id	sid	tid	token	lemma	pos
ejpam-5253	1	1	european	european	PROPN
ejpam-5253	1	2	journal	journal	PROPN
ejpam-5253	1	3	of	of	ADP
ejpam-5253	1	4	pure	pure	ADJ
ejpam-5253	1	5	and	and	CCONJ
ejpam-5253	1	6	applied	apply	VERB
ejpam-5253	1	7	mathematics	mathematic	NOUN
ejpam-5253	1	8	vol	vol	NOUN
ejpam-5253	1	9	.	.	PROPN
ejpam-5253	2	1	17	17	NUM
ejpam-5253	2	2	,	,	PUNCT
ejpam-5253	2	3	no	no	INTJ
ejpam-5253	2	4	.	.	NOUN
ejpam-5253	2	5	3	3	NUM
ejpam-5253	2	6	,	,	PUNCT
ejpam-5253	2	7	2024	2024	NUM
ejpam-5253	2	8	,	,	PUNCT
ejpam-5253	2	9	1804	1804	NUM
ejpam-5253	2	10	-	-	SYM
ejpam-5253	2	11	1817	1817	NUM
ejpam-5253	2	12	issn	issn	PROPN
ejpam-5253	2	13	1307	1307	NUM
ejpam-5253	2	14	-	-	SYM
ejpam-5253	2	15	5543	5543	NUM
ejpam-5253	2	16	–	–	PUNCT
ejpam-5253	2	17	ejpam.com	ejpam.com	X
ejpam-5253	2	18	published	publish	VERB
ejpam-5253	2	19	by	by	ADP
ejpam-5253	2	20	new	new	PROPN
ejpam-5253	2	21	york	york	PROPN
ejpam-5253	2	22	business	business	PROPN
ejpam-5253	2	23	global	global	ADJ
ejpam-5253	2	24	theoretical	theoretical	ADJ
ejpam-5253	2	25	foundations	foundation	NOUN
ejpam-5253	2	26	of	of	ADP
ejpam-5253	2	27	h	h	NOUN
ejpam-5253	2	28	-	-	PUNCT
ejpam-5253	2	29	rough	rough	ADJ
ejpam-5253	2	30	sets	set	NOUN
ejpam-5253	2	31	amjad	amjad	PROPN
ejpam-5253	2	32	a.	a.	PROPN
ejpam-5253	2	33	al	al	PROPN
ejpam-5253	2	34	-	-	PUNCT
ejpam-5253	2	35	rehili	rehili	PROPN
ejpam-5253	2	36	department	department	PROPN
ejpam-5253	2	37	of	of	ADP
ejpam-5253	2	38	mathematical	mathematical	ADJ
ejpam-5253	2	39	sciences	sciences	PROPN
ejpam-5253	2	40	,	,	PUNCT
ejpam-5253	2	41	college	college	NOUN
ejpam-5253	2	42	of	of	ADP
ejpam-5253	2	43	applied	apply	VERB
ejpam-5253	2	44	science	science	NOUN
ejpam-5253	2	45	,	,	PUNCT
ejpam-5253	2	46	al	al	PROPN
ejpam-5253	2	47	-	-	PROPN
ejpam-5253	2	48	madinah	madinah	PROPN
ejpam-5253	2	49	al	al	PROPN
ejpam-5253	2	50	-	-	PUNCT
ejpam-5253	2	51	munawarah	munawarah	PROPN
ejpam-5253	2	52	,	,	PUNCT
ejpam-5253	2	53	kingdom	kingdom	NOUN
ejpam-5253	2	54	of	of	ADP
ejpam-5253	2	55	saudi	saudi	PROPN
ejpam-5253	2	56	arabia	arabia	PROPN
ejpam-5253	2	57	abstract	abstract	NOUN
ejpam-5253	2	58	.	.	PUNCT
ejpam-5253	3	1	this	this	DET
ejpam-5253	3	2	paper	paper	NOUN
ejpam-5253	3	3	introduces	introduce	VERB
ejpam-5253	3	4	a	a	DET
ejpam-5253	3	5	pioneering	pioneer	VERB
ejpam-5253	3	6	advancement	advancement	NOUN
ejpam-5253	3	7	in	in	ADP
ejpam-5253	3	8	rough	rough	ADJ
ejpam-5253	3	9	set	set	NOUN
ejpam-5253	3	10	theory	theory	NOUN
ejpam-5253	3	11	by	by	ADP
ejpam-5253	3	12	presenting	present	VERB
ejpam-5253	3	13	a	a	DET
ejpam-5253	3	14	new	new	ADJ
ejpam-5253	3	15	class	class	NOUN
ejpam-5253	3	16	of	of	ADP
ejpam-5253	3	17	rough	rough	ADJ
ejpam-5253	3	18	sets	set	NOUN
ejpam-5253	3	19	termed	term	VERB
ejpam-5253	3	20	h	h	NOUN
ejpam-5253	3	21	-	-	PUNCT
ejpam-5253	3	22	rough	rough	ADJ
ejpam-5253	3	23	sets	set	NOUN
ejpam-5253	3	24	.	.	PUNCT
ejpam-5253	4	1	central	central	ADJ
ejpam-5253	4	2	to	to	ADP
ejpam-5253	4	3	this	this	DET
ejpam-5253	4	4	novel	novel	ADJ
ejpam-5253	4	5	approach	approach	NOUN
ejpam-5253	4	6	are	be	AUX
ejpam-5253	4	7	the	the	DET
ejpam-5253	4	8	concepts	concept	NOUN
ejpam-5253	4	9	of	of	ADP
ejpam-5253	4	10	h	h	NOUN
ejpam-5253	4	11	-	-	PUNCT
ejpam-5253	4	12	lower	low	ADJ
ejpam-5253	4	13	and	and	CCONJ
ejpam-5253	4	14	h	h	NOUN
ejpam-5253	4	15	-	-	PUNCT
ejpam-5253	4	16	upper	upper	ADJ
ejpam-5253	4	17	approximations	approximation	NOUN
ejpam-5253	4	18	,	,	PUNCT
ejpam-5253	4	19	intricately	intricately	ADV
ejpam-5253	4	20	tied	tie	VERB
ejpam-5253	4	21	to	to	ADP
ejpam-5253	4	22	the	the	DET
ejpam-5253	4	23	notion	notion	NOUN
ejpam-5253	4	24	of	of	ADP
ejpam-5253	4	25	h	h	NOUN
ejpam-5253	4	26	-	-	PUNCT
ejpam-5253	4	27	open	open	ADJ
ejpam-5253	4	28	sets	set	NOUN
ejpam-5253	4	29	.	.	PUNCT
ejpam-5253	5	1	we	we	PRON
ejpam-5253	5	2	delve	delve	VERB
ejpam-5253	5	3	into	into	ADP
ejpam-5253	5	4	the	the	DET
ejpam-5253	5	5	fundamental	fundamental	ADJ
ejpam-5253	5	6	properties	property	NOUN
ejpam-5253	5	7	of	of	ADP
ejpam-5253	5	8	h	h	NOUN
ejpam-5253	5	9	-	-	PUNCT
ejpam-5253	5	10	rough	rough	ADJ
ejpam-5253	5	11	sets	set	NOUN
ejpam-5253	5	12	and	and	CCONJ
ejpam-5253	5	13	establish	establish	VERB
ejpam-5253	5	14	the	the	DET
ejpam-5253	5	15	framework	framework	NOUN
ejpam-5253	5	16	of	of	ADP
ejpam-5253	5	17	h	h	NOUN
ejpam-5253	5	18	-	-	PUNCT
ejpam-5253	5	19	approximation	approximation	NOUN
ejpam-5253	5	20	spaces	space	NOUN
ejpam-5253	5	21	,	,	PUNCT
ejpam-5253	5	22	offering	offer	VERB
ejpam-5253	5	23	a	a	DET
ejpam-5253	5	24	comprehensive	comprehensive	ADJ
ejpam-5253	5	25	understanding	understanding	NOUN
ejpam-5253	5	26	of	of	ADP
ejpam-5253	5	27	their	their	PRON
ejpam-5253	5	28	theoretical	theoretical	ADJ
ejpam-5253	5	29	underpinnings	underpinning	NOUN
ejpam-5253	5	30	.	.	PUNCT
ejpam-5253	6	1	moreover	moreover	ADV
ejpam-5253	6	2	,	,	PUNCT
ejpam-5253	6	3	we	we	PRON
ejpam-5253	6	4	introduce	introduce	VERB
ejpam-5253	6	5	and	and	CCONJ
ejpam-5253	6	6	rigorously	rigorously	ADV
ejpam-5253	6	7	analyze	analyze	VERB
ejpam-5253	6	8	the	the	DET
ejpam-5253	6	9	concepts	concept	NOUN
ejpam-5253	6	10	of	of	ADP
ejpam-5253	6	11	h	h	NOUN
ejpam-5253	6	12	-	-	PUNCT
ejpam-5253	6	13	rough	rough	ADJ
ejpam-5253	6	14	equality	equality	NOUN
ejpam-5253	6	15	and	and	CCONJ
ejpam-5253	6	16	h	h	NOUN
ejpam-5253	6	17	-	-	PUNCT
ejpam-5253	6	18	rough	rough	ADJ
ejpam-5253	6	19	inclusion	inclusion	NOUN
ejpam-5253	6	20	,	,	PUNCT
ejpam-5253	6	21	providing	provide	VERB
ejpam-5253	6	22	formal	formal	ADJ
ejpam-5253	6	23	definitions	definition	NOUN
ejpam-5253	6	24	and	and	CCONJ
ejpam-5253	6	25	insightful	insightful	ADJ
ejpam-5253	6	26	examinations	examination	NOUN
ejpam-5253	6	27	of	of	ADP
ejpam-5253	6	28	their	their	PRON
ejpam-5253	6	29	implications	implication	NOUN
ejpam-5253	6	30	in	in	ADP
ejpam-5253	6	31	data	datum	NOUN
ejpam-5253	6	32	approximation	approximation	NOUN
ejpam-5253	6	33	tasks	task	NOUN
ejpam-5253	6	34	.	.	PUNCT
ejpam-5253	7	1	through	through	ADP
ejpam-5253	7	2	detailed	detailed	ADJ
ejpam-5253	7	3	examples	example	NOUN
ejpam-5253	7	4	and	and	CCONJ
ejpam-5253	7	5	thorough	thorough	ADJ
ejpam-5253	7	6	exploration	exploration	NOUN
ejpam-5253	7	7	,	,	PUNCT
ejpam-5253	7	8	this	this	DET
ejpam-5253	7	9	paper	paper	NOUN
ejpam-5253	7	10	showcases	showcase	VERB
ejpam-5253	7	11	how	how	SCONJ
ejpam-5253	7	12	h	h	NOUN
ejpam-5253	7	13	-	-	PUNCT
ejpam-5253	7	14	rough	rough	ADJ
ejpam-5253	7	15	sets	set	NOUN
ejpam-5253	7	16	extend	extend	VERB
ejpam-5253	7	17	rough	rough	ADJ
ejpam-5253	7	18	set	set	NOUN
ejpam-5253	7	19	theory	theory	NOUN
ejpam-5253	7	20	,	,	PUNCT
ejpam-5253	7	21	offering	offer	VERB
ejpam-5253	7	22	more	more	ADV
ejpam-5253	7	23	flexible	flexible	ADJ
ejpam-5253	7	24	and	and	CCONJ
ejpam-5253	7	25	precise	precise	ADJ
ejpam-5253	7	26	techniques	technique	NOUN
ejpam-5253	7	27	for	for	ADP
ejpam-5253	7	28	data	data	NOUN
ejpam-5253	7	29	approximation	approximation	NOUN
ejpam-5253	7	30	.	.	PUNCT
ejpam-5253	8	1	this	this	DET
ejpam-5253	8	2	study	study	NOUN
ejpam-5253	8	3	not	not	PART
ejpam-5253	8	4	only	only	ADV
ejpam-5253	8	5	contributes	contribute	VERB
ejpam-5253	8	6	to	to	ADP
ejpam-5253	8	7	the	the	DET
ejpam-5253	8	8	theoretical	theoretical	ADJ
ejpam-5253	8	9	development	development	NOUN
ejpam-5253	8	10	of	of	ADP
ejpam-5253	8	11	rough	rough	ADJ
ejpam-5253	8	12	set	set	NOUN
ejpam-5253	8	13	theory	theory	NOUN
ejpam-5253	8	14	but	but	CCONJ
ejpam-5253	8	15	also	also	ADV
ejpam-5253	8	16	opens	open	VERB
ejpam-5253	8	17	up	up	ADP
ejpam-5253	8	18	exciting	exciting	ADJ
ejpam-5253	8	19	possibilities	possibility	NOUN
ejpam-5253	8	20	for	for	ADP
ejpam-5253	8	21	practical	practical	ADJ
ejpam-5253	8	22	applications	application	NOUN
ejpam-5253	8	23	across	across	ADP
ejpam-5253	8	24	various	various	ADJ
ejpam-5253	8	25	domains	domain	NOUN
ejpam-5253	8	26	.	.	PUNCT
ejpam-5253	9	1	2020	2020	NUM
ejpam-5253	9	2	mathematics	mathematic	NOUN
ejpam-5253	9	3	subject	subject	NOUN
ejpam-5253	9	4	classifications	classification	NOUN
ejpam-5253	9	5	:	:	PUNCT
ejpam-5253	9	6	04a05,54a05,03e75	04a05,54a05,03e75	NUM
ejpam-5253	9	7	,	,	PUNCT
ejpam-5253	9	8	54c08	54c08	NUM
ejpam-5253	9	9	key	key	ADJ
ejpam-5253	9	10	words	word	NOUN
ejpam-5253	9	11	and	and	CCONJ
ejpam-5253	9	12	phrases	phrase	NOUN
ejpam-5253	9	13	:	:	PUNCT
ejpam-5253	9	14	rough	rough	ADJ
ejpam-5253	9	15	sets	set	NOUN
ejpam-5253	9	16	,	,	PUNCT
ejpam-5253	9	17	upper	upper	ADJ
ejpam-5253	9	18	and	and	CCONJ
ejpam-5253	9	19	lower	low	ADJ
ejpam-5253	9	20	approximations	approximation	NOUN
ejpam-5253	9	21	,	,	PUNCT
ejpam-5253	9	22	accuracy	accuracy	NOUN
ejpam-5253	9	23	measure	measure	NOUN
ejpam-5253	9	24	,	,	PUNCT
ejpam-5253	9	25	h	h	NOUN
ejpam-5253	9	26	-	-	PUNCT
ejpam-5253	9	27	open	open	ADJ
ejpam-5253	9	28	sets	set	NOUN
ejpam-5253	9	29	,	,	PUNCT
ejpam-5253	9	30	h	h	NOUN
ejpam-5253	9	31	-	-	PUNCT
ejpam-5253	9	32	rough	rough	ADJ
ejpam-5253	9	33	sets	set	NOUN
ejpam-5253	9	34	,	,	PUNCT
ejpam-5253	9	35	h	h	NOUN
ejpam-5253	9	36	-	-	PUNCT
ejpam-5253	9	37	upper	upper	ADJ
ejpam-5253	9	38	and	and	CCONJ
ejpam-5253	9	39	h	h	NOUN
ejpam-5253	9	40	-	-	PUNCT
ejpam-5253	9	41	lower	low	ADJ
ejpam-5253	9	42	approximations	approximation	NOUN
ejpam-5253	9	43	and	and	CCONJ
ejpam-5253	9	44	h	h	NOUN
ejpam-5253	9	45	-	-	PUNCT
ejpam-5253	9	46	accuracy	accuracy	NOUN
ejpam-5253	9	47	measure	measure	NOUN
ejpam-5253	9	48	1	1	NUM
ejpam-5253	9	49	.	.	PUNCT
ejpam-5253	10	1	introduction	introduction	NOUN
ejpam-5253	10	2	information	information	NOUN
ejpam-5253	10	3	technology	technology	NOUN
ejpam-5253	10	4	is	be	AUX
ejpam-5253	10	5	the	the	DET
ejpam-5253	10	6	most	most	ADV
ejpam-5253	10	7	significant	significant	ADJ
ejpam-5253	10	8	feature	feature	NOUN
ejpam-5253	10	9	of	of	ADP
ejpam-5253	10	10	the	the	DET
ejpam-5253	10	11	21st	21st	ADJ
ejpam-5253	10	12	century	century	NOUN
ejpam-5253	10	13	,	,	PUNCT
ejpam-5253	10	14	playing	play	VERB
ejpam-5253	10	15	a	a	DET
ejpam-5253	10	16	vital	vital	ADJ
ejpam-5253	10	17	role	role	NOUN
ejpam-5253	10	18	in	in	ADP
ejpam-5253	10	19	information	information	NOUN
ejpam-5253	10	20	discovery	discovery	NOUN
ejpam-5253	10	21	through	through	ADP
ejpam-5253	10	22	available	available	ADJ
ejpam-5253	10	23	knowledge	knowledge	NOUN
ejpam-5253	10	24	.	.	PUNCT
ejpam-5253	11	1	rough	rough	ADJ
ejpam-5253	11	2	set	set	NOUN
ejpam-5253	11	3	theory	theory	NOUN
ejpam-5253	11	4	[	[	X
ejpam-5253	11	5	12	12	NUM
ejpam-5253	11	6	]	]	X
ejpam-5253	11	7	,	,	PUNCT
ejpam-5253	11	8	a	a	DET
ejpam-5253	11	9	recent	recent	ADJ
ejpam-5253	11	10	approach	approach	NOUN
ejpam-5253	11	11	for	for	ADP
ejpam-5253	11	12	reasoning	reason	VERB
ejpam-5253	11	13	about	about	ADP
ejpam-5253	11	14	data	datum	NOUN
ejpam-5253	11	15	,	,	PUNCT
ejpam-5253	11	16	was	be	AUX
ejpam-5253	11	17	created	create	VERB
ejpam-5253	11	18	by	by	ADP
ejpam-5253	11	19	pawlak	pawlak	ADJ
ejpam-5253	11	20	.	.	PUNCT
ejpam-5253	12	1	this	this	DET
ejpam-5253	12	2	theory	theory	NOUN
ejpam-5253	12	3	extends	extend	VERB
ejpam-5253	12	4	set	set	VERB
ejpam-5253	12	5	theory	theory	NOUN
ejpam-5253	12	6	by	by	ADP
ejpam-5253	12	7	describing	describe	VERB
ejpam-5253	12	8	a	a	DET
ejpam-5253	12	9	subset	subset	NOUN
ejpam-5253	12	10	of	of	ADP
ejpam-5253	12	11	a	a	DET
ejpam-5253	12	12	universe	universe	NOUN
ejpam-5253	12	13	with	with	ADP
ejpam-5253	12	14	a	a	DET
ejpam-5253	12	15	pair	pair	NOUN
ejpam-5253	12	16	of	of	ADP
ejpam-5253	12	17	ordinary	ordinary	ADJ
ejpam-5253	12	18	sets	set	NOUN
ejpam-5253	12	19	known	know	VERB
ejpam-5253	12	20	as	as	ADP
ejpam-5253	12	21	the	the	DET
ejpam-5253	12	22	lower	low	ADJ
ejpam-5253	12	23	and	and	CCONJ
ejpam-5253	12	24	upper	upper	ADJ
ejpam-5253	12	25	approximation	approximation	NOUN
ejpam-5253	12	26	.	.	PUNCT
ejpam-5253	13	1	it	it	PRON
ejpam-5253	13	2	depends	depend	VERB
ejpam-5253	13	3	on	on	ADP
ejpam-5253	13	4	a	a	DET
ejpam-5253	13	5	specific	specific	ADJ
ejpam-5253	13	6	topological	topological	ADJ
ejpam-5253	13	7	structure	structure	NOUN
ejpam-5253	13	8	and	and	CCONJ
ejpam-5253	13	9	finds	find	VERB
ejpam-5253	13	10	many	many	ADJ
ejpam-5253	13	11	applications	application	NOUN
ejpam-5253	13	12	across	across	ADP
ejpam-5253	13	13	various	various	ADJ
ejpam-5253	13	14	real	real	ADJ
ejpam-5253	13	15	-	-	PUNCT
ejpam-5253	13	16	life	life	NOUN
ejpam-5253	13	17	fields	field	NOUN
ejpam-5253	13	18	.	.	PUNCT
ejpam-5253	14	1	the	the	DET
ejpam-5253	14	2	theory	theory	NOUN
ejpam-5253	14	3	and	and	CCONJ
ejpam-5253	14	4	applications	application	NOUN
ejpam-5253	14	5	of	of	ADP
ejpam-5253	14	6	rough	rough	ADJ
ejpam-5253	14	7	sets	set	NOUN
ejpam-5253	14	8	have	have	AUX
ejpam-5253	14	9	impressively	impressively	ADV
ejpam-5253	14	10	developed	develop	VERB
ejpam-5253	14	11	over	over	ADP
ejpam-5253	14	12	time	time	NOUN
ejpam-5253	14	13	.	.	PUNCT
ejpam-5253	15	1	numerous	numerous	ADJ
ejpam-5253	15	2	papers	paper	NOUN
ejpam-5253	15	3	have	have	AUX
ejpam-5253	15	4	been	be	AUX
ejpam-5253	15	5	written	write	VERB
ejpam-5253	15	6	to	to	PART
ejpam-5253	15	7	generalize	generalize	VERB
ejpam-5253	15	8	rough	rough	ADJ
ejpam-5253	15	9	sets	set	NOUN
ejpam-5253	15	10	(	(	PUNCT
ejpam-5253	15	11	[	[	X
ejpam-5253	15	12	3],[2],[4],[5],[7],[14],[12],[15],[16],[18	3],[2],[4],[5],[7],[14],[12],[15],[16],[18	NUM
ejpam-5253	15	13	]	]	PUNCT
ejpam-5253	15	14	,	,	PUNCT
ejpam-5253	15	15	[	[	X
ejpam-5253	15	16	19],[20	19],[20	NOUN
ejpam-5253	15	17	]	]	X
ejpam-5253	15	18	)	)	PUNCT
ejpam-5253	15	19	.	.	PUNCT
ejpam-5253	16	1	in	in	ADP
ejpam-5253	16	2	[	[	X
ejpam-5253	16	3	17	17	NUM
ejpam-5253	16	4	]	]	PUNCT
ejpam-5253	16	5	wiweger	wiweger	NOUN
ejpam-5253	16	6	introduced	introduce	VERB
ejpam-5253	16	7	the	the	DET
ejpam-5253	16	8	concept	concept	NOUN
ejpam-5253	16	9	of	of	ADP
ejpam-5253	16	10	topological	topological	ADJ
ejpam-5253	16	11	rough	rough	ADJ
ejpam-5253	16	12	sets	set	NOUN
ejpam-5253	16	13	,	,	PUNCT
ejpam-5253	16	14	one	one	NUM
ejpam-5253	16	15	of	of	ADP
ejpam-5253	16	16	the	the	DET
ejpam-5253	16	17	most	most	ADV
ejpam-5253	16	18	important	important	ADJ
ejpam-5253	16	19	generalizations	generalization	NOUN
ejpam-5253	16	20	of	of	ADP
ejpam-5253	16	21	rough	rough	ADJ
ejpam-5253	16	22	sets	set	NOUN
ejpam-5253	16	23	.	.	PUNCT
ejpam-5253	17	1	this	this	DET
ejpam-5253	17	2	generalization	generalization	NOUN
ejpam-5253	17	3	utilizes	utilize	VERB
ejpam-5253	17	4	an	an	DET
ejpam-5253	17	5	approach	approach	NOUN
ejpam-5253	17	6	starting	start	VERB
ejpam-5253	17	7	with	with	ADP
ejpam-5253	17	8	a	a	DET
ejpam-5253	17	9	topological	topological	ADJ
ejpam-5253	17	10	space	space	NOUN
ejpam-5253	17	11	and	and	CCONJ
ejpam-5253	17	12	defines	define	VERB
ejpam-5253	17	13	the	the	DET
ejpam-5253	17	14	approximation	approximation	NOUN
ejpam-5253	17	15	via	via	ADP
ejpam-5253	17	16	the	the	DET
ejpam-5253	17	17	interior	interior	ADJ
ejpam-5253	17	18	and	and	CCONJ
ejpam-5253	17	19	closure	closure	NOUN
ejpam-5253	17	20	operators	operator	NOUN
ejpam-5253	17	21	of	of	ADP
ejpam-5253	17	22	topological	topological	ADJ
ejpam-5253	17	23	spaces	space	NOUN
ejpam-5253	17	24	.	.	PUNCT
ejpam-5253	18	1	doi	doi	NOUN
ejpam-5253	18	2	:	:	PUNCT
ejpam-5253	18	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5253	https://doi.org/10.29020/nybg.ejpam.v17i3.5253	PROPN
ejpam-5253	18	4	email	email	NOUN
ejpam-5253	18	5	address	address	NOUN
ejpam-5253	18	6	:	:	PUNCT
ejpam-5253	18	7	dora	dora	PROPN
ejpam-5253	18	8	4008@hotmail.com	4008@hotmail.com	X
ejpam-5253	18	9	(	(	PUNCT
ejpam-5253	18	10	amjad	amjad	PROPN
ejpam-5253	18	11	a.	a.	PROPN
ejpam-5253	18	12	al	al	PROPN
ejpam-5253	18	13	-	-	PUNCT
ejpam-5253	18	14	rehili	rehili	NOUN
ejpam-5253	18	15	)	)	PUNCT
ejpam-5253	18	16	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5253	18	17	1804	1804	NUM
ejpam-5253	19	1	©	©	ADP
ejpam-5253	19	2	2024	2024	NUM
ejpam-5253	19	3	ejpam	ejpam	NOUN
ejpam-5253	19	4	all	all	DET
ejpam-5253	19	5	rights	right	NOUN
ejpam-5253	19	6	reserved	reserve	VERB
ejpam-5253	19	7	.	.	PUNCT
ejpam-5253	20	1	a.	a.	PROPN
ejpam-5253	20	2	al	al	PROPN
ejpam-5253	20	3	-	-	PUNCT
ejpam-5253	20	4	rehili	rehili	NOUN
ejpam-5253	20	5	/	/	SYM
ejpam-5253	20	6	eur	eur	PROPN
ejpam-5253	20	7	.	.	PUNCT
ejpam-5253	21	1	j.	j.	PROPN
ejpam-5253	21	2	pure	pure	PROPN
ejpam-5253	21	3	appl	appl	PROPN
ejpam-5253	21	4	.	.	PROPN
ejpam-5253	21	5	math	math	PROPN
ejpam-5253	21	6	,	,	PUNCT
ejpam-5253	21	7	17	17	NUM
ejpam-5253	21	8	(	(	PUNCT
ejpam-5253	21	9	3	3	NUM
ejpam-5253	21	10	)	)	PUNCT
ejpam-5253	21	11	(	(	PUNCT
ejpam-5253	21	12	2024	2024	NUM
ejpam-5253	21	13	)	)	PUNCT
ejpam-5253	21	14	,	,	PUNCT
ejpam-5253	21	15	1804	1804	NUM
ejpam-5253	21	16	-	-	SYM
ejpam-5253	21	17	1817	1817	NUM
ejpam-5253	21	18	1805	1805	NUM
ejpam-5253	21	19	in	in	ADP
ejpam-5253	21	20	[	[	X
ejpam-5253	21	21	1	1	NUM
ejpam-5253	21	22	]	]	X
ejpam-5253	21	23	abbas	abbas	PROPN
ejpam-5253	21	24	introduced	introduce	VERB
ejpam-5253	21	25	h	h	NOUN
ejpam-5253	21	26	-	-	PUNCT
ejpam-5253	21	27	open	open	ADJ
ejpam-5253	21	28	.	.	PUNCT
ejpam-5253	22	1	in	in	ADP
ejpam-5253	22	2	this	this	DET
ejpam-5253	22	3	paper	paper	NOUN
ejpam-5253	22	4	,	,	PUNCT
ejpam-5253	22	5	we	we	PRON
ejpam-5253	22	6	introduce	introduce	VERB
ejpam-5253	22	7	a	a	DET
ejpam-5253	22	8	new	new	ADJ
ejpam-5253	22	9	classification	classification	NOUN
ejpam-5253	22	10	for	for	ADP
ejpam-5253	22	11	the	the	DET
ejpam-5253	22	12	universe	universe	NOUN
ejpam-5253	22	13	called	call	VERB
ejpam-5253	22	14	h	h	NOUN
ejpam-5253	22	15	-	-	PUNCT
ejpam-5253	22	16	approximation	approximation	NOUN
ejpam-5253	22	17	space	space	NOUN
ejpam-5253	22	18	.	.	PUNCT
ejpam-5253	23	1	additionally	additionally	ADV
ejpam-5253	23	2	,	,	PUNCT
ejpam-5253	23	3	we	we	PRON
ejpam-5253	23	4	study	study	VERB
ejpam-5253	23	5	the	the	DET
ejpam-5253	23	6	concepts	concept	NOUN
ejpam-5253	23	7	of	of	ADP
ejpam-5253	23	8	h	h	NOUN
ejpam-5253	23	9	-	-	PUNCT
ejpam-5253	23	10	lower	low	ADJ
ejpam-5253	23	11	and	and	CCONJ
ejpam-5253	23	12	h	h	NOUN
ejpam-5253	23	13	-	-	PUNCT
ejpam-5253	23	14	upper	upper	ADJ
ejpam-5253	23	15	approximations	approximation	NOUN
ejpam-5253	23	16	,	,	PUNCT
ejpam-5253	23	17	investigate	investigate	VERB
ejpam-5253	23	18	h	h	NOUN
ejpam-5253	23	19	-	-	PUNCT
ejpam-5253	23	20	rough	rough	ADJ
ejpam-5253	23	21	sets	set	NOUN
ejpam-5253	23	22	,	,	PUNCT
ejpam-5253	23	23	compare	compare	VERB
ejpam-5253	23	24	this	this	DET
ejpam-5253	23	25	concept	concept	NOUN
ejpam-5253	23	26	with	with	ADP
ejpam-5253	23	27	rough	rough	ADJ
ejpam-5253	23	28	sets	set	NOUN
ejpam-5253	23	29	,	,	PUNCT
ejpam-5253	23	30	and	and	CCONJ
ejpam-5253	23	31	provide	provide	VERB
ejpam-5253	23	32	some	some	DET
ejpam-5253	23	33	properties	property	NOUN
ejpam-5253	23	34	and	and	CCONJ
ejpam-5253	23	35	examples	example	NOUN
ejpam-5253	23	36	.	.	PUNCT
ejpam-5253	24	1	2	2	X
ejpam-5253	24	2	.	.	NUM
ejpam-5253	24	3	preliminaries	preliminary	NOUN
ejpam-5253	24	4	rough	rough	ADJ
ejpam-5253	24	5	set	set	NOUN
ejpam-5253	24	6	theory	theory	NOUN
ejpam-5253	24	7	finds	find	VERB
ejpam-5253	24	8	its	its	PRON
ejpam-5253	24	9	roots	root	NOUN
ejpam-5253	24	10	in	in	ADP
ejpam-5253	24	11	the	the	DET
ejpam-5253	24	12	necessity	necessity	NOUN
ejpam-5253	24	13	to	to	PART
ejpam-5253	24	14	represent	represent	VERB
ejpam-5253	24	15	subsets	subset	NOUN
ejpam-5253	24	16	of	of	ADP
ejpam-5253	24	17	a	a	DET
ejpam-5253	24	18	universe	universe	NOUN
ejpam-5253	24	19	through	through	ADP
ejpam-5253	24	20	equivalence	equivalence	NOUN
ejpam-5253	24	21	classes	class	NOUN
ejpam-5253	24	22	within	within	ADP
ejpam-5253	24	23	a	a	DET
ejpam-5253	24	24	partition	partition	NOUN
ejpam-5253	24	25	of	of	ADP
ejpam-5253	24	26	that	that	DET
ejpam-5253	24	27	universe	universe	NOUN
ejpam-5253	24	28	,	,	PUNCT
ejpam-5253	24	29	which	which	PRON
ejpam-5253	24	30	defines	define	VERB
ejpam-5253	24	31	a	a	DET
ejpam-5253	24	32	topological	topological	ADJ
ejpam-5253	24	33	space	space	NOUN
ejpam-5253	24	34	,	,	PUNCT
ejpam-5253	24	35	denoted	denote	VERB
ejpam-5253	24	36	as	as	ADP
ejpam-5253	24	37	approximation	approximation	NOUN
ejpam-5253	24	38	space	space	NOUN
ejpam-5253	24	39	a	a	DET
ejpam-5253	24	40	=	=	SYM
ejpam-5253	24	41	(	(	PUNCT
ejpam-5253	24	42	m	m	PROPN
ejpam-5253	24	43	,	,	PUNCT
ejpam-5253	24	44	r	r	NOUN
ejpam-5253	24	45	)	)	PUNCT
ejpam-5253	24	46	.	.	PUNCT
ejpam-5253	25	1	here	here	ADV
ejpam-5253	25	2	,	,	PUNCT
ejpam-5253	25	3	m	m	VERB
ejpam-5253	25	4	denotes	denote	VERB
ejpam-5253	25	5	the	the	DET
ejpam-5253	25	6	universe	universe	NOUN
ejpam-5253	25	7	set	set	VERB
ejpam-5253	25	8	and	and	CCONJ
ejpam-5253	25	9	r	r	NOUN
ejpam-5253	25	10	stands	stand	VERB
ejpam-5253	25	11	for	for	ADP
ejpam-5253	25	12	an	an	DET
ejpam-5253	25	13	equivalence	equivalence	NOUN
ejpam-5253	25	14	relation	relation	NOUN
ejpam-5253	25	15	(	(	PUNCT
ejpam-5253	25	16	[	[	X
ejpam-5253	25	17	8],[13	8],[13	NOUN
ejpam-5253	25	18	]	]	PUNCT
ejpam-5253	25	19	)	)	PUNCT
ejpam-5253	25	20	.	.	PUNCT
ejpam-5253	26	1	the	the	DET
ejpam-5253	26	2	equivalence	equivalence	NOUN
ejpam-5253	26	3	classes	class	NOUN
ejpam-5253	26	4	of	of	ADP
ejpam-5253	26	5	r	r	NOUN
ejpam-5253	26	6	are	be	AUX
ejpam-5253	26	7	referred	refer	VERB
ejpam-5253	26	8	to	to	ADP
ejpam-5253	26	9	as	as	ADP
ejpam-5253	26	10	granules	granule	NOUN
ejpam-5253	26	11	,	,	PUNCT
ejpam-5253	26	12	elementary	elementary	ADJ
ejpam-5253	26	13	sets	set	NOUN
ejpam-5253	26	14	,	,	PUNCT
ejpam-5253	26	15	or	or	CCONJ
ejpam-5253	26	16	blocks	block	NOUN
ejpam-5253	26	17	,	,	PUNCT
ejpam-5253	26	18	denoted	denote	VERB
ejpam-5253	26	19	by	by	ADP
ejpam-5253	26	20	rm	rm	PROPN
ejpam-5253	26	21	⊆	⊆	NUM
ejpam-5253	26	22	m	m	NOUN
ejpam-5253	26	23	for	for	ADP
ejpam-5253	26	24	each	each	DET
ejpam-5253	26	25	m	m	NOUN
ejpam-5253	26	26	∈	∈	NOUN
ejpam-5253	26	27	m	m	NOUN
ejpam-5253	26	28	.	.	PUNCT
ejpam-5253	27	1	within	within	ADP
ejpam-5253	27	2	this	this	DET
ejpam-5253	27	3	approximation	approximation	NOUN
ejpam-5253	27	4	space	space	NOUN
ejpam-5253	27	5	,	,	PUNCT
ejpam-5253	27	6	two	two	NUM
ejpam-5253	27	7	operators	operator	NOUN
ejpam-5253	27	8	are	be	AUX
ejpam-5253	27	9	considered	consider	VERB
ejpam-5253	27	10	:	:	PUNCT
ejpam-5253	27	11	(	(	PUNCT
ejpam-5253	27	12	i	i	NOUN
ejpam-5253	27	13	)	)	PUNCT
ejpam-5253	27	14	r(k	r(k	PROPN
ejpam-5253	27	15	)	)	PUNCT
ejpam-5253	27	16	=	=	PRON
ejpam-5253	27	17	{	{	PUNCT
ejpam-5253	27	18	m	m	VERB
ejpam-5253	27	19	∈	∈	ADJ
ejpam-5253	27	20	m	m	VERB
ejpam-5253	27	21	:	:	PUNCT
ejpam-5253	27	22	rm	rm	NOUN
ejpam-5253	27	23	∩k	∩k	PROPN
ejpam-5253	27	24	̸=	̸=	PROPN
ejpam-5253	27	25	ϕ	ϕ	PROPN
ejpam-5253	27	26	}	}	PUNCT
ejpam-5253	27	27	is	be	AUX
ejpam-5253	27	28	called	call	VERB
ejpam-5253	27	29	upper	upper	ADJ
ejpam-5253	27	30	approximation	approximation	NOUN
ejpam-5253	27	31	of	of	ADP
ejpam-5253	27	32	k	k	PROPN
ejpam-5253	27	33	⊆	⊆	NUM
ejpam-5253	27	34	m	m	NOUN
ejpam-5253	27	35	.	.	PUNCT
ejpam-5253	28	1	(	(	PUNCT
ejpam-5253	28	2	ii	ii	NOUN
ejpam-5253	28	3	)	)	PUNCT
ejpam-5253	28	4	r(k	r(k	PROPN
ejpam-5253	28	5	)	)	PUNCT
ejpam-5253	29	1	=	=	PRON
ejpam-5253	29	2	{	{	PUNCT
ejpam-5253	29	3	m	m	VERB
ejpam-5253	29	4	∈	∈	ADJ
ejpam-5253	29	5	m	m	VERB
ejpam-5253	29	6	:	:	PUNCT
ejpam-5253	29	7	rm	rm	PROPN
ejpam-5253	29	8	⊆	⊆	NUM
ejpam-5253	29	9	k	k	X
ejpam-5253	29	10	}	}	PUNCT
ejpam-5253	29	11	is	be	AUX
ejpam-5253	29	12	called	call	VERB
ejpam-5253	29	13	lower	low	ADJ
ejpam-5253	29	14	approximation	approximation	NOUN
ejpam-5253	29	15	of	of	ADP
ejpam-5253	29	16	k	k	PROPN
ejpam-5253	29	17	⊆	⊆	NUM
ejpam-5253	29	18	m	m	NOUN
ejpam-5253	29	19	.	.	PUNCT
ejpam-5253	30	1	let	let	AUX
ejpam-5253	30	2	posr(k	posr(k	NOUN
ejpam-5253	30	3	)	)	PUNCT
ejpam-5253	30	4	=	=	SYM
ejpam-5253	30	5	r(k	r(k	PROPN
ejpam-5253	30	6	)	)	PUNCT
ejpam-5253	30	7	denote	denote	VERB
ejpam-5253	30	8	the	the	DET
ejpam-5253	30	9	positive	positive	ADJ
ejpam-5253	30	10	region	region	NOUN
ejpam-5253	30	11	of	of	ADP
ejpam-5253	30	12	k	k	PROPN
ejpam-5253	30	13	,	,	PUNCT
ejpam-5253	30	14	negr(k	negr(k	PROPN
ejpam-5253	30	15	)	)	PUNCT
ejpam-5253	30	16	=	=	SYM
ejpam-5253	31	1	m	m	VERB
ejpam-5253	31	2	−	−	PROPN
ejpam-5253	31	3	r(k	r(k	PROPN
ejpam-5253	31	4	)	)	PUNCT
ejpam-5253	31	5	denote	denote	VERB
ejpam-5253	31	6	the	the	DET
ejpam-5253	31	7	negative	negative	ADJ
ejpam-5253	31	8	region	region	NOUN
ejpam-5253	31	9	of	of	ADP
ejpam-5253	31	10	k	k	PROPN
ejpam-5253	31	11	,	,	PUNCT
ejpam-5253	31	12	and	and	CCONJ
ejpam-5253	31	13	bnr(k	bnr(k	PROPN
ejpam-5253	31	14	)	)	PUNCT
ejpam-5253	31	15	=	=	SYM
ejpam-5253	31	16	r(k	r(k	PROPN
ejpam-5253	31	17	)	)	PUNCT
ejpam-5253	31	18	−	−	PROPN
ejpam-5253	32	1	r(k	r(k	PROPN
ejpam-5253	32	2	)	)	PUNCT
ejpam-5253	32	3	denote	denote	VERB
ejpam-5253	32	4	the	the	DET
ejpam-5253	32	5	borderline	borderline	NOUN
ejpam-5253	32	6	region	region	NOUN
ejpam-5253	32	7	of	of	ADP
ejpam-5253	32	8	m	m	PROPN
ejpam-5253	32	9	.	.	PUNCT
ejpam-5253	33	1	the	the	DET
ejpam-5253	33	2	degree	degree	NOUN
ejpam-5253	33	3	of	of	ADP
ejpam-5253	33	4	completeness	completeness	NOUN
ejpam-5253	33	5	can	can	AUX
ejpam-5253	33	6	also	also	ADV
ejpam-5253	33	7	be	be	AUX
ejpam-5253	33	8	characterized	characterize	VERB
ejpam-5253	33	9	by	by	ADP
ejpam-5253	33	10	the	the	DET
ejpam-5253	33	11	accuracy	accuracy	NOUN
ejpam-5253	33	12	measure	measure	NOUN
ejpam-5253	33	13	,	,	PUNCT
ejpam-5253	33	14	in	in	ADP
ejpam-5253	33	15	which	which	PRON
ejpam-5253	33	16	|	|	ADV
ejpam-5253	33	17	r	r	NOUN
ejpam-5253	33	18	|	|	ADV
ejpam-5253	33	19	represents	represent	VERB
ejpam-5253	33	20	the	the	DET
ejpam-5253	33	21	cardinality	cardinality	NOUN
ejpam-5253	33	22	of	of	ADP
ejpam-5253	33	23	set	set	ADJ
ejpam-5253	33	24	r	r	NOUN
ejpam-5253	33	25	as	as	SCONJ
ejpam-5253	33	26	follows	follow	VERB
ejpam-5253	33	27	:	:	PUNCT
ejpam-5253	33	28	αr(k	αr(k	X
ejpam-5253	33	29	)	)	PUNCT
ejpam-5253	34	1	=	=	SYM
ejpam-5253	34	2	|	|	ADP
ejpam-5253	34	3	r(k	r(k	PROPN
ejpam-5253	34	4	)	)	PUNCT
ejpam-5253	34	5	|∣∣	|∣∣	ADP
ejpam-5253	34	6	r(k	r(k	PROPN
ejpam-5253	34	7	)	)	PUNCT
ejpam-5253	34	8	∣∣	∣∣	NUM
ejpam-5253	34	9	,	,	PUNCT
ejpam-5253	34	10	where	where	SCONJ
ejpam-5253	34	11	k	k	PROPN
ejpam-5253	34	12	̸=	̸=	PROPN
ejpam-5253	34	13	ϕ	ϕ	NOUN
ejpam-5253	34	14	accuracy	accuracy	NOUN
ejpam-5253	34	15	measures	measure	NOUN
ejpam-5253	34	16	aim	aim	VERB
ejpam-5253	34	17	to	to	PART
ejpam-5253	34	18	quantify	quantify	VERB
ejpam-5253	34	19	the	the	DET
ejpam-5253	34	20	completeness	completeness	NOUN
ejpam-5253	34	21	of	of	ADP
ejpam-5253	34	22	knowledge	knowledge	NOUN
ejpam-5253	34	23	.	.	PUNCT
ejpam-5253	35	1	αr(k	αr(k	NOUN
ejpam-5253	35	2	)	)	PUNCT
ejpam-5253	36	1	helps	helps	AUX
ejpam-5253	36	2	gauge	gauge	VERB
ejpam-5253	36	3	the	the	DET
ejpam-5253	36	4	size	size	NOUN
ejpam-5253	36	5	of	of	ADP
ejpam-5253	36	6	the	the	DET
ejpam-5253	36	7	boundary	boundary	ADJ
ejpam-5253	36	8	region	region	NOUN
ejpam-5253	36	9	of	of	ADP
ejpam-5253	36	10	datasets	dataset	NOUN
ejpam-5253	36	11	,	,	PUNCT
ejpam-5253	36	12	yet	yet	CCONJ
ejpam-5253	36	13	it	it	PRON
ejpam-5253	36	14	does	do	AUX
ejpam-5253	36	15	n’t	not	PART
ejpam-5253	36	16	readily	readily	ADV
ejpam-5253	36	17	capture	capture	VERB
ejpam-5253	36	18	knowledge	knowledge	NOUN
ejpam-5253	36	19	structure	structure	NOUN
ejpam-5253	36	20	.	.	PUNCT
ejpam-5253	37	1	one	one	NUM
ejpam-5253	37	2	key	key	ADJ
ejpam-5253	37	3	advantage	advantage	NOUN
ejpam-5253	37	4	of	of	ADP
ejpam-5253	37	5	rough	rough	ADJ
ejpam-5253	37	6	set	set	NOUN
ejpam-5253	37	7	theory	theory	NOUN
ejpam-5253	37	8	lies	lie	VERB
ejpam-5253	37	9	in	in	ADP
ejpam-5253	37	10	its	its	PRON
ejpam-5253	37	11	capability	capability	NOUN
ejpam-5253	37	12	to	to	PART
ejpam-5253	37	13	manage	manage	VERB
ejpam-5253	37	14	categories	category	NOUN
ejpam-5253	37	15	that	that	PRON
ejpam-5253	37	16	defy	defy	VERB
ejpam-5253	37	17	sharp	sharp	ADJ
ejpam-5253	37	18	definition	definition	NOUN
ejpam-5253	37	19	within	within	ADP
ejpam-5253	37	20	a	a	DET
ejpam-5253	37	21	knowledge	knowledge	NOUN
ejpam-5253	37	22	base	base	NOUN
ejpam-5253	37	23	.	.	PUNCT
ejpam-5253	38	1	the	the	DET
ejpam-5253	38	2	rough	rough	ADJ
ejpam-5253	38	3	sets	set	NOUN
ejpam-5253	38	4	framework	framework	NOUN
ejpam-5253	38	5	allows	allow	VERB
ejpam-5253	38	6	for	for	ADP
ejpam-5253	38	7	the	the	DET
ejpam-5253	38	8	measurement	measurement	NOUN
ejpam-5253	38	9	of	of	ADP
ejpam-5253	38	10	characteristics	characteristic	NOUN
ejpam-5253	38	11	in	in	ADP
ejpam-5253	38	12	potential	potential	ADJ
ejpam-5253	38	13	datasets	dataset	NOUN
ejpam-5253	38	14	,	,	PUNCT
ejpam-5253	38	15	facilitating	facilitate	VERB
ejpam-5253	38	16	the	the	DET
ejpam-5253	38	17	assessment	assessment	NOUN
ejpam-5253	38	18	of	of	ADP
ejpam-5253	38	19	inexactness	inexactness	NOUN
ejpam-5253	38	20	and	and	CCONJ
ejpam-5253	38	21	expression	expression	NOUN
ejpam-5253	38	22	of	of	ADP
ejpam-5253	38	23	topological	topological	ADJ
ejpam-5253	38	24	imprecision	imprecision	NOUN
ejpam-5253	38	25	characterization	characterization	NOUN
ejpam-5253	38	26	.	.	PUNCT
ejpam-5253	39	1	(	(	PUNCT
ejpam-5253	39	2	i	i	NOUN
ejpam-5253	39	3	)	)	PUNCT
ejpam-5253	39	4	if	if	SCONJ
ejpam-5253	39	5	r(k	r(k	NOUN
ejpam-5253	39	6	)	)	PUNCT
ejpam-5253	39	7	̸=	̸=	PROPN
ejpam-5253	39	8	ϕ	ϕ	PROPN
ejpam-5253	39	9	and	and	CCONJ
ejpam-5253	39	10	r(k	r(k	PROPN
ejpam-5253	39	11	)	)	PUNCT
ejpam-5253	39	12	̸=	̸=	PROPN
ejpam-5253	39	13	m	m	NOUN
ejpam-5253	39	14	,	,	PUNCT
ejpam-5253	39	15	then	then	ADV
ejpam-5253	39	16	k	k	PROPN
ejpam-5253	39	17	is	be	AUX
ejpam-5253	39	18	roughly	roughly	ADV
ejpam-5253	39	19	r	r	NOUN
ejpam-5253	39	20	-	-	ADJ
ejpam-5253	39	21	definable	definable	ADJ
ejpam-5253	39	22	.	.	PUNCT
ejpam-5253	40	1	(	(	PUNCT
ejpam-5253	40	2	ii	ii	NOUN
ejpam-5253	40	3	)	)	PUNCT
ejpam-5253	40	4	if	if	SCONJ
ejpam-5253	40	5	r(k	r(k	PROPN
ejpam-5253	40	6	)	)	PUNCT
