id	sid	tid	token	lemma	pos
ejpam-5475	1	1	european	european	PROPN
ejpam-5475	1	2	journal	journal	PROPN
ejpam-5475	1	3	of	of	ADP
ejpam-5475	1	4	pure	pure	ADJ
ejpam-5475	1	5	and	and	CCONJ
ejpam-5475	1	6	applied	apply	VERB
ejpam-5475	1	7	mathematics	mathematic	NOUN
ejpam-5475	1	8	vol	vol	NOUN
ejpam-5475	1	9	.	.	PROPN
ejpam-5475	2	1	17	17	NUM
ejpam-5475	2	2	,	,	PUNCT
ejpam-5475	2	3	no	no	INTJ
ejpam-5475	2	4	.	.	NOUN
ejpam-5475	2	5	4	4	NUM
ejpam-5475	2	6	,	,	PUNCT
ejpam-5475	2	7	2024	2024	NUM
ejpam-5475	2	8	,	,	PUNCT
ejpam-5475	2	9	3610	3610	NUM
ejpam-5475	2	10	-	-	SYM
ejpam-5475	2	11	3621	3621	NUM
ejpam-5475	2	12	issn	issn	PROPN
ejpam-5475	2	13	1307	1307	NUM
ejpam-5475	2	14	-	-	SYM
ejpam-5475	2	15	5543	5543	NUM
ejpam-5475	2	16	–	–	PUNCT
ejpam-5475	2	17	ejpam.com	ejpam.com	X
ejpam-5475	2	18	published	publish	VERB
ejpam-5475	2	19	by	by	ADP
ejpam-5475	2	20	new	new	PROPN
ejpam-5475	2	21	york	york	PROPN
ejpam-5475	2	22	business	business	PROPN
ejpam-5475	2	23	global	global	ADJ
ejpam-5475	2	24	modifications	modification	NOUN
ejpam-5475	2	25	to	to	ADP
ejpam-5475	2	26	mixed	mixed	ADJ
ejpam-5475	2	27	θ(ν1	θ(ν1	NOUN
ejpam-5475	2	28	,	,	PUNCT
ejpam-5475	3	1	ν2)-open	ν2)-open	ADJ
ejpam-5475	3	2	sets	set	NOUN
ejpam-5475	3	3	in	in	ADP
ejpam-5475	3	4	generalized	generalized	ADJ
ejpam-5475	3	5	topological	topological	ADJ
ejpam-5475	3	6	spaces	space	NOUN
ejpam-5475	3	7	abdo	abdo	PROPN
ejpam-5475	3	8	qahis1,∗	qahis1,∗	PROPN
ejpam-5475	3	9	,	,	PUNCT
ejpam-5475	3	10	awn	awn	VERB
ejpam-5475	3	11	alqahtani2	alqahtani2	PROPN
ejpam-5475	3	12	1	1	NUM
ejpam-5475	3	13	department	department	NOUN
ejpam-5475	3	14	of	of	ADP
ejpam-5475	3	15	mathematics	mathematic	NOUN
ejpam-5475	3	16	,	,	PUNCT
ejpam-5475	3	17	faculty	faculty	NOUN
ejpam-5475	3	18	of	of	ADP
ejpam-5475	3	19	science	science	NOUN
ejpam-5475	3	20	and	and	CCONJ
ejpam-5475	3	21	arts	art	NOUN
ejpam-5475	3	22	,	,	PUNCT
ejpam-5475	3	23	najran	najran	ADJ
ejpam-5475	3	24	university	university	NOUN
ejpam-5475	3	25	,	,	PUNCT
ejpam-5475	3	26	saudi	saudi	PROPN
ejpam-5475	3	27	arabia	arabia	PROPN
ejpam-5475	3	28	abstract	abstract	PROPN
ejpam-5475	3	29	.	.	PUNCT
ejpam-5475	4	1	á.	á.	PROPN
ejpam-5475	4	2	császár	császár	PROPN
ejpam-5475	4	3	and	and	CCONJ
ejpam-5475	4	4	makai	makai	PROPN
ejpam-5475	4	5	jr	jr	PROPN
ejpam-5475	4	6	.	.	PUNCT
ejpam-5475	5	1	[	[	X
ejpam-5475	5	2	5	5	NUM
ejpam-5475	5	3	]	]	PUNCT
ejpam-5475	5	4	introduced	introduce	VERB
ejpam-5475	5	5	the	the	DET
ejpam-5475	5	6	concepts	concept	NOUN
ejpam-5475	5	7	of	of	ADP
ejpam-5475	5	8	the	the	DET
ejpam-5475	5	9	mixed	mixed	ADJ
ejpam-5475	5	10	operation	operation	NOUN
ejpam-5475	5	11	γθ(ν1,ν2	γθ(ν1,ν2	ADP
ejpam-5475	5	12	)	)	PUNCT
ejpam-5475	5	13	and	and	CCONJ
ejpam-5475	5	14	mixed	mixed	ADJ
ejpam-5475	5	15	θ(ν1	θ(ν1	NOUN
ejpam-5475	5	16	,	,	PUNCT
ejpam-5475	5	17	ν2)-open	ν2)-open	ADJ
ejpam-5475	5	18	sets	set	NOUN
ejpam-5475	5	19	in	in	ADP
ejpam-5475	5	20	generalized	generalized	ADJ
ejpam-5475	5	21	topological	topological	ADJ
ejpam-5475	5	22	spaces	space	NOUN
ejpam-5475	5	23	.	.	PUNCT
ejpam-5475	6	1	in	in	ADP
ejpam-5475	6	2	this	this	DET
ejpam-5475	6	3	paper	paper	NOUN
ejpam-5475	6	4	,	,	PUNCT
ejpam-5475	6	5	we	we	PRON
ejpam-5475	6	6	extend	extend	VERB
ejpam-5475	6	7	this	this	DET
ejpam-5475	6	8	framework	framework	NOUN
ejpam-5475	6	9	by	by	ADP
ejpam-5475	6	10	introducing	introduce	VERB
ejpam-5475	6	11	the	the	DET
ejpam-5475	6	12	concepts	concept	NOUN
ejpam-5475	6	13	of	of	ADP
ejpam-5475	6	14	mixed	mixed	ADJ
ejpam-5475	6	15	operation	operation	NOUN
ejpam-5475	6	16	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	6	17	)	)	PUNCT
ejpam-5475	6	18	and	and	CCONJ
ejpam-5475	6	19	mixed	mixed	ADJ
ejpam-5475	6	20	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	6	21	,	,	PUNCT
ejpam-5475	6	22	ν2)-open	ν2)-open	ADJ
ejpam-5475	6	23	sets	set	NOUN
ejpam-5475	6	24	(	(	PUNCT
ejpam-5475	6	25	briefly	briefly	ADV
ejpam-5475	6	26	,	,	PUNCT
ejpam-5475	6	27	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	6	28	,	,	PUNCT
ejpam-5475	6	29	ν2)-open	ν2)-open	ADJ
ejpam-5475	6	30	sets	set	NOUN
ejpam-5475	6	31	)	)	PUNCT
ejpam-5475	6	32	and	and	CCONJ
ejpam-5475	6	33	investigate	investigate	VERB
ejpam-5475	6	34	their	their	PRON
ejpam-5475	6	35	fundamental	fundamental	ADJ
ejpam-5475	6	36	properties	property	NOUN
ejpam-5475	6	37	in	in	ADP
ejpam-5475	6	38	generalized	generalized	ADJ
ejpam-5475	6	39	topological	topological	ADJ
ejpam-5475	6	40	spaces	space	NOUN
ejpam-5475	6	41	.	.	PUNCT
ejpam-5475	7	1	we	we	PRON
ejpam-5475	7	2	explore	explore	VERB
ejpam-5475	7	3	the	the	DET
ejpam-5475	7	4	relationships	relationship	NOUN
ejpam-5475	7	5	among	among	ADP
ejpam-5475	7	6	γθ̃(ν1,ν2	γθ̃(ν1,ν2	ADJ
ejpam-5475	7	7	)	)	PUNCT
ejpam-5475	7	8	,	,	PUNCT
ejpam-5475	7	9	γθ(ν1,ν2	γθ(ν1,ν2	PROPN
ejpam-5475	7	10	)	)	PUNCT
ejpam-5475	7	11	,	,	PUNCT
ejpam-5475	7	12	and	and	CCONJ
ejpam-5475	7	13	γθ(ν	γθ(ν	NUM
ejpam-5475	7	14	)	)	PUNCT
ejpam-5475	7	15	,	,	PUNCT
ejpam-5475	7	16	as	as	ADV
ejpam-5475	7	17	well	well	ADV
ejpam-5475	7	18	as	as	ADP
ejpam-5475	7	19	the	the	DET
ejpam-5475	7	20	relationships	relationship	NOUN
ejpam-5475	7	21	among	among	ADP
ejpam-5475	7	22	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	7	23	,	,	PUNCT
ejpam-5475	7	24	ν2)-open	ν2)-open	ADJ
ejpam-5475	7	25	sets	set	NOUN
ejpam-5475	7	26	,	,	PUNCT
ejpam-5475	7	27	θ(ν1	θ(ν1	NOUN
ejpam-5475	7	28	,	,	PUNCT
ejpam-5475	7	29	ν2)-open	ν2)-open	ADJ
ejpam-5475	7	30	sets	set	NOUN
ejpam-5475	7	31	,	,	PUNCT
ejpam-5475	7	32	and	and	CCONJ
ejpam-5475	7	33	µ-open	µ-open	NOUN
ejpam-5475	7	34	sets	set	NOUN
ejpam-5475	7	35	.	.	PUNCT
ejpam-5475	8	1	additionally	additionally	ADV
ejpam-5475	8	2	,	,	PUNCT
ejpam-5475	8	3	we	we	PRON
ejpam-5475	8	4	introduce	introduce	VERB
ejpam-5475	8	5	the	the	DET
ejpam-5475	8	6	notion	notion	NOUN
ejpam-5475	8	7	of	of	ADP
ejpam-5475	8	8	g(ν1	g(ν1	NOUN
ejpam-5475	8	9	,	,	PUNCT
ejpam-5475	8	10	ν2)-regularity	ν2)-regularity	NOUN
ejpam-5475	8	11	in	in	ADP
ejpam-5475	8	12	generalized	generalized	ADJ
ejpam-5475	8	13	topological	topological	ADJ
ejpam-5475	8	14	spaces	space	NOUN
ejpam-5475	8	15	.	.	PUNCT
ejpam-5475	9	1	finally	finally	ADV
ejpam-5475	9	2	,	,	PUNCT
ejpam-5475	9	3	we	we	PRON
ejpam-5475	9	4	provide	provide	VERB
ejpam-5475	9	5	characterizations	characterization	NOUN
ejpam-5475	9	6	of	of	ADP
ejpam-5475	9	7	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	9	8	,	,	PUNCT
ejpam-5475	9	9	ν2)-open	ν2)-open	ADJ
ejpam-5475	9	10	sets	set	NOUN
ejpam-5475	9	11	using	use	VERB
ejpam-5475	9	12	mixed	mixed	ADJ
ejpam-5475	9	13	g(ν1	g(ν1	NOUN
ejpam-5475	9	14	,	,	PUNCT
ejpam-5475	9	15	ν)-regular	ν)-regular	ADJ
ejpam-5475	9	16	concept	concept	NOUN
ejpam-5475	9	17	.	.	PUNCT
ejpam-5475	10	1	2020	2020	NUM
ejpam-5475	10	2	mathematics	mathematic	NOUN
ejpam-5475	10	3	subject	subject	NOUN
ejpam-5475	10	4	classifications	classification	NOUN
ejpam-5475	10	5	:	:	PUNCT
ejpam-5475	10	6	54a05	54a05	NUM
ejpam-5475	10	7	,	,	PUNCT
ejpam-5475	10	8	54c08	54c08	NUM
ejpam-5475	10	9	key	key	ADJ
ejpam-5475	10	10	words	word	NOUN
ejpam-5475	10	11	and	and	CCONJ
ejpam-5475	10	12	phrases	phrase	NOUN
ejpam-5475	10	13	:	:	PUNCT
ejpam-5475	10	14	γθ(ν1,ν2	γθ(ν1,ν2	NUM
ejpam-5475	10	15	)	)	PUNCT
ejpam-5475	10	16	operation	operation	NOUN
ejpam-5475	10	17	,	,	PUNCT
ejpam-5475	10	18	θ(ν1	θ(ν1	NOUN
ejpam-5475	10	19	,	,	PUNCT
ejpam-5475	10	20	ν2)-open	ν2)-open	ADJ
ejpam-5475	10	21	sets	set	NOUN
ejpam-5475	10	22	,	,	PUNCT
ejpam-5475	10	23	γθ̃(ν1,ν2	γθ̃(ν1,ν2	ADJ
ejpam-5475	10	24	)	)	PUNCT
ejpam-5475	10	25	operation	operation	NOUN
ejpam-5475	10	26	,	,	PUNCT
ejpam-5475	10	27	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	10	28	,	,	PUNCT
ejpam-5475	10	29	ν2)open	ν2)open	ADJ
ejpam-5475	10	30	set	set	NOUN
ejpam-5475	10	31	,	,	PUNCT
ejpam-5475	10	32	g(ν1	g(ν1	NOUN
ejpam-5475	10	33	,	,	PUNCT
ejpam-5475	10	34	ν2)-regularity	ν2)-regularity	NOUN
ejpam-5475	10	35	1	1	NUM
ejpam-5475	10	36	.	.	PUNCT
ejpam-5475	11	1	introduction	introduction	NOUN
ejpam-5475	11	2	á.	á.	PROPN
ejpam-5475	11	3	császár	császár	PROPN
ejpam-5475	11	4	[	[	X
ejpam-5475	11	5	1	1	X
ejpam-5475	11	6	]	]	PUNCT
ejpam-5475	11	7	introduced	introduce	VERB
ejpam-5475	11	8	the	the	DET
ejpam-5475	11	9	concepts	concept	NOUN
ejpam-5475	11	10	of	of	ADP
ejpam-5475	11	11	generalized	generalized	ADJ
ejpam-5475	11	12	topology	topology	NOUN
ejpam-5475	11	13	and	and	CCONJ
ejpam-5475	11	14	generalized	generalize	VERB
ejpam-5475	11	15	open	open	ADJ
ejpam-5475	11	16	sets	set	NOUN
ejpam-5475	11	17	,	,	PUNCT
ejpam-5475	11	18	as	as	ADV
ejpam-5475	11	19	well	well	ADV
ejpam-5475	11	20	as	as	ADP
ejpam-5475	11	21	the	the	DET
ejpam-5475	11	22	interior	interior	ADJ
ejpam-5475	11	23	and	and	CCONJ
ejpam-5475	11	24	closure	closure	NOUN
ejpam-5475	11	25	operators	operator	NOUN
ejpam-5475	11	26	within	within	ADP
ejpam-5475	11	27	generalized	generalized	ADJ
ejpam-5475	11	28	topological	topological	ADJ
ejpam-5475	11	29	spaces	space	NOUN
ejpam-5475	11	30	..	..	PUNCT
ejpam-5475	11	31	for	for	ADP
ejpam-5475	11	32	further	further	ADJ
ejpam-5475	11	33	details	detail	NOUN
ejpam-5475	11	34	,	,	PUNCT
ejpam-5475	11	35	see	see	VERB
ejpam-5475	11	36	[	[	X
ejpam-5475	11	37	1	1	NUM
ejpam-5475	11	38	]	]	PUNCT
ejpam-5475	11	39	.	.	PUNCT
ejpam-5475	12	1	in	in	ADP
ejpam-5475	12	2	the	the	DET
ejpam-5475	12	3	same	same	ADJ
ejpam-5475	12	4	work	work	NOUN
ejpam-5475	12	5	,	,	PUNCT
ejpam-5475	12	6	he	he	PRON
ejpam-5475	12	7	also	also	ADV
ejpam-5475	12	8	introduced	introduce	VERB
ejpam-5475	12	9	the	the	DET
ejpam-5475	12	10	notion	notion	NOUN
ejpam-5475	12	11	of	of	ADP
ejpam-5475	12	12	θ(ν)-open	θ(ν)-open	PROPN
ejpam-5475	12	13	sets	set	NOUN
ejpam-5475	12	14	and	and	CCONJ
ejpam-5475	12	15	investigated	investigate	VERB
ejpam-5475	12	16	their	their	PRON
ejpam-5475	12	17	properties	property	NOUN
ejpam-5475	12	18	.	.	PUNCT
ejpam-5475	13	1	similarly	similarly	ADV
ejpam-5475	13	2	,	,	PUNCT
ejpam-5475	13	3	in	in	ADP
ejpam-5475	13	4	[	[	PUNCT
ejpam-5475	13	5	7	7	NUM
ejpam-5475	13	6	]	]	PUNCT
ejpam-5475	13	7	,	,	PUNCT
ejpam-5475	13	8	the	the	DET
ejpam-5475	13	9	author	author	NOUN
ejpam-5475	13	10	defined	define	VERB
ejpam-5475	13	11	a	a	DET
ejpam-5475	13	12	weaker	weak	ADJ
ejpam-5475	13	13	form	form	NOUN
ejpam-5475	13	14	of	of	ADP
ejpam-5475	13	15	θ(ν)-open	θ(ν)-open	PROPN
ejpam-5475	13	16	sets	set	NOUN
ejpam-5475	13	17	called	call	VERB
ejpam-5475	13	18	θ̃(ν)-open	θ̃(ν)-open	ADJ
ejpam-5475	13	19	sets	set	NOUN
ejpam-5475	13	20	in	in	ADP
ejpam-5475	13	21	generalized	generalized	ADJ
ejpam-5475	13	22	topological	topological	ADJ
ejpam-5475	13	23	spaces	space	NOUN
ejpam-5475	13	24	.	.	PUNCT
ejpam-5475	14	1	for	for	ADP
ejpam-5475	14	2	additional	additional	ADJ
ejpam-5475	14	3	details	detail	NOUN
ejpam-5475	14	4	,	,	PUNCT
ejpam-5475	14	5	see	see	VERB
ejpam-5475	14	6	[	[	X
ejpam-5475	14	7	6	6	NUM
ejpam-5475	14	8	,	,	PUNCT
ejpam-5475	14	9	12	12	NUM
ejpam-5475	14	10	]	]	PUNCT
ejpam-5475	14	11	.	.	PUNCT
ejpam-5475	15	1	furthermore	furthermore	ADV
ejpam-5475	15	2	,	,	PUNCT
ejpam-5475	15	3	in	in	ADP
ejpam-5475	15	4	[	[	PUNCT
ejpam-5475	15	5	5	5	NUM
ejpam-5475	15	6	]	]	PUNCT
ejpam-5475	15	7	,	,	PUNCT
ejpam-5475	15	8	á.	á.	PROPN
ejpam-5475	15	9	császár	császár	PROPN
ejpam-5475	15	10	and	and	CCONJ
ejpam-5475	15	11	makai	makai	PROPN
ejpam-5475	15	12	jr	jr	PROPN
ejpam-5475	15	13	.	.	PROPN
ejpam-5475	15	14	modified	modify	VERB
ejpam-5475	15	15	the	the	DET
ejpam-5475	15	16	concept	concept	NOUN
ejpam-5475	15	17	of	of	ADP
ejpam-5475	15	18	θ(ν)-open	θ(ν)-open	PROPN
ejpam-5475	15	19	sets	set	NOUN
ejpam-5475	15	20	by	by	ADP
ejpam-5475	15	21	considering	consider	VERB
ejpam-5475	15	22	two	two	NUM
ejpam-5475	15	23	generalized	generalize	VERB
ejpam-5475	15	24	topologies	topology	NOUN
ejpam-5475	15	25	ν1	ν1	NOUN
ejpam-5475	15	26	and	and	CCONJ
ejpam-5475	15	27	ν2	ν2	NOUN
ejpam-5475	15	28	on	on	ADP
ejpam-5475	15	29	a	a	DET
ejpam-5475	15	30	nonempty	nonempty	ADV
ejpam-5475	15	31	set	set	VERB
ejpam-5475	15	32	x	x	NOUN
ejpam-5475	15	33	,	,	PUNCT
ejpam-5475	15	34	introducing	introduce	VERB
ejpam-5475	15	35	the	the	DET
ejpam-5475	15	36	notion	notion	NOUN
ejpam-5475	15	37	of	of	ADP
ejpam-5475	15	38	mixed	mixed	ADJ
ejpam-5475	15	39	θ(ν1	θ(ν1	NOUN
ejpam-5475	15	40	,	,	PUNCT
ejpam-5475	15	41	ν2)-open	ν2)-open	ADJ
ejpam-5475	15	42	sets	set	NOUN
ejpam-5475	15	43	(	(	PUNCT
ejpam-5475	15	44	briefly	briefly	ADV
ejpam-5475	15	45	,	,	PUNCT
ejpam-5475	15	46	θ(ν1	θ(ν1	NOUN
ejpam-5475	15	47	,	,	PUNCT
ejpam-5475	15	48	ν2)-open	ν2)-open	NOUN
ejpam-5475	15	49	)	)	PUNCT
ejpam-5475	15	50	.	.	PUNCT
ejpam-5475	16	1	in	in	ADP
ejpam-5475	16	2	our	our	PRON
ejpam-5475	16	3	research	research	NOUN
ejpam-5475	16	4	,	,	PUNCT
ejpam-5475	16	5	inspired	inspire	VERB
ejpam-5475	16	6	by	by	ADP
ejpam-5475	16	7	the	the	DET
ejpam-5475	16	8	approach	approach	NOUN
ejpam-5475	16	9	in	in	ADP
ejpam-5475	16	10	[	[	X
ejpam-5475	16	11	4	4	NUM
ejpam-5475	16	12	,	,	PUNCT
ejpam-5475	16	13	5	5	NUM
ejpam-5475	16	14	]	]	PUNCT
ejpam-5475	16	15	,	,	PUNCT
ejpam-5475	16	16	we	we	PRON
ejpam-5475	16	17	extend	extend	VERB
ejpam-5475	16	18	the	the	DET
ejpam-5475	16	19	definitions	definition	NOUN
ejpam-5475	16	20	of	of	ADP
ejpam-5475	16	21	θ̃(ν)-open	θ̃(ν)-open	ADJ
ejpam-5475	16	22	sets	set	NOUN
ejpam-5475	16	23	and	and	CCONJ
ejpam-5475	16	24	the	the	DET
ejpam-5475	16	25	operation	operation	NOUN
ejpam-5475	16	26	γθ̃(ν	γθ̃(ν	NOUN
ejpam-5475	16	27	)	)	PUNCT
ejpam-5475	16	28	by	by	ADP
ejpam-5475	16	29	considering	consider	VERB
ejpam-5475	16	30	a	a	DET
ejpam-5475	16	31	mixture	mixture	NOUN
ejpam-5475	16	32	of	of	ADP
ejpam-5475	16	33	two	two	NUM
ejpam-5475	16	34	generalized	generalized	ADJ
ejpam-5475	16	35	topologies	topology	NOUN
ejpam-5475	16	36	ν1	ν1	NOUN
ejpam-5475	16	37	and	and	CCONJ
ejpam-5475	16	38	ν2	ν2	NOUN
ejpam-5475	16	39	.	.	PUNCT
ejpam-5475	17	1	in	in	ADP
ejpam-5475	17	2	section	section	NOUN
ejpam-5475	17	3	3	3	NUM
ejpam-5475	17	4	,	,	PUNCT
ejpam-5475	17	5	we	we	PRON
ejpam-5475	17	6	introduce	introduce	VERB
ejpam-5475	17	7	the	the	DET
ejpam-5475	17	8	mixed	mixed	ADJ
ejpam-5475	17	9	operation	operation	NOUN
ejpam-5475	17	10	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	17	11	)	)	PUNCT
ejpam-5475	17	12	(	(	PUNCT
ejpam-5475	17	13	briefly	briefly	ADV
ejpam-5475	17	14	,	,	PUNCT
ejpam-5475	17	15	γθ̃(ν1,ν2	γθ̃(ν1,ν2	ADJ
ejpam-5475	17	16	)	)	PUNCT
ejpam-5475	17	17	)	)	PUNCT
ejpam-5475	17	18	and	and	CCONJ
ejpam-5475	17	19	explore	explore	VERB
ejpam-5475	17	20	the	the	DET
ejpam-5475	17	21	relationships	relationship	NOUN
ejpam-5475	17	22	between	between	ADP
ejpam-5475	17	23	this	this	DET
ejpam-5475	17	24	new	new	ADJ
ejpam-5475	17	25	operation	operation	NOUN
ejpam-5475	17	26	and	and	CCONJ
ejpam-5475	17	27	the	the	DET
ejpam-5475	17	28	operation	operation	NOUN
ejpam-5475	17	29	γθ(ν1,ν2	γθ(ν1,ν2	PROPN
ejpam-5475	17	30	)	)	PUNCT
ejpam-5475	17	31	.	.	PUNCT
ejpam-5475	18	1	additionally	additionally	ADV
ejpam-5475	18	2	,	,	PUNCT
ejpam-5475	18	3	∗corresponding	∗corresponde	VERB
ejpam-5475	18	4	author	author	NOUN
ejpam-5475	18	5	.	.	PUNCT
ejpam-5475	19	1	doi	doi	NOUN
ejpam-5475	19	2	:	:	PUNCT
ejpam-5475	19	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5475	https://doi.org/10.29020/nybg.ejpam.v17i4.5475	ADJ
ejpam-5475	19	4	email	email	NOUN
ejpam-5475	19	5	addresses	address	NOUN
ejpam-5475	19	6	:	:	PUNCT
ejpam-5475	19	7	cahis82@gmail.com	cahis82@gmail.com	X
ejpam-5475	19	8	(	(	PUNCT
ejpam-5475	19	9	a.	a.	NOUN
ejpam-5475	19	10	qahis	qahis	PROPN
ejpam-5475	19	11	)	)	PUNCT
ejpam-5475	19	12	,	,	PUNCT
ejpam-5475	19	13	odalqahtani@nu.edu.sa	odalqahtani@nu.edu.sa	PROPN
ejpam-5475	19	14	(	(	PUNCT
ejpam-5475	19	15	a.	a.	NOUN
ejpam-5475	19	16	alqahtani	alqahtani	PROPN
ejpam-5475	19	17	)	)	PUNCT
ejpam-5475	19	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5475	19	19	3610	3610	NUM
ejpam-5475	19	20	copyright	copyright	NOUN
ejpam-5475	19	21	:	:	PUNCT
ejpam-5475	19	22	©	©	PROPN
ejpam-5475	19	23	2024	2024	NUM
ejpam-5475	19	24	the	the	DET
ejpam-5475	19	25	author(s	author(s	NOUN
ejpam-5475	19	26	)	)	PUNCT
ejpam-5475	19	27	.	.	PUNCT
ejpam-5475	20	1	(	(	PUNCT
ejpam-5475	20	2	cc	cc	NOUN
ejpam-5475	20	3	by	by	ADP
ejpam-5475	20	4	-	-	PUNCT
ejpam-5475	20	5	nc	nc	PROPN
ejpam-5475	20	6	4.0	4.0	NUM
ejpam-5475	20	7	)	)	PUNCT
ejpam-5475	20	8	a.	a.	NOUN
ejpam-5475	20	9	qahis	qahis	PROPN
ejpam-5475	20	10	,	,	PUNCT
ejpam-5475	20	11	a.	a.	NOUN
ejpam-5475	20	12	alqahtani	alqahtani	PROPN
ejpam-5475	20	13	/	/	SYM
ejpam-5475	20	14	eur	eur	PROPN
ejpam-5475	20	15	.	.	PUNCT
ejpam-5475	21	1	j.	j.	PROPN
ejpam-5475	21	2	pure	pure	PROPN
ejpam-5475	21	3	appl	appl	PROPN
ejpam-5475	21	4	.	.	PROPN
ejpam-5475	21	5	math	math	PROPN
ejpam-5475	21	6	,	,	PUNCT
ejpam-5475	21	7	17	17	NUM
ejpam-5475	21	8	(	(	PUNCT
ejpam-5475	21	9	4	4	NUM
ejpam-5475	21	10	)	)	PUNCT
ejpam-5475	21	11	(	(	PUNCT
ejpam-5475	21	12	2024	2024	NUM
ejpam-5475	21	13	)	)	PUNCT
ejpam-5475	21	14	,	,	PUNCT
ejpam-5475	21	15	3610	3610	NUM
ejpam-5475	21	16	-	-	SYM
ejpam-5475	21	17	3621	3621	NUM
ejpam-5475	21	18	3611	3611	NUM
ejpam-5475	21	19	we	we	PRON
ejpam-5475	21	20	establish	establish	VERB
ejpam-5475	21	21	sufficient	sufficient	ADJ
ejpam-5475	21	22	conditions	condition	NOUN
ejpam-5475	21	23	for	for	ADP
ejpam-5475	21	24	equivalence	equivalence	NOUN
ejpam-5475	21	25	between	between	ADP
ejpam-5475	21	26	the	the	DET
ejpam-5475	21	27	operation	operation	NOUN
ejpam-5475	21	28	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	21	29	)	)	PUNCT
ejpam-5475	21	30	and	and	CCONJ
ejpam-5475	21	31	the	the	DET
ejpam-5475	21	32	previous	previous	ADJ
ejpam-5475	21	33	operation	operation	NOUN
ejpam-5475	21	34	γθ(ν1,ν2	γθ(ν1,ν2	ADP
ejpam-5475	21	35	)	)	PUNCT
ejpam-5475	21	36	.	.	PUNCT
ejpam-5475	22	1	in	in	ADP
ejpam-5475	22	2	section	section	NOUN
ejpam-5475	22	3	4	4	NUM
ejpam-5475	22	4	,	,	PUNCT
ejpam-5475	22	5	we	we	PRON
ejpam-5475	22	6	define	define	VERB
ejpam-5475	22	7	the	the	DET
ejpam-5475	22	8	class	class	NOUN
ejpam-5475	22	9	of	of	ADP
ejpam-5475	22	10	mixed	mixed	ADJ
ejpam-5475	22	11	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	22	12	,	,	PUNCT
ejpam-5475	22	13	ν2)-open	ν2)-open	ADJ
ejpam-5475	22	14	sets	set	NOUN
ejpam-5475	22	15	(	(	PUNCT
ejpam-5475	22	16	briefly	briefly	ADV
ejpam-5475	22	17	,	,	PUNCT
ejpam-5475	22	18	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	22	19	,	,	PUNCT
ejpam-5475	22	20	ν2)-open	ν2)-open	ADJ
ejpam-5475	22	21	sets	set	NOUN
ejpam-5475	22	22	)	)	PUNCT
ejpam-5475	22	23	as	as	ADP
ejpam-5475	22	24	a	a	DET
ejpam-5475	22	25	new	new	ADJ
ejpam-5475	22	26	category	category	NOUN
ejpam-5475	22	27	lying	lie	VERB
ejpam-5475	22	28	strictly	strictly	ADV
ejpam-5475	22	29	between	between	ADP
ejpam-5475	22	30	the	the	DET
ejpam-5475	22	31	class	class	NOUN
ejpam-5475	22	32	of	of	ADP
ejpam-5475	22	33	ν1	ν1	NOUN
ejpam-5475	22	34	-	-	PUNCT
ejpam-5475	22	35	open	open	ADJ
ejpam-5475	22	36	sets	set	NOUN
ejpam-5475	22	37	and	and	CCONJ
ejpam-5475	22	38	the	the	DET
ejpam-5475	22	39	class	class	NOUN
ejpam-5475	22	40	of	of	ADP
ejpam-5475	22	41	θ(ν1	θ(ν1	NOUN
ejpam-5475	22	42	,	,	PUNCT
ejpam-5475	22	43	ν2)open	ν2)open	ADJ
ejpam-5475	22	44	sets	set	NOUN
ejpam-5475	22	45	.	.	PUNCT
ejpam-5475	23	1	as	as	ADP
ejpam-5475	23	2	the	the	DET
ejpam-5475	23	3	main	main	ADJ
ejpam-5475	23	4	results	result	NOUN
ejpam-5475	23	5	of	of	ADP
ejpam-5475	23	6	this	this	DET
ejpam-5475	23	7	section	section	NOUN
ejpam-5475	23	8	,	,	PUNCT
ejpam-5475	23	9	we	we	PRON
ejpam-5475	23	10	introduce	introduce	VERB
ejpam-5475	23	11	the	the	DET
ejpam-5475	23	12	concept	concept	NOUN
ejpam-5475	23	13	of	of	ADP
ejpam-5475	23	14	relative	relative	ADJ
ejpam-5475	23	15	mixed	mixed	ADJ
ejpam-5475	23	16	g(ν1	g(ν1	NOUN
ejpam-5475	23	17	,	,	PUNCT
ejpam-5475	23	18	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	23	19	(	(	PUNCT
ejpam-5475	23	20	briefly	briefly	ADV
ejpam-5475	23	21	,	,	PUNCT
ejpam-5475	23	22	g(ν1	g(ν1	NOUN
ejpam-5475	23	23	,	,	PUNCT
ejpam-5475	23	24	ν2)-regularity	ν2)-regularity	NOUN
ejpam-5475	23	25	)	)	PUNCT
ejpam-5475	23	26	as	as	ADP
ejpam-5475	23	27	a	a	DET
ejpam-5475	23	28	novel	novel	ADJ
ejpam-5475	23	29	separation	separation	NOUN
ejpam-5475	23	30	axiom	axiom	NOUN
ejpam-5475	23	31	in	in	ADP
ejpam-5475	23	32	generalized	generalized	ADJ
ejpam-5475	23	33	topological	topological	ADJ
ejpam-5475	23	34	spaces	space	NOUN
ejpam-5475	23	35	.	.	PUNCT
ejpam-5475	24	1	moreover	moreover	ADV
ejpam-5475	24	2	,	,	PUNCT
ejpam-5475	24	3	we	we	PRON
ejpam-5475	24	4	provide	provide	VERB
ejpam-5475	24	5	a	a	DET
ejpam-5475	24	6	characterization	characterization	NOUN
ejpam-5475	24	7	of	of	ADP
ejpam-5475	24	8	g(ν1	g(ν1	NOUN
ejpam-5475	24	9	,	,	PUNCT
ejpam-5475	24	10	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	24	11	spaces	space	NOUN
ejpam-5475	24	12	.	.	PUNCT
ejpam-5475	25	1	2	2	X
ejpam-5475	25	2	.	.	X
ejpam-5475	25	3	preliminaries	preliminary	NOUN
ejpam-5475	25	4	letx	letx	ADV
ejpam-5475	25	5	be	be	AUX
ejpam-5475	25	6	a	a	DET
ejpam-5475	25	7	nonempty	nonempty	ADV
ejpam-5475	25	8	set	set	VERB
ejpam-5475	25	9	and	and	CCONJ
ejpam-5475	25	10	ν	ν	X
ejpam-5475	25	11	a	a	DET
ejpam-5475	25	12	collection	collection	NOUN
ejpam-5475	25	13	of	of	ADP
ejpam-5475	25	14	subsets	subset	NOUN
ejpam-5475	25	15	ofx	ofx	NOUN
ejpam-5475	25	16	.	.	PUNCT
ejpam-5475	26	1	ν	ν	NOUN
ejpam-5475	26	2	is	be	AUX
ejpam-5475	26	3	defined	define	VERB
ejpam-5475	26	4	as	as	ADP
ejpam-5475	26	5	a	a	DET
ejpam-5475	26	6	generalized	generalized	ADJ
ejpam-5475	26	7	topology	topology	NOUN
ejpam-5475	26	8	(	(	PUNCT
ejpam-5475	26	9	gt	gt	PROPN
ejpam-5475	26	10	)	)	PUNCT
ejpam-5475	26	11	on	on	ADP
ejpam-5475	26	12	x	x	SYM
ejpam-5475	26	13	if	if	SCONJ
ejpam-5475	26	14	it	it	PRON
ejpam-5475	26	15	satisfies	satisfy	VERB
ejpam-5475	26	16	the	the	DET
ejpam-5475	26	17	following	follow	VERB
ejpam-5475	26	18	conditions	condition	NOUN
ejpam-5475	26	19	:	:	PUNCT
ejpam-5475	26	20	(	(	PUNCT
ejpam-5475	26	21	i	i	NOUN
ejpam-5475	26	22	)	)	PUNCT
ejpam-5475	27	1	∅	∅	NOUN
ejpam-5475	27	2	∈	∈	PROPN
ejpam-5475	27	3	ν	ν	X
ejpam-5475	27	4	.	.	PUNCT
ejpam-5475	27	5	(	(	PUNCT
ejpam-5475	27	6	ii	ii	NOUN
ejpam-5475	27	7	)	)	PUNCT
ejpam-5475	27	8	any	any	DET
ejpam-5475	27	9	union	union	NOUN
ejpam-5475	27	10	of	of	ADP
ejpam-5475	27	11	elements	element	NOUN
ejpam-5475	27	12	within	within	ADP
ejpam-5475	27	13	ν	ν	NOUN
ejpam-5475	27	14	is	be	AUX
ejpam-5475	27	15	also	also	ADV
ejpam-5475	27	16	an	an	DET
ejpam-5475	27	17	element	element	NOUN
ejpam-5475	27	18	of	of	ADP
ejpam-5475	27	19	ν	ν	NOUN
ejpam-5475	27	20	.	.	PUNCT
ejpam-5475	28	1	this	this	DET
ejpam-5475	28	2	concept	concept	NOUN
ejpam-5475	28	3	was	be	AUX
ejpam-5475	28	4	introduced	introduce	VERB
ejpam-5475	28	5	by	by	ADP
ejpam-5475	28	6	á.	á.	PROPN
ejpam-5475	28	7	császár	császár	PROPN
ejpam-5475	28	8	in	in	ADP
ejpam-5475	28	9	[	[	X
ejpam-5475	28	10	1	1	NUM
ejpam-5475	28	11	]	]	PUNCT
ejpam-5475	28	12	.	.	PUNCT
ejpam-5475	29	1	we	we	PRON
ejpam-5475	29	2	denote	denote	VERB
ejpam-5475	29	3	the	the	DET
ejpam-5475	29	4	pair	pair	NOUN
ejpam-5475	29	5	(	(	PUNCT
ejpam-5475	29	6	x	x	NOUN
ejpam-5475	29	7	,	,	PUNCT
ejpam-5475	29	8	ν	ν	NOUN
ejpam-5475	29	9	)	)	PUNCT
ejpam-5475	29	10	as	as	ADP
ejpam-5475	29	11	a	a	DET
ejpam-5475	29	12	generalized	generalized	ADJ
ejpam-5475	29	13	topological	topological	ADJ
ejpam-5475	29	14	space	space	NOUN
ejpam-5475	29	15	(	(	PUNCT
ejpam-5475	29	16	gts	gts	NOUN
ejpam-5475	29	17	)	)	PUNCT
ejpam-5475	29	18	on	on	ADP
ejpam-5475	29	19	x.	x.	NOUN
ejpam-5475	29	20	the	the	DET
ejpam-5475	29	21	subsets	subset	NOUN
ejpam-5475	29	22	in	in	ADP
ejpam-5475	29	23	ν	ν	NOUN
ejpam-5475	29	24	are	be	AUX
ejpam-5475	29	25	termed	term	VERB
ejpam-5475	29	26	ν	ν	ADJ
ejpam-5475	29	27	-	-	ADJ
ejpam-5475	29	28	open	open	ADJ
ejpam-5475	29	29	sets	set	NOUN
ejpam-5475	29	30	,	,	PUNCT
ejpam-5475	29	31	and	and	CCONJ
ejpam-5475	29	32	their	their	PRON
ejpam-5475	29	33	complements	complement	NOUN
ejpam-5475	29	34	are	be	AUX
ejpam-5475	29	35	ν	ν	ADJ
ejpam-5475	29	36	-	-	PUNCT
ejpam-5475	29	37	closed	closed	ADJ
ejpam-5475	29	38	sets	set	NOUN
ejpam-5475	29	39	,	,	PUNCT
ejpam-5475	29	40	as	as	SCONJ
ejpam-5475	29	41	defined	define	VERB
ejpam-5475	29	42	in	in	ADP
ejpam-5475	29	43	[	[	X
ejpam-5475	29	44	2	2	NUM
ejpam-5475	29	45	]	]	PUNCT
ejpam-5475	29	46	.	.	PUNCT
ejpam-5475	30	1	the	the	DET
ejpam-5475	30	2	union	union	NOUN
ejpam-5475	30	3	of	of	ADP
ejpam-5475	30	4	all	all	DET
ejpam-5475	30	5	elements	element	NOUN
ejpam-5475	30	6	of	of	ADP
ejpam-5475	30	7	ν	ν	NOUN
ejpam-5475	30	8	is	be	AUX
ejpam-5475	30	9	denoted	denote	VERB
ejpam-5475	30	10	by	by	ADP
ejpam-5475	30	11	mν	mν	PROPN
ejpam-5475	30	12	.	.	PUNCT
