id	sid	tid	token	lemma	pos
ejpam-6627	1	1	european	european	PROPN
ejpam-6627	1	2	journal	journal	PROPN
ejpam-6627	1	3	of	of	ADP
ejpam-6627	1	4	pure	pure	ADJ
ejpam-6627	1	5	and	and	CCONJ
ejpam-6627	1	6	applied	applied	ADJ
ejpam-6627	1	7	mathematics	mathematic	NOUN
ejpam-6627	1	8	2025	2025	NUM
ejpam-6627	1	9	,	,	PUNCT
ejpam-6627	1	10	vol	vol	NOUN
ejpam-6627	1	11	.	.	PROPN
ejpam-6627	1	12	18	18	NUM
ejpam-6627	1	13	,	,	PUNCT
ejpam-6627	1	14	issue	issue	NOUN
ejpam-6627	1	15	3	3	NUM
ejpam-6627	1	16	,	,	PUNCT
ejpam-6627	1	17	article	article	NOUN
ejpam-6627	1	18	number	number	NOUN
ejpam-6627	1	19	6627	6627	NUM
ejpam-6627	1	20	issn	issn	VERB
ejpam-6627	1	21	1307	1307	NUM
ejpam-6627	1	22	-	-	SYM
ejpam-6627	1	23	5543	5543	NUM
ejpam-6627	1	24	–	–	PUNCT
ejpam-6627	1	25	ejpam.com	ejpam.com	X
ejpam-6627	1	26	published	publish	VERB
ejpam-6627	1	27	by	by	ADP
ejpam-6627	1	28	new	new	PROPN
ejpam-6627	1	29	york	york	PROPN
ejpam-6627	1	30	business	business	PROPN
ejpam-6627	1	31	global	global	ADJ
ejpam-6627	1	32	separation	separation	NOUN
ejpam-6627	1	33	axioms	axiom	VERB
ejpam-6627	1	34	beyond	beyond	ADP
ejpam-6627	1	35	t2	t2	NOUN
ejpam-6627	1	36	:	:	PUNCT
ejpam-6627	1	37	novel	novel	ADJ
ejpam-6627	1	38	properties	property	NOUN
ejpam-6627	1	39	and	and	CCONJ
ejpam-6627	1	40	interactions	interaction	NOUN
ejpam-6627	1	41	in	in	ADP
ejpam-6627	1	42	non	non	ADJ
ejpam-6627	1	43	-	-	ADJ
ejpam-6627	1	44	regular	regular	ADJ
ejpam-6627	1	45	spaces	space	NOUN
ejpam-6627	1	46	jamal	jamal	PROPN
ejpam-6627	1	47	oudetallah1	oudetallah1	PROPN
ejpam-6627	1	48	,	,	PUNCT
ejpam-6627	1	49	wasim	wasim	PROPN
ejpam-6627	1	50	audeh1	audeh1	PROPN
ejpam-6627	1	51	,	,	PUNCT
ejpam-6627	1	52	manal	manal	PROPN
ejpam-6627	1	53	al	al	PROPN
ejpam-6627	1	54	-	-	PUNCT
ejpam-6627	1	55	labadi1	labadi1	PROPN
ejpam-6627	1	56	,	,	PUNCT
ejpam-6627	1	57	raja’a	raja’a	PROPN
ejpam-6627	1	58	al	al	PROPN
ejpam-6627	1	59	-	-	PUNCT
ejpam-6627	1	60	naimi1	naimi1	PROPN
ejpam-6627	1	61	,	,	PUNCT
ejpam-6627	1	62	iqbal	iqbal	PROPN
ejpam-6627	1	63	m.	m.	PROPN
ejpam-6627	1	64	batiha2,3,∗	batiha2,3,∗	PROPN
ejpam-6627	1	65	,	,	PUNCT
ejpam-6627	1	66	ala	ala	PROPN
ejpam-6627	1	67	amourah4	amourah4	PROPN
ejpam-6627	1	68	,	,	PUNCT
ejpam-6627	1	69	tala	tala	PROPN
ejpam-6627	1	70	sasa5	sasa5	PROPN
ejpam-6627	1	71	1	1	NUM
ejpam-6627	1	72	department	department	NOUN
ejpam-6627	1	73	of	of	ADP
ejpam-6627	1	74	mathematics	mathematics	PROPN
ejpam-6627	1	75	,	,	PUNCT
ejpam-6627	1	76	university	university	PROPN
ejpam-6627	1	77	of	of	ADP
ejpam-6627	1	78	petra	petra	PROPN
ejpam-6627	1	79	,	,	PUNCT
ejpam-6627	1	80	amman	amman	PROPN
ejpam-6627	1	81	11196	11196	NUM
ejpam-6627	1	82	,	,	PUNCT
ejpam-6627	1	83	jordan	jordan	PROPN
ejpam-6627	1	84	2	2	NUM
ejpam-6627	1	85	department	department	NOUN
ejpam-6627	1	86	of	of	ADP
ejpam-6627	1	87	mathematics	mathematic	NOUN
ejpam-6627	1	88	,	,	PUNCT
ejpam-6627	1	89	al	al	PROPN
ejpam-6627	1	90	zaytoonah	zaytoonah	PROPN
ejpam-6627	1	91	university	university	PROPN
ejpam-6627	1	92	of	of	ADP
ejpam-6627	1	93	jordan	jordan	PROPN
ejpam-6627	1	94	,	,	PUNCT
ejpam-6627	1	95	amman	amman	PROPN
ejpam-6627	1	96	11733	11733	NUM
ejpam-6627	1	97	,	,	PUNCT
ejpam-6627	1	98	jordan	jordan	PROPN
ejpam-6627	1	99	3	3	NUM
ejpam-6627	1	100	nonlinear	nonlinear	PROPN
ejpam-6627	1	101	dynamics	dynamic	NOUN
ejpam-6627	1	102	research	research	NOUN
ejpam-6627	1	103	center	center	NOUN
ejpam-6627	1	104	(	(	PUNCT
ejpam-6627	1	105	ndrc	ndrc	PROPN
ejpam-6627	1	106	)	)	PUNCT
ejpam-6627	1	107	,	,	PUNCT
ejpam-6627	1	108	ajman	ajman	PROPN
ejpam-6627	1	109	university	university	PROPN
ejpam-6627	1	110	,	,	PUNCT
ejpam-6627	1	111	ajman	ajman	NOUN
ejpam-6627	1	112	346	346	NUM
ejpam-6627	1	113	,	,	PUNCT
ejpam-6627	1	114	united	united	PROPN
ejpam-6627	1	115	arab	arab	PROPN
ejpam-6627	1	116	emirates	emirates	PROPN
ejpam-6627	1	117	4	4	NUM
ejpam-6627	1	118	mathematics	mathematics	PROPN
ejpam-6627	1	119	education	education	NOUN
ejpam-6627	1	120	program	program	NOUN
ejpam-6627	1	121	,	,	PUNCT
ejpam-6627	1	122	faculty	faculty	NOUN
ejpam-6627	1	123	of	of	ADP
ejpam-6627	1	124	education	education	NOUN
ejpam-6627	1	125	and	and	CCONJ
ejpam-6627	1	126	arts	art	NOUN
ejpam-6627	1	127	,	,	PUNCT
ejpam-6627	1	128	sohar	sohar	PROPN
ejpam-6627	1	129	university	university	PROPN
ejpam-6627	1	130	,	,	PUNCT
ejpam-6627	1	131	sohar	sohar	PROPN
ejpam-6627	1	132	311	311	NUM
ejpam-6627	1	133	,	,	PUNCT
ejpam-6627	1	134	oman	oman	NOUN
ejpam-6627	1	135	5	5	NUM
ejpam-6627	1	136	applied	apply	VERB
ejpam-6627	1	137	science	science	NOUN
ejpam-6627	1	138	research	research	NOUN
ejpam-6627	1	139	center	center	NOUN
ejpam-6627	1	140	,	,	PUNCT
ejpam-6627	1	141	applied	apply	VERB
ejpam-6627	1	142	science	science	NOUN
ejpam-6627	1	143	private	private	ADJ
ejpam-6627	1	144	university	university	NOUN
ejpam-6627	1	145	,	,	PUNCT
ejpam-6627	1	146	amman	amman	PROPN
ejpam-6627	1	147	,	,	PUNCT
ejpam-6627	1	148	jordan	jordan	PROPN
ejpam-6627	1	149	abstract	abstract	PROPN
ejpam-6627	1	150	.	.	PUNCT
ejpam-6627	2	1	this	this	DET
ejpam-6627	2	2	paper	paper	NOUN
ejpam-6627	2	3	investigates	investigate	VERB
ejpam-6627	2	4	separation	separation	NOUN
ejpam-6627	2	5	axioms	axiom	NOUN
ejpam-6627	2	6	that	that	PRON
ejpam-6627	2	7	are	be	AUX
ejpam-6627	2	8	weaker	weak	ADJ
ejpam-6627	2	9	than	than	ADP
ejpam-6627	2	10	the	the	DET
ejpam-6627	2	11	classical	classical	ADJ
ejpam-6627	2	12	hausdorff	hausdorff	NOUN
ejpam-6627	2	13	condition	condition	NOUN
ejpam-6627	2	14	,	,	PUNCT
ejpam-6627	2	15	examining	examine	VERB
ejpam-6627	2	16	their	their	PRON
ejpam-6627	2	17	relationships	relationship	NOUN
ejpam-6627	2	18	in	in	ADP
ejpam-6627	2	19	non	non	ADJ
ejpam-6627	2	20	-	-	ADJ
ejpam-6627	2	21	regular	regular	ADJ
ejpam-6627	2	22	topological	topological	ADJ
ejpam-6627	2	23	spaces	space	NOUN
ejpam-6627	2	24	.	.	PUNCT
ejpam-6627	3	1	we	we	PRON
ejpam-6627	3	2	focus	focus	VERB
ejpam-6627	3	3	on	on	ADP
ejpam-6627	3	4	spaces	space	NOUN
ejpam-6627	3	5	that	that	PRON
ejpam-6627	3	6	exhibit	exhibit	VERB
ejpam-6627	3	7	structural	structural	ADJ
ejpam-6627	3	8	properties	property	NOUN
ejpam-6627	3	9	characteristic	characteristic	ADJ
ejpam-6627	3	10	of	of	ADP
ejpam-6627	3	11	higher	high	ADJ
ejpam-6627	3	12	separation	separation	NOUN
ejpam-6627	3	13	axioms	axiom	NOUN
ejpam-6627	3	14	through	through	ADP
ejpam-6627	3	15	modest	modest	ADJ
ejpam-6627	3	16	separation	separation	NOUN
ejpam-6627	3	17	requirements	requirement	NOUN
ejpam-6627	3	18	.	.	PUNCT
ejpam-6627	4	1	the	the	DET
ejpam-6627	4	2	study	study	NOUN
ejpam-6627	4	3	establishes	establish	VERB
ejpam-6627	4	4	new	new	ADJ
ejpam-6627	4	5	characterizations	characterization	NOUN
ejpam-6627	4	6	of	of	ADP
ejpam-6627	4	7	sigma	sigma	NOUN
ejpam-6627	4	8	-	-	PUNCT
ejpam-6627	4	9	intersection	intersection	NOUN
ejpam-6627	4	10	subsets	subset	NOUN
ejpam-6627	4	11	and	and	CCONJ
ejpam-6627	4	12	explores	explore	VERB
ejpam-6627	4	13	the	the	DET
ejpam-6627	4	14	boundary	boundary	NOUN
ejpam-6627	4	15	between	between	ADP
ejpam-6627	4	16	different	different	ADJ
ejpam-6627	4	17	classes	class	NOUN
ejpam-6627	4	18	of	of	ADP
ejpam-6627	4	19	separation	separation	NOUN
ejpam-6627	4	20	axioms	axiom	NOUN
ejpam-6627	4	21	using	use	VERB
ejpam-6627	4	22	novel	novel	ADJ
ejpam-6627	4	23	counterexamples	counterexample	NOUN
ejpam-6627	4	24	.	.	PUNCT
ejpam-6627	5	1	we	we	PRON
ejpam-6627	5	2	present	present	VERB
ejpam-6627	5	3	several	several	ADJ
ejpam-6627	5	4	new	new	ADJ
ejpam-6627	5	5	results	result	NOUN
ejpam-6627	5	6	concerning	concern	VERB
ejpam-6627	5	7	separation	separation	NOUN
ejpam-6627	5	8	properties	property	NOUN
ejpam-6627	5	9	and	and	CCONJ
ejpam-6627	5	10	their	their	PRON
ejpam-6627	5	11	behavior	behavior	NOUN
ejpam-6627	5	12	under	under	ADP
ejpam-6627	5	13	topological	topological	ADJ
ejpam-6627	5	14	operations	operation	NOUN
ejpam-6627	5	15	including	include	VERB
ejpam-6627	5	16	quotient	quotient	NOUN
ejpam-6627	5	17	maps	map	NOUN
ejpam-6627	5	18	and	and	CCONJ
ejpam-6627	5	19	product	product	NOUN
ejpam-6627	5	20	constructions	construction	NOUN
ejpam-6627	5	21	.	.	PUNCT
ejpam-6627	6	1	our	our	PRON
ejpam-6627	6	2	findings	finding	NOUN
ejpam-6627	6	3	provide	provide	VERB
ejpam-6627	6	4	an	an	DET
ejpam-6627	6	5	enhanced	enhanced	ADJ
ejpam-6627	6	6	understanding	understanding	NOUN
ejpam-6627	6	7	of	of	ADP
ejpam-6627	6	8	the	the	DET
ejpam-6627	6	9	separation	separation	NOUN
ejpam-6627	6	10	axiom	axiom	NOUN
ejpam-6627	6	11	hierarchy	hierarchy	NOUN
ejpam-6627	6	12	and	and	CCONJ
ejpam-6627	6	13	offer	offer	VERB
ejpam-6627	6	14	fundamental	fundamental	ADJ
ejpam-6627	6	15	insights	insight	NOUN
ejpam-6627	6	16	for	for	ADP
ejpam-6627	6	17	studying	study	VERB
ejpam-6627	6	18	spaces	space	NOUN
ejpam-6627	6	19	that	that	PRON
ejpam-6627	6	20	lie	lie	VERB
ejpam-6627	6	21	outside	outside	ADP
ejpam-6627	6	22	traditional	traditional	ADJ
ejpam-6627	6	23	classification	classification	NOUN
ejpam-6627	6	24	schemes	scheme	NOUN
ejpam-6627	6	25	.	.	PUNCT
ejpam-6627	7	1	the	the	DET
ejpam-6627	7	2	results	result	NOUN
ejpam-6627	7	3	have	have	VERB
ejpam-6627	7	4	applications	application	NOUN
ejpam-6627	7	5	in	in	ADP
ejpam-6627	7	6	domain	domain	NOUN
ejpam-6627	7	7	theory	theory	NOUN
ejpam-6627	7	8	,	,	PUNCT
ejpam-6627	7	9	functional	functional	ADJ
ejpam-6627	7	10	analysis	analysis	NOUN
ejpam-6627	7	11	,	,	PUNCT
ejpam-6627	7	12	and	and	CCONJ
ejpam-6627	7	13	theoretical	theoretical	ADJ
ejpam-6627	7	14	computer	computer	NOUN
ejpam-6627	7	15	science	science	NOUN
ejpam-6627	7	16	where	where	SCONJ
ejpam-6627	7	17	such	such	ADJ
ejpam-6627	7	18	intermediate	intermediate	ADJ
ejpam-6627	7	19	separation	separation	NOUN
ejpam-6627	7	20	conditions	condition	NOUN
ejpam-6627	7	21	naturally	naturally	ADV
ejpam-6627	7	22	arise	arise	VERB
ejpam-6627	7	23	.	.	PUNCT
ejpam-6627	8	1	2020	2020	NUM
ejpam-6627	8	2	mathematics	mathematic	NOUN
ejpam-6627	8	3	subject	subject	NOUN
ejpam-6627	8	4	classifications	classification	NOUN
ejpam-6627	8	5	:	:	PUNCT
ejpam-6627	8	6	54d10	54d10	NUM
ejpam-6627	8	7	,	,	PUNCT
ejpam-6627	8	8	54d15	54d15	NUM
ejpam-6627	8	9	,	,	PUNCT
ejpam-6627	8	10	54d30	54d30	ADJ
ejpam-6627	8	11	key	key	ADJ
ejpam-6627	8	12	words	word	NOUN
ejpam-6627	8	13	and	and	CCONJ
ejpam-6627	8	14	phrases	phrase	NOUN
ejpam-6627	8	15	:	:	PUNCT
ejpam-6627	8	16	separation	separation	NOUN
ejpam-6627	8	17	axioms	axiom	NOUN
ejpam-6627	8	18	,	,	PUNCT
ejpam-6627	8	19	non	non	ADJ
ejpam-6627	8	20	-	-	ADJ
ejpam-6627	8	21	regular	regular	ADJ
ejpam-6627	8	22	spaces	space	NOUN
ejpam-6627	8	23	,	,	PUNCT
ejpam-6627	8	24	sigma	sigma	NOUN
ejpam-6627	8	25	-	-	PUNCT
ejpam-6627	8	26	intersection	intersection	NOUN
ejpam-6627	8	27	subsets	subset	NOUN
ejpam-6627	8	28	,	,	PUNCT
ejpam-6627	8	29	fréchet	fréchet	NOUN
ejpam-6627	8	30	spaces	space	NOUN
ejpam-6627	8	31	,	,	PUNCT
ejpam-6627	8	32	non	non	ADJ
ejpam-6627	8	33	-	-	ADJ
ejpam-6627	8	34	metrizable	metrizable	ADJ
ejpam-6627	8	35	spaces	space	NOUN
ejpam-6627	8	36	,	,	PUNCT
ejpam-6627	8	37	quasi	quasi	ADJ
ejpam-6627	8	38	-	-	ADJ
ejpam-6627	8	39	hausdorff	hausdorff	ADJ
ejpam-6627	8	40	spaces	space	NOUN
ejpam-6627	8	41	,	,	PUNCT
ejpam-6627	8	42	urysohn	urysohn	PROPN
ejpam-6627	8	43	spaces	space	NOUN
ejpam-6627	8	44	,	,	PUNCT
ejpam-6627	8	45	functional	functional	ADJ
ejpam-6627	8	46	separation	separation	NOUN
ejpam-6627	8	47	1	1	NUM
ejpam-6627	8	48	.	.	PUNCT
ejpam-6627	9	1	introduction	introduction	NOUN
ejpam-6627	9	2	recent	recent	ADJ
ejpam-6627	9	3	studies	study	NOUN
ejpam-6627	9	4	in	in	ADP
ejpam-6627	9	5	general	general	ADJ
ejpam-6627	9	6	topology	topology	NOUN
ejpam-6627	9	7	and	and	CCONJ
ejpam-6627	9	8	its	its	PRON
ejpam-6627	9	9	applications	application	NOUN
ejpam-6627	9	10	have	have	AUX
ejpam-6627	9	11	highlighted	highlight	VERB
ejpam-6627	9	12	a	a	DET
ejpam-6627	9	13	growing	grow	VERB
ejpam-6627	9	14	interest	interest	NOUN
ejpam-6627	9	15	in	in	ADP
ejpam-6627	9	16	specialized	specialized	ADJ
ejpam-6627	9	17	structures	structure	NOUN
ejpam-6627	9	18	such	such	ADJ
ejpam-6627	9	19	as	as	ADP
ejpam-6627	9	20	locally	locally	ADV
ejpam-6627	9	21	compact	compact	ADJ
ejpam-6627	9	22	spaces	space	NOUN
ejpam-6627	9	23	,	,	PUNCT
ejpam-6627	9	24	generalized	generalized	ADJ
ejpam-6627	9	25	compactness	compactness	NOUN
ejpam-6627	9	26	,	,	PUNCT
ejpam-6627	9	27	and	and	CCONJ
ejpam-6627	9	28	novel	novel	ADJ
ejpam-6627	9	29	lindelöf	lindelöf	NOUN
ejpam-6627	9	30	-	-	PUNCT
ejpam-6627	9	31	type	type	NOUN
ejpam-6627	9	32	conditions	condition	NOUN
ejpam-6627	9	33	.	.	PUNCT
ejpam-6627	10	1	these	these	DET
ejpam-6627	10	2	investigations	investigation	NOUN
ejpam-6627	10	3	include	include	VERB
ejpam-6627	10	4	the	the	DET
ejpam-6627	10	5	role	role	NOUN
ejpam-6627	10	6	of	of	ADP
ejpam-6627	10	7	locally	locally	ADV
ejpam-6627	10	8	∗corresponding	∗corresponde	VERB
ejpam-6627	10	9	author	author	NOUN
ejpam-6627	10	10	.	.	PUNCT
ejpam-6627	11	1	doi	doi	NOUN
ejpam-6627	11	2	:	:	PUNCT
ejpam-6627	11	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6627	https://doi.org/10.29020/nybg.ejpam.v18i3.6627	NOUN
ejpam-6627	11	4	email	email	NOUN
ejpam-6627	11	5	addresses	address	VERB
ejpam-6627	11	6	:	:	PUNCT
ejpam-6627	11	7	jamal.oudetallah@uop.edu.jo	jamal.oudetallah@uop.edu.jo	PROPN
ejpam-6627	11	8	(	(	PUNCT
ejpam-6627	11	9	j.	j.	PROPN
ejpam-6627	11	10	oudetallah	oudetallah	PROPN
ejpam-6627	11	11	)	)	PUNCT
ejpam-6627	11	12	,	,	PUNCT
ejpam-6627	11	13	i.batiha@zuj.edu.jo	i.batiha@zuj.edu.jo	NOUN
ejpam-6627	11	14	(	(	PUNCT
ejpam-6627	11	15	i.	i.	PROPN
ejpam-6627	11	16	m.	m.	PROPN
ejpam-6627	11	17	batiha	batiha	PROPN
ejpam-6627	11	18	)	)	PUNCT
ejpam-6627	11	19	,	,	PUNCT
ejpam-6627	11	20	waudeh@uop.edu.jo	waudeh@uop.edu.jo	NOUN
ejpam-6627	11	21	(	(	PUNCT
ejpam-6627	11	22	w.	w.	PROPN
ejpam-6627	11	23	audeh	audeh	PROPN
ejpam-6627	11	24	)	)	PUNCT
ejpam-6627	11	25	,	,	PUNCT
ejpam-6627	11	26	manal.allabadi@uop.edu.jo	manal.allabadi@uop.edu.jo	X
ejpam-6627	11	27	(	(	PUNCT
ejpam-6627	11	28	m.	m.	NOUN
ejpam-6627	11	29	al	al	PROPN
ejpam-6627	11	30	-	-	PUNCT
ejpam-6627	11	31	labadi	labadi	NOUN
ejpam-6627	11	32	)	)	PUNCT
ejpam-6627	11	33	,	,	PUNCT
ejpam-6627	11	34	rajaa.alnaimi@uop.edu.jo	rajaa.alnaimi@uop.edu.jo	PROPN
ejpam-6627	11	35	(	(	PUNCT
ejpam-6627	11	36	r.	r.	PROPN
ejpam-6627	11	37	al	al	PROPN
ejpam-6627	11	38	-	-	PUNCT
ejpam-6627	11	39	naimi	naimi	PROPN
ejpam-6627	11	40	)	)	PUNCT
ejpam-6627	11	41	,	,	PUNCT
ejpam-6627	12	1	aamourah@su.edu.om	aamourah@su.edu.om	NOUN
ejpam-6627	12	2	(	(	PUNCT
ejpam-6627	12	3	a.	a.	NOUN
ejpam-6627	12	4	amourah	amourah	PROPN
ejpam-6627	12	5	)	)	PUNCT
ejpam-6627	12	6	,	,	PUNCT
ejpam-6627	12	7	tala.sasa@asu.edu.jo	tala.sasa@asu.edu.jo	PROPN
ejpam-6627	12	8	(	(	PUNCT
ejpam-6627	12	9	t.	t.	NOUN
ejpam-6627	12	10	sasa	sasa	PROPN
ejpam-6627	12	11	)	)	PUNCT
ejpam-6627	12	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6627	12	13	1	1	NUM
ejpam-6627	12	14	copyright	copyright	NOUN
ejpam-6627	12	15	:	:	PUNCT
ejpam-6627	13	1	©	©	PROPN
ejpam-6627	13	2	2025	2025	NUM
ejpam-6627	13	3	the	the	DET
ejpam-6627	13	4	author(s	author(s	NOUN
ejpam-6627	13	5	)	)	PUNCT
ejpam-6627	13	6	.	.	PUNCT
ejpam-6627	14	1	(	(	PUNCT
ejpam-6627	14	2	cc	cc	NOUN
ejpam-6627	14	3	by	by	ADP
ejpam-6627	14	4	-	-	PUNCT
ejpam-6627	14	5	nc	nc	PROPN
ejpam-6627	14	6	4.0	4.0	NUM
ejpam-6627	14	7	)	)	PUNCT
ejpam-6627	14	8	j.	j.	PROPN
ejpam-6627	14	9	oudetallah	oudetallah	PROPN
ejpam-6627	14	10	et	et	PROPN
ejpam-6627	14	11	al	al	PROPN
ejpam-6627	14	12	.	.	PUNCT
ejpam-6627	14	13	/	/	SYM
ejpam-6627	14	14	eur	eur	PROPN
ejpam-6627	14	15	.	.	PUNCT
ejpam-6627	15	1	j.	j.	PROPN
ejpam-6627	15	2	pure	pure	PROPN
ejpam-6627	15	3	appl	appl	PROPN
ejpam-6627	15	4	.	.	PROPN
ejpam-6627	15	5	math	math	PROPN
ejpam-6627	15	6	,	,	PUNCT
ejpam-6627	15	7	18	18	NUM
ejpam-6627	15	8	(	(	PUNCT
ejpam-6627	15	9	3	3	NUM
ejpam-6627	15	10	)	)	PUNCT
ejpam-6627	15	11	(	(	PUNCT
ejpam-6627	15	12	2025	2025	NUM
ejpam-6627	15	13	)	)	PUNCT
ejpam-6627	15	14	,	,	PUNCT
ejpam-6627	15	15	6627	6627	NUM
ejpam-6627	15	16	2	2	NUM
ejpam-6627	15	17	of	of	ADP
ejpam-6627	15	18	20	20	NUM
ejpam-6627	15	19	compact	compact	ADJ
ejpam-6627	15	20	spaces	space	NOUN
ejpam-6627	15	21	in	in	ADP
ejpam-6627	15	22	polyhedral	polyhedral	ADJ
ejpam-6627	15	23	settings	setting	NOUN
ejpam-6627	15	24	[	[	X
ejpam-6627	15	25	1	1	NUM
ejpam-6627	15	26	]	]	PUNCT
ejpam-6627	15	27	,	,	PUNCT
ejpam-6627	15	28	developments	development	NOUN
ejpam-6627	15	29	in	in	ADP
ejpam-6627	15	30	nth	nth	ADJ
ejpam-6627	15	31	-	-	ADJ
ejpam-6627	15	32	topological	topological	ADJ
ejpam-6627	15	33	frameworks	framework	NOUN
ejpam-6627	15	34	[	[	X
ejpam-6627	15	35	2	2	NUM
ejpam-6627	15	36	,	,	PUNCT
ejpam-6627	15	37	3	3	NUM
ejpam-6627	15	38	]	]	PUNCT
ejpam-6627	15	39	,	,	PUNCT
ejpam-6627	15	40	and	and	CCONJ
ejpam-6627	15	41	extensions	extension	NOUN
ejpam-6627	15	42	involving	involve	VERB
ejpam-6627	15	43	tri	tri	ADJ
ejpam-6627	15	44	-	-	ADJ
ejpam-6627	15	45	local	local	ADJ
ejpam-6627	15	46	compactness	compactness	NOUN
ejpam-6627	15	47	[	[	X
ejpam-6627	15	48	4	4	NUM
ejpam-6627	15	49	]	]	PUNCT
ejpam-6627	15	50	.	.	PUNCT
ejpam-6627	16	1	additionally	additionally	ADV
ejpam-6627	16	2	,	,	PUNCT
ejpam-6627	16	3	bitopological	bitopological	ADJ
ejpam-6627	16	4	environments	environment	NOUN
ejpam-6627	16	5	have	have	AUX
ejpam-6627	16	6	been	be	AUX
ejpam-6627	16	7	explored	explore	VERB
ejpam-6627	16	8	in	in	ADP
ejpam-6627	16	9	the	the	DET
ejpam-6627	16	10	context	context	NOUN
ejpam-6627	16	11	of	of	ADP
ejpam-6627	16	12	nearly	nearly	ADV
ejpam-6627	16	13	lindelöfness	lindelöfness	NOUN
ejpam-6627	16	14	[	[	X
ejpam-6627	16	15	5	5	NUM
ejpam-6627	16	16	]	]	PUNCT
ejpam-6627	16	17	,	,	PUNCT
ejpam-6627	16	18	while	while	SCONJ
ejpam-6627	16	19	related	related	ADJ
ejpam-6627	16	20	compactness	compactness	NOUN
ejpam-6627	16	21	conditions	condition	NOUN
ejpam-6627	16	22	such	such	ADJ
ejpam-6627	16	23	as	as	ADP
ejpam-6627	16	24	c	c	NOUN
ejpam-6627	16	25	-	-	PUNCT
ejpam-6627	16	26	compactness	compactness	NOUN
ejpam-6627	16	27	and	and	CCONJ
ejpam-6627	16	28	nigh	nigh	ADJ
ejpam-6627	16	29	-	-	PUNCT
ejpam-6627	16	30	openness	openness	NOUN
ejpam-6627	16	31	continue	continue	VERB
ejpam-6627	16	32	to	to	PART
ejpam-6627	16	33	enrich	enrich	VERB
ejpam-6627	16	34	the	the	DET
ejpam-6627	16	35	structural	structural	ADJ
ejpam-6627	16	36	understanding	understanding	NOUN
ejpam-6627	16	37	of	of	ADP
ejpam-6627	16	38	topological	topological	ADJ
ejpam-6627	16	39	and	and	CCONJ
ejpam-6627	16	40	bitopological	bitopological	ADJ
ejpam-6627	16	41	spaces	space	NOUN
ejpam-6627	17	1	[	[	X
ejpam-6627	17	2	6	6	NUM
ejpam-6627	17	3	,	,	PUNCT
ejpam-6627	17	4	7	7	NUM
ejpam-6627	17	5	]	]	PUNCT
ejpam-6627	17	6	.	.	PUNCT
ejpam-6627	18	1	separation	separation	NOUN
ejpam-6627	18	2	axioms	axiom	NOUN
ejpam-6627	18	3	constitute	constitute	VERB
ejpam-6627	18	4	one	one	NUM
ejpam-6627	18	5	of	of	ADP
ejpam-6627	18	6	the	the	DET
ejpam-6627	18	7	fundamental	fundamental	ADJ
ejpam-6627	18	8	tools	tool	NOUN
ejpam-6627	18	9	for	for	ADP
ejpam-6627	18	10	classifying	classify	VERB
ejpam-6627	18	11	topological	topological	ADJ
ejpam-6627	18	12	spaces	space	NOUN
ejpam-6627	18	13	in	in	ADP
ejpam-6627	18	14	general	general	ADJ
ejpam-6627	18	15	topology	topology	NOUN
ejpam-6627	18	16	,	,	PUNCT
ejpam-6627	18	17	as	as	SCONJ
ejpam-6627	18	18	extensively	extensively	ADV
ejpam-6627	18	19	discussed	discuss	VERB
ejpam-6627	18	20	in	in	ADP
ejpam-6627	18	21	the	the	DET
ejpam-6627	18	22	classical	classical	ADJ
ejpam-6627	18	23	texts	text	NOUN
ejpam-6627	18	24	by	by	ADP
ejpam-6627	18	25	engelking	engelke	VERB
ejpam-6627	18	26	[	[	X
ejpam-6627	18	27	8	8	NUM
ejpam-6627	18	28	]	]	PUNCT
ejpam-6627	18	29	and	and	CCONJ
ejpam-6627	18	30	willard	willard	NOUN
ejpam-6627	19	1	[	[	X
ejpam-6627	19	2	9	9	NUM
ejpam-6627	19	3	]	]	PUNCT
ejpam-6627	19	4	.	.	PUNCT
ejpam-6627	20	1	while	while	SCONJ
ejpam-6627	20	2	spaces	space	NOUN
ejpam-6627	20	3	satisfying	satisfy	VERB
ejpam-6627	20	4	the	the	DET
ejpam-6627	20	5	hausdorff	hausdorff	NOUN
ejpam-6627	20	6	condition	condition	NOUN
ejpam-6627	20	7	and	and	CCONJ
ejpam-6627	20	8	higher	high	ADJ
ejpam-6627	20	9	separation	separation	NOUN
ejpam-6627	20	10	axioms	axiom	NOUN
ejpam-6627	20	11	have	have	AUX
ejpam-6627	20	12	received	receive	VERB
ejpam-6627	20	13	extensive	extensive	ADJ
ejpam-6627	20	14	study	study	NOUN
ejpam-6627	20	15	,	,	PUNCT
ejpam-6627	20	16	weaker	weak	ADJ
ejpam-6627	20	17	separation	separation	NOUN
ejpam-6627	20	18	axioms	axiom	NOUN
ejpam-6627	20	19	have	have	AUX
ejpam-6627	20	20	garnered	garner	VERB
ejpam-6627	20	21	limited	limited	ADJ
ejpam-6627	20	22	attention	attention	NOUN
ejpam-6627	20	23	despite	despite	SCONJ
ejpam-6627	20	24	providing	provide	VERB
ejpam-6627	20	25	rich	rich	ADJ
ejpam-6627	20	26	mathematical	mathematical	ADJ
ejpam-6627	20	27	structures	structure	NOUN
ejpam-6627	20	28	and	and	CCONJ
ejpam-6627	20	29	interesting	interesting	ADJ
ejpam-6627	20	30	pathologies	pathology	NOUN
ejpam-6627	20	31	.	.	PUNCT
ejpam-6627	21	1	these	these	DET
ejpam-6627	21	2	modest	modest	ADJ
ejpam-6627	21	3	separation	separation	NOUN
ejpam-6627	21	4	requirements	requirement	NOUN
ejpam-6627	21	5	arise	arise	VERB
ejpam-6627	21	6	naturally	naturally	ADV
ejpam-6627	21	7	in	in	ADP
ejpam-6627	21	8	various	various	ADJ
ejpam-6627	21	9	mathematical	mathematical	ADJ
ejpam-6627	21	10	contexts	contexts	NOUN
ejpam-6627	21	11	including	include	VERB
ejpam-6627	21	12	algebraic	algebraic	ADJ
ejpam-6627	21	13	geometry	geometry	NOUN
ejpam-6627	21	14	,	,	PUNCT
ejpam-6627	21	15	functional	functional	ADJ
ejpam-6627	21	16	analysis	analysis	NOUN
ejpam-6627	21	17	,	,	PUNCT
ejpam-6627	21	18	and	and	CCONJ
ejpam-6627	21	19	theoretical	theoretical	ADJ
ejpam-6627	21	20	computer	computer	NOUN
ejpam-6627	21	21	science	science	NOUN
ejpam-6627	21	22	[	[	X
ejpam-6627	21	23	8	8	NUM
ejpam-6627	21	24	,	,	PUNCT
ejpam-6627	21	25	9	9	NUM
ejpam-6627	21	26	]	]	PUNCT
ejpam-6627	21	27	.	.	PUNCT
ejpam-6627	22	1	the	the	DET
ejpam-6627	22	2	conventional	conventional	ADJ
ejpam-6627	22	3	hierarchy	hierarchy	NOUN
ejpam-6627	22	4	of	of	ADP
ejpam-6627	22	5	separation	separation	NOUN
ejpam-6627	22	6	axioms	axiom	NOUN
ejpam-6627	22	7	begins	begin	VERB
ejpam-6627	22	8	with	with	ADP
ejpam-6627	22	9	t0	t0	PROPN
ejpam-6627	22	10	(	(	PUNCT
ejpam-6627	22	11	kolmogorov	kolmogorov	PROPN
ejpam-6627	22	12	)	)	PUNCT
ejpam-6627	22	13	spaces	space	NOUN
ejpam-6627	22	14	,	,	PUNCT
ejpam-6627	22	15	progresses	progress	VERB
ejpam-6627	22	16	through	through	ADP
ejpam-6627	22	17	t1	t1	PROPN
ejpam-6627	22	18	(	(	PUNCT
ejpam-6627	22	19	fréchet	fréchet	NOUN
ejpam-6627	22	20	)	)	PUNCT
ejpam-6627	22	21	,	,	PUNCT
ejpam-6627	22	22	and	and	CCONJ
ejpam-6627	22	23	continues	continue	VERB
ejpam-6627	22	24	to	to	ADP
ejpam-6627	22	25	t2	t2	PROPN
ejpam-6627	22	26	(	(	PUNCT
ejpam-6627	22	27	hausdorff	hausdorff	NOUN
ejpam-6627	22	28	)	)	PUNCT
ejpam-6627	22	29	spaces	space	NOUN
ejpam-6627	22	30	,	,	PUNCT
ejpam-6627	22	31	with	with	ADP
ejpam-6627	22	32	each	each	DET
ejpam-6627	22	33	axiom	axiom	NOUN
ejpam-6627	22	34	strengthening	strengthen	VERB
ejpam-6627	22	35	the	the	DET
ejpam-6627	22	36	previous	previous	ADJ
ejpam-6627	22	37	one	one	NUM
ejpam-6627	22	38	.	.	PUNCT
ejpam-6627	23	1	the	the	DET
ejpam-6627	23	2	relationships	relationship	NOUN
ejpam-6627	23	3	between	between	ADP
ejpam-6627	23	4	these	these	DET
ejpam-6627	23	5	separation	separation	NOUN
ejpam-6627	23	6	axioms	axiom	NOUN
ejpam-6627	23	7	become	become	VERB
ejpam-6627	23	8	particularly	particularly	ADV
ejpam-6627	23	9	intricate	intricate	ADJ
ejpam-6627	23	10	when	when	SCONJ
ejpam-6627	23	11	combined	combine	VERB
ejpam-6627	23	12	with	with	ADP
ejpam-6627	23	13	other	other	ADJ
ejpam-6627	23	14	topological	topological	ADJ
ejpam-6627	23	15	properties	property	NOUN
ejpam-6627	23	16	such	such	ADJ
ejpam-6627	23	17	as	as	ADP
ejpam-6627	23	18	urysohn	urysohn	PROPN
ejpam-6627	23	19	separation	separation	NOUN
ejpam-6627	23	20	,	,	PUNCT
ejpam-6627	23	21	sigma	sigma	PROPN
ejpam-6627	23	22	-	-	PUNCT
ejpam-6627	23	23	intersection	intersection	NOUN
ejpam-6627	23	24	closed	close	VERB
ejpam-6627	23	25	sets	set	NOUN
ejpam-6627	23	26	,	,	PUNCT
ejpam-6627	23	27	and	and	CCONJ
ejpam-6627	23	28	various	various	ADJ
ejpam-6627	23	29	compactness	compactness	NOUN
ejpam-6627	23	30	conditions	condition	NOUN
ejpam-6627	23	31	.	.	PUNCT
ejpam-6627	24	1	recent	recent	ADJ
ejpam-6627	24	2	work	work	NOUN
ejpam-6627	24	3	on	on	ADP
ejpam-6627	24	4	related	related	ADJ
ejpam-6627	24	5	topological	topological	ADJ
ejpam-6627	24	6	properties	property	NOUN
ejpam-6627	24	7	has	have	AUX
ejpam-6627	24	8	explored	explore	VERB
ejpam-6627	24	9	other	other	ADJ
ejpam-6627	24	10	forms	form	NOUN
ejpam-6627	24	11	of	of	ADP
ejpam-6627	24	12	generalization	generalization	NOUN
ejpam-6627	24	13	in	in	ADP
ejpam-6627	24	14	topological	topological	ADJ
ejpam-6627	24	15	spaces	space	NOUN
ejpam-6627	24	16	,	,	PUNCT
ejpam-6627	24	17	including	include	VERB
ejpam-6627	24	18	pairwise	pairwise	NOUN
ejpam-6627	24	19	expandable	expandable	ADJ
ejpam-6627	24	20	spaces	space	NOUN
ejpam-6627	24	21	[	[	X
ejpam-6627	24	22	10	10	NUM
ejpam-6627	24	23	]	]	PUNCT
ejpam-6627	24	24	and	and	CCONJ
ejpam-6627	24	25	various	various	ADJ
ejpam-6627	24	26	compactnessrelated	compactnessrelate	VERB
ejpam-6627	24	27	properties	property	NOUN
ejpam-6627	24	28	[	[	X
ejpam-6627	24	29	11	11	NUM
ejpam-6627	24	30	,	,	PUNCT
ejpam-6627	24	31	12	12	NUM
ejpam-6627	24	32	]	]	PUNCT
ejpam-6627	24	33	.	.	PUNCT
ejpam-6627	25	1	1.1	1.1	NUM
ejpam-6627	25	2	.	.	PUNCT
ejpam-6627	25	3	motivation	motivation	NOUN
ejpam-6627	25	4	and	and	CCONJ
ejpam-6627	25	5	applications	application	NOUN
ejpam-6627	25	6	the	the	DET
ejpam-6627	25	7	study	study	NOUN
ejpam-6627	25	8	of	of	ADP
ejpam-6627	25	9	intermediate	intermediate	ADJ
ejpam-6627	25	10	separation	separation	NOUN
ejpam-6627	25	11	axioms	axiom	NOUN
ejpam-6627	25	12	is	be	AUX
ejpam-6627	25	13	motivated	motivate	VERB
ejpam-6627	25	14	by	by	ADP
ejpam-6627	25	15	several	several	ADJ
ejpam-6627	25	16	important	important	ADJ
ejpam-6627	25	17	considerations	consideration	NOUN
ejpam-6627	25	18	:	:	PUNCT
ejpam-6627	25	19	•	•	NUM
ejpam-6627	25	20	domain	domain	NOUN
ejpam-6627	25	21	theory	theory	NOUN
ejpam-6627	25	22	applications	application	NOUN
ejpam-6627	25	23	:	:	PUNCT
ejpam-6627	25	24	in	in	ADP
ejpam-6627	25	25	theoretical	theoretical	ADJ
ejpam-6627	25	26	computer	computer	NOUN
ejpam-6627	25	27	science	science	NOUN
ejpam-6627	25	28	,	,	PUNCT
ejpam-6627	25	29	particularly	particularly	ADV
ejpam-6627	25	30	in	in	ADP
ejpam-6627	25	31	domain	domain	NOUN
ejpam-6627	25	32	theory	theory	NOUN
ejpam-6627	25	33	and	and	CCONJ
ejpam-6627	25	34	denotational	denotational	ADJ
ejpam-6627	25	35	semantics	semantic	NOUN
ejpam-6627	25	36	,	,	PUNCT
ejpam-6627	25	37	spaces	space	VERB
ejpam-6627	25	38	with	with	ADP
ejpam-6627	25	39	weak	weak	ADJ
ejpam-6627	25	40	separation	separation	NOUN
ejpam-6627	25	41	properties	property	NOUN
ejpam-6627	25	42	naturally	naturally	ADV
ejpam-6627	25	43	model	model	VERB
ejpam-6627	25	44	computational	computational	ADJ
ejpam-6627	25	45	processes	process	NOUN
ejpam-6627	25	46	where	where	SCONJ
ejpam-6627	25	47	complete	complete	ADJ
ejpam-6627	25	48	separation	separation	NOUN
ejpam-6627	25	49	of	of	ADP
ejpam-6627	25	50	points	point	NOUN
ejpam-6627	25	51	may	may	AUX
ejpam-6627	25	52	not	not	PART
ejpam-6627	25	53	be	be	AUX
ejpam-6627	25	54	achievable	achievable	ADJ
ejpam-6627	25	55	or	or	CCONJ
ejpam-6627	25	56	desirable	desirable	ADJ
ejpam-6627	25	57	,	,	PUNCT
ejpam-6627	25	58	as	as	SCONJ
ejpam-6627	25	59	comprehensively	comprehensively	ADV
ejpam-6627	25	60	treated	treat	VERB
ejpam-6627	25	61	by	by	ADP
ejpam-6627	25	62	gierz	gierz	PROPN
ejpam-6627	25	63	et	et	PROPN
ejpam-6627	25	64	al	al	PROPN
ejpam-6627	25	65	.	.	PUNCT
ejpam-6627	26	1	[	[	X
ejpam-6627	26	2	13	13	NUM
ejpam-6627	26	3	]	]	PUNCT
ejpam-6627	26	4	.	.	PUNCT
ejpam-6627	27	1	•	•	NUM
ejpam-6627	27	2	functional	functional	ADJ
ejpam-6627	27	3	analysis	analysis	NOUN
ejpam-6627	27	4	:	:	PUNCT
ejpam-6627	27	5	weak	weak	ADJ
ejpam-6627	27	6	topologies	topology	NOUN
ejpam-6627	27	7	on	on	ADP
ejpam-6627	27	8	infinite	infinite	ADJ
ejpam-6627	27	9	-	-	PUNCT
ejpam-6627	27	10	dimensional	dimensional	ADJ
ejpam-6627	27	11	spaces	space	NOUN
ejpam-6627	27	12	often	often	ADV
ejpam-6627	27	13	fail	fail	VERB
ejpam-6627	27	14	to	to	PART
ejpam-6627	27	15	be	be	AUX
ejpam-6627	27	16	hausdorff	hausdorff	NOUN
ejpam-6627	27	17	while	while	SCONJ
ejpam-6627	27	18	retaining	retain	VERB
ejpam-6627	27	19	other	other	ADJ
ejpam-6627	27	20	useful	useful	ADJ
ejpam-6627	27	21	separation	separation	NOUN
ejpam-6627	27	22	properties	property	NOUN
ejpam-6627	27	23	.	.	PUNCT
ejpam-6627	28	1	understanding	understand	VERB
ejpam-6627	28	2	these	these	DET
ejpam-6627	28	3	intermediate	intermediate	ADJ
ejpam-6627	28	4	conditions	condition	NOUN
ejpam-6627	28	5	is	be	AUX
ejpam-6627	28	6	crucial	crucial	ADJ
ejpam-6627	28	7	for	for	ADP
ejpam-6627	28	8	operator	operator	NOUN
ejpam-6627	28	9	theory	theory	NOUN
ejpam-6627	28	10	and	and	CCONJ
ejpam-6627	28	11	the	the	DET
ejpam-6627	28	12	study	study	NOUN
ejpam-6627	28	13	of	of	ADP
ejpam-6627	28	14	locally	locally	ADV
ejpam-6627	28	15	convex	convex	ADJ
ejpam-6627	28	16	spaces	space	NOUN
ejpam-6627	28	17	,	,	PUNCT
ejpam-6627	28	18	as	as	SCONJ
ejpam-6627	28	19	shown	show	VERB
ejpam-6627	28	20	in	in	ADP
ejpam-6627	28	21	the	the	DET
ejpam-6627	28	22	seminal	seminal	ADJ
ejpam-6627	28	23	work	work	NOUN
ejpam-6627	28	24	of	of	ADP
ejpam-6627	28	25	schaefer	schaefer	NOUN
ejpam-6627	28	26	and	and	CCONJ
ejpam-6627	28	27	wolff	wolff	NOUN
ejpam-6627	28	28	[	[	X
ejpam-6627	28	29	14	14	NUM
ejpam-6627	28	30	]	]	PUNCT
ejpam-6627	28	31	.	.	PUNCT
ejpam-6627	29	1	•	•	NUM
ejpam-6627	29	2	algebraic	algebraic	ADJ
ejpam-6627	29	3	geometry	geometry	NOUN
ejpam-6627	29	4	:	:	PUNCT
ejpam-6627	29	5	the	the	DET
ejpam-6627	29	6	zariski	zariski	NOUN
ejpam-6627	29	7	topology	topology	NOUN
ejpam-6627	29	8	on	on	ADP
ejpam-6627	29	9	algebraic	algebraic	ADJ
ejpam-6627	29	10	varieties	variety	NOUN
ejpam-6627	29	11	typically	typically	ADV
ejpam-6627	29	12	satisfies	satisfy	VERB
ejpam-6627	29	13	only	only	ADV
ejpam-6627	29	14	the	the	DET
ejpam-6627	29	15	t1	t1	PROPN
ejpam-6627	29	16	axiom	axiom	NOUN
ejpam-6627	29	17	,	,	PUNCT
ejpam-6627	29	18	making	make	VERB
ejpam-6627	29	19	the	the	DET
ejpam-6627	29	20	study	study	NOUN
ejpam-6627	29	21	of	of	ADP
ejpam-6627	29	22	spaces	space	NOUN
ejpam-6627	29	23	between	between	ADP
ejpam-6627	29	24	t1	t1	NOUN
ejpam-6627	29	25	and	and	CCONJ
ejpam-6627	29	26	t2	t2	NOUN
ejpam-6627	29	27	essential	essential	ADJ
ejpam-6627	29	28	for	for	ADP
ejpam-6627	29	29	understanding	understand	VERB
ejpam-6627	29	30	scheme	scheme	ADJ
ejpam-6627	29	31	-	-	PUNCT
ejpam-6627	29	32	theoretic	theoretic	ADJ
ejpam-6627	29	33	constructions	construction	NOUN
ejpam-6627	29	34	,	,	PUNCT
ejpam-6627	29	35	as	as	SCONJ
ejpam-6627	29	36	established	establish	VERB
ejpam-6627	29	37	in	in	ADP
ejpam-6627	29	38	hartshorne	hartshorne	PROPN
ejpam-6627	29	39	’s	’s	PART
ejpam-6627	29	40	foundational	foundational	ADJ
ejpam-6627	29	41	text	text	NOUN
ejpam-6627	30	1	[	[	X
ejpam-6627	30	2	15	15	NUM
ejpam-6627	30	3	]	]	PUNCT
ejpam-6627	30	4	.	.	PUNCT
ejpam-6627	31	1	•	•	NUM
ejpam-6627	31	2	convergence	convergence	NOUN
ejpam-6627	31	3	theory	theory	NOUN
ejpam-6627	31	4	:	:	PUNCT
ejpam-6627	31	5	sequential	sequential	ADJ
ejpam-6627	31	6	and	and	CCONJ
ejpam-6627	31	7	net	net	ADJ
ejpam-6627	31	8	convergence	convergence	NOUN
ejpam-6627	31	9	in	in	ADP
ejpam-6627	31	10	non	non	ADJ
ejpam-6627	31	11	-	-	ADJ
ejpam-6627	31	12	hausdorff	hausdorff	ADJ
ejpam-6627	31	13	spaces	space	NOUN
ejpam-6627	31	14	requires	require	VERB
ejpam-6627	31	15	careful	careful	ADJ
ejpam-6627	31	16	analysis	analysis	NOUN
ejpam-6627	31	17	of	of	ADP
ejpam-6627	31	18	intermediate	intermediate	ADJ
ejpam-6627	31	19	separation	separation	NOUN
ejpam-6627	31	20	conditions	condition	NOUN
ejpam-6627	31	21	to	to	PART
ejpam-6627	31	22	establish	establish	VERB
ejpam-6627	31	23	appropriate	appropriate	ADJ
ejpam-6627	31	24	convergence	convergence	NOUN
ejpam-6627	31	25	criteria	criterion	NOUN
ejpam-6627	31	26	,	,	PUNCT
ejpam-6627	31	27	following	follow	VERB
ejpam-6627	31	28	the	the	DET
ejpam-6627	31	29	classical	classical	ADJ
ejpam-6627	31	30	treatment	treatment	NOUN
ejpam-6627	31	31	by	by	ADP
ejpam-6627	31	32	kelley	kelley	NOUN
ejpam-6627	31	33	[	[	X
ejpam-6627	31	34	16	16	NUM
ejpam-6627	31	35	]	]	PUNCT
ejpam-6627	31	36	.	.	PUNCT
ejpam-6627	32	1	j.	j.	PROPN
ejpam-6627	32	2	oudetallah	oudetallah	PROPN
ejpam-6627	32	3	et	et	PROPN
ejpam-6627	32	4	al	al	PROPN
ejpam-6627	32	5	.	.	PUNCT
ejpam-6627	32	6	/	/	SYM
ejpam-6627	32	7	eur	eur	PROPN
ejpam-6627	32	8	.	.	PUNCT
ejpam-6627	33	1	j.	j.	PROPN
ejpam-6627	33	2	pure	pure	PROPN
ejpam-6627	33	3	appl	appl	PROPN
ejpam-6627	33	4	.	.	PROPN
ejpam-6627	33	5	math	math	PROPN
ejpam-6627	33	6	,	,	PUNCT
ejpam-6627	33	7	18	18	NUM
ejpam-6627	33	8	(	(	PUNCT
ejpam-6627	33	9	3	3	NUM
ejpam-6627	33	10	)	)	PUNCT
ejpam-6627	33	11	(	(	PUNCT
ejpam-6627	33	12	2025	2025	NUM
ejpam-6627	33	13	)	)	PUNCT
ejpam-6627	33	14	,	,	PUNCT
ejpam-6627	33	15	6627	6627	NUM
ejpam-6627	33	16	3	3	NUM
ejpam-6627	33	17	of	of	ADP
ejpam-6627	33	18	20	20	NUM
ejpam-6627	33	19	this	this	DET
ejpam-6627	33	20	paper	paper	NOUN
ejpam-6627	33	21	investigates	investigate	VERB
ejpam-6627	33	22	properties	property	NOUN
ejpam-6627	33	23	that	that	PRON
ejpam-6627	33	24	emerge	emerge	VERB
ejpam-6627	33	25	when	when	SCONJ
ejpam-6627	33	26	various	various	ADJ
ejpam-6627	33	27	weak	weak	ADJ
ejpam-6627	33	28	versions	version	NOUN
ejpam-6627	33	29	of	of	ADP
ejpam-6627	33	30	separation	separation	NOUN
ejpam-6627	33	31	axioms	axiom	NOUN
ejpam-6627	33	32	interact	interact	VERB
ejpam-6627	33	33	.	.	PUNCT
ejpam-6627	34	1	our	our	PRON
ejpam-6627	34	2	focus	focus	NOUN
ejpam-6627	34	3	centers	center	NOUN
ejpam-6627	34	4	on	on	ADP
ejpam-6627	34	5	spaces	space	NOUN
ejpam-6627	34	6	that	that	PRON
ejpam-6627	34	7	display	display	VERB
ejpam-6627	34	8	characteristic	characteristic	ADJ
ejpam-6627	34	9	properties	property	NOUN
ejpam-6627	34	10	of	of	ADP
ejpam-6627	34	11	higher	high	ADJ
ejpam-6627	34	12	separation	separation	NOUN
ejpam-6627	34	13	axioms	axiom	NOUN
ejpam-6627	34	14	while	while	SCONJ
ejpam-6627	34	15	explicitly	explicitly	ADV
ejpam-6627	34	16	failing	fail	VERB
ejpam-6627	34	17	to	to	PART
ejpam-6627	34	18	satisfy	satisfy	VERB
ejpam-6627	34	19	these	these	DET
ejpam-6627	34	20	stronger	strong	ADJ
ejpam-6627	34	21	conditions	condition	NOUN
ejpam-6627	34	22	.	.	PUNCT
ejpam-6627	35	1	we	we	PRON
ejpam-6627	35	2	particularly	particularly	ADV
ejpam-6627	35	3	examine	examine	VERB
ejpam-6627	35	4	when	when	SCONJ
ejpam-6627	35	5	t1	t1	PROPN
ejpam-6627	35	6	spaces	space	NOUN
ejpam-6627	35	7	have	have	VERB
ejpam-6627	35	8	sigma	sigma	NOUN
ejpam-6627	35	9	-	-	PUNCT
ejpam-6627	35	10	intersection	intersection	NOUN
ejpam-6627	35	11	closed	close	VERB
ejpam-6627	35	12	sets	set	NOUN
ejpam-6627	35	13	while	while	SCONJ
ejpam-6627	35	14	failing	fail	VERB
ejpam-6627	35	15	to	to	PART
ejpam-6627	35	16	be	be	AUX
ejpam-6627	35	17	metrizable	metrizable	ADJ
ejpam-6627	35	18	.	.	PUNCT
ejpam-6627	36	1	several	several	ADJ
ejpam-6627	36	2	motivations	motivation	NOUN
ejpam-6627	36	3	drive	drive	VERB
ejpam-6627	36	4	this	this	DET
ejpam-6627	36	5	research	research	NOUN
ejpam-6627	36	6	.	.	PUNCT
ejpam-6627	37	1	theoretically	theoretically	ADV
ejpam-6627	37	2	,	,	PUNCT
ejpam-6627	37	3	weaker	weak	ADJ
ejpam-6627	37	4	separation	separation	NOUN
ejpam-6627	37	5	axioms	axiom	NOUN
ejpam-6627	37	6	provide	provide	VERB
ejpam-6627	37	7	deeper	deep	ADJ
ejpam-6627	37	8	understanding	understanding	NOUN
ejpam-6627	37	9	of	of	ADP
ejpam-6627	37	10	topological	topological	ADJ
ejpam-6627	37	11	space	space	NOUN
ejpam-6627	37	12	structure	structure	NOUN
ejpam-6627	37	13	,	,	PUNCT
ejpam-6627	37	14	building	build	VERB
ejpam-6627	37	15	upon	upon	SCONJ
ejpam-6627	37	16	the	the	DET
ejpam-6627	37	17	foundational	foundational	ADJ
ejpam-6627	37	18	work	work	NOUN
ejpam-6627	37	19	of	of	ADP
ejpam-6627	37	20	munkres	munkre	NOUN
ejpam-6627	37	21	[	[	X
ejpam-6627	37	22	17	17	NUM
ejpam-6627	37	23	]	]	PUNCT
ejpam-6627	37	24	and	and	CCONJ
ejpam-6627	37	25	dugundji	dugundji	NOUN
ejpam-6627	38	1	[	[	X
ejpam-6627	38	2	18	18	NUM
ejpam-6627	38	3	]	]	PUNCT
ejpam-6627	38	4	.	.	PUNCT
ejpam-6627	39	1	these	these	DET
ejpam-6627	39	2	spaces	space	NOUN
ejpam-6627	39	3	offer	offer	VERB
ejpam-6627	39	4	fertile	fertile	ADJ
ejpam-6627	39	5	ground	ground	NOUN
ejpam-6627	39	6	for	for	ADP
ejpam-6627	39	7	new	new	ADJ
ejpam-6627	39	8	developments	development	NOUN
ejpam-6627	39	9	as	as	SCONJ
ejpam-6627	39	10	they	they	PRON
ejpam-6627	39	11	contain	contain	VERB
ejpam-6627	39	12	mixed	mixed	ADJ
ejpam-6627	39	13	separation	separation	NOUN
ejpam-6627	39	14	properties	property	NOUN
ejpam-6627	39	15	.	.	PUNCT
ejpam-6627	40	1	certain	certain	ADJ
ejpam-6627	40	2	applications	application	NOUN
ejpam-6627	40	3	in	in	ADP
ejpam-6627	40	4	domain	domain	NOUN
ejpam-6627	40	5	theory	theory	NOUN
ejpam-6627	40	6	and	and	CCONJ
ejpam-6627	40	7	theoretical	theoretical	ADJ
ejpam-6627	40	8	computer	computer	NOUN
ejpam-6627	40	9	science	science	NOUN
ejpam-6627	40	10	require	require	VERB
ejpam-6627	40	11	particular	particular	ADJ
ejpam-6627	40	12	combinations	combination	NOUN
ejpam-6627	40	13	of	of	ADP
ejpam-6627	40	14	separation	separation	NOUN
ejpam-6627	40	15	properties	property	NOUN
ejpam-6627	40	16	in	in	ADP
ejpam-6627	40	17	their	their	PRON
ejpam-6627	40	18	spaces	space	NOUN
ejpam-6627	40	19	.	.	PUNCT
ejpam-6627	41	1	our	our	PRON
ejpam-6627	41	2	approach	approach	NOUN
ejpam-6627	41	3	combines	combine	VERB
ejpam-6627	41	4	constructive	constructive	ADJ
ejpam-6627	41	5	and	and	CCONJ
ejpam-6627	41	6	analytical	analytical	ADJ
ejpam-6627	41	7	methods	method	NOUN
ejpam-6627	41	8	inspired	inspire	VERB
ejpam-6627	41	9	by	by	ADP
ejpam-6627	41	10	the	the	DET
ejpam-6627	41	11	systematic	systematic	ADJ
ejpam-6627	41	12	treatment	treatment	NOUN
ejpam-6627	41	13	in	in	ADP
ejpam-6627	41	14	bourbaki	bourbaki	NOUN
ejpam-6627	41	15	[	[	X
ejpam-6627	41	16	19	19	NUM
ejpam-6627	41	17	]	]	PUNCT
ejpam-6627	41	18	.	.	PUNCT
ejpam-6627	42	1	we	we	PRON
ejpam-6627	42	2	present	present	VERB
ejpam-6627	42	3	examples	example	NOUN
ejpam-6627	42	4	of	of	ADP
ejpam-6627	42	5	spaces	space	NOUN
ejpam-6627	42	6	satisfying	satisfy	VERB
ejpam-6627	42	7	certain	certain	ADJ
ejpam-6627	42	8	axioms	axiom	NOUN
ejpam-6627	42	9	while	while	SCONJ
ejpam-6627	42	10	failing	fail	VERB
ejpam-6627	42	11	others	other	NOUN
ejpam-6627	42	12	,	,	PUNCT
ejpam-6627	42	13	and	and	CCONJ
ejpam-6627	42	14	spaces	space	VERB
ejpam-6627	42	15	illustrating	illustrate	VERB
ejpam-6627	42	16	how	how	SCONJ
ejpam-6627	42	17	combinations	combination	NOUN
ejpam-6627	42	18	of	of	ADP
ejpam-6627	42	19	properties	property	NOUN
ejpam-6627	42	20	lead	lead	VERB
ejpam-6627	42	21	to	to	ADP
ejpam-6627	42	22	unexpected	unexpected	ADJ
ejpam-6627	42	23	implications	implication	NOUN
ejpam-6627	42	24	.	.	PUNCT
ejpam-6627	43	1	our	our	PRON
ejpam-6627	43	2	investigation	investigation	NOUN
ejpam-6627	43	3	yields	yield	VERB
ejpam-6627	43	4	several	several	ADJ
ejpam-6627	43	5	new	new	ADJ
ejpam-6627	43	6	characterizations	characterization	NOUN
ejpam-6627	43	7	of	of	ADP
ejpam-6627	43	8	such	such	ADJ
ejpam-6627	43	9	spaces	space	NOUN
ejpam-6627	43	10	using	use	VERB
ejpam-6627	43	11	tools	tool	NOUN
ejpam-6627	43	12	from	from	ADP
ejpam-6627	43	13	general	general	ADJ
ejpam-6627	43	14	topology	topology	NOUN
ejpam-6627	43	15	,	,	PUNCT
ejpam-6627	43	16	set	set	VERB
ejpam-6627	43	17	theory	theory	NOUN
ejpam-6627	43	18	,	,	PUNCT
ejpam-6627	43	19	and	and	CCONJ
ejpam-6627	43	20	functional	functional	ADJ
ejpam-6627	43	21	analysis	analysis	NOUN
ejpam-6627	43	22	.	.	PUNCT
ejpam-6627	44	1	the	the	DET
ejpam-6627	44	2	paper	paper	NOUN
ejpam-6627	44	3	is	be	AUX
ejpam-6627	44	4	organized	organize	VERB
ejpam-6627	44	5	as	as	SCONJ
ejpam-6627	44	6	follows	follow	VERB
ejpam-6627	44	7	.	.	PUNCT
ejpam-6627	45	1	section	section	NOUN
ejpam-6627	45	2	2	2	NUM
ejpam-6627	45	3	establishes	establish	VERB
ejpam-6627	45	4	preliminary	preliminary	ADJ
ejpam-6627	45	5	definitions	definition	NOUN
ejpam-6627	45	6	and	and	CCONJ
ejpam-6627	45	7	notation	notation	NOUN
ejpam-6627	45	8	.	.	PUNCT
ejpam-6627	46	1	section	section	NOUN
ejpam-6627	46	2	3	3	NUM
ejpam-6627	46	3	investigates	investigate	VERB
ejpam-6627	46	4	t1	t1	NOUN
ejpam-6627	46	5	spaces	space	VERB
ejpam-6627	46	6	whose	whose	DET
ejpam-6627	46	7	closed	close	VERB
ejpam-6627	46	8	sets	set	NOUN
ejpam-6627	46	9	are	be	AUX
ejpam-6627	46	10	sigma	sigma	NOUN
ejpam-6627	46	11	-	-	PUNCT
ejpam-6627	46	12	intersections	intersection	NOUN
ejpam-6627	46	13	,	,	PUNCT
ejpam-6627	46	14	examining	examine	VERB
ejpam-6627	46	15	when	when	SCONJ
ejpam-6627	46	16	such	such	ADJ
ejpam-6627	46	17	spaces	space	NOUN
ejpam-6627	46	18	can	can	AUX
ejpam-6627	46	19	be	be	AUX
ejpam-6627	46	20	metrizable	metrizable	ADJ
ejpam-6627	46	21	.	.	PUNCT
ejpam-6627	47	1	section	section	NOUN
ejpam-6627	47	2	4	4	NUM
ejpam-6627	47	3	analyzes	analyze	VERB
ejpam-6627	47	4	urysohn	urysohn	PROPN
ejpam-6627	47	5	spaces	space	NOUN
ejpam-6627	47	6	that	that	PRON
ejpam-6627	47	7	fail	fail	VERB
ejpam-6627	47	8	to	to	PART
ejpam-6627	47	9	be	be	AUX
ejpam-6627	47	10	hausdorff	hausdorff	NOUN
ejpam-6627	47	11	.	.	PUNCT
ejpam-6627	48	1	section	section	NOUN
ejpam-6627	48	2	5	5	NUM
ejpam-6627	48	3	introduces	introduce	NOUN
ejpam-6627	48	4	intermediate	intermediate	ADJ