ejpam-5253	40	7	=	=	SYM
ejpam-5253	40	8	ϕ	ϕ	PROPN
ejpam-5253	40	9	and	and	CCONJ
ejpam-5253	40	10	r(k	r(k	PROPN
ejpam-5253	40	11	)	)	PUNCT
ejpam-5253	40	12	̸=	̸=	PROPN
ejpam-5253	40	13	m	m	NOUN
ejpam-5253	40	14	,	,	PUNCT
ejpam-5253	40	15	then	then	ADV
ejpam-5253	40	16	k	k	PROPN
ejpam-5253	40	17	is	be	AUX
ejpam-5253	40	18	internally	internally	ADV
ejpam-5253	40	19	r	r	NOUN
ejpam-5253	40	20	-	-	ADJ
ejpam-5253	40	21	undefinable	undefinable	ADJ
ejpam-5253	40	22	.	.	PUNCT
ejpam-5253	41	1	(	(	PUNCT
ejpam-5253	41	2	iii	iii	X
ejpam-5253	41	3	)	)	PUNCT
ejpam-5253	41	4	if	if	SCONJ
ejpam-5253	41	5	r(k	r(k	NOUN
ejpam-5253	41	6	)	)	PUNCT
ejpam-5253	41	7	̸=	̸=	PROPN
ejpam-5253	41	8	ϕ	ϕ	PROPN
ejpam-5253	41	9	and	and	CCONJ
ejpam-5253	41	10	r(k	r(k	PROPN
ejpam-5253	41	11	)	)	PUNCT
ejpam-5253	42	1	=	=	NUM
ejpam-5253	42	2	m	m	PROPN
ejpam-5253	42	3	,	,	PUNCT
ejpam-5253	42	4	then	then	ADV
ejpam-5253	42	5	k	k	PROPN
ejpam-5253	42	6	is	be	AUX
ejpam-5253	42	7	externally	externally	ADV
ejpam-5253	42	8	r	r	NOUN
ejpam-5253	42	9	-	-	ADJ
ejpam-5253	42	10	undefinable	undefinable	ADJ
ejpam-5253	42	11	.	.	PUNCT
ejpam-5253	43	1	(	(	PUNCT
ejpam-5253	43	2	iv	iv	X
ejpam-5253	43	3	)	)	PUNCT
ejpam-5253	43	4	if	if	SCONJ
ejpam-5253	43	5	r(k	r(k	PROPN
ejpam-5253	43	6	)	)	PUNCT
ejpam-5253	43	7	=	=	SYM
ejpam-5253	43	8	ϕ	ϕ	PROPN
ejpam-5253	43	9	and	and	CCONJ
ejpam-5253	43	10	r(k	r(k	PROPN
ejpam-5253	43	11	)	)	PUNCT
ejpam-5253	44	1	=	=	NUM
ejpam-5253	44	2	m	m	PROPN
ejpam-5253	44	3	,	,	PUNCT
ejpam-5253	44	4	then	then	ADV
ejpam-5253	44	5	k	k	PROPN
ejpam-5253	44	6	is	be	AUX
ejpam-5253	44	7	totally	totally	ADV
ejpam-5253	44	8	r	r	NOUN
ejpam-5253	44	9	-	-	ADJ
ejpam-5253	44	10	undefinable	undefinable	ADJ
ejpam-5253	44	11	.	.	PUNCT
ejpam-5253	45	1	we	we	PRON
ejpam-5253	45	2	denote	denote	VERB
ejpam-5253	45	3	the	the	DET
ejpam-5253	45	4	set	set	NOUN
ejpam-5253	45	5	of	of	ADP
ejpam-5253	45	6	all	all	DET
ejpam-5253	45	7	roughly	roughly	ADV
ejpam-5253	45	8	r	r	NOUN
ejpam-5253	45	9	-	-	ADJ
ejpam-5253	45	10	definable	definable	ADJ
ejpam-5253	45	11	(	(	PUNCT
ejpam-5253	45	12	resp	resp	NOUN
ejpam-5253	45	13	.	.	PUNCT
ejpam-5253	46	1	internally	internally	ADV
ejpam-5253	46	2	r	r	X
ejpam-5253	46	3	-	-	ADJ
ejpam-5253	46	4	undefinable	undefinable	ADJ
ejpam-5253	46	5	,	,	PUNCT
ejpam-5253	46	6	externally	externally	ADV
ejpam-5253	46	7	r	r	NOUN
ejpam-5253	46	8	-	-	ADJ
ejpam-5253	46	9	undefinable	undefinable	ADJ
ejpam-5253	46	10	and	and	CCONJ
ejpam-5253	46	11	totally	totally	ADV
ejpam-5253	46	12	r	r	NOUN
ejpam-5253	46	13	-	-	ADJ
ejpam-5253	46	14	undefinable	undefinable	ADJ
ejpam-5253	46	15	)	)	PUNCT
ejpam-5253	46	16	sets	set	NOUN
ejpam-5253	46	17	by	by	ADP
ejpam-5253	46	18	rd(m	rd(m	NOUN
ejpam-5253	46	19	)	)	PUNCT
ejpam-5253	46	20	(	(	PUNCT
ejpam-5253	46	21	resp	resp	NOUN
ejpam-5253	46	22	.	.	PUNCT
ejpam-5253	47	1	iud(m	iud(m	PROPN
ejpam-5253	47	2	)	)	PUNCT
ejpam-5253	47	3	,	,	PUNCT
ejpam-5253	47	4	eud(m	eud(m	PROPN
ejpam-5253	47	5	)	)	PUNCT
ejpam-5253	47	6	and	and	CCONJ
ejpam-5253	47	7	tud(m	tud(m	NOUN
ejpam-5253	47	8	)	)	PUNCT
ejpam-5253	47	9	)	)	PUNCT
ejpam-5253	48	1	(	(	PUNCT
ejpam-5253	48	2	[	[	X
ejpam-5253	48	3	8],[13	8],[13	NOUN
ejpam-5253	48	4	]	]	PUNCT
ejpam-5253	48	5	)	)	PUNCT
ejpam-5253	48	6	.	.	PUNCT
ejpam-5253	49	1	using	use	VERB
ejpam-5253	49	2	αr(k	αr(k	NOUN
ejpam-5253	49	3	)	)	PUNCT
ejpam-5253	49	4	and	and	CCONJ
ejpam-5253	49	5	these	these	DET
ejpam-5253	49	6	classifications	classification	NOUN
ejpam-5253	49	7	,	,	PUNCT
ejpam-5253	49	8	rough	rough	ADJ
ejpam-5253	49	9	sets	set	NOUN
ejpam-5253	49	10	can	can	AUX
ejpam-5253	49	11	be	be	AUX
ejpam-5253	49	12	characterized	characterize	VERB
ejpam-5253	49	13	by	by	ADP
ejpam-5253	49	14	their	their	PRON
ejpam-5253	49	15	boundary	boundary	ADJ
ejpam-5253	49	16	region	region	NOUN
ejpam-5253	49	17	size	size	NOUN
ejpam-5253	49	18	and	and	CCONJ
ejpam-5253	49	19	structure	structure	NOUN
ejpam-5253	49	20	.	.	PUNCT
ejpam-5253	50	1	rough	rough	ADJ
ejpam-5253	50	2	sets	set	NOUN
ejpam-5253	50	3	are	be	AUX
ejpam-5253	50	4	regarded	regard	VERB
ejpam-5253	50	5	as	as	ADP
ejpam-5253	50	6	a	a	DET
ejpam-5253	50	7	specific	specific	ADJ
ejpam-5253	50	8	subset	subset	NOUN
ejpam-5253	50	9	of	of	ADP
ejpam-5253	50	10	relative	relative	ADJ
ejpam-5253	50	11	sets	set	NOUN
ejpam-5253	50	12	and	and	CCONJ
ejpam-5253	50	13	are	be	AUX
ejpam-5253	50	14	incorporated	incorporate	VERB
ejpam-5253	50	15	into	into	ADP
ejpam-5253	50	16	the	the	DET
ejpam-5253	50	17	framework	framework	NOUN
ejpam-5253	50	18	of	of	ADP
ejpam-5253	50	19	belnap	belnap	NOUN
ejpam-5253	50	20	’s	’s	PART
ejpam-5253	50	21	logic	logic	NOUN
ejpam-5253	51	1	[	[	X
ejpam-5253	51	2	10	10	NUM
ejpam-5253	51	3	]	]	PUNCT
ejpam-5253	51	4	.	.	PUNCT
ejpam-5253	52	1	a.	a.	PROPN
ejpam-5253	52	2	al	al	PROPN
ejpam-5253	52	3	-	-	PUNCT
ejpam-5253	52	4	rehili	rehili	NOUN
ejpam-5253	52	5	/	/	SYM
ejpam-5253	52	6	eur	eur	PROPN
ejpam-5253	52	7	.	.	PUNCT
ejpam-5253	53	1	j.	j.	PROPN
ejpam-5253	53	2	pure	pure	PROPN
ejpam-5253	53	3	appl	appl	PROPN
ejpam-5253	53	4	.	.	PROPN
ejpam-5253	53	5	math	math	PROPN
ejpam-5253	53	6	,	,	PUNCT
ejpam-5253	53	7	17	17	NUM
ejpam-5253	53	8	(	(	PUNCT
ejpam-5253	53	9	3	3	NUM
ejpam-5253	53	10	)	)	PUNCT
ejpam-5253	53	11	(	(	PUNCT
ejpam-5253	53	12	2024	2024	NUM
ejpam-5253	53	13	)	)	PUNCT
ejpam-5253	53	14	,	,	PUNCT
ejpam-5253	53	15	1804	1804	NUM
ejpam-5253	53	16	-	-	SYM
ejpam-5253	53	17	1817	1817	NUM
ejpam-5253	53	18	1806	1806	NUM
ejpam-5253	53	19	definition	definition	NOUN
ejpam-5253	53	20	1	1	NUM
ejpam-5253	53	21	(	(	PUNCT
ejpam-5253	53	22	[	[	X
ejpam-5253	53	23	8],[13	8],[13	NOUN
ejpam-5253	53	24	]	]	PUNCT
ejpam-5253	53	25	)	)	PUNCT
ejpam-5253	53	26	.	.	PUNCT
ejpam-5253	54	1	if	if	SCONJ
ejpam-5253	54	2	(	(	PUNCT
ejpam-5253	54	3	m	m	NOUN
ejpam-5253	54	4	,	,	PUNCT
ejpam-5253	54	5	r	r	NOUN
ejpam-5253	54	6	)	)	PUNCT
ejpam-5253	54	7	be	be	AUX
ejpam-5253	54	8	an	an	DET
ejpam-5253	54	9	approximation	approximation	NOUN
ejpam-5253	54	10	space	space	NOUN
ejpam-5253	54	11	and	and	CCONJ
ejpam-5253	54	12	k	k	PROPN
ejpam-5253	54	13	⊆	⊆	NUM
ejpam-5253	54	14	m	m	NOUN
ejpam-5253	54	15	.	.	PUNCT
ejpam-5253	55	1	then	then	ADV
ejpam-5253	55	2	there	there	PRON
ejpam-5253	55	3	are	be	VERB
ejpam-5253	55	4	memberships	membership	NOUN
ejpam-5253	55	5	which	which	PRON
ejpam-5253	55	6	are	be	AUX
ejpam-5253	55	7	defined	define	VERB
ejpam-5253	55	8	by	by	ADP
ejpam-5253	55	9	:	:	PUNCT
ejpam-5253	55	10	(	(	PUNCT
ejpam-5253	55	11	i	i	NOUN
ejpam-5253	55	12	)	)	PUNCT
ejpam-5253	55	13	the	the	DET
ejpam-5253	55	14	strong	strong	ADJ
ejpam-5253	55	15	membership	membership	NOUN
ejpam-5253	55	16	is	be	AUX
ejpam-5253	55	17	denoted	denote	VERB
ejpam-5253	55	18	by	by	ADP
ejpam-5253	55	19	∈	∈	PROPN
ejpam-5253	55	20	,	,	PUNCT
ejpam-5253	55	21	(	(	PUNCT
ejpam-5253	55	22	m∈k	m∈k	NOUN
ejpam-5253	55	23	⇔	⇔	X
ejpam-5253	55	24	m	m	PROPN
ejpam-5253	55	25	∈	∈	PROPN
ejpam-5253	55	26	r(k	r(k	PROPN
ejpam-5253	55	27	)	)	PUNCT
ejpam-5253	55	28	)	)	PUNCT
ejpam-5253	55	29	.	.	PUNCT
ejpam-5253	56	1	(	(	PUNCT
ejpam-5253	56	2	ii	ii	X
ejpam-5253	56	3	)	)	PUNCT
ejpam-5253	56	4	the	the	DET
ejpam-5253	56	5	weak	weak	ADJ
ejpam-5253	56	6	membership	membership	NOUN
ejpam-5253	56	7	is	be	AUX
ejpam-5253	56	8	denoted	denote	VERB
ejpam-5253	56	9	by	by	ADP
ejpam-5253	56	10	∈	∈	PROPN
ejpam-5253	56	11	,	,	PUNCT
ejpam-5253	56	12	(	(	PUNCT
ejpam-5253	56	13	m∈k	m∈k	NOUN
ejpam-5253	56	14	⇔	⇔	X
ejpam-5253	56	15	m	m	PROPN
ejpam-5253	56	16	∈	∈	PROPN
ejpam-5253	56	17	r(k	r(k	PROPN
ejpam-5253	56	18	)	)	PUNCT
ejpam-5253	56	19	)	)	PUNCT
ejpam-5253	56	20	.	.	PUNCT
ejpam-5253	57	1	definition	definition	NOUN
ejpam-5253	57	2	2	2	NUM
ejpam-5253	57	3	(	(	PUNCT
ejpam-5253	57	4	[	[	X
ejpam-5253	57	5	6	6	NUM
ejpam-5253	57	6	]	]	NUM
ejpam-5253	57	7	)	)	PUNCT
ejpam-5253	57	8	.	.	PUNCT
ejpam-5253	58	1	a	a	DET
ejpam-5253	58	2	topological	topological	ADJ
ejpam-5253	58	3	space	space	NOUN
ejpam-5253	58	4	is	be	AUX
ejpam-5253	58	5	defined	define	VERB
ejpam-5253	58	6	as	as	ADP
ejpam-5253	58	7	a	a	DET
ejpam-5253	58	8	pair	pair	NOUN
ejpam-5253	58	9	(	(	PUNCT
ejpam-5253	58	10	m	m	PROPN
ejpam-5253	58	11	,	,	PUNCT
ejpam-5253	58	12	σ	σ	PROPN
ejpam-5253	58	13	)	)	PUNCT
ejpam-5253	58	14	,	,	PUNCT
ejpam-5253	58	15	where	where	SCONJ
ejpam-5253	58	16	m	m	NOUN
ejpam-5253	58	17	is	be	AUX
ejpam-5253	58	18	a	a	DET
ejpam-5253	58	19	set	set	NOUN
ejpam-5253	58	20	and	and	CCONJ
ejpam-5253	58	21	σ	σ	NOUN
ejpam-5253	58	22	is	be	AUX
ejpam-5253	58	23	a	a	DET
ejpam-5253	58	24	family	family	NOUN
ejpam-5253	58	25	of	of	ADP
ejpam-5253	58	26	subsets	subset	NOUN
ejpam-5253	58	27	of	of	ADP
ejpam-5253	58	28	m	m	PRON
ejpam-5253	58	29	satisfying	satisfy	VERB
ejpam-5253	58	30	the	the	DET
ejpam-5253	58	31	following	follow	VERB
ejpam-5253	58	32	conditions	condition	NOUN
ejpam-5253	58	33	:	:	PUNCT
ejpam-5253	58	34	(	(	PUNCT
ejpam-5253	58	35	i	i	NOUN
ejpam-5253	58	36	)	)	PUNCT
ejpam-5253	58	37	ϕ,m	ϕ,m	ADP
ejpam-5253	58	38	∈	∈	PROPN
ejpam-5253	58	39	σ	σ	PROPN
ejpam-5253	58	40	.	.	PUNCT
ejpam-5253	58	41	(	(	PUNCT
ejpam-5253	58	42	ii	ii	NOUN
ejpam-5253	58	43	)	)	PUNCT
ejpam-5253	58	44	σ	σ	PROPN
ejpam-5253	58	45	is	be	AUX
ejpam-5253	58	46	closed	close	VERB
ejpam-5253	58	47	under	under	ADP
ejpam-5253	58	48	arbitrary	arbitrary	ADJ
ejpam-5253	58	49	union	union	NOUN
ejpam-5253	58	50	.	.	PUNCT
ejpam-5253	59	1	(	(	PUNCT
ejpam-5253	59	2	iii	iii	X
ejpam-5253	59	3	)	)	PUNCT
ejpam-5253	59	4	σ	σ	NOUN
ejpam-5253	59	5	is	be	AUX
ejpam-5253	59	6	closed	close	VERB
ejpam-5253	59	7	under	under	ADP
ejpam-5253	59	8	finite	finite	ADJ
ejpam-5253	59	9	intersection	intersection	NOUN
ejpam-5253	59	10	.	.	PUNCT
ejpam-5253	60	1	the	the	DET
ejpam-5253	60	2	elements	element	NOUN
ejpam-5253	60	3	of	of	ADP
ejpam-5253	60	4	m	m	NOUN
ejpam-5253	60	5	are	be	AUX
ejpam-5253	60	6	referred	refer	VERB
ejpam-5253	60	7	to	to	ADP
ejpam-5253	60	8	as	as	ADP
ejpam-5253	60	9	the	the	DET
ejpam-5253	60	10	points	point	NOUN
ejpam-5253	60	11	of	of	ADP
ejpam-5253	60	12	the	the	DET
ejpam-5253	60	13	space	space	NOUN
ejpam-5253	60	14	,	,	PUNCT
ejpam-5253	60	15	while	while	SCONJ
ejpam-5253	60	16	the	the	DET
ejpam-5253	60	17	subsets	subset	NOUN
ejpam-5253	60	18	of	of	ADP
ejpam-5253	60	19	m	m	AUX
ejpam-5253	60	20	belonging	belong	VERB
ejpam-5253	60	21	to	to	ADP
ejpam-5253	60	22	σ	σ	PROPN
ejpam-5253	60	23	are	be	AUX
ejpam-5253	60	24	termed	term	VERB
ejpam-5253	60	25	open	open	ADJ
ejpam-5253	60	26	sets	set	NOUN
ejpam-5253	60	27	within	within	ADP
ejpam-5253	60	28	the	the	DET
ejpam-5253	60	29	space	space	NOUN
ejpam-5253	60	30	.	.	PUNCT
ejpam-5253	61	1	the	the	DET
ejpam-5253	61	2	complements	complement	NOUN
ejpam-5253	61	3	of	of	ADP
ejpam-5253	61	4	these	these	DET
ejpam-5253	61	5	subsets	subset	NOUN
ejpam-5253	61	6	,	,	PUNCT
ejpam-5253	61	7	belonging	belong	VERB
ejpam-5253	61	8	to	to	ADP
ejpam-5253	61	9	the	the	DET
ejpam-5253	61	10	complement	complement	NOUN
ejpam-5253	61	11	of	of	ADP
ejpam-5253	61	12	σ	σ	PROPN
ejpam-5253	61	13	,	,	PUNCT
ejpam-5253	61	14	are	be	AUX
ejpam-5253	61	15	known	know	VERB
ejpam-5253	61	16	as	as	ADP
ejpam-5253	61	17	closed	close	VERB
ejpam-5253	61	18	sets	set	NOUN
ejpam-5253	61	19	within	within	ADP
ejpam-5253	61	20	the	the	DET
ejpam-5253	61	21	space	space	NOUN
ejpam-5253	61	22	.	.	PUNCT
ejpam-5253	62	1	additionally	additionally	ADV
ejpam-5253	62	2	,	,	PUNCT
ejpam-5253	62	3	the	the	DET
ejpam-5253	62	4	family	family	NOUN
ejpam-5253	62	5	σ	σ	NOUN
ejpam-5253	62	6	of	of	ADP
ejpam-5253	62	7	open	open	ADJ
ejpam-5253	62	8	subsets	subset	NOUN
ejpam-5253	62	9	of	of	ADP
ejpam-5253	62	10	m	m	PROPN
ejpam-5253	62	11	is	be	AUX
ejpam-5253	62	12	referred	refer	VERB
ejpam-5253	62	13	to	to	ADP
ejpam-5253	62	14	as	as	ADP
ejpam-5253	62	15	the	the	DET
ejpam-5253	62	16	topology	topology	NOUN
ejpam-5253	62	17	for	for	ADP
ejpam-5253	62	18	m	m	PROPN
ejpam-5253	62	19	.the	.the	PRON
ejpam-5253	62	20	closure	closure	NOUN
ejpam-5253	62	21	of	of	ADP
ejpam-5253	62	22	k	k	PROPN
ejpam-5253	62	23	⊆	⊆	NUM
ejpam-5253	62	24	m	m	NOUN
ejpam-5253	62	25	is	be	AUX
ejpam-5253	62	26	the	the	DET
ejpam-5253	62	27	intersection	intersection	NOUN
ejpam-5253	62	28	of	of	ADP
ejpam-5253	62	29	all	all	DET
ejpam-5253	62	30	closed	closed	ADJ
ejpam-5253	62	31	sets	set	NOUN
ejpam-5253	62	32	containing	contain	VERB
ejpam-5253	62	33	k	k	NOUN
ejpam-5253	62	34	,	,	PUNCT
ejpam-5253	62	35	denoted	denote	VERB
ejpam-5253	62	36	as	as	ADP
ejpam-5253	62	37	(	(	PUNCT
ejpam-5253	62	38	k	k	NOUN
ejpam-5253	62	39	=	=	PUNCT
ejpam-5253	62	40	∩{f	∩{f	PROPN
ejpam-5253	62	41	⊆	⊆	NUM
ejpam-5253	62	42	m	m	NOUN
ejpam-5253	62	43	:	:	PUNCT
ejpam-5253	62	44	f	f	PROPN
ejpam-5253	62	45	is	be	AUX
ejpam-5253	62	46	closed	closed	ADJ
ejpam-5253	62	47	and	and	CCONJ
ejpam-5253	62	48	k	k	PROPN
ejpam-5253	62	49	⊆	⊆	NUM
ejpam-5253	62	50	f	f	NOUN
ejpam-5253	62	51	}	}	PUNCT
ejpam-5253	62	52	)	)	PUNCT
ejpam-5253	62	53	.	.	PUNCT
ejpam-5253	63	1	also	also	ADV
ejpam-5253	63	2	,	,	PUNCT
ejpam-5253	63	3	k	k	PROPN
ejpam-5253	63	4	is	be	AUX
ejpam-5253	63	5	closed	close	VERB
ejpam-5253	63	6	iff	iff	PROPN
ejpam-5253	63	7	k	k	PROPN
ejpam-5253	63	8	=	=	PROPN
ejpam-5253	63	9	k.	k.	PROPN
ejpam-5253	64	1	the	the	DET
ejpam-5253	64	2	interior	interior	PROPN
ejpam-5253	64	3	of	of	ADP
ejpam-5253	64	4	k	k	PROPN
ejpam-5253	64	5	in	in	ADP
ejpam-5253	64	6	m	m	PROPN
ejpam-5253	64	7	is	be	AUX
ejpam-5253	64	8	the	the	DET
ejpam-5253	64	9	union	union	NOUN
ejpam-5253	64	10	of	of	ADP
ejpam-5253	64	11	all	all	DET
ejpam-5253	64	12	open	open	ADJ
ejpam-5253	64	13	subsets	subset	NOUN
ejpam-5253	64	14	of	of	ADP
ejpam-5253	64	15	m	m	PROPN
ejpam-5253	64	16	contained	contain	VERB
ejpam-5253	64	17	in	in	ADP
ejpam-5253	64	18	k	k	PROPN
ejpam-5253	64	19	denoted	denote	VERB
ejpam-5253	64	20	as	as	ADP
ejpam-5253	64	21	(	(	PUNCT
ejpam-5253	64	22	k	k	X
ejpam-5253	64	23	◦	◦	NOUN
ejpam-5253	64	24	=	=	SYM
ejpam-5253	64	25	∪{g	∪{g	PROPN
ejpam-5253	64	26	⊆	⊆	NUM
ejpam-5253	64	27	m	m	NOUN
ejpam-5253	64	28	:	:	PUNCT
ejpam-5253	64	29	g	g	NOUN
ejpam-5253	64	30	is	be	AUX
ejpam-5253	64	31	open	open	ADJ
ejpam-5253	64	32	and	and	CCONJ
ejpam-5253	64	33	g	g	PROPN
ejpam-5253	64	34	⊆	⊆	NUM
ejpam-5253	64	35	k	k	NOUN
ejpam-5253	64	36	}	}	PUNCT
ejpam-5253	64	37	)	)	PUNCT
ejpam-5253	64	38	.	.	PUNCT
ejpam-5253	65	1	additionally	additionally	ADV
ejpam-5253	65	2	,	,	PUNCT
ejpam-5253	65	3	k	k	PROPN
ejpam-5253	65	4	is	be	AUX
ejpam-5253	65	5	open	open	ADJ
ejpam-5253	65	6	iff	iff	PROPN
ejpam-5253	65	7	k	k	PROPN
ejpam-5253	65	8	=	=	PUNCT
ejpam-5253	65	9	k	k	X
ejpam-5253	65	10	◦	◦	NOUN
ejpam-5253	65	11	.	.	PUNCT
ejpam-5253	66	1	the	the	DET
ejpam-5253	66	2	border	border	NOUN
ejpam-5253	66	3	of	of	ADP
ejpam-5253	66	4	k	k	PROPN
ejpam-5253	66	5	⊆	⊆	PROPN
ejpam-5253	66	6	m	m	AUX
ejpam-5253	66	7	denoted	denote	VERB
ejpam-5253	66	8	as	as	ADP
ejpam-5253	66	9	(	(	PUNCT
ejpam-5253	66	10	b(k	b(k	PROPN
ejpam-5253	66	11	)	)	PUNCT
ejpam-5253	66	12	=	=	SYM
ejpam-5253	67	1	k	k	PROPN
ejpam-5253	67	2	\k	\k	NOUN
ejpam-5253	67	3	◦	◦	NOUN
ejpam-5253	67	4	)	)	PUNCT
ejpam-5253	67	5	.	.	PUNCT
ejpam-5253	68	1	a	a	DET
ejpam-5253	68	2	subset	subset	NOUN
ejpam-5253	68	3	k	k	X
ejpam-5253	68	4	is	be	AUX
ejpam-5253	68	5	classified	classify	VERB
ejpam-5253	68	6	as	as	ADP
ejpam-5253	68	7	exact	exact	ADJ
ejpam-5253	68	8	if	if	SCONJ
ejpam-5253	68	9	b(k	b(k	NOUN
ejpam-5253	68	10	)	)	PUNCT
ejpam-5253	69	1	=	=	SYM
ejpam-5253	69	2	ϕ	ϕ	NOUN
ejpam-5253	69	3	;	;	PUNCT
ejpam-5253	69	4	otherwise	otherwise	ADV
ejpam-5253	69	5	,	,	PUNCT
ejpam-5253	69	6	it	it	PRON
ejpam-5253	69	7	’s	’s	AUX
ejpam-5253	69	8	considered	consider	VERB
ejpam-5253	69	9	rough	rough	ADJ
ejpam-5253	69	10	.	.	PUNCT
ejpam-5253	70	1	it	it	PRON
ejpam-5253	70	2	’s	’	VERB
ejpam-5253	70	3	evident	evident	ADJ
ejpam-5253	70	4	that	that	SCONJ
ejpam-5253	70	5	k	k	PROPN
ejpam-5253	70	6	is	be	AUX
ejpam-5253	70	7	exact	exact	ADJ
ejpam-5253	70	8	if	if	SCONJ
ejpam-5253	70	9	and	and	CCONJ
ejpam-5253	70	10	only	only	ADV
ejpam-5253	70	11	if	if	SCONJ
ejpam-5253	70	12	k	k	PROPN
ejpam-5253	70	13	=	=	SYM
ejpam-5253	70	14	k	k	X
ejpam-5253	70	15	◦	◦	NOUN
ejpam-5253	70	16	.	.	PUNCT
ejpam-5253	71	1	in	in	ADP
ejpam-5253	71	2	pawlak	pawlak	ADJ
ejpam-5253	71	3	space	space	NOUN
ejpam-5253	71	4	,	,	PUNCT
ejpam-5253	71	5	a	a	DET
ejpam-5253	71	6	subset	subset	NOUN
ejpam-5253	71	7	k	k	PROPN
ejpam-5253	71	8	⊆	⊆	NUM
ejpam-5253	71	9	m	m	VERB
ejpam-5253	71	10	can	can	AUX
ejpam-5253	71	11	either	either	CCONJ
ejpam-5253	71	12	be	be	AUX
ejpam-5253	71	13	rough	rough	ADJ
ejpam-5253	71	14	or	or	CCONJ
ejpam-5253	71	15	exact	exact	ADJ
ejpam-5253	71	16	.	.	PUNCT
ejpam-5253	72	1	definition	definition	NOUN
ejpam-5253	72	2	3	3	NUM
ejpam-5253	72	3	(	(	PUNCT
ejpam-5253	72	4	[	[	X
ejpam-5253	72	5	1	1	NUM
ejpam-5253	72	6	]	]	NUM
ejpam-5253	72	7	)	)	PUNCT
ejpam-5253	72	8	.	.	PUNCT
ejpam-5253	73	1	a	a	DET
ejpam-5253	73	2	subset	subset	NOUN
ejpam-5253	73	3	k	k	PROPN
ejpam-5253	73	4	of	of	ADP
ejpam-5253	73	5	the	the	DET
ejpam-5253	73	6	topological	topological	ADJ
ejpam-5253	73	7	space	space	NOUN
ejpam-5253	73	8	m	m	VERB
ejpam-5253	73	9	is	be	AUX
ejpam-5253	73	10	termed	term	VERB
ejpam-5253	73	11	h	h	ADJ
ejpam-5253	73	12	-	-	PUNCT
ejpam-5253	73	13	open	open	ADJ
ejpam-5253	73	14	set	set	NOUN
ejpam-5253	73	15	if	if	SCONJ
ejpam-5253	73	16	for	for	ADP
ejpam-5253	73	17	every	every	DET
ejpam-5253	73	18	non	non	ADJ
ejpam-5253	73	19	-	-	ADJ
ejpam-5253	73	20	empty	empty	ADJ
ejpam-5253	73	21	set	set	VERB
ejpam-5253	73	22	h	h	NOUN
ejpam-5253	73	23	∈	∈	PROPN
ejpam-5253	73	24	m	m	VERB
ejpam-5253	73	25	where	where	SCONJ
ejpam-5253	73	26	h	h	NOUN
ejpam-5253	73	27	̸=	̸=	PROPN
ejpam-5253	73	28	m	m	PROPN
ejpam-5253	73	29	and	and	CCONJ
ejpam-5253	73	30	h	h	NOUN
ejpam-5253	73	31	∈	∈	PROPN
ejpam-5253	73	32	σ	σ	PROPN
ejpam-5253	73	33	,	,	PUNCT
ejpam-5253	73	34	k	k	PROPN
ejpam-5253	73	35	⊆	⊆	X
ejpam-5253	73	36	(	(	PUNCT
ejpam-5253	73	37	k	k	NOUN
ejpam-5253	73	38	∪h	∪h	NUM
ejpam-5253	73	39	)	)	PUNCT
ejpam-5253	73	40	◦	◦	NOUN
ejpam-5253	73	41	.the	.the	DET
ejpam-5253	73	42	complement	complement	NOUN
ejpam-5253	73	43	of	of	ADP
ejpam-5253	73	44	the	the	DET
ejpam-5253	73	45	h	h	NOUN
ejpam-5253	73	46	-	-	PUNCT
ejpam-5253	73	47	open	open	ADJ
ejpam-5253	73	48	set	set	NOUN
ejpam-5253	73	49	is	be	AUX
ejpam-5253	73	50	referred	refer	VERB
ejpam-5253	73	51	to	to	ADP
ejpam-5253	73	52	as	as	SCONJ
ejpam-5253	73	53	h	h	NOUN
ejpam-5253	73	54	-	-	PUNCT
ejpam-5253	73	55	closed	closed	ADJ
ejpam-5253	73	56	.	.	PUNCT
ejpam-5253	74	1	we	we	PRON
ejpam-5253	74	2	denoted	denote	VERB
ejpam-5253	74	3	the	the	DET
ejpam-5253	74	4	collection	collection	NOUN
ejpam-5253	74	5	of	of	ADP
ejpam-5253	74	6	all	all	DET
ejpam-5253	74	7	h	h	NOUN
ejpam-5253	74	8	-	-	PUNCT
ejpam-5253	74	9	open	open	ADJ
ejpam-5253	74	10	sets	set	NOUN
ejpam-5253	74	11	of	of	ADP
ejpam-5253	74	12	a	a	DET
ejpam-5253	74	13	topological	topological	ADJ
ejpam-5253	74	14	space	space	NOUN
ejpam-5253	74	15	(	(	PUNCT
ejpam-5253	74	16	m	m	PROPN
ejpam-5253	74	17	,	,	PUNCT
ejpam-5253	74	18	σ	σ	PROPN
ejpam-5253	74	19	)	)	PUNCT
ejpam-5253	74	20	as	as	ADP
ejpam-5253	74	21	σh	σh	PROPN
ejpam-5253	74	22	.	.	PROPN
ejpam-5253	74	23	theorem	theorem	VERB
ejpam-5253	74	24	1	1	NUM
ejpam-5253	74	25	(	(	PUNCT
ejpam-5253	74	26	[	[	X
ejpam-5253	74	27	1	1	NUM
ejpam-5253	74	28	]	]	PUNCT
ejpam-5253	74	29	)	)	PUNCT
ejpam-5253	74	30	.	.	PUNCT
ejpam-5253	75	1	in	in	ADP
ejpam-5253	75	2	any	any	DET
ejpam-5253	75	3	topological	topological	ADJ
ejpam-5253	75	4	space	space	NOUN
ejpam-5253	75	5	(	(	PUNCT
ejpam-5253	75	6	k	k	X
ejpam-5253	75	7	,	,	PUNCT
ejpam-5253	75	8	σ	σ	PROPN
ejpam-5253	75	9	)	)	PUNCT
ejpam-5253	75	10	every	every	DET
ejpam-5253	75	11	open	open	ADJ
ejpam-5253	75	12	set	set	NOUN
ejpam-5253	75	13	is	be	AUX
ejpam-5253	75	14	h	h	NOUN
ejpam-5253	75	15	-	-	PUNCT
ejpam-5253	75	16	open	open	ADJ
ejpam-5253	75	17	set	set	NOUN
ejpam-5253	75	18	.	.	PUNCT
ejpam-5253	76	1	the	the	DET
ejpam-5253	76	2	converse	converse	NOUN
ejpam-5253	76	3	of	of	ADP
ejpam-5253	76	4	the	the	DET
ejpam-5253	76	5	theorem	theorem	NOUN
ejpam-5253	76	6	1	1	NUM
ejpam-5253	76	7	may	may	AUX
ejpam-5253	76	8	not	not	PART
ejpam-5253	76	9	hold	hold	VERB
ejpam-5253	76	10	,	,	PUNCT
ejpam-5253	76	11	as	as	SCONJ
ejpam-5253	76	12	demonstrated	demonstrate	VERB
ejpam-5253	76	13	in	in	ADP
ejpam-5253	76	14	the	the	DET
ejpam-5253	76	15	following	follow	VERB
ejpam-5253	76	16	example	example	NOUN
ejpam-5253	76	17	.	.	PUNCT
ejpam-5253	77	1	example	example	NOUN
ejpam-5253	78	1	1	1	NUM
ejpam-5253	78	2	.	.	PUNCT
ejpam-5253	78	3	let	let	VERB
ejpam-5253	78	4	m	m	VERB
ejpam-5253	78	5	=	=	PRON
ejpam-5253	78	6	{	{	PUNCT
ejpam-5253	78	7	k	k	NOUN
ejpam-5253	78	8	,	,	PUNCT
ejpam-5253	78	9	q	q	X
ejpam-5253	78	10	,	,	PUNCT
ejpam-5253	78	11	s	s	AUX
ejpam-5253	78	12	}	}	PUNCT
ejpam-5253	78	13	with	with	ADP
ejpam-5253	78	14	a	a	DET
ejpam-5253	78	15	topology	topology	NOUN
ejpam-5253	78	16	σ	σ	NOUN
ejpam-5253	78	17	=	=	SYM
ejpam-5253	78	18	{	{	PUNCT
ejpam-5253	78	19	φ	φ	PROPN
ejpam-5253	78	20	,	,	PUNCT
ejpam-5253	78	21	m	m	PRON
ejpam-5253	78	22	,	,	PUNCT
ejpam-5253	78	23	{	{	PUNCT
ejpam-5253	78	24	k	k	X
ejpam-5253	78	25	}	}	PUNCT
ejpam-5253	78	26	,	,	PUNCT
ejpam-5253	78	27	{	{	PUNCT
ejpam-5253	78	28	q	q	X
ejpam-5253	78	29	}	}	PUNCT
ejpam-5253	78	30	,	,	PUNCT
ejpam-5253	78	31	{	{	PUNCT
ejpam-5253	78	32	k	k	NOUN
ejpam-5253	78	33	,	,	PUNCT
ejpam-5253	78	34	q	q	NOUN
ejpam-5253	78	35	}	}	PUNCT
ejpam-5253	78	36	}	}	PUNCT
ejpam-5253	78	37	,	,	PUNCT
ejpam-5253	78	38	then	then	ADV
ejpam-5253	78	39	:	:	PUNCT
ejpam-5253	78	40	σh	σh	PROPN
ejpam-5253	78	41	=	=	PUNCT
ejpam-5253	78	42	{	{	PUNCT
ejpam-5253	78	43	m	m	PROPN
ejpam-5253	78	44	,	,	PUNCT
ejpam-5253	78	45	φ	φ	PROPN
ejpam-5253	78	46	,	,	PUNCT
ejpam-5253	78	47	{	{	PUNCT
ejpam-5253	78	48	k	k	X
ejpam-5253	78	49	}	}	PUNCT
ejpam-5253	78	50	,	,	PUNCT
ejpam-5253	78	51	{	{	PUNCT
ejpam-5253	78	52	q	q	X
ejpam-5253	78	53	}	}	PUNCT
ejpam-5253	78	54	,	,	PUNCT
ejpam-5253	78	55	{	{	PUNCT
ejpam-5253	78	56	s	s	X
ejpam-5253	78	57	}	}	PUNCT
ejpam-5253	78	58	,	,	PUNCT
ejpam-5253	78	59	{	{	PUNCT
ejpam-5253	78	60	k	k	NOUN
ejpam-5253	78	61	,	,	PUNCT
ejpam-5253	78	62	q	q	NOUN
ejpam-5253	78	63	}	}	PUNCT
ejpam-5253	78	64	,	,	PUNCT
ejpam-5253	78	65	{	{	PUNCT
ejpam-5253	78	66	k	k	X
ejpam-5253	78	67	,	,	PUNCT
ejpam-5253	78	68	s	s	PART
ejpam-5253	78	69	}	}	PUNCT
ejpam-5253	78	70	,	,	PUNCT
ejpam-5253	78	71	{	{	PUNCT
ejpam-5253	78	72	q	q	X
ejpam-5253	78	73	,	,	PUNCT
ejpam-5253	78	74	s	s	PART
ejpam-5253	78	75	}	}	PUNCT
ejpam-5253	78	76	}	}	PUNCT
ejpam-5253	78	77	.	.	PUNCT
ejpam-5253	79	1	definition	definition	NOUN
ejpam-5253	79	2	4	4	NUM
ejpam-5253	79	3	(	(	PUNCT
ejpam-5253	79	4	[	[	X
ejpam-5253	79	5	1	1	NUM
ejpam-5253	79	6	]	]	PUNCT
ejpam-5253	79	7	)	)	PUNCT
ejpam-5253	79	8	.	.	PUNCT
ejpam-5253	80	1	(	(	PUNCT
ejpam-5253	80	2	i	i	NOUN
ejpam-5253	80	3	)	)	PUNCT
ejpam-5253	80	4	the	the	DET
ejpam-5253	80	5	interior	interior	NOUN
ejpam-5253	80	6	of	of	ADP
ejpam-5253	80	7	k	k	PROPN
ejpam-5253	80	8	in	in	ADP
ejpam-5253	80	9	m	m	PROPN
ejpam-5253	80	10	is	be	AUX
ejpam-5253	80	11	the	the	DET
ejpam-5253	80	12	union	union	NOUN
ejpam-5253	80	13	of	of	ADP
ejpam-5253	80	14	all	all	DET
ejpam-5253	80	15	h	h	NOUN
ejpam-5253	80	16	-	-	PUNCT
ejpam-5253	80	17	open	open	ADJ
ejpam-5253	80	18	subsets	subset	NOUN
ejpam-5253	80	19	of	of	ADP
ejpam-5253	80	20	m	m	PROPN
ejpam-5253	80	21	contained	contain	VERB
ejpam-5253	80	22	in	in	ADP
ejpam-5253	80	23	k	k	PROPN
ejpam-5253	80	24	denoted	denote	VERB
ejpam-5253	80	25	as	as	ADP
ejpam-5253	80	26	(	(	PUNCT
ejpam-5253	80	27	inth(k	inth(k	PROPN
ejpam-5253	80	28	)	)	PUNCT
ejpam-5253	80	29	=	=	SYM
ejpam-5253	81	1	∪{g	∪{g	PROPN
ejpam-5253	81	2	⊆	⊆	NUM
ejpam-5253	81	3	m	m	NOUN
ejpam-5253	81	4	:	:	PUNCT
ejpam-5253	81	5	g	g	PROPN
ejpam-5253	81	6	is	be	AUX
ejpam-5253	81	7	h	h	NOUN
ejpam-5253	81	8	-	-	PUNCT
ejpam-5253	81	9	open	open	ADJ
ejpam-5253	81	10	and	and	CCONJ
ejpam-5253	81	11	g	g	PROPN
ejpam-5253	81	12	⊆	⊆	NUM
ejpam-5253	81	13	k	k	NOUN
ejpam-5253	81	14	}	}	PUNCT
ejpam-5253	81	15	)	)	PUNCT
ejpam-5253	81	16	.	.	PUNCT
ejpam-5253	82	1	(	(	PUNCT
ejpam-5253	82	2	ii	ii	X
ejpam-5253	82	3	)	)	PUNCT
ejpam-5253	82	4	the	the	DET
ejpam-5253	82	5	subset	subset	NOUN
ejpam-5253	82	6	bh(k	bh(k	PUNCT
ejpam-5253	82	7	)	)	PUNCT
ejpam-5253	82	8	=	=	SYM
ejpam-5253	83	1	k	k	X
ejpam-5253	83	2	\	\	PROPN
ejpam-5253	83	3	inth(k	inth(k	PROPN
ejpam-5253	83	4	)	)	PUNCT
ejpam-5253	83	5	is	be	AUX
ejpam-5253	83	6	said	say	VERB
ejpam-5253	83	7	to	to	PART
ejpam-5253	83	8	be	be	AUX
ejpam-5253	83	9	h	h	NOUN
ejpam-5253	83	10	-	-	PUNCT
ejpam-5253	83	11	border	border	NOUN
ejpam-5253	83	12	of	of	ADP
ejpam-5253	83	13	k.	k.	PROPN
ejpam-5253	83	14	(	(	PUNCT
ejpam-5253	83	15	iii	iii	PROPN
ejpam-5253	83	16	)	)	PUNCT
ejpam-5253	83	17	the	the	DET
ejpam-5253	83	18	subset	subset	NOUN
ejpam-5253	83	19	exth(k	exth(k	NOUN
ejpam-5253	83	20	)	)	PUNCT
ejpam-5253	83	21	=	=	SYM
ejpam-5253	83	22	inth	inth	NOUN
ejpam-5253	83	23	(	(	PUNCT
ejpam-5253	83	24	m	m	PROPN
ejpam-5253	83	25	\k	\k	NOUN
ejpam-5253	83	26	)	)	PUNCT
ejpam-5253	83	27	is	be	AUX
ejpam-5253	83	28	called	call	VERB
ejpam-5253	83	29	h	h	NOUN
ejpam-5253	83	30	-	-	PUNCT
ejpam-5253	83	31	exterior	exterior	NOUN
ejpam-5253	83	32	of	of	ADP
ejpam-5253	83	33	k.	k.	PROPN
ejpam-5253	83	34	a.	a.	PROPN
ejpam-5253	83	35	al	al	PROPN
ejpam-5253	83	36	-	-	PUNCT
ejpam-5253	83	37	rehili	rehili	NOUN
ejpam-5253	83	38	/	/	SYM
ejpam-5253	83	39	eur	eur	PROPN
ejpam-5253	83	40	.	.	PUNCT
ejpam-5253	84	1	j.	j.	PROPN
ejpam-5253	84	2	pure	pure	PROPN
ejpam-5253	84	3	appl	appl	PROPN
ejpam-5253	84	4	.	.	PROPN
ejpam-5253	84	5	math	math	PROPN
ejpam-5253	84	6	,	,	PUNCT
ejpam-5253	84	7	17	17	NUM
ejpam-5253	84	8	(	(	PUNCT
ejpam-5253	84	9	3	3	NUM
ejpam-5253	84	10	)	)	PUNCT
ejpam-5253	84	11	(	(	PUNCT
ejpam-5253	84	12	2024	2024	NUM
ejpam-5253	84	13	)	)	PUNCT
ejpam-5253	84	14	,	,	PUNCT
ejpam-5253	84	15	1804	1804	NUM
ejpam-5253	84	16	-	-	SYM
ejpam-5253	84	17	1817	1817	NUM
ejpam-5253	84	18	1807	1807	NUM
ejpam-5253	84	19	3	3	NUM
ejpam-5253	84	20	.	.	PUNCT
ejpam-5253	84	21	h	h	NOUN
ejpam-5253	84	22	-	-	PUNCT
ejpam-5253	84	23	rough	rough	ADJ
ejpam-5253	84	24	classification	classification	NOUN
ejpam-5253	84	25	in	in	ADP
ejpam-5253	84	26	this	this	DET
ejpam-5253	84	27	section	section	NOUN
ejpam-5253	84	28	we	we	PRON
ejpam-5253	84	29	introduce	introduce	VERB
ejpam-5253	84	30	the	the	DET
ejpam-5253	84	31	h	h	NOUN
ejpam-5253	84	32	-	-	PUNCT
ejpam-5253	84	33	approximation	approximation	NOUN
ejpam-5253	84	34	space	space	NOUN
ejpam-5253	84	35	which	which	PRON
ejpam-5253	84	36	is	be	AUX
ejpam-5253	84	37	a	a	DET
ejpam-5253	84	38	new	new	ADJ
ejpam-5253	84	39	class	class	NOUN
ejpam-5253	84	40	of	of	ADP
ejpam-5253	84	41	approximation	approximation	NOUN
ejpam-5253	84	42	space	space	NOUN
ejpam-5253	84	43	.	.	PUNCT
ejpam-5253	85	1	additionally	additionally	ADV
ejpam-5253	85	2	,	,	PUNCT
ejpam-5253	85	3	we	we	PRON
ejpam-5253	85	4	study	study	VERB
ejpam-5253	85	5	the	the	DET
ejpam-5253	85	6	concepts	concept	NOUN
ejpam-5253	85	7	of	of	ADP
ejpam-5253	85	8	h	h	NOUN
ejpam-5253	85	9	-	-	PUNCT
ejpam-5253	85	10	lower	low	ADJ
ejpam-5253	85	11	approximation	approximation	NOUN
ejpam-5253	85	12	and	and	CCONJ
ejpam-5253	85	13	h	h	NOUN
ejpam-5253	85	14	-	-	PUNCT
ejpam-5253	85	15	upper	upper	ADJ
ejpam-5253	85	16	approximation	approximation	NOUN
ejpam-5253	85	17	and	and	CCONJ
ejpam-5253	85	18	outline	outline	VERB
ejpam-5253	85	19	some	some	DET
ejpam-5253	85	20	properties	property	NOUN
ejpam-5253	85	21	of	of	ADP
ejpam-5253	85	22	h	h	NOUN
ejpam-5253	85	23	-	-	PUNCT
ejpam-5253	85	24	approximation	approximation	NOUN
ejpam-5253	85	25	.	.	PUNCT
ejpam-5253	86	1	definition	definition	NOUN
ejpam-5253	86	2	5	5	NUM
ejpam-5253	86	3	.	.	PUNCT
ejpam-5253	87	1	if	if	SCONJ
ejpam-5253	87	2	m	m	NOUN
ejpam-5253	87	3	is	be	AUX
ejpam-5253	87	4	a	a	DET
ejpam-5253	87	5	finite	finite	ADJ
ejpam-5253	87	6	non	non	ADJ
ejpam-5253	87	7	-	-	ADJ
ejpam-5253	87	8	empty	empty	ADJ
ejpam-5253	87	9	universe	universe	NOUN
ejpam-5253	87	10	.	.	PUNCT
ejpam-5253	88	1	the	the	DET
ejpam-5253	88	2	pair	pair	NOUN
ejpam-5253	88	3	(	(	PUNCT
ejpam-5253	88	4	m	m	PROPN
ejpam-5253	88	5	,	,	PUNCT
ejpam-5253	88	6	rh	rh	PROPN
ejpam-5253	88	7	)	)	PUNCT
ejpam-5253	88	8	is	be	AUX
ejpam-5253	88	9	referred	refer	VERB
ejpam-5253	88	10	to	to	ADP
ejpam-5253	88	11	as	as	ADP
ejpam-5253	88	12	an	an	DET
ejpam-5253	88	13	h	h	NOUN
ejpam-5253	88	14	-	-	PUNCT
ejpam-5253	88	15	approximation	approximation	NOUN
ejpam-5253	88	16	space	space	NOUN
ejpam-5253	88	17	,	,	PUNCT
ejpam-5253	88	18	where	where	SCONJ
ejpam-5253	88	19	rh	rh	PROPN
ejpam-5253	88	20	represents	represent	VERB
ejpam-5253	88	21	a	a	DET
ejpam-5253	88	22	general	general	ADJ
ejpam-5253	88	23	relation	relation	NOUN
ejpam-5253	88	24	used	use	VERB
ejpam-5253	88	25	to	to	PART
ejpam-5253	88	26	generate	generate	VERB
ejpam-5253	88	27	a	a	DET
ejpam-5253	88	28	subbase	subbase	NOUN
ejpam-5253	88	29	for	for	ADP
ejpam-5253	88	30	a	a	DET
ejpam-5253	88	31	topology	topology	NOUN
ejpam-5253	88	32	σ	σ	NOUN
ejpam-5253	88	33	on	on	ADP
ejpam-5253	88	34	m	m	PROPN
ejpam-5253	88	35	and	and	CCONJ
ejpam-5253	88	36	a	a	DET
ejpam-5253	88	37	class	class	NOUN
ejpam-5253	88	38	of	of	ADP
ejpam-5253	88	39	h	h	NOUN
ejpam-5253	88	40	-	-	PUNCT
ejpam-5253	88	41	open	open	ADJ
ejpam-5253	88	42	sets	set	NOUN
ejpam-5253	88	43	σh	σh	PROPN
ejpam-5253	88	44	.	.	PROPN
ejpam-5253	88	45	example	example	NOUN
ejpam-5253	89	1	2	2	NUM
ejpam-5253	89	2	.	.	PUNCT
ejpam-5253	89	3	let	let	VERB
ejpam-5253	89	4	m	m	VERB
ejpam-5253	89	5	=	=	PRON
ejpam-5253	89	6	{	{	PUNCT
ejpam-5253	89	7	k	k	NOUN
ejpam-5253	89	8	,	,	PUNCT
ejpam-5253	89	9	q	q	X
ejpam-5253	89	10	,	,	PUNCT
ejpam-5253	89	11	s	s	PROPN