ejpam-5475	31	1	additionally	additionally	ADV
ejpam-5475	31	2	,	,	PUNCT
ejpam-5475	31	3	a	a	DET
ejpam-5475	31	4	gts	gts	NOUN
ejpam-5475	31	5	(	(	PUNCT
ejpam-5475	31	6	x	x	NOUN
ejpam-5475	31	7	,	,	PUNCT
ejpam-5475	31	8	ν	ν	X
ejpam-5475	31	9	)	)	PUNCT
ejpam-5475	31	10	is	be	AUX
ejpam-5475	31	11	called	call	VERB
ejpam-5475	31	12	strong	strong	ADJ
ejpam-5475	31	13	[	[	X
ejpam-5475	31	14	11	11	NUM
ejpam-5475	31	15	]	]	PUNCT
ejpam-5475	31	16	if	if	SCONJ
ejpam-5475	31	17	x	x	PROPN
ejpam-5475	31	18	∈	∈	PROPN
ejpam-5475	31	19	ν	ν	NOUN
ejpam-5475	31	20	.	.	PROPN
ejpam-5475	31	21	for	for	ADP
ejpam-5475	31	22	a	a	DET
ejpam-5475	31	23	subset	subset	NOUN
ejpam-5475	31	24	a	a	DET
ejpam-5475	31	25	of	of	ADP
ejpam-5475	31	26	a	a	DET
ejpam-5475	31	27	gts	gts	NOUN
ejpam-5475	31	28	(	(	PUNCT
ejpam-5475	31	29	x	x	NOUN
ejpam-5475	31	30	,	,	PUNCT
ejpam-5475	31	31	ν	ν	NOUN
ejpam-5475	31	32	)	)	PUNCT
ejpam-5475	31	33	,	,	PUNCT
ejpam-5475	31	34	the	the	DET
ejpam-5475	31	35	ν	ν	NOUN
ejpam-5475	31	36	-	-	NOUN
ejpam-5475	31	37	closure	closure	NOUN
ejpam-5475	31	38	of	of	ADP
ejpam-5475	31	39	a	a	PRON
ejpam-5475	31	40	,	,	PUNCT
ejpam-5475	31	41	denoted	denote	VERB
ejpam-5475	31	42	cν(a	cν(a	NOUN
ejpam-5475	31	43	)	)	PUNCT
ejpam-5475	31	44	,	,	PUNCT
ejpam-5475	31	45	is	be	AUX
ejpam-5475	31	46	defined	define	VERB
ejpam-5475	31	47	as	as	ADP
ejpam-5475	31	48	the	the	DET
ejpam-5475	31	49	intersection	intersection	NOUN
ejpam-5475	31	50	of	of	ADP
ejpam-5475	31	51	all	all	DET
ejpam-5475	31	52	ν	ν	NOUN
ejpam-5475	31	53	-	-	PUNCT
ejpam-5475	31	54	closed	closed	ADJ
ejpam-5475	31	55	sets	set	NOUN
ejpam-5475	31	56	containing	contain	VERB
ejpam-5475	31	57	a.	a.	NOUN
ejpam-5475	31	58	the	the	DET
ejpam-5475	31	59	ν	ν	NOUN
ejpam-5475	31	60	-	-	NOUN
ejpam-5475	31	61	interior	interior	NOUN
ejpam-5475	31	62	of	of	ADP
ejpam-5475	31	63	a	a	PRON
ejpam-5475	31	64	,	,	PUNCT
ejpam-5475	31	65	denoted	denote	VERB
ejpam-5475	31	66	iν(a	iν(a	NOUN
ejpam-5475	31	67	)	)	PUNCT
ejpam-5475	31	68	,	,	PUNCT
ejpam-5475	31	69	is	be	AUX
ejpam-5475	31	70	defined	define	VERB
ejpam-5475	31	71	as	as	ADP
ejpam-5475	31	72	the	the	DET
ejpam-5475	31	73	union	union	NOUN
ejpam-5475	31	74	of	of	ADP
ejpam-5475	31	75	all	all	DET
ejpam-5475	31	76	ν	ν	PROPN
ejpam-5475	31	77	-	-	ADJ
ejpam-5475	31	78	open	open	ADJ
ejpam-5475	31	79	sets	set	NOUN
ejpam-5475	31	80	contained	contain	VERB
ejpam-5475	31	81	in	in	ADP
ejpam-5475	31	82	a	a	DET
ejpam-5475	31	83	(	(	PUNCT
ejpam-5475	31	84	see	see	PROPN
ejpam-5475	31	85	[	[	X
ejpam-5475	31	86	1	1	NUM
ejpam-5475	31	87	,	,	PUNCT
ejpam-5475	31	88	2	2	NUM
ejpam-5475	31	89	]	]	PUNCT
ejpam-5475	31	90	)	)	PUNCT
ejpam-5475	31	91	.	.	PUNCT
ejpam-5475	32	1	recalling	recall	VERB
ejpam-5475	32	2	from	from	ADP
ejpam-5475	32	3	[	[	X
ejpam-5475	32	4	3	3	NUM
ejpam-5475	32	5	]	]	PUNCT
ejpam-5475	32	6	,	,	PUNCT
ejpam-5475	32	7	let	let	VERB
ejpam-5475	32	8	ν	ν	NOUN
ejpam-5475	32	9	be	be	AUX
ejpam-5475	32	10	a	a	DET
ejpam-5475	32	11	gt	gt	PROPN
ejpam-5475	32	12	on	on	ADP
ejpam-5475	32	13	the	the	DET
ejpam-5475	32	14	nonempty	nonempty	ADJ
ejpam-5475	32	15	set	set	VERB
ejpam-5475	32	16	x	x	NOUN
ejpam-5475	32	17	,	,	PUNCT
ejpam-5475	32	18	and	and	CCONJ
ejpam-5475	32	19	p(x	p(x	PROPN
ejpam-5475	32	20	)	)	PUNCT
ejpam-5475	32	21	denote	denote	VERB
ejpam-5475	32	22	the	the	DET
ejpam-5475	32	23	power	power	NOUN
ejpam-5475	32	24	set	set	NOUN
ejpam-5475	32	25	of	of	ADP
ejpam-5475	32	26	x.	x.	NOUN
ejpam-5475	32	27	define	define	VERB
ejpam-5475	32	28	θ(ν	θ(ν	PROPN
ejpam-5475	32	29	)	)	PUNCT
ejpam-5475	32	30	⊆	⊆	NUM
ejpam-5475	32	31	p(x	p(x	NOUN
ejpam-5475	32	32	)	)	PUNCT
ejpam-5475	32	33	such	such	ADJ
ejpam-5475	32	34	that	that	SCONJ
ejpam-5475	32	35	a	a	DET
ejpam-5475	32	36	∈	∈	PROPN
ejpam-5475	32	37	θ(ν	θ(ν	NOUN
ejpam-5475	32	38	)	)	PUNCT
ejpam-5475	32	39	if	if	SCONJ
ejpam-5475	32	40	for	for	ADP
ejpam-5475	32	41	each	each	DET
ejpam-5475	32	42	x	x	SYM
ejpam-5475	32	43	∈	∈	PROPN
ejpam-5475	32	44	a	a	PRON
ejpam-5475	32	45	,	,	PUNCT
ejpam-5475	32	46	there	there	PRON
ejpam-5475	32	47	exists	exist	VERB
ejpam-5475	32	48	m	m	VERB
ejpam-5475	32	49	∈	∈	NOUN
ejpam-5475	32	50	ν	ν	NOUN
ejpam-5475	32	51	containing	contain	VERB
ejpam-5475	32	52	x	x	PUNCT
ejpam-5475	32	53	with	with	ADP
ejpam-5475	32	54	m	m	PROPN
ejpam-5475	32	55	⊆	⊆	NUM
ejpam-5475	32	56	cν(m	cν(m	NOUN
ejpam-5475	32	57	)	)	PUNCT
ejpam-5475	32	58	⊆	⊆	NUM
ejpam-5475	32	59	a.	a.	NOUN
ejpam-5475	32	60	then	then	ADV
ejpam-5475	32	61	θ(ν	θ(ν	VERB
ejpam-5475	32	62	)	)	PUNCT
ejpam-5475	32	63	forms	form	VERB
ejpam-5475	32	64	a	a	DET
ejpam-5475	32	65	gt	gt	PROPN
ejpam-5475	32	66	on	on	ADP
ejpam-5475	32	67	x	x	PRON
ejpam-5475	32	68	,	,	PUNCT
ejpam-5475	32	69	included	include	VERB
ejpam-5475	32	70	in	in	ADP
ejpam-5475	32	71	ν	ν	NOUN
ejpam-5475	32	72	.	.	PUNCT
ejpam-5475	33	1	the	the	DET
ejpam-5475	33	2	sets	set	NOUN
ejpam-5475	33	3	in	in	ADP
ejpam-5475	33	4	θ(ν	θ(ν	PROPN
ejpam-5475	33	5	)	)	PUNCT
ejpam-5475	33	6	are	be	AUX
ejpam-5475	33	7	known	know	VERB
ejpam-5475	33	8	as	as	ADP
ejpam-5475	33	9	θ(ν)-open	θ(ν)-open	PROPN
ejpam-5475	33	10	sets	set	NOUN
ejpam-5475	33	11	,	,	PUNCT
ejpam-5475	33	12	and	and	CCONJ
ejpam-5475	33	13	their	their	PRON
ejpam-5475	33	14	complements	complement	NOUN
ejpam-5475	33	15	are	be	AUX
ejpam-5475	33	16	referred	refer	VERB
ejpam-5475	33	17	to	to	ADP
ejpam-5475	33	18	as	as	ADP
ejpam-5475	33	19	θ(ν)-closed	θ(ν)-close	VERB
ejpam-5475	33	20	sets	set	NOUN
ejpam-5475	33	21	.	.	PUNCT
ejpam-5475	34	1	the	the	DET
ejpam-5475	34	2	operation	operation	NOUN
ejpam-5475	34	3	γθ	γθ	NOUN
ejpam-5475	34	4	:	:	PUNCT
ejpam-5475	34	5	p(x	p(x	PROPN
ejpam-5475	34	6	)	)	PUNCT
ejpam-5475	34	7	→	→	SYM
ejpam-5475	34	8	p(x	p(x	PROPN
ejpam-5475	34	9	)	)	PUNCT
ejpam-5475	34	10	is	be	AUX
ejpam-5475	34	11	defined	define	VERB
ejpam-5475	34	12	for	for	ADP
ejpam-5475	34	13	a	a	DET
ejpam-5475	34	14	⊆	⊆	NUM
ejpam-5475	34	15	x	x	SYM
ejpam-5475	34	16	by	by	ADP
ejpam-5475	34	17	γθ(a	γθ(a	NUM
ejpam-5475	34	18	)	)	PUNCT
ejpam-5475	34	19	=	=	PRON
ejpam-5475	35	1	{	{	PUNCT
ejpam-5475	35	2	x	x	PUNCT
ejpam-5475	35	3	∈	∈	PROPN
ejpam-5475	35	4	x	x	X
ejpam-5475	35	5	:	:	PUNCT
ejpam-5475	35	6	cν(m	cν(m	X
ejpam-5475	35	7	)	)	PUNCT
ejpam-5475	36	1	∩a	∩a	PROPN
ejpam-5475	36	2	̸=	̸=	PROPN
ejpam-5475	36	3	∅	∅	NOUN
ejpam-5475	36	4	,	,	PUNCT
ejpam-5475	36	5	∀m	∀m	PROPN
ejpam-5475	36	6	∈	∈	PROPN
ejpam-5475	36	7	ν	ν	PROPN
ejpam-5475	36	8	,	,	PUNCT
ejpam-5475	36	9	x	x	SYM
ejpam-5475	36	10	∈	∈	PROPN
ejpam-5475	36	11	m	m	PRON
ejpam-5475	36	12	}	}	PUNCT
ejpam-5475	36	13	.	.	PUNCT
ejpam-5475	37	1	in	in	ADP
ejpam-5475	37	2	[	[	X
ejpam-5475	37	3	7	7	NUM
ejpam-5475	37	4	]	]	PUNCT
ejpam-5475	37	5	,	,	PUNCT
ejpam-5475	37	6	min	min	PROPN
ejpam-5475	37	7	extended	extend	VERB
ejpam-5475	37	8	this	this	PRON
ejpam-5475	37	9	by	by	ADP
ejpam-5475	37	10	defining	define	VERB
ejpam-5475	37	11	θ̃(ν	θ̃(ν	PROPN
ejpam-5475	37	12	)	)	PUNCT
ejpam-5475	37	13	⊆	⊆	NUM
ejpam-5475	37	14	p(x	p(x	NOUN
ejpam-5475	37	15	)	)	PUNCT
ejpam-5475	37	16	such	such	ADJ
ejpam-5475	37	17	that	that	SCONJ
ejpam-5475	37	18	a	a	DET
ejpam-5475	37	19	∈	∈	PROPN
ejpam-5475	37	20	θ̃(ν	θ̃(ν	NOUN
ejpam-5475	37	21	)	)	PUNCT
ejpam-5475	37	22	if	if	SCONJ
ejpam-5475	37	23	for	for	ADP
ejpam-5475	37	24	each	each	DET
ejpam-5475	37	25	x	x	SYM
ejpam-5475	37	26	∈	∈	PROPN
ejpam-5475	37	27	a	a	PRON
ejpam-5475	37	28	,	,	PUNCT
ejpam-5475	37	29	there	there	PRON
ejpam-5475	37	30	exists	exist	VERB
ejpam-5475	37	31	m	m	VERB
ejpam-5475	37	32	∈	∈	NOUN
ejpam-5475	37	33	ν	ν	NOUN
ejpam-5475	37	34	containing	contain	VERB
ejpam-5475	37	35	x	x	PUNCT
ejpam-5475	37	36	with	with	ADP
ejpam-5475	37	37	m	m	PROPN
ejpam-5475	37	38	⊆	⊆	NUM
ejpam-5475	37	39	cν(m	cν(m	NOUN
ejpam-5475	37	40	)	)	PUNCT
ejpam-5475	37	41	∩mν	∩mν	PROPN
ejpam-5475	37	42	⊆	⊆	NUM
ejpam-5475	37	43	a.	a.	NOUN
ejpam-5475	37	44	θ̃(ν	θ̃(ν	PROPN
ejpam-5475	37	45	)	)	PUNCT
ejpam-5475	37	46	is	be	AUX
ejpam-5475	37	47	a	a	DET
ejpam-5475	37	48	gt	gt	PROPN
ejpam-5475	37	49	on	on	ADP
ejpam-5475	37	50	x	x	PRON
ejpam-5475	37	51	,	,	PUNCT
ejpam-5475	37	52	contained	contain	VERB
ejpam-5475	37	53	in	in	ADP
ejpam-5475	37	54	ν	ν	NOUN
ejpam-5475	37	55	,	,	PUNCT
ejpam-5475	37	56	and	and	CCONJ
ejpam-5475	37	57	θ(ν	θ(ν	PROPN
ejpam-5475	37	58	)	)	PUNCT
ejpam-5475	37	59	⊆	⊆	NUM
ejpam-5475	37	60	θ̃(ν	θ̃(ν	PROPN
ejpam-5475	37	61	)	)	PUNCT
ejpam-5475	37	62	.	.	PUNCT
ejpam-5475	38	1	the	the	DET
ejpam-5475	38	2	elements	element	NOUN
ejpam-5475	38	3	of	of	ADP
ejpam-5475	38	4	θ̃(ν	θ̃(ν	PROPN
ejpam-5475	38	5	)	)	PUNCT
ejpam-5475	38	6	are	be	AUX
ejpam-5475	38	7	referred	refer	VERB
ejpam-5475	38	8	to	to	ADP
ejpam-5475	38	9	as	as	ADV
ejpam-5475	38	10	θ̃(ν)-open	θ̃(ν)-open	VERB
ejpam-5475	38	11	sets	set	NOUN
ejpam-5475	38	12	,	,	PUNCT
ejpam-5475	38	13	while	while	SCONJ
ejpam-5475	38	14	their	their	PRON
ejpam-5475	38	15	complements	complement	NOUN
ejpam-5475	38	16	are	be	AUX
ejpam-5475	38	17	known	know	VERB
ejpam-5475	38	18	as	as	ADP
ejpam-5475	38	19	θ̃(ν)-closed	θ̃(ν)-closed	ADJ
ejpam-5475	38	20	sets	set	NOUN
ejpam-5475	38	21	.	.	PUNCT
ejpam-5475	39	1	the	the	DET
ejpam-5475	39	2	operation	operation	NOUN
ejpam-5475	39	3	γθ̃	γθ̃	PROPN
ejpam-5475	39	4	:	:	PUNCT
ejpam-5475	39	5	p(x	p(x	PROPN
ejpam-5475	39	6	)	)	PUNCT
ejpam-5475	39	7	→	→	SYM
ejpam-5475	39	8	p(x	p(x	PROPN
ejpam-5475	39	9	)	)	PUNCT
ejpam-5475	39	10	is	be	AUX
ejpam-5475	39	11	defined	define	VERB
ejpam-5475	39	12	for	for	ADP
ejpam-5475	39	13	a	a	DET
ejpam-5475	39	14	⊆	⊆	NUM
ejpam-5475	39	15	x	x	SYM
ejpam-5475	39	16	by	by	ADP
ejpam-5475	39	17	γθ̃(a	γθ̃(a	NOUN
ejpam-5475	39	18	)	)	PUNCT
ejpam-5475	40	1	=	=	PRON
ejpam-5475	40	2	{	{	PUNCT
ejpam-5475	40	3	x	x	PUNCT
ejpam-5475	40	4	∈	∈	PROPN
ejpam-5475	40	5	x	x	X
ejpam-5475	40	6	:	:	PUNCT
ejpam-5475	40	7	(	(	PUNCT
ejpam-5475	40	8	cν(m	cν(m	NOUN
ejpam-5475	40	9	)	)	PUNCT
ejpam-5475	40	10	∩mν	∩mν	NOUN
ejpam-5475	40	11	)	)	PUNCT
ejpam-5475	41	1	∩a	∩a	PROPN
ejpam-5475	41	2	̸=	̸=	PROPN
ejpam-5475	41	3	∅,∀m	∅,∀m	ADP
ejpam-5475	41	4	∈	∈	PROPN
ejpam-5475	41	5	ν	ν	NOUN
ejpam-5475	41	6	,	,	PUNCT
ejpam-5475	41	7	x	x	SYM
ejpam-5475	41	8	∈	∈	PROPN
ejpam-5475	41	9	m	m	NOUN
ejpam-5475	41	10	}	}	PUNCT
ejpam-5475	41	11	.	.	PUNCT
ejpam-5475	42	1	furthermore	furthermore	ADV
ejpam-5475	42	2	,	,	PUNCT
ejpam-5475	42	3	in	in	ADP
ejpam-5475	42	4	[	[	PUNCT
ejpam-5475	42	5	5	5	NUM
ejpam-5475	42	6	]	]	PUNCT
ejpam-5475	42	7	,	,	PUNCT
ejpam-5475	42	8	á.	á.	PROPN
ejpam-5475	42	9	császár	császár	PROPN
ejpam-5475	42	10	and	and	CCONJ
ejpam-5475	42	11	makai	makai	PROPN
ejpam-5475	42	12	jr	jr	PROPN
ejpam-5475	42	13	.	.	PROPN
ejpam-5475	42	14	introduced	introduce	VERB
ejpam-5475	42	15	θ(ν1	θ(ν1	NOUN
ejpam-5475	42	16	,	,	PUNCT
ejpam-5475	42	17	ν2	ν2	NOUN
ejpam-5475	42	18	)	)	PUNCT
ejpam-5475	42	19	for	for	ADP
ejpam-5475	42	20	combining	combine	VERB
ejpam-5475	42	21	two	two	NUM
ejpam-5475	42	22	gts	gts	NOUN
ejpam-5475	42	23	ν1	ν1	NOUN
ejpam-5475	42	24	and	and	CCONJ
ejpam-5475	42	25	ν2	ν2	NOUN
ejpam-5475	42	26	on	on	ADP
ejpam-5475	42	27	x.	x.	NOUN
ejpam-5475	42	28	a	a	DET
ejpam-5475	42	29	set	set	NOUN
ejpam-5475	42	30	a	a	DET
ejpam-5475	42	31	⊆	⊆	NUM
ejpam-5475	42	32	x	x	PUNCT
ejpam-5475	42	33	belongs	belong	VERB
ejpam-5475	42	34	to	to	ADP
ejpam-5475	42	35	θ(ν1	θ(ν1	NOUN
ejpam-5475	42	36	,	,	PUNCT
ejpam-5475	42	37	ν2	ν2	NOUN
ejpam-5475	42	38	)	)	PUNCT
ejpam-5475	42	39	if	if	SCONJ
ejpam-5475	42	40	x	x	SYM
ejpam-5475	42	41	∈	∈	PROPN
ejpam-5475	42	42	a	a	PRON
ejpam-5475	42	43	implies	imply	VERB
ejpam-5475	42	44	the	the	DET
ejpam-5475	42	45	existence	existence	NOUN
ejpam-5475	42	46	of	of	ADP
ejpam-5475	42	47	m	m	PROPN
ejpam-5475	42	48	∈	∈	PROPN
ejpam-5475	42	49	ν1	ν1	NOUN
ejpam-5475	42	50	with	with	ADP
ejpam-5475	42	51	x	x	PROPN
ejpam-5475	42	52	∈	∈	PROPN
ejpam-5475	42	53	m	m	ADJ
ejpam-5475	42	54	⊆	⊆	NUM
ejpam-5475	42	55	cν2(m	cν2(m	NOUN
ejpam-5475	42	56	)	)	PUNCT
ejpam-5475	42	57	⊆	⊆	NUM
ejpam-5475	42	58	a.	a.	NOUN
ejpam-5475	42	59	θ(ν1	θ(ν1	NOUN
ejpam-5475	42	60	,	,	PUNCT
ejpam-5475	42	61	ν2	ν2	NOUN
ejpam-5475	42	62	)	)	PUNCT
ejpam-5475	42	63	is	be	AUX
ejpam-5475	42	64	also	also	ADV
ejpam-5475	42	65	a	a	DET
ejpam-5475	42	66	gt	gt	PROPN
ejpam-5475	42	67	contained	contain	VERB
ejpam-5475	42	68	in	in	ADP
ejpam-5475	42	69	ν1	ν1	NOUN
ejpam-5475	42	70	on	on	ADP
ejpam-5475	42	71	x.	x.	NOUN
ejpam-5475	42	72	the	the	DET
ejpam-5475	42	73	a.	a.	NOUN
ejpam-5475	42	74	qahis	qahis	NOUN
ejpam-5475	42	75	,	,	PUNCT
ejpam-5475	42	76	a.	a.	NOUN
ejpam-5475	42	77	alqahtani	alqahtani	PROPN
ejpam-5475	42	78	/	/	SYM
ejpam-5475	42	79	eur	eur	PROPN
ejpam-5475	42	80	.	.	PUNCT
ejpam-5475	43	1	j.	j.	PROPN
ejpam-5475	43	2	pure	pure	PROPN
ejpam-5475	43	3	appl	appl	PROPN
ejpam-5475	43	4	.	.	PROPN
ejpam-5475	43	5	math	math	PROPN
ejpam-5475	43	6	,	,	PUNCT
ejpam-5475	43	7	17	17	NUM
ejpam-5475	43	8	(	(	PUNCT
ejpam-5475	43	9	4	4	NUM
ejpam-5475	43	10	)	)	PUNCT
ejpam-5475	43	11	(	(	PUNCT
ejpam-5475	43	12	2024	2024	NUM
ejpam-5475	43	13	)	)	PUNCT
ejpam-5475	43	14	,	,	PUNCT
ejpam-5475	43	15	3610	3610	NUM
ejpam-5475	43	16	-	-	SYM
ejpam-5475	43	17	3621	3621	NUM
ejpam-5475	43	18	3612	3612	NUM
ejpam-5475	43	19	elements	element	NOUN
ejpam-5475	43	20	of	of	ADP
ejpam-5475	43	21	θ(ν1	θ(ν1	NOUN
ejpam-5475	43	22	,	,	PUNCT
ejpam-5475	43	23	ν2	ν2	NOUN
ejpam-5475	43	24	)	)	PUNCT
ejpam-5475	43	25	are	be	AUX
ejpam-5475	43	26	called	call	VERB
ejpam-5475	43	27	θ(ν1	θ(ν1	NOUN
ejpam-5475	43	28	,	,	PUNCT
ejpam-5475	43	29	ν2)-open	ν2)-open	ADJ
ejpam-5475	43	30	sets	set	NOUN
ejpam-5475	43	31	,	,	PUNCT
ejpam-5475	43	32	and	and	CCONJ
ejpam-5475	43	33	their	their	PRON
ejpam-5475	43	34	complements	complement	NOUN
ejpam-5475	43	35	are	be	AUX
ejpam-5475	43	36	θ(ν1	θ(ν1	NOUN
ejpam-5475	43	37	,	,	PUNCT
ejpam-5475	43	38	ν2)closed	ν2)close	VERB
ejpam-5475	43	39	sets	set	NOUN
ejpam-5475	43	40	.	.	PUNCT
ejpam-5475	44	1	the	the	DET
ejpam-5475	44	2	operation	operation	NOUN
ejpam-5475	44	3	γθ(ν1,ν2	γθ(ν1,ν2	PROPN
ejpam-5475	44	4	)	)	PUNCT
ejpam-5475	44	5	:	:	PUNCT
ejpam-5475	44	6	p(x	p(x	PROPN
ejpam-5475	44	7	)	)	PUNCT
ejpam-5475	44	8	→	→	SYM
ejpam-5475	44	9	p(x	p(x	PROPN
ejpam-5475	44	10	)	)	PUNCT
ejpam-5475	44	11	is	be	AUX
ejpam-5475	44	12	defined	define	VERB
ejpam-5475	44	13	for	for	ADP
ejpam-5475	44	14	a	a	DET
ejpam-5475	44	15	⊆	⊆	NUM
ejpam-5475	44	16	x	x	SYM
ejpam-5475	44	17	by	by	ADP
ejpam-5475	44	18	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	44	19	)	)	PUNCT
ejpam-5475	45	1	=	=	PRON
ejpam-5475	45	2	{	{	PUNCT
ejpam-5475	45	3	x	x	PUNCT
ejpam-5475	45	4	∈	∈	NOUN
ejpam-5475	45	5	x	x	X
ejpam-5475	45	6	:	:	PUNCT
ejpam-5475	45	7	cν2(m	cν2(m	PROPN
ejpam-5475	45	8	)	)	PUNCT
ejpam-5475	46	1	∩a	∩a	PROPN
ejpam-5475	46	2	̸=	̸=	PROPN
ejpam-5475	46	3	∅,∀m	∅,∀m	ADP
ejpam-5475	46	4	∈	∈	PROPN
ejpam-5475	46	5	ν1	ν1	NOUN
ejpam-5475	46	6	,	,	PUNCT
ejpam-5475	46	7	x	x	SYM
ejpam-5475	46	8	∈	∈	PROPN
ejpam-5475	46	9	m	m	PRON
ejpam-5475	46	10	}	}	PUNCT
ejpam-5475	46	11	.	.	PUNCT
ejpam-5475	47	1	in	in	ADP
ejpam-5475	47	2	conclusion	conclusion	NOUN
ejpam-5475	47	3	,	,	PUNCT
ejpam-5475	47	4	we	we	PRON
ejpam-5475	47	5	revisit	revisit	VERB
ejpam-5475	47	6	the	the	DET
ejpam-5475	47	7	following	follow	VERB
ejpam-5475	47	8	definitions	definition	NOUN
ejpam-5475	47	9	and	and	CCONJ
ejpam-5475	47	10	facts	fact	NOUN
ejpam-5475	47	11	due	due	ADJ
ejpam-5475	47	12	to	to	ADP
ejpam-5475	47	13	their	their	PRON
ejpam-5475	47	14	significance	significance	NOUN
ejpam-5475	47	15	in	in	ADP
ejpam-5475	47	16	our	our	PRON
ejpam-5475	47	17	paper	paper	NOUN
ejpam-5475	47	18	’s	’s	PART
ejpam-5475	47	19	content	content	NOUN
ejpam-5475	47	20	.	.	PUNCT
ejpam-5475	48	1	lemma	lemma	PROPN
ejpam-5475	48	2	1	1	NUM
ejpam-5475	48	3	.	.	PUNCT
ejpam-5475	49	1	[	[	X
ejpam-5475	49	2	9	9	NUM
ejpam-5475	49	3	]	]	PUNCT
ejpam-5475	49	4	let	let	VERB
ejpam-5475	49	5	ν1	ν1	NOUN
ejpam-5475	49	6	and	and	CCONJ
ejpam-5475	49	7	ν2	ν2	NOUN
ejpam-5475	49	8	be	be	AUX
ejpam-5475	49	9	two	two	NUM
ejpam-5475	49	10	gts	gts	NOUN
ejpam-5475	49	11	on	on	ADP
ejpam-5475	49	12	a	a	DET
ejpam-5475	49	13	nonempty	nonempty	ADV
ejpam-5475	49	14	set	set	VERB
ejpam-5475	49	15	x	x	NOUN
ejpam-5475	49	16	,	,	PUNCT
ejpam-5475	49	17	and	and	CCONJ
ejpam-5475	49	18	let	let	VERB
ejpam-5475	49	19	a	a	DET
ejpam-5475	49	20	⊆	⊆	NUM
ejpam-5475	49	21	x.	x.	NOUN
ejpam-5475	49	22	if	if	SCONJ
ejpam-5475	49	23	a	a	DET
ejpam-5475	49	24	∈	∈	PROPN
ejpam-5475	49	25	ν2	ν2	NOUN
ejpam-5475	49	26	,	,	PUNCT
ejpam-5475	49	27	then	then	ADV
ejpam-5475	49	28	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	49	29	)	)	PUNCT
ejpam-5475	50	1	=	=	SYM
ejpam-5475	50	2	cν1(a	cν1(a	NOUN
ejpam-5475	50	3	)	)	PUNCT
ejpam-5475	50	4	.	.	PUNCT
ejpam-5475	51	1	definition	definition	NOUN
ejpam-5475	51	2	1	1	NUM
ejpam-5475	51	3	.	.	PUNCT
ejpam-5475	52	1	[	[	X
ejpam-5475	52	2	5	5	NUM
ejpam-5475	52	3	]	]	PUNCT
ejpam-5475	52	4	let	let	VERB
ejpam-5475	52	5	ν1	ν1	NOUN
ejpam-5475	52	6	and	and	CCONJ
ejpam-5475	52	7	ν2	ν2	NOUN
ejpam-5475	52	8	be	be	AUX
ejpam-5475	52	9	two	two	NUM
ejpam-5475	52	10	gts	gts	NOUN
ejpam-5475	52	11	on	on	ADP
ejpam-5475	52	12	a	a	DET
ejpam-5475	52	13	nonempty	nonempty	ADV
ejpam-5475	52	14	set	set	VERB
ejpam-5475	52	15	x.	x.	NOUN
ejpam-5475	52	16	a	a	DET
ejpam-5475	52	17	subset	subset	NOUN
ejpam-5475	52	18	a	a	PRON
ejpam-5475	52	19	of	of	ADP
ejpam-5475	52	20	x	x	PRON
ejpam-5475	52	21	is	be	AUX
ejpam-5475	52	22	called	call	VERB
ejpam-5475	52	23	(	(	PUNCT
ejpam-5475	52	24	ν1	ν1	NOUN
ejpam-5475	52	25	,	,	PUNCT
ejpam-5475	52	26	ν2)-regular	ν2)-regular	NOUN
ejpam-5475	52	27	-	-	PUNCT
ejpam-5475	52	28	open	open	ADJ
ejpam-5475	52	29	if	if	SCONJ
ejpam-5475	52	30	a	a	DET
ejpam-5475	52	31	=	=	NOUN
ejpam-5475	52	32	iν1	iν1	NOUN
ejpam-5475	52	33	(	(	PUNCT
ejpam-5475	52	34	cν2(a	cν2(a	NOUN
ejpam-5475	52	35	)	)	PUNCT
ejpam-5475	52	36	)	)	PUNCT
ejpam-5475	52	37	.	.	PUNCT
ejpam-5475	53	1	theorem	theorem	NOUN
ejpam-5475	53	2	1	1	NUM
ejpam-5475	53	3	.	.	PUNCT
ejpam-5475	54	1	[	[	X
ejpam-5475	54	2	5	5	NUM
ejpam-5475	54	3	]	]	PUNCT
ejpam-5475	54	4	let	let	VERB
ejpam-5475	54	5	ν1	ν1	NOUN
ejpam-5475	54	6	and	and	CCONJ
ejpam-5475	54	7	ν2	ν2	NOUN
ejpam-5475	54	8	be	be	AUX
ejpam-5475	54	9	two	two	NUM
ejpam-5475	54	10	gts	gts	NOUN
ejpam-5475	54	11	on	on	ADP
ejpam-5475	54	12	a	a	DET
ejpam-5475	54	13	nonempty	nonempty	ADV
ejpam-5475	54	14	set	set	VERB
ejpam-5475	54	15	x	x	NOUN
ejpam-5475	54	16	,	,	PUNCT
ejpam-5475	54	17	and	and	CCONJ
ejpam-5475	54	18	let	let	VERB
ejpam-5475	54	19	a	a	DET
ejpam-5475	54	20	⊆	⊆	NUM
ejpam-5475	54	21	x.	x.	NOUN
ejpam-5475	54	22	then	then	ADV
ejpam-5475	54	23	a	a	PRON
ejpam-5475	54	24	is	be	AUX
ejpam-5475	54	25	θ(ν1	θ(ν1	NOUN
ejpam-5475	54	26	,	,	PUNCT
ejpam-5475	54	27	ν2)-closed	ν2)-close	VERB
ejpam-5475	54	28	if	if	SCONJ
ejpam-5475	54	29	and	and	CCONJ
ejpam-5475	54	30	only	only	ADV
ejpam-5475	54	31	if	if	SCONJ
ejpam-5475	54	32	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	NOUN
ejpam-5475	54	33	)	)	PUNCT
ejpam-5475	55	1	=	=	PUNCT
ejpam-5475	55	2	a.	a.	NOUN
ejpam-5475	55	3	definition	definition	NOUN
ejpam-5475	55	4	2	2	NUM
ejpam-5475	55	5	.	.	PUNCT
ejpam-5475	56	1	[	[	X
ejpam-5475	56	2	8	8	NUM
ejpam-5475	56	3	]	]	X
ejpam-5475	56	4	let	let	AUX
ejpam-5475	56	5	(	(	PUNCT
ejpam-5475	56	6	x	x	NOUN
ejpam-5475	56	7	,	,	PUNCT
ejpam-5475	56	8	ν	ν	NOUN
ejpam-5475	56	9	)	)	PUNCT
ejpam-5475	56	10	be	be	AUX
ejpam-5475	56	11	a	a	DET
ejpam-5475	56	12	gts	gts	NOUN
ejpam-5475	56	13	.	.	PUNCT
ejpam-5475	57	1	we	we	PRON
ejpam-5475	57	2	say	say	VERB
ejpam-5475	57	3	that	that	SCONJ
ejpam-5475	57	4	x	x	PRON
ejpam-5475	57	5	is	be	AUX
ejpam-5475	57	6	g	g	NOUN
ejpam-5475	57	7	-	-	PUNCT
ejpam-5475	57	8	regular	regular	ADJ
ejpam-5475	57	9	with	with	ADP
ejpam-5475	57	10	respect	respect	NOUN
ejpam-5475	57	11	to	to	ADP
ejpam-5475	57	12	mν	mν	PRON
ejpam-5475	57	13	if	if	SCONJ
ejpam-5475	57	14	,	,	PUNCT
ejpam-5475	57	15	for	for	ADP
ejpam-5475	57	16	every	every	DET
ejpam-5475	57	17	point	point	NOUN
ejpam-5475	57	18	x	x	PUNCT
ejpam-5475	57	19	∈	∈	NOUN
ejpam-5475	57	20	mν	mν	ADV
ejpam-5475	57	21	and	and	CCONJ
ejpam-5475	57	22	every	every	PRON
ejpam-5475	57	23	ν	ν	NOUN
ejpam-5475	57	24	-	-	ADJ
ejpam-5475	57	25	closed	closed	ADJ
ejpam-5475	57	26	set	set	NOUN
ejpam-5475	57	27	f	f	PROPN
ejpam-5475	57	28	such	such	ADJ
ejpam-5475	57	29	that	that	PRON
ejpam-5475	57	30	x	x	X
ejpam-5475	57	31	/∈	/∈	PUNCT
ejpam-5475	58	1	f	f	PROPN
ejpam-5475	58	2	,	,	PUNCT
ejpam-5475	58	3	there	there	PRON
ejpam-5475	58	4	exist	exist	VERB
ejpam-5475	58	5	sets	set	NOUN
ejpam-5475	58	6	u	u	NOUN
ejpam-5475	58	7	and	and	CCONJ
ejpam-5475	58	8	v	v	NOUN
ejpam-5475	58	9	in	in	ADP
ejpam-5475	58	10	ν	ν	NOUN
ejpam-5475	58	11	satisfying	satisfy	VERB
ejpam-5475	58	12	the	the	DET
ejpam-5475	58	13	following	follow	VERB
ejpam-5475	58	14	conditions	condition	NOUN
ejpam-5475	58	15	:	:	PUNCT
ejpam-5475	58	16	x	x	SYM
ejpam-5475	58	17	∈	∈	PROPN
ejpam-5475	58	18	u	u	PROPN
ejpam-5475	58	19	,	,	PUNCT
ejpam-5475	58	20	f	f	PROPN
ejpam-5475	58	21	∩mν	∩mν	PROPN
ejpam-5475	58	22	⊆	⊆	NUM
ejpam-5475	58	23	v	v	NOUN
ejpam-5475	58	24	,	,	PUNCT
ejpam-5475	58	25	and	and	CCONJ
ejpam-5475	58	26	u	u	NOUN
ejpam-5475	58	27	∩	∩	NOUN
ejpam-5475	58	28	v	v	NOUN
ejpam-5475	58	29	=	=	PUNCT
ejpam-5475	58	30	∅.	∅.	NOUN
ejpam-5475	58	31	definition	definition	NOUN
ejpam-5475	58	32	3	3	NUM
ejpam-5475	58	33	.	.	PUNCT
ejpam-5475	59	1	[	[	X
ejpam-5475	59	2	9	9	NUM
ejpam-5475	59	3	]	]	PUNCT
ejpam-5475	59	4	let	let	VERB
ejpam-5475	59	5	ν1	ν1	NOUN
ejpam-5475	59	6	and	and	CCONJ
ejpam-5475	59	7	ν2	ν2	NOUN
ejpam-5475	59	8	be	be	AUX
ejpam-5475	59	9	two	two	NUM
ejpam-5475	59	10	gts	gts	NOUN
ejpam-5475	59	11	defined	define	VERB
ejpam-5475	59	12	on	on	ADP
ejpam-5475	59	13	a	a	DET
ejpam-5475	59	14	nonempty	nonempty	ADV
ejpam-5475	59	15	set	set	VERB
ejpam-5475	59	16	x.	x.	NOUN
ejpam-5475	59	17	we	we	PRON
ejpam-5475	59	18	say	say	VERB
ejpam-5475	59	19	that	that	SCONJ
ejpam-5475	59	20	x	x	PRON
ejpam-5475	59	21	is	be	AUX
ejpam-5475	59	22	(	(	PUNCT
ejpam-5475	59	23	ν1	ν1	NOUN
ejpam-5475	59	24	,	,	PUNCT
ejpam-5475	59	25	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	59	26	if	if	SCONJ
ejpam-5475	59	27	,	,	PUNCT
ejpam-5475	59	28	for	for	ADP
ejpam-5475	59	29	every	every	DET
ejpam-5475	59	30	point	point	NOUN
ejpam-5475	59	31	x	x	X
ejpam-5475	59	32	∈	∈	NOUN
ejpam-5475	59	33	x	x	X
ejpam-5475	59	34	and	and	CCONJ
ejpam-5475	59	35	every	every	DET
ejpam-5475	59	36	ν1	ν1	NOUN
ejpam-5475	59	37	-	-	PUNCT
ejpam-5475	59	38	closed	close	VERB
ejpam-5475	59	39	set	set	NOUN
ejpam-5475	59	40	f	f	PROPN
ejpam-5475	59	41	with	with	ADP
ejpam-5475	59	42	x	x	PROPN
ejpam-5475	59	43	/∈	/∈	PROPN
ejpam-5475	59	44	f	f	PROPN
ejpam-5475	59	45	,	,	PUNCT
ejpam-5475	59	46	there	there	PRON
ejpam-5475	59	47	exist	exist	VERB
ejpam-5475	59	48	open	open	ADJ
ejpam-5475	59	49	sets	set	NOUN
ejpam-5475	59	50	u	u	PROPN
ejpam-5475	59	51	∈	∈	PROPN
ejpam-5475	59	52	ν1	ν1	NOUN
ejpam-5475	59	53	and	and	CCONJ
ejpam-5475	59	54	v	v	NOUN
ejpam-5475	59	55	∈	∈	NOUN
ejpam-5475	59	56	ν2	ν2	NOUN
ejpam-5475	59	57	such	such	ADJ
ejpam-5475	59	58	that	that	SCONJ
ejpam-5475	59	59	x	x	SYM
ejpam-5475	59	60	∈	∈	PROPN
ejpam-5475	59	61	u	u	PROPN
ejpam-5475	59	62	,	,	PUNCT
ejpam-5475	59	63	f	f	PROPN
ejpam-5475	59	64	⊆	⊆	NUM
ejpam-5475	59	65	v	v	NOUN
ejpam-5475	59	66	,	,	PUNCT
ejpam-5475	59	67	and	and	CCONJ
ejpam-5475	59	68	u	u	NOUN
ejpam-5475	59	69	∩	∩	NOUN
ejpam-5475	59	70	v	v	NOUN
ejpam-5475	59	71	=	=	PUNCT
ejpam-5475	59	72	∅.	∅.	PRON
ejpam-5475	59	73	3	3	NUM
ejpam-5475	59	74	.	.	PUNCT
ejpam-5475	60	1	properties	property	NOUN
ejpam-5475	60	2	of	of	ADP
ejpam-5475	60	3	the	the	DET
ejpam-5475	60	4	mixed	mixed	ADJ
ejpam-5475	60	5	operation	operation	NOUN
ejpam-5475	60	6	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	60	7	)	)	PUNCT
ejpam-5475	60	8	we	we	PRON
ejpam-5475	60	9	begin	begin	VERB
ejpam-5475	60	10	this	this	DET
ejpam-5475	60	11	section	section	NOUN
ejpam-5475	60	12	by	by	ADP
ejpam-5475	60	13	introducing	introduce	VERB
ejpam-5475	60	14	our	our	PRON
ejpam-5475	60	15	primary	primary	ADJ
ejpam-5475	60	16	definition	definition	NOUN
ejpam-5475	60	17	of	of	ADP
ejpam-5475	60	18	the	the	DET
ejpam-5475	60	19	mixed	mixed	ADJ
ejpam-5475	60	20	operation	operation	NOUN
ejpam-5475	60	21	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	60	22	)	)	PUNCT
ejpam-5475	60	23	and	and	CCONJ
ejpam-5475	60	24	presenting	present	VERB
ejpam-5475	60	25	intriguing	intriguing	ADJ
ejpam-5475	60	26	results	result	NOUN
ejpam-5475	60	27	associated	associate	VERB
ejpam-5475	60	28	with	with	ADP
ejpam-5475	60	29	it	it	PRON
ejpam-5475	60	30	.	.	PUNCT
ejpam-5475	61	1	definition	definition	NOUN
ejpam-5475	61	2	4	4	NUM
ejpam-5475	61	3	.	.	PUNCT