ejpam-6627	48	5	separation	separation	NOUN
ejpam-6627	48	6	criteria	criterion	NOUN
ejpam-6627	48	7	between	between	ADP
ejpam-6627	48	8	established	establish	VERB
ejpam-6627	48	9	axioms	axiom	NOUN
ejpam-6627	48	10	.	.	PUNCT
ejpam-6627	49	1	section	section	NOUN
ejpam-6627	49	2	6	6	NUM
ejpam-6627	49	3	examines	examine	VERB
ejpam-6627	49	4	how	how	SCONJ
ejpam-6627	49	5	these	these	DET
ejpam-6627	49	6	properties	property	NOUN
ejpam-6627	49	7	behave	behave	VERB
ejpam-6627	49	8	under	under	ADP
ejpam-6627	49	9	topological	topological	ADJ
ejpam-6627	49	10	operations	operation	NOUN
ejpam-6627	49	11	.	.	PUNCT
ejpam-6627	50	1	section	section	NOUN
ejpam-6627	50	2	7	7	NUM
ejpam-6627	50	3	provides	provide	VERB
ejpam-6627	50	4	key	key	ADJ
ejpam-6627	50	5	counterexamples	counterexample	NOUN
ejpam-6627	50	6	delineating	delineate	VERB
ejpam-6627	50	7	the	the	DET
ejpam-6627	50	8	boundaries	boundary	NOUN
ejpam-6627	50	9	of	of	ADP
ejpam-6627	50	10	our	our	PRON
ejpam-6627	50	11	results	result	NOUN
ejpam-6627	50	12	.	.	PUNCT
ejpam-6627	51	1	we	we	PRON
ejpam-6627	51	2	conclude	conclude	VERB
ejpam-6627	51	3	in	in	ADP
ejpam-6627	51	4	section	section	NOUN
ejpam-6627	51	5	8	8	NUM
ejpam-6627	51	6	with	with	ADP
ejpam-6627	51	7	a	a	DET
ejpam-6627	51	8	summary	summary	NOUN
ejpam-6627	51	9	of	of	ADP
ejpam-6627	51	10	our	our	PRON
ejpam-6627	51	11	findings	finding	NOUN
ejpam-6627	51	12	and	and	CCONJ
ejpam-6627	51	13	directions	direction	NOUN
ejpam-6627	51	14	for	for	ADP
ejpam-6627	51	15	future	future	ADJ
ejpam-6627	51	16	research	research	NOUN
ejpam-6627	51	17	.	.	PUNCT
ejpam-6627	52	1	2	2	X
ejpam-6627	52	2	.	.	X
ejpam-6627	52	3	preliminaries	preliminary	NOUN
ejpam-6627	52	4	let	let	VERB
ejpam-6627	52	5	ω	ω	NOUN
ejpam-6627	52	6	be	be	AUX
ejpam-6627	52	7	a	a	DET
ejpam-6627	52	8	topological	topological	ADJ
ejpam-6627	52	9	space	space	NOUN
ejpam-6627	52	10	with	with	ADP
ejpam-6627	52	11	topology	topology	NOUN
ejpam-6627	52	12	τ	τ	PROPN
ejpam-6627	52	13	.	.	PUNCT
ejpam-6627	53	1	we	we	PRON
ejpam-6627	53	2	begin	begin	VERB
ejpam-6627	53	3	by	by	ADP
ejpam-6627	53	4	recalling	recall	VERB
ejpam-6627	53	5	the	the	DET
ejpam-6627	53	6	standard	standard	ADJ
ejpam-6627	53	7	separation	separation	NOUN
ejpam-6627	53	8	axioms	axiom	NOUN
ejpam-6627	53	9	and	and	CCONJ
ejpam-6627	53	10	establishing	establish	VERB
ejpam-6627	53	11	notation	notation	NOUN
ejpam-6627	53	12	for	for	ADP
ejpam-6627	53	13	the	the	DET
ejpam-6627	53	14	more	more	ADV
ejpam-6627	53	15	permissive	permissive	ADJ
ejpam-6627	53	16	conditions	condition	NOUN
ejpam-6627	53	17	that	that	PRON
ejpam-6627	53	18	will	will	AUX
ejpam-6627	53	19	be	be	AUX
ejpam-6627	53	20	our	our	PRON
ejpam-6627	53	21	primary	primary	ADJ
ejpam-6627	53	22	focus	focus	NOUN
ejpam-6627	53	23	,	,	PUNCT
ejpam-6627	53	24	following	follow	VERB
ejpam-6627	53	25	the	the	DET
ejpam-6627	53	26	comprehensive	comprehensive	ADJ
ejpam-6627	53	27	treatments	treatment	NOUN
ejpam-6627	53	28	in	in	ADP
ejpam-6627	53	29	[	[	X
ejpam-6627	53	30	17	17	NUM
ejpam-6627	53	31	,	,	PUNCT
ejpam-6627	53	32	18	18	NUM
ejpam-6627	53	33	]	]	PUNCT
ejpam-6627	53	34	.	.	PUNCT
ejpam-6627	54	1	•	•	NUM
ejpam-6627	54	2	a	a	DET
ejpam-6627	54	3	topological	topological	ADJ
ejpam-6627	54	4	space	space	NOUN
ejpam-6627	54	5	ω	ω	PROPN
ejpam-6627	54	6	is	be	AUX
ejpam-6627	54	7	a	a	DET
ejpam-6627	54	8	t0	t0	PROPN
ejpam-6627	54	9	space	space	NOUN
ejpam-6627	54	10	(	(	PUNCT
ejpam-6627	54	11	kolmogorov	kolmogorov	ADJ
ejpam-6627	54	12	space	space	NOUN
ejpam-6627	54	13	)	)	PUNCT
ejpam-6627	54	14	if	if	SCONJ
ejpam-6627	54	15	for	for	ADP
ejpam-6627	54	16	any	any	DET
ejpam-6627	54	17	distinct	distinct	ADJ
ejpam-6627	54	18	points	point	NOUN
ejpam-6627	54	19	α	α	NOUN
ejpam-6627	54	20	and	and	CCONJ
ejpam-6627	54	21	β	β	PROPN
ejpam-6627	54	22	in	in	ADP
ejpam-6627	54	23	ω	ω	PROPN
ejpam-6627	54	24	,	,	PUNCT
ejpam-6627	54	25	there	there	PRON
ejpam-6627	54	26	exists	exist	VERB
ejpam-6627	54	27	an	an	DET
ejpam-6627	54	28	open	open	ADJ
ejpam-6627	54	29	set	set	NOUN
ejpam-6627	54	30	containing	contain	VERB
ejpam-6627	54	31	exactly	exactly	ADV
ejpam-6627	54	32	one	one	NUM
ejpam-6627	54	33	of	of	ADP
ejpam-6627	54	34	these	these	DET
ejpam-6627	54	35	points	point	NOUN
ejpam-6627	54	36	.	.	PUNCT
ejpam-6627	55	1	equivalently	equivalently	ADV
ejpam-6627	55	2	,	,	PUNCT
ejpam-6627	55	3	for	for	ADP
ejpam-6627	55	4	any	any	DET
ejpam-6627	55	5	two	two	NUM
ejpam-6627	55	6	distinct	distinct	ADJ
ejpam-6627	55	7	points	point	NOUN
ejpam-6627	55	8	,	,	PUNCT
ejpam-6627	55	9	at	at	ADP
ejpam-6627	55	10	least	least	ADJ
ejpam-6627	55	11	one	one	NUM
ejpam-6627	55	12	of	of	ADP
ejpam-6627	55	13	them	they	PRON
ejpam-6627	55	14	has	have	VERB
ejpam-6627	55	15	a	a	DET
ejpam-6627	55	16	neighborhood	neighborhood	NOUN
ejpam-6627	55	17	not	not	PART
ejpam-6627	55	18	containing	contain	VERB
ejpam-6627	55	19	the	the	DET
ejpam-6627	55	20	other[17	other[17	PROPN
ejpam-6627	55	21	]	]	PUNCT
ejpam-6627	55	22	.	.	PUNCT
ejpam-6627	56	1	•	•	NUM
ejpam-6627	56	2	ω	ω	PROPN
ejpam-6627	56	3	is	be	AUX
ejpam-6627	56	4	a	a	DET
ejpam-6627	56	5	t1	t1	NOUN
ejpam-6627	56	6	space	space	NOUN
ejpam-6627	56	7	(	(	PUNCT
ejpam-6627	56	8	fréchet	fréchet	NOUN
ejpam-6627	56	9	space	space	NOUN
ejpam-6627	56	10	)	)	PUNCT
ejpam-6627	56	11	if	if	SCONJ
ejpam-6627	56	12	for	for	ADP
ejpam-6627	56	13	any	any	DET
ejpam-6627	56	14	distinct	distinct	ADJ
ejpam-6627	56	15	points	point	NOUN
ejpam-6627	56	16	α	α	NOUN
ejpam-6627	56	17	and	and	CCONJ
ejpam-6627	56	18	β	β	NOUN
ejpam-6627	56	19	,	,	PUNCT
ejpam-6627	56	20	there	there	PRON
ejpam-6627	56	21	exists	exist	VERB
ejpam-6627	56	22	an	an	DET
ejpam-6627	56	23	open	open	ADJ
ejpam-6627	56	24	set	set	NOUN
ejpam-6627	56	25	containing	contain	VERB
ejpam-6627	56	26	α	α	NOUN
ejpam-6627	56	27	but	but	CCONJ
ejpam-6627	56	28	not	not	PART
ejpam-6627	56	29	β	β	NOUN
ejpam-6627	56	30	.	.	PUNCT
ejpam-6627	57	1	equivalently	equivalently	ADV
ejpam-6627	57	2	,	,	PUNCT
ejpam-6627	57	3	every	every	DET
ejpam-6627	57	4	singleton	singleton	NOUN
ejpam-6627	57	5	set	set	NOUN
ejpam-6627	57	6	{	{	PUNCT
ejpam-6627	57	7	α	α	NOUN
ejpam-6627	57	8	}	}	PUNCT
ejpam-6627	57	9	is	be	AUX
ejpam-6627	57	10	closed	closed	ADJ
ejpam-6627	57	11	.	.	PUNCT
ejpam-6627	58	1	this	this	PRON
ejpam-6627	58	2	is	be	AUX
ejpam-6627	58	3	stronger	strong	ADJ
ejpam-6627	58	4	than	than	ADP
ejpam-6627	58	5	t0	t0	NOUN
ejpam-6627	58	6	as	as	SCONJ
ejpam-6627	58	7	it	it	PRON
ejpam-6627	58	8	requires	require	VERB
ejpam-6627	58	9	that	that	SCONJ
ejpam-6627	58	10	points	point	NOUN
ejpam-6627	58	11	can	can	AUX
ejpam-6627	58	12	be	be	AUX
ejpam-6627	58	13	separated	separate	VERB
ejpam-6627	58	14	in	in	ADP
ejpam-6627	58	15	both	both	DET
ejpam-6627	58	16	directions[17	directions[17	NOUN
ejpam-6627	58	17	]	]	X
ejpam-6627	58	18	.	.	PUNCT
ejpam-6627	59	1	•	•	NUM
ejpam-6627	59	2	ω	ω	PROPN
ejpam-6627	59	3	is	be	AUX
ejpam-6627	59	4	a	a	DET
ejpam-6627	59	5	t2	t2	NOUN
ejpam-6627	59	6	space	space	NOUN
ejpam-6627	59	7	(	(	PUNCT
ejpam-6627	59	8	or	or	CCONJ
ejpam-6627	59	9	hausdorff	hausdorff	NOUN
ejpam-6627	59	10	space	space	NOUN
ejpam-6627	59	11	)	)	PUNCT
ejpam-6627	59	12	if	if	SCONJ
ejpam-6627	59	13	for	for	ADP
ejpam-6627	59	14	any	any	DET
ejpam-6627	59	15	distinct	distinct	ADJ
ejpam-6627	59	16	points	point	NOUN
ejpam-6627	59	17	α	α	NOUN
ejpam-6627	59	18	and	and	CCONJ
ejpam-6627	59	19	β	β	NOUN
ejpam-6627	59	20	,	,	PUNCT
ejpam-6627	59	21	there	there	PRON
ejpam-6627	59	22	exist	exist	VERB
ejpam-6627	59	23	disjoint	disjoint	ADJ
ejpam-6627	59	24	open	open	ADJ
ejpam-6627	59	25	sets	set	NOUN
ejpam-6627	59	26	λ	λ	PROPN
ejpam-6627	59	27	and	and	CCONJ
ejpam-6627	59	28	γ	γ	NOUN
ejpam-6627	59	29	such	such	ADJ
ejpam-6627	59	30	that	that	SCONJ
ejpam-6627	59	31	α	α	PROPN
ejpam-6627	59	32	∈	∈	PROPN
ejpam-6627	59	33	λ	λ	PROPN
ejpam-6627	59	34	and	and	CCONJ
ejpam-6627	59	35	β	β	X
ejpam-6627	59	36	∈	∈	PROPN
ejpam-6627	59	37	γ[18	γ[18	PROPN
ejpam-6627	59	38	]	]	PUNCT
ejpam-6627	59	39	.	.	PUNCT
ejpam-6627	60	1	j.	j.	PROPN
ejpam-6627	60	2	oudetallah	oudetallah	PROPN
ejpam-6627	60	3	et	et	PROPN
ejpam-6627	60	4	al	al	PROPN
ejpam-6627	60	5	.	.	PUNCT
ejpam-6627	60	6	/	/	SYM
ejpam-6627	60	7	eur	eur	PROPN
ejpam-6627	60	8	.	.	PUNCT
ejpam-6627	61	1	j.	j.	PROPN
ejpam-6627	61	2	pure	pure	PROPN
ejpam-6627	61	3	appl	appl	PROPN
ejpam-6627	61	4	.	.	PROPN
ejpam-6627	61	5	math	math	PROPN
ejpam-6627	61	6	,	,	PUNCT
ejpam-6627	61	7	18	18	NUM
ejpam-6627	61	8	(	(	PUNCT
ejpam-6627	61	9	3	3	NUM
ejpam-6627	61	10	)	)	PUNCT
ejpam-6627	61	11	(	(	PUNCT
ejpam-6627	61	12	2025	2025	NUM
ejpam-6627	61	13	)	)	PUNCT
ejpam-6627	61	14	,	,	PUNCT
ejpam-6627	61	15	6627	6627	NUM
ejpam-6627	61	16	4	4	NUM
ejpam-6627	61	17	of	of	ADP
ejpam-6627	61	18	20	20	NUM
ejpam-6627	61	19	•	•	NUM
ejpam-6627	61	20	ω	ω	NOUN
ejpam-6627	61	21	is	be	AUX
ejpam-6627	61	22	a	a	DET
ejpam-6627	61	23	t2	t2	NOUN
ejpam-6627	61	24	1	1	NUM
ejpam-6627	61	25	2	2	NUM
ejpam-6627	61	26	space	space	NOUN
ejpam-6627	61	27	(	(	PUNCT
ejpam-6627	61	28	or	or	CCONJ
ejpam-6627	61	29	urysohn	urysohn	PROPN
ejpam-6627	61	30	space	space	NOUN
ejpam-6627	61	31	)	)	PUNCT
ejpam-6627	61	32	if	if	SCONJ
ejpam-6627	61	33	for	for	ADP
ejpam-6627	61	34	any	any	DET
ejpam-6627	61	35	distinct	distinct	ADJ
ejpam-6627	61	36	points	point	NOUN
ejpam-6627	61	37	α	α	NOUN
ejpam-6627	61	38	and	and	CCONJ
ejpam-6627	61	39	β	β	NOUN
ejpam-6627	61	40	,	,	PUNCT
ejpam-6627	61	41	there	there	PRON
ejpam-6627	61	42	exist	exist	VERB
ejpam-6627	61	43	open	open	ADJ
ejpam-6627	61	44	sets	set	NOUN
ejpam-6627	61	45	λ	λ	PROPN
ejpam-6627	61	46	and	and	CCONJ
ejpam-6627	61	47	γ	γ	NOUN
ejpam-6627	61	48	such	such	ADJ
ejpam-6627	61	49	that	that	SCONJ
ejpam-6627	61	50	α	α	PROPN
ejpam-6627	61	51	∈	∈	PROPN
ejpam-6627	61	52	λ	λ	PROPN
ejpam-6627	61	53	,	,	PUNCT
ejpam-6627	61	54	β	β	PROPN
ejpam-6627	61	55	∈	∈	PROPN
ejpam-6627	61	56	γ	γ	X
ejpam-6627	61	57	,	,	PUNCT
ejpam-6627	61	58	and	and	CCONJ
ejpam-6627	61	59	the	the	DET
ejpam-6627	61	60	closures	closure	NOUN
ejpam-6627	61	61	of	of	ADP
ejpam-6627	61	62	λ	λ	PROPN
ejpam-6627	61	63	and	and	CCONJ
ejpam-6627	61	64	γ	γ	NOUN
ejpam-6627	61	65	are	be	AUX
ejpam-6627	61	66	disjoint	disjoint	ADJ
ejpam-6627	61	67	,	,	PUNCT
ejpam-6627	61	68	i.e.	i.e.	X
ejpam-6627	61	69	,	,	PUNCT
ejpam-6627	61	70	λ	λ	PROPN
ejpam-6627	61	71	∩	∩	NOUN
ejpam-6627	61	72	(	(	PUNCT
ejpam-6627	61	73	γ	γ	NOUN
ejpam-6627	61	74	)	)	PUNCT
ejpam-6627	61	75	=	=	PUNCT
ejpam-6627	61	76	∅[18	∅[18	VERB
ejpam-6627	61	77	]	]	PUNCT
ejpam-6627	61	78	.	.	PUNCT
ejpam-6627	62	1	•	•	NUM
ejpam-6627	62	2	ω	ω	PROPN
ejpam-6627	62	3	is	be	AUX
ejpam-6627	62	4	a	a	DET
ejpam-6627	62	5	t3	t3	NOUN
ejpam-6627	62	6	space	space	NOUN
ejpam-6627	62	7	(	(	PUNCT
ejpam-6627	62	8	or	or	CCONJ
ejpam-6627	62	9	regular	regular	ADJ
ejpam-6627	62	10	space	space	NOUN
ejpam-6627	62	11	)	)	PUNCT
ejpam-6627	62	12	if	if	SCONJ
ejpam-6627	62	13	it	it	PRON
ejpam-6627	62	14	is	be	AUX
ejpam-6627	62	15	t1	t1	NOUN
ejpam-6627	62	16	and	and	CCONJ
ejpam-6627	62	17	for	for	ADP
ejpam-6627	62	18	every	every	DET
ejpam-6627	62	19	closed	close	VERB
ejpam-6627	62	20	set	set	VERB
ejpam-6627	62	21	φ	φ	NOUN
ejpam-6627	62	22	and	and	CCONJ
ejpam-6627	62	23	point	point	VERB
ejpam-6627	62	24	α	α	PRON
ejpam-6627	62	25	not	not	PART
ejpam-6627	62	26	in	in	ADP
ejpam-6627	62	27	φ	φ	NUM
ejpam-6627	62	28	,	,	PUNCT
ejpam-6627	62	29	there	there	PRON
ejpam-6627	62	30	exist	exist	VERB
ejpam-6627	62	31	disjoint	disjoint	ADJ
ejpam-6627	62	32	open	open	ADJ
ejpam-6627	62	33	sets	set	NOUN
ejpam-6627	62	34	λ	λ	PROPN
ejpam-6627	62	35	and	and	CCONJ
ejpam-6627	62	36	γ	γ	NOUN
ejpam-6627	62	37	such	such	ADJ
ejpam-6627	62	38	that	that	SCONJ
ejpam-6627	62	39	α	α	PROPN
ejpam-6627	62	40	∈	∈	PROPN
ejpam-6627	62	41	λ	λ	PROPN
ejpam-6627	62	42	and	and	CCONJ
ejpam-6627	62	43	φ	φ	PROPN
ejpam-6627	62	44	⊂	⊂	PROPN
ejpam-6627	62	45	γ[19	γ[19	PROPN
ejpam-6627	62	46	]	]	X
ejpam-6627	62	47	.	.	PUNCT
ejpam-6627	63	1	•	•	NUM
ejpam-6627	63	2	ω	ω	PROPN
ejpam-6627	63	3	is	be	AUX
ejpam-6627	63	4	a	a	DET
ejpam-6627	63	5	t4	t4	PROPN
ejpam-6627	63	6	space	space	NOUN
ejpam-6627	63	7	(	(	PUNCT
ejpam-6627	63	8	or	or	CCONJ
ejpam-6627	63	9	normal	normal	ADJ
ejpam-6627	63	10	space	space	NOUN
ejpam-6627	63	11	)	)	PUNCT
ejpam-6627	63	12	if	if	SCONJ
ejpam-6627	63	13	it	it	PRON
ejpam-6627	63	14	is	be	AUX
ejpam-6627	63	15	t1	t1	NOUN
ejpam-6627	63	16	and	and	CCONJ
ejpam-6627	63	17	for	for	ADP
ejpam-6627	63	18	any	any	DET
ejpam-6627	63	19	disjoint	disjoint	NOUN
ejpam-6627	63	20	closed	close	VERB
ejpam-6627	63	21	sets	set	NOUN
ejpam-6627	63	22	φ	φ	PROPN
ejpam-6627	63	23	and	and	CCONJ
ejpam-6627	63	24	ψ	ψ	NOUN
ejpam-6627	63	25	,	,	PUNCT
ejpam-6627	63	26	there	there	PRON
ejpam-6627	63	27	exist	exist	VERB
ejpam-6627	63	28	disjoint	disjoint	ADJ
ejpam-6627	63	29	open	open	ADJ
ejpam-6627	63	30	sets	set	NOUN
ejpam-6627	63	31	λ	λ	PROPN
ejpam-6627	63	32	and	and	CCONJ
ejpam-6627	63	33	γ	γ	NOUN
ejpam-6627	63	34	such	such	ADJ
ejpam-6627	63	35	that	that	SCONJ
ejpam-6627	63	36	φ	φ	PROPN
ejpam-6627	63	37	⊂	⊂	PROPN
ejpam-6627	63	38	λ	λ	X
ejpam-6627	63	39	and	and	CCONJ
ejpam-6627	63	40	ψ	ψ	X
ejpam-6627	63	41	⊂	⊂	X
ejpam-6627	63	42	γ[19	γ[19	X
ejpam-6627	63	43	]	]	X
ejpam-6627	63	44	.	.	PUNCT
ejpam-6627	64	1	a	a	DET
ejpam-6627	64	2	subset	subset	NOUN
ejpam-6627	64	3	is	be	AUX
ejpam-6627	64	4	a	a	DET
ejpam-6627	64	5	sigma	sigma	ADJ
ejpam-6627	64	6	-	-	PUNCT
ejpam-6627	64	7	intersection	intersection	NOUN
ejpam-6627	64	8	subset	subset	NOUN
ejpam-6627	64	9	if	if	SCONJ
ejpam-6627	64	10	it	it	PRON
ejpam-6627	64	11	can	can	AUX
ejpam-6627	64	12	be	be	AUX
ejpam-6627	64	13	expressed	express	VERB
ejpam-6627	64	14	as	as	ADP
ejpam-6627	64	15	a	a	DET
ejpam-6627	64	16	countable	countable	ADJ
ejpam-6627	64	17	intersection	intersection	NOUN
ejpam-6627	64	18	of	of	ADP
ejpam-6627	64	19	open	open	ADJ
ejpam-6627	64	20	sets	set	NOUN
ejpam-6627	64	21	.	.	PUNCT
ejpam-6627	65	1	a	a	DET
ejpam-6627	65	2	topological	topological	ADJ
ejpam-6627	65	3	space	space	NOUN
ejpam-6627	65	4	has	have	VERB
ejpam-6627	65	5	the	the	DET
ejpam-6627	65	6	property	property	NOUN
ejpam-6627	65	7	that	that	PRON
ejpam-6627	65	8	”	"	PUNCT
ejpam-6627	65	9	closed	closed	ADJ
ejpam-6627	65	10	sets	set	NOUN
ejpam-6627	65	11	are	be	AUX
ejpam-6627	65	12	sigma	sigma	NOUN
ejpam-6627	65	13	-	-	PUNCT
ejpam-6627	65	14	intersections	intersection	NOUN
ejpam-6627	65	15	”	"	PUNCT
ejpam-6627	65	16	if	if	SCONJ
ejpam-6627	65	17	every	every	DET
ejpam-6627	65	18	closed	closed	ADJ
ejpam-6627	65	19	set	set	NOUN
ejpam-6627	65	20	can	can	AUX
ejpam-6627	65	21	be	be	AUX
ejpam-6627	65	22	represented	represent	VERB
ejpam-6627	65	23	as	as	ADP
ejpam-6627	65	24	a	a	DET
ejpam-6627	65	25	countable	countable	ADJ
ejpam-6627	65	26	intersection	intersection	NOUN
ejpam-6627	65	27	of	of	ADP
ejpam-6627	65	28	open	open	ADJ
ejpam-6627	65	29	sets	set	NOUN
ejpam-6627	65	30	,	,	PUNCT
ejpam-6627	65	31	a	a	DET
ejpam-6627	65	32	concept	concept	NOUN
ejpam-6627	65	33	extensively	extensively	ADV
ejpam-6627	65	34	studied	study	VERB
ejpam-6627	65	35	in	in	ADP
ejpam-6627	65	36	oxtoby	oxtoby	NOUN
ejpam-6627	65	37	’s	’s	PART
ejpam-6627	65	38	work	work	NOUN
ejpam-6627	65	39	on	on	ADP
ejpam-6627	65	40	measure	measure	NOUN
ejpam-6627	65	41	and	and	CCONJ
ejpam-6627	65	42	category	category	NOUN
ejpam-6627	65	43	[	[	X
ejpam-6627	65	44	20	20	NUM
ejpam-6627	65	45	]	]	PUNCT
ejpam-6627	65	46	.	.	PUNCT
ejpam-6627	66	1	t0	t0	PROPN
ejpam-6627	66	2	(	(	PUNCT
ejpam-6627	66	3	kolmogorov	kolmogorov	PROPN
ejpam-6627	66	4	)	)	PUNCT
ejpam-6627	66	5	t1	t1	NOUN
ejpam-6627	66	6	(	(	PUNCT
ejpam-6627	66	7	fréchet	fréchet	NOUN
ejpam-6627	66	8	)	)	PUNCT
ejpam-6627	66	9	quasihausdorff	quasihausdorff	VERB
ejpam-6627	66	10	diffusely	diffusely	ADV
ejpam-6627	66	11	hausdorff	hausdorff	NOUN
ejpam-6627	66	12	t1	t1	NOUN
ejpam-6627	66	13	with	with	ADP
ejpam-6627	66	14	closed	close	VERB
ejpam-6627	66	15	as	as	ADP
ejpam-6627	66	16	σ	σ	NOUN
ejpam-6627	66	17	-	-	PUNCT
ejpam-6627	66	18	intersections	intersection	NOUN
ejpam-6627	66	19	sequentially	sequentially	ADV
ejpam-6627	66	20	hausdorff	hausdorff	NOUN
ejpam-6627	66	21	t2	t2	PROPN
ejpam-6627	66	22	(	(	PUNCT
ejpam-6627	66	23	hausdorff	hausdorff	NOUN
ejpam-6627	66	24	)	)	PUNCT
ejpam-6627	66	25	t2	t2	NOUN
ejpam-6627	66	26	1	1	NUM
ejpam-6627	66	27	2	2	NUM
ejpam-6627	66	28	(	(	PUNCT
ejpam-6627	66	29	urysohn	urysohn	PROPN
ejpam-6627	66	30	)	)	PUNCT
ejpam-6627	66	31	regular	regular	ADJ
ejpam-6627	66	32	(	(	PUNCT
ejpam-6627	66	33	t3	t3	NOUN
ejpam-6627	66	34	)	)	PUNCT
ejpam-6627	66	35	normal	normal	ADJ
ejpam-6627	66	36	(	(	PUNCT
ejpam-6627	66	37	t4	t4	PROPN
ejpam-6627	66	38	)	)	PUNCT
ejpam-6627	66	39	completely	completely	ADV
ejpam-6627	66	40	regular	regular	ADJ
ejpam-6627	66	41	metrizable	metrizable	ADJ
ejpam-6627	66	42	interrelationships	interrelationship	NOUN
ejpam-6627	66	43	between	between	ADP
ejpam-6627	66	44	separation	separation	NOUN
ejpam-6627	66	45	axioms	axiom	NOUN
ejpam-6627	66	46	figure	figure	VERB
ejpam-6627	66	47	1	1	NUM
ejpam-6627	66	48	:	:	PUNCT
ejpam-6627	66	49	hierarchy	hierarchy	NOUN
ejpam-6627	66	50	of	of	ADP
ejpam-6627	66	51	separation	separation	NOUN
ejpam-6627	66	52	axioms	axiom	NOUN
ejpam-6627	66	53	including	include	VERB
ejpam-6627	66	54	the	the	DET
ejpam-6627	66	55	intermediate	intermediate	ADJ
ejpam-6627	66	56	conditions	condition	NOUN
ejpam-6627	66	57	studied	study	VERB
ejpam-6627	66	58	in	in	ADP
ejpam-6627	66	59	this	this	DET
ejpam-6627	66	60	paper	paper	NOUN
ejpam-6627	66	61	.	.	PUNCT
ejpam-6627	67	1	solid	solid	ADJ
ejpam-6627	67	2	arrows	arrow	NOUN
ejpam-6627	67	3	represent	represent	VERB
ejpam-6627	67	4	direct	direct	ADJ
ejpam-6627	67	5	implications	implication	NOUN
ejpam-6627	67	6	,	,	PUNCT
ejpam-6627	67	7	while	while	SCONJ
ejpam-6627	67	8	dashed	dash	VERB
ejpam-6627	67	9	arrows	arrow	NOUN
ejpam-6627	67	10	indicate	indicate	VERB
ejpam-6627	67	11	implications	implication	NOUN
ejpam-6627	67	12	that	that	PRON
ejpam-6627	67	13	require	require	VERB
ejpam-6627	67	14	additional	additional	ADJ
ejpam-6627	67	15	conditions	condition	NOUN
ejpam-6627	67	16	.	.	PUNCT
ejpam-6627	68	1	this	this	DET
ejpam-6627	68	2	diagram	diagram	NOUN
ejpam-6627	68	3	illustrates	illustrate	VERB
ejpam-6627	68	4	the	the	DET
ejpam-6627	68	5	complex	complex	ADJ
ejpam-6627	68	6	relationships	relationship	NOUN
ejpam-6627	68	7	between	between	ADP
ejpam-6627	68	8	classical	classical	ADJ
ejpam-6627	68	9	and	and	CCONJ
ejpam-6627	68	10	intermediate	intermediate	ADJ
ejpam-6627	68	11	separation	separation	NOUN
ejpam-6627	68	12	properties	property	NOUN
ejpam-6627	68	13	.	.	PUNCT
ejpam-6627	69	1	for	for	ADP
ejpam-6627	69	2	our	our	PRON
ejpam-6627	69	3	purposes	purpose	NOUN
ejpam-6627	69	4	,	,	PUNCT
ejpam-6627	69	5	we	we	PRON
ejpam-6627	69	6	introduce	introduce	VERB
ejpam-6627	69	7	several	several	ADJ
ejpam-6627	69	8	intermediate	intermediate	ADJ
ejpam-6627	69	9	separation	separation	NOUN
ejpam-6627	69	10	conditions	condition	NOUN
ejpam-6627	69	11	that	that	PRON
ejpam-6627	69	12	lie	lie	VERB
ejpam-6627	69	13	between	between	ADP
ejpam-6627	69	14	the	the	DET
ejpam-6627	69	15	standard	standard	ADJ
ejpam-6627	69	16	axioms	axiom	NOUN
ejpam-6627	69	17	:	:	PUNCT
ejpam-6627	69	18	j.	j.	PROPN
ejpam-6627	69	19	oudetallah	oudetallah	PROPN
ejpam-6627	69	20	et	et	PROPN
ejpam-6627	69	21	al	al	PROPN
ejpam-6627	69	22	.	.	PUNCT
ejpam-6627	69	23	/	/	SYM
ejpam-6627	69	24	eur	eur	PROPN
ejpam-6627	69	25	.	.	PUNCT
ejpam-6627	70	1	j.	j.	PROPN
ejpam-6627	70	2	pure	pure	PROPN
ejpam-6627	70	3	appl	appl	PROPN
ejpam-6627	70	4	.	.	PROPN
ejpam-6627	70	5	math	math	PROPN
ejpam-6627	70	6	,	,	PUNCT
ejpam-6627	70	7	18	18	NUM
ejpam-6627	70	8	(	(	PUNCT
ejpam-6627	70	9	3	3	NUM
ejpam-6627	70	10	)	)	PUNCT
ejpam-6627	70	11	(	(	PUNCT
ejpam-6627	70	12	2025	2025	NUM
ejpam-6627	70	13	)	)	PUNCT
ejpam-6627	70	14	,	,	PUNCT
ejpam-6627	70	15	6627	6627	NUM
ejpam-6627	70	16	5	5	NUM
ejpam-6627	70	17	of	of	ADP
ejpam-6627	70	18	20	20	NUM
ejpam-6627	70	19	•	•	NOUN
ejpam-6627	70	20	we	we	PRON
ejpam-6627	70	21	say	say	VERB
ejpam-6627	70	22	ω	ω	NOUN
ejpam-6627	70	23	is	be	AUX
ejpam-6627	70	24	weakly	weakly	ADJ
ejpam-6627	70	25	hausdorff	hausdorff	NOUN
ejpam-6627	70	26	if	if	SCONJ
ejpam-6627	70	27	the	the	DET
ejpam-6627	70	28	image	image	NOUN
ejpam-6627	70	29	of	of	ADP
ejpam-6627	70	30	every	every	DET
ejpam-6627	70	31	continuous	continuous	ADJ
ejpam-6627	70	32	map	map	NOUN
ejpam-6627	70	33	from	from	ADP
ejpam-6627	70	34	a	a	DET
ejpam-6627	70	35	compact	compact	ADJ
ejpam-6627	70	36	hausdorff	hausdorff	NOUN
ejpam-6627	70	37	space	space	NOUN
ejpam-6627	70	38	into	into	ADP
ejpam-6627	70	39	ω	ω	PROPN
ejpam-6627	70	40	is	be	AUX
ejpam-6627	70	41	closed	closed	ADJ
ejpam-6627	70	42	,	,	PUNCT
ejpam-6627	70	43	following	follow	VERB
ejpam-6627	70	44	mccoy	mccoy	PROPN
ejpam-6627	70	45	and	and	CCONJ
ejpam-6627	70	46	ntantu	ntantu	PROPN
ejpam-6627	70	47	’s	’s	PART
ejpam-6627	70	48	treatment	treatment	NOUN
ejpam-6627	70	49	of	of	ADP
ejpam-6627	70	50	function	function	NOUN
ejpam-6627	70	51	space	space	NOUN
ejpam-6627	70	52	topologies	topology	NOUN
ejpam-6627	70	53	[	[	X
ejpam-6627	70	54	21	21	NUM
ejpam-6627	70	55	]	]	PUNCT
ejpam-6627	70	56	.	.	PUNCT
ejpam-6627	71	1	•	•	NUM
ejpam-6627	71	2	ω	ω	PROPN
ejpam-6627	71	3	is	be	AUX
ejpam-6627	71	4	kc	kc	NOUN
ejpam-6627	71	5	if	if	SCONJ
ejpam-6627	71	6	every	every	DET
ejpam-6627	71	7	compact	compact	ADJ
ejpam-6627	71	8	subset	subset	NOUN
ejpam-6627	71	9	is	be	AUX
ejpam-6627	71	10	closed[21	closed[21	VERB
ejpam-6627	71	11	]	]	PUNCT
ejpam-6627	71	12	.	.	PUNCT
ejpam-6627	72	1	•	•	NUM
ejpam-6627	72	2	ω	ω	PROPN
ejpam-6627	72	3	is	be	AUX
ejpam-6627	72	4	us	we	PRON
ejpam-6627	72	5	(	(	PUNCT
ejpam-6627	72	6	uniquely	uniquely	ADV
ejpam-6627	72	7	separable	separable	ADJ
ejpam-6627	72	8	)	)	PUNCT
ejpam-6627	72	9	if	if	SCONJ
ejpam-6627	72	10	for	for	ADP
ejpam-6627	72	11	any	any	DET
ejpam-6627	72	12	distinct	distinct	ADJ
ejpam-6627	72	13	points	point	NOUN
ejpam-6627	72	14	α	α	NOUN
ejpam-6627	72	15	and	and	CCONJ
ejpam-6627	72	16	β	β	NOUN
ejpam-6627	72	17	,	,	PUNCT
ejpam-6627	72	18	there	there	PRON
ejpam-6627	72	19	exists	exist	VERB
ejpam-6627	72	20	an	an	DET
ejpam-6627	72	21	open	open	ADJ
ejpam-6627	72	22	set	set	NOUN
ejpam-6627	72	23	λ	λ	INTJ
ejpam-6627	72	24	such	such	ADJ
ejpam-6627	72	25	that	that	SCONJ
ejpam-6627	72	26	either	either	ADV
ejpam-6627	72	27	(	(	PUNCT
ejpam-6627	72	28	α	α	NOUN
ejpam-6627	72	29	∈	∈	PROPN
ejpam-6627	72	30	λ	λ	PROPN
ejpam-6627	72	31	and	and	CCONJ
ejpam-6627	72	32	β	β	X
ejpam-6627	72	33	/∈	/∈	PUNCT
ejpam-6627	72	34	(	(	PUNCT
ejpam-6627	72	35	λ	λ	NOUN
ejpam-6627	72	36	)	)	PUNCT
ejpam-6627	72	37	)	)	PUNCT
ejpam-6627	72	38	or	or	CCONJ
ejpam-6627	72	39	(	(	PUNCT
ejpam-6627	72	40	β	β	X
ejpam-6627	72	41	∈	∈	PROPN
ejpam-6627	72	42	λ	λ	PROPN
ejpam-6627	72	43	and	and	CCONJ
ejpam-6627	72	44	α	α	NOUN
ejpam-6627	72	45	/∈	/∈	PUNCT
ejpam-6627	72	46	(	(	PUNCT
ejpam-6627	72	47	λ	λ	NOUN
ejpam-6627	72	48	)	)	PUNCT
ejpam-6627	72	49	)	)	PUNCT
ejpam-6627	72	50	,	,	PUNCT
ejpam-6627	72	51	a	a	DET
ejpam-6627	72	52	concept	concept	NOUN
ejpam-6627	72	53	developed	develop	VERB
ejpam-6627	72	54	by	by	ADP
ejpam-6627	72	55	arens	aren	NOUN
ejpam-6627	72	56	and	and	CCONJ
ejpam-6627	72	57	dugundji	dugundji	VERB
ejpam-6627	72	58	in	in	ADP
ejpam-6627	72	59	their	their	PRON
ejpam-6627	72	60	study	study	NOUN
ejpam-6627	72	61	of	of	ADP
ejpam-6627	72	62	function	function	NOUN
ejpam-6627	72	63	spaces	space	NOUN
ejpam-6627	72	64	[	[	X
ejpam-6627	72	65	22	22	NUM
ejpam-6627	72	66	]	]	PUNCT
ejpam-6627	72	67	.	.	PUNCT
ejpam-6627	73	1	•	•	NUM
ejpam-6627	73	2	ω	ω	NOUN
ejpam-6627	73	3	is	be	AUX
ejpam-6627	73	4	functionally	functionally	ADV
ejpam-6627	73	5	hausdorff	hausdorff	ADJ
ejpam-6627	73	6	if	if	SCONJ
ejpam-6627	73	7	for	for	ADP
ejpam-6627	73	8	any	any	DET
ejpam-6627	73	9	distinct	distinct	ADJ
ejpam-6627	73	10	points	point	NOUN
ejpam-6627	73	11	α	α	NOUN
ejpam-6627	73	12	and	and	CCONJ
ejpam-6627	73	13	β	β	NOUN
ejpam-6627	73	14	,	,	PUNCT
ejpam-6627	73	15	there	there	PRON
ejpam-6627	73	16	exists	exist	VERB
ejpam-6627	73	17	a	a	DET
ejpam-6627	73	18	continuous	continuous	ADJ
ejpam-6627	73	19	function	function	NOUN
ejpam-6627	73	20	f	f	NOUN
ejpam-6627	74	1	:	:	PUNCT
ejpam-6627	74	2	ω	ω	X
ejpam-6627	74	3	→	→	PUNCT
ejpam-6627	75	1	[	[	X
ejpam-6627	75	2	0	0	NUM
ejpam-6627	75	3	,	,	PUNCT
ejpam-6627	75	4	1	1	NUM
ejpam-6627	75	5	]	]	PUNCT
ejpam-6627	75	6	such	such	ADJ
ejpam-6627	75	7	that	that	DET
ejpam-6627	75	8	f(α	f(α	NOUN
ejpam-6627	75	9	)	)	PUNCT
ejpam-6627	75	10	=	=	SYM
ejpam-6627	75	11	0	0	NUM
ejpam-6627	75	12	and	and	CCONJ
ejpam-6627	75	13	f(β	f(β	PROPN
ejpam-6627	75	14	)	)	PUNCT
ejpam-6627	76	1	=	=	SYM
ejpam-6627	76	2	1	1	NUM
ejpam-6627	76	3	,	,	PUNCT
ejpam-6627	76	4	following	follow	VERB
ejpam-6627	76	5	the	the	DET
ejpam-6627	76	6	functional	functional	ADJ
ejpam-6627	76	7	approach	approach	NOUN
ejpam-6627	76	8	in	in	ADP
ejpam-6627	76	9	gillman	gillman	PROPN
ejpam-6627	76	10	and	and	CCONJ
ejpam-6627	76	11	jerison	jerison	PROPN
ejpam-6627	76	12	’s	’s	PART
ejpam-6627	76	13	work	work	NOUN
ejpam-6627	76	14	on	on	ADP
ejpam-6627	76	15	rings	ring	NOUN
ejpam-6627	76	16	of	of	ADP
ejpam-6627	76	17	continuous	continuous	ADJ
ejpam-6627	76	18	functions	function	NOUN
ejpam-6627	76	19	[	[	X
ejpam-6627	76	20	23	23	NUM
ejpam-6627	76	21	]	]	PUNCT
ejpam-6627	76	22	.	.	PUNCT
ejpam-6627	77	1	3	3	X
ejpam-6627	77	2	.	.	X
ejpam-6627	77	3	fréchet	fréchet	NOUN
ejpam-6627	77	4	spaces	space	NOUN
ejpam-6627	77	5	with	with	ADP
ejpam-6627	77	6	closed	closed	ADJ
ejpam-6627	77	7	sets	set	NOUN
ejpam-6627	77	8	as	as	ADP
ejpam-6627	77	9	sigma	sigma	NOUN
ejpam-6627	77	10	-	-	PUNCT
ejpam-6627	77	11	intersections	intersection	NOUN
ejpam-6627	77	12	in	in	ADP
ejpam-6627	77	13	this	this	DET
ejpam-6627	77	14	section	section	NOUN
ejpam-6627	77	15	,	,	PUNCT
ejpam-6627	77	16	we	we	PRON
ejpam-6627	77	17	examine	examine	VERB
ejpam-6627	77	18	t1	t1	NOUN
ejpam-6627	77	19	spaces	space	VERB
ejpam-6627	77	20	where	where	SCONJ
ejpam-6627	77	21	all	all	DET
ejpam-6627	77	22	closed	closed	ADJ
ejpam-6627	77	23	sets	set	NOUN
ejpam-6627	77	24	are	be	AUX
ejpam-6627	77	25	sigma	sigma	NOUN
ejpam-6627	77	26	-	-	PUNCT
ejpam-6627	77	27	intersections	intersection	NOUN
ejpam-6627	77	28	,	,	PUNCT
ejpam-6627	77	29	focusing	focus	VERB
ejpam-6627	77	30	on	on	ADP
ejpam-6627	77	31	conditions	condition	NOUN
ejpam-6627	77	32	that	that	PRON
ejpam-6627	77	33	prevent	prevent	VERB
ejpam-6627	77	34	these	these	DET
ejpam-6627	77	35	spaces	space	NOUN
ejpam-6627	77	36	from	from	ADP
ejpam-6627	77	37	being	be	AUX
ejpam-6627	77	38	metrizable	metrizable	ADJ
ejpam-6627	77	39	.	.	PUNCT
ejpam-6627	78	1	recall	recall	VERB
ejpam-6627	78	2	that	that	SCONJ
ejpam-6627	78	3	a	a	DET
ejpam-6627	78	4	space	space	NOUN
ejpam-6627	78	5	is	be	AUX
ejpam-6627	78	6	metrizable	metrizable	ADJ
ejpam-6627	78	7	if	if	SCONJ
ejpam-6627	78	8	its	its	PRON
ejpam-6627	78	9	topology	topology	NOUN
ejpam-6627	78	10	can	can	AUX
ejpam-6627	78	11	be	be	AUX
ejpam-6627	78	12	induced	induce	VERB
ejpam-6627	78	13	by	by	ADP
ejpam-6627	78	14	a	a	DET
ejpam-6627	78	15	metric	metric	NOUN
ejpam-6627	78	16	.	.	PUNCT
ejpam-6627	79	1	while	while	SCONJ
ejpam-6627	79	2	metrizable	metrizable	ADJ
ejpam-6627	79	3	spaces	space	NOUN
ejpam-6627	79	4	have	have	VERB
ejpam-6627	79	5	the	the	DET
ejpam-6627	79	6	property	property	NOUN
ejpam-6627	79	7	that	that	PRON
ejpam-6627	79	8	all	all	DET
ejpam-6627	79	9	closed	closed	ADJ
ejpam-6627	79	10	sets	set	NOUN
ejpam-6627	79	11	are	be	AUX
ejpam-6627	79	12	sigma	sigma	NOUN
ejpam-6627	79	13	-	-	PUNCT
ejpam-6627	79	14	intersections	intersection	NOUN
ejpam-6627	79	15	,	,	PUNCT
ejpam-6627	79	16	the	the	DET
ejpam-6627	79	17	converse	converse	NOUN
ejpam-6627	79	18	generally	generally	ADV
ejpam-6627	79	19	fails	fail	VERB
ejpam-6627	79	20	,	,	PUNCT
ejpam-6627	79	21	as	as	SCONJ
ejpam-6627	79	22	shown	show	VERB
ejpam-6627	79	23	in	in	ADP
ejpam-6627	79	24	various	various	ADJ
ejpam-6627	79	25	counterexamples	counterexample	NOUN
ejpam-6627	79	26	by	by	ADP
ejpam-6627	79	27	steen	steen	PROPN
ejpam-6627	79	28	and	and	CCONJ
ejpam-6627	79	29	seebach	seebach	NOUN
ejpam-6627	79	30	[	[	X
ejpam-6627	79	31	24	24	NUM
ejpam-6627	79	32	]	]	PUNCT
ejpam-6627	79	33	.	.	PUNCT
ejpam-6627	80	1	we	we	PRON
ejpam-6627	80	2	begin	begin	VERB
ejpam-6627	80	3	by	by	ADP
ejpam-6627	80	4	establishing	establish	VERB
ejpam-6627	80	5	the	the	DET
ejpam-6627	80	6	following	follow	VERB
ejpam-6627	80	7	fundamental	fundamental	ADJ
ejpam-6627	80	8	characterization	characterization	NOUN
ejpam-6627	80	9	.	.	PUNCT
ejpam-6627	81	1	theorem	theorem	NOUN
ejpam-6627	81	2	1	1	NUM
ejpam-6627	81	3	.	.	X
ejpam-6627	81	4	for	for	ADP
ejpam-6627	81	5	a	a	DET
ejpam-6627	81	6	t1	t1	PROPN
ejpam-6627	81	7	space	space	NOUN
ejpam-6627	81	8	ω	ω	PROPN
ejpam-6627	81	9	,	,	PUNCT
ejpam-6627	81	10	the	the	DET
ejpam-6627	81	11	following	follow	VERB
ejpam-6627	81	12	are	be	AUX
ejpam-6627	81	13	equivalent	equivalent	ADJ
ejpam-6627	81	14	:	:	PUNCT
ejpam-6627	81	15	(	(	PUNCT
ejpam-6627	81	16	i	i	NOUN
ejpam-6627	81	17	)	)	PUNCT
ejpam-6627	81	18	every	every	DET
ejpam-6627	81	19	closed	closed	ADJ
ejpam-6627	81	20	set	set	NOUN
ejpam-6627	81	21	of	of	ADP
ejpam-6627	81	22	ω	ω	PROPN
ejpam-6627	81	23	is	be	AUX
ejpam-6627	81	24	a	a	DET
ejpam-6627	81	25	sigma	sigma	NOUN
ejpam-6627	81	26	-	-	PUNCT
ejpam-6627	81	27	intersection	intersection	NOUN
ejpam-6627	81	28	.	.	PUNCT
ejpam-6627	82	1	(	(	PUNCT
ejpam-6627	82	2	ii	ii	NOUN
ejpam-6627	82	3	)	)	PUNCT
ejpam-6627	82	4	for	for	ADP
ejpam-6627	82	5	every	every	DET
ejpam-6627	82	6	closed	close	VERB
ejpam-6627	82	7	set	set	VERB
ejpam-6627	82	8	φ	φ	PROPN
ejpam-6627	82	9	of	of	ADP
ejpam-6627	82	10	ω	ω	PROPN
ejpam-6627	82	11	,	,	PUNCT
ejpam-6627	82	12	there	there	PRON
ejpam-6627	82	13	exists	exist	VERB
ejpam-6627	82	14	a	a	DET
ejpam-6627	82	15	sequence	sequence	NOUN
ejpam-6627	82	16	of	of	ADP
ejpam-6627	82	17	open	open	ADJ
ejpam-6627	82	18	sets	set	NOUN
ejpam-6627	82	19	{	{	PUNCT
ejpam-6627	82	20	λn	λn	NOUN
ejpam-6627	82	21	}	}	PUNCT
ejpam-6627	82	22	such	such	ADJ
ejpam-6627	82	23	that	that	SCONJ
ejpam-6627	82	24	φ	φ	PROPN
ejpam-6627	82	25	=	=	SYM
ejpam-6627	82	26	⋂	⋂	PROPN
ejpam-6627	82	27	n∈n	n∈n	ADJ
ejpam-6627	82	28	λn	λn	NOUN
ejpam-6627	82	29	.	.	PUNCT
ejpam-6627	83	1	(	(	PUNCT
ejpam-6627	83	2	iii	iii	X
ejpam-6627	83	3	)	)	PUNCT
ejpam-6627	83	4	the	the	DET
ejpam-6627	83	5	complement	complement	NOUN
ejpam-6627	83	6	of	of	ADP
ejpam-6627	83	7	every	every	DET
ejpam-6627	83	8	open	open	ADJ
ejpam-6627	83	9	set	set	NOUN
ejpam-6627	83	10	can	can	AUX
ejpam-6627	83	11	be	be	AUX
ejpam-6627	83	12	written	write	VERB
ejpam-6627	83	13	as	as	ADP
ejpam-6627	83	14	a	a	DET
ejpam-6627	83	15	countable	countable	ADJ
ejpam-6627	83	16	union	union	NOUN
ejpam-6627	83	17	of	of	ADP
ejpam-6627	83	18	closed	closed	ADJ
ejpam-6627	83	19	sets	set	NOUN
ejpam-6627	83	20	.	.	PUNCT
ejpam-6627	84	1	proof	proof	NOUN
ejpam-6627	84	2	.	.	PUNCT
ejpam-6627	85	1	(	(	PUNCT
ejpam-6627	85	2	1	1	X
ejpam-6627	85	3	)	)	PUNCT
ejpam-6627	85	4	⇒	⇒	NOUN
ejpam-6627	85	5	(	(	PUNCT
ejpam-6627	85	6	2	2	NUM
ejpam-6627	85	7	):	):	PUNCT
ejpam-6627	85	8	this	this	PRON
ejpam-6627	85	9	follows	follow	VERB
ejpam-6627	85	10	directly	directly	ADV
ejpam-6627	85	11	from	from	ADP
ejpam-6627	85	12	the	the	DET
ejpam-6627	85	13	definition	definition	NOUN
ejpam-6627	85	14	of	of	ADP
ejpam-6627	85	15	a	a	DET
ejpam-6627	85	16	sigma	sigma	NOUN
ejpam-6627	85	17	-	-	PUNCT
ejpam-6627	85	18	intersection	intersection	NOUN
ejpam-6627	85	19	.	.	PUNCT
ejpam-6627	86	1	(	(	PUNCT
ejpam-6627	86	2	2	2	X
ejpam-6627	86	3	)	)	PUNCT
ejpam-6627	86	4	⇒	⇒	NOUN
ejpam-6627	86	5	(	(	PUNCT
ejpam-6627	86	6	3	3	NUM
ejpam-6627	86	7	):	):	PUNCT
ejpam-6627	86	8	let	let	VERB
ejpam-6627	86	9	λ	λ	PRON
ejpam-6627	86	10	be	be	AUX
ejpam-6627	86	11	an	an	DET
ejpam-6627	86	12	open	open	ADJ
ejpam-6627	86	13	set	set	NOUN
ejpam-6627	86	14	in	in	ADP
ejpam-6627	86	15	ω	ω	PROPN
ejpam-6627	86	16	.	.	PUNCT
ejpam-6627	87	1	then	then	ADV
ejpam-6627	87	2	ω	ω	PROPN
ejpam-6627	87	3	\λ	\λ	PROPN
ejpam-6627	87	4	is	be	AUX
ejpam-6627	87	5	closed	close	VERB
ejpam-6627	87	6	,	,	PUNCT
ejpam-6627	87	7	so	so	ADV
ejpam-6627	87	8	by	by	ADP
ejpam-6627	87	9	assumption	assumption	NOUN
ejpam-6627	87	10	,	,	PUNCT
ejpam-6627	87	11	there	there	PRON
ejpam-6627	87	12	exists	exist	VERB
ejpam-6627	87	13	a	a	DET
ejpam-6627	87	14	sequence	sequence	NOUN
ejpam-6627	87	15	of	of	ADP
ejpam-6627	87	16	open	open	ADJ
ejpam-6627	87	17	sets	set	NOUN
ejpam-6627	87	18	{	{	PUNCT
ejpam-6627	87	19	γn	γn	NOUN
ejpam-6627	87	20	}	}	PUNCT
ejpam-6627	88	1	such	such	ADJ
ejpam-6627	88	2	that	that	SCONJ
ejpam-6627	88	3	ω	ω	NUM
ejpam-6627	88	4	\	\	NOUN
ejpam-6627	88	5	λ	λ	PROPN
ejpam-6627	88	6	=	=	SYM
ejpam-6627	88	7	⋂	⋂	PROPN
ejpam-6627	88	8	n∈n	n∈n	NOUN
ejpam-6627	88	9	γn	γn	NOUN
ejpam-6627	88	10	.	.	PUNCT
ejpam-6627	89	1	taking	take	VERB
ejpam-6627	89	2	complements	complement	NOUN
ejpam-6627	89	3	,	,	PUNCT
ejpam-6627	89	4	we	we	PRON
ejpam-6627	89	5	get	get	VERB
ejpam-6627	89	6	λ	λ	X
ejpam-6627	89	7	=	=	SYM
ejpam-6627	89	8	ω	ω	NUM
ejpam-6627	89	9	\	\	PROPN
ejpam-6627	89	10	⋂	⋂	PROPN
ejpam-6627	89	11	n∈n	n∈n	NOUN
ejpam-6627	89	12	γn	γn	ADP
ejpam-6627	89	13	=	=	PUNCT
ejpam-6627	89	14	⋃	⋃	PROPN
ejpam-6627	89	15	n∈n(ω	n∈n(ω	VERB
ejpam-6627	89	16	\	\	NOUN
ejpam-6627	89	17	γn	γn	NUM
ejpam-6627	89	18	)	)	PUNCT
ejpam-6627	89	19	.	.	PUNCT
ejpam-6627	90	1	each	each	DET
ejpam-6627	90	2	ω	ω	NUM
ejpam-6627	90	3	\	\	PROPN
ejpam-6627	90	4	γn	γn	PROPN
ejpam-6627	90	5	is	be	AUX
ejpam-6627	90	6	closed	closed	ADJ
ejpam-6627	90	7	,	,	PUNCT
ejpam-6627	90	8	so	so	SCONJ
ejpam-6627	90	9	ω	ω	NOUN
ejpam-6627	90	10	\	\	PROPN
ejpam-6627	90	11	λ	λ	NOUN
ejpam-6627	90	12	is	be	AUX
ejpam-6627	90	13	expressible	expressible	ADJ
ejpam-6627	90	14	as	as	ADP
ejpam-6627	90	15	a	a	DET
ejpam-6627	90	16	countable	countable	ADJ
ejpam-6627	90	17	union	union	NOUN
ejpam-6627	90	18	of	of	ADP
ejpam-6627	90	19	closed	closed	ADJ
ejpam-6627	90	20	sets	set	NOUN
ejpam-6627	90	21	.	.	PUNCT
ejpam-6627	91	1	(	(	PUNCT
ejpam-6627	91	2	3	3	X
ejpam-6627	91	3	)	)	PUNCT
ejpam-6627	91	4	⇒	⇒	NOUN
ejpam-6627	91	5	(	(	PUNCT
ejpam-6627	91	6	1	1	NUM
ejpam-6627	91	7	):	):	PUNCT
ejpam-6627	91	8	let	let	VERB
ejpam-6627	91	9	φ	φ	PROPN
ejpam-6627	91	10	be	be	AUX
ejpam-6627	91	11	a	a	DET
ejpam-6627	91	12	closed	closed	ADJ
ejpam-6627	91	13	set	set	NOUN
ejpam-6627	91	14	in	in	ADP
ejpam-6627	91	15	ω	ω	PROPN
ejpam-6627	91	16	.	.	PUNCT
ejpam-6627	92	1	then	then	ADV
ejpam-6627	92	2	ω	ω	PROPN
ejpam-6627	92	3	\	\	PROPN
ejpam-6627	92	4	φ	φ	PROPN
ejpam-6627	92	5	is	be	AUX
ejpam-6627	92	6	open	open	ADJ
ejpam-6627	92	7	,	,	PUNCT
ejpam-6627	92	8	and	and	CCONJ
ejpam-6627	92	9	by	by	ADP
ejpam-6627	92	10	assumption	assumption	NOUN
ejpam-6627	92	11	,	,	PUNCT
ejpam-6627	92	12	ω	ω	NUM
ejpam-6627	92	13	\	\	PROPN
ejpam-6627	92	14	(	(	PUNCT
ejpam-6627	92	15	ω	ω	PROPN
ejpam-6627	92	16	\	\	PROPN
ejpam-6627	92	17	φ	φ	NUM
ejpam-6627	92	18	)	)	PUNCT
ejpam-6627	92	19	=	=	PUNCT
ejpam-6627	93	1	φ	φ	PROPN
ejpam-6627	93	2	can	can	AUX
ejpam-6627	93	3	be	be	AUX
ejpam-6627	93	4	written	write	VERB
ejpam-6627	93	5	as	as	ADP
ejpam-6627	93	6	a	a	DET
ejpam-6627	93	7	countable	countable	ADJ
ejpam-6627	93	8	union	union	NOUN
ejpam-6627	93	9	of	of	ADP
ejpam-6627	93	10	closed	closed	ADJ
ejpam-6627	93	11	sets	set	NOUN
ejpam-6627	93	12	,	,	PUNCT
ejpam-6627	93	13	say	say	VERB
ejpam-6627	93	14	φ	φ	PROPN
ejpam-6627	93	15	=	=	PUNCT
ejpam-6627	94	1	⋃	⋃	VERB
ejpam-6627	94	2	n∈nφn	n∈nφn	NOUN
ejpam-6627	94	3	where	where	SCONJ
ejpam-6627	94	4	each	each	DET
ejpam-6627	94	5	φn	φn	NOUN
ejpam-6627	94	6	is	be	AUX
ejpam-6627	94	7	closed	closed	ADJ
ejpam-6627	94	8	.	.	PUNCT
ejpam-6627	95	1	taking	take	VERB
ejpam-6627	95	2	complements	complement	NOUN
ejpam-6627	95	3	,	,	PUNCT
ejpam-6627	95	4	we	we	PRON
ejpam-6627	95	5	get	get	VERB
ejpam-6627	95	6	ω	ω	NUM
ejpam-6627	95	7	\	\	PROPN
ejpam-6627	95	8	φ	φ	PROPN
ejpam-6627	95	9	=	=	SYM
ejpam-6627	95	10	⋂	⋂	PROPN
ejpam-6627	95	11	n∈n(ω	n∈n(ω	VERB
ejpam-6627	95	12	\	\	PROPN
ejpam-6627	95	13	φn	φn	NOUN
ejpam-6627	95	14	)	)	PUNCT
ejpam-6627	95	15	.	.	PUNCT
ejpam-6627	96	1	each	each	DET
ejpam-6627	96	2	ω	ω	NUM
ejpam-6627	96	3	\	\	PROPN
ejpam-6627	96	4	φn	φn	NOUN
ejpam-6627	96	5	is	be	AUX
ejpam-6627	96	6	open	open	ADJ
ejpam-6627	96	7	,	,	PUNCT
ejpam-6627	96	8	so	so	ADV
ejpam-6627	96	9	φ	φ	PROPN
ejpam-6627	96	10	is	be	AUX
ejpam-6627	96	11	expressible	expressible	ADJ
ejpam-6627	96	12	as	as	ADP
ejpam-6627	96	13	a	a	DET
ejpam-6627	96	14	countable	countable	ADJ
ejpam-6627	96	15	intersection	intersection	NOUN
ejpam-6627	96	16	of	of	ADP
ejpam-6627	96	17	open	open	ADJ
ejpam-6627	96	18	sets	set	NOUN
ejpam-6627	96	19	,	,	PUNCT
ejpam-6627	96	20	i.e.	i.e.	X
ejpam-6627	96	21	,	,	PUNCT
ejpam-6627	96	22	φ	φ	PROPN
ejpam-6627	96	23	is	be	AUX
ejpam-6627	96	24	a	a	DET
ejpam-6627	96	25	sigma	sigma	NOUN
ejpam-6627	96	26	-	-	PUNCT
ejpam-6627	96	27	intersection	intersection	NOUN
ejpam-6627	96	28	.	.	PUNCT
ejpam-6627	97	1	j.	j.	PROPN
ejpam-6627	97	2	oudetallah	oudetallah	PROPN
ejpam-6627	97	3	et	et	PROPN
ejpam-6627	97	4	al	al	PROPN
ejpam-6627	97	5	.	.	PUNCT
ejpam-6627	97	6	/	/	SYM
ejpam-6627	97	7	eur	eur	PROPN
ejpam-6627	97	8	.	.	PUNCT
ejpam-6627	98	1	j.	j.	PROPN
ejpam-6627	98	2	pure	pure	PROPN
ejpam-6627	98	3	appl	appl	PROPN
ejpam-6627	98	4	.	.	PROPN
ejpam-6627	98	5	math	math	PROPN
ejpam-6627	98	6	,	,	PUNCT
ejpam-6627	98	7	18	18	NUM
ejpam-6627	98	8	(	(	PUNCT
ejpam-6627	98	9	3	3	NUM
ejpam-6627	98	10	)	)	PUNCT
ejpam-6627	98	11	(	(	PUNCT
ejpam-6627	98	12	2025	2025	NUM
ejpam-6627	98	13	)	)	PUNCT
ejpam-6627	98	14	,	,	PUNCT
ejpam-6627	98	15	6627	6627	NUM
ejpam-6627	98	16	6	6	NUM
ejpam-6627	98	17	of	of	ADP
ejpam-6627	98	18	20	20	NUM
ejpam-6627	98	19	the	the	DET
ejpam-6627	98	20	interplay	interplay	NOUN
ejpam-6627	98	21	between	between	ADP
ejpam-6627	98	22	the	the	DET
ejpam-6627	98	23	t1	t1	NOUN
ejpam-6627	98	24	property	property	NOUN
ejpam-6627	98	25	and	and	CCONJ
ejpam-6627	98	26	the	the	DET
ejpam-6627	98	27	condition	condition	NOUN
ejpam-6627	98	28	that	that	PRON
ejpam-6627	98	29	closed	close	VERB
ejpam-6627	98	30	sets	set	NOUN
ejpam-6627	98	31	are	be	AUX
ejpam-6627	98	32	sigmaintersections	sigmaintersection	NOUN
ejpam-6627	98	33	produces	produce	VERB
ejpam-6627	98	34	spaces	space	NOUN
ejpam-6627	98	35	with	with	ADP
ejpam-6627	98	36	many	many	ADJ
ejpam-6627	98	37	desirable	desirable	ADJ
ejpam-6627	98	38	properties	property	NOUN
ejpam-6627	98	39	without	without	ADP
ejpam-6627	98	40	necessarily	necessarily	ADV
ejpam-6627	98	41	being	be	AUX
ejpam-6627	98	42	hausdorff	hausdorff	NOUN
ejpam-6627	98	43	.	.	PUNCT
ejpam-6627	99	1	this	this	DET
ejpam-6627	99	2	subtle	subtle	ADJ
ejpam-6627	99	3	interaction	interaction	NOUN
ejpam-6627	99	4	between	between	ADP
ejpam-6627	99	5	separation	separation	NOUN
ejpam-6627	99	6	conditions	condition	NOUN
ejpam-6627	99	7	and	and	CCONJ
ejpam-6627	99	8	other	other	ADJ
ejpam-6627	99	9	topological	topological	ADJ
ejpam-6627	99	10	properties	property	NOUN
ejpam-6627	99	11	is	be	AUX
ejpam-6627	99	12	evident	evident	ADJ
ejpam-6627	99	13	in	in	ADP
ejpam-6627	99	14	the	the	DET
ejpam-6627	99	15	following	follow	VERB
ejpam-6627	99	16	results	result	NOUN
ejpam-6627	99	17	.	.	PUNCT
ejpam-6627	100	1	theorem	theorem	NOUN
ejpam-6627	100	2	2	2	NUM
ejpam-6627	100	3	.	.	PUNCT
ejpam-6627	100	4	a	a	DET
ejpam-6627	100	5	t1	t1	PROPN
ejpam-6627	100	6	space	space	NOUN
ejpam-6627	100	7	ω	ω	PROPN
ejpam-6627	100	8	where	where	SCONJ
ejpam-6627	100	9	closed	closed	ADJ
ejpam-6627	100	10	sets	set	NOUN
ejpam-6627	100	11	are	be	AUX
ejpam-6627	100	12	sigma	sigma	NOUN
ejpam-6627	100	13	-	-	PUNCT
ejpam-6627	100	14	intersections	intersection	NOUN
ejpam-6627	100	15	is	be	AUX
ejpam-6627	100	16	metrizable	metrizable	ADJ
ejpam-6627	101	1	if	if	SCONJ
ejpam-6627	101	2	and	and	CCONJ
ejpam-6627	101	3	only	only	ADV
ejpam-6627	101	4	if	if	SCONJ
ejpam-6627	101	5	it	it	PRON
ejpam-6627	101	6	is	be	AUX
ejpam-6627	101	7	regular	regular	ADJ
ejpam-6627	101	8	and	and	CCONJ
ejpam-6627	101	9	has	have	VERB
ejpam-6627	101	10	a	a	DET
ejpam-6627	101	11	countable	countable	ADJ
ejpam-6627	101	12	basis	basis	NOUN
ejpam-6627	101	13	.	.	PUNCT
ejpam-6627	102	1	proof	proof	NOUN
ejpam-6627	102	2	.	.	PUNCT
ejpam-6627	103	1	the	the	DET
ejpam-6627	103	2	necessity	necessity	NOUN
ejpam-6627	103	3	is	be	AUX
ejpam-6627	103	4	clear	clear	ADJ
ejpam-6627	103	5	since	since	SCONJ
ejpam-6627	103	6	every	every	DET
ejpam-6627	103	7	metrizable	metrizable	ADJ
ejpam-6627	103	8	space	space	NOUN
ejpam-6627	103	9	is	be	AUX
ejpam-6627	103	10	regular	regular	ADJ
ejpam-6627	103	11	and	and	CCONJ
ejpam-6627	103	12	has	have	VERB
ejpam-6627	103	13	a	a	DET
ejpam-6627	103	14	countable	countable	ADJ
ejpam-6627	103	15	basis	basis	NOUN
ejpam-6627	103	16	,	,	PUNCT
ejpam-6627	103	17	as	as	SCONJ
ejpam-6627	103	18	established	establish	VERB
ejpam-6627	103	19	in	in	ADP
ejpam-6627	103	20	the	the	DET
ejpam-6627	103	21	classical	classical	ADJ
ejpam-6627	103	22	work	work	NOUN
ejpam-6627	103	23	of	of	ADP
ejpam-6627	103	24	urysohn	urysohn	NOUN
ejpam-6627	103	25	[	[	X
ejpam-6627	103	26	25	25	NUM
ejpam-6627	103	27	]	]	PUNCT
ejpam-6627	103	28	.	.	PUNCT
ejpam-6627	104	1	for	for	ADP
ejpam-6627	104	2	sufficiency	sufficiency	NOUN
ejpam-6627	104	3	,	,	PUNCT
ejpam-6627	104	4	we	we	PRON
ejpam-6627	104	5	apply	apply	VERB
ejpam-6627	104	6	the	the	DET
ejpam-6627	104	7	urysohn	urysohn	PROPN
ejpam-6627	104	8	metrization	metrization	NOUN
ejpam-6627	104	9	theorem	theorem	NOUN
ejpam-6627	104	10	.	.	PUNCT
ejpam-6627	105	1	since	since	SCONJ
ejpam-6627	105	2	ω	ω	PROPN
ejpam-6627	105	3	is	be	AUX
ejpam-6627	105	4	t1	t1	NOUN
ejpam-6627	105	5	and	and	CCONJ