ejpam-5253	89	12	,	,	PUNCT
ejpam-5253	89	13	t	t	PROPN
ejpam-5253	89	14	}	}	PUNCT
ejpam-5253	89	15	be	be	AUX
ejpam-5253	89	16	a	a	DET
ejpam-5253	89	17	universe	universe	NOUN
ejpam-5253	89	18	and	and	CCONJ
ejpam-5253	89	19	a	a	DET
ejpam-5253	89	20	relation	relation	NOUN
ejpam-5253	89	21	r	r	NOUN
ejpam-5253	89	22	defined	define	VERB
ejpam-5253	89	23	by	by	ADP
ejpam-5253	89	24	r	r	NOUN
ejpam-5253	89	25	=	=	SYM
ejpam-5253	89	26	{	{	PUNCT
ejpam-5253	89	27	(	(	PUNCT
ejpam-5253	89	28	k	k	X
ejpam-5253	89	29	,	,	PUNCT
ejpam-5253	89	30	k	k	NOUN
ejpam-5253	89	31	)	)	PUNCT
ejpam-5253	89	32	,	,	PUNCT
ejpam-5253	89	33	(	(	PUNCT
ejpam-5253	89	34	k	k	X
ejpam-5253	89	35	,	,	PUNCT
ejpam-5253	89	36	s	s	PART
ejpam-5253	89	37	)	)	PUNCT
ejpam-5253	89	38	,	,	PUNCT
ejpam-5253	89	39	(	(	PUNCT
ejpam-5253	89	40	k	k	X
ejpam-5253	89	41	,	,	PUNCT
ejpam-5253	89	42	t	t	PROPN
ejpam-5253	89	43	)	)	PUNCT
ejpam-5253	89	44	,	,	PUNCT
ejpam-5253	89	45	(	(	PUNCT
ejpam-5253	89	46	q	q	X
ejpam-5253	89	47	,	,	PUNCT
ejpam-5253	89	48	q	q	NOUN
ejpam-5253	89	49	)	)	PUNCT
ejpam-5253	89	50	,	,	PUNCT
ejpam-5253	89	51	(	(	PUNCT
ejpam-5253	89	52	q	q	X
ejpam-5253	89	53	,	,	PUNCT
ejpam-5253	89	54	t	t	PROPN
ejpam-5253	89	55	)	)	PUNCT
ejpam-5253	89	56	,	,	PUNCT
ejpam-5253	89	57	(	(	PUNCT
ejpam-5253	89	58	s	s	X
ejpam-5253	89	59	,	,	PUNCT
ejpam-5253	89	60	k	k	NOUN
ejpam-5253	89	61	)	)	PUNCT
ejpam-5253	89	62	,	,	PUNCT
ejpam-5253	89	63	(	(	PUNCT
ejpam-5253	89	64	s	s	X
ejpam-5253	89	65	,	,	PUNCT
ejpam-5253	89	66	q	q	NOUN
ejpam-5253	89	67	)	)	PUNCT
ejpam-5253	89	68	,	,	PUNCT
ejpam-5253	89	69	(	(	PUNCT
ejpam-5253	89	70	s	s	X
ejpam-5253	89	71	,	,	PUNCT
ejpam-5253	89	72	t	t	PROPN
ejpam-5253	89	73	)	)	PUNCT
ejpam-5253	89	74	,	,	PUNCT
ejpam-5253	89	75	(	(	PUNCT
ejpam-5253	89	76	t	t	PROPN
ejpam-5253	89	77	,	,	PUNCT
ejpam-5253	89	78	k	k	NOUN
ejpam-5253	89	79	)	)	PUNCT
ejpam-5253	89	80	}	}	PUNCT
ejpam-5253	89	81	,	,	PUNCT
ejpam-5253	89	82	thus	thus	ADV
ejpam-5253	89	83	kr	kr	PROPN
ejpam-5253	89	84	=	=	SYM
ejpam-5253	89	85	{	{	PUNCT
ejpam-5253	89	86	k	k	PROPN
ejpam-5253	89	87	,	,	PUNCT
ejpam-5253	89	88	s	s	PROPN
ejpam-5253	89	89	,	,	PUNCT
ejpam-5253	89	90	t	t	PROPN
ejpam-5253	89	91	}	}	PUNCT
ejpam-5253	89	92	,	,	PUNCT
ejpam-5253	89	93	qr	qr	PROPN
ejpam-5253	89	94	=	=	PRON
ejpam-5253	89	95	{	{	PUNCT
ejpam-5253	89	96	q	q	PROPN
ejpam-5253	89	97	,	,	PUNCT
ejpam-5253	89	98	t	t	PROPN
ejpam-5253	89	99	}	}	PUNCT
ejpam-5253	89	100	,	,	PUNCT
ejpam-5253	89	101	sr	sr	PROPN
ejpam-5253	89	102	=	=	PRON
ejpam-5253	89	103	{	{	PUNCT
ejpam-5253	89	104	k	k	NOUN
ejpam-5253	89	105	,	,	PUNCT
ejpam-5253	89	106	q	q	NOUN
ejpam-5253	89	107	,	,	PUNCT
ejpam-5253	89	108	t	t	PROPN
ejpam-5253	89	109	}	}	PUNCT
ejpam-5253	89	110	and	and	CCONJ
ejpam-5253	89	111	tr	tr	VERB
ejpam-5253	89	112	=	=	NOUN
ejpam-5253	89	113	{	{	PUNCT
ejpam-5253	89	114	k	k	NOUN
ejpam-5253	89	115	}	}	PUNCT
ejpam-5253	89	116	.	.	PUNCT
ejpam-5253	90	1	consequently	consequently	ADV
ejpam-5253	90	2	,	,	PUNCT
ejpam-5253	90	3	the	the	DET
ejpam-5253	90	4	topology	topology	NOUN
ejpam-5253	90	5	associated	associate	VERB
ejpam-5253	90	6	with	with	ADP
ejpam-5253	90	7	this	this	DET
ejpam-5253	90	8	relation	relation	NOUN
ejpam-5253	90	9	is	be	AUX
ejpam-5253	90	10	σ	σ	NOUN
ejpam-5253	90	11	=	=	PUNCT
ejpam-5253	90	12	{	{	PUNCT
ejpam-5253	90	13	m,ϕ	m,ϕ	NOUN
ejpam-5253	90	14	,	,	PUNCT
ejpam-5253	90	15	{	{	PUNCT
ejpam-5253	90	16	k	k	X
ejpam-5253	90	17	}	}	PUNCT
ejpam-5253	90	18	,	,	PUNCT
ejpam-5253	90	19	{	{	PUNCT
ejpam-5253	90	20	t	t	NOUN
ejpam-5253	90	21	}	}	PUNCT
ejpam-5253	90	22	,	,	PUNCT
ejpam-5253	90	23	{	{	PUNCT
ejpam-5253	90	24	k	k	X
ejpam-5253	90	25	,	,	PUNCT
ejpam-5253	90	26	t	t	PROPN
ejpam-5253	90	27	}	}	PUNCT
ejpam-5253	90	28	,	,	PUNCT
ejpam-5253	90	29	{	{	PUNCT
ejpam-5253	90	30	q	q	NOUN
ejpam-5253	90	31	,	,	PUNCT
ejpam-5253	90	32	t	t	PROPN
ejpam-5253	90	33	}	}	PUNCT
ejpam-5253	90	34	,	,	PUNCT
ejpam-5253	90	35	{	{	PUNCT
ejpam-5253	90	36	k	k	X
ejpam-5253	90	37	,	,	PUNCT
ejpam-5253	90	38	q	q	NOUN
ejpam-5253	90	39	,	,	PUNCT
ejpam-5253	90	40	t	t	PROPN
ejpam-5253	90	41	}	}	PUNCT
ejpam-5253	90	42	,	,	PUNCT
ejpam-5253	90	43	{	{	PUNCT
ejpam-5253	90	44	k	k	X
ejpam-5253	90	45	,	,	PUNCT
ejpam-5253	90	46	s	s	PROPN
ejpam-5253	90	47	,	,	PUNCT
ejpam-5253	90	48	t	t	PROPN
ejpam-5253	90	49	}	}	PUNCT
ejpam-5253	90	50	}	}	PUNCT
ejpam-5253	90	51	and	and	CCONJ
ejpam-5253	90	52	σh	σh	PROPN
ejpam-5253	90	53	=	=	PUNCT
ejpam-5253	90	54	{	{	PUNCT
ejpam-5253	90	55	m,ϕ	m,ϕ	NOUN
ejpam-5253	90	56	,	,	PUNCT
ejpam-5253	90	57	{	{	PUNCT
ejpam-5253	90	58	k	k	X
ejpam-5253	90	59	}	}	PUNCT
ejpam-5253	90	60	,	,	PUNCT
ejpam-5253	90	61	{	{	PUNCT
ejpam-5253	90	62	t	t	NOUN
ejpam-5253	90	63	}	}	PUNCT
ejpam-5253	90	64	,	,	PUNCT
ejpam-5253	90	65	{	{	PUNCT
ejpam-5253	90	66	k	k	X
ejpam-5253	90	67	,	,	PUNCT
ejpam-5253	90	68	t	t	PROPN
ejpam-5253	90	69	}	}	PUNCT
ejpam-5253	90	70	,	,	PUNCT
ejpam-5253	90	71	{	{	PUNCT
ejpam-5253	90	72	q	q	NOUN
ejpam-5253	90	73	,	,	PUNCT
ejpam-5253	90	74	t	t	PROPN
ejpam-5253	90	75	}	}	PUNCT
ejpam-5253	90	76	,	,	PUNCT
ejpam-5253	90	77	{	{	PUNCT
ejpam-5253	90	78	s	s	X
ejpam-5253	90	79	,	,	PUNCT
ejpam-5253	90	80	t	t	PROPN
ejpam-5253	90	81	}	}	PUNCT
ejpam-5253	90	82	,	,	PUNCT
ejpam-5253	90	83	{	{	PUNCT
ejpam-5253	90	84	k	k	X
ejpam-5253	90	85	,	,	PUNCT
ejpam-5253	90	86	s	s	PROPN
ejpam-5253	90	87	,	,	PUNCT
ejpam-5253	90	88	t	t	PROPN
ejpam-5253	90	89	}	}	PUNCT
ejpam-5253	90	90	,	,	PUNCT
ejpam-5253	90	91	{	{	PUNCT
ejpam-5253	90	92	q	q	X
ejpam-5253	90	93	,	,	PUNCT
ejpam-5253	90	94	s	s	PROPN
ejpam-5253	90	95	,	,	PUNCT
ejpam-5253	90	96	t	t	PROPN
ejpam-5253	90	97	}	}	PUNCT
ejpam-5253	90	98	,	,	PUNCT
ejpam-5253	90	99	{	{	PUNCT
ejpam-5253	90	100	k	k	X
ejpam-5253	90	101	,	,	PUNCT
ejpam-5253	90	102	q	q	NOUN
ejpam-5253	90	103	,	,	PUNCT
ejpam-5253	90	104	t	t	PROPN
ejpam-5253	90	105	}	}	PUNCT
ejpam-5253	90	106	}	}	PUNCT
ejpam-5253	90	107	.	.	PUNCT
ejpam-5253	91	1	so	so	ADV
ejpam-5253	91	2	(	(	PUNCT
ejpam-5253	91	3	m	m	PROPN
ejpam-5253	91	4	,	,	PUNCT
ejpam-5253	91	5	rh	rh	PROPN
ejpam-5253	91	6	)	)	PUNCT
ejpam-5253	91	7	is	be	AUX
ejpam-5253	91	8	a	a	DET
ejpam-5253	91	9	h	h	NOUN
ejpam-5253	91	10	-	-	PUNCT
ejpam-5253	91	11	approximation	approximation	NOUN
ejpam-5253	91	12	space	space	NOUN
ejpam-5253	91	13	.	.	PUNCT
ejpam-5253	92	1	example	example	NOUN
ejpam-5253	93	1	3	3	X
ejpam-5253	93	2	.	.	PUNCT
ejpam-5253	93	3	let	let	VERB
ejpam-5253	93	4	m	m	VERB
ejpam-5253	93	5	=	=	PRON
ejpam-5253	93	6	{	{	PUNCT
ejpam-5253	93	7	k	k	NOUN
ejpam-5253	93	8	,	,	PUNCT
ejpam-5253	93	9	q	q	X
ejpam-5253	93	10	,	,	PUNCT
ejpam-5253	93	11	s	s	AUX
ejpam-5253	93	12	}	}	PUNCT
ejpam-5253	93	13	be	be	AUX
ejpam-5253	93	14	a	a	DET
ejpam-5253	93	15	universe	universe	NOUN
ejpam-5253	93	16	and	and	CCONJ
ejpam-5253	93	17	a	a	DET
ejpam-5253	93	18	relation	relation	NOUN
ejpam-5253	93	19	r	r	NOUN
ejpam-5253	93	20	defined	define	VERB
ejpam-5253	93	21	by	by	ADP
ejpam-5253	93	22	kr	kr	PROPN
ejpam-5253	93	23	=	=	SYM
ejpam-5253	93	24	{	{	PUNCT
ejpam-5253	93	25	k	k	NOUN
ejpam-5253	93	26	,	,	PUNCT
ejpam-5253	93	27	q	q	NOUN
ejpam-5253	93	28	}	}	PUNCT
ejpam-5253	93	29	,	,	PUNCT
ejpam-5253	93	30	qr	qr	PROPN
ejpam-5253	93	31	=	=	PRON
ejpam-5253	93	32	{	{	PUNCT
ejpam-5253	93	33	q	q	X
ejpam-5253	93	34	}	}	PUNCT
ejpam-5253	93	35	,	,	PUNCT
ejpam-5253	93	36	sr	sr	PROPN
ejpam-5253	93	37	=	=	PRON
ejpam-5253	93	38	{	{	PUNCT
ejpam-5253	93	39	k	k	NOUN
ejpam-5253	93	40	,	,	PUNCT
ejpam-5253	93	41	q	q	NOUN
ejpam-5253	93	42	}	}	PUNCT
ejpam-5253	93	43	.	.	PUNCT
ejpam-5253	94	1	consequently	consequently	ADV
ejpam-5253	94	2	,	,	PUNCT
ejpam-5253	94	3	the	the	DET
ejpam-5253	94	4	topology	topology	NOUN
ejpam-5253	94	5	associated	associate	VERB
ejpam-5253	94	6	with	with	ADP
ejpam-5253	94	7	this	this	DET
ejpam-5253	94	8	relation	relation	NOUN
ejpam-5253	94	9	is	be	AUX
ejpam-5253	94	10	σ	σ	NOUN
ejpam-5253	94	11	=	=	PUNCT
ejpam-5253	94	12	{	{	PUNCT
ejpam-5253	94	13	m,ϕ	m,ϕ	NOUN
ejpam-5253	94	14	,	,	PUNCT
ejpam-5253	94	15	{	{	PUNCT
ejpam-5253	94	16	q	q	X
ejpam-5253	94	17	}	}	PUNCT
ejpam-5253	94	18	,	,	PUNCT
ejpam-5253	94	19	{	{	PUNCT
ejpam-5253	94	20	k	k	NOUN
ejpam-5253	94	21	,	,	PUNCT
ejpam-5253	94	22	q	q	NOUN
ejpam-5253	94	23	}	}	PUNCT
ejpam-5253	94	24	}	}	PUNCT
ejpam-5253	94	25	and	and	CCONJ
ejpam-5253	94	26	σh	σh	PROPN
ejpam-5253	94	27	=	=	PUNCT
ejpam-5253	94	28	{	{	PUNCT
ejpam-5253	94	29	m,ϕ	m,ϕ	NOUN
ejpam-5253	94	30	,	,	PUNCT
ejpam-5253	94	31	{	{	PUNCT
ejpam-5253	94	32	k	k	X
ejpam-5253	94	33	}	}	PUNCT
ejpam-5253	94	34	,	,	PUNCT
ejpam-5253	94	35	{	{	PUNCT
ejpam-5253	94	36	q	q	X
ejpam-5253	94	37	}	}	PUNCT
ejpam-5253	94	38	,	,	PUNCT
ejpam-5253	94	39	{	{	PUNCT
ejpam-5253	94	40	k	k	NOUN
ejpam-5253	94	41	,	,	PUNCT
ejpam-5253	94	42	q	q	NOUN
ejpam-5253	94	43	}	}	PUNCT
ejpam-5253	94	44	,	,	PUNCT
ejpam-5253	94	45	{	{	PUNCT
ejpam-5253	94	46	k	k	X
ejpam-5253	94	47	,	,	PUNCT
ejpam-5253	94	48	s	s	PART
ejpam-5253	94	49	}	}	PUNCT
ejpam-5253	94	50	}	}	PUNCT
ejpam-5253	94	51	.	.	PUNCT
ejpam-5253	95	1	so	so	ADV
ejpam-5253	95	2	(	(	PUNCT
ejpam-5253	95	3	m	m	PROPN
ejpam-5253	95	4	,	,	PUNCT
ejpam-5253	95	5	rh	rh	PROPN
ejpam-5253	95	6	)	)	PUNCT
ejpam-5253	95	7	is	be	AUX
ejpam-5253	95	8	a	a	DET
ejpam-5253	95	9	happroximation	happroximation	NOUN
ejpam-5253	95	10	space	space	NOUN
ejpam-5253	95	11	.	.	PUNCT
ejpam-5253	96	1	definition	definition	NOUN
ejpam-5253	96	2	6	6	NUM
ejpam-5253	96	3	.	.	PUNCT
ejpam-5253	97	1	if	if	SCONJ
ejpam-5253	97	2	(	(	PUNCT
ejpam-5253	97	3	m	m	PROPN
ejpam-5253	97	4	,	,	PUNCT
ejpam-5253	97	5	rh	rh	PROPN
ejpam-5253	97	6	)	)	PUNCT
ejpam-5253	97	7	is	be	AUX
ejpam-5253	97	8	a	a	DET
ejpam-5253	97	9	h	h	NOUN
ejpam-5253	97	10	-	-	PUNCT
ejpam-5253	97	11	approximation	approximation	NOUN
ejpam-5253	97	12	space	space	NOUN
ejpam-5253	97	13	and	and	CCONJ
ejpam-5253	97	14	k	k	PROPN
ejpam-5253	97	15	is	be	AUX
ejpam-5253	97	16	any	any	DET
ejpam-5253	97	17	non	non	ADJ
ejpam-5253	97	18	-	-	ADJ
ejpam-5253	97	19	empty	empty	ADJ
ejpam-5253	97	20	subset	subset	NOUN
ejpam-5253	97	21	of	of	ADP
ejpam-5253	97	22	m	m	PROPN
ejpam-5253	97	23	.	.	PUNCT
ejpam-5253	98	1	then	then	ADV
ejpam-5253	98	2	we	we	PRON
ejpam-5253	98	3	defined	define	VERB
ejpam-5253	98	4	,	,	PUNCT
ejpam-5253	98	5	(	(	PUNCT
ejpam-5253	98	6	i	i	NOUN
ejpam-5253	98	7	)	)	PUNCT
ejpam-5253	98	8	the	the	DET
ejpam-5253	98	9	h	h	NOUN
ejpam-5253	98	10	-	-	PUNCT
ejpam-5253	98	11	lower	low	ADJ
ejpam-5253	98	12	approximation	approximation	NOUN
ejpam-5253	98	13	,	,	PUNCT
ejpam-5253	98	14	rh(k	rh(k	ADJ
ejpam-5253	98	15	)	)	PUNCT
ejpam-5253	98	16	=	=	PUNCT
ejpam-5253	99	1	∪{h	∪{h	PROPN
ejpam-5253	99	2	∈	∈	NOUN
ejpam-5253	99	3	σh	σh	NOUN
ejpam-5253	99	4	:	:	PUNCT
ejpam-5253	99	5	h	h	PROPN
ejpam-5253	99	6	⊆	⊆	NUM
ejpam-5253	99	7	k	k	NOUN
ejpam-5253	99	8	}	}	PUNCT
ejpam-5253	99	9	.	.	PUNCT
ejpam-5253	100	1	(	(	PUNCT
ejpam-5253	100	2	ii	ii	X
ejpam-5253	100	3	)	)	PUNCT
ejpam-5253	100	4	the	the	DET
ejpam-5253	100	5	h	h	NOUN
ejpam-5253	100	6	-	-	PUNCT
ejpam-5253	100	7	upper	upper	ADJ
ejpam-5253	100	8	approximation	approximation	NOUN
ejpam-5253	100	9	,	,	PUNCT
ejpam-5253	100	10	rh(k	rh(k	ADJ
ejpam-5253	100	11	)	)	PUNCT
ejpam-5253	100	12	=	=	SYM
ejpam-5253	100	13	∩{f	∩{f	NOUN
ejpam-5253	100	14	∈	∈	PROPN
ejpam-5253	100	15	σhc	σhc	NOUN
ejpam-5253	100	16	:	:	PUNCT
ejpam-5253	100	17	f	f	PROPN
ejpam-5253	100	18	⊇	⊇	PROPN
ejpam-5253	100	19	k	k	PROPN
ejpam-5253	100	20	}	}	PUNCT
ejpam-5253	100	21	.	.	PUNCT
ejpam-5253	101	1	definition	definition	NOUN
ejpam-5253	101	2	7	7	NUM
ejpam-5253	101	3	.	.	PUNCT
ejpam-5253	102	1	if	if	SCONJ
ejpam-5253	102	2	(	(	PUNCT
ejpam-5253	102	3	m	m	PROPN
ejpam-5253	102	4	,	,	PUNCT
ejpam-5253	102	5	rh	rh	PROPN
ejpam-5253	102	6	)	)	PUNCT
ejpam-5253	102	7	is	be	AUX
ejpam-5253	102	8	a	a	DET
ejpam-5253	102	9	h	h	NOUN
ejpam-5253	102	10	-	-	PUNCT
ejpam-5253	102	11	approximation	approximation	NOUN
ejpam-5253	102	12	space	space	NOUN
ejpam-5253	102	13	and	and	CCONJ
ejpam-5253	102	14	from	from	ADP
ejpam-5253	102	15	the	the	DET
ejpam-5253	102	16	relation	relation	NOUN
ejpam-5253	102	17	int(k	int(k	NOUN
ejpam-5253	102	18	)	)	PUNCT
ejpam-5253	102	19	⊆	⊆	NUM
ejpam-5253	102	20	inth(k	inth(k	PROPN
ejpam-5253	102	21	)	)	PUNCT
ejpam-5253	102	22	⊆	⊆	NUM
ejpam-5253	102	23	k	k	PROPN
ejpam-5253	102	24	⊆	⊆	NUM
ejpam-5253	102	25	clh(k	clh(k	PROPN
ejpam-5253	102	26	)	)	PUNCT
ejpam-5253	102	27	⊆	⊆	NUM
ejpam-5253	102	28	cl(k	cl(k	NOUN
ejpam-5253	102	29	)	)	PUNCT
ejpam-5253	102	30	,	,	PUNCT
ejpam-5253	102	31	for	for	ADP
ejpam-5253	102	32	any	any	DET
ejpam-5253	102	33	k	k	PROPN
ejpam-5253	102	34	⊆	⊆	NUM
ejpam-5253	102	35	m	m	NOUN
ejpam-5253	102	36	.	.	PUNCT
ejpam-5253	103	1	then	then	ADV
ejpam-5253	103	2	the	the	DET
ejpam-5253	103	3	universe	universe	NOUN
ejpam-5253	103	4	m	m	VERB
ejpam-5253	103	5	can	can	AUX
ejpam-5253	103	6	be	be	AUX
ejpam-5253	103	7	divided	divide	VERB
ejpam-5253	103	8	into	into	ADP
ejpam-5253	103	9	12	12	NUM
ejpam-5253	103	10	regions	region	NOUN
ejpam-5253	103	11	with	with	ADP
ejpam-5253	103	12	respect	respect	NOUN
ejpam-5253	103	13	to	to	ADP
ejpam-5253	103	14	any	any	DET
ejpam-5253	103	15	k	k	PROPN
ejpam-5253	103	16	⊆	⊆	NUM
ejpam-5253	103	17	m	m	NOUN
ejpam-5253	103	18	as	as	SCONJ
ejpam-5253	103	19	follows	follow	VERB
ejpam-5253	103	20	:	:	PUNCT
ejpam-5253	103	21	(	(	PUNCT
ejpam-5253	103	22	i	i	NOUN
ejpam-5253	103	23	)	)	PUNCT
ejpam-5253	103	24	the	the	DET
ejpam-5253	103	25	internal	internal	ADJ
ejpam-5253	103	26	edg	edg	NOUN
ejpam-5253	103	27	of	of	ADP
ejpam-5253	103	28	k	k	PROPN
ejpam-5253	103	29	(	(	PUNCT
ejpam-5253	103	30	[	[	X
ejpam-5253	103	31	12	12	NUM
ejpam-5253	103	32	]	]	PUNCT
ejpam-5253	103	33	)	)	PUNCT
ejpam-5253	103	34	,	,	PUNCT
ejpam-5253	103	35	edg(k	edg(k	PROPN
ejpam-5253	103	36	)	)	PUNCT
ejpam-5253	103	37	=	=	SYM
ejpam-5253	104	1	k	k	PROPN
ejpam-5253	104	2	−r(k	−r(k	NOUN
ejpam-5253	104	3	)	)	PUNCT
ejpam-5253	104	4	.	.	PUNCT
ejpam-5253	105	1	(	(	PUNCT
ejpam-5253	105	2	ii	ii	X
ejpam-5253	105	3	)	)	PUNCT
ejpam-5253	105	4	the	the	DET
ejpam-5253	105	5	h	h	NOUN
ejpam-5253	105	6	-	-	PUNCT
ejpam-5253	105	7	internal	internal	ADJ
ejpam-5253	105	8	edg	edg	NOUN
ejpam-5253	105	9	of	of	ADP
ejpam-5253	105	10	k	k	PROPN
ejpam-5253	105	11	,	,	PUNCT
ejpam-5253	105	12	edg	edg	PROPN
ejpam-5253	105	13	h	h	PROPN
ejpam-5253	105	14	(	(	PUNCT
ejpam-5253	105	15	k	k	NOUN
ejpam-5253	105	16	)	)	PUNCT
ejpam-5253	105	17	=	=	SYM
ejpam-5253	106	1	k	k	PROPN
ejpam-5253	106	2	−rh(k	−rh(k	PROPN
ejpam-5253	106	3	)	)	PUNCT
ejpam-5253	106	4	.	.	PUNCT
ejpam-5253	107	1	(	(	PUNCT
ejpam-5253	107	2	iii	iii	X
ejpam-5253	107	3	)	)	PUNCT
ejpam-5253	107	4	the	the	DET
ejpam-5253	107	5	external	external	ADJ
ejpam-5253	107	6	edg	edg	NOUN
ejpam-5253	107	7	of	of	ADP
ejpam-5253	107	8	k	k	PROPN
ejpam-5253	107	9	(	(	PUNCT
ejpam-5253	107	10	[	[	X
ejpam-5253	107	11	12	12	NUM
ejpam-5253	107	12	]	]	NUM
ejpam-5253	107	13	)	)	PUNCT
ejpam-5253	107	14	,	,	PUNCT
ejpam-5253	107	15	edg(k	edg(k	PROPN
ejpam-5253	107	16	)	)	PUNCT
ejpam-5253	107	17	=	=	SYM
ejpam-5253	108	1	r(k	r(k	ADJ
ejpam-5253	108	2	)	)	PUNCT
ejpam-5253	108	3	−k	−k	NOUN
ejpam-5253	108	4	.	.	PUNCT
ejpam-5253	109	1	(	(	PUNCT
ejpam-5253	109	2	iv	iv	X
ejpam-5253	109	3	)	)	PUNCT
ejpam-5253	109	4	the	the	DET
ejpam-5253	109	5	h	h	NOUN
ejpam-5253	109	6	-	-	PUNCT
ejpam-5253	109	7	external	external	ADJ
ejpam-5253	109	8	edg	edg	NOUN
ejpam-5253	109	9	of	of	ADP
ejpam-5253	109	10	k	k	PROPN
ejpam-5253	109	11	,	,	PUNCT
ejpam-5253	109	12	edgh(k	edgh(k	PROPN
ejpam-5253	109	13	)	)	PUNCT
ejpam-5253	109	14	=	=	SYM
ejpam-5253	109	15	rh(k	rh(k	X
ejpam-5253	109	16	)	)	PUNCT
ejpam-5253	109	17	−k	−k	NOUN
ejpam-5253	109	18	.	.	PUNCT
ejpam-5253	110	1	(	(	PUNCT
ejpam-5253	110	2	v	v	X
ejpam-5253	110	3	)	)	PUNCT
ejpam-5253	110	4	the	the	DET
ejpam-5253	110	5	boundary	boundary	NOUN
ejpam-5253	110	6	of	of	ADP
ejpam-5253	110	7	k	k	PROPN
ejpam-5253	110	8	(	(	PUNCT
ejpam-5253	110	9	[	[	X
ejpam-5253	110	10	12	12	NUM
ejpam-5253	110	11	]	]	NUM
ejpam-5253	110	12	)	)	PUNCT
ejpam-5253	110	13	,	,	PUNCT
ejpam-5253	110	14	b(k	b(k	NOUN
ejpam-5253	110	15	)	)	PUNCT
ejpam-5253	111	1	=	=	SYM
ejpam-5253	111	2	r(k	r(k	ADJ
ejpam-5253	111	3	)	)	PUNCT
ejpam-5253	111	4	−r(k	−r(k	NOUN
ejpam-5253	111	5	)	)	PUNCT
ejpam-5253	111	6	.	.	PUNCT
ejpam-5253	112	1	(	(	PUNCT
ejpam-5253	112	2	vi	vi	X
ejpam-5253	112	3	)	)	PUNCT
ejpam-5253	112	4	the	the	DET
ejpam-5253	112	5	h	h	NOUN
ejpam-5253	112	6	-	-	PUNCT
ejpam-5253	112	7	boundary	boundary	NOUN
ejpam-5253	112	8	of	of	ADP
ejpam-5253	112	9	k	k	PROPN
ejpam-5253	112	10	,	,	PUNCT
ejpam-5253	112	11	bh(k	bh(k	ADJ
ejpam-5253	112	12	)	)	PUNCT
ejpam-5253	112	13	=	=	SYM
ejpam-5253	113	1	rh(k	rh(k	X
ejpam-5253	113	2	)	)	PUNCT
ejpam-5253	113	3	−rh(k	−rh(k	NOUN
ejpam-5253	113	4	)	)	PUNCT
ejpam-5253	113	5	.	.	PUNCT
ejpam-5253	114	1	(	(	PUNCT
ejpam-5253	114	2	vii	vii	PROPN
ejpam-5253	114	3	)	)	PUNCT
ejpam-5253	114	4	the	the	DET
ejpam-5253	114	5	exterior	exterior	NOUN
ejpam-5253	114	6	of	of	ADP
ejpam-5253	114	7	k	k	X
ejpam-5253	114	8	(	(	PUNCT
ejpam-5253	114	9	[	[	X
ejpam-5253	114	10	12	12	NUM
ejpam-5253	114	11	]	]	NUM
ejpam-5253	114	12	)	)	PUNCT
ejpam-5253	114	13	,	,	PUNCT
ejpam-5253	114	14	ext(k	ext(k	PROPN
ejpam-5253	114	15	)	)	PUNCT
ejpam-5253	114	16	=	=	PROPN
ejpam-5253	114	17	m	m	NOUN
ejpam-5253	114	18	−r(k	−r(k	NOUN
ejpam-5253	114	19	)	)	PUNCT
ejpam-5253	114	20	.	.	PUNCT
ejpam-5253	115	1	(	(	PUNCT
ejpam-5253	115	2	viii	viii	NOUN
ejpam-5253	115	3	)	)	PUNCT
ejpam-5253	115	4	the	the	DET
ejpam-5253	115	5	h	h	NOUN
ejpam-5253	115	6	-	-	PUNCT
ejpam-5253	115	7	exterior	exterior	NOUN
ejpam-5253	115	8	of	of	ADP
ejpam-5253	115	9	k	k	NOUN
ejpam-5253	115	10	,	,	PUNCT
ejpam-5253	115	11	exth(k	exth(k	NOUN
ejpam-5253	115	12	)	)	PUNCT
ejpam-5253	115	13	=	=	PROPN
ejpam-5253	115	14	m	m	VERB
ejpam-5253	115	15	−rh(k	−rh(k	NOUN
ejpam-5253	115	16	)	)	PUNCT
ejpam-5253	115	17	.	.	PUNCT
ejpam-5253	116	1	(	(	PUNCT
ejpam-5253	116	2	ix	ix	ADJ
ejpam-5253	116	3	)	)	PUNCT
ejpam-5253	116	4	r(k	r(k	PROPN
ejpam-5253	116	5	)	)	PUNCT
ejpam-5253	116	6	−rh(k	−rh(k	NOUN
ejpam-5253	116	7	)	)	PUNCT
ejpam-5253	116	8	.	.	PUNCT
ejpam-5253	117	1	a.	a.	PROPN
ejpam-5253	117	2	al	al	PROPN
ejpam-5253	117	3	-	-	PUNCT
ejpam-5253	117	4	rehili	rehili	NOUN
ejpam-5253	117	5	/	/	SYM
ejpam-5253	117	6	eur	eur	PROPN
ejpam-5253	117	7	.	.	PUNCT
ejpam-5253	118	1	j.	j.	PROPN
ejpam-5253	118	2	pure	pure	PROPN
ejpam-5253	118	3	appl	appl	PROPN
ejpam-5253	118	4	.	.	PROPN
ejpam-5253	118	5	math	math	PROPN
ejpam-5253	118	6	,	,	PUNCT
ejpam-5253	118	7	17	17	NUM
ejpam-5253	118	8	(	(	PUNCT
ejpam-5253	118	9	3	3	NUM
ejpam-5253	118	10	)	)	PUNCT
ejpam-5253	118	11	(	(	PUNCT
ejpam-5253	118	12	2024	2024	NUM
ejpam-5253	118	13	)	)	PUNCT
ejpam-5253	118	14	,	,	PUNCT
ejpam-5253	118	15	1804	1804	NUM
ejpam-5253	118	16	-	-	SYM
ejpam-5253	118	17	1817	1817	NUM
ejpam-5253	118	18	1808	1808	NUM
ejpam-5253	118	19	(	(	PUNCT
ejpam-5253	118	20	x	x	NOUN
ejpam-5253	118	21	)	)	PUNCT
ejpam-5253	118	22	rh(k	rh(k	ADJ
ejpam-5253	118	23	)	)	PUNCT
ejpam-5253	118	24	−r(k	−r(k	NOUN
ejpam-5253	118	25	)	)	PUNCT
ejpam-5253	118	26	.	.	PUNCT
ejpam-5253	119	1	(	(	PUNCT
ejpam-5253	119	2	xi	xi	NOUN
ejpam-5253	119	3	)	)	PUNCT
ejpam-5253	119	4	rh(k	rh(k	ADJ
ejpam-5253	119	5	)	)	PUNCT
ejpam-5253	119	6	−r(k	−r(k	NOUN
ejpam-5253	119	7	)	)	PUNCT
ejpam-5253	119	8	.	.	PUNCT
ejpam-5253	120	1	(	(	PUNCT
ejpam-5253	120	2	xii	xii	NOUN
ejpam-5253	120	3	)	)	PUNCT
ejpam-5253	120	4	r(k	r(k	PROPN
ejpam-5253	120	5	)	)	PUNCT
ejpam-5253	120	6	−rh(k	−rh(k	NOUN
ejpam-5253	120	7	)	)	PUNCT
ejpam-5253	120	8	.	.	PUNCT
ejpam-5253	121	1	remark	remark	PROPN
ejpam-5253	121	2	1	1	NUM
ejpam-5253	121	3	.	.	PUNCT
ejpam-5253	122	1	in	in	ADP
ejpam-5253	122	2	figure	figure	NOUN
ejpam-5253	122	3	1	1	NUM
ejpam-5253	122	4	,	,	PUNCT
ejpam-5253	122	5	the	the	DET
ejpam-5253	122	6	study	study	NOUN
ejpam-5253	122	7	of	of	ADP
ejpam-5253	122	8	h	h	NOUN
ejpam-5253	122	9	-	-	PUNCT
ejpam-5253	122	10	approximation	approximation	NOUN
ejpam-5253	122	11	spaces	space	NOUN
ejpam-5253	122	12	is	be	AUX
ejpam-5253	122	13	a	a	DET
ejpam-5253	122	14	generalization	generalization	NOUN
ejpam-5253	122	15	of	of	ADP
ejpam-5253	122	16	the	the	DET
ejpam-5253	122	17	study	study	NOUN
ejpam-5253	122	18	of	of	ADP
ejpam-5253	122	19	approximation	approximation	NOUN
ejpam-5253	122	20	spaces	space	NOUN
ejpam-5253	122	21	.	.	PUNCT
ejpam-5253	123	1	this	this	DET
ejpam-5253	123	2	extension	extension	NOUN
ejpam-5253	123	3	is	be	AUX
ejpam-5253	123	4	evident	evident	ADJ
ejpam-5253	123	5	as	as	ADP
ejpam-5253	123	6	elements	element	NOUN
ejpam-5253	123	7	within	within	ADP
ejpam-5253	123	8	the	the	DET
ejpam-5253	123	9	regions	region	NOUN
ejpam-5253	123	10	[	[	X
ejpam-5253	123	11	rh(k	rh(k	X
ejpam-5253	123	12	)	)	PUNCT
ejpam-5253	123	13	−	−	PROPN
ejpam-5253	124	1	r(k	r(k	PROPN
ejpam-5253	124	2	)	)	PUNCT
ejpam-5253	124	3	]	]	PUNCT
ejpam-5253	125	1	are	be	AUX
ejpam-5253	125	2	well	well	ADV
ejpam-5253	125	3	defined	define	VERB
ejpam-5253	125	4	within	within	ADP
ejpam-5253	125	5	k	k	PROPN
ejpam-5253	125	6	,	,	PUNCT
ejpam-5253	125	7	contrasting	contrast	VERB
ejpam-5253	125	8	with	with	ADP
ejpam-5253	125	9	their	their	PRON
ejpam-5253	125	10	undefined	undefined	ADJ
ejpam-5253	125	11	nature	nature	NOUN
ejpam-5253	125	12	in	in	ADP
ejpam-5253	125	13	pawlak	pawlak	ADJ
ejpam-5253	125	14	’s	’s	PART
ejpam-5253	125	15	approximation	approximation	NOUN
ejpam-5253	125	16	spaces	space	NOUN
ejpam-5253	125	17	.	.	PUNCT
ejpam-5253	126	1	furthermore	furthermore	ADV
ejpam-5253	126	2	,	,	PUNCT
ejpam-5253	126	3	elements	element	NOUN
ejpam-5253	126	4	within	within	ADP
ejpam-5253	126	5	the	the	DET
ejpam-5253	126	6	region	region	NOUN
ejpam-5253	126	7	[	[	X
ejpam-5253	126	8	r(k)−rh(k	r(k)−rh(k	X
ejpam-5253	126	9	)	)	PUNCT
ejpam-5253	126	10	]	]	PUNCT
ejpam-5253	126	11	lie	lie	VERB
ejpam-5253	126	12	outside	outside	ADV
ejpam-5253	126	13	of	of	ADP
ejpam-5253	126	14	k	k	PROPN
ejpam-5253	126	15	,	,	PUNCT
ejpam-5253	126	16	addressing	address	VERB
ejpam-5253	126	17	a	a	DET
ejpam-5253	126	18	prior	prior	ADJ
ejpam-5253	126	19	lack	lack	NOUN
ejpam-5253	126	20	of	of	ADP
ejpam-5253	126	21	clarity	clarity	NOUN
ejpam-5253	126	22	in	in	ADP
ejpam-5253	126	23	pawlak	pawlak	ADJ
ejpam-5253	126	24	’s	’s	PART
ejpam-5253	126	25	spaces	space	NOUN
ejpam-5253	126	26	.	.	PUNCT
ejpam-5253	127	1	our	our	PRON
ejpam-5253	127	2	paper	paper	NOUN
ejpam-5253	127	3	involves	involve	VERB
ejpam-5253	127	4	redefining	redefine	VERB
ejpam-5253	127	5	the	the	DET
ejpam-5253	127	6	boundary	boundary	NOUN
ejpam-5253	127	7	of	of	ADP
ejpam-5253	127	8	k	k	PROPN
ejpam-5253	127	9	in	in	ADP
ejpam-5253	127	10	pawlak	pawlak	ADJ
ejpam-5253	127	11	’s	’s	PART
ejpam-5253	127	12	approximation	approximation	NOUN
ejpam-5253	127	13	space	space	NOUN
ejpam-5253	127	14	as	as	ADP
ejpam-5253	127	15	the	the	DET
ejpam-5253	127	16	h	h	NOUN
ejpam-5253	127	17	-	-	PUNCT
ejpam-5253	127	18	boundary	boundary	NOUN
ejpam-5253	127	19	of	of	ADP
ejpam-5253	127	20	k.	k.	PROPN
ejpam-5253	127	21	additionally	additionally	ADV
ejpam-5253	127	22	,	,	PUNCT
ejpam-5253	127	23	we	we	PRON
ejpam-5253	127	24	expand	expand	VERB
ejpam-5253	127	25	the	the	DET
ejpam-5253	127	26	exterior	exterior	NOUN
ejpam-5253	127	27	of	of	ADP
ejpam-5253	127	28	k	k	PROPN
ejpam-5253	127	29	,	,	PUNCT
ejpam-5253	127	30	encompassing	encompass	VERB
ejpam-5253	127	31	elements	element	NOUN
ejpam-5253	127	32	not	not	PART
ejpam-5253	127	33	belonging	belong	VERB
ejpam-5253	127	34	to	to	ADP
ejpam-5253	127	35	k	k	PROPN
ejpam-5253	127	36	,	,	PUNCT
ejpam-5253	127	37	termed	term	VERB
ejpam-5253	127	38	as	as	ADP
ejpam-5253	127	39	the	the	DET
ejpam-5253	127	40	h	h	NOUN
ejpam-5253	127	41	-	-	PUNCT
ejpam-5253	127	42	exterior	exterior	NOUN
ejpam-5253	127	43	of	of	ADP
ejpam-5253	127	44	k.	k.	NOUN
ejpam-5253	127	45	figure	figure	PROPN
ejpam-5253	127	46	1	1	NUM
ejpam-5253	127	47	:	:	PUNCT
ejpam-5253	127	48	representation	representation	NOUN
ejpam-5253	127	49	of	of	ADP
ejpam-5253	127	50	h	h	NOUN
ejpam-5253	127	51	-	-	PUNCT
ejpam-5253	127	52	approximation	approximation	NOUN
ejpam-5253	127	53	spaces	space	NOUN
ejpam-5253	127	54	.	.	PUNCT
ejpam-5253	128	1	proposition	proposition	NOUN
ejpam-5253	128	2	1	1	NUM
ejpam-5253	128	3	.	.	PUNCT
ejpam-5253	129	1	let	let	VERB
ejpam-5253	129	2	(	(	PUNCT
ejpam-5253	129	3	m	m	PROPN
ejpam-5253	129	4	,	,	PUNCT
ejpam-5253	129	5	rh	rh	PROPN
ejpam-5253	129	6	)	)	PUNCT
ejpam-5253	129	7	be	be	VERB
ejpam-5253	129	8	h	h	NOUN
ejpam-5253	129	9	-	-	PUNCT
ejpam-5253	129	10	approximation	approximation	NOUN
ejpam-5253	129	11	spaces	space	NOUN
ejpam-5253	129	12	and	and	CCONJ
ejpam-5253	129	13	k	k	PROPN
ejpam-5253	129	14	⊆	⊆	NUM
ejpam-5253	129	15	m	m	NOUN
ejpam-5253	129	16	,	,	PUNCT
ejpam-5253	129	17	then	then	ADV
ejpam-5253	129	18	the	the	DET
ejpam-5253	129	19	following	following	ADJ
ejpam-5253	129	20	statements	statement	NOUN
ejpam-5253	129	21	hold	hold	VERB
ejpam-5253	129	22	:	:	PUNCT
ejpam-5253	129	23	(	(	PUNCT
ejpam-5253	129	24	i	i	NOUN
ejpam-5253	129	25	)	)	PUNCT
ejpam-5253	129	26	b(k	b(k	PROPN
ejpam-5253	129	27	)	)	PUNCT
ejpam-5253	130	1	=	=	SYM
ejpam-5253	130	2	edg(k	edg(k	PROPN
ejpam-5253	130	3	)	)	PUNCT
ejpam-5253	130	4	∪	∪	ADP
ejpam-5253	130	5	edg(k	edg(k	PROPN
ejpam-5253	130	6	)	)	PUNCT
ejpam-5253	130	7	.	.	PUNCT
ejpam-5253	131	1	(	(	PUNCT
ejpam-5253	131	2	ii	ii	NOUN
ejpam-5253	131	3	)	)	PUNCT
ejpam-5253	131	4	bh(k	bh(k	ADV
ejpam-5253	131	5	)	)	PUNCT
ejpam-5253	131	6	=	=	SYM
ejpam-5253	131	7	edg	edg	PROPN
ejpam-5253	131	8	h	h	NOUN
ejpam-5253	131	9	(	(	PUNCT
ejpam-5253	131	10	k	k	NOUN
ejpam-5253	131	11	)	)	PUNCT
ejpam-5253	131	12	∪	∪	ADP
ejpam-5253	131	13	edgh(k	edgh(k	NOUN
ejpam-5253	131	14	)	)	PUNCT
ejpam-5253	131	15	.	.	PUNCT
ejpam-5253	132	1	(	(	PUNCT
ejpam-5253	132	2	iii	iii	X
ejpam-5253	132	3	)	)	PUNCT
ejpam-5253	132	4	r(k	r(k	PROPN
ejpam-5253	132	5	)	)	PUNCT
ejpam-5253	132	6	−rh(k	−rh(k	NOUN
ejpam-5253	132	7	)	)	PUNCT
ejpam-5253	132	8	=	=	SYM
ejpam-5253	132	9	edg(k	edg(k	PROPN
ejpam-5253	132	10	)	)	PUNCT
ejpam-5253	132	11	∪	∪	ADP
ejpam-5253	132	12	edg	edg	PROPN
ejpam-5253	132	13	h	h	PROPN
ejpam-5253	132	14	(	(	PUNCT
ejpam-5253	132	15	k	k	NOUN
ejpam-5253	132	16	)	)	PUNCT
ejpam-5253	132	17	.	.	PUNCT
ejpam-5253	133	1	(	(	PUNCT
ejpam-5253	133	2	iv	iv	X
ejpam-5253	133	3	)	)	PUNCT
ejpam-5253	133	4	rh(k	rh(k	NOUN
ejpam-5253	133	5	)	)	PUNCT
ejpam-5253	133	6	−r(k	−r(k	NOUN
ejpam-5253	133	7	)	)	PUNCT
ejpam-5253	134	1	=	=	SYM
ejpam-5253	134	2	edgh(k	edgh(k	PROPN
ejpam-5253	134	3	)	)	PUNCT
ejpam-5253	134	4	∪	∪	ADP
ejpam-5253	134	5	edg(k	edg(k	PROPN
ejpam-5253	134	6	)	)	PUNCT
ejpam-5253	134	7	.	.	PUNCT
ejpam-5253	135	1	(	(	PUNCT
ejpam-5253	135	2	v	v	NOUN
ejpam-5253	135	3	)	)	PUNCT
ejpam-5253	135	4	edg(k	edg(k	PROPN
ejpam-5253	135	5	)	)	PUNCT
ejpam-5253	136	1	=	=	SYM
ejpam-5253	136	2	edg	edg	PROPN
ejpam-5253	136	3	h	h	NOUN
ejpam-5253	136	4	(	(	PUNCT
ejpam-5253	136	5	k	k	NOUN
ejpam-5253	136	6	)	)	PUNCT
ejpam-5253	136	7	∪	∪	NOUN
ejpam-5253	136	8	(	(	PUNCT
ejpam-5253	136	9	rh(k	rh(k	ADJ
ejpam-5253	136	10	)	)	PUNCT
ejpam-5253	136	11	−r(k	−r(k	NOUN
ejpam-5253	136	12	)	)	PUNCT
ejpam-5253	136	13	)	)	PUNCT
ejpam-5253	136	14	.	.	PUNCT
ejpam-5253	137	1	(	(	PUNCT
ejpam-5253	137	2	vi	vi	NOUN
ejpam-5253	137	3	)	)	PUNCT
ejpam-5253	137	4	edg(k	edg(k	PROPN
ejpam-5253	137	5	)	)	PUNCT
ejpam-5253	138	1	=	=	SYM
ejpam-5253	138	2	edgh(k	edgh(k	ADJ
ejpam-5253	138	3	)	)	PUNCT
ejpam-5253	138	4	∪	∪	NOUN
ejpam-5253	138	5	(	(	PUNCT
ejpam-5253	138	6	r(k	r(k	PROPN
ejpam-5253	138	7	)	)	PUNCT
ejpam-5253	138	8	−rh(k	−rh(k	NOUN
ejpam-5253	138	9	)	)	PUNCT
ejpam-5253	138	10	)	)	PUNCT
ejpam-5253	138	11	.	.	PUNCT
ejpam-5253	139	1	a.	a.	PROPN
ejpam-5253	139	2	al	al	PROPN
ejpam-5253	139	3	-	-	PUNCT
ejpam-5253	139	4	rehili	rehili	NOUN
ejpam-5253	139	5	/	/	SYM
ejpam-5253	139	6	eur	eur	PROPN
ejpam-5253	139	7	.	.	PUNCT
ejpam-5253	140	1	j.	j.	PROPN
ejpam-5253	140	2	pure	pure	PROPN
ejpam-5253	140	3	appl	appl	PROPN
ejpam-5253	140	4	.	.	PROPN
ejpam-5253	140	5	math	math	PROPN
ejpam-5253	140	6	,	,	PUNCT
ejpam-5253	140	7	17	17	NUM
ejpam-5253	140	8	(	(	PUNCT
ejpam-5253	140	9	3	3	NUM
ejpam-5253	140	10	)	)	PUNCT
ejpam-5253	140	11	(	(	PUNCT
ejpam-5253	140	12	2024	2024	NUM
ejpam-5253	140	13	)	)	PUNCT
ejpam-5253	140	14	,	,	PUNCT
ejpam-5253	140	15	1804	1804	NUM
ejpam-5253	140	16	-	-	SYM
ejpam-5253	140	17	1817	1817	NUM
ejpam-5253	140	18	1809	1809	NUM
ejpam-5253	140	19	proof	proof	NOUN
ejpam-5253	140	20	.	.	PUNCT
ejpam-5253	141	1	(	(	PUNCT
ejpam-5253	141	2	i	i	NOUN
ejpam-5253	141	3	)	)	PUNCT
ejpam-5253	141	4	clear	clear	ADJ
ejpam-5253	141	5	.	.	PUNCT
ejpam-5253	142	1	(	(	PUNCT
ejpam-5253	142	2	ii	ii	X
ejpam-5253	142	3	)	)	PUNCT
ejpam-5253	142	4	it	it	PRON
ejpam-5253	142	5	follows	follow	VERB
ejpam-5253	142	6	from	from	ADP
ejpam-5253	142	7	bh(k	bh(k	NOUN
ejpam-5253	142	8	)	)	PUNCT
ejpam-5253	142	9	=	=	SYM
ejpam-5253	142	10	rh(k	rh(k	X
ejpam-5253	142	11	)	)	PUNCT
ejpam-5253	142	12	−rh(k	−rh(k	NOUN
ejpam-5253	142	13	)	)	PUNCT
ejpam-5253	142	14	=	=	PUNCT
ejpam-5253	142	15	(	(	PUNCT
ejpam-5253	142	16	rh(k	rh(k	X
ejpam-5253	142	17	)	)	PUNCT
ejpam-5253	142	18	−k	−k	NOUN
ejpam-5253	142	19	)	)	PUNCT
ejpam-5253	142	20	∪	∪	NOUN
ejpam-5253	142	21	(	(	PUNCT
ejpam-5253	142	22	k	k	PROPN
ejpam-5253	142	23	−rh(k	−rh(k	PROPN
ejpam-5253	142	24	)	)	PUNCT
ejpam-5253	142	25	)	)	PUNCT
ejpam-5253	143	1	=	=	SYM
ejpam-5253	143	2	edg	edg	PROPN
ejpam-5253	143	3	h	h	NOUN
ejpam-5253	143	4	(	(	PUNCT
ejpam-5253	143	5	k	k	NOUN
ejpam-5253	143	6	)	)	PUNCT
ejpam-5253	143	7	∪	∪	ADP
ejpam-5253	143	8	edgh(k	edgh(k	NOUN
ejpam-5253	143	9	)	)	PUNCT
ejpam-5253	143	10	.	.	PUNCT
ejpam-5253	144	1	(	(	PUNCT
ejpam-5253	144	2	iii	iii	X
ejpam-5253	144	3	)	)	PUNCT
ejpam-5253	144	4	(	(	PUNCT
ejpam-5253	144	5	iv	iv	X
ejpam-5253	144	6	)	)	PUNCT
