpam-6375	537	9	z	z	PROPN
ejpam-6375	537	10	∩	∩	X
ejpam-6375	537	11	hα	hα	PART
ejpam-6375	537	12	)	)	PUNCT
ejpam-6375	537	13	=	=	SYM
ejpam-6375	537	14	z	z	NOUN
ejpam-6375	537	15	∩	∩	NOUN
ejpam-6375	537	16	(	(	PUNCT
ejpam-6375	537	17	f	f	PROPN
ejpam-6375	537	18	∩	∩	PROPN
ejpam-6375	537	19	hα	hα	PART
ejpam-6375	537	20	)	)	PUNCT
ejpam-6375	537	21	=	=	PUNCT
ejpam-6375	537	22	∅.	∅.	ADP
ejpam-6375	537	23	thus	thus	ADV
ejpam-6375	537	24	,	,	PUNCT
ejpam-6375	537	25	z	z	PROPN
ejpam-6375	537	26	is	be	AUX
ejpam-6375	537	27	gpt	gpt	NOUN
ejpam-6375	537	28	-	-	PUNCT
ejpam-6375	537	29	s∗	s∗	NOUN
ejpam-6375	537	30	g	g	PROPN
ejpam-6375	537	31	-normal	-normal	NOUN
ejpam-6375	537	32	.	.	PUNCT
ejpam-6375	538	1	theorem	theorem	VERB
ejpam-6375	538	2	3.31	3.31	NUM
ejpam-6375	538	3	.	.	PUNCT
ejpam-6375	539	1	the	the	DET
ejpam-6375	539	2	following	follow	VERB
ejpam-6375	539	3	conditions	condition	NOUN
ejpam-6375	539	4	in	in	ADP
ejpam-6375	539	5	(	(	PUNCT
ejpam-6375	539	6	v	v	NOUN
ejpam-6375	539	7	,	,	PUNCT
ejpam-6375	539	8	gτ	gτ	INTJ
ejpam-6375	539	9	,	,	PUNCT
ejpam-6375	539	10	p	p	X
ejpam-6375	539	11	)	)	PUNCT
ejpam-6375	539	12	are	be	AUX
ejpam-6375	539	13	equivalent	equivalent	ADJ
ejpam-6375	539	14	:	:	PUNCT
ejpam-6375	539	15	1	1	X
ejpam-6375	539	16	)	)	PUNCT
ejpam-6375	539	17	the	the	DET
ejpam-6375	539	18	space	space	NOUN
ejpam-6375	539	19	v	v	NOUN
ejpam-6375	539	20	is	be	AUX
ejpam-6375	539	21	gpt	gpt	NOUN
ejpam-6375	539	22	-	-	PUNCT
ejpam-6375	539	23	s∗	s∗	NOUN
ejpam-6375	539	24	g	g	NOUN
ejpam-6375	539	25	-normal	-normal	NOUN
ejpam-6375	539	26	.	.	PUNCT
ejpam-6375	540	1	2	2	NUM
ejpam-6375	540	2	)	)	PUNCT
ejpam-6375	540	3	for	for	ADP
ejpam-6375	540	4	each	each	DET
ejpam-6375	540	5	e	e	NOUN
ejpam-6375	540	6	belonging	belong	VERB
ejpam-6375	540	7	to	to	ADP
ejpam-6375	540	8	gpt	gpt	NOUN
ejpam-6375	540	9	-	-	PUNCT
ejpam-6375	540	10	s∗	s∗	NOUN
ejpam-6375	540	11	gc(v	gc(v	NOUN
ejpam-6375	540	12	)	)	PUNCT
ejpam-6375	540	13	,	,	PUNCT
ejpam-6375	540	14	there	there	PRON
ejpam-6375	540	15	exists	exist	VERB
ejpam-6375	540	16	an	an	DET
ejpam-6375	540	17	open	open	ADJ
ejpam-6375	540	18	set	set	NOUN
ejpam-6375	540	19	t1	t1	NOUN
ejpam-6375	540	20	such	such	ADJ
ejpam-6375	540	21	that	that	SCONJ
ejpam-6375	540	22	e	e	PROPN
ejpam-6375	540	23	⊆	⊆	NUM
ejpam-6375	540	24	t1	t1	NOUN
ejpam-6375	540	25	⊆	⊆	NUM
ejpam-6375	540	26	cl(t1	cl(t1	NOUN
ejpam-6375	540	27	)	)	PUNCT
ejpam-6375	540	28	⊆	⊆	NUM
ejpam-6375	540	29	t2	t2	NOUN
ejpam-6375	540	30	for	for	ADP
ejpam-6375	540	31	some	some	DET
ejpam-6375	540	32	t2	t2	NOUN
ejpam-6375	540	33	in	in	ADP
ejpam-6375	540	34	gpt	gpt	NOUN
ejpam-6375	540	35	-	-	PUNCT
ejpam-6375	540	36	s∗	s∗	PROPN
ejpam-6375	540	37	go(v	go(v	ADV
ejpam-6375	540	38	)	)	PUNCT
ejpam-6375	540	39	with	with	ADP
ejpam-6375	540	40	e	e	PROPN
ejpam-6375	540	41	⊆	⊆	NUM
ejpam-6375	540	42	t2	t2	NOUN
ejpam-6375	540	43	.	.	PUNCT
ejpam-6375	541	1	m.	m.	NOUN
ejpam-6375	541	2	shahbaz	shahbaz	PROPN
ejpam-6375	541	3	et	et	PROPN
ejpam-6375	541	4	al	al	PROPN
ejpam-6375	541	5	.	.	PUNCT
ejpam-6375	541	6	/	/	SYM
ejpam-6375	541	7	eur	eur	PROPN
ejpam-6375	541	8	.	.	PUNCT
ejpam-6375	542	1	j.	j.	PROPN
ejpam-6375	542	2	pure	pure	PROPN
ejpam-6375	542	3	appl	appl	PROPN
ejpam-6375	542	4	.	.	PROPN
ejpam-6375	542	5	math	math	PROPN
ejpam-6375	542	6	,	,	PUNCT
ejpam-6375	542	7	18	18	NUM
ejpam-6375	542	8	(	(	PUNCT
ejpam-6375	542	9	4	4	NUM
ejpam-6375	542	10	)	)	PUNCT
ejpam-6375	542	11	(	(	PUNCT
ejpam-6375	542	12	2025	2025	NUM
ejpam-6375	542	13	)	)	PUNCT
ejpam-6375	542	14	,	,	PUNCT
ejpam-6375	542	15	6375	6375	NUM
ejpam-6375	542	16	19	19	NUM
ejpam-6375	542	17	of	of	ADP
ejpam-6375	542	18	22	22	NUM
ejpam-6375	542	19	3	3	NUM
ejpam-6375	542	20	)	)	PUNCT
ejpam-6375	542	21	given	give	VERB
ejpam-6375	542	22	two	two	NUM
ejpam-6375	542	23	disjoint	disjoint	NOUN
ejpam-6375	542	24	sets	set	NOUN
ejpam-6375	542	25	e	e	NOUN
ejpam-6375	542	26	and	and	CCONJ
ejpam-6375	542	27	d	d	PROPN
ejpam-6375	542	28	in	in	ADP
ejpam-6375	542	29	gpt	gpt	NOUN
ejpam-6375	542	30	-	-	PUNCT
ejpam-6375	542	31	s∗	s∗	NOUN
ejpam-6375	542	32	gc(v	gc(v	NOUN
ejpam-6375	542	33	)	)	PUNCT
ejpam-6375	543	1	,	,	PUNCT
ejpam-6375	543	2	there	there	PRON
ejpam-6375	543	3	exists	exist	VERB
ejpam-6375	543	4	an	an	DET
ejpam-6375	543	5	open	open	ADJ
ejpam-6375	543	6	set	set	NOUN
ejpam-6375	543	7	t1	t1	NOUN
ejpam-6375	543	8	such	such	ADJ
ejpam-6375	543	9	that	that	SCONJ
ejpam-6375	543	10	e	e	PROPN
ejpam-6375	543	11	⊆	⊆	NUM
ejpam-6375	543	12	t1	t1	NOUN
ejpam-6375	543	13	and	and	CCONJ
ejpam-6375	543	14	cl(t1	cl(t1	NOUN
ejpam-6375	543	15	)	)	PUNCT
ejpam-6375	543	16	∩d	∩d	VERB
ejpam-6375	543	17	=	=	PUNCT
ejpam-6375	543	18	∅.	∅.	VERB
ejpam-6375	543	19	4	4	NUM
ejpam-6375	543	20	)	)	PUNCT
ejpam-6375	543	21	for	for	ADP
ejpam-6375	543	22	any	any	DET
ejpam-6375	543	23	two	two	NUM
ejpam-6375	543	24	disjoint	disjoint	NOUN
ejpam-6375	543	25	sets	set	NOUN
ejpam-6375	543	26	e	e	NOUN
ejpam-6375	543	27	,	,	PUNCT
ejpam-6375	543	28	d	d	NOUN
ejpam-6375	543	29	in	in	ADP
ejpam-6375	543	30	gpt	gpt	NOUN
ejpam-6375	543	31	-	-	PUNCT
ejpam-6375	543	32	s∗	s∗	NOUN
ejpam-6375	543	33	gc(v	gc(v	NOUN
ejpam-6375	543	34	)	)	PUNCT
ejpam-6375	543	35	,	,	PUNCT
ejpam-6375	543	36	there	there	PRON
ejpam-6375	543	37	exist	exist	VERB
ejpam-6375	543	38	open	open	ADJ
ejpam-6375	543	39	sets	set	NOUN
ejpam-6375	543	40	t1	t1	NOUN
ejpam-6375	543	41	and	and	CCONJ
ejpam-6375	543	42	t2	t2	NOUN
ejpam-6375	543	43	such	such	ADJ
ejpam-6375	543	44	that	that	SCONJ
ejpam-6375	543	45	e	e	PROPN
ejpam-6375	543	46	⊆	⊆	NUM
ejpam-6375	543	47	t2	t2	NOUN
ejpam-6375	543	48	,	,	PUNCT
ejpam-6375	543	49	d	d	PROPN
ejpam-6375	543	50	⊆	⊆	NUM
ejpam-6375	543	51	t1	t1	NOUN
ejpam-6375	543	52	,	,	PUNCT
ejpam-6375	543	53	and	and	CCONJ
ejpam-6375	543	54	cl(t2	cl(t2	NOUN
ejpam-6375	543	55	)	)	PUNCT
ejpam-6375	543	56	∩	∩	NOUN
ejpam-6375	543	57	cl(t1	cl(t1	X
ejpam-6375	543	58	)	)	PUNCT
ejpam-6375	543	59	=	=	PUNCT
ejpam-6375	543	60	∅.	∅.	NOUN
ejpam-6375	543	61	proof	proof	NOUN
ejpam-6375	543	62	.	.	PUNCT
ejpam-6375	544	1	(	(	PUNCT
ejpam-6375	544	2	1	1	X
ejpam-6375	544	3	)	)	PUNCT
ejpam-6375	544	4	implies	imply	VERB
ejpam-6375	544	5	(	(	PUNCT
ejpam-6375	544	6	2	2	NUM
ejpam-6375	544	7	):	):	PUNCT
ejpam-6375	544	8	suppose	suppose	VERB
ejpam-6375	544	9	e	e	NOUN
ejpam-6375	544	10	belongs	belong	VERB
ejpam-6375	544	11	to	to	ADP
ejpam-6375	544	12	gpt	gpt	NOUN
ejpam-6375	544	13	-	-	PUNCT
ejpam-6375	544	14	s∗	s∗	NOUN
ejpam-6375	544	15	gc(v	gc(v	NOUN
ejpam-6375	544	16	)	)	PUNCT
ejpam-6375	544	17	and	and	CCONJ
ejpam-6375	544	18	t2	t2	NOUN
ejpam-6375	544	19	is	be	AUX
ejpam-6375	544	20	an	an	DET
ejpam-6375	544	21	element	element	NOUN
ejpam-6375	544	22	of	of	ADP
ejpam-6375	544	23	gpts∗	gpts∗	PROPN
ejpam-6375	544	24	gc(v	gc(v	NOUN
ejpam-6375	544	25	)	)	PUNCT
ejpam-6375	544	26	with	with	ADP
ejpam-6375	544	27	e	e	PROPN
ejpam-6375	544	28	⊆	⊆	NUM
ejpam-6375	544	29	t2	t2	NOUN
ejpam-6375	544	30	.	.	PUNCT
ejpam-6375	545	1	since	since	SCONJ
ejpam-6375	545	2	e	e	PROPN
ejpam-6375	545	3	and	and	CCONJ
ejpam-6375	545	4	v−t2	v−t2	NOUN
ejpam-6375	545	5	are	be	AUX
ejpam-6375	545	6	disjoint	disjoint	ADJ
ejpam-6375	545	7	,	,	PUNCT
ejpam-6375	545	8	the	the	DET
ejpam-6375	545	9	assumption	assumption	NOUN
ejpam-6375	545	10	of	of	ADP
ejpam-6375	545	11	gpt	gpt	NOUN
ejpam-6375	545	12	-	-	PUNCT
ejpam-6375	545	13	s∗	s∗	PROPN
ejpam-6375	545	14	g	g	PROPN
ejpam-6375	545	15	-normality	-normality	PROPN
ejpam-6375	545	16	guarantees	guarantee	NOUN
ejpam-6375	545	17	open	open	ADJ
ejpam-6375	545	18	sets	set	NOUN
ejpam-6375	545	19	t1	t1	NOUN
ejpam-6375	545	20	and	and	CCONJ
ejpam-6375	545	21	t3	t3	NOUN
ejpam-6375	545	22	such	such	ADJ
ejpam-6375	545	23	that	that	SCONJ
ejpam-6375	545	24	e	e	PROPN
ejpam-6375	545	25	⊆	⊆	NUM
ejpam-6375	545	26	t1	t1	NOUN
ejpam-6375	545	27	and	and	CCONJ
ejpam-6375	545	28	v	v	ADP
ejpam-6375	545	29	−	−	PROPN
ejpam-6375	545	30	t2	t2	PROPN
ejpam-6375	545	31	⊆	⊆	NUM
ejpam-6375	545	32	t3	t3	NOUN
ejpam-6375	545	33	with	with	ADP
ejpam-6375	545	34	t1	t1	PROPN
ejpam-6375	545	35	∩	∩	ADJ
ejpam-6375	545	36	t3	t3	NOUN
ejpam-6375	545	37	=	=	PUNCT
ejpam-6375	545	38	∅.	∅.	NOUN
ejpam-6375	545	39	this	this	DET
ejpam-6375	545	40	ensures	ensure	NOUN
ejpam-6375	545	41	t1	t1	NOUN
ejpam-6375	545	42	⊆	⊆	NUM
ejpam-6375	545	43	v−	v−	NOUN
ejpam-6375	545	44	t3	t3	NOUN
ejpam-6375	545	45	and	and	CCONJ
ejpam-6375	545	46	further	further	ADJ
ejpam-6375	545	47	cl(t1	cl(t1	NOUN
ejpam-6375	545	48	)	)	PUNCT
ejpam-6375	545	49	⊆	⊆	NUM
ejpam-6375	545	50	v−	v−	PROPN
ejpam-6375	545	51	t3	t3	PROPN
ejpam-6375	545	52	⊆	⊆	NUM
ejpam-6375	545	53	t2	t2	NOUN
ejpam-6375	545	54	,	,	PUNCT
ejpam-6375	545	55	leading	lead	VERB
ejpam-6375	545	56	to	to	ADP
ejpam-6375	545	57	cl(t1	cl(t1	NOUN
ejpam-6375	545	58	)	)	PUNCT
ejpam-6375	545	59	⊆	⊆	NUM
ejpam-6375	545	60	t2	t2	NOUN
ejpam-6375	545	61	.	.	PUNCT
ejpam-6375	546	1	(	(	PUNCT
ejpam-6375	546	2	2	2	X
ejpam-6375	546	3	)	)	PUNCT
ejpam-6375	546	4	implies	imply	VERB
ejpam-6375	546	5	(	(	PUNCT
ejpam-6375	546	6	3	3	NUM
ejpam-6375	546	7	):	):	PUNCT
ejpam-6375	546	8	given	give	VERB
ejpam-6375	546	9	two	two	NUM
ejpam-6375	546	10	disjoint	disjoint	NOUN
ejpam-6375	546	11	sets	set	NOUN
ejpam-6375	546	12	e	e	NOUN
ejpam-6375	546	13	and	and	CCONJ
ejpam-6375	546	14	d	d	PROPN
