id	sid	tid	token	lemma	pos
ejpam-3764	1	1	european	european	PROPN
ejpam-3764	1	2	journal	journal	PROPN
ejpam-3764	1	3	of	of	ADP
ejpam-3764	1	4	pure	pure	ADJ
ejpam-3764	1	5	and	and	CCONJ
ejpam-3764	1	6	applied	apply	VERB
ejpam-3764	1	7	mathematics	mathematic	NOUN
ejpam-3764	1	8	vol	vol	NOUN
ejpam-3764	1	9	.	.	PROPN
ejpam-3764	2	1	13	13	NUM
ejpam-3764	2	2	,	,	PUNCT
ejpam-3764	2	3	no	no	INTJ
ejpam-3764	2	4	.	.	NOUN
ejpam-3764	2	5	3	3	NUM
ejpam-3764	2	6	,	,	PUNCT
ejpam-3764	2	7	2020	2020	NUM
ejpam-3764	2	8	,	,	PUNCT
ejpam-3764	2	9	513	513	NUM
ejpam-3764	2	10	-	-	SYM
ejpam-3764	2	11	528	528	NUM
ejpam-3764	2	12	issn	issn	PROPN
ejpam-3764	2	13	1307	1307	NUM
ejpam-3764	2	14	-	-	SYM
ejpam-3764	2	15	5543	5543	NUM
ejpam-3764	2	16	–	–	PUNCT
ejpam-3764	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3764	2	18	published	publish	VERB
ejpam-3764	2	19	by	by	ADP
ejpam-3764	2	20	new	new	PROPN
ejpam-3764	2	21	york	york	PROPN
ejpam-3764	2	22	business	business	PROPN
ejpam-3764	2	23	global	global	ADJ
ejpam-3764	2	24	new	new	ADJ
ejpam-3764	2	25	topology	topology	NOUN
ejpam-3764	2	26	on	on	ADP
ejpam-3764	2	27	omega	omega	NOUN
ejpam-3764	2	28	algebra	algebra	NOUN
ejpam-3764	2	29	mesfer	mesfer	VERB
ejpam-3764	2	30	alqahtani1,∗	alqahtani1,∗	NOUN
ejpam-3764	2	31	,	,	PUNCT
ejpam-3764	2	32	cenap	cenap	NOUN
ejpam-3764	2	33	özel1	özel1	NOUN
ejpam-3764	2	34	,	,	PUNCT
ejpam-3764	2	35	ibtesam	ibtesam	PROPN
ejpam-3764	2	36	alshammari2	alshammari2	PROPN
ejpam-3764	2	37	1	1	NUM
ejpam-3764	2	38	department	department	NOUN
ejpam-3764	2	39	of	of	ADP
ejpam-3764	2	40	mathematics	mathematic	NOUN
ejpam-3764	2	41	,	,	PUNCT
ejpam-3764	2	42	faculty	faculty	NOUN
ejpam-3764	2	43	of	of	ADP
ejpam-3764	2	44	science	science	NOUN
ejpam-3764	2	45	,	,	PUNCT
ejpam-3764	2	46	king	king	PROPN
ejpam-3764	2	47	abdulaziz	abdulaziz	PROPN
ejpam-3764	2	48	university	university	PROPN
ejpam-3764	2	49	,	,	PUNCT
ejpam-3764	2	50	jeddah	jeddah	PROPN
ejpam-3764	2	51	,	,	PUNCT
ejpam-3764	2	52	saudi	saudi	PROPN
ejpam-3764	2	53	arabia	arabia	PROPN
ejpam-3764	2	54	2	2	NUM
ejpam-3764	2	55	department	department	NOUN
ejpam-3764	2	56	of	of	ADP
ejpam-3764	2	57	mathematics	mathematic	NOUN
ejpam-3764	2	58	,	,	PUNCT
ejpam-3764	2	59	faculty	faculty	NOUN
ejpam-3764	2	60	of	of	ADP
ejpam-3764	2	61	science	science	NOUN
ejpam-3764	2	62	,	,	PUNCT
ejpam-3764	2	63	university	university	NOUN
ejpam-3764	2	64	of	of	ADP
ejpam-3764	2	65	hafr	hafr	PROPN
ejpam-3764	2	66	al	al	PROPN
ejpam-3764	2	67	batin	batin	PROPN
ejpam-3764	2	68	,	,	PUNCT
ejpam-3764	2	69	hafr	hafr	PROPN
ejpam-3764	2	70	al	al	PROPN
ejpam-3764	2	71	batin	batin	PROPN
ejpam-3764	2	72	,	,	PUNCT
ejpam-3764	2	73	saudi	saudi	PROPN
ejpam-3764	2	74	arabia	arabia	PROPN
ejpam-3764	2	75	abstract	abstract	NOUN
ejpam-3764	2	76	.	.	PUNCT
ejpam-3764	3	1	the	the	DET
ejpam-3764	3	2	purpose	purpose	NOUN
ejpam-3764	3	3	of	of	ADP
ejpam-3764	3	4	this	this	DET
ejpam-3764	3	5	paper	paper	NOUN
ejpam-3764	3	6	is	be	AUX
ejpam-3764	3	7	to	to	PART
ejpam-3764	3	8	define	define	VERB
ejpam-3764	3	9	a	a	DET
ejpam-3764	3	10	new	new	ADJ
ejpam-3764	3	11	topology	topology	NOUN
ejpam-3764	3	12	called	call	VERB
ejpam-3764	3	13	omega	omega	NOUN
ejpam-3764	3	14	topology	topology	NOUN
ejpam-3764	3	15	over	over	ADP
ejpam-3764	3	16	a	a	DET
ejpam-3764	3	17	new	new	ADJ
ejpam-3764	3	18	structure	structure	NOUN
ejpam-3764	3	19	called	call	VERB
ejpam-3764	3	20	omega	omega	NOUN
ejpam-3764	3	21	algebra	algebra	NOUN
ejpam-3764	3	22	and	and	CCONJ
ejpam-3764	3	23	discuss	discuss	VERB
ejpam-3764	3	24	some	some	PRON
ejpam-3764	3	25	of	of	ADP
ejpam-3764	3	26	its	its	PRON
ejpam-3764	3	27	topological	topological	ADJ
ejpam-3764	3	28	properties	property	NOUN
ejpam-3764	3	29	.	.	PUNCT
ejpam-3764	4	1	four	four	NUM
ejpam-3764	4	2	different	different	ADJ
ejpam-3764	4	3	examples	example	NOUN
ejpam-3764	4	4	of	of	ADP
ejpam-3764	4	5	omega	omega	NOUN
ejpam-3764	4	6	topology	topology	NOUN
ejpam-3764	4	7	are	be	AUX
ejpam-3764	4	8	introduced	introduce	VERB
ejpam-3764	4	9	.	.	PUNCT
ejpam-3764	5	1	furthermore	furthermore	ADV
ejpam-3764	5	2	,	,	PUNCT
ejpam-3764	5	3	we	we	PRON
ejpam-3764	5	4	define	define	VERB
ejpam-3764	5	5	a	a	DET
ejpam-3764	5	6	new	new	ADJ
ejpam-3764	5	7	topology	topology	NOUN
ejpam-3764	5	8	over	over	ADP
ejpam-3764	5	9	a	a	DET
ejpam-3764	5	10	semiring	semiring	NOUN
ejpam-3764	5	11	in	in	ADP
ejpam-3764	5	12	conventional	conventional	ADJ
ejpam-3764	5	13	algebra	algebra	NOUN
ejpam-3764	5	14	and	and	CCONJ
ejpam-3764	5	15	we	we	PRON
ejpam-3764	5	16	study	study	VERB
ejpam-3764	5	17	the	the	DET
ejpam-3764	5	18	relationship	relationship	NOUN
ejpam-3764	5	19	between	between	ADP
ejpam-3764	5	20	omega	omega	NOUN
ejpam-3764	5	21	topology	topology	NOUN
ejpam-3764	5	22	and	and	CCONJ
ejpam-3764	5	23	weaker	weak	ADJ
ejpam-3764	5	24	kinds	kind	NOUN
ejpam-3764	5	25	of	of	ADP
ejpam-3764	5	26	normality	normality	NOUN
ejpam-3764	5	27	.	.	PUNCT
ejpam-3764	6	1	2020	2020	NUM
ejpam-3764	6	2	mathematics	mathematic	NOUN
ejpam-3764	6	3	subject	subject	NOUN
ejpam-3764	6	4	classifications	classification	NOUN
ejpam-3764	6	5	:	:	PUNCT
ejpam-3764	6	6	16y60	16y60	NUM
ejpam-3764	6	7	,	,	PUNCT
ejpam-3764	6	8	54f15	54f15	NUM
ejpam-3764	6	9	key	key	ADJ
ejpam-3764	6	10	words	word	NOUN
ejpam-3764	6	11	and	and	CCONJ
ejpam-3764	6	12	phrases	phrase	NOUN
ejpam-3764	6	13	:	:	PUNCT
ejpam-3764	6	14	tropical	tropical	ADJ
ejpam-3764	6	15	geometry	geometry	NOUN
ejpam-3764	6	16	,	,	PUNCT
ejpam-3764	6	17	idempotent	idempotent	ADJ
ejpam-3764	6	18	semiring	semiring	NOUN
ejpam-3764	6	19	,	,	PUNCT
ejpam-3764	6	20	omega	omega	NOUN
ejpam-3764	6	21	algebra	algebra	NOUN
ejpam-3764	6	22	,	,	PUNCT
ejpam-3764	6	23	omega	omega	NOUN
ejpam-3764	6	24	topology	topology	NOUN
ejpam-3764	6	25	,	,	PUNCT
ejpam-3764	6	26	topological	topological	ADJ
ejpam-3764	6	27	space	space	NOUN
ejpam-3764	6	28	,	,	PUNCT
ejpam-3764	6	29	topological	topological	ADJ
ejpam-3764	6	30	properties	property	NOUN
ejpam-3764	6	31	,	,	PUNCT
ejpam-3764	6	32	homeomorphism	homeomorphism	X
ejpam-3764	6	33	1	1	X
ejpam-3764	6	34	.	.	X
ejpam-3764	6	35	introduction	introduction	NOUN
ejpam-3764	6	36	tropical	tropical	ADJ
ejpam-3764	6	37	geometry	geometry	NOUN
ejpam-3764	6	38	is	be	AUX
ejpam-3764	6	39	the	the	DET
ejpam-3764	6	40	most	most	ADV
ejpam-3764	6	41	recent	recent	ADJ
ejpam-3764	6	42	but	but	CCONJ
ejpam-3764	6	43	fast	fast	ADJ
ejpam-3764	6	44	growing	grow	VERB
ejpam-3764	6	45	branch	branch	NOUN
ejpam-3764	6	46	of	of	ADP
ejpam-3764	6	47	mathematical	mathematical	ADJ
ejpam-3764	6	48	science	science	NOUN
ejpam-3764	6	49	,	,	PUNCT
ejpam-3764	6	50	which	which	PRON
ejpam-3764	6	51	is	be	AUX
ejpam-3764	6	52	analytically	analytically	ADV
ejpam-3764	6	53	based	base	VERB
ejpam-3764	6	54	on	on	ADP
ejpam-3764	6	55	idempotent	idempotent	ADJ
ejpam-3764	6	56	analysis	analysis	NOUN
ejpam-3764	6	57	and	and	CCONJ
ejpam-3764	6	58	algebraically	algebraically	ADV
ejpam-3764	6	59	on	on	ADP
ejpam-3764	6	60	idempotent	idempotent	ADJ
ejpam-3764	6	61	semirings	semiring	NOUN
ejpam-3764	6	62	also	also	ADV
ejpam-3764	6	63	known	know	VERB
ejpam-3764	6	64	as	as	ADP
ejpam-3764	6	65	tropical	tropical	ADJ
ejpam-3764	6	66	semirings	semiring	NOUN
ejpam-3764	6	67	.	.	PUNCT
ejpam-3764	7	1	these	these	PRON
ejpam-3764	7	2	are	be	AUX
ejpam-3764	7	3	basically	basically	ADV
ejpam-3764	7	4	extended	extend	VERB
ejpam-3764	7	5	sets	set	NOUN
ejpam-3764	7	6	of	of	ADP
ejpam-3764	7	7	real	real	ADJ
ejpam-3764	7	8	numbers	number	NOUN
ejpam-3764	7	9	r∞	r∞	ADV
ejpam-3764	7	10	:	:	PUNCT
ejpam-3764	7	11	=	=	PUNCT
ejpam-3764	7	12	r∪{∞	r∪{∞	PROPN
ejpam-3764	7	13	}	}	PUNCT
ejpam-3764	7	14	and	and	CCONJ
ejpam-3764	7	15	r−∞	r−∞	NOUN
ejpam-3764	7	16	:	:	PUNCT
ejpam-3764	7	17	=	=	SYM
ejpam-3764	7	18	r∪{−∞	r∪{−∞	NOUN
ejpam-3764	7	19	}	}	PUNCT
ejpam-3764	7	20	which	which	PRON
ejpam-3764	7	21	are	be	AUX
ejpam-3764	7	22	given	give	VERB
ejpam-3764	7	23	monoidal	monoidal	ADJ
ejpam-3764	7	24	structures	structure	NOUN
ejpam-3764	7	25	by	by	ADP
ejpam-3764	7	26	using	use	VERB
ejpam-3764	7	27	min	min	NOUN
ejpam-3764	7	28	and	and	CCONJ
ejpam-3764	7	29	max	max	PROPN
ejpam-3764	7	30	operations	operation	NOUN
ejpam-3764	7	31	for	for	ADP
ejpam-3764	7	32	addition	addition	NOUN
ejpam-3764	7	33	,	,	PUNCT
ejpam-3764	7	34	respectively	respectively	ADV
ejpam-3764	7	35	.	.	PUNCT
ejpam-3764	8	1	in	in	ADP
ejpam-3764	8	2	order	order	NOUN
ejpam-3764	8	3	to	to	PART
ejpam-3764	8	4	adhere	adhere	VERB
ejpam-3764	8	5	the	the	DET
ejpam-3764	8	6	semiring	semire	VERB
ejpam-3764	8	7	structure	structure	NOUN
ejpam-3764	8	8	,	,	PUNCT
ejpam-3764	8	9	the	the	DET
ejpam-3764	8	10	additive	additive	ADJ
ejpam-3764	8	11	operation	operation	NOUN
ejpam-3764	8	12	of	of	ADP
ejpam-3764	8	13	r	r	NOUN
ejpam-3764	8	14	is	be	AUX
ejpam-3764	8	15	used	use	VERB
ejpam-3764	8	16	as	as	ADP
ejpam-3764	8	17	the	the	DET
ejpam-3764	8	18	multiplication	multiplication	NOUN
ejpam-3764	8	19	operation	operation	NOUN
ejpam-3764	8	20	.	.	PUNCT
ejpam-3764	9	1	by	by	ADP
ejpam-3764	9	2	these	these	DET
ejpam-3764	9	3	choices	choice	NOUN
ejpam-3764	9	4	,	,	PUNCT
ejpam-3764	9	5	both	both	PRON
ejpam-3764	9	6	r∞	r∞	NUM
ejpam-3764	9	7	and	and	CCONJ
ejpam-3764	9	8	r−∞	r−∞	NOUN
ejpam-3764	9	9	become	become	VERB
ejpam-3764	9	10	idempotent	idempotent	ADJ
ejpam-3764	9	11	semirings	semiring	NOUN
ejpam-3764	9	12	.	.	PUNCT
ejpam-3764	10	1	in	in	ADP
ejpam-3764	10	2	the	the	DET
ejpam-3764	10	3	literature	literature	NOUN
ejpam-3764	10	4	,	,	PUNCT
ejpam-3764	10	5	they	they	PRON
ejpam-3764	10	6	are	be	AUX
ejpam-3764	10	7	also	also	ADV
ejpam-3764	10	8	termed	term	VERB
ejpam-3764	10	9	as	as	ADP
ejpam-3764	10	10	min	min	NOUN
ejpam-3764	10	11	and	and	CCONJ
ejpam-3764	10	12	max	max	PROPN
ejpam-3764	10	13	plus	plus	CCONJ
ejpam-3764	10	14	algebras	algebra	NOUN
ejpam-3764	10	15	,	,	PUNCT
ejpam-3764	10	16	respectively	respectively	ADV
ejpam-3764	10	17	.	.	PUNCT
ejpam-3764	11	1	in	in	ADP
ejpam-3764	11	2	both	both	DET
ejpam-3764	11	3	cases	case	NOUN
ejpam-3764	11	4	,	,	PUNCT
ejpam-3764	11	5	0	0	NUM
ejpam-3764	11	6	of	of	ADP
ejpam-3764	11	7	r	r	NOUN
ejpam-3764	11	8	becomes	become	VERB
ejpam-3764	11	9	a	a	DET
ejpam-3764	11	10	multiplicative	multiplicative	ADJ
ejpam-3764	11	11	identity	identity	NOUN
ejpam-3764	11	12	and	and	CCONJ
ejpam-3764	11	13	∞	∞	NUM
ejpam-3764	11	14	and	and	CCONJ
ejpam-3764	11	15	−∞	−∞	ADP
ejpam-3764	11	16	become	become	VERB
ejpam-3764	11	17	additive	additive	ADJ
ejpam-3764	11	18	identities	identity	NOUN
ejpam-3764	11	19	of	of	ADP
ejpam-3764	11	20	these	these	DET
ejpam-3764	11	21	semirings	semiring	NOUN
ejpam-3764	11	22	,	,	PUNCT
ejpam-3764	11	23	respectively	respectively	ADV
ejpam-3764	11	24	.	.	PUNCT
ejpam-3764	12	1	interestingly	interestingly	ADV
ejpam-3764	12	2	,	,	PUNCT
ejpam-3764	12	3	some	some	DET
ejpam-3764	12	4	authors	author	NOUN
ejpam-3764	12	5	associated	associate	VERB
ejpam-3764	12	6	r−∞	r−∞	NOUN
ejpam-3764	12	7	to	to	ADP
ejpam-3764	12	8	tropical	tropical	ADJ
ejpam-3764	12	9	geometry	geometry	NOUN
ejpam-3764	12	10	,	,	PUNCT
ejpam-3764	12	11	while	while	SCONJ
ejpam-3764	12	12	other	other	ADJ
ejpam-3764	12	13	authors	author	NOUN
ejpam-3764	12	14	associated	associate	VERB
ejpam-3764	12	15	r∞	r∞	PROPN
ejpam-3764	12	16	to	to	ADP
ejpam-3764	12	17	tropical	tropical	ADJ
ejpam-3764	12	18	geometry	geometry	NOUN
ejpam-3764	12	19	,	,	PUNCT
ejpam-3764	12	20	see	see	VERB
ejpam-3764	12	21	for	for	ADP
ejpam-3764	12	22	instance	instance	NOUN
ejpam-3764	12	23	[	[	X
ejpam-3764	12	24	8	8	NUM
ejpam-3764	12	25	]	]	PUNCT
ejpam-3764	12	26	,	,	PUNCT
ejpam-3764	12	27	[	[	X
ejpam-3764	12	28	10	10	NUM
ejpam-3764	12	29	]	]	PUNCT
ejpam-3764	12	30	,	,	PUNCT
ejpam-3764	12	31	[	[	X
ejpam-3764	12	32	12	12	NUM
ejpam-3764	12	33	]	]	PUNCT
ejpam-3764	12	34	and	and	CCONJ
ejpam-3764	12	35	[	[	X
ejpam-3764	12	36	14	14	NUM
ejpam-3764	12	37	]	]	PUNCT
ejpam-3764	12	38	.	.	PUNCT
ejpam-3764	13	1	omega	omega	NOUN
ejpam-3764	13	2	algebra	algebra	PROPN
ejpam-3764	13	3	,	,	PUNCT
ejpam-3764	13	4	or	or	CCONJ
ejpam-3764	13	5	”	"	PUNCT
ejpam-3764	13	6	walgebra	walgebra	NOUN
ejpam-3764	13	7	”	"	PUNCT
ejpam-3764	13	8	for	for	ADP
ejpam-3764	13	9	short	short	ADJ
ejpam-3764	13	10	,	,	PUNCT
ejpam-3764	13	11	unifies	unify	VERB
ejpam-3764	13	12	the	the	DET
ejpam-3764	13	13	different	different	ADJ
ejpam-3764	13	14	terms	term	NOUN
ejpam-3764	13	15	and	and	CCONJ
ejpam-3764	13	16	introduces	introduce	VERB
ejpam-3764	13	17	an	an	DET
ejpam-3764	13	18	original	original	ADJ
ejpam-3764	13	19	structure	structure	NOUN
ejpam-3764	13	20	,	,	PUNCT
ejpam-3764	13	21	which	which	PRON
ejpam-3764	13	22	,	,	PUNCT
ejpam-3764	13	23	in	in	ADP
ejpam-3764	13	24	fact	fact	NOUN
ejpam-3764	13	25	,	,	PUNCT
ejpam-3764	13	26	is	be	AUX
ejpam-3764	13	27	an	an	DET
ejpam-3764	13	28	”	"	PUNCT
ejpam-3764	13	29	abstract	abstract	ADJ
ejpam-3764	13	30	tropical	tropical	ADJ
ejpam-3764	13	31	algebra	algebra	NOUN
ejpam-3764	13	32	”	"	PUNCT
ejpam-3764	13	33	.	.	PUNCT
ejpam-3764	14	1	the	the	DET
ejpam-3764	14	2	r−∞	r−∞	NOUN
ejpam-3764	14	3	and	and	CCONJ
ejpam-3764	14	4	r∞	r∞	PROPN
ejpam-3764	14	5	and	and	CCONJ
ejpam-3764	14	6	their	their	PRON
ejpam-3764	14	7	nearby	nearby	ADJ
ejpam-3764	14	8	structures	structure	NOUN
ejpam-3764	14	9	,	,	PUNCT
ejpam-3764	14	10	like	like	ADP
ejpam-3764	14	11	min−max	min−max	ADJ
ejpam-3764	14	12	and	and	CCONJ
ejpam-3764	14	13	max−	max−	NOUN
ejpam-3764	14	14	times	time	NOUN
ejpam-3764	14	15	algebras	algebra	NOUN
ejpam-3764	14	16	,	,	PUNCT
ejpam-3764	14	17	etc	etc	X
ejpam-3764	14	18	.	.	X
ejpam-3764	14	19	,	,	PUNCT
ejpam-3764	14	20	are	be	AUX
ejpam-3764	14	21	all	all	PRON
ejpam-3764	14	22	subsumed	subsume	VERB
ejpam-3764	14	23	under	under	ADP
ejpam-3764	14	24	omega	omega	NOUN
ejpam-3764	14	25	algebra	algebra	NOUN
ejpam-3764	14	26	.	.	PUNCT
ejpam-3764	15	1	∗corresponding	∗corresponde	VERB
ejpam-3764	15	2	author	author	NOUN
ejpam-3764	15	3	.	.	PUNCT
ejpam-3764	16	1	doi	doi	NOUN
ejpam-3764	16	2	:	:	PUNCT
ejpam-3764	16	3	https://doi.org/10.29020/nybg.ejpam.v13i3.3764	https://doi.org/10.29020/nybg.ejpam.v13i3.3764	NUM
ejpam-3764	16	4	email	email	NOUN
ejpam-3764	16	5	addresses	address	NOUN
ejpam-3764	16	6	:	:	PUNCT
ejpam-3764	16	7	mesfer	mesfer	VERB
ejpam-3764	16	8	¯	¯	PROPN
ejpam-3764	16	9	alqhtani@hotmail.com	alqhtani@hotmail.com	X
ejpam-3764	17	1	(	(	PUNCT
ejpam-3764	17	2	m.	m.	NOUN
ejpam-3764	17	3	alqahtani	alqahtani	PROPN
ejpam-3764	17	4	)	)	PUNCT
ejpam-3764	17	5	,	,	PUNCT
ejpam-3764	18	1	cozel@kau.edu.sa	cozel@kau.edu.sa	NOUN
ejpam-3764	18	2	(	(	PUNCT
ejpam-3764	18	3	c.	c.	PROPN
ejpam-3764	18	4	özel	özel	PROPN
ejpam-3764	18	5	)	)	PUNCT
ejpam-3764	18	6	,	,	PUNCT
ejpam-3764	18	7	iealshamri@uhb.edu.sa	iealshamri@uhb.edu.sa	NOUN
ejpam-3764	18	8	(	(	PUNCT
ejpam-3764	18	9	i.	i.	PROPN
ejpam-3764	18	10	alshammari	alshammari	PROPN
ejpam-3764	18	11	)	)	PUNCT
ejpam-3764	18	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-3764	19	1	513	513	NUM
ejpam-3764	19	2	c	c	NOUN
ejpam-3764	19	3	©	©	NOUN
ejpam-3764	19	4	2020	2020	NUM
ejpam-3764	19	5	ejpam	ejpam	VERB
ejpam-3764	19	6	all	all	DET
ejpam-3764	19	7	rights	right	NOUN
ejpam-3764	19	8	reserved	reserve	VERB
ejpam-3764	19	9	.	.	PUNCT
ejpam-3764	20	1	m.	m.	PROPN
ejpam-3764	20	2	alqahtani	alqahtani	PROPN
ejpam-3764	20	3	,	,	PUNCT
ejpam-3764	20	4	c.	c.	PROPN
ejpam-3764	20	5	özel	özel	PROPN
ejpam-3764	20	6	,	,	PUNCT
ejpam-3764	20	7	i.	i.	PROPN
ejpam-3764	20	8	alshammari	alshammari	PROPN
ejpam-3764	20	9	/	/	SYM
ejpam-3764	20	10	eur	eur	PROPN
ejpam-3764	20	11	.	.	PUNCT
ejpam-3764	21	1	j.	j.	PROPN
ejpam-3764	21	2	pure	pure	PROPN
ejpam-3764	21	3	appl	appl	PROPN
ejpam-3764	21	4	.	.	PROPN
ejpam-3764	21	5	math	math	PROPN
ejpam-3764	21	6	,	,	PUNCT
ejpam-3764	21	7	13	13	NUM
ejpam-3764	21	8	(	(	PUNCT
ejpam-3764	21	9	3	3	NUM
ejpam-3764	21	10	)	)	PUNCT
ejpam-3764	21	11	(	(	PUNCT
ejpam-3764	21	12	2020	2020	NUM
ejpam-3764	21	13	)	)	PUNCT
ejpam-3764	21	14	,	,	PUNCT
ejpam-3764	21	15	513	513	NUM
ejpam-3764	21	16	-	-	SYM
ejpam-3764	21	17	528	528	NUM
ejpam-3764	21	18	514	514	NUM
ejpam-3764	21	19	all	all	DET
ejpam-3764	21	20	these	these	PRON
ejpam-3764	21	21	are	be	AUX
ejpam-3764	21	22	idempotent	idempotent	ADJ
ejpam-3764	21	23	semirings	semiring	NOUN
ejpam-3764	21	24	,	,	PUNCT
ejpam-3764	21	25	which	which	PRON
ejpam-3764	21	26	are	be	AUX
ejpam-3764	21	27	also	also	ADV
ejpam-3764	21	28	called	call	VERB
ejpam-3764	21	29	dioids	dioid	NOUN
ejpam-3764	21	30	.	.	PUNCT
ejpam-3764	22	1	in	in	ADP
ejpam-3764	22	2	the	the	DET
ejpam-3764	22	3	previous	previous	ADJ
ejpam-3764	22	4	studies	study	NOUN
ejpam-3764	22	5	,	,	PUNCT
ejpam-3764	22	6	for	for	ADP
ejpam-3764	22	7	the	the	DET
ejpam-3764	22	8	construction	construction	NOUN
ejpam-3764	22	9	of	of	ADP
ejpam-3764	22	10	all	all	DET
ejpam-3764	22	11	such	such	ADJ
ejpam-3764	22	12	semirings	semiring	NOUN
ejpam-3764	22	13	,	,	PUNCT
ejpam-3764	22	14	an	an	DET
ejpam-3764	22	15	ordered	order	VERB
ejpam-3764	22	16	infinite	infinite	ADJ
ejpam-3764	22	17	abelian	abelian	ADJ
ejpam-3764	22	18	group	group	NOUN
ejpam-3764	22	19	is	be	AUX
ejpam-3764	22	20	mandatory	mandatory	ADJ
ejpam-3764	22	21	.	.	PUNCT
ejpam-3764	23	1	in	in	ADP
ejpam-3764	23	2	ω−	ω−	PROPN
ejpam-3764	23	3	algebra	algebra	NOUN
ejpam-3764	23	4	,	,	PUNCT
ejpam-3764	23	5	the	the	DET
ejpam-3764	23	6	definition	definition	NOUN
ejpam-3764	23	7	is	be	AUX
ejpam-3764	23	8	extended	extend	VERB
ejpam-3764	23	9	to	to	PART
ejpam-3764	23	10	cyclically	cyclically	ADV
ejpam-3764	23	11	ordered	order	VERB
ejpam-3764	23	12	abelian	abelian	ADJ
ejpam-3764	23	13	groups	group	NOUN
ejpam-3764	23	14	and	and	CCONJ
ejpam-3764	23	15	also	also	ADV
ejpam-3764	23	16	to	to	PART
ejpam-3764	23	17	finite	finite	VERB
ejpam-3764	23	18	sets	set	NOUN
ejpam-3764	23	19	under	under	ADP
ejpam-3764	23	20	some	some	DET
ejpam-3764	23	21	suitable	suitable	ADJ
ejpam-3764	23	22	ordering	ordering	NOUN
ejpam-3764	23	23	.	.	PUNCT
ejpam-3764	24	1	note	note	VERB
ejpam-3764	24	2	that	that	SCONJ
ejpam-3764	24	3	cyclically	cyclically	ADV
ejpam-3764	24	4	ordered	order	VERB
ejpam-3764	24	5	abelian	abelian	ADJ
ejpam-3764	24	6	groups	group	NOUN
ejpam-3764	24	7	are	be	AUX
ejpam-3764	24	8	more	more	ADV
ejpam-3764	24	9	general	general	ADJ
ejpam-3764	24	10	than	than	ADP
ejpam-3764	24	11	that	that	PRON
ejpam-3764	24	12	of	of	ADP
ejpam-3764	24	13	ordered	order	VERB
ejpam-3764	24	14	abelian	abelian	ADJ
ejpam-3764	24	15	groups	group	NOUN
ejpam-3764	25	1	[	[	X
ejpam-3764	25	2	16	16	NUM
ejpam-3764	25	3	]	]	PUNCT
ejpam-3764	25	4	.	.	PUNCT
ejpam-3764	26	1	the	the	DET
ejpam-3764	26	2	aim	aim	NOUN
ejpam-3764	26	3	of	of	ADP
ejpam-3764	26	4	this	this	DET
ejpam-3764	26	5	paper	paper	NOUN
ejpam-3764	26	6	is	be	AUX
ejpam-3764	26	7	to	to	PART
ejpam-3764	26	8	define	define	VERB
ejpam-3764	26	9	a	a	DET
ejpam-3764	26	10	new	new	ADJ
ejpam-3764	26	11	topology	topology	NOUN
ejpam-3764	26	12	,	,	PUNCT
ejpam-3764	26	13	which	which	PRON
ejpam-3764	26	14	is	be	AUX
ejpam-3764	26	15	called	call	VERB
ejpam-3764	26	16	omega	omega	NOUN
ejpam-3764	26	17	topology	topology	NOUN
ejpam-3764	26	18	over	over	ADP
ejpam-3764	26	19	omega	omega	NOUN
ejpam-3764	26	20	algebra	algebra	NOUN
ejpam-3764	26	21	,	,	PUNCT
ejpam-3764	26	22	and	and	CCONJ
ejpam-3764	26	23	discuss	discuss	VERB
ejpam-3764	26	24	some	some	PRON
ejpam-3764	26	25	of	of	ADP
ejpam-3764	26	26	its	its	PRON
ejpam-3764	26	27	topological	topological	ADJ
ejpam-3764	26	28	properties	property	NOUN
ejpam-3764	26	29	.	.	PUNCT
ejpam-3764	27	1	four	four	NUM
ejpam-3764	27	2	different	different	ADJ
ejpam-3764	27	3	examples	example	NOUN
ejpam-3764	27	4	of	of	ADP
ejpam-3764	27	5	omega	omega	NOUN
ejpam-3764	27	6	topology	topology	NOUN
ejpam-3764	27	7	are	be	AUX
ejpam-3764	27	8	introduced	introduce	VERB
ejpam-3764	27	9	.	.	PUNCT
ejpam-3764	28	1	furthermore	furthermore	ADV
ejpam-3764	28	2	,	,	PUNCT
ejpam-3764	28	3	we	we	PRON
ejpam-3764	28	4	defined	define	VERB
ejpam-3764	28	5	a	a	DET
ejpam-3764	28	6	new	new	ADJ
ejpam-3764	28	7	topology	topology	NOUN
ejpam-3764	28	8	over	over	ADP
ejpam-3764	28	9	a	a	DET
ejpam-3764	28	10	semiring	semiring	NOUN
ejpam-3764	28	11	in	in	ADP
ejpam-3764	28	12	conventional	conventional	ADJ
ejpam-3764	28	13	algebra	algebra	NOUN
ejpam-3764	28	14	and	and	CCONJ
ejpam-3764	28	15	study	study	VERB
ejpam-3764	28	16	the	the	DET
ejpam-3764	28	17	relationship	relationship	NOUN
ejpam-3764	28	18	between	between	ADP
ejpam-3764	28	19	omega	omega	NOUN
ejpam-3764	28	20	topology	topology	NOUN
ejpam-3764	28	21	and	and	CCONJ
ejpam-3764	28	22	weaker	weak	ADJ
ejpam-3764	28	23	kinds	kind	NOUN
ejpam-3764	28	24	of	of	ADP
ejpam-3764	28	25	normality	normality	NOUN
ejpam-3764	28	26	.	.	PUNCT
ejpam-3764	29	1	this	this	DET
ejpam-3764	29	2	paper	paper	NOUN
ejpam-3764	29	3	is	be	AUX
ejpam-3764	29	4	divided	divide	VERB
ejpam-3764	29	5	as	as	SCONJ
ejpam-3764	29	6	follows	follow	VERB
ejpam-3764	29	7	.	.	PUNCT
ejpam-3764	30	1	in	in	ADP
ejpam-3764	30	2	section	section	NOUN
ejpam-3764	30	3	2	2	NUM
ejpam-3764	30	4	,	,	PUNCT
ejpam-3764	30	5	we	we	PRON
ejpam-3764	30	6	review	review	VERB
ejpam-3764	30	7	an	an	DET
ejpam-3764	30	8	abstract	abstract	ADJ
ejpam-3764	30	9	definition	definition	NOUN
ejpam-3764	30	10	and	and	CCONJ
ejpam-3764	30	11	some	some	DET
ejpam-3764	30	12	basic	basic	ADJ
ejpam-3764	30	13	facts	fact	NOUN
ejpam-3764	30	14	about	about	ADP
ejpam-3764	30	15	abstract	abstract	ADJ
ejpam-3764	30	16	omega	omega	NOUN
ejpam-3764	30	17	algebras	algebra	NOUN
ejpam-3764	30	18	.	.	PUNCT
ejpam-3764	31	1	we	we	PRON
ejpam-3764	31	2	support	support	VERB
ejpam-3764	31	3	these	these	PRON
ejpam-3764	31	4	by	by	ADP
ejpam-3764	31	5	presenting	present	VERB
ejpam-3764	31	6	three	three	NUM
ejpam-3764	31	7	concrete	concrete	ADJ
ejpam-3764	31	8	examples	example	NOUN
ejpam-3764	31	9	.	.	PUNCT
ejpam-3764	32	1	in	in	ADP
ejpam-3764	32	2	section	section	NOUN
ejpam-3764	32	3	3	3	NUM
ejpam-3764	32	4	,	,	PUNCT
ejpam-3764	32	5	we	we	PRON
ejpam-3764	32	6	define	define	VERB
ejpam-3764	32	7	a	a	DET
ejpam-3764	32	8	new	new	ADJ
ejpam-3764	32	9	topology	topology	NOUN
ejpam-3764	32	10	on	on	ADP
ejpam-3764	32	11	omega	omega	NOUN
ejpam-3764	32	12	algebra	algebra	NOUN
ejpam-3764	32	13	and	and	CCONJ
ejpam-3764	32	14	discuss	discuss	VERB
ejpam-3764	32	15	some	some	PRON
ejpam-3764	32	16	of	of	ADP
ejpam-3764	32	17	its	its	PRON
ejpam-3764	32	18	topological	topological	ADJ
ejpam-3764	32	19	properties	property	NOUN
ejpam-3764	32	20	.	.	PUNCT
ejpam-3764	33	1	in	in	ADP
ejpam-3764	33	2	section	section	NOUN
ejpam-3764	33	3	4	4	NUM
ejpam-3764	33	4	,	,	PUNCT
ejpam-3764	33	5	we	we	PRON
ejpam-3764	33	6	provide	provide	VERB
ejpam-3764	33	7	four	four	NUM
ejpam-3764	33	8	different	different	ADJ
ejpam-3764	33	9	examples	example	NOUN
ejpam-3764	33	10	of	of	ADP
ejpam-3764	33	11	omega	omega	NOUN
ejpam-3764	33	12	topology	topology	NOUN
ejpam-3764	33	13	:	:	PUNCT
ejpam-3764	33	14	the	the	DET
ejpam-3764	33	15	first	first	ADJ
ejpam-3764	33	16	and	and	CCONJ
ejpam-3764	33	17	fourth	fourth	ADJ
ejpam-3764	33	18	examples	example	NOUN
ejpam-3764	33	19	are	be	AUX
ejpam-3764	33	20	from	from	ADP
ejpam-3764	33	21	an	an	DET
ejpam-3764	33	22	ordered	order	VERB
ejpam-3764	33	23	infinite	infinite	ADJ
ejpam-3764	33	24	sets	set	NOUN
ejpam-3764	33	25	,	,	PUNCT
ejpam-3764	33	26	the	the	DET
ejpam-3764	33	27	second	second	ADJ
ejpam-3764	33	28	example	example	NOUN
ejpam-3764	33	29	is	be	AUX
ejpam-3764	33	30	from	from	ADP
ejpam-3764	33	31	a	a	DET
ejpam-3764	33	32	cyclically	cyclically	ADV
ejpam-3764	33	33	ordered	order	VERB
ejpam-3764	33	34	infinite	infinite	ADJ
ejpam-3764	33	35	set	set	NOUN
ejpam-3764	33	36	,	,	PUNCT
ejpam-3764	33	37	and	and	CCONJ
ejpam-3764	33	38	the	the	DET
ejpam-3764	33	39	third	third	ADJ
ejpam-3764	33	40	example	example	NOUN
ejpam-3764	33	41	is	be	AUX
ejpam-3764	33	42	from	from	ADP
ejpam-3764	33	43	a	a	DET
ejpam-3764	33	44	finite	finite	ADJ
ejpam-3764	33	45	set	set	NOUN
ejpam-3764	33	46	.	.	PUNCT
ejpam-3764	34	1	furthermore	furthermore	ADV
ejpam-3764	34	2	,	,	PUNCT
ejpam-3764	34	3	we	we	PRON
ejpam-3764	34	4	define	define	VERB
ejpam-3764	34	5	a	a	DET
ejpam-3764	34	6	new	new	ADJ
ejpam-3764	34	7	topology	topology	NOUN
ejpam-3764	34	8	over	over	ADP
ejpam-3764	34	9	a	a	DET
ejpam-3764	34	10	semiring	semiring	NOUN
ejpam-3764	34	11	in	in	ADP
ejpam-3764	34	12	conventional	conventional	ADJ
ejpam-3764	34	13	algebra	algebra	NOUN
ejpam-3764	34	14	.	.	PUNCT
ejpam-3764	35	1	finally	finally	ADV
ejpam-3764	35	2	,	,	PUNCT
ejpam-3764	35	3	we	we	PRON
ejpam-3764	35	4	study	study	VERB
ejpam-3764	35	5	the	the	DET
ejpam-3764	35	6	relationship	relationship	NOUN
ejpam-3764	35	7	between	between	ADP
ejpam-3764	35	8	omega	omega	NOUN
ejpam-3764	35	9	topology	topology	NOUN
ejpam-3764	35	10	and	and	CCONJ
ejpam-3764	35	11	weaker	weak	ADJ
ejpam-3764	35	12	kinds	kind	NOUN
ejpam-3764	35	13	of	of	ADP
ejpam-3764	35	14	normality	normality	NOUN
ejpam-3764	35	15	in	in	ADP
ejpam-3764	35	16	section	section	NOUN
ejpam-3764	35	17	5	5	NUM
ejpam-3764	35	18	.	.	PUNCT
ejpam-3764	36	1	throughout	throughout	ADP
ejpam-3764	36	2	this	this	DET
ejpam-3764	36	3	paper	paper	NOUN
ejpam-3764	36	4	,	,	PUNCT
ejpam-3764	36	5	we	we	PRON
ejpam-3764	36	6	do	do	AUX
ejpam-3764	36	7	not	not	PART
ejpam-3764	36	8	assume	assume	VERB
ejpam-3764	36	9	t2	t2	NOUN
ejpam-3764	36	10	in	in	ADP
ejpam-3764	36	11	the	the	DET
ejpam-3764	36	12	definition	definition	NOUN
ejpam-3764	36	13	of	of	ADP
ejpam-3764	36	14	compactness	compactness	NOUN
ejpam-3764	36	15	.	.	PUNCT
ejpam-3764	37	1	we	we	PRON
ejpam-3764	37	2	also	also	ADV
ejpam-3764	37	3	do	do	AUX
ejpam-3764	37	4	not	not	PART
ejpam-3764	37	5	assume	assume	VERB
ejpam-3764	37	6	regularity	regularity	NOUN
ejpam-3764	37	7	in	in	ADP
ejpam-3764	37	8	the	the	DET
ejpam-3764	37	9	definition	definition	NOUN
ejpam-3764	37	10	of	of	ADP
ejpam-3764	37	11	lindelöfness	lindelöfness	PROPN
ejpam-3764	37	12	.	.	PUNCT
ejpam-3764	38	1	this	this	DET
ejpam-3764	38	2	paper	paper	NOUN
ejpam-3764	38	3	is	be	AUX
ejpam-3764	38	4	produced	produce	VERB
ejpam-3764	38	5	from	from	ADP
ejpam-3764	38	6	the	the	DET
ejpam-3764	38	7	phd	phd	NOUN
ejpam-3764	38	8	thesis	thesis	NOUN
ejpam-3764	38	9	of	of	ADP
ejpam-3764	38	10	mr	mr	PROPN
ejpam-3764	38	11	.	.	PROPN
ejpam-3764	38	12	mesfer	mesfer	PROPN
ejpam-3764	38	13	hayyan	hayyan	PROPN
ejpam-3764	38	14	alqahtani	alqahtani	PROPN
ejpam-3764	38	15	in	in	ADP
ejpam-3764	38	16	king	king	PROPN
ejpam-3764	38	17	abdulaziz	abdulaziz	PROPN
ejpam-3764	38	18	university	university	PROPN
ejpam-3764	38	19	.	.	PUNCT
ejpam-3764	39	1	2	2	X
ejpam-3764	39	2	.	.	X
ejpam-3764	39	3	preliminaries	preliminary	NOUN
ejpam-3764	39	4	in	in	ADP
ejpam-3764	39	5	this	this	DET
ejpam-3764	39	6	section	section	NOUN
ejpam-3764	39	7	,	,	PUNCT
ejpam-3764	39	8	we	we	PRON
ejpam-3764	39	9	provide	provide	VERB
ejpam-3764	39	10	an	an	DET
ejpam-3764	39	11	abstract	abstract	ADJ
ejpam-3764	39	12	definition	definition	NOUN
ejpam-3764	39	13	and	and	CCONJ
ejpam-3764	39	14	review	review	VERB
ejpam-3764	39	15	some	some	DET
ejpam-3764	39	16	basic	basic	ADJ
ejpam-3764	39	17	facts	fact	NOUN
ejpam-3764	39	18	about	about	ADP
ejpam-3764	39	19	abstract	abstract	ADJ
ejpam-3764	39	20	omega	omega	NOUN
ejpam-3764	39	21	algebras	algebra	NOUN
ejpam-3764	39	22	.	.	PUNCT
ejpam-3764	40	1	furthermore	furthermore	ADV
ejpam-3764	40	2	,	,	PUNCT
ejpam-3764	40	3	we	we	PRON
ejpam-3764	40	4	support	support	VERB
ejpam-3764	40	5	these	these	PRON
ejpam-3764	40	6	by	by	ADP
ejpam-3764	40	7	presenting	present	VERB
ejpam-3764	40	8	concrete	concrete	ADJ
ejpam-3764	40	9	examples	example	NOUN
ejpam-3764	40	10	:	:	PUNCT
ejpam-3764	40	11	one	one	NUM
ejpam-3764	40	12	from	from	ADP
ejpam-3764	40	13	an	an	DET
ejpam-3764	40	14	ordered	order	VERB
ejpam-3764	40	15	infinite	infinite	NOUN
ejpam-3764	40	16	set	set	NOUN
ejpam-3764	40	17	,	,	PUNCT
ejpam-3764	40	18	another	another	PRON
ejpam-3764	40	19	from	from	ADP
ejpam-3764	40	20	a	a	DET
ejpam-3764	40	21	cyclically	cyclically	ADV
ejpam-3764	40	22	ordered	order	VERB
ejpam-3764	40	23	infinite	infinite	ADJ
ejpam-3764	40	24	set	set	NOUN
ejpam-3764	40	25	,	,	PUNCT
ejpam-3764	40	26	and	and	CCONJ
ejpam-3764	40	27	a	a	DET
ejpam-3764	40	28	third	third	ADJ
ejpam-3764	40	29	one	one	NUM
ejpam-3764	40	30	from	from	ADP
ejpam-3764	40	31	a	a	DET
ejpam-3764	40	32	finite	finite	NOUN
ejpam-3764	40	33	set	set	NOUN
ejpam-3764	40	34	.	.	PUNCT
ejpam-3764	41	1	for	for	ADP
ejpam-3764	41	2	more	more	ADJ
ejpam-3764	41	3	details	detail	NOUN
ejpam-3764	41	4	,	,	PUNCT
ejpam-3764	41	5	see	see	VERB
ejpam-3764	41	6	[	[	X
ejpam-3764	41	7	11	11	NUM
ejpam-3764	41	8	]	]	PUNCT
ejpam-3764	41	9	.	.	PUNCT
ejpam-3764	42	1	let	let	AUX
ejpam-3764	42	2	(	(	PUNCT
ejpam-3764	42	3	g	g	NOUN
ejpam-3764	42	4	,	,	PUNCT
ejpam-3764	42	5	◦	◦	NOUN
ejpam-3764	42	6	,	,	PUNCT
ejpam-3764	42	7	e	e	NOUN
ejpam-3764	42	8	)	)	PUNCT
ejpam-3764	42	9	be	be	AUX
ejpam-3764	42	10	an	an	DET
ejpam-3764	42	11	abelian	abelian	ADJ
ejpam-3764	42	12	group	group	NOUN
ejpam-3764	42	13	.	.	PUNCT
ejpam-3764	43	1	let	let	VERB
ejpam-3764	43	2	a	a	DET
ejpam-3764	43	3	be	be	AUX
ejpam-3764	43	4	a	a	DET
ejpam-3764	43	5	closed	closed	ADJ
ejpam-3764	43	6	subset	subset	NOUN
ejpam-3764	43	7	of	of	ADP
ejpam-3764	43	8	g	g	PROPN
ejpam-3764	43	9	and	and	CCONJ
ejpam-3764	43	10	e	e	PROPN
ejpam-3764	43	11	∈	∈	PROPN
ejpam-3764	43	12	a.	a.	NOUN
ejpam-3764	43	13	then	then	ADV
ejpam-3764	43	14	(	(	PUNCT
ejpam-3764	43	15	a	a	DET
ejpam-3764	43	16	,	,	PUNCT
ejpam-3764	43	17	◦	◦	NOUN
ejpam-3764	43	18	,	,	PUNCT
ejpam-3764	43	19	e	e	NOUN
ejpam-3764	43	20	)	)	PUNCT
ejpam-3764	43	21	is	be	AUX
ejpam-3764	43	22	a	a	DET
ejpam-3764	43	23	submonoid	submonoid	NOUN
ejpam-3764	43	24	of	of	ADP
ejpam-3764	43	25	g.	g.	PROPN
ejpam-3764	43	26	assume	assume	VERB
ejpam-3764	43	27	that	that	SCONJ
ejpam-3764	43	28	ω	ω	PROPN
ejpam-3764	43	29	is	be	AUX
ejpam-3764	43	30	an	an	DET
ejpam-3764	43	31	indeterminate	indeterminate	NOUN
ejpam-3764	43	32	(	(	PUNCT
ejpam-3764	43	33	may	may	AUX
ejpam-3764	43	34	belong	belong	VERB
ejpam-3764	43	35	to	to	ADP
ejpam-3764	43	36	a	a	PRON
ejpam-3764	43	37	or	or	CCONJ
ejpam-3764	43	38	g	g	NOUN
ejpam-3764	43	39	,	,	PUNCT
ejpam-3764	43	40	as	as	SCONJ
ejpam-3764	43	41	we	we	PRON
ejpam-3764	43	42	will	will	AUX
ejpam-3764	43	43	see	see	VERB
ejpam-3764	43	44	in	in	ADP
ejpam-3764	43	45	examples	example	NOUN
ejpam-3764	43	46	1	1	NUM
ejpam-3764	43	47	and	and	CCONJ
ejpam-3764	43	48	2	2	NUM
ejpam-3764	43	49	)	)	PUNCT
ejpam-3764	43	50	.	.	PUNCT
ejpam-3764	44	1	obviously	obviously	ADV
ejpam-3764	44	2	,	,	PUNCT
ejpam-3764	44	3	in	in	ADP
ejpam-3764	44	4	this	this	DET
ejpam-3764	44	5	case	case	NOUN
ejpam-3764	44	6	ω	ω	NOUN
ejpam-3764	44	7	is	be	AUX
ejpam-3764	44	8	no	no	ADV
ejpam-3764	44	9	longer	long	ADV
ejpam-3764	44	10	an	an	DET
ejpam-3764	44	11	indeterminate	indeterminate	NOUN
ejpam-3764	44	12	.	.	PUNCT
ejpam-3764	45	1	because	because	SCONJ
ejpam-3764	45	2	the	the	DET
ejpam-3764	45	3	terms	term	NOUN
ejpam-3764	45	4	are	be	AUX
ejpam-3764	45	5	generated	generate	VERB
ejpam-3764	45	6	from	from	ADP
ejpam-3764	45	7	tropical	tropical	ADJ
ejpam-3764	45	8	geometry	geometry	NOUN
ejpam-3764	45	9	,	,	PUNCT
ejpam-3764	45	10	this	this	DET
ejpam-3764	45	11	indeterminate	indeterminate	NOUN
ejpam-3764	45	12	can	can	AUX
ejpam-3764	45	13	be	be	AUX
ejpam-3764	45	14	called	call	VERB
ejpam-3764	45	15	a	a	DET
ejpam-3764	45	16	tropical	tropical	ADJ
ejpam-3764	45	17	indeterminate	indeterminate	NOUN
ejpam-3764	45	18	.	.	PUNCT
ejpam-3764	46	1	definition	definition	NOUN
ejpam-3764	46	2	1	1	NUM
ejpam-3764	46	3	.	.	PUNCT
ejpam-3764	47	1	[	[	X
ejpam-3764	47	2	11	11	NUM
ejpam-3764	47	3	]	]	PUNCT
ejpam-3764	47	4	we	we	PRON
ejpam-3764	47	5	say	say	VERB
ejpam-3764	47	6	that	that	PRON
ejpam-3764	47	7	aω	aω	PROPN
ejpam-3764	47	8	=	=	PUNCT
ejpam-3764	47	9	a	a	DET
ejpam-3764	47	10	∪	∪	X
ejpam-3764	47	11	{	{	PUNCT
ejpam-3764	47	12	ω	ω	NOUN
ejpam-3764	47	13	}	}	PUNCT
ejpam-3764	47	14	is	be	AUX
ejpam-3764	47	15	an	an	DET
ejpam-3764	47	16	omega	omega	NOUN
ejpam-3764	47	17	algebra	algebra	NOUN
ejpam-3764	47	18	(	(	PUNCT
ejpam-3764	47	19	in	in	ADP
ejpam-3764	47	20	short	short	ADJ
ejpam-3764	47	21	ω−	ω−	ADJ
ejpam-3764	47	22	algebra	algebra	NOUN
ejpam-3764	47	23	)	)	PUNCT
ejpam-3764	47	24	over	over	ADP
ejpam-3764	47	25	the	the	DET
ejpam-3764	47	26	group	group	NOUN
ejpam-3764	47	27	g	g	PROPN
ejpam-3764	47	28	in	in	ADP
ejpam-3764	47	29	case	case	NOUN
ejpam-3764	47	30	aω	aω	NOUN
ejpam-3764	47	31	is	be	AUX
ejpam-3764	47	32	closed	close	VERB
ejpam-3764	47	33	under	under	ADP
ejpam-3764	47	34	two	two	NUM
ejpam-3764	47	35	binary	binary	ADJ
ejpam-3764	47	36	operations	operation	NOUN
ejpam-3764	47	37	,	,	PUNCT
ejpam-3764	47	38	⊕,⊗	⊕,⊗	X
ejpam-3764	47	39	:	:	PUNCT
ejpam-3764	47	40	aω	aω	PROPN
ejpam-3764	47	41	×aω	×aω	PROPN
ejpam-3764	47	42	−→	−→	PROPN
ejpam-3764	47	43	aω	aω	PROPN
ejpam-3764	47	44	,	,	PUNCT
ejpam-3764	47	45	then	then	ADV
ejpam-3764	47	46	for	for	ADP
ejpam-3764	47	47	all	all	DET
ejpam-3764	47	48	a1	a1	NOUN
ejpam-3764	47	49	,	,	PUNCT
ejpam-3764	47	50	a2	a2	PROPN
ejpam-3764	47	51	,	,	PUNCT
ejpam-3764	47	52	a3	a3	NOUN
ejpam-3764	47	53	∈	∈	PROPN
ejpam-3764	47	54	a	a	PRON
ejpam-3764	47	55	,	,	PUNCT
ejpam-3764	47	56	the	the	DET
ejpam-3764	47	57	following	following	ADJ
ejpam-3764	47	58	axioms	axiom	NOUN
ejpam-3764	47	59	are	be	AUX
ejpam-3764	47	60	satisfied	satisfied	ADJ
ejpam-3764	47	61	:	:	PUNCT
ejpam-3764	47	62	(	(	PUNCT
ejpam-3764	47	63	i	i	NOUN
ejpam-3764	47	64	)	)	PUNCT
ejpam-3764	47	65	a1	a1	PROPN
ejpam-3764	47	66	⊕	⊕	PROPN
ejpam-3764	47	67	a2	a2	PROPN
ejpam-3764	47	68	=	=	SYM
ejpam-3764	47	69	a1	a1	PROPN
ejpam-3764	47	70	or	or	CCONJ
ejpam-3764	47	71	a2	a2	PROPN
ejpam-3764	47	72	;	;	PUNCT
ejpam-3764	47	73	(	(	PUNCT
ejpam-3764	47	74	ii	ii	NOUN
ejpam-3764	47	75	)	)	PUNCT
ejpam-3764	47	76	a1	a1	PROPN
ejpam-3764	47	77	⊕	⊕	PROPN
ejpam-3764	47	78	ω	ω	PROPN
ejpam-3764	47	79	=	=	NOUN
ejpam-3764	47	80	a1	a1	PROPN
ejpam-3764	47	81	=	=	SYM
ejpam-3764	47	82	ω	ω	PROPN
ejpam-3764	47	83	⊕	⊕	PROPN
ejpam-3764	47	84	a1	a1	PROPN
ejpam-3764	47	85	;	;	PUNCT
ejpam-3764	47	86	m.	m.	NOUN
ejpam-3764	47	87	alqahtani	alqahtani	PROPN
ejpam-3764	47	88	,	,	PUNCT
ejpam-3764	47	89	c.	c.	PROPN
ejpam-3764	47	90	özel	özel	PROPN
ejpam-3764	47	91	,	,	PUNCT
ejpam-3764	47	92	i.	i.	PROPN
ejpam-3764	47	93	alshammari	alshammari	PROPN
ejpam-3764	47	94	/	/	SYM
ejpam-3764	47	95	eur	eur	PROPN
ejpam-3764	47	96	.	.	PUNCT
ejpam-3764	48	1	j.	j.	PROPN
ejpam-3764	48	2	pure	pure	PROPN
ejpam-3764	48	3	appl	appl	PROPN
ejpam-3764	48	4	.	.	PROPN
ejpam-3764	48	5	math	math	PROPN
ejpam-3764	48	6	,	,	PUNCT
ejpam-3764	48	7	13	13	NUM
ejpam-3764	48	8	(	(	PUNCT
ejpam-3764	48	9	3	3	NUM
ejpam-3764	48	10	)	)	PUNCT
ejpam-3764	48	11	(	(	PUNCT
ejpam-3764	48	12	2020	2020	NUM
ejpam-3764	48	13	)	)	PUNCT
ejpam-3764	48	14	,	,	PUNCT
ejpam-3764	48	15	513	513	NUM
ejpam-3764	48	16	-	-	SYM
ejpam-3764	48	17	528	528	NUM
ejpam-3764	48	18	515	515	NUM
ejpam-3764	48	19	(	(	PUNCT
ejpam-3764	48	20	iii	iii	X
ejpam-3764	48	21	)	)	PUNCT
ejpam-3764	49	1	ω	ω	PROPN
ejpam-3764	49	2	⊕	⊕	PROPN
ejpam-3764	49	3	ω	ω	PROPN
ejpam-3764	49	4	=	=	SYM
ejpam-3764	49	5	ω	ω	PROPN
ejpam-3764	49	6	;	;	PUNCT
ejpam-3764	49	7	(	(	PUNCT
ejpam-3764	49	8	iv	iv	X
ejpam-3764	49	9	)	)	PUNCT
ejpam-3764	49	10	a1	a1	NOUN
ejpam-3764	49	11	⊗	⊗	PROPN
ejpam-3764	49	12	a2	a2	PROPN
ejpam-3764	49	13	=	=	PROPN
ejpam-3764	49	14	a2	a2	PROPN
ejpam-3764	49	15	⊗	⊗	PROPN
ejpam-3764	49	16	a1	a1	PROPN
ejpam-3764	49	17	∈	∈	PROPN
ejpam-3764	49	18	a	a	PRON
ejpam-3764	49	19	;	;	PUNCT
ejpam-3764	49	20	(	(	PUNCT
ejpam-3764	49	21	v	v	NOUN
ejpam-3764	49	22	)	)	PUNCT
ejpam-3764	49	23	(	(	PUNCT
ejpam-3764	49	24	a1	a1	PROPN
ejpam-3764	49	25	⊗	⊗	PROPN
ejpam-3764	49	26	a2)⊗	a2)⊗	NOUN
ejpam-3764	49	27	a3	a3	NOUN
ejpam-3764	49	28	=	=	PUNCT
ejpam-3764	49	29	a1	a1	NOUN
ejpam-3764	49	30	⊗	⊗	PROPN
ejpam-3764	49	31	(	(	PUNCT
ejpam-3764	49	32	a2	a2	PROPN
ejpam-3764	49	33	⊗	⊗	PROPN
ejpam-3764	49	34	a3	a3	PROPN
ejpam-3764	49	35	)	)	PUNCT
ejpam-3764	49	36	;	;	PUNCT
ejpam-3764	49	37	(	(	PUNCT
ejpam-3764	49	38	vi	vi	NOUN
ejpam-3764	49	39	)	)	PUNCT
ejpam-3764	49	40	a1	a1	NOUN
ejpam-3764	49	41	⊗	⊗	PROPN
ejpam-3764	49	42	e	e	PROPN
ejpam-3764	49	43	=	=	NOUN
ejpam-3764	49	44	a1	a1	PROPN
ejpam-3764	49	45	;	;	PUNCT
