id	sid	tid	token	lemma	pos
ejpam-5618	1	1	european	european	PROPN
ejpam-5618	1	2	journal	journal	PROPN
ejpam-5618	1	3	of	of	ADP
ejpam-5618	1	4	pure	pure	ADJ
ejpam-5618	1	5	and	and	CCONJ
ejpam-5618	1	6	applied	applied	ADJ
ejpam-5618	1	7	mathematics	mathematic	NOUN
ejpam-5618	1	8	2025	2025	NUM
ejpam-5618	1	9	,	,	PUNCT
ejpam-5618	1	10	vol	vol	NOUN
ejpam-5618	1	11	.	.	PROPN
ejpam-5618	1	12	18	18	NUM
ejpam-5618	1	13	,	,	PUNCT
ejpam-5618	1	14	issue	issue	NOUN
ejpam-5618	1	15	1	1	NUM
ejpam-5618	1	16	,	,	PUNCT
ejpam-5618	1	17	article	article	NOUN
ejpam-5618	1	18	number	number	NOUN
ejpam-5618	1	19	5618	5618	NUM
ejpam-5618	1	20	issn	issn	VERB
ejpam-5618	1	21	1307	1307	NUM
ejpam-5618	1	22	-	-	SYM
ejpam-5618	1	23	5543	5543	NUM
ejpam-5618	1	24	–	–	PUNCT
ejpam-5618	1	25	ejpam.com	ejpam.com	X
ejpam-5618	1	26	published	publish	VERB
ejpam-5618	1	27	by	by	ADP
ejpam-5618	1	28	new	new	PROPN
ejpam-5618	1	29	york	york	PROPN
ejpam-5618	1	30	business	business	PROPN
ejpam-5618	1	31	global	global	PROPN
ejpam-5618	1	32	more	more	ADV
ejpam-5618	1	33	on	on	ADP
ejpam-5618	1	34	ideal	ideal	ADJ
ejpam-5618	1	35	topological	topological	ADJ
ejpam-5618	1	36	groups	group	NOUN
ejpam-5618	1	37	saud	saud	PROPN
ejpam-5618	1	38	m.	m.	PROPN
ejpam-5618	1	39	alammar1,∗	alammar1,∗	PROPN
ejpam-5618	1	40	,	,	PUNCT
ejpam-5618	1	41	m.	m.	NOUN
ejpam-5618	1	42	a.	a.	PROPN
ejpam-5618	1	43	al	al	PROPN
ejpam-5618	1	44	shumrani2	shumrani2	PROPN
ejpam-5618	1	45	,	,	PUNCT
ejpam-5618	1	46	cenap	cenap	ADJ
ejpam-5618	1	47	özel2	özel2	ADJ
ejpam-5618	1	48	1	1	NUM
ejpam-5618	1	49	department	department	NOUN
ejpam-5618	1	50	of	of	ADP
ejpam-5618	1	51	mathematics	mathematic	NOUN
ejpam-5618	1	52	,	,	PUNCT
ejpam-5618	1	53	college	college	NOUN
ejpam-5618	1	54	of	of	ADP
ejpam-5618	1	55	science	science	NOUN
ejpam-5618	1	56	,	,	PUNCT
ejpam-5618	1	57	university	university	NOUN
ejpam-5618	1	58	of	of	ADP
ejpam-5618	1	59	ha’il	ha’il	PROPN
ejpam-5618	1	60	,	,	PUNCT
ejpam-5618	1	61	ha’il	ha’il	PROPN
ejpam-5618	1	62	2440	2440	NUM
ejpam-5618	1	63	,	,	PUNCT
ejpam-5618	1	64	saudi	saudi	PROPN
ejpam-5618	1	65	arabia	arabia	PROPN
ejpam-5618	1	66	2	2	NUM
ejpam-5618	1	67	department	department	NOUN
ejpam-5618	1	68	of	of	ADP
ejpam-5618	1	69	mathematics	mathematic	NOUN
ejpam-5618	1	70	,	,	PUNCT
ejpam-5618	1	71	king	king	PROPN
ejpam-5618	1	72	abdulaziz	abdulaziz	PROPN
ejpam-5618	1	73	university	university	PROPN
ejpam-5618	1	74	,	,	PUNCT
ejpam-5618	1	75	p.o.box	p.o.box	NOUN
ejpam-5618	1	76	:	:	PUNCT
ejpam-5618	1	77	80203	80203	NUM
ejpam-5618	1	78	jeddah	jeddah	PROPN
ejpam-5618	1	79	21589	21589	NUM
ejpam-5618	1	80	,	,	PUNCT
ejpam-5618	1	81	saudi	saudi	PROPN
ejpam-5618	1	82	arabia	arabia	PROPN
ejpam-5618	1	83	abstract	abstract	NOUN
ejpam-5618	1	84	.	.	PUNCT
ejpam-5618	2	1	in	in	ADP
ejpam-5618	2	2	this	this	DET
ejpam-5618	2	3	article	article	NOUN
ejpam-5618	2	4	,	,	PUNCT
ejpam-5618	2	5	we	we	PRON
ejpam-5618	2	6	define	define	VERB
ejpam-5618	2	7	and	and	CCONJ
ejpam-5618	2	8	study	study	VERB
ejpam-5618	2	9	the	the	DET
ejpam-5618	2	10	concept	concept	NOUN
ejpam-5618	2	11	of	of	ADP
ejpam-5618	2	12	ideal	ideal	ADJ
ejpam-5618	2	13	topological	topological	ADJ
ejpam-5618	2	14	groups	group	NOUN
ejpam-5618	2	15	.	.	PUNCT
ejpam-5618	3	1	we	we	PRON
ejpam-5618	3	2	study	study	VERB
ejpam-5618	3	3	its	its	PRON
ejpam-5618	3	4	relation	relation	NOUN
ejpam-5618	3	5	to	to	ADP
ejpam-5618	3	6	topological	topological	ADJ
ejpam-5618	3	7	groups	group	NOUN
ejpam-5618	3	8	.	.	PUNCT
ejpam-5618	4	1	we	we	PRON
ejpam-5618	4	2	present	present	VERB
ejpam-5618	4	3	examples	example	NOUN
ejpam-5618	4	4	that	that	PRON
ejpam-5618	4	5	show	show	VERB
ejpam-5618	4	6	that	that	SCONJ
ejpam-5618	4	7	ideal	ideal	ADJ
ejpam-5618	4	8	topological	topological	ADJ
ejpam-5618	4	9	groups	group	NOUN
ejpam-5618	4	10	and	and	CCONJ
ejpam-5618	4	11	topological	topological	ADJ
ejpam-5618	4	12	groups	group	NOUN
ejpam-5618	4	13	are	be	AUX
ejpam-5618	4	14	independent	independent	ADJ
ejpam-5618	4	15	concepts	concept	NOUN
ejpam-5618	4	16	.	.	PUNCT
ejpam-5618	5	1	we	we	PRON
ejpam-5618	5	2	give	give	VERB
ejpam-5618	5	3	a	a	DET
ejpam-5618	5	4	sufficient	sufficient	ADJ
ejpam-5618	5	5	condition	condition	NOUN
ejpam-5618	5	6	for	for	ADP
ejpam-5618	5	7	a	a	DET
ejpam-5618	5	8	topological	topological	ADJ
ejpam-5618	5	9	group	group	NOUN
ejpam-5618	5	10	to	to	PART
ejpam-5618	5	11	be	be	AUX
ejpam-5618	5	12	an	an	DET
ejpam-5618	5	13	ideal	ideal	ADJ
ejpam-5618	5	14	topological	topological	ADJ
ejpam-5618	5	15	group	group	NOUN
ejpam-5618	5	16	as	as	ADV
ejpam-5618	5	17	well	well	ADV
ejpam-5618	5	18	as	as	SCONJ
ejpam-5618	5	19	we	we	PRON
ejpam-5618	5	20	give	give	VERB
ejpam-5618	5	21	a	a	DET
ejpam-5618	5	22	sufficient	sufficient	ADJ
ejpam-5618	5	23	condition	condition	NOUN
ejpam-5618	5	24	for	for	ADP
ejpam-5618	5	25	an	an	DET
ejpam-5618	5	26	ideal	ideal	ADJ
ejpam-5618	5	27	topological	topological	ADJ
ejpam-5618	5	28	group	group	NOUN
ejpam-5618	5	29	to	to	PART
ejpam-5618	5	30	be	be	AUX
ejpam-5618	5	31	a	a	DET
ejpam-5618	5	32	topological	topological	ADJ
ejpam-5618	5	33	group	group	NOUN
ejpam-5618	5	34	.	.	PUNCT
ejpam-5618	6	1	unlike	unlike	ADP
ejpam-5618	6	2	topological	topological	ADJ
ejpam-5618	6	3	groups	group	NOUN
ejpam-5618	6	4	,	,	PUNCT
ejpam-5618	6	5	ideal	ideal	ADJ
ejpam-5618	6	6	topological	topological	ADJ
ejpam-5618	6	7	groups	group	NOUN
ejpam-5618	6	8	are	be	AUX
ejpam-5618	6	9	not	not	PART
ejpam-5618	6	10	nicely	nicely	ADV
ejpam-5618	6	11	behaved	behave	VERB
ejpam-5618	6	12	with	with	ADP
ejpam-5618	6	13	regard	regard	NOUN
ejpam-5618	6	14	to	to	ADP
ejpam-5618	6	15	subgroups	subgroup	NOUN
ejpam-5618	6	16	.	.	PUNCT
ejpam-5618	7	1	we	we	PRON
ejpam-5618	7	2	give	give	VERB
ejpam-5618	7	3	an	an	DET
ejpam-5618	7	4	example	example	NOUN
ejpam-5618	7	5	of	of	ADP
ejpam-5618	7	6	a	a	DET
ejpam-5618	7	7	subgroup	subgroup	NOUN
ejpam-5618	7	8	of	of	ADP
ejpam-5618	7	9	an	an	DET
ejpam-5618	7	10	ideal	ideal	ADJ
ejpam-5618	7	11	topological	topological	ADJ
ejpam-5618	7	12	group	group	NOUN
ejpam-5618	7	13	that	that	PRON
ejpam-5618	7	14	is	be	AUX
ejpam-5618	7	15	not	not	PART
ejpam-5618	7	16	an	an	DET
ejpam-5618	7	17	ideal	ideal	ADJ
ejpam-5618	7	18	topological	topological	ADJ
ejpam-5618	7	19	group	group	NOUN
ejpam-5618	7	20	.	.	PUNCT
ejpam-5618	8	1	we	we	PRON
ejpam-5618	8	2	show	show	VERB
ejpam-5618	8	3	that	that	SCONJ
ejpam-5618	8	4	every	every	DET
ejpam-5618	8	5	open	open	ADJ
ejpam-5618	8	6	subgroup	subgroup	NOUN
ejpam-5618	8	7	of	of	ADP
ejpam-5618	8	8	an	an	DET
ejpam-5618	8	9	ideal	ideal	ADJ
ejpam-5618	8	10	topological	topological	ADJ
ejpam-5618	8	11	group	group	NOUN
ejpam-5618	8	12	is	be	AUX
ejpam-5618	8	13	also	also	ADV
ejpam-5618	8	14	an	an	DET
ejpam-5618	8	15	ideal	ideal	ADJ
ejpam-5618	8	16	topological	topological	ADJ
ejpam-5618	8	17	group	group	NOUN
ejpam-5618	8	18	.	.	PUNCT
ejpam-5618	9	1	moreover	moreover	ADV
ejpam-5618	9	2	,	,	PUNCT
ejpam-5618	9	3	we	we	PRON
ejpam-5618	9	4	investigate	investigate	VERB
ejpam-5618	9	5	i	i	NOUN
ejpam-5618	9	6	-	-	PUNCT
ejpam-5618	9	7	connectedness	connectedness	NOUN
ejpam-5618	9	8	of	of	ADP
ejpam-5618	9	9	ideal	ideal	ADJ
ejpam-5618	9	10	topological	topological	ADJ
ejpam-5618	9	11	groups	group	NOUN
ejpam-5618	9	12	.	.	PUNCT
ejpam-5618	10	1	2020	2020	NUM
ejpam-5618	10	2	mathematics	mathematic	NOUN
ejpam-5618	10	3	subject	subject	NOUN
ejpam-5618	10	4	classifications	classification	NOUN
ejpam-5618	10	5	:	:	PUNCT
ejpam-5618	10	6	54h11	54h11	NUM
ejpam-5618	10	7	,	,	PUNCT
ejpam-5618	10	8	22a05	22a05	NUM
ejpam-5618	10	9	key	key	ADJ
ejpam-5618	10	10	words	word	NOUN
ejpam-5618	10	11	and	and	CCONJ
ejpam-5618	10	12	phrases	phrase	NOUN
ejpam-5618	10	13	:	:	PUNCT
ejpam-5618	10	14	ideal	ideal	ADJ
ejpam-5618	10	15	topological	topological	ADJ
ejpam-5618	10	16	groups	group	NOUN
ejpam-5618	10	17	,	,	PUNCT
ejpam-5618	10	18	topological	topological	ADJ
ejpam-5618	10	19	groups	group	NOUN
ejpam-5618	10	20	,	,	PUNCT
ejpam-5618	10	21	ideal	ideal	ADJ
ejpam-5618	10	22	topological	topological	ADJ
ejpam-5618	10	23	spaces	space	NOUN
ejpam-5618	10	24	,	,	PUNCT
ejpam-5618	10	25	submaximal	submaximal	ADJ
ejpam-5618	10	26	spaces	space	NOUN
ejpam-5618	10	27	,	,	PUNCT
ejpam-5618	10	28	i	i	PRON
ejpam-5618	10	29	-	-	PUNCT
ejpam-5618	10	30	connectedness	connectedness	NOUN
ejpam-5618	10	31	1	1	NUM
ejpam-5618	10	32	.	.	PUNCT
ejpam-5618	11	1	introduction	introduction	NOUN
ejpam-5618	11	2	the	the	DET
ejpam-5618	11	3	notion	notion	NOUN
ejpam-5618	11	4	of	of	ADP
ejpam-5618	11	5	ideal	ideal	ADJ
ejpam-5618	11	6	topological	topological	ADJ
ejpam-5618	11	7	spaces	space	NOUN
ejpam-5618	11	8	was	be	AUX
ejpam-5618	11	9	studied	study	VERB
ejpam-5618	11	10	in	in	ADP
ejpam-5618	11	11	the	the	DET
ejpam-5618	11	12	classic	classic	ADJ
ejpam-5618	11	13	book	book	NOUN
ejpam-5618	12	1	[	[	X
ejpam-5618	12	2	11	11	NUM
ejpam-5618	12	3	]	]	PUNCT
ejpam-5618	12	4	and	and	CCONJ
ejpam-5618	12	5	also	also	ADV
ejpam-5618	12	6	in	in	ADP
ejpam-5618	12	7	[	[	X
ejpam-5618	12	8	13	13	NUM
ejpam-5618	12	9	]	]	PUNCT
ejpam-5618	12	10	.	.	PUNCT
ejpam-5618	13	1	in	in	ADP
ejpam-5618	13	2	1990	1990	NUM
ejpam-5618	13	3	,	,	PUNCT
ejpam-5618	13	4	jankovic	jankovic	PROPN
ejpam-5618	13	5	and	and	CCONJ
ejpam-5618	13	6	hamlett	hamlett	PROPN
ejpam-5618	13	7	[	[	X
ejpam-5618	13	8	9	9	NUM
ejpam-5618	13	9	]	]	PUNCT
ejpam-5618	13	10	introduced	introduce	VERB
ejpam-5618	13	11	i	i	PRON
ejpam-5618	13	12	-	-	PUNCT
ejpam-5618	13	13	open	open	ADJ
ejpam-5618	13	14	sets	set	NOUN
ejpam-5618	13	15	in	in	ADP
ejpam-5618	13	16	topological	topological	ADJ
ejpam-5618	13	17	spaces	space	NOUN
ejpam-5618	13	18	and	and	CCONJ
ejpam-5618	13	19	later	later	ADV
ejpam-5618	13	20	obtained	obtain	VERB
ejpam-5618	13	21	several	several	ADJ
ejpam-5618	13	22	properties	property	NOUN
ejpam-5618	13	23	of	of	ADP
ejpam-5618	13	24	ideal	ideal	ADJ
ejpam-5618	13	25	topological	topological	ADJ
ejpam-5618	13	26	spaces	space	NOUN
ejpam-5618	13	27	in	in	ADP
ejpam-5618	13	28	[	[	X
ejpam-5618	13	29	10	10	NUM
ejpam-5618	13	30	]	]	PUNCT
ejpam-5618	13	31	.	.	PUNCT
ejpam-5618	14	1	abd	abd	PROPN
ejpam-5618	14	2	el	el	PROPN
ejpam-5618	14	3	-	-	PROPN
ejpam-5618	14	4	monsef	monsef	PROPN
ejpam-5618	14	5	et	et	PROPN
ejpam-5618	14	6	.	.	PUNCT
ejpam-5618	15	1	al.[1	al.[1	PROPN
ejpam-5618	15	2	]	]	PUNCT
ejpam-5618	15	3	investigated	investigate	VERB
ejpam-5618	15	4	further	further	ADJ
ejpam-5618	15	5	properties	property	NOUN
ejpam-5618	15	6	of	of	ADP
ejpam-5618	15	7	i	i	PRON
ejpam-5618	15	8	-	-	PUNCT
ejpam-5618	15	9	open	open	ADJ
ejpam-5618	15	10	sets	set	NOUN
ejpam-5618	15	11	and	and	CCONJ
ejpam-5618	15	12	introduced	introduce	VERB
ejpam-5618	15	13	i	i	PROPN
ejpam-5618	15	14	-	-	PUNCT
ejpam-5618	15	15	closed	close	VERB
ejpam-5618	15	16	sets	set	NOUN
ejpam-5618	15	17	,	,	PUNCT
ejpam-5618	15	18	icontinuous	icontinuous	ADJ
ejpam-5618	15	19	mappings	mapping	NOUN
ejpam-5618	15	20	and	and	CCONJ
ejpam-5618	15	21	i	i	PRON
ejpam-5618	15	22	-	-	PUNCT
ejpam-5618	15	23	open	open	ADJ
ejpam-5618	15	24	(	(	PUNCT
ejpam-5618	15	25	closed	closed	ADJ
ejpam-5618	15	26	)	)	PUNCT
ejpam-5618	15	27	mappings	mapping	NOUN
ejpam-5618	15	28	and	and	CCONJ
ejpam-5618	15	29	studied	study	VERB
ejpam-5618	15	30	the	the	DET
ejpam-5618	15	31	relations	relation	NOUN
ejpam-5618	15	32	between	between	ADP
ejpam-5618	15	33	them	they	PRON
ejpam-5618	15	34	.	.	PUNCT
ejpam-5618	16	1	in	in	ADP
ejpam-5618	16	2	1943	1943	NUM
ejpam-5618	16	3	,	,	PUNCT
ejpam-5618	16	4	hewitt	hewitt	NOUN
ejpam-5618	17	1	[	[	X
ejpam-5618	17	2	7	7	X
ejpam-5618	17	3	]	]	PUNCT
ejpam-5618	17	4	introduced	introduce	VERB
ejpam-5618	17	5	the	the	DET
ejpam-5618	17	6	concept	concept	NOUN
ejpam-5618	17	7	of	of	ADP
ejpam-5618	17	8	submaximal	submaximal	ADJ
ejpam-5618	17	9	spaces	space	NOUN
ejpam-5618	17	10	.	.	PUNCT
ejpam-5618	18	1	arhangel’skii	arhangel’skii	VERB
ejpam-5618	18	2	and	and	CCONJ
ejpam-5618	18	3	collins	collin	VERB
ejpam-5618	19	1	[	[	X
ejpam-5618	19	2	3	3	NUM
ejpam-5618	19	3	]	]	PUNCT
ejpam-5618	19	4	studied	study	VERB
ejpam-5618	19	5	submaximal	submaximal	ADJ
ejpam-5618	19	6	spaces	space	NOUN
ejpam-5618	19	7	and	and	CCONJ
ejpam-5618	19	8	gave	give	VERB
ejpam-5618	19	9	characterizations	characterization	NOUN
ejpam-5618	19	10	of	of	ADP
ejpam-5618	19	11	it	it	PRON
ejpam-5618	19	12	.	.	PUNCT
ejpam-5618	20	1	dontchev	dontchev	ADJ
ejpam-5618	20	2	[	[	X
ejpam-5618	20	3	5	5	NUM
ejpam-5618	20	4	]	]	PUNCT
ejpam-5618	20	5	defined	define	VERB
ejpam-5618	20	6	the	the	DET
ejpam-5618	20	7	i	i	PROPN
ejpam-5618	20	8	-	-	PUNCT
ejpam-5618	20	9	irresolute	irresolute	ADJ
ejpam-5618	20	10	mapping	mapping	NOUN
ejpam-5618	20	11	and	and	CCONJ
ejpam-5618	20	12	investigated	investigate	VERB
ejpam-5618	20	13	the	the	DET
ejpam-5618	20	14	relationship	relationship	NOUN
ejpam-5618	20	15	between	between	ADP
ejpam-5618	20	16	i	i	NOUN
ejpam-5618	20	17	-	-	PUNCT
ejpam-5618	20	18	open	open	ADJ
ejpam-5618	20	19	classes	class	NOUN
ejpam-5618	20	20	and	and	CCONJ
ejpam-5618	20	21	preopen	preopen	ADJ
ejpam-5618	20	22	classes	class	NOUN
ejpam-5618	20	23	.	.	PUNCT
ejpam-5618	21	1	in	in	ADP
ejpam-5618	21	2	2020	2020	NUM
ejpam-5618	21	3	,	,	PUNCT
ejpam-5618	21	4	jafari	jafari	PROPN
ejpam-5618	21	5	and	and	CCONJ
ejpam-5618	21	6	rajesh	rajesh	PROPN
ejpam-5618	21	7	[	[	X
ejpam-5618	21	8	8	8	NUM
ejpam-5618	21	9	]	]	PUNCT
ejpam-5618	21	10	initiated	initiate	VERB
ejpam-5618	21	11	the	the	DET
ejpam-5618	21	12	study	study	NOUN
ejpam-5618	21	13	of	of	ADP
ejpam-5618	21	14	ideal	ideal	ADJ
ejpam-5618	21	15	topological	topological	ADJ
ejpam-5618	21	16	groups	group	NOUN
ejpam-5618	21	17	.	.	PUNCT
ejpam-5618	22	1	in	in	ADP
ejpam-5618	22	2	this	this	DET
ejpam-5618	22	3	paper	paper	NOUN
ejpam-5618	22	4	,	,	PUNCT
ejpam-5618	22	5	we	we	PRON
ejpam-5618	22	6	define	define	VERB
ejpam-5618	22	7	the	the	DET
ejpam-5618	22	8	notion	notion	NOUN
ejpam-5618	22	9	of	of	ADP
ejpam-5618	22	10	ideal	ideal	ADJ
ejpam-5618	22	11	topological	topological	ADJ
ejpam-5618	22	12	groups	group	NOUN
ejpam-5618	22	13	and	and	CCONJ
ejpam-5618	22	14	present	present	VERB
ejpam-5618	22	15	their	their	PRON
ejpam-5618	22	16	main	main	ADJ
ejpam-5618	22	17	properties	property	NOUN
ejpam-5618	22	18	.	.	PUNCT
ejpam-5618	23	1	also	also	ADV
ejpam-5618	23	2	,	,	PUNCT
ejpam-5618	23	3	we	we	PRON
ejpam-5618	23	4	study	study	VERB
ejpam-5618	23	5	the	the	DET
ejpam-5618	23	6	relation	relation	NOUN
ejpam-5618	23	7	between	between	ADP
ejpam-5618	23	8	topological	topological	ADJ
ejpam-5618	23	9	groups	group	NOUN
ejpam-5618	23	10	and	and	CCONJ
ejpam-5618	23	11	ideal	ideal	ADJ
ejpam-5618	23	12	topological	topological	ADJ
ejpam-5618	23	13	groups	group	NOUN
ejpam-5618	23	14	.	.	PUNCT
ejpam-5618	24	1	furthermore	furthermore	ADV
ejpam-5618	24	2	,	,	PUNCT
ejpam-5618	24	3	we	we	PRON
ejpam-5618	24	4	investigate	investigate	VERB
ejpam-5618	24	5	i	i	NOUN
ejpam-5618	24	6	-	-	PUNCT
ejpam-5618	24	7	connectedness	connectedness	NOUN
ejpam-5618	24	8	of	of	ADP
ejpam-5618	24	9	ideal	ideal	ADJ
ejpam-5618	24	10	topological	topological	ADJ
ejpam-5618	24	11	groups	group	NOUN
ejpam-5618	24	12	,	,	PUNCT
ejpam-5618	24	13	see	see	VERB
ejpam-5618	24	14	[	[	X
ejpam-5618	24	15	2	2	NUM
ejpam-5618	24	16	]	]	PUNCT
ejpam-5618	24	17	.	.	PUNCT
ejpam-5618	25	1	∗corresponding	∗corresponde	VERB
ejpam-5618	25	2	author	author	NOUN
ejpam-5618	25	3	.	.	PUNCT
ejpam-5618	26	1	doi	doi	NOUN
ejpam-5618	26	2	:	:	PUNCT
ejpam-5618	26	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5618	https://doi.org/10.29020/nybg.ejpam.v18i1.5618	NUM
ejpam-5618	26	4	email	email	NOUN
ejpam-5618	26	5	addresses	address	VERB
ejpam-5618	26	6	:	:	PUNCT
ejpam-5618	26	7	s.alammar@uoh.edu.sa	s.alammar@uoh.edu.sa	PROPN
ejpam-5618	26	8	(	(	PUNCT
ejpam-5618	26	9	s.	s.	PROPN
ejpam-5618	26	10	alammar	alammar	PROPN
ejpam-5618	26	11	)	)	PUNCT
ejpam-5618	26	12	,	,	PUNCT
ejpam-5618	26	13	maalshmrani1@kau.edu.sa	maalshmrani1@kau.edu.sa	PROPN
ejpam-5618	26	14	(	(	PUNCT
ejpam-5618	26	15	m.	m.	NOUN
ejpam-5618	26	16	al	al	PROPN
ejpam-5618	26	17	shumrani	shumrani	PROPN
ejpam-5618	26	18	)	)	PUNCT
ejpam-5618	26	19	,	,	PUNCT
ejpam-5618	26	20	cenap.ozel@gmail.com	cenap.ozel@gmail.com	X
ejpam-5618	26	21	(	(	PUNCT
ejpam-5618	26	22	c.	c.	PROPN
ejpam-5618	26	23	özel	özel	PROPN
ejpam-5618	26	24	)	)	PUNCT
ejpam-5618	26	25	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5618	27	1	1	1	NUM
ejpam-5618	27	2	copyright	copyright	NOUN
ejpam-5618	27	3	:	:	PUNCT
ejpam-5618	27	4	©	©	PROPN
ejpam-5618	27	5	2025	2025	NUM
ejpam-5618	27	6	the	the	DET
ejpam-5618	27	7	author(s	author(s	NOUN
ejpam-5618	27	8	)	)	PUNCT
ejpam-5618	27	9	.	.	PUNCT
ejpam-5618	28	1	(	(	PUNCT
ejpam-5618	28	2	cc	cc	NOUN
ejpam-5618	28	3	by	by	ADP
ejpam-5618	28	4	-	-	PUNCT
ejpam-5618	28	5	nc	nc	PROPN
ejpam-5618	28	6	4.0	4.0	NUM
ejpam-5618	28	7	)	)	PUNCT
ejpam-5618	28	8	s.	s.	PROPN
ejpam-5618	28	9	alammar	alammar	PROPN
ejpam-5618	28	10	,	,	PUNCT
ejpam-5618	28	11	m.	m.	NOUN
ejpam-5618	28	12	al	al	PROPN
ejpam-5618	28	13	shumrani	shumrani	PROPN
ejpam-5618	28	14	,	,	PUNCT
ejpam-5618	28	15	c.	c.	PROPN
ejpam-5618	28	16	özel	özel	PROPN
ejpam-5618	28	17	/	/	SYM
ejpam-5618	28	18	eur	eur	PROPN
ejpam-5618	28	19	.	.	PUNCT
ejpam-5618	29	1	j.	j.	PROPN
ejpam-5618	29	2	pure	pure	PROPN
ejpam-5618	29	3	appl	appl	PROPN
ejpam-5618	29	4	.	.	PROPN
ejpam-5618	29	5	math	math	PROPN
ejpam-5618	29	6	,	,	PUNCT
ejpam-5618	29	7	18	18	NUM
ejpam-5618	29	8	(	(	PUNCT
ejpam-5618	29	9	1	1	NUM
ejpam-5618	29	10	)	)	PUNCT
ejpam-5618	29	11	(	(	PUNCT
ejpam-5618	29	12	2025	2025	NUM
ejpam-5618	29	13	)	)	PUNCT
ejpam-5618	29	14	,	,	PUNCT
ejpam-5618	29	15	5618	5618	NUM
ejpam-5618	29	16	2	2	NUM
ejpam-5618	29	17	of	of	ADP
ejpam-5618	29	18	13	13	NUM
ejpam-5618	29	19	2	2	NUM
ejpam-5618	29	20	.	.	PUNCT
ejpam-5618	29	21	preliminaries	preliminary	NOUN
ejpam-5618	29	22	in	in	ADP
ejpam-5618	29	23	this	this	DET
ejpam-5618	29	24	section	section	NOUN
ejpam-5618	30	1	,	,	PUNCT
ejpam-5618	30	2	we	we	PRON
ejpam-5618	30	3	recall	recall	VERB
ejpam-5618	30	4	some	some	DET
ejpam-5618	30	5	definitions	definition	NOUN
ejpam-5618	30	6	and	and	CCONJ
ejpam-5618	30	7	results	result	NOUN
ejpam-5618	30	8	which	which	PRON
ejpam-5618	30	9	we	we	PRON
ejpam-5618	30	10	shall	shall	AUX
ejpam-5618	30	11	use	use	VERB
ejpam-5618	30	12	frequently	frequently	ADV
ejpam-5618	30	13	in	in	ADP
ejpam-5618	30	14	the	the	DET
ejpam-5618	30	15	following	follow	VERB
ejpam-5618	30	16	sections	section	NOUN
ejpam-5618	30	17	.	.	PUNCT
ejpam-5618	31	1	note	note	VERB
ejpam-5618	31	2	that	that	SCONJ
ejpam-5618	31	3	the	the	DET
ejpam-5618	31	4	fundamental	fundamental	ADJ
ejpam-5618	31	5	reference	reference	NOUN
ejpam-5618	31	6	for	for	ADP
ejpam-5618	31	7	topological	topological	ADJ
ejpam-5618	31	8	spaces	space	NOUN
ejpam-5618	31	9	is	be	AUX
ejpam-5618	31	10	[	[	X
ejpam-5618	31	11	6	6	NUM
ejpam-5618	31	12	]	]	PUNCT
ejpam-5618	31	13	;	;	PUNCT
ejpam-5618	31	14	see	see	VERB
ejpam-5618	31	15	also	also	ADV
ejpam-5618	31	16	[	[	X
ejpam-5618	31	17	12	12	NUM
ejpam-5618	31	18	]	]	PUNCT
ejpam-5618	31	19	and	and	CCONJ
ejpam-5618	31	20	the	the	DET
ejpam-5618	31	21	main	main	ADJ
ejpam-5618	31	22	reference	reference	NOUN
ejpam-5618	31	23	for	for	ADP
ejpam-5618	31	24	topological	topological	ADJ
ejpam-5618	31	25	groups	group	NOUN
ejpam-5618	31	26	and	and	CCONJ
ejpam-5618	31	27	their	their	PRON
ejpam-5618	31	28	properties	property	NOUN
ejpam-5618	31	29	is	be	AUX
ejpam-5618	31	30	[	[	X
ejpam-5618	31	31	4	4	NUM
ejpam-5618	31	32	]	]	PUNCT
ejpam-5618	31	33	.	.	PUNCT
ejpam-5618	32	1	definition	definition	NOUN
ejpam-5618	32	2	1	1	NUM
ejpam-5618	32	3	.	.	PUNCT
ejpam-5618	33	1	[	[	X
ejpam-5618	33	2	13	13	NUM
ejpam-5618	33	3	]	]	PUNCT
ejpam-5618	33	4	a	a	DET
ejpam-5618	33	5	collection	collection	NOUN
ejpam-5618	33	6	i	i	PRON
ejpam-5618	33	7	of	of	ADP
ejpam-5618	33	8	subsets	subset	NOUN
ejpam-5618	33	9	of	of	ADP
ejpam-5618	33	10	a	a	DET
ejpam-5618	33	11	set	set	NOUN
ejpam-5618	33	12	x	x	PUNCT
ejpam-5618	33	13	is	be	AUX
ejpam-5618	33	14	called	call	VERB
ejpam-5618	33	15	an	an	DET
ejpam-5618	33	16	ideal	ideal	NOUN
ejpam-5618	33	17	on	on	ADP
ejpam-5618	33	18	x	x	SYM
ejpam-5618	33	19	if	if	SCONJ
ejpam-5618	33	20	it	it	PRON
ejpam-5618	33	21	satisfies	satisfy	VERB
ejpam-5618	33	22	the	the	DET
ejpam-5618	33	23	following	follow	VERB
ejpam-5618	33	24	two	two	NUM
ejpam-5618	33	25	conditions	condition	NOUN
ejpam-5618	33	26	:	:	PUNCT
ejpam-5618	33	27	(	(	PUNCT
ejpam-5618	33	28	i	i	NOUN
ejpam-5618	33	29	)	)	PUNCT
ejpam-5618	34	1	if	if	SCONJ
ejpam-5618	34	2	a	a	DET
ejpam-5618	34	3	∈	∈	X
ejpam-5618	34	4	i	i	PRON
ejpam-5618	34	5	and	and	CCONJ
ejpam-5618	34	6	b	b	PROPN
ejpam-5618	34	7	⊂	⊂	PROPN
ejpam-5618	34	8	a	a	PROPN
ejpam-5618	34	9	,	,	PUNCT
ejpam-5618	34	10	then	then	ADV
ejpam-5618	34	11	b	b	PROPN
ejpam-5618	34	12	∈	∈	PROPN
ejpam-5618	34	13	i.	i.	NOUN
ejpam-5618	34	14	(	(	PUNCT
ejpam-5618	34	15	ii	ii	PROPN
ejpam-5618	34	16	)	)	PUNCT
ejpam-5618	34	17	if	if	SCONJ
ejpam-5618	34	18	a	a	PRON
ejpam-5618	34	19	,	,	PUNCT
ejpam-5618	34	20	b	b	X
ejpam-5618	34	21	∈	∈	PROPN
ejpam-5618	35	1	i	i	PRON
ejpam-5618	35	2	,	,	PUNCT
ejpam-5618	35	3	then	then	ADV
ejpam-5618	35	4	a	a	DET
ejpam-5618	35	5	∪b	∪b	X
ejpam-5618	35	6	∈	∈	PROPN
ejpam-5618	35	7	i.	i.	NOUN
ejpam-5618	35	8	recall	recall	VERB
ejpam-5618	35	9	that	that	SCONJ
ejpam-5618	35	10	if	if	SCONJ
ejpam-5618	35	11	(	(	PUNCT
ejpam-5618	35	12	x	x	NOUN
ejpam-5618	35	13	,	,	PUNCT
ejpam-5618	35	14	τ	τ	X
ejpam-5618	35	15	)	)	PUNCT
ejpam-5618	35	16	is	be	AUX
ejpam-5618	35	17	a	a	DET
ejpam-5618	35	18	topological	topological	ADJ
ejpam-5618	35	19	space	space	NOUN
ejpam-5618	35	20	and	and	CCONJ
ejpam-5618	35	21	i	i	PRON
ejpam-5618	35	22	is	be	AUX
ejpam-5618	35	23	an	an	DET
ejpam-5618	35	24	ideal	ideal	NOUN
ejpam-5618	35	25	on	on	ADP
ejpam-5618	35	26	x	x	NOUN
ejpam-5618	35	27	,	,	PUNCT
ejpam-5618	35	28	then	then	ADV
ejpam-5618	35	29	(	(	PUNCT
ejpam-5618	35	30	x	x	X
ejpam-5618	35	31	,	,	PUNCT
ejpam-5618	35	32	τ	τ	PROPN
ejpam-5618	35	33	,	,	PUNCT
ejpam-5618	35	34	i	i	PROPN
ejpam-5618	35	35	)	)	PUNCT
ejpam-5618	35	36	,	,	PUNCT
ejpam-5618	35	37	or	or	CCONJ
ejpam-5618	35	38	simply	simply	ADV
ejpam-5618	35	39	x	x	X
ejpam-5618	35	40	,	,	PUNCT
ejpam-5618	35	41	is	be	AUX
ejpam-5618	35	42	called	call	VERB
ejpam-5618	35	43	an	an	DET
ejpam-5618	35	44	ideal	ideal	ADJ
ejpam-5618	35	45	topological	topological	ADJ
ejpam-5618	35	46	space	space	NOUN
ejpam-5618	35	47	.	.	PUNCT
ejpam-5618	36	1	definition	definition	NOUN
ejpam-5618	36	2	2	2	NUM
ejpam-5618	36	3	.	.	PUNCT
ejpam-5618	37	1	[	[	X
ejpam-5618	37	2	11	11	NUM
ejpam-5618	37	3	]	]	X
ejpam-5618	37	4	let	let	AUX
ejpam-5618	37	5	(	(	PUNCT
ejpam-5618	37	6	x	x	NOUN
ejpam-5618	37	7	,	,	PUNCT
ejpam-5618	37	8	τ	τ	PROPN
ejpam-5618	37	9	,	,	PUNCT
ejpam-5618	37	10	i	i	PRON
ejpam-5618	37	11	)	)	PUNCT
ejpam-5618	37	12	be	be	VERB
ejpam-5618	37	13	an	an	DET
ejpam-5618	37	14	ideal	ideal	ADJ
ejpam-5618	37	15	topological	topological	ADJ
ejpam-5618	37	16	space	space	NOUN
ejpam-5618	37	17	.	.	PUNCT
ejpam-5618	38	1	then	then	ADV
ejpam-5618	38	2	a⋆	a⋆	ADV
ejpam-5618	38	3	=	=	SYM
ejpam-5618	38	4	{	{	PUNCT
ejpam-5618	38	5	x	x	PUNCT
ejpam-5618	38	6	∈	∈	PROPN
ejpam-5618	38	7	x	x	X
ejpam-5618	38	8	:	:	PUNCT
ejpam-5618	39	1	u∩a	u∩a	PROPN
ejpam-5618	39	2	/∈	/∈	PUNCT
ejpam-5618	40	1	i	i	PRON
ejpam-5618	40	2	for	for	ADP
ejpam-5618	40	3	every	every	DET
ejpam-5618	40	4	open	open	ADJ
ejpam-5618	40	5	neighborhood	neighborhood	NOUN
ejpam-5618	40	6	u	u	NOUN
ejpam-5618	40	7	of	of	ADP
ejpam-5618	40	8	x	x	NOUN
ejpam-5618	40	9	}	}	PUNCT
ejpam-5618	40	10	.	.	PUNCT
ejpam-5618	41	1	definition	definition	NOUN
ejpam-5618	41	2	3	3	NUM
ejpam-5618	41	3	.	.	PUNCT
ejpam-5618	42	1	[	[	X
ejpam-5618	42	2	10	10	NUM
ejpam-5618	42	3	]	]	X
ejpam-5618	42	4	let	let	VERB
ejpam-5618	42	5	(	(	PUNCT
ejpam-5618	42	6	x	x	NOUN
ejpam-5618	42	7	,	,	PUNCT
ejpam-5618	42	8	τ	τ	PROPN
ejpam-5618	42	9	,	,	PUNCT
ejpam-5618	42	10	i	i	PRON
ejpam-5618	42	11	)	)	PUNCT
ejpam-5618	42	12	be	be	VERB
ejpam-5618	42	13	an	an	DET
ejpam-5618	42	14	ideal	ideal	ADJ
ejpam-5618	42	15	topological	topological	ADJ
ejpam-5618	42	16	space	space	NOUN
ejpam-5618	42	17	.	.	PUNCT
ejpam-5618	43	1	we	we	PRON
ejpam-5618	43	2	say	say	VERB
ejpam-5618	43	3	that	that	SCONJ
ejpam-5618	43	4	a	a	DET
ejpam-5618	43	5	subset	subset	NOUN
ejpam-5618	43	6	a	a	PRON
ejpam-5618	43	7	of	of	ADP
ejpam-5618	43	8	x	x	PUNCT
ejpam-5618	43	9	is	be	AUX
ejpam-5618	43	10	an	an	DET
ejpam-5618	43	11	i	i	NOUN
ejpam-5618	43	12	-	-	PUNCT
ejpam-5618	43	13	open	open	ADJ
ejpam-5618	43	14	if	if	SCONJ
ejpam-5618	43	15	a	a	DET
ejpam-5618	43	16	⊆	⊆	NUM
ejpam-5618	43	17	int	int	NOUN
ejpam-5618	43	18	(	(	PUNCT
ejpam-5618	43	19	a∗	a∗	NOUN
ejpam-5618	43	20	)	)	PUNCT
ejpam-5618	43	21	.	.	PUNCT
ejpam-5618	44	1	let	let	VERB
ejpam-5618	44	2	(	(	PUNCT
ejpam-5618	44	3	x	x	X
ejpam-5618	44	4	,	,	PUNCT
ejpam-5618	44	5	τ	τ	PROPN
ejpam-5618	44	6	,	,	PUNCT
ejpam-5618	44	7	i	i	PRON
ejpam-5618	44	8	)	)	PUNCT
ejpam-5618	44	9	be	be	VERB
ejpam-5618	44	10	an	an	DET
ejpam-5618	44	11	ideal	ideal	ADJ
ejpam-5618	44	12	topological	topological	ADJ
ejpam-5618	44	13	space	space	NOUN
ejpam-5618	44	14	.	.	PUNCT
ejpam-5618	45	1	a	a	DET
ejpam-5618	45	2	subset	subset	NOUN
ejpam-5618	45	3	a	a	PRON
ejpam-5618	45	4	is	be	AUX
ejpam-5618	45	5	called	call	VERB
ejpam-5618	45	6	i	i	PRON
ejpam-5618	45	7	-	-	PUNCT
ejpam-5618	45	8	closed	close	VERB
ejpam-5618	45	9	if	if	SCONJ
ejpam-5618	45	10	the	the	DET
ejpam-5618	45	11	set	set	NOUN
ejpam-5618	45	12	x	x	PUNCT
ejpam-5618	45	13	−	−	PROPN
ejpam-5618	45	14	a	a	PRON
ejpam-5618	45	15	is	be	AUX
ejpam-5618	45	16	i	i	PRON
ejpam-5618	45	17	-	-	PUNCT
ejpam-5618	45	18	open	open	ADJ
ejpam-5618	45	19	.	.	PUNCT
ejpam-5618	46	1	remark	remark	NOUN
ejpam-5618	46	2	1	1	NUM
ejpam-5618	46	3	.	.	PUNCT
ejpam-5618	47	1	[	[	X
ejpam-5618	47	2	1	1	X
ejpam-5618	47	3	]	]	X
ejpam-5618	47	4	arbitrary	arbitrary	ADJ
ejpam-5618	47	5	union	union	NOUN
ejpam-5618	47	6	of	of	ADP
ejpam-5618	47	7	i	i	PROPN
ejpam-5618	47	8	-	-	PUNCT
ejpam-5618	47	9	open	open	ADJ
ejpam-5618	47	10	sets	set	NOUN
ejpam-5618	47	11	is	be	AUX
ejpam-5618	47	12	an	an	DET
ejpam-5618	47	13	i	i	NOUN
ejpam-5618	47	14	-	-	PUNCT
ejpam-5618	47	15	open	open	ADJ
ejpam-5618	47	16	set	set	NOUN
ejpam-5618	47	17	.	.	PUNCT
ejpam-5618	48	1	in	in	ADP
ejpam-5618	48	2	contrast	contrast	NOUN
ejpam-5618	48	3	,	,	PUNCT
ejpam-5618	48	4	the	the	DET
ejpam-5618	48	5	intersection	intersection	NOUN
ejpam-5618	48	6	of	of	ADP
ejpam-5618	48	7	two	two	NUM
ejpam-5618	48	8	i	i	NOUN
ejpam-5618	48	9	-	-	PUNCT
ejpam-5618	48	10	open	open	ADJ
ejpam-5618	48	11	sets	set	NOUN
ejpam-5618	48	12	may	may	AUX
ejpam-5618	48	13	not	not	PART
ejpam-5618	48	14	be	be	AUX
ejpam-5618	48	15	an	an	DET
ejpam-5618	48	16	i	i	NOUN
ejpam-5618	48	17	-	-	PUNCT
ejpam-5618	48	18	open	open	ADJ
ejpam-5618	48	19	set	set	NOUN
ejpam-5618	48	20	.	.	PUNCT
ejpam-5618	49	1	however	however	ADV
ejpam-5618	49	2	,	,	PUNCT
ejpam-5618	49	3	the	the	DET
ejpam-5618	49	4	intersection	intersection	NOUN
ejpam-5618	49	5	of	of	ADP
ejpam-5618	49	6	an	an	DET
ejpam-5618	49	7	open	open	ADJ
ejpam-5618	49	8	set	set	NOUN
ejpam-5618	49	9	with	with	ADP
ejpam-5618	49	10	an	an	DET
ejpam-5618	49	11	i	i	NOUN
ejpam-5618	49	12	-	-	PUNCT
ejpam-5618	49	13	open	open	ADJ
ejpam-5618	49	14	set	set	NOUN
ejpam-5618	49	15	is	be	AUX
ejpam-5618	49	16	an	an	DET
ejpam-5618	49	17	i	i	NOUN
ejpam-5618	49	18	-	-	PUNCT
ejpam-5618	49	19	open	open	ADJ
ejpam-5618	49	20	set	set	NOUN
ejpam-5618	49	21	.	.	PUNCT
ejpam-5618	50	1	definition	definition	NOUN
ejpam-5618	50	2	4	4	NUM
ejpam-5618	50	3	.	.	PUNCT
ejpam-5618	51	1	let	let	VERB
ejpam-5618	51	2	x	x	PRON
ejpam-5618	51	3	be	be	AUX
ejpam-5618	51	4	an	an	DET
ejpam-5618	51	5	ideal	ideal	ADJ
ejpam-5618	51	6	topological	topological	ADJ
ejpam-5618	51	7	space	space	NOUN
ejpam-5618	51	8	and	and	CCONJ
ejpam-5618	51	9	b	b	NOUN
ejpam-5618	51	10	be	be	AUX
ejpam-5618	51	11	a	a	DET
ejpam-5618	51	12	collection	collection	NOUN
ejpam-5618	51	13	of	of	ADP
ejpam-5618	51	14	i	i	NOUN
ejpam-5618	51	15	-	-	PUNCT
ejpam-5618	51	16	open	open	ADJ
ejpam-5618	51	17	subsets	subset	NOUN
ejpam-5618	51	18	of	of	ADP
ejpam-5618	51	19	x.	x.	NOUN
ejpam-5618	51	20	then	then	ADV
ejpam-5618	51	21	b	b	PROPN
ejpam-5618	51	22	is	be	AUX
ejpam-5618	51	23	called	call	VERB
ejpam-5618	51	24	an	an	DET
ejpam-5618	51	25	i	i	NOUN
ejpam-5618	51	26	-	-	PUNCT
ejpam-5618	51	27	open	open	ADJ
ejpam-5618	51	28	base	base	NOUN
ejpam-5618	51	29	for	for	ADP
ejpam-5618	51	30	x	x	PRON
ejpam-5618	51	31	if	if	SCONJ
ejpam-5618	51	32	every	every	DET
ejpam-5618	51	33	nonempty	nonempty	ADJ
ejpam-5618	51	34	i	i	PRON
ejpam-5618	51	35	-	-	PUNCT
ejpam-5618	51	36	open	open	ADJ
ejpam-5618	51	37	set	set	NOUN
ejpam-5618	51	38	is	be	AUX
ejpam-5618	51	39	a	a	DET
ejpam-5618	51	40	union	union	NOUN
ejpam-5618	51	41	of	of	ADP
ejpam-5618	51	42	members	member	NOUN
ejpam-5618	51	43	of	of	ADP
ejpam-5618	51	44	b.	b.	PROPN
ejpam-5618	51	45	definition	definition	NOUN
ejpam-5618	51	46	5	5	NUM
ejpam-5618	51	47	.	.	PUNCT
ejpam-5618	52	1	let	let	VERB
ejpam-5618	52	2	x	x	PRON
ejpam-5618	52	3	be	be	AUX
ejpam-5618	52	4	an	an	DET
ejpam-5618	52	5	ideal	ideal	ADJ
ejpam-5618	52	6	topological	topological	ADJ
ejpam-5618	52	7	space	space	NOUN
ejpam-5618	52	8	and	and	CCONJ
ejpam-5618	52	9	x	x	PUNCT
ejpam-5618	52	10	∈	∈	NOUN
ejpam-5618	52	11	x.	x.	NOUN
ejpam-5618	53	1	the	the	DET
ejpam-5618	53	2	collection	collection	NOUN
ejpam-5618	53	3	bx	bx	PROPN
ejpam-5618	53	4	of	of	ADP
ejpam-5618	53	5	i	i	PROPN
ejpam-5618	53	6	-	-	PUNCT
ejpam-5618	53	7	open	open	ADJ
ejpam-5618	53	8	neighborhoods	neighborhood	NOUN
ejpam-5618	53	9	of	of	ADP
ejpam-5618	53	10	x	x	PUNCT
ejpam-5618	53	11	in	in	ADP
ejpam-5618	53	12	x	x	PROPN
ejpam-5618	53	13	is	be	AUX
ejpam-5618	53	14	called	call	VERB
ejpam-5618	53	15	an	an	DET
ejpam-5618	53	16	i	i	NOUN
ejpam-5618	53	17	-	-	PUNCT
ejpam-5618	53	18	open	open	ADJ
ejpam-5618	53	19	base	base	NOUN
ejpam-5618	53	20	at	at	ADP
ejpam-5618	53	21	x	x	PUNCT
ejpam-5618	53	22	if	if	SCONJ
ejpam-5618	53	23	for	for	ADP
ejpam-5618	53	24	any	any	DET
ejpam-5618	53	25	i	i	NOUN
ejpam-5618	53	26	-	-	PUNCT
ejpam-5618	53	27	open	open	ADJ
ejpam-5618	53	28	neighborhood	neighborhood	NOUN
ejpam-5618	53	29	u	u	NOUN
ejpam-5618	53	30	of	of	ADP
ejpam-5618	53	31	x	x	SYM
ejpam-5618	53	32	,	,	PUNCT
ejpam-5618	53	33	there	there	PRON
ejpam-5618	53	34	is	be	VERB
ejpam-5618	53	35	v	v	ADP
ejpam-5618	53	36	∈	∈	NOUN
ejpam-5618	53	37	bx	bx	NOUN
ejpam-5618	54	1	such	such	ADJ
ejpam-5618	54	2	that	that	PRON
ejpam-5618	54	3	v	v	ADP
ejpam-5618	54	4	⊂	⊂	PROPN
ejpam-5618	54	5	u	u	PROPN
ejpam-5618	54	6	.	.	PUNCT
ejpam-5618	55	1	definition	definition	NOUN
ejpam-5618	55	2	6	6	NUM
ejpam-5618	55	3	.	.	PUNCT
ejpam-5618	56	1	[	[	X
ejpam-5618	56	2	1	1	X
ejpam-5618	56	3	]	]	PUNCT
ejpam-5618	56	4	a	a	DET
ejpam-5618	56	5	mapping	mapping	NOUN
ejpam-5618	56	6	f	f	NOUN
ejpam-5618	56	7	:	:	PUNCT
ejpam-5618	56	8	(	(	PUNCT
ejpam-5618	56	9	x	x	X
ejpam-5618	56	10	,	,	PUNCT
ejpam-5618	56	11	τ	τ	PROPN
ejpam-5618	56	12	,	,	PUNCT
ejpam-5618	56	13	i	i	NOUN
ejpam-5618	56	14	)	)	PUNCT
ejpam-5618	56	15	→	→	SYM
ejpam-5618	56	16	(	(	PUNCT
ejpam-5618	56	17	y	y	PROPN
ejpam-5618	56	18	,	,	PUNCT
ejpam-5618	56	19	σ	σ	PROPN
ejpam-5618	56	20	)	)	PUNCT
ejpam-5618	56	21	is	be	AUX
ejpam-5618	56	22	called	call	VERB
ejpam-5618	56	23	i	i	PRON
ejpam-5618	56	24	-	-	NOUN
ejpam-5618	56	25	continuous	continuous	ADJ
ejpam-5618	56	26	if	if	SCONJ
ejpam-5618	56	27	the	the	DET
ejpam-5618	56	28	inverse	inverse	ADJ
ejpam-5618	56	29	image	image	NOUN
ejpam-5618	56	30	of	of	ADP
ejpam-5618	56	31	any	any	DET
ejpam-5618	56	32	open	open	ADJ
ejpam-5618	56	33	set	set	NOUN
ejpam-5618	56	34	in	in	ADP
ejpam-5618	56	35	y	y	PROPN
ejpam-5618	56	36	is	be	AUX
ejpam-5618	56	37	i	i	PRON
ejpam-5618	56	38	-	-	PUNCT
ejpam-5618	56	39	open	open	ADJ
ejpam-5618	56	40	set	set	NOUN
ejpam-5618	56	41	in	in	ADP
ejpam-5618	56	42	x.	x.	NOUN
