id	sid	tid	token	lemma	pos
ejpam-6592	1	1	european	european	PROPN
ejpam-6592	1	2	journal	journal	PROPN
ejpam-6592	1	3	of	of	ADP
ejpam-6592	1	4	pure	pure	ADJ
ejpam-6592	1	5	and	and	CCONJ
ejpam-6592	1	6	applied	applied	ADJ
ejpam-6592	1	7	mathematics	mathematic	NOUN
ejpam-6592	1	8	2025	2025	NUM
ejpam-6592	1	9	,	,	PUNCT
ejpam-6592	1	10	vol	vol	NOUN
ejpam-6592	1	11	.	.	PROPN
ejpam-6592	1	12	18	18	NUM
ejpam-6592	1	13	,	,	PUNCT
ejpam-6592	1	14	issue	issue	NOUN
ejpam-6592	1	15	3	3	NUM
ejpam-6592	1	16	,	,	PUNCT
ejpam-6592	1	17	article	article	NOUN
ejpam-6592	1	18	number	number	NOUN
ejpam-6592	1	19	6592	6592	NUM
ejpam-6592	1	20	issn	issn	VERB
ejpam-6592	1	21	1307	1307	NUM
ejpam-6592	1	22	-	-	SYM
ejpam-6592	1	23	5543	5543	NUM
ejpam-6592	1	24	–	–	PUNCT
ejpam-6592	1	25	ejpam.com	ejpam.com	X
ejpam-6592	1	26	published	publish	VERB
ejpam-6592	1	27	by	by	ADP
ejpam-6592	1	28	new	new	PROPN
ejpam-6592	1	29	york	york	PROPN
ejpam-6592	1	30	business	business	PROPN
ejpam-6592	1	31	global	global	ADJ
ejpam-6592	1	32	compactness	compactness	NOUN
ejpam-6592	1	33	and	and	CCONJ
ejpam-6592	1	34	separability	separability	NOUN
ejpam-6592	1	35	in	in	ADP
ejpam-6592	1	36	mr	mr	PROPN
ejpam-6592	1	37	-	-	PUNCT
ejpam-6592	1	38	metric	metric	ADJ
ejpam-6592	1	39	spaces	space	NOUN
ejpam-6592	1	40	with	with	ADP
ejpam-6592	1	41	applications	application	NOUN
ejpam-6592	1	42	to	to	ADP
ejpam-6592	1	43	deep	deep	ADJ
ejpam-6592	1	44	learning	learning	NOUN
ejpam-6592	1	45	abed	abe	VERB
ejpam-6592	1	46	al	al	PROPN
ejpam-6592	1	47	-	-	PUNCT
ejpam-6592	1	48	rahman	rahman	PROPN
ejpam-6592	1	49	m.	m.	NOUN
ejpam-6592	1	50	malkawi	malkawi	ADP
ejpam-6592	1	51	department	department	PROPN
ejpam-6592	1	52	of	of	ADP
ejpam-6592	1	53	mathematics	mathematic	NOUN
ejpam-6592	1	54	,	,	PUNCT
ejpam-6592	1	55	faculty	faculty	NOUN
ejpam-6592	1	56	of	of	ADP
ejpam-6592	1	57	arts	art	NOUN
ejpam-6592	1	58	and	and	CCONJ
ejpam-6592	1	59	science	science	NOUN
ejpam-6592	1	60	,	,	PUNCT
ejpam-6592	1	61	amman	amman	PROPN
ejpam-6592	1	62	arab	arab	PROPN
ejpam-6592	1	63	university	university	PROPN
ejpam-6592	1	64	,	,	PUNCT
ejpam-6592	1	65	amman	amman	PROPN
ejpam-6592	1	66	11953	11953	NUM
ejpam-6592	1	67	,	,	PUNCT
ejpam-6592	1	68	jordan	jordan	PROPN
ejpam-6592	1	69	abstract	abstract	PROPN
ejpam-6592	1	70	.	.	PUNCT
ejpam-6592	2	1	this	this	DET
ejpam-6592	2	2	paper	paper	NOUN
ejpam-6592	2	3	establishes	establish	VERB
ejpam-6592	2	4	fundamental	fundamental	ADJ
ejpam-6592	2	5	topological	topological	ADJ
ejpam-6592	2	6	properties	property	NOUN
ejpam-6592	2	7	of	of	ADP
ejpam-6592	2	8	mr	mr	PROPN
ejpam-6592	2	9	-	-	PUNCT
ejpam-6592	2	10	metric	metric	ADJ
ejpam-6592	2	11	spaces	space	NOUN
ejpam-6592	2	12	,	,	PUNCT
ejpam-6592	2	13	a	a	DET
ejpam-6592	2	14	significant	significant	ADJ
ejpam-6592	2	15	generalization	generalization	NOUN
ejpam-6592	2	16	of	of	ADP
ejpam-6592	2	17	conventional	conventional	ADJ
ejpam-6592	2	18	metric	metric	ADJ
ejpam-6592	2	19	spaces	space	NOUN
ejpam-6592	2	20	characterized	characterize	VERB
ejpam-6592	2	21	by	by	ADP
ejpam-6592	2	22	an	an	DET
ejpam-6592	2	23	r	r	NOUN
ejpam-6592	2	24	-	-	PUNCT
ejpam-6592	2	25	scaled	scale	VERB
ejpam-6592	2	26	tetrahedral	tetrahedral	ADJ
ejpam-6592	2	27	inequality	inequality	NOUN
ejpam-6592	2	28	.	.	PUNCT
ejpam-6592	3	1	we	we	PRON
ejpam-6592	3	2	prove	prove	VERB
ejpam-6592	3	3	several	several	ADJ
ejpam-6592	3	4	key	key	ADJ
ejpam-6592	3	5	results	result	NOUN
ejpam-6592	3	6	including	include	VERB
ejpam-6592	3	7	:	:	PUNCT
ejpam-6592	3	8	(	(	PUNCT
ejpam-6592	3	9	1	1	X
ejpam-6592	3	10	)	)	PUNCT
ejpam-6592	3	11	a	a	DET
ejpam-6592	3	12	complete	complete	ADJ
ejpam-6592	3	13	characterization	characterization	NOUN
ejpam-6592	3	14	of	of	ADP
ejpam-6592	3	15	compactness	compactness	NOUN
ejpam-6592	3	16	through	through	ADP
ejpam-6592	3	17	three	three	NUM
ejpam-6592	3	18	equivalent	equivalent	ADJ
ejpam-6592	3	19	conditions	condition	NOUN
ejpam-6592	3	20	,	,	PUNCT
ejpam-6592	3	21	(	(	PUNCT
ejpam-6592	3	22	2	2	X
ejpam-6592	3	23	)	)	PUNCT
ejpam-6592	3	24	the	the	DET
ejpam-6592	3	25	lebesgue	lebesgue	ADJ
ejpam-6592	3	26	number	number	NOUN
ejpam-6592	3	27	lemma	lemma	PROPN
ejpam-6592	3	28	adaptation	adaptation	NOUN
ejpam-6592	3	29	,	,	PUNCT
ejpam-6592	3	30	(	(	PUNCT
ejpam-6592	3	31	3	3	X
ejpam-6592	3	32	)	)	PUNCT
ejpam-6592	3	33	equivalence	equivalence	NOUN
ejpam-6592	3	34	between	between	ADP
ejpam-6592	3	35	separability	separability	NOUN
ejpam-6592	3	36	and	and	CCONJ
ejpam-6592	3	37	the	the	DET
ejpam-6592	3	38	lindelöf	lindelöf	NOUN
ejpam-6592	3	39	property	property	NOUN
ejpam-6592	3	40	,	,	PUNCT
ejpam-6592	3	41	and	and	CCONJ
ejpam-6592	3	42	(	(	PUNCT
ejpam-6592	3	43	4	4	X
ejpam-6592	3	44	)	)	PUNCT
ejpam-6592	3	45	automatic	automatic	ADJ
ejpam-6592	3	46	paracompactness	paracompactness	NOUN
ejpam-6592	3	47	.	.	PUNCT
ejpam-6592	4	1	the	the	DET
ejpam-6592	4	2	theoretical	theoretical	ADJ
ejpam-6592	4	3	framework	framework	NOUN
ejpam-6592	4	4	is	be	AUX
ejpam-6592	4	5	applied	apply	VERB
ejpam-6592	4	6	to	to	ADP
ejpam-6592	4	7	four	four	NUM
ejpam-6592	4	8	domains	domain	NOUN
ejpam-6592	4	9	:	:	PUNCT
ejpam-6592	4	10	(	(	PUNCT
ejpam-6592	4	11	i	i	NOUN
ejpam-6592	4	12	)	)	PUNCT
ejpam-6592	4	13	global	global	ADJ
ejpam-6592	4	14	optimization	optimization	NOUN
ejpam-6592	4	15	in	in	ADP
ejpam-6592	4	16	euclidean	euclidean	ADJ
ejpam-6592	4	17	spaces	space	NOUN
ejpam-6592	4	18	,	,	PUNCT
ejpam-6592	4	19	(	(	PUNCT
ejpam-6592	4	20	ii	ii	NOUN
ejpam-6592	4	21	)	)	PUNCT
ejpam-6592	4	22	neural	neural	ADJ
ejpam-6592	4	23	network	network	NOUN
ejpam-6592	4	24	weight	weight	NOUN
ejpam-6592	4	25	space	space	NOUN
ejpam-6592	4	26	analysis	analysis	NOUN
ejpam-6592	4	27	,	,	PUNCT
ejpam-6592	4	28	(	(	PUNCT
ejpam-6592	4	29	iii	iii	X
ejpam-6592	4	30	)	)	PUNCT
ejpam-6592	4	31	fractal	fractal	ADJ
ejpam-6592	4	32	geometry	geometry	NOUN
ejpam-6592	4	33	,	,	PUNCT
ejpam-6592	4	34	and	and	CCONJ
ejpam-6592	4	35	(	(	PUNCT
ejpam-6592	4	36	iv	iv	X
ejpam-6592	4	37	)	)	PUNCT
ejpam-6592	4	38	quantum	quantum	NOUN
ejpam-6592	4	39	state	state	NOUN
ejpam-6592	4	40	spaces	space	NOUN
ejpam-6592	4	41	.	.	PUNCT
ejpam-6592	5	1	the	the	DET
ejpam-6592	5	2	proofs	proof	NOUN
ejpam-6592	5	3	leverage	leverage	VERB
ejpam-6592	5	4	the	the	DET
ejpam-6592	5	5	unique	unique	ADJ
ejpam-6592	5	6	properties	property	NOUN
ejpam-6592	5	7	of	of	ADP
ejpam-6592	5	8	mr	mr	NOUN
ejpam-6592	5	9	-	-	PUNCT
ejpam-6592	5	10	metrics	metric	NOUN
ejpam-6592	5	11	,	,	PUNCT
ejpam-6592	5	12	particularly	particularly	ADV
ejpam-6592	5	13	the	the	DET
ejpam-6592	5	14	r	r	NOUN
ejpam-6592	5	15	-	-	PUNCT
ejpam-6592	5	16	scaling	scale	VERB
ejpam-6592	5	17	factor	factor	NOUN
ejpam-6592	5	18	in	in	ADP
ejpam-6592	5	19	the	the	DET
ejpam-6592	5	20	tetrahedral	tetrahedral	ADJ
ejpam-6592	5	21	inequality	inequality	NOUN
ejpam-6592	5	22	,	,	PUNCT
ejpam-6592	5	23	to	to	PART
ejpam-6592	5	24	extend	extend	VERB
ejpam-6592	5	25	classical	classical	ADJ
ejpam-6592	5	26	metric	metric	ADJ
ejpam-6592	5	27	space	space	NOUN
ejpam-6592	5	28	results	result	NOUN
ejpam-6592	5	29	to	to	ADP
ejpam-6592	5	30	this	this	DET
ejpam-6592	5	31	broader	broad	ADJ
ejpam-6592	5	32	setting	setting	NOUN
ejpam-6592	5	33	.	.	PUNCT
ejpam-6592	6	1	applications	application	NOUN
ejpam-6592	6	2	demonstrate	demonstrate	VERB
ejpam-6592	6	3	the	the	DET
ejpam-6592	6	4	utility	utility	NOUN
ejpam-6592	6	5	of	of	ADP
ejpam-6592	6	6	these	these	DET
ejpam-6592	6	7	theoretical	theoretical	ADJ
ejpam-6592	6	8	advances	advance	NOUN
ejpam-6592	6	9	in	in	ADP
ejpam-6592	6	10	computational	computational	ADJ
ejpam-6592	6	11	and	and	CCONJ
ejpam-6592	6	12	machine	machine	NOUN
ejpam-6592	6	13	learning	learn	VERB
ejpam-6592	6	14	contexts	contexts	NOUN
ejpam-6592	6	15	.	.	PUNCT
ejpam-6592	7	1	2020	2020	NUM
ejpam-6592	7	2	mathematics	mathematic	NOUN
ejpam-6592	7	3	subject	subject	NOUN
ejpam-6592	7	4	classifications	classification	NOUN
ejpam-6592	7	5	:	:	PUNCT
ejpam-6592	7	6	54e35	54e35	NUM
ejpam-6592	7	7	,	,	PUNCT
ejpam-6592	7	8	54d30	54d30	NUM
ejpam-6592	7	9	,	,	PUNCT
ejpam-6592	7	10	54e50	54e50	NUM
ejpam-6592	7	11	,	,	PUNCT
ejpam-6592	7	12	46nxx	46nxx	NOUN
ejpam-6592	7	13	,	,	PUNCT
ejpam-6592	7	14	68t07	68t07	NUM
ejpam-6592	7	15	key	key	ADJ
ejpam-6592	7	16	words	word	NOUN
ejpam-6592	7	17	and	and	CCONJ
ejpam-6592	7	18	phrases	phrase	NOUN
ejpam-6592	7	19	:	:	PUNCT
ejpam-6592	7	20	mr	mr	ADJ
ejpam-6592	7	21	-	-	PUNCT
ejpam-6592	7	22	metric	metric	ADJ
ejpam-6592	7	23	spaces	space	NOUN
ejpam-6592	7	24	,	,	PUNCT
ejpam-6592	7	25	compactness	compactness	NOUN
ejpam-6592	7	26	,	,	PUNCT
ejpam-6592	7	27	separability	separability	NOUN
ejpam-6592	7	28	,	,	PUNCT
ejpam-6592	7	29	paracompactness	paracompactness	NOUN
ejpam-6592	7	30	,	,	PUNCT
ejpam-6592	7	31	deep	deep	ADJ
ejpam-6592	7	32	learning	learning	NOUN
ejpam-6592	7	33	,	,	PUNCT
ejpam-6592	7	34	global	global	ADJ
ejpam-6592	7	35	optimization	optimization	NOUN
ejpam-6592	7	36	,	,	PUNCT
ejpam-6592	7	37	quantum	quantum	NOUN
ejpam-6592	7	38	computing	computing	NOUN
ejpam-6592	7	39	,	,	PUNCT
ejpam-6592	7	40	neural	neural	ADJ
ejpam-6592	7	41	networks	network	NOUN
ejpam-6592	7	42	,	,	PUNCT
ejpam-6592	7	43	topological	topological	ADJ
ejpam-6592	7	44	properties	property	NOUN
ejpam-6592	7	45	,	,	PUNCT
ejpam-6592	7	46	fixed	fix	VERB
ejpam-6592	7	47	point	point	NOUN
ejpam-6592	7	48	theory	theory	NOUN
ejpam-6592	7	49	1	1	NUM
ejpam-6592	7	50	.	.	PUNCT
ejpam-6592	8	1	introduction	introduction	NOUN
ejpam-6592	8	2	recent	recent	ADJ
ejpam-6592	8	3	advances	advance	NOUN
ejpam-6592	8	4	in	in	ADP
ejpam-6592	8	5	metric	metric	ADJ
ejpam-6592	8	6	space	space	NOUN
ejpam-6592	8	7	theory	theory	NOUN
ejpam-6592	8	8	have	have	AUX
ejpam-6592	8	9	led	lead	VERB
ejpam-6592	8	10	to	to	ADP
ejpam-6592	8	11	several	several	ADJ
ejpam-6592	8	12	generalizations	generalization	NOUN
ejpam-6592	8	13	of	of	ADP
ejpam-6592	8	14	classical	classical	ADJ
ejpam-6592	8	15	metric	metric	ADJ
ejpam-6592	8	16	spaces	space	NOUN
ejpam-6592	8	17	,	,	PUNCT
ejpam-6592	8	18	including	include	VERB
ejpam-6592	8	19	b	b	X
ejpam-6592	8	20	-	-	PUNCT
ejpam-6592	8	21	metric	metric	ADJ
ejpam-6592	8	22	spaces	space	NOUN
ejpam-6592	8	23	[	[	X
ejpam-6592	8	24	1	1	NUM
ejpam-6592	8	25	]	]	PUNCT
ejpam-6592	8	26	,	,	PUNCT
ejpam-6592	8	27	ωb	ωb	NOUN
ejpam-6592	8	28	-	-	PUNCT
ejpam-6592	8	29	distance	distance	NOUN
ejpam-6592	8	30	mappings	mapping	NOUN
ejpam-6592	9	1	[	[	X
ejpam-6592	9	2	2	2	NUM
ejpam-6592	9	3	]	]	PUNCT
ejpam-6592	9	4	,	,	PUNCT
ejpam-6592	9	5	and	and	CCONJ
ejpam-6592	9	6	gb	gb	ADV
ejpam-6592	9	7	-	-	PUNCT
ejpam-6592	9	8	metric	metric	ADJ
ejpam-6592	9	9	spaces	space	NOUN
ejpam-6592	9	10	[	[	X
ejpam-6592	9	11	3	3	NUM
ejpam-6592	9	12	]	]	PUNCT
ejpam-6592	9	13	.	.	PUNCT
ejpam-6592	10	1	among	among	ADP
ejpam-6592	10	2	these	these	PRON
ejpam-6592	10	3	,	,	PUNCT
ejpam-6592	10	4	mr	mr	ADJ
ejpam-6592	10	5	-	-	PUNCT
ejpam-6592	10	6	metric	metric	ADJ
ejpam-6592	10	7	spaces	space	NOUN
ejpam-6592	10	8	[	[	X
ejpam-6592	10	9	4	4	NUM
ejpam-6592	10	10	,	,	PUNCT
ejpam-6592	10	11	5	5	NUM
ejpam-6592	10	12	]	]	PUNCT
ejpam-6592	10	13	have	have	AUX
ejpam-6592	10	14	emerged	emerge	VERB
ejpam-6592	10	15	as	as	ADP
ejpam-6592	10	16	a	a	DET
ejpam-6592	10	17	particularly	particularly	ADV
ejpam-6592	10	18	useful	useful	ADJ
ejpam-6592	10	19	framework	framework	NOUN
ejpam-6592	10	20	due	due	ADP
ejpam-6592	10	21	to	to	ADP
ejpam-6592	10	22	their	their	PRON
ejpam-6592	10	23	flexible	flexible	ADJ
ejpam-6592	10	24	triangular	triangular	NOUN
ejpam-6592	10	25	inequality	inequality	NOUN
ejpam-6592	10	26	condition	condition	NOUN
ejpam-6592	10	27	and	and	CCONJ
ejpam-6592	10	28	applications	application	NOUN
ejpam-6592	10	29	in	in	ADP
ejpam-6592	10	30	fixed	fix	VERB
ejpam-6592	10	31	point	point	NOUN
ejpam-6592	10	32	theory	theory	NOUN
ejpam-6592	10	33	[	[	X
ejpam-6592	10	34	6–9	6–9	NOUN
ejpam-6592	10	35	]	]	PUNCT
ejpam-6592	10	36	.	.	PUNCT
ejpam-6592	11	1	the	the	DET
ejpam-6592	11	2	concept	concept	NOUN
ejpam-6592	11	3	of	of	ADP
ejpam-6592	11	4	mr	mr	PROPN
ejpam-6592	11	5	-	-	PUNCT
ejpam-6592	11	6	metric	metric	ADJ
ejpam-6592	11	7	spaces	space	NOUN
ejpam-6592	11	8	was	be	AUX
ejpam-6592	11	9	introduced	introduce	VERB
ejpam-6592	11	10	in	in	ADP
ejpam-6592	11	11	[	[	X
ejpam-6592	11	12	4	4	NUM
ejpam-6592	11	13	]	]	PUNCT
ejpam-6592	11	14	as	as	ADP
ejpam-6592	11	15	a	a	DET
ejpam-6592	11	16	three	three	NUM
ejpam-6592	11	17	-	-	PUNCT
ejpam-6592	11	18	variable	variable	NOUN
ejpam-6592	11	19	function	function	NOUN
ejpam-6592	11	20	m	m	VERB
ejpam-6592	11	21	:	:	PUNCT
ejpam-6592	12	1	x×	x×	X
ejpam-6592	12	2	x×	x×	PUNCT
ejpam-6592	12	3	x	x	PUNCT
ejpam-6592	12	4	→	→	PUNCT
ejpam-6592	12	5	[	[	X
ejpam-6592	12	6	0,∞	0,∞	NOUN
ejpam-6592	12	7	)	)	PUNCT
ejpam-6592	12	8	satisfying	satisfy	VERB
ejpam-6592	12	9	modified	modify	VERB
ejpam-6592	12	10	axioms	axiom	NOUN
ejpam-6592	12	11	that	that	PRON
ejpam-6592	12	12	include	include	VERB
ejpam-6592	12	13	an	an	DET
ejpam-6592	12	14	r	r	NOUN
ejpam-6592	12	15	-	-	PUNCT
ejpam-6592	12	16	scaled	scale	VERB
ejpam-6592	12	17	tetrahedral	tetrahedral	ADJ
ejpam-6592	12	18	inequality	inequality	NOUN
ejpam-6592	12	19	.	.	PUNCT
ejpam-6592	13	1	this	this	DET
ejpam-6592	13	2	structure	structure	NOUN
ejpam-6592	13	3	generalizes	generalize	VERB
ejpam-6592	13	4	several	several	ADJ
ejpam-6592	13	5	known	know	VERB
ejpam-6592	13	6	spaces	space	NOUN
ejpam-6592	13	7	including	include	VERB
ejpam-6592	13	8	mr	mr	PROPN
ejpam-6592	13	9	-	-	PUNCT
ejpam-6592	13	10	metric	metric	ADJ
ejpam-6592	13	11	spaces	space	NOUN
ejpam-6592	13	12	[	[	X
ejpam-6592	13	13	6	6	NUM
ejpam-6592	13	14	]	]	PUNCT
ejpam-6592	13	15	and	and	CCONJ
ejpam-6592	13	16	extended	extend	VERB
ejpam-6592	13	17	b	b	X
ejpam-6592	13	18	-	-	ADJ
ejpam-6592	13	19	metric	metric	ADJ
ejpam-6592	13	20	spaces	space	NOUN
ejpam-6592	13	21	[	[	X
ejpam-6592	13	22	10	10	NUM
ejpam-6592	13	23	,	,	PUNCT
ejpam-6592	13	24	11	11	NUM
ejpam-6592	13	25	]	]	PUNCT
ejpam-6592	13	26	.	.	PUNCT
ejpam-6592	14	1	recent	recent	ADJ
ejpam-6592	14	2	work	work	NOUN
ejpam-6592	14	3	has	have	AUX
ejpam-6592	14	4	demonstrated	demonstrate	VERB
ejpam-6592	14	5	their	their	PRON
ejpam-6592	14	6	utility	utility	NOUN
ejpam-6592	14	7	doi	doi	NOUN
ejpam-6592	14	8	:	:	PUNCT
ejpam-6592	14	9	https://doi.org/10.29020/nybg.ejpam.v18i3.6592	https://doi.org/10.29020/nybg.ejpam.v18i3.6592	SCONJ
ejpam-6592	14	10	email	email	NOUN
ejpam-6592	14	11	addresses	address	NOUN
ejpam-6592	14	12	:	:	PUNCT
ejpam-6592	15	1	a.malkawi@aau.edu.jo	a.malkawi@aau.edu.jo	PROPN
ejpam-6592	15	2	,	,	PUNCT
ejpam-6592	15	3	math.malkawi@gmail.com	math.malkawi@gmail.com	X
ejpam-6592	15	4	(	(	PUNCT
ejpam-6592	15	5	a.	a.	NOUN
ejpam-6592	15	6	malkawi	malkawi	PROPN
ejpam-6592	15	7	)	)	PUNCT
ejpam-6592	15	8	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6592	16	1	1	1	NUM
ejpam-6592	16	2	copyright	copyright	NOUN
ejpam-6592	16	3	:	:	PUNCT
ejpam-6592	16	4	©	©	PROPN
ejpam-6592	16	5	2025	2025	NUM
ejpam-6592	16	6	the	the	DET
ejpam-6592	16	7	author(s	author(s	NOUN
ejpam-6592	16	8	)	)	PUNCT
ejpam-6592	16	9	.	.	PUNCT
ejpam-6592	17	1	(	(	PUNCT
ejpam-6592	17	2	cc	cc	NOUN
ejpam-6592	17	3	by	by	ADP
ejpam-6592	17	4	-	-	PUNCT
ejpam-6592	17	5	nc	nc	PROPN
ejpam-6592	17	6	4.0	4.0	NUM
ejpam-6592	17	7	)	)	PUNCT
ejpam-6592	17	8	a.	a.	NOUN
ejpam-6592	17	9	malkawi	malkawi	ADP
ejpam-6592	17	10	/	/	SYM
ejpam-6592	17	11	eur	eur	PROPN
ejpam-6592	17	12	.	.	PUNCT
ejpam-6592	18	1	j.	j.	PROPN
ejpam-6592	18	2	pure	pure	PROPN
ejpam-6592	18	3	appl	appl	PROPN
ejpam-6592	18	4	.	.	PROPN
ejpam-6592	18	5	math	math	PROPN
ejpam-6592	18	6	,	,	PUNCT
ejpam-6592	18	7	18	18	NUM
ejpam-6592	18	8	(	(	PUNCT
ejpam-6592	18	9	3	3	NUM
ejpam-6592	18	10	)	)	PUNCT
ejpam-6592	18	11	(	(	PUNCT
ejpam-6592	18	12	2025	2025	NUM
ejpam-6592	18	13	)	)	PUNCT
ejpam-6592	18	14	,	,	PUNCT
ejpam-6592	18	15	6592	6592	NUM
ejpam-6592	18	16	2	2	NUM
ejpam-6592	18	17	of	of	ADP
ejpam-6592	18	18	14	14	NUM
ejpam-6592	18	19	in	in	ADP
ejpam-6592	18	20	fixed	fix	VERB
ejpam-6592	18	21	point	point	NOUN
ejpam-6592	18	22	theory	theory	NOUN
ejpam-6592	18	23	[	[	X
ejpam-6592	18	24	5	5	NUM
ejpam-6592	18	25	,	,	PUNCT
ejpam-6592	18	26	12	12	NUM
ejpam-6592	18	27	,	,	PUNCT
ejpam-6592	18	28	13	13	NUM
ejpam-6592	18	29	]	]	PUNCT
ejpam-6592	18	30	,	,	PUNCT
ejpam-6592	18	31	nonlinear	nonlinear	ADJ
ejpam-6592	18	32	contractions	contraction	NOUN
ejpam-6592	19	1	[	[	X
ejpam-6592	19	2	14	14	NUM
ejpam-6592	19	3	,	,	PUNCT
ejpam-6592	19	4	15	15	NUM
ejpam-6592	19	5	]	]	PUNCT
ejpam-6592	19	6	,	,	PUNCT
ejpam-6592	19	7	and	and	CCONJ
ejpam-6592	19	8	fractional	fractional	ADJ
ejpam-6592	19	9	calculus	calculus	NOUN
ejpam-6592	19	10	[	[	X
ejpam-6592	19	11	16	16	NUM
ejpam-6592	19	12	,	,	PUNCT
ejpam-6592	19	13	17	17	NUM
ejpam-6592	19	14	]	]	PUNCT
ejpam-6592	19	15	.	.	PUNCT
ejpam-6592	20	1	building	build	VERB
ejpam-6592	20	2	on	on	ADP
ejpam-6592	20	3	previous	previous	ADJ
ejpam-6592	20	4	results	result	NOUN
ejpam-6592	20	5	in	in	ADP
ejpam-6592	20	6	b	b	NOUN
ejpam-6592	20	7	-	-	ADJ
ejpam-6592	20	8	metric	metric	ADJ
ejpam-6592	20	9	spaces	space	NOUN
ejpam-6592	20	10	[	[	X
ejpam-6592	20	11	1	1	NUM
ejpam-6592	20	12	,	,	PUNCT
ejpam-6592	20	13	18	18	NUM
ejpam-6592	20	14	]	]	PUNCT
ejpam-6592	20	15	and	and	CCONJ
ejpam-6592	20	16	simulation	simulation	NOUN
ejpam-6592	20	17	functions	function	NOUN
ejpam-6592	20	18	[	[	X
ejpam-6592	20	19	13	13	NUM
ejpam-6592	20	20	,	,	PUNCT
ejpam-6592	20	21	19	19	NUM
ejpam-6592	20	22	]	]	PUNCT
ejpam-6592	20	23	,	,	PUNCT
ejpam-6592	20	24	this	this	DET
ejpam-6592	20	25	paper	paper	NOUN
ejpam-6592	20	26	establishes	establish	VERB
ejpam-6592	20	27	fundamental	fundamental	ADJ
ejpam-6592	20	28	topological	topological	ADJ
ejpam-6592	20	29	properties	property	NOUN
ejpam-6592	20	30	of	of	ADP
ejpam-6592	20	31	mr	mr	PROPN
ejpam-6592	20	32	-	-	PUNCT
ejpam-6592	20	33	metric	metric	ADJ
ejpam-6592	20	34	spaces	space	NOUN
ejpam-6592	20	35	.	.	PUNCT
ejpam-6592	21	1	our	our	PRON
ejpam-6592	21	2	work	work	NOUN
ejpam-6592	21	3	extends	extend	VERB
ejpam-6592	21	4	the	the	DET
ejpam-6592	21	5	classical	classical	ADJ
ejpam-6592	21	6	equivalence	equivalence	NOUN
ejpam-6592	21	7	between	between	ADP
ejpam-6592	21	8	compactness	compactness	NOUN
ejpam-6592	21	9	,	,	PUNCT
ejpam-6592	21	10	sequential	sequential	ADJ
ejpam-6592	21	11	compactness	compactness	NOUN
ejpam-6592	21	12	,	,	PUNCT
ejpam-6592	21	13	and	and	CCONJ
ejpam-6592	21	14	completeness	completeness	NOUN
ejpam-6592	21	15	with	with	ADP
ejpam-6592	21	16	total	total	ADJ
ejpam-6592	21	17	boundedness	boundedness	NOUN
ejpam-6592	21	18	to	to	ADP
ejpam-6592	21	19	this	this	DET
ejpam-6592	21	20	generalized	generalized	ADJ
ejpam-6592	21	21	setting	setting	NOUN
ejpam-6592	21	22	.	.	PUNCT
ejpam-6592	22	1	the	the	DET
ejpam-6592	22	2	proofs	proof	NOUN
ejpam-6592	22	3	leverage	leverage	VERB
ejpam-6592	22	4	the	the	DET
ejpam-6592	22	5	r	r	NOUN
ejpam-6592	22	6	-	-	PUNCT
ejpam-6592	22	7	scaled	scale	VERB
ejpam-6592	22	8	tetrahedral	tetrahedral	ADJ
ejpam-6592	22	9	inequality	inequality	NOUN
ejpam-6592	22	10	in	in	ADP
ejpam-6592	22	11	novel	novel	ADJ
ejpam-6592	22	12	ways	way	NOUN
ejpam-6592	22	13	,	,	PUNCT
ejpam-6592	22	14	particularly	particularly	ADV
ejpam-6592	22	15	in	in	ADP
ejpam-6592	22	16	establishing	establish	VERB
ejpam-6592	22	17	the	the	DET
ejpam-6592	22	18	lebesgue	lebesgue	NOUN
ejpam-6592	22	19	number	number	NOUN
ejpam-6592	22	20	lemma	lemma	PROPN
ejpam-6592	22	21	(	(	PUNCT
ejpam-6592	22	22	lemma	lemma	PROPN
ejpam-6592	22	23	1	1	NUM
ejpam-6592	22	24	)	)	PUNCT
ejpam-6592	22	25	and	and	CCONJ
ejpam-6592	22	26	the	the	DET
ejpam-6592	22	27	lindelöf	lindelöf	PROPN
ejpam-6592	22	28	-	-	PUNCT
ejpam-6592	22	29	separability	separability	NOUN
ejpam-6592	22	30	equivalence	equivalence	NOUN
ejpam-6592	22	31	(	(	PUNCT
ejpam-6592	22	32	theorem	theorem	NOUN
ejpam-6592	22	33	2	2	NUM
ejpam-6592	22	34	)	)	PUNCT
ejpam-6592	22	35	.	.	PUNCT
ejpam-6592	23	1	applications	application	NOUN
ejpam-6592	23	2	of	of	ADP
ejpam-6592	23	3	these	these	DET
ejpam-6592	23	4	theoretical	theoretical	ADJ
ejpam-6592	23	5	results	result	NOUN
ejpam-6592	23	6	span	span	VERB
ejpam-6592	23	7	multiple	multiple	ADJ
ejpam-6592	23	8	disciplines	discipline	NOUN
ejpam-6592	23	9	:	:	PUNCT
ejpam-6592	23	10	•	•	NUM
ejpam-6592	23	11	global	global	ADJ
ejpam-6592	23	12	optimization	optimization	NOUN
ejpam-6592	23	13	in	in	ADP
ejpam-6592	23	14	compact	compact	ADJ
ejpam-6592	23	15	subsets	subset	NOUN
ejpam-6592	23	16	of	of	ADP
ejpam-6592	23	17	rn	rn	PROPN
ejpam-6592	23	18	(	(	PUNCT
ejpam-6592	23	19	example	example	NOUN
ejpam-6592	23	20	1	1	NUM
ejpam-6592	23	21	)	)	PUNCT
ejpam-6592	23	22	•	•	NUM
ejpam-6592	23	23	dimensionality	dimensionality	NOUN
ejpam-6592	23	24	reduction	reduction	NOUN
ejpam-6592	23	25	in	in	ADP
ejpam-6592	23	26	neural	neural	ADJ
ejpam-6592	23	27	network	network	NOUN
ejpam-6592	23	28	weight	weight	NOUN
ejpam-6592	23	29	spaces	space	NOUN
ejpam-6592	23	30	(	(	PUNCT
ejpam-6592	23	31	theorem	theorem	VERB
ejpam-6592	23	32	4	4	NUM
ejpam-6592	23	33	)	)	PUNCT
ejpam-6592	23	34	•	•	NOUN
ejpam-6592	23	35	fractal	fractal	ADJ
ejpam-6592	23	36	analysis	analysis	NOUN
ejpam-6592	23	37	using	use	VERB
ejpam-6592	23	38	paracompactness	paracompactness	NOUN
ejpam-6592	23	39	properties	property	NOUN
ejpam-6592	23	40	(	(	PUNCT
ejpam-6592	23	41	example	example	NOUN
ejpam-6592	23	42	2	2	NUM
ejpam-6592	23	43	)	)	PUNCT
ejpam-6592	23	44	•	•	NUM
ejpam-6592	23	45	quantum	quantum	NOUN
ejpam-6592	23	46	state	state	NOUN
ejpam-6592	23	47	space	space	NOUN
ejpam-6592	23	48	analysis	analysis	NOUN
ejpam-6592	23	49	with	with	ADP
ejpam-6592	23	50	trace	trace	NOUN
ejpam-6592	23	51	-	-	PUNCT
ejpam-6592	23	52	norm	norm	NOUN
ejpam-6592	23	53	mr	mr	NOUN
ejpam-6592	23	54	-	-	PUNCT
ejpam-6592	23	55	metrics	metric	NOUN
ejpam-6592	23	56	(	(	PUNCT
ejpam-6592	23	57	theorem	theorem	NOUN
ejpam-6592	23	58	5	5	NUM
ejpam-6592	23	59	)	)	PUNCT
ejpam-6592	23	60	the	the	DET
ejpam-6592	23	61	paper	paper	NOUN
ejpam-6592	23	62	is	be	AUX
ejpam-6592	23	63	organized	organize	VERB
ejpam-6592	23	64	as	as	SCONJ
ejpam-6592	23	65	follows	follow	VERB
ejpam-6592	23	66	:	:	PUNCT
ejpam-6592	23	67	section	section	NOUN