ejpam-3764	49	46	(	(	PUNCT
ejpam-3764	49	47	vii	vii	PROPN
ejpam-3764	49	48	)	)	PUNCT
ejpam-3764	49	49	a1	a1	NOUN
ejpam-3764	49	50	⊗	⊗	PROPN
ejpam-3764	49	51	ω	ω	PROPN
ejpam-3764	50	1	=	=	SYM
ejpam-3764	50	2	ω	ω	NUM
ejpam-3764	50	3	⊗	⊗	PROPN
ejpam-3764	50	4	a1	a1	NOUN
ejpam-3764	50	5	=	=	SYM
ejpam-3764	50	6	{	{	PUNCT
ejpam-3764	50	7	ω	ω	PROPN
ejpam-3764	50	8	if	if	SCONJ
ejpam-3764	50	9	ω	ω	PROPN
ejpam-3764	50	10	6=	6=	PROPN
ejpam-3764	50	11	e	e	X
ejpam-3764	50	12	a1	a1	NOUN
ejpam-3764	50	13	if	if	SCONJ
ejpam-3764	50	14	ω	ω	PROPN
ejpam-3764	50	15	=	=	SYM
ejpam-3764	50	16	e	e	X
ejpam-3764	50	17	;	;	PUNCT
ejpam-3764	50	18	(	(	PUNCT
ejpam-3764	50	19	viii	viii	NOUN
ejpam-3764	50	20	)	)	PUNCT
ejpam-3764	50	21	ω	ω	PROPN
ejpam-3764	51	1	⊗	⊗	PROPN
ejpam-3764	51	2	ω	ω	PROPN
ejpam-3764	51	3	=	=	SYM
ejpam-3764	51	4	ω	ω	PROPN
ejpam-3764	51	5	;	;	PUNCT
ejpam-3764	51	6	(	(	PUNCT
ejpam-3764	51	7	ix	ix	X
ejpam-3764	51	8	)	)	PUNCT
ejpam-3764	51	9	a1	a1	NOUN
ejpam-3764	51	10	⊗	⊗	PROPN
ejpam-3764	51	11	(	(	PUNCT
ejpam-3764	51	12	a2	a2	PROPN
ejpam-3764	51	13	⊕	⊕	PROPN
ejpam-3764	51	14	a3	a3	NOUN
ejpam-3764	51	15	)	)	PUNCT
ejpam-3764	51	16	=	=	PUNCT
ejpam-3764	52	1	(	(	PUNCT
ejpam-3764	52	2	a1	a1	PROPN
ejpam-3764	52	3	⊗	⊗	PROPN
ejpam-3764	52	4	a2)⊕	a2)⊕	PROPN
ejpam-3764	52	5	(	(	PUNCT
ejpam-3764	52	6	a1	a1	PROPN
ejpam-3764	52	7	⊗	⊗	PROPN
ejpam-3764	52	8	a3	a3	NOUN
ejpam-3764	52	9	)	)	PUNCT
ejpam-3764	52	10	.	.	PUNCT
ejpam-3764	53	1	remark	remark	PROPN
ejpam-3764	53	2	1	1	NUM
ejpam-3764	53	3	.	.	PUNCT
ejpam-3764	54	1	[	[	X
ejpam-3764	54	2	11	11	NUM
ejpam-3764	54	3	]	]	SYM
ejpam-3764	54	4	(	(	PUNCT
ejpam-3764	54	5	1	1	X
ejpam-3764	54	6	)	)	PUNCT
ejpam-3764	54	7	⊕	⊕	PROPN
ejpam-3764	54	8	is	be	AUX
ejpam-3764	54	9	a	a	DET
ejpam-3764	54	10	pairwise	pairwise	NOUN
ejpam-3764	54	11	comparison	comparison	NOUN
ejpam-3764	54	12	operation	operation	NOUN
ejpam-3764	54	13	such	such	ADJ
ejpam-3764	54	14	as	as	ADP
ejpam-3764	54	15	max	max	PROPN
ejpam-3764	54	16	,	,	PUNCT
ejpam-3764	54	17	min	min	PROPN
ejpam-3764	54	18	,	,	PUNCT
ejpam-3764	54	19	inf	inf	NOUN
ejpam-3764	54	20	,	,	PUNCT
ejpam-3764	54	21	sup	sup	NOUN
ejpam-3764	54	22	,	,	PUNCT
ejpam-3764	54	23	up	up	ADV
ejpam-3764	54	24	,	,	PUNCT
ejpam-3764	54	25	down	down	ADV
ejpam-3764	54	26	,	,	PUNCT
ejpam-3764	54	27	lexicographic	lexicographic	ADJ
ejpam-3764	54	28	ordering	ordering	NOUN
ejpam-3764	54	29	,	,	PUNCT
ejpam-3764	54	30	or	or	CCONJ
ejpam-3764	54	31	anything	anything	PRON
ejpam-3764	54	32	else	else	ADV
ejpam-3764	54	33	that	that	PRON
ejpam-3764	54	34	compairs	compair	VERB
ejpam-3764	54	35	two	two	NUM
ejpam-3764	54	36	elements	element	NOUN
ejpam-3764	54	37	of	of	ADP
ejpam-3764	54	38	aω	aω	PROPN
ejpam-3764	54	39	.	.	PUNCT
ejpam-3764	55	1	obviously	obviously	ADV
ejpam-3764	55	2	,	,	PUNCT
ejpam-3764	55	3	it	it	PRON
ejpam-3764	55	4	is	be	AUX
ejpam-3764	55	5	associative	associative	ADJ
ejpam-3764	55	6	and	and	CCONJ
ejpam-3764	55	7	commutative	commutative	ADJ
ejpam-3764	55	8	and	and	CCONJ
ejpam-3764	55	9	the	the	DET
ejpam-3764	55	10	tropical	tropical	ADJ
ejpam-3764	55	11	indeterminate	indeterminate	NOUN
ejpam-3764	55	12	ω	ω	NOUN
ejpam-3764	55	13	play	play	VERB
ejpam-3764	55	14	the	the	DET
ejpam-3764	55	15	role	role	NOUN
ejpam-3764	55	16	of	of	ADP
ejpam-3764	55	17	the	the	DET
ejpam-3764	55	18	identity	identity	NOUN
ejpam-3764	55	19	.	.	PUNCT
ejpam-3764	56	1	hence	hence	ADV
ejpam-3764	56	2	(	(	PUNCT
ejpam-3764	56	3	aω,⊕	aω,⊕	NUM
ejpam-3764	56	4	,	,	PUNCT
ejpam-3764	56	5	ω	ω	NUM
ejpam-3764	56	6	)	)	PUNCT
ejpam-3764	56	7	is	be	AUX
ejpam-3764	56	8	a	a	DET
ejpam-3764	56	9	commutative	commutative	ADJ
ejpam-3764	56	10	monoid	monoid	NOUN
ejpam-3764	56	11	.	.	PUNCT
ejpam-3764	57	1	(	(	PUNCT
ejpam-3764	57	2	2	2	X
ejpam-3764	57	3	)	)	PUNCT
ejpam-3764	57	4	⊗	⊗	PROPN
ejpam-3764	57	5	is	be	AUX
ejpam-3764	57	6	also	also	ADV
ejpam-3764	57	7	associative	associative	ADJ
ejpam-3764	57	8	and	and	CCONJ
ejpam-3764	57	9	commutative	commutative	ADJ
ejpam-3764	57	10	on	on	ADP
ejpam-3764	57	11	aω	aω	PROPN
ejpam-3764	57	12	,	,	PUNCT
ejpam-3764	57	13	and	and	CCONJ
ejpam-3764	57	14	e	e	NOUN
ejpam-3764	57	15	plays	play	VERB
ejpam-3764	57	16	the	the	DET
ejpam-3764	57	17	role	role	NOUN
ejpam-3764	57	18	of	of	ADP
ejpam-3764	57	19	the	the	DET
ejpam-3764	57	20	multiplicative	multiplicative	ADJ
ejpam-3764	57	21	identity	identity	NOUN
ejpam-3764	57	22	of	of	ADP
ejpam-3764	57	23	aω	aω	PROPN
ejpam-3764	57	24	.	.	PUNCT
ejpam-3764	58	1	hence	hence	ADV
ejpam-3764	58	2	,	,	PUNCT
ejpam-3764	58	3	(	(	PUNCT
ejpam-3764	58	4	aω,⊗	aω,⊗	PROPN
ejpam-3764	58	5	,	,	PUNCT
ejpam-3764	58	6	e	e	NOUN
ejpam-3764	58	7	)	)	PUNCT
ejpam-3764	58	8	is	be	AUX
ejpam-3764	58	9	also	also	ADV
ejpam-3764	58	10	a	a	DET
ejpam-3764	58	11	commutative	commutative	ADJ
ejpam-3764	58	12	monoid	monoid	NOUN
ejpam-3764	58	13	.	.	PUNCT
ejpam-3764	59	1	(	(	PUNCT
ejpam-3764	59	2	3	3	X
ejpam-3764	59	3	)	)	PUNCT
ejpam-3764	59	4	the	the	DET
ejpam-3764	59	5	left	left	ADJ
ejpam-3764	59	6	distributive	distributive	ADJ
ejpam-3764	59	7	law	law	NOUN
ejpam-3764	59	8	(	(	PUNCT
ejpam-3764	59	9	ix	ix	X
ejpam-3764	59	10	)	)	PUNCT
ejpam-3764	59	11	also	also	ADV
ejpam-3764	59	12	gives	give	VERB
ejpam-3764	59	13	the	the	DET
ejpam-3764	59	14	right	right	ADJ
ejpam-3764	59	15	distributive	distributive	ADJ
ejpam-3764	59	16	law	law	NOUN
ejpam-3764	59	17	.	.	PUNCT
ejpam-3764	60	1	(	(	PUNCT
ejpam-3764	60	2	4	4	X
ejpam-3764	60	3	)	)	PUNCT
ejpam-3764	60	4	every	every	DET
ejpam-3764	60	5	element	element	NOUN
ejpam-3764	60	6	of	of	ADP
ejpam-3764	60	7	aω	aω	PROPN
ejpam-3764	60	8	is	be	AUX
ejpam-3764	60	9	an	an	DET
ejpam-3764	60	10	idempotent	idempotent	NOUN
ejpam-3764	60	11	under	under	ADP
ejpam-3764	60	12	⊕.	⊕.	PROPN
ejpam-3764	60	13	(	(	PUNCT
ejpam-3764	60	14	5	5	NUM
ejpam-3764	60	15	)	)	PUNCT
ejpam-3764	60	16	altogether	altogether	ADV
ejpam-3764	60	17	,	,	PUNCT
ejpam-3764	60	18	we	we	PRON
ejpam-3764	60	19	write	write	VERB
ejpam-3764	60	20	both	both	DET
ejpam-3764	60	21	structures	structure	NOUN
ejpam-3764	60	22	as	as	ADP
ejpam-3764	60	23	:	:	PUNCT
ejpam-3764	60	24	aω	aω	X
ejpam-3764	60	25	=	=	PUNCT
ejpam-3764	60	26	(	(	PUNCT
ejpam-3764	60	27	aω,⊕,⊗	aω,⊕,⊗	PROPN
ejpam-3764	60	28	,	,	PUNCT
ejpam-3764	60	29	ω	ω	PROPN
ejpam-3764	60	30	,	,	PUNCT
ejpam-3764	60	31	e	e	NOUN
ejpam-3764	60	32	)	)	PUNCT
ejpam-3764	60	33	.	.	PUNCT
ejpam-3764	61	1	this	this	PRON
ejpam-3764	61	2	is	be	AUX
ejpam-3764	61	3	an	an	DET
ejpam-3764	61	4	idempotent	idempotent	ADJ
ejpam-3764	61	5	semiring	semiring	NOUN
ejpam-3764	61	6	,	,	PUNCT
ejpam-3764	61	7	which	which	PRON
ejpam-3764	61	8	is	be	AUX
ejpam-3764	61	9	also	also	ADV
ejpam-3764	61	10	called	call	VERB
ejpam-3764	61	11	”	"	PUNCT
ejpam-3764	61	12	dioid	dioid	NOUN
ejpam-3764	61	13	”	"	PUNCT
ejpam-3764	61	14	in	in	ADP
ejpam-3764	61	15	the	the	DET
ejpam-3764	61	16	literature	literature	NOUN
ejpam-3764	61	17	.	.	PUNCT
ejpam-3764	62	1	remark	remark	PROPN
ejpam-3764	62	2	2	2	NUM
ejpam-3764	62	3	.	.	PUNCT
ejpam-3764	63	1	[	[	X
ejpam-3764	63	2	11	11	NUM
ejpam-3764	63	3	]	]	PUNCT
ejpam-3764	63	4	ω−	ω−	ADP
ejpam-3764	63	5	algebra	algebra	NOUN
ejpam-3764	63	6	can	can	AUX
ejpam-3764	63	7	similarly	similarly	ADV
ejpam-3764	63	8	be	be	AUX
ejpam-3764	63	9	defined	define	VERB
ejpam-3764	63	10	over	over	ADP
ejpam-3764	63	11	a	a	DET
ejpam-3764	63	12	commutative	commutative	ADJ
ejpam-3764	63	13	monoid	monoid	NOUN
ejpam-3764	63	14	,	,	PUNCT
ejpam-3764	63	15	ring	ring	NOUN
ejpam-3764	63	16	,	,	PUNCT
ejpam-3764	63	17	or	or	CCONJ
ejpam-3764	63	18	even	even	ADV
ejpam-3764	63	19	a	a	DET
ejpam-3764	63	20	semiring	semiring	NOUN
ejpam-3764	63	21	.	.	PUNCT
ejpam-3764	64	1	more	more	ADV
ejpam-3764	64	2	generally	generally	ADV
ejpam-3764	64	3	,	,	PUNCT
ejpam-3764	64	4	one	one	PRON
ejpam-3764	64	5	may	may	AUX
ejpam-3764	64	6	construct	construct	VERB
ejpam-3764	64	7	analogously	analogously	ADV
ejpam-3764	64	8	such	such	ADJ
ejpam-3764	64	9	algebras	algebra	NOUN
ejpam-3764	64	10	on	on	ADP
ejpam-3764	64	11	other	other	ADJ
ejpam-3764	64	12	more	more	ADV
ejpam-3764	64	13	weaker	weak	ADJ
ejpam-3764	64	14	structures	structure	NOUN
ejpam-3764	64	15	.	.	PUNCT
ejpam-3764	65	1	in	in	ADP
ejpam-3764	65	2	this	this	DET
ejpam-3764	65	3	note	note	NOUN
ejpam-3764	65	4	,	,	PUNCT
ejpam-3764	65	5	we	we	PRON
ejpam-3764	65	6	confined	confine	VERB
ejpam-3764	65	7	ourselves	ourselves	PRON
ejpam-3764	65	8	to	to	PART
ejpam-3764	65	9	only	only	ADV
ejpam-3764	65	10	ω−	ω−	PROPN
ejpam-3764	65	11	algebras	algebra	NOUN
ejpam-3764	65	12	over	over	ADP
ejpam-3764	65	13	abelian	abelian	ADJ
ejpam-3764	65	14	groups	group	NOUN
ejpam-3764	65	15	and	and	CCONJ
ejpam-3764	65	16	rings	ring	NOUN
ejpam-3764	65	17	.	.	PUNCT
ejpam-3764	65	18	example	example	NOUN
ejpam-3764	66	1	1	1	NUM
ejpam-3764	66	2	.	.	PUNCT
ejpam-3764	67	1	[	[	X
ejpam-3764	67	2	11	11	NUM
ejpam-3764	67	3	]	]	X
ejpam-3764	67	4	max	max	PROPN
ejpam-3764	67	5	-	-	PUNCT
ejpam-3764	67	6	plus	plus	CCONJ
ejpam-3764	67	7	algebra	algebra	NOUN
ejpam-3764	67	8	,	,	PUNCT
ejpam-3764	67	9	min	min	NOUN
ejpam-3764	67	10	-	-	ADJ
ejpam-3764	67	11	plus	plus	ADJ
ejpam-3764	67	12	algebra	algebra	NOUN
ejpam-3764	67	13	and	and	CCONJ
ejpam-3764	67	14	all	all	DET
ejpam-3764	67	15	such	such	ADJ
ejpam-3764	67	16	”	"	PUNCT
ejpam-3764	67	17	so	so	ADV
ejpam-3764	67	18	called	call	VERB
ejpam-3764	67	19	”	"	PUNCT
ejpam-3764	67	20	algebras	algebra	NOUN
ejpam-3764	67	21	are	be	AUX
ejpam-3764	67	22	particular	particular	ADJ
ejpam-3764	67	23	cases	case	NOUN
ejpam-3764	67	24	of	of	ADP
ejpam-3764	67	25	the	the	DET
ejpam-3764	67	26	ω−	ω−	ADJ
ejpam-3764	67	27	algebra	algebra	NOUN
ejpam-3764	67	28	over	over	ADP
ejpam-3764	67	29	the	the	DET
ejpam-3764	67	30	ring	ring	NOUN
ejpam-3764	67	31	r	r	NOUN
ejpam-3764	67	32	or	or	CCONJ
ejpam-3764	67	33	its	its	PRON
ejpam-3764	67	34	associated	associated	ADJ
ejpam-3764	67	35	subrings	subring	NOUN
ejpam-3764	67	36	.	.	PUNCT
ejpam-3764	68	1	a	a	DET
ejpam-3764	68	2	simpler	simple	ADJ
ejpam-3764	68	3	example	example	NOUN
ejpam-3764	68	4	is	be	AUX
ejpam-3764	68	5	the	the	DET
ejpam-3764	68	6	following	following	NOUN
ejpam-3764	68	7	.	.	PUNCT
ejpam-3764	69	1	in	in	ADP
ejpam-3764	69	2	the	the	DET
ejpam-3764	69	3	abelian	abelian	ADJ
ejpam-3764	69	4	group	group	NOUN
ejpam-3764	69	5	(	(	PUNCT
ejpam-3764	69	6	z,+	z,+	NUM
ejpam-3764	69	7	)	)	PUNCT
ejpam-3764	69	8	,	,	PUNCT
ejpam-3764	69	9	for	for	ADP
ejpam-3764	69	10	any	any	DET
ejpam-3764	69	11	integer	integer	NOUN
ejpam-3764	69	12	m	m	NOUN
ejpam-3764	69	13	,	,	PUNCT
ejpam-3764	69	14	we	we	PRON
ejpam-3764	69	15	have	have	VERB
ejpam-3764	69	16	w	w	PROPN
ejpam-3764	69	17	(	(	PUNCT
ejpam-3764	69	18	m	m	NOUN
ejpam-3764	69	19	)	)	PUNCT
ejpam-3764	69	20	=	=	PRON
ejpam-3764	69	21	{	{	PUNCT
ejpam-3764	69	22	0,m	0,m	NOUN
ejpam-3764	69	23	,	,	PUNCT
ejpam-3764	69	24	2	2	NUM
ejpam-3764	69	25	m	m	NOUN
ejpam-3764	69	26	,	,	PUNCT
ejpam-3764	69	27	·	·	PUNCT
ejpam-3764	69	28	·	·	PUNCT
ejpam-3764	69	29	·	·	PUNCT
ejpam-3764	69	30	}	}	PUNCT
ejpam-3764	69	31	.	.	PUNCT
ejpam-3764	70	1	this	this	PRON
ejpam-3764	70	2	is	be	AUX
ejpam-3764	70	3	an	an	DET
ejpam-3764	70	4	additive	additive	ADJ
ejpam-3764	70	5	submonoid	submonoid	NOUN
ejpam-3764	70	6	of	of	ADP
ejpam-3764	70	7	(	(	PUNCT
ejpam-3764	70	8	z,+	z,+	NUM
ejpam-3764	70	9	)	)	PUNCT
ejpam-3764	70	10	.	.	PUNCT
ejpam-3764	71	1	let	let	VERB
ejpam-3764	71	2	ω	ω	PROPN
ejpam-3764	71	3	=	=	SYM
ejpam-3764	71	4	−∞	−∞	PROPN
ejpam-3764	71	5	,	,	PUNCT
ejpam-3764	71	6	a1	a1	PROPN
ejpam-3764	71	7	⊕	⊕	PROPN
ejpam-3764	71	8	a2	a2	PROPN
ejpam-3764	71	9	=	=	SYM
ejpam-3764	71	10	max(a1	max(a1	PROPN
ejpam-3764	71	11	,	,	PUNCT
ejpam-3764	71	12	a2	a2	PROPN
ejpam-3764	71	13	)	)	PUNCT
ejpam-3764	71	14	and	and	CCONJ
ejpam-3764	71	15	a1	a1	PROPN
ejpam-3764	71	16	⊗	⊗	PROPN
ejpam-3764	71	17	a2	a2	PROPN
ejpam-3764	71	18	=	=	SYM
ejpam-3764	71	19	a1	a1	PROPN
ejpam-3764	71	20	+	+	CCONJ
ejpam-3764	71	21	a2	a2	PROPN
ejpam-3764	71	22	,	,	PUNCT
ejpam-3764	71	23	∀a1	∀a1	PROPN
ejpam-3764	71	24	,	,	PUNCT
ejpam-3764	71	25	a2	a2	PROPN
ejpam-3764	71	26	∈w	∈w	PROPN
ejpam-3764	71	27	(	(	PUNCT
ejpam-3764	71	28	m	m	NOUN
ejpam-3764	71	29	)	)	PUNCT
ejpam-3764	71	30	.	.	PUNCT
ejpam-3764	72	1	then	then	ADV
ejpam-3764	72	2	.	.	PUNCT
ejpam-3764	73	1	w	w	NOUN
ejpam-3764	73	2	(	(	PUNCT
ejpam-3764	73	3	m)−∞	m)−∞	ADV
ejpam-3764	73	4	=	=	SYM
ejpam-3764	73	5	{	{	PUNCT
ejpam-3764	73	6	w	w	PROPN
ejpam-3764	73	7	(	(	PUNCT
ejpam-3764	73	8	m)−∞,⊕,⊗,−∞	m)−∞,⊕,⊗,−∞	PROPN
ejpam-3764	73	9	,	,	PUNCT
ejpam-3764	73	10	0	0	NUM
ejpam-3764	73	11	}	}	PUNCT
ejpam-3764	73	12	is	be	AUX
ejpam-3764	73	13	−∞	−∞	X
ejpam-3764	73	14	−	−	PROPN
ejpam-3764	73	15	algebra	algebra	NOUN
ejpam-3764	73	16	over	over	ADP
ejpam-3764	73	17	the	the	DET
ejpam-3764	73	18	abelian	abelian	ADJ
ejpam-3764	73	19	group	group	NOUN
ejpam-3764	73	20	of	of	ADP
ejpam-3764	74	1	integers	integer	NOUN
ejpam-3764	74	2	z.	z.	PROPN
ejpam-3764	74	3	hence	hence	ADV
ejpam-3764	74	4	,	,	PUNCT
ejpam-3764	74	5	we	we	PRON
ejpam-3764	74	6	have	have	VERB
ejpam-3764	74	7	a	a	DET
ejpam-3764	74	8	sequence	sequence	NOUN
ejpam-3764	74	9	of	of	ADP
ejpam-3764	74	10	ω−	ω−	INTJ
ejpam-3764	74	11	subalgebras	subalgebras	PROPN
ejpam-3764	74	12	w	w	PROPN
ejpam-3764	74	13	(	(	PUNCT
ejpam-3764	74	14	m	m	NOUN
ejpam-3764	74	15	)	)	PUNCT
ejpam-3764	74	16	≥w	≥w	NOUN
ejpam-3764	74	17	(	(	PUNCT
ejpam-3764	74	18	2	2	NUM
ejpam-3764	74	19	m	m	NOUN
ejpam-3764	74	20	)	)	PUNCT
ejpam-3764	74	21	≥	≥	X
ejpam-3764	74	22	·	·	PUNCT
ejpam-3764	74	23	·	·	PUNCT
ejpam-3764	74	24	·	·	PUNCT
ejpam-3764	74	25	.	.	PUNCT
ejpam-3764	75	1	m.	m.	PROPN
ejpam-3764	75	2	alqahtani	alqahtani	PROPN
ejpam-3764	75	3	,	,	PUNCT
ejpam-3764	75	4	c.	c.	PROPN
ejpam-3764	75	5	özel	özel	PROPN
ejpam-3764	75	6	,	,	PUNCT
ejpam-3764	75	7	i.	i.	PROPN
ejpam-3764	75	8	alshammari	alshammari	PROPN
ejpam-3764	75	9	/	/	SYM
ejpam-3764	75	10	eur	eur	PROPN
ejpam-3764	75	11	.	.	PUNCT
ejpam-3764	76	1	j.	j.	PROPN
ejpam-3764	76	2	pure	pure	PROPN
ejpam-3764	76	3	appl	appl	PROPN
ejpam-3764	76	4	.	.	PROPN
ejpam-3764	76	5	math	math	PROPN
ejpam-3764	76	6	,	,	PUNCT
ejpam-3764	76	7	13	13	NUM
ejpam-3764	76	8	(	(	PUNCT
ejpam-3764	76	9	3	3	NUM
ejpam-3764	76	10	)	)	PUNCT
ejpam-3764	76	11	(	(	PUNCT
ejpam-3764	76	12	2020	2020	NUM
ejpam-3764	76	13	)	)	PUNCT
ejpam-3764	76	14	,	,	PUNCT
ejpam-3764	76	15	513	513	NUM
ejpam-3764	76	16	-	-	SYM
ejpam-3764	76	17	528	528	NUM
ejpam-3764	76	18	516	516	NUM
ejpam-3764	76	19	example	example	NOUN
ejpam-3764	76	20	2	2	NUM
ejpam-3764	76	21	.	.	PUNCT
ejpam-3764	77	1	[	[	X
ejpam-3764	77	2	11	11	NUM
ejpam-3764	77	3	]	]	PUNCT
ejpam-3764	77	4	a	a	DET
ejpam-3764	77	5	cyclically	cyclically	ADV
ejpam-3764	77	6	ordered	order	VERB
ejpam-3764	77	7	abelian	abelian	PROPN
ejpam-3764	77	8	group	group	NOUN
ejpam-3764	77	9	.	.	PUNCT
ejpam-3764	78	1	this	this	DET
ejpam-3764	78	2	example	example	NOUN
ejpam-3764	78	3	is	be	AUX
ejpam-3764	78	4	constructed	construct	VERB
ejpam-3764	78	5	exclusively	exclusively	ADV
ejpam-3764	78	6	over	over	ADP
ejpam-3764	78	7	an	an	DET
ejpam-3764	78	8	abelian	abelian	ADJ
ejpam-3764	78	9	group	group	NOUN
ejpam-3764	78	10	.	.	PUNCT
ejpam-3764	79	1	a	a	DET
ejpam-3764	79	2	cyclically	cyclically	ADV
ejpam-3764	79	3	ordered	order	VERB
ejpam-3764	79	4	abelian	abelian	PROPN
ejpam-3764	79	5	group	group	NOUN
ejpam-3764	79	6	is	be	AUX
ejpam-3764	79	7	more	more	ADV
ejpam-3764	79	8	general	general	ADJ
ejpam-3764	79	9	than	than	ADP
ejpam-3764	79	10	that	that	PRON
ejpam-3764	79	11	of	of	ADP
ejpam-3764	79	12	a	a	DET
ejpam-3764	79	13	linearly	linearly	ADV
ejpam-3764	79	14	ordered	order	VERB
ejpam-3764	79	15	abelian	abelian	ADJ
ejpam-3764	79	16	group	group	NOUN
ejpam-3764	79	17	.	.	PUNCT
ejpam-3764	80	1	every	every	DET
ejpam-3764	80	2	linearly	linearly	ADV
ejpam-3764	80	3	ordered	order	VERB
ejpam-3764	80	4	abelian	abelian	ADJ
ejpam-3764	80	5	group	group	NOUN
ejpam-3764	80	6	is	be	AUX
ejpam-3764	80	7	cyclically	cyclically	ADV
ejpam-3764	80	8	ordered	order	VERB
ejpam-3764	80	9	,	,	PUNCT
ejpam-3764	80	10	but	but	CCONJ
ejpam-3764	80	11	the	the	DET
ejpam-3764	80	12	converse	converse	NOUN
ejpam-3764	80	13	,	,	PUNCT
ejpam-3764	80	14	in	in	ADP
ejpam-3764	80	15	general	general	ADJ
ejpam-3764	80	16	,	,	PUNCT
ejpam-3764	80	17	is	be	AUX
ejpam-3764	80	18	not	not	PART
ejpam-3764	80	19	true	true	ADJ
ejpam-3764	80	20	.	.	PUNCT
ejpam-3764	81	1	the	the	DET
ejpam-3764	81	2	following	following	ADJ
ejpam-3764	81	3	example	example	NOUN
ejpam-3764	81	4	is	be	AUX
ejpam-3764	81	5	that	that	PRON
ejpam-3764	81	6	of	of	ADP
ejpam-3764	81	7	a	a	DET
ejpam-3764	81	8	cyclically	cyclically	ADV
ejpam-3764	81	9	ordered	order	VERB
ejpam-3764	81	10	abelian	abelian	PROPN
ejpam-3764	81	11	group	group	NOUN
ejpam-3764	81	12	,	,	PUNCT
ejpam-3764	81	13	which	which	PRON
ejpam-3764	81	14	is	be	AUX
ejpam-3764	81	15	not	not	PART
ejpam-3764	81	16	an	an	DET
ejpam-3764	81	17	ordered	order	VERB
ejpam-3764	81	18	abelian	abelian	ADJ
ejpam-3764	81	19	group	group	NOUN
ejpam-3764	82	1	[	[	X
ejpam-3764	82	2	16	16	NUM
ejpam-3764	82	3	]	]	PUNCT
ejpam-3764	82	4	.	.	PUNCT
ejpam-3764	83	1	for	for	ADP
ejpam-3764	83	2	more	more	ADJ
ejpam-3764	83	3	details	detail	NOUN
ejpam-3764	83	4	about	about	ADP
ejpam-3764	83	5	a	a	DET
ejpam-3764	83	6	cyclically	cyclically	ADV
ejpam-3764	83	7	ordered	order	VERB
ejpam-3764	83	8	abelian	abelian	ADJ
ejpam-3764	83	9	groups	group	NOUN
ejpam-3764	83	10	.	.	PUNCT
ejpam-3764	84	1	consider	consider	VERB
ejpam-3764	84	2	the	the	DET
ejpam-3764	84	3	cyclically	cyclically	ADV
ejpam-3764	84	4	ordered	order	VERB
ejpam-3764	84	5	abelian	abelian	ADJ
ejpam-3764	84	6	group	group	NOUN
ejpam-3764	84	7	in	in	ADP
ejpam-3764	84	8	the	the	DET
ejpam-3764	84	9	form	form	NOUN
ejpam-3764	84	10	of	of	ADP
ejpam-3764	84	11	the	the	DET
ejpam-3764	84	12	unit	unit	NOUN
ejpam-3764	84	13	circle	circle	NOUN
ejpam-3764	84	14	c	c	PROPN
ejpam-3764	84	15	=	=	PRON
ejpam-3764	84	16	{	{	PUNCT
ejpam-3764	84	17	z	z	NOUN
ejpam-3764	84	18	∈	∈	PROPN
ejpam-3764	84	19	c	c	NOUN
ejpam-3764	84	20	|	|	ADV
ejpam-3764	84	21	|z|	|z|	VERB
ejpam-3764	84	22	=	=	SYM
ejpam-3764	84	23	1	1	NUM
ejpam-3764	84	24	}	}	PUNCT
ejpam-3764	84	25	.	.	PUNCT
ejpam-3764	85	1	let	let	VERB
ejpam-3764	85	2	w	w	VERB
ejpam-3764	85	3	=	=	PUNCT
ejpam-3764	85	4	{	{	PUNCT
ejpam-3764	85	5	0	0	NUM
ejpam-3764	85	6	,	,	PUNCT
ejpam-3764	85	7	1	1	NUM
ejpam-3764	85	8	,	,	PUNCT
ejpam-3764	85	9	2	2	NUM
ejpam-3764	85	10	,	,	PUNCT
ejpam-3764	85	11	·	·	PUNCT
ejpam-3764	85	12	·	·	PUNCT
ejpam-3764	85	13	·	·	PUNCT
ejpam-3764	85	14	}	}	PUNCT
ejpam-3764	85	15	.	.	PUNCT
ejpam-3764	86	1	for	for	ADP
ejpam-3764	86	2	some	some	DET
ejpam-3764	86	3	θ	θ	NOUN
ejpam-3764	86	4	∈	∈	PROPN
ejpam-3764	87	1	[	[	X
ejpam-3764	87	2	0	0	NUM
ejpam-3764	87	3	,	,	PUNCT
ejpam-3764	87	4	1	1	NUM
ejpam-3764	87	5	)	)	PUNCT
ejpam-3764	87	6	,	,	PUNCT
ejpam-3764	87	7	define	define	VERB
ejpam-3764	87	8	ρx	ρx	PROPN
ejpam-3764	87	9	=	=	SYM
ejpam-3764	87	10	e2πiθx	e2πiθx	PROPN
ejpam-3764	87	11	,	,	PUNCT
ejpam-3764	87	12	where	where	SCONJ
ejpam-3764	87	13	x	x	NOUN
ejpam-3764	87	14	∈w	∈w	NOUN
ejpam-3764	87	15	,	,	PUNCT
ejpam-3764	87	16	in	in	ADP
ejpam-3764	87	17	particular	particular	ADJ
ejpam-3764	87	18	,	,	PUNCT
ejpam-3764	87	19	ρ0	ρ0	PROPN
ejpam-3764	87	20	=	=	SYM
ejpam-3764	87	21	1	1	X
ejpam-3764	87	22	.	.	X
ejpam-3764	87	23	set	set	VERB
ejpam-3764	87	24	a	a	DET
ejpam-3764	87	25	:	:	PUNCT
ejpam-3764	87	26	=	=	SYM
ejpam-3764	87	27	{	{	PUNCT
ejpam-3764	87	28	ρx	ρx	PROPN
ejpam-3764	87	29	|x	|x	PROPN
ejpam-3764	87	30	∈w	∈w	PROPN
ejpam-3764	87	31	}	}	PUNCT
ejpam-3764	87	32	⊂	⊂	PROPN
ejpam-3764	87	33	c.	c.	PROPN
ejpam-3764	87	34	because	because	SCONJ
ejpam-3764	87	35	ρx1ρx2	ρx1ρx2	PROPN
ejpam-3764	87	36	=	=	SYM
ejpam-3764	87	37	ρx1+x2	ρx1+x2	PROPN
ejpam-3764	87	38	,	,	PUNCT
ejpam-3764	87	39	∀x1	∀x1	ADP
ejpam-3764	87	40	,	,	PUNCT
ejpam-3764	87	41	x2	x2	PROPN
ejpam-3764	87	42	∈w	∈w	NOUN
ejpam-3764	87	43	,	,	PUNCT
ejpam-3764	87	44	a	a	PRON
ejpam-3764	87	45	is	be	AUX
ejpam-3764	87	46	multiplicatively	multiplicatively	ADV
ejpam-3764	87	47	closed	close	VERB
ejpam-3764	87	48	.	.	PUNCT
ejpam-3764	88	1	theorem	theorem	NOUN
ejpam-3764	88	2	1	1	NUM
ejpam-3764	88	3	.	.	PUNCT
ejpam-3764	89	1	[	[	X
ejpam-3764	89	2	11	11	NUM
ejpam-3764	89	3	]	]	X
ejpam-3764	89	4	aρ0	aρ0	NOUN
ejpam-3764	89	5	=	=	PUNCT
ejpam-3764	89	6	a	a	DET
ejpam-3764	89	7	∪	∪	ADJ
ejpam-3764	89	8	{	{	PUNCT
ejpam-3764	89	9	ρ0	ρ0	PROPN
ejpam-3764	89	10	}	}	PUNCT
ejpam-3764	89	11	is	be	AUX
ejpam-3764	89	12	an	an	DET
ejpam-3764	89	13	omega	omega	NOUN
ejpam-3764	89	14	algebra	algebra	NOUN
ejpam-3764	89	15	with	with	ADP
ejpam-3764	89	16	the	the	DET
ejpam-3764	89	17	identical	identical	ADJ
ejpam-3764	89	18	additive	additive	NOUN
ejpam-3764	89	19	and	and	CCONJ
ejpam-3764	89	20	multiplicative	multiplicative	ADJ
ejpam-3764	89	21	identities	identity	NOUN
ejpam-3764	89	22	.	.	PUNCT
ejpam-3764	90	1	this	this	DET
ejpam-3764	90	2	omega	omega	NOUN
ejpam-3764	90	3	algebra	algebra	NOUN
ejpam-3764	90	4	contains	contain	VERB
ejpam-3764	90	5	infinite	infinite	ADJ
ejpam-3764	90	6	omega	omega	NOUN
ejpam-3764	90	7	subalgebras	subalgebra	NOUN
ejpam-3764	90	8	.	.	PUNCT
ejpam-3764	91	1	proof	proof	NOUN
ejpam-3764	91	2	.	.	PUNCT
ejpam-3764	92	1	define	define	VERB
ejpam-3764	92	2	⊕	⊕	PROPN
ejpam-3764	92	3	on	on	ADP
ejpam-3764	92	4	a	a	PRON
ejpam-3764	92	5	by	by	ADP
ejpam-3764	92	6	ρx1	ρx1	PROPN
ejpam-3764	92	7	⊕	⊕	PROPN
ejpam-3764	92	8	ρx2	ρx2	PROPN
ejpam-3764	93	1	=	=	PUNCT
ejpam-3764	93	2	ρx3	ρx3	NOUN
ejpam-3764	93	3	where	where	SCONJ
ejpam-3764	93	4	x1	x1	ADJ
ejpam-3764	93	5	,	,	PUNCT
ejpam-3764	93	6	x2	x2	PROPN
ejpam-3764	93	7	,	,	PUNCT
ejpam-3764	93	8	x3	x3	ADJ
ejpam-3764	93	9	∈w	∈w	NOUN
ejpam-3764	93	10	,	,	PUNCT
ejpam-3764	93	11	with	with	ADP
ejpam-3764	93	12	x3	x3	PROPN
ejpam-3764	93	13	=	=	SYM
ejpam-3764	93	14	max(x1	max(x1	PROPN
ejpam-3764	93	15	,	,	PUNCT
ejpam-3764	93	16	x2	x2	PROPN
ejpam-3764	93	17	)	)	PUNCT
ejpam-3764	93	18	and	and	CCONJ
ejpam-3764	93	19	define	define	VERB
ejpam-3764	93	20	⊗	⊗	PROPN
ejpam-3764	93	21	on	on	ADP
ejpam-3764	93	22	a	a	PRON
ejpam-3764	93	23	by	by	ADP
ejpam-3764	93	24	ρx1	ρx1	PROPN
ejpam-3764	93	25	⊗	⊗	PROPN
ejpam-3764	93	26	ρx2	ρx2	PROPN
ejpam-3764	94	1	=	=	PUNCT
ejpam-3764	94	2	ρx1+x2	ρx1+x2	PROPN
ejpam-3764	94	3	,	,	PUNCT
ejpam-3764	94	4	where	where	SCONJ
ejpam-3764	94	5	x1	x1	ADJ
ejpam-3764	94	6	,	,	PUNCT
ejpam-3764	94	7	x2	x2	PROPN
ejpam-3764	94	8	∈w	∈w	PROPN
ejpam-3764	94	9	.	.	PUNCT
ejpam-3764	95	1	clearly	clearly	ADV
ejpam-3764	95	2	,	,	PUNCT
ejpam-3764	95	3	both	both	DET
ejpam-3764	95	4	operations	operation	NOUN
ejpam-3764	95	5	are	be	AUX
ejpam-3764	95	6	associative	associative	ADJ
ejpam-3764	95	7	,	,	PUNCT
ejpam-3764	95	8	and	and	CCONJ
ejpam-3764	95	9	as	as	ADP
ejpam-3764	95	10	ρ0	ρ0	PROPN
ejpam-3764	95	11	⊕	⊕	PROPN
ejpam-3764	95	12	ρx1	ρx1	NOUN
ejpam-3764	96	1	=	=	PUNCT
ejpam-3764	96	2	ρx1	ρx1	NOUN
ejpam-3764	96	3	and	and	CCONJ
ejpam-3764	96	4	ρ0	ρ0	PROPN
ejpam-3764	97	1	⊗	⊗	PROPN
ejpam-3764	97	2	ρx1	ρx1	NOUN
ejpam-3764	98	1	=	=	NOUN
ejpam-3764	98	2	ρx1	ρx1	NOUN
ejpam-3764	98	3	,	,	PUNCT
ejpam-3764	98	4	so	so	CCONJ
ejpam-3764	98	5	(	(	PUNCT
ejpam-3764	98	6	a,⊕	a,⊕	PROPN
ejpam-3764	98	7	,	,	PUNCT
ejpam-3764	98	8	ρ0	ρ0	PROPN
ejpam-3764	98	9	)	)	PUNCT
ejpam-3764	98	10	and	and	CCONJ
ejpam-3764	98	11	(	(	PUNCT
ejpam-3764	98	12	a,⊗	a,⊗	PROPN
ejpam-3764	98	13	,	,	PUNCT
ejpam-3764	98	14	ρ0	ρ0	PROPN
ejpam-3764	98	15	)	)	PUNCT
ejpam-3764	98	16	are	be	AUX
ejpam-3764	98	17	monoids	monoid	NOUN
ejpam-3764	98	18	.	.	PUNCT
ejpam-3764	99	1	finally	finally	ADV
ejpam-3764	99	2	,	,	PUNCT
ejpam-3764	99	3	∀x1	∀x1	PROPN
ejpam-3764	99	4	,	,	PUNCT
ejpam-3764	99	5	x2	x2	PROPN
ejpam-3764	99	6	,	,	PUNCT
ejpam-3764	99	7	x3	x3	ADJ
ejpam-3764	99	8	∈w	∈w	NOUN
ejpam-3764	99	9	,	,	PUNCT
ejpam-3764	99	10	ρx1	ρx1	NOUN
ejpam-3764	99	11	⊗	⊗	PROPN
ejpam-3764	99	12	(	(	PUNCT
ejpam-3764	99	13	ρx2	ρx2	PROPN
ejpam-3764	99	14	⊕	⊕	PROPN
ejpam-3764	99	15	ρx3	ρx3	PROPN
ejpam-3764	99	16	)	)	PUNCT
ejpam-3764	100	1	=	=	PUNCT
ejpam-3764	101	1	ρx1	ρx1	NOUN
ejpam-3764	102	1	⊗	⊗	NUM
ejpam-3764	102	2	ρmax(x2,x3	ρmax(x2,x3	PROPN
ejpam-3764	102	3	)	)	PUNCT
ejpam-3764	102	4	=	=	SYM
ejpam-3764	102	5	ρx1+max(x2,x3	ρx1+max(x2,x3	PROPN
ejpam-3764	102	6	)	)	PUNCT
ejpam-3764	102	7	=	=	SYM
ejpam-3764	103	1	ρmax(x1+x2,x1+x3	ρmax(x1+x2,x1+x3	NOUN
ejpam-3764	103	2	)	)	PUNCT
ejpam-3764	103	3	=	=	NOUN
ejpam-3764	104	1	(	(	PUNCT
ejpam-3764	104	2	ρx1+x2	ρx1+x2	PROPN
ejpam-3764	104	3	⊕	⊕	PROPN
ejpam-3764	104	4	ρx1+x3	ρx1+x3	NOUN
ejpam-3764	104	5	)	)	PUNCT
ejpam-3764	104	6	=	=	SYM
ejpam-3764	105	1	(	(	PUNCT
ejpam-3764	105	2	ρx1	ρx1	NOUN
ejpam-3764	105	3	⊗	⊗	PROPN
ejpam-3764	105	4	ρx2)⊕	ρx2)⊕	PROPN
ejpam-3764	105	5	(	(	PUNCT
ejpam-3764	105	6	ρx1	ρx1	NOUN
ejpam-3764	105	7	⊗	⊗	NUM
ejpam-3764	105	8	ρx3	ρx3	PROPN
ejpam-3764	105	9	)	)	PUNCT
ejpam-3764	105	10	.	.	PUNCT
ejpam-3764	106	1	as	as	ADP
ejpam-3764	106	2	,	,	PUNCT
ejpam-3764	106	3	∀x1	∀x1	NUM
ejpam-3764	106	4	∈w	∈w	NOUN
ejpam-3764	106	5	,	,	PUNCT
ejpam-3764	107	1	ρx1	ρx1	NOUN
ejpam-3764	107	2	⊕	⊕	PROPN
ejpam-3764	107	3	ρx1	ρx1	NOUN
ejpam-3764	108	1	=	=	PUNCT
ejpam-3764	108	2	ρx1	ρx1	NOUN
ejpam-3764	108	3	and	and	CCONJ
ejpam-3764	108	4	ρ0	ρ0	PROPN
ejpam-3764	108	5	=	=	SYM
ejpam-3764	108	6	1	1	NUM
ejpam-3764	108	7	we	we	PRON
ejpam-3764	108	8	conclude	conclude	VERB
ejpam-3764	108	9	that	that	SCONJ
ejpam-3764	108	10	a	a	DET
ejpam-3764	108	11	=	=	X
ejpam-3764	108	12	(	(	PUNCT
ejpam-3764	108	13	a,⊕,⊗	a,⊕,⊗	PROPN
ejpam-3764	108	14	,	,	PUNCT
ejpam-3764	108	15	1	1	NUM
ejpam-3764	108	16	,	,	PUNCT
ejpam-3764	108	17	1	1	NUM
ejpam-3764	108	18	)	)	PUNCT
ejpam-3764	108	19	is	be	AUX
ejpam-3764	108	20	an	an	DET
ejpam-3764	108	21	omega	omega	NOUN
ejpam-3764	108	22	algebra	algebra	NOUN
ejpam-3764	108	23	.	.	PUNCT
ejpam-3764	109	1	finally	finally	ADV
ejpam-3764	109	2	,	,	PUNCT
ejpam-3764	109	3	consider	consider	VERB
ejpam-3764	109	4	w	w	PROPN
ejpam-3764	109	5	(	(	PUNCT
ejpam-3764	109	6	m	m	NOUN
ejpam-3764	109	7	)	)	PUNCT
ejpam-3764	109	8	=	=	PRON
ejpam-3764	109	9	{	{	PUNCT
ejpam-3764	109	10	0,m	0,m	NOUN
ejpam-3764	109	11	,	,	PUNCT
ejpam-3764	109	12	2	2	NUM
ejpam-3764	109	13	m	m	NOUN
ejpam-3764	109	14	,	,	PUNCT
ejpam-3764	109	15	·	·	PUNCT
ejpam-3764	109	16	·	·	PUNCT
ejpam-3764	109	17	·	·	PUNCT
ejpam-3764	110	1	}	}	PUNCT
ejpam-3764	110	2	,	,	PUNCT
ejpam-3764	110	3	where	where	SCONJ
ejpam-3764	110	4	m	m	VERB
ejpam-3764	110	5	=	=	SYM
ejpam-3764	110	6	1	1	NUM
ejpam-3764	110	7	,	,	PUNCT
ejpam-3764	110	8	2	2	NUM
ejpam-3764	110	9	,	,	PUNCT
ejpam-3764	110	10	·	·	PUNCT
ejpam-3764	110	11	·	·	PUNCT
ejpam-3764	110	12	·	·	PUNCT
ejpam-3764	110	13	.	.	PUNCT
ejpam-3764	111	1	for	for	ADP
ejpam-3764	111	2	each	each	DET
ejpam-3764	111	3	m	m	NOUN
ejpam-3764	111	4	,	,	PUNCT
ejpam-3764	111	5	one	one	PRON
ejpam-3764	111	6	can	can	AUX
ejpam-3764	111	7	construct	construct	VERB
ejpam-3764	111	8	an	an	DET
ejpam-3764	111	9	omega	omega	NOUN
ejpam-3764	111	10	subalgebra	subalgebra	NOUN
ejpam-3764	111	11	.	.	PUNCT
ejpam-3764	112	1	m.	m.	PROPN
ejpam-3764	112	2	alqahtani	alqahtani	PROPN
ejpam-3764	112	3	,	,	PUNCT
ejpam-3764	112	4	c.	c.	PROPN
ejpam-3764	112	5	özel	özel	PROPN
ejpam-3764	112	6	,	,	PUNCT
ejpam-3764	112	7	i.	i.	PROPN
ejpam-3764	112	8	alshammari	alshammari	PROPN
ejpam-3764	112	9	/	/	SYM
ejpam-3764	112	10	eur	eur	PROPN
ejpam-3764	112	11	.	.	PUNCT
ejpam-3764	113	1	j.	j.	PROPN
ejpam-3764	113	2	pure	pure	PROPN
ejpam-3764	113	3	appl	appl	PROPN
ejpam-3764	113	4	.	.	PROPN
ejpam-3764	113	5	math	math	PROPN
ejpam-3764	113	6	,	,	PUNCT
ejpam-3764	113	7	13	13	NUM
ejpam-3764	113	8	(	(	PUNCT
ejpam-3764	113	9	3	3	NUM
ejpam-3764	113	10	)	)	PUNCT
ejpam-3764	113	11	(	(	PUNCT
ejpam-3764	113	12	2020	2020	NUM
ejpam-3764	113	13	)	)	PUNCT
ejpam-3764	113	14	,	,	PUNCT
ejpam-3764	113	15	513	513	NUM
ejpam-3764	113	16	-	-	SYM
ejpam-3764	113	17	528	528	NUM
ejpam-3764	113	18	517	517	NUM
ejpam-3764	113	19	example	example	NOUN
ejpam-3764	113	20	3	3	NUM
ejpam-3764	113	21	.	.	PUNCT
ejpam-3764	114	1	[	[	X
ejpam-3764	114	2	11	11	NUM
ejpam-3764	114	3	]	]	PUNCT
ejpam-3764	114	4	a	a	DET
ejpam-3764	114	5	laxicographic	laxicographic	ADJ
ejpam-3764	114	6	ordering	ordering	NOUN
ejpam-3764	114	7	.	.	PUNCT
ejpam-3764	115	1	consider	consider	VERB
ejpam-3764	115	2	the	the	DET
ejpam-3764	115	3	binary	binary	PROPN
ejpam-3764	115	4	linear	linear	PROPN
ejpam-3764	115	5	code	code	NOUN
ejpam-3764	115	6	of	of	ADP
ejpam-3764	115	7	length	length	NOUN
ejpam-3764	115	8	2	2	NUM
ejpam-3764	115	9	;	;	PUNCT
ejpam-3764	115	10	z(2	z(2	NUM
ejpam-3764	115	11	)	)	PUNCT
ejpam-3764	115	12	2	2	NUM
ejpam-3764	115	13	=	=	SYM
ejpam-3764	115	14	{	{	PUNCT
ejpam-3764	115	15	00	00	NUM
ejpam-3764	115	16	,	,	PUNCT
ejpam-3764	115	17	01	01	NUM
ejpam-3764	115	18	,	,	PUNCT
ejpam-3764	115	19	10	10	NUM
ejpam-3764	115	20	,	,	PUNCT
ejpam-3764	115	21	11	11	NUM
ejpam-3764	115	22	}	}	PUNCT
ejpam-3764	115	23	.	.	PUNCT
ejpam-3764	116	1	under	under	ADP
ejpam-3764	116	2	componentwise	componentwise	NOUN
ejpam-3764	116	3	addition	addition	NOUN
ejpam-3764	116	4	+	+	CCONJ
ejpam-3764	116	5	and	and	CCONJ
ejpam-3764	116	6	componentwise	componentwise	VERB
ejpam-3764	116	7	multiplication	multiplication	NOUN
ejpam-3764	116	8	◦	◦	NOUN
ejpam-3764	116	9	,	,	PUNCT
ejpam-3764	116	10	(	(	PUNCT
ejpam-3764	116	11	z(2	z(2	PROPN
ejpam-3764	116	12	)	)	PUNCT
ejpam-3764	116	13	2	2	NUM
ejpam-3764	116	14	,	,	PUNCT
ejpam-3764	116	15	+	+	NOUN
ejpam-3764	116	16	,	,	PUNCT
ejpam-3764	116	17	◦	◦	NOUN
ejpam-3764	116	18	)	)	PUNCT
ejpam-3764	116	19	is	be	AUX
ejpam-3764	116	20	a	a	DET
ejpam-3764	116	21	ring	ring	NOUN
ejpam-3764	116	22	with	with	ADP
ejpam-3764	116	23	code	code	NOUN
ejpam-3764	116	24	-	-	PUNCT
ejpam-3764	116	25	words	word	NOUN
ejpam-3764	116	26	0	0	NUM
ejpam-3764	117	1	=	=	SYM
ejpam-3764	117	2	00	00	PUNCT
ejpam-3764	117	3	and	and	CCONJ
ejpam-3764	117	4	1	1	NUM
ejpam-3764	117	5	=	=	SYM
ejpam-3764	117	6	11	11	NUM
ejpam-3764	117	7	as	as	ADP
ejpam-3764	117	8	additive	additive	ADJ
ejpam-3764	117	9	and	and	CCONJ
ejpam-3764	117	10	multiplicative	multiplicative	ADJ
ejpam-3764	117	11	identities	identity	NOUN
ejpam-3764	117	12	.	.	PUNCT
ejpam-3764	118	1	we	we	PRON
ejpam-3764	118	2	define	define	VERB
ejpam-3764	118	3	the	the	DET
ejpam-3764	118	4	laxicographic	laxicographic	ADJ
ejpam-3764	118	5	odering	odering	NOUN
ejpam-3764	118	6	on	on	ADP
ejpam-3764	118	7	the	the	DET
ejpam-3764	118	8	elements	element	NOUN
ejpam-3764	118	9	of	of	ADP
ejpam-3764	118	10	z(2	z(2	PROPN
ejpam-3764	118	11	)	)	PUNCT
ejpam-3764	118	12	2	2	NUM
ejpam-3764	118	13	and	and	CCONJ
ejpam-3764	118	14	arrange	arrange	VERB
ejpam-3764	118	15	them	they	PRON
ejpam-3764	118	16	as	as	ADP
ejpam-3764	118	17	:	:	PUNCT
ejpam-3764	118	18	00	00	PUNCT
ejpam-3764	118	19	<	<	X
ejpam-3764	119	1	01	01	NUM
ejpam-3764	119	2	<	<	X
ejpam-3764	119	3	10	10	NUM
ejpam-3764	119	4	<	<	X
ejpam-3764	119	5	11	11	NUM
ejpam-3764	119	6	let	let	VERB
ejpam-3764	119	7	a	a	PRON
ejpam-3764	119	8	=	=	PUNCT
ejpam-3764	119	9	{	{	PUNCT
ejpam-3764	119	10	00	00	NUM
ejpam-3764	119	11	,	,	PUNCT
ejpam-3764	119	12	01	01	NUM
ejpam-3764	119	13	}	}	PUNCT
ejpam-3764	119	14	.	.	PUNCT
ejpam-3764	120	1	consider	consider	VERB
ejpam-3764	120	2	ω	ω	NOUN
ejpam-3764	120	3	=	=	SYM
ejpam-3764	120	4	11	11	NUM
ejpam-3764	120	5	.	.	PUNCT
ejpam-3764	121	1	note	note	VERB
ejpam-3764	121	2	that	that	SCONJ
ejpam-3764	121	3	,	,	PUNCT
ejpam-3764	121	4	in	in	ADP
ejpam-3764	121	5	this	this	DET
ejpam-3764	121	6	example	example	NOUN
ejpam-3764	121	7	,	,	PUNCT
ejpam-3764	121	8	ω	ω	PROPN
ejpam-3764	121	9	/∈	/∈	PUNCT
ejpam-3764	122	1	a	a	DET
ejpam-3764	122	2	but	but	CCONJ
ejpam-3764	122	3	ω	ω	NUM
ejpam-3764	122	4	∈	∈	PROPN
ejpam-3764	122	5	g.	g.	NOUN
ejpam-3764	122	6	we	we	PRON
ejpam-3764	122	7	define	define	VERB
ejpam-3764	122	8	addition	addition	NOUN
ejpam-3764	122	9	on	on	ADP
ejpam-3764	122	10	aω	aω	PROPN
ejpam-3764	122	11	=	=	PUNCT
ejpam-3764	122	12	{	{	PUNCT
ejpam-3764	122	13	00	00	NUM
ejpam-3764	122	14	,	,	PUNCT
ejpam-3764	122	15	01	01	NUM
ejpam-3764	122	16	,	,	PUNCT
ejpam-3764	122	17	11	11	NUM
ejpam-3764	122	18	}	}	PUNCT
ejpam-3764	122	19	by	by	ADP
ejpam-3764	122	20	:	:	PUNCT
ejpam-3764	122	21	a⊕	a⊕	PROPN
ejpam-3764	122	22	b	b	X
ejpam-3764	122	23	=	=	SYM
ejpam-3764	122	24	min(a	min(a	PROPN
ejpam-3764	122	25	,	,	PUNCT
ejpam-3764	122	26	b	b	NOUN
ejpam-3764	122	27	)	)	PUNCT
ejpam-3764	122	28	.	.	PUNCT
ejpam-3764	123	1	hence	hence	ADV
ejpam-3764	123	2	we	we	PRON
ejpam-3764	123	3	get	get	VERB
ejpam-3764	123	4	the	the	DET
ejpam-3764	123	5	table	table	NOUN
ejpam-3764	123	6	:	:	PUNCT
ejpam-3764	123	7	⊕	⊕	PROPN
ejpam-3764	123	8	00	00	PUNCT
ejpam-3764	123	9	01	01	NUM
ejpam-3764	123	10	11	11	NUM
ejpam-3764	123	11	00	00	NUM
ejpam-3764	123	12	00	00	NUM
ejpam-3764	123	13	00	00	NUM
ejpam-3764	123	14	00	00	NUM
ejpam-3764	123	15	01	01	NUM
ejpam-3764	123	16	00	00	NUM
ejpam-3764	123	17	01	01	NUM
ejpam-3764	123	18	01	01	NUM
ejpam-3764	123	19	11	11	NUM
ejpam-3764	123	20	00	00	NUM
ejpam-3764	123	21	01	01	NUM
ejpam-3764	123	22	11	11	NUM
ejpam-3764	123	23	.	.	PUNCT
ejpam-3764	124	1	define	define	VERB
ejpam-3764	124	2	multiplication	multiplication	NOUN
ejpam-3764	124	3	as	as	ADP
ejpam-3764	124	4	the	the	DET
ejpam-3764	124	5	boolean	boolean	ADJ
ejpam-3764	124	6	sum	sum	NOUN
ejpam-3764	124	7	,	,	PUNCT
ejpam-3764	124	8	namely	namely	ADV
ejpam-3764	124	9	,	,	PUNCT
ejpam-3764	124	10	0	0	NUM
ejpam-3764	125	1	+	+	CCONJ
ejpam-3764	125	2	0	0	NUM
ejpam-3764	125	3	=	=	SYM
ejpam-3764	125	4	0	0	NUM
ejpam-3764	125	5	,	,	PUNCT
ejpam-3764	125	6	0	0	NUM
ejpam-3764	126	1	+	+	CCONJ
ejpam-3764	126	2	1	1	NUM
ejpam-3764	126	3	=	=	SYM
ejpam-3764	126	4	1	1	NUM
ejpam-3764	126	5	,	,	PUNCT
ejpam-3764	126	6	1	1	NUM
ejpam-3764	126	7	+	+	SYM
ejpam-3764	126	8	1	1	NUM
ejpam-3764	126	9	=	=	SYM
ejpam-3764	126	10	1	1	NUM
ejpam-3764	126	11	.	.	PUNCT
ejpam-3764	127	1	hence	hence	ADV
ejpam-3764	127	2	we	we	PRON
ejpam-3764	127	3	get	get	VERB
ejpam-3764	127	4	the	the	DET
ejpam-3764	127	5	table	table	NOUN
ejpam-3764	127	6	:	:	PUNCT
ejpam-3764	127	7	⊗	⊗	PROPN
ejpam-3764	127	8	00	00	NUM
ejpam-3764	127	9	01	01	NUM
ejpam-3764	127	10	11	11	NUM
ejpam-3764	127	11	00	00	NUM
ejpam-3764	127	12	00	00	NUM
ejpam-3764	127	13	01	01	NUM
ejpam-3764	127	14	11	11	NUM
ejpam-3764	127	15	01	01	NUM
ejpam-3764	127	16	01	01	NUM
ejpam-3764	127	17	01	01	NUM
ejpam-3764	127	18	11	11	NUM
ejpam-3764	127	19	11	11	NUM
ejpam-3764	127	20	11	11	NUM
ejpam-3764	127	21	11	11	NUM
ejpam-3764	127	22	11	11	NUM
ejpam-3764	127	23	.	.	PUNCT
ejpam-3764	128	1	we	we	PRON
ejpam-3764	128	2	conclude	conclude	VERB
ejpam-3764	128	3	that	that	PRON
ejpam-3764	128	4	(	(	PUNCT
ejpam-3764	128	5	aω,⊕	aω,⊕	NUM
ejpam-3764	128	6	,	,	PUNCT
ejpam-3764	128	7	11	11	NUM
ejpam-3764	128	8	)	)	PUNCT
ejpam-3764	128	9	and	and	CCONJ
ejpam-3764	128	10	(	(	PUNCT
ejpam-3764	128	11	aω,⊗	aω,⊗	PROPN
ejpam-3764	128	12	,	,	PUNCT
ejpam-3764	128	13	00	00	NUM
ejpam-3764	128	14	)	)	PUNCT
ejpam-3764	128	15	are	be	AUX
ejpam-3764	128	16	the	the	DET
ejpam-3764	128	17	additive	additive	ADJ
ejpam-3764	128	18	and	and	CCONJ
ejpam-3764	128	19	multiplicative	multiplicative	ADJ
ejpam-3764	128	20	monoides	monoide	NOUN
ejpam-3764	128	21	.	.	PUNCT
ejpam-3764	129	1	clearly	clearly	ADV
ejpam-3764	129	2	,	,	PUNCT
ejpam-3764	129	3	this	this	PRON
ejpam-3764	129	4	is	be	AUX
ejpam-3764	129	5	a	a	DET
ejpam-3764	129	6	simple	simple	ADJ
ejpam-3764	129	7	ω−	ω−	ADJ
ejpam-3764	129	8	algebra	algebra	NOUN
ejpam-3764	129	9	.	.	PUNCT
ejpam-3764	130	1	3	3	X
ejpam-3764	130	2	.	.	X
ejpam-3764	130	3	omega	omega	NOUN
ejpam-3764	130	4	topology	topology	NOUN
ejpam-3764	130	5	in	in	ADP
ejpam-3764	130	6	this	this	DET
ejpam-3764	130	7	section	section	NOUN
ejpam-3764	130	8	,	,	PUNCT
ejpam-3764	130	9	we	we	PRON
ejpam-3764	130	10	define	define	VERB
ejpam-3764	130	11	a	a	DET
ejpam-3764	130	12	new	new	ADJ
ejpam-3764	130	13	topology	topology	NOUN
ejpam-3764	130	14	on	on	ADP
ejpam-3764	130	15	omega	omega	NOUN
ejpam-3764	130	16	algebra	algebra	NOUN
ejpam-3764	130	17	and	and	CCONJ
ejpam-3764	130	18	discuss	discuss	VERB
ejpam-3764	130	19	some	some	PRON
ejpam-3764	130	20	of	of	ADP
ejpam-3764	130	21	its	its	PRON
ejpam-3764	130	22	topological	topological	ADJ
ejpam-3764	130	23	properties	property	NOUN