ejpam-5475	62	1	let	let	VERB
ejpam-5475	62	2	ν1	ν1	NOUN
ejpam-5475	62	3	and	and	CCONJ
ejpam-5475	62	4	ν2	ν2	NOUN
ejpam-5475	62	5	be	be	AUX
ejpam-5475	62	6	two	two	NUM
ejpam-5475	62	7	gts	gts	NOUN
ejpam-5475	62	8	defined	define	VERB
ejpam-5475	62	9	on	on	ADP
ejpam-5475	62	10	a	a	DET
ejpam-5475	62	11	nonempty	nonempty	ADV
ejpam-5475	62	12	set	set	VERB
ejpam-5475	62	13	x	x	NOUN
ejpam-5475	62	14	,	,	PUNCT
ejpam-5475	62	15	and	and	CCONJ
ejpam-5475	62	16	let	let	VERB
ejpam-5475	62	17	a	a	DET
ejpam-5475	62	18	⊆	⊆	NUM
ejpam-5475	62	19	x.	x.	NOUN
ejpam-5475	62	20	define	define	VERB
ejpam-5475	62	21	γθ̃(ν1,ν2	γθ̃(ν1,ν2	NOUN
ejpam-5475	62	22	)	)	PUNCT
ejpam-5475	62	23	:	:	PUNCT
ejpam-5475	62	24	p(x	p(x	PROPN
ejpam-5475	62	25	)	)	PUNCT
ejpam-5475	62	26	→	→	SYM
ejpam-5475	62	27	p(x	p(x	PROPN
ejpam-5475	62	28	)	)	PUNCT
ejpam-5475	62	29	as	as	ADP
ejpam-5475	62	30	a	a	DET
ejpam-5475	62	31	mixed	mixed	ADJ
ejpam-5475	62	32	operation	operation	NOUN
ejpam-5475	62	33	by	by	ADP
ejpam-5475	62	34	:	:	PUNCT
ejpam-5475	62	35	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	62	36	)	)	PUNCT
ejpam-5475	63	1	=	=	PRON
ejpam-5475	63	2	{	{	PUNCT
ejpam-5475	63	3	x	x	PUNCT
ejpam-5475	63	4	∈	∈	PROPN
ejpam-5475	63	5	x	x	X
ejpam-5475	63	6	:	:	PUNCT
ejpam-5475	63	7	(	(	PUNCT
ejpam-5475	63	8	cν2(m	cν2(m	PROPN
ejpam-5475	63	9	)	)	PUNCT
ejpam-5475	63	10	∩mν1	∩mν1	ADJ
ejpam-5475	63	11	)	)	PUNCT
ejpam-5475	63	12	∩a	∩a	PROPN
ejpam-5475	63	13	̸=	̸=	PROPN
ejpam-5475	63	14	∅	∅	NOUN
ejpam-5475	63	15	,	,	PUNCT
ejpam-5475	63	16	for	for	ADP
ejpam-5475	63	17	all	all	DET
ejpam-5475	63	18	m	m	PROPN
ejpam-5475	63	19	∈	∈	PROPN
ejpam-5475	63	20	ν1	ν1	NOUN
ejpam-5475	63	21	,	,	PUNCT
ejpam-5475	63	22	x	x	SYM
ejpam-5475	63	23	∈	∈	PROPN
ejpam-5475	63	24	m	m	NOUN
ejpam-5475	63	25	}	}	PUNCT
ejpam-5475	63	26	.	.	PUNCT
ejpam-5475	64	1	if	if	SCONJ
ejpam-5475	64	2	x	x	SYM
ejpam-5475	64	3	∈	∈	PROPN
ejpam-5475	64	4	x	x	PUNCT
ejpam-5475	64	5	−mν1	−mν1	ADP
ejpam-5475	64	6	,	,	PUNCT
ejpam-5475	64	7	then	then	ADV
ejpam-5475	64	8	by	by	ADP
ejpam-5475	64	9	definition	definition	NOUN
ejpam-5475	64	10	,	,	PUNCT
ejpam-5475	64	11	x	x	X
ejpam-5475	64	12	∈	∈	PROPN
ejpam-5475	64	13	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	64	14	)	)	PUNCT
ejpam-5475	64	15	.	.	PUNCT
ejpam-5475	65	1	according	accord	VERB
ejpam-5475	65	2	to	to	ADP
ejpam-5475	65	3	this	this	DET
ejpam-5475	65	4	definition	definition	NOUN
ejpam-5475	65	5	,	,	PUNCT
ejpam-5475	65	6	x	x	X
ejpam-5475	65	7	/∈	/∈	PUNCT
ejpam-5475	65	8	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	65	9	)	)	PUNCT
ejpam-5475	66	1	if	if	SCONJ
ejpam-5475	66	2	and	and	CCONJ
ejpam-5475	66	3	only	only	ADV
ejpam-5475	66	4	if	if	SCONJ
ejpam-5475	66	5	there	there	PRON
ejpam-5475	66	6	exists	exist	VERB
ejpam-5475	66	7	m	m	PROPN
ejpam-5475	66	8	∈	∈	NOUN
ejpam-5475	66	9	ν1	ν1	NOUN
ejpam-5475	66	10	such	such	ADJ
ejpam-5475	66	11	that	that	SCONJ
ejpam-5475	66	12	(	(	PUNCT
ejpam-5475	66	13	cν2(m	cν2(m	PROPN
ejpam-5475	66	14	)	)	PUNCT
ejpam-5475	66	15	∩mν1	∩mν1	ADV
ejpam-5475	66	16	)	)	PUNCT
ejpam-5475	67	1	∩a	∩a	PROPN
ejpam-5475	68	1	=	=	PUNCT
ejpam-5475	68	2	∅.	∅.	PRON
ejpam-5475	68	3	remark	remark	NOUN
ejpam-5475	68	4	1	1	NUM
ejpam-5475	68	5	.	.	PUNCT
ejpam-5475	69	1	let	let	VERB
ejpam-5475	69	2	ν	ν	NOUN
ejpam-5475	69	3	be	be	AUX
ejpam-5475	69	4	a	a	DET
ejpam-5475	69	5	gt	gt	PROPN
ejpam-5475	69	6	on	on	ADP
ejpam-5475	69	7	a	a	DET
ejpam-5475	69	8	nonempty	nonempty	ADV
ejpam-5475	69	9	set	set	VERB
ejpam-5475	69	10	x.	x.	NOUN
ejpam-5475	69	11	for	for	ADP
ejpam-5475	69	12	any	any	DET
ejpam-5475	69	13	subset	subset	NOUN
ejpam-5475	69	14	a	a	DET
ejpam-5475	69	15	⊆	⊆	NUM
ejpam-5475	69	16	x	x	SYM
ejpam-5475	69	17	,	,	PUNCT
ejpam-5475	69	18	it	it	PRON
ejpam-5475	69	19	holds	hold	VERB
ejpam-5475	69	20	that	that	SCONJ
ejpam-5475	69	21	γθ̃(ν	γθ̃(ν	NOUN
ejpam-5475	69	22	,	,	PUNCT
ejpam-5475	69	23	ν)(a	ν)(a	NOUN
ejpam-5475	69	24	)	)	PUNCT
ejpam-5475	69	25	=	=	SYM
ejpam-5475	69	26	γθ̃(ν)(a	γθ̃(ν)(a	NOUN
ejpam-5475	69	27	)	)	PUNCT
ejpam-5475	69	28	.	.	PUNCT
ejpam-5475	70	1	in	in	ADP
ejpam-5475	70	2	remark	remark	NOUN
ejpam-5475	70	3	1	1	NUM
ejpam-5475	70	4	above	above	ADV
ejpam-5475	70	5	,	,	PUNCT
ejpam-5475	70	6	for	for	ADP
ejpam-5475	70	7	a	a	DET
ejpam-5475	70	8	strong	strong	ADJ
ejpam-5475	70	9	gts	gts	NOUN
ejpam-5475	70	10	(	(	PUNCT
ejpam-5475	70	11	x	x	NOUN
ejpam-5475	70	12	,	,	PUNCT
ejpam-5475	70	13	ν	ν	NOUN
ejpam-5475	70	14	)	)	PUNCT
ejpam-5475	70	15	,	,	PUNCT
ejpam-5475	70	16	the	the	DET
ejpam-5475	70	17	following	follow	VERB
ejpam-5475	70	18	equality	equality	NOUN
ejpam-5475	70	19	holds	hold	VERB
ejpam-5475	70	20	:	:	PUNCT
ejpam-5475	70	21	γθ̃(ν	γθ̃(ν	ADV
ejpam-5475	70	22	,	,	PUNCT
ejpam-5475	70	23	ν)(a	ν)(a	NOUN
ejpam-5475	70	24	)	)	PUNCT
ejpam-5475	70	25	=	=	SYM
ejpam-5475	71	1	γθ̃(ν)(a	γθ̃(ν)(a	NOUN
ejpam-5475	71	2	)	)	PUNCT
ejpam-5475	71	3	=	=	PUNCT
ejpam-5475	71	4	γθ(ν)(a	γθ(ν)(a	NUM
ejpam-5475	71	5	)	)	PUNCT
ejpam-5475	71	6	.	.	PUNCT
ejpam-5475	72	1	a.	a.	NOUN
ejpam-5475	72	2	qahis	qahis	PROPN
ejpam-5475	72	3	,	,	PUNCT
ejpam-5475	72	4	a.	a.	NOUN
ejpam-5475	72	5	alqahtani	alqahtani	PROPN
ejpam-5475	72	6	/	/	SYM
ejpam-5475	72	7	eur	eur	PROPN
ejpam-5475	72	8	.	.	PUNCT
ejpam-5475	73	1	j.	j.	PROPN
ejpam-5475	73	2	pure	pure	PROPN
ejpam-5475	73	3	appl	appl	PROPN
ejpam-5475	73	4	.	.	PROPN
ejpam-5475	73	5	math	math	PROPN
ejpam-5475	73	6	,	,	PUNCT
ejpam-5475	73	7	17	17	NUM
ejpam-5475	73	8	(	(	PUNCT
ejpam-5475	73	9	4	4	NUM
ejpam-5475	73	10	)	)	PUNCT
ejpam-5475	73	11	(	(	PUNCT
ejpam-5475	73	12	2024	2024	NUM
ejpam-5475	73	13	)	)	PUNCT
ejpam-5475	73	14	,	,	PUNCT
ejpam-5475	73	15	3610	3610	NUM
ejpam-5475	73	16	-	-	SYM
ejpam-5475	73	17	3621	3621	NUM
ejpam-5475	73	18	3613	3613	NUM
ejpam-5475	73	19	theorem	theorem	NOUN
ejpam-5475	73	20	2	2	NUM
ejpam-5475	73	21	.	.	PUNCT
ejpam-5475	74	1	let	let	VERB
ejpam-5475	74	2	ν1	ν1	NOUN
ejpam-5475	74	3	and	and	CCONJ
ejpam-5475	74	4	ν2	ν2	NOUN
ejpam-5475	74	5	be	be	AUX
ejpam-5475	74	6	two	two	NUM
ejpam-5475	74	7	gt	gt	NOUN
ejpam-5475	74	8	’s	’s	NOUN
ejpam-5475	74	9	on	on	ADP
ejpam-5475	74	10	a	a	DET
ejpam-5475	74	11	nonempty	nonempty	ADV
ejpam-5475	74	12	set	set	VERB
ejpam-5475	74	13	x.	x.	NOUN
ejpam-5475	74	14	then	then	ADV
ejpam-5475	74	15	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	74	16	)	)	PUNCT
ejpam-5475	74	17	⊆	⊆	NUM
ejpam-5475	74	18	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	NOUN
ejpam-5475	74	19	)	)	PUNCT
ejpam-5475	74	20	for	for	ADP
ejpam-5475	74	21	any	any	DET
ejpam-5475	74	22	a	a	DET
ejpam-5475	74	23	⊆	⊆	NUM
ejpam-5475	74	24	x.	x.	NOUN
ejpam-5475	74	25	proof	proof	NOUN
ejpam-5475	74	26	.	.	PUNCT
ejpam-5475	75	1	let	let	VERB
ejpam-5475	75	2	x	x	PUNCT
ejpam-5475	75	3	∈	∈	PROPN
ejpam-5475	75	4	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	75	5	)	)	PUNCT
ejpam-5475	75	6	andm	andm	PROPN
ejpam-5475	75	7	∈	∈	PROPN
ejpam-5475	75	8	ν1	ν1	NOUN
ejpam-5475	75	9	such	such	ADJ
ejpam-5475	75	10	that	that	SCONJ
ejpam-5475	75	11	x	x	SYM
ejpam-5475	75	12	∈	∈	NOUN
ejpam-5475	75	13	m	m	VERB
ejpam-5475	75	14	.	.	PUNCT
ejpam-5475	76	1	then	then	ADV
ejpam-5475	76	2	(	(	PUNCT
ejpam-5475	76	3	cν2(m)∩mν1)∩a	cν2(m)∩mν1)∩a	PROPN
ejpam-5475	76	4	̸=	̸=	PROPN
ejpam-5475	76	5	∅.	∅.	NOUN
ejpam-5475	76	6	since	since	SCONJ
ejpam-5475	76	7	(	(	PUNCT
ejpam-5475	76	8	cν2(m	cν2(m	PROPN
ejpam-5475	76	9	)	)	PUNCT
ejpam-5475	76	10	∩	∩	ADJ
ejpam-5475	76	11	mν1	mν1	NOUN
ejpam-5475	76	12	)	)	PUNCT
ejpam-5475	76	13	∩	∩	NOUN
ejpam-5475	76	14	a	a	DET
ejpam-5475	76	15	⊆	⊆	NUM
ejpam-5475	76	16	cν2(m	cν2(m	NOUN
ejpam-5475	76	17	)	)	PUNCT
ejpam-5475	76	18	∩	∩	NOUN
ejpam-5475	76	19	a	a	X
ejpam-5475	76	20	,	,	PUNCT
ejpam-5475	76	21	it	it	PRON
ejpam-5475	76	22	follows	follow	VERB
ejpam-5475	76	23	that	that	SCONJ
ejpam-5475	76	24	cν2(m	cν2(m	PROPN
ejpam-5475	76	25	)	)	PUNCT
ejpam-5475	76	26	∩	∩	NOUN
ejpam-5475	76	27	a	a	DET
ejpam-5475	76	28	̸=	̸=	PROPN
ejpam-5475	76	29	∅.	∅.	VERB
ejpam-5475	76	30	therefore	therefore	ADV
ejpam-5475	76	31	,	,	PUNCT
ejpam-5475	76	32	x	x	PROPN
ejpam-5475	76	33	∈	∈	PROPN
ejpam-5475	76	34	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	76	35	)	)	PUNCT
ejpam-5475	76	36	.	.	PUNCT
ejpam-5475	77	1	the	the	DET
ejpam-5475	77	2	following	follow	VERB
ejpam-5475	77	3	example	example	NOUN
ejpam-5475	77	4	shows	show	VERB
ejpam-5475	77	5	that	that	SCONJ
ejpam-5475	77	6	generally	generally	ADV
ejpam-5475	77	7	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	77	8	)	)	PUNCT
ejpam-5475	77	9	̸=	̸=	PROPN
ejpam-5475	77	10	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	NOUN
ejpam-5475	77	11	)	)	PUNCT
ejpam-5475	77	12	.	.	PUNCT
ejpam-5475	78	1	example	example	NOUN
ejpam-5475	79	1	1	1	X
ejpam-5475	79	2	.	.	X
ejpam-5475	79	3	consider	consider	VERB
ejpam-5475	79	4	the	the	DET
ejpam-5475	79	5	set	set	NOUN
ejpam-5475	79	6	x	x	PUNCT
ejpam-5475	79	7	=	=	X
ejpam-5475	79	8	{	{	PUNCT
ejpam-5475	79	9	a	a	PRON
ejpam-5475	79	10	,	,	PUNCT
ejpam-5475	79	11	b	b	NOUN
ejpam-5475	79	12	,	,	PUNCT
ejpam-5475	79	13	c	c	NOUN
ejpam-5475	79	14	,	,	PUNCT
ejpam-5475	79	15	d	d	NOUN
ejpam-5475	79	16	}	}	PUNCT
ejpam-5475	79	17	equipped	equip	VERB
ejpam-5475	79	18	with	with	ADP
ejpam-5475	79	19	two	two	NUM
ejpam-5475	79	20	generalized	generalized	ADJ
ejpam-5475	79	21	topologies	topology	NOUN
ejpam-5475	79	22	:	:	PUNCT
ejpam-5475	79	23	ν1	ν1	NOUN
ejpam-5475	79	24	=	=	SYM
ejpam-5475	79	25	{	{	PUNCT
ejpam-5475	79	26	∅	∅	NOUN
ejpam-5475	79	27	,	,	PUNCT
ejpam-5475	79	28	{	{	PUNCT
ejpam-5475	79	29	b	b	NOUN
ejpam-5475	79	30	,	,	PUNCT
ejpam-5475	79	31	d	d	NOUN
ejpam-5475	79	32	}	}	PUNCT
ejpam-5475	79	33	}	}	PUNCT
ejpam-5475	79	34	and	and	CCONJ
ejpam-5475	79	35	ν2	ν2	NOUN
ejpam-5475	79	36	=	=	SYM
ejpam-5475	79	37	{	{	PUNCT
ejpam-5475	79	38	∅	∅	NOUN
ejpam-5475	79	39	,	,	PUNCT
ejpam-5475	79	40	{	{	PUNCT
ejpam-5475	79	41	a	a	DET
ejpam-5475	79	42	,	,	PUNCT
ejpam-5475	79	43	b	b	NOUN
ejpam-5475	79	44	}	}	PUNCT
ejpam-5475	79	45	,	,	PUNCT
ejpam-5475	79	46	{	{	PUNCT
ejpam-5475	79	47	b	b	X
ejpam-5475	79	48	,	,	PUNCT
ejpam-5475	79	49	c	c	NOUN
ejpam-5475	79	50	}	}	PUNCT
ejpam-5475	79	51	,	,	PUNCT
ejpam-5475	79	52	{	{	PUNCT
ejpam-5475	79	53	a	a	PRON
ejpam-5475	79	54	,	,	PUNCT
ejpam-5475	79	55	b	b	NOUN
ejpam-5475	79	56	,	,	PUNCT
ejpam-5475	79	57	c	c	NOUN
ejpam-5475	79	58	}	}	PUNCT
ejpam-5475	79	59	}	}	PUNCT
ejpam-5475	79	60	.	.	PUNCT
ejpam-5475	80	1	let	let	VERB
ejpam-5475	80	2	a	a	DET
ejpam-5475	80	3	=	=	X
ejpam-5475	80	4	{	{	PUNCT
ejpam-5475	80	5	a	a	X
ejpam-5475	80	6	,	,	PUNCT
ejpam-5475	80	7	c	c	NOUN
ejpam-5475	80	8	}	}	PUNCT
ejpam-5475	80	9	.	.	PUNCT
ejpam-5475	81	1	observe	observe	VERB
ejpam-5475	81	2	the	the	DET
ejpam-5475	81	3	following	following	NOUN
ejpam-5475	81	4	:	:	PUNCT
ejpam-5475	81	5	cν2({b	cν2({b	ADJ
ejpam-5475	81	6	,	,	PUNCT
ejpam-5475	81	7	d	d	NOUN
ejpam-5475	81	8	}	}	PUNCT
ejpam-5475	81	9	)	)	PUNCT
ejpam-5475	81	10	=	=	SYM
ejpam-5475	81	11	x	x	PUNCT
ejpam-5475	81	12	and	and	CCONJ
ejpam-5475	81	13	mν1	mν1	ADJ
ejpam-5475	81	14	=	=	SYM
ejpam-5475	81	15	{	{	PUNCT
ejpam-5475	81	16	b	b	NOUN
ejpam-5475	81	17	,	,	PUNCT
ejpam-5475	81	18	d	d	NOUN
ejpam-5475	81	19	}	}	PUNCT
ejpam-5475	81	20	.	.	PUNCT
ejpam-5475	82	1	additionally	additionally	ADV
ejpam-5475	82	2	,	,	PUNCT
ejpam-5475	82	3	cν2({b	cν2({b	ADJ
ejpam-5475	82	4	,	,	PUNCT
ejpam-5475	82	5	d	d	NOUN
ejpam-5475	82	6	}	}	PUNCT
ejpam-5475	82	7	)	)	PUNCT
ejpam-5475	82	8	∩a	∩a	PROPN
ejpam-5475	82	9	̸=	̸=	PROPN
ejpam-5475	82	10	∅	∅	NOUN
ejpam-5475	82	11	and	and	CCONJ
ejpam-5475	82	12	(	(	PUNCT
ejpam-5475	82	13	cν2({b	cν2({b	X
ejpam-5475	82	14	,	,	PUNCT
ejpam-5475	82	15	d	d	NOUN
ejpam-5475	82	16	}	}	PUNCT
ejpam-5475	82	17	)	)	PUNCT
ejpam-5475	82	18	∩mν1	∩mν1	ADV
ejpam-5475	82	19	)	)	PUNCT
ejpam-5475	83	1	∩a	∩a	PROPN
ejpam-5475	84	1	=	=	PUNCT
ejpam-5475	84	2	∅.	∅.	VERB
ejpam-5475	84	3	therefore	therefore	ADV
ejpam-5475	84	4	,	,	PUNCT
ejpam-5475	84	5	b	b	X
ejpam-5475	84	6	,	,	PUNCT
ejpam-5475	84	7	d	d	PROPN
ejpam-5475	84	8	∈	∈	PROPN
ejpam-5475	84	9	γθ(ν1	γθ(ν1	NOUN
ejpam-5475	84	10	,	,	PUNCT
ejpam-5475	84	11	ν2)(a	ν2)(a	PROPN
ejpam-5475	84	12	)	)	PUNCT
ejpam-5475	84	13	and	and	CCONJ
ejpam-5475	84	14	b	b	X
ejpam-5475	84	15	,	,	PUNCT
ejpam-5475	84	16	d	d	PROPN
ejpam-5475	84	17	/∈	/∈	PUNCT
ejpam-5475	84	18	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	84	19	)	)	PUNCT
ejpam-5475	84	20	.	.	PUNCT
ejpam-5475	85	1	thus	thus	ADV
ejpam-5475	85	2	,	,	PUNCT
ejpam-5475	85	3	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	85	4	)	)	PUNCT
ejpam-5475	86	1	=	=	PRON
ejpam-5475	86	2	{	{	PUNCT
ejpam-5475	86	3	a	a	X
ejpam-5475	86	4	,	,	PUNCT
ejpam-5475	86	5	c	c	NOUN
ejpam-5475	86	6	}	}	PUNCT
ejpam-5475	86	7	and	and	CCONJ
ejpam-5475	86	8	γθ(ν1	γθ(ν1	NOUN
ejpam-5475	86	9	,	,	PUNCT
ejpam-5475	86	10	ν2)(a	ν2)(a	PROPN
ejpam-5475	86	11	)	)	PUNCT
ejpam-5475	87	1	=	=	PUNCT
ejpam-5475	88	1	x.	x.	NOUN
ejpam-5475	88	2	consequently	consequently	ADV
ejpam-5475	88	3	,	,	PUNCT
ejpam-5475	88	4	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	88	5	)	)	PUNCT
ejpam-5475	89	1	⊂	⊂	PROPN
ejpam-5475	89	2	γθ(ν1	γθ(ν1	NOUN
ejpam-5475	89	3	,	,	PUNCT
ejpam-5475	89	4	ν2)(a	ν2)(a	PROPN
ejpam-5475	89	5	)	)	PUNCT
ejpam-5475	89	6	.	.	PUNCT
ejpam-5475	90	1	corollary	corollary	ADJ
ejpam-5475	90	2	1	1	NUM
ejpam-5475	90	3	.	.	PUNCT
ejpam-5475	91	1	let	let	VERB
ejpam-5475	91	2	ν1	ν1	NOUN
ejpam-5475	91	3	and	and	CCONJ
ejpam-5475	91	4	ν2	ν2	NOUN
ejpam-5475	91	5	be	be	AUX
ejpam-5475	91	6	two	two	NUM
ejpam-5475	91	7	gt	gt	NOUN
ejpam-5475	91	8	’s	’s	NOUN
ejpam-5475	91	9	on	on	ADP
ejpam-5475	91	10	a	a	DET
ejpam-5475	91	11	nonempty	nonempty	ADV
ejpam-5475	91	12	set	set	VERB
ejpam-5475	91	13	x	x	PUNCT
ejpam-5475	91	14	and	and	CCONJ
ejpam-5475	91	15	let	let	VERB
ejpam-5475	91	16	a	a	DET
ejpam-5475	91	17	⊆	⊆	NUM
ejpam-5475	91	18	x.	x.	NOUN
ejpam-5475	91	19	if	if	SCONJ
ejpam-5475	91	20	(	(	PUNCT
ejpam-5475	91	21	x	x	NOUN
ejpam-5475	91	22	,	,	PUNCT
ejpam-5475	91	23	ν1	ν1	NOUN
ejpam-5475	91	24	)	)	PUNCT
ejpam-5475	91	25	is	be	AUX
ejpam-5475	91	26	a	a	DET
ejpam-5475	91	27	strong	strong	ADJ
ejpam-5475	91	28	gts	gts	NOUN
ejpam-5475	91	29	,	,	PUNCT
ejpam-5475	91	30	then	then	ADV
ejpam-5475	91	31	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	91	32	)	)	PUNCT
ejpam-5475	92	1	=	=	SYM
ejpam-5475	92	2	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	92	3	)	)	PUNCT
ejpam-5475	92	4	.	.	PUNCT
ejpam-5475	93	1	theorem	theorem	NOUN
ejpam-5475	93	2	3	3	X
ejpam-5475	93	3	.	.	PUNCT
ejpam-5475	94	1	let	let	VERB
ejpam-5475	94	2	ν1	ν1	NOUN
ejpam-5475	94	3	and	and	CCONJ
ejpam-5475	94	4	ν2	ν2	NOUN
ejpam-5475	94	5	be	be	AUX
ejpam-5475	94	6	two	two	NUM
ejpam-5475	94	7	gt	gt	NOUN
ejpam-5475	94	8	’s	’s	NOUN
ejpam-5475	94	9	on	on	ADP
ejpam-5475	94	10	a	a	DET
ejpam-5475	94	11	nonempty	nonempty	ADV
ejpam-5475	94	12	set	set	VERB
ejpam-5475	94	13	x	x	PUNCT
ejpam-5475	94	14	and	and	CCONJ
ejpam-5475	94	15	a	a	PRON
ejpam-5475	94	16	,	,	PUNCT
ejpam-5475	94	17	b	b	PROPN
ejpam-5475	94	18	⊆	⊆	NUM
ejpam-5475	94	19	x.	x.	NOUN
ejpam-5475	94	20	then	then	ADV
ejpam-5475	94	21	the	the	DET
ejpam-5475	94	22	operation	operation	NOUN
ejpam-5475	94	23	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	94	24	)	)	PUNCT
ejpam-5475	94	25	has	have	VERB
ejpam-5475	94	26	the	the	DET
ejpam-5475	94	27	following	follow	VERB
ejpam-5475	94	28	properties	property	NOUN
ejpam-5475	94	29	.	.	PUNCT
ejpam-5475	95	1	(	(	PUNCT
ejpam-5475	95	2	i	i	NOUN
ejpam-5475	95	3	)	)	PUNCT
ejpam-5475	95	4	if	if	SCONJ
ejpam-5475	95	5	a	a	DET
ejpam-5475	95	6	⊆	⊆	NUM
ejpam-5475	95	7	b	b	NOUN
ejpam-5475	95	8	,	,	PUNCT
ejpam-5475	95	9	then	then	ADV
ejpam-5475	95	10	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	95	11	)	)	PUNCT
ejpam-5475	95	12	⊆	⊆	X
ejpam-5475	95	13	γθ̃(ν1,ν2)(b	γθ̃(ν1,ν2)(b	PROPN
ejpam-5475	95	14	)	)	PUNCT
ejpam-5475	95	15	.	.	PUNCT
ejpam-5475	96	1	(	(	PUNCT
ejpam-5475	96	2	ii	ii	X
ejpam-5475	96	3	)	)	PUNCT
ejpam-5475	96	4	a	a	DET
ejpam-5475	96	5	⊆	⊆	NUM
ejpam-5475	96	6	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	96	7	)	)	PUNCT
ejpam-5475	96	8	.	.	PUNCT
ejpam-5475	97	1	(	(	PUNCT
ejpam-5475	97	2	iii	iii	X
ejpam-5475	97	3	)	)	PUNCT
ejpam-5475	97	4	if	if	SCONJ
ejpam-5475	97	5	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	97	6	)	)	PUNCT
ejpam-5475	97	7	⊆	⊆	NUM
ejpam-5475	97	8	a	a	PRON
ejpam-5475	97	9	,	,	PUNCT
ejpam-5475	97	10	then	then	ADV
ejpam-5475	97	11	a	a	DET
ejpam-5475	97	12	=	=	SYM
ejpam-5475	97	13	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	97	14	)	)	PUNCT
ejpam-5475	97	15	.	.	PUNCT
ejpam-5475	98	1	proof	proof	NOUN
ejpam-5475	98	2	.	.	PUNCT
ejpam-5475	99	1	(	(	PUNCT
ejpam-5475	99	2	i	i	NOUN
ejpam-5475	99	3	)	)	PUNCT
ejpam-5475	99	4	let	let	VERB
ejpam-5475	99	5	x	x	PUNCT
ejpam-5475	99	6	∈	∈	PROPN
ejpam-5475	99	7	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	99	8	)	)	PUNCT
ejpam-5475	99	9	and	and	CCONJ
ejpam-5475	99	10	m	m	PROPN
ejpam-5475	99	11	∈	∈	NOUN
ejpam-5475	99	12	ν1	ν1	NOUN
ejpam-5475	99	13	such	such	ADJ
ejpam-5475	99	14	that	that	SCONJ
ejpam-5475	99	15	x	x	SYM
ejpam-5475	99	16	∈	∈	NOUN
ejpam-5475	99	17	m	m	VERB
ejpam-5475	99	18	.	.	PUNCT
ejpam-5475	100	1	then	then	ADV
ejpam-5475	100	2	,	,	PUNCT
ejpam-5475	100	3	(	(	PUNCT
ejpam-5475	100	4	cν2(m)∩mν1	cν2(m)∩mν1	ADV
ejpam-5475	100	5	)	)	PUNCT
ejpam-5475	100	6	∩	∩	NOUN
ejpam-5475	100	7	a	a	DET
ejpam-5475	100	8	̸=	̸=	PROPN
ejpam-5475	100	9	∅.	∅.	NOUN
ejpam-5475	100	10	since	since	SCONJ
ejpam-5475	100	11	a	a	DET
ejpam-5475	100	12	⊆	⊆	NUM
ejpam-5475	100	13	b	b	NOUN
ejpam-5475	100	14	,	,	PUNCT
ejpam-5475	100	15	it	it	PRON
ejpam-5475	100	16	follows	follow	VERB
ejpam-5475	100	17	that	that	SCONJ
ejpam-5475	100	18	(	(	PUNCT
ejpam-5475	100	19	cν2(m	cν2(m	PROPN
ejpam-5475	100	20	)	)	PUNCT
ejpam-5475	100	21	∩mν1	∩mν1	ADV
ejpam-5475	100	22	)	)	PUNCT
ejpam-5475	101	1	∩b	∩b	NOUN
ejpam-5475	101	2	̸=	̸=	PROPN
ejpam-5475	101	3	∅	∅	NOUN
ejpam-5475	101	4	,	,	PUNCT
ejpam-5475	101	5	and	and	CCONJ
ejpam-5475	101	6	hence	hence	ADV
ejpam-5475	101	7	x	x	X
ejpam-5475	101	8	∈	∈	PROPN
ejpam-5475	101	9	γθ̃(ν1,ν2)(b	γθ̃(ν1,ν2)(b	PROPN
ejpam-5475	101	10	)	)	PUNCT
ejpam-5475	101	11	.	.	PUNCT
ejpam-5475	102	1	(	(	PUNCT
ejpam-5475	102	2	ii	ii	NOUN
ejpam-5475	102	3	)	)	PUNCT
ejpam-5475	102	4	case	case	NOUN
ejpam-5475	102	5	1	1	NUM
ejpam-5475	102	6	:	:	PUNCT
ejpam-5475	102	7	if	if	SCONJ
ejpam-5475	102	8	x	x	PROPN
ejpam-5475	102	9	∈	∈	PROPN
ejpam-5475	102	10	a	a	PRON
ejpam-5475	102	11	and	and	CCONJ
ejpam-5475	102	12	x	x	SYM
ejpam-5475	102	13	∈	∈	NOUN
ejpam-5475	102	14	mν1	mν1	NOUN
ejpam-5475	102	15	,	,	PUNCT
ejpam-5475	102	16	then	then	ADV
ejpam-5475	102	17	for	for	SCONJ
ejpam-5475	102	18	each	each	DET
ejpam-5475	102	19	ν1	ν1	NOUN
ejpam-5475	102	20	-	-	PUNCT
ejpam-5475	102	21	open	open	NOUN
ejpam-5475	102	22	set	set	NOUN
ejpam-5475	102	23	m	m	AUX
ejpam-5475	102	24	containing	contain	VERB
ejpam-5475	102	25	x	x	SYM
ejpam-5475	102	26	,	,	PUNCT
ejpam-5475	102	27	(	(	PUNCT
ejpam-5475	102	28	cν2(m	cν2(m	PROPN
ejpam-5475	102	29	)	)	PUNCT
ejpam-5475	102	30	∩mν1	∩mν1	ADV
ejpam-5475	102	31	)	)	PUNCT
ejpam-5475	103	1	∩a	∩a	PROPN
ejpam-5475	103	2	̸=	̸=	PROPN
ejpam-5475	103	3	∅	∅	NOUN
ejpam-5475	103	4	,	,	PUNCT
ejpam-5475	103	5	so	so	ADV
ejpam-5475	103	6	x	x	X
ejpam-5475	103	7	∈	∈	PROPN
ejpam-5475	103	8	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	103	9	)	)	PUNCT
ejpam-5475	103	10	.	.	PUNCT
ejpam-5475	104	1	case	case	NOUN
ejpam-5475	104	2	2	2	NUM
ejpam-5475	104	3	:	:	PUNCT
ejpam-5475	104	4	if	if	SCONJ
ejpam-5475	104	5	x	x	PROPN
ejpam-5475	104	6	∈	∈	PROPN
ejpam-5475	104	7	a	a	PRON
ejpam-5475	104	8	and	and	CCONJ
ejpam-5475	104	9	x	x	SYM
ejpam-5475	104	10	/∈	/∈	PUNCT
ejpam-5475	104	11	mν1	mν1	ADJ
ejpam-5475	104	12	,	,	PUNCT
ejpam-5475	104	13	then	then	ADV
ejpam-5475	104	14	by	by	ADP
ejpam-5475	104	15	definition	definition	NOUN
ejpam-5475	104	16	4	4	NUM
ejpam-5475	104	17	,	,	PUNCT
ejpam-5475	104	18	x	x	X
ejpam-5475	104	19	∈	∈	NOUN
ejpam-5475	104	20	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	104	21	)	)	PUNCT
ejpam-5475	104	22	.	.	PUNCT
ejpam-5475	105	1	therefore	therefore	ADV
ejpam-5475	105	2	,	,	PUNCT
ejpam-5475	105	3	a	a	DET
ejpam-5475	105	4	⊆	⊆	NUM
ejpam-5475	105	5	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	105	6	)	)	PUNCT
ejpam-5475	105	7	.	.	PUNCT
ejpam-5475	106	1	from	from	ADP
ejpam-5475	106	2	cases	case	NOUN
ejpam-5475	106	3	1	1	NUM
ejpam-5475	106	4	and	and	CCONJ
ejpam-5475	106	5	2	2	NUM
ejpam-5475	106	6	,	,	PUNCT
ejpam-5475	106	7	we	we	PRON
ejpam-5475	106	8	derive	derive	VERB
ejpam-5475	106	9	that	that	SCONJ
ejpam-5475	106	10	a	a	DET
ejpam-5475	106	11	⊆	⊆	NUM
ejpam-5475	106	12	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	106	13	)	)	PUNCT
ejpam-5475	106	14	.	.	PUNCT
ejpam-5475	107	1	(	(	PUNCT
ejpam-5475	107	2	iii	iii	X
ejpam-5475	107	3	)	)	PUNCT
ejpam-5475	107	4	let	let	VERB
ejpam-5475	107	5	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	107	6	)	)	PUNCT
ejpam-5475	108	1	⊆	⊆	NUM
ejpam-5475	108	2	a.	a.	NOUN
ejpam-5475	108	3	then	then	ADV
ejpam-5475	108	4	by	by	ADP
ejpam-5475	108	5	(	(	PUNCT
ejpam-5475	108	6	ii	ii	NOUN
ejpam-5475	108	7	)	)	PUNCT
ejpam-5475	108	8	,	,	PUNCT
ejpam-5475	108	9	a	a	DET
ejpam-5475	108	10	⊆	⊆	NUM
ejpam-5475	108	11	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	108	12	)	)	PUNCT
ejpam-5475	108	13	.	.	PUNCT
ejpam-5475	109	1	hence	hence	ADV
ejpam-5475	109	2	,	,	PUNCT
ejpam-5475	109	3	a	a	DET
ejpam-5475	109	4	=	=	ADJ
ejpam-5475	109	5	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	109	6	)	)	PUNCT
ejpam-5475	109	7	.	.	PUNCT
ejpam-5475	110	1	a.	a.	NOUN
ejpam-5475	110	2	qahis	qahis	PROPN
ejpam-5475	110	3	,	,	PUNCT
ejpam-5475	110	4	a.	a.	NOUN
ejpam-5475	110	5	alqahtani	alqahtani	PROPN
ejpam-5475	110	6	/	/	SYM
ejpam-5475	110	7	eur	eur	PROPN
ejpam-5475	110	8	.	.	PUNCT
ejpam-5475	111	1	j.	j.	PROPN
ejpam-5475	111	2	pure	pure	PROPN
ejpam-5475	111	3	appl	appl	PROPN
ejpam-5475	111	4	.	.	PROPN
ejpam-5475	111	5	math	math	PROPN
ejpam-5475	111	6	,	,	PUNCT
ejpam-5475	111	7	17	17	NUM
ejpam-5475	111	8	(	(	PUNCT
ejpam-5475	111	9	4	4	NUM
ejpam-5475	111	10	)	)	PUNCT
ejpam-5475	111	11	(	(	PUNCT
ejpam-5475	111	12	2024	2024	NUM
ejpam-5475	111	13	)	)	PUNCT
ejpam-5475	111	14	,	,	PUNCT
ejpam-5475	111	15	3610	3610	NUM
ejpam-5475	111	16	-	-	SYM
ejpam-5475	111	17	3621	3621	NUM
ejpam-5475	111	18	3614	3614	NUM
ejpam-5475	111	19	theorem	theorem	VERB
ejpam-5475	111	20	4	4	NUM
ejpam-5475	111	21	.	.	PUNCT
ejpam-5475	112	1	let	let	VERB
ejpam-5475	112	2	ν1	ν1	NOUN
ejpam-5475	112	3	and	and	CCONJ
ejpam-5475	112	4	ν2	ν2	NOUN
ejpam-5475	112	5	be	be	AUX
ejpam-5475	112	6	two	two	NUM
ejpam-5475	112	7	gt	gt	NOUN
ejpam-5475	112	8	’s	’s	NOUN
ejpam-5475	112	9	on	on	ADP
ejpam-5475	112	10	a	a	DET
ejpam-5475	112	11	nonempty	nonempty	ADV
ejpam-5475	112	12	set	set	VERB
ejpam-5475	112	13	x	x	PUNCT
ejpam-5475	112	14	and	and	CCONJ
ejpam-5475	112	15	let	let	VERB
ejpam-5475	112	16	a	a	DET
ejpam-5475	112	17	⊆	⊆	NUM
ejpam-5475	112	18	x.	x.	NOUN
ejpam-5475	112	19	then	then	ADV
ejpam-5475	112	20	the	the	DET
ejpam-5475	112	21	following	follow	VERB
ejpam-5475	112	22	hold	hold	NOUN
ejpam-5475	112	23	.	.	PUNCT
ejpam-5475	113	1	(	(	PUNCT
ejpam-5475	113	2	i	i	NOUN
ejpam-5475	113	3	)	)	PUNCT
ejpam-5475	113	4	if	if	SCONJ
ejpam-5475	113	5	a	a	DET
ejpam-5475	113	6	⊆	⊆	NUM
ejpam-5475	113	7	x	x	SYM
ejpam-5475	113	8	−mν1	−mν1	ADP
ejpam-5475	113	9	,	,	PUNCT
ejpam-5475	113	10	then	then	ADV
ejpam-5475	113	11	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	113	12	)	)	PUNCT
ejpam-5475	114	1	=	=	PUNCT
ejpam-5475	114	2	x	x	PUNCT
ejpam-5475	114	3	−mν1	−mν1	PROPN
ejpam-5475	114	4	.	.	PROPN
ejpam-5475	114	5	(	(	PUNCT
ejpam-5475	114	6	ii	ii	NOUN
ejpam-5475	114	7	)	)	PUNCT
ejpam-5475	114	8	x	x	PRON
ejpam-5475	114	9	−mν1	−mν1	VERB
ejpam-5475	114	10	⊆	⊆	NUM
ejpam-5475	114	11	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	114	12	)	)	PUNCT
ejpam-5475	114	13	.	.	PUNCT
ejpam-5475	115	1	proof	proof	NOUN
ejpam-5475	115	2	.	.	PUNCT
ejpam-5475	116	1	(	(	PUNCT
ejpam-5475	116	2	i	i	NOUN
ejpam-5475	116	3	)	)	PUNCT
ejpam-5475	116	4	let	let	VERB
ejpam-5475	116	5	a	a	DET
ejpam-5475	116	6	⊆	⊆	NUM
ejpam-5475	116	7	x	x	SYM
ejpam-5475	116	8	−	−	NOUN
ejpam-5475	116	9	mν1	mν1	ADJ
ejpam-5475	116	10	and	and	CCONJ
ejpam-5475	116	11	x	x	SYM
ejpam-5475	116	12	∈	∈	NOUN
ejpam-5475	116	13	x	x	X
ejpam-5475	116	14	−	−	NOUN
ejpam-5475	116	15	mν1	mν1	X
ejpam-5475	116	16	.	.	PUNCT
ejpam-5475	117	1	by	by	ADP
ejpam-5475	117	2	definition	definition	NOUN
ejpam-5475	117	3	4	4	NUM
ejpam-5475	117	4	,	,	PUNCT
ejpam-5475	117	5	x	x	X
ejpam-5475	117	6	∈	∈	NOUN
ejpam-5475	117	7	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	117	8	)	)	PUNCT