ejpam-5253	144	7	,	,	PUNCT
ejpam-5253	144	8	(	(	PUNCT
ejpam-5253	144	9	v	v	NOUN
ejpam-5253	144	10	)	)	PUNCT
ejpam-5253	144	11	,	,	PUNCT
ejpam-5253	144	12	and	and	CCONJ
ejpam-5253	144	13	(	(	PUNCT
ejpam-5253	144	14	vi	vi	X
ejpam-5253	144	15	)	)	PUNCT
ejpam-5253	144	16	are	be	AUX
ejpam-5253	144	17	obvious	obvious	ADJ
ejpam-5253	144	18	.	.	PUNCT
ejpam-5253	145	1	definition	definition	NOUN
ejpam-5253	145	2	8	8	NUM
ejpam-5253	145	3	.	.	PUNCT
ejpam-5253	146	1	if	if	SCONJ
ejpam-5253	146	2	(	(	PUNCT
ejpam-5253	146	3	m	m	PROPN
ejpam-5253	146	4	,	,	PUNCT
ejpam-5253	146	5	rh	rh	PROPN
ejpam-5253	146	6	)	)	PUNCT
ejpam-5253	146	7	be	be	VERB
ejpam-5253	146	8	a	a	DET
ejpam-5253	146	9	h	h	NOUN
ejpam-5253	146	10	-	-	PUNCT
ejpam-5253	146	11	approximation	approximation	NOUN
ejpam-5253	146	12	space	space	NOUN
ejpam-5253	146	13	and	and	CCONJ
ejpam-5253	146	14	k	k	PROPN
ejpam-5253	146	15	⊆	⊆	NUM
ejpam-5253	146	16	m	m	NOUN
ejpam-5253	146	17	.	.	PUNCT
ejpam-5253	147	1	then	then	ADV
ejpam-5253	147	2	there	there	PRON
ejpam-5253	147	3	are	be	VERB
ejpam-5253	147	4	memberships	membership	NOUN
ejpam-5253	147	5	which	which	PRON
ejpam-5253	147	6	are	be	AUX
ejpam-5253	147	7	defined	define	VERB
ejpam-5253	147	8	by	by	ADP
ejpam-5253	147	9	:	:	PUNCT
ejpam-5253	147	10	(	(	PUNCT
ejpam-5253	147	11	i	i	NOUN
ejpam-5253	147	12	)	)	PUNCT
ejpam-5253	147	13	the	the	DET
ejpam-5253	147	14	h	h	NOUN
ejpam-5253	147	15	-	-	PUNCT
ejpam-5253	147	16	strong	strong	ADJ
ejpam-5253	147	17	membership	membership	NOUN
ejpam-5253	147	18	is	be	AUX
ejpam-5253	147	19	denoted	denote	VERB
ejpam-5253	147	20	by	by	ADP
ejpam-5253	147	21	∈h	∈h	NOUN
ejpam-5253	147	22	,	,	PUNCT
ejpam-5253	147	23	(	(	PUNCT
ejpam-5253	147	24	m	m	PROPN
ejpam-5253	147	25	∈h	∈h	NOUN
ejpam-5253	147	26	k	k	PROPN
ejpam-5253	147	27	⇔	⇔	PROPN
ejpam-5253	147	28	m	m	PROPN
ejpam-5253	147	29	∈	∈	PROPN
ejpam-5253	147	30	rh(k	rh(k	NOUN
ejpam-5253	147	31	)	)	PUNCT
ejpam-5253	147	32	)	)	PUNCT
ejpam-5253	147	33	.	.	PUNCT
ejpam-5253	148	1	(	(	PUNCT
ejpam-5253	148	2	ii	ii	X
ejpam-5253	148	3	)	)	PUNCT
ejpam-5253	148	4	the	the	DET
ejpam-5253	148	5	h	h	NOUN
ejpam-5253	148	6	-	-	PUNCT
ejpam-5253	148	7	weak	weak	ADJ
ejpam-5253	148	8	membership	membership	NOUN
ejpam-5253	148	9	is	be	AUX
ejpam-5253	148	10	denoted	denote	VERB
ejpam-5253	148	11	by	by	ADP
ejpam-5253	148	12	∈h	∈h	NOUN
ejpam-5253	148	13	,	,	PUNCT
ejpam-5253	148	14	(	(	PUNCT
ejpam-5253	148	15	m	m	PROPN
ejpam-5253	148	16	∈h	∈h	NOUN
ejpam-5253	148	17	k	k	PROPN
ejpam-5253	148	18	⇔	⇔	PROPN
ejpam-5253	148	19	m	m	PROPN
ejpam-5253	148	20	∈	∈	PROPN
ejpam-5253	148	21	rh(k	rh(k	NOUN
ejpam-5253	148	22	)	)	PUNCT
ejpam-5253	148	23	)	)	PUNCT
ejpam-5253	148	24	.	.	PUNCT
ejpam-5253	149	1	remark	remark	PROPN
ejpam-5253	149	2	2	2	NUM
ejpam-5253	149	3	.	.	PUNCT
ejpam-5253	149	4	based	base	VERB
ejpam-5253	149	5	on	on	ADP
ejpam-5253	149	6	the	the	DET
ejpam-5253	149	7	definition	definition	NOUN
ejpam-5253	149	8	8	8	NUM
ejpam-5253	149	9	.	.	PUNCT
ejpam-5253	150	1	we	we	PRON
ejpam-5253	150	2	can	can	AUX
ejpam-5253	150	3	be	be	AUX
ejpam-5253	150	4	written	write	VERB
ejpam-5253	150	5	h	h	NOUN
ejpam-5253	150	6	-	-	PUNCT
ejpam-5253	150	7	lower	low	ADJ
ejpam-5253	150	8	and	and	CCONJ
ejpam-5253	150	9	h	h	ADJ
ejpam-5253	150	10	-	-	PUNCT
ejpam-5253	150	11	upper	upper	ADJ
ejpam-5253	150	12	approximations	approximation	NOUN
ejpam-5253	150	13	of	of	ADP
ejpam-5253	150	14	a	a	DET
ejpam-5253	150	15	set	set	NOUN
ejpam-5253	150	16	k	k	PROPN
ejpam-5253	150	17	⊆	⊆	NUM
ejpam-5253	150	18	m	m	NOUN
ejpam-5253	150	19	as	as	ADP
ejpam-5253	150	20	(	(	PUNCT
ejpam-5253	150	21	i	i	NOUN
ejpam-5253	150	22	)	)	PUNCT
ejpam-5253	150	23	rh(k	rh(k	NOUN
ejpam-5253	150	24	)	)	PUNCT
ejpam-5253	150	25	=	=	PRON
ejpam-5253	151	1	{	{	PUNCT
ejpam-5253	152	1	m	m	VERB
ejpam-5253	152	2	∈	∈	ADJ
ejpam-5253	152	3	k	k	NOUN
ejpam-5253	152	4	:	:	PUNCT
ejpam-5253	152	5	m	m	VERB
ejpam-5253	152	6	∈h	∈h	NOUN
ejpam-5253	152	7	k	k	NOUN
ejpam-5253	152	8	}	}	PUNCT
ejpam-5253	152	9	.	.	PUNCT
ejpam-5253	153	1	(	(	PUNCT
ejpam-5253	153	2	ii	ii	NOUN
ejpam-5253	153	3	)	)	PUNCT
ejpam-5253	153	4	rh(k	rh(k	NOUN
ejpam-5253	153	5	)	)	PUNCT
ejpam-5253	153	6	=	=	PRON
ejpam-5253	154	1	{	{	PUNCT
ejpam-5253	155	1	m	m	VERB
ejpam-5253	155	2	∈	∈	ADJ
ejpam-5253	155	3	k	k	NOUN
ejpam-5253	155	4	:	:	PUNCT
ejpam-5253	155	5	m	m	VERB
ejpam-5253	155	6	∈h	∈h	NOUN
ejpam-5253	155	7	k	k	NOUN
ejpam-5253	155	8	}	}	PUNCT
ejpam-5253	155	9	.	.	PUNCT
ejpam-5253	156	1	proposition	proposition	NOUN
ejpam-5253	156	2	2	2	NUM
ejpam-5253	156	3	.	.	PUNCT
ejpam-5253	157	1	if	if	SCONJ
ejpam-5253	157	2	(	(	PUNCT
ejpam-5253	157	3	m	m	PROPN
ejpam-5253	157	4	,	,	PUNCT
ejpam-5253	157	5	rh	rh	PROPN
ejpam-5253	157	6	)	)	PUNCT
ejpam-5253	157	7	is	be	AUX
ejpam-5253	157	8	an	an	DET
ejpam-5253	157	9	h	h	NOUN
ejpam-5253	157	10	-	-	PUNCT
ejpam-5253	157	11	approximation	approximation	NOUN
ejpam-5253	157	12	space	space	NOUN
ejpam-5253	157	13	and	and	CCONJ
ejpam-5253	157	14	k	k	PROPN
ejpam-5253	157	15	⊆	⊆	NUM
ejpam-5253	157	16	m	m	NOUN
ejpam-5253	157	17	.	.	PUNCT
ejpam-5253	158	1	then	then	ADV
ejpam-5253	158	2	(	(	PUNCT
ejpam-5253	158	3	i	i	NOUN
ejpam-5253	158	4	)	)	PUNCT
ejpam-5253	158	5	m	m	VERB
ejpam-5253	158	6	∈	∈	PROPN
ejpam-5253	158	7	k	k	PROPN
ejpam-5253	158	8	⇒	⇒	PROPN
ejpam-5253	158	9	m	m	VERB
ejpam-5253	158	10	∈h	∈h	PROPN
ejpam-5253	158	11	k.	k.	PROPN
ejpam-5253	158	12	(	(	PUNCT
ejpam-5253	158	13	ii	ii	PROPN
ejpam-5253	158	14	)	)	PUNCT
ejpam-5253	158	15	m	m	PROPN
ejpam-5253	158	16	∈h	∈h	NOUN
ejpam-5253	158	17	k	k	PROPN
ejpam-5253	158	18	⇒	⇒	PROPN
ejpam-5253	158	19	m	m	VERB
ejpam-5253	158	20	∈	∈	PROPN
ejpam-5253	158	21	k.	k.	NOUN
ejpam-5253	159	1	the	the	DET
ejpam-5253	159	2	converse	converse	NOUN
ejpam-5253	159	3	of	of	ADP
ejpam-5253	159	4	proposition	proposition	NOUN
ejpam-5253	159	5	2	2	NUM
ejpam-5253	159	6	may	may	AUX
ejpam-5253	159	7	not	not	PART
ejpam-5253	159	8	be	be	AUX
ejpam-5253	159	9	true	true	ADJ
ejpam-5253	159	10	in	in	ADP
ejpam-5253	159	11	general	general	ADJ
ejpam-5253	159	12	as	as	SCONJ
ejpam-5253	159	13	seen	see	VERB
ejpam-5253	159	14	in	in	ADP
ejpam-5253	159	15	the	the	DET
ejpam-5253	159	16	following	follow	VERB
ejpam-5253	159	17	example	example	NOUN
ejpam-5253	159	18	example	example	NOUN
ejpam-5253	159	19	4	4	X
ejpam-5253	159	20	.	.	PUNCT
ejpam-5253	160	1	let	let	VERB
ejpam-5253	160	2	m	m	VERB
ejpam-5253	160	3	=	=	PRON
ejpam-5253	160	4	{	{	PUNCT
ejpam-5253	160	5	k	k	NOUN
ejpam-5253	160	6	,	,	PUNCT
ejpam-5253	160	7	q	q	X
ejpam-5253	160	8	,	,	PUNCT
ejpam-5253	160	9	s	s	PROPN
ejpam-5253	160	10	,	,	PUNCT
ejpam-5253	160	11	t	t	PROPN
ejpam-5253	160	12	}	}	PUNCT
ejpam-5253	160	13	be	be	AUX
ejpam-5253	160	14	a	a	DET
ejpam-5253	160	15	universe	universe	NOUN
ejpam-5253	160	16	and	and	CCONJ
ejpam-5253	160	17	a	a	DET
ejpam-5253	160	18	relation	relation	NOUN
ejpam-5253	160	19	r	r	NOUN
ejpam-5253	160	20	defined	define	VERB
ejpam-5253	160	21	by	by	ADP
ejpam-5253	160	22	r	r	NOUN
ejpam-5253	160	23	=	=	SYM
ejpam-5253	160	24	{	{	PUNCT
ejpam-5253	160	25	(	(	PUNCT
ejpam-5253	160	26	k	k	X
ejpam-5253	160	27	,	,	PUNCT
ejpam-5253	160	28	k	k	NOUN
ejpam-5253	160	29	)	)	PUNCT
ejpam-5253	160	30	,	,	PUNCT
ejpam-5253	160	31	(	(	PUNCT
ejpam-5253	160	32	t	t	PROPN
ejpam-5253	160	33	,	,	PUNCT
ejpam-5253	160	34	s	s	PART
ejpam-5253	160	35	)	)	PUNCT
ejpam-5253	160	36	,	,	PUNCT
ejpam-5253	160	37	(	(	PUNCT
ejpam-5253	160	38	t	t	PROPN
ejpam-5253	160	39	,	,	PUNCT
ejpam-5253	160	40	t	t	PROPN
ejpam-5253	160	41	)	)	PUNCT
ejpam-5253	160	42	,	,	PUNCT
ejpam-5253	160	43	(	(	PUNCT
ejpam-5253	160	44	s	s	X
ejpam-5253	160	45	,	,	PUNCT
ejpam-5253	160	46	k	k	NOUN
ejpam-5253	160	47	)	)	PUNCT
ejpam-5253	160	48	,	,	PUNCT
ejpam-5253	160	49	(	(	PUNCT
ejpam-5253	160	50	s	s	X
ejpam-5253	160	51	,	,	PUNCT
ejpam-5253	160	52	t	t	PROPN
ejpam-5253	160	53	)	)	PUNCT
ejpam-5253	160	54	,	,	PUNCT
ejpam-5253	160	55	(	(	PUNCT
ejpam-5253	160	56	s	s	X
ejpam-5253	160	57	,	,	PUNCT
ejpam-5253	160	58	s	s	PART
ejpam-5253	160	59	)	)	PUNCT
ejpam-5253	160	60	}	}	PUNCT
ejpam-5253	160	61	,	,	PUNCT
ejpam-5253	160	62	thus	thus	ADV
ejpam-5253	160	63	kr	kr	PROPN
ejpam-5253	160	64	=	=	SYM
ejpam-5253	160	65	{	{	PUNCT
ejpam-5253	160	66	k	k	NOUN
ejpam-5253	160	67	}	}	PUNCT
ejpam-5253	160	68	,	,	PUNCT
ejpam-5253	160	69	qr	qr	PROPN
ejpam-5253	160	70	=	=	SYM
ejpam-5253	160	71	ϕ	ϕ	PROPN
ejpam-5253	160	72	,	,	PUNCT
ejpam-5253	160	73	sr	sr	PROPN
ejpam-5253	160	74	=	=	PRON
ejpam-5253	160	75	{	{	PUNCT
ejpam-5253	160	76	k	k	PROPN
ejpam-5253	160	77	,	,	PUNCT
ejpam-5253	160	78	s	s	PROPN
ejpam-5253	160	79	,	,	PUNCT
ejpam-5253	160	80	t	t	PROPN
ejpam-5253	160	81	}	}	PUNCT
ejpam-5253	160	82	and	and	CCONJ
ejpam-5253	160	83	tr	tr	VERB
ejpam-5253	160	84	=	=	NOUN
ejpam-5253	160	85	{	{	PUNCT
ejpam-5253	160	86	s	s	PROPN
ejpam-5253	160	87	,	,	PUNCT
ejpam-5253	160	88	t	t	PROPN
ejpam-5253	160	89	}	}	PUNCT
ejpam-5253	160	90	.	.	PUNCT
ejpam-5253	161	1	consequently	consequently	ADV
ejpam-5253	161	2	,	,	PUNCT
ejpam-5253	161	3	the	the	DET
ejpam-5253	161	4	topology	topology	NOUN
ejpam-5253	161	5	associated	associate	VERB
ejpam-5253	161	6	with	with	ADP
ejpam-5253	161	7	this	this	DET
ejpam-5253	161	8	relation	relation	NOUN
ejpam-5253	161	9	is	be	AUX
ejpam-5253	161	10	σ	σ	NOUN
ejpam-5253	161	11	=	=	PUNCT
ejpam-5253	161	12	{	{	PUNCT
ejpam-5253	161	13	m,ϕ	m,ϕ	NOUN
ejpam-5253	161	14	,	,	PUNCT
ejpam-5253	161	15	{	{	PUNCT
ejpam-5253	161	16	k	k	X
ejpam-5253	161	17	}	}	PUNCT
ejpam-5253	161	18	,	,	PUNCT
ejpam-5253	161	19	{	{	PUNCT
ejpam-5253	161	20	s	s	X
ejpam-5253	161	21	,	,	PUNCT
ejpam-5253	161	22	t	t	PROPN
ejpam-5253	161	23	}	}	PUNCT
ejpam-5253	161	24	,	,	PUNCT
ejpam-5253	161	25	{	{	PUNCT
ejpam-5253	161	26	k	k	X
ejpam-5253	161	27	,	,	PUNCT
ejpam-5253	161	28	s	s	PROPN
ejpam-5253	161	29	,	,	PUNCT
ejpam-5253	161	30	t	t	PROPN
ejpam-5253	161	31	}	}	PUNCT
ejpam-5253	161	32	}	}	PUNCT
ejpam-5253	161	33	.	.	PUNCT
ejpam-5253	162	1	so	so	ADV
ejpam-5253	162	2	(	(	PUNCT
ejpam-5253	162	3	m	m	PROPN
ejpam-5253	162	4	,	,	PUNCT
ejpam-5253	162	5	rh	rh	PROPN
ejpam-5253	162	6	)	)	PUNCT
ejpam-5253	162	7	is	be	AUX
ejpam-5253	162	8	a	a	DET
ejpam-5253	162	9	h	h	NOUN
ejpam-5253	162	10	-	-	PUNCT
ejpam-5253	162	11	approximation	approximation	NOUN
ejpam-5253	162	12	space	space	NOUN
ejpam-5253	162	13	.	.	PUNCT
ejpam-5253	163	1	let	let	VERB
ejpam-5253	163	2	k	k	NOUN
ejpam-5253	163	3	=	=	PUNCT
ejpam-5253	163	4	{	{	PUNCT
ejpam-5253	163	5	q	q	PROPN
ejpam-5253	163	6	,	,	PUNCT
ejpam-5253	163	7	s	s	PROPN
ejpam-5253	163	8	,	,	PUNCT
ejpam-5253	163	9	t	t	PROPN
ejpam-5253	163	10	}	}	PUNCT
ejpam-5253	163	11	,	,	PUNCT
ejpam-5253	163	12	we	we	PRON
ejpam-5253	163	13	have	have	VERB
ejpam-5253	163	14	q	q	NOUN
ejpam-5253	163	15	∈h	∈h	NOUN
ejpam-5253	163	16	k	k	PROPN
ejpam-5253	163	17	but	but	CCONJ
ejpam-5253	163	18	q	q	PROPN
ejpam-5253	163	19	/∈	/∈	PROPN
ejpam-5253	164	1	k.	k.	PROPN
ejpam-5253	165	1	also	also	ADV
ejpam-5253	165	2	,	,	PUNCT
ejpam-5253	165	3	let	let	VERB
ejpam-5253	165	4	r	r	NOUN
ejpam-5253	165	5	=	=	PRON
ejpam-5253	165	6	{	{	PUNCT
ejpam-5253	165	7	k	k	NOUN
ejpam-5253	165	8	}	}	PUNCT
ejpam-5253	165	9	.	.	PUNCT
ejpam-5253	166	1	we	we	PRON
ejpam-5253	166	2	have	have	VERB
ejpam-5253	166	3	q	q	PROPN
ejpam-5253	166	4	∈	∈	PROPN
ejpam-5253	166	5	r	r	NOUN
ejpam-5253	166	6	but	but	CCONJ
ejpam-5253	166	7	q	q	NOUN
ejpam-5253	166	8	/∈h	/∈h	PROPN
ejpam-5253	166	9	k.	k.	PROPN
ejpam-5253	166	10	definition	definition	NOUN
ejpam-5253	166	11	9	9	NUM
ejpam-5253	166	12	.	.	PUNCT
ejpam-5253	167	1	if	if	SCONJ
ejpam-5253	167	2	m	m	NOUN
ejpam-5253	167	3	is	be	AUX
ejpam-5253	167	4	a	a	DET
ejpam-5253	167	5	finite	finite	ADJ
ejpam-5253	167	6	none	none	NOUN
ejpam-5253	167	7	-	-	PUNCT
ejpam-5253	167	8	empty	empty	ADJ
ejpam-5253	167	9	universe	universe	NOUN
ejpam-5253	167	10	,	,	PUNCT
ejpam-5253	167	11	k	k	PROPN
ejpam-5253	167	12	⊆	⊆	NUM
ejpam-5253	167	13	m	m	NOUN
ejpam-5253	167	14	and	and	CCONJ
ejpam-5253	167	15	k	k	PROPN
ejpam-5253	167	16	̸=	̸=	PROPN
ejpam-5253	167	17	ϕ	ϕ	NOUN
ejpam-5253	167	18	,	,	PUNCT
ejpam-5253	167	19	then	then	ADV
ejpam-5253	167	20	we	we	PRON
ejpam-5253	167	21	can	can	AUX
ejpam-5253	167	22	express	express	VERB
ejpam-5253	167	23	the	the	DET
ejpam-5253	167	24	degree	degree	NOUN
ejpam-5253	167	25	of	of	ADP
ejpam-5253	167	26	completeness	completeness	NOUN
ejpam-5253	167	27	using	use	VERB
ejpam-5253	167	28	a	a	DET
ejpam-5253	167	29	novel	novel	ADJ
ejpam-5253	167	30	metric	metric	NOUN
ejpam-5253	167	31	termed	term	VERB
ejpam-5253	167	32	the	the	DET
ejpam-5253	167	33	h	h	NOUN
ejpam-5253	167	34	-	-	PUNCT
ejpam-5253	167	35	accuracy	accuracy	NOUN
ejpam-5253	167	36	measure	measure	NOUN
ejpam-5253	167	37	,	,	PUNCT
ejpam-5253	167	38	defined	define	VERB
ejpam-5253	167	39	as	as	SCONJ
ejpam-5253	167	40	follows	follow	VERB
ejpam-5253	167	41	:	:	PUNCT
ejpam-5253	167	42	αrh	αrh	PROPN
ejpam-5253	167	43	(	(	PUNCT
ejpam-5253	167	44	k	k	NOUN
ejpam-5253	167	45	)	)	PUNCT
ejpam-5253	167	46	=	=	SYM
ejpam-5253	167	47	|	|	ADV
ejpam-5253	167	48	rh(k	rh(k	NOUN
ejpam-5253	167	49	)	)	PUNCT
ejpam-5253	167	50	|∣∣	|∣∣	ADP
ejpam-5253	167	51	rh(k	rh(k	NOUN
ejpam-5253	167	52	)	)	PUNCT
ejpam-5253	167	53	∣∣	∣∣	X
ejpam-5253	167	54	a.	a.	PROPN
ejpam-5253	167	55	al	al	PROPN
ejpam-5253	167	56	-	-	PUNCT
ejpam-5253	167	57	rehili	rehili	NOUN
ejpam-5253	167	58	/	/	SYM
ejpam-5253	167	59	eur	eur	PROPN
ejpam-5253	167	60	.	.	PUNCT
ejpam-5253	168	1	j.	j.	PROPN
ejpam-5253	168	2	pure	pure	PROPN
ejpam-5253	168	3	appl	appl	PROPN
ejpam-5253	168	4	.	.	PROPN
ejpam-5253	168	5	math	math	PROPN
ejpam-5253	168	6	,	,	PUNCT
ejpam-5253	168	7	17	17	NUM
ejpam-5253	168	8	(	(	PUNCT
ejpam-5253	168	9	3	3	NUM
ejpam-5253	168	10	)	)	PUNCT
ejpam-5253	168	11	(	(	PUNCT
ejpam-5253	168	12	2024	2024	NUM
ejpam-5253	168	13	)	)	PUNCT
ejpam-5253	168	14	,	,	PUNCT
ejpam-5253	168	15	1804	1804	NUM
ejpam-5253	168	16	-	-	SYM
ejpam-5253	168	17	1817	1817	NUM
ejpam-5253	168	18	1810	1810	NUM
ejpam-5253	168	19	example	example	NOUN
ejpam-5253	168	20	5	5	NUM
ejpam-5253	168	21	.	.	PUNCT
ejpam-5253	169	1	in	in	ADP
ejpam-5253	169	2	example	example	NOUN
ejpam-5253	169	3	2	2	NUM
ejpam-5253	169	4	,	,	PUNCT
ejpam-5253	169	5	we	we	PRON
ejpam-5253	169	6	can	can	AUX
ejpam-5253	169	7	deduce	deduce	VERB
ejpam-5253	169	8	the	the	DET
ejpam-5253	169	9	following	follow	VERB
ejpam-5253	169	10	table	table	NOUN
ejpam-5253	169	11	showing	show	VERB
ejpam-5253	169	12	the	the	DET
ejpam-5253	169	13	degree	degree	NOUN
ejpam-5253	169	14	of	of	ADP
ejpam-5253	169	15	accuracy	accuracy	NOUN
ejpam-5253	169	16	measure	measure	NOUN
ejpam-5253	169	17	αr(k	αr(k	NUM
ejpam-5253	169	18	)	)	PUNCT
ejpam-5253	169	19	and	and	CCONJ
ejpam-5253	169	20	h	h	NOUN
ejpam-5253	169	21	-	-	PUNCT
ejpam-5253	169	22	accuracy	accuracy	NOUN
ejpam-5253	169	23	measure	measure	NOUN
ejpam-5253	169	24	αrh	αrh	INTJ
ejpam-5253	169	25	(	(	PUNCT
ejpam-5253	169	26	k	k	NOUN
ejpam-5253	169	27	)	)	PUNCT
ejpam-5253	169	28	for	for	ADP
ejpam-5253	169	29	some	some	DET
ejpam-5253	169	30	sets	set	NOUN
ejpam-5253	169	31	.	.	PUNCT
ejpam-5253	170	1	table	table	NOUN
ejpam-5253	170	2	1	1	NUM
ejpam-5253	170	3	:	:	PUNCT
ejpam-5253	170	4	the	the	DET
ejpam-5253	170	5	degree	degree	NOUN
ejpam-5253	170	6	of	of	ADP
ejpam-5253	170	7	accuracy	accuracy	NOUN
ejpam-5253	170	8	measure	measure	NOUN
ejpam-5253	170	9	and	and	CCONJ
ejpam-5253	170	10	h	h	NOUN
ejpam-5253	170	11	-	-	PUNCT
ejpam-5253	170	12	accuracy	accuracy	NOUN
ejpam-5253	170	13	measure	measure	NOUN
ejpam-5253	170	14	.	.	PUNCT
ejpam-5253	171	1	the	the	DET
ejpam-5253	171	2	set	set	NOUN
ejpam-5253	171	3	k	k	PROPN
ejpam-5253	171	4	⊆	⊆	NUM
ejpam-5253	171	5	m	m	NOUN
ejpam-5253	171	6	αr(k	αr(k	NOUN
ejpam-5253	171	7	)	)	PUNCT
ejpam-5253	171	8	αrh	αrh	NOUN
ejpam-5253	171	9	(	(	PUNCT
ejpam-5253	171	10	k	k	NOUN
ejpam-5253	171	11	)	)	PUNCT
ejpam-5253	171	12	{	{	PUNCT
ejpam-5253	171	13	k	k	NOUN
ejpam-5253	171	14	}	}	SYM
ejpam-5253	171	15	1	1	NUM
ejpam-5253	171	16	2	2	NUM
ejpam-5253	171	17	1	1	NUM
ejpam-5253	171	18	{	{	PUNCT
ejpam-5253	171	19	t	t	NOUN
ejpam-5253	171	20	}	}	PUNCT
ejpam-5253	171	21	1	1	NUM
ejpam-5253	171	22	3	3	NUM
ejpam-5253	171	23	1	1	NUM
ejpam-5253	171	24	{	{	PUNCT
ejpam-5253	171	25	k	k	NOUN
ejpam-5253	171	26	,	,	PUNCT
ejpam-5253	171	27	q	q	NOUN
ejpam-5253	171	28	}	}	PUNCT
ejpam-5253	171	29	1	1	NUM
ejpam-5253	171	30	3	3	NUM
ejpam-5253	171	31	1	1	NUM
ejpam-5253	171	32	2	2	NUM
ejpam-5253	171	33	{	{	PUNCT
ejpam-5253	171	34	k	k	NOUN
ejpam-5253	171	35	,	,	PUNCT
ejpam-5253	171	36	s	s	NOUN
ejpam-5253	171	37	}	}	PUNCT
ejpam-5253	171	38	1	1	NUM
ejpam-5253	171	39	2	2	NUM
ejpam-5253	171	40	1	1	NUM
ejpam-5253	171	41	2	2	NUM
ejpam-5253	171	42	{	{	PUNCT
ejpam-5253	171	43	k	k	PROPN
ejpam-5253	171	44	,	,	PUNCT
ejpam-5253	171	45	t	t	PROPN
ejpam-5253	171	46	}	}	PUNCT
ejpam-5253	171	47	1	1	NUM
ejpam-5253	171	48	2	2	NUM
ejpam-5253	171	49	1	1	NUM
ejpam-5253	171	50	2	2	NUM
ejpam-5253	171	51	{	{	PUNCT
ejpam-5253	171	52	q	q	NOUN
ejpam-5253	171	53	,	,	PUNCT
ejpam-5253	171	54	t	t	PROPN
ejpam-5253	171	55	}	}	PUNCT
ejpam-5253	171	56	2	2	NUM
ejpam-5253	171	57	3	3	NUM
ejpam-5253	171	58	2	2	NUM
ejpam-5253	171	59	3	3	NUM
ejpam-5253	171	60	{	{	PUNCT
ejpam-5253	171	61	s	s	PROPN
ejpam-5253	171	62	,	,	PUNCT
ejpam-5253	171	63	t	t	PROPN
ejpam-5253	171	64	}	}	PUNCT
ejpam-5253	171	65	1	1	NUM
ejpam-5253	171	66	3	3	NUM
ejpam-5253	171	67	2	2	NUM
ejpam-5253	171	68	3	3	NUM
ejpam-5253	171	69	{	{	PUNCT
ejpam-5253	171	70	k	k	NOUN
ejpam-5253	171	71	,	,	PUNCT
ejpam-5253	171	72	q	q	X
ejpam-5253	171	73	,	,	PUNCT
ejpam-5253	171	74	s	s	PART
ejpam-5253	171	75	}	}	PUNCT
ejpam-5253	171	76	1	1	NUM
ejpam-5253	171	77	3	3	NUM
ejpam-5253	171	78	1	1	NUM
ejpam-5253	171	79	3	3	NUM
ejpam-5253	171	80	{	{	PUNCT
ejpam-5253	171	81	k	k	NOUN
ejpam-5253	171	82	,	,	PUNCT
ejpam-5253	171	83	q	q	NOUN
ejpam-5253	171	84	,	,	PUNCT
ejpam-5253	171	85	t	t	PROPN
ejpam-5253	171	86	}	}	PUNCT
ejpam-5253	171	87	3	3	NUM
ejpam-5253	171	88	4	4	NUM
ejpam-5253	171	89	3	3	NUM
ejpam-5253	171	90	4	4	NUM
ejpam-5253	171	91	{	{	PUNCT
ejpam-5253	171	92	k	k	NOUN
ejpam-5253	171	93	,	,	PUNCT
ejpam-5253	171	94	s	s	PROPN
ejpam-5253	171	95	,	,	PUNCT
ejpam-5253	171	96	t	t	PROPN
ejpam-5253	171	97	}	}	PUNCT
ejpam-5253	171	98	3	3	NUM
ejpam-5253	171	99	4	4	NUM
ejpam-5253	171	100	3	3	NUM
ejpam-5253	171	101	4	4	NUM
ejpam-5253	171	102	{	{	PUNCT
ejpam-5253	171	103	q	q	NOUN
ejpam-5253	171	104	,	,	PUNCT
ejpam-5253	171	105	s	s	PROPN
ejpam-5253	171	106	,	,	PUNCT
ejpam-5253	171	107	t	t	PROPN
ejpam-5253	171	108	}	}	PUNCT
ejpam-5253	171	109	1	1	NUM
ejpam-5253	171	110	2	2	NUM
ejpam-5253	171	111	1	1	NUM
ejpam-5253	171	112	the	the	DET
ejpam-5253	171	113	degree	degree	NOUN
ejpam-5253	171	114	of	of	ADP
ejpam-5253	171	115	exactness	exactness	NOUN
ejpam-5253	171	116	of	of	ADP
ejpam-5253	171	117	set	set	NOUN
ejpam-5253	171	118	k	k	PROPN
ejpam-5253	171	119	=	=	PRON
ejpam-5253	171	120	{	{	PUNCT
ejpam-5253	171	121	k	k	NOUN
ejpam-5253	171	122	}	}	PUNCT
ejpam-5253	171	123	is	be	AUX
ejpam-5253	171	124	observed	observe	VERB
ejpam-5253	171	125	to	to	PART
ejpam-5253	171	126	be	be	AUX
ejpam-5253	171	127	50	50	NUM
ejpam-5253	171	128	%	%	NOUN
ejpam-5253	171	129	using	use	VERB
ejpam-5253	171	130	the	the	DET
ejpam-5253	171	131	accuracy	accuracy	NOUN
ejpam-5253	171	132	measure	measure	NOUN
ejpam-5253	171	133	and	and	CCONJ
ejpam-5253	171	134	100	100	NUM
ejpam-5253	171	135	%	%	NOUN
ejpam-5253	171	136	using	use	VERB
ejpam-5253	171	137	the	the	DET
ejpam-5253	171	138	h	h	NOUN
ejpam-5253	171	139	-	-	PUNCT
ejpam-5253	171	140	accuracy	accuracy	NOUN
ejpam-5253	171	141	measure	measure	NOUN
ejpam-5253	171	142	.	.	PUNCT
ejpam-5253	172	1	thus	thus	ADV
ejpam-5253	172	2	,	,	PUNCT
ejpam-5253	172	3	it	it	PRON
ejpam-5253	172	4	follows	follow	VERB
ejpam-5253	172	5	that	that	SCONJ
ejpam-5253	172	6	the	the	DET
ejpam-5253	172	7	h	h	NOUN
ejpam-5253	172	8	-	-	PUNCT
ejpam-5253	172	9	accuracy	accuracy	NOUN
ejpam-5253	172	10	measure	measure	NOUN
ejpam-5253	172	11	outperforms	outperform	VERB
ejpam-5253	172	12	the	the	DET
ejpam-5253	172	13	accuracy	accuracy	NOUN
ejpam-5253	172	14	measure	measure	NOUN
ejpam-5253	172	15	in	in	ADP
ejpam-5253	172	16	this	this	DET
ejpam-5253	172	17	particular	particular	ADJ
ejpam-5253	172	18	case	case	NOUN
ejpam-5253	172	19	.	.	PUNCT
ejpam-5253	173	1	4	4	X
ejpam-5253	173	2	.	.	X
ejpam-5253	173	3	h	h	NUM
ejpam-5253	173	4	-	-	PUNCT
ejpam-5253	173	5	rough	rough	ADJ
ejpam-5253	173	6	equality	equality	NOUN
ejpam-5253	173	7	and	and	CCONJ
ejpam-5253	173	8	h	h	NOUN
ejpam-5253	173	9	-	-	PUNCT
ejpam-5253	173	10	rough	rough	ADJ
ejpam-5253	173	11	inclusion	inclusion	NOUN
ejpam-5253	173	12	in	in	ADP
ejpam-5253	173	13	this	this	DET
ejpam-5253	173	14	section	section	NOUN
ejpam-5253	173	15	,	,	PUNCT
ejpam-5253	173	16	the	the	DET
ejpam-5253	173	17	focus	focus	NOUN
ejpam-5253	173	18	is	be	AUX
ejpam-5253	173	19	on	on	ADP
ejpam-5253	173	20	exploring	explore	VERB
ejpam-5253	173	21	h	h	NOUN
ejpam-5253	173	22	-	-	PUNCT
ejpam-5253	173	23	rough	rough	ADJ
ejpam-5253	173	24	equality	equality	NOUN
ejpam-5253	173	25	and	and	CCONJ
ejpam-5253	173	26	h	h	NOUN
ejpam-5253	173	27	-	-	PUNCT
ejpam-5253	173	28	rough	rough	ADJ
ejpam-5253	173	29	inclusion	inclusion	NOUN
ejpam-5253	173	30	,	,	PUNCT
ejpam-5253	173	31	drawing	draw	VERB
ejpam-5253	173	32	from	from	ADP
ejpam-5253	173	33	the	the	DET
ejpam-5253	173	34	groundwork	groundwork	NOUN
ejpam-5253	173	35	laid	lay	VERB
ejpam-5253	173	36	by	by	ADP
ejpam-5253	173	37	pawlak	pawlak	ADJ
ejpam-5253	173	38	and	and	CCONJ
ejpam-5253	173	39	novotny	novotny	PROPN
ejpam-5253	173	40	(	(	PUNCT
ejpam-5253	174	1	[	[	X
ejpam-5253	174	2	11],[9	11],[9	NUM
ejpam-5253	174	3	]	]	PUNCT
ejpam-5253	174	4	)	)	PUNCT
ejpam-5253	174	5	in	in	ADP
ejpam-5253	174	6	their	their	PRON
ejpam-5253	174	7	introduction	introduction	NOUN
ejpam-5253	174	8	of	of	ADP
ejpam-5253	174	9	rough	rough	ADJ
ejpam-5253	174	10	equality	equality	NOUN
ejpam-5253	174	11	and	and	CCONJ
ejpam-5253	174	12	inclusion	inclusion	NOUN
ejpam-5253	174	13	.	.	PUNCT
ejpam-5253	175	1	definition	definition	NOUN
ejpam-5253	175	2	10	10	NUM
ejpam-5253	175	3	.	.	PUNCT
ejpam-5253	176	1	if	if	SCONJ
ejpam-5253	176	2	(	(	PUNCT
ejpam-5253	176	3	m	m	PROPN
ejpam-5253	176	4	,	,	PUNCT
ejpam-5253	176	5	rh	rh	PROPN
ejpam-5253	176	6	)	)	PUNCT
ejpam-5253	176	7	is	be	AUX
ejpam-5253	176	8	a	a	DET
ejpam-5253	176	9	h	h	NOUN
ejpam-5253	176	10	-	-	PUNCT
ejpam-5253	176	11	approximation	approximation	NOUN
ejpam-5253	176	12	space	space	NOUN
ejpam-5253	176	13	and	and	CCONJ
ejpam-5253	176	14	k	k	NOUN
ejpam-5253	176	15	,	,	PUNCT
ejpam-5253	176	16	q	q	X
ejpam-5253	176	17	⊆	⊆	NUM
ejpam-5253	176	18	m	m	NOUN
ejpam-5253	176	19	.	.	PUNCT
ejpam-5253	177	1	then	then	ADV
ejpam-5253	177	2	k	k	PROPN
ejpam-5253	177	3	and	and	CCONJ
ejpam-5253	177	4	q	q	PROPN
ejpam-5253	177	5	are	be	AUX
ejpam-5253	177	6	called	call	VERB
ejpam-5253	177	7	:	:	PUNCT
ejpam-5253	177	8	(	(	PUNCT
ejpam-5253	177	9	i	i	NOUN
ejpam-5253	177	10	)	)	PUNCT
ejpam-5253	177	11	h	h	NOUN
ejpam-5253	177	12	-	-	PUNCT
ejpam-5253	177	13	roughly	roughly	ADV
ejpam-5253	177	14	bottom	bottom	ADJ
ejpam-5253	177	15	equal	equal	ADJ
ejpam-5253	177	16	(	(	PUNCT
ejpam-5253	177	17	k	k	PROPN
ejpam-5253	177	18	∼h	∼h	PROPN
ejpam-5253	177	19	q	q	NOUN
ejpam-5253	177	20	)	)	PUNCT
ejpam-5253	177	21	if	if	SCONJ
ejpam-5253	177	22	rh(k	rh(k	NOUN
ejpam-5253	177	23	)	)	PUNCT
ejpam-5253	177	24	=	=	SYM
ejpam-5253	177	25	rh(q	rh(q	NOUN
ejpam-5253	177	26	)	)	PUNCT
ejpam-5253	177	27	.	.	PUNCT
ejpam-5253	178	1	(	(	PUNCT
ejpam-5253	178	2	ii	ii	NOUN
ejpam-5253	178	3	)	)	PUNCT
ejpam-5253	178	4	h	h	NOUN
ejpam-5253	178	5	-	-	PUNCT
ejpam-5253	178	6	roughly	roughly	ADV
ejpam-5253	178	7	top	top	ADJ
ejpam-5253	178	8	equal	equal	ADJ
ejpam-5253	178	9	(	(	PUNCT
ejpam-5253	178	10	k	k	NOUN
ejpam-5253	178	11	≃h	≃h	PRON
ejpam-5253	178	12	q	q	NOUN
ejpam-5253	178	13	)	)	PUNCT
ejpam-5253	178	14	if	if	SCONJ
ejpam-5253	178	15	rh(k	rh(k	NOUN
ejpam-5253	178	16	)	)	PUNCT
ejpam-5253	178	17	=	=	SYM
ejpam-5253	178	18	rh(q	rh(q	NOUN
ejpam-5253	178	19	)	)	PUNCT
ejpam-5253	178	20	.	.	PUNCT
ejpam-5253	179	1	(	(	PUNCT
ejpam-5253	179	2	iii	iii	X
ejpam-5253	179	3	)	)	PUNCT
ejpam-5253	179	4	h	h	NOUN
ejpam-5253	179	5	-	-	PUNCT
ejpam-5253	179	6	roughly	roughly	ADV
ejpam-5253	179	7	equal	equal	ADJ
ejpam-5253	179	8	(	(	PUNCT
ejpam-5253	179	9	k	k	X
ejpam-5253	179	10	≈h	≈h	NOUN
ejpam-5253	179	11	q	q	NOUN
ejpam-5253	179	12	)	)	PUNCT
ejpam-5253	179	13	if	if	SCONJ
ejpam-5253	179	14	(	(	PUNCT
ejpam-5253	179	15	k	k	PROPN
ejpam-5253	179	16	∼h	∼h	PROPN
ejpam-5253	179	17	q	q	NOUN
ejpam-5253	179	18	)	)	PUNCT
ejpam-5253	179	19	and	and	CCONJ
ejpam-5253	179	20	(	(	PUNCT
ejpam-5253	179	21	k	k	NOUN
ejpam-5253	179	22	≃h	≃h	NOUN
ejpam-5253	179	23	q	q	NOUN
ejpam-5253	179	24	)	)	PUNCT
ejpam-5253	179	25	.	.	PUNCT
ejpam-5253	179	26	example	example	NOUN
ejpam-5253	180	1	6	6	NUM
ejpam-5253	180	2	.	.	PUNCT
ejpam-5253	181	1	in	in	ADP
ejpam-5253	181	2	example	example	NOUN
ejpam-5253	181	3	2	2	NUM
ejpam-5253	181	4	,	,	PUNCT
ejpam-5253	181	5	we	we	PRON
ejpam-5253	181	6	have	have	VERB
ejpam-5253	181	7	the	the	DET
ejpam-5253	181	8	sets	set	NOUN
ejpam-5253	181	9	{	{	PUNCT
ejpam-5253	181	10	k	k	X
ejpam-5253	181	11	,	,	PUNCT
ejpam-5253	181	12	s	s	PART
ejpam-5253	181	13	}	}	PUNCT
ejpam-5253	181	14	,	,	PUNCT
ejpam-5253	181	15	{	{	PUNCT
ejpam-5253	181	16	k	k	X
ejpam-5253	181	17	,	,	PUNCT
ejpam-5253	181	18	q	q	X
ejpam-5253	181	19	,	,	PUNCT
ejpam-5253	181	20	s	s	PART
ejpam-5253	181	21	}	}	PUNCT
ejpam-5253	181	22	are	be	AUX
ejpam-5253	181	23	h	h	ADJ
ejpam-5253	181	24	-	-	PUNCT
ejpam-5253	181	25	roughly	roughly	ADV
ejpam-5253	181	26	bottom	bottom	ADJ
ejpam-5253	181	27	equal	equal	ADJ
ejpam-5253	181	28	and	and	CCONJ
ejpam-5253	181	29	{	{	PUNCT
ejpam-5253	181	30	s	s	PROPN
ejpam-5253	181	31	,	,	PUNCT
ejpam-5253	181	32	t	t	PROPN
ejpam-5253	181	33	}	}	PUNCT
ejpam-5253	181	34	,	,	PUNCT
ejpam-5253	181	35	{	{	PUNCT
ejpam-5253	181	36	q	q	X
ejpam-5253	181	37	,	,	PUNCT
ejpam-5253	181	38	s	s	PROPN
ejpam-5253	181	39	,	,	PUNCT
ejpam-5253	181	40	t	t	PROPN
ejpam-5253	181	41	}	}	PUNCT
ejpam-5253	181	42	are	be	AUX
ejpam-5253	181	43	h	h	ADJ
ejpam-5253	181	44	-	-	PUNCT
ejpam-5253	181	45	roughly	roughly	ADV
ejpam-5253	181	46	top	top	ADJ
ejpam-5253	181	47	equal	equal	ADJ
ejpam-5253	181	48	.	.	PUNCT
ejpam-5253	182	1	it	it	PRON
ejpam-5253	182	2	’s	’	VERB
ejpam-5253	182	3	straightforward	straightforward	ADJ
ejpam-5253	182	4	to	to	PART
ejpam-5253	182	5	demonstrate	demonstrate	VERB
ejpam-5253	182	6	that	that	SCONJ
ejpam-5253	182	7	≈h	≈h	PROPN
ejpam-5253	182	8	forms	form	VERB
ejpam-5253	182	9	an	an	DET
ejpam-5253	182	10	equivalence	equivalence	NOUN
ejpam-5253	182	11	relation	relation	NOUN
ejpam-5253	182	12	on	on	ADP
ejpam-5253	182	13	p	p	PROPN
ejpam-5253	182	14	(	(	PUNCT
ejpam-5253	182	15	m	m	NOUN
ejpam-5253	182	16	)	)	PUNCT
ejpam-5253	182	17	,	,	PUNCT
ejpam-5253	182	18	making	make	VERB
ejpam-5253	182	19	the	the	DET
ejpam-5253	182	20	pair	pair	NOUN
ejpam-5253	182	21	(	(	PUNCT
ejpam-5253	182	22	p	p	X
ejpam-5253	182	23	(	(	PUNCT
ejpam-5253	182	24	m),≈h	m),≈h	NOUN
ejpam-5253	182	25	)	)	PUNCT
ejpam-5253	182	26	an	an	DET
ejpam-5253	182	27	approximation	approximation	NOUN
ejpam-5253	182	28	space	space	NOUN
ejpam-5253	182	29	.	.	PUNCT
ejpam-5253	183	1	additionally	additionally	ADV
ejpam-5253	183	2	,	,	PUNCT
ejpam-5253	183	3	this	this	DET
ejpam-5253	183	4	relation	relation	NOUN
ejpam-5253	183	5	,	,	PUNCT
ejpam-5253	183	6	≈h	≈h	PROPN
ejpam-5253	183	7	,	,	PUNCT
ejpam-5253	183	8	is	be	AUX
ejpam-5253	183	9	termed	term	VERB
ejpam-5253	183	10	as	as	ADP
ejpam-5253	183	11	the	the	DET
ejpam-5253	183	12	h	h	NOUN
ejpam-5253	183	13	-	-	PUNCT
ejpam-5253	183	14	rough	rough	ADJ
ejpam-5253	183	15	equality	equality	NOUN
ejpam-5253	183	16	within	within	ADP
ejpam-5253	183	17	the	the	DET
ejpam-5253	183	18	h	h	NOUN
ejpam-5253	183	19	-	-	PUNCT
ejpam-5253	183	20	approximation	approximation	NOUN
ejpam-5253	183	21	space	space	NOUN
ejpam-5253	183	22	(	(	PUNCT
ejpam-5253	183	23	m	m	PROPN
ejpam-5253	183	24	,	,	PUNCT
ejpam-5253	183	25	rh	rh	PROPN
ejpam-5253	183	26	)	)	PUNCT
ejpam-5253	183	27	.	.	PUNCT
ejpam-5253	184	1	definition	definition	NOUN
ejpam-5253	184	2	11	11	NUM
ejpam-5253	184	3	.	.	PUNCT
ejpam-5253	185	1	let	let	AUX
ejpam-5253	185	2	(	(	PUNCT
ejpam-5253	185	3	m	m	PROPN
ejpam-5253	185	4	,	,	PUNCT
ejpam-5253	185	5	rh	rh	PROPN
ejpam-5253	185	6	)	)	PUNCT
ejpam-5253	185	7	be	be	VERB
ejpam-5253	185	8	a	a	DET
ejpam-5253	185	9	h	h	NOUN
ejpam-5253	185	10	-	-	PUNCT
ejpam-5253	185	11	approximation	approximation	NOUN
ejpam-5253	185	12	space	space	NOUN
ejpam-5253	185	13	.	.	PUNCT
ejpam-5253	186	1	the	the	DET
ejpam-5253	186	2	equivalence	equivalence	NOUN
ejpam-5253	186	3	relation	relation	NOUN
ejpam-5253	186	4	eh	eh	INTJ
ejpam-5253	186	5	on	on	ADP
ejpam-5253	186	6	the	the	DET
ejpam-5253	186	7	set	set	NOUN
ejpam-5253	186	8	p	p	PROPN
ejpam-5253	186	9	(	(	PUNCT
ejpam-5253	186	10	m	m	NOUN
ejpam-5253	186	11	)	)	PUNCT
ejpam-5253	186	12	is	be	AUX
ejpam-5253	186	13	defined	define	VERB
ejpam-5253	186	14	by	by	ADP
ejpam-5253	186	15	the	the	DET
ejpam-5253	186	16	following	follow	VERB
ejpam-5253	186	17	condition	condition	NOUN
ejpam-5253	186	18	:	:	PUNCT
ejpam-5253	186	19	a.	a.	PROPN
ejpam-5253	186	20	al	al	PROPN
ejpam-5253	186	21	-	-	PUNCT
ejpam-5253	186	22	rehili	rehili	NOUN
ejpam-5253	186	23	/	/	SYM
ejpam-5253	186	24	eur	eur	PROPN
ejpam-5253	186	25	.	.	PUNCT
ejpam-5253	187	1	j.	j.	PROPN
ejpam-5253	187	2	pure	pure	PROPN
ejpam-5253	187	3	appl	appl	PROPN
ejpam-5253	187	4	.	.	PROPN
ejpam-5253	187	5	math	math	PROPN
ejpam-5253	187	6	,	,	PUNCT
ejpam-5253	187	7	17	17	NUM
ejpam-5253	187	8	(	(	PUNCT
ejpam-5253	187	9	3	3	NUM
ejpam-5253	187	10	)	)	PUNCT
ejpam-5253	187	11	(	(	PUNCT