ejpam-3764	130	24	.	.	PUNCT
ejpam-3764	131	1	proposition	proposition	NOUN
ejpam-3764	131	2	1	1	NUM
ejpam-3764	131	3	.	.	PUNCT
ejpam-3764	132	1	let	let	AUX
ejpam-3764	132	2	(	(	PUNCT
ejpam-3764	132	3	g	g	NOUN
ejpam-3764	132	4	,	,	PUNCT
ejpam-3764	132	5	◦	◦	NOUN
ejpam-3764	132	6	,	,	PUNCT
ejpam-3764	132	7	e	e	NOUN
ejpam-3764	132	8	)	)	PUNCT
ejpam-3764	132	9	be	be	AUX
ejpam-3764	132	10	an	an	DET
ejpam-3764	132	11	abelian	abelian	ADJ
ejpam-3764	132	12	group	group	NOUN
ejpam-3764	132	13	and	and	CCONJ
ejpam-3764	132	14	aω	aω	NOUN
ejpam-3764	132	15	=	=	PUNCT
ejpam-3764	132	16	(	(	PUNCT
ejpam-3764	132	17	aω,⊕,⊗	aω,⊕,⊗	PROPN
ejpam-3764	132	18	,	,	PUNCT
ejpam-3764	132	19	ω	ω	PROPN
ejpam-3764	132	20	,	,	PUNCT
ejpam-3764	132	21	e	e	NOUN
ejpam-3764	132	22	)	)	PUNCT
ejpam-3764	132	23	an	an	DET
ejpam-3764	132	24	ω−algebra	ω−algebra	PROPN
ejpam-3764	132	25	over	over	ADP
ejpam-3764	132	26	the	the	DET
ejpam-3764	132	27	group	group	NOUN
ejpam-3764	132	28	g.	g.	NOUN
ejpam-3764	133	1	we	we	PRON
ejpam-3764	133	2	define	define	VERB
ejpam-3764	133	3	a	a	DET
ejpam-3764	133	4	new	new	ADJ
ejpam-3764	133	5	topology	topology	NOUN
ejpam-3764	133	6	on	on	ADP
ejpam-3764	133	7	aω	aω	PROPN
ejpam-3764	133	8	is	be	AUX
ejpam-3764	133	9	called	call	VERB
ejpam-3764	133	10	an	an	DET
ejpam-3764	133	11	omega	omega	NOUN
ejpam-3764	133	12	topology	topology	NOUN
ejpam-3764	133	13	,	,	PUNCT
ejpam-3764	133	14	denoted	denote	VERB
ejpam-3764	133	15	by	by	ADP
ejpam-3764	133	16	τω	τω	INTJ
ejpam-3764	133	17	,	,	PUNCT
ejpam-3764	133	18	as	as	SCONJ
ejpam-3764	133	19	follow	follow	VERB
ejpam-3764	133	20	:	:	PUNCT
ejpam-3764	133	21	τω	τω	X
ejpam-3764	133	22	=	=	PRON
ejpam-3764	133	23	{	{	PUNCT
ejpam-3764	133	24	∅	∅	NOUN
ejpam-3764	133	25	,	,	PUNCT
ejpam-3764	133	26	aω	aω	NOUN
ejpam-3764	133	27	}	}	PUNCT
ejpam-3764	133	28	∪	∪	NOUN
ejpam-3764	133	29	{	{	PUNCT
ejpam-3764	133	30	u	u	NOUN
ejpam-3764	133	31	⊆	⊆	NUM
ejpam-3764	133	32	aω	aω	X
ejpam-3764	133	33	:	:	PUNCT
ejpam-3764	133	34	ω	ω	NUM
ejpam-3764	133	35	∈	∈	PROPN
ejpam-3764	133	36	u	u	NOUN
ejpam-3764	133	37	and	and	CCONJ
ejpam-3764	133	38	for	for	ADP
ejpam-3764	133	39	any	any	DET
ejpam-3764	133	40	a	a	DET
ejpam-3764	133	41	∈	∈	PROPN
ejpam-3764	133	42	u	u	NOUN
ejpam-3764	133	43	\	\	PROPN
ejpam-3764	133	44	{	{	PUNCT
ejpam-3764	133	45	ω	ω	NOUN
ejpam-3764	133	46	}	}	PUNCT
ejpam-3764	133	47	,	,	PUNCT
ejpam-3764	133	48	the	the	DET
ejpam-3764	133	49	multiplicative	multiplicative	ADJ
ejpam-3764	133	50	inverse	inverse	NOUN
ejpam-3764	133	51	of	of	ADP
ejpam-3764	133	52	exists	exist	NOUN
ejpam-3764	133	53	in	in	ADP
ejpam-3764	133	54	u	u	NOUN
ejpam-3764	133	55	}	}	PUNCT
ejpam-3764	133	56	m.	m.	NOUN
ejpam-3764	133	57	alqahtani	alqahtani	PROPN
ejpam-3764	133	58	,	,	PUNCT
ejpam-3764	133	59	c.	c.	PROPN
ejpam-3764	133	60	özel	özel	PROPN
ejpam-3764	133	61	,	,	PUNCT
ejpam-3764	133	62	i.	i.	PROPN
ejpam-3764	133	63	alshammari	alshammari	PROPN
ejpam-3764	133	64	/	/	SYM
ejpam-3764	133	65	eur	eur	PROPN
ejpam-3764	133	66	.	.	PUNCT
ejpam-3764	134	1	j.	j.	PROPN
ejpam-3764	134	2	pure	pure	PROPN
ejpam-3764	134	3	appl	appl	PROPN
ejpam-3764	134	4	.	.	PROPN
ejpam-3764	134	5	math	math	PROPN
ejpam-3764	134	6	,	,	PUNCT
ejpam-3764	134	7	13	13	NUM
ejpam-3764	134	8	(	(	PUNCT
ejpam-3764	134	9	3	3	NUM
ejpam-3764	134	10	)	)	PUNCT
ejpam-3764	134	11	(	(	PUNCT
ejpam-3764	134	12	2020	2020	NUM
ejpam-3764	134	13	)	)	PUNCT
ejpam-3764	134	14	,	,	PUNCT
ejpam-3764	134	15	513	513	NUM
ejpam-3764	134	16	-	-	SYM
ejpam-3764	134	17	528	528	NUM
ejpam-3764	134	18	518	518	NUM
ejpam-3764	134	19	proof	proof	NOUN
ejpam-3764	134	20	.	.	PUNCT
ejpam-3764	135	1	condition	condition	NOUN
ejpam-3764	135	2	∅	∅	NOUN
ejpam-3764	135	3	,	,	PUNCT
ejpam-3764	135	4	aω	aω	PROPN
ejpam-3764	135	5	∈	∈	NOUN
ejpam-3764	135	6	τω	τω	NOUN
ejpam-3764	135	7	is	be	AUX
ejpam-3764	135	8	satisfied	satisfied	ADJ
ejpam-3764	135	9	from	from	ADP
ejpam-3764	135	10	the	the	DET
ejpam-3764	135	11	definition	definition	NOUN
ejpam-3764	135	12	of	of	ADP
ejpam-3764	135	13	τω	τω	INTJ
ejpam-3764	135	14	.	.	PUNCT
ejpam-3764	135	15	now	now	ADV
ejpam-3764	135	16	let	let	VERB
ejpam-3764	135	17	v1	v1	NOUN
ejpam-3764	135	18	,	,	PUNCT
ejpam-3764	135	19	v2	v2	PROPN
ejpam-3764	135	20	∈	∈	PROPN
ejpam-3764	135	21	τω	τω	NOUN
ejpam-3764	135	22	be	be	AUX
ejpam-3764	135	23	arbitrary	arbitrary	ADJ
ejpam-3764	135	24	.	.	PUNCT
ejpam-3764	136	1	if	if	SCONJ
ejpam-3764	136	2	either	either	CCONJ
ejpam-3764	136	3	v1	v1	VERB
ejpam-3764	136	4	or	or	CCONJ
ejpam-3764	136	5	v2	v2	NOUN
ejpam-3764	136	6	is	be	AUX
ejpam-3764	136	7	empty	empty	ADJ
ejpam-3764	136	8	,	,	PUNCT
ejpam-3764	136	9	then	then	ADV
ejpam-3764	136	10	v1∩v2	v1∩v2	PROPN
ejpam-3764	136	11	=	=	SYM
ejpam-3764	136	12	∅	∅	NOUN
ejpam-3764	136	13	∈	∈	PROPN
ejpam-3764	136	14	τω	τω	INTJ
ejpam-3764	136	15	.	.	PROPN
ejpam-3764	136	16	assume	assume	VERB
ejpam-3764	136	17	now	now	ADV
ejpam-3764	136	18	,	,	PUNCT
ejpam-3764	136	19	v1	v1	VERB
ejpam-3764	136	20	6=	6=	SYM
ejpam-3764	136	21	∅	∅	NOUN
ejpam-3764	136	22	6=	6=	ADP
ejpam-3764	137	1	v2	v2	NOUN
ejpam-3764	137	2	.	.	PUNCT
ejpam-3764	138	1	if	if	SCONJ
ejpam-3764	138	2	either	either	CCONJ
ejpam-3764	138	3	v1	v1	VERB
ejpam-3764	138	4	or	or	CCONJ
ejpam-3764	138	5	v2	v2	NOUN
ejpam-3764	138	6	is	be	AUX
ejpam-3764	138	7	a	a	DET
ejpam-3764	138	8	whole	whole	ADJ
ejpam-3764	138	9	set	set	NOUN
ejpam-3764	138	10	aω	aω	PROPN
ejpam-3764	138	11	,	,	PUNCT
ejpam-3764	138	12	then	then	ADV
ejpam-3764	138	13	v1	v1	VERB
ejpam-3764	138	14	∩	∩	ADJ
ejpam-3764	138	15	v2	v2	NOUN
ejpam-3764	138	16	=	=	SYM
ejpam-3764	138	17	v1	v1	NOUN
ejpam-3764	138	18	or	or	CCONJ
ejpam-3764	138	19	v2	v2	NOUN
ejpam-3764	138	20	∈	∈	PROPN
ejpam-3764	138	21	τω	τω	INTJ
ejpam-3764	138	22	.	.	PUNCT
ejpam-3764	139	1	so	so	ADV
ejpam-3764	139	2	,	,	PUNCT
ejpam-3764	139	3	assume	assume	VERB
ejpam-3764	139	4	that	that	SCONJ
ejpam-3764	139	5	v1	v1	PROPN
ejpam-3764	139	6	6=	6=	ADP
ejpam-3764	139	7	aω	aω	PROPN
ejpam-3764	139	8	6=	6=	PROPN
ejpam-3764	139	9	v2	v2	PROPN
ejpam-3764	139	10	,	,	PUNCT
ejpam-3764	139	11	then	then	ADV
ejpam-3764	139	12	v1∩v2	v1∩v2	PROPN
ejpam-3764	139	13	∈	∈	PROPN
ejpam-3764	139	14	τω	τω	ADP
ejpam-3764	139	15	,	,	PUNCT
ejpam-3764	139	16	because	because	SCONJ
ejpam-3764	139	17	ω	ω	PROPN
ejpam-3764	139	18	∈	∈	PROPN
ejpam-3764	139	19	v1	v1	NOUN
ejpam-3764	139	20	and	and	CCONJ
ejpam-3764	139	21	ω	ω	NUM
ejpam-3764	139	22	∈	∈	PROPN
ejpam-3764	139	23	v2	v2	NOUN
ejpam-3764	139	24	.	.	PUNCT
ejpam-3764	140	1	hence	hence	ADV
ejpam-3764	140	2	ω	ω	PROPN
ejpam-3764	140	3	∈	∈	PROPN
ejpam-3764	140	4	v1∩v2	v1∩v2	PROPN
ejpam-3764	140	5	.	.	PUNCT
ejpam-3764	141	1	also	also	ADV
ejpam-3764	141	2	for	for	ADP
ejpam-3764	141	3	any	any	DET
ejpam-3764	141	4	element	element	NOUN
ejpam-3764	141	5	(	(	PUNCT
ejpam-3764	141	6	a	a	DET
ejpam-3764	141	7	6=	6=	PROPN
ejpam-3764	141	8	ω	ω	NUM
ejpam-3764	141	9	)	)	PUNCT
ejpam-3764	141	10	∈	∈	PROPN
ejpam-3764	141	11	v1	v1	NOUN
ejpam-3764	141	12	∩	∩	ADJ
ejpam-3764	141	13	v2	v2	NOUN
ejpam-3764	141	14	,	,	PUNCT
ejpam-3764	141	15	we	we	PRON
ejpam-3764	141	16	have	have	VERB
ejpam-3764	141	17	a	a	DET
ejpam-3764	141	18	∈	∈	NOUN
ejpam-3764	141	19	v1	v1	NOUN
ejpam-3764	141	20	and	and	CCONJ
ejpam-3764	141	21	a	a	DET
ejpam-3764	141	22	∈	∈	PROPN
ejpam-3764	141	23	v2	v2	NOUN
ejpam-3764	141	24	,	,	PUNCT
ejpam-3764	141	25	then	then	ADV
ejpam-3764	141	26	a	a	PRON
ejpam-3764	141	27	and	and	CCONJ
ejpam-3764	141	28	the	the	DET
ejpam-3764	141	29	multiplicative	multiplicative	ADJ
ejpam-3764	141	30	inverse	inverse	NOUN
ejpam-3764	141	31	of	of	ADP
ejpam-3764	141	32	a	a	PRON
ejpam-3764	141	33	must	must	AUX
ejpam-3764	141	34	belong	belong	VERB
ejpam-3764	141	35	to	to	ADP
ejpam-3764	141	36	v1	v1	NOUN
ejpam-3764	141	37	and	and	CCONJ
ejpam-3764	141	38	v2	v2	NOUN
ejpam-3764	141	39	.	.	PUNCT
ejpam-3764	142	1	hence	hence	ADV
ejpam-3764	142	2	a	a	PRON
ejpam-3764	142	3	and	and	CCONJ
ejpam-3764	142	4	the	the	DET
ejpam-3764	142	5	multiplicative	multiplicative	ADJ
ejpam-3764	142	6	inverse	inverse	NOUN
ejpam-3764	142	7	of	of	ADP
ejpam-3764	142	8	a	a	DET
ejpam-3764	142	9	belong	belong	NOUN
ejpam-3764	142	10	to	to	ADP
ejpam-3764	142	11	v1	v1	VERB
ejpam-3764	142	12	∩	∩	ADJ
ejpam-3764	142	13	v2	v2	NOUN
ejpam-3764	142	14	,	,	PUNCT
ejpam-3764	142	15	then	then	ADV
ejpam-3764	142	16	v1	v1	VERB
ejpam-3764	142	17	∩	∩	ADJ
ejpam-3764	142	18	v2	v2	NOUN
ejpam-3764	142	19	∈	∈	PROPN
ejpam-3764	142	20	τω	τω	INTJ
ejpam-3764	142	21	.	.	PROPN
ejpam-3764	143	1	for	for	ADP
ejpam-3764	143	2	the	the	DET
ejpam-3764	143	3	third	third	ADJ
ejpam-3764	143	4	condition	condition	NOUN
ejpam-3764	143	5	,	,	PUNCT
ejpam-3764	143	6	let	let	VERB
ejpam-3764	143	7	sγ	sγ	PRON
ejpam-3764	143	8	∈	∈	PROPN
ejpam-3764	143	9	τω	τω	NOUN
ejpam-3764	143	10	for	for	ADP
ejpam-3764	143	11	any	any	DET
ejpam-3764	143	12	γ	γ	PROPN
ejpam-3764	143	13	∈	∈	PROPN
ejpam-3764	143	14	i.	i.	NOUN
ejpam-3764	143	15	if	if	SCONJ
ejpam-3764	143	16	sγ	sγ	NOUN
ejpam-3764	143	17	=	=	PUNCT
ejpam-3764	143	18	∅	∅	NOUN
ejpam-3764	143	19	for	for	ADP
ejpam-3764	143	20	all	all	DET
ejpam-3764	143	21	γ	γ	PROPN
ejpam-3764	143	22	∈	∈	PROPN
ejpam-3764	143	23	i	i	PRON
ejpam-3764	143	24	,	,	PUNCT
ejpam-3764	143	25	then	then	ADV
ejpam-3764	143	26	⋃	⋃	NOUN
ejpam-3764	143	27	γ∈i	γ∈i	ADV
ejpam-3764	143	28	sγ	sγ	NOUN
ejpam-3764	143	29	=	=	SYM
ejpam-3764	143	30	∅	∅	NOUN
ejpam-3764	143	31	∈	∈	PROPN
ejpam-3764	143	32	τω	τω	INTJ
ejpam-3764	143	33	.	.	PUNCT
ejpam-3764	144	1	so	so	ADV
ejpam-3764	144	2	,	,	PUNCT
ejpam-3764	144	3	assume	assume	VERB
ejpam-3764	144	4	that	that	SCONJ
ejpam-3764	144	5	some	some	DET
ejpam-3764	144	6	member	member	NOUN
ejpam-3764	144	7	is	be	AUX
ejpam-3764	144	8	non	non	ADJ
ejpam-3764	144	9	-	-	ADJ
ejpam-3764	144	10	empty	empty	ADJ
ejpam-3764	144	11	,	,	PUNCT
ejpam-3764	144	12	but	but	CCONJ
ejpam-3764	144	13	since	since	SCONJ
ejpam-3764	144	14	the	the	DET
ejpam-3764	144	15	empty	empty	ADJ
ejpam-3764	144	16	set	set	NOUN
ejpam-3764	144	17	does	do	AUX
ejpam-3764	144	18	not	not	PART
ejpam-3764	144	19	affect	affect	VERB
ejpam-3764	144	20	any	any	DET
ejpam-3764	144	21	union	union	NOUN
ejpam-3764	144	22	,	,	PUNCT
ejpam-3764	144	23	assume	assume	VERB
ejpam-3764	144	24	that	that	SCONJ
ejpam-3764	144	25	,	,	PUNCT
ejpam-3764	144	26	without	without	ADP
ejpam-3764	144	27	loss	loss	NOUN
ejpam-3764	144	28	of	of	ADP
ejpam-3764	144	29	generality	generality	NOUN
ejpam-3764	144	30	sγ	sγ	ADP
ejpam-3764	144	31	6=	6=	NOUN
ejpam-3764	144	32	∅	∅	NOUN
ejpam-3764	144	33	for	for	ADP
ejpam-3764	144	34	all	all	DET
ejpam-3764	144	35	γ	γ	PROPN
ejpam-3764	144	36	∈	∈	PROPN
ejpam-3764	144	37	i.	i.	NOUN
ejpam-3764	144	38	if	if	SCONJ
ejpam-3764	144	39	there	there	PRON
ejpam-3764	144	40	exist	exist	VERB
ejpam-3764	144	41	a	a	DET
ejpam-3764	144	42	γ1	γ1	NOUN
ejpam-3764	144	43	∈	∈	PROPN
ejpam-3764	144	44	i	i	PRON
ejpam-3764	144	45	where	where	SCONJ
ejpam-3764	144	46	sγ1	sγ1	ADV
ejpam-3764	144	47	=	=	SYM
ejpam-3764	144	48	aω	aω	PROPN
ejpam-3764	144	49	,	,	PUNCT
ejpam-3764	144	50	then	then	ADV
ejpam-3764	144	51	⋃	⋃	NOUN
ejpam-3764	144	52	γ∈i	γ∈i	ADV
ejpam-3764	144	53	sγ	sγ	NOUN
ejpam-3764	144	54	=	=	PUNCT
ejpam-3764	144	55	aω	aω	PROPN
ejpam-3764	144	56	∈	∈	PROPN
ejpam-3764	145	1	τω	τω	INTJ
ejpam-3764	145	2	.	.	PUNCT
ejpam-3764	146	1	so	so	ADV
ejpam-3764	146	2	,	,	PUNCT
ejpam-3764	146	3	assume	assume	VERB
ejpam-3764	146	4	now	now	ADV
ejpam-3764	146	5	that	that	SCONJ
ejpam-3764	146	6	sγ	sγ	PROPN
ejpam-3764	146	7	6=	6=	PROPN
ejpam-3764	146	8	aω	aω	PROPN
ejpam-3764	146	9	for	for	ADP
ejpam-3764	146	10	all	all	DET
ejpam-3764	146	11	γ	γ	PROPN
ejpam-3764	146	12	∈	∈	PROPN
ejpam-3764	146	13	i	i	PRON
ejpam-3764	146	14	,	,	PUNCT
ejpam-3764	146	15	then	then	ADV
ejpam-3764	146	16	⋃	⋃	NOUN
ejpam-3764	146	17	γ∈i	γ∈i	ADV
ejpam-3764	146	18	sγ	sγ	ADP
ejpam-3764	146	19	∈	∈	PROPN
ejpam-3764	147	1	τω	τω	INTJ
ejpam-3764	147	2	,	,	PUNCT
ejpam-3764	147	3	because	because	SCONJ
ejpam-3764	147	4	ω	ω	PROPN
ejpam-3764	147	5	∈	∈	NOUN
ejpam-3764	147	6	sγ	sγ	NOUN
ejpam-3764	147	7	for	for	ADP
ejpam-3764	147	8	all	all	DET
ejpam-3764	147	9	γ	γ	PROPN
ejpam-3764	147	10	∈	∈	PROPN
ejpam-3764	147	11	i.	i.	NOUN
ejpam-3764	147	12	hence	hence	ADV
ejpam-3764	147	13	ω	ω	PROPN
ejpam-3764	147	14	∈	∈	PROPN
ejpam-3764	147	15	⋃	⋃	NOUN
ejpam-3764	147	16	γ∈i	γ∈i	ADJ
ejpam-3764	147	17	sγ	sγ	NOUN
ejpam-3764	147	18	.	.	PUNCT
ejpam-3764	148	1	also	also	ADV
ejpam-3764	148	2	for	for	ADP
ejpam-3764	148	3	any	any	DET
ejpam-3764	148	4	(	(	PUNCT
ejpam-3764	148	5	a	a	PRON
ejpam-3764	148	6	6=	6=	PROPN
ejpam-3764	148	7	ω	ω	NUM
ejpam-3764	148	8	)	)	PUNCT
ejpam-3764	148	9	∈	∈	PROPN
ejpam-3764	148	10	⋃	⋃	NOUN
ejpam-3764	148	11	γ∈i	γ∈i	ADV
ejpam-3764	148	12	sγ	sγ	NOUN
ejpam-3764	148	13	,	,	PUNCT
ejpam-3764	148	14	there	there	PRON
ejpam-3764	148	15	exists	exist	VERB
ejpam-3764	148	16	γa	γa	PROPN
ejpam-3764	148	17	∈	∈	PROPN
ejpam-3764	149	1	i	i	PRON
ejpam-3764	149	2	such	such	ADJ
ejpam-3764	149	3	that	that	SCONJ
ejpam-3764	149	4	a	a	DET
ejpam-3764	149	5	∈	∈	ADJ
ejpam-3764	149	6	sγa	sγa	NOUN
ejpam-3764	149	7	,	,	PUNCT
ejpam-3764	149	8	hence	hence	ADV
ejpam-3764	149	9	a	a	PRON
ejpam-3764	149	10	and	and	CCONJ
ejpam-3764	149	11	the	the	DET
ejpam-3764	149	12	multiplicative	multiplicative	ADJ
ejpam-3764	149	13	inverse	inverse	NOUN
ejpam-3764	149	14	of	of	ADP
ejpam-3764	149	15	a	a	DET
ejpam-3764	149	16	belong	belong	NOUN
ejpam-3764	149	17	to	to	ADP
ejpam-3764	149	18	sγa	sγa	NOUN
ejpam-3764	149	19	,	,	PUNCT
ejpam-3764	149	20	then	then	ADV
ejpam-3764	149	21	a	a	PRON
ejpam-3764	149	22	and	and	CCONJ
ejpam-3764	149	23	the	the	DET
ejpam-3764	149	24	multiplicative	multiplicative	ADJ
ejpam-3764	149	25	inverse	inverse	NOUN
ejpam-3764	149	26	of	of	ADP
ejpam-3764	149	27	a	a	DET
ejpam-3764	149	28	belong	belong	NOUN
ejpam-3764	149	29	to	to	ADP
ejpam-3764	149	30	⋃	⋃	NOUN
ejpam-3764	149	31	γ∈i	γ∈i	ADJ
ejpam-3764	149	32	sγ	sγ	NOUN
ejpam-3764	149	33	.	.	PUNCT
ejpam-3764	150	1	hence	hence	ADV
ejpam-3764	150	2	⋃	⋃	NOUN
ejpam-3764	150	3	γ∈i	γ∈i	ADV
ejpam-3764	150	4	sγ	sγ	ADP
ejpam-3764	150	5	∈	∈	PROPN
ejpam-3764	150	6	τω	τω	INTJ
ejpam-3764	150	7	.	.	PUNCT
ejpam-3764	150	8	therefore	therefore	ADV
ejpam-3764	150	9	,	,	PUNCT
ejpam-3764	150	10	(	(	PUNCT
ejpam-3764	150	11	aω	aω	INTJ
ejpam-3764	150	12	,	,	PUNCT
ejpam-3764	150	13	τω	τω	INTJ
ejpam-3764	150	14	)	)	PUNCT
ejpam-3764	150	15	is	be	AUX
ejpam-3764	150	16	topological	topological	ADJ
ejpam-3764	150	17	space	space	NOUN
ejpam-3764	150	18	.	.	PUNCT
ejpam-3764	151	1	corollary	corollary	ADJ
ejpam-3764	151	2	1	1	NUM
ejpam-3764	151	3	.	.	PUNCT
ejpam-3764	152	1	if	if	SCONJ
ejpam-3764	152	2	a	a	DET
ejpam-3764	152	3	∈	∈	NOUN
ejpam-3764	152	4	aω	aω	X
ejpam-3764	152	5	\	\	PROPN
ejpam-3764	152	6	{	{	PUNCT
ejpam-3764	152	7	ω	ω	NOUN
ejpam-3764	152	8	}	}	PUNCT
ejpam-3764	152	9	has	have	VERB
ejpam-3764	152	10	no	no	DET
ejpam-3764	152	11	multiplicative	multiplicative	ADJ
ejpam-3764	152	12	inverse	inverse	NOUN
ejpam-3764	152	13	,	,	PUNCT
ejpam-3764	152	14	then	then	ADV
ejpam-3764	152	15	aω	aω	PROPN
ejpam-3764	152	16	is	be	AUX
ejpam-3764	152	17	the	the	DET
ejpam-3764	152	18	only	only	ADJ
ejpam-3764	152	19	open	open	ADJ
ejpam-3764	152	20	set	set	VERB
ejpam-3764	152	21	in	in	ADP
ejpam-3764	152	22	(	(	PUNCT
ejpam-3764	152	23	aω	aω	PROPN
ejpam-3764	152	24	,	,	PUNCT
ejpam-3764	152	25	τω	τω	NOUN
ejpam-3764	152	26	)	)	PUNCT
ejpam-3764	152	27	containing	contain	VERB
ejpam-3764	152	28	a.	a.	NOUN
ejpam-3764	152	29	let	let	VERB
ejpam-3764	152	30	us	we	PRON
ejpam-3764	152	31	denoted	denote	VERB
ejpam-3764	152	32	for	for	ADP
ejpam-3764	152	33	the	the	DET
ejpam-3764	152	34	multiplicative	multiplicative	ADJ
ejpam-3764	152	35	inverse	inverse	NOUN
ejpam-3764	152	36	of	of	ADP
ejpam-3764	152	37	a	a	DET
ejpam-3764	152	38	∈	∈	PROPN
ejpam-3764	152	39	aω	aω	NOUN
ejpam-3764	152	40	by	by	ADP
ejpam-3764	152	41	a−1	a−1	PROPN
ejpam-3764	152	42	.	.	PUNCT
ejpam-3764	153	1	if	if	SCONJ
ejpam-3764	153	2	(	(	PUNCT
ejpam-3764	153	3	aω	aω	PROPN
ejpam-3764	153	4	\	\	PROPN
ejpam-3764	153	5	{	{	PUNCT
ejpam-3764	153	6	ω},⊗	ω},⊗	NOUN
ejpam-3764	153	7	)	)	PUNCT
ejpam-3764	153	8	is	be	AUX
ejpam-3764	153	9	a	a	DET
ejpam-3764	153	10	group	group	NOUN
ejpam-3764	153	11	,	,	PUNCT
ejpam-3764	153	12	where	where	SCONJ
ejpam-3764	153	13	ω	ω	NUM
ejpam-3764	153	14	and	and	CCONJ
ejpam-3764	153	15	e	e	NOUN
ejpam-3764	153	16	are	be	AUX
ejpam-3764	153	17	the	the	DET
ejpam-3764	153	18	zero	zero	NUM
ejpam-3764	153	19	and	and	CCONJ
ejpam-3764	153	20	multiplicative	multiplicative	ADJ
ejpam-3764	153	21	identity	identity	NOUN
ejpam-3764	153	22	elements	element	NOUN
ejpam-3764	153	23	,	,	PUNCT
ejpam-3764	153	24	respectively	respectively	ADV
ejpam-3764	153	25	,	,	PUNCT
ejpam-3764	153	26	then	then	ADV
ejpam-3764	153	27	for	for	ADP
ejpam-3764	153	28	any	any	DET
ejpam-3764	153	29	a	a	DET
ejpam-3764	153	30	∈	∈	NOUN
ejpam-3764	153	31	aω	aω	X
ejpam-3764	153	32	\	\	PROPN
ejpam-3764	153	33	{	{	PUNCT
ejpam-3764	153	34	ω	ω	NOUN
ejpam-3764	153	35	}	}	PUNCT
ejpam-3764	153	36	,	,	PUNCT
ejpam-3764	153	37	we	we	PRON
ejpam-3764	153	38	have	have	VERB
ejpam-3764	153	39	a−1	a−1	PROPN
ejpam-3764	153	40	∈	∈	PROPN
ejpam-3764	153	41	a	a	PRON
ejpam-3764	153	42	.	.	PUNCT
ejpam-3764	154	1	proposition	proposition	NOUN
ejpam-3764	154	2	2	2	NUM
ejpam-3764	154	3	.	.	PUNCT
ejpam-3764	155	1	the	the	DET
ejpam-3764	155	2	omega	omega	NOUN
ejpam-3764	155	3	topological	topological	ADJ
ejpam-3764	155	4	space	space	NOUN
ejpam-3764	155	5	(	(	PUNCT
ejpam-3764	155	6	aω	aω	PROPN
ejpam-3764	155	7	,	,	PUNCT
ejpam-3764	155	8	τω	τω	X
ejpam-3764	155	9	)	)	PUNCT
ejpam-3764	155	10	has	have	VERB
ejpam-3764	155	11	a	a	DET
ejpam-3764	155	12	base	base	NOUN
ejpam-3764	155	13	b	b	NOUN
ejpam-3764	155	14	=	=	SYM
ejpam-3764	155	15	{	{	PUNCT
ejpam-3764	155	16	aω	aω	PROPN
ejpam-3764	155	17	,	,	PUNCT
ejpam-3764	155	18	{	{	PUNCT
ejpam-3764	155	19	ω	ω	NOUN
ejpam-3764	155	20	}	}	PUNCT
ejpam-3764	155	21	,	,	PUNCT
ejpam-3764	155	22	{	{	PUNCT
ejpam-3764	155	23	ω	ω	NOUN
ejpam-3764	155	24	,	,	PUNCT
ejpam-3764	155	25	a	a	PRON
ejpam-3764	155	26	,	,	PUNCT
ejpam-3764	155	27	a−1	a−1	PROPN
ejpam-3764	155	28	}	}	PUNCT
ejpam-3764	155	29	:	:	PUNCT
ejpam-3764	155	30	a	a	DET
ejpam-3764	155	31	∈	∈	NOUN
ejpam-3764	155	32	aω	aω	X
ejpam-3764	155	33	\	\	PROPN
ejpam-3764	155	34	{	{	PUNCT
ejpam-3764	155	35	ω	ω	NOUN
ejpam-3764	155	36	}	}	PUNCT
ejpam-3764	155	37	,	,	PUNCT
ejpam-3764	155	38	which	which	PRON
ejpam-3764	155	39	has	have	VERB
ejpam-3764	155	40	a	a	DET
ejpam-3764	155	41	multiplicative	multiplicative	ADJ
ejpam-3764	155	42	inverse	inverse	NOUN
ejpam-3764	155	43	}	}	PUNCT
ejpam-3764	155	44	.	.	PUNCT
ejpam-3764	156	1	proof	proof	NOUN
ejpam-3764	156	2	.	.	PUNCT
ejpam-3764	157	1	for	for	ADP
ejpam-3764	157	2	the	the	DET
ejpam-3764	157	3	first	first	ADJ
ejpam-3764	157	4	condition	condition	NOUN
ejpam-3764	157	5	,	,	PUNCT
ejpam-3764	157	6	let	let	VERB
ejpam-3764	157	7	b	b	X
ejpam-3764	157	8	∈	∈	PROPN
ejpam-3764	157	9	b	b	NOUN
ejpam-3764	157	10	be	be	AUX
ejpam-3764	157	11	arbitrary	arbitrary	ADJ
ejpam-3764	157	12	,	,	PUNCT
ejpam-3764	157	13	if	if	SCONJ
ejpam-3764	157	14	b	b	X
ejpam-3764	157	15	=	=	SYM
ejpam-3764	157	16	{	{	PUNCT
ejpam-3764	157	17	ω	ω	NOUN
ejpam-3764	157	18	}	}	PUNCT
ejpam-3764	157	19	or	or	CCONJ
ejpam-3764	157	20	aω	aω	NOUN
ejpam-3764	157	21	,	,	PUNCT
ejpam-3764	158	1	then	then	ADV
ejpam-3764	158	2	b	b	X
ejpam-3764	158	3	∈	∈	PROPN
ejpam-3764	158	4	τω	τω	NOUN
ejpam-3764	158	5	is	be	AUX
ejpam-3764	158	6	satisfied	satisfied	ADJ
ejpam-3764	158	7	from	from	ADP
ejpam-3764	158	8	the	the	DET
ejpam-3764	158	9	definition	definition	NOUN
ejpam-3764	158	10	of	of	ADP
ejpam-3764	158	11	τω	τω	PUNCT
ejpam-3764	158	12	.	.	PUNCT
ejpam-3764	158	13	assuming	assume	VERB
ejpam-3764	158	14	that	that	SCONJ
ejpam-3764	158	15	b	b	NOUN
ejpam-3764	158	16	=	=	SYM
ejpam-3764	158	17	{	{	PUNCT
ejpam-3764	158	18	ω	ω	PROPN
ejpam-3764	158	19	,	,	PUNCT
ejpam-3764	158	20	a	a	PRON
ejpam-3764	158	21	,	,	PUNCT
ejpam-3764	158	22	a−1	a−1	PROPN
ejpam-3764	158	23	}	}	PUNCT
ejpam-3764	158	24	for	for	ADP
ejpam-3764	158	25	any	any	DET
ejpam-3764	158	26	a	a	DET
ejpam-3764	158	27	∈	∈	NOUN
ejpam-3764	158	28	aω	aω	X
ejpam-3764	158	29	\	\	PROPN
ejpam-3764	158	30	{	{	PUNCT
ejpam-3764	158	31	ω	ω	NOUN
ejpam-3764	158	32	}	}	PUNCT
ejpam-3764	158	33	which	which	PRON
ejpam-3764	158	34	has	have	VERB
ejpam-3764	158	35	a	a	DET
ejpam-3764	158	36	multiplicative	multiplicative	ADJ
ejpam-3764	158	37	inverse	inverse	NOUN
ejpam-3764	158	38	,	,	PUNCT
ejpam-3764	158	39	then	then	ADV
ejpam-3764	158	40	b	b	X
ejpam-3764	158	41	∈	∈	PROPN
ejpam-3764	158	42	τω	τω	ADP
ejpam-3764	158	43	,	,	PUNCT
ejpam-3764	158	44	because	because	SCONJ
ejpam-3764	158	45	ω	ω	PROPN
ejpam-3764	158	46	∈	∈	PROPN
ejpam-3764	158	47	b	b	PROPN
ejpam-3764	158	48	,	,	PUNCT
ejpam-3764	158	49	and	and	CCONJ
ejpam-3764	158	50	the	the	DET
ejpam-3764	158	51	element	element	NOUN
ejpam-3764	158	52	a	a	PRON
ejpam-3764	158	53	in	in	ADP
ejpam-3764	158	54	b	b	NOUN
ejpam-3764	158	55	its	its	PRON
ejpam-3764	158	56	multiplicative	multiplicative	ADJ
ejpam-3764	158	57	inverse	inverse	NOUN
ejpam-3764	158	58	exists	exist	VERB
ejpam-3764	158	59	in	in	ADP
ejpam-3764	158	60	b.	b.	PROPN
ejpam-3764	158	61	hence	hence	PROPN
ejpam-3764	158	62	,	,	PUNCT
ejpam-3764	158	63	b	b	PROPN
ejpam-3764	158	64	⊆	⊆	NUM
ejpam-3764	158	65	τω	τω	PROPN
ejpam-3764	158	66	.	.	PROPN
ejpam-3764	159	1	for	for	ADP
ejpam-3764	159	2	the	the	DET
ejpam-3764	159	3	second	second	ADJ
ejpam-3764	159	4	condition	condition	NOUN
ejpam-3764	159	5	,	,	PUNCT
ejpam-3764	159	6	let	let	VERB
ejpam-3764	159	7	a	a	DET
ejpam-3764	159	8	∈	∈	NOUN
ejpam-3764	159	9	aω	aω	AUX
ejpam-3764	159	10	be	be	AUX
ejpam-3764	159	11	arbitrary	arbitrary	ADJ
ejpam-3764	159	12	and	and	CCONJ
ejpam-3764	159	13	let	let	VERB
ejpam-3764	159	14	u	u	PRON
ejpam-3764	159	15	be	be	AUX
ejpam-3764	159	16	any	any	DET
ejpam-3764	159	17	open	open	ADJ
ejpam-3764	159	18	neighborhood	neighborhood	NOUN
ejpam-3764	159	19	of	of	ADP
ejpam-3764	159	20	a	a	PRON
ejpam-3764	159	21	in	in	ADP
ejpam-3764	159	22	aω	aω	PROPN
ejpam-3764	159	23	.	.	PUNCT
ejpam-3764	160	1	then	then	ADV
ejpam-3764	160	2	we	we	PRON
ejpam-3764	160	3	have	have	VERB
ejpam-3764	160	4	two	two	NUM
ejpam-3764	160	5	cases	case	NOUN
ejpam-3764	160	6	:	:	PUNCT
ejpam-3764	160	7	case	case	NOUN
ejpam-3764	160	8	1	1	NUM
ejpam-3764	160	9	:	:	PUNCT
ejpam-3764	160	10	if	if	SCONJ
ejpam-3764	160	11	a	a	DET
ejpam-3764	160	12	=	=	SYM
ejpam-3764	160	13	ω	ω	PROPN
ejpam-3764	160	14	,	,	PUNCT
ejpam-3764	160	15	then	then	ADV
ejpam-3764	160	16	we	we	PRON
ejpam-3764	160	17	have	have	VERB
ejpam-3764	160	18	b	b	NOUN
ejpam-3764	160	19	=	=	SYM
ejpam-3764	160	20	{	{	PUNCT
ejpam-3764	160	21	ω	ω	PROPN
ejpam-3764	160	22	}	}	PUNCT
ejpam-3764	160	23	∈	∈	PROPN
ejpam-3764	160	24	b	b	NOUN
ejpam-3764	160	25	,	,	PUNCT
ejpam-3764	160	26	where	where	SCONJ
ejpam-3764	160	27	ω	ω	PROPN
ejpam-3764	160	28	∈	∈	PROPN
ejpam-3764	160	29	b	b	PROPN
ejpam-3764	160	30	⊆	⊆	NUM
ejpam-3764	160	31	u	u	NOUN
ejpam-3764	160	32	,	,	PUNCT
ejpam-3764	160	33	because	because	SCONJ
ejpam-3764	160	34	the	the	DET
ejpam-3764	160	35	smallest	small	ADJ
ejpam-3764	160	36	open	open	ADJ
ejpam-3764	160	37	neighborhood	neighborhood	NOUN
ejpam-3764	160	38	in	in	ADP
ejpam-3764	160	39	aω	aω	NOUN
ejpam-3764	160	40	containing	contain	VERB
ejpam-3764	160	41	ω	ω	PROPN
ejpam-3764	160	42	is	be	AUX
ejpam-3764	160	43	{	{	PUNCT
ejpam-3764	160	44	ω	ω	NOUN
ejpam-3764	160	45	}	}	PUNCT
ejpam-3764	160	46	.	.	PUNCT
ejpam-3764	161	1	case	case	NOUN
ejpam-3764	161	2	2	2	NUM
ejpam-3764	161	3	:	:	PUNCT
ejpam-3764	161	4	let	let	VERB
ejpam-3764	161	5	a	a	DET
ejpam-3764	161	6	6=	6=	PROPN
ejpam-3764	161	7	ω	ω	NUM
ejpam-3764	161	8	subcase	subcase	NOUN
ejpam-3764	161	9	2.1	2.1	NUM
ejpam-3764	161	10	:	:	PUNCT
ejpam-3764	161	11	if	if	SCONJ
ejpam-3764	161	12	a	a	PRON
ejpam-3764	161	13	has	have	VERB
ejpam-3764	161	14	no	no	DET
ejpam-3764	161	15	multiplicative	multiplicative	ADJ
ejpam-3764	161	16	inverse	inverse	NOUN
ejpam-3764	161	17	,	,	PUNCT
ejpam-3764	161	18	then	then	ADV
ejpam-3764	161	19	there	there	PRON
ejpam-3764	161	20	exists	exist	VERB
ejpam-3764	161	21	b	b	NOUN
ejpam-3764	161	22	=	=	SYM
ejpam-3764	161	23	aω	aω	PROPN
ejpam-3764	161	24	∈	∈	PROPN
ejpam-3764	161	25	b	b	PROPN
ejpam-3764	161	26	,	,	PUNCT
ejpam-3764	161	27	such	such	ADJ
ejpam-3764	161	28	that	that	SCONJ
ejpam-3764	161	29	a	a	DET
ejpam-3764	161	30	∈	∈	PROPN
ejpam-3764	161	31	b	b	X
ejpam-3764	161	32	⊆	⊆	NUM
ejpam-3764	161	33	u	u	NOUN
ejpam-3764	161	34	,	,	PUNCT
ejpam-3764	161	35	because	because	SCONJ
ejpam-3764	161	36	the	the	DET
ejpam-3764	161	37	smallest	small	ADJ
ejpam-3764	161	38	open	open	ADJ
ejpam-3764	161	39	neighborhood	neighborhood	NOUN
ejpam-3764	161	40	in	in	ADP
ejpam-3764	161	41	aω	aω	NOUN
ejpam-3764	161	42	containing	contain	VERB
ejpam-3764	161	43	a	a	PRON
ejpam-3764	161	44	is	be	AUX
ejpam-3764	161	45	aω	aω	PROPN
ejpam-3764	161	46	.	.	PROPN
ejpam-3764	161	47	subcase	subcase	PROPN
ejpam-3764	161	48	2.2	2.2	NUM
ejpam-3764	161	49	:	:	PUNCT
ejpam-3764	161	50	if	if	SCONJ
ejpam-3764	161	51	a	a	PRON
ejpam-3764	161	52	has	have	VERB
ejpam-3764	161	53	a	a	DET
ejpam-3764	161	54	multiplicative	multiplicative	ADJ
ejpam-3764	161	55	inverse	inverse	NOUN
ejpam-3764	161	56	,	,	PUNCT
ejpam-3764	161	57	then	then	ADV
ejpam-3764	161	58	there	there	PRON
ejpam-3764	161	59	exists	exist	VERB
ejpam-3764	161	60	b	b	NOUN
ejpam-3764	161	61	=	=	SYM
ejpam-3764	161	62	{	{	PUNCT
ejpam-3764	161	63	ω	ω	PROPN
ejpam-3764	161	64	,	,	PUNCT
ejpam-3764	161	65	a	a	PRON
ejpam-3764	161	66	,	,	PUNCT
ejpam-3764	161	67	a−1	a−1	PROPN
ejpam-3764	161	68	}	}	PUNCT
ejpam-3764	161	69	∈	∈	PROPN
ejpam-3764	161	70	b	b	NOUN
ejpam-3764	161	71	,	,	PUNCT
ejpam-3764	161	72	such	such	ADJ
ejpam-3764	161	73	that	that	SCONJ
ejpam-3764	161	74	a	a	DET
ejpam-3764	161	75	∈	∈	PROPN
ejpam-3764	161	76	b	b	X
ejpam-3764	161	77	⊆	⊆	NUM
ejpam-3764	161	78	u	u	NOUN
ejpam-3764	161	79	,	,	PUNCT
ejpam-3764	161	80	because	because	SCONJ
ejpam-3764	161	81	the	the	DET
ejpam-3764	161	82	smallest	small	ADJ
ejpam-3764	161	83	open	open	ADJ
ejpam-3764	161	84	neighborhood	neighborhood	NOUN
ejpam-3764	161	85	in	in	ADP
ejpam-3764	161	86	aω	aω	NOUN
ejpam-3764	161	87	containing	contain	VERB
ejpam-3764	161	88	a	a	PRON
ejpam-3764	161	89	is	be	AUX
ejpam-3764	161	90	{	{	PUNCT
ejpam-3764	161	91	ω	ω	PROPN
ejpam-3764	161	92	,	,	PUNCT
ejpam-3764	161	93	a	a	PRON
ejpam-3764	161	94	,	,	PUNCT
ejpam-3764	161	95	a−1	a−1	PROPN
ejpam-3764	161	96	}	}	PUNCT
ejpam-3764	161	97	.	.	PUNCT
ejpam-3764	162	1	therefore	therefore	ADV
ejpam-3764	162	2	,	,	PUNCT
ejpam-3764	162	3	b	b	PROPN
ejpam-3764	162	4	is	be	AUX
ejpam-3764	162	5	a	a	DET
ejpam-3764	162	6	base	base	NOUN
ejpam-3764	162	7	for	for	ADP
ejpam-3764	162	8	the	the	DET
ejpam-3764	162	9	omega	omega	NOUN
ejpam-3764	162	10	topological	topological	ADJ
ejpam-3764	162	11	space	space	NOUN
ejpam-3764	162	12	(	(	PUNCT
ejpam-3764	162	13	aω	aω	PROPN
ejpam-3764	162	14	,	,	PUNCT
ejpam-3764	162	15	τω	τω	NOUN
ejpam-3764	162	16	)	)	PUNCT
ejpam-3764	162	17	.	.	PUNCT
ejpam-3764	163	1	corollary	corollary	ADJ
ejpam-3764	163	2	2	2	NUM
ejpam-3764	163	3	.	.	PUNCT
ejpam-3764	164	1	if	if	SCONJ
ejpam-3764	164	2	(	(	PUNCT
ejpam-3764	164	3	aω	aω	PROPN
ejpam-3764	164	4	\	\	PROPN
ejpam-3764	164	5	{	{	PUNCT
ejpam-3764	164	6	ω},⊗	ω},⊗	NOUN
ejpam-3764	164	7	)	)	PUNCT
ejpam-3764	164	8	be	be	VERB
ejpam-3764	164	9	a	a	DET
ejpam-3764	164	10	group	group	NOUN
ejpam-3764	164	11	,	,	PUNCT
ejpam-3764	164	12	then	then	ADV
ejpam-3764	164	13	the	the	DET
ejpam-3764	164	14	omega	omega	NOUN
ejpam-3764	164	15	topological	topological	ADJ
ejpam-3764	164	16	space	space	NOUN
ejpam-3764	164	17	(	(	PUNCT
ejpam-3764	164	18	aω	aω	PROPN
ejpam-3764	164	19	,	,	PUNCT
ejpam-3764	164	20	τω	τω	X
ejpam-3764	164	21	)	)	PUNCT
ejpam-3764	164	22	has	have	VERB
ejpam-3764	164	23	a	a	DET
ejpam-3764	164	24	base	base	NOUN
ejpam-3764	164	25	b	b	NOUN
ejpam-3764	164	26	=	=	PRON
ejpam-3764	164	27	{	{	PUNCT
ejpam-3764	164	28	{	{	PUNCT
ejpam-3764	164	29	ω	ω	NOUN
ejpam-3764	164	30	}	}	PUNCT
ejpam-3764	164	31	,	,	PUNCT
ejpam-3764	164	32	{	{	PUNCT
ejpam-3764	164	33	ω	ω	NOUN
ejpam-3764	164	34	,	,	PUNCT
ejpam-3764	164	35	a	a	PRON
ejpam-3764	164	36	,	,	PUNCT
ejpam-3764	164	37	a−1	a−1	PROPN
ejpam-3764	164	38	}	}	PUNCT
ejpam-3764	164	39	:	:	PUNCT
ejpam-3764	164	40	a	a	DET
ejpam-3764	164	41	∈	∈	NOUN
ejpam-3764	164	42	aω	aω	X
ejpam-3764	164	43	\	\	PROPN
ejpam-3764	164	44	{	{	PUNCT
ejpam-3764	164	45	ω	ω	NOUN
ejpam-3764	164	46	}	}	PUNCT
ejpam-3764	164	47	}	}	PUNCT
ejpam-3764	164	48	.	.	PUNCT
ejpam-3764	165	1	m.	m.	PROPN
ejpam-3764	165	2	alqahtani	alqahtani	PROPN
ejpam-3764	165	3	,	,	PUNCT
ejpam-3764	165	4	c.	c.	PROPN
ejpam-3764	165	5	özel	özel	PROPN
ejpam-3764	165	6	,	,	PUNCT
ejpam-3764	165	7	i.	i.	PROPN
ejpam-3764	165	8	alshammari	alshammari	PROPN
ejpam-3764	165	9	/	/	SYM
ejpam-3764	165	10	eur	eur	PROPN
ejpam-3764	165	11	.	.	PUNCT
ejpam-3764	166	1	j.	j.	PROPN
ejpam-3764	166	2	pure	pure	PROPN
ejpam-3764	166	3	appl	appl	PROPN
ejpam-3764	166	4	.	.	PROPN
ejpam-3764	166	5	math	math	PROPN
ejpam-3764	166	6	,	,	PUNCT
ejpam-3764	166	7	13	13	NUM
ejpam-3764	166	8	(	(	PUNCT
ejpam-3764	166	9	3	3	NUM
ejpam-3764	166	10	)	)	PUNCT
ejpam-3764	166	11	(	(	PUNCT
ejpam-3764	166	12	2020	2020	NUM
ejpam-3764	166	13	)	)	PUNCT
ejpam-3764	166	14	,	,	PUNCT
ejpam-3764	166	15	513	513	NUM
ejpam-3764	166	16	-	-	SYM
ejpam-3764	166	17	528	528	NUM
ejpam-3764	166	18	519	519	NUM
ejpam-3764	166	19	proposition	proposition	NOUN
ejpam-3764	166	20	3	3	NUM
ejpam-3764	166	21	.	.	PUNCT
ejpam-3764	167	1	if	if	SCONJ
ejpam-3764	167	2	aω	aω	PROPN
ejpam-3764	167	3	has	have	VERB
ejpam-3764	167	4	a	a	DET
ejpam-3764	167	5	finite	finite	ADJ
ejpam-3764	167	6	number	number	NOUN
ejpam-3764	167	7	of	of	ADP
ejpam-3764	167	8	elements	element	NOUN
ejpam-3764	167	9	,	,	PUNCT
ejpam-3764	167	10	which	which	PRON
ejpam-3764	167	11	have	have	VERB
ejpam-3764	167	12	a	a	DET
ejpam-3764	167	13	multiplicative	multiplicative	ADJ
ejpam-3764	167	14	inverses	inverse	NOUN
ejpam-3764	167	15	,	,	PUNCT
ejpam-3764	167	16	then	then	ADV
ejpam-3764	167	17	the	the	DET
ejpam-3764	167	18	omega	omega	NOUN
ejpam-3764	167	19	topological	topological	ADJ
ejpam-3764	167	20	space	space	NOUN
ejpam-3764	167	21	(	(	PUNCT
ejpam-3764	167	22	aω	aω	PROPN
ejpam-3764	167	23	,	,	PUNCT
ejpam-3764	167	24	τω	τω	INTJ
ejpam-3764	167	25	)	)	PUNCT
ejpam-3764	167	26	is	be	AUX
ejpam-3764	167	27	second	second	ADV
ejpam-3764	167	28	countable	countable	ADJ
ejpam-3764	167	29	.	.	PUNCT
ejpam-3764	168	1	proof	proof	NOUN
ejpam-3764	168	2	.	.	PUNCT
ejpam-3764	169	1	suppose	suppose	VERB
ejpam-3764	169	2	that	that	SCONJ
ejpam-3764	169	3	a1	a1	NOUN
ejpam-3764	169	4	,	,	PUNCT
ejpam-3764	169	5	a2	a2	PROPN
ejpam-3764	169	6	,	,	PUNCT
ejpam-3764	169	7	·	·	PUNCT
ejpam-3764	169	8	·	·	PUNCT
ejpam-3764	169	9	·	·	PUNCT
ejpam-3764	169	10	,	,	PUNCT
ejpam-3764	169	11	am	be	AUX
ejpam-3764	169	12	,	,	PUNCT
ejpam-3764	169	13	where	where	SCONJ
ejpam-3764	169	14	m	m	PROPN
ejpam-3764	169	15	∈	∈	NOUN
ejpam-3764	169	16	z+	z+	NUM
ejpam-3764	169	17	are	be	AUX
ejpam-3764	169	18	the	the	DET
ejpam-3764	169	19	finite	finite	ADJ
ejpam-3764	169	20	number	number	NOUN
ejpam-3764	169	21	of	of	ADP
ejpam-3764	169	22	elements	element	NOUN
ejpam-3764	169	23	in	in	ADP
ejpam-3764	169	24	aω	aω	PROPN
ejpam-3764	169	25	,	,	PUNCT
ejpam-3764	169	26	which	which	PRON
ejpam-3764	169	27	have	have	VERB
ejpam-3764	169	28	a	a	DET
ejpam-3764	169	29	multiplicative	multiplicative	ADJ
ejpam-3764	169	30	inverses	inverse	NOUN
ejpam-3764	169	31	.	.	PUNCT
ejpam-3764	170	1	then	then	ADV
ejpam-3764	170	2	b	b	X
ejpam-3764	170	3	=	=	PRON
ejpam-3764	170	4	{	{	PUNCT
ejpam-3764	170	5	aω	aω	PROPN
ejpam-3764	170	6	,	,	PUNCT
ejpam-3764	170	7	{	{	PUNCT
ejpam-3764	170	8	ω	ω	NOUN
ejpam-3764	170	9	}	}	PUNCT
ejpam-3764	170	10	,	,	PUNCT
ejpam-3764	170	11	{	{	PUNCT
ejpam-3764	170	12	ω	ω	NOUN
ejpam-3764	170	13	,	,	PUNCT
ejpam-3764	170	14	a1	a1	NOUN
ejpam-3764	170	15	,	,	PUNCT
ejpam-3764	170	16	a−11	a−11	PUNCT
ejpam-3764	170	17	}	}	PUNCT
ejpam-3764	170	18	,	,	PUNCT
ejpam-3764	170	19	·	·	PUNCT
ejpam-3764	170	20	·	·	PUNCT
ejpam-3764	170	21	·	·	PUNCT
ejpam-3764	170	22	,	,	PUNCT
ejpam-3764	170	23	{	{	PUNCT
ejpam-3764	170	24	ω	ω	NOUN
ejpam-3764	170	25	,	,	PUNCT
ejpam-3764	170	26	am	be	AUX
ejpam-3764	170	27	,	,	PUNCT
ejpam-3764	170	28	a−1	a−1	PROPN
ejpam-3764	170	29	m	m	PRON
ejpam-3764	170	30	}	}	PUNCT
ejpam-3764	170	31	}	}	PUNCT
ejpam-3764	170	32	is	be	AUX
ejpam-3764	170	33	a	a	DET
ejpam-3764	170	34	countable	countable	ADJ
ejpam-3764	170	35	base	base	NOUN
ejpam-3764	170	36	for	for	ADP
ejpam-3764	170	37	(	(	PUNCT
ejpam-3764	170	38	aω	aω	PROPN
ejpam-3764	170	39	,	,	PUNCT
ejpam-3764	170	40	τω	τω	NOUN
ejpam-3764	170	41	)	)	PUNCT
ejpam-3764	170	42	.	.	PUNCT
ejpam-3764	171	1	proposition	proposition	NOUN
ejpam-3764	171	2	4	4	NUM
ejpam-3764	171	3	.	.	PUNCT
ejpam-3764	172	1	the	the	DET
ejpam-3764	172	2	omega	omega	NOUN
ejpam-3764	172	3	topological	topological	ADJ
ejpam-3764	172	4	space	space	NOUN
ejpam-3764	172	5	(	(	PUNCT
ejpam-3764	172	6	aω	aω	PROPN
ejpam-3764	172	7	,	,	PUNCT
ejpam-3764	172	8	τω	τω	INTJ
ejpam-3764	172	9	)	)	PUNCT
ejpam-3764	172	10	is	be	AUX
ejpam-3764	172	11	first	first	ADV
ejpam-3764	172	12	countable	countable	ADJ
ejpam-3764	172	13	.	.	PUNCT
ejpam-3764	173	1	proof	proof	NOUN
ejpam-3764	173	2	.	.	PUNCT
ejpam-3764	174	1	let	let	VERB
ejpam-3764	174	2	a	a	DET
ejpam-3764	174	3	∈	∈	NOUN
ejpam-3764	174	4	aω	aω	AUX
ejpam-3764	174	5	be	be	AUX
ejpam-3764	174	6	arbitrary	arbitrary	ADJ
ejpam-3764	174	7	.	.	PUNCT
ejpam-3764	175	1	if	if	SCONJ
ejpam-3764	175	2	a	a	DET
ejpam-3764	175	3	=	=	SYM
ejpam-3764	175	4	ω	ω	PROPN
ejpam-3764	175	5	,	,	PUNCT
ejpam-3764	175	6	then	then	ADV
ejpam-3764	175	7	b(ω	b(ω	ADV
ejpam-3764	175	8	)	)	PUNCT
ejpam-3764	175	9	=	=	PRON
ejpam-3764	175	10	{	{	PUNCT
ejpam-3764	175	11	{	{	PUNCT
ejpam-3764	175	12	ω	ω	NOUN
ejpam-3764	175	13	}	}	PUNCT
ejpam-3764	175	14	}	}	PUNCT
ejpam-3764	175	15	is	be	AUX
ejpam-3764	175	16	a	a	DET
ejpam-3764	175	17	countable	countable	ADJ
ejpam-3764	175	18	local	local	ADJ
ejpam-3764	175	19	base	base	NOUN
ejpam-3764	175	20	at	at	ADP
ejpam-3764	175	21	ω	ω	PROPN
ejpam-3764	175	22	.	.	PUNCT
ejpam-3764	176	1	assume	assume	VERB
ejpam-3764	177	1	that	that	SCONJ
ejpam-3764	177	2	,	,	PUNCT
ejpam-3764	177	3	a	a	DET
ejpam-3764	177	4	6=	6=	PROPN
ejpam-3764	177	5	ω	ω	PROPN
ejpam-3764	177	6	.	.	PUNCT
ejpam-3764	177	7	case	case	NOUN
ejpam-3764	177	8	1	1	NUM
ejpam-3764	177	9	:	:	PUNCT
ejpam-3764	177	10	if	if	SCONJ
ejpam-3764	177	11	a	a	PRON
ejpam-3764	177	12	has	have	VERB
ejpam-3764	177	13	a	a	DET
ejpam-3764	177	14	multiplicative	multiplicative	ADJ
ejpam-3764	177	15	inverse	inverse	NOUN
ejpam-3764	177	16	,	,	PUNCT
ejpam-3764	177	17	then	then	ADV
ejpam-3764	177	18	b(a	b(a	NOUN
ejpam-3764	177	19	)	)	PUNCT
ejpam-3764	177	20	=	=	PRON
ejpam-3764	177	21	{	{	PUNCT
ejpam-3764	177	22	{	{	PUNCT
ejpam-3764	177	23	ω	ω	PROPN
ejpam-3764	177	24	,	,	PUNCT
ejpam-3764	177	25	a	a	PRON
ejpam-3764	177	26	,	,	PUNCT
ejpam-3764	177	27	a−1	a−1	PROPN
ejpam-3764	177	28	}	}	PUNCT
ejpam-3764	177	29	}	}	PUNCT
ejpam-3764	177	30	is	be	AUX
ejpam-3764	177	31	a	a	DET
ejpam-3764	177	32	countable	countable	ADJ
ejpam-3764	177	33	local	local	ADJ
ejpam-3764	177	34	base	base	NOUN
ejpam-3764	177	35	at	at	ADP
ejpam-3764	177	36	a.	a.	NOUN
ejpam-3764	177	37	case	case	NOUN
ejpam-3764	177	38	2	2	NUM
ejpam-3764	177	39	:	:	PUNCT
ejpam-3764	177	40	if	if	SCONJ
ejpam-3764	177	41	a	a	PRON
ejpam-3764	177	42	has	have	VERB
ejpam-3764	177	43	no	no	DET
ejpam-3764	177	44	multiplicative	multiplicative	ADJ
ejpam-3764	177	45	inverse	inverse	NOUN
ejpam-3764	177	46	,	,	PUNCT
ejpam-3764	177	47	then	then	ADV
ejpam-3764	177	48	b(a	b(a	NOUN
ejpam-3764	177	49	)	)	PUNCT
ejpam-3764	177	50	=	=	PRON
ejpam-3764	177	51	{	{	PUNCT
ejpam-3764	177	52	aω	aω	NOUN
ejpam-3764	177	53	}	}	PUNCT
ejpam-3764	177	54	is	be	AUX
ejpam-3764	177	55	a	a	DET
ejpam-3764	177	56	countable	countable	ADJ
ejpam-3764	177	57	local	local	ADJ
ejpam-3764	177	58	base	base	NOUN
ejpam-3764	177	59	at	at	ADP
ejpam-3764	177	60	a.	a.	NOUN
ejpam-3764	177	61	hence	hence	ADV
ejpam-3764	177	62	for	for	ADP