ejpam-6375	546	15	in	in	ADP
ejpam-6375	546	16	gpt	gpt	NOUN
ejpam-6375	546	17	-	-	PUNCT
ejpam-6375	546	18	s∗	s∗	NOUN
ejpam-6375	546	19	gc(v	gc(v	NOUN
ejpam-6375	546	20	)	)	PUNCT
ejpam-6375	546	21	,	,	PUNCT
ejpam-6375	546	22	we	we	PRON
ejpam-6375	546	23	note	note	VERB
ejpam-6375	546	24	that	that	SCONJ
ejpam-6375	546	25	e	e	PROPN
ejpam-6375	546	26	⊆	⊆	NUM
ejpam-6375	546	27	v−d	v−d	NOUN
ejpam-6375	546	28	.	.	PUNCT
ejpam-6375	547	1	by	by	ADP
ejpam-6375	547	2	(	(	PUNCT
ejpam-6375	547	3	2	2	NUM
ejpam-6375	547	4	)	)	PUNCT
ejpam-6375	547	5	,	,	PUNCT
ejpam-6375	547	6	there	there	PRON
ejpam-6375	547	7	exists	exist	VERB
ejpam-6375	547	8	an	an	DET
ejpam-6375	547	9	open	open	ADJ
ejpam-6375	547	10	set	set	NOUN
ejpam-6375	547	11	t1	t1	NOUN
ejpam-6375	547	12	such	such	ADJ
ejpam-6375	547	13	that	that	SCONJ
ejpam-6375	547	14	e	e	PROPN
ejpam-6375	547	15	⊆	⊆	NUM
ejpam-6375	547	16	t1	t1	NOUN
ejpam-6375	547	17	and	and	CCONJ
ejpam-6375	547	18	cl(t1	cl(t1	NOUN
ejpam-6375	547	19	)	)	PUNCT
ejpam-6375	547	20	⊆	⊆	NUM
ejpam-6375	547	21	v−d	v−d	NOUN
ejpam-6375	547	22	,	,	PUNCT
ejpam-6375	547	23	ensuring	ensure	VERB
ejpam-6375	547	24	that	that	SCONJ
ejpam-6375	547	25	cl(t1	cl(t1	NOUN
ejpam-6375	547	26	)	)	PUNCT
ejpam-6375	547	27	∩d	∩d	VERB
ejpam-6375	548	1	=	=	PUNCT
ejpam-6375	548	2	∅.	∅.	X
ejpam-6375	548	3	(	(	PUNCT
ejpam-6375	548	4	3	3	NUM
ejpam-6375	548	5	)	)	PUNCT
ejpam-6375	548	6	implies	imply	VERB
ejpam-6375	548	7	(	(	PUNCT
ejpam-6375	548	8	4	4	NUM
ejpam-6375	548	9	):	):	PUNCT
ejpam-6375	548	10	given	give	VERB
ejpam-6375	548	11	two	two	NUM
ejpam-6375	548	12	disjoint	disjoint	NOUN
ejpam-6375	548	13	sets	set	NOUN
ejpam-6375	548	14	e	e	NOUN
ejpam-6375	548	15	and	and	CCONJ
ejpam-6375	548	16	d	d	PROPN
ejpam-6375	548	17	in	in	ADP
ejpam-6375	548	18	gpt	gpt	NOUN
ejpam-6375	548	19	-	-	PUNCT
ejpam-6375	548	20	s∗	s∗	NOUN
ejpam-6375	548	21	gc(v	gc(v	NOUN
ejpam-6375	548	22	)	)	PUNCT
ejpam-6375	548	23	,	,	PUNCT
ejpam-6375	548	24	(	(	PUNCT
ejpam-6375	548	25	3	3	X
ejpam-6375	548	26	)	)	PUNCT
ejpam-6375	548	27	guarantees	guarantee	VERB
ejpam-6375	548	28	an	an	DET
ejpam-6375	548	29	open	open	ADJ
ejpam-6375	548	30	set	set	VERB
ejpam-6375	548	31	t2	t2	NOUN
ejpam-6375	548	32	such	such	ADJ
ejpam-6375	548	33	that	that	SCONJ
ejpam-6375	548	34	e	e	PROPN
ejpam-6375	548	35	⊆	⊆	NUM
ejpam-6375	548	36	t2	t2	NOUN
ejpam-6375	548	37	and	and	CCONJ
ejpam-6375	548	38	cl(t2	cl(t2	NOUN
ejpam-6375	548	39	)	)	PUNCT
ejpam-6375	548	40	∩d	∩d	VERB
ejpam-6375	549	1	=	=	PUNCT
ejpam-6375	549	2	∅.	∅.	NOUN
ejpam-6375	549	3	since	since	SCONJ
ejpam-6375	549	4	cl(t2	cl(t2	NOUN
ejpam-6375	549	5	)	)	PUNCT
ejpam-6375	549	6	is	be	AUX
ejpam-6375	549	7	closed	close	VERB
ejpam-6375	549	8	,	,	PUNCT
ejpam-6375	549	9	applying	apply	VERB
ejpam-6375	549	10	(	(	PUNCT
ejpam-6375	549	11	3	3	NUM
ejpam-6375	549	12	)	)	PUNCT
ejpam-6375	549	13	again	again	ADV
ejpam-6375	549	14	ensures	ensure	VERB
ejpam-6375	549	15	the	the	DET
ejpam-6375	549	16	existence	existence	NOUN
ejpam-6375	549	17	of	of	ADP
ejpam-6375	549	18	an	an	DET
ejpam-6375	549	19	open	open	ADJ
ejpam-6375	549	20	set	set	NOUN
ejpam-6375	549	21	t1	t1	NOUN
ejpam-6375	549	22	containing	contain	VERB
ejpam-6375	549	23	d	d	PROPN
ejpam-6375	549	24	such	such	ADJ
ejpam-6375	549	25	that	that	DET
ejpam-6375	549	26	cl(t2	cl(t2	NOUN
ejpam-6375	549	27	)	)	PUNCT
ejpam-6375	549	28	∩	∩	NOUN
ejpam-6375	549	29	cl(t1	cl(t1	X
ejpam-6375	549	30	)	)	PUNCT
ejpam-6375	549	31	=	=	SYM
ejpam-6375	549	32	∅.	∅.	X
ejpam-6375	549	33	(	(	PUNCT
ejpam-6375	549	34	4	4	NUM
ejpam-6375	549	35	)	)	PUNCT
ejpam-6375	549	36	implies	imply	VERB
ejpam-6375	549	37	(	(	PUNCT
ejpam-6375	549	38	1	1	NUM
ejpam-6375	549	39	):	):	PUNCT
ejpam-6375	549	40	suppose	suppose	VERB
ejpam-6375	549	41	e	e	NOUN
ejpam-6375	549	42	and	and	CCONJ
ejpam-6375	549	43	d	d	PROPN
ejpam-6375	549	44	are	be	AUX
ejpam-6375	549	45	two	two	NUM
ejpam-6375	549	46	disjoint	disjoint	NOUN
ejpam-6375	549	47	sets	set	NOUN
ejpam-6375	549	48	in	in	ADP
ejpam-6375	549	49	gpt	gpt	NOUN
ejpam-6375	549	50	-	-	PUNCT
ejpam-6375	549	51	s∗	s∗	NOUN
ejpam-6375	549	52	gc(v	gc(v	NOUN
ejpam-6375	549	53	)	)	PUNCT
ejpam-6375	549	54	.	.	PUNCT
ejpam-6375	550	1	by	by	ADP
ejpam-6375	550	2	(	(	PUNCT
ejpam-6375	550	3	4	4	NUM
ejpam-6375	550	4	)	)	PUNCT
ejpam-6375	550	5	,	,	PUNCT
ejpam-6375	550	6	there	there	PRON
ejpam-6375	550	7	exist	exist	VERB
ejpam-6375	550	8	disjoint	disjoint	ADJ
ejpam-6375	550	9	open	open	ADJ
ejpam-6375	550	10	sets	set	NOUN
ejpam-6375	550	11	t1	t1	NOUN
ejpam-6375	550	12	and	and	CCONJ
ejpam-6375	550	13	t2	t2	NOUN
ejpam-6375	550	14	such	such	ADJ
ejpam-6375	550	15	that	that	SCONJ
ejpam-6375	550	16	e	e	PROPN
ejpam-6375	550	17	⊆	⊆	NUM
ejpam-6375	550	18	t2	t2	NOUN
ejpam-6375	550	19	and	and	CCONJ
ejpam-6375	550	20	d	d	NOUN
ejpam-6375	550	21	⊆	⊆	NUM
ejpam-6375	550	22	t1	t1	NOUN
ejpam-6375	550	23	.	.	PUNCT
ejpam-6375	551	1	this	this	PRON
ejpam-6375	551	2	confirms	confirm	VERB
ejpam-6375	551	3	that	that	SCONJ
ejpam-6375	551	4	v	v	X
ejpam-6375	551	5	satisfies	satisfy	VERB
ejpam-6375	551	6	the	the	DET
ejpam-6375	551	7	definition	definition	NOUN
ejpam-6375	551	8	of	of	ADP
ejpam-6375	551	9	gpt	gpt	NOUN
ejpam-6375	551	10	-	-	PUNCT
ejpam-6375	551	11	s∗	s∗	PROPN
ejpam-6375	551	12	g	g	PROPN
ejpam-6375	551	13	-normality	-normality	NOUN
ejpam-6375	551	14	.	.	PUNCT
ejpam-6375	552	1	theorem	theorem	VERB
ejpam-6375	552	2	3.32	3.32	NUM
ejpam-6375	552	3	.	.	PUNCT
ejpam-6375	553	1	assume	assume	VERB
ejpam-6375	553	2	a	a	DET
ejpam-6375	553	3	mapping	mapping	NOUN
ejpam-6375	554	1	f	f	X
ejpam-6375	554	2	:	:	PUNCT
ejpam-6375	554	3	v	v	X
ejpam-6375	554	4	→	→	SYM
ejpam-6375	554	5	z.	z.	PROPN
ejpam-6375	554	6	if	if	SCONJ
ejpam-6375	554	7	f	f	PROPN
ejpam-6375	554	8	is	be	AUX
ejpam-6375	554	9	bijective	bijective	ADJ
ejpam-6375	554	10	open	open	ADJ
ejpam-6375	554	11	,	,	PUNCT
ejpam-6375	554	12	gpt	gpt	NOUN
ejpam-6375	554	13	-	-	PUNCT
ejpam-6375	554	14	s∗	s∗	PROPN
ejpam-6375	554	15	g	g	PROPN
ejpam-6375	554	16	-irresolute	-irresolute	PROPN
ejpam-6375	554	17	from	from	ADP
ejpam-6375	554	18	gpt	gpt	NOUN
ejpam-6375	554	19	-	-	PUNCT
ejpam-6375	554	20	s∗	s∗	PROPN
ejpam-6375	554	21	g	g	PROPN
ejpam-6375	554	22	-normal	-normal	ADJ
ejpam-6375	554	23	v	v	NOUN
ejpam-6375	554	24	onto	onto	ADP
ejpam-6375	554	25	z	z	NOUN
ejpam-6375	554	26	,	,	PUNCT
ejpam-6375	554	27	then	then	ADV
ejpam-6375	554	28	z	z	PROPN
ejpam-6375	554	29	is	be	AUX
ejpam-6375	554	30	gpt	gpt	NOUN
ejpam-6375	554	31	-	-	PUNCT
ejpam-6375	554	32	s∗	s∗	NOUN
ejpam-6375	554	33	g	g	NOUN
ejpam-6375	554	34	-normal	-normal	NOUN
ejpam-6375	554	35	.	.	PUNCT
ejpam-6375	555	1	proof	proof	NOUN
ejpam-6375	555	2	.	.	PUNCT
ejpam-6375	556	1	assume	assume	VERB
ejpam-6375	556	2	disjoint	disjoint	NOUN
ejpam-6375	556	3	sets	set	NOUN
ejpam-6375	556	4	e	e	NOUN
ejpam-6375	556	5	,	,	PUNCT
ejpam-6375	556	6	d	d	PROPN
ejpam-6375	556	7	∈	∈	PROPN
ejpam-6375	556	8	gpt	gpt	NOUN
ejpam-6375	556	9	-	-	PUNCT
ejpam-6375	556	10	s∗	s∗	NOUN
ejpam-6375	556	11	gc(v	gc(v	NOUN
ejpam-6375	556	12	)	)	PUNCT
ejpam-6375	556	13	.	.	PUNCT
ejpam-6375	557	1	since	since	SCONJ
ejpam-6375	557	2	f	f	PROPN
ejpam-6375	557	3	is	be	AUX
ejpam-6375	557	4	gpt	gpt	NOUN
ejpam-6375	557	5	-	-	PUNCT
ejpam-6375	557	6	s∗	s∗	PROPN
ejpam-6375	557	7	g	g	PROPN
ejpam-6375	557	8	-irresolute	-irresolute	PROPN
ejpam-6375	557	9	then	then	ADV
ejpam-6375	557	10	f−1(e	f−1(e	NOUN
ejpam-6375	557	11	)	)	PUNCT
ejpam-6375	557	12	and	and	CCONJ
ejpam-6375	557	13	f−1(d	f−1(d	PROPN
ejpam-6375	557	14	)	)	PUNCT
ejpam-6375	557	15	are	be	AUX
ejpam-6375	557	16	in	in	ADP
ejpam-6375	557	17	gpt	gpt	NOUN
ejpam-6375	557	18	-	-	PUNCT
ejpam-6375	557	19	s∗	s∗	NOUN
ejpam-6375	557	20	gc(v	gc(v	NOUN
ejpam-6375	557	21	)	)	PUNCT
ejpam-6375	557	22	.	.	PUNCT
ejpam-6375	558	1	as	as	SCONJ
ejpam-6375	558	2	v	v	NOUN
ejpam-6375	558	3	is	be	AUX
ejpam-6375	558	4	gpt	gpt	NOUN
ejpam-6375	558	5	-	-	PUNCT
ejpam-6375	558	6	s∗	s∗	NOUN
ejpam-6375	558	7	g	g	PROPN
ejpam-6375	558	8	-normal	-normal	ADJ
ejpam-6375	558	9	implies	imply	VERB
ejpam-6375	558	10	f−1(e	f−1(e	NOUN
ejpam-6375	558	11	)	)	PUNCT
ejpam-6375	558	12	⊆	⊆	NUM
ejpam-6375	558	13	t2	t2	NOUN
ejpam-6375	558	14	and	and	CCONJ
ejpam-6375	558	15	f−1(d	f−1(d	PROPN
ejpam-6375	558	16	)	)	PUNCT
ejpam-6375	559	1	⊆	⊆	NUM
ejpam-6375	559	2	t1	t1	NOUN
ejpam-6375	559	3	where	where	SCONJ
ejpam-6375	559	4	t1	t1	NOUN
ejpam-6375	559	5	and	and	CCONJ
ejpam-6375	559	6	t2	t2	NOUN
ejpam-6375	559	7	are	be	AUX
ejpam-6375	559	8	open	open	ADJ
ejpam-6375	559	9	in	in	ADP
ejpam-6375	559	10	v.	v.	ADP
ejpam-6375	559	11	also	also	ADV
ejpam-6375	559	12	,	,	PUNCT
ejpam-6375	559	13	as	as	SCONJ
ejpam-6375	559	14	f	f	PROPN
ejpam-6375	559	15	is	be	AUX
ejpam-6375	559	16	bijective	bijective	ADJ
ejpam-6375	559	17	and	and	CCONJ
ejpam-6375	559	18	open	open	ADJ
ejpam-6375	559	19	,	,	PUNCT
ejpam-6375	559	20	f(t2	f(t2	ADJ
ejpam-6375	559	21	)	)	PUNCT
ejpam-6375	559	22	and	and	CCONJ
ejpam-6375	559	23	f(t1	f(t1	NOUN
ejpam-6375	559	24	)	)	PUNCT
ejpam-6375	559	25	are	be	AUX
ejpam-6375	559	26	open	open	ADJ
ejpam-6375	559	27	and	and	CCONJ
ejpam-6375	559	28	e	e	NOUN
ejpam-6375	559	29	⊆	⊆	NUM
ejpam-6375	559	30	f(t2	f(t2	NOUN
ejpam-6375	559	31	)	)	PUNCT
ejpam-6375	559	32	,	,	PUNCT
ejpam-6375	560	1	d	d	NOUN
ejpam-6375	560	2	⊆	⊆	NUM
ejpam-6375	560	3	f(t1	f(t1	NOUN