ejpam-5253	187	12	2024	2024	NUM
ejpam-5253	187	13	)	)	PUNCT
ejpam-5253	187	14	,	,	PUNCT
ejpam-5253	187	15	1804	1804	NUM
ejpam-5253	187	16	-	-	SYM
ejpam-5253	187	17	1817	1817	NUM
ejpam-5253	187	18	1811	1811	NUM
ejpam-5253	187	19	(	(	PUNCT
ejpam-5253	187	20	k	k	X
ejpam-5253	187	21	,	,	PUNCT
ejpam-5253	187	22	q	q	NOUN
ejpam-5253	187	23	)	)	PUNCT
ejpam-5253	187	24	∈	∈	PROPN
ejpam-5253	187	25	eh	eh	INTJ
ejpam-5253	187	26	if	if	SCONJ
ejpam-5253	187	27	inth(k	inth(k	PROPN
ejpam-5253	187	28	)	)	PUNCT
ejpam-5253	187	29	=	=	SYM
ejpam-5253	187	30	inth(q	inth(q	PROPN
ejpam-5253	187	31	)	)	PUNCT
ejpam-5253	187	32	and	and	CCONJ
ejpam-5253	187	33	clh(k	clh(k	PROPN
ejpam-5253	187	34	)	)	PUNCT
ejpam-5253	187	35	=	=	SYM
ejpam-5253	187	36	clh(q	clh(q	NOUN
ejpam-5253	187	37	)	)	PUNCT
ejpam-5253	187	38	.	.	PUNCT
ejpam-5253	188	1	the	the	DET
ejpam-5253	188	2	equivalence	equivalence	NOUN
ejpam-5253	188	3	relation	relation	NOUN
ejpam-5253	188	4	eh	eh	INTJ
ejpam-5253	188	5	is	be	AUX
ejpam-5253	188	6	identical	identical	ADJ
ejpam-5253	188	7	to	to	ADP
ejpam-5253	188	8	≈h	≈h	PROPN
ejpam-5253	188	9	,	,	PUNCT
ejpam-5253	188	10	given	give	VERB
ejpam-5253	188	11	that	that	PRON
ejpam-5253	188	12	rh(k	rh(k	NOUN
ejpam-5253	188	13	)	)	PUNCT
ejpam-5253	188	14	=	=	SYM
ejpam-5253	188	15	inth(k	inth(k	PROPN
ejpam-5253	188	16	)	)	PUNCT
ejpam-5253	188	17	and	and	CCONJ
ejpam-5253	188	18	rh(k	rh(k	NOUN
ejpam-5253	188	19	)	)	PUNCT
ejpam-5253	188	20	=	=	SYM
ejpam-5253	188	21	clh(k	clh(k	PROPN
ejpam-5253	188	22	)	)	PUNCT
ejpam-5253	188	23	remark	remark	NOUN
ejpam-5253	188	24	3	3	NUM
ejpam-5253	188	25	.	.	PUNCT
ejpam-5253	189	1	denoting	denote	VERB
ejpam-5253	189	2	the	the	DET
ejpam-5253	189	3	equivalence	equivalence	NOUN
ejpam-5253	189	4	class	class	NOUN
ejpam-5253	189	5	of	of	ADP
ejpam-5253	189	6	the	the	DET
ejpam-5253	189	7	relation	relation	NOUN
ejpam-5253	189	8	(	(	PUNCT
ejpam-5253	189	9	≈h	≈h	PROPN
ejpam-5253	189	10	or	or	CCONJ
ejpam-5253	189	11	eh	eh	INTJ
ejpam-5253	189	12	)	)	PUNCT
ejpam-5253	189	13	containing	contain	VERB
ejpam-5253	189	14	any	any	DET
ejpam-5253	189	15	subset	subset	NOUN
ejpam-5253	189	16	k	k	PROPN
ejpam-5253	189	17	of	of	ADP
ejpam-5253	189	18	m	m	PROPN
ejpam-5253	189	19	as	as	ADP
ejpam-5253	189	20	[	[	X
ejpam-5253	189	21	k]≈h	k]≈h	NOUN
ejpam-5253	189	22	or	or	CCONJ
ejpam-5253	189	23	[	[	X
ejpam-5253	189	24	k]eh	k]eh	PROPN
ejpam-5253	189	25	.	.	PUNCT
ejpam-5253	190	1	we	we	PRON
ejpam-5253	190	2	can	can	AUX
ejpam-5253	190	3	conclude	conclude	VERB
ejpam-5253	190	4	that	that	PRON
ejpam-5253	190	5	:	:	PUNCT
ejpam-5253	191	1	[	[	X
ejpam-5253	191	2	k]≈h	k]≈h	NOUN
ejpam-5253	191	3	=	=	X
ejpam-5253	191	4	{	{	PUNCT
ejpam-5253	191	5	q	q	X
ejpam-5253	191	6	⊂	⊂	X
ejpam-5253	191	7	m	m	VERB
ejpam-5253	191	8	:	:	PUNCT
ejpam-5253	191	9	rh(q	rh(q	NUM
ejpam-5253	191	10	)	)	PUNCT
ejpam-5253	191	11	=	=	SYM
ejpam-5253	191	12	rh(k	rh(k	X
ejpam-5253	191	13	)	)	PUNCT
ejpam-5253	191	14	and	and	CCONJ
ejpam-5253	191	15	rh(q	rh(q	NUM
ejpam-5253	191	16	)	)	PUNCT
ejpam-5253	191	17	=	=	SYM
ejpam-5253	191	18	rh(k	rh(k	X
ejpam-5253	191	19	)	)	PUNCT
ejpam-5253	191	20	}	}	PUNCT
ejpam-5253	191	21	.	.	PUNCT
ejpam-5253	192	1	we	we	PRON
ejpam-5253	192	2	denote	denote	VERB
ejpam-5253	192	3	by	by	ADP
ejpam-5253	192	4	rh(m	rh(m	NOUN
ejpam-5253	192	5	)	)	PUNCT
ejpam-5253	192	6	the	the	DET
ejpam-5253	192	7	family	family	NOUN
ejpam-5253	192	8	of	of	ADP
ejpam-5253	192	9	h	h	NOUN
ejpam-5253	192	10	-	-	PUNCT
ejpam-5253	192	11	rough	rough	ADJ
ejpam-5253	192	12	classes	class	NOUN
ejpam-5253	192	13	in	in	ADP
ejpam-5253	192	14	a	a	DET
ejpam-5253	192	15	h	h	NOUN
ejpam-5253	192	16	-	-	PUNCT
ejpam-5253	192	17	approximation	approximation	NOUN
ejpam-5253	192	18	space	space	NOUN
ejpam-5253	192	19	(	(	PUNCT
ejpam-5253	192	20	m	m	PROPN
ejpam-5253	192	21	,	,	PUNCT
ejpam-5253	192	22	rh	rh	PROPN
ejpam-5253	192	23	)	)	PUNCT
ejpam-5253	192	24	.	.	PUNCT
ejpam-5253	193	1	definition	definition	NOUN
ejpam-5253	193	2	12	12	NUM
ejpam-5253	193	3	.	.	PUNCT
ejpam-5253	194	1	if	if	SCONJ
ejpam-5253	194	2	(	(	PUNCT
ejpam-5253	194	3	m	m	PROPN
ejpam-5253	194	4	,	,	PUNCT
ejpam-5253	194	5	rh	rh	PROPN
ejpam-5253	194	6	)	)	PUNCT
ejpam-5253	194	7	be	be	VERB
ejpam-5253	194	8	a	a	DET
ejpam-5253	194	9	h	h	NOUN
ejpam-5253	194	10	-	-	PUNCT
ejpam-5253	194	11	approximation	approximation	NOUN
ejpam-5253	194	12	space	space	NOUN
ejpam-5253	194	13	and	and	CCONJ
ejpam-5253	194	14	k	k	NOUN
ejpam-5253	194	15	,	,	PUNCT
ejpam-5253	194	16	q	q	X
ejpam-5253	194	17	⊆	⊆	NUM
ejpam-5253	194	18	m	m	NOUN
ejpam-5253	194	19	.	.	PUNCT
ejpam-5253	195	1	then	then	ADV
ejpam-5253	195	2	:	:	PUNCT
ejpam-5253	195	3	(	(	PUNCT
ejpam-5253	195	4	i	i	NOUN
ejpam-5253	195	5	)	)	PUNCT
ejpam-5253	195	6	k	k	PROPN
ejpam-5253	195	7	is	be	AUX
ejpam-5253	195	8	h	h	NOUN
ejpam-5253	195	9	-	-	PUNCT
ejpam-5253	195	10	roughly	roughly	ADV
ejpam-5253	195	11	bottom	bottom	NOUN
ejpam-5253	195	12	included	include	VERB
ejpam-5253	195	13	in	in	ADP
ejpam-5253	195	14	q	q	PROPN
ejpam-5253	195	15	(	(	PUNCT
ejpam-5253	195	16	k	k	X
ejpam-5253	195	17	⊂h˜	⊂h˜	X
ejpam-5253	195	18	q	q	NOUN
ejpam-5253	195	19	)	)	PUNCT
ejpam-5253	195	20	if	if	SCONJ
ejpam-5253	195	21	rh(k	rh(k	NOUN
ejpam-5253	195	22	)	)	PUNCT
ejpam-5253	195	23	⊆	⊆	NUM
ejpam-5253	195	24	rh(q	rh(q	NUM
ejpam-5253	195	25	)	)	PUNCT
ejpam-5253	195	26	.	.	PUNCT
ejpam-5253	196	1	(	(	PUNCT
ejpam-5253	196	2	ii	ii	X
ejpam-5253	196	3	)	)	PUNCT
ejpam-5253	196	4	k	k	PROPN
ejpam-5253	196	5	is	be	AUX
ejpam-5253	196	6	h	h	NOUN
ejpam-5253	196	7	-	-	PUNCT
ejpam-5253	196	8	roughly	roughly	ADV
ejpam-5253	196	9	top	top	NOUN
ejpam-5253	196	10	included	include	VERB
ejpam-5253	196	11	in	in	ADP
ejpam-5253	196	12	q	q	PROPN
ejpam-5253	196	13	(	(	PUNCT
ejpam-5253	196	14	k	k	X
ejpam-5253	196	15	⊂̃h	⊂̃h	X
ejpam-5253	196	16	q	q	PROPN
ejpam-5253	196	17	)	)	PUNCT
ejpam-5253	196	18	if	if	SCONJ
ejpam-5253	196	19	rh(k	rh(k	NOUN
ejpam-5253	196	20	)	)	PUNCT
ejpam-5253	196	21	⊆	⊆	NUM
ejpam-5253	196	22	rh(q	rh(q	NUM
ejpam-5253	196	23	)	)	PUNCT
ejpam-5253	196	24	.	.	PUNCT
ejpam-5253	197	1	(	(	PUNCT
ejpam-5253	197	2	iii	iii	X
ejpam-5253	197	3	)	)	PUNCT
ejpam-5253	197	4	k	k	NOUN
ejpam-5253	197	5	is	be	AUX
ejpam-5253	197	6	h	h	ADV
ejpam-5253	197	7	-	-	PUNCT
ejpam-5253	197	8	roughly	roughly	ADV
ejpam-5253	197	9	included	include	VERB
ejpam-5253	197	10	in	in	ADP
ejpam-5253	197	11	q	q	PROPN
ejpam-5253	197	12	(	(	PUNCT
ejpam-5253	197	13	k	k	PROPN
ejpam-5253	197	14	⊂̃h˜	⊂̃h˜	NOUN
ejpam-5253	197	15	q	q	PROPN
ejpam-5253	197	16	)	)	PUNCT
ejpam-5253	197	17	if	if	SCONJ
ejpam-5253	197	18	(	(	PUNCT
ejpam-5253	197	19	k	k	X
ejpam-5253	197	20	⊂h˜	⊂h˜	X
ejpam-5253	197	21	q	q	NOUN
ejpam-5253	197	22	)	)	PUNCT
ejpam-5253	197	23	and	and	CCONJ
ejpam-5253	197	24	(	(	PUNCT
ejpam-5253	197	25	k	k	X
ejpam-5253	197	26	⊂̃h	⊂̃h	X
ejpam-5253	197	27	q	q	PROPN
ejpam-5253	197	28	)	)	PUNCT
ejpam-5253	197	29	.	.	PUNCT
ejpam-5253	198	1	example	example	NOUN
ejpam-5253	199	1	7	7	NUM
ejpam-5253	199	2	.	.	PUNCT
ejpam-5253	200	1	in	in	ADP
ejpam-5253	200	2	example	example	NOUN
ejpam-5253	200	3	2	2	NUM
ejpam-5253	200	4	,	,	PUNCT
ejpam-5253	200	5	we	we	PRON
ejpam-5253	200	6	have	have	VERB
ejpam-5253	200	7	{	{	PUNCT
ejpam-5253	200	8	k	k	X
ejpam-5253	200	9	,	,	PUNCT
ejpam-5253	200	10	s	s	PART
ejpam-5253	200	11	}	}	PUNCT
ejpam-5253	200	12	is	be	AUX
ejpam-5253	200	13	h	h	ADJ
ejpam-5253	200	14	-	-	PUNCT
ejpam-5253	200	15	roughly	roughly	ADV
ejpam-5253	200	16	bottom	bottom	NOUN
ejpam-5253	200	17	included	include	VERB
ejpam-5253	200	18	in	in	ADP
ejpam-5253	200	19	{	{	PUNCT
ejpam-5253	200	20	k	k	NOUN
ejpam-5253	200	21	,	,	PUNCT
ejpam-5253	200	22	q	q	X
ejpam-5253	200	23	,	,	PUNCT
ejpam-5253	200	24	s	s	PART
ejpam-5253	200	25	}	}	PUNCT
ejpam-5253	200	26	.	.	PUNCT
ejpam-5253	201	1	also	also	ADV
ejpam-5253	201	2	,	,	PUNCT
ejpam-5253	201	3	{	{	PUNCT
ejpam-5253	201	4	s	s	X
ejpam-5253	201	5	,	,	PUNCT
ejpam-5253	201	6	t	t	PROPN
ejpam-5253	201	7	}	}	PUNCT
ejpam-5253	201	8	is	be	AUX
ejpam-5253	201	9	h	h	NOUN
ejpam-5253	201	10	-	-	PUNCT
ejpam-5253	201	11	roughly	roughly	ADV
ejpam-5253	201	12	top	top	NOUN
ejpam-5253	201	13	included	include	VERB
ejpam-5253	201	14	in	in	ADP
ejpam-5253	201	15	{	{	PUNCT
ejpam-5253	201	16	q	q	PROPN
ejpam-5253	201	17	,	,	PUNCT
ejpam-5253	201	18	s	s	PROPN
ejpam-5253	201	19	,	,	PUNCT
ejpam-5253	201	20	t	t	PROPN
ejpam-5253	201	21	}	}	PUNCT
ejpam-5253	201	22	.	.	PUNCT
ejpam-5253	202	1	5	5	X
ejpam-5253	202	2	.	.	X
ejpam-5253	202	3	h	h	NOUN
ejpam-5253	202	4	-	-	PUNCT
ejpam-5253	202	5	rough	rough	ADJ
ejpam-5253	202	6	sets	set	NOUN
ejpam-5253	202	7	in	in	ADP
ejpam-5253	202	8	this	this	DET
ejpam-5253	202	9	section	section	NOUN
ejpam-5253	202	10	,	,	PUNCT
ejpam-5253	202	11	we	we	PRON
ejpam-5253	202	12	introduce	introduce	VERB
ejpam-5253	202	13	a	a	DET
ejpam-5253	202	14	new	new	ADJ
ejpam-5253	202	15	concept	concept	NOUN
ejpam-5253	202	16	known	know	VERB
ejpam-5253	202	17	as	as	ADP
ejpam-5253	202	18	the	the	DET
ejpam-5253	202	19	h	h	NOUN
ejpam-5253	202	20	-	-	PUNCT
ejpam-5253	202	21	rough	rough	ADJ
ejpam-5253	202	22	set	set	NOUN
ejpam-5253	202	23	,	,	PUNCT
ejpam-5253	202	24	and	and	CCONJ
ejpam-5253	202	25	we	we	PRON
ejpam-5253	202	26	illustrate	illustrate	VERB
ejpam-5253	202	27	its	its	PRON
ejpam-5253	202	28	properties	property	NOUN
ejpam-5253	202	29	and	and	CCONJ
ejpam-5253	202	30	provide	provide	VERB
ejpam-5253	202	31	examples	example	NOUN
ejpam-5253	202	32	.	.	PUNCT
ejpam-5253	203	1	definition	definition	NOUN
ejpam-5253	203	2	13	13	NUM
ejpam-5253	203	3	.	.	PUNCT
ejpam-5253	204	1	let	let	VERB
ejpam-5253	204	2	(	(	PUNCT
ejpam-5253	204	3	m	m	PROPN
ejpam-5253	204	4	,	,	PUNCT
ejpam-5253	204	5	rh	rh	PROPN
ejpam-5253	204	6	)	)	PUNCT
ejpam-5253	204	7	be	be	VERB
ejpam-5253	204	8	h	h	NOUN
ejpam-5253	204	9	-	-	PUNCT
ejpam-5253	204	10	approximation	approximation	NOUN
ejpam-5253	204	11	space	space	NOUN
ejpam-5253	204	12	and	and	CCONJ
ejpam-5253	204	13	the	the	DET
ejpam-5253	204	14	set	set	NOUN
ejpam-5253	205	1	k	k	PROPN
ejpam-5253	205	2	⊆	⊆	NUM
ejpam-5253	205	3	m	m	VERB
ejpam-5253	205	4	is	be	AUX
ejpam-5253	205	5	called	call	VERB
ejpam-5253	205	6	:	:	PUNCT
ejpam-5253	205	7	(	(	PUNCT
ejpam-5253	205	8	i	i	NOUN
ejpam-5253	205	9	)	)	PUNCT
ejpam-5253	205	10	rh	rh	PROPN
ejpam-5253	205	11	-	-	PUNCT
ejpam-5253	205	12	definable	definable	ADJ
ejpam-5253	205	13	(	(	PUNCT
ejpam-5253	205	14	h	h	NOUN
ejpam-5253	205	15	-	-	PUNCT
ejpam-5253	205	16	exact	exact	ADJ
ejpam-5253	205	17	)	)	PUNCT
ejpam-5253	205	18	if	if	SCONJ
ejpam-5253	205	19	rh(k	rh(k	NOUN
ejpam-5253	205	20	)	)	PUNCT
ejpam-5253	205	21	=	=	SYM
ejpam-5253	205	22	rh(k	rh(k	X
ejpam-5253	205	23	)	)	PUNCT
ejpam-5253	205	24	or	or	CCONJ
ejpam-5253	205	25	bh(k	bh(k	NUM
ejpam-5253	205	26	)	)	PUNCT
ejpam-5253	206	1	=	=	SYM
ejpam-5253	206	2	ϕ.	ϕ.	PROPN
ejpam-5253	206	3	(	(	PUNCT
ejpam-5253	206	4	ii	ii	PROPN
ejpam-5253	206	5	)	)	PUNCT
ejpam-5253	206	6	h	h	NOUN
ejpam-5253	206	7	-	-	PUNCT
ejpam-5253	206	8	rough	rough	ADJ
ejpam-5253	206	9	if	if	SCONJ
ejpam-5253	206	10	rh(k	rh(k	NOUN
ejpam-5253	206	11	)	)	PUNCT
ejpam-5253	206	12	̸=	̸=	PROPN
ejpam-5253	206	13	rh(k	rh(k	NOUN
ejpam-5253	206	14	)	)	PUNCT
ejpam-5253	206	15	or	or	CCONJ
ejpam-5253	206	16	bh(k	bh(k	NUM
ejpam-5253	206	17	)	)	PUNCT
ejpam-5253	206	18	̸=	̸=	PROPN
ejpam-5253	206	19	ϕ.	ϕ.	PROPN
ejpam-5253	206	20	example	example	NOUN
ejpam-5253	206	21	8	8	NUM
ejpam-5253	206	22	.	.	PUNCT
ejpam-5253	207	1	in	in	ADP
ejpam-5253	207	2	example	example	NOUN
ejpam-5253	207	3	4	4	NUM
ejpam-5253	207	4	,	,	PUNCT
ejpam-5253	207	5	consider	consider	VERB
ejpam-5253	207	6	the	the	DET
ejpam-5253	207	7	h	h	NOUN
ejpam-5253	207	8	-	-	PUNCT
ejpam-5253	207	9	approximation	approximation	NOUN
ejpam-5253	207	10	space	space	NOUN
ejpam-5253	207	11	(	(	PUNCT
ejpam-5253	207	12	m	m	PROPN
ejpam-5253	207	13	,	,	PUNCT
ejpam-5253	207	14	rh	rh	PROPN
ejpam-5253	207	15	)	)	PUNCT
ejpam-5253	207	16	.	.	PUNCT
ejpam-5253	208	1	here	here	ADV
ejpam-5253	208	2	,	,	PUNCT
ejpam-5253	208	3	the	the	DET
ejpam-5253	208	4	set	set	NOUN
ejpam-5253	208	5	{	{	PUNCT
ejpam-5253	208	6	q	q	NOUN
ejpam-5253	208	7	,	,	PUNCT
ejpam-5253	208	8	s	s	PROPN
ejpam-5253	208	9	,	,	PUNCT
ejpam-5253	208	10	t	t	PROPN
ejpam-5253	208	11	}	}	PUNCT
ejpam-5253	208	12	is	be	AUX
ejpam-5253	208	13	h	h	NOUN
ejpam-5253	208	14	-	-	PUNCT
ejpam-5253	208	15	exact	exact	ADJ
ejpam-5253	208	16	,	,	PUNCT
ejpam-5253	208	17	whereas	whereas	SCONJ
ejpam-5253	208	18	{	{	PUNCT
ejpam-5253	208	19	q	q	NOUN
ejpam-5253	208	20	}	}	PUNCT
ejpam-5253	208	21	is	be	AUX
ejpam-5253	208	22	h	h	NOUN
ejpam-5253	208	23	-	-	PUNCT
ejpam-5253	208	24	rough	rough	ADJ
ejpam-5253	208	25	set	set	NOUN
ejpam-5253	208	26	.	.	PUNCT
ejpam-5253	209	1	proposition	proposition	NOUN
ejpam-5253	209	2	3	3	X
ejpam-5253	209	3	.	.	PUNCT
ejpam-5253	210	1	let	let	AUX
ejpam-5253	210	2	(	(	PUNCT
ejpam-5253	210	3	m	m	PROPN
ejpam-5253	210	4	,	,	PUNCT
ejpam-5253	210	5	rh	rh	PROPN
ejpam-5253	210	6	)	)	PUNCT
ejpam-5253	210	7	be	be	VERB
ejpam-5253	210	8	a	a	DET
ejpam-5253	210	9	h	h	NOUN
ejpam-5253	210	10	-	-	PUNCT
ejpam-5253	210	11	approximation	approximation	NOUN
ejpam-5253	210	12	space	space	NOUN
ejpam-5253	210	13	.	.	PUNCT
ejpam-5253	211	1	then	then	ADV
ejpam-5253	211	2	:	:	PUNCT
ejpam-5253	211	3	(	(	PUNCT
ejpam-5253	211	4	i	i	NOUN
ejpam-5253	211	5	)	)	PUNCT
ejpam-5253	211	6	every	every	DET
ejpam-5253	211	7	exact	exact	NOUN
ejpam-5253	211	8	set	set	VERB
ejpam-5253	211	9	in	in	ADP
ejpam-5253	211	10	m	m	PROPN
ejpam-5253	211	11	is	be	AUX
ejpam-5253	211	12	h	h	NOUN
ejpam-5253	211	13	-	-	PUNCT
ejpam-5253	211	14	exact	exact	ADJ
ejpam-5253	211	15	.	.	PUNCT
ejpam-5253	212	1	(	(	PUNCT
ejpam-5253	212	2	ii	ii	NOUN
ejpam-5253	212	3	)	)	PUNCT
ejpam-5253	212	4	every	every	DET
ejpam-5253	212	5	h	h	NOUN
ejpam-5253	212	6	-	-	PUNCT
ejpam-5253	212	7	rough	rough	ADJ
ejpam-5253	212	8	set	set	NOUN
ejpam-5253	212	9	in	in	ADP
ejpam-5253	212	10	m	m	PROPN
ejpam-5253	212	11	is	be	AUX
ejpam-5253	212	12	rough	rough	ADJ
ejpam-5253	212	13	.	.	PUNCT
ejpam-5253	213	1	proof	proof	NOUN
ejpam-5253	213	2	.	.	PUNCT
ejpam-5253	214	1	clear	clear	ADJ
ejpam-5253	214	2	.	.	PUNCT
ejpam-5253	215	1	the	the	DET
ejpam-5253	215	2	converse	converse	NOUN
ejpam-5253	215	3	of	of	ADP
ejpam-5253	215	4	all	all	DET
ejpam-5253	215	5	parts	part	NOUN
ejpam-5253	215	6	of	of	ADP
ejpam-5253	215	7	proposition	proposition	NOUN
ejpam-5253	215	8	3	3	NUM
ejpam-5253	215	9	may	may	AUX
ejpam-5253	215	10	not	not	PART
ejpam-5253	215	11	hold	hold	VERB
ejpam-5253	215	12	in	in	ADP
ejpam-5253	215	13	general	general	ADJ
ejpam-5253	215	14	as	as	SCONJ
ejpam-5253	215	15	demonstrated	demonstrate	VERB
ejpam-5253	215	16	in	in	ADP
ejpam-5253	215	17	the	the	DET
ejpam-5253	215	18	following	follow	VERB
ejpam-5253	215	19	example	example	NOUN
ejpam-5253	215	20	.	.	PUNCT
ejpam-5253	216	1	example	example	NOUN
ejpam-5253	217	1	9	9	NUM
ejpam-5253	217	2	.	.	PUNCT
ejpam-5253	218	1	in	in	ADP
ejpam-5253	218	2	example	example	NOUN
ejpam-5253	218	3	4	4	NUM
ejpam-5253	218	4	,	,	PUNCT
ejpam-5253	218	5	if	if	SCONJ
ejpam-5253	218	6	we	we	PRON
ejpam-5253	218	7	consider	consider	VERB
ejpam-5253	218	8	the	the	DET
ejpam-5253	218	9	h	h	NOUN
ejpam-5253	218	10	-	-	PUNCT
ejpam-5253	218	11	approximation	approximation	NOUN
ejpam-5253	218	12	space	space	NOUN
ejpam-5253	218	13	(	(	PUNCT
ejpam-5253	218	14	m	m	PROPN
ejpam-5253	218	15	,	,	PUNCT
ejpam-5253	218	16	rh	rh	PROPN
ejpam-5253	218	17	)	)	PUNCT
ejpam-5253	218	18	.	.	PUNCT
ejpam-5253	219	1	then	then	ADV
ejpam-5253	219	2	the	the	DET
ejpam-5253	219	3	set	set	NOUN
ejpam-5253	219	4	{	{	PUNCT
ejpam-5253	219	5	q	q	NOUN
ejpam-5253	219	6	,	,	PUNCT
ejpam-5253	219	7	s	s	PROPN
ejpam-5253	219	8	,	,	PUNCT
ejpam-5253	219	9	t	t	PROPN
ejpam-5253	219	10	}	}	PUNCT
ejpam-5253	219	11	is	be	AUX
ejpam-5253	219	12	h	h	NOUN
ejpam-5253	219	13	-	-	PUNCT
ejpam-5253	219	14	exact	exact	ADJ
ejpam-5253	219	15	but	but	CCONJ
ejpam-5253	219	16	not	not	PART
ejpam-5253	219	17	exact	exact	ADJ
ejpam-5253	219	18	and	and	CCONJ
ejpam-5253	219	19	the	the	DET
ejpam-5253	219	20	set	set	NOUN
ejpam-5253	219	21	{	{	PUNCT
ejpam-5253	219	22	k	k	NOUN
ejpam-5253	219	23	}	}	PUNCT
ejpam-5253	219	24	is	be	AUX
ejpam-5253	219	25	rough	rough	ADJ
ejpam-5253	219	26	but	but	CCONJ
ejpam-5253	219	27	not	not	PART
ejpam-5253	219	28	h	h	NOUN
ejpam-5253	219	29	-	-	PUNCT
ejpam-5253	219	30	rough	rough	ADJ
ejpam-5253	219	31	.	.	PUNCT
ejpam-5253	220	1	a.	a.	PROPN
ejpam-5253	220	2	al	al	PROPN
ejpam-5253	220	3	-	-	PUNCT
ejpam-5253	220	4	rehili	rehili	NOUN
ejpam-5253	220	5	/	/	SYM
ejpam-5253	220	6	eur	eur	PROPN
ejpam-5253	220	7	.	.	PUNCT
ejpam-5253	221	1	j.	j.	PROPN
ejpam-5253	221	2	pure	pure	PROPN
ejpam-5253	221	3	appl	appl	PROPN
ejpam-5253	221	4	.	.	PROPN
ejpam-5253	221	5	math	math	PROPN
ejpam-5253	221	6	,	,	PUNCT
ejpam-5253	221	7	17	17	NUM
ejpam-5253	221	8	(	(	PUNCT
ejpam-5253	221	9	3	3	NUM
ejpam-5253	221	10	)	)	PUNCT
ejpam-5253	221	11	(	(	PUNCT
ejpam-5253	221	12	2024	2024	NUM
ejpam-5253	221	13	)	)	PUNCT
ejpam-5253	221	14	,	,	PUNCT
ejpam-5253	221	15	1804	1804	NUM
ejpam-5253	221	16	-	-	SYM
ejpam-5253	221	17	1817	1817	NUM
ejpam-5253	221	18	1812	1812	NUM
ejpam-5253	221	19	remark	remark	NOUN
ejpam-5253	221	20	4	4	NUM
ejpam-5253	221	21	.	.	PUNCT
ejpam-5253	222	1	the	the	DET
ejpam-5253	222	2	intersection	intersection	NOUN
ejpam-5253	222	3	of	of	ADP
ejpam-5253	222	4	two	two	NUM
ejpam-5253	222	5	h	h	ADJ
ejpam-5253	222	6	-	-	PUNCT
ejpam-5253	222	7	exact	exact	ADJ
ejpam-5253	222	8	sets	set	NOUN
ejpam-5253	222	9	may	may	AUX
ejpam-5253	222	10	not	not	PART
ejpam-5253	222	11	necessarily	necessarily	ADV
ejpam-5253	222	12	result	result	VERB
ejpam-5253	222	13	in	in	ADP
ejpam-5253	222	14	a	a	DET
ejpam-5253	222	15	h	h	NOUN
ejpam-5253	222	16	-	-	PUNCT
ejpam-5253	222	17	exact	exact	ADJ
ejpam-5253	222	18	set	set	NOUN
ejpam-5253	222	19	.	.	PUNCT
ejpam-5253	222	20	example	example	NOUN
ejpam-5253	223	1	10	10	NUM
ejpam-5253	223	2	.	.	PUNCT
ejpam-5253	224	1	in	in	ADP
ejpam-5253	224	2	example	example	NOUN
ejpam-5253	224	3	4	4	NUM
ejpam-5253	224	4	,	,	PUNCT
ejpam-5253	224	5	consider	consider	VERB
ejpam-5253	224	6	the	the	DET
ejpam-5253	224	7	h	h	NOUN
ejpam-5253	224	8	-	-	PUNCT
ejpam-5253	224	9	approximation	approximation	NOUN
ejpam-5253	224	10	space	space	NOUN
ejpam-5253	224	11	(	(	PUNCT
ejpam-5253	224	12	m	m	PROPN
ejpam-5253	224	13	,	,	PUNCT
ejpam-5253	224	14	rh	rh	PROPN
ejpam-5253	224	15	)	)	PUNCT
ejpam-5253	224	16	.	.	PUNCT
ejpam-5253	225	1	we	we	PRON
ejpam-5253	225	2	have	have	VERB
ejpam-5253	225	3	{	{	PUNCT
ejpam-5253	225	4	q	q	NOUN
ejpam-5253	225	5	,	,	PUNCT
ejpam-5253	225	6	s	s	PROPN
ejpam-5253	225	7	,	,	PUNCT
ejpam-5253	225	8	t	t	PROPN
ejpam-5253	225	9	}	}	PUNCT
ejpam-5253	225	10	and	and	CCONJ
ejpam-5253	225	11	{	{	PUNCT
ejpam-5253	225	12	k	k	NOUN
ejpam-5253	225	13	}	}	PUNCT
ejpam-5253	225	14	are	be	AUX
ejpam-5253	225	15	two	two	NUM
ejpam-5253	225	16	h	h	ADJ
ejpam-5253	225	17	-	-	PUNCT
ejpam-5253	225	18	exact	exact	ADJ
ejpam-5253	225	19	sets	set	NOUN
ejpam-5253	225	20	but	but	CCONJ
ejpam-5253	226	1	{	{	PUNCT
ejpam-5253	226	2	q	q	X
ejpam-5253	226	3	,	,	PUNCT
ejpam-5253	226	4	s	s	PROPN
ejpam-5253	226	5	,	,	PUNCT
ejpam-5253	226	6	t	t	PROPN
ejpam-5253	226	7	}	}	PUNCT
ejpam-5253	226	8	∩	∩	NOUN
ejpam-5253	226	9	{	{	PUNCT
ejpam-5253	226	10	k	k	NOUN
ejpam-5253	226	11	}	}	PUNCT
ejpam-5253	226	12	=	=	SYM
ejpam-5253	226	13	ϕ	ϕ	NOUN
ejpam-5253	226	14	does	do	AUX
ejpam-5253	226	15	not	not	PART
ejpam-5253	226	16	h	h	NOUN
ejpam-5253	226	17	-	-	PUNCT
ejpam-5253	226	18	exact	exact	ADJ
ejpam-5253	226	19	.	.	PUNCT
ejpam-5253	227	1	definition	definition	NOUN
ejpam-5253	227	2	14	14	NUM
ejpam-5253	227	3	.	.	PUNCT
ejpam-5253	228	1	if	if	SCONJ
ejpam-5253	228	2	(	(	PUNCT
ejpam-5253	228	3	m	m	PROPN
ejpam-5253	228	4	,	,	PUNCT
ejpam-5253	228	5	rh	rh	PROPN
ejpam-5253	228	6	)	)	PUNCT
ejpam-5253	228	7	is	be	AUX
ejpam-5253	228	8	a	a	DET
ejpam-5253	228	9	h	h	NOUN
ejpam-5253	228	10	-	-	PUNCT
ejpam-5253	228	11	approximation	approximation	NOUN
ejpam-5253	228	12	space	space	NOUN
ejpam-5253	228	13	,	,	PUNCT
ejpam-5253	228	14	then	then	ADV
ejpam-5253	228	15	the	the	DET
ejpam-5253	228	16	set	set	NOUN
ejpam-5253	228	17	k	k	PROPN
ejpam-5253	228	18	⊆	⊆	NUM
ejpam-5253	228	19	m	m	VERB
ejpam-5253	228	20	is	be	AUX
ejpam-5253	228	21	called	call	VERB
ejpam-5253	228	22	:	:	PUNCT
ejpam-5253	228	23	(	(	PUNCT
ejpam-5253	228	24	i	i	NOUN
ejpam-5253	228	25	)	)	PUNCT
ejpam-5253	228	26	roughly	roughly	ADV
ejpam-5253	228	27	rh	rh	PROPN
ejpam-5253	228	28	-	-	PUNCT
ejpam-5253	228	29	definable	definable	ADJ
ejpam-5253	228	30	,	,	PUNCT
ejpam-5253	228	31	if	if	SCONJ
ejpam-5253	228	32	rh(k	rh(k	NOUN
ejpam-5253	228	33	)	)	PUNCT
ejpam-5253	228	34	̸=	̸=	PROPN
ejpam-5253	228	35	ϕ	ϕ	NOUN
ejpam-5253	228	36	and	and	CCONJ
ejpam-5253	228	37	rh(k	rh(k	NOUN
ejpam-5253	228	38	)	)	PUNCT
ejpam-5253	228	39	̸=	̸=	PROPN
ejpam-5253	228	40	m	m	NOUN
ejpam-5253	228	41	.	.	PUNCT
ejpam-5253	229	1	(	(	PUNCT
ejpam-5253	229	2	ii	ii	NOUN
ejpam-5253	229	3	)	)	PUNCT
ejpam-5253	229	4	internally	internally	ADV
ejpam-5253	229	5	rh	rh	PROPN
ejpam-5253	229	6	-	-	PUNCT
ejpam-5253	229	7	undefinable	undefinable	ADJ
ejpam-5253	229	8	,	,	PUNCT
ejpam-5253	229	9	if	if	SCONJ
ejpam-5253	229	10	rh(k	rh(k	ADJ
ejpam-5253	229	11	)	)	PUNCT
ejpam-5253	229	12	=	=	SYM
ejpam-5253	229	13	ϕ	ϕ	NOUN
ejpam-5253	229	14	and	and	CCONJ
ejpam-5253	229	15	rh(k	rh(k	ADJ
ejpam-5253	229	16	)	)	PUNCT
ejpam-5253	229	17	̸=	̸=	PROPN
ejpam-5253	229	18	m	m	NOUN
ejpam-5253	229	19	.	.	PUNCT
ejpam-5253	230	1	(	(	PUNCT
ejpam-5253	230	2	iii	iii	X
ejpam-5253	230	3	)	)	PUNCT
ejpam-5253	230	4	externally	externally	ADV
ejpam-5253	230	5	rh	rh	PROPN
ejpam-5253	230	6	-	-	PUNCT
ejpam-5253	230	7	undefinable	undefinable	ADJ
ejpam-5253	230	8	,	,	PUNCT
ejpam-5253	230	9	if	if	SCONJ
ejpam-5253	230	10	rh(k	rh(k	NOUN
ejpam-5253	230	11	)	)	PUNCT
ejpam-5253	230	12	̸=	̸=	PROPN
ejpam-5253	230	13	ϕ	ϕ	NOUN
ejpam-5253	230	14	and	and	CCONJ
ejpam-5253	230	15	rh(k	rh(k	NOUN
ejpam-5253	230	16	)	)	PUNCT
ejpam-5253	231	1	=	=	SYM
ejpam-5253	231	2	m	m	NOUN
ejpam-5253	231	3	.	.	PUNCT
ejpam-5253	232	1	(	(	PUNCT
ejpam-5253	232	2	iv	iv	X
ejpam-5253	232	3	)	)	PUNCT
ejpam-5253	232	4	totally	totally	ADV
ejpam-5253	232	5	rh	rh	NOUN
ejpam-5253	232	6	-	-	PUNCT
ejpam-5253	232	7	undefinable	undefinable	ADJ
ejpam-5253	232	8	,	,	PUNCT
ejpam-5253	232	9	if	if	SCONJ
ejpam-5253	232	10	rh(k	rh(k	ADJ
ejpam-5253	232	11	)	)	PUNCT
ejpam-5253	232	12	=	=	SYM
ejpam-5253	232	13	ϕ	ϕ	NOUN
ejpam-5253	232	14	and	and	CCONJ
ejpam-5253	232	15	rh(k	rh(k	NOUN
ejpam-5253	232	16	)	)	PUNCT
ejpam-5253	233	1	=	=	SYM
ejpam-5253	233	2	m	m	NOUN
ejpam-5253	233	3	.	.	PUNCT
ejpam-5253	234	1	the	the	DET
ejpam-5253	234	2	set	set	NOUN
ejpam-5253	234	3	of	of	ADP
ejpam-5253	234	4	all	all	DET
ejpam-5253	234	5	roughly	roughly	ADV
ejpam-5253	234	6	rh	rh	NOUN
ejpam-5253	234	7	-	-	PUNCT
ejpam-5253	234	8	definable	definable	ADJ
ejpam-5253	234	9	(	(	PUNCT
ejpam-5253	234	10	resp	resp	NOUN
ejpam-5253	234	11	.	.	PUNCT
ejpam-5253	235	1	internally	internally	ADV
ejpam-5253	235	2	rh	rh	VERB
ejpam-5253	235	3	-	-	PUNCT
ejpam-5253	235	4	undefinable	undefinable	ADJ
ejpam-5253	235	5	,	,	PUNCT
ejpam-5253	235	6	externally	externally	ADV
ejpam-5253	235	7	rhundefinable	rhundefinable	ADJ
ejpam-5253	235	8	and	and	CCONJ
ejpam-5253	235	9	totally	totally	ADV
ejpam-5253	235	10	rh	rh	PROPN
ejpam-5253	235	11	-	-	PUNCT
ejpam-5253	235	12	undefinable	undefinable	ADJ
ejpam-5253	235	13	)	)	PUNCT
ejpam-5253	235	14	sets	set	NOUN
ejpam-5253	235	15	is	be	AUX
ejpam-5253	235	16	denoted	denote	VERB
ejpam-5253	235	17	by	by	ADP
ejpam-5253	235	18	rdh(m	rdh(m	PROPN
ejpam-5253	235	19	)	)	PUNCT
ejpam-5253	235	20	(	(	PUNCT
ejpam-5253	235	21	resp	resp	NOUN
ejpam-5253	235	22	.	.	PUNCT
ejpam-5253	236	1	iudh(m	iudh(m	NOUN
ejpam-5253	236	2	)	)	PUNCT
ejpam-5253	236	3	,	,	PUNCT
ejpam-5253	236	4	eudh(m	eudh(m	NOUN
ejpam-5253	236	5	)	)	PUNCT
ejpam-5253	236	6	and	and	CCONJ
ejpam-5253	236	7	tudh(m	tudh(m	NOUN
ejpam-5253	236	8	)	)	PUNCT
ejpam-5253	236	9	)	)	PUNCT
ejpam-5253	236	10	.	.	PUNCT
ejpam-5253	237	1	remark	remark	NOUN
ejpam-5253	237	2	5	5	NUM
ejpam-5253	237	3	.	.	PUNCT
ejpam-5253	238	1	let	let	AUX
ejpam-5253	238	2	(	(	PUNCT
ejpam-5253	238	3	m	m	PROPN
ejpam-5253	238	4	,	,	PUNCT
ejpam-5253	238	5	rh	rh	PROPN
ejpam-5253	238	6	)	)	PUNCT
ejpam-5253	238	7	be	be	VERB
ejpam-5253	238	8	any	any	DET
ejpam-5253	238	9	h	h	NOUN
ejpam-5253	238	10	-	-	PUNCT
ejpam-5253	238	11	approximation	approximation	NOUN
ejpam-5253	238	12	space	space	NOUN
ejpam-5253	238	13	.	.	PUNCT
ejpam-5253	239	1	then	then	ADV
ejpam-5253	239	2	the	the	DET
ejpam-5253	239	3	following	follow	VERB
ejpam-5253	239	4	are	be	AUX
ejpam-5253	239	5	hold	hold	ADJ
ejpam-5253	239	6	:	:	PUNCT
ejpam-5253	239	7	(	(	PUNCT
ejpam-5253	239	8	i	i	NOUN
ejpam-5253	239	9	)	)	PUNCT
ejpam-5253	239	10	rdh(m	rdh(m	PROPN
ejpam-5253	239	11	)	)	PUNCT
ejpam-5253	239	12	⊇	⊇	NOUN
ejpam-5253	239	13	rd(m	rd(m	NOUN
ejpam-5253	239	14	)	)	PUNCT
ejpam-5253	239	15	.	.	PUNCT
ejpam-5253	240	1	(	(	PUNCT
ejpam-5253	240	2	ii	ii	X
ejpam-5253	240	3	)	)	PUNCT
ejpam-5253	240	4	iudh(m	iudh(m	PROPN
ejpam-5253	240	5	)	)	PUNCT
ejpam-5253	240	6	⊆	⊆	NUM
ejpam-5253	240	7	iud(m	iud(m	PROPN
ejpam-5253	240	8	)	)	PUNCT
ejpam-5253	240	9	.	.	PUNCT
ejpam-5253	241	1	(	(	PUNCT
ejpam-5253	241	2	iii	iii	X
ejpam-5253	241	3	)	)	PUNCT
ejpam-5253	241	4	eudh(m	eudh(m	NOUN
ejpam-5253	241	5	)	)	PUNCT
ejpam-5253	241	6	⊆	⊆	NUM
ejpam-5253	241	7	eud(m	eud(m	PROPN
ejpam-5253	241	8	)	)	PUNCT
ejpam-5253	241	9	.	.	PUNCT
ejpam-5253	242	1	(	(	PUNCT
ejpam-5253	242	2	iv	iv	X
ejpam-5253	242	3	)	)	PUNCT
ejpam-5253	242	4	tudh(m	tudh(m	NOUN
ejpam-5253	242	5	)	)	PUNCT
ejpam-5253	242	6	⊆	⊆	NUM
ejpam-5253	242	7	tud(m	tud(m	PROPN
ejpam-5253	242	8	)	)	PUNCT
ejpam-5253	242	9	.	.	PUNCT
ejpam-5253	243	1	example	example	NOUN
ejpam-5253	244	1	11	11	NUM
ejpam-5253	244	2	.	.	PUNCT
ejpam-5253	245	1	in	in	ADP
ejpam-5253	245	2	example	example	NOUN
ejpam-5253	245	3	3	3	NUM
ejpam-5253	245	4	,	,	PUNCT
ejpam-5253	245	5	we	we	PRON
ejpam-5253	245	6	have	have	VERB
ejpam-5253	245	7	the	the	DET
ejpam-5253	245	8	set	set	NOUN
ejpam-5253	245	9	{	{	PUNCT
ejpam-5253	245	10	k	k	NOUN
ejpam-5253	245	11	,	,	PUNCT
ejpam-5253	245	12	s	s	NOUN
ejpam-5253	245	13	}	}	PUNCT
ejpam-5253	245	14	∈	∈	PROPN
ejpam-5253	245	15	rdh(m	rdh(m	PROPN
ejpam-5253	245	16	)	)	PUNCT
ejpam-5253	245	17	but	but	CCONJ
ejpam-5253	245	18	{	{	PUNCT
ejpam-5253	245	19	k	k	X
ejpam-5253	245	20	,	,	PUNCT
ejpam-5253	245	21	s	s	NOUN
ejpam-5253	245	22	}	}	PUNCT
ejpam-5253	245	23	/∈	/∈	PUNCT
ejpam-5253	245	24	rd(m	rd(m	NOUN
ejpam-5253	245	25	)	)	PUNCT
ejpam-5253	245	26	.	.	PUNCT
ejpam-5253	246	1	the	the	DET
ejpam-5253	246	2	set	set	NOUN
ejpam-5253	246	3	{	{	PUNCT
ejpam-5253	246	4	s	s	NOUN
ejpam-5253	246	5	}	}	PUNCT
ejpam-5253	246	6	∈	∈	PROPN
ejpam-5253	246	7	iud(m	iud(m	PROPN
ejpam-5253	246	8	)	)	PUNCT
ejpam-5253	246	9	but	but	CCONJ
ejpam-5253	246	10	{	{	PUNCT
ejpam-5253	246	11	s	s	NOUN
ejpam-5253	246	12	}	}	PUNCT
ejpam-5253	246	13	/∈	/∈	PUNCT
ejpam-5253	246	14	iudh(m	iudh(m	NOUN
ejpam-5253	246	15	)	)	PUNCT
ejpam-5253	246	16	.	.	PUNCT
ejpam-5253	247	1	also	also	ADV
ejpam-5253	247	2	,	,	PUNCT
ejpam-5253	247	3	the	the	DET
ejpam-5253	247	4	set	set	NOUN
ejpam-5253	247	5	{	{	PUNCT
ejpam-5253	247	6	q	q	NOUN
ejpam-5253	247	7	,	,	PUNCT
ejpam-5253	247	8	s	s	NOUN
ejpam-5253	247	9	}	}	PUNCT
ejpam-5253	247	10	∈	∈	PROPN
ejpam-5253	247	11	eud(m	eud(m	PROPN
ejpam-5253	247	12	)	)	PUNCT
ejpam-5253	247	13	but	but	CCONJ
ejpam-5253	247	14	{	{	PUNCT
ejpam-5253	247	15	q	q	X
ejpam-5253	247	16	,	,	PUNCT
ejpam-5253	247	17	s	s	NOUN
ejpam-5253	247	18	}	}	PUNCT
ejpam-5253	247	19	∈	∈	PROPN
ejpam-5253	247	20	eudh(m	eudh(m	NOUN
ejpam-5253	247	21	)	)	PUNCT
ejpam-5253	247	22	.	.	PUNCT
ejpam-5253	248	1	proposition	proposition	NOUN
ejpam-5253	248	2	4	4	NUM
ejpam-5253	248	3	.	.	PUNCT
ejpam-5253	249	1	let	let	AUX
ejpam-5253	249	2	(	(	PUNCT
ejpam-5253	249	3	m	m	PROPN
ejpam-5253	249	4	,	,	PUNCT
ejpam-5253	249	5	rh	rh	PROPN
ejpam-5253	249	6	)	)	PUNCT
ejpam-5253	249	7	be	be	VERB
ejpam-5253	249	8	any	any	DET
ejpam-5253	249	9	h	h	NOUN
ejpam-5253	249	10	-	-	PUNCT
ejpam-5253	249	11	approximation	approximation	NOUN
ejpam-5253	249	12	space	space	NOUN
ejpam-5253	249	13	and	and	CCONJ
ejpam-5253	249	14	for	for	ADP
ejpam-5253	249	15	all	all	DET
ejpam-5253	249	16	m	m	PROPN
ejpam-5253	249	17	,	,	PUNCT
ejpam-5253	249	18	n	n	PROPN
ejpam-5253	249	19	∈	∈	NOUN
ejpam-5253	249	20	m	m	NOUN
ejpam-5253	249	21	,	,	PUNCT
ejpam-5253	249	22	if	if	SCONJ
ejpam-5253	249	23	m	m	VERB
ejpam-5253	249	24	∈	∈	NOUN
ejpam-5253	249	25	rh({n	rh({n	NOUN
ejpam-5253	249	26	}	}	PUNCT
ejpam-5253	249	27	)	)	PUNCT
ejpam-5253	249	28	and	and	CCONJ
ejpam-5253	249	29	n	n	DET
ejpam-5253	249	30	∈	∈	PROPN
ejpam-5253	249	31	rh({m	rh({m	NOUN
ejpam-5253	249	32	}	}	PUNCT
ejpam-5253	249	33	)	)	PUNCT
ejpam-5253	249	34	,	,	PUNCT
ejpam-5253	249	35	then	then	ADV
ejpam-5253	249	36	it	it	PRON
ejpam-5253	249	37	implies	imply	VERB
ejpam-5253	249	38	that	that	SCONJ
ejpam-5253	249	39	rh({m	rh({m	VERB
ejpam-5253	249	40	}	}	PUNCT
ejpam-5253	249	41	)	)	PUNCT
ejpam-5253	250	1	=	=	SYM
ejpam-5253	250	2	rh({n	rh({n	NOUN
ejpam-5253	250	3	}	}	PUNCT
ejpam-5253	250	4	)	)	PUNCT
ejpam-5253	250	5	.	.	PUNCT
ejpam-5253	251	1	proof	proof	NOUN
ejpam-5253	251	2	.	.	PUNCT
ejpam-5253	252	1	according	accord	VERB
ejpam-5253	252	2	to	to	ADP
ejpam-5253	252	3	the	the	DET
ejpam-5253	252	4	definition	definition	NOUN
ejpam-5253	252	5	,	,	PUNCT
ejpam-5253	252	6	the	the	DET
ejpam-5253	252	7	h	h	NOUN
ejpam-5253	252	8	-	-	PUNCT
ejpam-5253	252	9	upper	upper	ADJ
ejpam-5253	252	10	approximation	approximation	NOUN
ejpam-5253	252	11	of	of	ADP
ejpam-5253	252	12	a	a	DET
ejpam-5253	252	13	set	set	NOUN
