id	sid	tid	token	lemma	pos
ejpam-4298	1	1	european	european	PROPN
ejpam-4298	1	2	journal	journal	PROPN
ejpam-4298	1	3	of	of	ADP
ejpam-4298	1	4	pure	pure	ADJ
ejpam-4298	1	5	and	and	CCONJ
ejpam-4298	1	6	applied	apply	VERB
ejpam-4298	1	7	mathematics	mathematic	NOUN
ejpam-4298	1	8	vol	vol	NOUN
ejpam-4298	1	9	.	.	PROPN
ejpam-4298	2	1	15	15	NUM
ejpam-4298	2	2	,	,	PUNCT
ejpam-4298	2	3	no	no	INTJ
ejpam-4298	2	4	.	.	NOUN
ejpam-4298	2	5	2	2	NUM
ejpam-4298	2	6	,	,	PUNCT
ejpam-4298	2	7	2022	2022	NUM
ejpam-4298	2	8	,	,	PUNCT
ejpam-4298	2	9	589	589	NUM
ejpam-4298	2	10	-	-	SYM
ejpam-4298	2	11	601	601	NUM
ejpam-4298	2	12	issn	issn	PROPN
ejpam-4298	2	13	1307	1307	NUM
ejpam-4298	2	14	-	-	SYM
ejpam-4298	2	15	5543	5543	NUM
ejpam-4298	2	16	–	–	PUNCT
ejpam-4298	2	17	ejpam.com	ejpam.com	X
ejpam-4298	2	18	published	publish	VERB
ejpam-4298	2	19	by	by	ADP
ejpam-4298	2	20	new	new	PROPN
ejpam-4298	2	21	york	york	PROPN
ejpam-4298	2	22	business	business	PROPN
ejpam-4298	2	23	global	global	VERB
ejpam-4298	2	24	some	some	DET
ejpam-4298	2	25	properties	property	NOUN
ejpam-4298	2	26	of	of	ADP
ejpam-4298	2	27	weak	weak	ADJ
ejpam-4298	2	28	separation	separation	NOUN
ejpam-4298	2	29	axioms	axiom	NOUN
ejpam-4298	2	30	in	in	ADP
ejpam-4298	2	31	coc	coc	ADJ
ejpam-4298	2	32	-	-	ADJ
ejpam-4298	2	33	compact	compact	ADJ
ejpam-4298	2	34	sets	set	NOUN
ejpam-4298	2	35	fuad	fuad	PROPN
ejpam-4298	2	36	a.	a.	NOUN
ejpam-4298	2	37	abushaheen	abushaheen	VERB
ejpam-4298	2	38	basic	basic	ADJ
ejpam-4298	2	39	science	science	NOUN
ejpam-4298	2	40	department	department	NOUN
ejpam-4298	2	41	,	,	PUNCT
ejpam-4298	2	42	middle	middle	PROPN
ejpam-4298	2	43	east	east	PROPN
ejpam-4298	2	44	university	university	PROPN
ejpam-4298	2	45	,	,	PUNCT
ejpam-4298	2	46	amman	amman	PROPN
ejpam-4298	2	47	,	,	PUNCT
ejpam-4298	2	48	jordan	jordan	PROPN
ejpam-4298	2	49	abstract	abstract	PROPN
ejpam-4298	2	50	.	.	PUNCT
ejpam-4298	3	1	in	in	ADP
ejpam-4298	3	2	this	this	DET
ejpam-4298	3	3	paper	paper	NOUN
ejpam-4298	3	4	,	,	PUNCT
ejpam-4298	3	5	we	we	PRON
ejpam-4298	3	6	introduce	introduce	VERB
ejpam-4298	3	7	some	some	DET
ejpam-4298	3	8	separation	separation	NOUN
ejpam-4298	3	9	axioms	axiom	NOUN
ejpam-4298	3	10	in	in	ADP
ejpam-4298	3	11	coc	coc	ADJ
ejpam-4298	3	12	-	-	ADJ
ejpam-4298	3	13	compact	compact	ADJ
ejpam-4298	3	14	set	set	NOUN
ejpam-4298	3	15	,	,	PUNCT
ejpam-4298	3	16	namely	namely	ADV
ejpam-4298	3	17	coc	coc	NOUN
ejpam-4298	3	18	-	-	PUNCT
ejpam-4298	3	19	t0space	t0space	NOUN
ejpam-4298	3	20	,	,	PUNCT
ejpam-4298	3	21	coc	coc	NOUN
ejpam-4298	3	22	-	-	PUNCT
ejpam-4298	3	23	t	t	PROPN
ejpam-4298	3	24	1	1	NUM
ejpam-4298	3	25	4	4	NUM
ejpam-4298	3	26	space	space	NOUN
ejpam-4298	3	27	,	,	PUNCT
ejpam-4298	3	28	coc	coc	PROPN
ejpam-4298	3	29	-	-	PUNCT
ejpam-4298	3	30	t	t	PROPN
ejpam-4298	3	31	3	3	NUM
ejpam-4298	3	32	8	8	NUM
ejpam-4298	3	33	space	space	NOUN
ejpam-4298	3	34	,	,	PUNCT
ejpam-4298	3	35	coc	coc	PROPN
ejpam-4298	3	36	-	-	PUNCT
ejpam-4298	3	37	t	t	PROPN
ejpam-4298	3	38	1	1	NUM
ejpam-4298	3	39	2	2	NUM
ejpam-4298	3	40	space	space	NOUN
ejpam-4298	3	41	,	,	PUNCT
ejpam-4298	3	42	coc	coc	PROPN
ejpam-4298	3	43	-	-	PUNCT
ejpam-4298	3	44	t	t	PROPN
ejpam-4298	3	45	5	5	NUM
ejpam-4298	3	46	8	8	NUM
ejpam-4298	3	47	space	space	NOUN
ejpam-4298	3	48	,	,	PUNCT
ejpam-4298	3	49	coc	coc	NOUN
ejpam-4298	3	50	-	-	PUNCT
ejpam-4298	3	51	dispace	dispace	NOUN
ejpam-4298	3	52	,	,	PUNCT
ejpam-4298	3	53	coc	coc	NOUN
ejpam-4298	3	54	-	-	PUNCT
ejpam-4298	3	55	rispace	rispace	NOUN
ejpam-4298	3	56	for	for	ADP
ejpam-4298	3	57	i	i	PROPN
ejpam-4298	3	58	=	=	SYM
ejpam-4298	3	59	0	0	NUM
ejpam-4298	3	60	,	,	PUNCT
ejpam-4298	3	61	1	1	NUM
ejpam-4298	3	62	,	,	PUNCT
ejpam-4298	3	63	weak	weak	ADJ
ejpam-4298	3	64	coc	coc	NOUN
ejpam-4298	3	65	-	-	PUNCT
ejpam-4298	3	66	d1space	d1space	NOUN
ejpam-4298	3	67	and	and	CCONJ
ejpam-4298	3	68	weak	weak	ADJ
ejpam-4298	3	69	coc	coc	NOUN
ejpam-4298	3	70	-	-	PUNCT
ejpam-4298	3	71	r0space	r0space	NOUN
ejpam-4298	3	72	,	,	PUNCT
ejpam-4298	3	73	and	and	CCONJ
ejpam-4298	3	74	we	we	PRON
ejpam-4298	3	75	study	study	VERB
ejpam-4298	3	76	some	some	DET
ejpam-4298	3	77	relations	relation	NOUN
ejpam-4298	3	78	between	between	ADP
ejpam-4298	3	79	them	they	PRON
ejpam-4298	3	80	,	,	PUNCT
ejpam-4298	3	81	also	also	ADV
ejpam-4298	3	82	we	we	PRON
ejpam-4298	3	83	prove	prove	VERB
ejpam-4298	3	84	that	that	SCONJ
ejpam-4298	3	85	some	some	PRON
ejpam-4298	3	86	of	of	ADP
ejpam-4298	3	87	these	these	DET
ejpam-4298	3	88	separation	separation	NOUN
ejpam-4298	3	89	axioms	axiom	NOUN
ejpam-4298	3	90	have	have	VERB
ejpam-4298	3	91	”	"	PUNCT
ejpam-4298	3	92	hereditary	hereditary	ADJ
ejpam-4298	3	93	property	property	NOUN
ejpam-4298	3	94	”	"	PUNCT
ejpam-4298	3	95	.	.	PUNCT
ejpam-4298	4	1	2020	2020	NUM
ejpam-4298	4	2	mathematics	mathematic	NOUN
ejpam-4298	4	3	subject	subject	NOUN
ejpam-4298	4	4	classifications	classification	NOUN
ejpam-4298	4	5	:	:	PUNCT
ejpam-4298	4	6	54b05	54b05	NUM
ejpam-4298	4	7	,	,	PUNCT
ejpam-4298	4	8	54b10	54b10	NUM
ejpam-4298	4	9	,	,	PUNCT
ejpam-4298	4	10	54d20	54d20	NUM
ejpam-4298	4	11	.	.	PUNCT
ejpam-4298	5	1	key	key	ADJ
ejpam-4298	5	2	words	word	NOUN
ejpam-4298	5	3	and	and	CCONJ
ejpam-4298	5	4	phrases	phrase	NOUN
ejpam-4298	5	5	:	:	PUNCT
ejpam-4298	5	6	coc	coc	NOUN
ejpam-4298	5	7	-	-	PUNCT
ejpam-4298	5	8	tispace	tispace	NOUN
ejpam-4298	5	9	for	for	ADP
ejpam-4298	5	10	i	i	PROPN
ejpam-4298	5	11	=	=	SYM
ejpam-4298	5	12	0	0	NUM
ejpam-4298	5	13	,	,	PUNCT
ejpam-4298	5	14	1	1	NUM
ejpam-4298	5	15	4	4	NUM
ejpam-4298	5	16	,	,	PUNCT
ejpam-4298	5	17	1	1	NUM
ejpam-4298	5	18	2	2	NUM
ejpam-4298	5	19	,	,	PUNCT
ejpam-4298	5	20	5	5	NUM
ejpam-4298	5	21	8	8	NUM
ejpam-4298	5	22	,	,	PUNCT
ejpam-4298	5	23	3	3	NUM
ejpam-4298	5	24	4	4	NUM
ejpam-4298	5	25	,	,	PUNCT
ejpam-4298	5	26	coc	coc	NOUN
ejpam-4298	5	27	-	-	PUNCT
ejpam-4298	5	28	dispace	dispace	NOUN
ejpam-4298	5	29	,	,	PUNCT
ejpam-4298	5	30	coc	coc	NOUN
ejpam-4298	5	31	-	-	PUNCT
ejpam-4298	5	32	rispace	rispace	NOUN
ejpam-4298	5	33	for	for	ADP
ejpam-4298	5	34	i	i	PROPN
ejpam-4298	5	35	=	=	SYM
ejpam-4298	5	36	0	0	NUM
ejpam-4298	5	37	,	,	PUNCT
ejpam-4298	5	38	1	1	NUM
ejpam-4298	5	39	,	,	PUNCT
ejpam-4298	5	40	weak	weak	ADJ
ejpam-4298	5	41	coc	coc	NOUN
ejpam-4298	5	42	-	-	PUNCT
ejpam-4298	5	43	d1space	d1space	NOUN
ejpam-4298	5	44	and	and	CCONJ
ejpam-4298	5	45	weak	weak	ADJ
ejpam-4298	5	46	coc	coc	NOUN
ejpam-4298	5	47	-	-	PUNCT
ejpam-4298	5	48	r0space	r0space	NOUN
ejpam-4298	5	49	1	1	NUM
ejpam-4298	5	50	.	.	PUNCT
ejpam-4298	5	51	introduction	introduction	NOUN
ejpam-4298	5	52	and	and	CCONJ
ejpam-4298	5	53	preliminaries	preliminary	NOUN
ejpam-4298	5	54	in	in	ADP
ejpam-4298	5	55	[	[	X
ejpam-4298	5	56	4	4	NUM
ejpam-4298	5	57	]	]	PUNCT
ejpam-4298	5	58	,	,	PUNCT
ejpam-4298	5	59	the	the	DET
ejpam-4298	5	60	authors	author	NOUN
ejpam-4298	5	61	defined	define	VERB
ejpam-4298	5	62	a	a	DET
ejpam-4298	5	63	new	new	ADJ
ejpam-4298	5	64	type	type	NOUN
ejpam-4298	5	65	of	of	ADP
ejpam-4298	5	66	open	open	ADJ
ejpam-4298	5	67	sets	set	NOUN
ejpam-4298	5	68	called	call	VERB
ejpam-4298	5	69	coc	coc	ADJ
ejpam-4298	5	70	-	-	ADJ
ejpam-4298	5	71	compact	compact	ADJ
ejpam-4298	5	72	set	set	NOUN
ejpam-4298	5	73	as	as	ADP
ejpam-4298	5	74	a	a	DET
ejpam-4298	5	75	generalizations	generalization	NOUN
ejpam-4298	5	76	of	of	ADP
ejpam-4298	5	77	open	open	ADJ
ejpam-4298	5	78	sets	set	NOUN
ejpam-4298	5	79	.	.	PUNCT
ejpam-4298	6	1	after	after	ADP
ejpam-4298	6	2	this	this	DET
ejpam-4298	6	3	paper	paper	NOUN
ejpam-4298	6	4	many	many	ADJ
ejpam-4298	6	5	papers	paper	NOUN
ejpam-4298	6	6	in	in	ADP
ejpam-4298	6	7	this	this	DET
ejpam-4298	6	8	concept	concept	NOUN
ejpam-4298	6	9	were	be	AUX
ejpam-4298	6	10	appeared	appear	VERB
ejpam-4298	6	11	,	,	PUNCT
ejpam-4298	6	12	see	see	VERB
ejpam-4298	6	13	[	[	X
ejpam-4298	6	14	1–3	1–3	NOUN
ejpam-4298	6	15	]	]	X
ejpam-4298	6	16	.	.	PUNCT
ejpam-4298	7	1	also	also	ADV
ejpam-4298	7	2	many	many	ADJ
ejpam-4298	7	3	authors	author	NOUN
ejpam-4298	7	4	studied	study	VERB
ejpam-4298	7	5	weak	weak	ADJ
ejpam-4298	7	6	separation	separation	NOUN
ejpam-4298	7	7	axioms	axiom	NOUN
ejpam-4298	7	8	in	in	ADP
ejpam-4298	7	9	different	different	ADJ
ejpam-4298	7	10	types	type	NOUN
ejpam-4298	7	11	of	of	ADP
ejpam-4298	7	12	open	open	ADJ
ejpam-4298	7	13	sets	set	NOUN
ejpam-4298	7	14	,	,	PUNCT
ejpam-4298	7	15	for	for	ADP
ejpam-4298	7	16	example	example	NOUN
ejpam-4298	7	17	[	[	X
ejpam-4298	7	18	5	5	NUM
ejpam-4298	7	19	,	,	PUNCT
ejpam-4298	7	20	8	8	NUM
ejpam-4298	7	21	,	,	PUNCT
ejpam-4298	7	22	9	9	NUM
ejpam-4298	7	23	]	]	PUNCT
ejpam-4298	7	24	.	.	PUNCT
ejpam-4298	8	1	definition	definition	NOUN
ejpam-4298	8	2	1	1	NUM
ejpam-4298	8	3	.	.	PUNCT
ejpam-4298	9	1	[	[	X
ejpam-4298	9	2	4	4	X
ejpam-4298	9	3	]	]	X
ejpam-4298	9	4	a	a	DET
ejpam-4298	9	5	subset	subset	NOUN
ejpam-4298	9	6	a	a	PRON
ejpam-4298	9	7	of	of	ADP
ejpam-4298	9	8	a	a	DET
ejpam-4298	9	9	topological	topological	ADJ
ejpam-4298	9	10	space	space	NOUN
ejpam-4298	9	11	(	(	PUNCT
ejpam-4298	9	12	x	x	X
ejpam-4298	9	13	,	,	PUNCT
ejpam-4298	9	14	τ	τ	X
ejpam-4298	9	15	)	)	PUNCT
ejpam-4298	9	16	is	be	AUX
ejpam-4298	9	17	called	call	VERB
ejpam-4298	9	18	co	co	ADJ
ejpam-4298	9	19	-	-	ADJ
ejpam-4298	9	20	compact	compact	ADJ
ejpam-4298	9	21	open	open	ADJ
ejpam-4298	9	22	set	set	NOUN
ejpam-4298	9	23	(	(	PUNCT
ejpam-4298	9	24	notation	notation	NOUN
ejpam-4298	9	25	:	:	PUNCT
ejpam-4298	9	26	coc	coc	NOUN
ejpam-4298	9	27	-	-	PUNCT
ejpam-4298	9	28	open	open	ADJ
ejpam-4298	9	29	)	)	PUNCT
ejpam-4298	9	30	if	if	SCONJ
ejpam-4298	9	31	for	for	ADP
ejpam-4298	9	32	every	every	DET
ejpam-4298	9	33	x	x	PROPN
ejpam-4298	9	34	∈	∈	PROPN
ejpam-4298	9	35	a	a	PRON
ejpam-4298	9	36	,	,	PUNCT
ejpam-4298	9	37	there	there	PRON
ejpam-4298	9	38	exists	exist	VERB
ejpam-4298	9	39	an	an	DET
ejpam-4298	9	40	open	open	ADJ
ejpam-4298	9	41	set	set	NOUN
ejpam-4298	9	42	u	u	NOUN
ejpam-4298	9	43	⊆	⊆	NUM
ejpam-4298	9	44	x	x	PUNCT
ejpam-4298	9	45	and	and	CCONJ
ejpam-4298	9	46	a	a	DET
ejpam-4298	9	47	compact	compact	ADJ
ejpam-4298	9	48	subset	subset	NOUN
ejpam-4298	9	49	k	k	PROPN
ejpam-4298	9	50	of	of	ADP
ejpam-4298	9	51	x	x	INTJ
ejpam-4298	9	52	such	such	ADJ
ejpam-4298	9	53	that	that	SCONJ
ejpam-4298	9	54	x	x	SYM
ejpam-4298	9	55	∈	∈	NOUN
ejpam-4298	9	56	u	u	NOUN
ejpam-4298	9	57	−k	−k	NOUN
ejpam-4298	9	58	⊆	⊆	NUM
ejpam-4298	9	59	a.	a.	NOUN
ejpam-4298	9	60	the	the	DET
ejpam-4298	9	61	complement	complement	NOUN
ejpam-4298	9	62	of	of	ADP
ejpam-4298	9	63	a	a	DET
ejpam-4298	9	64	coc	coc	NOUN
ejpam-4298	9	65	-	-	PUNCT
ejpam-4298	9	66	open	open	ADJ
ejpam-4298	9	67	subset	subset	NOUN
ejpam-4298	9	68	is	be	AUX
ejpam-4298	9	69	called	call	VERB
ejpam-4298	9	70	coc	coc	NOUN
ejpam-4298	9	71	-	-	PUNCT
ejpam-4298	9	72	closed	closed	ADJ
ejpam-4298	9	73	.	.	PUNCT
ejpam-4298	10	1	the	the	DET
ejpam-4298	10	2	family	family	NOUN
ejpam-4298	10	3	of	of	ADP
ejpam-4298	10	4	all	all	DET
ejpam-4298	10	5	coc	coc	ADJ
ejpam-4298	10	6	-	-	PUNCT
ejpam-4298	10	7	open	open	ADJ
ejpam-4298	10	8	subsets	subset	NOUN
ejpam-4298	10	9	of	of	ADP
ejpam-4298	10	10	a	a	DET
ejpam-4298	10	11	topological	topological	ADJ
ejpam-4298	10	12	space	space	NOUN
ejpam-4298	10	13	x	x	PRON
ejpam-4298	10	14	will	will	AUX
ejpam-4298	10	15	be	be	AUX
ejpam-4298	10	16	denoted	denote	VERB
ejpam-4298	10	17	by	by	ADP
ejpam-4298	10	18	τk	τk	NOUN
ejpam-4298	10	19	.	.	PROPN
ejpam-4298	10	20	theorem	theorem	NOUN
ejpam-4298	10	21	1	1	NUM
ejpam-4298	10	22	.	.	PUNCT
ejpam-4298	11	1	[	[	X
ejpam-4298	11	2	4	4	X
ejpam-4298	11	3	]	]	X
ejpam-4298	11	4	let	let	VERB
ejpam-4298	11	5	(	(	PUNCT
ejpam-4298	11	6	x	x	NOUN
ejpam-4298	11	7	,	,	PUNCT
ejpam-4298	11	8	τ	τ	X
ejpam-4298	11	9	)	)	PUNCT
ejpam-4298	11	10	be	be	VERB
ejpam-4298	11	11	a	a	DET
ejpam-4298	11	12	topological	topological	ADJ
ejpam-4298	11	13	space	space	NOUN
ejpam-4298	11	14	.	.	PUNCT
ejpam-4298	12	1	then	then	ADV
ejpam-4298	12	2	(	(	PUNCT
ejpam-4298	12	3	i	i	NOUN
ejpam-4298	12	4	)	)	PUNCT
ejpam-4298	12	5	the	the	DET
ejpam-4298	12	6	collection	collection	NOUN
ejpam-4298	12	7	τk	τk	SCONJ
ejpam-4298	12	8	forms	form	VERB
ejpam-4298	12	9	a	a	DET
ejpam-4298	12	10	topology	topology	NOUN
ejpam-4298	12	11	on	on	ADP
ejpam-4298	12	12	x	x	PUNCT
ejpam-4298	12	13	with	with	ADP
ejpam-4298	12	14	τ	τ	PROPN
ejpam-4298	12	15	⊆	⊆	NUM
ejpam-4298	12	16	τk	τk	NOUN
ejpam-4298	12	17	.	.	PUNCT
ejpam-4298	13	1	(	(	PUNCT
ejpam-4298	13	2	ii	ii	X
ejpam-4298	13	3	)	)	PUNCT
ejpam-4298	13	4	the	the	DET
ejpam-4298	13	5	set	set	NOUN
ejpam-4298	13	6	{	{	PUNCT
ejpam-4298	13	7	u	u	NOUN
ejpam-4298	13	8	−k	−k	PROPN
ejpam-4298	13	9	:	:	PUNCT
ejpam-4298	13	10	u	u	X
ejpam-4298	13	11	∈	∈	PROPN
ejpam-4298	13	12	τ	τ	X
ejpam-4298	13	13	and	and	CCONJ
ejpam-4298	13	14	k	k	PROPN
ejpam-4298	13	15	is	be	AUX
ejpam-4298	13	16	compact	compact	ADJ
ejpam-4298	13	17	in	in	ADP
ejpam-4298	13	18	x	x	NOUN
ejpam-4298	13	19	}	}	PUNCT
ejpam-4298	13	20	forms	form	VERB
ejpam-4298	13	21	a	a	DET
ejpam-4298	13	22	base	base	NOUN
ejpam-4298	13	23	for	for	ADP
ejpam-4298	13	24	τk	τk	NOUN
ejpam-4298	13	25	.	.	PUNCT
ejpam-4298	14	1	lemma	lemma	PROPN
ejpam-4298	14	2	1	1	NUM
ejpam-4298	14	3	.	.	PUNCT
ejpam-4298	15	1	[	[	X
ejpam-4298	15	2	4	4	X
ejpam-4298	15	3	]	]	X
ejpam-4298	15	4	let	let	VERB
ejpam-4298	15	5	(	(	PUNCT
ejpam-4298	15	6	x	x	NOUN
ejpam-4298	15	7	,	,	PUNCT
ejpam-4298	15	8	τ	τ	X
ejpam-4298	15	9	)	)	PUNCT
ejpam-4298	15	10	be	be	VERB
ejpam-4298	15	11	a	a	DET
ejpam-4298	15	12	topological	topological	ADJ
ejpam-4298	15	13	space	space	NOUN
ejpam-4298	15	14	and	and	CCONJ
ejpam-4298	15	15	a	a	DET
ejpam-4298	15	16	be	be	AUX
ejpam-4298	15	17	a	a	DET
ejpam-4298	15	18	closed	closed	ADJ
ejpam-4298	15	19	subset	subset	NOUN
ejpam-4298	15	20	of	of	ADP
ejpam-4298	15	21	x.	x.	NOUN
ejpam-4298	15	22	then	then	ADV
ejpam-4298	15	23	(	(	PUNCT
ejpam-4298	15	24	τ	τ	PROPN
ejpam-4298	15	25	|a	|a	VERB
ejpam-4298	15	26	)	)	PUNCT
ejpam-4298	16	1	k	k	PROPN
ejpam-4298	17	1	=	=	NOUN
ejpam-4298	17	2	τk	τk	ADP
ejpam-4298	17	3	|a	|a	X
ejpam-4298	17	4	.	.	PUNCT
ejpam-4298	18	1	doi	doi	NOUN
ejpam-4298	18	2	:	:	PUNCT
ejpam-4298	18	3	https://doi.org/10.29020/nybg.ejpam.v15i2.4298	https://doi.org/10.29020/nybg.ejpam.v15i2.4298	NOUN
ejpam-4298	18	4	email	email	NOUN
ejpam-4298	18	5	address	address	NOUN
ejpam-4298	18	6	:	:	PUNCT
ejpam-4298	18	7	fshaheen@meu.edu.jo	fshaheen@meu.edu.jo	PROPN
ejpam-4298	18	8	(	(	PUNCT
ejpam-4298	18	9	f.a	f.a	PROPN
ejpam-4298	18	10	.	.	PROPN
ejpam-4298	18	11	abushaheen	abushaheen	PROPN
ejpam-4298	18	12	)	)	PUNCT
ejpam-4298	18	13	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4298	19	1	589	589	NUM
ejpam-4298	20	1	©	©	ADP
ejpam-4298	20	2	2022	2022	NUM
ejpam-4298	20	3	ejpam	ejpam	VERB
ejpam-4298	20	4	all	all	DET
ejpam-4298	20	5	rights	right	NOUN
ejpam-4298	20	6	reserved	reserve	VERB
ejpam-4298	20	7	.	.	PUNCT
ejpam-4298	21	1	f.a	f.a	PROPN
ejpam-4298	21	2	.	.	PROPN
ejpam-4298	21	3	abushaheen	abushaheen	PROPN
ejpam-4298	21	4	/	/	SYM
ejpam-4298	21	5	eur	eur	PROPN
ejpam-4298	21	6	.	.	PUNCT
ejpam-4298	22	1	j.	j.	PROPN
ejpam-4298	22	2	pure	pure	PROPN
ejpam-4298	22	3	appl	appl	PROPN
ejpam-4298	22	4	.	.	PROPN
ejpam-4298	22	5	math	math	PROPN
ejpam-4298	22	6	,	,	PUNCT
ejpam-4298	22	7	15	15	NUM
ejpam-4298	22	8	(	(	PUNCT
ejpam-4298	22	9	2	2	NUM
ejpam-4298	22	10	)	)	PUNCT
ejpam-4298	22	11	(	(	PUNCT
ejpam-4298	22	12	2022	2022	NUM
ejpam-4298	22	13	)	)	PUNCT
ejpam-4298	22	14	,	,	PUNCT
ejpam-4298	22	15	589	589	NUM
ejpam-4298	22	16	-	-	SYM
ejpam-4298	22	17	601	601	NUM
ejpam-4298	22	18	590	590	NUM
ejpam-4298	22	19	throughout	throughout	ADP
ejpam-4298	22	20	this	this	DET
ejpam-4298	22	21	paper	paper	NOUN
ejpam-4298	23	1	,	,	PUNCT
ejpam-4298	23	2	we	we	PRON
ejpam-4298	23	3	use	use	VERB
ejpam-4298	23	4	r	r	NOUN
ejpam-4298	23	5	,	,	PUNCT
ejpam-4298	23	6	q	q	NOUN
ejpam-4298	23	7	and	and	CCONJ
ejpam-4298	23	8	n	n	ADP
ejpam-4298	23	9	to	to	PART
ejpam-4298	23	10	denote	denote	VERB
ejpam-4298	23	11	the	the	DET
ejpam-4298	23	12	set	set	NOUN
ejpam-4298	23	13	of	of	ADP
ejpam-4298	23	14	real	real	ADJ
ejpam-4298	23	15	numbers	number	NOUN
ejpam-4298	23	16	,	,	PUNCT
ejpam-4298	23	17	rational	rational	ADJ
ejpam-4298	23	18	numbers	number	NOUN
ejpam-4298	23	19	and	and	CCONJ
ejpam-4298	23	20	natural	natural	ADJ
ejpam-4298	23	21	numbers	number	NOUN
ejpam-4298	23	22	,	,	PUNCT
ejpam-4298	23	23	respectively	respectively	ADV
ejpam-4298	23	24	.	.	PUNCT
ejpam-4298	24	1	the	the	DET
ejpam-4298	24	2	coc	coc	NOUN
ejpam-4298	24	3	-	-	PUNCT
ejpam-4298	24	4	closure	closure	NOUN
ejpam-4298	24	5	of	of	ADP
ejpam-4298	24	6	a	a	PRON
ejpam-4298	24	7	and	and	CCONJ
ejpam-4298	24	8	the	the	DET
ejpam-4298	24	9	cocinterior	cocinterior	NOUN
ejpam-4298	24	10	of	of	ADP
ejpam-4298	24	11	a	a	PRON
ejpam-4298	24	12	will	will	AUX
ejpam-4298	24	13	be	be	AUX
ejpam-4298	24	14	denoted	denote	VERB
ejpam-4298	24	15	by	by	ADP
ejpam-4298	24	16	a	a	DET
ejpam-4298	24	17	cocand	cocand	NOUN
ejpam-4298	24	18	intcoc(a	intcoc(a	NOUN
ejpam-4298	24	19	)	)	PUNCT
ejpam-4298	24	20	,	,	PUNCT
ejpam-4298	24	21	respectively	respectively	ADV
ejpam-4298	24	22	.	.	PUNCT
ejpam-4298	25	1	terms	term	NOUN
ejpam-4298	25	2	and	and	CCONJ
ejpam-4298	25	3	notations	notation	NOUN
ejpam-4298	25	4	not	not	PART
ejpam-4298	25	5	explained	explain	VERB