ejpam-6375	560	4	)	)	PUNCT
ejpam-6375	560	5	.	.	PUNCT
ejpam-6375	561	1	thus	thus	ADV
ejpam-6375	561	2	,	,	PUNCT
ejpam-6375	561	3	z	z	PROPN
ejpam-6375	561	4	is	be	AUX
ejpam-6375	561	5	gpt	gpt	NOUN
ejpam-6375	561	6	-	-	PUNCT
ejpam-6375	561	7	s∗	s∗	NOUN
ejpam-6375	561	8	g	g	PROPN
ejpam-6375	561	9	-normal	-normal	NOUN
ejpam-6375	561	10	.	.	PUNCT
ejpam-6375	562	1	theorem	theorem	VERB
ejpam-6375	562	2	3.33	3.33	NUM
ejpam-6375	562	3	.	.	PUNCT
ejpam-6375	563	1	the	the	DET
ejpam-6375	563	2	following	follow	VERB
ejpam-6375	563	3	statements	statement	NOUN
ejpam-6375	563	4	hold	hold	VERB
ejpam-6375	563	5	equivalently	equivalently	ADV
ejpam-6375	563	6	in	in	ADP
ejpam-6375	563	7	(	(	PUNCT
ejpam-6375	563	8	v	v	NOUN
ejpam-6375	563	9	,	,	PUNCT
ejpam-6375	563	10	gτ	gτ	INTJ
ejpam-6375	563	11	,	,	PUNCT
ejpam-6375	563	12	p	p	X
ejpam-6375	563	13	):	):	PUNCT
ejpam-6375	563	14	1	1	X
ejpam-6375	563	15	)	)	PUNCT
ejpam-6375	563	16	v	v	NOUN
ejpam-6375	563	17	is	be	AUX
ejpam-6375	563	18	gpt	gpt	NOUN
ejpam-6375	563	19	-	-	PUNCT
ejpam-6375	563	20	g	g	NOUN
ejpam-6375	563	21	-	-	PUNCT
ejpam-6375	563	22	normal	normal	ADJ
ejpam-6375	563	23	.	.	PUNCT
ejpam-6375	564	1	2	2	X
ejpam-6375	564	2	)	)	PUNCT
ejpam-6375	564	3	there	there	PRON
ejpam-6375	564	4	exist	exist	VERB
ejpam-6375	564	5	disjoint	disjoint	ADJ
ejpam-6375	564	6	open	open	ADJ
ejpam-6375	564	7	sets	set	NOUN
ejpam-6375	564	8	t1	t1	PROPN
ejpam-6375	564	9	,	,	PUNCT
ejpam-6375	564	10	t2	t2	PROPN
ejpam-6375	564	11	∈	∈	PROPN
ejpam-6375	564	12	gpt	gpt	NOUN
ejpam-6375	564	13	-	-	PUNCT
ejpam-6375	564	14	s∗	s∗	PROPN
ejpam-6375	564	15	go(v	go(v	NOUN
ejpam-6375	564	16	)	)	PUNCT
ejpam-6375	564	17	such	such	ADJ
ejpam-6375	564	18	that	that	SCONJ
ejpam-6375	564	19	e	e	PROPN
ejpam-6375	564	20	⊆	⊆	NUM
ejpam-6375	564	21	t2	t2	NOUN
ejpam-6375	564	22	and	and	CCONJ
ejpam-6375	564	23	d	d	NOUN
ejpam-6375	564	24	⊆	⊆	NUM
ejpam-6375	564	25	t1	t1	NOUN
ejpam-6375	564	26	for	for	ADP
ejpam-6375	564	27	any	any	DET
ejpam-6375	564	28	disjoint	disjoint	NOUN
ejpam-6375	564	29	sets	set	NOUN
ejpam-6375	564	30	e	e	NOUN
ejpam-6375	564	31	and	and	CCONJ
ejpam-6375	564	32	d.	d.	PROPN
ejpam-6375	564	33	proof	proof	NOUN
ejpam-6375	564	34	.	.	PUNCT
ejpam-6375	565	1	(	(	PUNCT
ejpam-6375	565	2	1	1	X
ejpam-6375	565	3	)	)	PUNCT
ejpam-6375	565	4	implies	imply	VERB
ejpam-6375	565	5	(	(	PUNCT
ejpam-6375	565	6	2	2	X
ejpam-6375	565	7	)	)	PUNCT
ejpam-6375	565	8	suppose	suppose	VERB
ejpam-6375	565	9	that	that	SCONJ
ejpam-6375	565	10	v	v	NOUN
ejpam-6375	565	11	is	be	AUX
ejpam-6375	565	12	gpt	gpt	NOUN
ejpam-6375	565	13	-	-	PUNCT
ejpam-6375	565	14	g	g	NOUN
ejpam-6375	565	15	-	-	PUNCT
ejpam-6375	565	16	normal	normal	ADJ
ejpam-6375	565	17	,	,	PUNCT
ejpam-6375	565	18	and	and	CCONJ
ejpam-6375	565	19	let	let	VERB
ejpam-6375	565	20	e	e	NOUN
ejpam-6375	565	21	and	and	CCONJ
ejpam-6375	565	22	d	d	AUX
ejpam-6375	565	23	be	be	AUX
ejpam-6375	565	24	two	two	NUM
ejpam-6375	565	25	disjoint	disjoint	ADJ
ejpam-6375	565	26	subsets	subset	NOUN
ejpam-6375	565	27	of	of	ADP
ejpam-6375	565	28	v.	v.	ADP
ejpam-6375	565	29	by	by	ADP
ejpam-6375	565	30	the	the	DET
ejpam-6375	565	31	assumption	assumption	NOUN
ejpam-6375	565	32	that	that	SCONJ
ejpam-6375	565	33	(	(	PUNCT
ejpam-6375	565	34	v	v	NOUN
ejpam-6375	565	35	,	,	PUNCT
ejpam-6375	565	36	gτ	gτ	INTJ
ejpam-6375	565	37	,	,	PUNCT
ejpam-6375	565	38	p	p	X
ejpam-6375	565	39	)	)	PUNCT
ejpam-6375	565	40	is	be	AUX
ejpam-6375	565	41	gpt	gpt	NOUN
ejpam-6375	565	42	-	-	PUNCT
ejpam-6375	565	43	g	g	NOUN
ejpam-6375	565	44	-	-	PUNCT
ejpam-6375	565	45	normal	normal	ADJ
ejpam-6375	565	46	,	,	PUNCT
ejpam-6375	565	47	there	there	PRON
ejpam-6375	565	48	exist	exist	VERB
ejpam-6375	565	49	disjoint	disjoint	NOUN
ejpam-6375	565	50	gptg	gptg	NOUN
ejpam-6375	565	51	-	-	PUNCT
ejpam-6375	565	52	open	open	ADJ
ejpam-6375	565	53	sets	set	NOUN
ejpam-6375	565	54	t1	t1	VERB
ejpam-6375	565	55	and	and	CCONJ
ejpam-6375	565	56	t2	t2	NOUN
ejpam-6375	565	57	such	such	ADJ
ejpam-6375	565	58	that	that	SCONJ
ejpam-6375	565	59	e	e	PROPN
ejpam-6375	565	60	⊆	⊆	NUM
ejpam-6375	565	61	t2	t2	NOUN
ejpam-6375	565	62	and	and	CCONJ
ejpam-6375	565	63	d	d	NOUN
ejpam-6375	565	64	⊆	⊆	NUM
ejpam-6375	565	65	t1	t1	NOUN
ejpam-6375	565	66	.	.	PUNCT
ejpam-6375	566	1	consequently	consequently	ADV
ejpam-6375	566	2	,	,	PUNCT
ejpam-6375	566	3	t1	t1	PROPN
ejpam-6375	566	4	,	,	PUNCT
ejpam-6375	566	5	t2	t2	PROPN
ejpam-6375	566	6	∈	∈	PROPN
ejpam-6375	566	7	gpt	gpt	NOUN
ejpam-6375	566	8	-	-	PUNCT
ejpam-6375	566	9	s∗	s∗	PROPN
ejpam-6375	566	10	go(v	go(v	NOUN
ejpam-6375	566	11	)	)	PUNCT
ejpam-6375	566	12	,	,	PUNCT
ejpam-6375	566	13	satisfying	satisfy	VERB
ejpam-6375	566	14	e	e	PROPN
ejpam-6375	566	15	⊆	⊆	NUM
ejpam-6375	566	16	t2	t2	NOUN
ejpam-6375	566	17	and	and	CCONJ
ejpam-6375	566	18	d	d	NOUN
ejpam-6375	566	19	⊆	⊆	NUM
ejpam-6375	566	20	t1	t1	NOUN
ejpam-6375	566	21	while	while	SCONJ
ejpam-6375	566	22	ensuring	ensure	VERB
ejpam-6375	566	23	t1	t1	NOUN
ejpam-6375	566	24	∩	∩	ADJ
ejpam-6375	566	25	t2	t2	NOUN
ejpam-6375	566	26	=	=	PUNCT
ejpam-6375	566	27	∅.	∅.	X
ejpam-6375	566	28	(	(	PUNCT
ejpam-6375	566	29	2	2	NUM
ejpam-6375	566	30	)	)	PUNCT
ejpam-6375	566	31	implies	imply	VERB
ejpam-6375	566	32	(	(	PUNCT
ejpam-6375	566	33	1	1	X
ejpam-6375	566	34	)	)	PUNCT
ejpam-6375	566	35	consider	consider	VERB
ejpam-6375	566	36	that	that	PRON
ejpam-6375	566	37	for	for	ADP
ejpam-6375	566	38	any	any	DET
ejpam-6375	566	39	two	two	NUM
ejpam-6375	566	40	disjoint	disjoint	NOUN
ejpam-6375	566	41	gpt	gpt	NOUN
ejpam-6375	566	42	-	-	PUNCT
ejpam-6375	566	43	s∗	s∗	PROPN
ejpam-6375	566	44	g	g	PROPN
ejpam-6375	566	45	-closed	-close	VERB
ejpam-6375	566	46	sets	set	NOUN
ejpam-6375	566	47	e	e	NOUN
ejpam-6375	566	48	,	,	PUNCT
ejpam-6375	566	49	d	d	PROPN
ejpam-6375	566	50	∈	∈	PROPN
ejpam-6375	566	51	gpt	gpt	NOUN
ejpam-6375	566	52	-	-	PUNCT
ejpam-6375	566	53	s∗	s∗	NOUN
ejpam-6375	566	54	gc(v	gc(v	NOUN
ejpam-6375	566	55	)	)	PUNCT
ejpam-6375	566	56	,	,	PUNCT
ejpam-6375	566	57	there	there	PRON
ejpam-6375	566	58	exist	exist	VERB
ejpam-6375	566	59	disjoint	disjoint	ADJ
ejpam-6375	566	60	open	open	ADJ
ejpam-6375	566	61	sets	set	NOUN
ejpam-6375	566	62	t1	t1	NOUN
ejpam-6375	566	63	and	and	CCONJ
ejpam-6375	566	64	t2	t2	NOUN
ejpam-6375	566	65	such	such	ADJ
ejpam-6375	566	66	that	that	SCONJ
ejpam-6375	566	67	e	e	PROPN
ejpam-6375	566	68	⊆	⊆	NUM
ejpam-6375	566	69	t2	t2	NOUN
ejpam-6375	566	70	,	,	PUNCT
ejpam-6375	566	71	d	d	PROPN
ejpam-6375	566	72	⊆	⊆	NUM
ejpam-6375	566	73	t1	t1	NOUN
ejpam-6375	566	74	,	,	PUNCT
ejpam-6375	566	75	and	and	CCONJ
ejpam-6375	566	76	t1	t1	NOUN
ejpam-6375	566	77	∩	∩	ADJ
ejpam-6375	566	78	t2	t2	NOUN
ejpam-6375	566	79	=	=	SYM
ejpam-6375	566	80	∅	∅	NOUN
ejpam-6375	566	81	where	where	SCONJ
ejpam-6375	566	82	t1	t1	NOUN
ejpam-6375	566	83	,	,	PUNCT
ejpam-6375	566	84	t2	t2	PROPN
ejpam-6375	566	85	∈	∈	PROPN
ejpam-6375	566	86	gpt	gpt	NOUN
ejpam-6375	566	87	-	-	PUNCT
ejpam-6375	566	88	s∗	s∗	PROPN
ejpam-6375	566	89	go(v	go(v	NOUN
ejpam-6375	566	90	)	)	PUNCT
ejpam-6375	566	91	.	.	PUNCT
ejpam-6375	567	1	since	since	SCONJ
ejpam-6375	567	2	e	e	PROPN
ejpam-6375	567	3	is	be	AUX
ejpam-6375	567	4	contained	contain	VERB
ejpam-6375	567	5	in	in	ADP
ejpam-6375	567	6	gpt	gpt	NOUN
ejpam-6375	567	7	-	-	PUNCT
ejpam-6375	567	8	gint(t2	gint(t2	NOUN
ejpam-6375	567	9	)	)	PUNCT
ejpam-6375	567	10	and	and	CCONJ
ejpam-6375	567	11	d	d	X
ejpam-6375	567	12	in	in	ADP
ejpam-6375	567	13	gpt	gpt	NOUN
ejpam-6375	567	14	-	-	PUNCT
ejpam-6375	567	15	gint(t1	gint(t1	NOUN
ejpam-6375	567	16	)	)	PUNCT
ejpam-6375	567	17	,	,	PUNCT
ejpam-6375	567	18	and	and	CCONJ
ejpam-6375	567	19	their	their	PRON
ejpam-6375	567	20	interiors	interior	NOUN
ejpam-6375	567	21	remain	remain	VERB
ejpam-6375	567	22	disjoint	disjoint	ADJ
ejpam-6375	567	23	,	,	PUNCT
ejpam-6375	567	24	it	it	PRON
ejpam-6375	567	25	follows	follow	VERB
ejpam-6375	567	26	that	that	SCONJ
ejpam-6375	567	27	(	(	PUNCT
ejpam-6375	567	28	v	v	NOUN
ejpam-6375	567	29	,	,	PUNCT
ejpam-6375	567	30	gτ	gτ	INTJ
ejpam-6375	567	31	,	,	PUNCT
ejpam-6375	567	32	p	p	X
ejpam-6375	567	33	)	)	PUNCT
ejpam-6375	567	34	is	be	AUX
ejpam-6375	567	35	gpt	gpt	NOUN
ejpam-6375	567	36	-	-	PUNCT
ejpam-6375	567	37	g	g	NOUN
ejpam-6375	567	38	-	-	PUNCT
ejpam-6375	567	39	normal	normal	ADJ
ejpam-6375	567	40	.	.	PUNCT
ejpam-6375	568	1	m.	m.	PROPN
ejpam-6375	568	2	shahbaz	shahbaz	PROPN
ejpam-6375	568	3	et	et	PROPN
ejpam-6375	568	4	al	al	PROPN
ejpam-6375	568	5	.	.	PUNCT
ejpam-6375	568	6	/	/	SYM
ejpam-6375	568	7	eur	eur	PROPN
ejpam-6375	568	8	.	.	PUNCT
ejpam-6375	569	1	j.	j.	PROPN
ejpam-6375	569	2	pure	pure	PROPN
ejpam-6375	569	3	appl	appl	PROPN
ejpam-6375	569	4	.	.	PROPN
ejpam-6375	569	5	math	math	PROPN
ejpam-6375	569	6	,	,	PUNCT
ejpam-6375	569	7	18	18	NUM
ejpam-6375	569	8	(	(	PUNCT
ejpam-6375	569	9	4	4	NUM
ejpam-6375	569	10	)	)	PUNCT
ejpam-6375	569	11	(	(	PUNCT
ejpam-6375	569	12	2025	2025	NUM
ejpam-6375	569	13	)	)	PUNCT
ejpam-6375	569	14	,	,	PUNCT
ejpam-6375	569	15	6375	6375	NUM
ejpam-6375	569	16	20	20	NUM
ejpam-6375	569	17	of	of	ADP
ejpam-6375	569	18	22	22	NUM
ejpam-6375	569	19	4	4	NUM
ejpam-6375	569	20	.	.	PUNCT
ejpam-6375	569	21	methodology	methodology	NOUN
ejpam-6375	569	22	a	a	DET
ejpam-6375	569	23	theoretical	theoretical	ADJ
ejpam-6375	569	24	framework	framework	NOUN