ejpam-6627	105	6	regular	regular	ADJ
ejpam-6627	105	7	,	,	PUNCT
ejpam-6627	105	8	it	it	PRON
ejpam-6627	105	9	is	be	AUX
ejpam-6627	105	10	t3	t3	PROPN
ejpam-6627	105	11	.	.	PUNCT
ejpam-6627	106	1	moreover	moreover	ADV
ejpam-6627	106	2	,	,	PUNCT
ejpam-6627	106	3	since	since	SCONJ
ejpam-6627	106	4	ω	ω	PROPN
ejpam-6627	106	5	has	have	VERB
ejpam-6627	106	6	a	a	DET
ejpam-6627	106	7	countable	countable	ADJ
ejpam-6627	106	8	basis	basis	NOUN
ejpam-6627	106	9	,	,	PUNCT
ejpam-6627	106	10	it	it	PRON
ejpam-6627	106	11	is	be	AUX
ejpam-6627	106	12	second	second	ADV
ejpam-6627	106	13	-	-	PUNCT
ejpam-6627	106	14	countable	countable	ADJ
ejpam-6627	106	15	.	.	PUNCT
ejpam-6627	107	1	it	it	PRON
ejpam-6627	107	2	remains	remain	VERB
ejpam-6627	107	3	to	to	PART
ejpam-6627	107	4	show	show	VERB
ejpam-6627	107	5	that	that	SCONJ
ejpam-6627	107	6	ω	ω	PROPN
ejpam-6627	107	7	is	be	AUX
ejpam-6627	107	8	normal	normal	ADJ
ejpam-6627	107	9	(	(	PUNCT
ejpam-6627	107	10	t4	t4	PROPN
ejpam-6627	107	11	)	)	PUNCT
ejpam-6627	107	12	.	.	PUNCT
ejpam-6627	108	1	let	let	VERB
ejpam-6627	108	2	φ	φ	PROPN
ejpam-6627	108	3	and	and	CCONJ
ejpam-6627	108	4	ψ	ψ	X
ejpam-6627	108	5	be	be	AUX
ejpam-6627	108	6	disjoint	disjoint	NOUN
ejpam-6627	108	7	closed	close	VERB
ejpam-6627	108	8	sets	set	NOUN
ejpam-6627	108	9	in	in	ADP
ejpam-6627	108	10	ω	ω	PROPN
ejpam-6627	108	11	.	.	PUNCT
ejpam-6627	109	1	since	since	SCONJ
ejpam-6627	109	2	closed	closed	ADJ
ejpam-6627	109	3	sets	set	NOUN
ejpam-6627	109	4	are	be	AUX
ejpam-6627	109	5	sigma	sigma	NOUN
ejpam-6627	109	6	-	-	PUNCT
ejpam-6627	109	7	intersections	intersection	NOUN
ejpam-6627	109	8	,	,	PUNCT
ejpam-6627	109	9	we	we	PRON
ejpam-6627	109	10	can	can	AUX
ejpam-6627	109	11	write	write	VERB
ejpam-6627	109	12	φ	φ	PROPN
ejpam-6627	109	13	=	=	SYM
ejpam-6627	109	14	⋂	⋂	PROPN
ejpam-6627	109	15	n∈n	n∈n	VERB
ejpam-6627	109	16	λn	λn	NOUN
ejpam-6627	109	17	and	and	CCONJ
ejpam-6627	109	18	ψ	ψ	X
ejpam-6627	109	19	=	=	SYM
ejpam-6627	109	20	⋂	⋂	PROPN
ejpam-6627	109	21	n∈n	n∈n	VERB
ejpam-6627	109	22	γn	γn	NOUN
ejpam-6627	109	23	,	,	PUNCT
ejpam-6627	109	24	where	where	SCONJ
ejpam-6627	109	25	each	each	DET
ejpam-6627	109	26	λn	λn	NOUN
ejpam-6627	109	27	and	and	CCONJ
ejpam-6627	109	28	γn	γn	NOUN
ejpam-6627	109	29	is	be	AUX
ejpam-6627	109	30	open	open	ADJ
ejpam-6627	109	31	.	.	PUNCT
ejpam-6627	110	1	by	by	ADP
ejpam-6627	110	2	regularity	regularity	NOUN
ejpam-6627	110	3	and	and	CCONJ
ejpam-6627	110	4	utilizing	utilize	VERB
ejpam-6627	110	5	the	the	DET
ejpam-6627	110	6	fact	fact	NOUN
ejpam-6627	110	7	that	that	SCONJ
ejpam-6627	110	8	ω	ω	PROPN
ejpam-6627	110	9	has	have	AUX
ejpam-6627	110	10	a	a	DET
ejpam-6627	110	11	countable	countable	ADJ
ejpam-6627	110	12	basis	basis	NOUN
ejpam-6627	110	13	,	,	PUNCT
ejpam-6627	110	14	we	we	PRON
ejpam-6627	110	15	can	can	AUX
ejpam-6627	110	16	construct	construct	VERB
ejpam-6627	110	17	disjoint	disjoint	ADJ
ejpam-6627	110	18	open	open	ADJ
ejpam-6627	110	19	sets	set	NOUN
ejpam-6627	110	20	containing	contain	VERB
ejpam-6627	110	21	φ	φ	PROPN
ejpam-6627	110	22	and	and	CCONJ
ejpam-6627	110	23	ψ	ψ	X
ejpam-6627	110	24	respectively	respectively	ADV
ejpam-6627	110	25	,	,	PUNCT
ejpam-6627	110	26	establishing	establish	VERB
ejpam-6627	110	27	that	that	SCONJ
ejpam-6627	110	28	ω	ω	PROPN
ejpam-6627	110	29	is	be	AUX
ejpam-6627	110	30	normal	normal	ADJ
ejpam-6627	110	31	.	.	PUNCT
ejpam-6627	111	1	by	by	ADP
ejpam-6627	111	2	the	the	DET
ejpam-6627	111	3	urysohn	urysohn	PROPN
ejpam-6627	111	4	metrization	metrization	NOUN
ejpam-6627	111	5	theorem	theorem	PROPN
ejpam-6627	111	6	,	,	PUNCT
ejpam-6627	111	7	a	a	DET
ejpam-6627	111	8	regular	regular	ADJ
ejpam-6627	111	9	space	space	NOUN
ejpam-6627	111	10	with	with	ADP
ejpam-6627	111	11	a	a	DET
ejpam-6627	111	12	countable	countable	ADJ
ejpam-6627	111	13	basis	basis	NOUN
ejpam-6627	111	14	is	be	AUX
ejpam-6627	111	15	metrizable[25	metrizable[25	NOUN
ejpam-6627	111	16	]	]	PUNCT
ejpam-6627	111	17	,	,	PUNCT
ejpam-6627	111	18	completing	complete	VERB
ejpam-6627	111	19	the	the	DET
ejpam-6627	111	20	proof	proof	NOUN
ejpam-6627	111	21	.	.	PUNCT
ejpam-6627	112	1	this	this	DET
ejpam-6627	112	2	theorem	theorem	NOUN
ejpam-6627	112	3	identifies	identify	VERB
ejpam-6627	112	4	two	two	NUM
ejpam-6627	112	5	key	key	ADJ
ejpam-6627	112	6	obstacles	obstacle	NOUN
ejpam-6627	112	7	preventing	prevent	VERB
ejpam-6627	112	8	a	a	DET
ejpam-6627	112	9	t1	t1	NOUN
ejpam-6627	112	10	space	space	NOUN
ejpam-6627	112	11	with	with	ADP
ejpam-6627	112	12	closed	closed	ADJ
ejpam-6627	112	13	sets	set	NOUN
ejpam-6627	112	14	as	as	ADP
ejpam-6627	112	15	sigma	sigma	NOUN
ejpam-6627	112	16	-	-	PUNCT
ejpam-6627	112	17	intersections	intersection	NOUN
ejpam-6627	112	18	from	from	ADP
ejpam-6627	112	19	being	be	AUX
ejpam-6627	112	20	metrizable	metrizable	ADJ
ejpam-6627	112	21	:	:	PUNCT
ejpam-6627	112	22	the	the	DET
ejpam-6627	112	23	lack	lack	NOUN
ejpam-6627	112	24	of	of	ADP
ejpam-6627	112	25	regularity	regularity	NOUN
ejpam-6627	112	26	or	or	CCONJ
ejpam-6627	112	27	the	the	DET
ejpam-6627	112	28	absence	absence	NOUN
ejpam-6627	112	29	of	of	ADP
ejpam-6627	112	30	a	a	DET
ejpam-6627	112	31	countable	countable	ADJ
ejpam-6627	112	32	basis	basis	NOUN
ejpam-6627	112	33	.	.	PUNCT
ejpam-6627	113	1	we	we	PRON
ejpam-6627	113	2	now	now	ADV
ejpam-6627	113	3	illustrate	illustrate	VERB
ejpam-6627	113	4	these	these	DET
ejpam-6627	113	5	obstacles	obstacle	NOUN
ejpam-6627	113	6	with	with	ADP
ejpam-6627	113	7	concrete	concrete	ADJ
ejpam-6627	113	8	examples	example	NOUN
ejpam-6627	113	9	,	,	PUNCT
ejpam-6627	113	10	following	follow	VERB
ejpam-6627	113	11	the	the	DET
ejpam-6627	113	12	systematic	systematic	ADJ
ejpam-6627	113	13	approach	approach	NOUN
ejpam-6627	113	14	to	to	ADP
ejpam-6627	113	15	counterexamples	counterexample	NOUN
ejpam-6627	113	16	in	in	ADP
ejpam-6627	113	17	topology	topology	NOUN
ejpam-6627	113	18	presented	present	VERB
ejpam-6627	113	19	by	by	ADP
ejpam-6627	113	20	steen	steen	PROPN
ejpam-6627	113	21	and	and	CCONJ
ejpam-6627	113	22	seebach	seebach	NOUN
ejpam-6627	113	23	[	[	X
ejpam-6627	113	24	24	24	NUM
ejpam-6627	113	25	]	]	PUNCT
ejpam-6627	113	26	.	.	PUNCT
ejpam-6627	114	1	example	example	NOUN
ejpam-6627	115	1	1	1	NUM
ejpam-6627	115	2	.	.	PUNCT
ejpam-6627	116	1	let	let	VERB
ejpam-6627	116	2	ω	ω	PRON
ejpam-6627	116	3	be	be	AUX
ejpam-6627	116	4	an	an	DET
ejpam-6627	116	5	uncountable	uncountable	ADJ
ejpam-6627	116	6	set	set	NOUN
ejpam-6627	116	7	with	with	ADP
ejpam-6627	116	8	the	the	DET
ejpam-6627	116	9	co	co	ADJ
ejpam-6627	116	10	-	-	ADJ
ejpam-6627	116	11	countable	countable	ADJ
ejpam-6627	116	12	topology	topology	NOUN
ejpam-6627	116	13	(	(	PUNCT
ejpam-6627	116	14	a	a	DET
ejpam-6627	116	15	set	set	NOUN
ejpam-6627	116	16	is	be	AUX
ejpam-6627	116	17	open	open	ADJ
ejpam-6627	116	18	if	if	SCONJ
ejpam-6627	116	19	its	its	PRON
ejpam-6627	116	20	complement	complement	NOUN
ejpam-6627	116	21	is	be	AUX
ejpam-6627	116	22	countable	countable	ADJ
ejpam-6627	116	23	or	or	CCONJ
ejpam-6627	116	24	it	it	PRON
ejpam-6627	116	25	is	be	AUX
ejpam-6627	116	26	the	the	DET
ejpam-6627	116	27	empty	empty	ADJ
ejpam-6627	116	28	set	set	NOUN
ejpam-6627	116	29	)	)	PUNCT
ejpam-6627	116	30	.	.	PUNCT
ejpam-6627	117	1	then	then	ADV
ejpam-6627	117	2	ω	ω	PROPN
ejpam-6627	117	3	is	be	AUX
ejpam-6627	117	4	t1	t1	NOUN
ejpam-6627	117	5	since	since	SCONJ
ejpam-6627	117	6	every	every	DET
ejpam-6627	117	7	finite	finite	NOUN
ejpam-6627	117	8	set	set	NOUN
ejpam-6627	117	9	is	be	AUX
ejpam-6627	117	10	closed	closed	ADJ
ejpam-6627	117	11	.	.	PUNCT
ejpam-6627	118	1	moreover	moreover	ADV
ejpam-6627	118	2	,	,	PUNCT
ejpam-6627	118	3	every	every	DET
ejpam-6627	118	4	closed	closed	ADJ
ejpam-6627	118	5	set	set	NOUN
ejpam-6627	118	6	is	be	AUX
ejpam-6627	118	7	a	a	DET
ejpam-6627	118	8	sigma	sigma	NOUN
ejpam-6627	118	9	-	-	PUNCT
ejpam-6627	118	10	intersection	intersection	NOUN
ejpam-6627	118	11	because	because	SCONJ
ejpam-6627	118	12	:	:	PUNCT
ejpam-6627	118	13	•	•	ADP
ejpam-6627	118	14	each	each	DET
ejpam-6627	118	15	non	non	ADJ
ejpam-6627	118	16	-	-	ADJ
ejpam-6627	118	17	empty	empty	ADJ
ejpam-6627	118	18	closed	closed	ADJ
ejpam-6627	118	19	set	set	VERB
ejpam-6627	118	20	φ	φ	PROPN
ejpam-6627	118	21	is	be	AUX
ejpam-6627	118	22	either	either	CCONJ
ejpam-6627	118	23	countable	countable	ADJ
ejpam-6627	118	24	or	or	CCONJ
ejpam-6627	118	25	ω	ω	NOUN
ejpam-6627	118	26	itself	itself	PRON
ejpam-6627	118	27	.	.	PUNCT
ejpam-6627	119	1	•	•	INTJ
ejpam-6627	119	2	if	if	SCONJ
ejpam-6627	119	3	φ	φ	PROPN
ejpam-6627	119	4	is	be	AUX
ejpam-6627	119	5	countable	countable	ADJ
ejpam-6627	119	6	,	,	PUNCT
ejpam-6627	119	7	then	then	ADV
ejpam-6627	119	8	φ	φ	PROPN
ejpam-6627	119	9	=	=	SYM
ejpam-6627	119	10	⋂	⋂	PROPN
ejpam-6627	119	11	n∈n(ω	n∈n(ω	PROPN
ejpam-6627	119	12	\	\	PROPN
ejpam-6627	119	13	an	an	PRON
ejpam-6627	119	14	)	)	PUNCT
ejpam-6627	119	15	where	where	SCONJ
ejpam-6627	119	16	{	{	PUNCT
ejpam-6627	119	17	an	an	PRON
ejpam-6627	119	18	}	}	PUNCT
ejpam-6627	119	19	is	be	AUX
ejpam-6627	119	20	a	a	DET
ejpam-6627	119	21	sequence	sequence	NOUN
ejpam-6627	119	22	of	of	ADP
ejpam-6627	119	23	finite	finite	ADJ
ejpam-6627	119	24	sets	set	NOUN
ejpam-6627	119	25	whose	whose	DET
ejpam-6627	119	26	union	union	NOUN
ejpam-6627	119	27	is	be	AUX
ejpam-6627	119	28	exactly	exactly	ADV
ejpam-6627	119	29	ω	ω	NUM
ejpam-6627	119	30	\	\	PROPN
ejpam-6627	119	31	φ	φ	PROPN
ejpam-6627	119	32	.	.	PROPN
ejpam-6627	119	33	•	•	NUM
ejpam-6627	119	34	if	if	SCONJ
ejpam-6627	119	35	φ	φ	PROPN
ejpam-6627	119	36	=	=	SYM
ejpam-6627	119	37	ω	ω	PROPN
ejpam-6627	119	38	,	,	PUNCT
ejpam-6627	119	39	then	then	ADV
ejpam-6627	119	40	φ	φ	PROPN
ejpam-6627	119	41	is	be	AUX
ejpam-6627	119	42	trivially	trivially	ADV
ejpam-6627	119	43	a	a	DET
ejpam-6627	119	44	sigma	sigma	NOUN
ejpam-6627	119	45	-	-	PUNCT
ejpam-6627	119	46	intersection	intersection	NOUN
ejpam-6627	119	47	.	.	PUNCT
ejpam-6627	120	1	however	however	ADV
ejpam-6627	120	2	,	,	PUNCT
ejpam-6627	120	3	ω	ω	PROPN
ejpam-6627	120	4	is	be	AUX
ejpam-6627	120	5	not	not	PART
ejpam-6627	120	6	metrizable	metrizable	ADJ
ejpam-6627	120	7	because	because	SCONJ
ejpam-6627	120	8	it	it	PRON
ejpam-6627	120	9	is	be	AUX
ejpam-6627	120	10	not	not	PART
ejpam-6627	120	11	even	even	ADV
ejpam-6627	120	12	t2	t2	NOUN
ejpam-6627	120	13	.	.	PUNCT
ejpam-6627	121	1	for	for	ADP
ejpam-6627	121	2	any	any	DET
ejpam-6627	121	3	two	two	NUM
ejpam-6627	121	4	open	open	ADJ
ejpam-6627	121	5	sets	set	NOUN
ejpam-6627	121	6	λ	λ	PROPN
ejpam-6627	121	7	and	and	CCONJ
ejpam-6627	121	8	γ	γ	PROPN
ejpam-6627	121	9	in	in	ADP
ejpam-6627	121	10	ω	ω	NUM
ejpam-6627	121	11	,	,	PUNCT
ejpam-6627	121	12	both	both	PRON
ejpam-6627	121	13	complements	complement	VERB
ejpam-6627	121	14	ω	ω	X
ejpam-6627	121	15	\λ	\λ	PROPN
ejpam-6627	121	16	and	and	CCONJ
ejpam-6627	121	17	ω	ω	NUM
ejpam-6627	121	18	\	\	PROPN
ejpam-6627	121	19	γ	γ	NOUN
ejpam-6627	121	20	are	be	AUX
ejpam-6627	121	21	countable	countable	ADJ
ejpam-6627	121	22	,	,	PUNCT
ejpam-6627	121	23	so	so	CCONJ
ejpam-6627	121	24	(	(	PUNCT
ejpam-6627	121	25	ω	ω	NUM
ejpam-6627	121	26	\λ)∪	\λ)∪	PROPN
ejpam-6627	121	27	(	(	PUNCT
ejpam-6627	121	28	ω	ω	NOUN
ejpam-6627	121	29	\	\	PROPN
ejpam-6627	121	30	γ	γ	X
ejpam-6627	121	31	)	)	PUNCT
ejpam-6627	121	32	is	be	AUX
ejpam-6627	121	33	countable	countable	ADJ
ejpam-6627	121	34	,	,	PUNCT
ejpam-6627	121	35	which	which	PRON
ejpam-6627	121	36	means	mean	VERB
ejpam-6627	121	37	λ	λ	PROPN
ejpam-6627	121	38	∩	∩	NOUN
ejpam-6627	121	39	γ	γ	NOUN
ejpam-6627	121	40	is	be	AUX
ejpam-6627	121	41	uncountable	uncountable	ADJ
ejpam-6627	121	42	and	and	CCONJ
ejpam-6627	121	43	hence	hence	ADV
ejpam-6627	121	44	non	non	ADJ
ejpam-6627	121	45	-	-	ADJ
ejpam-6627	121	46	empty	empty	ADJ
ejpam-6627	121	47	.	.	PUNCT
ejpam-6627	122	1	this	this	DET
ejpam-6627	122	2	example	example	NOUN
ejpam-6627	122	3	demonstrates	demonstrate	VERB
ejpam-6627	122	4	that	that	SCONJ
ejpam-6627	122	5	the	the	DET
ejpam-6627	122	6	condition	condition	NOUN
ejpam-6627	122	7	”	"	PUNCT
ejpam-6627	122	8	closed	closed	ADJ
ejpam-6627	122	9	sets	set	NOUN
ejpam-6627	122	10	are	be	AUX
ejpam-6627	122	11	sigma	sigma	NOUN
ejpam-6627	122	12	-	-	PUNCT
ejpam-6627	122	13	intersections	intersection	NOUN
ejpam-6627	122	14	”	"	PUNCT
ejpam-6627	122	15	combined	combine	VERB
ejpam-6627	122	16	with	with	ADP
ejpam-6627	122	17	t1	t1	PROPN
ejpam-6627	122	18	does	do	AUX
ejpam-6627	122	19	not	not	PART
ejpam-6627	122	20	imply	imply	VERB
ejpam-6627	122	21	metrizability	metrizability	NOUN
ejpam-6627	122	22	without	without	ADP
ejpam-6627	122	23	additional	additional	ADJ
ejpam-6627	122	24	separation	separation	NOUN
ejpam-6627	122	25	or	or	CCONJ
ejpam-6627	122	26	countability	countability	NOUN
ejpam-6627	122	27	assumptions	assumption	NOUN
ejpam-6627	122	28	.	.	PUNCT
ejpam-6627	123	1	j.	j.	PROPN
ejpam-6627	123	2	oudetallah	oudetallah	PROPN
ejpam-6627	123	3	et	et	PROPN
ejpam-6627	123	4	al	al	PROPN
ejpam-6627	123	5	.	.	PUNCT
ejpam-6627	123	6	/	/	SYM
ejpam-6627	123	7	eur	eur	PROPN
ejpam-6627	123	8	.	.	PUNCT
ejpam-6627	124	1	j.	j.	PROPN
ejpam-6627	124	2	pure	pure	PROPN
ejpam-6627	124	3	appl	appl	PROPN
ejpam-6627	124	4	.	.	PROPN
ejpam-6627	124	5	math	math	PROPN
ejpam-6627	124	6	,	,	PUNCT
ejpam-6627	124	7	18	18	NUM
ejpam-6627	124	8	(	(	PUNCT
ejpam-6627	124	9	3	3	NUM
ejpam-6627	124	10	)	)	PUNCT
ejpam-6627	124	11	(	(	PUNCT
ejpam-6627	124	12	2025	2025	NUM
ejpam-6627	124	13	)	)	PUNCT
ejpam-6627	124	14	,	,	PUNCT
ejpam-6627	124	15	6627	6627	NUM
ejpam-6627	124	16	7	7	NUM
ejpam-6627	124	17	of	of	ADP
ejpam-6627	124	18	20	20	NUM
ejpam-6627	124	19	ω	ω	NOUN
ejpam-6627	124	20	an	an	DET
ejpam-6627	124	21	open	open	ADJ
ejpam-6627	124	22	set	set	ADJ
ejpam-6627	124	23	λ	λ	PROPN
ejpam-6627	124	24	(	(	PUNCT
ejpam-6627	124	25	complement	complement	NOUN
ejpam-6627	124	26	of	of	ADP
ejpam-6627	124	27	a	a	DET
ejpam-6627	124	28	countable	countable	ADJ
ejpam-6627	124	29	set	set	NOUN
ejpam-6627	124	30	)	)	PUNCT
ejpam-6627	124	31	a	a	DET
ejpam-6627	124	32	countable	countable	ADJ
ejpam-6627	124	33	closed	close	VERB
ejpam-6627	124	34	set	set	VERB
ejpam-6627	124	35	φ	φ	PROPN
ejpam-6627	124	36	φ	φ	PROPN
ejpam-6627	124	37	=	=	SYM
ejpam-6627	125	1	⋂	⋂	PROPN
ejpam-6627	125	2	n∈n	n∈n	NOUN
ejpam-6627	125	3	λn	λn	NOUN
ejpam-6627	125	4	where	where	SCONJ
ejpam-6627	125	5	each	each	DET
ejpam-6627	125	6	λn	λn	NOUN
ejpam-6627	125	7	is	be	AUX
ejpam-6627	125	8	open	open	ADJ
ejpam-6627	125	9	figure	figure	NOUN
ejpam-6627	125	10	2	2	NUM
ejpam-6627	125	11	:	:	PUNCT
ejpam-6627	125	12	representation	representation	NOUN
ejpam-6627	125	13	of	of	ADP
ejpam-6627	125	14	a	a	DET
ejpam-6627	125	15	co	co	ADJ
ejpam-6627	125	16	-	-	ADJ
ejpam-6627	125	17	countable	countable	ADJ
ejpam-6627	125	18	topology	topology	NOUN
ejpam-6627	125	19	on	on	ADP
ejpam-6627	125	20	an	an	DET
ejpam-6627	125	21	uncountable	uncountable	ADJ
ejpam-6627	125	22	set	set	NOUN
ejpam-6627	125	23	ω	ω	NOUN
ejpam-6627	125	24	.	.	PUNCT
ejpam-6627	126	1	this	this	DET
ejpam-6627	126	2	space	space	NOUN
ejpam-6627	126	3	is	be	AUX
ejpam-6627	126	4	t1	t1	NOUN
ejpam-6627	126	5	with	with	ADP
ejpam-6627	126	6	closed	closed	ADJ
ejpam-6627	126	7	sets	set	NOUN
ejpam-6627	126	8	as	as	ADP
ejpam-6627	126	9	sigma	sigma	NOUN
ejpam-6627	126	10	-	-	PUNCT
ejpam-6627	126	11	intersections	intersection	NOUN
ejpam-6627	126	12	but	but	CCONJ
ejpam-6627	126	13	fails	fail	VERB
ejpam-6627	126	14	to	to	PART
ejpam-6627	126	15	be	be	AUX
ejpam-6627	126	16	hausdorff	hausdorff	NOUN
ejpam-6627	126	17	.	.	PUNCT
ejpam-6627	127	1	the	the	DET
ejpam-6627	127	2	diagram	diagram	NOUN
ejpam-6627	127	3	illustrates	illustrate	VERB
ejpam-6627	127	4	how	how	SCONJ
ejpam-6627	127	5	countable	countable	ADJ
ejpam-6627	127	6	closed	closed	ADJ
ejpam-6627	127	7	sets	set	NOUN
ejpam-6627	127	8	can	can	AUX
ejpam-6627	127	9	be	be	AUX
ejpam-6627	127	10	expressed	express	VERB
ejpam-6627	127	11	as	as	ADP
ejpam-6627	127	12	countable	countable	ADJ
ejpam-6627	127	13	intersections	intersection	NOUN
ejpam-6627	127	14	of	of	ADP
ejpam-6627	127	15	open	open	ADJ
ejpam-6627	127	16	sets	set	NOUN
ejpam-6627	127	17	.	.	PUNCT
ejpam-6627	128	1	another	another	DET
ejpam-6627	128	2	illuminating	illuminating	ADJ
ejpam-6627	128	3	example	example	NOUN
ejpam-6627	128	4	comes	come	VERB
ejpam-6627	128	5	from	from	ADP
ejpam-6627	128	6	the	the	DET
ejpam-6627	128	7	arens	arens	PROPN
ejpam-6627	128	8	-	-	PUNCT
ejpam-6627	128	9	fort	fort	NOUN
ejpam-6627	128	10	space	space	NOUN
ejpam-6627	128	11	,	,	PUNCT
ejpam-6627	128	12	which	which	PRON
ejpam-6627	128	13	demonstrates	demonstrate	VERB
ejpam-6627	128	14	how	how	SCONJ
ejpam-6627	128	15	the	the	DET
ejpam-6627	128	16	failure	failure	NOUN
ejpam-6627	128	17	of	of	ADP
ejpam-6627	128	18	regularity	regularity	NOUN
ejpam-6627	128	19	can	can	AUX
ejpam-6627	128	20	be	be	AUX
ejpam-6627	128	21	the	the	DET
ejpam-6627	128	22	sole	sole	ADJ
ejpam-6627	128	23	obstacle	obstacle	NOUN
ejpam-6627	128	24	to	to	ADP
ejpam-6627	128	25	metrizability	metrizability	NOUN
ejpam-6627	128	26	,	,	PUNCT
ejpam-6627	128	27	as	as	SCONJ
ejpam-6627	128	28	documented	document	VERB
ejpam-6627	128	29	in	in	ADP
ejpam-6627	128	30	steen	steen	PROPN
ejpam-6627	128	31	and	and	CCONJ
ejpam-6627	128	32	seebach	seebach	NOUN
ejpam-6627	128	33	’s	’s	PART
ejpam-6627	128	34	collection	collection	NOUN
ejpam-6627	129	1	[	[	X
ejpam-6627	129	2	24	24	NUM
ejpam-6627	129	3	]	]	PUNCT
ejpam-6627	129	4	.	.	PUNCT
ejpam-6627	130	1	example	example	NOUN
ejpam-6627	131	1	2	2	NUM
ejpam-6627	131	2	.	.	PUNCT
ejpam-6627	131	3	let	let	VERB
ejpam-6627	131	4	ω	ω	NOUN
ejpam-6627	131	5	=	=	NOUN
ejpam-6627	131	6	r2	r2	PROPN
ejpam-6627	131	7	and	and	CCONJ
ejpam-6627	131	8	define	define	VERB
ejpam-6627	131	9	a	a	DET
ejpam-6627	131	10	topology	topology	NOUN
ejpam-6627	131	11	as	as	SCONJ
ejpam-6627	131	12	follows	follow	VERB
ejpam-6627	131	13	:	:	PUNCT
ejpam-6627	131	14	every	every	DET
ejpam-6627	131	15	point	point	NOUN
ejpam-6627	131	16	except	except	SCONJ
ejpam-6627	131	17	the	the	DET
ejpam-6627	131	18	origin	origin	NOUN
ejpam-6627	131	19	(	(	PUNCT
ejpam-6627	131	20	0	0	NUM
ejpam-6627	131	21	,	,	PUNCT
ejpam-6627	131	22	0	0	NUM
ejpam-6627	131	23	)	)	PUNCT
ejpam-6627	131	24	is	be	AUX
ejpam-6627	131	25	isolated	isolate	VERB
ejpam-6627	131	26	,	,	PUNCT
ejpam-6627	131	27	and	and	CCONJ
ejpam-6627	131	28	a	a	DET
ejpam-6627	131	29	neighborhood	neighborhood	NOUN
ejpam-6627	131	30	basis	basis	NOUN
ejpam-6627	131	31	at	at	ADP
ejpam-6627	131	32	(	(	PUNCT
ejpam-6627	131	33	0	0	NUM
ejpam-6627	131	34	,	,	PUNCT
ejpam-6627	131	35	0	0	NUM
ejpam-6627	131	36	)	)	PUNCT
ejpam-6627	131	37	consists	consist	VERB
ejpam-6627	131	38	of	of	ADP
ejpam-6627	131	39	sets	set	NOUN
ejpam-6627	131	40	of	of	ADP
ejpam-6627	131	41	the	the	DET
ejpam-6627	131	42	form	form	NOUN
ejpam-6627	131	43	λ	λ	PROPN
ejpam-6627	131	44	\	\	PROPN
ejpam-6627	131	45	φ	φ	PROPN
ejpam-6627	131	46	,	,	PUNCT
ejpam-6627	131	47	where	where	SCONJ
ejpam-6627	131	48	λ	λ	PROPN
ejpam-6627	131	49	is	be	AUX
ejpam-6627	131	50	a	a	DET
ejpam-6627	131	51	neighborhood	neighborhood	NOUN
ejpam-6627	131	52	of	of	ADP
ejpam-6627	131	53	(	(	PUNCT
ejpam-6627	131	54	0	0	NUM
ejpam-6627	131	55	,	,	PUNCT
ejpam-6627	131	56	0	0	NUM
ejpam-6627	131	57	)	)	PUNCT
ejpam-6627	131	58	in	in	ADP
ejpam-6627	131	59	the	the	DET
ejpam-6627	131	60	usual	usual	ADJ
ejpam-6627	131	61	topology	topology	NOUN
ejpam-6627	131	62	and	and	CCONJ
ejpam-6627	131	63	φ	φ	PROPN
ejpam-6627	131	64	is	be	AUX
ejpam-6627	131	65	a	a	DET
ejpam-6627	131	66	finite	finite	NOUN
ejpam-6627	131	67	set	set	VERB
ejpam-6627	131	68	not	not	PART
ejpam-6627	131	69	containing	contain	VERB
ejpam-6627	131	70	(	(	PUNCT
ejpam-6627	131	71	0	0	NUM
ejpam-6627	131	72	,	,	PUNCT
ejpam-6627	131	73	0	0	NUM
ejpam-6627	131	74	)	)	PUNCT
ejpam-6627	131	75	.	.	PUNCT
ejpam-6627	132	1	this	this	DET
ejpam-6627	132	2	space	space	NOUN
ejpam-6627	132	3	is	be	AUX
ejpam-6627	132	4	t1	t1	NOUN
ejpam-6627	132	5	and	and	CCONJ
ejpam-6627	132	6	has	have	VERB
ejpam-6627	132	7	the	the	DET
ejpam-6627	132	8	property	property	NOUN
ejpam-6627	132	9	that	that	PRON
ejpam-6627	132	10	closed	close	VERB
ejpam-6627	132	11	sets	set	NOUN
ejpam-6627	132	12	are	be	AUX
ejpam-6627	132	13	sigmaintersections	sigmaintersection	NOUN
ejpam-6627	132	14	.	.	PUNCT
ejpam-6627	133	1	however	however	ADV
ejpam-6627	133	2	,	,	PUNCT
ejpam-6627	133	3	it	it	PRON
ejpam-6627	133	4	is	be	AUX
ejpam-6627	133	5	not	not	PART
ejpam-6627	133	6	regular	regular	ADJ
ejpam-6627	133	7	at	at	ADP
ejpam-6627	133	8	the	the	DET
ejpam-6627	133	9	origin	origin	NOUN
ejpam-6627	133	10	,	,	PUNCT
ejpam-6627	133	11	and	and	CCONJ
ejpam-6627	133	12	hence	hence	ADV
ejpam-6627	133	13	not	not	PART
ejpam-6627	133	14	metrizable	metrizable	ADJ
ejpam-6627	133	15	.	.	PUNCT
ejpam-6627	134	1	this	this	DET
ejpam-6627	134	2	example	example	NOUN
ejpam-6627	134	3	demonstrates	demonstrate	VERB
ejpam-6627	134	4	how	how	SCONJ
ejpam-6627	134	5	the	the	DET
ejpam-6627	134	6	failure	failure	NOUN
ejpam-6627	134	7	of	of	ADP
ejpam-6627	134	8	regularity	regularity	NOUN
ejpam-6627	134	9	can	can	AUX
ejpam-6627	134	10	be	be	AUX
ejpam-6627	134	11	the	the	DET
ejpam-6627	134	12	sole	sole	ADJ
ejpam-6627	134	13	obstacle	obstacle	NOUN
ejpam-6627	134	14	to	to	ADP
ejpam-6627	134	15	metrizability	metrizability	NOUN
ejpam-6627	134	16	in	in	ADP
ejpam-6627	134	17	a	a	DET
ejpam-6627	134	18	t1	t1	NOUN
ejpam-6627	134	19	space	space	NOUN
ejpam-6627	134	20	with	with	ADP
ejpam-6627	134	21	closed	closed	ADJ
ejpam-6627	134	22	sets	set	NOUN
ejpam-6627	134	23	as	as	ADP
ejpam-6627	134	24	sigma	sigma	NOUN
ejpam-6627	134	25	-	-	PUNCT
ejpam-6627	134	26	intersections	intersection	NOUN
ejpam-6627	134	27	.	.	PUNCT
ejpam-6627	135	1	we	we	PRON
ejpam-6627	135	2	now	now	ADV
ejpam-6627	135	3	provide	provide	VERB
ejpam-6627	135	4	a	a	DET
ejpam-6627	135	5	new	new	ADJ
ejpam-6627	135	6	characterization	characterization	NOUN
ejpam-6627	135	7	linking	link	VERB
ejpam-6627	135	8	functional	functional	ADJ
ejpam-6627	135	9	separation	separation	NOUN
ejpam-6627	135	10	with	with	ADP
ejpam-6627	135	11	the	the	DET
ejpam-6627	135	12	hausdorff	hausdorff	NOUN
ejpam-6627	135	13	property	property	NOUN
ejpam-6627	135	14	.	.	PUNCT
ejpam-6627	136	1	this	this	DET
ejpam-6627	136	2	result	result	NOUN
ejpam-6627	136	3	builds	build	VERB
ejpam-6627	136	4	upon	upon	SCONJ
ejpam-6627	136	5	recent	recent	ADJ
ejpam-6627	136	6	work	work	NOUN
ejpam-6627	136	7	on	on	ADP
ejpam-6627	136	8	h	h	NOUN
ejpam-6627	136	9	-	-	PUNCT
ejpam-6627	136	10	convexity	convexity	NOUN
ejpam-6627	136	11	in	in	ADP
ejpam-6627	136	12	metric	metric	ADJ
ejpam-6627	136	13	linear	linear	NOUN
ejpam-6627	136	14	spaces	space	NOUN
ejpam-6627	136	15	[	[	X
ejpam-6627	136	16	26	26	NUM
ejpam-6627	136	17	]	]	PUNCT
ejpam-6627	136	18	,	,	PUNCT
ejpam-6627	136	19	which	which	PRON
ejpam-6627	136	20	explores	explore	VERB
ejpam-6627	136	21	functional	functional	ADJ
ejpam-6627	136	22	properties	property	NOUN
ejpam-6627	136	23	in	in	ADP
ejpam-6627	136	24	topological	topological	ADJ
ejpam-6627	136	25	settings	setting	NOUN
ejpam-6627	136	26	,	,	PUNCT
ejpam-6627	136	27	and	and	CCONJ
ejpam-6627	136	28	relates	relate	VERB
ejpam-6627	136	29	to	to	ADP
ejpam-6627	136	30	the	the	DET
ejpam-6627	136	31	classical	classical	ADJ
ejpam-6627	136	32	tietze	tietze	NOUN
ejpam-6627	136	33	extension	extension	NOUN
ejpam-6627	136	34	theorem	theorem	VERB
ejpam-6627	136	35	[	[	X
ejpam-6627	136	36	27	27	NUM
ejpam-6627	136	37	]	]	PUNCT
ejpam-6627	136	38	.	.	PUNCT
ejpam-6627	137	1	theorem	theorem	NOUN
ejpam-6627	137	2	3	3	NUM
ejpam-6627	137	3	.	.	PUNCT
ejpam-6627	137	4	a	a	DET
ejpam-6627	137	5	t1	t1	PROPN
ejpam-6627	137	6	space	space	NOUN
ejpam-6627	137	7	ω	ω	PROPN
ejpam-6627	137	8	where	where	SCONJ
ejpam-6627	137	9	closed	closed	ADJ
ejpam-6627	137	10	sets	set	NOUN
ejpam-6627	137	11	are	be	AUX
ejpam-6627	137	12	sigma	sigma	NOUN
ejpam-6627	137	13	-	-	PUNCT
ejpam-6627	137	14	intersections	intersection	NOUN
ejpam-6627	137	15	is	be	AUX
ejpam-6627	137	16	hausdorff	hausdorff	NOUN
ejpam-6627	137	17	if	if	SCONJ
ejpam-6627	138	1	and	and	CCONJ
ejpam-6627	138	2	only	only	ADV
ejpam-6627	138	3	if	if	SCONJ
ejpam-6627	138	4	for	for	ADP
ejpam-6627	138	5	every	every	DET
ejpam-6627	138	6	pair	pair	NOUN
ejpam-6627	138	7	of	of	ADP
ejpam-6627	138	8	distinct	distinct	ADJ
ejpam-6627	138	9	points	point	NOUN
ejpam-6627	138	10	α	α	NOUN
ejpam-6627	138	11	and	and	CCONJ
ejpam-6627	138	12	β	β	NOUN
ejpam-6627	138	13	,	,	PUNCT
ejpam-6627	138	14	there	there	PRON
ejpam-6627	138	15	exists	exist	VERB
ejpam-6627	138	16	a	a	DET
ejpam-6627	138	17	continuous	continuous	ADJ
ejpam-6627	138	18	real	real	ADV
ejpam-6627	138	19	-	-	PUNCT
ejpam-6627	138	20	valued	value	VERB
ejpam-6627	138	21	function	function	NOUN
ejpam-6627	138	22	f	f	PROPN
ejpam-6627	138	23	such	such	ADJ
ejpam-6627	138	24	that	that	DET
ejpam-6627	138	25	f(α	f(α	NOUN
ejpam-6627	138	26	)	)	PUNCT
ejpam-6627	138	27	̸=	̸=	PROPN
ejpam-6627	138	28	f(β	f(β	NOUN
ejpam-6627	138	29	)	)	PUNCT
ejpam-6627	138	30	.	.	PUNCT
ejpam-6627	139	1	proof	proof	NOUN
ejpam-6627	139	2	.	.	PUNCT
ejpam-6627	140	1	first	first	ADV
ejpam-6627	140	2	,	,	PUNCT
ejpam-6627	140	3	assume	assume	VERB
ejpam-6627	140	4	ω	ω	PROPN
ejpam-6627	140	5	is	be	AUX
ejpam-6627	140	6	hausdorff	hausdorff	NOUN
ejpam-6627	140	7	.	.	PUNCT
ejpam-6627	141	1	let	let	VERB
ejpam-6627	141	2	α	α	PRON
ejpam-6627	141	3	,	,	PUNCT
ejpam-6627	141	4	β	β	X
ejpam-6627	141	5	∈	∈	PROPN
ejpam-6627	141	6	ω	ω	PROPN
ejpam-6627	141	7	with	with	ADP
ejpam-6627	141	8	α	α	PROPN
ejpam-6627	141	9	̸=	̸=	PROPN
ejpam-6627	141	10	β	β	NOUN
ejpam-6627	141	11	.	.	PUNCT
ejpam-6627	142	1	since	since	SCONJ
ejpam-6627	142	2	ω	ω	PROPN
ejpam-6627	142	3	is	be	AUX
ejpam-6627	142	4	hausdorff	hausdorff	NOUN
ejpam-6627	142	5	,	,	PUNCT
ejpam-6627	142	6	there	there	PRON
ejpam-6627	142	7	exist	exist	VERB
ejpam-6627	142	8	disjoint	disjoint	ADJ
ejpam-6627	142	9	open	open	ADJ
ejpam-6627	142	10	sets	set	NOUN
ejpam-6627	142	11	λ	λ	PROPN
ejpam-6627	142	12	and	and	CCONJ
ejpam-6627	142	13	γ	γ	NOUN
ejpam-6627	142	14	such	such	ADJ
ejpam-6627	142	15	that	that	SCONJ
ejpam-6627	142	16	α	α	PROPN
ejpam-6627	142	17	∈	∈	PROPN
ejpam-6627	142	18	λ	λ	PROPN
ejpam-6627	142	19	and	and	CCONJ
ejpam-6627	142	20	β	β	X
ejpam-6627	142	21	∈	∈	PROPN
ejpam-6627	142	22	γ	γ	PROPN
ejpam-6627	142	23	.	.	PUNCT
ejpam-6627	143	1	since	since	SCONJ
ejpam-6627	143	2	ω\λ	ω\λ	PROPN
ejpam-6627	143	3	is	be	AUX
ejpam-6627	143	4	closed	close	VERB
ejpam-6627	143	5	and	and	CCONJ
ejpam-6627	143	6	contains	contain	VERB
ejpam-6627	143	7	β	β	X
ejpam-6627	143	8	but	but	CCONJ
ejpam-6627	143	9	not	not	PART
ejpam-6627	143	10	α	α	NOUN
ejpam-6627	143	11	,	,	PUNCT
ejpam-6627	143	12	and	and	CCONJ
ejpam-6627	143	13	since	since	SCONJ
ejpam-6627	143	14	closed	close	VERB
ejpam-6627	143	15	sets	set	NOUN
ejpam-6627	143	16	are	be	AUX
ejpam-6627	143	17	sigma	sigma	NOUN
ejpam-6627	143	18	-	-	PUNCT
ejpam-6627	143	19	intersections	intersection	NOUN
ejpam-6627	143	20	,	,	PUNCT
ejpam-6627	143	21	we	we	PRON
ejpam-6627	143	22	can	can	AUX
ejpam-6627	143	23	use	use	VERB
ejpam-6627	143	24	urysohn	urysohn	PROPN
ejpam-6627	143	25	’s	’s	PART
ejpam-6627	143	26	lemma	lemma	PROPN
ejpam-6627	143	27	for	for	ADP
ejpam-6627	143	28	sigma	sigma	ADJ
ejpam-6627	143	29	-	-	PUNCT
ejpam-6627	143	30	intersection	intersection	NOUN
ejpam-6627	143	31	sets	set	NOUN
ejpam-6627	143	32	to	to	PART
ejpam-6627	143	33	construct	construct	VERB
ejpam-6627	143	34	a	a	DET
ejpam-6627	143	35	continuous	continuous	ADJ
ejpam-6627	143	36	function	function	NOUN
ejpam-6627	143	37	f	f	NOUN
ejpam-6627	144	1	:	:	PUNCT
ejpam-6627	144	2	ω	ω	X
ejpam-6627	144	3	→	→	PUNCT
ejpam-6627	145	1	[	[	X
ejpam-6627	145	2	0	0	NUM
ejpam-6627	145	3	,	,	PUNCT
ejpam-6627	145	4	1	1	NUM
ejpam-6627	145	5	]	]	PUNCT
ejpam-6627	145	6	such	such	ADJ
ejpam-6627	145	7	that	that	DET
ejpam-6627	145	8	f(α	f(α	NOUN
ejpam-6627	145	9	)	)	PUNCT
ejpam-6627	145	10	=	=	SYM
ejpam-6627	145	11	0	0	NUM
ejpam-6627	145	12	and	and	CCONJ
ejpam-6627	145	13	f(β	f(β	PROPN
ejpam-6627	145	14	)	)	PUNCT
ejpam-6627	146	1	=	=	SYM
ejpam-6627	146	2	1	1	NUM
ejpam-6627	146	3	,	,	PUNCT
ejpam-6627	146	4	following	follow	VERB
ejpam-6627	146	5	the	the	DET
ejpam-6627	146	6	functional	functional	ADJ
ejpam-6627	146	7	approach	approach	NOUN
ejpam-6627	146	8	pioneered	pioneer	VERB
ejpam-6627	146	9	by	by	ADP
ejpam-6627	146	10	tietze	tietze	NOUN
ejpam-6627	146	11	[	[	X
ejpam-6627	146	12	27	27	NUM
ejpam-6627	146	13	]	]	PUNCT
ejpam-6627	146	14	.	.	PUNCT
ejpam-6627	147	1	conversely	conversely	ADV
ejpam-6627	147	2	,	,	PUNCT
ejpam-6627	147	3	assume	assume	VERB
ejpam-6627	147	4	that	that	SCONJ
ejpam-6627	147	5	for	for	ADP
ejpam-6627	147	6	every	every	DET
ejpam-6627	147	7	pair	pair	NOUN
ejpam-6627	147	8	of	of	ADP
ejpam-6627	147	9	distinct	distinct	ADJ
ejpam-6627	147	10	points	point	NOUN
ejpam-6627	147	11	α	α	NOUN
ejpam-6627	147	12	and	and	CCONJ
ejpam-6627	147	13	β	β	NOUN
ejpam-6627	147	14	,	,	PUNCT
ejpam-6627	147	15	there	there	PRON
ejpam-6627	147	16	exists	exist	VERB
ejpam-6627	147	17	a	a	DET
ejpam-6627	147	18	continuous	continuous	ADJ
ejpam-6627	147	19	function	function	NOUN
ejpam-6627	147	20	f	f	PRON
ejpam-6627	147	21	such	such	ADJ
ejpam-6627	147	22	that	that	DET
ejpam-6627	147	23	f(α	f(α	NOUN
ejpam-6627	147	24	)	)	PUNCT
ejpam-6627	147	25	̸=	̸=	PROPN
ejpam-6627	147	26	f(β	f(β	NOUN
ejpam-6627	147	27	)	)	PUNCT
ejpam-6627	147	28	.	.	PUNCT
ejpam-6627	148	1	let	let	VERB
ejpam-6627	148	2	α	α	PRON
ejpam-6627	148	3	,	,	PUNCT
ejpam-6627	148	4	β	β	X
ejpam-6627	148	5	∈	∈	PROPN
ejpam-6627	148	6	ω	ω	PROPN
ejpam-6627	148	7	with	with	ADP
ejpam-6627	148	8	α	α	PROPN
ejpam-6627	148	9	̸=	̸=	PROPN
ejpam-6627	148	10	β	β	NOUN
ejpam-6627	148	11	.	.	PUNCT
ejpam-6627	149	1	by	by	ADP
ejpam-6627	149	2	assumption	assumption	NOUN
ejpam-6627	149	3	,	,	PUNCT
ejpam-6627	149	4	there	there	PRON
ejpam-6627	149	5	exists	exist	VERB
ejpam-6627	149	6	a	a	DET
ejpam-6627	149	7	continuous	continuous	ADJ
ejpam-6627	149	8	function	function	NOUN
ejpam-6627	149	9	g	g	ADP
ejpam-6627	149	10	such	such	ADJ
ejpam-6627	149	11	that	that	SCONJ
ejpam-6627	149	12	g(α	g(α	PROPN
ejpam-6627	149	13	)	)	PUNCT
ejpam-6627	149	14	̸=	̸=	PROPN
ejpam-6627	149	15	g(β	g(β	PROPN
ejpam-6627	149	16	)	)	PUNCT
ejpam-6627	149	17	.	.	PUNCT
ejpam-6627	150	1	without	without	ADP
ejpam-6627	150	2	loss	loss	NOUN
ejpam-6627	150	3	of	of	ADP
ejpam-6627	150	4	generality	generality	NOUN
ejpam-6627	150	5	,	,	PUNCT
ejpam-6627	150	6	assume	assume	VERB
ejpam-6627	150	7	g(α	g(α	PROPN
ejpam-6627	150	8	)	)	PUNCT
ejpam-6627	150	9	<	<	X
ejpam-6627	150	10	g(β	g(β	PROPN
ejpam-6627	150	11	)	)	PUNCT
ejpam-6627	150	12	.	.	PUNCT
ejpam-6627	151	1	choose	choose	VERB
ejpam-6627	151	2	c	c	NOUN
ejpam-6627	151	3	such	such	ADJ
ejpam-6627	151	4	that	that	DET
ejpam-6627	151	5	g(α	g(α	PROPN
ejpam-6627	151	6	)	)	PUNCT
ejpam-6627	151	7	<	<	X
ejpam-6627	151	8	c	c	X
ejpam-6627	151	9	<	<	X
ejpam-6627	151	10	g(β	g(β	PROPN
ejpam-6627	151	11	)	)	PUNCT
ejpam-6627	151	12	.	.	PUNCT
ejpam-6627	152	1	then	then	ADV
ejpam-6627	152	2	g−1((−∞	g−1((−∞	NUM
ejpam-6627	152	3	,	,	PUNCT
ejpam-6627	152	4	c	c	NOUN
ejpam-6627	152	5	)	)	PUNCT
ejpam-6627	152	6	)	)	PUNCT
ejpam-6627	152	7	and	and	CCONJ
ejpam-6627	152	8	j.	j.	PROPN
ejpam-6627	152	9	oudetallah	oudetallah	PROPN
ejpam-6627	152	10	et	et	PROPN
ejpam-6627	152	11	al	al	PROPN
ejpam-6627	152	12	.	.	PUNCT
ejpam-6627	152	13	/	/	SYM
ejpam-6627	152	14	eur	eur	PROPN
ejpam-6627	152	15	.	.	PUNCT
ejpam-6627	153	1	j.	j.	PROPN
ejpam-6627	153	2	pure	pure	PROPN
ejpam-6627	153	3	appl	appl	PROPN
ejpam-6627	153	4	.	.	PROPN
ejpam-6627	153	5	math	math	PROPN
ejpam-6627	153	6	,	,	PUNCT
ejpam-6627	153	7	18	18	NUM
ejpam-6627	153	8	(	(	PUNCT
ejpam-6627	153	9	3	3	NUM
ejpam-6627	153	10	)	)	PUNCT
ejpam-6627	153	11	(	(	PUNCT
ejpam-6627	153	12	2025	2025	NUM
ejpam-6627	153	13	)	)	PUNCT
ejpam-6627	153	14	,	,	PUNCT
ejpam-6627	153	15	6627	6627	NUM
ejpam-6627	153	16	8	8	NUM
ejpam-6627	153	17	of	of	ADP
ejpam-6627	153	18	20	20	NUM
ejpam-6627	153	19	g−1((c,∞	g−1((c,∞	PROPN
ejpam-6627	153	20	)	)	PUNCT
ejpam-6627	153	21	)	)	PUNCT
ejpam-6627	153	22	are	be	AUX
ejpam-6627	153	23	disjoint	disjoint	ADJ
ejpam-6627	153	24	open	open	ADJ
ejpam-6627	153	25	sets	set	NOUN
ejpam-6627	153	26	containing	contain	VERB
ejpam-6627	153	27	α	α	NOUN
ejpam-6627	153	28	and	and	CCONJ
ejpam-6627	153	29	β	β	X
ejpam-6627	153	30	respectively	respectively	ADV
ejpam-6627	153	31	,	,	PUNCT
ejpam-6627	153	32	showing	show	VERB
ejpam-6627	153	33	that	that	SCONJ
ejpam-6627	153	34	ω	ω	PROPN
ejpam-6627	153	35	is	be	AUX
ejpam-6627	153	36	hausdorff	hausdorff	NOUN
ejpam-6627	153	37	.	.	PUNCT
ejpam-6627	154	1	this	this	DET
ejpam-6627	154	2	result	result	NOUN
ejpam-6627	154	3	reveals	reveal	VERB
ejpam-6627	154	4	the	the	DET
ejpam-6627	154	5	deep	deep	ADJ
ejpam-6627	154	6	connection	connection	NOUN
ejpam-6627	154	7	between	between	ADP
ejpam-6627	154	8	separation	separation	NOUN
ejpam-6627	154	9	properties	property	NOUN
ejpam-6627	154	10	of	of	ADP
ejpam-6627	154	11	point	point	NOUN
ejpam-6627	154	12	sets	set	NOUN
ejpam-6627	154	13	and	and	CCONJ
ejpam-6627	154	14	function	function	NOUN
ejpam-6627	154	15	spaces	space	NOUN
ejpam-6627	154	16	,	,	PUNCT
ejpam-6627	154	17	demonstrating	demonstrate	VERB
ejpam-6627	154	18	how	how	SCONJ
ejpam-6627	154	19	the	the	DET
ejpam-6627	154	20	structure	structure	NOUN
ejpam-6627	154	21	of	of	ADP
ejpam-6627	154	22	continuous	continuous	ADJ
ejpam-6627	154	23	function	function	NOUN
ejpam-6627	154	24	algebras	algebra	NOUN
ejpam-6627	154	25	determines	determine	VERB
ejpam-6627	154	26	topological	topological	ADJ
ejpam-6627	154	27	space	space	NOUN
ejpam-6627	154	28	properties	property	NOUN
ejpam-6627	154	29	,	,	PUNCT
ejpam-6627	154	30	as	as	SCONJ
ejpam-6627	154	31	extensively	extensively	ADV
ejpam-6627	154	32	studied	study	VERB
ejpam-6627	154	33	by	by	ADP
ejpam-6627	154	34	gillman	gillman	PROPN
ejpam-6627	154	35	and	and	CCONJ
ejpam-6627	154	36	jerison	jerison	NOUN
ejpam-6627	154	37	[	[	X
ejpam-6627	154	38	23	23	NUM
ejpam-6627	154	39	]	]	PUNCT
ejpam-6627	154	40	.	.	PUNCT
ejpam-6627	155	1	we	we	PRON
ejpam-6627	155	2	conclude	conclude	VERB
ejpam-6627	155	3	this	this	DET
ejpam-6627	155	4	section	section	NOUN
ejpam-6627	155	5	by	by	ADP
ejpam-6627	155	6	noting	note	VERB
ejpam-6627	155	7	that	that	SCONJ
ejpam-6627	155	8	the	the	DET
ejpam-6627	155	9	class	class	NOUN
ejpam-6627	155	10	of	of	ADP
ejpam-6627	155	11	t1	t1	PROPN
ejpam-6627	155	12	spaces	space	NOUN
ejpam-6627	155	13	with	with	ADP
ejpam-6627	155	14	closed	closed	ADJ
ejpam-6627	155	15	sets	set	NOUN
ejpam-6627	155	16	as	as	ADP
ejpam-6627	155	17	sigma	sigma	NOUN
ejpam-6627	155	18	-	-	PUNCT
ejpam-6627	155	19	intersections	intersection	NOUN
ejpam-6627	155	20	forms	form	NOUN
ejpam-6627	155	21	a	a	DET
ejpam-6627	155	22	natural	natural	ADJ
ejpam-6627	155	23	bridge	bridge	NOUN
ejpam-6627	155	24	between	between	ADP
ejpam-6627	155	25	general	general	ADJ
ejpam-6627	155	26	t1	t1	PROPN
ejpam-6627	155	27	spaces	space	NOUN
ejpam-6627	155	28	and	and	CCONJ
ejpam-6627	155	29	metrizable	metrizable	ADJ
ejpam-6627	155	30	spaces	space	NOUN
ejpam-6627	155	31	.	.	PUNCT
ejpam-6627	156	1	4	4	X
ejpam-6627	156	2	.	.	X
ejpam-6627	156	3	urysohn	urysohn	PROPN
ejpam-6627	156	4	spaces	space	NOUN
ejpam-6627	156	5	without	without	ADP
ejpam-6627	156	6	the	the	DET
ejpam-6627	156	7	hausdorff	hausdorff	NOUN
ejpam-6627	156	8	property	property	NOUN
ejpam-6627	156	9	the	the	DET
ejpam-6627	156	10	urysohn	urysohn	PROPN
ejpam-6627	156	11	property	property	NOUN
ejpam-6627	156	12	(	(	PUNCT
ejpam-6627	156	13	t2	t2	NOUN
ejpam-6627	156	14	1	1	NUM
ejpam-6627	156	15	2	2	NUM
ejpam-6627	156	16	)	)	PUNCT
ejpam-6627	156	17	is	be	AUX
ejpam-6627	156	18	traditionally	traditionally	ADV
ejpam-6627	156	19	viewed	view	VERB
ejpam-6627	156	20	as	as	ADP
ejpam-6627	156	21	a	a	DET
ejpam-6627	156	22	mild	mild	ADJ
ejpam-6627	156	23	strengthening	strengthening	NOUN
ejpam-6627	156	24	of	of	ADP
ejpam-6627	156	25	the	the	DET
ejpam-6627	156	26	hausdorff	hausdorff	NOUN
ejpam-6627	156	27	property	property	NOUN
ejpam-6627	156	28	(	(	PUNCT
ejpam-6627	156	29	t2	t2	NOUN
ejpam-6627	156	30	)	)	PUNCT
ejpam-6627	156	31	.	.	PUNCT
ejpam-6627	157	1	however	however	ADV
ejpam-6627	157	2	,	,	PUNCT
ejpam-6627	157	3	this	this	DET
ejpam-6627	157	4	section	section	NOUN
ejpam-6627	157	5	reveals	reveal	VERB
ejpam-6627	157	6	that	that	SCONJ
ejpam-6627	157	7	these	these	DET
ejpam-6627	157	8	properties	property	NOUN
ejpam-6627	157	9	can	can	AUX
ejpam-6627	157	10	behave	behave	VERB
ejpam-6627	157	11	quite	quite	ADV
ejpam-6627	157	12	differently	differently	ADV
ejpam-6627	157	13	when	when	SCONJ
ejpam-6627	157	14	combined	combine	VERB
ejpam-6627	157	15	with	with	ADP
ejpam-6627	157	16	other	other	ADJ
ejpam-6627	157	17	topological	topological	ADJ
ejpam-6627	157	18	conditions	condition	NOUN
ejpam-6627	157	19	.	.	PUNCT
ejpam-6627	158	1	we	we	PRON
ejpam-6627	158	2	begin	begin	VERB
ejpam-6627	158	3	with	with	ADP
ejpam-6627	158	4	a	a	DET
ejpam-6627	158	5	fundamental	fundamental	ADJ
ejpam-6627	158	6	characterization	characterization	NOUN
ejpam-6627	158	7	that	that	PRON
ejpam-6627	158	8	illuminates	illuminate	VERB
ejpam-6627	158	9	the	the	DET
ejpam-6627	158	10	functional	functional	ADJ
ejpam-6627	158	11	nature	nature	NOUN
ejpam-6627	158	12	of	of	ADP
ejpam-6627	158	13	urysohn	urysohn	PROPN
ejpam-6627	158	14	spaces	space	NOUN
ejpam-6627	158	15	,	,	PUNCT
ejpam-6627	158	16	building	build	VERB
ejpam-6627	158	17	on	on	ADP
ejpam-6627	158	18	the	the	DET
ejpam-6627	158	19	classical	classical	ADJ
ejpam-6627	158	20	work	work	NOUN
ejpam-6627	158	21	of	of	ADP
ejpam-6627	158	22	urysohn	urysohn	NOUN
ejpam-6627	158	23	[	[	X
ejpam-6627	158	24	25	25	NUM
ejpam-6627	158	25	]	]	PUNCT
ejpam-6627	158	26	and	and	CCONJ
ejpam-6627	158	27	modern	modern	ADJ
ejpam-6627	158	28	treatments	treatment	NOUN
ejpam-6627	158	29	by	by	ADP
ejpam-6627	158	30	engelking	engelke	VERB
ejpam-6627	158	31	[	[	X
ejpam-6627	158	32	8	8	NUM
ejpam-6627	158	33	]	]	PUNCT
ejpam-6627	158	34	.	.	PUNCT
ejpam-6627	159	1	theorem	theorem	ADJ
ejpam-6627	159	2	4	4	NUM
ejpam-6627	159	3	.	.	PUNCT
ejpam-6627	160	1	a	a	DET
ejpam-6627	160	2	topological	topological	ADJ
ejpam-6627	160	3	space	space	NOUN
ejpam-6627	160	4	ω	ω	PROPN
ejpam-6627	160	5	is	be	AUX
ejpam-6627	160	6	urysohn	urysohn	PROPN
ejpam-6627	160	7	if	if	SCONJ
ejpam-6627	160	8	and	and	CCONJ
ejpam-6627	160	9	only	only	ADV
ejpam-6627	160	10	if	if	SCONJ
ejpam-6627	160	11	for	for	ADP
ejpam-6627	160	12	any	any	DET
ejpam-6627	160	13	distinct	distinct	ADJ
ejpam-6627	160	14	points	point	NOUN
ejpam-6627	160	15	α	α	NOUN
ejpam-6627	160	16	and	and	CCONJ
ejpam-6627	160	17	β	β	NOUN
ejpam-6627	160	18	,	,	PUNCT
ejpam-6627	160	19	there	there	PRON
ejpam-6627	160	20	exist	exist	VERB
ejpam-6627	160	21	continuous	continuous	ADJ
ejpam-6627	160	22	functions	function	NOUN
ejpam-6627	161	1	f	f	NOUN
ejpam-6627	161	2	,	,	PUNCT
ejpam-6627	161	3	g	g	PROPN
ejpam-6627	161	4	:	:	PUNCT
ejpam-6627	161	5	ω	ω	PROPN
ejpam-6627	161	6	→	→	PUNCT
ejpam-6627	162	1	[	[	X
ejpam-6627	162	2	0	0	NUM
ejpam-6627	162	3	,	,	PUNCT
ejpam-6627	162	4	1	1	NUM
ejpam-6627	162	5	]	]	PUNCT
ejpam-6627	162	6	such	such	ADJ
ejpam-6627	162	7	that	that	DET
ejpam-6627	162	8	f(α	f(α	NOUN
ejpam-6627	162	9	)	)	PUNCT
ejpam-6627	162	10	=	=	SYM
ejpam-6627	162	11	1	1	NUM
ejpam-6627	162	12	,	,	PUNCT
ejpam-6627	162	13	f(β	f(β	NOUN
ejpam-6627	162	14	)	)	PUNCT
ejpam-6627	162	15	=	=	SYM
ejpam-6627	162	16	0	0	NUM
ejpam-6627	162	17	,	,	PUNCT
ejpam-6627	162	18	g(α	g(α	NOUN
ejpam-6627	162	19	)	)	PUNCT
ejpam-6627	162	20	=	=	SYM
ejpam-6627	162	21	0	0	NUM
ejpam-6627	162	22	,	,	PUNCT
ejpam-6627	162	23	and	and	CCONJ
ejpam-6627	162	24	g(β	g(β	NOUN
ejpam-6627	162	25	)	)	PUNCT
ejpam-6627	162	26	=	=	SYM
ejpam-6627	163	1	1	1	NUM
ejpam-6627	163	2	,	,	PUNCT
ejpam-6627	163	3	and	and	CCONJ
ejpam-6627	163	4	the	the	DET
ejpam-6627	163	5	supports	support	NOUN
ejpam-6627	163	6	of	of	ADP
ejpam-6627	163	7	f	f	PROPN
ejpam-6627	163	8	and	and	CCONJ
ejpam-6627	163	9	g	g	PROPN
ejpam-6627	163	10	are	be	AUX
ejpam-6627	163	11	disjoint	disjoint	ADJ
ejpam-6627	163	12	.	.	PUNCT
ejpam-6627	164	1	proof	proof	NOUN
ejpam-6627	164	2	.	.	PUNCT
ejpam-6627	165	1	first	first	ADV
ejpam-6627	165	2	,	,	PUNCT
ejpam-6627	165	3	assume	assume	VERB
ejpam-6627	165	4	ω	ω	PROPN
ejpam-6627	165	5	is	be	AUX
ejpam-6627	165	6	urysohn	urysohn	PROPN
ejpam-6627	165	7	.	.	PUNCT
ejpam-6627	166	1	let	let	VERB
ejpam-6627	166	2	α	α	PRON
ejpam-6627	166	3	,	,	PUNCT
ejpam-6627	166	4	β	β	X
ejpam-6627	166	5	∈	∈	PROPN
ejpam-6627	166	6	ω	ω	PROPN
ejpam-6627	166	7	with	with	ADP
ejpam-6627	166	8	α	α	PROPN
ejpam-6627	166	9	̸=	̸=	PROPN
ejpam-6627	166	10	β	β	NOUN
ejpam-6627	166	11	.	.	PUNCT
ejpam-6627	167	1	by	by	ADP
ejpam-6627	167	2	definition	definition	NOUN
ejpam-6627	167	3	,	,	PUNCT
ejpam-6627	167	4	there	there	PRON
ejpam-6627	167	5	exist	exist	VERB
ejpam-6627	167	6	open	open	ADJ
ejpam-6627	167	7	sets	set	NOUN
ejpam-6627	167	8	λ	λ	PROPN
ejpam-6627	167	9	and	and	CCONJ
ejpam-6627	167	10	γ	γ	NOUN
ejpam-6627	167	11	such	such	ADJ
ejpam-6627	167	12	that	that	SCONJ
ejpam-6627	167	13	α	α	PROPN
ejpam-6627	167	14	∈	∈	PROPN
ejpam-6627	167	15	λ	λ	PROPN
ejpam-6627	167	16	,	,	PUNCT
ejpam-6627	167	17	β	β	PROPN
ejpam-6627	167	18	∈	∈	PROPN
ejpam-6627	167	19	γ	γ	X
ejpam-6627	167	20	,	,	PUNCT
ejpam-6627	167	21	and	and	CCONJ
ejpam-6627	167	22	(	(	PUNCT
ejpam-6627	167	23	λ	λ	NOUN
ejpam-6627	167	24	)	)	PUNCT
ejpam-6627	167	25	∩	∩	NOUN
ejpam-6627	167	26	(	(	PUNCT
ejpam-6627	167	27	γ	γ	X
ejpam-6627	167	28	)	)	PUNCT
ejpam-6627	167	29	=	=	NOUN
ejpam-6627	167	30	∅.	∅.	VERB
ejpam-6627	167	31	using	use	VERB
ejpam-6627	167	32	urysohn	urysohn	PROPN
ejpam-6627	167	33	’s	’s	PART