ejpam-5253	252	14	is	be	AUX
ejpam-5253	252	15	the	the	DET
ejpam-5253	252	16	h	h	NOUN
ejpam-5253	252	17	-	-	PUNCT
ejpam-5253	252	18	closure	closure	NOUN
ejpam-5253	252	19	of	of	ADP
ejpam-5253	252	20	that	that	DET
ejpam-5253	252	21	set	set	NOUN
ejpam-5253	252	22	.	.	PUNCT
ejpam-5253	253	1	given	give	VERB
ejpam-5253	253	2	that	that	PRON
ejpam-5253	253	3	clh({n	clh({n	NOUN
ejpam-5253	253	4	}	}	PUNCT
ejpam-5253	253	5	)	)	PUNCT
ejpam-5253	253	6	is	be	AUX
ejpam-5253	253	7	a	a	DET
ejpam-5253	253	8	h	h	NOUN
ejpam-5253	253	9	-	-	PUNCT
ejpam-5253	253	10	closed	closed	ADJ
ejpam-5253	253	11	set	set	NOUN
ejpam-5253	253	12	containing	contain	VERB
ejpam-5253	253	13	m	m	PROPN
ejpam-5253	253	14	(	(	PUNCT
ejpam-5253	253	15	based	base	VERB
ejpam-5253	253	16	on	on	ADP
ejpam-5253	253	17	the	the	DET
ejpam-5253	253	18	condition	condition	NOUN
ejpam-5253	253	19	)	)	PUNCT
ejpam-5253	253	20	and	and	CCONJ
ejpam-5253	253	21	clh({m	clh({m	PROPN
ejpam-5253	253	22	}	}	PUNCT
ejpam-5253	253	23	)	)	PUNCT
ejpam-5253	254	1	is	be	AUX
ejpam-5253	254	2	the	the	DET
ejpam-5253	254	3	smallest	small	ADJ
ejpam-5253	254	4	h	h	NOUN
ejpam-5253	254	5	-	-	PUNCT
ejpam-5253	254	6	closed	closed	ADJ
ejpam-5253	254	7	set	set	NOUN
ejpam-5253	254	8	containing	contain	VERB
ejpam-5253	254	9	m	m	PRON
ejpam-5253	254	10	,	,	PUNCT
ejpam-5253	254	11	it	it	PRON
ejpam-5253	254	12	follows	follow	VERB
ejpam-5253	254	13	that	that	SCONJ
ejpam-5253	254	14	clh({m	clh({m	PROPN
ejpam-5253	254	15	}	}	PUNCT
ejpam-5253	254	16	)	)	PUNCT
ejpam-5253	254	17	⊆	⊆	NUM
ejpam-5253	254	18	clh({n	clh({n	NOUN
ejpam-5253	254	19	}	}	PUNCT
ejpam-5253	254	20	)	)	PUNCT
ejpam-5253	254	21	.	.	PUNCT
ejpam-5253	255	1	consequently	consequently	ADV
ejpam-5253	255	2	,	,	PUNCT
ejpam-5253	255	3	rh({m	rh({m	PROPN
ejpam-5253	255	4	}	}	PUNCT
ejpam-5253	255	5	)	)	PUNCT
ejpam-5253	256	1	⊆	⊆	X
ejpam-5253	256	2	rh({n	rh({n	NOUN
ejpam-5253	256	3	}	}	PUNCT
ejpam-5253	256	4	)	)	PUNCT
ejpam-5253	256	5	.	.	PUNCT
ejpam-5253	257	1	symmetrically	symmetrically	ADV
ejpam-5253	257	2	,	,	PUNCT
ejpam-5253	257	3	the	the	DET
ejpam-5253	257	4	reverse	reverse	ADJ
ejpam-5253	257	5	inclusion	inclusion	NOUN
ejpam-5253	257	6	holds	hold	VERB
ejpam-5253	257	7	:	:	PUNCT
ejpam-5253	257	8	clh({n	clh({n	NOUN
ejpam-5253	257	9	}	}	PUNCT
ejpam-5253	257	10	)	)	PUNCT
ejpam-5253	257	11	⊆	⊆	NUM
ejpam-5253	257	12	clh({m	clh({m	PROPN
ejpam-5253	257	13	}	}	PUNCT
ejpam-5253	257	14	)	)	PUNCT
ejpam-5253	257	15	.	.	PUNCT
ejpam-5253	258	1	thus	thus	ADV
ejpam-5253	258	2	,	,	PUNCT
ejpam-5253	258	3	rh({n	rh({n	VERB
ejpam-5253	258	4	}	}	PUNCT
ejpam-5253	258	5	)	)	PUNCT
ejpam-5253	258	6	⊆	⊆	NUM
ejpam-5253	258	7	rh({m	rh({m	NOUN
ejpam-5253	258	8	}	}	PUNCT
ejpam-5253	258	9	)	)	PUNCT
ejpam-5253	258	10	,	,	PUNCT
ejpam-5253	258	11	completing	complete	VERB
ejpam-5253	258	12	the	the	DET
ejpam-5253	258	13	proof	proof	NOUN
ejpam-5253	258	14	.	.	PUNCT
ejpam-5253	259	1	proposition	proposition	NOUN
ejpam-5253	259	2	5	5	NUM
ejpam-5253	259	3	.	.	PUNCT
ejpam-5253	260	1	let	let	AUX
ejpam-5253	260	2	(	(	PUNCT
ejpam-5253	260	3	m	m	PROPN
ejpam-5253	260	4	,	,	PUNCT
ejpam-5253	260	5	rh	rh	PROPN
ejpam-5253	260	6	)	)	PUNCT
ejpam-5253	260	7	be	be	VERB
ejpam-5253	260	8	a	a	DET
ejpam-5253	260	9	h	h	NOUN
ejpam-5253	260	10	-	-	PUNCT
ejpam-5253	260	11	approximation	approximation	NOUN
ejpam-5253	260	12	space	space	NOUN
ejpam-5253	260	13	,	,	PUNCT
ejpam-5253	260	14	where	where	SCONJ
ejpam-5253	260	15	every	every	DET
ejpam-5253	260	16	h	h	NOUN
ejpam-5253	260	17	-	-	PUNCT
ejpam-5253	260	18	open	open	ADJ
ejpam-5253	260	19	subset	subset	NOUN
ejpam-5253	260	20	k	k	PROPN
ejpam-5253	260	21	of	of	ADP
ejpam-5253	260	22	m	m	PROPN
ejpam-5253	260	23	is	be	AUX
ejpam-5253	260	24	h	h	NOUN
ejpam-5253	260	25	-	-	PUNCT
ejpam-5253	260	26	closed	closed	ADJ
ejpam-5253	260	27	.	.	PUNCT
ejpam-5253	261	1	if	if	SCONJ
ejpam-5253	261	2	n	n	NUM
ejpam-5253	261	3	∈	∈	PROPN
ejpam-5253	261	4	rh({m	rh({m	NOUN
ejpam-5253	261	5	}	}	PUNCT
ejpam-5253	261	6	)	)	PUNCT
ejpam-5253	261	7	,	,	PUNCT
ejpam-5253	261	8	then	then	ADV
ejpam-5253	261	9	it	it	PRON
ejpam-5253	261	10	implies	imply	VERB
ejpam-5253	261	11	that	that	SCONJ
ejpam-5253	261	12	m	m	PROPN
ejpam-5253	261	13	∈	∈	NOUN
ejpam-5253	261	14	rh({n	rh({n	NOUN
ejpam-5253	261	15	}	}	PUNCT
ejpam-5253	261	16	)	)	PUNCT
ejpam-5253	261	17	for	for	ADP
ejpam-5253	261	18	all	all	DET
ejpam-5253	261	19	m	m	PROPN
ejpam-5253	261	20	,	,	PUNCT
ejpam-5253	261	21	n	n	PROPN
ejpam-5253	261	22	∈	∈	PROPN
ejpam-5253	261	23	m	m	NOUN
ejpam-5253	261	24	.	.	PUNCT
ejpam-5253	262	1	proof	proof	NOUN
ejpam-5253	262	2	.	.	PUNCT
ejpam-5253	263	1	if	if	SCONJ
ejpam-5253	263	2	m	m	AUX
ejpam-5253	263	3	/∈	/∈	VERB
ejpam-5253	263	4	rh({n	rh({n	ADJ
ejpam-5253	263	5	}	}	PUNCT
ejpam-5253	263	6	)	)	PUNCT
ejpam-5253	263	7	,	,	PUNCT
ejpam-5253	263	8	then	then	ADV
ejpam-5253	263	9	there	there	PRON
ejpam-5253	263	10	exists	exist	VERB
ejpam-5253	263	11	a	a	DET
ejpam-5253	263	12	h	h	NOUN
ejpam-5253	263	13	-	-	PUNCT
ejpam-5253	263	14	open	open	ADJ
ejpam-5253	263	15	set	set	NOUN
ejpam-5253	263	16	h	h	NOUN
ejpam-5253	263	17	containing	contain	VERB
ejpam-5253	263	18	m	m	VERB
ejpam-5253	263	19	such	such	ADJ
ejpam-5253	263	20	that	that	SCONJ
ejpam-5253	263	21	h	h	NOUN
ejpam-5253	263	22	∩	∩	NOUN
ejpam-5253	263	23	{	{	PUNCT
ejpam-5253	263	24	m	m	NOUN
ejpam-5253	263	25	}	}	PUNCT
ejpam-5253	263	26	=	=	SYM
ejpam-5253	263	27	ϕ	ϕ	NOUN
ejpam-5253	263	28	,	,	PUNCT
ejpam-5253	263	29	implying	imply	VERB
ejpam-5253	263	30	that	that	SCONJ
ejpam-5253	263	31	{	{	PUNCT
ejpam-5253	263	32	n	n	CCONJ
ejpam-5253	263	33	}	}	PUNCT
ejpam-5253	263	34	⊆	⊆	NUM
ejpam-5253	263	35	(	(	PUNCT
ejpam-5253	263	36	m	m	PROPN
ejpam-5253	263	37	\	\	PROPN
ejpam-5253	263	38	h	h	NOUN
ejpam-5253	263	39	)	)	PUNCT
ejpam-5253	263	40	.	.	PUNCT
ejpam-5253	264	1	however	however	ADV
ejpam-5253	264	2	,	,	PUNCT
ejpam-5253	264	3	(	(	PUNCT
ejpam-5253	264	4	m	m	PROPN
ejpam-5253	264	5	\	\	ADJ
ejpam-5253	264	6	h	h	NOUN
ejpam-5253	264	7	)	)	PUNCT
ejpam-5253	264	8	is	be	AUX
ejpam-5253	264	9	both	both	CCONJ
ejpam-5253	264	10	a	a	DET
ejpam-5253	264	11	h	h	NOUN
ejpam-5253	264	12	-	-	PUNCT
ejpam-5253	264	13	closed	closed	ADJ
ejpam-5253	264	14	set	set	NOUN
ejpam-5253	264	15	and	and	CCONJ
ejpam-5253	264	16	also	also	ADV
ejpam-5253	264	17	is	be	AUX
ejpam-5253	264	18	a	a	DET
ejpam-5253	264	19	h	h	NOUN
ejpam-5253	264	20	-	-	PUNCT
ejpam-5253	264	21	open	open	ADJ
ejpam-5253	264	22	set	set	NOUN
ejpam-5253	264	23	that	that	PRON
ejpam-5253	264	24	does	do	AUX
ejpam-5253	264	25	not	not	PART
ejpam-5253	264	26	contain	contain	VERB
ejpam-5253	264	27	m.	m.	NOUN
ejpam-5253	264	28	therefore	therefore	ADV
ejpam-5253	264	29	,	,	PUNCT
ejpam-5253	264	30	(	(	PUNCT
ejpam-5253	264	31	m	m	PROPN
ejpam-5253	264	32	\h	\h	NUM
ejpam-5253	264	33	)	)	PUNCT
ejpam-5253	264	34	∩	∩	NOUN
ejpam-5253	264	35	{	{	PUNCT
ejpam-5253	264	36	m	m	NOUN
ejpam-5253	264	37	}	}	PUNCT
ejpam-5253	264	38	=	=	SYM
ejpam-5253	264	39	ϕ	ϕ	NOUN
ejpam-5253	264	40	,	,	PUNCT
ejpam-5253	264	41	which	which	PRON
ejpam-5253	264	42	means	mean	VERB
ejpam-5253	264	43	n	n	X
ejpam-5253	264	44	̸=	̸=	PROPN
ejpam-5253	264	45	rh({m	rh({m	VERB
ejpam-5253	264	46	}	}	PUNCT
ejpam-5253	264	47	)	)	PUNCT
ejpam-5253	264	48	.	.	PUNCT
ejpam-5253	265	1	a.	a.	PROPN
ejpam-5253	265	2	al	al	PROPN
ejpam-5253	265	3	-	-	PUNCT
ejpam-5253	265	4	rehili	rehili	NOUN
ejpam-5253	265	5	/	/	SYM
ejpam-5253	265	6	eur	eur	PROPN
ejpam-5253	265	7	.	.	PUNCT
ejpam-5253	266	1	j.	j.	PROPN
ejpam-5253	266	2	pure	pure	PROPN
ejpam-5253	266	3	appl	appl	PROPN
ejpam-5253	266	4	.	.	PROPN
ejpam-5253	266	5	math	math	PROPN
ejpam-5253	266	6	,	,	PUNCT
ejpam-5253	266	7	17	17	NUM
ejpam-5253	266	8	(	(	PUNCT
ejpam-5253	266	9	3	3	NUM
ejpam-5253	266	10	)	)	PUNCT
ejpam-5253	266	11	(	(	PUNCT
ejpam-5253	266	12	2024	2024	NUM
ejpam-5253	266	13	)	)	PUNCT
ejpam-5253	266	14	,	,	PUNCT
ejpam-5253	266	15	1804	1804	NUM
ejpam-5253	266	16	-	-	SYM
ejpam-5253	266	17	1817	1817	NUM
ejpam-5253	266	18	1813	1813	NUM
ejpam-5253	266	19	proposition	proposition	NOUN
ejpam-5253	266	20	6	6	NUM
ejpam-5253	266	21	.	.	PUNCT
ejpam-5253	267	1	let	let	AUX
ejpam-5253	267	2	(	(	PUNCT
ejpam-5253	267	3	m	m	PROPN
ejpam-5253	267	4	,	,	PUNCT
ejpam-5253	267	5	rh	rh	PROPN
ejpam-5253	267	6	)	)	PUNCT
ejpam-5253	267	7	be	be	VERB
ejpam-5253	267	8	a	a	DET
ejpam-5253	267	9	h	h	NOUN
ejpam-5253	267	10	-	-	PUNCT
ejpam-5253	267	11	approximation	approximation	NOUN
ejpam-5253	267	12	space	space	NOUN
ejpam-5253	267	13	,	,	PUNCT
ejpam-5253	267	14	where	where	SCONJ
ejpam-5253	267	15	every	every	DET
ejpam-5253	267	16	h	h	NOUN
ejpam-5253	267	17	-	-	PUNCT
ejpam-5253	267	18	open	open	ADJ
ejpam-5253	267	19	subset	subset	NOUN
ejpam-5253	267	20	k	k	PROPN
ejpam-5253	267	21	of	of	ADP
ejpam-5253	267	22	m	m	PROPN
ejpam-5253	267	23	is	be	AUX
ejpam-5253	267	24	h	h	NOUN
ejpam-5253	267	25	-	-	PUNCT
ejpam-5253	267	26	closed	closed	ADJ
ejpam-5253	267	27	.	.	PUNCT
ejpam-5253	268	1	then	then	ADV
ejpam-5253	268	2	the	the	DET
ejpam-5253	268	3	family	family	NOUN
ejpam-5253	268	4	of	of	ADP
ejpam-5253	268	5	sets	set	NOUN
ejpam-5253	268	6	{	{	PUNCT
ejpam-5253	268	7	rh({m	rh({m	NOUN
ejpam-5253	268	8	}	}	PUNCT
ejpam-5253	268	9	)	)	PUNCT
ejpam-5253	268	10	:	:	PUNCT
ejpam-5253	269	1	m	m	VERB
ejpam-5253	269	2	∈	∈	PROPN
ejpam-5253	269	3	k	k	NOUN
ejpam-5253	269	4	}	}	PUNCT
ejpam-5253	269	5	is	be	AUX
ejpam-5253	269	6	a	a	DET
ejpam-5253	269	7	partition	partition	NOUN
ejpam-5253	269	8	of	of	ADP
ejpam-5253	269	9	the	the	DET
ejpam-5253	269	10	set	set	NOUN
ejpam-5253	269	11	m	m	NOUN
ejpam-5253	269	12	.	.	PUNCT
ejpam-5253	270	1	proof	proof	NOUN
ejpam-5253	270	2	.	.	PUNCT
ejpam-5253	271	1	if	if	SCONJ
ejpam-5253	271	2	m	m	PROPN
ejpam-5253	271	3	,	,	PUNCT
ejpam-5253	271	4	n	n	CCONJ
ejpam-5253	271	5	,	,	PUNCT
ejpam-5253	271	6	p	p	PROPN
ejpam-5253	271	7	∈	∈	PROPN
ejpam-5253	271	8	k	k	PROPN
ejpam-5253	271	9	and	and	CCONJ
ejpam-5253	271	10	p	p	PROPN
ejpam-5253	271	11	∈	∈	PROPN
ejpam-5253	271	12	rh({m})∩rh({n	rh({m})∩rh({n	NOUN
ejpam-5253	271	13	}	}	PUNCT
ejpam-5253	271	14	)	)	PUNCT
ejpam-5253	271	15	,	,	PUNCT
ejpam-5253	271	16	then	then	ADV
ejpam-5253	271	17	p	p	PROPN
ejpam-5253	271	18	∈	∈	PROPN
ejpam-5253	271	19	rh({m	rh({m	NOUN
ejpam-5253	271	20	}	}	PUNCT
ejpam-5253	271	21	)	)	PUNCT
ejpam-5253	271	22	and	and	CCONJ
ejpam-5253	271	23	p	p	PROPN
ejpam-5253	271	24	∈	∈	PROPN
ejpam-5253	271	25	rh({n	rh({n	VERB
ejpam-5253	271	26	}	}	PUNCT
ejpam-5253	271	27	)	)	PUNCT
ejpam-5253	271	28	.	.	PUNCT
ejpam-5253	272	1	consequently	consequently	ADV
ejpam-5253	272	2	,	,	PUNCT
ejpam-5253	272	3	by	by	ADP
ejpam-5253	272	4	proposition	proposition	NOUN
ejpam-5253	272	5	5	5	NUM
ejpam-5253	272	6	,	,	PUNCT
ejpam-5253	272	7	m	m	PROPN
ejpam-5253	272	8	∈	∈	NOUN
ejpam-5253	272	9	rh({p	rh({p	NOUN
ejpam-5253	272	10	}	}	PUNCT
ejpam-5253	272	11	)	)	PUNCT
ejpam-5253	272	12	and	and	CCONJ
ejpam-5253	272	13	n	n	DET
ejpam-5253	272	14	∈	∈	PROPN
ejpam-5253	272	15	rh({p	rh({p	NOUN
ejpam-5253	272	16	}	}	PUNCT
ejpam-5253	272	17	)	)	PUNCT
ejpam-5253	272	18	.	.	PUNCT
ejpam-5253	273	1	by	by	ADP
ejpam-5253	273	2	proposition	proposition	NOUN
ejpam-5253	273	3	4	4	NUM
ejpam-5253	273	4	,	,	PUNCT
ejpam-5253	273	5	it	it	PRON
ejpam-5253	273	6	follows	follow	VERB
ejpam-5253	273	7	that	that	PRON
ejpam-5253	273	8	rh({m	rh({m	VERB
ejpam-5253	273	9	}	}	PUNCT
ejpam-5253	273	10	)	)	PUNCT
ejpam-5253	274	1	=	=	SYM
ejpam-5253	274	2	rh({p	rh({p	NOUN
ejpam-5253	274	3	}	}	PUNCT
ejpam-5253	274	4	)	)	PUNCT
ejpam-5253	274	5	and	and	CCONJ
ejpam-5253	274	6	rh({n	rh({n	NOUN
ejpam-5253	274	7	}	}	PUNCT
ejpam-5253	274	8	)	)	PUNCT
ejpam-5253	274	9	=	=	SYM
ejpam-5253	274	10	rh({p	rh({p	NOUN
ejpam-5253	274	11	}	}	PUNCT
ejpam-5253	274	12	)	)	PUNCT
ejpam-5253	274	13	.	.	PUNCT
ejpam-5253	275	1	therefore	therefore	ADV
ejpam-5253	275	2	rh({m	rh({m	VERB
ejpam-5253	275	3	}	}	PUNCT
ejpam-5253	275	4	)	)	PUNCT
ejpam-5253	276	1	=	=	SYM
ejpam-5253	276	2	rh({n	rh({n	NOUN
ejpam-5253	276	3	}	}	PUNCT
ejpam-5253	276	4	)	)	PUNCT
ejpam-5253	276	5	=	=	SYM
ejpam-5253	276	6	rh({p	rh({p	NOUN
ejpam-5253	276	7	}	}	PUNCT
ejpam-5253	276	8	)	)	PUNCT
ejpam-5253	276	9	.	.	PUNCT
ejpam-5253	277	1	hence	hence	ADV
ejpam-5253	277	2	either	either	CCONJ
ejpam-5253	277	3	rh({m	rh({m	NOUN
ejpam-5253	277	4	}	}	PUNCT
ejpam-5253	277	5	)	)	PUNCT
ejpam-5253	278	1	=	=	SYM
ejpam-5253	278	2	rh({n	rh({n	NOUN
ejpam-5253	278	3	}	}	PUNCT
ejpam-5253	278	4	)	)	PUNCT
ejpam-5253	278	5	or	or	CCONJ
ejpam-5253	278	6	rh({m	rh({m	NOUN
ejpam-5253	278	7	}	}	PUNCT
ejpam-5253	278	8	)	)	PUNCT
ejpam-5253	278	9	∩rh({n	∩rh({n	PROPN
ejpam-5253	278	10	}	}	PUNCT
ejpam-5253	278	11	)	)	PUNCT
ejpam-5253	279	1	=	=	SYM
ejpam-5253	279	2	ϕ.	ϕ.	PROPN
ejpam-5253	279	3	6	6	NUM
ejpam-5253	279	4	.	.	PUNCT
ejpam-5253	280	1	properties	property	NOUN
ejpam-5253	280	2	of	of	ADP
ejpam-5253	280	3	h	h	NOUN
ejpam-5253	280	4	-	-	PUNCT
ejpam-5253	280	5	approximation	approximation	NOUN
ejpam-5253	280	6	spaces	space	NOUN
ejpam-5253	280	7	in	in	ADP
ejpam-5253	280	8	this	this	DET
ejpam-5253	280	9	section	section	NOUN
ejpam-5253	280	10	,	,	PUNCT
ejpam-5253	280	11	we	we	PRON
ejpam-5253	280	12	introduce	introduce	VERB
ejpam-5253	280	13	some	some	DET
ejpam-5253	280	14	properties	property	NOUN
ejpam-5253	280	15	of	of	ADP
ejpam-5253	280	16	h	h	NOUN
ejpam-5253	280	17	-	-	PUNCT
ejpam-5253	280	18	approximation	approximation	NOUN
ejpam-5253	280	19	spaces	space	NOUN
ejpam-5253	280	20	and	and	CCONJ
ejpam-5253	280	21	provide	provide	VERB
ejpam-5253	280	22	counterexamples	counterexample	NOUN
ejpam-5253	280	23	.	.	PUNCT
ejpam-5253	281	1	proposition	proposition	NOUN
ejpam-5253	281	2	7	7	NUM
ejpam-5253	281	3	.	.	PUNCT
ejpam-5253	282	1	let	let	VERB
ejpam-5253	282	2	(	(	PUNCT
ejpam-5253	282	3	m	m	PROPN
ejpam-5253	282	4	,	,	PUNCT
ejpam-5253	282	5	rh	rh	PROPN
ejpam-5253	282	6	)	)	PUNCT
ejpam-5253	282	7	be	be	VERB
ejpam-5253	282	8	h	h	NOUN
ejpam-5253	282	9	-	-	PUNCT
ejpam-5253	282	10	approximation	approximation	NOUN
ejpam-5253	282	11	space	space	NOUN
ejpam-5253	282	12	and	and	CCONJ
ejpam-5253	282	13	k	k	NOUN
ejpam-5253	282	14	,	,	PUNCT
ejpam-5253	282	15	q	q	X
ejpam-5253	282	16	⊆	⊆	NUM
ejpam-5253	282	17	m	m	NOUN
ejpam-5253	282	18	.	.	PUNCT
ejpam-5253	283	1	then	then	ADV
ejpam-5253	283	2	(	(	PUNCT
ejpam-5253	283	3	i	i	NOUN
ejpam-5253	283	4	)	)	PUNCT
ejpam-5253	283	5	rh(k	rh(k	X
ejpam-5253	283	6	)	)	PUNCT
ejpam-5253	284	1	⊆	⊆	NUM
ejpam-5253	284	2	k	k	NOUN
ejpam-5253	284	3	⊆	⊆	NUM
ejpam-5253	284	4	rh(k	rh(k	NOUN
ejpam-5253	284	5	)	)	PUNCT
ejpam-5253	284	6	.	.	PUNCT
ejpam-5253	285	1	(	(	PUNCT
ejpam-5253	285	2	ii	ii	NOUN
ejpam-5253	285	3	)	)	PUNCT
ejpam-5253	285	4	rh(ϕ	rh(ϕ	PUNCT
ejpam-5253	285	5	)	)	PUNCT
ejpam-5253	285	6	=	=	SYM
ejpam-5253	285	7	rh(ϕ	rh(ϕ	X
ejpam-5253	285	8	)	)	PUNCT
ejpam-5253	285	9	=	=	SYM
ejpam-5253	285	10	ϕ	ϕ	NOUN
ejpam-5253	285	11	,	,	PUNCT
ejpam-5253	285	12	rh(m	rh(m	NOUN
ejpam-5253	285	13	)	)	PUNCT
ejpam-5253	285	14	=	=	SYM
ejpam-5253	285	15	rh(m	rh(m	X
ejpam-5253	285	16	)	)	PUNCT
ejpam-5253	285	17	=	=	PUNCT
ejpam-5253	285	18	m	m	NOUN
ejpam-5253	285	19	.	.	PUNCT
ejpam-5253	286	1	(	(	PUNCT
ejpam-5253	286	2	iii	iii	X
ejpam-5253	286	3	)	)	PUNCT
ejpam-5253	286	4	if	if	SCONJ
ejpam-5253	286	5	k	k	PROPN
ejpam-5253	287	1	⊆	⊆	NUM
ejpam-5253	287	2	q	q	NOUN
ejpam-5253	287	3	then	then	ADV
ejpam-5253	287	4	rh(k	rh(k	NOUN
ejpam-5253	287	5	)	)	PUNCT
ejpam-5253	287	6	⊆	⊆	NUM
ejpam-5253	287	7	rh(q	rh(q	NUM
ejpam-5253	287	8	)	)	PUNCT
ejpam-5253	287	9	and	and	CCONJ
ejpam-5253	287	10	rh(k	rh(k	NOUN
ejpam-5253	287	11	)	)	PUNCT
ejpam-5253	287	12	⊆	⊆	NUM
ejpam-5253	287	13	rh(q	rh(q	NUM
ejpam-5253	287	14	)	)	PUNCT
ejpam-5253	287	15	.	.	PUNCT
ejpam-5253	288	1	proof	proof	NOUN
ejpam-5253	288	2	.	.	PUNCT
ejpam-5253	289	1	(	(	PUNCT
ejpam-5253	289	2	i	i	NOUN
ejpam-5253	289	3	)	)	PUNCT
ejpam-5253	289	4	let	let	VERB
ejpam-5253	289	5	m	m	PRON
ejpam-5253	289	6	∈	∈	PROPN
ejpam-5253	289	7	rh(k	rh(k	NOUN
ejpam-5253	289	8	)	)	PUNCT
ejpam-5253	289	9	which	which	PRON
ejpam-5253	289	10	mean	mean	VERB
ejpam-5253	289	11	that	that	SCONJ
ejpam-5253	289	12	m	m	VERB
ejpam-5253	289	13	∈	∈	VERB
ejpam-5253	289	14	∪{h	∪{h	PROPN
ejpam-5253	289	15	∈	∈	NOUN
ejpam-5253	289	16	σh	σh	PROPN
ejpam-5253	289	17	,	,	PUNCT
ejpam-5253	289	18	h	h	NOUN
ejpam-5253	289	19	⊆	⊆	NUM
ejpam-5253	289	20	k	k	NOUN
ejpam-5253	289	21	}	}	PUNCT
ejpam-5253	289	22	.	.	PUNCT
ejpam-5253	290	1	then	then	ADV
ejpam-5253	290	2	there	there	PRON
ejpam-5253	290	3	exists	exist	VERB
ejpam-5253	290	4	h0	h0	PROPN
ejpam-5253	290	5	∈	∈	PROPN
ejpam-5253	290	6	σh	σh	ADP
ejpam-5253	290	7	such	such	ADJ
ejpam-5253	290	8	that	that	SCONJ
ejpam-5253	290	9	m	m	PROPN
ejpam-5253	290	10	∈	∈	PROPN
ejpam-5253	290	11	h0	h0	NOUN
ejpam-5253	290	12	⊆	⊆	PROPN
ejpam-5253	290	13	k.	k.	PROPN
ejpam-5253	290	14	thus	thus	ADV
ejpam-5253	290	15	m	m	PROPN
ejpam-5253	290	16	∈	∈	PROPN
ejpam-5253	290	17	k.	k.	NOUN
ejpam-5253	290	18	hence	hence	ADV
ejpam-5253	290	19	rh(k	rh(k	NOUN
ejpam-5253	290	20	)	)	PUNCT
ejpam-5253	290	21	⊆	⊆	NUM
ejpam-5253	290	22	k.	k.	PROPN
ejpam-5253	290	23	also	also	ADV
ejpam-5253	290	24	,	,	PUNCT
ejpam-5253	290	25	let	let	VERB
ejpam-5253	290	26	m	m	PRON
ejpam-5253	290	27	∈	∈	VERB
ejpam-5253	290	28	m	m	NOUN
ejpam-5253	290	29	and	and	CCONJ
ejpam-5253	290	30	by	by	ADP
ejpam-5253	290	31	definition	definition	NOUN
ejpam-5253	290	32	of	of	ADP
ejpam-5253	290	33	rh(k	rh(k	NOUN
ejpam-5253	290	34	)	)	PUNCT
ejpam-5253	290	35	=	=	SYM
ejpam-5253	290	36	∩{f	∩{f	NOUN
ejpam-5253	290	37	∈	∈	PROPN
ejpam-5253	290	38	σhc	σhc	NOUN
ejpam-5253	290	39	,	,	PUNCT
ejpam-5253	290	40	k	k	PROPN
ejpam-5253	290	41	⊆	⊆	NUM
ejpam-5253	290	42	f	f	X
ejpam-5253	290	43	}	}	PUNCT
ejpam-5253	290	44	,	,	PUNCT
ejpam-5253	290	45	then	then	ADV
ejpam-5253	290	46	m	m	VERB
ejpam-5253	290	47	∈	∈	PROPN
ejpam-5253	290	48	f	f	PROPN
ejpam-5253	290	49	for	for	ADP
ejpam-5253	290	50	all	all	DET
ejpam-5253	290	51	f	f	PROPN
ejpam-5253	290	52	∈	∈	PROPN
ejpam-5253	290	53	σhc	σhc	NOUN
ejpam-5253	290	54	.	.	PUNCT
ejpam-5253	291	1	hence	hence	ADV
ejpam-5253	291	2	k	k	X
ejpam-5253	291	3	⊆	⊆	NUM
ejpam-5253	291	4	rh(k	rh(k	NOUN
ejpam-5253	291	5	)	)	PUNCT
ejpam-5253	291	6	.	.	PUNCT
ejpam-5253	292	1	(	(	PUNCT
ejpam-5253	292	2	ii	ii	X
ejpam-5253	292	3	)	)	PUNCT
ejpam-5253	292	4	it	it	PRON
ejpam-5253	292	5	directly	directly	ADV
ejpam-5253	292	6	follows	follow	VERB
ejpam-5253	292	7	.	.	PUNCT
ejpam-5253	293	1	(	(	PUNCT
ejpam-5253	293	2	iii	iii	X
ejpam-5253	293	3	)	)	PUNCT
ejpam-5253	293	4	let	let	VERB
ejpam-5253	293	5	m	m	PRON
ejpam-5253	293	6	∈	∈	PROPN
ejpam-5253	293	7	rh(k	rh(k	NOUN
ejpam-5253	293	8	)	)	PUNCT
ejpam-5253	293	9	,	,	PUNCT
ejpam-5253	293	10	by	by	ADP
ejpam-5253	293	11	definition	definition	NOUN
ejpam-5253	293	12	of	of	ADP
ejpam-5253	293	13	h	h	NOUN
ejpam-5253	293	14	-	-	PUNCT
ejpam-5253	293	15	lower	low	ADJ
ejpam-5253	293	16	approximation	approximation	NOUN
ejpam-5253	293	17	of	of	ADP
ejpam-5253	293	18	k	k	PROPN
ejpam-5253	293	19	,	,	PUNCT
ejpam-5253	293	20	we	we	PRON
ejpam-5253	293	21	have	have	VERB
ejpam-5253	293	22	m	m	NOUN
ejpam-5253	293	23	∈	∈	ADJ
ejpam-5253	293	24	∪{h	∪{h	X
ejpam-5253	293	25	∈	∈	PROPN
ejpam-5253	293	26	σh	σh	PROPN
ejpam-5253	293	27	,	,	PUNCT
ejpam-5253	293	28	h	h	NOUN
ejpam-5253	294	1	⊆	⊆	NUM
ejpam-5253	294	2	k	k	NOUN
ejpam-5253	294	3	}	}	PUNCT
ejpam-5253	294	4	but	but	CCONJ
ejpam-5253	294	5	k	k	PROPN
ejpam-5253	294	6	⊆	⊆	NUM
ejpam-5253	294	7	q	q	NOUN
ejpam-5253	294	8	,	,	PUNCT
ejpam-5253	294	9	thus	thus	ADV
ejpam-5253	294	10	h	h	NOUN
ejpam-5253	295	1	⊆	⊆	NUM
ejpam-5253	295	2	q	q	NOUN
ejpam-5253	295	3	and	and	CCONJ
ejpam-5253	295	4	m	m	PROPN
ejpam-5253	295	5	∈	∈	PROPN
ejpam-5253	295	6	h	h	NOUN
ejpam-5253	295	7	,	,	PUNCT
ejpam-5253	295	8	then	then	ADV
ejpam-5253	295	9	m	m	NOUN
ejpam-5253	295	10	∈	∈	PROPN
ejpam-5253	295	11	rh(q	rh(q	NOUN
ejpam-5253	295	12	)	)	PUNCT
ejpam-5253	295	13	.	.	PUNCT
ejpam-5253	296	1	also	also	ADV
ejpam-5253	296	2	,	,	PUNCT
ejpam-5253	296	3	let	let	VERB
ejpam-5253	296	4	m	m	PRON
ejpam-5253	296	5	̸=	̸=	PROPN
ejpam-5253	296	6	rh(q	rh(q	PUNCT
ejpam-5253	296	7	)	)	PUNCT
ejpam-5253	296	8	this	this	PRON
ejpam-5253	296	9	means	mean	VERB
ejpam-5253	296	10	that	that	SCONJ
ejpam-5253	296	11	m	m	PROPN
ejpam-5253	296	12	̸=	̸=	PROPN
ejpam-5253	296	13	∩{f	∩{f	NOUN
ejpam-5253	296	14	∈	∈	PROPN
ejpam-5253	296	15	σhc	σhc	NOUN
ejpam-5253	296	16	,	,	PUNCT
ejpam-5253	296	17	q	q	NOUN
ejpam-5253	296	18	⊆	⊆	NUM
ejpam-5253	296	19	f	f	X
ejpam-5253	296	20	}	}	PUNCT
ejpam-5253	296	21	then	then	ADV
ejpam-5253	296	22	,	,	PUNCT
ejpam-5253	296	23	there	there	PRON
ejpam-5253	296	24	exists	exist	VERB
ejpam-5253	296	25	f	f	PROPN
ejpam-5253	296	26	∈	∈	PROPN
ejpam-5253	296	27	σhc	σhc	NOUN
ejpam-5253	296	28	,	,	PUNCT
ejpam-5253	296	29	q	q	NOUN
ejpam-5253	296	30	⊆	⊆	NUM
ejpam-5253	296	31	f	f	NOUN
ejpam-5253	296	32	and	and	CCONJ
ejpam-5253	296	33	m	m	PROPN
ejpam-5253	296	34	/∈	/∈	PUNCT
ejpam-5253	297	1	f	f	PROPN
ejpam-5253	297	2	which	which	PRON
ejpam-5253	297	3	means	mean	VERB
ejpam-5253	297	4	that	that	SCONJ
ejpam-5253	297	5	,	,	PUNCT
ejpam-5253	297	6	there	there	PRON
ejpam-5253	297	7	exists	exist	VERB
ejpam-5253	297	8	f	f	PROPN
ejpam-5253	297	9	∈	∈	PROPN
ejpam-5253	297	10	σhc	σhc	PROPN
ejpam-5253	297	11	,	,	PUNCT
ejpam-5253	297	12	k	k	PROPN
ejpam-5253	297	13	⊆	⊆	NUM
ejpam-5253	297	14	q	q	SYM
ejpam-5253	297	15	⊆	⊆	NUM
ejpam-5253	297	16	f	f	NOUN
ejpam-5253	297	17	and	and	CCONJ
ejpam-5253	297	18	m	m	PROPN
ejpam-5253	297	19	/∈	/∈	PUNCT
ejpam-5253	298	1	f	f	PROPN
ejpam-5253	298	2	which	which	PRON
ejpam-5253	298	3	implies	imply	VERB
ejpam-5253	298	4	m	m	VERB
ejpam-5253	298	5	/∈	/∈	PROPN
ejpam-5253	298	6	∩{f	∩{f	PROPN
ejpam-5253	298	7	∈	∈	PROPN
ejpam-5253	298	8	σhc	σhc	NOUN
ejpam-5253	298	9	,	,	PUNCT
ejpam-5253	298	10	k	k	PROPN
ejpam-5253	298	11	⊆	⊆	NUM
ejpam-5253	298	12	f	f	X
ejpam-5253	298	13	}	}	PUNCT
ejpam-5253	298	14	,	,	PUNCT
ejpam-5253	298	15	thus	thus	ADV
ejpam-5253	298	16	m	m	ADJ
ejpam-5253	298	17	/∈	/∈	INTJ
ejpam-5253	298	18	rh(k	rh(k	ADJ
ejpam-5253	298	19	)	)	PUNCT
ejpam-5253	298	20	.	.	PUNCT
ejpam-5253	299	1	therefore	therefore	ADV
ejpam-5253	299	2	rh(k	rh(k	NOUN
ejpam-5253	299	3	)	)	PUNCT
ejpam-5253	299	4	⊆	⊆	NUM
ejpam-5253	299	5	rh(q	rh(q	NUM
ejpam-5253	299	6	)	)	PUNCT
ejpam-5253	299	7	.	.	PUNCT
ejpam-5253	300	1	proposition	proposition	NOUN
ejpam-5253	300	2	8	8	NUM
ejpam-5253	300	3	.	.	PUNCT
ejpam-5253	301	1	let	let	AUX
ejpam-5253	301	2	(	(	PUNCT
ejpam-5253	301	3	m	m	PROPN
ejpam-5253	301	4	,	,	PUNCT
ejpam-5253	301	5	rh	rh	PROPN
ejpam-5253	301	6	)	)	PUNCT
ejpam-5253	301	7	be	be	VERB
ejpam-5253	301	8	a	a	DET
ejpam-5253	301	9	h	h	NOUN
ejpam-5253	301	10	-	-	PUNCT
ejpam-5253	301	11	approximation	approximation	NOUN
ejpam-5253	301	12	space	space	NOUN
ejpam-5253	301	13	and	and	CCONJ
ejpam-5253	301	14	k	k	NOUN
ejpam-5253	301	15	,	,	PUNCT
ejpam-5253	301	16	q	q	X
ejpam-5253	301	17	⊆	⊆	NUM
ejpam-5253	301	18	m	m	NOUN
ejpam-5253	301	19	.	.	PUNCT
ejpam-5253	302	1	then	then	ADV
ejpam-5253	302	2	(	(	PUNCT
ejpam-5253	302	3	i	i	NOUN
ejpam-5253	302	4	)	)	PUNCT
ejpam-5253	302	5	rh(m	rh(m	VERB
ejpam-5253	302	6	\k	\k	NOUN
ejpam-5253	302	7	)	)	PUNCT
ejpam-5253	303	1	=	=	PUNCT
ejpam-5253	303	2	m	m	NOUN
ejpam-5253	303	3	\rh(k	\rh(k	PROPN
ejpam-5253	303	4	)	)	PUNCT
ejpam-5253	303	5	.	.	PUNCT
ejpam-5253	304	1	(	(	PUNCT
ejpam-5253	304	2	ii	ii	NOUN
ejpam-5253	304	3	)	)	PUNCT
ejpam-5253	304	4	rh(x	rh(x	PUNCT
ejpam-5253	304	5	\k	\k	NOUN
ejpam-5253	304	6	)	)	PUNCT
ejpam-5253	305	1	=	=	PUNCT
ejpam-5253	305	2	m	m	NOUN
ejpam-5253	305	3	\rh(k	\rh(k	PROPN
ejpam-5253	305	4	)	)	PUNCT
ejpam-5253	305	5	.	.	PUNCT
ejpam-5253	306	1	(	(	PUNCT
ejpam-5253	306	2	iii	iii	NOUN
ejpam-5253	306	3	)	)	PUNCT
ejpam-5253	306	4	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	306	5	)	)	PUNCT
ejpam-5253	306	6	)	)	PUNCT
ejpam-5253	307	1	=	=	SYM
ejpam-5253	307	2	rh(k	rh(k	X
ejpam-5253	307	3	)	)	PUNCT
ejpam-5253	307	4	.	.	PUNCT
ejpam-5253	308	1	(	(	PUNCT
ejpam-5253	308	2	iv	iv	X
ejpam-5253	308	3	)	)	PUNCT
ejpam-5253	308	4	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	308	5	)	)	PUNCT
ejpam-5253	308	6	)	)	PUNCT
ejpam-5253	309	1	=	=	SYM
ejpam-5253	309	2	rh(k	rh(k	X
ejpam-5253	309	3	)	)	PUNCT
ejpam-5253	309	4	.	.	PUNCT
ejpam-5253	310	1	(	(	PUNCT
ejpam-5253	310	2	v	v	NOUN
ejpam-5253	310	3	)	)	PUNCT
ejpam-5253	310	4	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	310	5	)	)	PUNCT
ejpam-5253	310	6	)	)	PUNCT
ejpam-5253	311	1	⊆	⊆	NUM
ejpam-5253	311	2	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	311	3	)	)	PUNCT
ejpam-5253	311	4	)	)	PUNCT
ejpam-5253	311	5	.	.	PUNCT
ejpam-5253	312	1	a.	a.	PROPN
ejpam-5253	312	2	al	al	PROPN
ejpam-5253	312	3	-	-	PUNCT
ejpam-5253	312	4	rehili	rehili	NOUN
ejpam-5253	312	5	/	/	SYM
ejpam-5253	312	6	eur	eur	PROPN
ejpam-5253	312	7	.	.	PUNCT
ejpam-5253	313	1	j.	j.	PROPN
ejpam-5253	313	2	pure	pure	PROPN
ejpam-5253	313	3	appl	appl	PROPN
ejpam-5253	313	4	.	.	PROPN
ejpam-5253	313	5	math	math	PROPN
ejpam-5253	313	6	,	,	PUNCT
ejpam-5253	313	7	17	17	NUM
ejpam-5253	313	8	(	(	PUNCT
ejpam-5253	313	9	3	3	NUM
ejpam-5253	313	10	)	)	PUNCT
ejpam-5253	313	11	(	(	PUNCT
ejpam-5253	313	12	2024	2024	NUM
ejpam-5253	313	13	)	)	PUNCT
ejpam-5253	313	14	,	,	PUNCT
ejpam-5253	313	15	1804	1804	NUM
ejpam-5253	313	16	-	-	SYM
ejpam-5253	313	17	1817	1817	NUM
ejpam-5253	313	18	1814	1814	NUM
ejpam-5253	313	19	(	(	PUNCT
ejpam-5253	313	20	vi	vi	NOUN
ejpam-5253	313	21	)	)	PUNCT
ejpam-5253	313	22	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	313	23	)	)	PUNCT
ejpam-5253	313	24	)	)	PUNCT
ejpam-5253	314	1	⊆	⊆	NUM
ejpam-5253	314	2	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	314	3	)	)	PUNCT
ejpam-5253	314	4	)	)	PUNCT
ejpam-5253	314	5	.	.	PUNCT
ejpam-5253	315	1	proof	proof	NOUN
ejpam-5253	315	2	.	.	PUNCT
ejpam-5253	316	1	(	(	PUNCT
ejpam-5253	316	2	i	i	NOUN
ejpam-5253	316	3	)	)	PUNCT
ejpam-5253	316	4	let	let	VERB
ejpam-5253	316	5	m	m	PRON
ejpam-5253	316	6	∈	∈	NOUN
ejpam-5253	316	7	rh(m	rh(m	NOUN
ejpam-5253	316	8	\	\	PUNCT
ejpam-5253	316	9	k	k	X
ejpam-5253	316	10	)	)	PUNCT
ejpam-5253	316	11	which	which	PRON
ejpam-5253	316	12	is	be	AUX
ejpam-5253	316	13	equivalent	equivalent	ADJ
ejpam-5253	316	14	to	to	ADP
ejpam-5253	316	15	m	m	PROPN
ejpam-5253	316	16	∈	∈	NOUN
ejpam-5253	316	17	∪{h	∪{h	X
ejpam-5253	316	18	∈	∈	PROPN
ejpam-5253	316	19	σh	σh	PROPN
ejpam-5253	316	20	,	,	PUNCT
ejpam-5253	316	21	h	h	NOUN
ejpam-5253	316	22	⊆	⊆	NUM
ejpam-5253	316	23	m	m	NOUN
ejpam-5253	316	24	\	\	X
ejpam-5253	316	25	k	k	NOUN
ejpam-5253	316	26	}	}	PUNCT
ejpam-5253	316	27	.	.	PUNCT
ejpam-5253	317	1	so	so	ADV
ejpam-5253	317	2	there	there	PRON
ejpam-5253	317	3	exists	exist	VERB
ejpam-5253	317	4	h0	h0	PROPN
ejpam-5253	317	5	∈	∈	PROPN
ejpam-5253	317	6	σh	σh	ADP
ejpam-5253	317	7	such	such	ADJ
ejpam-5253	317	8	that	that	SCONJ
ejpam-5253	317	9	m	m	PROPN
ejpam-5253	317	10	∈	∈	PROPN
ejpam-5253	317	11	h0	h0	NOUN
ejpam-5253	317	12	⊆	⊆	NUM
ejpam-5253	317	13	m	m	NOUN
ejpam-5253	317	14	\k	\k	NOUN
ejpam-5253	317	15	.	.	PUNCT
ejpam-5253	318	1	then	then	ADV
ejpam-5253	318	2	there	there	PRON
ejpam-5253	318	3	exists	exist	VERB
ejpam-5253	318	4	hc	hc	PROPN
ejpam-5253	318	5	0	0	NUM
ejpam-5253	318	6	such	such	ADJ
ejpam-5253	318	7	that	that	SCONJ
ejpam-5253	319	1	k	k	PROPN
ejpam-5253	319	2	⊂	⊂	PROPN
ejpam-5253	319	3	hc	hc	PROPN
ejpam-5253	319	4	0	0	NUM
ejpam-5253	319	5	and	and	CCONJ
ejpam-5253	319	6	m	m	NOUN
ejpam-5253	319	7	/∈	/∈	PUNCT
ejpam-5253	320	1	hc	hc	PROPN
ejpam-5253	320	2	0	0	PROPN
ejpam-5253	320	3	,	,	PUNCT
ejpam-5253	320	4	hc	hc	PROPN
ejpam-5253	320	5	0	0	NUM
ejpam-5253	320	6	∈	∈	PROPN
ejpam-5253	320	7	σhc	σhc	NOUN
ejpam-5253	320	8	.	.	PUNCT
ejpam-5253	321	1	thus	thus	ADV
ejpam-5253	321	2	,	,	PUNCT
ejpam-5253	321	3	m	m	NOUN
ejpam-5253	321	4	/∈	/∈	INTJ
ejpam-5253	321	5	rh(k	rh(k	ADJ
ejpam-5253	321	6	)	)	PUNCT
ejpam-5253	321	7	.	.	PUNCT
ejpam-5253	322	1	so	so	ADV
ejpam-5253	322	2	m	m	VERB
ejpam-5253	322	3	∈	∈	PROPN
ejpam-5253	322	4	m	m	NOUN
ejpam-5253	322	5	\rh(k	\rh(k	X
ejpam-5253	322	6	)	)	PUNCT
ejpam-5253	322	7	.	.	PUNCT
ejpam-5253	323	1	therefore	therefore	ADV
ejpam-5253	323	2	rh(m	rh(m	VERB
ejpam-5253	323	3	\k	\k	NOUN
ejpam-5253	323	4	)	)	PUNCT
ejpam-5253	324	1	=	=	PUNCT
ejpam-5253	324	2	m	m	NOUN
ejpam-5253	324	3	\rh(k	\rh(k	PROPN
ejpam-5253	324	4	)	)	PUNCT
ejpam-5253	324	5	.	.	PUNCT
ejpam-5253	325	1	(	(	PUNCT
ejpam-5253	325	2	ii	ii	NOUN
ejpam-5253	325	3	)	)	PUNCT
ejpam-5253	325	4	comparable	comparable	ADJ
ejpam-5253	325	5	to	to	ADP
ejpam-5253	325	6	(	(	PUNCT
ejpam-5253	325	7	i	i	NOUN
ejpam-5253	325	8	)	)	PUNCT
ejpam-5253	325	9	(	(	PUNCT
ejpam-5253	325	10	iii	iii	NOUN
ejpam-5253	325	11	)	)	PUNCT
ejpam-5253	325	12	since	since	SCONJ
ejpam-5253	325	13	rh(k	rh(k	NOUN
ejpam-5253	325	14	)	)	PUNCT
ejpam-5253	325	15	=	=	PUNCT
ejpam-5253	325	16	∪{h	∪{h	PROPN
ejpam-5253	325	17	∈	∈	PROPN
ejpam-5253	325	18	σh	σh	PROPN
ejpam-5253	325	19	,	,	PUNCT
ejpam-5253	325	20	h	h	NOUN
ejpam-5253	325	21	⊆	⊆	NUM
ejpam-5253	325	22	k	k	NOUN
ejpam-5253	325	23	}	}	PUNCT
ejpam-5253	325	24	.	.	PUNCT
ejpam-5253	326	1	this	this	PRON
ejpam-5253	326	2	implies	imply	VERB
ejpam-5253	326	3	that	that	SCONJ
ejpam-5253	326	4	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	326	5	)	)	PUNCT
ejpam-5253	326	6	)	)	PUNCT
ejpam-5253	327	1	=	=	SYM
ejpam-5253	327	2	∪{∪{h	∪{∪{h	PROPN
ejpam-5253	327	3	∈	∈	PROPN
ejpam-5253	327	4	σh	σh	PROPN
ejpam-5253	327	5	,	,	PUNCT
ejpam-5253	327	6	h	h	NOUN
ejpam-5253	327	7	⊆	⊆	NUM
ejpam-5253	327	8	k	k	NOUN
ejpam-5253	327	9	}	}	PUNCT
ejpam-5253	327	10	}	}	PUNCT
ejpam-5253	327	11	=	=	PUNCT
ejpam-5253	327	12	∪{h	∪{h	PROPN
ejpam-5253	327	13	∈	∈	PROPN
ejpam-5253	327	14	σh	σh	PROPN
ejpam-5253	327	15	,	,	PUNCT
ejpam-5253	327	16	h	h	NOUN
ejpam-5253	327	17	⊆	⊆	NUM
ejpam-5253	327	18	k	k	NOUN