ejpam-6375	569	25	demonstrates	demonstrate	VERB
ejpam-6375	569	26	the	the	DET
ejpam-6375	569	27	examination	examination	NOUN
ejpam-6375	569	28	of	of	ADP
ejpam-6375	569	29	compactness	compactness	NOUN
ejpam-6375	569	30	as	as	ADV
ejpam-6375	569	31	well	well	ADV
ejpam-6375	569	32	as	as	ADP
ejpam-6375	569	33	connectedness	connectedness	NOUN
ejpam-6375	569	34	and	and	CCONJ
ejpam-6375	569	35	separation	separation	NOUN
ejpam-6375	569	36	axioms	axiom	NOUN
ejpam-6375	569	37	in	in	ADP
ejpam-6375	569	38	generalized	generalized	ADJ
ejpam-6375	569	39	primal	primal	ADJ
ejpam-6375	569	40	topological	topological	ADJ
ejpam-6375	569	41	spaces	space	NOUN
ejpam-6375	569	42	.	.	PUNCT
ejpam-6375	570	1	the	the	DET
ejpam-6375	570	2	research	research	NOUN
ejpam-6375	570	3	method	method	NOUN
ejpam-6375	570	4	includes	include	VERB
ejpam-6375	570	5	the	the	DET
ejpam-6375	570	6	following	follow	VERB
ejpam-6375	570	7	main	main	ADJ
ejpam-6375	570	8	stages	stage	NOUN
ejpam-6375	570	9	of	of	ADP
ejpam-6375	570	10	development	development	NOUN
ejpam-6375	570	11	:	:	PUNCT
ejpam-6375	570	12	◦	◦	NOUN
ejpam-6375	570	13	review	review	NOUN
ejpam-6375	570	14	of	of	ADP
ejpam-6375	570	15	relevant	relevant	ADJ
ejpam-6375	570	16	literature	literature	NOUN
ejpam-6375	570	17	:	:	PUNCT
ejpam-6375	570	18	this	this	DET
ejpam-6375	570	19	research	research	NOUN
ejpam-6375	570	20	explores	explore	VERB
ejpam-6375	570	21	the	the	DET
ejpam-6375	570	22	developmental	developmental	ADJ
ejpam-6375	570	23	history	history	NOUN
ejpam-6375	570	24	of	of	ADP
ejpam-6375	570	25	compactness	compactness	NOUN
ejpam-6375	570	26	and	and	CCONJ
ejpam-6375	570	27	connectedness	connectedness	NOUN
ejpam-6375	570	28	and	and	CCONJ
ejpam-6375	570	29	separation	separation	NOUN
ejpam-6375	570	30	properties	property	NOUN
ejpam-6375	570	31	through	through	ADP
ejpam-6375	570	32	extensive	extensive	ADJ
ejpam-6375	570	33	study	study	NOUN
ejpam-6375	570	34	of	of	ADP
ejpam-6375	570	35	published	publish	VERB
ejpam-6375	570	36	literature	literature	NOUN
ejpam-6375	570	37	across	across	ADP
ejpam-6375	570	38	generalized	generalized	ADJ
ejpam-6375	570	39	contexts	contexts	NOUN
ejpam-6375	570	40	.	.	PUNCT
ejpam-6375	571	1	the	the	DET
ejpam-6375	571	2	research	research	NOUN
ejpam-6375	571	3	focuses	focus	VERB
ejpam-6375	571	4	on	on	ADP
ejpam-6375	571	5	reviewing	review	VERB
ejpam-6375	571	6	studies	study	NOUN
ejpam-6375	571	7	that	that	PRON
ejpam-6375	571	8	created	create	VERB
ejpam-6375	571	9	fundamental	fundamental	ADJ
ejpam-6375	571	10	educational	educational	ADJ
ejpam-6375	571	11	frameworks	framework	NOUN
ejpam-6375	571	12	to	to	PART
ejpam-6375	571	13	transfer	transfer	VERB
ejpam-6375	571	14	the	the	DET
ejpam-6375	571	15	principles	principle	NOUN
ejpam-6375	571	16	of	of	ADP
ejpam-6375	571	17	s∗	s∗	PROPN
ejpam-6375	571	18	g	g	PROPN
ejpam-6375	571	19	-compactness	-compactness	NOUN
ejpam-6375	571	20	and	and	CCONJ
ejpam-6375	571	21	s∗	s∗	VERB
ejpam-6375	571	22	g	g	PROPN
ejpam-6375	571	23	-connectedness	-connectedness	ADJ
ejpam-6375	571	24	to	to	ADP
ejpam-6375	571	25	generalized	generalize	VERB
ejpam-6375	571	26	primal	primal	ADJ
ejpam-6375	571	27	topological	topological	ADJ
ejpam-6375	571	28	spaces	space	NOUN
ejpam-6375	571	29	.	.	PUNCT
ejpam-6375	572	1	◦	◦	NOUN
ejpam-6375	572	2	definition	definition	NOUN
ejpam-6375	572	3	development	development	NOUN
ejpam-6375	572	4	:	:	PUNCT
ejpam-6375	572	5	the	the	DET
ejpam-6375	572	6	article	article	NOUN
ejpam-6375	572	7	introduces	introduce	VERB
ejpam-6375	572	8	new	new	ADJ
ejpam-6375	572	9	definitions	definition	NOUN
ejpam-6375	572	10	of	of	ADP
ejpam-6375	572	11	s∗	s∗	PROPN
ejpam-6375	572	12	g	g	PROPN
ejpam-6375	572	13	-compactness	-compactness	NOUN
ejpam-6375	572	14	and	and	CCONJ
ejpam-6375	572	15	s∗	s∗	VERB
ejpam-6375	572	16	g	g	PROPN
ejpam-6375	572	17	-connectedness	-connectedness	NOUN
ejpam-6375	572	18	under	under	ADP
ejpam-6375	572	19	generalized	generalized	ADJ
ejpam-6375	572	20	primal	primal	ADJ
ejpam-6375	572	21	topology	topology	NOUN
ejpam-6375	572	22	.	.	PUNCT
ejpam-6375	573	1	new	new	ADJ
ejpam-6375	573	2	definitions	definition	NOUN
ejpam-6375	573	3	for	for	ADP
ejpam-6375	573	4	generalized	generalized	ADJ
ejpam-6375	573	5	primal	primal	ADJ
ejpam-6375	573	6	structures	structure	NOUN
ejpam-6375	573	7	are	be	AUX
ejpam-6375	573	8	developed	develop	VERB
ejpam-6375	573	9	but	but	CCONJ
ejpam-6375	573	10	go	go	VERB
ejpam-6375	573	11	through	through	ADP
ejpam-6375	573	12	extensive	extensive	ADJ
ejpam-6375	573	13	evaluation	evaluation	NOUN
ejpam-6375	573	14	to	to	PART
ejpam-6375	573	15	confirm	confirm	VERB
ejpam-6375	573	16	their	their	PRON
ejpam-6375	573	17	alignment	alignment	NOUN
ejpam-6375	573	18	with	with	ADP
ejpam-6375	573	19	core	core	ADJ
ejpam-6375	573	20	principles	principle	NOUN
ejpam-6375	573	21	of	of	ADP
ejpam-6375	573	22	primal	primal	ADJ
ejpam-6375	573	23	topology	topology	NOUN
ejpam-6375	573	24	.	.	PUNCT
ejpam-6375	574	1	◦	◦	NOUN
ejpam-6375	574	2	analyzing	analyze	VERB
ejpam-6375	574	3	the	the	DET
ejpam-6375	574	4	separation	separation	NOUN
ejpam-6375	574	5	axioms	axiom	NOUN
ejpam-6375	574	6	:	:	PUNCT
ejpam-6375	574	7	generalized	generalized	ADJ
ejpam-6375	574	8	primal	primal	ADJ
ejpam-6375	574	9	topology	topology	NOUN
ejpam-6375	574	10	utilizes	utilize	VERB
ejpam-6375	574	11	s∗	s∗	PROPN
ejpam-6375	574	12	g	g	PROPN
ejpam-6375	574	13	open	open	ADJ
ejpam-6375	574	14	sets	set	NOUN
ejpam-6375	574	15	to	to	PART
ejpam-6375	574	16	study	study	VERB
ejpam-6375	574	17	the	the	DET
ejpam-6375	574	18	classical	classical	ADJ
ejpam-6375	574	19	separation	separation	NOUN
ejpam-6375	574	20	conditions	condition	NOUN
ejpam-6375	574	21	t0	t0	PROPN
ejpam-6375	574	22	,	,	PUNCT
ejpam-6375	574	23	t1	t1	NOUN
ejpam-6375	574	24	and	and	CCONJ
ejpam-6375	574	25	t2	t2	NOUN
ejpam-6375	574	26	.	.	PUNCT
ejpam-6375	575	1	this	this	DET
ejpam-6375	575	2	study	study	NOUN
ejpam-6375	575	3	establishes	establish	VERB
ejpam-6375	575	4	the	the	DET
ejpam-6375	575	5	s∗	s∗	PROPN
ejpam-6375	575	6	g	g	PROPN
ejpam-6375	575	7	-t0	-t0	PROPN
ejpam-6375	575	8	,	,	PUNCT
ejpam-6375	575	9	s∗	s∗	PROPN
ejpam-6375	575	10	g	g	PROPN
ejpam-6375	575	11	-t1	-t1	VERB
ejpam-6375	575	12	together	together	ADV
ejpam-6375	575	13	with	with	ADP
ejpam-6375	575	14	the	the	DET
ejpam-6375	575	15	s∗	s∗	PROPN
ejpam-6375	575	16	g	g	PROPN
ejpam-6375	575	17	-t2	-t2	PROPN
ejpam-6375	575	18	conditions	condition	NOUN
ejpam-6375	575	19	before	before	ADP
ejpam-6375	575	20	performing	perform	VERB
ejpam-6375	575	21	their	their	PRON
ejpam-6375	575	22	respective	respective	ADJ
ejpam-6375	575	23	analyses	analysis	NOUN
ejpam-6375	575	24	.	.	PUNCT
ejpam-6375	576	1	◦	◦	NOUN
ejpam-6375	576	2	establishing	establish	VERB
ejpam-6375	576	3	theoretical	theoretical	ADJ
ejpam-6375	576	4	proofs	proof	NOUN
ejpam-6375	576	5	:	:	PUNCT
ejpam-6375	576	6	the	the	DET
ejpam-6375	576	7	formal	formal	ADJ
ejpam-6375	576	8	verification	verification	NOUN
ejpam-6375	576	9	process	process	NOUN
ejpam-6375	576	10	proves	prove	VERB
ejpam-6375	576	11	the	the	DET
ejpam-6375	576	12	various	various	ADJ
ejpam-6375	576	13	attributes	attribute	NOUN
ejpam-6375	576	14	defined	define	VERB
ejpam-6375	576	15	in	in	ADP
ejpam-6375	576	16	the	the	DET
ejpam-6375	576	17	proposed	propose	VERB
ejpam-6375	576	18	constructions	construction	NOUN
ejpam-6375	576	19	.	.	PUNCT
ejpam-6375	577	1	mathematical	mathematical	ADJ
ejpam-6375	577	2	proofs	proof	NOUN
ejpam-6375	577	3	establish	establish	VERB
ejpam-6375	577	4	internal	internal	ADJ
ejpam-6375	577	5	validity	validity	NOUN
ejpam-6375	577	6	while	while	SCONJ
ejpam-6375	577	7	ensuring	ensure	VERB
ejpam-6375	577	8	logical	logical	ADJ
ejpam-6375	577	9	consistency	consistency	NOUN
ejpam-6375	577	10	of	of	ADP
ejpam-6375	577	11	new	new	ADJ
ejpam-6375	577	12	concepts	concept	NOUN
ejpam-6375	577	13	in	in	ADP
ejpam-6375	577	14	order	order	NOUN
ejpam-6375	577	15	to	to	PART
ejpam-6375	577	16	provide	provide	VERB
ejpam-6375	577	17	respectable	respectable	ADJ
ejpam-6375	577	18	grounds	ground	NOUN
ejpam-6375	577	19	for	for	ADP
ejpam-6375	577	20	theoretical	theoretical	ADJ
ejpam-6375	577	21	future	future	ADJ
ejpam-6375	577	22	research	research	NOUN
ejpam-6375	577	23	.	.	PUNCT
ejpam-6375	578	1	◦	◦	VERB
ejpam-6375	578	2	comparative	comparative	ADJ
ejpam-6375	578	3	analysis	analysis	NOUN
ejpam-6375	578	4	:	:	PUNCT
ejpam-6375	578	5	the	the	DET
ejpam-6375	578	6	paper	paper	NOUN
ejpam-6375	578	7	evaluates	evaluate	VERB
ejpam-6375	578	8	the	the	DET
ejpam-6375	578	9	new	new	ADJ
ejpam-6375	578	10	developed	develop	VERB
ejpam-6375	578	11	theory	theory	NOUN
ejpam-6375	578	12	by	by	ADP
ejpam-6375	578	13	showing	show	VERB
ejpam-6375	578	14	its	its	PRON
ejpam-6375	578	15	distinctions	distinction	NOUN
ejpam-6375	578	16	and	and	CCONJ
ejpam-6375	578	17	correspondences	correspondence	NOUN
ejpam-6375	578	18	to	to	ADP
ejpam-6375	578	19	traditional	traditional	ADJ
ejpam-6375	578	20	theories	theory	NOUN
ejpam-6375	578	21	alongside	alongside	ADP
ejpam-6375	578	22	demonstrating	demonstrate	VERB
ejpam-6375	578	23	advantages	advantage	NOUN
ejpam-6375	578	24	for	for	ADP
ejpam-6375	578	25	accepting	accept	VERB
ejpam-6375	578	26	generalized	generalize	VERB
ejpam-6375	578	27	primal	primal	ADJ