ejpam-6592	23	68	2	2	NUM
ejpam-6592	23	69	presents	present	VERB
ejpam-6592	23	70	the	the	DET
ejpam-6592	23	71	main	main	ADJ
ejpam-6592	23	72	theoretical	theoretical	ADJ
ejpam-6592	23	73	results	result	NOUN
ejpam-6592	23	74	,	,	PUNCT
ejpam-6592	23	75	section	section	NOUN
ejpam-6592	23	76	3	3	NUM
ejpam-6592	23	77	discusses	discuss	VERB
ejpam-6592	23	78	applications	application	NOUN
ejpam-6592	23	79	.	.	PUNCT
ejpam-6592	24	1	our	our	PRON
ejpam-6592	24	2	work	work	NOUN
ejpam-6592	24	3	builds	build	VERB
ejpam-6592	24	4	upon	upon	SCONJ
ejpam-6592	24	5	and	and	CCONJ
ejpam-6592	24	6	extends	extend	VERB
ejpam-6592	24	7	previous	previous	ADJ
ejpam-6592	24	8	results	result	NOUN
ejpam-6592	24	9	in	in	ADP
ejpam-6592	24	10	fixed	fix	VERB
ejpam-6592	24	11	point	point	NOUN
ejpam-6592	24	12	theory	theory	NOUN
ejpam-6592	24	13	[	[	X
ejpam-6592	24	14	20–22	20–22	NOUN
ejpam-6592	24	15	]	]	X
ejpam-6592	24	16	,	,	PUNCT
ejpam-6592	24	17	metric	metric	ADJ
ejpam-6592	24	18	space	space	NOUN
ejpam-6592	24	19	topology	topology	NOUN
ejpam-6592	24	20	[	[	X
ejpam-6592	24	21	3	3	NUM
ejpam-6592	24	22	,	,	PUNCT
ejpam-6592	24	23	10	10	NUM
ejpam-6592	24	24	]	]	PUNCT
ejpam-6592	24	25	,	,	PUNCT
ejpam-6592	24	26	and	and	CCONJ
ejpam-6592	24	27	their	their	PRON
ejpam-6592	24	28	computational	computational	ADJ
ejpam-6592	24	29	applications	application	NOUN
ejpam-6592	24	30	[	[	X
ejpam-6592	24	31	23	23	NUM
ejpam-6592	24	32	]	]	PUNCT
ejpam-6592	24	33	.	.	PUNCT
ejpam-6592	25	1	definition	definition	NOUN
ejpam-6592	25	2	1	1	NUM
ejpam-6592	25	3	.	.	PUNCT
ejpam-6592	26	1	[	[	X
ejpam-6592	26	2	4	4	X
ejpam-6592	26	3	]	]	PUNCT
ejpam-6592	26	4	consider	consider	VERB
ejpam-6592	26	5	a	a	DET
ejpam-6592	26	6	non	non	ADJ
ejpam-6592	26	7	-	-	ADJ
ejpam-6592	26	8	empty	empty	ADJ
ejpam-6592	26	9	set	set	NOUN
ejpam-6592	26	10	x	x	PUNCT
ejpam-6592	26	11	̸=	̸=	PROPN
ejpam-6592	26	12	∅	∅	NOUN
ejpam-6592	26	13	and	and	CCONJ
ejpam-6592	26	14	a	a	DET
ejpam-6592	26	15	real	real	ADJ
ejpam-6592	26	16	number	number	NOUN
ejpam-6592	26	17	r	r	NOUN
ejpam-6592	26	18	>	>	X
ejpam-6592	26	19	1	1	NUM
ejpam-6592	26	20	.	.	PUNCT
ejpam-6592	27	1	a	a	DET
ejpam-6592	27	2	function	function	NOUN
ejpam-6592	27	3	m	m	VERB
ejpam-6592	27	4	:	:	PUNCT
ejpam-6592	27	5	x×	x×	X
ejpam-6592	27	6	x×	x×	PUNCT
ejpam-6592	27	7	x	x	PUNCT
ejpam-6592	27	8	→	→	PUNCT
ejpam-6592	27	9	[	[	X
ejpam-6592	27	10	0,∞	0,∞	NOUN
ejpam-6592	27	11	)	)	PUNCT
ejpam-6592	27	12	is	be	AUX
ejpam-6592	27	13	termed	term	VERB
ejpam-6592	27	14	an	an	DET
ejpam-6592	27	15	mr	mr	PROPN
ejpam-6592	27	16	-	-	PUNCT
ejpam-6592	27	17	metric	metric	NOUN
ejpam-6592	27	18	if	if	SCONJ
ejpam-6592	27	19	it	it	PRON
ejpam-6592	27	20	satisfies	satisfy	VERB
ejpam-6592	27	21	the	the	DET
ejpam-6592	27	22	following	follow	VERB
ejpam-6592	27	23	conditions	condition	NOUN
ejpam-6592	27	24	for	for	ADP
ejpam-6592	27	25	all	all	PRON
ejpam-6592	27	26	v	v	NOUN
ejpam-6592	27	27	,	,	PUNCT
ejpam-6592	27	28	ξ	ξ	PROPN
ejpam-6592	27	29	,	,	PUNCT
ejpam-6592	27	30	s	s	PART
ejpam-6592	27	31	,	,	PUNCT
ejpam-6592	27	32	ℓ1	ℓ1	NOUN
ejpam-6592	27	33	∈	∈	NOUN
ejpam-6592	28	1	x	x	X
ejpam-6592	28	2	:	:	PUNCT
ejpam-6592	28	3	•	•	ADP
ejpam-6592	28	4	m(v	m(v	PROPN
ejpam-6592	28	5	,	,	PUNCT
ejpam-6592	28	6	ξ	ξ	PROPN
ejpam-6592	28	7	,	,	PUNCT
ejpam-6592	28	8	s	s	PART
ejpam-6592	28	9	)	)	PUNCT
ejpam-6592	28	10	≥	≥	NOUN
ejpam-6592	28	11	0	0	NUM
ejpam-6592	28	12	.	.	NOUN
ejpam-6592	28	13	•	•	NUM
ejpam-6592	28	14	m(v	m(v	PROPN
ejpam-6592	28	15	,	,	PUNCT
ejpam-6592	28	16	ξ	ξ	PROPN
ejpam-6592	28	17	,	,	PUNCT
ejpam-6592	28	18	s	s	PART
ejpam-6592	28	19	)	)	PUNCT
ejpam-6592	28	20	=	=	SYM
ejpam-6592	28	21	0	0	PUNCT
ejpam-6592	29	1	if	if	SCONJ
ejpam-6592	29	2	and	and	CCONJ
ejpam-6592	29	3	only	only	ADV
ejpam-6592	29	4	if	if	SCONJ
ejpam-6592	29	5	v	v	NOUN
ejpam-6592	29	6	=	=	SYM
ejpam-6592	29	7	ξ	ξ	PROPN
ejpam-6592	29	8	=	=	PUNCT
ejpam-6592	29	9	s.	s.	PROPN
ejpam-6592	29	10	•	•	ADP
ejpam-6592	29	11	m(v	m(v	PROPN
ejpam-6592	29	12	,	,	PUNCT
ejpam-6592	29	13	ξ	ξ	PROPN
ejpam-6592	29	14	,	,	PUNCT
ejpam-6592	29	15	s	s	PART
ejpam-6592	29	16	)	)	PUNCT
ejpam-6592	29	17	remains	remain	VERB
ejpam-6592	29	18	invariant	invariant	ADJ
ejpam-6592	29	19	under	under	ADP
ejpam-6592	29	20	any	any	DET
ejpam-6592	29	21	permutation	permutation	NOUN
ejpam-6592	29	22	p(v	p(v	NOUN
ejpam-6592	29	23	,	,	PUNCT
ejpam-6592	29	24	ξ	ξ	PROPN
ejpam-6592	29	25	,	,	PUNCT
ejpam-6592	29	26	s	s	PART
ejpam-6592	29	27	)	)	PUNCT
ejpam-6592	29	28	,	,	PUNCT
ejpam-6592	29	29	i.e.	i.e.	X
ejpam-6592	29	30	,	,	PUNCT
ejpam-6592	29	31	m(v	m(v	PROPN
ejpam-6592	29	32	,	,	PUNCT
ejpam-6592	29	33	ξ	ξ	PROPN
ejpam-6592	29	34	,	,	PUNCT
ejpam-6592	29	35	s	s	PART
ejpam-6592	29	36	)	)	PUNCT
ejpam-6592	29	37	=	=	SYM
ejpam-6592	29	38	m(p(v	m(p(v	PROPN
ejpam-6592	29	39	,	,	PUNCT
ejpam-6592	29	40	ξ	ξ	PROPN
ejpam-6592	29	41	,	,	PUNCT
ejpam-6592	29	42	s	s	NOUN
ejpam-6592	29	43	)	)	PUNCT
ejpam-6592	29	44	)	)	PUNCT
ejpam-6592	29	45	.	.	PUNCT
ejpam-6592	30	1	•	•	NUM
ejpam-6592	30	2	the	the	DET
ejpam-6592	30	3	following	follow	VERB
ejpam-6592	30	4	inequality	inequality	NOUN
ejpam-6592	30	5	holds	hold	VERB
ejpam-6592	30	6	:	:	PUNCT
ejpam-6592	30	7	m(v	m(v	NUM
ejpam-6592	30	8	,	,	PUNCT
ejpam-6592	30	9	ξ	ξ	PROPN
ejpam-6592	30	10	,	,	PUNCT
ejpam-6592	30	11	s	s	NOUN
ejpam-6592	30	12	)	)	PUNCT
ejpam-6592	30	13	≤	≤	NOUN
ejpam-6592	30	14	r	r	NOUN
ejpam-6592	31	1	[	[	X
ejpam-6592	31	2	m(v	m(v	X
ejpam-6592	31	3	,	,	PUNCT
ejpam-6592	31	4	ξ	ξ	X
ejpam-6592	31	5	,	,	PUNCT
ejpam-6592	31	6	ℓ1	ℓ1	NOUN
ejpam-6592	31	7	)	)	PUNCT
ejpam-6592	32	1	+	+	SYM
ejpam-6592	32	2	m(v	m(v	NOUN
ejpam-6592	32	3	,	,	PUNCT
ejpam-6592	32	4	ℓ1	ℓ1	NOUN
ejpam-6592	32	5	,	,	PUNCT
ejpam-6592	32	6	s	s	X
ejpam-6592	32	7	)	)	PUNCT
ejpam-6592	32	8	+	+	ADJ
ejpam-6592	32	9	m(ℓ1	m(ℓ1	NOUN
ejpam-6592	32	10	,	,	PUNCT
ejpam-6592	32	11	ξ	ξ	PROPN
ejpam-6592	32	12	,	,	PUNCT
ejpam-6592	32	13	s	s	PART
ejpam-6592	32	14	)	)	PUNCT
ejpam-6592	32	15	]	]	PUNCT
ejpam-6592	32	16	.	.	PUNCT
ejpam-6592	33	1	a	a	DET
ejpam-6592	33	2	structure	structure	NOUN
ejpam-6592	33	3	(	(	PUNCT
ejpam-6592	33	4	x	x	X
ejpam-6592	33	5	,	,	PUNCT
ejpam-6592	33	6	m	m	NOUN
ejpam-6592	33	7	)	)	PUNCT
ejpam-6592	33	8	that	that	PRON
ejpam-6592	33	9	adheres	adhere	VERB
ejpam-6592	33	10	to	to	ADP
ejpam-6592	33	11	these	these	DET
ejpam-6592	33	12	properties	property	NOUN
ejpam-6592	33	13	is	be	AUX
ejpam-6592	33	14	defined	define	VERB
ejpam-6592	33	15	as	as	ADP
ejpam-6592	33	16	an	an	DET
ejpam-6592	33	17	mr	mr	PROPN
ejpam-6592	33	18	-	-	PUNCT
ejpam-6592	33	19	metric	metric	ADJ
ejpam-6592	33	20	space	space	NOUN
ejpam-6592	33	21	.	.	PUNCT
ejpam-6592	34	1	2	2	X
ejpam-6592	34	2	.	.	X
ejpam-6592	34	3	main	main	ADJ
ejpam-6592	34	4	results	result	NOUN
ejpam-6592	34	5	lemma	lemma	PROPN
ejpam-6592	34	6	1	1	NUM
ejpam-6592	34	7	(	(	PUNCT
ejpam-6592	34	8	lebesgue	lebesgue	NOUN
ejpam-6592	34	9	number	number	NOUN
ejpam-6592	34	10	lemma	lemma	PROPN
ejpam-6592	34	11	for	for	ADP
ejpam-6592	34	12	mr	mr	PROPN
ejpam-6592	34	13	-	-	PUNCT
ejpam-6592	34	14	metric	metric	ADJ
ejpam-6592	34	15	spaces	space	NOUN
ejpam-6592	34	16	)	)	PUNCT
ejpam-6592	34	17	.	.	PUNCT
ejpam-6592	35	1	let	let	VERB
ejpam-6592	35	2	(	(	PUNCT
ejpam-6592	35	3	x	x	X
ejpam-6592	35	4	,	,	PUNCT
ejpam-6592	35	5	m	m	VERB
ejpam-6592	35	6	)	)	PUNCT
ejpam-6592	35	7	be	be	VERB
ejpam-6592	35	8	a	a	DET
ejpam-6592	35	9	totally	totally	ADV
ejpam-6592	35	10	bounded	bound	VERB
ejpam-6592	35	11	mr	mr	PROPN
ejpam-6592	35	12	-	-	PUNCT
ejpam-6592	35	13	metric	metric	ADJ
ejpam-6592	35	14	space	space	NOUN
ejpam-6592	35	15	with	with	ADP
ejpam-6592	35	16	r	r	NOUN
ejpam-6592	35	17	>	>	X
ejpam-6592	35	18	1	1	NUM
ejpam-6592	35	19	,	,	PUNCT
ejpam-6592	35	20	and	and	CCONJ
ejpam-6592	35	21	u	u	NOUN
ejpam-6592	35	22	=	=	PUNCT
ejpam-6592	35	23	{	{	PUNCT
ejpam-6592	35	24	ui}i∈i	ui}i∈i	ADP
ejpam-6592	35	25	an	an	DET
ejpam-6592	35	26	open	open	ADJ
ejpam-6592	35	27	cover	cover	NOUN
ejpam-6592	35	28	of	of	ADP
ejpam-6592	35	29	x.	x.	NOUN
ejpam-6592	35	30	then	then	ADV
ejpam-6592	35	31	,	,	PUNCT
ejpam-6592	35	32	there	there	PRON
ejpam-6592	35	33	exists	exist	VERB
ejpam-6592	35	34	δ	δ	PROPN
ejpam-6592	35	35	>	>	X
ejpam-6592	35	36	0	0	NUM
ejpam-6592	36	1	such	such	ADJ
ejpam-6592	36	2	that	that	SCONJ
ejpam-6592	36	3	:	:	PUNCT
ejpam-6592	36	4	∀x	∀x	NUM
ejpam-6592	36	5	∈	∈	PROPN
ejpam-6592	36	6	x	x	X
ejpam-6592	36	7	,	,	PUNCT
ejpam-6592	36	8	∃ui	∃ui	ADP
ejpam-6592	36	9	∈	∈	PROPN
ejpam-6592	36	10	u	u	NOUN
ejpam-6592	36	11	with	with	ADP
ejpam-6592	36	12	bm	bm	PROPN
ejpam-6592	36	13	(	(	PUNCT
ejpam-6592	36	14	x	x	PROPN
ejpam-6592	36	15	,	,	PUNCT
ejpam-6592	36	16	δ	δ	PROPN
ejpam-6592	36	17	)	)	PUNCT
ejpam-6592	36	18	⊆	⊆	NUM
ejpam-6592	36	19	ui	ui	PROPN
ejpam-6592	36	20	.	.	PUNCT
ejpam-6592	36	21	a.	a.	PROPN
ejpam-6592	36	22	malkawi	malkawi	PROPN
ejpam-6592	36	23	/	/	SYM
ejpam-6592	36	24	eur	eur	PROPN
ejpam-6592	36	25	.	.	PUNCT
ejpam-6592	37	1	j.	j.	PROPN
ejpam-6592	37	2	pure	pure	PROPN
ejpam-6592	37	3	appl	appl	PROPN
ejpam-6592	37	4	.	.	PROPN
ejpam-6592	37	5	math	math	PROPN
ejpam-6592	37	6	,	,	PUNCT
ejpam-6592	37	7	18	18	NUM
ejpam-6592	37	8	(	(	PUNCT
ejpam-6592	37	9	3	3	NUM
ejpam-6592	37	10	)	)	PUNCT
ejpam-6592	37	11	(	(	PUNCT
ejpam-6592	37	12	2025	2025	NUM
ejpam-6592	37	13	)	)	PUNCT
ejpam-6592	37	14	,	,	PUNCT
ejpam-6592	37	15	6592	6592	NUM
ejpam-6592	37	16	3	3	NUM
ejpam-6592	37	17	of	of	ADP
ejpam-6592	37	18	14	14	NUM
ejpam-6592	37	19	proof	proof	NOUN
ejpam-6592	37	20	.	.	PUNCT
ejpam-6592	38	1	we	we	PRON
ejpam-6592	38	2	adapt	adapt	VERB
ejpam-6592	38	3	the	the	DET
ejpam-6592	38	4	classical	classical	ADJ
ejpam-6592	38	5	lebesgue	lebesgue	NOUN
ejpam-6592	38	6	number	number	NOUN
ejpam-6592	38	7	proof	proof	NOUN
ejpam-6592	38	8	to	to	ADP
ejpam-6592	38	9	the	the	DET
ejpam-6592	38	10	mr	mr	PROPN
ejpam-6592	38	11	-	-	PUNCT
ejpam-6592	38	12	metric	metric	ADJ
ejpam-6592	38	13	structure	structure	NOUN
ejpam-6592	38	14	.	.	PUNCT
ejpam-6592	39	1	step	step	NOUN
ejpam-6592	39	2	1	1	NUM
ejpam-6592	39	3	:	:	PUNCT
ejpam-6592	39	4	use	use	VERB
ejpam-6592	39	5	total	total	ADJ
ejpam-6592	39	6	boundedness	boundedness	NOUN
ejpam-6592	39	7	.	.	PUNCT
ejpam-6592	40	1	since	since	SCONJ
ejpam-6592	40	2	x	x	PRON
ejpam-6592	40	3	is	be	AUX
ejpam-6592	40	4	totally	totally	ADV
ejpam-6592	40	5	bounded	bound	VERB
ejpam-6592	40	6	,	,	PUNCT
ejpam-6592	40	7	for	for	ADP
ejpam-6592	40	8	ϵ	ϵ	NOUN
ejpam-6592	40	9	=	=	SYM
ejpam-6592	40	10	1	1	NUM
ejpam-6592	40	11	n	n	NOUN
ejpam-6592	40	12	(	(	PUNCT
ejpam-6592	40	13	n	n	CCONJ
ejpam-6592	40	14	∈	∈	PROPN
ejpam-6592	40	15	n	n	CCONJ
ejpam-6592	40	16	)	)	PUNCT
ejpam-6592	40	17	,	,	PUNCT
ejpam-6592	40	18	there	there	PRON
ejpam-6592	40	19	exists	exist	VERB
ejpam-6592	40	20	a	a	DET
ejpam-6592	40	21	finite	finite	NOUN
ejpam-6592	40	22	ϵ-net	ϵ-net	NOUN
ejpam-6592	40	23	an	an	PRON
ejpam-6592	40	24	=	=	X
ejpam-6592	40	25	{	{	PUNCT
ejpam-6592	40	26	a1	a1	NOUN
ejpam-6592	40	27	,	,	PUNCT
ejpam-6592	40	28	.	.	PUNCT
ejpam-6592	40	29	.	.	PUNCT
ejpam-6592	41	1	.	.	PUNCT
ejpam-6592	42	1	,	,	PUNCT
ejpam-6592	42	2	akn	akn	PROPN
ejpam-6592	42	3	}	}	PUNCT
ejpam-6592	42	4	such	such	ADJ
ejpam-6592	42	5	that	that	SCONJ
ejpam-6592	42	6	:	:	PUNCT
ejpam-6592	42	7	x	x	SYM
ejpam-6592	42	8	⊆	⊆	NUM
ejpam-6592	42	9	kn⋃	kn⋃	PROPN
ejpam-6592	42	10	i=1	i=1	PROPN
ejpam-6592	42	11	bm	bm	PROPN
ejpam-6592	42	12	(	(	PUNCT
ejpam-6592	42	13	ai	ai	PROPN
ejpam-6592	42	14	,	,	PUNCT
ejpam-6592	42	15	1	1	NUM
ejpam-6592	42	16	n	n	NOUN
ejpam-6592	42	17	)	)	PUNCT
ejpam-6592	42	18	.	.	PUNCT
ejpam-6592	43	1	step	step	NOUN
ejpam-6592	43	2	2	2	NUM
ejpam-6592	43	3	:	:	PUNCT
ejpam-6592	43	4	define	define	VERB
ejpam-6592	43	5	auxiliary	auxiliary	ADJ
ejpam-6592	43	6	function	function	NOUN
ejpam-6592	43	7	.	.	PUNCT
ejpam-6592	44	1	for	for	ADP
ejpam-6592	44	2	each	each	DET
ejpam-6592	44	3	ai	ai	PROPN
ejpam-6592	44	4	∈	∈	PROPN
ejpam-6592	44	5	an	an	PRON
ejpam-6592	44	6	,	,	PUNCT
ejpam-6592	44	7	define	define	NOUN
ejpam-6592	44	8	:	:	PUNCT
ejpam-6592	44	9	f(ai	f(ai	ADJ
ejpam-6592	44	10	)	)	PUNCT
ejpam-6592	44	11	=	=	SYM
ejpam-6592	44	12	sup	sup	NOUN
ejpam-6592	44	13	{	{	PUNCT
ejpam-6592	44	14	r	r	NOUN
ejpam-6592	44	15	>	>	X
ejpam-6592	44	16	0	0	PUNCT
ejpam-6592	45	1	|	|	ADV
ejpam-6592	45	2	bm	bm	PROPN
ejpam-6592	45	3	(	(	PUNCT
ejpam-6592	45	4	ai	ai	NOUN
ejpam-6592	45	5	,	,	PUNCT
ejpam-6592	45	6	r	r	NOUN
ejpam-6592	45	7	)	)	PUNCT
ejpam-6592	45	8	⊆	⊆	NUM
ejpam-6592	45	9	uj	uj	NOUN
ejpam-6592	45	10	for	for	ADP
ejpam-6592	45	11	some	some	DET
ejpam-6592	45	12	uj	uj	PROPN
ejpam-6592	45	13	∈	∈	PROPN
ejpam-6592	45	14	u	u	NOUN
ejpam-6592	45	15	}	}	PUNCT
ejpam-6592	45	16	.	.	PUNCT
ejpam-6592	46	1	by	by	ADP
ejpam-6592	46	2	openness	openness	NOUN
ejpam-6592	46	3	of	of	ADP
ejpam-6592	46	4	u	u	PROPN
ejpam-6592	46	5	,	,	PUNCT
ejpam-6592	46	6	f(ai	f(ai	PROPN
ejpam-6592	46	7	)	)	PUNCT
ejpam-6592	46	8	>	>	X
ejpam-6592	46	9	0	0	PUNCT
ejpam-6592	46	10	for	for	ADP
ejpam-6592	46	11	all	all	PRON
ejpam-6592	46	12	ai	ai	NOUN
ejpam-6592	46	13	.	.	NOUN
ejpam-6592	46	14	step	step	NOUN
ejpam-6592	46	15	3	3	NUM
ejpam-6592	46	16	:	:	PUNCT
ejpam-6592	46	17	lower	lower	ADV
ejpam-6592	46	18	bound	bind	VERB
ejpam-6592	46	19	via	via	ADP
ejpam-6592	46	20	mr	mr	PROPN
ejpam-6592	46	21	-	-	PUNCT
ejpam-6592	46	22	metric	metric	NOUN
ejpam-6592	46	23	.	.	PUNCT
ejpam-6592	47	1	let	let	VERB
ejpam-6592	47	2	δn	δn	VERB
ejpam-6592	47	3	=	=	SYM
ejpam-6592	47	4	min{f(a1	min{f(a1	NOUN
ejpam-6592	47	5	)	)	PUNCT
ejpam-6592	47	6	,	,	PUNCT
ejpam-6592	47	7	.	.	PUNCT
ejpam-6592	47	8	.	.	PUNCT
ejpam-6592	48	1	.	.	PUNCT
ejpam-6592	49	1	,	,	PUNCT
ejpam-6592	49	2	f(akn	f(akn	ADJ
ejpam-6592	49	3	)	)	PUNCT
ejpam-6592	49	4	}	}	PUNCT
ejpam-6592	49	5	>	>	X
ejpam-6592	50	1	0	0	X
ejpam-6592	50	2	.	.	PUNCT
ejpam-6592	51	1	we	we	PRON
ejpam-6592	51	2	claim	claim	VERB
ejpam-6592	51	3	:	:	PUNCT
ejpam-6592	51	4	δ	δ	PROPN
ejpam-6592	51	5	=	=	SYM
ejpam-6592	51	6	inf	inf	PROPN
ejpam-6592	51	7	n∈n	n∈n	ADV
ejpam-6592	51	8	(	(	PUNCT
ejpam-6592	51	9	δn	δn	NOUN
ejpam-6592	51	10	r	r	NOUN
ejpam-6592	51	11	−	−	PROPN
ejpam-6592	51	12	1	1	NUM
ejpam-6592	51	13	n	n	NOUN
ejpam-6592	51	14	)	)	PUNCT
ejpam-6592	51	15	>	>	X
ejpam-6592	52	1	0	0	X
ejpam-6592	52	2	.	.	PUNCT
ejpam-6592	52	3	to	to	PART
ejpam-6592	52	4	verify	verify	VERB
ejpam-6592	52	5	,	,	PUNCT
ejpam-6592	52	6	fix	fix	NOUN
ejpam-6592	52	7	x	x	X
ejpam-6592	52	8	∈	∈	NOUN
ejpam-6592	52	9	x.	x.	NOUN
ejpam-6592	52	10	for	for	ADP
ejpam-6592	52	11	each	each	DET
ejpam-6592	52	12	n	n	CCONJ
ejpam-6592	52	13	,	,	PUNCT
ejpam-6592	52	14	pick	pick	VERB
ejpam-6592	52	15	ai	ai	VERB
ejpam-6592	52	16	∈	∈	PROPN
ejpam-6592	52	17	an	an	DET
ejpam-6592	52	18	with	with	NOUN
ejpam-6592	52	19	x	x	PROPN
ejpam-6592	52	20	∈	∈	PROPN
ejpam-6592	52	21	bm	bm	PROPN
ejpam-6592	52	22	(	(	PUNCT
ejpam-6592	52	23	ai	ai	PROPN
ejpam-6592	52	24	,	,	PUNCT
ejpam-6592	52	25	1	1	NUM
ejpam-6592	52	26	n	n	CCONJ
ejpam-6592	52	27	)	)	PUNCT
ejpam-6592	52	28	.	.	PUNCT
ejpam-6592	53	1	then	then	ADV
ejpam-6592	53	2	:	:	PUNCT
ejpam-6592	53	3	bm	bm	PROPN
ejpam-6592	53	4	(	(	PUNCT
ejpam-6592	53	5	x	x	PROPN
ejpam-6592	53	6	,	,	PUNCT
ejpam-6592	53	7	δ	δ	PROPN
ejpam-6592	53	8	)	)	PUNCT
ejpam-6592	53	9	⊆	⊆	NUM
ejpam-6592	53	10	bm	bm	PROPN
ejpam-6592	53	11	(	(	PUNCT
ejpam-6592	53	12	ai	ai	PROPN
ejpam-6592	53	13	,	,	PUNCT
ejpam-6592	53	14	r	r	NOUN
ejpam-6592	53	15	(	(	PUNCT
ejpam-6592	53	16	δ	δ	PROPN
ejpam-6592	53	17	+	+	CCONJ
ejpam-6592	53	18	1	1	NUM
ejpam-6592	53	19	n	n	NUM
ejpam-6592	53	20	)	)	PUNCT
ejpam-6592	53	21	)	)	PUNCT
ejpam-6592	54	1	⊆	⊆	X
ejpam-6592	54	2	bm	bm	PROPN
ejpam-6592	54	3	(	(	PUNCT
ejpam-6592	54	4	ai	ai	PROPN
ejpam-6592	54	5	,	,	PUNCT
ejpam-6592	54	6	δn	δn	ADJ
ejpam-6592	54	7	)	)	PUNCT
ejpam-6592	54	8	⊆	⊆	NUM
ejpam-6592	54	9	uj	uj	NOUN
ejpam-6592	54	10	,	,	PUNCT
ejpam-6592	54	11	where	where	SCONJ
ejpam-6592	54	12	the	the	DET
ejpam-6592	54	13	first	first	ADJ
ejpam-6592	54	14	inclusion	inclusion	NOUN
ejpam-6592	54	15	uses	use	VERB
ejpam-6592	54	16	the	the	DET
ejpam-6592	54	17	mr	mr	PROPN
ejpam-6592	54	18	-	-	PUNCT
ejpam-6592	54	19	metric	metric	ADJ
ejpam-6592	54	20	inequality	inequality	NOUN
ejpam-6592	54	21	(	(	PUNCT
ejpam-6592	54	22	m4	m4	PROPN
ejpam-6592	54	23	)	)	PUNCT
ejpam-6592	54	24	.	.	PUNCT
ejpam-6592	55	1	for	for	ADP
ejpam-6592	55	2	n	n	CCONJ
ejpam-6592	55	3	large	large	ADJ
ejpam-6592	55	4	enough	enough	ADV
ejpam-6592	55	5	,	,	PUNCT
ejpam-6592	55	6	δn	δn	ADJ
ejpam-6592	55	7	r	r	NOUN
ejpam-6592	55	8	−	−	PROPN
ejpam-6592	55	9	1	1	NUM
ejpam-6592	55	10	n	n	NOUN
ejpam-6592	55	11	>	>	ADP
ejpam-6592	55	12	0	0	NUM
ejpam-6592	55	13	,	,	PUNCT
ejpam-6592	55	14	ensuring	ensure	VERB
ejpam-6592	55	15	δ	δ	PROPN
ejpam-6592	55	16	>	>	X
ejpam-6592	55	17	0	0	X
ejpam-6592	55	18	.	.	PUNCT
ejpam-6592	56	1	lemma	lemma	PROPN
ejpam-6592	56	2	2	2	PROPN
ejpam-6592	56	3	(	(	PUNCT
ejpam-6592	56	4	finite	finite	VERB
ejpam-6592	56	5	ϵ-nets	ϵ-net	NOUN
ejpam-6592	56	6	and	and	CCONJ
ejpam-6592	56	7	sequential	sequential	ADJ
ejpam-6592	56	8	compactness	compactness	NOUN
ejpam-6592	56	9	)	)	PUNCT
ejpam-6592	56	10	.	.	PUNCT
ejpam-6592	57	1	in	in	ADP
ejpam-6592	57	2	an	an	DET
ejpam-6592	57	3	mr	mr	PROPN
ejpam-6592	57	4	-	-	PUNCT
ejpam-6592	57	5	metric	metric	ADJ
ejpam-6592	57	6	space	space	NOUN
ejpam-6592	57	7	(	(	PUNCT
ejpam-6592	57	8	x	x	X
ejpam-6592	57	9	,	,	PUNCT
ejpam-6592	57	10	m	m	NOUN
ejpam-6592	57	11	)	)	PUNCT
ejpam-6592	57	12	,	,	PUNCT
ejpam-6592	57	13	sequential	sequential	ADJ
ejpam-6592	57	14	compactness	compactness	NOUN
ejpam-6592	57	15	implies	imply	VERB
ejpam-6592	57	16	:	:	PUNCT
ejpam-6592	57	17	(	(	PUNCT
ejpam-6592	57	18	i	i	NOUN
ejpam-6592	57	19	)	)	PUNCT
ejpam-6592	57	20	every	every	DET
ejpam-6592	57	21	infinite	infinite	NOUN
ejpam-6592	57	22	subset	subset	NOUN
ejpam-6592	57	23	s	s	PART
ejpam-6592	57	24	⊆	⊆	NUM
ejpam-6592	57	25	x	x	PUNCT
ejpam-6592	57	26	has	have	VERB
ejpam-6592	57	27	an	an	DET
ejpam-6592	57	28	accumulation	accumulation	NOUN
ejpam-6592	57	29	point	point	NOUN
ejpam-6592	57	30	.	.	PUNCT
ejpam-6592	58	1	(	(	PUNCT
ejpam-6592	58	2	ii	ii	NOUN
ejpam-6592	58	3	)	)	PUNCT
ejpam-6592	58	4	for	for	ADP
ejpam-6592	58	5	every	every	DET
ejpam-6592	58	6	ϵ	ϵ	PROPN
ejpam-6592	58	7	>	>	X
ejpam-6592	58	8	0	0	NUM
ejpam-6592	58	9	,	,	PUNCT
ejpam-6592	58	10	there	there	PRON
ejpam-6592	58	11	exists	exist	VERB
ejpam-6592	58	12	a	a	DET
ejpam-6592	58	13	finite	finite	ADJ
ejpam-6592	58	14	ϵ-net	ϵ-net	NOUN
ejpam-6592	58	15	.	.	PUNCT
ejpam-6592	59	1	proof	proof	NOUN
ejpam-6592	59	2	.	.	PUNCT
ejpam-6592	60	1	part	part	NOUN
ejpam-6592	60	2	(	(	PUNCT
ejpam-6592	60	3	a	a	NOUN
ejpam-6592	60	4	):	):	PUNCT
ejpam-6592	60	5	accumulation	accumulation	NOUN
ejpam-6592	60	6	points	point	NOUN
ejpam-6592	60	7	.	.	PUNCT
ejpam-6592	61	1	let	let	VERB
ejpam-6592	61	2	s	s	PRON
ejpam-6592	61	3	⊆	⊆	NUM
ejpam-6592	61	4	x	x	VERB
ejpam-6592	61	5	be	be	AUX
ejpam-6592	61	6	infinite	infinite	ADJ
ejpam-6592	61	7	.	.	PUNCT
ejpam-6592	61	8	construct	construct	VERB
ejpam-6592	61	9	a	a	DET
ejpam-6592	61	10	sequence	sequence	NOUN
ejpam-6592	61	11	(	(	PUNCT
ejpam-6592	61	12	sn)n∈n	sn)n∈n	PROPN
ejpam-6592	61	13	of	of	ADP
ejpam-6592	61	14	distinct	distinct	ADJ
ejpam-6592	61	15	points	point	NOUN
ejpam-6592	61	16	in	in	ADP
ejpam-6592	61	17	s.	s.	PROPN
ejpam-6592	61	18	by	by	ADP
ejpam-6592	61	19	sequential	sequential	ADJ
ejpam-6592	61	20	compactness	compactness	NOUN
ejpam-6592	61	21	,	,	PUNCT
ejpam-6592	61	22	(	(	PUNCT
ejpam-6592	61	23	sn	sn	INTJ
ejpam-6592	61	24	)	)	PUNCT
ejpam-6592	61	25	has	have	VERB
ejpam-6592	61	26	a	a	DET
ejpam-6592	61	27	convergent	convergent	ADJ
ejpam-6592	61	28	subsequence	subsequence	NOUN
ejpam-6592	61	29	(	(	PUNCT
ejpam-6592	61	30	snk	snk	NOUN
ejpam-6592	61	31	)	)	PUNCT
ejpam-6592	61	32	→	→	PUNCT
ejpam-6592	61	33	x.	x.	NOUN
ejpam-6592	61	34	then	then	ADV
ejpam-6592	61	35	,	,	PUNCT
ejpam-6592	61	36	x	x	X
ejpam-6592	61	37	is	be	AUX
ejpam-6592	61	38	an	an	DET
ejpam-6592	61	39	accumulation	accumulation	NOUN
ejpam-6592	61	40	point	point	NOUN
ejpam-6592	61	41	of	of	ADP
ejpam-6592	61	42	s	s	PROPN
ejpam-6592	61	43	,	,	PUNCT
ejpam-6592	61	44	as	as	ADP
ejpam-6592	61	45	every	every	DET
ejpam-6592	61	46	neighborhood	neighborhood	NOUN
ejpam-6592	61	47	of	of	ADP
ejpam-6592	61	48	x	x	PUNCT
ejpam-6592	61	49	contains	contain	VERB
ejpam-6592	61	50	infinitely	infinitely	ADV
ejpam-6592	61	51	many	many	ADJ
ejpam-6592	61	52	snk	snk	NOUN
ejpam-6592	61	53	.	.	PUNCT
ejpam-6592	62	1	part	part	NOUN
ejpam-6592	62	2	(	(	PUNCT
ejpam-6592	62	3	b	b	NOUN
ejpam-6592	62	4	):	):	PUNCT
ejpam-6592	62	5	construction	construction	NOUN
ejpam-6592	62	6	of	of	ADP
ejpam-6592	62	7	ϵ-nets	ϵ-net	NOUN
ejpam-6592	62	8	.	.	PUNCT
ejpam-6592	63	1	fix	fix	VERB
ejpam-6592	63	2	ϵ	ϵ	X
ejpam-6592	63	3	>	>	X
ejpam-6592	63	4	0	0	X
ejpam-6592	63	5	.	.	PUNCT
ejpam-6592	63	6	suppose	suppose	VERB
ejpam-6592	63	7	no	no	DET
ejpam-6592	63	8	finite	finite	NOUN
ejpam-6592	63	9	ϵ-net	ϵ-net	NOUN
ejpam-6592	63	10	exists	exist	VERB
ejpam-6592	63	11	.	.	PUNCT
ejpam-6592	64	1	inductively	inductively	ADV
ejpam-6592	64	2	build	build	VERB
ejpam-6592	64	3	a	a	DET
ejpam-6592	64	4	sequence	sequence	NOUN
ejpam-6592	64	5	(	(	PUNCT
ejpam-6592	64	6	xn	xn	X
ejpam-6592	64	7	)	)	PUNCT
ejpam-6592	64	8	such	such	ADJ
ejpam-6592	64	9	that	that	SCONJ
ejpam-6592	64	10	:	:	PUNCT
ejpam-6592	64	11	m(xn	m(xn	NUM
ejpam-6592	64	12	,	,	PUNCT
ejpam-6592	64	13	xi	xi	PROPN
ejpam-6592	64	14	,	,	PUNCT
ejpam-6592	64	15	xi	xi	PROPN
ejpam-6592	64	16	)	)	PUNCT
ejpam-6592	64	17	≥	≥	NOUN
ejpam-6592	64	18	ϵ	ϵ	ADP
ejpam-6592	64	19	∀i	∀i	NOUN
ejpam-6592	64	20	<	<	X
ejpam-6592	64	21	n.	n.	NOUN
ejpam-6592	64	22	this	this	DET
ejpam-6592	64	23	sequence	sequence	NOUN
ejpam-6592	64	24	has	have	VERB
ejpam-6592	64	25	no	no	DET
ejpam-6592	64	26	cauchy	cauchy	ADJ
ejpam-6592	64	27	subsequence	subsequence	NOUN
ejpam-6592	64	28	(	(	PUNCT
ejpam-6592	64	29	since	since	SCONJ
ejpam-6592	64	30	r	r	NOUN
ejpam-6592	64	31	-	-	PUNCT
ejpam-6592	64	32	scaled	scale	VERB
ejpam-6592	64	33	tetrahedral	tetrahedral	ADJ
ejpam-6592	64	34	inequality	inequality	NOUN
ejpam-6592	64	35	prevents	prevent	VERB
ejpam-6592	64	36	clustering	cluster	VERB
ejpam-6592	64	37	)	)	PUNCT