ejpam-5475	117	9	implies	imply	VERB
ejpam-5475	117	10	x	x	NOUN
ejpam-5475	117	11	−mν1	−mν1	ADP
ejpam-5475	117	12	⊆	⊆	NUM
ejpam-5475	117	13	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	117	14	)	)	PUNCT
ejpam-5475	117	15	.	.	PUNCT
ejpam-5475	118	1	conversely	conversely	ADV
ejpam-5475	118	2	,	,	PUNCT
ejpam-5475	118	3	if	if	SCONJ
ejpam-5475	118	4	x	x	SYM
ejpam-5475	118	5	∈	∈	NOUN
ejpam-5475	118	6	mν1	mν1	NOUN
ejpam-5475	118	7	,	,	PUNCT
ejpam-5475	118	8	then	then	ADV
ejpam-5475	118	9	for	for	ADP
ejpam-5475	118	10	any	any	DET
ejpam-5475	118	11	m	m	NOUN
ejpam-5475	118	12	∈	∈	NOUN
ejpam-5475	118	13	ν1	ν1	NOUN
ejpam-5475	118	14	containing	contain	VERB
ejpam-5475	118	15	x	x	SYM
ejpam-5475	118	16	,	,	PUNCT
ejpam-5475	118	17	(	(	PUNCT
ejpam-5475	118	18	cν2(m	cν2(m	PROPN
ejpam-5475	118	19	)	)	PUNCT
ejpam-5475	118	20	∩mν1	∩mν1	ADJ
ejpam-5475	118	21	)	)	PUNCT
ejpam-5475	118	22	∩	∩	NOUN
ejpam-5475	118	23	a	a	DET
ejpam-5475	118	24	=	=	SYM
ejpam-5475	118	25	∅	∅	NOUN
ejpam-5475	118	26	,	,	PUNCT
ejpam-5475	118	27	hence	hence	ADV
ejpam-5475	118	28	x	x	X
ejpam-5475	118	29	/∈	/∈	PUNCT
ejpam-5475	118	30	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	118	31	)	)	PUNCT
ejpam-5475	118	32	.	.	PUNCT
ejpam-5475	119	1	this	this	PRON
ejpam-5475	119	2	implies	imply	VERB
ejpam-5475	119	3	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	119	4	)	)	PUNCT
ejpam-5475	120	1	=	=	PUNCT
ejpam-5475	121	1	x	x	PUNCT
ejpam-5475	121	2	\mν1	\mν1	ADV
ejpam-5475	121	3	.	.	PUNCT
ejpam-5475	122	1	(	(	PUNCT
ejpam-5475	122	2	ii	ii	X
ejpam-5475	122	3	)	)	PUNCT
ejpam-5475	122	4	this	this	PRON
ejpam-5475	122	5	follows	follow	VERB
ejpam-5475	122	6	directly	directly	ADV
ejpam-5475	122	7	from	from	ADP
ejpam-5475	122	8	the	the	DET
ejpam-5475	122	9	definition	definition	NOUN
ejpam-5475	122	10	of	of	ADP
ejpam-5475	122	11	the	the	DET
ejpam-5475	122	12	operation	operation	NOUN
ejpam-5475	122	13	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	122	14	)	)	PUNCT
ejpam-5475	122	15	.	.	PUNCT
ejpam-5475	123	1	theorem	theorem	NOUN
ejpam-5475	123	2	5	5	NUM
ejpam-5475	123	3	.	.	PUNCT
ejpam-5475	124	1	let	let	VERB
ejpam-5475	124	2	ν1	ν1	NOUN
ejpam-5475	124	3	and	and	CCONJ
ejpam-5475	124	4	ν2	ν2	NOUN
ejpam-5475	124	5	be	be	AUX
ejpam-5475	124	6	two	two	NUM
ejpam-5475	124	7	gt	gt	NOUN
ejpam-5475	124	8	’s	’s	NOUN
ejpam-5475	124	9	on	on	ADP
ejpam-5475	124	10	a	a	DET
ejpam-5475	124	11	nonempty	nonempty	ADV
ejpam-5475	124	12	set	set	VERB
ejpam-5475	124	13	x	x	PUNCT
ejpam-5475	124	14	and	and	CCONJ
ejpam-5475	124	15	a	a	DET
ejpam-5475	124	16	⊆	⊆	NUM
ejpam-5475	124	17	x.	x.	NOUN
ejpam-5475	124	18	then	then	ADV
ejpam-5475	124	19	the	the	DET
ejpam-5475	124	20	following	follow	VERB
ejpam-5475	124	21	hold	hold	NOUN
ejpam-5475	124	22	.	.	PUNCT
ejpam-5475	125	1	(	(	PUNCT
ejpam-5475	125	2	i	i	NOUN
ejpam-5475	125	3	)	)	PUNCT
ejpam-5475	125	4	if	if	SCONJ
ejpam-5475	125	5	a	a	DET
ejpam-5475	125	6	∈	∈	PROPN
ejpam-5475	125	7	ν1	ν1	NOUN
ejpam-5475	125	8	,	,	PUNCT
ejpam-5475	125	9	then	then	ADV
ejpam-5475	125	10	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	125	11	)	)	PUNCT
ejpam-5475	125	12	=	=	SYM
ejpam-5475	125	13	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	125	14	)	)	PUNCT
ejpam-5475	125	15	.	.	PUNCT
ejpam-5475	126	1	(	(	PUNCT
ejpam-5475	126	2	ii	ii	NOUN
ejpam-5475	126	3	)	)	PUNCT
ejpam-5475	126	4	if	if	SCONJ
ejpam-5475	126	5	a	a	DET
ejpam-5475	126	6	∈	∈	PROPN
ejpam-5475	126	7	ν2	ν2	NOUN
ejpam-5475	126	8	,	,	PUNCT
ejpam-5475	126	9	then	then	ADV
ejpam-5475	126	10	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	126	11	)	)	PUNCT
ejpam-5475	126	12	=	=	SYM
ejpam-5475	126	13	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	126	14	)	)	PUNCT
ejpam-5475	126	15	.	.	PUNCT
ejpam-5475	127	1	proof	proof	NOUN
ejpam-5475	127	2	.	.	PUNCT
ejpam-5475	128	1	(	(	PUNCT
ejpam-5475	128	2	i	i	NOUN
ejpam-5475	128	3	)	)	PUNCT
ejpam-5475	128	4	this	this	PRON
ejpam-5475	128	5	follows	follow	VERB
ejpam-5475	128	6	directly	directly	ADV
ejpam-5475	128	7	from	from	ADP
ejpam-5475	128	8	definition	definition	NOUN
ejpam-5475	128	9	4	4	NUM
ejpam-5475	128	10	.	.	PUNCT
ejpam-5475	128	11	(	(	PUNCT
ejpam-5475	128	12	ii	ii	NOUN
ejpam-5475	128	13	)	)	PUNCT
ejpam-5475	128	14	by	by	ADP
ejpam-5475	128	15	theorem	theorem	NOUN
ejpam-5475	128	16	2	2	NUM
ejpam-5475	128	17	,	,	PUNCT
ejpam-5475	128	18	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	128	19	)	)	PUNCT
ejpam-5475	128	20	⊆	⊆	NUM
ejpam-5475	128	21	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	NOUN
ejpam-5475	128	22	)	)	PUNCT
ejpam-5475	128	23	.	.	PUNCT
ejpam-5475	129	1	for	for	ADP
ejpam-5475	129	2	the	the	DET
ejpam-5475	129	3	converse	converse	NOUN
ejpam-5475	129	4	inclusion	inclusion	NOUN
ejpam-5475	129	5	,	,	PUNCT
ejpam-5475	129	6	let	let	VERB
ejpam-5475	129	7	x	x	X
ejpam-5475	129	8	∈	∈	PROPN
ejpam-5475	129	9	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	129	10	)	)	PUNCT
ejpam-5475	129	11	and	and	CCONJ
ejpam-5475	129	12	m	m	PROPN
ejpam-5475	129	13	∈	∈	NOUN
ejpam-5475	129	14	ν1	ν1	NOUN
ejpam-5475	129	15	such	such	ADJ
ejpam-5475	129	16	that	that	SCONJ
ejpam-5475	129	17	x	x	SYM
ejpam-5475	129	18	∈	∈	NOUN
ejpam-5475	129	19	m	m	VERB
ejpam-5475	129	20	.	.	PUNCT
ejpam-5475	130	1	then	then	ADV
ejpam-5475	130	2	cν2(m)∩a	cν2(m)∩a	VERB
ejpam-5475	130	3	̸=	̸=	PROPN
ejpam-5475	130	4	∅.	∅.	PRON
ejpam-5475	130	5	hence	hence	ADV
ejpam-5475	130	6	,	,	PUNCT
ejpam-5475	130	7	there	there	PRON
ejpam-5475	130	8	exists	exist	VERB
ejpam-5475	130	9	z	z	NOUN
ejpam-5475	130	10	∈	∈	PROPN
ejpam-5475	130	11	cν2(m)∩a	cν2(m)∩a	NOUN
ejpam-5475	130	12	.	.	PUNCT
ejpam-5475	131	1	since	since	SCONJ
ejpam-5475	131	2	a	a	PRON
ejpam-5475	131	3	is	be	AUX
ejpam-5475	131	4	a	a	DET
ejpam-5475	131	5	ν2	ν2	NOUN
ejpam-5475	131	6	-	-	PUNCT
ejpam-5475	131	7	open	open	ADJ
ejpam-5475	131	8	set	set	NOUN
ejpam-5475	131	9	containing	contain	VERB
ejpam-5475	131	10	z	z	PROPN
ejpam-5475	131	11	,	,	PUNCT
ejpam-5475	131	12	it	it	PRON
ejpam-5475	131	13	follows	follow	VERB
ejpam-5475	131	14	that	that	SCONJ
ejpam-5475	131	15	m	m	VERB
ejpam-5475	131	16	∩a	∩a	PROPN
ejpam-5475	131	17	̸=	̸=	PROPN
ejpam-5475	131	18	∅.	∅.	ADV
ejpam-5475	131	19	as	as	ADP
ejpam-5475	131	20	m	m	NOUN
ejpam-5475	131	21	∩a	∩a	NOUN
ejpam-5475	131	22	=	=	PUNCT
ejpam-5475	131	23	(	(	PUNCT
ejpam-5475	131	24	m	m	INTJ
ejpam-5475	131	25	∩mν1)∩a	∩mν1)∩a	X
ejpam-5475	131	26	,	,	PUNCT
ejpam-5475	131	27	we	we	PRON
ejpam-5475	131	28	have	have	VERB
ejpam-5475	131	29	(	(	PUNCT
ejpam-5475	131	30	m	m	NOUN
ejpam-5475	131	31	∩mν1)∩a	∩mν1)∩a	X
ejpam-5475	131	32	̸=	̸=	PROPN
ejpam-5475	131	33	∅.	∅.	ADP
ejpam-5475	131	34	thus	thus	ADV
ejpam-5475	131	35	,	,	PUNCT
ejpam-5475	131	36	(	(	PUNCT
ejpam-5475	131	37	cν2(m	cν2(m	NOUN
ejpam-5475	131	38	)	)	PUNCT
ejpam-5475	131	39	∩	∩	ADJ
ejpam-5475	131	40	mν1	mν1	NOUN
ejpam-5475	131	41	)	)	PUNCT
ejpam-5475	131	42	∩	∩	NOUN
ejpam-5475	131	43	a	a	DET
ejpam-5475	131	44	̸=	̸=	PROPN
ejpam-5475	131	45	∅.	∅.	ADP
ejpam-5475	131	46	this	this	DET
ejpam-5475	131	47	implies	imply	VERB
ejpam-5475	131	48	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	NOUN
ejpam-5475	131	49	)	)	PUNCT
ejpam-5475	132	1	⊆	⊆	NUM
ejpam-5475	132	2	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	132	3	)	)	PUNCT
ejpam-5475	132	4	.	.	PUNCT
ejpam-5475	133	1	finally	finally	ADV
ejpam-5475	133	2	,	,	PUNCT
ejpam-5475	133	3	we	we	PRON
ejpam-5475	133	4	conclude	conclude	VERB
ejpam-5475	133	5	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	133	6	)	)	PUNCT
ejpam-5475	134	1	=	=	SYM
ejpam-5475	134	2	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	PROPN
ejpam-5475	134	3	)	)	PUNCT
ejpam-5475	134	4	.	.	PUNCT
ejpam-5475	135	1	based	base	VERB
ejpam-5475	135	2	on	on	ADP
ejpam-5475	135	3	lemma	lemma	PROPN
ejpam-5475	135	4	1	1	NUM
ejpam-5475	135	5	and	and	CCONJ
ejpam-5475	135	6	the	the	DET
ejpam-5475	135	7	implication	implication	NOUN
ejpam-5475	135	8	(	(	PUNCT
ejpam-5475	135	9	ii	ii	NOUN
ejpam-5475	135	10	)	)	PUNCT
ejpam-5475	135	11	from	from	ADP
ejpam-5475	135	12	theorem	theorem	ADJ
ejpam-5475	135	13	5	5	NUM
ejpam-5475	135	14	above	above	ADV
ejpam-5475	135	15	,	,	PUNCT
ejpam-5475	135	16	we	we	PRON
ejpam-5475	135	17	derive	derive	VERB
ejpam-5475	135	18	the	the	DET
ejpam-5475	135	19	following	follow	VERB
ejpam-5475	135	20	corollary	corollary	NOUN
ejpam-5475	135	21	.	.	PUNCT
ejpam-5475	136	1	corollary	corollary	ADJ
ejpam-5475	136	2	2	2	NUM
ejpam-5475	136	3	.	.	PUNCT
ejpam-5475	137	1	let	let	VERB
ejpam-5475	137	2	ν1	ν1	NOUN
ejpam-5475	137	3	and	and	CCONJ
ejpam-5475	137	4	ν2	ν2	NOUN
ejpam-5475	137	5	be	be	AUX
ejpam-5475	137	6	two	two	NUM
ejpam-5475	137	7	gt	gt	NOUN
ejpam-5475	137	8	’s	’s	NOUN
ejpam-5475	137	9	on	on	ADP
ejpam-5475	137	10	a	a	DET
ejpam-5475	137	11	nonempty	nonempty	ADJ
ejpam-5475	137	12	set	set	VERB
ejpam-5475	137	13	.	.	PUNCT
ejpam-5475	138	1	if	if	SCONJ
ejpam-5475	138	2	a	a	DET
ejpam-5475	138	3	∈	∈	PROPN
ejpam-5475	138	4	ν2	ν2	NOUN
ejpam-5475	138	5	,	,	PUNCT
ejpam-5475	138	6	then	then	ADV
ejpam-5475	138	7	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	138	8	)	)	PUNCT
ejpam-5475	138	9	=	=	SYM
ejpam-5475	138	10	γθ(ν1,ν2)(a	γθ(ν1,ν2)(a	NOUN
ejpam-5475	138	11	)	)	PUNCT
ejpam-5475	138	12	=	=	SYM
ejpam-5475	138	13	cν1(a	cν1(a	NOUN
ejpam-5475	138	14	)	)	PUNCT
ejpam-5475	138	15	.	.	PUNCT
ejpam-5475	139	1	theorem	theorem	VERB
ejpam-5475	139	2	6	6	NUM
ejpam-5475	139	3	.	.	PUNCT
ejpam-5475	140	1	let	let	VERB
ejpam-5475	140	2	ν1	ν1	NOUN
ejpam-5475	140	3	and	and	CCONJ
ejpam-5475	140	4	ν2	ν2	NOUN
ejpam-5475	140	5	be	be	AUX
ejpam-5475	140	6	two	two	NUM
ejpam-5475	140	7	gt	gt	NOUN
ejpam-5475	140	8	’s	’s	NOUN
ejpam-5475	140	9	on	on	ADP
ejpam-5475	140	10	a	a	DET
ejpam-5475	140	11	nonempty	nonempty	ADV
ejpam-5475	140	12	set	set	VERB
ejpam-5475	140	13	x	x	PUNCT
ejpam-5475	140	14	and	and	CCONJ
ejpam-5475	140	15	a	a	DET
ejpam-5475	140	16	⊆	⊆	NUM
ejpam-5475	140	17	x.	x.	NOUN
ejpam-5475	140	18	then	then	ADV
ejpam-5475	140	19	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	140	20	)	)	PUNCT
ejpam-5475	141	1	=	=	SYM
ejpam-5475	141	2	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	141	3	∩mν1	∩mν1	ADJ
ejpam-5475	141	4	)	)	PUNCT
ejpam-5475	141	5	proof	proof	NOUN
ejpam-5475	141	6	.	.	PUNCT
ejpam-5475	142	1	since	since	SCONJ
ejpam-5475	142	2	a∩mν1	a∩mν1	PROPN
ejpam-5475	142	3	⊆	⊆	NUM
ejpam-5475	142	4	a	a	PRON
ejpam-5475	142	5	,	,	PUNCT
ejpam-5475	142	6	by	by	ADP
ejpam-5475	142	7	theorem	theorem	NOUN
ejpam-5475	142	8	3(i	3(i	NUM
ejpam-5475	142	9	)	)	PUNCT
ejpam-5475	142	10	,	,	PUNCT
ejpam-5475	142	11	we	we	PRON
ejpam-5475	142	12	have	have	VERB
ejpam-5475	142	13	γθ̃(ν1,ν2)(a∩mν1	γθ̃(ν1,ν2)(a∩mν1	PRON
ejpam-5475	142	14	)	)	PUNCT
ejpam-5475	142	15	⊆	⊆	NUM
ejpam-5475	142	16	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	142	17	)	)	PUNCT
ejpam-5475	142	18	.	.	PUNCT
ejpam-5475	143	1	for	for	ADP
ejpam-5475	143	2	the	the	DET
ejpam-5475	143	3	converse	converse	NOUN
ejpam-5475	143	4	inclusion	inclusion	NOUN
ejpam-5475	143	5	,	,	PUNCT
ejpam-5475	143	6	suppose	suppose	VERB
ejpam-5475	143	7	x	x	X
ejpam-5475	143	8	∈	∈	PROPN
ejpam-5475	143	9	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	143	10	)	)	PUNCT
ejpam-5475	143	11	and	and	CCONJ
ejpam-5475	143	12	m	m	PROPN
ejpam-5475	143	13	∈	∈	NOUN
ejpam-5475	143	14	ν1	ν1	NOUN
ejpam-5475	143	15	contains	contain	VERB
ejpam-5475	143	16	x.	x.	NOUN
ejpam-5475	143	17	then	then	ADV
ejpam-5475	143	18	(	(	PUNCT
ejpam-5475	143	19	cν2(m	cν2(m	PROPN
ejpam-5475	143	20	)	)	PUNCT
ejpam-5475	143	21	∩mν1	∩mν1	ADV
ejpam-5475	143	22	)	)	PUNCT
ejpam-5475	144	1	∩a	∩a	PROPN
ejpam-5475	144	2	̸=	̸=	PROPN
ejpam-5475	144	3	∅.	∅.	ADV
ejpam-5475	144	4	by	by	ADP
ejpam-5475	144	5	the	the	DET
ejpam-5475	144	6	equality	equality	NOUN
ejpam-5475	144	7	:(	:(	PUNCT
ejpam-5475	144	8	cν2(m	cν2(m	PROPN
ejpam-5475	144	9	)	)	PUNCT
ejpam-5475	144	10	∩mν1	∩mν1	ADV
ejpam-5475	144	11	)	)	PUNCT
ejpam-5475	145	1	∩a	∩a	NOUN
ejpam-5475	145	2	=	=	PUNCT
ejpam-5475	145	3	(	(	PUNCT
ejpam-5475	145	4	cν2(m	cν2(m	PROPN
ejpam-5475	145	5	)	)	PUNCT
ejpam-5475	145	6	∩mν1	∩mν1	ADJ
ejpam-5475	145	7	)	)	PUNCT
ejpam-5475	145	8	∩	∩	NOUN
ejpam-5475	145	9	[	[	PUNCT
ejpam-5475	145	10	(	(	PUNCT
ejpam-5475	145	11	a	a	DET
ejpam-5475	145	12	∩mν1	∩mν1	ADJ
ejpam-5475	145	13	)	)	PUNCT
ejpam-5475	145	14	∪	∪	NOUN
ejpam-5475	145	15	(	(	PUNCT
ejpam-5475	145	16	a	a	DET
ejpam-5475	145	17	∩	∩	NOUN
ejpam-5475	145	18	(	(	PUNCT
ejpam-5475	145	19	x	x	SYM
ejpam-5475	145	20	−mν1	−mν1	ADP
ejpam-5475	145	21	)	)	PUNCT
ejpam-5475	145	22	)	)	PUNCT
ejpam-5475	145	23	]	]	PUNCT
ejpam-5475	145	24	,	,	PUNCT
ejpam-5475	145	25	it	it	PRON
ejpam-5475	145	26	follows	follow	VERB
ejpam-5475	145	27	that	that	SCONJ
ejpam-5475	145	28	(	(	PUNCT
ejpam-5475	145	29	cν2(m)∩mν1	cν2(m)∩mν1	ADV
ejpam-5475	145	30	)	)	PUNCT
ejpam-5475	145	31	∩	∩	NOUN
ejpam-5475	145	32	(	(	PUNCT
ejpam-5475	145	33	a∩mν1	a∩mν1	ADJ
ejpam-5475	145	34	)	)	PUNCT
ejpam-5475	145	35	̸=	̸=	PROPN
ejpam-5475	145	36	∅.	∅.	PRON
ejpam-5475	145	37	hence	hence	ADV
ejpam-5475	145	38	,	,	PUNCT
ejpam-5475	145	39	x	x	PUNCT
ejpam-5475	145	40	∈	∈	PROPN
ejpam-5475	145	41	γθ̃(ν1,ν2)(a∩mν1	γθ̃(ν1,ν2)(a∩mν1	NOUN
ejpam-5475	145	42	)	)	PUNCT
ejpam-5475	145	43	,	,	PUNCT
ejpam-5475	145	44	implying	imply	VERB
ejpam-5475	145	45	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	145	46	)	)	PUNCT
ejpam-5475	146	1	⊆	⊆	NUM
ejpam-5475	146	2	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	146	3	∩mν1	∩mν1	ADJ
ejpam-5475	146	4	)	)	PUNCT
ejpam-5475	146	5	.	.	PUNCT
ejpam-5475	147	1	therefore	therefore	ADV
ejpam-5475	147	2	,	,	PUNCT
ejpam-5475	147	3	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	147	4	)	)	PUNCT
ejpam-5475	147	5	=	=	SYM
ejpam-5475	147	6	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	147	7	∩mν1	∩mν1	NOUN
ejpam-5475	147	8	)	)	PUNCT
ejpam-5475	147	9	,	,	PUNCT
ejpam-5475	147	10	completing	complete	VERB
ejpam-5475	147	11	the	the	DET
ejpam-5475	147	12	proof	proof	NOUN
ejpam-5475	147	13	.	.	PUNCT
ejpam-5475	148	1	a.	a.	NOUN
ejpam-5475	148	2	qahis	qahis	PROPN
ejpam-5475	148	3	,	,	PUNCT
ejpam-5475	148	4	a.	a.	NOUN
ejpam-5475	148	5	alqahtani	alqahtani	PROPN
ejpam-5475	148	6	/	/	SYM
ejpam-5475	148	7	eur	eur	PROPN
ejpam-5475	148	8	.	.	PUNCT
ejpam-5475	149	1	j.	j.	PROPN
ejpam-5475	149	2	pure	pure	PROPN
ejpam-5475	149	3	appl	appl	PROPN
ejpam-5475	149	4	.	.	PROPN
ejpam-5475	149	5	math	math	PROPN
ejpam-5475	149	6	,	,	PUNCT
ejpam-5475	149	7	17	17	NUM
ejpam-5475	149	8	(	(	PUNCT
ejpam-5475	149	9	4	4	NUM
ejpam-5475	149	10	)	)	PUNCT
ejpam-5475	149	11	(	(	PUNCT
ejpam-5475	149	12	2024	2024	NUM
ejpam-5475	149	13	)	)	PUNCT
ejpam-5475	149	14	,	,	PUNCT
ejpam-5475	149	15	3610	3610	NUM
ejpam-5475	149	16	-	-	SYM
ejpam-5475	149	17	3621	3621	NUM
ejpam-5475	149	18	3615	3615	NUM
ejpam-5475	149	19	definition	definition	NOUN
ejpam-5475	149	20	5	5	NUM
ejpam-5475	149	21	.	.	PUNCT
ejpam-5475	150	1	let	let	VERB
ejpam-5475	150	2	ν1	ν1	NOUN
ejpam-5475	150	3	and	and	CCONJ
ejpam-5475	150	4	ν2	ν2	NOUN
ejpam-5475	150	5	be	be	AUX
ejpam-5475	150	6	two	two	NUM
ejpam-5475	150	7	gts	gts	NOUN
ejpam-5475	150	8	defined	define	VERB
ejpam-5475	150	9	on	on	ADP
ejpam-5475	150	10	a	a	DET
ejpam-5475	150	11	nonempty	nonempty	ADV
ejpam-5475	150	12	set	set	VERB
ejpam-5475	150	13	x	x	NOUN
ejpam-5475	150	14	,	,	PUNCT
ejpam-5475	150	15	and	and	CCONJ
ejpam-5475	150	16	let	let	VERB
ejpam-5475	150	17	a	a	DET
ejpam-5475	150	18	⊆	⊆	NUM
ejpam-5475	150	19	mν1	mν1	NOUN
ejpam-5475	150	20	.	.	PUNCT
ejpam-5475	151	1	we	we	PRON
ejpam-5475	151	2	define	define	VERB
ejpam-5475	151	3	the	the	DET
ejpam-5475	151	4	restriction	restriction	NOUN
ejpam-5475	151	5	operation	operation	NOUN
ejpam-5475	151	6	with	with	ADP
ejpam-5475	151	7	respect	respect	NOUN
ejpam-5475	151	8	to	to	ADP
ejpam-5475	151	9	mν1	mν1	PROPN
ejpam-5475	151	10	as	as	SCONJ
ejpam-5475	151	11	follows	follow	VERB
ejpam-5475	151	12	:	:	PUNCT
ejpam-5475	151	13	(	(	PUNCT
ejpam-5475	151	14	γ	γ	PROPN
ejpam-5475	151	15	|mν1	|mν1	NOUN
ejpam-5475	151	16	)	)	PUNCT
ejpam-5475	151	17	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	PUNCT
ejpam-5475	151	18	)	)	PUNCT
ejpam-5475	152	1	=	=	PRON
ejpam-5475	152	2	{	{	PUNCT
ejpam-5475	152	3	x	x	PUNCT
ejpam-5475	152	4	∈	∈	PROPN
ejpam-5475	152	5	mν1	mν1	X
ejpam-5475	152	6	:	:	PUNCT
ejpam-5475	152	7	cν2(m	cν2(m	X
ejpam-5475	152	8	)	)	PUNCT
ejpam-5475	152	9	∩a	∩a	PROPN
ejpam-5475	152	10	̸=	̸=	PROPN
ejpam-5475	152	11	∅	∅	NOUN
ejpam-5475	152	12	,	,	PUNCT
ejpam-5475	152	13	∀m	∀m	PROPN
ejpam-5475	152	14	∈	∈	PROPN
ejpam-5475	152	15	ν1	ν1	NOUN
ejpam-5475	152	16	,	,	PUNCT
ejpam-5475	152	17	x	x	X
ejpam-5475	152	18	∈	∈	PROPN
ejpam-5475	152	19	m	m	NOUN
ejpam-5475	152	20	}	}	PUNCT
ejpam-5475	152	21	.	.	PUNCT
ejpam-5475	153	1	the	the	DET
ejpam-5475	153	2	following	follow	VERB
ejpam-5475	153	3	lemma	lemma	PROPN
ejpam-5475	153	4	is	be	AUX
ejpam-5475	153	5	crucial	crucial	ADJ
ejpam-5475	153	6	for	for	ADP
ejpam-5475	153	7	proving	prove	VERB
ejpam-5475	153	8	the	the	DET
ejpam-5475	153	9	next	next	ADJ
ejpam-5475	153	10	theorem	theorem	NOUN
ejpam-5475	153	11	.	.	PUNCT
ejpam-5475	154	1	lemma	lemma	PROPN
ejpam-5475	154	2	2	2	X
ejpam-5475	154	3	.	.	PUNCT
ejpam-5475	154	4	let	let	VERB
ejpam-5475	154	5	ν1	ν1	NOUN
ejpam-5475	154	6	and	and	CCONJ
ejpam-5475	154	7	ν2	ν2	NOUN
ejpam-5475	154	8	be	be	AUX
ejpam-5475	154	9	two	two	NUM
ejpam-5475	154	10	gt	gt	NOUN
ejpam-5475	154	11	’s	’s	NOUN
ejpam-5475	154	12	on	on	ADP
ejpam-5475	154	13	a	a	DET
ejpam-5475	154	14	nonempty	nonempty	ADV
ejpam-5475	154	15	set	set	VERB
ejpam-5475	154	16	x	x	PUNCT
ejpam-5475	154	17	and	and	CCONJ
ejpam-5475	154	18	a	a	DET
ejpam-5475	154	19	⊆	⊆	NUM
ejpam-5475	154	20	x.	x.	NOUN
ejpam-5475	154	21	then	then	ADV
ejpam-5475	154	22	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	154	23	)	)	PUNCT
ejpam-5475	155	1	=	=	PRON
ejpam-5475	155	2	(	(	PUNCT
ejpam-5475	155	3	x	x	SYM
ejpam-5475	155	4	−mν1	−mν1	ADP
ejpam-5475	155	5	)	)	PUNCT
ejpam-5475	155	6	∪	∪	NOUN
ejpam-5475	155	7	(	(	PUNCT
ejpam-5475	155	8	γ	γ	PROPN
ejpam-5475	155	9	|mν1	|mν1	NOUN
ejpam-5475	155	10	)	)	PUNCT
ejpam-5475	155	11	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	ADV
ejpam-5475	155	12	∩mν1	∩mν1	ADJ
ejpam-5475	155	13	)	)	PUNCT
ejpam-5475	155	14	.	.	PUNCT
ejpam-5475	156	1	proof	proof	NOUN
ejpam-5475	156	2	.	.	PUNCT
ejpam-5475	157	1	let	let	VERB
ejpam-5475	157	2	x	x	PUNCT
ejpam-5475	157	3	∈	∈	PROPN
ejpam-5475	157	4	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	157	5	)	)	PUNCT
ejpam-5475	157	6	and	and	CCONJ
ejpam-5475	157	7	m	m	PROPN
ejpam-5475	157	8	∈	∈	NOUN
ejpam-5475	157	9	ν1	ν1	NOUN
ejpam-5475	157	10	such	such	ADJ
ejpam-5475	157	11	that	that	SCONJ
ejpam-5475	157	12	x	x	SYM
ejpam-5475	157	13	∈	∈	NOUN
ejpam-5475	157	14	m	m	VERB
ejpam-5475	157	15	.	.	PUNCT
ejpam-5475	158	1	by	by	ADP
ejpam-5475	158	2	the	the	DET
ejpam-5475	158	3	definition	definition	NOUN
ejpam-5475	158	4	of	of	ADP
ejpam-5475	158	5	γθ̃(ν1,ν2	γθ̃(ν1,ν2	ADJ
ejpam-5475	158	6	)	)	PUNCT
ejpam-5475	158	7	,	,	PUNCT
ejpam-5475	158	8	(	(	PUNCT
ejpam-5475	158	9	cν2(m	cν2(m	PROPN
ejpam-5475	158	10	)	)	PUNCT
ejpam-5475	158	11	∩mν1	∩mν1	ADV
ejpam-5475	158	12	)	)	PUNCT
ejpam-5475	159	1	∩a	∩a	PROPN
ejpam-5475	159	2	̸=	̸=	PROPN
ejpam-5475	159	3	∅	∅	NOUN
ejpam-5475	159	4	and	and	CCONJ
ejpam-5475	159	5	(	(	PUNCT
ejpam-5475	159	6	cν2(m	cν2(m	PROPN
ejpam-5475	159	7	)	)	PUNCT
ejpam-5475	159	8	∩mν1	∩mν1	ADV
ejpam-5475	159	9	)	)	PUNCT
ejpam-5475	160	1	∩a	∩a	NOUN
ejpam-5475	160	2	=	=	PUNCT
ejpam-5475	160	3	cν2(m	cν2(m	PROPN
ejpam-5475	160	4	)	)	PUNCT
ejpam-5475	161	1	∩	∩	NOUN
ejpam-5475	161	2	(	(	PUNCT
ejpam-5475	161	3	mν1	mν1	X
ejpam-5475	161	4	∩a	∩a	NOUN
ejpam-5475	161	5	)	)	PUNCT
ejpam-5475	161	6	.	.	PUNCT
ejpam-5475	162	1	since	since	SCONJ
ejpam-5475	162	2	mν1	mν1	ADJ
ejpam-5475	162	3	∩a	∩a	PROPN
ejpam-5475	162	4	⊆	⊆	NUM
ejpam-5475	162	5	mν1	mν1	X
ejpam-5475	162	6	,	,	PUNCT
ejpam-5475	162	7	by	by	ADP
ejpam-5475	162	8	definition	definition	NOUN
ejpam-5475	162	9	5	5	NUM
ejpam-5475	162	10	,	,	PUNCT
ejpam-5475	162	11	x	x	SYM
ejpam-5475	162	12	∈	∈	PROPN
ejpam-5475	162	13	(	(	PUNCT
ejpam-5475	162	14	γ	γ	NOUN
ejpam-5475	162	15	|mν1	|mν1	NOUN
ejpam-5475	162	16	)	)	PUNCT
ejpam-5475	162	17	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	ADV
ejpam-5475	162	18	∩mν1	∩mν1	ADJ
ejpam-5475	162	19	)	)	PUNCT
ejpam-5475	162	20	,	,	PUNCT
ejpam-5475	162	21	hence	hence	ADV
ejpam-5475	162	22	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	162	23	)	)	PUNCT
ejpam-5475	163	1	⊆	⊆	NUM
ejpam-5475	163	2	(	(	PUNCT
ejpam-5475	163	3	γ	γ	NOUN
ejpam-5475	163	4	|mν1	|mν1	NOUN
ejpam-5475	163	5	)	)	PUNCT
ejpam-5475	163	6	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	ADV
ejpam-5475	163	7	∩mν1	∩mν1	ADJ
ejpam-5475	163	8	)	)	PUNCT
ejpam-5475	163	9	.	.	PUNCT
ejpam-5475	164	1	obviously	obviously	ADV
ejpam-5475	164	2	,	,	PUNCT
ejpam-5475	164	3	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	164	4	)	)	PUNCT
ejpam-5475	165	1	⊆	⊆	NUM
ejpam-5475	165	2	(	(	PUNCT
ejpam-5475	165	3	x	x	NOUN
ejpam-5475	165	4	−mν1	−mν1	ADP
ejpam-5475	165	5	)	)	PUNCT
ejpam-5475	165	6	∪	∪	NOUN
ejpam-5475	165	7	(	(	PUNCT
ejpam-5475	165	8	γ	γ	PROPN
ejpam-5475	165	9	|mν1	|mν1	NOUN
ejpam-5475	165	10	)	)	PUNCT
ejpam-5475	165	11	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	ADV
ejpam-5475	165	12	∩mν1	∩mν1	ADJ
ejpam-5475	165	13	)	)	PUNCT
ejpam-5475	165	14	.	.	PUNCT
ejpam-5475	166	1	(	(	PUNCT
ejpam-5475	166	2	1	1	X
ejpam-5475	166	3	)	)	PUNCT
ejpam-5475	166	4	for	for	ADP
ejpam-5475	166	5	the	the	DET
ejpam-5475	166	6	other	other	ADJ
ejpam-5475	166	7	inclusion	inclusion	NOUN
ejpam-5475	166	8	,	,	PUNCT
ejpam-5475	166	9	from	from	ADP
ejpam-5475	166	10	theorem	theorem	ADJ
ejpam-5475	166	11	4(ii	4(ii	NUM
ejpam-5475	166	12	)	)	PUNCT
ejpam-5475	166	13	,	,	PUNCT
ejpam-5475	166	14	x	x	PRON
ejpam-5475	166	15	−mν1	−mν1	VERB
ejpam-5475	166	16	⊆	⊆	NUM
ejpam-5475	166	17	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	166	18	)	)	PUNCT
ejpam-5475	166	19	.	.	PUNCT
ejpam-5475	167	1	let	let	VERB
ejpam-5475	167	2	x	x	X
ejpam-5475	167	3	∈	∈	PROPN
ejpam-5475	167	4	(	(	PUNCT
ejpam-5475	167	5	γ	γ	NOUN
ejpam-5475	167	6	|mν1	|mν1	NOUN
ejpam-5475	167	7	)	)	PUNCT
ejpam-5475	167	8	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	ADV
ejpam-5475	167	9	∩mν1	∩mν1	ADJ
ejpam-5475	167	10	)	)	PUNCT
ejpam-5475	167	11	.	.	PUNCT
ejpam-5475	168	1	then	then	ADV
ejpam-5475	168	2	for	for	SCONJ
ejpam-5475	168	3	each	each	DET
ejpam-5475	168	4	ν1	ν1	NOUN
ejpam-5475	168	5	-	-	PUNCT
ejpam-5475	168	6	open	open	NOUN
ejpam-5475	168	7	set	set	NOUN
ejpam-5475	168	8	m	m	AUX
ejpam-5475	168	9	containing	contain	VERB
ejpam-5475	168	10	x	x	PROPN
ejpam-5475	168	11	,	,	PUNCT
ejpam-5475	168	12	cν2(m	cν2(m	PROPN
ejpam-5475	168	13	)	)	PUNCT
ejpam-5475	168	14	∩	∩	NOUN
ejpam-5475	168	15	(	(	PUNCT
ejpam-5475	168	16	mν1	mν1	X
ejpam-5475	168	17	∩a	∩a	PROPN
ejpam-5475	168	18	)	)	PUNCT
ejpam-5475	168	19	̸=	̸=	PROPN
ejpam-5475	168	20	∅.	∅.	ADV
ejpam-5475	168	21	since	since	SCONJ
ejpam-5475	168	22	cν2(m	cν2(m	PROPN
ejpam-5475	168	23	)	)	PUNCT
ejpam-5475	168	24	∩	∩	NOUN
ejpam-5475	168	25	(	(	PUNCT
ejpam-5475	168	26	mν1	mν1	X
ejpam-5475	168	27	∩a	∩a	PROPN
ejpam-5475	168	28	)	)	PUNCT
ejpam-5475	169	1	=	=	PRON
ejpam-5475	169	2	(	(	PUNCT
ejpam-5475	169	3	cν2(m	cν2(m	PROPN
ejpam-5475	169	4	)	)	PUNCT
ejpam-5475	169	5	∩mν1	∩mν1	ADV
ejpam-5475	169	6	)	)	PUNCT
ejpam-5475	170	1	∩a	∩a	PROPN
ejpam-5475	170	2	,	,	PUNCT
ejpam-5475	170	3	it	it	PRON
ejpam-5475	170	4	follows	follow	VERB
ejpam-5475	170	5	that	that	SCONJ
ejpam-5475	170	6	x	x	SYM
ejpam-5475	170	7	∈	∈	NOUN
ejpam-5475	170	8	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	170	9	)	)	PUNCT
ejpam-5475	170	10	and	and	CCONJ
ejpam-5475	170	11	thus	thus	ADV
ejpam-5475	170	12	(	(	PUNCT
ejpam-5475	170	13	γ	γ	PROPN
ejpam-5475	170	14	|mν1	|mν1	NOUN
ejpam-5475	170	15	)	)	PUNCT
ejpam-5475	170	16	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	NOUN
ejpam-5475	170	17	∩mν1	∩mν1	ADJ
ejpam-5475	170	18	)	)	PUNCT
ejpam-5475	170	19	⊆	⊆	NUM
ejpam-5475	170	20	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	170	21	)	)	PUNCT
ejpam-5475	170	22	.	.	PUNCT
ejpam-5475	171	1	thus	thus	ADV
ejpam-5475	171	2	,	,	PUNCT
ejpam-5475	171	3	(	(	PUNCT
ejpam-5475	171	4	x	x	SYM
ejpam-5475	171	5	−mν1	−mν1	ADP
ejpam-5475	171	6	)	)	PUNCT
ejpam-5475	171	7	∪	∪	NOUN
ejpam-5475	171	8	(	(	PUNCT
ejpam-5475	171	9	γ	γ	PROPN
ejpam-5475	171	10	|mν1	|mν1	NOUN
ejpam-5475	171	11	)	)	PUNCT
ejpam-5475	171	12	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	NOUN
ejpam-5475	171	13	∩mν1	∩mν1	ADJ