ejpam-5618	56	43	definition	definition	NOUN
ejpam-5618	56	44	7	7	NUM
ejpam-5618	56	45	.	.	PUNCT
ejpam-5618	57	1	[	[	X
ejpam-5618	57	2	1	1	X
ejpam-5618	57	3	]	]	PUNCT
ejpam-5618	57	4	a	a	DET
ejpam-5618	57	5	mapping	mapping	NOUN
ejpam-5618	57	6	f	f	NOUN
ejpam-5618	57	7	:	:	PUNCT
ejpam-5618	57	8	(	(	PUNCT
ejpam-5618	57	9	x	x	X
ejpam-5618	57	10	,	,	PUNCT
ejpam-5618	57	11	τ	τ	PROPN
ejpam-5618	57	12	,	,	PUNCT
ejpam-5618	57	13	i	i	NOUN
ejpam-5618	57	14	)	)	PUNCT
ejpam-5618	57	15	−→	−→	NOUN
ejpam-5618	57	16	(	(	PUNCT
ejpam-5618	57	17	y	y	PROPN
ejpam-5618	57	18	,	,	PUNCT
ejpam-5618	57	19	σ	σ	PROPN
ejpam-5618	57	20	)	)	PUNCT
ejpam-5618	57	21	is	be	AUX
ejpam-5618	57	22	said	say	VERB
ejpam-5618	57	23	to	to	PART
ejpam-5618	57	24	be	be	AUX
ejpam-5618	57	25	i	i	NOUN
ejpam-5618	57	26	-	-	NOUN
ejpam-5618	57	27	continuous	continuous	ADJ
ejpam-5618	57	28	at	at	ADP
ejpam-5618	57	29	a	a	DET
ejpam-5618	57	30	point	point	NOUN
ejpam-5618	57	31	x	x	PUNCT
ejpam-5618	57	32	in	in	ADP
ejpam-5618	57	33	x	x	PRON
ejpam-5618	57	34	if	if	SCONJ
ejpam-5618	57	35	for	for	ADP
ejpam-5618	57	36	each	each	DET
ejpam-5618	57	37	open	open	ADJ
ejpam-5618	57	38	neighborhood	neighborhood	NOUN
ejpam-5618	57	39	v	v	NOUN
ejpam-5618	57	40	of	of	ADP
ejpam-5618	57	41	f(x	f(x	PROPN
ejpam-5618	57	42	)	)	PUNCT
ejpam-5618	57	43	in	in	ADP
ejpam-5618	57	44	y	y	PROPN
ejpam-5618	57	45	,	,	PUNCT
ejpam-5618	57	46	there	there	PRON
ejpam-5618	57	47	is	be	VERB
ejpam-5618	57	48	an	an	DET
ejpam-5618	57	49	i	i	NOUN
ejpam-5618	57	50	-	-	PUNCT
ejpam-5618	57	51	open	open	ADJ
ejpam-5618	57	52	neighborhood	neighborhood	NOUN
ejpam-5618	57	53	u	u	NOUN
ejpam-5618	57	54	of	of	ADP
ejpam-5618	57	55	x	x	PUNCT
ejpam-5618	57	56	in	in	ADP
ejpam-5618	57	57	x	x	X
ejpam-5618	57	58	such	such	ADJ
ejpam-5618	57	59	that	that	DET
ejpam-5618	57	60	f(u	f(u	PROPN
ejpam-5618	57	61	)	)	PUNCT
ejpam-5618	57	62	⊂	⊂	PROPN
ejpam-5618	57	63	v	v	PROPN
ejpam-5618	57	64	.	.	PUNCT
ejpam-5618	58	1	theorem	theorem	NOUN
ejpam-5618	58	2	1	1	NUM
ejpam-5618	58	3	.	.	PUNCT
ejpam-5618	59	1	[	[	X
ejpam-5618	59	2	1	1	X
ejpam-5618	59	3	]	]	PUNCT
ejpam-5618	59	4	let	let	VERB
ejpam-5618	59	5	f	f	X
ejpam-5618	59	6	:	:	PUNCT
ejpam-5618	59	7	(	(	PUNCT
ejpam-5618	59	8	x	x	X
ejpam-5618	59	9	,	,	PUNCT
ejpam-5618	59	10	τ	τ	PROPN
ejpam-5618	59	11	,	,	PUNCT
ejpam-5618	59	12	i	i	NOUN
ejpam-5618	59	13	)	)	PUNCT
ejpam-5618	59	14	→	→	SYM
ejpam-5618	59	15	(	(	PUNCT
ejpam-5618	59	16	y	y	PROPN
ejpam-5618	59	17	,	,	PUNCT
ejpam-5618	59	18	σ	σ	PROPN
ejpam-5618	59	19	)	)	PUNCT
ejpam-5618	59	20	be	be	AUX
ejpam-5618	59	21	a	a	DET
ejpam-5618	59	22	mapping	mapping	NOUN
ejpam-5618	59	23	.	.	PUNCT
ejpam-5618	60	1	then	then	ADV
ejpam-5618	60	2	the	the	DET
ejpam-5618	60	3	following	follow	VERB
ejpam-5618	60	4	are	be	AUX
ejpam-5618	60	5	equivalent	equivalent	ADJ
ejpam-5618	60	6	:	:	PUNCT
ejpam-5618	60	7	(	(	PUNCT
ejpam-5618	60	8	i	i	NOUN
ejpam-5618	60	9	)	)	PUNCT
ejpam-5618	60	10	f	f	PROPN
ejpam-5618	60	11	is	be	AUX
ejpam-5618	60	12	i	i	PRON
ejpam-5618	60	13	-	-	PUNCT
ejpam-5618	60	14	continuous	continuous	ADJ
ejpam-5618	60	15	.	.	PUNCT
ejpam-5618	61	1	s.	s.	PROPN
ejpam-5618	61	2	alammar	alammar	PROPN
ejpam-5618	61	3	,	,	PUNCT
ejpam-5618	61	4	m.	m.	NOUN
ejpam-5618	61	5	al	al	PROPN
ejpam-5618	61	6	shumrani	shumrani	PROPN
ejpam-5618	61	7	,	,	PUNCT
ejpam-5618	61	8	c.	c.	PROPN
ejpam-5618	61	9	özel	özel	PROPN
ejpam-5618	61	10	/	/	SYM
ejpam-5618	61	11	eur	eur	PROPN
ejpam-5618	61	12	.	.	PUNCT
ejpam-5618	62	1	j.	j.	PROPN
ejpam-5618	62	2	pure	pure	PROPN
ejpam-5618	62	3	appl	appl	PROPN
ejpam-5618	62	4	.	.	PROPN
ejpam-5618	62	5	math	math	PROPN
ejpam-5618	62	6	,	,	PUNCT
ejpam-5618	62	7	18	18	NUM
ejpam-5618	62	8	(	(	PUNCT
ejpam-5618	62	9	1	1	NUM
ejpam-5618	62	10	)	)	PUNCT
ejpam-5618	62	11	(	(	PUNCT
ejpam-5618	62	12	2025	2025	NUM
ejpam-5618	62	13	)	)	PUNCT
ejpam-5618	62	14	,	,	PUNCT
ejpam-5618	62	15	5618	5618	NUM
ejpam-5618	62	16	3	3	NUM
ejpam-5618	62	17	of	of	ADP
ejpam-5618	62	18	13	13	NUM
ejpam-5618	62	19	(	(	PUNCT
ejpam-5618	62	20	ii	ii	NOUN
ejpam-5618	62	21	)	)	PUNCT
ejpam-5618	62	22	f	f	PROPN
ejpam-5618	62	23	is	be	AUX
ejpam-5618	62	24	i	i	PRON
ejpam-5618	62	25	-	-	ADJ
ejpam-5618	62	26	continuous	continuous	ADJ
ejpam-5618	62	27	at	at	ADP
ejpam-5618	62	28	each	each	DET
ejpam-5618	62	29	point	point	NOUN
ejpam-5618	62	30	x	x	PUNCT
ejpam-5618	62	31	in	in	ADP
ejpam-5618	62	32	x.	x.	NOUN
ejpam-5618	62	33	definition	definition	NOUN
ejpam-5618	62	34	8	8	NUM
ejpam-5618	62	35	.	.	PUNCT
ejpam-5618	63	1	[	[	X
ejpam-5618	63	2	1	1	X
ejpam-5618	63	3	]	]	PUNCT
ejpam-5618	63	4	a	a	DET
ejpam-5618	63	5	mapping	mapping	NOUN
ejpam-5618	63	6	f	f	NOUN
ejpam-5618	63	7	:	:	PUNCT
ejpam-5618	63	8	(	(	PUNCT
ejpam-5618	63	9	x	x	X
ejpam-5618	63	10	,	,	PUNCT
ejpam-5618	63	11	τ	τ	NOUN
ejpam-5618	63	12	)	)	PUNCT
ejpam-5618	63	13	−→	−→	NOUN
ejpam-5618	63	14	(	(	PUNCT
ejpam-5618	63	15	y	y	PROPN
ejpam-5618	63	16	,	,	PUNCT
ejpam-5618	63	17	σ	σ	PROPN
ejpam-5618	63	18	,	,	PUNCT
ejpam-5618	63	19	i	i	PROPN
ejpam-5618	63	20	)	)	PUNCT
ejpam-5618	63	21	is	be	AUX
ejpam-5618	63	22	called	call	VERB
ejpam-5618	63	23	i	i	PRON
ejpam-5618	63	24	-	-	PUNCT
ejpam-5618	63	25	open	open	ADJ
ejpam-5618	63	26	if	if	SCONJ
ejpam-5618	63	27	for	for	ADP
ejpam-5618	63	28	each	each	DET
ejpam-5618	63	29	open	open	ADJ
ejpam-5618	63	30	set	set	VERB
ejpam-5618	63	31	u	u	NOUN
ejpam-5618	63	32	in	in	ADP
ejpam-5618	63	33	x	x	PROPN
ejpam-5618	63	34	,	,	PUNCT
ejpam-5618	63	35	f(u	f(u	PROPN
ejpam-5618	63	36	)	)	PUNCT
ejpam-5618	63	37	is	be	AUX
ejpam-5618	63	38	i	i	PRON
ejpam-5618	63	39	-	-	PUNCT
ejpam-5618	63	40	open	open	ADJ
ejpam-5618	63	41	in	in	ADP
ejpam-5618	63	42	y	y	PROPN
ejpam-5618	63	43	.	.	PUNCT
ejpam-5618	64	1	definition	definition	NOUN
ejpam-5618	64	2	9	9	NUM
ejpam-5618	64	3	.	.	PUNCT
ejpam-5618	65	1	[	[	X
ejpam-5618	65	2	5	5	NUM
ejpam-5618	65	3	]	]	PUNCT
ejpam-5618	65	4	a	a	DET
ejpam-5618	65	5	mapping	mapping	NOUN
ejpam-5618	65	6	f	f	NOUN
ejpam-5618	65	7	:	:	PUNCT
ejpam-5618	65	8	(	(	PUNCT
ejpam-5618	65	9	x	x	X
ejpam-5618	65	10	,	,	PUNCT
ejpam-5618	65	11	τ	τ	PROPN
ejpam-5618	65	12	,	,	PUNCT
ejpam-5618	65	13	i	i	NOUN
ejpam-5618	65	14	)	)	PUNCT
ejpam-5618	65	15	−→	−→	NOUN
ejpam-5618	65	16	(	(	PUNCT
ejpam-5618	65	17	y	y	PROPN
ejpam-5618	65	18	,	,	PUNCT
ejpam-5618	65	19	σ	σ	PROPN
ejpam-5618	65	20	,	,	PUNCT
ejpam-5618	65	21	j	j	PROPN
ejpam-5618	65	22	)	)	PUNCT
ejpam-5618	65	23	is	be	AUX
ejpam-5618	65	24	called	call	VERB
ejpam-5618	65	25	i	i	PRON
ejpam-5618	65	26	-	-	PUNCT
ejpam-5618	65	27	irresolute	irresolute	ADJ
ejpam-5618	65	28	if	if	SCONJ
ejpam-5618	65	29	the	the	DET
ejpam-5618	65	30	inverse	inverse	ADJ
ejpam-5618	65	31	image	image	NOUN
ejpam-5618	65	32	of	of	ADP
ejpam-5618	65	33	each	each	DET
ejpam-5618	65	34	j	j	PROPN
ejpam-5618	65	35	-open	-open	PROPN
ejpam-5618	65	36	set	set	VERB
ejpam-5618	65	37	in	in	ADP
ejpam-5618	65	38	y	y	PROPN
ejpam-5618	65	39	is	be	AUX
ejpam-5618	65	40	i	i	PRON
ejpam-5618	65	41	-	-	PUNCT
ejpam-5618	65	42	open	open	ADJ
ejpam-5618	65	43	set	set	NOUN
ejpam-5618	65	44	in	in	ADP
ejpam-5618	65	45	x.	x.	NOUN
ejpam-5618	65	46	definition	definition	NOUN
ejpam-5618	65	47	10	10	NUM
ejpam-5618	65	48	.	.	PUNCT
ejpam-5618	66	1	[	[	X
ejpam-5618	66	2	8	8	NUM
ejpam-5618	66	3	]	]	PUNCT
ejpam-5618	66	4	a	a	DET
ejpam-5618	66	5	bijective	bijective	ADJ
ejpam-5618	66	6	mapping	mapping	NOUN
ejpam-5618	66	7	f	f	NOUN
ejpam-5618	66	8	:	:	PUNCT
ejpam-5618	66	9	(	(	PUNCT
ejpam-5618	66	10	x	x	X
ejpam-5618	66	11	,	,	PUNCT
ejpam-5618	66	12	τ	τ	PROPN
ejpam-5618	66	13	,	,	PUNCT
ejpam-5618	66	14	i	i	NOUN
ejpam-5618	66	15	)	)	PUNCT
ejpam-5618	66	16	−→	−→	NOUN
ejpam-5618	66	17	(	(	PUNCT
ejpam-5618	66	18	y	y	PROPN
ejpam-5618	66	19	,	,	PUNCT
ejpam-5618	66	20	σ	σ	PROPN
ejpam-5618	66	21	,	,	PUNCT
ejpam-5618	66	22	j	j	PROPN
ejpam-5618	66	23	)	)	PUNCT
ejpam-5618	66	24	is	be	AUX
ejpam-5618	66	25	called	call	VERB
ejpam-5618	66	26	i	i	PRON
ejpam-5618	66	27	-	-	PUNCT
ejpam-5618	66	28	homeomorphism	homeomorphism	PROPN
ejpam-5618	66	29	if	if	SCONJ
ejpam-5618	66	30	f	f	PROPN
ejpam-5618	66	31	and	and	CCONJ
ejpam-5618	66	32	the	the	DET
ejpam-5618	66	33	inverse	inverse	NOUN
ejpam-5618	66	34	mapping	mapping	NOUN
ejpam-5618	66	35	f−1	f−1	PROPN
ejpam-5618	66	36	are	be	AUX
ejpam-5618	66	37	i	i	PRON
ejpam-5618	66	38	-	-	PUNCT
ejpam-5618	66	39	continuous	continuous	ADJ
ejpam-5618	66	40	.	.	PUNCT
ejpam-5618	67	1	definition	definition	NOUN
ejpam-5618	67	2	11	11	NUM
ejpam-5618	67	3	.	.	PUNCT
ejpam-5618	68	1	[	[	X
ejpam-5618	68	2	8	8	X
ejpam-5618	68	3	]	]	X
ejpam-5618	68	4	an	an	DET
ejpam-5618	68	5	ideal	ideal	ADJ
ejpam-5618	68	6	topological	topological	ADJ
ejpam-5618	68	7	space	space	NOUN
ejpam-5618	68	8	(	(	PUNCT
ejpam-5618	68	9	x	x	X
ejpam-5618	68	10	,	,	PUNCT
ejpam-5618	68	11	τ	τ	PROPN
ejpam-5618	68	12	,	,	PUNCT
ejpam-5618	68	13	i	i	PROPN
ejpam-5618	68	14	)	)	PUNCT
ejpam-5618	68	15	is	be	AUX
ejpam-5618	68	16	said	say	VERB
ejpam-5618	68	17	to	to	PART
ejpam-5618	68	18	be	be	AUX
ejpam-5618	68	19	i	i	NOUN
ejpam-5618	68	20	-	-	PUNCT
ejpam-5618	68	21	homogeneous	homogeneous	ADJ
ejpam-5618	68	22	if	if	SCONJ
ejpam-5618	68	23	for	for	ADP
ejpam-5618	68	24	all	all	DET
ejpam-5618	68	25	x	x	NOUN
ejpam-5618	68	26	,	,	PUNCT
ejpam-5618	68	27	y	y	PROPN
ejpam-5618	68	28	∈	∈	PROPN
ejpam-5618	68	29	x	x	PUNCT
ejpam-5618	68	30	there	there	PRON
ejpam-5618	68	31	is	be	VERB
ejpam-5618	68	32	an	an	DET
ejpam-5618	68	33	i	i	PROPN
ejpam-5618	68	34	-	-	PUNCT
ejpam-5618	68	35	homeomorphism	homeomorphism	PROPN
ejpam-5618	68	36	f	f	PROPN
ejpam-5618	68	37	of	of	ADP
ejpam-5618	68	38	the	the	DET
ejpam-5618	68	39	space	space	NOUN
ejpam-5618	68	40	x	x	X
ejpam-5618	68	41	onto	onto	ADP
ejpam-5618	68	42	itself	itself	PRON
ejpam-5618	68	43	such	such	ADJ
ejpam-5618	68	44	that	that	SCONJ
ejpam-5618	68	45	f(x	f(x	NOUN
ejpam-5618	68	46	)	)	PUNCT
ejpam-5618	69	1	=	=	PUNCT
ejpam-5618	70	1	y.	y.	NOUN
ejpam-5618	70	2	definition	definition	NOUN
ejpam-5618	70	3	12	12	NUM
ejpam-5618	70	4	.	.	PUNCT
ejpam-5618	71	1	an	an	DET
ejpam-5618	71	2	ideal	ideal	ADJ
ejpam-5618	71	3	topological	topological	ADJ
ejpam-5618	71	4	space	space	NOUN
ejpam-5618	71	5	(	(	PUNCT
ejpam-5618	71	6	x	x	X
ejpam-5618	71	7	,	,	PUNCT
ejpam-5618	71	8	τ	τ	PROPN
ejpam-5618	71	9	,	,	PUNCT
ejpam-5618	71	10	i	i	PROPN
ejpam-5618	71	11	)	)	PUNCT
ejpam-5618	71	12	is	be	AUX
ejpam-5618	71	13	said	say	VERB
ejpam-5618	71	14	to	to	PART
ejpam-5618	71	15	be	be	AUX
ejpam-5618	71	16	i	i	NOUN
ejpam-5618	71	17	t1	t1	NOUN
ejpam-5618	71	18	-	-	PUNCT
ejpam-5618	71	19	space	space	NOUN
ejpam-5618	71	20	if	if	SCONJ
ejpam-5618	71	21	given	give	VERB
ejpam-5618	71	22	any	any	DET
ejpam-5618	71	23	two	two	NUM
ejpam-5618	71	24	distinct	distinct	ADJ
ejpam-5618	71	25	points	point	NOUN
ejpam-5618	71	26	x	x	NOUN
ejpam-5618	71	27	,	,	PUNCT
ejpam-5618	71	28	y	y	PROPN
ejpam-5618	71	29	∈	∈	PROPN
ejpam-5618	71	30	x	x	PRON
ejpam-5618	71	31	,	,	PUNCT
ejpam-5618	71	32	there	there	PRON
ejpam-5618	71	33	are	be	VERB
ejpam-5618	71	34	two	two	NUM
ejpam-5618	71	35	i	i	NOUN
ejpam-5618	71	36	-	-	PUNCT
ejpam-5618	71	37	open	open	ADJ
ejpam-5618	71	38	sets	set	VERB
ejpam-5618	71	39	u	u	NOUN
ejpam-5618	71	40	and	and	CCONJ
ejpam-5618	71	41	v	v	ADP
ejpam-5618	71	42	containing	contain	VERB
ejpam-5618	71	43	x	x	PROPN
ejpam-5618	71	44	and	and	CCONJ
ejpam-5618	71	45	y	y	PROPN
ejpam-5618	71	46	,	,	PUNCT
ejpam-5618	71	47	respectively	respectively	ADV
ejpam-5618	71	48	,	,	PUNCT
ejpam-5618	71	49	such	such	ADJ
ejpam-5618	71	50	that	that	SCONJ
ejpam-5618	71	51	y	y	PROPN
ejpam-5618	71	52	/∈	/∈	PUNCT
ejpam-5618	71	53	u	u	PROPN
ejpam-5618	71	54	and	and	CCONJ
ejpam-5618	71	55	x	x	NOUN
ejpam-5618	71	56	/∈	/∈	NOUN
ejpam-5618	71	57	v	v	INTJ
ejpam-5618	71	58	.	.	PUNCT
ejpam-5618	72	1	we	we	PRON
ejpam-5618	72	2	recall	recall	VERB
ejpam-5618	72	3	that	that	SCONJ
ejpam-5618	72	4	if	if	SCONJ
ejpam-5618	72	5	(	(	PUNCT
ejpam-5618	72	6	x	x	NOUN
ejpam-5618	72	7	,	,	PUNCT
ejpam-5618	72	8	τ	τ	PROPN
ejpam-5618	72	9	,	,	PUNCT
ejpam-5618	72	10	i	i	PROPN
ejpam-5618	72	11	)	)	PUNCT
ejpam-5618	72	12	is	be	AUX
ejpam-5618	72	13	an	an	DET
ejpam-5618	72	14	ideal	ideal	ADJ
ejpam-5618	72	15	topological	topological	ADJ
ejpam-5618	72	16	space	space	NOUN
ejpam-5618	72	17	,	,	PUNCT
ejpam-5618	72	18	then	then	ADV
ejpam-5618	72	19	open	open	VERB
ejpam-5618	72	20	sets	set	NOUN
ejpam-5618	72	21	and	and	CCONJ
ejpam-5618	72	22	i	i	PRON
ejpam-5618	72	23	-	-	PUNCT
ejpam-5618	72	24	open	open	ADJ
ejpam-5618	72	25	sets	set	NOUN
ejpam-5618	72	26	in	in	ADP
ejpam-5618	72	27	x	x	SYM
ejpam-5618	72	28	are	be	AUX
ejpam-5618	72	29	independent	independent	ADJ
ejpam-5618	72	30	[	[	X
ejpam-5618	72	31	1	1	NUM
ejpam-5618	72	32	]	]	PUNCT
ejpam-5618	72	33	.	.	PUNCT
ejpam-5618	73	1	however	however	ADV
ejpam-5618	73	2	,	,	PUNCT
ejpam-5618	73	3	if	if	SCONJ
ejpam-5618	73	4	x	x	PRON
ejpam-5618	73	5	is	be	AUX
ejpam-5618	73	6	an	an	DET
ejpam-5618	73	7	i	i	NOUN
ejpam-5618	73	8	t1	t1	NOUN
ejpam-5618	73	9	-	-	NOUN
ejpam-5618	73	10	space	space	NOUN
ejpam-5618	73	11	,	,	PUNCT
ejpam-5618	73	12	then	then	ADV
ejpam-5618	73	13	we	we	PRON
ejpam-5618	73	14	have	have	VERB
ejpam-5618	73	15	the	the	DET
ejpam-5618	73	16	following	follow	VERB
ejpam-5618	73	17	lemma	lemma	PROPN
ejpam-5618	73	18	which	which	PRON
ejpam-5618	73	19	we	we	PRON
ejpam-5618	73	20	shall	shall	AUX
ejpam-5618	73	21	use	use	VERB
ejpam-5618	73	22	in	in	ADP
ejpam-5618	73	23	next	next	ADJ
ejpam-5618	73	24	section	section	NOUN
ejpam-5618	73	25	.	.	PUNCT
ejpam-5618	74	1	we	we	PRON
ejpam-5618	74	2	do	do	AUX
ejpam-5618	74	3	not	not	PART
ejpam-5618	74	4	have	have	VERB
ejpam-5618	74	5	a	a	DET
ejpam-5618	74	6	reference	reference	NOUN
ejpam-5618	74	7	for	for	ADP
ejpam-5618	74	8	this	this	DET
ejpam-5618	74	9	lemma	lemma	PROPN
ejpam-5618	74	10	and	and	CCONJ
ejpam-5618	74	11	thus	thus	ADV
ejpam-5618	74	12	we	we	PRON
ejpam-5618	74	13	give	give	VERB
ejpam-5618	74	14	a	a	DET
ejpam-5618	74	15	proof	proof	NOUN
ejpam-5618	74	16	of	of	ADP
ejpam-5618	74	17	it	it	PRON
ejpam-5618	74	18	.	.	PUNCT
ejpam-5618	75	1	lemma	lemma	PROPN
ejpam-5618	75	2	1	1	X
ejpam-5618	75	3	.	.	PUNCT
ejpam-5618	76	1	let	let	VERB
ejpam-5618	76	2	x	x	PRON
ejpam-5618	76	3	be	be	AUX
ejpam-5618	76	4	an	an	DET
ejpam-5618	76	5	it1	it1	PROPN
ejpam-5618	76	6	ideal	ideal	ADJ
ejpam-5618	76	7	topological	topological	ADJ
ejpam-5618	76	8	space	space	NOUN
ejpam-5618	76	9	.	.	PUNCT
ejpam-5618	77	1	then	then	ADV
ejpam-5618	77	2	every	every	DET
ejpam-5618	77	3	nonempty	nonempty	ADV
ejpam-5618	77	4	open	open	NOUN
ejpam-5618	77	5	set	set	VERB
ejpam-5618	77	6	a	a	PRON
ejpam-5618	77	7	of	of	ADP
ejpam-5618	77	8	x	x	SYM
ejpam-5618	77	9	is	be	AUX
ejpam-5618	77	10	i	i	PRON
ejpam-5618	77	11	-	-	PUNCT
ejpam-5618	77	12	open	open	ADJ
ejpam-5618	77	13	set	set	NOUN
ejpam-5618	77	14	.	.	PUNCT
ejpam-5618	78	1	proof	proof	NOUN
ejpam-5618	78	2	.	.	PUNCT
ejpam-5618	79	1	fix	fix	VERB
ejpam-5618	79	2	x	x	X
ejpam-5618	79	3	∈	∈	NOUN
ejpam-5618	79	4	x	x	X
ejpam-5618	79	5	such	such	ADJ
ejpam-5618	79	6	that	that	SCONJ
ejpam-5618	79	7	x	x	X
ejpam-5618	79	8	/∈	/∈	PUNCT
ejpam-5618	79	9	a.	a.	NOUN
ejpam-5618	79	10	since	since	SCONJ
ejpam-5618	79	11	x	x	PRON
ejpam-5618	79	12	is	be	AUX
ejpam-5618	79	13	an	an	DET
ejpam-5618	79	14	it1	it1	NOUN
ejpam-5618	79	15	-	-	PUNCT
ejpam-5618	79	16	space	space	NOUN
ejpam-5618	79	17	,	,	PUNCT
ejpam-5618	79	18	then	then	ADV
ejpam-5618	79	19	for	for	ADP
ejpam-5618	79	20	each	each	DET
ejpam-5618	79	21	y	y	PROPN
ejpam-5618	79	22	∈	∈	PROPN
ejpam-5618	79	23	a	a	PRON
ejpam-5618	79	24	there	there	PRON
ejpam-5618	79	25	are	be	VERB
ejpam-5618	79	26	i	i	NOUN
ejpam-5618	79	27	-	-	PUNCT
ejpam-5618	79	28	open	open	ADJ
ejpam-5618	79	29	sets	set	VERB
ejpam-5618	79	30	u	u	NOUN
ejpam-5618	79	31	and	and	CCONJ
ejpam-5618	79	32	vy	vy	ADV
ejpam-5618	79	33	containing	contain	VERB
ejpam-5618	79	34	x	x	PROPN
ejpam-5618	79	35	and	and	CCONJ
ejpam-5618	79	36	y	y	PROPN
ejpam-5618	79	37	,	,	PUNCT
ejpam-5618	79	38	respectively	respectively	ADV
ejpam-5618	79	39	,	,	PUNCT
ejpam-5618	79	40	such	such	ADJ
ejpam-5618	79	41	that	that	SCONJ
ejpam-5618	79	42	y	y	PROPN
ejpam-5618	79	43	/∈	/∈	PUNCT
ejpam-5618	79	44	u	u	PROPN
ejpam-5618	79	45	and	and	CCONJ
ejpam-5618	79	46	x	x	PROPN
ejpam-5618	79	47	/∈	/∈	PUNCT
ejpam-5618	80	1	vy	vy	X
ejpam-5618	80	2	.	.	PUNCT
ejpam-5618	81	1	let	let	VERB
ejpam-5618	81	2	wy	wy	PROPN
ejpam-5618	81	3	=	=	PUNCT
ejpam-5618	81	4	vy	vy	PROPN
ejpam-5618	81	5	∩	∩	PROPN
ejpam-5618	81	6	a.	a.	NOUN
ejpam-5618	81	7	thus	thus	ADV
ejpam-5618	81	8	,	,	PUNCT
ejpam-5618	81	9	wy	wy	PROPN
ejpam-5618	81	10	is	be	AUX
ejpam-5618	81	11	i	i	PRON
ejpam-5618	81	12	-	-	PUNCT
ejpam-5618	81	13	open	open	ADJ
ejpam-5618	81	14	and	and	CCONJ
ejpam-5618	81	15	wy	wy	PROPN
ejpam-5618	81	16	⊂	⊂	PROPN
ejpam-5618	81	17	a.	a.	NOUN
ejpam-5618	81	18	let	let	VERB
ejpam-5618	81	19	w	w	PROPN
ejpam-5618	81	20	=	=	SYM
ejpam-5618	81	21	∪y∈awy	∪y∈awy	PROPN
ejpam-5618	81	22	.	.	PUNCT
ejpam-5618	82	1	clearly	clearly	ADV
ejpam-5618	82	2	,	,	PUNCT
ejpam-5618	82	3	w	w	PROPN
ejpam-5618	82	4	is	be	AUX
ejpam-5618	82	5	i	i	PRON
ejpam-5618	82	6	-	-	PUNCT
ejpam-5618	82	7	open	open	ADJ
ejpam-5618	82	8	being	be	AUX
ejpam-5618	82	9	the	the	DET
ejpam-5618	82	10	union	union	NOUN
ejpam-5618	82	11	of	of	ADP
ejpam-5618	82	12	i	i	PROPN
ejpam-5618	82	13	-	-	PUNCT
ejpam-5618	82	14	open	open	ADJ
ejpam-5618	82	15	sets	set	NOUN
ejpam-5618	82	16	.	.	PUNCT
ejpam-5618	83	1	but	but	CCONJ
ejpam-5618	83	2	w	w	NOUN
ejpam-5618	83	3	=	=	NOUN
ejpam-5618	83	4	a.	a.	NOUN
ejpam-5618	83	5	hence	hence	ADV
ejpam-5618	83	6	,	,	PUNCT
ejpam-5618	83	7	a	a	PRON
ejpam-5618	83	8	is	be	AUX
ejpam-5618	83	9	i	i	PRON
ejpam-5618	83	10	-	-	PUNCT
ejpam-5618	83	11	open	open	ADJ
ejpam-5618	83	12	.	.	PUNCT
ejpam-5618	84	1	definition	definition	NOUN
ejpam-5618	84	2	13	13	NUM
ejpam-5618	84	3	.	.	PUNCT
ejpam-5618	85	1	[	[	X
ejpam-5618	85	2	3	3	X
ejpam-5618	85	3	]	]	PUNCT
ejpam-5618	85	4	a	a	DET
ejpam-5618	85	5	topological	topological	ADJ
ejpam-5618	85	6	space	space	NOUN
ejpam-5618	85	7	x	x	PUNCT
ejpam-5618	85	8	is	be	AUX
ejpam-5618	85	9	submaximal	submaximal	ADJ
ejpam-5618	85	10	space	space	NOUN
ejpam-5618	85	11	if	if	SCONJ
ejpam-5618	85	12	it	it	PRON
ejpam-5618	85	13	satisfies	satisfy	VERB
ejpam-5618	85	14	one	one	NUM
ejpam-5618	85	15	of	of	ADP
ejpam-5618	85	16	the	the	DET
ejpam-5618	85	17	following	follow	VERB
ejpam-5618	85	18	equivalent	equivalent	ADJ
ejpam-5618	85	19	conditions	condition	NOUN
ejpam-5618	85	20	:	:	PUNCT
ejpam-5618	85	21	(	(	PUNCT
ejpam-5618	85	22	i	i	NOUN
ejpam-5618	85	23	)	)	PUNCT
ejpam-5618	85	24	every	every	DET
ejpam-5618	85	25	subset	subset	NOUN
ejpam-5618	85	26	of	of	ADP
ejpam-5618	85	27	it	it	PRON
ejpam-5618	85	28	is	be	AUX
ejpam-5618	85	29	locally	locally	ADV
ejpam-5618	85	30	closed	closed	ADJ
ejpam-5618	85	31	,	,	PUNCT
ejpam-5618	85	32	that	that	ADV
ejpam-5618	85	33	is	is	ADV
ejpam-5618	85	34	,	,	PUNCT
ejpam-5618	85	35	an	an	DET
ejpam-5618	85	36	intersection	intersection	NOUN
ejpam-5618	85	37	of	of	ADP
ejpam-5618	85	38	an	an	DET
ejpam-5618	85	39	open	open	ADJ
ejpam-5618	85	40	subset	subset	NOUN
ejpam-5618	85	41	and	and	CCONJ
ejpam-5618	85	42	a	a	DET
ejpam-5618	85	43	closed	closed	ADJ
ejpam-5618	85	44	subset	subset	NOUN
ejpam-5618	85	45	.	.	PUNCT
ejpam-5618	86	1	(	(	PUNCT
ejpam-5618	86	2	ii	ii	NOUN
ejpam-5618	86	3	)	)	PUNCT
ejpam-5618	86	4	every	every	DET
ejpam-5618	86	5	dense	dense	ADJ
ejpam-5618	86	6	subset	subset	NOUN
ejpam-5618	86	7	is	be	AUX
ejpam-5618	86	8	open	open	ADJ
ejpam-5618	86	9	.	.	PUNCT
ejpam-5618	87	1	(	(	PUNCT
ejpam-5618	87	2	iii	iii	X
ejpam-5618	87	3	)	)	PUNCT
ejpam-5618	87	4	every	every	DET
ejpam-5618	87	5	preopen	preopen	NOUN
ejpam-5618	87	6	subset	subset	NOUN
ejpam-5618	87	7	is	be	AUX
ejpam-5618	87	8	open	open	ADJ
ejpam-5618	87	9	.	.	PUNCT
ejpam-5618	88	1	remark	remark	NOUN
ejpam-5618	88	2	2	2	NUM
ejpam-5618	88	3	.	.	X
ejpam-5618	88	4	observe	observe	VERB
ejpam-5618	88	5	that	that	SCONJ
ejpam-5618	88	6	if	if	SCONJ
ejpam-5618	88	7	(	(	PUNCT
ejpam-5618	88	8	x	x	NOUN
ejpam-5618	88	9	,	,	PUNCT
ejpam-5618	88	10	τ	τ	PROPN
ejpam-5618	88	11	,	,	PUNCT
ejpam-5618	88	12	i	i	PROPN
ejpam-5618	88	13	)	)	PUNCT
ejpam-5618	88	14	is	be	AUX
ejpam-5618	88	15	an	an	DET
ejpam-5618	88	16	ideal	ideal	ADJ
ejpam-5618	88	17	topological	topological	ADJ
ejpam-5618	88	18	space	space	NOUN
ejpam-5618	88	19	,	,	PUNCT
ejpam-5618	88	20	then	then	ADV
ejpam-5618	88	21	every	every	DET
ejpam-5618	88	22	i	i	NOUN
ejpam-5618	88	23	-	-	PUNCT
ejpam-5618	88	24	open	open	ADJ
ejpam-5618	88	25	set	set	NOUN
ejpam-5618	88	26	is	be	AUX
ejpam-5618	88	27	preopen	preopen	ADJ
ejpam-5618	88	28	in	in	ADP
ejpam-5618	88	29	x	x	PUNCT
ejpam-5618	89	1	[	[	X
ejpam-5618	89	2	1	1	NUM
ejpam-5618	89	3	]	]	PUNCT
ejpam-5618	89	4	.	.	PUNCT
ejpam-5618	90	1	and	and	CCONJ
ejpam-5618	90	2	if	if	SCONJ
ejpam-5618	90	3	x	x	PRON
ejpam-5618	90	4	is	be	AUX
ejpam-5618	90	5	submaximal	submaximal	ADJ
ejpam-5618	90	6	,	,	PUNCT
ejpam-5618	90	7	then	then	ADV
ejpam-5618	90	8	every	every	DET
ejpam-5618	90	9	i	i	NOUN
ejpam-5618	90	10	-	-	PUNCT
ejpam-5618	90	11	open	open	ADJ
ejpam-5618	90	12	set	set	NOUN
ejpam-5618	90	13	in	in	ADP
ejpam-5618	90	14	x	x	PUNCT
ejpam-5618	90	15	is	be	AUX
ejpam-5618	90	16	open	open	ADJ
ejpam-5618	90	17	.	.	PUNCT
ejpam-5618	91	1	theorem	theorem	NOUN
ejpam-5618	91	2	2	2	NUM
ejpam-5618	91	3	.	.	PUNCT
ejpam-5618	92	1	[	[	X
ejpam-5618	92	2	1	1	X
ejpam-5618	92	3	]	]	PUNCT
ejpam-5618	92	4	let	let	VERB
ejpam-5618	92	5	f	f	X
ejpam-5618	92	6	:	:	PUNCT
ejpam-5618	92	7	(	(	PUNCT
ejpam-5618	92	8	x	x	X
ejpam-5618	92	9	,	,	PUNCT
ejpam-5618	92	10	τ	τ	PROPN
ejpam-5618	92	11	,	,	PUNCT
ejpam-5618	92	12	i	i	NOUN
ejpam-5618	92	13	)	)	PUNCT
ejpam-5618	92	14	→	→	SYM
ejpam-5618	92	15	(	(	PUNCT
ejpam-5618	92	16	y	y	PROPN
ejpam-5618	92	17	,	,	PUNCT
ejpam-5618	92	18	σ	σ	PROPN
ejpam-5618	92	19	)	)	PUNCT
ejpam-5618	92	20	and	and	CCONJ
ejpam-5618	92	21	g	g	NOUN
ejpam-5618	92	22	:	:	PUNCT
ejpam-5618	92	23	(	(	PUNCT
ejpam-5618	92	24	y	y	PROPN
ejpam-5618	92	25	,	,	PUNCT
ejpam-5618	92	26	σ	σ	PROPN
ejpam-5618	92	27	,	,	PUNCT
ejpam-5618	92	28	j	j	PROPN
ejpam-5618	92	29	)	)	PUNCT
ejpam-5618	92	30	→	→	PUNCT
ejpam-5618	92	31	(	(	PUNCT
ejpam-5618	92	32	z	z	NOUN
ejpam-5618	92	33	,	,	PUNCT
ejpam-5618	92	34	µ	µ	NOUN
ejpam-5618	92	35	)	)	PUNCT
ejpam-5618	92	36	be	be	VERB
ejpam-5618	92	37	two	two	NUM
ejpam-5618	92	38	mappings	mapping	NOUN
ejpam-5618	92	39	.	.	PUNCT
ejpam-5618	93	1	if	if	SCONJ
ejpam-5618	93	2	f	f	PROPN
ejpam-5618	93	3	is	be	AUX
ejpam-5618	93	4	i	i	PRON
ejpam-5618	93	5	-	-	PUNCT
ejpam-5618	93	6	continuous	continuous	ADJ
ejpam-5618	93	7	and	and	CCONJ
ejpam-5618	93	8	g	g	NOUN
ejpam-5618	93	9	is	be	AUX
ejpam-5618	93	10	continuous	continuous	ADJ
ejpam-5618	93	11	,	,	PUNCT
ejpam-5618	93	12	then	then	ADV
ejpam-5618	93	13	g	g	PROPN
ejpam-5618	93	14	◦	◦	PROPN
ejpam-5618	93	15	f	f	PROPN
ejpam-5618	93	16	is	be	AUX
ejpam-5618	93	17	i	i	PRON
ejpam-5618	93	18	-	-	PUNCT
ejpam-5618	93	19	continuous	continuous	ADJ
ejpam-5618	93	20	.	.	PUNCT
ejpam-5618	94	1	theorem	theorem	NOUN
ejpam-5618	94	2	3	3	NUM
ejpam-5618	94	3	.	.	PUNCT
ejpam-5618	95	1	[	[	X
ejpam-5618	95	2	5	5	X
ejpam-5618	95	3	]	]	PUNCT
ejpam-5618	95	4	let	let	VERB
ejpam-5618	95	5	f	f	X
ejpam-5618	95	6	:	:	PUNCT
ejpam-5618	95	7	(	(	PUNCT
ejpam-5618	95	8	x	x	X
ejpam-5618	95	9	,	,	PUNCT
ejpam-5618	95	10	τ	τ	PROPN
ejpam-5618	95	11	,	,	PUNCT
ejpam-5618	95	12	i	i	NOUN
ejpam-5618	95	13	)	)	PUNCT
ejpam-5618	95	14	→	→	SYM
ejpam-5618	95	15	(	(	PUNCT
ejpam-5618	95	16	y	y	PROPN
ejpam-5618	95	17	,	,	PUNCT
ejpam-5618	95	18	σ	σ	PROPN
ejpam-5618	95	19	,	,	PUNCT
ejpam-5618	95	20	j	j	PROPN
ejpam-5618	95	21	)	)	PUNCT
ejpam-5618	95	22	and	and	CCONJ
ejpam-5618	95	23	g	g	NOUN
ejpam-5618	95	24	:	:	PUNCT
ejpam-5618	95	25	(	(	PUNCT
ejpam-5618	95	26	y	y	PROPN
ejpam-5618	95	27	,	,	PUNCT
ejpam-5618	95	28	σ	σ	PROPN
ejpam-5618	95	29	,	,	PUNCT
ejpam-5618	95	30	j	j	PROPN
ejpam-5618	95	31	)	)	PUNCT
ejpam-5618	95	32	→	→	PUNCT
ejpam-5618	95	33	(	(	PUNCT
ejpam-5618	95	34	z	z	NOUN
ejpam-5618	95	35	,	,	PUNCT
ejpam-5618	95	36	µ,k	µ,k	NOUN
ejpam-5618	95	37	)	)	PUNCT
ejpam-5618	95	38	be	be	VERB
ejpam-5618	95	39	two	two	NUM
ejpam-5618	95	40	mappings	mapping	NOUN
ejpam-5618	95	41	.	.	PUNCT
ejpam-5618	96	1	if	if	SCONJ
ejpam-5618	96	2	f	f	PROPN
ejpam-5618	96	3	is	be	AUX
ejpam-5618	96	4	i	i	NOUN
ejpam-5618	96	5	-	-	PUNCT
ejpam-5618	96	6	irresolute	irresolute	ADJ
ejpam-5618	96	7	and	and	CCONJ
ejpam-5618	96	8	g	g	NOUN
ejpam-5618	96	9	is	be	AUX
ejpam-5618	96	10	i	i	NOUN
ejpam-5618	96	11	-	-	PUNCT
ejpam-5618	96	12	continuous	continuous	ADJ
ejpam-5618	96	13	,	,	PUNCT
ejpam-5618	96	14	then	then	ADV
ejpam-5618	96	15	g	g	PROPN
ejpam-5618	96	16	◦	◦	PROPN
ejpam-5618	96	17	f	f	PROPN
ejpam-5618	96	18	is	be	AUX
ejpam-5618	96	19	i	i	PRON
ejpam-5618	96	20	-	-	PUNCT
ejpam-5618	96	21	continuous	continuous	ADJ
ejpam-5618	96	22	.	.	PUNCT
ejpam-5618	97	1	s.	s.	PROPN
ejpam-5618	97	2	alammar	alammar	PROPN
ejpam-5618	97	3	,	,	PUNCT
ejpam-5618	97	4	m.	m.	NOUN
ejpam-5618	97	5	al	al	PROPN
ejpam-5618	97	6	shumrani	shumrani	PROPN
ejpam-5618	97	7	,	,	PUNCT
ejpam-5618	97	8	c.	c.	PROPN
ejpam-5618	97	9	özel	özel	PROPN
ejpam-5618	97	10	/	/	SYM
ejpam-5618	97	11	eur	eur	PROPN
ejpam-5618	97	12	.	.	PUNCT
ejpam-5618	98	1	j.	j.	PROPN
ejpam-5618	98	2	pure	pure	PROPN
ejpam-5618	98	3	appl	appl	PROPN
ejpam-5618	98	4	.	.	PROPN
ejpam-5618	98	5	math	math	PROPN
ejpam-5618	98	6	,	,	PUNCT
ejpam-5618	98	7	18	18	NUM
ejpam-5618	98	8	(	(	PUNCT
ejpam-5618	98	9	1	1	NUM
ejpam-5618	98	10	)	)	PUNCT
ejpam-5618	98	11	(	(	PUNCT
ejpam-5618	98	12	2025	2025	NUM
ejpam-5618	98	13	)	)	PUNCT
ejpam-5618	98	14	,	,	PUNCT
ejpam-5618	98	15	5618	5618	NUM
ejpam-5618	98	16	4	4	NUM
ejpam-5618	98	17	of	of	ADP
ejpam-5618	98	18	13	13	NUM
ejpam-5618	98	19	theorem	theorem	NOUN
ejpam-5618	98	20	4	4	NUM
ejpam-5618	98	21	.	.	PUNCT
ejpam-5618	99	1	[	[	X
ejpam-5618	99	2	5	5	X
ejpam-5618	99	3	]	]	PUNCT
ejpam-5618	99	4	let	let	VERB
ejpam-5618	99	5	f	f	X
ejpam-5618	99	6	:	:	PUNCT
ejpam-5618	99	7	(	(	PUNCT
ejpam-5618	99	8	x	x	X
ejpam-5618	99	9	,	,	PUNCT
ejpam-5618	99	10	τ	τ	PROPN
ejpam-5618	99	11	,	,	PUNCT
ejpam-5618	99	12	i	i	NOUN
ejpam-5618	99	13	)	)	PUNCT
ejpam-5618	99	14	→	→	SYM
ejpam-5618	99	15	(	(	PUNCT
ejpam-5618	99	16	y	y	PROPN
ejpam-5618	99	17	,	,	PUNCT
ejpam-5618	99	18	σ	σ	PROPN
ejpam-5618	99	19	,	,	PUNCT
ejpam-5618	99	20	j	j	PROPN
ejpam-5618	99	21	)	)	PUNCT
ejpam-5618	99	22	be	be	AUX
ejpam-5618	99	23	an	an	DET
ejpam-5618	99	24	i	i	NOUN
ejpam-5618	99	25	-	-	PUNCT
ejpam-5618	99	26	continuous	continuous	ADJ
ejpam-5618	99	27	mapping	mapping	NOUN
ejpam-5618	99	28	.	.	PUNCT
ejpam-5618	100	1	if	if	SCONJ
ejpam-5618	100	2	y	y	PROPN
ejpam-5618	100	3	is	be	AUX
ejpam-5618	100	4	submaximal	submaximal	ADJ
ejpam-5618	100	5	space	space	NOUN
ejpam-5618	100	6	,	,	PUNCT
ejpam-5618	100	7	then	then	ADV
ejpam-5618	100	8	f	f	PROPN
ejpam-5618	100	9	is	be	AUX
ejpam-5618	100	10	i	i	NOUN
ejpam-5618	100	11	-	-	PUNCT
ejpam-5618	100	12	irresolute	irresolute	PROPN
ejpam-5618	100	13	.	.	PUNCT
ejpam-5618	101	1	theorem	theorem	ADJ
ejpam-5618	101	2	5	5	NUM
ejpam-5618	101	3	.	.	PUNCT
ejpam-5618	102	1	[	[	X
ejpam-5618	102	2	1	1	X
ejpam-5618	102	3	]	]	X
ejpam-5618	102	4	let	let	VERB
ejpam-5618	102	5	{	{	PUNCT
ejpam-5618	102	6	xα	xα	INTJ
ejpam-5618	102	7	:	:	PUNCT
ejpam-5618	102	8	α	α	PROPN
ejpam-5618	102	9	∈	∈	PROPN
ejpam-5618	102	10	∆	∆	PROPN
ejpam-5618	102	11	}	}	PUNCT
ejpam-5618	102	12	be	be	AUX
ejpam-5618	102	13	a	a	DET
ejpam-5618	102	14	family	family	NOUN
ejpam-5618	102	15	of	of	ADP
ejpam-5618	102	16	spaces	space	NOUN
ejpam-5618	102	17	,	,	PUNCT
ejpam-5618	102	18	x	x	PUNCT
ejpam-5618	102	19	=	=	SYM
ejpam-5618	102	20	∏	∏	NUM
ejpam-5618	102	21	xα	xα	INTJ
ejpam-5618	102	22	be	be	AUX
ejpam-5618	102	23	the	the	DET
ejpam-5618	102	24	product	product	NOUN
ejpam-5618	102	25	space	space	NOUN
ejpam-5618	102	26	and	and	CCONJ
ejpam-5618	102	27	a	a	DET
ejpam-5618	102	28	=	=	NOUN
ejpam-5618	102	29	∏n	∏n	ADJ
ejpam-5618	102	30	α=1aα	α=1aα	NUM
ejpam-5618	102	31	×	×	PROPN
ejpam-5618	102	32	∏	∏	PROPN
ejpam-5618	102	33	α	α	NOUN
ejpam-5618	102	34	̸=β	̸=β	NOUN
ejpam-5618	102	35	xβ	xβ	ADP
ejpam-5618	102	36	a	a	DET
ejpam-5618	102	37	non	non	X
ejpam-5618	102	38	empty	empty	ADJ
ejpam-5618	102	39	subset	subset	NOUN
ejpam-5618	102	40	of	of	ADP
ejpam-5618	102	41	x	x	X
ejpam-5618	102	42	,	,	PUNCT
ejpam-5618	102	43	where	where	SCONJ
ejpam-5618	102	44	n	n	PRON
ejpam-5618	102	45	is	be	AUX
ejpam-5618	102	46	a	a	DET
ejpam-5618	102	47	positive	positive	ADJ
ejpam-5618	102	48	integer	integer	NOUN
ejpam-5618	102	49	and	and	CCONJ
ejpam-5618	102	50	aα	aα	NOUN
ejpam-5618	102	51	⊂	⊂	PROPN
ejpam-5618	102	52	xα	xα	PROPN
ejpam-5618	102	53	.	.	PUNCT
ejpam-5618	103	1	then	then	ADV
ejpam-5618	103	2	aα	aα	NOUN
ejpam-5618	103	3	is	be	AUX
ejpam-5618	103	4	i	i	PRON
ejpam-5618	103	5	-	-	PUNCT
ejpam-5618	103	6	open	open	ADJ
ejpam-5618	103	7	in	in	ADP
ejpam-5618	103	8	xα	xα	INTJ
ejpam-5618	103	9	for	for	ADP
ejpam-5618	103	10	each	each	DET
ejpam-5618	103	11	1	1	NUM
ejpam-5618	103	12	≤	≤	NUM
ejpam-5618	103	13	α	α	NOUN
ejpam-5618	103	14	≤	≤	NOUN
ejpam-5618	103	15	n	n	CCONJ
ejpam-5618	103	16	if	if	SCONJ
ejpam-5618	104	1	and	and	CCONJ
ejpam-5618	104	2	only	only	ADV
ejpam-5618	104	3	if	if	SCONJ
ejpam-5618	104	4	a	a	PRON
ejpam-5618	104	5	is	be	AUX
ejpam-5618	104	6	i	i	PRON
ejpam-5618	104	7	-	-	PUNCT
ejpam-5618	104	8	open	open	ADJ
ejpam-5618	104	9	in	in	ADP
ejpam-5618	104	10	x.	x.	NOUN
ejpam-5618	104	11	theorem	theorem	VERB
ejpam-5618	104	12	6	6	NUM
ejpam-5618	104	13	.	.	PUNCT
ejpam-5618	105	1	[	[	X
ejpam-5618	105	2	1	1	X
ejpam-5618	105	3	]	]	PUNCT
ejpam-5618	105	4	let	let	VERB
ejpam-5618	105	5	f	f	X
ejpam-5618	105	6	:	:	PUNCT
ejpam-5618	105	7	(	(	PUNCT
ejpam-5618	105	8	x	x	X
ejpam-5618	105	9	,	,	PUNCT
ejpam-5618	105	10	τ	τ	PROPN
ejpam-5618	105	11	,	,	PUNCT
ejpam-5618	105	12	i	i	NOUN
ejpam-5618	105	13	)	)	PUNCT
ejpam-5618	105	14	→	→	SYM
ejpam-5618	105	15	(	(	PUNCT
ejpam-5618	105	16	y	y	PROPN
ejpam-5618	105	17	,	,	PUNCT
ejpam-5618	105	18	σ	σ	PROPN
ejpam-5618	105	19	)	)	PUNCT
ejpam-5618	105	20	be	be	VERB
ejpam-5618	105	21	an	an	DET
ejpam-5618	105	22	i	i	NOUN
ejpam-5618	105	23	-	-	PUNCT
ejpam-5618	105	24	continuous	continuous	ADJ
ejpam-5618	105	25	and	and	CCONJ
ejpam-5618	105	26	u	u	NOUN
ejpam-5618	105	27	∈	∈	PROPN
ejpam-5618	105	28	τ	τ	X
ejpam-5618	105	29	.	.	PUNCT
ejpam-5618	106	1	then	then	ADV
ejpam-5618	106	2	the	the	DET
ejpam-5618	106	3	restriction	restriction	NOUN
ejpam-5618	106	4	f	f	PROPN
ejpam-5618	106	5	|u	|u	PROPN
ejpam-5618	106	6	is	be	AUX
ejpam-5618	106	7	an	an	DET
ejpam-5618	106	8	i	i	NOUN
ejpam-5618	106	9	-	-	PUNCT
ejpam-5618	106	10	continuous	continuous	ADJ
ejpam-5618	106	11	.	.	PUNCT
ejpam-5618	107	1	3	3	X
ejpam-5618	107	2	.	.	X
ejpam-5618	107	3	ideal	ideal	ADJ
ejpam-5618	107	4	topological	topological	ADJ
ejpam-5618	107	5	groups	group	NOUN
ejpam-5618	107	6	in	in	ADP
ejpam-5618	107	7	this	this	DET
ejpam-5618	107	8	section	section	NOUN
ejpam-5618	107	9	,	,	PUNCT
ejpam-5618	107	10	we	we	PRON
ejpam-5618	107	11	define	define	VERB
ejpam-5618	107	12	ideal	ideal	ADJ
ejpam-5618	107	13	topological	topological	ADJ
ejpam-5618	107	14	groups	group	NOUN
ejpam-5618	107	15	and	and	CCONJ
ejpam-5618	107	16	give	give	VERB
ejpam-5618	107	17	their	their	PRON
ejpam-5618	107	18	basic	basic	ADJ
ejpam-5618	107	19	properties	property	NOUN
ejpam-5618	107	20	.	.	PUNCT
ejpam-5618	108	1	also	also	ADV
ejpam-5618	108	2	,	,	PUNCT
ejpam-5618	108	3	we	we	PRON
ejpam-5618	108	4	study	study	VERB
ejpam-5618	108	5	the	the	DET
ejpam-5618	108	6	relation	relation	NOUN
ejpam-5618	108	7	between	between	ADP
ejpam-5618	108	8	ideal	ideal	ADJ
ejpam-5618	108	9	topological	topological	ADJ
ejpam-5618	108	10	groups	group	NOUN
ejpam-5618	108	11	and	and	CCONJ
ejpam-5618	108	12	topological	topological	ADJ
ejpam-5618	108	13	groups	group	NOUN
ejpam-5618	108	14	.	.	PUNCT
ejpam-5618	109	1	definition	definition	NOUN