ejpam-4298	25	6	in	in	ADP
ejpam-4298	25	7	this	this	DET
ejpam-4298	25	8	paper	paper	NOUN
ejpam-4298	25	9	are	be	AUX
ejpam-4298	25	10	taken	take	VERB
ejpam-4298	25	11	from	from	ADP
ejpam-4298	25	12	[	[	X
ejpam-4298	25	13	4	4	NUM
ejpam-4298	25	14	,	,	PUNCT
ejpam-4298	25	15	7	7	NUM
ejpam-4298	25	16	]	]	PUNCT
ejpam-4298	25	17	.	.	PUNCT
ejpam-4298	26	1	2	2	X
ejpam-4298	26	2	.	.	X
ejpam-4298	26	3	coc	coc	PROPN
ejpam-4298	26	4	-	-	PUNCT
ejpam-4298	26	5	t	t	NOUN
ejpam-4298	26	6	-spaces	-space	NOUN
ejpam-4298	26	7	and	and	CCONJ
ejpam-4298	26	8	coc	coc	PROPN
ejpam-4298	26	9	-	-	PUNCT
ejpam-4298	26	10	d	d	NOUN
ejpam-4298	26	11	-	-	PUNCT
ejpam-4298	26	12	spaces	spaces	ADJ
ejpam-4298	26	13	definition	definition	NOUN
ejpam-4298	26	14	2	2	NUM
ejpam-4298	26	15	.	.	PUNCT
ejpam-4298	27	1	a	a	DET
ejpam-4298	27	2	space	space	NOUN
ejpam-4298	27	3	(	(	PUNCT
ejpam-4298	27	4	x	x	X
ejpam-4298	27	5	,	,	PUNCT
ejpam-4298	27	6	τ	τ	X
ejpam-4298	27	7	)	)	PUNCT
ejpam-4298	27	8	is	be	AUX
ejpam-4298	27	9	called	call	VERB
ejpam-4298	27	10	co	co	ADJ
ejpam-4298	27	11	-	-	ADJ
ejpam-4298	27	12	compact	compact	ADJ
ejpam-4298	27	13	-	-	PUNCT
ejpam-4298	27	14	t0	t0	NOUN
ejpam-4298	27	15	-	-	PUNCT
ejpam-4298	27	16	space	space	NOUN
ejpam-4298	27	17	(	(	PUNCT
ejpam-4298	27	18	coc	coc	NOUN
ejpam-4298	27	19	-	-	PUNCT
ejpam-4298	27	20	t0	t0	NOUN
ejpam-4298	27	21	-	-	NOUN
ejpam-4298	27	22	space	space	NOUN
ejpam-4298	27	23	)	)	PUNCT
ejpam-4298	27	24	if	if	SCONJ
ejpam-4298	27	25	for	for	ADP
ejpam-4298	27	26	all	all	DET
ejpam-4298	27	27	x	x	SYM
ejpam-4298	27	28	6=	6=	ADP
ejpam-4298	27	29	y	y	PROPN
ejpam-4298	27	30	∈	∈	PROPN
ejpam-4298	28	1	x	x	PRON
ejpam-4298	28	2	,	,	PUNCT
ejpam-4298	28	3	there	there	PRON
ejpam-4298	28	4	exists	exist	VERB
ejpam-4298	28	5	a	a	DET
ejpam-4298	28	6	coc	coc	NOUN
ejpam-4298	28	7	-	-	PUNCT
ejpam-4298	28	8	open	open	ADJ
ejpam-4298	28	9	set	set	NOUN
ejpam-4298	28	10	u	u	NOUN
ejpam-4298	28	11	contains	contain	VERB
ejpam-4298	28	12	one	one	NUM
ejpam-4298	28	13	point	point	NOUN
ejpam-4298	28	14	but	but	CCONJ
ejpam-4298	28	15	not	not	PART
ejpam-4298	28	16	other	other	ADJ
ejpam-4298	28	17	.	.	PUNCT
ejpam-4298	29	1	definition	definition	NOUN
ejpam-4298	29	2	3	3	NUM
ejpam-4298	29	3	.	.	PUNCT
ejpam-4298	30	1	a	a	DET
ejpam-4298	30	2	subset	subset	NOUN
ejpam-4298	30	3	a	a	PRON
ejpam-4298	30	4	of	of	ADP
ejpam-4298	30	5	a	a	DET
ejpam-4298	30	6	topological	topological	ADJ
ejpam-4298	30	7	space	space	NOUN
ejpam-4298	30	8	(	(	PUNCT
ejpam-4298	30	9	x	x	X
ejpam-4298	30	10	,	,	PUNCT
ejpam-4298	30	11	τ	τ	X
ejpam-4298	30	12	)	)	PUNCT
ejpam-4298	30	13	is	be	AUX
ejpam-4298	30	14	called	call	VERB
ejpam-4298	30	15	coc	coc	PROPN
ejpam-4298	30	16	-	-	PUNCT
ejpam-4298	31	1	d	d	NOUN
ejpam-4298	31	2	-	-	PUNCT
ejpam-4298	31	3	set	set	VERB
ejpam-4298	31	4	if	if	SCONJ
ejpam-4298	31	5	a	a	PRON
ejpam-4298	31	6	=	=	X
ejpam-4298	31	7	u	u	NOUN
ejpam-4298	31	8	−	−	PROPN
ejpam-4298	31	9	v	v	NOUN
ejpam-4298	31	10	,	,	PUNCT
ejpam-4298	31	11	for	for	ADP
ejpam-4298	31	12	some	some	DET
ejpam-4298	31	13	u	u	NOUN
ejpam-4298	31	14	,	,	PUNCT
ejpam-4298	31	15	v	v	PROPN
ejpam-4298	31	16	∈	∈	PROPN
ejpam-4298	31	17	τk	τk	NOUN
ejpam-4298	31	18	.	.	PUNCT
ejpam-4298	31	19	definition	definition	NOUN
ejpam-4298	31	20	4	4	NUM
ejpam-4298	31	21	.	.	PUNCT
ejpam-4298	32	1	a	a	DET
ejpam-4298	32	2	space	space	NOUN
ejpam-4298	32	3	(	(	PUNCT
ejpam-4298	32	4	x	x	X
ejpam-4298	32	5	,	,	PUNCT
ejpam-4298	32	6	τ	τ	X
ejpam-4298	32	7	)	)	PUNCT
ejpam-4298	32	8	is	be	AUX
ejpam-4298	32	9	called	call	VERB
ejpam-4298	32	10	co	co	ADJ
ejpam-4298	32	11	-	-	ADJ
ejpam-4298	32	12	compact	compact	ADJ
ejpam-4298	32	13	-	-	PUNCT
ejpam-4298	32	14	d0	d0	NOUN
ejpam-4298	32	15	-	-	PUNCT
ejpam-4298	32	16	space	space	NOUN
ejpam-4298	32	17	(	(	PUNCT
ejpam-4298	32	18	coc	coc	NOUN
ejpam-4298	32	19	-	-	PUNCT
ejpam-4298	32	20	d0	d0	NOUN
ejpam-4298	32	21	-	-	PUNCT
ejpam-4298	32	22	space	space	NOUN
ejpam-4298	32	23	)	)	PUNCT
ejpam-4298	32	24	if	if	SCONJ
ejpam-4298	32	25	for	for	ADP
ejpam-4298	32	26	all	all	DET
ejpam-4298	32	27	x	x	SYM
ejpam-4298	32	28	6=	6=	ADP
ejpam-4298	32	29	y	y	PROPN
ejpam-4298	32	30	∈	∈	PROPN
ejpam-4298	33	1	x	x	PRON
ejpam-4298	33	2	,	,	PUNCT
ejpam-4298	33	3	there	there	PRON
ejpam-4298	33	4	exists	exist	VERB
ejpam-4298	33	5	a	a	DET
ejpam-4298	33	6	coc	coc	PROPN
ejpam-4298	33	7	-	-	PUNCT
ejpam-4298	33	8	d	d	NOUN
ejpam-4298	33	9	-	-	PUNCT
ejpam-4298	33	10	set	set	VERB
ejpam-4298	33	11	u	u	NOUN
ejpam-4298	33	12	contains	contain	VERB
ejpam-4298	33	13	one	one	NUM
ejpam-4298	33	14	point	point	NOUN
ejpam-4298	33	15	but	but	CCONJ
ejpam-4298	33	16	not	not	PART
ejpam-4298	33	17	other	other	ADJ
ejpam-4298	33	18	.	.	PUNCT
ejpam-4298	34	1	theorem	theorem	NOUN
ejpam-4298	34	2	2	2	NUM
ejpam-4298	34	3	.	.	PUNCT
ejpam-4298	34	4	a	a	DET
ejpam-4298	34	5	coc	coc	NOUN
ejpam-4298	34	6	-	-	PUNCT
ejpam-4298	34	7	closed	closed	ADJ
ejpam-4298	34	8	subspace	subspace	NOUN
ejpam-4298	34	9	of	of	ADP
ejpam-4298	34	10	a	a	DET
ejpam-4298	34	11	coc	coc	NOUN
ejpam-4298	34	12	-	-	PUNCT
ejpam-4298	34	13	d0	d0	NOUN
ejpam-4298	34	14	-	-	PUNCT
ejpam-4298	34	15	space	space	NOUN
ejpam-4298	34	16	(	(	PUNCT
ejpam-4298	34	17	x	x	X
ejpam-4298	34	18	,	,	PUNCT
ejpam-4298	34	19	τ	τ	X
ejpam-4298	34	20	)	)	PUNCT
ejpam-4298	34	21	is	be	AUX
ejpam-4298	34	22	coc	coc	NOUN
ejpam-4298	34	23	-	-	PUNCT
ejpam-4298	34	24	d0	d0	NOUN
ejpam-4298	34	25	-	-	PUNCT
ejpam-4298	34	26	space	space	NOUN
ejpam-4298	34	27	.	.	PUNCT
ejpam-4298	35	1	proof	proof	NOUN
ejpam-4298	35	2	.	.	PUNCT
ejpam-4298	36	1	let	let	VERB
ejpam-4298	36	2	a	a	DET
ejpam-4298	36	3	be	be	AUX
ejpam-4298	36	4	a	a	DET
ejpam-4298	36	5	coc	coc	NOUN
ejpam-4298	36	6	-	-	PUNCT
ejpam-4298	36	7	closed	closed	ADJ
ejpam-4298	36	8	subset	subset	NOUN
ejpam-4298	36	9	x	x	PUNCT
ejpam-4298	36	10	and	and	CCONJ
ejpam-4298	36	11	let	let	VERB
ejpam-4298	36	12	x	x	PRON
ejpam-4298	36	13	6=	6=	ADP
ejpam-4298	36	14	y	y	PROPN
ejpam-4298	36	15	∈	∈	PROPN
ejpam-4298	36	16	a.	a.	NOUN
ejpam-4298	37	1	so	so	ADV
ejpam-4298	37	2	there	there	PRON
ejpam-4298	37	3	exists	exist	VERB
ejpam-4298	37	4	a	a	DET
ejpam-4298	37	5	coc	coc	PROPN
ejpam-4298	37	6	-	-	PUNCT
ejpam-4298	37	7	d	d	NOUN
ejpam-4298	37	8	-	-	PUNCT
ejpam-4298	37	9	set	set	VERB
ejpam-4298	37	10	d	d	NOUN
ejpam-4298	37	11	=	=	SYM
ejpam-4298	37	12	u	u	PROPN
ejpam-4298	37	13	−	−	PROPN
ejpam-4298	37	14	v	v	NOUN
ejpam-4298	37	15	with	with	ADP
ejpam-4298	37	16	u	u	NOUN
ejpam-4298	37	17	,	,	PUNCT
ejpam-4298	37	18	v	v	PROPN
ejpam-4298	37	19	∈	∈	NOUN
ejpam-4298	37	20	τk	τk	ADP
ejpam-4298	37	21	such	such	ADJ
ejpam-4298	37	22	that	that	SCONJ
ejpam-4298	37	23	x	x	SYM
ejpam-4298	37	24	∈	∈	PROPN
ejpam-4298	37	25	d	d	NOUN
ejpam-4298	37	26	and	and	CCONJ
ejpam-4298	37	27	y	y	PROPN
ejpam-4298	37	28	/∈	/∈	PUNCT
ejpam-4298	38	1	d.	d.	PROPN
ejpam-4298	38	2	now	now	ADV
ejpam-4298	38	3	x	x	X
ejpam-4298	38	4	∈	∈	PROPN
ejpam-4298	39	1	d	d	X
ejpam-4298	39	2	∩a	∩a	NOUN
ejpam-4298	39	3	=	=	PUNCT
ejpam-4298	39	4	(	(	PUNCT
ejpam-4298	39	5	u	u	NOUN
ejpam-4298	39	6	−	−	PROPN
ejpam-4298	39	7	v	v	NOUN
ejpam-4298	39	8	)	)	PUNCT
ejpam-4298	39	9	∩a	∩a	PROPN
ejpam-4298	39	10	=	=	PUNCT
ejpam-4298	39	11	(	(	PUNCT
ejpam-4298	39	12	a∩u)−	a∩u)−	NOUN
ejpam-4298	39	13	(	(	PUNCT
ejpam-4298	39	14	a∩	a∩	PROPN
ejpam-4298	39	15	v	v	NOUN
ejpam-4298	39	16	)	)	PUNCT
ejpam-4298	39	17	,	,	PUNCT
ejpam-4298	39	18	so	so	ADV
ejpam-4298	39	19	by	by	ADP
ejpam-4298	39	20	lemma	lemma	PROPN
ejpam-4298	39	21	1	1	NUM
ejpam-4298	39	22	we	we	PRON
ejpam-4298	39	23	have	have	VERB
ejpam-4298	39	24	a∩u	a∩u	PROPN
ejpam-4298	39	25	and	and	CCONJ
ejpam-4298	39	26	a∩	a∩	PROPN
ejpam-4298	39	27	v	v	X
ejpam-4298	39	28	∈	∈	PROPN
ejpam-4298	39	29	(	(	PUNCT
ejpam-4298	39	30	τ	τ	X
ejpam-4298	39	31	|a)k	|a)k	NOUN
ejpam-4298	39	32	=	=	PUNCT
ejpam-4298	39	33	τk	τk	ADP
ejpam-4298	39	34	|a	|a	NOUN
ejpam-4298	39	35	,	,	PUNCT
ejpam-4298	39	36	hence	hence	ADV
ejpam-4298	39	37	the	the	DET
ejpam-4298	39	38	result	result	NOUN
ejpam-4298	39	39	.	.	PUNCT
ejpam-4298	40	1	theorem	theorem	VERB
ejpam-4298	40	2	3	3	NUM
ejpam-4298	40	3	.	.	PUNCT
ejpam-4298	40	4	a	a	DET
ejpam-4298	40	5	space	space	NOUN
ejpam-4298	40	6	(	(	PUNCT
ejpam-4298	40	7	x	x	X
ejpam-4298	40	8	,	,	PUNCT
ejpam-4298	40	9	τ	τ	X
ejpam-4298	40	10	)	)	PUNCT
ejpam-4298	40	11	is	be	AUX
ejpam-4298	41	1	coc	coc	NOUN
ejpam-4298	41	2	-	-	PUNCT
ejpam-4298	41	3	t0	t0	NOUN
ejpam-4298	41	4	-	-	PUNCT
ejpam-4298	41	5	space	space	NOUN
ejpam-4298	41	6	if	if	SCONJ
ejpam-4298	41	7	and	and	CCONJ
ejpam-4298	41	8	only	only	ADV
ejpam-4298	41	9	if	if	SCONJ
ejpam-4298	41	10	it	it	PRON
ejpam-4298	41	11	is	be	AUX
ejpam-4298	41	12	coc	coc	NOUN
ejpam-4298	41	13	-	-	PUNCT
ejpam-4298	41	14	d0	d0	NOUN
ejpam-4298	41	15	-	-	PUNCT
ejpam-4298	41	16	space	space	NOUN
ejpam-4298	41	17	.	.	PUNCT
ejpam-4298	42	1	proof	proof	NOUN
ejpam-4298	42	2	.	.	PUNCT
ejpam-4298	43	1	(	(	PUNCT
ejpam-4298	43	2	⇒	⇒	PROPN
ejpam-4298	43	3	)	)	PUNCT
ejpam-4298	43	4	it	it	PRON
ejpam-4298	43	5	is	be	AUX
ejpam-4298	43	6	clear	clear	ADJ
ejpam-4298	43	7	since	since	SCONJ
ejpam-4298	43	8	every	every	DET
ejpam-4298	43	9	proper	proper	ADJ
ejpam-4298	43	10	coc	coc	NOUN
ejpam-4298	43	11	-	-	PUNCT
ejpam-4298	43	12	open	open	ADJ
ejpam-4298	43	13	subset	subset	NOUN
ejpam-4298	43	14	of	of	ADP
ejpam-4298	43	15	x	x	PUNCT
ejpam-4298	43	16	is	be	AUX
ejpam-4298	43	17	coc	coc	ADJ
ejpam-4298	43	18	-	-	PUNCT
ejpam-4298	43	19	d	d	NOUN
ejpam-4298	43	20	-	-	PUNCT
ejpam-4298	43	21	set	set	NOUN
ejpam-4298	43	22	.	.	PUNCT
ejpam-4298	44	1	(	(	PUNCT
ejpam-4298	44	2	⇐	⇐	NOUN
ejpam-4298	44	3	)	)	PUNCT
ejpam-4298	44	4	let	let	VERB
ejpam-4298	44	5	x	x	X
ejpam-4298	44	6	6=	6=	ADP
ejpam-4298	44	7	y	y	PROPN
ejpam-4298	44	8	∈	∈	PROPN
ejpam-4298	44	9	x	x	PRON
ejpam-4298	44	10	,	,	PUNCT
ejpam-4298	44	11	so	so	SCONJ
ejpam-4298	44	12	there	there	PRON
ejpam-4298	44	13	exists	exist	VERB
ejpam-4298	44	14	a	a	DET
ejpam-4298	44	15	coc	coc	NOUN
ejpam-4298	44	16	-	-	PUNCT
ejpam-4298	44	17	d0set	d0set	VERB
ejpam-4298	44	18	u	u	NOUN
ejpam-4298	44	19	contains	contain	VERB
ejpam-4298	44	20	x	x	PUNCT
ejpam-4298	44	21	with	with	ADP
ejpam-4298	44	22	u	u	NOUN
ejpam-4298	44	23	=	=	NOUN
ejpam-4298	44	24	u1	u1	PROPN
ejpam-4298	44	25	−	−	PROPN
ejpam-4298	44	26	u2	u2	PROPN
ejpam-4298	44	27	where	where	SCONJ
ejpam-4298	44	28	u1	u1	NOUN
ejpam-4298	44	29	,	,	PUNCT
ejpam-4298	44	30	u2	u2	PROPN
ejpam-4298	44	31	∈	∈	PROPN
ejpam-4298	44	32	τk	τk	ADP
ejpam-4298	44	33	i.e.	i.e.	X
ejpam-4298	44	34	x	x	SYM
ejpam-4298	44	35	∈	∈	NOUN
ejpam-4298	44	36	u1	u1	NOUN
ejpam-4298	44	37	and	and	CCONJ
ejpam-4298	44	38	x	x	NOUN
ejpam-4298	44	39	/∈	/∈	PUNCT
ejpam-4298	45	1	u2	u2	PROPN
ejpam-4298	45	2	.	.	PROPN
ejpam-4298	46	1	for	for	ADP
ejpam-4298	46	2	y	y	PROPN
ejpam-4298	46	3	,	,	PUNCT
ejpam-4298	46	4	we	we	PRON
ejpam-4298	46	5	have	have	VERB
ejpam-4298	46	6	the	the	DET
ejpam-4298	46	7	following	follow	VERB
ejpam-4298	46	8	cases:(1	cases:(1	PROPN
ejpam-4298	46	9	)	)	PUNCT
ejpam-4298	47	1	if	if	SCONJ
ejpam-4298	47	2	y	y	PROPN
ejpam-4298	47	3	/∈	/∈	PUNCT
ejpam-4298	47	4	u1	u1	PROPN
ejpam-4298	47	5	,	,	PUNCT
ejpam-4298	47	6	we	we	PRON
ejpam-4298	47	7	are	be	AUX
ejpam-4298	47	8	done	do	VERB
ejpam-4298	47	9	.	.	PUNCT
ejpam-4298	48	1	(	(	PUNCT
ejpam-4298	48	2	2	2	X
ejpam-4298	48	3	)	)	PUNCT
ejpam-4298	48	4	if	if	SCONJ
ejpam-4298	48	5	y	y	PROPN
ejpam-4298	48	6	∈	∈	PROPN
ejpam-4298	48	7	u1	u1	NOUN
ejpam-4298	48	8	and	and	CCONJ
ejpam-4298	48	9	y	y	PROPN
ejpam-4298	48	10	∈	∈	PROPN
ejpam-4298	48	11	u2	u2	PROPN
ejpam-4298	48	12	,	,	PUNCT
ejpam-4298	48	13	so	so	SCONJ
ejpam-4298	48	14	u2	u2	PROPN
ejpam-4298	48	15	contains	contain	VERB
ejpam-4298	48	16	y	y	PROPN
ejpam-4298	48	17	but	but	CCONJ
ejpam-4298	48	18	not	not	PART
ejpam-4298	48	19	x.	x.	NOUN
ejpam-4298	48	20	theorem	theorem	VERB
ejpam-4298	48	21	4	4	NUM
ejpam-4298	48	22	.	.	PUNCT
ejpam-4298	48	23	a	a	DET
ejpam-4298	48	24	space	space	NOUN
ejpam-4298	48	25	(	(	PUNCT
ejpam-4298	48	26	x	x	X
ejpam-4298	48	27	,	,	PUNCT
ejpam-4298	48	28	τ	τ	X
ejpam-4298	48	29	)	)	PUNCT
ejpam-4298	48	30	is	be	AUX
ejpam-4298	48	31	coc	coc	NOUN
ejpam-4298	48	32	-	-	PUNCT
ejpam-4298	48	33	t0	t0	NOUN
ejpam-4298	48	34	-	-	PUNCT
ejpam-4298	48	35	space	space	NOUN
ejpam-4298	48	36	if	if	SCONJ
ejpam-4298	48	37	and	and	CCONJ
ejpam-4298	48	38	only	only	ADV
ejpam-4298	48	39	if	if	SCONJ
ejpam-4298	48	40	for	for	ADP
ejpam-4298	48	41	all	all	DET
ejpam-4298	48	42	x	x	SYM
ejpam-4298	48	43	6=	6=	ADP
ejpam-4298	48	44	y	y	PROPN
ejpam-4298	48	45	∈	∈	PROPN
ejpam-4298	49	1	x	x	PRON
ejpam-4298	49	2	,	,	PUNCT
ejpam-4298	49	3	we	we	PRON
ejpam-4298	49	4	have	have	VERB
ejpam-4298	49	5	{	{	PUNCT
ejpam-4298	49	6	x}coc	x}coc	PROPN
ejpam-4298	49	7	=	=	PROPN
ejpam-4298	49	8	{	{	PUNCT
ejpam-4298	49	9	y}coc	y}coc	NOUN
ejpam-4298	49	10	.	.	PUNCT
ejpam-4298	50	1	proof	proof	NOUN
ejpam-4298	50	2	.	.	PUNCT
ejpam-4298	51	1	(	(	PUNCT
ejpam-4298	51	2	⇒	⇒	NOUN
ejpam-4298	51	3	)	)	PUNCT
ejpam-4298	51	4	let	let	VERB
ejpam-4298	51	5	x	x	X
ejpam-4298	51	6	6=	6=	ADP
ejpam-4298	51	7	y	y	PROPN
ejpam-4298	51	8	∈	∈	PROPN
ejpam-4298	51	9	x	x	PRON
ejpam-4298	51	10	,	,	PUNCT
ejpam-4298	51	11	there	there	PRON
ejpam-4298	51	12	exists	exist	VERB
ejpam-4298	51	13	a	a	DET
ejpam-4298	51	14	cocopen	cocopen	NOUN
ejpam-4298	51	15	set	set	VERB
ejpam-4298	51	16	u	u	NOUN
ejpam-4298	51	17	contains	contain	VERB
ejpam-4298	51	18	one	one	NUM
ejpam-4298	51	19	point	point	NOUN
ejpam-4298	51	20	but	but	CCONJ
ejpam-4298	51	21	not	not	PART
ejpam-4298	51	22	other	other	ADJ
ejpam-4298	51	23	,	,	PUNCT
ejpam-4298	51	24	say	say	VERB
ejpam-4298	51	25	x	x	X
ejpam-4298	51	26	∈	∈	PROPN
ejpam-4298	51	27	u	u	NOUN
ejpam-4298	51	28	and	and	CCONJ
ejpam-4298	51	29	y	y	PROPN
ejpam-4298	51	30	/∈	/∈	PUNCT
ejpam-4298	52	1	u	u	PROPN
ejpam-4298	52	2	.	.	PUNCT
ejpam-4298	53	1	then	then	ADV
ejpam-4298	53	2	x−u	x−u	PROPN
ejpam-4298	53	3	is	be	AUX
ejpam-4298	53	4	a	a	DET
ejpam-4298	53	5	coc	coc	NOUN
ejpam-4298	53	6	-	-	PUNCT
ejpam-4298	53	7	closed	close	VERB
ejpam-4298	53	8	set	set	NOUN
ejpam-4298	53	9	contains	contain	VERB
ejpam-4298	53	10	y	y	PROPN
ejpam-4298	53	11	and	and	CCONJ
ejpam-4298	53	12	{	{	PUNCT
ejpam-4298	53	13	y}coc	y}coc	PROPN
ejpam-4298	53	14	⊆	⊆	NUM
ejpam-4298	53	15	x−u	x−u	NOUN
ejpam-4298	53	16	,	,	PUNCT
ejpam-4298	53	17	so	so	ADV
ejpam-4298	53	18	x	x	X
ejpam-4298	53	19	/∈	/∈	PUNCT
ejpam-4298	53	20	{	{	PUNCT
ejpam-4298	53	21	y}coc	y}coc	ADJ
ejpam-4298	53	22	,	,	PUNCT
ejpam-4298	53	23	hence	hence	ADV
ejpam-4298	53	24	{	{	PUNCT
ejpam-4298	53	25	x}coc	x}coc	PROPN
ejpam-4298	53	26	6=	6=	PROPN
ejpam-4298	53	27	{	{	PUNCT
ejpam-4298	53	28	y}coc	y}coc	PROPN
ejpam-4298	53	29	.	.	PUNCT
ejpam-4298	54	1	(	(	PUNCT
ejpam-4298	54	2	⇐	⇐	NOUN
ejpam-4298	54	3	)	)	PUNCT
ejpam-4298	54	4	let	let	VERB
ejpam-4298	54	5	x	x	X
ejpam-4298	54	6	6=	6=	ADP
ejpam-4298	54	7	y	y	PROPN
ejpam-4298	54	8	∈	∈	PROPN
ejpam-4298	55	1	x.	x.	NOUN
ejpam-4298	56	1	then	then	ADV
ejpam-4298	56	2	it	it	PRON
ejpam-4298	56	3	is	be	AUX
ejpam-4298	56	4	clear	clear	ADJ
ejpam-4298	56	5	that	that	SCONJ
ejpam-4298	56	6	x	x	X
ejpam-4298	56	7	−	−	X
ejpam-4298	56	8	{	{	PUNCT
ejpam-4298	56	9	y}coc	y}coc	PROPN
ejpam-4298	56	10	is	be	AUX
ejpam-4298	56	11	coc	coc	ADJ
ejpam-4298	56	12	-	-	PUNCT
ejpam-4298	56	13	open	open	ADJ
ejpam-4298	56	14	set	set	NOUN
ejpam-4298	56	15	contains	contain	VERB
ejpam-4298	56	16	x	x	PUNCT
ejpam-4298	56	17	but	but	CCONJ
ejpam-4298	56	18	not	not	PART
ejpam-4298	56	19	y	y	NOUN
ejpam-4298	56	20	,	,	PUNCT
ejpam-4298	56	21	hence	hence	ADV
ejpam-4298	56	22	x	x	VERB
ejpam-4298	56	23	is	be	AUX
ejpam-4298	56	24	coc	coc	NOUN
ejpam-4298	56	25	-	-	PUNCT
ejpam-4298	56	26	t0	t0	NOUN
ejpam-4298	56	27	-	-	NOUN
ejpam-4298	56	28	space	space	NOUN
ejpam-4298	56	29	.	.	PUNCT
ejpam-4298	57	1	definition	definition	NOUN
ejpam-4298	57	2	5	5	NUM
ejpam-4298	57	3	.	.	PUNCT
ejpam-4298	58	1	a	a	DET
ejpam-4298	58	2	space	space	NOUN
ejpam-4298	58	3	(	(	PUNCT
ejpam-4298	58	4	x	x	X
ejpam-4298	58	5	,	,	PUNCT
ejpam-4298	58	6	τ	τ	X
ejpam-4298	58	7	)	)	PUNCT
ejpam-4298	58	8	is	be	AUX
ejpam-4298	58	9	called	call	VERB
ejpam-4298	58	10	co	co	ADJ
ejpam-4298	58	11	-	-	ADJ
ejpam-4298	58	12	compact	compact	ADJ
ejpam-4298	58	13	-t1	-t1	NOUN
ejpam-4298	58	14	-	-	PUNCT
ejpam-4298	58	15	space	space	NOUN
ejpam-4298	58	16	(	(	PUNCT
ejpam-4298	58	17	coc	coc	NOUN
ejpam-4298	58	18	-	-	PUNCT
ejpam-4298	58	19	t1	t1	NOUN
ejpam-4298	58	20	-	-	PUNCT
ejpam-4298	58	21	space	space	NOUN
ejpam-4298	58	22	)	)	PUNCT
ejpam-4298	58	23	if	if	SCONJ
ejpam-4298	58	24	for	for	ADP
ejpam-4298	58	25	all	all	DET
ejpam-4298	58	26	x	x	SYM
ejpam-4298	58	27	6=	6=	ADP
ejpam-4298	58	28	y	y	PROPN
ejpam-4298	58	29	∈	∈	PROPN
ejpam-4298	59	1	x	x	PRON
ejpam-4298	59	2	,	,	PUNCT
ejpam-4298	59	3	there	there	PRON
ejpam-4298	59	4	exist	exist	VERB
ejpam-4298	59	5	coc	coc	ADJ
ejpam-4298	59	6	-	-	PUNCT
ejpam-4298	59	7	open	open	ADJ
ejpam-4298	59	8	sets	set	NOUN
ejpam-4298	59	9	ux	ux	PROPN
ejpam-4298	59	10	,	,	PUNCT
ejpam-4298	59	11	vy	vy	NOUN
ejpam-4298	59	12	with	with	ADP
ejpam-4298	59	13	{	{	PUNCT
ejpam-4298	59	14	ux	ux	PROPN
ejpam-4298	59	15	,	,	PUNCT
ejpam-4298	59	16	vy}∩	vy}∩	PROPN
ejpam-4298	59	17	τ	τ	PROPN
ejpam-4298	59	18	6=	6=	ADP
ejpam-4298	59	19	φ	φ	NUM