ejpam-6375	578	28	topological	topological	ADJ
ejpam-6375	578	29	methods	method	NOUN
ejpam-6375	578	30	.	.	PUNCT
ejpam-6375	579	1	◦	◦	NOUN
ejpam-6375	579	2	results	result	NOUN
ejpam-6375	579	3	synthesis	synthesis	NOUN
ejpam-6375	579	4	:	:	PUNCT
ejpam-6375	579	5	the	the	DET
ejpam-6375	579	6	research	research	NOUN
ejpam-6375	579	7	findings	finding	NOUN
ejpam-6375	579	8	transform	transform	VERB
ejpam-6375	579	9	into	into	ADP
ejpam-6375	579	10	structural	structural	ADJ
ejpam-6375	579	11	models	model	NOUN
ejpam-6375	579	12	for	for	ADP
ejpam-6375	579	13	describing	describe	VERB
ejpam-6375	579	14	s∗	s∗	PROPN
ejpam-6375	579	15	g	g	PROPN
ejpam-6375	579	16	-compact	-compact	NOUN
ejpam-6375	579	17	as	as	ADV
ejpam-6375	579	18	well	well	ADV
ejpam-6375	579	19	as	as	ADP
ejpam-6375	579	20	s∗	s∗	PROPN
ejpam-6375	579	21	g	g	PROPN
ejpam-6375	579	22	-connected	-connected	ADJ
ejpam-6375	579	23	spaces	space	NOUN
ejpam-6375	579	24	through	through	ADP
ejpam-6375	579	25	the	the	DET
ejpam-6375	579	26	generalized	generalize	VERB
ejpam-6375	579	27	primal	primal	ADJ
ejpam-6375	579	28	topological	topological	ADJ
ejpam-6375	579	29	framework	framework	NOUN
ejpam-6375	579	30	,	,	PUNCT
ejpam-6375	579	31	together	together	ADV
ejpam-6375	579	32	with	with	ADP
ejpam-6375	579	33	separation	separation	NOUN
ejpam-6375	579	34	axiom	axiom	NOUN
ejpam-6375	579	35	analysis	analysis	NOUN
ejpam-6375	579	36	.	.	PUNCT
ejpam-6375	580	1	furthermore	furthermore	ADV
ejpam-6375	580	2	,	,	PUNCT
ejpam-6375	580	3	the	the	DET
ejpam-6375	580	4	work	work	NOUN
ejpam-6375	580	5	discusses	discuss	VERB
ejpam-6375	580	6	what	what	PRON
ejpam-6375	580	7	aspects	aspect	VERB
ejpam-6375	580	8	the	the	DET
ejpam-6375	580	9	advancement	advancement	NOUN
ejpam-6375	580	10	means	mean	VERB
ejpam-6375	580	11	for	for	ADP
ejpam-6375	580	12	developing	develop	VERB
ejpam-6375	580	13	topological	topological	ADJ
ejpam-6375	580	14	theory	theory	NOUN
ejpam-6375	580	15	.	.	PUNCT
ejpam-6375	581	1	5	5	X
ejpam-6375	581	2	.	.	X
ejpam-6375	581	3	conclusions	conclusion	NOUN
ejpam-6375	581	4	researchers	researcher	NOUN
ejpam-6375	581	5	perform	perform	VERB
ejpam-6375	581	6	an	an	DET
ejpam-6375	581	7	extensive	extensive	ADJ
ejpam-6375	581	8	investigation	investigation	NOUN
ejpam-6375	581	9	into	into	ADP
ejpam-6375	581	10	separation	separation	NOUN
ejpam-6375	581	11	axioms	axiom	NOUN
ejpam-6375	581	12	together	together	ADV
ejpam-6375	581	13	with	with	ADP
ejpam-6375	581	14	connectedness	connectedness	NOUN
ejpam-6375	581	15	and	and	CCONJ
ejpam-6375	581	16	compactness	compactness	NOUN
ejpam-6375	581	17	phenomena	phenomenon	NOUN
ejpam-6375	581	18	in	in	ADP
ejpam-6375	581	19	generalized	generalized	ADJ
ejpam-6375	581	20	primal	primal	ADJ
ejpam-6375	581	21	topological	topological	ADJ
ejpam-6375	581	22	spaces	space	NOUN
ejpam-6375	581	23	.	.	PUNCT
ejpam-6375	582	1	this	this	DET
ejpam-6375	582	2	paper	paper	NOUN
ejpam-6375	582	3	extends	extend	VERB
ejpam-6375	582	4	traditional	traditional	ADJ
ejpam-6375	582	5	concepts	concept	NOUN
ejpam-6375	582	6	to	to	ADP
ejpam-6375	582	7	the	the	DET
ejpam-6375	582	8	new	new	ADJ
ejpam-6375	582	9	framework	framework	NOUN
ejpam-6375	582	10	,	,	PUNCT
ejpam-6375	582	11	which	which	PRON
ejpam-6375	582	12	increases	increase	VERB
ejpam-6375	582	13	our	our	PRON
ejpam-6375	582	14	understanding	understanding	NOUN
ejpam-6375	582	15	of	of	ADP
ejpam-6375	582	16	s∗	s∗	PROPN
ejpam-6375	582	17	g	g	PROPN
ejpam-6375	582	18	-compactness	-compactness	NOUN
ejpam-6375	582	19	and	and	CCONJ
ejpam-6375	582	20	s∗	s∗	VERB
ejpam-6375	582	21	g	g	PROPN
ejpam-6375	582	22	-connectedness	-connectedness	NOUN
ejpam-6375	582	23	at	at	ADP
ejpam-6375	582	24	a	a	DET
ejpam-6375	582	25	theoretical	theoretical	ADJ
ejpam-6375	582	26	level	level	NOUN
ejpam-6375	582	27	.	.	PUNCT
ejpam-6375	583	1	the	the	DET
ejpam-6375	583	2	new	new	ADJ
ejpam-6375	583	3	definitions	definition	NOUN
ejpam-6375	583	4	specifically	specifically	ADV
ejpam-6375	583	5	designed	design	VERB
ejpam-6375	583	6	for	for	ADP
ejpam-6375	583	7	generalized	generalized	ADJ
ejpam-6375	583	8	primal	primal	ADJ
ejpam-6375	583	9	spaces	space	NOUN
ejpam-6375	583	10	combined	combine	VERB
ejpam-6375	583	11	with	with	ADP
ejpam-6375	583	12	their	their	PRON
ejpam-6375	583	13	formal	formal	ADJ
ejpam-6375	583	14	support	support	NOUN
ejpam-6375	583	15	system	system	NOUN
ejpam-6375	583	16	enhances	enhance	VERB
ejpam-6375	583	17	these	these	DET
ejpam-6375	583	18	concepts	concept	NOUN
ejpam-6375	583	19	for	for	ADP
ejpam-6375	583	20	application	application	NOUN
ejpam-6375	583	21	throughout	throughout	ADP
ejpam-6375	583	22	nonclassical	nonclassical	ADJ
ejpam-6375	583	23	spaces	space	NOUN
ejpam-6375	583	24	.	.	PUNCT
ejpam-6375	584	1	m.	m.	NOUN
ejpam-6375	584	2	shahbaz	shahbaz	PROPN
ejpam-6375	584	3	et	et	PROPN
ejpam-6375	584	4	al	al	PROPN
ejpam-6375	584	5	.	.	PUNCT
ejpam-6375	584	6	/	/	SYM
ejpam-6375	584	7	eur	eur	PROPN
ejpam-6375	584	8	.	.	PUNCT
ejpam-6375	585	1	j.	j.	PROPN
ejpam-6375	585	2	pure	pure	PROPN
ejpam-6375	585	3	appl	appl	PROPN
ejpam-6375	585	4	.	.	PROPN
ejpam-6375	585	5	math	math	PROPN
ejpam-6375	585	6	,	,	PUNCT
ejpam-6375	585	7	18	18	NUM
ejpam-6375	585	8	(	(	PUNCT
ejpam-6375	585	9	4	4	NUM
ejpam-6375	585	10	)	)	PUNCT
ejpam-6375	585	11	(	(	PUNCT
ejpam-6375	585	12	2025	2025	NUM
ejpam-6375	585	13	)	)	PUNCT
ejpam-6375	585	14	,	,	PUNCT
ejpam-6375	585	15	6375	6375	NUM
ejpam-6375	585	16	21	21	NUM
ejpam-6375	585	17	of	of	ADP
ejpam-6375	585	18	22	22	NUM
ejpam-6375	585	19	the	the	DET
ejpam-6375	585	20	study	study	NOUN
ejpam-6375	585	21	introduces	introduce	VERB
ejpam-6375	585	22	a	a	DET
ejpam-6375	585	23	new	new	ADJ
ejpam-6375	585	24	set	set	NOUN
ejpam-6375	585	25	of	of	ADP
ejpam-6375	585	26	classification	classification	NOUN
ejpam-6375	585	27	tools	tool	NOUN
ejpam-6375	585	28	through	through	ADP
ejpam-6375	585	29	s∗	s∗	PROPN
ejpam-6375	585	30	g	g	PROPN
ejpam-6375	585	31	-t0	-t0	PROPN
ejpam-6375	585	32	,	,	PUNCT
ejpam-6375	585	33	s∗	s∗	PROPN
ejpam-6375	585	34	g	g	PROPN
ejpam-6375	585	35	-t1	-t1	PROPN
ejpam-6375	585	36	,	,	PUNCT
ejpam-6375	585	37	and	and	CCONJ
ejpam-6375	585	38	s∗	s∗	PROPN
ejpam-6375	585	39	g	g	PROPN
ejpam-6375	585	40	-t2	-t2	PROPN
ejpam-6375	585	41	separation	separation	NOUN
ejpam-6375	585	42	axiom	axiom	NOUN
ejpam-6375	585	43	reinterpretations	reinterpretation	NOUN
ejpam-6375	585	44	to	to	PART
ejpam-6375	585	45	analyze	analyze	VERB
ejpam-6375	585	46	spaces	space	NOUN
ejpam-6375	585	47	based	base	VERB
ejpam-6375	585	48	on	on	ADP
ejpam-6375	585	49	their	their	PRON
ejpam-6375	585	50	topological	topological	ADJ
ejpam-6375	585	51	properties	property	NOUN
ejpam-6375	585	52	.	.	PUNCT
ejpam-6375	586	1	in	in	ADP
ejpam-6375	586	2	their	their	PRON
ejpam-6375	586	3	independent	independent	ADJ
ejpam-6375	586	4	studies	study	NOUN
ejpam-6375	586	5	,	,	PUNCT
ejpam-6375	586	6	s∗	s∗	VERB
ejpam-6375	586	7	g	g	PROPN
ejpam-6375	586	8	-compactness	-compactness	NOUN
ejpam-6375	586	9	and	and	CCONJ
ejpam-6375	586	10	s∗	s∗	PROPN
ejpam-6375	586	11	g	g	PROPN
ejpam-6375	586	12	-connectedness	-connectedness	NOUN
ejpam-6375	586	13	do	do	AUX
ejpam-6375	586	14	not	not	PART
ejpam-6375	586	15	strictly	strictly	ADV
ejpam-6375	586	16	mirror	mirror	VERB
ejpam-6375	586	17	traditional	traditional	ADJ
ejpam-6375	586	18	separation	separation	NOUN
ejpam-6375	586	19	axioms	axiom	VERB
ejpam-6375	586	20	,	,	PUNCT
ejpam-6375	586	21	yet	yet	CCONJ
ejpam-6375	586	22	their	their	PRON
ejpam-6375	586	23	examination	examination	NOUN
ejpam-6375	586	24	broadens	broaden	VERB
ejpam-6375	586	25	the	the	DET
ejpam-6375	586	26	fundamental	fundamental	ADJ
ejpam-6375	586	27	understanding	understanding	NOUN
ejpam-6375	586	28	of	of	ADP
ejpam-6375	586	29	generalized	generalized	ADJ
ejpam-6375	586	30	primal	primal	ADJ
ejpam-6375	586	31	spaces	space	NOUN
ejpam-6375	586	32	.	.	PUNCT
ejpam-6375	587	1	this	this	DET
ejpam-6375	587	2	research	research	NOUN
ejpam-6375	587	3	provides	provide	VERB
ejpam-6375	587	4	significant	significant	ADJ
ejpam-6375	587	5	value	value	NOUN
ejpam-6375	587	6	to	to	ADP
ejpam-6375	587	7	the	the	DET
ejpam-6375	587	8	expansion	expansion	NOUN
ejpam-6375	587	9	of	of	ADP
ejpam-6375	587	10	generalized	generalized	ADJ
ejpam-6375	587	11	primal	primal	ADJ
ejpam-6375	587	12	topology	topology	NOUN
ejpam-6375	587	13	as	as	ADP
ejpam-6375	587	14	a	a	DET
ejpam-6375	587	15	field	field	NOUN
ejpam-6375	587	16	of	of	ADP
ejpam-6375	587	17	exploration	exploration	NOUN
ejpam-6375	587	18	.	.	PUNCT
ejpam-6375	588	1	author	author	NOUN
ejpam-6375	588	2	contributions	contribution	NOUN
ejpam-6375	588	3	conceptualization	conceptualization	NOUN
ejpam-6375	588	4	,	,	PUNCT
ejpam-6375	588	5	m.s	m.s	PROPN
ejpam-6375	588	6	.	.	PROPN
ejpam-6375	588	7	,	,	PUNCT
ejpam-6375	588	8	u.i	u.i	PROPN
ejpam-6375	588	9	.	.	PROPN
ejpam-6375	588	10	and	and	CCONJ
ejpam-6375	588	11	i.l.p	i.l.p	NOUN
ejpam-6375	588	12	.	.	PUNCT
ejpam-6375	588	13	;	;	PUNCT
ejpam-6375	589	1	methodology	methodology	PROPN
ejpam-6375	589	2	,	,	PUNCT
ejpam-6375	589	3	m.i	m.i	PROPN
ejpam-6375	589	4	.	.	PROPN
ejpam-6375	589	5	and	and	CCONJ
ejpam-6375	589	6	i.l.p	i.l.p	NOUN
ejpam-6375	589	7	.	.	PUNCT
ejpam-6375	589	8	;	;	PUNCT
ejpam-6375	589	9	software	software	NOUN
ejpam-6375	589	10	,	,	PUNCT