ejpam-5618	109	2	14	14	NUM
ejpam-5618	109	3	.	.	PUNCT
ejpam-5618	110	1	an	an	DET
ejpam-5618	110	2	ideal	ideal	ADJ
ejpam-5618	110	3	topological	topological	ADJ
ejpam-5618	110	4	group	group	NOUN
ejpam-5618	110	5	g	g	PROPN
ejpam-5618	110	6	is	be	AUX
ejpam-5618	110	7	a	a	DET
ejpam-5618	110	8	group	group	NOUN
ejpam-5618	110	9	that	that	PRON
ejpam-5618	110	10	is	be	AUX
ejpam-5618	110	11	also	also	ADV
ejpam-5618	110	12	an	an	DET
ejpam-5618	110	13	ideal	ideal	ADJ
ejpam-5618	110	14	topological	topological	ADJ
ejpam-5618	110	15	space	space	NOUN
ejpam-5618	110	16	such	such	ADJ
ejpam-5618	110	17	that	that	SCONJ
ejpam-5618	110	18	the	the	DET
ejpam-5618	110	19	multiplication	multiplication	NOUN
ejpam-5618	110	20	mapping	mapping	NOUN
ejpam-5618	110	21	m	m	VERB
ejpam-5618	110	22	:	:	PUNCT
ejpam-5618	110	23	g	g	ADP
ejpam-5618	110	24	×	×	PROPN
ejpam-5618	110	25	g	g	PROPN
ejpam-5618	110	26	→	→	SYM
ejpam-5618	110	27	g	g	PROPN
ejpam-5618	110	28	and	and	CCONJ
ejpam-5618	110	29	the	the	DET
ejpam-5618	110	30	inverse	inverse	NOUN
ejpam-5618	110	31	mapping	mapping	NOUN
ejpam-5618	110	32	inv	inv	VERB
ejpam-5618	110	33	:	:	PUNCT
ejpam-5618	110	34	g	g	NOUN
ejpam-5618	110	35	→	→	SYM
ejpam-5618	110	36	g	g	NOUN
ejpam-5618	110	37	both	both	PRON
ejpam-5618	110	38	are	be	AUX
ejpam-5618	110	39	i	i	PRON
ejpam-5618	110	40	-	-	NOUN
ejpam-5618	110	41	continuous	continuous	ADJ
ejpam-5618	110	42	.	.	PUNCT
ejpam-5618	111	1	we	we	PRON
ejpam-5618	111	2	present	present	VERB
ejpam-5618	111	3	some	some	DET
ejpam-5618	111	4	examples	example	NOUN
ejpam-5618	111	5	of	of	ADP
ejpam-5618	111	6	ideal	ideal	ADJ
ejpam-5618	111	7	topological	topological	ADJ
ejpam-5618	111	8	groups	group	NOUN
ejpam-5618	111	9	.	.	PUNCT
ejpam-5618	112	1	example	example	NOUN
ejpam-5618	113	1	1	1	NUM
ejpam-5618	113	2	.	.	PUNCT
ejpam-5618	113	3	r	r	NOUN
ejpam-5618	113	4	under	under	ADP
ejpam-5618	113	5	addition	addition	NOUN
ejpam-5618	113	6	with	with	ADP
ejpam-5618	113	7	its	its	PRON
ejpam-5618	113	8	usual	usual	ADJ
ejpam-5618	113	9	topology	topology	NOUN
ejpam-5618	113	10	is	be	AUX
ejpam-5618	113	11	a	a	DET
ejpam-5618	113	12	topological	topological	ADJ
ejpam-5618	113	13	group	group	NOUN
ejpam-5618	113	14	.	.	PUNCT
ejpam-5618	114	1	if	if	SCONJ
ejpam-5618	114	2	we	we	PRON
ejpam-5618	114	3	consider	consider	VERB
ejpam-5618	114	4	the	the	DET
ejpam-5618	114	5	ideal	ideal	NOUN
ejpam-5618	114	6	of	of	ADP
ejpam-5618	114	7	finite	finite	ADJ
ejpam-5618	114	8	subsets	subset	NOUN
ejpam-5618	114	9	of	of	ADP
ejpam-5618	114	10	r	r	NOUN
ejpam-5618	114	11	,	,	PUNCT
ejpam-5618	114	12	then	then	ADV
ejpam-5618	114	13	it	it	PRON
ejpam-5618	114	14	can	can	AUX
ejpam-5618	114	15	be	be	AUX
ejpam-5618	114	16	shown	show	VERB
ejpam-5618	114	17	that	that	SCONJ
ejpam-5618	114	18	any	any	DET
ejpam-5618	114	19	open	open	ADJ
ejpam-5618	114	20	interval	interval	NOUN
ejpam-5618	114	21	is	be	AUX
ejpam-5618	114	22	i	i	NOUN
ejpam-5618	114	23	-	-	PUNCT
ejpam-5618	114	24	open	open	ADJ
ejpam-5618	114	25	.	.	PUNCT
ejpam-5618	115	1	therefore	therefore	ADV
ejpam-5618	115	2	,	,	PUNCT
ejpam-5618	115	3	we	we	PRON
ejpam-5618	115	4	deduce	deduce	VERB
ejpam-5618	115	5	that	that	SCONJ
ejpam-5618	115	6	r	r	NOUN
ejpam-5618	115	7	is	be	AUX
ejpam-5618	115	8	ideal	ideal	ADJ
ejpam-5618	115	9	topological	topological	ADJ
ejpam-5618	115	10	group	group	NOUN
ejpam-5618	115	11	.	.	PUNCT
ejpam-5618	116	1	example	example	NOUN
ejpam-5618	117	1	2	2	NUM
ejpam-5618	117	2	.	.	X
ejpam-5618	117	3	consider	consider	VERB
ejpam-5618	117	4	r	r	NOUN
ejpam-5618	117	5	under	under	ADP
ejpam-5618	117	6	addition	addition	NOUN
ejpam-5618	117	7	with	with	ADP
ejpam-5618	117	8	its	its	PRON
ejpam-5618	117	9	usual	usual	ADJ
ejpam-5618	117	10	topology	topology	NOUN
ejpam-5618	117	11	and	and	CCONJ
ejpam-5618	117	12	with	with	ADP
ejpam-5618	117	13	the	the	DET
ejpam-5618	117	14	ideal	ideal	NOUN
ejpam-5618	117	15	of	of	ADP
ejpam-5618	117	16	countable	countable	ADJ
ejpam-5618	117	17	subsets	subset	NOUN
ejpam-5618	117	18	of	of	ADP
ejpam-5618	117	19	r.	r.	PROPN
ejpam-5618	117	20	then	then	ADV
ejpam-5618	117	21	it	it	PRON
ejpam-5618	117	22	is	be	AUX
ejpam-5618	117	23	not	not	PART
ejpam-5618	117	24	difficult	difficult	ADJ
ejpam-5618	117	25	to	to	PART
ejpam-5618	117	26	show	show	VERB
ejpam-5618	117	27	that	that	SCONJ
ejpam-5618	117	28	any	any	DET
ejpam-5618	117	29	open	open	ADJ
ejpam-5618	117	30	interval	interval	NOUN
ejpam-5618	117	31	is	be	AUX
ejpam-5618	117	32	i	i	NOUN
ejpam-5618	117	33	-	-	PUNCT
ejpam-5618	117	34	open	open	ADJ
ejpam-5618	117	35	.	.	PUNCT
ejpam-5618	118	1	therefore	therefore	ADV
ejpam-5618	118	2	,	,	PUNCT
ejpam-5618	118	3	we	we	PRON
ejpam-5618	118	4	deduce	deduce	VERB
ejpam-5618	118	5	that	that	SCONJ
ejpam-5618	118	6	r	r	NOUN
ejpam-5618	118	7	is	be	AUX
ejpam-5618	118	8	ideal	ideal	ADJ
ejpam-5618	118	9	topological	topological	ADJ
ejpam-5618	118	10	group	group	NOUN
ejpam-5618	118	11	.	.	PUNCT
ejpam-5618	119	1	example	example	NOUN
ejpam-5618	120	1	3	3	X
ejpam-5618	120	2	.	.	PUNCT
ejpam-5618	120	3	let	let	VERB
ejpam-5618	120	4	g	g	NOUN
ejpam-5618	120	5	be	be	AUX
ejpam-5618	120	6	any	any	DET
ejpam-5618	120	7	group	group	NOUN
ejpam-5618	120	8	with	with	ADP
ejpam-5618	120	9	the	the	DET
ejpam-5618	120	10	discrete	discrete	ADJ
ejpam-5618	120	11	topology	topology	NOUN
ejpam-5618	120	12	.	.	PUNCT
ejpam-5618	121	1	then	then	ADV
ejpam-5618	121	2	it	it	PRON
ejpam-5618	121	3	is	be	AUX
ejpam-5618	121	4	known	know	VERB
ejpam-5618	121	5	that	that	SCONJ
ejpam-5618	121	6	g	g	PROPN
ejpam-5618	121	7	is	be	AUX
ejpam-5618	121	8	a	a	DET
ejpam-5618	121	9	topological	topological	ADJ
ejpam-5618	121	10	group	group	NOUN
ejpam-5618	121	11	.	.	PUNCT
ejpam-5618	122	1	if	if	SCONJ
ejpam-5618	122	2	we	we	PRON
ejpam-5618	122	3	consider	consider	VERB
ejpam-5618	122	4	the	the	DET
ejpam-5618	122	5	ideal	ideal	NOUN
ejpam-5618	122	6	of	of	ADP
ejpam-5618	122	7	nowhere	nowhere	ADJ
ejpam-5618	122	8	dense	dense	ADJ
ejpam-5618	122	9	subsets	subset	NOUN
ejpam-5618	122	10	in	in	ADP
ejpam-5618	122	11	g	g	NOUN
ejpam-5618	122	12	,	,	PUNCT
ejpam-5618	122	13	then	then	ADV
ejpam-5618	122	14	the	the	DET
ejpam-5618	122	15	class	class	NOUN
ejpam-5618	122	16	of	of	ADP
ejpam-5618	122	17	i	i	PROPN
ejpam-5618	122	18	-	-	PUNCT
ejpam-5618	122	19	open	open	ADJ
ejpam-5618	122	20	sets	set	NOUN
ejpam-5618	122	21	is	be	AUX
ejpam-5618	122	22	the	the	DET
ejpam-5618	122	23	power	power	NOUN
ejpam-5618	122	24	set	set	NOUN
ejpam-5618	122	25	of	of	ADP
ejpam-5618	122	26	g.	g.	PROPN
ejpam-5618	122	27	therefore	therefore	ADV
ejpam-5618	122	28	,	,	PUNCT
ejpam-5618	122	29	g	g	PROPN
ejpam-5618	122	30	is	be	AUX
ejpam-5618	122	31	ideal	ideal	ADJ
ejpam-5618	122	32	topological	topological	ADJ
ejpam-5618	122	33	group	group	NOUN
ejpam-5618	122	34	.	.	PUNCT
ejpam-5618	123	1	in	in	ADP
ejpam-5618	123	2	particular	particular	ADJ
ejpam-5618	123	3	,	,	PUNCT
ejpam-5618	123	4	r	r	NOUN
ejpam-5618	123	5	under	under	ADP
ejpam-5618	123	6	addition	addition	NOUN
ejpam-5618	123	7	with	with	ADP
ejpam-5618	123	8	the	the	DET
ejpam-5618	123	9	discrete	discrete	ADJ
ejpam-5618	123	10	topology	topology	NOUN
ejpam-5618	123	11	and	and	CCONJ
ejpam-5618	123	12	with	with	ADP
ejpam-5618	123	13	the	the	DET
ejpam-5618	123	14	ideal	ideal	NOUN
ejpam-5618	123	15	of	of	ADP
ejpam-5618	123	16	nowhere	nowhere	ADJ
ejpam-5618	123	17	dense	dense	ADJ
ejpam-5618	123	18	subsets	subset	NOUN
ejpam-5618	123	19	is	be	AUX
ejpam-5618	123	20	ideal	ideal	ADJ
ejpam-5618	123	21	topological	topological	ADJ
ejpam-5618	123	22	group	group	NOUN
ejpam-5618	123	23	.	.	PUNCT
ejpam-5618	124	1	it	it	PRON
ejpam-5618	124	2	is	be	AUX
ejpam-5618	124	3	natural	natural	ADJ
ejpam-5618	124	4	to	to	PART
ejpam-5618	124	5	ask	ask	VERB
ejpam-5618	124	6	about	about	ADP
ejpam-5618	124	7	the	the	DET
ejpam-5618	124	8	relation	relation	NOUN
ejpam-5618	124	9	between	between	ADP
ejpam-5618	124	10	topological	topological	ADJ
ejpam-5618	124	11	groups	group	NOUN
ejpam-5618	124	12	and	and	CCONJ
ejpam-5618	124	13	ideal	ideal	ADJ
ejpam-5618	124	14	topological	topological	ADJ
ejpam-5618	124	15	groups	group	NOUN
ejpam-5618	124	16	.	.	PUNCT
ejpam-5618	125	1	we	we	PRON
ejpam-5618	125	2	find	find	VERB
ejpam-5618	125	3	that	that	SCONJ
ejpam-5618	125	4	the	the	DET
ejpam-5618	125	5	two	two	NUM
ejpam-5618	125	6	concepts	concept	NOUN
ejpam-5618	125	7	are	be	AUX
ejpam-5618	125	8	independent	independent	ADJ
ejpam-5618	125	9	,	,	PUNCT
ejpam-5618	125	10	as	as	SCONJ
ejpam-5618	125	11	the	the	DET
ejpam-5618	125	12	following	follow	VERB
ejpam-5618	125	13	examples	example	NOUN
ejpam-5618	125	14	show	show	NOUN
ejpam-5618	125	15	.	.	PUNCT
ejpam-5618	126	1	example	example	NOUN
ejpam-5618	127	1	4	4	X
ejpam-5618	127	2	.	.	PUNCT
ejpam-5618	127	3	let	let	VERB
ejpam-5618	127	4	g	g	PRON
ejpam-5618	127	5	be	be	AUX
ejpam-5618	127	6	a	a	DET
ejpam-5618	127	7	group	group	NOUN
ejpam-5618	127	8	with	with	ADP
ejpam-5618	127	9	a	a	DET
ejpam-5618	127	10	topology	topology	NOUN
ejpam-5618	127	11	and	and	CCONJ
ejpam-5618	127	12	with	with	ADP
ejpam-5618	127	13	the	the	DET
ejpam-5618	127	14	ideal	ideal	NOUN
ejpam-5618	127	15	of	of	ADP
ejpam-5618	127	16	of	of	ADP
ejpam-5618	127	17	all	all	DET
ejpam-5618	127	18	subsets	subset	NOUN
ejpam-5618	127	19	of	of	ADP
ejpam-5618	127	20	g.	g.	PROPN
ejpam-5618	127	21	then	then	ADV
ejpam-5618	127	22	the	the	DET
ejpam-5618	127	23	class	class	NOUN
ejpam-5618	127	24	of	of	ADP
ejpam-5618	127	25	i	i	PROPN
ejpam-5618	127	26	-	-	PUNCT
ejpam-5618	127	27	open	open	ADJ
ejpam-5618	127	28	sets	set	NOUN
ejpam-5618	127	29	contains	contain	VERB
ejpam-5618	127	30	only	only	ADV
ejpam-5618	127	31	the	the	DET
ejpam-5618	127	32	empty	empty	ADJ
ejpam-5618	127	33	set	set	NOUN
ejpam-5618	127	34	∅.	∅.	PRON
ejpam-5618	127	35	therefore	therefore	ADV
ejpam-5618	127	36	,	,	PUNCT
ejpam-5618	127	37	g	g	PROPN
ejpam-5618	127	38	can	can	AUX
ejpam-5618	127	39	not	not	PART
ejpam-5618	127	40	be	be	AUX
ejpam-5618	127	41	an	an	DET
ejpam-5618	127	42	ideal	ideal	ADJ
ejpam-5618	127	43	topological	topological	ADJ
ejpam-5618	127	44	group	group	NOUN
ejpam-5618	127	45	since	since	SCONJ
ejpam-5618	127	46	the	the	DET
ejpam-5618	127	47	multiplication	multiplication	NOUN
ejpam-5618	127	48	mapping	mapping	NOUN
ejpam-5618	127	49	and	and	CCONJ
ejpam-5618	127	50	the	the	DET
ejpam-5618	127	51	inverse	inverse	NOUN
ejpam-5618	127	52	mapping	mapping	NOUN
ejpam-5618	127	53	are	be	AUX
ejpam-5618	127	54	not	not	PART
ejpam-5618	127	55	i	i	NOUN
ejpam-5618	127	56	-	-	NOUN
ejpam-5618	127	57	continuous	continuous	ADJ
ejpam-5618	127	58	.	.	PUNCT
ejpam-5618	128	1	in	in	ADP
ejpam-5618	128	2	particular	particular	ADJ
ejpam-5618	128	3	,	,	PUNCT
ejpam-5618	128	4	r	r	NOUN
ejpam-5618	128	5	under	under	ADP
ejpam-5618	128	6	addition	addition	NOUN
ejpam-5618	128	7	with	with	ADP
ejpam-5618	128	8	its	its	PRON
ejpam-5618	128	9	usual	usual	ADJ
ejpam-5618	128	10	topology	topology	NOUN
ejpam-5618	128	11	is	be	AUX
ejpam-5618	128	12	a	a	DET
ejpam-5618	128	13	topological	topological	ADJ
ejpam-5618	128	14	group	group	NOUN
ejpam-5618	128	15	.	.	PUNCT
ejpam-5618	129	1	if	if	SCONJ
ejpam-5618	129	2	we	we	PRON
ejpam-5618	129	3	consider	consider	VERB
ejpam-5618	129	4	the	the	DET
ejpam-5618	129	5	ideal	ideal	NOUN
ejpam-5618	129	6	of	of	ADP
ejpam-5618	129	7	all	all	DET
ejpam-5618	129	8	subsets	subset	NOUN
ejpam-5618	129	9	of	of	ADP
ejpam-5618	129	10	r	r	NOUN
ejpam-5618	129	11	,	,	PUNCT
ejpam-5618	129	12	then	then	ADV
ejpam-5618	129	13	∅	∅	NOUN
ejpam-5618	129	14	is	be	AUX
ejpam-5618	129	15	the	the	DET
ejpam-5618	129	16	only	only	ADJ
ejpam-5618	129	17	i	i	NOUN
ejpam-5618	129	18	-	-	PUNCT
ejpam-5618	129	19	open	open	ADJ
ejpam-5618	129	20	set	set	NOUN
ejpam-5618	129	21	.	.	PUNCT
ejpam-5618	130	1	thus	thus	ADV
ejpam-5618	130	2	,	,	PUNCT
ejpam-5618	130	3	r	r	NOUN
ejpam-5618	130	4	is	be	AUX
ejpam-5618	130	5	not	not	PART
ejpam-5618	130	6	an	an	DET
ejpam-5618	130	7	ideal	ideal	ADJ
ejpam-5618	130	8	topological	topological	ADJ
ejpam-5618	130	9	group	group	NOUN
ejpam-5618	130	10	since	since	SCONJ
ejpam-5618	130	11	the	the	DET
ejpam-5618	130	12	multiplication	multiplication	NOUN
ejpam-5618	130	13	mapping	mapping	NOUN
ejpam-5618	130	14	and	and	CCONJ
ejpam-5618	130	15	the	the	DET
ejpam-5618	130	16	inverse	inverse	NOUN
ejpam-5618	130	17	mapping	mapping	NOUN
ejpam-5618	130	18	are	be	AUX
ejpam-5618	130	19	not	not	PART
ejpam-5618	130	20	i	i	NOUN
ejpam-5618	130	21	-	-	PUNCT
ejpam-5618	130	22	continuous	continuous	ADJ
ejpam-5618	130	23	.	.	PUNCT
ejpam-5618	131	1	s.	s.	PROPN
ejpam-5618	131	2	alammar	alammar	PROPN
ejpam-5618	131	3	,	,	PUNCT
ejpam-5618	131	4	m.	m.	NOUN
ejpam-5618	131	5	al	al	PROPN
ejpam-5618	131	6	shumrani	shumrani	PROPN
ejpam-5618	131	7	,	,	PUNCT
ejpam-5618	131	8	c.	c.	PROPN
ejpam-5618	131	9	özel	özel	PROPN
ejpam-5618	131	10	/	/	SYM
ejpam-5618	131	11	eur	eur	PROPN
ejpam-5618	131	12	.	.	PUNCT
ejpam-5618	132	1	j.	j.	PROPN
ejpam-5618	132	2	pure	pure	PROPN
ejpam-5618	132	3	appl	appl	PROPN
ejpam-5618	132	4	.	.	PROPN
ejpam-5618	132	5	math	math	PROPN
ejpam-5618	132	6	,	,	PUNCT
ejpam-5618	132	7	18	18	NUM
ejpam-5618	132	8	(	(	PUNCT
ejpam-5618	132	9	1	1	NUM
ejpam-5618	132	10	)	)	PUNCT
ejpam-5618	132	11	(	(	PUNCT
ejpam-5618	132	12	2025	2025	NUM
ejpam-5618	132	13	)	)	PUNCT
ejpam-5618	132	14	,	,	PUNCT
ejpam-5618	132	15	5618	5618	NUM
ejpam-5618	132	16	5	5	NUM
ejpam-5618	132	17	of	of	ADP
ejpam-5618	132	18	13	13	NUM
ejpam-5618	132	19	example	example	NOUN
ejpam-5618	132	20	5	5	NUM
ejpam-5618	132	21	.	.	X
ejpam-5618	132	22	consider	consider	VERB
ejpam-5618	132	23	z3	z3	PROPN
ejpam-5618	132	24	=	=	PUNCT
ejpam-5618	132	25	{	{	PUNCT
ejpam-5618	132	26	0	0	NUM
ejpam-5618	132	27	,	,	PUNCT
ejpam-5618	132	28	1	1	NUM
ejpam-5618	132	29	,	,	PUNCT
ejpam-5618	132	30	2	2	NUM
ejpam-5618	132	31	}	}	PUNCT
ejpam-5618	132	32	,	,	PUNCT
ejpam-5618	132	33	the	the	DET
ejpam-5618	132	34	group	group	NOUN
ejpam-5618	132	35	of	of	ADP
ejpam-5618	132	36	integers	integer	NOUN
ejpam-5618	132	37	mod	mod	PROPN
ejpam-5618	132	38	3	3	NUM
ejpam-5618	132	39	,	,	PUNCT
ejpam-5618	132	40	with	with	ADP
ejpam-5618	132	41	a	a	DET
ejpam-5618	132	42	topology	topology	NOUN
ejpam-5618	132	43	τ	τ	X
ejpam-5618	132	44	=	=	SYM
ejpam-5618	132	45	{	{	PUNCT
ejpam-5618	132	46	z3	z3	PROPN
ejpam-5618	132	47	,	,	PUNCT
ejpam-5618	132	48	∅	∅	NOUN
ejpam-5618	132	49	,	,	PUNCT
ejpam-5618	132	50	{	{	PUNCT
ejpam-5618	132	51	1	1	NUM
ejpam-5618	132	52	,	,	PUNCT
ejpam-5618	132	53	2	2	NUM
ejpam-5618	132	54	}	}	PUNCT
ejpam-5618	132	55	}	}	PUNCT
ejpam-5618	132	56	on	on	ADP
ejpam-5618	132	57	z3	z3	PROPN
ejpam-5618	132	58	.	.	PUNCT
ejpam-5618	133	1	let	let	VERB
ejpam-5618	133	2	i	i	PRON
ejpam-5618	133	3	=	=	PUNCT
ejpam-5618	133	4	{	{	PUNCT
ejpam-5618	133	5	∅	∅	NOUN
ejpam-5618	133	6	,	,	PUNCT
ejpam-5618	133	7	0	0	NUM
ejpam-5618	133	8	}	}	PUNCT
ejpam-5618	133	9	be	be	AUX
ejpam-5618	133	10	an	an	DET
ejpam-5618	133	11	ideal	ideal	NOUN
ejpam-5618	133	12	on	on	ADP
ejpam-5618	133	13	z3	z3	PROPN
ejpam-5618	133	14	.	.	PUNCT
ejpam-5618	134	1	then	then	ADV
ejpam-5618	134	2	it	it	PRON
ejpam-5618	134	3	is	be	AUX
ejpam-5618	134	4	clear	clear	ADJ
ejpam-5618	134	5	that	that	SCONJ
ejpam-5618	134	6	the	the	DET
ejpam-5618	134	7	class	class	NOUN
ejpam-5618	134	8	of	of	ADP
ejpam-5618	134	9	i	i	PROPN
ejpam-5618	134	10	-	-	PUNCT
ejpam-5618	134	11	open	open	ADJ
ejpam-5618	134	12	sets	set	NOUN
ejpam-5618	134	13	is	be	AUX
ejpam-5618	134	14	the	the	DET
ejpam-5618	134	15	power	power	NOUN
ejpam-5618	134	16	set	set	NOUN
ejpam-5618	134	17	of	of	ADP
ejpam-5618	134	18	z3	z3	PROPN
ejpam-5618	134	19	except	except	SCONJ
ejpam-5618	134	20	{	{	PUNCT
ejpam-5618	134	21	0	0	NUM
ejpam-5618	134	22	}	}	PUNCT
ejpam-5618	134	23	.	.	PUNCT
ejpam-5618	135	1	it	it	PRON
ejpam-5618	135	2	is	be	AUX
ejpam-5618	135	3	not	not	PART
ejpam-5618	135	4	difficult	difficult	ADJ
ejpam-5618	135	5	to	to	PART
ejpam-5618	135	6	check	check	VERB
ejpam-5618	135	7	that	that	SCONJ
ejpam-5618	135	8	the	the	DET
ejpam-5618	135	9	multiplication	multiplication	NOUN
ejpam-5618	135	10	mapping	mapping	NOUN
ejpam-5618	135	11	and	and	CCONJ
ejpam-5618	135	12	the	the	DET
ejpam-5618	135	13	inverse	inverse	NOUN
ejpam-5618	135	14	mapping	mapping	NOUN
ejpam-5618	135	15	are	be	AUX
ejpam-5618	135	16	i	i	PRON
ejpam-5618	135	17	-	-	NOUN
ejpam-5618	135	18	continuous	continuous	ADJ
ejpam-5618	135	19	.	.	PUNCT
ejpam-5618	136	1	therefore	therefore	ADV
ejpam-5618	136	2	,	,	PUNCT
ejpam-5618	136	3	z3	z3	PROPN
ejpam-5618	136	4	is	be	AUX
ejpam-5618	136	5	an	an	DET
ejpam-5618	136	6	ideal	ideal	ADJ
ejpam-5618	136	7	topological	topological	ADJ
ejpam-5618	136	8	group	group	NOUN
ejpam-5618	136	9	.	.	PUNCT
ejpam-5618	137	1	however	however	ADV
ejpam-5618	137	2	,	,	PUNCT
ejpam-5618	137	3	z3	z3	PROPN
ejpam-5618	137	4	is	be	AUX
ejpam-5618	137	5	not	not	PART
ejpam-5618	137	6	a	a	DET
ejpam-5618	137	7	topological	topological	ADJ
ejpam-5618	137	8	group	group	NOUN
ejpam-5618	137	9	since	since	SCONJ
ejpam-5618	137	10	the	the	DET
ejpam-5618	137	11	multiplication	multiplication	NOUN
ejpam-5618	137	12	mapping	mapping	NOUN
ejpam-5618	137	13	is	be	AUX
ejpam-5618	137	14	not	not	PART
ejpam-5618	137	15	continuous	continuous	ADJ
ejpam-5618	137	16	at	at	ADP
ejpam-5618	137	17	the	the	DET
ejpam-5618	137	18	element	element	NOUN
ejpam-5618	137	19	(	(	PUNCT
ejpam-5618	137	20	0	0	NUM
ejpam-5618	137	21	,	,	PUNCT
ejpam-5618	137	22	1	1	NUM
ejpam-5618	137	23	)	)	PUNCT
ejpam-5618	137	24	.	.	PUNCT
ejpam-5618	138	1	using	use	VERB
ejpam-5618	138	2	lemma	lemma	PROPN
ejpam-5618	138	3	1	1	NUM
ejpam-5618	138	4	,	,	PUNCT
ejpam-5618	138	5	a	a	DET
ejpam-5618	138	6	sufficient	sufficient	ADJ
ejpam-5618	138	7	condition	condition	NOUN
ejpam-5618	138	8	for	for	ADP
ejpam-5618	138	9	a	a	DET
ejpam-5618	138	10	topological	topological	ADJ
ejpam-5618	138	11	group	group	NOUN
ejpam-5618	138	12	to	to	PART
ejpam-5618	138	13	be	be	AUX
ejpam-5618	138	14	an	an	DET
ejpam-5618	138	15	ideal	ideal	ADJ
ejpam-5618	138	16	topological	topological	ADJ
ejpam-5618	138	17	group	group	NOUN
ejpam-5618	138	18	is	be	AUX
ejpam-5618	138	19	presented	present	VERB
ejpam-5618	138	20	in	in	ADP
ejpam-5618	138	21	the	the	DET
ejpam-5618	138	22	following	follow	VERB
ejpam-5618	138	23	result	result	NOUN
ejpam-5618	138	24	.	.	PUNCT
ejpam-5618	139	1	theorem	theorem	ADJ
ejpam-5618	139	2	7	7	NUM
ejpam-5618	139	3	.	.	PUNCT
ejpam-5618	140	1	let	let	VERB
ejpam-5618	140	2	g	g	NOUN
ejpam-5618	140	3	be	be	AUX
ejpam-5618	140	4	an	an	DET
ejpam-5618	140	5	ideal	ideal	ADJ
ejpam-5618	140	6	topological	topological	ADJ
ejpam-5618	140	7	space	space	NOUN
ejpam-5618	140	8	.	.	PUNCT
ejpam-5618	141	1	if	if	SCONJ
ejpam-5618	141	2	g	g	PROPN
ejpam-5618	141	3	is	be	AUX
ejpam-5618	141	4	an	an	DET
ejpam-5618	141	5	it1	it1	PROPN
ejpam-5618	141	6	topological	topological	PROPN
ejpam-5618	141	7	group	group	NOUN
ejpam-5618	141	8	,	,	PUNCT
ejpam-5618	141	9	then	then	ADV
ejpam-5618	141	10	g	g	PROPN
ejpam-5618	141	11	is	be	AUX
ejpam-5618	141	12	an	an	DET
ejpam-5618	141	13	ideal	ideal	ADJ
ejpam-5618	141	14	topological	topological	ADJ
ejpam-5618	141	15	group	group	NOUN
ejpam-5618	141	16	.	.	PUNCT
ejpam-5618	142	1	proof	proof	NOUN
ejpam-5618	142	2	.	.	PUNCT
ejpam-5618	143	1	suppose	suppose	VERB
ejpam-5618	143	2	that	that	SCONJ
ejpam-5618	143	3	g	g	PROPN
ejpam-5618	143	4	is	be	AUX
ejpam-5618	143	5	an	an	DET
ejpam-5618	143	6	it1	it1	PROPN
ejpam-5618	143	7	topological	topological	PROPN
ejpam-5618	143	8	group	group	NOUN
ejpam-5618	143	9	.	.	PUNCT
ejpam-5618	144	1	we	we	PRON
ejpam-5618	144	2	shall	shall	AUX
ejpam-5618	144	3	show	show	VERB
ejpam-5618	144	4	that	that	SCONJ
ejpam-5618	144	5	the	the	DET
ejpam-5618	144	6	multiplication	multiplication	NOUN
ejpam-5618	144	7	mapping	mapping	NOUN
ejpam-5618	144	8	m	m	VERB
ejpam-5618	144	9	:	:	PUNCT
ejpam-5618	144	10	g	g	ADP
ejpam-5618	144	11	×	×	PROPN
ejpam-5618	144	12	g	g	PROPN
ejpam-5618	144	13	→	→	SYM
ejpam-5618	144	14	g	g	PROPN
ejpam-5618	144	15	and	and	CCONJ
ejpam-5618	144	16	the	the	DET
ejpam-5618	144	17	inverse	inverse	NOUN
ejpam-5618	144	18	mapping	mapping	NOUN
ejpam-5618	144	19	inv	inv	VERB
ejpam-5618	144	20	:	:	PUNCT
ejpam-5618	144	21	g	g	NOUN
ejpam-5618	144	22	→	→	SYM
ejpam-5618	144	23	g	g	NOUN
ejpam-5618	144	24	both	both	PRON
ejpam-5618	144	25	are	be	AUX
ejpam-5618	144	26	i	i	PRON
ejpam-5618	144	27	-	-	NOUN
ejpam-5618	144	28	continuous	continuous	ADJ
ejpam-5618	144	29	.	.	PUNCT
ejpam-5618	145	1	first	first	ADV
ejpam-5618	145	2	we	we	PRON
ejpam-5618	145	3	show	show	VERB
ejpam-5618	145	4	that	that	SCONJ
ejpam-5618	145	5	m	m	PROPN
ejpam-5618	145	6	is	be	AUX
ejpam-5618	145	7	i	i	PRON
ejpam-5618	145	8	-	-	PUNCT
ejpam-5618	145	9	continuous	continuous	ADJ
ejpam-5618	145	10	.	.	PUNCT
ejpam-5618	146	1	let	let	VERB
ejpam-5618	146	2	w	w	NOUN
ejpam-5618	146	3	be	be	AUX
ejpam-5618	146	4	an	an	DET
ejpam-5618	146	5	open	open	ADJ
ejpam-5618	146	6	set	set	NOUN
ejpam-5618	146	7	in	in	ADP
ejpam-5618	146	8	g.	g.	PROPN
ejpam-5618	146	9	then	then	ADV
ejpam-5618	146	10	there	there	PRON
ejpam-5618	146	11	exist	exist	VERB
ejpam-5618	146	12	open	open	ADJ
ejpam-5618	146	13	sets	set	NOUN
ejpam-5618	146	14	u	u	NOUN
ejpam-5618	146	15	and	and	CCONJ
ejpam-5618	146	16	v	v	NOUN
ejpam-5618	146	17	in	in	ADP
ejpam-5618	146	18	g	g	PROPN
ejpam-5618	147	1	such	such	DET
ejpam-5618	147	2	that	that	PRON
ejpam-5618	147	3	uv	uv	PROPN
ejpam-5618	148	1	⊂	⊂	PROPN
ejpam-5618	148	2	w	w	NOUN
ejpam-5618	148	3	since	since	SCONJ
ejpam-5618	148	4	g	g	PROPN
ejpam-5618	148	5	is	be	AUX
ejpam-5618	148	6	a	a	DET
ejpam-5618	148	7	topological	topological	ADJ
ejpam-5618	148	8	group	group	NOUN
ejpam-5618	148	9	.	.	PUNCT
ejpam-5618	149	1	using	use	VERB
ejpam-5618	149	2	lemma	lemma	PROPN
ejpam-5618	149	3	1	1	NUM
ejpam-5618	149	4	,	,	PUNCT
ejpam-5618	149	5	u	u	NOUN
ejpam-5618	149	6	and	and	CCONJ
ejpam-5618	149	7	v	v	NOUN
ejpam-5618	149	8	are	be	AUX
ejpam-5618	149	9	i	i	PRON
ejpam-5618	149	10	-	-	PUNCT
ejpam-5618	149	11	open	open	ADJ
ejpam-5618	149	12	in	in	ADP
ejpam-5618	149	13	g.	g.	PROPN
ejpam-5618	149	14	therefore	therefore	ADV
ejpam-5618	149	15	,	,	PUNCT
ejpam-5618	149	16	m	m	PROPN
ejpam-5618	149	17	is	be	AUX
ejpam-5618	149	18	i	i	PRON
ejpam-5618	149	19	-	-	PUNCT
ejpam-5618	149	20	continuous	continuous	ADJ
ejpam-5618	149	21	.	.	PUNCT
ejpam-5618	150	1	similarly	similarly	ADV
ejpam-5618	150	2	,	,	PUNCT
ejpam-5618	150	3	we	we	PRON
ejpam-5618	150	4	can	can	AUX
ejpam-5618	150	5	show	show	VERB
ejpam-5618	150	6	that	that	SCONJ
ejpam-5618	150	7	inv	inv	NOUN
ejpam-5618	150	8	is	be	AUX
ejpam-5618	150	9	i	i	PRON
ejpam-5618	150	10	-	-	PUNCT
ejpam-5618	150	11	continuous	continuous	ADJ
ejpam-5618	150	12	.	.	PUNCT
ejpam-5618	151	1	hence	hence	ADV
ejpam-5618	151	2	,	,	PUNCT
ejpam-5618	151	3	g	g	PROPN
ejpam-5618	151	4	is	be	AUX
ejpam-5618	151	5	an	an	DET
ejpam-5618	151	6	ideal	ideal	ADJ
ejpam-5618	151	7	topological	topological	ADJ
ejpam-5618	151	8	group	group	NOUN
ejpam-5618	151	9	.	.	PUNCT
ejpam-5618	152	1	note	note	VERB
ejpam-5618	152	2	that	that	SCONJ
ejpam-5618	152	3	if	if	SCONJ
ejpam-5618	152	4	g	g	PROPN
ejpam-5618	152	5	is	be	AUX
ejpam-5618	152	6	a	a	DET
ejpam-5618	152	7	submaximal	submaximal	ADJ
ejpam-5618	152	8	space	space	NOUN
ejpam-5618	152	9	,	,	PUNCT
ejpam-5618	152	10	then	then	ADV
ejpam-5618	152	11	every	every	DET
ejpam-5618	152	12	an	an	DET
ejpam-5618	152	13	i	i	PRON
ejpam-5618	152	14	-	-	PUNCT
ejpam-5618	152	15	open	open	ADJ
ejpam-5618	152	16	set	set	NOUN
ejpam-5618	152	17	of	of	ADP
ejpam-5618	152	18	g	g	PROPN
ejpam-5618	152	19	is	be	AUX
ejpam-5618	152	20	open	open	ADJ
ejpam-5618	152	21	.	.	PUNCT
ejpam-5618	153	1	therefore	therefore	ADV
ejpam-5618	153	2	,	,	PUNCT
ejpam-5618	153	3	we	we	PRON
ejpam-5618	153	4	have	have	VERB
ejpam-5618	153	5	the	the	DET
ejpam-5618	153	6	following	follow	VERB
ejpam-5618	153	7	straightforward	straightforward	ADJ
ejpam-5618	153	8	result	result	NOUN
ejpam-5618	153	9	.	.	PUNCT
ejpam-5618	154	1	theorem	theorem	ADJ
ejpam-5618	154	2	8	8	NUM
ejpam-5618	154	3	.	.	PUNCT
ejpam-5618	155	1	let	let	VERB
ejpam-5618	155	2	g	g	PRON
ejpam-5618	155	3	be	be	AUX
ejpam-5618	155	4	an	an	DET
ejpam-5618	155	5	ideal	ideal	ADJ
ejpam-5618	155	6	topological	topological	ADJ
ejpam-5618	155	7	group	group	NOUN
ejpam-5618	155	8	.	.	PUNCT
ejpam-5618	156	1	if	if	SCONJ
ejpam-5618	156	2	g	g	PROPN
ejpam-5618	156	3	is	be	AUX
ejpam-5618	156	4	submaximal	submaximal	ADJ
ejpam-5618	156	5	,	,	PUNCT
ejpam-5618	156	6	then	then	ADV
ejpam-5618	156	7	g	g	PROPN
ejpam-5618	156	8	is	be	AUX
ejpam-5618	156	9	a	a	DET
ejpam-5618	156	10	topological	topological	ADJ
ejpam-5618	156	11	group	group	NOUN
ejpam-5618	156	12	.	.	PUNCT
ejpam-5618	157	1	proof	proof	NOUN
ejpam-5618	157	2	.	.	PUNCT
ejpam-5618	158	1	suppose	suppose	VERB
ejpam-5618	158	2	that	that	SCONJ
ejpam-5618	158	3	g	g	PROPN
ejpam-5618	158	4	is	be	AUX
ejpam-5618	158	5	submaximal	submaximal	ADJ
ejpam-5618	158	6	.	.	PUNCT
ejpam-5618	159	1	we	we	PRON
ejpam-5618	159	2	shall	shall	AUX
ejpam-5618	159	3	show	show	VERB
ejpam-5618	159	4	that	that	SCONJ
ejpam-5618	159	5	the	the	DET
ejpam-5618	159	6	multiplication	multiplication	NOUN
ejpam-5618	159	7	mapping	mapping	NOUN
ejpam-5618	159	8	m	m	VERB
ejpam-5618	159	9	:	:	PUNCT
ejpam-5618	159	10	g×g	g×g	PROPN
ejpam-5618	159	11	→	→	SYM
ejpam-5618	159	12	g	g	PROPN
ejpam-5618	159	13	and	and	CCONJ
ejpam-5618	159	14	the	the	DET
ejpam-5618	159	15	inverse	inverse	NOUN
ejpam-5618	159	16	mapping	mapping	NOUN
ejpam-5618	159	17	inv	inv	VERB
ejpam-5618	159	18	:	:	PUNCT
ejpam-5618	159	19	g	g	NOUN
ejpam-5618	159	20	→	→	SYM
ejpam-5618	159	21	g	g	NOUN
ejpam-5618	159	22	both	both	PRON
ejpam-5618	159	23	are	be	AUX
ejpam-5618	159	24	continuous	continuous	ADJ
ejpam-5618	159	25	.	.	PUNCT
ejpam-5618	160	1	first	first	ADV
ejpam-5618	160	2	we	we	PRON
ejpam-5618	160	3	show	show	VERB
ejpam-5618	160	4	that	that	SCONJ
ejpam-5618	160	5	m	m	NOUN
ejpam-5618	160	6	is	be	AUX
ejpam-5618	160	7	continuous	continuous	ADJ
ejpam-5618	160	8	.	.	PUNCT
ejpam-5618	161	1	let	let	VERB
ejpam-5618	161	2	w	w	NOUN
ejpam-5618	161	3	be	be	AUX
ejpam-5618	161	4	an	an	DET
ejpam-5618	161	5	open	open	ADJ
ejpam-5618	161	6	set	set	NOUN
ejpam-5618	161	7	in	in	ADP
ejpam-5618	161	8	g.	g.	PROPN
ejpam-5618	161	9	then	then	ADV
ejpam-5618	161	10	there	there	PRON
ejpam-5618	161	11	exist	exist	VERB
ejpam-5618	161	12	i	i	PRON
ejpam-5618	161	13	-	-	PUNCT
ejpam-5618	161	14	open	open	ADJ
ejpam-5618	161	15	sets	set	VERB
ejpam-5618	161	16	u	u	NOUN
ejpam-5618	161	17	and	and	CCONJ
ejpam-5618	161	18	v	v	NOUN
ejpam-5618	161	19	in	in	ADP
ejpam-5618	161	20	g	g	PROPN
ejpam-5618	161	21	such	such	DET
ejpam-5618	161	22	that	that	PRON
ejpam-5618	161	23	uv	uv	PROPN
ejpam-5618	162	1	⊂	⊂	PROPN
ejpam-5618	162	2	w	w	NOUN
ejpam-5618	162	3	since	since	SCONJ
ejpam-5618	162	4	g	g	PROPN
ejpam-5618	162	5	is	be	AUX
ejpam-5618	162	6	an	an	DET
ejpam-5618	162	7	ideal	ideal	ADJ
ejpam-5618	162	8	topological	topological	ADJ
ejpam-5618	162	9	group	group	NOUN
ejpam-5618	162	10	.	.	PUNCT
ejpam-5618	163	1	since	since	SCONJ
ejpam-5618	163	2	g	g	PROPN
ejpam-5618	163	3	is	be	AUX
ejpam-5618	163	4	submaximal	submaximal	ADJ
ejpam-5618	163	5	,	,	PUNCT
ejpam-5618	163	6	u	u	NOUN
ejpam-5618	163	7	and	and	CCONJ
ejpam-5618	163	8	v	v	NOUN
ejpam-5618	163	9	are	be	AUX
ejpam-5618	163	10	open	open	ADJ
ejpam-5618	163	11	in	in	ADP
ejpam-5618	163	12	g.	g.	PROPN
ejpam-5618	163	13	therefore	therefore	ADV
ejpam-5618	163	14	,	,	PUNCT
ejpam-5618	163	15	m	m	VERB
ejpam-5618	163	16	is	be	AUX
ejpam-5618	163	17	continuous	continuous	ADJ
ejpam-5618	163	18	.	.	PUNCT
ejpam-5618	164	1	similarly	similarly	ADV
ejpam-5618	164	2	,	,	PUNCT
ejpam-5618	164	3	we	we	PRON
ejpam-5618	164	4	can	can	AUX
ejpam-5618	164	5	show	show	VERB
ejpam-5618	164	6	that	that	SCONJ
ejpam-5618	164	7	inv	inv	NOUN
ejpam-5618	164	8	is	be	AUX
ejpam-5618	164	9	continuous	continuous	ADJ
ejpam-5618	164	10	.	.	PUNCT
ejpam-5618	165	1	hence	hence	ADV
ejpam-5618	165	2	,	,	PUNCT
ejpam-5618	165	3	g	g	PROPN
ejpam-5618	165	4	is	be	AUX
ejpam-5618	165	5	a	a	DET
ejpam-5618	165	6	topological	topological	ADJ
ejpam-5618	165	7	group	group	NOUN
ejpam-5618	165	8	.	.	PUNCT
ejpam-5618	166	1	theorem	theorem	VERB
ejpam-5618	166	2	9	9	NUM
ejpam-5618	166	3	.	.	PUNCT
ejpam-5618	167	1	let	let	VERB
ejpam-5618	167	2	g	g	PRON
ejpam-5618	167	3	be	be	AUX
ejpam-5618	167	4	an	an	DET
ejpam-5618	167	5	ideal	ideal	ADJ
ejpam-5618	167	6	topological	topological	ADJ
ejpam-5618	167	7	group	group	NOUN
ejpam-5618	167	8	and	and	CCONJ
ejpam-5618	168	1	g	g	PROPN
ejpam-5618	168	2	∈	∈	PROPN
ejpam-5618	168	3	g.	g.	NOUN
ejpam-5618	168	4	then	then	ADV
ejpam-5618	168	5	each	each	DET
ejpam-5618	168	6	left	left	ADJ
ejpam-5618	168	7	(	(	PUNCT
ejpam-5618	168	8	right	right	ADJ
ejpam-5618	168	9	)	)	PUNCT
ejpam-5618	168	10	translation	translation	NOUN
ejpam-5618	168	11	map	map	NOUN
ejpam-5618	168	12	lg	lg	NOUN
ejpam-5618	168	13	:	:	PUNCT
ejpam-5618	168	14	g	g	PROPN
ejpam-5618	168	15	→	→	SYM
ejpam-5618	168	16	g	g	PROPN
ejpam-5618	168	17	(	(	PUNCT
ejpam-5618	168	18	rg	rg	X
ejpam-5618	168	19	:	:	PUNCT
ejpam-5618	168	20	g	g	PROPN
ejpam-5618	168	21	→	→	SYM
ejpam-5618	168	22	g	g	NOUN
ejpam-5618	168	23	)	)	PUNCT
ejpam-5618	168	24	is	be	AUX
ejpam-5618	168	25	an	an	DET
ejpam-5618	168	26	i	i	PROPN
ejpam-5618	168	27	-	-	PUNCT
ejpam-5618	168	28	homeomorphism	homeomorphism	PROPN
ejpam-5618	168	29	.	.	PUNCT
ejpam-5618	169	1	proof	proof	NOUN
ejpam-5618	169	2	.	.	PUNCT
ejpam-5618	170	1	we	we	PRON
ejpam-5618	170	2	prove	prove	VERB
ejpam-5618	170	3	that	that	SCONJ
ejpam-5618	170	4	left	leave	VERB
ejpam-5618	170	5	translation	translation	NOUN
ejpam-5618	170	6	map	map	NOUN
ejpam-5618	170	7	lg	lg	NOUN
ejpam-5618	170	8	is	be	AUX
ejpam-5618	170	9	an	an	DET
ejpam-5618	170	10	i	i	PROPN
ejpam-5618	170	11	-	-	PUNCT
ejpam-5618	170	12	homeomorphism	homeomorphism	PROPN
ejpam-5618	170	13	.	.	PUNCT
ejpam-5618	171	1	obviously	obviously	ADV
ejpam-5618	171	2	,	,	PUNCT
ejpam-5618	171	3	lg	lg	NOUN
ejpam-5618	171	4	is	be	AUX
ejpam-5618	171	5	a	a	DET
ejpam-5618	171	6	bijective	bijective	ADJ
ejpam-5618	171	7	mapping	mapping	NOUN
ejpam-5618	171	8	.	.	PUNCT
ejpam-5618	172	1	let	let	VERB
ejpam-5618	172	2	x	x	PRON
ejpam-5618	172	3	be	be	AUX
ejpam-5618	172	4	an	an	DET
ejpam-5618	172	5	element	element	NOUN
ejpam-5618	172	6	in	in	ADP
ejpam-5618	172	7	g	g	NOUN
ejpam-5618	172	8	;	;	PUNCT
ejpam-5618	172	9	let	let	VERB
ejpam-5618	172	10	w	w	NOUN
ejpam-5618	172	11	be	be	AUX
ejpam-5618	172	12	an	an	DET
ejpam-5618	172	13	open	open	ADJ
ejpam-5618	172	14	neighborhood	neighborhood	NOUN
ejpam-5618	172	15	of	of	ADP
ejpam-5618	172	16	lg(x	lg(x	PUNCT
ejpam-5618	173	1	)	)	PUNCT
ejpam-5618	173	2	=	=	SYM
ejpam-5618	173	3	gx	gx	PROPN
ejpam-5618	173	4	.	.	PUNCT
ejpam-5618	174	1	since	since	SCONJ
ejpam-5618	174	2	g	g	PROPN
ejpam-5618	174	3	is	be	AUX
ejpam-5618	174	4	an	an	DET
ejpam-5618	174	5	ideal	ideal	ADJ
ejpam-5618	174	6	topological	topological	ADJ
ejpam-5618	174	7	group	group	NOUN
ejpam-5618	174	8	,	,	PUNCT
ejpam-5618	174	9	there	there	PRON
ejpam-5618	174	10	are	be	VERB
ejpam-5618	174	11	i	i	NOUN
ejpam-5618	174	12	-	-	PUNCT
ejpam-5618	174	13	open	open	ADJ
ejpam-5618	174	14	sets	set	VERB
ejpam-5618	174	15	u	u	NOUN
ejpam-5618	174	16	and	and	CCONJ
ejpam-5618	174	17	v	v	ADP
ejpam-5618	174	18	containing	contain	VERB
ejpam-5618	174	19	g	g	NOUN
ejpam-5618	174	20	and	and	CCONJ
ejpam-5618	174	21	x	x	NOUN
ejpam-5618	174	22	,	,	PUNCT
ejpam-5618	174	23	respectively	respectively	ADV
ejpam-5618	174	24	,	,	PUNCT
ejpam-5618	174	25	such	such	ADJ
ejpam-5618	174	26	that	that	SCONJ
ejpam-5618	174	27	uv	uv	PROPN
ejpam-5618	174	28	⊂	⊂	PROPN
ejpam-5618	174	29	w	w	PROPN
ejpam-5618	174	30	.	.	PUNCT
ejpam-5618	175	1	this	this	PRON
ejpam-5618	175	2	shows	show	VERB
ejpam-5618	175	3	that	that	SCONJ
ejpam-5618	175	4	lg	lg	PROPN
ejpam-5618	175	5	is	be	AUX
ejpam-5618	175	6	i	i	NOUN
ejpam-5618	175	7	-	-	PUNCT
ejpam-5618	175	8	continuous	continuous	ADJ
ejpam-5618	175	9	.	.	PUNCT