ejpam-4298	59	20	such	such	ADJ
ejpam-4298	59	21	that	that	SCONJ
ejpam-4298	59	22	x	x	SYM
ejpam-4298	59	23	∈	∈	PROPN
ejpam-4298	59	24	ux	ux	PROPN
ejpam-4298	59	25	,	,	PUNCT
ejpam-4298	59	26	y	y	PROPN
ejpam-4298	59	27	∈	∈	PROPN
ejpam-4298	59	28	vy	vy	NOUN
ejpam-4298	59	29	and	and	CCONJ
ejpam-4298	59	30	y	y	PROPN
ejpam-4298	59	31	/∈	/∈	PUNCT
ejpam-4298	60	1	ux	ux	PROPN
ejpam-4298	60	2	,	,	PUNCT
ejpam-4298	60	3	x	x	PROPN
ejpam-4298	60	4	/∈	/∈	PUNCT
ejpam-4298	61	1	vy	vy	PROPN
ejpam-4298	61	2	.	.	PUNCT
ejpam-4298	61	3	definition	definition	NOUN
ejpam-4298	61	4	6	6	NUM
ejpam-4298	61	5	.	.	PUNCT
ejpam-4298	62	1	[	[	X
ejpam-4298	62	2	3	3	X
ejpam-4298	62	3	]	]	PUNCT
ejpam-4298	62	4	a	a	DET
ejpam-4298	62	5	space	space	NOUN
ejpam-4298	62	6	(	(	PUNCT
ejpam-4298	62	7	x	x	X
ejpam-4298	62	8	,	,	PUNCT
ejpam-4298	62	9	τ	τ	X
ejpam-4298	62	10	)	)	PUNCT
ejpam-4298	62	11	is	be	AUX
ejpam-4298	62	12	called	call	VERB
ejpam-4298	62	13	co	co	ADJ
ejpam-4298	62	14	-	-	ADJ
ejpam-4298	62	15	compact	compact	ADJ
ejpam-4298	62	16	-t2	-t2	NOUN
ejpam-4298	62	17	-	-	PUNCT
ejpam-4298	62	18	space	space	NOUN
ejpam-4298	62	19	(	(	PUNCT
ejpam-4298	62	20	coc	coc	NOUN
ejpam-4298	62	21	-	-	PUNCT
ejpam-4298	62	22	t2	t2	NOUN
ejpam-4298	62	23	-	-	PUNCT
ejpam-4298	62	24	space	space	NOUN
ejpam-4298	62	25	)	)	PUNCT
ejpam-4298	62	26	if	if	SCONJ
ejpam-4298	62	27	for	for	ADP
ejpam-4298	62	28	all	all	DET
ejpam-4298	62	29	x	x	SYM
ejpam-4298	62	30	6=	6=	ADP
ejpam-4298	62	31	y	y	PROPN
ejpam-4298	62	32	∈	∈	PROPN
ejpam-4298	63	1	x	x	PRON
ejpam-4298	63	2	,	,	PUNCT
ejpam-4298	63	3	there	there	PRON
ejpam-4298	63	4	exist	exist	VERB
ejpam-4298	63	5	coc	coc	ADJ
ejpam-4298	63	6	-	-	PUNCT
ejpam-4298	63	7	open	open	ADJ
ejpam-4298	63	8	sets	set	NOUN
ejpam-4298	63	9	ux	ux	PROPN
ejpam-4298	63	10	,	,	PUNCT
ejpam-4298	63	11	vy	vy	NOUN
ejpam-4298	63	12	with	with	ADP
ejpam-4298	63	13	{	{	PUNCT
ejpam-4298	63	14	ux	ux	PROPN
ejpam-4298	63	15	,	,	PUNCT
ejpam-4298	63	16	vy}∩	vy}∩	PROPN
ejpam-4298	63	17	τ	τ	PROPN
ejpam-4298	63	18	6=	6=	ADP
ejpam-4298	63	19	φ	φ	NUM
ejpam-4298	63	20	such	such	ADJ
ejpam-4298	63	21	that	that	SCONJ
ejpam-4298	63	22	x	x	SYM
ejpam-4298	63	23	∈	∈	PROPN
ejpam-4298	63	24	ux	ux	PROPN
ejpam-4298	63	25	,	,	PUNCT
ejpam-4298	63	26	y	y	PROPN
ejpam-4298	63	27	∈	∈	PROPN
ejpam-4298	63	28	vy	vy	NOUN
ejpam-4298	63	29	and	and	CCONJ
ejpam-4298	63	30	ux	ux	PROPN
ejpam-4298	63	31	∩	∩	PROPN
ejpam-4298	63	32	vy	vy	PROPN
ejpam-4298	63	33	=	=	PROPN
ejpam-4298	63	34	φ	φ	PROPN
ejpam-4298	63	35	.	.	PUNCT
ejpam-4298	64	1	f.a	f.a	PROPN
ejpam-4298	64	2	.	.	PROPN
ejpam-4298	64	3	abushaheen	abushaheen	PROPN
ejpam-4298	64	4	/	/	SYM
ejpam-4298	64	5	eur	eur	PROPN
ejpam-4298	64	6	.	.	PUNCT
ejpam-4298	65	1	j.	j.	PROPN
ejpam-4298	65	2	pure	pure	PROPN
ejpam-4298	65	3	appl	appl	PROPN
ejpam-4298	65	4	.	.	PROPN
ejpam-4298	65	5	math	math	PROPN
ejpam-4298	65	6	,	,	PUNCT
ejpam-4298	65	7	15	15	NUM
ejpam-4298	65	8	(	(	PUNCT
ejpam-4298	65	9	2	2	NUM
ejpam-4298	65	10	)	)	PUNCT
ejpam-4298	65	11	(	(	PUNCT
ejpam-4298	65	12	2022	2022	NUM
ejpam-4298	65	13	)	)	PUNCT
ejpam-4298	65	14	,	,	PUNCT
ejpam-4298	65	15	589	589	NUM
ejpam-4298	65	16	-	-	SYM
ejpam-4298	65	17	601	601	NUM
ejpam-4298	65	18	591	591	NUM
ejpam-4298	65	19	it	it	PRON
ejpam-4298	65	20	is	be	AUX
ejpam-4298	65	21	clear	clear	ADJ
ejpam-4298	65	22	that	that	SCONJ
ejpam-4298	65	23	if	if	SCONJ
ejpam-4298	65	24	(	(	PUNCT
ejpam-4298	65	25	x	x	NOUN
ejpam-4298	65	26	,	,	PUNCT
ejpam-4298	65	27	τ	τ	X
ejpam-4298	65	28	)	)	PUNCT
ejpam-4298	65	29	is	be	AUX
ejpam-4298	65	30	coc	coc	NOUN
ejpam-4298	65	31	-	-	PUNCT
ejpam-4298	65	32	t1	t1	NOUN
ejpam-4298	65	33	-	-	PUNCT
ejpam-4298	65	34	space	space	NOUN
ejpam-4298	65	35	,	,	PUNCT
ejpam-4298	65	36	then	then	ADV
ejpam-4298	65	37	(	(	PUNCT
ejpam-4298	65	38	x	x	X
ejpam-4298	65	39	,	,	PUNCT
ejpam-4298	65	40	τk	τk	ADV
ejpam-4298	65	41	)	)	PUNCT
ejpam-4298	65	42	is	be	AUX
ejpam-4298	65	43	t1	t1	NOUN
ejpam-4298	65	44	-	-	PUNCT
ejpam-4298	65	45	space	space	NOUN
ejpam-4298	65	46	.	.	PUNCT
ejpam-4298	66	1	and	and	CCONJ
ejpam-4298	66	2	every	every	DET
ejpam-4298	66	3	t1	t1	NOUN
ejpam-4298	66	4	-	-	PUNCT
ejpam-4298	66	5	space	space	NOUN
ejpam-4298	66	6	is	be	AUX
ejpam-4298	66	7	coc	coc	NOUN
ejpam-4298	66	8	-	-	PUNCT
ejpam-4298	66	9	t1	t1	NOUN
ejpam-4298	66	10	-	-	PUNCT
ejpam-4298	66	11	space	space	NOUN
ejpam-4298	66	12	,	,	PUNCT
ejpam-4298	66	13	but	but	CCONJ
ejpam-4298	66	14	the	the	DET
ejpam-4298	66	15	converse	converse	NOUN
ejpam-4298	66	16	need	need	AUX
ejpam-4298	66	17	not	not	PART
ejpam-4298	66	18	be	be	AUX
ejpam-4298	66	19	true	true	ADJ
ejpam-4298	66	20	,	,	PUNCT
ejpam-4298	66	21	consider	consider	VERB
ejpam-4298	66	22	the	the	DET
ejpam-4298	66	23	following	follow	VERB
ejpam-4298	66	24	example	example	NOUN
ejpam-4298	66	25	.	.	PUNCT
ejpam-4298	67	1	example	example	NOUN
ejpam-4298	68	1	1	1	NUM
ejpam-4298	68	2	.	.	PUNCT
ejpam-4298	68	3	let	let	VERB
ejpam-4298	68	4	x	x	PUNCT
ejpam-4298	68	5	=	=	SYM
ejpam-4298	68	6	r	r	NOUN
ejpam-4298	68	7	and	and	CCONJ
ejpam-4298	68	8	τ	τ	X
ejpam-4298	68	9	=	=	PUNCT
ejpam-4298	68	10	{	{	PUNCT
ejpam-4298	68	11	φ	φ	NOUN
ejpam-4298	68	12	}	}	PUNCT
ejpam-4298	68	13	∪	∪	NOUN
ejpam-4298	68	14	{	{	PUNCT
ejpam-4298	68	15	u	u	NOUN
ejpam-4298	68	16	⊆	⊆	NUM
ejpam-4298	68	17	r	r	NOUN
ejpam-4298	68	18	,	,	PUNCT
ejpam-4298	68	19	0	0	NUM
ejpam-4298	68	20	∈	∈	PROPN
ejpam-4298	68	21	u	u	NOUN
ejpam-4298	68	22	}	}	PUNCT
ejpam-4298	68	23	.	.	PUNCT
ejpam-4298	69	1	proof	proof	NOUN
ejpam-4298	69	2	.	.	PUNCT
ejpam-4298	70	1	a	a	DET
ejpam-4298	70	2	space	space	NOUN
ejpam-4298	70	3	(	(	PUNCT
ejpam-4298	70	4	x	x	X
ejpam-4298	70	5	,	,	PUNCT
ejpam-4298	70	6	τ	τ	X
ejpam-4298	70	7	)	)	PUNCT
ejpam-4298	70	8	is	be	AUX
ejpam-4298	70	9	coc	coc	NOUN
ejpam-4298	70	10	-	-	PUNCT
ejpam-4298	70	11	t1space	t1space	NOUN
ejpam-4298	70	12	,	,	PUNCT
ejpam-4298	70	13	to	to	PART
ejpam-4298	70	14	prove	prove	VERB
ejpam-4298	70	15	this	this	PRON
ejpam-4298	70	16	let	let	NOUN
ejpam-4298	70	17	x	x	PRON
ejpam-4298	70	18	6=	6=	ADP
ejpam-4298	70	19	y	y	PROPN
ejpam-4298	70	20	∈	∈	PROPN
ejpam-4298	70	21	x	x	PRON
ejpam-4298	70	22	,	,	PUNCT
ejpam-4298	70	23	so	so	SCONJ
ejpam-4298	70	24	we	we	PRON
ejpam-4298	70	25	have	have	VERB
ejpam-4298	70	26	the	the	DET
ejpam-4298	70	27	following	follow	VERB
ejpam-4298	70	28	cases	case	NOUN
ejpam-4298	70	29	:	:	PUNCT
ejpam-4298	70	30	(	(	PUNCT
ejpam-4298	70	31	i	i	NOUN
ejpam-4298	70	32	)	)	PUNCT
ejpam-4298	70	33	for	for	ADP
ejpam-4298	70	34	x	x	X
ejpam-4298	70	35	=	=	SYM
ejpam-4298	70	36	0	0	PROPN
ejpam-4298	70	37	,	,	PUNCT
ejpam-4298	70	38	y	y	PROPN
ejpam-4298	70	39	6=	6=	PROPN
ejpam-4298	70	40	0	0	NUM
ejpam-4298	70	41	,	,	PUNCT
ejpam-4298	70	42	let	let	VERB
ejpam-4298	70	43	u	u	PRON
ejpam-4298	70	44	=	=	X
ejpam-4298	70	45	{	{	PUNCT
ejpam-4298	70	46	0	0	NUM
ejpam-4298	70	47	}	}	PUNCT
ejpam-4298	70	48	and	and	CCONJ
ejpam-4298	70	49	v	v	NOUN
ejpam-4298	70	50	=	=	SYM
ejpam-4298	70	51	{	{	PUNCT
ejpam-4298	70	52	y	y	NOUN
ejpam-4298	70	53	,	,	PUNCT
ejpam-4298	70	54	0}−{0	0}−{0	NOUN
ejpam-4298	70	55	}	}	PUNCT
ejpam-4298	70	56	,	,	PUNCT
ejpam-4298	70	57	then	then	ADV
ejpam-4298	70	58	u	u	NOUN
ejpam-4298	70	59	,	,	PUNCT
ejpam-4298	70	60	v	v	PROPN
ejpam-4298	70	61	∈	∈	NOUN
ejpam-4298	70	62	τk	τk	ADP
ejpam-4298	70	63	and	and	CCONJ
ejpam-4298	70	64	{	{	PUNCT
ejpam-4298	70	65	u	u	NOUN
ejpam-4298	70	66	,	,	PUNCT
ejpam-4298	70	67	v	v	NOUN
ejpam-4298	70	68	}	}	PUNCT
ejpam-4298	70	69	∩τ	∩τ	NOUN
ejpam-4298	70	70	=	=	PUNCT
ejpam-4298	70	71	{	{	PUNCT
ejpam-4298	70	72	v	v	NOUN
ejpam-4298	70	73	}	}	PUNCT
ejpam-4298	70	74	with	with	ADP
ejpam-4298	70	75	x	x	PROPN
ejpam-4298	70	76	/∈	/∈	PUNCT
ejpam-4298	70	77	v	v	NOUN
ejpam-4298	70	78	and	and	CCONJ
ejpam-4298	70	79	y	y	PROPN
ejpam-4298	70	80	/∈	/∈	PUNCT
ejpam-4298	71	1	u	u	INTJ
ejpam-4298	71	2	.	.	PUNCT
ejpam-4298	72	1	(	(	PUNCT
ejpam-4298	72	2	ii	ii	NOUN
ejpam-4298	72	3	)	)	PUNCT
ejpam-4298	72	4	for	for	ADP
ejpam-4298	72	5	y	y	PROPN
ejpam-4298	72	6	=	=	SYM
ejpam-4298	72	7	0	0	PROPN
ejpam-4298	72	8	,	,	PUNCT
ejpam-4298	72	9	x	x	PUNCT
ejpam-4298	72	10	6=	6=	ADP
ejpam-4298	72	11	0	0	NUM
ejpam-4298	72	12	,	,	PUNCT
ejpam-4298	72	13	same	same	ADJ
ejpam-4298	72	14	as	as	ADP
ejpam-4298	72	15	(	(	PUNCT
ejpam-4298	72	16	i	i	NOUN
ejpam-4298	72	17	)	)	PUNCT
ejpam-4298	72	18	.	.	PUNCT
ejpam-4298	73	1	(	(	PUNCT
ejpam-4298	73	2	iii	iii	NOUN
ejpam-4298	73	3	)	)	PUNCT
ejpam-4298	73	4	for	for	ADP
ejpam-4298	73	5	x	x	SYM
ejpam-4298	73	6	6=	6=	ADP
ejpam-4298	73	7	0	0	NUM
ejpam-4298	73	8	,	,	PUNCT
ejpam-4298	73	9	y	y	PROPN
ejpam-4298	73	10	6=	6=	PROPN
ejpam-4298	73	11	0	0	NUM
ejpam-4298	73	12	,	,	PUNCT
ejpam-4298	73	13	let	let	VERB
ejpam-4298	73	14	u	u	PRON
ejpam-4298	73	15	=	=	PUNCT
ejpam-4298	73	16	{	{	PUNCT
ejpam-4298	73	17	x	x	NOUN
ejpam-4298	73	18	,	,	PUNCT
ejpam-4298	73	19	0	0	NUM
ejpam-4298	73	20	}	}	PUNCT
ejpam-4298	73	21	,	,	PUNCT
ejpam-4298	73	22	v	v	X
ejpam-4298	73	23	=	=	SYM
ejpam-4298	73	24	{	{	PUNCT
ejpam-4298	73	25	y	y	PROPN
ejpam-4298	73	26	,	,	PUNCT
ejpam-4298	73	27	0	0	NUM
ejpam-4298	73	28	}	}	PUNCT
ejpam-4298	73	29	−	−	PROPN
ejpam-4298	73	30	{	{	PUNCT
ejpam-4298	73	31	0	0	NUM
ejpam-4298	73	32	}	}	PUNCT
ejpam-4298	73	33	,	,	PUNCT
ejpam-4298	73	34	then	then	ADV
ejpam-4298	73	35	u	u	NOUN
ejpam-4298	73	36	,	,	PUNCT
ejpam-4298	73	37	v	v	PROPN
ejpam-4298	73	38	∈	∈	NOUN
ejpam-4298	73	39	τk	τk	ADP
ejpam-4298	73	40	and	and	CCONJ
ejpam-4298	73	41	{	{	PUNCT
ejpam-4298	73	42	u	u	NOUN
ejpam-4298	73	43	,	,	PUNCT
ejpam-4298	73	44	v	v	NOUN
ejpam-4298	73	45	}	}	PUNCT
ejpam-4298	73	46	∩	∩	NOUN
ejpam-4298	73	47	τ	τ	X
ejpam-4298	73	48	=	=	SYM
ejpam-4298	73	49	{	{	PUNCT
ejpam-4298	73	50	v	v	NOUN
ejpam-4298	73	51	}	}	PUNCT
ejpam-4298	73	52	,	,	PUNCT
ejpam-4298	73	53	then	then	ADV
ejpam-4298	74	1	x	x	X
ejpam-4298	74	2	/∈	/∈	PUNCT
ejpam-4298	74	3	v	v	NOUN
ejpam-4298	74	4	and	and	CCONJ
ejpam-4298	74	5	y	y	PROPN
ejpam-4298	74	6	/∈	/∈	PUNCT
ejpam-4298	74	7	u	u	PROPN
ejpam-4298	74	8	.	.	PUNCT
ejpam-4298	75	1	but	but	CCONJ
ejpam-4298	75	2	(	(	PUNCT
ejpam-4298	75	3	x	x	X
ejpam-4298	75	4	,	,	PUNCT
ejpam-4298	75	5	τ	τ	X
ejpam-4298	75	6	)	)	PUNCT
ejpam-4298	75	7	is	be	AUX
ejpam-4298	75	8	not	not	PART
ejpam-4298	75	9	t1space	t1space	NOUN
ejpam-4298	75	10	,	,	PUNCT
ejpam-4298	75	11	for	for	ADP
ejpam-4298	75	12	instance	instance	NOUN
ejpam-4298	75	13	take	take	VERB
ejpam-4298	75	14	x	x	NOUN
ejpam-4298	75	15	=	=	SYM
ejpam-4298	75	16	0	0	NUM
ejpam-4298	75	17	,	,	PUNCT
ejpam-4298	75	18	y	y	PROPN
ejpam-4298	75	19	=	=	SYM
ejpam-4298	75	20	1	1	NUM
ejpam-4298	75	21	,	,	PUNCT
ejpam-4298	75	22	then	then	ADV
ejpam-4298	75	23	there	there	PRON
ejpam-4298	75	24	is	be	VERB
ejpam-4298	75	25	no	no	DET
ejpam-4298	75	26	open	open	ADJ
ejpam-4298	75	27	set	set	NOUN
ejpam-4298	75	28	contains	contain	VERB
ejpam-4298	75	29	y	y	PROPN
ejpam-4298	75	30	but	but	CCONJ
ejpam-4298	75	31	not	not	PART
ejpam-4298	75	32	x.	x.	NOUN
ejpam-4298	75	33	theorem	theorem	VERB
ejpam-4298	75	34	5	5	NUM
ejpam-4298	75	35	.	.	PUNCT
ejpam-4298	75	36	a	a	DET
ejpam-4298	75	37	space	space	NOUN
ejpam-4298	75	38	(	(	PUNCT
ejpam-4298	75	39	x	x	X
ejpam-4298	75	40	,	,	PUNCT
ejpam-4298	75	41	τ	τ	X
ejpam-4298	75	42	)	)	PUNCT
ejpam-4298	75	43	is	be	AUX
ejpam-4298	75	44	coc	coc	NOUN
ejpam-4298	75	45	-	-	PUNCT
ejpam-4298	75	46	t1	t1	NOUN
ejpam-4298	75	47	-	-	PUNCT
ejpam-4298	75	48	space	space	NOUN
ejpam-4298	75	49	if	if	SCONJ
ejpam-4298	75	50	and	and	CCONJ
ejpam-4298	75	51	only	only	ADV
ejpam-4298	75	52	if	if	SCONJ
ejpam-4298	75	53	every	every	DET
ejpam-4298	75	54	singleton	singleton	NOUN
ejpam-4298	75	55	is	be	AUX
ejpam-4298	75	56	coc	coc	NOUN
ejpam-4298	75	57	-	-	PUNCT
ejpam-4298	75	58	closed	closed	ADJ
ejpam-4298	75	59	.	.	PUNCT
ejpam-4298	76	1	definition	definition	NOUN
ejpam-4298	76	2	7	7	NUM
ejpam-4298	76	3	.	.	PUNCT
ejpam-4298	77	1	a	a	DET
ejpam-4298	77	2	space	space	NOUN
ejpam-4298	77	3	(	(	PUNCT
ejpam-4298	77	4	x	x	X
ejpam-4298	77	5	,	,	PUNCT
ejpam-4298	77	6	τ	τ	X
ejpam-4298	77	7	)	)	PUNCT
ejpam-4298	77	8	is	be	AUX
ejpam-4298	77	9	called	call	VERB
ejpam-4298	77	10	co	co	ADJ
ejpam-4298	77	11	-	-	ADJ
ejpam-4298	77	12	compact	compact	ADJ
ejpam-4298	77	13	-	-	PUNCT
ejpam-4298	77	14	d1	d1	NOUN
ejpam-4298	77	15	-	-	PUNCT
ejpam-4298	77	16	space	space	NOUN
ejpam-4298	77	17	(	(	PUNCT
ejpam-4298	77	18	coc	coc	NOUN
ejpam-4298	77	19	-	-	PUNCT
ejpam-4298	77	20	d1	d1	NOUN
ejpam-4298	77	21	-	-	PUNCT
ejpam-4298	77	22	space	space	NOUN
ejpam-4298	77	23	)	)	PUNCT
ejpam-4298	77	24	if	if	SCONJ
ejpam-4298	77	25	for	for	ADP
ejpam-4298	77	26	all	all	DET
ejpam-4298	77	27	x	x	SYM
ejpam-4298	77	28	6=	6=	ADP
ejpam-4298	77	29	y	y	PROPN
ejpam-4298	77	30	∈	∈	PROPN
ejpam-4298	78	1	x	x	PRON
ejpam-4298	78	2	,	,	PUNCT
ejpam-4298	78	3	there	there	PRON
ejpam-4298	78	4	exist	exist	VERB
ejpam-4298	78	5	coc	coc	PROPN
ejpam-4298	78	6	-	-	PUNCT
ejpam-4298	78	7	d	d	NOUN
ejpam-4298	78	8	-	-	PUNCT
ejpam-4298	78	9	sets	set	NOUN
ejpam-4298	78	10	ux	ux	NOUN
ejpam-4298	78	11	,	,	PUNCT
ejpam-4298	78	12	vy	vy	NOUN
ejpam-4298	78	13	such	such	ADJ
ejpam-4298	78	14	that	that	SCONJ
ejpam-4298	78	15	x	x	SYM
ejpam-4298	78	16	∈	∈	PROPN
ejpam-4298	78	17	ux	ux	NOUN
ejpam-4298	78	18	,	,	PUNCT
ejpam-4298	78	19	y	y	PROPN
ejpam-4298	78	20	∈	∈	PROPN
ejpam-4298	78	21	vy	vy	NOUN
ejpam-4298	78	22	and	and	CCONJ
ejpam-4298	78	23	y	y	PROPN
ejpam-4298	78	24	/∈	/∈	PUNCT
ejpam-4298	79	1	ux	ux	PROPN
ejpam-4298	79	2	,	,	PUNCT
ejpam-4298	79	3	x	x	PROPN
ejpam-4298	79	4	/∈	/∈	PUNCT
ejpam-4298	80	1	vy	vy	PROPN
ejpam-4298	80	2	.	.	PUNCT
ejpam-4298	80	3	theorem	theorem	VERB
ejpam-4298	80	4	6	6	NUM
ejpam-4298	80	5	.	.	PUNCT
ejpam-4298	81	1	a	a	DET
ejpam-4298	81	2	coc	coc	NOUN
ejpam-4298	81	3	-	-	PUNCT
ejpam-4298	81	4	closed	closed	ADJ
ejpam-4298	81	5	subspace	subspace	NOUN
ejpam-4298	81	6	of	of	ADP
ejpam-4298	81	7	a	a	DET
ejpam-4298	81	8	coc	coc	ADJ
ejpam-4298	81	9	-	-	PUNCT
ejpam-4298	81	10	d1	d1	NOUN
ejpam-4298	81	11	-	-	PUNCT
ejpam-4298	81	12	space	space	NOUN
ejpam-4298	81	13	(	(	PUNCT
ejpam-4298	81	14	x	x	X
ejpam-4298	81	15	,	,	PUNCT
ejpam-4298	81	16	τ	τ	X
ejpam-4298	81	17	)	)	PUNCT
ejpam-4298	81	18	is	be	AUX
ejpam-4298	81	19	coc	coc	ADJ
ejpam-4298	81	20	-	-	PUNCT
ejpam-4298	81	21	d1	d1	NOUN
ejpam-4298	81	22	-	-	PUNCT
ejpam-4298	81	23	space	space	NOUN
ejpam-4298	81	24	.	.	PUNCT
ejpam-4298	82	1	definition	definition	NOUN
ejpam-4298	82	2	8	8	NUM
ejpam-4298	82	3	.	.	PUNCT
ejpam-4298	83	1	a	a	DET
ejpam-4298	83	2	space	space	NOUN
ejpam-4298	83	3	(	(	PUNCT
ejpam-4298	83	4	x	x	X
ejpam-4298	83	5	,	,	PUNCT
ejpam-4298	83	6	τ	τ	X
ejpam-4298	83	7	)	)	PUNCT
ejpam-4298	83	8	is	be	AUX
ejpam-4298	83	9	called	call	VERB
ejpam-4298	83	10	co	co	ADJ
ejpam-4298	83	11	-	-	ADJ
ejpam-4298	83	12	compact	compact	ADJ
ejpam-4298	83	13	-	-	PUNCT
ejpam-4298	83	14	d2	d2	NOUN
ejpam-4298	83	15	-	-	PUNCT
ejpam-4298	83	16	space	space	NOUN
ejpam-4298	83	17	(	(	PUNCT
ejpam-4298	83	18	coc	coc	NOUN
ejpam-4298	83	19	-	-	PUNCT
ejpam-4298	83	20	d2	d2	NOUN
ejpam-4298	83	21	-	-	PUNCT
ejpam-4298	83	22	space	space	NOUN
ejpam-4298	83	23	)	)	PUNCT
ejpam-4298	83	24	if	if	SCONJ
ejpam-4298	83	25	for	for	ADP
ejpam-4298	83	26	all	all	DET
ejpam-4298	83	27	x	x	SYM
ejpam-4298	83	28	6=	6=	ADP
ejpam-4298	83	29	y	y	PROPN
ejpam-4298	83	30	∈	∈	PROPN
ejpam-4298	84	1	x	x	PRON
ejpam-4298	84	2	,	,	PUNCT
ejpam-4298	84	3	there	there	PRON
ejpam-4298	84	4	exist	exist	VERB
ejpam-4298	84	5	disjoint	disjoint	ADJ
ejpam-4298	84	6	coc	coc	PROPN
ejpam-4298	84	7	-	-	PUNCT
ejpam-4298	84	8	d	d	NOUN
ejpam-4298	84	9	-	-	PUNCT
ejpam-4298	84	10	sets	set	NOUN
ejpam-4298	84	11	ux	ux	NOUN
ejpam-4298	84	12	,	,	PUNCT
ejpam-4298	84	13	vy	vy	NOUN
ejpam-4298	84	14	such	such	ADJ
ejpam-4298	84	15	that	that	SCONJ
ejpam-4298	84	16	x	x	SYM
ejpam-4298	84	17	∈	∈	PROPN
ejpam-4298	84	18	ux	ux	PROPN
ejpam-4298	84	19	,	,	PUNCT
ejpam-4298	84	20	y	y	PROPN
ejpam-4298	84	21	∈	∈	PROPN
ejpam-4298	84	22	vy	vy	NOUN
ejpam-4298	84	23	and	and	CCONJ
ejpam-4298	84	24	y	y	PROPN
ejpam-4298	84	25	/∈	/∈	PUNCT
ejpam-4298	85	1	ux	ux	PROPN
ejpam-4298	85	2	,	,	PUNCT
ejpam-4298	85	3	x	x	PROPN
ejpam-4298	85	4	/∈	/∈	PUNCT
ejpam-4298	86	1	vy	vy	PROPN
ejpam-4298	86	2	.	.	PUNCT
ejpam-4298	86	3	theorem	theorem	VERB
ejpam-4298	86	4	7	7	NUM
ejpam-4298	86	5	.	.	PUNCT
ejpam-4298	87	1	let	let	AUX
ejpam-4298	87	2	(	(	PUNCT
ejpam-4298	87	3	x	x	NOUN
ejpam-4298	87	4	,	,	PUNCT
ejpam-4298	87	5	τ	τ	X
ejpam-4298	87	6	)	)	PUNCT
ejpam-4298	87	7	be	be	VERB
ejpam-4298	87	8	a	a	DET
ejpam-4298	87	9	topological	topological	ADJ
ejpam-4298	87	10	space	space	NOUN
ejpam-4298	87	11	.	.	PUNCT
ejpam-4298	88	1	then	then	ADV
ejpam-4298	88	2	:	:	PUNCT
ejpam-4298	88	3	(	(	PUNCT
ejpam-4298	88	4	i	i	NOUN
ejpam-4298	88	5	)	)	PUNCT
ejpam-4298	88	6	if	if	SCONJ
ejpam-4298	88	7	x	x	PRON
ejpam-4298	88	8	is	be	AUX
ejpam-4298	88	9	coc	coc	ADJ
ejpam-4298	88	10	-	-	PUNCT
ejpam-4298	88	11	ti	ti	NOUN
ejpam-4298	88	12	-	-	NOUN
ejpam-4298	88	13	space	space	NOUN
ejpam-4298	88	14	,	,	PUNCT
ejpam-4298	88	15	then	then	ADV
ejpam-4298	88	16	x	x	PUNCT
ejpam-4298	88	17	is	be	AUX
ejpam-4298	88	18	coc	coc	ADJ
ejpam-4298	88	19	-	-	PUNCT
ejpam-4298	88	20	ti−1	ti−1	NOUN
ejpam-4298	88	21	-	-	PUNCT
ejpam-4298	88	22	space	space	NOUN
ejpam-4298	88	23	for	for	ADP
ejpam-4298	88	24	i	i	PRON