ejpam-5475	171	14	)	)	PUNCT
ejpam-5475	171	15	⊆	⊆	NUM
ejpam-5475	171	16	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	171	17	)	)	PUNCT
ejpam-5475	171	18	.	.	PUNCT
ejpam-5475	172	1	(	(	PUNCT
ejpam-5475	172	2	2	2	X
ejpam-5475	172	3	)	)	PUNCT
ejpam-5475	172	4	from	from	ADP
ejpam-5475	172	5	equalities	equality	NOUN
ejpam-5475	172	6	(	(	PUNCT
ejpam-5475	172	7	1	1	NUM
ejpam-5475	172	8	)	)	PUNCT
ejpam-5475	172	9	and	and	CCONJ
ejpam-5475	172	10	(	(	PUNCT
ejpam-5475	172	11	2	2	NUM
ejpam-5475	172	12	)	)	PUNCT
ejpam-5475	172	13	,	,	PUNCT
ejpam-5475	172	14	we	we	PRON
ejpam-5475	172	15	conclude	conclude	VERB
ejpam-5475	172	16	that	that	PRON
ejpam-5475	172	17	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	172	18	)	)	PUNCT
ejpam-5475	173	1	=	=	PRON
ejpam-5475	173	2	(	(	PUNCT
ejpam-5475	173	3	x	x	SYM
ejpam-5475	173	4	−mν1	−mν1	ADP
ejpam-5475	173	5	)	)	PUNCT
ejpam-5475	173	6	∪	∪	NOUN
ejpam-5475	173	7	(	(	PUNCT
ejpam-5475	173	8	γ	γ	PROPN
ejpam-5475	173	9	|mν1	|mν1	NOUN
ejpam-5475	173	10	)	)	PUNCT
ejpam-5475	173	11	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	ADV
ejpam-5475	173	12	∩mν1	∩mν1	ADJ
ejpam-5475	173	13	)	)	PUNCT
ejpam-5475	173	14	.	.	PUNCT
ejpam-5475	174	1	theorem	theorem	VERB
ejpam-5475	174	2	7	7	NUM
ejpam-5475	174	3	.	.	PUNCT
ejpam-5475	175	1	let	let	VERB
ejpam-5475	175	2	ν1	ν1	NOUN
ejpam-5475	175	3	and	and	CCONJ
ejpam-5475	175	4	ν2	ν2	NOUN
ejpam-5475	175	5	be	be	AUX
ejpam-5475	175	6	two	two	NUM
ejpam-5475	175	7	gt	gt	NOUN
ejpam-5475	175	8	’s	’s	NOUN
ejpam-5475	175	9	on	on	ADP
ejpam-5475	175	10	a	a	DET
ejpam-5475	175	11	nonempty	nonempty	ADV
ejpam-5475	175	12	set	set	VERB
ejpam-5475	175	13	x	x	NOUN
ejpam-5475	175	14	,	,	PUNCT
ejpam-5475	175	15	and	and	CCONJ
ejpam-5475	175	16	let	let	VERB
ejpam-5475	175	17	a	a	DET
ejpam-5475	175	18	⊆	⊆	NUM
ejpam-5475	175	19	x.	x.	NOUN
ejpam-5475	175	20	the	the	DET
ejpam-5475	175	21	following	follow	VERB
ejpam-5475	175	22	properties	property	NOUN
ejpam-5475	175	23	then	then	ADV
ejpam-5475	175	24	hold	hold	VERB
ejpam-5475	175	25	:	:	PUNCT
ejpam-5475	175	26	(	(	PUNCT
ejpam-5475	175	27	i	i	NOUN
ejpam-5475	175	28	)	)	PUNCT
ejpam-5475	175	29	γθ̃(ν1,ν2)(x	γθ̃(ν1,ν2)(x	PROPN
ejpam-5475	175	30	)	)	PUNCT
ejpam-5475	175	31	=	=	PUNCT
ejpam-5475	176	1	x.	x.	NOUN
ejpam-5475	176	2	a.	a.	NOUN
ejpam-5475	176	3	qahis	qahis	PROPN
ejpam-5475	176	4	,	,	PUNCT
ejpam-5475	176	5	a.	a.	NOUN
ejpam-5475	176	6	alqahtani	alqahtani	PROPN
ejpam-5475	176	7	/	/	SYM
ejpam-5475	176	8	eur	eur	PROPN
ejpam-5475	176	9	.	.	PUNCT
ejpam-5475	177	1	j.	j.	PROPN
ejpam-5475	177	2	pure	pure	PROPN
ejpam-5475	177	3	appl	appl	PROPN
ejpam-5475	177	4	.	.	PROPN
ejpam-5475	177	5	math	math	PROPN
ejpam-5475	177	6	,	,	PUNCT
ejpam-5475	177	7	17	17	NUM
ejpam-5475	177	8	(	(	PUNCT
ejpam-5475	177	9	4	4	NUM
ejpam-5475	177	10	)	)	PUNCT
ejpam-5475	177	11	(	(	PUNCT
ejpam-5475	177	12	2024	2024	NUM
ejpam-5475	177	13	)	)	PUNCT
ejpam-5475	177	14	,	,	PUNCT
ejpam-5475	177	15	3610	3610	NUM
ejpam-5475	177	16	-	-	SYM
ejpam-5475	177	17	3621	3621	NUM
ejpam-5475	177	18	3616	3616	NUM
ejpam-5475	177	19	(	(	PUNCT
ejpam-5475	177	20	ii	ii	NOUN
ejpam-5475	177	21	)	)	PUNCT
ejpam-5475	177	22	γθ̃(ν1,ν2)(x	γθ̃(ν1,ν2)(x	PROPN
ejpam-5475	177	23	−mν1	−mν1	PROPN
ejpam-5475	177	24	)	)	PUNCT
ejpam-5475	177	25	=	=	PUNCT
ejpam-5475	177	26	x	x	PUNCT
ejpam-5475	178	1	−mν1	−mν1	PROPN
ejpam-5475	178	2	.	.	PROPN
ejpam-5475	178	3	(	(	PUNCT
ejpam-5475	178	4	iii	iii	NOUN
ejpam-5475	178	5	)	)	PUNCT
ejpam-5475	178	6	γθ̃(ν1,ν2)(∅	γθ̃(ν1,ν2)(∅	NOUN
ejpam-5475	178	7	)	)	PUNCT
ejpam-5475	178	8	=	=	SYM
ejpam-5475	178	9	x	x	PUNCT
ejpam-5475	178	10	−mν1	−mν1	PROPN
ejpam-5475	178	11	.	.	PROPN
ejpam-5475	178	12	(	(	PUNCT
ejpam-5475	178	13	iv	iv	X
ejpam-5475	178	14	)	)	PUNCT
ejpam-5475	178	15	if	if	SCONJ
ejpam-5475	178	16	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	178	17	)	)	PUNCT
ejpam-5475	178	18	=	=	SYM
ejpam-5475	179	1	a	a	PRON
ejpam-5475	179	2	,	,	PUNCT
ejpam-5475	179	3	then	then	ADV
ejpam-5475	179	4	x	x	PRON
ejpam-5475	179	5	−mν1	−mν1	CCONJ
ejpam-5475	179	6	⊆	⊆	NUM
ejpam-5475	179	7	a.	a.	NOUN
ejpam-5475	179	8	proof	proof	NOUN
ejpam-5475	179	9	.	.	PUNCT
ejpam-5475	180	1	(	(	PUNCT
ejpam-5475	180	2	i	i	NOUN
ejpam-5475	180	3	)	)	PUNCT
ejpam-5475	180	4	by	by	ADP
ejpam-5475	180	5	theorem	theorem	ADJ
ejpam-5475	180	6	3(ii	3(ii	NUM
ejpam-5475	180	7	)	)	PUNCT
ejpam-5475	180	8	,	,	PUNCT
ejpam-5475	180	9	x	x	X
ejpam-5475	180	10	⊆	⊆	NUM
ejpam-5475	180	11	γθ̃(ν1,ν2)(x	γθ̃(ν1,ν2)(x	PROPN
ejpam-5475	180	12	)	)	PUNCT
ejpam-5475	180	13	,	,	PUNCT
ejpam-5475	180	14	implying	imply	VERB
ejpam-5475	180	15	γθ̃(ν1,ν2)(x	γθ̃(ν1,ν2)(x	PROPN
ejpam-5475	180	16	)	)	PUNCT
ejpam-5475	180	17	=	=	PUNCT
ejpam-5475	181	1	x.	x.	NOUN
ejpam-5475	181	2	(	(	PUNCT
ejpam-5475	181	3	ii	ii	PROPN
ejpam-5475	181	4	)	)	PUNCT
ejpam-5475	181	5	from	from	ADP
ejpam-5475	181	6	lemma	lemma	PROPN
ejpam-5475	181	7	2	2	NUM
ejpam-5475	181	8	,	,	PUNCT
ejpam-5475	181	9	we	we	PRON
ejpam-5475	181	10	obtain	obtain	VERB
ejpam-5475	181	11	γθ̃(ν1,ν2)(x	γθ̃(ν1,ν2)(x	PROPN
ejpam-5475	181	12	−mν1	−mν1	ADV
ejpam-5475	181	13	)	)	PUNCT
ejpam-5475	181	14	=	=	SYM
ejpam-5475	181	15	(	(	PUNCT
ejpam-5475	181	16	x	x	SYM
ejpam-5475	181	17	−mν1	−mν1	ADP
ejpam-5475	181	18	)	)	PUNCT
ejpam-5475	181	19	∪	∪	NOUN
ejpam-5475	181	20	(	(	PUNCT
ejpam-5475	181	21	γ	γ	PROPN
ejpam-5475	181	22	|mν1	|mν1	NOUN
ejpam-5475	181	23	)	)	PUNCT
ejpam-5475	181	24	θ̃(ν1,ν2	θ̃(ν1,ν2	NUM
ejpam-5475	181	25	)	)	PUNCT
ejpam-5475	182	1	(	(	PUNCT
ejpam-5475	182	2	(	(	PUNCT
ejpam-5475	182	3	x	x	SYM
ejpam-5475	182	4	−mν1	−mν1	ADJ
ejpam-5475	182	5	)	)	PUNCT
ejpam-5475	182	6	∩mν1	∩mν1	ADV
ejpam-5475	182	7	)	)	PUNCT
ejpam-5475	183	1	=	=	SYM
ejpam-5475	183	2	(	(	PUNCT
ejpam-5475	183	3	x	x	SYM
ejpam-5475	183	4	−mν1	−mν1	ADP
ejpam-5475	183	5	)	)	PUNCT
ejpam-5475	183	6	∪	∪	NOUN
ejpam-5475	183	7	(	(	PUNCT
ejpam-5475	183	8	γ	γ	NOUN
ejpam-5475	183	9	|mν1	|mν1	NOUN
ejpam-5475	183	10	)	)	PUNCT
ejpam-5475	183	11	θ̃(ν1,ν2)(∅	θ̃(ν1,ν2)(∅	NOUN
ejpam-5475	183	12	)	)	PUNCT
ejpam-5475	183	13	=	=	SYM
ejpam-5475	183	14	(	(	PUNCT
ejpam-5475	183	15	x	x	SYM
ejpam-5475	183	16	−mν1	−mν1	ADP
ejpam-5475	183	17	)	)	PUNCT
ejpam-5475	183	18	∪	∪	NOUN
ejpam-5475	183	19	∅	∅	NOUN
ejpam-5475	183	20	=	=	PUNCT
ejpam-5475	183	21	x	x	X
ejpam-5475	183	22	−mν1	−mν1	ADJ
ejpam-5475	183	23	.	.	PUNCT
ejpam-5475	184	1	(	(	PUNCT
ejpam-5475	184	2	iii	iii	X
ejpam-5475	184	3	)	)	PUNCT
ejpam-5475	184	4	this	this	PRON
ejpam-5475	184	5	follows	follow	VERB
ejpam-5475	184	6	directly	directly	ADV
ejpam-5475	184	7	from	from	ADP
ejpam-5475	184	8	lemma	lemma	PROPN
ejpam-5475	184	9	2	2	NUM
ejpam-5475	184	10	and	and	CCONJ
ejpam-5475	184	11	definition	definition	NOUN
ejpam-5475	184	12	5	5	NUM
ejpam-5475	184	13	.	.	PUNCT
ejpam-5475	184	14	(	(	PUNCT
ejpam-5475	184	15	iv	iv	AUX
ejpam-5475	184	16	)	)	PUNCT
ejpam-5475	184	17	let	let	VERB
ejpam-5475	184	18	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	184	19	)	)	PUNCT
ejpam-5475	185	1	=	=	PUNCT
ejpam-5475	185	2	a.	a.	NOUN
ejpam-5475	185	3	then	then	ADV
ejpam-5475	185	4	by	by	ADP
ejpam-5475	185	5	lemma	lemma	PROPN
ejpam-5475	185	6	2	2	NUM
ejpam-5475	185	7	,	,	PUNCT
ejpam-5475	185	8	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	185	9	)	)	PUNCT
ejpam-5475	185	10	=	=	PRON
ejpam-5475	186	1	(	(	PUNCT
ejpam-5475	186	2	x	x	SYM
ejpam-5475	186	3	−mν1	−mν1	ADP
ejpam-5475	186	4	)	)	PUNCT
ejpam-5475	186	5	∪	∪	NOUN
ejpam-5475	186	6	(	(	PUNCT
ejpam-5475	186	7	γ	γ	PROPN
ejpam-5475	186	8	|mν1	|mν1	NOUN
ejpam-5475	186	9	)	)	PUNCT
ejpam-5475	186	10	θ̃(ν1,ν2)(a	θ̃(ν1,ν2)(a	NOUN
ejpam-5475	186	11	∩mν1	∩mν1	ADJ
ejpam-5475	186	12	)	)	PUNCT
ejpam-5475	187	1	=	=	SYM
ejpam-5475	187	2	a	a	PRON
ejpam-5475	187	3	,	,	PUNCT
ejpam-5475	187	4	which	which	PRON
ejpam-5475	187	5	implies	imply	VERB
ejpam-5475	187	6	x	x	PUNCT
ejpam-5475	187	7	−mν1	−mν1	ADP
ejpam-5475	187	8	⊆	⊆	NUM
ejpam-5475	187	9	a.	a.	NOUN
ejpam-5475	187	10	4	4	NUM
ejpam-5475	187	11	.	.	PUNCT
ejpam-5475	187	12	mixed	mixed	ADJ
ejpam-5475	187	13	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	187	14	,	,	PUNCT
ejpam-5475	187	15	ν2)-open	ν2)-open	ADJ
ejpam-5475	187	16	sets	set	VERB
ejpam-5475	187	17	definition	definition	NOUN
ejpam-5475	187	18	6	6	NUM
ejpam-5475	187	19	.	.	PUNCT
ejpam-5475	188	1	let	let	VERB
ejpam-5475	188	2	ν1	ν1	NOUN
ejpam-5475	188	3	and	and	CCONJ
ejpam-5475	188	4	ν2	ν2	NOUN
ejpam-5475	188	5	be	be	AUX
ejpam-5475	188	6	two	two	NUM
ejpam-5475	188	7	gts	gts	NOUN
ejpam-5475	188	8	defined	define	VERB
ejpam-5475	188	9	on	on	ADP
ejpam-5475	188	10	a	a	DET
ejpam-5475	188	11	nonempty	nonempty	ADV
ejpam-5475	188	12	set	set	VERB
ejpam-5475	188	13	x.	x.	NOUN
ejpam-5475	188	14	a	a	DET
ejpam-5475	188	15	subset	subset	NOUN
ejpam-5475	188	16	a	a	PRON
ejpam-5475	188	17	of	of	ADP
ejpam-5475	188	18	x	x	PUNCT
ejpam-5475	188	19	is	be	AUX
ejpam-5475	188	20	mixed	mixed	ADJ
ejpam-5475	188	21	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	188	22	,	,	PUNCT
ejpam-5475	188	23	ν2)-open	ν2)-open	VERB
ejpam-5475	188	24	(	(	PUNCT
ejpam-5475	188	25	briefly	briefly	ADV
ejpam-5475	188	26	,	,	PUNCT
ejpam-5475	188	27	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	188	28	,	,	PUNCT
ejpam-5475	188	29	ν2)-open	ν2)-open	ADJ
ejpam-5475	188	30	)	)	PUNCT
ejpam-5475	188	31	if	if	SCONJ
ejpam-5475	188	32	for	for	ADP
ejpam-5475	188	33	every	every	DET
ejpam-5475	188	34	x	x	PROPN
ejpam-5475	188	35	∈	∈	PROPN
ejpam-5475	188	36	a	a	PRON
ejpam-5475	188	37	,	,	PUNCT
ejpam-5475	188	38	there	there	PRON
ejpam-5475	188	39	exists	exist	VERB
ejpam-5475	188	40	m	m	PROPN
ejpam-5475	188	41	∈	∈	NOUN
ejpam-5475	188	42	ν1	ν1	NOUN
ejpam-5475	188	43	such	such	ADJ
ejpam-5475	188	44	that	that	SCONJ
ejpam-5475	188	45	x	x	SYM
ejpam-5475	188	46	∈	∈	PROPN
ejpam-5475	188	47	m	m	NOUN
ejpam-5475	188	48	and	and	CCONJ
ejpam-5475	188	49	m	m	PROPN
ejpam-5475	188	50	⊆	⊆	NUM
ejpam-5475	188	51	cν2(m	cν2(m	NOUN
ejpam-5475	188	52	)	)	PUNCT
ejpam-5475	188	53	∩mν1	∩mν1	NOUN
ejpam-5475	189	1	⊆	⊆	NUM
ejpam-5475	189	2	a.	a.	NOUN
ejpam-5475	189	3	the	the	DET
ejpam-5475	189	4	complement	complement	NOUN
ejpam-5475	189	5	of	of	ADP
ejpam-5475	189	6	a	a	DET
ejpam-5475	189	7	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	189	8	,	,	PUNCT
ejpam-5475	189	9	ν2)-open	ν2)-open	ADJ
ejpam-5475	189	10	set	set	NOUN
ejpam-5475	189	11	is	be	AUX
ejpam-5475	189	12	called	call	VERB
ejpam-5475	189	13	a	a	DET
ejpam-5475	189	14	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	189	15	,	,	PUNCT
ejpam-5475	189	16	ν2)-closed	ν2)-closed	ADJ
ejpam-5475	189	17	set	set	NOUN
ejpam-5475	189	18	.	.	PUNCT
ejpam-5475	190	1	the	the	DET
ejpam-5475	190	2	family	family	NOUN
ejpam-5475	190	3	of	of	ADP
ejpam-5475	190	4	all	all	DET
ejpam-5475	190	5	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	190	6	,	,	PUNCT
ejpam-5475	190	7	ν2)-open	ν2)-open	ADJ
ejpam-5475	190	8	sets	set	NOUN
ejpam-5475	190	9	in	in	ADP
ejpam-5475	190	10	x	x	PROPN
ejpam-5475	190	11	is	be	AUX
ejpam-5475	190	12	denoted	denote	VERB
ejpam-5475	190	13	by	by	ADP
ejpam-5475	190	14	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	190	15	,	,	PUNCT
ejpam-5475	190	16	ν2	ν2	PROPN
ejpam-5475	190	17	)	)	PUNCT
ejpam-5475	190	18	.	.	PUNCT
ejpam-5475	191	1	remark	remark	PROPN
ejpam-5475	191	2	2	2	NUM
ejpam-5475	191	3	.	.	PUNCT
ejpam-5475	192	1	let	let	VERB
ejpam-5475	192	2	ν	ν	NOUN
ejpam-5475	192	3	be	be	AUX
ejpam-5475	192	4	a	a	DET
ejpam-5475	192	5	gt	gt	PROPN
ejpam-5475	192	6	on	on	ADP
ejpam-5475	192	7	a	a	DET
ejpam-5475	192	8	nonempty	nonempty	ADV
ejpam-5475	192	9	set	set	VERB
ejpam-5475	192	10	x.	x.	NOUN
ejpam-5475	192	11	then	then	ADV
ejpam-5475	192	12	every	every	DET
ejpam-5475	192	13	θ̃(ν	θ̃(ν	PROPN
ejpam-5475	192	14	,	,	PUNCT
ejpam-5475	192	15	ν)-open	ν)-open	PUNCT
ejpam-5475	192	16	set	set	VERB
ejpam-5475	192	17	in	in	ADP
ejpam-5475	192	18	x	x	PUNCT
ejpam-5475	192	19	is	be	AUX
ejpam-5475	192	20	θ̃(ν)-open	θ̃(ν)-open	ADJ
ejpam-5475	192	21	.	.	PUNCT
ejpam-5475	193	1	theorem	theorem	NOUN
ejpam-5475	193	2	8	8	NUM
ejpam-5475	193	3	.	.	PUNCT
ejpam-5475	194	1	let	let	VERB
ejpam-5475	194	2	ν1	ν1	NOUN
ejpam-5475	194	3	and	and	CCONJ
ejpam-5475	194	4	ν2	ν2	NOUN
ejpam-5475	194	5	be	be	AUX
ejpam-5475	194	6	two	two	NUM
ejpam-5475	194	7	gt	gt	NOUN
ejpam-5475	194	8	’s	’s	NOUN
ejpam-5475	194	9	on	on	ADP
ejpam-5475	194	10	a	a	DET
ejpam-5475	194	11	nonempty	nonempty	ADV
ejpam-5475	194	12	set	set	VERB
ejpam-5475	194	13	x.	x.	NOUN
ejpam-5475	194	14	then	then	ADV
ejpam-5475	194	15	θ(ν1	θ(ν1	NOUN
ejpam-5475	194	16	,	,	PUNCT
ejpam-5475	194	17	ν2	ν2	NOUN
ejpam-5475	194	18	)	)	PUNCT
ejpam-5475	195	1	⊆	⊆	NUM
ejpam-5475	195	2	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	195	3	,	,	PUNCT
ejpam-5475	195	4	ν2	ν2	NOUN
ejpam-5475	195	5	)	)	PUNCT
ejpam-5475	195	6	⊆	⊆	NUM
ejpam-5475	195	7	ν1	ν1	NOUN
ejpam-5475	195	8	.	.	PUNCT
ejpam-5475	196	1	proof	proof	NOUN
ejpam-5475	196	2	.	.	PUNCT
ejpam-5475	197	1	to	to	PART
ejpam-5475	197	2	show	show	VERB
ejpam-5475	197	3	θ(ν1	θ(ν1	NOUN
ejpam-5475	197	4	,	,	PUNCT
ejpam-5475	197	5	ν2	ν2	NOUN
ejpam-5475	197	6	)	)	PUNCT
ejpam-5475	197	7	⊆	⊆	NUM
ejpam-5475	197	8	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	197	9	,	,	PUNCT
ejpam-5475	197	10	ν2	ν2	NOUN
ejpam-5475	197	11	)	)	PUNCT
ejpam-5475	197	12	,	,	PUNCT
ejpam-5475	197	13	take	take	VERB
ejpam-5475	197	14	a	a	DET
ejpam-5475	197	15	∈	∈	PROPN
ejpam-5475	197	16	θ(ν1	θ(ν1	NOUN
ejpam-5475	197	17	,	,	PUNCT
ejpam-5475	197	18	ν2	ν2	NOUN
ejpam-5475	197	19	)	)	PUNCT
ejpam-5475	197	20	and	and	CCONJ
ejpam-5475	197	21	x	x	AUX
ejpam-5475	197	22	∈	∈	NOUN
ejpam-5475	197	23	a.	a.	NOUN
ejpam-5475	197	24	there	there	PRON
ejpam-5475	197	25	exists	exist	VERB
ejpam-5475	197	26	m	m	PROPN
ejpam-5475	197	27	∈	∈	NOUN
ejpam-5475	197	28	ν1	ν1	NOUN
ejpam-5475	198	1	such	such	ADJ
ejpam-5475	198	2	that	that	SCONJ
ejpam-5475	198	3	m	m	PROPN
ejpam-5475	198	4	⊆	⊆	NUM
ejpam-5475	198	5	cν2(m	cν2(m	NOUN
ejpam-5475	198	6	)	)	PUNCT
ejpam-5475	198	7	⊆	⊆	NUM
ejpam-5475	198	8	a.	a.	NOUN
ejpam-5475	198	9	since	since	SCONJ
ejpam-5475	198	10	cν2(m	cν2(m	PROPN
ejpam-5475	198	11	)	)	PUNCT
ejpam-5475	198	12	∩	∩	NOUN
ejpam-5475	198	13	mν1	mν1	ADJ
ejpam-5475	198	14	⊆	⊆	NUM
ejpam-5475	198	15	cν2(m	cν2(m	NOUN
ejpam-5475	198	16	)	)	PUNCT
ejpam-5475	198	17	,	,	PUNCT
ejpam-5475	198	18	we	we	PRON
ejpam-5475	198	19	have	have	VERB
ejpam-5475	198	20	m	m	PROPN
ejpam-5475	198	21	⊆	⊆	NUM
ejpam-5475	198	22	cν2(m	cν2(m	NOUN
ejpam-5475	198	23	)	)	PUNCT
ejpam-5475	198	24	∩mν1	∩mν1	NOUN
ejpam-5475	198	25	⊆	⊆	NUM
ejpam-5475	198	26	a.	a.	NOUN
ejpam-5475	198	27	hence	hence	ADV
ejpam-5475	198	28	,	,	PUNCT
ejpam-5475	198	29	a	a	PRON
ejpam-5475	198	30	is	be	AUX
ejpam-5475	198	31	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	198	32	,	,	PUNCT
ejpam-5475	198	33	ν2)-open	ν2)-open	ADJ
ejpam-5475	198	34	,	,	PUNCT
ejpam-5475	198	35	implying	imply	VERB
ejpam-5475	198	36	a	a	DET
ejpam-5475	198	37	∈	∈	PROPN
ejpam-5475	198	38	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	198	39	,	,	PUNCT
ejpam-5475	198	40	ν2	ν2	NOUN
ejpam-5475	198	41	)	)	PUNCT
ejpam-5475	198	42	.	.	PUNCT
ejpam-5475	199	1	next	next	ADV
ejpam-5475	199	2	,	,	PUNCT
ejpam-5475	199	3	to	to	PART
ejpam-5475	199	4	show	show	VERB
ejpam-5475	199	5	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	199	6	,	,	PUNCT
ejpam-5475	199	7	ν2	ν2	NOUN
ejpam-5475	199	8	)	)	PUNCT
ejpam-5475	199	9	⊆	⊆	NUM
ejpam-5475	199	10	ν1	ν1	NOUN
ejpam-5475	199	11	,	,	PUNCT
ejpam-5475	199	12	suppose	suppose	VERB
ejpam-5475	199	13	a	a	DET
ejpam-5475	199	14	∈	∈	PROPN
ejpam-5475	199	15	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	199	16	,	,	PUNCT
ejpam-5475	199	17	ν2	ν2	NOUN
ejpam-5475	199	18	)	)	PUNCT
ejpam-5475	199	19	and	and	CCONJ
ejpam-5475	199	20	x	x	SYM
ejpam-5475	199	21	∈	∈	NOUN
ejpam-5475	199	22	a.	a.	NOUN
ejpam-5475	199	23	there	there	PRON
ejpam-5475	199	24	exists	exist	VERB
ejpam-5475	199	25	m	m	PROPN
ejpam-5475	199	26	∈	∈	NOUN
ejpam-5475	199	27	ν1	ν1	NOUN
ejpam-5475	199	28	such	such	ADJ
ejpam-5475	199	29	that	that	SCONJ
ejpam-5475	199	30	x	x	SYM
ejpam-5475	199	31	∈	∈	PROPN
ejpam-5475	199	32	m	m	VERB
ejpam-5475	199	33	⊆	⊆	NUM
ejpam-5475	199	34	cν2(m	cν2(m	NOUN
ejpam-5475	199	35	)	)	PUNCT
ejpam-5475	199	36	∩mν1	∩mν1	NOUN
ejpam-5475	200	1	⊆	⊆	NUM
ejpam-5475	200	2	a.	a.	NOUN
ejpam-5475	200	3	therefore	therefore	ADV
ejpam-5475	200	4	,	,	PUNCT
ejpam-5475	200	5	a	a	DET
ejpam-5475	200	6	=	=	X
ejpam-5475	200	7	⋃	⋃	PROPN
ejpam-5475	200	8	x∈amx	x∈amx	PROPN
ejpam-5475	200	9	∈	∈	PROPN
ejpam-5475	200	10	ν1	ν1	NOUN
ejpam-5475	200	11	.	.	PUNCT
ejpam-5475	201	1	remark	remark	PROPN
ejpam-5475	201	2	3	3	NUM
ejpam-5475	201	3	.	.	PUNCT
ejpam-5475	202	1	according	accord	VERB
ejpam-5475	202	2	to	to	ADP
ejpam-5475	202	3	theorem	theorem	NOUN
ejpam-5475	202	4	8	8	NUM
ejpam-5475	202	5	,	,	PUNCT
ejpam-5475	202	6	the	the	DET
ejpam-5475	202	7	diagram	diagram	NOUN
ejpam-5475	202	8	below	below	ADV
ejpam-5475	202	9	illustrates	illustrate	VERB
ejpam-5475	202	10	the	the	DET
ejpam-5475	202	11	relationship	relationship	NOUN
ejpam-5475	202	12	.	.	PUNCT
ejpam-5475	203	1	a.	a.	NOUN
ejpam-5475	203	2	qahis	qahis	PROPN
ejpam-5475	203	3	,	,	PUNCT
ejpam-5475	203	4	a.	a.	NOUN
ejpam-5475	203	5	alqahtani	alqahtani	PROPN
ejpam-5475	203	6	/	/	SYM
ejpam-5475	203	7	eur	eur	PROPN
ejpam-5475	203	8	.	.	PUNCT
ejpam-5475	204	1	j.	j.	PROPN
ejpam-5475	204	2	pure	pure	PROPN
ejpam-5475	204	3	appl	appl	PROPN
ejpam-5475	204	4	.	.	PROPN
ejpam-5475	204	5	math	math	PROPN
ejpam-5475	204	6	,	,	PUNCT
ejpam-5475	204	7	17	17	NUM
ejpam-5475	204	8	(	(	PUNCT
ejpam-5475	204	9	4	4	NUM
ejpam-5475	204	10	)	)	PUNCT
ejpam-5475	204	11	(	(	PUNCT
ejpam-5475	204	12	2024	2024	NUM
ejpam-5475	204	13	)	)	PUNCT
ejpam-5475	204	14	,	,	PUNCT
ejpam-5475	204	15	3610	3610	NUM
ejpam-5475	204	16	-	-	SYM
ejpam-5475	204	17	3621	3621	NUM
ejpam-5475	204	18	3617	3617	NUM
ejpam-5475	204	19	the	the	DET
ejpam-5475	204	20	implications	implication	NOUN
ejpam-5475	204	21	stated	state	VERB
ejpam-5475	204	22	above	above	ADV
ejpam-5475	204	23	do	do	AUX
ejpam-5475	204	24	not	not	PART
ejpam-5475	204	25	work	work	VERB
ejpam-5475	204	26	in	in	ADP
ejpam-5475	204	27	reverse	reverse	NOUN
ejpam-5475	204	28	,	,	PUNCT
ejpam-5475	204	29	as	as	SCONJ
ejpam-5475	204	30	illustrated	illustrate	VERB
ejpam-5475	204	31	by	by	ADP
ejpam-5475	204	32	the	the	DET
ejpam-5475	204	33	following	follow	VERB
ejpam-5475	204	34	example	example	NOUN
ejpam-5475	204	35	.	.	PUNCT
ejpam-5475	205	1	example	example	NOUN
ejpam-5475	206	1	2	2	NUM
ejpam-5475	206	2	.	.	PUNCT
ejpam-5475	206	3	let	let	VERB
ejpam-5475	206	4	x	x	PUNCT
ejpam-5475	206	5	=	=	PRON
ejpam-5475	206	6	{	{	PUNCT
ejpam-5475	206	7	a	a	PRON
ejpam-5475	206	8	,	,	PUNCT
ejpam-5475	206	9	b	b	NOUN
ejpam-5475	206	10	,	,	PUNCT
ejpam-5475	206	11	c	c	NOUN
ejpam-5475	206	12	,	,	PUNCT
ejpam-5475	206	13	d	d	NOUN
ejpam-5475	206	14	}	}	PUNCT
ejpam-5475	206	15	.	.	PUNCT
ejpam-5475	207	1	consider	consider	VERB
ejpam-5475	207	2	two	two	NUM
ejpam-5475	207	3	generalized	generalized	ADJ
ejpam-5475	207	4	topologies	topology	NOUN
ejpam-5475	207	5	ν1	ν1	NOUN
ejpam-5475	207	6	=	=	SYM
ejpam-5475	207	7	{	{	PUNCT
ejpam-5475	207	8	∅	∅	NOUN
ejpam-5475	207	9	,	,	PUNCT
ejpam-5475	207	10	{	{	PUNCT
ejpam-5475	207	11	a	a	DET
ejpam-5475	207	12	,	,	PUNCT
ejpam-5475	207	13	b	b	NOUN
ejpam-5475	207	14	}	}	PUNCT
ejpam-5475	207	15	,	,	PUNCT
ejpam-5475	207	16	{	{	PUNCT
ejpam-5475	207	17	b	b	X
ejpam-5475	207	18	,	,	PUNCT
ejpam-5475	207	19	c	c	NOUN
ejpam-5475	207	20	}	}	PUNCT
ejpam-5475	207	21	,	,	PUNCT
ejpam-5475	207	22	{	{	PUNCT
ejpam-5475	207	23	a	a	PRON
ejpam-5475	207	24	,	,	PUNCT
ejpam-5475	207	25	b	b	NOUN
ejpam-5475	207	26	,	,	PUNCT
ejpam-5475	207	27	c	c	NOUN
ejpam-5475	207	28	}	}	PUNCT
ejpam-5475	207	29	}	}	PUNCT
ejpam-5475	207	30	and	and	CCONJ
ejpam-5475	207	31	ν2	ν2	NOUN
ejpam-5475	207	32	=	=	SYM
ejpam-5475	207	33	{	{	PUNCT
ejpam-5475	207	34	∅	∅	NOUN
ejpam-5475	207	35	,	,	PUNCT
ejpam-5475	207	36	{	{	PUNCT
ejpam-5475	207	37	b	b	NOUN
ejpam-5475	207	38	,	,	PUNCT
ejpam-5475	207	39	d	d	NOUN
ejpam-5475	207	40	}	}	PUNCT
ejpam-5475	207	41	}	}	PUNCT
ejpam-5475	207	42	on	on	ADP
ejpam-5475	207	43	x.	x.	NOUN
ejpam-5475	207	44	it	it	PRON
ejpam-5475	207	45	can	can	AUX
ejpam-5475	207	46	be	be	AUX
ejpam-5475	207	47	verified	verify	VERB
ejpam-5475	207	48	that	that	SCONJ
ejpam-5475	207	49	:	:	PUNCT
ejpam-5475	207	50	the	the	DET
ejpam-5475	207	51	set	set	NOUN
ejpam-5475	207	52	{	{	PUNCT
ejpam-5475	207	53	a	a	PRON
ejpam-5475	207	54	,	,	PUNCT
ejpam-5475	207	55	b	b	NOUN
ejpam-5475	207	56	,	,	PUNCT
ejpam-5475	207	57	c	c	NOUN
ejpam-5475	207	58	}	}	PUNCT
ejpam-5475	207	59	is	be	AUX
ejpam-5475	207	60	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	207	61	,	,	PUNCT
ejpam-5475	207	62	ν2)-open	ν2)-open	ADJ
ejpam-5475	207	63	but	but	CCONJ
ejpam-5475	207	64	not	not	PART
ejpam-5475	207	65	θ(ν1	θ(ν1	NOUN
ejpam-5475	207	66	,	,	PUNCT
ejpam-5475	207	67	ν2)-open	ν2)-open	NOUN
ejpam-5475	207	68	.	.	PUNCT
ejpam-5475	208	1	the	the	DET
ejpam-5475	208	2	set	set	NOUN
ejpam-5475	208	3	{	{	PUNCT
ejpam-5475	208	4	a	a	PRON
ejpam-5475	208	5	,	,	PUNCT
ejpam-5475	208	6	b	b	NOUN
ejpam-5475	208	7	}	}	PUNCT
ejpam-5475	208	8	is	be	AUX
ejpam-5475	208	9	ν1	ν1	NOUN
ejpam-5475	208	10	-	-	PUNCT
ejpam-5475	208	11	open	open	ADJ
ejpam-5475	208	12	but	but	CCONJ
ejpam-5475	208	13	not	not	PART
ejpam-5475	208	14	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	208	15	,	,	PUNCT
ejpam-5475	208	16	ν2)-open	ν2)-open	ADJ
ejpam-5475	208	17	.	.	PUNCT
ejpam-5475	209	1	remark	remark	PROPN
ejpam-5475	209	2	4	4	NUM
ejpam-5475	209	3	.	.	PUNCT
ejpam-5475	210	1	let	let	VERB
ejpam-5475	210	2	ν1	ν1	NOUN
ejpam-5475	210	3	and	and	CCONJ
ejpam-5475	210	4	ν2	ν2	NOUN
ejpam-5475	210	5	be	be	AUX
ejpam-5475	210	6	two	two	NUM
ejpam-5475	210	7	gt	gt	NOUN
ejpam-5475	210	8	’s	’s	NOUN
ejpam-5475	210	9	on	on	ADP
ejpam-5475	210	10	a	a	DET
ejpam-5475	210	11	nonempty	nonempty	ADV
ejpam-5475	210	12	set	set	VERB
ejpam-5475	210	13	x.	x.	NOUN
ejpam-5475	210	14	if	if	SCONJ
ejpam-5475	210	15	the	the	DET
ejpam-5475	210	16	gts	gts	NOUN
ejpam-5475	210	17	(	(	PUNCT
ejpam-5475	210	18	x	x	NOUN
ejpam-5475	210	19	,	,	PUNCT
ejpam-5475	210	20	ν1	ν1	NOUN
ejpam-5475	210	21	)	)	PUNCT
ejpam-5475	210	22	is	be	AUX
ejpam-5475	210	23	strong	strong	ADJ
ejpam-5475	210	24	,	,	PUNCT
ejpam-5475	210	25	then	then	ADV
ejpam-5475	210	26	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	210	27	,	,	PUNCT
ejpam-5475	210	28	ν2	ν2	NOUN
ejpam-5475	210	29	)	)	PUNCT
ejpam-5475	210	30	=	=	SYM
ejpam-5475	210	31	θ(ν1	θ(ν1	NOUN
ejpam-5475	210	32	,	,	PUNCT
ejpam-5475	210	33	ν2	ν2	NOUN
ejpam-5475	210	34	)	)	PUNCT
ejpam-5475	210	35	.	.	PUNCT
ejpam-5475	211	1	theorem	theorem	VERB
ejpam-5475	211	2	9	9	NUM
ejpam-5475	211	3	.	.	PUNCT
ejpam-5475	212	1	let	let	VERB
ejpam-5475	212	2	ν1	ν1	NOUN
ejpam-5475	212	3	and	and	CCONJ
ejpam-5475	212	4	ν2	ν2	NOUN
ejpam-5475	212	5	be	be	AUX
ejpam-5475	212	6	two	two	NUM
ejpam-5475	212	7	gt	gt	NOUN
ejpam-5475	212	8	’s	’s	NOUN
ejpam-5475	212	9	on	on	ADP
ejpam-5475	212	10	a	a	DET
ejpam-5475	212	11	nonempty	nonempty	ADV
ejpam-5475	212	12	set	set	VERB
ejpam-5475	212	13	x.	x.	NOUN
ejpam-5475	212	14	then	then	ADV
ejpam-5475	212	15	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	212	16	,	,	PUNCT
ejpam-5475	212	17	ν2	ν2	NOUN
ejpam-5475	212	18	)	)	PUNCT
ejpam-5475	212	19	is	be	AUX
ejpam-5475	212	20	also	also	ADV
ejpam-5475	212	21	a	a	DET
ejpam-5475	212	22	generalized	generalized	ADJ
ejpam-5475	212	23	topology	topology	NOUN
ejpam-5475	212	24	on	on	ADP
ejpam-5475	212	25	x	x	PUNCT
ejpam-5475	212	26	contained	contain	VERB
ejpam-5475	212	27	in	in	ADP
ejpam-5475	212	28	ν1	ν1	NOUN
ejpam-5475	212	29	.	.	PUNCT
ejpam-5475	213	1	proof	proof	NOUN
ejpam-5475	213	2	.	.	PUNCT
ejpam-5475	214	1	it	it	PRON
ejpam-5475	214	2	is	be	AUX
ejpam-5475	214	3	evident	evident	ADJ
ejpam-5475	214	4	that	that	SCONJ
ejpam-5475	214	5	∅	∅	NOUN
ejpam-5475	214	6	∈	∈	PROPN
ejpam-5475	214	7	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	214	8	,	,	PUNCT
ejpam-5475	214	9	ν2	ν2	NOUN
ejpam-5475	214	10	)	)	PUNCT
ejpam-5475	214	11	.	.	PUNCT
ejpam-5475	215	1	let	let	VERB
ejpam-5475	215	2	{	{	PUNCT
ejpam-5475	215	3	aα	aα	NOUN
ejpam-5475	215	4	:	:	PUNCT
ejpam-5475	215	5	α	α	PROPN
ejpam-5475	215	6	∈	∈	PROPN
ejpam-5475	215	7	λ	λ	NOUN
ejpam-5475	215	8	}	}	PUNCT
ejpam-5475	215	9	be	be	VERB
ejpam-5475	215	10	a	a	DET
ejpam-5475	215	11	collection	collection	NOUN
ejpam-5475	215	12	of	of	ADP
ejpam-5475	215	13	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	215	14	,	,	PUNCT
ejpam-5475	215	15	ν2)open	ν2)open	ADJ
ejpam-5475	215	16	sets	set	NOUN
ejpam-5475	215	17	in	in	ADP