ejpam-5618	176	1	on	on	ADP
ejpam-5618	176	2	the	the	DET
ejpam-5618	176	3	other	other	ADJ
ejpam-5618	176	4	hand	hand	NOUN
ejpam-5618	176	5	,	,	PUNCT
ejpam-5618	176	6	the	the	DET
ejpam-5618	176	7	inverse	inverse	NOUN
ejpam-5618	176	8	mapping	mapping	NOUN
ejpam-5618	176	9	of	of	ADP
ejpam-5618	176	10	lg	lg	NOUN
ejpam-5618	176	11	is	be	AUX
ejpam-5618	176	12	defined	define	VERB
ejpam-5618	176	13	as	as	ADP
ejpam-5618	176	14	(	(	PUNCT
ejpam-5618	176	15	lg(x	lg(x	PUNCT
ejpam-5618	176	16	)	)	PUNCT
ejpam-5618	176	17	)	)	PUNCT
ejpam-5618	177	1	−1	−1	NOUN
ejpam-5618	177	2	=	=	SYM
ejpam-5618	177	3	g−1x	g−1x	NOUN
ejpam-5618	177	4	=	=	SYM
ejpam-5618	177	5	lg−1(x	lg−1(x	NOUN
ejpam-5618	177	6	)	)	PUNCT
ejpam-5618	177	7	.	.	PUNCT
ejpam-5618	178	1	this	this	PRON
ejpam-5618	178	2	indicates	indicate	VERB
ejpam-5618	178	3	that	that	SCONJ
ejpam-5618	178	4	(	(	PUNCT
ejpam-5618	178	5	lg(x	lg(x	PUNCT
ejpam-5618	178	6	)	)	PUNCT
ejpam-5618	178	7	)	)	PUNCT
ejpam-5618	178	8	−1	−1	NOUN
ejpam-5618	178	9	is	be	AUX
ejpam-5618	178	10	i	i	NOUN
ejpam-5618	178	11	-	-	PUNCT
ejpam-5618	178	12	continuous	continuous	ADJ
ejpam-5618	178	13	.	.	PUNCT
ejpam-5618	179	1	we	we	PRON
ejpam-5618	179	2	get	get	VERB
ejpam-5618	179	3	,	,	PUNCT
ejpam-5618	179	4	lg	lg	NOUN
ejpam-5618	179	5	is	be	AUX
ejpam-5618	179	6	an	an	DET
ejpam-5618	179	7	i	i	PROPN
ejpam-5618	179	8	-	-	PUNCT
ejpam-5618	179	9	homeomorphism	homeomorphism	PROPN
ejpam-5618	179	10	.	.	PUNCT
ejpam-5618	180	1	similarly	similarly	ADV
ejpam-5618	180	2	,	,	PUNCT
ejpam-5618	180	3	we	we	PRON
ejpam-5618	180	4	can	can	AUX
ejpam-5618	180	5	show	show	VERB
ejpam-5618	180	6	that	that	SCONJ
ejpam-5618	180	7	right	right	ADJ
ejpam-5618	180	8	translation	translation	NOUN
ejpam-5618	180	9	map	map	NOUN
ejpam-5618	180	10	rg	rg	PROPN
ejpam-5618	180	11	is	be	AUX
ejpam-5618	180	12	an	an	DET
ejpam-5618	180	13	i	i	PROPN
ejpam-5618	180	14	-	-	PUNCT
ejpam-5618	180	15	homeomorphism	homeomorphism	PROPN
ejpam-5618	180	16	.	.	PUNCT
ejpam-5618	181	1	theorem	theorem	NOUN
ejpam-5618	181	2	9	9	NUM
ejpam-5618	181	3	above	above	ADV
ejpam-5618	181	4	implies	imply	VERB
ejpam-5618	181	5	that	that	SCONJ
ejpam-5618	181	6	lg	lg	PROPN
ejpam-5618	181	7	and	and	CCONJ
ejpam-5618	181	8	rg	rg	PROPN
ejpam-5618	181	9	are	be	AUX
ejpam-5618	181	10	i	i	NOUN
ejpam-5618	181	11	-	-	PUNCT
ejpam-5618	181	12	open	open	ADJ
ejpam-5618	181	13	mappings	mapping	NOUN
ejpam-5618	181	14	for	for	ADP
ejpam-5618	181	15	each	each	DET
ejpam-5618	181	16	g	g	PROPN
ejpam-5618	181	17	∈	∈	PROPN
ejpam-5618	181	18	g.	g.	NOUN
ejpam-5618	182	1	hence	hence	ADV
ejpam-5618	182	2	,	,	PUNCT
ejpam-5618	182	3	we	we	PRON
ejpam-5618	182	4	have	have	VERB
ejpam-5618	182	5	the	the	DET
ejpam-5618	182	6	following	follow	VERB
ejpam-5618	182	7	immediate	immediate	ADJ
ejpam-5618	182	8	result	result	NOUN
ejpam-5618	182	9	.	.	PUNCT
ejpam-5618	183	1	s.	s.	PROPN
ejpam-5618	183	2	alammar	alammar	PROPN
ejpam-5618	183	3	,	,	PUNCT
ejpam-5618	183	4	m.	m.	NOUN
ejpam-5618	183	5	al	al	PROPN
ejpam-5618	183	6	shumrani	shumrani	PROPN
ejpam-5618	183	7	,	,	PUNCT
ejpam-5618	183	8	c.	c.	PROPN
ejpam-5618	183	9	özel	özel	PROPN
ejpam-5618	183	10	/	/	SYM
ejpam-5618	183	11	eur	eur	PROPN
ejpam-5618	183	12	.	.	PUNCT
ejpam-5618	184	1	j.	j.	PROPN
ejpam-5618	184	2	pure	pure	PROPN
ejpam-5618	184	3	appl	appl	PROPN
ejpam-5618	184	4	.	.	PROPN
ejpam-5618	184	5	math	math	PROPN
ejpam-5618	184	6	,	,	PUNCT
ejpam-5618	184	7	18	18	NUM
ejpam-5618	184	8	(	(	PUNCT
ejpam-5618	184	9	1	1	NUM
ejpam-5618	184	10	)	)	PUNCT
ejpam-5618	184	11	(	(	PUNCT
ejpam-5618	184	12	2025	2025	NUM
ejpam-5618	184	13	)	)	PUNCT
ejpam-5618	184	14	,	,	PUNCT
ejpam-5618	184	15	5618	5618	NUM
ejpam-5618	184	16	6	6	NUM
ejpam-5618	184	17	of	of	ADP
ejpam-5618	184	18	13	13	NUM
ejpam-5618	184	19	corollary	corollary	ADJ
ejpam-5618	184	20	1	1	NUM
ejpam-5618	184	21	.	.	PUNCT
ejpam-5618	185	1	in	in	ADP
ejpam-5618	185	2	any	any	DET
ejpam-5618	185	3	ideal	ideal	ADJ
ejpam-5618	185	4	topological	topological	ADJ
ejpam-5618	185	5	group	group	NOUN
ejpam-5618	185	6	,	,	PUNCT
ejpam-5618	185	7	every	every	DET
ejpam-5618	185	8	open	open	ADJ
ejpam-5618	185	9	set	set	NOUN
ejpam-5618	185	10	is	be	AUX
ejpam-5618	185	11	i	i	PRON
ejpam-5618	185	12	-	-	PUNCT
ejpam-5618	185	13	open	open	ADJ
ejpam-5618	185	14	.	.	PUNCT
ejpam-5618	186	1	proposition	proposition	NOUN
ejpam-5618	186	2	1	1	NUM
ejpam-5618	186	3	.	.	PUNCT
ejpam-5618	187	1	let	let	VERB
ejpam-5618	187	2	g	g	PRON
ejpam-5618	187	3	be	be	AUX
ejpam-5618	187	4	an	an	DET
ejpam-5618	187	5	ideal	ideal	ADJ
ejpam-5618	187	6	topological	topological	ADJ
ejpam-5618	187	7	group	group	NOUN
ejpam-5618	187	8	.	.	PUNCT
ejpam-5618	188	1	let	let	VERB
ejpam-5618	188	2	a	a	PRON
ejpam-5618	188	3	and	and	CCONJ
ejpam-5618	188	4	b	b	NOUN
ejpam-5618	188	5	be	be	AUX
ejpam-5618	188	6	subsets	subset	NOUN
ejpam-5618	188	7	in	in	ADP
ejpam-5618	188	8	g	g	PROPN
ejpam-5618	188	9	and	and	CCONJ
ejpam-5618	188	10	g	g	PROPN
ejpam-5618	188	11	∈	∈	PROPN
ejpam-5618	189	1	g.	g.	NOUN
ejpam-5618	190	1	then	then	ADV
ejpam-5618	190	2	(	(	PUNCT
ejpam-5618	190	3	i	i	NOUN
ejpam-5618	190	4	)	)	PUNCT
ejpam-5618	190	5	if	if	SCONJ
ejpam-5618	190	6	a	a	PRON
ejpam-5618	190	7	is	be	AUX
ejpam-5618	190	8	an	an	DET
ejpam-5618	190	9	open	open	ADJ
ejpam-5618	190	10	set	set	NOUN
ejpam-5618	190	11	,	,	PUNCT
ejpam-5618	190	12	then	then	ADV
ejpam-5618	190	13	ag	ag	PROPN
ejpam-5618	190	14	and	and	CCONJ
ejpam-5618	190	15	ga	ga	PROPN
ejpam-5618	190	16	both	both	PRON
ejpam-5618	190	17	are	be	AUX
ejpam-5618	190	18	i	i	NOUN
ejpam-5618	190	19	-	-	PUNCT
ejpam-5618	190	20	open	open	ADJ
ejpam-5618	190	21	sets	set	NOUN
ejpam-5618	190	22	.	.	PUNCT
ejpam-5618	191	1	(	(	PUNCT
ejpam-5618	191	2	ii	ii	NOUN
ejpam-5618	191	3	)	)	PUNCT
ejpam-5618	191	4	if	if	SCONJ
ejpam-5618	191	5	a	a	PRON
ejpam-5618	191	6	is	be	AUX
ejpam-5618	191	7	a	a	DET
ejpam-5618	191	8	closed	closed	ADJ
ejpam-5618	191	9	set	set	NOUN
ejpam-5618	191	10	,	,	PUNCT
ejpam-5618	191	11	then	then	ADV
ejpam-5618	191	12	ag	ag	PROPN
ejpam-5618	191	13	and	and	CCONJ
ejpam-5618	191	14	ga	ga	PROPN
ejpam-5618	191	15	both	both	PRON
ejpam-5618	191	16	are	be	AUX
ejpam-5618	191	17	i	i	NOUN
ejpam-5618	191	18	-	-	PUNCT
ejpam-5618	191	19	closed	close	VERB
ejpam-5618	191	20	sets	set	NOUN
ejpam-5618	191	21	.	.	PUNCT
ejpam-5618	192	1	(	(	PUNCT
ejpam-5618	192	2	iii	iii	X
ejpam-5618	192	3	)	)	PUNCT
ejpam-5618	192	4	if	if	SCONJ
ejpam-5618	192	5	a	a	PRON
ejpam-5618	192	6	is	be	AUX
ejpam-5618	192	7	an	an	DET
ejpam-5618	192	8	open	open	ADJ
ejpam-5618	192	9	set	set	NOUN
ejpam-5618	192	10	,	,	PUNCT
ejpam-5618	192	11	then	then	ADV
ejpam-5618	192	12	ab	ab	PROPN
ejpam-5618	192	13	and	and	CCONJ
ejpam-5618	192	14	ba	ba	PROPN
ejpam-5618	192	15	both	both	PRON
ejpam-5618	192	16	are	be	AUX
ejpam-5618	192	17	i	i	NOUN
ejpam-5618	192	18	-	-	PUNCT
ejpam-5618	192	19	open	open	ADJ
ejpam-5618	192	20	sets	set	NOUN
ejpam-5618	192	21	.	.	PUNCT
ejpam-5618	193	1	proof	proof	NOUN
ejpam-5618	193	2	.	.	PUNCT
ejpam-5618	194	1	(	(	PUNCT
ejpam-5618	194	2	i	i	NOUN
ejpam-5618	194	3	)	)	PUNCT
ejpam-5618	194	4	and	and	CCONJ
ejpam-5618	194	5	(	(	PUNCT
ejpam-5618	194	6	ii	ii	AUX
ejpam-5618	194	7	)	)	PUNCT
ejpam-5618	194	8	follow	follow	VERB
ejpam-5618	194	9	using	use	VERB
ejpam-5618	194	10	theorem	theorem	NOUN
ejpam-5618	194	11	9	9	NUM
ejpam-5618	194	12	.	.	PUNCT
ejpam-5618	195	1	since	since	SCONJ
ejpam-5618	195	2	ab	ab	PROPN
ejpam-5618	195	3	=	=	PROPN
ejpam-5618	195	4	∪b∈bab	∪b∈bab	PROPN
ejpam-5618	195	5	and	and	CCONJ
ejpam-5618	195	6	the	the	DET
ejpam-5618	195	7	union	union	NOUN
ejpam-5618	195	8	of	of	ADP
ejpam-5618	195	9	i	i	PROPN
ejpam-5618	195	10	-	-	PUNCT
ejpam-5618	195	11	open	open	ADJ
ejpam-5618	195	12	sets	set	NOUN
ejpam-5618	195	13	is	be	AUX
ejpam-5618	195	14	i	i	NOUN
ejpam-5618	195	15	-	-	PUNCT
ejpam-5618	195	16	open	open	ADJ
ejpam-5618	195	17	,	,	PUNCT
ejpam-5618	195	18	ab	ab	PROPN
ejpam-5618	195	19	is	be	AUX
ejpam-5618	195	20	i	i	NOUN
ejpam-5618	195	21	-	-	PUNCT
ejpam-5618	195	22	open	open	ADJ
ejpam-5618	195	23	and	and	CCONJ
ejpam-5618	195	24	similarly	similarly	ADV
ejpam-5618	195	25	,	,	PUNCT
ejpam-5618	195	26	ba	ba	PROPN
ejpam-5618	195	27	is	be	AUX
ejpam-5618	195	28	i	i	PRON
ejpam-5618	195	29	-	-	PUNCT
ejpam-5618	195	30	open	open	ADJ
ejpam-5618	195	31	.	.	PUNCT
ejpam-5618	196	1	hence	hence	ADV
ejpam-5618	196	2	,	,	PUNCT
ejpam-5618	196	3	(	(	PUNCT
ejpam-5618	196	4	iii	iii	X
ejpam-5618	196	5	)	)	PUNCT
ejpam-5618	196	6	is	be	AUX
ejpam-5618	196	7	proved	prove	VERB
ejpam-5618	196	8	.	.	PUNCT
ejpam-5618	197	1	theorem	theorem	ADJ
ejpam-5618	197	2	10	10	NUM
ejpam-5618	197	3	.	.	PUNCT
ejpam-5618	198	1	let	let	VERB
ejpam-5618	198	2	g	g	PRON
ejpam-5618	198	3	be	be	AUX
ejpam-5618	198	4	an	an	DET
ejpam-5618	198	5	ideal	ideal	ADJ
ejpam-5618	198	6	topological	topological	ADJ
ejpam-5618	198	7	group	group	NOUN
ejpam-5618	198	8	.	.	PUNCT
ejpam-5618	199	1	then	then	ADV
ejpam-5618	199	2	the	the	DET
ejpam-5618	199	3	inverse	inverse	NOUN
ejpam-5618	199	4	mapping	mapping	NOUN
ejpam-5618	199	5	inv	inv	VERB
ejpam-5618	199	6	:	:	PUNCT
ejpam-5618	199	7	g	g	NOUN
ejpam-5618	199	8	→	→	SYM
ejpam-5618	199	9	g	g	NOUN
ejpam-5618	199	10	defined	define	VERB
ejpam-5618	199	11	by	by	ADP
ejpam-5618	199	12	inv(x	inv(x	PROPN
ejpam-5618	199	13	)	)	PUNCT
ejpam-5618	199	14	=	=	PUNCT
ejpam-5618	200	1	x−1	x−1	PROPN
ejpam-5618	200	2	is	be	AUX
ejpam-5618	200	3	an	an	DET
ejpam-5618	200	4	i	i	PROPN
ejpam-5618	200	5	-	-	PUNCT
ejpam-5618	200	6	homeomorphism	homeomorphism	PROPN
ejpam-5618	200	7	.	.	PUNCT
ejpam-5618	201	1	proof	proof	NOUN
ejpam-5618	201	2	.	.	PUNCT
ejpam-5618	202	1	it	it	PRON
ejpam-5618	202	2	is	be	AUX
ejpam-5618	202	3	clear	clear	ADJ
ejpam-5618	202	4	that	that	SCONJ
ejpam-5618	202	5	inv	inv	VERB
ejpam-5618	202	6	mapping	mapping	NOUN
ejpam-5618	202	7	is	be	AUX
ejpam-5618	202	8	bijective	bijective	ADJ
ejpam-5618	202	9	.	.	PUNCT
ejpam-5618	203	1	since	since	SCONJ
ejpam-5618	203	2	g	g	PROPN
ejpam-5618	203	3	is	be	AUX
ejpam-5618	203	4	an	an	DET
ejpam-5618	203	5	ideal	ideal	ADJ
ejpam-5618	203	6	topological	topological	ADJ
ejpam-5618	203	7	group	group	NOUN
ejpam-5618	203	8	,	,	PUNCT
ejpam-5618	203	9	inv	inv	VERB
ejpam-5618	203	10	is	be	AUX
ejpam-5618	203	11	i	i	PRON
ejpam-5618	203	12	-	-	PUNCT
ejpam-5618	203	13	continuous	continuous	ADJ
ejpam-5618	203	14	.	.	PUNCT
ejpam-5618	204	1	since	since	SCONJ
ejpam-5618	204	2	(	(	PUNCT
ejpam-5618	204	3	inv)−1(x	inv)−1(x	NOUN
ejpam-5618	204	4	)	)	PUNCT
ejpam-5618	204	5	=	=	SYM
ejpam-5618	204	6	x−1	x−1	PROPN
ejpam-5618	204	7	,	,	PUNCT
ejpam-5618	204	8	we	we	PRON
ejpam-5618	204	9	have	have	VERB
ejpam-5618	204	10	that	that	PRON
ejpam-5618	204	11	(	(	PUNCT
ejpam-5618	204	12	inv)−1	inv)−1	NOUN
ejpam-5618	204	13	is	be	AUX
ejpam-5618	204	14	i	i	NOUN
ejpam-5618	204	15	-	-	PUNCT
ejpam-5618	204	16	continuous	continuous	ADJ
ejpam-5618	204	17	.	.	PUNCT
ejpam-5618	205	1	hence	hence	ADV
ejpam-5618	205	2	,	,	PUNCT
ejpam-5618	205	3	inv	inv	VERB
ejpam-5618	205	4	is	be	AUX
ejpam-5618	205	5	an	an	DET
ejpam-5618	205	6	i	i	PROPN
ejpam-5618	205	7	-	-	PUNCT
ejpam-5618	205	8	homeomorphism	homeomorphism	PROPN
ejpam-5618	205	9	.	.	PUNCT
ejpam-5618	206	1	corollary	corollary	ADJ
ejpam-5618	206	2	2	2	NUM
ejpam-5618	206	3	.	.	PUNCT
ejpam-5618	207	1	let	let	VERB
ejpam-5618	207	2	g	g	PRON
ejpam-5618	207	3	be	be	AUX
ejpam-5618	207	4	an	an	DET
ejpam-5618	207	5	ideal	ideal	ADJ
ejpam-5618	207	6	topological	topological	ADJ
ejpam-5618	207	7	group	group	NOUN
ejpam-5618	207	8	.	.	PUNCT
ejpam-5618	208	1	if	if	SCONJ
ejpam-5618	208	2	a	a	PRON
ejpam-5618	208	3	is	be	AUX
ejpam-5618	208	4	an	an	DET
ejpam-5618	208	5	open	open	ADJ
ejpam-5618	208	6	subset	subset	NOUN
ejpam-5618	208	7	of	of	ADP
ejpam-5618	208	8	g	g	PROPN
ejpam-5618	208	9	,	,	PUNCT
ejpam-5618	208	10	then	then	ADV
ejpam-5618	208	11	a−1	a−1	PROPN
ejpam-5618	208	12	is	be	AUX
ejpam-5618	208	13	an	an	DET
ejpam-5618	208	14	i	i	NOUN
ejpam-5618	208	15	-	-	PUNCT
ejpam-5618	208	16	open	open	ADJ
ejpam-5618	208	17	set	set	NOUN
ejpam-5618	208	18	.	.	PUNCT
ejpam-5618	209	1	corollary	corollary	ADJ
ejpam-5618	209	2	3	3	X
ejpam-5618	209	3	.	.	PUNCT
ejpam-5618	210	1	let	let	VERB
ejpam-5618	210	2	g	g	PRON
ejpam-5618	210	3	be	be	AUX
ejpam-5618	210	4	an	an	DET
ejpam-5618	210	5	ideal	ideal	ADJ
ejpam-5618	210	6	topological	topological	ADJ
ejpam-5618	210	7	group	group	NOUN
ejpam-5618	210	8	.	.	PUNCT
ejpam-5618	211	1	let	let	VERB
ejpam-5618	211	2	a	a	DET
ejpam-5618	211	3	be	be	AUX
ejpam-5618	211	4	an	an	DET
ejpam-5618	211	5	open	open	ADJ
ejpam-5618	211	6	subset	subset	NOUN
ejpam-5618	211	7	of	of	ADP
ejpam-5618	211	8	g.	g.	PROPN
ejpam-5618	211	9	then	then	ADV
ejpam-5618	211	10	there	there	PRON
ejpam-5618	211	11	exists	exist	VERB
ejpam-5618	211	12	a	a	DET
ejpam-5618	211	13	symmetric	symmetric	ADJ
ejpam-5618	211	14	i	i	NOUN
ejpam-5618	211	15	-	-	PUNCT
ejpam-5618	211	16	open	open	VERB
ejpam-5618	211	17	set	set	NOUN
ejpam-5618	211	18	u	u	NOUN
ejpam-5618	211	19	of	of	ADP
ejpam-5618	211	20	g	g	PROPN
ejpam-5618	211	21	such	such	ADJ
ejpam-5618	211	22	that	that	SCONJ
ejpam-5618	211	23	u	u	PROPN
ejpam-5618	211	24	⊂	⊂	PROPN
ejpam-5618	211	25	a.	a.	NOUN
ejpam-5618	211	26	proof	proof	NOUN
ejpam-5618	211	27	.	.	PUNCT
ejpam-5618	212	1	by	by	ADP
ejpam-5618	212	2	corollary	corollary	ADJ
ejpam-5618	212	3	2	2	NUM
ejpam-5618	212	4	,	,	PUNCT
ejpam-5618	212	5	a−1	a−1	PROPN
ejpam-5618	212	6	is	be	AUX
ejpam-5618	212	7	i	i	PRON
ejpam-5618	212	8	-	-	PUNCT
ejpam-5618	212	9	open	open	ADJ
ejpam-5618	212	10	.	.	PUNCT
ejpam-5618	213	1	let	let	VERB
ejpam-5618	213	2	u	u	NOUN
ejpam-5618	213	3	=	=	PROPN
ejpam-5618	213	4	a	a	DET
ejpam-5618	213	5	∩a−1	∩a−1	PROPN
ejpam-5618	213	6	.	.	PUNCT
ejpam-5618	214	1	then	then	ADV
ejpam-5618	214	2	u	u	NOUN
ejpam-5618	214	3	is	be	AUX
ejpam-5618	214	4	i	i	PRON
ejpam-5618	214	5	-	-	PUNCT
ejpam-5618	214	6	open	open	ADJ
ejpam-5618	214	7	being	be	AUX
ejpam-5618	214	8	the	the	DET
ejpam-5618	214	9	intersection	intersection	NOUN
ejpam-5618	214	10	of	of	ADP
ejpam-5618	214	11	open	open	ADJ
ejpam-5618	214	12	set	set	NOUN
ejpam-5618	214	13	with	with	ADP
ejpam-5618	214	14	i	i	PRON
ejpam-5618	214	15	-	-	PUNCT
ejpam-5618	214	16	open	open	ADJ
ejpam-5618	214	17	set	set	NOUN
ejpam-5618	214	18	.	.	PUNCT
ejpam-5618	215	1	u	u	NOUN
ejpam-5618	215	2	is	be	AUX
ejpam-5618	215	3	symmetric	symmetric	ADJ
ejpam-5618	215	4	and	and	CCONJ
ejpam-5618	215	5	u	u	NOUN
ejpam-5618	215	6	⊂	⊂	PROPN
ejpam-5618	215	7	a.	a.	NOUN
ejpam-5618	215	8	theorem	theorem	VERB
ejpam-5618	215	9	11	11	NUM
ejpam-5618	215	10	.	.	PUNCT
ejpam-5618	216	1	let	let	VERB
ejpam-5618	216	2	g	g	PRON
ejpam-5618	216	3	be	be	AUX
ejpam-5618	216	4	an	an	DET
ejpam-5618	216	5	ideal	ideal	ADJ
ejpam-5618	216	6	topological	topological	ADJ
ejpam-5618	216	7	group	group	NOUN
ejpam-5618	216	8	,	,	PUNCT
ejpam-5618	216	9	and	and	CCONJ
ejpam-5618	216	10	let	let	VERB
ejpam-5618	216	11	a	a	PRON
ejpam-5618	216	12	be	be	AUX
ejpam-5618	216	13	an	an	DET
ejpam-5618	216	14	i	i	NOUN
ejpam-5618	216	15	-	-	PUNCT
ejpam-5618	216	16	open	open	ADJ
ejpam-5618	216	17	set	set	NOUN
ejpam-5618	216	18	in	in	ADP
ejpam-5618	216	19	g.	g.	PROPN
ejpam-5618	216	20	if	if	SCONJ
ejpam-5618	216	21	g	g	PROPN
ejpam-5618	216	22	is	be	AUX
ejpam-5618	216	23	submaximal	submaximal	ADJ
ejpam-5618	216	24	,	,	PUNCT
ejpam-5618	216	25	then	then	ADV
ejpam-5618	216	26	(	(	PUNCT
ejpam-5618	216	27	i	i	NOUN
ejpam-5618	216	28	)	)	PUNCT
ejpam-5618	216	29	ag	ag	PROPN
ejpam-5618	216	30	and	and	CCONJ
ejpam-5618	216	31	ga	ga	PROPN
ejpam-5618	216	32	both	both	PRON
ejpam-5618	216	33	are	be	AUX
ejpam-5618	216	34	i	i	PRON
ejpam-5618	216	35	-	-	PUNCT
ejpam-5618	216	36	open	open	ADJ
ejpam-5618	216	37	set	set	NOUN
ejpam-5618	216	38	for	for	ADP
ejpam-5618	216	39	any	any	DET
ejpam-5618	216	40	g	g	PROPN
ejpam-5618	216	41	∈	∈	PROPN
ejpam-5618	216	42	g.	g.	PROPN
ejpam-5618	216	43	(	(	PUNCT
ejpam-5618	216	44	ii	ii	PROPN
ejpam-5618	216	45	)	)	PUNCT
ejpam-5618	216	46	ab	ab	PROPN
ejpam-5618	216	47	and	and	CCONJ
ejpam-5618	216	48	ba	ba	PROPN
ejpam-5618	216	49	both	both	PRON
ejpam-5618	216	50	are	be	AUX
ejpam-5618	216	51	i	i	PRON
ejpam-5618	216	52	-	-	PUNCT
ejpam-5618	216	53	open	open	ADJ
ejpam-5618	216	54	set	set	NOUN
ejpam-5618	216	55	for	for	ADP
ejpam-5618	216	56	any	any	DET
ejpam-5618	216	57	b	b	PROPN
ejpam-5618	216	58	⊂	⊂	PROPN
ejpam-5618	216	59	g.	g.	PROPN
ejpam-5618	216	60	(	(	PUNCT
ejpam-5618	216	61	iii	iii	X
ejpam-5618	216	62	)	)	PUNCT
ejpam-5618	216	63	a	a	PRON
ejpam-5618	216	64	is	be	AUX
ejpam-5618	216	65	i	i	PRON
ejpam-5618	216	66	-	-	PUNCT
ejpam-5618	216	67	open	open	ADJ
ejpam-5618	216	68	if	if	SCONJ
ejpam-5618	217	1	and	and	CCONJ
ejpam-5618	217	2	only	only	ADV
ejpam-5618	217	3	if	if	SCONJ
ejpam-5618	217	4	a−1	a−1	PROPN
ejpam-5618	217	5	is	be	AUX
ejpam-5618	217	6	i	i	PRON
ejpam-5618	217	7	-	-	PUNCT
ejpam-5618	217	8	open	open	ADJ
ejpam-5618	217	9	.	.	PUNCT
ejpam-5618	218	1	proof	proof	NOUN
ejpam-5618	218	2	.	.	PUNCT
ejpam-5618	219	1	since	since	SCONJ
ejpam-5618	219	2	g	g	PROPN
ejpam-5618	219	3	is	be	AUX
ejpam-5618	219	4	submaximal	submaximal	ADJ
ejpam-5618	219	5	,	,	PUNCT
ejpam-5618	219	6	a	a	PRON
ejpam-5618	219	7	is	be	AUX
ejpam-5618	219	8	open	open	ADJ
ejpam-5618	219	9	.	.	PUNCT
ejpam-5618	220	1	using	use	VERB
ejpam-5618	220	2	proposition	proposition	NOUN
ejpam-5618	220	3	1	1	NUM
ejpam-5618	220	4	,	,	PUNCT
ejpam-5618	220	5	(	(	PUNCT
ejpam-5618	220	6	i	i	NOUN
ejpam-5618	220	7	)	)	PUNCT
ejpam-5618	220	8	and	and	CCONJ
ejpam-5618	220	9	(	(	PUNCT
ejpam-5618	220	10	ii	ii	NOUN
ejpam-5618	220	11	)	)	PUNCT
ejpam-5618	220	12	follow	follow	VERB
ejpam-5618	220	13	.	.	PUNCT
ejpam-5618	221	1	(	(	PUNCT
ejpam-5618	221	2	iii	iii	X
ejpam-5618	221	3	)	)	PUNCT
ejpam-5618	221	4	follows	follow	VERB
ejpam-5618	221	5	immediately	immediately	ADV
ejpam-5618	221	6	using	use	VERB
ejpam-5618	221	7	corollary	corollary	ADJ
ejpam-5618	221	8	2	2	NUM
ejpam-5618	221	9	.	.	PUNCT
ejpam-5618	221	10	theorem	theorem	NOUN
ejpam-5618	221	11	12	12	NUM
ejpam-5618	221	12	.	.	PUNCT
ejpam-5618	222	1	let	let	VERB
ejpam-5618	222	2	g	g	PRON
ejpam-5618	222	3	be	be	AUX
ejpam-5618	222	4	an	an	DET
ejpam-5618	222	5	ideal	ideal	ADJ
ejpam-5618	222	6	topological	topological	ADJ
ejpam-5618	222	7	group	group	NOUN
ejpam-5618	222	8	.	.	PUNCT
ejpam-5618	223	1	suppose	suppose	VERB
ejpam-5618	223	2	g	g	PROPN
ejpam-5618	223	3	is	be	AUX
ejpam-5618	223	4	a	a	DET
ejpam-5618	223	5	submaximal	submaximal	ADJ
ejpam-5618	223	6	space	space	NOUN
ejpam-5618	223	7	such	such	ADJ
ejpam-5618	223	8	that	that	SCONJ
ejpam-5618	223	9	g	g	PROPN
ejpam-5618	223	10	×	×	NOUN
ejpam-5618	223	11	g	g	PROPN
ejpam-5618	223	12	is	be	AUX
ejpam-5618	223	13	submaximal	submaximal	ADJ
ejpam-5618	223	14	.	.	PUNCT
ejpam-5618	224	1	then	then	ADV
ejpam-5618	224	2	the	the	DET
ejpam-5618	224	3	mapping	mapping	NOUN
ejpam-5618	224	4	f	f	X
ejpam-5618	224	5	:	:	PUNCT
ejpam-5618	224	6	g	g	PROPN
ejpam-5618	224	7	×	×	PROPN
ejpam-5618	224	8	g	g	NOUN
ejpam-5618	224	9	→	→	SYM
ejpam-5618	224	10	g	g	PROPN
ejpam-5618	224	11	sending	send	VERB
ejpam-5618	224	12	(	(	PUNCT
ejpam-5618	224	13	x	x	NOUN
ejpam-5618	224	14	,	,	PUNCT
ejpam-5618	224	15	y	y	NOUN
ejpam-5618	224	16	)	)	PUNCT
ejpam-5618	224	17	to	to	ADP
ejpam-5618	224	18	xy−1	xy−1	PROPN
ejpam-5618	224	19	is	be	AUX
ejpam-5618	224	20	i	i	NOUN
ejpam-5618	224	21	-	-	PUNCT
ejpam-5618	224	22	continuous	continuous	ADJ
ejpam-5618	224	23	.	.	PUNCT
ejpam-5618	225	1	s.	s.	PROPN
ejpam-5618	225	2	alammar	alammar	PROPN
ejpam-5618	225	3	,	,	PUNCT
ejpam-5618	225	4	m.	m.	NOUN
ejpam-5618	225	5	al	al	PROPN
ejpam-5618	225	6	shumrani	shumrani	PROPN
ejpam-5618	225	7	,	,	PUNCT
ejpam-5618	225	8	c.	c.	PROPN
ejpam-5618	225	9	özel	özel	PROPN
ejpam-5618	225	10	/	/	SYM
ejpam-5618	225	11	eur	eur	PROPN
ejpam-5618	225	12	.	.	PUNCT
ejpam-5618	226	1	j.	j.	PROPN
ejpam-5618	226	2	pure	pure	PROPN
ejpam-5618	226	3	appl	appl	PROPN
ejpam-5618	226	4	.	.	PROPN
ejpam-5618	226	5	math	math	PROPN
ejpam-5618	226	6	,	,	PUNCT
ejpam-5618	226	7	18	18	NUM
ejpam-5618	226	8	(	(	PUNCT
ejpam-5618	226	9	1	1	NUM
ejpam-5618	226	10	)	)	PUNCT
ejpam-5618	226	11	(	(	PUNCT
ejpam-5618	226	12	2025	2025	NUM
ejpam-5618	226	13	)	)	PUNCT
ejpam-5618	226	14	,	,	PUNCT
ejpam-5618	226	15	5618	5618	NUM
ejpam-5618	226	16	7	7	NUM
ejpam-5618	226	17	of	of	ADP
ejpam-5618	226	18	13	13	NUM
ejpam-5618	226	19	proof	proof	NOUN
ejpam-5618	226	20	.	.	PUNCT
ejpam-5618	227	1	consider	consider	VERB
ejpam-5618	227	2	the	the	DET
ejpam-5618	227	3	mapping	mapping	NOUN
ejpam-5618	227	4	h	h	NOUN
ejpam-5618	227	5	:	:	PUNCT
ejpam-5618	227	6	g	g	ADP
ejpam-5618	227	7	×	×	NOUN
ejpam-5618	227	8	g	g	NOUN
ejpam-5618	227	9	→	→	SYM
ejpam-5618	227	10	g	g	PROPN
ejpam-5618	227	11	×	×	NOUN
ejpam-5618	227	12	g	g	NOUN
ejpam-5618	227	13	such	such	ADJ
ejpam-5618	227	14	that	that	SCONJ
ejpam-5618	227	15	h(x	h(x	PROPN
ejpam-5618	227	16	,	,	PUNCT
ejpam-5618	227	17	y	y	PROPN
ejpam-5618	227	18	)	)	PUNCT
ejpam-5618	227	19	=	=	SYM
ejpam-5618	227	20	(	(	PUNCT
ejpam-5618	227	21	x	x	X
ejpam-5618	227	22	,	,	PUNCT
ejpam-5618	227	23	inv(y	inv(y	PROPN
ejpam-5618	227	24	)	)	PUNCT
ejpam-5618	227	25	)	)	PUNCT
ejpam-5618	227	26	.	.	PUNCT
ejpam-5618	228	1	first	first	ADV
ejpam-5618	228	2	we	we	PRON
ejpam-5618	228	3	show	show	VERB
ejpam-5618	228	4	that	that	SCONJ
ejpam-5618	228	5	h	h	NOUN
ejpam-5618	228	6	is	be	AUX
ejpam-5618	228	7	i	i	NOUN
ejpam-5618	228	8	-	-	PUNCT
ejpam-5618	228	9	irresolute	irresolute	ADJ
ejpam-5618	228	10	.	.	PUNCT
ejpam-5618	229	1	take	take	VERB
ejpam-5618	229	2	an	an	DET
ejpam-5618	229	3	i	i	NOUN
ejpam-5618	229	4	-	-	PUNCT
ejpam-5618	229	5	open	open	ADJ
ejpam-5618	229	6	set	set	PROPN
ejpam-5618	229	7	u×v	u×v	PROPN
ejpam-5618	229	8	.	.	PUNCT
ejpam-5618	230	1	from	from	ADP
ejpam-5618	230	2	the	the	DET
ejpam-5618	230	3	submaximality	submaximality	NOUN
ejpam-5618	230	4	of	of	ADP
ejpam-5618	230	5	g×g	g×g	PROPN
ejpam-5618	230	6	,	,	PUNCT
ejpam-5618	230	7	u×v	u×v	PROPN
ejpam-5618	230	8	is	be	AUX
ejpam-5618	230	9	open	open	ADJ
ejpam-5618	230	10	.	.	PUNCT
ejpam-5618	231	1	h−1(u	h−1(u	PROPN
ejpam-5618	231	2	,	,	PUNCT
ejpam-5618	231	3	v	v	NOUN
ejpam-5618	231	4	)	)	PUNCT
ejpam-5618	231	5	=	=	SYM
ejpam-5618	231	6	(	(	PUNCT
ejpam-5618	231	7	u	u	NOUN
ejpam-5618	231	8	,	,	PUNCT
ejpam-5618	231	9	inv−1(v	inv−1(v	PROPN
ejpam-5618	231	10	)	)	PUNCT
ejpam-5618	231	11	)	)	PUNCT
ejpam-5618	231	12	.	.	PUNCT
ejpam-5618	232	1	hence	hence	ADV
ejpam-5618	232	2	,	,	PUNCT
ejpam-5618	232	3	(	(	PUNCT
ejpam-5618	232	4	u	u	NOUN
ejpam-5618	232	5	,	,	PUNCT
ejpam-5618	232	6	inv−1(v	inv−1(v	PROPN
ejpam-5618	232	7	)	)	PUNCT
ejpam-5618	232	8	)	)	PUNCT
ejpam-5618	232	9	is	be	AUX
ejpam-5618	232	10	i	i	PRON
ejpam-5618	232	11	-	-	PUNCT
ejpam-5618	232	12	open	open	ADJ
ejpam-5618	232	13	since	since	SCONJ
ejpam-5618	232	14	inv	inv	NOUN
ejpam-5618	232	15	is	be	AUX
ejpam-5618	232	16	i	i	PRON
ejpam-5618	232	17	-	-	PUNCT
ejpam-5618	232	18	continuous	continuous	ADJ
ejpam-5618	232	19	.	.	PUNCT
ejpam-5618	233	1	note	note	VERB
ejpam-5618	233	2	that	that	SCONJ
ejpam-5618	233	3	f(x	f(x	PROPN
ejpam-5618	233	4	,	,	PUNCT
ejpam-5618	233	5	y	y	NOUN
ejpam-5618	233	6	)	)	PUNCT
ejpam-5618	233	7	=	=	SYM
ejpam-5618	233	8	m(h(x	m(h(x	PROPN
ejpam-5618	233	9	,	,	PUNCT
ejpam-5618	233	10	y	y	PROPN
ejpam-5618	233	11	)	)	PUNCT
ejpam-5618	233	12	)	)	PUNCT
ejpam-5618	234	1	=	=	SYM
ejpam-5618	234	2	m((x	m((x	NOUN
ejpam-5618	234	3	,	,	PUNCT
ejpam-5618	234	4	inv(y	inv(y	PROPN
ejpam-5618	234	5	)	)	PUNCT
ejpam-5618	234	6	)	)	PUNCT
ejpam-5618	235	1	=	=	SYM
ejpam-5618	235	2	xinv(y	xinv(y	NUM
ejpam-5618	235	3	)	)	PUNCT
ejpam-5618	235	4	=	=	SYM
ejpam-5618	236	1	xy−1	xy−1	PROPN
ejpam-5618	236	2	,	,	PUNCT
ejpam-5618	236	3	that	that	ADV
ejpam-5618	236	4	is	be	AUX
ejpam-5618	236	5	,	,	PUNCT
ejpam-5618	236	6	f	f	PROPN
ejpam-5618	236	7	=	=	PUNCT
ejpam-5618	236	8	m	m	VERB
ejpam-5618	236	9	◦	◦	NOUN
ejpam-5618	236	10	h.	h.	NOUN
ejpam-5618	236	11	since	since	SCONJ
ejpam-5618	236	12	the	the	DET
ejpam-5618	236	13	multiplication	multiplication	NOUN
ejpam-5618	236	14	mapping	mapping	NOUN
ejpam-5618	236	15	m	m	NOUN
ejpam-5618	236	16	is	be	AUX
ejpam-5618	236	17	i	i	PRON
ejpam-5618	236	18	-	-	PUNCT
ejpam-5618	236	19	continuous	continuous	ADJ
ejpam-5618	236	20	and	and	CCONJ
ejpam-5618	236	21	h	h	NOUN
ejpam-5618	236	22	is	be	AUX
ejpam-5618	236	23	i	i	NOUN
ejpam-5618	236	24	-	-	PUNCT
ejpam-5618	236	25	irresolute	irresolute	ADJ
ejpam-5618	236	26	,	,	PUNCT
ejpam-5618	236	27	theorem	theorem	ADJ
ejpam-5618	236	28	3	3	NUM
ejpam-5618	236	29	implies	imply	VERB
ejpam-5618	236	30	that	that	SCONJ
ejpam-5618	236	31	f	f	PROPN
ejpam-5618	236	32	is	be	AUX
ejpam-5618	236	33	i	i	PRON
ejpam-5618	236	34	-	-	PUNCT
ejpam-5618	236	35	continuous	continuous	ADJ
ejpam-5618	236	36	.	.	PUNCT
ejpam-5618	237	1	theorem	theorem	NOUN
ejpam-5618	237	2	13	13	NUM
ejpam-5618	237	3	.	.	PUNCT
ejpam-5618	238	1	let	let	VERB
ejpam-5618	238	2	g	g	PRON
ejpam-5618	238	3	be	be	AUX
ejpam-5618	238	4	an	an	DET
ejpam-5618	238	5	ideal	ideal	ADJ
ejpam-5618	238	6	topological	topological	ADJ
ejpam-5618	238	7	group	group	NOUN
ejpam-5618	238	8	.	.	PUNCT
ejpam-5618	239	1	let	let	VERB
ejpam-5618	239	2	βe	βe	PRON
ejpam-5618	239	3	be	be	AUX
ejpam-5618	239	4	an	an	DET
ejpam-5618	239	5	open	open	ADJ
ejpam-5618	239	6	base	base	NOUN
ejpam-5618	239	7	at	at	ADP
ejpam-5618	239	8	the	the	DET
ejpam-5618	239	9	identity	identity	NOUN
ejpam-5618	239	10	element	element	NOUN
ejpam-5618	239	11	e	e	PROPN
ejpam-5618	239	12	of	of	ADP
ejpam-5618	239	13	g.	g.	PROPN
ejpam-5618	240	1	then	then	ADV
ejpam-5618	240	2	(	(	PUNCT
ejpam-5618	240	3	i	i	NOUN
ejpam-5618	240	4	)	)	PUNCT
ejpam-5618	240	5	for	for	ADP
ejpam-5618	240	6	every	every	DET
ejpam-5618	240	7	u	u	PROPN
ejpam-5618	240	8	∈	∈	PROPN
ejpam-5618	240	9	βe	βe	PRON
ejpam-5618	240	10	,	,	PUNCT
ejpam-5618	240	11	there	there	PRON
ejpam-5618	240	12	is	be	VERB
ejpam-5618	240	13	v	v	ADP
ejpam-5618	240	14	an	an	DET
ejpam-5618	240	15	i	i	NOUN
ejpam-5618	240	16	-	-	PUNCT
ejpam-5618	240	17	open	open	ADJ
ejpam-5618	240	18	neighborhood	neighborhood	NOUN
ejpam-5618	240	19	of	of	ADP
ejpam-5618	240	20	e	e	NOUN
ejpam-5618	240	21	such	such	ADJ
ejpam-5618	240	22	that	that	PRON
ejpam-5618	240	23	v	v	ADP
ejpam-5618	240	24	2	2	NUM
ejpam-5618	240	25	⊂	⊂	NOUN
ejpam-5618	240	26	u	u	PROPN
ejpam-5618	240	27	.	.	PUNCT
ejpam-5618	241	1	(	(	PUNCT
ejpam-5618	241	2	ii	ii	NOUN
ejpam-5618	241	3	)	)	PUNCT
ejpam-5618	241	4	for	for	ADP
ejpam-5618	241	5	every	every	DET
ejpam-5618	241	6	u	u	PROPN
ejpam-5618	241	7	∈	∈	PROPN
ejpam-5618	241	8	βe	βe	PRON
ejpam-5618	241	9	,	,	PUNCT
ejpam-5618	241	10	there	there	PRON
ejpam-5618	241	11	is	be	VERB
ejpam-5618	241	12	v	v	ADP
ejpam-5618	241	13	an	an	DET
ejpam-5618	241	14	i	i	NOUN
ejpam-5618	241	15	-	-	PUNCT
ejpam-5618	241	16	open	open	ADJ
ejpam-5618	241	17	neighborhood	neighborhood	NOUN
ejpam-5618	241	18	of	of	ADP
ejpam-5618	241	19	e	e	NOUN
ejpam-5618	241	20	such	such	ADJ
ejpam-5618	241	21	that	that	DET
ejpam-5618	241	22	v	v	ADP
ejpam-5618	241	23	−1	−1	NOUN
ejpam-5618	241	24	⊂	⊂	X
ejpam-5618	241	25	u	u	PROPN
ejpam-5618	241	26	.	.	PUNCT
ejpam-5618	242	1	(	(	PUNCT
ejpam-5618	242	2	iii	iii	NOUN
ejpam-5618	242	3	)	)	PUNCT
ejpam-5618	242	4	for	for	ADP
ejpam-5618	242	5	every	every	DET
ejpam-5618	242	6	u	u	PROPN
ejpam-5618	242	7	∈	∈	PROPN
ejpam-5618	243	1	βe	βe	X
ejpam-5618	243	2	and	and	CCONJ
ejpam-5618	243	3	g	g	PROPN
ejpam-5618	243	4	∈	∈	PROPN
ejpam-5618	243	5	g	g	PROPN
ejpam-5618	243	6	,	,	PUNCT
ejpam-5618	243	7	there	there	PRON
ejpam-5618	243	8	is	be	VERB
ejpam-5618	243	9	v	v	ADP
ejpam-5618	243	10	an	an	DET
ejpam-5618	243	11	i	i	NOUN
ejpam-5618	243	12	-	-	PUNCT
ejpam-5618	243	13	open	open	ADJ
ejpam-5618	243	14	neighborhood	neighborhood	NOUN
ejpam-5618	243	15	of	of	ADP
ejpam-5618	243	16	e	e	NOUN
ejpam-5618	243	17	such	such	ADJ
ejpam-5618	243	18	that	that	SCONJ
ejpam-5618	243	19	gv	gv	ADP
ejpam-5618	243	20	⊂	⊂	PROPN
ejpam-5618	243	21	u	u	PROPN
ejpam-5618	243	22	.	.	PUNCT
ejpam-5618	244	1	(	(	PUNCT
ejpam-5618	244	2	iv	iv	X
ejpam-5618	244	3	)	)	PUNCT
ejpam-5618	244	4	if	if	SCONJ
ejpam-5618	244	5	g	g	PROPN
ejpam-5618	244	6	is	be	AUX
ejpam-5618	244	7	it1	it1	NOUN
ejpam-5618	244	8	-	-	PUNCT
ejpam-5618	244	9	space	space	NOUN
ejpam-5618	244	10	,	,	PUNCT
ejpam-5618	244	11	then	then	ADV
ejpam-5618	244	12	for	for	ADP
ejpam-5618	244	13	every	every	DET
ejpam-5618	244	14	u	u	NOUN
ejpam-5618	244	15	,	,	PUNCT
ejpam-5618	244	16	v	v	X
ejpam-5618	244	17	∈	∈	PROPN
ejpam-5618	244	18	βe	βe	PRON
ejpam-5618	244	19	,	,	PUNCT
ejpam-5618	244	20	there	there	PRON
ejpam-5618	244	21	is	be	VERB
ejpam-5618	244	22	w	w	ADP
ejpam-5618	244	23	an	an	DET
ejpam-5618	244	24	i	i	NOUN
ejpam-5618	244	25	-	-	PUNCT
ejpam-5618	244	26	open	open	ADJ
ejpam-5618	244	27	neighborhood	neighborhood	NOUN
ejpam-5618	244	28	of	of	ADP
ejpam-5618	244	29	e	e	NOUN
ejpam-5618	244	30	such	such	ADJ
ejpam-5618	244	31	that	that	SCONJ
ejpam-5618	244	32	w	w	PROPN
ejpam-5618	244	33	⊂	⊂	PROPN
ejpam-5618	244	34	u	u	PROPN
ejpam-5618	244	35	∩	∩	ADJ
ejpam-5618	244	36	v	v	NOUN
ejpam-5618	244	37	.	.	PUNCT
ejpam-5618	245	1	proof	proof	NOUN
ejpam-5618	245	2	.	.	PUNCT
ejpam-5618	246	1	(	(	PUNCT
ejpam-5618	246	2	i	i	NOUN
ejpam-5618	246	3	)	)	PUNCT
ejpam-5618	246	4	follows	follow	VERB
ejpam-5618	246	5	from	from	ADP
ejpam-5618	246	6	the	the	DET
ejpam-5618	246	7	i	i	NOUN
ejpam-5618	246	8	-	-	PUNCT
ejpam-5618	246	9	continuity	continuity	NOUN
ejpam-5618	246	10	of	of	ADP
ejpam-5618	246	11	the	the	DET
ejpam-5618	246	12	multiplication	multiplication	NOUN
ejpam-5618	246	13	mapping	mapping	NOUN
ejpam-5618	246	14	at	at	ADP
ejpam-5618	246	15	the	the	DET
ejpam-5618	246	16	identity	identity	NOUN
ejpam-5618	246	17	element	element	NOUN
ejpam-5618	246	18	e.	e.	PROPN
ejpam-5618	246	19	(	(	PUNCT
ejpam-5618	246	20	ii	ii	PROPN
ejpam-5618	246	21	)	)	PUNCT
ejpam-5618	246	22	follows	follow	VERB
ejpam-5618	246	23	from	from	ADP
ejpam-5618	246	24	the	the	DET
ejpam-5618	246	25	i	i	NOUN
ejpam-5618	246	26	-	-	PUNCT
ejpam-5618	246	27	continuity	continuity	NOUN
ejpam-5618	246	28	of	of	ADP
ejpam-5618	246	29	the	the	DET
ejpam-5618	246	30	inverse	inverse	NOUN
ejpam-5618	246	31	mapping	mapping	NOUN
ejpam-5618	246	32	at	at	ADP
ejpam-5618	246	33	the	the	DET
ejpam-5618	246	34	identity	identity	NOUN
ejpam-5618	246	35	element	element	NOUN
ejpam-5618	246	36	e.	e.	PROPN
ejpam-5618	246	37	(	(	PUNCT
ejpam-5618	246	38	iii	iii	NOUN
ejpam-5618	246	39	)	)	PUNCT
ejpam-5618	246	40	follows	follow	VERB
ejpam-5618	246	41	from	from	ADP
ejpam-5618	246	42	the	the	DET
ejpam-5618	246	43	i	i	NOUN
ejpam-5618	246	44	-	-	PUNCT
ejpam-5618	246	45	continuity	continuity	NOUN
ejpam-5618	246	46	of	of	ADP
ejpam-5618	246	47	the	the	DET
ejpam-5618	246	48	left	left	ADJ
ejpam-5618	246	49	translation	translation	NOUN
ejpam-5618	246	50	mapping	mapping	NOUN
ejpam-5618	246	51	in	in	ADP
ejpam-5618	246	52	g.	g.	PROPN
ejpam-5618	246	53	for	for	ADP
ejpam-5618	246	54	(	(	PUNCT
ejpam-5618	246	55	iv	iv	X
ejpam-5618	246	56	)	)	PUNCT
ejpam-5618	246	57	,	,	PUNCT
ejpam-5618	246	58	u	u	PROPN
ejpam-5618	246	59	∩	∩	NOUN
ejpam-5618	246	60	v	v	NOUN
ejpam-5618	246	61	is	be	AUX
ejpam-5618	246	62	a	a	DET
ejpam-5618	246	63	neighborhood	neighborhood	NOUN
ejpam-5618	246	64	of	of	ADP
ejpam-5618	246	65	e.	e.	PROPN
ejpam-5618	246	66	therefore	therefore	ADV
ejpam-5618	246	67	,	,	PUNCT
ejpam-5618	246	68	there	there	PRON
ejpam-5618	246	69	is	be	VERB
ejpam-5618	246	70	w	w	ADP
ejpam-5618	246	71	∈	∈	PROPN
ejpam-5618	246	72	βe	βe	ADP
ejpam-5618	246	73	such	such	ADJ
ejpam-5618	246	74	that	that	SCONJ
ejpam-5618	246	75	w	w	PROPN
ejpam-5618	246	76	⊂	⊂	PROPN
ejpam-5618	246	77	u	u	PROPN
ejpam-5618	247	1	∩v	∩v	NOUN
ejpam-5618	247	2	.	.	PUNCT
ejpam-5618	248	1	w	w	PROPN
ejpam-5618	248	2	is	be	AUX
ejpam-5618	248	3	an	an	DET
ejpam-5618	248	4	i	i	NOUN
ejpam-5618	248	5	-	-	PUNCT
ejpam-5618	248	6	open	open	NOUN
ejpam-5618	248	7	set	set	NOUN
ejpam-5618	248	8	since	since	SCONJ
ejpam-5618	248	9	g	g	PROPN