ejpam-3764	177	63	any	any	DET
ejpam-3764	177	64	a	a	DET
ejpam-3764	177	65	∈	∈	PROPN
ejpam-3764	177	66	aω	aω	NOUN
ejpam-3764	177	67	,	,	PUNCT
ejpam-3764	177	68	there	there	PRON
ejpam-3764	177	69	exists	exist	VERB
ejpam-3764	177	70	a	a	DET
ejpam-3764	177	71	countable	countable	ADJ
ejpam-3764	177	72	local	local	ADJ
ejpam-3764	177	73	base	base	NOUN
ejpam-3764	177	74	at	at	ADP
ejpam-3764	177	75	a.	a.	NOUN
ejpam-3764	177	76	then	then	ADV
ejpam-3764	177	77	(	(	PUNCT
ejpam-3764	177	78	aω	aω	INTJ
ejpam-3764	177	79	,	,	PUNCT
ejpam-3764	177	80	τω	τω	INTJ
ejpam-3764	177	81	)	)	PUNCT
ejpam-3764	177	82	is	be	AUX
ejpam-3764	177	83	first	first	ADV
ejpam-3764	177	84	countable	countable	ADJ
ejpam-3764	177	85	.	.	PUNCT
ejpam-3764	178	1	proposition	proposition	NOUN
ejpam-3764	178	2	5	5	NUM
ejpam-3764	178	3	.	.	PUNCT
ejpam-3764	179	1	the	the	DET
ejpam-3764	179	2	omega	omega	NOUN
ejpam-3764	179	3	topological	topological	ADJ
ejpam-3764	179	4	space	space	NOUN
ejpam-3764	179	5	(	(	PUNCT
ejpam-3764	179	6	aω	aω	PROPN
ejpam-3764	179	7	,	,	PUNCT
ejpam-3764	179	8	τω	τω	INTJ
ejpam-3764	179	9	)	)	PUNCT
ejpam-3764	179	10	is	be	AUX
ejpam-3764	179	11	separable	separable	ADJ
ejpam-3764	179	12	.	.	PUNCT
ejpam-3764	180	1	proof	proof	NOUN
ejpam-3764	180	2	.	.	PUNCT
ejpam-3764	181	1	there	there	PRON
ejpam-3764	181	2	exist	exist	VERB
ejpam-3764	181	3	{	{	PUNCT
ejpam-3764	181	4	ω	ω	NOUN
ejpam-3764	181	5	}	}	PUNCT
ejpam-3764	181	6	⊆	⊆	NUM
ejpam-3764	181	7	aω	aω	NOUN
ejpam-3764	181	8	,	,	PUNCT
ejpam-3764	181	9	such	such	ADJ
ejpam-3764	181	10	that	that	SCONJ
ejpam-3764	181	11	for	for	ADP
ejpam-3764	181	12	any	any	DET
ejpam-3764	181	13	nonempty	nonempty	ADJ
ejpam-3764	181	14	u	u	NOUN
ejpam-3764	181	15	∈	∈	NOUN
ejpam-3764	181	16	τω	τω	INTJ
ejpam-3764	181	17	,	,	PUNCT
ejpam-3764	181	18	we	we	PRON
ejpam-3764	181	19	have	have	VERB
ejpam-3764	181	20	u∩{ω	u∩{ω	PROPN
ejpam-3764	181	21	}	}	PUNCT
ejpam-3764	181	22	6=	6=	NOUN
ejpam-3764	181	23	∅	∅	NOUN
ejpam-3764	181	24	,	,	PUNCT
ejpam-3764	181	25	because	because	SCONJ
ejpam-3764	181	26	any	any	DET
ejpam-3764	181	27	nonempty	nonempty	ADJ
ejpam-3764	181	28	open	open	ADJ
ejpam-3764	181	29	set	set	VERB
ejpam-3764	181	30	in	in	ADP
ejpam-3764	181	31	(	(	PUNCT
ejpam-3764	181	32	aω	aω	PROPN
ejpam-3764	181	33	,	,	PUNCT
ejpam-3764	181	34	τω	τω	X
ejpam-3764	181	35	)	)	PUNCT
ejpam-3764	181	36	must	must	AUX
ejpam-3764	181	37	be	be	AUX
ejpam-3764	181	38	contaning	contane	VERB
ejpam-3764	181	39	ω	ω	NUM
ejpam-3764	181	40	.	.	PUNCT
ejpam-3764	182	1	hence	hence	ADV
ejpam-3764	182	2	,	,	PUNCT
ejpam-3764	182	3	{	{	PUNCT
ejpam-3764	182	4	ω	ω	NOUN
ejpam-3764	182	5	}	}	PUNCT
ejpam-3764	182	6	=	=	SYM
ejpam-3764	182	7	aω	aω	X
ejpam-3764	182	8	(	(	PUNCT
ejpam-3764	182	9	which	which	PRON
ejpam-3764	182	10	means	mean	VERB
ejpam-3764	182	11	{	{	PUNCT
ejpam-3764	182	12	ω	ω	NOUN
ejpam-3764	182	13	}	}	PUNCT
ejpam-3764	182	14	is	be	AUX
ejpam-3764	182	15	a	a	DET
ejpam-3764	182	16	dense	dense	ADJ
ejpam-3764	182	17	subset	subset	NOUN
ejpam-3764	182	18	of	of	ADP
ejpam-3764	182	19	aω	aω	PROPN
ejpam-3764	182	20	)	)	PUNCT
ejpam-3764	182	21	.	.	PUNCT
ejpam-3764	183	1	then	then	ADV
ejpam-3764	183	2	aω	aω	INTJ
ejpam-3764	183	3	has	have	VERB
ejpam-3764	183	4	a	a	DET
ejpam-3764	183	5	countable	countable	ADJ
ejpam-3764	183	6	dense	dense	ADJ
ejpam-3764	183	7	subset	subset	NOUN
ejpam-3764	183	8	.	.	PUNCT
ejpam-3764	184	1	therefore	therefore	ADV
ejpam-3764	184	2	,	,	PUNCT
ejpam-3764	184	3	(	(	PUNCT
ejpam-3764	184	4	aω	aω	INTJ
ejpam-3764	184	5	,	,	PUNCT
ejpam-3764	184	6	τω	τω	INTJ
ejpam-3764	184	7	)	)	PUNCT
ejpam-3764	184	8	is	be	AUX
ejpam-3764	184	9	separable	separable	ADJ
ejpam-3764	184	10	.	.	PUNCT
ejpam-3764	185	1	let	let	VERB
ejpam-3764	185	2	us	we	PRON
ejpam-3764	185	3	recall	recall	VERB
ejpam-3764	185	4	this	this	DET
ejpam-3764	185	5	definition	definition	NOUN
ejpam-3764	185	6	.	.	PUNCT
ejpam-3764	186	1	definition	definition	NOUN
ejpam-3764	186	2	2	2	NUM
ejpam-3764	186	3	.	.	PUNCT
ejpam-3764	187	1	a	a	DET
ejpam-3764	187	2	topological	topological	ADJ
ejpam-3764	187	3	space	space	NOUN
ejpam-3764	187	4	x	x	PRON
ejpam-3764	187	5	is	be	AUX
ejpam-3764	187	6	called	call	VERB
ejpam-3764	187	7	hyperconnected	hyperconnecte	VERB
ejpam-3764	187	8	if	if	SCONJ
ejpam-3764	187	9	every	every	DET
ejpam-3764	187	10	non	non	ADJ
ejpam-3764	187	11	-	-	ADJ
ejpam-3764	187	12	empty	empty	ADJ
ejpam-3764	187	13	open	open	ADJ
ejpam-3764	187	14	subset	subset	NOUN
ejpam-3764	187	15	is	be	AUX
ejpam-3764	187	16	dense	dense	ADJ
ejpam-3764	187	17	in	in	ADP
ejpam-3764	187	18	x.	x.	NOUN
ejpam-3764	187	19	proposition	proposition	NOUN
ejpam-3764	187	20	6	6	NUM
ejpam-3764	187	21	.	.	PUNCT
ejpam-3764	188	1	the	the	DET
ejpam-3764	188	2	omega	omega	NOUN
ejpam-3764	188	3	topological	topological	ADJ
ejpam-3764	188	4	space	space	NOUN
ejpam-3764	188	5	(	(	PUNCT
ejpam-3764	188	6	aω	aω	PROPN
ejpam-3764	188	7	,	,	PUNCT
ejpam-3764	188	8	τω	τω	INTJ
ejpam-3764	188	9	)	)	PUNCT
ejpam-3764	188	10	is	be	AUX
ejpam-3764	188	11	hyperconnected	hyperconnecte	VERB
ejpam-3764	188	12	.	.	PUNCT
ejpam-3764	189	1	proof	proof	NOUN
ejpam-3764	189	2	.	.	PUNCT
ejpam-3764	190	1	if	if	SCONJ
ejpam-3764	190	2	aω	aω	PROPN
ejpam-3764	190	3	is	be	AUX
ejpam-3764	190	4	singleton	singleton	NOUN
ejpam-3764	190	5	,	,	PUNCT
ejpam-3764	190	6	then	then	ADV
ejpam-3764	190	7	(	(	PUNCT
ejpam-3764	190	8	aω	aω	INTJ
ejpam-3764	190	9	,	,	PUNCT
ejpam-3764	190	10	τω	τω	INTJ
ejpam-3764	190	11	)	)	PUNCT
ejpam-3764	190	12	is	be	AUX
ejpam-3764	190	13	hyperconnected	hyperconnecte	VERB
ejpam-3764	190	14	.	.	PUNCT
ejpam-3764	191	1	suppose	suppose	VERB
ejpam-3764	191	2	that	that	SCONJ
ejpam-3764	191	3	aω	aω	PROPN
ejpam-3764	191	4	,	,	PUNCT
ejpam-3764	191	5	which	which	PRON
ejpam-3764	191	6	has	have	VERB
ejpam-3764	191	7	more	more	ADJ
ejpam-3764	191	8	than	than	ADP
ejpam-3764	191	9	one	one	NUM
ejpam-3764	191	10	element	element	NOUN
ejpam-3764	191	11	.	.	PUNCT
ejpam-3764	192	1	let	let	VERB
ejpam-3764	192	2	u	u	PRON
ejpam-3764	192	3	be	be	AUX
ejpam-3764	192	4	arbitrary	arbitrary	ADJ
ejpam-3764	192	5	non	non	ADJ
ejpam-3764	192	6	-	-	ADJ
ejpam-3764	192	7	empty	empty	ADJ
ejpam-3764	192	8	open	open	ADJ
ejpam-3764	192	9	subset	subset	NOUN
ejpam-3764	192	10	of	of	ADP
ejpam-3764	192	11	aω	aω	PROPN
ejpam-3764	192	12	,	,	PUNCT
ejpam-3764	192	13	then	then	ADV
ejpam-3764	192	14	u	u	NOUN
ejpam-3764	192	15	intersects	intersect	VERB
ejpam-3764	192	16	every	every	DET
ejpam-3764	192	17	non	non	ADJ
ejpam-3764	192	18	-	-	ADJ
ejpam-3764	192	19	empty	empty	ADJ
ejpam-3764	192	20	open	open	ADJ
ejpam-3764	192	21	subset	subset	NOUN
ejpam-3764	192	22	of	of	ADP
ejpam-3764	192	23	aω	aω	PROPN
ejpam-3764	192	24	,	,	PUNCT
ejpam-3764	192	25	because	because	SCONJ
ejpam-3764	192	26	any	any	DET
ejpam-3764	192	27	non	non	ADJ
ejpam-3764	192	28	-	-	ADJ
ejpam-3764	192	29	empty	empty	ADJ
ejpam-3764	192	30	open	open	ADJ
ejpam-3764	192	31	subset	subset	NOUN
ejpam-3764	192	32	of	of	ADP
ejpam-3764	192	33	aω	aω	PROPN
ejpam-3764	192	34	contained	contain	VERB
ejpam-3764	192	35	ω	ω	NOUN
ejpam-3764	192	36	.	.	PUNCT
ejpam-3764	193	1	hence	hence	ADV
ejpam-3764	193	2	u	u	NOUN
ejpam-3764	193	3	is	be	AUX
ejpam-3764	193	4	dense	dense	ADJ
ejpam-3764	193	5	in	in	ADP
ejpam-3764	193	6	aω	aω	PROPN
ejpam-3764	193	7	.	.	PUNCT
ejpam-3764	194	1	since	since	SCONJ
ejpam-3764	194	2	u	u	PRON
ejpam-3764	194	3	was	be	AUX
ejpam-3764	194	4	chosen	choose	VERB
ejpam-3764	194	5	arbitrary	arbitrary	ADJ
ejpam-3764	194	6	,	,	PUNCT
ejpam-3764	194	7	then	then	ADV
ejpam-3764	194	8	every	every	DET
ejpam-3764	194	9	non	non	ADJ
ejpam-3764	194	10	-	-	ADJ
ejpam-3764	194	11	empty	empty	ADJ
ejpam-3764	194	12	open	open	ADJ
ejpam-3764	194	13	subset	subset	NOUN
ejpam-3764	194	14	of	of	ADP
ejpam-3764	194	15	aω	aω	PROPN
ejpam-3764	194	16	is	be	AUX
ejpam-3764	194	17	dense	dense	ADJ
ejpam-3764	194	18	.	.	PUNCT
ejpam-3764	195	1	therefore	therefore	ADV
ejpam-3764	195	2	(	(	PUNCT
ejpam-3764	195	3	aω	aω	INTJ
ejpam-3764	195	4	,	,	PUNCT
ejpam-3764	195	5	τω	τω	INTJ
ejpam-3764	195	6	)	)	PUNCT
ejpam-3764	195	7	is	be	AUX
ejpam-3764	195	8	hyperconnected	hyperconnecte	VERB
ejpam-3764	195	9	.	.	PUNCT
ejpam-3764	196	1	since	since	SCONJ
ejpam-3764	196	2	any	any	DET
ejpam-3764	196	3	hyperconnected	hyperconnecte	VERB
ejpam-3764	196	4	space	space	NOUN
ejpam-3764	196	5	is	be	AUX
ejpam-3764	196	6	connected	connect	VERB
ejpam-3764	196	7	and	and	CCONJ
ejpam-3764	196	8	locally	locally	ADV
ejpam-3764	196	9	connected	connect	VERB
ejpam-3764	196	10	,	,	PUNCT
ejpam-3764	196	11	then	then	ADV
ejpam-3764	196	12	we	we	PRON
ejpam-3764	196	13	conclude	conclude	VERB
ejpam-3764	196	14	the	the	DET
ejpam-3764	196	15	following	follow	VERB
ejpam-3764	196	16	corollaries	corollary	NOUN
ejpam-3764	196	17	.	.	PUNCT
ejpam-3764	197	1	corollary	corollary	ADJ
ejpam-3764	197	2	3	3	NUM
ejpam-3764	197	3	.	.	PUNCT
ejpam-3764	198	1	the	the	DET
ejpam-3764	198	2	omega	omega	NOUN
ejpam-3764	198	3	topological	topological	ADJ
ejpam-3764	198	4	space	space	NOUN
ejpam-3764	198	5	(	(	PUNCT
ejpam-3764	198	6	aω	aω	PROPN
ejpam-3764	198	7	,	,	PUNCT
ejpam-3764	198	8	τω	τω	INTJ
ejpam-3764	198	9	)	)	PUNCT
ejpam-3764	198	10	is	be	AUX
ejpam-3764	198	11	connected	connect	VERB
ejpam-3764	198	12	.	.	PUNCT
ejpam-3764	199	1	corollary	corollary	ADJ
ejpam-3764	199	2	4	4	NUM
ejpam-3764	199	3	.	.	PUNCT
ejpam-3764	200	1	the	the	DET
ejpam-3764	200	2	omega	omega	NOUN
ejpam-3764	200	3	topological	topological	ADJ
ejpam-3764	200	4	space	space	NOUN
ejpam-3764	200	5	(	(	PUNCT
ejpam-3764	200	6	aω	aω	PROPN
ejpam-3764	200	7	,	,	PUNCT
ejpam-3764	200	8	τω	τω	INTJ
ejpam-3764	200	9	)	)	PUNCT
ejpam-3764	200	10	is	be	AUX
ejpam-3764	200	11	locally	locally	ADV
ejpam-3764	200	12	connected	connect	VERB
ejpam-3764	200	13	.	.	PUNCT
ejpam-3764	201	1	proposition	proposition	NOUN
ejpam-3764	201	2	7	7	NUM
ejpam-3764	201	3	.	.	PUNCT
ejpam-3764	202	1	the	the	DET
ejpam-3764	202	2	omega	omega	NOUN
ejpam-3764	202	3	topological	topological	ADJ
ejpam-3764	202	4	space	space	NOUN
ejpam-3764	202	5	(	(	PUNCT
ejpam-3764	202	6	aω	aω	PROPN
ejpam-3764	202	7	,	,	PUNCT
ejpam-3764	202	8	τω	τω	INTJ
ejpam-3764	202	9	)	)	PUNCT
ejpam-3764	202	10	is	be	AUX
ejpam-3764	202	11	not	not	PART
ejpam-3764	202	12	t1	t1	NOUN
ejpam-3764	202	13	.	.	PUNCT
ejpam-3764	203	1	m.	m.	PROPN
ejpam-3764	203	2	alqahtani	alqahtani	PROPN
ejpam-3764	203	3	,	,	PUNCT
ejpam-3764	203	4	c.	c.	PROPN
ejpam-3764	203	5	özel	özel	PROPN
ejpam-3764	203	6	,	,	PUNCT
ejpam-3764	203	7	i.	i.	PROPN
ejpam-3764	203	8	alshammari	alshammari	PROPN
ejpam-3764	203	9	/	/	SYM
ejpam-3764	203	10	eur	eur	PROPN
ejpam-3764	203	11	.	.	PUNCT
ejpam-3764	204	1	j.	j.	PROPN
ejpam-3764	204	2	pure	pure	PROPN
ejpam-3764	204	3	appl	appl	PROPN
ejpam-3764	204	4	.	.	PROPN
ejpam-3764	204	5	math	math	PROPN
ejpam-3764	204	6	,	,	PUNCT
ejpam-3764	204	7	13	13	NUM
ejpam-3764	204	8	(	(	PUNCT
ejpam-3764	204	9	3	3	NUM
ejpam-3764	204	10	)	)	PUNCT
ejpam-3764	204	11	(	(	PUNCT
ejpam-3764	204	12	2020	2020	NUM
ejpam-3764	204	13	)	)	PUNCT
ejpam-3764	204	14	,	,	PUNCT
ejpam-3764	204	15	513	513	NUM
ejpam-3764	204	16	-	-	SYM
ejpam-3764	204	17	528	528	NUM
ejpam-3764	204	18	520	520	NUM
ejpam-3764	204	19	proof	proof	NOUN
ejpam-3764	204	20	.	.	PUNCT
ejpam-3764	205	1	if	if	SCONJ
ejpam-3764	205	2	aω	aω	PROPN
ejpam-3764	205	3	is	be	AUX
ejpam-3764	205	4	singleton	singleton	NOUN
ejpam-3764	205	5	,	,	PUNCT
ejpam-3764	205	6	then	then	ADV
ejpam-3764	205	7	it	it	PRON
ejpam-3764	205	8	is	be	AUX
ejpam-3764	205	9	t1	t1	PROPN
ejpam-3764	205	10	.	.	PUNCT
ejpam-3764	206	1	assume	assume	VERB
ejpam-3764	206	2	that	that	SCONJ
ejpam-3764	206	3	aω	aω	PROPN
ejpam-3764	206	4	,	,	PUNCT
ejpam-3764	206	5	which	which	PRON
ejpam-3764	206	6	has	have	VERB
ejpam-3764	206	7	more	more	ADJ
ejpam-3764	206	8	than	than	ADP
ejpam-3764	206	9	one	one	NUM
ejpam-3764	206	10	element	element	NOUN
ejpam-3764	206	11	.	.	PUNCT
ejpam-3764	207	1	there	there	PRON
ejpam-3764	207	2	exists	exist	VERB
ejpam-3764	207	3	a	a	DET
ejpam-3764	207	4	,	,	PUNCT
ejpam-3764	207	5	ω	ω	PROPN
ejpam-3764	207	6	∈	∈	NOUN
ejpam-3764	207	7	aω	aω	INTJ
ejpam-3764	207	8	such	such	ADJ
ejpam-3764	207	9	that	that	SCONJ
ejpam-3764	207	10	a	a	DET
ejpam-3764	207	11	6=	6=	PROPN
ejpam-3764	207	12	ω	ω	PROPN
ejpam-3764	207	13	.	.	PUNCT
ejpam-3764	208	1	since	since	SCONJ
ejpam-3764	208	2	any	any	DET
ejpam-3764	208	3	nonempty	nonempty	ADJ
ejpam-3764	208	4	open	open	ADJ
ejpam-3764	208	5	set	set	VERB
ejpam-3764	208	6	must	must	AUX
ejpam-3764	208	7	be	be	AUX
ejpam-3764	208	8	containing	contain	VERB
ejpam-3764	208	9	ω	ω	PROPN
ejpam-3764	208	10	,	,	PUNCT
ejpam-3764	208	11	then	then	ADV
ejpam-3764	208	12	we	we	PRON
ejpam-3764	208	13	can	can	AUX
ejpam-3764	208	14	not	not	PART
ejpam-3764	208	15	find	find	VERB
ejpam-3764	208	16	two	two	NUM
ejpam-3764	208	17	open	open	ADJ
ejpam-3764	208	18	sets	set	NOUN
ejpam-3764	208	19	u	u	NOUN
ejpam-3764	208	20	and	and	CCONJ
ejpam-3764	208	21	v	v	ADP
ejpam-3764	208	22	such	such	ADJ
ejpam-3764	208	23	that	that	SCONJ
ejpam-3764	208	24	a	a	DET
ejpam-3764	208	25	∈	∈	PROPN
ejpam-3764	208	26	u	u	NOUN
ejpam-3764	208	27	,	,	PUNCT
ejpam-3764	208	28	ω	ω	PROPN
ejpam-3764	208	29	/∈	/∈	PUNCT
ejpam-3764	209	1	u	u	NOUN
ejpam-3764	209	2	,	,	PUNCT
ejpam-3764	209	3	a	a	DET
ejpam-3764	209	4	/∈	/∈	NOUN
ejpam-3764	209	5	v	v	NOUN
ejpam-3764	209	6	and	and	CCONJ
ejpam-3764	209	7	ω	ω	NUM
ejpam-3764	209	8	∈	∈	PROPN
ejpam-3764	209	9	v.	v.	CCONJ
ejpam-3764	209	10	therefore	therefore	ADV
ejpam-3764	209	11	,	,	PUNCT
ejpam-3764	209	12	(	(	PUNCT
ejpam-3764	209	13	aω	aω	INTJ
ejpam-3764	209	14	,	,	PUNCT
ejpam-3764	209	15	τω	τω	INTJ
ejpam-3764	209	16	)	)	PUNCT
ejpam-3764	209	17	is	be	AUX
ejpam-3764	209	18	not	not	PART
ejpam-3764	209	19	t1	t1	NOUN
ejpam-3764	209	20	.	.	PUNCT
ejpam-3764	210	1	proposition	proposition	NOUN
ejpam-3764	210	2	8	8	NUM
ejpam-3764	210	3	.	.	PUNCT
ejpam-3764	211	1	let	let	VERB
ejpam-3764	211	2	(	(	PUNCT
ejpam-3764	211	3	aω	aω	PROPN
ejpam-3764	211	4	\	\	PROPN
ejpam-3764	211	5	{	{	PUNCT
ejpam-3764	211	6	ω},⊗	ω},⊗	NOUN
ejpam-3764	211	7	)	)	PUNCT
ejpam-3764	211	8	be	be	VERB
ejpam-3764	211	9	a	a	DET
ejpam-3764	211	10	group	group	NOUN
ejpam-3764	211	11	,	,	PUNCT
ejpam-3764	211	12	which	which	PRON
ejpam-3764	211	13	has	have	VERB
ejpam-3764	211	14	more	more	ADJ
ejpam-3764	211	15	than	than	ADP
ejpam-3764	211	16	two	two	NUM
ejpam-3764	211	17	elements	element	NOUN
ejpam-3764	211	18	.	.	PUNCT
ejpam-3764	212	1	then	then	ADV
ejpam-3764	212	2	the	the	DET
ejpam-3764	212	3	omega	omega	NOUN
ejpam-3764	212	4	topological	topological	ADJ
ejpam-3764	212	5	space	space	NOUN
ejpam-3764	212	6	(	(	PUNCT
ejpam-3764	212	7	aω	aω	PROPN
ejpam-3764	212	8	,	,	PUNCT
ejpam-3764	212	9	τω	τω	INTJ
ejpam-3764	212	10	)	)	PUNCT
ejpam-3764	212	11	is	be	AUX
ejpam-3764	212	12	not	not	PART
ejpam-3764	212	13	t0	t0	NOUN
ejpam-3764	212	14	.	.	PUNCT
ejpam-3764	213	1	proof	proof	NOUN
ejpam-3764	213	2	.	.	PUNCT
ejpam-3764	214	1	if	if	SCONJ
ejpam-3764	214	2	aω	aω	PROPN
ejpam-3764	214	3	has	have	VERB
ejpam-3764	214	4	only	only	ADV
ejpam-3764	214	5	two	two	NUM
ejpam-3764	214	6	elements	element	NOUN
ejpam-3764	214	7	,	,	PUNCT
ejpam-3764	214	8	then	then	ADV
ejpam-3764	214	9	it	it	PRON
ejpam-3764	214	10	is	be	AUX
ejpam-3764	214	11	t0	t0	PROPN
ejpam-3764	214	12	.	.	PUNCT
ejpam-3764	215	1	assume	assume	VERB
ejpam-3764	215	2	that	that	SCONJ
ejpam-3764	215	3	aω	aω	PROPN
ejpam-3764	215	4	,	,	PUNCT
ejpam-3764	215	5	which	which	PRON
ejpam-3764	215	6	has	have	VERB
ejpam-3764	215	7	more	more	ADJ
ejpam-3764	215	8	than	than	ADP
ejpam-3764	215	9	two	two	NUM
ejpam-3764	215	10	elements	element	NOUN
ejpam-3764	215	11	.	.	PUNCT
ejpam-3764	216	1	if	if	SCONJ
ejpam-3764	216	2	a	a	DET
ejpam-3764	216	3	∈	∈	PROPN
ejpam-3764	216	4	aω	aω	NOUN
ejpam-3764	216	5	is	be	AUX
ejpam-3764	216	6	arbitrary	arbitrary	ADJ
ejpam-3764	216	7	,	,	PUNCT
ejpam-3764	216	8	such	such	ADJ
ejpam-3764	216	9	that	that	SCONJ
ejpam-3764	216	10	a	a	DET
ejpam-3764	216	11	6=	6=	PROPN
ejpam-3764	216	12	ω	ω	PROPN
ejpam-3764	216	13	and	and	CCONJ
ejpam-3764	216	14	a	a	DET
ejpam-3764	216	15	6=	6=	ADP
ejpam-3764	216	16	e	e	NOUN
ejpam-3764	216	17	,	,	PUNCT
ejpam-3764	216	18	then	then	ADV
ejpam-3764	216	19	a	a	PRON
ejpam-3764	216	20	6=	6=	ADP
ejpam-3764	216	21	a−1	a−1	PROPN
ejpam-3764	216	22	in	in	ADP
ejpam-3764	216	23	aω	aω	PROPN
ejpam-3764	216	24	and	and	CCONJ
ejpam-3764	216	25	any	any	DET
ejpam-3764	216	26	open	open	ADJ
ejpam-3764	216	27	set	set	NOUN
ejpam-3764	216	28	containing	contain	VERB
ejpam-3764	216	29	a	a	PRON
ejpam-3764	216	30	must	must	AUX
ejpam-3764	216	31	be	be	AUX
ejpam-3764	216	32	containing	contain	VERB
ejpam-3764	216	33	a−1	a−1	PROPN
ejpam-3764	216	34	.	.	PUNCT
ejpam-3764	217	1	hence	hence	ADV
ejpam-3764	217	2	,	,	PUNCT
ejpam-3764	217	3	(	(	PUNCT
ejpam-3764	217	4	aω	aω	INTJ
ejpam-3764	217	5	,	,	PUNCT
ejpam-3764	217	6	τω	τω	INTJ
ejpam-3764	217	7	)	)	PUNCT
ejpam-3764	217	8	is	be	AUX
ejpam-3764	217	9	not	not	PART
ejpam-3764	217	10	t0	t0	NOUN
ejpam-3764	217	11	.	.	PUNCT
ejpam-3764	218	1	proposition	proposition	NOUN
ejpam-3764	218	2	9	9	NUM
ejpam-3764	218	3	.	.	PUNCT
ejpam-3764	219	1	if	if	SCONJ
ejpam-3764	219	2	aω	aω	PROPN
ejpam-3764	219	3	,	,	PUNCT
ejpam-3764	219	4	has	have	VERB
ejpam-3764	219	5	more	more	ADJ
ejpam-3764	219	6	than	than	ADP
ejpam-3764	219	7	one	one	NUM
ejpam-3764	219	8	element	element	NOUN
ejpam-3764	219	9	,	,	PUNCT
ejpam-3764	219	10	then	then	ADV
ejpam-3764	219	11	the	the	DET
ejpam-3764	219	12	omega	omega	NOUN
ejpam-3764	219	13	topological	topological	ADJ
ejpam-3764	219	14	space	space	NOUN
ejpam-3764	219	15	(	(	PUNCT
ejpam-3764	219	16	aω	aω	PROPN
ejpam-3764	219	17	,	,	PUNCT
ejpam-3764	219	18	τω	τω	INTJ
ejpam-3764	219	19	)	)	PUNCT
ejpam-3764	219	20	is	be	AUX
ejpam-3764	219	21	not	not	PART
ejpam-3764	219	22	regular	regular	ADJ
ejpam-3764	219	23	.	.	PUNCT
ejpam-3764	220	1	proof	proof	NOUN
ejpam-3764	220	2	.	.	PUNCT
ejpam-3764	221	1	if	if	SCONJ
ejpam-3764	221	2	aω	aω	PROPN
ejpam-3764	221	3	is	be	AUX
ejpam-3764	221	4	singleton	singleton	PROPN
ejpam-3764	221	5	then	then	ADV
ejpam-3764	221	6	it	it	PRON
ejpam-3764	221	7	is	be	AUX
ejpam-3764	221	8	regular	regular	ADJ
ejpam-3764	221	9	.	.	PUNCT
ejpam-3764	222	1	there	there	PRON
ejpam-3764	222	2	exists	exist	VERB
ejpam-3764	222	3	k	k	PROPN
ejpam-3764	222	4	=	=	PUNCT
ejpam-3764	222	5	aω	aω	X
ejpam-3764	222	6	\	\	PROPN
ejpam-3764	222	7	{	{	PUNCT
ejpam-3764	222	8	e	e	NOUN
ejpam-3764	222	9	,	,	PUNCT
ejpam-3764	222	10	ω	ω	PROPN
ejpam-3764	222	11	}	}	PUNCT
ejpam-3764	222	12	be	be	AUX
ejpam-3764	222	13	a	a	DET
ejpam-3764	222	14	closed	closed	ADJ
ejpam-3764	222	15	subset	subset	NOUN
ejpam-3764	222	16	of	of	ADP
ejpam-3764	222	17	aω	aω	PROPN
ejpam-3764	222	18	and	and	CCONJ
ejpam-3764	222	19	ω	ω	NUM
ejpam-3764	222	20	/∈	/∈	PUNCT
ejpam-3764	223	1	k.	k.	PROPN
ejpam-3764	224	1	we	we	PRON
ejpam-3764	224	2	can	can	AUX
ejpam-3764	224	3	not	not	PART
ejpam-3764	224	4	separated	separate	VERB
ejpam-3764	224	5	ω	ω	PROPN
ejpam-3764	224	6	and	and	CCONJ
ejpam-3764	224	7	k	k	X
ejpam-3764	224	8	by	by	ADP
ejpam-3764	224	9	any	any	DET
ejpam-3764	224	10	open	open	ADJ
ejpam-3764	224	11	sets	set	NOUN
ejpam-3764	224	12	(	(	PUNCT
ejpam-3764	224	13	because	because	SCONJ
ejpam-3764	224	14	any	any	DET
ejpam-3764	224	15	open	open	ADJ
ejpam-3764	224	16	set	set	NOUN
ejpam-3764	224	17	in	in	ADP
ejpam-3764	224	18	aω	aω	PROPN
ejpam-3764	224	19	is	be	AUX
ejpam-3764	224	20	containing	contain	VERB
ejpam-3764	224	21	ω	ω	NUM
ejpam-3764	224	22	)	)	PUNCT
ejpam-3764	224	23	.	.	PUNCT
ejpam-3764	225	1	hence	hence	ADV
ejpam-3764	225	2	,	,	PUNCT
ejpam-3764	225	3	(	(	PUNCT
ejpam-3764	225	4	aω	aω	INTJ
ejpam-3764	225	5	,	,	PUNCT
ejpam-3764	225	6	τω	τω	INTJ
ejpam-3764	225	7	)	)	PUNCT
ejpam-3764	225	8	is	be	AUX
ejpam-3764	225	9	not	not	PART
ejpam-3764	225	10	regular	regular	ADJ
ejpam-3764	225	11	.	.	PUNCT
ejpam-3764	226	1	proposition	proposition	NOUN
ejpam-3764	226	2	10	10	NUM
ejpam-3764	226	3	.	.	PUNCT
ejpam-3764	227	1	if	if	SCONJ
ejpam-3764	227	2	(	(	PUNCT
ejpam-3764	227	3	aω	aω	PROPN
ejpam-3764	227	4	\	\	PROPN
ejpam-3764	227	5	{	{	PUNCT
ejpam-3764	227	6	ω},⊗	ω},⊗	NOUN
ejpam-3764	227	7	)	)	PUNCT
ejpam-3764	227	8	be	be	VERB
ejpam-3764	227	9	a	a	DET
ejpam-3764	227	10	group	group	NOUN
ejpam-3764	227	11	,	,	PUNCT
ejpam-3764	227	12	which	which	PRON
ejpam-3764	227	13	has	have	VERB
ejpam-3764	227	14	more	more	ADJ
ejpam-3764	227	15	than	than	ADP
ejpam-3764	227	16	two	two	NUM
ejpam-3764	227	17	elements	element	NOUN
ejpam-3764	227	18	,	,	PUNCT
ejpam-3764	227	19	then	then	ADV
ejpam-3764	227	20	the	the	DET
ejpam-3764	227	21	omega	omega	NOUN
ejpam-3764	227	22	topological	topological	ADJ
ejpam-3764	227	23	space	space	NOUN
ejpam-3764	227	24	(	(	PUNCT
ejpam-3764	227	25	aω	aω	PROPN
ejpam-3764	227	26	,	,	PUNCT
ejpam-3764	227	27	τω	τω	INTJ
ejpam-3764	227	28	)	)	PUNCT
ejpam-3764	227	29	is	be	AUX
ejpam-3764	227	30	not	not	PART
ejpam-3764	227	31	normal	normal	ADJ
ejpam-3764	227	32	.	.	PUNCT
ejpam-3764	228	1	proof	proof	NOUN
ejpam-3764	228	2	.	.	PUNCT
ejpam-3764	229	1	if	if	SCONJ
ejpam-3764	229	2	aω	aω	PROPN
ejpam-3764	229	3	=	=	SYM
ejpam-3764	229	4	{	{	PUNCT
ejpam-3764	229	5	e	e	NOUN
ejpam-3764	229	6	=	=	SYM
ejpam-3764	229	7	ω	ω	PROPN
ejpam-3764	229	8	}	}	PUNCT
ejpam-3764	229	9	or	or	CCONJ
ejpam-3764	229	10	{	{	PUNCT
ejpam-3764	229	11	e	e	NOUN
ejpam-3764	229	12	,	,	PUNCT
ejpam-3764	229	13	ω	ω	PROPN
ejpam-3764	229	14	}	}	PUNCT
ejpam-3764	229	15	,	,	PUNCT
ejpam-3764	229	16	then	then	ADV
ejpam-3764	229	17	it	it	PRON
ejpam-3764	229	18	is	be	AUX
ejpam-3764	229	19	normal	normal	ADJ
ejpam-3764	229	20	.	.	PUNCT
ejpam-3764	230	1	assume	assume	VERB
ejpam-3764	230	2	that	that	SCONJ
ejpam-3764	230	3	aω	aω	PROPN
ejpam-3764	230	4	has	have	VERB
ejpam-3764	230	5	more	more	ADJ
ejpam-3764	230	6	than	than	ADP
ejpam-3764	230	7	two	two	NUM
ejpam-3764	230	8	elements	element	NOUN
ejpam-3764	230	9	,	,	PUNCT
ejpam-3764	230	10	then	then	ADV
ejpam-3764	230	11	there	there	PRON
ejpam-3764	230	12	exists	exist	VERB
ejpam-3764	230	13	a	a	DET
ejpam-3764	230	14	∈	∈	NOUN
ejpam-3764	230	15	aω	aω	ADP
ejpam-3764	230	16	such	such	ADJ
ejpam-3764	230	17	that	that	SCONJ
ejpam-3764	230	18	a	a	DET
ejpam-3764	230	19	6=	6=	PROPN
ejpam-3764	230	20	e	e	NOUN
ejpam-3764	230	21	,	,	PUNCT
ejpam-3764	230	22	a	a	DET
ejpam-3764	230	23	6=	6=	NOUN
ejpam-3764	230	24	ω	ω	PROPN
ejpam-3764	230	25	and	and	CCONJ
ejpam-3764	230	26	its	its	PRON
ejpam-3764	230	27	multiplicative	multiplicative	ADJ
ejpam-3764	230	28	inverse	inverse	NOUN
ejpam-3764	230	29	exists	exist	VERB
ejpam-3764	230	30	in	in	ADP
ejpam-3764	230	31	aω	aω	PROPN
ejpam-3764	230	32	.	.	PUNCT
ejpam-3764	231	1	then	then	ADV
ejpam-3764	231	2	,	,	PUNCT
ejpam-3764	231	3	there	there	PRON
ejpam-3764	231	4	exists	exist	VERB
ejpam-3764	231	5	k	k	PROPN
ejpam-3764	232	1	=	=	X
ejpam-3764	232	2	{	{	PUNCT
ejpam-3764	232	3	a	a	NOUN
ejpam-3764	232	4	,	,	PUNCT
ejpam-3764	232	5	a−1	a−1	PROPN
ejpam-3764	232	6	}	}	PUNCT
ejpam-3764	232	7	and	and	CCONJ
ejpam-3764	232	8	h	h	NOUN
ejpam-3764	232	9	=	=	SYM
ejpam-3764	232	10	{	{	PUNCT
ejpam-3764	232	11	e	e	NOUN
ejpam-3764	232	12	}	}	PUNCT
ejpam-3764	232	13	are	be	AUX
ejpam-3764	232	14	two	two	NUM
ejpam-3764	232	15	disjoint	disjoint	NOUN
ejpam-3764	232	16	nonempty	nonempty	X
ejpam-3764	232	17	closed	close	VERB
ejpam-3764	232	18	subsets	subset	NOUN
ejpam-3764	232	19	of	of	ADP
ejpam-3764	232	20	aω	aω	PROPN
ejpam-3764	232	21	,	,	PUNCT
ejpam-3764	232	22	such	such	ADJ
ejpam-3764	232	23	that	that	SCONJ
ejpam-3764	232	24	we	we	PRON
ejpam-3764	232	25	can	can	AUX
ejpam-3764	232	26	not	not	PART
ejpam-3764	232	27	separat	separat	VERB
ejpam-3764	232	28	them	they	PRON
ejpam-3764	232	29	by	by	ADP
ejpam-3764	232	30	any	any	DET
ejpam-3764	232	31	open	open	ADJ
ejpam-3764	232	32	sets	set	NOUN
ejpam-3764	232	33	(	(	PUNCT
ejpam-3764	232	34	because	because	SCONJ
ejpam-3764	232	35	any	any	DET
ejpam-3764	232	36	open	open	ADJ
ejpam-3764	232	37	set	set	NOUN
ejpam-3764	232	38	in	in	ADP
ejpam-3764	232	39	aω	aω	PROPN
ejpam-3764	232	40	is	be	AUX
ejpam-3764	232	41	containing	contain	VERB
ejpam-3764	232	42	ω	ω	NUM
ejpam-3764	232	43	)	)	PUNCT
ejpam-3764	232	44	.	.	PUNCT
ejpam-3764	233	1	hence	hence	ADV
ejpam-3764	233	2	,	,	PUNCT
ejpam-3764	233	3	(	(	PUNCT
ejpam-3764	233	4	aω	aω	INTJ
ejpam-3764	233	5	,	,	PUNCT
ejpam-3764	233	6	τω	τω	INTJ
ejpam-3764	233	7	)	)	PUNCT
ejpam-3764	233	8	is	be	AUX
ejpam-3764	233	9	not	not	PART
ejpam-3764	233	10	normal	normal	ADJ
ejpam-3764	233	11	.	.	PUNCT
ejpam-3764	234	1	proposition	proposition	NOUN
ejpam-3764	234	2	11	11	NUM
ejpam-3764	234	3	.	.	PUNCT
ejpam-3764	235	1	if	if	SCONJ
ejpam-3764	235	2	(	(	PUNCT
ejpam-3764	235	3	aω	aω	PROPN
ejpam-3764	235	4	\	\	PROPN
ejpam-3764	235	5	{	{	PUNCT
ejpam-3764	235	6	ω},⊗	ω},⊗	NOUN
ejpam-3764	235	7	)	)	PUNCT
ejpam-3764	235	8	be	be	VERB
ejpam-3764	235	9	a	a	DET
ejpam-3764	235	10	group	group	NOUN
ejpam-3764	235	11	and	and	CCONJ
ejpam-3764	235	12	a	a	PRON
ejpam-3764	235	13	is	be	AUX
ejpam-3764	235	14	uncountable	uncountable	ADJ
ejpam-3764	235	15	infinite	infinite	ADJ
ejpam-3764	235	16	set	set	NOUN
ejpam-3764	235	17	,	,	PUNCT
ejpam-3764	235	18	then	then	ADV
ejpam-3764	235	19	the	the	DET
ejpam-3764	235	20	omega	omega	NOUN
ejpam-3764	235	21	topological	topological	ADJ
ejpam-3764	235	22	space	space	NOUN
ejpam-3764	235	23	(	(	PUNCT
ejpam-3764	235	24	aω	aω	PROPN
ejpam-3764	235	25	,	,	PUNCT
ejpam-3764	235	26	τω	τω	INTJ
ejpam-3764	235	27	)	)	PUNCT
ejpam-3764	235	28	is	be	AUX
ejpam-3764	235	29	not	not	PART
ejpam-3764	235	30	compact	compact	ADJ
ejpam-3764	235	31	(	(	PUNCT
ejpam-3764	235	32	lindelöf	lindelöf	PROPN
ejpam-3764	235	33	)	)	PUNCT
ejpam-3764	235	34	.	.	PUNCT
ejpam-3764	236	1	proof	proof	NOUN
ejpam-3764	236	2	.	.	PUNCT
ejpam-3764	237	1	there	there	PRON
ejpam-3764	237	2	exists	exist	VERB
ejpam-3764	237	3	{	{	PUNCT
ejpam-3764	237	4	{	{	PUNCT
ejpam-3764	237	5	ω	ω	NOUN
ejpam-3764	237	6	}	}	PUNCT
ejpam-3764	237	7	,	,	PUNCT
ejpam-3764	237	8	{	{	PUNCT
ejpam-3764	237	9	ω	ω	NOUN
ejpam-3764	237	10	,	,	PUNCT
ejpam-3764	237	11	a	a	PRON
ejpam-3764	237	12	,	,	PUNCT
ejpam-3764	237	13	a−1	a−1	PROPN
ejpam-3764	237	14	}	}	PUNCT
ejpam-3764	237	15	:	:	PUNCT
ejpam-3764	237	16	a	a	DET
ejpam-3764	237	17	∈	∈	NOUN
ejpam-3764	237	18	aω	aω	X
ejpam-3764	237	19	\	\	PROPN
ejpam-3764	237	20	{	{	PUNCT
ejpam-3764	237	21	ω	ω	NOUN
ejpam-3764	237	22	}	}	PUNCT
ejpam-3764	237	23	}	}	PUNCT
ejpam-3764	237	24	,	,	PUNCT
ejpam-3764	237	25	which	which	PRON
ejpam-3764	237	26	s	s	VERB
ejpam-3764	237	27	an	an	DET
ejpam-3764	237	28	open	open	ADJ
ejpam-3764	237	29	cover	cover	NOUN
ejpam-3764	237	30	of	of	ADP
ejpam-3764	237	31	aω	aω	PROPN
ejpam-3764	237	32	,	,	PUNCT
ejpam-3764	237	33	and	and	CCONJ
ejpam-3764	237	34	has	have	VERB
ejpam-3764	237	35	no	no	DET
ejpam-3764	237	36	finite	finite	NOUN
ejpam-3764	237	37	(	(	PUNCT
ejpam-3764	237	38	countable	countable	ADJ
ejpam-3764	237	39	)	)	PUNCT
ejpam-3764	237	40	subcover	subcover	NOUN
ejpam-3764	237	41	of	of	ADP
ejpam-3764	237	42	aω	aω	PROPN
ejpam-3764	237	43	.	.	PUNCT
ejpam-3764	237	44	proposition	proposition	NOUN
ejpam-3764	237	45	12	12	NUM
ejpam-3764	237	46	.	.	PUNCT
ejpam-3764	238	1	let	let	VERB
ejpam-3764	238	2	a	a	DET
ejpam-3764	238	3	∈	∈	NOUN
ejpam-3764	238	4	aω	aω	X
ejpam-3764	238	5	\	\	PROPN
ejpam-3764	238	6	{	{	PUNCT
ejpam-3764	238	7	ω	ω	NOUN
ejpam-3764	238	8	}	}	PUNCT
ejpam-3764	238	9	has	have	VERB
ejpam-3764	238	10	no	no	DET
ejpam-3764	238	11	multiplicative	multiplicative	ADJ
ejpam-3764	238	12	inverse	inverse	NOUN
ejpam-3764	238	13	.	.	PUNCT
ejpam-3764	239	1	then	then	ADV
ejpam-3764	239	2	the	the	DET
ejpam-3764	239	3	omega	omega	NOUN
ejpam-3764	239	4	topological	topological	ADJ
ejpam-3764	239	5	space	space	NOUN
ejpam-3764	239	6	(	(	PUNCT
ejpam-3764	239	7	aω	aω	PROPN
ejpam-3764	239	8	,	,	PUNCT
ejpam-3764	239	9	τω	τω	INTJ
ejpam-3764	239	10	)	)	PUNCT
ejpam-3764	239	11	is	be	AUX
ejpam-3764	239	12	compact	compact	ADJ
ejpam-3764	239	13	.	.	PUNCT
ejpam-3764	240	1	proof	proof	NOUN
ejpam-3764	240	2	.	.	PUNCT
ejpam-3764	241	1	let	let	VERB
ejpam-3764	241	2	{	{	PUNCT
ejpam-3764	241	3	cα	cα	NOUN
ejpam-3764	241	4	:	:	PUNCT
ejpam-3764	241	5	α	α	PROPN
ejpam-3764	241	6	∈	∈	PROPN
ejpam-3764	241	7	λ	λ	NOUN
ejpam-3764	241	8	}	}	PUNCT
ejpam-3764	241	9	be	be	VERB
ejpam-3764	241	10	any	any	DET
ejpam-3764	241	11	open	open	ADJ
ejpam-3764	241	12	cover	cover	NOUN
ejpam-3764	241	13	of	of	ADP
ejpam-3764	241	14	aω	aω	PROPN
ejpam-3764	241	15	.	.	PUNCT
ejpam-3764	242	1	since	since	SCONJ
ejpam-3764	242	2	a	a	DET
ejpam-3764	242	3	∈	∈	PROPN
ejpam-3764	242	4	aω	aω	NOUN
ejpam-3764	242	5	,	,	PUNCT
ejpam-3764	242	6	then	then	ADV
ejpam-3764	242	7	for	for	ADP
ejpam-3764	242	8	some	some	DET
ejpam-3764	242	9	β	β	NOUN
ejpam-3764	242	10	∈	∈	PROPN
ejpam-3764	242	11	λ	λ	PROPN
ejpam-3764	242	12	,	,	PUNCT
ejpam-3764	242	13	there	there	PRON
ejpam-3764	242	14	exists	exist	VERB
ejpam-3764	242	15	cβ	cβ	NOUN
ejpam-3764	242	16	containing	contain	VERB
ejpam-3764	242	17	a.	a.	NOUN
ejpam-3764	242	18	but	but	CCONJ
ejpam-3764	242	19	cβ	cβ	PROPN
ejpam-3764	242	20	=	=	SYM
ejpam-3764	242	21	aω	aω	PROPN
ejpam-3764	242	22	,	,	PUNCT
ejpam-3764	242	23	because	because	SCONJ
ejpam-3764	242	24	aω	aω	PROPN
ejpam-3764	242	25	is	be	AUX
ejpam-3764	242	26	the	the	DET
ejpam-3764	242	27	only	only	ADJ
ejpam-3764	242	28	open	open	ADJ
ejpam-3764	242	29	set	set	NOUN
ejpam-3764	242	30	containing	contain	VERB
ejpam-3764	242	31	a.	a.	NOUN
ejpam-3764	242	32	hence	hence	ADV
ejpam-3764	242	33	,	,	PUNCT
ejpam-3764	242	34	{	{	PUNCT
ejpam-3764	242	35	cβ	cβ	NOUN
ejpam-3764	242	36	}	}	PUNCT
ejpam-3764	242	37	is	be	AUX
ejpam-3764	242	38	a	a	DET
ejpam-3764	242	39	finite	finite	ADJ
ejpam-3764	242	40	subcover	subcover	NOUN
ejpam-3764	242	41	of	of	ADP
ejpam-3764	242	42	{	{	PUNCT
ejpam-3764	242	43	cα	cα	X
ejpam-3764	242	44	:	:	PUNCT
ejpam-3764	242	45	α	α	PROPN
ejpam-3764	242	46	∈	∈	PROPN
ejpam-3764	242	47	λ	λ	NOUN
ejpam-3764	242	48	}	}	PUNCT
ejpam-3764	242	49	,	,	PUNCT
ejpam-3764	242	50	which	which	PRON
ejpam-3764	242	51	cover	cover	VERB
ejpam-3764	242	52	aω	aω	PROPN
ejpam-3764	242	53	.	.	PUNCT
ejpam-3764	243	1	therefore	therefore	ADV
ejpam-3764	243	2	,	,	PUNCT
ejpam-3764	243	3	(	(	PUNCT
ejpam-3764	243	4	aω	aω	INTJ
ejpam-3764	243	5	,	,	PUNCT
ejpam-3764	243	6	τω	τω	INTJ
ejpam-3764	243	7	)	)	PUNCT
ejpam-3764	243	8	is	be	AUX
ejpam-3764	243	9	a	a	DET
ejpam-3764	243	10	compact	compact	ADJ
ejpam-3764	243	11	space	space	NOUN
ejpam-3764	243	12	.	.	PUNCT
ejpam-3764	244	1	since	since	SCONJ
ejpam-3764	244	2	any	any	DET
ejpam-3764	244	3	compact	compact	ADJ
ejpam-3764	244	4	space	space	NOUN
ejpam-3764	244	5	is	be	AUX
ejpam-3764	244	6	lindelöf	lindelöf	NOUN
ejpam-3764	244	7	and	and	CCONJ
ejpam-3764	244	8	countably	countably	ADV
ejpam-3764	244	9	compact	compact	ADJ
ejpam-3764	244	10	,	,	PUNCT
ejpam-3764	244	11	then	then	ADV
ejpam-3764	244	12	we	we	PRON
ejpam-3764	244	13	conclude	conclude	VERB
ejpam-3764	244	14	the	the	DET
ejpam-3764	244	15	following	follow	VERB
ejpam-3764	244	16	corollaries	corollary	NOUN
ejpam-3764	244	17	.	.	PUNCT
ejpam-3764	245	1	m.	m.	NOUN
ejpam-3764	245	2	alqahtani	alqahtani	PROPN
ejpam-3764	245	3	,	,	PUNCT
ejpam-3764	245	4	c.	c.	PROPN
ejpam-3764	245	5	özel	özel	PROPN
ejpam-3764	245	6	,	,	PUNCT
ejpam-3764	245	7	i.	i.	PROPN
ejpam-3764	245	8	alshammari	alshammari	PROPN
ejpam-3764	245	9	/	/	SYM
ejpam-3764	245	10	eur	eur	PROPN
ejpam-3764	245	11	.	.	PUNCT
ejpam-3764	246	1	j.	j.	PROPN
ejpam-3764	246	2	pure	pure	PROPN
ejpam-3764	246	3	appl	appl	PROPN
ejpam-3764	246	4	.	.	PROPN
ejpam-3764	246	5	math	math	PROPN
ejpam-3764	246	6	,	,	PUNCT
ejpam-3764	246	7	13	13	NUM
ejpam-3764	246	8	(	(	PUNCT
ejpam-3764	246	9	3	3	NUM
ejpam-3764	246	10	)	)	PUNCT
ejpam-3764	246	11	(	(	PUNCT
ejpam-3764	246	12	2020	2020	NUM
ejpam-3764	246	13	)	)	PUNCT
ejpam-3764	246	14	,	,	PUNCT
ejpam-3764	246	15	513	513	NUM
ejpam-3764	246	16	-	-	SYM
ejpam-3764	246	17	528	528	NUM
ejpam-3764	246	18	521	521	NUM
ejpam-3764	246	19	corollary	corollary	ADJ
ejpam-3764	246	20	5	5	NUM
ejpam-3764	246	21	.	.	PUNCT
ejpam-3764	247	1	if	if	SCONJ
ejpam-3764	247	2	a	a	DET
ejpam-3764	247	3	∈	∈	NOUN
ejpam-3764	247	4	aω	aω	X
ejpam-3764	247	5	\	\	PROPN
ejpam-3764	247	6	{	{	PUNCT
ejpam-3764	247	7	ω	ω	NOUN
ejpam-3764	247	8	}	}	PUNCT
ejpam-3764	247	9	has	have	VERB
ejpam-3764	247	10	no	no	DET
ejpam-3764	247	11	multiplicative	multiplicative	ADJ
ejpam-3764	247	12	inverse	inverse	NOUN
ejpam-3764	247	13	,	,	PUNCT
ejpam-3764	247	14	then	then	ADV
ejpam-3764	247	15	the	the	DET
ejpam-3764	247	16	omega	omega	NOUN
ejpam-3764	247	17	topological	topological	ADJ
ejpam-3764	247	18	space	space	NOUN
ejpam-3764	247	19	(	(	PUNCT
ejpam-3764	247	20	aω	aω	PROPN
ejpam-3764	247	21	,	,	PUNCT
ejpam-3764	247	22	τω	τω	INTJ
ejpam-3764	247	23	)	)	PUNCT
ejpam-3764	247	24	is	be	AUX
ejpam-3764	247	25	lindelöf	lindelöf	PROPN
ejpam-3764	247	26	.	.	PUNCT
ejpam-3764	248	1	corollary	corollary	ADJ
ejpam-3764	248	2	6	6	NUM
ejpam-3764	248	3	.	.	PUNCT
ejpam-3764	249	1	if	if	SCONJ
ejpam-3764	249	2	a	a	DET
ejpam-3764	249	3	∈	∈	NOUN
ejpam-3764	249	4	aω	aω	X
ejpam-3764	249	5	\	\	PROPN
ejpam-3764	249	6	{	{	PUNCT
ejpam-3764	249	7	ω	ω	NOUN
ejpam-3764	249	8	}	}	PUNCT
ejpam-3764	249	9	has	have	VERB
ejpam-3764	249	10	no	no	DET
ejpam-3764	249	11	multiplicative	multiplicative	ADJ
ejpam-3764	249	12	inverse	inverse	NOUN
ejpam-3764	249	13	,	,	PUNCT
ejpam-3764	249	14	then	then	ADV
ejpam-3764	249	15	the	the	DET
ejpam-3764	249	16	omega	omega	NOUN
ejpam-3764	249	17	topological	topological	ADJ
ejpam-3764	249	18	space	space	NOUN
ejpam-3764	249	19	(	(	PUNCT
ejpam-3764	249	20	aω	aω	PROPN
ejpam-3764	249	21	,	,	PUNCT
ejpam-3764	249	22	τω	τω	INTJ
ejpam-3764	249	23	)	)	PUNCT
ejpam-3764	249	24	is	be	AUX
ejpam-3764	249	25	countably	countably	ADV
ejpam-3764	249	26	compact	compact	ADJ
ejpam-3764	249	27	.	.	PUNCT
ejpam-3764	250	1	remark	remark	NOUN
ejpam-3764	250	2	3	3	NUM
ejpam-3764	250	3	.	.	PUNCT
ejpam-3764	251	1	since	since	SCONJ
ejpam-3764	251	2	every	every	DET
ejpam-3764	251	3	nonempty	nonempty	ADJ
ejpam-3764	251	4	open	open	ADJ
ejpam-3764	251	5	set	set	VERB
ejpam-3764	251	6	in	in	ADP
ejpam-3764	251	7	(	(	PUNCT
ejpam-3764	251	8	aω	aω	PROPN
ejpam-3764	251	9	,	,	PUNCT
ejpam-3764	251	10	τω	τω	NOUN
ejpam-3764	251	11	)	)	PUNCT
ejpam-3764	251	12	containing	contain	VERB
ejpam-3764	251	13	ω	ω	PROPN
ejpam-3764	251	14	,	,	PUNCT
ejpam-3764	251	15	then	then	ADV
ejpam-3764	251	16	the	the	DET
ejpam-3764	251	17	closure	closure	NOUN
ejpam-3764	251	18	of	of	ADP
ejpam-3764	251	19	any	any	DET
ejpam-3764	251	20	nonempty	nonempty	ADJ
ejpam-3764	251	21	open	open	ADJ
ejpam-3764	251	22	set	set	NOUN
ejpam-3764	251	23	is	be	AUX
ejpam-3764	251	24	equal	equal	ADJ
ejpam-3764	251	25	aω	aω	PROPN
ejpam-3764	251	26	.	.	PROPN
ejpam-3764	251	27	4	4	NUM
ejpam-3764	251	28	.	.	X
ejpam-3764	252	1	some	some	PRON
ejpam-3764	252	2	of	of	ADP
ejpam-3764	252	3	the	the	DET
ejpam-3764	252	4	fundamental	fundamental	ADJ
ejpam-3764	252	5	properties	property	NOUN
ejpam-3764	252	6	for	for	ADP
ejpam-3764	252	7	different	different	ADJ
ejpam-3764	252	8	examples	example	NOUN
ejpam-3764	252	9	on	on	ADP
ejpam-3764	252	10	omega	omega	NOUN
ejpam-3764	252	11	topology	topology	NOUN
ejpam-3764	252	12	in	in	ADP
ejpam-3764	252	13	this	this	DET
ejpam-3764	252	14	section	section	NOUN
ejpam-3764	252	15	,	,	PUNCT
ejpam-3764	252	16	we	we	PRON
ejpam-3764	252	17	give	give	VERB
ejpam-3764	252	18	four	four	NUM
ejpam-3764	252	19	different	different	ADJ
ejpam-3764	252	20	examples	example	NOUN
ejpam-3764	252	21	of	of	ADP
ejpam-3764	252	22	omega	omega	NOUN
ejpam-3764	252	23	topologies	topology	NOUN
ejpam-3764	252	24	.	.	PUNCT
ejpam-3764	253	1	the	the	DET
ejpam-3764	253	2	first	first	ADJ
ejpam-3764	253	3	and	and	CCONJ
ejpam-3764	253	4	fourth	fourth	ADJ
ejpam-3764	253	5	examples	example	NOUN
ejpam-3764	253	6	are	be	AUX
ejpam-3764	253	7	from	from	ADP
ejpam-3764	253	8	an	an	DET
ejpam-3764	253	9	ordered	order	VERB
ejpam-3764	253	10	infinite	infinite	NOUN
ejpam-3764	253	11	set	set	NOUN
ejpam-3764	253	12	,	,	PUNCT
ejpam-3764	253	13	the	the	DET
ejpam-3764	253	14	second	second	NOUN
ejpam-3764	253	15	is	be	AUX
ejpam-3764	253	16	from	from	ADP
ejpam-3764	253	17	a	a	DET
ejpam-3764	253	18	cyclically	cyclically	ADV