ejpam-4298	88	25	=	=	SYM
ejpam-4298	88	26	1	1	NUM
ejpam-4298	88	27	,	,	PUNCT
ejpam-4298	88	28	2	2	NUM
ejpam-4298	88	29	.	.	PUNCT
ejpam-4298	88	30	(	(	PUNCT
ejpam-4298	88	31	ii	ii	NOUN
ejpam-4298	88	32	)	)	PUNCT
ejpam-4298	88	33	if	if	SCONJ
ejpam-4298	88	34	x	x	PRON
ejpam-4298	88	35	is	be	AUX
ejpam-4298	88	36	coc	coc	ADJ
ejpam-4298	88	37	-	-	PUNCT
ejpam-4298	88	38	ti	ti	NOUN
ejpam-4298	88	39	-	-	NOUN
ejpam-4298	88	40	space	space	NOUN
ejpam-4298	88	41	,	,	PUNCT
ejpam-4298	88	42	then	then	ADV
ejpam-4298	88	43	x	x	PUNCT
ejpam-4298	88	44	is	be	AUX
ejpam-4298	88	45	coc	coc	ADJ
ejpam-4298	88	46	-	-	PUNCT
ejpam-4298	88	47	di	di	NOUN
ejpam-4298	88	48	-	-	NOUN
ejpam-4298	88	49	space	space	NOUN
ejpam-4298	88	50	for	for	ADP
ejpam-4298	88	51	i	i	PRON
ejpam-4298	88	52	=	=	SYM
ejpam-4298	88	53	1	1	NUM
ejpam-4298	88	54	,	,	PUNCT
ejpam-4298	88	55	2	2	NUM
ejpam-4298	88	56	.	.	PUNCT
ejpam-4298	88	57	(	(	PUNCT
ejpam-4298	88	58	iii	iii	X
ejpam-4298	88	59	)	)	PUNCT
ejpam-4298	88	60	if	if	SCONJ
ejpam-4298	88	61	x	x	PRON
ejpam-4298	88	62	is	be	AUX
ejpam-4298	88	63	coc	coc	ADJ
ejpam-4298	88	64	-	-	PUNCT
ejpam-4298	88	65	di	di	NOUN
ejpam-4298	88	66	-	-	NOUN
ejpam-4298	88	67	space	space	NOUN
ejpam-4298	88	68	,	,	PUNCT
ejpam-4298	88	69	then	then	ADV
ejpam-4298	88	70	x	x	PUNCT
ejpam-4298	88	71	is	be	AUX
ejpam-4298	88	72	coc	coc	PROPN
ejpam-4298	88	73	-	-	PUNCT
ejpam-4298	88	74	di−1	di−1	NOUN
ejpam-4298	88	75	-	-	PUNCT
ejpam-4298	88	76	space	space	NOUN
ejpam-4298	88	77	for	for	ADP
ejpam-4298	88	78	i	i	PRON
ejpam-4298	88	79	=	=	SYM
ejpam-4298	88	80	1	1	NUM
ejpam-4298	88	81	,	,	PUNCT
ejpam-4298	88	82	2	2	NUM
ejpam-4298	88	83	.	.	PUNCT
ejpam-4298	88	84	(	(	PUNCT
ejpam-4298	88	85	iv	iv	X
ejpam-4298	88	86	)	)	PUNCT
ejpam-4298	88	87	if	if	SCONJ
ejpam-4298	88	88	x	x	PRON
ejpam-4298	88	89	is	be	AUX
ejpam-4298	88	90	coc	coc	ADJ
ejpam-4298	88	91	-	-	PUNCT
ejpam-4298	88	92	d1	d1	NOUN
ejpam-4298	88	93	-	-	PUNCT
ejpam-4298	88	94	space	space	NOUN
ejpam-4298	88	95	,	,	PUNCT
ejpam-4298	88	96	then	then	ADV
ejpam-4298	88	97	x	x	PUNCT
ejpam-4298	88	98	is	be	AUX
ejpam-4298	88	99	coc	coc	NOUN
ejpam-4298	88	100	-	-	PUNCT
ejpam-4298	88	101	t0	t0	NOUN
ejpam-4298	88	102	-	-	NOUN
ejpam-4298	88	103	space	space	NOUN
ejpam-4298	88	104	.	.	PUNCT
ejpam-4298	89	1	(	(	PUNCT
ejpam-4298	89	2	v	v	NOUN
ejpam-4298	89	3	)	)	PUNCT
ejpam-4298	89	4	x	x	X
ejpam-4298	89	5	is	be	AUX
ejpam-4298	89	6	coc	coc	ADJ
ejpam-4298	89	7	-	-	PUNCT
ejpam-4298	89	8	d1	d1	NOUN
ejpam-4298	89	9	-	-	PUNCT
ejpam-4298	89	10	space	space	NOUN
ejpam-4298	89	11	if	if	SCONJ
ejpam-4298	89	12	and	and	CCONJ
ejpam-4298	89	13	only	only	ADV
ejpam-4298	89	14	if	if	SCONJ
ejpam-4298	89	15	x	x	PRON
ejpam-4298	89	16	is	be	AUX
ejpam-4298	89	17	coc	coc	ADJ
ejpam-4298	89	18	-	-	PUNCT
ejpam-4298	89	19	d2	d2	NOUN
ejpam-4298	89	20	-	-	PUNCT
ejpam-4298	89	21	space	space	NOUN
ejpam-4298	89	22	.	.	PUNCT
ejpam-4298	90	1	proof	proof	NOUN
ejpam-4298	90	2	.	.	PUNCT
ejpam-4298	91	1	we	we	PRON
ejpam-4298	91	2	will	will	AUX
ejpam-4298	91	3	prove	prove	VERB
ejpam-4298	91	4	(	(	PUNCT
ejpam-4298	91	5	v	v	NOUN
ejpam-4298	91	6	)	)	PUNCT
ejpam-4298	91	7	only	only	ADV
ejpam-4298	91	8	.	.	PUNCT
ejpam-4298	92	1	(	(	PUNCT
ejpam-4298	92	2	⇐	⇐	ADJ
ejpam-4298	92	3	)	)	PUNCT
ejpam-4298	92	4	obvious	obvious	ADJ
ejpam-4298	92	5	.	.	PUNCT
ejpam-4298	93	1	(	(	PUNCT
ejpam-4298	93	2	⇒	⇒	NOUN
ejpam-4298	93	3	)	)	PUNCT
ejpam-4298	93	4	for	for	ADP
ejpam-4298	93	5	x	x	SYM
ejpam-4298	93	6	6=	6=	ADP
ejpam-4298	93	7	y	y	PROPN
ejpam-4298	93	8	∈	∈	PROPN
ejpam-4298	94	1	x	x	PRON
ejpam-4298	94	2	,	,	PUNCT
ejpam-4298	94	3	there	there	PRON
ejpam-4298	94	4	exist	exist	VERB
ejpam-4298	94	5	coc	coc	PROPN
ejpam-4298	94	6	-	-	PUNCT
ejpam-4298	94	7	d	d	NOUN
ejpam-4298	94	8	-	-	PUNCT
ejpam-4298	94	9	sets	set	NOUN
ejpam-4298	94	10	u1	u1	NOUN
ejpam-4298	94	11	,	,	PUNCT
ejpam-4298	94	12	u2	u2	NOUN
ejpam-4298	94	13	with	with	ADP
ejpam-4298	94	14	x	x	PART
ejpam-4298	94	15	∈	∈	PROPN
ejpam-4298	94	16	u1	u1	NOUN
ejpam-4298	94	17	,	,	PUNCT
ejpam-4298	94	18	y	y	PROPN
ejpam-4298	94	19	/∈	/∈	PUNCT
ejpam-4298	94	20	u1	u1	NOUN
ejpam-4298	94	21	and	and	CCONJ
ejpam-4298	94	22	y	y	PROPN
ejpam-4298	94	23	∈	∈	PROPN
ejpam-4298	94	24	u2	u2	PROPN
ejpam-4298	94	25	,	,	PUNCT
ejpam-4298	94	26	x	x	NOUN
ejpam-4298	94	27	/∈	/∈	SYM
ejpam-4298	94	28	u1	u1	NOUN
ejpam-4298	94	29	,	,	PUNCT
ejpam-4298	94	30	assume	assume	VERB
ejpam-4298	94	31	u1	u1	NOUN
ejpam-4298	94	32	=	=	SYM
ejpam-4298	94	33	v1−w1	v1−w1	NOUN
ejpam-4298	94	34	,	,	PUNCT
ejpam-4298	94	35	u2	u2	NOUN
ejpam-4298	94	36	=	=	PUNCT
ejpam-4298	94	37	v2−w2	v2−w2	NUM
ejpam-4298	94	38	where	where	SCONJ
ejpam-4298	94	39	v1,w1	v1,w1	PROPN
ejpam-4298	94	40	,	,	PUNCT
ejpam-4298	94	41	v2,w2	v2,w2	PROPN
ejpam-4298	94	42	∈	∈	PROPN
ejpam-4298	94	43	τk	τk	ADP
ejpam-4298	94	44	.	.	PUNCT
ejpam-4298	95	1	then	then	ADV
ejpam-4298	95	2	for	for	ADP
ejpam-4298	95	3	x	x	PROPN
ejpam-4298	95	4	/∈	/∈	PROPN
ejpam-4298	95	5	u2	u2	PROPN
ejpam-4298	95	6	,	,	PUNCT
ejpam-4298	95	7	we	we	PRON
ejpam-4298	95	8	have	have	VERB
ejpam-4298	95	9	the	the	DET
ejpam-4298	95	10	following	follow	VERB
ejpam-4298	95	11	cases	case	NOUN
ejpam-4298	95	12	:	:	PUNCT
ejpam-4298	95	13	(	(	PUNCT
ejpam-4298	95	14	1	1	X
ejpam-4298	95	15	)	)	PUNCT
ejpam-4298	95	16	x	x	SYM
ejpam-4298	95	17	/∈	/∈	PUNCT
ejpam-4298	96	1	v2	v2	NOUN
ejpam-4298	96	2	(	(	PUNCT
ejpam-4298	96	3	2	2	NUM
ejpam-4298	96	4	)	)	PUNCT
ejpam-4298	96	5	x	x	SYM
ejpam-4298	96	6	∈	∈	PROPN
ejpam-4298	96	7	v2	v2	NOUN
ejpam-4298	96	8	and	and	CCONJ
ejpam-4298	96	9	x	x	PROPN
ejpam-4298	96	10	∈	∈	PROPN
ejpam-4298	96	11	w2	w2	NOUN
ejpam-4298	96	12	.	.	PUNCT
ejpam-4298	97	1	for	for	ADP
ejpam-4298	97	2	(	(	PUNCT
ejpam-4298	97	3	1	1	X
ejpam-4298	97	4	)	)	PUNCT
ejpam-4298	97	5	if	if	SCONJ
ejpam-4298	97	6	x	x	NOUN
ejpam-4298	97	7	/∈	/∈	PUNCT
ejpam-4298	98	1	v2	v2	PROPN
ejpam-4298	98	2	,	,	PUNCT
ejpam-4298	98	3	we	we	PRON
ejpam-4298	98	4	have	have	VERB
ejpam-4298	98	5	:	:	PUNCT
ejpam-4298	98	6	(	(	PUNCT
ejpam-4298	98	7	i	i	NOUN
ejpam-4298	98	8	)	)	PUNCT
ejpam-4298	98	9	if	if	SCONJ
ejpam-4298	98	10	y	y	PROPN
ejpam-4298	98	11	/∈	/∈	PUNCT
ejpam-4298	98	12	v1	v1	PROPN
ejpam-4298	98	13	,	,	PUNCT
ejpam-4298	98	14	x	x	SYM
ejpam-4298	98	15	∈	∈	NOUN
ejpam-4298	98	16	v1	v1	NOUN
ejpam-4298	98	17	−	−	PROPN
ejpam-4298	98	18	w1	w1	NOUN
ejpam-4298	98	19	,	,	PUNCT
ejpam-4298	98	20	then	then	ADV
ejpam-4298	98	21	x	x	PART
ejpam-4298	98	22	∈	∈	PROPN
ejpam-4298	98	23	v1	v1	NOUN
ejpam-4298	98	24	−	−	PROPN
ejpam-4298	99	1	(	(	PUNCT
ejpam-4298	99	2	v2	v2	PROPN
ejpam-4298	99	3	∪	∪	NOUN
ejpam-4298	99	4	w1	w1	NOUN
ejpam-4298	99	5	)	)	PUNCT
ejpam-4298	99	6	and	and	CCONJ
ejpam-4298	99	7	y	y	PROPN
ejpam-4298	99	8	∈	∈	PROPN
ejpam-4298	99	9	v2	v2	PROPN
ejpam-4298	99	10	−	−	PROPN
ejpam-4298	99	11	w2	w2	NOUN
ejpam-4298	99	12	,	,	PUNCT
ejpam-4298	99	13	so	so	ADV
ejpam-4298	99	14	y	y	PROPN
ejpam-4298	99	15	∈	∈	PROPN
ejpam-4298	99	16	v2	v2	PROPN
ejpam-4298	99	17	−	−	PROPN
ejpam-4298	99	18	(	(	PUNCT
ejpam-4298	99	19	v1	v1	VERB
ejpam-4298	99	20	∪	∪	ADJ
ejpam-4298	99	21	w2	w2	NOUN
ejpam-4298	99	22	)	)	PUNCT
ejpam-4298	99	23	and	and	CCONJ
ejpam-4298	99	24	f.a	f.a	PROPN
ejpam-4298	99	25	.	.	PROPN
ejpam-4298	99	26	abushaheen	abushaheen	PROPN
ejpam-4298	99	27	/	/	SYM
ejpam-4298	99	28	eur	eur	PROPN
ejpam-4298	99	29	.	.	PUNCT
ejpam-4298	100	1	j.	j.	PROPN
ejpam-4298	100	2	pure	pure	PROPN
ejpam-4298	100	3	appl	appl	PROPN
ejpam-4298	100	4	.	.	PROPN
ejpam-4298	100	5	math	math	PROPN
ejpam-4298	100	6	,	,	PUNCT
ejpam-4298	100	7	15	15	NUM
ejpam-4298	100	8	(	(	PUNCT
ejpam-4298	100	9	2	2	NUM
ejpam-4298	100	10	)	)	PUNCT
ejpam-4298	100	11	(	(	PUNCT
ejpam-4298	100	12	2022	2022	NUM
ejpam-4298	100	13	)	)	PUNCT
ejpam-4298	100	14	,	,	PUNCT
ejpam-4298	100	15	589	589	NUM
ejpam-4298	100	16	-	-	SYM
ejpam-4298	100	17	601	601	NUM
ejpam-4298	100	18	592	592	NUM
ejpam-4298	100	19	(	(	PUNCT
ejpam-4298	100	20	v1−(v2∪w1	v1−(v2∪w1	NOUN
ejpam-4298	100	21	)	)	PUNCT
ejpam-4298	100	22	)	)	PUNCT
ejpam-4298	101	1	∩	∩	NOUN
ejpam-4298	101	2	(	(	PUNCT
ejpam-4298	101	3	v2−(v1∪w2	v2−(v1∪w2	NOUN
ejpam-4298	101	4	)	)	PUNCT
ejpam-4298	101	5	)	)	PUNCT
ejpam-4298	102	1	=	=	SYM
ejpam-4298	102	2	φ	φ	PROPN
ejpam-4298	102	3	.	.	PUNCT
ejpam-4298	102	4	(	(	PUNCT
ejpam-4298	102	5	ii	ii	NOUN
ejpam-4298	102	6	)	)	PUNCT
ejpam-4298	102	7	if	if	SCONJ
ejpam-4298	102	8	y	y	PROPN
ejpam-4298	102	9	∈	∈	PROPN
ejpam-4298	102	10	v1	v1	PROPN
ejpam-4298	102	11	and	and	CCONJ
ejpam-4298	102	12	y	y	PROPN
ejpam-4298	102	13	∈	∈	PROPN
ejpam-4298	102	14	w1	w1	NOUN
ejpam-4298	102	15	,	,	PUNCT
ejpam-4298	102	16	we	we	PRON
ejpam-4298	102	17	have	have	VERB
ejpam-4298	102	18	x	x	PROPN
ejpam-4298	102	19	∈	∈	PROPN
ejpam-4298	102	20	u1−u2	u1−u2	PROPN
ejpam-4298	102	21	,	,	PUNCT
ejpam-4298	102	22	y	y	PROPN
ejpam-4298	102	23	∈	∈	PROPN
ejpam-4298	102	24	u2	u2	PROPN
ejpam-4298	102	25	and	and	CCONJ
ejpam-4298	102	26	(	(	PUNCT
ejpam-4298	102	27	u1	u1	NOUN
ejpam-4298	102	28	−	−	PROPN
ejpam-4298	102	29	u2	u2	PROPN
ejpam-4298	102	30	)	)	PUNCT
ejpam-4298	102	31	∩	∩	ADJ
ejpam-4298	102	32	u2	u2	PROPN
ejpam-4298	102	33	=	=	SYM
ejpam-4298	102	34	φ	φ	PROPN
ejpam-4298	102	35	.	.	PUNCT
ejpam-4298	103	1	for	for	ADP
ejpam-4298	103	2	(	(	PUNCT
ejpam-4298	103	3	2	2	X
ejpam-4298	103	4	)	)	PUNCT
ejpam-4298	103	5	if	if	SCONJ
ejpam-4298	103	6	y	y	PROPN
ejpam-4298	103	7	∈	∈	PROPN
ejpam-4298	103	8	u2	u2	PROPN
ejpam-4298	103	9	=	=	PROPN
ejpam-4298	103	10	v2	v2	PROPN
ejpam-4298	103	11	−w2	−w2	PROPN
ejpam-4298	103	12	,	,	PUNCT
ejpam-4298	103	13	then	then	ADV
ejpam-4298	103	14	x	x	PROPN
ejpam-4298	103	15	∈	∈	PROPN
ejpam-4298	103	16	w2	w2	NOUN
ejpam-4298	103	17	and	and	CCONJ
ejpam-4298	103	18	(	(	PUNCT
ejpam-4298	103	19	v2	v2	PROPN
ejpam-4298	103	20	−w2	−w2	PROPN
ejpam-4298	103	21	)	)	PUNCT
ejpam-4298	103	22	∩w2	∩w2	PUNCT
ejpam-4298	103	23	=	=	SYM
ejpam-4298	103	24	φ	φ	PROPN
ejpam-4298	103	25	.	.	PUNCT
ejpam-4298	104	1	from	from	ADP
ejpam-4298	104	2	(	(	PUNCT
ejpam-4298	104	3	1	1	NUM
ejpam-4298	104	4	)	)	PUNCT
ejpam-4298	104	5	and	and	CCONJ
ejpam-4298	104	6	(	(	PUNCT
ejpam-4298	104	7	2	2	NUM
ejpam-4298	104	8	)	)	PUNCT
ejpam-4298	104	9	,	,	PUNCT
ejpam-4298	104	10	x	x	X
ejpam-4298	104	11	is	be	AUX
ejpam-4298	104	12	coc	coc	ADJ
ejpam-4298	104	13	-	-	PUNCT
ejpam-4298	104	14	d2	d2	NOUN
ejpam-4298	104	15	-	-	PUNCT
ejpam-4298	104	16	space	space	NOUN
ejpam-4298	104	17	.	.	PUNCT
ejpam-4298	105	1	the	the	DET
ejpam-4298	105	2	following	follow	VERB
ejpam-4298	105	3	theorem	theorem	NOUN
ejpam-4298	105	4	gives	give	VERB
ejpam-4298	105	5	improvement	improvement	NOUN
ejpam-4298	105	6	of	of	ADP
ejpam-4298	105	7	theorem	theorem	ADJ
ejpam-4298	105	8	7(iv	7(iv	NUM
ejpam-4298	105	9	)	)	PUNCT
ejpam-4298	105	10	.	.	PUNCT
ejpam-4298	106	1	theorem	theorem	VERB
ejpam-4298	106	2	8	8	NUM
ejpam-4298	106	3	.	.	PUNCT
ejpam-4298	107	1	a	a	DET
ejpam-4298	107	2	space	space	NOUN
ejpam-4298	107	3	(	(	PUNCT
ejpam-4298	107	4	x	x	X
ejpam-4298	107	5	,	,	PUNCT
ejpam-4298	107	6	τ	τ	X
ejpam-4298	107	7	)	)	PUNCT
ejpam-4298	107	8	is	be	AUX
ejpam-4298	107	9	coc	coc	ADJ
ejpam-4298	107	10	-	-	PUNCT
ejpam-4298	107	11	d1	d1	NOUN
ejpam-4298	107	12	-	-	PUNCT
ejpam-4298	107	13	space	space	NOUN
ejpam-4298	107	14	if	if	SCONJ
ejpam-4298	107	15	and	and	CCONJ
ejpam-4298	107	16	only	only	ADV
ejpam-4298	107	17	if	if	SCONJ
ejpam-4298	107	18	x	x	PRON
ejpam-4298	107	19	is	be	AUX
ejpam-4298	107	20	coc	coc	NOUN
ejpam-4298	107	21	-	-	PUNCT
ejpam-4298	107	22	t0	t0	NOUN
ejpam-4298	107	23	-	-	PUNCT
ejpam-4298	107	24	space	space	NOUN
ejpam-4298	107	25	and	and	CCONJ
ejpam-4298	107	26	intcoc(ax	intcoc(ax	NOUN
ejpam-4298	107	27	)	)	PUNCT
ejpam-4298	107	28	6=	6=	ADP
ejpam-4298	107	29	x	x	PUNCT
ejpam-4298	107	30	for	for	ADP
ejpam-4298	107	31	all	all	PRON
ejpam-4298	107	32	x	x	SYM
ejpam-4298	107	33	∈	∈	NOUN
ejpam-4298	107	34	ax	ax	NOUN
ejpam-4298	107	35	⊆	⊆	NUM
ejpam-4298	107	36	x.	x.	NOUN
ejpam-4298	107	37	proof	proof	NOUN
ejpam-4298	107	38	.	.	PUNCT
ejpam-4298	108	1	(	(	PUNCT
ejpam-4298	108	2	⇒	⇒	NOUN
ejpam-4298	108	3	)	)	PUNCT
ejpam-4298	108	4	for	for	ADP
ejpam-4298	108	5	x	x	PROPN
ejpam-4298	108	6	∈	∈	PROPN
ejpam-4298	108	7	x	x	NOUN
ejpam-4298	108	8	,	,	PUNCT
ejpam-4298	108	9	there	there	PRON
ejpam-4298	108	10	exists	exist	VERB
ejpam-4298	108	11	a	a	DET
ejpam-4298	108	12	coc	coc	PROPN
ejpam-4298	108	13	-	-	PUNCT
ejpam-4298	108	14	d	d	NOUN
ejpam-4298	108	15	-	-	PUNCT
ejpam-4298	108	16	set	set	VERB
ejpam-4298	108	17	ox	ox	NOUN
ejpam-4298	108	18	=	=	PUNCT
ejpam-4298	108	19	u	u	PROPN
ejpam-4298	108	20	−	−	PROPN
ejpam-4298	108	21	v	v	NOUN
ejpam-4298	108	22	with	with	ADP
ejpam-4298	108	23	u	u	NOUN
ejpam-4298	108	24	,	,	PUNCT
ejpam-4298	108	25	v	v	PROPN
ejpam-4298	108	26	∈	∈	NOUN
ejpam-4298	108	27	τk	τk	ADP
ejpam-4298	108	28	and	and	CCONJ
ejpam-4298	108	29	x	x	PROPN
ejpam-4298	108	30	∈	∈	PROPN
ejpam-4298	108	31	ox	ox	NOUN
ejpam-4298	108	32	,	,	PUNCT
ejpam-4298	108	33	but	but	CCONJ
ejpam-4298	108	34	u	u	PROPN
ejpam-4298	108	35	6=	6=	PROPN
ejpam-4298	108	36	x	x	PROPN
ejpam-4298	108	37	,	,	PUNCT
ejpam-4298	108	38	so	so	ADV
ejpam-4298	108	39	intcoc(u	intcoc(u	NOUN
ejpam-4298	108	40	)	)	PUNCT
ejpam-4298	108	41	6=	6=	PUNCT
ejpam-4298	109	1	x	x	SYM
ejpam-4298	109	2	,	,	PUNCT
ejpam-4298	109	3	hence	hence	ADV
ejpam-4298	109	4	the	the	DET
ejpam-4298	109	5	result	result	NOUN
ejpam-4298	109	6	.	.	PUNCT
ejpam-4298	110	1	(	(	PUNCT
ejpam-4298	110	2	⇐	⇐	NOUN
ejpam-4298	110	3	)	)	PUNCT
ejpam-4298	110	4	for	for	ADP
ejpam-4298	110	5	x	x	SYM
ejpam-4298	110	6	6=	6=	ADP
ejpam-4298	110	7	y	y	PROPN
ejpam-4298	110	8	∈	∈	PROPN
ejpam-4298	110	9	x	x	X
ejpam-4298	110	10	,	,	PUNCT
ejpam-4298	110	11	with	with	ADP
ejpam-4298	110	12	out	out	ADP
ejpam-4298	110	13	loss	loss	NOUN
ejpam-4298	110	14	of	of	ADP
ejpam-4298	110	15	generality	generality	NOUN
ejpam-4298	110	16	there	there	PRON
ejpam-4298	110	17	exists	exist	VERB
ejpam-4298	110	18	a	a	DET
ejpam-4298	110	19	coc	coc	NOUN
ejpam-4298	110	20	-	-	PUNCT
ejpam-4298	110	21	open	open	ADJ
ejpam-4298	110	22	set	set	NOUN
ejpam-4298	110	23	u	u	NOUN
ejpam-4298	110	24	contains	contain	VERB
ejpam-4298	110	25	x	x	PUNCT
ejpam-4298	110	26	but	but	CCONJ
ejpam-4298	110	27	not	not	PART
ejpam-4298	110	28	y	y	PROPN
ejpam-4298	110	29	and	and	CCONJ
ejpam-4298	110	30	there	there	PRON
ejpam-4298	110	31	exists	exist	VERB
ejpam-4298	110	32	coc	coc	ADJ
ejpam-4298	110	33	-	-	PUNCT
ejpam-4298	110	34	open	open	ADJ
ejpam-4298	110	35	set	set	VERB
ejpam-4298	110	36	v	v	NOUN
ejpam-4298	110	37	contains	contain	VERB
ejpam-4298	110	38	y	y	PROPN
ejpam-4298	110	39	and	and	CCONJ
ejpam-4298	110	40	intcoc(v	intcoc(v	NOUN
ejpam-4298	110	41	)	)	PUNCT
ejpam-4298	110	42	6=	6=	PUNCT
ejpam-4298	111	1	x	x	SYM
ejpam-4298	111	2	,	,	PUNCT
ejpam-4298	111	3	hence	hence	ADV
ejpam-4298	111	4	y	y	PROPN
ejpam-4298	111	5	∈	∈	PROPN
ejpam-4298	111	6	v	v	ADP
ejpam-4298	111	7	−u	−u	PROPN
ejpam-4298	111	8	,	,	PUNCT
ejpam-4298	111	9	therefore	therefore	ADV
ejpam-4298	111	10	x	x	X
ejpam-4298	111	11	is	be	AUX
ejpam-4298	111	12	coc	coc	ADJ
ejpam-4298	111	13	-	-	PUNCT
ejpam-4298	111	14	d1	d1	NOUN
ejpam-4298	111	15	-	-	PUNCT
ejpam-4298	111	16	space	space	NOUN
ejpam-4298	111	17	.	.	PUNCT
ejpam-4298	112	1	3	3	X
ejpam-4298	112	2	.	.	X
ejpam-4298	112	3	coc	coc	ADJ
ejpam-4298	112	4	-	-	PUNCT
ejpam-4298	112	5	r0	r0	NOUN
ejpam-4298	112	6	and	and	CCONJ
ejpam-4298	112	7	coc	coc	NOUN
ejpam-4298	112	8	-	-	PUNCT
ejpam-4298	112	9	r1	r1	NOUN
ejpam-4298	112	10	-	-	PUNCT
ejpam-4298	112	11	spaces	space	NOUN
ejpam-4298	112	12	definition	definition	NOUN
ejpam-4298	112	13	9	9	NUM
ejpam-4298	112	14	.	.	PUNCT
ejpam-4298	113	1	a	a	DET
ejpam-4298	113	2	space	space	NOUN
ejpam-4298	113	3	(	(	PUNCT
ejpam-4298	113	4	x	x	X
ejpam-4298	113	5	,	,	PUNCT
ejpam-4298	113	6	τ	τ	X
ejpam-4298	113	7	)	)	PUNCT
ejpam-4298	113	8	is	be	AUX
ejpam-4298	113	9	called	call	VERB
ejpam-4298	113	10	co	co	ADJ
ejpam-4298	113	11	-	-	ADJ
ejpam-4298	113	12	compact	compact	ADJ
ejpam-4298	113	13	-	-	PUNCT
ejpam-4298	113	14	r0	r0	NOUN
ejpam-4298	113	15	-	-	PUNCT
ejpam-4298	113	16	space	space	NOUN
ejpam-4298	113	17	(	(	PUNCT
ejpam-4298	113	18	coc	coc	NOUN
ejpam-4298	113	19	-	-	PUNCT
ejpam-4298	113	20	r0	r0	NOUN
ejpam-4298	113	21	-	-	PUNCT
ejpam-4298	113	22	space	space	NOUN
ejpam-4298	113	23	)	)	PUNCT
ejpam-4298	113	24	if	if	SCONJ
ejpam-4298	113	25	every	every	DET
ejpam-4298	113	26	cocopen	cocopen	NOUN
ejpam-4298	113	27	set	set	NOUN
ejpam-4298	113	28	contains	contain	VERB
ejpam-4298	113	29	the	the	DET
ejpam-4298	113	30	coc	coc	NOUN
ejpam-4298	113	31	-	-	PUNCT
ejpam-4298	113	32	closure	closure	NOUN
ejpam-4298	113	33	of	of	ADP
ejpam-4298	113	34	its	its	PRON
ejpam-4298	113	35	singletons	singleton	NOUN
ejpam-4298	113	36	,	,	PUNCT
ejpam-4298	113	37	i.e.	i.e.	X
ejpam-4298	113	38	for	for	ADP
ejpam-4298	113	39	each	each	PRON
ejpam-4298	113	40	coc	coc	NOUN
ejpam-4298	113	41	-	-	PUNCT
ejpam-4298	113	42	open	open	ADJ
ejpam-4298	113	43	set	set	NOUN
ejpam-4298	113	44	o	o	NOUN
ejpam-4298	113	45	we	we	PRON
ejpam-4298	113	46	have	have	VERB
ejpam-4298	113	47	{	{	PUNCT
ejpam-4298	113	48	x}coc	x}coc	PROPN
ejpam-4298	113	49	⊆	⊆	NUM
ejpam-4298	113	50	o	o	NOUN
ejpam-4298	113	51	for	for	ADP
ejpam-4298	113	52	all	all	DET
ejpam-4298	113	53	x	x	SYM