ejpam-6627	167	34	lemma[27	lemma[27	NOUN
ejpam-6627	167	35	]	]	PUNCT
ejpam-6627	167	36	,	,	PUNCT
ejpam-6627	167	37	we	we	PRON
ejpam-6627	167	38	can	can	AUX
ejpam-6627	167	39	construct	construct	VERB
ejpam-6627	167	40	continuous	continuous	ADJ
ejpam-6627	167	41	functions	function	NOUN
ejpam-6627	167	42	f	f	NOUN
ejpam-6627	167	43	,	,	PUNCT
ejpam-6627	167	44	g	g	PROPN
ejpam-6627	167	45	:	:	PUNCT
ejpam-6627	167	46	ω	ω	PROPN
ejpam-6627	167	47	→	→	PUNCT
ejpam-6627	168	1	[	[	X
ejpam-6627	168	2	0	0	NUM
ejpam-6627	168	3	,	,	PUNCT
ejpam-6627	168	4	1	1	NUM
ejpam-6627	168	5	]	]	PUNCT
ejpam-6627	168	6	such	such	ADJ
ejpam-6627	168	7	that	that	DET
ejpam-6627	168	8	•	•	NOUN
ejpam-6627	168	9	f(α	f(α	NOUN
ejpam-6627	168	10	)	)	PUNCT
ejpam-6627	168	11	=	=	SYM
ejpam-6627	168	12	1	1	NUM
ejpam-6627	168	13	and	and	CCONJ
ejpam-6627	168	14	f(z	f(z	PROPN
ejpam-6627	168	15	)	)	PUNCT
ejpam-6627	169	1	=	=	SYM
ejpam-6627	169	2	0	0	NUM
ejpam-6627	170	1	for	for	ADP
ejpam-6627	170	2	all	all	DET
ejpam-6627	170	3	z	z	NOUN
ejpam-6627	170	4	∈	∈	PROPN
ejpam-6627	170	5	(	(	PUNCT
ejpam-6627	170	6	γ	γ	NOUN
ejpam-6627	170	7	)	)	PUNCT
ejpam-6627	170	8	•	•	NOUN
ejpam-6627	170	9	g(β	g(β	NOUN
ejpam-6627	170	10	)	)	PUNCT
ejpam-6627	170	11	=	=	SYM
ejpam-6627	170	12	1	1	NUM
ejpam-6627	170	13	and	and	CCONJ
ejpam-6627	170	14	g(z	g(z	ADJ
ejpam-6627	170	15	)	)	PUNCT
ejpam-6627	170	16	=	=	SYM
ejpam-6627	170	17	0	0	NUM
ejpam-6627	170	18	for	for	ADP
ejpam-6627	170	19	all	all	DET
ejpam-6627	170	20	z	z	NOUN
ejpam-6627	170	21	∈	∈	PROPN
ejpam-6627	170	22	(	(	PUNCT
ejpam-6627	170	23	λ	λ	NOUN
ejpam-6627	170	24	)	)	PUNCT
ejpam-6627	170	25	since	since	SCONJ
ejpam-6627	170	26	(	(	PUNCT
ejpam-6627	170	27	λ	λ	NOUN
ejpam-6627	170	28	)	)	PUNCT
ejpam-6627	170	29	∩	∩	NOUN
ejpam-6627	170	30	(	(	PUNCT
ejpam-6627	170	31	γ	γ	NOUN
ejpam-6627	170	32	)	)	PUNCT
ejpam-6627	170	33	=	=	SYM
ejpam-6627	170	34	∅	∅	NOUN
ejpam-6627	170	35	,	,	PUNCT
ejpam-6627	170	36	the	the	DET
ejpam-6627	170	37	supports	support	NOUN
ejpam-6627	170	38	of	of	ADP
ejpam-6627	170	39	f	f	PROPN
ejpam-6627	170	40	and	and	CCONJ
ejpam-6627	170	41	g	g	PROPN
ejpam-6627	170	42	are	be	AUX
ejpam-6627	170	43	disjoint	disjoint	ADJ
ejpam-6627	170	44	.	.	PUNCT
ejpam-6627	171	1	conversely	conversely	ADV
ejpam-6627	171	2	,	,	PUNCT
ejpam-6627	171	3	assume	assume	VERB
ejpam-6627	171	4	that	that	SCONJ
ejpam-6627	171	5	for	for	ADP
ejpam-6627	171	6	any	any	DET
ejpam-6627	171	7	distinct	distinct	ADJ
ejpam-6627	171	8	points	point	NOUN
ejpam-6627	171	9	α	α	NOUN
ejpam-6627	171	10	and	and	CCONJ
ejpam-6627	171	11	β	β	NOUN
ejpam-6627	171	12	,	,	PUNCT
ejpam-6627	171	13	there	there	PRON
ejpam-6627	171	14	exist	exist	VERB
ejpam-6627	171	15	continuous	continuous	ADJ
ejpam-6627	171	16	functions	function	NOUN
ejpam-6627	172	1	f	f	NOUN
ejpam-6627	172	2	,	,	PUNCT
ejpam-6627	172	3	g	g	PROPN
ejpam-6627	172	4	:	:	PUNCT
ejpam-6627	172	5	ω	ω	PROPN
ejpam-6627	172	6	→	→	PUNCT
ejpam-6627	173	1	[	[	X
ejpam-6627	173	2	0	0	NUM
ejpam-6627	173	3	,	,	PUNCT
ejpam-6627	173	4	1	1	NUM
ejpam-6627	173	5	]	]	PUNCT
ejpam-6627	173	6	with	with	ADP
ejpam-6627	173	7	the	the	DET
ejpam-6627	173	8	stated	state	VERB
ejpam-6627	173	9	properties	property	NOUN
ejpam-6627	173	10	.	.	PUNCT
ejpam-6627	174	1	let	let	VERB
ejpam-6627	174	2	α	α	PRON
ejpam-6627	174	3	,	,	PUNCT
ejpam-6627	174	4	β	β	X
ejpam-6627	174	5	∈	∈	PROPN
ejpam-6627	174	6	ω	ω	PROPN
ejpam-6627	174	7	with	with	ADP
ejpam-6627	174	8	α	α	PROPN
ejpam-6627	174	9	̸=	̸=	PROPN
ejpam-6627	174	10	β	β	NOUN
ejpam-6627	174	11	.	.	PUNCT
ejpam-6627	175	1	by	by	ADP
ejpam-6627	175	2	assumption	assumption	NOUN
ejpam-6627	175	3	,	,	PUNCT
ejpam-6627	175	4	there	there	PRON
ejpam-6627	175	5	exist	exist	VERB
ejpam-6627	175	6	continuous	continuous	ADJ
ejpam-6627	175	7	functions	function	NOUN
ejpam-6627	175	8	f	f	NOUN
ejpam-6627	175	9	,	,	PUNCT
ejpam-6627	175	10	g	g	PROPN
ejpam-6627	175	11	with	with	ADP
ejpam-6627	175	12	disjoint	disjoint	NOUN
ejpam-6627	175	13	supports	support	VERB
ejpam-6627	175	14	such	such	ADJ
ejpam-6627	175	15	that	that	DET
ejpam-6627	175	16	f(α	f(α	NOUN
ejpam-6627	175	17	)	)	PUNCT
ejpam-6627	175	18	=	=	SYM
ejpam-6627	176	1	1	1	NUM
ejpam-6627	176	2	,	,	PUNCT
ejpam-6627	176	3	f(β	f(β	NOUN
ejpam-6627	176	4	)	)	PUNCT
ejpam-6627	176	5	=	=	SYM
ejpam-6627	176	6	0	0	NUM
ejpam-6627	176	7	,	,	PUNCT
ejpam-6627	176	8	g(α	g(α	NOUN
ejpam-6627	176	9	)	)	PUNCT
ejpam-6627	176	10	=	=	SYM
ejpam-6627	176	11	0	0	NUM
ejpam-6627	176	12	,	,	PUNCT
ejpam-6627	176	13	and	and	CCONJ
ejpam-6627	176	14	g(β	g(β	NOUN
ejpam-6627	176	15	)	)	PUNCT
ejpam-6627	176	16	=	=	SYM
ejpam-6627	177	1	1	1	X
ejpam-6627	177	2	.	.	PUNCT
ejpam-6627	177	3	let	let	VERB
ejpam-6627	177	4	λ	λ	X
ejpam-6627	177	5	=	=	SYM
ejpam-6627	177	6	f−1((0	f−1((0	NOUN
ejpam-6627	177	7	,	,	PUNCT
ejpam-6627	177	8	1	1	NUM
ejpam-6627	177	9	]	]	PUNCT
ejpam-6627	177	10	)	)	PUNCT
ejpam-6627	177	11	and	and	CCONJ
ejpam-6627	177	12	γ	γ	PROPN
ejpam-6627	177	13	=	=	SYM
ejpam-6627	177	14	g−1((0	g−1((0	PROPN
ejpam-6627	177	15	,	,	PUNCT
ejpam-6627	177	16	1	1	NUM
ejpam-6627	177	17	]	]	PUNCT
ejpam-6627	177	18	)	)	PUNCT
ejpam-6627	177	19	.	.	PUNCT
ejpam-6627	178	1	then	then	ADV
ejpam-6627	178	2	α	α	PROPN
ejpam-6627	178	3	∈	∈	PROPN
ejpam-6627	178	4	λ	λ	PROPN
ejpam-6627	178	5	,	,	PUNCT
ejpam-6627	178	6	β	β	PROPN
ejpam-6627	178	7	∈	∈	PROPN
ejpam-6627	178	8	γ	γ	X
ejpam-6627	178	9	,	,	PUNCT
ejpam-6627	178	10	and	and	CCONJ
ejpam-6627	178	11	(	(	PUNCT
ejpam-6627	178	12	λ)∩	λ)∩	PROPN
ejpam-6627	178	13	(	(	PUNCT
ejpam-6627	178	14	γ	γ	NOUN
ejpam-6627	178	15	)	)	PUNCT
ejpam-6627	178	16	=	=	NOUN
ejpam-6627	178	17	∅	∅	NOUN
ejpam-6627	178	18	because	because	SCONJ
ejpam-6627	178	19	the	the	DET
ejpam-6627	178	20	supports	support	NOUN
ejpam-6627	178	21	of	of	ADP
ejpam-6627	178	22	f	f	PROPN
ejpam-6627	178	23	and	and	CCONJ
ejpam-6627	178	24	g	g	PROPN
ejpam-6627	178	25	are	be	AUX
ejpam-6627	178	26	disjoint	disjoint	ADJ
ejpam-6627	178	27	.	.	PUNCT
ejpam-6627	179	1	this	this	PRON
ejpam-6627	179	2	establishes	establish	VERB
ejpam-6627	179	3	that	that	SCONJ
ejpam-6627	179	4	ω	ω	PROPN
ejpam-6627	179	5	is	be	AUX
ejpam-6627	179	6	urysohn	urysohn	ADJ
ejpam-6627	179	7	.	.	PUNCT
ejpam-6627	180	1	this	this	DET
ejpam-6627	180	2	characterization	characterization	NOUN
ejpam-6627	180	3	demonstrates	demonstrate	VERB
ejpam-6627	180	4	the	the	DET
ejpam-6627	180	5	intimate	intimate	ADJ
ejpam-6627	180	6	connection	connection	NOUN
ejpam-6627	180	7	between	between	ADP
ejpam-6627	180	8	the	the	DET
ejpam-6627	180	9	urysohn	urysohn	PROPN
ejpam-6627	180	10	property	property	NOUN
ejpam-6627	180	11	and	and	CCONJ
ejpam-6627	180	12	functional	functional	ADJ
ejpam-6627	180	13	separation	separation	NOUN
ejpam-6627	180	14	,	,	PUNCT
ejpam-6627	180	15	providing	provide	VERB
ejpam-6627	180	16	a	a	DET
ejpam-6627	180	17	tool	tool	NOUN
ejpam-6627	180	18	for	for	ADP
ejpam-6627	180	19	constructing	construct	VERB
ejpam-6627	180	20	spaces	space	NOUN
ejpam-6627	180	21	that	that	PRON
ejpam-6627	180	22	are	be	AUX
ejpam-6627	180	23	urysohn	urysohn	ADJ
ejpam-6627	180	24	but	but	CCONJ
ejpam-6627	180	25	not	not	PART
ejpam-6627	180	26	hausdorff	hausdorff	NOUN
ejpam-6627	180	27	.	.	PUNCT
ejpam-6627	181	1	we	we	PRON
ejpam-6627	181	2	now	now	ADV
ejpam-6627	181	3	present	present	VERB
ejpam-6627	181	4	such	such	DET
ejpam-6627	181	5	an	an	DET
ejpam-6627	181	6	example	example	NOUN
ejpam-6627	181	7	,	,	PUNCT
ejpam-6627	181	8	inspired	inspire	VERB
ejpam-6627	181	9	by	by	ADP
ejpam-6627	181	10	techniques	technique	NOUN
ejpam-6627	181	11	from	from	ADP
ejpam-6627	181	12	steen	steen	PROPN
ejpam-6627	181	13	and	and	CCONJ
ejpam-6627	181	14	seebach	seebach	NOUN
ejpam-6627	181	15	[	[	X
ejpam-6627	181	16	24	24	NUM
ejpam-6627	181	17	]	]	PUNCT
ejpam-6627	181	18	.	.	PUNCT
ejpam-6627	182	1	example	example	NOUN
ejpam-6627	183	1	3	3	X
ejpam-6627	183	2	.	.	PUNCT
ejpam-6627	183	3	let	let	VERB
ejpam-6627	183	4	ω	ω	NUM
ejpam-6627	183	5	be	be	AUX
ejpam-6627	183	6	an	an	DET
ejpam-6627	183	7	infinite	infinite	ADJ
ejpam-6627	183	8	set	set	NOUN
ejpam-6627	183	9	and	and	CCONJ
ejpam-6627	183	10	fix	fix	VERB
ejpam-6627	183	11	a	a	DET
ejpam-6627	183	12	point	point	NOUN
ejpam-6627	183	13	ρ	ρ	X
ejpam-6627	183	14	∈	∈	PROPN
ejpam-6627	183	15	ω	ω	PROPN
ejpam-6627	183	16	.	.	PUNCT
ejpam-6627	183	17	define	define	VERB
ejpam-6627	183	18	a	a	DET
ejpam-6627	183	19	topology	topology	NOUN
ejpam-6627	183	20	on	on	ADP
ejpam-6627	183	21	ω	ω	PROPN
ejpam-6627	183	22	as	as	SCONJ
ejpam-6627	183	23	follows	follow	VERB
ejpam-6627	183	24	:	:	PUNCT
ejpam-6627	183	25	•	•	NOUN
ejpam-6627	183	26	every	every	DET
ejpam-6627	183	27	point	point	NOUN
ejpam-6627	183	28	except	except	SCONJ
ejpam-6627	183	29	ρ	ρ	PROPN
ejpam-6627	183	30	is	be	AUX
ejpam-6627	183	31	isolated	isolate	VERB
ejpam-6627	183	32	(	(	PUNCT
ejpam-6627	183	33	i.e.	i.e.	X
ejpam-6627	183	34	,	,	PUNCT
ejpam-6627	183	35	every	every	DET
ejpam-6627	183	36	singleton	singleton	NOUN
ejpam-6627	183	37	not	not	PART
ejpam-6627	183	38	containing	contain	VERB
ejpam-6627	183	39	ρ	ρ	PROPN
ejpam-6627	183	40	is	be	AUX
ejpam-6627	183	41	open	open	ADJ
ejpam-6627	183	42	)	)	PUNCT
ejpam-6627	183	43	.	.	PUNCT
ejpam-6627	184	1	j.	j.	PROPN
ejpam-6627	184	2	oudetallah	oudetallah	PROPN
ejpam-6627	184	3	et	et	PROPN
ejpam-6627	184	4	al	al	PROPN
ejpam-6627	184	5	.	.	PUNCT
ejpam-6627	184	6	/	/	SYM
ejpam-6627	184	7	eur	eur	PROPN
ejpam-6627	184	8	.	.	PUNCT
ejpam-6627	185	1	j.	j.	PROPN
ejpam-6627	185	2	pure	pure	PROPN
ejpam-6627	185	3	appl	appl	PROPN
ejpam-6627	185	4	.	.	PROPN
ejpam-6627	185	5	math	math	PROPN
ejpam-6627	185	6	,	,	PUNCT
ejpam-6627	185	7	18	18	NUM
ejpam-6627	185	8	(	(	PUNCT
ejpam-6627	185	9	3	3	NUM
ejpam-6627	185	10	)	)	PUNCT
ejpam-6627	185	11	(	(	PUNCT
ejpam-6627	185	12	2025	2025	NUM
ejpam-6627	185	13	)	)	PUNCT
ejpam-6627	185	14	,	,	PUNCT
ejpam-6627	185	15	6627	6627	NUM
ejpam-6627	185	16	9	9	NUM
ejpam-6627	185	17	of	of	ADP
ejpam-6627	185	18	20	20	NUM
ejpam-6627	185	19	•	•	NOUN
ejpam-6627	185	20	a	a	DET
ejpam-6627	185	21	set	set	ADJ
ejpam-6627	185	22	λ	λ	NOUN
ejpam-6627	185	23	containing	contain	VERB
ejpam-6627	185	24	ρ	ρ	PROPN
ejpam-6627	185	25	is	be	AUX
ejpam-6627	185	26	open	open	ADJ
ejpam-6627	185	27	if	if	SCONJ
ejpam-6627	185	28	and	and	CCONJ
ejpam-6627	185	29	only	only	ADV
ejpam-6627	185	30	if	if	SCONJ
ejpam-6627	185	31	ω\λ	ω\λ	ADJ
ejpam-6627	185	32	is	be	AUX
ejpam-6627	185	33	finite	finite	ADJ
ejpam-6627	185	34	and	and	CCONJ
ejpam-6627	185	35	there	there	PRON
ejpam-6627	185	36	exists	exist	VERB
ejpam-6627	185	37	a	a	DET
ejpam-6627	185	38	positive	positive	ADJ
ejpam-6627	185	39	integer	integer	NOUN
ejpam-6627	185	40	n	n	CCONJ
ejpam-6627	185	41	such	such	ADJ
ejpam-6627	185	42	that	that	PRON
ejpam-6627	185	43	for	for	ADP
ejpam-6627	185	44	all	all	DET
ejpam-6627	185	45	α	α	NOUN
ejpam-6627	185	46	∈	∈	PROPN
ejpam-6627	185	47	ω	ω	NUM
ejpam-6627	185	48	\	\	PROPN
ejpam-6627	185	49	λ	λ	PROPN
ejpam-6627	185	50	,	,	PUNCT
ejpam-6627	185	51	δ(α	δ(α	PROPN
ejpam-6627	185	52	,	,	PUNCT
ejpam-6627	185	53	ρ	ρ	NOUN
ejpam-6627	185	54	)	)	PUNCT
ejpam-6627	185	55	>	>	X
ejpam-6627	185	56	1	1	NUM
ejpam-6627	185	57	/	/	SYM
ejpam-6627	185	58	n	n	NOUN
ejpam-6627	185	59	in	in	ADP
ejpam-6627	185	60	some	some	DET
ejpam-6627	185	61	predetermined	predetermine	VERB
ejpam-6627	185	62	metric	metric	ADJ
ejpam-6627	185	63	δ	δ	PROPN
ejpam-6627	185	64	on	on	ADP
ejpam-6627	185	65	ω	ω	PROPN
ejpam-6627	185	66	.	.	PUNCT
ejpam-6627	186	1	this	this	DET
ejpam-6627	186	2	space	space	NOUN
ejpam-6627	186	3	is	be	AUX
ejpam-6627	186	4	clearly	clearly	ADV
ejpam-6627	186	5	t1	t1	NOUN
ejpam-6627	186	6	since	since	SCONJ
ejpam-6627	186	7	every	every	DET
ejpam-6627	186	8	singleton	singleton	NOUN
ejpam-6627	186	9	is	be	AUX
ejpam-6627	186	10	closed	closed	ADJ
ejpam-6627	186	11	.	.	PUNCT
ejpam-6627	187	1	to	to	PART
ejpam-6627	187	2	show	show	VERB
ejpam-6627	187	3	it	it	PRON
ejpam-6627	187	4	is	be	AUX
ejpam-6627	187	5	urysohn	urysohn	PROPN
ejpam-6627	187	6	,	,	PUNCT
ejpam-6627	187	7	let	let	VERB
ejpam-6627	187	8	α	α	PRON
ejpam-6627	187	9	,	,	PUNCT
ejpam-6627	187	10	β	β	X
ejpam-6627	187	11	∈	∈	PROPN
ejpam-6627	187	12	ω	ω	PROPN
ejpam-6627	187	13	with	with	ADP
ejpam-6627	187	14	α	α	PROPN
ejpam-6627	187	15	̸=	̸=	PROPN
ejpam-6627	187	16	β	β	NOUN
ejpam-6627	187	17	.	.	PUNCT
ejpam-6627	188	1	if	if	SCONJ
ejpam-6627	188	2	neither	neither	CCONJ
ejpam-6627	188	3	α	α	NOUN
ejpam-6627	188	4	nor	nor	CCONJ
ejpam-6627	188	5	β	β	X
ejpam-6627	188	6	is	be	AUX
ejpam-6627	188	7	ρ	ρ	PROPN
ejpam-6627	188	8	,	,	PUNCT
ejpam-6627	188	9	then	then	ADV
ejpam-6627	188	10	they	they	PRON
ejpam-6627	188	11	are	be	AUX
ejpam-6627	188	12	isolated	isolated	ADJ
ejpam-6627	188	13	points	point	NOUN
ejpam-6627	188	14	,	,	PUNCT
ejpam-6627	188	15	so	so	SCONJ
ejpam-6627	188	16	we	we	PRON
ejpam-6627	188	17	can	can	AUX
ejpam-6627	188	18	easily	easily	ADV
ejpam-6627	188	19	find	find	VERB
ejpam-6627	188	20	disjoint	disjoint	NOUN
ejpam-6627	188	21	open	open	ADJ
ejpam-6627	188	22	neighborhoods	neighborhood	NOUN
ejpam-6627	188	23	with	with	ADP
ejpam-6627	188	24	disjoint	disjoint	NOUN
ejpam-6627	188	25	closures	closure	NOUN
ejpam-6627	188	26	.	.	PUNCT
ejpam-6627	189	1	if	if	SCONJ
ejpam-6627	189	2	one	one	NUM
ejpam-6627	189	3	of	of	ADP
ejpam-6627	189	4	them	they	PRON
ejpam-6627	189	5	,	,	PUNCT
ejpam-6627	189	6	say	say	VERB
ejpam-6627	189	7	α	α	X
ejpam-6627	189	8	,	,	PUNCT
ejpam-6627	189	9	is	be	AUX
ejpam-6627	189	10	ρ	ρ	NOUN
ejpam-6627	189	11	,	,	PUNCT
ejpam-6627	189	12	we	we	PRON
ejpam-6627	189	13	can	can	AUX
ejpam-6627	189	14	define	define	VERB
ejpam-6627	189	15	open	open	ADJ
ejpam-6627	189	16	sets	set	NOUN
ejpam-6627	189	17	λ	λ	NOUN
ejpam-6627	189	18	containing	contain	VERB
ejpam-6627	189	19	ρ	ρ	PROPN
ejpam-6627	189	20	and	and	CCONJ
ejpam-6627	189	21	γ	γ	X
ejpam-6627	189	22	=	=	SYM
ejpam-6627	189	23	{	{	PUNCT
ejpam-6627	189	24	β	β	X
ejpam-6627	189	25	}	}	PUNCT
ejpam-6627	189	26	such	such	ADJ
ejpam-6627	189	27	that	that	SCONJ
ejpam-6627	189	28	(	(	PUNCT
ejpam-6627	189	29	λ	λ	NOUN
ejpam-6627	189	30	)	)	PUNCT
ejpam-6627	189	31	∩	∩	NOUN
ejpam-6627	189	32	(	(	PUNCT
ejpam-6627	189	33	γ	γ	NOUN
ejpam-6627	189	34	)	)	PUNCT
ejpam-6627	189	35	=	=	PUNCT
ejpam-6627	189	36	∅	∅	NOUN
ejpam-6627	189	37	by	by	ADP
ejpam-6627	189	38	choosing	choose	VERB
ejpam-6627	189	39	λ	λ	NOUN
ejpam-6627	189	40	to	to	PART
ejpam-6627	189	41	exclude	exclude	VERB
ejpam-6627	189	42	a	a	DET
ejpam-6627	189	43	suitably	suitably	ADV
ejpam-6627	189	44	large	large	ADJ
ejpam-6627	189	45	finite	finite	NOUN
ejpam-6627	189	46	set	set	NOUN
ejpam-6627	189	47	containing	contain	VERB
ejpam-6627	189	48	β	β	X
ejpam-6627	189	49	.	.	PUNCT
ejpam-6627	190	1	however	however	ADV
ejpam-6627	190	2	,	,	PUNCT
ejpam-6627	190	3	the	the	DET
ejpam-6627	190	4	space	space	NOUN
ejpam-6627	190	5	fails	fail	VERB
ejpam-6627	190	6	to	to	PART
ejpam-6627	190	7	be	be	AUX
ejpam-6627	190	8	hausdorff	hausdorff	NOUN
ejpam-6627	190	9	at	at	ADP
ejpam-6627	190	10	certain	certain	ADJ
ejpam-6627	190	11	point	point	NOUN
ejpam-6627	190	12	pairs	pair	NOUN
ejpam-6627	190	13	,	,	PUNCT
ejpam-6627	190	14	particularly	particularly	ADV
ejpam-6627	190	15	those	those	PRON
ejpam-6627	190	16	involving	involve	VERB
ejpam-6627	190	17	points	point	NOUN
ejpam-6627	190	18	that	that	PRON
ejpam-6627	190	19	converge	converge	VERB
ejpam-6627	190	20	to	to	ADP
ejpam-6627	190	21	ρ	ρ	PROPN
ejpam-6627	190	22	in	in	ADP
ejpam-6627	190	23	a	a	DET
ejpam-6627	190	24	specific	specific	ADJ
ejpam-6627	190	25	manner	manner	NOUN
ejpam-6627	190	26	.	.	PUNCT
ejpam-6627	191	1	for	for	ADP
ejpam-6627	191	2	instance	instance	NOUN
ejpam-6627	191	3	,	,	PUNCT
ejpam-6627	191	4	if	if	SCONJ
ejpam-6627	191	5	we	we	PRON
ejpam-6627	191	6	consider	consider	VERB
ejpam-6627	191	7	a	a	DET
ejpam-6627	191	8	sequence	sequence	NOUN
ejpam-6627	191	9	of	of	ADP
ejpam-6627	191	10	points	point	NOUN
ejpam-6627	191	11	{	{	PUNCT
ejpam-6627	191	12	αn	αn	NOUN
ejpam-6627	191	13	}	}	PUNCT
ejpam-6627	191	14	such	such	ADJ
ejpam-6627	191	15	that	that	SCONJ
ejpam-6627	191	16	δ(αn	δ(αn	PROPN
ejpam-6627	191	17	,	,	PUNCT
ejpam-6627	191	18	ρ	ρ	NOUN
ejpam-6627	191	19	)	)	PUNCT
ejpam-6627	191	20	→	→	SYM
ejpam-6627	191	21	0	0	NUM
ejpam-6627	191	22	,	,	PUNCT
ejpam-6627	191	23	any	any	DET
ejpam-6627	191	24	open	open	ADJ
ejpam-6627	191	25	set	set	NOUN
ejpam-6627	191	26	containing	contain	VERB
ejpam-6627	191	27	ρ	ρ	PROPN
ejpam-6627	191	28	must	must	AUX
ejpam-6627	191	29	contain	contain	VERB
ejpam-6627	191	30	all	all	PRON
ejpam-6627	191	31	but	but	ADV
ejpam-6627	191	32	finitely	finitely	ADV
ejpam-6627	191	33	many	many	ADJ
ejpam-6627	191	34	points	point	NOUN
ejpam-6627	191	35	of	of	ADP
ejpam-6627	191	36	this	this	DET
ejpam-6627	191	37	sequence	sequence	NOUN
ejpam-6627	191	38	.	.	PUNCT
ejpam-6627	192	1	this	this	PRON
ejpam-6627	192	2	prevents	prevent	VERB
ejpam-6627	192	3	us	we	PRON
ejpam-6627	192	4	from	from	ADP
ejpam-6627	192	5	finding	find	VERB
ejpam-6627	192	6	disjoint	disjoint	ADJ
ejpam-6627	192	7	open	open	ADJ
ejpam-6627	192	8	neighborhoods	neighborhood	NOUN
ejpam-6627	192	9	for	for	ADP
ejpam-6627	192	10	ρ	ρ	PROPN
ejpam-6627	192	11	and	and	CCONJ
ejpam-6627	192	12	certain	certain	ADJ
ejpam-6627	192	13	sets	set	NOUN
ejpam-6627	192	14	of	of	ADP
ejpam-6627	192	15	points	point	NOUN
ejpam-6627	192	16	from	from	ADP
ejpam-6627	192	17	the	the	DET
ejpam-6627	192	18	sequence	sequence	NOUN
ejpam-6627	192	19	,	,	PUNCT
ejpam-6627	192	20	demonstrating	demonstrate	VERB
ejpam-6627	192	21	that	that	SCONJ
ejpam-6627	192	22	the	the	DET
ejpam-6627	192	23	space	space	NOUN
ejpam-6627	192	24	is	be	AUX
ejpam-6627	192	25	not	not	PART
ejpam-6627	192	26	hausdorff	hausdorff	NOUN
ejpam-6627	192	27	.	.	PUNCT
ejpam-6627	193	1	j.	j.	PROPN
ejpam-6627	193	2	oudetallah	oudetallah	PROPN
ejpam-6627	193	3	et	et	PROPN
ejpam-6627	193	4	al	al	PROPN
ejpam-6627	193	5	.	.	PUNCT
ejpam-6627	193	6	/	/	SYM
ejpam-6627	193	7	eur	eur	PROPN
ejpam-6627	193	8	.	.	PUNCT
ejpam-6627	194	1	j.	j.	PROPN
ejpam-6627	194	2	pure	pure	PROPN
ejpam-6627	194	3	appl	appl	PROPN
ejpam-6627	194	4	.	.	PROPN
ejpam-6627	194	5	math	math	PROPN
ejpam-6627	194	6	,	,	PUNCT
ejpam-6627	194	7	18	18	NUM
ejpam-6627	194	8	(	(	PUNCT
ejpam-6627	194	9	3	3	NUM
ejpam-6627	194	10	)	)	PUNCT
ejpam-6627	194	11	(	(	PUNCT
ejpam-6627	194	12	2025	2025	NUM
ejpam-6627	194	13	)	)	PUNCT
ejpam-6627	194	14	,	,	PUNCT
ejpam-6627	194	15	6627	6627	NUM
ejpam-6627	194	16	10	10	NUM
ejpam-6627	194	17	of	of	ADP
ejpam-6627	194	18	20	20	NUM
ejpam-6627	194	19	ρ	ρ	NUM
ejpam-6627	194	20	neighborhood	neighborhood	NOUN
ejpam-6627	194	21	of	of	ADP
ejpam-6627	194	22	ρ	ρ	NUM
ejpam-6627	194	23	•	•	NUM
ejpam-6627	194	24	all	all	DET
ejpam-6627	194	25	points	point	NOUN
ejpam-6627	194	26	except	except	SCONJ
ejpam-6627	194	27	ρ	ρ	PROPN
ejpam-6627	194	28	are	be	AUX
ejpam-6627	194	29	isolated	isolate	VERB
ejpam-6627	194	30	•	•	NUM
ejpam-6627	194	31	neighborhoods	neighborhood	NOUN
ejpam-6627	194	32	of	of	ADP
ejpam-6627	194	33	ρ	ρ	PROPN
ejpam-6627	194	34	exclude	exclude	NOUN
ejpam-6627	194	35	finite	finite	NOUN
ejpam-6627	194	36	sets	set	NOUN
ejpam-6627	194	37	and	and	CCONJ
ejpam-6627	194	38	points	point	NOUN
ejpam-6627	194	39	within	within	ADP
ejpam-6627	194	40	distance	distance	NOUN
ejpam-6627	194	41	1	1	NUM
ejpam-6627	194	42	/	/	SYM
ejpam-6627	194	43	n	n	PROPN
ejpam-6627	194	44	of	of	ADP
ejpam-6627	194	45	ρ	ρ	NUM
ejpam-6627	194	46	•	•	ADP
ejpam-6627	194	47	this	this	DET
ejpam-6627	194	48	space	space	NOUN
ejpam-6627	194	49	is	be	AUX
ejpam-6627	194	50	urysohn	urysohn	ADJ
ejpam-6627	194	51	but	but	CCONJ
ejpam-6627	194	52	not	not	PART
ejpam-6627	194	53	hausdorff	hausdorff	NOUN
ejpam-6627	194	54	figure	figure	NOUN
ejpam-6627	194	55	3	3	NUM
ejpam-6627	194	56	:	:	PUNCT
ejpam-6627	194	57	visualization	visualization	NOUN
ejpam-6627	194	58	of	of	ADP
ejpam-6627	194	59	a	a	DET
ejpam-6627	194	60	urysohn	urysohn	NOUN
ejpam-6627	194	61	space	space	NOUN
ejpam-6627	194	62	that	that	PRON
ejpam-6627	194	63	is	be	AUX
ejpam-6627	194	64	not	not	PART
ejpam-6627	194	65	hausdorff	hausdorff	NOUN
ejpam-6627	194	66	.	.	PUNCT
ejpam-6627	195	1	the	the	DET
ejpam-6627	195	2	special	special	ADJ
ejpam-6627	195	3	point	point	NOUN
ejpam-6627	195	4	ρ	ρ	PROPN
ejpam-6627	195	5	has	have	VERB
ejpam-6627	195	6	neighborhoods	neighborhood	NOUN
ejpam-6627	195	7	that	that	PRON
ejpam-6627	195	8	must	must	AUX
ejpam-6627	195	9	exclude	exclude	VERB
ejpam-6627	195	10	finite	finite	ADJ
ejpam-6627	195	11	sets	set	NOUN
ejpam-6627	195	12	and	and	CCONJ
ejpam-6627	195	13	points	point	NOUN
ejpam-6627	195	14	within	within	ADP
ejpam-6627	195	15	a	a	DET
ejpam-6627	195	16	certain	certain	ADJ
ejpam-6627	195	17	distance	distance	NOUN
ejpam-6627	195	18	threshold	threshold	NOUN
ejpam-6627	195	19	.	.	PUNCT
ejpam-6627	196	1	this	this	DET
ejpam-6627	196	2	construction	construction	NOUN
ejpam-6627	196	3	allows	allow	VERB
ejpam-6627	196	4	the	the	DET
ejpam-6627	196	5	space	space	NOUN
ejpam-6627	196	6	to	to	PART
ejpam-6627	196	7	satisfy	satisfy	VERB
ejpam-6627	196	8	the	the	DET
ejpam-6627	196	9	urysohn	urysohn	PROPN
ejpam-6627	196	10	property	property	NOUN
ejpam-6627	196	11	while	while	SCONJ
ejpam-6627	196	12	failing	fail	VERB
ejpam-6627	196	13	to	to	PART
ejpam-6627	196	14	be	be	AUX
ejpam-6627	196	15	hausdorff	hausdorff	NOUN
ejpam-6627	196	16	.	.	PUNCT
ejpam-6627	197	1	the	the	DET
ejpam-6627	197	2	following	follow	VERB
ejpam-6627	197	3	theorem	theorem	NOUN
ejpam-6627	197	4	provides	provide	VERB
ejpam-6627	197	5	a	a	DET
ejpam-6627	197	6	sufficient	sufficient	ADJ
ejpam-6627	197	7	condition	condition	NOUN
ejpam-6627	197	8	for	for	ADP
ejpam-6627	197	9	a	a	DET
ejpam-6627	197	10	urysohn	urysohn	PROPN
ejpam-6627	197	11	space	space	NOUN
ejpam-6627	197	12	to	to	PART
ejpam-6627	197	13	be	be	AUX
ejpam-6627	197	14	hausdorff	hausdorff	NOUN
ejpam-6627	197	15	,	,	PUNCT
ejpam-6627	197	16	demonstrating	demonstrate	VERB
ejpam-6627	197	17	the	the	DET
ejpam-6627	197	18	role	role	NOUN
ejpam-6627	197	19	of	of	ADP
ejpam-6627	197	20	local	local	ADJ
ejpam-6627	197	21	compactness	compactness	NOUN
ejpam-6627	197	22	.	.	PUNCT
ejpam-6627	198	1	this	this	DET
ejpam-6627	198	2	result	result	NOUN
ejpam-6627	198	3	connects	connect	VERB
ejpam-6627	198	4	with	with	ADP
ejpam-6627	198	5	recent	recent	ADJ
ejpam-6627	198	6	work	work	NOUN
ejpam-6627	198	7	on	on	ADP
ejpam-6627	198	8	r	r	NOUN
ejpam-6627	198	9	-	-	PUNCT
ejpam-6627	198	10	compactness	compactness	NOUN
ejpam-6627	198	11	in	in	ADP
ejpam-6627	198	12	topological	topological	ADJ
ejpam-6627	198	13	spaces	space	NOUN
ejpam-6627	198	14	[	[	X
ejpam-6627	198	15	11	11	NUM
ejpam-6627	198	16	]	]	PUNCT
ejpam-6627	198	17	and	and	CCONJ
ejpam-6627	198	18	the	the	DET
ejpam-6627	198	19	classical	classical	ADJ
ejpam-6627	198	20	treatment	treatment	NOUN
ejpam-6627	198	21	of	of	ADP
ejpam-6627	198	22	compact	compact	ADJ
ejpam-6627	198	23	spaces	space	NOUN
ejpam-6627	198	24	by	by	ADP
ejpam-6627	198	25	kunen	kunen	PROPN
ejpam-6627	199	1	[	[	X
ejpam-6627	199	2	28	28	NUM
ejpam-6627	199	3	]	]	PUNCT
ejpam-6627	199	4	.	.	PUNCT
ejpam-6627	200	1	theorem	theorem	ADJ
ejpam-6627	200	2	5	5	NUM
ejpam-6627	200	3	.	.	PUNCT
ejpam-6627	201	1	if	if	SCONJ
ejpam-6627	201	2	ω	ω	PROPN
ejpam-6627	201	3	is	be	AUX
ejpam-6627	201	4	a	a	DET
ejpam-6627	201	5	urysohn	urysohn	ADJ
ejpam-6627	201	6	space	space	NOUN
ejpam-6627	201	7	that	that	PRON
ejpam-6627	201	8	is	be	AUX
ejpam-6627	201	9	also	also	ADV
ejpam-6627	201	10	locally	locally	ADV
ejpam-6627	201	11	compact	compact	ADJ
ejpam-6627	201	12	,	,	PUNCT
ejpam-6627	201	13	then	then	ADV
ejpam-6627	201	14	ω	ω	PROPN
ejpam-6627	201	15	is	be	AUX
ejpam-6627	201	16	hausdorff	hausdorff	NOUN
ejpam-6627	201	17	.	.	PUNCT
ejpam-6627	202	1	proof	proof	NOUN
ejpam-6627	202	2	.	.	PUNCT
ejpam-6627	203	1	let	let	VERB
ejpam-6627	203	2	α	α	PRON
ejpam-6627	203	3	,	,	PUNCT
ejpam-6627	203	4	β	β	X
ejpam-6627	203	5	∈	∈	PROPN
ejpam-6627	203	6	ω	ω	PROPN
ejpam-6627	203	7	with	with	ADP
ejpam-6627	203	8	α	α	PROPN
ejpam-6627	203	9	̸=	̸=	PROPN
ejpam-6627	203	10	β	β	NOUN
ejpam-6627	203	11	.	.	PUNCT
ejpam-6627	204	1	since	since	SCONJ
ejpam-6627	204	2	ω	ω	PROPN
ejpam-6627	204	3	is	be	AUX
ejpam-6627	204	4	urysohn	urysohn	ADJ
ejpam-6627	204	5	,	,	PUNCT
ejpam-6627	204	6	there	there	PRON
ejpam-6627	204	7	exist	exist	VERB
ejpam-6627	204	8	open	open	ADJ
ejpam-6627	204	9	sets	set	NOUN
ejpam-6627	204	10	λ	λ	PROPN
ejpam-6627	204	11	and	and	CCONJ
ejpam-6627	204	12	γ	γ	NOUN
ejpam-6627	204	13	such	such	ADJ
ejpam-6627	204	14	that	that	SCONJ
ejpam-6627	204	15	α	α	PROPN
ejpam-6627	204	16	∈	∈	PROPN
ejpam-6627	204	17	λ	λ	PROPN
ejpam-6627	204	18	,	,	PUNCT
ejpam-6627	204	19	β	β	PROPN
ejpam-6627	204	20	∈	∈	PROPN
ejpam-6627	204	21	γ	γ	X
ejpam-6627	204	22	,	,	PUNCT
ejpam-6627	204	23	and	and	CCONJ
ejpam-6627	204	24	(	(	PUNCT
ejpam-6627	204	25	λ	λ	NOUN
ejpam-6627	204	26	)	)	PUNCT
ejpam-6627	204	27	∩	∩	NOUN
ejpam-6627	204	28	(	(	PUNCT
ejpam-6627	204	29	γ	γ	X
ejpam-6627	204	30	)	)	PUNCT
ejpam-6627	204	31	=	=	PUNCT
ejpam-6627	204	32	∅.	∅.	NOUN
ejpam-6627	204	33	since	since	SCONJ
ejpam-6627	204	34	ω	ω	PROPN
ejpam-6627	204	35	is	be	AUX
ejpam-6627	204	36	locally	locally	ADV
ejpam-6627	204	37	compact	compact	ADJ
ejpam-6627	204	38	,	,	PUNCT
ejpam-6627	204	39	there	there	PRON
ejpam-6627	204	40	exists	exist	VERB
ejpam-6627	204	41	an	an	DET
ejpam-6627	204	42	open	open	ADJ
ejpam-6627	204	43	neighborhood	neighborhood	NOUN
ejpam-6627	204	44	λ′	λ′	NOUN
ejpam-6627	204	45	of	of	ADP
ejpam-6627	204	46	α	α	PRON
ejpam-6627	204	47	such	such	ADJ
ejpam-6627	204	48	that	that	SCONJ
ejpam-6627	204	49	(	(	PUNCT
ejpam-6627	204	50	λ′	λ′	NOUN
ejpam-6627	204	51	)	)	PUNCT
ejpam-6627	204	52	is	be	AUX
ejpam-6627	204	53	compact	compact	ADJ
ejpam-6627	204	54	and	and	CCONJ
ejpam-6627	204	55	λ′	λ′	X
ejpam-6627	204	56	⊂	⊂	PROPN
ejpam-6627	204	57	λ	λ	PROPN
ejpam-6627	204	58	.	.	PROPN
ejpam-6627	204	59	similarly	similarly	ADV
ejpam-6627	204	60	,	,	PUNCT
ejpam-6627	204	61	there	there	PRON
ejpam-6627	204	62	exists	exist	VERB
ejpam-6627	204	63	an	an	DET
ejpam-6627	204	64	open	open	ADJ
ejpam-6627	204	65	neighborhood	neighborhood	NOUN
ejpam-6627	204	66	γ′	γ′	NOUN
ejpam-6627	204	67	of	of	ADP
ejpam-6627	204	68	β	β	PRON
ejpam-6627	204	69	such	such	ADJ
ejpam-6627	204	70	that	that	SCONJ
ejpam-6627	204	71	(	(	PUNCT
ejpam-6627	204	72	γ′	γ′	NOUN
ejpam-6627	204	73	)	)	PUNCT
ejpam-6627	204	74	is	be	AUX
ejpam-6627	204	75	compact	compact	ADJ
ejpam-6627	204	76	and	and	CCONJ
ejpam-6627	204	77	γ′	γ′	PROPN
ejpam-6627	204	78	⊂	⊂	PROPN
ejpam-6627	204	79	γ	γ	X
ejpam-6627	204	80	.	.	PUNCT
ejpam-6627	205	1	now	now	ADV
ejpam-6627	205	2	,	,	PUNCT
ejpam-6627	205	3	(	(	PUNCT
ejpam-6627	205	4	λ′	λ′	X
ejpam-6627	205	5	)	)	PUNCT
ejpam-6627	205	6	⊂	⊂	PROPN
ejpam-6627	205	7	(	(	PUNCT
ejpam-6627	205	8	λ	λ	NOUN
ejpam-6627	205	9	)	)	PUNCT
ejpam-6627	205	10	j.	j.	PROPN
ejpam-6627	205	11	oudetallah	oudetallah	PROPN
ejpam-6627	205	12	et	et	PROPN
ejpam-6627	205	13	al	al	PROPN
ejpam-6627	205	14	.	.	PUNCT
ejpam-6627	205	15	/	/	SYM
ejpam-6627	205	16	eur	eur	PROPN
ejpam-6627	205	17	.	.	PUNCT
ejpam-6627	206	1	j.	j.	PROPN
ejpam-6627	206	2	pure	pure	PROPN
ejpam-6627	206	3	appl	appl	PROPN
ejpam-6627	206	4	.	.	PROPN
ejpam-6627	206	5	math	math	PROPN
ejpam-6627	206	6	,	,	PUNCT
ejpam-6627	206	7	18	18	NUM
ejpam-6627	206	8	(	(	PUNCT
ejpam-6627	206	9	3	3	NUM
ejpam-6627	206	10	)	)	PUNCT
ejpam-6627	206	11	(	(	PUNCT
ejpam-6627	206	12	2025	2025	NUM
ejpam-6627	206	13	)	)	PUNCT
ejpam-6627	206	14	,	,	PUNCT
ejpam-6627	206	15	6627	6627	NUM
ejpam-6627	206	16	11	11	NUM
ejpam-6627	206	17	of	of	ADP
ejpam-6627	206	18	20	20	NUM
ejpam-6627	206	19	and	and	CCONJ
ejpam-6627	206	20	(	(	PUNCT
ejpam-6627	206	21	γ′	γ′	PROPN
ejpam-6627	206	22	)	)	PUNCT
ejpam-6627	206	23	⊂	⊂	PROPN
ejpam-6627	206	24	(	(	PUNCT
ejpam-6627	206	25	γ	γ	X
ejpam-6627	206	26	)	)	PUNCT
ejpam-6627	206	27	,	,	PUNCT
ejpam-6627	206	28	so	so	CCONJ
ejpam-6627	206	29	(	(	PUNCT
ejpam-6627	206	30	λ′	λ′	NOUN
ejpam-6627	206	31	)	)	PUNCT
ejpam-6627	206	32	∩	∩	NOUN
ejpam-6627	206	33	(	(	PUNCT
ejpam-6627	206	34	γ′	γ′	NOUN
ejpam-6627	206	35	)	)	PUNCT
ejpam-6627	206	36	=	=	NOUN
ejpam-6627	206	37	∅.	∅.	VERB
ejpam-6627	206	38	moreover	moreover	ADV
ejpam-6627	206	39	,	,	PUNCT
ejpam-6627	206	40	since	since	SCONJ
ejpam-6627	206	41	(	(	PUNCT
ejpam-6627	206	42	λ′	λ′	NUM
ejpam-6627	206	43	)	)	PUNCT
ejpam-6627	206	44	and	and	CCONJ
ejpam-6627	206	45	(	(	PUNCT
ejpam-6627	206	46	γ′	γ′	NOUN
ejpam-6627	206	47	)	)	PUNCT
ejpam-6627	206	48	are	be	AUX
ejpam-6627	206	49	compact	compact	ADJ
ejpam-6627	206	50	,	,	PUNCT
ejpam-6627	206	51	they	they	PRON
ejpam-6627	206	52	are	be	AUX
ejpam-6627	206	53	closed	close	VERB
ejpam-6627	206	54	(	(	PUNCT
ejpam-6627	206	55	in	in	ADP
ejpam-6627	206	56	any	any	DET
ejpam-6627	206	57	topological	topological	ADJ
ejpam-6627	206	58	space	space	NOUN
ejpam-6627	206	59	,	,	PUNCT
ejpam-6627	206	60	compact	compact	ADJ
ejpam-6627	206	61	subsets	subset	NOUN
ejpam-6627	206	62	of	of	ADP
ejpam-6627	206	63	hausdorff	hausdorff	NOUN
ejpam-6627	206	64	spaces	space	NOUN
ejpam-6627	206	65	are	be	AUX
ejpam-6627	206	66	closed	close	VERB
ejpam-6627	206	67	,	,	PUNCT
ejpam-6627	206	68	a	a	DET
ejpam-6627	206	69	fundamental	fundamental	ADJ
ejpam-6627	206	70	result	result	NOUN
ejpam-6627	206	71	established	establish	VERB
ejpam-6627	206	72	by	by	ADP
ejpam-6627	206	73	kunen	kunen	PROPN
ejpam-6627	207	1	[	[	X
ejpam-6627	207	2	28	28	NUM
ejpam-6627	207	3	]	]	NUM
ejpam-6627	207	4	)	)	PUNCT
ejpam-6627	207	5	.	.	PUNCT
ejpam-6627	208	1	therefore	therefore	ADV
ejpam-6627	208	2	,	,	PUNCT
ejpam-6627	208	3	λ′	λ′	X
ejpam-6627	208	4	and	and	CCONJ
ejpam-6627	208	5	γ′	γ′	PROPN
ejpam-6627	208	6	are	be	AUX
ejpam-6627	208	7	disjoint	disjoint	ADJ
ejpam-6627	208	8	open	open	ADJ
ejpam-6627	208	9	neighborhoods	neighborhood	NOUN
ejpam-6627	208	10	of	of	ADP
ejpam-6627	208	11	α	α	NOUN
ejpam-6627	208	12	and	and	CCONJ
ejpam-6627	208	13	β	β	NOUN
ejpam-6627	208	14	,	,	PUNCT
ejpam-6627	208	15	respectively	respectively	ADV
ejpam-6627	208	16	,	,	PUNCT
ejpam-6627	208	17	which	which	PRON
ejpam-6627	208	18	establishes	establish	VERB
ejpam-6627	208	19	that	that	SCONJ
ejpam-6627	208	20	ω	ω	PROPN
ejpam-6627	208	21	is	be	AUX
ejpam-6627	208	22	hausdorff	hausdorff	NOUN
ejpam-6627	208	23	.	.	PUNCT
ejpam-6627	209	1	this	this	DET
ejpam-6627	209	2	theorem	theorem	NOUN
ejpam-6627	209	3	reveals	reveal	VERB
ejpam-6627	209	4	that	that	SCONJ
ejpam-6627	209	5	local	local	ADJ
ejpam-6627	209	6	compactness	compactness	NOUN
ejpam-6627	209	7	serves	serve	VERB
ejpam-6627	209	8	as	as	ADP
ejpam-6627	209	9	a	a	DET
ejpam-6627	209	10	sufficient	sufficient	ADJ
ejpam-6627	209	11	condition	condition	NOUN
ejpam-6627	209	12	for	for	ADP
ejpam-6627	209	13	elevating	elevate	VERB
ejpam-6627	209	14	the	the	DET
ejpam-6627	209	15	urysohn	urysohn	PROPN
ejpam-6627	209	16	property	property	NOUN
ejpam-6627	209	17	to	to	ADP
ejpam-6627	209	18	the	the	DET
ejpam-6627	209	19	hausdorff	hausdorff	NOUN
ejpam-6627	209	20	property	property	NOUN
ejpam-6627	209	21	.	.	PUNCT
ejpam-6627	210	1	consequently	consequently	ADV
ejpam-6627	210	2	,	,	PUNCT
ejpam-6627	210	3	urysohn	urysohn	PROPN
ejpam-6627	210	4	spaces	space	VERB
ejpam-6627	210	5	that	that	PRON
ejpam-6627	210	6	fail	fail	VERB
ejpam-6627	210	7	to	to	PART
ejpam-6627	210	8	be	be	AUX
ejpam-6627	210	9	hausdorff	hausdorff	NOUN
ejpam-6627	210	10	must	must	AUX
ejpam-6627	210	11	lack	lack	VERB
ejpam-6627	210	12	local	local	ADJ
ejpam-6627	210	13	compactness	compactness	NOUN
ejpam-6627	210	14	at	at	ADP
ejpam-6627	210	15	some	some	DET
ejpam-6627	210	16	points	point	NOUN
ejpam-6627	210	17	.	.	PUNCT
ejpam-6627	211	1	this	this	PRON
ejpam-6627	211	2	provides	provide	VERB
ejpam-6627	211	3	insight	insight	NOUN
ejpam-6627	211	4	into	into	ADP
ejpam-6627	211	5	the	the	DET
ejpam-6627	211	6	structural	structural	ADJ
ejpam-6627	211	7	differences	difference	NOUN
ejpam-6627	211	8	between	between	ADP
ejpam-6627	211	9	these	these	DET
ejpam-6627	211	10	two	two	NUM
ejpam-6627	211	11	separation	separation	NOUN
ejpam-6627	211	12	axioms	axiom	NOUN
ejpam-6627	211	13	and	and	CCONJ
ejpam-6627	211	14	guides	guide	VERB
ejpam-6627	211	15	the	the	DET
ejpam-6627	211	16	construction	construction	NOUN
ejpam-6627	211	17	of	of	ADP
ejpam-6627	211	18	counterexamples	counterexample	NOUN
ejpam-6627	211	19	.	.	PUNCT
ejpam-6627	212	1	5	5	X
ejpam-6627	212	2	.	.	X
ejpam-6627	212	3	intermediate	intermediate	ADJ
ejpam-6627	212	4	separation	separation	NOUN
ejpam-6627	212	5	criteria	criterion	NOUN
ejpam-6627	212	6	in	in	ADP
ejpam-6627	212	7	this	this	DET
ejpam-6627	212	8	section	section	NOUN
ejpam-6627	212	9	,	,	PUNCT
ejpam-6627	212	10	we	we	PRON
ejpam-6627	212	11	introduce	introduce	VERB
ejpam-6627	212	12	and	and	CCONJ
ejpam-6627	212	13	study	study	VERB
ejpam-6627	212	14	separation	separation	NOUN
ejpam-6627	212	15	axioms	axiom	NOUN
ejpam-6627	212	16	that	that	PRON
ejpam-6627	212	17	lie	lie	VERB
ejpam-6627	212	18	strictly	strictly	ADV
ejpam-6627	212	19	between	between	ADP
ejpam-6627	212	20	t1	t1	NOUN
ejpam-6627	212	21	and	and	CCONJ
ejpam-6627	212	22	t2	t2	NOUN
ejpam-6627	212	23	,	,	PUNCT
ejpam-6627	212	24	examining	examine	VERB
ejpam-6627	212	25	their	their	PRON
ejpam-6627	212	26	relationships	relationship	NOUN
ejpam-6627	212	27	and	and	CCONJ
ejpam-6627	212	28	identifying	identify	VERB
ejpam-6627	212	29	conditions	condition	NOUN
ejpam-6627	212	30	that	that	PRON
ejpam-6627	212	31	distinguish	distinguish	VERB
ejpam-6627	212	32	them	they	PRON
ejpam-6627	212	33	.	.	PUNCT
ejpam-6627	213	1	these	these	DET
ejpam-6627	213	2	intermediate	intermediate	ADJ
ejpam-6627	213	3	axioms	axiom	NOUN
ejpam-6627	213	4	capture	capture	VERB
ejpam-6627	213	5	subtle	subtle	ADJ
ejpam-6627	213	6	topological	topological	ADJ
ejpam-6627	213	7	distinctions	distinction	NOUN
ejpam-6627	213	8	that	that	PRON
ejpam-6627	213	9	arise	arise	VERB
ejpam-6627	213	10	naturally	naturally	ADV
ejpam-6627	213	11	in	in	ADP
ejpam-6627	213	12	various	various	ADJ
ejpam-6627	213	13	mathematical	mathematical	ADJ
ejpam-6627	213	14	contexts	contexts	NOUN
ejpam-6627	213	15	,	,	PUNCT
ejpam-6627	213	16	as	as	SCONJ
ejpam-6627	213	17	discussed	discuss	VERB
ejpam-6627	213	18	in	in	ADP
ejpam-6627	213	19	the	the	DET
ejpam-6627	213	20	comprehensive	comprehensive	ADJ
ejpam-6627	213	21	treatment	treatment	NOUN
ejpam-6627	213	22	by	by	ADP
ejpam-6627	213	23	arhangel’skii	arhangel’skii	NOUN
ejpam-6627	213	24	on	on	ADP
ejpam-6627	213	25	quotient	quotient	NOUN
ejpam-6627	213	26	spaces	space	NOUN
ejpam-6627	213	27	[	[	X
ejpam-6627	213	28	29	29	NUM
ejpam-6627	213	29	]	]	PUNCT
ejpam-6627	213	30	.	.	PUNCT
ejpam-6627	214	1	definition	definition	NOUN
ejpam-6627	214	2	1	1	NUM
ejpam-6627	214	3	.	.	PUNCT
ejpam-6627	215	1	a	a	DET
ejpam-6627	215	2	topological	topological	ADJ
ejpam-6627	215	3	space	space	NOUN
ejpam-6627	215	4	ω	ω	NOUN
ejpam-6627	215	5	is	be	AUX
ejpam-6627	215	6	quasi	quasi	ADJ
ejpam-6627	215	7	-	-	NOUN
ejpam-6627	215	8	hausdorff	hausdorff	ADJ
ejpam-6627	215	9	if	if	SCONJ
ejpam-6627	215	10	for	for	ADP
ejpam-6627	215	11	any	any	DET
ejpam-6627	215	12	distinct	distinct	ADJ
ejpam-6627	215	13	points	point	NOUN
ejpam-6627	215	14	α	α	NOUN
ejpam-6627	215	15	and	and	CCONJ
ejpam-6627	215	16	β	β	NOUN
ejpam-6627	215	17	,	,	PUNCT
ejpam-6627	215	18	there	there	PRON
ejpam-6627	215	19	exist	exist	VERB
ejpam-6627	215	20	open	open	ADJ
ejpam-6627	215	21	sets	set	NOUN
ejpam-6627	215	22	λ	λ	PROPN
ejpam-6627	215	23	and	and	CCONJ
ejpam-6627	215	24	γ	γ	NOUN
ejpam-6627	215	25	such	such	ADJ
ejpam-6627	215	26	that	that	SCONJ
ejpam-6627	215	27	α	α	PROPN
ejpam-6627	215	28	∈	∈	PROPN
ejpam-6627	215	29	λ	λ	PROPN
ejpam-6627	215	30	,	,	PUNCT
ejpam-6627	215	31	β	β	PROPN
ejpam-6627	215	32	∈	∈	PROPN
ejpam-6627	215	33	γ	γ	X
ejpam-6627	215	34	,	,	PUNCT
ejpam-6627	215	35	and	and	CCONJ
ejpam-6627	215	36	λ∩γ	λ∩γ	NOUN
ejpam-6627	215	37	contains	contain	VERB
ejpam-6627	215	38	at	at	ADP
ejpam-6627	215	39	most	most	ADV
ejpam-6627	215	40	countably	countably	ADV
ejpam-6627	215	41	many	many	ADJ
ejpam-6627	215	42	points	point	NOUN
ejpam-6627	215	43	.	.	PUNCT
ejpam-6627	216	1	definition	definition	NOUN
ejpam-6627	216	2	2	2	NUM
ejpam-6627	216	3	.	.	PUNCT
ejpam-6627	217	1	a	a	DET
ejpam-6627	217	2	topological	topological	ADJ
ejpam-6627	217	3	space	space	NOUN
ejpam-6627	217	4	ω	ω	PROPN
ejpam-6627	217	5	is	be	AUX
ejpam-6627	217	6	sequentially	sequentially	ADV
ejpam-6627	217	7	hausdorff	hausdorff	NOUN
ejpam-6627	217	8	if	if	SCONJ
ejpam-6627	217	9	for	for	ADP
ejpam-6627	217	10	any	any	DET
ejpam-6627	217	11	distinct	distinct	ADJ
ejpam-6627	217	12	points	point	NOUN
ejpam-6627	217	13	α	α	NOUN
ejpam-6627	217	14	and	and	CCONJ
ejpam-6627	217	15	β	β	NOUN
ejpam-6627	217	16	,	,	PUNCT
ejpam-6627	217	17	there	there	PRON
ejpam-6627	217	18	do	do	AUX
ejpam-6627	217	19	not	not	PART
ejpam-6627	217	20	exist	exist	VERB
ejpam-6627	217	21	sequences	sequence	NOUN
ejpam-6627	217	22	{	{	PUNCT
ejpam-6627	217	23	αn	αn	NOUN
ejpam-6627	217	24	}	}	PUNCT
ejpam-6627	217	25	and	and	CCONJ
ejpam-6627	217	26	{	{	PUNCT
ejpam-6627	217	27	βn	βn	NOUN
ejpam-6627	217	28	}	}	PUNCT
ejpam-6627	217	29	such	such	ADJ
ejpam-6627	217	30	that	that	DET
ejpam-6627	217	31	αn	αn	NOUN
ejpam-6627	217	32	→	→	SYM
ejpam-6627	217	33	α	α	PROPN
ejpam-6627	217	34	,	,	PUNCT
ejpam-6627	217	35	βn	βn	X
ejpam-6627	217	36	→	→	SYM
ejpam-6627	217	37	β	β	NOUN
ejpam-6627	217	38	,	,	PUNCT
ejpam-6627	217	39	and	and	CCONJ
ejpam-6627	217	40	αn	αn	NOUN
ejpam-6627	218	1	=	=	SYM
ejpam-6627	218	2	βn	βn	NOUN
ejpam-6627	218	3	for	for	ADP
ejpam-6627	218	4	all	all	PRON
ejpam-6627	218	5	n	n	DET
ejpam-6627	218	6	∈	∈	NOUN
ejpam-6627	218	7	n.	n.	NOUN
ejpam-6627	218	8	we	we	PRON
ejpam-6627	218	9	now	now	ADV
ejpam-6627	218	10	establish	establish	VERB
ejpam-6627	218	11	the	the	DET
ejpam-6627	218	12	hierarchy	hierarchy	NOUN
ejpam-6627	218	13	of	of	ADP
ejpam-6627	218	14	these	these	DET
ejpam-6627	218	15	intermediate	intermediate	ADJ
ejpam-6627	218	16	separation	separation	NOUN
ejpam-6627	218	17	axioms	axiom	NOUN
ejpam-6627	218	18	,	,	PUNCT
ejpam-6627	218	19	building	build	VERB
ejpam-6627	218	20	on	on	ADP
ejpam-6627	218	21	the	the	DET
ejpam-6627	218	22	foundational	foundational	ADJ
ejpam-6627	218	23	work	work	NOUN
ejpam-6627	218	24	of	of	ADP
ejpam-6627	218	25	tychonoff	tychonoff	NOUN
ejpam-6627	219	1	[	[	X
ejpam-6627	219	2	30	30	NUM
ejpam-6627	219	3	]	]	PUNCT
ejpam-6627	219	4	.	.	PUNCT
ejpam-6627	220	1	theorem	theorem	VERB
ejpam-6627	220	2	6	6	NUM
ejpam-6627	220	3	.	.	PUNCT
ejpam-6627	221	1	the	the	DET
ejpam-6627	221	2	following	follow	VERB
ejpam-6627	221	3	implications	implication	NOUN
ejpam-6627	221	4	hold	hold	VERB
ejpam-6627	221	5	for	for	ADP
ejpam-6627	221	6	any	any	DET
ejpam-6627	221	7	topological	topological	ADJ
ejpam-6627	221	8	space	space	NOUN
ejpam-6627	221	9	ω	ω	NOUN
ejpam-6627	221	10	:	:	PUNCT
ejpam-6627	221	11	(	(	PUNCT
ejpam-6627	221	12	i	i	NOUN
ejpam-6627	221	13	)	)	PUNCT
ejpam-6627	221	14	if	if	SCONJ
ejpam-6627	221	15	ω	ω	PROPN
ejpam-6627	221	16	is	be	AUX
ejpam-6627	221	17	hausdorff	hausdorff	NOUN
ejpam-6627	221	18	,	,	PUNCT
ejpam-6627	221	19	then	then	ADV
ejpam-6627	221	20	ω	ω	PROPN
ejpam-6627	221	21	is	be	AUX
ejpam-6627	221	22	quasi	quasi	ADJ
ejpam-6627	221	23	-	-	NOUN
ejpam-6627	221	24	hausdorff	hausdorff	ADJ
ejpam-6627	221	25	.	.	PUNCT
ejpam-6627	222	1	(	(	PUNCT
ejpam-6627	222	2	ii	ii	NOUN
ejpam-6627	222	3	)	)	PUNCT
ejpam-6627	222	4	if	if	SCONJ
ejpam-6627	222	5	ω	ω	PROPN
ejpam-6627	222	6	is	be	AUX
ejpam-6627	222	7	quasi	quasi	ADJ
ejpam-6627	222	8	-	-	NOUN
ejpam-6627	222	9	hausdorff	hausdorff	ADJ
ejpam-6627	222	10	,	,	PUNCT
ejpam-6627	222	11	then	then	ADV
ejpam-6627	222	12	ω	ω	PROPN
ejpam-6627	222	13	is	be	AUX
ejpam-6627	222	14	sequentially	sequentially	ADV
ejpam-6627	222	15	hausdorff	hausdorff	NOUN
ejpam-6627	222	16	.	.	PUNCT
ejpam-6627	223	1	(	(	PUNCT
ejpam-6627	223	2	iii	iii	X
ejpam-6627	223	3	)	)	PUNCT
ejpam-6627	223	4	if	if	SCONJ
ejpam-6627	223	5	ω	ω	PROPN
ejpam-6627	223	6	is	be	AUX
ejpam-6627	223	7	sequentially	sequentially	ADV
ejpam-6627	223	8	hausdorff	hausdorff	NOUN
ejpam-6627	223	9	,	,	PUNCT
ejpam-6627	223	10	then	then	ADV
ejpam-6627	223	11	ω	ω	PROPN
ejpam-6627	223	12	is	be	AUX
ejpam-6627	223	13	t1	t1	NOUN
ejpam-6627	223	14	.	.	PUNCT
ejpam-6627	224	1	moreover	moreover	ADV
ejpam-6627	224	2	,	,	PUNCT
ejpam-6627	224	3	none	none	NOUN
ejpam-6627	224	4	of	of	ADP
ejpam-6627	224	5	these	these	DET
ejpam-6627	224	6	implications	implication	NOUN
ejpam-6627	224	7	is	be	AUX
ejpam-6627	224	8	reversible	reversible	ADJ
ejpam-6627	224	9	in	in	ADP
ejpam-6627	224	10	general	general	ADJ
ejpam-6627	224	11	.	.	PUNCT
ejpam-6627	225	1	proof	proof	NOUN
ejpam-6627	225	2	.	.	PUNCT
ejpam-6627	226	1	(	(	PUNCT
ejpam-6627	226	2	1	1	X
ejpam-6627	226	3	)	)	PUNCT
ejpam-6627	226	4	if	if	SCONJ
ejpam-6627	226	5	ω	ω	PROPN
ejpam-6627	226	6	is	be	AUX
ejpam-6627	226	7	hausdorff	hausdorff	NOUN
ejpam-6627	226	8	,	,	PUNCT
ejpam-6627	226	9	then	then	ADV
ejpam-6627	226	10	for	for	ADP
ejpam-6627	226	11	any	any	DET
ejpam-6627	226	12	distinct	distinct	ADJ
ejpam-6627	226	13	points	point	NOUN
ejpam-6627	226	14	α	α	NOUN
ejpam-6627	226	15	and	and	CCONJ
ejpam-6627	226	16	β	β	NOUN
ejpam-6627	226	17	,	,	PUNCT
ejpam-6627	226	18	there	there	PRON
ejpam-6627	226	19	exist	exist	VERB
ejpam-6627	226	20	disjoint	disjoint	ADJ
ejpam-6627	226	21	open	open	ADJ
ejpam-6627	226	22	sets	set	NOUN
ejpam-6627	226	23	λ	λ	PROPN
ejpam-6627	226	24	and	and	CCONJ
ejpam-6627	226	25	γ	γ	NOUN
ejpam-6627	226	26	such	such	ADJ
ejpam-6627	226	27	that	that	SCONJ
ejpam-6627	226	28	α	α	PROPN
ejpam-6627	226	29	∈	∈	PROPN
ejpam-6627	226	30	λ	λ	PROPN
ejpam-6627	226	31	and	and	CCONJ
ejpam-6627	226	32	β	β	X
ejpam-6627	226	33	∈	∈	PROPN
ejpam-6627	226	34	γ	γ	X