ejpam-3764	253	19	ordered	order	VERB
ejpam-3764	253	20	infinite	infinite	ADJ
ejpam-3764	253	21	set	set	NOUN
ejpam-3764	253	22	,	,	PUNCT
ejpam-3764	253	23	and	and	CCONJ
ejpam-3764	253	24	the	the	DET
ejpam-3764	253	25	third	third	NOUN
ejpam-3764	253	26	is	be	AUX
ejpam-3764	253	27	from	from	ADP
ejpam-3764	253	28	a	a	DET
ejpam-3764	253	29	finite	finite	ADJ
ejpam-3764	253	30	set	set	NOUN
ejpam-3764	253	31	.	.	PUNCT
ejpam-3764	254	1	furthermore	furthermore	ADV
ejpam-3764	254	2	,	,	PUNCT
ejpam-3764	254	3	we	we	PRON
ejpam-3764	254	4	define	define	VERB
ejpam-3764	254	5	a	a	DET
ejpam-3764	254	6	new	new	ADJ
ejpam-3764	254	7	topology	topology	NOUN
ejpam-3764	254	8	over	over	ADP
ejpam-3764	254	9	semiring	semire	VERB
ejpam-3764	254	10	in	in	ADP
ejpam-3764	254	11	conventional	conventional	ADJ
ejpam-3764	254	12	algebra	algebra	NOUN
ejpam-3764	254	13	.	.	PUNCT
ejpam-3764	255	1	example	example	NOUN
ejpam-3764	256	1	4	4	NUM
ejpam-3764	256	2	.	.	PUNCT
ejpam-3764	256	3	by	by	ADP
ejpam-3764	256	4	example	example	NOUN
ejpam-3764	256	5	1	1	NUM
ejpam-3764	256	6	,	,	PUNCT
ejpam-3764	256	7	we	we	PRON
ejpam-3764	256	8	have	have	VERB
ejpam-3764	256	9	(	(	PUNCT
ejpam-3764	256	10	w−∞	w−∞	X
ejpam-3764	256	11	,	,	PUNCT
ejpam-3764	256	12	τ−∞	τ−∞	NOUN
ejpam-3764	256	13	)	)	PUNCT
ejpam-3764	256	14	which	which	PRON
ejpam-3764	256	15	is	be	AUX
ejpam-3764	256	16	a	a	DET
ejpam-3764	256	17	topological	topological	ADJ
ejpam-3764	256	18	space	space	NOUN
ejpam-3764	256	19	,	,	PUNCT
ejpam-3764	256	20	where	where	SCONJ
ejpam-3764	256	21	w	w	NOUN
ejpam-3764	256	22	=	=	SYM
ejpam-3764	256	23	{	{	PUNCT
ejpam-3764	256	24	0	0	NUM
ejpam-3764	256	25	,	,	PUNCT
ejpam-3764	256	26	1	1	NUM
ejpam-3764	256	27	,	,	PUNCT
ejpam-3764	256	28	2	2	NUM
ejpam-3764	256	29	,	,	PUNCT
ejpam-3764	256	30	3	3	NUM
ejpam-3764	256	31	,	,	PUNCT
ejpam-3764	256	32	·	·	PUNCT
ejpam-3764	256	33	·	·	PUNCT
ejpam-3764	256	34	·	·	PUNCT
ejpam-3764	256	35	}	}	PUNCT
ejpam-3764	256	36	.	.	PUNCT
ejpam-3764	257	1	if	if	SCONJ
ejpam-3764	257	2	a	a	DET
ejpam-3764	257	3	∈w	∈w	NOUN
ejpam-3764	257	4	\{0	\{0	VERB
ejpam-3764	257	5	}	}	PUNCT
ejpam-3764	257	6	be	be	AUX
ejpam-3764	257	7	arbitrary	arbitrary	ADJ
ejpam-3764	257	8	,	,	PUNCT
ejpam-3764	257	9	then	then	ADV
ejpam-3764	257	10	a−1	a−1	PROPN
ejpam-3764	257	11	does	do	AUX
ejpam-3764	257	12	not	not	PART
ejpam-3764	257	13	exists	exist	VERB
ejpam-3764	257	14	in	in	ADP
ejpam-3764	257	15	(	(	PUNCT
ejpam-3764	257	16	w−∞	w−∞	NOUN
ejpam-3764	257	17	\{−∞},⊗	\{−∞},⊗	NUM
ejpam-3764	257	18	)	)	PUNCT
ejpam-3764	257	19	,	,	PUNCT
ejpam-3764	257	20	where	where	SCONJ
ejpam-3764	257	21	a−1	a−1	PROPN
ejpam-3764	257	22	is	be	AUX
ejpam-3764	257	23	the	the	DET
ejpam-3764	257	24	multiplicative	multiplicative	ADJ
ejpam-3764	257	25	inverse	inverse	NOUN
ejpam-3764	257	26	of	of	ADP
ejpam-3764	257	27	a.	a.	NOUN
ejpam-3764	257	28	hence	hence	ADV
ejpam-3764	257	29	,	,	PUNCT
ejpam-3764	257	30	we	we	PRON
ejpam-3764	257	31	have	have	AUX
ejpam-3764	257	32	τ−∞	τ−∞	VERB
ejpam-3764	257	33	=	=	SYM
ejpam-3764	257	34	{	{	PUNCT
ejpam-3764	257	35	w−∞	w−∞	X
ejpam-3764	257	36	,	,	PUNCT
ejpam-3764	257	37	∅	∅	NOUN
ejpam-3764	257	38	,	,	PUNCT
ejpam-3764	257	39	{	{	PUNCT
ejpam-3764	257	40	−∞	−∞	NOUN
ejpam-3764	257	41	}	}	PUNCT
ejpam-3764	257	42	,	,	PUNCT
ejpam-3764	257	43	{	{	PUNCT
ejpam-3764	257	44	−∞	−∞	NOUN
ejpam-3764	257	45	,	,	PUNCT
ejpam-3764	257	46	0	0	NUM
ejpam-3764	257	47	}	}	PUNCT
ejpam-3764	257	48	}	}	PUNCT
ejpam-3764	257	49	.	.	PUNCT
ejpam-3764	258	1	a	a	DET
ejpam-3764	258	2	direct	direct	ADJ
ejpam-3764	258	3	check	check	NOUN
ejpam-3764	258	4	shows	show	VERB
ejpam-3764	258	5	that	that	SCONJ
ejpam-3764	258	6	(	(	PUNCT
ejpam-3764	258	7	w−∞	w−∞	X
ejpam-3764	258	8	,	,	PUNCT
ejpam-3764	258	9	τ−∞	τ−∞	NOUN
ejpam-3764	258	10	)	)	PUNCT
ejpam-3764	258	11	is	be	AUX
ejpam-3764	258	12	a	a	DET
ejpam-3764	258	13	topological	topological	ADJ
ejpam-3764	258	14	space	space	NOUN
ejpam-3764	258	15	.	.	PUNCT
ejpam-3764	259	1	proposition	proposition	NOUN
ejpam-3764	259	2	13	13	NUM
ejpam-3764	259	3	.	.	PUNCT
ejpam-3764	260	1	the	the	DET
ejpam-3764	260	2	omega	omega	NOUN
ejpam-3764	260	3	topological	topological	ADJ
ejpam-3764	260	4	space	space	NOUN
ejpam-3764	260	5	(	(	PUNCT
ejpam-3764	260	6	w−∞	w−∞	X
ejpam-3764	260	7	,	,	PUNCT
ejpam-3764	260	8	τ−∞	τ−∞	NOUN
ejpam-3764	260	9	)	)	PUNCT
ejpam-3764	260	10	is	be	AUX
ejpam-3764	260	11	second	second	ADV
ejpam-3764	260	12	countable	countable	ADJ
ejpam-3764	260	13	.	.	PUNCT
ejpam-3764	261	1	proof	proof	NOUN
ejpam-3764	261	2	.	.	PUNCT
ejpam-3764	262	1	there	there	PRON
ejpam-3764	262	2	exists	exist	VERB
ejpam-3764	262	3	an	an	DET
ejpam-3764	262	4	element	element	NOUN
ejpam-3764	262	5	2	2	NUM
ejpam-3764	262	6	∈w−∞	∈w−∞	NUM
ejpam-3764	262	7	\	\	NOUN
ejpam-3764	262	8	{	{	PUNCT
ejpam-3764	262	9	−∞	−∞	NOUN
ejpam-3764	262	10	}	}	PUNCT
ejpam-3764	262	11	,	,	PUNCT
ejpam-3764	262	12	which	which	PRON
ejpam-3764	262	13	has	have	VERB
ejpam-3764	262	14	no	no	DET
ejpam-3764	262	15	multiplicative	multiplicative	ADJ
ejpam-3764	262	16	inverse	inverse	NOUN
ejpam-3764	262	17	,	,	PUNCT
ejpam-3764	262	18	then	then	ADV
ejpam-3764	262	19	by	by	ADP
ejpam-3764	262	20	proposition	proposition	NOUN
ejpam-3764	262	21	3	3	NUM
ejpam-3764	262	22	,	,	PUNCT
ejpam-3764	262	23	(	(	PUNCT
ejpam-3764	262	24	w−∞	w−∞	X
ejpam-3764	262	25	,	,	PUNCT
ejpam-3764	262	26	τ−∞	τ−∞	NOUN
ejpam-3764	262	27	)	)	PUNCT
ejpam-3764	262	28	is	be	AUX
ejpam-3764	262	29	second	second	ADJ
ejpam-3764	262	30	countable	countable	ADJ
ejpam-3764	262	31	.	.	PUNCT
ejpam-3764	263	1	corollary	corollary	ADJ
ejpam-3764	263	2	7	7	NUM
ejpam-3764	263	3	.	.	PUNCT
ejpam-3764	264	1	the	the	DET
ejpam-3764	264	2	omega	omega	NOUN
ejpam-3764	264	3	topological	topological	ADJ
ejpam-3764	264	4	space	space	NOUN
ejpam-3764	264	5	(	(	PUNCT
ejpam-3764	264	6	w−∞	w−∞	X
ejpam-3764	264	7	,	,	PUNCT
ejpam-3764	264	8	τ−∞	τ−∞	NOUN
ejpam-3764	264	9	)	)	PUNCT
ejpam-3764	264	10	is	be	AUX
ejpam-3764	264	11	first	first	ADV
ejpam-3764	264	12	countable	countable	ADJ
ejpam-3764	264	13	.	.	PUNCT
ejpam-3764	265	1	corollary	corollary	ADJ
ejpam-3764	265	2	8	8	NUM
ejpam-3764	265	3	.	.	PUNCT
ejpam-3764	266	1	the	the	DET
ejpam-3764	266	2	omega	omega	NOUN
ejpam-3764	266	3	topological	topological	ADJ
ejpam-3764	266	4	space	space	NOUN
ejpam-3764	266	5	(	(	PUNCT
ejpam-3764	266	6	w−∞	w−∞	X
ejpam-3764	266	7	,	,	PUNCT
ejpam-3764	266	8	τ−∞	τ−∞	NOUN
ejpam-3764	266	9	)	)	PUNCT
ejpam-3764	266	10	is	be	AUX
ejpam-3764	266	11	separable	separable	ADJ
ejpam-3764	266	12	.	.	PUNCT
ejpam-3764	267	1	proposition	proposition	NOUN
ejpam-3764	267	2	14	14	NUM
ejpam-3764	267	3	.	.	PUNCT
ejpam-3764	268	1	the	the	DET
ejpam-3764	268	2	omega	omega	NOUN
ejpam-3764	268	3	topological	topological	ADJ
ejpam-3764	268	4	space	space	NOUN
ejpam-3764	268	5	(	(	PUNCT
ejpam-3764	268	6	w−∞	w−∞	X
ejpam-3764	268	7	,	,	PUNCT
ejpam-3764	268	8	τ−∞	τ−∞	NOUN
ejpam-3764	268	9	)	)	PUNCT
ejpam-3764	268	10	is	be	AUX
ejpam-3764	268	11	not	not	PART
ejpam-3764	268	12	t0	t0	NOUN
ejpam-3764	268	13	.	.	PUNCT
ejpam-3764	269	1	proof	proof	NOUN
ejpam-3764	269	2	.	.	PUNCT
ejpam-3764	270	1	there	there	PRON
ejpam-3764	270	2	exists	exist	VERB
ejpam-3764	270	3	(	(	PUNCT
ejpam-3764	270	4	2	2	NUM
ejpam-3764	270	5	6=	6=	SYM
ejpam-3764	270	6	3	3	NUM
ejpam-3764	270	7	)	)	PUNCT
ejpam-3764	270	8	in	in	ADP
ejpam-3764	270	9	w−∞.	w−∞.	NOUN
ejpam-3764	270	10	let	let	VERB
ejpam-3764	270	11	u	u	PRON
ejpam-3764	270	12	be	be	AUX
ejpam-3764	270	13	any	any	DET
ejpam-3764	270	14	open	open	ADJ
ejpam-3764	270	15	set	set	NOUN
ejpam-3764	270	16	,	,	PUNCT
ejpam-3764	270	17	which	which	PRON
ejpam-3764	270	18	either	either	CCONJ
ejpam-3764	270	19	contains	contain	VERB
ejpam-3764	270	20	2	2	NUM
ejpam-3764	270	21	or	or	CCONJ
ejpam-3764	270	22	3	3	NUM
ejpam-3764	270	23	.	.	PUNCT
ejpam-3764	271	1	however	however	ADV
ejpam-3764	271	2	,	,	PUNCT
ejpam-3764	271	3	there	there	PRON
ejpam-3764	271	4	exists	exist	VERB
ejpam-3764	271	5	only	only	ADV
ejpam-3764	271	6	one	one	NUM
ejpam-3764	271	7	open	open	ADJ
ejpam-3764	271	8	set	set	VERB
ejpam-3764	271	9	u	u	NOUN
ejpam-3764	271	10	=	=	NOUN
ejpam-3764	271	11	w−∞	w−∞	NOUN
ejpam-3764	271	12	containing	contain	VERB
ejpam-3764	271	13	2	2	NUM
ejpam-3764	271	14	and	and	CCONJ
ejpam-3764	271	15	3	3	NUM
ejpam-3764	271	16	.	.	PUNCT
ejpam-3764	271	17	hence	hence	ADV
ejpam-3764	271	18	,	,	PUNCT
ejpam-3764	271	19	(	(	PUNCT
ejpam-3764	271	20	w−∞	w−∞	X
ejpam-3764	271	21	,	,	PUNCT
ejpam-3764	271	22	τ−∞	τ−∞	NOUN
ejpam-3764	271	23	)	)	PUNCT
ejpam-3764	271	24	is	be	AUX
ejpam-3764	271	25	not	not	PART
ejpam-3764	271	26	t0	t0	NOUN
ejpam-3764	271	27	.	.	PUNCT
ejpam-3764	272	1	proposition	proposition	NOUN
ejpam-3764	272	2	15	15	NUM
ejpam-3764	272	3	.	.	PUNCT
ejpam-3764	273	1	the	the	DET
ejpam-3764	273	2	omega	omega	NOUN
ejpam-3764	273	3	topological	topological	ADJ
ejpam-3764	273	4	space	space	NOUN
ejpam-3764	273	5	(	(	PUNCT
ejpam-3764	273	6	w−∞	w−∞	X
ejpam-3764	273	7	,	,	PUNCT
ejpam-3764	273	8	τ−∞	τ−∞	NOUN
ejpam-3764	273	9	)	)	PUNCT
ejpam-3764	273	10	is	be	AUX
ejpam-3764	273	11	not	not	PART
ejpam-3764	273	12	regular	regular	ADJ
ejpam-3764	273	13	.	.	PUNCT
ejpam-3764	274	1	proof	proof	NOUN
ejpam-3764	274	2	.	.	PUNCT
ejpam-3764	275	1	we	we	PRON
ejpam-3764	275	2	have	have	VERB
ejpam-3764	275	3	a	a	DET
ejpam-3764	275	4	closed	closed	ADJ
ejpam-3764	275	5	set	set	NOUN
ejpam-3764	275	6	c	c	NOUN
ejpam-3764	275	7	=	=	SYM
ejpam-3764	275	8	w−∞	w−∞	X
ejpam-3764	275	9	\	\	PROPN
ejpam-3764	275	10	{	{	PUNCT
ejpam-3764	275	11	−∞	−∞	NOUN
ejpam-3764	275	12	,	,	PUNCT
ejpam-3764	275	13	0	0	NUM
ejpam-3764	275	14	}	}	PUNCT
ejpam-3764	275	15	and	and	CCONJ
ejpam-3764	275	16	0	0	NUM
ejpam-3764	275	17	/∈	/∈	PUNCT
ejpam-3764	276	1	c	c	X
ejpam-3764	276	2	,	,	PUNCT
ejpam-3764	276	3	such	such	ADJ
ejpam-3764	276	4	that	that	PRON
ejpam-3764	276	5	for	for	SCONJ
ejpam-3764	276	6	any	any	DET
ejpam-3764	276	7	open	open	ADJ
ejpam-3764	276	8	sets	set	NOUN
ejpam-3764	276	9	v1	v1	VERB
ejpam-3764	276	10	and	and	CCONJ
ejpam-3764	276	11	v2	v2	NOUN
ejpam-3764	276	12	containing	contain	VERB
ejpam-3764	276	13	0	0	NUM
ejpam-3764	276	14	and	and	CCONJ
ejpam-3764	276	15	c	c	NOUN
ejpam-3764	276	16	,	,	PUNCT
ejpam-3764	276	17	respectively	respectively	ADV
ejpam-3764	276	18	,	,	PUNCT
ejpam-3764	276	19	we	we	PRON
ejpam-3764	276	20	have	have	VERB
ejpam-3764	276	21	v1	v1	NOUN
ejpam-3764	276	22	∩v2	∩v2	PROPN
ejpam-3764	277	1	6=	6=	ADP
ejpam-3764	277	2	∅.	∅.	ADP
ejpam-3764	277	3	hence	hence	ADV
ejpam-3764	277	4	,	,	PUNCT
ejpam-3764	277	5	(	(	PUNCT
ejpam-3764	277	6	w−∞	w−∞	X
ejpam-3764	277	7	,	,	PUNCT
ejpam-3764	277	8	τ−∞	τ−∞	NOUN
ejpam-3764	277	9	)	)	PUNCT
ejpam-3764	277	10	is	be	AUX
ejpam-3764	277	11	not	not	PART
ejpam-3764	277	12	regular	regular	ADJ
ejpam-3764	277	13	.	.	PUNCT
ejpam-3764	278	1	proposition	proposition	NOUN
ejpam-3764	278	2	16	16	NUM
ejpam-3764	278	3	.	.	PUNCT
ejpam-3764	279	1	the	the	DET
ejpam-3764	279	2	omega	omega	NOUN
ejpam-3764	279	3	topological	topological	ADJ
ejpam-3764	279	4	space	space	NOUN
ejpam-3764	279	5	(	(	PUNCT
ejpam-3764	279	6	w−∞	w−∞	X
ejpam-3764	279	7	,	,	PUNCT
ejpam-3764	279	8	τ−∞	τ−∞	NOUN
ejpam-3764	279	9	)	)	PUNCT
ejpam-3764	279	10	is	be	AUX
ejpam-3764	279	11	normal	normal	ADJ
ejpam-3764	279	12	.	.	PUNCT
ejpam-3764	280	1	m.	m.	PROPN
ejpam-3764	280	2	alqahtani	alqahtani	PROPN
ejpam-3764	280	3	,	,	PUNCT
ejpam-3764	280	4	c.	c.	PROPN
ejpam-3764	280	5	özel	özel	PROPN
ejpam-3764	280	6	,	,	PUNCT
ejpam-3764	280	7	i.	i.	PROPN
ejpam-3764	280	8	alshammari	alshammari	PROPN
ejpam-3764	280	9	/	/	SYM
ejpam-3764	280	10	eur	eur	PROPN
ejpam-3764	280	11	.	.	PUNCT
ejpam-3764	281	1	j.	j.	PROPN
ejpam-3764	281	2	pure	pure	PROPN
ejpam-3764	281	3	appl	appl	PROPN
ejpam-3764	281	4	.	.	PROPN
ejpam-3764	281	5	math	math	PROPN
ejpam-3764	281	6	,	,	PUNCT
ejpam-3764	281	7	13	13	NUM
ejpam-3764	281	8	(	(	PUNCT
ejpam-3764	281	9	3	3	NUM
ejpam-3764	281	10	)	)	PUNCT
ejpam-3764	281	11	(	(	PUNCT
ejpam-3764	281	12	2020	2020	NUM
ejpam-3764	281	13	)	)	PUNCT
ejpam-3764	281	14	,	,	PUNCT
ejpam-3764	281	15	513	513	NUM
ejpam-3764	281	16	-	-	SYM
ejpam-3764	281	17	528	528	NUM
ejpam-3764	281	18	522	522	NUM
ejpam-3764	281	19	proof	proof	NOUN
ejpam-3764	281	20	.	.	PUNCT
ejpam-3764	282	1	let	let	VERB
ejpam-3764	282	2	k1	k1	NOUN
ejpam-3764	282	3	and	and	CCONJ
ejpam-3764	282	4	k2	k2	PROPN
ejpam-3764	282	5	be	be	VERB
ejpam-3764	282	6	any	any	DET
ejpam-3764	282	7	two	two	NUM
ejpam-3764	282	8	closed	closed	ADJ
ejpam-3764	282	9	sets	set	NOUN
ejpam-3764	282	10	,	,	PUNCT
ejpam-3764	282	11	where	where	SCONJ
ejpam-3764	282	12	k1∩k2	k1∩k2	NOUN
ejpam-3764	282	13	=	=	PUNCT
ejpam-3764	282	14	∅	∅	NOUN
ejpam-3764	282	15	and	and	CCONJ
ejpam-3764	282	16	k1,k2	k1,k2	PROPN
ejpam-3764	282	17	⊆w−∞.	⊆w−∞.	PROPN
ejpam-3764	282	18	since	since	SCONJ
ejpam-3764	282	19	all	all	DET
ejpam-3764	282	20	closed	close	VERB
ejpam-3764	282	21	subsets	subset	NOUN
ejpam-3764	282	22	of	of	ADP
ejpam-3764	282	23	w−∞	w−∞	NOUN
ejpam-3764	282	24	are	be	AUX
ejpam-3764	282	25	w−∞	w−∞	NOUN
ejpam-3764	282	26	,	,	PUNCT
ejpam-3764	282	27	∅,w−∞	∅,w−∞	ADP
ejpam-3764	282	28	\	\	PROPN
ejpam-3764	282	29	{	{	PUNCT
ejpam-3764	282	30	−∞	−∞	NOUN
ejpam-3764	282	31	}	}	PUNCT
ejpam-3764	282	32	and	and	CCONJ
ejpam-3764	282	33	w−∞	w−∞	NOUN
ejpam-3764	282	34	\	\	PROPN
ejpam-3764	282	35	{	{	PUNCT
ejpam-3764	282	36	−∞	−∞	NOUN
ejpam-3764	282	37	,	,	PUNCT
ejpam-3764	282	38	0	0	NUM
ejpam-3764	282	39	}	}	PUNCT
ejpam-3764	282	40	,	,	PUNCT
ejpam-3764	282	41	then	then	ADV
ejpam-3764	282	42	k1	k1	PROPN
ejpam-3764	282	43	or	or	CCONJ
ejpam-3764	282	44	k2	k2	NOUN
ejpam-3764	282	45	is	be	AUX
ejpam-3764	282	46	equal	equal	ADJ
ejpam-3764	282	47	∅.	∅.	PRON
ejpam-3764	282	48	if	if	SCONJ
ejpam-3764	282	49	k1	k1	NOUN
ejpam-3764	282	50	=	=	SYM
ejpam-3764	282	51	∅	∅	NOUN
ejpam-3764	282	52	,	,	PUNCT
ejpam-3764	282	53	then	then	ADV
ejpam-3764	282	54	there	there	PRON
ejpam-3764	282	55	exists	exist	VERB
ejpam-3764	282	56	u1	u1	NOUN
ejpam-3764	282	57	=	=	SYM
ejpam-3764	282	58	∅	∅	NOUN
ejpam-3764	282	59	and	and	CCONJ
ejpam-3764	282	60	u2	u2	NOUN
ejpam-3764	282	61	=	=	PUNCT
ejpam-3764	282	62	w−∞	w−∞	NOUN
ejpam-3764	282	63	are	be	AUX
ejpam-3764	282	64	open	open	ADJ
ejpam-3764	282	65	sets	set	NOUN
ejpam-3764	282	66	and	and	CCONJ
ejpam-3764	282	67	u1	u1	NOUN
ejpam-3764	282	68	∩	∩	ADJ
ejpam-3764	282	69	u2	u2	NOUN
ejpam-3764	282	70	=	=	NOUN
ejpam-3764	282	71	∅	∅	NOUN
ejpam-3764	282	72	in	in	ADP
ejpam-3764	282	73	w−∞	w−∞	NOUN
ejpam-3764	282	74	,	,	PUNCT
ejpam-3764	282	75	where	where	SCONJ
ejpam-3764	282	76	k1	k1	NOUN
ejpam-3764	282	77	⊆	⊆	NUM
ejpam-3764	282	78	u1	u1	NOUN
ejpam-3764	282	79	and	and	CCONJ
ejpam-3764	282	80	k2	k2	ADJ
ejpam-3764	282	81	⊆	⊆	NUM
ejpam-3764	282	82	u2	u2	NOUN
ejpam-3764	282	83	.	.	PUNCT
ejpam-3764	283	1	if	if	SCONJ
ejpam-3764	283	2	k2	k2	PROPN
ejpam-3764	283	3	=	=	SYM
ejpam-3764	283	4	∅	∅	NOUN
ejpam-3764	283	5	,	,	PUNCT
ejpam-3764	283	6	then	then	ADV
ejpam-3764	283	7	there	there	PRON
ejpam-3764	283	8	exists	exist	VERB
ejpam-3764	283	9	u1	u1	NOUN
ejpam-3764	283	10	=	=	SYM
ejpam-3764	283	11	∅	∅	NOUN
ejpam-3764	283	12	and	and	CCONJ
ejpam-3764	283	13	u2	u2	NOUN
ejpam-3764	283	14	=	=	PUNCT
ejpam-3764	283	15	w−∞	w−∞	NOUN
ejpam-3764	283	16	are	be	AUX
ejpam-3764	283	17	open	open	ADJ
ejpam-3764	283	18	sets	set	NOUN
ejpam-3764	283	19	and	and	CCONJ
ejpam-3764	283	20	u1	u1	NOUN
ejpam-3764	283	21	∩	∩	ADJ
ejpam-3764	283	22	u2	u2	NOUN
ejpam-3764	283	23	=	=	NOUN
ejpam-3764	283	24	∅	∅	NOUN
ejpam-3764	283	25	in	in	ADP
ejpam-3764	283	26	w−∞	w−∞	NOUN
ejpam-3764	283	27	,	,	PUNCT
ejpam-3764	283	28	where	where	SCONJ
ejpam-3764	283	29	k2	k2	ADJ
ejpam-3764	283	30	⊆	⊆	NUM
ejpam-3764	283	31	u1	u1	NOUN
ejpam-3764	283	32	and	and	CCONJ
ejpam-3764	283	33	k1	k1	NOUN
ejpam-3764	283	34	⊆	⊆	NUM
ejpam-3764	283	35	u2	u2	NOUN
ejpam-3764	283	36	.	.	PUNCT
ejpam-3764	284	1	hence	hence	ADV
ejpam-3764	284	2	,	,	PUNCT
ejpam-3764	284	3	(	(	PUNCT
ejpam-3764	284	4	w−∞	w−∞	X
ejpam-3764	284	5	,	,	PUNCT
ejpam-3764	284	6	τ−∞	τ−∞	NOUN
ejpam-3764	284	7	)	)	PUNCT
ejpam-3764	284	8	is	be	AUX
ejpam-3764	284	9	normal	normal	ADJ
ejpam-3764	284	10	.	.	PUNCT
ejpam-3764	285	1	proposition	proposition	NOUN
ejpam-3764	285	2	17	17	NUM
ejpam-3764	285	3	.	.	PUNCT
ejpam-3764	286	1	the	the	DET
ejpam-3764	286	2	omega	omega	NOUN
ejpam-3764	286	3	topological	topological	ADJ
ejpam-3764	286	4	space	space	NOUN
ejpam-3764	286	5	(	(	PUNCT
ejpam-3764	286	6	w−∞	w−∞	X
ejpam-3764	286	7	,	,	PUNCT
ejpam-3764	286	8	τ−∞	τ−∞	NOUN
ejpam-3764	286	9	)	)	PUNCT
ejpam-3764	286	10	is	be	AUX
ejpam-3764	286	11	hyperconnected	hyperconnecte	VERB
ejpam-3764	286	12	.	.	PUNCT
ejpam-3764	287	1	proof	proof	NOUN
ejpam-3764	287	2	.	.	PUNCT
ejpam-3764	288	1	using	use	VERB
ejpam-3764	288	2	the	the	DET
ejpam-3764	288	3	same	same	ADJ
ejpam-3764	288	4	proof	proof	NOUN
ejpam-3764	288	5	of	of	ADP
ejpam-3764	288	6	proposition	proposition	NOUN
ejpam-3764	288	7	6	6	NUM
ejpam-3764	288	8	.	.	PUNCT
ejpam-3764	288	9	corollary	corollary	ADJ
ejpam-3764	288	10	9	9	NUM
ejpam-3764	288	11	.	.	PUNCT
ejpam-3764	289	1	the	the	DET
ejpam-3764	289	2	omega	omega	NOUN
ejpam-3764	289	3	topological	topological	ADJ
ejpam-3764	289	4	space	space	NOUN
ejpam-3764	289	5	(	(	PUNCT
ejpam-3764	289	6	w−∞	w−∞	X
ejpam-3764	289	7	,	,	PUNCT
ejpam-3764	289	8	τ−∞	τ−∞	NOUN
ejpam-3764	289	9	)	)	PUNCT
ejpam-3764	289	10	is	be	AUX
ejpam-3764	289	11	connected	connect	VERB
ejpam-3764	289	12	.	.	PUNCT
ejpam-3764	290	1	corollary	corollary	ADJ
ejpam-3764	290	2	10	10	NUM
ejpam-3764	290	3	.	.	PUNCT
ejpam-3764	291	1	the	the	DET
ejpam-3764	291	2	omega	omega	NOUN
ejpam-3764	291	3	topological	topological	ADJ
ejpam-3764	291	4	space	space	NOUN
ejpam-3764	291	5	(	(	PUNCT
ejpam-3764	291	6	w−∞	w−∞	X
ejpam-3764	291	7	,	,	PUNCT
ejpam-3764	291	8	τ−∞	τ−∞	NOUN
ejpam-3764	291	9	)	)	PUNCT
ejpam-3764	291	10	is	be	AUX
ejpam-3764	291	11	locally	locally	ADV
ejpam-3764	291	12	connected	connect	VERB
ejpam-3764	291	13	.	.	PUNCT
ejpam-3764	292	1	proposition	proposition	NOUN
ejpam-3764	292	2	18	18	NUM
ejpam-3764	292	3	.	.	PUNCT
ejpam-3764	293	1	the	the	DET
ejpam-3764	293	2	omega	omega	NOUN
ejpam-3764	293	3	topological	topological	ADJ
ejpam-3764	293	4	space	space	NOUN
ejpam-3764	293	5	(	(	PUNCT
ejpam-3764	293	6	w−∞	w−∞	X
ejpam-3764	293	7	,	,	PUNCT
ejpam-3764	293	8	τ−∞	τ−∞	NOUN
ejpam-3764	293	9	)	)	PUNCT
ejpam-3764	293	10	is	be	AUX
ejpam-3764	293	11	compact	compact	ADJ
ejpam-3764	293	12	.	.	PUNCT
ejpam-3764	294	1	proof	proof	NOUN
ejpam-3764	294	2	.	.	PUNCT
ejpam-3764	295	1	there	there	PRON
ejpam-3764	295	2	exists	exist	VERB
ejpam-3764	295	3	an	an	DET
ejpam-3764	295	4	element	element	NOUN
ejpam-3764	295	5	2	2	NUM
ejpam-3764	295	6	∈w−∞	∈w−∞	NUM
ejpam-3764	295	7	\	\	NOUN
ejpam-3764	295	8	{	{	PUNCT
ejpam-3764	295	9	−∞	−∞	NOUN
ejpam-3764	295	10	}	}	PUNCT
ejpam-3764	295	11	,	,	PUNCT
ejpam-3764	295	12	which	which	PRON
ejpam-3764	295	13	has	have	VERB
ejpam-3764	295	14	no	no	DET
ejpam-3764	295	15	multiplicative	multiplicative	ADJ
ejpam-3764	295	16	inverse	inverse	NOUN
ejpam-3764	295	17	.	.	PUNCT
ejpam-3764	296	1	hence	hence	ADV
ejpam-3764	296	2	,	,	PUNCT
ejpam-3764	296	3	by	by	ADP
ejpam-3764	296	4	proposition	proposition	NOUN
ejpam-3764	296	5	12	12	NUM
ejpam-3764	296	6	,	,	PUNCT
ejpam-3764	296	7	(	(	PUNCT
ejpam-3764	296	8	w−∞	w−∞	X
ejpam-3764	296	9	,	,	PUNCT
ejpam-3764	296	10	τ−∞	τ−∞	NOUN
ejpam-3764	296	11	)	)	PUNCT
ejpam-3764	296	12	is	be	AUX
ejpam-3764	296	13	compact	compact	ADJ
ejpam-3764	296	14	.	.	PUNCT
ejpam-3764	297	1	corollary	corollary	ADJ
ejpam-3764	297	2	11	11	NUM
ejpam-3764	297	3	.	.	PUNCT
ejpam-3764	298	1	the	the	DET
ejpam-3764	298	2	omega	omega	NOUN
ejpam-3764	298	3	topological	topological	ADJ
ejpam-3764	298	4	space	space	NOUN
ejpam-3764	298	5	(	(	PUNCT
ejpam-3764	298	6	w−∞	w−∞	X
ejpam-3764	298	7	,	,	PUNCT
ejpam-3764	298	8	τ−∞	τ−∞	NOUN
ejpam-3764	298	9	)	)	PUNCT
ejpam-3764	298	10	is	be	AUX
ejpam-3764	298	11	countably	countably	ADV
ejpam-3764	298	12	compact	compact	ADJ
ejpam-3764	298	13	.	.	PUNCT
ejpam-3764	299	1	corollary	corollary	ADJ
ejpam-3764	299	2	12	12	NUM
ejpam-3764	299	3	.	.	PUNCT
ejpam-3764	300	1	the	the	DET
ejpam-3764	300	2	omega	omega	NOUN
ejpam-3764	300	3	topological	topological	ADJ
ejpam-3764	300	4	space	space	NOUN
ejpam-3764	300	5	(	(	PUNCT
ejpam-3764	300	6	w−∞	w−∞	X
ejpam-3764	300	7	,	,	PUNCT
ejpam-3764	300	8	τ−∞	τ−∞	NOUN
ejpam-3764	300	9	)	)	PUNCT
ejpam-3764	300	10	is	be	AUX
ejpam-3764	300	11	lindelöf	lindelöf	PROPN
ejpam-3764	300	12	.	.	PUNCT
ejpam-3764	300	13	example	example	NOUN
ejpam-3764	301	1	5	5	NUM
ejpam-3764	301	2	.	.	PUNCT
ejpam-3764	301	3	by	by	ADP
ejpam-3764	301	4	example	example	NOUN
ejpam-3764	301	5	2	2	NUM
ejpam-3764	301	6	,	,	PUNCT
ejpam-3764	301	7	(	(	PUNCT
ejpam-3764	301	8	aρ0	aρ0	INTJ
ejpam-3764	301	9	,	,	PUNCT
ejpam-3764	301	10	τρ0	τρ0	PROPN
ejpam-3764	301	11	)	)	PUNCT
ejpam-3764	301	12	is	be	AUX
ejpam-3764	301	13	a	a	DET
ejpam-3764	301	14	topological	topological	ADJ
ejpam-3764	301	15	space	space	NOUN
ejpam-3764	301	16	.	.	PUNCT
ejpam-3764	302	1	if	if	SCONJ
ejpam-3764	302	2	ρx	ρx	VERB
ejpam-3764	302	3	∈	∈	PROPN
ejpam-3764	302	4	aρ0\{ρ0	aρ0\{ρ0	ADJ
ejpam-3764	302	5	}	}	PUNCT
ejpam-3764	302	6	is	be	AUX
ejpam-3764	302	7	arbitrary	arbitrary	ADJ
ejpam-3764	302	8	,	,	PUNCT
ejpam-3764	302	9	then	then	ADV
ejpam-3764	302	10	ρ−1x	ρ−1x	NOUN
ejpam-3764	302	11	does	do	AUX
ejpam-3764	302	12	not	not	PART
ejpam-3764	302	13	exists	exist	VERB
ejpam-3764	302	14	in	in	ADP
ejpam-3764	302	15	(	(	PUNCT
ejpam-3764	302	16	aρ0	aρ0	PROPN
ejpam-3764	302	17	\	\	NOUN
ejpam-3764	302	18	{	{	PUNCT
ejpam-3764	302	19	ρ0},⊗	ρ0},⊗	NOUN
ejpam-3764	302	20	)	)	PUNCT
ejpam-3764	302	21	,	,	PUNCT
ejpam-3764	302	22	where	where	SCONJ
ejpam-3764	302	23	ρ−1x	ρ−1x	NOUN
ejpam-3764	302	24	is	be	AUX
ejpam-3764	302	25	the	the	DET
ejpam-3764	302	26	multiplicative	multiplicative	ADJ
ejpam-3764	302	27	inverse	inverse	NOUN
ejpam-3764	302	28	of	of	ADP
ejpam-3764	302	29	ρx	ρx	PROPN
ejpam-3764	302	30	(	(	PUNCT
ejpam-3764	302	31	because	because	SCONJ
ejpam-3764	302	32	ρx	ρx	PROPN
ejpam-3764	302	33	⊗	⊗	PROPN
ejpam-3764	302	34	ρ−1x	ρ−1x	NOUN
ejpam-3764	303	1	=	=	SYM
ejpam-3764	303	2	ρx	ρx	VERB
ejpam-3764	303	3	⊗	⊗	PROPN
ejpam-3764	303	4	ρx−1	ρx−1	PROPN
ejpam-3764	303	5	=	=	SYM
ejpam-3764	303	6	ρx+x−1	ρx+x−1	NOUN
ejpam-3764	303	7	=	=	PROPN
ejpam-3764	303	8	ρx+(−x	ρx+(−x	PROPN
ejpam-3764	303	9	)	)	PUNCT
ejpam-3764	304	1	=	=	SYM
ejpam-3764	304	2	ρ0	ρ0	PROPN
ejpam-3764	304	3	and	and	CCONJ
ejpam-3764	304	4	−x	−x	NOUN
ejpam-3764	304	5	/∈w	/∈w	PUNCT
ejpam-3764	304	6	)	)	PUNCT
ejpam-3764	304	7	.	.	PUNCT
ejpam-3764	305	1	then	then	ADV
ejpam-3764	305	2	we	we	PRON
ejpam-3764	305	3	have	have	VERB
ejpam-3764	305	4	τρ0	τρ0	PUNCT
ejpam-3764	306	1	=	=	PUNCT
ejpam-3764	306	2	{	{	PUNCT
ejpam-3764	306	3	aρ0	aρ0	INTJ
ejpam-3764	306	4	,	,	PUNCT
ejpam-3764	306	5	∅	∅	NOUN
ejpam-3764	306	6	,	,	PUNCT
ejpam-3764	306	7	{	{	PUNCT
ejpam-3764	306	8	ρ0	ρ0	PROPN
ejpam-3764	306	9	}	}	PUNCT
ejpam-3764	306	10	}	}	PUNCT
ejpam-3764	306	11	.	.	PUNCT
ejpam-3764	307	1	a	a	DET
ejpam-3764	307	2	direct	direct	ADJ
ejpam-3764	307	3	check	check	NOUN
ejpam-3764	307	4	shows	show	VERB
ejpam-3764	307	5	that	that	SCONJ
ejpam-3764	307	6	(	(	PUNCT
ejpam-3764	307	7	aρ0	aρ0	INTJ
ejpam-3764	307	8	,	,	PUNCT
ejpam-3764	307	9	τρ0	τρ0	PROPN
ejpam-3764	307	10	)	)	PUNCT
ejpam-3764	307	11	is	be	AUX
ejpam-3764	307	12	a	a	DET
ejpam-3764	307	13	topological	topological	ADJ
ejpam-3764	307	14	space	space	NOUN
ejpam-3764	307	15	.	.	PUNCT
ejpam-3764	308	1	remark	remark	PROPN
ejpam-3764	308	2	4	4	NUM
ejpam-3764	308	3	.	.	PUNCT
ejpam-3764	309	1	the	the	DET
ejpam-3764	309	2	omega	omega	NOUN
ejpam-3764	309	3	topological	topological	ADJ
ejpam-3764	309	4	space	space	NOUN
ejpam-3764	309	5	(	(	PUNCT
ejpam-3764	309	6	aρ0	aρ0	INTJ
ejpam-3764	309	7	,	,	PUNCT
ejpam-3764	309	8	τρ0	τρ0	PROPN
ejpam-3764	309	9	)	)	PUNCT
ejpam-3764	309	10	is	be	AUX
ejpam-3764	309	11	second	second	ADV
ejpam-3764	309	12	countable	countable	ADJ
ejpam-3764	309	13	,	,	PUNCT
ejpam-3764	309	14	first	first	ADV
ejpam-3764	309	15	countable	countable	ADJ
ejpam-3764	309	16	,	,	PUNCT
ejpam-3764	309	17	separable	separable	ADJ
ejpam-3764	309	18	,	,	PUNCT
ejpam-3764	309	19	normal	normal	ADJ
ejpam-3764	309	20	,	,	PUNCT
ejpam-3764	309	21	hyperconnected	hyperconnecte	VERB
ejpam-3764	309	22	,	,	PUNCT
ejpam-3764	309	23	connected	connect	VERB
ejpam-3764	309	24	,	,	PUNCT
ejpam-3764	309	25	locally	locally	ADV
ejpam-3764	309	26	connected	connect	VERB
ejpam-3764	309	27	,	,	PUNCT
ejpam-3764	309	28	compact	compact	ADJ
ejpam-3764	309	29	,	,	PUNCT
ejpam-3764	309	30	countably	countably	ADV
ejpam-3764	309	31	compact	compact	ADJ
ejpam-3764	309	32	,	,	PUNCT
ejpam-3764	309	33	lindelöf	lindelöf	PROPN
ejpam-3764	309	34	,	,	PUNCT
ejpam-3764	309	35	does	do	AUX
ejpam-3764	309	36	not	not	PART
ejpam-3764	309	37	satisfy	satisfy	VERB
ejpam-3764	309	38	t0	t0	PROPN
ejpam-3764	309	39	and	and	CCONJ
ejpam-3764	309	40	regular	regular	ADJ
ejpam-3764	309	41	.	.	PUNCT
ejpam-3764	310	1	proposition	proposition	NOUN
ejpam-3764	310	2	19	19	NUM
ejpam-3764	310	3	.	.	PUNCT
ejpam-3764	311	1	the	the	DET
ejpam-3764	311	2	omega	omega	NOUN
ejpam-3764	311	3	topological	topological	ADJ
ejpam-3764	311	4	space	space	NOUN
ejpam-3764	311	5	(	(	PUNCT
ejpam-3764	311	6	aρ0	aρ0	INTJ
ejpam-3764	311	7	,	,	PUNCT
ejpam-3764	311	8	τρ0	τρ0	PROPN
ejpam-3764	311	9	)	)	PUNCT
ejpam-3764	311	10	is	be	AUX
ejpam-3764	311	11	not	not	PART
ejpam-3764	311	12	homeomorphic	homeomorphic	ADJ
ejpam-3764	311	13	to	to	ADP
ejpam-3764	311	14	(	(	PUNCT
ejpam-3764	311	15	w−∞	w−∞	X
ejpam-3764	311	16	,	,	PUNCT
ejpam-3764	311	17	τ−∞	τ−∞	NOUN
ejpam-3764	311	18	)	)	PUNCT
ejpam-3764	311	19	.	.	PUNCT
ejpam-3764	312	1	proof	proof	NOUN
ejpam-3764	312	2	.	.	PUNCT
ejpam-3764	313	1	there	there	PRON
ejpam-3764	313	2	exists	exist	VERB
ejpam-3764	313	3	an	an	DET
ejpam-3764	313	4	open	open	ADJ
ejpam-3764	313	5	set	set	NOUN
ejpam-3764	313	6	in	in	ADP
ejpam-3764	313	7	w−∞	w−∞	NOUN
ejpam-3764	313	8	,	,	PUNCT
ejpam-3764	313	9	which	which	PRON
ejpam-3764	313	10	consists	consist	VERB
ejpam-3764	313	11	two	two	NUM
ejpam-3764	313	12	elements	element	NOUN
ejpam-3764	313	13	,	,	PUNCT
ejpam-3764	313	14	and	and	CCONJ
ejpam-3764	313	15	such	such	ADJ
ejpam-3764	313	16	that	that	SCONJ
ejpam-3764	313	17	an	an	DET
ejpam-3764	313	18	open	open	ADJ
ejpam-3764	313	19	set	set	NOUN
ejpam-3764	313	20	does	do	AUX
ejpam-3764	313	21	not	not	PART
ejpam-3764	313	22	exists	exist	VERB
ejpam-3764	313	23	in	in	ADP
ejpam-3764	313	24	aρ0	aρ0	PROPN
ejpam-3764	313	25	.	.	PUNCT
ejpam-3764	314	1	example	example	NOUN
ejpam-3764	315	1	6	6	NUM
ejpam-3764	315	2	.	.	PUNCT
ejpam-3764	315	3	by	by	ADP
ejpam-3764	315	4	example	example	NOUN
ejpam-3764	315	5	3	3	NUM
ejpam-3764	315	6	,	,	PUNCT
ejpam-3764	315	7	(	(	PUNCT
ejpam-3764	315	8	a11	a11	PROPN
ejpam-3764	315	9	,	,	PUNCT
ejpam-3764	315	10	τ11	τ11	PROPN
ejpam-3764	315	11	)	)	PUNCT
ejpam-3764	315	12	is	be	AUX
ejpam-3764	315	13	a	a	DET
ejpam-3764	315	14	topological	topological	ADJ
ejpam-3764	315	15	space	space	NOUN
ejpam-3764	315	16	.	.	PUNCT
ejpam-3764	316	1	let	let	VERB
ejpam-3764	316	2	a	a	DET
ejpam-3764	316	3	∈	∈	PROPN
ejpam-3764	316	4	a11	a11	PROPN
ejpam-3764	316	5	\	\	PROPN
ejpam-3764	316	6	{	{	PUNCT
ejpam-3764	316	7	11	11	NUM
ejpam-3764	316	8	}	}	PUNCT
ejpam-3764	316	9	be	be	AUX
ejpam-3764	316	10	arbitrary	arbitrary	ADJ
ejpam-3764	316	11	.	.	PUNCT
ejpam-3764	317	1	if	if	SCONJ
ejpam-3764	317	2	a	a	DET
ejpam-3764	317	3	=	=	NOUN
ejpam-3764	317	4	00	00	NUM
ejpam-3764	317	5	,	,	PUNCT
ejpam-3764	317	6	then	then	ADV
ejpam-3764	317	7	the	the	DET
ejpam-3764	317	8	multiplicative	multiplicative	ADJ
ejpam-3764	317	9	inverse	inverse	NOUN
ejpam-3764	317	10	of	of	ADP
ejpam-3764	317	11	a	a	PRON
ejpam-3764	317	12	in	in	ADP
ejpam-3764	317	13	a11	a11	PROPN
ejpam-3764	317	14	is	be	AUX
ejpam-3764	317	15	00	00	NUM
ejpam-3764	317	16	.	.	PUNCT
ejpam-3764	318	1	if	if	SCONJ
ejpam-3764	318	2	a	a	DET
ejpam-3764	318	3	=	=	NOUN
ejpam-3764	318	4	01	01	NUM
ejpam-3764	318	5	,	,	PUNCT
ejpam-3764	318	6	then	then	ADV
ejpam-3764	318	7	the	the	DET
ejpam-3764	318	8	multiplicative	multiplicative	ADJ
ejpam-3764	318	9	inverse	inverse	NOUN
ejpam-3764	318	10	of	of	ADP
ejpam-3764	318	11	a	a	PRON
ejpam-3764	318	12	in	in	ADP
ejpam-3764	318	13	a11	a11	PROPN
ejpam-3764	318	14	does	do	AUX
ejpam-3764	318	15	not	not	PART
ejpam-3764	318	16	exists	exist	VERB
ejpam-3764	318	17	.	.	PUNCT
ejpam-3764	319	1	then	then	ADV
ejpam-3764	319	2	we	we	PRON
ejpam-3764	319	3	have	have	VERB
ejpam-3764	319	4	τ11	τ11	NUM
ejpam-3764	319	5	=	=	SYM
ejpam-3764	319	6	{	{	PUNCT
ejpam-3764	319	7	a11	a11	PROPN
ejpam-3764	319	8	,	,	PUNCT
ejpam-3764	319	9	∅	∅	NOUN
ejpam-3764	319	10	,	,	PUNCT
ejpam-3764	319	11	{	{	PUNCT
ejpam-3764	319	12	11	11	NUM
ejpam-3764	319	13	}	}	PUNCT
ejpam-3764	319	14	,	,	PUNCT
ejpam-3764	319	15	{	{	PUNCT
ejpam-3764	319	16	11	11	NUM
ejpam-3764	319	17	,	,	PUNCT
ejpam-3764	319	18	00	00	NUM
ejpam-3764	319	19	}	}	PUNCT
ejpam-3764	319	20	}	}	PUNCT
ejpam-3764	319	21	.	.	PUNCT
ejpam-3764	320	1	a	a	DET
ejpam-3764	320	2	direct	direct	ADJ
ejpam-3764	320	3	check	check	NOUN
ejpam-3764	320	4	shows	show	VERB
ejpam-3764	320	5	that	that	SCONJ
ejpam-3764	320	6	(	(	PUNCT
ejpam-3764	320	7	a11	a11	PROPN
ejpam-3764	320	8	,	,	PUNCT
ejpam-3764	320	9	τ11	τ11	PROPN
ejpam-3764	320	10	)	)	PUNCT
ejpam-3764	320	11	is	be	AUX
ejpam-3764	320	12	a	a	DET
ejpam-3764	320	13	topological	topological	ADJ
ejpam-3764	320	14	space	space	NOUN
ejpam-3764	320	15	.	.	PUNCT
ejpam-3764	321	1	m.	m.	PROPN
ejpam-3764	321	2	alqahtani	alqahtani	PROPN
ejpam-3764	321	3	,	,	PUNCT
ejpam-3764	321	4	c.	c.	PROPN
ejpam-3764	321	5	özel	özel	PROPN
ejpam-3764	321	6	,	,	PUNCT
ejpam-3764	321	7	i.	i.	PROPN
ejpam-3764	321	8	alshammari	alshammari	PROPN
ejpam-3764	321	9	/	/	SYM
ejpam-3764	321	10	eur	eur	PROPN
ejpam-3764	321	11	.	.	PUNCT
ejpam-3764	322	1	j.	j.	PROPN
ejpam-3764	322	2	pure	pure	PROPN
ejpam-3764	322	3	appl	appl	PROPN
ejpam-3764	322	4	.	.	PROPN
ejpam-3764	322	5	math	math	PROPN
ejpam-3764	322	6	,	,	PUNCT
ejpam-3764	322	7	13	13	NUM
ejpam-3764	322	8	(	(	PUNCT
ejpam-3764	322	9	3	3	NUM
ejpam-3764	322	10	)	)	PUNCT
ejpam-3764	322	11	(	(	PUNCT
ejpam-3764	322	12	2020	2020	NUM
ejpam-3764	322	13	)	)	PUNCT
ejpam-3764	322	14	,	,	PUNCT
ejpam-3764	322	15	513	513	NUM
ejpam-3764	322	16	-	-	SYM
ejpam-3764	322	17	528	528	NUM
ejpam-3764	322	18	523	523	NUM
ejpam-3764	322	19	proposition	proposition	NOUN
ejpam-3764	322	20	20	20	NUM
ejpam-3764	322	21	.	.	PUNCT
ejpam-3764	323	1	the	the	DET
ejpam-3764	323	2	omega	omega	NOUN
ejpam-3764	323	3	topological	topological	ADJ
ejpam-3764	323	4	space	space	NOUN
ejpam-3764	323	5	(	(	PUNCT
ejpam-3764	323	6	a11	a11	PROPN
ejpam-3764	323	7	,	,	PUNCT
ejpam-3764	323	8	τ11	τ11	PROPN
ejpam-3764	323	9	)	)	PUNCT
ejpam-3764	323	10	is	be	AUX
ejpam-3764	323	11	t0	t0	PROPN
ejpam-3764	323	12	and	and	CCONJ
ejpam-3764	323	13	does	do	AUX
ejpam-3764	323	14	not	not	PART
ejpam-3764	323	15	satisfy	satisfy	VERB
ejpam-3764	323	16	t1	t1	NOUN
ejpam-3764	323	17	.	.	PUNCT
ejpam-3764	324	1	proof	proof	NOUN
ejpam-3764	324	2	.	.	PUNCT
ejpam-3764	325	1	if	if	SCONJ
ejpam-3764	325	2	the	the	DET
ejpam-3764	325	3	space	space	NOUN
ejpam-3764	325	4	a11	a11	PROPN
ejpam-3764	325	5	consists	consist	VERB
ejpam-3764	325	6	of	of	ADP
ejpam-3764	325	7	three	three	NUM
ejpam-3764	325	8	elements	element	NOUN
ejpam-3764	325	9	00	00	NUM
ejpam-3764	325	10	,	,	PUNCT
ejpam-3764	325	11	01	01	NUM
ejpam-3764	325	12	and	and	CCONJ
ejpam-3764	325	13	11	11	NUM
ejpam-3764	325	14	then	then	ADV
ejpam-3764	325	15	we	we	PRON
ejpam-3764	325	16	have	have	VERB
ejpam-3764	325	17	three	three	NUM
ejpam-3764	325	18	cases	case	NOUN
ejpam-3764	325	19	:	:	PUNCT
ejpam-3764	325	20	case	case	NOUN
ejpam-3764	325	21	1	1	NUM
ejpam-3764	325	22	:	:	PUNCT
ejpam-3764	325	23	if	if	SCONJ
ejpam-3764	325	24	00	00	NUM
ejpam-3764	325	25	6=	6=	NUM
ejpam-3764	325	26	01	01	NUM
ejpam-3764	325	27	in	in	ADP
ejpam-3764	325	28	a11	a11	PROPN
ejpam-3764	325	29	,	,	PUNCT
ejpam-3764	325	30	then	then	ADV
ejpam-3764	325	31	we	we	PRON
ejpam-3764	325	32	have	have	VERB
ejpam-3764	325	33	{	{	PUNCT
ejpam-3764	325	34	00	00	NUM
ejpam-3764	325	35	,	,	PUNCT
ejpam-3764	325	36	11	11	NUM
ejpam-3764	325	37	}	}	PUNCT
ejpam-3764	325	38	an	an	DET
ejpam-3764	325	39	open	open	ADJ
ejpam-3764	325	40	set	set	NOUN
ejpam-3764	325	41	,	,	PUNCT
ejpam-3764	325	42	where	where	SCONJ
ejpam-3764	325	43	00	00	PUNCT
ejpam-3764	325	44	∈	∈	PROPN
ejpam-3764	325	45	{	{	PUNCT
ejpam-3764	325	46	00	00	NUM
ejpam-3764	325	47	,	,	PUNCT
ejpam-3764	325	48	11	11	NUM
ejpam-3764	325	49	}	}	PUNCT
ejpam-3764	325	50	and	and	CCONJ
ejpam-3764	325	51	01	01	NUM
ejpam-3764	325	52	/∈	/∈	PUNCT
ejpam-3764	325	53	{	{	PUNCT
ejpam-3764	325	54	00	00	NUM
ejpam-3764	325	55	,	,	PUNCT
ejpam-3764	325	56	11	11	NUM
ejpam-3764	325	57	}	}	PUNCT
ejpam-3764	325	58	.	.	PUNCT
ejpam-3764	326	1	case	case	NOUN
ejpam-3764	326	2	2	2	NUM
ejpam-3764	326	3	:	:	PUNCT
ejpam-3764	326	4	if	if	SCONJ
ejpam-3764	326	5	00	00	NUM
ejpam-3764	326	6	6=	6=	NUM
ejpam-3764	326	7	11	11	NUM
ejpam-3764	326	8	in	in	ADP
ejpam-3764	326	9	a11	a11	PROPN
ejpam-3764	326	10	,	,	PUNCT
ejpam-3764	326	11	then	then	ADV
ejpam-3764	326	12	we	we	PRON
ejpam-3764	326	13	have	have	VERB
ejpam-3764	326	14	{	{	PUNCT
ejpam-3764	326	15	11	11	NUM
ejpam-3764	326	16	}	}	PUNCT
ejpam-3764	326	17	an	an	DET
ejpam-3764	326	18	open	open	ADJ
ejpam-3764	326	19	set	set	NOUN
ejpam-3764	326	20	,	,	PUNCT
ejpam-3764	326	21	where	where	SCONJ
ejpam-3764	326	22	11	11	NUM
ejpam-3764	326	23	∈	∈	NOUN
ejpam-3764	326	24	{	{	PUNCT
ejpam-3764	326	25	11	11	NUM
ejpam-3764	326	26	}	}	PUNCT
ejpam-3764	326	27	and	and	CCONJ
ejpam-3764	326	28	00	00	NUM
ejpam-3764	326	29	/∈	/∈	PUNCT
ejpam-3764	327	1	{	{	PUNCT
ejpam-3764	327	2	11	11	NUM
ejpam-3764	327	3	}	}	PUNCT
ejpam-3764	327	4	.	.	PUNCT
ejpam-3764	328	1	case	case	NOUN
ejpam-3764	328	2	3	3	NUM
ejpam-3764	328	3	:	:	PUNCT
ejpam-3764	328	4	if	if	SCONJ
ejpam-3764	328	5	01	01	NUM
ejpam-3764	328	6	6=	6=	NUM
ejpam-3764	328	7	11	11	NUM
ejpam-3764	328	8	in	in	ADP
ejpam-3764	328	9	a11	a11	PROPN
ejpam-3764	328	10	,	,	PUNCT
ejpam-3764	328	11	then	then	ADV
ejpam-3764	328	12	we	we	PRON
ejpam-3764	328	13	have	have	VERB
ejpam-3764	328	14	{	{	PUNCT
ejpam-3764	328	15	11	11	NUM
ejpam-3764	328	16	}	}	PUNCT
ejpam-3764	328	17	an	an	DET
ejpam-3764	328	18	open	open	ADJ
ejpam-3764	328	19	set	set	NOUN
ejpam-3764	328	20	,	,	PUNCT
ejpam-3764	328	21	where	where	SCONJ
ejpam-3764	328	22	11	11	NUM
ejpam-3764	328	23	∈	∈	NOUN
ejpam-3764	328	24	{	{	PUNCT
ejpam-3764	328	25	11	11	NUM
ejpam-3764	328	26	}	}	PUNCT
ejpam-3764	328	27	and	and	CCONJ
ejpam-3764	328	28	01	01	NUM
ejpam-3764	328	29	/∈	/∈	PUNCT
ejpam-3764	328	30	{	{	PUNCT
ejpam-3764	328	31	11	11	NUM
ejpam-3764	328	32	}	}	PUNCT
ejpam-3764	328	33	.	.	PUNCT
ejpam-3764	329	1	hence	hence	ADV
ejpam-3764	329	2	,	,	PUNCT
ejpam-3764	329	3	(	(	PUNCT
ejpam-3764	329	4	a11	a11	PROPN
ejpam-3764	329	5	,	,	PUNCT
ejpam-3764	329	6	τ11	τ11	PROPN
ejpam-3764	329	7	)	)	PUNCT
ejpam-3764	329	8	is	be	AUX
ejpam-3764	329	9	t0	t0	PROPN
ejpam-3764	329	10	.	.	PUNCT
ejpam-3764	330	1	suppose	suppose	VERB
ejpam-3764	330	2	that	that	SCONJ
ejpam-3764	330	3	(	(	PUNCT
ejpam-3764	330	4	a11	a11	PROPN
ejpam-3764	330	5	,	,	PUNCT
ejpam-3764	330	6	τ11	τ11	PROPN
ejpam-3764	330	7	)	)	PUNCT
ejpam-3764	330	8	is	be	AUX
ejpam-3764	330	9	t1	t1	NOUN
ejpam-3764	330	10	,	,	PUNCT
ejpam-3764	330	11	then	then	ADV
ejpam-3764	330	12	{	{	PUNCT
ejpam-3764	330	13	11	11	NUM
ejpam-3764	330	14	}	}	PUNCT
ejpam-3764	330	15	is	be	AUX
ejpam-3764	330	16	closed	closed	ADJ
ejpam-3764	330	17	.	.	PUNCT
ejpam-3764	331	1	however	however	ADV
ejpam-3764	331	2	,	,	PUNCT
ejpam-3764	331	3	a11	a11	PROPN
ejpam-3764	331	4	\	\	PROPN
ejpam-3764	331	5	{	{	PUNCT
ejpam-3764	331	6	11	11	NUM
ejpam-3764	331	7	}	}	PUNCT
ejpam-3764	331	8	=	=	PRON
ejpam-3764	331	9	{	{	PUNCT
ejpam-3764	331	10	00	00	NUM
ejpam-3764	331	11	,	,	PUNCT
ejpam-3764	331	12	01	01	NUM
ejpam-3764	331	13	}	}	PUNCT
ejpam-3764	331	14	is	be	AUX
ejpam-3764	331	15	not	not	PART
ejpam-3764	331	16	open	open	ADJ
ejpam-3764	331	17	,	,	PUNCT
ejpam-3764	331	18	thus	thus	ADV
ejpam-3764	331	19	a	a	DET
ejpam-3764	331	20	contradiction	contradiction	NOUN
ejpam-3764	331	21	.	.	PUNCT