ejpam-5618	248	10	is	be	AUX
ejpam-5618	248	11	an	an	DET
ejpam-5618	248	12	it1	it1	NOUN
ejpam-5618	248	13	-	-	PUNCT
ejpam-5618	248	14	space	space	NOUN
ejpam-5618	248	15	.	.	PUNCT
ejpam-5618	249	1	theorem	theorem	NOUN
ejpam-5618	249	2	14	14	NUM
ejpam-5618	249	3	.	.	PUNCT
ejpam-5618	250	1	let	let	VERB
ejpam-5618	250	2	g	g	PRON
ejpam-5618	250	3	be	be	AUX
ejpam-5618	250	4	an	an	DET
ejpam-5618	250	5	ideal	ideal	ADJ
ejpam-5618	250	6	topological	topological	ADJ
ejpam-5618	250	7	group	group	NOUN
ejpam-5618	250	8	.	.	PUNCT
ejpam-5618	251	1	let	let	VERB
ejpam-5618	251	2	βe	βe	PRON
ejpam-5618	251	3	be	be	AUX
ejpam-5618	251	4	an	an	DET
ejpam-5618	251	5	open	open	ADJ
ejpam-5618	251	6	base	base	NOUN
ejpam-5618	251	7	at	at	ADP
ejpam-5618	251	8	the	the	DET
ejpam-5618	251	9	identity	identity	NOUN
ejpam-5618	251	10	element	element	NOUN
ejpam-5618	251	11	e	e	PROPN
ejpam-5618	251	12	of	of	ADP
ejpam-5618	251	13	g.	g.	PROPN
ejpam-5618	251	14	if	if	SCONJ
ejpam-5618	251	15	g	g	PROPN
ejpam-5618	251	16	is	be	AUX
ejpam-5618	251	17	submaximal	submaximal	ADJ
ejpam-5618	251	18	,	,	PUNCT
ejpam-5618	251	19	then	then	ADV
ejpam-5618	251	20	(	(	PUNCT
ejpam-5618	251	21	i	i	NOUN
ejpam-5618	251	22	)	)	PUNCT
ejpam-5618	251	23	for	for	ADP
ejpam-5618	251	24	every	every	DET
ejpam-5618	251	25	u	u	PROPN
ejpam-5618	251	26	∈	∈	PROPN
ejpam-5618	251	27	βe	βe	PRON
ejpam-5618	251	28	,	,	PUNCT
ejpam-5618	251	29	there	there	PRON
ejpam-5618	251	30	is	be	VERB
ejpam-5618	251	31	v	v	ADP
ejpam-5618	251	32	an	an	DET
ejpam-5618	251	33	i	i	NOUN
ejpam-5618	251	34	-	-	PUNCT
ejpam-5618	251	35	open	open	ADJ
ejpam-5618	251	36	neighborhood	neighborhood	NOUN
ejpam-5618	251	37	of	of	ADP
ejpam-5618	251	38	e	e	NOUN
ejpam-5618	251	39	such	such	ADJ
ejpam-5618	251	40	that	that	PRON
ejpam-5618	251	41	v	v	ADP
ejpam-5618	251	42	2	2	NUM
ejpam-5618	251	43	⊂	⊂	NOUN
ejpam-5618	251	44	u	u	PROPN
ejpam-5618	251	45	.	.	PUNCT
ejpam-5618	252	1	(	(	PUNCT
ejpam-5618	252	2	ii	ii	NOUN
ejpam-5618	252	3	)	)	PUNCT
ejpam-5618	252	4	for	for	ADP
ejpam-5618	252	5	every	every	DET
ejpam-5618	252	6	u	u	PROPN
ejpam-5618	252	7	∈	∈	PROPN
ejpam-5618	252	8	βe	βe	PRON
ejpam-5618	252	9	,	,	PUNCT
ejpam-5618	252	10	there	there	PRON
ejpam-5618	252	11	is	be	VERB
ejpam-5618	252	12	v	v	ADP
ejpam-5618	252	13	an	an	DET
ejpam-5618	252	14	i	i	NOUN
ejpam-5618	252	15	-	-	PUNCT
ejpam-5618	252	16	open	open	ADJ
ejpam-5618	252	17	neighborhood	neighborhood	NOUN
ejpam-5618	252	18	of	of	ADP
ejpam-5618	252	19	e	e	NOUN
ejpam-5618	252	20	such	such	ADJ
ejpam-5618	252	21	that	that	DET
ejpam-5618	252	22	v	v	ADP
ejpam-5618	252	23	−1	−1	NOUN
ejpam-5618	252	24	⊂	⊂	X
ejpam-5618	252	25	u	u	PROPN
ejpam-5618	252	26	.	.	PUNCT
ejpam-5618	253	1	(	(	PUNCT
ejpam-5618	253	2	iii	iii	NOUN
ejpam-5618	253	3	)	)	PUNCT
ejpam-5618	253	4	for	for	ADP
ejpam-5618	253	5	every	every	DET
ejpam-5618	253	6	u	u	PROPN
ejpam-5618	253	7	∈	∈	PROPN
ejpam-5618	254	1	βe	βe	X
ejpam-5618	254	2	and	and	CCONJ
ejpam-5618	254	3	g	g	PROPN
ejpam-5618	254	4	∈	∈	PROPN
ejpam-5618	254	5	g	g	PROPN
ejpam-5618	254	6	,	,	PUNCT
ejpam-5618	254	7	there	there	PRON
ejpam-5618	254	8	is	be	VERB
ejpam-5618	254	9	v	v	ADP
ejpam-5618	254	10	an	an	DET
ejpam-5618	254	11	i	i	NOUN
ejpam-5618	254	12	-	-	PUNCT
ejpam-5618	254	13	open	open	ADJ
ejpam-5618	254	14	neighborhood	neighborhood	NOUN
ejpam-5618	254	15	of	of	ADP
ejpam-5618	254	16	e	e	NOUN
ejpam-5618	254	17	such	such	ADJ
ejpam-5618	254	18	that	that	SCONJ
ejpam-5618	254	19	gv	gv	ADP
ejpam-5618	254	20	⊂	⊂	PROPN
ejpam-5618	254	21	u	u	PROPN
ejpam-5618	254	22	.	.	PUNCT
ejpam-5618	255	1	(	(	PUNCT
ejpam-5618	255	2	iv	iv	X
ejpam-5618	255	3	)	)	PUNCT
ejpam-5618	255	4	assume	assume	VERB
ejpam-5618	255	5	that	that	SCONJ
ejpam-5618	255	6	g×g	g×g	PROPN
ejpam-5618	255	7	is	be	AUX
ejpam-5618	255	8	submaximal	submaximal	ADJ
ejpam-5618	255	9	.	.	PUNCT
ejpam-5618	256	1	for	for	ADP
ejpam-5618	256	2	every	every	DET
ejpam-5618	256	3	u	u	PROPN
ejpam-5618	256	4	∈	∈	PROPN
ejpam-5618	256	5	βe	βe	PRON
ejpam-5618	256	6	,	,	PUNCT
ejpam-5618	256	7	there	there	PRON
ejpam-5618	256	8	is	be	VERB
ejpam-5618	256	9	v	v	ADP
ejpam-5618	256	10	an	an	DET
ejpam-5618	256	11	i	i	NOUN
ejpam-5618	256	12	-	-	PUNCT
ejpam-5618	256	13	open	open	ADJ
ejpam-5618	256	14	neighborhood	neighborhood	NOUN
ejpam-5618	256	15	of	of	ADP
ejpam-5618	256	16	e	e	NOUN
ejpam-5618	256	17	such	such	ADJ
ejpam-5618	256	18	that	that	SCONJ
ejpam-5618	256	19	v	v	NOUN
ejpam-5618	256	20	v	v	NUM
ejpam-5618	256	21	−1	−1	NOUN
ejpam-5618	256	22	⊂	⊂	X
ejpam-5618	256	23	u	u	PROPN
ejpam-5618	256	24	.	.	PUNCT
ejpam-5618	257	1	(	(	PUNCT
ejpam-5618	257	2	v	v	NOUN
ejpam-5618	257	3	)	)	PUNCT
ejpam-5618	257	4	for	for	ADP
ejpam-5618	257	5	every	every	DET
ejpam-5618	257	6	u	u	PROPN
ejpam-5618	257	7	∈	∈	PROPN
ejpam-5618	257	8	βe	βe	X
ejpam-5618	257	9	and	and	CCONJ
ejpam-5618	257	10	g	g	PROPN
ejpam-5618	257	11	∈	∈	PROPN
ejpam-5618	257	12	g	g	PROPN
ejpam-5618	257	13	,	,	PUNCT
ejpam-5618	257	14	there	there	PRON
ejpam-5618	257	15	is	be	VERB
ejpam-5618	257	16	v	v	ADP
ejpam-5618	257	17	an	an	DET
ejpam-5618	257	18	i	i	NOUN
ejpam-5618	257	19	-	-	PUNCT
ejpam-5618	257	20	open	open	ADJ
ejpam-5618	257	21	neighborhood	neighborhood	NOUN
ejpam-5618	257	22	of	of	ADP
ejpam-5618	257	23	e	e	NOUN
ejpam-5618	257	24	such	such	ADJ
ejpam-5618	257	25	that	that	SCONJ
ejpam-5618	257	26	gv	gv	ADP
ejpam-5618	257	27	g−1	g−1	PROPN
ejpam-5618	257	28	⊂	⊂	PROPN
ejpam-5618	257	29	u	u	PROPN
ejpam-5618	257	30	.	.	PUNCT
ejpam-5618	258	1	proof	proof	NOUN
ejpam-5618	258	2	.	.	PUNCT
ejpam-5618	259	1	(	(	PUNCT
ejpam-5618	259	2	i),(ii	i),(ii	PROPN
ejpam-5618	259	3	)	)	PUNCT
ejpam-5618	259	4	and	and	CCONJ
ejpam-5618	259	5	(	(	PUNCT
ejpam-5618	259	6	iii	iii	X
ejpam-5618	259	7	)	)	PUNCT
ejpam-5618	259	8	are	be	AUX
ejpam-5618	259	9	proved	prove	VERB
ejpam-5618	259	10	in	in	ADP
ejpam-5618	259	11	theorem	theorem	NOUN
ejpam-5618	259	12	13	13	NUM
ejpam-5618	259	13	.	.	PUNCT
ejpam-5618	260	1	we	we	PRON
ejpam-5618	260	2	show	show	VERB
ejpam-5618	260	3	(	(	PUNCT
ejpam-5618	260	4	iv	iv	NUM
ejpam-5618	260	5	)	)	PUNCT
ejpam-5618	260	6	.	.	PUNCT
ejpam-5618	261	1	using	use	VERB
ejpam-5618	261	2	theorem	theorem	NOUN
ejpam-5618	261	3	12	12	NUM
ejpam-5618	261	4	,	,	PUNCT
ejpam-5618	261	5	the	the	DET
ejpam-5618	261	6	mapping	mapping	NOUN
ejpam-5618	261	7	f	f	X
ejpam-5618	261	8	:	:	PUNCT
ejpam-5618	261	9	g	g	ADP
ejpam-5618	261	10	×	×	NOUN
ejpam-5618	261	11	g	g	NOUN
ejpam-5618	261	12	−→	−→	NOUN
ejpam-5618	261	13	g	g	PROPN
ejpam-5618	261	14	defined	define	VERB
ejpam-5618	261	15	by	by	ADP
ejpam-5618	261	16	f(x	f(x	PROPN
ejpam-5618	261	17	,	,	PUNCT
ejpam-5618	261	18	y	y	NOUN
ejpam-5618	261	19	)	)	PUNCT
ejpam-5618	261	20	=	=	PUNCT
ejpam-5618	261	21	xy−1	xy−1	PROPN
ejpam-5618	261	22	is	be	AUX
ejpam-5618	261	23	i	i	PRON
ejpam-5618	261	24	-	-	PUNCT
ejpam-5618	261	25	continuous	continuous	ADJ
ejpam-5618	261	26	.	.	PUNCT
ejpam-5618	262	1	therefore	therefore	ADV
ejpam-5618	262	2	,	,	PUNCT
ejpam-5618	262	3	for	for	ADP
ejpam-5618	262	4	u	u	PROPN
ejpam-5618	262	5	∈	∈	PROPN
ejpam-5618	262	6	βe	βe	PRON
ejpam-5618	262	7	,	,	PUNCT
ejpam-5618	262	8	there	there	PRON
ejpam-5618	262	9	is	be	VERB
ejpam-5618	262	10	i	i	NOUN
ejpam-5618	262	11	-	-	PUNCT
ejpam-5618	262	12	open	open	ADJ
ejpam-5618	262	13	neighborhood	neighborhood	NOUN
ejpam-5618	262	14	a×b	a×b	PUNCT
ejpam-5618	262	15	of	of	ADP
ejpam-5618	262	16	(	(	PUNCT
ejpam-5618	262	17	e	e	NOUN
ejpam-5618	262	18	,	,	PUNCT
ejpam-5618	262	19	e	e	NOUN
ejpam-5618	262	20	)	)	PUNCT
ejpam-5618	262	21	in	in	ADP
ejpam-5618	262	22	g×g	g×g	PROPN
ejpam-5618	262	23	such	such	ADJ
ejpam-5618	262	24	that	that	PRON
ejpam-5618	262	25	f(a×b	f(a×b	PROPN
ejpam-5618	262	26	)	)	PUNCT
ejpam-5618	263	1	⊂	⊂	PROPN
ejpam-5618	263	2	u	u	PROPN
ejpam-5618	263	3	.	.	PUNCT
ejpam-5618	264	1	but	but	CCONJ
ejpam-5618	264	2	a∩b	a∩b	PROPN
ejpam-5618	264	3	is	be	AUX
ejpam-5618	264	4	an	an	DET
ejpam-5618	264	5	open	open	ADJ
ejpam-5618	264	6	neighborhood	neighborhood	NOUN
ejpam-5618	264	7	of	of	ADP
ejpam-5618	264	8	e.	e.	PROPN
ejpam-5618	264	9	thus	thus	ADV
ejpam-5618	264	10	,	,	PUNCT
ejpam-5618	264	11	there	there	PRON
ejpam-5618	264	12	is	be	VERB
ejpam-5618	264	13	v	v	ADP
ejpam-5618	264	14	∈	∈	NOUN
ejpam-5618	264	15	βe	βe	PRON
ejpam-5618	264	16	such	such	ADJ
ejpam-5618	265	1	that	that	PRON
ejpam-5618	265	2	v	v	X
ejpam-5618	265	3	⊂	⊂	PROPN
ejpam-5618	265	4	a∩b	a∩b	PROPN
ejpam-5618	265	5	.	.	PUNCT
ejpam-5618	265	6	note	note	VERB
ejpam-5618	265	7	that	that	SCONJ
ejpam-5618	265	8	v	v	NOUN
ejpam-5618	265	9	is	be	AUX
ejpam-5618	265	10	i	i	PRON
ejpam-5618	265	11	-	-	PUNCT
ejpam-5618	265	12	open	open	ADJ
ejpam-5618	265	13	by	by	ADP
ejpam-5618	265	14	corollary	corollary	ADJ
ejpam-5618	265	15	1	1	NUM
ejpam-5618	265	16	.	.	PUNCT
ejpam-5618	266	1	moreover	moreover	ADV
ejpam-5618	266	2	,	,	PUNCT
ejpam-5618	266	3	v	v	PRON
ejpam-5618	266	4	×v	×v	NOUN
ejpam-5618	266	5	⊂	⊂	PUNCT
ejpam-5618	266	6	a×b	a×b	PROPN
ejpam-5618	266	7	.	.	PUNCT
ejpam-5618	267	1	then	then	ADV
ejpam-5618	267	2	f(v	f(v	PROPN
ejpam-5618	267	3	×v	×v	VERB
ejpam-5618	267	4	)	)	PUNCT
ejpam-5618	267	5	⊂	⊂	PROPN
ejpam-5618	267	6	f(a×b	f(a×b	PROPN
ejpam-5618	267	7	)	)	PUNCT
ejpam-5618	268	1	⊂	⊂	PROPN
ejpam-5618	268	2	u	u	PROPN
ejpam-5618	268	3	.	.	PUNCT
ejpam-5618	269	1	s.	s.	PROPN
ejpam-5618	269	2	alammar	alammar	PROPN
ejpam-5618	269	3	,	,	PUNCT
ejpam-5618	269	4	m.	m.	NOUN
ejpam-5618	269	5	al	al	PROPN
ejpam-5618	269	6	shumrani	shumrani	PROPN
ejpam-5618	269	7	,	,	PUNCT
ejpam-5618	269	8	c.	c.	PROPN
ejpam-5618	269	9	özel	özel	PROPN
ejpam-5618	269	10	/	/	SYM
ejpam-5618	269	11	eur	eur	PROPN
ejpam-5618	269	12	.	.	PUNCT
ejpam-5618	270	1	j.	j.	PROPN
ejpam-5618	270	2	pure	pure	PROPN
ejpam-5618	270	3	appl	appl	PROPN
ejpam-5618	270	4	.	.	PROPN
ejpam-5618	270	5	math	math	PROPN
ejpam-5618	270	6	,	,	PUNCT
ejpam-5618	270	7	18	18	NUM
ejpam-5618	270	8	(	(	PUNCT
ejpam-5618	270	9	1	1	NUM
ejpam-5618	270	10	)	)	PUNCT
ejpam-5618	270	11	(	(	PUNCT
ejpam-5618	270	12	2025	2025	NUM
ejpam-5618	270	13	)	)	PUNCT
ejpam-5618	270	14	,	,	PUNCT
ejpam-5618	270	15	5618	5618	NUM
ejpam-5618	270	16	8	8	NUM
ejpam-5618	270	17	of	of	ADP
ejpam-5618	270	18	13	13	NUM
ejpam-5618	270	19	that	that	PRON
ejpam-5618	270	20	is	be	AUX
ejpam-5618	270	21	,	,	PUNCT
ejpam-5618	270	22	v	v	NOUN
ejpam-5618	270	23	v	v	X
ejpam-5618	270	24	−1	−1	NOUN
ejpam-5618	270	25	⊂	⊂	X
ejpam-5618	270	26	u	u	NOUN
ejpam-5618	270	27	.	.	PUNCT
ejpam-5618	271	1	we	we	PRON
ejpam-5618	271	2	show	show	VERB
ejpam-5618	271	3	(	(	PUNCT
ejpam-5618	271	4	v	v	NOUN
ejpam-5618	271	5	)	)	PUNCT
ejpam-5618	271	6	.	.	PUNCT
ejpam-5618	272	1	consider	consider	VERB
ejpam-5618	272	2	the	the	DET
ejpam-5618	272	3	mapping	mapping	NOUN
ejpam-5618	272	4	rg−1	rg−1	PROPN
ejpam-5618	272	5	◦	◦	NOUN
ejpam-5618	272	6	lg	lg	NOUN
ejpam-5618	272	7	:	:	PUNCT
ejpam-5618	273	1	g	g	PROPN
ejpam-5618	273	2	−→	−→	NOUN
ejpam-5618	273	3	g	g	PROPN
ejpam-5618	273	4	defined	define	VERB
ejpam-5618	273	5	by	by	ADP
ejpam-5618	273	6	rg−1	rg−1	PROPN
ejpam-5618	273	7	◦	◦	NOUN
ejpam-5618	273	8	lg(a	lg(a	PUNCT
ejpam-5618	273	9	)	)	PUNCT
ejpam-5618	274	1	=	=	SYM
ejpam-5618	275	1	gag−1	gag−1	PROPN
ejpam-5618	275	2	.	.	PUNCT
ejpam-5618	276	1	since	since	SCONJ
ejpam-5618	276	2	lx	lx	NOUN
ejpam-5618	276	3	is	be	AUX
ejpam-5618	276	4	i	i	PRON
ejpam-5618	276	5	-	-	PUNCT
ejpam-5618	276	6	continuous	continuous	ADJ
ejpam-5618	276	7	and	and	CCONJ
ejpam-5618	276	8	rx−1	rx−1	NOUN
ejpam-5618	276	9	is	be	AUX
ejpam-5618	276	10	continuous	continuous	ADJ
ejpam-5618	276	11	by	by	ADP
ejpam-5618	276	12	the	the	DET
ejpam-5618	276	13	submaximality	submaximality	NOUN
ejpam-5618	276	14	of	of	ADP
ejpam-5618	276	15	g	g	PROPN
ejpam-5618	276	16	,	,	PUNCT
ejpam-5618	276	17	we	we	PRON
ejpam-5618	276	18	have	have	VERB
ejpam-5618	276	19	that	that	DET
ejpam-5618	276	20	rx−1	rx−1	NOUN
ejpam-5618	276	21	◦	◦	NOUN
ejpam-5618	276	22	lx	lx	ADP
ejpam-5618	276	23	is	be	AUX
ejpam-5618	276	24	i	i	PRON
ejpam-5618	276	25	-	-	PUNCT
ejpam-5618	276	26	continuous	continuous	ADJ
ejpam-5618	276	27	by	by	ADP
ejpam-5618	276	28	theorem	theorem	NOUN
ejpam-5618	276	29	2	2	NUM
ejpam-5618	276	30	.	.	PUNCT
ejpam-5618	277	1	therefore	therefore	ADV
ejpam-5618	277	2	,	,	PUNCT
ejpam-5618	277	3	if	if	SCONJ
ejpam-5618	277	4	u	u	PROPN
ejpam-5618	277	5	∈	∈	PROPN
ejpam-5618	277	6	βe	βe	PRON
ejpam-5618	277	7	,	,	PUNCT
ejpam-5618	277	8	then	then	ADV
ejpam-5618	277	9	there	there	PRON
ejpam-5618	277	10	is	be	VERB
ejpam-5618	277	11	i	i	PRON
ejpam-5618	277	12	-	-	PUNCT
ejpam-5618	277	13	open	open	ADJ
ejpam-5618	277	14	set	set	VERB
ejpam-5618	277	15	v	v	NOUN
ejpam-5618	277	16	containing	contain	VERB
ejpam-5618	277	17	e	e	NOUN
ejpam-5618	277	18	such	such	ADJ
ejpam-5618	277	19	that	that	SCONJ
ejpam-5618	277	20	rg−1	rg−1	PROPN
ejpam-5618	277	21	◦	◦	NOUN
ejpam-5618	277	22	lg(v	lg(v	PUNCT
ejpam-5618	277	23	)	)	PUNCT
ejpam-5618	278	1	⊂	⊂	PROPN
ejpam-5618	278	2	u	u	PROPN
ejpam-5618	278	3	.	.	PUNCT
ejpam-5618	279	1	hence	hence	ADV
ejpam-5618	279	2	,	,	PUNCT
ejpam-5618	279	3	gv	gv	ADP
ejpam-5618	279	4	g−1	g−1	PROPN
ejpam-5618	279	5	⊂	⊂	PROPN
ejpam-5618	279	6	u	u	PROPN
ejpam-5618	279	7	.	.	PUNCT
ejpam-5618	279	8	theorem	theorem	VERB
ejpam-5618	279	9	15	15	NUM
ejpam-5618	279	10	.	.	PUNCT
ejpam-5618	280	1	every	every	DET
ejpam-5618	280	2	ideal	ideal	ADJ
ejpam-5618	280	3	topological	topological	ADJ
ejpam-5618	280	4	group	group	NOUN
ejpam-5618	280	5	g	g	PROPN
ejpam-5618	280	6	is	be	AUX
ejpam-5618	280	7	an	an	DET
ejpam-5618	280	8	i	i	NOUN
ejpam-5618	280	9	-	-	PUNCT
ejpam-5618	280	10	homogeneous	homogeneous	ADJ
ejpam-5618	280	11	space	space	NOUN
ejpam-5618	280	12	.	.	PUNCT
ejpam-5618	281	1	proof	proof	NOUN
ejpam-5618	281	2	.	.	PUNCT
ejpam-5618	282	1	we	we	PRON
ejpam-5618	282	2	know	know	VERB
ejpam-5618	282	3	that	that	SCONJ
ejpam-5618	282	4	the	the	DET
ejpam-5618	282	5	right	right	ADJ
ejpam-5618	282	6	translation	translation	NOUN
ejpam-5618	282	7	mapping	mapping	NOUN
ejpam-5618	282	8	rg	rg	INTJ
ejpam-5618	282	9	:	:	PUNCT
ejpam-5618	282	10	g	g	PROPN
ejpam-5618	282	11	−→	−→	NOUN
ejpam-5618	282	12	g	g	PROPN
ejpam-5618	282	13	given	give	VERB
ejpam-5618	282	14	by	by	ADP
ejpam-5618	282	15	rg(x	rg(x	NOUN
ejpam-5618	282	16	)	)	PUNCT
ejpam-5618	282	17	=	=	SYM
ejpam-5618	282	18	xg	xg	PROPN
ejpam-5618	282	19	is	be	AUX
ejpam-5618	282	20	i	i	PROPN
ejpam-5618	282	21	-	-	PUNCT
ejpam-5618	282	22	homeomorphism	homeomorphism	PROPN
ejpam-5618	282	23	.	.	PUNCT
ejpam-5618	283	1	take	take	VERB
ejpam-5618	283	2	x	x	PUNCT
ejpam-5618	283	3	and	and	CCONJ
ejpam-5618	283	4	y	y	PROPN
ejpam-5618	283	5	in	in	ADP
ejpam-5618	283	6	g.	g.	PROPN
ejpam-5618	283	7	let	let	VERB
ejpam-5618	283	8	z	z	NOUN
ejpam-5618	283	9	=	=	SYM
ejpam-5618	283	10	x−1y	x−1y	PROPN
ejpam-5618	283	11	.	.	PUNCT
ejpam-5618	284	1	therefore	therefore	ADV
ejpam-5618	284	2	,	,	PUNCT
ejpam-5618	284	3	rz	rz	NOUN
ejpam-5618	284	4	:	:	PUNCT
ejpam-5618	284	5	g	g	PROPN
ejpam-5618	284	6	−→	−→	NOUN
ejpam-5618	284	7	g	g	PROPN
ejpam-5618	284	8	defined	define	VERB
ejpam-5618	284	9	by	by	ADP
ejpam-5618	284	10	rz(x	rz(x	NOUN
ejpam-5618	284	11	)	)	PUNCT
ejpam-5618	284	12	=	=	SYM
ejpam-5618	285	1	xz	xz	PROPN
ejpam-5618	285	2	=	=	PUNCT
ejpam-5618	286	1	x	x	X
ejpam-5618	286	2	(	(	PUNCT
ejpam-5618	286	3	x−1y	x−1y	PROPN
ejpam-5618	286	4	)	)	PUNCT
ejpam-5618	286	5	=	=	PUNCT
ejpam-5618	287	1	y	y	PROPN
ejpam-5618	287	2	is	be	AUX
ejpam-5618	287	3	i	i	PROPN
ejpam-5618	287	4	-	-	PUNCT
ejpam-5618	287	5	homeomorphism	homeomorphism	PROPN
ejpam-5618	287	6	.	.	PUNCT
ejpam-5618	288	1	hence	hence	ADV
ejpam-5618	288	2	,	,	PUNCT
ejpam-5618	288	3	g	g	PROPN
ejpam-5618	288	4	is	be	AUX
ejpam-5618	288	5	an	an	DET
ejpam-5618	288	6	i	i	NOUN
ejpam-5618	288	7	-	-	PUNCT
ejpam-5618	288	8	homogeneous	homogeneous	ADJ
ejpam-5618	288	9	space	space	NOUN
ejpam-5618	288	10	.	.	PUNCT
ejpam-5618	289	1	theorem	theorem	VERB
ejpam-5618	289	2	16	16	NUM
ejpam-5618	289	3	.	.	PUNCT
ejpam-5618	290	1	let	let	VERB
ejpam-5618	290	2	g	g	PRON
ejpam-5618	290	3	be	be	AUX
ejpam-5618	290	4	an	an	DET
ejpam-5618	290	5	ideal	ideal	ADJ
ejpam-5618	290	6	topological	topological	ADJ
ejpam-5618	290	7	group	group	NOUN
ejpam-5618	290	8	.	.	PUNCT
ejpam-5618	291	1	suppose	suppose	VERB
ejpam-5618	291	2	that	that	SCONJ
ejpam-5618	291	3	g	g	PROPN
ejpam-5618	291	4	is	be	AUX
ejpam-5618	291	5	submaximal	submaximal	ADJ
ejpam-5618	291	6	and	and	CCONJ
ejpam-5618	291	7	βe	βe	PRON
ejpam-5618	291	8	is	be	AUX
ejpam-5618	291	9	an	an	DET
ejpam-5618	291	10	i	i	NOUN
ejpam-5618	291	11	-	-	PUNCT
ejpam-5618	291	12	open	open	ADJ
ejpam-5618	291	13	base	base	NOUN
ejpam-5618	291	14	at	at	ADP
ejpam-5618	291	15	the	the	DET
ejpam-5618	291	16	identity	identity	NOUN
ejpam-5618	291	17	element	element	NOUN
ejpam-5618	291	18	e	e	PROPN
ejpam-5618	291	19	of	of	ADP
ejpam-5618	291	20	g.	g.	PROPN
ejpam-5618	291	21	then	then	ADV
ejpam-5618	291	22	the	the	DET
ejpam-5618	291	23	family	family	NOUN
ejpam-5618	291	24	βg	βg	VERB
ejpam-5618	291	25	=	=	PUNCT
ejpam-5618	291	26	{	{	PUNCT
ejpam-5618	291	27	ug	ug	X
ejpam-5618	291	28	:	:	PUNCT
ejpam-5618	291	29	u	u	PROPN
ejpam-5618	291	30	∈	∈	PROPN
ejpam-5618	291	31	βe	βe	PRON
ejpam-5618	291	32	}	}	PUNCT
ejpam-5618	291	33	forms	form	VERB
ejpam-5618	291	34	an	an	DET
ejpam-5618	291	35	i	i	NOUN
ejpam-5618	291	36	-	-	PUNCT
ejpam-5618	291	37	open	open	ADJ
ejpam-5618	291	38	base	base	NOUN
ejpam-5618	291	39	at	at	ADP
ejpam-5618	291	40	the	the	DET
ejpam-5618	291	41	element	element	NOUN
ejpam-5618	291	42	g	g	PROPN
ejpam-5618	291	43	of	of	ADP
ejpam-5618	291	44	g.	g.	PROPN
ejpam-5618	291	45	proof	proof	NOUN
ejpam-5618	291	46	.	.	PUNCT
ejpam-5618	292	1	consider	consider	VERB
ejpam-5618	292	2	the	the	DET
ejpam-5618	292	3	right	right	ADJ
ejpam-5618	292	4	translation	translation	NOUN
ejpam-5618	292	5	mapping	mapping	NOUN
ejpam-5618	292	6	rg	rg	INTJ
ejpam-5618	292	7	:	:	PUNCT
ejpam-5618	292	8	g	g	PROPN
ejpam-5618	292	9	−→	−→	NOUN
ejpam-5618	292	10	g.	g.	NOUN
ejpam-5618	292	11	let	let	VERB
ejpam-5618	292	12	v	v	PART
ejpam-5618	292	13	be	be	AUX
ejpam-5618	292	14	an	an	DET
ejpam-5618	292	15	i	i	NOUN
ejpam-5618	292	16	-	-	PUNCT
ejpam-5618	292	17	open	open	ADJ
ejpam-5618	292	18	neighborhood	neighborhood	NOUN
ejpam-5618	292	19	of	of	ADP
ejpam-5618	292	20	g.	g.	NOUN
ejpam-5618	292	21	by	by	ADP
ejpam-5618	292	22	submaximality	submaximality	NOUN
ejpam-5618	292	23	of	of	ADP
ejpam-5618	292	24	g	g	PROPN
ejpam-5618	292	25	,	,	PUNCT
ejpam-5618	292	26	v	v	NOUN
ejpam-5618	292	27	is	be	AUX
ejpam-5618	292	28	an	an	DET
ejpam-5618	292	29	open	open	ADJ
ejpam-5618	292	30	neighborhood	neighborhood	NOUN
ejpam-5618	292	31	of	of	ADP
ejpam-5618	292	32	g.	g.	PROPN
ejpam-5618	292	33	since	since	SCONJ
ejpam-5618	292	34	the	the	DET
ejpam-5618	292	35	right	right	ADJ
ejpam-5618	292	36	translation	translation	NOUN
ejpam-5618	292	37	mapping	mapping	NOUN
ejpam-5618	292	38	is	be	AUX
ejpam-5618	292	39	i	i	PROPN
ejpam-5618	292	40	-	-	PUNCT
ejpam-5618	292	41	homeomorphism	homeomorphism	PROPN
ejpam-5618	292	42	,	,	PUNCT
ejpam-5618	292	43	there	there	PRON
ejpam-5618	292	44	is	be	VERB
ejpam-5618	292	45	an	an	DET
ejpam-5618	292	46	i	i	NOUN
ejpam-5618	292	47	-	-	PUNCT
ejpam-5618	292	48	open	open	ADJ
ejpam-5618	292	49	neighborhood	neighborhood	NOUN
ejpam-5618	292	50	w	w	NOUN
ejpam-5618	292	51	of	of	ADP
ejpam-5618	292	52	e	e	NOUN
ejpam-5618	292	53	such	such	ADJ
ejpam-5618	292	54	that	that	PRON
ejpam-5618	292	55	rg(w	rg(w	NUM
ejpam-5618	292	56	)	)	PUNCT
ejpam-5618	293	1	⊂	⊂	PROPN
ejpam-5618	293	2	v	v	X
ejpam-5618	293	3	.	.	PUNCT
ejpam-5618	294	1	since	since	SCONJ
ejpam-5618	294	2	βe	βe	PRON
ejpam-5618	294	3	is	be	AUX
ejpam-5618	294	4	an	an	DET
ejpam-5618	294	5	i	i	NOUN
ejpam-5618	294	6	-	-	PUNCT
ejpam-5618	294	7	open	open	ADJ
ejpam-5618	294	8	base	base	NOUN
ejpam-5618	294	9	at	at	ADP
ejpam-5618	294	10	e	e	NOUN
ejpam-5618	294	11	,	,	PUNCT
ejpam-5618	294	12	there	there	PRON
ejpam-5618	294	13	is	be	VERB
ejpam-5618	294	14	u	u	NOUN
ejpam-5618	294	15	∈	∈	PROPN
ejpam-5618	294	16	βe	βe	PRON
ejpam-5618	294	17	such	such	ADJ
ejpam-5618	294	18	that	that	SCONJ
ejpam-5618	294	19	u	u	PROPN
ejpam-5618	294	20	⊂	⊂	PROPN
ejpam-5618	294	21	w	w	PROPN
ejpam-5618	294	22	.	.	PUNCT
ejpam-5618	295	1	but	but	CCONJ
ejpam-5618	295	2	ug	ug	ADP
ejpam-5618	295	3	⊂	⊂	PROPN
ejpam-5618	295	4	wg	wg	PROPN
ejpam-5618	295	5	=	=	X
ejpam-5618	295	6	rg(w	rg(w	PRON
ejpam-5618	295	7	)	)	PUNCT
ejpam-5618	296	1	⊂	⊂	PROPN
ejpam-5618	296	2	v	v	X
ejpam-5618	296	3	.	.	PUNCT
ejpam-5618	297	1	therefore	therefore	ADV
ejpam-5618	297	2	,	,	PUNCT
ejpam-5618	297	3	βg	βg	ADV
ejpam-5618	297	4	is	be	AUX
ejpam-5618	297	5	an	an	DET
ejpam-5618	297	6	i	i	NOUN
ejpam-5618	297	7	-	-	PUNCT
ejpam-5618	297	8	open	open	ADJ
ejpam-5618	297	9	base	base	NOUN
ejpam-5618	297	10	at	at	ADP
ejpam-5618	297	11	the	the	DET
ejpam-5618	297	12	element	element	NOUN
ejpam-5618	297	13	g.	g.	PROPN
ejpam-5618	297	14	theorem	theorem	VERB
ejpam-5618	297	15	17	17	NUM
ejpam-5618	297	16	.	.	PUNCT
ejpam-5618	298	1	let	let	VERB
ejpam-5618	298	2	g	g	PRON
ejpam-5618	298	3	be	be	AUX
ejpam-5618	298	4	an	an	DET
ejpam-5618	298	5	ideal	ideal	ADJ
ejpam-5618	298	6	topological	topological	ADJ
ejpam-5618	298	7	group	group	NOUN
ejpam-5618	298	8	.	.	PUNCT
ejpam-5618	299	1	suppose	suppose	VERB
ejpam-5618	299	2	that	that	SCONJ
ejpam-5618	299	3	g	g	PROPN
ejpam-5618	299	4	is	be	AUX
ejpam-5618	299	5	submaximal	submaximal	ADJ
ejpam-5618	299	6	and	and	CCONJ
ejpam-5618	299	7	βe	βe	PRON
ejpam-5618	299	8	is	be	AUX
ejpam-5618	299	9	an	an	DET
ejpam-5618	299	10	i	i	NOUN
ejpam-5618	299	11	-	-	PUNCT
ejpam-5618	299	12	open	open	ADJ
ejpam-5618	299	13	base	base	NOUN
ejpam-5618	299	14	at	at	ADP
ejpam-5618	299	15	the	the	DET
ejpam-5618	299	16	identity	identity	NOUN
ejpam-5618	299	17	element	element	NOUN
ejpam-5618	299	18	e	e	PROPN
ejpam-5618	299	19	of	of	ADP
ejpam-5618	299	20	g.	g.	PROPN
ejpam-5618	299	21	then	then	ADV
ejpam-5618	299	22	the	the	DET
ejpam-5618	299	23	family	family	NOUN
ejpam-5618	299	24	β⋆	β⋆	X
ejpam-5618	300	1	=	=	CCONJ
ejpam-5618	300	2	{	{	PUNCT
ejpam-5618	300	3	u−1;u	u−1;u	PROPN
ejpam-5618	300	4	∈	∈	NOUN
ejpam-5618	300	5	βe	βe	PRON
ejpam-5618	300	6	}	}	PUNCT
ejpam-5618	300	7	forms	form	VERB
ejpam-5618	300	8	an	an	DET
ejpam-5618	300	9	i	i	NOUN
ejpam-5618	300	10	-	-	PUNCT
ejpam-5618	300	11	open	open	ADJ
ejpam-5618	300	12	base	base	NOUN
ejpam-5618	300	13	at	at	ADP
ejpam-5618	300	14	e.	e.	PROPN
ejpam-5618	300	15	proof	proof	PROPN
ejpam-5618	300	16	.	.	PUNCT
ejpam-5618	301	1	consider	consider	VERB
ejpam-5618	301	2	the	the	DET
ejpam-5618	301	3	inverse	inverse	NOUN
ejpam-5618	301	4	mapping	mapping	NOUN
ejpam-5618	301	5	inv	inv	VERB
ejpam-5618	301	6	:	:	PUNCT
ejpam-5618	301	7	g	g	PROPN
ejpam-5618	301	8	−→	−→	NOUN
ejpam-5618	301	9	g.	g.	NOUN
ejpam-5618	301	10	let	let	VERB
ejpam-5618	301	11	v	v	PART
ejpam-5618	301	12	be	be	AUX
ejpam-5618	301	13	an	an	DET
ejpam-5618	301	14	i	i	NOUN
ejpam-5618	301	15	-	-	PUNCT
ejpam-5618	301	16	open	open	ADJ
ejpam-5618	301	17	neighborhood	neighborhood	NOUN
ejpam-5618	301	18	of	of	ADP
ejpam-5618	301	19	e.	e.	PROPN
ejpam-5618	301	20	by	by	ADP
ejpam-5618	301	21	submaximality	submaximality	NOUN
ejpam-5618	301	22	of	of	ADP
ejpam-5618	301	23	g	g	PROPN
ejpam-5618	301	24	,	,	PUNCT
ejpam-5618	301	25	v	v	NOUN
ejpam-5618	301	26	is	be	AUX
ejpam-5618	301	27	an	an	DET
ejpam-5618	301	28	open	open	ADJ
ejpam-5618	301	29	neighborhood	neighborhood	NOUN
ejpam-5618	301	30	of	of	ADP
ejpam-5618	301	31	e.	e.	PROPN
ejpam-5618	301	32	since	since	SCONJ
ejpam-5618	301	33	inv	inv	NOUN
ejpam-5618	301	34	is	be	AUX
ejpam-5618	301	35	an	an	DET
ejpam-5618	301	36	ihomeomorphism	ihomeomorphism	NOUN
ejpam-5618	301	37	,	,	PUNCT
ejpam-5618	301	38	there	there	PRON
ejpam-5618	301	39	is	be	VERB
ejpam-5618	301	40	an	an	DET
ejpam-5618	301	41	i	i	NOUN
ejpam-5618	301	42	-	-	PUNCT
ejpam-5618	301	43	open	open	ADJ
ejpam-5618	301	44	neighborhood	neighborhood	NOUN
ejpam-5618	301	45	w	w	NOUN
ejpam-5618	301	46	of	of	ADP
ejpam-5618	301	47	e	e	NOUN
ejpam-5618	301	48	such	such	ADJ
ejpam-5618	301	49	that	that	SCONJ
ejpam-5618	301	50	inv(w	inv(w	NOUN
ejpam-5618	301	51	)	)	PUNCT
ejpam-5618	302	1	⊂	⊂	PROPN
ejpam-5618	302	2	v	v	INTJ
ejpam-5618	302	3	.	.	PUNCT
ejpam-5618	303	1	since	since	SCONJ
ejpam-5618	303	2	βe	βe	PRON
ejpam-5618	303	3	is	be	AUX
ejpam-5618	303	4	an	an	DET
ejpam-5618	303	5	i	i	NOUN
ejpam-5618	303	6	-	-	PUNCT
ejpam-5618	303	7	open	open	ADJ
ejpam-5618	303	8	base	base	NOUN
ejpam-5618	303	9	at	at	ADP
ejpam-5618	303	10	e	e	NOUN
ejpam-5618	303	11	,	,	PUNCT
ejpam-5618	303	12	there	there	PRON
ejpam-5618	303	13	is	be	VERB
ejpam-5618	303	14	u	u	NOUN
ejpam-5618	303	15	∈	∈	PROPN
ejpam-5618	303	16	βe	βe	PRON
ejpam-5618	303	17	such	such	ADJ
ejpam-5618	303	18	that	that	SCONJ
ejpam-5618	303	19	u	u	PROPN
ejpam-5618	303	20	⊂	⊂	PROPN
ejpam-5618	303	21	w	w	PROPN
ejpam-5618	303	22	.	.	PUNCT
ejpam-5618	304	1	but	but	CCONJ
ejpam-5618	304	2	u−1	u−1	PROPN
ejpam-5618	304	3	⊂	⊂	PROPN
ejpam-5618	304	4	w−1	w−1	PROPN
ejpam-5618	304	5	=	=	SYM
ejpam-5618	304	6	inv(w	inv(w	X
ejpam-5618	304	7	)	)	PUNCT
ejpam-5618	305	1	⊂	⊂	PROPN
ejpam-5618	305	2	v	v	PROPN
ejpam-5618	305	3	.	.	PUNCT
ejpam-5618	306	1	therefore	therefore	ADV
ejpam-5618	306	2	,	,	PUNCT
ejpam-5618	306	3	β⋆	β⋆	NUM
ejpam-5618	306	4	is	be	AUX
ejpam-5618	306	5	an	an	DET
ejpam-5618	306	6	i	i	NOUN
ejpam-5618	306	7	-	-	PUNCT
ejpam-5618	306	8	open	open	ADJ
ejpam-5618	306	9	base	base	NOUN
ejpam-5618	306	10	at	at	ADP
ejpam-5618	306	11	e.	e.	PROPN
ejpam-5618	306	12	theorem	theorem	PROPN
ejpam-5618	306	13	18	18	NUM
ejpam-5618	306	14	.	.	PUNCT
ejpam-5618	307	1	let	let	VERB
ejpam-5618	307	2	g	g	NOUN
ejpam-5618	307	3	and	and	CCONJ
ejpam-5618	307	4	h	h	NOUN
ejpam-5618	307	5	be	be	AUX
ejpam-5618	307	6	ideal	ideal	ADJ
ejpam-5618	307	7	topological	topological	ADJ
ejpam-5618	307	8	groups	group	NOUN
ejpam-5618	307	9	.	.	PUNCT
ejpam-5618	308	1	suppose	suppose	VERB
ejpam-5618	308	2	that	that	SCONJ
ejpam-5618	308	3	f	f	X
ejpam-5618	308	4	:	:	PUNCT
ejpam-5618	308	5	g	g	PROPN
ejpam-5618	308	6	→	→	SYM
ejpam-5618	308	7	h	h	NOUN
ejpam-5618	308	8	is	be	AUX
ejpam-5618	308	9	a	a	DET
ejpam-5618	308	10	homomorphism	homomorphism	NOUN
ejpam-5618	308	11	such	such	ADJ
ejpam-5618	308	12	that	that	DET
ejpam-5618	308	13	for	for	SCONJ
ejpam-5618	308	14	every	every	DET
ejpam-5618	308	15	i	i	NOUN
ejpam-5618	308	16	-	-	PUNCT
ejpam-5618	308	17	open	open	ADJ
ejpam-5618	308	18	set	set	VERB
ejpam-5618	308	19	v	v	NOUN
ejpam-5618	308	20	containing	contain	VERB
ejpam-5618	308	21	the	the	DET
ejpam-5618	308	22	identity	identity	NOUN
ejpam-5618	308	23	eh	eh	INTJ
ejpam-5618	308	24	in	in	ADP
ejpam-5618	308	25	h	h	NOUN
ejpam-5618	308	26	,	,	PUNCT
ejpam-5618	308	27	there	there	PRON
ejpam-5618	308	28	is	be	VERB
ejpam-5618	308	29	an	an	DET
ejpam-5618	308	30	open	open	ADJ
ejpam-5618	308	31	set	set	NOUN
ejpam-5618	308	32	u	u	NOUN
ejpam-5618	308	33	containing	contain	VERB
ejpam-5618	308	34	the	the	DET
ejpam-5618	308	35	identity	identity	NOUN
ejpam-5618	308	36	element	element	NOUN
ejpam-5618	308	37	eg	eg	NOUN
ejpam-5618	308	38	in	in	ADP
ejpam-5618	308	39	g	g	NOUN
ejpam-5618	308	40	with	with	ADP
ejpam-5618	308	41	f(u	f(u	PROPN
ejpam-5618	308	42	)	)	PUNCT
ejpam-5618	308	43	⊂	⊂	PROPN
ejpam-5618	308	44	v	v	PROPN
ejpam-5618	308	45	.	.	PUNCT
ejpam-5618	309	1	then	then	ADV
ejpam-5618	309	2	f	f	PROPN
ejpam-5618	309	3	is	be	AUX
ejpam-5618	309	4	i	i	NOUN
ejpam-5618	309	5	-	-	PUNCT
ejpam-5618	309	6	continuous	continuous	ADJ
ejpam-5618	309	7	.	.	PUNCT
ejpam-5618	310	1	proof	proof	NOUN
ejpam-5618	310	2	.	.	PUNCT
ejpam-5618	311	1	given	give	VERB
ejpam-5618	311	2	x	x	SYM
ejpam-5618	311	3	∈	∈	PROPN
ejpam-5618	311	4	g.	g.	NOUN
ejpam-5618	311	5	we	we	PRON
ejpam-5618	311	6	show	show	VERB
ejpam-5618	311	7	that	that	SCONJ
ejpam-5618	311	8	f	f	PROPN
ejpam-5618	311	9	is	be	AUX
ejpam-5618	311	10	i	i	PRON
ejpam-5618	311	11	-	-	ADJ
ejpam-5618	311	12	continuous	continuous	ADJ
ejpam-5618	311	13	at	at	ADP
ejpam-5618	311	14	x.	x.	NOUN
ejpam-5618	311	15	suppose	suppose	VERB
ejpam-5618	311	16	that	that	SCONJ
ejpam-5618	311	17	o	o	PROPN
ejpam-5618	311	18	is	be	AUX
ejpam-5618	311	19	an	an	DET
ejpam-5618	311	20	open	open	ADJ
ejpam-5618	311	21	neighborhood	neighborhood	NOUN
ejpam-5618	311	22	of	of	ADP
ejpam-5618	311	23	f(x	f(x	PROPN
ejpam-5618	311	24	)	)	PUNCT
ejpam-5618	312	1	=	=	SYM
ejpam-5618	312	2	y	y	PROPN
ejpam-5618	312	3	in	in	ADP
ejpam-5618	312	4	h.	h.	PROPN
ejpam-5618	312	5	since	since	SCONJ
ejpam-5618	312	6	the	the	DET
ejpam-5618	312	7	left	left	ADJ
ejpam-5618	312	8	translation	translation	NOUN
ejpam-5618	312	9	mapping	mapping	NOUN
ejpam-5618	312	10	ly	ly	X
ejpam-5618	312	11	is	be	AUX
ejpam-5618	312	12	i	i	PROPN
ejpam-5618	312	13	-	-	PUNCT
ejpam-5618	312	14	homeomorphism	homeomorphism	PROPN
ejpam-5618	312	15	in	in	ADP
ejpam-5618	312	16	h	h	NOUN
ejpam-5618	312	17	,	,	PUNCT
ejpam-5618	312	18	there	there	PRON
ejpam-5618	312	19	is	be	VERB
ejpam-5618	312	20	i	i	PRON
ejpam-5618	312	21	-	-	PUNCT
ejpam-5618	312	22	open	open	ADJ
ejpam-5618	312	23	set	set	VERB
ejpam-5618	312	24	v	v	NOUN
ejpam-5618	312	25	containing	contain	VERB
ejpam-5618	312	26	the	the	DET
ejpam-5618	312	27	identity	identity	NOUN
ejpam-5618	312	28	element	element	NOUN
ejpam-5618	312	29	eh	eh	INTJ
ejpam-5618	312	30	of	of	ADP
ejpam-5618	312	31	h	h	NOUN
ejpam-5618	312	32	such	such	ADJ
ejpam-5618	312	33	that	that	SCONJ
ejpam-5618	312	34	yv	yv	PROPN
ejpam-5618	312	35	⊂	⊂	PROPN
ejpam-5618	312	36	o.	o.	PROPN
ejpam-5618	312	37	by	by	ADP
ejpam-5618	312	38	assumption	assumption	NOUN
ejpam-5618	312	39	,	,	PUNCT
ejpam-5618	312	40	there	there	PRON
ejpam-5618	312	41	is	be	VERB
ejpam-5618	312	42	an	an	DET
ejpam-5618	312	43	open	open	ADJ
ejpam-5618	312	44	set	set	NOUN
ejpam-5618	312	45	u	u	NOUN
ejpam-5618	312	46	containing	contain	VERB
ejpam-5618	312	47	the	the	DET
ejpam-5618	312	48	identity	identity	NOUN
ejpam-5618	312	49	element	element	NOUN
ejpam-5618	312	50	eg	eg	NOUN
ejpam-5618	312	51	in	in	ADP
ejpam-5618	312	52	g	g	PROPN
ejpam-5618	312	53	such	such	ADJ
ejpam-5618	312	54	that	that	DET
ejpam-5618	312	55	f(u	f(u	PROPN
ejpam-5618	312	56	)	)	PUNCT
ejpam-5618	313	1	⊂	⊂	PROPN
ejpam-5618	313	2	v	v	PROPN
ejpam-5618	313	3	.	.	PUNCT
ejpam-5618	314	1	therefore	therefore	ADV
ejpam-5618	314	2	,	,	PUNCT
ejpam-5618	314	3	f(xu	f(xu	NUM
ejpam-5618	314	4	)	)	PUNCT
ejpam-5618	314	5	=	=	SYM
ejpam-5618	314	6	f(x)f(u	f(x)f(u	NOUN
ejpam-5618	314	7	)	)	PUNCT
ejpam-5618	314	8	=	=	SYM
ejpam-5618	314	9	yf(u	yf(u	X
ejpam-5618	314	10	)	)	PUNCT
ejpam-5618	315	1	⊂	⊂	PROPN
ejpam-5618	315	2	yv	yv	PROPN
ejpam-5618	315	3	⊂	⊂	PROPN
ejpam-5618	315	4	o.	o.	PROPN
ejpam-5618	315	5	note	note	VERB
ejpam-5618	315	6	that	that	SCONJ
ejpam-5618	315	7	xu	xu	PROPN
ejpam-5618	315	8	is	be	AUX
ejpam-5618	315	9	an	an	DET
ejpam-5618	315	10	i	i	NOUN
ejpam-5618	315	11	-	-	PUNCT
ejpam-5618	315	12	open	open	ADJ
ejpam-5618	315	13	set	set	NOUN
ejpam-5618	315	14	containing	contain	VERB
ejpam-5618	315	15	x	x	PUNCT
ejpam-5618	315	16	by	by	ADP
ejpam-5618	315	17	proposition	proposition	NOUN
ejpam-5618	315	18	1	1	NUM
ejpam-5618	315	19	.	.	PUNCT
ejpam-5618	316	1	hence	hence	ADV
ejpam-5618	316	2	,	,	PUNCT
ejpam-5618	316	3	f	f	PROPN
ejpam-5618	316	4	is	be	AUX
ejpam-5618	316	5	i	i	PRON
ejpam-5618	316	6	-	-	ADJ
ejpam-5618	316	7	continuous	continuous	ADJ
ejpam-5618	316	8	at	at	ADP
ejpam-5618	316	9	x.	x.	PROPN
ejpam-5618	316	10	s.	s.	PROPN
ejpam-5618	316	11	alammar	alammar	PROPN
ejpam-5618	316	12	,	,	PUNCT
ejpam-5618	316	13	m.	m.	NOUN
ejpam-5618	316	14	al	al	PROPN
ejpam-5618	316	15	shumrani	shumrani	PROPN
ejpam-5618	316	16	,	,	PUNCT
ejpam-5618	316	17	c.	c.	PROPN
ejpam-5618	316	18	özel	özel	PROPN
ejpam-5618	316	19	/	/	SYM
ejpam-5618	316	20	eur	eur	PROPN
ejpam-5618	316	21	.	.	PUNCT
ejpam-5618	317	1	j.	j.	PROPN
ejpam-5618	317	2	pure	pure	PROPN
ejpam-5618	317	3	appl	appl	PROPN
ejpam-5618	317	4	.	.	PROPN
ejpam-5618	317	5	math	math	PROPN
ejpam-5618	317	6	,	,	PUNCT