ejpam-5475	215	18	x	x	NOUN
ejpam-5475	215	19	,	,	PUNCT
ejpam-5475	215	20	and	and	CCONJ
ejpam-5475	215	21	let	let	VERB
ejpam-5475	215	22	x	x	X
ejpam-5475	215	23	∈	∈	PROPN
ejpam-5475	215	24	⋃	⋃	PROPN
ejpam-5475	215	25	α∈λaα	α∈λaα	PROPN
ejpam-5475	215	26	.	.	PUNCT
ejpam-5475	216	1	there	there	PRON
ejpam-5475	216	2	exists	exist	VERB
ejpam-5475	216	3	α0	α0	PROPN
ejpam-5475	216	4	∈	∈	PROPN
ejpam-5475	216	5	λ	λ	NOUN
ejpam-5475	216	6	such	such	ADJ
ejpam-5475	216	7	that	that	SCONJ
ejpam-5475	216	8	x	x	SYM
ejpam-5475	216	9	∈	∈	PROPN
ejpam-5475	216	10	aα0	aα0	NOUN
ejpam-5475	216	11	.	.	PUNCT
ejpam-5475	217	1	since	since	SCONJ
ejpam-5475	217	2	aα0	aα0	PROPN
ejpam-5475	217	3	is	be	AUX
ejpam-5475	217	4	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	217	5	,	,	PUNCT
ejpam-5475	217	6	ν2)-open	ν2)-open	ADJ
ejpam-5475	217	7	,	,	PUNCT
ejpam-5475	217	8	there	there	PRON
ejpam-5475	217	9	exists	exist	VERB
ejpam-5475	217	10	m	m	PROPN
ejpam-5475	217	11	∈	∈	NOUN
ejpam-5475	217	12	ν1	ν1	NOUN
ejpam-5475	217	13	such	such	ADJ
ejpam-5475	217	14	that	that	SCONJ
ejpam-5475	217	15	x	x	SYM
ejpam-5475	217	16	∈	∈	PROPN
ejpam-5475	217	17	m	m	NOUN
ejpam-5475	217	18	and	and	CCONJ
ejpam-5475	217	19	m	m	PROPN
ejpam-5475	217	20	⊆	⊆	NUM
ejpam-5475	217	21	cν2(m	cν2(m	NOUN
ejpam-5475	217	22	)	)	PUNCT
ejpam-5475	217	23	∩mν1	∩mν1	NOUN
ejpam-5475	217	24	⊆	⊆	NUM
ejpam-5475	217	25	aα0	aα0	NOUN
ejpam-5475	217	26	⊆⋃	⊆⋃	PROPN
ejpam-5475	217	27	α∈λaα	α∈λaα	PROPN
ejpam-5475	217	28	.	.	PUNCT
ejpam-5475	218	1	therefore	therefore	ADV
ejpam-5475	218	2	,	,	PUNCT
ejpam-5475	218	3	⋃	⋃	PROPN
ejpam-5475	218	4	α∈λaα	α∈λaα	PROPN
ejpam-5475	218	5	is	be	AUX
ejpam-5475	218	6	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	218	7	,	,	PUNCT
ejpam-5475	218	8	ν2)-open	ν2)-open	ADJ
ejpam-5475	218	9	.	.	PUNCT
ejpam-5475	219	1	theorem	theorem	NOUN
ejpam-5475	219	2	10	10	NUM
ejpam-5475	219	3	.	.	PUNCT
ejpam-5475	220	1	let	let	VERB
ejpam-5475	220	2	ν1	ν1	NOUN
ejpam-5475	220	3	and	and	CCONJ
ejpam-5475	220	4	ν2	ν2	NOUN
ejpam-5475	220	5	be	be	AUX
ejpam-5475	220	6	two	two	NUM
ejpam-5475	220	7	gt	gt	NOUN
ejpam-5475	220	8	’s	’s	NOUN
ejpam-5475	220	9	on	on	ADP
ejpam-5475	220	10	a	a	DET
ejpam-5475	220	11	nonempty	nonempty	ADV
ejpam-5475	220	12	set	set	VERB
ejpam-5475	220	13	x	x	NOUN
ejpam-5475	220	14	,	,	PUNCT
ejpam-5475	220	15	and	and	CCONJ
ejpam-5475	220	16	let	let	VERB
ejpam-5475	220	17	a	a	DET
ejpam-5475	220	18	⊆	⊆	NUM
ejpam-5475	220	19	x.	x.	NOUN
ejpam-5475	220	20	then	then	ADV
ejpam-5475	220	21	a	a	PRON
ejpam-5475	220	22	is	be	AUX
ejpam-5475	220	23	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	220	24	,	,	PUNCT
ejpam-5475	220	25	ν2)-closed	ν2)-close	VERB
ejpam-5475	220	26	if	if	SCONJ
ejpam-5475	220	27	and	and	CCONJ
ejpam-5475	220	28	only	only	ADV
ejpam-5475	220	29	if	if	SCONJ
ejpam-5475	220	30	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	220	31	)	)	PUNCT
ejpam-5475	221	1	=	=	NOUN
ejpam-5475	221	2	a.	a.	NOUN
ejpam-5475	221	3	proof	proof	NOUN
ejpam-5475	221	4	.	.	PUNCT
ejpam-5475	222	1	let	let	VERB
ejpam-5475	222	2	a	a	DET
ejpam-5475	222	3	be	be	AUX
ejpam-5475	222	4	a	a	DET
ejpam-5475	222	5	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	222	6	,	,	PUNCT
ejpam-5475	222	7	ν2)-closed	ν2)-closed	ADJ
ejpam-5475	222	8	set	set	NOUN
ejpam-5475	222	9	.	.	PUNCT
ejpam-5475	223	1	assume	assume	VERB
ejpam-5475	223	2	x	x	SYM
ejpam-5475	223	3	∈	∈	PROPN
ejpam-5475	223	4	x−a	x−a	PROPN
ejpam-5475	223	5	.	.	PUNCT
ejpam-5475	224	1	thenx−a	thenx−a	PROPN
ejpam-5475	224	2	is	be	AUX
ejpam-5475	224	3	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	224	4	,	,	PUNCT
ejpam-5475	224	5	ν2)-open	ν2)-open	ADJ
ejpam-5475	224	6	.	.	PUNCT
ejpam-5475	225	1	according	accord	VERB
ejpam-5475	225	2	to	to	ADP
ejpam-5475	225	3	definition	definition	NOUN
ejpam-5475	225	4	6	6	NUM
ejpam-5475	225	5	,	,	PUNCT
ejpam-5475	225	6	there	there	PRON
ejpam-5475	225	7	exists	exist	VERB
ejpam-5475	225	8	m	m	PROPN
ejpam-5475	225	9	∈	∈	NOUN
ejpam-5475	225	10	ν1	ν1	NOUN
ejpam-5475	225	11	such	such	ADJ
ejpam-5475	225	12	that	that	SCONJ
ejpam-5475	225	13	x	x	SYM
ejpam-5475	225	14	∈	∈	PROPN
ejpam-5475	225	15	m	m	VERB
ejpam-5475	225	16	⊆	⊆	NUM
ejpam-5475	225	17	cν2(m)∩mν1	cν2(m)∩mν1	ADV
ejpam-5475	225	18	⊆	⊆	NUM
ejpam-5475	225	19	x−a	x−a	NOUN
ejpam-5475	225	20	.	.	PUNCT
ejpam-5475	226	1	hence	hence	ADV
ejpam-5475	226	2	,	,	PUNCT
ejpam-5475	226	3	(	(	PUNCT
ejpam-5475	226	4	cν2(m	cν2(m	PROPN
ejpam-5475	226	5	)	)	PUNCT
ejpam-5475	226	6	∩mν1	∩mν1	ADJ
ejpam-5475	226	7	)	)	PUNCT
ejpam-5475	226	8	∩	∩	NOUN
ejpam-5475	226	9	a	a	DET
ejpam-5475	226	10	=	=	SYM
ejpam-5475	226	11	∅	∅	NOUN
ejpam-5475	226	12	,	,	PUNCT
ejpam-5475	226	13	implying	imply	VERB
ejpam-5475	226	14	x	x	X
ejpam-5475	226	15	/∈	/∈	PUNCT
ejpam-5475	226	16	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	226	17	)	)	PUNCT
ejpam-5475	226	18	.	.	PUNCT
ejpam-5475	227	1	therefore	therefore	ADV
ejpam-5475	227	2	,	,	PUNCT
ejpam-5475	227	3	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	227	4	)	)	PUNCT
ejpam-5475	227	5	⊆	⊆	NUM
ejpam-5475	227	6	a.	a.	NOUN
ejpam-5475	227	7	by	by	ADP
ejpam-5475	227	8	theorem	theorem	ADJ
ejpam-5475	227	9	3	3	NUM
ejpam-5475	227	10	(	(	PUNCT
ejpam-5475	227	11	ii	ii	NOUN
ejpam-5475	227	12	)	)	PUNCT
ejpam-5475	227	13	,	,	PUNCT
ejpam-5475	227	14	we	we	PRON
ejpam-5475	227	15	conclude	conclude	VERB
ejpam-5475	227	16	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	227	17	)	)	PUNCT
ejpam-5475	228	1	=	=	VERB
ejpam-5475	228	2	a.	a.	NOUN
ejpam-5475	228	3	conversely	conversely	ADV
ejpam-5475	228	4	,	,	PUNCT
ejpam-5475	228	5	suppose	suppose	VERB
ejpam-5475	228	6	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NOUN
ejpam-5475	228	7	)	)	PUNCT
ejpam-5475	229	1	=	=	PUNCT
ejpam-5475	229	2	a.	a.	NOUN
ejpam-5475	229	3	if	if	SCONJ
ejpam-5475	229	4	x	x	SYM
ejpam-5475	229	5	∈	∈	PROPN
ejpam-5475	229	6	x−a	x−a	PROPN
ejpam-5475	229	7	,	,	PUNCT
ejpam-5475	229	8	then	then	ADV
ejpam-5475	229	9	there	there	PRON
ejpam-5475	229	10	existsm	existsm	NOUN
ejpam-5475	229	11	∈	∈	PROPN
ejpam-5475	229	12	ν1	ν1	NOUN
ejpam-5475	229	13	containing	contain	VERB
ejpam-5475	229	14	x	x	PUNCT
ejpam-5475	229	15	such	such	ADJ
ejpam-5475	229	16	that	that	SCONJ
ejpam-5475	229	17	(	(	PUNCT
ejpam-5475	229	18	cν2(m	cν2(m	PROPN
ejpam-5475	229	19	)	)	PUNCT
ejpam-5475	229	20	∩	∩	NOUN
ejpam-5475	229	21	mν1	mν1	X
ejpam-5475	229	22	)	)	PUNCT
ejpam-5475	229	23	∩	∩	NOUN
ejpam-5475	229	24	a	a	DET
ejpam-5475	229	25	=	=	SYM
ejpam-5475	229	26	∅	∅	NOUN
ejpam-5475	229	27	,	,	PUNCT
ejpam-5475	229	28	implying	imply	VERB
ejpam-5475	229	29	m	m	NOUN
ejpam-5475	229	30	⊆	⊆	NUM
ejpam-5475	229	31	cν2(m	cν2(m	NOUN
ejpam-5475	229	32	)	)	PUNCT
ejpam-5475	229	33	∩	∩	NOUN
ejpam-5475	229	34	mν1	mν1	ADJ
ejpam-5475	229	35	⊆	⊆	NUM
ejpam-5475	229	36	x	x	SYM
ejpam-5475	229	37	−	−	NOUN
ejpam-5475	229	38	a.	a.	NOUN
ejpam-5475	229	39	hence	hence	ADV
ejpam-5475	229	40	,	,	PUNCT
ejpam-5475	229	41	x	x	PRON
ejpam-5475	229	42	−a	−a	NOUN
ejpam-5475	229	43	is	be	AUX
ejpam-5475	229	44	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	229	45	,	,	PUNCT
ejpam-5475	229	46	ν2)-open	ν2)-open	ADJ
ejpam-5475	229	47	,	,	PUNCT
ejpam-5475	229	48	showing	show	VERB
ejpam-5475	229	49	a	a	DET
ejpam-5475	229	50	is	be	AUX
ejpam-5475	229	51	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	229	52	,	,	PUNCT
ejpam-5475	229	53	ν2)-closed	ν2)-closed	PROPN
ejpam-5475	229	54	.	.	PUNCT
ejpam-5475	230	1	based	base	VERB
ejpam-5475	230	2	on	on	ADP
ejpam-5475	230	3	(	(	PUNCT
ejpam-5475	230	4	ii	ii	NOUN
ejpam-5475	230	5	)	)	PUNCT
ejpam-5475	230	6	of	of	ADP
ejpam-5475	230	7	theorems	theorem	NOUN
ejpam-5475	230	8	5	5	NUM
ejpam-5475	230	9	and	and	CCONJ
ejpam-5475	230	10	10	10	NUM
ejpam-5475	230	11	,	,	PUNCT
ejpam-5475	230	12	the	the	DET
ejpam-5475	230	13	following	follow	VERB
ejpam-5475	230	14	corollary	corollary	NOUN
ejpam-5475	230	15	follows	follow	VERB
ejpam-5475	230	16	.	.	PUNCT
ejpam-5475	231	1	corollary	corollary	ADJ
ejpam-5475	231	2	3	3	X
ejpam-5475	231	3	.	.	PUNCT
ejpam-5475	232	1	let	let	VERB
ejpam-5475	232	2	ν1	ν1	NOUN
ejpam-5475	232	3	and	and	CCONJ
ejpam-5475	232	4	ν2	ν2	NOUN
ejpam-5475	232	5	be	be	AUX
ejpam-5475	232	6	two	two	NUM
ejpam-5475	232	7	topologies	topology	NOUN
ejpam-5475	232	8	on	on	ADP
ejpam-5475	232	9	a	a	DET
ejpam-5475	232	10	nonempty	nonempty	ADV
ejpam-5475	232	11	set	set	VERB
ejpam-5475	232	12	x	x	NOUN
ejpam-5475	232	13	,	,	PUNCT
ejpam-5475	232	14	and	and	CCONJ
ejpam-5475	232	15	let	let	VERB
ejpam-5475	232	16	a	a	DET
ejpam-5475	232	17	⊆	⊆	NUM
ejpam-5475	232	18	x.	x.	NOUN
ejpam-5475	232	19	if	if	SCONJ
ejpam-5475	232	20	a	a	DET
ejpam-5475	232	21	∈	∈	NOUN
ejpam-5475	232	22	ν2	ν2	NOUN
ejpam-5475	232	23	and	and	CCONJ
ejpam-5475	232	24	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	232	25	)	)	PUNCT
ejpam-5475	233	1	=	=	SYM
ejpam-5475	234	1	a	a	PRON
ejpam-5475	234	2	,	,	PUNCT
ejpam-5475	234	3	then	then	ADV
ejpam-5475	234	4	a	a	PRON
ejpam-5475	234	5	is	be	AUX
ejpam-5475	234	6	θ(ν1	θ(ν1	NOUN
ejpam-5475	234	7	,	,	PUNCT
ejpam-5475	234	8	ν2)-closed	ν2)-closed	ADJ
ejpam-5475	234	9	set	set	NOUN
ejpam-5475	234	10	.	.	PUNCT
ejpam-5475	235	1	theorem	theorem	VERB
ejpam-5475	235	2	11	11	NUM
ejpam-5475	235	3	.	.	PUNCT
ejpam-5475	236	1	let	let	VERB
ejpam-5475	236	2	ν1	ν1	NOUN
ejpam-5475	236	3	and	and	CCONJ
ejpam-5475	236	4	ν2	ν2	NOUN
ejpam-5475	236	5	be	be	AUX
ejpam-5475	236	6	two	two	NUM
ejpam-5475	236	7	gt	gt	NOUN
ejpam-5475	236	8	’s	’s	NOUN
ejpam-5475	236	9	on	on	ADP
ejpam-5475	236	10	a	a	DET
ejpam-5475	236	11	nonempty	nonempty	ADV
ejpam-5475	236	12	set	set	VERB
ejpam-5475	236	13	x	x	NOUN
ejpam-5475	236	14	,	,	PUNCT
ejpam-5475	236	15	and	and	CCONJ
ejpam-5475	236	16	let	let	VERB
ejpam-5475	236	17	a	a	DET
ejpam-5475	236	18	⊆	⊆	NUM
ejpam-5475	236	19	x.	x.	NOUN
ejpam-5475	236	20	if	if	SCONJ
ejpam-5475	236	21	a	a	PRON
ejpam-5475	236	22	is	be	AUX
ejpam-5475	236	23	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	236	24	,	,	PUNCT
ejpam-5475	236	25	ν2)-open	ν2)-open	ADJ
ejpam-5475	236	26	and	and	CCONJ
ejpam-5475	236	27	x	x	PUNCT
ejpam-5475	236	28	∈	∈	PROPN
ejpam-5475	236	29	a	a	PRON
ejpam-5475	236	30	,	,	PUNCT
ejpam-5475	236	31	then	then	ADV
ejpam-5475	236	32	there	there	PRON
ejpam-5475	236	33	exists	exist	VERB
ejpam-5475	236	34	a	a	DET
ejpam-5475	236	35	(	(	PUNCT
ejpam-5475	236	36	ν1	ν1	NOUN
ejpam-5475	236	37	,	,	PUNCT
ejpam-5475	236	38	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	236	39	-	-	PUNCT
ejpam-5475	236	40	open	open	ADJ
ejpam-5475	236	41	set	set	NOUN
ejpam-5475	236	42	u	u	NOUN
ejpam-5475	236	43	containing	contain	VERB
ejpam-5475	236	44	x	x	PUNCT
ejpam-5475	236	45	such	such	ADJ
ejpam-5475	236	46	that	that	SCONJ
ejpam-5475	236	47	u	u	PROPN
ejpam-5475	236	48	⊆	⊆	NUM
ejpam-5475	236	49	cν2(u	cν2(u	NOUN
ejpam-5475	236	50	)	)	PUNCT
ejpam-5475	236	51	∩mν1	∩mν1	NOUN
ejpam-5475	236	52	⊆	⊆	NUM
ejpam-5475	236	53	a.	a.	NOUN
ejpam-5475	236	54	proof	proof	NOUN
ejpam-5475	236	55	.	.	PUNCT
ejpam-5475	237	1	let	let	VERB
ejpam-5475	237	2	a	a	DET
ejpam-5475	237	3	be	be	AUX
ejpam-5475	237	4	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	237	5	,	,	PUNCT
ejpam-5475	237	6	ν2)-open	ν2)-open	VERB
ejpam-5475	237	7	in	in	ADP
ejpam-5475	237	8	x	x	NOUN
ejpam-5475	237	9	,	,	PUNCT
ejpam-5475	237	10	and	and	CCONJ
ejpam-5475	237	11	suppose	suppose	VERB
ejpam-5475	237	12	x	x	X
ejpam-5475	237	13	∈	∈	PROPN
ejpam-5475	237	14	a.	a.	NOUN
ejpam-5475	237	15	thus	thus	ADV
ejpam-5475	237	16	,	,	PUNCT
ejpam-5475	237	17	there	there	PRON
ejpam-5475	237	18	exists	exist	VERB
ejpam-5475	237	19	a	a	DET
ejpam-5475	237	20	ν1	ν1	NOUN
ejpam-5475	237	21	-	-	PUNCT
ejpam-5475	237	22	open	open	NOUN
ejpam-5475	237	23	set	set	NOUN
ejpam-5475	237	24	m	m	VERB
ejpam-5475	237	25	such	such	ADJ
ejpam-5475	237	26	that	that	SCONJ
ejpam-5475	237	27	x	x	SYM
ejpam-5475	237	28	∈	∈	PROPN
ejpam-5475	237	29	m	m	VERB
ejpam-5475	237	30	⊆	⊆	NUM
ejpam-5475	237	31	cν2(m	cν2(m	NOUN
ejpam-5475	237	32	)	)	PUNCT
ejpam-5475	237	33	∩	∩	NOUN
ejpam-5475	237	34	mν1	mν1	ADJ
ejpam-5475	237	35	⊆	⊆	NUM
ejpam-5475	237	36	a.	a.	NOUN
ejpam-5475	237	37	define	define	NOUN
ejpam-5475	237	38	u	u	NOUN
ejpam-5475	237	39	=	=	NOUN
ejpam-5475	237	40	iν1	iν1	NOUN
ejpam-5475	237	41	(	(	PUNCT
ejpam-5475	237	42	cν2(m	cν2(m	PROPN
ejpam-5475	237	43	)	)	PUNCT
ejpam-5475	237	44	)	)	PUNCT
ejpam-5475	237	45	.	.	PUNCT
ejpam-5475	238	1	then	then	ADV
ejpam-5475	238	2	u	u	NOUN
ejpam-5475	238	3	is	be	AUX
ejpam-5475	238	4	(	(	PUNCT
ejpam-5475	238	5	ν1	ν1	NOUN
ejpam-5475	238	6	,	,	PUNCT
ejpam-5475	238	7	ν2)-regular	ν2)-regular	NOUN
ejpam-5475	238	8	-	-	PUNCT
ejpam-5475	238	9	open	open	ADJ
ejpam-5475	238	10	,	,	PUNCT
ejpam-5475	238	11	with	with	ADP
ejpam-5475	238	12	m	m	PROPN
ejpam-5475	238	13	⊆	⊆	NUM
ejpam-5475	238	14	u	u	NOUN
ejpam-5475	238	15	⊆	⊆	NUM
ejpam-5475	238	16	cν2(u	cν2(u	NOUN
ejpam-5475	238	17	)	)	PUNCT
ejpam-5475	238	18	=	=	SYM
ejpam-5475	239	1	cν2	cν2	NOUN
ejpam-5475	239	2	(	(	PUNCT
ejpam-5475	239	3	iν1	iν1	NOUN
ejpam-5475	239	4	(	(	PUNCT
ejpam-5475	239	5	cν2(m	cν2(m	PROPN
ejpam-5475	239	6	)	)	PUNCT
ejpam-5475	239	7	)	)	PUNCT
ejpam-5475	239	8	)	)	PUNCT
ejpam-5475	240	1	⊆	⊆	NUM
ejpam-5475	240	2	cν2(m	cν2(m	NOUN
ejpam-5475	240	3	)	)	PUNCT
ejpam-5475	240	4	.	.	PUNCT
ejpam-5475	241	1	a.	a.	NOUN
ejpam-5475	241	2	qahis	qahis	PROPN
ejpam-5475	241	3	,	,	PUNCT
ejpam-5475	241	4	a.	a.	NOUN
ejpam-5475	241	5	alqahtani	alqahtani	PROPN
ejpam-5475	241	6	/	/	SYM
ejpam-5475	241	7	eur	eur	PROPN
ejpam-5475	241	8	.	.	PUNCT
ejpam-5475	242	1	j.	j.	PROPN
ejpam-5475	242	2	pure	pure	PROPN
ejpam-5475	242	3	appl	appl	PROPN
ejpam-5475	242	4	.	.	PROPN
ejpam-5475	242	5	math	math	PROPN
ejpam-5475	242	6	,	,	PUNCT
ejpam-5475	242	7	17	17	NUM
ejpam-5475	242	8	(	(	PUNCT
ejpam-5475	242	9	4	4	NUM
ejpam-5475	242	10	)	)	PUNCT
ejpam-5475	242	11	(	(	PUNCT
ejpam-5475	242	12	2024	2024	NUM
ejpam-5475	242	13	)	)	PUNCT
ejpam-5475	242	14	,	,	PUNCT
ejpam-5475	242	15	3610	3610	NUM
ejpam-5475	242	16	-	-	SYM
ejpam-5475	242	17	3621	3621	NUM
ejpam-5475	242	18	3618	3618	NUM
ejpam-5475	242	19	this	this	PRON
ejpam-5475	242	20	implies	imply	VERB
ejpam-5475	242	21	x	x	X
ejpam-5475	242	22	∈	∈	PROPN
ejpam-5475	242	23	m	m	VERB
ejpam-5475	242	24	⊆	⊆	NUM
ejpam-5475	242	25	u	u	NOUN
ejpam-5475	242	26	⊆	⊆	NUM
ejpam-5475	242	27	cν2(u	cν2(u	NOUN
ejpam-5475	242	28	)	)	PUNCT
ejpam-5475	242	29	∩mν1	∩mν1	NOUN
ejpam-5475	242	30	⊆	⊆	NUM
ejpam-5475	242	31	cν2(m	cν2(m	NOUN
ejpam-5475	242	32	)	)	PUNCT
ejpam-5475	242	33	∩mν1	∩mν1	NOUN
ejpam-5475	243	1	⊆	⊆	NUM
ejpam-5475	243	2	a.	a.	NOUN
ejpam-5475	243	3	therefore	therefore	ADV
ejpam-5475	243	4	,	,	PUNCT
ejpam-5475	243	5	x	x	PUNCT
ejpam-5475	243	6	∈	∈	PROPN
ejpam-5475	243	7	u	u	NOUN
ejpam-5475	243	8	⊆	⊆	NUM
ejpam-5475	243	9	cν2(u	cν2(u	NOUN
ejpam-5475	243	10	)	)	PUNCT
ejpam-5475	243	11	∩mν1	∩mν1	NOUN
ejpam-5475	243	12	⊆	⊆	NUM
ejpam-5475	243	13	a	a	PRON
ejpam-5475	243	14	for	for	ADP
ejpam-5475	243	15	some	some	DET
ejpam-5475	243	16	(	(	PUNCT
ejpam-5475	243	17	ν1	ν1	NOUN
ejpam-5475	243	18	,	,	PUNCT
ejpam-5475	243	19	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	243	20	-	-	PUNCT
ejpam-5475	243	21	open	open	ADJ
ejpam-5475	243	22	set	set	NOUN
ejpam-5475	243	23	u	u	PROPN
ejpam-5475	243	24	.	.	PUNCT
ejpam-5475	244	1	since	since	SCONJ
ejpam-5475	244	2	every	every	DET
ejpam-5475	244	3	set	set	NOUN
ejpam-5475	244	4	that	that	PRON
ejpam-5475	244	5	is	be	AUX
ejpam-5475	244	6	(	(	PUNCT
ejpam-5475	244	7	ν1	ν1	NOUN
ejpam-5475	244	8	,	,	PUNCT
ejpam-5475	244	9	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	244	10	-	-	PUNCT
ejpam-5475	244	11	open	open	ADJ
ejpam-5475	244	12	is	be	AUX
ejpam-5475	244	13	ν1	ν1	NOUN
ejpam-5475	244	14	-	-	PUNCT
ejpam-5475	244	15	open	open	ADJ
ejpam-5475	244	16	in	in	ADP
ejpam-5475	244	17	x	x	X
ejpam-5475	244	18	,	,	PUNCT
ejpam-5475	244	19	the	the	DET
ejpam-5475	244	20	following	follow	VERB
ejpam-5475	244	21	corollary	corollary	NOUN
ejpam-5475	244	22	is	be	AUX
ejpam-5475	244	23	clearly	clearly	ADV
ejpam-5475	244	24	derived	derive	VERB
ejpam-5475	244	25	.	.	PUNCT
ejpam-5475	245	1	corollary	corollary	ADJ
ejpam-5475	245	2	4	4	NUM
ejpam-5475	245	3	.	.	PUNCT
ejpam-5475	246	1	let	let	VERB
ejpam-5475	246	2	ν1	ν1	NOUN
ejpam-5475	246	3	and	and	CCONJ
ejpam-5475	246	4	ν2	ν2	NOUN
ejpam-5475	246	5	be	be	AUX
ejpam-5475	246	6	two	two	NUM
ejpam-5475	246	7	gt	gt	NOUN
ejpam-5475	246	8	’s	’s	NOUN
ejpam-5475	246	9	on	on	ADP
ejpam-5475	246	10	a	a	DET
ejpam-5475	246	11	nonempty	nonempty	ADV
ejpam-5475	246	12	set	set	VERB
ejpam-5475	246	13	x	x	NOUN
ejpam-5475	246	14	,	,	PUNCT
ejpam-5475	246	15	and	and	CCONJ
ejpam-5475	246	16	let	let	VERB
ejpam-5475	246	17	a	a	DET
ejpam-5475	246	18	⊆	⊆	NUM
ejpam-5475	246	19	x.	x.	NOUN
ejpam-5475	246	20	then	then	ADV
ejpam-5475	246	21	a	a	PRON
ejpam-5475	246	22	is	be	AUX
ejpam-5475	246	23	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	246	24	,	,	PUNCT
ejpam-5475	246	25	ν2)-open	ν2)-open	VERB
ejpam-5475	246	26	if	if	SCONJ
ejpam-5475	246	27	and	and	CCONJ
ejpam-5475	246	28	only	only	ADV
ejpam-5475	246	29	if	if	SCONJ
ejpam-5475	246	30	there	there	PRON
ejpam-5475	246	31	exists	exist	VERB
ejpam-5475	246	32	a	a	DET
ejpam-5475	246	33	(	(	PUNCT
ejpam-5475	246	34	ν1	ν1	NOUN
ejpam-5475	246	35	,	,	PUNCT
ejpam-5475	246	36	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	246	37	-	-	PUNCT
ejpam-5475	246	38	open	open	ADJ
ejpam-5475	246	39	set	set	NOUN
ejpam-5475	246	40	u	u	NOUN
ejpam-5475	246	41	containing	contain	VERB
ejpam-5475	246	42	x	x	PUNCT
ejpam-5475	246	43	such	such	ADJ
ejpam-5475	246	44	that	that	SCONJ
ejpam-5475	246	45	u	u	PROPN
ejpam-5475	246	46	⊆	⊆	NUM
ejpam-5475	246	47	cν2(u	cν2(u	NOUN
ejpam-5475	246	48	)	)	PUNCT
ejpam-5475	246	49	∩mν1	∩mν1	NOUN
ejpam-5475	246	50	⊆	⊆	NUM
ejpam-5475	246	51	a.	a.	NOUN
ejpam-5475	246	52	definition	definition	NOUN
ejpam-5475	246	53	7	7	NUM
ejpam-5475	246	54	.	.	PUNCT
ejpam-5475	247	1	let	let	VERB
ejpam-5475	247	2	ν1	ν1	NOUN
ejpam-5475	247	3	and	and	CCONJ
ejpam-5475	247	4	ν2	ν2	NOUN
ejpam-5475	247	5	be	be	AUX
ejpam-5475	247	6	two	two	NUM
ejpam-5475	247	7	gts	gts	NOUN
ejpam-5475	247	8	defined	define	VERB
ejpam-5475	247	9	on	on	ADP
ejpam-5475	247	10	a	a	DET
ejpam-5475	247	11	nonempty	nonempty	ADV
ejpam-5475	247	12	set	set	VERB
ejpam-5475	247	13	x.	x.	NOUN
ejpam-5475	248	1	we	we	PRON
ejpam-5475	248	2	say	say	VERB
ejpam-5475	248	3	that	that	SCONJ
ejpam-5475	248	4	x	x	PRON
ejpam-5475	248	5	is	be	AUX
ejpam-5475	248	6	g(ν1	g(ν1	NOUN
ejpam-5475	248	7	,	,	PUNCT
ejpam-5475	248	8	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	248	9	with	with	ADP
ejpam-5475	248	10	respect	respect	NOUN
ejpam-5475	248	11	to	to	ADP
ejpam-5475	248	12	mν1	mν1	PROPN
ejpam-5475	248	13	(	(	PUNCT
ejpam-5475	248	14	or	or	CCONJ
ejpam-5475	248	15	simply	simply	ADV
ejpam-5475	248	16	g(ν1	g(ν1	VERB
ejpam-5475	248	17	,	,	PUNCT
ejpam-5475	248	18	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	248	19	)	)	PUNCT
ejpam-5475	248	20	if	if	SCONJ
ejpam-5475	248	21	,	,	PUNCT
ejpam-5475	248	22	for	for	ADP
ejpam-5475	248	23	every	every	DET
ejpam-5475	248	24	x	x	SYM
ejpam-5475	248	25	∈	∈	PROPN
ejpam-5475	248	26	mν1	mν1	NOUN
ejpam-5475	248	27	and	and	CCONJ
ejpam-5475	248	28	every	every	DET
ejpam-5475	248	29	ν1	ν1	NOUN
ejpam-5475	248	30	-	-	PUNCT
ejpam-5475	248	31	closed	close	VERB
ejpam-5475	248	32	set	set	NOUN
ejpam-5475	248	33	f	f	PROPN
ejpam-5475	248	34	with	with	ADP
ejpam-5475	248	35	x	x	PROPN
ejpam-5475	248	36	/∈	/∈	PROPN
ejpam-5475	248	37	f	f	PROPN
ejpam-5475	248	38	,	,	PUNCT
ejpam-5475	248	39	there	there	PRON
ejpam-5475	248	40	exist	exist	VERB
ejpam-5475	248	41	open	open	ADJ
ejpam-5475	248	42	sets	set	NOUN
ejpam-5475	248	43	u	u	PROPN
ejpam-5475	248	44	∈	∈	PROPN
ejpam-5475	248	45	ν1	ν1	NOUN
ejpam-5475	248	46	and	and	CCONJ
ejpam-5475	248	47	v	v	NOUN
ejpam-5475	248	48	∈	∈	NOUN
ejpam-5475	248	49	ν2	ν2	NOUN
ejpam-5475	248	50	such	such	ADJ
ejpam-5475	248	51	that	that	SCONJ
ejpam-5475	248	52	:	:	PUNCT
ejpam-5475	248	53	x	x	X
ejpam-5475	248	54	∈	∈	PROPN
ejpam-5475	248	55	u	u	PROPN
ejpam-5475	248	56	,	,	PUNCT
ejpam-5475	248	57	f	f	PROPN
ejpam-5475	248	58	∩mν1	∩mν1	PROPN
ejpam-5475	248	59	⊆	⊆	NUM
ejpam-5475	248	60	v	v	NOUN
ejpam-5475	248	61	,	,	PUNCT
ejpam-5475	248	62	and	and	CCONJ
ejpam-5475	248	63	u	u	NOUN
ejpam-5475	248	64	∩	∩	NOUN
ejpam-5475	248	65	v	v	NOUN
ejpam-5475	248	66	=	=	PUNCT
ejpam-5475	248	67	∅.	∅.	NOUN
ejpam-5475	248	68	proposition	proposition	NOUN
ejpam-5475	248	69	1	1	NUM
ejpam-5475	248	70	.	.	PUNCT
ejpam-5475	249	1	let	let	VERB
ejpam-5475	249	2	ν1	ν1	NOUN
ejpam-5475	249	3	and	and	CCONJ
ejpam-5475	249	4	ν2	ν2	NOUN
ejpam-5475	249	5	be	be	AUX
ejpam-5475	249	6	two	two	NUM
ejpam-5475	249	7	gt	gt	NOUN
ejpam-5475	249	8	’s	’s	NOUN
ejpam-5475	249	9	on	on	ADP
ejpam-5475	249	10	a	a	DET
ejpam-5475	249	11	nonempty	nonempty	ADV
ejpam-5475	249	12	set	set	VERB
ejpam-5475	249	13	x	x	PUNCT
ejpam-5475	249	14	such	such	ADJ
ejpam-5475	249	15	that	that	DET
ejpam-5475	249	16	ν1	ν1	NOUN
ejpam-5475	249	17	=	=	NOUN
ejpam-5475	249	18	ν2	ν2	NOUN
ejpam-5475	249	19	.	.	PUNCT
ejpam-5475	250	1	if	if	SCONJ
ejpam-5475	250	2	x	x	PRON
ejpam-5475	250	3	is	be	AUX
ejpam-5475	250	4	(	(	PUNCT
ejpam-5475	250	5	ν1	ν1	NOUN
ejpam-5475	250	6	,	,	PUNCT
ejpam-5475	250	7	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	250	8	,	,	PUNCT
ejpam-5475	250	9	then	then	ADV
ejpam-5475	250	10	x	x	PUNCT
ejpam-5475	250	11	is	be	AUX
ejpam-5475	250	12	either	either	DET
ejpam-5475	250	13	ν1	ν1	NOUN
ejpam-5475	250	14	-	-	PUNCT
ejpam-5475	250	15	regular	regular	ADJ
ejpam-5475	250	16	or	or	CCONJ
ejpam-5475	250	17	ν2	ν2	NOUN
ejpam-5475	250	18	-	-	PUNCT
ejpam-5475	250	19	regular	regular	NOUN
ejpam-5475	250	20	.	.	PUNCT
ejpam-5475	251	1	proposition	proposition	NOUN
ejpam-5475	251	2	2	2	NUM
ejpam-5475	251	3	.	.	PUNCT
ejpam-5475	252	1	let	let	VERB
ejpam-5475	252	2	ν1	ν1	NOUN
ejpam-5475	252	3	and	and	CCONJ
ejpam-5475	252	4	ν2	ν2	NOUN
ejpam-5475	252	5	be	be	AUX
ejpam-5475	252	6	two	two	NUM
ejpam-5475	252	7	gt	gt	NOUN
ejpam-5475	252	8	’s	’s	NOUN
ejpam-5475	252	9	on	on	ADP
ejpam-5475	252	10	a	a	DET
ejpam-5475	252	11	nonempty	nonempty	ADV
ejpam-5475	252	12	set	set	VERB
ejpam-5475	252	13	x.	x.	NOUN
ejpam-5475	253	1	if	if	SCONJ
ejpam-5475	253	2	x	x	PRON
ejpam-5475	253	3	is	be	AUX
ejpam-5475	253	4	a	a	DET
ejpam-5475	253	5	(	(	PUNCT
ejpam-5475	253	6	ν1	ν1	NOUN
ejpam-5475	253	7	,	,	PUNCT
ejpam-5475	253	8	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	253	9	,	,	PUNCT
ejpam-5475	253	10	then	then	ADV
ejpam-5475	253	11	x	x	PUNCT
ejpam-5475	253	12	is	be	AUX
ejpam-5475	253	13	g(ν1	g(ν1	NOUN
ejpam-5475	253	14	,	,	PUNCT
ejpam-5475	253	15	ν2)-regular	ν2)-regular	ADJ
ejpam-5475	253	16	.	.	PUNCT
ejpam-5475	254	1	proof	proof	NOUN
ejpam-5475	254	2	.	.	PUNCT
ejpam-5475	255	1	let	let	VERB
ejpam-5475	255	2	x	x	PRON
ejpam-5475	255	3	be	be	AUX
ejpam-5475	255	4	(	(	PUNCT
ejpam-5475	255	5	ν1	ν1	NOUN
ejpam-5475	255	6	,	,	PUNCT
ejpam-5475	255	7	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	255	8	.	.	PUNCT
ejpam-5475	256	1	take	take	VERB
ejpam-5475	256	2	x	x	SYM
ejpam-5475	256	3	∈	∈	NOUN
ejpam-5475	256	4	mν1	mν1	NOUN
ejpam-5475	256	5	and	and	CCONJ
ejpam-5475	256	6	consider	consider	VERB
ejpam-5475	256	7	any	any	DET
ejpam-5475	256	8	ν1	ν1	NOUN
ejpam-5475	256	9	-	-	PUNCT
ejpam-5475	256	10	closed	close	VERB
ejpam-5475	256	11	set	set	NOUN
ejpam-5475	256	12	f	f	PROPN
ejpam-5475	256	13	such	such	ADJ
ejpam-5475	256	14	that	that	PRON
ejpam-5475	256	15	x	x	X
ejpam-5475	256	16	/∈	/∈	PROPN
ejpam-5475	256	17	f	f	PROPN
ejpam-5475	256	18	.	.	PUNCT
ejpam-5475	257	1	by	by	ADP
ejpam-5475	257	2	the	the	DET
ejpam-5475	257	3	definition	definition	NOUN
ejpam-5475	257	4	of	of	ADP
ejpam-5475	257	5	(	(	PUNCT
ejpam-5475	257	6	ν1	ν1	NOUN
ejpam-5475	257	7	,	,	PUNCT
ejpam-5475	257	8	ν2)-regularity	ν2)-regularity	NOUN
ejpam-5475	257	9	,	,	PUNCT
ejpam-5475	257	10	there	there	PRON
ejpam-5475	257	11	exist	exist	VERB
ejpam-5475	257	12	u	u	PROPN
ejpam-5475	257	13	∈	∈	PROPN
ejpam-5475	257	14	ν1	ν1	NOUN
ejpam-5475	257	15	,	,	PUNCT
ejpam-5475	257	16	v	v	NOUN
ejpam-5475	257	17	∈	∈	NOUN
ejpam-5475	257	18	ν2	ν2	NOUN
ejpam-5475	257	19	such	such	ADJ
ejpam-5475	257	20	that	that	SCONJ
ejpam-5475	257	21	x	x	SYM
ejpam-5475	257	22	∈	∈	PROPN
ejpam-5475	257	23	u	u	PROPN
ejpam-5475	257	24	,	,	PUNCT
ejpam-5475	257	25	f	f	PROPN
ejpam-5475	257	26	⊆	⊆	NUM
ejpam-5475	257	27	v	v	NOUN
ejpam-5475	257	28	,	,	PUNCT
ejpam-5475	257	29	and	and	CCONJ
ejpam-5475	257	30	u	u	NOUN
ejpam-5475	257	31	∩	∩	NOUN
ejpam-5475	257	32	v	v	NOUN
ejpam-5475	257	33	=	=	PUNCT
ejpam-5475	257	34	∅.	∅.	NOUN
ejpam-5475	257	35	since	since	SCONJ
ejpam-5475	257	36	f	f	PROPN