ejpam-3764	332	1	then	then	ADV
ejpam-3764	332	2	(	(	PUNCT
ejpam-3764	332	3	a11	a11	PROPN
ejpam-3764	332	4	,	,	PUNCT
ejpam-3764	332	5	τ11	τ11	PROPN
ejpam-3764	332	6	)	)	PUNCT
ejpam-3764	332	7	is	be	AUX
ejpam-3764	332	8	not	not	PART
ejpam-3764	332	9	t1	t1	NOUN
ejpam-3764	332	10	.	.	PUNCT
ejpam-3764	333	1	remark	remark	PROPN
ejpam-3764	333	2	5	5	NUM
ejpam-3764	333	3	.	.	PUNCT
ejpam-3764	334	1	the	the	DET
ejpam-3764	334	2	omega	omega	NOUN
ejpam-3764	334	3	topological	topological	ADJ
ejpam-3764	334	4	space	space	NOUN
ejpam-3764	334	5	(	(	PUNCT
ejpam-3764	334	6	a11	a11	PROPN
ejpam-3764	334	7	,	,	PUNCT
ejpam-3764	334	8	τ11	τ11	PROPN
ejpam-3764	334	9	)	)	PUNCT
ejpam-3764	334	10	is	be	AUX
ejpam-3764	334	11	second	second	ADV
ejpam-3764	334	12	countable	countable	ADJ
ejpam-3764	334	13	,	,	PUNCT
ejpam-3764	334	14	first	first	ADV
ejpam-3764	334	15	countable	countable	ADJ
ejpam-3764	334	16	,	,	PUNCT
ejpam-3764	334	17	separable	separable	ADJ
ejpam-3764	334	18	,	,	PUNCT
ejpam-3764	334	19	not	not	PART
ejpam-3764	334	20	regular	regular	ADJ
ejpam-3764	334	21	,	,	PUNCT
ejpam-3764	334	22	normal	normal	ADJ
ejpam-3764	334	23	,	,	PUNCT
ejpam-3764	334	24	hyperconnected	hyperconnecte	VERB
ejpam-3764	334	25	,	,	PUNCT
ejpam-3764	334	26	connected	connect	VERB
ejpam-3764	334	27	,	,	PUNCT
ejpam-3764	334	28	locally	locally	ADV
ejpam-3764	334	29	connected	connect	VERB
ejpam-3764	334	30	,	,	PUNCT
ejpam-3764	334	31	compact	compact	ADJ
ejpam-3764	334	32	,	,	PUNCT
ejpam-3764	334	33	countably	countably	ADV
ejpam-3764	334	34	compact	compact	ADJ
ejpam-3764	334	35	and	and	CCONJ
ejpam-3764	334	36	lindelöf	lindelöf	PROPN
ejpam-3764	334	37	.	.	PUNCT
ejpam-3764	334	38	example	example	NOUN
ejpam-3764	335	1	7	7	NUM
ejpam-3764	335	2	.	.	PUNCT
ejpam-3764	336	1	in	in	ADP
ejpam-3764	336	2	the	the	DET
ejpam-3764	336	3	ring	ring	NOUN
ejpam-3764	336	4	(	(	PUNCT
ejpam-3764	336	5	r,+	r,+	NUM
ejpam-3764	336	6	,	,	PUNCT
ejpam-3764	336	7	·	·	PUNCT
ejpam-3764	336	8	)	)	PUNCT
ejpam-3764	336	9	,	,	PUNCT
ejpam-3764	336	10	(	(	PUNCT
ejpam-3764	336	11	r,+	r,+	X
ejpam-3764	336	12	)	)	PUNCT
ejpam-3764	336	13	is	be	AUX
ejpam-3764	336	14	an	an	DET
ejpam-3764	336	15	additive	additive	ADJ
ejpam-3764	336	16	submonoid	submonoid	NOUN
ejpam-3764	336	17	of	of	ADP
ejpam-3764	336	18	an	an	DET
ejpam-3764	336	19	abelian	abelian	ADJ
ejpam-3764	336	20	group	group	NOUN
ejpam-3764	336	21	(	(	PUNCT
ejpam-3764	336	22	r,+	r,+	NUM
ejpam-3764	336	23	)	)	PUNCT
ejpam-3764	336	24	.	.	PUNCT
ejpam-3764	337	1	let	let	VERB
ejpam-3764	337	2	ω	ω	PROPN
ejpam-3764	337	3	=	=	SYM
ejpam-3764	337	4	−∞	−∞	PROPN
ejpam-3764	337	5	,	,	PUNCT
ejpam-3764	337	6	a1	a1	PROPN
ejpam-3764	337	7	⊕	⊕	PROPN
ejpam-3764	337	8	a2	a2	PROPN
ejpam-3764	337	9	=	=	SYM
ejpam-3764	337	10	max(a1	max(a1	PROPN
ejpam-3764	337	11	,	,	PUNCT
ejpam-3764	337	12	a2	a2	PROPN
ejpam-3764	337	13	)	)	PUNCT
ejpam-3764	337	14	and	and	CCONJ
ejpam-3764	337	15	a1	a1	PROPN
ejpam-3764	337	16	⊗	⊗	PROPN
ejpam-3764	337	17	a2	a2	PROPN
ejpam-3764	337	18	=	=	SYM
ejpam-3764	337	19	a1	a1	PROPN
ejpam-3764	337	20	+	+	SYM
ejpam-3764	337	21	a2,∀a1	a2,∀a1	PROPN
ejpam-3764	337	22	,	,	PUNCT
ejpam-3764	337	23	a2	a2	PROPN
ejpam-3764	337	24	∈	∈	PROPN
ejpam-3764	337	25	r.	r.	PROPN
ejpam-3764	337	26	then	then	ADV
ejpam-3764	337	27	,	,	PUNCT
ejpam-3764	337	28	r−∞	r−∞	X
ejpam-3764	337	29	=	=	SYM
ejpam-3764	337	30	(	(	PUNCT
ejpam-3764	337	31	r−∞,⊕,⊗,−∞	r−∞,⊕,⊗,−∞	PROPN
ejpam-3764	337	32	,	,	PUNCT
ejpam-3764	337	33	0	0	NUM
ejpam-3764	337	34	)	)	PUNCT
ejpam-3764	337	35	is	be	AUX
ejpam-3764	337	36	−∞	−∞	X
ejpam-3764	337	37	−	−	PROPN
ejpam-3764	337	38	algebra	algebra	NOUN
ejpam-3764	337	39	over	over	ADP
ejpam-3764	337	40	the	the	DET
ejpam-3764	337	41	ring	ring	NOUN
ejpam-3764	337	42	(	(	PUNCT
ejpam-3764	337	43	r,+	r,+	NUM
ejpam-3764	337	44	,	,	PUNCT
ejpam-3764	337	45	·	·	PUNCT
ejpam-3764	337	46	)	)	PUNCT
ejpam-3764	337	47	.	.	PUNCT
ejpam-3764	338	1	then	then	ADV
ejpam-3764	338	2	,	,	PUNCT
ejpam-3764	338	3	using	use	VERB
ejpam-3764	338	4	the	the	DET
ejpam-3764	338	5	same	same	ADJ
ejpam-3764	338	6	proof	proof	NOUN
ejpam-3764	338	7	as	as	ADP
ejpam-3764	338	8	that	that	PRON
ejpam-3764	338	9	of	of	ADP
ejpam-3764	338	10	proposition	proposition	NOUN
ejpam-3764	338	11	1	1	NUM
ejpam-3764	338	12	(	(	PUNCT
ejpam-3764	338	13	r−∞	r−∞	NOUN
ejpam-3764	338	14	,	,	PUNCT
ejpam-3764	338	15	τ−∞	τ−∞	NOUN
ejpam-3764	338	16	)	)	PUNCT
ejpam-3764	338	17	is	be	AUX
ejpam-3764	338	18	a	a	DET
ejpam-3764	338	19	topological	topological	ADJ
ejpam-3764	338	20	space	space	NOUN
ejpam-3764	338	21	.	.	PUNCT
ejpam-3764	339	1	remark	remark	PROPN
ejpam-3764	339	2	6	6	NUM
ejpam-3764	339	3	.	.	PUNCT
ejpam-3764	340	1	the	the	DET
ejpam-3764	340	2	omega	omega	NOUN
ejpam-3764	340	3	topological	topological	ADJ
ejpam-3764	340	4	space	space	NOUN
ejpam-3764	340	5	(	(	PUNCT
ejpam-3764	340	6	r−∞	r−∞	NOUN
ejpam-3764	340	7	,	,	PUNCT
ejpam-3764	340	8	τ−∞	τ−∞	NOUN
ejpam-3764	340	9	)	)	PUNCT
ejpam-3764	340	10	is	be	AUX
ejpam-3764	340	11	first	first	ADV
ejpam-3764	340	12	countable	countable	ADJ
ejpam-3764	340	13	,	,	PUNCT
ejpam-3764	340	14	separable	separable	ADJ
ejpam-3764	340	15	,	,	PUNCT
ejpam-3764	340	16	hyperconnected	hyperconnecte	VERB
ejpam-3764	340	17	,	,	PUNCT
ejpam-3764	340	18	connected	connected	ADJ
ejpam-3764	340	19	and	and	CCONJ
ejpam-3764	340	20	locally	locally	ADV
ejpam-3764	340	21	connected	connect	VERB
ejpam-3764	340	22	and	and	CCONJ
ejpam-3764	340	23	does	do	AUX
ejpam-3764	340	24	not	not	PART
ejpam-3764	340	25	satisfy	satisfy	VERB
ejpam-3764	340	26	any	any	PRON
ejpam-3764	340	27	of	of	ADP
ejpam-3764	340	28	these	these	DET
ejpam-3764	340	29	t0	t0	NOUN
ejpam-3764	340	30	;	;	PUNCT
ejpam-3764	340	31	regular	regular	ADJ
ejpam-3764	340	32	and	and	CCONJ
ejpam-3764	340	33	normal	normal	ADJ
ejpam-3764	340	34	.	.	PUNCT
ejpam-3764	340	35	example	example	NOUN
ejpam-3764	340	36	8	8	NUM
ejpam-3764	340	37	.	.	PUNCT
ejpam-3764	341	1	in	in	ADP
ejpam-3764	341	2	the	the	DET
ejpam-3764	341	3	ring	ring	NOUN
ejpam-3764	341	4	(	(	PUNCT
ejpam-3764	341	5	r,+	r,+	NUM
ejpam-3764	341	6	,	,	PUNCT
ejpam-3764	341	7	·	·	PUNCT
ejpam-3764	341	8	)	)	PUNCT
ejpam-3764	341	9	,	,	PUNCT
ejpam-3764	341	10	(	(	PUNCT
ejpam-3764	341	11	r,+	r,+	X
ejpam-3764	341	12	)	)	PUNCT
ejpam-3764	341	13	is	be	AUX
ejpam-3764	341	14	an	an	DET
ejpam-3764	341	15	additive	additive	ADJ
ejpam-3764	341	16	submonoid	submonoid	NOUN
ejpam-3764	341	17	of	of	ADP
ejpam-3764	341	18	an	an	DET
ejpam-3764	341	19	abelian	abelian	ADJ
ejpam-3764	341	20	group	group	NOUN
ejpam-3764	341	21	(	(	PUNCT
ejpam-3764	341	22	r,+	r,+	NUM
ejpam-3764	341	23	)	)	PUNCT
ejpam-3764	341	24	.	.	PUNCT
ejpam-3764	342	1	let	let	VERB
ejpam-3764	342	2	ω	ω	NOUN
ejpam-3764	342	3	=	=	PUNCT
ejpam-3764	343	1	+	+	NUM
ejpam-3764	343	2	∞	∞	PROPN
ejpam-3764	343	3	,	,	PUNCT
ejpam-3764	343	4	a1	a1	PROPN
ejpam-3764	343	5	⊕	⊕	PROPN
ejpam-3764	343	6	a2	a2	PROPN
ejpam-3764	343	7	=	=	SYM
ejpam-3764	343	8	min(a1	min(a1	PROPN
ejpam-3764	343	9	,	,	PUNCT
ejpam-3764	343	10	a2	a2	PROPN
ejpam-3764	343	11	)	)	PUNCT
ejpam-3764	343	12	and	and	CCONJ
ejpam-3764	343	13	a1	a1	PROPN
ejpam-3764	343	14	⊗	⊗	PROPN
ejpam-3764	343	15	a2	a2	PROPN
ejpam-3764	343	16	=	=	SYM
ejpam-3764	343	17	a1	a1	PROPN
ejpam-3764	343	18	+	+	CCONJ
ejpam-3764	343	19	a2	a2	PROPN
ejpam-3764	343	20	,	,	PUNCT
ejpam-3764	343	21	∀a1	∀a1	PROPN
ejpam-3764	343	22	,	,	PUNCT
ejpam-3764	343	23	a2	a2	PROPN
ejpam-3764	343	24	∈	∈	PROPN
ejpam-3764	343	25	r.	r.	PROPN
ejpam-3764	343	26	then	then	ADV
ejpam-3764	343	27	,	,	PUNCT
ejpam-3764	343	28	r+∞	r+∞	PROPN
ejpam-3764	343	29	=	=	PUNCT
ejpam-3764	343	30	(	(	PUNCT
ejpam-3764	343	31	r+∞,⊕,⊗,+∞	r+∞,⊕,⊗,+∞	X
ejpam-3764	343	32	,	,	PUNCT
ejpam-3764	343	33	0	0	NUM
ejpam-3764	343	34	)	)	PUNCT
ejpam-3764	343	35	is	be	AUX
ejpam-3764	343	36	+	+	ADJ
ejpam-3764	343	37	∞	∞	NUM
ejpam-3764	343	38	−	−	PROPN
ejpam-3764	343	39	algebra	algebra	NOUN
ejpam-3764	343	40	over	over	ADP
ejpam-3764	343	41	the	the	DET
ejpam-3764	343	42	ring	ring	NOUN
ejpam-3764	343	43	(	(	PUNCT
ejpam-3764	343	44	r,+	r,+	NUM
ejpam-3764	343	45	,	,	PUNCT
ejpam-3764	343	46	·	·	PUNCT
ejpam-3764	343	47	)	)	PUNCT
ejpam-3764	343	48	.	.	PUNCT
ejpam-3764	344	1	then	then	ADV
ejpam-3764	344	2	,	,	PUNCT
ejpam-3764	344	3	using	use	VERB
ejpam-3764	344	4	the	the	DET
ejpam-3764	344	5	same	same	ADJ
ejpam-3764	344	6	proof	proof	NOUN
ejpam-3764	344	7	as	as	ADP
ejpam-3764	344	8	that	that	PRON
ejpam-3764	344	9	of	of	ADP
ejpam-3764	344	10	proposition	proposition	NOUN
ejpam-3764	344	11	1	1	NUM
ejpam-3764	344	12	(	(	PUNCT
ejpam-3764	344	13	r+∞	r+∞	PROPN
ejpam-3764	344	14	,	,	PUNCT
ejpam-3764	344	15	τ+∞	τ+∞	NUM
ejpam-3764	344	16	)	)	PUNCT
ejpam-3764	344	17	is	be	AUX
ejpam-3764	344	18	a	a	DET
ejpam-3764	344	19	topological	topological	ADJ
ejpam-3764	344	20	space	space	NOUN
ejpam-3764	344	21	.	.	PUNCT
ejpam-3764	345	1	proposition	proposition	NOUN
ejpam-3764	345	2	21	21	NUM
ejpam-3764	345	3	.	.	PUNCT
ejpam-3764	346	1	the	the	DET
ejpam-3764	346	2	omega	omega	NOUN
ejpam-3764	346	3	topological	topological	ADJ
ejpam-3764	346	4	spaces	space	NOUN
ejpam-3764	346	5	(	(	PUNCT
ejpam-3764	346	6	r−∞	r−∞	NOUN
ejpam-3764	346	7	,	,	PUNCT
ejpam-3764	346	8	τ−∞	τ−∞	NOUN
ejpam-3764	346	9	)	)	PUNCT
ejpam-3764	346	10	and	and	CCONJ
ejpam-3764	346	11	(	(	PUNCT
ejpam-3764	346	12	r+∞	r+∞	PROPN
ejpam-3764	346	13	,	,	PUNCT
ejpam-3764	346	14	τ+∞	τ+∞	NUM
ejpam-3764	346	15	)	)	PUNCT
ejpam-3764	346	16	are	be	AUX
ejpam-3764	346	17	homeomorphic	homeomorphic	ADJ
ejpam-3764	346	18	,	,	PUNCT
ejpam-3764	346	19	where	where	SCONJ
ejpam-3764	346	20	r−∞	r−∞	NOUN
ejpam-3764	346	21	is	be	AUX
ejpam-3764	346	22	a	a	DET
ejpam-3764	346	23	max−plus	max−plus	ADJ
ejpam-3764	346	24	algebra	algebra	NOUN
ejpam-3764	346	25	and	and	CCONJ
ejpam-3764	346	26	r+∞	r+∞	PROPN
ejpam-3764	346	27	is	be	AUX
ejpam-3764	346	28	a	a	DET
ejpam-3764	346	29	min−plus	min−plus	ADJ
ejpam-3764	346	30	algebra	algebra	NOUN
ejpam-3764	346	31	.	.	PUNCT
ejpam-3764	347	1	these	these	PRON
ejpam-3764	347	2	are	be	AUX
ejpam-3764	347	3	special	special	ADJ
ejpam-3764	347	4	cases	case	NOUN
ejpam-3764	347	5	of	of	ADP
ejpam-3764	347	6	omega	omega	NOUN
ejpam-3764	347	7	algebra	algebra	NOUN
ejpam-3764	347	8	.	.	PUNCT
ejpam-3764	348	1	proof	proof	NOUN
ejpam-3764	348	2	.	.	PUNCT
ejpam-3764	349	1	we	we	PRON
ejpam-3764	349	2	have	have	VERB
ejpam-3764	349	3	a	a	DET
ejpam-3764	349	4	map	map	NOUN
ejpam-3764	349	5	h	h	NOUN
ejpam-3764	349	6	:	:	PUNCT
ejpam-3764	349	7	(	(	PUNCT
ejpam-3764	349	8	r−∞	r−∞	NOUN
ejpam-3764	349	9	,	,	PUNCT
ejpam-3764	349	10	τ−∞)→	τ−∞)→	X
ejpam-3764	349	11	(	(	PUNCT
ejpam-3764	349	12	r+∞	r+∞	NOUN
ejpam-3764	349	13	,	,	PUNCT
ejpam-3764	349	14	τ+∞	τ+∞	NUM
ejpam-3764	349	15	)	)	PUNCT
ejpam-3764	349	16	is	be	AUX
ejpam-3764	349	17	defined	define	VERB
ejpam-3764	349	18	by	by	ADP
ejpam-3764	349	19	:	:	PUNCT
ejpam-3764	349	20	h	h	PROPN
ejpam-3764	349	21	(	(	PUNCT
ejpam-3764	349	22	x1	x1	PROPN
ejpam-3764	349	23	)	)	PUNCT
ejpam-3764	350	1	=	=	PRON
ejpam-3764	350	2	{	{	PUNCT
ejpam-3764	350	3	x1	x1	INTJ
ejpam-3764	350	4	if	if	SCONJ
ejpam-3764	350	5	x1	x1	PROPN
ejpam-3764	350	6	∈	∈	PROPN
ejpam-3764	350	7	r	r	NOUN
ejpam-3764	350	8	+	+	NOUN
ejpam-3764	350	9	∞	∞	PROPN
ejpam-3764	350	10	if	if	SCONJ
ejpam-3764	350	11	x1	x1	PROPN
ejpam-3764	350	12	=	=	SYM
ejpam-3764	350	13	−∞	−∞	NOUN
ejpam-3764	350	14	;	;	PUNCT
ejpam-3764	350	15	let	let	VERB
ejpam-3764	350	16	x1	x1	NUM
ejpam-3764	350	17	,	,	PUNCT
ejpam-3764	350	18	x2	x2	PROPN
ejpam-3764	350	19	∈	∈	PROPN
ejpam-3764	350	20	r−∞	r−∞	NOUN
ejpam-3764	350	21	be	be	VERB
ejpam-3764	350	22	arbitrary	arbitrary	ADJ
ejpam-3764	350	23	.	.	PUNCT
ejpam-3764	351	1	let	let	VERB
ejpam-3764	351	2	h(x1	h(x1	NOUN
ejpam-3764	351	3	)	)	PUNCT
ejpam-3764	351	4	=	=	SYM
ejpam-3764	351	5	h(x2	h(x2	NOUN
ejpam-3764	351	6	)	)	PUNCT
ejpam-3764	352	1	,	,	PUNCT
ejpam-3764	352	2	then	then	ADV
ejpam-3764	352	3	x1	x1	PROPN
ejpam-3764	352	4	=	=	SYM
ejpam-3764	352	5	x2	x2	PROPN
ejpam-3764	352	6	.	.	PUNCT
ejpam-3764	353	1	hence	hence	ADV
ejpam-3764	353	2	,	,	PUNCT
ejpam-3764	353	3	h	h	PROPN
ejpam-3764	353	4	is	be	AUX
ejpam-3764	353	5	an	an	DET
ejpam-3764	353	6	injective	injective	ADJ
ejpam-3764	353	7	.	.	PUNCT
ejpam-3764	354	1	if	if	SCONJ
ejpam-3764	354	2	x1	x1	PROPN
ejpam-3764	354	3	∈	∈	PROPN
ejpam-3764	354	4	r+∞	r+∞	PROPN
ejpam-3764	354	5	is	be	AUX
ejpam-3764	354	6	arbitrary	arbitrary	ADJ
ejpam-3764	354	7	,	,	PUNCT
ejpam-3764	354	8	then	then	ADV
ejpam-3764	354	9	we	we	PRON
ejpam-3764	354	10	have	have	VERB
ejpam-3764	354	11	two	two	NUM
ejpam-3764	354	12	cases	case	NOUN
ejpam-3764	354	13	:	:	PUNCT
ejpam-3764	354	14	case	case	NOUN
ejpam-3764	354	15	1	1	NUM
ejpam-3764	354	16	:	:	PUNCT
ejpam-3764	354	17	if	if	SCONJ
ejpam-3764	354	18	x1	x1	PROPN
ejpam-3764	354	19	6=	6=	PROPN
ejpam-3764	354	20	+	+	ADJ
ejpam-3764	354	21	∞	∞	PROPN
ejpam-3764	354	22	,	,	PUNCT
ejpam-3764	354	23	then	then	ADV
ejpam-3764	354	24	there	there	PRON
ejpam-3764	354	25	exists	exist	VERB
ejpam-3764	354	26	x1	x1	PROPN
ejpam-3764	354	27	∈	∈	PROPN
ejpam-3764	354	28	r−∞	r−∞	NOUN
ejpam-3764	354	29	\	\	NOUN
ejpam-3764	354	30	{	{	PUNCT
ejpam-3764	354	31	−∞	−∞	NOUN
ejpam-3764	354	32	}	}	PUNCT
ejpam-3764	354	33	,	,	PUNCT
ejpam-3764	354	34	such	such	ADJ
ejpam-3764	354	35	that	that	DET
ejpam-3764	354	36	h(x1	h(x1	NOUN
ejpam-3764	354	37	)	)	PUNCT
ejpam-3764	354	38	=	=	SYM
ejpam-3764	355	1	x1	x1	PROPN
ejpam-3764	355	2	.	.	PUNCT
ejpam-3764	355	3	case	case	NOUN
ejpam-3764	355	4	2	2	NUM
ejpam-3764	355	5	:	:	PUNCT
ejpam-3764	355	6	if	if	SCONJ
ejpam-3764	355	7	x1	x1	PROPN
ejpam-3764	355	8	=	=	SYM
ejpam-3764	355	9	+	+	NOUN
ejpam-3764	355	10	∞	∞	PROPN
ejpam-3764	355	11	,	,	PUNCT
ejpam-3764	355	12	then	then	ADV
ejpam-3764	355	13	there	there	PRON
ejpam-3764	355	14	exists	exist	VERB
ejpam-3764	355	15	x1	x1	PROPN
ejpam-3764	355	16	=	=	PUNCT
ejpam-3764	355	17	−∞	−∞	ADP
ejpam-3764	355	18	∈	∈	PROPN
ejpam-3764	355	19	r−∞	r−∞	NOUN
ejpam-3764	355	20	,	,	PUNCT
ejpam-3764	355	21	such	such	ADJ
ejpam-3764	355	22	that	that	DET
ejpam-3764	355	23	h(−∞	h(−∞	NOUN
ejpam-3764	355	24	)	)	PUNCT
ejpam-3764	355	25	=	=	PUNCT
ejpam-3764	356	1	+	+	NUM
ejpam-3764	356	2	∞.	∞.	PROPN
ejpam-3764	356	3	hence	hence	ADV
ejpam-3764	356	4	,	,	PUNCT
ejpam-3764	356	5	h	h	PROPN
ejpam-3764	356	6	is	be	AUX
ejpam-3764	356	7	surjective	surjective	ADJ
ejpam-3764	356	8	.	.	PUNCT
ejpam-3764	357	1	let	let	VERB
ejpam-3764	357	2	b	b	X
ejpam-3764	357	3	∈	∈	PROPN
ejpam-3764	357	4	τ+∞	τ+∞	PUNCT
ejpam-3764	357	5	be	be	AUX
ejpam-3764	357	6	any	any	DET
ejpam-3764	357	7	basic	basic	ADJ
ejpam-3764	357	8	open	open	ADJ
ejpam-3764	357	9	set	set	NOUN
ejpam-3764	357	10	.	.	PUNCT
ejpam-3764	358	1	since	since	SCONJ
ejpam-3764	358	2	(	(	PUNCT
ejpam-3764	358	3	r−∞	r−∞	X
ejpam-3764	358	4	\	\	NOUN
ejpam-3764	358	5	{	{	PUNCT
ejpam-3764	358	6	−∞},⊗	−∞},⊗	PROPN
ejpam-3764	358	7	)	)	PUNCT
ejpam-3764	358	8	and	and	CCONJ
ejpam-3764	358	9	(	(	PUNCT
ejpam-3764	358	10	r+∞	r+∞	PROPN
ejpam-3764	358	11	\	\	X
ejpam-3764	358	12	{	{	PUNCT
ejpam-3764	358	13	+	+	NOUN
ejpam-3764	358	14	∞},⊗	∞},⊗	ADJ
ejpam-3764	358	15	)	)	PUNCT
ejpam-3764	358	16	are	be	AUX
ejpam-3764	358	17	groups	group	NOUN
ejpam-3764	358	18	,	,	PUNCT
ejpam-3764	358	19	then	then	ADV
ejpam-3764	358	20	by	by	ADP
ejpam-3764	358	21	problem	problem	NOUN
ejpam-3764	358	22	2	2	NUM
ejpam-3764	358	23	,	,	PUNCT
ejpam-3764	358	24	we	we	PRON
ejpam-3764	358	25	have	have	VERB
ejpam-3764	358	26	b	b	NOUN
ejpam-3764	358	27	=	=	PRON
ejpam-3764	358	28	{	{	PUNCT
ejpam-3764	358	29	{	{	PUNCT
ejpam-3764	358	30	−∞	−∞	NOUN
ejpam-3764	358	31	}	}	PUNCT
ejpam-3764	358	32	,	,	PUNCT
ejpam-3764	358	33	{	{	PUNCT
ejpam-3764	358	34	−∞	−∞	NOUN
ejpam-3764	358	35	,	,	PUNCT
ejpam-3764	358	36	c	c	X
ejpam-3764	358	37	,	,	PUNCT
ejpam-3764	358	38	c−1	c−1	PROPN
ejpam-3764	358	39	}	}	PUNCT
ejpam-3764	358	40	:	:	PUNCT
ejpam-3764	358	41	c	c	X
ejpam-3764	358	42	∈	∈	PROPN
ejpam-3764	358	43	r	r	NOUN
ejpam-3764	358	44	}	}	PUNCT
ejpam-3764	358	45	and	and	CCONJ
ejpam-3764	358	46	m.	m.	NOUN
ejpam-3764	358	47	alqahtani	alqahtani	PROPN
ejpam-3764	358	48	,	,	PUNCT
ejpam-3764	358	49	c.	c.	PROPN
ejpam-3764	358	50	özel	özel	PROPN
ejpam-3764	358	51	,	,	PUNCT
ejpam-3764	358	52	i.	i.	PROPN
ejpam-3764	358	53	alshammari	alshammari	PROPN
ejpam-3764	358	54	/	/	SYM
ejpam-3764	358	55	eur	eur	PROPN
ejpam-3764	358	56	.	.	PUNCT
ejpam-3764	359	1	j.	j.	PROPN
ejpam-3764	359	2	pure	pure	PROPN
ejpam-3764	359	3	appl	appl	PROPN
ejpam-3764	359	4	.	.	PROPN
ejpam-3764	359	5	math	math	PROPN
ejpam-3764	359	6	,	,	PUNCT
ejpam-3764	359	7	13	13	NUM
ejpam-3764	359	8	(	(	PUNCT
ejpam-3764	359	9	3	3	NUM
ejpam-3764	359	10	)	)	PUNCT
ejpam-3764	359	11	(	(	PUNCT
ejpam-3764	359	12	2020	2020	NUM
ejpam-3764	359	13	)	)	PUNCT
ejpam-3764	359	14	,	,	PUNCT
ejpam-3764	359	15	513	513	NUM
ejpam-3764	359	16	-	-	SYM
ejpam-3764	359	17	528	528	NUM
ejpam-3764	359	18	524	524	NUM
ejpam-3764	359	19	b	b	X
ejpam-3764	359	20	=	=	PRON
ejpam-3764	359	21	{	{	PUNCT
ejpam-3764	359	22	{	{	PUNCT
ejpam-3764	359	23	+	+	NOUN
ejpam-3764	359	24	∞	∞	NOUN
ejpam-3764	359	25	}	}	PUNCT
ejpam-3764	359	26	,	,	PUNCT
ejpam-3764	359	27	{	{	PUNCT
ejpam-3764	359	28	+	+	NOUN
ejpam-3764	359	29	∞	∞	NOUN
ejpam-3764	359	30	,	,	PUNCT
ejpam-3764	359	31	c	c	X
ejpam-3764	359	32	,	,	PUNCT
ejpam-3764	359	33	c−1	c−1	PROPN
ejpam-3764	359	34	}	}	PUNCT
ejpam-3764	359	35	:	:	PUNCT
ejpam-3764	359	36	c	c	X
ejpam-3764	359	37	∈	∈	PROPN
ejpam-3764	359	38	r	r	NOUN
ejpam-3764	359	39	}	}	PUNCT
ejpam-3764	359	40	are	be	AUX
ejpam-3764	359	41	a	a	DET
ejpam-3764	359	42	base	base	NOUN
ejpam-3764	359	43	for	for	ADP
ejpam-3764	359	44	r−∞	r−∞	NOUN
ejpam-3764	359	45	and	and	CCONJ
ejpam-3764	359	46	r+∞	r+∞	PROPN
ejpam-3764	359	47	,	,	PUNCT
ejpam-3764	359	48	respectively	respectively	ADV
ejpam-3764	359	49	.	.	PUNCT
ejpam-3764	360	1	to	to	PART
ejpam-3764	360	2	prove	prove	VERB
ejpam-3764	360	3	that	that	SCONJ
ejpam-3764	360	4	h	h	NOUN
ejpam-3764	360	5	is	be	AUX
ejpam-3764	360	6	continuous	continuous	ADJ
ejpam-3764	360	7	,	,	PUNCT
ejpam-3764	360	8	we	we	PRON
ejpam-3764	360	9	have	have	VERB
ejpam-3764	360	10	two	two	NUM
ejpam-3764	360	11	cases	case	NOUN
ejpam-3764	360	12	:	:	PUNCT
ejpam-3764	360	13	case	case	NOUN
ejpam-3764	360	14	1	1	NUM
ejpam-3764	360	15	:	:	PUNCT
ejpam-3764	360	16	if	if	SCONJ
ejpam-3764	360	17	b	b	X
ejpam-3764	360	18	=	=	PRON
ejpam-3764	360	19	{	{	PUNCT
ejpam-3764	360	20	+	+	NOUN
ejpam-3764	360	21	∞	∞	NOUN
ejpam-3764	360	22	}	}	PUNCT
ejpam-3764	360	23	,	,	PUNCT
ejpam-3764	360	24	then	then	ADV
ejpam-3764	360	25	h−1(b	h−1(b	PROPN
ejpam-3764	360	26	)	)	PUNCT
ejpam-3764	360	27	=	=	PUNCT
ejpam-3764	360	28	h−1({+∞	h−1({+∞	NOUN
ejpam-3764	360	29	}	}	PUNCT
ejpam-3764	360	30	)	)	PUNCT
ejpam-3764	361	1	=	=	SYM
ejpam-3764	361	2	{	{	PUNCT
ejpam-3764	361	3	−∞	−∞	NOUN
ejpam-3764	361	4	}	}	PUNCT
ejpam-3764	361	5	∈	∈	PROPN
ejpam-3764	361	6	τ−∞.	τ−∞.	NOUN
ejpam-3764	361	7	case	case	NOUN
ejpam-3764	361	8	2	2	NUM
ejpam-3764	361	9	:	:	PUNCT
ejpam-3764	361	10	if	if	SCONJ
ejpam-3764	361	11	b	b	X
ejpam-3764	361	12	=	=	PRON
ejpam-3764	361	13	{	{	PUNCT
ejpam-3764	361	14	+	+	NOUN
ejpam-3764	361	15	∞	∞	PROPN
ejpam-3764	361	16	,	,	PUNCT
ejpam-3764	361	17	c	c	X
ejpam-3764	361	18	,	,	PUNCT
ejpam-3764	361	19	c−1	c−1	PROPN
ejpam-3764	361	20	}	}	PUNCT
ejpam-3764	361	21	,	,	PUNCT
ejpam-3764	361	22	then	then	ADV
ejpam-3764	361	23	h−1(b	h−1(b	PROPN
ejpam-3764	361	24	)	)	PUNCT
ejpam-3764	361	25	=	=	PUNCT
ejpam-3764	362	1	h−1({+∞	h−1({+∞	NOUN
ejpam-3764	362	2	,	,	PUNCT
ejpam-3764	362	3	c	c	X
ejpam-3764	362	4	,	,	PUNCT
ejpam-3764	362	5	c−1	c−1	PROPN
ejpam-3764	362	6	}	}	PUNCT
ejpam-3764	362	7	)	)	PUNCT
ejpam-3764	362	8	=	=	SYM
ejpam-3764	362	9	{	{	PUNCT
ejpam-3764	362	10	−∞	−∞	NOUN
ejpam-3764	362	11	,	,	PUNCT
ejpam-3764	362	12	c	c	X
ejpam-3764	362	13	,	,	PUNCT
ejpam-3764	362	14	c−1	c−1	PROPN
ejpam-3764	362	15	}	}	PUNCT
ejpam-3764	362	16	∈	∈	PROPN
ejpam-3764	362	17	τ−∞.	τ−∞.	NOUN
ejpam-3764	362	18	hence	hence	ADV
ejpam-3764	362	19	,	,	PUNCT
ejpam-3764	362	20	h	h	PROPN
ejpam-3764	362	21	is	be	AUX
ejpam-3764	362	22	continuous	continuous	ADJ
ejpam-3764	362	23	.	.	PUNCT
ejpam-3764	363	1	to	to	PART
ejpam-3764	363	2	prove	prove	VERB
ejpam-3764	363	3	that	that	SCONJ
ejpam-3764	363	4	h−1	h−1	PROPN
ejpam-3764	363	5	is	be	AUX
ejpam-3764	363	6	continuous	continuous	ADJ
ejpam-3764	363	7	,	,	PUNCT
ejpam-3764	363	8	we	we	PRON
ejpam-3764	363	9	have	have	VERB
ejpam-3764	363	10	two	two	NUM
ejpam-3764	363	11	cases	case	NOUN
ejpam-3764	363	12	:	:	PUNCT
ejpam-3764	363	13	(	(	PUNCT
ejpam-3764	363	14	since	since	SCONJ
ejpam-3764	363	15	h	h	NOUN
ejpam-3764	363	16	is	be	AUX
ejpam-3764	363	17	one	one	NUM
ejpam-3764	363	18	to	to	ADP
ejpam-3764	363	19	one	one	NUM
ejpam-3764	363	20	and	and	CCONJ
ejpam-3764	363	21	onto	onto	ADP
ejpam-3764	363	22	,	,	PUNCT
ejpam-3764	363	23	then	then	ADV
ejpam-3764	363	24	(	(	PUNCT
ejpam-3764	363	25	h−1)−1(b	h−1)−1(b	PROPN
ejpam-3764	363	26	)	)	PUNCT
ejpam-3764	363	27	=	=	SYM
ejpam-3764	363	28	h(b	h(b	PROPN
ejpam-3764	363	29	)	)	PUNCT
ejpam-3764	363	30	)	)	PUNCT
ejpam-3764	363	31	.	.	PUNCT
ejpam-3764	364	1	case	case	NOUN
ejpam-3764	364	2	1	1	NUM
ejpam-3764	364	3	:	:	PUNCT
ejpam-3764	364	4	if	if	SCONJ
ejpam-3764	364	5	b	b	X
ejpam-3764	364	6	=	=	SYM
ejpam-3764	364	7	{	{	PUNCT
ejpam-3764	364	8	−∞	−∞	NOUN
ejpam-3764	364	9	}	}	PUNCT
ejpam-3764	364	10	,	,	PUNCT
ejpam-3764	364	11	then	then	ADV
ejpam-3764	364	12	(	(	PUNCT
ejpam-3764	364	13	h−1)−1(b	h−1)−1(b	PROPN
ejpam-3764	364	14	)	)	PUNCT
ejpam-3764	364	15	=	=	SYM
ejpam-3764	364	16	h(b	h(b	ADJ
ejpam-3764	364	17	)	)	PUNCT
ejpam-3764	364	18	=	=	SYM
ejpam-3764	364	19	h({−∞	h({−∞	NOUN
ejpam-3764	364	20	}	}	PUNCT
ejpam-3764	364	21	)	)	PUNCT
ejpam-3764	365	1	=	=	PRON
ejpam-3764	365	2	{	{	PUNCT
ejpam-3764	365	3	+	+	NOUN
ejpam-3764	365	4	∞	∞	ADJ
ejpam-3764	365	5	}	}	PUNCT
ejpam-3764	365	6	∈	∈	PROPN
ejpam-3764	365	7	τ+∞.	τ+∞.	NOUN
ejpam-3764	365	8	case	case	NOUN
ejpam-3764	365	9	2	2	NUM
ejpam-3764	365	10	:	:	PUNCT
ejpam-3764	365	11	ifb	ifb	PROPN
ejpam-3764	365	12	=	=	SYM
ejpam-3764	365	13	{	{	PUNCT
ejpam-3764	365	14	−∞	−∞	NOUN
ejpam-3764	365	15	,	,	PUNCT
ejpam-3764	365	16	c	c	X
ejpam-3764	365	17	,	,	PUNCT
ejpam-3764	365	18	c−1	c−1	PROPN
ejpam-3764	365	19	}	}	PUNCT
ejpam-3764	365	20	,	,	PUNCT
ejpam-3764	365	21	then	then	ADV
ejpam-3764	365	22	(	(	PUNCT
ejpam-3764	365	23	h−1)−1(b	h−1)−1(b	PROPN
ejpam-3764	365	24	)	)	PUNCT
ejpam-3764	365	25	=	=	SYM
ejpam-3764	365	26	h(b	h(b	ADJ
ejpam-3764	365	27	)	)	PUNCT
ejpam-3764	365	28	=	=	SYM
ejpam-3764	366	1	h({−∞	h({−∞	NOUN
ejpam-3764	366	2	,	,	PUNCT
ejpam-3764	366	3	c	c	X
ejpam-3764	366	4	,	,	PUNCT
ejpam-3764	366	5	c−1	c−1	PROPN
ejpam-3764	366	6	}	}	PUNCT
ejpam-3764	366	7	)	)	PUNCT
ejpam-3764	366	8	=	=	PRON
ejpam-3764	366	9	{	{	PUNCT
ejpam-3764	366	10	+	+	NOUN
ejpam-3764	366	11	∞	∞	PROPN
ejpam-3764	366	12	,	,	PUNCT
ejpam-3764	366	13	c	c	X
ejpam-3764	366	14	,	,	PUNCT
ejpam-3764	366	15	c−1	c−1	PROPN
ejpam-3764	366	16	}	}	PUNCT
ejpam-3764	366	17	∈	∈	PROPN
ejpam-3764	366	18	τ+∞.	τ+∞.	NOUN
ejpam-3764	366	19	hence	hence	ADV
ejpam-3764	366	20	,	,	PUNCT
ejpam-3764	366	21	h−1	h−1	PROPN
ejpam-3764	366	22	is	be	AUX
ejpam-3764	366	23	continuous	continuous	ADJ
ejpam-3764	366	24	(	(	PUNCT
ejpam-3764	366	25	which	which	PRON
ejpam-3764	366	26	means	mean	VERB
ejpam-3764	366	27	h	h	NOUN
ejpam-3764	366	28	is	be	AUX
ejpam-3764	366	29	open	open	ADJ
ejpam-3764	366	30	)	)	PUNCT
ejpam-3764	366	31	.	.	PUNCT
ejpam-3764	367	1	in	in	ADP
ejpam-3764	367	2	conclusion	conclusion	NOUN
ejpam-3764	367	3	,	,	PUNCT
ejpam-3764	367	4	if	if	SCONJ
ejpam-3764	367	5	h	h	NOUN
ejpam-3764	367	6	is	be	AUX
ejpam-3764	367	7	homeomorphism	homeomorphism	PROPN
ejpam-3764	367	8	,	,	PUNCT
ejpam-3764	367	9	then	then	ADV
ejpam-3764	367	10	(	(	PUNCT
ejpam-3764	367	11	r−∞	r−∞	NOUN
ejpam-3764	367	12	,	,	PUNCT
ejpam-3764	367	13	τ−∞	τ−∞	NOUN
ejpam-3764	367	14	)	)	PUNCT
ejpam-3764	367	15	and	and	CCONJ
ejpam-3764	367	16	(	(	PUNCT
ejpam-3764	367	17	r+∞	r+∞	PROPN
ejpam-3764	367	18	,	,	PUNCT
ejpam-3764	367	19	τ+∞	τ+∞	NUM
ejpam-3764	367	20	)	)	PUNCT
ejpam-3764	367	21	are	be	AUX
ejpam-3764	367	22	homeomorphic	homeomorphic	ADJ
ejpam-3764	367	23	.	.	PUNCT
ejpam-3764	368	1	theorem	theorem	NOUN
ejpam-3764	368	2	2	2	NUM
ejpam-3764	368	3	.	.	PUNCT
ejpam-3764	369	1	let	let	VERB
ejpam-3764	369	2	x	x	PRON
ejpam-3764	369	3	be	be	AUX
ejpam-3764	369	4	any	any	DET
ejpam-3764	369	5	semiring	semiring	NOUN
ejpam-3764	369	6	in	in	ADP
ejpam-3764	369	7	conventional	conventional	ADJ
ejpam-3764	369	8	algebra	algebra	NOUN
ejpam-3764	369	9	,	,	PUNCT
ejpam-3764	369	10	such	such	ADJ
ejpam-3764	369	11	that	that	SCONJ
ejpam-3764	369	12	e	e	NOUN
ejpam-3764	369	13	is	be	AUX
ejpam-3764	369	14	the	the	DET
ejpam-3764	369	15	zero	zero	NUM
ejpam-3764	369	16	element	element	NOUN
ejpam-3764	369	17	.	.	PUNCT
ejpam-3764	370	1	we	we	PRON
ejpam-3764	370	2	define	define	VERB
ejpam-3764	370	3	a	a	DET
ejpam-3764	370	4	topology	topology	NOUN
ejpam-3764	370	5	on	on	ADP
ejpam-3764	370	6	x	x	PROPN
ejpam-3764	370	7	is	be	AUX
ejpam-3764	370	8	called	call	VERB
ejpam-3764	370	9	zero	zero	NUM
ejpam-3764	370	10	element	element	NOUN
ejpam-3764	370	11	topology	topology	NOUN
ejpam-3764	370	12	,	,	PUNCT
ejpam-3764	370	13	as	as	SCONJ
ejpam-3764	370	14	follows	follow	VERB
ejpam-3764	370	15	:	:	PUNCT
ejpam-3764	370	16	τe	τe	ADP
ejpam-3764	370	17	=	=	SYM
ejpam-3764	370	18	{	{	PUNCT
ejpam-3764	370	19	∅	∅	NOUN
ejpam-3764	370	20	,	,	PUNCT
ejpam-3764	370	21	x	x	NOUN
ejpam-3764	370	22	}	}	PUNCT
ejpam-3764	370	23	∪	∪	VERB
ejpam-3764	370	24	{	{	PUNCT
ejpam-3764	370	25	u	u	NOUN
ejpam-3764	370	26	⊆	⊆	NUM
ejpam-3764	370	27	x	x	SYM
ejpam-3764	370	28	:	:	PUNCT
ejpam-3764	370	29	e	e	X
ejpam-3764	370	30	∈	∈	PROPN
ejpam-3764	370	31	u	u	NOUN
ejpam-3764	370	32	and	and	CCONJ
ejpam-3764	370	33	for	for	ADP
ejpam-3764	370	34	any	any	DET
ejpam-3764	370	35	a	a	DET
ejpam-3764	370	36	∈	∈	PROPN
ejpam-3764	370	37	u	u	NOUN
ejpam-3764	370	38	\	\	NOUN
ejpam-3764	370	39	{	{	PUNCT
ejpam-3764	370	40	e	e	NOUN
ejpam-3764	370	41	}	}	PUNCT
ejpam-3764	370	42	,	,	PUNCT
ejpam-3764	370	43	the	the	DET
ejpam-3764	370	44	multiplicative	multiplicative	ADJ
ejpam-3764	370	45	inverse	inverse	NOUN
ejpam-3764	370	46	of	of	ADP
ejpam-3764	370	47	a	a	DET
ejpam-3764	370	48	exists	exist	NOUN
ejpam-3764	370	49	in	in	ADP
ejpam-3764	370	50	u	u	NOUN
ejpam-3764	370	51	}	}	PUNCT
ejpam-3764	370	52	.	.	PUNCT
ejpam-3764	371	1	then	then	ADV
ejpam-3764	371	2	(	(	PUNCT
ejpam-3764	371	3	x	x	X
ejpam-3764	371	4	,	,	PUNCT
ejpam-3764	371	5	τe	τe	NOUN
ejpam-3764	371	6	)	)	PUNCT
ejpam-3764	371	7	is	be	AUX
ejpam-3764	371	8	a	a	DET
ejpam-3764	371	9	topological	topological	ADJ
ejpam-3764	371	10	space	space	NOUN
ejpam-3764	371	11	.	.	PUNCT
ejpam-3764	372	1	proof	proof	NOUN
ejpam-3764	372	2	.	.	PUNCT
ejpam-3764	373	1	condition	condition	NOUN
ejpam-3764	373	2	∅	∅	NOUN
ejpam-3764	373	3	,	,	PUNCT
ejpam-3764	373	4	x	x	SYM
ejpam-3764	373	5	∈	∈	NOUN
ejpam-3764	374	1	τe	τe	VERB
ejpam-3764	374	2	is	be	AUX
ejpam-3764	374	3	satisfied	satisfied	ADJ
ejpam-3764	374	4	from	from	ADP
ejpam-3764	374	5	the	the	DET
ejpam-3764	374	6	definition	definition	NOUN
ejpam-3764	374	7	of	of	ADP
ejpam-3764	374	8	τe	τe	PROPN
ejpam-3764	374	9	.	.	PUNCT
ejpam-3764	375	1	now	now	ADV
ejpam-3764	375	2	let	let	VERB
ejpam-3764	375	3	u1	u1	NOUN
ejpam-3764	375	4	,	,	PUNCT
ejpam-3764	375	5	u2	u2	PROPN
ejpam-3764	375	6	∈	∈	PROPN
ejpam-3764	375	7	τe	τe	PART
ejpam-3764	375	8	be	be	AUX
ejpam-3764	375	9	arbitrary	arbitrary	ADJ
ejpam-3764	375	10	.	.	PUNCT
ejpam-3764	376	1	if	if	SCONJ
ejpam-3764	376	2	either	either	PRON
ejpam-3764	376	3	u1	u1	NOUN
ejpam-3764	376	4	=	=	SYM
ejpam-3764	376	5	∅	∅	NOUN
ejpam-3764	376	6	or	or	CCONJ
ejpam-3764	376	7	u2	u2	NOUN
ejpam-3764	376	8	=	=	PUNCT
ejpam-3764	376	9	∅	∅	NOUN
ejpam-3764	376	10	,	,	PUNCT
ejpam-3764	376	11	then	then	ADV
ejpam-3764	376	12	u1	u1	VERB
ejpam-3764	376	13	∩u2	∩u2	NOUN
ejpam-3764	376	14	=	=	NOUN
ejpam-3764	376	15	∅	∅	NOUN
ejpam-3764	376	16	∈	∈	PROPN
ejpam-3764	376	17	τe	τe	PROPN
ejpam-3764	376	18	.	.	PUNCT
ejpam-3764	376	19	assume	assume	VERB
ejpam-3764	376	20	,	,	PUNCT
ejpam-3764	376	21	u1	u1	NOUN
ejpam-3764	376	22	6=	6=	NOUN
ejpam-3764	376	23	∅	∅	NOUN
ejpam-3764	376	24	and	and	CCONJ
ejpam-3764	376	25	u2	u2	PROPN
ejpam-3764	376	26	6=	6=	PROPN
ejpam-3764	376	27	∅.	∅.	NOUN
ejpam-3764	376	28	if	if	SCONJ
ejpam-3764	376	29	either	either	CCONJ
ejpam-3764	376	30	u1	u1	NOUN
ejpam-3764	376	31	=	=	SYM
ejpam-3764	376	32	x	x	PROPN
ejpam-3764	376	33	or	or	CCONJ
ejpam-3764	376	34	u2	u2	NOUN
ejpam-3764	376	35	=	=	SYM
ejpam-3764	376	36	x	x	NOUN
ejpam-3764	376	37	,	,	PUNCT
ejpam-3764	376	38	then	then	ADV
ejpam-3764	376	39	u1	u1	NOUN
ejpam-3764	376	40	∩	∩	ADJ
ejpam-3764	376	41	u2	u2	NOUN
ejpam-3764	376	42	=	=	SYM
ejpam-3764	376	43	u1	u1	NOUN
ejpam-3764	376	44	or	or	CCONJ
ejpam-3764	376	45	u2	u2	PROPN
ejpam-3764	376	46	∈	∈	PROPN
ejpam-3764	376	47	τe	τe	NOUN
ejpam-3764	376	48	.	.	PUNCT
ejpam-3764	377	1	so	so	ADV
ejpam-3764	377	2	assume	assume	VERB
ejpam-3764	377	3	,	,	PUNCT
ejpam-3764	377	4	u1	u1	VERB
ejpam-3764	377	5	6=	6=	NUM
ejpam-3764	377	6	x	x	NOUN
ejpam-3764	377	7	and	and	CCONJ
ejpam-3764	377	8	u2	u2	PROPN
ejpam-3764	378	1	6=	6=	PROPN
ejpam-3764	378	2	x	x	PROPN
ejpam-3764	378	3	,	,	PUNCT
ejpam-3764	378	4	then	then	ADV
ejpam-3764	378	5	u1	u1	NOUN
ejpam-3764	379	1	∩	∩	ADJ
ejpam-3764	379	2	u2	u2	PROPN
ejpam-3764	379	3	∈	∈	PROPN
ejpam-3764	379	4	τe	τe	ADP
ejpam-3764	379	5	,	,	PUNCT
ejpam-3764	379	6	because	because	SCONJ
ejpam-3764	379	7	e	e	PROPN
ejpam-3764	379	8	∈	∈	PROPN
ejpam-3764	379	9	u1	u1	NOUN
ejpam-3764	379	10	and	and	CCONJ
ejpam-3764	379	11	e	e	NOUN
ejpam-3764	379	12	∈	∈	PROPN
ejpam-3764	379	13	u2	u2	PROPN
ejpam-3764	379	14	.	.	PUNCT
ejpam-3764	380	1	hence	hence	ADV
ejpam-3764	380	2	e	e	PROPN
ejpam-3764	380	3	∈	∈	PROPN
ejpam-3764	380	4	u1	u1	NOUN
ejpam-3764	380	5	∩	∩	NOUN
ejpam-3764	380	6	u2	u2	NOUN
ejpam-3764	380	7	,	,	PUNCT
ejpam-3764	380	8	also	also	ADV
ejpam-3764	380	9	for	for	ADP
ejpam-3764	380	10	any	any	DET
ejpam-3764	380	11	element	element	NOUN
ejpam-3764	380	12	(	(	PUNCT
ejpam-3764	380	13	a	a	DET
ejpam-3764	380	14	6=	6=	NUM
ejpam-3764	380	15	e	e	NOUN
ejpam-3764	380	16	)	)	PUNCT
ejpam-3764	380	17	∈	∈	PROPN
ejpam-3764	381	1	u1∩u2	u1∩u2	PROPN
ejpam-3764	381	2	,	,	PUNCT
ejpam-3764	381	3	if	if	SCONJ
ejpam-3764	381	4	a	a	DET
ejpam-3764	381	5	∈	∈	PROPN
ejpam-3764	381	6	u1	u1	NOUN
ejpam-3764	381	7	and	and	CCONJ
ejpam-3764	381	8	a	a	DET
ejpam-3764	381	9	∈	∈	PROPN
ejpam-3764	381	10	u2	u2	NOUN
ejpam-3764	381	11	,	,	PUNCT
ejpam-3764	381	12	then	then	ADV
ejpam-3764	381	13	a	a	PRON
ejpam-3764	381	14	and	and	CCONJ
ejpam-3764	381	15	the	the	DET
ejpam-3764	381	16	multiplicative	multiplicative	ADJ
ejpam-3764	381	17	inverse	inverse	NOUN
ejpam-3764	381	18	of	of	ADP
ejpam-3764	381	19	a	a	PRON
ejpam-3764	381	20	must	must	AUX
ejpam-3764	381	21	belong	belong	VERB
ejpam-3764	381	22	to	to	ADP
ejpam-3764	381	23	u1	u1	NOUN
ejpam-3764	381	24	and	and	CCONJ
ejpam-3764	381	25	u2	u2	PROPN
ejpam-3764	381	26	.	.	PROPN
ejpam-3764	382	1	hence	hence	ADV
ejpam-3764	382	2	a	a	PRON
ejpam-3764	382	3	and	and	CCONJ
ejpam-3764	382	4	the	the	DET
ejpam-3764	382	5	multiplicative	multiplicative	ADJ
ejpam-3764	382	6	inverse	inverse	NOUN
ejpam-3764	382	7	of	of	ADP
ejpam-3764	382	8	a	a	DET
ejpam-3764	382	9	belong	belong	NOUN
ejpam-3764	382	10	to	to	PART
ejpam-3764	382	11	u1	u1	PROPN
ejpam-3764	382	12	∩u2	∩u2	PROPN
ejpam-3764	382	13	,	,	PUNCT
ejpam-3764	382	14	then	then	ADV
ejpam-3764	382	15	u1	u1	VERB
ejpam-3764	382	16	∩	∩	ADJ
ejpam-3764	382	17	u2	u2	PROPN
ejpam-3764	382	18	∈	∈	PROPN
ejpam-3764	382	19	τe	τe	PROPN
ejpam-3764	382	20	.	.	PUNCT
ejpam-3764	383	1	for	for	ADP
ejpam-3764	383	2	the	the	DET
ejpam-3764	383	3	third	third	ADJ
ejpam-3764	383	4	condition	condition	NOUN
ejpam-3764	383	5	,	,	PUNCT
ejpam-3764	383	6	let	let	VERB
ejpam-3764	383	7	sγ	sγ	PRON
ejpam-3764	383	8	∈	∈	VERB
ejpam-3764	383	9	τe	τe	ADP
ejpam-3764	383	10	for	for	ADP
ejpam-3764	383	11	any	any	DET
ejpam-3764	383	12	γ	γ	PROPN
ejpam-3764	383	13	∈	∈	PROPN
ejpam-3764	383	14	i.	i.	NOUN
ejpam-3764	383	15	if	if	SCONJ
ejpam-3764	383	16	sγ	sγ	NOUN
ejpam-3764	383	17	=	=	PUNCT
ejpam-3764	383	18	∅	∅	NOUN
ejpam-3764	383	19	for	for	ADP
ejpam-3764	383	20	all	all	DET
ejpam-3764	383	21	γ	γ	PROPN
ejpam-3764	383	22	∈	∈	PROPN
ejpam-3764	383	23	i	i	PRON
ejpam-3764	383	24	,	,	PUNCT
ejpam-3764	383	25	then	then	ADV
ejpam-3764	383	26	⋃	⋃	NOUN
ejpam-3764	383	27	γ∈i	γ∈i	ADV
ejpam-3764	383	28	sγ	sγ	NOUN
ejpam-3764	383	29	=	=	SYM
ejpam-3764	383	30	∅	∅	NOUN
ejpam-3764	383	31	∈	∈	PROPN
ejpam-3764	383	32	τe	τe	NOUN
ejpam-3764	383	33	.	.	PUNCT
ejpam-3764	384	1	so	so	ADV
ejpam-3764	384	2	,	,	PUNCT
ejpam-3764	384	3	assume	assume	VERB
ejpam-3764	384	4	that	that	SCONJ
ejpam-3764	384	5	some	some	DET
ejpam-3764	384	6	member	member	NOUN
ejpam-3764	384	7	is	be	AUX
ejpam-3764	384	8	non	non	ADJ
ejpam-3764	384	9	-	-	ADJ
ejpam-3764	384	10	empty	empty	ADJ
ejpam-3764	384	11	.	.	PUNCT
ejpam-3764	385	1	however	however	ADV
ejpam-3764	385	2	,	,	PUNCT
ejpam-3764	385	3	since	since	SCONJ
ejpam-3764	385	4	the	the	DET
ejpam-3764	385	5	empty	empty	ADJ
ejpam-3764	385	6	set	set	NOUN
ejpam-3764	385	7	does	do	AUX
ejpam-3764	385	8	not	not	PART
ejpam-3764	385	9	affect	affect	VERB
ejpam-3764	385	10	any	any	DET
ejpam-3764	385	11	union	union	NOUN
ejpam-3764	385	12	,	,	PUNCT
ejpam-3764	385	13	assume	assume	VERB
ejpam-3764	385	14	that	that	SCONJ
ejpam-3764	385	15	,	,	PUNCT
ejpam-3764	385	16	without	without	ADP
ejpam-3764	385	17	loss	loss	NOUN
ejpam-3764	385	18	of	of	ADP
ejpam-3764	385	19	generality	generality	NOUN
ejpam-3764	385	20	sγ	sγ	ADP
ejpam-3764	385	21	6=	6=	NOUN
ejpam-3764	385	22	∅	∅	NOUN
ejpam-3764	385	23	for	for	ADP
ejpam-3764	385	24	all	all	DET
ejpam-3764	385	25	γ	γ	PROPN
ejpam-3764	385	26	∈	∈	PROPN
ejpam-3764	385	27	i.	i.	NOUN
ejpam-3764	385	28	if	if	SCONJ
ejpam-3764	385	29	there	there	PRON
ejpam-3764	385	30	exists	exist	VERB
ejpam-3764	385	31	a	a	DET
ejpam-3764	385	32	γ1	γ1	NOUN
ejpam-3764	385	33	∈	∈	NOUN
ejpam-3764	385	34	i	i	PRON
ejpam-3764	385	35	such	such	ADJ
ejpam-3764	386	1	that	that	SCONJ
ejpam-3764	386	2	sγ1	sγ1	PROPN
ejpam-3764	386	3	=	=	SYM
ejpam-3764	386	4	x	x	NOUN
ejpam-3764	386	5	,	,	PUNCT
ejpam-3764	386	6	then	then	ADV
ejpam-3764	386	7	⋃	⋃	NOUN
ejpam-3764	386	8	γ∈i	γ∈i	ADV
ejpam-3764	386	9	sγ1	sγ1	VERB
ejpam-3764	386	10	=	=	PUNCT
ejpam-3764	387	1	x	x	SYM
ejpam-3764	387	2	∈	∈	PROPN
ejpam-3764	387	3	τe	τe	NOUN
ejpam-3764	387	4	.	.	PUNCT
ejpam-3764	388	1	so	so	ADV
ejpam-3764	388	2	,	,	PUNCT
ejpam-3764	388	3	assume	assume	VERB
ejpam-3764	388	4	now	now	ADV
ejpam-3764	388	5	that	that	SCONJ
ejpam-3764	388	6	sγ	sγ	PRON
ejpam-3764	388	7	6=	6=	NOUN
ejpam-3764	388	8	x	x	PUNCT
ejpam-3764	388	9	for	for	ADP
ejpam-3764	388	10	all	all	DET
ejpam-3764	388	11	γ	γ	PROPN
ejpam-3764	388	12	∈	∈	PROPN
ejpam-3764	388	13	i	i	PRON
ejpam-3764	388	14	,	,	PUNCT
ejpam-3764	388	15	then	then	ADV
ejpam-3764	388	16	⋃	⋃	NOUN
ejpam-3764	388	17	γ∈i	γ∈i	ADV
ejpam-3764	388	18	sγ	sγ	ADP
ejpam-3764	388	19	∈	∈	PROPN
ejpam-3764	388	20	τe	τe	ADP
ejpam-3764	388	21	,	,	PUNCT
ejpam-3764	388	22	because	because	SCONJ
ejpam-3764	388	23	e	e	PROPN
ejpam-3764	388	24	∈	∈	PROPN
ejpam-3764	388	25	sγ	sγ	VERB
ejpam-3764	388	26	for	for	ADP
ejpam-3764	388	27	all	all	DET
ejpam-3764	388	28	γ	γ	PROPN
ejpam-3764	388	29	∈	∈	PROPN
ejpam-3764	388	30	i.	i.	NOUN
ejpam-3764	388	31	hence	hence	ADV
ejpam-3764	388	32	e	e	PROPN
ejpam-3764	388	33	∈	∈	PROPN
ejpam-3764	388	34	⋃	⋃	NOUN
ejpam-3764	388	35	γ∈i	γ∈i	ADJ
ejpam-3764	388	36	sγ	sγ	NOUN
ejpam-3764	388	37	.	.	PUNCT
ejpam-3764	389	1	also	also	ADV
ejpam-3764	389	2	,	,	PUNCT
ejpam-3764	389	3	for	for	ADP
ejpam-3764	389	4	any	any	DET
ejpam-3764	389	5	(	(	PUNCT
ejpam-3764	389	6	a	a	PRON
ejpam-3764	389	7	6=	6=	NUM
ejpam-3764	389	8	e	e	NOUN
ejpam-3764	389	9	)	)	PUNCT
ejpam-3764	389	10	∈	∈	PROPN
ejpam-3764	389	11	⋃	⋃	NOUN
ejpam-3764	389	12	γ∈i	γ∈i	ADV
ejpam-3764	389	13	sγ	sγ	NOUN
ejpam-3764	389	14	,	,	PUNCT
ejpam-3764	389	15	there	there	PRON
ejpam-3764	389	16	exists	exist	VERB
ejpam-3764	389	17	γa	γa	PROPN
ejpam-3764	389	18	∈	∈	PROPN
ejpam-3764	390	1	i	i	PRON
ejpam-3764	390	2	such	such	ADJ
ejpam-3764	390	3	that	that	SCONJ
ejpam-3764	390	4	a	a	DET