ejpam-6592	64	38	,	,	PUNCT
ejpam-6592	64	39	contradicting	contradict	VERB
ejpam-6592	64	40	sequential	sequential	ADJ
ejpam-6592	64	41	compactness	compactness	NOUN
ejpam-6592	64	42	.	.	PUNCT
ejpam-6592	65	1	thus	thus	ADV
ejpam-6592	65	2	,	,	PUNCT
ejpam-6592	65	3	a	a	DET
ejpam-6592	65	4	finite	finite	NOUN
ejpam-6592	65	5	ϵ-net	ϵ-net	NOUN
ejpam-6592	65	6	must	must	AUX
ejpam-6592	65	7	exist	exist	VERB
ejpam-6592	65	8	.	.	PUNCT
ejpam-6592	66	1	theorem	theorem	ADJ
ejpam-6592	66	2	1	1	NUM
ejpam-6592	66	3	(	(	PUNCT
ejpam-6592	66	4	characterization	characterization	NOUN
ejpam-6592	66	5	of	of	ADP
ejpam-6592	66	6	compactness	compactness	NOUN
ejpam-6592	66	7	in	in	ADP
ejpam-6592	66	8	mr	mr	PROPN
ejpam-6592	66	9	-	-	PUNCT
ejpam-6592	66	10	metric	metric	ADJ
ejpam-6592	66	11	spaces	space	NOUN
ejpam-6592	66	12	)	)	PUNCT
ejpam-6592	66	13	.	.	PUNCT
ejpam-6592	67	1	let	let	VERB
ejpam-6592	67	2	(	(	PUNCT
ejpam-6592	67	3	x	x	X
ejpam-6592	67	4	,	,	PUNCT
ejpam-6592	67	5	m	m	VERB
ejpam-6592	67	6	)	)	PUNCT
ejpam-6592	67	7	be	be	VERB
ejpam-6592	67	8	an	an	DET
ejpam-6592	67	9	mr	mr	ADJ
ejpam-6592	67	10	-	-	PUNCT
ejpam-6592	67	11	metric	metric	ADJ
ejpam-6592	67	12	space	space	NOUN
ejpam-6592	67	13	with	with	ADP
ejpam-6592	67	14	r	r	NOUN
ejpam-6592	67	15	>	>	X
ejpam-6592	67	16	1	1	NUM
ejpam-6592	67	17	.	.	PUNCT
ejpam-6592	68	1	the	the	DET
ejpam-6592	68	2	following	follow	VERB
ejpam-6592	68	3	are	be	AUX
ejpam-6592	68	4	equivalent	equivalent	ADJ
ejpam-6592	68	5	:	:	PUNCT
ejpam-6592	68	6	a.	a.	NOUN
ejpam-6592	68	7	malkawi	malkawi	PROPN
ejpam-6592	68	8	/	/	SYM
ejpam-6592	68	9	eur	eur	PROPN
ejpam-6592	68	10	.	.	PUNCT
ejpam-6592	69	1	j.	j.	PROPN
ejpam-6592	69	2	pure	pure	PROPN
ejpam-6592	69	3	appl	appl	PROPN
ejpam-6592	69	4	.	.	PROPN
ejpam-6592	69	5	math	math	PROPN
ejpam-6592	69	6	,	,	PUNCT
ejpam-6592	69	7	18	18	NUM
ejpam-6592	69	8	(	(	PUNCT
ejpam-6592	69	9	3	3	NUM
ejpam-6592	69	10	)	)	PUNCT
ejpam-6592	69	11	(	(	PUNCT
ejpam-6592	69	12	2025	2025	NUM
ejpam-6592	69	13	)	)	PUNCT
ejpam-6592	69	14	,	,	PUNCT
ejpam-6592	69	15	6592	6592	NUM
ejpam-6592	69	16	4	4	NUM
ejpam-6592	69	17	of	of	ADP
ejpam-6592	69	18	14	14	NUM
ejpam-6592	69	19	(	(	PUNCT
ejpam-6592	69	20	i	i	NOUN
ejpam-6592	69	21	)	)	PUNCT
ejpam-6592	69	22	x	x	X
ejpam-6592	69	23	is	be	AUX
ejpam-6592	69	24	compact	compact	ADJ
ejpam-6592	69	25	(	(	PUNCT
ejpam-6592	69	26	every	every	DET
ejpam-6592	69	27	open	open	ADJ
ejpam-6592	69	28	cover	cover	NOUN
ejpam-6592	69	29	has	have	VERB
ejpam-6592	69	30	a	a	DET
ejpam-6592	69	31	finite	finite	ADJ
ejpam-6592	69	32	subcover	subcover	PROPN
ejpam-6592	69	33	)	)	PUNCT
ejpam-6592	69	34	.	.	PUNCT
ejpam-6592	70	1	(	(	PUNCT
ejpam-6592	70	2	ii	ii	NOUN
ejpam-6592	70	3	)	)	PUNCT
ejpam-6592	70	4	x	x	VERB
ejpam-6592	70	5	is	be	AUX
ejpam-6592	70	6	sequentially	sequentially	ADV
ejpam-6592	70	7	compact	compact	ADJ
ejpam-6592	70	8	(	(	PUNCT
ejpam-6592	70	9	every	every	DET
ejpam-6592	70	10	sequence	sequence	NOUN
ejpam-6592	70	11	has	have	VERB
ejpam-6592	70	12	a	a	DET
ejpam-6592	70	13	convergent	convergent	NOUN
ejpam-6592	70	14	subsequence	subsequence	NOUN
ejpam-6592	70	15	)	)	PUNCT
ejpam-6592	70	16	.	.	PUNCT
ejpam-6592	71	1	(	(	PUNCT
ejpam-6592	71	2	iii	iii	X
ejpam-6592	71	3	)	)	PUNCT
ejpam-6592	71	4	x	x	PRON
ejpam-6592	71	5	is	be	AUX
ejpam-6592	71	6	complete	complete	ADJ
ejpam-6592	71	7	and	and	CCONJ
ejpam-6592	71	8	totally	totally	ADV
ejpam-6592	71	9	bounded	bounded	ADJ
ejpam-6592	71	10	(	(	PUNCT
ejpam-6592	71	11	for	for	ADP
ejpam-6592	71	12	every	every	PRON
ejpam-6592	71	13	ϵ	ϵ	X
ejpam-6592	71	14	>	>	X
ejpam-6592	71	15	0	0	NUM
ejpam-6592	71	16	,	,	PUNCT
ejpam-6592	71	17	there	there	PRON
ejpam-6592	71	18	exists	exist	VERB
ejpam-6592	71	19	a	a	DET
ejpam-6592	71	20	finite	finite	ADJ
ejpam-6592	71	21	ϵ-net	ϵ-net	NOUN
ejpam-6592	71	22	)	)	PUNCT
ejpam-6592	71	23	.	.	PUNCT
ejpam-6592	72	1	proof	proof	NOUN
ejpam-6592	72	2	.	.	PUNCT
ejpam-6592	73	1	we	we	PRON
ejpam-6592	73	2	prove	prove	VERB
ejpam-6592	73	3	the	the	DET
ejpam-6592	73	4	equivalences	equivalence	NOUN
ejpam-6592	73	5	via	via	ADP
ejpam-6592	73	6	the	the	DET
ejpam-6592	73	7	cycle	cycle	NOUN
ejpam-6592	73	8	(	(	PUNCT
ejpam-6592	73	9	i	i	NOUN
ejpam-6592	73	10	)	)	PUNCT
ejpam-6592	73	11	⇒	⇒	PROPN
ejpam-6592	73	12	(	(	PUNCT
ejpam-6592	73	13	ii	ii	NOUN
ejpam-6592	73	14	)	)	PUNCT
ejpam-6592	73	15	⇒	⇒	NOUN
ejpam-6592	73	16	(	(	PUNCT
ejpam-6592	73	17	iii	iii	NOUN
ejpam-6592	73	18	)	)	PUNCT
ejpam-6592	73	19	⇒	⇒	NOUN
ejpam-6592	73	20	(	(	PUNCT
ejpam-6592	73	21	i	i	NOUN
ejpam-6592	73	22	)	)	PUNCT
ejpam-6592	73	23	.	.	PUNCT
ejpam-6592	74	1	part	part	NOUN
ejpam-6592	74	2	1	1	NUM
ejpam-6592	74	3	:	:	PUNCT
ejpam-6592	74	4	(	(	PUNCT
ejpam-6592	74	5	i	i	NOUN
ejpam-6592	74	6	)	)	PUNCT
ejpam-6592	74	7	⇒	⇒	PROPN
ejpam-6592	74	8	(	(	PUNCT
ejpam-6592	74	9	ii	ii	NOUN
ejpam-6592	74	10	)	)	PUNCT
ejpam-6592	74	11	(	(	PUNCT
ejpam-6592	74	12	compactness	compactness	NOUN
ejpam-6592	74	13	implies	imply	VERB
ejpam-6592	74	14	sequential	sequential	ADJ
ejpam-6592	74	15	compactness	compactness	NOUN
ejpam-6592	74	16	)	)	PUNCT
ejpam-6592	74	17	.	.	PUNCT
ejpam-6592	75	1	•	•	NUM
ejpam-6592	75	2	let	let	VERB
ejpam-6592	75	3	(	(	PUNCT
ejpam-6592	75	4	xn)n∈n	xn)n∈n	NUM
ejpam-6592	75	5	be	be	AUX
ejpam-6592	75	6	a	a	DET
ejpam-6592	75	7	sequence	sequence	NOUN
ejpam-6592	75	8	in	in	ADP
ejpam-6592	75	9	x.	x.	PROPN
ejpam-6592	75	10	suppose	suppose	VERB
ejpam-6592	75	11	for	for	ADP
ejpam-6592	75	12	contradiction	contradiction	NOUN
ejpam-6592	75	13	that	that	PRON
ejpam-6592	75	14	no	no	DET
ejpam-6592	75	15	subsequence	subsequence	NOUN
ejpam-6592	75	16	converges	converge	VERB
ejpam-6592	75	17	.	.	PUNCT
ejpam-6592	76	1	•	•	NUM
ejpam-6592	76	2	for	for	ADP
ejpam-6592	76	3	each	each	DET
ejpam-6592	76	4	x	x	SYM
ejpam-6592	76	5	∈	∈	PROPN
ejpam-6592	76	6	x	x	X
ejpam-6592	76	7	,	,	PUNCT
ejpam-6592	76	8	there	there	PRON
ejpam-6592	76	9	exists	exist	VERB
ejpam-6592	76	10	an	an	DET
ejpam-6592	76	11	open	open	ADJ
ejpam-6592	76	12	ball	ball	NOUN
ejpam-6592	76	13	bx	bx	NOUN
ejpam-6592	76	14	=	=	SYM
ejpam-6592	76	15	bm	bm	PROPN
ejpam-6592	76	16	(	(	PUNCT
ejpam-6592	76	17	x	x	NOUN
ejpam-6592	76	18	,	,	PUNCT
ejpam-6592	76	19	ϵx	ϵx	NOUN
ejpam-6592	76	20	)	)	PUNCT
ejpam-6592	76	21	containing	contain	VERB
ejpam-6592	76	22	only	only	ADV
ejpam-6592	76	23	finitely	finitely	ADV
ejpam-6592	76	24	many	many	ADJ
ejpam-6592	76	25	xn	xn	PUNCT
ejpam-6592	77	1	(	(	PUNCT
ejpam-6592	77	2	otherwise	otherwise	ADV
ejpam-6592	77	3	,	,	PUNCT
ejpam-6592	77	4	a	a	DET
ejpam-6592	77	5	convergent	convergent	NOUN
ejpam-6592	77	6	subsequence	subsequence	NOUN
ejpam-6592	77	7	exists	exist	VERB
ejpam-6592	77	8	)	)	PUNCT
ejpam-6592	77	9	.	.	PUNCT
ejpam-6592	78	1	•	•	NUM
ejpam-6592	78	2	the	the	DET
ejpam-6592	78	3	collection	collection	NOUN
ejpam-6592	78	4	{	{	PUNCT
ejpam-6592	78	5	bx	bx	NOUN
ejpam-6592	79	1	|	|	ADV
ejpam-6592	79	2	x	x	SYM
ejpam-6592	79	3	∈	∈	PROPN
ejpam-6592	79	4	x	x	X
ejpam-6592	79	5	}	}	PUNCT
ejpam-6592	79	6	is	be	AUX
ejpam-6592	79	7	an	an	DET
ejpam-6592	79	8	open	open	ADJ
ejpam-6592	79	9	cover	cover	NOUN
ejpam-6592	79	10	of	of	ADP
ejpam-6592	79	11	x.	x.	NOUN
ejpam-6592	79	12	by	by	ADP
ejpam-6592	79	13	compactness	compactness	NOUN
ejpam-6592	79	14	,	,	PUNCT
ejpam-6592	79	15	there	there	PRON
ejpam-6592	79	16	exists	exist	VERB
ejpam-6592	79	17	a	a	DET
ejpam-6592	79	18	finite	finite	ADJ
ejpam-6592	79	19	subcover	subcover	PROPN
ejpam-6592	79	20	{	{	PUNCT
ejpam-6592	79	21	bx1	bx1	VERB
ejpam-6592	79	22	,	,	PUNCT
ejpam-6592	79	23	.	.	PUNCT
ejpam-6592	79	24	.	.	PUNCT
ejpam-6592	80	1	.	.	PUNCT
ejpam-6592	81	1	,	,	PUNCT
ejpam-6592	81	2	bxk	bxk	NOUN
ejpam-6592	81	3	}	}	PUNCT
ejpam-6592	81	4	.	.	PUNCT
ejpam-6592	82	1	•	•	INTJ
ejpam-6592	82	2	but	but	CCONJ
ejpam-6592	82	3	each	each	DET
ejpam-6592	82	4	bxi	bxi	NOUN
ejpam-6592	82	5	contains	contain	VERB
ejpam-6592	82	6	finitely	finitely	ADV
ejpam-6592	82	7	many	many	ADJ
ejpam-6592	82	8	xn	xn	NOUN
ejpam-6592	82	9	,	,	PUNCT
ejpam-6592	82	10	so	so	ADV
ejpam-6592	82	11	x	x	PUNCT
ejpam-6592	82	12	contains	contain	VERB
ejpam-6592	82	13	finitely	finitely	ADV
ejpam-6592	82	14	many	many	ADJ
ejpam-6592	82	15	xn	xn	NOUN
ejpam-6592	82	16	,	,	PUNCT
ejpam-6592	82	17	a	a	DET
ejpam-6592	82	18	contradiction	contradiction	NOUN
ejpam-6592	82	19	.	.	PUNCT
ejpam-6592	83	1	part	part	NOUN
ejpam-6592	83	2	2	2	NUM
ejpam-6592	83	3	:	:	PUNCT
ejpam-6592	83	4	(	(	PUNCT
ejpam-6592	83	5	ii	ii	NOUN
ejpam-6592	83	6	)	)	PUNCT
ejpam-6592	83	7	⇒	⇒	NOUN
ejpam-6592	83	8	(	(	PUNCT
ejpam-6592	83	9	iii	iii	NOUN
ejpam-6592	83	10	)	)	PUNCT
ejpam-6592	83	11	(	(	PUNCT
ejpam-6592	83	12	sequential	sequential	ADJ
ejpam-6592	83	13	compactness	compactness	NOUN
ejpam-6592	83	14	implies	imply	VERB
ejpam-6592	83	15	completeness	completeness	NOUN
ejpam-6592	83	16	and	and	CCONJ
ejpam-6592	83	17	total	total	ADJ
ejpam-6592	83	18	boundedness	boundedness	NOUN
ejpam-6592	83	19	)	)	PUNCT
ejpam-6592	83	20	.	.	PUNCT
ejpam-6592	84	1	•	•	NUM
ejpam-6592	84	2	completeness	completeness	NOUN
ejpam-6592	84	3	:	:	PUNCT
ejpam-6592	84	4	let	let	VERB
ejpam-6592	84	5	(	(	PUNCT
ejpam-6592	84	6	xn	xn	X
ejpam-6592	84	7	)	)	PUNCT
ejpam-6592	84	8	be	be	VERB
ejpam-6592	84	9	a	a	DET
ejpam-6592	84	10	cauchy	cauchy	ADJ
ejpam-6592	84	11	sequence	sequence	NOUN
ejpam-6592	84	12	.	.	PUNCT
ejpam-6592	85	1	by	by	ADP
ejpam-6592	85	2	sequential	sequential	ADJ
ejpam-6592	85	3	compactness	compactness	NOUN
ejpam-6592	85	4	,	,	PUNCT
ejpam-6592	85	5	it	it	PRON
ejpam-6592	85	6	has	have	VERB
ejpam-6592	85	7	a	a	DET
ejpam-6592	85	8	convergent	convergent	ADJ
ejpam-6592	85	9	subsequence	subsequence	NOUN
ejpam-6592	85	10	(	(	PUNCT
ejpam-6592	85	11	xnk	xnk	PROPN
ejpam-6592	85	12	)	)	PUNCT
ejpam-6592	85	13	→	→	PUNCT
ejpam-6592	85	14	x.	x.	NOUN
ejpam-6592	85	15	then	then	ADV
ejpam-6592	85	16	,	,	PUNCT
ejpam-6592	85	17	the	the	DET
ejpam-6592	85	18	entire	entire	ADJ
ejpam-6592	85	19	sequence	sequence	NOUN
ejpam-6592	85	20	(	(	PUNCT
ejpam-6592	85	21	xn	xn	X
ejpam-6592	85	22	)	)	PUNCT
ejpam-6592	85	23	converges	converge	VERB
ejpam-6592	85	24	to	to	ADP
ejpam-6592	85	25	x	x	SYM
ejpam-6592	85	26	(	(	PUNCT
ejpam-6592	85	27	standard	standard	ADJ
ejpam-6592	85	28	argument	argument	NOUN
ejpam-6592	85	29	)	)	PUNCT
ejpam-6592	85	30	.	.	PUNCT
ejpam-6592	86	1	•	•	NUM
ejpam-6592	86	2	total	total	ADJ
ejpam-6592	86	3	boundedness	boundedness	NOUN
ejpam-6592	86	4	:	:	PUNCT
ejpam-6592	86	5	fix	fix	VERB
ejpam-6592	86	6	ϵ	ϵ	X
ejpam-6592	86	7	>	>	X
ejpam-6592	86	8	0	0	X
ejpam-6592	86	9	.	.	PUNCT
ejpam-6592	86	10	suppose	suppose	VERB
ejpam-6592	86	11	x	x	PRON
ejpam-6592	86	12	has	have	VERB
ejpam-6592	86	13	no	no	DET
ejpam-6592	86	14	finite	finite	ADJ
ejpam-6592	86	15	ϵ-net	ϵ-net	NOUN
ejpam-6592	86	16	.	.	PUNCT
ejpam-6592	87	1	inductively	inductively	ADV
ejpam-6592	87	2	construct	construct	VERB
ejpam-6592	87	3	a	a	DET
ejpam-6592	87	4	sequence	sequence	NOUN
ejpam-6592	87	5	(	(	PUNCT
ejpam-6592	87	6	xn	xn	X
ejpam-6592	87	7	)	)	PUNCT
ejpam-6592	87	8	such	such	ADJ
ejpam-6592	87	9	that	that	SCONJ
ejpam-6592	87	10	:	:	PUNCT
ejpam-6592	87	11	m(xn	m(xn	NUM
ejpam-6592	87	12	,	,	PUNCT
ejpam-6592	87	13	xi	xi	PROPN
ejpam-6592	87	14	,	,	PUNCT
ejpam-6592	87	15	xi	xi	PROPN
ejpam-6592	87	16	)	)	PUNCT
ejpam-6592	87	17	≥	≥	NOUN
ejpam-6592	87	18	ϵ	ϵ	ADP
ejpam-6592	87	19	∀i	∀i	NOUN
ejpam-6592	87	20	<	<	X
ejpam-6592	87	21	n.	n.	NOUN
ejpam-6592	87	22	this	this	DET
ejpam-6592	87	23	sequence	sequence	NOUN
ejpam-6592	87	24	has	have	VERB
ejpam-6592	87	25	no	no	DET
ejpam-6592	87	26	convergent	convergent	NOUN
ejpam-6592	87	27	subsequence	subsequence	NOUN
ejpam-6592	87	28	,	,	PUNCT
ejpam-6592	87	29	contradicting	contradict	VERB
ejpam-6592	87	30	sequential	sequential	ADJ
ejpam-6592	87	31	compactness	compactness	NOUN
ejpam-6592	87	32	.	.	PUNCT
ejpam-6592	88	1	part	part	NOUN
ejpam-6592	88	2	3	3	NUM
ejpam-6592	88	3	:	:	PUNCT
ejpam-6592	88	4	(	(	PUNCT
ejpam-6592	88	5	iii	iii	X
ejpam-6592	88	6	)	)	PUNCT
ejpam-6592	88	7	⇒	⇒	NOUN
ejpam-6592	88	8	(	(	PUNCT
ejpam-6592	88	9	i	i	NOUN
ejpam-6592	88	10	)	)	PUNCT
ejpam-6592	88	11	(	(	PUNCT
ejpam-6592	88	12	complete	complete	ADJ
ejpam-6592	88	13	+	+	CCONJ
ejpam-6592	88	14	totally	totally	ADV
ejpam-6592	88	15	bounded	bounded	ADJ
ejpam-6592	88	16	implies	imply	VERB
ejpam-6592	88	17	compactness	compactness	NOUN
ejpam-6592	88	18	)	)	PUNCT
ejpam-6592	88	19	.	.	PUNCT
ejpam-6592	89	1	•	•	INTJ
ejpam-6592	89	2	let	let	VERB
ejpam-6592	89	3	u	u	PRON
ejpam-6592	89	4	=	=	PUNCT
ejpam-6592	89	5	{	{	PUNCT
ejpam-6592	89	6	ui}i∈i	ui}i∈i	INTJ
ejpam-6592	89	7	be	be	AUX
ejpam-6592	89	8	an	an	DET
ejpam-6592	89	9	open	open	ADJ
ejpam-6592	89	10	cover	cover	NOUN
ejpam-6592	89	11	of	of	ADP
ejpam-6592	89	12	x.	x.	NOUN
ejpam-6592	89	13	we	we	PRON
ejpam-6592	89	14	construct	construct	VERB
ejpam-6592	89	15	a	a	DET
ejpam-6592	89	16	finite	finite	PROPN
ejpam-6592	89	17	subcover	subcover	PROPN
ejpam-6592	89	18	.	.	PUNCT
ejpam-6592	90	1	•	•	NUM
ejpam-6592	90	2	step	step	NOUN
ejpam-6592	90	3	1	1	NUM
ejpam-6592	90	4	:	:	PUNCT
ejpam-6592	90	5	lebesgue	lebesgue	NOUN
ejpam-6592	90	6	number	number	NOUN
ejpam-6592	90	7	lemma	lemma	PROPN
ejpam-6592	90	8	.	.	PUNCT
ejpam-6592	91	1	by	by	ADP
ejpam-6592	91	2	total	total	ADJ
ejpam-6592	91	3	boundedness	boundedness	NOUN
ejpam-6592	91	4	,	,	PUNCT
ejpam-6592	91	5	for	for	ADP
ejpam-6592	91	6	each	each	DET
ejpam-6592	91	7	n	n	PRON
ejpam-6592	91	8	∈	∈	PROPN
ejpam-6592	91	9	n	n	CCONJ
ejpam-6592	91	10	,	,	PUNCT
ejpam-6592	91	11	there	there	PRON
ejpam-6592	91	12	exists	exist	VERB
ejpam-6592	91	13	a	a	DET
ejpam-6592	91	14	finite	finite	ADJ
ejpam-6592	91	15	1	1	NUM
ejpam-6592	91	16	/	/	SYM
ejpam-6592	91	17	n	n	CCONJ
ejpam-6592	91	18	-	-	PUNCT
ejpam-6592	91	19	net	net	NOUN
ejpam-6592	91	20	an	an	PROPN
ejpam-6592	91	21	.	.	PUNCT
ejpam-6592	91	22	define	define	NOUN
ejpam-6592	91	23	:	:	PUNCT
ejpam-6592	91	24	δn	δn	PROPN
ejpam-6592	91	25	=	=	PUNCT
ejpam-6592	91	26	inf	inf	PROPN
ejpam-6592	91	27	x∈x	x∈x	PROPN
ejpam-6592	91	28	sup	sup	PROPN
ejpam-6592	91	29	a∈an	a∈an	PROPN
ejpam-6592	91	30	m(x	m(x	PROPN
ejpam-6592	91	31	,	,	PUNCT
ejpam-6592	91	32	a	a	PRON
ejpam-6592	91	33	,	,	PUNCT
ejpam-6592	91	34	a	a	NOUN
ejpam-6592	91	35	)	)	PUNCT
ejpam-6592	91	36	.	.	PUNCT
ejpam-6592	92	1	using	use	VERB
ejpam-6592	92	2	the	the	DET
ejpam-6592	92	3	mr	mr	PROPN
ejpam-6592	92	4	-	-	PUNCT
ejpam-6592	92	5	metric	metric	ADJ
ejpam-6592	92	6	axioms	axiom	NOUN
ejpam-6592	92	7	,	,	PUNCT
ejpam-6592	92	8	we	we	PRON
ejpam-6592	92	9	show	show	VERB
ejpam-6592	92	10	δn	δn	NOUN
ejpam-6592	92	11	→	→	SYM
ejpam-6592	92	12	0	0	NUM
ejpam-6592	92	13	.	.	PUNCT
ejpam-6592	93	1	thus	thus	ADV
ejpam-6592	93	2	,	,	PUNCT
ejpam-6592	93	3	there	there	PRON
ejpam-6592	93	4	exists	exist	VERB
ejpam-6592	93	5	a	a	DET
ejpam-6592	93	6	lebesgue	lebesgue	NOUN
ejpam-6592	93	7	number	number	NOUN
ejpam-6592	93	8	δ	δ	PROPN
ejpam-6592	93	9	>	>	X
ejpam-6592	93	10	0	0	NUM
ejpam-6592	94	1	such	such	ADJ
ejpam-6592	94	2	that	that	SCONJ
ejpam-6592	94	3	every	every	DET
ejpam-6592	94	4	δ	δ	PROPN
ejpam-6592	94	5	-	-	PUNCT
ejpam-6592	94	6	ball	ball	NOUN
ejpam-6592	94	7	lies	lie	VERB
ejpam-6592	94	8	in	in	ADP
ejpam-6592	94	9	some	some	DET
ejpam-6592	94	10	ui	ui	NOUN
ejpam-6592	94	11	.	.	PROPN
ejpam-6592	95	1	•	•	NUM
ejpam-6592	95	2	step	step	NOUN
ejpam-6592	95	3	2	2	NUM
ejpam-6592	95	4	:	:	PUNCT
ejpam-6592	95	5	finite	finite	PROPN
ejpam-6592	95	6	subcover	subcover	PROPN
ejpam-6592	95	7	.	.	PUNCT
ejpam-6592	96	1	for	for	ADP
ejpam-6592	96	2	δ	δ	PROPN
ejpam-6592	96	3	as	as	ADP
ejpam-6592	96	4	above	above	ADV
ejpam-6592	96	5	,	,	PUNCT
ejpam-6592	96	6	total	total	ADJ
ejpam-6592	96	7	boundedness	boundedness	NOUN
ejpam-6592	96	8	yields	yield	VERB
ejpam-6592	96	9	a	a	DET
ejpam-6592	96	10	finite	finite	ADJ
ejpam-6592	96	11	δ/2	δ/2	NOUN
ejpam-6592	96	12	-	-	PUNCT
ejpam-6592	96	13	net	net	ADJ
ejpam-6592	96	14	{	{	PUNCT
ejpam-6592	96	15	y1	y1	NOUN
ejpam-6592	96	16	,	,	PUNCT
ejpam-6592	96	17	.	.	PUNCT
ejpam-6592	96	18	.	.	PUNCT
ejpam-6592	97	1	.	.	PUNCT
ejpam-6592	98	1	,	,	PUNCT
ejpam-6592	98	2	yk	yk	PROPN
ejpam-6592	98	3	}	}	PUNCT
ejpam-6592	98	4	.	.	PUNCT
ejpam-6592	99	1	each	each	DET
ejpam-6592	99	2	bm	bm	PROPN
ejpam-6592	99	3	(	(	PUNCT
ejpam-6592	99	4	yi	yi	PROPN
ejpam-6592	99	5	,	,	PUNCT
ejpam-6592	99	6	δ/2	δ/2	NUM
ejpam-6592	99	7	)	)	PUNCT
ejpam-6592	99	8	is	be	AUX
ejpam-6592	99	9	contained	contain	VERB
ejpam-6592	99	10	in	in	ADP
ejpam-6592	99	11	some	some	DET
ejpam-6592	99	12	ui	ui	NOUN
ejpam-6592	99	13	.	.	PUNCT
ejpam-6592	100	1	the	the	DET
ejpam-6592	100	2	union	union	NOUN
ejpam-6592	100	3	of	of	ADP
ejpam-6592	100	4	these	these	DET
ejpam-6592	100	5	ui	ui	PROPN
ejpam-6592	100	6	covers	cover	VERB
ejpam-6592	100	7	x.	x.	NOUN
ejpam-6592	100	8	a.	a.	NOUN
ejpam-6592	100	9	malkawi	malkawi	PROPN
ejpam-6592	100	10	/	/	SYM
ejpam-6592	100	11	eur	eur	PROPN
ejpam-6592	100	12	.	.	PUNCT
ejpam-6592	101	1	j.	j.	PROPN
ejpam-6592	101	2	pure	pure	PROPN
ejpam-6592	101	3	appl	appl	PROPN
ejpam-6592	101	4	.	.	PROPN
ejpam-6592	101	5	math	math	PROPN
ejpam-6592	101	6	,	,	PUNCT
ejpam-6592	101	7	18	18	NUM
ejpam-6592	101	8	(	(	PUNCT
ejpam-6592	101	9	3	3	NUM
ejpam-6592	101	10	)	)	PUNCT
ejpam-6592	101	11	(	(	PUNCT
ejpam-6592	101	12	2025	2025	NUM
ejpam-6592	101	13	)	)	PUNCT
ejpam-6592	101	14	,	,	PUNCT
ejpam-6592	101	15	6592	6592	NUM
ejpam-6592	101	16	5	5	NUM
ejpam-6592	101	17	of	of	ADP
ejpam-6592	101	18	14	14	NUM
ejpam-6592	101	19	corollary	corollary	ADJ
ejpam-6592	101	20	1	1	NUM
ejpam-6592	101	21	(	(	PUNCT
ejpam-6592	101	22	uniform	uniform	ADJ
ejpam-6592	101	23	continuity	continuity	NOUN
ejpam-6592	101	24	in	in	ADP
ejpam-6592	101	25	compact	compact	ADJ
ejpam-6592	101	26	mr	mr	ADJ
ejpam-6592	101	27	-	-	PUNCT
ejpam-6592	101	28	metric	metric	ADJ
ejpam-6592	101	29	spaces	space	NOUN
ejpam-6592	101	30	)	)	PUNCT
ejpam-6592	101	31	.	.	PUNCT
ejpam-6592	102	1	let	let	VERB
ejpam-6592	102	2	(	(	PUNCT
ejpam-6592	102	3	x	x	X
ejpam-6592	102	4	,	,	PUNCT
ejpam-6592	102	5	m	m	VERB
ejpam-6592	102	6	)	)	PUNCT
ejpam-6592	102	7	be	be	AUX
ejpam-6592	102	8	a	a	DET
ejpam-6592	102	9	compact	compact	ADJ
ejpam-6592	102	10	mr	mr	ADJ
ejpam-6592	102	11	-	-	PUNCT
ejpam-6592	102	12	metric	metric	ADJ
ejpam-6592	102	13	space	space	NOUN
ejpam-6592	102	14	with	with	ADP
ejpam-6592	102	15	r	r	NOUN
ejpam-6592	102	16	>	>	X
ejpam-6592	102	17	1	1	NUM
ejpam-6592	102	18	.	.	PUNCT
ejpam-6592	103	1	if	if	SCONJ
ejpam-6592	103	2	f	f	PROPN
ejpam-6592	103	3	:	:	PUNCT
ejpam-6592	103	4	x	x	X
ejpam-6592	103	5	→	→	SYM
ejpam-6592	103	6	r	r	NOUN
ejpam-6592	103	7	is	be	AUX
ejpam-6592	103	8	continuous	continuous	ADJ
ejpam-6592	103	9	,	,	PUNCT
ejpam-6592	103	10	then	then	ADV
ejpam-6592	103	11	f	f	PROPN
ejpam-6592	103	12	is	be	AUX
ejpam-6592	103	13	uniformly	uniformly	ADV
ejpam-6592	103	14	continuous	continuous	ADJ
ejpam-6592	103	15	.	.	PUNCT
ejpam-6592	104	1	proof	proof	NOUN
ejpam-6592	104	2	.	.	PUNCT
ejpam-6592	105	1	we	we	PRON
ejpam-6592	105	2	prove	prove	VERB
ejpam-6592	105	3	that	that	SCONJ
ejpam-6592	105	4	for	for	ADP
ejpam-6592	105	5	every	every	DET
ejpam-6592	105	6	ϵ	ϵ	X
ejpam-6592	105	7	>	>	X
ejpam-6592	105	8	0	0	NUM
ejpam-6592	105	9	,	,	PUNCT
ejpam-6592	105	10	there	there	PRON
ejpam-6592	105	11	exists	exist	VERB
ejpam-6592	105	12	δ	δ	PROPN
ejpam-6592	105	13	>	>	X
ejpam-6592	105	14	0	0	NUM
ejpam-6592	105	15	such	such	ADJ
ejpam-6592	105	16	that	that	SCONJ
ejpam-6592	105	17	:	:	PUNCT
ejpam-6592	105	18	m(x	m(x	PROPN
ejpam-6592	105	19	,	,	PUNCT
ejpam-6592	105	20	y	y	PROPN
ejpam-6592	105	21	,	,	PUNCT
ejpam-6592	105	22	y	y	PROPN
ejpam-6592	105	23	)	)	PUNCT
ejpam-6592	105	24	<	<	X
ejpam-6592	105	25	δ	δ	PROPN
ejpam-6592	106	1	=	=	NOUN
ejpam-6592	106	2	⇒	⇒	VERB
ejpam-6592	106	3	|f(x)−	|f(x)−	NOUN
ejpam-6592	106	4	f(y)|	f(y)|	PROPN
ejpam-6592	106	5	<	<	X
ejpam-6592	106	6	ϵ	ϵ	X
ejpam-6592	106	7	∀x	∀x	PROPN
ejpam-6592	106	8	,	,	PUNCT
ejpam-6592	106	9	y	y	PROPN
ejpam-6592	106	10	∈	∈	PROPN
ejpam-6592	106	11	x.	x.	NOUN
ejpam-6592	106	12	step	step	NOUN
ejpam-6592	106	13	1	1	NUM
ejpam-6592	106	14	:	:	PUNCT
ejpam-6592	106	15	leverage	leverage	NOUN
ejpam-6592	106	16	pointwise	pointwise	VERB
ejpam-6592	106	17	continuity	continuity	NOUN
ejpam-6592	106	18	since	since	SCONJ
ejpam-6592	106	19	f	f	PROPN
ejpam-6592	106	20	is	be	AUX
ejpam-6592	106	21	continuous	continuous	ADJ
ejpam-6592	106	22	,	,	PUNCT
ejpam-6592	106	23	for	for	ADP
ejpam-6592	106	24	each	each	DET
ejpam-6592	106	25	p	p	NOUN
ejpam-6592	106	26	∈	∈	PROPN
ejpam-6592	106	27	x	x	PUNCT
ejpam-6592	106	28	there	there	PRON
ejpam-6592	106	29	exists	exist	VERB
ejpam-6592	106	30	δp	δp	ADV
ejpam-6592	106	31	>	>	X
ejpam-6592	106	32	0	0	NUM
ejpam-6592	107	1	such	such	ADJ
ejpam-6592	107	2	that	that	SCONJ
ejpam-6592	107	3	:	:	PUNCT
ejpam-6592	107	4	x	x	X
ejpam-6592	107	5	∈	∈	X
ejpam-6592	107	6	bm	bm	PROPN
ejpam-6592	107	7	(	(	PUNCT
ejpam-6592	107	8	p	p	X
ejpam-6592	107	9	,	,	PUNCT
ejpam-6592	107	10	δp	δp	NOUN
ejpam-6592	107	11	)	)	PUNCT
ejpam-6592	107	12	=	=	AUX
ejpam-6592	107	13	⇒	⇒	VERB
ejpam-6592	107	14	|f(x)−	|f(x)−	PROPN
ejpam-6592	107	15	f(p)|	f(p)|	NOUN
ejpam-6592	107	16	<	<	X
ejpam-6592	107	17	ϵ	ϵ	X
ejpam-6592	107	18	2	2	NUM
ejpam-6592	107	19	.	.	PUNCT
ejpam-6592	108	1	step	step	NOUN
ejpam-6592	108	2	2	2	NUM
ejpam-6592	108	3	:	:	PUNCT
ejpam-6592	108	4	construct	construct	VERB
ejpam-6592	108	5	open	open	ADJ
ejpam-6592	108	6	cover	cover	VERB
ejpam-6592	108	7	the	the	DET
ejpam-6592	108	8	collection	collection	NOUN
ejpam-6592	108	9	{	{	PUNCT
ejpam-6592	108	10	bm	bm	PROPN
ejpam-6592	108	11	(	(	PUNCT
ejpam-6592	108	12	p	p	X
ejpam-6592	108	13	,	,	PUNCT
ejpam-6592	108	14	δp	δp	PRON
ejpam-6592	108	15	2r	2r	NUM
ejpam-6592	108	16	)	)	PUNCT
ejpam-6592	108	17	}	}	PUNCT
ejpam-6592	108	18	p∈x	p∈x	ADJ
ejpam-6592	108	19	forms	form	NOUN
ejpam-6592	108	20	an	an	DET
ejpam-6592	108	21	open	open	ADJ
ejpam-6592	108	22	cover	cover	NOUN
ejpam-6592	108	23	of	of	ADP
ejpam-6592	108	24	the	the	DET
ejpam-6592	108	25	compact	compact	ADJ
ejpam-6592	108	26	space	space	NOUN
ejpam-6592	108	27	x.	x.	NOUN
ejpam-6592	108	28	by	by	ADP
ejpam-6592	108	29	compactness	compactness	NOUN
ejpam-6592	108	30	,	,	PUNCT
ejpam-6592	108	31	there	there	PRON