ejpam-4298	113	54	∈	∈	PROPN
ejpam-4298	113	55	o.	o.	NOUN
ejpam-4298	113	56	definition	definition	NOUN
ejpam-4298	113	57	10	10	NUM
ejpam-4298	113	58	.	.	PUNCT
ejpam-4298	114	1	a	a	DET
ejpam-4298	114	2	space	space	NOUN
ejpam-4298	114	3	(	(	PUNCT
ejpam-4298	114	4	x	x	X
ejpam-4298	114	5	,	,	PUNCT
ejpam-4298	114	6	τ	τ	X
ejpam-4298	114	7	)	)	PUNCT
ejpam-4298	114	8	is	be	AUX
ejpam-4298	114	9	called	call	VERB
ejpam-4298	114	10	co	co	ADJ
ejpam-4298	114	11	-	-	ADJ
ejpam-4298	114	12	compact	compact	ADJ
ejpam-4298	114	13	-	-	PUNCT
ejpam-4298	114	14	r1	r1	NOUN
ejpam-4298	114	15	-	-	PUNCT
ejpam-4298	114	16	space	space	NOUN
ejpam-4298	114	17	(	(	PUNCT
ejpam-4298	114	18	coc	coc	NOUN
ejpam-4298	114	19	-	-	PUNCT
ejpam-4298	114	20	r1	r1	NOUN
ejpam-4298	114	21	-	-	PUNCT
ejpam-4298	114	22	space	space	NOUN
ejpam-4298	114	23	)	)	PUNCT
ejpam-4298	114	24	if	if	SCONJ
ejpam-4298	114	25	for	for	ADP
ejpam-4298	114	26	x	x	SYM
ejpam-4298	114	27	6=	6=	ADP
ejpam-4298	114	28	y	y	PROPN
ejpam-4298	114	29	∈	∈	PROPN
ejpam-4298	114	30	x	x	PUNCT
ejpam-4298	114	31	with	with	ADP
ejpam-4298	114	32	{	{	PUNCT
ejpam-4298	114	33	x}coc	x}coc	PROPN
ejpam-4298	114	34	6=	6=	PROPN
ejpam-4298	114	35	{	{	PUNCT
ejpam-4298	114	36	y}coc	y}coc	NOUN
ejpam-4298	114	37	,	,	PUNCT
ejpam-4298	114	38	then	then	ADV
ejpam-4298	114	39	there	there	PRON
ejpam-4298	114	40	exist	exist	VERB
ejpam-4298	114	41	disjoint	disjoint	ADJ
ejpam-4298	114	42	coc	coc	ADJ
ejpam-4298	114	43	-	-	PUNCT
ejpam-4298	114	44	open	open	ADJ
ejpam-4298	114	45	sets	set	NOUN
ejpam-4298	114	46	u	u	NOUN
ejpam-4298	114	47	,	,	PUNCT
ejpam-4298	114	48	v	v	NOUN
ejpam-4298	114	49	with	with	ADP
ejpam-4298	114	50	{	{	PUNCT
ejpam-4298	114	51	x}coc	x}coc	PROPN
ejpam-4298	114	52	⊆	⊆	NUM
ejpam-4298	114	53	u	u	NOUN
ejpam-4298	114	54	,	,	PUNCT
ejpam-4298	114	55	{	{	PUNCT
ejpam-4298	114	56	y}coc	y}coc	PROPN
ejpam-4298	114	57	⊆	⊆	NUM
ejpam-4298	114	58	v	v	NOUN
ejpam-4298	114	59	.	.	PUNCT
ejpam-4298	115	1	the	the	DET
ejpam-4298	115	2	following	follow	VERB
ejpam-4298	115	3	theorem	theorem	NOUN
ejpam-4298	115	4	is	be	AUX
ejpam-4298	115	5	obvious	obvious	ADJ
ejpam-4298	115	6	.	.	PUNCT
ejpam-4298	116	1	theorem	theorem	NOUN
ejpam-4298	116	2	9	9	NUM
ejpam-4298	116	3	.	.	PUNCT
ejpam-4298	117	1	let	let	AUX
ejpam-4298	117	2	(	(	PUNCT
ejpam-4298	117	3	x	x	NOUN
ejpam-4298	117	4	,	,	PUNCT
ejpam-4298	117	5	τ	τ	X
ejpam-4298	117	6	)	)	PUNCT
ejpam-4298	117	7	be	be	VERB
ejpam-4298	117	8	a	a	DET
ejpam-4298	117	9	topological	topological	ADJ
ejpam-4298	117	10	space	space	NOUN
ejpam-4298	117	11	.	.	PUNCT
ejpam-4298	118	1	then	then	ADV
ejpam-4298	118	2	:	:	PUNCT
ejpam-4298	118	3	(	(	PUNCT
ejpam-4298	118	4	i	i	NOUN
ejpam-4298	118	5	)	)	PUNCT
ejpam-4298	118	6	a	a	DET
ejpam-4298	118	7	coc	coc	NOUN
ejpam-4298	118	8	-	-	PUNCT
ejpam-4298	118	9	closed	closed	ADJ
ejpam-4298	118	10	subspace	subspace	NOUN
ejpam-4298	118	11	of	of	ADP
ejpam-4298	118	12	a	a	DET
ejpam-4298	118	13	coc	coc	ADJ
ejpam-4298	118	14	-	-	PUNCT
ejpam-4298	118	15	r0	r0	NOUN
ejpam-4298	118	16	-	-	PUNCT
ejpam-4298	118	17	space	space	NOUN
ejpam-4298	118	18	x	x	PUNCT
ejpam-4298	118	19	is	be	AUX
ejpam-4298	118	20	coc	coc	ADJ
ejpam-4298	118	21	-	-	PUNCT
ejpam-4298	118	22	r0	r0	NOUN
ejpam-4298	118	23	-	-	PUNCT
ejpam-4298	118	24	space	space	NOUN
ejpam-4298	118	25	.	.	PUNCT
ejpam-4298	119	1	(	(	PUNCT
ejpam-4298	119	2	ii	ii	NOUN
ejpam-4298	119	3	)	)	PUNCT
ejpam-4298	119	4	a	a	DET
ejpam-4298	119	5	coc	coc	NOUN
ejpam-4298	119	6	-	-	PUNCT
ejpam-4298	119	7	closed	closed	ADJ
ejpam-4298	119	8	subspace	subspace	NOUN
ejpam-4298	119	9	of	of	ADP
ejpam-4298	119	10	a	a	DET
ejpam-4298	119	11	coc	coc	PROPN
ejpam-4298	119	12	-	-	PUNCT
ejpam-4298	119	13	r1	r1	NOUN
ejpam-4298	119	14	-	-	PUNCT
ejpam-4298	119	15	space	space	NOUN
ejpam-4298	119	16	x	x	PUNCT
ejpam-4298	119	17	is	be	AUX
ejpam-4298	119	18	coc	coc	ADJ
ejpam-4298	119	19	-	-	PUNCT
ejpam-4298	119	20	r1	r1	NOUN
ejpam-4298	119	21	-	-	PUNCT
ejpam-4298	119	22	space	space	NOUN
ejpam-4298	119	23	.	.	PUNCT
ejpam-4298	120	1	theorem	theorem	ADJ
ejpam-4298	120	2	10	10	NUM
ejpam-4298	120	3	.	.	PUNCT
ejpam-4298	121	1	every	every	DET
ejpam-4298	121	2	coc	coc	PROPN
ejpam-4298	121	3	-	-	PUNCT
ejpam-4298	121	4	r1	r1	NOUN
ejpam-4298	121	5	-	-	PUNCT
ejpam-4298	121	6	space	space	NOUN
ejpam-4298	121	7	(	(	PUNCT
ejpam-4298	121	8	x	x	X
ejpam-4298	121	9	,	,	PUNCT
ejpam-4298	121	10	τ	τ	X
ejpam-4298	121	11	)	)	PUNCT
ejpam-4298	121	12	is	be	AUX
ejpam-4298	121	13	coc	coc	ADJ
ejpam-4298	121	14	-	-	PUNCT
ejpam-4298	121	15	r0	r0	NOUN
ejpam-4298	121	16	-	-	PUNCT
ejpam-4298	121	17	space	space	NOUN
ejpam-4298	121	18	.	.	PUNCT
ejpam-4298	122	1	proof	proof	NOUN
ejpam-4298	122	2	.	.	PUNCT
ejpam-4298	123	1	let	let	VERB
ejpam-4298	123	2	u	u	PRON
ejpam-4298	123	3	be	be	AUX
ejpam-4298	123	4	a	a	DET
ejpam-4298	123	5	coc	coc	NOUN
ejpam-4298	123	6	-	-	PUNCT
ejpam-4298	123	7	open	open	ADJ
ejpam-4298	123	8	set	set	NOUN
ejpam-4298	123	9	in	in	ADP
ejpam-4298	123	10	x	x	PUNCT
ejpam-4298	123	11	with	with	ADP
ejpam-4298	123	12	x	x	PROPN
ejpam-4298	123	13	∈	∈	PROPN
ejpam-4298	123	14	u	u	NOUN
ejpam-4298	123	15	.	.	PUNCT
ejpam-4298	124	1	for	for	ADP
ejpam-4298	124	2	y	y	PROPN
ejpam-4298	124	3	/∈	/∈	PUNCT
ejpam-4298	124	4	u	u	PROPN
ejpam-4298	124	5	,	,	PUNCT
ejpam-4298	124	6	we	we	PRON
ejpam-4298	124	7	have	have	VERB
ejpam-4298	124	8	x	x	X
ejpam-4298	124	9	/∈	/∈	PUNCT
ejpam-4298	124	10	{	{	PUNCT
ejpam-4298	124	11	y}coc	y}coc	ADJ
ejpam-4298	124	12	,	,	PUNCT
ejpam-4298	124	13	thus	thus	ADV
ejpam-4298	124	14	{	{	PUNCT
ejpam-4298	124	15	x}coc	x}coc	PROPN
ejpam-4298	124	16	6=	6=	PROPN
ejpam-4298	124	17	{	{	PUNCT
ejpam-4298	124	18	y}coc	y}coc	ADJ
ejpam-4298	124	19	,	,	PUNCT
ejpam-4298	124	20	but	but	CCONJ
ejpam-4298	124	21	x	x	X
ejpam-4298	124	22	is	be	AUX
ejpam-4298	124	23	coc	coc	ADJ
ejpam-4298	124	24	-	-	PUNCT
ejpam-4298	124	25	r1	r1	NOUN
ejpam-4298	124	26	-	-	PUNCT
ejpam-4298	124	27	space	space	NOUN
ejpam-4298	124	28	,	,	PUNCT
ejpam-4298	124	29	so	so	SCONJ
ejpam-4298	124	30	there	there	PRON
ejpam-4298	124	31	exits	exit	VERB
ejpam-4298	124	32	a	a	DET
ejpam-4298	124	33	coc	coc	NOUN
ejpam-4298	124	34	-	-	PUNCT
ejpam-4298	124	35	open	open	ADJ
ejpam-4298	124	36	set	set	NOUN
ejpam-4298	124	37	vy	vy	NOUN
ejpam-4298	124	38	contains	contain	VERB
ejpam-4298	124	39	y	y	PRON
ejpam-4298	124	40	such	such	ADJ
ejpam-4298	124	41	that	that	SCONJ
ejpam-4298	124	42	{	{	PUNCT
ejpam-4298	124	43	y}coc	y}coc	ADJ
ejpam-4298	124	44	⊆	⊆	NUM
ejpam-4298	124	45	vy	vy	NOUN
ejpam-4298	124	46	and	and	CCONJ
ejpam-4298	124	47	x	x	PROPN
ejpam-4298	124	48	/∈	/∈	PUNCT
ejpam-4298	125	1	vy	vy	NOUN
ejpam-4298	125	2	,	,	PUNCT
ejpam-4298	125	3	hence	hence	ADV
ejpam-4298	125	4	{	{	PUNCT
ejpam-4298	125	5	x}coc	x}coc	PROPN
ejpam-4298	125	6	⊆	⊆	NUM
ejpam-4298	125	7	u	u	NOUN
ejpam-4298	125	8	,	,	PUNCT
ejpam-4298	125	9	thus	thus	ADV
ejpam-4298	125	10	x	x	X
ejpam-4298	125	11	is	be	AUX
ejpam-4298	125	12	coc	coc	ADJ
ejpam-4298	125	13	-	-	PUNCT
ejpam-4298	125	14	r0	r0	NOUN
ejpam-4298	125	15	-	-	PUNCT
ejpam-4298	125	16	space	space	NOUN
ejpam-4298	125	17	.	.	PUNCT
ejpam-4298	126	1	theorem	theorem	VERB
ejpam-4298	126	2	11	11	NUM
ejpam-4298	126	3	.	.	PUNCT
ejpam-4298	127	1	a	a	DET
ejpam-4298	127	2	space	space	NOUN
ejpam-4298	127	3	(	(	PUNCT
ejpam-4298	127	4	x	x	X
ejpam-4298	127	5	,	,	PUNCT
ejpam-4298	127	6	τ	τ	X
ejpam-4298	127	7	)	)	PUNCT
ejpam-4298	127	8	is	be	AUX
ejpam-4298	127	9	coc	coc	NOUN
ejpam-4298	127	10	-	-	PUNCT
ejpam-4298	127	11	t1	t1	NOUN
ejpam-4298	127	12	-	-	PUNCT
ejpam-4298	127	13	space	space	NOUN
ejpam-4298	127	14	if	if	SCONJ
ejpam-4298	127	15	and	and	CCONJ
ejpam-4298	127	16	only	only	ADV
ejpam-4298	127	17	if	if	SCONJ
ejpam-4298	127	18	it	it	PRON
ejpam-4298	127	19	is	be	AUX
ejpam-4298	127	20	coc	coc	NOUN
ejpam-4298	127	21	-	-	PUNCT
ejpam-4298	127	22	t0	t0	NOUN
ejpam-4298	127	23	-	-	PUNCT
ejpam-4298	127	24	space	space	NOUN
ejpam-4298	127	25	and	and	CCONJ
ejpam-4298	127	26	coc	coc	NOUN
ejpam-4298	127	27	-	-	PUNCT
ejpam-4298	127	28	r0space	r0space	NOUN
ejpam-4298	127	29	.	.	PUNCT
ejpam-4298	128	1	proof	proof	NOUN
ejpam-4298	128	2	.	.	PUNCT
ejpam-4298	129	1	(	(	PUNCT
ejpam-4298	129	2	⇒	⇒	NOUN
ejpam-4298	129	3	)	)	PUNCT
ejpam-4298	129	4	notes	note	VERB
ejpam-4298	129	5	that	that	SCONJ
ejpam-4298	129	6	{	{	PUNCT
ejpam-4298	129	7	x	x	X
ejpam-4298	129	8	}	}	PUNCT
ejpam-4298	129	9	is	be	AUX
ejpam-4298	129	10	coc	coc	ADJ
ejpam-4298	129	11	-	-	PUNCT
ejpam-4298	129	12	closed	closed	ADJ
ejpam-4298	129	13	subset	subset	NOUN
ejpam-4298	129	14	of	of	ADP
ejpam-4298	129	15	x	x	PRON
ejpam-4298	129	16	for	for	ADP
ejpam-4298	129	17	all	all	PRON
ejpam-4298	129	18	x	x	SYM
ejpam-4298	129	19	∈	∈	ADJ
ejpam-4298	129	20	x.	x.	NOUN
ejpam-4298	129	21	(	(	PUNCT
ejpam-4298	129	22	⇐	⇐	ADJ
ejpam-4298	129	23	)	)	PUNCT
ejpam-4298	129	24	let	let	VERB
ejpam-4298	129	25	x	x	X
ejpam-4298	129	26	6=	6=	ADP
ejpam-4298	129	27	y	y	PROPN
ejpam-4298	129	28	∈	∈	PROPN
ejpam-4298	129	29	x	x	X
ejpam-4298	129	30	,	,	PUNCT
ejpam-4298	129	31	with	with	ADP
ejpam-4298	129	32	out	out	ADP
ejpam-4298	129	33	loss	loss	NOUN
ejpam-4298	129	34	of	of	ADP
ejpam-4298	129	35	generality	generality	NOUN
ejpam-4298	129	36	there	there	PRON
ejpam-4298	129	37	exists	exist	VERB
ejpam-4298	129	38	a	a	DET
ejpam-4298	129	39	coc	coc	NOUN
ejpam-4298	129	40	-	-	PUNCT
ejpam-4298	129	41	open	open	NOUN
ejpam-4298	129	42	set	set	NOUN
ejpam-4298	129	43	o	o	NOUN
ejpam-4298	129	44	with	with	ADP
ejpam-4298	129	45	x	x	PROPN
ejpam-4298	129	46	∈	∈	NOUN
ejpam-4298	129	47	o	o	NOUN
ejpam-4298	130	1	⊆	⊆	NUM
ejpam-4298	130	2	x	x	SYM
ejpam-4298	130	3	−	−	PROPN
ejpam-4298	130	4	{	{	PUNCT
ejpam-4298	130	5	y	y	NOUN
ejpam-4298	130	6	}	}	PUNCT
ejpam-4298	130	7	.	.	PUNCT
ejpam-4298	131	1	thus	thus	ADV
ejpam-4298	131	2	x	x	X
ejpam-4298	131	3	/∈	/∈	PUNCT
ejpam-4298	131	4	{	{	PUNCT
ejpam-4298	131	5	y}coc	y}coc	ADJ
ejpam-4298	131	6	,	,	PUNCT
ejpam-4298	131	7	so	so	SCONJ
ejpam-4298	131	8	y	y	PROPN
ejpam-4298	131	9	/∈	/∈	PUNCT
ejpam-4298	131	10	{	{	PUNCT
ejpam-4298	132	1	x}coc	x}coc	PROPN
ejpam-4298	132	2	,	,	PUNCT
ejpam-4298	132	3	hence	hence	ADV
ejpam-4298	132	4	x	x	X
ejpam-4298	132	5	−	−	PROPN
ejpam-4298	132	6	{	{	PUNCT
ejpam-4298	132	7	x}coc	x}coc	PROPN
ejpam-4298	132	8	is	be	AUX
ejpam-4298	132	9	coc	coc	ADJ
ejpam-4298	132	10	-	-	PUNCT
ejpam-4298	132	11	open	open	ADJ
ejpam-4298	132	12	set	set	NOUN
ejpam-4298	132	13	contains	contain	VERB
ejpam-4298	132	14	y	y	PROPN
ejpam-4298	132	15	but	but	CCONJ
ejpam-4298	132	16	not	not	PART
ejpam-4298	132	17	x.	x.	NOUN
ejpam-4298	132	18	f.a	f.a	PROPN
ejpam-4298	132	19	.	.	PROPN
ejpam-4298	132	20	abushaheen	abushaheen	PROPN
ejpam-4298	132	21	/	/	SYM
ejpam-4298	132	22	eur	eur	PROPN
ejpam-4298	132	23	.	.	PUNCT
ejpam-4298	133	1	j.	j.	PROPN
ejpam-4298	133	2	pure	pure	PROPN
ejpam-4298	133	3	appl	appl	PROPN
ejpam-4298	133	4	.	.	PROPN
ejpam-4298	133	5	math	math	PROPN
ejpam-4298	133	6	,	,	PUNCT
ejpam-4298	133	7	15	15	NUM
ejpam-4298	133	8	(	(	PUNCT
ejpam-4298	133	9	2	2	NUM
ejpam-4298	133	10	)	)	PUNCT
ejpam-4298	133	11	(	(	PUNCT
ejpam-4298	133	12	2022	2022	NUM
ejpam-4298	133	13	)	)	PUNCT
ejpam-4298	133	14	,	,	PUNCT
ejpam-4298	133	15	589	589	NUM
ejpam-4298	133	16	-	-	SYM
ejpam-4298	133	17	601	601	NUM
ejpam-4298	133	18	593	593	NUM
ejpam-4298	133	19	corollary	corollary	ADJ
ejpam-4298	133	20	1	1	NUM
ejpam-4298	133	21	.	.	PUNCT
ejpam-4298	134	1	let	let	AUX
ejpam-4298	134	2	(	(	PUNCT
ejpam-4298	134	3	x	x	NOUN
ejpam-4298	134	4	,	,	PUNCT
ejpam-4298	134	5	τ	τ	X
ejpam-4298	134	6	)	)	PUNCT
ejpam-4298	134	7	be	be	VERB
ejpam-4298	134	8	a	a	DET
ejpam-4298	134	9	coc	coc	ADJ
ejpam-4298	134	10	-	-	PUNCT
ejpam-4298	134	11	r0	r0	NOUN
ejpam-4298	134	12	-	-	PUNCT
ejpam-4298	134	13	space	space	NOUN
ejpam-4298	134	14	.	.	PUNCT
ejpam-4298	135	1	then	then	ADV
ejpam-4298	135	2	the	the	DET
ejpam-4298	135	3	following	follow	VERB
ejpam-4298	135	4	are	be	AUX
ejpam-4298	135	5	equivalent	equivalent	ADJ
ejpam-4298	135	6	:	:	PUNCT
ejpam-4298	135	7	(	(	PUNCT
ejpam-4298	135	8	i	i	NOUN
ejpam-4298	135	9	)	)	PUNCT
ejpam-4298	135	10	x	x	X
ejpam-4298	135	11	is	be	AUX
ejpam-4298	135	12	coc	coc	ADJ
ejpam-4298	135	13	-	-	PUNCT
ejpam-4298	135	14	t2	t2	NOUN
ejpam-4298	135	15	-	-	PUNCT
ejpam-4298	135	16	space	space	NOUN
ejpam-4298	135	17	,	,	PUNCT
ejpam-4298	135	18	(	(	PUNCT
ejpam-4298	135	19	ii	ii	NOUN
ejpam-4298	135	20	)	)	PUNCT
ejpam-4298	135	21	x	x	X
ejpam-4298	135	22	is	be	AUX
ejpam-4298	135	23	coc	coc	NOUN
ejpam-4298	135	24	-	-	PUNCT
ejpam-4298	135	25	t1	t1	NOUN
ejpam-4298	135	26	-	-	PUNCT
ejpam-4298	135	27	space	space	NOUN
ejpam-4298	135	28	,	,	PUNCT
ejpam-4298	135	29	(	(	PUNCT
ejpam-4298	135	30	iii	iii	X
ejpam-4298	135	31	)	)	PUNCT
ejpam-4298	135	32	x	x	X
ejpam-4298	135	33	is	be	AUX
ejpam-4298	135	34	coc	coc	NOUN
ejpam-4298	135	35	-	-	PUNCT
ejpam-4298	135	36	t0	t0	NOUN
ejpam-4298	135	37	-	-	NOUN
ejpam-4298	135	38	space	space	NOUN
ejpam-4298	135	39	.	.	PUNCT
ejpam-4298	136	1	definition	definition	NOUN
ejpam-4298	136	2	11	11	NUM
ejpam-4298	136	3	.	.	PUNCT
ejpam-4298	137	1	let	let	VERB
ejpam-4298	137	2	(	(	PUNCT
ejpam-4298	137	3	x	x	NOUN
ejpam-4298	137	4	,	,	PUNCT
ejpam-4298	137	5	τ	τ	X
ejpam-4298	137	6	)	)	PUNCT
ejpam-4298	137	7	be	be	VERB
ejpam-4298	137	8	a	a	DET
ejpam-4298	137	9	topological	topological	ADJ
ejpam-4298	137	10	space	space	NOUN
ejpam-4298	137	11	and	and	CCONJ
ejpam-4298	137	12	a	a	DET
ejpam-4298	137	13	⊆	⊆	NUM
ejpam-4298	137	14	x.	x.	NOUN
ejpam-4298	137	15	then	then	ADV
ejpam-4298	137	16	the	the	DET
ejpam-4298	137	17	coc	coc	NOUN
ejpam-4298	137	18	-	-	PUNCT
ejpam-4298	137	19	kernal	kernal	NOUN
ejpam-4298	137	20	of	of	ADP
ejpam-4298	137	21	a	a	DET
ejpam-4298	137	22	define	define	NOUN
ejpam-4298	137	23	by	by	ADP
ejpam-4298	137	24	:	:	PUNCT
ejpam-4298	137	25	coc	coc	PROPN
ejpam-4298	137	26	-	-	PUNCT
ejpam-4298	137	27	ker(a	ker(a	PROPN
ejpam-4298	137	28	)	)	PUNCT
ejpam-4298	137	29	=	=	SYM
ejpam-4298	138	1	∩{u	∩{u	PROPN
ejpam-4298	138	2	∈	∈	PROPN
ejpam-4298	138	3	τk	τk	ADP
ejpam-4298	138	4	:	:	PUNCT
ejpam-4298	138	5	a	a	DET
ejpam-4298	138	6	⊆	⊆	NUM
ejpam-4298	138	7	u	u	NOUN
ejpam-4298	138	8	}	}	PUNCT
ejpam-4298	138	9	,	,	PUNCT
ejpam-4298	138	10	if	if	SCONJ
ejpam-4298	138	11	there	there	PRON
ejpam-4298	138	12	no	no	DET
ejpam-4298	138	13	coc	coc	NOUN
ejpam-4298	138	14	-	-	PUNCT
ejpam-4298	138	15	open	open	ADJ
ejpam-4298	138	16	set	set	NOUN
ejpam-4298	138	17	contains	contain	VERB
ejpam-4298	138	18	a	a	DET
ejpam-4298	138	19	,	,	PUNCT
ejpam-4298	138	20	then	then	ADV
ejpam-4298	138	21	coc	coc	NOUN
ejpam-4298	138	22	-	-	PUNCT
ejpam-4298	138	23	ker(a	ker(a	PROPN
ejpam-4298	138	24	)	)	PUNCT
ejpam-4298	138	25	=	=	PUNCT
ejpam-4298	139	1	x.	x.	NOUN
ejpam-4298	139	2	lemma	lemma	PROPN
ejpam-4298	140	1	2	2	X
ejpam-4298	140	2	.	.	PUNCT
ejpam-4298	141	1	if	if	SCONJ
ejpam-4298	141	2	(	(	PUNCT
ejpam-4298	141	3	x	x	NOUN
ejpam-4298	141	4	,	,	PUNCT
ejpam-4298	141	5	τ	τ	X
ejpam-4298	141	6	)	)	PUNCT
ejpam-4298	141	7	is	be	AUX
ejpam-4298	141	8	a	a	DET
ejpam-4298	141	9	topological	topological	ADJ
ejpam-4298	141	10	space	space	NOUN
ejpam-4298	141	11	and	and	CCONJ
ejpam-4298	141	12	a	a	PRON
ejpam-4298	141	13	is	be	AUX
ejpam-4298	141	14	a	a	DET
ejpam-4298	141	15	subset	subset	NOUN
ejpam-4298	141	16	of	of	ADP
ejpam-4298	141	17	x	x	X
ejpam-4298	141	18	,	,	PUNCT
ejpam-4298	141	19	then	then	ADV
ejpam-4298	141	20	coc	coc	NOUN
ejpam-4298	141	21	-	-	PUNCT
ejpam-4298	141	22	ker(a	ker(a	PROPN
ejpam-4298	141	23	)	)	PUNCT
ejpam-4298	141	24	=	=	PRON
ejpam-4298	142	1	{	{	PUNCT
ejpam-4298	142	2	x	x	PUNCT
ejpam-4298	142	3	∈	∈	PROPN
ejpam-4298	142	4	x	x	X
ejpam-4298	142	5	:	:	PUNCT
ejpam-4298	142	6	{	{	PUNCT
ejpam-4298	142	7	x}coc	x}coc	PROPN
ejpam-4298	142	8	∩a	∩a	PROPN
ejpam-4298	142	9	6=	6=	PROPN
ejpam-4298	142	10	φ	φ	NUM
ejpam-4298	142	11	}	}	PUNCT
ejpam-4298	142	12	.	.	PUNCT
ejpam-4298	143	1	proof	proof	NOUN
ejpam-4298	143	2	.	.	PUNCT
ejpam-4298	144	1	for	for	ADP
ejpam-4298	144	2	x	x	PROPN
ejpam-4298	144	3	/∈	/∈	SYM
ejpam-4298	144	4	coc	coc	PROPN
ejpam-4298	144	5	-	-	PUNCT
ejpam-4298	144	6	ker(a	ker(a	PROPN
ejpam-4298	144	7	)	)	PUNCT
ejpam-4298	144	8	,	,	PUNCT
ejpam-4298	144	9	there	there	PRON
ejpam-4298	144	10	exists	exist	VERB
ejpam-4298	144	11	a	a	DET
ejpam-4298	144	12	coc	coc	NOUN
ejpam-4298	144	13	-	-	PUNCT
ejpam-4298	144	14	open	open	ADJ
ejpam-4298	144	15	set	set	NOUN
ejpam-4298	144	16	u	u	NOUN
ejpam-4298	144	17	contains	contain	VERB
ejpam-4298	144	18	a	a	PRON
ejpam-4298	144	19	and	and	CCONJ
ejpam-4298	144	20	x	x	SYM
ejpam-4298	144	21	/∈	/∈	PUNCT
ejpam-4298	144	22	u	u	NOUN
ejpam-4298	144	23	,	,	PUNCT
ejpam-4298	144	24	then	then	ADV
ejpam-4298	144	25	{	{	PUNCT
ejpam-4298	144	26	x}coc	x}coc	PROPN
ejpam-4298	144	27	∩	∩	PROPN
ejpam-4298	144	28	u	u	PROPN
ejpam-4298	144	29	=	=	PROPN
ejpam-4298	144	30	φ	φ	PROPN
ejpam-4298	144	31	.	.	PUNCT
ejpam-4298	145	1	for	for	ADP
ejpam-4298	145	2	{	{	PUNCT
ejpam-4298	145	3	x}coc	x}coc	PROPN
ejpam-4298	145	4	∩	∩	PROPN
ejpam-4298	145	5	u	u	PROPN
ejpam-4298	145	6	=	=	SYM
ejpam-4298	145	7	φ	φ	PROPN
ejpam-4298	145	8	,	,	PUNCT
ejpam-4298	145	9	we	we	PRON
ejpam-4298	145	10	have	have	VERB
ejpam-4298	145	11	x	x	X
ejpam-4298	145	12	/∈	/∈	PUNCT
ejpam-4298	146	1	x	x	SYM
ejpam-4298	146	2	−	−	PROPN