ejpam-5618	317	7	18	18	NUM
ejpam-5618	317	8	(	(	PUNCT
ejpam-5618	317	9	1	1	NUM
ejpam-5618	317	10	)	)	PUNCT
ejpam-5618	317	11	(	(	PUNCT
ejpam-5618	317	12	2025	2025	NUM
ejpam-5618	317	13	)	)	PUNCT
ejpam-5618	317	14	,	,	PUNCT
ejpam-5618	317	15	5618	5618	NUM
ejpam-5618	317	16	9	9	NUM
ejpam-5618	317	17	of	of	ADP
ejpam-5618	317	18	13	13	NUM
ejpam-5618	317	19	theorem	theorem	NOUN
ejpam-5618	317	20	19	19	NUM
ejpam-5618	317	21	.	.	PUNCT
ejpam-5618	318	1	let	let	VERB
ejpam-5618	318	2	g	g	NOUN
ejpam-5618	318	3	and	and	CCONJ
ejpam-5618	318	4	h	h	NOUN
ejpam-5618	318	5	be	be	AUX
ejpam-5618	318	6	ideal	ideal	ADJ
ejpam-5618	318	7	topological	topological	ADJ
ejpam-5618	318	8	groups	group	NOUN
ejpam-5618	318	9	.	.	PUNCT
ejpam-5618	319	1	suppose	suppose	VERB
ejpam-5618	319	2	that	that	SCONJ
ejpam-5618	319	3	g	g	PROPN
ejpam-5618	319	4	and	and	CCONJ
ejpam-5618	319	5	h	h	NOUN
ejpam-5618	319	6	both	both	PRON
ejpam-5618	319	7	are	be	AUX
ejpam-5618	319	8	submaximal	submaximal	ADJ
ejpam-5618	319	9	and	and	CCONJ
ejpam-5618	319	10	f	f	X
ejpam-5618	319	11	:	:	PUNCT
ejpam-5618	319	12	g	g	PROPN
ejpam-5618	319	13	→	→	SYM
ejpam-5618	319	14	h	h	NOUN
ejpam-5618	319	15	is	be	AUX
ejpam-5618	319	16	a	a	DET
ejpam-5618	319	17	homomorphism	homomorphism	NOUN
ejpam-5618	319	18	.	.	PUNCT
ejpam-5618	320	1	if	if	SCONJ
ejpam-5618	320	2	f	f	PROPN
ejpam-5618	320	3	is	be	AUX
ejpam-5618	320	4	i	i	PRON
ejpam-5618	320	5	-	-	NOUN
ejpam-5618	320	6	continuous	continuous	ADJ
ejpam-5618	320	7	at	at	ADP
ejpam-5618	320	8	the	the	DET
ejpam-5618	320	9	identity	identity	NOUN
ejpam-5618	320	10	element	element	NOUN
ejpam-5618	320	11	eg	eg	NOUN
ejpam-5618	320	12	of	of	ADP
ejpam-5618	320	13	g	g	PROPN
ejpam-5618	320	14	,	,	PUNCT
ejpam-5618	320	15	then	then	ADV
ejpam-5618	320	16	f	f	PROPN
ejpam-5618	320	17	is	be	AUX
ejpam-5618	320	18	i	i	NOUN
ejpam-5618	320	19	-	-	PUNCT
ejpam-5618	320	20	continuous	continuous	ADJ
ejpam-5618	320	21	.	.	PUNCT
ejpam-5618	321	1	proof	proof	NOUN
ejpam-5618	321	2	.	.	PUNCT
ejpam-5618	322	1	given	give	VERB
ejpam-5618	322	2	x	x	SYM
ejpam-5618	322	3	∈	∈	PROPN
ejpam-5618	322	4	g.	g.	NOUN
ejpam-5618	322	5	we	we	PRON
ejpam-5618	322	6	show	show	VERB
ejpam-5618	322	7	that	that	SCONJ
ejpam-5618	322	8	f	f	PROPN
ejpam-5618	322	9	is	be	AUX
ejpam-5618	322	10	i	i	PRON
ejpam-5618	322	11	-	-	ADJ
ejpam-5618	322	12	continuous	continuous	ADJ
ejpam-5618	322	13	at	at	ADP
ejpam-5618	322	14	x.	x.	NOUN
ejpam-5618	322	15	suppose	suppose	VERB
ejpam-5618	322	16	that	that	SCONJ
ejpam-5618	322	17	o	o	PROPN
ejpam-5618	322	18	is	be	AUX
ejpam-5618	322	19	an	an	DET
ejpam-5618	322	20	open	open	ADJ
ejpam-5618	322	21	neighborhood	neighborhood	NOUN
ejpam-5618	322	22	of	of	ADP
ejpam-5618	322	23	f(x	f(x	PROPN
ejpam-5618	322	24	)	)	PUNCT
ejpam-5618	323	1	=	=	SYM
ejpam-5618	323	2	y	y	PROPN
ejpam-5618	323	3	in	in	ADP
ejpam-5618	323	4	h.	h.	PROPN
ejpam-5618	323	5	since	since	SCONJ
ejpam-5618	323	6	the	the	DET
ejpam-5618	323	7	left	left	ADJ
ejpam-5618	323	8	translation	translation	NOUN
ejpam-5618	323	9	map	map	NOUN
ejpam-5618	323	10	ly	ly	X
ejpam-5618	323	11	is	be	AUX
ejpam-5618	323	12	an	an	DET
ejpam-5618	323	13	i	i	PROPN
ejpam-5618	323	14	-	-	PUNCT
ejpam-5618	323	15	homeomorphism	homeomorphism	PROPN
ejpam-5618	323	16	ofh	ofh	PROPN
ejpam-5618	323	17	,	,	PUNCT
ejpam-5618	323	18	there	there	PRON
ejpam-5618	323	19	is	be	VERB
ejpam-5618	323	20	an	an	DET
ejpam-5618	323	21	i	i	NOUN
ejpam-5618	323	22	-	-	PUNCT
ejpam-5618	323	23	open	open	ADJ
ejpam-5618	323	24	neighborhood	neighborhood	NOUN
ejpam-5618	323	25	v	v	NOUN
ejpam-5618	323	26	of	of	ADP
ejpam-5618	323	27	the	the	DET
ejpam-5618	323	28	identity	identity	NOUN
ejpam-5618	323	29	element	element	NOUN
ejpam-5618	323	30	eh	eh	INTJ
ejpam-5618	323	31	ofh	ofh	NUM
ejpam-5618	323	32	such	such	ADJ
ejpam-5618	323	33	that	that	SCONJ
ejpam-5618	323	34	yv	yv	PROPN
ejpam-5618	323	35	⊂	⊂	PROPN
ejpam-5618	323	36	o.	o.	PROPN
ejpam-5618	323	37	since	since	SCONJ
ejpam-5618	323	38	h	h	PROPN
ejpam-5618	323	39	is	be	AUX
ejpam-5618	323	40	submaximal	submaximal	ADJ
ejpam-5618	323	41	,	,	PUNCT
ejpam-5618	323	42	v	v	NOUN
ejpam-5618	323	43	is	be	AUX
ejpam-5618	323	44	an	an	DET
ejpam-5618	323	45	open	open	ADJ
ejpam-5618	323	46	neighborhood	neighborhood	NOUN
ejpam-5618	323	47	of	of	ADP
ejpam-5618	323	48	eh	eh	INTJ
ejpam-5618	323	49	.	.	PUNCT
ejpam-5618	324	1	but	but	CCONJ
ejpam-5618	324	2	f	f	PROPN
ejpam-5618	324	3	is	be	AUX
ejpam-5618	324	4	i	i	PRON
ejpam-5618	324	5	-	-	NOUN
ejpam-5618	324	6	continuous	continuous	ADJ
ejpam-5618	324	7	at	at	ADP
ejpam-5618	324	8	the	the	DET
ejpam-5618	324	9	identity	identity	NOUN
ejpam-5618	324	10	element	element	NOUN
ejpam-5618	324	11	eg	eg	NOUN
ejpam-5618	324	12	of	of	ADP
ejpam-5618	324	13	g.	g.	PROPN
ejpam-5618	324	14	therefore	therefore	ADV
ejpam-5618	324	15	,	,	PUNCT
ejpam-5618	324	16	there	there	PRON
ejpam-5618	324	17	is	be	VERB
ejpam-5618	324	18	an	an	DET
ejpam-5618	324	19	i	i	NOUN
ejpam-5618	324	20	-	-	PUNCT
ejpam-5618	324	21	open	open	ADJ
ejpam-5618	324	22	neighborhood	neighborhood	NOUN
ejpam-5618	324	23	u	u	NOUN
ejpam-5618	324	24	of	of	ADP
ejpam-5618	324	25	eg	eg	NOUN
ejpam-5618	324	26	such	such	ADJ
ejpam-5618	324	27	that	that	DET
ejpam-5618	324	28	f(u	f(u	PROPN
ejpam-5618	324	29	)	)	PUNCT
ejpam-5618	324	30	⊂	⊂	PROPN
ejpam-5618	324	31	v	v	X
ejpam-5618	324	32	.	.	PUNCT
ejpam-5618	325	1	note	note	VERB
ejpam-5618	325	2	that	that	SCONJ
ejpam-5618	325	3	the	the	DET
ejpam-5618	325	4	set	set	PROPN
ejpam-5618	325	5	xu	xu	PROPN
ejpam-5618	325	6	is	be	AUX
ejpam-5618	325	7	an	an	DET
ejpam-5618	325	8	i	i	NOUN
ejpam-5618	325	9	-	-	PUNCT
ejpam-5618	325	10	open	open	ADJ
ejpam-5618	325	11	neighborhood	neighborhood	NOUN
ejpam-5618	325	12	of	of	ADP
ejpam-5618	325	13	x	x	PUNCT
ejpam-5618	325	14	since	since	SCONJ
ejpam-5618	325	15	g	g	PROPN
ejpam-5618	325	16	is	be	AUX
ejpam-5618	325	17	submaximal	submaximal	ADJ
ejpam-5618	325	18	.	.	PUNCT
ejpam-5618	326	1	thus	thus	ADV
ejpam-5618	326	2	,	,	PUNCT
ejpam-5618	326	3	f(xu	f(xu	NUM
ejpam-5618	326	4	)	)	PUNCT
ejpam-5618	326	5	=	=	SYM
ejpam-5618	326	6	f(x)f(u	f(x)f(u	NOUN
ejpam-5618	326	7	)	)	PUNCT
ejpam-5618	326	8	=	=	SYM
ejpam-5618	326	9	yf(u	yf(u	X
ejpam-5618	326	10	)	)	PUNCT
ejpam-5618	327	1	⊂	⊂	PROPN
ejpam-5618	327	2	yv	yv	PROPN
ejpam-5618	327	3	⊂	⊂	PROPN
ejpam-5618	327	4	o.	o.	PROPN
ejpam-5618	327	5	hence	hence	ADV
ejpam-5618	327	6	,	,	PUNCT
ejpam-5618	327	7	f	f	PROPN
ejpam-5618	327	8	is	be	AUX
ejpam-5618	327	9	i	i	PRON
ejpam-5618	327	10	-	-	ADJ
ejpam-5618	327	11	continuous	continuous	ADJ
ejpam-5618	327	12	at	at	ADP
ejpam-5618	327	13	x.	x.	NOUN
ejpam-5618	327	14	next	next	ADV
ejpam-5618	327	15	,	,	PUNCT
ejpam-5618	327	16	we	we	PRON
ejpam-5618	327	17	will	will	AUX
ejpam-5618	327	18	investigate	investigate	VERB
ejpam-5618	327	19	subgroups	subgroup	NOUN
ejpam-5618	327	20	in	in	ADP
ejpam-5618	327	21	ideal	ideal	ADJ
ejpam-5618	327	22	topological	topological	ADJ
ejpam-5618	327	23	groups	group	NOUN
ejpam-5618	327	24	.	.	PUNCT
ejpam-5618	328	1	theorem	theorem	NOUN
ejpam-5618	328	2	20	20	NUM
ejpam-5618	328	3	.	.	PUNCT
ejpam-5618	329	1	let	let	VERB
ejpam-5618	329	2	g	g	PRON
ejpam-5618	329	3	be	be	AUX
ejpam-5618	329	4	an	an	DET
ejpam-5618	329	5	ideal	ideal	ADJ
ejpam-5618	329	6	topological	topological	ADJ
ejpam-5618	329	7	group	group	NOUN
ejpam-5618	329	8	and	and	CCONJ
ejpam-5618	329	9	h	h	NOUN
ejpam-5618	329	10	be	be	AUX
ejpam-5618	329	11	a	a	DET
ejpam-5618	329	12	subgroup	subgroup	NOUN
ejpam-5618	329	13	of	of	ADP
ejpam-5618	329	14	g.	g.	PROPN
ejpam-5618	329	15	if	if	SCONJ
ejpam-5618	329	16	h	h	NOUN
ejpam-5618	329	17	contains	contain	VERB
ejpam-5618	329	18	a	a	DET
ejpam-5618	329	19	nonempty	nonempty	ADJ
ejpam-5618	329	20	open	open	ADJ
ejpam-5618	329	21	set	set	NOUN
ejpam-5618	329	22	,	,	PUNCT
ejpam-5618	329	23	then	then	ADV
ejpam-5618	329	24	h	h	PROPN
ejpam-5618	329	25	is	be	AUX
ejpam-5618	329	26	i	i	PRON
ejpam-5618	329	27	-	-	PUNCT
ejpam-5618	329	28	open	open	ADJ
ejpam-5618	329	29	in	in	ADP
ejpam-5618	329	30	g.	g.	PROPN
ejpam-5618	329	31	proof	proof	PROPN
ejpam-5618	329	32	.	.	PUNCT
ejpam-5618	330	1	suppose	suppose	VERB
ejpam-5618	330	2	that	that	SCONJ
ejpam-5618	330	3	u	u	PROPN
ejpam-5618	330	4	is	be	AUX
ejpam-5618	330	5	a	a	DET
ejpam-5618	330	6	nonempty	nonempty	ADJ
ejpam-5618	330	7	open	open	ADJ
ejpam-5618	330	8	subset	subset	NOUN
ejpam-5618	330	9	of	of	ADP
ejpam-5618	330	10	g	g	PROPN
ejpam-5618	330	11	such	such	ADJ
ejpam-5618	330	12	that	that	SCONJ
ejpam-5618	330	13	u	u	PROPN
ejpam-5618	330	14	⊂	⊂	PROPN
ejpam-5618	330	15	h.	h.	PROPN
ejpam-5618	330	16	for	for	ADP
ejpam-5618	330	17	any	any	DET
ejpam-5618	330	18	h	h	NOUN
ejpam-5618	330	19	∈	∈	PROPN
ejpam-5618	330	20	h	h	NOUN
ejpam-5618	330	21	,	,	PUNCT
ejpam-5618	330	22	the	the	DET
ejpam-5618	330	23	set	set	NOUN
ejpam-5618	330	24	lh(u	lh(u	PUNCT
ejpam-5618	330	25	)	)	PUNCT
ejpam-5618	330	26	=	=	PUNCT
ejpam-5618	330	27	hu	hu	PROPN
ejpam-5618	330	28	is	be	AUX
ejpam-5618	330	29	an	an	DET
ejpam-5618	330	30	i	i	NOUN
ejpam-5618	330	31	-	-	PUNCT
ejpam-5618	330	32	open	open	ADJ
ejpam-5618	330	33	set	set	NOUN
ejpam-5618	330	34	in	in	ADP
ejpam-5618	330	35	g	g	NOUN
ejpam-5618	330	36	since	since	SCONJ
ejpam-5618	330	37	the	the	DET
ejpam-5618	330	38	left	left	ADJ
ejpam-5618	330	39	translation	translation	NOUN
ejpam-5618	330	40	mapping	mapping	NOUN
ejpam-5618	330	41	is	be	AUX
ejpam-5618	330	42	an	an	DET
ejpam-5618	330	43	i	i	PROPN
ejpam-5618	330	44	-	-	PUNCT
ejpam-5618	330	45	homeomorphism	homeomorphism	PROPN
ejpam-5618	330	46	.	.	PUNCT
ejpam-5618	331	1	therefore	therefore	ADV
ejpam-5618	331	2	,	,	PUNCT
ejpam-5618	331	3	the	the	DET
ejpam-5618	331	4	set	set	NOUN
ejpam-5618	331	5	h	h	NOUN
ejpam-5618	331	6	=	=	PUNCT
ejpam-5618	331	7	⋃	⋃	NOUN
ejpam-5618	331	8	h∈h(hu	h∈h(hu	NOUN
ejpam-5618	331	9	)	)	PUNCT
ejpam-5618	331	10	is	be	AUX
ejpam-5618	331	11	i	i	PRON
ejpam-5618	331	12	-	-	PUNCT
ejpam-5618	331	13	open	open	ADJ
ejpam-5618	331	14	in	in	ADP
ejpam-5618	331	15	g.	g.	PROPN
ejpam-5618	331	16	corollary	corollary	PROPN
ejpam-5618	331	17	4	4	NUM
ejpam-5618	331	18	.	.	PUNCT
ejpam-5618	332	1	let	let	VERB
ejpam-5618	332	2	g	g	PRON
ejpam-5618	332	3	be	be	AUX
ejpam-5618	332	4	an	an	DET
ejpam-5618	332	5	ideal	ideal	ADJ
ejpam-5618	332	6	topological	topological	ADJ
ejpam-5618	332	7	group	group	NOUN
ejpam-5618	332	8	and	and	CCONJ
ejpam-5618	332	9	h	h	NOUN
ejpam-5618	332	10	be	be	AUX
ejpam-5618	332	11	a	a	DET
ejpam-5618	332	12	subgroup	subgroup	NOUN
ejpam-5618	332	13	of	of	ADP
ejpam-5618	332	14	g.	g.	PROPN
ejpam-5618	332	15	if	if	SCONJ
ejpam-5618	332	16	g	g	PROPN
ejpam-5618	332	17	is	be	AUX
ejpam-5618	332	18	submaximal	submaximal	ADJ
ejpam-5618	332	19	and	and	CCONJ
ejpam-5618	332	20	h	h	NOUN
ejpam-5618	332	21	contains	contain	VERB
ejpam-5618	332	22	a	a	DET
ejpam-5618	332	23	nonempty	nonempty	ADJ
ejpam-5618	332	24	i	i	PRON
ejpam-5618	332	25	-	-	PUNCT
ejpam-5618	332	26	open	open	ADJ
ejpam-5618	332	27	set	set	NOUN
ejpam-5618	332	28	,	,	PUNCT
ejpam-5618	332	29	then	then	ADV
ejpam-5618	332	30	h	h	PROPN
ejpam-5618	332	31	is	be	AUX
ejpam-5618	332	32	i	i	PRON
ejpam-5618	332	33	-	-	PUNCT
ejpam-5618	332	34	open	open	ADJ
ejpam-5618	332	35	in	in	ADP
ejpam-5618	332	36	g.	g.	PROPN
ejpam-5618	332	37	unlike	unlike	ADP
ejpam-5618	332	38	topological	topological	ADJ
ejpam-5618	332	39	groups	group	NOUN
ejpam-5618	332	40	,	,	PUNCT
ejpam-5618	332	41	ideal	ideal	ADJ
ejpam-5618	332	42	topological	topological	ADJ
ejpam-5618	332	43	groups	group	NOUN
ejpam-5618	332	44	are	be	AUX
ejpam-5618	332	45	not	not	PART
ejpam-5618	332	46	well	well	ADV
ejpam-5618	332	47	behaved	behave	VERB
ejpam-5618	332	48	with	with	ADP
ejpam-5618	332	49	respect	respect	NOUN
ejpam-5618	332	50	to	to	ADP
ejpam-5618	332	51	subgroups	subgroup	NOUN
ejpam-5618	332	52	.	.	PUNCT
ejpam-5618	333	1	the	the	DET
ejpam-5618	333	2	following	follow	VERB
ejpam-5618	333	3	example	example	NOUN
ejpam-5618	333	4	demonstrates	demonstrate	VERB
ejpam-5618	333	5	that	that	SCONJ
ejpam-5618	333	6	a	a	DET
ejpam-5618	333	7	subgroup	subgroup	NOUN
ejpam-5618	333	8	of	of	ADP
ejpam-5618	333	9	an	an	DET
ejpam-5618	333	10	ideal	ideal	ADJ
ejpam-5618	333	11	topological	topological	ADJ
ejpam-5618	333	12	group	group	NOUN
ejpam-5618	333	13	is	be	AUX
ejpam-5618	333	14	not	not	PART
ejpam-5618	333	15	necessarily	necessarily	ADV
ejpam-5618	333	16	an	an	DET
ejpam-5618	333	17	ideal	ideal	ADJ
ejpam-5618	333	18	topological	topological	ADJ
ejpam-5618	333	19	group	group	NOUN
ejpam-5618	333	20	.	.	PUNCT
ejpam-5618	334	1	example	example	NOUN
ejpam-5618	335	1	6	6	NUM
ejpam-5618	335	2	.	.	PUNCT
ejpam-5618	336	1	in	in	ADP
ejpam-5618	336	2	example	example	NOUN
ejpam-5618	336	3	1	1	NUM
ejpam-5618	336	4	,	,	PUNCT
ejpam-5618	336	5	r	r	NOUN
ejpam-5618	336	6	is	be	AUX
ejpam-5618	336	7	an	an	DET
ejpam-5618	336	8	ideal	ideal	ADJ
ejpam-5618	336	9	topological	topological	ADJ
ejpam-5618	336	10	group	group	NOUN
ejpam-5618	336	11	.	.	PUNCT
ejpam-5618	337	1	we	we	PRON
ejpam-5618	337	2	know	know	VERB
ejpam-5618	337	3	that	that	SCONJ
ejpam-5618	337	4	z	z	PROPN
ejpam-5618	337	5	is	be	AUX
ejpam-5618	337	6	a	a	DET
ejpam-5618	337	7	subgroup	subgroup	NOUN
ejpam-5618	337	8	of	of	ADP
ejpam-5618	337	9	r.	r.	PROPN
ejpam-5618	337	10	moreover	moreover	ADV
ejpam-5618	337	11	,	,	PUNCT
ejpam-5618	337	12	z	z	PROPN
ejpam-5618	337	13	is	be	AUX
ejpam-5618	337	14	closed	close	VERB
ejpam-5618	337	15	in	in	ADP
ejpam-5618	337	16	r	r	NOUN
ejpam-5618	337	17	and	and	CCONJ
ejpam-5618	337	18	z	z	NOUN
ejpam-5618	337	19	has	have	VERB
ejpam-5618	337	20	the	the	DET
ejpam-5618	337	21	discrete	discrete	ADJ
ejpam-5618	337	22	topology	topology	NOUN
ejpam-5618	337	23	.	.	PUNCT
ejpam-5618	338	1	therefore	therefore	ADV
ejpam-5618	338	2	,	,	PUNCT
ejpam-5618	338	3	the	the	DET
ejpam-5618	338	4	class	class	NOUN
ejpam-5618	338	5	of	of	ADP
ejpam-5618	338	6	i	i	NOUN
ejpam-5618	338	7	-	-	PUNCT
ejpam-5618	338	8	open	open	ADJ
ejpam-5618	338	9	sets	set	NOUN
ejpam-5618	338	10	in	in	ADP
ejpam-5618	338	11	z	z	NOUN
ejpam-5618	338	12	contains	contain	VERB
ejpam-5618	338	13	only	only	ADV
ejpam-5618	338	14	∅.	∅.	PRON
ejpam-5618	338	15	this	this	PRON
ejpam-5618	338	16	implies	imply	VERB
ejpam-5618	338	17	that	that	SCONJ
ejpam-5618	338	18	z	z	NOUN
ejpam-5618	338	19	is	be	AUX
ejpam-5618	338	20	not	not	PART
ejpam-5618	338	21	ideal	ideal	ADJ
ejpam-5618	338	22	topological	topological	ADJ
ejpam-5618	338	23	group	group	NOUN
ejpam-5618	338	24	since	since	SCONJ
ejpam-5618	338	25	neither	neither	CCONJ
ejpam-5618	338	26	the	the	DET
ejpam-5618	338	27	multiplication	multiplication	NOUN
ejpam-5618	338	28	mapping	mapping	NOUN
ejpam-5618	338	29	nor	nor	CCONJ
ejpam-5618	338	30	the	the	DET
ejpam-5618	338	31	inverse	inverse	NOUN
ejpam-5618	338	32	mapping	mapping	NOUN
ejpam-5618	338	33	is	be	AUX
ejpam-5618	338	34	i	i	PRON
ejpam-5618	338	35	-	-	PUNCT
ejpam-5618	338	36	continuous	continuous	ADJ
ejpam-5618	338	37	.	.	PUNCT
ejpam-5618	339	1	theorem	theorem	NOUN
ejpam-5618	339	2	21	21	NUM
ejpam-5618	339	3	.	.	PUNCT
ejpam-5618	340	1	every	every	DET
ejpam-5618	340	2	open	open	ADJ
ejpam-5618	340	3	subgroup	subgroup	NOUN
ejpam-5618	340	4	h	h	NOUN
ejpam-5618	340	5	of	of	ADP
ejpam-5618	340	6	an	an	DET
ejpam-5618	340	7	ideal	ideal	ADJ
ejpam-5618	340	8	topological	topological	ADJ
ejpam-5618	340	9	group	group	NOUN
ejpam-5618	340	10	g	g	PROPN
ejpam-5618	340	11	is	be	AUX
ejpam-5618	340	12	also	also	ADV
ejpam-5618	340	13	an	an	DET
ejpam-5618	340	14	ideal	ideal	ADJ
ejpam-5618	340	15	topological	topological	ADJ
ejpam-5618	340	16	group	group	NOUN
ejpam-5618	340	17	.	.	PUNCT
ejpam-5618	341	1	proof	proof	NOUN
ejpam-5618	341	2	.	.	PUNCT
ejpam-5618	342	1	we	we	PRON
ejpam-5618	342	2	shall	shall	AUX
ejpam-5618	342	3	show	show	VERB
ejpam-5618	342	4	that	that	SCONJ
ejpam-5618	342	5	the	the	DET
ejpam-5618	342	6	multiplication	multiplication	NOUN
ejpam-5618	342	7	mappingmh	mappingmh	NOUN
ejpam-5618	342	8	:	:	PUNCT
ejpam-5618	342	9	h×h	h×h	PROPN
ejpam-5618	342	10	→	→	SYM
ejpam-5618	342	11	h	h	NOUN
ejpam-5618	342	12	and	and	CCONJ
ejpam-5618	342	13	the	the	DET
ejpam-5618	342	14	inverse	inverse	NOUN
ejpam-5618	342	15	mapping	mapping	NOUN
ejpam-5618	342	16	invh	invh	VERB
ejpam-5618	342	17	:	:	PUNCT
ejpam-5618	342	18	h	h	NOUN
ejpam-5618	342	19	→	→	SYM
ejpam-5618	342	20	h	h	NOUN
ejpam-5618	342	21	both	both	PRON
ejpam-5618	342	22	are	be	AUX
ejpam-5618	342	23	i	i	PRON
ejpam-5618	342	24	-	-	NOUN
ejpam-5618	342	25	continuous	continuous	ADJ
ejpam-5618	342	26	.	.	PUNCT
ejpam-5618	343	1	first	first	ADV
ejpam-5618	343	2	we	we	PRON
ejpam-5618	343	3	show	show	VERB
ejpam-5618	343	4	that	that	SCONJ
ejpam-5618	343	5	mh	mh	PROPN
ejpam-5618	343	6	is	be	AUX
ejpam-5618	343	7	i	i	PRON
ejpam-5618	343	8	-	-	PUNCT
ejpam-5618	343	9	continuous	continuous	ADJ
ejpam-5618	343	10	.	.	PUNCT
ejpam-5618	344	1	let	let	VERB
ejpam-5618	344	2	w	w	NOUN
ejpam-5618	344	3	be	be	AUX
ejpam-5618	344	4	an	an	DET
ejpam-5618	344	5	open	open	ADJ
ejpam-5618	344	6	set	set	NOUN
ejpam-5618	344	7	in	in	ADP
ejpam-5618	344	8	h.	h.	PROPN
ejpam-5618	344	9	then	then	ADV
ejpam-5618	344	10	w	w	PROPN
ejpam-5618	344	11	is	be	AUX
ejpam-5618	344	12	open	open	ADJ
ejpam-5618	344	13	in	in	ADP
ejpam-5618	344	14	g.	g.	PROPN
ejpam-5618	344	15	since	since	SCONJ
ejpam-5618	344	16	g	g	PROPN
ejpam-5618	344	17	is	be	AUX
ejpam-5618	344	18	an	an	DET
ejpam-5618	344	19	ideal	ideal	ADJ
ejpam-5618	344	20	topological	topological	ADJ
ejpam-5618	344	21	group	group	NOUN
ejpam-5618	344	22	,	,	PUNCT
ejpam-5618	344	23	the	the	DET
ejpam-5618	344	24	multiplication	multiplication	NOUN
ejpam-5618	344	25	mapping	mapping	NOUN
ejpam-5618	345	1	mg	mg	NOUN
ejpam-5618	346	1	:	:	PUNCT
ejpam-5618	347	1	g	g	ADP
ejpam-5618	347	2	×	×	PROPN
ejpam-5618	347	3	g	g	PROPN
ejpam-5618	347	4	→	→	SYM
ejpam-5618	347	5	g	g	PROPN
ejpam-5618	347	6	is	be	AUX
ejpam-5618	347	7	i	i	NOUN
ejpam-5618	347	8	-	-	PUNCT
ejpam-5618	347	9	continuous	continuous	ADJ
ejpam-5618	347	10	.	.	PUNCT
ejpam-5618	348	1	but	but	CCONJ
ejpam-5618	348	2	h	h	NOUN
ejpam-5618	348	3	is	be	AUX
ejpam-5618	348	4	open	open	ADJ
ejpam-5618	348	5	in	in	ADP
ejpam-5618	348	6	g	g	PROPN
ejpam-5618	348	7	and	and	CCONJ
ejpam-5618	348	8	h	h	NOUN
ejpam-5618	348	9	×	×	NOUN
ejpam-5618	348	10	h	h	NOUN
ejpam-5618	348	11	is	be	AUX
ejpam-5618	348	12	open	open	ADJ
ejpam-5618	348	13	in	in	ADP
ejpam-5618	348	14	g	g	PROPN
ejpam-5618	348	15	×	×	PROPN
ejpam-5618	348	16	g.	g.	NOUN
ejpam-5618	348	17	using	use	VERB
ejpam-5618	348	18	theorem	theorem	NOUN
ejpam-5618	348	19	6	6	NUM
ejpam-5618	348	20	,	,	PUNCT
ejpam-5618	349	1	the	the	DET
ejpam-5618	349	2	restriction	restriction	NOUN
ejpam-5618	349	3	mg|h×h	mg|h×h	NOUN
ejpam-5618	349	4	:	:	PUNCT
ejpam-5618	349	5	h	h	PROPN
ejpam-5618	349	6	×	×	NOUN
ejpam-5618	349	7	h	h	NOUN
ejpam-5618	349	8	→	→	SYM
ejpam-5618	349	9	g	g	PROPN
ejpam-5618	349	10	is	be	AUX
ejpam-5618	349	11	i	i	NOUN
ejpam-5618	349	12	-	-	PUNCT
ejpam-5618	349	13	continuous	continuous	ADJ
ejpam-5618	349	14	.	.	PUNCT
ejpam-5618	350	1	thus	thus	ADV
ejpam-5618	350	2	,	,	PUNCT
ejpam-5618	350	3	there	there	PRON
ejpam-5618	350	4	exists	exist	VERB
ejpam-5618	350	5	an	an	DET
ejpam-5618	350	6	i	i	NOUN
ejpam-5618	350	7	-	-	PUNCT
ejpam-5618	350	8	open	open	ADJ
ejpam-5618	350	9	set	set	VERB
ejpam-5618	350	10	u	u	PRON
ejpam-5618	350	11	×	×	PROPN
ejpam-5618	350	12	v	v	NOUN
ejpam-5618	350	13	in	in	ADP
ejpam-5618	350	14	h	h	NOUN
ejpam-5618	350	15	×h	×h	PROPN
ejpam-5618	351	1	and	and	CCONJ
ejpam-5618	351	2	uv	uv	PRON
ejpam-5618	351	3	⊂	⊂	PROPN
ejpam-5618	351	4	w	w	PROPN
ejpam-5618	351	5	.	.	PUNCT
ejpam-5618	352	1	note	note	VERB
ejpam-5618	352	2	that	that	SCONJ
ejpam-5618	352	3	u	u	PROPN
ejpam-5618	352	4	and	and	CCONJ
ejpam-5618	352	5	v	v	PROPN
ejpam-5618	352	6	both	both	PRON
ejpam-5618	352	7	are	be	AUX
ejpam-5618	352	8	i	i	PRON
ejpam-5618	352	9	-	-	PUNCT
ejpam-5618	352	10	open	open	ADJ
ejpam-5618	352	11	in	in	ADP
ejpam-5618	352	12	h	h	NOUN
ejpam-5618	352	13	by	by	ADP
ejpam-5618	352	14	theorem	theorem	NOUN
ejpam-5618	352	15	5	5	NUM
ejpam-5618	352	16	.	.	PUNCT
ejpam-5618	353	1	therefore	therefore	ADV
ejpam-5618	353	2	,	,	PUNCT
ejpam-5618	353	3	mh	mh	PROPN
ejpam-5618	353	4	is	be	AUX
ejpam-5618	353	5	i	i	PRON
ejpam-5618	353	6	-	-	PUNCT
ejpam-5618	353	7	continuous	continuous	ADJ
ejpam-5618	353	8	.	.	PUNCT
ejpam-5618	354	1	similarly	similarly	ADV
ejpam-5618	354	2	,	,	PUNCT
ejpam-5618	354	3	we	we	PRON
ejpam-5618	354	4	can	can	AUX
ejpam-5618	354	5	show	show	VERB
ejpam-5618	354	6	that	that	SCONJ
ejpam-5618	354	7	the	the	DET
ejpam-5618	354	8	inverse	inverse	NOUN
ejpam-5618	354	9	mapping	mapping	NOUN
ejpam-5618	354	10	invh	invh	VERB
ejpam-5618	354	11	is	be	AUX
ejpam-5618	354	12	i	i	NOUN
ejpam-5618	354	13	-	-	PUNCT
ejpam-5618	354	14	continuous	continuous	ADJ
ejpam-5618	354	15	.	.	PUNCT
ejpam-5618	355	1	hence	hence	ADV
ejpam-5618	355	2	,	,	PUNCT
ejpam-5618	355	3	h	h	NOUN
ejpam-5618	355	4	is	be	AUX
ejpam-5618	355	5	ideal	ideal	ADJ
ejpam-5618	355	6	topological	topological	ADJ
ejpam-5618	355	7	group	group	NOUN
ejpam-5618	355	8	.	.	PUNCT
ejpam-5618	356	1	s.	s.	PROPN
ejpam-5618	356	2	alammar	alammar	PROPN
ejpam-5618	356	3	,	,	PUNCT
ejpam-5618	356	4	m.	m.	NOUN
ejpam-5618	356	5	al	al	PROPN
ejpam-5618	356	6	shumrani	shumrani	PROPN
ejpam-5618	356	7	,	,	PUNCT
ejpam-5618	356	8	c.	c.	PROPN
ejpam-5618	356	9	özel	özel	PROPN
ejpam-5618	356	10	/	/	SYM
ejpam-5618	356	11	eur	eur	PROPN
ejpam-5618	356	12	.	.	PUNCT
ejpam-5618	357	1	j.	j.	PROPN
ejpam-5618	357	2	pure	pure	PROPN
ejpam-5618	357	3	appl	appl	PROPN
ejpam-5618	357	4	.	.	PROPN
ejpam-5618	357	5	math	math	PROPN
ejpam-5618	357	6	,	,	PUNCT
ejpam-5618	357	7	18	18	NUM
ejpam-5618	357	8	(	(	PUNCT
ejpam-5618	357	9	1	1	NUM
ejpam-5618	357	10	)	)	PUNCT
ejpam-5618	357	11	(	(	PUNCT
ejpam-5618	357	12	2025	2025	NUM
ejpam-5618	357	13	)	)	PUNCT
ejpam-5618	357	14	,	,	PUNCT
ejpam-5618	357	15	5618	5618	NUM
ejpam-5618	357	16	10	10	NUM
ejpam-5618	357	17	of	of	ADP
ejpam-5618	357	18	13	13	NUM
ejpam-5618	357	19	theorem	theorem	NOUN
ejpam-5618	357	20	22	22	NUM
ejpam-5618	357	21	.	.	PUNCT
ejpam-5618	358	1	every	every	DET
ejpam-5618	358	2	open	open	ADJ
ejpam-5618	358	3	subgroup	subgroup	NOUN
ejpam-5618	358	4	h	h	NOUN
ejpam-5618	358	5	of	of	ADP
ejpam-5618	358	6	an	an	DET
ejpam-5618	358	7	ideal	ideal	ADJ
ejpam-5618	358	8	topological	topological	ADJ
ejpam-5618	358	9	group	group	NOUN
ejpam-5618	358	10	g	g	PROPN
ejpam-5618	358	11	is	be	AUX
ejpam-5618	358	12	i	i	PRON
ejpam-5618	358	13	-	-	PUNCT
ejpam-5618	358	14	closed	close	VERB
ejpam-5618	358	15	in	in	ADP
ejpam-5618	358	16	g.	g.	PROPN
ejpam-5618	358	17	proof	proof	NOUN
ejpam-5618	358	18	.	.	PUNCT
ejpam-5618	359	1	since	since	SCONJ
ejpam-5618	359	2	h	h	NOUN
ejpam-5618	359	3	is	be	AUX
ejpam-5618	359	4	open	open	ADJ
ejpam-5618	359	5	,	,	PUNCT
ejpam-5618	359	6	hg	hg	PROPN
ejpam-5618	359	7	is	be	AUX
ejpam-5618	359	8	i	i	PRON
ejpam-5618	359	9	-	-	PUNCT
ejpam-5618	359	10	open	open	ADJ
ejpam-5618	359	11	for	for	ADP
ejpam-5618	359	12	each	each	DET
ejpam-5618	359	13	g	g	PROPN
ejpam-5618	359	14	∈	∈	PROPN
ejpam-5618	359	15	g	g	NOUN
ejpam-5618	359	16	by	by	ADP
ejpam-5618	359	17	proposition	proposition	NOUN
ejpam-5618	359	18	1	1	NUM
ejpam-5618	359	19	.	.	PUNCT
ejpam-5618	360	1	consider	consider	VERB
ejpam-5618	360	2	the	the	DET
ejpam-5618	360	3	family	family	NOUN
ejpam-5618	360	4	α	α	NOUN
ejpam-5618	360	5	=	=	PUNCT
ejpam-5618	360	6	{	{	PUNCT
ejpam-5618	360	7	hg	hg	NOUN
ejpam-5618	360	8	:	:	PUNCT
ejpam-5618	360	9	g	g	PROPN
ejpam-5618	360	10	∈	∈	PROPN
ejpam-5618	360	11	g	g	PROPN
ejpam-5618	360	12	}	}	PUNCT
ejpam-5618	360	13	of	of	ADP
ejpam-5618	360	14	all	all	DET
ejpam-5618	360	15	right	right	ADJ
ejpam-5618	360	16	cosets	coset	NOUN
ejpam-5618	360	17	of	of	ADP
ejpam-5618	360	18	h	h	PROPN
ejpam-5618	360	19	in	in	ADP
ejpam-5618	360	20	g.	g.	PROPN
ejpam-5618	360	21	note	note	VERB
ejpam-5618	360	22	that	that	SCONJ
ejpam-5618	360	23	α	α	PRON
ejpam-5618	360	24	is	be	AUX
ejpam-5618	360	25	a	a	DET
ejpam-5618	360	26	disjoint	disjoint	NOUN
ejpam-5618	360	27	i	i	NOUN
ejpam-5618	360	28	-	-	PUNCT
ejpam-5618	360	29	open	open	ADJ
ejpam-5618	360	30	covering	covering	NOUN
ejpam-5618	360	31	of	of	ADP
ejpam-5618	360	32	g.	g.	PROPN
ejpam-5618	360	33	therefore	therefore	ADV
ejpam-5618	360	34	,	,	PUNCT
ejpam-5618	360	35	each	each	DET
ejpam-5618	360	36	element	element	NOUN
ejpam-5618	360	37	of	of	ADP
ejpam-5618	360	38	α	α	PROPN
ejpam-5618	360	39	is	be	AUX
ejpam-5618	360	40	i	i	PRON
ejpam-5618	360	41	-	-	PUNCT
ejpam-5618	360	42	closed	close	VERB
ejpam-5618	360	43	in	in	ADP
ejpam-5618	360	44	g.	g.	PROPN
ejpam-5618	360	45	in	in	ADP
ejpam-5618	360	46	particular	particular	ADJ
ejpam-5618	360	47	,	,	PUNCT
ejpam-5618	360	48	h	h	NOUN
ejpam-5618	361	1	=	=	PRON
ejpam-5618	361	2	he	he	PRON
ejpam-5618	361	3	is	be	AUX
ejpam-5618	361	4	i	i	PROPN
ejpam-5618	361	5	-	-	PUNCT
ejpam-5618	361	6	closed	close	VERB
ejpam-5618	361	7	in	in	ADP
ejpam-5618	361	8	g.	g.	PROPN
ejpam-5618	361	9	corollary	corollary	PROPN
ejpam-5618	361	10	5	5	NUM
ejpam-5618	361	11	.	.	PUNCT
ejpam-5618	362	1	let	let	VERB
ejpam-5618	362	2	g	g	PRON
ejpam-5618	362	3	be	be	AUX
ejpam-5618	362	4	an	an	DET
ejpam-5618	362	5	ideal	ideal	ADJ
ejpam-5618	362	6	topological	topological	ADJ
ejpam-5618	362	7	group	group	NOUN
ejpam-5618	362	8	and	and	CCONJ
ejpam-5618	362	9	h	h	NOUN
ejpam-5618	362	10	be	be	AUX
ejpam-5618	362	11	a	a	DET
ejpam-5618	362	12	subgroup	subgroup	NOUN
ejpam-5618	362	13	of	of	ADP
ejpam-5618	362	14	g.	g.	PROPN
ejpam-5618	362	15	if	if	SCONJ
ejpam-5618	362	16	h	h	NOUN
ejpam-5618	362	17	is	be	AUX
ejpam-5618	362	18	i	i	PRON
ejpam-5618	362	19	-	-	PUNCT
ejpam-5618	362	20	open	open	ADJ
ejpam-5618	362	21	and	and	CCONJ
ejpam-5618	362	22	g	g	NOUN
ejpam-5618	362	23	is	be	AUX
ejpam-5618	362	24	submaximal	submaximal	ADJ
ejpam-5618	362	25	,	,	PUNCT
ejpam-5618	362	26	then	then	ADV
ejpam-5618	362	27	h	h	PROPN
ejpam-5618	362	28	is	be	AUX
ejpam-5618	362	29	i	i	NOUN
ejpam-5618	362	30	-	-	PUNCT
ejpam-5618	362	31	closed	closed	ADJ
ejpam-5618	362	32	in	in	ADP
ejpam-5618	362	33	g.	g.	PROPN
ejpam-5618	362	34	4	4	NUM
ejpam-5618	362	35	.	.	PUNCT
ejpam-5618	363	1	i	i	PRON
ejpam-5618	363	2	-	-	PUNCT
ejpam-5618	363	3	connectedness	connectedness	NOUN
ejpam-5618	363	4	in	in	ADP
ejpam-5618	363	5	ideal	ideal	ADJ
ejpam-5618	363	6	topological	topological	ADJ
ejpam-5618	363	7	groups	group	NOUN
ejpam-5618	363	8	in	in	ADP
ejpam-5618	363	9	this	this	DET
ejpam-5618	363	10	section	section	NOUN
ejpam-5618	363	11	,	,	PUNCT
ejpam-5618	363	12	first	first	ADV
ejpam-5618	363	13	we	we	PRON
ejpam-5618	363	14	review	review	VERB
ejpam-5618	363	15	the	the	DET
ejpam-5618	363	16	definition	definition	NOUN
ejpam-5618	363	17	of	of	ADP
ejpam-5618	363	18	i	i	NOUN
ejpam-5618	363	19	-	-	PUNCT
ejpam-5618	363	20	connectedness	connectedness	NOUN
ejpam-5618	363	21	in	in	ADP
ejpam-5618	363	22	ideal	ideal	ADJ
ejpam-5618	363	23	topological	topological	ADJ
ejpam-5618	363	24	spaces	space	NOUN
ejpam-5618	363	25	.	.	PUNCT
ejpam-5618	364	1	we	we	PRON
ejpam-5618	364	2	give	give	VERB
ejpam-5618	364	3	examples	example	NOUN
ejpam-5618	364	4	of	of	ADP
ejpam-5618	364	5	i	i	PRON
ejpam-5618	364	6	-	-	PUNCT
ejpam-5618	364	7	connected	connect	VERB
ejpam-5618	364	8	and	and	CCONJ
ejpam-5618	364	9	i	i	PRON
ejpam-5618	364	10	-	-	PUNCT
ejpam-5618	364	11	disconnected	disconnected	ADJ
ejpam-5618	364	12	ideal	ideal	ADJ
ejpam-5618	364	13	topological	topological	ADJ
ejpam-5618	364	14	spaces	space	NOUN
ejpam-5618	364	15	and	and	CCONJ
ejpam-5618	364	16	ideal	ideal	ADJ
ejpam-5618	364	17	topological	topological	ADJ
ejpam-5618	364	18	groups	group	NOUN
ejpam-5618	364	19	.	.	PUNCT
ejpam-5618	365	1	furthermore	furthermore	ADV
ejpam-5618	365	2	,	,	PUNCT
ejpam-5618	365	3	some	some	DET
ejpam-5618	365	4	properties	property	NOUN
ejpam-5618	365	5	of	of	ADP
ejpam-5618	365	6	i	i	NOUN
ejpam-5618	365	7	-	-	PUNCT
ejpam-5618	365	8	connectedness	connectedness	NOUN
ejpam-5618	365	9	of	of	ADP
ejpam-5618	365	10	ideal	ideal	ADJ
ejpam-5618	365	11	topological	topological	ADJ
ejpam-5618	365	12	groups	group	NOUN
ejpam-5618	365	13	are	be	AUX
ejpam-5618	365	14	studied	study	VERB
ejpam-5618	365	15	.	.	PUNCT
ejpam-5618	366	1	definition	definition	NOUN
ejpam-5618	366	2	15	15	NUM
ejpam-5618	366	3	.	.	PUNCT
ejpam-5618	367	1	let	let	VERB
ejpam-5618	367	2	x	x	PRON
ejpam-5618	367	3	be	be	AUX
ejpam-5618	367	4	an	an	DET
ejpam-5618	367	5	ideal	ideal	ADJ
ejpam-5618	367	6	topological	topological	ADJ
ejpam-5618	367	7	space	space	NOUN
ejpam-5618	367	8	.	.	PUNCT
ejpam-5618	368	1	an	an	DET
ejpam-5618	368	2	i	i	NOUN
ejpam-5618	368	3	-	-	PUNCT
ejpam-5618	368	4	separation	separation	NOUN
ejpam-5618	368	5	of	of	ADP
ejpam-5618	368	6	x	x	X
ejpam-5618	368	7	is	be	AUX
ejpam-5618	368	8	a	a	DET
ejpam-5618	368	9	pair	pair	NOUN
ejpam-5618	368	10	a	a	DET
ejpam-5618	368	11	,	,	PUNCT
ejpam-5618	368	12	b	b	PROPN
ejpam-5618	368	13	of	of	ADP
ejpam-5618	368	14	disjoint	disjoint	NOUN
ejpam-5618	368	15	nonempty	nonempty	NOUN
ejpam-5618	368	16	i	i	PRON
ejpam-5618	368	17	-	-	PUNCT
ejpam-5618	368	18	open	open	ADJ
ejpam-5618	368	19	subsets	subset	NOUN
ejpam-5618	368	20	of	of	ADP
ejpam-5618	368	21	x	x	PUNCT
ejpam-5618	368	22	whose	whose	DET
ejpam-5618	368	23	union	union	NOUN
ejpam-5618	368	24	is	be	AUX
ejpam-5618	368	25	x.	x.	NOUN
ejpam-5618	368	26	definition	definition	NOUN
ejpam-5618	368	27	16	16	NUM
ejpam-5618	368	28	.	.	PUNCT
ejpam-5618	369	1	an	an	DET
ejpam-5618	369	2	ideal	ideal	ADJ
ejpam-5618	369	3	topological	topological	ADJ
ejpam-5618	369	4	space	space	NOUN
ejpam-5618	369	5	x	x	PRON
ejpam-5618	369	6	is	be	AUX
ejpam-5618	369	7	said	say	VERB
ejpam-5618	369	8	to	to	PART
ejpam-5618	369	9	be	be	AUX
ejpam-5618	369	10	an	an	DET
ejpam-5618	369	11	i	i	NOUN
ejpam-5618	369	12	-	-	PUNCT
ejpam-5618	369	13	connected	connect	VERB
ejpam-5618	369	14	if	if	SCONJ
ejpam-5618	369	15	there	there	PRON
ejpam-5618	369	16	does	do	AUX
ejpam-5618	369	17	not	not	PART
ejpam-5618	369	18	exist	exist	VERB
ejpam-5618	369	19	an	an	DET
ejpam-5618	369	20	i	i	NOUN
ejpam-5618	369	21	-	-	PUNCT
ejpam-5618	369	22	separation	separation	NOUN
ejpam-5618	369	23	of	of	ADP
ejpam-5618	369	24	x.	x.	PROPN
ejpam-5618	369	25	example	example	NOUN
ejpam-5618	370	1	7	7	NUM
ejpam-5618	370	2	.	.	PUNCT
ejpam-5618	371	1	any	any	DET
ejpam-5618	371	2	group	group	NOUN
ejpam-5618	371	3	g	g	NOUN
ejpam-5618	371	4	with	with	ADP
ejpam-5618	371	5	the	the	DET
ejpam-5618	371	6	discrete	discrete	ADJ
ejpam-5618	371	7	topology	topology	NOUN
ejpam-5618	371	8	and	and	CCONJ
ejpam-5618	371	9	with	with	ADP
ejpam-5618	371	10	the	the	DET
ejpam-5618	371	11	ideal	ideal	NOUN
ejpam-5618	371	12	of	of	ADP
ejpam-5618	371	13	nowhere	nowhere	ADJ
ejpam-5618	371	14	dense	dense	ADJ
ejpam-5618	371	15	subsets	subset	NOUN
ejpam-5618	371	16	in	in	ADP
ejpam-5618	371	17	g	g	PROPN
ejpam-5618	371	18	is	be	AUX
ejpam-5618	371	19	an	an	DET
ejpam-5618	371	20	ideal	ideal	ADJ
ejpam-5618	371	21	topological	topological	ADJ
ejpam-5618	371	22	space	space	NOUN
ejpam-5618	371	23	in	in	ADP
ejpam-5618	371	24	which	which	PRON
ejpam-5618	371	25	the	the	DET
ejpam-5618	371	26	class	class	NOUN
ejpam-5618	371	27	of	of	ADP
ejpam-5618	371	28	i	i	PROPN
ejpam-5618	371	29	-	-	PUNCT
ejpam-5618	371	30	open	open	ADJ
ejpam-5618	371	31	sets	set	NOUN
ejpam-5618	371	32	is	be	AUX
ejpam-5618	371	33	the	the	DET
ejpam-5618	371	34	power	power	NOUN
ejpam-5618	371	35	set	set	NOUN
ejpam-5618	371	36	of	of	ADP
ejpam-5618	371	37	g.	g.	PROPN
ejpam-5618	371	38	thus	thus	ADV
ejpam-5618	371	39	,	,	PUNCT
ejpam-5618	371	40	g	g	PROPN
ejpam-5618	371	41	is	be	AUX
ejpam-5618	371	42	an	an	DET
ejpam-5618	371	43	i	i	NOUN
ejpam-5618	371	44	-	-	PUNCT
ejpam-5618	371	45	disconnected	disconnected	ADJ
ejpam-5618	371	46	space	space	NOUN
ejpam-5618	371	47	.	.	PUNCT
ejpam-5618	372	1	in	in	ADP
ejpam-5618	372	2	particular	particular	ADJ
ejpam-5618	372	3	,	,	PUNCT
ejpam-5618	372	4	r	r	NOUN
ejpam-5618	372	5	with	with	ADP