ejpam-6592	108	32	exists	exist	VERB
ejpam-6592	108	33	a	a	DET
ejpam-6592	108	34	finite	finite	ADJ
ejpam-6592	108	35	subcover	subcover	NOUN
ejpam-6592	108	36	:	:	PUNCT
ejpam-6592	108	37	x	x	SYM
ejpam-6592	109	1	⊆	⊆	NUM
ejpam-6592	109	2	n⋃	n⋃	PRON
ejpam-6592	109	3	i=1	i=1	PROPN
ejpam-6592	109	4	bm	bm	PROPN
ejpam-6592	109	5	(	(	PUNCT
ejpam-6592	109	6	pi	pi	NOUN
ejpam-6592	109	7	,	,	PUNCT
ejpam-6592	109	8	δpi	δpi	NOUN
ejpam-6592	109	9	2r	2r	NUM
ejpam-6592	109	10	)	)	PUNCT
ejpam-6592	109	11	.	.	PUNCT
ejpam-6592	110	1	step	step	NOUN
ejpam-6592	110	2	3	3	NUM
ejpam-6592	110	3	:	:	PUNCT
ejpam-6592	110	4	determine	determine	VERB
ejpam-6592	110	5	uniform	uniform	PROPN
ejpam-6592	110	6	δ	δ	PROPN
ejpam-6592	110	7	let	let	VERB
ejpam-6592	110	8	δ	δ	PROPN
ejpam-6592	110	9	=	=	SYM
ejpam-6592	110	10	min	min	PROPN
ejpam-6592	110	11	{	{	PUNCT
ejpam-6592	110	12	δpi	δpi	NOUN
ejpam-6592	110	13	2r	2r	NUM
ejpam-6592	110	14	:	:	PUNCT
ejpam-6592	111	1	1	1	NUM
ejpam-6592	111	2	≤	≤	NUM
ejpam-6592	111	3	i	i	PRON
ejpam-6592	111	4	≤	≤	NOUN
ejpam-6592	111	5	n	n	CCONJ
ejpam-6592	111	6	}	}	PUNCT
ejpam-6592	111	7	>	>	X
ejpam-6592	111	8	0	0	X
ejpam-6592	111	9	.	.	PUNCT
ejpam-6592	112	1	step	step	NOUN
ejpam-6592	112	2	4	4	NUM
ejpam-6592	112	3	:	:	PUNCT
ejpam-6592	112	4	verify	verify	VERB
ejpam-6592	112	5	uniform	uniform	ADJ
ejpam-6592	112	6	continuity	continuity	NOUN
ejpam-6592	112	7	for	for	ADP
ejpam-6592	112	8	any	any	DET
ejpam-6592	112	9	x	x	NOUN
ejpam-6592	112	10	,	,	PUNCT
ejpam-6592	112	11	y	y	PROPN
ejpam-6592	112	12	∈	∈	PROPN
ejpam-6592	112	13	x	x	PUNCT
ejpam-6592	112	14	with	with	ADP
ejpam-6592	112	15	m(x	m(x	PROPN
ejpam-6592	112	16	,	,	PUNCT
ejpam-6592	112	17	y	y	PROPN
ejpam-6592	112	18	,	,	PUNCT
ejpam-6592	112	19	y	y	PROPN
ejpam-6592	112	20	)	)	PUNCT
ejpam-6592	112	21	<	<	X
ejpam-6592	112	22	δ	δ	PROPN
ejpam-6592	112	23	:	:	PUNCT
ejpam-6592	112	24	(	(	PUNCT
ejpam-6592	112	25	i	i	NOUN
ejpam-6592	112	26	)	)	PUNCT
ejpam-6592	112	27	choose	choose	VERB
ejpam-6592	112	28	pi	pi	NOUN
ejpam-6592	112	29	such	such	ADJ
ejpam-6592	112	30	that	that	SCONJ
ejpam-6592	112	31	x	x	SYM
ejpam-6592	112	32	∈	∈	PROPN
ejpam-6592	112	33	bm	bm	PROPN
ejpam-6592	112	34	(	(	PUNCT
ejpam-6592	112	35	pi	pi	NOUN
ejpam-6592	112	36	,	,	PUNCT
ejpam-6592	112	37	δpi	δpi	NOUN
ejpam-6592	112	38	2r	2r	NUM
ejpam-6592	112	39	)	)	PUNCT
ejpam-6592	112	40	(	(	PUNCT
ejpam-6592	112	41	possible	possible	ADJ
ejpam-6592	112	42	by	by	ADP
ejpam-6592	112	43	subcover	subcover	PROPN
ejpam-6592	112	44	)	)	PUNCT
ejpam-6592	112	45	.	.	PUNCT
ejpam-6592	113	1	(	(	PUNCT
ejpam-6592	113	2	ii	ii	NOUN
ejpam-6592	113	3	)	)	PUNCT
ejpam-6592	113	4	by	by	ADP
ejpam-6592	113	5	the	the	DET
ejpam-6592	113	6	mr	mr	PROPN
ejpam-6592	113	7	-	-	PUNCT
ejpam-6592	113	8	metric	metric	ADJ
ejpam-6592	113	9	inequality	inequality	NOUN
ejpam-6592	113	10	(	(	PUNCT
ejpam-6592	113	11	axiom	axiom	PROPN
ejpam-6592	113	12	m4	m4	PROPN
ejpam-6592	113	13	):	):	PUNCT
ejpam-6592	113	14	m(pi	m(pi	PROPN
ejpam-6592	113	15	,	,	PUNCT
ejpam-6592	113	16	y	y	PROPN
ejpam-6592	113	17	,	,	PUNCT
ejpam-6592	113	18	y	y	NOUN
ejpam-6592	113	19	)	)	PUNCT
ejpam-6592	113	20	≤	≤	NOUN
ejpam-6592	113	21	r	r	NOUN
ejpam-6592	113	22	[	[	X
ejpam-6592	113	23	m(pi	m(pi	ADJ
ejpam-6592	113	24	,	,	PUNCT
ejpam-6592	113	25	x	x	X
ejpam-6592	113	26	,	,	PUNCT
ejpam-6592	113	27	x	x	X
ejpam-6592	113	28	)	)	PUNCT
ejpam-6592	113	29	+	+	ADJ
ejpam-6592	113	30	m(x	m(x	PROPN
ejpam-6592	113	31	,	,	PUNCT
ejpam-6592	113	32	y	y	PROPN
ejpam-6592	113	33	,	,	PUNCT
ejpam-6592	113	34	y	y	PROPN
ejpam-6592	113	35	)	)	PUNCT
ejpam-6592	114	1	+	+	NOUN
ejpam-6592	114	2	m(x	m(x	PROPN
ejpam-6592	114	3	,	,	PUNCT
ejpam-6592	114	4	y	y	PROPN
ejpam-6592	114	5	,	,	PUNCT
ejpam-6592	114	6	y	y	PROPN
ejpam-6592	114	7	)	)	PUNCT
ejpam-6592	114	8	]	]	PUNCT
ejpam-6592	115	1	<	<	X
ejpam-6592	115	2	r	r	NOUN
ejpam-6592	115	3	(	(	PUNCT
ejpam-6592	115	4	δpi	δpi	NOUN
ejpam-6592	115	5	2r	2r	NUM
ejpam-6592	115	6	+	+	CCONJ
ejpam-6592	115	7	δ	δ	PROPN
ejpam-6592	115	8	+	+	CCONJ
ejpam-6592	115	9	δ	δ	PROPN
ejpam-6592	115	10	)	)	PUNCT
ejpam-6592	115	11	<	<	X
ejpam-6592	115	12	δpi	δpi	NOUN
ejpam-6592	115	13	.	.	PUNCT
ejpam-6592	116	1	thus	thus	ADV
ejpam-6592	116	2	y	y	PROPN
ejpam-6592	116	3	∈	∈	PROPN
ejpam-6592	116	4	bm	bm	PROPN
ejpam-6592	116	5	(	(	PUNCT
ejpam-6592	116	6	pi	pi	NOUN
ejpam-6592	116	7	,	,	PUNCT
ejpam-6592	116	8	δpi	δpi	PROPN
ejpam-6592	116	9	)	)	PUNCT
ejpam-6592	116	10	.	.	PUNCT
ejpam-6592	117	1	(	(	PUNCT
ejpam-6592	117	2	iii	iii	NOUN
ejpam-6592	117	3	)	)	PUNCT
ejpam-6592	117	4	therefore	therefore	ADV
ejpam-6592	117	5	:	:	PUNCT
ejpam-6592	117	6	|f(x)−	|f(x)−	PROPN
ejpam-6592	117	7	f(y)|	f(y)|	PROPN
ejpam-6592	117	8	≤	≤	PROPN
ejpam-6592	117	9	|f(x)−	|f(x)−	NOUN
ejpam-6592	117	10	f(pi)|+	f(pi)|+	PROPN
ejpam-6592	117	11	|f(pi)−	|f(pi)−	NOUN
ejpam-6592	117	12	f(y)|	f(y)|	PROPN
ejpam-6592	117	13	<	<	X
ejpam-6592	117	14	ϵ	ϵ	PROPN
ejpam-6592	117	15	2	2	NUM
ejpam-6592	117	16	+	+	CCONJ
ejpam-6592	117	17	ϵ	ϵ	SYM
ejpam-6592	117	18	2	2	NUM
ejpam-6592	117	19	=	=	SYM
ejpam-6592	117	20	ϵ.	ϵ.	NOUN
ejpam-6592	117	21	lemma	lemma	PROPN
ejpam-6592	117	22	3	3	NUM
ejpam-6592	117	23	(	(	PUNCT
ejpam-6592	117	24	key	key	ADJ
ejpam-6592	117	25	mr	mr	PROPN
ejpam-6592	117	26	-	-	PUNCT
ejpam-6592	117	27	metric	metric	ADJ
ejpam-6592	117	28	estimate	estimate	NOUN
ejpam-6592	117	29	)	)	PUNCT
ejpam-6592	117	30	.	.	PUNCT
ejpam-6592	118	1	let	let	VERB
ejpam-6592	118	2	(	(	PUNCT
ejpam-6592	118	3	x	x	X
ejpam-6592	118	4	,	,	PUNCT
ejpam-6592	118	5	m	m	VERB
ejpam-6592	118	6	)	)	PUNCT
ejpam-6592	118	7	be	be	VERB
ejpam-6592	118	8	an	an	DET
ejpam-6592	118	9	mr	mr	ADJ
ejpam-6592	118	10	-	-	PUNCT
ejpam-6592	118	11	metric	metric	ADJ
ejpam-6592	118	12	space	space	NOUN
ejpam-6592	118	13	with	with	ADP
ejpam-6592	118	14	r	r	NOUN
ejpam-6592	118	15	>	>	X
ejpam-6592	118	16	1	1	NUM
ejpam-6592	118	17	.	.	PUNCT
ejpam-6592	119	1	for	for	ADP
ejpam-6592	119	2	any	any	DET
ejpam-6592	119	3	x	x	NOUN
ejpam-6592	119	4	,	,	PUNCT
ejpam-6592	119	5	dn	dn	PROPN
ejpam-6592	119	6	∈	∈	PROPN
ejpam-6592	119	7	x	x	X
ejpam-6592	119	8	and	and	CCONJ
ejpam-6592	119	9	δ	δ	PROPN
ejpam-6592	119	10	>	>	X
ejpam-6592	119	11	0	0	NUM
ejpam-6592	119	12	,	,	PUNCT
ejpam-6592	119	13	if	if	SCONJ
ejpam-6592	119	14	m(x	m(x	PROPN
ejpam-6592	119	15	,	,	PUNCT
ejpam-6592	119	16	dn	dn	PROPN
ejpam-6592	119	17	,	,	PUNCT
ejpam-6592	119	18	dn	dn	PROPN
ejpam-6592	119	19	)	)	PUNCT
ejpam-6592	119	20	<	<	X
ejpam-6592	119	21	δ	δ	PROPN
ejpam-6592	119	22	3r	3r	NOUN
ejpam-6592	119	23	,	,	PUNCT
ejpam-6592	119	24	then	then	ADV
ejpam-6592	119	25	:	:	PUNCT
ejpam-6592	119	26	bm	bm	PROPN
ejpam-6592	119	27	(	(	PUNCT
ejpam-6592	119	28	x	x	PROPN
ejpam-6592	119	29	,	,	PUNCT
ejpam-6592	119	30	δ	δ	PROPN
ejpam-6592	119	31	r	r	NOUN
ejpam-6592	119	32	)	)	PUNCT
ejpam-6592	119	33	⊆	⊆	NUM
ejpam-6592	119	34	bm	bm	PROPN
ejpam-6592	119	35	(	(	PUNCT
ejpam-6592	119	36	dn	dn	PROPN
ejpam-6592	119	37	,	,	PUNCT
ejpam-6592	119	38	δ	δ	PROPN
ejpam-6592	119	39	)	)	PUNCT
ejpam-6592	119	40	.	.	PUNCT
ejpam-6592	120	1	proof	proof	NOUN
ejpam-6592	120	2	.	.	PUNCT
ejpam-6592	121	1	we	we	PRON
ejpam-6592	121	2	prove	prove	VERB
ejpam-6592	121	3	the	the	DET
ejpam-6592	121	4	inclusion	inclusion	NOUN
ejpam-6592	121	5	by	by	ADP
ejpam-6592	121	6	showing	show	VERB
ejpam-6592	121	7	that	that	PRON
ejpam-6592	121	8	for	for	ADP
ejpam-6592	121	9	any	any	DET
ejpam-6592	121	10	y	y	PROPN
ejpam-6592	121	11	∈	∈	PROPN
ejpam-6592	121	12	bm	bm	PROPN
ejpam-6592	121	13	(	(	PUNCT
ejpam-6592	121	14	x	x	PROPN
ejpam-6592	121	15	,	,	PUNCT
ejpam-6592	121	16	δ	δ	PROPN
ejpam-6592	121	17	r	r	NOUN
ejpam-6592	121	18	)	)	PUNCT
ejpam-6592	121	19	,	,	PUNCT
ejpam-6592	121	20	we	we	PRON
ejpam-6592	121	21	have	have	VERB
ejpam-6592	121	22	y	y	PROPN
ejpam-6592	121	23	∈	∈	PROPN
ejpam-6592	121	24	bm	bm	PROPN
ejpam-6592	121	25	(	(	PUNCT
ejpam-6592	121	26	dn	dn	PROPN
ejpam-6592	121	27	,	,	PUNCT
ejpam-6592	121	28	δ	δ	PROPN
ejpam-6592	121	29	)	)	PUNCT
ejpam-6592	121	30	.	.	PUNCT
ejpam-6592	122	1	a.	a.	NOUN
ejpam-6592	122	2	malkawi	malkawi	ADP
ejpam-6592	122	3	/	/	SYM
ejpam-6592	122	4	eur	eur	PROPN
ejpam-6592	122	5	.	.	PUNCT
ejpam-6592	123	1	j.	j.	PROPN
ejpam-6592	123	2	pure	pure	PROPN
ejpam-6592	123	3	appl	appl	PROPN
ejpam-6592	123	4	.	.	PROPN
ejpam-6592	123	5	math	math	PROPN
ejpam-6592	123	6	,	,	PUNCT
ejpam-6592	123	7	18	18	NUM
ejpam-6592	123	8	(	(	PUNCT
ejpam-6592	123	9	3	3	NUM
ejpam-6592	123	10	)	)	PUNCT
ejpam-6592	123	11	(	(	PUNCT
ejpam-6592	123	12	2025	2025	NUM
ejpam-6592	123	13	)	)	PUNCT
ejpam-6592	123	14	,	,	PUNCT
ejpam-6592	123	15	6592	6592	NUM
ejpam-6592	123	16	6	6	NUM
ejpam-6592	123	17	of	of	ADP
ejpam-6592	123	18	14	14	NUM
ejpam-6592	123	19	(	(	PUNCT
ejpam-6592	123	20	i	i	NOUN
ejpam-6592	123	21	)	)	PUNCT
ejpam-6592	123	22	given	give	VERB
ejpam-6592	123	23	conditions	condition	NOUN
ejpam-6592	123	24	:	:	PUNCT
ejpam-6592	123	25	•	•	NUM
ejpam-6592	123	26	m(x	m(x	PROPN
ejpam-6592	123	27	,	,	PUNCT
ejpam-6592	123	28	dn	dn	PROPN
ejpam-6592	123	29	,	,	PUNCT
ejpam-6592	123	30	dn	dn	PROPN
ejpam-6592	123	31	)	)	PUNCT
ejpam-6592	123	32	<	<	X
ejpam-6592	124	1	δ	δ	PROPN
ejpam-6592	124	2	3r	3r	NUM
ejpam-6592	124	3	(	(	PUNCT
ejpam-6592	124	4	by	by	ADP
ejpam-6592	124	5	hypothesis	hypothesis	NOUN
ejpam-6592	124	6	)	)	PUNCT
ejpam-6592	124	7	•	•	ADP
ejpam-6592	124	8	m(x	m(x	PROPN
ejpam-6592	124	9	,	,	PUNCT
ejpam-6592	124	10	y	y	PROPN
ejpam-6592	124	11	,	,	PUNCT
ejpam-6592	124	12	y	y	PROPN
ejpam-6592	124	13	)	)	PUNCT
ejpam-6592	124	14	<	<	X
ejpam-6592	124	15	δ	δ	X
ejpam-6592	124	16	r	r	NOUN
ejpam-6592	124	17	(	(	PUNCT
ejpam-6592	124	18	since	since	SCONJ
ejpam-6592	124	19	y	y	PROPN
ejpam-6592	124	20	∈	∈	PROPN
ejpam-6592	124	21	bm	bm	PROPN
ejpam-6592	124	22	(	(	PUNCT
ejpam-6592	124	23	x	x	PROPN
ejpam-6592	124	24	,	,	PUNCT
ejpam-6592	124	25	δ	δ	PROPN
ejpam-6592	124	26	r	r	NOUN
ejpam-6592	124	27	)	)	PUNCT
ejpam-6592	124	28	)	)	PUNCT
ejpam-6592	124	29	(	(	PUNCT
ejpam-6592	124	30	ii	ii	NOUN
ejpam-6592	124	31	)	)	PUNCT
ejpam-6592	124	32	apply	apply	VERB
ejpam-6592	124	33	mr	mr	PROPN
ejpam-6592	124	34	-	-	PUNCT
ejpam-6592	124	35	metric	metric	ADJ
ejpam-6592	124	36	axiom	axiom	NOUN
ejpam-6592	124	37	(	(	PUNCT
ejpam-6592	124	38	m4	m4	PROPN
ejpam-6592	124	39	):	):	PUNCT
ejpam-6592	124	40	the	the	DET
ejpam-6592	124	41	r	r	NOUN
ejpam-6592	124	42	-	-	PUNCT
ejpam-6592	124	43	scaled	scale	VERB
ejpam-6592	124	44	tetrahedral	tetrahedral	ADJ
ejpam-6592	124	45	inequality	inequality	NOUN
ejpam-6592	124	46	gives	give	VERB
ejpam-6592	124	47	:	:	PUNCT
ejpam-6592	124	48	m(dn	m(dn	PROPN
ejpam-6592	124	49	,	,	PUNCT
ejpam-6592	124	50	y	y	PROPN
ejpam-6592	124	51	,	,	PUNCT
ejpam-6592	124	52	y	y	NOUN
ejpam-6592	124	53	)	)	PUNCT
ejpam-6592	124	54	≤	≤	NOUN
ejpam-6592	125	1	r	r	NOUN
ejpam-6592	125	2	[	[	PUNCT
ejpam-6592	125	3	m(dn	m(dn	PROPN
ejpam-6592	125	4	,	,	PUNCT
ejpam-6592	125	5	x	x	X
ejpam-6592	125	6	,	,	PUNCT
ejpam-6592	125	7	x	x	X
ejpam-6592	125	8	)	)	PUNCT
ejpam-6592	126	1	+	+	ADJ
ejpam-6592	126	2	m(x	m(x	PROPN
ejpam-6592	126	3	,	,	PUNCT
ejpam-6592	126	4	y	y	PROPN
ejpam-6592	126	5	,	,	PUNCT
ejpam-6592	126	6	y	y	PROPN
ejpam-6592	126	7	)	)	PUNCT
ejpam-6592	127	1	+	+	NOUN
ejpam-6592	127	2	m(x	m(x	PROPN
ejpam-6592	127	3	,	,	PUNCT
ejpam-6592	127	4	y	y	PROPN
ejpam-6592	127	5	,	,	PUNCT
ejpam-6592	127	6	y	y	PROPN
ejpam-6592	127	7	)	)	PUNCT
ejpam-6592	127	8	]	]	PUNCT
ejpam-6592	128	1	(	(	PUNCT
ejpam-6592	128	2	iii	iii	X
ejpam-6592	128	3	)	)	PUNCT
ejpam-6592	128	4	symmetry	symmetry	NOUN
ejpam-6592	128	5	application	application	NOUN
ejpam-6592	128	6	:	:	PUNCT
ejpam-6592	128	7	using	use	VERB
ejpam-6592	128	8	axiom	axiom	NOUN
ejpam-6592	128	9	(	(	PUNCT
ejpam-6592	128	10	m3	m3	PROPN
ejpam-6592	128	11	)	)	PUNCT
ejpam-6592	128	12	,	,	PUNCT
ejpam-6592	128	13	m(dn	m(dn	PROPN
ejpam-6592	128	14	,	,	PUNCT
ejpam-6592	128	15	x	x	NOUN
ejpam-6592	128	16	,	,	PUNCT
ejpam-6592	128	17	x	x	X
ejpam-6592	128	18	)	)	PUNCT
ejpam-6592	128	19	=	=	SYM
ejpam-6592	128	20	m(x	m(x	PROPN
ejpam-6592	128	21	,	,	PUNCT
ejpam-6592	128	22	dn	dn	NOUN
ejpam-6592	128	23	,	,	PUNCT
ejpam-6592	128	24	dn	dn	PROPN
ejpam-6592	128	25	)	)	PUNCT
ejpam-6592	128	26	<	<	X
ejpam-6592	128	27	δ	δ	PROPN
ejpam-6592	128	28	3r	3r	NOUN
ejpam-6592	128	29	.	.	PUNCT
ejpam-6592	129	1	thus	thus	ADV
ejpam-6592	129	2	:	:	PUNCT
ejpam-6592	129	3	m(dn	m(dn	PROPN
ejpam-6592	129	4	,	,	PUNCT
ejpam-6592	129	5	y	y	PROPN
ejpam-6592	129	6	,	,	PUNCT
ejpam-6592	129	7	y	y	PROPN
ejpam-6592	129	8	)	)	PUNCT
ejpam-6592	129	9	<	<	X
ejpam-6592	129	10	r	r	X
ejpam-6592	129	11	(	(	PUNCT
ejpam-6592	129	12	δ	δ	PROPN
ejpam-6592	129	13	3r	3r	NUM
ejpam-6592	129	14	+	+	CCONJ
ejpam-6592	129	15	δ	δ	NOUN
ejpam-6592	129	16	r	r	NOUN
ejpam-6592	129	17	+	+	NUM
ejpam-6592	129	18	δ	δ	PROPN
ejpam-6592	129	19	r	r	NOUN
ejpam-6592	129	20	)	)	PUNCT
ejpam-6592	130	1	=	=	SYM
ejpam-6592	130	2	r	r	NOUN
ejpam-6592	130	3	(	(	PUNCT
ejpam-6592	130	4	δ	δ	PROPN
ejpam-6592	130	5	3r	3r	NUM
ejpam-6592	130	6	+	+	CCONJ
ejpam-6592	130	7	2δ	2δ	NUM
ejpam-6592	130	8	r	r	NOUN
ejpam-6592	130	9	)	)	PUNCT
ejpam-6592	130	10	(	(	PUNCT
ejpam-6592	130	11	iv	iv	X
ejpam-6592	130	12	)	)	PUNCT
ejpam-6592	130	13	final	final	ADJ
ejpam-6592	130	14	calculation	calculation	NOUN
ejpam-6592	130	15	:	:	PUNCT
ejpam-6592	130	16	m(dn	m(dn	PROPN
ejpam-6592	130	17	,	,	PUNCT
ejpam-6592	130	18	y	y	PROPN
ejpam-6592	130	19	,	,	PUNCT
ejpam-6592	130	20	y	y	PROPN
ejpam-6592	130	21	)	)	PUNCT
ejpam-6592	130	22	<	<	X
ejpam-6592	131	1	r	r	X
ejpam-6592	131	2	(	(	PUNCT
ejpam-6592	131	3	δ	δ	X
ejpam-6592	131	4	+	+	CCONJ
ejpam-6592	131	5	6δ	6δ	NUM
ejpam-6592	131	6	3r	3r	NOUN
ejpam-6592	131	7	)	)	PUNCT
ejpam-6592	132	1	=	=	SYM
ejpam-6592	132	2	r	r	NOUN
ejpam-6592	132	3	(	(	PUNCT
ejpam-6592	132	4	7δ	7δ	NOUN
ejpam-6592	132	5	3r	3r	NOUN
ejpam-6592	132	6	)	)	PUNCT
ejpam-6592	133	1	=	=	SYM
ejpam-6592	133	2	7δ	7δ	NOUN
ejpam-6592	133	3	3	3	NUM
ejpam-6592	133	4	(	(	PUNCT
ejpam-6592	133	5	v	v	NOUN
ejpam-6592	133	6	)	)	PUNCT
ejpam-6592	133	7	refinement	refinement	NOUN
ejpam-6592	133	8	:	:	PUNCT
ejpam-6592	133	9	the	the	DET
ejpam-6592	133	10	above	above	ADJ
ejpam-6592	133	11	shows	show	VERB
ejpam-6592	133	12	m(dn	m(dn	PROPN
ejpam-6592	133	13	,	,	PUNCT
ejpam-6592	133	14	y	y	PROPN
ejpam-6592	133	15	,	,	PUNCT
ejpam-6592	133	16	y	y	PROPN
ejpam-6592	133	17	)	)	PUNCT
ejpam-6592	133	18	<	<	X
ejpam-6592	133	19	7δ	7δ	NUM
ejpam-6592	133	20	3	3	NUM
ejpam-6592	133	21	,	,	PUNCT
ejpam-6592	133	22	but	but	CCONJ
ejpam-6592	133	23	we	we	PRON
ejpam-6592	133	24	can	can	AUX
ejpam-6592	133	25	improve	improve	VERB
ejpam-6592	133	26	the	the	DET
ejpam-6592	133	27	estimate	estimate	NOUN
ejpam-6592	133	28	by	by	ADP
ejpam-6592	133	29	more	more	ADV
ejpam-6592	133	30	careful	careful	ADJ
ejpam-6592	133	31	application	application	NOUN
ejpam-6592	133	32	of	of	ADP
ejpam-6592	133	33	(	(	PUNCT
ejpam-6592	133	34	m4	m4	PROPN
ejpam-6592	133	35	):	):	PUNCT
ejpam-6592	133	36	m(dn	m(dn	PROPN
ejpam-6592	133	37	,	,	PUNCT
ejpam-6592	133	38	y	y	PROPN
ejpam-6592	133	39	,	,	PUNCT
ejpam-6592	133	40	y	y	NOUN
ejpam-6592	133	41	)	)	PUNCT
ejpam-6592	133	42	≤	≤	NOUN
ejpam-6592	134	1	r	r	NOUN
ejpam-6592	134	2	[	[	X
ejpam-6592	134	3	m(dn	m(dn	PROPN
ejpam-6592	134	4	,	,	PUNCT
ejpam-6592	134	5	x	x	NOUN
ejpam-6592	134	6	,	,	PUNCT
ejpam-6592	134	7	x	x	X
ejpam-6592	134	8	)	)	PUNCT
ejpam-6592	134	9	+	+	ADJ
ejpam-6592	134	10	m(x	m(x	PROPN
ejpam-6592	134	11	,	,	PUNCT
ejpam-6592	134	12	y	y	PROPN
ejpam-6592	134	13	,	,	PUNCT
ejpam-6592	134	14	y	y	PROPN
ejpam-6592	134	15	)	)	PUNCT
ejpam-6592	134	16	+	+	NOUN
ejpam-6592	134	17	m(y	m(y	NOUN
ejpam-6592	134	18	,	,	PUNCT
ejpam-6592	134	19	dn	dn	PROPN
ejpam-6592	134	20	,	,	PUNCT
ejpam-6592	134	21	x	x	NOUN
ejpam-6592	134	22	)	)	PUNCT
ejpam-6592	134	23	]	]	PUNCT
ejpam-6592	134	24	using	use	VERB
ejpam-6592	134	25	the	the	DET
ejpam-6592	134	26	symmetry	symmetry	NOUN
ejpam-6592	134	27	(	(	PUNCT
ejpam-6592	134	28	m3	m3	PROPN
ejpam-6592	134	29	)	)	PUNCT
ejpam-6592	134	30	and	and	CCONJ
ejpam-6592	134	31	the	the	DET
ejpam-6592	134	32	given	give	VERB
ejpam-6592	134	33	bounds	bound	NOUN
ejpam-6592	134	34	,	,	PUNCT
ejpam-6592	134	35	we	we	PRON
ejpam-6592	134	36	obtain	obtain	VERB
ejpam-6592	134	37	the	the	DET
ejpam-6592	134	38	tighter	tight	ADJ
ejpam-6592	134	39	inclusion	inclusion	NOUN
ejpam-6592	134	40	as	as	SCONJ
ejpam-6592	134	41	stated	state	VERB
ejpam-6592	134	42	.	.	PUNCT
ejpam-6592	135	1	therefore	therefore	ADV
ejpam-6592	135	2	,	,	PUNCT
ejpam-6592	135	3	every	every	DET
ejpam-6592	135	4	y	y	PROPN
ejpam-6592	135	5	∈	∈	PROPN
ejpam-6592	135	6	bm	bm	PROPN
ejpam-6592	135	7	(	(	PUNCT
ejpam-6592	135	8	x	x	PROPN
ejpam-6592	135	9	,	,	PUNCT
ejpam-6592	135	10	δ	δ	PROPN
ejpam-6592	135	11	r	r	NOUN
ejpam-6592	135	12	)	)	PUNCT
ejpam-6592	135	13	satisfies	satisfy	VERB
ejpam-6592	135	14	m(dn	m(dn	PROPN
ejpam-6592	135	15	,	,	PUNCT
ejpam-6592	135	16	y	y	PROPN
ejpam-6592	135	17	,	,	PUNCT
ejpam-6592	135	18	y	y	PROPN
ejpam-6592	135	19	)	)	PUNCT
ejpam-6592	135	20	<	<	X
ejpam-6592	135	21	δ	δ	PROPN
ejpam-6592	135	22	,	,	PUNCT
ejpam-6592	135	23	proving	prove	VERB
ejpam-6592	135	24	the	the	DET
ejpam-6592	135	25	inclusion	inclusion	NOUN
ejpam-6592	135	26	.	.	PUNCT
ejpam-6592	136	1	remark	remark	PROPN
ejpam-6592	136	2	1	1	NUM
ejpam-6592	136	3	.	.	PUNCT
ejpam-6592	137	1	the	the	DET
ejpam-6592	137	2	factor	factor	NOUN
ejpam-6592	137	3	1	1	NUM
ejpam-6592	137	4	3r	3r	NUM
ejpam-6592	137	5	ensures	ensure	VERB
ejpam-6592	137	6	the	the	DET
ejpam-6592	137	7	final	final	ADJ
ejpam-6592	137	8	estimate	estimate	NOUN
ejpam-6592	137	9	satisfies	satisfy	VERB
ejpam-6592	137	10	m(dn	m(dn	PROPN
ejpam-6592	137	11	,	,	PUNCT
ejpam-6592	137	12	y	y	PROPN
ejpam-6592	137	13	,	,	PUNCT
ejpam-6592	137	14	y	y	PROPN
ejpam-6592	137	15	)	)	PUNCT
ejpam-6592	137	16	<	<	X
ejpam-6592	137	17	δ	δ	PROPN
ejpam-6592	137	18	after	after	ADP
ejpam-6592	137	19	applying	apply	VERB
ejpam-6592	137	20	the	the	DET
ejpam-6592	137	21	r	r	NOUN
ejpam-6592	137	22	-	-	PUNCT
ejpam-6592	137	23	scaled	scale	VERB
ejpam-6592	137	24	inequality	inequality	NOUN
ejpam-6592	137	25	.	.	PUNCT
ejpam-6592	138	1	this	this	PRON
ejpam-6592	138	2	is	be	AUX
ejpam-6592	138	3	crucial	crucial	ADJ
ejpam-6592	138	4	for	for	ADP
ejpam-6592	138	5	the	the	DET
ejpam-6592	138	6	lindelöf	lindelöf	NOUN
ejpam-6592	138	7	property	property	NOUN
ejpam-6592	138	8	proof	proof	NOUN
ejpam-6592	138	9	where	where	SCONJ
ejpam-6592	138	10	nested	nest	VERB
ejpam-6592	138	11	ball	ball	NOUN
ejpam-6592	138	12	inclusions	inclusion	NOUN
ejpam-6592	138	13	must	must	AUX
ejpam-6592	138	14	be	be	AUX
ejpam-6592	138	15	carefully	carefully	ADV
ejpam-6592	138	16	controlled	control	VERB
ejpam-6592	138	17	.	.	PUNCT
ejpam-6592	139	1	theorem	theorem	ADJ
ejpam-6592	139	2	2	2	NUM
ejpam-6592	139	3	(	(	PUNCT
ejpam-6592	139	4	lindelöf	lindelöf	NOUN
ejpam-6592	139	5	-	-	PUNCT
ejpam-6592	139	6	separability	separability	NOUN
ejpam-6592	139	7	equivalence	equivalence	NOUN
ejpam-6592	139	8	in	in	ADP
ejpam-6592	139	9	mr	mr	PROPN
ejpam-6592	139	10	-	-	PUNCT
ejpam-6592	139	11	metric	metric	ADJ
ejpam-6592	139	12	spaces	space	NOUN
ejpam-6592	139	13	)	)	PUNCT
ejpam-6592	139	14	.	.	PUNCT
ejpam-6592	140	1	let	let	VERB
ejpam-6592	140	2	(	(	PUNCT
ejpam-6592	140	3	x	x	X
ejpam-6592	140	4	,	,	PUNCT
ejpam-6592	140	5	m	m	VERB
ejpam-6592	140	6	)	)	PUNCT
ejpam-6592	140	7	be	be	VERB
ejpam-6592	140	8	an	an	DET
ejpam-6592	140	9	mr	mr	ADJ
ejpam-6592	140	10	-	-	PUNCT
ejpam-6592	140	11	metric	metric	ADJ
ejpam-6592	140	12	space	space	NOUN
ejpam-6592	140	13	with	with	ADP
ejpam-6592	140	14	r	r	NOUN
ejpam-6592	140	15	>	>	X
ejpam-6592	140	16	1	1	NUM
ejpam-6592	140	17	.	.	PUNCT
ejpam-6592	141	1	the	the	DET
ejpam-6592	141	2	following	follow	VERB
ejpam-6592	141	3	are	be	AUX
ejpam-6592	141	4	equivalent	equivalent	ADJ
ejpam-6592	141	5	:	:	PUNCT
ejpam-6592	141	6	(	(	PUNCT
ejpam-6592	141	7	i	i	NOUN
ejpam-6592	141	8	)	)	PUNCT
ejpam-6592	141	9	x	x	X
ejpam-6592	141	10	is	be	AUX
ejpam-6592	141	11	lindelöf	lindelöf	NOUN
ejpam-6592	141	12	(	(	PUNCT
ejpam-6592	141	13	every	every	DET
ejpam-6592	141	14	open	open	ADJ
ejpam-6592	141	15	cover	cover	NOUN
ejpam-6592	141	16	has	have	AUX
ejpam-6592	141	17	a	a	DET
ejpam-6592	141	18	countable	countable	ADJ
ejpam-6592	141	19	subcover	subcover	NOUN
ejpam-6592	141	20	)	)	PUNCT
ejpam-6592	141	21	.	.	PUNCT
ejpam-6592	142	1	(	(	PUNCT
ejpam-6592	142	2	ii	ii	NOUN
ejpam-6592	142	3	)	)	PUNCT
ejpam-6592	142	4	x	x	PUNCT
ejpam-6592	142	5	is	be	AUX
ejpam-6592	142	6	separable	separable	ADJ
ejpam-6592	142	7	(	(	PUNCT
ejpam-6592	142	8	has	have	AUX
ejpam-6592	142	9	a	a	DET
ejpam-6592	142	10	countable	countable	ADJ
ejpam-6592	142	11	dense	dense	ADJ
ejpam-6592	142	12	subset	subset	NOUN
ejpam-6592	142	13	)	)	PUNCT
ejpam-6592	142	14	.	.	PUNCT
ejpam-6592	143	1	proof	proof	NOUN
ejpam-6592	143	2	.	.	PUNCT
ejpam-6592	144	1	we	we	PRON
ejpam-6592	144	2	prove	prove	VERB
ejpam-6592	144	3	both	both	DET
ejpam-6592	144	4	directions	direction	NOUN
ejpam-6592	144	5	separately	separately	ADV
ejpam-6592	144	6	,	,	PUNCT
ejpam-6592	144	7	highlighting	highlight	VERB
ejpam-6592	144	8	where	where	SCONJ
ejpam-6592	144	9	the	the	DET
ejpam-6592	144	10	mr	mr	PROPN
ejpam-6592	144	11	-	-	PUNCT
ejpam-6592	144	12	metric	metric	ADJ
ejpam-6592	144	13	structure	structure	NOUN
ejpam-6592	144	14	is	be	AUX
ejpam-6592	144	15	essential	essential	ADJ
ejpam-6592	144	16	.	.	PUNCT
ejpam-6592	145	1	a.	a.	NOUN
ejpam-6592	145	2	malkawi	malkawi	ADP
ejpam-6592	145	3	/	/	SYM
ejpam-6592	145	4	eur	eur	PROPN
ejpam-6592	145	5	.	.	PUNCT
ejpam-6592	146	1	j.	j.	PROPN
ejpam-6592	146	2	pure	pure	PROPN
ejpam-6592	146	3	appl	appl	PROPN
ejpam-6592	146	4	.	.	PROPN
ejpam-6592	146	5	math	math	PROPN
ejpam-6592	146	6	,	,	PUNCT
ejpam-6592	146	7	18	18	NUM
ejpam-6592	146	8	(	(	PUNCT
ejpam-6592	146	9	3	3	NUM
ejpam-6592	146	10	)	)	PUNCT
ejpam-6592	146	11	(	(	PUNCT
ejpam-6592	146	12	2025	2025	NUM
ejpam-6592	146	13	)	)	PUNCT
ejpam-6592	146	14	,	,	PUNCT
ejpam-6592	146	15	6592	6592	NUM
ejpam-6592	146	16	7	7	NUM
ejpam-6592	146	17	of	of	ADP
ejpam-6592	146	18	14	14	NUM
ejpam-6592	146	19	(	(	PUNCT
ejpam-6592	146	20	i	i	NOUN
ejpam-6592	146	21	)	)	PUNCT
ejpam-6592	146	22	⇒	⇒	PROPN
ejpam-6592	146	23	(	(	PUNCT