ejpam-4298	146	3	{	{	PUNCT
ejpam-4298	146	4	x}coc	x}coc	PROPN
ejpam-4298	146	5	,	,	PUNCT
ejpam-4298	146	6	thus	thus	ADV
ejpam-4298	146	7	x	x	X
ejpam-4298	146	8	/∈	/∈	SYM
ejpam-4298	146	9	coc	coc	PROPN
ejpam-4298	146	10	-	-	PUNCT
ejpam-4298	146	11	ker(a	ker(a	NOUN
ejpam-4298	146	12	)	)	PUNCT
ejpam-4298	146	13	.	.	PUNCT
ejpam-4298	147	1	lemma	lemma	PROPN
ejpam-4298	147	2	3	3	X
ejpam-4298	147	3	.	.	PUNCT
ejpam-4298	148	1	let	let	AUX
ejpam-4298	148	2	(	(	PUNCT
ejpam-4298	148	3	x	x	NOUN
ejpam-4298	148	4	,	,	PUNCT
ejpam-4298	148	5	τ	τ	X
ejpam-4298	148	6	)	)	PUNCT
ejpam-4298	148	7	be	be	VERB
ejpam-4298	148	8	a	a	DET
ejpam-4298	148	9	topological	topological	ADJ
ejpam-4298	148	10	space	space	NOUN
ejpam-4298	148	11	and	and	CCONJ
ejpam-4298	148	12	x	x	PUNCT
ejpam-4298	148	13	∈	∈	PROPN
ejpam-4298	148	14	x.	x.	NOUN
ejpam-4298	148	15	then	then	ADV
ejpam-4298	148	16	y	y	PROPN
ejpam-4298	148	17	∈	∈	PROPN
ejpam-4298	148	18	coc	coc	PROPN
ejpam-4298	148	19	-	-	PUNCT
ejpam-4298	148	20	ker({x	ker({x	NOUN
ejpam-4298	148	21	}	}	PUNCT
ejpam-4298	148	22	)	)	PUNCT
ejpam-4298	149	1	if	if	SCONJ
ejpam-4298	149	2	and	and	CCONJ
ejpam-4298	149	3	only	only	ADV
ejpam-4298	149	4	if	if	SCONJ
ejpam-4298	149	5	x	x	SYM
ejpam-4298	149	6	∈	∈	PROPN
ejpam-4298	149	7	{	{	PUNCT
ejpam-4298	149	8	y}coc	y}coc	PROPN
ejpam-4298	149	9	.	.	PUNCT
ejpam-4298	149	10	theorem	theorem	VERB
ejpam-4298	149	11	12	12	NUM
ejpam-4298	149	12	.	.	PUNCT
ejpam-4298	150	1	let	let	VERB
ejpam-4298	150	2	(	(	PUNCT
ejpam-4298	150	3	x	x	NOUN
ejpam-4298	150	4	,	,	PUNCT
ejpam-4298	150	5	τ	τ	X
ejpam-4298	150	6	)	)	PUNCT
ejpam-4298	150	7	be	be	VERB
ejpam-4298	150	8	a	a	DET
ejpam-4298	150	9	topological	topological	ADJ
ejpam-4298	150	10	space	space	NOUN
ejpam-4298	150	11	and	and	CCONJ
ejpam-4298	150	12	x	x	SYM
ejpam-4298	150	13	6=	6=	NUM
ejpam-4298	150	14	y	y	PROPN
ejpam-4298	150	15	∈	∈	PROPN
ejpam-4298	150	16	x.	x.	NOUN
ejpam-4298	150	17	then	then	ADV
ejpam-4298	150	18	coc	coc	NOUN
ejpam-4298	150	19	-	-	PUNCT
ejpam-4298	150	20	ker({x	ker({x	NOUN
ejpam-4298	150	21	}	}	PUNCT
ejpam-4298	150	22	)	)	PUNCT
ejpam-4298	151	1	6=	6=	X
ejpam-4298	151	2	coc	coc	PROPN
ejpam-4298	151	3	-	-	PUNCT
ejpam-4298	151	4	ker({y	ker({y	PROPN
ejpam-4298	151	5	}	}	PUNCT
ejpam-4298	151	6	)	)	PUNCT
ejpam-4298	152	1	if	if	SCONJ
ejpam-4298	152	2	and	and	CCONJ
ejpam-4298	152	3	only	only	ADV
ejpam-4298	152	4	if	if	SCONJ
ejpam-4298	152	5	{	{	PUNCT
ejpam-4298	152	6	x}coc	x}coc	PROPN
ejpam-4298	152	7	6=	6=	PROPN
ejpam-4298	152	8	{	{	PUNCT
ejpam-4298	152	9	y}coc	y}coc	ADJ
ejpam-4298	152	10	.	.	PUNCT
ejpam-4298	152	11	proof	proof	NOUN
ejpam-4298	152	12	.	.	PUNCT
ejpam-4298	153	1	(	(	PUNCT
ejpam-4298	153	2	⇒	⇒	PROPN
ejpam-4298	153	3	)	)	PUNCT
ejpam-4298	153	4	let	let	VERB
ejpam-4298	153	5	w	w	PROPN
ejpam-4298	153	6	∈	∈	PROPN
ejpam-4298	153	7	coc	coc	PROPN
ejpam-4298	153	8	-	-	PUNCT
ejpam-4298	153	9	ker({x	ker({x	NOUN
ejpam-4298	153	10	}	}	PUNCT
ejpam-4298	153	11	)	)	PUNCT
ejpam-4298	154	1	and	and	CCONJ
ejpam-4298	154	2	w	w	PROPN
ejpam-4298	154	3	/∈	/∈	INTJ
ejpam-4298	154	4	coc	coc	PROPN
ejpam-4298	154	5	-	-	PUNCT
ejpam-4298	154	6	ker({y	ker({y	PROPN
ejpam-4298	154	7	}	}	PUNCT
ejpam-4298	154	8	)	)	PUNCT
ejpam-4298	154	9	.	.	PUNCT
ejpam-4298	155	1	then	then	ADV
ejpam-4298	155	2	{	{	PUNCT
ejpam-4298	155	3	w}coc	w}coc	ADJ
ejpam-4298	155	4	∩	∩	ADJ
ejpam-4298	155	5	{	{	PUNCT
ejpam-4298	155	6	x	x	X
ejpam-4298	155	7	}	}	PUNCT
ejpam-4298	155	8	6=	6=	NUM
ejpam-4298	155	9	φ	φ	PROPN
ejpam-4298	155	10	and	and	CCONJ
ejpam-4298	155	11	{	{	PUNCT
ejpam-4298	155	12	w}coc	w}coc	ADJ
ejpam-4298	155	13	∩	∩	ADJ
ejpam-4298	155	14	{	{	PUNCT
ejpam-4298	155	15	y	y	NOUN
ejpam-4298	155	16	}	}	PUNCT
ejpam-4298	155	17	=	=	SYM
ejpam-4298	155	18	φ	φ	NUM
ejpam-4298	155	19	,	,	PUNCT
ejpam-4298	155	20	so	so	ADV
ejpam-4298	155	21	x	x	SYM
ejpam-4298	155	22	∈	∈	PROPN
ejpam-4298	155	23	{	{	PUNCT
ejpam-4298	155	24	w}coc	w}coc	ADJ
ejpam-4298	155	25	,	,	PUNCT
ejpam-4298	155	26	and	and	CCONJ
ejpam-4298	155	27	hence	hence	ADV
ejpam-4298	155	28	{	{	PUNCT
ejpam-4298	155	29	x}coc	x}coc	PROPN
ejpam-4298	155	30	⊆	⊆	NUM
ejpam-4298	155	31	{	{	PUNCT
ejpam-4298	155	32	w}coc	w}coc	ADJ
ejpam-4298	155	33	,	,	PUNCT
ejpam-4298	155	34	therefore	therefore	ADV
ejpam-4298	155	35	{	{	PUNCT
ejpam-4298	155	36	w}coc	w}coc	ADJ
ejpam-4298	155	37	∩	∩	ADJ
ejpam-4298	155	38	{	{	PUNCT
ejpam-4298	155	39	y	y	NOUN
ejpam-4298	155	40	}	}	PUNCT
ejpam-4298	155	41	=	=	SYM
ejpam-4298	155	42	φ	φ	PROPN
ejpam-4298	155	43	and	and	CCONJ
ejpam-4298	155	44	hence	hence	ADV
ejpam-4298	155	45	y	y	PROPN
ejpam-4298	155	46	/∈	/∈	PUNCT
ejpam-4298	155	47	{	{	PUNCT
ejpam-4298	155	48	x}coc	x}coc	PROPN
ejpam-4298	155	49	.	.	PUNCT
ejpam-4298	156	1	(	(	PUNCT
ejpam-4298	156	2	⇐	⇐	NOUN
ejpam-4298	156	3	)	)	PUNCT
ejpam-4298	156	4	since	since	SCONJ
ejpam-4298	156	5	coc	coc	NOUN
ejpam-4298	156	6	-	-	PUNCT
ejpam-4298	156	7	ker({x	ker({x	NOUN
ejpam-4298	156	8	}	}	PUNCT
ejpam-4298	156	9	)	)	PUNCT
ejpam-4298	157	1	6=	6=	X
ejpam-4298	157	2	coc	coc	PROPN
ejpam-4298	157	3	-	-	PUNCT
ejpam-4298	157	4	ker({y	ker({y	PROPN
ejpam-4298	157	5	}	}	PUNCT
ejpam-4298	157	6	)	)	PUNCT
ejpam-4298	157	7	,	,	PUNCT
ejpam-4298	157	8	there	there	PRON
ejpam-4298	157	9	is	be	VERB
ejpam-4298	157	10	z	z	PROPN
ejpam-4298	157	11	∈	∈	PROPN
ejpam-4298	157	12	{	{	PUNCT
ejpam-4298	157	13	x}coc	x}coc	PROPN
ejpam-4298	157	14	and	and	CCONJ
ejpam-4298	157	15	z	z	PROPN
ejpam-4298	157	16	/∈	/∈	PUNCT
ejpam-4298	157	17	{	{	PUNCT
ejpam-4298	157	18	y}coc	y}coc	ADJ
ejpam-4298	157	19	,	,	PUNCT
ejpam-4298	157	20	hence	hence	ADV
ejpam-4298	157	21	there	there	PRON
ejpam-4298	157	22	exists	exist	VERB
ejpam-4298	157	23	a	a	DET
ejpam-4298	157	24	coc	coc	NOUN
ejpam-4298	157	25	-	-	PUNCT
ejpam-4298	157	26	open	open	ADJ
ejpam-4298	157	27	set	set	NOUN
ejpam-4298	157	28	uz	uz	NOUN
ejpam-4298	157	29	with	with	ADP
ejpam-4298	157	30	x	x	PROPN
ejpam-4298	157	31	∈	∈	PROPN
ejpam-4298	157	32	uz	uz	NOUN
ejpam-4298	157	33	and	and	CCONJ
ejpam-4298	157	34	y	y	PROPN
ejpam-4298	157	35	/∈	/∈	PUNCT
ejpam-4298	158	1	uz	uz	PROPN
ejpam-4298	158	2	,	,	PUNCT
ejpam-4298	158	3	so	so	ADV
ejpam-4298	158	4	y	y	PROPN
ejpam-4298	158	5	/∈	/∈	PUNCT
ejpam-4298	158	6	coc	coc	PROPN
ejpam-4298	158	7	-	-	PUNCT
ejpam-4298	158	8	ker({x	ker({x	NOUN
ejpam-4298	158	9	}	}	PUNCT
ejpam-4298	158	10	)	)	PUNCT
ejpam-4298	158	11	.	.	PUNCT
ejpam-4298	159	1	theorem	theorem	VERB
ejpam-4298	159	2	13	13	NUM
ejpam-4298	159	3	.	.	PUNCT
ejpam-4298	160	1	a	a	DET
ejpam-4298	160	2	space	space	NOUN
ejpam-4298	160	3	(	(	PUNCT
ejpam-4298	160	4	x	x	X
ejpam-4298	160	5	,	,	PUNCT
ejpam-4298	160	6	τ	τ	X
ejpam-4298	160	7	)	)	PUNCT
ejpam-4298	160	8	is	be	AUX
ejpam-4298	160	9	coc	coc	ADJ
ejpam-4298	160	10	-	-	PUNCT
ejpam-4298	160	11	r0	r0	NOUN
ejpam-4298	160	12	-	-	PUNCT
ejpam-4298	160	13	space	space	NOUN
ejpam-4298	160	14	if	if	SCONJ
ejpam-4298	160	15	and	and	CCONJ
ejpam-4298	160	16	only	only	ADV
ejpam-4298	160	17	if	if	SCONJ
ejpam-4298	160	18	for	for	ADP
ejpam-4298	160	19	x	x	SYM
ejpam-4298	160	20	6=	6=	ADP
ejpam-4298	160	21	y	y	PROPN
ejpam-4298	160	22	∈	∈	PROPN
ejpam-4298	160	23	x	x	X
ejpam-4298	160	24	,	,	PUNCT
ejpam-4298	160	25	{	{	PUNCT
ejpam-4298	160	26	x}coc	x}coc	PROPN
ejpam-4298	160	27	6=	6=	PROPN
ejpam-4298	160	28	{	{	PUNCT
ejpam-4298	160	29	y}coc	y}coc	PROPN
ejpam-4298	160	30	gives	give	VERB
ejpam-4298	160	31	{	{	PUNCT
ejpam-4298	160	32	x}coc	x}coc	PROPN
ejpam-4298	160	33	∩	∩	PROPN
ejpam-4298	160	34	{	{	PUNCT
ejpam-4298	160	35	y}coc	y}coc	PROPN
ejpam-4298	160	36	=	=	SYM
ejpam-4298	160	37	φ	φ	PROPN
ejpam-4298	160	38	.	.	PUNCT
ejpam-4298	161	1	proof	proof	NOUN
ejpam-4298	161	2	.	.	PUNCT
ejpam-4298	162	1	(	(	PUNCT
ejpam-4298	162	2	⇐	⇐	NOUN
ejpam-4298	162	3	)	)	PUNCT
ejpam-4298	162	4	let	let	VERB
ejpam-4298	162	5	x	x	SYM
ejpam-4298	162	6	∈	∈	PROPN
ejpam-4298	162	7	ox	ox	NOUN
ejpam-4298	162	8	∈	∈	PROPN
ejpam-4298	162	9	τk	τk	ADP
ejpam-4298	162	10	and	and	CCONJ
ejpam-4298	162	11	assume	assume	VERB
ejpam-4298	162	12	that	that	SCONJ
ejpam-4298	162	13	y	y	PROPN
ejpam-4298	162	14	/∈	/∈	PUNCT
ejpam-4298	162	15	ox	ox	PROPN
ejpam-4298	162	16	.	.	PUNCT
ejpam-4298	163	1	then	then	ADV
ejpam-4298	163	2	x	x	X
ejpam-4298	163	3	/∈	/∈	PUNCT
ejpam-4298	163	4	{	{	PUNCT
ejpam-4298	163	5	y}coc	y}coc	ADJ
ejpam-4298	163	6	,	,	PUNCT
ejpam-4298	163	7	hence	hence	ADV
ejpam-4298	163	8	{	{	PUNCT
ejpam-4298	163	9	x}coc	x}coc	PROPN
ejpam-4298	163	10	6=	6=	PROPN
ejpam-4298	163	11	{	{	PUNCT
ejpam-4298	163	12	y}coc	y}coc	ADJ
ejpam-4298	163	13	,	,	PUNCT
ejpam-4298	163	14	so	so	ADV
ejpam-4298	163	15	{	{	PUNCT
ejpam-4298	163	16	x}coc	x}coc	PROPN
ejpam-4298	163	17	∩	∩	PROPN
ejpam-4298	163	18	{	{	PUNCT
ejpam-4298	163	19	y}coc	y}coc	PROPN
ejpam-4298	163	20	=	=	SYM
ejpam-4298	163	21	φ	φ	PROPN
ejpam-4298	163	22	,	,	PUNCT
ejpam-4298	163	23	therefore	therefore	ADV
ejpam-4298	163	24	y	y	PROPN
ejpam-4298	163	25	/∈	/∈	PUNCT
ejpam-4298	163	26	{	{	PUNCT
ejpam-4298	163	27	x}coc	x}coc	PROPN
ejpam-4298	163	28	and	and	CCONJ
ejpam-4298	163	29	{	{	PUNCT
ejpam-4298	163	30	x}coc	x}coc	PROPN
ejpam-4298	163	31	⊆	⊆	NUM
ejpam-4298	163	32	ox	ox	NOUN
ejpam-4298	163	33	,	,	PUNCT
ejpam-4298	163	34	so	so	ADV
ejpam-4298	163	35	x	x	PUNCT
ejpam-4298	163	36	is	be	AUX
ejpam-4298	163	37	coc	coc	ADJ
ejpam-4298	163	38	-	-	PUNCT
ejpam-4298	163	39	r0	r0	NOUN
ejpam-4298	163	40	-	-	PUNCT
ejpam-4298	163	41	space	space	NOUN
ejpam-4298	163	42	.	.	PUNCT
ejpam-4298	164	1	(	(	PUNCT
ejpam-4298	164	2	⇒	⇒	PROPN
ejpam-4298	164	3	)	)	PUNCT
ejpam-4298	164	4	let	let	VERB
ejpam-4298	164	5	x	x	X
ejpam-4298	164	6	6=	6=	ADP
ejpam-4298	164	7	y	y	PROPN
ejpam-4298	164	8	∈	∈	PROPN
ejpam-4298	164	9	x	x	PUNCT
ejpam-4298	164	10	with	with	ADP
ejpam-4298	164	11	{	{	PUNCT
ejpam-4298	164	12	x}coc	x}coc	PROPN
ejpam-4298	164	13	6=	6=	PROPN
ejpam-4298	164	14	{	{	PUNCT
ejpam-4298	164	15	y}coc	y}coc	PROPN
ejpam-4298	164	16	.	.	PUNCT
ejpam-4298	165	1	so	so	ADV
ejpam-4298	165	2	there	there	PRON
ejpam-4298	165	3	exists	exist	VERB
ejpam-4298	165	4	z	z	PROPN
ejpam-4298	165	5	∈	∈	PROPN
ejpam-4298	165	6	{	{	PUNCT
ejpam-4298	165	7	x}coc	x}coc	PROPN
ejpam-4298	165	8	and	and	CCONJ
ejpam-4298	165	9	z	z	PROPN
ejpam-4298	165	10	/∈	/∈	PUNCT
ejpam-4298	165	11	{	{	PUNCT
ejpam-4298	165	12	y}coc	y}coc	ADJ
ejpam-4298	165	13	,	,	PUNCT
ejpam-4298	165	14	then	then	ADV
ejpam-4298	165	15	z	z	PROPN
ejpam-4298	165	16	∈	∈	PROPN
ejpam-4298	165	17	x−{y}coc	x−{y}coc	PROPN
ejpam-4298	165	18	,	,	PUNCT
ejpam-4298	165	19	so	so	SCONJ
ejpam-4298	165	20	there	there	PRON
ejpam-4298	165	21	exists	exist	VERB
ejpam-4298	165	22	a	a	DET
ejpam-4298	165	23	coc	coc	NOUN
ejpam-4298	165	24	-	-	PUNCT
ejpam-4298	165	25	open	open	ADJ
ejpam-4298	165	26	set	set	NOUN
ejpam-4298	165	27	u	u	NOUN
ejpam-4298	165	28	contains	contain	VERB
ejpam-4298	165	29	z	z	NOUN
ejpam-4298	165	30	but	but	CCONJ
ejpam-4298	165	31	not	not	PART
ejpam-4298	165	32	y	y	NOUN
ejpam-4298	165	33	,	,	PUNCT
ejpam-4298	165	34	but	but	CCONJ
ejpam-4298	165	35	z	z	NOUN
ejpam-4298	165	36	∈	∈	PROPN
ejpam-4298	165	37	{	{	PUNCT
ejpam-4298	165	38	x}coc	x}coc	PROPN
ejpam-4298	165	39	,	,	PUNCT
ejpam-4298	165	40	so	so	ADV
ejpam-4298	165	41	x	x	SYM
ejpam-4298	165	42	∈	∈	PROPN
ejpam-4298	165	43	u	u	NOUN
ejpam-4298	165	44	and	and	CCONJ
ejpam-4298	165	45	x	x	PROPN
ejpam-4298	165	46	/∈	/∈	PUNCT
ejpam-4298	165	47	{	{	PUNCT
ejpam-4298	165	48	y}coc	y}coc	ADJ
ejpam-4298	165	49	,	,	PUNCT
ejpam-4298	165	50	hence	hence	ADV
ejpam-4298	165	51	{	{	PUNCT
ejpam-4298	165	52	x}coc	x}coc	PROPN
ejpam-4298	165	53	⊆	⊆	NUM
ejpam-4298	165	54	x	x	SYM
ejpam-4298	165	55	−	−	PROPN
ejpam-4298	165	56	{	{	PUNCT
ejpam-4298	165	57	y}coc	y}coc	NOUN
ejpam-4298	165	58	,	,	PUNCT
ejpam-4298	165	59	therefore	therefore	ADV
ejpam-4298	165	60	{	{	PUNCT
ejpam-4298	165	61	x}coc	x}coc	PROPN
ejpam-4298	165	62	∩	∩	PROPN
ejpam-4298	165	63	{	{	PUNCT
ejpam-4298	165	64	y}coc	y}coc	PROPN
ejpam-4298	165	65	=	=	SYM
ejpam-4298	165	66	φ	φ	PROPN
ejpam-4298	165	67	.	.	PUNCT
ejpam-4298	165	68	theorem	theorem	VERB
ejpam-4298	165	69	14	14	NUM
ejpam-4298	165	70	.	.	PUNCT
ejpam-4298	166	1	a	a	DET
ejpam-4298	166	2	space	space	NOUN
ejpam-4298	166	3	(	(	PUNCT
ejpam-4298	166	4	x	x	X
ejpam-4298	166	5	,	,	PUNCT
ejpam-4298	166	6	τ	τ	X
ejpam-4298	166	7	)	)	PUNCT
ejpam-4298	166	8	is	be	AUX
ejpam-4298	166	9	coc	coc	ADJ
ejpam-4298	166	10	-	-	PUNCT
ejpam-4298	166	11	r0	r0	NOUN
ejpam-4298	166	12	-	-	PUNCT
ejpam-4298	166	13	space	space	NOUN
ejpam-4298	166	14	if	if	SCONJ
ejpam-4298	166	15	and	and	CCONJ
ejpam-4298	166	16	only	only	ADV
ejpam-4298	166	17	if	if	SCONJ
ejpam-4298	166	18	for	for	ADP
ejpam-4298	166	19	x	x	SYM
ejpam-4298	166	20	6=	6=	ADP
ejpam-4298	166	21	y	y	PROPN
ejpam-4298	166	22	∈	∈	PROPN
ejpam-4298	166	23	x	x	SYM
ejpam-4298	166	24	,	,	PUNCT
ejpam-4298	166	25	coc	coc	ADJ
ejpam-4298	166	26	-	-	PUNCT
ejpam-4298	166	27	ker({x	ker({x	NOUN
ejpam-4298	166	28	}	}	PUNCT
ejpam-4298	166	29	)	)	PUNCT
ejpam-4298	166	30	6=	6=	X
ejpam-4298	166	31	coc	coc	PROPN
ejpam-4298	166	32	-	-	PUNCT
ejpam-4298	166	33	ker({y	ker({y	PROPN
ejpam-4298	166	34	}	}	PUNCT
ejpam-4298	166	35	)	)	PUNCT
ejpam-4298	166	36	gives	give	VERB
ejpam-4298	166	37	coc	coc	NOUN
ejpam-4298	166	38	-	-	PUNCT
ejpam-4298	166	39	ker({x	ker({x	NOUN
ejpam-4298	166	40	}	}	PUNCT
ejpam-4298	166	41	)	)	PUNCT
ejpam-4298	166	42	∩	∩	PROPN
ejpam-4298	166	43	coc	coc	PROPN
ejpam-4298	166	44	-	-	PUNCT
ejpam-4298	166	45	ker({y	ker({y	PROPN
ejpam-4298	166	46	}	}	PUNCT
ejpam-4298	166	47	)	)	PUNCT
ejpam-4298	167	1	=	=	SYM
ejpam-4298	167	2	φ	φ	PROPN
ejpam-4298	167	3	.	.	PUNCT
ejpam-4298	168	1	f.a	f.a	PROPN
ejpam-4298	168	2	.	.	PROPN
ejpam-4298	168	3	abushaheen	abushaheen	PROPN
ejpam-4298	168	4	/	/	SYM
ejpam-4298	168	5	eur	eur	PROPN
ejpam-4298	168	6	.	.	PUNCT
ejpam-4298	169	1	j.	j.	PROPN
ejpam-4298	169	2	pure	pure	PROPN
ejpam-4298	169	3	appl	appl	PROPN
ejpam-4298	169	4	.	.	PROPN
ejpam-4298	169	5	math	math	PROPN
ejpam-4298	169	6	,	,	PUNCT
ejpam-4298	169	7	15	15	NUM
ejpam-4298	169	8	(	(	PUNCT
ejpam-4298	169	9	2	2	NUM
ejpam-4298	169	10	)	)	PUNCT
ejpam-4298	169	11	(	(	PUNCT
ejpam-4298	169	12	2022	2022	NUM
ejpam-4298	169	13	)	)	PUNCT
ejpam-4298	169	14	,	,	PUNCT
ejpam-4298	169	15	589	589	NUM
ejpam-4298	169	16	-	-	SYM
ejpam-4298	169	17	601	601	NUM
ejpam-4298	169	18	594	594	NUM
ejpam-4298	169	19	proof	proof	NOUN
ejpam-4298	169	20	.	.	PUNCT
ejpam-4298	170	1	(	(	PUNCT
ejpam-4298	170	2	⇒	⇒	NOUN
ejpam-4298	170	3	)	)	PUNCT
ejpam-4298	170	4	let	let	VERB
ejpam-4298	170	5	x	x	PRON
ejpam-4298	170	6	be	be	AUX
ejpam-4298	170	7	a	a	DET
ejpam-4298	170	8	coc	coc	ADJ
ejpam-4298	170	9	-	-	PUNCT
ejpam-4298	170	10	r0	r0	NOUN
ejpam-4298	170	11	-	-	PUNCT
ejpam-4298	170	12	space	space	NOUN
ejpam-4298	170	13	and	and	CCONJ
ejpam-4298	170	14	for	for	ADP
ejpam-4298	170	15	x	x	SYM
ejpam-4298	170	16	6=	6=	ADP
ejpam-4298	170	17	y	y	PROPN
ejpam-4298	170	18	∈	∈	PROPN
ejpam-4298	170	19	x	x	PUNCT
ejpam-4298	170	20	with	with	ADP
ejpam-4298	170	21	coc	coc	NOUN
ejpam-4298	170	22	-	-	PUNCT
ejpam-4298	170	23	ker({x	ker({x	NOUN
ejpam-4298	170	24	}	}	PUNCT
ejpam-4298	170	25	)	)	PUNCT
ejpam-4298	170	26	6=	6=	X
ejpam-4298	170	27	coc	coc	PROPN
ejpam-4298	170	28	-	-	PUNCT
ejpam-4298	170	29	ker({y	ker({y	PROPN
ejpam-4298	170	30	}	}	PUNCT
ejpam-4298	170	31	)	)	PUNCT
ejpam-4298	170	32	.	.	PUNCT
ejpam-4298	171	1	let	let	VERB
ejpam-4298	171	2	w	w	PROPN
ejpam-4298	171	3	∈	∈	PROPN
ejpam-4298	171	4	coc	coc	PROPN
ejpam-4298	171	5	-	-	PUNCT
ejpam-4298	171	6	ker({x})∩coc	ker({x})∩coc	PROPN
ejpam-4298	171	7	-	-	PUNCT
ejpam-4298	171	8	ker({y	ker({y	PROPN
ejpam-4298	171	9	}	}	PUNCT
ejpam-4298	171	10	)	)	PUNCT
ejpam-4298	171	11	.	.	PUNCT
ejpam-4298	172	1	then	then	ADV
ejpam-4298	172	2	w	w	PROPN
ejpam-4298	172	3	∈	∈	PROPN
ejpam-4298	172	4	coc	coc	PROPN
ejpam-4298	172	5	-	-	PUNCT
ejpam-4298	172	6	ker({x	ker({x	NOUN
ejpam-4298	172	7	}	}	PUNCT
ejpam-4298	172	8	)	)	PUNCT
ejpam-4298	173	1	so	so	ADV
ejpam-4298	173	2	x	x	PUNCT
ejpam-4298	173	3	∈	∈	PROPN
ejpam-4298	173	4	{	{	PUNCT
ejpam-4298	173	5	w}coc	w}coc	ADJ
ejpam-4298	173	6	,	,	PUNCT
ejpam-4298	173	7	and	and	CCONJ
ejpam-4298	173	8	then	then	ADV
ejpam-4298	173	9	by	by	ADP
ejpam-4298	173	10	lemma	lemma	PROPN
ejpam-4298	173	11	3	3	PROPN
ejpam-4298	173	12	{	{	PUNCT
ejpam-4298	173	13	x}coc	x}coc	PROPN
ejpam-4298	173	14	=	=	PROPN
ejpam-4298	173	15	{	{	PUNCT
ejpam-4298	173	16	w}coc	w}coc	ADJ
ejpam-4298	173	17	,	,	PUNCT
ejpam-4298	173	18	in	in	ADP
ejpam-4298	173	19	same	same	ADJ
ejpam-4298	173	20	method	method	NOUN
ejpam-4298	173	21	we	we	PRON
ejpam-4298	173	22	have	have	VERB
ejpam-4298	173	23	{	{	PUNCT
ejpam-4298	173	24	y}coc	y}coc	NOUN
ejpam-4298	173	25	=	=	SYM
ejpam-4298	173	26	{	{	PUNCT
ejpam-4298	173	27	w}coc	w}coc	ADJ
ejpam-4298	173	28	,	,	PUNCT
ejpam-4298	173	29	and	and	CCONJ
ejpam-4298	173	30	this	this	PRON
ejpam-4298	173	31	is	be	AUX
ejpam-4298	173	32	a	a	DET
ejpam-4298	173	33	contradiction	contradiction	NOUN
ejpam-4298	173	34	which	which	PRON
ejpam-4298	173	35	completes	complete	VERB
ejpam-4298	173	36	the	the	DET
ejpam-4298	173	37	proof	proof	NOUN
ejpam-4298	173	38	.	.	PUNCT
ejpam-4298	174	1	(	(	PUNCT
ejpam-4298	174	2	⇐	⇐	ADJ
ejpam-4298	174	3	)	)	PUNCT
ejpam-4298	174	4	assume	assume	VERB
ejpam-4298	174	5	{	{	PUNCT
ejpam-4298	174	6	x}coc	x}coc	PROPN