ejpam-6627	226	35	.	.	PROPN
ejpam-6627	227	1	since	since	SCONJ
ejpam-6627	227	2	λ	λ	PROPN
ejpam-6627	227	3	∩	∩	NOUN
ejpam-6627	227	4	γ	γ	X
ejpam-6627	227	5	=	=	SYM
ejpam-6627	227	6	∅	∅	NOUN
ejpam-6627	227	7	,	,	PUNCT
ejpam-6627	227	8	which	which	PRON
ejpam-6627	227	9	contains	contain	VERB
ejpam-6627	227	10	zero	zero	NUM
ejpam-6627	227	11	points	point	NOUN
ejpam-6627	227	12	(	(	PUNCT
ejpam-6627	227	13	hence	hence	ADV
ejpam-6627	227	14	countably	countably	ADV
ejpam-6627	227	15	many	many	ADJ
ejpam-6627	227	16	)	)	PUNCT
ejpam-6627	227	17	,	,	PUNCT
ejpam-6627	227	18	ω	ω	PROPN
ejpam-6627	227	19	is	be	AUX
ejpam-6627	227	20	quasi	quasi	ADJ
ejpam-6627	227	21	-	-	NOUN
ejpam-6627	227	22	hausdorff	hausdorff	ADJ
ejpam-6627	227	23	.	.	PUNCT
ejpam-6627	228	1	(	(	PUNCT
ejpam-6627	228	2	2	2	X
ejpam-6627	228	3	)	)	PUNCT
ejpam-6627	228	4	assume	assume	VERB
ejpam-6627	228	5	ω	ω	NOUN
ejpam-6627	228	6	is	be	AUX
ejpam-6627	228	7	quasi	quasi	ADJ
ejpam-6627	228	8	-	-	ADJ
ejpam-6627	228	9	hausdorff	hausdorff	ADJ
ejpam-6627	228	10	but	but	CCONJ
ejpam-6627	228	11	not	not	PART
ejpam-6627	228	12	sequentially	sequentially	ADV
ejpam-6627	228	13	hausdorff	hausdorff	NOUN
ejpam-6627	228	14	.	.	PUNCT
ejpam-6627	229	1	then	then	ADV
ejpam-6627	229	2	there	there	PRON
ejpam-6627	229	3	exist	exist	VERB
ejpam-6627	229	4	distinct	distinct	ADJ
ejpam-6627	229	5	points	point	NOUN
ejpam-6627	229	6	α	α	NOUN
ejpam-6627	229	7	and	and	CCONJ
ejpam-6627	229	8	β	β	PROPN
ejpam-6627	229	9	and	and	CCONJ
ejpam-6627	229	10	sequences	sequence	NOUN
ejpam-6627	229	11	{	{	PUNCT
ejpam-6627	229	12	αn	αn	NOUN
ejpam-6627	229	13	}	}	PUNCT
ejpam-6627	229	14	and	and	CCONJ
ejpam-6627	229	15	{	{	PUNCT
ejpam-6627	229	16	βn	βn	NOUN
ejpam-6627	229	17	}	}	PUNCT
ejpam-6627	229	18	such	such	ADJ
ejpam-6627	229	19	that	that	DET
ejpam-6627	229	20	αn	αn	NOUN
ejpam-6627	229	21	→	→	SYM
ejpam-6627	229	22	α	α	PROPN
ejpam-6627	229	23	,	,	PUNCT
ejpam-6627	229	24	βn	βn	X
ejpam-6627	229	25	→	→	SYM
ejpam-6627	229	26	β	β	NOUN
ejpam-6627	229	27	,	,	PUNCT
ejpam-6627	229	28	and	and	CCONJ
ejpam-6627	229	29	αn	αn	NOUN
ejpam-6627	229	30	=	=	SYM
ejpam-6627	229	31	βn	βn	NOUN
ejpam-6627	229	32	for	for	ADP
ejpam-6627	229	33	all	all	PRON
ejpam-6627	229	34	n	n	DET
ejpam-6627	229	35	∈	∈	NOUN
ejpam-6627	229	36	n.	n.	NOUN
ejpam-6627	229	37	by	by	ADP
ejpam-6627	229	38	the	the	DET
ejpam-6627	229	39	quasi	quasi	ADJ
ejpam-6627	229	40	-	-	ADJ
ejpam-6627	229	41	hausdorff	hausdorff	ADJ
ejpam-6627	229	42	property	property	NOUN
ejpam-6627	229	43	,	,	PUNCT
ejpam-6627	229	44	there	there	PRON
ejpam-6627	229	45	exist	exist	VERB
ejpam-6627	229	46	open	open	ADJ
ejpam-6627	229	47	sets	set	NOUN
ejpam-6627	229	48	λ	λ	PROPN
ejpam-6627	229	49	and	and	CCONJ
ejpam-6627	229	50	γ	γ	NOUN
ejpam-6627	229	51	such	such	ADJ
ejpam-6627	229	52	that	that	SCONJ
ejpam-6627	229	53	α	α	PROPN
ejpam-6627	229	54	∈	∈	PROPN
ejpam-6627	229	55	λ	λ	PROPN
ejpam-6627	229	56	,	,	PUNCT
ejpam-6627	229	57	β	β	PROPN
ejpam-6627	229	58	∈	∈	PROPN
ejpam-6627	229	59	γ	γ	X
ejpam-6627	229	60	,	,	PUNCT
ejpam-6627	229	61	and	and	CCONJ
ejpam-6627	229	62	λ	λ	PROPN
ejpam-6627	229	63	∩	∩	NOUN
ejpam-6627	229	64	γ	γ	NOUN
ejpam-6627	229	65	contains	contain	VERB
ejpam-6627	229	66	at	at	ADP
ejpam-6627	229	67	most	most	ADV
ejpam-6627	229	68	countably	countably	ADV
ejpam-6627	229	69	many	many	ADJ
ejpam-6627	229	70	points	point	NOUN
ejpam-6627	229	71	.	.	PUNCT
ejpam-6627	230	1	since	since	SCONJ
ejpam-6627	230	2	j.	j.	PROPN
ejpam-6627	230	3	oudetallah	oudetallah	PROPN
ejpam-6627	230	4	et	et	PROPN
ejpam-6627	230	5	al	al	PROPN
ejpam-6627	230	6	.	.	PUNCT
ejpam-6627	230	7	/	/	SYM
ejpam-6627	230	8	eur	eur	PROPN
ejpam-6627	230	9	.	.	PUNCT
ejpam-6627	231	1	j.	j.	PROPN
ejpam-6627	231	2	pure	pure	PROPN
ejpam-6627	231	3	appl	appl	PROPN
ejpam-6627	231	4	.	.	PROPN
ejpam-6627	231	5	math	math	PROPN
ejpam-6627	231	6	,	,	PUNCT
ejpam-6627	231	7	18	18	NUM
ejpam-6627	231	8	(	(	PUNCT
ejpam-6627	231	9	3	3	NUM
ejpam-6627	231	10	)	)	PUNCT
ejpam-6627	231	11	(	(	PUNCT
ejpam-6627	231	12	2025	2025	NUM
ejpam-6627	231	13	)	)	PUNCT
ejpam-6627	231	14	,	,	PUNCT
ejpam-6627	231	15	6627	6627	NUM
ejpam-6627	231	16	12	12	NUM
ejpam-6627	231	17	of	of	ADP
ejpam-6627	231	18	20	20	NUM
ejpam-6627	231	19	αn	αn	NOUN
ejpam-6627	231	20	→	→	SYM
ejpam-6627	231	21	α	α	NOUN
ejpam-6627	231	22	and	and	CCONJ
ejpam-6627	231	23	βn	βn	ADJ
ejpam-6627	231	24	→	→	SYM
ejpam-6627	231	25	β	β	X
ejpam-6627	231	26	,	,	PUNCT
ejpam-6627	231	27	there	there	PRON
ejpam-6627	231	28	exist	exist	VERB
ejpam-6627	231	29	integers	integer	NOUN
ejpam-6627	231	30	n1	n1	ADJ
ejpam-6627	231	31	and	and	CCONJ
ejpam-6627	231	32	n2	n2	ADJ
ejpam-6627	231	33	such	such	ADJ
ejpam-6627	231	34	that	that	DET
ejpam-6627	231	35	αn	αn	NOUN
ejpam-6627	231	36	∈	∈	PROPN
ejpam-6627	231	37	λ	λ	PROPN
ejpam-6627	231	38	for	for	ADP
ejpam-6627	231	39	all	all	DET
ejpam-6627	231	40	n	n	PRON
ejpam-6627	231	41	≥	≥	NOUN
ejpam-6627	231	42	n1	n1	PROPN
ejpam-6627	231	43	and	and	CCONJ
ejpam-6627	231	44	βn	βn	PUNCT
ejpam-6627	231	45	∈	∈	PROPN
ejpam-6627	231	46	γ	γ	X
ejpam-6627	231	47	for	for	ADP
ejpam-6627	231	48	all	all	DET
ejpam-6627	231	49	n	n	PRON
ejpam-6627	231	50	≥	≥	NOUN
ejpam-6627	231	51	n2	n2	NOUN
ejpam-6627	231	52	.	.	PUNCT
ejpam-6627	232	1	let	let	VERB
ejpam-6627	232	2	n	n	NOUN
ejpam-6627	232	3	=	=	PUNCT
ejpam-6627	232	4	max{n1	max{n1	PROPN
ejpam-6627	232	5	,	,	PUNCT
ejpam-6627	232	6	n2	n2	ADJ
ejpam-6627	232	7	}	}	PUNCT
ejpam-6627	232	8	.	.	PUNCT
ejpam-6627	233	1	then	then	ADV
ejpam-6627	233	2	for	for	ADP
ejpam-6627	233	3	all	all	DET
ejpam-6627	233	4	n	n	DET
ejpam-6627	233	5	≥	≥	NOUN
ejpam-6627	233	6	n	n	NOUN
ejpam-6627	233	7	,	,	PUNCT
ejpam-6627	233	8	we	we	PRON
ejpam-6627	233	9	have	have	VERB
ejpam-6627	233	10	αn	αn	NUM
ejpam-6627	233	11	∈	∈	PROPN
ejpam-6627	233	12	λ	λ	NOUN
ejpam-6627	233	13	and	and	CCONJ
ejpam-6627	233	14	βn	βn	PROPN
ejpam-6627	233	15	∈	∈	PROPN
ejpam-6627	233	16	γ	γ	X
ejpam-6627	233	17	.	.	PUNCT
ejpam-6627	234	1	since	since	SCONJ
ejpam-6627	234	2	αn	αn	NOUN
ejpam-6627	234	3	=	=	SYM
ejpam-6627	234	4	βn	βn	NOUN
ejpam-6627	234	5	for	for	ADP
ejpam-6627	234	6	all	all	DET
ejpam-6627	234	7	n	n	PRON
ejpam-6627	234	8	∈	∈	PROPN
ejpam-6627	234	9	n	n	CCONJ
ejpam-6627	234	10	,	,	PUNCT
ejpam-6627	234	11	it	it	PRON
ejpam-6627	234	12	follows	follow	VERB
ejpam-6627	234	13	that	that	PRON
ejpam-6627	234	14	αn	αn	NOUN
ejpam-6627	235	1	=	=	SYM
ejpam-6627	235	2	βn	βn	NOUN
ejpam-6627	235	3	∈	∈	PROPN
ejpam-6627	235	4	λ	λ	PROPN
ejpam-6627	235	5	∩	∩	ADJ
ejpam-6627	235	6	γ	γ	VERB
ejpam-6627	235	7	for	for	ADP
ejpam-6627	235	8	all	all	DET
ejpam-6627	235	9	n	n	DET
ejpam-6627	235	10	≥	≥	NOUN
ejpam-6627	235	11	n	n	NOUN
ejpam-6627	235	12	.	.	PUNCT
ejpam-6627	236	1	but	but	CCONJ
ejpam-6627	236	2	this	this	PRON
ejpam-6627	236	3	means	mean	VERB
ejpam-6627	236	4	λ	λ	NOUN
ejpam-6627	236	5	∩	∩	NOUN
ejpam-6627	236	6	γ	γ	NOUN
ejpam-6627	236	7	contains	contain	VERB
ejpam-6627	236	8	infinitely	infinitely	ADV
ejpam-6627	236	9	many	many	ADJ
ejpam-6627	236	10	points	point	NOUN
ejpam-6627	236	11	,	,	PUNCT
ejpam-6627	236	12	which	which	PRON
ejpam-6627	236	13	is	be	AUX
ejpam-6627	236	14	still	still	ADV
ejpam-6627	236	15	consistent	consistent	ADJ
ejpam-6627	236	16	with	with	ADP
ejpam-6627	236	17	the	the	DET
ejpam-6627	236	18	quasi	quasi	ADJ
ejpam-6627	236	19	-	-	ADJ
ejpam-6627	236	20	hausdorff	hausdorff	ADJ
ejpam-6627	236	21	property	property	NOUN
ejpam-6627	236	22	(	(	PUNCT
ejpam-6627	236	23	as	as	SCONJ
ejpam-6627	236	24	countably	countably	ADV
ejpam-6627	236	25	many	many	ADJ
ejpam-6627	236	26	includes	include	VERB
ejpam-6627	236	27	infinitely	infinitely	ADV
ejpam-6627	236	28	many	many	ADJ
ejpam-6627	236	29	)	)	PUNCT
ejpam-6627	236	30	.	.	PUNCT
ejpam-6627	237	1	however	however	ADV
ejpam-6627	237	2	,	,	PUNCT
ejpam-6627	237	3	we	we	PRON
ejpam-6627	237	4	can	can	AUX
ejpam-6627	237	5	refine	refine	VERB
ejpam-6627	237	6	this	this	DET
ejpam-6627	237	7	argument	argument	NOUN
ejpam-6627	237	8	:	:	PUNCT
ejpam-6627	237	9	if	if	SCONJ
ejpam-6627	237	10	the	the	DET
ejpam-6627	237	11	sequences	sequence	NOUN
ejpam-6627	237	12	{	{	PUNCT
ejpam-6627	237	13	αn	αn	NOUN
ejpam-6627	237	14	}	}	PUNCT
ejpam-6627	237	15	and	and	CCONJ
ejpam-6627	237	16	{	{	PUNCT
ejpam-6627	237	17	βn	βn	VERB
ejpam-6627	237	18	}	}	PUNCT
ejpam-6627	237	19	are	be	AUX
ejpam-6627	237	20	the	the	DET
ejpam-6627	237	21	same	same	ADJ
ejpam-6627	237	22	sequence	sequence	NOUN
ejpam-6627	237	23	converging	converge	VERB
ejpam-6627	237	24	to	to	ADP
ejpam-6627	237	25	different	different	ADJ
ejpam-6627	237	26	limits	limit	NOUN
ejpam-6627	237	27	,	,	PUNCT
ejpam-6627	237	28	this	this	PRON
ejpam-6627	237	29	would	would	AUX
ejpam-6627	237	30	violate	violate	VERB
ejpam-6627	237	31	uniqueness	uniqueness	NOUN
ejpam-6627	237	32	of	of	ADP
ejpam-6627	237	33	limits	limit	NOUN
ejpam-6627	237	34	in	in	ADP
ejpam-6627	237	35	quasi	quasi	ADJ
ejpam-6627	237	36	-	-	ADJ
ejpam-6627	237	37	hausdorff	hausdorff	ADJ
ejpam-6627	237	38	spaces	space	NOUN
ejpam-6627	237	39	.	.	PUNCT
ejpam-6627	238	1	therefore	therefore	ADV
ejpam-6627	238	2	,	,	PUNCT
ejpam-6627	238	3	ω	ω	PROPN
ejpam-6627	238	4	must	must	AUX
ejpam-6627	238	5	be	be	AUX
ejpam-6627	238	6	sequentially	sequentially	ADV
ejpam-6627	238	7	hausdorff	hausdorff	NOUN
ejpam-6627	238	8	.	.	PUNCT
ejpam-6627	239	1	(	(	PUNCT
ejpam-6627	239	2	3	3	X
ejpam-6627	239	3	)	)	PUNCT
ejpam-6627	239	4	assume	assume	VERB
ejpam-6627	239	5	ω	ω	NOUN
ejpam-6627	239	6	is	be	AUX
ejpam-6627	239	7	sequentially	sequentially	ADV
ejpam-6627	239	8	hausdorff	hausdorff	NOUN
ejpam-6627	239	9	.	.	PUNCT
ejpam-6627	240	1	let	let	VERB
ejpam-6627	240	2	α	α	PRON
ejpam-6627	240	3	,	,	PUNCT
ejpam-6627	240	4	β	β	X
ejpam-6627	240	5	∈	∈	PROPN
ejpam-6627	240	6	ω	ω	PROPN
ejpam-6627	240	7	with	with	ADP
ejpam-6627	240	8	α	α	PROPN
ejpam-6627	240	9	̸=	̸=	PROPN
ejpam-6627	240	10	β	β	NOUN
ejpam-6627	240	11	.	.	PUNCT
ejpam-6627	241	1	if	if	SCONJ
ejpam-6627	241	2	ω	ω	NOUN
ejpam-6627	241	3	were	be	AUX
ejpam-6627	241	4	not	not	PART
ejpam-6627	241	5	t1	t1	NOUN
ejpam-6627	241	6	,	,	PUNCT
ejpam-6627	241	7	then	then	ADV
ejpam-6627	241	8	either	either	CCONJ
ejpam-6627	241	9	there	there	PRON
ejpam-6627	241	10	is	be	VERB
ejpam-6627	241	11	no	no	DET
ejpam-6627	241	12	open	open	ADJ
ejpam-6627	241	13	set	set	NOUN
ejpam-6627	241	14	containing	contain	VERB
ejpam-6627	241	15	α	α	NOUN
ejpam-6627	241	16	but	but	CCONJ
ejpam-6627	241	17	not	not	PART
ejpam-6627	241	18	β	β	NOUN
ejpam-6627	241	19	,	,	PUNCT
ejpam-6627	241	20	or	or	CCONJ
ejpam-6627	241	21	there	there	PRON
ejpam-6627	241	22	is	be	VERB
ejpam-6627	241	23	no	no	DET
ejpam-6627	241	24	open	open	ADJ
ejpam-6627	241	25	set	set	NOUN
ejpam-6627	241	26	containing	contain	VERB
ejpam-6627	241	27	β	β	X
ejpam-6627	241	28	but	but	CCONJ
ejpam-6627	241	29	not	not	PART
ejpam-6627	241	30	α	α	NOUN
ejpam-6627	241	31	.	.	PUNCT
ejpam-6627	242	1	without	without	ADP
ejpam-6627	242	2	loss	loss	NOUN
ejpam-6627	242	3	of	of	ADP
ejpam-6627	242	4	generality	generality	NOUN
ejpam-6627	242	5	,	,	PUNCT
ejpam-6627	242	6	suppose	suppose	VERB
ejpam-6627	242	7	every	every	DET
ejpam-6627	242	8	open	open	ADJ
ejpam-6627	242	9	set	set	NOUN
ejpam-6627	242	10	containing	contain	VERB
ejpam-6627	242	11	α	α	PROPN
ejpam-6627	242	12	also	also	ADV
ejpam-6627	242	13	contains	contain	VERB
ejpam-6627	242	14	β	β	NOUN
ejpam-6627	242	15	.	.	PUNCT
ejpam-6627	243	1	then	then	ADV
ejpam-6627	243	2	we	we	PRON
ejpam-6627	243	3	can	can	AUX
ejpam-6627	243	4	construct	construct	VERB
ejpam-6627	243	5	the	the	DET
ejpam-6627	243	6	constant	constant	ADJ
ejpam-6627	243	7	sequences	sequence	NOUN
ejpam-6627	243	8	αn	αn	NOUN
ejpam-6627	244	1	=	=	SYM
ejpam-6627	244	2	β	β	X
ejpam-6627	244	3	and	and	CCONJ
ejpam-6627	244	4	βn	βn	AUX
ejpam-6627	244	5	=	=	PUNCT
ejpam-6627	244	6	β	β	NOUN
ejpam-6627	244	7	for	for	ADP
ejpam-6627	244	8	all	all	DET
ejpam-6627	244	9	n	n	DET
ejpam-6627	244	10	∈	∈	PROPN
ejpam-6627	244	11	n.	n.	NOUN
ejpam-6627	244	12	these	these	DET
ejpam-6627	244	13	sequences	sequence	NOUN
ejpam-6627	244	14	satisfy	satisfy	VERB
ejpam-6627	244	15	αn	αn	NOUN
ejpam-6627	245	1	=	=	SYM
ejpam-6627	245	2	βn	βn	PROPN
ejpam-6627	245	3	for	for	ADP
ejpam-6627	245	4	all	all	DET
ejpam-6627	245	5	n	n	CCONJ
ejpam-6627	245	6	,	,	PUNCT
ejpam-6627	245	7	and	and	CCONJ
ejpam-6627	245	8	αn	αn	NOUN
ejpam-6627	245	9	→	→	SYM
ejpam-6627	245	10	β	β	X
ejpam-6627	245	11	and	and	CCONJ
ejpam-6627	245	12	βn	βn	VERB
ejpam-6627	245	13	→	→	SYM
ejpam-6627	245	14	β	β	X
ejpam-6627	245	15	.	.	PUNCT
ejpam-6627	246	1	but	but	CCONJ
ejpam-6627	246	2	since	since	SCONJ
ejpam-6627	246	3	every	every	DET
ejpam-6627	246	4	neighborhood	neighborhood	NOUN
ejpam-6627	246	5	of	of	ADP
ejpam-6627	246	6	α	α	PROPN
ejpam-6627	246	7	contains	contain	VERB
ejpam-6627	246	8	β	β	X
ejpam-6627	246	9	,	,	PUNCT
ejpam-6627	246	10	we	we	PRON
ejpam-6627	246	11	also	also	ADV
ejpam-6627	246	12	have	have	VERB
ejpam-6627	246	13	αn	αn	NOUN
ejpam-6627	246	14	→	→	SYM
ejpam-6627	246	15	α	α	NOUN
ejpam-6627	246	16	,	,	PUNCT
ejpam-6627	246	17	contradicting	contradict	VERB
ejpam-6627	246	18	the	the	DET
ejpam-6627	246	19	sequentially	sequentially	ADV
ejpam-6627	246	20	hausdorff	hausdorff	NOUN
ejpam-6627	246	21	property	property	NOUN
ejpam-6627	246	22	.	.	PUNCT
ejpam-6627	247	1	therefore	therefore	ADV
ejpam-6627	247	2	,	,	PUNCT
ejpam-6627	247	3	ω	ω	PROPN
ejpam-6627	247	4	must	must	AUX
ejpam-6627	247	5	be	be	AUX
ejpam-6627	247	6	t1	t1	NOUN
ejpam-6627	247	7	.	.	PUNCT
ejpam-6627	248	1	to	to	PART
ejpam-6627	248	2	show	show	VERB
ejpam-6627	248	3	that	that	SCONJ
ejpam-6627	248	4	none	none	NOUN
ejpam-6627	248	5	of	of	ADP
ejpam-6627	248	6	these	these	DET
ejpam-6627	248	7	implications	implication	NOUN
ejpam-6627	248	8	is	be	AUX
ejpam-6627	248	9	reversible	reversible	ADJ
ejpam-6627	248	10	,	,	PUNCT
ejpam-6627	248	11	we	we	PRON
ejpam-6627	248	12	provide	provide	VERB
ejpam-6627	248	13	the	the	DET
ejpam-6627	248	14	following	following	ADJ
ejpam-6627	248	15	counterexamples	counterexample	NOUN
ejpam-6627	248	16	,	,	PUNCT
ejpam-6627	248	17	following	follow	VERB
ejpam-6627	248	18	the	the	DET
ejpam-6627	248	19	systematic	systematic	ADJ
ejpam-6627	248	20	approach	approach	NOUN
ejpam-6627	248	21	in	in	ADP
ejpam-6627	248	22	steen	steen	PROPN
ejpam-6627	248	23	and	and	CCONJ
ejpam-6627	248	24	seebach	seebach	NOUN
ejpam-6627	248	25	[	[	X
ejpam-6627	248	26	24	24	NUM
ejpam-6627	248	27	]	]	X
ejpam-6627	248	28	:	:	PUNCT
ejpam-6627	248	29	for	for	ADP
ejpam-6627	248	30	(	(	PUNCT
ejpam-6627	248	31	1	1	NUM
ejpam-6627	248	32	)	)	PUNCT
ejpam-6627	248	33	,	,	PUNCT
ejpam-6627	248	34	consider	consider	VERB
ejpam-6627	248	35	the	the	DET
ejpam-6627	248	36	space	space	NOUN
ejpam-6627	248	37	r	r	NOUN
ejpam-6627	248	38	with	with	ADP
ejpam-6627	248	39	the	the	DET
ejpam-6627	248	40	lower	low	ADJ
ejpam-6627	248	41	limit	limit	NOUN
ejpam-6627	248	42	topology	topology	NOUN
ejpam-6627	248	43	.	.	PUNCT
ejpam-6627	249	1	this	this	DET
ejpam-6627	249	2	space	space	NOUN
ejpam-6627	249	3	is	be	AUX
ejpam-6627	249	4	t1	t1	NOUN
ejpam-6627	249	5	and	and	CCONJ
ejpam-6627	249	6	quasi	quasi	NOUN
ejpam-6627	249	7	-	-	NOUN
ejpam-6627	249	8	hausdorff	hausdorff	ADJ
ejpam-6627	249	9	(	(	PUNCT
ejpam-6627	249	10	since	since	SCONJ
ejpam-6627	249	11	any	any	DET
ejpam-6627	249	12	two	two	NUM
ejpam-6627	249	13	distinct	distinct	ADJ
ejpam-6627	249	14	points	point	NOUN
ejpam-6627	249	15	can	can	AUX
ejpam-6627	249	16	be	be	AUX
ejpam-6627	249	17	separated	separate	VERB
ejpam-6627	249	18	by	by	ADP
ejpam-6627	249	19	half	half	ADJ
ejpam-6627	249	20	-	-	PUNCT
ejpam-6627	249	21	open	open	ADJ
ejpam-6627	249	22	intervals	interval	NOUN
ejpam-6627	249	23	with	with	ADP
ejpam-6627	249	24	countable	countable	ADJ
ejpam-6627	249	25	intersection	intersection	NOUN
ejpam-6627	249	26	)	)	PUNCT
ejpam-6627	249	27	but	but	CCONJ
ejpam-6627	249	28	not	not	PART
ejpam-6627	249	29	hausdorff	hausdorff	NOUN
ejpam-6627	249	30	[	[	X
ejpam-6627	249	31	24	24	NUM
ejpam-6627	249	32	]	]	PUNCT
ejpam-6627	249	33	.	.	PUNCT
ejpam-6627	250	1	for	for	ADP
ejpam-6627	250	2	(	(	PUNCT
ejpam-6627	250	3	2	2	NUM
ejpam-6627	250	4	)	)	PUNCT
ejpam-6627	250	5	,	,	PUNCT
ejpam-6627	250	6	consider	consider	VERB
ejpam-6627	250	7	a	a	DET
ejpam-6627	250	8	co	co	ADJ
ejpam-6627	250	9	-	-	ADJ
ejpam-6627	250	10	countable	countable	ADJ
ejpam-6627	250	11	topology	topology	NOUN
ejpam-6627	250	12	on	on	ADP
ejpam-6627	250	13	an	an	DET
ejpam-6627	250	14	uncountable	uncountable	ADJ
ejpam-6627	250	15	set	set	NOUN
ejpam-6627	250	16	.	.	PUNCT
ejpam-6627	251	1	this	this	DET
ejpam-6627	251	2	space	space	NOUN
ejpam-6627	251	3	is	be	AUX
ejpam-6627	251	4	t1	t1	NOUN
ejpam-6627	251	5	but	but	CCONJ
ejpam-6627	251	6	any	any	DET
ejpam-6627	251	7	two	two	NUM
ejpam-6627	251	8	non	non	ADJ
ejpam-6627	251	9	-	-	ADJ
ejpam-6627	251	10	empty	empty	ADJ
ejpam-6627	251	11	open	open	ADJ
ejpam-6627	251	12	sets	set	NOUN
ejpam-6627	251	13	have	have	VERB
ejpam-6627	251	14	uncountable	uncountable	ADJ
ejpam-6627	251	15	intersection	intersection	NOUN
ejpam-6627	251	16	,	,	PUNCT
ejpam-6627	251	17	so	so	CCONJ
ejpam-6627	251	18	it	it	PRON
ejpam-6627	251	19	is	be	AUX
ejpam-6627	251	20	not	not	PART
ejpam-6627	251	21	quasihausdorff	quasihausdorff	NOUN
ejpam-6627	251	22	.	.	PUNCT
ejpam-6627	252	1	however	however	ADV
ejpam-6627	252	2	,	,	PUNCT
ejpam-6627	252	3	it	it	PRON
ejpam-6627	252	4	is	be	AUX
ejpam-6627	252	5	sequentially	sequentially	ADV
ejpam-6627	252	6	hausdorff	hausdorff	NOUN
ejpam-6627	252	7	because	because	SCONJ
ejpam-6627	252	8	any	any	DET
ejpam-6627	252	9	convergent	convergent	NOUN
ejpam-6627	252	10	sequence	sequence	NOUN
ejpam-6627	252	11	must	must	AUX
ejpam-6627	252	12	be	be	AUX
ejpam-6627	252	13	eventually	eventually	ADV
ejpam-6627	252	14	constant	constant	ADJ
ejpam-6627	252	15	[	[	X
ejpam-6627	252	16	24	24	NUM
ejpam-6627	252	17	]	]	PUNCT
ejpam-6627	252	18	.	.	PUNCT
ejpam-6627	253	1	for	for	ADP
ejpam-6627	253	2	(	(	PUNCT
ejpam-6627	253	3	3	3	NUM
ejpam-6627	253	4	)	)	PUNCT
ejpam-6627	253	5	,	,	PUNCT
ejpam-6627	253	6	consider	consider	VERB
ejpam-6627	253	7	the	the	DET
ejpam-6627	253	8	trivial	trivial	ADJ
ejpam-6627	253	9	topology	topology	NOUN
ejpam-6627	253	10	on	on	ADP
ejpam-6627	253	11	a	a	DET
ejpam-6627	253	12	set	set	NOUN
ejpam-6627	253	13	with	with	ADP
ejpam-6627	253	14	at	at	ADV
ejpam-6627	253	15	least	least	ADV
ejpam-6627	253	16	two	two	NUM
ejpam-6627	253	17	elements	element	NOUN
ejpam-6627	253	18	.	.	PUNCT
ejpam-6627	254	1	this	this	DET
ejpam-6627	254	2	space	space	NOUN
ejpam-6627	254	3	is	be	AUX
ejpam-6627	254	4	not	not	PART
ejpam-6627	254	5	t1	t1	NOUN
ejpam-6627	254	6	but	but	CCONJ
ejpam-6627	254	7	is	be	AUX
ejpam-6627	254	8	vacuously	vacuously	ADV
ejpam-6627	254	9	sequentially	sequentially	ADV
ejpam-6627	254	10	hausdorff	hausdorff	NOUN
ejpam-6627	254	11	since	since	SCONJ
ejpam-6627	254	12	no	no	DET
ejpam-6627	254	13	sequence	sequence	NOUN
ejpam-6627	254	14	converges	converge	VERB
ejpam-6627	254	15	unless	unless	SCONJ
ejpam-6627	254	16	it	it	PRON
ejpam-6627	254	17	is	be	AUX
ejpam-6627	254	18	eventually	eventually	ADV
ejpam-6627	254	19	constant	constant	ADJ
ejpam-6627	254	20	.	.	PUNCT
ejpam-6627	255	1	the	the	DET
ejpam-6627	255	2	following	follow	VERB
ejpam-6627	255	3	visualization	visualization	NOUN
ejpam-6627	255	4	illustrates	illustrate	VERB
ejpam-6627	255	5	these	these	DET
ejpam-6627	255	6	intermediate	intermediate	ADJ
ejpam-6627	255	7	separation	separation	NOUN
ejpam-6627	255	8	axioms	axiom	NOUN
ejpam-6627	255	9	.	.	PUNCT
ejpam-6627	256	1	j.	j.	PROPN
ejpam-6627	256	2	oudetallah	oudetallah	PROPN
ejpam-6627	256	3	et	et	PROPN
ejpam-6627	256	4	al	al	PROPN
ejpam-6627	256	5	.	.	PUNCT
ejpam-6627	256	6	/	/	SYM
ejpam-6627	256	7	eur	eur	PROPN
ejpam-6627	256	8	.	.	PUNCT
ejpam-6627	257	1	j.	j.	PROPN
ejpam-6627	257	2	pure	pure	PROPN
ejpam-6627	257	3	appl	appl	PROPN
ejpam-6627	257	4	.	.	PROPN
ejpam-6627	257	5	math	math	PROPN
ejpam-6627	257	6	,	,	PUNCT
ejpam-6627	257	7	18	18	NUM
ejpam-6627	257	8	(	(	PUNCT
ejpam-6627	257	9	3	3	NUM
ejpam-6627	257	10	)	)	PUNCT
ejpam-6627	257	11	(	(	PUNCT
ejpam-6627	257	12	2025	2025	NUM
ejpam-6627	257	13	)	)	PUNCT
ejpam-6627	257	14	,	,	PUNCT
ejpam-6627	257	15	6627	6627	NUM
ejpam-6627	257	16	13	13	NUM
ejpam-6627	257	17	of	of	ADP
ejpam-6627	257	18	20	20	NUM
ejpam-6627	257	19	t2	t2	NOUN
ejpam-6627	257	20	(	(	PUNCT
ejpam-6627	257	21	hausdorff	hausdorff	NOUN
ejpam-6627	257	22	)	)	PUNCT
ejpam-6627	257	23	quasi	quasi	NOUN
ejpam-6627	257	24	-	-	ADJ
ejpam-6627	257	25	hausdorff	hausdorff	ADJ
ejpam-6627	257	26	sequentially	sequentially	ADV
ejpam-6627	257	27	hausdorff	hausdorff	PROPN
ejpam-6627	257	28	t1	t1	PROPN
ejpam-6627	257	29	(	(	PUNCT
ejpam-6627	257	30	fréchet	fréchet	NOUN
ejpam-6627	257	31	)	)	PUNCT
ejpam-6627	257	32	r	r	NOUN
ejpam-6627	257	33	with	with	ADP
ejpam-6627	257	34	lower	low	ADJ
ejpam-6627	257	35	limit	limit	NOUN
ejpam-6627	257	36	topology	topology	NOUN
ejpam-6627	257	37	co	co	ADJ
ejpam-6627	257	38	-	-	ADJ
ejpam-6627	257	39	countable	countable	ADJ
ejpam-6627	257	40	topology	topology	NOUN
ejpam-6627	257	41	on	on	ADP
ejpam-6627	257	42	uncountable	uncountable	ADJ
ejpam-6627	257	43	set	set	ADJ
ejpam-6627	257	44	trivial	trivial	ADJ
ejpam-6627	257	45	topology	topology	NOUN
ejpam-6627	257	46	on	on	ADP
ejpam-6627	257	47	set	set	VERB
ejpam-6627	257	48	with	with	ADP
ejpam-6627	257	49	≥	≥	NOUN
ejpam-6627	257	50	2	2	NUM
ejpam-6627	257	51	elements	element	NOUN
ejpam-6627	257	52	counterexamples	counterexample	NOUN
ejpam-6627	257	53	showing	show	VERB
ejpam-6627	257	54	non	non	ADJ
ejpam-6627	257	55	-	-	NOUN
ejpam-6627	257	56	reversibility	reversibility	NOUN
ejpam-6627	257	57	of	of	ADP
ejpam-6627	257	58	implications	implication	NOUN
ejpam-6627	257	59	figure	figure	VERB
ejpam-6627	257	60	4	4	NUM
ejpam-6627	257	61	:	:	PUNCT
ejpam-6627	257	62	hierarchy	hierarchy	NOUN
ejpam-6627	257	63	of	of	ADP
ejpam-6627	257	64	intermediate	intermediate	ADJ
ejpam-6627	257	65	separation	separation	NOUN
ejpam-6627	257	66	axioms	axiom	NOUN
ejpam-6627	257	67	between	between	ADP
ejpam-6627	257	68	t1	t1	NOUN
ejpam-6627	257	69	and	and	CCONJ
ejpam-6627	257	70	t2	t2	NOUN
ejpam-6627	257	71	with	with	ADP
ejpam-6627	257	72	counterexamples	counterexample	NOUN
ejpam-6627	257	73	demonstrating	demonstrate	VERB
ejpam-6627	257	74	that	that	SCONJ
ejpam-6627	257	75	implications	implication	NOUN
ejpam-6627	257	76	are	be	AUX
ejpam-6627	257	77	not	not	PART
ejpam-6627	257	78	reversible	reversible	ADJ
ejpam-6627	257	79	.	.	PUNCT
ejpam-6627	258	1	each	each	DET
ejpam-6627	258	2	level	level	NOUN
ejpam-6627	258	3	represents	represent	VERB
ejpam-6627	258	4	a	a	DET
ejpam-6627	258	5	genuinely	genuinely	ADV
ejpam-6627	258	6	distinct	distinct	ADJ
ejpam-6627	258	7	class	class	NOUN
ejpam-6627	258	8	of	of	ADP
ejpam-6627	258	9	topological	topological	ADJ
ejpam-6627	258	10	spaces	space	NOUN
ejpam-6627	258	11	with	with	ADP
ejpam-6627	258	12	unique	unique	ADJ
ejpam-6627	258	13	properties	property	NOUN
ejpam-6627	258	14	.	.	PUNCT
ejpam-6627	259	1	we	we	PRON
ejpam-6627	259	2	now	now	ADV
ejpam-6627	259	3	establish	establish	VERB
ejpam-6627	259	4	a	a	DET
ejpam-6627	259	5	new	new	ADJ
ejpam-6627	259	6	result	result	NOUN
ejpam-6627	259	7	connecting	connect	VERB
ejpam-6627	259	8	quasi	quasi	ADJ
ejpam-6627	259	9	-	-	ADJ
ejpam-6627	259	10	hausdorff	hausdorff	ADJ
ejpam-6627	259	11	spaces	space	NOUN
ejpam-6627	259	12	with	with	ADP
ejpam-6627	259	13	sigma	sigma	ADJ
ejpam-6627	259	14	-	-	PUNCT
ejpam-6627	259	15	intersection	intersection	NOUN
ejpam-6627	259	16	properties	property	NOUN
ejpam-6627	259	17	.	.	PUNCT
ejpam-6627	260	1	this	this	DET
ejpam-6627	260	2	result	result	NOUN
ejpam-6627	260	3	relates	relate	VERB
ejpam-6627	260	4	to	to	ADP
ejpam-6627	260	5	recent	recent	ADJ
ejpam-6627	260	6	work	work	NOUN
ejpam-6627	260	7	on	on	ADP
ejpam-6627	260	8	d	d	NOUN
ejpam-6627	260	9	-	-	NOUN
ejpam-6627	260	10	metacompactness	metacompactness	NOUN
ejpam-6627	260	11	[	[	X
ejpam-6627	260	12	12	12	NUM
ejpam-6627	260	13	]	]	PUNCT
ejpam-6627	260	14	and	and	CCONJ
ejpam-6627	260	15	lindelöf	lindelöf	NOUN
ejpam-6627	260	16	properties	property	NOUN
ejpam-6627	260	17	[	[	X
ejpam-6627	260	18	31	31	NUM
ejpam-6627	260	19	]	]	PUNCT
ejpam-6627	260	20	in	in	ADP
ejpam-6627	260	21	topological	topological	ADJ
ejpam-6627	260	22	spaces	space	NOUN
ejpam-6627	260	23	,	,	PUNCT
ejpam-6627	260	24	as	as	ADV
ejpam-6627	260	25	well	well	ADV
ejpam-6627	260	26	as	as	ADP
ejpam-6627	260	27	the	the	DET
ejpam-6627	260	28	theory	theory	NOUN
ejpam-6627	260	29	of	of	ADP
ejpam-6627	260	30	ultrafilters	ultrafilter	NOUN
ejpam-6627	260	31	developed	develop	VERB
ejpam-6627	260	32	by	by	ADP
ejpam-6627	260	33	comfort	comfort	NOUN
ejpam-6627	260	34	and	and	CCONJ
ejpam-6627	260	35	negrepontis	negrepontis	ADV
ejpam-6627	260	36	[	[	X
ejpam-6627	260	37	32	32	NUM
ejpam-6627	260	38	]	]	PUNCT
ejpam-6627	260	39	.	.	PUNCT
ejpam-6627	261	1	theorem	theorem	ADJ
ejpam-6627	261	2	7	7	NUM
ejpam-6627	261	3	.	.	PUNCT
ejpam-6627	262	1	let	let	VERB
ejpam-6627	262	2	ω	ω	NUM
ejpam-6627	262	3	be	be	AUX
ejpam-6627	262	4	a	a	DET
ejpam-6627	262	5	quasi	quasi	ADJ
ejpam-6627	262	6	-	-	ADJ
ejpam-6627	262	7	hausdorff	hausdorff	ADJ
ejpam-6627	262	8	space	space	NOUN
ejpam-6627	262	9	where	where	SCONJ
ejpam-6627	262	10	every	every	DET
ejpam-6627	262	11	point	point	NOUN
ejpam-6627	262	12	has	have	VERB
ejpam-6627	262	13	a	a	DET
ejpam-6627	262	14	basis	basis	NOUN
ejpam-6627	262	15	of	of	ADP
ejpam-6627	262	16	neighborhoods	neighborhood	NOUN
ejpam-6627	262	17	with	with	ADP
ejpam-6627	262	18	countable	countable	ADJ
ejpam-6627	262	19	boundary	boundary	NOUN
ejpam-6627	262	20	.	.	PUNCT
ejpam-6627	263	1	then	then	ADV
ejpam-6627	263	2	every	every	DET
ejpam-6627	263	3	closed	close	VERB
ejpam-6627	263	4	set	set	VERB
ejpam-6627	263	5	in	in	ADP
ejpam-6627	263	6	ω	ω	PROPN
ejpam-6627	263	7	is	be	AUX
ejpam-6627	263	8	a	a	DET
ejpam-6627	263	9	sigma	sigma	NOUN
ejpam-6627	263	10	-	-	PUNCT
ejpam-6627	263	11	intersection	intersection	NOUN
ejpam-6627	263	12	.	.	PUNCT
ejpam-6627	264	1	proof	proof	NOUN
ejpam-6627	264	2	.	.	PUNCT
ejpam-6627	265	1	let	let	VERB
ejpam-6627	265	2	φ	φ	PROPN
ejpam-6627	265	3	be	be	AUX
ejpam-6627	265	4	a	a	DET
ejpam-6627	265	5	closed	closed	ADJ
ejpam-6627	265	6	set	set	NOUN
ejpam-6627	265	7	of	of	ADP
ejpam-6627	265	8	ω	ω	PROPN
ejpam-6627	265	9	and	and	CCONJ
ejpam-6627	265	10	let	let	VERB
ejpam-6627	265	11	α	α	PRON
ejpam-6627	265	12	∈	∈	PROPN
ejpam-6627	265	13	ω	ω	X
ejpam-6627	265	14	\	\	PROPN
ejpam-6627	265	15	φ	φ	PROPN
ejpam-6627	265	16	.	.	PUNCT
ejpam-6627	266	1	for	for	ADP
ejpam-6627	266	2	each	each	DET
ejpam-6627	266	3	β	β	X
ejpam-6627	266	4	∈	∈	PROPN
ejpam-6627	266	5	φ	φ	PROPN
ejpam-6627	266	6	,	,	PUNCT
ejpam-6627	266	7	since	since	SCONJ
ejpam-6627	266	8	ω	ω	PROPN
ejpam-6627	266	9	is	be	AUX
ejpam-6627	266	10	quasi	quasi	ADJ
ejpam-6627	266	11	-	-	NOUN
ejpam-6627	266	12	hausdorff	hausdorff	ADJ
ejpam-6627	266	13	,	,	PUNCT
ejpam-6627	266	14	there	there	PRON
ejpam-6627	266	15	exist	exist	VERB
ejpam-6627	266	16	open	open	ADJ
ejpam-6627	266	17	sets	set	NOUN
ejpam-6627	266	18	λβ	λβ	INTJ
ejpam-6627	266	19	and	and	CCONJ
ejpam-6627	266	20	γβ	γβ	PRON
ejpam-6627	267	1	such	such	ADJ
ejpam-6627	267	2	that	that	SCONJ
ejpam-6627	267	3	α	α	PROPN
ejpam-6627	267	4	∈	∈	PROPN
ejpam-6627	267	5	λβ	λβ	PROPN
ejpam-6627	267	6	,	,	PUNCT
ejpam-6627	267	7	β	β	X
ejpam-6627	267	8	∈	∈	PROPN
ejpam-6627	267	9	γβ	γβ	NOUN
ejpam-6627	267	10	,	,	PUNCT
ejpam-6627	267	11	and	and	CCONJ
ejpam-6627	267	12	λβ	λβ	ADP
ejpam-6627	267	13	∩γβ	∩γβ	PROPN
ejpam-6627	267	14	is	be	AUX
ejpam-6627	267	15	countable	countable	ADJ
ejpam-6627	267	16	.	.	PUNCT
ejpam-6627	268	1	since	since	SCONJ
ejpam-6627	268	2	every	every	DET
ejpam-6627	268	3	point	point	NOUN
ejpam-6627	268	4	has	have	VERB
ejpam-6627	268	5	a	a	DET
ejpam-6627	268	6	basis	basis	NOUN
ejpam-6627	268	7	of	of	ADP
ejpam-6627	268	8	neighborhoods	neighborhood	NOUN
ejpam-6627	268	9	with	with	ADP
ejpam-6627	268	10	countable	countable	ADJ
ejpam-6627	268	11	boundary	boundary	NOUN
ejpam-6627	268	12	,	,	PUNCT
ejpam-6627	268	13	for	for	ADP
ejpam-6627	268	14	each	each	DET
ejpam-6627	268	15	β	β	X
ejpam-6627	268	16	∈	∈	PROPN
ejpam-6627	268	17	φ	φ	PROPN
ejpam-6627	268	18	,	,	PUNCT
ejpam-6627	268	19	we	we	PRON
ejpam-6627	268	20	can	can	AUX
ejpam-6627	268	21	choose	choose	VERB
ejpam-6627	268	22	γ′	γ′	PROPN
ejpam-6627	268	23	β	β	X
ejpam-6627	268	24	⊂	⊂	PROPN
ejpam-6627	268	25	γβ	γβ	NOUN
ejpam-6627	268	26	such	such	ADJ
ejpam-6627	268	27	that	that	SCONJ
ejpam-6627	268	28	β	β	X
ejpam-6627	268	29	∈	∈	PROPN
ejpam-6627	268	30	γ′	γ′	PUNCT
ejpam-6627	268	31	β	β	X
ejpam-6627	268	32	and	and	CCONJ
ejpam-6627	268	33	∂γ′	∂γ′	PROPN
ejpam-6627	268	34	β	β	PROPN
ejpam-6627	268	35	is	be	AUX
ejpam-6627	268	36	countable	countable	ADJ
ejpam-6627	268	37	.	.	PUNCT
ejpam-6627	269	1	the	the	DET
ejpam-6627	269	2	set	set	NOUN
ejpam-6627	269	3	{	{	PUNCT
ejpam-6627	269	4	γ′	γ′	PROPN
ejpam-6627	269	5	β	β	X
ejpam-6627	269	6	:	:	PUNCT
ejpam-6627	269	7	β	β	X
ejpam-6627	269	8	∈	∈	PROPN
ejpam-6627	269	9	φ	φ	PROPN
ejpam-6627	269	10	}	}	PUNCT
ejpam-6627	269	11	forms	form	VERB
ejpam-6627	269	12	an	an	DET
ejpam-6627	269	13	open	open	ADJ
ejpam-6627	269	14	cover	cover	NOUN
ejpam-6627	269	15	of	of	ADP
ejpam-6627	269	16	φ	φ	PROPN
ejpam-6627	269	17	.	.	PUNCT
ejpam-6627	270	1	since	since	SCONJ
ejpam-6627	270	2	we	we	PRON
ejpam-6627	270	3	’re	’re	AUX
ejpam-6627	270	4	not	not	PART
ejpam-6627	270	5	assuming	assume	VERB
ejpam-6627	270	6	φ	φ	PROPN
ejpam-6627	270	7	is	be	AUX
ejpam-6627	270	8	compact	compact	ADJ
ejpam-6627	270	9	,	,	PUNCT
ejpam-6627	270	10	we	we	PRON
ejpam-6627	270	11	can	can	AUX
ejpam-6627	270	12	not	not	PART
ejpam-6627	270	13	extract	extract	VERB
ejpam-6627	270	14	a	a	DET
ejpam-6627	270	15	finite	finite	NOUN
ejpam-6627	270	16	subcover	subcover	PROPN
ejpam-6627	270	17	directly	directly	ADV
ejpam-6627	270	18	.	.	PUNCT
ejpam-6627	271	1	instead	instead	ADV
ejpam-6627	271	2	,	,	PUNCT
ejpam-6627	271	3	we	we	PRON
ejpam-6627	271	4	exploit	exploit	VERB
ejpam-6627	271	5	the	the	DET
ejpam-6627	271	6	structure	structure	NOUN
ejpam-6627	271	7	of	of	ADP
ejpam-6627	271	8	ω	ω	PROPN
ejpam-6627	271	9	as	as	SCONJ
ejpam-6627	271	10	follows	follow	VERB
ejpam-6627	271	11	:	:	PUNCT
ejpam-6627	271	12	for	for	SCONJ
ejpam-6627	271	13	each	each	DET
ejpam-6627	271	14	β	β	PROPN
ejpam-6627	271	15	∈	∈	PROPN
ejpam-6627	271	16	φ	φ	PROPN
ejpam-6627	271	17	,	,	PUNCT
ejpam-6627	271	18	let	let	VERB
ejpam-6627	271	19	cβ	cβ	NOUN
ejpam-6627	271	20	=	=	NOUN
ejpam-6627	271	21	λβ	λβ	ADP
ejpam-6627	271	22	∩	∩	NOUN
ejpam-6627	271	23	γ′	γ′	X
ejpam-6627	271	24	β	β	NOUN
ejpam-6627	271	25	,	,	PUNCT
ejpam-6627	271	26	which	which	PRON
ejpam-6627	271	27	is	be	AUX
ejpam-6627	271	28	countable	countable	ADJ
ejpam-6627	271	29	by	by	ADP
ejpam-6627	271	30	the	the	DET
ejpam-6627	271	31	quasi	quasi	ADJ
ejpam-6627	271	32	-	-	ADJ
ejpam-6627	271	33	hausdorff	hausdorff	ADJ
ejpam-6627	271	34	property	property	NOUN
ejpam-6627	271	35	.	.	PUNCT
ejpam-6627	272	1	let	let	VERB
ejpam-6627	272	2	c	c	NOUN
ejpam-6627	272	3	=	=	PUNCT
ejpam-6627	272	4	⋃	⋃	PROPN
ejpam-6627	272	5	β∈φcβ	β∈φcβ	PROPN
ejpam-6627	272	6	,	,	PUNCT
ejpam-6627	272	7	which	which	PRON
ejpam-6627	272	8	may	may	AUX
ejpam-6627	272	9	be	be	AUX
ejpam-6627	272	10	uncountable	uncountable	ADJ
ejpam-6627	272	11	.	.	PUNCT
ejpam-6627	273	1	however	however	ADV
ejpam-6627	273	2	,	,	PUNCT
ejpam-6627	273	3	we	we	PRON
ejpam-6627	273	4	can	can	AUX
ejpam-6627	273	5	extract	extract	VERB
ejpam-6627	273	6	a	a	DET
ejpam-6627	273	7	countable	countable	ADJ
ejpam-6627	273	8	subset	subset	NOUN
ejpam-6627	273	9	{	{	PUNCT
ejpam-6627	273	10	βn}n∈n	βn}n∈n	PUNCT
ejpam-6627	273	11	⊂	⊂	PROPN
ejpam-6627	273	12	φ	φ	NUM
ejpam-6627	273	13	such	such	ADJ
ejpam-6627	273	14	that	that	SCONJ
ejpam-6627	273	15	α	α	NOUN
ejpam-6627	273	16	/∈	/∈	PUNCT
ejpam-6627	274	1	(	(	PUNCT
ejpam-6627	274	2	φ	φ	NOUN
ejpam-6627	274	3	\	\	X
ejpam-6627	274	4	⋃	⋃	PUNCT
ejpam-6627	274	5	n∈n	n∈n	NOUN
ejpam-6627	274	6	γ′	γ′	NUM
ejpam-6627	274	7	βn	βn	NOUN
ejpam-6627	274	8	)	)	PUNCT
ejpam-6627	274	9	.	.	PUNCT
ejpam-6627	275	1	now	now	ADV
ejpam-6627	275	2	,	,	PUNCT
ejpam-6627	275	3	let	let	VERB
ejpam-6627	275	4	λ	λ	X
ejpam-6627	275	5	=	=	SYM
ejpam-6627	275	6	ω	ω	X
ejpam-6627	275	7	\	\	PROPN
ejpam-6627	275	8	(	(	PUNCT
ejpam-6627	275	9	φ	φ	PROPN
ejpam-6627	275	10	\	\	X
ejpam-6627	276	1	⋃	⋃	PUNCT
ejpam-6627	276	2	n∈n	n∈n	NOUN
ejpam-6627	276	3	γ′	γ′	NUM
ejpam-6627	276	4	βn	βn	NOUN
ejpam-6627	276	5	)	)	PUNCT
ejpam-6627	276	6	.	.	PUNCT
ejpam-6627	277	1	then	then	ADV
ejpam-6627	277	2	λ	λ	PROPN
ejpam-6627	277	3	is	be	AUX
ejpam-6627	277	4	open	open	ADJ
ejpam-6627	277	5	,	,	PUNCT
ejpam-6627	277	6	α	α	PROPN
ejpam-6627	277	7	∈	∈	PROPN
ejpam-6627	277	8	λ	λ	PROPN
ejpam-6627	277	9	,	,	PUNCT
ejpam-6627	277	10	and	and	CCONJ
ejpam-6627	277	11	λ	λ	PROPN
ejpam-6627	277	12	∩	∩	PROPN
ejpam-6627	277	13	φ	φ	PROPN
ejpam-6627	277	14	⊂	⊂	PROPN
ejpam-6627	277	15	⋃	⋃	PROPN
ejpam-6627	277	16	n∈n	n∈n	NOUN
ejpam-6627	277	17	γ′	γ′	NUM
ejpam-6627	277	18	βn	βn	NOUN
ejpam-6627	277	19	.	.	PUNCT
ejpam-6627	278	1	since	since	SCONJ
ejpam-6627	278	2	this	this	DET
ejpam-6627	278	3	construction	construction	NOUN
ejpam-6627	278	4	can	can	AUX
ejpam-6627	278	5	be	be	AUX
ejpam-6627	278	6	applied	apply	VERB
ejpam-6627	278	7	to	to	ADP
ejpam-6627	278	8	each	each	DET
ejpam-6627	278	9	α	α	PROPN
ejpam-6627	278	10	∈	∈	PROPN
ejpam-6627	278	11	ω	ω	NUM
ejpam-6627	278	12	\	\	PROPN
ejpam-6627	278	13	φ	φ	PROPN
ejpam-6627	278	14	,	,	PUNCT
ejpam-6627	278	15	we	we	PRON
ejpam-6627	278	16	obtain	obtain	VERB
ejpam-6627	278	17	a	a	DET
ejpam-6627	278	18	set	set	NOUN
ejpam-6627	278	19	of	of	ADP
ejpam-6627	278	20	open	open	ADJ
ejpam-6627	278	21	sets	set	NOUN
ejpam-6627	278	22	{	{	PUNCT
ejpam-6627	278	23	λα	λα	X
ejpam-6627	278	24	:	:	PUNCT
ejpam-6627	278	25	α	α	PROPN
ejpam-6627	278	26	∈	∈	PROPN
ejpam-6627	278	27	ω	ω	X
ejpam-6627	278	28	\	\	PROPN
ejpam-6627	278	29	φ	φ	PROPN
ejpam-6627	278	30	}	}	PUNCT
ejpam-6627	278	31	such	such	ADJ
ejpam-6627	278	32	that	that	SCONJ
ejpam-6627	278	33	ω	ω	NUM
ejpam-6627	278	34	\	\	PROPN
ejpam-6627	278	35	φ	φ	PROPN
ejpam-6627	278	36	=	=	SYM
ejpam-6627	278	37	⋃	⋃	PROPN
ejpam-6627	278	38	α∈ω\φ	α∈ω\φ	PROPN
ejpam-6627	278	39	λα	λα	NOUN
ejpam-6627	278	40	.	.	PUNCT
ejpam-6627	279	1	using	use	VERB
ejpam-6627	279	2	the	the	DET
ejpam-6627	279	3	fact	fact	NOUN
ejpam-6627	279	4	that	that	SCONJ
ejpam-6627	279	5	each	each	DET
ejpam-6627	279	6	∂γ′	∂γ′	PROPN
ejpam-6627	279	7	βn	βn	NOUN
ejpam-6627	279	8	is	be	AUX
ejpam-6627	279	9	countable	countable	ADJ
ejpam-6627	279	10	,	,	PUNCT
ejpam-6627	279	11	we	we	PRON
ejpam-6627	279	12	can	can	AUX
ejpam-6627	279	13	construct	construct	VERB
ejpam-6627	279	14	a	a	DET
ejpam-6627	279	15	countable	countable	ADJ
ejpam-6627	279	16	collection	collection	NOUN
ejpam-6627	279	17	of	of	ADP
ejpam-6627	279	18	open	open	ADJ
ejpam-6627	279	19	sets	set	NOUN
ejpam-6627	279	20	{	{	PUNCT
ejpam-6627	279	21	θm}m∈n	θm}m∈n	ADP
ejpam-6627	279	22	such	such	ADJ
ejpam-6627	279	23	that	that	SCONJ
ejpam-6627	279	24	φ	φ	PROPN
ejpam-6627	279	25	=	=	SYM
ejpam-6627	279	26	⋂	⋂	PROPN
ejpam-6627	279	27	m∈nθm	m∈nθm	PROPN
ejpam-6627	279	28	,	,	PUNCT
ejpam-6627	279	29	establishing	establish	VERB
ejpam-6627	279	30	that	that	SCONJ
ejpam-6627	279	31	φ	φ	PROPN
ejpam-6627	279	32	is	be	AUX
ejpam-6627	279	33	a	a	DET
ejpam-6627	279	34	sigma	sigma	NOUN
ejpam-6627	279	35	-	-	PUNCT
ejpam-6627	279	36	intersection	intersection	NOUN
ejpam-6627	279	37	.	.	PUNCT
ejpam-6627	280	1	this	this	DET
ejpam-6627	280	2	theorem	theorem	NOUN
ejpam-6627	280	3	connects	connect	VERB
ejpam-6627	280	4	quasi	quasi	ADJ
ejpam-6627	280	5	-	-	ADJ
ejpam-6627	280	6	hausdorff	hausdorff	ADJ
ejpam-6627	280	7	spaces	space	NOUN
ejpam-6627	280	8	with	with	ADP
ejpam-6627	280	9	the	the	DET
ejpam-6627	280	10	property	property	NOUN
ejpam-6627	280	11	that	that	PRON
ejpam-6627	280	12	closed	close	VERB
ejpam-6627	280	13	sets	set	NOUN
ejpam-6627	280	14	are	be	AUX
ejpam-6627	280	15	sigma	sigma	NOUN
ejpam-6627	280	16	-	-	PUNCT
ejpam-6627	280	17	intersections	intersection	NOUN
ejpam-6627	280	18	,	,	PUNCT
ejpam-6627	280	19	providing	provide	VERB
ejpam-6627	280	20	a	a	DET
ejpam-6627	280	21	broader	broad	ADJ
ejpam-6627	280	22	context	context	NOUN
ejpam-6627	280	23	for	for	ADP
ejpam-6627	280	24	the	the	DET
ejpam-6627	280	25	results	result	NOUN
ejpam-6627	280	26	in	in	ADP
ejpam-6627	280	27	section	section	NOUN
ejpam-6627	280	28	3	3	NUM
ejpam-6627	280	29	.	.	PUNCT
ejpam-6627	281	1	it	it	PRON
ejpam-6627	281	2	demonstrates	demonstrate	VERB
ejpam-6627	281	3	how	how	SCONJ
ejpam-6627	281	4	intermediate	intermediate	ADJ
ejpam-6627	281	5	separation	separation	NOUN
ejpam-6627	281	6	axioms	axiom	NOUN
ejpam-6627	281	7	can	can	AUX
ejpam-6627	281	8	yield	yield	VERB
ejpam-6627	281	9	functionally	functionally	ADV
ejpam-6627	281	10	significant	significant	ADJ
ejpam-6627	281	11	topological	topological	ADJ
ejpam-6627	281	12	properties	property	NOUN
ejpam-6627	281	13	,	,	PUNCT
ejpam-6627	281	14	bridging	bridge	VERB
ejpam-6627	281	15	the	the	DET
ejpam-6627	281	16	gap	gap	NOUN
ejpam-6627	281	17	between	between	ADP
ejpam-6627	281	18	purely	purely	ADV
ejpam-6627	281	19	set	set	VERB
ejpam-6627	281	20	-	-	PUNCT
ejpam-6627	281	21	theoretic	theoretic	NOUN
ejpam-6627	281	22	conditions	condition	NOUN
ejpam-6627	281	23	and	and	CCONJ
ejpam-6627	281	24	analytically	analytically	ADV
ejpam-6627	281	25	useful	useful	ADJ
ejpam-6627	281	26	properties	property	NOUN
ejpam-6627	281	27	.	.	PUNCT
ejpam-6627	282	1	j.	j.	PROPN
ejpam-6627	282	2	oudetallah	oudetallah	PROPN
ejpam-6627	282	3	et	et	PROPN
ejpam-6627	282	4	al	al	PROPN
ejpam-6627	282	5	.	.	PUNCT
ejpam-6627	282	6	/	/	SYM
ejpam-6627	282	7	eur	eur	PROPN
ejpam-6627	282	8	.	.	PUNCT
ejpam-6627	283	1	j.	j.	PROPN
ejpam-6627	283	2	pure	pure	PROPN
ejpam-6627	283	3	appl	appl	PROPN
ejpam-6627	283	4	.	.	PROPN
ejpam-6627	283	5	math	math	PROPN
ejpam-6627	283	6	,	,	PUNCT
ejpam-6627	283	7	18	18	NUM
ejpam-6627	283	8	(	(	PUNCT
ejpam-6627	283	9	3	3	NUM
ejpam-6627	283	10	)	)	PUNCT
ejpam-6627	283	11	(	(	PUNCT
ejpam-6627	283	12	2025	2025	NUM
ejpam-6627	283	13	)	)	PUNCT
ejpam-6627	283	14	,	,	PUNCT
ejpam-6627	283	15	6627	6627	NUM
ejpam-6627	283	16	14	14	NUM
ejpam-6627	283	17	of	of	ADP
ejpam-6627	283	18	20	20	NUM
ejpam-6627	283	19	6	6	NUM
ejpam-6627	283	20	.	.	PUNCT
ejpam-6627	284	1	behavior	behavior	NOUN
ejpam-6627	284	2	under	under	ADP
ejpam-6627	284	3	topological	topological	ADJ
ejpam-6627	284	4	operations	operation	NOUN
ejpam-6627	284	5	understanding	understand	VERB
ejpam-6627	284	6	how	how	SCONJ
ejpam-6627	284	7	separation	separation	NOUN
ejpam-6627	284	8	axioms	axiom	NOUN
ejpam-6627	284	9	behave	behave	VERB
ejpam-6627	284	10	under	under	ADP
ejpam-6627	284	11	standard	standard	ADJ
ejpam-6627	284	12	topological	topological	ADJ
ejpam-6627	284	13	operations	operation	NOUN
ejpam-6627	284	14	is	be	AUX
ejpam-6627	284	15	crucial	crucial	ADJ
ejpam-6627	284	16	for	for	ADP
ejpam-6627	284	17	constructing	construct	VERB
ejpam-6627	284	18	new	new	ADJ
ejpam-6627	284	19	spaces	space	NOUN
ejpam-6627	284	20	and	and	CCONJ
ejpam-6627	284	21	determining	determine	VERB
ejpam-6627	284	22	when	when	SCONJ
ejpam-6627	284	23	properties	property	NOUN
ejpam-6627	284	24	are	be	AUX
ejpam-6627	284	25	preserved	preserve	VERB
ejpam-6627	284	26	.	.	PUNCT
ejpam-6627	285	1	in	in	ADP
ejpam-6627	285	2	this	this	DET
ejpam-6627	285	3	section	section	NOUN
ejpam-6627	285	4	,	,	PUNCT
ejpam-6627	285	5	we	we	PRON
ejpam-6627	285	6	investigate	investigate	VERB
ejpam-6627	285	7	the	the	DET
ejpam-6627	285	8	behavior	behavior	NOUN
ejpam-6627	285	9	of	of	ADP
ejpam-6627	285	10	various	various	ADJ
ejpam-6627	285	11	separation	separation	NOUN
ejpam-6627	285	12	axioms	axiom	NOUN
ejpam-6627	285	13	under	under	ADP
ejpam-6627	285	14	quotient	quotient	NOUN
ejpam-6627	285	15	maps	map	NOUN
ejpam-6627	285	16	and	and	CCONJ
ejpam-6627	285	17	product	product	NOUN
ejpam-6627	285	18	constructions	construction	NOUN
ejpam-6627	285	19	,	,	PUNCT
ejpam-6627	285	20	following	follow	VERB
ejpam-6627	285	21	the	the	DET
ejpam-6627	285	22	systematic	systematic	ADJ
ejpam-6627	285	23	approach	approach	NOUN
ejpam-6627	285	24	of	of	ADP
ejpam-6627	285	25	arhangel’skii	arhangel’skii	NOUN
ejpam-6627	285	26	[	[	X
ejpam-6627	285	27	29	29	NUM
ejpam-6627	285	28	]	]	PUNCT
ejpam-6627	285	29	and	and	CCONJ
ejpam-6627	285	30	tychonoff	tychonoff	X
ejpam-6627	286	1	[	[	X
ejpam-6627	286	2	30	30	NUM
ejpam-6627	286	3	]	]	PUNCT
ejpam-6627	286	4	.	.	PUNCT
ejpam-6627	287	1	theorem	theorem	ADJ
ejpam-6627	287	2	8	8	NUM
ejpam-6627	287	3	.	.	PUNCT
ejpam-6627	288	1	let	let	VERB
ejpam-6627	288	2	ω	ω	NUM
ejpam-6627	288	3	be	be	AUX
ejpam-6627	288	4	a	a	DET
ejpam-6627	288	5	topological	topological	ADJ
ejpam-6627	288	6	space	space	NOUN
ejpam-6627	288	7	and	and	CCONJ
ejpam-6627	288	8	∼	∼	NOUN
ejpam-6627	288	9	be	be	AUX
ejpam-6627	288	10	an	an	DET
ejpam-6627	288	11	equivalence	equivalence	NOUN
ejpam-6627	288	12	relation	relation	NOUN
ejpam-6627	288	13	on	on	ADP
ejpam-6627	288	14	ω	ω	PROPN
ejpam-6627	288	15	.	.	PUNCT
ejpam-6627	289	1	the	the	DET
ejpam-6627	289	2	following	follow	VERB
ejpam-6627	289	3	statements	statement	NOUN
ejpam-6627	289	4	hold	hold	VERB
ejpam-6627	289	5	regarding	regard	VERB
ejpam-6627	289	6	the	the	DET
ejpam-6627	289	7	quotient	quotient	NOUN
ejpam-6627	289	8	space	space	NOUN
ejpam-6627	289	9	ω/∼	ω/∼	PROPN
ejpam-6627	289	10	:	:	PUNCT
ejpam-6627	289	11	(	(	PUNCT
ejpam-6627	289	12	i	i	NOUN
ejpam-6627	289	13	)	)	PUNCT
ejpam-6627	289	14	if	if	SCONJ
ejpam-6627	289	15	ω	ω	PROPN
ejpam-6627	289	16	is	be	AUX
ejpam-6627	289	17	t0	t0	NOUN
ejpam-6627	289	18	,	,	PUNCT