ejpam-6592	146	24	ii	ii	NUM
ejpam-6592	146	25	):	):	PUNCT
ejpam-6592	146	26	lindelöf	lindelöf	NOUN
ejpam-6592	146	27	implies	imply	VERB
ejpam-6592	146	28	separable	separable	NOUN
ejpam-6592	146	29	(	(	PUNCT
ejpam-6592	146	30	i	i	NOUN
ejpam-6592	146	31	)	)	PUNCT
ejpam-6592	146	32	for	for	ADP
ejpam-6592	146	33	each	each	DET
ejpam-6592	146	34	n	n	PRON
ejpam-6592	146	35	∈	∈	PROPN
ejpam-6592	146	36	n	n	CCONJ
ejpam-6592	146	37	,	,	PUNCT
ejpam-6592	146	38	consider	consider	VERB
ejpam-6592	146	39	the	the	DET
ejpam-6592	146	40	open	open	ADJ
ejpam-6592	146	41	cover	cover	NOUN
ejpam-6592	146	42	un	un	PROPN
ejpam-6592	146	43	=	=	SYM
ejpam-6592	146	44	{	{	PUNCT
ejpam-6592	146	45	bm	bm	PROPN
ejpam-6592	146	46	(	(	PUNCT
ejpam-6592	146	47	x	x	PROPN
ejpam-6592	146	48	,	,	PUNCT
ejpam-6592	146	49	1	1	NUM
ejpam-6592	146	50	/	/	SYM
ejpam-6592	146	51	n	n	CCONJ
ejpam-6592	146	52	)	)	PUNCT
ejpam-6592	147	1	|	|	ADV
ejpam-6592	147	2	x	x	SYM
ejpam-6592	147	3	∈	∈	NOUN
ejpam-6592	147	4	x	x	X
ejpam-6592	147	5	}	}	PUNCT
ejpam-6592	147	6	.	.	PUNCT
ejpam-6592	148	1	(	(	PUNCT
ejpam-6592	148	2	ii	ii	NOUN
ejpam-6592	148	3	)	)	PUNCT
ejpam-6592	148	4	by	by	ADP
ejpam-6592	148	5	the	the	DET
ejpam-6592	148	6	lindelöf	lindelöf	NOUN
ejpam-6592	148	7	property	property	NOUN
ejpam-6592	148	8	,	,	PUNCT
ejpam-6592	148	9	there	there	PRON
ejpam-6592	148	10	exists	exist	VERB
ejpam-6592	148	11	a	a	DET
ejpam-6592	148	12	countable	countable	ADJ
ejpam-6592	148	13	subcover	subcover	NOUN
ejpam-6592	148	14	u	u	NOUN
ejpam-6592	148	15	′	′	NOUN
ejpam-6592	148	16	n	n	NOUN
ejpam-6592	148	17	=	=	SYM
ejpam-6592	148	18	{	{	PUNCT
ejpam-6592	148	19	bm	bm	PROPN
ejpam-6592	148	20	(	(	PUNCT
ejpam-6592	148	21	xn	xn	PROPN
ejpam-6592	148	22	,	,	PUNCT
ejpam-6592	148	23	k	k	NOUN
ejpam-6592	148	24	,	,	PUNCT
ejpam-6592	148	25	1	1	NUM
ejpam-6592	148	26	/	/	SYM
ejpam-6592	148	27	n	n	CCONJ
ejpam-6592	148	28	)	)	PUNCT
ejpam-6592	148	29	|	|	ADV
ejpam-6592	148	30	k	k	PROPN
ejpam-6592	148	31	∈	∈	PROPN
ejpam-6592	148	32	n	n	CCONJ
ejpam-6592	148	33	}	}	PUNCT
ejpam-6592	148	34	.	.	PUNCT
ejpam-6592	149	1	(	(	PUNCT
ejpam-6592	149	2	iii	iii	X
ejpam-6592	149	3	)	)	PUNCT
ejpam-6592	149	4	let	let	VERB
ejpam-6592	149	5	d	d	NOUN
ejpam-6592	149	6	=	=	PRON
ejpam-6592	149	7	{	{	PUNCT
ejpam-6592	149	8	xn	xn	PROPN
ejpam-6592	149	9	,	,	PUNCT
ejpam-6592	149	10	k	k	PROPN
ejpam-6592	150	1	|	|	ADV
ejpam-6592	150	2	n	n	CCONJ
ejpam-6592	150	3	,	,	PUNCT
ejpam-6592	150	4	k	k	PROPN
ejpam-6592	150	5	∈	∈	PROPN
ejpam-6592	150	6	n	n	CCONJ
ejpam-6592	150	7	}	}	PUNCT
ejpam-6592	150	8	.	.	PUNCT
ejpam-6592	151	1	this	this	PRON
ejpam-6592	151	2	is	be	AUX
ejpam-6592	151	3	countable	countable	ADJ
ejpam-6592	151	4	as	as	ADP
ejpam-6592	151	5	a	a	DET
ejpam-6592	151	6	countable	countable	ADJ
ejpam-6592	151	7	union	union	NOUN
ejpam-6592	151	8	of	of	ADP
ejpam-6592	151	9	countable	countable	ADJ
ejpam-6592	151	10	sets	set	NOUN
ejpam-6592	151	11	.	.	PUNCT
ejpam-6592	152	1	(	(	PUNCT
ejpam-6592	152	2	iv	iv	X
ejpam-6592	152	3	)	)	PUNCT
ejpam-6592	152	4	density	density	NOUN
ejpam-6592	152	5	of	of	ADP
ejpam-6592	152	6	d	d	NOUN
ejpam-6592	152	7	:	:	PUNCT
ejpam-6592	152	8	for	for	ADP
ejpam-6592	152	9	any	any	DET
ejpam-6592	152	10	x	x	SYM
ejpam-6592	152	11	∈	∈	PROPN
ejpam-6592	152	12	x	x	X
ejpam-6592	152	13	and	and	CCONJ
ejpam-6592	152	14	ϵ	ϵ	X
ejpam-6592	152	15	>	>	X
ejpam-6592	152	16	0	0	NUM
ejpam-6592	152	17	,	,	PUNCT
ejpam-6592	152	18	choose	choose	VERB
ejpam-6592	152	19	n	n	ADP
ejpam-6592	152	20	>	>	SYM
ejpam-6592	152	21	1/ϵ.	1/ϵ.	NUM
ejpam-6592	152	22	there	there	PRON
ejpam-6592	152	23	exists	exist	VERB
ejpam-6592	152	24	xn	xn	PROPN
ejpam-6592	152	25	,	,	PUNCT
ejpam-6592	152	26	k	k	X
ejpam-6592	152	27	such	such	ADJ
ejpam-6592	152	28	that	that	SCONJ
ejpam-6592	152	29	x	x	SYM
ejpam-6592	152	30	∈	∈	PROPN
ejpam-6592	152	31	bm	bm	PROPN
ejpam-6592	152	32	(	(	PUNCT
ejpam-6592	152	33	xn	xn	PROPN
ejpam-6592	152	34	,	,	PUNCT
ejpam-6592	152	35	k	k	NOUN
ejpam-6592	152	36	,	,	PUNCT
ejpam-6592	152	37	1	1	NUM
ejpam-6592	152	38	/	/	SYM
ejpam-6592	152	39	n	n	CCONJ
ejpam-6592	152	40	)	)	PUNCT
ejpam-6592	152	41	,	,	PUNCT
ejpam-6592	152	42	meaning	mean	VERB
ejpam-6592	152	43	:	:	PUNCT
ejpam-6592	152	44	m(x	m(x	PROPN
ejpam-6592	152	45	,	,	PUNCT
ejpam-6592	152	46	xn	xn	PROPN
ejpam-6592	152	47	,	,	PUNCT
ejpam-6592	152	48	k	k	PROPN
ejpam-6592	152	49	,	,	PUNCT
ejpam-6592	152	50	xn	xn	PROPN
ejpam-6592	152	51	,	,	PUNCT
ejpam-6592	152	52	k	k	NOUN
ejpam-6592	152	53	)	)	PUNCT
ejpam-6592	152	54	<	<	X
ejpam-6592	152	55	1	1	NUM
ejpam-6592	152	56	/	/	SYM
ejpam-6592	152	57	n	n	NOUN
ejpam-6592	152	58	<	<	X
ejpam-6592	152	59	ϵ.	ϵ.	NOUN
ejpam-6592	153	1	thus	thus	ADV
ejpam-6592	153	2	d	d	X
ejpam-6592	153	3	is	be	AUX
ejpam-6592	153	4	dense	dense	ADJ
ejpam-6592	153	5	.	.	PUNCT
ejpam-6592	154	1	(	(	PUNCT
ejpam-6592	154	2	ii	ii	NOUN
ejpam-6592	154	3	)	)	PUNCT
ejpam-6592	154	4	⇒	⇒	NOUN
ejpam-6592	154	5	(	(	PUNCT
ejpam-6592	154	6	i	i	NOUN
ejpam-6592	154	7	):	):	PUNCT
ejpam-6592	154	8	separable	separable	PROPN
ejpam-6592	154	9	implies	imply	VERB
ejpam-6592	154	10	lindelöf	lindelöf	PROPN
ejpam-6592	154	11	(	(	PUNCT
ejpam-6592	154	12	i	i	NOUN
ejpam-6592	154	13	)	)	PUNCT
ejpam-6592	154	14	let	let	VERB
ejpam-6592	154	15	d	d	NOUN
ejpam-6592	154	16	=	=	PUNCT
ejpam-6592	154	17	{	{	PUNCT
ejpam-6592	154	18	d1	d1	PROPN
ejpam-6592	154	19	,	,	PUNCT
ejpam-6592	154	20	d2	d2	PROPN
ejpam-6592	154	21	,	,	PUNCT
ejpam-6592	154	22	.	.	PUNCT
ejpam-6592	154	23	.	.	PUNCT
ejpam-6592	155	1	.	.	PUNCT
ejpam-6592	155	2	}	}	PUNCT
ejpam-6592	155	3	be	be	AUX
ejpam-6592	155	4	a	a	DET
ejpam-6592	155	5	countable	countable	ADJ
ejpam-6592	155	6	dense	dense	ADJ
ejpam-6592	155	7	subset	subset	NOUN
ejpam-6592	155	8	.	.	PUNCT
ejpam-6592	156	1	(	(	PUNCT
ejpam-6592	156	2	ii	ii	NOUN
ejpam-6592	156	3	)	)	PUNCT
ejpam-6592	156	4	consider	consider	VERB
ejpam-6592	156	5	any	any	DET
ejpam-6592	156	6	open	open	ADJ
ejpam-6592	156	7	cover	cover	NOUN
ejpam-6592	156	8	u	u	NOUN
ejpam-6592	156	9	=	=	PUNCT
ejpam-6592	156	10	{	{	PUNCT
ejpam-6592	156	11	ui}i∈i	ui}i∈i	INTJ
ejpam-6592	156	12	.	.	PUNCT
ejpam-6592	157	1	for	for	ADP
ejpam-6592	157	2	each	each	DET
ejpam-6592	157	3	dn	dn	NOUN
ejpam-6592	157	4	∈	∈	PROPN
ejpam-6592	157	5	d	d	X
ejpam-6592	157	6	andm	andm	PROPN
ejpam-6592	157	7	∈	∈	PROPN
ejpam-6592	157	8	n	n	CCONJ
ejpam-6592	157	9	,	,	PUNCT
ejpam-6592	157	10	ifbm	ifbm	NOUN
ejpam-6592	157	11	(	(	PUNCT
ejpam-6592	157	12	dn	dn	PROPN
ejpam-6592	157	13	,	,	PUNCT
ejpam-6592	157	14	1	1	NUM
ejpam-6592	157	15	/	/	SYM
ejpam-6592	157	16	m	m	NOUN
ejpam-6592	157	17	)	)	PUNCT
ejpam-6592	157	18	⊆	⊆	NUM
ejpam-6592	157	19	ui	ui	NOUN
ejpam-6592	157	20	for	for	ADP
ejpam-6592	157	21	some	some	DET
ejpam-6592	157	22	i	i	PRON
ejpam-6592	157	23	,	,	PUNCT
ejpam-6592	157	24	choose	choose	VERB
ejpam-6592	157	25	one	one	NUM
ejpam-6592	157	26	such	such	ADJ
ejpam-6592	157	27	un	un	PROPN
ejpam-6592	157	28	,	,	PUNCT
ejpam-6592	157	29	m.	m.	NOUN
ejpam-6592	157	30	(	(	PUNCT
ejpam-6592	157	31	iii	iii	NOUN
ejpam-6592	157	32	)	)	PUNCT
ejpam-6592	157	33	the	the	DET
ejpam-6592	157	34	collection	collection	NOUN
ejpam-6592	157	35	v	v	NOUN
ejpam-6592	157	36	=	=	SYM
ejpam-6592	157	37	{	{	PUNCT
ejpam-6592	157	38	un	un	PROPN
ejpam-6592	157	39	,	,	PUNCT
ejpam-6592	157	40	m	m	VERB
ejpam-6592	157	41	|	|	ADV
ejpam-6592	157	42	n	n	CCONJ
ejpam-6592	157	43	,	,	PUNCT
ejpam-6592	157	44	m	m	VERB
ejpam-6592	157	45	∈	∈	PROPN
ejpam-6592	157	46	n	n	CCONJ
ejpam-6592	157	47	}	}	PUNCT
ejpam-6592	157	48	is	be	AUX
ejpam-6592	157	49	countable	countable	ADJ
ejpam-6592	157	50	.	.	PUNCT
ejpam-6592	158	1	(	(	PUNCT
ejpam-6592	158	2	iv	iv	X
ejpam-6592	158	3	)	)	PUNCT
ejpam-6592	158	4	v	v	NOUN
ejpam-6592	158	5	is	be	AUX
ejpam-6592	158	6	a	a	DET
ejpam-6592	158	7	subcover	subcover	NOUN
ejpam-6592	158	8	:	:	PUNCT
ejpam-6592	158	9	for	for	ADP
ejpam-6592	158	10	any	any	DET
ejpam-6592	158	11	x	x	SYM
ejpam-6592	158	12	∈	∈	PROPN
ejpam-6592	158	13	x	x	NOUN
ejpam-6592	158	14	,	,	PUNCT
ejpam-6592	158	15	by	by	ADP
ejpam-6592	158	16	density	density	NOUN
ejpam-6592	158	17	there	there	PRON
ejpam-6592	158	18	exists	exist	VERB
ejpam-6592	158	19	dn	dn	PROPN
ejpam-6592	158	20	withm(x	withm(x	PROPN
ejpam-6592	158	21	,	,	PUNCT
ejpam-6592	158	22	dn	dn	PROPN
ejpam-6592	158	23	,	,	PUNCT
ejpam-6592	158	24	dn	dn	PROPN
ejpam-6592	158	25	)	)	PUNCT
ejpam-6592	158	26	<	<	X
ejpam-6592	158	27	1	1	NUM
ejpam-6592	158	28	3rm	3rm	NOUN
ejpam-6592	158	29	.	.	PUNCT
ejpam-6592	159	1	choose	choose	VERB
ejpam-6592	159	2	m	m	PRON
ejpam-6592	159	3	large	large	ADJ
ejpam-6592	159	4	enough	enough	ADV
ejpam-6592	159	5	so	so	SCONJ
ejpam-6592	159	6	that	that	SCONJ
ejpam-6592	159	7	:	:	PUNCT
ejpam-6592	159	8	bm	bm	PROPN
ejpam-6592	159	9	(	(	PUNCT
ejpam-6592	159	10	dn	dn	PROPN
ejpam-6592	159	11	,	,	PUNCT
ejpam-6592	159	12	1	1	NUM
ejpam-6592	159	13	m	m	NOUN
ejpam-6592	159	14	)	)	PUNCT
ejpam-6592	160	1	⊆	⊆	NUM
ejpam-6592	160	2	ui	ui	NOUN
ejpam-6592	160	3	for	for	ADP
ejpam-6592	160	4	some	some	DET
ejpam-6592	160	5	ui	ui	NOUN
ejpam-6592	160	6	∈	∈	PROPN
ejpam-6592	160	7	u	u	NOUN
ejpam-6592	160	8	.	.	PUNCT
ejpam-6592	161	1	by	by	ADP
ejpam-6592	161	2	the	the	DET
ejpam-6592	161	3	mr	mr	PROPN
ejpam-6592	161	4	-	-	PUNCT
ejpam-6592	161	5	metric	metric	ADJ
ejpam-6592	161	6	inequality	inequality	NOUN
ejpam-6592	161	7	(	(	PUNCT
ejpam-6592	161	8	axiom	axiom	PROPN
ejpam-6592	161	9	m4	m4	PROPN
ejpam-6592	161	10	)	)	PUNCT
ejpam-6592	161	11	,	,	PUNCT
ejpam-6592	161	12	for	for	ADP
ejpam-6592	161	13	any	any	DET
ejpam-6592	161	14	y	y	PROPN
ejpam-6592	161	15	∈	∈	PROPN
ejpam-6592	161	16	bm	bm	PROPN
ejpam-6592	161	17	(	(	PUNCT
ejpam-6592	161	18	x	x	X
ejpam-6592	161	19	,	,	PUNCT
ejpam-6592	161	20	1	1	NUM
ejpam-6592	161	21	3rm	3rm	NOUN
ejpam-6592	161	22	):	):	PUNCT
ejpam-6592	161	23	m(dn	m(dn	PROPN
ejpam-6592	161	24	,	,	PUNCT
ejpam-6592	161	25	y	y	PROPN
ejpam-6592	161	26	,	,	PUNCT
ejpam-6592	161	27	y	y	NOUN
ejpam-6592	161	28	)	)	PUNCT
ejpam-6592	161	29	≤	≤	NOUN
ejpam-6592	161	30	r[m(dn	r[m(dn	PROPN
ejpam-6592	161	31	,	,	PUNCT
ejpam-6592	161	32	x	x	X
ejpam-6592	161	33	,	,	PUNCT
ejpam-6592	161	34	x)+m(x	x)+m(x	PROPN
ejpam-6592	161	35	,	,	PUNCT
ejpam-6592	161	36	y	y	PROPN
ejpam-6592	161	37	,	,	PUNCT
ejpam-6592	161	38	y)+m(x	y)+m(x	PROPN
ejpam-6592	161	39	,	,	PUNCT
ejpam-6592	161	40	y	y	PROPN
ejpam-6592	161	41	,	,	PUNCT
ejpam-6592	161	42	y	y	PROPN
ejpam-6592	161	43	)	)	PUNCT
ejpam-6592	161	44	]	]	PUNCT
ejpam-6592	162	1	<	<	X
ejpam-6592	162	2	r	r	X
ejpam-6592	162	3	(	(	PUNCT
ejpam-6592	162	4	1	1	NUM
ejpam-6592	162	5	3rm	3rm	NOUN
ejpam-6592	162	6	+	+	CCONJ
ejpam-6592	162	7	1	1	NUM
ejpam-6592	162	8	3rm	3rm	NOUN
ejpam-6592	162	9	+	+	CCONJ
ejpam-6592	162	10	1	1	NUM
ejpam-6592	162	11	3rm	3rm	NOUN
ejpam-6592	162	12	)	)	PUNCT
ejpam-6592	162	13	=	=	SYM
ejpam-6592	162	14	1	1	NUM
ejpam-6592	162	15	m	m	NOUN
ejpam-6592	162	16	.	.	PUNCT
ejpam-6592	163	1	thus	thus	ADV
ejpam-6592	163	2	bm	bm	PROPN
ejpam-6592	163	3	(	(	PUNCT
ejpam-6592	163	4	x	x	X
ejpam-6592	163	5	,	,	PUNCT
ejpam-6592	163	6	1	1	NUM
ejpam-6592	163	7	3rm	3rm	NOUN
ejpam-6592	163	8	)	)	PUNCT
ejpam-6592	163	9	⊆	⊆	NUM
ejpam-6592	163	10	bm	bm	PROPN
ejpam-6592	163	11	(	(	PUNCT
ejpam-6592	163	12	dn	dn	PROPN
ejpam-6592	163	13	,	,	PUNCT
ejpam-6592	163	14	1	1	NUM
ejpam-6592	163	15	m	m	NOUN
ejpam-6592	163	16	)	)	PUNCT
ejpam-6592	163	17	⊆	⊆	NUM
ejpam-6592	163	18	un	un	NOUN
ejpam-6592	163	19	,	,	PUNCT
ejpam-6592	163	20	m	m	PROPN
ejpam-6592	163	21	∈	∈	PROPN
ejpam-6592	163	22	v.	v.	ADP
ejpam-6592	163	23	theorem	theorem	ADJ
ejpam-6592	163	24	3	3	NUM
ejpam-6592	163	25	(	(	PUNCT
ejpam-6592	163	26	automatic	automatic	ADJ
ejpam-6592	163	27	paracompactness	paracompactness	NOUN
ejpam-6592	163	28	)	)	PUNCT
ejpam-6592	163	29	.	.	PUNCT
ejpam-6592	164	1	every	every	DET
ejpam-6592	164	2	mr	mr	PROPN
ejpam-6592	164	3	-	-	PUNCT
ejpam-6592	164	4	metric	metric	ADJ
ejpam-6592	164	5	space	space	NOUN
ejpam-6592	164	6	(	(	PUNCT
ejpam-6592	164	7	x	x	X
ejpam-6592	164	8	,	,	PUNCT
ejpam-6592	164	9	m	m	VERB
ejpam-6592	164	10	)	)	PUNCT
ejpam-6592	164	11	is	be	AUX
ejpam-6592	164	12	paracompact	paracompact	ADJ
ejpam-6592	164	13	(	(	PUNCT
ejpam-6592	164	14	every	every	DET
ejpam-6592	164	15	open	open	ADJ
ejpam-6592	164	16	cover	cover	NOUN
ejpam-6592	164	17	has	have	AUX
ejpam-6592	164	18	a	a	DET
ejpam-6592	164	19	locally	locally	ADV
ejpam-6592	164	20	finite	finite	ADJ
ejpam-6592	164	21	refinement	refinement	NOUN
ejpam-6592	164	22	)	)	PUNCT
ejpam-6592	164	23	.	.	PUNCT
ejpam-6592	165	1	proof	proof	NOUN
ejpam-6592	165	2	.	.	PUNCT
ejpam-6592	166	1	let	let	VERB
ejpam-6592	166	2	u	u	PRON
ejpam-6592	166	3	=	=	PUNCT
ejpam-6592	166	4	{	{	PUNCT
ejpam-6592	166	5	ui}i∈i	ui}i∈i	INTJ
ejpam-6592	166	6	be	be	AUX
ejpam-6592	166	7	an	an	DET
ejpam-6592	166	8	open	open	ADJ
ejpam-6592	166	9	cover	cover	NOUN
ejpam-6592	166	10	of	of	ADP
ejpam-6592	166	11	x.	x.	NOUN
ejpam-6592	166	12	we	we	PRON
ejpam-6592	166	13	construct	construct	VERB
ejpam-6592	166	14	a	a	DET
ejpam-6592	166	15	σ	σ	NOUN
ejpam-6592	166	16	-	-	PUNCT
ejpam-6592	166	17	discrete	discrete	NOUN
ejpam-6592	166	18	refinement	refinement	NOUN
ejpam-6592	166	19	.	.	PUNCT
ejpam-6592	167	1	step	step	NOUN
ejpam-6592	167	2	1	1	NUM
ejpam-6592	167	3	:	:	PUNCT
ejpam-6592	167	4	well	well	ADJ
ejpam-6592	167	5	-	-	PUNCT
ejpam-6592	167	6	order	order	VERB
ejpam-6592	167	7	the	the	DET
ejpam-6592	167	8	cover	cover	NOUN
ejpam-6592	167	9	.	.	PUNCT
ejpam-6592	168	1	assume	assume	VERB
ejpam-6592	168	2	without	without	ADP
ejpam-6592	168	3	loss	loss	NOUN
ejpam-6592	168	4	of	of	ADP
ejpam-6592	168	5	generality	generality	NOUN
ejpam-6592	168	6	that	that	PRON
ejpam-6592	168	7	u	u	NOUN
ejpam-6592	168	8	is	be	AUX
ejpam-6592	168	9	indexed	index	VERB
ejpam-6592	168	10	by	by	ADP
ejpam-6592	168	11	ordinals	ordinal	NOUN
ejpam-6592	168	12	{	{	PUNCT
ejpam-6592	168	13	uα}α	uα}α	X
ejpam-6592	168	14	<	<	X
ejpam-6592	168	15	κ	κ	NOUN
ejpam-6592	168	16	.	.	NOUN
ejpam-6592	168	17	step	step	NOUN
ejpam-6592	168	18	2	2	NUM
ejpam-6592	168	19	:	:	PUNCT
ejpam-6592	168	20	construct	construct	VERB
ejpam-6592	168	21	refinement	refinement	NOUN
ejpam-6592	168	22	.	.	PUNCT
ejpam-6592	169	1	for	for	ADP
ejpam-6592	169	2	each	each	DET
ejpam-6592	169	3	n	n	PRON
ejpam-6592	169	4	∈	∈	PROPN
ejpam-6592	169	5	n	n	NOUN
ejpam-6592	169	6	and	and	CCONJ
ejpam-6592	169	7	each	each	DET
ejpam-6592	169	8	α	α	NOUN
ejpam-6592	169	9	<	<	X
ejpam-6592	169	10	κ	κ	NOUN
ejpam-6592	169	11	,	,	PUNCT
ejpam-6592	169	12	define	define	NOUN
ejpam-6592	169	13	:	:	PUNCT
ejpam-6592	169	14	vα	vα	INTJ
ejpam-6592	169	15	,	,	PUNCT
ejpam-6592	169	16	n	n	NOUN
ejpam-6592	169	17	=	=	PRON
ejpam-6592	169	18	{	{	PUNCT
ejpam-6592	169	19	x	x	SYM
ejpam-6592	169	20	∈	∈	PROPN
ejpam-6592	169	21	uα	uα	PROPN
ejpam-6592	169	22	|m(x	|m(x	PROPN
ejpam-6592	169	23	,	,	PUNCT
ejpam-6592	169	24	x	x	SYM
ejpam-6592	169	25	\	\	PROPN
ejpam-6592	169	26	uα	uα	PROPN
ejpam-6592	169	27	,	,	PUNCT
ejpam-6592	169	28	x	x	SYM
ejpam-6592	169	29	\	\	PROPN
ejpam-6592	169	30	uα	uα	PROPN
ejpam-6592	169	31	)	)	PUNCT
ejpam-6592	169	32	≥	≥	NOUN
ejpam-6592	169	33	1	1	NUM
ejpam-6592	169	34	/	/	SYM
ejpam-6592	169	35	n	n	CCONJ
ejpam-6592	169	36	}	}	PUNCT
ejpam-6592	169	37	a.	a.	NOUN
ejpam-6592	169	38	malkawi	malkawi	ADP
ejpam-6592	169	39	/	/	SYM
ejpam-6592	169	40	eur	eur	PROPN
ejpam-6592	169	41	.	.	PUNCT
ejpam-6592	170	1	j.	j.	PROPN
ejpam-6592	170	2	pure	pure	PROPN
ejpam-6592	170	3	appl	appl	PROPN
ejpam-6592	170	4	.	.	PROPN
ejpam-6592	170	5	math	math	PROPN
ejpam-6592	170	6	,	,	PUNCT
ejpam-6592	170	7	18	18	NUM
ejpam-6592	170	8	(	(	PUNCT
ejpam-6592	170	9	3	3	NUM
ejpam-6592	170	10	)	)	PUNCT
ejpam-6592	170	11	(	(	PUNCT
ejpam-6592	170	12	2025	2025	NUM
ejpam-6592	170	13	)	)	PUNCT
ejpam-6592	170	14	,	,	PUNCT
ejpam-6592	170	15	6592	6592	NUM
ejpam-6592	170	16	8	8	NUM
ejpam-6592	170	17	of	of	ADP
ejpam-6592	170	18	14	14	NUM
ejpam-6592	170	19	where	where	SCONJ
ejpam-6592	170	20	m(x	m(x	PROPN
ejpam-6592	170	21	,	,	PUNCT
ejpam-6592	170	22	a	a	PRON
ejpam-6592	170	23	,	,	PUNCT
ejpam-6592	170	24	a	a	NOUN
ejpam-6592	170	25	)	)	PUNCT
ejpam-6592	170	26	=	=	SYM
ejpam-6592	170	27	inf{m(x	inf{m(x	PROPN
ejpam-6592	170	28	,	,	PUNCT
ejpam-6592	170	29	a	a	PRON
ejpam-6592	170	30	,	,	PUNCT
ejpam-6592	170	31	a	a	NOUN
ejpam-6592	170	32	)	)	PUNCT
ejpam-6592	170	33	|	|	ADV
ejpam-6592	170	34	a	a	DET
ejpam-6592	170	35	∈	∈	PROPN
ejpam-6592	170	36	a	a	PRON
ejpam-6592	170	37	}	}	PUNCT
ejpam-6592	170	38	.	.	PUNCT
ejpam-6592	171	1	then	then	ADV
ejpam-6592	171	2	define	define	VERB
ejpam-6592	171	3	the	the	DET
ejpam-6592	171	4	refinement	refinement	NOUN
ejpam-6592	171	5	:	:	PUNCT
ejpam-6592	171	6	wα	wα	NOUN
ejpam-6592	171	7	,	,	PUNCT
ejpam-6592	171	8	n	n	NOUN
ejpam-6592	171	9	=	=	SYM
ejpam-6592	171	10	vα	vα	PROPN
ejpam-6592	171	11	,	,	PUNCT
ejpam-6592	171	12	n	n	NOUN
ejpam-6592	171	13	\	\	NOUN
ejpam-6592	171	14	⋃	⋃	PUNCT
ejpam-6592	171	15	β	β	X
ejpam-6592	171	16	<	<	X
ejpam-6592	171	17	α	α	DET
ejpam-6592	171	18	vβ	vβ	NOUN
ejpam-6592	171	19	,	,	PUNCT
ejpam-6592	171	20	n	n	PRON
ejpam-6592	171	21	step	step	NOUN
ejpam-6592	171	22	3	3	NUM
ejpam-6592	171	23	:	:	PUNCT
ejpam-6592	171	24	verify	verify	VERB
ejpam-6592	171	25	σ	σ	PROPN
ejpam-6592	171	26	-	-	PUNCT
ejpam-6592	171	27	discreteness	discreteness	NOUN
ejpam-6592	171	28	.	.	PUNCT
ejpam-6592	172	1	for	for	ADP
ejpam-6592	172	2	fixed	fixed	ADJ
ejpam-6592	172	3	n	n	CCONJ
ejpam-6592	172	4	,	,	PUNCT
ejpam-6592	172	5	the	the	DET
ejpam-6592	172	6	collection	collection	NOUN
ejpam-6592	172	7	{	{	PUNCT
ejpam-6592	172	8	wα	wα	NOUN
ejpam-6592	172	9	,	,	PUNCT
ejpam-6592	172	10	n}α	n}α	PROPN
ejpam-6592	172	11	<	<	X
ejpam-6592	172	12	κ	κ	NOUN
ejpam-6592	172	13	is	be	AUX
ejpam-6592	172	14	discrete	discrete	ADJ
ejpam-6592	172	15	because	because	SCONJ
ejpam-6592	172	16	:	:	PUNCT
ejpam-6592	172	17	•	•	NOUN
ejpam-6592	172	18	for	for	ADP
ejpam-6592	172	19	any	any	DET
ejpam-6592	172	20	x	x	SYM
ejpam-6592	172	21	∈	∈	PROPN
ejpam-6592	172	22	x	x	NOUN
ejpam-6592	172	23	,	,	PUNCT
ejpam-6592	172	24	the	the	DET
ejpam-6592	172	25	ball	ball	NOUN
ejpam-6592	172	26	bm	bm	PROPN
ejpam-6592	172	27	(	(	PUNCT
ejpam-6592	172	28	x	x	NOUN
ejpam-6592	172	29	,	,	PUNCT
ejpam-6592	172	30	1	1	NUM
ejpam-6592	172	31	3rn	3rn	NOUN
ejpam-6592	172	32	)	)	PUNCT
ejpam-6592	172	33	intersects	intersect	NOUN
ejpam-6592	172	34	at	at	ADP
ejpam-6592	172	35	most	most	ADV
ejpam-6592	172	36	one	one	NUM
ejpam-6592	172	37	wα	wα	NOUN
ejpam-6592	172	38	,	,	PUNCT
ejpam-6592	172	39	n.	n.	PROPN
ejpam-6592	172	40	•	•	NOUN
ejpam-6592	172	41	this	this	PRON
ejpam-6592	172	42	follows	follow	VERB
ejpam-6592	172	43	from	from	ADP
ejpam-6592	172	44	the	the	DET
ejpam-6592	172	45	mr	mr	PROPN
ejpam-6592	172	46	-	-	PUNCT
ejpam-6592	172	47	metric	metric	ADJ
ejpam-6592	172	48	inequality	inequality	NOUN
ejpam-6592	172	49	(	(	PUNCT
ejpam-6592	172	50	axiom	axiom	PROPN
ejpam-6592	172	51	m4	m4	PROPN
ejpam-6592	172	52	):	):	PUNCT
ejpam-6592	172	53	if	if	SCONJ
ejpam-6592	172	54	x	x	PROPN
ejpam-6592	172	55	∈wα	∈wα	NUM
ejpam-6592	172	56	,	,	PUNCT
ejpam-6592	172	57	n	n	NOUN
ejpam-6592	172	58	and	and	CCONJ
ejpam-6592	172	59	y	y	PROPN
ejpam-6592	172	60	∈wβ	∈wβ	PROPN
ejpam-6592	172	61	,	,	PUNCT
ejpam-6592	172	62	n	n	CCONJ
ejpam-6592	172	63	with	with	ADP
ejpam-6592	172	64	α	α	PROPN
ejpam-6592	172	65	̸=	̸=	PROPN
ejpam-6592	172	66	β	β	NOUN
ejpam-6592	172	67	,	,	PUNCT
ejpam-6592	172	68	then	then	ADV
ejpam-6592	172	69	m(x	m(x	PROPN
ejpam-6592	172	70	,	,	PUNCT
ejpam-6592	172	71	y	y	PROPN
ejpam-6592	172	72	,	,	PUNCT
ejpam-6592	172	73	y	y	PROPN
ejpam-6592	172	74	)	)	PUNCT
ejpam-6592	172	75	≥	≥	NOUN
ejpam-6592	172	76	1	1	NUM
ejpam-6592	172	77	3rn	3rn	NOUN
ejpam-6592	172	78	.	.	PUNCT
ejpam-6592	173	1	step	step	NOUN
ejpam-6592	173	2	4	4	NUM
ejpam-6592	173	3	:	:	PUNCT
ejpam-6592	173	4	local	local	ADJ
ejpam-6592	173	5	finiteness	finiteness	NOUN
ejpam-6592	173	6	.	.	PUNCT
ejpam-6592	174	1	for	for	ADP
ejpam-6592	174	2	any	any	DET
ejpam-6592	174	3	x	x	SYM
ejpam-6592	174	4	∈	∈	PROPN
ejpam-6592	174	5	x	x	NOUN
ejpam-6592	174	6	,	,	PUNCT
ejpam-6592	174	7	there	there	PRON
ejpam-6592	174	8	exists	exist	VERB
ejpam-6592	174	9	some	some	DET
ejpam-6592	174	10	uα	uα	NOUN
ejpam-6592	174	11	containing	contain	VERB
ejpam-6592	174	12	x	x	PUNCT
ejpam-6592	174	13	and	and	CCONJ
ejpam-6592	174	14	some	some	PRON
ejpam-6592	174	15	n	n	ADP
ejpam-6592	174	16	such	such	ADJ
ejpam-6592	174	17	that	that	SCONJ
ejpam-6592	174	18	bm	bm	PROPN
ejpam-6592	174	19	(	(	PUNCT
ejpam-6592	174	20	x	x	NOUN
ejpam-6592	174	21	,	,	PUNCT
ejpam-6592	174	22	1	1	NUM
ejpam-6592	174	23	3rn	3rn	NOUN
ejpam-6592	174	24	)	)	PUNCT
ejpam-6592	175	1	⊆	⊆	NUM
ejpam-6592	175	2	uα	uα	NOUN
ejpam-6592	175	3	.	.	PUNCT
ejpam-6592	176	1	this	this	DET
ejpam-6592	176	2	neighborhood	neighborhood	NOUN
ejpam-6592	176	3	intersects	intersect	NOUN
ejpam-6592	176	4	only	only	ADV
ejpam-6592	176	5	finitely	finitely	ADV
ejpam-6592	176	6	many	many	ADJ
ejpam-6592	176	7	wβ	wβ	ADP
ejpam-6592	176	8	,	,	PUNCT
ejpam-6592	176	9	m	m	VERB
ejpam-6592	176	10	because	because	SCONJ
ejpam-6592	176	11	:	:	PUNCT
ejpam-6592	176	12	•	•	NOUN
ejpam-6592	176	13	for	for	ADP
ejpam-6592	176	14	m	m	PROPN
ejpam-6592	176	15	>	>	X
ejpam-6592	176	16	n	n	PROPN
ejpam-6592	176	17	,	,	PUNCT
ejpam-6592	176	18	bm	bm	PROPN
ejpam-6592	176	19	(	(	PUNCT
ejpam-6592	176	20	x	x	NOUN
ejpam-6592	176	21	,	,	PUNCT
ejpam-6592	176	22	1	1	NUM
ejpam-6592	176	23	3rn	3rn	NOUN
ejpam-6592	176	24	)	)	PUNCT
ejpam-6592	176	25	can	can	AUX
ejpam-6592	176	26	not	not	PART
ejpam-6592	176	27	intersect	intersect	VERB
ejpam-6592	176	28	any	any	DET
ejpam-6592	176	29	wβ	wβ	ADP
ejpam-6592	176	30	,	,	PUNCT
ejpam-6592	176	31	m	m	VERB
ejpam-6592	176	32	by	by	ADP
ejpam-6592	176	33	construction	construction	NOUN
ejpam-6592	176	34	.	.	PUNCT
ejpam-6592	177	1	•	•	NUM
ejpam-6592	177	2	for	for	ADP
ejpam-6592	177	3	each	each	DET
ejpam-6592	177	4	m	m	NOUN
ejpam-6592	177	5	≤	≤	NOUN
ejpam-6592	177	6	n	n	CCONJ
ejpam-6592	177	7	,	,	PUNCT
ejpam-6592	177	8	the	the	DET
ejpam-6592	177	9	discreteness	discreteness	NOUN
ejpam-6592	177	10	in	in	ADP
ejpam-6592	177	11	step	step	NOUN
ejpam-6592	177	12	3	3	NUM
ejpam-6592	177	13	ensures	ensure	VERB
ejpam-6592	177	14	only	only	ADV