ejpam-5475	257	37	∩	∩	PROPN
ejpam-5475	257	38	mν1	mν1	ADJ
ejpam-5475	257	39	⊆	⊆	NUM
ejpam-5475	257	40	f	f	PROPN
ejpam-5475	257	41	⊆	⊆	NUM
ejpam-5475	257	42	v	v	NOUN
ejpam-5475	257	43	,	,	PUNCT
ejpam-5475	257	44	we	we	PRON
ejpam-5475	257	45	conclude	conclude	VERB
ejpam-5475	257	46	that	that	SCONJ
ejpam-5475	257	47	x	x	PUNCT
ejpam-5475	257	48	is	be	AUX
ejpam-5475	257	49	g(ν1	g(ν1	NOUN
ejpam-5475	257	50	,	,	PUNCT
ejpam-5475	257	51	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	257	52	.	.	PUNCT
ejpam-5475	258	1	theorem	theorem	VERB
ejpam-5475	258	2	12	12	NUM
ejpam-5475	258	3	.	.	PUNCT
ejpam-5475	259	1	let	let	VERB
ejpam-5475	259	2	x	x	PRON
ejpam-5475	259	3	be	be	AUX
ejpam-5475	259	4	a	a	DET
ejpam-5475	259	5	nonempty	nonempty	ADV
ejpam-5475	259	6	set	set	VERB
ejpam-5475	259	7	and	and	CCONJ
ejpam-5475	259	8	ν1	ν1	NOUN
ejpam-5475	259	9	,	,	PUNCT
ejpam-5475	259	10	ν2	ν2	NOUN
ejpam-5475	259	11	be	be	AUX
ejpam-5475	259	12	two	two	NUM
ejpam-5475	259	13	gt	gt	NOUN
ejpam-5475	259	14	’s	’s	NOUN
ejpam-5475	259	15	on	on	ADP
ejpam-5475	259	16	x.	x.	NOUN
ejpam-5475	259	17	the	the	DET
ejpam-5475	259	18	following	follow	VERB
ejpam-5475	259	19	statements	statement	NOUN
ejpam-5475	259	20	are	be	AUX
ejpam-5475	259	21	equivalent	equivalent	ADJ
ejpam-5475	259	22	:	:	PUNCT
ejpam-5475	259	23	(	(	PUNCT
ejpam-5475	259	24	i	i	NOUN
ejpam-5475	259	25	)	)	PUNCT
ejpam-5475	259	26	x	x	VERB
ejpam-5475	259	27	is	be	AUX
ejpam-5475	259	28	g(ν1	g(ν1	NOUN
ejpam-5475	259	29	,	,	PUNCT
ejpam-5475	259	30	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	259	31	.	.	PUNCT
ejpam-5475	260	1	(	(	PUNCT
ejpam-5475	260	2	ii	ii	NOUN
ejpam-5475	260	3	)	)	PUNCT
ejpam-5475	260	4	for	for	ADP
ejpam-5475	260	5	every	every	DET
ejpam-5475	260	6	x	x	SYM
ejpam-5475	260	7	∈	∈	PROPN
ejpam-5475	260	8	x	x	X
ejpam-5475	260	9	and	and	CCONJ
ejpam-5475	260	10	every	every	DET
ejpam-5475	260	11	ν1	ν1	NOUN
ejpam-5475	260	12	-	-	PUNCT
ejpam-5475	260	13	open	open	NOUN
ejpam-5475	260	14	set	set	NOUN
ejpam-5475	260	15	u	u	NOUN
ejpam-5475	260	16	containing	contain	VERB
ejpam-5475	260	17	x	x	PRON
ejpam-5475	260	18	,	,	PUNCT
ejpam-5475	260	19	there	there	PRON
ejpam-5475	260	20	exists	exist	VERB
ejpam-5475	260	21	a	a	DET
ejpam-5475	260	22	ν1	ν1	NOUN
ejpam-5475	260	23	-	-	PUNCT
ejpam-5475	260	24	open	open	NOUN
ejpam-5475	260	25	set	set	VERB
ejpam-5475	260	26	v	v	NOUN
ejpam-5475	260	27	containing	contain	VERB
ejpam-5475	260	28	x	x	PUNCT
ejpam-5475	260	29	such	such	ADJ
ejpam-5475	260	30	that	that	DET
ejpam-5475	260	31	v	v	ADP
ejpam-5475	260	32	⊆	⊆	NUM
ejpam-5475	260	33	cν2(v	cν2(v	PROPN
ejpam-5475	260	34	)	)	PUNCT
ejpam-5475	261	1	∩mν1	∩mν1	VERB
ejpam-5475	261	2	⊆	⊆	NUM
ejpam-5475	261	3	u	u	NOUN
ejpam-5475	261	4	.	.	PUNCT
ejpam-5475	262	1	proof	proof	NOUN
ejpam-5475	262	2	.	.	PUNCT
ejpam-5475	263	1	(	(	PUNCT
ejpam-5475	263	2	i	i	NOUN
ejpam-5475	263	3	)	)	PUNCT
ejpam-5475	263	4	⇒	⇒	PROPN
ejpam-5475	263	5	(	(	PUNCT
ejpam-5475	263	6	ii	ii	PROPN
ejpam-5475	263	7	):	):	PUNCT
ejpam-5475	263	8	assume	assume	VERB
ejpam-5475	263	9	x	x	PUNCT
ejpam-5475	263	10	is	be	AUX
ejpam-5475	263	11	g(ν1	g(ν1	NOUN
ejpam-5475	263	12	,	,	PUNCT
ejpam-5475	263	13	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	263	14	.	.	PUNCT
ejpam-5475	264	1	for	for	ADP
ejpam-5475	264	2	x	x	SYM
ejpam-5475	264	3	∈	∈	PROPN
ejpam-5475	264	4	mν1	mν1	NOUN
ejpam-5475	264	5	and	and	CCONJ
ejpam-5475	264	6	a	a	DET
ejpam-5475	264	7	ν1	ν1	NOUN
ejpam-5475	264	8	-	-	PUNCT
ejpam-5475	264	9	open	open	NOUN
ejpam-5475	264	10	set	set	NOUN
ejpam-5475	264	11	u	u	NOUN
ejpam-5475	264	12	containing	contain	VERB
ejpam-5475	264	13	x	x	X
ejpam-5475	264	14	,	,	PUNCT
ejpam-5475	264	15	there	there	PRON
ejpam-5475	264	16	exist	exist	VERB
ejpam-5475	264	17	v	v	ADP
ejpam-5475	264	18	∈	∈	PROPN
ejpam-5475	264	19	ν1	ν1	NOUN
ejpam-5475	264	20	and	and	CCONJ
ejpam-5475	264	21	w	w	NOUN
ejpam-5475	264	22	∈	∈	NOUN
ejpam-5475	264	23	ν2	ν2	NOUN
ejpam-5475	264	24	such	such	ADJ
ejpam-5475	264	25	that	that	SCONJ
ejpam-5475	264	26	x	x	SYM
ejpam-5475	264	27	∈	∈	PROPN
ejpam-5475	264	28	v	v	NOUN
ejpam-5475	264	29	,	,	PUNCT
ejpam-5475	264	30	(	(	PUNCT
ejpam-5475	264	31	x	x	X
ejpam-5475	264	32	−	−	PROPN
ejpam-5475	264	33	u	u	NOUN
ejpam-5475	264	34	)	)	PUNCT
ejpam-5475	264	35	∩mν1	∩mν1	PROPN
ejpam-5475	264	36	⊆	⊆	NUM
ejpam-5475	264	37	w	w	NOUN
ejpam-5475	264	38	,	,	PUNCT
ejpam-5475	264	39	and	and	CCONJ
ejpam-5475	264	40	v	v	ADP
ejpam-5475	264	41	⊆	⊆	NUM
ejpam-5475	264	42	x−w	x−w	PROPN
ejpam-5475	264	43	.	.	PUNCT
ejpam-5475	265	1	sincex−w	sincex−w	PROPN
ejpam-5475	265	2	is	be	AUX
ejpam-5475	265	3	ν2	ν2	ADV
ejpam-5475	265	4	-	-	PUNCT
ejpam-5475	265	5	closed	closed	ADJ
ejpam-5475	265	6	,	,	PUNCT
ejpam-5475	265	7	cν2(v	cν2(v	PROPN
ejpam-5475	265	8	)	)	PUNCT
ejpam-5475	266	1	⊆	⊆	NUM
ejpam-5475	266	2	x−w	x−w	PROPN
ejpam-5475	266	3	.	.	PUNCT
ejpam-5475	267	1	hence	hence	ADV
ejpam-5475	267	2	,	,	PUNCT
ejpam-5475	267	3	cν2(v	cν2(v	PROPN
ejpam-5475	267	4	)	)	PUNCT
ejpam-5475	267	5	∩	∩	NOUN
ejpam-5475	267	6	(	(	PUNCT
ejpam-5475	267	7	(	(	PUNCT
ejpam-5475	267	8	x−u)∩mν1	x−u)∩mν1	PROPN
ejpam-5475	267	9	)	)	PUNCT
ejpam-5475	267	10	⊆	⊆	NUM
ejpam-5475	267	11	cν2(v	cν2(v	NOUN
ejpam-5475	267	12	)	)	PUNCT
ejpam-5475	267	13	∩w	∩w	NOUN
ejpam-5475	268	1	=	=	VERB
ejpam-5475	268	2	∅	∅	NOUN
ejpam-5475	268	3	,	,	PUNCT
ejpam-5475	268	4	implying	imply	VERB
ejpam-5475	268	5	v	v	ADP
ejpam-5475	268	6	⊆	⊆	NUM
ejpam-5475	268	7	cν2(v	cν2(v	PROPN
ejpam-5475	268	8	)	)	PUNCT
ejpam-5475	268	9	∩mν1	∩mν1	VERB
ejpam-5475	268	10	⊆	⊆	NUM
ejpam-5475	268	11	u	u	NOUN
ejpam-5475	268	12	.	.	PUNCT
ejpam-5475	269	1	(	(	PUNCT
ejpam-5475	269	2	ii	ii	NOUN
ejpam-5475	269	3	)	)	PUNCT
ejpam-5475	269	4	⇒	⇒	NOUN
ejpam-5475	269	5	(	(	PUNCT
ejpam-5475	269	6	i	i	NOUN
ejpam-5475	269	7	):	):	PUNCT
ejpam-5475	269	8	let	let	VERB
ejpam-5475	269	9	f	f	PRON
ejpam-5475	269	10	be	be	AUX
ejpam-5475	269	11	a	a	DET
ejpam-5475	269	12	ν1	ν1	NOUN
ejpam-5475	269	13	-	-	PUNCT
ejpam-5475	269	14	closed	close	VERB
ejpam-5475	269	15	set	set	NOUN
ejpam-5475	269	16	and	and	CCONJ
ejpam-5475	269	17	x	x	SYM
ejpam-5475	269	18	∈	∈	NOUN
ejpam-5475	269	19	mν1	mν1	X
ejpam-5475	269	20	with	with	ADP
ejpam-5475	269	21	x	x	PROPN
ejpam-5475	269	22	/∈	/∈	PROPN
ejpam-5475	269	23	f	f	PROPN
ejpam-5475	269	24	.	.	PUNCT
ejpam-5475	270	1	since	since	SCONJ
ejpam-5475	270	2	x	x	PRON
ejpam-5475	270	3	−	−	PROPN
ejpam-5475	270	4	f	f	PROPN
ejpam-5475	270	5	is	be	AUX
ejpam-5475	270	6	a	a	DET
ejpam-5475	270	7	ν1	ν1	NOUN
ejpam-5475	270	8	-	-	PUNCT
ejpam-5475	270	9	open	open	ADJ
ejpam-5475	270	10	set	set	NOUN
ejpam-5475	270	11	containing	contain	VERB
ejpam-5475	270	12	x	x	X
ejpam-5475	270	13	,	,	PUNCT
ejpam-5475	270	14	by	by	ADP
ejpam-5475	270	15	hypothesis	hypothesis	NOUN
ejpam-5475	270	16	,	,	PUNCT
ejpam-5475	270	17	there	there	PRON
ejpam-5475	270	18	exists	exist	VERB
ejpam-5475	270	19	a	a	DET
ejpam-5475	270	20	ν1	ν1	NOUN
ejpam-5475	270	21	-	-	PUNCT
ejpam-5475	270	22	open	open	NOUN
ejpam-5475	270	23	set	set	VERB
ejpam-5475	270	24	v	v	NOUN
ejpam-5475	270	25	containing	contain	VERB
ejpam-5475	270	26	x	x	PUNCT
ejpam-5475	270	27	such	such	ADJ
ejpam-5475	270	28	that	that	SCONJ
ejpam-5475	270	29	x	x	SYM
ejpam-5475	270	30	∈	∈	NOUN
ejpam-5475	270	31	v	v	ADP
ejpam-5475	270	32	⊆	⊆	NUM
ejpam-5475	270	33	cν2(v	cν2(v	PROPN
ejpam-5475	270	34	)	)	PUNCT
ejpam-5475	270	35	∩	∩	PROPN
ejpam-5475	270	36	mν1	mν1	ADJ
ejpam-5475	270	37	⊆	⊆	NUM
ejpam-5475	271	1	x	x	SYM
ejpam-5475	271	2	−	−	PROPN
ejpam-5475	271	3	f	f	NOUN
ejpam-5475	271	4	.	.	PUNCT
ejpam-5475	272	1	this	this	PRON
ejpam-5475	272	2	implies	imply	VERB
ejpam-5475	272	3	cν2(v	cν2(v	PROPN
ejpam-5475	272	4	)	)	PUNCT
ejpam-5475	272	5	∩	∩	PROPN
ejpam-5475	272	6	mν1	mν1	ADJ
ejpam-5475	272	7	∩	∩	NOUN
ejpam-5475	272	8	f	f	X
ejpam-5475	272	9	=	=	PUNCT
ejpam-5475	272	10	∅.	∅.	VERB
ejpam-5475	272	11	hence	hence	ADV
ejpam-5475	272	12	,	,	PUNCT
ejpam-5475	272	13	f	f	PROPN
ejpam-5475	272	14	∩mν1	∩mν1	PROPN
ejpam-5475	272	15	⊆	⊆	NUM
ejpam-5475	272	16	x−	x−	PROPN
ejpam-5475	272	17	cν2(v	cν2(v	PROPN
ejpam-5475	272	18	)	)	PUNCT
ejpam-5475	272	19	,	,	PUNCT
ejpam-5475	272	20	and	and	CCONJ
ejpam-5475	272	21	since	since	SCONJ
ejpam-5475	272	22	x−	x−	PROPN
ejpam-5475	272	23	cν2(v	cν2(v	PROPN
ejpam-5475	272	24	)	)	PUNCT
ejpam-5475	272	25	∈	∈	PROPN
ejpam-5475	272	26	ν2	ν2	NOUN
ejpam-5475	272	27	and	and	CCONJ
ejpam-5475	272	28	v	v	ADP
ejpam-5475	272	29	∩	∩	NOUN
ejpam-5475	272	30	(	(	PUNCT
ejpam-5475	272	31	x−	x−	PROPN
ejpam-5475	272	32	cν2(v	cν2(v	PROPN
ejpam-5475	272	33	)	)	PUNCT
ejpam-5475	272	34	)	)	PUNCT
ejpam-5475	273	1	=	=	NOUN
ejpam-5475	273	2	∅	∅	NOUN
ejpam-5475	273	3	,	,	PUNCT
ejpam-5475	273	4	we	we	PRON
ejpam-5475	273	5	conclude	conclude	VERB
ejpam-5475	273	6	x	x	VERB
ejpam-5475	273	7	is	be	AUX
ejpam-5475	273	8	g(ν1	g(ν1	NOUN
ejpam-5475	273	9	,	,	PUNCT
ejpam-5475	273	10	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	273	11	.	.	PROPN
ejpam-5475	273	12	theorem	theorem	VERB
ejpam-5475	273	13	13	13	NUM
ejpam-5475	273	14	.	.	PUNCT
ejpam-5475	274	1	let	let	VERB
ejpam-5475	274	2	ν1	ν1	NOUN
ejpam-5475	274	3	and	and	CCONJ
ejpam-5475	274	4	ν2	ν2	NOUN
ejpam-5475	274	5	be	be	AUX
ejpam-5475	274	6	two	two	NUM
ejpam-5475	274	7	gt	gt	NOUN
ejpam-5475	274	8	’s	’s	NOUN
ejpam-5475	274	9	on	on	ADP
ejpam-5475	274	10	a	a	DET
ejpam-5475	274	11	nonempty	nonempty	ADV
ejpam-5475	274	12	set	set	VERB
ejpam-5475	274	13	x.	x.	NOUN
ejpam-5475	274	14	if	if	SCONJ
ejpam-5475	274	15	x	x	PRON
ejpam-5475	274	16	is	be	AUX
ejpam-5475	274	17	g(ν1	g(ν1	NOUN
ejpam-5475	274	18	,	,	PUNCT
ejpam-5475	274	19	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	274	20	,	,	PUNCT
ejpam-5475	274	21	then	then	ADV
ejpam-5475	274	22	every	every	DET
ejpam-5475	274	23	ν1	ν1	NOUN
ejpam-5475	274	24	-	-	PUNCT
ejpam-5475	274	25	open	open	ADJ
ejpam-5475	274	26	set	set	NOUN
ejpam-5475	274	27	is	be	AUX
ejpam-5475	274	28	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	274	29	,	,	PUNCT
ejpam-5475	274	30	ν2)-open	ν2)-open	NOUN
ejpam-5475	274	31	.	.	PUNCT
ejpam-5475	275	1	a.	a.	NOUN
ejpam-5475	275	2	qahis	qahis	PROPN
ejpam-5475	275	3	,	,	PUNCT
ejpam-5475	275	4	a.	a.	NOUN
ejpam-5475	275	5	alqahtani	alqahtani	PROPN
ejpam-5475	275	6	/	/	SYM
ejpam-5475	275	7	eur	eur	PROPN
ejpam-5475	275	8	.	.	PUNCT
ejpam-5475	276	1	j.	j.	PROPN
ejpam-5475	276	2	pure	pure	PROPN
ejpam-5475	276	3	appl	appl	PROPN
ejpam-5475	276	4	.	.	PROPN
ejpam-5475	276	5	math	math	PROPN
ejpam-5475	276	6	,	,	PUNCT
ejpam-5475	276	7	17	17	NUM
ejpam-5475	276	8	(	(	PUNCT
ejpam-5475	276	9	4	4	NUM
ejpam-5475	276	10	)	)	PUNCT
ejpam-5475	276	11	(	(	PUNCT
ejpam-5475	276	12	2024	2024	NUM
ejpam-5475	276	13	)	)	PUNCT
ejpam-5475	276	14	,	,	PUNCT
ejpam-5475	276	15	3610	3610	NUM
ejpam-5475	276	16	-	-	SYM
ejpam-5475	276	17	3621	3621	NUM
ejpam-5475	276	18	3619	3619	NUM
ejpam-5475	276	19	proof	proof	NOUN
ejpam-5475	276	20	.	.	PUNCT
ejpam-5475	277	1	let	let	VERB
ejpam-5475	277	2	x	x	PRON
ejpam-5475	277	3	be	be	AUX
ejpam-5475	277	4	g(ν1	g(ν1	NOUN
ejpam-5475	277	5	,	,	PUNCT
ejpam-5475	277	6	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	277	7	,	,	PUNCT
ejpam-5475	277	8	and	and	CCONJ
ejpam-5475	277	9	consider	consider	VERB
ejpam-5475	277	10	any	any	DET
ejpam-5475	277	11	ν1	ν1	NOUN
ejpam-5475	277	12	-	-	PUNCT
ejpam-5475	277	13	open	open	NOUN
ejpam-5475	277	14	set	set	VERB
ejpam-5475	277	15	a	a	PRON
ejpam-5475	277	16	in	in	ADP
ejpam-5475	277	17	x.	x.	NOUN
ejpam-5475	277	18	for	for	ADP
ejpam-5475	277	19	each	each	DET
ejpam-5475	277	20	x	x	PROPN
ejpam-5475	277	21	∈	∈	PROPN
ejpam-5475	277	22	a	a	PRON
ejpam-5475	277	23	,	,	PUNCT
ejpam-5475	277	24	by	by	ADP
ejpam-5475	277	25	theorem	theorem	NOUN
ejpam-5475	277	26	12	12	NUM
ejpam-5475	277	27	,	,	PUNCT
ejpam-5475	277	28	there	there	PRON
ejpam-5475	277	29	exists	exist	VERB
ejpam-5475	277	30	a	a	DET
ejpam-5475	277	31	ν1	ν1	NOUN
ejpam-5475	277	32	-	-	PUNCT
ejpam-5475	277	33	open	open	NOUN
ejpam-5475	277	34	set	set	VERB
ejpam-5475	277	35	v	v	ADP
ejpam-5475	277	36	such	such	ADJ
ejpam-5475	277	37	that	that	SCONJ
ejpam-5475	277	38	x	x	SYM
ejpam-5475	277	39	∈	∈	NOUN
ejpam-5475	277	40	v	v	ADP
ejpam-5475	277	41	⊆	⊆	NUM
ejpam-5475	277	42	cν2(v	cν2(v	PROPN
ejpam-5475	277	43	)	)	PUNCT
ejpam-5475	277	44	∩mν1	∩mν1	VERB
ejpam-5475	278	1	⊆	⊆	NUM
ejpam-5475	278	2	a.	a.	NOUN
ejpam-5475	278	3	hence	hence	ADV
ejpam-5475	278	4	,	,	PUNCT
ejpam-5475	278	5	a	a	PRON
ejpam-5475	278	6	is	be	AUX
ejpam-5475	278	7	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	278	8	,	,	PUNCT
ejpam-5475	278	9	ν2)-open	ν2)-open	ADJ
ejpam-5475	278	10	.	.	PUNCT
ejpam-5475	279	1	corollary	corollary	ADJ
ejpam-5475	279	2	5	5	NUM
ejpam-5475	279	3	.	.	PUNCT
ejpam-5475	280	1	let	let	VERB
ejpam-5475	280	2	ν1	ν1	NOUN
ejpam-5475	280	3	and	and	CCONJ
ejpam-5475	280	4	ν2	ν2	NOUN
ejpam-5475	280	5	be	be	VERB
ejpam-5475	280	6	twogt	twogt	NOUN
ejpam-5475	280	7	’s	’s	NOUN
ejpam-5475	280	8	on	on	ADP
ejpam-5475	280	9	a	a	DET
ejpam-5475	280	10	nonempty	nonempty	ADV
ejpam-5475	280	11	set	set	VERB
ejpam-5475	280	12	x.	x.	NOUN
ejpam-5475	280	13	if	if	SCONJ
ejpam-5475	280	14	x	x	PRON
ejpam-5475	280	15	is	be	AUX
ejpam-5475	280	16	g(ν1	g(ν1	NOUN
ejpam-5475	280	17	,	,	PUNCT
ejpam-5475	280	18	ν2)-regular	ν2)-regular	PROPN
ejpam-5475	280	19	,	,	PUNCT
ejpam-5475	280	20	then	then	ADV
ejpam-5475	280	21	ν1	ν1	PROPN
ejpam-5475	280	22	=	=	SYM
ejpam-5475	280	23	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	280	24	,	,	PUNCT
ejpam-5475	280	25	ν2	ν2	NOUN
ejpam-5475	280	26	)	)	PUNCT
ejpam-5475	280	27	.	.	PUNCT
ejpam-5475	281	1	proof	proof	NOUN
ejpam-5475	281	2	.	.	PUNCT
ejpam-5475	282	1	it	it	PRON
ejpam-5475	282	2	can	can	AUX
ejpam-5475	282	3	be	be	AUX
ejpam-5475	282	4	deduced	deduce	VERB
ejpam-5475	282	5	from	from	ADP
ejpam-5475	282	6	theorem	theorem	ADJ
ejpam-5475	282	7	8	8	NUM
ejpam-5475	282	8	and	and	CCONJ
ejpam-5475	282	9	theorem	theorem	VERB
ejpam-5475	282	10	13	13	NUM
ejpam-5475	282	11	.	.	PUNCT
ejpam-5475	283	1	definition	definition	NOUN
ejpam-5475	283	2	8	8	NUM
ejpam-5475	283	3	.	.	PUNCT
ejpam-5475	284	1	let	let	VERB
ejpam-5475	284	2	ν1	ν1	NOUN
ejpam-5475	284	3	and	and	CCONJ
ejpam-5475	284	4	ν2	ν2	NOUN
ejpam-5475	284	5	be	be	AUX
ejpam-5475	284	6	two	two	NUM
ejpam-5475	284	7	gts	gts	NOUN
ejpam-5475	284	8	defined	define	VERB
ejpam-5475	284	9	on	on	ADP
ejpam-5475	284	10	a	a	DET
ejpam-5475	284	11	nonempty	nonempty	ADV
ejpam-5475	284	12	set	set	VERB
ejpam-5475	284	13	x	x	NOUN
ejpam-5475	284	14	,	,	PUNCT
ejpam-5475	284	15	and	and	CCONJ
ejpam-5475	284	16	let	let	VERB
ejpam-5475	284	17	a	a	DET
ejpam-5475	284	18	⊆	⊆	NUM
ejpam-5475	284	19	x.	x.	NOUN
ejpam-5475	284	20	define	define	VERB
ejpam-5475	284	21	the	the	DET
ejpam-5475	284	22	following	follow	VERB
ejpam-5475	284	23	notions	notion	NOUN
ejpam-5475	284	24	:	:	PUNCT
ejpam-5475	284	25	cθ̃(ν1,ν2)(a	cθ̃(ν1,ν2)(a	NOUN
ejpam-5475	284	26	)	)	PUNCT
ejpam-5475	285	1	=	=	SYM
ejpam-5475	285	2	⋂	⋂	PROPN
ejpam-5475	285	3	{	{	PUNCT
ejpam-5475	285	4	f	f	NOUN
ejpam-5475	285	5	⊆	⊆	NUM
ejpam-5475	285	6	x	x	PUNCT
ejpam-5475	285	7	|	|	ADV
ejpam-5475	285	8	a	a	DET
ejpam-5475	285	9	⊆	⊆	NUM
ejpam-5475	285	10	f	f	NOUN
ejpam-5475	285	11	for	for	ADP
ejpam-5475	285	12	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	285	13	,	,	PUNCT
ejpam-5475	285	14	ν2)-closed	ν2)-close	VERB
ejpam-5475	285	15	set	set	VERB
ejpam-5475	285	16	f	f	PROPN
ejpam-5475	285	17	in	in	ADP
ejpam-5475	285	18	x	x	X
ejpam-5475	285	19	}	}	PUNCT
ejpam-5475	285	20	;	;	PUNCT
ejpam-5475	285	21	iθ̃(ν1,ν2)(a	iθ̃(ν1,ν2)(a	PROPN
ejpam-5475	285	22	)	)	PUNCT
ejpam-5475	285	23	=	=	SYM
ejpam-5475	286	1	⋃	⋃	NOUN
ejpam-5475	286	2	{	{	PUNCT
ejpam-5475	286	3	v	v	ADP
ejpam-5475	286	4	⊆	⊆	NUM
ejpam-5475	286	5	x	x	NOUN
ejpam-5475	286	6	|	|	ADV
ejpam-5475	286	7	v	v	ADP
ejpam-5475	286	8	⊆	⊆	NUM
ejpam-5475	286	9	a	a	PRON
ejpam-5475	286	10	for	for	ADP
ejpam-5475	286	11	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	286	12	,	,	PUNCT
ejpam-5475	286	13	ν2)-open	ν2)-open	VERB
ejpam-5475	286	14	set	set	VERB
ejpam-5475	286	15	v	v	NOUN
ejpam-5475	286	16	in	in	ADP
ejpam-5475	286	17	x	x	X
ejpam-5475	286	18	}	}	PUNCT
ejpam-5475	286	19	;	;	PUNCT
ejpam-5475	286	20	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	PROPN
ejpam-5475	286	21	)	)	PUNCT
ejpam-5475	286	22	=	=	PRON
ejpam-5475	286	23	{	{	PUNCT
ejpam-5475	286	24	x	x	PUNCT
ejpam-5475	286	25	∈	∈	PROPN
ejpam-5475	286	26	x	x	PUNCT
ejpam-5475	286	27	|	|	ADV
ejpam-5475	286	28	cν2(m	cν2(m	NOUN
ejpam-5475	286	29	)	)	PUNCT
ejpam-5475	286	30	∩mν1	∩mν1	NOUN
ejpam-5475	287	1	⊆	⊆	NUM
ejpam-5475	287	2	a	a	DET
ejpam-5475	287	3	for	for	ADP
ejpam-5475	287	4	some	some	DET
ejpam-5475	287	5	ν1	ν1	NOUN
ejpam-5475	287	6	-	-	PUNCT
ejpam-5475	287	7	open	open	NOUN
ejpam-5475	287	8	set	set	NOUN
ejpam-5475	287	9	m	m	AUX
ejpam-5475	287	10	containing	contain	VERB
ejpam-5475	287	11	x	x	X
ejpam-5475	287	12	}	}	PUNCT
ejpam-5475	287	13	.	.	PUNCT
ejpam-5475	288	1	note	note	VERB
ejpam-5475	288	2	that	that	SCONJ
ejpam-5475	288	3	x	x	X
ejpam-5475	288	4	∈	∈	PROPN
ejpam-5475	288	5	cθ̃(ν1,ν2)(a	cθ̃(ν1,ν2)(a	NOUN
ejpam-5475	288	6	)	)	PUNCT
ejpam-5475	288	7	if	if	SCONJ
ejpam-5475	288	8	and	and	CCONJ
ejpam-5475	288	9	only	only	ADV
ejpam-5475	288	10	if	if	SCONJ
ejpam-5475	288	11	∀u	∀u	NOUN
ejpam-5475	288	12	∈	∈	PROPN
ejpam-5475	288	13	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	288	14	,	,	PUNCT
ejpam-5475	288	15	ν2	ν2	NOUN
ejpam-5475	288	16	)	)	PUNCT
ejpam-5475	288	17	,	,	PUNCT
ejpam-5475	288	18	(	(	PUNCT
ejpam-5475	288	19	x	x	SYM
ejpam-5475	288	20	∈	∈	NOUN
ejpam-5475	288	21	u	u	NOUN
ejpam-5475	288	22	⇒	⇒	X
ejpam-5475	288	23	u	u	NOUN
ejpam-5475	288	24	∩a	∩a	PROPN
ejpam-5475	288	25	̸=	̸=	PROPN
ejpam-5475	288	26	∅	∅	NOUN
ejpam-5475	288	27	)	)	PUNCT
ejpam-5475	288	28	.	.	PUNCT
ejpam-5475	289	1	theorem	theorem	NOUN
ejpam-5475	289	2	14	14	NUM
ejpam-5475	289	3	.	.	PUNCT
ejpam-5475	290	1	let	let	VERB
ejpam-5475	290	2	ν1	ν1	NOUN
ejpam-5475	290	3	and	and	CCONJ
ejpam-5475	290	4	ν2	ν2	NOUN
ejpam-5475	290	5	be	be	AUX
ejpam-5475	290	6	two	two	NUM
ejpam-5475	290	7	gt	gt	NOUN
ejpam-5475	290	8	’s	’s	NOUN
ejpam-5475	290	9	on	on	ADP
ejpam-5475	290	10	a	a	DET
ejpam-5475	290	11	nonempty	nonempty	ADV
ejpam-5475	290	12	set	set	VERB
ejpam-5475	290	13	x	x	NOUN
ejpam-5475	290	14	,	,	PUNCT
ejpam-5475	290	15	and	and	CCONJ
ejpam-5475	290	16	let	let	VERB
ejpam-5475	290	17	a	a	DET
ejpam-5475	290	18	⊆	⊆	NUM
ejpam-5475	290	19	x.	x.	NOUN
ejpam-5475	290	20	then	then	ADV
ejpam-5475	290	21	the	the	DET
ejpam-5475	290	22	following	follow	VERB
ejpam-5475	290	23	hold	hold	NOUN
ejpam-5475	290	24	:	:	PUNCT
ejpam-5475	290	25	(	(	PUNCT
ejpam-5475	290	26	i	i	NOUN
ejpam-5475	290	27	)	)	PUNCT
ejpam-5475	290	28	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	PROPN
ejpam-5475	290	29	)	)	PUNCT
ejpam-5475	291	1	⊆	⊆	NUM
ejpam-5475	291	2	cθ̃(ν1,ν2)(a	cθ̃(ν1,ν2)(a	NOUN
ejpam-5475	291	3	)	)	PUNCT
ejpam-5475	291	4	⊆	⊆	NUM
ejpam-5475	291	5	cθ(ν1,ν2)(a	cθ(ν1,ν2)(a	NOUN
ejpam-5475	291	6	)	)	PUNCT
ejpam-5475	291	7	.	.	PUNCT
ejpam-5475	292	1	(	(	PUNCT
ejpam-5475	292	2	ii	ii	NOUN
ejpam-5475	292	3	)	)	PUNCT
ejpam-5475	292	4	for	for	ADP
ejpam-5475	292	5	any	any	DET
ejpam-5475	292	6	x	x	SYM
ejpam-5475	292	7	∈	∈	PROPN
ejpam-5475	292	8	x	x	NOUN
ejpam-5475	292	9	,	,	PUNCT
ejpam-5475	292	10	x	x	PROPN
ejpam-5475	292	11	∈	∈	PROPN
ejpam-5475	292	12	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	PROPN
ejpam-5475	292	13	)	)	PUNCT
ejpam-5475	292	14	if	if	SCONJ
ejpam-5475	292	15	and	and	CCONJ
ejpam-5475	292	16	only	only	ADV
ejpam-5475	292	17	if	if	SCONJ
ejpam-5475	292	18	there	there	PRON
ejpam-5475	292	19	exists	exist	VERB
ejpam-5475	292	20	a	a	DET
ejpam-5475	292	21	ν1	ν1	NOUN
ejpam-5475	292	22	-	-	PUNCT
ejpam-5475	292	23	open	open	NOUN
ejpam-5475	292	24	set	set	NOUN
ejpam-5475	292	25	m	m	VERB
ejpam-5475	293	1	such	such	ADJ
ejpam-5475	293	2	that	that	SCONJ
ejpam-5475	293	3	x	x	SYM
ejpam-5475	293	4	∈	∈	PROPN
ejpam-5475	293	5	m	m	NOUN
ejpam-5475	293	6	and	and	CCONJ
ejpam-5475	293	7	m	m	PROPN
ejpam-5475	293	8	⊆	⊆	NUM
ejpam-5475	293	9	cν2(m	cν2(m	NOUN
ejpam-5475	293	10	)	)	PUNCT
ejpam-5475	293	11	∩mν1	∩mν1	NOUN
ejpam-5475	293	12	⊆	⊆	NUM
ejpam-5475	293	13	a.	a.	NOUN
ejpam-5475	293	14	proof	proof	NOUN
ejpam-5475	293	15	.	.	PUNCT
ejpam-5475	294	1	(	(	PUNCT
ejpam-5475	294	2	i	i	NOUN
ejpam-5475	294	3	)	)	PUNCT
ejpam-5475	294	4	let	let	VERB
ejpam-5475	294	5	x	x	PRON
ejpam-5475	294	6	/∈	/∈	PUNCT
ejpam-5475	295	1	cθ̃(ν1,ν2)(a	cθ̃(ν1,ν2)(a	NOUN
ejpam-5475	295	2	)	)	PUNCT
ejpam-5475	295	3	.	.	PUNCT
ejpam-5475	296	1	this	this	PRON
ejpam-5475	296	2	implies	imply	VERB
ejpam-5475	296	3	there	there	PRON
ejpam-5475	296	4	exists	exist	VERB
ejpam-5475	296	5	a	a	DET
ejpam-5475	296	6	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	296	7	,	,	PUNCT
ejpam-5475	296	8	ν2)-open	ν2)-open	VERB
ejpam-5475	296	9	set	set	VERB
ejpam-5475	296	10	v	v	ADP
ejpam-5475	296	11	such	such	ADJ
ejpam-5475	296	12	that	that	SCONJ
ejpam-5475	296	13	x	x	SYM
ejpam-5475	296	14	∈	∈	PROPN
ejpam-5475	296	15	v	v	NOUN
ejpam-5475	296	16	and	and	CCONJ
ejpam-5475	296	17	v	v	ADP
ejpam-5475	296	18	∩a	∩a	NOUN
ejpam-5475	296	19	=	=	PUNCT
ejpam-5475	296	20	∅.	∅.	NOUN
ejpam-5475	296	21	since	since	SCONJ
ejpam-5475	296	22	v	v	NOUN
ejpam-5475	296	23	is	be	AUX
ejpam-5475	296	24	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	296	25	,	,	PUNCT
ejpam-5475	296	26	ν2)-open	ν2)-open	ADJ
ejpam-5475	296	27	,	,	PUNCT
ejpam-5475	296	28	there	there	PRON
ejpam-5475	296	29	exists	exist	VERB
ejpam-5475	296	30	m	m	PROPN
ejpam-5475	296	31	∈	∈	NOUN
ejpam-5475	296	32	ν1	ν1	NOUN
ejpam-5475	296	33	such	such	ADJ
ejpam-5475	296	34	that	that	SCONJ
ejpam-5475	296	35	x	x	SYM
ejpam-5475	296	36	∈	∈	PROPN
ejpam-5475	296	37	m	m	NOUN
ejpam-5475	296	38	and	and	CCONJ
ejpam-5475	296	39	m	m	PROPN
ejpam-5475	296	40	⊆	⊆	NUM
ejpam-5475	296	41	cν2(m	cν2(m	NOUN
ejpam-5475	296	42	)	)	PUNCT
ejpam-5475	296	43	∩	∩	NOUN
ejpam-5475	296	44	mν1	mν1	ADJ
ejpam-5475	296	45	⊆	⊆	NUM
ejpam-5475	296	46	v	v	ADP
ejpam-5475	296	47	⊆	⊆	NUM
ejpam-5475	296	48	x	x	SYM
ejpam-5475	296	49	−	−	NOUN
ejpam-5475	296	50	a.	a.	NOUN
ejpam-5475	296	51	this	this	PRON
ejpam-5475	296	52	implies	imply	VERB
ejpam-5475	296	53	(	(	PUNCT
ejpam-5475	296	54	cν2(m	cν2(m	NOUN
ejpam-5475	296	55	)	)	PUNCT
ejpam-5475	296	56	∩	∩	NOUN
ejpam-5475	296	57	mν1	mν1	X
ejpam-5475	296	58	)	)	PUNCT
ejpam-5475	296	59	∩	∩	NOUN
ejpam-5475	296	60	a	a	DET
ejpam-5475	296	61	=	=	SYM
ejpam-5475	296	62	∅	∅	NOUN
ejpam-5475	296	63	,	,	PUNCT
ejpam-5475	296	64	hence	hence	ADV
ejpam-5475	296	65	x	x	X
ejpam-5475	296	66	/∈	/∈	PUNCT
ejpam-5475	296	67	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	296	68	)	)	PUNCT
ejpam-5475	296	69	.	.	PUNCT
ejpam-5475	297	1	thus	thus	ADV
ejpam-5475	297	2	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	297	3	)	)	PUNCT
ejpam-5475	298	1	⊆	⊆	NUM
ejpam-5475	298	2	cθ̃(ν1,ν2)(a	cθ̃(ν1,ν2)(a	NOUN
ejpam-5475	298	3	)	)	PUNCT
ejpam-5475	298	4	.	.	PUNCT
ejpam-5475	299	1	since	since	SCONJ
ejpam-5475	299	2	every	every	DET
ejpam-5475	299	3	θ(ν1	θ(ν1	NOUN
ejpam-5475	299	4	,	,	PUNCT
ejpam-5475	299	5	ν2)-open	ν2)-open	ADJ
ejpam-5475	299	6	set	set	VERB
ejpam-5475	299	7	in	in	ADP
ejpam-5475	299	8	x	x	PROPN
ejpam-5475	299	9	is	be	AUX
ejpam-5475	299	10	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	299	11	,	,	PUNCT
ejpam-5475	299	12	ν2)-open	ν2)-open	ADJ
ejpam-5475	299	13	,	,	PUNCT
ejpam-5475	299	14	it	it	PRON
ejpam-5475	299	15	follows	follow	VERB
ejpam-5475	299	16	that	that	SCONJ
ejpam-5475	299	17	cθ̃(ν1,ν2)(a	cθ̃(ν1,ν2)(a	NOUN
ejpam-5475	299	18	)	)	PUNCT
ejpam-5475	299	19	⊆	⊆	NUM
ejpam-5475	299	20	cθ(ν1,ν2)(a	cθ(ν1,ν2)(a	NOUN
ejpam-5475	299	21	)	)	PUNCT
ejpam-5475	299	22	.	.	PUNCT
ejpam-5475	300	1	(	(	PUNCT
ejpam-5475	300	2	ii	ii	X
ejpam-5475	300	3	)	)	PUNCT
ejpam-5475	300	4	the	the	DET
ejpam-5475	300	5	proof	proof	NOUN
ejpam-5475	300	6	is	be	AUX
ejpam-5475	300	7	clear	clear	ADJ
ejpam-5475	300	8	from	from	ADP
ejpam-5475	300	9	the	the	DET
ejpam-5475	300	10	definition	definition	NOUN
ejpam-5475	300	11	.	.	PUNCT
ejpam-5475	301	1	corollary	corollary	ADJ
ejpam-5475	301	2	6	6	NUM
ejpam-5475	301	3	.	.	PUNCT
ejpam-5475	302	1	let	let	VERB
ejpam-5475	302	2	ν1	ν1	NOUN
ejpam-5475	302	3	and	and	CCONJ
ejpam-5475	302	4	ν2	ν2	NOUN
ejpam-5475	302	5	be	be	AUX
ejpam-5475	302	6	two	two	NUM
ejpam-5475	302	7	generalized	generalized	ADJ
ejpam-5475	302	8	topologies	topology	NOUN
ejpam-5475	302	9	on	on	ADP
ejpam-5475	302	10	a	a	DET
ejpam-5475	302	11	nonempty	nonempty	ADV
ejpam-5475	302	12	set	set	VERB
ejpam-5475	302	13	x	x	NOUN
ejpam-5475	302	14	,	,	PUNCT
ejpam-5475	302	15	and	and	CCONJ
ejpam-5475	302	16	let	let	VERB