ejpam-3764	390	5	∈	∈	PROPN
ejpam-3764	390	6	sγa	sγa	NOUN
ejpam-3764	390	7	.	.	PUNCT
ejpam-3764	391	1	hence	hence	ADV
ejpam-3764	391	2	a	a	PRON
ejpam-3764	391	3	and	and	CCONJ
ejpam-3764	391	4	the	the	DET
ejpam-3764	391	5	multiplicative	multiplicative	ADJ
ejpam-3764	391	6	inverse	inverse	NOUN
ejpam-3764	391	7	of	of	ADP
ejpam-3764	391	8	a	a	DET
ejpam-3764	391	9	belong	belong	NOUN
ejpam-3764	391	10	to	to	ADP
ejpam-3764	391	11	sγa	sγa	NOUN
ejpam-3764	391	12	.	.	PUNCT
ejpam-3764	392	1	furthermore	furthermore	ADV
ejpam-3764	392	2	,	,	PUNCT
ejpam-3764	392	3	a	a	PRON
ejpam-3764	392	4	and	and	CCONJ
ejpam-3764	392	5	the	the	DET
ejpam-3764	392	6	multiplicative	multiplicative	ADJ
ejpam-3764	392	7	inverse	inverse	NOUN
ejpam-3764	392	8	of	of	ADP
ejpam-3764	392	9	a	a	DET
ejpam-3764	392	10	belong	belong	NOUN
ejpam-3764	392	11	to	to	ADP
ejpam-3764	392	12	⋃	⋃	NOUN
ejpam-3764	392	13	γ∈i	γ∈i	ADJ
ejpam-3764	392	14	sγ	sγ	NOUN
ejpam-3764	392	15	.	.	PUNCT
ejpam-3764	393	1	hence	hence	ADV
ejpam-3764	393	2	⋃	⋃	NOUN
ejpam-3764	393	3	γ∈i	γ∈i	ADV
ejpam-3764	393	4	sγ	sγ	ADP
ejpam-3764	393	5	∈	∈	PROPN
ejpam-3764	393	6	τe	τe	NOUN
ejpam-3764	393	7	.	.	PUNCT
ejpam-3764	394	1	therefore	therefore	ADV
ejpam-3764	394	2	,	,	PUNCT
ejpam-3764	394	3	(	(	PUNCT
ejpam-3764	394	4	x	x	X
ejpam-3764	394	5	,	,	PUNCT
ejpam-3764	394	6	τe	τe	NOUN
ejpam-3764	394	7	)	)	PUNCT
ejpam-3764	394	8	is	be	AUX
ejpam-3764	394	9	topological	topological	ADJ
ejpam-3764	394	10	space	space	NOUN
ejpam-3764	394	11	.	.	PUNCT
ejpam-3764	394	12	example	example	NOUN
ejpam-3764	395	1	9	9	NUM
ejpam-3764	395	2	.	.	PUNCT
ejpam-3764	396	1	the	the	DET
ejpam-3764	396	2	space	space	NOUN
ejpam-3764	396	3	(	(	PUNCT
ejpam-3764	396	4	r,+	r,+	NUM
ejpam-3764	396	5	,	,	PUNCT
ejpam-3764	396	6	·	·	PUNCT
ejpam-3764	396	7	)	)	PUNCT
ejpam-3764	396	8	is	be	AUX
ejpam-3764	396	9	a	a	DET
ejpam-3764	396	10	ring	ring	NOUN
ejpam-3764	396	11	,	,	PUNCT
ejpam-3764	396	12	where	where	SCONJ
ejpam-3764	396	13	0	0	NUM
ejpam-3764	396	14	and	and	CCONJ
ejpam-3764	396	15	1	1	NUM
ejpam-3764	396	16	are	be	AUX
ejpam-3764	396	17	the	the	DET
ejpam-3764	396	18	zero	zero	NUM
ejpam-3764	396	19	and	and	CCONJ
ejpam-3764	396	20	unit	unit	NOUN
ejpam-3764	396	21	elements	element	NOUN
ejpam-3764	396	22	,	,	PUNCT
ejpam-3764	396	23	respectively	respectively	ADV
ejpam-3764	396	24	.	.	PUNCT
ejpam-3764	397	1	then	then	ADV
ejpam-3764	397	2	(	(	PUNCT
ejpam-3764	397	3	r	r	NOUN
ejpam-3764	397	4	,	,	PUNCT
ejpam-3764	397	5	τ0	τ0	NOUN
ejpam-3764	397	6	)	)	PUNCT
ejpam-3764	397	7	is	be	AUX
ejpam-3764	397	8	a	a	DET
ejpam-3764	397	9	topological	topological	ADJ
ejpam-3764	397	10	space	space	NOUN
ejpam-3764	397	11	.	.	PUNCT
ejpam-3764	398	1	the	the	DET
ejpam-3764	398	2	proof	proof	NOUN
ejpam-3764	398	3	is	be	AUX
ejpam-3764	398	4	similar	similar	ADJ
ejpam-3764	398	5	to	to	ADP
ejpam-3764	398	6	theorem	theorem	NOUN
ejpam-3764	398	7	2	2	NUM
ejpam-3764	398	8	;	;	PUNCT
ejpam-3764	398	9	just	just	ADV
ejpam-3764	398	10	replacing	replace	VERB
ejpam-3764	398	11	x	x	PRON
ejpam-3764	398	12	,	,	PUNCT
ejpam-3764	398	13	τe	τe	ADV
ejpam-3764	398	14	and	and	CCONJ
ejpam-3764	398	15	e	e	X
ejpam-3764	398	16	by	by	ADP
ejpam-3764	398	17	r	r	PROPN
ejpam-3764	398	18	,	,	PUNCT
ejpam-3764	398	19	τ0	τ0	NOUN
ejpam-3764	398	20	and	and	CCONJ
ejpam-3764	398	21	0	0	NUM
ejpam-3764	398	22	,	,	PUNCT
ejpam-3764	398	23	respectively	respectively	ADV
ejpam-3764	398	24	.	.	PUNCT
ejpam-3764	399	1	remark	remark	PROPN
ejpam-3764	399	2	7	7	NUM
ejpam-3764	399	3	.	.	PUNCT
ejpam-3764	400	1	the	the	DET
ejpam-3764	400	2	topological	topological	ADJ
ejpam-3764	400	3	space	space	NOUN
ejpam-3764	400	4	(	(	PUNCT
ejpam-3764	400	5	r	r	NOUN
ejpam-3764	400	6	,	,	PUNCT
ejpam-3764	400	7	τ0	τ0	NOUN
ejpam-3764	400	8	)	)	PUNCT
ejpam-3764	400	9	is	be	AUX
ejpam-3764	400	10	first	first	ADV
ejpam-3764	400	11	countable	countable	ADJ
ejpam-3764	400	12	,	,	PUNCT
ejpam-3764	400	13	separable	separable	ADJ
ejpam-3764	400	14	,	,	PUNCT
ejpam-3764	400	15	hyperconnected	hyperconnecte	VERB
ejpam-3764	400	16	,	,	PUNCT
ejpam-3764	400	17	connected	connected	ADJ
ejpam-3764	400	18	and	and	CCONJ
ejpam-3764	400	19	locally	locally	ADV
ejpam-3764	400	20	connected	connect	VERB
ejpam-3764	400	21	,	,	PUNCT
ejpam-3764	400	22	and	and	CCONJ
ejpam-3764	400	23	does	do	AUX
ejpam-3764	400	24	not	not	PART
ejpam-3764	400	25	satisfy	satisfy	VERB
ejpam-3764	400	26	any	any	PRON
ejpam-3764	400	27	of	of	ADP
ejpam-3764	400	28	these	these	DET
ejpam-3764	400	29	t0	t0	NOUN
ejpam-3764	400	30	,	,	PUNCT
ejpam-3764	400	31	regular	regular	ADJ
ejpam-3764	400	32	,	,	PUNCT
ejpam-3764	400	33	normal	normal	ADJ
ejpam-3764	400	34	,	,	PUNCT
ejpam-3764	400	35	compact	compact	ADJ
ejpam-3764	400	36	and	and	CCONJ
ejpam-3764	400	37	lindelöf	lindelöf	PROPN
ejpam-3764	400	38	.	.	PUNCT
ejpam-3764	401	1	m.	m.	PROPN
ejpam-3764	401	2	alqahtani	alqahtani	PROPN
ejpam-3764	401	3	,	,	PUNCT
ejpam-3764	401	4	c.	c.	PROPN
ejpam-3764	401	5	özel	özel	PROPN
ejpam-3764	401	6	,	,	PUNCT
ejpam-3764	401	7	i.	i.	PROPN
ejpam-3764	401	8	alshammari	alshammari	PROPN
ejpam-3764	401	9	/	/	SYM
ejpam-3764	401	10	eur	eur	PROPN
ejpam-3764	401	11	.	.	PUNCT
ejpam-3764	402	1	j.	j.	PROPN
ejpam-3764	402	2	pure	pure	PROPN
ejpam-3764	402	3	appl	appl	PROPN
ejpam-3764	402	4	.	.	PROPN
ejpam-3764	402	5	math	math	PROPN
ejpam-3764	402	6	,	,	PUNCT
ejpam-3764	402	7	13	13	NUM
ejpam-3764	402	8	(	(	PUNCT
ejpam-3764	402	9	3	3	NUM
ejpam-3764	402	10	)	)	PUNCT
ejpam-3764	402	11	(	(	PUNCT
ejpam-3764	402	12	2020	2020	NUM
ejpam-3764	402	13	)	)	PUNCT
ejpam-3764	402	14	,	,	PUNCT
ejpam-3764	402	15	513	513	NUM
ejpam-3764	402	16	-	-	SYM
ejpam-3764	402	17	528	528	NUM
ejpam-3764	402	18	525	525	NUM
ejpam-3764	402	19	5	5	NUM
ejpam-3764	402	20	.	.	PUNCT
ejpam-3764	402	21	omega	omega	NOUN
ejpam-3764	402	22	topology	topology	NOUN
ejpam-3764	402	23	and	and	CCONJ
ejpam-3764	402	24	other	other	ADJ
ejpam-3764	402	25	properties	property	NOUN
ejpam-3764	402	26	recall	recall	VERB
ejpam-3764	402	27	that	that	SCONJ
ejpam-3764	402	28	,	,	PUNCT
ejpam-3764	402	29	a	a	DET
ejpam-3764	402	30	subset	subset	NOUN
ejpam-3764	402	31	a	a	PRON
ejpam-3764	402	32	of	of	ADP
ejpam-3764	402	33	a	a	DET
ejpam-3764	402	34	space	space	NOUN
ejpam-3764	402	35	x	x	PUNCT
ejpam-3764	402	36	is	be	AUX
ejpam-3764	402	37	said	say	VERB
ejpam-3764	402	38	to	to	PART
ejpam-3764	402	39	be	be	AUX
ejpam-3764	402	40	regularly	regularly	ADV
ejpam-3764	402	41	-	-	PUNCT
ejpam-3764	402	42	open	open	ADJ
ejpam-3764	402	43	or	or	CCONJ
ejpam-3764	402	44	an	an	DET
ejpam-3764	402	45	open	open	ADJ
ejpam-3764	402	46	domain	domain	NOUN
ejpam-3764	402	47	if	if	SCONJ
ejpam-3764	402	48	it	it	PRON
ejpam-3764	402	49	is	be	AUX
ejpam-3764	402	50	the	the	DET
ejpam-3764	402	51	interior	interior	NOUN
ejpam-3764	402	52	of	of	ADP
ejpam-3764	402	53	its	its	PRON
ejpam-3764	402	54	own	own	ADJ
ejpam-3764	402	55	closure	closure	NOUN
ejpam-3764	402	56	,	,	PUNCT
ejpam-3764	402	57	see	see	VERB
ejpam-3764	402	58	[	[	X
ejpam-3764	402	59	7	7	NUM
ejpam-3764	402	60	]	]	PUNCT
ejpam-3764	402	61	.	.	PUNCT
ejpam-3764	403	1	a	a	DET
ejpam-3764	403	2	set	set	NOUN
ejpam-3764	403	3	a	a	PRON
ejpam-3764	403	4	is	be	AUX
ejpam-3764	403	5	said	say	VERB
ejpam-3764	403	6	to	to	PART
ejpam-3764	403	7	be	be	AUX
ejpam-3764	403	8	a	a	DET
ejpam-3764	403	9	regularly	regularly	ADV
ejpam-3764	403	10	-	-	PUNCT
ejpam-3764	403	11	closed	close	VERB
ejpam-3764	403	12	or	or	CCONJ
ejpam-3764	403	13	a	a	DET
ejpam-3764	403	14	closed	closed	ADJ
ejpam-3764	403	15	domain	domain	NOUN
ejpam-3764	403	16	if	if	SCONJ
ejpam-3764	403	17	its	its	PRON
ejpam-3764	403	18	complement	complement	NOUN
ejpam-3764	403	19	is	be	AUX
ejpam-3764	403	20	an	an	DET
ejpam-3764	403	21	open	open	ADJ
ejpam-3764	403	22	domain	domain	NOUN
ejpam-3764	403	23	.	.	PUNCT
ejpam-3764	404	1	a	a	DET
ejpam-3764	404	2	subset	subset	NOUN
ejpam-3764	404	3	a	a	PRON
ejpam-3764	404	4	of	of	ADP
ejpam-3764	404	5	a	a	DET
ejpam-3764	404	6	space	space	NOUN
ejpam-3764	404	7	x	x	PUNCT
ejpam-3764	404	8	is	be	AUX
ejpam-3764	404	9	called	call	VERB
ejpam-3764	404	10	a	a	DET
ejpam-3764	404	11	π	π	PROPN
ejpam-3764	404	12	-	-	VERB
ejpam-3764	404	13	closed	closed	ADJ
ejpam-3764	404	14	if	if	SCONJ
ejpam-3764	404	15	it	it	PRON
ejpam-3764	404	16	is	be	AUX
ejpam-3764	404	17	a	a	DET
ejpam-3764	404	18	finite	finite	ADJ
ejpam-3764	404	19	intersection	intersection	NOUN
ejpam-3764	404	20	of	of	ADP
ejpam-3764	404	21	closed	closed	ADJ
ejpam-3764	404	22	domain	domain	NOUN
ejpam-3764	404	23	sets	set	NOUN
ejpam-3764	404	24	,	,	PUNCT
ejpam-3764	404	25	see	see	VERB
ejpam-3764	404	26	[	[	X
ejpam-3764	404	27	17	17	NUM
ejpam-3764	404	28	]	]	PUNCT
ejpam-3764	404	29	.	.	PUNCT
ejpam-3764	405	1	a	a	DET
ejpam-3764	405	2	subset	subset	NOUN
ejpam-3764	405	3	a	a	PRON
ejpam-3764	405	4	is	be	AUX
ejpam-3764	405	5	called	call	VERB
ejpam-3764	405	6	a	a	DET
ejpam-3764	405	7	π	π	NOUN
ejpam-3764	405	8	-	-	NOUN
ejpam-3764	405	9	open	open	ADJ
ejpam-3764	405	10	if	if	SCONJ
ejpam-3764	405	11	its	its	PRON
ejpam-3764	405	12	complement	complement	NOUN
ejpam-3764	405	13	is	be	AUX
ejpam-3764	405	14	a	a	DET
ejpam-3764	405	15	π	π	NOUN
ejpam-3764	405	16	-	-	VERB
ejpam-3764	405	17	closed	closed	ADJ
ejpam-3764	405	18	.	.	PUNCT
ejpam-3764	406	1	if	if	SCONJ
ejpam-3764	406	2	t	t	PROPN
ejpam-3764	406	3	and	and	CCONJ
ejpam-3764	406	4	t	t	PROPN
ejpam-3764	406	5	′	′	NOUN
ejpam-3764	406	6	are	be	AUX
ejpam-3764	406	7	two	two	NUM
ejpam-3764	406	8	topologies	topology	NOUN
ejpam-3764	406	9	on	on	ADP
ejpam-3764	406	10	a	a	DET
ejpam-3764	406	11	set	set	NOUN
ejpam-3764	406	12	x	x	PUNCT
ejpam-3764	406	13	such	such	ADJ
ejpam-3764	406	14	that	that	SCONJ
ejpam-3764	406	15	t	t	PROPN
ejpam-3764	406	16	′	′	NUM
ejpam-3764	406	17	⊆	⊆	NUM
ejpam-3764	406	18	t	t	NOUN
ejpam-3764	406	19	,	,	PUNCT
ejpam-3764	406	20	then	then	ADV
ejpam-3764	406	21	t	t	PROPN
ejpam-3764	406	22	′	′	NUM
ejpam-3764	406	23	is	be	AUX
ejpam-3764	406	24	called	call	VERB
ejpam-3764	406	25	the	the	DET
ejpam-3764	406	26	coarser	coarse	ADJ
ejpam-3764	406	27	topology	topology	NOUN
ejpam-3764	406	28	than	than	ADP
ejpam-3764	406	29	t	t	PROPN
ejpam-3764	406	30	and	and	CCONJ
ejpam-3764	406	31	t	t	PROPN
ejpam-3764	406	32	is	be	AUX
ejpam-3764	406	33	called	call	VERB
ejpam-3764	406	34	the	the	DET
ejpam-3764	406	35	finer	finer	NOUN
ejpam-3764	406	36	.	.	PUNCT
ejpam-3764	407	1	a	a	DET
ejpam-3764	407	2	space	space	NOUN
ejpam-3764	407	3	x	x	PUNCT
ejpam-3764	407	4	is	be	AUX
ejpam-3764	407	5	said	say	VERB
ejpam-3764	407	6	to	to	PART
ejpam-3764	407	7	be	be	AUX
ejpam-3764	407	8	a	a	DET
ejpam-3764	407	9	π	π	NOUN
ejpam-3764	407	10	-	-	NOUN
ejpam-3764	407	11	normal	normal	ADJ
ejpam-3764	407	12	,	,	PUNCT
ejpam-3764	407	13	[	[	X
ejpam-3764	407	14	3	3	NUM
ejpam-3764	407	15	]	]	PUNCT
ejpam-3764	407	16	,	,	PUNCT
ejpam-3764	407	17	if	if	SCONJ
ejpam-3764	407	18	any	any	DET
ejpam-3764	407	19	pair	pair	NOUN
ejpam-3764	407	20	of	of	ADP
ejpam-3764	407	21	disjoint	disjoint	NOUN
ejpam-3764	407	22	closed	closed	ADJ
ejpam-3764	407	23	subsets	subset	NOUN
ejpam-3764	407	24	a	a	PRON
ejpam-3764	407	25	and	and	CCONJ
ejpam-3764	407	26	b	b	NOUN
ejpam-3764	407	27	of	of	ADP
ejpam-3764	407	28	x	x	PRON
ejpam-3764	407	29	,	,	PUNCT
ejpam-3764	407	30	one	one	NUM
ejpam-3764	407	31	of	of	ADP
ejpam-3764	407	32	which	which	PRON
ejpam-3764	407	33	is	be	AUX
ejpam-3764	407	34	π	π	PROPN
ejpam-3764	407	35	-	-	VERB
ejpam-3764	407	36	closed	closed	ADJ
ejpam-3764	407	37	,	,	PUNCT
ejpam-3764	407	38	can	can	AUX
ejpam-3764	407	39	be	be	AUX
ejpam-3764	407	40	separated	separate	VERB
ejpam-3764	407	41	by	by	ADP
ejpam-3764	407	42	two	two	NUM
ejpam-3764	407	43	disjoint	disjoint	ADJ
ejpam-3764	407	44	open	open	ADJ
ejpam-3764	407	45	subsets	subset	NOUN
ejpam-3764	407	46	.	.	PUNCT
ejpam-3764	408	1	a	a	DET
ejpam-3764	408	2	space	space	NOUN
ejpam-3764	408	3	x	x	PUNCT
ejpam-3764	408	4	is	be	AUX
ejpam-3764	408	5	said	say	VERB
ejpam-3764	408	6	to	to	PART
ejpam-3764	408	7	be	be	AUX
ejpam-3764	408	8	a	a	DET
ejpam-3764	408	9	almost	almost	ADV
ejpam-3764	408	10	-	-	PUNCT
ejpam-3764	408	11	normal	normal	ADJ
ejpam-3764	408	12	,	,	PUNCT
ejpam-3764	408	13	[	[	X
ejpam-3764	408	14	3	3	NUM
ejpam-3764	408	15	]	]	PUNCT
ejpam-3764	408	16	,	,	PUNCT
ejpam-3764	408	17	if	if	SCONJ
ejpam-3764	408	18	any	any	DET
ejpam-3764	408	19	pair	pair	NOUN
ejpam-3764	408	20	of	of	ADP
ejpam-3764	408	21	disjoint	disjoint	NOUN
ejpam-3764	408	22	closed	closed	ADJ
ejpam-3764	408	23	subsets	subset	NOUN
ejpam-3764	408	24	a	a	PRON
ejpam-3764	408	25	and	and	CCONJ
ejpam-3764	408	26	b	b	NOUN
ejpam-3764	408	27	of	of	ADP
ejpam-3764	408	28	x	x	PRON
ejpam-3764	408	29	,	,	PUNCT
ejpam-3764	408	30	one	one	NUM
ejpam-3764	408	31	of	of	ADP
ejpam-3764	408	32	which	which	PRON
ejpam-3764	408	33	is	be	AUX
ejpam-3764	408	34	closed	closed	ADJ
ejpam-3764	408	35	domain	domain	NOUN
ejpam-3764	408	36	,	,	PUNCT
ejpam-3764	408	37	can	can	AUX
ejpam-3764	408	38	be	be	AUX
ejpam-3764	408	39	separated	separate	VERB
ejpam-3764	408	40	by	by	ADP
ejpam-3764	408	41	two	two	NUM
ejpam-3764	408	42	disjoint	disjoint	ADJ
ejpam-3764	408	43	open	open	ADJ
ejpam-3764	408	44	subsets	subset	NOUN
ejpam-3764	408	45	.	.	PUNCT
ejpam-3764	409	1	a	a	DET
ejpam-3764	409	2	space	space	NOUN
ejpam-3764	409	3	x	x	PUNCT
ejpam-3764	409	4	is	be	AUX
ejpam-3764	409	5	said	say	VERB
ejpam-3764	409	6	to	to	PART
ejpam-3764	409	7	be	be	AUX
ejpam-3764	409	8	a	a	DET
ejpam-3764	409	9	mildly	mildly	ADV
ejpam-3764	409	10	normal	normal	ADJ
ejpam-3764	409	11	,	,	PUNCT
ejpam-3764	409	12	[	[	X
ejpam-3764	409	13	15	15	NUM
ejpam-3764	409	14	]	]	X
ejpam-3764	409	15	,	,	PUNCT
ejpam-3764	409	16	if	if	SCONJ
ejpam-3764	409	17	any	any	DET
ejpam-3764	409	18	pair	pair	NOUN
ejpam-3764	409	19	of	of	ADP
ejpam-3764	409	20	disjoint	disjoint	NOUN
ejpam-3764	409	21	closed	closed	ADJ
ejpam-3764	409	22	domain	domain	NOUN
ejpam-3764	409	23	subsets	subset	NOUN
ejpam-3764	409	24	a	a	PRON
ejpam-3764	409	25	and	and	CCONJ
ejpam-3764	409	26	b	b	NOUN
ejpam-3764	409	27	of	of	ADP
ejpam-3764	409	28	x	x	PRON
ejpam-3764	409	29	can	can	AUX
ejpam-3764	409	30	be	be	AUX
ejpam-3764	409	31	separated	separate	VERB
ejpam-3764	409	32	by	by	ADP
ejpam-3764	409	33	two	two	NUM
ejpam-3764	409	34	disjoint	disjoint	ADJ
ejpam-3764	409	35	open	open	ADJ
ejpam-3764	409	36	subsets	subset	NOUN
ejpam-3764	409	37	.	.	PUNCT
ejpam-3764	410	1	a	a	DET
ejpam-3764	410	2	space	space	NOUN
ejpam-3764	410	3	(	(	PUNCT
ejpam-3764	410	4	x	x	X
ejpam-3764	410	5	,	,	PUNCT
ejpam-3764	410	6	t	t	PROPN
ejpam-3764	410	7	)	)	PUNCT
ejpam-3764	410	8	is	be	AUX
ejpam-3764	410	9	said	say	VERB
ejpam-3764	410	10	to	to	PART
ejpam-3764	410	11	be	be	AUX
ejpam-3764	410	12	a	a	DET
ejpam-3764	410	13	epi	epi	NOUN
ejpam-3764	410	14	-	-	NOUN
ejpam-3764	410	15	normal	normal	ADJ
ejpam-3764	410	16	,	,	PUNCT
ejpam-3764	410	17	[	[	X
ejpam-3764	410	18	5	5	NUM
ejpam-3764	410	19	]	]	PUNCT
ejpam-3764	410	20	,	,	PUNCT
ejpam-3764	410	21	if	if	SCONJ
ejpam-3764	410	22	there	there	PRON
ejpam-3764	410	23	exists	exist	VERB
ejpam-3764	410	24	a	a	DET
ejpam-3764	410	25	coarser	coarse	ADJ
ejpam-3764	410	26	topology	topology	NOUN
ejpam-3764	410	27	t	t	NOUN
ejpam-3764	410	28	′	′	NUM
ejpam-3764	410	29	on	on	ADP
ejpam-3764	410	30	x	x	INTJ
ejpam-3764	410	31	such	such	ADJ
ejpam-3764	410	32	that	that	SCONJ
ejpam-3764	410	33	(	(	PUNCT
ejpam-3764	410	34	x	x	X
ejpam-3764	410	35	,	,	PUNCT
ejpam-3764	410	36	t	t	PROPN
ejpam-3764	410	37	′	′	NUM
ejpam-3764	410	38	)	)	PUNCT
ejpam-3764	410	39	is	be	AUX
ejpam-3764	410	40	t4space	t4space	NOUN
ejpam-3764	410	41	(	(	PUNCT
ejpam-3764	410	42	normal	normal	ADJ
ejpam-3764	410	43	and	and	CCONJ
ejpam-3764	410	44	t1	t1	NOUN
ejpam-3764	410	45	-	-	NOUN
ejpam-3764	410	46	space	space	NOUN
ejpam-3764	410	47	)	)	PUNCT
ejpam-3764	410	48	.	.	PUNCT
ejpam-3764	411	1	a	a	DET
ejpam-3764	411	2	space	space	NOUN
ejpam-3764	411	3	(	(	PUNCT
ejpam-3764	411	4	x	x	X
ejpam-3764	411	5	,	,	PUNCT
ejpam-3764	411	6	t	t	PROPN
ejpam-3764	411	7	)	)	PUNCT
ejpam-3764	411	8	is	be	AUX
ejpam-3764	411	9	said	say	VERB
ejpam-3764	411	10	to	to	PART
ejpam-3764	411	11	be	be	AUX
ejpam-3764	411	12	a	a	DET
ejpam-3764	411	13	epi	epi	NOUN
ejpam-3764	411	14	-	-	ADJ
ejpam-3764	411	15	mildly	mildly	ADV
ejpam-3764	411	16	normal	normal	ADJ
ejpam-3764	411	17	,	,	PUNCT
ejpam-3764	411	18	[	[	X
ejpam-3764	411	19	9	9	NUM
ejpam-3764	411	20	]	]	PUNCT
ejpam-3764	411	21	,	,	PUNCT
ejpam-3764	411	22	if	if	SCONJ
ejpam-3764	411	23	there	there	PRON
ejpam-3764	411	24	exists	exist	VERB
ejpam-3764	411	25	a	a	DET
ejpam-3764	411	26	coarser	coarse	ADJ
ejpam-3764	411	27	topology	topology	NOUN
ejpam-3764	411	28	t	t	NOUN
ejpam-3764	411	29	′	′	NUM
ejpam-3764	411	30	on	on	ADP
ejpam-3764	411	31	x	x	INTJ
ejpam-3764	411	32	such	such	ADJ
ejpam-3764	411	33	that	that	SCONJ
ejpam-3764	411	34	(	(	PUNCT
ejpam-3764	411	35	x	x	X
ejpam-3764	411	36	,	,	PUNCT
ejpam-3764	411	37	t	t	PROPN
ejpam-3764	411	38	′	′	NUM
ejpam-3764	411	39	)	)	PUNCT
ejpam-3764	411	40	is	be	AUX
ejpam-3764	411	41	t2	t2	NOUN
ejpam-3764	411	42	and	and	CCONJ
ejpam-3764	411	43	almost	almost	ADV
ejpam-3764	411	44	normal	normal	ADJ
ejpam-3764	411	45	space	space	NOUN
ejpam-3764	411	46	.	.	PUNCT
ejpam-3764	412	1	theorem	theorem	NOUN
ejpam-3764	412	2	3	3	NUM
ejpam-3764	412	3	.	.	PUNCT
ejpam-3764	413	1	if	if	SCONJ
ejpam-3764	413	2	(	(	PUNCT
ejpam-3764	413	3	aω\{ω},⊗	aω\{ω},⊗	NOUN
ejpam-3764	413	4	)	)	PUNCT
ejpam-3764	413	5	be	be	VERB
ejpam-3764	413	6	a	a	DET
ejpam-3764	413	7	group	group	NOUN
ejpam-3764	413	8	has	have	VERB
ejpam-3764	413	9	more	more	ADJ
ejpam-3764	413	10	than	than	ADP
ejpam-3764	413	11	one	one	NUM
ejpam-3764	413	12	element	element	NOUN
ejpam-3764	413	13	,	,	PUNCT
ejpam-3764	413	14	then	then	ADV
ejpam-3764	413	15	omega	omega	NOUN
ejpam-3764	413	16	topological	topological	ADJ
ejpam-3764	413	17	space	space	NOUN
ejpam-3764	413	18	(	(	PUNCT
ejpam-3764	413	19	aω	aω	PROPN
ejpam-3764	413	20	,	,	PUNCT
ejpam-3764	413	21	τω	τω	INTJ
ejpam-3764	413	22	)	)	PUNCT
ejpam-3764	413	23	is	be	AUX
ejpam-3764	413	24	π	π	NOUN
ejpam-3764	413	25	-	-	ADJ
ejpam-3764	413	26	normal	normal	ADJ
ejpam-3764	413	27	.	.	PUNCT
ejpam-3764	414	1	proof	proof	NOUN
ejpam-3764	414	2	.	.	PUNCT
ejpam-3764	415	1	since	since	SCONJ
ejpam-3764	415	2	the	the	DET
ejpam-3764	415	3	only	only	ADJ
ejpam-3764	415	4	π	π	PROPN
ejpam-3764	415	5	-	-	PUNCT
ejpam-3764	415	6	closed	closed	ADJ
ejpam-3764	415	7	sets	set	NOUN
ejpam-3764	415	8	are	be	AUX
ejpam-3764	415	9	the	the	DET
ejpam-3764	415	10	ground	ground	NOUN
ejpam-3764	415	11	set	set	VERB
ejpam-3764	415	12	aω	aω	INTJ
ejpam-3764	415	13	and	and	CCONJ
ejpam-3764	415	14	the	the	DET
ejpam-3764	415	15	empty	empty	ADJ
ejpam-3764	415	16	set	set	NOUN
ejpam-3764	415	17	,	,	PUNCT
ejpam-3764	415	18	then	then	ADV
ejpam-3764	415	19	(	(	PUNCT
ejpam-3764	415	20	aω	aω	INTJ
ejpam-3764	415	21	,	,	PUNCT
ejpam-3764	415	22	τω	τω	INTJ
ejpam-3764	415	23	)	)	PUNCT
ejpam-3764	415	24	is	be	AUX
ejpam-3764	415	25	a	a	DET
ejpam-3764	415	26	π	π	NOUN
ejpam-3764	415	27	-	-	NOUN
ejpam-3764	415	28	normal	normal	ADJ
ejpam-3764	415	29	.	.	PUNCT
ejpam-3764	416	1	it	it	PRON
ejpam-3764	416	2	is	be	AUX
ejpam-3764	416	3	clear	clear	ADJ
ejpam-3764	416	4	from	from	ADP
ejpam-3764	416	5	the	the	DET
ejpam-3764	416	6	definitions	definition	NOUN
ejpam-3764	416	7	that	that	PRON
ejpam-3764	416	8	normal⇒	normal⇒	PROPN
ejpam-3764	416	9	π	π	NOUN
ejpam-3764	416	10	−	−	PUNCT
ejpam-3764	416	11	normal⇒	normal⇒	PROPN
ejpam-3764	416	12	almost	almost	ADV
ejpam-3764	416	13	normal⇒	normal⇒	ADJ
ejpam-3764	416	14	mildly	mildly	ADV
ejpam-3764	416	15	normal	normal	ADJ
ejpam-3764	416	16	.	.	PUNCT
ejpam-3764	417	1	(	(	PUNCT
ejpam-3764	417	2	1	1	X
ejpam-3764	417	3	)	)	PUNCT
ejpam-3764	417	4	by	by	ADP
ejpam-3764	417	5	(	(	PUNCT
ejpam-3764	417	6	1	1	NUM
ejpam-3764	417	7	)	)	PUNCT
ejpam-3764	417	8	and	and	CCONJ
ejpam-3764	417	9	theorem	theorem	VERB
ejpam-3764	417	10	3	3	NUM
ejpam-3764	417	11	,	,	PUNCT
ejpam-3764	417	12	we	we	PRON
ejpam-3764	417	13	conclude	conclude	VERB
ejpam-3764	417	14	the	the	DET
ejpam-3764	417	15	following	follow	VERB
ejpam-3764	417	16	corollaries	corollary	NOUN
ejpam-3764	417	17	.	.	PUNCT
ejpam-3764	418	1	corollary	corollary	ADJ
ejpam-3764	418	2	13	13	NUM
ejpam-3764	418	3	.	.	PUNCT
ejpam-3764	419	1	if	if	SCONJ
ejpam-3764	419	2	(	(	PUNCT
ejpam-3764	419	3	aω	aω	PROPN
ejpam-3764	419	4	\	\	PROPN
ejpam-3764	419	5	{	{	PUNCT
ejpam-3764	419	6	ω},⊗	ω},⊗	NOUN
ejpam-3764	419	7	)	)	PUNCT
ejpam-3764	419	8	be	be	VERB
ejpam-3764	419	9	a	a	DET
ejpam-3764	419	10	group	group	NOUN
ejpam-3764	419	11	has	have	VERB
ejpam-3764	419	12	more	more	ADJ
ejpam-3764	419	13	than	than	ADP
ejpam-3764	419	14	one	one	NUM
ejpam-3764	419	15	element	element	NOUN
ejpam-3764	419	16	,	,	PUNCT
ejpam-3764	419	17	then	then	ADV
ejpam-3764	419	18	omega	omega	NOUN
ejpam-3764	419	19	topological	topological	ADJ
ejpam-3764	419	20	space	space	NOUN
ejpam-3764	419	21	(	(	PUNCT
ejpam-3764	419	22	aω	aω	PROPN
ejpam-3764	419	23	,	,	PUNCT
ejpam-3764	419	24	τω	τω	INTJ
ejpam-3764	419	25	)	)	PUNCT
ejpam-3764	419	26	is	be	AUX
ejpam-3764	419	27	almost	almost	ADV
ejpam-3764	419	28	normal	normal	ADJ
ejpam-3764	419	29	.	.	PUNCT
ejpam-3764	420	1	corollary	corollary	ADJ
ejpam-3764	420	2	14	14	NUM
ejpam-3764	420	3	.	.	PUNCT
ejpam-3764	421	1	if	if	SCONJ
ejpam-3764	421	2	(	(	PUNCT
ejpam-3764	421	3	aω	aω	PROPN
ejpam-3764	421	4	\	\	PROPN
ejpam-3764	421	5	{	{	PUNCT
ejpam-3764	421	6	ω},⊗	ω},⊗	NOUN
ejpam-3764	421	7	)	)	PUNCT
ejpam-3764	421	8	be	be	VERB
ejpam-3764	421	9	a	a	DET
ejpam-3764	421	10	group	group	NOUN
ejpam-3764	421	11	has	have	VERB
ejpam-3764	421	12	more	more	ADJ
ejpam-3764	421	13	than	than	ADP
ejpam-3764	421	14	one	one	NUM
ejpam-3764	421	15	element	element	NOUN
ejpam-3764	421	16	,	,	PUNCT
ejpam-3764	421	17	then	then	ADV
ejpam-3764	421	18	omega	omega	NOUN
ejpam-3764	421	19	topological	topological	ADJ
ejpam-3764	421	20	space	space	NOUN
ejpam-3764	421	21	(	(	PUNCT
ejpam-3764	421	22	aω	aω	PROPN
ejpam-3764	421	23	,	,	PUNCT
ejpam-3764	421	24	τω	τω	INTJ
ejpam-3764	421	25	)	)	PUNCT
ejpam-3764	421	26	is	be	AUX
ejpam-3764	421	27	mildly	mildly	ADV
ejpam-3764	421	28	normal	normal	ADJ
ejpam-3764	421	29	.	.	PUNCT
ejpam-3764	422	1	proposition	proposition	NOUN
ejpam-3764	422	2	22	22	NUM
ejpam-3764	422	3	.	.	PUNCT
ejpam-3764	423	1	any	any	DET
ejpam-3764	423	2	omega	omega	NOUN
ejpam-3764	423	3	topological	topological	ADJ
ejpam-3764	423	4	space	space	NOUN
ejpam-3764	423	5	(	(	PUNCT
ejpam-3764	423	6	aω	aω	PROPN
ejpam-3764	423	7	,	,	PUNCT
ejpam-3764	423	8	τω	τω	INTJ
ejpam-3764	423	9	)	)	PUNCT
ejpam-3764	423	10	is	be	AUX
ejpam-3764	423	11	not	not	PART
ejpam-3764	423	12	epi	epi	ADJ
ejpam-3764	423	13	-	-	ADJ
ejpam-3764	423	14	mildly	mildly	ADV
ejpam-3764	423	15	normal	normal	ADJ
ejpam-3764	423	16	.	.	PUNCT
ejpam-3764	424	1	proof	proof	NOUN
ejpam-3764	424	2	.	.	PUNCT
ejpam-3764	425	1	suppose	suppose	VERB
ejpam-3764	425	2	that	that	SCONJ
ejpam-3764	425	3	,	,	PUNCT
ejpam-3764	425	4	(	(	PUNCT
ejpam-3764	425	5	aω	aω	INTJ
ejpam-3764	425	6	,	,	PUNCT
ejpam-3764	425	7	τω	τω	INTJ
ejpam-3764	425	8	)	)	PUNCT
ejpam-3764	425	9	is	be	AUX
ejpam-3764	425	10	epi	epi	NOUN
ejpam-3764	425	11	-	-	ADJ
ejpam-3764	425	12	mildly	mildly	ADV
ejpam-3764	425	13	normal	normal	ADJ
ejpam-3764	425	14	.	.	PUNCT
ejpam-3764	426	1	then	then	ADV
ejpam-3764	426	2	there	there	PRON
ejpam-3764	426	3	exists	exist	VERB
ejpam-3764	426	4	a	a	DET
ejpam-3764	426	5	coarser	coarse	ADJ
ejpam-3764	426	6	topology	topology	NOUN
ejpam-3764	426	7	t	t	NOUN
ejpam-3764	426	8	′	′	NUM
ejpam-3764	426	9	on	on	ADP
ejpam-3764	426	10	aω	aω	INTJ
ejpam-3764	426	11	such	such	ADJ
ejpam-3764	426	12	that	that	PRON
ejpam-3764	426	13	(	(	PUNCT
ejpam-3764	426	14	aω	aω	PROPN
ejpam-3764	426	15	,	,	PUNCT
ejpam-3764	426	16	t	t	PROPN
ejpam-3764	426	17	′	′	NUM
ejpam-3764	426	18	)	)	PUNCT
ejpam-3764	426	19	is	be	AUX
ejpam-3764	426	20	t2	t2	NOUN
ejpam-3764	426	21	and	and	CCONJ
ejpam-3764	426	22	mildly	mildly	ADV
ejpam-3764	426	23	normal	normal	ADJ
ejpam-3764	426	24	space	space	NOUN
ejpam-3764	426	25	.	.	PUNCT
ejpam-3764	427	1	hence	hence	ADV
ejpam-3764	427	2	(	(	PUNCT
ejpam-3764	427	3	aω	aω	INTJ
ejpam-3764	427	4	,	,	PUNCT
ejpam-3764	427	5	τω	τω	INTJ
ejpam-3764	427	6	)	)	PUNCT
ejpam-3764	427	7	is	be	AUX
ejpam-3764	427	8	t2	t2	NOUN
ejpam-3764	427	9	,	,	PUNCT
ejpam-3764	427	10	thus	thus	ADV
ejpam-3764	427	11	a	a	DET
ejpam-3764	427	12	contradiction	contradiction	NOUN
ejpam-3764	427	13	,	,	PUNCT
ejpam-3764	427	14	because	because	SCONJ
ejpam-3764	427	15	(	(	PUNCT
ejpam-3764	427	16	aω	aω	INTJ
ejpam-3764	427	17	,	,	PUNCT
ejpam-3764	427	18	τω	τω	INTJ
ejpam-3764	427	19	)	)	PUNCT
ejpam-3764	427	20	is	be	AUX
ejpam-3764	427	21	not	not	PART
ejpam-3764	427	22	t1	t1	NOUN
ejpam-3764	427	23	(	(	PUNCT
ejpam-3764	427	24	see	see	VERB
ejpam-3764	427	25	proposition	proposition	NOUN
ejpam-3764	427	26	7	7	NUM
ejpam-3764	427	27	)	)	PUNCT
ejpam-3764	427	28	.	.	PUNCT
ejpam-3764	428	1	then	then	ADV
ejpam-3764	428	2	(	(	PUNCT
ejpam-3764	428	3	aω	aω	INTJ
ejpam-3764	428	4	,	,	PUNCT
ejpam-3764	428	5	τω	τω	INTJ
ejpam-3764	428	6	)	)	PUNCT
ejpam-3764	428	7	is	be	AUX
ejpam-3764	428	8	not	not	PART
ejpam-3764	428	9	epi	epi	ADJ
ejpam-3764	428	10	-	-	ADJ
ejpam-3764	428	11	mildly	mildly	ADV
ejpam-3764	428	12	normal	normal	ADJ
ejpam-3764	428	13	.	.	PUNCT
ejpam-3764	429	1	proposition	proposition	NOUN
ejpam-3764	429	2	23	23	NUM
ejpam-3764	429	3	.	.	PUNCT
ejpam-3764	430	1	any	any	DET
ejpam-3764	430	2	omega	omega	NOUN
ejpam-3764	430	3	topological	topological	ADJ
ejpam-3764	430	4	space	space	NOUN
ejpam-3764	430	5	(	(	PUNCT
ejpam-3764	430	6	aω	aω	PROPN
ejpam-3764	430	7	,	,	PUNCT
ejpam-3764	430	8	τω	τω	INTJ
ejpam-3764	430	9	)	)	PUNCT
ejpam-3764	430	10	is	be	AUX
ejpam-3764	430	11	not	not	PART
ejpam-3764	430	12	epi	epi	NOUN
ejpam-3764	430	13	-	-	ADJ
ejpam-3764	430	14	almost	almost	ADV
ejpam-3764	430	15	normal	normal	ADJ
ejpam-3764	430	16	.	.	PUNCT
ejpam-3764	431	1	proof	proof	NOUN
ejpam-3764	431	2	.	.	PUNCT
ejpam-3764	432	1	using	use	VERB
ejpam-3764	432	2	the	the	DET
ejpam-3764	432	3	same	same	ADJ
ejpam-3764	432	4	proof	proof	NOUN
ejpam-3764	432	5	of	of	ADP
ejpam-3764	432	6	proposition	proposition	NOUN
ejpam-3764	432	7	22	22	NUM
ejpam-3764	432	8	.	.	PUNCT
ejpam-3764	433	1	m.	m.	PROPN
ejpam-3764	433	2	alqahtani	alqahtani	PROPN
ejpam-3764	433	3	,	,	PUNCT
ejpam-3764	433	4	c.	c.	PROPN
ejpam-3764	433	5	özel	özel	PROPN
ejpam-3764	433	6	,	,	PUNCT
ejpam-3764	433	7	i.	i.	PROPN
ejpam-3764	433	8	alshammari	alshammari	PROPN
ejpam-3764	433	9	/	/	SYM
ejpam-3764	433	10	eur	eur	PROPN
ejpam-3764	433	11	.	.	PUNCT
ejpam-3764	434	1	j.	j.	PROPN
ejpam-3764	434	2	pure	pure	PROPN
ejpam-3764	434	3	appl	appl	PROPN
ejpam-3764	434	4	.	.	PROPN
ejpam-3764	434	5	math	math	PROPN
ejpam-3764	434	6	,	,	PUNCT
ejpam-3764	434	7	13	13	NUM
ejpam-3764	434	8	(	(	PUNCT
ejpam-3764	434	9	3	3	NUM
ejpam-3764	434	10	)	)	PUNCT
ejpam-3764	434	11	(	(	PUNCT
ejpam-3764	434	12	2020	2020	NUM
ejpam-3764	434	13	)	)	PUNCT
ejpam-3764	434	14	,	,	PUNCT
ejpam-3764	434	15	513	513	NUM
ejpam-3764	434	16	-	-	SYM
ejpam-3764	434	17	528	528	NUM
ejpam-3764	434	18	526	526	NUM
ejpam-3764	434	19	definition	definition	NOUN
ejpam-3764	434	20	3	3	X
ejpam-3764	434	21	.	.	PUNCT
ejpam-3764	435	1	let	let	VERB
ejpam-3764	435	2	x	x	PRON
ejpam-3764	435	3	be	be	AUX
ejpam-3764	435	4	a	a	DET
ejpam-3764	435	5	space	space	NOUN
ejpam-3764	435	6	.	.	PUNCT
ejpam-3764	436	1	then	then	ADV
ejpam-3764	436	2	:	:	PUNCT
ejpam-3764	436	3	1	1	X
ejpam-3764	436	4	)	)	PUNCT
ejpam-3764	436	5	a	a	DET
ejpam-3764	436	6	space	space	NOUN
ejpam-3764	436	7	x	x	PUNCT
ejpam-3764	436	8	is	be	AUX
ejpam-3764	436	9	called	call	VERB
ejpam-3764	436	10	a	a	DET
ejpam-3764	436	11	c	c	NOUN
ejpam-3764	436	12	-	-	NOUN
ejpam-3764	436	13	normal	normal	ADJ
ejpam-3764	436	14	if	if	SCONJ
ejpam-3764	436	15	there	there	PRON
ejpam-3764	436	16	exist	exist	VERB
ejpam-3764	436	17	a	a	DET
ejpam-3764	436	18	normal	normal	ADJ
ejpam-3764	436	19	space	space	NOUN
ejpam-3764	436	20	y	y	PROPN
ejpam-3764	436	21	and	and	CCONJ
ejpam-3764	436	22	a	a	DET
ejpam-3764	436	23	bijective	bijective	ADJ
ejpam-3764	436	24	function	function	NOUN
ejpam-3764	437	1	f	f	NOUN
ejpam-3764	437	2	:	:	PUNCT
ejpam-3764	437	3	x	x	X
ejpam-3764	437	4	→	→	SYM
ejpam-3764	437	5	y	y	PROPN
ejpam-3764	437	6	such	such	ADJ
ejpam-3764	437	7	that	that	SCONJ
ejpam-3764	437	8	the	the	DET
ejpam-3764	437	9	restriction	restriction	NOUN
ejpam-3764	437	10	function	function	NOUN
ejpam-3764	437	11	f	f	PROPN
ejpam-3764	437	12	|a	|a	VERB
ejpam-3764	437	13	:	:	PUNCT
ejpam-3764	437	14	a	a	DET
ejpam-3764	437	15	→	→	SYM
ejpam-3764	437	16	f(a	f(a	NOUN
ejpam-3764	437	17	)	)	PUNCT
ejpam-3764	437	18	is	be	AUX
ejpam-3764	437	19	a	a	DET
ejpam-3764	437	20	homeomorphism	homeomorphism	NOUN
ejpam-3764	437	21	for	for	ADP
ejpam-3764	437	22	each	each	DET
ejpam-3764	437	23	compact	compact	ADJ
ejpam-3764	437	24	subspace	subspace	NOUN
ejpam-3764	437	25	a	a	DET
ejpam-3764	437	26	⊆	⊆	NUM
ejpam-3764	437	27	x	x	SYM
ejpam-3764	437	28	,	,	PUNCT
ejpam-3764	437	29	[	[	X
ejpam-3764	437	30	2	2	NUM
ejpam-3764	437	31	]	]	PUNCT
ejpam-3764	437	32	.	.	PUNCT
ejpam-3764	438	1	2	2	X
ejpam-3764	438	2	)	)	PUNCT
ejpam-3764	438	3	a	a	DET
ejpam-3764	438	4	space	space	NOUN
ejpam-3764	438	5	x	x	PUNCT
ejpam-3764	438	6	is	be	AUX
ejpam-3764	438	7	called	call	VERB
ejpam-3764	438	8	a	a	DET
ejpam-3764	438	9	cc	cc	NOUN
ejpam-3764	438	10	-	-	NOUN
ejpam-3764	438	11	normal	normal	ADJ
ejpam-3764	438	12	if	if	SCONJ
ejpam-3764	438	13	there	there	PRON
ejpam-3764	438	14	exist	exist	VERB
ejpam-3764	438	15	a	a	DET
ejpam-3764	438	16	normal	normal	ADJ
ejpam-3764	438	17	space	space	NOUN
ejpam-3764	438	18	y	y	PROPN
ejpam-3764	438	19	and	and	CCONJ
ejpam-3764	438	20	a	a	DET
ejpam-3764	438	21	bijective	bijective	ADJ
ejpam-3764	438	22	function	function	NOUN
ejpam-3764	439	1	f	f	NOUN
ejpam-3764	439	2	:	:	PUNCT
ejpam-3764	439	3	x	x	X
ejpam-3764	439	4	→	→	SYM
ejpam-3764	439	5	y	y	PROPN
ejpam-3764	439	6	such	such	ADJ
ejpam-3764	439	7	that	that	SCONJ
ejpam-3764	439	8	the	the	DET
ejpam-3764	439	9	restriction	restriction	NOUN
ejpam-3764	439	10	function	function	NOUN
ejpam-3764	439	11	f	f	PROPN
ejpam-3764	439	12	|a	|a	VERB
ejpam-3764	439	13	:	:	PUNCT
ejpam-3764	439	14	a	a	DET
ejpam-3764	439	15	→	→	SYM
ejpam-3764	439	16	f(a	f(a	NOUN
ejpam-3764	439	17	)	)	PUNCT
ejpam-3764	439	18	is	be	AUX
ejpam-3764	439	19	a	a	DET
ejpam-3764	439	20	homeomorphism	homeomorphism	NOUN
ejpam-3764	439	21	for	for	ADP
ejpam-3764	439	22	each	each	DET
ejpam-3764	439	23	countably	countably	ADV
ejpam-3764	439	24	compact	compact	ADJ
ejpam-3764	439	25	subspace	subspace	NOUN
ejpam-3764	439	26	a	a	DET
ejpam-3764	439	27	⊆	⊆	NUM
ejpam-3764	439	28	x.[4	x.[4	NUM
ejpam-3764	439	29	]	]	X
ejpam-3764	439	30	.	.	PUNCT
ejpam-3764	440	1	3	3	X
ejpam-3764	440	2	)	)	PUNCT
ejpam-3764	440	3	a	a	DET
ejpam-3764	440	4	spacex	spacex	NOUN
ejpam-3764	440	5	is	be	AUX
ejpam-3764	440	6	called	call	VERB
ejpam-3764	440	7	a	a	DET
ejpam-3764	440	8	l	l	NOUN
ejpam-3764	440	9	-	-	ADJ
ejpam-3764	440	10	normal	normal	ADJ
ejpam-3764	440	11	if	if	SCONJ
ejpam-3764	440	12	there	there	PRON
ejpam-3764	440	13	exist	exist	VERB
ejpam-3764	440	14	a	a	DET
ejpam-3764	440	15	normal	normal	ADJ
ejpam-3764	440	16	space	space	NOUN
ejpam-3764	440	17	y	y	PROPN
ejpam-3764	440	18	and	and	CCONJ
ejpam-3764	440	19	a	a	DET
ejpam-3764	440	20	bijective	bijective	ADJ
ejpam-3764	440	21	function	function	NOUN
ejpam-3764	441	1	f	f	NOUN
ejpam-3764	441	2	:	:	PUNCT
ejpam-3764	441	3	x	x	X
ejpam-3764	441	4	→	→	SYM
ejpam-3764	441	5	y	y	PROPN
ejpam-3764	441	6	such	such	ADJ
ejpam-3764	441	7	that	that	SCONJ
ejpam-3764	441	8	the	the	DET
ejpam-3764	441	9	restriction	restriction	NOUN
ejpam-3764	441	10	function	function	NOUN
ejpam-3764	441	11	f	f	PROPN
ejpam-3764	441	12	|a	|a	VERB
ejpam-3764	441	13	:	:	PUNCT
ejpam-3764	441	14	a	a	DET
ejpam-3764	441	15	→	→	SYM
ejpam-3764	441	16	f(a	f(a	NOUN
ejpam-3764	441	17	)	)	PUNCT
ejpam-3764	441	18	is	be	AUX
ejpam-3764	441	19	a	a	DET
ejpam-3764	441	20	homeomorphism	homeomorphism	NOUN
ejpam-3764	441	21	for	for	SCONJ
ejpam-3764	441	22	each	each	DET
ejpam-3764	441	23	lindelöf	lindelöf	NOUN
ejpam-3764	441	24	subspace	subspace	VERB
ejpam-3764	441	25	a	a	DET
ejpam-3764	441	26	⊆	⊆	NUM
ejpam-3764	441	27	x.[6	x.[6	NUM
ejpam-3764	441	28	]	]	PUNCT
ejpam-3764	441	29	.	.	PUNCT
ejpam-3764	442	1	4	4	X
ejpam-3764	442	2	)	)	PUNCT
ejpam-3764	442	3	a	a	DET
ejpam-3764	442	4	space	space	NOUN
ejpam-3764	442	5	x	x	PUNCT
ejpam-3764	442	6	is	be	AUX
ejpam-3764	442	7	called	call	VERB
ejpam-3764	442	8	a	a	DET
ejpam-3764	442	9	snormal	snormal	ADJ
ejpam-3764	442	10	if	if	SCONJ
ejpam-3764	442	11	there	there	PRON
ejpam-3764	442	12	exist	exist	VERB
ejpam-3764	442	13	a	a	DET
ejpam-3764	442	14	normal	normal	ADJ
ejpam-3764	442	15	space	space	NOUN
ejpam-3764	442	16	y	y	PROPN
ejpam-3764	442	17	and	and	CCONJ
ejpam-3764	442	18	a	a	DET
ejpam-3764	442	19	bijective	bijective	ADJ
ejpam-3764	442	20	function	function	NOUN
ejpam-3764	443	1	f	f	NOUN
ejpam-3764	443	2	:	:	PUNCT
ejpam-3764	443	3	x	x	X
ejpam-3764	443	4	→	→	SYM
ejpam-3764	443	5	y	y	PROPN
ejpam-3764	443	6	such	such	ADJ
ejpam-3764	443	7	that	that	SCONJ
ejpam-3764	443	8	the	the	DET
ejpam-3764	443	9	restriction	restriction	NOUN
ejpam-3764	443	10	function	function	NOUN
ejpam-3764	443	11	f	f	PROPN
ejpam-3764	443	12	|a	|a	VERB
ejpam-3764	443	13	:	:	PUNCT
ejpam-3764	443	14	a	a	DET
ejpam-3764	443	15	→	→	SYM
ejpam-3764	443	16	f(a	f(a	NOUN
ejpam-3764	443	17	)	)	PUNCT
ejpam-3764	443	18	is	be	AUX
ejpam-3764	443	19	a	a	DET
ejpam-3764	443	20	homeomorphism	homeomorphism	NOUN
ejpam-3764	443	21	for	for	ADP
ejpam-3764	443	22	each	each	DET
ejpam-3764	443	23	seprable	seprable	ADJ
ejpam-3764	443	24	subspace	subspace	NOUN
ejpam-3764	443	25	a	a	DET
ejpam-3764	443	26	⊆	⊆	NUM
ejpam-3764	443	27	x.[1	x.[1	NOUN
ejpam-3764	443	28	]	]	X
ejpam-3764	443	29	.	.	PUNCT
ejpam-3764	444	1	5	5	X
ejpam-3764	444	2	)	)	PUNCT
ejpam-3764	444	3	a	a	DET
ejpam-3764	444	4	space	space	NOUN
ejpam-3764	444	5	x	x	PUNCT
ejpam-3764	444	6	is	be	AUX
ejpam-3764	444	7	called	call	VERB
ejpam-3764	444	8	a	a	DET
ejpam-3764	444	9	c	c	NOUN
ejpam-3764	444	10	-	-	PUNCT
ejpam-3764	444	11	paracompact	paracompact	ADJ
ejpam-3764	444	12	(	(	PUNCT
ejpam-3764	444	13	c2	c2	NOUN
ejpam-3764	444	14	-	-	PUNCT
ejpam-3764	444	15	paracompact	paracompact	NOUN
ejpam-3764	444	16	)	)	PUNCT
ejpam-3764	444	17	if	if	SCONJ
ejpam-3764	444	18	there	there	PRON
ejpam-3764	444	19	exist	exist	VERB
ejpam-3764	444	20	a	a	DET
ejpam-3764	444	21	paracompact	paracompact	NOUN
ejpam-3764	444	22	(	(	PUNCT
ejpam-3764	444	23	hausdorff	hausdorff	NOUN
ejpam-3764	444	24	paracompact	paracompact	NOUN
ejpam-3764	444	25	)	)	PUNCT
ejpam-3764	444	26	space	space	NOUN
ejpam-3764	444	27	y	y	PROPN
ejpam-3764	444	28	and	and	CCONJ
ejpam-3764	444	29	a	a	DET
ejpam-3764	444	30	bijective	bijective	ADJ
ejpam-3764	444	31	function	function	NOUN
ejpam-3764	445	1	f	f	NOUN
ejpam-3764	445	2	:	:	PUNCT
ejpam-3764	445	3	x	x	X
ejpam-3764	445	4	→	→	SYM
ejpam-3764	445	5	y	y	PROPN
ejpam-3764	445	6	such	such	ADJ
ejpam-3764	445	7	that	that	SCONJ
ejpam-3764	445	8	the	the	DET
ejpam-3764	445	9	restriction	restriction	NOUN
ejpam-3764	445	10	function	function	NOUN
ejpam-3764	445	11	f	f	PROPN
ejpam-3764	445	12	|a	|a	VERB
ejpam-3764	445	13	:	:	PUNCT
ejpam-3764	445	14	a→	a→	PROPN
ejpam-3764	445	15	f(a	f(a	NOUN
ejpam-3764	445	16	)	)	PUNCT
ejpam-3764	445	17	is	be	AUX
ejpam-3764	445	18	a	a	DET
ejpam-3764	445	19	homeomorphism	homeomorphism	NOUN
ejpam-3764	445	20	for	for	ADP
ejpam-3764	445	21	each	each	DET
ejpam-3764	445	22	compact	compact	ADJ
ejpam-3764	445	23	subspace	subspace	NOUN
ejpam-3764	445	24	a	a	DET
ejpam-3764	445	25	⊆	⊆	NUM
ejpam-3764	445	26	x.[13	x.[13	NOUN
ejpam-3764	445	27	]	]	PUNCT
ejpam-3764	445	28	.	.	PUNCT
ejpam-3764	446	1	theorem	theorem	ADJ
ejpam-3764	446	2	4	4	NUM
ejpam-3764	446	3	.	.	PUNCT
ejpam-3764	447	1	if	if	SCONJ
ejpam-3764	447	2	a	a	DET
ejpam-3764	447	3	∈	∈	NOUN
ejpam-3764	447	4	aω	aω	X
ejpam-3764	447	5	\	\	PROPN
ejpam-3764	447	6	{	{	PUNCT
ejpam-3764	447	7	ω	ω	NOUN
ejpam-3764	447	8	}	}	PUNCT
ejpam-3764	447	9	has	have	VERB
ejpam-3764	447	10	no	no	DET
ejpam-3764	447	11	multiplicative	multiplicative	ADJ
ejpam-3764	447	12	inverse	inverse	NOUN