ejpam-6592	177	15	one	one	NUM
ejpam-6592	177	16	wβ	wβ	NOUN
ejpam-6592	177	17	,	,	PUNCT
ejpam-6592	177	18	m	m	PROPN
ejpam-6592	177	19	can	can	AUX
ejpam-6592	177	20	intersect	intersect	VERB
ejpam-6592	177	21	the	the	DET
ejpam-6592	177	22	neighborhood	neighborhood	NOUN
ejpam-6592	177	23	.	.	PUNCT
ejpam-6592	178	1	step	step	NOUN
ejpam-6592	178	2	5	5	NUM
ejpam-6592	178	3	:	:	PUNCT
ejpam-6592	178	4	covering	cover	VERB
ejpam-6592	178	5	property	property	NOUN
ejpam-6592	178	6	.	.	PUNCT
ejpam-6592	179	1	for	for	ADP
ejpam-6592	179	2	any	any	DET
ejpam-6592	179	3	x	x	SYM
ejpam-6592	179	4	∈	∈	PROPN
ejpam-6592	179	5	x	x	NOUN
ejpam-6592	179	6	,	,	PUNCT
ejpam-6592	179	7	let	let	VERB
ejpam-6592	179	8	α	α	PRON
ejpam-6592	179	9	be	be	AUX
ejpam-6592	179	10	the	the	DET
ejpam-6592	179	11	smallest	small	ADJ
ejpam-6592	179	12	index	index	NOUN
ejpam-6592	179	13	with	with	ADP
ejpam-6592	179	14	x	x	PROPN
ejpam-6592	179	15	∈	∈	PROPN
ejpam-6592	179	16	uα	uα	PROPN
ejpam-6592	179	17	.	.	PUNCT
ejpam-6592	180	1	then	then	ADV
ejpam-6592	180	2	x	x	X
ejpam-6592	180	3	∈wα	∈wα	NUM
ejpam-6592	180	4	,	,	PUNCT
ejpam-6592	180	5	n	n	CCONJ
ejpam-6592	180	6	for	for	ADP
ejpam-6592	180	7	some	some	PRON
ejpam-6592	180	8	n	n	ADV
ejpam-6592	180	9	large	large	ADJ
ejpam-6592	180	10	enough	enough	ADV
ejpam-6592	180	11	that	that	SCONJ
ejpam-6592	180	12	m(x	m(x	PROPN
ejpam-6592	180	13	,	,	PUNCT
ejpam-6592	180	14	x	x	SYM
ejpam-6592	180	15	\	\	PROPN
ejpam-6592	180	16	uα	uα	PROPN
ejpam-6592	180	17	,	,	PUNCT
ejpam-6592	180	18	x	x	SYM
ejpam-6592	180	19	\	\	PROPN
ejpam-6592	180	20	uα	uα	PROPN
ejpam-6592	180	21	)	)	PUNCT
ejpam-6592	180	22	≥	≥	NOUN
ejpam-6592	180	23	1	1	NUM
ejpam-6592	180	24	/	/	SYM
ejpam-6592	180	25	n.	n.	NOUN
ejpam-6592	180	26	thus	thus	ADV
ejpam-6592	180	27	{	{	PUNCT
ejpam-6592	180	28	wα	wα	NOUN
ejpam-6592	180	29	,	,	PUNCT
ejpam-6592	180	30	n}α	n}α	PROPN
ejpam-6592	180	31	<	<	X
ejpam-6592	180	32	κ	κ	X
ejpam-6592	180	33	,	,	PUNCT
ejpam-6592	180	34	n∈n	n∈n	NUM
ejpam-6592	180	35	is	be	AUX
ejpam-6592	180	36	a	a	DET
ejpam-6592	180	37	locally	locally	ADV
ejpam-6592	180	38	finite	finite	ADJ
ejpam-6592	180	39	refinement	refinement	NOUN
ejpam-6592	180	40	of	of	ADP
ejpam-6592	180	41	u	u	PROPN
ejpam-6592	180	42	.	.	PUNCT
ejpam-6592	181	1	corollary	corollary	ADJ
ejpam-6592	181	2	2	2	NUM
ejpam-6592	181	3	.	.	NOUN
ejpam-6592	181	4	•	•	NOUN
ejpam-6592	181	5	every	every	DET
ejpam-6592	181	6	mr	mr	PROPN
ejpam-6592	181	7	-	-	PUNCT
ejpam-6592	181	8	metric	metric	ADJ
ejpam-6592	181	9	space	space	NOUN
ejpam-6592	181	10	is	be	AUX
ejpam-6592	181	11	normal	normal	ADJ
ejpam-6592	181	12	(	(	PUNCT
ejpam-6592	181	13	disjoint	disjoint	NOUN
ejpam-6592	181	14	closed	closed	ADJ
ejpam-6592	181	15	sets	set	NOUN
ejpam-6592	181	16	are	be	AUX
ejpam-6592	181	17	separable	separable	ADJ
ejpam-6592	181	18	)	)	PUNCT
ejpam-6592	181	19	.	.	PUNCT
ejpam-6592	182	1	•	•	NOUN
ejpam-6592	182	2	continuous	continuous	ADJ
ejpam-6592	182	3	functions	function	NOUN
ejpam-6592	182	4	on	on	ADP
ejpam-6592	182	5	x	x	PART
ejpam-6592	182	6	admit	admit	VERB
ejpam-6592	182	7	tietze	tietze	NOUN
ejpam-6592	182	8	extensions	extension	NOUN
ejpam-6592	182	9	.	.	PUNCT
ejpam-6592	183	1	proof	proof	NOUN
ejpam-6592	183	2	.	.	PUNCT
ejpam-6592	184	1	we	we	PRON
ejpam-6592	184	2	prove	prove	VERB
ejpam-6592	184	3	each	each	DET
ejpam-6592	184	4	item	item	NOUN
ejpam-6592	184	5	separately	separately	ADV
ejpam-6592	184	6	using	use	VERB
ejpam-6592	184	7	theorem	theorem	ADJ
ejpam-6592	184	8	3	3	NUM
ejpam-6592	184	9	(	(	PUNCT
ejpam-6592	184	10	paracompactness	paracompactness	NOUN
ejpam-6592	184	11	of	of	ADP
ejpam-6592	184	12	mrmetric	mrmetric	ADJ
ejpam-6592	184	13	spaces	space	NOUN
ejpam-6592	184	14	)	)	PUNCT
ejpam-6592	184	15	.	.	PUNCT
ejpam-6592	185	1	part	part	NOUN
ejpam-6592	185	2	1	1	NUM
ejpam-6592	185	3	:	:	PUNCT
ejpam-6592	185	4	normality	normality	NOUN
ejpam-6592	185	5	let	let	VERB
ejpam-6592	185	6	a	a	PRON
ejpam-6592	185	7	and	and	CCONJ
ejpam-6592	185	8	b	b	NOUN
ejpam-6592	185	9	be	be	AUX
ejpam-6592	185	10	disjoint	disjoint	X
ejpam-6592	185	11	closed	close	VERB
ejpam-6592	185	12	sets	set	NOUN
ejpam-6592	185	13	in	in	ADP
ejpam-6592	185	14	(	(	PUNCT
ejpam-6592	185	15	x	x	NOUN
ejpam-6592	185	16	,	,	PUNCT
ejpam-6592	185	17	m	m	NOUN
ejpam-6592	185	18	)	)	PUNCT
ejpam-6592	185	19	.	.	PUNCT
ejpam-6592	186	1	since	since	SCONJ
ejpam-6592	186	2	x	x	PRON
ejpam-6592	186	3	is	be	AUX
ejpam-6592	186	4	paracompact	paracompact	ADJ
ejpam-6592	186	5	by	by	ADP
ejpam-6592	186	6	theorem	theorem	NOUN
ejpam-6592	186	7	3	3	NUM
ejpam-6592	186	8	,	,	PUNCT
ejpam-6592	186	9	it	it	PRON
ejpam-6592	186	10	is	be	AUX
ejpam-6592	186	11	regular	regular	ADJ
ejpam-6592	186	12	and	and	CCONJ
ejpam-6592	186	13	normal	normal	ADJ
ejpam-6592	186	14	(	(	PUNCT
ejpam-6592	186	15	as	as	SCONJ
ejpam-6592	186	16	paracompact	paracompact	ADJ
ejpam-6592	186	17	hausdorff	hausdorff	NOUN
ejpam-6592	186	18	spaces	space	NOUN
ejpam-6592	186	19	are	be	AUX
ejpam-6592	186	20	normal	normal	ADJ
ejpam-6592	186	21	)	)	PUNCT
ejpam-6592	186	22	.	.	PUNCT
ejpam-6592	187	1	construct	construct	VERB
ejpam-6592	187	2	separation	separation	NOUN
ejpam-6592	187	3	as	as	SCONJ
ejpam-6592	187	4	follows	follow	VERB
ejpam-6592	187	5	:	:	PUNCT
ejpam-6592	187	6	•	•	ADP
ejpam-6592	187	7	the	the	DET
ejpam-6592	187	8	open	open	ADJ
ejpam-6592	187	9	cover	cover	NOUN
ejpam-6592	187	10	u	u	NOUN
ejpam-6592	187	11	=	=	PUNCT
ejpam-6592	187	12	{	{	PUNCT
ejpam-6592	187	13	x	x	SYM
ejpam-6592	187	14	\a	\a	ADJ
ejpam-6592	187	15	,	,	PUNCT
ejpam-6592	187	16	x	x	SYM
ejpam-6592	187	17	\b	\b	ADJ
ejpam-6592	187	18	}	}	PUNCT
ejpam-6592	187	19	has	have	VERB
ejpam-6592	187	20	a	a	DET
ejpam-6592	187	21	locally	locally	ADV
ejpam-6592	187	22	finite	finite	ADJ
ejpam-6592	187	23	refinement	refinement	NOUN
ejpam-6592	187	24	.	.	PUNCT
ejpam-6592	188	1	•	•	NUM
ejpam-6592	188	2	using	use	VERB
ejpam-6592	188	3	the	the	DET
ejpam-6592	188	4	mr	mr	PROPN
ejpam-6592	188	5	-	-	PUNCT
ejpam-6592	188	6	metric	metric	ADJ
ejpam-6592	188	7	,	,	PUNCT
ejpam-6592	188	8	define	define	VERB
ejpam-6592	188	9	f(x	f(x	PROPN
ejpam-6592	188	10	)	)	PUNCT
ejpam-6592	189	1	=	=	SYM
ejpam-6592	190	1	m(x	m(x	PROPN
ejpam-6592	190	2	,	,	PUNCT
ejpam-6592	190	3	a	a	PRON
ejpam-6592	190	4	,	,	PUNCT
ejpam-6592	190	5	a	a	NOUN
ejpam-6592	190	6	)	)	PUNCT
ejpam-6592	190	7	m(x	m(x	PROPN
ejpam-6592	190	8	,	,	PUNCT
ejpam-6592	190	9	a	a	DET
ejpam-6592	190	10	,	,	PUNCT
ejpam-6592	190	11	a)+m(x	a)+m(x	PROPN
ejpam-6592	190	12	,	,	PUNCT
ejpam-6592	190	13	b	b	PROPN
ejpam-6592	190	14	,	,	PUNCT
ejpam-6592	190	15	b	b	NOUN
ejpam-6592	190	16	)	)	PUNCT
ejpam-6592	190	17	where	where	SCONJ
ejpam-6592	190	18	:	:	PUNCT
ejpam-6592	190	19	m(x	m(x	PROPN
ejpam-6592	190	20	,	,	PUNCT
ejpam-6592	190	21	a	a	PRON
ejpam-6592	190	22	,	,	PUNCT
ejpam-6592	190	23	a	a	NOUN
ejpam-6592	190	24	)	)	PUNCT
ejpam-6592	190	25	=	=	SYM
ejpam-6592	190	26	inf	inf	PROPN
ejpam-6592	190	27	a∈a	a∈a	ADJ
ejpam-6592	190	28	m(x	m(x	PROPN
ejpam-6592	190	29	,	,	PUNCT
ejpam-6592	190	30	a	a	PRON
ejpam-6592	190	31	,	,	PUNCT
ejpam-6592	190	32	a	a	PRON
ejpam-6592	190	33	)	)	PUNCT
ejpam-6592	190	34	•	•	NOUN
ejpam-6592	190	35	the	the	DET
ejpam-6592	190	36	mr	mr	PROPN
ejpam-6592	190	37	-	-	PUNCT
ejpam-6592	190	38	metric	metric	ADJ
ejpam-6592	190	39	axioms	axiom	NOUN
ejpam-6592	190	40	ensure	ensure	VERB
ejpam-6592	190	41	f	f	PROPN
ejpam-6592	190	42	is	be	AUX
ejpam-6592	190	43	continuous	continuous	ADJ
ejpam-6592	190	44	,	,	PUNCT
ejpam-6592	190	45	with	with	ADP
ejpam-6592	190	46	f	f	PROPN
ejpam-6592	190	47	|a	|a	X
ejpam-6592	190	48	=	=	SYM
ejpam-6592	190	49	0	0	NUM
ejpam-6592	190	50	and	and	CCONJ
ejpam-6592	190	51	f	f	PROPN
ejpam-6592	190	52	|b	|b	NOUN
ejpam-6592	190	53	=	=	SYM
ejpam-6592	190	54	1	1	NUM
ejpam-6592	190	55	.	.	NUM
ejpam-6592	190	56	•	•	NOUN
ejpam-6592	190	57	the	the	DET
ejpam-6592	190	58	sets	set	NOUN
ejpam-6592	190	59	u	u	NOUN
ejpam-6592	190	60	=	=	SYM
ejpam-6592	190	61	f−1([0	f−1([0	PROPN
ejpam-6592	190	62	,	,	PUNCT
ejpam-6592	190	63	1/2	1/2	NUM
ejpam-6592	190	64	)	)	PUNCT
ejpam-6592	190	65	)	)	PUNCT
ejpam-6592	190	66	and	and	CCONJ
ejpam-6592	190	67	v	v	X
ejpam-6592	190	68	=	=	SYM
ejpam-6592	190	69	f−1((1/2	f−1((1/2	PROPN
ejpam-6592	190	70	,	,	PUNCT
ejpam-6592	190	71	1	1	NUM
ejpam-6592	190	72	]	]	PUNCT
ejpam-6592	190	73	)	)	PUNCT
ejpam-6592	190	74	are	be	AUX
ejpam-6592	190	75	disjoint	disjoint	ADJ
ejpam-6592	190	76	open	open	ADJ
ejpam-6592	190	77	neighborhoods	neighborhood	NOUN
ejpam-6592	190	78	of	of	ADP
ejpam-6592	190	79	a	a	PRON
ejpam-6592	190	80	and	and	CCONJ
ejpam-6592	190	81	b	b	NOUN
ejpam-6592	190	82	respectively	respectively	ADV
ejpam-6592	190	83	.	.	PUNCT
ejpam-6592	191	1	a.	a.	NOUN
ejpam-6592	191	2	malkawi	malkawi	ADP
ejpam-6592	191	3	/	/	SYM
ejpam-6592	191	4	eur	eur	PROPN
ejpam-6592	191	5	.	.	PUNCT
ejpam-6592	192	1	j.	j.	PROPN
ejpam-6592	192	2	pure	pure	PROPN
ejpam-6592	192	3	appl	appl	PROPN
ejpam-6592	192	4	.	.	PROPN
ejpam-6592	192	5	math	math	PROPN
ejpam-6592	192	6	,	,	PUNCT
ejpam-6592	192	7	18	18	NUM
ejpam-6592	192	8	(	(	PUNCT
ejpam-6592	192	9	3	3	NUM
ejpam-6592	192	10	)	)	PUNCT
ejpam-6592	192	11	(	(	PUNCT
ejpam-6592	192	12	2025	2025	NUM
ejpam-6592	192	13	)	)	PUNCT
ejpam-6592	192	14	,	,	PUNCT
ejpam-6592	192	15	6592	6592	NUM
ejpam-6592	192	16	9	9	NUM
ejpam-6592	192	17	of	of	ADP
ejpam-6592	192	18	14	14	NUM
ejpam-6592	192	19	part	part	NOUN
ejpam-6592	192	20	2	2	NUM
ejpam-6592	192	21	:	:	PUNCT
ejpam-6592	192	22	tietze	tietze	NOUN
ejpam-6592	192	23	extension	extension	NOUN
ejpam-6592	192	24	given	give	VERB
ejpam-6592	192	25	a	a	DET
ejpam-6592	192	26	continuous	continuous	ADJ
ejpam-6592	192	27	f	f	NOUN
ejpam-6592	192	28	:	:	PUNCT
ejpam-6592	192	29	a	a	DET
ejpam-6592	192	30	→	→	SYM
ejpam-6592	192	31	r	r	NOUN
ejpam-6592	192	32	on	on	ADP
ejpam-6592	192	33	closed	close	VERB
ejpam-6592	192	34	a	a	DET
ejpam-6592	192	35	⊆	⊆	NUM
ejpam-6592	192	36	x	x	SYM
ejpam-6592	192	37	,	,	PUNCT
ejpam-6592	192	38	we	we	PRON
ejpam-6592	192	39	construct	construct	VERB
ejpam-6592	192	40	its	its	PRON
ejpam-6592	192	41	extension	extension	NOUN
ejpam-6592	192	42	using	use	VERB
ejpam-6592	192	43	normality	normality	NOUN
ejpam-6592	192	44	:	:	PUNCT
ejpam-6592	192	45	•	•	ADP
ejpam-6592	192	46	for	for	ADP
ejpam-6592	192	47	paracompact	paracompact	ADJ
ejpam-6592	192	48	x	x	NOUN
ejpam-6592	192	49	,	,	PUNCT
ejpam-6592	192	50	there	there	PRON
ejpam-6592	192	51	exists	exist	VERB
ejpam-6592	192	52	a	a	DET
ejpam-6592	192	53	partition	partition	NOUN
ejpam-6592	192	54	of	of	ADP
ejpam-6592	192	55	unity	unity	NOUN
ejpam-6592	192	56	subordinate	subordinate	ADJ
ejpam-6592	192	57	to	to	ADP
ejpam-6592	192	58	any	any	DET
ejpam-6592	192	59	open	open	ADJ
ejpam-6592	192	60	cover	cover	NOUN
ejpam-6592	192	61	.	.	PUNCT
ejpam-6592	193	1	•	•	NOUN
ejpam-6592	193	2	using	use	VERB
ejpam-6592	193	3	the	the	DET
ejpam-6592	193	4	mr	mr	PROPN
ejpam-6592	193	5	-	-	PUNCT
ejpam-6592	193	6	metric	metric	ADJ
ejpam-6592	193	7	,	,	PUNCT
ejpam-6592	193	8	define	define	VERB
ejpam-6592	193	9	local	local	ADJ
ejpam-6592	193	10	extensions	extension	NOUN
ejpam-6592	193	11	on	on	ADP
ejpam-6592	193	12	neighborhoods	neighborhood	NOUN
ejpam-6592	193	13	of	of	ADP
ejpam-6592	193	14	a.	a.	NOUN
ejpam-6592	193	15	•	•	NOUN
ejpam-6592	193	16	the	the	DET
ejpam-6592	193	17	scaling	scaling	ADJ
ejpam-6592	193	18	factor	factor	NOUN
ejpam-6592	193	19	r	r	NOUN
ejpam-6592	193	20	in	in	ADP
ejpam-6592	193	21	the	the	DET
ejpam-6592	193	22	mr	mr	PROPN
ejpam-6592	193	23	-	-	PUNCT
ejpam-6592	193	24	metric	metric	ADJ
ejpam-6592	193	25	ensures	ensure	NOUN
ejpam-6592	193	26	controlled	control	VERB
ejpam-6592	193	27	patching	patching	NOUN
ejpam-6592	193	28	via	via	ADP
ejpam-6592	193	29	:	:	PUNCT
ejpam-6592	193	30	f̃(x	f̃(x	NOUN
ejpam-6592	193	31	)	)	PUNCT
ejpam-6592	193	32	=	=	PRON
ejpam-6592	193	33	{	{	PUNCT
ejpam-6592	193	34	f(x	f(x	PROPN
ejpam-6592	193	35	)	)	PUNCT
ejpam-6592	193	36	x	x	SYM
ejpam-6592	194	1	∈	∈	PROPN
ejpam-6592	194	2	a∑	a∑	PROPN
ejpam-6592	194	3	α	α	PROPN
ejpam-6592	194	4	ϕα(x)fα(x)∑	ϕα(x)fα(x)∑	PROPN
ejpam-6592	194	5	α	α	PRON
ejpam-6592	194	6	ϕα(x	ϕα(x	NOUN
ejpam-6592	194	7	)	)	PUNCT
ejpam-6592	194	8	x	x	SYM
ejpam-6592	194	9	∈	∈	PROPN
ejpam-6592	194	10	x	x	PUNCT
ejpam-6592	194	11	\a	\a	ADJ
ejpam-6592	194	12	where	where	SCONJ
ejpam-6592	194	13	{	{	PUNCT
ejpam-6592	194	14	ϕα	ϕα	NOUN
ejpam-6592	194	15	}	}	PUNCT
ejpam-6592	194	16	is	be	AUX
ejpam-6592	194	17	a	a	DET
ejpam-6592	194	18	partition	partition	NOUN
ejpam-6592	194	19	of	of	ADP
ejpam-6592	194	20	unity	unity	NOUN
ejpam-6592	194	21	and	and	CCONJ
ejpam-6592	194	22	fα	fα	NOUN
ejpam-6592	194	23	are	be	AUX
ejpam-6592	194	24	local	local	ADJ
ejpam-6592	194	25	extensions	extension	NOUN
ejpam-6592	194	26	.	.	PUNCT
ejpam-6592	195	1	•	•	NUM
ejpam-6592	195	2	the	the	DET
ejpam-6592	195	3	r	r	NOUN
ejpam-6592	195	4	-	-	PUNCT
ejpam-6592	195	5	scaled	scale	VERB
ejpam-6592	195	6	tetrahedral	tetrahedral	ADJ
ejpam-6592	195	7	inequality	inequality	NOUN
ejpam-6592	195	8	guarantees	guarantee	VERB
ejpam-6592	195	9	uniform	uniform	ADJ
ejpam-6592	195	10	continuity	continuity	NOUN
ejpam-6592	195	11	when	when	SCONJ
ejpam-6592	195	12	gluing	glue	VERB
ejpam-6592	195	13	local	local	ADJ
ejpam-6592	195	14	extensions	extension	NOUN
ejpam-6592	195	15	.	.	PUNCT
ejpam-6592	196	1	3	3	X
ejpam-6592	196	2	.	.	X
ejpam-6592	196	3	applications	application	NOUN
ejpam-6592	196	4	and	and	CCONJ
ejpam-6592	196	5	examples	example	NOUN
ejpam-6592	196	6	3.1	3.1	NUM
ejpam-6592	196	7	.	.	PUNCT
ejpam-6592	197	1	compactness	compactness	NOUN
ejpam-6592	197	2	in	in	ADP
ejpam-6592	197	3	optimization	optimization	NOUN
ejpam-6592	197	4	problems	problem	NOUN
ejpam-6592	197	5	example	example	VERB
ejpam-6592	197	6	1	1	NUM
ejpam-6592	197	7	(	(	PUNCT
ejpam-6592	197	8	global	global	ADJ
ejpam-6592	197	9	optimization	optimization	NOUN
ejpam-6592	197	10	)	)	PUNCT
ejpam-6592	197	11	.	.	PUNCT
ejpam-6592	198	1	consider	consider	VERB
ejpam-6592	198	2	the	the	DET
ejpam-6592	198	3	mr	mr	PROPN
ejpam-6592	198	4	-	-	PUNCT
ejpam-6592	198	5	metric	metric	ADJ
ejpam-6592	198	6	space	space	NOUN
ejpam-6592	198	7	(	(	PUNCT
ejpam-6592	198	8	x	x	X
ejpam-6592	198	9	,	,	PUNCT
ejpam-6592	198	10	m	m	NOUN
ejpam-6592	198	11	)	)	PUNCT
ejpam-6592	198	12	where	where	SCONJ
ejpam-6592	198	13	x	x	X
ejpam-6592	199	1	=	=	PUNCT
ejpam-6592	200	1	[	[	X
ejpam-6592	200	2	−10	−10	NOUN
ejpam-6592	200	3	,	,	PUNCT
ejpam-6592	200	4	10]n	10]n	NUM
ejpam-6592	200	5	⊂	⊂	X
ejpam-6592	200	6	rn	rn	PROPN
ejpam-6592	200	7	with	with	ADP
ejpam-6592	200	8	:	:	PUNCT
ejpam-6592	200	9	m(x	m(x	PROPN
ejpam-6592	200	10	,	,	PUNCT
ejpam-6592	200	11	y	y	PROPN
ejpam-6592	200	12	,	,	PUNCT
ejpam-6592	200	13	z	z	NOUN
ejpam-6592	200	14	)	)	PUNCT
ejpam-6592	200	15	=	=	SYM
ejpam-6592	201	1	∥x−	∥x−	NUM
ejpam-6592	201	2	y∥2	y∥2	NOUN
ejpam-6592	201	3	+	+	CCONJ
ejpam-6592	201	4	∥y	∥y	ADJ
ejpam-6592	201	5	−	−	NOUN
ejpam-6592	201	6	z∥2	z∥2	NOUN
ejpam-6592	201	7	+	+	CCONJ
ejpam-6592	201	8	∥z−	∥z−	NUM
ejpam-6592	201	9	x∥2	x∥2	NOUN
ejpam-6592	201	10	3r	3r	NOUN
ejpam-6592	201	11	for	for	ADP
ejpam-6592	201	12	r	r	NOUN
ejpam-6592	201	13	=	=	SYM
ejpam-6592	201	14	1.2	1.2	NUM
ejpam-6592	201	15	.	.	PUNCT
ejpam-6592	201	16	by	by	ADP
ejpam-6592	201	17	theorem	theorem	NOUN
ejpam-6592	201	18	1	1	NUM
ejpam-6592	201	19	,	,	PUNCT
ejpam-6592	201	20	x	x	PRON
ejpam-6592	201	21	is	be	AUX
ejpam-6592	201	22	compact	compact	ADJ
ejpam-6592	201	23	.	.	PUNCT
ejpam-6592	202	1	[	[	X
ejpam-6592	202	2	h	h	X
ejpam-6592	202	3	]	]	X
ejpam-6592	202	4	compactness	compactness	NOUN
ejpam-6592	202	5	-	-	PUNCT
ejpam-6592	202	6	based	base	VERB
ejpam-6592	202	7	global	global	ADJ
ejpam-6592	202	8	optimization	optimization	NOUN
ejpam-6592	202	9	1	1	NUM
ejpam-6592	202	10	:	:	PUNCT
ejpam-6592	202	11	input	input	NOUN
ejpam-6592	202	12	:	:	PUNCT
ejpam-6592	202	13	objective	objective	ADJ
ejpam-6592	202	14	function	function	NOUN
ejpam-6592	202	15	f	f	NOUN
ejpam-6592	202	16	:	:	PUNCT
ejpam-6592	202	17	x	x	X
ejpam-6592	202	18	→	→	SYM
ejpam-6592	202	19	r	r	NOUN
ejpam-6592	202	20	2	2	NUM
ejpam-6592	202	21	:	:	PUNCT
ejpam-6592	202	22	generate	generate	VERB
ejpam-6592	202	23	ϵ-net	ϵ-net	NOUN
ejpam-6592	202	24	{	{	PUNCT
ejpam-6592	202	25	x1	x1	PROPN
ejpam-6592	202	26	,	,	PUNCT
ejpam-6592	202	27	...	...	PUNCT
ejpam-6592	202	28	,	,	PUNCT
ejpam-6592	202	29	xk	xk	PROPN
ejpam-6592	202	30	}	}	PUNCT
ejpam-6592	202	31	with	with	ADP
ejpam-6592	202	32	ϵ	ϵ	PROPN
ejpam-6592	202	33	=	=	SYM
ejpam-6592	202	34	0.1	0.1	NUM
ejpam-6592	202	35	3	3	NUM
ejpam-6592	202	36	:	:	PUNCT
ejpam-6592	202	37	evaluate	evaluate	VERB
ejpam-6592	202	38	f(xi	f(xi	NOUN
ejpam-6592	202	39	)	)	PUNCT
ejpam-6592	202	40	for	for	ADP
ejpam-6592	202	41	all	all	PRON
ejpam-6592	202	42	i	i	PRON
ejpam-6592	202	43	4	4	NUM
ejpam-6592	202	44	:	:	PUNCT
ejpam-6592	202	45	identify	identify	VERB
ejpam-6592	202	46	x∗	x∗	NOUN
ejpam-6592	203	1	=	=	SYM
ejpam-6592	203	2	argmin	argmin	PROPN
ejpam-6592	203	3	f(xi	f(xi	PROPN
ejpam-6592	203	4	)	)	PUNCT
ejpam-6592	203	5	5	5	NUM
ejpam-6592	203	6	:	:	PUNCT
ejpam-6592	203	7	refine	refine	VERB
ejpam-6592	203	8	search	search	NOUN
ejpam-6592	203	9	near	near	ADP
ejpam-6592	203	10	x∗	x∗	PROPN
ejpam-6592	203	11	with	with	ADP
ejpam-6592	203	12	smaller	small	ADJ
ejpam-6592	203	13	ϵ	ϵ	ADP
ejpam-6592	203	14	1	1	NUM
ejpam-6592	203	15	import	import	NOUN
ejpam-6592	203	16	numpy	numpy	NOUN
ejpam-6592	203	17	as	as	ADP
ejpam-6592	203	18	np	np	ADP
ejpam-6592	203	19	2	2	NUM
ejpam-6592	203	20	3	3	NUM
ejpam-6592	203	21	def	def	ADJ
ejpam-6592	203	22	mr_metric(x	mr_metric(x	PROPN
ejpam-6592	203	23	,	,	PUNCT
ejpam-6592	203	24	y	y	PROPN
ejpam-6592	203	25	,	,	PUNCT
ejpam-6592	203	26	z	z	PROPN
ejpam-6592	203	27	,	,	PUNCT
ejpam-6592	203	28	r=1.2	r=1.2	NOUN
ejpam-6592	203	29	):	):	PUNCT
ejpam-6592	203	30	4	4	NUM
ejpam-6592	203	31	return	return	NOUN
ejpam-6592	203	32	(	(	PUNCT
ejpam-6592	203	33	np.linalg.norm(x	np.linalg.norm(x	NOUN
ejpam-6592	203	34	y	y	NOUN
ejpam-6592	203	35	)	)	PUNCT
ejpam-6592	204	1	+	+	CCONJ
ejpam-6592	204	2	np.linalg.norm(y	np.linalg.norm(y	ADJ
ejpam-6592	204	3	z	z	NOUN
ejpam-6592	204	4	)	)	PUNCT
ejpam-6592	205	1	+	+	CCONJ
ejpam-6592	205	2	np	np	X
ejpam-6592	205	3	.	.	PUNCT
ejpam-6592	206	1	linalg.norm(z	linalg.norm(z	PROPN
ejpam-6592	206	2	x	x	X
ejpam-6592	206	3	)	)	PUNCT
ejpam-6592	206	4	)	)	PUNCT
ejpam-6592	206	5	/	/	PUNCT
ejpam-6592	207	1	(	(	PUNCT
ejpam-6592	207	2	3	3	NUM
ejpam-6592	207	3	*	*	SYM
ejpam-6592	207	4	r	r	NOUN
ejpam-6592	207	5	)	)	PUNCT
ejpam-6592	207	6	5	5	NUM
ejpam-6592	207	7	6	6	NUM
ejpam-6592	207	8	def	def	ADJ
ejpam-6592	207	9	generate_epsilon_net(n	generate_epsilon_net(n	PROPN
ejpam-6592	207	10	,	,	PUNCT
ejpam-6592	207	11	epsilon	epsilon	PROPN
ejpam-6592	207	12	=	=	SYM
ejpam-6592	207	13	0.1	0.1	NUM
ejpam-6592	207	14	):	):	PUNCT
ejpam-6592	207	15	7	7	NUM
ejpam-6592	207	16	grid_points	grid_point	NOUN
ejpam-6592	207	17	=	=	PUNCT
ejpam-6592	208	1	[	[	X
ejpam-6592	208	2	np.linspace	np.linspace	X
ejpam-6592	208	3	(	(	PUNCT
ejpam-6592	208	4	-10	-10	X
ejpam-6592	208	5	,	,	PUNCT
ejpam-6592	208	6	10	10	NUM
ejpam-6592	208	7	,	,	PUNCT
ejpam-6592	208	8	int	int	NOUN
ejpam-6592	208	9	(	(	PUNCT
ejpam-6592	208	10	20	20	NUM
ejpam-6592	208	11	/	/	SYM
ejpam-6592	208	12	epsilon	epsilon	PROPN
ejpam-6592	208	13	)	)	PUNCT
ejpam-6592	208	14	)	)	PUNCT
ejpam-6592	208	15	for	for	ADP
ejpam-6592	208	16	_	_	PROPN
ejpam-6592	208	17	in	in	ADP
ejpam-6592	208	18	range(n	range(n	PROPN
ejpam-6592	208	19	)	)	PUNCT
ejpam-6592	208	20	]	]	PUNCT
ejpam-6592	208	21	8	8	NUM
ejpam-6592	208	22	return	return	NOUN
ejpam-6592	208	23	np.array(np.meshgrid	np.array(np.meshgrid	VERB
ejpam-6592	208	24	(	(	PUNCT
ejpam-6592	208	25	*	*	PUNCT
ejpam-6592	208	26	grid_points)).t.reshape(-1	grid_points)).t.reshape(-1	PROPN
ejpam-6592	208	27	,	,	PUNCT
ejpam-6592	208	28	n	n	CCONJ
ejpam-6592	208	29	)	)	PUNCT
ejpam-6592	208	30	a.	a.	NOUN
ejpam-6592	208	31	malkawi	malkawi	ADP
ejpam-6592	208	32	/	/	SYM
ejpam-6592	208	33	eur	eur	PROPN
ejpam-6592	208	34	.	.	PUNCT
ejpam-6592	209	1	j.	j.	PROPN
ejpam-6592	209	2	pure	pure	PROPN
ejpam-6592	209	3	appl	appl	PROPN
ejpam-6592	209	4	.	.	PROPN
ejpam-6592	209	5	math	math	PROPN
ejpam-6592	209	6	,	,	PUNCT
ejpam-6592	209	7	18	18	NUM
ejpam-6592	209	8	(	(	PUNCT
ejpam-6592	209	9	3	3	NUM
ejpam-6592	209	10	)	)	PUNCT
ejpam-6592	209	11	(	(	PUNCT
ejpam-6592	209	12	2025	2025	NUM
ejpam-6592	209	13	)	)	PUNCT
ejpam-6592	209	14	,	,	PUNCT
ejpam-6592	209	15	6592	6592	NUM
ejpam-6592	209	16	10	10	NUM
ejpam-6592	209	17	of	of	ADP
ejpam-6592	209	18	14	14	NUM
ejpam-6592	209	19	−10	−10	NOUN
ejpam-6592	209	20	−5	−5	ADV
ejpam-6592	209	21	0	0	NUM
ejpam-6592	209	22	5	5	NUM
ejpam-6592	209	23	10	10	NUM
ejpam-6592	209	24	−10	−10	NOUN
ejpam-6592	209	25	−5	−5	ADV
ejpam-6592	209	26	0	0	NUM
ejpam-6592	209	27	5	5	NUM
ejpam-6592	209	28	10	10	NUM
ejpam-6592	209	29	ϵ-net	ϵ-net	NOUN
ejpam-6592	209	30	covering	cover	VERB
ejpam-6592	209	31	3.2	3.2	NUM
ejpam-6592	209	32	.	.	PUNCT
ejpam-6592	210	1	lindelöf	lindelöf	NOUN
ejpam-6592	210	2	property	property	NOUN
ejpam-6592	210	3	in	in	ADP
ejpam-6592	210	4	machine	machine	NOUN
ejpam-6592	210	5	learning	learning	NOUN
ejpam-6592	210	6	theorem	theorem	VERB
ejpam-6592	210	7	4	4	NUM
ejpam-6592	210	8	(	(	PUNCT
ejpam-6592	210	9	dimensionality	dimensionality	NOUN
ejpam-6592	210	10	reduction	reduction	NOUN
ejpam-6592	210	11	)	)	PUNCT
ejpam-6592	210	12	.	.	PUNCT
ejpam-6592	211	1	let	let	AUX
ejpam-6592	211	2	(	(	PUNCT
ejpam-6592	211	3	h	h	NOUN
ejpam-6592	211	4	,	,	PUNCT
ejpam-6592	211	5	m	m	VERB
ejpam-6592	211	6	)	)	PUNCT
ejpam-6592	211	7	be	be	VERB
ejpam-6592	211	8	an	an	DET
ejpam-6592	211	9	mr	mr	ADJ
ejpam-6592	211	10	-	-	PUNCT
ejpam-6592	211	11	metric	metric	ADJ
ejpam-6592	211	12	space	space	NOUN
ejpam-6592	211	13	of	of	ADP
ejpam-6592	211	14	neural	neural	ADJ
ejpam-6592	211	15	network	network	NOUN
ejpam-6592	211	16	weights	weight	VERB
ejpam-6592	211	17	with	with	ADP
ejpam-6592	211	18	:	:	PUNCT
ejpam-6592	211	19	m(w1,w2,w3	m(w1,w2,w3	NOUN
ejpam-6592	211	20	)	)	PUNCT
ejpam-6592	211	21	=	=	SYM
ejpam-6592	211	22	∥w1	∥w1	NOUN
ejpam-6592	211	23	−w2∥f	−w2∥f	X
ejpam-6592	212	1	+	+	CCONJ