ejpam-5475	302	17	a	a	DET
ejpam-5475	302	18	⊆	⊆	NUM
ejpam-5475	302	19	x.	x.	NOUN
ejpam-5475	302	20	then	then	ADV
ejpam-5475	302	21	iθ̃(ν1,ν2)(a	iθ̃(ν1,ν2)(a	PROPN
ejpam-5475	302	22	)	)	PUNCT
ejpam-5475	302	23	⊆	⊆	NUM
ejpam-5475	302	24	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	NOUN
ejpam-5475	302	25	)	)	PUNCT
ejpam-5475	302	26	.	.	PUNCT
ejpam-5475	303	1	proof	proof	NOUN
ejpam-5475	303	2	.	.	PUNCT
ejpam-5475	304	1	let	let	VERB
ejpam-5475	304	2	x	x	PUNCT
ejpam-5475	304	3	∈	∈	PROPN
ejpam-5475	304	4	iθ̃(ν1,ν2)(a	iθ̃(ν1,ν2)(a	PROPN
ejpam-5475	304	5	)	)	PUNCT
ejpam-5475	304	6	.	.	PUNCT
ejpam-5475	305	1	this	this	PRON
ejpam-5475	305	2	means	mean	VERB
ejpam-5475	305	3	there	there	PRON
ejpam-5475	305	4	exists	exist	VERB
ejpam-5475	305	5	a	a	DET
ejpam-5475	305	6	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	305	7	,	,	PUNCT
ejpam-5475	305	8	ν2)-open	ν2)-open	VERB
ejpam-5475	305	9	set	set	VERB
ejpam-5475	305	10	v	v	NOUN
ejpam-5475	305	11	in	in	ADP
ejpam-5475	305	12	x	x	PUNCT
ejpam-5475	305	13	containing	contain	VERB
ejpam-5475	305	14	x	x	NOUN
ejpam-5475	305	15	,	,	PUNCT
ejpam-5475	305	16	such	such	ADJ
ejpam-5475	305	17	that	that	SCONJ
ejpam-5475	305	18	x	x	SYM
ejpam-5475	305	19	∈	∈	NOUN
ejpam-5475	305	20	v	v	ADP
ejpam-5475	305	21	⊆	⊆	NUM
ejpam-5475	305	22	a.	a.	NOUN
ejpam-5475	305	23	since	since	SCONJ
ejpam-5475	305	24	v	v	NOUN
ejpam-5475	305	25	is	be	AUX
ejpam-5475	305	26	θ̃(ν1	θ̃(ν1	NOUN
ejpam-5475	305	27	,	,	PUNCT
ejpam-5475	305	28	ν2)-open	ν2)-open	ADJ
ejpam-5475	305	29	,	,	PUNCT
ejpam-5475	305	30	there	there	PRON
ejpam-5475	305	31	exists	exist	VERB
ejpam-5475	305	32	m	m	PROPN
ejpam-5475	305	33	∈	∈	NOUN
ejpam-5475	305	34	ν1	ν1	NOUN
ejpam-5475	305	35	such	such	ADJ
ejpam-5475	305	36	that	that	SCONJ
ejpam-5475	305	37	x	x	SYM
ejpam-5475	305	38	∈	∈	PROPN
ejpam-5475	305	39	m	m	NOUN
ejpam-5475	305	40	and	and	CCONJ
ejpam-5475	305	41	m	m	PROPN
ejpam-5475	305	42	⊆	⊆	NUM
ejpam-5475	305	43	cν2(m	cν2(m	NOUN
ejpam-5475	305	44	)	)	PUNCT
ejpam-5475	305	45	∩mν1	∩mν1	NOUN
ejpam-5475	305	46	⊆	⊆	NUM
ejpam-5475	305	47	v	v	ADP
ejpam-5475	305	48	⊆	⊆	NUM
ejpam-5475	305	49	a.	a.	NOUN
ejpam-5475	305	50	therefore	therefore	ADV
ejpam-5475	305	51	,	,	PUNCT
ejpam-5475	305	52	x	x	PROPN
ejpam-5475	305	53	∈	∈	PROPN
ejpam-5475	305	54	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	PROPN
ejpam-5475	305	55	)	)	PUNCT
ejpam-5475	305	56	.	.	PUNCT
ejpam-5475	306	1	let	let	VERB
ejpam-5475	306	2	ν1	ν1	NOUN
ejpam-5475	306	3	and	and	CCONJ
ejpam-5475	306	4	ν2	ν2	NOUN
ejpam-5475	306	5	be	be	AUX
ejpam-5475	306	6	two	two	NUM
ejpam-5475	306	7	gt	gt	NOUN
ejpam-5475	306	8	’s	’s	NOUN
ejpam-5475	306	9	on	on	ADP
ejpam-5475	306	10	a	a	DET
ejpam-5475	306	11	nonempty	nonempty	ADV
ejpam-5475	306	12	set	set	VERB
ejpam-5475	306	13	x.	x.	NOUN
ejpam-5475	306	14	the	the	DET
ejpam-5475	306	15	notations	notation	NOUN
ejpam-5475	306	16	are	be	AUX
ejpam-5475	306	17	defined	define	VERB
ejpam-5475	306	18	as	as	SCONJ
ejpam-5475	306	19	follows	follow	VERB
ejpam-5475	306	20	:	:	PUNCT
ejpam-5475	306	21	lθ(ν1,ν2)(a	lθ(ν1,ν2)(a	ADJ
ejpam-5475	306	22	)	)	PUNCT
ejpam-5475	307	1	=	=	PRON
ejpam-5475	307	2	{	{	PUNCT
ejpam-5475	307	3	x	x	PUNCT
ejpam-5475	307	4	∈	∈	NOUN
ejpam-5475	307	5	x	x	X
ejpam-5475	307	6	:	:	PUNCT
ejpam-5475	307	7	cν2(m	cν2(m	PROPN
ejpam-5475	307	8	)	)	PUNCT
ejpam-5475	307	9	⊆	⊆	NUM
ejpam-5475	307	10	a	a	DET
ejpam-5475	307	11	for	for	ADP
ejpam-5475	307	12	some	some	DET
ejpam-5475	307	13	ν1	ν1	NOUN
ejpam-5475	307	14	-	-	PUNCT
ejpam-5475	307	15	open	open	NOUN
ejpam-5475	307	16	set	set	NOUN
ejpam-5475	307	17	m	m	AUX
ejpam-5475	307	18	containing	contain	VERB
ejpam-5475	307	19	x	x	X
ejpam-5475	307	20	}	}	PUNCT
ejpam-5475	307	21	;	;	PUNCT
ejpam-5475	307	22	lθ̃(ν1)(a	lθ̃(ν1)(a	X
ejpam-5475	307	23	)	)	PUNCT
ejpam-5475	307	24	=	=	PRON
ejpam-5475	308	1	{	{	PUNCT
ejpam-5475	308	2	x	x	PUNCT
ejpam-5475	308	3	∈	∈	PROPN
ejpam-5475	308	4	x	x	X
ejpam-5475	308	5	:	:	PUNCT
ejpam-5475	308	6	cν1(m	cν1(m	X
ejpam-5475	308	7	)	)	PUNCT
ejpam-5475	308	8	∩mν1	∩mν1	ADV
ejpam-5475	308	9	⊆	⊆	NUM
ejpam-5475	308	10	a	a	PRON
ejpam-5475	308	11	for	for	ADP
ejpam-5475	308	12	some	some	DET
ejpam-5475	308	13	ν1	ν1	NOUN
ejpam-5475	308	14	-	-	PUNCT
ejpam-5475	308	15	open	open	NOUN
ejpam-5475	308	16	set	set	NOUN
ejpam-5475	308	17	m	m	AUX
ejpam-5475	308	18	containing	contain	VERB
ejpam-5475	308	19	x	x	X
ejpam-5475	308	20	}	}	PUNCT
ejpam-5475	308	21	[	[	X
ejpam-5475	308	22	10	10	NUM
ejpam-5475	308	23	]	]	PUNCT
ejpam-5475	308	24	.	.	PUNCT
ejpam-5475	309	1	references	reference	NOUN
ejpam-5475	309	2	3620	3620	NUM
ejpam-5475	309	3	corollary	corollary	ADJ
ejpam-5475	309	4	7	7	NUM
ejpam-5475	309	5	.	.	PUNCT
ejpam-5475	310	1	let	let	VERB
ejpam-5475	310	2	ν1	ν1	NOUN
ejpam-5475	310	3	and	and	CCONJ
ejpam-5475	310	4	ν2	ν2	NOUN
ejpam-5475	310	5	be	be	AUX
ejpam-5475	310	6	two	two	NUM
ejpam-5475	310	7	gt	gt	NOUN
ejpam-5475	310	8	’s	’s	NOUN
ejpam-5475	310	9	on	on	ADP
ejpam-5475	310	10	a	a	DET
ejpam-5475	310	11	nonempty	nonempty	ADV
ejpam-5475	310	12	set	set	VERB
ejpam-5475	310	13	x.	x.	NOUN
ejpam-5475	310	14	then	then	ADV
ejpam-5475	310	15	for	for	ADP
ejpam-5475	310	16	any	any	DET
ejpam-5475	310	17	subset	subset	NOUN
ejpam-5475	310	18	a	a	DET
ejpam-5475	310	19	⊆	⊆	NUM
ejpam-5475	310	20	x	x	SYM
ejpam-5475	310	21	,	,	PUNCT
ejpam-5475	310	22	lθ(ν1,ν2)(a	lθ(ν1,ν2)(a	NOUN
ejpam-5475	310	23	)	)	PUNCT
ejpam-5475	310	24	⊆	⊆	NUM
ejpam-5475	310	25	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	NOUN
ejpam-5475	310	26	)	)	PUNCT
ejpam-5475	310	27	.	.	PUNCT
ejpam-5475	311	1	proof	proof	NOUN
ejpam-5475	311	2	.	.	PUNCT
ejpam-5475	312	1	let	let	VERB
ejpam-5475	312	2	x	x	X
ejpam-5475	312	3	∈	∈	PROPN
ejpam-5475	312	4	lθ(ν1,ν2)(a	lθ(ν1,ν2)(a	NOUN
ejpam-5475	312	5	)	)	PUNCT
ejpam-5475	312	6	.	.	PUNCT
ejpam-5475	313	1	this	this	PRON
ejpam-5475	313	2	means	mean	VERB
ejpam-5475	313	3	there	there	PRON
ejpam-5475	313	4	exists	exist	VERB
ejpam-5475	313	5	a	a	DET
ejpam-5475	313	6	ν1	ν1	NOUN
ejpam-5475	313	7	-	-	PUNCT
ejpam-5475	313	8	open	open	NOUN
ejpam-5475	313	9	set	set	NOUN
ejpam-5475	313	10	m	m	AUX
ejpam-5475	313	11	containing	contain	VERB
ejpam-5475	313	12	x	x	PUNCT
ejpam-5475	313	13	such	such	ADJ
ejpam-5475	313	14	that	that	DET
ejpam-5475	313	15	cν2(m	cν2(m	NOUN
ejpam-5475	313	16	)	)	PUNCT
ejpam-5475	313	17	⊆	⊆	NUM
ejpam-5475	313	18	a.	a.	NOUN
ejpam-5475	313	19	since	since	SCONJ
ejpam-5475	313	20	cν2(m	cν2(m	PROPN
ejpam-5475	313	21	)	)	PUNCT
ejpam-5475	313	22	∩	∩	NOUN
ejpam-5475	313	23	mν1	mν1	ADJ
ejpam-5475	313	24	⊆	⊆	NUM
ejpam-5475	313	25	cν2(m	cν2(m	NOUN
ejpam-5475	313	26	)	)	PUNCT
ejpam-5475	313	27	⊆	⊆	NUM
ejpam-5475	313	28	a	a	PRON
ejpam-5475	313	29	,	,	PUNCT
ejpam-5475	313	30	it	it	PRON
ejpam-5475	313	31	follows	follow	VERB
ejpam-5475	313	32	that	that	SCONJ
ejpam-5475	313	33	x	x	PUNCT
ejpam-5475	313	34	∈	∈	PROPN
ejpam-5475	313	35	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	PROPN
ejpam-5475	313	36	)	)	PUNCT
ejpam-5475	313	37	.	.	PUNCT
ejpam-5475	314	1	therefore	therefore	ADV
ejpam-5475	314	2	,	,	PUNCT
ejpam-5475	314	3	lθ(ν1,ν2)(a	lθ(ν1,ν2)(a	NOUN
ejpam-5475	314	4	)	)	PUNCT
ejpam-5475	314	5	⊆	⊆	NUM
ejpam-5475	314	6	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	NOUN
ejpam-5475	314	7	)	)	PUNCT
ejpam-5475	314	8	.	.	PUNCT
ejpam-5475	315	1	remark	remark	NOUN
ejpam-5475	315	2	5	5	NUM
ejpam-5475	315	3	.	.	PUNCT
ejpam-5475	316	1	let	let	VERB
ejpam-5475	316	2	ν	ν	NOUN
ejpam-5475	316	3	be	be	AUX
ejpam-5475	316	4	a	a	DET
ejpam-5475	316	5	gt	gt	PROPN
ejpam-5475	316	6	on	on	ADP
ejpam-5475	316	7	a	a	DET
ejpam-5475	316	8	nonempty	nonempty	ADJ
ejpam-5475	316	9	set	set	VERB
ejpam-5475	316	10	x	x	NOUN
ejpam-5475	316	11	,	,	PUNCT
ejpam-5475	316	12	and	and	CCONJ
ejpam-5475	316	13	let	let	VERB
ejpam-5475	316	14	a	a	DET
ejpam-5475	316	15	⊆	⊆	NUM
ejpam-5475	316	16	x.	x.	NOUN
ejpam-5475	316	17	then	then	ADV
ejpam-5475	316	18	lθ̃(ν	lθ̃(ν	NOUN
ejpam-5475	316	19	,	,	PUNCT
ejpam-5475	316	20	ν)(a	ν)(a	NUM
ejpam-5475	316	21	)	)	PUNCT
ejpam-5475	316	22	=	=	SYM
ejpam-5475	317	1	lθ̃(ν)(a	lθ̃(ν)(a	NUM
ejpam-5475	317	2	)	)	PUNCT
ejpam-5475	317	3	.	.	PUNCT
ejpam-5475	318	1	theorem	theorem	VERB
ejpam-5475	318	2	15	15	NUM
ejpam-5475	318	3	.	.	PUNCT
ejpam-5475	319	1	let	let	VERB
ejpam-5475	319	2	ν1	ν1	NOUN
ejpam-5475	319	3	and	and	CCONJ
ejpam-5475	319	4	ν2	ν2	NOUN
ejpam-5475	319	5	be	be	AUX
ejpam-5475	319	6	two	two	NUM
ejpam-5475	319	7	gts	gts	NOUN
ejpam-5475	319	8	on	on	ADP
ejpam-5475	319	9	a	a	DET
ejpam-5475	319	10	nonempty	nonempty	ADV
ejpam-5475	319	11	set	set	VERB
ejpam-5475	319	12	x	x	PUNCT
ejpam-5475	319	13	and	and	CCONJ
ejpam-5475	319	14	let	let	VERB
ejpam-5475	319	15	a	a	DET
ejpam-5475	319	16	⊆	⊆	NUM
ejpam-5475	319	17	x.	x.	NOUN
ejpam-5475	319	18	then	then	ADV
ejpam-5475	319	19	the	the	DET
ejpam-5475	319	20	following	follow	VERB
ejpam-5475	319	21	properties	property	NOUN
ejpam-5475	319	22	hold	hold	VERB
ejpam-5475	319	23	:	:	PUNCT
ejpam-5475	319	24	(	(	PUNCT
ejpam-5475	319	25	i	i	NOUN
ejpam-5475	319	26	)	)	PUNCT
ejpam-5475	319	27	iθ̃(ν1,ν2)(a	iθ̃(ν1,ν2)(a	PROPN
ejpam-5475	319	28	)	)	PUNCT
ejpam-5475	319	29	=	=	NOUN
ejpam-5475	320	1	x	x	PUNCT
ejpam-5475	320	2	−	−	NOUN
ejpam-5475	320	3	cθ̃(ν1,ν2)(x	cθ̃(ν1,ν2)(x	NOUN
ejpam-5475	320	4	−a	−a	NOUN
ejpam-5475	320	5	)	)	PUNCT
ejpam-5475	320	6	and	and	CCONJ
ejpam-5475	320	7	cθ̃(ν1,ν2)(a	cθ̃(ν1,ν2)(a	NOUN
ejpam-5475	320	8	)	)	PUNCT
ejpam-5475	321	1	=	=	PUNCT
ejpam-5475	321	2	x	x	PUNCT
ejpam-5475	321	3	−	−	PROPN
ejpam-5475	321	4	iθ̃(ν1,ν2)(x	iθ̃(ν1,ν2)(x	PROPN
ejpam-5475	321	5	−a	−a	NOUN
ejpam-5475	321	6	)	)	PUNCT
ejpam-5475	321	7	.	.	PUNCT
ejpam-5475	322	1	(	(	PUNCT
ejpam-5475	322	2	ii	ii	X
ejpam-5475	322	3	)	)	PUNCT
ejpam-5475	322	4	lθ̃(ν1,ν2)(a	lθ̃(ν1,ν2)(a	PROPN
ejpam-5475	322	5	)	)	PUNCT
ejpam-5475	323	1	=	=	PUNCT
ejpam-5475	324	1	x	x	PUNCT
ejpam-5475	324	2	−	−	PROPN
ejpam-5475	324	3	γθ̃(ν1,ν2)(x	γθ̃(ν1,ν2)(x	PROPN
ejpam-5475	324	4	−a	−a	NOUN
ejpam-5475	324	5	)	)	PUNCT
ejpam-5475	324	6	and	and	CCONJ
ejpam-5475	324	7	γθ̃(ν1,ν2)(a	γθ̃(ν1,ν2)(a	NUM
ejpam-5475	324	8	)	)	PUNCT
ejpam-5475	325	1	=	=	PUNCT
ejpam-5475	325	2	x	x	PUNCT
ejpam-5475	326	1	−	−	DET
ejpam-5475	326	2	lθ̃(ν1,ν2)(x	lθ̃(ν1,ν2)(x	PROPN
ejpam-5475	326	3	−a	−a	NOUN
ejpam-5475	326	4	)	)	PUNCT
ejpam-5475	326	5	.	.	PUNCT
ejpam-5475	327	1	proof	proof	NOUN
ejpam-5475	327	2	.	.	PUNCT
ejpam-5475	328	1	the	the	DET
ejpam-5475	328	2	proof	proof	NOUN
ejpam-5475	328	3	is	be	AUX
ejpam-5475	328	4	straightforward	straightforward	ADJ
ejpam-5475	328	5	and	and	CCONJ
ejpam-5475	328	6	hence	hence	ADV
ejpam-5475	328	7	omitted	omit	VERB
ejpam-5475	328	8	.	.	PUNCT
ejpam-5475	329	1	conclusion	conclusion	NOUN
ejpam-5475	329	2	in	in	ADP
ejpam-5475	329	3	this	this	DET
ejpam-5475	329	4	work	work	NOUN
ejpam-5475	329	5	,	,	PUNCT
ejpam-5475	329	6	we	we	PRON
ejpam-5475	329	7	have	have	AUX
ejpam-5475	329	8	introduced	introduce	VERB
ejpam-5475	329	9	and	and	CCONJ
ejpam-5475	329	10	studied	study	VERB
ejpam-5475	329	11	the	the	DET
ejpam-5475	329	12	operation	operation	NOUN
ejpam-5475	329	13	γθ̃(ν1,ν2	γθ̃(ν1,ν2	PROPN
ejpam-5475	329	14	)	)	PUNCT
ejpam-5475	329	15	and	and	CCONJ
ejpam-5475	329	16	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	329	17	,	,	PUNCT
ejpam-5475	329	18	ν2)open	ν2)open	ADJ
ejpam-5475	329	19	sets	set	NOUN
ejpam-5475	329	20	in	in	ADP
ejpam-5475	329	21	generalized	generalized	ADJ
ejpam-5475	329	22	topological	topological	ADJ
ejpam-5475	329	23	spaces	space	NOUN
ejpam-5475	329	24	.	.	PUNCT
ejpam-5475	330	1	we	we	PRON
ejpam-5475	330	2	have	have	AUX
ejpam-5475	330	3	established	establish	VERB
ejpam-5475	330	4	several	several	ADJ
ejpam-5475	330	5	significant	significant	ADJ
ejpam-5475	330	6	results	result	NOUN
ejpam-5475	330	7	concerning	concern	VERB
ejpam-5475	330	8	these	these	DET
ejpam-5475	330	9	concepts	concept	NOUN
ejpam-5475	330	10	.	.	PUNCT
ejpam-5475	331	1	the	the	DET
ejpam-5475	331	2	relationships	relationship	NOUN
ejpam-5475	331	3	among	among	ADP
ejpam-5475	331	4	γθ̃(ν1,ν2	γθ̃(ν1,ν2	ADJ
ejpam-5475	331	5	)	)	PUNCT
ejpam-5475	331	6	,	,	PUNCT
ejpam-5475	331	7	γθ(ν1,ν2	γθ(ν1,ν2	PROPN
ejpam-5475	331	8	)	)	PUNCT
ejpam-5475	331	9	,	,	PUNCT
ejpam-5475	331	10	and	and	CCONJ
ejpam-5475	331	11	γθ(ν	γθ(ν	NUM
ejpam-5475	331	12	)	)	PUNCT
ejpam-5475	331	13	,	,	PUNCT
ejpam-5475	331	14	as	as	ADV
ejpam-5475	331	15	well	well	ADV
ejpam-5475	331	16	as	as	ADP
ejpam-5475	331	17	the	the	DET
ejpam-5475	331	18	relationships	relationship	NOUN
ejpam-5475	331	19	among	among	ADP
ejpam-5475	331	20	θ̃(ν1	θ̃(ν1	PROPN
ejpam-5475	331	21	,	,	PUNCT
ejpam-5475	331	22	ν2)-open	ν2)-open	ADJ
ejpam-5475	331	23	sets	set	NOUN
ejpam-5475	331	24	,	,	PUNCT
ejpam-5475	331	25	θ(ν1	θ(ν1	NOUN
ejpam-5475	331	26	,	,	PUNCT
ejpam-5475	331	27	ν2)-open	ν2)-open	ADJ
ejpam-5475	331	28	sets	set	NOUN
ejpam-5475	331	29	,	,	PUNCT
ejpam-5475	331	30	and	and	CCONJ
ejpam-5475	331	31	µ-open	µ-open	NOUN
ejpam-5475	331	32	sets	set	NOUN
ejpam-5475	331	33	have	have	AUX
ejpam-5475	331	34	been	be	AUX
ejpam-5475	331	35	thoroughly	thoroughly	ADV
ejpam-5475	331	36	investigated	investigate	VERB
ejpam-5475	331	37	.	.	PUNCT
ejpam-5475	332	1	finally	finally	ADV
ejpam-5475	332	2	,	,	PUNCT
ejpam-5475	332	3	we	we	PRON
ejpam-5475	332	4	have	have	AUX
ejpam-5475	332	5	derived	derive	VERB
ejpam-5475	332	6	various	various	ADJ
ejpam-5475	332	7	properties	property	NOUN
ejpam-5475	332	8	and	and	CCONJ
ejpam-5475	332	9	characterizations	characterization	NOUN
ejpam-5475	332	10	in	in	ADP
ejpam-5475	332	11	terms	term	NOUN
ejpam-5475	332	12	of	of	ADP
ejpam-5475	332	13	the	the	DET
ejpam-5475	332	14	concept	concept	NOUN
ejpam-5475	332	15	of	of	ADP
ejpam-5475	332	16	g(ν1	g(ν1	NOUN
ejpam-5475	332	17	,	,	PUNCT
ejpam-5475	332	18	ν2)-regularity	ν2)-regularity	NOUN
ejpam-5475	332	19	.	.	PUNCT
ejpam-5475	333	1	acknowledgements	acknowledgement	VERB
ejpam-5475	333	2	the	the	DET
ejpam-5475	333	3	authors	author	NOUN
ejpam-5475	333	4	thank	thank	VERB
ejpam-5475	333	5	the	the	DET
ejpam-5475	333	6	readers	reader	NOUN
ejpam-5475	333	7	of	of	ADP
ejpam-5475	333	8	european	european	PROPN
ejpam-5475	333	9	journal	journal	PROPN
ejpam-5475	333	10	of	of	ADP
ejpam-5475	333	11	pure	pure	ADJ
ejpam-5475	333	12	and	and	CCONJ
ejpam-5475	333	13	applied	applied	ADJ
ejpam-5475	333	14	mathematics	mathematic	NOUN
ejpam-5475	333	15	,	,	PUNCT
ejpam-5475	333	16	for	for	ADP
ejpam-5475	333	17	making	make	VERB
ejpam-5475	333	18	our	our	PRON
ejpam-5475	333	19	journal	journal	NOUN
ejpam-5475	333	20	successful	successful	ADJ
ejpam-5475	333	21	.	.	PUNCT
ejpam-5475	334	1	references	reference	NOUN
ejpam-5475	334	2	[	[	X
ejpam-5475	334	3	1	1	X
ejpam-5475	334	4	]	]	PUNCT
ejpam-5475	334	5	á	á	NOUN
ejpam-5475	334	6	császár	császár	NOUN
ejpam-5475	334	7	.	.	PUNCT
ejpam-5475	335	1	generalized	generalize	VERB
ejpam-5475	335	2	topology	topology	NOUN
ejpam-5475	335	3	,	,	PUNCT
ejpam-5475	335	4	generized	generize	VERB
ejpam-5475	335	5	continuity	continuity	NOUN
ejpam-5475	335	6	.	.	PUNCT
ejpam-5475	336	1	acta	acta	PROPN
ejpam-5475	336	2	mathematica	mathematica	PROPN
ejpam-5475	336	3	hungarica	hungarica	PROPN
ejpam-5475	336	4	,	,	PUNCT
ejpam-5475	336	5	96(4):351–357	96(4):351–357	NOUN
ejpam-5475	336	6	,	,	PUNCT
ejpam-5475	336	7	2002	2002	NUM
ejpam-5475	336	8	.	.	PUNCT
ejpam-5475	337	1	[	[	X
ejpam-5475	337	2	2	2	X
ejpam-5475	337	3	]	]	PUNCT
ejpam-5475	337	4	á	á	NOUN
ejpam-5475	337	5	császár	császár	PROPN
ejpam-5475	337	6	.	.	PUNCT
ejpam-5475	338	1	generalized	generalize	VERB
ejpam-5475	338	2	open	open	ADJ
ejpam-5475	338	3	sets	set	NOUN
ejpam-5475	338	4	in	in	ADP
ejpam-5475	338	5	generalized	generalized	ADJ
ejpam-5475	338	6	topologies	topology	NOUN
ejpam-5475	338	7	.	.	PUNCT
ejpam-5475	339	1	acta	acta	PROPN
ejpam-5475	339	2	mathematica	mathematica	PROPN
ejpam-5475	339	3	hungarica	hungarica	PROPN
ejpam-5475	339	4	,	,	PUNCT
ejpam-5475	339	5	106:53–66	106:53–66	NUM
ejpam-5475	339	6	,	,	PUNCT
ejpam-5475	339	7	2005	2005	NUM
ejpam-5475	339	8	.	.	PUNCT
ejpam-5475	340	1	[	[	X
ejpam-5475	340	2	3	3	X
ejpam-5475	340	3	]	]	PUNCT
ejpam-5475	340	4	á	á	NOUN
ejpam-5475	340	5	császár	császár	NOUN
ejpam-5475	340	6	.	.	PUNCT
ejpam-5475	341	1	δ	δ	PROPN
ejpam-5475	341	2	-	-	PUNCT
ejpam-5475	341	3	and	and	CCONJ
ejpam-5475	341	4	θ	θ	NOUN
ejpam-5475	341	5	-	-	PUNCT
ejpam-5475	341	6	modifications	modification	NOUN
ejpam-5475	341	7	of	of	ADP
ejpam-5475	341	8	generalized	generalized	ADJ
ejpam-5475	341	9	topologies	topology	NOUN
ejpam-5475	341	10	.	.	PUNCT
ejpam-5475	342	1	acta	acta	PROPN
ejpam-5475	342	2	mathematica	mathematica	PROPN
ejpam-5475	342	3	hungarica	hungarica	PROPN
ejpam-5475	342	4	,	,	PUNCT
ejpam-5475	342	5	120(3):275–279	120(3):275–279	NUM
ejpam-5475	342	6	,	,	PUNCT
ejpam-5475	342	7	2008	2008	NUM
ejpam-5475	342	8	.	.	PUNCT
ejpam-5475	343	1	references	reference	NOUN
ejpam-5475	343	2	3621	3621	NUM
ejpam-5475	343	3	[	[	X
ejpam-5475	343	4	4	4	NUM
ejpam-5475	343	5	]	]	PUNCT
ejpam-5475	343	6	á	á	NOUN
ejpam-5475	343	7	császár	császár	NOUN
ejpam-5475	343	8	.	.	PUNCT
ejpam-5475	344	1	mixed	mixed	ADJ
ejpam-5475	344	2	constructions	construction	NOUN
ejpam-5475	344	3	for	for	ADP
ejpam-5475	344	4	generalized	generalized	ADJ
ejpam-5475	344	5	topologies	topology	NOUN
ejpam-5475	344	6	.	.	PUNCT
ejpam-5475	345	1	acta	acta	PROPN
ejpam-5475	345	2	mathematica	mathematica	PROPN
ejpam-5475	345	3	hungarica	hungarica	PROPN
ejpam-5475	345	4	,	,	PUNCT
ejpam-5475	345	5	122(1):153–159	122(1):153–159	NUM
ejpam-5475	345	6	,	,	PUNCT
ejpam-5475	345	7	2009	2009	NUM
ejpam-5475	345	8	.	.	PUNCT
ejpam-5475	346	1	[	[	X
ejpam-5475	346	2	5	5	X
ejpam-5475	346	3	]	]	PUNCT
ejpam-5475	346	4	á	á	NOUN
ejpam-5475	346	5	császár	császár	NOUN
ejpam-5475	346	6	and	and	CCONJ
ejpam-5475	346	7	e	e	PROPN
ejpam-5475	346	8	makai	makai	PROPN
ejpam-5475	346	9	jr	jr	PROPN
ejpam-5475	346	10	.	.	PROPN
ejpam-5475	346	11	further	further	ADJ
ejpam-5475	346	12	remarks	remark	NOUN
ejpam-5475	346	13	on	on	ADP
ejpam-5475	346	14	δ	δ	PROPN
ejpam-5475	346	15	-	-	PUNCT
ejpam-5475	346	16	and	and	CCONJ
ejpam-5475	346	17	θ	θ	NOUN
ejpam-5475	346	18	-	-	NOUN
ejpam-5475	346	19	modifications	modification	NOUN
ejpam-5475	346	20	.	.	PUNCT
ejpam-5475	347	1	acta	acta	PROPN
ejpam-5475	347	2	mathematica	mathematica	PROPN
ejpam-5475	347	3	hungarica	hungarica	PROPN
ejpam-5475	347	4	,	,	PUNCT
ejpam-5475	347	5	123(3):223–228	123(3):223–228	NUM
ejpam-5475	347	6	,	,	PUNCT
ejpam-5475	347	7	2009	2009	NUM
ejpam-5475	347	8	.	.	PUNCT
ejpam-5475	348	1	[	[	X
ejpam-5475	348	2	6	6	NUM
ejpam-5475	348	3	]	]	X
ejpam-5475	348	4	y.	y.	PROPN
ejpam-5475	348	5	k.kim	k.kim	PROPN
ejpam-5475	348	6	and	and	CCONJ
ejpam-5475	348	7	w.k	w.k	PROPN
ejpam-5475	348	8	.	.	PROPN
ejpam-5475	348	9	min	min	PROPN
ejpam-5475	348	10	.	.	PUNCT
ejpam-5475	348	11	h(θ)-open	h(θ)-open	NOUN
ejpam-5475	348	12	sets	set	NOUN
ejpam-5475	348	13	induced	induce	VERB
ejpam-5475	348	14	by	by	ADP
ejpam-5475	348	15	hereditary	hereditary	ADJ
ejpam-5475	348	16	classes	class	NOUN
ejpam-5475	348	17	on	on	ADP
ejpam-5475	348	18	generalized	generalized	ADJ
ejpam-5475	348	19	topological	topological	ADJ
ejpam-5475	348	20	spaces	space	NOUN
ejpam-5475	348	21	.	.	PUNCT
ejpam-5475	349	1	international	international	ADJ
ejpam-5475	349	2	journal	journal	NOUN
ejpam-5475	349	3	of	of	ADP
ejpam-5475	349	4	pure	pure	ADJ
ejpam-5475	349	5	and	and	CCONJ
ejpam-5475	349	6	applied	applied	ADJ
ejpam-5475	349	7	mathematics	mathematic	NOUN
ejpam-5475	349	8	,	,	PUNCT
ejpam-5475	349	9	93:307–315	93:307–315	PROPN
ejpam-5475	349	10	,	,	PUNCT
ejpam-5475	349	11	may	may	AUX
ejpam-5475	349	12	2014	2014	NUM
ejpam-5475	349	13	.	.	PUNCT
ejpam-5475	350	1	[	[	X
ejpam-5475	350	2	7	7	X
ejpam-5475	350	3	]	]	X
ejpam-5475	350	4	w	w	PROPN
ejpam-5475	350	5	k	k	PROPN
ejpam-5475	350	6	min	min	PROPN
ejpam-5475	350	7	.	.	PROPN
ejpam-5475	350	8	remarks	remark	NOUN
ejpam-5475	350	9	on	on	ADP
ejpam-5475	350	10	θ	θ	ADJ
ejpam-5475	350	11	-	-	ADJ
ejpam-5475	350	12	open	open	ADJ
ejpam-5475	350	13	sets	set	NOUN
ejpam-5475	350	14	in	in	ADP
ejpam-5475	350	15	generalized	generalized	ADJ
ejpam-5475	350	16	topological	topological	ADJ
ejpam-5475	350	17	spaces	space	NOUN
ejpam-5475	350	18	.	.	PUNCT
ejpam-5475	351	1	applied	apply	VERB
ejpam-5475	351	2	mathematics	mathematics	NOUN
ejpam-5475	351	3	letters	letter	NOUN
ejpam-5475	351	4	,	,	PUNCT
ejpam-5475	351	5	24(2):165–168	24(2):165–168	PROPN
ejpam-5475	351	6	,	,	PUNCT
ejpam-5475	351	7	2011	2011	NUM
ejpam-5475	351	8	.	.	PUNCT
ejpam-5475	352	1	[	[	X
ejpam-5475	352	2	8	8	NUM
ejpam-5475	352	3	]	]	PUNCT
ejpam-5475	352	4	won	win	VERB
ejpam-5475	352	5	keun	keun	PROPN
ejpam-5475	352	6	min	min	PROPN
ejpam-5475	352	7	.	.	PROPN
ejpam-5475	352	8	continuity	continuity	NOUN
ejpam-5475	352	9	on	on	ADP
ejpam-5475	352	10	generalized	generalized	ADJ
ejpam-5475	352	11	topological	topological	ADJ
ejpam-5475	352	12	spaces	space	NOUN
ejpam-5475	352	13	.	.	PUNCT
ejpam-5475	353	1	acta	acta	PROPN
ejpam-5475	353	2	mathematica	mathematica	PROPN
ejpam-5475	353	3	hungarica	hungarica	PROPN
ejpam-5475	353	4	,	,	PUNCT
ejpam-5475	353	5	129:350–356	129:350–356	NUM
ejpam-5475	353	6	,	,	PUNCT
ejpam-5475	353	7	2010	2010	NUM
ejpam-5475	353	8	.	.	PUNCT
ejpam-5475	354	1	[	[	X
ejpam-5475	354	2	9	9	NUM
ejpam-5475	354	3	]	]	PUNCT
ejpam-5475	354	4	won	win	VERB
ejpam-5475	354	5	keun	keun	PROPN
ejpam-5475	354	6	min	min	PROPN
ejpam-5475	354	7	.	.	PROPN
ejpam-5475	354	8	mixed	mixed	ADJ
ejpam-5475	354	9	weak	weak	ADJ
ejpam-5475	354	10	continuity	continuity	NOUN
ejpam-5475	354	11	on	on	ADP
ejpam-5475	354	12	generalized	generalized	ADJ
ejpam-5475	354	13	topological	topological	ADJ
ejpam-5475	354	14	spaces	space	NOUN
ejpam-5475	354	15	.	.	PUNCT
ejpam-5475	355	1	acta	acta	PROPN
ejpam-5475	355	2	mathematica	mathematica	PROPN
ejpam-5475	355	3	hungarica	hungarica	PROPN
ejpam-5475	355	4	,	,	PUNCT
ejpam-5475	355	5	132(4):339–347	132(4):339–347	NUM
ejpam-5475	355	6	,	,	PUNCT
ejpam-5475	355	7	2011	2011	NUM
ejpam-5475	355	8	.	.	PUNCT
ejpam-5475	356	1	[	[	X
ejpam-5475	356	2	10	10	NUM
ejpam-5475	356	3	]	]	X
ejpam-5475	356	4	abdo	abdo	PROPN
ejpam-5475	356	5	qahis	qahis	PROPN
ejpam-5475	356	6	and	and	CCONJ
ejpam-5475	356	7	fatimah	fatimah	PROPN
ejpam-5475	356	8	al	al	PROPN
ejpam-5475	356	9	mahri	mahri	PROPN
ejpam-5475	356	10	.	.	PUNCT
ejpam-5475	357	1	a	a	DET
ejpam-5475	357	2	new	new	ADJ
ejpam-5475	357	3	class	class	NOUN
ejpam-5475	357	4	between	between	ADP
ejpam-5475	357	5	θ̃µ-open	θ̃µ-open	PROPN
ejpam-5475	357	6	sets	set	NOUN
ejpam-5475	357	7	and	and	CCONJ
ejpam-5475	357	8	µ-open	µ-open	NOUN
ejpam-5475	357	9	sets	set	NOUN
ejpam-5475	357	10	in	in	ADP
ejpam-5475	357	11	generalized	generalized	ADJ
ejpam-5475	357	12	topological	topological	ADJ
ejpam-5475	357	13	spaces	space	NOUN
ejpam-5475	357	14	.	.	PUNCT
ejpam-5475	358	1	missouri	missouri	PROPN
ejpam-5475	358	2	journal	journal	PROPN
ejpam-5475	358	3	of	of	ADP
ejpam-5475	358	4	mathematical	mathematical	ADJ
ejpam-5475	358	5	sciences	sciences	PROPN
ejpam-5475	358	6	,	,	PUNCT
ejpam-5475	358	7	36(1):98–110	36(1):98–110	NUM
ejpam-5475	358	8	,	,	PUNCT
ejpam-5475	358	9	2024	2024	NUM
ejpam-5475	358	10	.	.	PUNCT
ejpam-5475	359	1	[	[	X
ejpam-5475	359	2	11	11	NUM
ejpam-5475	359	3	]	]	PUNCT
ejpam-5475	359	4	ratna	ratna	PROPN
ejpam-5475	359	5	dev	dev	PROPN
ejpam-5475	359	6	sarma	sarma	PROPN
ejpam-5475	359	7	.	.	PUNCT
ejpam-5475	360	1	on	on	ADP
ejpam-5475	360	2	extremally	extremally	ADV
ejpam-5475	360	3	disconnected	disconnect	VERB
ejpam-5475	360	4	generalized	generalized	ADJ
ejpam-5475	360	5	topologies	topology	NOUN
ejpam-5475	360	6	.	.	PUNCT
ejpam-5475	361	1	acta	acta	PROPN
ejpam-5475	361	2	mathematica	mathematica	PROPN
ejpam-5475	361	3	hungarica	hungarica	PROPN
ejpam-5475	361	4	,	,	PUNCT
ejpam-5475	361	5	134(4):583–588	134(4):583–588	NUM
ejpam-5475	361	6	,	,	PUNCT
ejpam-5475	361	7	2012	2012	NUM
ejpam-5475	361	8	.	.	PUNCT
ejpam-5475	362	1	[	[	X
ejpam-5475	362	2	12	12	NUM
ejpam-5475	362	3	]	]	PUNCT
ejpam-5475	362	4	ugur	ugur	ADJ
ejpam-5475	362	5	sengul	sengul	NOUN
ejpam-5475	362	6	.	.	PUNCT
ejpam-5475	363	1	more	more	ADV
ejpam-5475	363	2	on	on	ADP
ejpam-5475	363	3	δand	δand	NOUN
ejpam-5475	363	4	θ	θ	NOUN
ejpam-5475	363	5	-	-	NOUN
ejpam-5475	363	6	modifications	modification	NOUN
ejpam-5475	363	7	.	.	PUNCT
ejpam-5475	364	1	creative	creative	ADJ
ejpam-5475	364	2	mathematics	mathematic	NOUN
ejpam-5475	364	3	and	and	CCONJ
ejpam-5475	364	4	informatics	informatic	NOUN
ejpam-5475	364	5	,	,	PUNCT
ejpam-5475	364	6	30(1):89–96	30(1):89–96	NUM
ejpam-5475	364	7	,	,	PUNCT
ejpam-5475	364	8	02	02	NUM
ejpam-5475	364	9	2021	2021	NUM
ejpam-5475	364	10	.	.	PUNCT
ejpam-5475	364	11	.	.	PUNCT