ejpam-6627	289	19	then	then	ADV
ejpam-6627	289	20	ω/∼	ω/∼	PROPN
ejpam-6627	289	21	is	be	AUX
ejpam-6627	289	22	t0	t0	PROPN
ejpam-6627	289	23	if	if	SCONJ
ejpam-6627	289	24	and	and	CCONJ
ejpam-6627	289	25	only	only	ADV
ejpam-6627	289	26	if	if	SCONJ
ejpam-6627	289	27	for	for	ADP
ejpam-6627	289	28	any	any	DET
ejpam-6627	289	29	distinct	distinct	ADJ
ejpam-6627	289	30	equivalence	equivalence	NOUN
ejpam-6627	289	31	classes	class	NOUN
ejpam-6627	289	32	[	[	X
ejpam-6627	289	33	α	α	X
ejpam-6627	289	34	]	]	X
ejpam-6627	289	35	and	and	CCONJ
ejpam-6627	289	36	[	[	X
ejpam-6627	289	37	β	β	X
ejpam-6627	289	38	]	]	X
ejpam-6627	289	39	,	,	PUNCT
ejpam-6627	289	40	there	there	PRON
ejpam-6627	289	41	exists	exist	VERB
ejpam-6627	289	42	an	an	DET
ejpam-6627	289	43	open	open	ADJ
ejpam-6627	289	44	set	set	NOUN
ejpam-6627	289	45	λ	λ	INTJ
ejpam-6627	289	46	such	such	ADJ
ejpam-6627	289	47	that	that	SCONJ
ejpam-6627	289	48	λ	λ	PROPN
ejpam-6627	289	49	contains	contain	VERB
ejpam-6627	289	50	exactly	exactly	ADV
ejpam-6627	289	51	one	one	NUM
ejpam-6627	289	52	of	of	ADP
ejpam-6627	289	53	[	[	X
ejpam-6627	289	54	α	α	X
ejpam-6627	289	55	]	]	X
ejpam-6627	289	56	or	or	CCONJ
ejpam-6627	289	57	[	[	X
ejpam-6627	289	58	β	β	X
ejpam-6627	289	59	]	]	X
ejpam-6627	289	60	.	.	PUNCT
ejpam-6627	290	1	(	(	PUNCT
ejpam-6627	290	2	ii	ii	NOUN
ejpam-6627	290	3	)	)	PUNCT
ejpam-6627	290	4	if	if	SCONJ
ejpam-6627	290	5	ω	ω	PROPN
ejpam-6627	290	6	is	be	AUX
ejpam-6627	290	7	t1	t1	NOUN
ejpam-6627	290	8	,	,	PUNCT
ejpam-6627	290	9	then	then	ADV
ejpam-6627	290	10	ω/∼	ω/∼	PROPN
ejpam-6627	290	11	is	be	AUX
ejpam-6627	290	12	t1	t1	NOUN
ejpam-6627	290	13	if	if	SCONJ
ejpam-6627	290	14	and	and	CCONJ
ejpam-6627	290	15	only	only	ADV
ejpam-6627	290	16	if	if	SCONJ
ejpam-6627	290	17	each	each	DET
ejpam-6627	290	18	equivalence	equivalence	NOUN
ejpam-6627	290	19	class	class	NOUN
ejpam-6627	290	20	is	be	AUX
ejpam-6627	290	21	closed	close	VERB
ejpam-6627	290	22	in	in	ADP
ejpam-6627	290	23	ω	ω	PROPN
ejpam-6627	290	24	.	.	PUNCT
ejpam-6627	291	1	(	(	PUNCT
ejpam-6627	291	2	iii	iii	X
ejpam-6627	291	3	)	)	PUNCT
ejpam-6627	291	4	if	if	SCONJ
ejpam-6627	291	5	ω	ω	PROPN
ejpam-6627	291	6	is	be	AUX
ejpam-6627	291	7	t2	t2	NOUN
ejpam-6627	291	8	,	,	PUNCT
ejpam-6627	291	9	then	then	ADV
ejpam-6627	291	10	ω/∼	ω/∼	PROPN
ejpam-6627	291	11	is	be	AUX
ejpam-6627	291	12	t2	t2	NOUN
ejpam-6627	291	13	if	if	SCONJ
ejpam-6627	291	14	and	and	CCONJ
ejpam-6627	291	15	only	only	ADV
ejpam-6627	291	16	if	if	SCONJ
ejpam-6627	291	17	the	the	DET
ejpam-6627	291	18	equivalence	equivalence	NOUN
ejpam-6627	291	19	relation	relation	NOUN
ejpam-6627	291	20	∼	∼	NOUN
ejpam-6627	291	21	is	be	AUX
ejpam-6627	291	22	closed	close	VERB
ejpam-6627	291	23	as	as	ADP
ejpam-6627	291	24	a	a	DET
ejpam-6627	291	25	subset	subset	NOUN
ejpam-6627	291	26	of	of	ADP
ejpam-6627	291	27	ω×	ω×	PROPN
ejpam-6627	291	28	ω	ω	PROPN
ejpam-6627	291	29	.	.	PUNCT
ejpam-6627	292	1	(	(	PUNCT
ejpam-6627	292	2	iv	iv	X
ejpam-6627	292	3	)	)	PUNCT
ejpam-6627	292	4	if	if	SCONJ
ejpam-6627	292	5	ω	ω	PROPN
ejpam-6627	292	6	is	be	AUX
ejpam-6627	292	7	quasi	quasi	ADJ
ejpam-6627	292	8	-	-	NOUN
ejpam-6627	292	9	hausdorff	hausdorff	ADJ
ejpam-6627	292	10	,	,	PUNCT
ejpam-6627	292	11	then	then	ADV
ejpam-6627	292	12	ω/∼	ω/∼	PROPN
ejpam-6627	292	13	is	be	AUX
ejpam-6627	292	14	quasi	quasi	ADJ
ejpam-6627	292	15	-	-	ADJ
ejpam-6627	292	16	hausdorff	hausdorff	ADJ
ejpam-6627	292	17	if	if	SCONJ
ejpam-6627	292	18	and	and	CCONJ
ejpam-6627	292	19	only	only	ADV
ejpam-6627	292	20	if	if	SCONJ
ejpam-6627	292	21	for	for	ADP
ejpam-6627	292	22	any	any	DET
ejpam-6627	292	23	distinct	distinct	ADJ
ejpam-6627	292	24	equivalence	equivalence	NOUN
ejpam-6627	292	25	classes	class	NOUN
ejpam-6627	292	26	[	[	X
ejpam-6627	292	27	α	α	X
ejpam-6627	292	28	]	]	X
ejpam-6627	292	29	and	and	CCONJ
ejpam-6627	292	30	[	[	X
ejpam-6627	292	31	β	β	X
ejpam-6627	292	32	]	]	X
ejpam-6627	292	33	,	,	PUNCT
ejpam-6627	292	34	there	there	PRON
ejpam-6627	292	35	exist	exist	VERB
ejpam-6627	292	36	open	open	ADJ
ejpam-6627	292	37	sets	set	NOUN
ejpam-6627	292	38	λ	λ	PROPN
ejpam-6627	292	39	and	and	CCONJ
ejpam-6627	292	40	γ	γ	NOUN
ejpam-6627	293	1	such	such	ADJ
ejpam-6627	293	2	that	that	SCONJ
ejpam-6627	293	3	[	[	X
ejpam-6627	293	4	α	α	X
ejpam-6627	293	5	]	]	X
ejpam-6627	293	6	⊂	⊂	PROPN
ejpam-6627	293	7	λ	λ	PROPN
ejpam-6627	293	8	,	,	PUNCT
ejpam-6627	293	9	[	[	X
ejpam-6627	293	10	β	β	X
ejpam-6627	293	11	]	]	X
ejpam-6627	293	12	⊂	⊂	PROPN
ejpam-6627	293	13	γ	γ	X
ejpam-6627	293	14	,	,	PUNCT
ejpam-6627	293	15	and	and	CCONJ
ejpam-6627	293	16	λ	λ	PROPN
ejpam-6627	293	17	∩	∩	NOUN
ejpam-6627	293	18	γ	γ	NOUN
ejpam-6627	293	19	intersects	intersect	NOUN
ejpam-6627	293	20	at	at	ADP
ejpam-6627	293	21	most	most	ADV
ejpam-6627	293	22	countably	countably	ADV
ejpam-6627	293	23	many	many	ADJ
ejpam-6627	293	24	equivalence	equivalence	NOUN
ejpam-6627	293	25	classes	class	NOUN
ejpam-6627	293	26	.	.	PUNCT
ejpam-6627	294	1	proof	proof	NOUN
ejpam-6627	294	2	.	.	PUNCT
ejpam-6627	295	1	(	(	PUNCT
ejpam-6627	295	2	1	1	X
ejpam-6627	295	3	)	)	PUNCT
ejpam-6627	295	4	assume	assume	VERB
ejpam-6627	295	5	ω	ω	PROPN
ejpam-6627	295	6	is	be	AUX
ejpam-6627	295	7	t0	t0	NOUN
ejpam-6627	295	8	.	.	PUNCT
ejpam-6627	296	1	let	let	VERB
ejpam-6627	297	1	π	π	NOUN
ejpam-6627	297	2	:	:	PUNCT
ejpam-6627	297	3	ω	ω	X
ejpam-6627	297	4	→	→	SYM
ejpam-6627	297	5	ω/∼	ω/∼	PUNCT
ejpam-6627	297	6	be	be	AUX
ejpam-6627	297	7	the	the	DET
ejpam-6627	297	8	quotient	quotient	NOUN
ejpam-6627	297	9	map	map	NOUN
ejpam-6627	297	10	.	.	PUNCT
ejpam-6627	298	1	ω/∼	ω/∼	NUM
ejpam-6627	298	2	is	be	AUX
ejpam-6627	298	3	t0	t0	PROPN
ejpam-6627	298	4	if	if	SCONJ
ejpam-6627	298	5	and	and	CCONJ
ejpam-6627	298	6	only	only	ADV
ejpam-6627	298	7	if	if	SCONJ
ejpam-6627	298	8	for	for	ADP
ejpam-6627	298	9	any	any	DET
ejpam-6627	298	10	distinct	distinct	ADJ
ejpam-6627	298	11	equivalence	equivalence	NOUN
ejpam-6627	298	12	classes	class	NOUN
ejpam-6627	298	13	[	[	X
ejpam-6627	298	14	α	α	X
ejpam-6627	298	15	]	]	X
ejpam-6627	298	16	and	and	CCONJ
ejpam-6627	298	17	[	[	X
ejpam-6627	298	18	β	β	X
ejpam-6627	298	19	]	]	X
ejpam-6627	298	20	,	,	PUNCT
ejpam-6627	298	21	there	there	PRON
ejpam-6627	298	22	exists	exist	VERB
ejpam-6627	298	23	an	an	DET
ejpam-6627	298	24	open	open	ADJ
ejpam-6627	298	25	set	set	NOUN
ejpam-6627	298	26	u	u	NOUN
ejpam-6627	298	27	in	in	ADP
ejpam-6627	298	28	ω/∼	ω/∼	PROPN
ejpam-6627	298	29	such	such	ADJ
ejpam-6627	298	30	that	that	SCONJ
ejpam-6627	298	31	u	u	PROPN
ejpam-6627	298	32	contains	contain	VERB
ejpam-6627	298	33	exactly	exactly	ADV
ejpam-6627	298	34	one	one	NUM
ejpam-6627	298	35	of	of	ADP
ejpam-6627	298	36	[	[	X
ejpam-6627	298	37	α	α	X
ejpam-6627	298	38	]	]	X
ejpam-6627	298	39	or	or	CCONJ
ejpam-6627	298	40	[	[	X
ejpam-6627	298	41	β	β	X
ejpam-6627	298	42	]	]	X
ejpam-6627	298	43	.	.	PUNCT
ejpam-6627	299	1	this	this	PRON
ejpam-6627	299	2	is	be	AUX
ejpam-6627	299	3	equivalent	equivalent	ADJ
ejpam-6627	299	4	to	to	ADP
ejpam-6627	299	5	the	the	DET
ejpam-6627	299	6	existence	existence	NOUN
ejpam-6627	299	7	of	of	ADP
ejpam-6627	299	8	an	an	DET
ejpam-6627	299	9	open	open	ADJ
ejpam-6627	299	10	set	set	ADJ
ejpam-6627	299	11	λ	λ	PROPN
ejpam-6627	299	12	in	in	ADP
ejpam-6627	299	13	ω	ω	NUM
ejpam-6627	299	14	such	such	ADJ
ejpam-6627	299	15	that	that	DET
ejpam-6627	299	16	π−1(u	π−1(u	NOUN
ejpam-6627	299	17	)	)	PUNCT
ejpam-6627	300	1	=	=	SYM
ejpam-6627	300	2	λ	λ	NOUN
ejpam-6627	300	3	and	and	CCONJ
ejpam-6627	300	4	λ	λ	PROPN
ejpam-6627	300	5	contains	contain	VERB
ejpam-6627	300	6	exactly	exactly	ADV
ejpam-6627	300	7	one	one	NUM
ejpam-6627	300	8	of	of	ADP
ejpam-6627	300	9	the	the	DET
ejpam-6627	300	10	equivalence	equivalence	NOUN
ejpam-6627	300	11	classes	class	NOUN
ejpam-6627	300	12	[	[	X
ejpam-6627	300	13	α	α	X
ejpam-6627	300	14	]	]	X
ejpam-6627	300	15	or	or	CCONJ
ejpam-6627	300	16	[	[	X
ejpam-6627	300	17	β	β	X
ejpam-6627	300	18	]	]	X
ejpam-6627	300	19	.	.	PUNCT
ejpam-6627	301	1	(	(	PUNCT
ejpam-6627	301	2	2	2	X
ejpam-6627	301	3	)	)	PUNCT
ejpam-6627	301	4	assume	assume	VERB
ejpam-6627	301	5	ω	ω	PROPN
ejpam-6627	301	6	is	be	AUX
ejpam-6627	301	7	t1	t1	NOUN
ejpam-6627	301	8	.	.	PUNCT
ejpam-6627	302	1	ω/∼	ω/∼	NUM
ejpam-6627	302	2	is	be	AUX
ejpam-6627	302	3	t1	t1	NOUN
ejpam-6627	302	4	if	if	SCONJ
ejpam-6627	302	5	and	and	CCONJ
ejpam-6627	302	6	only	only	ADV
ejpam-6627	302	7	if	if	SCONJ
ejpam-6627	302	8	for	for	ADP
ejpam-6627	302	9	any	any	DET
ejpam-6627	302	10	equivalence	equivalence	NOUN
ejpam-6627	302	11	class	class	NOUN
ejpam-6627	303	1	[	[	X
ejpam-6627	303	2	α	α	X
ejpam-6627	303	3	]	]	X
ejpam-6627	303	4	,	,	PUNCT
ejpam-6627	303	5	the	the	DET
ejpam-6627	303	6	singleton	singleton	NOUN
ejpam-6627	303	7	{	{	PUNCT
ejpam-6627	304	1	[	[	X
ejpam-6627	304	2	α	α	X
ejpam-6627	304	3	]	]	X
ejpam-6627	304	4	}	}	PUNCT
ejpam-6627	304	5	is	be	AUX
ejpam-6627	304	6	closed	close	VERB
ejpam-6627	304	7	in	in	ADP
ejpam-6627	304	8	ω/∼.	ω/∼.	NOUN
ejpam-6627	304	9	by	by	ADP
ejpam-6627	304	10	the	the	DET
ejpam-6627	304	11	properties	property	NOUN
ejpam-6627	304	12	of	of	ADP
ejpam-6627	304	13	quotient	quotient	NOUN
ejpam-6627	304	14	topology	topology	NOUN
ejpam-6627	304	15	,	,	PUNCT
ejpam-6627	304	16	this	this	PRON
ejpam-6627	304	17	is	be	AUX
ejpam-6627	304	18	equivalent	equivalent	ADJ
ejpam-6627	304	19	to	to	ADP
ejpam-6627	304	20	π−1({[α	π−1({[α	PROPN
ejpam-6627	304	21	]	]	PUNCT
ejpam-6627	304	22	}	}	PUNCT
ejpam-6627	304	23	)	)	PUNCT
ejpam-6627	305	1	=	=	PUNCT
ejpam-6627	306	1	[	[	X
ejpam-6627	306	2	α	α	X
ejpam-6627	306	3	]	]	X
ejpam-6627	306	4	being	be	AUX
ejpam-6627	306	5	closed	close	VERB
ejpam-6627	306	6	in	in	ADP
ejpam-6627	306	7	ω	ω	PROPN
ejpam-6627	306	8	.	.	PUNCT
ejpam-6627	307	1	(	(	PUNCT
ejpam-6627	307	2	3	3	X
ejpam-6627	307	3	)	)	PUNCT
ejpam-6627	307	4	assume	assume	VERB
ejpam-6627	307	5	ω	ω	PROPN
ejpam-6627	307	6	is	be	AUX
ejpam-6627	307	7	t2	t2	NOUN
ejpam-6627	307	8	.	.	PUNCT
ejpam-6627	308	1	ω/∼	ω/∼	NUM
ejpam-6627	308	2	is	be	AUX
ejpam-6627	308	3	t2	t2	NOUN
ejpam-6627	308	4	if	if	SCONJ
ejpam-6627	308	5	and	and	CCONJ
ejpam-6627	308	6	only	only	ADV
ejpam-6627	308	7	if	if	SCONJ
ejpam-6627	308	8	for	for	ADP
ejpam-6627	308	9	any	any	DET
ejpam-6627	308	10	distinct	distinct	ADJ
ejpam-6627	308	11	equivalence	equivalence	NOUN
ejpam-6627	308	12	classes	class	NOUN
ejpam-6627	308	13	[	[	X
ejpam-6627	308	14	α	α	X
ejpam-6627	308	15	]	]	X
ejpam-6627	308	16	and	and	CCONJ
ejpam-6627	308	17	[	[	X
ejpam-6627	308	18	β	β	X
ejpam-6627	308	19	]	]	X
ejpam-6627	308	20	,	,	PUNCT
ejpam-6627	308	21	there	there	PRON
ejpam-6627	308	22	exist	exist	VERB
ejpam-6627	308	23	disjoint	disjoint	ADJ
ejpam-6627	308	24	open	open	ADJ
ejpam-6627	308	25	sets	set	NOUN
ejpam-6627	308	26	u	u	NOUN
ejpam-6627	308	27	and	and	CCONJ
ejpam-6627	308	28	v	v	NOUN
ejpam-6627	308	29	in	in	ADP
ejpam-6627	308	30	ω/∼	ω/∼	NUM
ejpam-6627	308	31	such	such	ADJ
ejpam-6627	308	32	that	that	SCONJ
ejpam-6627	308	33	[	[	X
ejpam-6627	308	34	α	α	X
ejpam-6627	308	35	]	]	X
ejpam-6627	308	36	∈	∈	PROPN
ejpam-6627	308	37	u	u	NOUN
ejpam-6627	308	38	and	and	CCONJ
ejpam-6627	308	39	[	[	X
ejpam-6627	308	40	β	β	X
ejpam-6627	308	41	]	]	X
ejpam-6627	308	42	∈	∈	PROPN
ejpam-6627	308	43	v	v	NOUN
ejpam-6627	308	44	.	.	PUNCT
ejpam-6627	309	1	this	this	PRON
ejpam-6627	309	2	is	be	AUX
ejpam-6627	309	3	equivalent	equivalent	ADJ
ejpam-6627	309	4	to	to	ADP
ejpam-6627	309	5	the	the	DET
ejpam-6627	309	6	existence	existence	NOUN
ejpam-6627	309	7	of	of	ADP
ejpam-6627	309	8	disjoint	disjoint	NOUN
ejpam-6627	309	9	open	open	ADJ
ejpam-6627	309	10	sets	set	NOUN
ejpam-6627	309	11	λ	λ	PROPN
ejpam-6627	309	12	and	and	CCONJ
ejpam-6627	309	13	γ	γ	PROPN
ejpam-6627	309	14	in	in	ADP
ejpam-6627	309	15	ω	ω	NUM
ejpam-6627	309	16	such	such	ADJ
ejpam-6627	309	17	that	that	SCONJ
ejpam-6627	310	1	[	[	X
ejpam-6627	310	2	α	α	X
ejpam-6627	310	3	]	]	X
ejpam-6627	310	4	⊂	⊂	PROPN
ejpam-6627	310	5	λ	λ	X
ejpam-6627	310	6	and	and	CCONJ
ejpam-6627	310	7	[	[	X
ejpam-6627	310	8	β	β	X
ejpam-6627	310	9	]	]	X
ejpam-6627	310	10	⊂	⊂	PROPN
ejpam-6627	310	11	γ	γ	X
ejpam-6627	310	12	.	.	PUNCT
ejpam-6627	310	13	such	such	ADJ
ejpam-6627	310	14	sets	set	NOUN
ejpam-6627	310	15	exist	exist	VERB
ejpam-6627	310	16	if	if	SCONJ
ejpam-6627	310	17	and	and	CCONJ
ejpam-6627	310	18	only	only	ADV
ejpam-6627	310	19	if	if	SCONJ
ejpam-6627	310	20	the	the	DET
ejpam-6627	310	21	equivalence	equivalence	NOUN
ejpam-6627	310	22	relation	relation	NOUN
ejpam-6627	310	23	∼	∼	NOUN
ejpam-6627	310	24	is	be	AUX
ejpam-6627	310	25	closed	close	VERB
ejpam-6627	310	26	as	as	ADP
ejpam-6627	310	27	a	a	DET
ejpam-6627	310	28	subset	subset	NOUN
ejpam-6627	310	29	of	of	ADP
ejpam-6627	310	30	ω×	ω×	PROPN
ejpam-6627	310	31	ω	ω	PROPN
ejpam-6627	310	32	,	,	PUNCT
ejpam-6627	310	33	a	a	DET
ejpam-6627	310	34	fundamental	fundamental	ADJ
ejpam-6627	310	35	result	result	NOUN
ejpam-6627	310	36	established	establish	VERB
ejpam-6627	310	37	by	by	ADP
ejpam-6627	310	38	arhangel’skii	arhangel’skii	PROPN
ejpam-6627	311	1	[	[	X
ejpam-6627	311	2	29	29	NUM
ejpam-6627	311	3	]	]	PUNCT
ejpam-6627	311	4	.	.	PUNCT
ejpam-6627	312	1	(	(	PUNCT
ejpam-6627	312	2	4	4	X
ejpam-6627	312	3	)	)	PUNCT
ejpam-6627	312	4	the	the	DET
ejpam-6627	312	5	proof	proof	NOUN
ejpam-6627	312	6	for	for	ADP
ejpam-6627	312	7	the	the	DET
ejpam-6627	312	8	quasi	quasi	ADJ
ejpam-6627	312	9	-	-	ADJ
ejpam-6627	312	10	hausdorff	hausdorff	ADJ
ejpam-6627	312	11	case	case	NOUN
ejpam-6627	312	12	follows	follow	VERB
ejpam-6627	312	13	a	a	DET
ejpam-6627	312	14	similar	similar	ADJ
ejpam-6627	312	15	pattern	pattern	NOUN
ejpam-6627	312	16	,	,	PUNCT
ejpam-6627	312	17	utilizing	utilize	VERB
ejpam-6627	312	18	the	the	DET
ejpam-6627	312	19	definition	definition	NOUN
ejpam-6627	312	20	of	of	ADP
ejpam-6627	312	21	quasi	quasi	ADJ
ejpam-6627	312	22	-	-	ADJ
ejpam-6627	312	23	hausdorff	hausdorff	ADJ
ejpam-6627	312	24	and	and	CCONJ
ejpam-6627	312	25	the	the	DET
ejpam-6627	312	26	properties	property	NOUN
ejpam-6627	312	27	of	of	ADP
ejpam-6627	312	28	quotient	quotient	NOUN
ejpam-6627	312	29	topology	topology	NOUN
ejpam-6627	312	30	.	.	PUNCT
ejpam-6627	313	1	this	this	DET
ejpam-6627	313	2	theorem	theorem	NOUN
ejpam-6627	313	3	provides	provide	VERB
ejpam-6627	313	4	precise	precise	ADJ
ejpam-6627	313	5	conditions	condition	NOUN
ejpam-6627	313	6	under	under	ADP
ejpam-6627	313	7	which	which	PRON
ejpam-6627	313	8	separation	separation	NOUN
ejpam-6627	313	9	axioms	axiom	NOUN
ejpam-6627	313	10	are	be	AUX
ejpam-6627	313	11	preserved	preserve	VERB
ejpam-6627	313	12	by	by	ADP
ejpam-6627	313	13	quotient	quotient	NOUN
ejpam-6627	313	14	operations	operation	NOUN
ejpam-6627	313	15	,	,	PUNCT
ejpam-6627	313	16	essential	essential	ADJ
ejpam-6627	313	17	for	for	ADP
ejpam-6627	313	18	constructing	construct	VERB
ejpam-6627	313	19	sophisticated	sophisticated	ADJ
ejpam-6627	313	20	topological	topological	ADJ
ejpam-6627	313	21	spaces	space	NOUN
ejpam-6627	313	22	through	through	ADP
ejpam-6627	313	23	quotient	quotient	NOUN
ejpam-6627	313	24	constructions	construction	NOUN
ejpam-6627	313	25	.	.	PUNCT
ejpam-6627	314	1	next	next	ADV
ejpam-6627	314	2	,	,	PUNCT
ejpam-6627	314	3	we	we	PRON
ejpam-6627	314	4	examine	examine	VERB
ejpam-6627	314	5	the	the	DET
ejpam-6627	314	6	behavior	behavior	NOUN
ejpam-6627	314	7	of	of	ADP
ejpam-6627	314	8	these	these	DET
ejpam-6627	314	9	separation	separation	NOUN
ejpam-6627	314	10	axioms	axiom	NOUN
ejpam-6627	314	11	under	under	ADP
ejpam-6627	314	12	product	product	NOUN
ejpam-6627	314	13	constructions	construction	NOUN
ejpam-6627	314	14	,	,	PUNCT
ejpam-6627	314	15	following	follow	VERB
ejpam-6627	314	16	the	the	DET
ejpam-6627	314	17	classical	classical	ADJ
ejpam-6627	314	18	work	work	NOUN
ejpam-6627	314	19	of	of	ADP
ejpam-6627	314	20	tychonoff	tychonoff	NOUN
ejpam-6627	314	21	[	[	X
ejpam-6627	314	22	30	30	NUM
ejpam-6627	314	23	]	]	PUNCT
ejpam-6627	314	24	.	.	PUNCT
ejpam-6627	315	1	theorem	theorem	VERB
ejpam-6627	315	2	9	9	NUM
ejpam-6627	315	3	.	.	PUNCT
ejpam-6627	316	1	let	let	AUX
ejpam-6627	316	2	{	{	PUNCT
ejpam-6627	316	3	ωi}i∈i	ωi}i∈i	NOUN
ejpam-6627	316	4	be	be	AUX
ejpam-6627	316	5	a	a	DET
ejpam-6627	316	6	family	family	NOUN
ejpam-6627	316	7	of	of	ADP
ejpam-6627	316	8	topological	topological	ADJ
ejpam-6627	316	9	spaces	space	NOUN
ejpam-6627	316	10	and	and	CCONJ
ejpam-6627	316	11	let	let	VERB
ejpam-6627	316	12	ω	ω	PROPN
ejpam-6627	316	13	=	=	SYM
ejpam-6627	316	14	∏	∏	PROPN
ejpam-6627	316	15	i∈i	i∈i	ADJ
ejpam-6627	316	16	ωi	ωi	AUX
ejpam-6627	316	17	be	be	AUX
ejpam-6627	316	18	their	their	PRON
ejpam-6627	316	19	product	product	NOUN
ejpam-6627	316	20	with	with	ADP
ejpam-6627	316	21	the	the	DET
ejpam-6627	316	22	product	product	NOUN
ejpam-6627	316	23	topology	topology	NOUN
ejpam-6627	316	24	.	.	PUNCT
ejpam-6627	317	1	the	the	DET
ejpam-6627	317	2	following	follow	VERB
ejpam-6627	317	3	statements	statement	NOUN
ejpam-6627	317	4	hold	hold	VERB
ejpam-6627	317	5	:	:	PUNCT
ejpam-6627	317	6	(	(	PUNCT
ejpam-6627	317	7	i	i	NOUN
ejpam-6627	317	8	)	)	PUNCT
ejpam-6627	317	9	ω	ω	PROPN
ejpam-6627	317	10	is	be	AUX
ejpam-6627	317	11	t0	t0	PROPN
ejpam-6627	317	12	if	if	SCONJ
ejpam-6627	318	1	and	and	CCONJ
ejpam-6627	318	2	only	only	ADV
ejpam-6627	318	3	if	if	SCONJ
ejpam-6627	318	4	each	each	DET
ejpam-6627	318	5	ωi	ωi	X
ejpam-6627	318	6	is	be	AUX
ejpam-6627	318	7	t0	t0	PROPN
ejpam-6627	318	8	.	.	PUNCT
ejpam-6627	319	1	j.	j.	PROPN
ejpam-6627	319	2	oudetallah	oudetallah	PROPN
ejpam-6627	319	3	et	et	PROPN
ejpam-6627	319	4	al	al	PROPN
ejpam-6627	319	5	.	.	PUNCT
ejpam-6627	319	6	/	/	SYM
ejpam-6627	319	7	eur	eur	PROPN
ejpam-6627	319	8	.	.	PUNCT
ejpam-6627	320	1	j.	j.	PROPN
ejpam-6627	320	2	pure	pure	PROPN
ejpam-6627	320	3	appl	appl	PROPN
ejpam-6627	320	4	.	.	PROPN
ejpam-6627	320	5	math	math	PROPN
ejpam-6627	320	6	,	,	PUNCT
ejpam-6627	320	7	18	18	NUM
ejpam-6627	320	8	(	(	PUNCT
ejpam-6627	320	9	3	3	NUM
ejpam-6627	320	10	)	)	PUNCT
ejpam-6627	320	11	(	(	PUNCT
ejpam-6627	320	12	2025	2025	NUM
ejpam-6627	320	13	)	)	PUNCT
ejpam-6627	320	14	,	,	PUNCT
ejpam-6627	320	15	6627	6627	NUM
ejpam-6627	320	16	15	15	NUM
ejpam-6627	320	17	of	of	ADP
ejpam-6627	320	18	20	20	NUM
ejpam-6627	320	19	(	(	PUNCT
ejpam-6627	320	20	ii	ii	NOUN
ejpam-6627	320	21	)	)	PUNCT
ejpam-6627	320	22	ω	ω	PROPN
ejpam-6627	320	23	is	be	AUX
ejpam-6627	320	24	t1	t1	NOUN
ejpam-6627	320	25	if	if	SCONJ
ejpam-6627	320	26	and	and	CCONJ
ejpam-6627	320	27	only	only	ADV
ejpam-6627	320	28	if	if	SCONJ
ejpam-6627	320	29	each	each	DET
ejpam-6627	320	30	ωi	ωi	X
ejpam-6627	320	31	is	be	AUX
ejpam-6627	320	32	t1	t1	PROPN
ejpam-6627	320	33	.	.	PUNCT
ejpam-6627	321	1	(	(	PUNCT
ejpam-6627	321	2	iii	iii	X
ejpam-6627	321	3	)	)	PUNCT
ejpam-6627	321	4	ω	ω	NOUN
ejpam-6627	321	5	is	be	AUX
ejpam-6627	321	6	t2	t2	NOUN
ejpam-6627	321	7	if	if	SCONJ
ejpam-6627	321	8	and	and	CCONJ
ejpam-6627	321	9	only	only	ADV
ejpam-6627	321	10	if	if	SCONJ
ejpam-6627	321	11	each	each	DET
ejpam-6627	321	12	ωi	ωi	X
ejpam-6627	321	13	is	be	AUX
ejpam-6627	321	14	t2	t2	NOUN
ejpam-6627	321	15	.	.	PUNCT
ejpam-6627	322	1	(	(	PUNCT
ejpam-6627	322	2	iv	iv	X
ejpam-6627	322	3	)	)	PUNCT
ejpam-6627	322	4	ω	ω	NOUN
ejpam-6627	322	5	is	be	AUX
ejpam-6627	322	6	quasi	quasi	ADJ
ejpam-6627	322	7	-	-	ADJ
ejpam-6627	322	8	hausdorff	hausdorff	ADJ
ejpam-6627	322	9	if	if	SCONJ
ejpam-6627	323	1	and	and	CCONJ
ejpam-6627	323	2	only	only	ADV
ejpam-6627	323	3	if	if	SCONJ
ejpam-6627	323	4	each	each	DET
ejpam-6627	323	5	ωi	ωi	X
ejpam-6627	323	6	is	be	AUX
ejpam-6627	323	7	quasi	quasi	ADJ
ejpam-6627	323	8	-	-	NOUN
ejpam-6627	323	9	hausdorff	hausdorff	ADJ
ejpam-6627	323	10	.	.	PUNCT
ejpam-6627	324	1	(	(	PUNCT
ejpam-6627	324	2	v	v	NOUN
ejpam-6627	324	3	)	)	PUNCT
ejpam-6627	324	4	if	if	SCONJ
ejpam-6627	324	5	each	each	DET
ejpam-6627	324	6	ωi	ωi	X
ejpam-6627	324	7	has	have	VERB
ejpam-6627	324	8	the	the	DET
ejpam-6627	324	9	property	property	NOUN
ejpam-6627	324	10	that	that	PRON
ejpam-6627	324	11	closed	close	VERB
ejpam-6627	324	12	sets	set	NOUN
ejpam-6627	324	13	are	be	AUX
ejpam-6627	324	14	sigma	sigma	NOUN
ejpam-6627	324	15	-	-	PUNCT
ejpam-6627	324	16	intersections	intersection	NOUN
ejpam-6627	324	17	,	,	PUNCT
ejpam-6627	324	18	then	then	ADV
ejpam-6627	324	19	ω	ω	PROPN
ejpam-6627	324	20	has	have	VERB
ejpam-6627	324	21	this	this	DET
ejpam-6627	324	22	property	property	NOUN
ejpam-6627	324	23	if	if	SCONJ
ejpam-6627	324	24	and	and	CCONJ
ejpam-6627	324	25	only	only	ADV
ejpam-6627	324	26	if	if	SCONJ
ejpam-6627	324	27	all	all	PRON
ejpam-6627	324	28	but	but	ADV
ejpam-6627	324	29	countably	countably	ADV
ejpam-6627	324	30	many	many	ADJ
ejpam-6627	324	31	ωi	ωi	NUM
ejpam-6627	324	32	are	be	AUX
ejpam-6627	324	33	discrete	discrete	ADJ
ejpam-6627	324	34	or	or	CCONJ
ejpam-6627	324	35	the	the	DET
ejpam-6627	324	36	index	index	NOUN
ejpam-6627	324	37	set	set	NOUN
ejpam-6627	324	38	i	i	PRON
ejpam-6627	324	39	is	be	AUX
ejpam-6627	324	40	countable	countable	ADJ
ejpam-6627	324	41	.	.	PUNCT
ejpam-6627	325	1	proof	proof	NOUN
ejpam-6627	325	2	.	.	PUNCT
ejpam-6627	326	1	the	the	DET
ejpam-6627	326	2	proofs	proof	NOUN
ejpam-6627	326	3	for	for	ADP
ejpam-6627	326	4	properties	property	NOUN
ejpam-6627	326	5	(	(	PUNCT
ejpam-6627	326	6	1	1	NUM
ejpam-6627	326	7	)	)	PUNCT
ejpam-6627	326	8	,	,	PUNCT
ejpam-6627	326	9	(	(	PUNCT
ejpam-6627	326	10	2	2	NUM
ejpam-6627	326	11	)	)	PUNCT
ejpam-6627	326	12	,	,	PUNCT
ejpam-6627	326	13	and	and	CCONJ
ejpam-6627	326	14	(	(	PUNCT
ejpam-6627	326	15	3	3	X
ejpam-6627	326	16	)	)	PUNCT
ejpam-6627	326	17	are	be	AUX
ejpam-6627	326	18	standard	standard	ADJ
ejpam-6627	326	19	results	result	NOUN
ejpam-6627	326	20	in	in	ADP
ejpam-6627	326	21	general	general	ADJ
ejpam-6627	326	22	topology	topology	NOUN
ejpam-6627	326	23	,	,	PUNCT
ejpam-6627	326	24	established	establish	VERB
ejpam-6627	326	25	in	in	ADP
ejpam-6627	326	26	tychonoff	tychonoff	NOUN
ejpam-6627	326	27	’s	’s	PART
ejpam-6627	326	28	fundamental	fundamental	ADJ
ejpam-6627	326	29	paper	paper	NOUN
ejpam-6627	326	30	[	[	X
ejpam-6627	326	31	30	30	NUM
ejpam-6627	326	32	]	]	PUNCT
ejpam-6627	326	33	.	.	PUNCT
ejpam-6627	327	1	(	(	PUNCT
ejpam-6627	327	2	4	4	X
ejpam-6627	327	3	)	)	PUNCT
ejpam-6627	327	4	assume	assume	VERB
ejpam-6627	327	5	each	each	DET
ejpam-6627	327	6	ωi	ωi	NOUN
ejpam-6627	327	7	is	be	AUX
ejpam-6627	327	8	quasi	quasi	ADJ
ejpam-6627	327	9	-	-	NOUN
ejpam-6627	327	10	hausdorff	hausdorff	ADJ
ejpam-6627	327	11	.	.	PUNCT
ejpam-6627	328	1	let	let	VERB
ejpam-6627	328	2	α	α	NOUN
ejpam-6627	328	3	=	=	PUNCT
ejpam-6627	328	4	(	(	PUNCT
ejpam-6627	328	5	αi)i∈i	αi)i∈i	NUM
ejpam-6627	328	6	and	and	CCONJ
ejpam-6627	328	7	β	β	X
ejpam-6627	328	8	=	=	SYM
ejpam-6627	328	9	(	(	PUNCT
ejpam-6627	328	10	βi)i∈i	βi)i∈i	NUM
ejpam-6627	328	11	be	be	VERB
ejpam-6627	328	12	distinct	distinct	ADJ
ejpam-6627	328	13	points	point	NOUN
ejpam-6627	328	14	in	in	ADP
ejpam-6627	328	15	ω	ω	PROPN
ejpam-6627	328	16	.	.	PUNCT
ejpam-6627	329	1	then	then	ADV
ejpam-6627	329	2	there	there	PRON
ejpam-6627	329	3	exists	exist	VERB
ejpam-6627	329	4	at	at	ADP
ejpam-6627	329	5	least	least	ADV
ejpam-6627	329	6	one	one	NUM
ejpam-6627	329	7	index	index	NOUN
ejpam-6627	329	8	j	j	PROPN
ejpam-6627	329	9	∈	∈	PROPN
ejpam-6627	329	10	i	i	PRON
ejpam-6627	329	11	such	such	VERB
ejpam-6627	329	12	that	that	SCONJ
ejpam-6627	329	13	αj	αj	X
ejpam-6627	329	14	̸=	̸=	PROPN
ejpam-6627	329	15	βj	βj	PRON
ejpam-6627	329	16	.	.	PUNCT
ejpam-6627	330	1	since	since	SCONJ
ejpam-6627	330	2	ωj	ωj	ADV
ejpam-6627	330	3	is	be	AUX
ejpam-6627	330	4	quasi	quasi	ADJ
ejpam-6627	330	5	-	-	NOUN
ejpam-6627	330	6	hausdorff	hausdorff	ADJ
ejpam-6627	330	7	,	,	PUNCT
ejpam-6627	330	8	there	there	PRON
ejpam-6627	330	9	exist	exist	VERB
ejpam-6627	330	10	open	open	ADJ
ejpam-6627	330	11	sets	set	NOUN
ejpam-6627	330	12	λj	λj	X
ejpam-6627	330	13	and	and	CCONJ
ejpam-6627	330	14	γj	γj	ADP
ejpam-6627	330	15	in	in	ADP
ejpam-6627	330	16	ωj	ωj	ADP
ejpam-6627	330	17	such	such	ADJ
ejpam-6627	330	18	that	that	SCONJ
ejpam-6627	330	19	αj	αj	PROPN
ejpam-6627	330	20	∈	∈	PROPN
ejpam-6627	330	21	λj	λj	X
ejpam-6627	330	22	,	,	PUNCT
ejpam-6627	330	23	βj	βj	PROPN
ejpam-6627	330	24	∈	∈	PROPN
ejpam-6627	330	25	γj	γj	NOUN
ejpam-6627	330	26	,	,	PUNCT
ejpam-6627	330	27	and	and	CCONJ
ejpam-6627	330	28	λj	λj	PROPN
ejpam-6627	330	29	∩	∩	NOUN
ejpam-6627	330	30	γj	γj	PROPN
ejpam-6627	330	31	contains	contain	VERB
ejpam-6627	330	32	at	at	ADP
ejpam-6627	330	33	most	most	ADV
ejpam-6627	330	34	countably	countably	ADV
ejpam-6627	330	35	many	many	ADJ
ejpam-6627	330	36	points	point	NOUN
ejpam-6627	330	37	.	.	PUNCT
ejpam-6627	331	1	consider	consider	VERB
ejpam-6627	331	2	the	the	DET
ejpam-6627	331	3	open	open	ADJ
ejpam-6627	331	4	sets	set	NOUN
ejpam-6627	331	5	λ	λ	X
ejpam-6627	331	6	=	=	SYM
ejpam-6627	331	7	π−1	π−1	PROPN
ejpam-6627	331	8	j	j	PROPN
ejpam-6627	331	9	(	(	PUNCT
ejpam-6627	331	10	λj	λj	PROPN
ejpam-6627	331	11	)	)	PUNCT
ejpam-6627	331	12	and	and	CCONJ
ejpam-6627	331	13	γ	γ	X
ejpam-6627	331	14	=	=	SYM
ejpam-6627	331	15	π−1	π−1	PROPN
ejpam-6627	331	16	j	j	PROPN
ejpam-6627	331	17	(	(	PUNCT
ejpam-6627	331	18	γj	γj	PROPN
ejpam-6627	331	19	)	)	PUNCT
ejpam-6627	331	20	in	in	ADP
ejpam-6627	331	21	ω	ω	PROPN
ejpam-6627	331	22	,	,	PUNCT
ejpam-6627	331	23	where	where	SCONJ
ejpam-6627	331	24	πj	πj	ADP
ejpam-6627	331	25	:	:	PUNCT
ejpam-6627	331	26	ω	ω	X
ejpam-6627	331	27	→	→	PUNCT
ejpam-6627	331	28	ωj	ωj	X
ejpam-6627	331	29	is	be	AUX
ejpam-6627	331	30	the	the	DET
ejpam-6627	331	31	projection	projection	NOUN
ejpam-6627	331	32	onto	onto	ADP
ejpam-6627	331	33	the	the	DET
ejpam-6627	331	34	j	j	PROPN
ejpam-6627	331	35	-	-	PUNCT
ejpam-6627	331	36	th	th	VERB
ejpam-6627	331	37	coordinate	coordinate	NOUN
ejpam-6627	331	38	.	.	PUNCT
ejpam-6627	332	1	then	then	ADV
ejpam-6627	332	2	α	α	PROPN
ejpam-6627	332	3	∈	∈	PROPN
ejpam-6627	332	4	λ	λ	PROPN
ejpam-6627	332	5	,	,	PUNCT
ejpam-6627	332	6	β	β	PROPN
ejpam-6627	332	7	∈	∈	PROPN
ejpam-6627	332	8	γ	γ	X
ejpam-6627	332	9	,	,	PUNCT
ejpam-6627	332	10	and	and	CCONJ
ejpam-6627	332	11	λ∩γ	λ∩γ	X
ejpam-6627	332	12	=	=	PUNCT
ejpam-6627	332	13	π−1	π−1	PROPN
ejpam-6627	332	14	j	j	PROPN
ejpam-6627	332	15	(	(	PUNCT
ejpam-6627	332	16	λj	λj	PROPN
ejpam-6627	332	17	∩γj	∩γj	PROPN
ejpam-6627	332	18	)	)	PUNCT
ejpam-6627	332	19	intersects	intersect	NOUN
ejpam-6627	332	20	at	at	ADP
ejpam-6627	332	21	most	most	ADV
ejpam-6627	332	22	countably	countably	ADV
ejpam-6627	332	23	many	many	ADJ
ejpam-6627	332	24	points	point	NOUN
ejpam-6627	332	25	in	in	ADP
ejpam-6627	332	26	each	each	DET
ejpam-6627	332	27	fiber	fiber	NOUN
ejpam-6627	332	28	,	,	PUNCT
ejpam-6627	332	29	resulting	result	VERB
ejpam-6627	332	30	in	in	ADP
ejpam-6627	332	31	at	at	ADP
ejpam-6627	332	32	most	most	ADJ
ejpam-6627	332	33	countably	countably	ADV
ejpam-6627	332	34	many	many	ADJ
ejpam-6627	332	35	points	point	NOUN
ejpam-6627	332	36	overall	overall	ADV
ejpam-6627	332	37	.	.	PUNCT
ejpam-6627	333	1	thus	thus	ADV
ejpam-6627	333	2	,	,	PUNCT
ejpam-6627	333	3	ω	ω	PROPN
ejpam-6627	333	4	is	be	AUX
ejpam-6627	333	5	quasihausdorff	quasihausdorff	NOUN
ejpam-6627	333	6	.	.	PUNCT
ejpam-6627	334	1	conversely	conversely	ADV
ejpam-6627	334	2	,	,	PUNCT
ejpam-6627	334	3	if	if	SCONJ
ejpam-6627	334	4	ω	ω	PROPN
ejpam-6627	334	5	is	be	AUX
ejpam-6627	334	6	quasi	quasi	ADJ
ejpam-6627	334	7	-	-	NOUN
ejpam-6627	334	8	hausdorff	hausdorff	ADJ
ejpam-6627	334	9	,	,	PUNCT
ejpam-6627	334	10	then	then	ADV
ejpam-6627	334	11	each	each	DET
ejpam-6627	334	12	ωi	ωi	NUM
ejpam-6627	334	13	must	must	AUX
ejpam-6627	334	14	be	be	AUX
ejpam-6627	334	15	quasi	quasi	ADJ
ejpam-6627	334	16	-	-	ADJ
ejpam-6627	334	17	hausdorff	hausdorff	ADJ
ejpam-6627	334	18	by	by	ADP
ejpam-6627	334	19	considering	consider	VERB
ejpam-6627	334	20	points	point	NOUN
ejpam-6627	334	21	that	that	PRON
ejpam-6627	334	22	differ	differ	VERB
ejpam-6627	334	23	only	only	ADV
ejpam-6627	334	24	in	in	ADP
ejpam-6627	334	25	the	the	DET
ejpam-6627	334	26	i	i	PROPN
ejpam-6627	334	27	-	-	PUNCT
ejpam-6627	334	28	th	th	VERB
ejpam-6627	334	29	coordinate	coordinate	NOUN
ejpam-6627	334	30	.	.	PUNCT
ejpam-6627	335	1	(	(	PUNCT
ejpam-6627	335	2	5	5	X
ejpam-6627	335	3	)	)	PUNCT
ejpam-6627	335	4	assume	assume	VERB
ejpam-6627	335	5	each	each	DET
ejpam-6627	335	6	ωi	ωi	PROPN
ejpam-6627	335	7	has	have	VERB
ejpam-6627	335	8	the	the	DET
ejpam-6627	335	9	property	property	NOUN
ejpam-6627	335	10	that	that	PRON
ejpam-6627	335	11	closed	close	VERB
ejpam-6627	335	12	sets	set	NOUN
ejpam-6627	335	13	are	be	AUX
ejpam-6627	335	14	sigma	sigma	NOUN
ejpam-6627	335	15	-	-	PUNCT
ejpam-6627	335	16	intersections	intersection	NOUN
ejpam-6627	335	17	.	.	PUNCT
ejpam-6627	336	1	if	if	SCONJ
ejpam-6627	336	2	all	all	DET
ejpam-6627	336	3	but	but	ADV
ejpam-6627	336	4	countably	countably	ADV
ejpam-6627	336	5	many	many	ADJ
ejpam-6627	336	6	ωi	ωi	NUM
ejpam-6627	336	7	are	be	AUX
ejpam-6627	336	8	discrete	discrete	ADJ
ejpam-6627	336	9	or	or	CCONJ
ejpam-6627	336	10	the	the	DET
ejpam-6627	336	11	index	index	NOUN
ejpam-6627	336	12	set	set	NOUN
ejpam-6627	336	13	i	i	PRON
ejpam-6627	336	14	is	be	AUX
ejpam-6627	336	15	countable	countable	ADJ
ejpam-6627	336	16	,	,	PUNCT
ejpam-6627	336	17	then	then	ADV
ejpam-6627	336	18	every	every	DET
ejpam-6627	336	19	basic	basic	ADJ
ejpam-6627	336	20	closed	close	VERB
ejpam-6627	336	21	set	set	VERB
ejpam-6627	336	22	in	in	ADP
ejpam-6627	336	23	ω	ω	PROPN
ejpam-6627	336	24	is	be	AUX
ejpam-6627	336	25	a	a	DET
ejpam-6627	336	26	sigma	sigma	NOUN
ejpam-6627	336	27	-	-	PUNCT
ejpam-6627	336	28	intersection	intersection	NOUN
ejpam-6627	336	29	,	,	PUNCT
ejpam-6627	336	30	and	and	CCONJ
ejpam-6627	336	31	since	since	SCONJ
ejpam-6627	336	32	the	the	DET
ejpam-6627	336	33	collection	collection	NOUN
ejpam-6627	336	34	of	of	ADP
ejpam-6627	336	35	sigma	sigma	PROPN
ejpam-6627	336	36	-	-	PUNCT
ejpam-6627	336	37	intersection	intersection	NOUN
ejpam-6627	336	38	sets	set	NOUN
ejpam-6627	336	39	is	be	AUX
ejpam-6627	336	40	closed	close	VERB
ejpam-6627	336	41	under	under	ADP
ejpam-6627	336	42	finite	finite	ADJ
ejpam-6627	336	43	unions	union	NOUN
ejpam-6627	336	44	and	and	CCONJ
ejpam-6627	336	45	countable	countable	ADJ
ejpam-6627	336	46	intersections	intersection	NOUN
ejpam-6627	336	47	,	,	PUNCT
ejpam-6627	336	48	all	all	DET
ejpam-6627	336	49	closed	closed	ADJ
ejpam-6627	336	50	sets	set	NOUN
ejpam-6627	336	51	in	in	ADP
ejpam-6627	336	52	ω	ω	PROPN
ejpam-6627	336	53	are	be	AUX
ejpam-6627	336	54	sigmaintersections	sigmaintersection	NOUN
ejpam-6627	336	55	.	.	PUNCT
ejpam-6627	337	1	conversely	conversely	ADV
ejpam-6627	337	2	,	,	PUNCT
ejpam-6627	337	3	if	if	SCONJ
ejpam-6627	337	4	all	all	DET
ejpam-6627	337	5	closed	close	VERB
ejpam-6627	337	6	sets	set	NOUN
ejpam-6627	337	7	in	in	ADP
ejpam-6627	337	8	ω	ω	PROPN
ejpam-6627	337	9	are	be	AUX
ejpam-6627	337	10	sigma	sigma	NOUN
ejpam-6627	337	11	-	-	PUNCT
ejpam-6627	337	12	intersections	intersection	NOUN
ejpam-6627	337	13	and	and	CCONJ
ejpam-6627	337	14	the	the	DET
ejpam-6627	337	15	index	index	NOUN
ejpam-6627	337	16	set	set	NOUN
ejpam-6627	337	17	i	i	PRON
ejpam-6627	337	18	is	be	AUX
ejpam-6627	337	19	uncountable	uncountable	ADJ
ejpam-6627	337	20	with	with	ADP
ejpam-6627	337	21	uncountably	uncountably	ADV
ejpam-6627	337	22	many	many	ADJ
ejpam-6627	337	23	non	non	ADJ
ejpam-6627	337	24	-	-	ADJ
ejpam-6627	337	25	discrete	discrete	ADJ
ejpam-6627	337	26	ωi	ωi	NOUN
ejpam-6627	337	27	,	,	PUNCT
ejpam-6627	337	28	then	then	ADV
ejpam-6627	337	29	we	we	PRON
ejpam-6627	337	30	can	can	AUX
ejpam-6627	337	31	construct	construct	VERB
ejpam-6627	337	32	a	a	DET
ejpam-6627	337	33	closed	closed	ADJ
ejpam-6627	337	34	set	set	NOUN
ejpam-6627	337	35	that	that	PRON
ejpam-6627	337	36	is	be	AUX
ejpam-6627	337	37	not	not	PART
ejpam-6627	337	38	a	a	DET
ejpam-6627	337	39	sigma	sigma	NOUN
ejpam-6627	337	40	-	-	PUNCT
ejpam-6627	337	41	intersection	intersection	NOUN
ejpam-6627	337	42	,	,	PUNCT
ejpam-6627	337	43	yielding	yield	VERB
ejpam-6627	337	44	a	a	DET
ejpam-6627	337	45	contradiction	contradiction	NOUN
ejpam-6627	337	46	,	,	PUNCT
ejpam-6627	337	47	following	follow	VERB
ejpam-6627	337	48	the	the	DET
ejpam-6627	337	49	ultrafilter	ultrafilter	NOUN
ejpam-6627	337	50	techniques	technique	NOUN
ejpam-6627	337	51	developed	develop	VERB
ejpam-6627	337	52	by	by	ADP
ejpam-6627	337	53	comfort	comfort	NOUN
ejpam-6627	337	54	and	and	CCONJ
ejpam-6627	337	55	negrepontis	negrepontis	ADV
ejpam-6627	337	56	[	[	X
ejpam-6627	337	57	32	32	NUM
ejpam-6627	337	58	]	]	PUNCT
ejpam-6627	337	59	.	.	PUNCT
ejpam-6627	338	1	this	this	DET
ejpam-6627	338	2	theorem	theorem	NOUN
ejpam-6627	338	3	reveals	reveal	VERB
ejpam-6627	338	4	the	the	DET
ejpam-6627	338	5	different	different	ADJ
ejpam-6627	338	6	behaviors	behavior	NOUN
ejpam-6627	338	7	of	of	ADP
ejpam-6627	338	8	various	various	ADJ
ejpam-6627	338	9	separation	separation	NOUN
ejpam-6627	338	10	axioms	axiom	NOUN
ejpam-6627	338	11	under	under	ADP
ejpam-6627	338	12	product	product	NOUN
ejpam-6627	338	13	constructions	construction	NOUN
ejpam-6627	338	14	,	,	PUNCT
ejpam-6627	338	15	with	with	ADP
ejpam-6627	338	16	property	property	NOUN
ejpam-6627	338	17	(	(	PUNCT
ejpam-6627	338	18	5	5	X
ejpam-6627	338	19	)	)	PUNCT
ejpam-6627	338	20	being	be	AUX
ejpam-6627	338	21	particularly	particularly	ADV
ejpam-6627	338	22	subtle	subtle	ADJ
ejpam-6627	338	23	,	,	PUNCT
ejpam-6627	338	24	connecting	connect	VERB
ejpam-6627	338	25	sigmaintersection	sigmaintersection	NOUN
ejpam-6627	338	26	properties	property	NOUN
ejpam-6627	338	27	with	with	ADP
ejpam-6627	338	28	countability	countability	NOUN
ejpam-6627	338	29	conditions	condition	NOUN
ejpam-6627	338	30	.	.	PUNCT
ejpam-6627	339	1	these	these	DET
ejpam-6627	339	2	results	result	NOUN
ejpam-6627	339	3	are	be	AUX
ejpam-6627	339	4	essential	essential	ADJ
ejpam-6627	339	5	for	for	ADP
ejpam-6627	339	6	understanding	understand	VERB
ejpam-6627	339	7	how	how	SCONJ
ejpam-6627	339	8	topological	topological	ADJ
ejpam-6627	339	9	properties	property	NOUN
ejpam-6627	339	10	propagate	propagate	VERB
ejpam-6627	339	11	through	through	ADP
ejpam-6627	339	12	standard	standard	ADJ
ejpam-6627	339	13	constructions	construction	NOUN
ejpam-6627	339	14	and	and	CCONJ
ejpam-6627	339	15	for	for	ADP
ejpam-6627	339	16	identifying	identify	VERB
ejpam-6627	339	17	when	when	SCONJ
ejpam-6627	339	18	additional	additional	ADJ
ejpam-6627	339	19	conditions	condition	NOUN
ejpam-6627	339	20	are	be	AUX
ejpam-6627	339	21	necessary	necessary	ADJ
ejpam-6627	339	22	to	to	PART
ejpam-6627	339	23	preserve	preserve	VERB
ejpam-6627	339	24	desired	desire	VERB
ejpam-6627	339	25	properties	property	NOUN
ejpam-6627	339	26	.	.	PUNCT
ejpam-6627	340	1	7	7	X
ejpam-6627	340	2	.	.	X
ejpam-6627	340	3	critical	critical	ADJ
ejpam-6627	340	4	counterexamples	counterexample	NOUN
ejpam-6627	340	5	in	in	ADP
ejpam-6627	340	6	this	this	DET
ejpam-6627	340	7	section	section	NOUN
ejpam-6627	340	8	,	,	PUNCT
ejpam-6627	340	9	we	we	PRON
ejpam-6627	340	10	present	present	VERB
ejpam-6627	340	11	carefully	carefully	ADV
ejpam-6627	340	12	constructed	construct	VERB
ejpam-6627	340	13	counterexamples	counterexample	NOUN
ejpam-6627	340	14	that	that	PRON
ejpam-6627	340	15	delineate	delineate	VERB
ejpam-6627	340	16	the	the	DET
ejpam-6627	340	17	boundaries	boundary	NOUN
ejpam-6627	340	18	between	between	ADP
ejpam-6627	340	19	different	different	ADJ
ejpam-6627	340	20	separation	separation	NOUN
ejpam-6627	340	21	axioms	axiom	NOUN
ejpam-6627	340	22	and	and	CCONJ
ejpam-6627	340	23	demonstrate	demonstrate	VERB
ejpam-6627	340	24	the	the	DET
ejpam-6627	340	25	sharpness	sharpness	NOUN
ejpam-6627	340	26	of	of	ADP
ejpam-6627	340	27	our	our	PRON
ejpam-6627	340	28	results	result	NOUN
ejpam-6627	340	29	.	.	PUNCT
ejpam-6627	341	1	these	these	DET
ejpam-6627	341	2	examples	example	NOUN
ejpam-6627	341	3	not	not	PART
ejpam-6627	341	4	only	only	ADV
ejpam-6627	341	5	show	show	VERB
ejpam-6627	341	6	that	that	SCONJ
ejpam-6627	341	7	our	our	PRON
ejpam-6627	341	8	theorems	theorem	NOUN
ejpam-6627	341	9	can	can	AUX
ejpam-6627	341	10	not	not	PART
ejpam-6627	341	11	be	be	AUX
ejpam-6627	341	12	strengthened	strengthen	VERB
ejpam-6627	341	13	but	but	CCONJ
ejpam-6627	341	14	also	also	ADV
ejpam-6627	341	15	provide	provide	VERB
ejpam-6627	341	16	insight	insight	NOUN
ejpam-6627	341	17	into	into	ADP
ejpam-6627	341	18	the	the	DET
ejpam-6627	341	19	subtle	subtle	ADJ
ejpam-6627	341	20	distinctions	distinction	NOUN
ejpam-6627	341	21	between	between	ADP
ejpam-6627	341	22	various	various	ADJ
ejpam-6627	341	23	separation	separation	NOUN
ejpam-6627	341	24	conditions	condition	NOUN
ejpam-6627	341	25	,	,	PUNCT
ejpam-6627	341	26	following	follow	VERB
ejpam-6627	341	27	the	the	DET
ejpam-6627	341	28	systematic	systematic	ADJ
ejpam-6627	341	29	approach	approach	NOUN
ejpam-6627	341	30	to	to	ADP
ejpam-6627	341	31	counterexamples	counterexample	NOUN
ejpam-6627	341	32	pioneered	pioneer	VERB
ejpam-6627	341	33	by	by	ADP
ejpam-6627	341	34	steen	steen	PROPN
ejpam-6627	341	35	and	and	CCONJ
ejpam-6627	341	36	seebach	seebach	NOUN
ejpam-6627	342	1	[	[	X
ejpam-6627	342	2	24	24	NUM
ejpam-6627	342	3	]	]	PUNCT
ejpam-6627	342	4	.	.	PUNCT
ejpam-6627	343	1	first	first	ADV
ejpam-6627	343	2	,	,	PUNCT
ejpam-6627	343	3	we	we	PRON
ejpam-6627	343	4	present	present	VERB
ejpam-6627	343	5	an	an	DET
ejpam-6627	343	6	example	example	NOUN
ejpam-6627	343	7	of	of	ADP
ejpam-6627	343	8	a	a	DET
ejpam-6627	343	9	t1	t1	NOUN
ejpam-6627	343	10	space	space	NOUN
ejpam-6627	343	11	that	that	PRON
ejpam-6627	343	12	is	be	AUX
ejpam-6627	343	13	not	not	PART
ejpam-6627	343	14	sequentially	sequentially	ADV
ejpam-6627	343	15	hausdorff	hausdorff	NOUN
ejpam-6627	343	16	,	,	PUNCT
ejpam-6627	343	17	illustrating	illustrate	VERB
ejpam-6627	343	18	the	the	DET
ejpam-6627	343	19	weakness	weakness	NOUN
ejpam-6627	343	20	of	of	ADP
ejpam-6627	343	21	this	this	DET
ejpam-6627	343	22	intermediate	intermediate	ADJ
ejpam-6627	343	23	separation	separation	NOUN
ejpam-6627	343	24	axiom	axiom	NOUN
ejpam-6627	343	25	.	.	PUNCT
ejpam-6627	344	1	j.	j.	PROPN
ejpam-6627	344	2	oudetallah	oudetallah	PROPN
ejpam-6627	344	3	et	et	PROPN
ejpam-6627	344	4	al	al	PROPN
ejpam-6627	344	5	.	.	PUNCT
ejpam-6627	344	6	/	/	SYM
ejpam-6627	344	7	eur	eur	PROPN
ejpam-6627	344	8	.	.	PUNCT
ejpam-6627	345	1	j.	j.	PROPN
ejpam-6627	345	2	pure	pure	PROPN
ejpam-6627	345	3	appl	appl	PROPN
ejpam-6627	345	4	.	.	PROPN
ejpam-6627	345	5	math	math	PROPN
ejpam-6627	345	6	,	,	PUNCT
ejpam-6627	345	7	18	18	NUM
ejpam-6627	345	8	(	(	PUNCT
ejpam-6627	345	9	3	3	NUM
ejpam-6627	345	10	)	)	PUNCT
ejpam-6627	345	11	(	(	PUNCT
ejpam-6627	345	12	2025	2025	NUM
ejpam-6627	345	13	)	)	PUNCT
ejpam-6627	345	14	,	,	PUNCT
ejpam-6627	345	15	6627	6627	NUM
ejpam-6627	345	16	16	16	NUM
ejpam-6627	345	17	of	of	ADP
ejpam-6627	345	18	20	20	NUM
ejpam-6627	345	19	example	example	NOUN
ejpam-6627	345	20	4	4	NUM
ejpam-6627	345	21	.	.	PUNCT
ejpam-6627	346	1	let	let	VERB
ejpam-6627	346	2	ω	ω	NOUN
ejpam-6627	346	3	=	=	NOUN
ejpam-6627	346	4	r2	r2	PROPN
ejpam-6627	346	5	and	and	CCONJ
ejpam-6627	346	6	define	define	VERB
ejpam-6627	346	7	a	a	DET
ejpam-6627	346	8	topology	topology	NOUN
ejpam-6627	346	9	as	as	SCONJ
ejpam-6627	346	10	follows	follow	VERB
ejpam-6627	346	11	:	:	PUNCT
ejpam-6627	346	12	a	a	DET
ejpam-6627	346	13	set	set	NOUN
ejpam-6627	346	14	λ	λ	NOUN
ejpam-6627	346	15	is	be	AUX
ejpam-6627	346	16	open	open	ADJ
ejpam-6627	346	17	if	if	SCONJ
ejpam-6627	346	18	and	and	CCONJ
ejpam-6627	346	19	only	only	ADV
ejpam-6627	346	20	if	if	SCONJ
ejpam-6627	346	21	for	for	ADP
ejpam-6627	346	22	each	each	DET
ejpam-6627	346	23	point	point	NOUN
ejpam-6627	346	24	(	(	PUNCT
ejpam-6627	346	25	α	α	X
ejpam-6627	347	1	,	,	PUNCT
ejpam-6627	347	2	β	β	NOUN
ejpam-6627	347	3	)	)	PUNCT
ejpam-6627	347	4	∈	∈	PROPN
ejpam-6627	347	5	λ	λ	PROPN
ejpam-6627	347	6	,	,	PUNCT
ejpam-6627	347	7	there	there	PRON
ejpam-6627	347	8	exists	exist	VERB
ejpam-6627	347	9	ϵ	ϵ	X
ejpam-6627	347	10	>	>	X
ejpam-6627	347	11	0	0	NUM
ejpam-6627	348	1	such	such	ADJ
ejpam-6627	348	2	that	that	SCONJ
ejpam-6627	348	3	{	{	PUNCT
ejpam-6627	348	4	(	(	PUNCT
ejpam-6627	348	5	α	α	NOUN
ejpam-6627	348	6	,	,	PUNCT
ejpam-6627	348	7	γ	γ	NOUN
ejpam-6627	348	8	)	)	PUNCT
ejpam-6627	348	9	:	:	PUNCT
ejpam-6627	348	10	|γ	|γ	ADP
ejpam-6627	348	11	−	−	PROPN
ejpam-6627	348	12	β|	β|	ADP
ejpam-6627	348	13	<	<	X
ejpam-6627	348	14	ϵ	ϵ	X
ejpam-6627	348	15	}	}	PUNCT
ejpam-6627	348	16	⊂	⊂	PROPN
ejpam-6627	348	17	λ	λ	PROPN
ejpam-6627	348	18	.	.	PUNCT
ejpam-6627	349	1	this	this	DET
ejpam-6627	349	2	space	space	NOUN
ejpam-6627	349	3	is	be	AUX
ejpam-6627	349	4	clearly	clearly	ADV
ejpam-6627	349	5	t1	t1	ADJ
ejpam-6627	349	6	since	since	SCONJ
ejpam-6627	349	7	for	for	ADP
ejpam-6627	349	8	any	any	DET
ejpam-6627	349	9	point	point	NOUN
ejpam-6627	349	10	(	(	PUNCT
ejpam-6627	349	11	α	α	NOUN
ejpam-6627	349	12	,	,	PUNCT
ejpam-6627	349	13	β	β	NOUN
ejpam-6627	349	14	)	)	PUNCT
ejpam-6627	349	15	,	,	PUNCT
ejpam-6627	349	16	the	the	DET
ejpam-6627	349	17	set	set	PROPN
ejpam-6627	349	18	ω	ω	PROPN
ejpam-6627	349	19	\	\	PROPN
ejpam-6627	349	20	{	{	PUNCT
ejpam-6627	349	21	(	(	PUNCT
ejpam-6627	349	22	α	α	X
ejpam-6627	349	23	,	,	PUNCT
ejpam-6627	349	24	β	β	NOUN
ejpam-6627	349	25	)	)	PUNCT
ejpam-6627	349	26	}	}	PUNCT
ejpam-6627	349	27	is	be	AUX
ejpam-6627	349	28	open	open	ADJ
ejpam-6627	349	29	.	.	PUNCT
ejpam-6627	350	1	however	however	ADV
ejpam-6627	350	2	,	,	PUNCT
ejpam-6627	350	3	it	it	PRON
ejpam-6627	350	4	is	be	AUX
ejpam-6627	350	5	not	not	PART
ejpam-6627	350	6	sequentially	sequentially	ADV
ejpam-6627	350	7	hausdorff	hausdorff	NOUN
ejpam-6627	350	8	.	.	PUNCT
ejpam-6627	351	1	consider	consider	VERB
ejpam-6627	351	2	the	the	DET
ejpam-6627	351	3	distinct	distinct	ADJ
ejpam-6627	351	4	points	point	NOUN
ejpam-6627	351	5	(	(	PUNCT
ejpam-6627	351	6	0	0	NUM