ejpam-5253	327	19	}	}	PUNCT
ejpam-5253	327	20	=	=	SYM
ejpam-5253	327	21	rh(k	rh(k	X
ejpam-5253	327	22	)	)	PUNCT
ejpam-5253	327	23	.	.	PUNCT
ejpam-5253	328	1	(	(	PUNCT
ejpam-5253	328	2	iv	iv	X
ejpam-5253	328	3	)	)	PUNCT
ejpam-5253	328	4	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	328	5	)	)	PUNCT
ejpam-5253	328	6	)	)	PUNCT
ejpam-5253	329	1	=	=	PUNCT
ejpam-5253	329	2	rh(m	rh(m	NOUN
ejpam-5253	329	3	\	\	NOUN
ejpam-5253	329	4	rh(m	rh(m	NOUN
ejpam-5253	329	5	\k	\k	NOUN
ejpam-5253	329	6	)	)	PUNCT
ejpam-5253	329	7	)	)	PUNCT
ejpam-5253	330	1	=	=	PUNCT
ejpam-5253	330	2	m	m	VERB
ejpam-5253	330	3	\	\	NOUN
ejpam-5253	330	4	rh(m	rh(m	X
ejpam-5253	330	5	\	\	NOUN
ejpam-5253	330	6	rh(m	rh(m	NOUN
ejpam-5253	330	7	\k	\k	NOUN
ejpam-5253	330	8	)	)	PUNCT
ejpam-5253	330	9	)	)	PUNCT
ejpam-5253	330	10	.	.	PUNCT
ejpam-5253	331	1	from	from	ADP
ejpam-5253	331	2	(	(	PUNCT
ejpam-5253	331	3	i	i	NOUN
ejpam-5253	331	4	)	)	PUNCT
ejpam-5253	331	5	,	,	PUNCT
ejpam-5253	331	6	(	(	PUNCT
ejpam-5253	331	7	ii	ii	NOUN
ejpam-5253	331	8	)	)	PUNCT
ejpam-5253	331	9	and	and	CCONJ
ejpam-5253	331	10	(	(	PUNCT
ejpam-5253	331	11	iii	iii	NOUN
ejpam-5253	331	12	)	)	PUNCT
ejpam-5253	331	13	,	,	PUNCT
ejpam-5253	331	14	we	we	PRON
ejpam-5253	331	15	get	get	VERB
ejpam-5253	331	16	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	331	17	)	)	PUNCT
ejpam-5253	331	18	)	)	PUNCT
ejpam-5253	332	1	=	=	PUNCT
ejpam-5253	332	2	m	m	VERB
ejpam-5253	332	3	\rh(m	\rh(m	NOUN
ejpam-5253	332	4	\k	\k	NOUN
ejpam-5253	332	5	)	)	PUNCT
ejpam-5253	332	6	=	=	PUNCT
ejpam-5253	333	1	m	m	VERB
ejpam-5253	333	2	\	\	NOUN
ejpam-5253	333	3	(	(	PUNCT
ejpam-5253	333	4	m	m	PROPN
ejpam-5253	333	5	\rh(k	\rh(k	PROPN
ejpam-5253	333	6	)	)	PUNCT
ejpam-5253	333	7	)	)	PUNCT
ejpam-5253	334	1	=	=	SYM
ejpam-5253	334	2	rh(k	rh(k	X
ejpam-5253	334	3	)	)	PUNCT
ejpam-5253	334	4	.	.	PUNCT
ejpam-5253	335	1	(	(	PUNCT
ejpam-5253	335	2	v	v	NOUN
ejpam-5253	335	3	)	)	PUNCT
ejpam-5253	335	4	since	since	SCONJ
ejpam-5253	335	5	rh(k	rh(k	NOUN
ejpam-5253	335	6	)	)	PUNCT
ejpam-5253	335	7	⊆	⊆	NUM
ejpam-5253	335	8	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	335	9	)	)	PUNCT
ejpam-5253	335	10	)	)	PUNCT
ejpam-5253	336	1	and	and	CCONJ
ejpam-5253	336	2	by	by	ADP
ejpam-5253	336	3	(	(	PUNCT
ejpam-5253	336	4	iii	iii	X
ejpam-5253	336	5	)	)	PUNCT
ejpam-5253	336	6	we	we	PRON
ejpam-5253	336	7	have	have	VERB
ejpam-5253	336	8	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	336	9	)	)	PUNCT
ejpam-5253	336	10	)	)	PUNCT
ejpam-5253	337	1	=	=	SYM
ejpam-5253	337	2	rh(k	rh(k	X
ejpam-5253	337	3	)	)	PUNCT
ejpam-5253	337	4	,	,	PUNCT
ejpam-5253	337	5	then	then	ADV
ejpam-5253	337	6	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	337	7	)	)	PUNCT
ejpam-5253	337	8	)	)	PUNCT
ejpam-5253	338	1	⊆	⊆	NUM
ejpam-5253	338	2	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	338	3	)	)	PUNCT
ejpam-5253	338	4	)	)	PUNCT
ejpam-5253	338	5	.	.	PUNCT
ejpam-5253	339	1	(	(	PUNCT
ejpam-5253	339	2	vi	vi	NOUN
ejpam-5253	339	3	)	)	PUNCT
ejpam-5253	339	4	since	since	SCONJ
ejpam-5253	339	5	rh(rh(k	rh(rh(k	PROPN
ejpam-5253	339	6	)	)	PUNCT
ejpam-5253	339	7	)	)	PUNCT
ejpam-5253	340	1	⊆	⊆	NUM
ejpam-5253	340	2	rh(k	rh(k	NOUN
ejpam-5253	340	3	)	)	PUNCT
ejpam-5253	340	4	and	and	CCONJ
ejpam-5253	340	5	by	by	ADP
ejpam-5253	340	6	(	(	PUNCT
ejpam-5253	340	7	iv	iv	X
ejpam-5253	340	8	)	)	PUNCT
ejpam-5253	340	9	,	,	PUNCT
ejpam-5253	340	10	we	we	PRON
ejpam-5253	340	11	have	have	VERB
ejpam-5253	340	12	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	340	13	)	)	PUNCT
ejpam-5253	340	14	)	)	PUNCT
ejpam-5253	341	1	=	=	SYM
ejpam-5253	341	2	rh(k	rh(k	X
ejpam-5253	341	3	)	)	PUNCT
ejpam-5253	341	4	,	,	PUNCT
ejpam-5253	341	5	then	then	ADV
ejpam-5253	341	6	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	341	7	)	)	PUNCT
ejpam-5253	341	8	)	)	PUNCT
ejpam-5253	342	1	⊆	⊆	NUM
ejpam-5253	342	2	rh(rh(k	rh(rh(k	NOUN
ejpam-5253	342	3	)	)	PUNCT
ejpam-5253	342	4	)	)	PUNCT
ejpam-5253	342	5	.	.	PUNCT
ejpam-5253	343	1	proposition	proposition	NOUN
ejpam-5253	343	2	9	9	NUM
ejpam-5253	343	3	.	.	PUNCT
ejpam-5253	344	1	let	let	AUX
ejpam-5253	344	2	(	(	PUNCT
ejpam-5253	344	3	m	m	PROPN
ejpam-5253	344	4	,	,	PUNCT
ejpam-5253	344	5	rh	rh	PROPN
ejpam-5253	344	6	)	)	PUNCT
ejpam-5253	344	7	be	be	VERB
ejpam-5253	344	8	a	a	DET
ejpam-5253	344	9	h	h	NOUN
ejpam-5253	344	10	-	-	PUNCT
ejpam-5253	344	11	approximation	approximation	NOUN
ejpam-5253	344	12	space	space	NOUN
ejpam-5253	344	13	and	and	CCONJ
ejpam-5253	344	14	k	k	NOUN
ejpam-5253	344	15	,	,	PUNCT
ejpam-5253	344	16	q	q	X
ejpam-5253	344	17	⊆	⊆	NUM
ejpam-5253	344	18	m	m	NOUN
ejpam-5253	344	19	.	.	PUNCT
ejpam-5253	345	1	then	then	ADV
ejpam-5253	345	2	(	(	PUNCT
ejpam-5253	345	3	i	i	NOUN
ejpam-5253	345	4	)	)	PUNCT
ejpam-5253	345	5	rh(k	rh(k	X
ejpam-5253	345	6	∪q	∪q	NUM
ejpam-5253	345	7	)	)	PUNCT
ejpam-5253	345	8	⊇	⊇	NOUN
ejpam-5253	345	9	rh(k	rh(k	X
ejpam-5253	345	10	)	)	PUNCT
ejpam-5253	345	11	∪rh(q	∪rh(q	CCONJ
ejpam-5253	345	12	)	)	PUNCT
ejpam-5253	345	13	.	.	PUNCT
ejpam-5253	346	1	(	(	PUNCT
ejpam-5253	346	2	ii	ii	NOUN
ejpam-5253	346	3	)	)	PUNCT
ejpam-5253	346	4	rh(k	rh(k	X
ejpam-5253	346	5	∪q	∪q	NUM
ejpam-5253	346	6	)	)	PUNCT
ejpam-5253	346	7	⊇	⊇	NOUN
ejpam-5253	346	8	rh(k	rh(k	X
ejpam-5253	346	9	)	)	PUNCT
ejpam-5253	346	10	∪rh(q	∪rh(q	CCONJ
ejpam-5253	346	11	)	)	PUNCT
ejpam-5253	346	12	.	.	PUNCT
ejpam-5253	347	1	(	(	PUNCT
ejpam-5253	347	2	iii	iii	NOUN
ejpam-5253	347	3	)	)	PUNCT
ejpam-5253	347	4	rh(k	rh(k	PUNCT
ejpam-5253	347	5	∩q	∩q	NOUN
ejpam-5253	347	6	)	)	PUNCT
ejpam-5253	348	1	⊆	⊆	NUM
ejpam-5253	348	2	rh(k	rh(k	X
ejpam-5253	348	3	)	)	PUNCT
ejpam-5253	348	4	∩rh(q	∩rh(q	NOUN
ejpam-5253	348	5	)	)	PUNCT
ejpam-5253	348	6	.	.	PUNCT
ejpam-5253	349	1	(	(	PUNCT
ejpam-5253	349	2	iv	iv	X
ejpam-5253	349	3	)	)	PUNCT
ejpam-5253	349	4	rh(k	rh(k	PUNCT
ejpam-5253	349	5	∩q	∩q	NOUN
ejpam-5253	349	6	)	)	PUNCT
ejpam-5253	350	1	⊆	⊆	NUM
ejpam-5253	350	2	rh(k	rh(k	X
ejpam-5253	350	3	)	)	PUNCT
ejpam-5253	350	4	∩rh(q	∩rh(q	NOUN
ejpam-5253	350	5	)	)	PUNCT
ejpam-5253	350	6	.	.	PUNCT
ejpam-5253	351	1	proof	proof	NOUN
ejpam-5253	351	2	.	.	PUNCT
ejpam-5253	352	1	(	(	PUNCT
ejpam-5253	352	2	i	i	NOUN
ejpam-5253	352	3	)	)	PUNCT
ejpam-5253	352	4	since	since	SCONJ
ejpam-5253	352	5	we	we	PRON
ejpam-5253	352	6	have	have	VERB
ejpam-5253	352	7	k	k	PROPN
ejpam-5253	352	8	⊆	⊆	NUM
ejpam-5253	352	9	k	k	X
ejpam-5253	352	10	∪	∪	X
ejpam-5253	352	11	q	q	PROPN
ejpam-5253	352	12	and	and	CCONJ
ejpam-5253	352	13	q	q	PROPN
ejpam-5253	352	14	⊆	⊆	NUM
ejpam-5253	352	15	k	k	SYM
ejpam-5253	352	16	∪	∪	PROPN
ejpam-5253	352	17	q.	q.	PROPN
ejpam-5253	352	18	then	then	ADV
ejpam-5253	352	19	rh(k	rh(k	NOUN
ejpam-5253	352	20	)	)	PUNCT
ejpam-5253	352	21	⊆	⊆	NUM
ejpam-5253	352	22	rh(k	rh(k	NOUN
ejpam-5253	352	23	∪	∪	X
ejpam-5253	352	24	q	q	PROPN
ejpam-5253	352	25	)	)	PUNCT
ejpam-5253	352	26	and	and	CCONJ
ejpam-5253	352	27	rh(q	rh(q	NUM
ejpam-5253	352	28	)	)	PUNCT
ejpam-5253	352	29	⊆	⊆	NUM
ejpam-5253	352	30	rh(k	rh(k	NOUN
ejpam-5253	352	31	∪q	∪q	NUM
ejpam-5253	352	32	)	)	PUNCT
ejpam-5253	352	33	by	by	ADP
ejpam-5253	352	34	(	(	PUNCT
ejpam-5253	352	35	iii	iii	NOUN
ejpam-5253	352	36	)	)	PUNCT
ejpam-5253	352	37	in	in	ADP
ejpam-5253	352	38	proposition	proposition	NOUN
ejpam-5253	352	39	7	7	NUM
ejpam-5253	352	40	,	,	PUNCT
ejpam-5253	352	41	then	then	ADV
ejpam-5253	352	42	rh(k	rh(k	X
ejpam-5253	352	43	∪q	∪q	NUM
ejpam-5253	352	44	)	)	PUNCT
ejpam-5253	352	45	⊇	⊇	NOUN
ejpam-5253	352	46	rh(k	rh(k	X
ejpam-5253	352	47	)	)	PUNCT
ejpam-5253	352	48	∪rh(q	∪rh(q	CCONJ
ejpam-5253	352	49	)	)	PUNCT
ejpam-5253	352	50	.	.	PUNCT
ejpam-5253	353	1	(	(	PUNCT
ejpam-5253	353	2	ii	ii	NOUN
ejpam-5253	353	3	)	)	PUNCT
ejpam-5253	353	4	(	(	PUNCT
ejpam-5253	353	5	iii	iii	NOUN
ejpam-5253	353	6	)	)	PUNCT
ejpam-5253	353	7	and	and	CCONJ
ejpam-5253	353	8	(	(	PUNCT
ejpam-5253	353	9	iv	iv	X
ejpam-5253	353	10	)	)	PUNCT
ejpam-5253	353	11	similar	similar	ADJ
ejpam-5253	353	12	to	to	ADP
ejpam-5253	353	13	(	(	PUNCT
ejpam-5253	353	14	i	i	NOUN
ejpam-5253	353	15	)	)	PUNCT
ejpam-5253	353	16	.	.	PUNCT
ejpam-5253	354	1	the	the	DET
ejpam-5253	354	2	equality	equality	NOUN
ejpam-5253	354	3	of	of	ADP
ejpam-5253	354	4	all	all	DET
ejpam-5253	354	5	parts	part	NOUN
ejpam-5253	354	6	in	in	ADP
ejpam-5253	354	7	proposition	proposition	NOUN
ejpam-5253	354	8	9	9	NUM
ejpam-5253	354	9	does	do	AUX
ejpam-5253	354	10	not	not	PART
ejpam-5253	354	11	hold	hold	VERB
ejpam-5253	354	12	,	,	PUNCT
ejpam-5253	354	13	as	as	SCONJ
ejpam-5253	354	14	demonstrated	demonstrate	VERB
ejpam-5253	354	15	in	in	ADP
ejpam-5253	354	16	the	the	DET
ejpam-5253	354	17	following	follow	VERB
ejpam-5253	354	18	example	example	NOUN
ejpam-5253	354	19	.	.	PUNCT
ejpam-5253	355	1	example	example	NOUN
ejpam-5253	355	2	12	12	NUM
ejpam-5253	355	3	.	.	PUNCT
ejpam-5253	356	1	in	in	ADP
ejpam-5253	356	2	example	example	NOUN
ejpam-5253	356	3	2	2	NUM
ejpam-5253	356	4	:	:	PUNCT
ejpam-5253	356	5	(	(	PUNCT
ejpam-5253	356	6	i	i	NOUN
ejpam-5253	356	7	)	)	PUNCT
ejpam-5253	356	8	if	if	SCONJ
ejpam-5253	356	9	k	k	PROPN
ejpam-5253	356	10	=	=	X
ejpam-5253	356	11	{	{	PUNCT
ejpam-5253	356	12	t	t	PROPN
ejpam-5253	356	13	}	}	PUNCT
ejpam-5253	356	14	,	,	PUNCT
ejpam-5253	356	15	q	q	NOUN
ejpam-5253	356	16	=	=	PUNCT
ejpam-5253	356	17	{	{	PUNCT
ejpam-5253	356	18	k	k	NOUN
ejpam-5253	356	19	,	,	PUNCT
ejpam-5253	356	20	q	q	NOUN
ejpam-5253	356	21	}	}	PUNCT
ejpam-5253	356	22	,	,	PUNCT
ejpam-5253	356	23	then	then	ADV
ejpam-5253	356	24	we	we	PRON
ejpam-5253	356	25	have	have	VERB
ejpam-5253	356	26	rh(k	rh(k	NOUN
ejpam-5253	356	27	∪q	∪q	NUM
ejpam-5253	356	28	)	)	PUNCT
ejpam-5253	356	29	=	=	SYM
ejpam-5253	356	30	{	{	PUNCT
ejpam-5253	356	31	k	k	NOUN
ejpam-5253	356	32	,	,	PUNCT
ejpam-5253	356	33	q	q	NOUN
ejpam-5253	356	34	,	,	PUNCT
ejpam-5253	356	35	t	t	PROPN
ejpam-5253	356	36	}	}	PUNCT
ejpam-5253	356	37	,	,	PUNCT
ejpam-5253	356	38	rh(k	rh(k	X
ejpam-5253	356	39	)	)	PUNCT
ejpam-5253	356	40	=	=	SYM
ejpam-5253	356	41	{	{	PUNCT
ejpam-5253	356	42	t	t	NOUN
ejpam-5253	356	43	}	}	PUNCT
ejpam-5253	356	44	,	,	PUNCT
ejpam-5253	356	45	rh(q	rh(q	NOUN
ejpam-5253	356	46	)	)	PUNCT
ejpam-5253	356	47	=	=	PRON
ejpam-5253	356	48	{	{	PUNCT
ejpam-5253	357	1	k}.therefore	k}.therefore	INTJ
ejpam-5253	357	2	rh(k	rh(k	X
ejpam-5253	357	3	∪q	∪q	NUM
ejpam-5253	357	4	)	)	PUNCT
ejpam-5253	357	5	̸=	̸=	PROPN
ejpam-5253	357	6	rh(k	rh(k	X
ejpam-5253	357	7	)	)	PUNCT
ejpam-5253	357	8	∪rh(q	∪rh(q	CCONJ
ejpam-5253	357	9	)	)	PUNCT
ejpam-5253	357	10	.	.	PUNCT
ejpam-5253	358	1	a.	a.	PROPN
ejpam-5253	358	2	al	al	PROPN
ejpam-5253	358	3	-	-	PUNCT
ejpam-5253	358	4	rehili	rehili	NOUN
ejpam-5253	358	5	/	/	SYM
ejpam-5253	358	6	eur	eur	PROPN
ejpam-5253	358	7	.	.	PUNCT
ejpam-5253	359	1	j.	j.	PROPN
ejpam-5253	359	2	pure	pure	PROPN
ejpam-5253	359	3	appl	appl	PROPN
ejpam-5253	359	4	.	.	PROPN
ejpam-5253	359	5	math	math	PROPN
ejpam-5253	359	6	,	,	PUNCT
ejpam-5253	359	7	17	17	NUM
ejpam-5253	359	8	(	(	PUNCT
ejpam-5253	359	9	3	3	NUM
ejpam-5253	359	10	)	)	PUNCT
ejpam-5253	359	11	(	(	PUNCT
ejpam-5253	359	12	2024	2024	NUM
ejpam-5253	359	13	)	)	PUNCT
ejpam-5253	359	14	,	,	PUNCT
ejpam-5253	359	15	1804	1804	NUM
ejpam-5253	359	16	-	-	SYM
ejpam-5253	359	17	1817	1817	NUM
ejpam-5253	359	18	1815	1815	NUM
ejpam-5253	359	19	(	(	PUNCT
ejpam-5253	359	20	ii	ii	NOUN
ejpam-5253	359	21	)	)	PUNCT
ejpam-5253	359	22	if	if	SCONJ
ejpam-5253	359	23	k	k	PROPN
ejpam-5253	359	24	=	=	X
ejpam-5253	359	25	{	{	PUNCT
ejpam-5253	359	26	t	t	PROPN
ejpam-5253	359	27	}	}	PUNCT
ejpam-5253	359	28	,	,	PUNCT
ejpam-5253	359	29	q	q	NOUN
ejpam-5253	360	1	=	=	PUNCT
ejpam-5253	360	2	{	{	PUNCT
ejpam-5253	360	3	k	k	NOUN
ejpam-5253	360	4	,	,	PUNCT
ejpam-5253	360	5	q	q	NOUN
ejpam-5253	360	6	}	}	PUNCT
ejpam-5253	360	7	,	,	PUNCT
ejpam-5253	360	8	then	then	ADV
ejpam-5253	360	9	we	we	PRON
ejpam-5253	360	10	have	have	VERB
ejpam-5253	360	11	rh(k	rh(k	NOUN
ejpam-5253	360	12	∪q	∪q	NUM
ejpam-5253	360	13	)	)	PUNCT
ejpam-5253	360	14	=	=	SYM
ejpam-5253	360	15	m	m	NOUN
ejpam-5253	360	16	,	,	PUNCT
ejpam-5253	360	17	rh(k	rh(k	ADJ
ejpam-5253	360	18	)	)	PUNCT
ejpam-5253	360	19	=	=	SYM
ejpam-5253	360	20	{	{	PUNCT
ejpam-5253	360	21	q	q	X
ejpam-5253	360	22	,	,	PUNCT
ejpam-5253	360	23	s	s	PROPN
ejpam-5253	360	24	,	,	PUNCT
ejpam-5253	360	25	t	t	PROPN
ejpam-5253	360	26	}	}	PUNCT
ejpam-5253	360	27	,	,	PUNCT
ejpam-5253	360	28	rh(q	rh(q	NOUN
ejpam-5253	360	29	)	)	PUNCT
ejpam-5253	360	30	=	=	PRON
ejpam-5253	360	31	{	{	PUNCT
ejpam-5253	360	32	k	k	NOUN
ejpam-5253	360	33	,	,	PUNCT
ejpam-5253	360	34	q}.therefore	q}.therefore	ADP
ejpam-5253	360	35	rh(k	rh(k	ADJ
ejpam-5253	360	36	)	)	PUNCT
ejpam-5253	360	37	∪rh(q	∪rh(q	CCONJ
ejpam-5253	360	38	)	)	PUNCT
ejpam-5253	360	39	̸=	̸=	PROPN
ejpam-5253	360	40	rh(k	rh(k	X
ejpam-5253	360	41	∪q	∪q	NUM
ejpam-5253	360	42	)	)	PUNCT
ejpam-5253	360	43	.	.	PUNCT
ejpam-5253	361	1	(	(	PUNCT
ejpam-5253	361	2	iii	iii	X
ejpam-5253	361	3	)	)	PUNCT
ejpam-5253	361	4	if	if	SCONJ
ejpam-5253	361	5	k	k	PROPN
ejpam-5253	361	6	=	=	PRON
ejpam-5253	361	7	{	{	PUNCT
ejpam-5253	361	8	k	k	NOUN
ejpam-5253	361	9	,	,	PUNCT
ejpam-5253	361	10	q	q	X
ejpam-5253	361	11	,	,	PUNCT
ejpam-5253	361	12	s	s	PART
ejpam-5253	361	13	}	}	PUNCT
ejpam-5253	361	14	,	,	PUNCT
ejpam-5253	361	15	q	q	NOUN
ejpam-5253	361	16	=	=	PUNCT
ejpam-5253	361	17	{	{	PUNCT
ejpam-5253	361	18	q	q	PROPN
ejpam-5253	361	19	,	,	PUNCT
ejpam-5253	361	20	s	s	PROPN
ejpam-5253	361	21	,	,	PUNCT
ejpam-5253	361	22	t	t	PROPN
ejpam-5253	361	23	}	}	PUNCT
ejpam-5253	361	24	,	,	PUNCT
ejpam-5253	361	25	then	then	ADV
ejpam-5253	361	26	we	we	PRON
ejpam-5253	361	27	have	have	VERB
ejpam-5253	361	28	rh(k	rh(k	NOUN
ejpam-5253	361	29	∩	∩	ADJ
ejpam-5253	361	30	q	q	NOUN
ejpam-5253	361	31	)	)	PUNCT
ejpam-5253	361	32	=	=	SYM
ejpam-5253	361	33	ϕ	ϕ	NOUN
ejpam-5253	361	34	,	,	PUNCT
ejpam-5253	361	35	rh(k	rh(k	ADJ
ejpam-5253	361	36	)	)	PUNCT
ejpam-5253	361	37	=	=	PRON
ejpam-5253	361	38	{	{	PUNCT
ejpam-5253	361	39	k	k	NOUN
ejpam-5253	361	40	}	}	PUNCT
ejpam-5253	361	41	and	and	CCONJ
ejpam-5253	361	42	rh(q	rh(q	NOUN
ejpam-5253	361	43	)	)	PUNCT
ejpam-5253	362	1	=	=	PRON
ejpam-5253	362	2	{	{	PUNCT
ejpam-5253	362	3	q	q	X
ejpam-5253	362	4	,	,	PUNCT
ejpam-5253	362	5	s	s	PROPN
ejpam-5253	362	6	,	,	PUNCT
ejpam-5253	362	7	t	t	PROPN
ejpam-5253	362	8	}	}	PUNCT
ejpam-5253	362	9	.	.	PUNCT
ejpam-5253	363	1	therefore	therefore	ADV
ejpam-5253	363	2	rh(k	rh(k	PUNCT
ejpam-5253	363	3	∩q	∩q	NOUN
ejpam-5253	363	4	)	)	PUNCT
ejpam-5253	364	1	̸=	̸=	PROPN
ejpam-5253	364	2	rh(k	rh(k	X
ejpam-5253	364	3	)	)	PUNCT
ejpam-5253	364	4	∩rh(q	∩rh(q	NOUN
ejpam-5253	364	5	)	)	PUNCT
ejpam-5253	364	6	.	.	PUNCT
ejpam-5253	365	1	(	(	PUNCT
ejpam-5253	365	2	iv	iv	X
ejpam-5253	365	3	)	)	PUNCT
ejpam-5253	365	4	if	if	SCONJ
ejpam-5253	365	5	k	k	PROPN
ejpam-5253	365	6	=	=	PRON
ejpam-5253	365	7	{	{	PUNCT
ejpam-5253	365	8	k	k	NOUN
ejpam-5253	365	9	}	}	PUNCT
ejpam-5253	365	10	,	,	PUNCT
ejpam-5253	365	11	q	q	NOUN
ejpam-5253	365	12	=	=	PUNCT
ejpam-5253	365	13	{	{	PUNCT
ejpam-5253	365	14	q	q	PROPN
ejpam-5253	365	15	,	,	PUNCT
ejpam-5253	365	16	t	t	PROPN
ejpam-5253	365	17	}	}	PUNCT
ejpam-5253	365	18	,	,	PUNCT
ejpam-5253	365	19	then	then	ADV
ejpam-5253	365	20	we	we	PRON
ejpam-5253	365	21	have	have	VERB
ejpam-5253	365	22	rh(k	rh(k	NOUN
ejpam-5253	365	23	∩	∩	ADJ
ejpam-5253	365	24	q	q	NOUN
ejpam-5253	365	25	)	)	PUNCT
ejpam-5253	365	26	=	=	SYM
ejpam-5253	365	27	ϕ	ϕ	NOUN
ejpam-5253	365	28	,	,	PUNCT
ejpam-5253	365	29	rh(k	rh(k	ADJ
ejpam-5253	365	30	)	)	PUNCT
ejpam-5253	365	31	=	=	PRON
ejpam-5253	365	32	{	{	PUNCT
ejpam-5253	365	33	k	k	NOUN
ejpam-5253	365	34	}	}	PUNCT
ejpam-5253	365	35	,	,	PUNCT
ejpam-5253	365	36	rh(q	rh(q	NOUN
ejpam-5253	365	37	)	)	PUNCT
ejpam-5253	366	1	=	=	PRON
ejpam-5253	366	2	{	{	PUNCT
ejpam-5253	366	3	q	q	X
ejpam-5253	366	4	,	,	PUNCT
ejpam-5253	366	5	s	s	PART
ejpam-5253	366	6	,	,	PUNCT
ejpam-5253	366	7	t}.therefore	t}.therefore	PROPN
ejpam-5253	366	8	rh(k	rh(k	NOUN
ejpam-5253	366	9	)	)	PUNCT
ejpam-5253	366	10	∩rh(q	∩rh(q	PROPN
ejpam-5253	366	11	)	)	PUNCT
ejpam-5253	366	12	̸=	̸=	PROPN
ejpam-5253	366	13	rh(k	rh(k	PUNCT
ejpam-5253	366	14	∩q	∩q	PROPN
ejpam-5253	366	15	)	)	PUNCT
ejpam-5253	366	16	.	.	PUNCT
ejpam-5253	367	1	the	the	DET
ejpam-5253	367	2	following	follow	VERB
ejpam-5253	367	3	theorems	theorem	NOUN
ejpam-5253	367	4	are	be	AUX
ejpam-5253	367	5	generalization	generalization	NOUN
ejpam-5253	367	6	of	of	ADP
ejpam-5253	367	7	proposition	proposition	NOUN
ejpam-5253	367	8	9	9	NUM
ejpam-5253	367	9	.	.	PUNCT
ejpam-5253	367	10	proposition	proposition	NOUN
ejpam-5253	367	11	10	10	NUM
ejpam-5253	367	12	.	.	PUNCT
ejpam-5253	368	1	let	let	AUX
ejpam-5253	368	2	(	(	PUNCT
ejpam-5253	368	3	m	m	PROPN
ejpam-5253	368	4	,	,	PUNCT
ejpam-5253	368	5	rh	rh	PROPN
ejpam-5253	368	6	)	)	PUNCT
ejpam-5253	368	7	be	be	VERB
ejpam-5253	368	8	a	a	DET
ejpam-5253	368	9	h	h	NOUN
ejpam-5253	368	10	-	-	PUNCT
ejpam-5253	368	11	approximation	approximation	NOUN
ejpam-5253	368	12	space	space	NOUN
ejpam-5253	368	13	and	and	CCONJ
ejpam-5253	368	14	k	k	NOUN
ejpam-5253	368	15	,	,	PUNCT
ejpam-5253	368	16	q	q	X
ejpam-5253	368	17	⊆	⊆	NUM
ejpam-5253	368	18	m	m	NOUN
ejpam-5253	368	19	.	.	PUNCT
ejpam-5253	369	1	if	if	SCONJ
ejpam-5253	369	2	k	k	PROPN
ejpam-5253	369	3	is	be	AUX
ejpam-5253	369	4	rhdefinable	rhdefinable	ADJ
ejpam-5253	369	5	.	.	PUNCT
ejpam-5253	370	1	then	then	ADV
ejpam-5253	370	2	the	the	DET
ejpam-5253	370	3	following	follow	VERB
ejpam-5253	370	4	are	be	AUX
ejpam-5253	370	5	hold	hold	ADJ
ejpam-5253	370	6	.	.	PUNCT
ejpam-5253	371	1	(	(	PUNCT
ejpam-5253	371	2	i	i	NOUN
ejpam-5253	371	3	)	)	PUNCT
ejpam-5253	371	4	rh(k	rh(k	X
ejpam-5253	371	5	∪q	∪q	NUM
ejpam-5253	371	6	)	)	PUNCT
ejpam-5253	371	7	=	=	SYM
ejpam-5253	371	8	rh(k	rh(k	X
ejpam-5253	371	9	)	)	PUNCT
ejpam-5253	371	10	∪rh(q	∪rh(q	CCONJ
ejpam-5253	371	11	)	)	PUNCT
ejpam-5253	371	12	.	.	PUNCT
ejpam-5253	372	1	(	(	PUNCT
ejpam-5253	372	2	ii	ii	NOUN
ejpam-5253	372	3	)	)	PUNCT
ejpam-5253	372	4	rh(k	rh(k	PUNCT
ejpam-5253	372	5	∩q	∩q	NOUN
ejpam-5253	372	6	)	)	PUNCT
ejpam-5253	373	1	=	=	SYM
ejpam-5253	373	2	rh(k	rh(k	X
ejpam-5253	373	3	)	)	PUNCT
ejpam-5253	373	4	∩rh(q	∩rh(q	NOUN
ejpam-5253	373	5	)	)	PUNCT
ejpam-5253	373	6	.	.	PUNCT
ejpam-5253	374	1	proof	proof	NOUN
ejpam-5253	374	2	.	.	PUNCT
ejpam-5253	375	1	(	(	PUNCT
ejpam-5253	375	2	i	i	NOUN
ejpam-5253	375	3	)	)	PUNCT
ejpam-5253	375	4	it	it	PRON
ejpam-5253	375	5	is	be	AUX
ejpam-5253	375	6	evident	evident	ADJ
ejpam-5253	375	7	that	that	SCONJ
ejpam-5253	375	8	rh(k	rh(k	NOUN
ejpam-5253	375	9	)	)	PUNCT
ejpam-5253	375	10	∪	∪	ADP
ejpam-5253	375	11	rh(q	rh(q	NOUN
ejpam-5253	375	12	)	)	PUNCT
ejpam-5253	375	13	⊆	⊆	NUM
ejpam-5253	375	14	rh(k	rh(k	NOUN
ejpam-5253	375	15	∪	∪	ADP
ejpam-5253	375	16	q	q	NOUN
ejpam-5253	375	17	)	)	PUNCT
ejpam-5253	375	18	.	.	PUNCT
ejpam-5253	376	1	for	for	ADP
ejpam-5253	376	2	the	the	DET
ejpam-5253	376	3	converse	converse	NOUN
ejpam-5253	376	4	inclusion	inclusion	NOUN
ejpam-5253	376	5	,	,	PUNCT
ejpam-5253	376	6	let	let	VERB
ejpam-5253	376	7	m	m	PRON
ejpam-5253	376	8	∈	∈	PROPN
ejpam-5253	376	9	rh(k	rh(k	NOUN
ejpam-5253	376	10	∪	∪	NOUN
ejpam-5253	376	11	q	q	NOUN
ejpam-5253	376	12	)	)	PUNCT
ejpam-5253	376	13	,	,	PUNCT
ejpam-5253	376	14	that	that	PRON
ejpam-5253	376	15	means	mean	VERB
ejpam-5253	376	16	,	,	PUNCT
ejpam-5253	376	17	m	m	VERB
ejpam-5253	376	18	∈	∈	ADJ
ejpam-5253	376	19	∪{h	∪{h	NOUN
ejpam-5253	376	20	∈	∈	NOUN
ejpam-5253	376	21	σh	σh	PROPN
ejpam-5253	376	22	,	,	PUNCT
ejpam-5253	376	23	h	h	NOUN
ejpam-5253	377	1	⊆	⊆	NUM
ejpam-5253	377	2	k	k	X
ejpam-5253	377	3	∪	∪	X
ejpam-5253	377	4	q	q	PROPN
ejpam-5253	377	5	}	}	PUNCT
ejpam-5253	377	6	.	.	PUNCT
ejpam-5253	378	1	then	then	ADV
ejpam-5253	378	2	there	there	PRON
ejpam-5253	378	3	exists	exist	VERB
ejpam-5253	378	4	h0	h0	PROPN
ejpam-5253	378	5	∈	∈	PROPN
ejpam-5253	378	6	σh	σh	ADP
ejpam-5253	378	7	such	such	ADJ
ejpam-5253	378	8	that	that	SCONJ
ejpam-5253	378	9	m	m	PROPN
ejpam-5253	378	10	∈	∈	PROPN
ejpam-5253	378	11	h0	h0	NOUN
ejpam-5253	378	12	⊂	⊂	PROPN
ejpam-5253	378	13	k	k	PROPN
ejpam-5253	378	14	∪q	∪q	PROPN
ejpam-5253	378	15	.	.	PUNCT
ejpam-5253	379	1	we	we	PRON
ejpam-5253	379	2	distinguish	distinguish	VERB
ejpam-5253	379	3	three	three	NUM
ejpam-5253	379	4	cases	case	NOUN
ejpam-5253	379	5	:	:	PUNCT
ejpam-5253	379	6	case	case	NOUN
ejpam-5253	379	7	(	(	PUNCT
ejpam-5253	379	8	1	1	X
ejpam-5253	379	9	)	)	PUNCT
ejpam-5253	379	10	if	if	SCONJ
ejpam-5253	379	11	h0	h0	PROPN
ejpam-5253	379	12	⊂	⊂	PROPN
ejpam-5253	379	13	k	k	PROPN
ejpam-5253	379	14	,	,	PUNCT
ejpam-5253	379	15	m	m	PROPN
ejpam-5253	379	16	∈	∈	NOUN
ejpam-5253	379	17	h0	h0	NOUN
ejpam-5253	379	18	and	and	CCONJ
ejpam-5253	379	19	h0	h0	PROPN
ejpam-5253	379	20	is	be	AUX
ejpam-5253	379	21	a	a	DET
ejpam-5253	379	22	h	h	NOUN
ejpam-5253	379	23	-	-	PUNCT
ejpam-5253	379	24	open	open	ADJ
ejpam-5253	379	25	set	set	NOUN
ejpam-5253	379	26	,	,	PUNCT
ejpam-5253	379	27	then	then	ADV
ejpam-5253	379	28	m	m	NOUN
ejpam-5253	379	29	∈	∈	NOUN
ejpam-5253	379	30	rh(k	rh(k	NOUN
ejpam-5253	379	31	)	)	PUNCT
ejpam-5253	379	32	.	.	PUNCT
ejpam-5253	380	1	case	case	NOUN
ejpam-5253	380	2	(	(	PUNCT
ejpam-5253	380	3	2	2	X
ejpam-5253	380	4	)	)	PUNCT
ejpam-5253	380	5	if	if	SCONJ
ejpam-5253	380	6	h0	h0	NOUN
ejpam-5253	380	7	∩k	∩k	NOUN
ejpam-5253	380	8	=	=	SYM
ejpam-5253	380	9	ϕ	ϕ	NOUN
ejpam-5253	380	10	,	,	PUNCT
ejpam-5253	380	11	then	then	ADV
ejpam-5253	380	12	h0	h0	VERB
ejpam-5253	380	13	⊆	⊆	NUM
ejpam-5253	380	14	q	q	NOUN
ejpam-5253	380	15	and	and	CCONJ
ejpam-5253	380	16	m	m	PROPN
ejpam-5253	380	17	∈	∈	PROPN
ejpam-5253	380	18	h0	h0	NOUN
ejpam-5253	380	19	,	,	PUNCT
ejpam-5253	380	20	thus	thus	ADV
ejpam-5253	380	21	m	m	NOUN
ejpam-5253	380	22	∈	∈	NOUN
ejpam-5253	380	23	rh(q	rh(q	NOUN
ejpam-5253	380	24	)	)	PUNCT
ejpam-5253	380	25	.	.	PUNCT
ejpam-5253	381	1	case	case	NOUN
ejpam-5253	381	2	(	(	PUNCT
ejpam-5253	381	3	3	3	X
ejpam-5253	381	4	)	)	PUNCT
ejpam-5253	381	5	if	if	SCONJ
ejpam-5253	381	6	h0∩k	h0∩k	PROPN
ejpam-5253	381	7	̸=	̸=	PROPN
ejpam-5253	381	8	ϕ.	ϕ.	VERB
ejpam-5253	381	9	since	since	SCONJ
ejpam-5253	381	10	m	m	PROPN
ejpam-5253	381	11	∈	∈	PROPN
ejpam-5253	381	12	h0	h0	NOUN
ejpam-5253	381	13	and	and	CCONJ
ejpam-5253	381	14	h0	h0	PROPN
ejpam-5253	381	15	is	be	AUX
ejpam-5253	381	16	an	an	DET
ejpam-5253	381	17	h	h	NOUN
ejpam-5253	381	18	-	-	PUNCT
ejpam-5253	381	19	open	open	ADJ
ejpam-5253	381	20	set	set	NOUN
ejpam-5253	381	21	,	,	PUNCT
ejpam-5253	381	22	then	then	ADV
ejpam-5253	381	23	m	m	VERB
ejpam-5253	381	24	∈	∈	PROPN
ejpam-5253	381	25	clh(k	clh(k	PROPN
ejpam-5253	381	26	)	)	PUNCT
ejpam-5253	381	27	,	,	PUNCT
ejpam-5253	381	28	for	for	ADP
ejpam-5253	381	29	every	every	DET
ejpam-5253	381	30	h0	h0	NOUN
ejpam-5253	381	31	which	which	PRON
ejpam-5253	381	32	satisfies	satisfy	VERB
ejpam-5253	381	33	the	the	DET
ejpam-5253	381	34	aforementioned	aforementioned	ADJ
ejpam-5253	381	35	condition	condition	NOUN
ejpam-5253	381	36	,	,	PUNCT
ejpam-5253	381	37	thus	thus	ADV
ejpam-5253	381	38	m	m	NOUN
ejpam-5253	381	39	∈	∈	NOUN
ejpam-5253	381	40	rh(k	rh(k	NOUN
ejpam-5253	381	41	)	)	PUNCT
ejpam-5253	381	42	,	,	PUNCT
ejpam-5253	381	43	then	then	ADV
ejpam-5253	381	44	m	m	VERB
ejpam-5253	381	45	∈	∈	NOUN
ejpam-5253	381	46	rh(k	rh(k	NOUN
ejpam-5253	381	47	)	)	PUNCT
ejpam-5253	381	48	,	,	PUNCT
ejpam-5253	381	49	because	because	SCONJ
ejpam-5253	381	50	k	k	PROPN
ejpam-5253	381	51	is	be	AUX
ejpam-5253	381	52	rhdefinable	rhdefinable	ADJ
ejpam-5253	381	53	.	.	PUNCT
ejpam-5253	382	1	therefore	therefore	ADV
ejpam-5253	382	2	,	,	PUNCT
ejpam-5253	382	3	in	in	ADP
ejpam-5253	382	4	three	three	NUM
ejpam-5253	382	5	cases	case	NOUN
ejpam-5253	382	6	m	m	VERB
ejpam-5253	382	7	∈	∈	NOUN
ejpam-5253	382	8	rh(k	rh(k	NOUN
ejpam-5253	382	9	)	)	PUNCT
ejpam-5253	382	10	∪rh(q	∪rh(q	CCONJ
ejpam-5253	382	11	)	)	PUNCT
ejpam-5253	382	12	.	.	PUNCT
ejpam-5253	383	1	(	(	PUNCT
ejpam-5253	383	2	ii	ii	X
ejpam-5253	383	3	)	)	PUNCT
ejpam-5253	383	4	it	it	PRON
ejpam-5253	383	5	is	be	AUX
ejpam-5253	383	6	evident	evident	ADJ
ejpam-5253	383	7	that	that	SCONJ
ejpam-5253	383	8	rh(k∩q	rh(k∩q	NOUN
ejpam-5253	383	9	)	)	PUNCT
ejpam-5253	383	10	⊆	⊆	NUM
ejpam-5253	383	11	rh(k)∩rh(q	rh(k)∩rh(q	PROPN
ejpam-5253	383	12	)	)	PUNCT
ejpam-5253	383	13	.	.	PUNCT
ejpam-5253	384	1	we	we	PRON
ejpam-5253	384	2	prove	prove	VERB
ejpam-5253	384	3	the	the	DET
ejpam-5253	384	4	converse	converse	NOUN
ejpam-5253	384	5	inclusion	inclusion	NOUN
ejpam-5253	384	6	.	.	PUNCT
ejpam-5253	385	1	let	let	VERB
ejpam-5253	385	2	m	m	PRON
ejpam-5253	385	3	∈	∈	PROPN
ejpam-5253	385	4	rh(k	rh(k	X
ejpam-5253	385	5	)	)	PUNCT
ejpam-5253	385	6	∩rh(q	∩rh(q	NOUN
ejpam-5253	385	7	)	)	PUNCT
ejpam-5253	385	8	,	,	PUNCT
ejpam-5253	385	9	then	then	ADV
ejpam-5253	385	10	m	m	VERB
ejpam-5253	385	11	∈	∈	PROPN
ejpam-5253	385	12	rh(k	rh(k	PRON
ejpam-5253	385	13	)	)	PUNCT
ejpam-5253	385	14	implies	imply	VERB
ejpam-5253	385	15	m	m	PROPN
ejpam-5253	385	16	∈	∈	NOUN
ejpam-5253	385	17	rh(k	rh(k	NOUN
ejpam-5253	385	18	)	)	PUNCT
ejpam-5253	385	19	and	and	CCONJ
ejpam-5253	385	20	m	m	PROPN
ejpam-5253	385	21	∈	∈	NOUN
ejpam-5253	385	22	h	h	NOUN
ejpam-5253	385	23	⊆	⊆	NUM
ejpam-5253	385	24	m	m	NOUN
ejpam-5253	385	25	,	,	PUNCT
ejpam-5253	385	26	where	where	SCONJ
ejpam-5253	385	27	h	h	NOUN
ejpam-5253	385	28	is	be	AUX
ejpam-5253	385	29	an	an	DET
ejpam-5253	385	30	h	h	NOUN
ejpam-5253	385	31	-	-	PUNCT
ejpam-5253	385	32	open	open	ADJ
ejpam-5253	385	33	set	set	NOUN
ejpam-5253	385	34	and	and	CCONJ
ejpam-5253	385	35	m	m	NOUN
ejpam-5253	385	36	∈	∈	NOUN
ejpam-5253	385	37	rh(q	rh(q	PRON
ejpam-5253	385	38	)	)	PUNCT
ejpam-5253	385	39	implies	imply	VERB
ejpam-5253	385	40	for	for	ADP
ejpam-5253	385	41	all	all	DET
ejpam-5253	385	42	h	h	NOUN
ejpam-5253	385	43	∈	∈	PROPN
ejpam-5253	385	44	σh	σh	PROPN
ejpam-5253	385	45	,	,	PUNCT
ejpam-5253	385	46	h	h	NOUN
ejpam-5253	385	47	∩	∩	NOUN
ejpam-5253	385	48	q	q	PROPN
ejpam-5253	385	49	̸=	̸=	PROPN
ejpam-5253	385	50	ϕ.	ϕ.	PROPN
ejpam-5253	385	51	therefore	therefore	ADV
ejpam-5253	385	52	h	h	PROPN
ejpam-5253	385	53	∩	∩	NOUN
ejpam-5253	385	54	(	(	PUNCT
ejpam-5253	385	55	k	k	X
ejpam-5253	385	56	∩q	∩q	PROPN
ejpam-5253	385	57	)	)	PUNCT
ejpam-5253	386	1	=	=	PRON
ejpam-5253	386	2	(	(	PUNCT
ejpam-5253	386	3	h	h	NOUN
ejpam-5253	386	4	∩k	∩k	PROPN
ejpam-5253	386	5	)	)	PUNCT
ejpam-5253	386	6	∩q	∩q	PUNCT
ejpam-5253	387	1	=	=	PUNCT
ejpam-5253	387	2	h	h	NOUN
ejpam-5253	387	3	∩n	∩n	PROPN
ejpam-5253	387	4	̸=	̸=	PROPN
ejpam-5253	387	5	ϕ.	ϕ.	PROPN
ejpam-5253	387	6	hence	hence	ADV
ejpam-5253	387	7	m	m	NOUN
ejpam-5253	387	8	∈	∈	PROPN
ejpam-5253	387	9	rh(k	rh(k	NOUN
ejpam-5253	387	10	∩q	∩q	PROPN
ejpam-5253	387	11	)	)	PUNCT
ejpam-5253	387	12	.	.	PUNCT
ejpam-5253	388	1	proposition	proposition	NOUN
ejpam-5253	388	2	11	11	NUM
ejpam-5253	388	3	.	.	PUNCT
ejpam-5253	389	1	if	if	SCONJ
ejpam-5253	389	2	(	(	PUNCT
ejpam-5253	389	3	m	m	PROPN
ejpam-5253	389	4	,	,	PUNCT
ejpam-5253	389	5	rh	rh	PROPN
ejpam-5253	389	6	)	)	PUNCT
ejpam-5253	389	7	is	be	AUX
ejpam-5253	389	8	a	a	DET
ejpam-5253	389	9	h	h	NOUN
ejpam-5253	389	10	-	-	PUNCT
ejpam-5253	389	11	approximation	approximation	NOUN
ejpam-5253	389	12	space	space	NOUN
ejpam-5253	389	13	and	and	CCONJ
ejpam-5253	389	14	k	k	NOUN
ejpam-5253	389	15	,	,	PUNCT
ejpam-5253	389	16	q	q	X
ejpam-5253	389	17	⊆	⊆	NUM
ejpam-5253	389	18	m	m	NOUN
ejpam-5253	389	19	.	.	PUNCT
ejpam-5253	390	1	then	then	ADV
ejpam-5253	390	2	the	the	DET
ejpam-5253	390	3	following	follow	VERB
ejpam-5253	390	4	are	be	AUX
ejpam-5253	390	5	hold	hold	ADJ
ejpam-5253	390	6	.	.	PUNCT
ejpam-5253	391	1	(	(	PUNCT
ejpam-5253	391	2	i	i	NOUN
ejpam-5253	391	3	)	)	PUNCT
ejpam-5253	391	4	rh(cl(k	rh(cl(k	PROPN
ejpam-5253	391	5	)	)	PUNCT
ejpam-5253	392	1	∪q	∪q	NUM
ejpam-5253	392	2	)	)	PUNCT
ejpam-5253	392	3	=	=	SYM
ejpam-5253	392	4	cl(k	cl(k	X
ejpam-5253	392	5	)	)	PUNCT
ejpam-5253	392	6	∪rh(q	∪rh(q	CCONJ
ejpam-5253	392	7	)	)	PUNCT
ejpam-5253	392	8	.	.	PUNCT
ejpam-5253	393	1	(	(	PUNCT
ejpam-5253	393	2	ii	ii	NOUN
ejpam-5253	393	3	)	)	PUNCT
ejpam-5253	393	4	rh(int(k	rh(int(k	PROPN
ejpam-5253	393	5	)	)	PUNCT
ejpam-5253	393	6	∩q	∩q	NOUN
ejpam-5253	393	7	)	)	PUNCT
ejpam-5253	394	1	=	=	SYM
ejpam-5253	394	2	int(k	int(k	PROPN
ejpam-5253	394	3	)	)	PUNCT
ejpam-5253	394	4	∩rh(q	∩rh(q	NOUN
ejpam-5253	394	5	)	)	PUNCT
ejpam-5253	394	6	.	.	PUNCT
ejpam-5253	395	1	proof	proof	NOUN
ejpam-5253	395	2	.	.	PUNCT
ejpam-5253	396	1	(	(	PUNCT
ejpam-5253	396	2	i	i	NOUN
ejpam-5253	396	3	)	)	PUNCT
ejpam-5253	396	4	based	base	VERB
ejpam-5253	396	5	on	on	ADP
ejpam-5253	396	6	proposition	proposition	NOUN
ejpam-5253	396	7	7	7	NUM
ejpam-5253	396	8	(	(	PUNCT
ejpam-5253	396	9	i	i	NOUN
ejpam-5253	396	10	)	)	PUNCT
ejpam-5253	396	11	and	and	CCONJ
ejpam-5253	396	12	proposition	proposition	NOUN
ejpam-5253	396	13	9	9	NUM
ejpam-5253	396	14	(	(	PUNCT
ejpam-5253	396	15	ii	ii	NOUN
ejpam-5253	396	16	)	)	PUNCT
ejpam-5253	396	17	,	,	PUNCT
ejpam-5253	396	18	we	we	PRON
ejpam-5253	396	19	have	have	VERB
ejpam-5253	396	20	cl(k	cl(k	ADV
ejpam-5253	396	21	)	)	PUNCT
ejpam-5253	397	1	⊂	⊂	PROPN
ejpam-5253	397	2	rh(cl(k)).then	rh(cl(k)).then	ADV
ejpam-5253	397	3	cl(k)∪rh(q	cl(k)∪rh(q	NOUN
ejpam-5253	397	4	)	)	PUNCT
ejpam-5253	397	5	⊂	⊂	PROPN
ejpam-5253	397	6	rh(cl(k))∪rh(q	rh(cl(k))∪rh(q	NUM
ejpam-5253	397	7	)	)	PUNCT
ejpam-5253	398	1	⊂	⊂	PROPN
ejpam-5253	398	2	rh(cl(k)∪q	rh(cl(k)∪q	PROPN
ejpam-5253	398	3	)	)	PUNCT
ejpam-5253	398	4	.	.	PUNCT
ejpam-5253	399	1	conversely	conversely	ADV
ejpam-5253	399	2	,	,	PUNCT
ejpam-5253	399	3	since	since	SCONJ
ejpam-5253	399	4	cl(k)∪	cl(k)∪	VERB
ejpam-5253	399	5	q	q	PROPN
ejpam-5253	399	6	⊂	⊂	PROPN
ejpam-5253	399	7	cl(k)∪rh(q	cl(k)∪rh(q	PROPN
ejpam-5253	399	8	)	)	PUNCT
ejpam-5253	399	9	and	and	CCONJ
ejpam-5253	399	10	the	the	DET
ejpam-5253	399	11	union	union	NOUN