ejpam-6375	589	11	t.k	t.k	PROPN
ejpam-6375	589	12	.	.	PROPN
ejpam-6375	589	13	,	,	PUNCT
ejpam-6375	589	14	u.i	u.i	PROPN
ejpam-6375	589	15	.	.	PROPN
ejpam-6375	589	16	and	and	CCONJ
ejpam-6375	589	17	i.l.p	i.l.p	NOUN
ejpam-6375	589	18	.	.	PROPN
ejpam-6375	589	19	;	;	PUNCT
ejpam-6375	590	1	validation	validation	NOUN
ejpam-6375	590	2	,	,	PUNCT
ejpam-6375	590	3	m.a	m.a	PROPN
ejpam-6375	590	4	.	.	PROPN
ejpam-6375	590	5	and	and	CCONJ
ejpam-6375	590	6	i.l.p	i.l.p	NOUN
ejpam-6375	590	7	.	.	PUNCT
ejpam-6375	590	8	;	;	PUNCT
ejpam-6375	590	9	formal	formal	ADJ
ejpam-6375	590	10	analysis	analysis	NOUN
ejpam-6375	590	11	,	,	PUNCT
ejpam-6375	590	12	u.i	u.i	PROPN
ejpam-6375	590	13	.	.	PROPN
ejpam-6375	590	14	and	and	CCONJ
ejpam-6375	590	15	m.a	m.a	PROPN
ejpam-6375	590	16	.	.	PROPN
ejpam-6375	590	17	;	;	PUNCT
ejpam-6375	590	18	investigation	investigation	NOUN
ejpam-6375	590	19	,	,	PUNCT
ejpam-6375	590	20	m.a	m.a	PROPN
ejpam-6375	590	21	.	.	PROPN
ejpam-6375	590	22	;	;	PUNCT
ejpam-6375	590	23	resources	resource	NOUN
ejpam-6375	590	24	,	,	PUNCT
ejpam-6375	590	25	t.k	t.k	PROPN
ejpam-6375	590	26	.	.	PROPN
ejpam-6375	590	27	;	;	PUNCT
ejpam-6375	590	28	data	datum	NOUN
ejpam-6375	590	29	curation	curation	NOUN
ejpam-6375	590	30	,	,	PUNCT
ejpam-6375	590	31	u.i	u.i	PROPN
ejpam-6375	590	32	.	.	PROPN
ejpam-6375	590	33	;	;	PUNCT
ejpam-6375	590	34	writing	writing	NOUN
ejpam-6375	590	35	—	—	PUNCT
ejpam-6375	590	36	original	original	ADJ
ejpam-6375	590	37	draft	draft	NOUN
ejpam-6375	590	38	preparation	preparation	NOUN
ejpam-6375	590	39	,	,	PUNCT
ejpam-6375	590	40	m.s	m.s	PROPN
ejpam-6375	590	41	.	.	PROPN
ejpam-6375	590	42	,	,	PUNCT
ejpam-6375	590	43	m.i	m.i	PROPN
ejpam-6375	590	44	.	.	PROPN
ejpam-6375	590	45	,	,	PUNCT
ejpam-6375	590	46	and	and	CCONJ
ejpam-6375	590	47	u.i	u.i	PROPN
ejpam-6375	590	48	.	.	PROPN
ejpam-6375	590	49	;	;	PUNCT
ejpam-6375	590	50	writing	writing	NOUN
ejpam-6375	590	51	—	—	PUNCT
ejpam-6375	590	52	review	review	NOUN
ejpam-6375	590	53	and	and	CCONJ
ejpam-6375	590	54	editing	editing	NOUN
ejpam-6375	590	55	,	,	PUNCT
ejpam-6375	590	56	t.k	t.k	PROPN
ejpam-6375	590	57	.	.	PROPN
ejpam-6375	590	58	;	;	PUNCT
ejpam-6375	590	59	visualization	visualization	NOUN
ejpam-6375	590	60	,	,	PUNCT
ejpam-6375	590	61	m.a	m.a	PROPN
ejpam-6375	590	62	.	.	PROPN
ejpam-6375	590	63	supervision	supervision	PROPN
ejpam-6375	590	64	,	,	PUNCT
ejpam-6375	590	65	t.k	t.k	PROPN
ejpam-6375	590	66	.	.	PROPN
ejpam-6375	590	67	and	and	CCONJ
ejpam-6375	590	68	i.l.p	i.l.p	NOUN
ejpam-6375	590	69	.	.	PUNCT
ejpam-6375	590	70	;	;	PUNCT
ejpam-6375	590	71	project	project	NOUN
ejpam-6375	590	72	administration	administration	NOUN
ejpam-6375	590	73	,	,	PUNCT
ejpam-6375	590	74	u.i	u.i	PROPN
ejpam-6375	590	75	.	.	PROPN
ejpam-6375	590	76	and	and	CCONJ
ejpam-6375	590	77	i.l.p	i.l.p	PROPN
ejpam-6375	590	78	.	.	PROPN
ejpam-6375	590	79	;	;	PUNCT
ejpam-6375	590	80	funding	funding	NOUN
ejpam-6375	590	81	acquisition	acquisition	NOUN
ejpam-6375	590	82	,	,	PUNCT
ejpam-6375	590	83	m.a	m.a	PROPN
ejpam-6375	590	84	.	.	PROPN
ejpam-6375	591	1	all	all	DET
ejpam-6375	591	2	authors	author	NOUN
ejpam-6375	591	3	have	have	AUX
ejpam-6375	591	4	read	read	VERB
ejpam-6375	591	5	and	and	CCONJ
ejpam-6375	591	6	agreed	agree	VERB
ejpam-6375	591	7	to	to	ADP
ejpam-6375	591	8	the	the	DET
ejpam-6375	591	9	published	publish	VERB
ejpam-6375	591	10	version	version	NOUN
ejpam-6375	591	11	of	of	ADP
ejpam-6375	591	12	the	the	DET
ejpam-6375	591	13	manuscript	manuscript	NOUN
ejpam-6375	591	14	.	.	PUNCT
ejpam-6375	592	1	acknowledgements	acknowledgement	NOUN
ejpam-6375	592	2	the	the	DET
ejpam-6375	592	3	authors	author	NOUN
ejpam-6375	592	4	extend	extend	VERB
ejpam-6375	592	5	their	their	PRON
ejpam-6375	592	6	gratitude	gratitude	NOUN
ejpam-6375	592	7	to	to	ADP
ejpam-6375	592	8	the	the	DET
ejpam-6375	592	9	deanship	deanship	NOUN
ejpam-6375	592	10	of	of	ADP
ejpam-6375	592	11	graduate	graduate	NOUN
ejpam-6375	592	12	studies	study	NOUN
ejpam-6375	592	13	and	and	CCONJ
ejpam-6375	592	14	scientific	scientific	ADJ
ejpam-6375	592	15	research	research	NOUN
ejpam-6375	592	16	of	of	ADP
ejpam-6375	592	17	the	the	DET
ejpam-6375	592	18	islamic	islamic	PROPN
ejpam-6375	592	19	university	university	PROPN
ejpam-6375	592	20	of	of	ADP
ejpam-6375	592	21	madinah	madinah	PROPN
ejpam-6375	592	22	for	for	ADP
ejpam-6375	592	23	the	the	DET
ejpam-6375	592	24	support	support	NOUN
ejpam-6375	592	25	provided	provide	VERB
ejpam-6375	592	26	to	to	ADP
ejpam-6375	592	27	the	the	DET
ejpam-6375	592	28	postpublication	postpublication	NOUN
ejpam-6375	592	29	program	program	NOUN
ejpam-6375	592	30	4	4	NUM
ejpam-6375	592	31	.	.	PUNCT
ejpam-6375	592	32	competing	compete	VERB
ejpam-6375	592	33	interests	interest	NOUN
ejpam-6375	592	34	the	the	DET
ejpam-6375	592	35	authors	author	NOUN
ejpam-6375	592	36	declare	declare	VERB
ejpam-6375	592	37	that	that	SCONJ
ejpam-6375	592	38	they	they	PRON
ejpam-6375	592	39	have	have	VERB
ejpam-6375	592	40	no	no	DET
ejpam-6375	592	41	competing	compete	VERB
ejpam-6375	592	42	interests	interest	NOUN
ejpam-6375	592	43	.	.	PUNCT
ejpam-6375	593	1	references	reference	NOUN
ejpam-6375	593	2	[	[	X
ejpam-6375	593	3	1	1	NUM
ejpam-6375	593	4	]	]	PUNCT
ejpam-6375	593	5	m.	m.	NOUN
ejpam-6375	593	6	shahbaz	shahbaz	PROPN
ejpam-6375	593	7	,	,	PUNCT
ejpam-6375	593	8	t.	t.	PROPN
ejpam-6375	593	9	kamran	kamran	PROPN
ejpam-6375	593	10	,	,	PUNCT
ejpam-6375	593	11	u.	u.	PROPN
ejpam-6375	593	12	ishtiaq	ishtiaq	PROPN
ejpam-6375	593	13	,	,	PUNCT
ejpam-6375	593	14	m.	m.	PROPN
ejpam-6375	593	15	imtiaz	imtiaz	PROPN
ejpam-6375	593	16	,	,	PUNCT
ejpam-6375	593	17	i.l	i.l	PROPN
ejpam-6375	593	18	.	.	PROPN
ejpam-6375	593	19	popa	popa	PROPN
ejpam-6375	593	20	,	,	PUNCT
ejpam-6375	593	21	and	and	CCONJ
ejpam-6375	593	22	f.m	f.m	PROPN
ejpam-6375	593	23	.	.	PROPN
ejpam-6375	593	24	maiz	maiz	PROPN
ejpam-6375	593	25	.	.	PUNCT
ejpam-6375	594	1	some	some	DET
ejpam-6375	594	2	new	new	ADJ
ejpam-6375	594	3	notions	notion	NOUN
ejpam-6375	594	4	of	of	ADP
ejpam-6375	594	5	continuity	continuity	NOUN
ejpam-6375	594	6	in	in	ADP
ejpam-6375	594	7	generalized	generalized	ADJ
ejpam-6375	594	8	primal	primal	ADJ
ejpam-6375	594	9	topological	topological	ADJ
ejpam-6375	594	10	space	space	NOUN
ejpam-6375	594	11	.	.	PUNCT
ejpam-6375	595	1	mathematics	mathematic	NOUN
ejpam-6375	595	2	,	,	PUNCT
ejpam-6375	595	3	12(24):3995	12(24):3995	NUM
ejpam-6375	595	4	,	,	PUNCT
ejpam-6375	595	5	2024	2024	NUM
ejpam-6375	595	6	.	.	PUNCT
ejpam-6375	596	1	[	[	X
ejpam-6375	596	2	2	2	NUM
ejpam-6375	596	3	]	]	PUNCT
ejpam-6375	596	4	m.	m.	NOUN
ejpam-6375	596	5	shahbaz	shahbaz	PROPN
ejpam-6375	596	6	,	,	PUNCT
ejpam-6375	596	7	t.	t.	PROPN
ejpam-6375	596	8	kamran	kamran	PROPN
ejpam-6375	596	9	,	,	PUNCT
ejpam-6375	596	10	u.	u.	PROPN
ejpam-6375	596	11	ishtiaq	ishtiaq	PROPN
ejpam-6375	596	12	,	,	PUNCT
ejpam-6375	596	13	m.	m.	PROPN
ejpam-6375	596	14	imtiaz	imtiaz	PROPN
ejpam-6375	596	15	,	,	PUNCT
ejpam-6375	596	16	i.l	i.l	PROPN
ejpam-6375	596	17	.	.	PROPN
ejpam-6375	596	18	popa	popa	PROPN
ejpam-6375	596	19	,	,	PUNCT
ejpam-6375	596	20	and	and	CCONJ
ejpam-6375	596	21	f.m	f.m	PROPN
ejpam-6375	596	22	.	.	PROPN
ejpam-6375	596	23	maiz	maiz	PROPN
ejpam-6375	596	24	.	.	PUNCT
ejpam-6375	597	1	some	some	DET
ejpam-6375	597	2	categories	category	NOUN
ejpam-6375	597	3	of	of	ADP
ejpam-6375	597	4	compactness	compactness	NOUN
ejpam-6375	597	5	and	and	CCONJ
ejpam-6375	597	6	connectedness	connectedness	NOUN
ejpam-6375	597	7	in	in	ADP
ejpam-6375	597	8	generalized	generalized	ADJ
ejpam-6375	597	9	topological	topological	ADJ
ejpam-6375	597	10	spaces	space	NOUN
ejpam-6375	597	11	.	.	PUNCT
ejpam-6375	598	1	axioms	axiom	NOUN
ejpam-6375	598	2	,	,	PUNCT
ejpam-6375	598	3	14(2):141	14(2):141	NUM
ejpam-6375	598	4	,	,	PUNCT
ejpam-6375	598	5	2025	2025	NUM
ejpam-6375	598	6	.	.	PUNCT
ejpam-6375	599	1	[	[	X
ejpam-6375	599	2	3	3	NUM
ejpam-6375	599	3	]	]	SYM
ejpam-6375	599	4	á	á	PROPN
ejpam-6375	599	5	.	.	PUNCT
ejpam-6375	599	6	császár	császár	PROPN
ejpam-6375	599	7	.	.	PUNCT
ejpam-6375	600	1	generalized	generalize	VERB
ejpam-6375	600	2	open	open	ADJ
ejpam-6375	600	3	sets	set	NOUN
ejpam-6375	600	4	in	in	ADP
ejpam-6375	600	5	generalized	generalized	ADJ
ejpam-6375	600	6	topologies	topology	NOUN
ejpam-6375	600	7	.	.	PUNCT
ejpam-6375	601	1	acta	acta	PROPN
ejpam-6375	601	2	math	math	PROPN
ejpam-6375	601	3	.	.	PUNCT
ejpam-6375	602	1	hung	hung	PROPN
ejpam-6375	602	2	.	.	PROPN
ejpam-6375	602	3	,	,	PUNCT
ejpam-6375	602	4	106(3):233–241	106(3):233–241	PROPN
ejpam-6375	602	5	,	,	PUNCT
ejpam-6375	602	6	2005	2005	NUM
ejpam-6375	602	7	.	.	PUNCT
ejpam-6375	603	1	[	[	X
ejpam-6375	603	2	4	4	X
ejpam-6375	603	3	]	]	X
ejpam-6375	603	4	h.	h.	PROPN
ejpam-6375	603	5	maki	maki	PROPN
ejpam-6375	603	6	,	,	PUNCT
ejpam-6375	603	7	k.	k.	PROPN
ejpam-6375	603	8	balachandran	balachandran	PROPN
ejpam-6375	603	9	,	,	PUNCT
ejpam-6375	603	10	and	and	CCONJ
ejpam-6375	603	11	r.	r.	PROPN
ejpam-6375	603	12	devi	devi	PROPN
ejpam-6375	603	13	.	.	PUNCT
ejpam-6375	604	1	remarks	remark	NOUN
ejpam-6375	604	2	on	on	ADP
ejpam-6375	604	3	semi	semi	ADJ
ejpam-6375	604	4	-	-	ADJ
ejpam-6375	604	5	generalized	generalized	ADJ
ejpam-6375	604	6	closed	closed	ADJ
ejpam-6375	604	7	sets	set	NOUN
ejpam-6375	604	8	and	and	CCONJ