ejpam-6627	351	7	,	,	PUNCT
ejpam-6627	351	8	0	0	NUM
ejpam-6627	351	9	)	)	PUNCT
ejpam-6627	351	10	and	and	CCONJ
ejpam-6627	351	11	(	(	PUNCT
ejpam-6627	351	12	1	1	NUM
ejpam-6627	351	13	,	,	PUNCT
ejpam-6627	351	14	0	0	NUM
ejpam-6627	351	15	)	)	PUNCT
ejpam-6627	351	16	.	.	PUNCT
ejpam-6627	352	1	the	the	DET
ejpam-6627	352	2	sequences	sequence	NOUN
ejpam-6627	352	3	αn	αn	NOUN
ejpam-6627	352	4	=	=	SYM
ejpam-6627	352	5	(	(	PUNCT
ejpam-6627	352	6	0	0	NUM
ejpam-6627	352	7	,	,	PUNCT
ejpam-6627	352	8	1	1	NUM
ejpam-6627	352	9	/	/	SYM
ejpam-6627	352	10	n	n	CCONJ
ejpam-6627	352	11	)	)	PUNCT
ejpam-6627	352	12	and	and	CCONJ
ejpam-6627	352	13	βn	βn	NOUN
ejpam-6627	352	14	=	=	SYM
ejpam-6627	352	15	(	(	PUNCT
ejpam-6627	352	16	1	1	NUM
ejpam-6627	352	17	,	,	PUNCT
ejpam-6627	352	18	1	1	NUM
ejpam-6627	352	19	/	/	SYM
ejpam-6627	352	20	n	n	CCONJ
ejpam-6627	352	21	)	)	PUNCT
ejpam-6627	352	22	satisfy	satisfy	VERB
ejpam-6627	352	23	αn	αn	NOUN
ejpam-6627	352	24	→	→	SYM
ejpam-6627	352	25	(	(	PUNCT
ejpam-6627	352	26	0	0	NUM
ejpam-6627	352	27	,	,	PUNCT
ejpam-6627	352	28	0	0	NUM
ejpam-6627	352	29	)	)	PUNCT
ejpam-6627	352	30	and	and	CCONJ
ejpam-6627	352	31	βn	βn	VERB
ejpam-6627	352	32	→	→	SYM
ejpam-6627	352	33	(	(	PUNCT
ejpam-6627	352	34	1	1	NUM
ejpam-6627	352	35	,	,	PUNCT
ejpam-6627	352	36	0	0	NUM
ejpam-6627	352	37	)	)	PUNCT
ejpam-6627	352	38	,	,	PUNCT
ejpam-6627	352	39	but	but	CCONJ
ejpam-6627	352	40	there	there	PRON
ejpam-6627	352	41	is	be	VERB
ejpam-6627	352	42	no	no	DET
ejpam-6627	352	43	way	way	NOUN
ejpam-6627	352	44	to	to	PART
ejpam-6627	352	45	separate	separate	VERB
ejpam-6627	352	46	these	these	DET
ejpam-6627	352	47	sequences	sequence	NOUN
ejpam-6627	352	48	with	with	ADP
ejpam-6627	352	49	open	open	ADJ
ejpam-6627	352	50	sets	set	NOUN
ejpam-6627	352	51	,	,	PUNCT
ejpam-6627	352	52	demonstrating	demonstrate	VERB
ejpam-6627	352	53	the	the	DET
ejpam-6627	352	54	failure	failure	NOUN
ejpam-6627	352	55	of	of	ADP
ejpam-6627	352	56	the	the	DET
ejpam-6627	352	57	sequentially	sequentially	ADV
ejpam-6627	352	58	hausdorff	hausdorff	NOUN
ejpam-6627	352	59	property	property	NOUN
ejpam-6627	352	60	.	.	PUNCT
ejpam-6627	353	1	next	next	ADV
ejpam-6627	353	2	,	,	PUNCT
ejpam-6627	353	3	we	we	PRON
ejpam-6627	353	4	present	present	VERB
ejpam-6627	353	5	an	an	DET
ejpam-6627	353	6	example	example	NOUN
ejpam-6627	353	7	of	of	ADP
ejpam-6627	353	8	a	a	DET
ejpam-6627	353	9	sequentially	sequentially	ADV
ejpam-6627	353	10	hausdorff	hausdorff	NOUN
ejpam-6627	353	11	space	space	NOUN
ejpam-6627	353	12	that	that	PRON
ejpam-6627	353	13	is	be	AUX
ejpam-6627	353	14	not	not	PART
ejpam-6627	353	15	quasihausdorff	quasihausdorff	ADJ
ejpam-6627	353	16	,	,	PUNCT
ejpam-6627	353	17	inspired	inspire	VERB
ejpam-6627	353	18	by	by	ADP
ejpam-6627	353	19	constructions	construction	NOUN
ejpam-6627	353	20	in	in	ADP
ejpam-6627	353	21	[	[	X
ejpam-6627	353	22	24	24	NUM
ejpam-6627	353	23	]	]	PUNCT
ejpam-6627	353	24	.	.	PUNCT
ejpam-6627	354	1	example	example	NOUN
ejpam-6627	355	1	5	5	NUM
ejpam-6627	355	2	.	.	PUNCT
ejpam-6627	355	3	let	let	VERB
ejpam-6627	355	4	ω	ω	NOUN
ejpam-6627	355	5	=	=	PUNCT
ejpam-6627	356	1	[	[	X
ejpam-6627	356	2	0	0	NUM
ejpam-6627	356	3	,	,	PUNCT
ejpam-6627	356	4	1	1	NUM
ejpam-6627	356	5	]	]	PUNCT
ejpam-6627	356	6	with	with	ADP
ejpam-6627	356	7	the	the	DET
ejpam-6627	356	8	topology	topology	NOUN
ejpam-6627	356	9	generated	generate	VERB
ejpam-6627	356	10	by	by	ADP
ejpam-6627	356	11	the	the	DET
ejpam-6627	356	12	standard	standard	ADJ
ejpam-6627	356	13	open	open	ADJ
ejpam-6627	356	14	intervals	interval	NOUN
ejpam-6627	356	15	together	together	ADV
ejpam-6627	356	16	with	with	ADP
ejpam-6627	356	17	sets	set	NOUN
ejpam-6627	356	18	of	of	ADP
ejpam-6627	356	19	the	the	DET
ejpam-6627	356	20	form	form	NOUN
ejpam-6627	356	21	(	(	PUNCT
ejpam-6627	356	22	a	a	PRON
ejpam-6627	356	23	,	,	PUNCT
ejpam-6627	356	24	b	b	NOUN
ejpam-6627	356	25	)	)	PUNCT
ejpam-6627	356	26	\c	\c	NOUN
ejpam-6627	356	27	,	,	PUNCT
ejpam-6627	356	28	where	where	SCONJ
ejpam-6627	356	29	c	c	PROPN
ejpam-6627	356	30	is	be	AUX
ejpam-6627	356	31	countable	countable	ADJ
ejpam-6627	356	32	.	.	PUNCT
ejpam-6627	357	1	this	this	DET
ejpam-6627	357	2	space	space	NOUN
ejpam-6627	357	3	is	be	AUX
ejpam-6627	357	4	sequentially	sequentially	ADV
ejpam-6627	357	5	hausdorff	hausdorff	NOUN
ejpam-6627	357	6	because	because	SCONJ
ejpam-6627	357	7	any	any	DET
ejpam-6627	357	8	convergent	convergent	NOUN
ejpam-6627	357	9	sequence	sequence	NOUN
ejpam-6627	357	10	in	in	ADP
ejpam-6627	357	11	this	this	DET
ejpam-6627	357	12	topology	topology	NOUN
ejpam-6627	357	13	must	must	AUX
ejpam-6627	357	14	be	be	AUX
ejpam-6627	357	15	eventually	eventually	ADV
ejpam-6627	357	16	constant	constant	ADJ
ejpam-6627	357	17	.	.	PUNCT
ejpam-6627	358	1	however	however	ADV
ejpam-6627	358	2	,	,	PUNCT
ejpam-6627	358	3	it	it	PRON
ejpam-6627	358	4	is	be	AUX
ejpam-6627	358	5	not	not	PART
ejpam-6627	358	6	quasi	quasi	ADJ
ejpam-6627	358	7	-	-	NOUN
ejpam-6627	358	8	hausdorff	hausdorff	ADJ
ejpam-6627	358	9	because	because	SCONJ
ejpam-6627	358	10	any	any	DET
ejpam-6627	358	11	two	two	NUM
ejpam-6627	358	12	non	non	ADJ
ejpam-6627	358	13	-	-	ADJ
ejpam-6627	358	14	empty	empty	ADJ
ejpam-6627	358	15	open	open	ADJ
ejpam-6627	358	16	sets	set	NOUN
ejpam-6627	358	17	have	have	VERB
ejpam-6627	358	18	uncountable	uncountable	ADJ
ejpam-6627	358	19	intersection	intersection	NOUN
ejpam-6627	358	20	.	.	PUNCT
ejpam-6627	359	1	finally	finally	ADV
ejpam-6627	359	2	,	,	PUNCT
ejpam-6627	359	3	we	we	PRON
ejpam-6627	359	4	present	present	VERB
ejpam-6627	359	5	an	an	DET
ejpam-6627	359	6	example	example	NOUN
ejpam-6627	359	7	of	of	ADP
ejpam-6627	359	8	a	a	DET
ejpam-6627	359	9	quasi	quasi	ADJ
ejpam-6627	359	10	-	-	ADJ
ejpam-6627	359	11	hausdorff	hausdorff	ADJ
ejpam-6627	359	12	space	space	NOUN
ejpam-6627	359	13	that	that	PRON
ejpam-6627	359	14	is	be	AUX
ejpam-6627	359	15	not	not	PART
ejpam-6627	359	16	hausdorff	hausdorff	NOUN
ejpam-6627	359	17	,	,	PUNCT
ejpam-6627	359	18	completing	complete	VERB
ejpam-6627	359	19	our	our	PRON
ejpam-6627	359	20	hierarchy	hierarchy	NOUN
ejpam-6627	359	21	.	.	PUNCT
ejpam-6627	360	1	example	example	NOUN
ejpam-6627	361	1	6	6	NUM
ejpam-6627	361	2	.	.	PUNCT
ejpam-6627	362	1	let	let	VERB
ejpam-6627	362	2	ω	ω	NOUN
ejpam-6627	362	3	=	=	VERB
ejpam-6627	362	4	r	r	NOUN
ejpam-6627	362	5	with	with	ADP
ejpam-6627	362	6	the	the	DET
ejpam-6627	362	7	topology	topology	NOUN
ejpam-6627	362	8	where	where	SCONJ
ejpam-6627	362	9	a	a	DET
ejpam-6627	362	10	set	set	NOUN
ejpam-6627	362	11	λ	λ	NOUN
ejpam-6627	362	12	is	be	AUX
ejpam-6627	362	13	open	open	ADJ
ejpam-6627	362	14	if	if	SCONJ
ejpam-6627	362	15	and	and	CCONJ
ejpam-6627	362	16	only	only	ADV
ejpam-6627	362	17	if	if	SCONJ
ejpam-6627	362	18	for	for	ADP
ejpam-6627	362	19	each	each	DET
ejpam-6627	362	20	α	α	NOUN
ejpam-6627	362	21	∈	∈	PROPN
ejpam-6627	362	22	λ	λ	NOUN
ejpam-6627	362	23	,	,	PUNCT
ejpam-6627	362	24	there	there	PRON
ejpam-6627	362	25	exists	exist	VERB
ejpam-6627	362	26	ϵ	ϵ	X
ejpam-6627	362	27	>	>	X
ejpam-6627	362	28	0	0	NUM
ejpam-6627	363	1	such	such	ADJ
ejpam-6627	363	2	that	that	SCONJ
ejpam-6627	363	3	(	(	PUNCT
ejpam-6627	363	4	α	α	NOUN
ejpam-6627	363	5	,	,	PUNCT
ejpam-6627	363	6	α+	α+	NOUN
ejpam-6627	363	7	ϵ	ϵ	X
ejpam-6627	363	8	)	)	PUNCT
ejpam-6627	363	9	⊂	⊂	PROPN
ejpam-6627	364	1	λ	λ	X
ejpam-6627	364	2	.	.	PUNCT
ejpam-6627	365	1	this	this	PRON
ejpam-6627	365	2	is	be	AUX
ejpam-6627	365	3	the	the	DET
ejpam-6627	365	4	so	so	ADV
ejpam-6627	365	5	-	-	PUNCT
ejpam-6627	365	6	called	call	VERB
ejpam-6627	365	7	”	"	PUNCT
ejpam-6627	365	8	right	right	ADJ
ejpam-6627	365	9	half	half	ADJ
ejpam-6627	365	10	-	-	PUNCT
ejpam-6627	365	11	open	open	ADJ
ejpam-6627	365	12	interval	interval	NOUN
ejpam-6627	365	13	topology	topology	NOUN
ejpam-6627	365	14	”	"	PUNCT
ejpam-6627	365	15	or	or	CCONJ
ejpam-6627	365	16	”	"	PUNCT
ejpam-6627	365	17	lower	low	ADJ
ejpam-6627	365	18	limit	limit	NOUN
ejpam-6627	365	19	topology	topology	NOUN
ejpam-6627	365	20	.	.	PUNCT
ejpam-6627	365	21	”	"	PUNCT
ejpam-6627	366	1	this	this	DET
ejpam-6627	366	2	space	space	NOUN
ejpam-6627	366	3	is	be	AUX
ejpam-6627	366	4	t1	t1	NOUN
ejpam-6627	366	5	since	since	SCONJ
ejpam-6627	366	6	for	for	ADP
ejpam-6627	366	7	any	any	DET
ejpam-6627	366	8	α	α	NOUN
ejpam-6627	366	9	∈	∈	NOUN
ejpam-6627	366	10	r	r	NOUN
ejpam-6627	366	11	,	,	PUNCT
ejpam-6627	366	12	the	the	DET
ejpam-6627	366	13	set	set	NOUN
ejpam-6627	366	14	r	r	NOUN
ejpam-6627	366	15	\	\	NOUN
ejpam-6627	366	16	{	{	PUNCT
ejpam-6627	366	17	α	α	NOUN
ejpam-6627	366	18	}	}	PUNCT
ejpam-6627	366	19	is	be	AUX
ejpam-6627	366	20	open	open	ADJ
ejpam-6627	366	21	.	.	PUNCT
ejpam-6627	367	1	it	it	PRON
ejpam-6627	367	2	is	be	AUX
ejpam-6627	367	3	also	also	ADV
ejpam-6627	367	4	quasi	quasi	ADJ
ejpam-6627	367	5	-	-	NOUN
ejpam-6627	367	6	hausdorff	hausdorff	ADJ
ejpam-6627	367	7	because	because	SCONJ
ejpam-6627	367	8	for	for	ADP
ejpam-6627	367	9	any	any	DET
ejpam-6627	367	10	distinct	distinct	ADJ
ejpam-6627	367	11	points	point	NOUN
ejpam-6627	367	12	α	α	PRON
ejpam-6627	367	13	<	<	X
ejpam-6627	367	14	β	β	X
ejpam-6627	367	15	,	,	PUNCT
ejpam-6627	367	16	the	the	DET
ejpam-6627	367	17	sets	set	NOUN
ejpam-6627	367	18	[	[	X
ejpam-6627	367	19	α	α	X
ejpam-6627	367	20	,	,	PUNCT
ejpam-6627	367	21	β	β	NOUN
ejpam-6627	367	22	)	)	PUNCT
ejpam-6627	367	23	and	and	CCONJ
ejpam-6627	367	24	[	[	X
ejpam-6627	367	25	β	β	X
ejpam-6627	367	26	,	,	PUNCT
ejpam-6627	367	27	γ	γ	X
ejpam-6627	367	28	)	)	PUNCT
ejpam-6627	367	29	(	(	PUNCT
ejpam-6627	367	30	for	for	ADP
ejpam-6627	367	31	some	some	DET
ejpam-6627	367	32	γ	γ	NOUN
ejpam-6627	367	33	>	>	X
ejpam-6627	367	34	β	β	NOUN
ejpam-6627	367	35	)	)	PUNCT
ejpam-6627	367	36	are	be	AUX
ejpam-6627	367	37	open	open	ADJ
ejpam-6627	367	38	,	,	PUNCT
ejpam-6627	367	39	contain	contain	VERB
ejpam-6627	367	40	α	α	NOUN
ejpam-6627	367	41	and	and	CCONJ
ejpam-6627	367	42	β	β	X
ejpam-6627	367	43	respectively	respectively	ADV
ejpam-6627	367	44	,	,	PUNCT
ejpam-6627	367	45	and	and	CCONJ
ejpam-6627	367	46	their	their	PRON
ejpam-6627	367	47	intersection	intersection	NOUN
ejpam-6627	367	48	is	be	AUX
ejpam-6627	367	49	empty	empty	ADJ
ejpam-6627	367	50	,	,	PUNCT
ejpam-6627	367	51	which	which	PRON
ejpam-6627	367	52	is	be	AUX
ejpam-6627	367	53	certainly	certainly	ADV
ejpam-6627	367	54	countable	countable	ADJ
ejpam-6627	367	55	.	.	PUNCT
ejpam-6627	368	1	however	however	ADV
ejpam-6627	368	2	,	,	PUNCT
ejpam-6627	368	3	it	it	PRON
ejpam-6627	368	4	is	be	AUX
ejpam-6627	368	5	not	not	PART
ejpam-6627	368	6	hausdorff	hausdorff	ADJ
ejpam-6627	368	7	.	.	PUNCT
ejpam-6627	369	1	for	for	ADP
ejpam-6627	369	2	any	any	DET
ejpam-6627	369	3	distinct	distinct	ADJ
ejpam-6627	369	4	points	point	NOUN
ejpam-6627	369	5	α	α	PRON
ejpam-6627	369	6	<	<	X
ejpam-6627	369	7	β	β	X
ejpam-6627	369	8	,	,	PUNCT
ejpam-6627	369	9	any	any	DET
ejpam-6627	369	10	open	open	ADJ
ejpam-6627	369	11	set	set	NOUN
ejpam-6627	369	12	containing	contain	VERB
ejpam-6627	369	13	α	α	PRON
ejpam-6627	369	14	must	must	AUX
ejpam-6627	369	15	contain	contain	VERB
ejpam-6627	369	16	points	point	NOUN
ejpam-6627	369	17	arbitrarily	arbitrarily	ADV
ejpam-6627	369	18	close	close	ADJ
ejpam-6627	369	19	to	to	ADP
ejpam-6627	369	20	α	α	NOUN
ejpam-6627	369	21	from	from	ADP
ejpam-6627	369	22	the	the	DET
ejpam-6627	369	23	right	right	NOUN
ejpam-6627	369	24	,	,	PUNCT
ejpam-6627	369	25	and	and	CCONJ
ejpam-6627	369	26	any	any	DET
ejpam-6627	369	27	open	open	ADJ
ejpam-6627	369	28	set	set	NOUN
ejpam-6627	369	29	containing	contain	VERB
ejpam-6627	369	30	β	β	X
ejpam-6627	369	31	must	must	AUX
ejpam-6627	369	32	contain	contain	VERB
ejpam-6627	369	33	points	point	NOUN
ejpam-6627	369	34	arbitrarily	arbitrarily	ADV
ejpam-6627	369	35	close	close	ADJ
ejpam-6627	369	36	to	to	ADP
ejpam-6627	369	37	β	β	NOUN
ejpam-6627	369	38	from	from	ADP
ejpam-6627	369	39	the	the	DET
ejpam-6627	369	40	right	right	NOUN
ejpam-6627	369	41	.	.	PUNCT
ejpam-6627	370	1	no	no	ADV
ejpam-6627	370	2	matter	matter	ADV
ejpam-6627	370	3	how	how	SCONJ
ejpam-6627	370	4	small	small	ADJ
ejpam-6627	370	5	these	these	DET
ejpam-6627	370	6	open	open	ADJ
ejpam-6627	370	7	sets	set	NOUN
ejpam-6627	370	8	are	be	AUX
ejpam-6627	370	9	chosen	choose	VERB
ejpam-6627	370	10	,	,	PUNCT
ejpam-6627	370	11	their	their	PRON
ejpam-6627	370	12	intersection	intersection	NOUN
ejpam-6627	370	13	will	will	AUX
ejpam-6627	370	14	always	always	ADV
ejpam-6627	370	15	be	be	AUX
ejpam-6627	370	16	non	non	ADJ
ejpam-6627	370	17	-	-	ADJ
ejpam-6627	370	18	empty	empty	ADJ
ejpam-6627	370	19	.	.	PUNCT
ejpam-6627	371	1	these	these	DET
ejpam-6627	371	2	examples	example	NOUN
ejpam-6627	371	3	demonstrate	demonstrate	VERB
ejpam-6627	371	4	that	that	SCONJ
ejpam-6627	371	5	the	the	DET
ejpam-6627	371	6	hierarchy	hierarchy	NOUN
ejpam-6627	371	7	of	of	ADP
ejpam-6627	371	8	separation	separation	NOUN
ejpam-6627	371	9	axioms	axiom	NOUN
ejpam-6627	371	10	established	establish	VERB
ejpam-6627	371	11	in	in	ADP
ejpam-6627	371	12	this	this	DET
ejpam-6627	371	13	paper	paper	NOUN
ejpam-6627	371	14	is	be	AUX
ejpam-6627	371	15	strict	strict	ADJ
ejpam-6627	371	16	,	,	PUNCT
ejpam-6627	371	17	with	with	ADP
ejpam-6627	371	18	each	each	DET
ejpam-6627	371	19	level	level	NOUN
ejpam-6627	371	20	representing	represent	VERB
ejpam-6627	371	21	a	a	DET
ejpam-6627	371	22	genuinely	genuinely	ADV
ejpam-6627	371	23	different	different	ADJ
ejpam-6627	371	24	class	class	NOUN
ejpam-6627	371	25	of	of	ADP
ejpam-6627	371	26	topological	topological	ADJ
ejpam-6627	371	27	spaces	space	NOUN
ejpam-6627	371	28	.	.	PUNCT
ejpam-6627	372	1	the	the	DET
ejpam-6627	372	2	careful	careful	ADJ
ejpam-6627	372	3	construction	construction	NOUN
ejpam-6627	372	4	of	of	ADP
ejpam-6627	372	5	these	these	DET
ejpam-6627	372	6	counterexamples	counterexample	NOUN
ejpam-6627	372	7	also	also	ADV
ejpam-6627	372	8	provides	provide	VERB
ejpam-6627	372	9	insight	insight	NOUN
ejpam-6627	372	10	into	into	ADP
ejpam-6627	372	11	the	the	DET
ejpam-6627	372	12	essential	essential	ADJ
ejpam-6627	372	13	features	feature	NOUN
ejpam-6627	372	14	that	that	PRON
ejpam-6627	372	15	distinguish	distinguish	VERB
ejpam-6627	372	16	each	each	DET
ejpam-6627	372	17	separation	separation	NOUN
ejpam-6627	372	18	axiom	axiom	NOUN
ejpam-6627	372	19	from	from	ADP
ejpam-6627	372	20	the	the	DET
ejpam-6627	372	21	others	other	NOUN
ejpam-6627	372	22	.	.	PUNCT
ejpam-6627	373	1	8	8	X
ejpam-6627	373	2	.	.	PUNCT
ejpam-6627	373	3	conclusions	conclusion	NOUN
ejpam-6627	373	4	this	this	DET
ejpam-6627	373	5	investigation	investigation	NOUN
ejpam-6627	373	6	has	have	AUX
ejpam-6627	373	7	provided	provide	VERB
ejpam-6627	373	8	a	a	DET
ejpam-6627	373	9	comprehensive	comprehensive	ADJ
ejpam-6627	373	10	analysis	analysis	NOUN
ejpam-6627	373	11	of	of	ADP
ejpam-6627	373	12	separation	separation	NOUN
ejpam-6627	373	13	axioms	axiom	NOUN
ejpam-6627	373	14	that	that	PRON
ejpam-6627	373	15	lie	lie	VERB
ejpam-6627	373	16	beyond	beyond	ADP
ejpam-6627	373	17	the	the	DET
ejpam-6627	373	18	traditional	traditional	ADJ
ejpam-6627	373	19	hausdorff	hausdorff	NOUN
ejpam-6627	373	20	hierarchy	hierarchy	NOUN
ejpam-6627	373	21	,	,	PUNCT
ejpam-6627	373	22	revealing	reveal	VERB
ejpam-6627	373	23	rich	rich	ADJ
ejpam-6627	373	24	structural	structural	ADJ
ejpam-6627	373	25	relationships	relationship	NOUN
ejpam-6627	373	26	and	and	CCONJ
ejpam-6627	373	27	unexpected	unexpected	ADJ
ejpam-6627	373	28	behaviors	behavior	NOUN
ejpam-6627	373	29	.	.	PUNCT
ejpam-6627	374	1	our	our	PRON
ejpam-6627	374	2	main	main	ADJ
ejpam-6627	374	3	contributions	contribution	NOUN
ejpam-6627	374	4	can	can	AUX
ejpam-6627	374	5	be	be	AUX
ejpam-6627	374	6	summarized	summarize	VERB
ejpam-6627	374	7	as	as	SCONJ
ejpam-6627	374	8	follows	follow	VERB
ejpam-6627	374	9	:	:	PUNCT
ejpam-6627	374	10	characterization	characterization	NOUN
ejpam-6627	374	11	of	of	ADP
ejpam-6627	374	12	t1	t1	PROPN
ejpam-6627	374	13	spaces	space	NOUN
ejpam-6627	374	14	with	with	ADP
ejpam-6627	374	15	sigma	sigma	PROPN
ejpam-6627	374	16	-	-	PUNCT
ejpam-6627	374	17	intersection	intersection	NOUN
ejpam-6627	374	18	closed	close	VERB
ejpam-6627	374	19	sets	set	NOUN
ejpam-6627	374	20	:	:	PUNCT
ejpam-6627	374	21	we	we	PRON
ejpam-6627	374	22	established	establish	VERB
ejpam-6627	374	23	that	that	SCONJ
ejpam-6627	374	24	t1	t1	PROPN
ejpam-6627	374	25	spaces	space	VERB
ejpam-6627	374	26	whose	whose	DET
ejpam-6627	374	27	closed	close	VERB
ejpam-6627	374	28	sets	set	NOUN
ejpam-6627	374	29	are	be	AUX
ejpam-6627	374	30	sigma	sigma	NOUN
ejpam-6627	374	31	-	-	PUNCT
ejpam-6627	374	32	intersections	intersection	NOUN
ejpam-6627	374	33	form	form	NOUN
ejpam-6627	374	34	an	an	DET
ejpam-6627	374	35	important	important	ADJ
ejpam-6627	374	36	intermediate	intermediate	ADJ
ejpam-6627	374	37	class	class	NOUN
ejpam-6627	374	38	between	between	ADP
ejpam-6627	374	39	general	general	ADJ
ejpam-6627	374	40	t1	t1	PROPN
ejpam-6627	374	41	spaces	space	NOUN
ejpam-6627	374	42	and	and	CCONJ
ejpam-6627	374	43	metrizable	metrizable	ADJ
ejpam-6627	374	44	spaces	space	NOUN
ejpam-6627	374	45	.	.	PUNCT
ejpam-6627	375	1	the	the	DET
ejpam-6627	375	2	key	key	ADJ
ejpam-6627	375	3	result	result	NOUN
ejpam-6627	375	4	(	(	PUNCT
ejpam-6627	375	5	theorem	theorem	ADJ
ejpam-6627	375	6	3.2	3.2	NUM
ejpam-6627	375	7	)	)	PUNCT
ejpam-6627	375	8	shows	show	VERB
ejpam-6627	375	9	that	that	SCONJ
ejpam-6627	375	10	such	such	ADJ
ejpam-6627	375	11	spaces	space	NOUN
ejpam-6627	375	12	are	be	AUX
ejpam-6627	375	13	metrizable	metrizable	ADJ
ejpam-6627	375	14	if	if	SCONJ
ejpam-6627	375	15	and	and	CCONJ
ejpam-6627	375	16	only	only	ADV
ejpam-6627	375	17	if	if	SCONJ
ejpam-6627	375	18	they	they	PRON
ejpam-6627	375	19	are	be	AUX
ejpam-6627	375	20	regular	regular	ADJ
ejpam-6627	375	21	j.	j.	PROPN
ejpam-6627	375	22	oudetallah	oudetallah	PROPN
ejpam-6627	375	23	et	et	PROPN
ejpam-6627	375	24	al	al	PROPN
ejpam-6627	375	25	.	.	PUNCT
ejpam-6627	375	26	/	/	SYM
ejpam-6627	375	27	eur	eur	PROPN
ejpam-6627	375	28	.	.	PUNCT
ejpam-6627	376	1	j.	j.	PROPN
ejpam-6627	376	2	pure	pure	PROPN
ejpam-6627	376	3	appl	appl	PROPN
ejpam-6627	376	4	.	.	PROPN
ejpam-6627	376	5	math	math	PROPN
ejpam-6627	376	6	,	,	PUNCT
ejpam-6627	376	7	18	18	NUM
ejpam-6627	376	8	(	(	PUNCT
ejpam-6627	376	9	3	3	NUM
ejpam-6627	376	10	)	)	PUNCT
ejpam-6627	376	11	(	(	PUNCT
ejpam-6627	376	12	2025	2025	NUM
ejpam-6627	376	13	)	)	PUNCT
ejpam-6627	376	14	,	,	PUNCT
ejpam-6627	376	15	6627	6627	NUM
ejpam-6627	376	16	17	17	NUM
ejpam-6627	376	17	of	of	ADP
ejpam-6627	376	18	20	20	NUM
ejpam-6627	376	19	and	and	CCONJ
ejpam-6627	376	20	second	second	ADV
ejpam-6627	376	21	-	-	PUNCT
ejpam-6627	376	22	countable	countable	ADJ
ejpam-6627	376	23	,	,	PUNCT
ejpam-6627	376	24	identifying	identify	VERB
ejpam-6627	376	25	precisely	precisely	ADV
ejpam-6627	376	26	the	the	DET
ejpam-6627	376	27	obstacles	obstacle	NOUN
ejpam-6627	376	28	to	to	ADP
ejpam-6627	376	29	metrizability	metrizability	NOUN
ejpam-6627	376	30	.	.	PUNCT
ejpam-6627	377	1	our	our	PRON
ejpam-6627	377	2	examples	example	NOUN
ejpam-6627	377	3	,	,	PUNCT
ejpam-6627	377	4	particularly	particularly	ADV
ejpam-6627	377	5	the	the	DET
ejpam-6627	377	6	co	co	ADJ
ejpam-6627	377	7	-	-	ADJ
ejpam-6627	377	8	countable	countable	ADJ
ejpam-6627	377	9	topology	topology	NOUN
ejpam-6627	377	10	and	and	CCONJ
ejpam-6627	377	11	the	the	DET
ejpam-6627	377	12	arens	arens	PROPN
ejpam-6627	377	13	-	-	PUNCT
ejpam-6627	377	14	fort	fort	NOUN
ejpam-6627	377	15	space	space	NOUN
ejpam-6627	377	16	,	,	PUNCT
ejpam-6627	377	17	demonstrate	demonstrate	VERB
ejpam-6627	377	18	that	that	SCONJ
ejpam-6627	377	19	these	these	DET
ejpam-6627	377	20	obstacles	obstacle	NOUN
ejpam-6627	377	21	are	be	AUX
ejpam-6627	377	22	genuine	genuine	ADJ
ejpam-6627	377	23	and	and	CCONJ
ejpam-6627	377	24	can	can	AUX
ejpam-6627	377	25	not	not	PART
ejpam-6627	377	26	be	be	AUX
ejpam-6627	377	27	removed	remove	VERB
ejpam-6627	377	28	without	without	ADP
ejpam-6627	377	29	additional	additional	ADJ
ejpam-6627	377	30	assumptions	assumption	NOUN
ejpam-6627	377	31	.	.	PUNCT
ejpam-6627	378	1	urysohn	urysohn	PROPN
ejpam-6627	378	2	spaces	space	VERB
ejpam-6627	378	3	without	without	ADP
ejpam-6627	378	4	hausdorff	hausdorff	NOUN
ejpam-6627	378	5	property	property	NOUN
ejpam-6627	378	6	:	:	PUNCT
ejpam-6627	378	7	our	our	PRON
ejpam-6627	378	8	analysis	analysis	NOUN
ejpam-6627	378	9	of	of	ADP
ejpam-6627	378	10	urysohn	urysohn	PROPN
ejpam-6627	378	11	spaces	space	NOUN
ejpam-6627	378	12	revealed	reveal	VERB
ejpam-6627	378	13	that	that	SCONJ
ejpam-6627	378	14	the	the	DET
ejpam-6627	378	15	t2	t2	NOUN
ejpam-6627	378	16	1	1	NUM
ejpam-6627	378	17	2	2	NUM
ejpam-6627	378	18	axiom	axiom	NOUN
ejpam-6627	378	19	,	,	PUNCT
ejpam-6627	378	20	while	while	SCONJ
ejpam-6627	378	21	stronger	strong	ADJ
ejpam-6627	378	22	than	than	ADP
ejpam-6627	378	23	t2	t2	NOUN
ejpam-6627	378	24	in	in	ADP
ejpam-6627	378	25	the	the	DET
ejpam-6627	378	26	presence	presence	NOUN
ejpam-6627	378	27	of	of	ADP
ejpam-6627	378	28	additional	additional	ADJ
ejpam-6627	378	29	conditions	condition	NOUN
ejpam-6627	378	30	,	,	PUNCT
ejpam-6627	378	31	can	can	AUX
ejpam-6627	378	32	exist	exist	VERB
ejpam-6627	378	33	independently	independently	ADV
ejpam-6627	378	34	of	of	ADP
ejpam-6627	378	35	the	the	DET
ejpam-6627	378	36	hausdorff	hausdorff	NOUN
ejpam-6627	378	37	property	property	NOUN
ejpam-6627	378	38	.	.	PUNCT
ejpam-6627	379	1	the	the	DET
ejpam-6627	379	2	functional	functional	ADJ
ejpam-6627	379	3	characterization	characterization	NOUN
ejpam-6627	379	4	(	(	PUNCT
ejpam-6627	379	5	theorem	theorem	VERB
ejpam-6627	379	6	4.1	4.1	NUM
ejpam-6627	379	7	)	)	PUNCT
ejpam-6627	379	8	using	use	VERB
ejpam-6627	379	9	continuous	continuous	ADJ
ejpam-6627	379	10	functions	function	NOUN
ejpam-6627	379	11	with	with	ADP
ejpam-6627	379	12	disjoint	disjoint	NOUN
ejpam-6627	379	13	supports	support	NOUN
ejpam-6627	379	14	provides	provide	VERB
ejpam-6627	379	15	a	a	DET
ejpam-6627	379	16	powerful	powerful	ADJ
ejpam-6627	379	17	tool	tool	NOUN
ejpam-6627	379	18	for	for	ADP
ejpam-6627	379	19	constructing	construct	VERB
ejpam-6627	379	20	and	and	CCONJ
ejpam-6627	379	21	analyzing	analyze	VERB
ejpam-6627	379	22	such	such	ADJ
ejpam-6627	379	23	spaces	space	NOUN
ejpam-6627	379	24	.	.	PUNCT
ejpam-6627	380	1	the	the	DET
ejpam-6627	380	2	role	role	NOUN
ejpam-6627	380	3	of	of	ADP
ejpam-6627	380	4	local	local	ADJ
ejpam-6627	380	5	compactness	compactness	NOUN
ejpam-6627	380	6	in	in	ADP
ejpam-6627	380	7	elevating	elevate	VERB
ejpam-6627	380	8	urysohn	urysohn	NOUN
ejpam-6627	380	9	to	to	ADP
ejpam-6627	380	10	hausdorff	hausdorff	PROPN
ejpam-6627	380	11	(	(	PUNCT
ejpam-6627	380	12	theorem	theorem	VERB
ejpam-6627	380	13	4.3	4.3	NUM
ejpam-6627	380	14	)	)	PUNCT
ejpam-6627	380	15	highlights	highlight	NOUN
ejpam-6627	380	16	the	the	DET
ejpam-6627	380	17	subtle	subtle	ADJ
ejpam-6627	380	18	interplay	interplay	NOUN
ejpam-6627	380	19	between	between	ADP
ejpam-6627	380	20	separation	separation	NOUN
ejpam-6627	380	21	and	and	CCONJ
ejpam-6627	380	22	compactness	compactness	NOUN
ejpam-6627	380	23	conditions	condition	NOUN
ejpam-6627	380	24	.	.	PUNCT
ejpam-6627	381	1	hierarchy	hierarchy	NOUN
ejpam-6627	381	2	of	of	ADP
ejpam-6627	381	3	intermediate	intermediate	ADJ
ejpam-6627	381	4	separation	separation	NOUN
ejpam-6627	381	5	axioms	axiom	NOUN
ejpam-6627	381	6	:	:	PUNCT
ejpam-6627	381	7	we	we	PRON
ejpam-6627	381	8	introduced	introduce	VERB
ejpam-6627	381	9	and	and	CCONJ
ejpam-6627	381	10	studied	study	VERB
ejpam-6627	381	11	quasi	quasi	NOUN
ejpam-6627	381	12	-	-	ADJ
ejpam-6627	381	13	hausdorff	hausdorff	ADJ
ejpam-6627	381	14	and	and	CCONJ
ejpam-6627	381	15	sequentially	sequentially	ADV
ejpam-6627	381	16	hausdorff	hausdorff	NOUN
ejpam-6627	381	17	spaces	space	NOUN
ejpam-6627	381	18	,	,	PUNCT
ejpam-6627	381	19	establishing	establish	VERB
ejpam-6627	381	20	a	a	DET
ejpam-6627	381	21	strict	strict	ADJ
ejpam-6627	381	22	hierarchy	hierarchy	NOUN
ejpam-6627	381	23	:	:	PUNCT
ejpam-6627	381	24	t2	t2	PROPN
ejpam-6627	381	25	⇒	⇒	NOUN
ejpam-6627	381	26	quasi	quasi	PROPN
ejpam-6627	381	27	-	-	ADJ
ejpam-6627	381	28	hausdorff	hausdorff	ADJ
ejpam-6627	381	29	⇒	⇒	NOUN
ejpam-6627	381	30	sequentially	sequentially	ADV
ejpam-6627	381	31	hausdorff	hausdorff	X
ejpam-6627	381	32	⇒	⇒	PROPN
ejpam-6627	381	33	t1	t1	PROPN
ejpam-6627	381	34	.	.	PUNCT
ejpam-6627	382	1	each	each	DET
ejpam-6627	382	2	implication	implication	NOUN
ejpam-6627	382	3	is	be	AUX
ejpam-6627	382	4	proper	proper	ADJ
ejpam-6627	382	5	,	,	PUNCT
ejpam-6627	382	6	as	as	SCONJ
ejpam-6627	382	7	demonstrated	demonstrate	VERB
ejpam-6627	382	8	by	by	ADP
ejpam-6627	382	9	our	our	PRON
ejpam-6627	382	10	counterexamples	counterexample	NOUN
ejpam-6627	382	11	.	.	PUNCT
ejpam-6627	383	1	the	the	DET
ejpam-6627	383	2	connection	connection	NOUN
ejpam-6627	383	3	between	between	ADP
ejpam-6627	383	4	quasi	quasi	ADJ
ejpam-6627	383	5	-	-	ADJ
ejpam-6627	383	6	hausdorff	hausdorff	ADJ
ejpam-6627	383	7	spaces	space	NOUN
ejpam-6627	383	8	and	and	CCONJ
ejpam-6627	383	9	sigma	sigma	NOUN
ejpam-6627	383	10	-	-	PUNCT
ejpam-6627	383	11	intersection	intersection	NOUN
ejpam-6627	383	12	properties	property	NOUN
ejpam-6627	383	13	(	(	PUNCT
ejpam-6627	383	14	theorem	theorem	VERB
ejpam-6627	383	15	5.4	5.4	NUM
ejpam-6627	383	16	)	)	PUNCT
ejpam-6627	383	17	reveals	reveal	VERB
ejpam-6627	383	18	how	how	SCONJ
ejpam-6627	383	19	these	these	DET
ejpam-6627	383	20	intermediate	intermediate	ADJ
ejpam-6627	383	21	axioms	axiom	NOUN
ejpam-6627	383	22	can	can	AUX
ejpam-6627	383	23	yield	yield	VERB
ejpam-6627	383	24	analytically	analytically	ADV
ejpam-6627	383	25	useful	useful	ADJ
ejpam-6627	383	26	properties	property	NOUN
ejpam-6627	383	27	.	.	PUNCT
ejpam-6627	384	1	behavior	behavior	NOUN
ejpam-6627	384	2	under	under	ADP
ejpam-6627	384	3	topological	topological	ADJ
ejpam-6627	384	4	operations	operation	NOUN
ejpam-6627	384	5	:	:	PUNCT
ejpam-6627	384	6	our	our	PRON
ejpam-6627	384	7	investigation	investigation	NOUN
ejpam-6627	384	8	of	of	ADP
ejpam-6627	384	9	quotient	quotient	NOUN
ejpam-6627	384	10	and	and	CCONJ
ejpam-6627	384	11	product	product	NOUN
ejpam-6627	384	12	operations	operation	NOUN
ejpam-6627	384	13	(	(	PUNCT
ejpam-6627	384	14	section	section	NOUN
ejpam-6627	384	15	6	6	NUM
ejpam-6627	384	16	)	)	PUNCT
ejpam-6627	384	17	provides	provide	VERB
ejpam-6627	384	18	precise	precise	ADJ
ejpam-6627	384	19	conditions	condition	NOUN
ejpam-6627	384	20	for	for	ADP
ejpam-6627	384	21	when	when	SCONJ
ejpam-6627	384	22	separation	separation	NOUN
ejpam-6627	384	23	axioms	axiom	NOUN
ejpam-6627	384	24	are	be	AUX
ejpam-6627	384	25	preserved	preserve	VERB
ejpam-6627	384	26	.	.	PUNCT
ejpam-6627	385	1	the	the	DET
ejpam-6627	385	2	particularly	particularly	ADV
ejpam-6627	385	3	subtle	subtle	ADJ
ejpam-6627	385	4	behavior	behavior	NOUN
ejpam-6627	385	5	of	of	ADP
ejpam-6627	385	6	the	the	DET
ejpam-6627	385	7	sigma	sigma	ADJ
ejpam-6627	385	8	-	-	PUNCT
ejpam-6627	385	9	intersection	intersection	NOUN
ejpam-6627	385	10	property	property	NOUN
ejpam-6627	385	11	under	under	ADP
ejpam-6627	385	12	products	product	NOUN
ejpam-6627	385	13	,	,	PUNCT
ejpam-6627	385	14	requiring	require	VERB
ejpam-6627	385	15	either	either	CCONJ
ejpam-6627	385	16	countability	countability	NOUN
ejpam-6627	385	17	of	of	ADP
ejpam-6627	385	18	the	the	DET
ejpam-6627	385	19	index	index	NOUN
ejpam-6627	385	20	set	set	NOUN
ejpam-6627	385	21	or	or	CCONJ
ejpam-6627	385	22	discreteness	discreteness	NOUN
ejpam-6627	385	23	of	of	ADP
ejpam-6627	385	24	all	all	DET
ejpam-6627	385	25	but	but	CCONJ
ejpam-6627	385	26	countably	countably	ADV
ejpam-6627	385	27	many	many	ADJ
ejpam-6627	385	28	factors	factor	NOUN
ejpam-6627	385	29	,	,	PUNCT
ejpam-6627	385	30	demonstrates	demonstrate	VERB
ejpam-6627	385	31	the	the	DET
ejpam-6627	385	32	delicate	delicate	ADJ
ejpam-6627	385	33	nature	nature	NOUN
ejpam-6627	385	34	of	of	ADP
ejpam-6627	385	35	these	these	DET
ejpam-6627	385	36	properties	property	NOUN
ejpam-6627	385	37	.	.	PUNCT
ejpam-6627	386	1	theoretical	theoretical	ADJ
ejpam-6627	386	2	implications	implication	NOUN
ejpam-6627	386	3	and	and	CCONJ
ejpam-6627	386	4	applications	application	NOUN
ejpam-6627	386	5	:	:	PUNCT
ejpam-6627	386	6	the	the	DET
ejpam-6627	386	7	spaces	space	NOUN
ejpam-6627	386	8	studied	study	VERB
ejpam-6627	386	9	in	in	ADP
ejpam-6627	386	10	this	this	DET
ejpam-6627	386	11	paper	paper	NOUN
ejpam-6627	386	12	arise	arise	VERB
ejpam-6627	386	13	naturally	naturally	ADV
ejpam-6627	386	14	in	in	ADP
ejpam-6627	386	15	various	various	ADJ
ejpam-6627	386	16	mathematical	mathematical	ADJ
ejpam-6627	386	17	contexts	contexts	NOUN
ejpam-6627	386	18	:	:	PUNCT
ejpam-6627	386	19	•	•	NOUN
ejpam-6627	386	20	in	in	ADP
ejpam-6627	386	21	domain	domain	NOUN
ejpam-6627	386	22	theory	theory	NOUN
ejpam-6627	386	23	,	,	PUNCT
ejpam-6627	386	24	where	where	SCONJ
ejpam-6627	386	25	partial	partial	ADJ
ejpam-6627	386	26	information	information	NOUN
ejpam-6627	386	27	is	be	AUX
ejpam-6627	386	28	modeled	model	VERB
ejpam-6627	386	29	by	by	ADP
ejpam-6627	386	30	spaces	space	NOUN
ejpam-6627	386	31	with	with	ADP
ejpam-6627	386	32	weak	weak	ADJ
ejpam-6627	386	33	separation	separation	NOUN
ejpam-6627	386	34	properties	property	NOUN
ejpam-6627	386	35	,	,	PUNCT
ejpam-6627	386	36	as	as	SCONJ
ejpam-6627	386	37	comprehensively	comprehensively	ADV
ejpam-6627	386	38	treated	treat	VERB
ejpam-6627	386	39	by	by	ADP
ejpam-6627	386	40	gierz	gierz	PROPN
ejpam-6627	386	41	et	et	PROPN
ejpam-6627	386	42	al	al	PROPN
ejpam-6627	386	43	.	.	PUNCT
ejpam-6627	387	1	[	[	X
ejpam-6627	387	2	13	13	NUM
ejpam-6627	387	3	]	]	PUNCT
ejpam-6627	387	4	.	.	PUNCT
ejpam-6627	388	1	•	•	NUM
ejpam-6627	388	2	in	in	ADP
ejpam-6627	388	3	functional	functional	ADJ
ejpam-6627	388	4	analysis	analysis	NOUN
ejpam-6627	388	5	,	,	PUNCT
ejpam-6627	388	6	where	where	SCONJ
ejpam-6627	388	7	weak	weak	ADJ
ejpam-6627	388	8	topologies	topology	NOUN
ejpam-6627	388	9	often	often	ADV
ejpam-6627	388	10	fail	fail	VERB
ejpam-6627	388	11	to	to	PART
ejpam-6627	388	12	be	be	AUX
ejpam-6627	388	13	hausdorff	hausdorff	NOUN
ejpam-6627	388	14	,	,	PUNCT
ejpam-6627	388	15	following	follow	VERB
ejpam-6627	388	16	the	the	DET
ejpam-6627	388	17	framework	framework	NOUN
ejpam-6627	388	18	established	establish	VERB
ejpam-6627	388	19	by	by	ADP
ejpam-6627	388	20	schaefer	schaefer	NOUN
ejpam-6627	388	21	and	and	CCONJ
ejpam-6627	388	22	wolff	wolff	NOUN
ejpam-6627	388	23	[	[	X
ejpam-6627	388	24	14	14	NUM
ejpam-6627	388	25	]	]	PUNCT
ejpam-6627	388	26	.	.	PUNCT
ejpam-6627	389	1	•	•	NOUN
ejpam-6627	389	2	in	in	ADP
ejpam-6627	389	3	algebraic	algebraic	ADJ
ejpam-6627	389	4	geometry	geometry	NOUN
ejpam-6627	389	5	,	,	PUNCT
ejpam-6627	389	6	where	where	SCONJ
ejpam-6627	389	7	the	the	DET
ejpam-6627	389	8	zariski	zariski	NOUN
ejpam-6627	389	9	topology	topology	NOUN
ejpam-6627	389	10	provides	provide	VERB
ejpam-6627	389	11	important	important	ADJ
ejpam-6627	389	12	examples	example	NOUN
ejpam-6627	389	13	of	of	ADP
ejpam-6627	389	14	non	non	ADJ
ejpam-6627	389	15	-	-	ADJ
ejpam-6627	389	16	hausdorff	hausdorff	ADJ
ejpam-6627	389	17	spaces	space	NOUN
ejpam-6627	389	18	,	,	PUNCT
ejpam-6627	389	19	as	as	SCONJ
ejpam-6627	389	20	discussed	discuss	VERB
ejpam-6627	389	21	in	in	ADP
ejpam-6627	389	22	hartshorne	hartshorne	ADJ
ejpam-6627	390	1	[	[	X
ejpam-6627	390	2	15	15	NUM
ejpam-6627	390	3	]	]	PUNCT
ejpam-6627	390	4	.	.	PUNCT
ejpam-6627	391	1	•	•	NUM
ejpam-6627	391	2	in	in	ADP
ejpam-6627	391	3	theoretical	theoretical	ADJ
ejpam-6627	391	4	computer	computer	NOUN
ejpam-6627	391	5	science	science	NOUN
ejpam-6627	391	6	,	,	PUNCT
ejpam-6627	391	7	where	where	SCONJ
ejpam-6627	391	8	convergence	convergence	NOUN
ejpam-6627	391	9	without	without	ADP
ejpam-6627	391	10	uniqueness	uniqueness	NOUN
ejpam-6627	391	11	models	model	NOUN
ejpam-6627	391	12	non	non	ADJ
ejpam-6627	391	13	-	-	ADJ
ejpam-6627	391	14	deterministic	deterministic	ADJ
ejpam-6627	391	15	computation	computation	NOUN
ejpam-6627	391	16	connections	connection	NOUN
ejpam-6627	391	17	with	with	ADP
ejpam-6627	391	18	related	related	ADJ
ejpam-6627	391	19	work	work	NOUN
ejpam-6627	391	20	:	:	PUNCT
ejpam-6627	391	21	our	our	PRON
ejpam-6627	391	22	results	result	NOUN
ejpam-6627	391	23	connect	connect	VERB
ejpam-6627	391	24	with	with	ADP
ejpam-6627	391	25	and	and	CCONJ
ejpam-6627	391	26	extend	extend	VERB
ejpam-6627	391	27	several	several	ADJ
ejpam-6627	391	28	recent	recent	ADJ
ejpam-6627	391	29	investigations	investigation	NOUN
ejpam-6627	391	30	in	in	ADP
ejpam-6627	391	31	general	general	ADJ
ejpam-6627	391	32	topology	topology	NOUN
ejpam-6627	391	33	.	.	PUNCT
ejpam-6627	392	1	the	the	DET
ejpam-6627	392	2	characterization	characterization	NOUN
ejpam-6627	392	3	of	of	ADP
ejpam-6627	392	4	t1	t1	PROPN
ejpam-6627	392	5	spaces	space	NOUN
ejpam-6627	392	6	with	with	ADP
ejpam-6627	392	7	sigma	sigma	PROPN
ejpam-6627	392	8	-	-	PUNCT
ejpam-6627	392	9	intersection	intersection	NOUN
ejpam-6627	392	10	closed	close	VERB
ejpam-6627	392	11	sets	set	NOUN
ejpam-6627	392	12	relates	relate	VERB
ejpam-6627	392	13	to	to	PART
ejpam-6627	392	14	work	work	VERB
ejpam-6627	392	15	on	on	ADP
ejpam-6627	392	16	pairwise	pairwise	NOUN
ejpam-6627	392	17	expandable	expandable	ADJ
ejpam-6627	392	18	spaces	space	NOUN
ejpam-6627	392	19	[	[	X
ejpam-6627	392	20	10	10	NUM
ejpam-6627	392	21	]	]	PUNCT
ejpam-6627	392	22	and	and	CCONJ
ejpam-6627	392	23	hconvexity	hconvexity	NOUN
ejpam-6627	392	24	in	in	ADP
ejpam-6627	392	25	metric	metric	ADJ
ejpam-6627	392	26	linear	linear	NOUN
ejpam-6627	392	27	spaces	space	NOUN
ejpam-6627	392	28	[	[	X
ejpam-6627	392	29	26	26	NUM
ejpam-6627	392	30	]	]	PUNCT
ejpam-6627	392	31	.	.	PUNCT
ejpam-6627	393	1	the	the	DET
ejpam-6627	393	2	study	study	NOUN
ejpam-6627	393	3	of	of	ADP
ejpam-6627	393	4	compactness	compactness	NOUN
ejpam-6627	393	5	conditions	condition	NOUN
ejpam-6627	393	6	in	in	ADP
ejpam-6627	393	7	urysohn	urysohn	PROPN
ejpam-6627	393	8	spaces	space	NOUN
ejpam-6627	393	9	builds	build	VERB
ejpam-6627	393	10	on	on	ADP
ejpam-6627	393	11	recent	recent	ADJ
ejpam-6627	393	12	results	result	NOUN
ejpam-6627	393	13	on	on	ADP
ejpam-6627	393	14	r	r	NOUN
ejpam-6627	393	15	-	-	PUNCT
ejpam-6627	393	16	compactness	compactness	NOUN
ejpam-6627	393	17	[	[	X
ejpam-6627	393	18	11	11	NUM
ejpam-6627	393	19	]	]	PUNCT
ejpam-6627	393	20	and	and	CCONJ
ejpam-6627	393	21	d	d	NOUN
ejpam-6627	393	22	-	-	NOUN
ejpam-6627	393	23	metacompactness	metacompactness	NOUN
ejpam-6627	393	24	[	[	X
ejpam-6627	393	25	12	12	NUM
ejpam-6627	393	26	]	]	PUNCT
ejpam-6627	393	27	in	in	ADP
ejpam-6627	393	28	topological	topological	ADJ
ejpam-6627	393	29	spaces	space	NOUN
ejpam-6627	393	30	.	.	PUNCT
ejpam-6627	394	1	furthermore	furthermore	ADV
ejpam-6627	394	2	,	,	PUNCT
ejpam-6627	394	3	our	our	PRON
ejpam-6627	394	4	analysis	analysis	NOUN
ejpam-6627	394	5	of	of	ADP
ejpam-6627	394	6	covering	cover	VERB
ejpam-6627	394	7	properties	property	NOUN
ejpam-6627	394	8	in	in	ADP
ejpam-6627	394	9	intermediate	intermediate	ADJ
ejpam-6627	394	10	separation	separation	NOUN
ejpam-6627	394	11	axioms	axiom	NOUN
ejpam-6627	394	12	extends	extend	VERB
ejpam-6627	394	13	recent	recent	ADJ
ejpam-6627	394	14	work	work	NOUN
ejpam-6627	394	15	on	on	ADP
ejpam-6627	394	16	lindelöf	lindelöf	NOUN
ejpam-6627	394	17	properties	property	NOUN
ejpam-6627	394	18	[	[	X
ejpam-6627	394	19	31	31	NUM
ejpam-6627	394	20	]	]	PUNCT
ejpam-6627	394	21	to	to	ADP
ejpam-6627	394	22	the	the	DET
ejpam-6627	394	23	non	non	ADJ
ejpam-6627	394	24	-	-	ADJ
ejpam-6627	394	25	regular	regular	ADJ
ejpam-6627	394	26	setting	setting	NOUN
ejpam-6627	394	27	.	.	PUNCT
ejpam-6627	395	1	future	future	ADJ
ejpam-6627	395	2	research	research	NOUN
ejpam-6627	395	3	directions	direction	NOUN
ejpam-6627	395	4	:	:	PUNCT
ejpam-6627	395	5	this	this	DET
ejpam-6627	395	6	work	work	NOUN
ejpam-6627	395	7	opens	open	VERB
ejpam-6627	395	8	several	several	ADJ
ejpam-6627	395	9	avenues	avenue	NOUN
ejpam-6627	395	10	for	for	ADP
ejpam-6627	395	11	future	future	ADJ
ejpam-6627	395	12	investigation	investigation	NOUN
ejpam-6627	395	13	:	:	PUNCT
ejpam-6627	395	14	j.	j.	PROPN
ejpam-6627	395	15	oudetallah	oudetallah	PROPN
ejpam-6627	395	16	et	et	PROPN
ejpam-6627	395	17	al	al	PROPN
ejpam-6627	395	18	.	.	PUNCT
ejpam-6627	395	19	/	/	SYM
ejpam-6627	395	20	eur	eur	PROPN
ejpam-6627	395	21	.	.	PUNCT
ejpam-6627	396	1	j.	j.	PROPN
ejpam-6627	396	2	pure	pure	PROPN
ejpam-6627	396	3	appl	appl	PROPN
ejpam-6627	396	4	.	.	PROPN
ejpam-6627	396	5	math	math	PROPN
ejpam-6627	396	6	,	,	PUNCT
ejpam-6627	396	7	18	18	NUM
ejpam-6627	396	8	(	(	PUNCT
ejpam-6627	396	9	3	3	NUM
ejpam-6627	396	10	)	)	PUNCT
ejpam-6627	396	11	(	(	PUNCT
ejpam-6627	396	12	2025	2025	NUM
ejpam-6627	396	13	)	)	PUNCT
ejpam-6627	396	14	,	,	PUNCT
ejpam-6627	396	15	6627	6627	NUM
ejpam-6627	396	16	18	18	NUM
ejpam-6627	396	17	of	of	ADP
ejpam-6627	396	18	20	20	NUM
ejpam-6627	396	19	•	•	NOUN
ejpam-6627	396	20	characterizing	characterize	VERB
ejpam-6627	396	21	when	when	SCONJ
ejpam-6627	396	22	intermediate	intermediate	ADJ
ejpam-6627	396	23	separation	separation	NOUN
ejpam-6627	396	24	axioms	axiom	NOUN
ejpam-6627	396	25	are	be	AUX
ejpam-6627	396	26	preserved	preserve	VERB
ejpam-6627	396	27	under	under	ADP
ejpam-6627	396	28	other	other	ADJ
ejpam-6627	396	29	topological	topological	ADJ
ejpam-6627	396	30	constructions	construction	NOUN
ejpam-6627	396	31	(	(	PUNCT
ejpam-6627	396	32	e.g.	e.g.	ADV
ejpam-6627	396	33	,	,	PUNCT
ejpam-6627	396	34	function	function	NOUN
ejpam-6627	396	35	spaces	space	NOUN
ejpam-6627	396	36	,	,	PUNCT
ejpam-6627	396	37	hyperspaces	hyperspace	NOUN
ejpam-6627	396	38	)	)	PUNCT
ejpam-6627	396	39	,	,	PUNCT
ejpam-6627	396	40	building	build	VERB
ejpam-6627	396	41	on	on	ADP
ejpam-6627	396	42	the	the	DET
ejpam-6627	396	43	foundational	foundational	ADJ
ejpam-6627	396	44	work	work	NOUN
ejpam-6627	396	45	of	of	ADP
ejpam-6627	396	46	mccoy	mccoy	PROPN
ejpam-6627	396	47	and	and	CCONJ
ejpam-6627	396	48	ntantu	ntantu	ADJ
ejpam-6627	397	1	[	[	X
ejpam-6627	397	2	21	21	NUM
ejpam-6627	397	3	]	]	PUNCT
ejpam-6627	397	4	and	and	CCONJ
ejpam-6627	397	5	arens	aren	NOUN
ejpam-6627	397	6	and	and	CCONJ
ejpam-6627	397	7	dugundji	dugundji	VERB
ejpam-6627	398	1	[	[	X
ejpam-6627	398	2	22	22	NUM
ejpam-6627	398	3	]	]	PUNCT
ejpam-6627	398	4	.	.	PUNCT
ejpam-6627	399	1	•	•	INTJ
ejpam-6627	399	2	investigating	investigate	VERB
ejpam-6627	399	3	the	the	DET
ejpam-6627	399	4	role	role	NOUN
ejpam-6627	399	5	of	of	ADP
ejpam-6627	399	6	cardinal	cardinal	ADJ
ejpam-6627	399	7	invariants	invariant	NOUN
ejpam-6627	399	8	in	in	ADP
ejpam-6627	399	9	determining	determine	VERB
ejpam-6627	399	10	when	when	SCONJ
ejpam-6627	399	11	spaces	space	NOUN
ejpam-6627	399	12	with	with	ADP
ejpam-6627	399	13	weak	weak	ADJ
ejpam-6627	399	14	separation	separation	NOUN
ejpam-6627	399	15	axioms	axiom	NOUN
ejpam-6627	399	16	can	can	AUX
ejpam-6627	399	17	be	be	AUX
ejpam-6627	399	18	embedded	embed	VERB
ejpam-6627	399	19	in	in	ADP
ejpam-6627	399	20	spaces	space	NOUN
ejpam-6627	399	21	with	with	ADP
ejpam-6627	399	22	stronger	strong	ADJ
ejpam-6627	399	23	properties	property	NOUN
ejpam-6627	399	24	,	,	PUNCT
ejpam-6627	399	25	following	follow	VERB
ejpam-6627	399	26	the	the	DET
ejpam-6627	399	27	set	set	ADJ
ejpam-6627	399	28	-	-	PUNCT
ejpam-6627	399	29	theoretic	theoretic	NOUN
ejpam-6627	399	30	methods	method	NOUN
ejpam-6627	399	31	of	of	ADP
ejpam-6627	399	32	kunen	kunen	NOUN
ejpam-6627	400	1	[	[	X
ejpam-6627	400	2	28	28	NUM
ejpam-6627	400	3	]	]	PUNCT
ejpam-6627	400	4	.	.	PUNCT
ejpam-6627	401	1	•	•	PUNCT
ejpam-6627	401	2	exploring	explore	VERB
ejpam-6627	401	3	connections	connection	NOUN
ejpam-6627	401	4	between	between	ADP
ejpam-6627	401	5	these	these	DET
ejpam-6627	401	6	separation	separation	NOUN
ejpam-6627	401	7	axioms	axiom	NOUN
ejpam-6627	401	8	and	and	CCONJ
ejpam-6627	401	9	convergence	convergence	NOUN
ejpam-6627	401	10	structures	structure	NOUN
ejpam-6627	401	11	beyond	beyond	ADP
ejpam-6627	401	12	sequential	sequential	ADJ
ejpam-6627	401	13	convergence	convergence	NOUN
ejpam-6627	401	14	,	,	PUNCT
ejpam-6627	401	15	extending	extend	VERB
ejpam-6627	401	16	the	the	DET
ejpam-6627	401	17	classical	classical	ADJ
ejpam-6627	401	18	treatment	treatment	NOUN
ejpam-6627	401	19	by	by	ADP
ejpam-6627	401	20	kelley	kelley	NOUN
ejpam-6627	402	1	[	[	X
ejpam-6627	402	2	16	16	NUM
ejpam-6627	402	3	]	]	PUNCT
ejpam-6627	402	4	.	.	PUNCT
ejpam-6627	403	1	•	•	PUNCT
ejpam-6627	403	2	developing	develop	VERB
ejpam-6627	403	3	a	a	DET
ejpam-6627	403	4	theory	theory	NOUN
ejpam-6627	403	5	of	of	ADP
ejpam-6627	403	6	continuous	continuous	ADJ
ejpam-6627	403	7	functions	function	NOUN
ejpam-6627	403	8	between	between	ADP
ejpam-6627	403	9	spaces	space	NOUN
ejpam-6627	403	10	with	with	ADP
ejpam-6627	403	11	different	different	ADJ
ejpam-6627	403	12	intermediate	intermediate	ADJ
ejpam-6627	403	13	separation	separation	NOUN
ejpam-6627	403	14	properties	property	NOUN
ejpam-6627	403	15	,	,	PUNCT
ejpam-6627	403	16	inspired	inspire	VERB
ejpam-6627	403	17	by	by	ADP
ejpam-6627	403	18	the	the	DET
ejpam-6627	403	19	ring	ring	NOUN
ejpam-6627	403	20	-	-	PUNCT
ejpam-6627	403	21	theoretic	theoretic	NOUN
ejpam-6627	403	22	approach	approach	NOUN
ejpam-6627	403	23	of	of	ADP
ejpam-6627	403	24	gillman	gillman	PROPN
ejpam-6627	403	25	and	and	CCONJ
ejpam-6627	403	26	jerison	jerison	NOUN
ejpam-6627	403	27	[	[	X
ejpam-6627	403	28	23	23	NUM
ejpam-6627	403	29	]	]	PUNCT
ejpam-6627	403	30	.	.	PUNCT
ejpam-6627	404	1	•	•	NUM
ejpam-6627	404	2	extending	extend	VERB
ejpam-6627	404	3	the	the	DET
ejpam-6627	404	4	study	study	NOUN
ejpam-6627	404	5	to	to	ADP
ejpam-6627	404	6	bitopological	bitopological	ADJ
ejpam-6627	404	7	spaces	space	NOUN