ejpam-5618	372	6	the	the	DET
ejpam-5618	372	7	discrete	discrete	ADJ
ejpam-5618	372	8	topology	topology	NOUN
ejpam-5618	372	9	and	and	CCONJ
ejpam-5618	372	10	with	with	ADP
ejpam-5618	372	11	the	the	DET
ejpam-5618	372	12	ideal	ideal	NOUN
ejpam-5618	372	13	of	of	ADP
ejpam-5618	372	14	nowhere	nowhere	ADJ
ejpam-5618	372	15	dense	dense	ADJ
ejpam-5618	372	16	subsets	subset	NOUN
ejpam-5618	372	17	in	in	ADP
ejpam-5618	372	18	r	r	NOUN
ejpam-5618	372	19	is	be	AUX
ejpam-5618	372	20	an	an	DET
ejpam-5618	372	21	i	i	NOUN
ejpam-5618	372	22	-	-	PUNCT
ejpam-5618	372	23	disconnected	disconnect	VERB
ejpam-5618	372	24	ideal	ideal	ADJ
ejpam-5618	372	25	topological	topological	ADJ
ejpam-5618	372	26	space	space	NOUN
ejpam-5618	372	27	.	.	PUNCT
ejpam-5618	373	1	example	example	NOUN
ejpam-5618	373	2	8	8	NUM
ejpam-5618	373	3	.	.	PUNCT
ejpam-5618	374	1	any	any	DET
ejpam-5618	374	2	group	group	NOUN
ejpam-5618	374	3	g	g	NOUN
ejpam-5618	374	4	with	with	ADP
ejpam-5618	374	5	a	a	DET
ejpam-5618	374	6	topology	topology	NOUN
ejpam-5618	374	7	and	and	CCONJ
ejpam-5618	374	8	with	with	ADP
ejpam-5618	374	9	the	the	DET
ejpam-5618	374	10	ideal	ideal	NOUN
ejpam-5618	374	11	of	of	ADP
ejpam-5618	374	12	all	all	DET
ejpam-5618	374	13	subsets	subset	NOUN
ejpam-5618	374	14	in	in	ADP
ejpam-5618	374	15	g	g	PROPN
ejpam-5618	374	16	is	be	AUX
ejpam-5618	374	17	an	an	DET
ejpam-5618	374	18	ideal	ideal	ADJ
ejpam-5618	374	19	topological	topological	ADJ
ejpam-5618	374	20	space	space	NOUN
ejpam-5618	374	21	in	in	ADP
ejpam-5618	374	22	which	which	PRON
ejpam-5618	374	23	the	the	DET
ejpam-5618	374	24	class	class	NOUN
ejpam-5618	374	25	of	of	ADP
ejpam-5618	374	26	i	i	PROPN
ejpam-5618	374	27	-	-	PUNCT
ejpam-5618	374	28	open	open	ADJ
ejpam-5618	374	29	sets	set	NOUN
ejpam-5618	374	30	contains	contain	VERB
ejpam-5618	374	31	only	only	ADV
ejpam-5618	374	32	the	the	DET
ejpam-5618	374	33	empty	empty	ADJ
ejpam-5618	374	34	set	set	NOUN
ejpam-5618	374	35	∅.	∅.	PRON
ejpam-5618	374	36	thus	thus	ADV
ejpam-5618	374	37	,	,	PUNCT
ejpam-5618	374	38	g	g	PROPN
ejpam-5618	374	39	is	be	AUX
ejpam-5618	374	40	an	an	DET
ejpam-5618	374	41	i	i	NOUN
ejpam-5618	374	42	-	-	PUNCT
ejpam-5618	374	43	connected	connect	VERB
ejpam-5618	374	44	space	space	NOUN
ejpam-5618	374	45	.	.	PUNCT
ejpam-5618	375	1	in	in	ADP
ejpam-5618	375	2	particular	particular	ADJ
ejpam-5618	375	3	,	,	PUNCT
ejpam-5618	375	4	r	r	NOUN
ejpam-5618	375	5	with	with	ADP
ejpam-5618	375	6	its	its	PRON
ejpam-5618	375	7	usual	usual	ADJ
ejpam-5618	375	8	topology	topology	NOUN
ejpam-5618	375	9	and	and	CCONJ
ejpam-5618	375	10	with	with	ADP
ejpam-5618	375	11	the	the	DET
ejpam-5618	375	12	ideal	ideal	NOUN
ejpam-5618	375	13	of	of	ADP
ejpam-5618	375	14	all	all	DET
ejpam-5618	375	15	subsets	subset	NOUN
ejpam-5618	375	16	in	in	ADP
ejpam-5618	375	17	r	r	NOUN
ejpam-5618	375	18	is	be	AUX
ejpam-5618	375	19	an	an	DET
ejpam-5618	375	20	i	i	NOUN
ejpam-5618	375	21	-	-	PUNCT
ejpam-5618	375	22	connected	connect	VERB
ejpam-5618	375	23	ideal	ideal	ADJ
ejpam-5618	375	24	topological	topological	ADJ
ejpam-5618	375	25	space	space	NOUN
ejpam-5618	375	26	.	.	PUNCT
ejpam-5618	376	1	the	the	DET
ejpam-5618	376	2	following	follow	VERB
ejpam-5618	376	3	is	be	AUX
ejpam-5618	376	4	an	an	DET
ejpam-5618	376	5	example	example	NOUN
ejpam-5618	376	6	of	of	ADP
ejpam-5618	376	7	i	i	NOUN
ejpam-5618	376	8	-	-	PUNCT
ejpam-5618	376	9	connected	connect	VERB
ejpam-5618	376	10	ideal	ideal	ADJ
ejpam-5618	376	11	topological	topological	ADJ
ejpam-5618	376	12	group	group	NOUN
ejpam-5618	376	13	.	.	PUNCT
ejpam-5618	376	14	example	example	NOUN
ejpam-5618	377	1	9	9	NUM
ejpam-5618	377	2	.	.	X
ejpam-5618	377	3	consider	consider	VERB
ejpam-5618	377	4	z2	z2	NOUN
ejpam-5618	377	5	=	=	SYM
ejpam-5618	377	6	{	{	PUNCT
ejpam-5618	377	7	0	0	NUM
ejpam-5618	377	8	,	,	PUNCT
ejpam-5618	377	9	1	1	NUM
ejpam-5618	377	10	}	}	PUNCT
ejpam-5618	377	11	,	,	PUNCT
ejpam-5618	377	12	the	the	DET
ejpam-5618	377	13	group	group	NOUN
ejpam-5618	377	14	of	of	ADP
ejpam-5618	377	15	integers	integer	NOUN
ejpam-5618	377	16	mod	mod	PROPN
ejpam-5618	377	17	2	2	NUM
ejpam-5618	377	18	,	,	PUNCT
ejpam-5618	377	19	with	with	ADP
ejpam-5618	377	20	the	the	DET
ejpam-5618	377	21	trivial	trivial	ADJ
ejpam-5618	377	22	topology	topology	NOUN
ejpam-5618	377	23	τ	τ	X
ejpam-5618	377	24	:	:	PUNCT
ejpam-5618	377	25	{	{	PUNCT
ejpam-5618	377	26	∅,z2	∅,z2	NOUN
ejpam-5618	377	27	}	}	PUNCT
ejpam-5618	377	28	.	.	PUNCT
ejpam-5618	378	1	consider	consider	VERB
ejpam-5618	378	2	the	the	DET
ejpam-5618	378	3	ideal	ideal	NOUN
ejpam-5618	378	4	i	i	PRON
ejpam-5618	378	5	=	=	SYM
ejpam-5618	378	6	{	{	PUNCT
ejpam-5618	378	7	∅	∅	NOUN
ejpam-5618	378	8	,	,	PUNCT
ejpam-5618	378	9	{	{	PUNCT
ejpam-5618	378	10	1	1	NUM
ejpam-5618	378	11	}	}	PUNCT
ejpam-5618	378	12	}	}	PUNCT
ejpam-5618	378	13	.	.	PUNCT
ejpam-5618	379	1	then	then	ADV
ejpam-5618	379	2	it	it	PRON
ejpam-5618	379	3	is	be	AUX
ejpam-5618	379	4	not	not	PART
ejpam-5618	379	5	difficult	difficult	ADJ
ejpam-5618	379	6	to	to	PART
ejpam-5618	379	7	show	show	VERB
ejpam-5618	379	8	that	that	SCONJ
ejpam-5618	379	9	the	the	DET
ejpam-5618	379	10	i	i	NOUN
ejpam-5618	379	11	-	-	PUNCT
ejpam-5618	379	12	open	open	ADJ
ejpam-5618	379	13	sets	set	NOUN
ejpam-5618	379	14	are	be	AUX
ejpam-5618	379	15	∅	∅	NOUN
ejpam-5618	379	16	,	,	PUNCT
ejpam-5618	379	17	z2	z2	NOUN
ejpam-5618	379	18	,	,	PUNCT
ejpam-5618	379	19	and	and	CCONJ
ejpam-5618	379	20	{	{	PUNCT
ejpam-5618	379	21	0	0	NOUN
ejpam-5618	379	22	}	}	PUNCT
ejpam-5618	379	23	and	and	CCONJ
ejpam-5618	379	24	z2	z2	PROPN
ejpam-5618	379	25	is	be	AUX
ejpam-5618	379	26	an	an	DET
ejpam-5618	379	27	ideal	ideal	ADJ
ejpam-5618	379	28	topological	topological	ADJ
ejpam-5618	379	29	group	group	NOUN
ejpam-5618	379	30	.	.	PUNCT
ejpam-5618	380	1	it	it	PRON
ejpam-5618	380	2	is	be	AUX
ejpam-5618	380	3	obvious	obvious	ADJ
ejpam-5618	380	4	that	that	SCONJ
ejpam-5618	380	5	z2	z2	PROPN
ejpam-5618	380	6	is	be	AUX
ejpam-5618	380	7	i	i	PRON
ejpam-5618	380	8	-	-	PUNCT
ejpam-5618	380	9	connected	connect	VERB
ejpam-5618	380	10	.	.	PUNCT
ejpam-5618	381	1	the	the	DET
ejpam-5618	381	2	following	follow	VERB
ejpam-5618	381	3	is	be	AUX
ejpam-5618	381	4	an	an	DET
ejpam-5618	381	5	example	example	NOUN
ejpam-5618	381	6	of	of	ADP
ejpam-5618	381	7	i	i	NOUN
ejpam-5618	381	8	-	-	PUNCT
ejpam-5618	381	9	disconnected	disconnect	VERB
ejpam-5618	381	10	ideal	ideal	ADJ
ejpam-5618	381	11	topological	topological	ADJ
ejpam-5618	381	12	group	group	NOUN
ejpam-5618	381	13	.	.	PUNCT
ejpam-5618	381	14	example	example	NOUN
ejpam-5618	382	1	10	10	NUM
ejpam-5618	382	2	.	.	PUNCT
ejpam-5618	383	1	in	in	ADP
ejpam-5618	383	2	example	example	NOUN
ejpam-5618	383	3	1	1	NUM
ejpam-5618	383	4	,	,	PUNCT
ejpam-5618	383	5	r	r	NOUN
ejpam-5618	383	6	is	be	AUX
ejpam-5618	383	7	an	an	DET
ejpam-5618	383	8	ideal	ideal	ADJ
ejpam-5618	383	9	topological	topological	ADJ
ejpam-5618	383	10	group	group	NOUN
ejpam-5618	383	11	.	.	PUNCT
ejpam-5618	384	1	it	it	PRON
ejpam-5618	384	2	can	can	AUX
ejpam-5618	384	3	be	be	AUX
ejpam-5618	384	4	shown	show	VERB
ejpam-5618	384	5	that	that	SCONJ
ejpam-5618	384	6	the	the	DET
ejpam-5618	384	7	set	set	NOUN
ejpam-5618	384	8	of	of	ADP
ejpam-5618	384	9	rationals	rational	NOUN
ejpam-5618	384	10	and	and	CCONJ
ejpam-5618	384	11	the	the	DET
ejpam-5618	384	12	set	set	NOUN
ejpam-5618	384	13	of	of	ADP
ejpam-5618	384	14	irrationals	irrational	NOUN
ejpam-5618	384	15	are	be	AUX
ejpam-5618	384	16	i	i	NOUN
ejpam-5618	384	17	-	-	PUNCT
ejpam-5618	384	18	open	open	ADJ
ejpam-5618	384	19	sets	set	NOUN
ejpam-5618	384	20	.	.	PUNCT
ejpam-5618	385	1	therefore	therefore	ADV
ejpam-5618	385	2	,	,	PUNCT
ejpam-5618	385	3	r	r	NOUN
ejpam-5618	385	4	is	be	AUX
ejpam-5618	385	5	i	i	PRON
ejpam-5618	385	6	-	-	PUNCT
ejpam-5618	385	7	disconnected	disconnected	ADJ
ejpam-5618	385	8	.	.	PUNCT
ejpam-5618	386	1	definition	definition	NOUN
ejpam-5618	386	2	17	17	NUM
ejpam-5618	386	3	.	.	PUNCT
ejpam-5618	387	1	let	let	VERB
ejpam-5618	387	2	x	x	PRON
ejpam-5618	387	3	be	be	AUX
ejpam-5618	387	4	an	an	DET
ejpam-5618	387	5	ideal	ideal	ADJ
ejpam-5618	387	6	topological	topological	ADJ
ejpam-5618	387	7	space	space	NOUN
ejpam-5618	387	8	;	;	PUNCT
ejpam-5618	387	9	let	let	VERB
ejpam-5618	387	10	s	s	PRON
ejpam-5618	387	11	⊂	⊂	X
ejpam-5618	387	12	x.	x.	NOUN
ejpam-5618	387	13	for	for	ADP
ejpam-5618	387	14	x	x	PROPN
ejpam-5618	387	15	∈	∈	PROPN
ejpam-5618	387	16	s	s	PROPN
ejpam-5618	387	17	,	,	PUNCT
ejpam-5618	387	18	the	the	DET
ejpam-5618	387	19	set	set	VERB
ejpam-5618	387	20	sx	sx	NOUN
ejpam-5618	387	21	=	=	NOUN
ejpam-5618	387	22	⋃	⋃	PROPN
ejpam-5618	387	23	x∈c⊂s	x∈c⊂s	PROPN
ejpam-5618	388	1	c	c	NOUN
ejpam-5618	388	2	,	,	PUNCT
ejpam-5618	388	3	where	where	SCONJ
ejpam-5618	388	4	c	c	NOUN
ejpam-5618	388	5	is	be	AUX
ejpam-5618	388	6	i	i	PRON
ejpam-5618	388	7	-	-	PUNCT
ejpam-5618	388	8	connected	connect	VERB
ejpam-5618	388	9	in	in	ADP
ejpam-5618	388	10	s	s	PROPN
ejpam-5618	388	11	,	,	PUNCT
ejpam-5618	388	12	is	be	AUX
ejpam-5618	388	13	called	call	VERB
ejpam-5618	388	14	the	the	DET
ejpam-5618	388	15	i	i	PROPN
ejpam-5618	388	16	-	-	NOUN
ejpam-5618	388	17	component	component	NOUN
ejpam-5618	388	18	of	of	ADP
ejpam-5618	388	19	s	s	PRON
ejpam-5618	388	20	belonging	belong	VERB
ejpam-5618	388	21	to	to	ADP
ejpam-5618	388	22	x.	x.	PROPN
ejpam-5618	388	23	s.	s.	PROPN
ejpam-5618	388	24	alammar	alammar	PROPN
ejpam-5618	388	25	,	,	PUNCT
ejpam-5618	388	26	m.	m.	NOUN
ejpam-5618	388	27	al	al	PROPN
ejpam-5618	388	28	shumrani	shumrani	PROPN
ejpam-5618	388	29	,	,	PUNCT
ejpam-5618	388	30	c.	c.	PROPN
ejpam-5618	388	31	özel	özel	PROPN
ejpam-5618	388	32	/	/	SYM
ejpam-5618	388	33	eur	eur	PROPN
ejpam-5618	388	34	.	.	PUNCT
ejpam-5618	389	1	j.	j.	PROPN
ejpam-5618	389	2	pure	pure	PROPN
ejpam-5618	389	3	appl	appl	PROPN
ejpam-5618	389	4	.	.	PROPN
ejpam-5618	389	5	math	math	PROPN
ejpam-5618	389	6	,	,	PUNCT
ejpam-5618	389	7	18	18	NUM
ejpam-5618	389	8	(	(	PUNCT
ejpam-5618	389	9	1	1	NUM
ejpam-5618	389	10	)	)	PUNCT
ejpam-5618	389	11	(	(	PUNCT
ejpam-5618	389	12	2025	2025	NUM
ejpam-5618	389	13	)	)	PUNCT
ejpam-5618	389	14	,	,	PUNCT
ejpam-5618	389	15	5618	5618	NUM
ejpam-5618	389	16	11	11	NUM
ejpam-5618	389	17	of	of	ADP
ejpam-5618	389	18	13	13	NUM
ejpam-5618	389	19	definition	definition	NOUN
ejpam-5618	389	20	18	18	NUM
ejpam-5618	389	21	.	.	PUNCT
ejpam-5618	390	1	let	let	VERB
ejpam-5618	390	2	g	g	PRON
ejpam-5618	390	3	be	be	AUX
ejpam-5618	390	4	an	an	DET
ejpam-5618	390	5	ideal	ideal	ADJ
ejpam-5618	390	6	topological	topological	ADJ
ejpam-5618	390	7	group	group	NOUN
ejpam-5618	390	8	with	with	ADP
ejpam-5618	390	9	identity	identity	NOUN
ejpam-5618	390	10	element	element	NOUN
ejpam-5618	390	11	e	e	PROPN
ejpam-5618	390	12	of	of	ADP
ejpam-5618	390	13	g.	g.	PROPN
ejpam-5618	390	14	the	the	DET
ejpam-5618	390	15	i	i	PROPN
ejpam-5618	390	16	-	-	NOUN
ejpam-5618	390	17	component	component	NOUN
ejpam-5618	390	18	of	of	ADP
ejpam-5618	390	19	g	g	PROPN
ejpam-5618	390	20	is	be	AUX
ejpam-5618	390	21	the	the	DET
ejpam-5618	390	22	union	union	NOUN
ejpam-5618	390	23	of	of	ADP
ejpam-5618	390	24	all	all	DET
ejpam-5618	390	25	i	i	PRON
ejpam-5618	390	26	-	-	PUNCT
ejpam-5618	390	27	connected	connect	VERB
ejpam-5618	390	28	subsets	subset	NOUN
ejpam-5618	390	29	of	of	ADP
ejpam-5618	390	30	g	g	NOUN
ejpam-5618	390	31	containing	contain	VERB
ejpam-5618	390	32	e.	e.	PROPN
ejpam-5618	390	33	theorem	theorem	PROPN
ejpam-5618	390	34	23	23	NUM
ejpam-5618	390	35	.	.	PUNCT
ejpam-5618	391	1	let	let	VERB
ejpam-5618	391	2	x	x	PRON
ejpam-5618	391	3	and	and	CCONJ
ejpam-5618	391	4	y	y	PROPN
ejpam-5618	391	5	be	be	AUX
ejpam-5618	391	6	ideal	ideal	ADJ
ejpam-5618	391	7	topological	topological	ADJ
ejpam-5618	391	8	spaces	space	NOUN
ejpam-5618	391	9	.	.	PUNCT
ejpam-5618	392	1	let	let	VERB
ejpam-5618	392	2	f	f	NOUN
ejpam-5618	392	3	:	:	PUNCT
ejpam-5618	392	4	x	x	X
ejpam-5618	392	5	→	→	SYM
ejpam-5618	392	6	y	y	X
ejpam-5618	392	7	be	be	AUX
ejpam-5618	392	8	an	an	DET
ejpam-5618	392	9	i	i	NOUN
ejpam-5618	392	10	-	-	PUNCT
ejpam-5618	392	11	continuous	continuous	ADJ
ejpam-5618	392	12	surjective	surjective	ADJ
ejpam-5618	392	13	mapping	mapping	NOUN
ejpam-5618	392	14	.	.	PUNCT
ejpam-5618	393	1	assume	assume	VERB
ejpam-5618	393	2	that	that	SCONJ
ejpam-5618	393	3	y	y	PROPN
ejpam-5618	393	4	is	be	AUX
ejpam-5618	393	5	submaximal	submaximal	ADJ
ejpam-5618	393	6	space	space	NOUN
ejpam-5618	393	7	.	.	PUNCT
ejpam-5618	394	1	if	if	SCONJ
ejpam-5618	394	2	x	x	PRON
ejpam-5618	394	3	is	be	AUX
ejpam-5618	394	4	i	i	PRON
ejpam-5618	394	5	-	-	PUNCT
ejpam-5618	394	6	connected	connect	VERB
ejpam-5618	394	7	,	,	PUNCT
ejpam-5618	394	8	then	then	ADV
ejpam-5618	394	9	y	y	PROPN
ejpam-5618	394	10	is	be	AUX
ejpam-5618	394	11	i	i	PROPN
ejpam-5618	394	12	-	-	PUNCT
ejpam-5618	394	13	connected	connect	VERB
ejpam-5618	394	14	.	.	PUNCT
ejpam-5618	395	1	proof	proof	NOUN
ejpam-5618	395	2	.	.	PUNCT
ejpam-5618	396	1	assume	assume	VERB
ejpam-5618	396	2	y	y	PROPN
ejpam-5618	396	3	is	be	AUX
ejpam-5618	396	4	i	i	PRON
ejpam-5618	396	5	-	-	PUNCT
ejpam-5618	396	6	disconnected	disconnected	ADJ
ejpam-5618	396	7	.	.	PUNCT
ejpam-5618	397	1	then	then	ADV
ejpam-5618	397	2	,	,	PUNCT
ejpam-5618	397	3	there	there	PRON
ejpam-5618	397	4	are	be	VERB
ejpam-5618	397	5	two	two	NUM
ejpam-5618	397	6	i	i	NOUN
ejpam-5618	397	7	-	-	PUNCT
ejpam-5618	397	8	open	open	ADJ
ejpam-5618	397	9	sets	set	VERB
ejpam-5618	397	10	u	u	NOUN
ejpam-5618	397	11	and	and	CCONJ
ejpam-5618	397	12	v	v	NOUN
ejpam-5618	397	13	of	of	ADP
ejpam-5618	397	14	y	y	PRON
ejpam-5618	397	15	such	such	ADJ
ejpam-5618	397	16	that	that	SCONJ
ejpam-5618	397	17	u	u	PROPN
ejpam-5618	397	18	∩	∩	NOUN
ejpam-5618	397	19	v	v	NOUN
ejpam-5618	397	20	=	=	SYM
ejpam-5618	397	21	∅	∅	NOUN
ejpam-5618	397	22	,	,	PUNCT
ejpam-5618	397	23	and	and	CCONJ
ejpam-5618	397	24	y	y	PROPN
ejpam-5618	397	25	=	=	SYM
ejpam-5618	397	26	u	u	PROPN
ejpam-5618	397	27	∪	∪	VERB
ejpam-5618	397	28	v	v	NOUN
ejpam-5618	397	29	.	.	PUNCT
ejpam-5618	398	1	we	we	PRON
ejpam-5618	398	2	can	can	AUX
ejpam-5618	398	3	deduce	deduce	VERB
ejpam-5618	398	4	that	that	DET
ejpam-5618	398	5	,	,	PUNCT
ejpam-5618	398	6	f−1(u	f−1(u	PROPN
ejpam-5618	398	7	)	)	PUNCT
ejpam-5618	398	8	∩	∩	ADJ
ejpam-5618	398	9	f−1(v	f−1(v	NOUN
ejpam-5618	398	10	)	)	PUNCT
ejpam-5618	399	1	=	=	NOUN
ejpam-5618	399	2	∅	∅	NOUN
ejpam-5618	399	3	and	and	CCONJ
ejpam-5618	399	4	x	x	X
ejpam-5618	399	5	=	=	SYM
ejpam-5618	399	6	f−1(u	f−1(u	PROPN
ejpam-5618	399	7	)	)	PUNCT
ejpam-5618	399	8	∪	∪	NOUN
ejpam-5618	399	9	f−1(v	f−1(v	PROPN
ejpam-5618	399	10	)	)	PUNCT
ejpam-5618	399	11	.	.	PUNCT
ejpam-5618	400	1	but	but	CCONJ
ejpam-5618	400	2	f−1(u	f−1(u	PROPN
ejpam-5618	400	3	)	)	PUNCT
ejpam-5618	400	4	and	and	CCONJ
ejpam-5618	400	5	f−1(v	f−1(v	PROPN
ejpam-5618	400	6	)	)	PUNCT
ejpam-5618	400	7	are	be	AUX
ejpam-5618	400	8	i	i	PRON
ejpam-5618	400	9	-	-	PUNCT
ejpam-5618	400	10	open	open	ADJ
ejpam-5618	400	11	sets	set	NOUN
ejpam-5618	400	12	by	by	ADP
ejpam-5618	400	13	theorem	theorem	NOUN
ejpam-5618	400	14	4	4	NUM
ejpam-5618	400	15	which	which	PRON
ejpam-5618	400	16	is	be	AUX
ejpam-5618	400	17	a	a	DET
ejpam-5618	400	18	contradiction	contradiction	NOUN
ejpam-5618	400	19	since	since	SCONJ
ejpam-5618	400	20	x	x	PRON
ejpam-5618	400	21	is	be	AUX
ejpam-5618	400	22	i	i	PRON
ejpam-5618	400	23	-	-	PUNCT
ejpam-5618	400	24	connected	connect	VERB
ejpam-5618	400	25	.	.	PUNCT
ejpam-5618	401	1	hence	hence	ADV
ejpam-5618	401	2	,	,	PUNCT
ejpam-5618	401	3	y	y	PROPN
ejpam-5618	401	4	is	be	AUX
ejpam-5618	401	5	i	i	PRON
ejpam-5618	401	6	-	-	PUNCT
ejpam-5618	401	7	connected	connect	VERB
ejpam-5618	401	8	.	.	PUNCT
ejpam-5618	402	1	theorem	theorem	VERB
ejpam-5618	402	2	24	24	NUM
ejpam-5618	402	3	.	.	PUNCT
ejpam-5618	403	1	let	let	VERB
ejpam-5618	403	2	g	g	PRON
ejpam-5618	403	3	be	be	AUX
ejpam-5618	403	4	a	a	DET
ejpam-5618	403	5	submaximal	submaximal	ADJ
ejpam-5618	403	6	spcae	spcae	ADJ
ejpam-5618	403	7	and	and	CCONJ
ejpam-5618	403	8	ideal	ideal	ADJ
ejpam-5618	403	9	topological	topological	ADJ
ejpam-5618	403	10	group	group	NOUN
ejpam-5618	403	11	.	.	PUNCT
ejpam-5618	404	1	the	the	DET
ejpam-5618	404	2	i	i	PROPN
ejpam-5618	404	3	-	-	PUNCT
ejpam-5618	404	4	component	component	NOUN
ejpam-5618	404	5	h	h	NOUN
ejpam-5618	404	6	is	be	AUX
ejpam-5618	404	7	an	an	DET
ejpam-5618	404	8	invariant	invariant	ADJ
ejpam-5618	404	9	subgroup	subgroup	NOUN
ejpam-5618	404	10	of	of	ADP
ejpam-5618	404	11	g.	g.	PROPN
ejpam-5618	404	12	proof	proof	PROPN
ejpam-5618	404	13	.	.	PUNCT
ejpam-5618	405	1	recall	recall	VERB
ejpam-5618	405	2	that	that	SCONJ
ejpam-5618	405	3	a	a	DET
ejpam-5618	405	4	subset	subset	ADJ
ejpam-5618	405	5	h	h	NOUN
ejpam-5618	405	6	of	of	ADP
ejpam-5618	405	7	a	a	DET
ejpam-5618	405	8	group	group	NOUN
ejpam-5618	405	9	g	g	NOUN
ejpam-5618	405	10	is	be	AUX
ejpam-5618	405	11	called	call	VERB
ejpam-5618	405	12	invariant	invariant	ADJ
ejpam-5618	405	13	subgroup	subgroup	NOUN
ejpam-5618	405	14	of	of	ADP
ejpam-5618	405	15	g	g	PROPN
ejpam-5618	405	16	,	,	PUNCT
ejpam-5618	405	17	if	if	SCONJ
ejpam-5618	405	18	aha−1	aha−1	PROPN
ejpam-5618	405	19	=	=	SYM
ejpam-5618	405	20	h	h	PROPN
ejpam-5618	405	21	for	for	ADP
ejpam-5618	405	22	all	all	DET
ejpam-5618	405	23	a	a	DET
ejpam-5618	405	24	∈	∈	PROPN
ejpam-5618	405	25	g.	g.	NOUN
ejpam-5618	405	26	since	since	SCONJ
ejpam-5618	405	27	the	the	DET
ejpam-5618	405	28	left	left	ADJ
ejpam-5618	405	29	translation	translation	NOUN
ejpam-5618	405	30	la	la	NOUN
ejpam-5618	405	31	:	:	PUNCT
ejpam-5618	405	32	g	g	ADP
ejpam-5618	405	33	−→	−→	NOUN
ejpam-5618	405	34	g	g	PROPN
ejpam-5618	405	35	is	be	AUX
ejpam-5618	405	36	an	an	DET
ejpam-5618	405	37	i	i	NOUN
ejpam-5618	405	38	-	-	PUNCT
ejpam-5618	405	39	homeomorphism	homeomorphism	PROPN
ejpam-5618	405	40	.	.	PUNCT
ejpam-5618	406	1	so	so	ADV
ejpam-5618	406	2	,	,	PUNCT
ejpam-5618	406	3	we	we	PRON
ejpam-5618	406	4	have	have	AUX
ejpam-5618	406	5	ah	ah	INTJ
ejpam-5618	406	6	is	be	AUX
ejpam-5618	406	7	i	i	PRON
ejpam-5618	406	8	-	-	PUNCT
ejpam-5618	406	9	connected	connect	VERB
ejpam-5618	406	10	,	,	PUNCT
ejpam-5618	406	11	and	and	CCONJ
ejpam-5618	406	12	hence	hence	ADV
ejpam-5618	406	13	,	,	PUNCT
ejpam-5618	406	14	aha−1	aha−1	PROPN
ejpam-5618	406	15	is	be	AUX
ejpam-5618	406	16	i	i	PRON
ejpam-5618	406	17	-	-	PUNCT
ejpam-5618	406	18	connected	connect	VERB
ejpam-5618	406	19	,	,	PUNCT
ejpam-5618	406	20	from	from	ADP
ejpam-5618	406	21	theorem	theorem	NOUN
ejpam-5618	406	22	23	23	NUM
ejpam-5618	406	23	.	.	PUNCT
ejpam-5618	407	1	since	since	SCONJ
ejpam-5618	407	2	e	e	PROPN
ejpam-5618	407	3	∈	∈	PROPN
ejpam-5618	407	4	aha−1	aha−1	NOUN
ejpam-5618	407	5	and	and	CCONJ
ejpam-5618	407	6	h	h	NOUN
ejpam-5618	407	7	is	be	AUX
ejpam-5618	407	8	the	the	DET
ejpam-5618	407	9	union	union	NOUN
ejpam-5618	407	10	of	of	ADP
ejpam-5618	407	11	all	all	DET
ejpam-5618	407	12	i	i	PRON
ejpam-5618	407	13	-	-	PUNCT
ejpam-5618	407	14	connected	connect	VERB
ejpam-5618	407	15	subset	subset	NOUN
ejpam-5618	407	16	of	of	ADP
ejpam-5618	407	17	g	g	NOUN
ejpam-5618	407	18	containing	contain	VERB
ejpam-5618	407	19	e	e	NOUN
ejpam-5618	407	20	,	,	PUNCT
ejpam-5618	407	21	it	it	PRON
ejpam-5618	407	22	follows	follow	VERB
ejpam-5618	407	23	that	that	SCONJ
ejpam-5618	407	24	,	,	PUNCT
ejpam-5618	407	25	aha−1	aha−1	PROPN
ejpam-5618	407	26	⊂	⊂	PROPN
ejpam-5618	407	27	h.	h.	PROPN
ejpam-5618	407	28	replacing	replace	VERB
ejpam-5618	407	29	a	a	PRON
ejpam-5618	407	30	by	by	ADP
ejpam-5618	407	31	a−1	a−1	PROPN
ejpam-5618	407	32	in	in	ADP
ejpam-5618	407	33	this	this	DET
ejpam-5618	407	34	inclusion	inclusion	NOUN
ejpam-5618	407	35	,	,	PUNCT
ejpam-5618	407	36	we	we	PRON
ejpam-5618	407	37	obtain	obtain	VERB
ejpam-5618	407	38	that	that	SCONJ
ejpam-5618	407	39	a−1	a−1	PROPN
ejpam-5618	407	40	ha	ha	PROPN
ejpam-5618	407	41	⊂	⊂	PROPN
ejpam-5618	407	42	h	h	NOUN
ejpam-5618	407	43	or	or	CCONJ
ejpam-5618	407	44	,	,	PUNCT
ejpam-5618	407	45	equivalently	equivalently	ADV
ejpam-5618	407	46	,	,	PUNCT
ejpam-5618	407	47	h	h	PROPN
ejpam-5618	407	48	⊂	⊂	PROPN
ejpam-5618	407	49	aha−1.thus	aha−1.thus	X
ejpam-5618	407	50	,	,	PUNCT
ejpam-5618	407	51	aha−1	aha−1	PROPN
ejpam-5618	407	52	=	=	SYM
ejpam-5618	407	53	h.	h.	PROPN
ejpam-5618	407	54	hence	hence	ADV
ejpam-5618	407	55	,	,	PUNCT
ejpam-5618	407	56	h	h	NOUN
ejpam-5618	407	57	is	be	AUX
ejpam-5618	407	58	an	an	DET
ejpam-5618	407	59	invariant	invariant	ADJ
ejpam-5618	407	60	subgroup	subgroup	NOUN
ejpam-5618	407	61	of	of	ADP
ejpam-5618	407	62	g.	g.	PROPN
ejpam-5618	407	63	theorem	theorem	VERB
ejpam-5618	407	64	25	25	NUM
ejpam-5618	407	65	.	.	PUNCT
ejpam-5618	408	1	let	let	VERB
ejpam-5618	408	2	g	g	PRON
ejpam-5618	408	3	be	be	AUX
ejpam-5618	408	4	an	an	DET
ejpam-5618	408	5	ideal	ideal	ADJ
ejpam-5618	408	6	topological	topological	ADJ
ejpam-5618	408	7	group	group	NOUN
ejpam-5618	408	8	.	.	PUNCT
ejpam-5618	409	1	suppose	suppose	VERB
ejpam-5618	409	2	u	u	NOUN
ejpam-5618	409	3	is	be	AUX
ejpam-5618	409	4	an	an	DET
ejpam-5618	409	5	open	open	ADJ
ejpam-5618	409	6	set	set	NOUN
ejpam-5618	409	7	in	in	ADP
ejpam-5618	409	8	g.	g.	PROPN
ejpam-5618	409	9	then	then	ADV
ejpam-5618	409	10	the	the	DET
ejpam-5618	409	11	set	set	ADJ
ejpam-5618	409	12	l	l	NOUN
ejpam-5618	409	13	=	=	PUNCT
ejpam-5618	409	14	⋃∞	⋃∞	NUM
ejpam-5618	409	15	n=1	n=1	PART
ejpam-5618	409	16	u	u	PROPN
ejpam-5618	409	17	n	n	PRON
ejpam-5618	409	18	is	be	AUX
ejpam-5618	409	19	an	an	DET
ejpam-5618	409	20	i	i	NOUN
ejpam-5618	409	21	-	-	PUNCT
ejpam-5618	409	22	open	open	ADJ
ejpam-5618	409	23	set	set	NOUN
ejpam-5618	409	24	.	.	PUNCT
ejpam-5618	410	1	proof	proof	NOUN
ejpam-5618	410	2	.	.	PUNCT
ejpam-5618	411	1	since	since	SCONJ
ejpam-5618	411	2	u	u	NOUN
ejpam-5618	411	3	is	be	AUX
ejpam-5618	411	4	open	open	ADJ
ejpam-5618	411	5	set	set	VERB
ejpam-5618	411	6	in	in	ADP
ejpam-5618	411	7	an	an	DET
ejpam-5618	411	8	ideal	ideal	ADJ
ejpam-5618	411	9	topological	topological	ADJ
ejpam-5618	411	10	group	group	NOUN
ejpam-5618	411	11	,	,	PUNCT
ejpam-5618	411	12	then	then	ADV
ejpam-5618	411	13	,	,	PUNCT
ejpam-5618	411	14	by	by	ADP
ejpam-5618	411	15	proposition	proposition	NOUN
ejpam-5618	411	16	1	1	NUM
ejpam-5618	411	17	,	,	PUNCT
ejpam-5618	411	18	uu	uu	ADJ
ejpam-5618	411	19	=	=	PROPN
ejpam-5618	411	20	u2	u2	PROPN
ejpam-5618	411	21	is	be	AUX
ejpam-5618	411	22	i	i	NOUN
ejpam-5618	411	23	-	-	PUNCT
ejpam-5618	411	24	open	open	ADJ
ejpam-5618	411	25	set	set	NOUN
ejpam-5618	411	26	,	,	PUNCT
ejpam-5618	411	27	u2u	u2u	ADJ
ejpam-5618	411	28	=	=	SYM
ejpam-5618	411	29	u3	u3	NOUN
ejpam-5618	411	30	is	be	AUX
ejpam-5618	411	31	i	i	PRON
ejpam-5618	411	32	-	-	PUNCT
ejpam-5618	411	33	open	open	ADJ
ejpam-5618	411	34	set	set	NOUN
ejpam-5618	411	35	and	and	CCONJ
ejpam-5618	411	36	similarly	similarly	ADV
ejpam-5618	411	37	u4	u4	PROPN
ejpam-5618	411	38	,	,	PUNCT
ejpam-5618	411	39	u5	u5	PROPN
ejpam-5618	411	40	,	,	PUNCT
ejpam-5618	411	41	.	.	PUNCT
ejpam-5618	411	42	.	.	PUNCT
ejpam-5618	411	43	.	.	PUNCT
ejpam-5618	412	1	all	all	PRON
ejpam-5618	412	2	are	be	AUX
ejpam-5618	412	3	i	i	PRON
ejpam-5618	412	4	-	-	PUNCT
ejpam-5618	412	5	open	open	ADJ
ejpam-5618	412	6	sets	set	NOUN
ejpam-5618	412	7	in	in	ADP
ejpam-5618	412	8	g.	g.	PROPN
ejpam-5618	412	9	therefore	therefore	ADV
ejpam-5618	412	10	,	,	PUNCT
ejpam-5618	412	11	the	the	DET
ejpam-5618	412	12	set	set	NOUN
ejpam-5618	412	13	l	l	NOUN
ejpam-5618	412	14	=	=	PUNCT
ejpam-5618	412	15	⋃∞	⋃∞	NUM
ejpam-5618	412	16	n=1	n=1	PART
ejpam-5618	412	17	u	u	PROPN
ejpam-5618	412	18	n	n	ADV
ejpam-5618	412	19	is	be	AUX
ejpam-5618	412	20	i	i	PRON
ejpam-5618	412	21	-	-	PUNCT
ejpam-5618	412	22	open	open	ADJ
ejpam-5618	412	23	,	,	PUNCT
ejpam-5618	412	24	since	since	SCONJ
ejpam-5618	412	25	the	the	DET
ejpam-5618	412	26	union	union	NOUN
ejpam-5618	412	27	of	of	ADP
ejpam-5618	412	28	i	i	PROPN
ejpam-5618	412	29	-	-	PUNCT
ejpam-5618	412	30	open	open	ADJ
ejpam-5618	412	31	sets	set	NOUN
ejpam-5618	412	32	is	be	AUX
ejpam-5618	412	33	an	an	DET
ejpam-5618	412	34	i	i	NOUN
ejpam-5618	412	35	-	-	PUNCT
ejpam-5618	412	36	open	open	ADJ
ejpam-5618	412	37	set	set	NOUN
ejpam-5618	412	38	.	.	PUNCT
ejpam-5618	413	1	theorem	theorem	PROPN
ejpam-5618	413	2	26	26	NUM
ejpam-5618	413	3	.	.	PUNCT
ejpam-5618	414	1	let	let	VERB
ejpam-5618	414	2	g	g	PRON
ejpam-5618	414	3	be	be	AUX
ejpam-5618	414	4	an	an	DET
ejpam-5618	414	5	ideal	ideal	ADJ
ejpam-5618	414	6	topological	topological	ADJ
ejpam-5618	414	7	group	group	NOUN
ejpam-5618	414	8	.	.	PUNCT
ejpam-5618	415	1	suppose	suppose	VERB
ejpam-5618	415	2	u	u	NOUN
ejpam-5618	415	3	is	be	AUX
ejpam-5618	415	4	any	any	DET
ejpam-5618	415	5	symmetric	symmetric	ADJ
ejpam-5618	415	6	open	open	ADJ
ejpam-5618	415	7	neighborhood	neighborhood	NOUN
ejpam-5618	415	8	of	of	ADP
ejpam-5618	415	9	identity	identity	NOUN
ejpam-5618	415	10	element	element	NOUN
ejpam-5618	415	11	e.	e.	PROPN
ejpam-5618	416	1	then	then	ADV
ejpam-5618	416	2	the	the	DET
ejpam-5618	416	3	set	set	ADJ
ejpam-5618	416	4	l	l	NOUN
ejpam-5618	416	5	=	=	PUNCT
ejpam-5618	416	6	⋃∞	⋃∞	NUM
ejpam-5618	416	7	n=1	n=1	PART
ejpam-5618	416	8	u	u	PROPN
ejpam-5618	416	9	n	n	PRON
ejpam-5618	416	10	is	be	AUX
ejpam-5618	416	11	an	an	DET
ejpam-5618	416	12	i	i	NOUN
ejpam-5618	416	13	-	-	PUNCT
ejpam-5618	416	14	open	open	ADJ
ejpam-5618	416	15	subgroup	subgroup	NOUN
ejpam-5618	416	16	of	of	ADP
ejpam-5618	416	17	g.	g.	PROPN
ejpam-5618	416	18	proof	proof	PROPN
ejpam-5618	416	19	.	.	PUNCT
ejpam-5618	417	1	we	we	PRON
ejpam-5618	417	2	need	need	VERB
ejpam-5618	417	3	to	to	PART
ejpam-5618	417	4	prove	prove	VERB
ejpam-5618	417	5	that	that	SCONJ
ejpam-5618	417	6	l	l	NOUN
ejpam-5618	417	7	is	be	AUX
ejpam-5618	417	8	a	a	DET
ejpam-5618	417	9	subgroup	subgroup	NOUN
ejpam-5618	417	10	of	of	ADP
ejpam-5618	417	11	g.	g.	PROPN
ejpam-5618	417	12	take	take	VERB
ejpam-5618	417	13	x	x	PRON
ejpam-5618	417	14	,	,	PUNCT
ejpam-5618	417	15	y	y	PROPN
ejpam-5618	417	16	∈	∈	PROPN
ejpam-5618	417	17	l	l	NOUN
ejpam-5618	417	18	,	,	PUNCT
ejpam-5618	417	19	and	and	CCONJ
ejpam-5618	417	20	if	if	SCONJ
ejpam-5618	417	21	x	x	SYM
ejpam-5618	417	22	=	=	SYM
ejpam-5618	417	23	ul	ul	INTJ
ejpam-5618	417	24	,	,	PUNCT
ejpam-5618	417	25	y	y	PROPN
ejpam-5618	417	26	=	=	SYM
ejpam-5618	417	27	ut	ut	PROPN
ejpam-5618	417	28	,	,	PUNCT
ejpam-5618	417	29	hence	hence	ADV
ejpam-5618	417	30	we	we	PRON
ejpam-5618	417	31	get	get	VERB
ejpam-5618	417	32	x	x	X
ejpam-5618	417	33	·	·	PUNCT
ejpam-5618	417	34	y	y	X
ejpam-5618	417	35	=	=	SYM
ejpam-5618	417	36	ul	ul	PROPN
ejpam-5618	417	37	·	·	PUNCT
ejpam-5618	417	38	ut	ut	PROPN
ejpam-5618	417	39	=	=	PUNCT
ejpam-5618	417	40	ul+t	ul+t	PROPN
ejpam-5618	417	41	,	,	PUNCT
ejpam-5618	417	42	and	and	CCONJ
ejpam-5618	417	43	x−1	x−1	PROPN
ejpam-5618	417	44	=	=	PUNCT
ejpam-5618	417	45	(	(	PUNCT
ejpam-5618	417	46	ul	ul	INTJ
ejpam-5618	417	47	)	)	PUNCT
ejpam-5618	417	48	−1	−1	NOUN
ejpam-5618	417	49	=	=	SYM
ejpam-5618	417	50	(	(	PUNCT
ejpam-5618	417	51	u−1	u−1	PROPN
ejpam-5618	417	52	)	)	PUNCT
ejpam-5618	417	53	l	l	NOUN
ejpam-5618	418	1	=	=	PUNCT
ejpam-5618	418	2	ul	ul	INTJ
ejpam-5618	418	3	.	.	PUNCT
ejpam-5618	419	1	we	we	PRON
ejpam-5618	419	2	have	have	VERB
ejpam-5618	419	3	that	that	PRON
ejpam-5618	419	4	x	x	X
ejpam-5618	419	5	·	·	PUNCT
ejpam-5618	419	6	y	y	PROPN
ejpam-5618	419	7	and	and	CCONJ
ejpam-5618	419	8	x−1	x−1	PROPN
ejpam-5618	419	9	both	both	PRON
ejpam-5618	419	10	in	in	ADP
ejpam-5618	419	11	l.	l.	PROPN
ejpam-5618	419	12	hence	hence	ADV
ejpam-5618	419	13	l	l	PROPN
ejpam-5618	419	14	is	be	AUX
ejpam-5618	419	15	a	a	DET
ejpam-5618	419	16	subgroup	subgroup	NOUN
ejpam-5618	419	17	of	of	ADP
ejpam-5618	419	18	g.	g.	PROPN
ejpam-5618	419	19	therefore	therefore	ADV
ejpam-5618	419	20	,	,	PUNCT
ejpam-5618	419	21	we	we	PRON
ejpam-5618	419	22	get	get	VERB
ejpam-5618	419	23	l	l	NOUN
ejpam-5618	419	24	=	=	PRON
ejpam-5618	419	25	⋃∞	⋃∞	X
ejpam-5618	419	26	n=1	n=1	PART
ejpam-5618	419	27	u	u	PROPN
ejpam-5618	419	28	n	n	PRON
ejpam-5618	419	29	is	be	AUX
ejpam-5618	419	30	an	an	DET
ejpam-5618	419	31	i	i	NOUN
ejpam-5618	419	32	-	-	PUNCT
ejpam-5618	419	33	open	open	ADJ
ejpam-5618	419	34	subgroup	subgroup	NOUN
ejpam-5618	419	35	of	of	ADP
ejpam-5618	419	36	g.	g.	PROPN
ejpam-5618	419	37	corollary	corollary	PROPN
ejpam-5618	419	38	6	6	NUM
ejpam-5618	419	39	.	.	PUNCT
ejpam-5618	420	1	let	let	VERB
ejpam-5618	420	2	g	g	PRON
ejpam-5618	420	3	be	be	AUX
ejpam-5618	420	4	a	a	DET
ejpam-5618	420	5	submaximal	submaximal	ADJ
ejpam-5618	420	6	space	space	NOUN
ejpam-5618	420	7	and	and	CCONJ
ejpam-5618	420	8	ideal	ideal	ADJ
ejpam-5618	420	9	topological	topological	ADJ
ejpam-5618	420	10	group	group	NOUN
ejpam-5618	420	11	.	.	PUNCT
ejpam-5618	421	1	suppose	suppose	VERB
ejpam-5618	421	2	u	u	NOUN
ejpam-5618	421	3	is	be	AUX
ejpam-5618	421	4	any	any	DET
ejpam-5618	421	5	symmetric	symmetric	ADJ
ejpam-5618	421	6	i	i	NOUN
ejpam-5618	421	7	-	-	PUNCT
ejpam-5618	421	8	open	open	ADJ
ejpam-5618	421	9	neighborhood	neighborhood	NOUN
ejpam-5618	421	10	of	of	ADP
ejpam-5618	421	11	e.	e.	PROPN
ejpam-5618	421	12	then	then	ADV
ejpam-5618	421	13	the	the	DET
ejpam-5618	421	14	set	set	ADJ
ejpam-5618	421	15	l	l	NOUN
ejpam-5618	421	16	=	=	PUNCT
ejpam-5618	421	17	⋃∞	⋃∞	NUM
ejpam-5618	421	18	n=1	n=1	PART
ejpam-5618	421	19	u	u	PROPN
ejpam-5618	421	20	n	n	PRON
ejpam-5618	421	21	is	be	AUX
ejpam-5618	421	22	an	an	DET
ejpam-5618	421	23	i	i	NOUN
ejpam-5618	421	24	-	-	PUNCT
ejpam-5618	421	25	open	open	ADJ
ejpam-5618	421	26	and	and	CCONJ
ejpam-5618	421	27	i	i	PRON
ejpam-5618	421	28	-	-	PUNCT
ejpam-5618	421	29	closed	close	VERB
ejpam-5618	421	30	subgroup	subgroup	NOUN
ejpam-5618	421	31	of	of	ADP
ejpam-5618	421	32	g.	g.	PROPN
ejpam-5618	421	33	proof	proof	PROPN
ejpam-5618	421	34	.	.	PUNCT
ejpam-5618	422	1	the	the	DET
ejpam-5618	422	2	set	set	NOUN
ejpam-5618	422	3	u	u	NOUN
ejpam-5618	422	4	is	be	AUX
ejpam-5618	422	5	open	open	ADJ
ejpam-5618	422	6	since	since	SCONJ
ejpam-5618	422	7	g	g	PROPN
ejpam-5618	422	8	is	be	AUX
ejpam-5618	422	9	submaximal	submaximal	ADJ
ejpam-5618	422	10	.	.	PUNCT
ejpam-5618	423	1	by	by	ADP
ejpam-5618	423	2	theorem	theorem	NOUN
ejpam-5618	423	3	26	26	NUM
ejpam-5618	423	4	,	,	PUNCT
ejpam-5618	423	5	we	we	PRON
ejpam-5618	423	6	have	have	VERB
ejpam-5618	423	7	that	that	DET
ejpam-5618	423	8	l	l	NOUN
ejpam-5618	423	9	is	be	AUX
ejpam-5618	423	10	an	an	DET
ejpam-5618	423	11	i	i	NOUN
ejpam-5618	423	12	-	-	PUNCT
ejpam-5618	423	13	open	open	ADJ
ejpam-5618	423	14	subgroup	subgroup	NOUN
ejpam-5618	423	15	of	of	ADP
ejpam-5618	423	16	g.	g.	PROPN
ejpam-5618	423	17	using	use	VERB
ejpam-5618	423	18	corollary	corollary	ADJ
ejpam-5618	423	19	5	5	NUM
ejpam-5618	423	20	,	,	PUNCT
ejpam-5618	423	21	l	l	NOUN
ejpam-5618	423	22	is	be	AUX
ejpam-5618	423	23	i	i	NOUN
ejpam-5618	423	24	-	-	PUNCT
ejpam-5618	423	25	closed	closed	ADJ
ejpam-5618	423	26	.	.	PUNCT
ejpam-5618	424	1	s.	s.	PROPN
ejpam-5618	424	2	alammar	alammar	PROPN
ejpam-5618	424	3	,	,	PUNCT
ejpam-5618	424	4	m.	m.	NOUN
ejpam-5618	424	5	al	al	PROPN
ejpam-5618	424	6	shumrani	shumrani	PROPN
ejpam-5618	424	7	,	,	PUNCT
ejpam-5618	424	8	c.	c.	PROPN
ejpam-5618	424	9	özel	özel	PROPN
ejpam-5618	424	10	/	/	SYM
ejpam-5618	424	11	eur	eur	PROPN
ejpam-5618	424	12	.	.	PUNCT
ejpam-5618	425	1	j.	j.	PROPN
ejpam-5618	425	2	pure	pure	PROPN
ejpam-5618	425	3	appl	appl	PROPN
ejpam-5618	425	4	.	.	PROPN
ejpam-5618	425	5	math	math	PROPN
ejpam-5618	425	6	,	,	PUNCT
ejpam-5618	425	7	18	18	NUM
ejpam-5618	425	8	(	(	PUNCT
ejpam-5618	425	9	1	1	NUM
ejpam-5618	425	10	)	)	PUNCT
ejpam-5618	425	11	(	(	PUNCT
ejpam-5618	425	12	2025	2025	NUM
ejpam-5618	425	13	)	)	PUNCT
ejpam-5618	425	14	,	,	PUNCT
ejpam-5618	425	15	5618	5618	NUM
ejpam-5618	425	16	12	12	NUM
ejpam-5618	425	17	of	of	ADP
ejpam-5618	425	18	13	13	NUM
ejpam-5618	425	19	lemma	lemma	PROPN
ejpam-5618	425	20	2	2	NUM
ejpam-5618	425	21	.	.	PUNCT
ejpam-5618	426	1	let	let	VERB
ejpam-5618	426	2	g	g	PRON
ejpam-5618	426	3	be	be	AUX
ejpam-5618	426	4	a	a	DET
ejpam-5618	426	5	submaximal	submaximal	ADJ
ejpam-5618	426	6	space	space	NOUN
ejpam-5618	426	7	and	and	CCONJ