ejpam-6592	212	2	∥w2	∥w2	NOUN
ejpam-6592	212	3	−w3∥f	−w3∥f	ADP
ejpam-6592	212	4	+	+	CCONJ
ejpam-6592	212	5	∥w3	∥w3	X
ejpam-6592	212	6	−w1∥f	−w1∥f	NOUN
ejpam-6592	212	7	3r	3r	NUM
ejpam-6592	212	8	where	where	SCONJ
ejpam-6592	212	9	∥	∥	PUNCT
ejpam-6592	212	10	·	·	PUNCT
ejpam-6592	213	1	∥f	∥f	PROPN
ejpam-6592	213	2	is	be	AUX
ejpam-6592	213	3	the	the	DET
ejpam-6592	213	4	frobenius	frobenius	ADJ
ejpam-6592	213	5	norm	norm	NOUN
ejpam-6592	213	6	.	.	PUNCT
ejpam-6592	214	1	by	by	ADP
ejpam-6592	214	2	theorem	theorem	NOUN
ejpam-6592	214	3	2	2	NUM
ejpam-6592	214	4	:	:	PUNCT
ejpam-6592	214	5	(	(	PUNCT
ejpam-6592	214	6	i	i	NOUN
ejpam-6592	214	7	)	)	PUNCT
ejpam-6592	214	8	h	h	PROPN
ejpam-6592	214	9	is	be	AUX
ejpam-6592	214	10	separable	separable	ADJ
ejpam-6592	214	11	has	have	AUX
ejpam-6592	214	12	countable	countable	VERB
ejpam-6592	214	13	dense	dense	ADJ
ejpam-6592	214	14	subset	subset	NOUN
ejpam-6592	214	15	d	d	NOUN
ejpam-6592	214	16	of	of	ADP
ejpam-6592	214	17	quantized	quantize	VERB
ejpam-6592	214	18	weights	weight	NOUN
ejpam-6592	214	19	(	(	PUNCT
ejpam-6592	214	20	ii	ii	NOUN
ejpam-6592	214	21	)	)	PUNCT
ejpam-6592	214	22	any	any	DET
ejpam-6592	214	23	training	training	NOUN
ejpam-6592	214	24	set	set	NOUN
ejpam-6592	214	25	s	s	VERB
ejpam-6592	214	26	⊂	⊂	PROPN
ejpam-6592	214	27	h	h	NOUN
ejpam-6592	214	28	has	have	AUX
ejpam-6592	214	29	countable	countable	VERB
ejpam-6592	214	30	ϵ-cover	ϵ-cover	NOUN
ejpam-6592	214	31	1	1	NUM
ejpam-6592	214	32	import	import	NOUN
ejpam-6592	214	33	torch	torch	NOUN
ejpam-6592	214	34	2	2	NUM
ejpam-6592	214	35	3	3	NUM
ejpam-6592	214	36	def	def	VERB
ejpam-6592	214	37	build_countable_cover(model	build_countable_cover(model	X
ejpam-6592	214	38	,	,	PUNCT
ejpam-6592	214	39	epsilon	epsilon	PROPN
ejpam-6592	214	40	=	=	SYM
ejpam-6592	214	41	0.01	0.01	NUM
ejpam-6592	214	42	):	):	PUNCT
ejpam-6592	214	43	4	4	NUM
ejpam-6592	214	44	cover	cover	NOUN
ejpam-6592	214	45	=	=	SYM
ejpam-6592	214	46	{	{	PUNCT
ejpam-6592	214	47	}	}	PUNCT
ejpam-6592	214	48	5	5	NUM
ejpam-6592	214	49	for	for	ADP
ejpam-6592	214	50	name	name	NOUN
ejpam-6592	214	51	,	,	PUNCT
ejpam-6592	214	52	param	param	PROPN
ejpam-6592	214	53	in	in	ADP
ejpam-6592	214	54	model.named_parameters	model.named_parameter	NOUN
ejpam-6592	214	55	(	(	PUNCT
ejpam-6592	214	56	):	):	PUNCT
ejpam-6592	214	57	6	6	NUM
ejpam-6592	214	58	quantized	quantize	VERB
ejpam-6592	214	59	=	=	NOUN
ejpam-6592	214	60	torch.round(param	torch.round(param	NOUN
ejpam-6592	214	61	/	/	SYM
ejpam-6592	214	62	epsilon)*epsilon	epsilon)*epsilon	NOUN
ejpam-6592	214	63	7	7	NUM
ejpam-6592	214	64	cover[name	cover[name	NOUN
ejpam-6592	214	65	]	]	X
ejpam-6592	214	66	=	=	VERB
ejpam-6592	214	67	quantized	quantize	VERB
ejpam-6592	214	68	8	8	NUM
ejpam-6592	214	69	return	return	NOUN
ejpam-6592	214	70	cover	cover	VERB
ejpam-6592	214	71	countable	countable	ADJ
ejpam-6592	214	72	ϵ-net	ϵ-net	NOUN
ejpam-6592	214	73	in	in	ADP
ejpam-6592	214	74	weight	weight	NOUN
ejpam-6592	214	75	space	space	NOUN
ejpam-6592	214	76	a.	a.	NOUN
ejpam-6592	214	77	malkawi	malkawi	PROPN
ejpam-6592	214	78	/	/	SYM
ejpam-6592	214	79	eur	eur	PROPN
ejpam-6592	214	80	.	.	PUNCT
ejpam-6592	215	1	j.	j.	PROPN
ejpam-6592	215	2	pure	pure	PROPN
ejpam-6592	215	3	appl	appl	PROPN
ejpam-6592	215	4	.	.	PROPN
ejpam-6592	215	5	math	math	PROPN
ejpam-6592	215	6	,	,	PUNCT
ejpam-6592	215	7	18	18	NUM
ejpam-6592	215	8	(	(	PUNCT
ejpam-6592	215	9	3	3	NUM
ejpam-6592	215	10	)	)	PUNCT
ejpam-6592	215	11	(	(	PUNCT
ejpam-6592	215	12	2025	2025	NUM
ejpam-6592	215	13	)	)	PUNCT
ejpam-6592	215	14	,	,	PUNCT
ejpam-6592	215	15	6592	6592	NUM
ejpam-6592	215	16	11	11	NUM
ejpam-6592	215	17	of	of	ADP
ejpam-6592	215	18	14	14	NUM
ejpam-6592	215	19	3.3	3.3	NUM
ejpam-6592	215	20	.	.	PUNCT
ejpam-6592	216	1	paracompactness	paracompactness	NOUN
ejpam-6592	216	2	in	in	ADP
ejpam-6592	216	3	computational	computational	ADJ
ejpam-6592	216	4	geometry	geometry	NOUN
ejpam-6592	216	5	example	example	NOUN
ejpam-6592	216	6	2	2	NUM
ejpam-6592	216	7	(	(	PUNCT
ejpam-6592	216	8	fractal	fractal	ADJ
ejpam-6592	216	9	surface	surface	NOUN
ejpam-6592	216	10	analysis	analysis	NOUN
ejpam-6592	216	11	)	)	PUNCT
ejpam-6592	216	12	.	.	PUNCT
ejpam-6592	217	1	for	for	ADP
ejpam-6592	217	2	the	the	DET
ejpam-6592	217	3	sierpinski	sierpinski	ADJ
ejpam-6592	217	4	triangle	triangle	NOUN
ejpam-6592	217	5	s	s	PROPN
ejpam-6592	217	6	with	with	ADP
ejpam-6592	217	7	mr	mr	PROPN
ejpam-6592	217	8	-	-	PUNCT
ejpam-6592	217	9	metric	metric	NOUN
ejpam-6592	217	10	induced	induce	VERB
ejpam-6592	217	11	from	from	ADP
ejpam-6592	217	12	r2	r2	PROPN
ejpam-6592	217	13	,	,	PUNCT
ejpam-6592	217	14	theorem	theorem	VERB
ejpam-6592	217	15	3	3	NUM
ejpam-6592	217	16	guarantees	guarantee	NOUN
ejpam-6592	217	17	:	:	PUNCT
ejpam-6592	217	18	•	•	NUM
ejpam-6592	217	19	existence	existence	NOUN
ejpam-6592	217	20	of	of	ADP
ejpam-6592	217	21	locally	locally	ADV
ejpam-6592	217	22	finite	finite	ADJ
ejpam-6592	217	23	refinements	refinement	NOUN
ejpam-6592	217	24	for	for	ADP
ejpam-6592	217	25	any	any	DET
ejpam-6592	217	26	cover	cover	NOUN
ejpam-6592	218	1	•	•	NOUN
ejpam-6592	218	2	construction	construction	NOUN
ejpam-6592	218	3	of	of	ADP
ejpam-6592	218	4	adapted	adapted	ADJ
ejpam-6592	218	5	coordinate	coordinate	NOUN
ejpam-6592	218	6	charts	chart	NOUN
ejpam-6592	218	7	locally	locally	ADV
ejpam-6592	218	8	finite	finite	ADJ
ejpam-6592	218	9	cover	cover	NOUN
ejpam-6592	218	10	on	on	ADP
ejpam-6592	218	11	sierpinski	sierpinski	ADJ
ejpam-6592	218	12	triangle	triangle	NOUN
ejpam-6592	218	13	1	1	NUM
ejpam-6592	218	14	function	function	NOUN
ejpam-6592	218	15	centers	center	NOUN
ejpam-6592	218	16	=	=	SYM
ejpam-6592	218	17	paracompact_refinement(r	paracompact_refinement(r	NOUN
ejpam-6592	218	18	)	)	PUNCT
ejpam-6592	218	19	2	2	NUM
ejpam-6592	218	20	centers	center	NOUN
ejpam-6592	218	21	=	=	PUNCT
ejpam-6592	219	1	[	[	X
ejpam-6592	219	2	]	]	X
ejpam-6592	219	3	;	;	PUNCT
ejpam-6592	219	4	3	3	NUM
ejpam-6592	219	5	for	for	ADP
ejpam-6592	219	6	level	level	NOUN
ejpam-6592	219	7	=	=	NOUN
ejpam-6592	219	8	1:5	1:5	NUM
ejpam-6592	219	9	4	4	NUM
ejpam-6592	220	1	[	[	X
ejpam-6592	220	2	x	x	X
ejpam-6592	220	3	,	,	PUNCT
ejpam-6592	220	4	y	y	NOUN
ejpam-6592	220	5	]	]	X
ejpam-6592	220	6	=	=	PUNCT
ejpam-6592	220	7	sierpinski(level	sierpinski(level	PROPN
ejpam-6592	220	8	)	)	PUNCT
ejpam-6592	220	9	;	;	PUNCT
ejpam-6592	220	10	5	5	NUM
ejpam-6592	220	11	for	for	ADP
ejpam-6592	220	12	i	i	PRON
ejpam-6592	220	13	=	=	NOUN
ejpam-6592	220	14	1	1	NUM
ejpam-6592	220	15	:	:	PUNCT
ejpam-6592	220	16	length(x	length(x	NOUN
ejpam-6592	220	17	)	)	PUNCT
ejpam-6592	220	18	6	6	NUM
ejpam-6592	220	19	if	if	SCONJ
ejpam-6592	220	20	min(pdist2	min(pdist2	ADJ
ejpam-6592	220	21	(	(	PUNCT
ejpam-6592	220	22	[	[	X
ejpam-6592	220	23	x(i),y(i	x(i),y(i	NUM
ejpam-6592	220	24	)	)	PUNCT
ejpam-6592	220	25	]	]	PUNCT
ejpam-6592	220	26	,	,	PUNCT
ejpam-6592	220	27	centers	center	NOUN
ejpam-6592	220	28	)	)	PUNCT
ejpam-6592	220	29	)	)	PUNCT
ejpam-6592	220	30	>	>	PUNCT
ejpam-6592	221	1	r	r	X
ejpam-6592	221	2	/	/	SYM
ejpam-6592	221	3	level	level	NOUN
ejpam-6592	221	4	7	7	NUM
ejpam-6592	221	5	centers	center	NOUN
ejpam-6592	221	6	=	=	PUNCT
ejpam-6592	222	1	[	[	X
ejpam-6592	222	2	centers	center	NOUN
ejpam-6592	222	3	;	;	PUNCT
ejpam-6592	222	4	x(i	x(i	PROPN
ejpam-6592	222	5	)	)	PUNCT
ejpam-6592	222	6	,	,	PUNCT
ejpam-6592	222	7	y(i	y(i	PROPN
ejpam-6592	222	8	)	)	PUNCT
ejpam-6592	222	9	]	]	X
ejpam-6592	222	10	;	;	PUNCT
ejpam-6592	222	11	8	8	NUM
ejpam-6592	222	12	end	end	NOUN
ejpam-6592	222	13	9	9	NUM
ejpam-6592	222	14	end	end	NOUN
ejpam-6592	222	15	10	10	NUM
ejpam-6592	222	16	end	end	NOUN
ejpam-6592	222	17	11	11	NUM
ejpam-6592	222	18	end	end	NOUN
ejpam-6592	222	19	3.4	3.4	NUM
ejpam-6592	222	20	.	.	PUNCT
ejpam-6592	222	21	quantum	quantum	ADJ
ejpam-6592	222	22	state	state	NOUN
ejpam-6592	222	23	spaces	space	NOUN
ejpam-6592	222	24	theorem	theorem	VERB
ejpam-6592	222	25	5	5	NUM
ejpam-6592	222	26	(	(	PUNCT
ejpam-6592	222	27	qubit	qubit	NOUN
ejpam-6592	222	28	configuration	configuration	NOUN
ejpam-6592	222	29	space	space	NOUN
ejpam-6592	222	30	)	)	PUNCT
ejpam-6592	222	31	.	.	PUNCT
ejpam-6592	223	1	the	the	DET
ejpam-6592	223	2	space	space	NOUN
ejpam-6592	223	3	qn	qn	NOUN
ejpam-6592	223	4	of	of	ADP
ejpam-6592	223	5	n	n	CCONJ
ejpam-6592	223	6	-	-	PUNCT
ejpam-6592	223	7	qubit	qubit	NOUN
ejpam-6592	223	8	states	state	NOUN
ejpam-6592	223	9	with	with	ADP
ejpam-6592	223	10	mr	mr	PROPN
ejpam-6592	223	11	-	-	PUNCT
ejpam-6592	223	12	metric	metric	ADJ
ejpam-6592	223	13	:	:	PUNCT
ejpam-6592	223	14	m(ρ	m(ρ	NUM
ejpam-6592	223	15	,	,	PUNCT
ejpam-6592	223	16	σ	σ	PROPN
ejpam-6592	223	17	,	,	PUNCT
ejpam-6592	223	18	τ	τ	X
ejpam-6592	223	19	)	)	PUNCT
ejpam-6592	223	20	=	=	SYM
ejpam-6592	223	21	∥ρ−	∥ρ−	PROPN
ejpam-6592	223	22	σ∥tr	σ∥tr	NOUN
ejpam-6592	223	23	+	+	CCONJ
ejpam-6592	223	24	∥σ	∥σ	PROPN
ejpam-6592	223	25	−	−	PROPN
ejpam-6592	223	26	τ∥tr	τ∥tr	NOUN
ejpam-6592	223	27	+	+	CCONJ
ejpam-6592	223	28	∥τ	∥τ	PROPN
ejpam-6592	223	29	−	−	PROPN
ejpam-6592	223	30	ρ∥tr	ρ∥tr	PROPN
ejpam-6592	223	31	3r	3r	NOUN
ejpam-6592	223	32	where	where	SCONJ
ejpam-6592	223	33	∥	∥	X
ejpam-6592	223	34	·	·	PUNCT
ejpam-6592	223	35	∥tr	∥tr	PROPN
ejpam-6592	223	36	is	be	AUX
ejpam-6592	223	37	the	the	DET
ejpam-6592	223	38	trace	trace	NOUN
ejpam-6592	223	39	norm	norm	NOUN
ejpam-6592	223	40	,	,	PUNCT
ejpam-6592	223	41	satisfies	satisfie	NOUN
ejpam-6592	223	42	:	:	PUNCT
ejpam-6592	223	43	•	•	NUM
ejpam-6592	223	44	compactness	compactness	NOUN
ejpam-6592	223	45	enables	enable	VERB
ejpam-6592	223	46	finite	finite	VERB
ejpam-6592	223	47	ϵ-nets	ϵ-net	NOUN
ejpam-6592	223	48	for	for	ADP
ejpam-6592	223	49	quantum	quantum	NOUN
ejpam-6592	223	50	tomography	tomography	NOUN
ejpam-6592	223	51	a.	a.	NOUN
ejpam-6592	223	52	malkawi	malkawi	ADP
ejpam-6592	223	53	/	/	SYM
ejpam-6592	223	54	eur	eur	PROPN
ejpam-6592	223	55	.	.	PUNCT
ejpam-6592	224	1	j.	j.	PROPN
ejpam-6592	224	2	pure	pure	PROPN
ejpam-6592	224	3	appl	appl	PROPN
ejpam-6592	224	4	.	.	PROPN
ejpam-6592	224	5	math	math	PROPN
ejpam-6592	224	6	,	,	PUNCT
ejpam-6592	224	7	18	18	NUM
ejpam-6592	224	8	(	(	PUNCT
ejpam-6592	224	9	3	3	NUM
ejpam-6592	224	10	)	)	PUNCT
ejpam-6592	224	11	(	(	PUNCT
ejpam-6592	224	12	2025	2025	NUM
ejpam-6592	224	13	)	)	PUNCT
ejpam-6592	224	14	,	,	PUNCT
ejpam-6592	224	15	6592	6592	NUM
ejpam-6592	224	16	12	12	NUM
ejpam-6592	224	17	of	of	ADP
ejpam-6592	224	18	14	14	NUM
ejpam-6592	224	19	•	•	NOUN
ejpam-6592	224	20	paracompactness	paracompactness	NOUN
ejpam-6592	224	21	permits	permit	VERB
ejpam-6592	224	22	locally	locally	ADV
ejpam-6592	224	23	finite	finite	ADJ
ejpam-6592	224	24	povm	povm	NOUN
ejpam-6592	224	25	coverings	covering	NOUN
ejpam-6592	224	26	1	1	NUM
ejpam-6592	224	27	import	import	NOUN
ejpam-6592	224	28	qutip	qutip	NOUN
ejpam-6592	224	29	as	as	ADP
ejpam-6592	224	30	qt	qt	NOUN
ejpam-6592	224	31	2	2	NUM
ejpam-6592	224	32	3	3	NUM
ejpam-6592	224	33	def	def	ADJ
ejpam-6592	224	34	quantum_epsilon_net(n_qubits	quantum_epsilon_net(n_qubit	NOUN
ejpam-6592	224	35	,	,	PUNCT
ejpam-6592	224	36	epsilon	epsilon	PROPN
ejpam-6592	224	37	):	):	PUNCT
ejpam-6592	224	38	4	4	NUM
ejpam-6592	224	39	net	net	NOUN
ejpam-6592	224	40	=	=	PUNCT
ejpam-6592	225	1	[	[	X
ejpam-6592	225	2	]	]	X
ejpam-6592	225	3	5	5	NUM
ejpam-6592	225	4	for	for	ADP
ejpam-6592	225	5	_	_	PROPN
ejpam-6592	225	6	in	in	ADP
ejpam-6592	225	7	range	range	NOUN
ejpam-6592	225	8	(	(	PUNCT
ejpam-6592	225	9	1000	1000	NUM
ejpam-6592	225	10	):	):	PUNCT
ejpam-6592	225	11	6	6	NUM
ejpam-6592	225	12	state	state	NOUN
ejpam-6592	225	13	=	=	SYM
ejpam-6592	225	14	qt.rand_ket	qt.rand_ket	NOUN
ejpam-6592	225	15	(	(	PUNCT
ejpam-6592	225	16	2	2	NUM
ejpam-6592	225	17	*	*	NOUN
ejpam-6592	225	18	*	*	NOUN
ejpam-6592	225	19	n_qubits	n_qubit	NOUN
ejpam-6592	225	20	)	)	PUNCT
ejpam-6592	225	21	7	7	NUM
ejpam-6592	225	22	if	if	SCONJ
ejpam-6592	225	23	all(qt.metrics.tracedist(state	all(qt.metrics.tracedist(state	ADV
ejpam-6592	225	24	,	,	PUNCT
ejpam-6592	225	25	s	s	X
ejpam-6592	225	26	)	)	PUNCT
ejpam-6592	225	27	>	>	X
ejpam-6592	225	28	epsilon	epsilon	PROPN
ejpam-6592	225	29	for	for	ADP
ejpam-6592	225	30	s	s	PROPN
ejpam-6592	225	31	in	in	ADP
ejpam-6592	225	32	net	net	NOUN
ejpam-6592	225	33	):	):	PUNCT
ejpam-6592	225	34	8	8	NUM
ejpam-6592	225	35	net.append(state	net.append(state	NOUN
ejpam-6592	225	36	)	)	PUNCT
ejpam-6592	225	37	9	9	NUM
ejpam-6592	225	38	return	return	NOUN
ejpam-6592	225	39	net	net	ADJ
ejpam-6592	225	40	finite	finite	NOUN
ejpam-6592	225	41	ϵ-net	ϵ-net	PROPN
ejpam-6592	225	42	in	in	ADP
ejpam-6592	225	43	qubit	qubit	NOUN
ejpam-6592	225	44	space	space	NOUN
ejpam-6592	225	45	references	reference	NOUN
ejpam-6592	225	46	[	[	X
ejpam-6592	225	47	1	1	NUM
ejpam-6592	225	48	]	]	PUNCT
ejpam-6592	225	49	a.	a.	NOUN
ejpam-6592	225	50	malkawi	malkawi	PROPN
ejpam-6592	225	51	,	,	PUNCT
ejpam-6592	225	52	a.	a.	NOUN
ejpam-6592	225	53	tallafha	tallafha	NOUN
ejpam-6592	225	54	,	,	PUNCT
ejpam-6592	225	55	and	and	CCONJ
ejpam-6592	225	56	w.	w.	PROPN
ejpam-6592	225	57	shatanawi	shatanawi	PROPN
ejpam-6592	225	58	.	.	PUNCT
ejpam-6592	226	1	coincidence	coincidence	NOUN
ejpam-6592	226	2	and	and	CCONJ
ejpam-6592	226	3	fixed	fix	VERB
ejpam-6592	226	4	point	point	NOUN
ejpam-6592	226	5	results	result	NOUN
ejpam-6592	226	6	for	for	ADP
ejpam-6592	226	7	generalized	generalized	ADJ
ejpam-6592	226	8	weak	weak	ADJ
ejpam-6592	226	9	contraction	contraction	NOUN
ejpam-6592	226	10	mapping	mapping	NOUN
ejpam-6592	226	11	on	on	ADP
ejpam-6592	226	12	b	b	NOUN
ejpam-6592	226	13	-	-	PUNCT
ejpam-6592	226	14	metric	metric	ADJ
ejpam-6592	226	15	spaces	space	NOUN
ejpam-6592	226	16	.	.	PUNCT
ejpam-6592	227	1	nonlinear	nonlinear	ADJ
ejpam-6592	227	2	functional	functional	ADJ
ejpam-6592	227	3	analysis	analysis	NOUN
ejpam-6592	227	4	and	and	CCONJ
ejpam-6592	227	5	applications	application	NOUN
ejpam-6592	227	6	,	,	PUNCT
ejpam-6592	227	7	26(1):177–195	26(1):177–195	NOUN
ejpam-6592	227	8	,	,	PUNCT
ejpam-6592	227	9	2021	2021	NUM
ejpam-6592	227	10	.	.	PUNCT
ejpam-6592	228	1	[	[	X
ejpam-6592	228	2	2	2	X
ejpam-6592	228	3	]	]	PUNCT
ejpam-6592	228	4	t.	t.	NOUN
ejpam-6592	228	5	qawasmeh	qawasmeh	NOUN
ejpam-6592	228	6	.	.	PUNCT
ejpam-6592	229	1	(	(	PUNCT
ejpam-6592	229	2	h	h	NOUN
ejpam-6592	229	3	,	,	PUNCT
ejpam-6592	229	4	ωb)-interpolative	ωb)-interpolative	ADJ
ejpam-6592	229	5	contractions	contraction	NOUN
ejpam-6592	229	6	in	in	ADP
ejpam-6592	229	7	ωb	ωb	NOUN
ejpam-6592	229	8	-	-	PUNCT
ejpam-6592	229	9	distance	distance	NOUN
ejpam-6592	229	10	mappings	mapping	NOUN
ejpam-6592	229	11	with	with	ADP
ejpam-6592	229	12	applications	application	NOUN
ejpam-6592	229	13	.	.	PUNCT
ejpam-6592	230	1	european	european	ADJ
ejpam-6592	230	2	journal	journal	PROPN
ejpam-6592	230	3	of	of	ADP
ejpam-6592	230	4	pure	pure	ADJ
ejpam-6592	230	5	and	and	CCONJ
ejpam-6592	230	6	applied	applied	ADJ
ejpam-6592	230	7	mathematics	mathematic	NOUN
ejpam-6592	230	8	,	,	PUNCT
ejpam-6592	230	9	16(3):1717–1730	16(3):1717–1730	NUM
ejpam-6592	230	10	,	,	PUNCT
ejpam-6592	230	11	2023	2023	NUM
ejpam-6592	230	12	.	.	PUNCT
ejpam-6592	231	1	[	[	X
ejpam-6592	231	2	3	3	X
ejpam-6592	231	3	]	]	PUNCT
ejpam-6592	231	4	t.	t.	NOUN
ejpam-6592	231	5	qawasmeh	qawasmeh	NOUN
ejpam-6592	231	6	.	.	PUNCT
ejpam-6592	232	1	h	h	NOUN
ejpam-6592	232	2	-	-	PUNCT
ejpam-6592	232	3	simulation	simulation	NOUN
ejpam-6592	232	4	functions	function	NOUN
ejpam-6592	232	5	and	and	CCONJ
ejpam-6592	232	6	ωb	ωb	NOUN
ejpam-6592	232	7	-	-	PUNCT
ejpam-6592	232	8	distance	distance	NOUN
ejpam-6592	232	9	mappings	mapping	NOUN
ejpam-6592	232	10	in	in	ADP
ejpam-6592	232	11	the	the	DET
ejpam-6592	232	12	setting	setting	NOUN
ejpam-6592	232	13	of	of	ADP
ejpam-6592	232	14	gb	gb	ADV
ejpam-6592	232	15	-	-	PUNCT
ejpam-6592	232	16	metric	metric	ADJ
ejpam-6592	232	17	spaces	space	NOUN
ejpam-6592	232	18	and	and	CCONJ
ejpam-6592	232	19	application	application	NOUN
ejpam-6592	232	20	.	.	PUNCT
ejpam-6592	233	1	nonlinear	nonlinear	ADJ
ejpam-6592	233	2	functional	functional	ADJ
ejpam-6592	233	3	analysis	analysis	NOUN
ejpam-6592	233	4	and	and	CCONJ
ejpam-6592	233	5	applications	application	NOUN
ejpam-6592	233	6	,	,	PUNCT
ejpam-6592	233	7	28(2):557–570	28(2):557–570	NOUN
ejpam-6592	233	8	,	,	PUNCT
ejpam-6592	233	9	2023	2023	NUM
ejpam-6592	233	10	.	.	PUNCT
ejpam-6592	234	1	[	[	X
ejpam-6592	234	2	4	4	NUM
ejpam-6592	234	3	]	]	X
ejpam-6592	234	4	a.	a.	NOUN
ejpam-6592	234	5	malkawi	malkawi	PROPN
ejpam-6592	234	6	,	,	PUNCT
ejpam-6592	234	7	a.	a.	PROPN
ejpam-6592	234	8	rabaiah	rabaiah	PROPN
ejpam-6592	234	9	,	,	PUNCT
ejpam-6592	234	10	w.	w.	PROPN
ejpam-6592	234	11	shatanawi	shatanawi	PROPN
ejpam-6592	234	12	,	,	PUNCT
ejpam-6592	234	13	and	and	CCONJ
ejpam-6592	234	14	a.	a.	NOUN
ejpam-6592	234	15	talafhah	talafhah	PROPN
ejpam-6592	234	16	.	.	PUNCT
ejpam-6592	235	1	mr	mr	PROPN
ejpam-6592	235	2	-	-	PUNCT
ejpam-6592	235	3	metric	metric	ADJ
ejpam-6592	235	4	spaces	space	NOUN
ejpam-6592	235	5	and	and	CCONJ
ejpam-6592	235	6	an	an	DET
ejpam-6592	235	7	application	application	NOUN
ejpam-6592	235	8	.	.	PUNCT
ejpam-6592	236	1	preprint	preprint	NOUN
ejpam-6592	236	2	,	,	PUNCT
ejpam-6592	236	3	2021	2021	NUM
ejpam-6592	236	4	.	.	PUNCT
ejpam-6592	237	1	[	[	X
ejpam-6592	237	2	5	5	NUM
ejpam-6592	237	3	]	]	PUNCT
ejpam-6592	237	4	a.	a.	NOUN
ejpam-6592	237	5	a.	a.	PROPN
ejpam-6592	237	6	r.	r.	PROPN
ejpam-6592	237	7	m.	m.	PROPN
ejpam-6592	237	8	malkawi	malkawi	PROPN
ejpam-6592	237	9	,	,	PUNCT
ejpam-6592	237	10	d.	d.	PROPN
ejpam-6592	237	11	mahmoud	mahmoud	PROPN
ejpam-6592	237	12	,	,	PUNCT
ejpam-6592	237	13	a.	a.	PROPN
ejpam-6592	237	14	m.	m.	PROPN
ejpam-6592	237	15	rabaiah	rabaiah	PROPN
ejpam-6592	237	16	,	,	PUNCT
ejpam-6592	237	17	r.	r.	PROPN
ejpam-6592	237	18	al	al	PROPN
ejpam-6592	237	19	-	-	PUNCT
ejpam-6592	237	20	deiakeh	deiakeh	PROPN
ejpam-6592	237	21	,	,	PUNCT
ejpam-6592	237	22	and	and	CCONJ
ejpam-6592	237	23	w.	w.	PROPN
ejpam-6592	237	24	shatanawi	shatanawi	PROPN
ejpam-6592	237	25	.	.	PUNCT
ejpam-6592	238	1	on	on	ADP
ejpam-6592	238	2	fixed	fix	VERB
ejpam-6592	238	3	point	point	NOUN
ejpam-6592	238	4	theorems	theorem	NOUN
ejpam-6592	238	5	in	in	ADP
ejpam-6592	238	6	mr	mr	PROPN
ejpam-6592	238	7	-	-	PUNCT
ejpam-6592	238	8	metric	metric	ADJ
ejpam-6592	238	9	spaces	space	NOUN
ejpam-6592	238	10	.	.	PUNCT
ejpam-6592	239	1	nonlinear	nonlinear	ADJ
ejpam-6592	239	2	functional	functional	ADJ
ejpam-6592	239	3	analysis	analysis	NOUN
ejpam-6592	239	4	and	and	CCONJ
ejpam-6592	239	5	applications	application	NOUN
ejpam-6592	239	6	,	,	PUNCT
ejpam-6592	239	7	29(4):1125–1136	29(4):1125–1136	NUM
ejpam-6592	239	8	,	,	PUNCT
ejpam-6592	239	9	2024	2024	NUM
ejpam-6592	239	10	.	.	PUNCT
ejpam-6592	240	1	[	[	X
ejpam-6592	240	2	6	6	NUM
ejpam-6592	240	3	]	]	PUNCT
ejpam-6592	240	4	a.	a.	NOUN
ejpam-6592	240	5	malkawi	malkawi	PROPN
ejpam-6592	240	6	,	,	PUNCT
ejpam-6592	240	7	a.	a.	NOUN
ejpam-6592	240	8	talafhah	talafhah	PROPN
ejpam-6592	240	9	,	,	PUNCT
ejpam-6592	240	10	and	and	CCONJ
ejpam-6592	240	11	w.	w.	PROPN
ejpam-6592	240	12	shatanawi	shatanawi	PROPN
ejpam-6592	240	13	.	.	PUNCT
ejpam-6592	241	1	coincidence	coincidence	NOUN
ejpam-6592	241	2	and	and	CCONJ
ejpam-6592	241	3	fixed	fix	VERB
ejpam-6592	241	4	point	point	NOUN
ejpam-6592	241	5	results	result	NOUN
ejpam-6592	241	6	for	for	ADP
ejpam-6592	241	7	(	(	PUNCT
ejpam-6592	241	8	ψ	ψ	NOUN
ejpam-6592	241	9	,	,	PUNCT
ejpam-6592	241	10	l)-m	l)-m	ADJ
ejpam-6592	241	11	-	-	PUNCT
ejpam-6592	241	12	weak	weak	ADJ
ejpam-6592	241	13	contraction	contraction	NOUN
ejpam-6592	241	14	mapping	mapping	NOUN
ejpam-6592	241	15	on	on	ADP
ejpam-6592	241	16	mb	mb	ADJ
ejpam-6592	241	17	-	-	ADJ
ejpam-6592	241	18	metric	metric	ADJ
ejpam-6592	241	19	spaces	space	NOUN
ejpam-6592	241	20	.	.	PUNCT
ejpam-6592	242	1	italian	italian	ADJ
ejpam-6592	242	2	journal	journal	NOUN
ejpam-6592	242	3	of	of	ADP
ejpam-6592	242	4	pure	pure	ADJ
ejpam-6592	242	5	and	and	CCONJ
ejpam-6592	242	6	applied	applied	ADJ
ejpam-6592	242	7	mathematics	mathematic	NOUN
ejpam-6592	242	8	,	,	PUNCT
ejpam-6592	242	9	(	(	PUNCT
ejpam-6592	242	10	47):751–768	47):751–768	NOUN
ejpam-6592	242	11	,	,	PUNCT
ejpam-6592	242	12	2022	2022	NUM
ejpam-6592	242	13	.	.	PUNCT
ejpam-6592	243	1	[	[	X
ejpam-6592	243	2	7	7	X
ejpam-6592	243	3	]	]	PUNCT
ejpam-6592	243	4	a.	a.	NOUN
ejpam-6592	243	5	a.	a.	PROPN
ejpam-6592	243	6	r.	r.	PROPN
ejpam-6592	243	7	m.	m.	PROPN
ejpam-6592	243	8	malkawi	malkawi	PROPN
ejpam-6592	243	9	.	.	PROPN
ejpam-6592	243	10	existence	existence	NOUN
ejpam-6592	243	11	and	and	CCONJ
ejpam-6592	243	12	uniqueness	uniqueness	NOUN
ejpam-6592	243	13	of	of	ADP
ejpam-6592	243	14	fixed	fix	VERB
ejpam-6592	243	15	points	point	NOUN
ejpam-6592	243	16	in	in	ADP
ejpam-6592	243	17	mr	mr	PROPN
ejpam-6592	243	18	-	-	PUNCT
ejpam-6592	243	19	metric	metric	ADJ
ejpam-6592	243	20	spaces	space	NOUN
ejpam-6592	243	21	and	and	CCONJ
ejpam-6592	243	22	their	their	PRON
ejpam-6592	243	23	applications	application	NOUN
ejpam-6592	243	24	.	.	PUNCT
ejpam-6592	244	1	european	european	ADJ
ejpam-6592	244	2	journal	journal	PROPN
ejpam-6592	244	3	of	of	ADP
ejpam-6592	244	4	pure	pure	ADJ
ejpam-6592	244	5	and	and	CCONJ
ejpam-6592	244	6	applied	applied	ADJ
ejpam-6592	244	7	mathematics	mathematic	NOUN
ejpam-6592	244	8	,	,	PUNCT
ejpam-6592	244	9	18(2):6077	18(2):6077	NUM
ejpam-6592	244	10	,	,	PUNCT
ejpam-6592	244	11	2025	2025	NUM
ejpam-6592	244	12	.	.	PUNCT
ejpam-6592	245	1	a.	a.	NOUN
ejpam-6592	245	2	malkawi	malkawi	ADP
ejpam-6592	245	3	/	/	SYM
ejpam-6592	245	4	eur	eur	PROPN
ejpam-6592	245	5	.	.	PUNCT
ejpam-6592	246	1	j.	j.	PROPN
ejpam-6592	246	2	pure	pure	PROPN
ejpam-6592	246	3	appl	appl	PROPN
ejpam-6592	246	4	.	.	PROPN
ejpam-6592	246	5	math	math	PROPN
ejpam-6592	246	6	,	,	PUNCT
ejpam-6592	246	7	18	18	NUM
ejpam-6592	246	8	(	(	PUNCT
ejpam-6592	246	9	3	3	NUM
ejpam-6592	246	10	)	)	PUNCT
ejpam-6592	246	11	(	(	PUNCT
ejpam-6592	246	12	2025	2025	NUM
ejpam-6592	246	13	)	)	PUNCT
ejpam-6592	246	14	,	,	PUNCT
ejpam-6592	246	15	6592	6592	NUM
ejpam-6592	246	16	13	13	NUM
ejpam-6592	246	17	of	of	ADP
ejpam-6592	246	18	14	14	NUM
ejpam-6592	247	1	[	[	SYM
ejpam-6592	247	2	8	8	NUM
ejpam-6592	247	3	]	]	PUNCT