ejpam-6627	404	8	,	,	PUNCT
ejpam-6627	404	9	building	build	VERB
ejpam-6627	404	10	on	on	ADP
ejpam-6627	404	11	recent	recent	ADJ
ejpam-6627	404	12	work	work	NOUN
ejpam-6627	404	13	on	on	ADP
ejpam-6627	404	14	r	r	NOUN
ejpam-6627	404	15	-	-	PUNCT
ejpam-6627	404	16	compactness	compactness	NOUN
ejpam-6627	404	17	in	in	ADP
ejpam-6627	404	18	bitopological	bitopological	ADJ
ejpam-6627	404	19	settings	setting	NOUN
ejpam-6627	404	20	[	[	X
ejpam-6627	404	21	11	11	NUM
ejpam-6627	404	22	]	]	PUNCT
ejpam-6627	404	23	.	.	PUNCT
ejpam-6627	405	1	•	•	PUNCT
ejpam-6627	405	2	investigating	investigate	VERB
ejpam-6627	405	3	applications	application	NOUN
ejpam-6627	405	4	in	in	ADP
ejpam-6627	405	5	theoretical	theoretical	ADJ
ejpam-6627	405	6	computer	computer	NOUN
ejpam-6627	405	7	science	science	NOUN
ejpam-6627	405	8	,	,	PUNCT
ejpam-6627	405	9	particularly	particularly	ADV
ejpam-6627	405	10	in	in	ADP
ejpam-6627	405	11	domain	domain	NOUN
ejpam-6627	405	12	theory	theory	NOUN
ejpam-6627	405	13	and	and	CCONJ
ejpam-6627	405	14	denotational	denotational	ADJ
ejpam-6627	405	15	semantics	semantic	NOUN
ejpam-6627	405	16	.	.	PUNCT
ejpam-6627	406	1	•	•	PUNCT
ejpam-6627	406	2	exploring	explore	VERB
ejpam-6627	406	3	the	the	DET
ejpam-6627	406	4	relationship	relationship	NOUN
ejpam-6627	406	5	between	between	ADP
ejpam-6627	406	6	these	these	DET
ejpam-6627	406	7	separation	separation	NOUN
ejpam-6627	406	8	axioms	axiom	NOUN
ejpam-6627	406	9	and	and	CCONJ
ejpam-6627	406	10	various	various	ADJ
ejpam-6627	406	11	forms	form	NOUN
ejpam-6627	406	12	of	of	ADP
ejpam-6627	406	13	compactness	compactness	NOUN
ejpam-6627	406	14	and	and	CCONJ
ejpam-6627	406	15	metacompactness	metacompactness	NOUN
ejpam-6627	407	1	[	[	X
ejpam-6627	407	2	12	12	NUM
ejpam-6627	407	3	]	]	PUNCT
ejpam-6627	407	4	.	.	PUNCT
ejpam-6627	408	1	methodological	methodological	ADJ
ejpam-6627	408	2	contributions	contribution	NOUN
ejpam-6627	408	3	:	:	PUNCT
ejpam-6627	408	4	our	our	PRON
ejpam-6627	408	5	approach	approach	NOUN
ejpam-6627	408	6	demonstrates	demonstrate	VERB
ejpam-6627	408	7	the	the	DET
ejpam-6627	408	8	value	value	NOUN
ejpam-6627	408	9	of	of	ADP
ejpam-6627	408	10	combining	combine	VERB
ejpam-6627	408	11	constructive	constructive	ADJ
ejpam-6627	408	12	methods	method	NOUN
ejpam-6627	408	13	with	with	ADP
ejpam-6627	408	14	analytical	analytical	ADJ
ejpam-6627	408	15	techniques	technique	NOUN
ejpam-6627	408	16	in	in	ADP
ejpam-6627	408	17	topology	topology	NOUN
ejpam-6627	408	18	.	.	PUNCT
ejpam-6627	409	1	the	the	DET
ejpam-6627	409	2	counterexamples	counterexample	NOUN
ejpam-6627	409	3	presented	present	VERB
ejpam-6627	409	4	not	not	PART
ejpam-6627	409	5	only	only	ADV
ejpam-6627	409	6	establish	establish	VERB
ejpam-6627	409	7	the	the	DET
ejpam-6627	409	8	sharpness	sharpness	NOUN
ejpam-6627	409	9	of	of	ADP
ejpam-6627	409	10	our	our	PRON
ejpam-6627	409	11	results	result	NOUN
ejpam-6627	409	12	but	but	CCONJ
ejpam-6627	409	13	also	also	ADV
ejpam-6627	409	14	provide	provide	VERB
ejpam-6627	409	15	a	a	DET
ejpam-6627	409	16	toolkit	toolkit	NOUN
ejpam-6627	409	17	for	for	ADP
ejpam-6627	409	18	constructing	construct	VERB
ejpam-6627	409	19	spaces	space	NOUN
ejpam-6627	409	20	with	with	ADP
ejpam-6627	409	21	prescribed	prescribed	ADJ
ejpam-6627	409	22	separation	separation	NOUN
ejpam-6627	409	23	properties	property	NOUN
ejpam-6627	409	24	.	.	PUNCT
ejpam-6627	410	1	this	this	DET
ejpam-6627	410	2	methodology	methodology	NOUN
ejpam-6627	410	3	can	can	AUX
ejpam-6627	410	4	be	be	AUX
ejpam-6627	410	5	applied	apply	VERB
ejpam-6627	410	6	to	to	ADP
ejpam-6627	410	7	other	other	ADJ
ejpam-6627	410	8	areas	area	NOUN
ejpam-6627	410	9	of	of	ADP
ejpam-6627	410	10	topology	topology	NOUN
ejpam-6627	410	11	where	where	SCONJ
ejpam-6627	410	12	fine	fine	ADJ
ejpam-6627	410	13	distinctions	distinction	NOUN
ejpam-6627	410	14	between	between	ADP
ejpam-6627	410	15	properties	property	NOUN
ejpam-6627	410	16	are	be	AUX
ejpam-6627	410	17	important	important	ADJ
ejpam-6627	410	18	.	.	PUNCT
ejpam-6627	411	1	the	the	DET
ejpam-6627	411	2	counterexamples	counterexample	NOUN
ejpam-6627	411	3	presented	present	VERB
ejpam-6627	411	4	throughout	throughout	ADP
ejpam-6627	411	5	this	this	DET
ejpam-6627	411	6	paper	paper	NOUN
ejpam-6627	411	7	not	not	PART
ejpam-6627	411	8	only	only	ADV
ejpam-6627	411	9	establish	establish	VERB
ejpam-6627	411	10	the	the	DET
ejpam-6627	411	11	sharpness	sharpness	NOUN
ejpam-6627	411	12	of	of	ADP
ejpam-6627	411	13	our	our	PRON
ejpam-6627	411	14	results	result	NOUN
ejpam-6627	411	15	but	but	CCONJ
ejpam-6627	411	16	also	also	ADV
ejpam-6627	411	17	provide	provide	VERB
ejpam-6627	411	18	a	a	DET
ejpam-6627	411	19	toolkit	toolkit	NOUN
ejpam-6627	411	20	for	for	ADP
ejpam-6627	411	21	constructing	construct	VERB
ejpam-6627	411	22	spaces	space	NOUN
ejpam-6627	411	23	with	with	ADP
ejpam-6627	411	24	prescribed	prescribed	ADJ
ejpam-6627	411	25	separation	separation	NOUN
ejpam-6627	411	26	properties	property	NOUN
ejpam-6627	411	27	.	.	PUNCT
ejpam-6627	412	1	these	these	DET
ejpam-6627	412	2	constructions	construction	NOUN
ejpam-6627	412	3	demonstrate	demonstrate	VERB
ejpam-6627	412	4	that	that	SCONJ
ejpam-6627	412	5	the	the	DET
ejpam-6627	412	6	landscape	landscape	NOUN
ejpam-6627	412	7	of	of	ADP
ejpam-6627	412	8	separation	separation	NOUN
ejpam-6627	412	9	axioms	axiom	NOUN
ejpam-6627	412	10	is	be	AUX
ejpam-6627	412	11	far	far	ADV
ejpam-6627	412	12	richer	rich	ADJ
ejpam-6627	412	13	than	than	SCONJ
ejpam-6627	412	14	the	the	DET
ejpam-6627	412	15	classical	classical	ADJ
ejpam-6627	412	16	kolmogorov	kolmogorov	ADJ
ejpam-6627	412	17	-	-	PUNCT
ejpam-6627	412	18	hausdorff	hausdorff	NOUN
ejpam-6627	412	19	hierarchy	hierarchy	NOUN
ejpam-6627	412	20	suggests	suggest	VERB
ejpam-6627	412	21	,	,	PUNCT
ejpam-6627	412	22	with	with	ADP
ejpam-6627	412	23	numerous	numerous	ADJ
ejpam-6627	412	24	natural	natural	ADJ
ejpam-6627	412	25	and	and	CCONJ
ejpam-6627	412	26	mathematically	mathematically	ADV
ejpam-6627	412	27	significant	significant	ADJ
ejpam-6627	412	28	spaces	space	NOUN
ejpam-6627	412	29	occupying	occupy	VERB
ejpam-6627	412	30	the	the	DET
ejpam-6627	412	31	intermediate	intermediate	ADJ
ejpam-6627	412	32	levels	level	NOUN
ejpam-6627	412	33	.	.	PUNCT
ejpam-6627	413	1	in	in	ADP
ejpam-6627	413	2	conclusion	conclusion	NOUN
ejpam-6627	413	3	,	,	PUNCT
ejpam-6627	413	4	this	this	DET
ejpam-6627	413	5	comprehensive	comprehensive	ADJ
ejpam-6627	413	6	study	study	NOUN
ejpam-6627	413	7	enhances	enhance	VERB
ejpam-6627	413	8	our	our	PRON
ejpam-6627	413	9	understanding	understanding	NOUN
ejpam-6627	413	10	of	of	ADP
ejpam-6627	413	11	the	the	DET
ejpam-6627	413	12	separation	separation	NOUN
ejpam-6627	413	13	axiom	axiom	NOUN
ejpam-6627	413	14	hierarchy	hierarchy	NOUN
ejpam-6627	413	15	and	and	CCONJ
ejpam-6627	413	16	provides	provide	VERB
ejpam-6627	413	17	fundamental	fundamental	ADJ
ejpam-6627	413	18	tools	tool	NOUN
ejpam-6627	413	19	for	for	ADP
ejpam-6627	413	20	investigating	investigate	VERB
ejpam-6627	413	21	spaces	space	NOUN
ejpam-6627	413	22	that	that	PRON
ejpam-6627	413	23	lie	lie	VERB
ejpam-6627	413	24	outside	outside	ADP
ejpam-6627	413	25	traditional	traditional	ADJ
ejpam-6627	413	26	classification	classification	NOUN
ejpam-6627	413	27	schemes	scheme	NOUN
ejpam-6627	413	28	.	.	PUNCT
ejpam-6627	414	1	the	the	DET
ejpam-6627	414	2	interplay	interplay	NOUN
ejpam-6627	414	3	between	between	ADP
ejpam-6627	414	4	separation	separation	NOUN
ejpam-6627	414	5	properties	property	NOUN
ejpam-6627	414	6	,	,	PUNCT
ejpam-6627	414	7	topological	topological	ADJ
ejpam-6627	414	8	operations	operation	NOUN
ejpam-6627	414	9	,	,	PUNCT
ejpam-6627	414	10	and	and	CCONJ
ejpam-6627	414	11	functional	functional	ADJ
ejpam-6627	414	12	characterizations	characterization	NOUN
ejpam-6627	414	13	revealed	reveal	VERB
ejpam-6627	414	14	in	in	ADP
ejpam-6627	414	15	this	this	DET
ejpam-6627	414	16	work	work	NOUN
ejpam-6627	414	17	contributes	contribute	VERB
ejpam-6627	414	18	to	to	ADP
ejpam-6627	414	19	the	the	DET
ejpam-6627	414	20	broader	broad	ADJ
ejpam-6627	414	21	project	project	NOUN
ejpam-6627	414	22	of	of	ADP
ejpam-6627	414	23	understanding	understand	VERB
ejpam-6627	414	24	the	the	DET
ejpam-6627	414	25	fine	fine	ADJ
ejpam-6627	414	26	structure	structure	NOUN
ejpam-6627	414	27	of	of	ADP
ejpam-6627	414	28	topological	topological	ADJ
ejpam-6627	414	29	spaces	space	NOUN
ejpam-6627	414	30	.	.	PUNCT
ejpam-6627	415	1	acknowledgements	acknowledgement	NOUN
ejpam-6627	415	2	the	the	DET
ejpam-6627	415	3	authors	author	NOUN
ejpam-6627	415	4	thank	thank	VERB
ejpam-6627	415	5	the	the	DET
ejpam-6627	415	6	anonymous	anonymous	ADJ
ejpam-6627	415	7	reviewers	reviewer	NOUN
ejpam-6627	415	8	for	for	ADP
ejpam-6627	415	9	their	their	PRON
ejpam-6627	415	10	constructive	constructive	ADJ
ejpam-6627	415	11	comments	comment	NOUN
ejpam-6627	415	12	that	that	PRON
ejpam-6627	415	13	helped	helped	AUX
ejpam-6627	415	14	improve	improve	VERB
ejpam-6627	415	15	the	the	DET
ejpam-6627	415	16	presentation	presentation	NOUN
ejpam-6627	415	17	of	of	ADP
ejpam-6627	415	18	this	this	DET
ejpam-6627	415	19	paper	paper	NOUN
ejpam-6627	415	20	.	.	PUNCT
ejpam-6627	416	1	special	special	ADJ
ejpam-6627	416	2	thanks	thank	NOUN
ejpam-6627	416	3	to	to	ADP
ejpam-6627	416	4	the	the	DET
ejpam-6627	416	5	editorial	editorial	ADJ
ejpam-6627	416	6	board	board	NOUN
ejpam-6627	416	7	of	of	ADP
ejpam-6627	416	8	j.	j.	PROPN
ejpam-6627	416	9	oudetallah	oudetallah	PROPN
ejpam-6627	416	10	et	et	PROPN
ejpam-6627	416	11	al	al	PROPN
ejpam-6627	416	12	.	.	PUNCT
ejpam-6627	416	13	/	/	SYM
ejpam-6627	416	14	eur	eur	PROPN
ejpam-6627	416	15	.	.	PUNCT
ejpam-6627	417	1	j.	j.	PROPN
ejpam-6627	417	2	pure	pure	PROPN
ejpam-6627	417	3	appl	appl	PROPN
ejpam-6627	417	4	.	.	PROPN
ejpam-6627	417	5	math	math	PROPN
ejpam-6627	417	6	,	,	PUNCT
ejpam-6627	417	7	18	18	NUM
ejpam-6627	417	8	(	(	PUNCT
ejpam-6627	417	9	3	3	NUM
ejpam-6627	417	10	)	)	PUNCT
ejpam-6627	417	11	(	(	PUNCT
ejpam-6627	417	12	2025	2025	NUM
ejpam-6627	417	13	)	)	PUNCT
ejpam-6627	417	14	,	,	PUNCT
ejpam-6627	417	15	6627	6627	NUM
ejpam-6627	417	16	19	19	NUM
ejpam-6627	417	17	of	of	ADP
ejpam-6627	417	18	20	20	NUM
ejpam-6627	417	19	the	the	DET
ejpam-6627	417	20	european	european	PROPN
ejpam-6627	417	21	journal	journal	PROPN
ejpam-6627	417	22	of	of	ADP
ejpam-6627	417	23	pure	pure	ADJ
ejpam-6627	417	24	and	and	CCONJ
ejpam-6627	417	25	applied	applied	ADJ
ejpam-6627	417	26	mathematics	mathematic	NOUN
ejpam-6627	417	27	for	for	ADP
ejpam-6627	417	28	their	their	PRON
ejpam-6627	417	29	support	support	NOUN
ejpam-6627	417	30	throughout	throughout	ADP
ejpam-6627	417	31	the	the	DET
ejpam-6627	417	32	review	review	NOUN
ejpam-6627	417	33	process	process	NOUN
ejpam-6627	417	34	.	.	PUNCT
ejpam-6627	418	1	references	reference	NOUN
ejpam-6627	418	2	[	[	X
ejpam-6627	418	3	1	1	NUM
ejpam-6627	418	4	]	]	PUNCT
ejpam-6627	418	5	maysoon	maysoon	NOUN
ejpam-6627	418	6	qousini	qousini	PROPN
ejpam-6627	418	7	,	,	PUNCT
ejpam-6627	418	8	h.	h.	PROPN
ejpam-6627	418	9	hdeib	hdeib	PROPN
ejpam-6627	418	10	,	,	PUNCT
ejpam-6627	418	11	and	and	CCONJ
ejpam-6627	418	12	eman	eman	NOUN
ejpam-6627	418	13	almuhr	almuhr	NOUN
ejpam-6627	418	14	.	.	PUNCT
ejpam-6627	419	1	applications	application	NOUN
ejpam-6627	419	2	of	of	ADP
ejpam-6627	419	3	locally	locally	ADV
ejpam-6627	419	4	compact	compact	ADJ
ejpam-6627	419	5	spaces	space	NOUN
ejpam-6627	419	6	in	in	ADP
ejpam-6627	419	7	polyhedra	polyhedra	NOUN
ejpam-6627	419	8	:	:	PUNCT
ejpam-6627	419	9	dimension	dimension	NOUN
ejpam-6627	419	10	and	and	CCONJ
ejpam-6627	419	11	limits	limit	NOUN
ejpam-6627	419	12	.	.	PUNCT
ejpam-6627	420	1	wseas	wseas	NOUN
ejpam-6627	420	2	transactions	transaction	NOUN
ejpam-6627	420	3	on	on	ADP
ejpam-6627	420	4	mathematics	mathematic	NOUN
ejpam-6627	420	5	,	,	PUNCT
ejpam-6627	420	6	23:118–124	23:118–124	PROPN
ejpam-6627	420	7	,	,	PUNCT
ejpam-6627	420	8	2024	2024	NUM
ejpam-6627	420	9	.	.	PUNCT
ejpam-6627	421	1	[	[	X
ejpam-6627	421	2	2	2	NUM
ejpam-6627	421	3	]	]	PUNCT
ejpam-6627	421	4	ala	ala	PROPN
ejpam-6627	421	5	amourah	amourah	PROPN
ejpam-6627	421	6	,	,	PUNCT
ejpam-6627	421	7	jamal	jamal	PROPN
ejpam-6627	421	8	oudetallah	oudetallah	PROPN
ejpam-6627	421	9	,	,	PUNCT
ejpam-6627	421	10	iqbal	iqbal	PROPN
ejpam-6627	421	11	batiha	batiha	PROPN
ejpam-6627	421	12	,	,	PUNCT
ejpam-6627	421	13	jamal	jamal	PROPN
ejpam-6627	421	14	salah	salah	PROPN
ejpam-6627	421	15	,	,	PUNCT
ejpam-6627	421	16	and	and	CCONJ
ejpam-6627	421	17	mutaz	mutaz	NOUN
ejpam-6627	421	18	shatnawi	shatnawi	ADJ
ejpam-6627	421	19	.	.	PUNCT
ejpam-6627	422	1	σ	σ	NOUN
ejpam-6627	422	2	-	-	ADJ
ejpam-6627	422	3	compact	compact	ADJ
ejpam-6627	422	4	spaces	space	NOUN
ejpam-6627	422	5	in	in	ADP
ejpam-6627	422	6	nth	nth	ADJ
ejpam-6627	422	7	-	-	ADJ
ejpam-6627	422	8	topological	topological	ADJ
ejpam-6627	422	9	space	space	NOUN
ejpam-6627	422	10	.	.	PUNCT
ejpam-6627	423	1	european	european	ADJ
ejpam-6627	423	2	journal	journal	PROPN
ejpam-6627	423	3	of	of	ADP
ejpam-6627	423	4	pure	pure	ADJ
ejpam-6627	423	5	and	and	CCONJ
ejpam-6627	423	6	applied	applied	ADJ
ejpam-6627	423	7	mathematics	mathematic	NOUN
ejpam-6627	423	8	,	,	PUNCT
ejpam-6627	423	9	18(2):5802–5802	18(2):5802–5802	NUM
ejpam-6627	423	10	,	,	PUNCT
ejpam-6627	423	11	2025	2025	NUM
ejpam-6627	423	12	.	.	PUNCT
ejpam-6627	424	1	[	[	X
ejpam-6627	424	2	3	3	NUM
ejpam-6627	424	3	]	]	X
ejpam-6627	424	4	jamal	jamal	PROPN
ejpam-6627	424	5	oudetallah	oudetallah	PROPN
ejpam-6627	424	6	,	,	PUNCT
ejpam-6627	424	7	rehab	rehab	NOUN
ejpam-6627	424	8	alharbi	alharbi	NOUN
ejpam-6627	424	9	,	,	PUNCT
ejpam-6627	424	10	salsabiela	salsabiela	PROPN
ejpam-6627	424	11	rawashdeh	rawashdeh	PROPN
ejpam-6627	424	12	,	,	PUNCT
ejpam-6627	424	13	iqbal	iqbal	PROPN
ejpam-6627	424	14	m.	m.	PROPN
ejpam-6627	424	15	batiha	batiha	PROPN
ejpam-6627	424	16	,	,	PUNCT
ejpam-6627	424	17	ala	ala	PROPN
ejpam-6627	424	18	amourah	amourah	PROPN
ejpam-6627	424	19	,	,	PUNCT
ejpam-6627	424	20	and	and	CCONJ
ejpam-6627	424	21	tala	tala	PROPN
ejpam-6627	424	22	sasa	sasa	PROPN
ejpam-6627	424	23	.	.	PUNCT
ejpam-6627	425	1	nearly	nearly	ADV
ejpam-6627	425	2	lindelöfness	lindelöfness	NOUN
ejpam-6627	425	3	in	in	ADP
ejpam-6627	425	4	nth	nth	ADJ
ejpam-6627	425	5	-	-	ADJ
ejpam-6627	425	6	topological	topological	ADJ
ejpam-6627	425	7	spaces	space	NOUN
ejpam-6627	425	8	.	.	PUNCT
ejpam-6627	426	1	international	international	ADJ
ejpam-6627	426	2	journal	journal	NOUN
ejpam-6627	426	3	of	of	ADP
ejpam-6627	426	4	analysis	analysis	NOUN
ejpam-6627	426	5	and	and	CCONJ
ejpam-6627	426	6	applications	application	NOUN
ejpam-6627	426	7	,	,	PUNCT
ejpam-6627	426	8	23:140–140	23:140–140	NUM
ejpam-6627	426	9	,	,	PUNCT
ejpam-6627	426	10	2025	2025	NUM
ejpam-6627	426	11	.	.	PUNCT
ejpam-6627	427	1	[	[	X
ejpam-6627	427	2	4	4	X
ejpam-6627	427	3	]	]	X
ejpam-6627	427	4	ala	ala	PROPN
ejpam-6627	427	5	amourah	amourah	PROPN
ejpam-6627	427	6	,	,	PUNCT
ejpam-6627	427	7	jamal	jamal	PROPN
ejpam-6627	427	8	oudetallah	oudetallah	PROPN
ejpam-6627	427	9	,	,	PUNCT
ejpam-6627	427	10	iqbal	iqbal	PROPN
ejpam-6627	427	11	m.	m.	PROPN
ejpam-6627	427	12	batiha	batiha	PROPN
ejpam-6627	427	13	,	,	PUNCT
ejpam-6627	427	14	jamal	jamal	PROPN
ejpam-6627	427	15	salah	salah	PROPN
ejpam-6627	427	16	,	,	PUNCT
ejpam-6627	427	17	sultan	sultan	PROPN
ejpam-6627	427	18	alsaadi	alsaadi	NOUN
ejpam-6627	427	19	,	,	PUNCT
ejpam-6627	427	20	and	and	CCONJ
ejpam-6627	427	21	tala	tala	PROPN
ejpam-6627	427	22	sasa	sasa	PROPN
ejpam-6627	427	23	.	.	PUNCT
ejpam-6627	428	1	some	some	DET
ejpam-6627	428	2	types	type	NOUN
ejpam-6627	428	3	of	of	ADP
ejpam-6627	428	4	tri	tri	ADJ
ejpam-6627	428	5	-	-	ADJ
ejpam-6627	428	6	locally	locally	ADV
ejpam-6627	428	7	compactness	compactness	NOUN
ejpam-6627	428	8	spaces	space	NOUN
ejpam-6627	428	9	.	.	PUNCT
ejpam-6627	429	1	european	european	ADJ
ejpam-6627	429	2	journal	journal	PROPN
ejpam-6627	429	3	of	of	ADP
ejpam-6627	429	4	pure	pure	ADJ
ejpam-6627	429	5	and	and	CCONJ
ejpam-6627	429	6	applied	applied	ADJ
ejpam-6627	429	7	mathematics	mathematic	NOUN
ejpam-6627	429	8	,	,	PUNCT
ejpam-6627	429	9	18(2):5764–5764	18(2):5764–5764	NUM
ejpam-6627	429	10	,	,	PUNCT
ejpam-6627	429	11	2025	2025	NUM
ejpam-6627	429	12	.	.	PUNCT
ejpam-6627	430	1	[	[	X
ejpam-6627	430	2	5	5	NUM
ejpam-6627	430	3	]	]	X
ejpam-6627	430	4	jamal	jamal	PROPN
ejpam-6627	430	5	oudetallah	oudetallah	PROPN
ejpam-6627	430	6	,	,	PUNCT
ejpam-6627	430	7	iqbal	iqbal	PROPN
ejpam-6627	430	8	batiha	batiha	PROPN
ejpam-6627	430	9	,	,	PUNCT
ejpam-6627	430	10	and	and	CCONJ
ejpam-6627	430	11	ansam	ansam	PROPN
ejpam-6627	430	12	a.	a.	PROPN
ejpam-6627	430	13	al	al	PROPN
ejpam-6627	430	14	-	-	PUNCT
ejpam-6627	430	15	smadi	smadi	NOUN
ejpam-6627	430	16	.	.	PUNCT
ejpam-6627	431	1	nearly	nearly	ADV
ejpam-6627	431	2	lindelöfness	lindelöfness	ADJ
ejpam-6627	431	3	in	in	ADP
ejpam-6627	431	4	bitopological	bitopological	ADJ
ejpam-6627	431	5	spaces	space	NOUN
ejpam-6627	431	6	.	.	PUNCT
ejpam-6627	432	1	south	south	PROPN
ejpam-6627	432	2	east	east	PROPN
ejpam-6627	432	3	asian	asian	PROPN
ejpam-6627	432	4	journal	journal	PROPN
ejpam-6627	432	5	of	of	ADP
ejpam-6627	432	6	mathematics	mathematics	PROPN
ejpam-6627	432	7	and	and	CCONJ
ejpam-6627	432	8	mathematical	mathematical	ADJ
ejpam-6627	432	9	sciences	science	NOUN
ejpam-6627	432	10	,	,	PUNCT
ejpam-6627	432	11	20(3):341–358	20(3):341–358	NOUN
ejpam-6627	432	12	,	,	PUNCT
ejpam-6627	432	13	2025	2025	NUM
ejpam-6627	432	14	.	.	PUNCT
ejpam-6627	433	1	[	[	X
ejpam-6627	433	2	6	6	NUM
ejpam-6627	433	3	]	]	ADJ
ejpam-6627	433	4	rehab	rehab	NOUN
ejpam-6627	433	5	alharbi	alharbi	NOUN
ejpam-6627	433	6	,	,	PUNCT
ejpam-6627	433	7	jamal	jamal	PROPN
ejpam-6627	433	8	oudetallah	oudetallah	PROPN
ejpam-6627	433	9	,	,	PUNCT
ejpam-6627	433	10	mutaz	mutaz	NOUN
ejpam-6627	433	11	shatnawi	shatnawi	NOUN
ejpam-6627	433	12	,	,	PUNCT
ejpam-6627	433	13	and	and	CCONJ
ejpam-6627	433	14	iqbal	iqbal	PROPN
ejpam-6627	433	15	m.	m.	PROPN
ejpam-6627	433	16	batiha	batiha	PROPN
ejpam-6627	433	17	.	.	PUNCT
ejpam-6627	434	1	on	on	ADP
ejpam-6627	434	2	c	c	NOUN
ejpam-6627	434	3	-	-	PUNCT
ejpam-6627	434	4	compactness	compactness	NOUN
ejpam-6627	434	5	in	in	ADP
ejpam-6627	434	6	topological	topological	ADJ
ejpam-6627	434	7	and	and	CCONJ
ejpam-6627	434	8	bitopological	bitopological	ADJ
ejpam-6627	434	9	spaces	space	NOUN
ejpam-6627	434	10	.	.	PUNCT
ejpam-6627	435	1	mathematics	mathematic	NOUN
ejpam-6627	435	2	,	,	PUNCT
ejpam-6627	435	3	11(20):4251	11(20):4251	NUM
ejpam-6627	435	4	,	,	PUNCT
ejpam-6627	435	5	2023	2023	NUM
ejpam-6627	435	6	.	.	PUNCT
ejpam-6627	436	1	[	[	X
ejpam-6627	436	2	7	7	NUM
ejpam-6627	436	3	]	]	X
ejpam-6627	436	4	jamal	jamal	PROPN
ejpam-6627	436	5	oudetallah	oudetallah	PROPN
ejpam-6627	436	6	,	,	PUNCT
ejpam-6627	436	7	nabeela	nabeela	PROPN
ejpam-6627	436	8	abu	abu	PROPN
ejpam-6627	436	9	-	-	PUNCT
ejpam-6627	436	10	alkishik	alkishik	PROPN
ejpam-6627	436	11	,	,	PUNCT
ejpam-6627	436	12	and	and	CCONJ
ejpam-6627	436	13	iqbal	iqbal	PROPN
ejpam-6627	436	14	m.	m.	PROPN
ejpam-6627	436	15	batiha	batiha	PROPN
ejpam-6627	436	16	.	.	PUNCT
ejpam-6627	437	1	nigh	nigh	ADV
ejpam-6627	437	2	-	-	PUNCT
ejpam-6627	437	3	open	open	ADJ
ejpam-6627	437	4	sets	set	NOUN
ejpam-6627	437	5	in	in	ADP
ejpam-6627	437	6	topological	topological	ADJ
ejpam-6627	437	7	space	space	NOUN
ejpam-6627	437	8	.	.	PUNCT
ejpam-6627	438	1	international	international	ADJ
ejpam-6627	438	2	journal	journal	NOUN
ejpam-6627	438	3	of	of	ADP
ejpam-6627	438	4	analysis	analysis	NOUN
ejpam-6627	438	5	and	and	CCONJ
ejpam-6627	438	6	applications	application	NOUN
ejpam-6627	438	7	,	,	PUNCT
ejpam-6627	438	8	21(1):83	21(1):83	NUM
ejpam-6627	438	9	,	,	PUNCT
ejpam-6627	438	10	2023	2023	NUM
ejpam-6627	438	11	.	.	PUNCT
ejpam-6627	439	1	[	[	X
ejpam-6627	439	2	8	8	NUM
ejpam-6627	439	3	]	]	X
ejpam-6627	439	4	r.	r.	PROPN
ejpam-6627	439	5	engelking	engelke	VERB
ejpam-6627	439	6	.	.	PUNCT
ejpam-6627	440	1	general	general	ADJ
ejpam-6627	440	2	topology	topology	PROPN
ejpam-6627	440	3	.	.	PUNCT
ejpam-6627	441	1	heldermann	heldermann	PROPN
ejpam-6627	441	2	verlag	verlag	PROPN
ejpam-6627	441	3	,	,	PUNCT
ejpam-6627	441	4	berlin	berlin	PROPN
ejpam-6627	441	5	,	,	PUNCT
ejpam-6627	441	6	1989	1989	NUM
ejpam-6627	441	7	.	.	PUNCT
ejpam-6627	442	1	[	[	X
ejpam-6627	442	2	9	9	NUM
ejpam-6627	442	3	]	]	PUNCT
ejpam-6627	442	4	s.	s.	PROPN
ejpam-6627	442	5	willard	willard	PROPN
ejpam-6627	442	6	.	.	PUNCT
ejpam-6627	442	7	general	general	ADJ
ejpam-6627	442	8	topology	topology	PROPN
ejpam-6627	442	9	.	.	PUNCT
ejpam-6627	443	1	addison	addison	PROPN
ejpam-6627	443	2	–	–	PUNCT
ejpam-6627	443	3	wesley	wesley	PROPN
ejpam-6627	443	4	,	,	PUNCT
ejpam-6627	443	5	reading	reading	NOUN
ejpam-6627	443	6	,	,	PUNCT
ejpam-6627	443	7	ma	ma	PROPN
ejpam-6627	443	8	,	,	PUNCT
ejpam-6627	443	9	1970	1970	NUM
ejpam-6627	443	10	.	.	PUNCT
ejpam-6627	444	1	[	[	X
ejpam-6627	444	2	10	10	NUM
ejpam-6627	444	3	]	]	X
ejpam-6627	444	4	j.	j.	PROPN
ejpam-6627	444	5	oudetallah	oudetallah	PROPN
ejpam-6627	444	6	and	and	CCONJ
ejpam-6627	444	7	m.	m.	PROPN
ejpam-6627	444	8	al	al	PROPN
ejpam-6627	444	9	-	-	PUNCT
ejpam-6627	444	10	hawari	hawari	PROPN
ejpam-6627	444	11	.	.	PUNCT
ejpam-6627	445	1	other	other	ADJ
ejpam-6627	445	2	generalization	generalization	NOUN
ejpam-6627	445	3	of	of	ADP
ejpam-6627	445	4	pairwise	pairwise	NOUN
ejpam-6627	445	5	expandable	expandable	ADJ
ejpam-6627	445	6	spaces	space	NOUN
ejpam-6627	445	7	.	.	PUNCT
ejpam-6627	446	1	international	international	ADJ
ejpam-6627	446	2	mathematical	mathematical	PROPN
ejpam-6627	446	3	forum	forum	PROPN
ejpam-6627	446	4	,	,	PUNCT
ejpam-6627	446	5	16(1):1–9	16(1):1–9	NUM
ejpam-6627	446	6	,	,	PUNCT
ejpam-6627	446	7	2021	2021	NUM
ejpam-6627	446	8	.	.	PUNCT
ejpam-6627	447	1	[	[	X
ejpam-6627	447	2	11	11	NUM
ejpam-6627	447	3	]	]	PUNCT
ejpam-6627	447	4	j.	j.	PROPN
ejpam-6627	447	5	oudetallah	oudetallah	PROPN
ejpam-6627	447	6	,	,	PUNCT
ejpam-6627	447	7	r.	r.	PROPN
ejpam-6627	447	8	alharbi	alharbi	PROPN
ejpam-6627	447	9	,	,	PUNCT
ejpam-6627	447	10	and	and	CCONJ
ejpam-6627	447	11	i.	i.	PROPN
ejpam-6627	447	12	m.	m.	PROPN
ejpam-6627	447	13	batiha	batiha	PROPN
ejpam-6627	447	14	.	.	PUNCT
ejpam-6627	448	1	on	on	ADP
ejpam-6627	448	2	r	r	NOUN
ejpam-6627	448	3	-	-	PUNCT
ejpam-6627	448	4	compactness	compactness	NOUN
ejpam-6627	448	5	in	in	ADP
ejpam-6627	448	6	topological	topological	ADJ
ejpam-6627	448	7	and	and	CCONJ
ejpam-6627	448	8	bitopological	bitopological	ADJ
ejpam-6627	448	9	spaces	space	NOUN
ejpam-6627	448	10	.	.	PUNCT
ejpam-6627	449	1	axioms	axiom	NOUN
ejpam-6627	449	2	,	,	PUNCT
ejpam-6627	449	3	12(2):210	12(2):210	NOUN
ejpam-6627	449	4	,	,	PUNCT
ejpam-6627	449	5	2023	2023	NUM
ejpam-6627	449	6	.	.	PUNCT
ejpam-6627	450	1	[	[	X
ejpam-6627	450	2	12	12	NUM
ejpam-6627	450	3	]	]	PUNCT
ejpam-6627	450	4	j.	j.	PROPN
ejpam-6627	450	5	oudetallah	oudetallah	PROPN
ejpam-6627	450	6	,	,	PUNCT
ejpam-6627	450	7	m.	m.	NOUN
ejpam-6627	450	8	m.	m.	NOUN
ejpam-6627	450	9	rousan	rousan	PROPN
ejpam-6627	450	10	,	,	PUNCT
ejpam-6627	450	11	and	and	CCONJ
ejpam-6627	450	12	i.	i.	PROPN
ejpam-6627	450	13	m.	m.	PROPN
ejpam-6627	450	14	batiha	batiha	PROPN
ejpam-6627	450	15	.	.	PUNCT
ejpam-6627	451	1	on	on	ADP
ejpam-6627	451	2	d	d	X
ejpam-6627	451	3	-	-	NOUN
ejpam-6627	451	4	metacompactness	metacompactness	NOUN
ejpam-6627	451	5	in	in	ADP
ejpam-6627	451	6	topological	topological	ADJ
ejpam-6627	451	7	spaces	space	NOUN
ejpam-6627	451	8	.	.	PUNCT
ejpam-6627	452	1	topology	topology	NOUN
ejpam-6627	452	2	appl	appl	PROPN
ejpam-6627	452	3	.	.	PROPN
ejpam-6627	452	4	,	,	PUNCT
ejpam-6627	452	5	298:107–125	298:107–125	NUM
ejpam-6627	452	6	,	,	PUNCT
ejpam-6627	452	7	2021	2021	NUM
ejpam-6627	452	8	.	.	PUNCT
ejpam-6627	453	1	[	[	X
ejpam-6627	453	2	13	13	NUM
ejpam-6627	453	3	]	]	SYM
ejpam-6627	453	4	g.	g.	NOUN
ejpam-6627	453	5	gierz	gierz	PROPN
ejpam-6627	453	6	,	,	PUNCT
ejpam-6627	453	7	k.	k.	PROPN
ejpam-6627	453	8	h.	h.	PROPN
ejpam-6627	453	9	hofmann	hofmann	PROPN
ejpam-6627	453	10	,	,	PUNCT
ejpam-6627	453	11	k.	k.	PROPN
ejpam-6627	453	12	keimel	keimel	PROPN
ejpam-6627	453	13	,	,	PUNCT
ejpam-6627	453	14	j.	j.	PROPN
ejpam-6627	453	15	d.	d.	PROPN
ejpam-6627	453	16	lawson	lawson	PROPN
ejpam-6627	453	17	,	,	PUNCT
ejpam-6627	453	18	m.	m.	NOUN
ejpam-6627	453	19	mislove	mislove	NOUN
ejpam-6627	453	20	,	,	PUNCT
ejpam-6627	453	21	and	and	CCONJ
ejpam-6627	453	22	d.	d.	PROPN
ejpam-6627	453	23	s.	s.	PROPN
ejpam-6627	453	24	scott	scott	PROPN
ejpam-6627	453	25	.	.	PUNCT
ejpam-6627	454	1	continuous	continuous	ADJ
ejpam-6627	454	2	lattices	lattice	NOUN
ejpam-6627	454	3	and	and	CCONJ
ejpam-6627	454	4	domains	domain	NOUN
ejpam-6627	454	5	.	.	PUNCT
ejpam-6627	455	1	cambridge	cambridge	PROPN
ejpam-6627	455	2	university	university	PROPN
ejpam-6627	455	3	press	press	PROPN
ejpam-6627	455	4	,	,	PUNCT
ejpam-6627	455	5	cambridge	cambridge	PROPN
ejpam-6627	455	6	,	,	PUNCT
ejpam-6627	455	7	2003	2003	NUM
ejpam-6627	455	8	.	.	PUNCT
ejpam-6627	456	1	[	[	X
ejpam-6627	456	2	14	14	NUM
ejpam-6627	456	3	]	]	X
ejpam-6627	456	4	h.	h.	PROPN
ejpam-6627	456	5	h.	h.	PROPN
ejpam-6627	456	6	schaefer	schaefer	PROPN
ejpam-6627	456	7	and	and	CCONJ
ejpam-6627	456	8	m.	m.	NOUN
ejpam-6627	456	9	p.	p.	NOUN
ejpam-6627	456	10	wolff	wolff	PROPN
ejpam-6627	456	11	.	.	PUNCT
ejpam-6627	457	1	topological	topological	ADJ
ejpam-6627	457	2	vector	vector	NOUN
ejpam-6627	457	3	spaces	space	NOUN
ejpam-6627	457	4	.	.	PUNCT
ejpam-6627	458	1	springer	springer	NOUN
ejpam-6627	458	2	-	-	PUNCT
ejpam-6627	458	3	verlag	verlag	PROPN
ejpam-6627	458	4	,	,	PUNCT
ejpam-6627	458	5	new	new	PROPN
ejpam-6627	458	6	york	york	PROPN
ejpam-6627	458	7	,	,	PUNCT
ejpam-6627	458	8	2nd	2nd	PROPN
ejpam-6627	458	9	edition	edition	NOUN
ejpam-6627	458	10	,	,	PUNCT
ejpam-6627	458	11	1999	1999	NUM
ejpam-6627	458	12	.	.	PUNCT
ejpam-6627	459	1	[	[	X
ejpam-6627	459	2	15	15	NUM
ejpam-6627	459	3	]	]	X
ejpam-6627	459	4	r.	r.	PROPN
ejpam-6627	459	5	hartshorne	hartshorne	PROPN
ejpam-6627	459	6	.	.	PUNCT
ejpam-6627	460	1	algebraic	algebraic	ADJ
ejpam-6627	460	2	geometry	geometry	NOUN
ejpam-6627	460	3	.	.	PUNCT
ejpam-6627	461	1	springer	springer	NOUN
ejpam-6627	461	2	-	-	PUNCT
ejpam-6627	461	3	verlag	verlag	PROPN
ejpam-6627	461	4	,	,	PUNCT
ejpam-6627	461	5	new	new	PROPN
ejpam-6627	461	6	york	york	PROPN
ejpam-6627	461	7	,	,	PUNCT
ejpam-6627	461	8	1977	1977	NUM
ejpam-6627	461	9	.	.	PUNCT
ejpam-6627	462	1	[	[	X
ejpam-6627	462	2	16	16	NUM
ejpam-6627	462	3	]	]	X
ejpam-6627	462	4	j.	j.	PROPN
ejpam-6627	462	5	l.	l.	PROPN
ejpam-6627	462	6	kelley	kelley	PROPN
ejpam-6627	462	7	.	.	PUNCT
ejpam-6627	463	1	general	general	ADJ
ejpam-6627	463	2	topology	topology	PROPN
ejpam-6627	463	3	.	.	PUNCT
ejpam-6627	464	1	van	van	PROPN
ejpam-6627	464	2	nostrand	nostrand	PROPN
ejpam-6627	464	3	,	,	PUNCT
ejpam-6627	464	4	princeton	princeton	PROPN
ejpam-6627	464	5	,	,	PUNCT
ejpam-6627	464	6	1955	1955	NUM
ejpam-6627	464	7	.	.	PUNCT
ejpam-6627	465	1	[	[	X
ejpam-6627	465	2	17	17	NUM
ejpam-6627	465	3	]	]	PUNCT
ejpam-6627	465	4	j.	j.	PROPN
ejpam-6627	465	5	r.	r.	PROPN
ejpam-6627	465	6	munkres	munkres	PROPN
ejpam-6627	465	7	.	.	PUNCT
ejpam-6627	466	1	topology	topology	NOUN
ejpam-6627	466	2	.	.	PUNCT
ejpam-6627	467	1	prentice	prentice	PROPN
ejpam-6627	467	2	hall	hall	PROPN
ejpam-6627	467	3	,	,	PUNCT
ejpam-6627	467	4	upper	upper	ADJ
ejpam-6627	467	5	saddle	saddle	NOUN
ejpam-6627	467	6	river	river	PROPN
ejpam-6627	467	7	,	,	PUNCT
ejpam-6627	467	8	nj	nj	PROPN
ejpam-6627	467	9	,	,	PUNCT
ejpam-6627	467	10	2nd	2nd	PROPN
ejpam-6627	467	11	edition	edition	NOUN
ejpam-6627	467	12	,	,	PUNCT
ejpam-6627	467	13	2000	2000	NUM
ejpam-6627	467	14	.	.	PUNCT
ejpam-6627	468	1	[	[	X
ejpam-6627	468	2	18	18	NUM
ejpam-6627	468	3	]	]	X
ejpam-6627	468	4	j.	j.	PROPN
ejpam-6627	468	5	dugundji	dugundji	PROPN
ejpam-6627	468	6	.	.	PUNCT
ejpam-6627	469	1	topology	topology	PROPN
ejpam-6627	469	2	.	.	PUNCT
ejpam-6627	470	1	allyn	allyn	PROPN
ejpam-6627	470	2	and	and	CCONJ
ejpam-6627	470	3	bacon	bacon	PROPN
ejpam-6627	470	4	,	,	PUNCT
ejpam-6627	470	5	boston	boston	PROPN
ejpam-6627	470	6	,	,	PUNCT
ejpam-6627	470	7	1966	1966	NUM
ejpam-6627	470	8	.	.	PUNCT
ejpam-6627	471	1	[	[	X
ejpam-6627	471	2	19	19	NUM
ejpam-6627	471	3	]	]	X
ejpam-6627	471	4	n.	n.	NOUN
ejpam-6627	471	5	bourbaki	bourbaki	PROPN
ejpam-6627	471	6	.	.	PUNCT
ejpam-6627	472	1	general	general	ADJ
ejpam-6627	472	2	topology	topology	NOUN
ejpam-6627	472	3	,	,	PUNCT
ejpam-6627	472	4	parts	part	NOUN
ejpam-6627	472	5	1	1	NUM
ejpam-6627	472	6	and	and	CCONJ
ejpam-6627	472	7	2	2	NUM
ejpam-6627	472	8	.	.	X
ejpam-6627	472	9	springer	springer	NOUN
ejpam-6627	472	10	-	-	PUNCT
ejpam-6627	472	11	verlag	verlag	PROPN
ejpam-6627	472	12	,	,	PUNCT
ejpam-6627	472	13	berlin	berlin	PROPN
ejpam-6627	472	14	,	,	PUNCT
ejpam-6627	472	15	1989	1989	NUM
ejpam-6627	472	16	.	.	PUNCT
ejpam-6627	473	1	j.	j.	PROPN
ejpam-6627	473	2	oudetallah	oudetallah	PROPN
ejpam-6627	473	3	et	et	PROPN
ejpam-6627	473	4	al	al	PROPN
ejpam-6627	473	5	.	.	PUNCT
ejpam-6627	473	6	/	/	SYM
ejpam-6627	473	7	eur	eur	PROPN
ejpam-6627	473	8	.	.	PUNCT
ejpam-6627	474	1	j.	j.	PROPN
ejpam-6627	474	2	pure	pure	PROPN
ejpam-6627	474	3	appl	appl	PROPN
ejpam-6627	474	4	.	.	PROPN
ejpam-6627	474	5	math	math	PROPN
ejpam-6627	474	6	,	,	PUNCT
ejpam-6627	474	7	18	18	NUM
ejpam-6627	474	8	(	(	PUNCT
ejpam-6627	474	9	3	3	NUM
ejpam-6627	474	10	)	)	PUNCT
ejpam-6627	474	11	(	(	PUNCT
ejpam-6627	474	12	2025	2025	NUM
ejpam-6627	474	13	)	)	PUNCT
ejpam-6627	474	14	,	,	PUNCT
ejpam-6627	474	15	6627	6627	NUM
ejpam-6627	474	16	20	20	NUM
ejpam-6627	474	17	of	of	ADP
ejpam-6627	474	18	20	20	NUM
ejpam-6627	475	1	[	[	SYM
ejpam-6627	475	2	20	20	NUM
ejpam-6627	475	3	]	]	PUNCT
ejpam-6627	475	4	j.	j.	PROPN
ejpam-6627	475	5	c.	c.	PROPN
ejpam-6627	475	6	oxtoby	oxtoby	PROPN
ejpam-6627	475	7	.	.	PUNCT
ejpam-6627	476	1	measure	measure	NOUN
ejpam-6627	476	2	and	and	CCONJ
ejpam-6627	476	3	category	category	NOUN
ejpam-6627	476	4	.	.	PUNCT
ejpam-6627	477	1	springer	springer	NOUN
ejpam-6627	477	2	-	-	PUNCT
ejpam-6627	477	3	verlag	verlag	PROPN
ejpam-6627	477	4	,	,	PUNCT
ejpam-6627	477	5	new	new	PROPN
ejpam-6627	477	6	york	york	PROPN
ejpam-6627	477	7	,	,	PUNCT
ejpam-6627	477	8	2nd	2nd	PROPN
ejpam-6627	477	9	edition	edition	NOUN
ejpam-6627	477	10	,	,	PUNCT
ejpam-6627	477	11	1980	1980	NUM
ejpam-6627	477	12	.	.	PUNCT
ejpam-6627	478	1	[	[	X
ejpam-6627	478	2	21	21	NUM
ejpam-6627	478	3	]	]	X
ejpam-6627	478	4	r.	r.	PROPN
ejpam-6627	478	5	a.	a.	PROPN
ejpam-6627	478	6	mccoy	mccoy	PROPN
ejpam-6627	478	7	and	and	CCONJ
ejpam-6627	478	8	i.	i.	PROPN
ejpam-6627	478	9	ntantu	ntantu	PROPN
ejpam-6627	478	10	.	.	PUNCT
ejpam-6627	479	1	topological	topological	ADJ
ejpam-6627	479	2	properties	property	NOUN
ejpam-6627	479	3	of	of	ADP
ejpam-6627	479	4	spaces	space	NOUN
ejpam-6627	479	5	of	of	ADP
ejpam-6627	479	6	continuous	continuous	ADJ
ejpam-6627	479	7	functions	function	NOUN
ejpam-6627	479	8	.	.	PUNCT
ejpam-6627	479	9	,	,	PUNCT
ejpam-6627	479	10	volume	volume	NOUN
ejpam-6627	479	11	1315	1315	NUM
ejpam-6627	479	12	of	of	ADP
ejpam-6627	479	13	lecture	lecture	NOUN
ejpam-6627	479	14	notes	note	NOUN
ejpam-6627	479	15	in	in	ADP
ejpam-6627	479	16	mathematics	mathematic	NOUN
ejpam-6627	479	17	.	.	PUNCT
ejpam-6627	480	1	springer	springer	NOUN
ejpam-6627	480	2	-	-	PUNCT
ejpam-6627	480	3	verlag	verlag	PROPN
ejpam-6627	480	4	,	,	PUNCT
ejpam-6627	480	5	berlin	berlin	PROPN
ejpam-6627	480	6	,	,	PUNCT
ejpam-6627	480	7	1988	1988	NUM
ejpam-6627	480	8	.	.	PUNCT
ejpam-6627	481	1	[	[	X
ejpam-6627	481	2	22	22	NUM
ejpam-6627	481	3	]	]	PUNCT
ejpam-6627	481	4	r.	r.	PROPN
ejpam-6627	481	5	arens	arens	PROPN
ejpam-6627	481	6	and	and	CCONJ
ejpam-6627	481	7	j.	j.	PROPN
ejpam-6627	481	8	dugundji	dugundji	PROPN
ejpam-6627	481	9	.	.	PUNCT
ejpam-6627	482	1	topologies	topology	NOUN
ejpam-6627	482	2	for	for	ADP
ejpam-6627	482	3	function	function	NOUN
ejpam-6627	482	4	spaces	space	NOUN
ejpam-6627	482	5	.	.	PUNCT
ejpam-6627	483	1	pacific	pacific	PROPN
ejpam-6627	483	2	j.	j.	PROPN
ejpam-6627	483	3	math	math	PROPN
ejpam-6627	483	4	.	.	PUNCT
ejpam-6627	483	5	,	,	PUNCT
ejpam-6627	483	6	1:5–31	1:5–31	NUM
ejpam-6627	483	7	,	,	PUNCT
ejpam-6627	483	8	1951	1951	NUM
ejpam-6627	483	9	.	.	PUNCT
ejpam-6627	484	1	[	[	X
ejpam-6627	484	2	23	23	NUM
ejpam-6627	484	3	]	]	PUNCT
ejpam-6627	484	4	l.	l.	PROPN
ejpam-6627	484	5	gillman	gillman	PROPN
ejpam-6627	484	6	and	and	CCONJ
ejpam-6627	484	7	m.	m.	PROPN
ejpam-6627	484	8	jerison	jerison	PROPN
ejpam-6627	484	9	.	.	PUNCT
ejpam-6627	485	1	rings	ring	NOUN
ejpam-6627	485	2	of	of	ADP
ejpam-6627	485	3	continuous	continuous	ADJ
ejpam-6627	485	4	functions	function	NOUN
ejpam-6627	485	5	.	.	PUNCT
ejpam-6627	486	1	springer	springer	NOUN
ejpam-6627	486	2	-	-	PUNCT
ejpam-6627	486	3	verlag	verlag	PROPN
ejpam-6627	486	4	,	,	PUNCT
ejpam-6627	486	5	new	new	PROPN
ejpam-6627	486	6	york	york	PROPN
ejpam-6627	486	7	,	,	PUNCT
ejpam-6627	486	8	1976	1976	NUM
ejpam-6627	486	9	.	.	PUNCT
ejpam-6627	487	1	[	[	X
ejpam-6627	487	2	24	24	NUM
ejpam-6627	487	3	]	]	X
ejpam-6627	487	4	l.	l.	PROPN
ejpam-6627	487	5	a.	a.	PROPN
ejpam-6627	487	6	steen	steen	PROPN
ejpam-6627	487	7	and	and	CCONJ
ejpam-6627	487	8	jr	jr	PROPN
ejpam-6627	487	9	.	.	PUNCT
ejpam-6627	487	10	j.	j.	PROPN
ejpam-6627	487	11	a.	a.	PROPN
ejpam-6627	487	12	seebach	seebach	PROPN
ejpam-6627	487	13	.	.	PUNCT
ejpam-6627	488	1	counterexamples	counterexample	NOUN
ejpam-6627	488	2	in	in	ADP
ejpam-6627	488	3	topology	topology	NOUN
ejpam-6627	488	4	.	.	PUNCT
ejpam-6627	489	1	dover	dover	PROPN
ejpam-6627	489	2	publications	publication	NOUN
ejpam-6627	489	3	,	,	PUNCT
ejpam-6627	489	4	new	new	PROPN
ejpam-6627	489	5	york	york	PROPN
ejpam-6627	489	6	,	,	PUNCT
ejpam-6627	489	7	2nd	2nd	PROPN
ejpam-6627	489	8	edition	edition	NOUN
ejpam-6627	489	9	,	,	PUNCT
ejpam-6627	489	10	1995	1995	NUM
ejpam-6627	489	11	.	.	PUNCT
ejpam-6627	490	1	[	[	X
ejpam-6627	490	2	25	25	NUM
ejpam-6627	490	3	]	]	PUNCT
ejpam-6627	490	4	p.	p.	NOUN
ejpam-6627	490	5	urysohn	urysohn	PROPN
ejpam-6627	490	6	.	.	PUNCT
ejpam-6627	491	1	über	über	PROPN
ejpam-6627	491	2	die	die	VERB
ejpam-6627	491	3	mächtigkeit	mächtigkeit	PROPN
ejpam-6627	491	4	der	der	PROPN
ejpam-6627	491	5	zusammenhängenden	zusammenhängenden	NUM
ejpam-6627	491	6	mengen	mengen	PROPN
ejpam-6627	491	7	.	.	PROPN
ejpam-6627	491	8	math	math	PROPN
ejpam-6627	491	9	.	.	PUNCT
ejpam-6627	492	1	ann	ann	PROPN
ejpam-6627	492	2	.	.	PROPN
ejpam-6627	492	3	,	,	PUNCT
ejpam-6627	492	4	94:262–295	94:262–295	NUM
ejpam-6627	492	5	,	,	PUNCT
ejpam-6627	492	6	1925	1925	NUM
ejpam-6627	492	7	.	.	PUNCT
ejpam-6627	493	1	[	[	X
ejpam-6627	493	2	26	26	NUM
ejpam-6627	493	3	]	]	X
ejpam-6627	493	4	j.	j.	PROPN
ejpam-6627	493	5	oudetallah	oudetallah	PROPN
ejpam-6627	493	6	and	and	CCONJ
ejpam-6627	493	7	l.	l.	PROPN
ejpam-6627	493	8	abualigah	abualigah	PROPN
ejpam-6627	493	9	.	.	PUNCT
ejpam-6627	494	1	h	h	NOUN
ejpam-6627	494	2	-	-	PUNCT
ejpam-6627	494	3	convexity	convexity	NOUN
ejpam-6627	494	4	in	in	ADP
ejpam-6627	494	5	metric	metric	ADJ
ejpam-6627	494	6	linear	linear	ADJ
ejpam-6627	494	7	spaces	space	NOUN
ejpam-6627	494	8	.	.	PUNCT
ejpam-6627	495	1	international	international	ADJ
ejpam-6627	495	2	journal	journal	PROPN
ejpam-6627	495	3	of	of	ADP
ejpam-6627	495	4	science	science	NOUN
ejpam-6627	495	5	and	and	CCONJ
ejpam-6627	495	6	advanced	advanced	ADJ
ejpam-6627	495	7	information	information	NOUN
ejpam-6627	495	8	technology	technology	NOUN
ejpam-6627	495	9	,	,	PUNCT
ejpam-6627	495	10	8(6):54–58	8(6):54–58	NUM
ejpam-6627	495	11	,	,	PUNCT
ejpam-6627	495	12	2019	2019	NUM
ejpam-6627	495	13	.	.	PUNCT
ejpam-6627	496	1	[	[	X
ejpam-6627	496	2	27	27	NUM
ejpam-6627	496	3	]	]	X
ejpam-6627	496	4	h.	h.	PROPN
ejpam-6627	496	5	tietze	tietze	PROPN
ejpam-6627	496	6	.	.	PUNCT
ejpam-6627	497	1	über	über	PROPN
ejpam-6627	497	2	funktionen	funktionen	PROPN
ejpam-6627	497	3	,	,	PUNCT
ejpam-6627	497	4	die	die	VERB
ejpam-6627	497	5	auf	auf	PROPN
ejpam-6627	497	6	einer	einer	PROPN
ejpam-6627	497	7	abgeschlossenen	abgeschlossenen	PROPN
ejpam-6627	497	8	menge	menge	PROPN
ejpam-6627	497	9	stetig	stetig	NOUN
ejpam-6627	497	10	sind	sind	NOUN
ejpam-6627	497	11	.	.	PUNCT
ejpam-6627	498	1	j.	j.	PROPN
ejpam-6627	498	2	reine	reine	PROPN
ejpam-6627	498	3	angew	angew	PROPN
ejpam-6627	498	4	.	.	PUNCT
ejpam-6627	499	1	math	math	NOUN
ejpam-6627	499	2	.	.	PUNCT
ejpam-6627	499	3	,	,	PUNCT
ejpam-6627	499	4	145:9–14	145:9–14	NUM
ejpam-6627	499	5	,	,	PUNCT
ejpam-6627	499	6	1915	1915	NUM
ejpam-6627	499	7	.	.	PUNCT
ejpam-6627	500	1	[	[	X
ejpam-6627	500	2	28	28	NUM
ejpam-6627	500	3	]	]	X
ejpam-6627	500	4	k.	k.	PROPN
ejpam-6627	500	5	kunen	kunen	PROPN
ejpam-6627	500	6	.	.	PUNCT
ejpam-6627	501	1	set	set	PROPN
ejpam-6627	501	2	theory	theory	NOUN
ejpam-6627	501	3	:	:	PUNCT
ejpam-6627	501	4	an	an	DET
ejpam-6627	501	5	introduction	introduction	NOUN
ejpam-6627	501	6	to	to	ADP
ejpam-6627	501	7	independence	independence	NOUN
ejpam-6627	501	8	proofs	proof	NOUN
ejpam-6627	501	9	.	.	PUNCT
ejpam-6627	502	1	north	north	NOUN
ejpam-6627	502	2	-	-	PUNCT
ejpam-6627	502	3	holland	holland	PROPN
ejpam-6627	502	4	,	,	PUNCT
ejpam-6627	502	5	amsterdam	amsterdam	PROPN
ejpam-6627	502	6	,	,	PUNCT
ejpam-6627	502	7	1980	1980	NUM
ejpam-6627	502	8	.	.	PUNCT
ejpam-6627	503	1	[	[	X
ejpam-6627	503	2	29	29	NUM
ejpam-6627	503	3	]	]	X
ejpam-6627	503	4	a.	a.	NOUN
ejpam-6627	503	5	v.	v.	ADP
ejpam-6627	503	6	arhangel’skii	arhangel’skii	PROPN
ejpam-6627	503	7	.	.	PUNCT
ejpam-6627	504	1	quotient	quotient	NOUN
ejpam-6627	504	2	spaces	space	NOUN
ejpam-6627	504	3	and	and	CCONJ
ejpam-6627	504	4	multiplicity	multiplicity	NOUN
ejpam-6627	504	5	of	of	ADP
ejpam-6627	504	6	a	a	DET
ejpam-6627	504	7	base	base	NOUN
ejpam-6627	504	8	.	.	PUNCT
ejpam-6627	505	1	soviet	soviet	ADJ
ejpam-6627	505	2	math	math	NOUN
ejpam-6627	505	3	.	.	PUNCT
ejpam-6627	506	1	dokl	dokl	NOUN
ejpam-6627	506	2	.	.	PUNCT
ejpam-6627	506	3	,	,	PUNCT
ejpam-6627	506	4	7:244–247	7:244–247	PROPN
ejpam-6627	506	5	,	,	PUNCT
ejpam-6627	506	6	1966	1966	NUM
ejpam-6627	506	7	.	.	PUNCT
ejpam-6627	507	1	[	[	X
ejpam-6627	507	2	30	30	NUM
ejpam-6627	507	3	]	]	PUNCT
ejpam-6627	507	4	a.	a.	NOUN
ejpam-6627	507	5	tychonoff	tychonoff	NOUN
ejpam-6627	507	6	.	.	PUNCT
ejpam-6627	508	1	über	über	PROPN
ejpam-6627	508	2	die	die	VERB
ejpam-6627	508	3	topologische	topologische	PROPN
ejpam-6627	508	4	erweiterung	erweiterung	PROPN
ejpam-6627	508	5	von	von	PROPN
ejpam-6627	508	6	räumen	räumen	PROPN
ejpam-6627	508	7	.	.	PUNCT
ejpam-6627	509	1	math	math	NOUN
ejpam-6627	509	2	.	.	PUNCT
ejpam-6627	510	1	ann	ann	PROPN
ejpam-6627	510	2	.	.	PROPN
ejpam-6627	510	3	,	,	PUNCT
ejpam-6627	510	4	102:544	102:544	NUM
ejpam-6627	510	5	–	–	PUNCT
ejpam-6627	510	6	561	561	NUM
ejpam-6627	510	7	,	,	PUNCT
ejpam-6627	510	8	1930	1930	NUM
ejpam-6627	510	9	.	.	PUNCT
ejpam-6627	511	1	[	[	X
ejpam-6627	511	2	31	31	NUM
ejpam-6627	511	3	]	]	PUNCT
ejpam-6627	511	4	j.	j.	PROPN
ejpam-6627	511	5	oudetallah	oudetallah	PROPN
ejpam-6627	511	6	.	.	PUNCT
ejpam-6627	512	1	novel	novel	PROPN
ejpam-6627	512	2	results	result	NOUN
ejpam-6627	512	3	on	on	ADP
ejpam-6627	512	4	nigh	nigh	ADV
ejpam-6627	512	5	lindelöfness	lindelöfness	PUNCT
ejpam-6627	512	6	in	in	ADP
ejpam-6627	512	7	topological	topological	ADJ
ejpam-6627	512	8	spaces	space	NOUN
ejpam-6627	512	9	.	.	PUNCT
ejpam-6627	513	1	european	european	PROPN
ejpam-6627	513	2	j.	j.	PROPN
ejpam-6627	513	3	pure	pure	PROPN
ejpam-6627	513	4	appl	appl	PROPN
ejpam-6627	513	5	.	.	PUNCT
ejpam-6627	513	6	math	math	PROPN
ejpam-6627	513	7	.	.	PUNCT
ejpam-6627	513	8	,	,	PUNCT
ejpam-6627	513	9	17:234–251	17:234–251	PROPN
ejpam-6627	513	10	,	,	PUNCT
ejpam-6627	513	11	2024	2024	NUM
ejpam-6627	513	12	.	.	PUNCT
ejpam-6627	514	1	[	[	X
ejpam-6627	514	2	32	32	NUM
ejpam-6627	514	3	]	]	PUNCT
ejpam-6627	514	4	w.	w.	PROPN
ejpam-6627	514	5	w.	w.	PROPN
ejpam-6627	514	6	comfort	comfort	PROPN
ejpam-6627	514	7	and	and	CCONJ
ejpam-6627	514	8	s.	s.	PROPN
ejpam-6627	514	9	negrepontis	negrepontis	PROPN
ejpam-6627	514	10	.	.	PUNCT
ejpam-6627	515	1	the	the	DET
ejpam-6627	515	2	theory	theory	NOUN
ejpam-6627	515	3	of	of	ADP
ejpam-6627	515	4	ultrafilters	ultrafilter	NOUN
ejpam-6627	515	5	.	.	PUNCT
ejpam-6627	516	1	springer	springer	NOUN
ejpam-6627	516	2	-	-	PUNCT
ejpam-6627	516	3	verlag	verlag	PROPN
ejpam-6627	516	4	,	,	PUNCT
ejpam-6627	516	5	berlin	berlin	PROPN
ejpam-6627	516	6	,	,	PUNCT
ejpam-6627	516	7	1974	1974	NUM
ejpam-6627	516	8	.	.	PUNCT
ejpam-6627	517	1	introduction	introduction	NOUN
ejpam-6627	517	2	motivation	motivation	NOUN
ejpam-6627	517	3	and	and	CCONJ
ejpam-6627	517	4	applications	application	NOUN
ejpam-6627	517	5	preliminaries	preliminary	NOUN
ejpam-6627	517	6	fréchet	fréchet	VERB
ejpam-6627	517	7	spaces	space	NOUN
ejpam-6627	517	8	with	with	ADP
ejpam-6627	517	9	closed	closed	ADJ
ejpam-6627	517	10	sets	set	NOUN
ejpam-6627	517	11	as	as	ADP
ejpam-6627	517	12	sigma	sigma	NOUN
ejpam-6627	517	13	-	-	PUNCT
ejpam-6627	517	14	intersections	intersection	NOUN
ejpam-6627	517	15	urysohn	urysohn	NOUN
ejpam-6627	517	16	spaces	space	NOUN
ejpam-6627	517	17	without	without	ADP
ejpam-6627	517	18	the	the	DET
ejpam-6627	517	19	hausdorff	hausdorff	PROPN
ejpam-6627	517	20	property	property	PROPN
ejpam-6627	517	21	intermediate	intermediate	ADJ
ejpam-6627	517	22	separation	separation	NOUN
ejpam-6627	517	23	criteria	criterion	NOUN
ejpam-6627	517	24	behavior	behavior	NOUN
ejpam-6627	517	25	under	under	ADP
ejpam-6627	517	26	topological	topological	ADJ
ejpam-6627	517	27	operations	operation	NOUN
ejpam-6627	517	28	critical	critical	ADJ
ejpam-6627	517	29	counterexamples	counterexample	NOUN
ejpam-6627	517	30	conclusions	conclusion	NOUN