ejpam-5618	426	8	ideal	ideal	ADJ
ejpam-5618	426	9	topological	topological	ADJ
ejpam-5618	426	10	group	group	NOUN
ejpam-5618	426	11	.	.	PUNCT
ejpam-5618	427	1	suppose	suppose	VERB
ejpam-5618	427	2	u	u	NOUN
ejpam-5618	427	3	is	be	AUX
ejpam-5618	427	4	a	a	DET
ejpam-5618	427	5	neighborhood	neighborhood	NOUN
ejpam-5618	427	6	of	of	ADP
ejpam-5618	427	7	the	the	DET
ejpam-5618	427	8	identity	identity	NOUN
ejpam-5618	427	9	element	element	NOUN
ejpam-5618	427	10	e	e	NOUN
ejpam-5618	427	11	,	,	PUNCT
ejpam-5618	427	12	and	and	CCONJ
ejpam-5618	427	13	se	se	PROPN
ejpam-5618	427	14	is	be	AUX
ejpam-5618	427	15	an	an	DET
ejpam-5618	427	16	i	i	NOUN
ejpam-5618	427	17	-	-	NOUN
ejpam-5618	427	18	component	component	NOUN
ejpam-5618	427	19	of	of	ADP
ejpam-5618	427	20	s	s	PRON
ejpam-5618	427	21	belonging	belong	VERB
ejpam-5618	427	22	to	to	ADP
ejpam-5618	427	23	identity	identity	NOUN
ejpam-5618	427	24	element	element	NOUN
ejpam-5618	427	25	e.	e.	PROPN
ejpam-5618	427	26	then	then	ADV
ejpam-5618	427	27	se	se	PROPN
ejpam-5618	427	28	⊂	⊂	PROPN
ejpam-5618	427	29	⋃∞	⋃∞	X
ejpam-5618	427	30	n=1	n=1	PROPN
ejpam-5618	427	31	u	u	PROPN
ejpam-5618	427	32	n.	n.	VERB
ejpam-5618	427	33	in	in	ADP
ejpam-5618	427	34	particular	particular	ADJ
ejpam-5618	427	35	,	,	PUNCT
ejpam-5618	427	36	if	if	SCONJ
ejpam-5618	427	37	g	g	PROPN
ejpam-5618	427	38	is	be	AUX
ejpam-5618	427	39	i	i	PRON
ejpam-5618	427	40	-	-	PUNCT
ejpam-5618	427	41	connected	connect	VERB
ejpam-5618	427	42	,	,	PUNCT
ejpam-5618	427	43	then	then	ADV
ejpam-5618	427	44	we	we	PRON
ejpam-5618	427	45	have	have	VERB
ejpam-5618	427	46	g	g	NOUN
ejpam-5618	427	47	=	=	NOUN
ejpam-5618	427	48	⋃∞	⋃∞	X
ejpam-5618	427	49	n=1	n=1	PROPN
ejpam-5618	427	50	u	u	PROPN
ejpam-5618	427	51	n.	n.	NOUN
ejpam-5618	427	52	proof	proof	NOUN
ejpam-5618	427	53	.	.	PUNCT
ejpam-5618	428	1	let	let	VERB
ejpam-5618	428	2	u	u	PRON
ejpam-5618	428	3	be	be	AUX
ejpam-5618	428	4	a	a	DET
ejpam-5618	428	5	neighborhood	neighborhood	NOUN
ejpam-5618	428	6	of	of	ADP
ejpam-5618	428	7	the	the	DET
ejpam-5618	428	8	identity	identity	NOUN
ejpam-5618	428	9	element	element	NOUN
ejpam-5618	428	10	e	e	PROPN
ejpam-5618	428	11	of	of	ADP
ejpam-5618	428	12	g.	g.	PROPN
ejpam-5618	428	13	from	from	ADP
ejpam-5618	428	14	corollary	corollary	ADJ
ejpam-5618	428	15	3	3	NUM
ejpam-5618	428	16	,	,	PUNCT
ejpam-5618	428	17	there	there	PRON
ejpam-5618	428	18	is	be	VERB
ejpam-5618	428	19	a	a	DET
ejpam-5618	428	20	symmetric	symmetric	ADJ
ejpam-5618	428	21	i	i	NOUN
ejpam-5618	428	22	-	-	PUNCT
ejpam-5618	428	23	open	open	ADJ
ejpam-5618	428	24	neighborhood	neighborhood	NOUN
ejpam-5618	428	25	v	v	NOUN
ejpam-5618	428	26	of	of	ADP
ejpam-5618	428	27	the	the	DET
ejpam-5618	428	28	identity	identity	NOUN
ejpam-5618	428	29	element	element	NOUN
ejpam-5618	428	30	e	e	NOUN
ejpam-5618	428	31	such	such	ADJ
ejpam-5618	428	32	that	that	PRON
ejpam-5618	428	33	v	v	ADP
ejpam-5618	428	34	⊂	⊂	PROPN
ejpam-5618	428	35	u	u	PROPN
ejpam-5618	428	36	.	.	PUNCT
ejpam-5618	429	1	clearly	clearly	ADV
ejpam-5618	429	2	,	,	PUNCT
ejpam-5618	429	3	we	we	PRON
ejpam-5618	429	4	have	have	VERB
ejpam-5618	429	5	h	h	NOUN
ejpam-5618	429	6	=	=	PRON
ejpam-5618	429	7	⋃∞	⋃∞	X
ejpam-5618	429	8	n=1	n=1	PROPN
ejpam-5618	429	9	v	v	PROPN
ejpam-5618	429	10	n	n	PROPN
ejpam-5618	429	11	is	be	AUX
ejpam-5618	429	12	an	an	DET
ejpam-5618	429	13	i	i	NOUN
ejpam-5618	429	14	-	-	PUNCT
ejpam-5618	429	15	open	open	ADJ
ejpam-5618	429	16	and	and	CCONJ
ejpam-5618	429	17	i	i	PRON
ejpam-5618	429	18	-	-	PUNCT
ejpam-5618	429	19	closed	closed	ADJ
ejpam-5618	429	20	subgroup	subgroup	NOUN
ejpam-5618	429	21	of	of	ADP
ejpam-5618	429	22	g	g	PROPN
ejpam-5618	429	23	,	,	PUNCT
ejpam-5618	429	24	from	from	ADP
ejpam-5618	429	25	corollary	corollary	ADJ
ejpam-5618	429	26	6	6	NUM
ejpam-5618	429	27	.	.	PUNCT
ejpam-5618	430	1	since	since	SCONJ
ejpam-5618	430	2	se	se	PROPN
ejpam-5618	430	3	is	be	AUX
ejpam-5618	430	4	i	i	PROPN
ejpam-5618	430	5	-	-	PUNCT
ejpam-5618	430	6	connected	connect	VERB
ejpam-5618	430	7	of	of	ADP
ejpam-5618	430	8	s	s	AUX
ejpam-5618	430	9	belonging	belong	VERB
ejpam-5618	430	10	to	to	ADP
ejpam-5618	430	11	identity	identity	NOUN
ejpam-5618	430	12	element	element	NOUN
ejpam-5618	430	13	e	e	NOUN
ejpam-5618	430	14	,	,	PUNCT
ejpam-5618	430	15	we	we	PRON
ejpam-5618	430	16	have	have	AUX
ejpam-5618	430	17	se	se	PROPN
ejpam-5618	430	18	⊂	⊂	PROPN
ejpam-5618	430	19	⋃∞	⋃∞	X
ejpam-5618	430	20	n=1	n=1	PROPN
ejpam-5618	430	21	v	v	PROPN
ejpam-5618	430	22	n	n	PROPN
ejpam-5618	430	23	⊂	⊂	PRON
ejpam-5618	430	24	⋃∞	⋃∞	PUNCT
ejpam-5618	430	25	n=1	n=1	PROPN
ejpam-5618	430	26	u	u	PROPN
ejpam-5618	430	27	n.	n.	VERB
ejpam-5618	430	28	to	to	PART
ejpam-5618	430	29	conclude	conclude	VERB
ejpam-5618	430	30	,	,	PUNCT
ejpam-5618	430	31	if	if	SCONJ
ejpam-5618	430	32	g	g	PROPN
ejpam-5618	430	33	is	be	AUX
ejpam-5618	430	34	i	i	PRON
ejpam-5618	430	35	-	-	PUNCT
ejpam-5618	430	36	connected	connect	VERB
ejpam-5618	430	37	,	,	PUNCT
ejpam-5618	430	38	we	we	PRON
ejpam-5618	430	39	have	have	VERB
ejpam-5618	430	40	that	that	PRON
ejpam-5618	430	41	g	g	PROPN
ejpam-5618	430	42	=	=	SYM
ejpam-5618	430	43	⋃∞	⋃∞	NUM
ejpam-5618	430	44	n=1	n=1	PROPN
ejpam-5618	430	45	u	u	PROPN
ejpam-5618	430	46	n.	n.	PROPN
ejpam-5618	430	47	theorem	theorem	VERB
ejpam-5618	430	48	27	27	NUM
ejpam-5618	430	49	.	.	PUNCT
ejpam-5618	431	1	let	let	VERB
ejpam-5618	431	2	g	g	PRON
ejpam-5618	431	3	be	be	AUX
ejpam-5618	431	4	an	an	DET
ejpam-5618	431	5	ideal	ideal	ADJ
ejpam-5618	431	6	topological	topological	ADJ
ejpam-5618	431	7	group	group	NOUN
ejpam-5618	431	8	,	,	PUNCT
ejpam-5618	431	9	and	and	CCONJ
ejpam-5618	431	10	h	h	NOUN
ejpam-5618	431	11	be	be	AUX
ejpam-5618	431	12	an	an	DET
ejpam-5618	431	13	open	open	ADJ
ejpam-5618	431	14	subgroup	subgroup	NOUN
ejpam-5618	431	15	of	of	ADP
ejpam-5618	431	16	g.	g.	PROPN
ejpam-5618	432	1	then	then	ADV
ejpam-5618	432	2	(	(	PUNCT
ejpam-5618	432	3	i	i	NOUN
ejpam-5618	432	4	)	)	PUNCT
ejpam-5618	432	5	h	h	PROPN
ejpam-5618	432	6	is	be	AUX
ejpam-5618	432	7	an	an	DET
ejpam-5618	432	8	ideal	ideal	ADJ
ejpam-5618	432	9	topological	topological	ADJ
ejpam-5618	432	10	group	group	NOUN
ejpam-5618	432	11	.	.	PUNCT
ejpam-5618	433	1	(	(	PUNCT
ejpam-5618	433	2	ii	ii	NOUN
ejpam-5618	433	3	)	)	PUNCT
ejpam-5618	433	4	h	h	NOUN
ejpam-5618	433	5	is	be	AUX
ejpam-5618	433	6	an	an	DET
ejpam-5618	433	7	i	i	NOUN
ejpam-5618	433	8	-	-	PUNCT
ejpam-5618	433	9	open	open	ADJ
ejpam-5618	433	10	and	and	CCONJ
ejpam-5618	433	11	i	i	PRON
ejpam-5618	433	12	-	-	PUNCT
ejpam-5618	433	13	closed	close	VERB
ejpam-5618	433	14	in	in	ADP
ejpam-5618	433	15	g.	g.	PROPN
ejpam-5618	433	16	(	(	PUNCT
ejpam-5618	433	17	iii	iii	X
ejpam-5618	433	18	)	)	PUNCT
ejpam-5618	433	19	g	g	NOUN
ejpam-5618	433	20	is	be	AUX
ejpam-5618	433	21	not	not	PART
ejpam-5618	433	22	i	i	PRON
ejpam-5618	433	23	-	-	PUNCT
ejpam-5618	433	24	connected	connect	VERB
ejpam-5618	433	25	.	.	PUNCT
ejpam-5618	434	1	proof	proof	NOUN
ejpam-5618	434	2	.	.	PUNCT
ejpam-5618	435	1	(	(	PUNCT
ejpam-5618	435	2	i	i	NOUN
ejpam-5618	435	3	)	)	PUNCT
ejpam-5618	435	4	and	and	CCONJ
ejpam-5618	435	5	(	(	PUNCT
ejpam-5618	435	6	ii	ii	NOUN
ejpam-5618	435	7	)	)	PUNCT
ejpam-5618	435	8	are	be	AUX
ejpam-5618	435	9	proved	prove	VERB
ejpam-5618	435	10	.	.	PUNCT
ejpam-5618	436	1	since	since	SCONJ
ejpam-5618	436	2	h	h	PROPN
ejpam-5618	436	3	and	and	CCONJ
ejpam-5618	436	4	g−h	g−h	PROPN
ejpam-5618	436	5	are	be	AUX
ejpam-5618	436	6	disjoint	disjoint	NOUN
ejpam-5618	436	7	i	i	NOUN
ejpam-5618	436	8	-	-	PUNCT
ejpam-5618	436	9	open	open	ADJ
ejpam-5618	436	10	sets	set	NOUN
ejpam-5618	436	11	,	,	PUNCT
ejpam-5618	436	12	(	(	PUNCT
ejpam-5618	436	13	iii	iii	NOUN
ejpam-5618	436	14	)	)	PUNCT
ejpam-5618	436	15	holds	hold	VERB
ejpam-5618	436	16	.	.	PUNCT
ejpam-5618	437	1	theorem	theorem	PROPN
ejpam-5618	437	2	28	28	NUM
ejpam-5618	437	3	.	.	PUNCT
ejpam-5618	438	1	let	let	VERB
ejpam-5618	438	2	g	g	PRON
ejpam-5618	438	3	be	be	AUX
ejpam-5618	438	4	an	an	DET
ejpam-5618	438	5	ideal	ideal	ADJ
ejpam-5618	438	6	topological	topological	ADJ
ejpam-5618	438	7	group	group	NOUN
ejpam-5618	438	8	.	.	PUNCT
ejpam-5618	439	1	suppose	suppose	VERB
ejpam-5618	439	2	that	that	SCONJ
ejpam-5618	439	3	g	g	PROPN
ejpam-5618	439	4	is	be	AUX
ejpam-5618	439	5	an	an	DET
ejpam-5618	439	6	i	i	NOUN
ejpam-5618	439	7	-	-	PUNCT
ejpam-5618	439	8	connected	connect	VERB
ejpam-5618	439	9	space	space	NOUN
ejpam-5618	439	10	.	.	PUNCT
ejpam-5618	440	1	then	then	ADV
ejpam-5618	440	2	there	there	PRON
ejpam-5618	440	3	is	be	VERB
ejpam-5618	440	4	no	no	DET
ejpam-5618	440	5	open	open	ADJ
ejpam-5618	440	6	proper	proper	ADJ
ejpam-5618	440	7	subgroup	subgroup	NOUN
ejpam-5618	440	8	of	of	ADP
ejpam-5618	440	9	the	the	DET
ejpam-5618	440	10	group	group	NOUN
ejpam-5618	440	11	g.	g.	PROPN
ejpam-5618	440	12	proof	proof	PROPN
ejpam-5618	440	13	.	.	PUNCT
ejpam-5618	441	1	suppose	suppose	VERB
ejpam-5618	441	2	that	that	SCONJ
ejpam-5618	441	3	g	g	PROPN
ejpam-5618	441	4	is	be	AUX
ejpam-5618	441	5	an	an	DET
ejpam-5618	441	6	i	i	NOUN
ejpam-5618	441	7	-	-	PUNCT
ejpam-5618	441	8	connected	connect	VERB
ejpam-5618	441	9	space	space	NOUN
ejpam-5618	441	10	and	and	CCONJ
ejpam-5618	441	11	there	there	PRON
ejpam-5618	441	12	is	be	VERB
ejpam-5618	441	13	an	an	DET
ejpam-5618	441	14	open	open	ADJ
ejpam-5618	441	15	proper	proper	ADJ
ejpam-5618	441	16	subgroup	subgroup	NOUN
ejpam-5618	441	17	h	h	NOUN
ejpam-5618	441	18	of	of	ADP
ejpam-5618	441	19	the	the	DET
ejpam-5618	441	20	group	group	NOUN
ejpam-5618	441	21	g.	g.	PROPN
ejpam-5618	441	22	we	we	PRON
ejpam-5618	441	23	prove	prove	VERB
ejpam-5618	441	24	that	that	SCONJ
ejpam-5618	441	25	h	h	NOUN
ejpam-5618	442	1	=	=	SYM
ejpam-5618	442	2	icl(h	icl(h	PROPN
ejpam-5618	442	3	)	)	PUNCT
ejpam-5618	442	4	.	.	PUNCT
ejpam-5618	443	1	it	it	PRON
ejpam-5618	443	2	is	be	AUX
ejpam-5618	443	3	known	know	VERB
ejpam-5618	443	4	that	that	SCONJ
ejpam-5618	443	5	h	h	PROPN
ejpam-5618	443	6	⊂	⊂	X
ejpam-5618	443	7	icl(h	icl(h	PROPN
ejpam-5618	443	8	)	)	PUNCT
ejpam-5618	443	9	for	for	ADP
ejpam-5618	443	10	arbitrary	arbitrary	ADJ
ejpam-5618	443	11	subset	subset	ADJ
ejpam-5618	443	12	h	h	NOUN
ejpam-5618	443	13	of	of	ADP
ejpam-5618	443	14	g.	g.	PROPN
ejpam-5618	443	15	take	take	VERB
ejpam-5618	443	16	a	a	DET
ejpam-5618	443	17	∈	∈	NOUN
ejpam-5618	443	18	icl(h	icl(h	PROPN
ejpam-5618	443	19	)	)	PUNCT
ejpam-5618	443	20	.	.	PUNCT
ejpam-5618	444	1	the	the	DET
ejpam-5618	444	2	set	set	NOUN
ejpam-5618	444	3	ah	ah	INTJ
ejpam-5618	444	4	is	be	AUX
ejpam-5618	444	5	an	an	DET
ejpam-5618	444	6	i	i	NOUN
ejpam-5618	444	7	-	-	PUNCT
ejpam-5618	444	8	open	open	ADJ
ejpam-5618	444	9	set	set	NOUN
ejpam-5618	444	10	,	,	PUNCT
ejpam-5618	444	11	from	from	ADP
ejpam-5618	444	12	i	i	PROPN
ejpam-5618	444	13	-	-	PUNCT
ejpam-5618	444	14	homeomorphism	homeomorphism	PROPN
ejpam-5618	444	15	of	of	ADP
ejpam-5618	444	16	the	the	DET
ejpam-5618	444	17	left	left	ADJ
ejpam-5618	444	18	translation	translation	NOUN
ejpam-5618	444	19	map	map	NOUN
ejpam-5618	444	20	.	.	PUNCT
ejpam-5618	445	1	then	then	ADV
ejpam-5618	445	2	,	,	PUNCT
ejpam-5618	445	3	we	we	PRON
ejpam-5618	445	4	have	have	AUX
ejpam-5618	445	5	ah	ah	INTJ
ejpam-5618	445	6	∩	∩	ADJ
ejpam-5618	445	7	h	h	NOUN
ejpam-5618	445	8	̸=	̸=	PROPN
ejpam-5618	445	9	ϕ.	ϕ.	NOUN
ejpam-5618	445	10	suppose	suppose	VERB
ejpam-5618	445	11	that	that	SCONJ
ejpam-5618	445	12	b	b	X
ejpam-5618	445	13	∈	∈	ADJ
ejpam-5618	445	14	ah	ah	INTJ
ejpam-5618	445	15	and	and	CCONJ
ejpam-5618	445	16	b	b	X
ejpam-5618	445	17	∈	∈	PROPN
ejpam-5618	445	18	h	h	NOUN
ejpam-5618	445	19	,	,	PUNCT
ejpam-5618	445	20	then	then	ADV
ejpam-5618	445	21	there	there	PRON
ejpam-5618	445	22	is	be	VERB
ejpam-5618	445	23	h	h	PRON
ejpam-5618	445	24	∈	∈	NOUN
ejpam-5618	445	25	h	h	NOUN
ejpam-5618	445	26	such	such	ADJ
ejpam-5618	445	27	that	that	PRON
ejpam-5618	445	28	b	b	X
ejpam-5618	445	29	=	=	PUNCT
ejpam-5618	445	30	ah	ah	INTJ
ejpam-5618	445	31	∈	∈	PROPN
ejpam-5618	445	32	h.	h.	NOUN
ejpam-5618	445	33	we	we	PRON
ejpam-5618	445	34	get	get	VERB
ejpam-5618	445	35	a	a	DET
ejpam-5618	445	36	∈	∈	NOUN
ejpam-5618	445	37	hh−1	hh−1	NOUN
ejpam-5618	445	38	⊂	⊂	PROPN
ejpam-5618	445	39	hh−1	hh−1	PROPN
ejpam-5618	445	40	=	=	SYM
ejpam-5618	445	41	h.	h.	PROPN
ejpam-5618	445	42	therefore	therefore	ADV
ejpam-5618	445	43	,	,	PUNCT
ejpam-5618	445	44	we	we	PRON
ejpam-5618	445	45	have	have	VERB
ejpam-5618	445	46	h	h	NOUN
ejpam-5618	445	47	=	=	SYM
ejpam-5618	445	48	icl(h	icl(h	PROPN
ejpam-5618	445	49	)	)	PUNCT
ejpam-5618	445	50	.	.	PUNCT
ejpam-5618	446	1	this	this	PRON
ejpam-5618	446	2	indicates	indicate	VERB
ejpam-5618	446	3	that	that	SCONJ
ejpam-5618	446	4	the	the	DET
ejpam-5618	446	5	subgroup	subgroup	NOUN
ejpam-5618	446	6	h	h	NOUN
ejpam-5618	446	7	is	be	AUX
ejpam-5618	446	8	i	i	PRON
ejpam-5618	446	9	-	-	PUNCT
ejpam-5618	446	10	open	open	ADJ
ejpam-5618	446	11	and	and	CCONJ
ejpam-5618	446	12	i	i	PRON
ejpam-5618	446	13	-	-	PUNCT
ejpam-5618	446	14	closed	close	VERB
ejpam-5618	446	15	set	set	NOUN
ejpam-5618	446	16	.	.	PUNCT
ejpam-5618	447	1	by	by	ADP
ejpam-5618	447	2	theorem	theorem	NOUN
ejpam-5618	447	3	27	27	NUM
ejpam-5618	447	4	,	,	PUNCT
ejpam-5618	447	5	the	the	DET
ejpam-5618	447	6	ideal	ideal	ADJ
ejpam-5618	447	7	topological	topological	PROPN
ejpam-5618	447	8	group	group	NOUN
ejpam-5618	447	9	g	g	PROPN
ejpam-5618	447	10	is	be	AUX
ejpam-5618	447	11	not	not	PART
ejpam-5618	447	12	i	i	PRON
ejpam-5618	447	13	-	-	PUNCT
ejpam-5618	447	14	connected	connect	VERB
ejpam-5618	447	15	which	which	PRON
ejpam-5618	447	16	is	be	AUX
ejpam-5618	447	17	a	a	DET
ejpam-5618	447	18	contradiction	contradiction	NOUN
ejpam-5618	447	19	.	.	PUNCT
ejpam-5618	448	1	5	5	X
ejpam-5618	448	2	.	.	X
ejpam-5618	448	3	conclusions	conclusion	NOUN
ejpam-5618	448	4	in	in	ADP
ejpam-5618	448	5	this	this	DET
ejpam-5618	448	6	paper	paper	NOUN
ejpam-5618	448	7	,	,	PUNCT
ejpam-5618	448	8	we	we	PRON
ejpam-5618	448	9	defined	define	VERB
ejpam-5618	448	10	the	the	DET
ejpam-5618	448	11	notion	notion	NOUN
ejpam-5618	448	12	of	of	ADP
ejpam-5618	448	13	ideal	ideal	ADJ
ejpam-5618	448	14	topological	topological	ADJ
ejpam-5618	448	15	groups	group	NOUN
ejpam-5618	448	16	.	.	PUNCT
ejpam-5618	449	1	we	we	PRON
ejpam-5618	449	2	studied	study	VERB
ejpam-5618	449	3	its	its	PRON
ejpam-5618	449	4	main	main	ADJ
ejpam-5618	449	5	fundamental	fundamental	ADJ
ejpam-5618	449	6	properties	property	NOUN
ejpam-5618	449	7	.	.	PUNCT
ejpam-5618	450	1	we	we	PRON
ejpam-5618	450	2	presented	present	VERB
ejpam-5618	450	3	examples	example	NOUN
ejpam-5618	450	4	that	that	PRON
ejpam-5618	450	5	show	show	VERB
ejpam-5618	450	6	that	that	SCONJ
ejpam-5618	450	7	ideal	ideal	ADJ
ejpam-5618	450	8	topological	topological	ADJ
ejpam-5618	450	9	groups	group	NOUN
ejpam-5618	450	10	and	and	CCONJ
ejpam-5618	450	11	topological	topological	ADJ
ejpam-5618	450	12	groups	group	NOUN
ejpam-5618	450	13	are	be	AUX
ejpam-5618	450	14	independent	independent	ADJ
ejpam-5618	450	15	concepts	concept	NOUN
ejpam-5618	450	16	.	.	PUNCT
ejpam-5618	451	1	we	we	PRON
ejpam-5618	451	2	gave	give	VERB
ejpam-5618	451	3	a	a	DET
ejpam-5618	451	4	sufficient	sufficient	ADJ
ejpam-5618	451	5	condition	condition	NOUN
ejpam-5618	451	6	for	for	ADP
ejpam-5618	451	7	a	a	DET
ejpam-5618	451	8	topological	topological	ADJ
ejpam-5618	451	9	group	group	NOUN
ejpam-5618	451	10	to	to	PART
ejpam-5618	451	11	be	be	AUX
ejpam-5618	451	12	an	an	DET
ejpam-5618	451	13	ideal	ideal	ADJ
ejpam-5618	451	14	topological	topological	ADJ
ejpam-5618	451	15	group	group	NOUN
ejpam-5618	451	16	as	as	ADV
ejpam-5618	451	17	well	well	ADV
ejpam-5618	451	18	as	as	SCONJ
ejpam-5618	451	19	we	we	PRON
ejpam-5618	451	20	gave	give	VERB
ejpam-5618	451	21	a	a	DET
ejpam-5618	451	22	sufficient	sufficient	ADJ
ejpam-5618	451	23	condition	condition	NOUN
ejpam-5618	451	24	for	for	SCONJ
ejpam-5618	451	25	an	an	DET
ejpam-5618	451	26	ideal	ideal	ADJ
ejpam-5618	451	27	topological	topological	ADJ
ejpam-5618	451	28	group	group	NOUN
ejpam-5618	451	29	to	to	PART
ejpam-5618	451	30	be	be	AUX
ejpam-5618	451	31	a	a	DET
ejpam-5618	451	32	topological	topological	ADJ
ejpam-5618	451	33	group	group	NOUN
ejpam-5618	451	34	.	.	PUNCT
ejpam-5618	452	1	in	in	ADP
ejpam-5618	452	2	contrast	contrast	NOUN
ejpam-5618	452	3	to	to	ADP
ejpam-5618	452	4	the	the	DET
ejpam-5618	452	5	case	case	NOUN
ejpam-5618	452	6	of	of	ADP
ejpam-5618	452	7	topological	topological	ADJ
ejpam-5618	452	8	groups	group	NOUN
ejpam-5618	452	9	,	,	PUNCT
ejpam-5618	452	10	not	not	PART
ejpam-5618	452	11	every	every	DET
ejpam-5618	452	12	subgroup	subgroup	NOUN
ejpam-5618	452	13	of	of	ADP
ejpam-5618	452	14	an	an	DET
ejpam-5618	452	15	ideal	ideal	ADJ
ejpam-5618	452	16	topological	topological	ADJ
ejpam-5618	452	17	group	group	NOUN
ejpam-5618	452	18	is	be	AUX
ejpam-5618	452	19	an	an	DET
ejpam-5618	452	20	ideal	ideal	ADJ
ejpam-5618	452	21	topological	topological	ADJ
ejpam-5618	452	22	group	group	NOUN
ejpam-5618	452	23	.	.	PUNCT
ejpam-5618	453	1	we	we	PRON
ejpam-5618	453	2	showed	show	VERB
ejpam-5618	453	3	that	that	SCONJ
ejpam-5618	453	4	every	every	DET
ejpam-5618	453	5	open	open	ADJ
ejpam-5618	453	6	subgroup	subgroup	NOUN
ejpam-5618	453	7	of	of	ADP
ejpam-5618	453	8	an	an	DET
ejpam-5618	453	9	ideal	ideal	ADJ
ejpam-5618	453	10	topological	topological	ADJ
ejpam-5618	453	11	group	group	NOUN
ejpam-5618	453	12	is	be	AUX
ejpam-5618	453	13	an	an	DET
ejpam-5618	453	14	ideal	ideal	ADJ
ejpam-5618	453	15	topological	topological	ADJ
ejpam-5618	453	16	group	group	NOUN
ejpam-5618	453	17	.	.	PUNCT
ejpam-5618	454	1	moreover	moreover	ADV
ejpam-5618	454	2	,	,	PUNCT
ejpam-5618	454	3	we	we	PRON
ejpam-5618	454	4	investigated	investigate	VERB
ejpam-5618	454	5	i	i	NOUN
ejpam-5618	454	6	-	-	PUNCT
ejpam-5618	454	7	connectedness	connectedness	NOUN
ejpam-5618	454	8	of	of	ADP
ejpam-5618	454	9	ideal	ideal	ADJ
ejpam-5618	454	10	topological	topological	ADJ
ejpam-5618	454	11	groups	group	NOUN
ejpam-5618	454	12	.	.	PUNCT
ejpam-5618	455	1	in	in	ADP
ejpam-5618	455	2	future	future	ADJ
ejpam-5618	455	3	studies	study	NOUN
ejpam-5618	455	4	,	,	PUNCT
ejpam-5618	455	5	the	the	DET
ejpam-5618	455	6	operation	operation	NOUN
ejpam-5618	455	7	of	of	ADP
ejpam-5618	455	8	taking	take	VERB
ejpam-5618	455	9	quotient	quotient	NOUN
ejpam-5618	455	10	of	of	ADP
ejpam-5618	455	11	ideal	ideal	ADJ
ejpam-5618	455	12	topological	topological	ADJ
ejpam-5618	455	13	groups	group	NOUN
ejpam-5618	455	14	will	will	AUX
ejpam-5618	455	15	be	be	AUX
ejpam-5618	455	16	the	the	DET
ejpam-5618	455	17	subject	subject	NOUN
ejpam-5618	455	18	of	of	ADP
ejpam-5618	455	19	our	our	PRON
ejpam-5618	455	20	study	study	NOUN
ejpam-5618	455	21	.	.	PUNCT
ejpam-5618	456	1	in	in	ADP
ejpam-5618	456	2	addition	addition	NOUN
ejpam-5618	456	3	,	,	PUNCT
ejpam-5618	456	4	we	we	PRON
ejpam-5618	456	5	will	will	AUX
ejpam-5618	456	6	study	study	VERB
ejpam-5618	456	7	separation	separation	NOUN
ejpam-5618	456	8	axioms	axiom	NOUN
ejpam-5618	456	9	and	and	CCONJ
ejpam-5618	456	10	ideal	ideal	ADJ
ejpam-5618	456	11	topological	topological	ADJ
ejpam-5618	456	12	groups	group	NOUN
ejpam-5618	456	13	action	action	NOUN
ejpam-5618	456	14	on	on	ADP
ejpam-5618	456	15	ideal	ideal	ADJ
ejpam-5618	456	16	topological	topological	ADJ
ejpam-5618	456	17	spaces	space	NOUN
ejpam-5618	456	18	.	.	PUNCT
ejpam-5618	457	1	s.	s.	PROPN
ejpam-5618	457	2	alammar	alammar	PROPN
ejpam-5618	457	3	,	,	PUNCT
ejpam-5618	457	4	m.	m.	NOUN
ejpam-5618	457	5	al	al	PROPN
ejpam-5618	457	6	shumrani	shumrani	PROPN
ejpam-5618	457	7	,	,	PUNCT
ejpam-5618	457	8	c.	c.	PROPN
ejpam-5618	457	9	özel	özel	PROPN
ejpam-5618	457	10	/	/	SYM
ejpam-5618	457	11	eur	eur	PROPN
ejpam-5618	457	12	.	.	PUNCT
ejpam-5618	458	1	j.	j.	PROPN
ejpam-5618	458	2	pure	pure	PROPN
ejpam-5618	458	3	appl	appl	PROPN
ejpam-5618	458	4	.	.	PROPN
ejpam-5618	458	5	math	math	PROPN
ejpam-5618	458	6	,	,	PUNCT
ejpam-5618	458	7	18	18	NUM
ejpam-5618	458	8	(	(	PUNCT
ejpam-5618	458	9	1	1	NUM
ejpam-5618	458	10	)	)	PUNCT
ejpam-5618	458	11	(	(	PUNCT
ejpam-5618	458	12	2025	2025	NUM
ejpam-5618	458	13	)	)	PUNCT
ejpam-5618	458	14	,	,	PUNCT
ejpam-5618	458	15	5618	5618	NUM
ejpam-5618	458	16	13	13	NUM
ejpam-5618	458	17	of	of	ADP
ejpam-5618	458	18	13	13	NUM
ejpam-5618	458	19	acknowledgements	acknowledgement	NOUN
ejpam-5618	458	20	the	the	DET
ejpam-5618	458	21	authors	author	NOUN
ejpam-5618	458	22	express	express	VERB
ejpam-5618	458	23	their	their	PRON
ejpam-5618	458	24	sincere	sincere	ADJ
ejpam-5618	458	25	thanks	thank	NOUN
ejpam-5618	458	26	to	to	ADP
ejpam-5618	458	27	the	the	DET
ejpam-5618	458	28	referees	referee	NOUN
ejpam-5618	458	29	for	for	ADP
ejpam-5618	458	30	the	the	DET
ejpam-5618	458	31	careful	careful	ADJ
ejpam-5618	458	32	reading	reading	NOUN
ejpam-5618	458	33	of	of	ADP
ejpam-5618	458	34	the	the	DET
ejpam-5618	458	35	manuscript	manuscript	NOUN
ejpam-5618	458	36	and	and	CCONJ
ejpam-5618	458	37	for	for	ADP
ejpam-5618	458	38	several	several	ADJ
ejpam-5618	458	39	comments	comment	NOUN
ejpam-5618	458	40	and	and	CCONJ
ejpam-5618	458	41	suggestions	suggestion	NOUN
ejpam-5618	458	42	that	that	PRON
ejpam-5618	458	43	improved	improve	VERB
ejpam-5618	458	44	the	the	DET
ejpam-5618	458	45	manuscript	manuscript	NOUN
ejpam-5618	458	46	substantially	substantially	ADV
ejpam-5618	458	47	.	.	PUNCT
ejpam-5618	459	1	references	reference	NOUN
ejpam-5618	459	2	[	[	X
ejpam-5618	459	3	1	1	X
ejpam-5618	459	4	]	]	PUNCT
ejpam-5618	459	5	me	i	PRON
ejpam-5618	459	6	abd	abd	PROPN
ejpam-5618	459	7	el	el	PROPN
ejpam-5618	459	8	-	-	PROPN
ejpam-5618	459	9	monsef	monsef	PROPN
ejpam-5618	459	10	,	,	PUNCT
ejpam-5618	459	11	ef	ef	PROPN
ejpam-5618	459	12	lashien	lashien	PROPN
ejpam-5618	459	13	,	,	PUNCT
ejpam-5618	459	14	and	and	CCONJ
ejpam-5618	459	15	aa	aa	PROPN
ejpam-5618	459	16	nasef	nasef	PROPN
ejpam-5618	459	17	.	.	PUNCT
ejpam-5618	460	1	on	on	ADP
ejpam-5618	460	2	i	i	NOUN
ejpam-5618	460	3	-	-	PUNCT
ejpam-5618	460	4	open	open	ADJ
ejpam-5618	460	5	sets	set	NOUN
ejpam-5618	460	6	and	and	CCONJ
ejpam-5618	460	7	i	i	NOUN
ejpam-5618	460	8	-	-	PUNCT
ejpam-5618	460	9	continuous	continuous	ADJ
ejpam-5618	460	10	functions	function	NOUN
ejpam-5618	460	11	.	.	PUNCT
ejpam-5618	461	1	kyungpook	kyungpook	PROPN
ejpam-5618	461	2	mathematical	mathematical	PROPN
ejpam-5618	461	3	journal	journal	NOUN
ejpam-5618	461	4	,	,	PUNCT
ejpam-5618	461	5	32(1):21–30	32(1):21–30	NUM
ejpam-5618	461	6	,	,	PUNCT
ejpam-5618	461	7	1992	1992	NUM
ejpam-5618	461	8	.	.	PUNCT
ejpam-5618	462	1	[	[	X
ejpam-5618	462	2	2	2	NUM
ejpam-5618	462	3	]	]	SYM
ejpam-5618	462	4	s	s	X
ejpam-5618	462	5	alammar	alammar	PROPN
ejpam-5618	462	6	,	,	PUNCT
ejpam-5618	462	7	ma	ma	PROPN
ejpam-5618	462	8	alshumrani	alshumrani	PROPN
ejpam-5618	462	9	,	,	PUNCT
ejpam-5618	462	10	and	and	CCONJ
ejpam-5618	462	11	cenap	cenap	VERB
ejpam-5618	462	12	ozel	ozel	NOUN
ejpam-5618	462	13	.	.	PUNCT
ejpam-5618	463	1	more	more	ADJ
ejpam-5618	463	2	on	on	ADP
ejpam-5618	463	3	ideal	ideal	ADJ
ejpam-5618	463	4	topological	topological	ADJ
ejpam-5618	463	5	groups	group	NOUN
ejpam-5618	463	6	.	.	PUNCT
ejpam-5618	464	1	phd	phd	NOUN
ejpam-5618	464	2	thesis	thesis	PROPN
ejpam-5618	464	3	,	,	PUNCT
ejpam-5618	464	4	msc	msc	PROPN
ejpam-5618	464	5	thesis	thesis	NOUN
ejpam-5618	464	6	in	in	ADP
ejpam-5618	464	7	king	king	PROPN
ejpam-5618	464	8	abdulaziz	abdulaziz	PROPN
ejpam-5618	464	9	university	university	PROPN
ejpam-5618	464	10	,	,	PUNCT
ejpam-5618	464	11	2024	2024	NUM
ejpam-5618	464	12	.	.	PUNCT
ejpam-5618	465	1	[	[	X
ejpam-5618	465	2	3	3	X
ejpam-5618	465	3	]	]	X
ejpam-5618	465	4	a.v	a.v	PROPN
ejpam-5618	465	5	.	.	PROPN
ejpam-5618	465	6	arhangel’skĭı	arhangel’skĭı	PROPN
ejpam-5618	465	7	and	and	CCONJ
ejpam-5618	465	8	p.j	p.j	PROPN
ejpam-5618	465	9	.	.	PROPN
ejpam-5618	465	10	collins	collins	PROPN
ejpam-5618	465	11	.	.	PUNCT
ejpam-5618	466	1	on	on	ADP
ejpam-5618	466	2	submaximal	submaximal	ADJ
ejpam-5618	466	3	spaces	space	NOUN
ejpam-5618	466	4	.	.	PUNCT
ejpam-5618	467	1	topology	topology	NOUN
ejpam-5618	467	2	and	and	CCONJ
ejpam-5618	467	3	its	its	PRON
ejpam-5618	467	4	applications	application	NOUN
ejpam-5618	467	5	,	,	PUNCT
ejpam-5618	467	6	64(3):219–241	64(3):219–241	PROPN
ejpam-5618	467	7	,	,	PUNCT
ejpam-5618	467	8	1995	1995	NUM
ejpam-5618	467	9	.	.	PUNCT
ejpam-5618	468	1	[	[	X
ejpam-5618	468	2	4	4	X
ejpam-5618	468	3	]	]	PUNCT
ejpam-5618	468	4	av	av	PROPN
ejpam-5618	468	5	arkhangel’skĭı	arkhangel’skĭı	PROPN
ejpam-5618	468	6	and	and	CCONJ
ejpam-5618	468	7	mikhail	mikhail	PROPN
ejpam-5618	468	8	tkachenko	tkachenko	PROPN
ejpam-5618	468	9	.	.	PUNCT
ejpam-5618	469	1	topological	topological	ADJ
ejpam-5618	469	2	groups	group	NOUN
ejpam-5618	469	3	and	and	CCONJ
ejpam-5618	469	4	related	related	ADJ
ejpam-5618	469	5	structures	structure	NOUN
ejpam-5618	469	6	,	,	PUNCT
ejpam-5618	469	7	volume	volume	NOUN
ejpam-5618	469	8	1	1	NUM
ejpam-5618	469	9	.	.	PUNCT
ejpam-5618	469	10	atlantis	atlantis	PROPN
ejpam-5618	469	11	press	press	PROPN
ejpam-5618	469	12	,	,	PUNCT
ejpam-5618	469	13	2008	2008	NUM
ejpam-5618	469	14	.	.	PUNCT
ejpam-5618	470	1	[	[	X
ejpam-5618	470	2	5	5	X
ejpam-5618	470	3	]	]	X
ejpam-5618	470	4	julian	julian	PROPN
ejpam-5618	470	5	dontchev	dontchev	PROPN
ejpam-5618	470	6	.	.	PUNCT
ejpam-5618	471	1	on	on	ADP
ejpam-5618	471	2	hausdorff	hausdorff	NOUN
ejpam-5618	471	3	spaces	space	NOUN
ejpam-5618	471	4	via	via	ADP
ejpam-5618	471	5	topological	topological	ADJ
ejpam-5618	471	6	ideals	ideal	NOUN
ejpam-5618	471	7	and	and	CCONJ
ejpam-5618	471	8	i	i	NOUN
ejpam-5618	471	9	-	-	PUNCT
ejpam-5618	471	10	irresolute	irresolute	ADJ
ejpam-5618	471	11	functions	function	NOUN
ejpam-5618	471	12	.	.	PUNCT
ejpam-5618	472	1	annals	annal	NOUN
ejpam-5618	472	2	of	of	ADP
ejpam-5618	472	3	the	the	DET
ejpam-5618	472	4	new	new	PROPN
ejpam-5618	472	5	york	york	PROPN
ejpam-5618	472	6	academy	academy	PROPN
ejpam-5618	472	7	of	of	ADP
ejpam-5618	472	8	sciences	science	NOUN
ejpam-5618	472	9	,	,	PUNCT
ejpam-5618	472	10	767(1):28–38	767(1):28–38	NUM
ejpam-5618	472	11	,	,	PUNCT
ejpam-5618	472	12	1995	1995	NUM
ejpam-5618	472	13	.	.	PUNCT
ejpam-5618	473	1	[	[	X
ejpam-5618	473	2	6	6	NUM
ejpam-5618	473	3	]	]	PUNCT
ejpam-5618	473	4	ryszard	ryszard	NOUN
ejpam-5618	473	5	engelking	engelke	VERB
ejpam-5618	473	6	.	.	PUNCT
ejpam-5618	474	1	general	general	ADJ
ejpam-5618	474	2	topology	topology	NOUN
ejpam-5618	474	3	,	,	PUNCT
ejpam-5618	474	4	pwn	pwn	NOUN
ejpam-5618	474	5	-	-	PUNCT
ejpam-5618	474	6	polish	polish	PROPN
ejpam-5618	474	7	sci	sci	PROPN
ejpam-5618	474	8	.	.	PROPN
ejpam-5618	474	9	publ	publ	PROPN
ejpam-5618	474	10	.	.	PROPN
ejpam-5618	474	11	,	,	PUNCT
ejpam-5618	474	12	warszawa	warszawa	PROPN
ejpam-5618	474	13	,	,	PUNCT
ejpam-5618	474	14	1977	1977	NUM
ejpam-5618	474	15	.	.	PUNCT
ejpam-5618	475	1	[	[	X
ejpam-5618	475	2	7	7	X
ejpam-5618	475	3	]	]	PUNCT
ejpam-5618	475	4	edwin	edwin	PROPN
ejpam-5618	475	5	hewitt	hewitt	PROPN
ejpam-5618	475	6	.	.	PUNCT
ejpam-5618	476	1	a	a	DET
ejpam-5618	476	2	problem	problem	NOUN
ejpam-5618	476	3	of	of	ADP
ejpam-5618	476	4	set	set	NOUN
ejpam-5618	476	5	-	-	PUNCT
ejpam-5618	476	6	theoretic	theoretic	NOUN
ejpam-5618	476	7	topology	topology	NOUN
ejpam-5618	476	8	.	.	PUNCT
ejpam-5618	477	1	duke	duke	PROPN
ejpam-5618	477	2	math	math	PROPN
ejpam-5618	477	3	.	.	PUNCT
ejpam-5618	478	1	j.	j.	PROPN
ejpam-5618	478	2	,	,	PUNCT
ejpam-5618	478	3	10:309–333	10:309–333	NUM
ejpam-5618	478	4	,	,	PUNCT
ejpam-5618	478	5	1943	1943	NUM
ejpam-5618	478	6	.	.	PUNCT
ejpam-5618	479	1	[	[	X
ejpam-5618	479	2	8	8	NUM
ejpam-5618	479	3	]	]	PUNCT
ejpam-5618	479	4	s.	s.	PROPN
ejpam-5618	479	5	hadi	hadi	PROPN
ejpam-5618	479	6	jafari	jafari	PROPN
ejpam-5618	479	7	and	and	CCONJ
ejpam-5618	479	8	neelamegarajan	neelamegarajan	PROPN
ejpam-5618	479	9	rajesh	rajesh	PROPN
ejpam-5618	479	10	.	.	PUNCT
ejpam-5618	480	1	on	on	ADP
ejpam-5618	480	2	ideal	ideal	ADJ
ejpam-5618	480	3	topological	topological	ADJ
ejpam-5618	480	4	groups	group	NOUN
ejpam-5618	480	5	.	.	PUNCT
ejpam-5618	481	1	vixra	vixra	NOUN
ejpam-5618	481	2	,	,	PUNCT
ejpam-5618	481	3	2020	2020	NUM
ejpam-5618	481	4	.	.	PUNCT
ejpam-5618	482	1	[	[	X
ejpam-5618	482	2	9	9	NUM
ejpam-5618	482	3	]	]	X
ejpam-5618	482	4	dragan	dragan	NOUN
ejpam-5618	482	5	janković	janković	PROPN
ejpam-5618	482	6	and	and	CCONJ
ejpam-5618	482	7	tr	tr	VERB
ejpam-5618	482	8	hamlett	hamlett	PROPN
ejpam-5618	482	9	.	.	PUNCT
ejpam-5618	483	1	new	new	ADJ
ejpam-5618	483	2	topologies	topology	NOUN
ejpam-5618	483	3	from	from	ADP
ejpam-5618	483	4	old	old	ADJ
ejpam-5618	483	5	via	via	ADP
ejpam-5618	483	6	ideals	ideal	NOUN
ejpam-5618	483	7	.	.	PUNCT
ejpam-5618	484	1	the	the	DET
ejpam-5618	484	2	american	american	PROPN
ejpam-5618	484	3	mathematical	mathematical	PROPN
ejpam-5618	484	4	monthly	monthly	PROPN
ejpam-5618	484	5	,	,	PUNCT
ejpam-5618	484	6	97(4):295–310	97(4):295–310	PROPN
ejpam-5618	484	7	,	,	PUNCT
ejpam-5618	484	8	1990	1990	NUM
ejpam-5618	484	9	.	.	PUNCT
ejpam-5618	485	1	[	[	X
ejpam-5618	485	2	10	10	NUM
ejpam-5618	485	3	]	]	X
ejpam-5618	485	4	dragan	dragan	NOUN
ejpam-5618	485	5	janković	janković	PROPN
ejpam-5618	485	6	and	and	CCONJ
ejpam-5618	485	7	tr	tr	VERB
ejpam-5618	485	8	hamlett	hamlett	PROPN
ejpam-5618	485	9	.	.	PUNCT
ejpam-5618	486	1	compatible	compatible	ADJ
ejpam-5618	486	2	extensions	extension	NOUN
ejpam-5618	486	3	of	of	ADP
ejpam-5618	486	4	ideals	ideal	NOUN
ejpam-5618	486	5	.	.	PUNCT
ejpam-5618	487	1	boll	boll	NOUN
ejpam-5618	487	2	.	.	PUNCT
ejpam-5618	488	1	un	un	PROPN
ejpam-5618	488	2	.	.	PROPN
ejpam-5618	488	3	mat	mat	PROPN
ejpam-5618	488	4	.	.	PUNCT
ejpam-5618	488	5	ital	ital	PROPN
ejpam-5618	488	6	.	.	PROPN
ejpam-5618	488	7	,	,	PUNCT
ejpam-5618	488	8	7(6	7(6	NUM
ejpam-5618	488	9	-	-	PUNCT
ejpam-5618	488	10	b):453–465	b):453–465	NOUN
ejpam-5618	488	11	,	,	PUNCT
ejpam-5618	488	12	1992	1992	NUM
ejpam-5618	488	13	.	.	PUNCT
ejpam-5618	489	1	[	[	X
ejpam-5618	489	2	11	11	NUM
ejpam-5618	489	3	]	]	PUNCT
ejpam-5618	489	4	k.	k.	PROPN
ejpam-5618	489	5	kuratowski	kuratowski	PROPN
ejpam-5618	489	6	.	.	PUNCT
ejpam-5618	490	1	topology	topology	NOUN
ejpam-5618	490	2	,	,	PUNCT
ejpam-5618	490	3	volume	volume	NOUN
ejpam-5618	490	4	1	1	NUM
ejpam-5618	490	5	.	.	PUNCT
ejpam-5618	491	1	new	new	PROPN
ejpam-5618	491	2	york	york	PROPN
ejpam-5618	491	3	,	,	PUNCT
ejpam-5618	491	4	academic	academic	ADJ
ejpam-5618	491	5	press	press	NOUN
ejpam-5618	491	6	,	,	PUNCT
ejpam-5618	491	7	1966	1966	NUM
ejpam-5618	491	8	.	.	PUNCT
ejpam-5618	492	1	[	[	X
ejpam-5618	492	2	12	12	NUM
ejpam-5618	492	3	]	]	X
ejpam-5618	492	4	j.r	j.r	PROPN
ejpam-5618	492	5	munkres	munkre	NOUN
ejpam-5618	492	6	.	.	PUNCT
ejpam-5618	493	1	topology	topology	NOUN
ejpam-5618	493	2	,	,	PUNCT
ejpam-5618	493	3	2000	2000	NUM
ejpam-5618	493	4	.	.	PUNCT
ejpam-5618	494	1	[	[	X
ejpam-5618	494	2	13	13	NUM
ejpam-5618	494	3	]	]	SYM
ejpam-5618	494	4	r	r	NOUN
ejpam-5618	494	5	vaidyanathaswamy	vaidyanathaswamy	NOUN
ejpam-5618	494	6	.	.	PUNCT
ejpam-5618	495	1	the	the	DET
ejpam-5618	495	2	localisation	localisation	NOUN
ejpam-5618	495	3	theory	theory	NOUN
ejpam-5618	495	4	in	in	ADP
ejpam-5618	495	5	set	set	NOUN
ejpam-5618	495	6	-	-	PUNCT
ejpam-5618	495	7	topology	topology	NOUN
ejpam-5618	495	8	.	.	PUNCT
ejpam-5618	496	1	in	in	ADP
ejpam-5618	496	2	proceedings	proceeding	NOUN
ejpam-5618	496	3	of	of	ADP
ejpam-5618	496	4	the	the	DET
ejpam-5618	496	5	indian	indian	PROPN
ejpam-5618	496	6	academy	academy	PROPN
ejpam-5618	496	7	of	of	ADP
ejpam-5618	496	8	sciences	science	NOUN
ejpam-5618	496	9	-	-	PUNCT
ejpam-5618	496	10	section	section	NOUN
ejpam-5618	496	11	a	a	PRON
ejpam-5618	496	12	,	,	PUNCT
ejpam-5618	496	13	volume	volume	NOUN
ejpam-5618	496	14	20	20	NUM
ejpam-5618	496	15	,	,	PUNCT
ejpam-5618	496	16	pages	page	NOUN
ejpam-5618	496	17	51–61	51–61	NUM
ejpam-5618	496	18	.	.	PUNCT
ejpam-5618	496	19	springer	springer	PROPN
ejpam-5618	496	20	india	india	PROPN
ejpam-5618	496	21	,	,	PUNCT
ejpam-5618	496	22	1944	1944	NUM
ejpam-5618	496	23	.	.	PUNCT