ejpam-6375	604	9	generalized	generalized	ADJ
ejpam-6375	604	10	semi	semi	ADJ
ejpam-6375	604	11	-	-	ADJ
ejpam-6375	604	12	closed	closed	ADJ
ejpam-6375	604	13	sets	set	NOUN
ejpam-6375	604	14	.	.	PUNCT
ejpam-6375	605	1	kyungpook	kyungpook	PROPN
ejpam-6375	605	2	math	math	PROPN
ejpam-6375	605	3	.	.	PUNCT
ejpam-6375	606	1	j.	j.	PROPN
ejpam-6375	606	2	,	,	PUNCT
ejpam-6375	606	3	36(1):155–160	36(1):155–160	PROPN
ejpam-6375	606	4	,	,	PUNCT
ejpam-6375	606	5	1996	1996	NUM
ejpam-6375	606	6	.	.	PUNCT
ejpam-6375	607	1	[	[	X
ejpam-6375	607	2	5	5	X
ejpam-6375	607	3	]	]	PUNCT
ejpam-6375	607	4	h.	h.	PROPN
ejpam-6375	607	5	maki	maki	PROPN
ejpam-6375	607	6	,	,	PUNCT
ejpam-6375	607	7	k.c	k.c	PROPN
ejpam-6375	607	8	.	.	PROPN
ejpam-6375	607	9	rao	rao	PROPN
ejpam-6375	607	10	,	,	PUNCT
ejpam-6375	607	11	and	and	CCONJ
ejpam-6375	607	12	a.n	a.n	PROPN
ejpam-6375	607	13	.	.	PROPN
ejpam-6375	607	14	gani	gani	PROPN
ejpam-6375	607	15	.	.	PUNCT
ejpam-6375	608	1	on	on	ADP
ejpam-6375	608	2	generalizing	generalize	VERB
ejpam-6375	608	3	semi	semi	ADJ
ejpam-6375	608	4	-	-	ADJ
ejpam-6375	608	5	open	open	ADJ
ejpam-6375	608	6	sets	set	NOUN
ejpam-6375	608	7	and	and	CCONJ
ejpam-6375	608	8	preopen	preopen	ADJ
ejpam-6375	608	9	sets	set	NOUN
ejpam-6375	608	10	.	.	PUNCT
ejpam-6375	609	1	pure	pure	ADJ
ejpam-6375	609	2	appl	appl	PROPN
ejpam-6375	609	3	.	.	PUNCT
ejpam-6375	609	4	math	math	PROPN
ejpam-6375	609	5	.	.	PUNCT
ejpam-6375	610	1	sci	sci	PROPN
ejpam-6375	610	2	.	.	PROPN
ejpam-6375	610	3	,	,	PUNCT
ejpam-6375	610	4	49(1/2):17–30	49(1/2):17–30	NUM
ejpam-6375	610	5	,	,	PUNCT
ejpam-6375	610	6	1999	1999	NUM
ejpam-6375	610	7	.	.	PUNCT
ejpam-6375	611	1	[	[	X
ejpam-6375	611	2	6	6	NUM
ejpam-6375	611	3	]	]	PUNCT
ejpam-6375	611	4	g.	g.	NOUN
ejpam-6375	611	5	navalagi	navalagi	PROPN
ejpam-6375	611	6	and	and	CCONJ
ejpam-6375	611	7	h.	h.	PROPN
ejpam-6375	611	8	page	page	NOUN
ejpam-6375	611	9	.	.	PUNCT
ejpam-6375	612	1	ϑ-generalized	ϑ-generalize	VERB
ejpam-6375	612	2	semi	semi	ADJ
ejpam-6375	612	3	-	-	ADJ
ejpam-6375	612	4	open	open	ADJ
ejpam-6375	612	5	and	and	CCONJ
ejpam-6375	612	6	ϑ-generalized	ϑ-generalize	VERB
ejpam-6375	612	7	semi	semi	ADJ
ejpam-6375	612	8	-	-	ADJ
ejpam-6375	612	9	closed	closed	ADJ
ejpam-6375	612	10	functions	function	NOUN
ejpam-6375	612	11	.	.	PUNCT
ejpam-6375	613	1	proy	proy	PROPN
ejpam-6375	613	2	.	.	PUNCT
ejpam-6375	614	1	j.	j.	PROPN
ejpam-6375	614	2	math	math	PROPN
ejpam-6375	614	3	.	.	PUNCT
ejpam-6375	614	4	,	,	PUNCT
ejpam-6375	614	5	28(2):123–134	28(2):123–134	PROPN
ejpam-6375	614	6	,	,	PUNCT
ejpam-6375	614	7	2009	2009	NUM
ejpam-6375	614	8	.	.	PUNCT
ejpam-6375	615	1	[	[	X
ejpam-6375	615	2	7	7	X
ejpam-6375	615	3	]	]	PUNCT
ejpam-6375	615	4	s.	s.	PROPN
ejpam-6375	615	5	acharjee	acharjee	PROPN
ejpam-6375	615	6	,	,	PUNCT
ejpam-6375	615	7	m.	m.	NOUN
ejpam-6375	615	8	özkoç	özkoç	NOUN
ejpam-6375	615	9	,	,	PUNCT
ejpam-6375	615	10	and	and	CCONJ
ejpam-6375	615	11	f.	f.	PROPN
ejpam-6375	615	12	y.	y.	PROPN
ejpam-6375	615	13	issaka	issaka	PROPN
ejpam-6375	615	14	.	.	PUNCT
ejpam-6375	616	1	primal	primal	ADJ
ejpam-6375	616	2	topological	topological	ADJ
ejpam-6375	616	3	spaces	space	NOUN
ejpam-6375	616	4	.	.	PUNCT
ejpam-6375	617	1	bol	bol	NOUN
ejpam-6375	617	2	.	.	PUNCT
ejpam-6375	618	1	soc	soc	PROPN
ejpam-6375	618	2	.	.	PUNCT
ejpam-6375	619	1	paran	paran	PROPN
ejpam-6375	619	2	.	.	PUNCT
ejpam-6375	619	3	math	math	PROPN
ejpam-6375	619	4	.	.	PUNCT
ejpam-6375	620	1	(	(	PUNCT
ejpam-6375	620	2	3s	3s	NOUN
ejpam-6375	620	3	.	.	PUNCT
ejpam-6375	620	4	)	)	PUNCT
ejpam-6375	620	5	,	,	PUNCT
ejpam-6375	620	6	43:1–9	43:1–9	NUM
ejpam-6375	620	7	,	,	PUNCT
ejpam-6375	620	8	2025	2025	NUM
ejpam-6375	620	9	.	.	PUNCT
ejpam-6375	621	1	[	[	X
ejpam-6375	621	2	8	8	NUM
ejpam-6375	621	3	]	]	PUNCT
ejpam-6375	621	4	m.	m.	NOUN
ejpam-6375	621	5	özkoç	özkoç	NOUN
ejpam-6375	621	6	and	and	CCONJ
ejpam-6375	621	7	b.	b.	PROPN
ejpam-6375	621	8	köstel	köstel	PROPN
ejpam-6375	621	9	.	.	PUNCT
ejpam-6375	622	1	on	on	ADP
ejpam-6375	622	2	the	the	DET
ejpam-6375	622	3	topology	topology	NOUN
ejpam-6375	622	4	τ⋄r	τ⋄r	NUM
ejpam-6375	622	5	of	of	ADP
ejpam-6375	622	6	primal	primal	ADJ
ejpam-6375	622	7	topological	topological	ADJ
ejpam-6375	622	8	spaces	space	NOUN
ejpam-6375	622	9	.	.	PUNCT
ejpam-6375	623	1	aims	aim	VERB
ejpam-6375	623	2	math	math	NOUN
ejpam-6375	623	3	.	.	PUNCT
ejpam-6375	623	4	,	,	PUNCT
ejpam-6375	623	5	9(7):17171–17183	9(7):17171–17183	NUM
ejpam-6375	623	6	,	,	PUNCT
ejpam-6375	623	7	2024	2024	NUM
ejpam-6375	623	8	.	.	PUNCT
ejpam-6375	624	1	m.	m.	NOUN
ejpam-6375	624	2	shahbaz	shahbaz	PROPN
ejpam-6375	624	3	et	et	PROPN
ejpam-6375	624	4	al	al	PROPN
ejpam-6375	624	5	.	.	PUNCT
ejpam-6375	624	6	/	/	SYM
ejpam-6375	624	7	eur	eur	PROPN
ejpam-6375	624	8	.	.	PUNCT
ejpam-6375	625	1	j.	j.	PROPN
ejpam-6375	625	2	pure	pure	PROPN
ejpam-6375	625	3	appl	appl	PROPN
ejpam-6375	625	4	.	.	PROPN
ejpam-6375	625	5	math	math	PROPN
ejpam-6375	625	6	,	,	PUNCT
ejpam-6375	625	7	18	18	NUM
ejpam-6375	625	8	(	(	PUNCT
ejpam-6375	625	9	4	4	NUM
ejpam-6375	625	10	)	)	PUNCT
ejpam-6375	625	11	(	(	PUNCT
ejpam-6375	625	12	2025	2025	NUM
ejpam-6375	625	13	)	)	PUNCT
ejpam-6375	625	14	,	,	PUNCT
ejpam-6375	625	15	6375	6375	NUM
ejpam-6375	625	16	22	22	NUM
ejpam-6375	625	17	of	of	ADP
ejpam-6375	625	18	22	22	NUM
ejpam-6375	626	1	[	[	X
ejpam-6375	626	2	9	9	NUM
ejpam-6375	626	3	]	]	X
ejpam-6375	626	4	h.	h.	PROPN
ejpam-6375	626	5	al	al	PROPN
ejpam-6375	626	6	-	-	PUNCT
ejpam-6375	626	7	saadi	saadi	PROPN
ejpam-6375	626	8	and	and	CCONJ
ejpam-6375	626	9	h.	h.	PROPN
ejpam-6375	626	10	al	al	PROPN
ejpam-6375	626	11	-	-	PUNCT
ejpam-6375	626	12	malki	malki	PROPN
ejpam-6375	626	13	.	.	PUNCT
ejpam-6375	627	1	generalized	generalize	VERB
ejpam-6375	627	2	primal	primal	ADJ
ejpam-6375	627	3	topological	topological	ADJ
ejpam-6375	627	4	spaces	space	NOUN
ejpam-6375	627	5	.	.	PUNCT
ejpam-6375	628	1	aims	aim	VERB
ejpam-6375	628	2	math	math	NOUN
ejpam-6375	628	3	.	.	PUNCT
ejpam-6375	628	4	,	,	PUNCT
ejpam-6375	628	5	8(10):24162–24175	8(10):24162–24175	NUM
ejpam-6375	628	6	,	,	PUNCT
ejpam-6375	628	7	2023	2023	NUM
ejpam-6375	628	8	.	.	PUNCT
ejpam-6375	629	1	[	[	X
ejpam-6375	629	2	10	10	NUM
ejpam-6375	629	3	]	]	X
ejpam-6375	629	4	h.	h.	PROPN
ejpam-6375	629	5	al	al	PROPN
ejpam-6375	629	6	-	-	PUNCT
ejpam-6375	629	7	saadi	saadi	PROPN
ejpam-6375	629	8	and	and	CCONJ
ejpam-6375	629	9	h.	h.	PROPN
ejpam-6375	629	10	al	al	PROPN
ejpam-6375	629	11	-	-	PUNCT
ejpam-6375	629	12	malki	malki	NOUN
ejpam-6375	629	13	.	.	PUNCT
ejpam-6375	630	1	categories	category	NOUN
ejpam-6375	630	2	of	of	ADP
ejpam-6375	630	3	open	open	ADJ
ejpam-6375	630	4	sets	set	NOUN
ejpam-6375	630	5	in	in	ADP
ejpam-6375	630	6	generalized	generalized	ADJ
ejpam-6375	630	7	primal	primal	ADJ
ejpam-6375	630	8	topological	topological	ADJ
ejpam-6375	630	9	spaces	space	NOUN
ejpam-6375	630	10	.	.	PUNCT
ejpam-6375	631	1	mathematics	mathematic	NOUN
ejpam-6375	631	2	,	,	PUNCT
ejpam-6375	631	3	12(2):207	12(2):207	NOUN
ejpam-6375	631	4	,	,	PUNCT
ejpam-6375	631	5	2024	2024	NUM
ejpam-6375	631	6	.	.	PUNCT
ejpam-6375	632	1	[	[	X
ejpam-6375	632	2	11	11	NUM
ejpam-6375	632	3	]	]	PUNCT
ejpam-6375	632	4	t.	t.	PROPN
ejpam-6375	632	5	m.	m.	PROPN
ejpam-6375	632	6	al	al	PROPN
ejpam-6375	632	7	-	-	PUNCT
ejpam-6375	632	8	shami	shami	PROPN
ejpam-6375	632	9	,	,	PUNCT
ejpam-6375	632	10	z.	z.	PROPN
ejpam-6375	632	11	a.	a.	PROPN
ejpam-6375	632	12	ameen	ameen	PROPN
ejpam-6375	632	13	,	,	PUNCT
ejpam-6375	632	14	r.	r.	PROPN
ejpam-6375	632	15	a.	a.	PROPN
ejpam-6375	632	16	gdairi	gdairi	PROPN
ejpam-6375	632	17	,	,	PUNCT
ejpam-6375	632	18	and	and	CCONJ
ejpam-6375	632	19	a.	a.	NOUN
ejpam-6375	632	20	mhemdi	mhemdi	PROPN
ejpam-6375	632	21	.	.	PUNCT
ejpam-6375	633	1	on	on	ADP
ejpam-6375	633	2	primal	primal	ADJ
ejpam-6375	633	3	soft	soft	ADJ
ejpam-6375	633	4	topology	topology	NOUN
ejpam-6375	633	5	.	.	PUNCT
ejpam-6375	634	1	mathematics	mathematic	NOUN
ejpam-6375	634	2	,	,	PUNCT
ejpam-6375	634	3	11:2329	11:2329	NUM
ejpam-6375	634	4	,	,	PUNCT
ejpam-6375	634	5	2023	2023	NUM
ejpam-6375	634	6	.	.	PUNCT
ejpam-6375	635	1	[	[	X
ejpam-6375	635	2	12	12	NUM
ejpam-6375	635	3	]	]	PUNCT
ejpam-6375	635	4	p.	p.	NOUN
ejpam-6375	635	5	şaşmaz	şaşmaz	NOUN
ejpam-6375	635	6	and	and	CCONJ
ejpam-6375	635	7	m.	m.	NOUN
ejpam-6375	635	8	özkoç	özkoç	NOUN
ejpam-6375	635	9	.	.	PUNCT
ejpam-6375	636	1	on	on	ADP
ejpam-6375	636	2	a	a	DET
ejpam-6375	636	3	new	new	ADJ
ejpam-6375	636	4	operator	operator	NOUN
ejpam-6375	636	5	based	base	VERB
ejpam-6375	636	6	on	on	ADP
ejpam-6375	636	7	a	a	DET
ejpam-6375	636	8	primal	primal	ADJ
ejpam-6375	636	9	and	and	CCONJ
ejpam-6375	636	10	its	its	PRON
ejpam-6375	636	11	associated	associated	ADJ
ejpam-6375	636	12	topology	topology	NOUN
ejpam-6375	636	13	.	.	PUNCT
ejpam-6375	637	1	european	european	PROPN
ejpam-6375	637	2	journal	journal	PROPN
ejpam-6375	637	3	of	of	ADP
ejpam-6375	637	4	pure	pure	ADJ
ejpam-6375	637	5	and	and	CCONJ
ejpam-6375	637	6	applied	applied	ADJ
ejpam-6375	637	7	mathematics	mathematic	NOUN
ejpam-6375	637	8	,	,	PUNCT
ejpam-6375	637	9	17(4):2800–2811	17(4):2800–2811	NUM
ejpam-6375	637	10	,	,	PUNCT
ejpam-6375	637	11	2024	2024	NUM
ejpam-6375	637	12	.	.	PUNCT
ejpam-6375	638	1	[	[	X
ejpam-6375	638	2	13	13	NUM
ejpam-6375	638	3	]	]	SYM