ejpam-5253	399	12	of	of	ADP
ejpam-5253	399	13	an	an	DET
ejpam-5253	399	14	h	h	NOUN
ejpam-5253	399	15	-	-	PUNCT
ejpam-5253	399	16	open	open	ADJ
ejpam-5253	399	17	set	set	NOUN
ejpam-5253	399	18	and	and	CCONJ
ejpam-5253	399	19	a	a	DET
ejpam-5253	399	20	closed	closed	ADJ
ejpam-5253	399	21	set	set	NOUN
ejpam-5253	399	22	is	be	AUX
ejpam-5253	399	23	h	h	NOUN
ejpam-5253	399	24	-	-	PUNCT
ejpam-5253	399	25	closed	closed	ADJ
ejpam-5253	399	26	,	,	PUNCT
ejpam-5253	399	27	then	then	ADV
ejpam-5253	399	28	rh(cl(k)∪q	rh(cl(k)∪q	PROPN
ejpam-5253	399	29	)	)	PUNCT
ejpam-5253	399	30	⊂	⊂	PROPN
ejpam-5253	399	31	rh(cl(k)∪rh(q	rh(cl(k)∪rh(q	NUM
ejpam-5253	399	32	)	)	PUNCT
ejpam-5253	399	33	)	)	PUNCT
ejpam-5253	400	1	=	=	SYM
ejpam-5253	400	2	cl(k)∪rh(q	cl(k)∪rh(q	NOUN
ejpam-5253	400	3	)	)	PUNCT
ejpam-5253	400	4	.	.	PUNCT
ejpam-5253	401	1	therefore	therefore	ADV
ejpam-5253	401	2	,	,	PUNCT
ejpam-5253	401	3	rh(cl(k)∪q	rh(cl(k)∪q	PROPN
ejpam-5253	401	4	)	)	PUNCT
ejpam-5253	401	5	=	=	SYM
ejpam-5253	401	6	cl(k	cl(k	X
ejpam-5253	401	7	)	)	PUNCT
ejpam-5253	401	8	∪rh(q	∪rh(q	CCONJ
ejpam-5253	401	9	)	)	PUNCT
ejpam-5253	401	10	.	.	PUNCT
ejpam-5253	402	1	references	reference	NOUN
ejpam-5253	402	2	1816	1816	NUM
ejpam-5253	402	3	(	(	PUNCT
ejpam-5253	402	4	ii	ii	PROPN
ejpam-5253	402	5	)	)	PUNCT
ejpam-5253	402	6	considering	consider	VERB
ejpam-5253	402	7	the	the	DET
ejpam-5253	402	8	intersection	intersection	NOUN
ejpam-5253	402	9	of	of	ADP
ejpam-5253	402	10	an	an	DET
ejpam-5253	402	11	open	open	ADJ
ejpam-5253	402	12	set	set	NOUN
ejpam-5253	402	13	int(k	int(k	NOUN
ejpam-5253	402	14	)	)	PUNCT
ejpam-5253	402	15	and	and	CCONJ
ejpam-5253	402	16	an	an	DET
ejpam-5253	402	17	h	h	NOUN
ejpam-5253	402	18	-	-	PUNCT
ejpam-5253	402	19	open	open	ADJ
ejpam-5253	402	20	set	set	NOUN
ejpam-5253	402	21	rh(q	rh(q	PUNCT
ejpam-5253	402	22	)	)	PUNCT
ejpam-5253	402	23	is	be	AUX
ejpam-5253	402	24	hopen	hopen	NOUN
ejpam-5253	402	25	,	,	PUNCT
ejpam-5253	402	26	int(k	int(k	NOUN
ejpam-5253	402	27	)	)	PUNCT
ejpam-5253	402	28	∩	∩	NOUN
ejpam-5253	402	29	rh(q	rh(q	NUM
ejpam-5253	402	30	)	)	PUNCT
ejpam-5253	402	31	=	=	SYM
ejpam-5253	402	32	rh(int(k	rh(int(k	ADJ
ejpam-5253	402	33	)	)	PUNCT
ejpam-5253	402	34	∩	∩	NOUN
ejpam-5253	402	35	rh(q	rh(q	NUM
ejpam-5253	402	36	)	)	PUNCT
ejpam-5253	402	37	)	)	PUNCT
ejpam-5253	403	1	⊂	⊂	PROPN
ejpam-5253	403	2	rh(int(k	rh(int(k	PROPN
ejpam-5253	403	3	)	)	PUNCT
ejpam-5253	403	4	∩	∩	ADJ
ejpam-5253	403	5	q	q	NOUN
ejpam-5253	403	6	)	)	PUNCT
ejpam-5253	403	7	.	.	PUNCT
ejpam-5253	404	1	conversely	conversely	ADV
ejpam-5253	404	2	,	,	PUNCT
ejpam-5253	404	3	by	by	ADP
ejpam-5253	404	4	using	use	VERB
ejpam-5253	404	5	proposition	proposition	NOUN
ejpam-5253	404	6	9	9	NUM
ejpam-5253	404	7	(	(	PUNCT
ejpam-5253	404	8	iii	iii	NOUN
ejpam-5253	404	9	)	)	PUNCT
ejpam-5253	404	10	,	,	PUNCT
ejpam-5253	404	11	rh(int(k)∩q	rh(int(k)∩q	PROPN
ejpam-5253	404	12	)	)	PUNCT
ejpam-5253	404	13	⊂	⊂	PROPN
ejpam-5253	404	14	rh(int(k))∩rh(q	rh(int(k))∩rh(q	NOUN
ejpam-5253	404	15	)	)	PUNCT
ejpam-5253	404	16	⊂	⊂	PROPN
ejpam-5253	404	17	int(k)∩rh(q	int(k)∩rh(q	PROPN
ejpam-5253	404	18	)	)	PUNCT
ejpam-5253	404	19	.	.	PUNCT
ejpam-5253	405	1	therefore	therefore	ADV
ejpam-5253	405	2	rh(int(k	rh(int(k	PROPN
ejpam-5253	405	3	)	)	PUNCT
ejpam-5253	405	4	∩q	∩q	NOUN
ejpam-5253	405	5	)	)	PUNCT
ejpam-5253	406	1	=	=	SYM
ejpam-5253	406	2	int(k	int(k	PROPN
ejpam-5253	406	3	)	)	PUNCT
ejpam-5253	406	4	∩rh(q	∩rh(q	NOUN
ejpam-5253	406	5	)	)	PUNCT
ejpam-5253	406	6	.	.	PUNCT
ejpam-5253	407	1	7	7	X
ejpam-5253	407	2	.	.	X
ejpam-5253	407	3	conclusions	conclusion	NOUN
ejpam-5253	407	4	this	this	DET
ejpam-5253	407	5	paper	paper	NOUN
ejpam-5253	407	6	introduced	introduce	VERB
ejpam-5253	407	7	h	h	NOUN
ejpam-5253	407	8	-	-	PUNCT
ejpam-5253	407	9	rough	rough	ADJ
ejpam-5253	407	10	sets	set	NOUN
ejpam-5253	407	11	,	,	PUNCT
ejpam-5253	407	12	an	an	DET
ejpam-5253	407	13	extension	extension	NOUN
ejpam-5253	407	14	of	of	ADP
ejpam-5253	407	15	rough	rough	ADJ
ejpam-5253	407	16	set	set	NOUN
ejpam-5253	407	17	theory	theory	NOUN
ejpam-5253	407	18	,	,	PUNCT
ejpam-5253	407	19	by	by	ADP
ejpam-5253	407	20	incorporating	incorporate	VERB
ejpam-5253	407	21	h	h	ADJ
ejpam-5253	407	22	-	-	PUNCT
ejpam-5253	407	23	open	open	ADJ
ejpam-5253	407	24	sets	set	NOUN
ejpam-5253	407	25	to	to	PART
ejpam-5253	407	26	define	define	VERB
ejpam-5253	407	27	h	h	NOUN
ejpam-5253	407	28	-	-	PUNCT
ejpam-5253	407	29	lower	low	ADJ
ejpam-5253	407	30	and	and	CCONJ
ejpam-5253	407	31	h	h	NOUN
ejpam-5253	407	32	-	-	PUNCT
ejpam-5253	407	33	upper	upper	ADJ
ejpam-5253	407	34	approximations	approximation	NOUN
ejpam-5253	407	35	.	.	PUNCT
ejpam-5253	408	1	alongside	alongside	ADP
ejpam-5253	408	2	these	these	DET
ejpam-5253	408	3	new	new	ADJ
ejpam-5253	408	4	approximations	approximation	NOUN
ejpam-5253	408	5	,	,	PUNCT
ejpam-5253	408	6	we	we	PRON
ejpam-5253	408	7	explored	explore	VERB
ejpam-5253	408	8	h	h	NOUN
ejpam-5253	408	9	-	-	PUNCT
ejpam-5253	408	10	rough	rough	ADJ
ejpam-5253	408	11	equality	equality	NOUN
ejpam-5253	408	12	and	and	CCONJ
ejpam-5253	408	13	h	h	NOUN
ejpam-5253	408	14	-	-	PUNCT
ejpam-5253	408	15	rough	rough	ADJ
ejpam-5253	408	16	inclusion	inclusion	NOUN
ejpam-5253	408	17	,	,	PUNCT
ejpam-5253	408	18	thoroughly	thoroughly	ADV
ejpam-5253	408	19	examining	examine	VERB
ejpam-5253	408	20	the	the	DET
ejpam-5253	408	21	properties	property	NOUN
ejpam-5253	408	22	of	of	ADP
ejpam-5253	408	23	h	h	NOUN
ejpam-5253	408	24	-	-	PUNCT
ejpam-5253	408	25	approximation	approximation	NOUN
ejpam-5253	408	26	spaces	space	NOUN
ejpam-5253	408	27	.	.	PUNCT
ejpam-5253	409	1	our	our	PRON
ejpam-5253	409	2	findings	finding	NOUN
ejpam-5253	409	3	illustrate	illustrate	VERB
ejpam-5253	409	4	that	that	SCONJ
ejpam-5253	409	5	h	h	NOUN
ejpam-5253	409	6	-	-	PUNCT
ejpam-5253	409	7	rough	rough	ADJ
ejpam-5253	409	8	sets	set	NOUN
ejpam-5253	409	9	provide	provide	VERB
ejpam-5253	409	10	improved	improved	ADJ
ejpam-5253	409	11	precision	precision	NOUN
ejpam-5253	409	12	and	and	CCONJ
ejpam-5253	409	13	flexibility	flexibility	NOUN
ejpam-5253	409	14	in	in	ADP
ejpam-5253	409	15	data	data	NOUN
ejpam-5253	409	16	approximation	approximation	NOUN
ejpam-5253	409	17	and	and	CCONJ
ejpam-5253	409	18	analysis	analysis	NOUN
ejpam-5253	409	19	.	.	PUNCT
ejpam-5253	410	1	future	future	ADJ
ejpam-5253	410	2	research	research	NOUN
ejpam-5253	410	3	should	should	AUX
ejpam-5253	410	4	focus	focus	VERB
ejpam-5253	410	5	on	on	ADP
ejpam-5253	410	6	integrating	integrate	VERB
ejpam-5253	410	7	h	h	NOUN
ejpam-5253	410	8	-	-	PUNCT
ejpam-5253	410	9	rough	rough	ADJ
ejpam-5253	410	10	sets	set	NOUN
ejpam-5253	410	11	with	with	ADP
ejpam-5253	410	12	fuzzy	fuzzy	ADJ
ejpam-5253	410	13	set	set	NOUN
ejpam-5253	410	14	theory	theory	NOUN
ejpam-5253	410	15	to	to	PART
ejpam-5253	410	16	better	well	ADV
ejpam-5253	410	17	manage	manage	VERB
ejpam-5253	410	18	uncertainty	uncertainty	NOUN
ejpam-5253	410	19	,	,	PUNCT
ejpam-5253	410	20	developing	develop	VERB
ejpam-5253	410	21	efficient	efficient	ADJ
ejpam-5253	410	22	algorithms	algorithm	NOUN
ejpam-5253	410	23	for	for	ADP
ejpam-5253	410	24	processing	process	VERB
ejpam-5253	410	25	large	large	ADJ
ejpam-5253	410	26	-	-	PUNCT
ejpam-5253	410	27	scale	scale	NOUN
ejpam-5253	410	28	data	datum	NOUN
ejpam-5253	410	29	,	,	PUNCT
ejpam-5253	410	30	and	and	CCONJ
ejpam-5253	410	31	combining	combine	VERB
ejpam-5253	410	32	h	h	NOUN
ejpam-5253	410	33	-	-	PUNCT
ejpam-5253	410	34	rough	rough	ADJ
ejpam-5253	410	35	sets	set	NOUN
ejpam-5253	410	36	with	with	ADP
ejpam-5253	410	37	neural	neural	ADJ
ejpam-5253	410	38	networks	network	NOUN
ejpam-5253	410	39	and	and	CCONJ
ejpam-5253	410	40	decision	decision	NOUN
ejpam-5253	410	41	trees	tree	NOUN
ejpam-5253	410	42	for	for	ADP
ejpam-5253	410	43	enhanced	enhanced	ADJ
ejpam-5253	410	44	decision	decision	NOUN
ejpam-5253	410	45	-	-	PUNCT
ejpam-5253	410	46	making	make	VERB
ejpam-5253	410	47	processes	process	NOUN
ejpam-5253	410	48	.	.	PUNCT
ejpam-5253	411	1	additionally	additionally	ADV
ejpam-5253	411	2	,	,	PUNCT
ejpam-5253	411	3	applying	apply	VERB
ejpam-5253	411	4	h	h	NOUN
ejpam-5253	411	5	-	-	PUNCT
ejpam-5253	411	6	rough	rough	ADJ
ejpam-5253	411	7	sets	set	NOUN
ejpam-5253	411	8	to	to	ADP
ejpam-5253	411	9	machine	machine	NOUN
ejpam-5253	411	10	learning	learning	NOUN
ejpam-5253	411	11	tasks	task	NOUN
ejpam-5253	411	12	such	such	ADJ
ejpam-5253	411	13	as	as	ADP
ejpam-5253	411	14	feature	feature	NOUN
ejpam-5253	411	15	selection	selection	NOUN
ejpam-5253	411	16	,	,	PUNCT
ejpam-5253	411	17	clustering	clustering	NOUN
ejpam-5253	411	18	,	,	PUNCT
ejpam-5253	411	19	and	and	CCONJ
ejpam-5253	411	20	classification	classification	NOUN
ejpam-5253	411	21	holds	hold	VERB
ejpam-5253	411	22	significant	significant	ADJ
ejpam-5253	411	23	potential	potential	NOUN
ejpam-5253	411	24	.	.	PUNCT
ejpam-5253	412	1	further	further	ADJ
ejpam-5253	412	2	theoretical	theoretical	ADJ
ejpam-5253	412	3	advancements	advancement	NOUN
ejpam-5253	412	4	,	,	PUNCT
ejpam-5253	412	5	particularly	particularly	ADV
ejpam-5253	412	6	in	in	ADP
ejpam-5253	412	7	exploring	explore	VERB
ejpam-5253	412	8	the	the	DET
ejpam-5253	412	9	topological	topological	ADJ
ejpam-5253	412	10	properties	property	NOUN
ejpam-5253	412	11	of	of	ADP
ejpam-5253	412	12	h	h	NOUN
ejpam-5253	412	13	-	-	PUNCT
ejpam-5253	412	14	rough	rough	ADJ
ejpam-5253	412	15	sets	set	NOUN
ejpam-5253	412	16	,	,	PUNCT
ejpam-5253	412	17	will	will	AUX
ejpam-5253	412	18	continue	continue	VERB
ejpam-5253	412	19	to	to	PART
ejpam-5253	412	20	expand	expand	VERB
ejpam-5253	412	21	and	and	CCONJ
ejpam-5253	412	22	deepen	deepen	VERB
ejpam-5253	412	23	the	the	DET
ejpam-5253	412	24	utility	utility	NOUN
ejpam-5253	412	25	of	of	ADP
ejpam-5253	412	26	rough	rough	ADJ
ejpam-5253	412	27	set	set	NOUN
ejpam-5253	412	28	theory	theory	NOUN
ejpam-5253	412	29	,	,	PUNCT
ejpam-5253	412	30	ensuring	ensure	VERB
ejpam-5253	412	31	its	its	PRON
ejpam-5253	412	32	ongoing	ongoing	ADJ
ejpam-5253	412	33	relevance	relevance	NOUN
ejpam-5253	412	34	and	and	CCONJ
ejpam-5253	412	35	effectiveness	effectiveness	NOUN
ejpam-5253	412	36	in	in	ADP
ejpam-5253	412	37	modern	modern	ADJ
ejpam-5253	412	38	data	datum	NOUN
ejpam-5253	412	39	analysis	analysis	NOUN
ejpam-5253	412	40	.	.	PUNCT
ejpam-5253	413	1	acknowledgements	acknowledgement	NOUN
ejpam-5253	413	2	gratitude	gratitude	NOUN
ejpam-5253	413	3	is	be	AUX
ejpam-5253	413	4	extended	extend	VERB
ejpam-5253	413	5	to	to	ADP
ejpam-5253	413	6	my	my	PRON
ejpam-5253	413	7	late	late	ADJ
ejpam-5253	413	8	parents	parent	NOUN
ejpam-5253	413	9	,	,	PUNCT
ejpam-5253	413	10	whose	whose	DET
ejpam-5253	413	11	unwavering	unwavering	ADJ
ejpam-5253	413	12	support	support	NOUN
ejpam-5253	413	13	and	and	CCONJ
ejpam-5253	413	14	encouragement	encouragement	NOUN
ejpam-5253	413	15	continue	continue	VERB
ejpam-5253	413	16	to	to	PART
ejpam-5253	413	17	inspire	inspire	VERB
ejpam-5253	413	18	and	and	CCONJ
ejpam-5253	413	19	drive	drive	VERB
ejpam-5253	413	20	my	my	PRON
ejpam-5253	413	21	pursuit	pursuit	NOUN
ejpam-5253	413	22	of	of	ADP
ejpam-5253	413	23	scientific	scientific	ADJ
ejpam-5253	413	24	inquiry	inquiry	NOUN
ejpam-5253	413	25	,	,	PUNCT
ejpam-5253	413	26	forever	forever	ADV
ejpam-5253	413	27	embedded	embed	VERB
ejpam-5253	413	28	in	in	ADP
ejpam-5253	413	29	the	the	DET
ejpam-5253	413	30	foundation	foundation	NOUN
ejpam-5253	413	31	of	of	ADP
ejpam-5253	413	32	this	this	DET
ejpam-5253	413	33	research	research	NOUN
ejpam-5253	413	34	.	.	PUNCT
ejpam-5253	414	1	references	reference	NOUN
ejpam-5253	414	2	[	[	X
ejpam-5253	414	3	1	1	NUM
ejpam-5253	414	4	]	]	PUNCT
ejpam-5253	414	5	fadhil	fadhil	NOUN
ejpam-5253	414	6	abbas	abbas	NOUN
ejpam-5253	414	7	.	.	PUNCT
ejpam-5253	415	1	on	on	ADP
ejpam-5253	415	2	h	h	ADJ
ejpam-5253	415	3	-	-	PUNCT
ejpam-5253	415	4	open	open	ADJ
ejpam-5253	415	5	sets	set	NOUN
ejpam-5253	415	6	and	and	CCONJ
ejpam-5253	415	7	h	h	NOUN
ejpam-5253	415	8	-	-	PUNCT
ejpam-5253	415	9	continuous	continuous	ADJ
ejpam-5253	415	10	functions	function	NOUN
ejpam-5253	415	11	.	.	PUNCT
ejpam-5253	416	1	j.	j.	PROPN
ejpam-5253	416	2	appl	appl	PROPN
ejpam-5253	416	3	computat	computat	PROPN
ejpam-5253	416	4	math	math	PROPN
ejpam-5253	416	5	,	,	PUNCT
ejpam-5253	416	6	9(1	9(1	NUM
ejpam-5253	416	7	)	)	PUNCT
ejpam-5253	416	8	,	,	PUNCT
ejpam-5253	416	9	2020	2020	NUM
ejpam-5253	416	10	.	.	PUNCT
ejpam-5253	417	1	[	[	X
ejpam-5253	417	2	2	2	X
ejpam-5253	417	3	]	]	PUNCT
ejpam-5253	417	4	me	i	PRON
ejpam-5253	417	5	abd	abd	PROPN
ejpam-5253	417	6	el	el	PROPN
ejpam-5253	417	7	-	-	PROPN
ejpam-5253	417	8	monsef	monsef	ADJ
ejpam-5253	417	9	,	,	PUNCT
ejpam-5253	417	10	am	be	AUX
ejpam-5253	417	11	kozae	kozae	NOUN
ejpam-5253	417	12	,	,	PUNCT
ejpam-5253	417	13	and	and	CCONJ
ejpam-5253	417	14	ai	ai	VERB
ejpam-5253	417	15	el	el	PROPN
ejpam-5253	417	16	-	-	PUNCT
ejpam-5253	417	17	maghrabi	maghrabi	NOUN
ejpam-5253	417	18	.	.	PUNCT
ejpam-5253	418	1	some	some	DET
ejpam-5253	418	2	semi	semi	ADJ
ejpam-5253	418	3	-	-	ADJ
ejpam-5253	418	4	topological	topological	ADJ
ejpam-5253	418	5	applications	application	NOUN
ejpam-5253	418	6	on	on	ADP
ejpam-5253	418	7	rough	rough	ADJ
ejpam-5253	418	8	sets	set	NOUN
ejpam-5253	418	9	.	.	PUNCT
ejpam-5253	419	1	j.	j.	PROPN
ejpam-5253	419	2	egypt	egypt	PROPN
ejpam-5253	419	3	.	.	PUNCT
ejpam-5253	420	1	math	math	PROPN
ejpam-5253	420	2	.	.	PUNCT
ejpam-5253	421	1	soc	soc	PROPN
ejpam-5253	421	2	,	,	PUNCT
ejpam-5253	421	3	12(1):45–53	12(1):45–53	NUM
ejpam-5253	421	4	,	,	PUNCT
ejpam-5253	421	5	2004	2004	NUM
ejpam-5253	421	6	.	.	PUNCT
ejpam-5253	422	1	[	[	X
ejpam-5253	422	2	3	3	NUM
ejpam-5253	422	3	]	]	X
ejpam-5253	422	4	hm	hm	INTJ
ejpam-5253	422	5	abu	abu	PROPN
ejpam-5253	422	6	-	-	PUNCT
ejpam-5253	422	7	donia	donia	PROPN
ejpam-5253	422	8	,	,	PUNCT
ejpam-5253	422	9	aa	aa	PROPN
ejpam-5253	422	10	nasef	nasef	PROPN
ejpam-5253	422	11	,	,	PUNCT
ejpam-5253	422	12	and	and	CCONJ
ejpam-5253	422	13	ea	ea	X
ejpam-5253	422	14	marai	marai	ADJ
ejpam-5253	422	15	.	.	PUNCT
ejpam-5253	423	1	finite	finite	PROPN
ejpam-5253	423	2	information	information	NOUN
ejpam-5253	423	3	systems	system	NOUN
ejpam-5253	423	4	.	.	PUNCT
ejpam-5253	424	1	applied	apply	VERB
ejpam-5253	424	2	mathematics	mathematics	PROPN
ejpam-5253	424	3	&	&	CCONJ
ejpam-5253	424	4	information	information	NOUN
ejpam-5253	424	5	sciences	sciences	PROPN
ejpam-5253	424	6	,	,	PUNCT
ejpam-5253	424	7	1(1):13–21	1(1):13–21	NUM
ejpam-5253	424	8	,	,	PUNCT
ejpam-5253	424	9	2007	2007	NUM
ejpam-5253	424	10	.	.	PUNCT
ejpam-5253	425	1	[	[	X
ejpam-5253	425	2	4	4	NUM
ejpam-5253	425	3	]	]	X
ejpam-5253	425	4	marek	marek	PROPN
ejpam-5253	425	5	chuchro	chuchro	PROPN
ejpam-5253	425	6	.	.	PUNCT
ejpam-5253	426	1	a	a	DET
ejpam-5253	426	2	certain	certain	ADJ
ejpam-5253	426	3	conception	conception	NOUN
ejpam-5253	426	4	of	of	ADP
ejpam-5253	426	5	rough	rough	ADJ
ejpam-5253	426	6	sets	set	NOUN
ejpam-5253	426	7	in	in	ADP
ejpam-5253	426	8	topological	topological	ADJ
ejpam-5253	426	9	boolean	boolean	ADJ
ejpam-5253	426	10	algebras	algebra	NOUN
ejpam-5253	426	11	.	.	PUNCT
ejpam-5253	426	12	1993	1993	NUM
ejpam-5253	426	13	.	.	PUNCT
ejpam-5253	427	1	[	[	X
ejpam-5253	427	2	5	5	X
ejpam-5253	427	3	]	]	X
ejpam-5253	427	4	j	j	PROPN
ejpam-5253	427	5	james	james	PROPN
ejpam-5253	427	6	and	and	CCONJ
ejpam-5253	427	7	f	f	PROPN
ejpam-5253	427	8	james	james	PROPN
ejpam-5253	427	9	alpigini	alpigini	PROPN
ejpam-5253	427	10	.	.	PUNCT
ejpam-5253	428	1	peters	peters	PROPN
ejpam-5253	428	2	,	,	PUNCT
ejpam-5253	428	3	andrzej	andrzej	PROPN
ejpam-5253	428	4	skowron	skowron	PROPN
ejpam-5253	428	5	and	and	CCONJ
ejpam-5253	428	6	ning	ning	PROPN
ejpam-5253	428	7	zhong	zhong	PROPN
ejpam-5253	428	8	,	,	PUNCT
ejpam-5253	428	9	rough	rough	ADJ
ejpam-5253	428	10	set	set	NOUN
ejpam-5253	428	11	elements	element	NOUN
ejpam-5253	428	12	,	,	PUNCT
ejpam-5253	428	13	rough	rough	ADJ
ejpam-5253	428	14	sets	set	NOUN
ejpam-5253	428	15	and	and	CCONJ
ejpam-5253	428	16	current	current	ADJ
ejpam-5253	428	17	trends	trend	NOUN
ejpam-5253	428	18	in	in	ADP
ejpam-5253	428	19	compuring	compure	VERB
ejpam-5253	428	20	,	,	PUNCT
ejpam-5253	428	21	5th	5th	ADJ
ejpam-5253	428	22	int	int	NOUN
ejpam-5253	428	23	.	.	PUNCT
ejpam-5253	429	1	conf	conf	PROPN
ejpam-5253	429	2	.	.	PUNCT
ejpam-5253	430	1	malvern	malvern	PROPN
ejpam-5253	430	2	,	,	PUNCT
ejpam-5253	430	3	pa	pa	PROPN
ejpam-5253	430	4	,	,	PUNCT
ejpam-5253	430	5	usa	usa	PROPN
ejpam-5253	430	6	.	.	PROPN
ejpam-5253	430	7	proc	proc	PROPN
ejpam-5253	430	8	.	.	PUNCT
ejpam-5253	431	1	springer	springer	NOUN
ejpam-5253	431	2	,	,	PUNCT
ejpam-5253	431	3	pages	page	NOUN
ejpam-5253	431	4	12–16	12–16	NUM
ejpam-5253	431	5	,	,	PUNCT
ejpam-5253	431	6	2002	2002	NUM
ejpam-5253	431	7	.	.	PUNCT
ejpam-5253	432	1	references	reference	NOUN
ejpam-5253	432	2	1817	1817	NUM
ejpam-5253	432	3	[	[	X
ejpam-5253	432	4	6	6	NUM
ejpam-5253	432	5	]	]	PUNCT
ejpam-5253	432	6	john	john	PROPN
ejpam-5253	432	7	l	l	PROPN
ejpam-5253	432	8	kelley	kelley	PROPN
ejpam-5253	432	9	.	.	PUNCT
ejpam-5253	433	1	general	general	ADJ
ejpam-5253	433	2	topology	topology	PROPN
ejpam-5253	433	3	.	.	PUNCT
ejpam-5253	434	1	courier	courier	PROPN
ejpam-5253	434	2	dover	dover	PROPN
ejpam-5253	434	3	publications	publication	NOUN
ejpam-5253	434	4	,	,	PUNCT
ejpam-5253	434	5	2017	2017	NUM
ejpam-5253	434	6	.	.	PUNCT
ejpam-5253	435	1	[	[	X
ejpam-5253	435	2	7	7	X
ejpam-5253	435	3	]	]	PUNCT
ejpam-5253	435	4	ef	ef	X
ejpam-5253	435	5	lashin	lashin	NOUN
ejpam-5253	435	6	,	,	PUNCT
ejpam-5253	435	7	am	be	AUX
ejpam-5253	435	8	kozae	kozae	NOUN
ejpam-5253	435	9	,	,	PUNCT
ejpam-5253	435	10	aa	aa	NOUN
ejpam-5253	435	11	abo	abo	NOUN
ejpam-5253	435	12	khadra	khadra	NOUN
ejpam-5253	435	13	,	,	PUNCT
ejpam-5253	435	14	and	and	CCONJ
ejpam-5253	435	15	tamer	tame	ADJ
ejpam-5253	435	16	medhat	medhat	PROPN
ejpam-5253	435	17	.	.	PUNCT
ejpam-5253	436	1	rough	rough	ADJ
ejpam-5253	436	2	set	set	NOUN
ejpam-5253	436	3	theory	theory	NOUN
ejpam-5253	436	4	for	for	ADP
ejpam-5253	436	5	topological	topological	ADJ
ejpam-5253	436	6	spaces	space	NOUN
ejpam-5253	436	7	.	.	PUNCT
ejpam-5253	437	1	international	international	ADJ
ejpam-5253	437	2	journal	journal	PROPN
ejpam-5253	437	3	of	of	ADP
ejpam-5253	437	4	approximate	approximate	ADJ
ejpam-5253	437	5	reasoning	reasoning	NOUN
ejpam-5253	437	6	,	,	PUNCT
ejpam-5253	437	7	40(1	40(1	NUM
ejpam-5253	437	8	-	-	SYM
ejpam-5253	437	9	2):35–43	2):35–43	NUM
ejpam-5253	437	10	,	,	PUNCT
ejpam-5253	437	11	2005	2005	NUM
ejpam-5253	437	12	.	.	PUNCT
ejpam-5253	438	1	[	[	X
ejpam-5253	438	2	8	8	NUM
ejpam-5253	438	3	]	]	X
ejpam-5253	438	4	tsau	tsau	NOUN
ejpam-5253	438	5	y	y	PROPN
ejpam-5253	438	6	lin	lin	PROPN
ejpam-5253	438	7	.	.	PUNCT
ejpam-5253	439	1	topological	topological	ADJ
ejpam-5253	439	2	and	and	CCONJ
ejpam-5253	439	3	fuzzy	fuzzy	ADJ
ejpam-5253	439	4	rough	rough	ADJ
ejpam-5253	439	5	sets	set	NOUN
ejpam-5253	439	6	.	.	PUNCT
ejpam-5253	440	1	in	in	ADP
ejpam-5253	440	2	intelligent	intelligent	ADJ
ejpam-5253	440	3	decision	decision	NOUN
ejpam-5253	440	4	support	support	NOUN
ejpam-5253	440	5	:	:	PUNCT
ejpam-5253	440	6	handbook	handbook	NOUN
ejpam-5253	440	7	of	of	ADP
ejpam-5253	440	8	applications	application	NOUN
ejpam-5253	440	9	and	and	CCONJ
ejpam-5253	440	10	advances	advance	NOUN
ejpam-5253	440	11	of	of	ADP
ejpam-5253	440	12	the	the	DET
ejpam-5253	440	13	rough	rough	ADJ
ejpam-5253	440	14	sets	set	NOUN
ejpam-5253	440	15	theory	theory	NOUN
ejpam-5253	440	16	,	,	PUNCT
ejpam-5253	440	17	pages	page	NOUN
ejpam-5253	440	18	287–304	287–304	NUM
ejpam-5253	440	19	.	.	PUNCT
ejpam-5253	440	20	springer	springer	NOUN
ejpam-5253	440	21	,	,	PUNCT
ejpam-5253	440	22	1992	1992	NUM
ejpam-5253	440	23	.	.	PUNCT
ejpam-5253	441	1	[	[	X
ejpam-5253	441	2	9	9	NUM
ejpam-5253	441	3	]	]	PUNCT
ejpam-5253	441	4	z.	z.	PROPN
ejpam-5253	441	5	pawlak	pawlak	PROPN
ejpam-5253	441	6	m.	m.	PROPN
ejpam-5253	441	7	novotny	novotny	PROPN
ejpam-5253	441	8	.	.	PUNCT
ejpam-5253	442	1	on	on	ADP
ejpam-5253	442	2	rough	rough	ADJ
ejpam-5253	442	3	equalities	equality	NOUN
ejpam-5253	442	4	.	.	PUNCT
ejpam-5253	443	1	33(99	33(99	NUM
ejpam-5253	443	2	-	-	SYM
ejpam-5253	443	3	104	104	NUM
ejpam-5253	443	4	)	)	PUNCT
ejpam-5253	443	5	,	,	PUNCT
ejpam-5253	443	6	1985	1985	NUM
ejpam-5253	443	7	.	.	PUNCT
ejpam-5253	444	1	[	[	X
ejpam-5253	444	2	10	10	NUM
ejpam-5253	444	3	]	]	X
ejpam-5253	444	4	amin	amin	NOUN
ejpam-5253	444	5	mousavi	mousavi	PROPN
ejpam-5253	444	6	and	and	CCONJ
ejpam-5253	444	7	parviz	parviz	ADJ
ejpam-5253	444	8	jabedar	jabedar	ADJ
ejpam-5253	444	9	-	-	PUNCT
ejpam-5253	444	10	maralani	maralani	ADJ
ejpam-5253	444	11	.	.	PUNCT
ejpam-5253	445	1	relative	relative	ADJ
ejpam-5253	445	2	sets	set	NOUN
ejpam-5253	445	3	and	and	CCONJ
ejpam-5253	445	4	rough	rough	ADJ
ejpam-5253	445	5	sets	set	NOUN
ejpam-5253	445	6	.	.	PUNCT
ejpam-5253	446	1	2001	2001	NUM
ejpam-5253	446	2	.	.	PUNCT
ejpam-5253	447	1	[	[	X
ejpam-5253	447	2	11	11	NUM
ejpam-5253	447	3	]	]	X
ejpam-5253	447	4	m	m	PROPN
ejpam-5253	447	5	novotny	novotny	PROPN
ejpam-5253	447	6	and	and	CCONJ
ejpam-5253	447	7	z	z	PROPN
ejpam-5253	447	8	pawlak	pawlak	ADJ
ejpam-5253	447	9	.	.	PUNCT
ejpam-5253	448	1	characterization	characterization	NOUN
ejpam-5253	448	2	of	of	ADP
ejpam-5253	448	3	rough	rough	ADJ
ejpam-5253	448	4	top	top	ADJ
ejpam-5253	448	5	equalities	equality	NOUN
ejpam-5253	448	6	and	and	CCONJ
ejpam-5253	448	7	rough	rough	ADJ
ejpam-5253	448	8	bottom	bottom	ADJ
ejpam-5253	448	9	equalities	equality	NOUN
ejpam-5253	448	10	.	.	PUNCT
ejpam-5253	449	1	bulletin	bulletin	NOUN
ejpam-5253	449	2	of	of	ADP
ejpam-5253	449	3	the	the	DET
ejpam-5253	449	4	polish	polish	PROPN
ejpam-5253	449	5	academy	academy	PROPN
ejpam-5253	449	6	of	of	ADP
ejpam-5253	449	7	sciences	sciences	PROPN
ejpam-5253	449	8	.	.	PUNCT
ejpam-5253	450	1	mathematics	mathematic	NOUN
ejpam-5253	450	2	,	,	PUNCT
ejpam-5253	450	3	33(1	33(1	PROPN
ejpam-5253	450	4	-	-	PUNCT
ejpam-5253	450	5	2):91–97	2):91–97	NUM
ejpam-5253	450	6	,	,	PUNCT
ejpam-5253	450	7	1985	1985	NUM
ejpam-5253	450	8	.	.	PUNCT
ejpam-5253	451	1	[	[	X
ejpam-5253	451	2	12	12	NUM
ejpam-5253	451	3	]	]	PUNCT
ejpam-5253	451	4	zdzis	zdzis	NOUN
ejpam-5253	451	5	law	law	NOUN
ejpam-5253	451	6	pawlak	pawlak	NOUN
ejpam-5253	451	7	.	.	PUNCT
ejpam-5253	452	1	rough	rough	ADJ
ejpam-5253	452	2	sets	set	NOUN
ejpam-5253	452	3	:	:	PUNCT
ejpam-5253	452	4	theoretical	theoretical	ADJ
ejpam-5253	452	5	aspects	aspect	NOUN
ejpam-5253	452	6	of	of	ADP
ejpam-5253	452	7	reasoning	reasoning	NOUN
ejpam-5253	452	8	about	about	ADP
ejpam-5253	452	9	data	datum	NOUN
ejpam-5253	452	10	,	,	PUNCT
ejpam-5253	452	11	volume	volume	NOUN
ejpam-5253	452	12	9	9	NUM
ejpam-5253	452	13	.	.	PUNCT
ejpam-5253	452	14	springer	springer	NOUN
ejpam-5253	452	15	science	science	PROPN
ejpam-5253	452	16	&	&	CCONJ
ejpam-5253	452	17	business	business	NOUN
ejpam-5253	452	18	media	medium	NOUN
ejpam-5253	452	19	,	,	PUNCT
ejpam-5253	452	20	2012	2012	NUM
ejpam-5253	452	21	.	.	PUNCT
ejpam-5253	453	1	[	[	X
ejpam-5253	453	2	13	13	NUM
ejpam-5253	453	3	]	]	PUNCT
ejpam-5253	453	4	zdzis	zdzis	NOUN
ejpam-5253	453	5	law	law	NOUN
ejpam-5253	453	6	pawlak	pawlak	NOUN
ejpam-5253	453	7	.	.	PUNCT
ejpam-5253	454	1	rough	rough	ADJ
ejpam-5253	454	2	sets	set	NOUN
ejpam-5253	454	3	:	:	PUNCT
ejpam-5253	454	4	theoretical	theoretical	ADJ
ejpam-5253	454	5	aspects	aspect	NOUN
ejpam-5253	454	6	of	of	ADP
ejpam-5253	454	7	reasoning	reasoning	NOUN
ejpam-5253	454	8	about	about	ADP
ejpam-5253	454	9	data	datum	NOUN
ejpam-5253	454	10	,	,	PUNCT
ejpam-5253	454	11	volume	volume	NOUN
ejpam-5253	454	12	9	9	NUM
ejpam-5253	454	13	.	.	PUNCT
ejpam-5253	454	14	springer	springer	NOUN
ejpam-5253	454	15	science	science	PROPN
ejpam-5253	454	16	&	&	CCONJ
ejpam-5253	454	17	business	business	NOUN
ejpam-5253	454	18	media	medium	NOUN
ejpam-5253	454	19	,	,	PUNCT
ejpam-5253	454	20	2012	2012	NUM
ejpam-5253	454	21	.	.	PUNCT
ejpam-5253	455	1	[	[	X
ejpam-5253	455	2	14	14	NUM
ejpam-5253	455	3	]	]	PUNCT
ejpam-5253	455	4	viara	viara	NOUN
ejpam-5253	455	5	popova	popova	NOUN
ejpam-5253	455	6	.	.	PUNCT
ejpam-5253	456	1	knowledge	knowledge	NOUN
ejpam-5253	456	2	discovery	discovery	PROPN
ejpam-5253	456	3	and	and	CCONJ
ejpam-5253	456	4	monotonicity	monotonicity	NOUN
ejpam-5253	456	5	.	.	PUNCT
ejpam-5253	456	6	number	number	PROPN
ejpam-5253	456	7	erim	erim	PROPN
ejpam-5253	456	8	phd	phd	PROPN
ejpam-5253	456	9	series	series	PROPN
ejpam-5253	456	10	;	;	PUNCT
ejpam-5253	456	11	eps-2004	eps-2004	NOUN
ejpam-5253	456	12	-	-	PUNCT
ejpam-5253	456	13	037	037	NUM
ejpam-5253	456	14	-	-	PUNCT
ejpam-5253	456	15	lis	li	NOUN
ejpam-5253	456	16	.	.	PUNCT
ejpam-5253	456	17	2004	2004	NUM
ejpam-5253	456	18	.	.	PUNCT
ejpam-5253	457	1	[	[	X
ejpam-5253	457	2	15	15	NUM
ejpam-5253	457	3	]	]	X
ejpam-5253	457	4	keyun	keyun	VERB
ejpam-5253	457	5	qin	qin	PROPN
ejpam-5253	457	6	and	and	CCONJ
ejpam-5253	457	7	zheng	zheng	PROPN
ejpam-5253	457	8	pei	pei	PROPN
ejpam-5253	457	9	.	.	PUNCT
ejpam-5253	458	1	on	on	ADP
ejpam-5253	458	2	the	the	DET
ejpam-5253	458	3	topological	topological	ADJ
ejpam-5253	458	4	properties	property	NOUN
ejpam-5253	458	5	of	of	ADP
ejpam-5253	458	6	fuzzy	fuzzy	ADJ
ejpam-5253	458	7	rough	rough	ADJ
ejpam-5253	458	8	sets	set	NOUN
ejpam-5253	458	9	.	.	PUNCT
ejpam-5253	459	1	fuzzy	fuzzy	ADJ
ejpam-5253	459	2	sets	set	NOUN
ejpam-5253	459	3	and	and	CCONJ
ejpam-5253	459	4	systems	system	NOUN
ejpam-5253	459	5	,	,	PUNCT
ejpam-5253	459	6	151(3):601–613	151(3):601–613	NUM
ejpam-5253	459	7	,	,	PUNCT
ejpam-5253	459	8	2005	2005	NUM
ejpam-5253	459	9	.	.	PUNCT
ejpam-5253	460	1	[	[	X
ejpam-5253	460	2	16	16	NUM
ejpam-5253	460	3	]	]	X
ejpam-5253	460	4	anita	anita	PROPN
ejpam-5253	460	5	wasilewska	wasilewska	PROPN
ejpam-5253	460	6	and	and	CCONJ
ejpam-5253	460	7	laurent	laurent	NOUN
ejpam-5253	460	8	vigneron	vigneron	NOUN
ejpam-5253	460	9	.	.	PUNCT
ejpam-5253	461	1	on	on	ADP
ejpam-5253	461	2	generalized	generalize	VERB
ejpam-5253	461	3	rough	rough	ADJ
ejpam-5253	461	4	sets	set	NOUN
ejpam-5253	461	5	.	.	PUNCT
ejpam-5253	462	1	preprint	preprint	NOUN
ejpam-5253	462	2	,	,	PUNCT
ejpam-5253	462	3	1997	1997	NUM
ejpam-5253	462	4	.	.	PUNCT
ejpam-5253	463	1	[	[	X
ejpam-5253	463	2	17	17	NUM
ejpam-5253	463	3	]	]	PUNCT
ejpam-5253	463	4	a.	a.	NOUN
ejpam-5253	463	5	wiweger	wiweger	NOUN
ejpam-5253	463	6	.	.	PUNCT
ejpam-5253	464	1	on	on	ADP
ejpam-5253	464	2	topological	topological	ADJ
ejpam-5253	464	3	rough	rough	ADJ
ejpam-5253	464	4	sets	set	NOUN
ejpam-5253	464	5	.	.	PUNCT
ejpam-5253	465	1	bull	bull	NOUN
ejpam-5253	465	2	,	,	PUNCT
ejpam-5253	465	3	pol	pol	NOUN
ejpam-5253	465	4	.	.	PUNCT
ejpam-5253	466	1	acad	acad	PROPN
ejpam-5253	466	2	.	.	PROPN
ejpam-5253	466	3	,	,	PUNCT
ejpam-5253	466	4	math	math	NOUN
ejpam-5253	466	5	.	.	PROPN
ejpam-5253	466	6	,	,	PUNCT
ejpam-5253	466	7	37:89–93	37:89–93	NUM
ejpam-5253	466	8	,	,	PUNCT
ejpam-5253	466	9	1989	1989	NUM
ejpam-5253	466	10	.	.	PUNCT
ejpam-5253	467	1	[	[	X
ejpam-5253	467	2	18	18	NUM
ejpam-5253	467	3	]	]	X
ejpam-5253	467	4	yy	yy	PROPN
ejpam-5253	467	5	yao	yao	PROPN
ejpam-5253	467	6	.	.	PUNCT
ejpam-5253	468	1	two	two	NUM
ejpam-5253	468	2	views	view	NOUN
ejpam-5253	468	3	of	of	ADP
ejpam-5253	468	4	the	the	DET
ejpam-5253	468	5	theory	theory	NOUN
ejpam-5253	468	6	of	of	ADP
ejpam-5253	468	7	rough	rough	ADJ
ejpam-5253	468	8	sets	set	NOUN
ejpam-5253	468	9	in	in	ADP
ejpam-5253	468	10	finite	finite	ADJ
ejpam-5253	468	11	universes	universe	NOUN
ejpam-5253	468	12	.	.	PUNCT
ejpam-5253	469	1	international	international	ADJ
ejpam-5253	469	2	journal	journal	PROPN
ejpam-5253	469	3	of	of	ADP
ejpam-5253	469	4	approximate	approximate	ADJ
ejpam-5253	469	5	reasoning	reasoning	NOUN
ejpam-5253	469	6	,	,	PUNCT
ejpam-5253	469	7	15(4):291–317	15(4):291–317	NUM
ejpam-5253	469	8	,	,	PUNCT
ejpam-5253	469	9	1996	1996	NUM
ejpam-5253	469	10	.	.	PUNCT
ejpam-5253	470	1	[	[	X
ejpam-5253	470	2	19	19	NUM
ejpam-5253	470	3	]	]	X
ejpam-5253	470	4	yy	yy	PROPN
ejpam-5253	470	5	yao	yao	PROPN
ejpam-5253	470	6	.	.	PUNCT
ejpam-5253	471	1	generalized	generalize	VERB
ejpam-5253	471	2	rough	rough	ADJ
ejpam-5253	471	3	set	set	NOUN
ejpam-5253	471	4	models	model	NOUN
ejpam-5253	471	5	.	.	PUNCT
ejpam-5253	472	1	rough	rough	ADJ
ejpam-5253	472	2	sets	set	NOUN
ejpam-5253	472	3	in	in	ADP
ejpam-5253	472	4	knowledge	knowledge	NOUN
ejpam-5253	472	5	discovery	discovery	NOUN
ejpam-5253	472	6	,	,	PUNCT
ejpam-5253	472	7	1:286–318	1:286–318	NUM
ejpam-5253	472	8	,	,	PUNCT
ejpam-5253	472	9	1998	1998	NUM
ejpam-5253	472	10	.	.	PUNCT
ejpam-5253	473	1	[	[	X
ejpam-5253	473	2	20	20	NUM
ejpam-5253	473	3	]	]	X
ejpam-5253	473	4	yy	yy	PROPN
ejpam-5253	473	5	yao	yao	PROPN
ejpam-5253	473	6	.	.	PUNCT
ejpam-5253	474	1	on	on	ADP
ejpam-5253	474	2	generalizing	generalize	VERB
ejpam-5253	474	3	rough	rough	ADJ
ejpam-5253	474	4	set	set	NOUN
ejpam-5253	474	5	theory	theory	NOUN
ejpam-5253	474	6	.	.	PUNCT
ejpam-5253	475	1	in	in	ADP
ejpam-5253	475	2	international	international	ADJ
ejpam-5253	475	3	workshop	workshop	NOUN
ejpam-5253	475	4	on	on	ADP
ejpam-5253	475	5	rough	rough	ADJ
ejpam-5253	475	6	sets	set	NOUN
ejpam-5253	475	7	,	,	PUNCT
ejpam-5253	475	8	fuzzy	fuzzy	ADJ
ejpam-5253	475	9	sets	set	NOUN
ejpam-5253	475	10	,	,	PUNCT
ejpam-5253	475	11	data	datum	NOUN
ejpam-5253	475	12	mining	mining	NOUN
ejpam-5253	475	13	,	,	PUNCT
ejpam-5253	475	14	and	and	CCONJ
ejpam-5253	475	15	granular	granular	ADJ
ejpam-5253	475	16	-	-	PUNCT
ejpam-5253	475	17	soft	soft	ADJ
ejpam-5253	475	18	computing	computing	NOUN
ejpam-5253	475	19	,	,	PUNCT
ejpam-5253	475	20	pages	page	NOUN
ejpam-5253	475	21	44–51	44–51	NUM
ejpam-5253	475	22	.	.	PUNCT
ejpam-5253	475	23	springer	springer	NOUN
ejpam-5253	475	24	,	,	PUNCT
ejpam-5253	475	25	2003	2003	NUM
ejpam-5253	475	26	.	.	PUNCT