ejpam-4298	174	7	6=	6=	PROPN
ejpam-4298	174	8	{	{	PUNCT
ejpam-4298	174	9	y}coc	y}coc	ADJ
ejpam-4298	174	10	,	,	PUNCT
ejpam-4298	174	11	then	then	ADV
ejpam-4298	174	12	coc	coc	NOUN
ejpam-4298	174	13	-	-	PUNCT
ejpam-4298	174	14	ker({x	ker({x	NOUN
ejpam-4298	174	15	}	}	PUNCT
ejpam-4298	174	16	)	)	PUNCT
ejpam-4298	175	1	6=	6=	X
ejpam-4298	175	2	coc	coc	PROPN
ejpam-4298	175	3	-	-	PUNCT
ejpam-4298	175	4	ker({y	ker({y	PROPN
ejpam-4298	175	5	}	}	PUNCT
ejpam-4298	175	6	)	)	PUNCT
ejpam-4298	175	7	,	,	PUNCT
ejpam-4298	176	1	so	so	ADV
ejpam-4298	176	2	coc	coc	ADJ
ejpam-4298	176	3	-	-	PUNCT
ejpam-4298	176	4	ker({x	ker({x	NOUN
ejpam-4298	176	5	}	}	PUNCT
ejpam-4298	176	6	)	)	PUNCT
ejpam-4298	176	7	∩	∩	PROPN
ejpam-4298	176	8	coc	coc	PROPN
ejpam-4298	176	9	-	-	PUNCT
ejpam-4298	176	10	ker({y	ker({y	PROPN
ejpam-4298	176	11	}	}	PUNCT
ejpam-4298	176	12	)	)	PUNCT
ejpam-4298	177	1	=	=	SYM
ejpam-4298	177	2	φ	φ	X
ejpam-4298	177	3	.	.	PUNCT
ejpam-4298	178	1	if	if	SCONJ
ejpam-4298	178	2	z	z	PROPN
ejpam-4298	178	3	∈	∈	PROPN
ejpam-4298	178	4	{	{	PUNCT
ejpam-4298	178	5	x}coc	x}coc	PROPN
ejpam-4298	178	6	,	,	PUNCT
ejpam-4298	178	7	then	then	ADV
ejpam-4298	178	8	x	x	PROPN
ejpam-4298	178	9	∈	∈	PROPN
ejpam-4298	178	10	coc	coc	PROPN
ejpam-4298	178	11	-	-	PUNCT
ejpam-4298	178	12	ker({z	ker({z	PROPN
ejpam-4298	178	13	}	}	PUNCT
ejpam-4298	178	14	)	)	PUNCT
ejpam-4298	178	15	and	and	CCONJ
ejpam-4298	178	16	coc	coc	PROPN
ejpam-4298	178	17	-	-	PUNCT
ejpam-4298	178	18	ker({x})∩	ker({x})∩	NOUN
ejpam-4298	178	19	coc	coc	PROPN
ejpam-4298	178	20	-	-	PUNCT
ejpam-4298	178	21	ker({z	ker({z	PROPN
ejpam-4298	178	22	}	}	PUNCT
ejpam-4298	178	23	)	)	PUNCT
ejpam-4298	178	24	=	=	SYM
ejpam-4298	178	25	φ	φ	NUM
ejpam-4298	178	26	,	,	PUNCT
ejpam-4298	178	27	so	so	ADV
ejpam-4298	178	28	coc	coc	ADJ
ejpam-4298	178	29	-	-	PUNCT
ejpam-4298	178	30	ker({x	ker({x	NOUN
ejpam-4298	178	31	}	}	PUNCT
ejpam-4298	178	32	)	)	PUNCT
ejpam-4298	178	33	=	=	SYM
ejpam-4298	178	34	coc	coc	NOUN
ejpam-4298	178	35	-	-	PUNCT
ejpam-4298	178	36	ker({z	ker({z	PROPN
ejpam-4298	178	37	}	}	PUNCT
ejpam-4298	178	38	)	)	PUNCT
ejpam-4298	178	39	.	.	PUNCT
ejpam-4298	179	1	now	now	ADV
ejpam-4298	179	2	for	for	ADP
ejpam-4298	179	3	z	z	PROPN
ejpam-4298	179	4	∈	∈	PROPN
ejpam-4298	179	5	{	{	PUNCT
ejpam-4298	179	6	x}coc	x}coc	PROPN
ejpam-4298	179	7	∩	∩	PROPN
ejpam-4298	179	8	{	{	PUNCT
ejpam-4298	179	9	y}coc	y}coc	PROPN
ejpam-4298	179	10	,	,	PUNCT
ejpam-4298	179	11	we	we	PRON
ejpam-4298	179	12	have	have	VERB
ejpam-4298	179	13	coc	coc	ADJ
ejpam-4298	179	14	-	-	PUNCT
ejpam-4298	179	15	ker({x	ker({x	NOUN
ejpam-4298	179	16	}	}	PUNCT
ejpam-4298	179	17	)	)	PUNCT
ejpam-4298	180	1	=	=	SYM
ejpam-4298	180	2	coc	coc	PROPN
ejpam-4298	180	3	-	-	PUNCT
ejpam-4298	180	4	ker({y	ker({y	PROPN
ejpam-4298	180	5	}	}	PUNCT
ejpam-4298	180	6	)	)	PUNCT
ejpam-4298	181	1	=	=	SYM
ejpam-4298	181	2	coc	coc	NOUN
ejpam-4298	181	3	-	-	PUNCT
ejpam-4298	181	4	ker({z	ker({z	PROPN
ejpam-4298	181	5	}	}	PUNCT
ejpam-4298	181	6	)	)	PUNCT
ejpam-4298	181	7	,	,	PUNCT
ejpam-4298	181	8	and	and	CCONJ
ejpam-4298	181	9	this	this	PRON
ejpam-4298	181	10	is	be	AUX
ejpam-4298	181	11	a	a	DET
ejpam-4298	181	12	contradiction	contradiction	NOUN
ejpam-4298	181	13	,	,	PUNCT
ejpam-4298	181	14	hence	hence	ADV
ejpam-4298	181	15	{	{	PUNCT
ejpam-4298	181	16	x}coc	x}coc	PROPN
ejpam-4298	181	17	∩	∩	PROPN
ejpam-4298	181	18	{	{	PUNCT
ejpam-4298	181	19	y}coc	y}coc	PROPN
ejpam-4298	181	20	=	=	SYM
ejpam-4298	181	21	φ	φ	PROPN
ejpam-4298	181	22	.	.	PUNCT
ejpam-4298	181	23	theorem	theorem	VERB
ejpam-4298	181	24	15	15	NUM
ejpam-4298	181	25	.	.	PUNCT
ejpam-4298	182	1	for	for	ADP
ejpam-4298	182	2	a	a	DET
ejpam-4298	182	3	topological	topological	ADJ
ejpam-4298	182	4	space	space	NOUN
ejpam-4298	182	5	(	(	PUNCT
ejpam-4298	182	6	x	x	X
ejpam-4298	182	7	,	,	PUNCT
ejpam-4298	182	8	τ	τ	PROPN
ejpam-4298	182	9	)	)	PUNCT
ejpam-4298	182	10	.	.	PUNCT
ejpam-4298	183	1	the	the	DET
ejpam-4298	183	2	following	follow	VERB
ejpam-4298	183	3	are	be	AUX
ejpam-4298	183	4	equivalent	equivalent	ADJ
ejpam-4298	183	5	:	:	PUNCT
ejpam-4298	183	6	(	(	PUNCT
ejpam-4298	183	7	i	i	NOUN
ejpam-4298	183	8	)	)	PUNCT
ejpam-4298	183	9	x	x	X
ejpam-4298	183	10	is	be	AUX
ejpam-4298	183	11	a	a	DET
ejpam-4298	183	12	coc	coc	ADJ
ejpam-4298	183	13	-	-	PUNCT
ejpam-4298	183	14	r0	r0	NOUN
ejpam-4298	183	15	-	-	PUNCT
ejpam-4298	183	16	space	space	NOUN
ejpam-4298	183	17	,	,	PUNCT
ejpam-4298	183	18	(	(	PUNCT
ejpam-4298	183	19	ii	ii	NOUN
ejpam-4298	183	20	)	)	PUNCT
ejpam-4298	183	21	for	for	ADP
ejpam-4298	183	22	a	a	DET
ejpam-4298	183	23	subset	subset	NOUN
ejpam-4298	183	24	a	a	PRON
ejpam-4298	183	25	of	of	ADP
ejpam-4298	183	26	x	x	X
ejpam-4298	183	27	and	and	CCONJ
ejpam-4298	183	28	g	g	PROPN
ejpam-4298	183	29	coc	coc	NOUN
ejpam-4298	183	30	-	-	PUNCT
ejpam-4298	183	31	open	open	ADJ
ejpam-4298	183	32	set	set	NOUN
ejpam-4298	183	33	of	of	ADP
ejpam-4298	183	34	x	x	INTJ
ejpam-4298	183	35	such	such	ADJ
ejpam-4298	183	36	that	that	SCONJ
ejpam-4298	183	37	a	a	DET
ejpam-4298	183	38	∩	∩	NOUN
ejpam-4298	183	39	g	g	PROPN
ejpam-4298	183	40	6=	6=	PROPN
ejpam-4298	183	41	φ	φ	NUM
ejpam-4298	183	42	,	,	PUNCT
ejpam-4298	183	43	there	there	PRON
ejpam-4298	183	44	exists	exist	VERB
ejpam-4298	183	45	a	a	DET
ejpam-4298	183	46	coc	coc	NOUN
ejpam-4298	183	47	-	-	PUNCT
ejpam-4298	183	48	closed	close	VERB
ejpam-4298	183	49	subset	subset	NOUN
ejpam-4298	183	50	f	f	PROPN
ejpam-4298	183	51	of	of	ADP
ejpam-4298	183	52	x	x	INTJ
ejpam-4298	183	53	such	such	ADJ
ejpam-4298	183	54	that	that	SCONJ
ejpam-4298	183	55	a	a	DET
ejpam-4298	183	56	∩	∩	ADJ
ejpam-4298	183	57	f	f	PROPN
ejpam-4298	183	58	6=	6=	PROPN
ejpam-4298	183	59	φ	φ	PROPN
ejpam-4298	183	60	and	and	CCONJ
ejpam-4298	183	61	f	f	PROPN
ejpam-4298	183	62	⊆	⊆	NUM
ejpam-4298	183	63	g	g	NOUN
ejpam-4298	183	64	,	,	PUNCT
ejpam-4298	183	65	(	(	PUNCT
ejpam-4298	183	66	iii	iii	NOUN
ejpam-4298	183	67	)	)	PUNCT
ejpam-4298	183	68	for	for	ADP
ejpam-4298	183	69	any	any	DET
ejpam-4298	183	70	coc	coc	NOUN
ejpam-4298	183	71	-	-	PUNCT
ejpam-4298	183	72	open	open	NOUN
ejpam-4298	183	73	set	set	NOUN
ejpam-4298	183	74	g	g	NOUN
ejpam-4298	183	75	of	of	ADP
ejpam-4298	183	76	x	x	PROPN
ejpam-4298	183	77	,	,	PUNCT
ejpam-4298	183	78	g	g	PROPN
ejpam-4298	183	79	=	=	SYM
ejpam-4298	183	80	∪{f	∪{f	PROPN
ejpam-4298	183	81	:	:	PUNCT
ejpam-4298	183	82	f	f	PROPN
ejpam-4298	183	83	is	be	AUX
ejpam-4298	183	84	coc	coc	ADJ
ejpam-4298	183	85	-	-	PUNCT
ejpam-4298	183	86	closed	close	VERB
ejpam-4298	183	87	subset	subset	NOUN
ejpam-4298	183	88	with	with	ADP
ejpam-4298	183	89	f	f	PROPN
ejpam-4298	184	1	⊆	⊆	NUM
ejpam-4298	184	2	g	g	NOUN
ejpam-4298	184	3	}	}	PUNCT
ejpam-4298	184	4	,	,	PUNCT
ejpam-4298	184	5	(	(	PUNCT
ejpam-4298	184	6	iv	iv	X
ejpam-4298	184	7	)	)	PUNCT
ejpam-4298	184	8	for	for	ADP
ejpam-4298	184	9	any	any	DET
ejpam-4298	184	10	coc	coc	NOUN
ejpam-4298	184	11	-	-	PUNCT
ejpam-4298	184	12	closed	close	VERB
ejpam-4298	184	13	subset	subset	NOUN
ejpam-4298	184	14	f	f	PROPN
ejpam-4298	184	15	of	of	ADP
ejpam-4298	184	16	x	x	PROPN
ejpam-4298	184	17	,	,	PUNCT
ejpam-4298	184	18	f	f	PROPN
ejpam-4298	184	19	=	=	SYM
ejpam-4298	184	20	coc	coc	PROPN
ejpam-4298	184	21	-	-	PUNCT
ejpam-4298	184	22	ker(f	ker(f	PROPN
ejpam-4298	184	23	)	)	PUNCT
ejpam-4298	184	24	,	,	PUNCT
ejpam-4298	184	25	(	(	PUNCT
ejpam-4298	184	26	v	v	NOUN
ejpam-4298	184	27	)	)	PUNCT
ejpam-4298	184	28	for	for	ADP
ejpam-4298	184	29	any	any	DET
ejpam-4298	184	30	x	x	SYM
ejpam-4298	184	31	∈	∈	PROPN
ejpam-4298	184	32	x	x	NOUN
ejpam-4298	184	33	,	,	PUNCT
ejpam-4298	184	34	{	{	PUNCT
ejpam-4298	184	35	x}coc	x}coc	PROPN
ejpam-4298	184	36	⊆	⊆	NUM
ejpam-4298	184	37	coc	coc	PROPN
ejpam-4298	184	38	-	-	PUNCT
ejpam-4298	184	39	ker({x	ker({x	NOUN
ejpam-4298	184	40	}	}	PUNCT
ejpam-4298	184	41	)	)	PUNCT
ejpam-4298	184	42	.	.	PUNCT
ejpam-4298	185	1	proof	proof	NOUN
ejpam-4298	185	2	.	.	PUNCT
ejpam-4298	186	1	(	(	PUNCT
ejpam-4298	186	2	iii	iii	X
ejpam-4298	186	3	)	)	PUNCT
ejpam-4298	186	4	⇒	⇒	NOUN
ejpam-4298	186	5	(	(	PUNCT
ejpam-4298	186	6	iv	iv	NUM
ejpam-4298	186	7	)	)	PUNCT
ejpam-4298	186	8	,	,	PUNCT
ejpam-4298	186	9	(	(	PUNCT
ejpam-4298	186	10	v	v	NOUN
ejpam-4298	186	11	)	)	PUNCT
ejpam-4298	186	12	⇒	⇒	NOUN
ejpam-4298	186	13	(	(	PUNCT
ejpam-4298	186	14	i	i	NOUN
ejpam-4298	186	15	)	)	PUNCT
ejpam-4298	186	16	obvious	obvious	ADJ
ejpam-4298	186	17	.	.	PUNCT
ejpam-4298	187	1	(	(	PUNCT
ejpam-4298	187	2	i	i	NOUN
ejpam-4298	187	3	)	)	PUNCT
ejpam-4298	187	4	⇒	⇒	PROPN
ejpam-4298	187	5	(	(	PUNCT
ejpam-4298	187	6	ii	ii	NOUN
ejpam-4298	187	7	)	)	PUNCT
ejpam-4298	187	8	let	let	VERB
ejpam-4298	187	9	a	a	DET
ejpam-4298	187	10	⊆	⊆	NUM
ejpam-4298	187	11	x	x	NOUN
ejpam-4298	187	12	and	and	CCONJ
ejpam-4298	187	13	g	g	PROPN
ejpam-4298	187	14	is	be	AUX
ejpam-4298	187	15	a	a	DET
ejpam-4298	187	16	coc	coc	NOUN
ejpam-4298	187	17	-	-	PUNCT
ejpam-4298	187	18	open	open	NOUN
ejpam-4298	187	19	set	set	NOUN
ejpam-4298	187	20	and	and	CCONJ
ejpam-4298	187	21	let	let	VERB
ejpam-4298	187	22	x	x	SYM
ejpam-4298	187	23	∈	∈	PROPN
ejpam-4298	187	24	a	a	DET
ejpam-4298	187	25	∩	∩	ADJ
ejpam-4298	187	26	g.	g.	NOUN
ejpam-4298	188	1	then	then	ADV
ejpam-4298	188	2	the	the	DET
ejpam-4298	188	3	needed	need	VERB
ejpam-4298	188	4	coc	coc	NOUN
ejpam-4298	188	5	-	-	PUNCT
ejpam-4298	188	6	closed	close	VERB
ejpam-4298	188	7	subset	subset	NOUN
ejpam-4298	188	8	f	f	X
ejpam-4298	188	9	is	be	AUX
ejpam-4298	188	10	{	{	PUNCT
ejpam-4298	188	11	x}coc	x}coc	PROPN
ejpam-4298	188	12	.	.	PUNCT
ejpam-4298	189	1	(	(	PUNCT
ejpam-4298	189	2	ii	ii	NOUN
ejpam-4298	189	3	)	)	PUNCT
ejpam-4298	189	4	⇒	⇒	NOUN
ejpam-4298	189	5	(	(	PUNCT
ejpam-4298	189	6	iii	iii	NOUN
ejpam-4298	189	7	)	)	PUNCT
ejpam-4298	189	8	for	for	ADP
ejpam-4298	189	9	a	a	DET
ejpam-4298	189	10	coc	coc	NOUN
ejpam-4298	189	11	-	-	PUNCT
ejpam-4298	189	12	open	open	NOUN
ejpam-4298	189	13	set	set	NOUN
ejpam-4298	189	14	g	g	PROPN
ejpam-4298	189	15	⊇	⊇	NOUN
ejpam-4298	189	16	∪{f	∪{f	PROPN
ejpam-4298	189	17	:	:	PUNCT
ejpam-4298	189	18	f	f	PROPN
ejpam-4298	189	19	is	be	AUX
ejpam-4298	189	20	a	a	DET
ejpam-4298	189	21	coc	coc	NOUN
ejpam-4298	189	22	-	-	PUNCT
ejpam-4298	189	23	closed	close	VERB
ejpam-4298	189	24	with	with	ADP
ejpam-4298	189	25	f	f	PROPN
ejpam-4298	189	26	⊆	⊆	NUM
ejpam-4298	189	27	g	g	NOUN
ejpam-4298	189	28	}	}	PUNCT
ejpam-4298	189	29	,	,	PUNCT
ejpam-4298	189	30	let	let	VERB
ejpam-4298	189	31	x	x	PUNCT
ejpam-4298	189	32	∈	∈	PROPN
ejpam-4298	189	33	g	g	PROPN
ejpam-4298	189	34	,	,	PUNCT
ejpam-4298	189	35	then	then	ADV
ejpam-4298	189	36	there	there	PRON
ejpam-4298	189	37	exists	exist	VERB
ejpam-4298	189	38	a	a	DET
ejpam-4298	189	39	coc	coc	NOUN
ejpam-4298	189	40	-	-	PUNCT
ejpam-4298	189	41	closed	close	VERB
ejpam-4298	189	42	set	set	NOUN
ejpam-4298	189	43	f	f	PROPN
ejpam-4298	189	44	such	such	ADJ
ejpam-4298	189	45	that	that	SCONJ
ejpam-4298	189	46	x	x	SYM
ejpam-4298	189	47	∈	∈	PROPN
ejpam-4298	189	48	f	f	PROPN
ejpam-4298	189	49	and	and	CCONJ
ejpam-4298	189	50	f	f	PROPN
ejpam-4298	189	51	⊆	⊆	NUM
ejpam-4298	189	52	g	g	NOUN
ejpam-4298	189	53	,	,	PUNCT
ejpam-4298	189	54	so	so	ADV
ejpam-4298	190	1	x	x	SYM
ejpam-4298	190	2	∈	∈	PROPN
ejpam-4298	190	3	f	f	PROPN
ejpam-4298	190	4	⊆	⊆	NUM
ejpam-4298	190	5	∪{f	∪{f	PROPN
ejpam-4298	190	6	:	:	PUNCT
ejpam-4298	190	7	f	f	PROPN
ejpam-4298	190	8	is	be	AUX
ejpam-4298	190	9	a	a	DET
ejpam-4298	190	10	coc	coc	NOUN
ejpam-4298	190	11	-	-	PUNCT
ejpam-4298	190	12	closed	closed	ADJ
ejpam-4298	190	13	,	,	PUNCT
ejpam-4298	190	14	f	f	PROPN
ejpam-4298	190	15	⊆	⊆	NUM
ejpam-4298	190	16	g	g	NOUN
ejpam-4298	190	17	}	}	PUNCT
ejpam-4298	190	18	,	,	PUNCT
ejpam-4298	190	19	hence	hence	ADV
ejpam-4298	190	20	the	the	DET
ejpam-4298	190	21	result	result	NOUN
ejpam-4298	190	22	.	.	PUNCT
ejpam-4298	191	1	(	(	PUNCT
ejpam-4298	191	2	iv	iv	X
ejpam-4298	191	3	)	)	PUNCT
ejpam-4298	191	4	⇒	⇒	NOUN
ejpam-4298	191	5	(	(	PUNCT
ejpam-4298	191	6	v	v	NOUN
ejpam-4298	191	7	)	)	PUNCT
ejpam-4298	191	8	let	let	VERB
ejpam-4298	191	9	x	x	SYM
ejpam-4298	191	10	∈	∈	PROPN
ejpam-4298	191	11	x	x	X
ejpam-4298	191	12	and	and	CCONJ
ejpam-4298	191	13	y	y	PROPN
ejpam-4298	191	14	/∈	/∈	PUNCT
ejpam-4298	191	15	coc	coc	PROPN
ejpam-4298	191	16	-	-	PUNCT
ejpam-4298	191	17	ker({x	ker({x	NOUN
ejpam-4298	191	18	}	}	PUNCT
ejpam-4298	191	19	)	)	PUNCT
ejpam-4298	191	20	,	,	PUNCT
ejpam-4298	191	21	there	there	PRON
ejpam-4298	191	22	exists	exist	VERB
ejpam-4298	191	23	a	a	DET
ejpam-4298	191	24	coc	coc	NOUN
ejpam-4298	191	25	-	-	PUNCT
ejpam-4298	191	26	open	open	ADJ
ejpam-4298	191	27	set	set	VERB
ejpam-4298	191	28	ux	ux	PROPN
ejpam-4298	191	29	contains	contain	VERB
ejpam-4298	191	30	x	x	PUNCT
ejpam-4298	191	31	with	with	ADP
ejpam-4298	191	32	y	y	PROPN
ejpam-4298	191	33	/∈	/∈	PUNCT
ejpam-4298	192	1	u	u	PROPN
ejpam-4298	192	2	,	,	PUNCT
ejpam-4298	192	3	so	so	ADV
ejpam-4298	192	4	{	{	PUNCT
ejpam-4298	192	5	y}coc	y}coc	ADJ
ejpam-4298	192	6	∩	∩	ADJ
ejpam-4298	192	7	u	u	NOUN
ejpam-4298	192	8	=	=	PROPN
ejpam-4298	192	9	φ	φ	PROPN
ejpam-4298	192	10	and	and	CCONJ
ejpam-4298	192	11	hence	hence	ADV
ejpam-4298	192	12	coc	coc	NOUN
ejpam-4298	192	13	-	-	PUNCT
ejpam-4298	192	14	ker	ker	NOUN
ejpam-4298	192	15	(	(	PUNCT
ejpam-4298	192	16	{	{	PUNCT
ejpam-4298	192	17	y}coc	y}coc	ADJ
ejpam-4298	192	18	)	)	PUNCT
ejpam-4298	192	19	∩	∩	NOUN
ejpam-4298	192	20	u	u	NOUN
ejpam-4298	192	21	=	=	SYM
ejpam-4298	192	22	φ	φ	PROPN
ejpam-4298	192	23	,	,	PUNCT
ejpam-4298	192	24	therefore	therefore	ADV
ejpam-4298	192	25	there	there	PRON
ejpam-4298	192	26	exists	exist	VERB
ejpam-4298	192	27	a	a	DET
ejpam-4298	192	28	coc	coc	NOUN
ejpam-4298	192	29	-	-	PUNCT
ejpam-4298	192	30	open	open	ADJ
ejpam-4298	192	31	set	set	NOUN
ejpam-4298	192	32	oy	oy	ADP
ejpam-4298	192	33	such	such	ADJ
ejpam-4298	192	34	that	that	PRON
ejpam-4298	192	35	x	x	SYM
ejpam-4298	192	36	/∈	/∈	PUNCT
ejpam-4298	192	37	oy	oy	NOUN
ejpam-4298	192	38	and	and	CCONJ
ejpam-4298	192	39	{	{	PUNCT
ejpam-4298	192	40	y}coc	y}coc	ADJ
ejpam-4298	192	41	⊆	⊆	NUM
ejpam-4298	192	42	oy	oy	NOUN
ejpam-4298	192	43	,	,	PUNCT
ejpam-4298	192	44	so	so	CCONJ
ejpam-4298	192	45	{	{	PUNCT
ejpam-4298	192	46	x}coc	x}coc	PROPN
ejpam-4298	192	47	∩oy	∩oy	PROPN
ejpam-4298	192	48	=	=	SYM
ejpam-4298	192	49	φ	φ	PROPN
ejpam-4298	192	50	and	and	CCONJ
ejpam-4298	192	51	y	y	PROPN
ejpam-4298	192	52	/∈	/∈	PUNCT
ejpam-4298	193	1	{	{	PUNCT
ejpam-4298	193	2	x}coc	x}coc	PROPN
ejpam-4298	193	3	,	,	PUNCT
ejpam-4298	193	4	hence	hence	ADV
ejpam-4298	193	5	the	the	DET
ejpam-4298	193	6	result	result	NOUN
ejpam-4298	193	7	.	.	PUNCT
ejpam-4298	194	1	lemma	lemma	PROPN
ejpam-4298	194	2	4	4	NUM
ejpam-4298	194	3	.	.	PUNCT
ejpam-4298	195	1	a	a	DET
ejpam-4298	195	2	topological	topological	ADJ
ejpam-4298	195	3	space	space	NOUN
ejpam-4298	195	4	(	(	PUNCT
ejpam-4298	195	5	x	x	X
ejpam-4298	195	6	,	,	PUNCT
ejpam-4298	195	7	τ	τ	X
ejpam-4298	195	8	)	)	PUNCT
ejpam-4298	195	9	is	be	AUX
ejpam-4298	195	10	coc	coc	ADJ
ejpam-4298	195	11	-	-	PUNCT
ejpam-4298	195	12	r0	r0	NOUN
ejpam-4298	195	13	-	-	PUNCT
ejpam-4298	195	14	space	space	NOUN
ejpam-4298	195	15	if	if	SCONJ
ejpam-4298	195	16	and	and	CCONJ
ejpam-4298	195	17	only	only	ADV
ejpam-4298	195	18	if	if	SCONJ
ejpam-4298	195	19	for	for	ADP
ejpam-4298	195	20	each	each	DET
ejpam-4298	195	21	x	x	PUNCT
ejpam-4298	195	22	6=	6=	ADP
ejpam-4298	195	23	y	y	PROPN
ejpam-4298	195	24	∈	∈	PROPN
ejpam-4298	195	25	x	x	PUNCT
ejpam-4298	195	26	with	with	ADP
ejpam-4298	195	27	x	x	PROPN
ejpam-4298	195	28	∈	∈	PROPN
ejpam-4298	195	29	{	{	PUNCT
ejpam-4298	195	30	y}coc	y}coc	PROPN
ejpam-4298	195	31	gives	give	VERB
ejpam-4298	195	32	y	y	PROPN
ejpam-4298	195	33	∈	∈	PROPN
ejpam-4298	195	34	{	{	PUNCT
ejpam-4298	195	35	x}coc	x}coc	PROPN
ejpam-4298	195	36	proof	proof	NOUN
ejpam-4298	195	37	.	.	PUNCT
ejpam-4298	196	1	(	(	PUNCT
ejpam-4298	196	2	⇒	⇒	NOUN
ejpam-4298	196	3	)	)	PUNCT
ejpam-4298	196	4	let	let	VERB
ejpam-4298	196	5	x	x	PRON
ejpam-4298	196	6	be	be	AUX
ejpam-4298	196	7	a	a	DET
ejpam-4298	196	8	coc	coc	ADJ
ejpam-4298	196	9	-	-	PUNCT
ejpam-4298	196	10	r0	r0	NOUN
ejpam-4298	196	11	-	-	PUNCT
ejpam-4298	196	12	space	space	NOUN
ejpam-4298	196	13	and	and	CCONJ
ejpam-4298	196	14	x	x	PUNCT
ejpam-4298	196	15	∈	∈	PROPN
ejpam-4298	196	16	{	{	PUNCT
ejpam-4298	196	17	y}coc	y}coc	PROPN
ejpam-4298	196	18	.	.	PUNCT
ejpam-4298	197	1	if	if	SCONJ
ejpam-4298	197	2	u	u	NOUN
ejpam-4298	197	3	is	be	AUX
ejpam-4298	197	4	any	any	DET
ejpam-4298	197	5	coc	coc	NOUN
ejpam-4298	197	6	-	-	PUNCT
ejpam-4298	197	7	open	open	NOUN
ejpam-4298	197	8	set	set	NOUN
ejpam-4298	197	9	with	with	ADP
ejpam-4298	197	10	y	y	PROPN
ejpam-4298	197	11	∈	∈	PROPN
ejpam-4298	197	12	u	u	PROPN
ejpam-4298	197	13	,	,	PUNCT
ejpam-4298	197	14	then	then	ADV
ejpam-4298	197	15	x	x	PART
ejpam-4298	197	16	∈	∈	PROPN
ejpam-4298	197	17	u	u	NOUN
ejpam-4298	197	18	and	and	CCONJ
ejpam-4298	197	19	any	any	DET
ejpam-4298	197	20	coc	coc	NOUN
ejpam-4298	197	21	-	-	PUNCT
ejpam-4298	197	22	open	open	ADJ
ejpam-4298	197	23	set	set	NOUN
ejpam-4298	197	24	contains	contain	VERB
ejpam-4298	197	25	y	y	PROPN
ejpam-4298	197	26	must	must	AUX
ejpam-4298	197	27	contains	contain	VERB
ejpam-4298	197	28	x	x	PRON
ejpam-4298	197	29	,	,	PUNCT
ejpam-4298	197	30	hence	hence	ADV
ejpam-4298	197	31	y	y	PROPN