ejpam-3764	447	13	,	,	PUNCT
ejpam-3764	447	14	then	then	ADV
ejpam-3764	447	15	omega	omega	NOUN
ejpam-3764	447	16	topological	topological	ADJ
ejpam-3764	447	17	space	space	NOUN
ejpam-3764	447	18	(	(	PUNCT
ejpam-3764	447	19	aω	aω	PROPN
ejpam-3764	447	20	,	,	PUNCT
ejpam-3764	447	21	τω	τω	INTJ
ejpam-3764	447	22	)	)	PUNCT
ejpam-3764	447	23	is	be	AUX
ejpam-3764	447	24	c	c	NOUN
ejpam-3764	447	25	-	-	ADJ
ejpam-3764	447	26	normal	normal	ADJ
ejpam-3764	447	27	.	.	PUNCT
ejpam-3764	448	1	proof	proof	NOUN
ejpam-3764	448	2	.	.	PUNCT
ejpam-3764	449	1	let	let	VERB
ejpam-3764	449	2	a	a	DET
ejpam-3764	449	3	∈	∈	NOUN
ejpam-3764	449	4	aω	aω	X
ejpam-3764	449	5	\	\	PROPN
ejpam-3764	449	6	{	{	PUNCT
ejpam-3764	449	7	ω	ω	NOUN
ejpam-3764	449	8	}	}	PUNCT
ejpam-3764	449	9	has	have	VERB
ejpam-3764	449	10	no	no	DET
ejpam-3764	449	11	multiplicative	multiplicative	ADJ
ejpam-3764	449	12	inverse	inverse	NOUN
ejpam-3764	449	13	.	.	PUNCT
ejpam-3764	450	1	let	let	VERB
ejpam-3764	450	2	v	v	PART
ejpam-3764	450	3	be	be	AUX
ejpam-3764	450	4	any	any	DET
ejpam-3764	450	5	non	non	ADJ
ejpam-3764	450	6	-	-	ADJ
ejpam-3764	450	7	empty	empty	ADJ
ejpam-3764	450	8	closed	closed	ADJ
ejpam-3764	450	9	subset	subset	NOUN
ejpam-3764	450	10	of	of	ADP
ejpam-3764	450	11	aω	aω	PROPN
ejpam-3764	450	12	.	.	PUNCT
ejpam-3764	451	1	then	then	ADV
ejpam-3764	451	2	a	a	DET
ejpam-3764	451	3	∈	∈	NOUN
ejpam-3764	451	4	v.	v.	CCONJ
ejpam-3764	451	5	suppose	suppose	VERB
ejpam-3764	451	6	not	not	PART
ejpam-3764	451	7	,	,	PUNCT
ejpam-3764	451	8	a	a	DET
ejpam-3764	451	9	/∈	/∈	NOUN
ejpam-3764	451	10	v	v	NOUN
ejpam-3764	451	11	,	,	PUNCT
ejpam-3764	451	12	then	then	ADV
ejpam-3764	451	13	a	a	DET
ejpam-3764	451	14	∈	∈	NOUN
ejpam-3764	451	15	aω	aω	ADP
ejpam-3764	451	16	\	\	PROPN
ejpam-3764	452	1	v.	v.	ADP
ejpam-3764	452	2	by	by	ADP
ejpam-3764	452	3	the	the	DET
ejpam-3764	452	4	definition	definition	NOUN
ejpam-3764	452	5	of	of	ADP
ejpam-3764	452	6	τω	τω	INTJ
ejpam-3764	452	7	,	,	PUNCT
ejpam-3764	452	8	aω	aω	PROPN
ejpam-3764	452	9	\v	\v	PROPN
ejpam-3764	452	10	is	be	AUX
ejpam-3764	452	11	not	not	PART
ejpam-3764	452	12	open	open	ADJ
ejpam-3764	452	13	,	,	PUNCT
ejpam-3764	452	14	thus	thus	ADV
ejpam-3764	452	15	a	a	DET
ejpam-3764	452	16	contradiction	contradiction	NOUN
ejpam-3764	452	17	.	.	PUNCT
ejpam-3764	453	1	hence	hence	ADV
ejpam-3764	453	2	,	,	PUNCT
ejpam-3764	453	3	a	a	DET
ejpam-3764	453	4	belong	belong	NOUN
ejpam-3764	453	5	to	to	ADP
ejpam-3764	453	6	any	any	DET
ejpam-3764	453	7	non	non	ADJ
ejpam-3764	453	8	-	-	ADJ
ejpam-3764	453	9	empty	empty	ADJ
ejpam-3764	453	10	closed	closed	ADJ
ejpam-3764	453	11	subsets	subset	NOUN
ejpam-3764	453	12	of	of	ADP
ejpam-3764	453	13	aω	aω	PROPN
ejpam-3764	453	14	.	.	PUNCT
ejpam-3764	454	1	let	let	VERB
ejpam-3764	454	2	k	k	NOUN
ejpam-3764	454	3	and	and	CCONJ
ejpam-3764	454	4	h	h	PROPN
ejpam-3764	454	5	be	be	VERB
ejpam-3764	454	6	any	any	DET
ejpam-3764	454	7	two	two	NUM
ejpam-3764	454	8	disjoint	disjoint	NOUN
ejpam-3764	454	9	closed	closed	ADJ
ejpam-3764	454	10	subsets	subset	NOUN
ejpam-3764	454	11	of	of	ADP
ejpam-3764	454	12	aω	aω	PROPN
ejpam-3764	454	13	.	.	PUNCT
ejpam-3764	455	1	then	then	ADV
ejpam-3764	455	2	k	k	PROPN
ejpam-3764	455	3	or	or	CCONJ
ejpam-3764	455	4	h	h	NOUN
ejpam-3764	455	5	is	be	AUX
ejpam-3764	455	6	equal	equal	ADJ
ejpam-3764	455	7	∅.	∅.	ADV
ejpam-3764	455	8	if	if	SCONJ
ejpam-3764	455	9	k	k	NOUN
ejpam-3764	455	10	=	=	SYM
ejpam-3764	455	11	∅	∅	NOUN
ejpam-3764	455	12	,	,	PUNCT
ejpam-3764	455	13	then	then	ADV
ejpam-3764	455	14	there	there	PRON
ejpam-3764	455	15	exists	exist	VERB
ejpam-3764	455	16	u	u	NOUN
ejpam-3764	455	17	=	=	NOUN
ejpam-3764	455	18	∅	∅	NOUN
ejpam-3764	455	19	and	and	CCONJ
ejpam-3764	455	20	v	v	NOUN
ejpam-3764	455	21	=	=	SYM
ejpam-3764	455	22	aω	aω	NOUN
ejpam-3764	455	23	are	be	AUX
ejpam-3764	455	24	two	two	NUM
ejpam-3764	455	25	disjoint	disjoint	ADJ
ejpam-3764	455	26	open	open	ADJ
ejpam-3764	455	27	sets	set	NOUN
ejpam-3764	455	28	in	in	ADP
ejpam-3764	455	29	aω	aω	NOUN
ejpam-3764	455	30	containing	contain	VERB
ejpam-3764	455	31	k	k	PROPN
ejpam-3764	455	32	and	and	CCONJ
ejpam-3764	455	33	h	h	NOUN
ejpam-3764	455	34	,	,	PUNCT
ejpam-3764	455	35	respectively	respectively	ADV
ejpam-3764	455	36	.	.	PUNCT
ejpam-3764	456	1	if	if	SCONJ
ejpam-3764	456	2	h	h	NOUN
ejpam-3764	456	3	=	=	NOUN
ejpam-3764	456	4	∅	∅	NOUN
ejpam-3764	456	5	,	,	PUNCT
ejpam-3764	456	6	then	then	ADV
ejpam-3764	456	7	there	there	PRON
ejpam-3764	456	8	exists	exist	VERB
ejpam-3764	456	9	u	u	NOUN
ejpam-3764	456	10	=	=	NOUN
ejpam-3764	456	11	∅	∅	NOUN
ejpam-3764	456	12	and	and	CCONJ
ejpam-3764	456	13	v	v	NOUN
ejpam-3764	456	14	=	=	SYM
ejpam-3764	456	15	aω	aω	NOUN
ejpam-3764	456	16	are	be	AUX
ejpam-3764	456	17	two	two	NUM
ejpam-3764	456	18	disjoint	disjoint	ADJ
ejpam-3764	456	19	open	open	ADJ
ejpam-3764	456	20	sets	set	NOUN
ejpam-3764	456	21	in	in	ADP
ejpam-3764	456	22	aω	aω	NOUN
ejpam-3764	456	23	containing	contain	VERB
ejpam-3764	456	24	h	h	NOUN
ejpam-3764	456	25	and	and	CCONJ
ejpam-3764	456	26	k	k	NOUN
ejpam-3764	456	27	,	,	PUNCT
ejpam-3764	456	28	respectively	respectively	ADV
ejpam-3764	456	29	.	.	PUNCT
ejpam-3764	457	1	therefore	therefore	ADV
ejpam-3764	457	2	,	,	PUNCT
ejpam-3764	457	3	(	(	PUNCT
ejpam-3764	457	4	aω	aω	INTJ
ejpam-3764	457	5	,	,	PUNCT
ejpam-3764	457	6	τω	τω	INTJ
ejpam-3764	457	7	)	)	PUNCT
ejpam-3764	457	8	is	be	AUX
ejpam-3764	457	9	normal	normal	ADJ
ejpam-3764	457	10	.	.	PUNCT
ejpam-3764	458	1	then	then	ADV
ejpam-3764	458	2	there	there	PRON
ejpam-3764	458	3	exist	exist	VERB
ejpam-3764	458	4	y	y	PROPN
ejpam-3764	458	5	=	=	PUNCT
ejpam-3764	458	6	aω	aω	PROPN
ejpam-3764	458	7	is	be	AUX
ejpam-3764	458	8	a	a	DET
ejpam-3764	458	9	normal	normal	ADJ
ejpam-3764	458	10	space	space	NOUN
ejpam-3764	458	11	and	and	CCONJ
ejpam-3764	458	12	the	the	DET
ejpam-3764	458	13	identity	identity	NOUN
ejpam-3764	458	14	function	function	NOUN
ejpam-3764	459	1	i	i	NOUN
ejpam-3764	459	2	d	d	NOUN
ejpam-3764	459	3	:	:	PUNCT
ejpam-3764	459	4	aω	aω	PROPN
ejpam-3764	459	5	→	→	SYM
ejpam-3764	459	6	aω	aω	PROPN
ejpam-3764	459	7	is	be	AUX
ejpam-3764	459	8	bijective	bijective	ADJ
ejpam-3764	459	9	.	.	PUNCT
ejpam-3764	460	1	let	let	VERB
ejpam-3764	460	2	c	c	PRON
ejpam-3764	460	3	be	be	AUX
ejpam-3764	460	4	any	any	DET
ejpam-3764	460	5	compact	compact	ADJ
ejpam-3764	460	6	subset	subset	NOUN
ejpam-3764	460	7	of	of	ADP
ejpam-3764	460	8	(	(	PUNCT
ejpam-3764	460	9	aω	aω	PROPN
ejpam-3764	460	10	,	,	PUNCT
ejpam-3764	460	11	τω	τω	NOUN
ejpam-3764	460	12	)	)	PUNCT
ejpam-3764	460	13	.	.	PUNCT
ejpam-3764	461	1	then	then	ADV
ejpam-3764	461	2	the	the	DET
ejpam-3764	461	3	restriction	restriction	NOUN
ejpam-3764	461	4	function	function	VERB
ejpam-3764	461	5	i	i	PROPN
ejpam-3764	461	6	d	d	PROPN
ejpam-3764	461	7	�	�	PROPN
ejpam-3764	461	8	c	c	PROPN
ejpam-3764	461	9	:	:	PUNCT
ejpam-3764	461	10	c	c	X
ejpam-3764	461	11	→	→	SYM
ejpam-3764	461	12	f(c	f(c	PROPN
ejpam-3764	461	13	)	)	PUNCT
ejpam-3764	461	14	is	be	AUX
ejpam-3764	461	15	a	a	DET
ejpam-3764	461	16	homeomorphism	homeomorphism	NOUN
ejpam-3764	461	17	.	.	PUNCT
ejpam-3764	462	1	therefore	therefore	ADV
ejpam-3764	462	2	,	,	PUNCT
ejpam-3764	462	3	(	(	PUNCT
ejpam-3764	462	4	aω	aω	INTJ
ejpam-3764	462	5	,	,	PUNCT
ejpam-3764	462	6	τω	τω	INTJ
ejpam-3764	462	7	)	)	PUNCT
ejpam-3764	462	8	is	be	AUX
ejpam-3764	462	9	a	a	DET
ejpam-3764	462	10	c−normal	c−normal	PROPN
ejpam-3764	462	11	.	.	PUNCT
ejpam-3764	463	1	since	since	SCONJ
ejpam-3764	463	2	any	any	DET
ejpam-3764	463	3	normal	normal	ADJ
ejpam-3764	463	4	space	space	NOUN
ejpam-3764	463	5	is	be	AUX
ejpam-3764	463	6	cc	cc	VERB
ejpam-3764	463	7	-	-	ADJ
ejpam-3764	463	8	normal	normal	ADJ
ejpam-3764	463	9	,	,	PUNCT
ejpam-3764	463	10	l	l	NOUN
ejpam-3764	463	11	-	-	ADJ
ejpam-3764	463	12	normal	normal	ADJ
ejpam-3764	463	13	and	and	CCONJ
ejpam-3764	463	14	s	s	NOUN
ejpam-3764	463	15	-	-	ADJ
ejpam-3764	463	16	normal	normal	ADJ
ejpam-3764	463	17	,	,	PUNCT
ejpam-3764	463	18	just	just	ADV
ejpam-3764	463	19	by	by	ADP
ejpam-3764	463	20	taking	take	VERB
ejpam-3764	463	21	x	x	PUNCT
ejpam-3764	463	22	=	=	PUNCT
ejpam-3764	463	23	y	y	PROPN
ejpam-3764	463	24	and	and	CCONJ
ejpam-3764	463	25	f	f	PROPN
ejpam-3764	463	26	to	to	PART
ejpam-3764	463	27	be	be	AUX
ejpam-3764	463	28	the	the	DET
ejpam-3764	463	29	identity	identity	NOUN
ejpam-3764	463	30	function	function	NOUN
ejpam-3764	463	31	.	.	PUNCT
ejpam-3764	464	1	hence	hence	ADV
ejpam-3764	464	2	,	,	PUNCT
ejpam-3764	464	3	we	we	PRON
ejpam-3764	464	4	conclude	conclude	VERB
ejpam-3764	464	5	the	the	DET
ejpam-3764	464	6	following	follow	VERB
ejpam-3764	464	7	theorems	theorem	NOUN
ejpam-3764	464	8	.	.	PUNCT
ejpam-3764	465	1	theorem	theorem	NOUN
ejpam-3764	465	2	5	5	NUM
ejpam-3764	465	3	.	.	PUNCT
ejpam-3764	466	1	if	if	SCONJ
ejpam-3764	466	2	a	a	DET
ejpam-3764	466	3	∈	∈	NOUN
ejpam-3764	466	4	aω	aω	X
ejpam-3764	466	5	\	\	PROPN
ejpam-3764	466	6	{	{	PUNCT
ejpam-3764	466	7	ω	ω	NOUN
ejpam-3764	466	8	}	}	PUNCT
ejpam-3764	466	9	has	have	VERB
ejpam-3764	466	10	no	no	DET
ejpam-3764	466	11	multiplicative	multiplicative	ADJ
ejpam-3764	466	12	inverse	inverse	NOUN
ejpam-3764	466	13	,	,	PUNCT
ejpam-3764	466	14	then	then	ADV
ejpam-3764	466	15	omega	omega	NOUN
ejpam-3764	466	16	topological	topological	ADJ
ejpam-3764	466	17	space	space	NOUN
ejpam-3764	466	18	(	(	PUNCT
ejpam-3764	466	19	aω	aω	PROPN
ejpam-3764	466	20	,	,	PUNCT
ejpam-3764	466	21	τω	τω	INTJ
ejpam-3764	466	22	)	)	PUNCT
ejpam-3764	466	23	is	be	AUX
ejpam-3764	466	24	cc	cc	VERB
ejpam-3764	466	25	-	-	ADJ
ejpam-3764	466	26	normal	normal	ADJ
ejpam-3764	466	27	.	.	PUNCT
ejpam-3764	467	1	proof	proof	NOUN
ejpam-3764	467	2	.	.	PUNCT
ejpam-3764	468	1	using	use	VERB
ejpam-3764	468	2	the	the	DET
ejpam-3764	468	3	same	same	ADJ
ejpam-3764	468	4	proof	proof	NOUN
ejpam-3764	468	5	of	of	ADP
ejpam-3764	468	6	theorem	theorem	ADJ
ejpam-3764	468	7	4	4	NUM
ejpam-3764	468	8	.	.	PUNCT
ejpam-3764	468	9	theorem	theorem	NOUN
ejpam-3764	468	10	6	6	NUM
ejpam-3764	468	11	.	.	PUNCT
ejpam-3764	469	1	if	if	SCONJ
ejpam-3764	469	2	a	a	DET
ejpam-3764	469	3	∈	∈	NOUN
ejpam-3764	469	4	aω	aω	X
ejpam-3764	469	5	\	\	PROPN
ejpam-3764	469	6	{	{	PUNCT
ejpam-3764	469	7	ω	ω	NOUN
ejpam-3764	469	8	}	}	PUNCT
ejpam-3764	469	9	has	have	VERB
ejpam-3764	469	10	no	no	DET
ejpam-3764	469	11	multiplicative	multiplicative	ADJ
ejpam-3764	469	12	inverse	inverse	NOUN
ejpam-3764	469	13	,	,	PUNCT
ejpam-3764	469	14	then	then	ADV
ejpam-3764	469	15	omega	omega	NOUN
ejpam-3764	469	16	topological	topological	ADJ
ejpam-3764	469	17	space	space	NOUN
ejpam-3764	469	18	(	(	PUNCT
ejpam-3764	469	19	aω	aω	PROPN
ejpam-3764	469	20	,	,	PUNCT
ejpam-3764	469	21	τω	τω	INTJ
ejpam-3764	469	22	)	)	PUNCT
ejpam-3764	469	23	is	be	AUX
ejpam-3764	469	24	l	l	NOUN
ejpam-3764	469	25	-	-	ADJ
ejpam-3764	469	26	normal	normal	ADJ
ejpam-3764	469	27	.	.	PUNCT
ejpam-3764	470	1	references	reference	NOUN
ejpam-3764	470	2	527	527	NUM
ejpam-3764	470	3	proof	proof	NOUN
ejpam-3764	470	4	.	.	PUNCT
ejpam-3764	471	1	using	use	VERB
ejpam-3764	471	2	the	the	DET
ejpam-3764	471	3	same	same	ADJ
ejpam-3764	471	4	proof	proof	NOUN
ejpam-3764	471	5	of	of	ADP
ejpam-3764	471	6	theorem	theorem	ADJ
ejpam-3764	471	7	4	4	NUM
ejpam-3764	471	8	.	.	PUNCT
ejpam-3764	471	9	theorem	theorem	VERB
ejpam-3764	471	10	7	7	NUM
ejpam-3764	471	11	.	.	PUNCT
ejpam-3764	472	1	if	if	SCONJ
ejpam-3764	472	2	a	a	DET
ejpam-3764	472	3	∈	∈	NOUN
ejpam-3764	472	4	aω	aω	X
ejpam-3764	472	5	\	\	PROPN
ejpam-3764	472	6	{	{	PUNCT
ejpam-3764	472	7	ω	ω	NOUN
ejpam-3764	472	8	}	}	PUNCT
ejpam-3764	472	9	has	have	VERB
ejpam-3764	472	10	no	no	DET
ejpam-3764	472	11	multiplicative	multiplicative	ADJ
ejpam-3764	472	12	inverse	inverse	NOUN
ejpam-3764	472	13	,	,	PUNCT
ejpam-3764	472	14	then	then	ADV
ejpam-3764	472	15	omega	omega	NOUN
ejpam-3764	472	16	topological	topological	ADJ
ejpam-3764	472	17	space	space	NOUN
ejpam-3764	472	18	(	(	PUNCT
ejpam-3764	472	19	aω	aω	PROPN
ejpam-3764	472	20	,	,	PUNCT
ejpam-3764	472	21	τω	τω	INTJ
ejpam-3764	472	22	)	)	PUNCT
ejpam-3764	472	23	is	be	AUX
ejpam-3764	472	24	s	s	NOUN
ejpam-3764	472	25	-	-	ADJ
ejpam-3764	472	26	normal	normal	ADJ
ejpam-3764	472	27	.	.	PUNCT
ejpam-3764	473	1	proof	proof	NOUN
ejpam-3764	473	2	.	.	PUNCT
ejpam-3764	474	1	using	use	VERB
ejpam-3764	474	2	the	the	DET
ejpam-3764	474	3	same	same	ADJ
ejpam-3764	474	4	proof	proof	NOUN
ejpam-3764	474	5	of	of	ADP
ejpam-3764	474	6	theorem	theorem	ADJ
ejpam-3764	474	7	4	4	NUM
ejpam-3764	474	8	.	.	NOUN
ejpam-3764	474	9	example	example	NOUN
ejpam-3764	474	10	10	10	NUM
ejpam-3764	474	11	.	.	PUNCT
ejpam-3764	475	1	by	by	ADP
ejpam-3764	475	2	example	example	NOUN
ejpam-3764	475	3	4	4	NUM
ejpam-3764	475	4	,	,	PUNCT
ejpam-3764	475	5	(	(	PUNCT
ejpam-3764	475	6	a−∞	a−∞	ADJ
ejpam-3764	475	7	,	,	PUNCT
ejpam-3764	475	8	τ−∞	τ−∞	PUNCT
ejpam-3764	475	9	)	)	PUNCT
ejpam-3764	475	10	is	be	AUX
ejpam-3764	475	11	c	c	NOUN
ejpam-3764	475	12	-	-	ADJ
ejpam-3764	475	13	normal	normal	ADJ
ejpam-3764	475	14	,	,	PUNCT
ejpam-3764	475	15	cc	cc	NOUN
ejpam-3764	475	16	-	-	ADJ
ejpam-3764	475	17	normal	normal	ADJ
ejpam-3764	475	18	,	,	PUNCT
ejpam-3764	475	19	l	l	NOUN
ejpam-3764	475	20	-	-	ADJ
ejpam-3764	475	21	normal	normal	ADJ
ejpam-3764	475	22	and	and	CCONJ
ejpam-3764	475	23	snormal	snormal	ADJ
ejpam-3764	475	24	.	.	PUNCT
ejpam-3764	476	1	theorem	theorem	VERB
ejpam-3764	476	2	8	8	NUM
ejpam-3764	476	3	.	.	PUNCT
ejpam-3764	477	1	if	if	SCONJ
ejpam-3764	477	2	(	(	PUNCT
ejpam-3764	477	3	aω\{ω},⊗	aω\{ω},⊗	NOUN
ejpam-3764	477	4	)	)	PUNCT
ejpam-3764	477	5	be	be	VERB
ejpam-3764	477	6	a	a	DET
ejpam-3764	477	7	group	group	NOUN
ejpam-3764	477	8	has	have	VERB
ejpam-3764	477	9	more	more	ADJ
ejpam-3764	477	10	than	than	ADP
ejpam-3764	477	11	one	one	NUM
ejpam-3764	477	12	element	element	NOUN
ejpam-3764	477	13	,	,	PUNCT
ejpam-3764	477	14	then	then	ADV
ejpam-3764	477	15	omega	omega	NOUN
ejpam-3764	477	16	topological	topological	ADJ
ejpam-3764	477	17	space	space	NOUN
ejpam-3764	477	18	(	(	PUNCT
ejpam-3764	477	19	aω	aω	PROPN
ejpam-3764	477	20	,	,	PUNCT
ejpam-3764	477	21	τω	τω	INTJ
ejpam-3764	477	22	)	)	PUNCT
ejpam-3764	477	23	is	be	AUX
ejpam-3764	477	24	not	not	PART
ejpam-3764	477	25	s	s	NOUN
ejpam-3764	477	26	-	-	NOUN
ejpam-3764	477	27	normal	normal	ADJ
ejpam-3764	477	28	.	.	PUNCT
ejpam-3764	478	1	proof	proof	NOUN
ejpam-3764	478	2	.	.	PUNCT
ejpam-3764	479	1	from	from	ADP
ejpam-3764	479	2	the	the	DET
ejpam-3764	479	3	proposition	proposition	NOUN
ejpam-3764	479	4	any	any	DET
ejpam-3764	479	5	separable	separable	ADJ
ejpam-3764	479	6	s	s	NOUN
ejpam-3764	479	7	-	-	ADJ
ejpam-3764	479	8	normal	normal	ADJ
ejpam-3764	479	9	must	must	AUX
ejpam-3764	479	10	be	be	AUX
ejpam-3764	479	11	normal	normal	ADJ
ejpam-3764	479	12	(	(	PUNCT
ejpam-3764	479	13	see	see	VERB
ejpam-3764	479	14	[	[	X
ejpam-3764	479	15	1	1	NUM
ejpam-3764	479	16	]	]	PUNCT
ejpam-3764	479	17	)	)	PUNCT
ejpam-3764	479	18	and	and	CCONJ
ejpam-3764	479	19	since	since	SCONJ
ejpam-3764	479	20	(	(	PUNCT
ejpam-3764	479	21	aω	aω	INTJ
ejpam-3764	479	22	,	,	PUNCT
ejpam-3764	479	23	τω	τω	INTJ
ejpam-3764	479	24	)	)	PUNCT
ejpam-3764	479	25	is	be	AUX
ejpam-3764	479	26	separable	separable	ADJ
ejpam-3764	479	27	and	and	CCONJ
ejpam-3764	479	28	not	not	PART
ejpam-3764	479	29	normal	normal	ADJ
ejpam-3764	479	30	(	(	PUNCT
ejpam-3764	479	31	see	see	VERB
ejpam-3764	479	32	proposition	proposition	NOUN
ejpam-3764	479	33	5	5	NUM
ejpam-3764	479	34	and	and	CCONJ
ejpam-3764	479	35	proposition	proposition	NOUN
ejpam-3764	479	36	10	10	NUM
ejpam-3764	479	37	,	,	PUNCT
ejpam-3764	479	38	repectively	repectively	ADV
ejpam-3764	479	39	)	)	PUNCT
ejpam-3764	479	40	,	,	PUNCT
ejpam-3764	479	41	then	then	ADV
ejpam-3764	479	42	(	(	PUNCT
ejpam-3764	479	43	aω	aω	INTJ
ejpam-3764	479	44	,	,	PUNCT
ejpam-3764	479	45	τω	τω	INTJ
ejpam-3764	479	46	)	)	PUNCT
ejpam-3764	479	47	is	be	AUX
ejpam-3764	479	48	not	not	PART
ejpam-3764	479	49	s	s	NOUN
ejpam-3764	479	50	-	-	NOUN
ejpam-3764	479	51	normal	normal	ADJ
ejpam-3764	479	52	.	.	PUNCT
ejpam-3764	480	1	example	example	NOUN
ejpam-3764	480	2	11	11	NUM
ejpam-3764	480	3	.	.	PUNCT
ejpam-3764	481	1	by	by	ADP
ejpam-3764	481	2	example	example	NOUN
ejpam-3764	481	3	7	7	NUM
ejpam-3764	481	4	,	,	PUNCT
ejpam-3764	481	5	(	(	PUNCT
ejpam-3764	481	6	r−∞	r−∞	NOUN
ejpam-3764	481	7	,	,	PUNCT
ejpam-3764	481	8	τ−∞	τ−∞	NOUN
ejpam-3764	481	9	)	)	PUNCT
ejpam-3764	481	10	is	be	AUX
ejpam-3764	481	11	not	not	PART
ejpam-3764	481	12	a	a	DET
ejpam-3764	481	13	s	s	NOUN
ejpam-3764	481	14	-	-	ADJ
ejpam-3764	481	15	normal	normal	ADJ
ejpam-3764	481	16	.	.	PUNCT
ejpam-3764	482	1	theorem	theorem	VERB
ejpam-3764	482	2	9	9	NUM
ejpam-3764	482	3	.	.	PUNCT
ejpam-3764	483	1	every	every	DET
ejpam-3764	483	2	omega	omega	NOUN
ejpam-3764	483	3	topological	topological	ADJ
ejpam-3764	483	4	space	space	NOUN
ejpam-3764	483	5	(	(	PUNCT
ejpam-3764	483	6	aω	aω	PROPN
ejpam-3764	483	7	,	,	PUNCT
ejpam-3764	483	8	τω	τω	INTJ
ejpam-3764	483	9	)	)	PUNCT
ejpam-3764	483	10	is	be	AUX
ejpam-3764	483	11	not	not	PART
ejpam-3764	483	12	c2	c2	NOUN
ejpam-3764	483	13	-	-	PUNCT
ejpam-3764	483	14	paracompact	paracompact	NOUN
ejpam-3764	483	15	.	.	PUNCT
ejpam-3764	484	1	proof	proof	NOUN
ejpam-3764	484	2	.	.	PUNCT
ejpam-3764	485	1	since	since	SCONJ
ejpam-3764	485	2	any	any	DET
ejpam-3764	485	3	c2	c2	PROPN
ejpam-3764	485	4	-	-	PUNCT
ejpam-3764	485	5	paracompact	paracompact	NOUN
ejpam-3764	485	6	fréchet	fréchet	NOUN
ejpam-3764	485	7	space	space	NOUN
ejpam-3764	485	8	is	be	AUX
ejpam-3764	485	9	housdorff	housdorff	NOUN
ejpam-3764	485	10	,	,	PUNCT
ejpam-3764	485	11	see	see	VERB
ejpam-3764	485	12	[	[	X
ejpam-3764	485	13	13	13	NUM
ejpam-3764	485	14	]	]	PUNCT
ejpam-3764	485	15	,	,	PUNCT
ejpam-3764	485	16	and	and	CCONJ
ejpam-3764	485	17	(	(	PUNCT
ejpam-3764	485	18	aω	aω	INTJ
ejpam-3764	485	19	,	,	PUNCT
ejpam-3764	485	20	τω	τω	INTJ
ejpam-3764	485	21	)	)	PUNCT
ejpam-3764	485	22	is	be	AUX
ejpam-3764	485	23	first	first	ADV
ejpam-3764	485	24	countable	countable	ADJ
ejpam-3764	485	25	not	not	PART
ejpam-3764	485	26	housdorff	housdorff	NOUN
ejpam-3764	485	27	space	space	NOUN
ejpam-3764	485	28	,	,	PUNCT
ejpam-3764	485	29	then	then	ADV
ejpam-3764	485	30	(	(	PUNCT
ejpam-3764	485	31	aω	aω	INTJ
ejpam-3764	485	32	,	,	PUNCT
ejpam-3764	485	33	τω	τω	INTJ
ejpam-3764	485	34	)	)	PUNCT
ejpam-3764	485	35	can	can	AUX
ejpam-3764	485	36	not	not	PART
ejpam-3764	485	37	be	be	AUX
ejpam-3764	485	38	c2	c2	NOUN
ejpam-3764	485	39	-	-	PUNCT
ejpam-3764	485	40	paracompact	paracompact	NOUN
ejpam-3764	485	41	.	.	PUNCT
ejpam-3764	486	1	theorem	theorem	NOUN
ejpam-3764	486	2	10	10	NUM
ejpam-3764	486	3	.	.	PUNCT
ejpam-3764	487	1	let	let	VERB
ejpam-3764	487	2	a	a	DET
ejpam-3764	487	3	∈	∈	NOUN
ejpam-3764	487	4	aω	aω	X
ejpam-3764	487	5	\	\	PROPN
ejpam-3764	487	6	{	{	PUNCT
ejpam-3764	487	7	ω	ω	NOUN
ejpam-3764	487	8	}	}	PUNCT
ejpam-3764	487	9	has	have	VERB
ejpam-3764	487	10	no	no	DET
ejpam-3764	487	11	multiplicative	multiplicative	ADJ
ejpam-3764	487	12	inverse	inverse	NOUN
ejpam-3764	487	13	.	.	PUNCT
ejpam-3764	488	1	then	then	ADV
ejpam-3764	488	2	omega	omega	NOUN
ejpam-3764	488	3	topological	topological	ADJ
ejpam-3764	488	4	space	space	NOUN
ejpam-3764	488	5	(	(	PUNCT
ejpam-3764	488	6	aω	aω	PROPN
ejpam-3764	488	7	,	,	PUNCT
ejpam-3764	488	8	τω	τω	INTJ
ejpam-3764	488	9	)	)	PUNCT
ejpam-3764	488	10	is	be	AUX
ejpam-3764	488	11	not	not	PART
ejpam-3764	488	12	c	c	NOUN
ejpam-3764	488	13	-	-	PUNCT
ejpam-3764	488	14	paracompact	paracompact	ADJ
ejpam-3764	488	15	.	.	PUNCT
ejpam-3764	489	1	proof	proof	NOUN
ejpam-3764	489	2	.	.	PUNCT
ejpam-3764	490	1	assume	assume	VERB
ejpam-3764	490	2	that	that	SCONJ
ejpam-3764	490	3	(	(	PUNCT
ejpam-3764	490	4	aω	aω	INTJ
ejpam-3764	490	5	,	,	PUNCT
ejpam-3764	490	6	τω	τω	INTJ
ejpam-3764	490	7	)	)	PUNCT
ejpam-3764	490	8	is	be	AUX
ejpam-3764	490	9	c	c	NOUN
ejpam-3764	490	10	-	-	PUNCT
ejpam-3764	490	11	paracompact	paracompact	ADJ
ejpam-3764	490	12	.	.	PUNCT
ejpam-3764	491	1	let	let	VERB
ejpam-3764	491	2	y	y	PRON
ejpam-3764	491	3	be	be	AUX
ejpam-3764	491	4	a	a	DET
ejpam-3764	491	5	paracompact	paracompact	ADJ
ejpam-3764	491	6	space	space	NOUN
ejpam-3764	491	7	and	and	CCONJ
ejpam-3764	491	8	f	f	NOUN
ejpam-3764	491	9	:	:	PUNCT
ejpam-3764	491	10	aω	aω	PROPN
ejpam-3764	491	11	→	→	SYM
ejpam-3764	491	12	y	y	PROPN
ejpam-3764	491	13	be	be	AUX
ejpam-3764	491	14	bijective	bijective	ADJ
ejpam-3764	491	15	such	such	ADJ
ejpam-3764	491	16	that	that	SCONJ
ejpam-3764	491	17	the	the	DET
ejpam-3764	491	18	restriction	restriction	NOUN
ejpam-3764	491	19	f	f	PROPN
ejpam-3764	491	20	�	�	PROPN
ejpam-3764	491	21	c	c	PROPN
ejpam-3764	491	22	:	:	PUNCT
ejpam-3764	491	23	c	c	X
ejpam-3764	491	24	→	→	SYM
ejpam-3764	491	25	f(c	f(c	PROPN
ejpam-3764	491	26	)	)	PUNCT
ejpam-3764	491	27	is	be	AUX
ejpam-3764	491	28	a	a	DET
ejpam-3764	491	29	homeomorphism	homeomorphism	NOUN
ejpam-3764	491	30	for	for	ADP
ejpam-3764	491	31	all	all	DET
ejpam-3764	491	32	compact	compact	ADJ
ejpam-3764	491	33	subspace	subspace	NOUN
ejpam-3764	491	34	c	c	PROPN
ejpam-3764	491	35	of	of	ADP
ejpam-3764	491	36	(	(	PUNCT
ejpam-3764	491	37	aω	aω	PROPN
ejpam-3764	491	38	,	,	PUNCT
ejpam-3764	491	39	τω	τω	NOUN
ejpam-3764	491	40	)	)	PUNCT
ejpam-3764	491	41	.	.	PUNCT
ejpam-3764	492	1	hence	hence	ADV
ejpam-3764	492	2	,	,	PUNCT
ejpam-3764	492	3	aω	aω	PROPN
ejpam-3764	492	4	≡	≡	PROPN
ejpam-3764	492	5	y	y	PROPN
ejpam-3764	492	6	,	,	PUNCT
ejpam-3764	492	7	since	since	SCONJ
ejpam-3764	492	8	aω	aω	PROPN
ejpam-3764	492	9	is	be	AUX
ejpam-3764	492	10	compact	compact	ADJ
ejpam-3764	492	11	(	(	PUNCT
ejpam-3764	492	12	see	see	VERB
ejpam-3764	492	13	proposition	proposition	NOUN
ejpam-3764	492	14	12	12	NUM
ejpam-3764	492	15	)	)	PUNCT
ejpam-3764	492	16	.	.	PUNCT
ejpam-3764	493	1	however	however	ADV
ejpam-3764	493	2	,	,	PUNCT
ejpam-3764	493	3	aω	aω	PROPN
ejpam-3764	493	4	is	be	AUX
ejpam-3764	493	5	paracompact	paracompact	ADJ
ejpam-3764	493	6	,	,	PUNCT
ejpam-3764	493	7	thus	thus	ADV
ejpam-3764	493	8	a	a	DET
ejpam-3764	493	9	contradiction	contradiction	NOUN
ejpam-3764	493	10	.	.	PUNCT
ejpam-3764	494	1	because	because	SCONJ
ejpam-3764	494	2	any	any	DET
ejpam-3764	494	3	paracompact	paracompact	ADJ
ejpam-3764	494	4	space	space	NOUN
ejpam-3764	494	5	is	be	AUX
ejpam-3764	494	6	hausdorff	hausdorff	NOUN
ejpam-3764	494	7	space	space	NOUN
ejpam-3764	494	8	and	and	CCONJ
ejpam-3764	494	9	aω	aω	PROPN
ejpam-3764	494	10	is	be	AUX
ejpam-3764	494	11	not	not	PART
ejpam-3764	494	12	a	a	DET
ejpam-3764	494	13	hausdorff	hausdorff	NOUN
ejpam-3764	494	14	space	space	NOUN
ejpam-3764	494	15	.	.	PUNCT
ejpam-3764	495	1	therefore	therefore	ADV
ejpam-3764	495	2	,	,	PUNCT
ejpam-3764	495	3	(	(	PUNCT
ejpam-3764	495	4	aω	aω	INTJ
ejpam-3764	495	5	,	,	PUNCT
ejpam-3764	495	6	τω	τω	INTJ
ejpam-3764	495	7	)	)	PUNCT
ejpam-3764	495	8	is	be	AUX
ejpam-3764	495	9	not	not	PART
ejpam-3764	495	10	a	a	DET
ejpam-3764	495	11	cparacompact	cparacompact	NOUN
ejpam-3764	495	12	.	.	PUNCT
ejpam-3764	496	1	references	reference	NOUN
ejpam-3764	496	2	[	[	X
ejpam-3764	496	3	1	1	NUM
ejpam-3764	496	4	]	]	PUNCT
ejpam-3764	496	5	m.	m.	NOUN
ejpam-3764	496	6	alhomieyed	alhomieye	VERB
ejpam-3764	496	7	and	and	CCONJ
ejpam-3764	496	8	l.	l.	PROPN
ejpam-3764	496	9	kalantan	kalantan	PROPN
ejpam-3764	496	10	.	.	PUNCT
ejpam-3764	497	1	s	s	X
ejpam-3764	497	2	-	-	NOUN
ejpam-3764	497	3	normality	normality	NOUN
ejpam-3764	497	4	.	.	PUNCT
ejpam-3764	498	1	journal	journal	NOUN
ejpam-3764	498	2	of	of	ADP
ejpam-3764	498	3	mathematical	mathematical	ADJ
ejpam-3764	498	4	analysis	analysis	NOUN
ejpam-3764	498	5	,	,	PUNCT
ejpam-3764	498	6	9(3):48–54	9(3):48–54	NUM
ejpam-3764	498	7	,	,	PUNCT
ejpam-3764	498	8	2018	2018	NUM
ejpam-3764	498	9	.	.	PUNCT
ejpam-3764	499	1	[	[	X
ejpam-3764	499	2	2	2	X
ejpam-3764	499	3	]	]	PUNCT
ejpam-3764	499	4	s.	s.	PROPN
ejpam-3764	499	5	alzahrani	alzahrani	PROPN
ejpam-3764	499	6	and	and	CCONJ
ejpam-3764	499	7	l.	l.	PROPN
ejpam-3764	499	8	kalantan	kalantan	PROPN
ejpam-3764	499	9	.	.	PUNCT
ejpam-3764	500	1	c	c	X
ejpam-3764	500	2	-	-	PUNCT
ejpam-3764	500	3	normal	normal	ADJ
ejpam-3764	500	4	topological	topological	ADJ
ejpam-3764	500	5	property	property	NOUN
ejpam-3764	500	6	.	.	PUNCT
ejpam-3764	501	1	filomat	filomat	NOUN
ejpam-3764	501	2	,	,	PUNCT
ejpam-3764	501	3	31:2:407–411	31:2:407–411	NUM
ejpam-3764	501	4	,	,	PUNCT
ejpam-3764	501	5	2017	2017	NUM
ejpam-3764	501	6	.	.	PUNCT
ejpam-3764	502	1	[	[	X
ejpam-3764	502	2	3	3	X
ejpam-3764	502	3	]	]	X
ejpam-3764	502	4	l.	l.	PROPN
ejpam-3764	502	5	kalantan	kalantan	PROPN
ejpam-3764	502	6	.	.	PUNCT
ejpam-3764	503	1	π	π	X
ejpam-3764	503	2	-	-	ADJ
ejpam-3764	503	3	normal	normal	ADJ
ejpam-3764	503	4	topological	topological	ADJ
ejpam-3764	503	5	spaces	space	NOUN
ejpam-3764	503	6	.	.	PUNCT
ejpam-3764	504	1	filomat	filomat	NOUN
ejpam-3764	504	2	,	,	PUNCT
ejpam-3764	504	3	22	22	NUM
ejpam-3764	504	4	-	-	SYM
ejpam-3764	504	5	1:173–181	1:173–181	NUM
ejpam-3764	504	6	,	,	PUNCT
ejpam-3764	504	7	2008	2008	NUM
ejpam-3764	504	8	.	.	PUNCT
ejpam-3764	505	1	[	[	X
ejpam-3764	505	2	4	4	X
ejpam-3764	505	3	]	]	PUNCT
ejpam-3764	505	4	l.	l.	PROPN
ejpam-3764	505	5	kalantan	kalantan	PROPN
ejpam-3764	505	6	and	and	CCONJ
ejpam-3764	505	7	m.	m.	NOUN
ejpam-3764	505	8	alhomieyed	alhomieye	VERB
ejpam-3764	505	9	.	.	PUNCT
ejpam-3764	506	1	cc	cc	NOUN
ejpam-3764	506	2	-	-	ADJ
ejpam-3764	506	3	normal	normal	ADJ
ejpam-3764	506	4	topological	topological	ADJ
ejpam-3764	506	5	spaces	space	NOUN
ejpam-3764	506	6	.	.	PUNCT
ejpam-3764	507	1	turk	turk	PROPN
ejpam-3764	507	2	.	.	PUNCT
ejpam-3764	508	1	j.	j.	PROPN
ejpam-3764	508	2	math	math	PROPN
ejpam-3764	508	3	.	.	PUNCT
ejpam-3764	508	4	,	,	PUNCT
ejpam-3764	509	1	41:749–755	41:749–755	NUM
ejpam-3764	509	2	,	,	PUNCT
ejpam-3764	509	3	2017	2017	NUM
ejpam-3764	509	4	.	.	PUNCT
ejpam-3764	510	1	references	reference	NOUN
ejpam-3764	510	2	528	528	NUM
ejpam-3764	510	3	[	[	X
ejpam-3764	510	4	5	5	NUM
ejpam-3764	510	5	]	]	PUNCT
ejpam-3764	510	6	l.	l.	PROPN
ejpam-3764	510	7	kalantan	kalantan	PROPN
ejpam-3764	510	8	and	and	CCONJ
ejpam-3764	510	9	s.	s.	PROPN
ejpam-3764	510	10	alzahrani	alzahrani	PROPN
ejpam-3764	510	11	.	.	PUNCT
ejpam-3764	511	1	epinormality	epinormality	PROPN
ejpam-3764	511	2	.	.	PUNCT
ejpam-3764	512	1	j.	j.	PROPN
ejpam-3764	512	2	nonlinear	nonlinear	PROPN
ejpam-3764	512	3	sci	sci	PROPN
ejpam-3764	512	4	.	.	PUNCT
ejpam-3764	512	5	appl	appl	PROPN
ejpam-3764	512	6	.	.	PROPN
ejpam-3764	512	7	,	,	PUNCT
ejpam-3764	512	8	9:5398–5402	9:5398–5402	PROPN
ejpam-3764	512	9	,	,	PUNCT
ejpam-3764	512	10	2016	2016	NUM
ejpam-3764	512	11	.	.	PUNCT
ejpam-3764	513	1	[	[	X
ejpam-3764	513	2	6	6	NUM
ejpam-3764	513	3	]	]	PUNCT
ejpam-3764	513	4	l.	l.	PROPN
ejpam-3764	513	5	kalantan	kalantan	PROPN
ejpam-3764	513	6	and	and	CCONJ
ejpam-3764	513	7	m.	m.	PROPN
ejpam-3764	513	8	saeed	saeed	PROPN
ejpam-3764	513	9	.	.	PUNCT
ejpam-3764	514	1	l	l	NOUN
ejpam-3764	514	2	-	-	NOUN
ejpam-3764	514	3	normality	normality	NOUN
ejpam-3764	514	4	.	.	PUNCT
ejpam-3764	515	1	topology	topology	NOUN
ejpam-3764	515	2	proceedings	proceeding	NOUN
ejpam-3764	515	3	,	,	PUNCT
ejpam-3764	515	4	50:141–149	50:141–149	NUM
ejpam-3764	515	5	,	,	PUNCT
ejpam-3764	515	6	2017	2017	NUM
ejpam-3764	515	7	.	.	PUNCT
ejpam-3764	516	1	[	[	X
ejpam-3764	516	2	7	7	X
ejpam-3764	516	3	]	]	X
ejpam-3764	516	4	c.	c.	PROPN
ejpam-3764	516	5	kuratowski	kuratowski	PROPN
ejpam-3764	516	6	.	.	PUNCT
ejpam-3764	517	1	topology	topology	PROPN
ejpam-3764	517	2	i.	i.	PROPN
ejpam-3764	517	3	hafner	hafner	PROPN
ejpam-3764	517	4	,	,	PUNCT
ejpam-3764	517	5	new	new	ADJ
ejpam-3764	517	6	yor	yor	NOUN
ejpam-3764	517	7	,	,	PUNCT
ejpam-3764	517	8	1958	1958	NUM
ejpam-3764	517	9	.	.	PUNCT
ejpam-3764	518	1	[	[	X
ejpam-3764	518	2	8	8	NUM
ejpam-3764	518	3	]	]	X
ejpam-3764	518	4	g.	g.	PROPN
ejpam-3764	518	5	l.	l.	PROPN
ejpam-3764	518	6	litvinov	litvinov	PROPN
ejpam-3764	518	7	.	.	PUNCT
ejpam-3764	519	1	the	the	DET
ejpam-3764	519	2	maslov	maslov	ADJ
ejpam-3764	519	3	dequantization	dequantization	NOUN
ejpam-3764	519	4	idempotent	idempotent	NOUN
ejpam-3764	519	5	and	and	CCONJ
ejpam-3764	519	6	tropical	tropical	ADJ
ejpam-3764	519	7	mathematics	mathematic	NOUN
ejpam-3764	519	8	.	.	PUNCT
ejpam-3764	520	1	journal	journal	PROPN
ejpam-3764	520	2	of	of	ADP
ejpam-3764	520	3	mathematical	mathematical	ADJ
ejpam-3764	520	4	sciences	sciences	PROPN
ejpam-3764	520	5	,	,	PUNCT
ejpam-3764	520	6	3:426–444	3:426–444	NUM
ejpam-3764	520	7	,	,	PUNCT
ejpam-3764	520	8	2007	2007	NUM
ejpam-3764	520	9	.	.	PUNCT
ejpam-3764	521	1	[	[	X
ejpam-3764	521	2	9	9	NUM
ejpam-3764	521	3	]	]	X
ejpam-3764	521	4	l.kalantan	l.kalantan	ADJ
ejpam-3764	521	5	and	and	CCONJ
ejpam-3764	521	6	i.	i.	PROPN
ejpam-3764	521	7	alshammari	alshammari	PROPN
ejpam-3764	521	8	.	.	PUNCT
ejpam-3764	522	1	epi	epi	ADJ
ejpam-3764	522	2	-	-	ADJ
ejpam-3764	522	3	mild	mild	ADJ
ejpam-3764	522	4	normality	normality	NOUN
ejpam-3764	522	5	.	.	PUNCT
ejpam-3764	523	1	open	open	ADJ
ejpam-3764	523	2	mat	mat	PROPN
ejpam-3764	523	3	.	.	PUNCT
ejpam-3764	524	1	j.	j.	PROPN
ejpam-3764	524	2	,	,	PUNCT
ejpam-3764	524	3	16:1170–1175	16:1170–1175	PROPN
ejpam-3764	524	4	,	,	PUNCT
ejpam-3764	524	5	2018	2018	NUM
ejpam-3764	524	6	.	.	PUNCT
ejpam-3764	525	1	[	[	X
ejpam-3764	525	2	10	10	NUM
ejpam-3764	525	3	]	]	X
ejpam-3764	525	4	d.	d.	PROPN
ejpam-3764	525	5	maclagan	maclagan	VERB
ejpam-3764	525	6	and	and	CCONJ
ejpam-3764	525	7	b.	b.	PROPN
ejpam-3764	525	8	sturmfels	sturmfel	NOUN
ejpam-3764	525	9	.	.	PUNCT
ejpam-3764	526	1	introduction	introduction	NOUN
ejpam-3764	526	2	to	to	ADP
ejpam-3764	526	3	tropical	tropical	ADJ
ejpam-3764	526	4	geometry	geometry	NOUN
ejpam-3764	526	5	,	,	PUNCT
ejpam-3764	526	6	graduate	graduate	NOUN
ejpam-3764	526	7	studies	study	NOUN
ejpam-3764	526	8	in	in	ADP
ejpam-3764	526	9	mathematics	mathematic	NOUN
ejpam-3764	526	10	.	.	PUNCT
ejpam-3764	526	11	,	,	PUNCT
ejpam-3764	526	12	volume	volume	NOUN
ejpam-3764	526	13	161	161	NUM
ejpam-3764	526	14	.	.	PUNCT
ejpam-3764	527	1	american	american	PROPN
ejpam-3764	527	2	mathematical	mathematical	PROPN
ejpam-3764	527	3	society	society	NOUN
ejpam-3764	527	4	,	,	PUNCT
ejpam-3764	527	5	2015	2015	NUM
ejpam-3764	527	6	.	.	PUNCT
ejpam-3764	528	1	[	[	X
ejpam-3764	528	2	11	11	NUM
ejpam-3764	528	3	]	]	PUNCT
ejpam-3764	528	4	s.	s.	PROPN
ejpam-3764	528	5	khalid	khalid	PROPN
ejpam-3764	528	6	nauman	nauman	PROPN
ejpam-3764	528	7	,	,	PUNCT
ejpam-3764	528	8	c.	c.	PROPN
ejpam-3764	528	9	ozel	ozel	PROPN
ejpam-3764	528	10	,	,	PUNCT
ejpam-3764	528	11	and	and	CCONJ
ejpam-3764	528	12	h.	h.	PROPN
ejpam-3764	528	13	zekraoui	zekraoui	PROPN
ejpam-3764	528	14	.	.	PUNCT
ejpam-3764	529	1	abstract	abstract	ADJ
ejpam-3764	529	2	omega	omega	NOUN
ejpam-3764	529	3	algebra	algebra	NOUN
ejpam-3764	529	4	that	that	PRON
ejpam-3764	529	5	subsumes	subsume	VERB
ejpam-3764	529	6	min	min	NOUN
ejpam-3764	529	7	and	and	CCONJ
ejpam-3764	529	8	max	max	PROPN
ejpam-3764	529	9	plus	plus	CCONJ
ejpam-3764	529	10	algebras	algebra	NOUN
ejpam-3764	529	11	.	.	PUNCT
ejpam-3764	530	1	turkish	turkish	ADJ
ejpam-3764	530	2	journal	journal	NOUN
ejpam-3764	530	3	of	of	ADP
ejpam-3764	530	4	mathematics	mathematic	NOUN
ejpam-3764	530	5	and	and	CCONJ
ejpam-3764	530	6	computer	computer	NOUN
ejpam-3764	530	7	,	,	PUNCT
ejpam-3764	530	8	11(special	11(special	ADJ
ejpam-3764	530	9	issue):1–10	issue):1–10	NOUN
ejpam-3764	530	10	,	,	PUNCT
ejpam-3764	530	11	2019	2019	NUM
ejpam-3764	530	12	.	.	PUNCT
ejpam-3764	531	1	[	[	X
ejpam-3764	531	2	12	12	NUM
ejpam-3764	531	3	]	]	PUNCT
ejpam-3764	531	4	j.-e	j.-e	NOUN
ejpam-3764	531	5	.	.	PUNCT
ejpam-3764	532	1	pin	pin	NOUN
ejpam-3764	532	2	.	.	PUNCT
ejpam-3764	533	1	tropical	tropical	ADJ
ejpam-3764	533	2	semirings	semiring	NOUN
ejpam-3764	533	3	,	,	PUNCT
ejpam-3764	533	4	idempotency	idempotency	NOUN
ejpam-3764	533	5	.	.	PUNCT
ejpam-3764	533	6	,	,	PUNCT
ejpam-3764	533	7	volume	volume	NOUN
ejpam-3764	533	8	11	11	NUM
ejpam-3764	533	9	.	.	PUNCT
ejpam-3764	534	1	cambridge	cambridge	PROPN
ejpam-3764	534	2	univ	univ	PROPN
ejpam-3764	534	3	,	,	PUNCT
ejpam-3764	534	4	cambridge	cambridge	PROPN
ejpam-3764	534	5	,	,	PUNCT
ejpam-3764	534	6	1998	1998	NUM
ejpam-3764	534	7	.	.	PUNCT
ejpam-3764	535	1	[	[	X
ejpam-3764	535	2	13	13	NUM
ejpam-3764	535	3	]	]	PUNCT
ejpam-3764	535	4	maha	maha	PROPN
ejpam-3764	535	5	mohammed	mohammed	PROPN
ejpam-3764	535	6	saeed	saeed	PROPN
ejpam-3764	535	7	,	,	PUNCT
ejpam-3764	535	8	lutfi	lutfi	PROPN
ejpam-3764	535	9	kalantan	kalantan	PROPN
ejpam-3764	535	10	,	,	PUNCT
ejpam-3764	535	11	and	and	CCONJ
ejpam-3764	535	12	hala	hala	PROPN
ejpam-3764	535	13	alzumi	alzumi	PROPN
ejpam-3764	535	14	.	.	PUNCT
ejpam-3764	536	1	c	c	X
ejpam-3764	536	2	-	-	PUNCT
ejpam-3764	536	3	paracompactness	paracompactness	NOUN
ejpam-3764	536	4	and	and	CCONJ
ejpam-3764	536	5	c2	c2	PROPN
ejpam-3764	536	6	-paracompactness	-paracompactness	PROPN
ejpam-3764	536	7	.	.	PUNCT
ejpam-3764	537	1	turk	turk	PROPN
ejpam-3764	537	2	.	.	PUNCT
ejpam-3764	538	1	j.	j.	PROPN
ejpam-3764	538	2	math	math	PROPN
ejpam-3764	538	3	.	.	PUNCT
ejpam-3764	538	4	,	,	PUNCT
ejpam-3764	538	5	43:9–20	43:9–20	NUM
ejpam-3764	538	6	,	,	PUNCT
ejpam-3764	538	7	2019	2019	NUM
ejpam-3764	538	8	.	.	PUNCT
ejpam-3764	539	1	[	[	X
ejpam-3764	539	2	14	14	NUM
ejpam-3764	539	3	]	]	X
ejpam-3764	539	4	i.	i.	PROPN
ejpam-3764	539	5	simon	simon	PROPN
ejpam-3764	539	6	.	.	PUNCT
ejpam-3764	540	1	recognizable	recognizable	ADJ
ejpam-3764	540	2	sets	set	NOUN
ejpam-3764	540	3	with	with	ADP
ejpam-3764	540	4	multiplicities	multiplicity	NOUN
ejpam-3764	540	5	in	in	ADP
ejpam-3764	540	6	the	the	DET
ejpam-3764	540	7	tropical	tropical	ADJ
ejpam-3764	540	8	semiring	semiring	NOUN
ejpam-3764	540	9	.	.	PUNCT
ejpam-3764	540	10	,	,	PUNCT
ejpam-3764	540	11	volume	volume	NOUN
ejpam-3764	540	12	324	324	NUM
ejpam-3764	540	13	.	.	PUNCT
ejpam-3764	541	1	springer	springer	NOUN
ejpam-3764	541	2	,	,	PUNCT
ejpam-3764	541	3	berlin	berlin	PROPN
ejpam-3764	541	4	,	,	PUNCT
ejpam-3764	541	5	1988	1988	NUM
ejpam-3764	541	6	.	.	PUNCT
ejpam-3764	542	1	[	[	X
ejpam-3764	542	2	15	15	NUM
ejpam-3764	542	3	]	]	X
ejpam-3764	542	4	m.k	m.k	PROPN
ejpam-3764	542	5	.	.	PROPN
ejpam-3764	542	6	singal	singal	PROPN
ejpam-3764	542	7	and	and	CCONJ
ejpam-3764	542	8	a.r	a.r	PROPN
ejpam-3764	542	9	.	.	PROPN
ejpam-3764	542	10	singal	singal	PROPN
ejpam-3764	542	11	.	.	PUNCT
ejpam-3764	543	1	mildly	mildly	ADV
ejpam-3764	543	2	normal	normal	ADJ
ejpam-3764	543	3	spaces	space	NOUN
ejpam-3764	543	4	.	.	PUNCT
ejpam-3764	544	1	kyungpook	kyungpook	PROPN
ejpam-3764	544	2	math	math	PROPN
ejpam-3764	544	3	j.	j.	PROPN
ejpam-3764	544	4	,	,	PUNCT
ejpam-3764	544	5	13:27–31	13:27–31	PROPN
ejpam-3764	544	6	,	,	PUNCT
ejpam-3764	544	7	1973	1973	NUM
ejpam-3764	544	8	.	.	PUNCT
ejpam-3764	545	1	[	[	X
ejpam-3764	545	2	16	16	NUM
ejpam-3764	545	3	]	]	X
ejpam-3764	545	4	s.	s.	PROPN
ejpam-3764	545	5	świerczkowski	świerczkowski	PROPN
ejpam-3764	545	6	.	.	PUNCT
ejpam-3764	546	1	on	on	ADP
ejpam-3764	546	2	cyclically	cyclically	ADV
ejpam-3764	546	3	ordered	order	VERB
ejpam-3764	546	4	groups	group	NOUN
ejpam-3764	546	5	.	.	PUNCT
ejpam-3764	547	1	fundamenta	fundamenta	PROPN
ejpam-3764	547	2	mathematicae	mathematicae	PROPN
ejpam-3764	547	3	,	,	PUNCT
ejpam-3764	547	4	47:161	47:161	NUM
ejpam-3764	547	5	–	–	PUNCT
ejpam-3764	547	6	166	166	NUM
ejpam-3764	547	7	,	,	PUNCT
ejpam-3764	547	8	1959	1959	NUM
ejpam-3764	547	9	.	.	PUNCT
ejpam-3764	548	1	[	[	X
ejpam-3764	548	2	17	17	NUM
ejpam-3764	548	3	]	]	X
ejpam-3764	548	4	v.	v.	NOUN
ejpam-3764	548	5	zaitsev	zaitsev	NOUN
ejpam-3764	548	6	.	.	PUNCT
ejpam-3764	549	1	on	on	ADP
ejpam-3764	549	2	certain	certain	ADJ
ejpam-3764	549	3	classes	class	NOUN
ejpam-3764	549	4	of	of	ADP
ejpam-3764	549	5	topological	topological	ADJ
ejpam-3764	549	6	spaces	space	NOUN
ejpam-3764	549	7	and	and	CCONJ
ejpam-3764	549	8	their	their	PRON
ejpam-3764	549	9	bicompactifications	bicompactification	NOUN
ejpam-3764	549	10	.	.	PUNCT
ejpam-3764	550	1	dokl	dokl	NOUN
ejpam-3764	550	2	.	.	PUNCT
ejpam-3764	551	1	akad	akad	PROPN
ejpam-3764	551	2	.	.	PUNCT
ejpam-3764	552	1	nauk	nauk	PROPN
ejpam-3764	552	2	sssr	sssr	PROPN
ejpam-3764	552	3	,	,	PUNCT
ejpam-3764	552	4	178:778–779	178:778–779	NUM
ejpam-3764	552	5	,	,	PUNCT
ejpam-3764	552	6	1968	1968	NUM
ejpam-3764	552	7	.	.	PUNCT