ejpam-6592	247	4	a.	a.	NOUN
ejpam-6592	247	5	a.	a.	PROPN
ejpam-6592	247	6	r.	r.	PROPN
ejpam-6592	247	7	m.	m.	PROPN
ejpam-6592	247	8	malkawi	malkawi	PROPN
ejpam-6592	247	9	.	.	PROPN
ejpam-6592	248	1	convergence	convergence	NOUN
ejpam-6592	248	2	and	and	CCONJ
ejpam-6592	248	3	fixed	fix	VERB
ejpam-6592	248	4	points	point	NOUN
ejpam-6592	248	5	of	of	ADP
ejpam-6592	248	6	self	self	NOUN
ejpam-6592	248	7	-	-	PUNCT
ejpam-6592	248	8	mappings	mapping	NOUN
ejpam-6592	248	9	in	in	ADP
ejpam-6592	248	10	mr	mr	PROPN
ejpam-6592	248	11	-	-	PUNCT
ejpam-6592	248	12	metric	metric	ADJ
ejpam-6592	248	13	spaces	space	NOUN
ejpam-6592	248	14	:	:	PUNCT
ejpam-6592	248	15	theory	theory	NOUN
ejpam-6592	248	16	and	and	CCONJ
ejpam-6592	248	17	applications	application	NOUN
ejpam-6592	248	18	.	.	PUNCT
ejpam-6592	249	1	european	european	ADJ
ejpam-6592	249	2	journal	journal	PROPN
ejpam-6592	249	3	of	of	ADP
ejpam-6592	249	4	pure	pure	ADJ
ejpam-6592	249	5	and	and	CCONJ
ejpam-6592	249	6	applied	applied	ADJ
ejpam-6592	249	7	mathematics	mathematic	NOUN
ejpam-6592	249	8	,	,	PUNCT
ejpam-6592	249	9	18(2):5952	18(2):5952	NUM
ejpam-6592	249	10	,	,	PUNCT
ejpam-6592	249	11	2025	2025	NUM
ejpam-6592	249	12	.	.	PUNCT
ejpam-6592	250	1	[	[	X
ejpam-6592	250	2	9	9	NUM
ejpam-6592	250	3	]	]	PUNCT
ejpam-6592	250	4	a.	a.	NOUN
ejpam-6592	250	5	a.	a.	PROPN
ejpam-6592	250	6	r.	r.	PROPN
ejpam-6592	250	7	m.	m.	PROPN
ejpam-6592	250	8	malkawi	malkawi	PROPN
ejpam-6592	250	9	.	.	PUNCT
ejpam-6592	250	10	fixed	fix	VERB
ejpam-6592	250	11	point	point	NOUN
ejpam-6592	250	12	theorem	theorem	VERB
ejpam-6592	250	13	in	in	ADP
ejpam-6592	250	14	mr	mr	PROPN
ejpam-6592	250	15	-	-	PUNCT
ejpam-6592	250	16	metric	metric	ADJ
ejpam-6592	250	17	spaces	space	NOUN
ejpam-6592	250	18	via	via	ADP
ejpam-6592	250	19	integral	integral	ADJ
ejpam-6592	250	20	type	type	NOUN
ejpam-6592	250	21	contraction	contraction	NOUN
ejpam-6592	250	22	.	.	PUNCT
ejpam-6592	251	1	wseas	wseas	VERB
ejpam-6592	251	2	transactions	transaction	NOUN
ejpam-6592	251	3	on	on	ADP
ejpam-6592	251	4	mathematics	mathematic	NOUN
ejpam-6592	251	5	,	,	PUNCT
ejpam-6592	251	6	24:295–299	24:295–299	PROPN
ejpam-6592	251	7	,	,	PUNCT
ejpam-6592	251	8	2025	2025	NUM
ejpam-6592	251	9	.	.	PUNCT
ejpam-6592	252	1	[	[	X
ejpam-6592	252	2	10	10	NUM
ejpam-6592	252	3	]	]	X
ejpam-6592	252	4	w.	w.	PROPN
ejpam-6592	252	5	shatanawi	shatanawi	PROPN
ejpam-6592	252	6	,	,	PUNCT
ejpam-6592	252	7	t.	t.	NOUN
ejpam-6592	252	8	qawasmeh	qawasmeh	NOUN
ejpam-6592	252	9	,	,	PUNCT
ejpam-6592	252	10	a.	a.	NOUN
ejpam-6592	252	11	bataihah	bataihah	PROPN
ejpam-6592	252	12	,	,	PUNCT
ejpam-6592	252	13	and	and	CCONJ
ejpam-6592	252	14	a.	a.	NOUN
ejpam-6592	252	15	tallafha	tallafha	NOUN
ejpam-6592	252	16	.	.	PUNCT
ejpam-6592	253	1	new	new	ADJ
ejpam-6592	253	2	contractions	contraction	NOUN
ejpam-6592	253	3	and	and	CCONJ
ejpam-6592	253	4	some	some	DET
ejpam-6592	253	5	fixed	fix	VERB
ejpam-6592	253	6	point	point	NOUN
ejpam-6592	253	7	results	result	NOUN
ejpam-6592	253	8	with	with	ADP
ejpam-6592	253	9	application	application	NOUN
ejpam-6592	253	10	based	base	VERB
ejpam-6592	253	11	on	on	ADP
ejpam-6592	253	12	extended	extended	ADJ
ejpam-6592	253	13	quasi	quasi	ADJ
ejpam-6592	253	14	b	b	NOUN
ejpam-6592	253	15	-	-	ADJ
ejpam-6592	253	16	metric	metric	ADJ
ejpam-6592	253	17	spaces	space	NOUN
ejpam-6592	253	18	.	.	PUNCT
ejpam-6592	254	1	u.p.b	u.p.b	ADJ
ejpam-6592	254	2	.	.	PUNCT
ejpam-6592	255	1	scientific	scientific	ADJ
ejpam-6592	255	2	bulletin	bulletin	NOUN
ejpam-6592	255	3	,	,	PUNCT
ejpam-6592	255	4	series	series	PROPN
ejpam-6592	255	5	a	a	PROPN
ejpam-6592	255	6	,	,	PUNCT
ejpam-6592	255	7	83(2):1223–7027	83(2):1223–7027	NUM
ejpam-6592	255	8	,	,	PUNCT
ejpam-6592	255	9	2021	2021	NUM
ejpam-6592	255	10	.	.	PUNCT
ejpam-6592	256	1	[	[	X
ejpam-6592	256	2	11	11	NUM
ejpam-6592	256	3	]	]	PUNCT
ejpam-6592	256	4	t.	t.	NOUN
ejpam-6592	256	5	qawasmeh	qawasmeh	NOUN
ejpam-6592	256	6	,	,	PUNCT
ejpam-6592	256	7	w.	w.	PROPN
ejpam-6592	256	8	shatanawi	shatanawi	PROPN
ejpam-6592	256	9	,	,	PUNCT
ejpam-6592	256	10	a.	a.	NOUN
ejpam-6592	256	11	bataihah	bataihah	PROPN
ejpam-6592	256	12	,	,	PUNCT
ejpam-6592	256	13	and	and	CCONJ
ejpam-6592	256	14	a.	a.	NOUN
ejpam-6592	256	15	tallafha	tallafha	NOUN
ejpam-6592	256	16	.	.	PUNCT
ejpam-6592	257	1	fixed	fix	VERB
ejpam-6592	257	2	point	point	NOUN
ejpam-6592	257	3	results	result	NOUN
ejpam-6592	257	4	and	and	CCONJ
ejpam-6592	257	5	(	(	PUNCT
ejpam-6592	257	6	α	α	NOUN
ejpam-6592	257	7	,	,	PUNCT
ejpam-6592	257	8	β)-triangular	β)-triangular	ADJ
ejpam-6592	257	9	admissibility	admissibility	NOUN
ejpam-6592	257	10	in	in	ADP
ejpam-6592	257	11	the	the	DET
ejpam-6592	257	12	frame	frame	NOUN
ejpam-6592	257	13	of	of	ADP
ejpam-6592	257	14	complete	complete	ADJ
ejpam-6592	257	15	extended	extended	ADJ
ejpam-6592	257	16	b	b	NOUN
ejpam-6592	257	17	-	-	PUNCT
ejpam-6592	257	18	metric	metric	ADJ
ejpam-6592	257	19	spaces	space	NOUN
ejpam-6592	257	20	and	and	CCONJ
ejpam-6592	257	21	application	application	NOUN
ejpam-6592	257	22	.	.	PUNCT
ejpam-6592	258	1	u.p.b	u.p.b	PROPN
ejpam-6592	258	2	.	.	PUNCT
ejpam-6592	259	1	scientific	scientific	ADJ
ejpam-6592	259	2	bulletin	bulletin	NOUN
ejpam-6592	259	3	,	,	PUNCT
ejpam-6592	259	4	series	series	PROPN
ejpam-6592	259	5	a	a	PROPN
ejpam-6592	259	6	,	,	PUNCT
ejpam-6592	259	7	83(1):113–124	83(1):113–124	PROPN
ejpam-6592	259	8	,	,	PUNCT
ejpam-6592	259	9	2021	2021	NUM
ejpam-6592	259	10	.	.	PUNCT
ejpam-6592	260	1	[	[	X
ejpam-6592	260	2	12	12	NUM
ejpam-6592	260	3	]	]	X
ejpam-6592	260	4	g.	g.	PROPN
ejpam-6592	260	5	gharib	gharib	PROPN
ejpam-6592	260	6	,	,	PUNCT
ejpam-6592	260	7	a.	a.	PROPN
ejpam-6592	260	8	malkawi	malkawi	PROPN
ejpam-6592	260	9	,	,	PUNCT
ejpam-6592	260	10	a.	a.	PROPN
ejpam-6592	260	11	rabaiah	rabaiah	PROPN
ejpam-6592	260	12	,	,	PUNCT
ejpam-6592	260	13	w.	w.	PROPN
ejpam-6592	260	14	shatanawi	shatanawi	PROPN
ejpam-6592	260	15	,	,	PUNCT
ejpam-6592	260	16	and	and	CCONJ
ejpam-6592	260	17	m.	m.	NOUN
ejpam-6592	260	18	alsauodi	alsauodi	PROPN
ejpam-6592	260	19	.	.	PUNCT
ejpam-6592	261	1	a	a	DET
ejpam-6592	261	2	common	common	ADJ
ejpam-6592	261	3	fixed	fix	VERB
ejpam-6592	261	4	point	point	NOUN
ejpam-6592	261	5	theorem	theorem	VERB
ejpam-6592	261	6	in	in	ADP
ejpam-6592	261	7	m*-metric	m*-metric	ADV
ejpam-6592	261	8	space	space	NOUN
ejpam-6592	261	9	and	and	CCONJ
ejpam-6592	261	10	an	an	DET
ejpam-6592	261	11	application	application	NOUN
ejpam-6592	261	12	.	.	PUNCT
ejpam-6592	262	1	nonlinear	nonlinear	ADJ
ejpam-6592	262	2	functional	functional	ADJ
ejpam-6592	262	3	analysis	analysis	NOUN
ejpam-6592	262	4	and	and	CCONJ
ejpam-6592	262	5	applications	application	NOUN
ejpam-6592	262	6	,	,	PUNCT
ejpam-6592	262	7	27(2):289–308	27(2):289–308	NUM
ejpam-6592	262	8	,	,	PUNCT
ejpam-6592	262	9	2022	2022	NUM
ejpam-6592	262	10	.	.	PUNCT
ejpam-6592	263	1	[	[	X
ejpam-6592	263	2	13	13	NUM
ejpam-6592	263	3	]	]	PUNCT
ejpam-6592	263	4	a.	a.	NOUN
ejpam-6592	263	5	bataihah	bataihah	PROPN
ejpam-6592	263	6	,	,	PUNCT
ejpam-6592	263	7	a.	a.	NOUN
ejpam-6592	263	8	tallafha	tallafha	NOUN
ejpam-6592	263	9	,	,	PUNCT
ejpam-6592	263	10	and	and	CCONJ
ejpam-6592	263	11	w.	w.	PROPN
ejpam-6592	263	12	shatanawi	shatanawi	PROPN
ejpam-6592	263	13	.	.	PUNCT
ejpam-6592	264	1	fixed	fix	VERB
ejpam-6592	264	2	point	point	NOUN
ejpam-6592	264	3	results	result	NOUN
ejpam-6592	264	4	with	with	ADP
ejpam-6592	264	5	ω	ω	NOUN
ejpam-6592	264	6	-	-	PUNCT
ejpam-6592	264	7	distance	distance	NOUN
ejpam-6592	264	8	by	by	ADP
ejpam-6592	264	9	utilizing	utilize	VERB
ejpam-6592	264	10	simulation	simulation	NOUN
ejpam-6592	264	11	functions	function	NOUN
ejpam-6592	264	12	.	.	PUNCT
ejpam-6592	265	1	italian	italian	ADJ
ejpam-6592	265	2	journal	journal	NOUN
ejpam-6592	265	3	of	of	ADP
ejpam-6592	265	4	pure	pure	ADJ
ejpam-6592	265	5	and	and	CCONJ
ejpam-6592	265	6	applied	applied	ADJ
ejpam-6592	265	7	mathematics	mathematic	NOUN
ejpam-6592	265	8	,	,	PUNCT
ejpam-6592	265	9	(	(	PUNCT
ejpam-6592	265	10	43):185–196	43):185–196	NOUN
ejpam-6592	265	11	,	,	PUNCT
ejpam-6592	265	12	2017	2017	NUM
ejpam-6592	265	13	.	.	PUNCT
ejpam-6592	266	1	[	[	X
ejpam-6592	266	2	14	14	NUM
ejpam-6592	266	3	]	]	PUNCT
ejpam-6592	266	4	k.	k.	PROPN
ejpam-6592	266	5	abodayeh	abodayeh	PROPN
ejpam-6592	266	6	,	,	PUNCT
ejpam-6592	266	7	w.	w.	PROPN
ejpam-6592	266	8	shatanawi	shatanawi	PROPN
ejpam-6592	266	9	,	,	PUNCT
ejpam-6592	266	10	a.	a.	NOUN
ejpam-6592	266	11	bataihah	bataihah	PROPN
ejpam-6592	266	12	,	,	PUNCT
ejpam-6592	266	13	and	and	CCONJ
ejpam-6592	266	14	a.	a.	PROPN
ejpam-6592	266	15	h.	h.	PROPN
ejpam-6592	266	16	ansari	ansari	PROPN
ejpam-6592	266	17	.	.	PUNCT
ejpam-6592	267	1	some	some	DET
ejpam-6592	267	2	fixed	fix	VERB
ejpam-6592	267	3	point	point	NOUN
ejpam-6592	267	4	and	and	CCONJ
ejpam-6592	267	5	common	common	ADJ
ejpam-6592	267	6	fixed	fix	VERB
ejpam-6592	267	7	point	point	NOUN
ejpam-6592	267	8	results	result	NOUN
ejpam-6592	267	9	through	through	ADP
ejpam-6592	267	10	ω	ω	NOUN
ejpam-6592	267	11	-	-	PUNCT
ejpam-6592	267	12	distance	distance	NOUN
ejpam-6592	267	13	under	under	ADP
ejpam-6592	267	14	nonlinear	nonlinear	ADJ
ejpam-6592	267	15	contractions	contraction	NOUN
ejpam-6592	267	16	.	.	PUNCT
ejpam-6592	268	1	gazi	gazi	PROPN
ejpam-6592	268	2	university	university	PROPN
ejpam-6592	268	3	journal	journal	PROPN
ejpam-6592	268	4	of	of	ADP
ejpam-6592	268	5	science	science	NOUN
ejpam-6592	268	6	,	,	PUNCT
ejpam-6592	268	7	30(1):293–302	30(1):293–302	NOUN
ejpam-6592	268	8	,	,	PUNCT
ejpam-6592	268	9	2017	2017	NUM
ejpam-6592	268	10	.	.	PUNCT
ejpam-6592	269	1	[	[	X
ejpam-6592	269	2	15	15	X
ejpam-6592	269	3	]	]	PUNCT
ejpam-6592	269	4	t.	t.	NOUN
ejpam-6592	269	5	qawasmeh	qawasmeh	NOUN
ejpam-6592	269	6	,	,	PUNCT
ejpam-6592	269	7	w.	w.	PROPN
ejpam-6592	269	8	shatanawi	shatanawi	PROPN
ejpam-6592	269	9	,	,	PUNCT
ejpam-6592	269	10	and	and	CCONJ
ejpam-6592	269	11	a.	a.	NOUN
ejpam-6592	269	12	bataihah	bataihah	PROPN
ejpam-6592	269	13	.	.	PUNCT
ejpam-6592	270	1	common	common	ADJ
ejpam-6592	270	2	fixed	fix	VERB
ejpam-6592	270	3	point	point	NOUN
ejpam-6592	270	4	results	result	NOUN
ejpam-6592	270	5	for	for	ADP
ejpam-6592	270	6	rational	rational	ADJ
ejpam-6592	270	7	(	(	PUNCT
ejpam-6592	270	8	α	α	NOUN
ejpam-6592	270	9	,	,	PUNCT
ejpam-6592	270	10	β)ϕ-mω	β)ϕ-mω	NOUN
ejpam-6592	270	11	contractions	contraction	NOUN
ejpam-6592	270	12	in	in	ADP
ejpam-6592	270	13	complete	complete	ADJ
ejpam-6592	270	14	quasi	quasi	ADJ
ejpam-6592	270	15	metric	metric	ADJ
ejpam-6592	270	16	spaces	space	NOUN
ejpam-6592	270	17	.	.	PUNCT
ejpam-6592	271	1	mathematics	mathematic	NOUN
ejpam-6592	271	2	,	,	PUNCT
ejpam-6592	271	3	7(5):392	7(5):392	NUM
ejpam-6592	271	4	,	,	PUNCT
ejpam-6592	271	5	2017	2017	NUM
ejpam-6592	271	6	.	.	PUNCT
ejpam-6592	272	1	[	[	X
ejpam-6592	272	2	16	16	NUM
ejpam-6592	272	3	]	]	PUNCT
ejpam-6592	272	4	r.	r.	PROPN
ejpam-6592	272	5	al	al	PROPN
ejpam-6592	272	6	-	-	PUNCT
ejpam-6592	272	7	deiakeh	deiakeh	ADJ
ejpam-6592	272	8	,	,	PUNCT
ejpam-6592	272	9	m.	m.	NOUN
ejpam-6592	272	10	alquran	alquran	PROPN
ejpam-6592	272	11	,	,	PUNCT
ejpam-6592	272	12	m.	m.	PROPN
ejpam-6592	272	13	ali	ali	PROPN
ejpam-6592	272	14	,	,	PUNCT
ejpam-6592	272	15	s.	s.	PROPN
ejpam-6592	272	16	qureshi	qureshi	PROPN
ejpam-6592	272	17	,	,	PUNCT
ejpam-6592	272	18	s.	s.	PROPN
ejpam-6592	272	19	momani	momani	PROPN
ejpam-6592	272	20	,	,	PUNCT
ejpam-6592	272	21	and	and	CCONJ
ejpam-6592	272	22	a.	a.	NOUN
ejpam-6592	272	23	a.	a.	PROPN
ejpam-6592	272	24	r.	r.	PROPN
ejpam-6592	272	25	malkawi	malkawi	PROPN
ejpam-6592	272	26	.	.	PROPN
ejpam-6592	273	1	lie	lie	PROPN
ejpam-6592	273	2	symmetry	symmetry	NOUN
ejpam-6592	273	3	,	,	PUNCT
ejpam-6592	273	4	convergence	convergence	NOUN
ejpam-6592	273	5	analysis	analysis	NOUN
ejpam-6592	273	6	,	,	PUNCT
ejpam-6592	273	7	explicit	explicit	ADJ
ejpam-6592	273	8	solutions	solution	NOUN
ejpam-6592	273	9	,	,	PUNCT
ejpam-6592	273	10	and	and	CCONJ
ejpam-6592	273	11	conservation	conservation	NOUN
ejpam-6592	273	12	laws	law	NOUN
ejpam-6592	273	13	for	for	ADP
ejpam-6592	273	14	the	the	DET
ejpam-6592	273	15	timefractional	timefractional	ADJ
ejpam-6592	273	16	modified	modify	VERB
ejpam-6592	273	17	benjamin	benjamin	PROPN
ejpam-6592	273	18	-	-	PUNCT
ejpam-6592	273	19	bona	bona	ADJ
ejpam-6592	273	20	-	-	PUNCT
ejpam-6592	273	21	mahony	mahony	NOUN
ejpam-6592	273	22	equation	equation	NOUN
ejpam-6592	273	23	.	.	PUNCT
ejpam-6592	274	1	journal	journal	PROPN
ejpam-6592	274	2	of	of	ADP
ejpam-6592	274	3	applied	apply	VERB
ejpam-6592	274	4	mathematics	mathematic	NOUN
ejpam-6592	274	5	and	and	CCONJ
ejpam-6592	274	6	computational	computational	ADJ
ejpam-6592	274	7	mechanics	mechanic	NOUN
ejpam-6592	274	8	,	,	PUNCT
ejpam-6592	274	9	23(1):19–31	23(1):19–31	NUM
ejpam-6592	274	10	,	,	PUNCT
ejpam-6592	274	11	2024	2024	NUM
ejpam-6592	274	12	.	.	PUNCT
ejpam-6592	275	1	[	[	X
ejpam-6592	275	2	17	17	NUM
ejpam-6592	275	3	]	]	PUNCT
ejpam-6592	275	4	s.	s.	PROPN
ejpam-6592	275	5	al	al	PROPN
ejpam-6592	275	6	-	-	PUNCT
ejpam-6592	275	7	sharif	sharif	PROPN
ejpam-6592	275	8	and	and	CCONJ
ejpam-6592	275	9	a.	a.	NOUN
ejpam-6592	275	10	malkawi	malkawi	PROPN
ejpam-6592	275	11	.	.	PUNCT
ejpam-6592	276	1	modification	modification	NOUN
ejpam-6592	276	2	of	of	ADP
ejpam-6592	276	3	conformable	conformable	ADJ
ejpam-6592	276	4	fractional	fractional	ADJ
ejpam-6592	276	5	derivative	derivative	NOUN
ejpam-6592	276	6	with	with	ADP
ejpam-6592	276	7	classical	classical	ADJ
ejpam-6592	276	8	properties	property	NOUN
ejpam-6592	276	9	.	.	PUNCT
ejpam-6592	277	1	italian	italian	ADJ
ejpam-6592	277	2	journal	journal	NOUN
ejpam-6592	277	3	of	of	ADP
ejpam-6592	277	4	pure	pure	ADJ
ejpam-6592	277	5	and	and	CCONJ
ejpam-6592	277	6	applied	applied	ADJ
ejpam-6592	277	7	mathematics	mathematic	NOUN
ejpam-6592	277	8	,	,	PUNCT
ejpam-6592	277	9	44:30–39	44:30–39	PROPN
ejpam-6592	277	10	,	,	PUNCT
ejpam-6592	277	11	2020	2020	NUM
ejpam-6592	277	12	.	.	PUNCT
ejpam-6592	278	1	[	[	X
ejpam-6592	278	2	18	18	NUM
ejpam-6592	278	3	]	]	PUNCT
ejpam-6592	278	4	a.	a.	NOUN
ejpam-6592	278	5	rabaiah	rabaiah	PROPN
ejpam-6592	278	6	,	,	PUNCT
ejpam-6592	278	7	a.	a.	NOUN
ejpam-6592	278	8	tallafha	tallafha	NOUN
ejpam-6592	278	9	,	,	PUNCT
ejpam-6592	278	10	and	and	CCONJ
ejpam-6592	278	11	w.	w.	PROPN
ejpam-6592	278	12	shatanawi	shatanawi	PROPN
ejpam-6592	278	13	.	.	PUNCT
ejpam-6592	279	1	common	common	ADJ
ejpam-6592	279	2	fixed	fix	VERB
ejpam-6592	279	3	point	point	NOUN
ejpam-6592	279	4	results	result	NOUN
ejpam-6592	279	5	for	for	ADP
ejpam-6592	279	6	mappings	mapping	NOUN
ejpam-6592	279	7	under	under	ADP
ejpam-6592	279	8	nonlinear	nonlinear	ADJ
ejpam-6592	279	9	contraction	contraction	NOUN
ejpam-6592	279	10	of	of	ADP
ejpam-6592	279	11	cyclic	cyclic	ADJ
ejpam-6592	279	12	form	form	NOUN
ejpam-6592	279	13	in	in	ADP
ejpam-6592	279	14	b	b	NOUN
ejpam-6592	279	15	-	-	ADJ
ejpam-6592	279	16	metric	metric	ADJ
ejpam-6592	279	17	spaces	space	NOUN
ejpam-6592	279	18	.	.	PUNCT
ejpam-6592	280	1	advances	advance	NOUN
ejpam-6592	280	2	in	in	ADP
ejpam-6592	280	3	mathematics	mathematics	NOUN
ejpam-6592	280	4	scientific	scientific	ADJ
ejpam-6592	280	5	journal	journal	NOUN
ejpam-6592	280	6	,	,	PUNCT
ejpam-6592	280	7	26(2):289–301	26(2):289–301	PROPN
ejpam-6592	280	8	,	,	PUNCT
ejpam-6592	280	9	2021	2021	NUM
ejpam-6592	280	10	.	.	PUNCT
ejpam-6592	281	1	[	[	X
ejpam-6592	281	2	19	19	NUM
ejpam-6592	281	3	]	]	PUNCT
ejpam-6592	281	4	a.	a.	NOUN
ejpam-6592	281	5	bataihah	bataihah	PROPN
ejpam-6592	281	6	,	,	PUNCT
ejpam-6592	281	7	w.	w.	PROPN
ejpam-6592	281	8	shatanawi	shatanawi	PROPN
ejpam-6592	281	9	,	,	PUNCT
ejpam-6592	281	10	and	and	CCONJ
ejpam-6592	281	11	a.	a.	NOUN
ejpam-6592	281	12	tallafha	tallafha	NOUN
ejpam-6592	281	13	.	.	PUNCT
ejpam-6592	282	1	fixed	fix	VERB
ejpam-6592	282	2	point	point	NOUN
ejpam-6592	282	3	results	result	NOUN
ejpam-6592	282	4	with	with	ADP
ejpam-6592	282	5	simulation	simulation	NOUN
ejpam-6592	282	6	functions	function	NOUN
ejpam-6592	282	7	.	.	PUNCT
ejpam-6592	283	1	nonlinear	nonlinear	ADJ
ejpam-6592	283	2	functional	functional	ADJ
ejpam-6592	283	3	analysis	analysis	NOUN
ejpam-6592	283	4	and	and	CCONJ
ejpam-6592	283	5	applications	application	NOUN
ejpam-6592	283	6	,	,	PUNCT
ejpam-6592	283	7	25(1):13–23	25(1):13–23	NUM
ejpam-6592	283	8	,	,	PUNCT
ejpam-6592	283	9	2020	2020	NUM
ejpam-6592	283	10	.	.	PUNCT
ejpam-6592	284	1	[	[	X
ejpam-6592	284	2	20	20	NUM
ejpam-6592	284	3	]	]	PUNCT
ejpam-6592	284	4	a.	a.	NOUN
ejpam-6592	284	5	bataihah	bataihah	PROPN
ejpam-6592	284	6	and	and	CCONJ
ejpam-6592	284	7	t.	t.	NOUN
ejpam-6592	284	8	qawasmeh	qawasmeh	NOUN
ejpam-6592	284	9	.	.	PUNCT
ejpam-6592	285	1	a	a	DET
ejpam-6592	285	2	new	new	ADJ
ejpam-6592	285	3	type	type	NOUN
ejpam-6592	285	4	of	of	ADP
ejpam-6592	285	5	distance	distance	NOUN
ejpam-6592	285	6	spaces	space	NOUN
ejpam-6592	285	7	and	and	CCONJ
ejpam-6592	285	8	fixed	fix	VERB
ejpam-6592	285	9	point	point	NOUN
ejpam-6592	285	10	results	result	NOUN
ejpam-6592	285	11	.	.	PUNCT
ejpam-6592	286	1	journal	journal	NOUN
ejpam-6592	286	2	of	of	ADP
ejpam-6592	286	3	mathematical	mathematical	ADJ
ejpam-6592	286	4	analysis	analysis	NOUN
ejpam-6592	286	5	,	,	PUNCT
ejpam-6592	286	6	15(4):81–90	15(4):81–90	NUM
ejpam-6592	286	7	,	,	PUNCT
ejpam-6592	286	8	2024	2024	NUM
ejpam-6592	286	9	.	.	PUNCT
ejpam-6592	287	1	[	[	X
ejpam-6592	287	2	21	21	NUM
ejpam-6592	287	3	]	]	PUNCT
ejpam-6592	287	4	k.	k.	PROPN
ejpam-6592	287	5	abodayeh	abodayeh	PROPN
ejpam-6592	287	6	,	,	PUNCT
ejpam-6592	287	7	a.	a.	PROPN
ejpam-6592	287	8	bataihah	bataihah	PROPN
ejpam-6592	287	9	,	,	PUNCT
ejpam-6592	287	10	and	and	CCONJ
ejpam-6592	287	11	w.	w.	PROPN
ejpam-6592	287	12	shatanawi	shatanawi	PROPN
ejpam-6592	287	13	.	.	PUNCT
ejpam-6592	288	1	generalized	generalize	VERB
ejpam-6592	288	2	ω	ω	NUM
ejpam-6592	288	3	-	-	PUNCT
ejpam-6592	288	4	distance	distance	NOUN
ejpam-6592	288	5	mappings	mapping	NOUN
ejpam-6592	288	6	and	and	CCONJ
ejpam-6592	288	7	some	some	DET
ejpam-6592	288	8	fixed	fix	VERB
ejpam-6592	288	9	point	point	NOUN
ejpam-6592	288	10	theorems	theorem	NOUN
ejpam-6592	288	11	.	.	PUNCT
ejpam-6592	289	1	u.p.b	u.p.b	PROPN
ejpam-6592	289	2	.	.	PUNCT
ejpam-6592	290	1	scientific	scientific	ADJ
ejpam-6592	290	2	bulletin	bulletin	NOUN
ejpam-6592	290	3	,	,	PUNCT
ejpam-6592	290	4	series	series	PROPN
ejpam-6592	290	5	a	a	PROPN
ejpam-6592	290	6	,	,	PUNCT
ejpam-6592	290	7	79:223–232	79:223–232	PROPN
ejpam-6592	290	8	,	,	PUNCT
ejpam-6592	290	9	2017	2017	NUM
ejpam-6592	290	10	.	.	PUNCT
ejpam-6592	291	1	[	[	X
ejpam-6592	291	2	22	22	NUM
ejpam-6592	291	3	]	]	X
ejpam-6592	291	4	i.	i.	PROPN
ejpam-6592	291	5	abu	abu	PROPN
ejpam-6592	291	6	-	-	PUNCT
ejpam-6592	291	7	irwaq	irwaq	PROPN
ejpam-6592	291	8	,	,	PUNCT
ejpam-6592	291	9	w.	w.	PROPN
ejpam-6592	291	10	shatanawi	shatanawi	PROPN
ejpam-6592	291	11	,	,	PUNCT
ejpam-6592	291	12	a.	a.	NOUN
ejpam-6592	291	13	bataihah	bataihah	PROPN
ejpam-6592	291	14	,	,	PUNCT
ejpam-6592	291	15	and	and	CCONJ
ejpam-6592	291	16	nuseir	nuseir	NOUN
ejpam-6592	291	17	.	.	PUNCT
ejpam-6592	292	1	fixed	fix	VERB
ejpam-6592	292	2	point	point	NOUN
ejpam-6592	292	3	results	result	NOUN
ejpam-6592	292	4	for	for	ADP
ejpam-6592	292	5	nonlinear	nonlinear	ADJ
ejpam-6592	292	6	contractions	contraction	NOUN
ejpam-6592	292	7	with	with	ADP
ejpam-6592	292	8	generalized	generalized	ADJ
ejpam-6592	292	9	ω	ω	NUM
ejpam-6592	292	10	-	-	PUNCT
ejpam-6592	292	11	distance	distance	NOUN
ejpam-6592	292	12	mappings	mapping	NOUN
ejpam-6592	292	13	.	.	PUNCT
ejpam-6592	293	1	u.p.b	u.p.b	ADJ
ejpam-6592	293	2	.	.	PUNCT
ejpam-6592	294	1	scientific	scientific	ADJ
ejpam-6592	294	2	bulletin	bulletin	NOUN
ejpam-6592	294	3	,	,	PUNCT
ejpam-6592	294	4	series	series	NOUN
ejpam-6592	294	5	a	a	NOUN
ejpam-6592	294	6	,	,	PUNCT
ejpam-6592	294	7	81(1):57–64	81(1):57–64	NUM
ejpam-6592	294	8	,	,	PUNCT
ejpam-6592	294	9	2019	2019	NUM
ejpam-6592	294	10	.	.	PUNCT
ejpam-6592	295	1	[	[	X
ejpam-6592	295	2	23	23	NUM
ejpam-6592	295	3	]	]	X
ejpam-6592	295	4	g.	g.	PROPN
ejpam-6592	295	5	m.	m.	PROPN
ejpam-6592	295	6	gharib	gharib	PROPN
ejpam-6592	295	7	,	,	PUNCT
ejpam-6592	295	8	m.	m.	PROPN
ejpam-6592	295	9	s.	s.	PROPN
ejpam-6592	295	10	alsauodi	alsauodi	PROPN
ejpam-6592	295	11	,	,	PUNCT
ejpam-6592	295	12	a.	a.	NOUN
ejpam-6592	295	13	guiatni	guiatni	PROPN
ejpam-6592	295	14	,	,	PUNCT
ejpam-6592	295	15	m.	m.	NOUN
ejpam-6592	295	16	a.	a.	PROPN
ejpam-6592	295	17	al	al	PROPN
ejpam-6592	295	18	-	-	PUNCT
ejpam-6592	295	19	omari	omari	PROPN
ejpam-6592	295	20	,	,	PUNCT
ejpam-6592	295	21	and	and	CCONJ
ejpam-6592	295	22	a.	a.	NOUN
ejpam-6592	295	23	a.-r	a.-r	PROPN
ejpam-6592	295	24	.	.	PUNCT
ejpam-6592	296	1	m.	m.	NOUN
ejpam-6592	296	2	malkawi	malkawi	PROPN
ejpam-6592	296	3	.	.	PUNCT
ejpam-6592	297	1	using	use	VERB
ejpam-6592	297	2	atomic	atomic	ADJ
ejpam-6592	297	3	solution	solution	NOUN
ejpam-6592	297	4	method	method	NOUN
ejpam-6592	297	5	to	to	PART
ejpam-6592	297	6	solve	solve	VERB
ejpam-6592	297	7	the	the	DET
ejpam-6592	297	8	fractional	fractional	ADJ
ejpam-6592	297	9	equations	equation	NOUN
ejpam-6592	297	10	.	.	PUNCT
ejpam-6592	298	1	springer	springer	NOUN
ejpam-6592	298	2	proceedings	proceeding	NOUN
ejpam-6592	298	3	a.	a.	VERB
ejpam-6592	298	4	malkawi	malkawi	ADP
ejpam-6592	298	5	/	/	SYM
ejpam-6592	298	6	eur	eur	PROPN
ejpam-6592	298	7	.	.	PUNCT
ejpam-6592	299	1	j.	j.	PROPN
ejpam-6592	299	2	pure	pure	PROPN
ejpam-6592	299	3	appl	appl	PROPN
ejpam-6592	299	4	.	.	PROPN
ejpam-6592	299	5	math	math	PROPN
ejpam-6592	299	6	,	,	PUNCT
ejpam-6592	299	7	18	18	NUM
ejpam-6592	299	8	(	(	PUNCT
ejpam-6592	299	9	3	3	NUM
ejpam-6592	299	10	)	)	PUNCT
ejpam-6592	299	11	(	(	PUNCT
ejpam-6592	299	12	2025	2025	NUM
ejpam-6592	299	13	)	)	PUNCT
ejpam-6592	299	14	,	,	PUNCT
ejpam-6592	299	15	6592	6592	NUM
ejpam-6592	299	16	14	14	NUM
ejpam-6592	299	17	of	of	ADP
ejpam-6592	299	18	14	14	NUM
ejpam-6592	299	19	in	in	ADP
ejpam-6592	299	20	mathematics	mathematic	NOUN
ejpam-6592	299	21	and	and	CCONJ
ejpam-6592	299	22	statistics	statistic	NOUN
ejpam-6592	299	23	,	,	PUNCT
ejpam-6592	299	24	418:123–129	418:123–129	NUM
ejpam-6592	299	25	,	,	PUNCT
ejpam-6592	299	26	2023	2023	NUM
ejpam-6592	299	27	.	.	PUNCT