ejpam-6375	638	4	s.p	s.p	PROPN
ejpam-6375	638	5	.	.	PROPN
ejpam-6375	638	6	missier	missier	PROPN
ejpam-6375	638	7	and	and	CCONJ
ejpam-6375	638	8	j.	j.	PROPN
ejpam-6375	638	9	son	son	PROPN
ejpam-6375	638	10	.	.	PUNCT
ejpam-6375	639	1	a	a	DET
ejpam-6375	639	2	new	new	ADJ
ejpam-6375	639	3	notion	notion	NOUN
ejpam-6375	639	4	of	of	ADP
ejpam-6375	639	5	generalized	generalized	ADJ
ejpam-6375	639	6	closed	close	VERB
ejpam-6375	639	7	sets	set	NOUN
ejpam-6375	639	8	in	in	ADP
ejpam-6375	639	9	topological	topological	ADJ
ejpam-6375	639	10	spaces	space	NOUN
ejpam-6375	639	11	.	.	PUNCT
ejpam-6375	640	1	iosr	iosr	PROPN
ejpam-6375	640	2	j.	j.	PROPN
ejpam-6375	640	3	math	math	PROPN
ejpam-6375	640	4	.	.	PUNCT
ejpam-6375	640	5	,	,	PUNCT
ejpam-6375	640	6	10(4):122–128	10(4):122–128	NUM
ejpam-6375	640	7	,	,	PUNCT
ejpam-6375	640	8	2014	2014	NUM
ejpam-6375	640	9	.	.	PUNCT
ejpam-6375	641	1	[	[	X
ejpam-6375	641	2	14	14	NUM
ejpam-6375	641	3	]	]	PUNCT
ejpam-6375	641	4	j.	j.	PROPN
ejpam-6375	641	5	thomas	thomas	PROPN
ejpam-6375	641	6	and	and	CCONJ
ejpam-6375	641	7	s.j	s.j	PROPN
ejpam-6375	641	8	.	.	PROPN
ejpam-6375	641	9	john	john	PROPN
ejpam-6375	641	10	.	.	PUNCT
ejpam-6375	642	1	µ-compactness	µ-compactness	NOUN
ejpam-6375	642	2	in	in	ADP
ejpam-6375	642	3	generalized	generalized	ADJ
ejpam-6375	642	4	topological	topological	ADJ
ejpam-6375	642	5	spaces	space	NOUN
ejpam-6375	642	6	.	.	PUNCT
ejpam-6375	643	1	j.	j.	PROPN
ejpam-6375	643	2	adv	adv	PROPN
ejpam-6375	643	3	.	.	PUNCT
ejpam-6375	643	4	stud	stud	PROPN
ejpam-6375	643	5	.	.	PUNCT
ejpam-6375	644	1	topol	topol	PROPN
ejpam-6375	644	2	.	.	PROPN
ejpam-6375	644	3	,	,	PUNCT
ejpam-6375	644	4	3(3):18–22	3(3):18–22	NUM
ejpam-6375	644	5	,	,	PUNCT
ejpam-6375	644	6	2012	2012	NUM
ejpam-6375	644	7	.	.	PUNCT
ejpam-6375	645	1	[	[	X
ejpam-6375	645	2	15	15	NUM
ejpam-6375	645	3	]	]	PUNCT
ejpam-6375	645	4	p.	p.	NOUN
ejpam-6375	645	5	urysohn	urysohn	PROPN
ejpam-6375	645	6	.	.	PUNCT
ejpam-6375	646	1	über	über	PROPN
ejpam-6375	646	2	die	die	PROPN
ejpam-6375	646	3	mächtigkeit	mächtigkeit	PROPN
ejpam-6375	646	4	der	der	PROPN
ejpam-6375	646	5	zusammenhängenden	zusammenhängenden	PROPN
ejpam-6375	646	6	mengen	mengen	PROPN
ejpam-6375	646	7	.	.	PROPN
ejpam-6375	646	8	math	math	PROPN
ejpam-6375	646	9	.	.	PUNCT
ejpam-6375	647	1	ann	ann	PROPN
ejpam-6375	647	2	.	.	PROPN
ejpam-6375	647	3	,	,	PUNCT
ejpam-6375	647	4	94(1):262–295	94(1):262–295	PROPN
ejpam-6375	647	5	,	,	PUNCT
ejpam-6375	647	6	1925	1925	NUM
ejpam-6375	647	7	.	.	PUNCT
ejpam-6375	648	1	[	[	X
ejpam-6375	648	2	16	16	NUM
ejpam-6375	648	3	]	]	X
ejpam-6375	648	4	w.t	w.t	PROPN
ejpam-6375	648	5	.	.	PUNCT
ejpam-6375	648	6	van	van	PROPN
ejpam-6375	648	7	est	est	PROPN
ejpam-6375	648	8	and	and	CCONJ
ejpam-6375	648	9	h.	h.	PROPN
ejpam-6375	648	10	freudenthal	freudenthal	PROPN
ejpam-6375	648	11	.	.	PUNCT
ejpam-6375	648	12	trennung	trennung	ADJ
ejpam-6375	648	13	durch	durch	NOUN
ejpam-6375	648	14	stetige	stetige	NOUN
ejpam-6375	648	15	funktionen	funktionen	PROPN
ejpam-6375	648	16	in	in	ADP
ejpam-6375	648	17	topologischen	topologischen	PROPN
ejpam-6375	648	18	räumen	räuman	NOUN
ejpam-6375	648	19	.	.	PUNCT
ejpam-6375	649	1	in	in	ADP
ejpam-6375	649	2	indag	indag	PROPN
ejpam-6375	649	3	.	.	PUNCT
ejpam-6375	650	1	math	math	NOUN
ejpam-6375	650	2	.	.	PUNCT
ejpam-6375	651	1	(	(	PUNCT
ejpam-6375	651	2	proc	proc	NOUN
ejpam-6375	651	3	.	.	PUNCT
ejpam-6375	651	4	)	)	PUNCT
ejpam-6375	652	1	,	,	PUNCT
ejpam-6375	652	2	volume	volume	NOUN
ejpam-6375	652	3	54	54	NUM
ejpam-6375	652	4	,	,	PUNCT
ejpam-6375	652	5	pages	page	NOUN
ejpam-6375	652	6	359–368	359–368	NUM
ejpam-6375	652	7	,	,	PUNCT
ejpam-6375	652	8	1951	1951	NUM
ejpam-6375	652	9	.	.	PUNCT
ejpam-6375	653	1	[	[	X
ejpam-6375	653	2	17	17	NUM
ejpam-6375	653	3	]	]	X
ejpam-6375	653	4	m.h	m.h	PROPN
ejpam-6375	653	5	.	.	PROPN
ejpam-6375	653	6	stone	stone	PROPN
ejpam-6375	653	7	.	.	PUNCT
ejpam-6375	654	1	applications	application	NOUN
ejpam-6375	654	2	of	of	ADP
ejpam-6375	654	3	boolean	boolean	ADJ
ejpam-6375	654	4	algebras	algebra	NOUN
ejpam-6375	654	5	to	to	ADP
ejpam-6375	654	6	topology	topology	NOUN
ejpam-6375	654	7	.	.	PUNCT
ejpam-6375	655	1	trans	trans	PROPN
ejpam-6375	655	2	.	.	PROPN
ejpam-6375	656	1	am	be	AUX
ejpam-6375	656	2	.	.	PUNCT
ejpam-6375	657	1	math	math	NOUN
ejpam-6375	657	2	.	.	PUNCT
ejpam-6375	658	1	soc	soc	PROPN
ejpam-6375	658	2	.	.	PUNCT
ejpam-6375	658	3	,	,	PUNCT
ejpam-6375	658	4	41(3):375–481	41(3):375–481	NUM
ejpam-6375	658	5	,	,	PUNCT
ejpam-6375	658	6	1936	1936	NUM
ejpam-6375	658	7	.	.	PUNCT
ejpam-6375	659	1	[	[	X
ejpam-6375	659	2	18	18	NUM
ejpam-6375	659	3	]	]	X
ejpam-6375	659	4	j.w.t	j.w.t	X
ejpam-6375	659	5	.	.	PUNCT
ejpam-6375	659	6	youngs	young	NOUN
ejpam-6375	659	7	.	.	PUNCT
ejpam-6375	660	1	a	a	DET
ejpam-6375	660	2	note	note	NOUN
ejpam-6375	660	3	on	on	ADP
ejpam-6375	660	4	separation	separation	NOUN
ejpam-6375	660	5	axioms	axiom	NOUN
ejpam-6375	660	6	and	and	CCONJ
ejpam-6375	660	7	their	their	PRON
ejpam-6375	660	8	application	application	NOUN
ejpam-6375	660	9	in	in	ADP
ejpam-6375	660	10	the	the	DET
ejpam-6375	660	11	theory	theory	NOUN
ejpam-6375	660	12	of	of	ADP
ejpam-6375	660	13	a	a	DET
ejpam-6375	660	14	locally	locally	ADV
ejpam-6375	660	15	connected	connect	VERB
ejpam-6375	660	16	topological	topological	ADJ
ejpam-6375	660	17	space	space	NOUN
ejpam-6375	660	18	.	.	PUNCT
ejpam-6375	661	1	bull	bull	NOUN
ejpam-6375	661	2	.	.	PUNCT
ejpam-6375	662	1	am	be	AUX
ejpam-6375	662	2	.	.	PUNCT
ejpam-6375	663	1	math	math	NOUN
ejpam-6375	663	2	.	.	PUNCT
ejpam-6375	664	1	soc	soc	PROPN
ejpam-6375	664	2	.	.	PUNCT
ejpam-6375	664	3	,	,	PUNCT
ejpam-6375	664	4	49(9):713–719	49(9):713–719	PROPN
ejpam-6375	664	5	,	,	PUNCT
ejpam-6375	664	6	1943	1943	NUM
ejpam-6375	664	7	.	.	PUNCT
ejpam-6375	665	1	[	[	X
ejpam-6375	665	2	19	19	NUM
ejpam-6375	665	3	]	]	X
ejpam-6375	665	4	r.	r.	PROPN
ejpam-6375	665	5	khayyeri	khayyeri	PROPN
ejpam-6375	665	6	and	and	CCONJ
ejpam-6375	665	7	r.	r.	PROPN
ejpam-6375	665	8	mohamadian	mohamadian	PROPN
ejpam-6375	665	9	.	.	PUNCT
ejpam-6375	666	1	on	on	ADP
ejpam-6375	666	2	base	base	NOUN
ejpam-6375	666	3	for	for	ADP
ejpam-6375	666	4	generalized	generalized	ADJ
ejpam-6375	666	5	topological	topological	ADJ
ejpam-6375	666	6	spaces	space	NOUN
ejpam-6375	666	7	.	.	PUNCT
ejpam-6375	667	1	int	int	NOUN
ejpam-6375	667	2	.	.	PUNCT
ejpam-6375	668	1	j.	j.	PROPN
ejpam-6375	668	2	contemp	contemp	PROPN
ejpam-6375	668	3	.	.	PUNCT
ejpam-6375	669	1	math	math	NOUN
ejpam-6375	669	2	.	.	PUNCT
ejpam-6375	670	1	sci	sci	PROPN
ejpam-6375	670	2	.	.	PROPN
ejpam-6375	670	3	,	,	PUNCT
ejpam-6375	670	4	6(48):2377–2383	6(48):2377–2383	NUM
ejpam-6375	670	5	,	,	PUNCT
ejpam-6375	670	6	2011	2011	NUM
ejpam-6375	670	7	.	.	PUNCT
ejpam-6375	671	1	[	[	X
ejpam-6375	671	2	20	20	NUM
ejpam-6375	671	3	]	]	X
ejpam-6375	671	4	m.s	m.s	PROPN
ejpam-6375	671	5	.	.	PROPN
ejpam-6375	671	6	sarsak	sarsak	PROPN
ejpam-6375	671	7	.	.	PUNCT
ejpam-6375	672	1	on	on	ADP
ejpam-6375	672	2	some	some	DET
ejpam-6375	672	3	properties	property	NOUN
ejpam-6375	672	4	of	of	ADP
ejpam-6375	672	5	generalized	generalized	ADJ
ejpam-6375	672	6	open	open	ADJ
ejpam-6375	672	7	sets	set	NOUN
ejpam-6375	672	8	in	in	ADP
ejpam-6375	672	9	generalized	generalized	ADJ
ejpam-6375	672	10	topological	topological	ADJ
ejpam-6375	672	11	spaces	space	NOUN
ejpam-6375	672	12	.	.	PUNCT
ejpam-6375	673	1	demonstr	demonstr	NOUN
ejpam-6375	673	2	.	.	PUNCT
ejpam-6375	674	1	math	math	NOUN
ejpam-6375	674	2	.	.	PUNCT
ejpam-6375	675	1	,	,	PUNCT
ejpam-6375	675	2	46(2):415–427	46(2):415–427	NOUN
ejpam-6375	675	3	,	,	PUNCT
ejpam-6375	675	4	2013	2013	NUM
ejpam-6375	675	5	.	.	PUNCT
ejpam-6375	676	1	[	[	X
ejpam-6375	676	2	21	21	NUM
ejpam-6375	676	3	]	]	X
ejpam-6375	676	4	g.	g.	NOUN
ejpam-6375	676	5	choquet	choquet	PROPN
ejpam-6375	676	6	.	.	PUNCT
ejpam-6375	677	1	sur	sur	PROPN
ejpam-6375	677	2	les	les	PROPN
ejpam-6375	677	3	notions	notion	NOUN
ejpam-6375	677	4	de	de	X
ejpam-6375	677	5	filtre	filtre	NOUN
ejpam-6375	677	6	et	et	NOUN
ejpam-6375	677	7	de	de	NOUN
ejpam-6375	677	8	grille	grille	NOUN
ejpam-6375	677	9	.	.	PUNCT
ejpam-6375	678	1	c.	c.	PROPN
ejpam-6375	678	2	r.	r.	PROPN
ejpam-6375	678	3	acad	acad	PROPN
ejpam-6375	678	4	.	.	PUNCT
ejpam-6375	679	1	sci	sci	PROPN
ejpam-6375	679	2	.	.	PROPN
ejpam-6375	679	3	paris	paris	PROPN
ejpam-6375	679	4	,	,	PUNCT
ejpam-6375	679	5	224:171–173	224:171–173	NUM
ejpam-6375	679	6	,	,	PUNCT
ejpam-6375	679	7	1947	1947	NUM
ejpam-6375	679	8	.	.	PUNCT
ejpam-6375	680	1	[	[	X
ejpam-6375	680	2	22	22	NUM
ejpam-6375	680	3	]	]	X
ejpam-6375	680	4	l.a	l.a	PROPN
ejpam-6375	680	5	.	.	PROPN
ejpam-6375	680	6	steen	steen	PROPN
ejpam-6375	680	7	and	and	CCONJ
ejpam-6375	680	8	j.a	j.a	PROPN
ejpam-6375	680	9	.	.	PROPN
ejpam-6375	680	10	seebach	seebach	PROPN
ejpam-6375	680	11	.	.	PUNCT
ejpam-6375	681	1	counterexamples	counterexample	NOUN
ejpam-6375	681	2	in	in	ADP
ejpam-6375	681	3	topology	topology	NOUN
ejpam-6375	681	4	.	.	PUNCT
ejpam-6375	682	1	springer	springer	NOUN
ejpam-6375	682	2	,	,	PUNCT
ejpam-6375	682	3	1978	1978	NUM
ejpam-6375	682	4	.	.	PUNCT
ejpam-6375	683	1	introduction	introduction	NOUN
ejpam-6375	683	2	preliminaries	preliminary	NOUN
ejpam-6375	683	3	methodology	methodology	NOUN
ejpam-6375	683	4	conclusions	conclusion	NOUN