ejpam-4298	197	32	∈	∈	PROPN
ejpam-4298	197	33	{	{	PUNCT
ejpam-4298	197	34	x}coc	x}coc	PROPN
ejpam-4298	197	35	.	.	PUNCT
ejpam-4298	198	1	(	(	PUNCT
ejpam-4298	198	2	⇐	⇐	NOUN
ejpam-4298	198	3	)	)	PUNCT
ejpam-4298	198	4	let	let	VERB
ejpam-4298	198	5	u	u	PRON
ejpam-4298	198	6	be	be	AUX
ejpam-4298	198	7	a	a	DET
ejpam-4298	198	8	coc	coc	NOUN
ejpam-4298	198	9	-	-	PUNCT
ejpam-4298	198	10	open	open	NOUN
ejpam-4298	198	11	set	set	VERB
ejpam-4298	198	12	with	with	ADP
ejpam-4298	198	13	x	x	PROPN
ejpam-4298	198	14	∈	∈	PROPN
ejpam-4298	198	15	u	u	NOUN
ejpam-4298	198	16	.	.	PUNCT
ejpam-4298	199	1	for	for	ADP
ejpam-4298	199	2	x	x	PROPN
ejpam-4298	199	3	∈	∈	PROPN
ejpam-4298	199	4	{	{	PUNCT
ejpam-4298	199	5	y}coc	y}coc	PROPN
ejpam-4298	199	6	,	,	PUNCT
ejpam-4298	199	7	we	we	PRON
ejpam-4298	199	8	have	have	VERB
ejpam-4298	199	9	y	y	PROPN
ejpam-4298	199	10	∈	∈	PROPN
ejpam-4298	199	11	{	{	PUNCT
ejpam-4298	199	12	x}coc	x}coc	PROPN
ejpam-4298	199	13	,	,	PUNCT
ejpam-4298	199	14	therefore	therefore	ADV
ejpam-4298	199	15	{	{	PUNCT
ejpam-4298	199	16	x}coc	x}coc	PROPN
ejpam-4298	199	17	⊆	⊆	NUM
ejpam-4298	199	18	u	u	NOUN
ejpam-4298	199	19	,	,	PUNCT
ejpam-4298	199	20	hence	hence	ADV
ejpam-4298	199	21	x	x	VERB
ejpam-4298	199	22	is	be	AUX
ejpam-4298	199	23	coc	coc	ADJ
ejpam-4298	199	24	-	-	PUNCT
ejpam-4298	199	25	r0	r0	NOUN
ejpam-4298	199	26	-	-	PUNCT
ejpam-4298	199	27	space	space	NOUN
ejpam-4298	199	28	.	.	PUNCT
ejpam-4298	200	1	theorem	theorem	VERB
ejpam-4298	200	2	16	16	NUM
ejpam-4298	200	3	.	.	PUNCT
ejpam-4298	201	1	for	for	ADP
ejpam-4298	201	2	a	a	DET
ejpam-4298	201	3	topological	topological	ADJ
ejpam-4298	201	4	space	space	NOUN
ejpam-4298	201	5	(	(	PUNCT
ejpam-4298	201	6	x	x	X
ejpam-4298	201	7	,	,	PUNCT
ejpam-4298	201	8	τ	τ	PROPN
ejpam-4298	201	9	)	)	PUNCT
ejpam-4298	201	10	.	.	PUNCT
ejpam-4298	202	1	the	the	DET
ejpam-4298	202	2	following	follow	VERB
ejpam-4298	202	3	are	be	AUX
ejpam-4298	202	4	equivalent	equivalent	ADJ
ejpam-4298	202	5	:	:	PUNCT
ejpam-4298	202	6	(	(	PUNCT
ejpam-4298	202	7	i	i	NOUN
ejpam-4298	202	8	)	)	PUNCT
ejpam-4298	202	9	x	x	X
ejpam-4298	202	10	is	be	AUX
ejpam-4298	202	11	a	a	DET
ejpam-4298	202	12	coc	coc	ADJ
ejpam-4298	202	13	-	-	PUNCT
ejpam-4298	202	14	r0	r0	NOUN
ejpam-4298	202	15	-	-	PUNCT
ejpam-4298	202	16	space	space	NOUN
ejpam-4298	202	17	,	,	PUNCT
ejpam-4298	202	18	(	(	PUNCT
ejpam-4298	202	19	ii	ii	NOUN
ejpam-4298	202	20	)	)	PUNCT
ejpam-4298	202	21	if	if	SCONJ
ejpam-4298	202	22	f	f	PROPN
ejpam-4298	202	23	is	be	AUX
ejpam-4298	202	24	a	a	DET
ejpam-4298	202	25	coc	coc	NOUN
ejpam-4298	202	26	-	-	PUNCT
ejpam-4298	202	27	closed	closed	ADJ
ejpam-4298	202	28	subset	subset	NOUN
ejpam-4298	202	29	of	of	ADP
ejpam-4298	202	30	x	x	PUNCT
ejpam-4298	202	31	with	with	ADP
ejpam-4298	202	32	x	x	PROPN
ejpam-4298	202	33	∈	∈	PROPN
ejpam-4298	202	34	f	f	PROPN
ejpam-4298	202	35	,	,	PUNCT
ejpam-4298	202	36	then	then	ADV
ejpam-4298	202	37	coc	coc	NOUN
ejpam-4298	202	38	-	-	PUNCT
ejpam-4298	202	39	ker({x	ker({x	NOUN
ejpam-4298	202	40	}	}	PUNCT
ejpam-4298	202	41	)	)	PUNCT
ejpam-4298	203	1	⊆	⊆	NUM
ejpam-4298	203	2	f	f	PROPN
ejpam-4298	203	3	,	,	PUNCT
ejpam-4298	203	4	f.a	f.a	PROPN
ejpam-4298	203	5	.	.	PROPN
ejpam-4298	203	6	abushaheen	abushaheen	PROPN
ejpam-4298	203	7	/	/	SYM
ejpam-4298	203	8	eur	eur	PROPN
ejpam-4298	203	9	.	.	PUNCT
ejpam-4298	204	1	j.	j.	PROPN
ejpam-4298	204	2	pure	pure	PROPN
ejpam-4298	204	3	appl	appl	PROPN
ejpam-4298	204	4	.	.	PROPN
ejpam-4298	204	5	math	math	PROPN
ejpam-4298	204	6	,	,	PUNCT
ejpam-4298	204	7	15	15	NUM
ejpam-4298	204	8	(	(	PUNCT
ejpam-4298	204	9	2	2	NUM
ejpam-4298	204	10	)	)	PUNCT
ejpam-4298	204	11	(	(	PUNCT
ejpam-4298	204	12	2022	2022	NUM
ejpam-4298	204	13	)	)	PUNCT
ejpam-4298	204	14	,	,	PUNCT
ejpam-4298	204	15	589	589	NUM
ejpam-4298	204	16	-	-	SYM
ejpam-4298	204	17	601	601	NUM
ejpam-4298	204	18	595	595	NUM
ejpam-4298	204	19	(	(	PUNCT
ejpam-4298	204	20	iii	iii	NOUN
ejpam-4298	204	21	)	)	PUNCT
ejpam-4298	205	1	if	if	SCONJ
ejpam-4298	205	2	x	x	SYM
ejpam-4298	205	3	∈	∈	PROPN
ejpam-4298	205	4	x	x	NOUN
ejpam-4298	205	5	,	,	PUNCT
ejpam-4298	205	6	then	then	ADV
ejpam-4298	205	7	coc	coc	NOUN
ejpam-4298	205	8	-	-	PUNCT
ejpam-4298	205	9	ker({x	ker({x	NOUN
ejpam-4298	205	10	}	}	PUNCT
ejpam-4298	205	11	)	)	PUNCT
ejpam-4298	205	12	⊆	⊆	X
ejpam-4298	205	13	{	{	PUNCT
ejpam-4298	205	14	x}coc	x}coc	PROPN
ejpam-4298	205	15	.	.	PUNCT
ejpam-4298	206	1	proof	proof	NOUN
ejpam-4298	206	2	.	.	PUNCT
ejpam-4298	207	1	(	(	PUNCT
ejpam-4298	207	2	ii	ii	NOUN
ejpam-4298	207	3	)	)	PUNCT
ejpam-4298	207	4	⇒	⇒	NOUN
ejpam-4298	207	5	(	(	PUNCT
ejpam-4298	207	6	iii	iii	NOUN
ejpam-4298	207	7	)	)	PUNCT
ejpam-4298	207	8	obvious	obvious	ADJ
ejpam-4298	207	9	.	.	PUNCT
ejpam-4298	208	1	(	(	PUNCT
ejpam-4298	208	2	i	i	NOUN
ejpam-4298	208	3	)	)	PUNCT
ejpam-4298	208	4	⇒	⇒	PROPN
ejpam-4298	208	5	(	(	PUNCT
ejpam-4298	208	6	ii	ii	NOUN
ejpam-4298	208	7	)	)	PUNCT
ejpam-4298	208	8	let	let	VERB
ejpam-4298	208	9	f	f	PRON
ejpam-4298	208	10	be	be	AUX
ejpam-4298	208	11	a	a	DET
ejpam-4298	208	12	coc	coc	NOUN
ejpam-4298	208	13	-	-	PUNCT
ejpam-4298	208	14	closed	closed	ADJ
ejpam-4298	208	15	and	and	CCONJ
ejpam-4298	209	1	x	x	SYM
ejpam-4298	209	2	∈	∈	PROPN
ejpam-4298	209	3	f	f	X
ejpam-4298	209	4	.	.	PUNCT
ejpam-4298	210	1	so	so	ADV
ejpam-4298	210	2	coc	coc	ADJ
ejpam-4298	210	3	-	-	PUNCT
ejpam-4298	210	4	ker({x	ker({x	NOUN
ejpam-4298	210	5	}	}	PUNCT
ejpam-4298	210	6	)	)	PUNCT
ejpam-4298	211	1	⊆	⊆	NUM
ejpam-4298	211	2	coc	coc	PROPN
ejpam-4298	211	3	-	-	PUNCT
ejpam-4298	211	4	ker(f	ker(f	PROPN
ejpam-4298	211	5	)	)	PUNCT
ejpam-4298	211	6	,	,	PUNCT
ejpam-4298	211	7	then	then	ADV
ejpam-4298	211	8	by	by	ADP
ejpam-4298	211	9	theorem	theorem	NOUN
ejpam-4298	211	10	15	15	NUM
ejpam-4298	211	11	we	we	PRON
ejpam-4298	211	12	have	have	VERB
ejpam-4298	211	13	coc	coc	ADJ
ejpam-4298	211	14	-	-	PUNCT
ejpam-4298	211	15	ker({x	ker({x	NOUN
ejpam-4298	211	16	}	}	PUNCT
ejpam-4298	211	17	)	)	PUNCT
ejpam-4298	212	1	⊆	⊆	NUM
ejpam-4298	212	2	f	f	NOUN
ejpam-4298	212	3	.	.	PUNCT
ejpam-4298	213	1	(	(	PUNCT
ejpam-4298	213	2	iii	iii	X
ejpam-4298	213	3	)	)	PUNCT
ejpam-4298	213	4	⇒	⇒	NOUN
ejpam-4298	213	5	(	(	PUNCT
ejpam-4298	213	6	i	i	NOUN
ejpam-4298	213	7	)	)	PUNCT
ejpam-4298	213	8	let	let	VERB
ejpam-4298	213	9	x	x	X
ejpam-4298	213	10	∈	∈	PROPN
ejpam-4298	213	11	{	{	PUNCT
ejpam-4298	213	12	y}coc	y}coc	PROPN
ejpam-4298	213	13	.	.	PUNCT
ejpam-4298	214	1	so	so	ADV
ejpam-4298	214	2	y	y	PROPN
ejpam-4298	214	3	∈	∈	PROPN
ejpam-4298	214	4	coc	coc	PROPN
ejpam-4298	214	5	-	-	PUNCT
ejpam-4298	214	6	ker({x	ker({x	NOUN
ejpam-4298	214	7	}	}	PUNCT
ejpam-4298	214	8	)	)	PUNCT
ejpam-4298	214	9	,	,	PUNCT
ejpam-4298	214	10	therefore	therefore	ADV
ejpam-4298	214	11	by	by	ADP
ejpam-4298	214	12	(	(	PUNCT
ejpam-4298	214	13	iii	iii	X
ejpam-4298	214	14	)	)	PUNCT
ejpam-4298	214	15	y	y	PROPN
ejpam-4298	214	16	∈	∈	PROPN
ejpam-4298	214	17	{	{	PUNCT
ejpam-4298	214	18	x}coc	x}coc	PROPN
ejpam-4298	214	19	,	,	PUNCT
ejpam-4298	214	20	and	and	CCONJ
ejpam-4298	214	21	the	the	DET
ejpam-4298	214	22	result	result	NOUN
ejpam-4298	214	23	comes	come	VERB
ejpam-4298	214	24	from	from	ADP
ejpam-4298	214	25	lemma	lemma	PROPN
ejpam-4298	214	26	4	4	NUM
ejpam-4298	214	27	.	.	PUNCT
ejpam-4298	214	28	corollary	corollary	ADJ
ejpam-4298	214	29	2	2	NUM
ejpam-4298	214	30	.	.	PUNCT
ejpam-4298	215	1	a	a	DET
ejpam-4298	215	2	topological	topological	ADJ
ejpam-4298	215	3	space	space	NOUN
ejpam-4298	215	4	(	(	PUNCT
ejpam-4298	215	5	x	x	X
ejpam-4298	215	6	,	,	PUNCT
ejpam-4298	215	7	τ	τ	X
ejpam-4298	215	8	)	)	PUNCT
ejpam-4298	215	9	is	be	AUX
ejpam-4298	215	10	coc	coc	ADJ
ejpam-4298	215	11	-	-	PUNCT
ejpam-4298	215	12	r0	r0	NOUN
ejpam-4298	215	13	-	-	PUNCT
ejpam-4298	215	14	space	space	NOUN
ejpam-4298	215	15	if	if	SCONJ
ejpam-4298	215	16	and	and	CCONJ
ejpam-4298	215	17	only	only	ADV
ejpam-4298	215	18	if	if	SCONJ
ejpam-4298	215	19	coc	coc	ADJ
ejpam-4298	215	20	-	-	PUNCT
ejpam-4298	215	21	ker({x	ker({x	NOUN
ejpam-4298	215	22	}	}	PUNCT
ejpam-4298	215	23	)	)	PUNCT
ejpam-4298	216	1	=	=	SYM
ejpam-4298	216	2	{	{	PUNCT
ejpam-4298	216	3	x}coc	x}coc	PROPN
ejpam-4298	216	4	for	for	ADP
ejpam-4298	216	5	all	all	DET
ejpam-4298	216	6	x	x	SYM
ejpam-4298	216	7	∈	∈	NOUN
ejpam-4298	216	8	x.	x.	NOUN
ejpam-4298	216	9	4	4	X
ejpam-4298	216	10	.	.	X
ejpam-4298	216	11	coc	coc	PROPN
ejpam-4298	216	12	-	-	PUNCT
ejpam-4298	216	13	t	t	PROPN
ejpam-4298	216	14	1	1	NUM
ejpam-4298	216	15	2	2	NUM
ejpam-4298	216	16	-space	-space	NOUN
ejpam-4298	216	17	,	,	PUNCT
ejpam-4298	216	18	coc	coc	NOUN
ejpam-4298	216	19	-	-	PUNCT
ejpam-4298	216	20	t	t	PROPN
ejpam-4298	216	21	3	3	NUM
ejpam-4298	216	22	8	8	NUM
ejpam-4298	216	23	-space	-space	NOUN
ejpam-4298	216	24	and	and	CCONJ
ejpam-4298	216	25	coc	coc	NOUN
ejpam-4298	216	26	-	-	PUNCT
ejpam-4298	216	27	t	t	PROPN
ejpam-4298	216	28	1	1	NUM
ejpam-4298	216	29	4	4	NUM
ejpam-4298	216	30	-space	-space	NOUN
ejpam-4298	216	31	in	in	ADP
ejpam-4298	216	32	this	this	DET
ejpam-4298	216	33	section	section	NOUN
ejpam-4298	216	34	we	we	PRON
ejpam-4298	216	35	define	define	VERB
ejpam-4298	216	36	more	more	ADV
ejpam-4298	216	37	weak	weak	ADJ
ejpam-4298	216	38	separation	separation	NOUN
ejpam-4298	216	39	axioms	axiom	NOUN
ejpam-4298	216	40	in	in	ADP
ejpam-4298	216	41	coc	coc	NOUN
ejpam-4298	216	42	-	-	PUNCT
ejpam-4298	216	43	open	open	ADJ
ejpam-4298	216	44	set	set	NOUN
ejpam-4298	216	45	,	,	PUNCT
ejpam-4298	216	46	but	but	CCONJ
ejpam-4298	216	47	before	before	ADP
ejpam-4298	216	48	this	this	PRON
ejpam-4298	216	49	we	we	PRON
ejpam-4298	216	50	need	need	VERB
ejpam-4298	216	51	some	some	DET
ejpam-4298	216	52	definitions	definition	NOUN
ejpam-4298	216	53	and	and	CCONJ
ejpam-4298	216	54	lemmas	lemma	NOUN
ejpam-4298	216	55	.	.	PUNCT
ejpam-4298	217	1	definition	definition	NOUN
ejpam-4298	217	2	12	12	NUM
ejpam-4298	217	3	.	.	PUNCT
ejpam-4298	218	1	let	let	VERB
ejpam-4298	218	2	a	a	DET
ejpam-4298	218	3	be	be	AUX
ejpam-4298	218	4	a	a	DET
ejpam-4298	218	5	subset	subset	NOUN
ejpam-4298	218	6	of	of	ADP
ejpam-4298	218	7	a	a	DET
ejpam-4298	218	8	topological	topological	ADJ
ejpam-4298	218	9	space	space	NOUN
ejpam-4298	218	10	(	(	PUNCT
ejpam-4298	218	11	x	x	X
ejpam-4298	218	12	,	,	PUNCT
ejpam-4298	218	13	τ	τ	PROPN
ejpam-4298	218	14	)	)	PUNCT
ejpam-4298	218	15	.	.	PUNCT
ejpam-4298	219	1	then	then	ADV
ejpam-4298	219	2	a	a	PRON
ejpam-4298	219	3	is	be	AUX
ejpam-4298	219	4	called	call	VERB
ejpam-4298	219	5	coc	coc	NOUN
ejpam-4298	219	6	-	-	PUNCT
ejpam-4298	219	7	gclosed	gclose	VERB
ejpam-4298	219	8	if	if	SCONJ
ejpam-4298	219	9	{	{	PUNCT
ejpam-4298	219	10	a}coc	a}coc	NOUN
ejpam-4298	219	11	⊆	⊆	NUM
ejpam-4298	219	12	u	u	NOUN
ejpam-4298	219	13	,	,	PUNCT
ejpam-4298	219	14	whenever	whenever	SCONJ
ejpam-4298	219	15	a	a	DET
ejpam-4298	219	16	⊆	⊆	NUM
ejpam-4298	219	17	u	u	NOUN
ejpam-4298	219	18	and	and	CCONJ
ejpam-4298	219	19	u	u	NOUN
ejpam-4298	219	20	is	be	AUX
ejpam-4298	219	21	coc	coc	ADJ
ejpam-4298	219	22	-	-	PUNCT
ejpam-4298	219	23	open	open	ADJ
ejpam-4298	219	24	set	set	NOUN
ejpam-4298	219	25	.	.	PUNCT
ejpam-4298	220	1	a	a	PRON
ejpam-4298	220	2	is	be	AUX
ejpam-4298	220	3	called	call	VERB
ejpam-4298	220	4	coc	coc	NOUN
ejpam-4298	220	5	-	-	PUNCT
ejpam-4298	220	6	g	g	NOUN
ejpam-4298	220	7	-	-	PUNCT
ejpam-4298	220	8	open	open	ADJ
ejpam-4298	220	9	if	if	SCONJ
ejpam-4298	220	10	x	x	PRON
ejpam-4298	220	11	−a	−a	NOUN
ejpam-4298	220	12	is	be	AUX
ejpam-4298	220	13	coc	coc	ADJ
ejpam-4298	220	14	-	-	PUNCT
ejpam-4298	220	15	g	g	NOUN
ejpam-4298	220	16	-	-	PUNCT
ejpam-4298	220	17	closed	closed	ADJ
ejpam-4298	220	18	.	.	PUNCT
ejpam-4298	221	1	clearly	clearly	ADV
ejpam-4298	221	2	,	,	PUNCT
ejpam-4298	221	3	a	a	PRON
ejpam-4298	221	4	is	be	AUX
ejpam-4298	221	5	a	a	DET
ejpam-4298	221	6	coc	coc	VERB
ejpam-4298	221	7	-	-	PUNCT
ejpam-4298	221	8	g	g	NOUN
ejpam-4298	221	9	-	-	PUNCT
ejpam-4298	221	10	closed	close	VERB
ejpam-4298	221	11	of	of	ADP
ejpam-4298	221	12	(	(	PUNCT
ejpam-4298	221	13	x	x	X
ejpam-4298	221	14	,	,	PUNCT
ejpam-4298	221	15	τ	τ	X
ejpam-4298	221	16	)	)	PUNCT
ejpam-4298	221	17	if	if	SCONJ
ejpam-4298	221	18	f	f	PROPN
ejpam-4298	221	19	⊆	⊆	NUM
ejpam-4298	221	20	intcoc(a	intcoc(a	NOUN
ejpam-4298	221	21	)	)	PUNCT
ejpam-4298	221	22	,	,	PUNCT
ejpam-4298	221	23	whenever	whenever	SCONJ
ejpam-4298	221	24	f	f	PROPN
ejpam-4298	221	25	⊆	⊆	PROPN
ejpam-4298	221	26	a	a	PRON
ejpam-4298	221	27	and	and	CCONJ
ejpam-4298	221	28	f	f	PROPN
ejpam-4298	221	29	is	be	AUX
ejpam-4298	221	30	coc	coc	ADJ
ejpam-4298	221	31	-	-	PUNCT
ejpam-4298	221	32	closed	close	VERB
ejpam-4298	221	33	set	set	NOUN
ejpam-4298	221	34	of	of	ADP
ejpam-4298	221	35	x.	x.	NOUN
ejpam-4298	221	36	definition	definition	NOUN
ejpam-4298	221	37	13	13	NUM
ejpam-4298	221	38	.	.	PUNCT
ejpam-4298	222	1	let	let	VERB
ejpam-4298	222	2	a	a	DET
ejpam-4298	222	3	be	be	AUX
ejpam-4298	222	4	a	a	DET
ejpam-4298	222	5	subset	subset	NOUN
ejpam-4298	222	6	of	of	ADP
ejpam-4298	222	7	a	a	DET
ejpam-4298	222	8	topological	topological	ADJ
ejpam-4298	222	9	space	space	NOUN
ejpam-4298	222	10	(	(	PUNCT
ejpam-4298	222	11	x	x	X
ejpam-4298	222	12	,	,	PUNCT
ejpam-4298	222	13	τ	τ	PROPN
ejpam-4298	222	14	)	)	PUNCT
ejpam-4298	222	15	.	.	PUNCT
ejpam-4298	223	1	then	then	ADV
ejpam-4298	223	2	coc	coc	PROPN
ejpam-4298	223	3	-	-	PROPN
ejpam-4298	223	4	a∨	a∨	PROPN
ejpam-4298	223	5	=	=	PUNCT
ejpam-4298	223	6	∪{f	∪{f	NOUN
ejpam-4298	223	7	:	:	PUNCT
ejpam-4298	223	8	x	x	SYM
ejpam-4298	223	9	−	−	PUNCT
ejpam-4298	223	10	f	f	PROPN
ejpam-4298	223	11	∈	∈	PROPN
ejpam-4298	223	12	τk	τk	ADP
ejpam-4298	223	13	:	:	PUNCT
ejpam-4298	223	14	f	f	PROPN
ejpam-4298	223	15	⊆	⊆	NUM
ejpam-4298	223	16	a	a	PRON
ejpam-4298	223	17	}	}	PUNCT
ejpam-4298	223	18	,	,	PUNCT
ejpam-4298	223	19	if	if	SCONJ
ejpam-4298	223	20	there	there	PRON
ejpam-4298	223	21	is	be	VERB
ejpam-4298	223	22	no	no	DET
ejpam-4298	223	23	coc	coc	NOUN
ejpam-4298	223	24	-	-	PUNCT
ejpam-4298	223	25	closed	close	VERB
ejpam-4298	223	26	set	set	NOUN
ejpam-4298	223	27	contains	contain	VERB
ejpam-4298	223	28	in	in	ADP
ejpam-4298	223	29	a	a	DET
ejpam-4298	223	30	,	,	PUNCT
ejpam-4298	223	31	then	then	ADV
ejpam-4298	223	32	coc	coc	PROPN
ejpam-4298	223	33	-	-	PROPN
ejpam-4298	223	34	a∨	a∨	PROPN
ejpam-4298	223	35	=	=	SYM
ejpam-4298	223	36	φ	φ	PROPN
ejpam-4298	223	37	.	.	PUNCT
ejpam-4298	224	1	lemma	lemma	PROPN
ejpam-4298	224	2	5	5	X
ejpam-4298	224	3	.	.	PUNCT
ejpam-4298	224	4	let	let	VERB
ejpam-4298	224	5	a	a	DET
ejpam-4298	224	6	be	be	AUX
ejpam-4298	224	7	a	a	DET
ejpam-4298	224	8	subset	subset	NOUN
ejpam-4298	224	9	of	of	ADP
ejpam-4298	224	10	a	a	DET
ejpam-4298	224	11	topological	topological	ADJ
ejpam-4298	224	12	space	space	NOUN
ejpam-4298	224	13	(	(	PUNCT
ejpam-4298	224	14	x	x	X
ejpam-4298	224	15	,	,	PUNCT
ejpam-4298	224	16	τ	τ	PROPN
ejpam-4298	224	17	)	)	PUNCT
ejpam-4298	224	18	.	.	PUNCT
ejpam-4298	225	1	then	then	ADV
ejpam-4298	225	2	a	a	PRON
ejpam-4298	225	3	is	be	AUX
ejpam-4298	225	4	coc	coc	ADJ
ejpam-4298	225	5	-	-	PUNCT
ejpam-4298	225	6	g	g	NOUN
ejpam-4298	225	7	-	-	PUNCT
ejpam-4298	225	8	closed	closed	ADJ
ejpam-4298	225	9	(	(	PUNCT
ejpam-4298	225	10	cocg	cocg	NOUN
ejpam-4298	225	11	-	-	PUNCT
ejpam-4298	225	12	open	open	ADJ
ejpam-4298	225	13	)	)	PUNCT
ejpam-4298	225	14	if	if	SCONJ
ejpam-4298	225	15	and	and	CCONJ
ejpam-4298	225	16	only	only	ADV
ejpam-4298	225	17	if	if	SCONJ
ejpam-4298	225	18	{	{	PUNCT
ejpam-4298	225	19	a}coc	a}coc	NOUN
ejpam-4298	225	20	⊆	⊆	NUM
ejpam-4298	225	21	coc	coc	PROPN
ejpam-4298	225	22	-	-	PUNCT
ejpam-4298	225	23	ker(a	ker(a	NOUN
ejpam-4298	225	24	)	)	PUNCT
ejpam-4298	225	25	(	(	PUNCT
ejpam-4298	225	26	coc	coc	PROPN
ejpam-4298	225	27	-	-	PROPN
ejpam-4298	225	28	a∨	a∨	PROPN
ejpam-4298	225	29	⊆	⊆	NUM
ejpam-4298	225	30	intcoc(a	intcoc(a	NOUN
ejpam-4298	225	31	)	)	PUNCT
ejpam-4298	225	32	)	)	PUNCT
ejpam-4298	225	33	.	.	PUNCT
ejpam-4298	226	1	definition	definition	NOUN
ejpam-4298	226	2	14	14	NUM
ejpam-4298	226	3	.	.	PUNCT
ejpam-4298	227	1	let	let	VERB
ejpam-4298	227	2	a	a	DET
ejpam-4298	227	3	be	be	AUX
ejpam-4298	227	4	a	a	DET
ejpam-4298	227	5	subset	subset	NOUN
ejpam-4298	227	6	of	of	ADP
ejpam-4298	227	7	a	a	DET
ejpam-4298	227	8	topological	topological	ADJ
ejpam-4298	227	9	space	space	NOUN
ejpam-4298	227	10	(	(	PUNCT
ejpam-4298	227	11	x	x	X
ejpam-4298	227	12	,	,	PUNCT
ejpam-4298	227	13	τ	τ	X
ejpam-4298	227	14	)	)	PUNCT
ejpam-4298	227	15	.	.	PUNCT
ejpam-4298	228	1	then	then	ADV
ejpam-4298	228	2	a	a	PRON
ejpam-4298	228	3	is	be	AUX
ejpam-4298	228	4	called	call	VERB
ejpam-4298	228	5	coc-∧set	coc-∧set	PROPN
ejpam-4298	228	6	(	(	PUNCT
ejpam-4298	228	7	coc-∨-set	coc-∨-set	VERB
ejpam-4298	228	8	)	)	PUNCT
ejpam-4298	229	1	if	if	SCONJ
ejpam-4298	229	2	a	a	DET
ejpam-4298	229	3	=	=	X
ejpam-4298	229	4	coc	coc	NOUN
ejpam-4298	229	5	-	-	PUNCT
ejpam-4298	229	6	ker(a)(a	ker(a)(a	NUM
ejpam-4298	229	7	=	=	SYM
ejpam-4298	229	8	coc	coc	PROPN
ejpam-4298	229	9	-	-	PROPN
ejpam-4298	229	10	a∨	a∨	PROPN
ejpam-4298	229	11	)	)	PUNCT
ejpam-4298	229	12	,	,	PUNCT
ejpam-4298	229	13	or	or	CCONJ
ejpam-4298	229	14	equivalently	equivalently	ADV
ejpam-4298	229	15	,	,	PUNCT
ejpam-4298	229	16	a	a	PRON
ejpam-4298	229	17	is	be	AUX
ejpam-4298	229	18	the	the	DET
ejpam-4298	229	19	intersection	intersection	NOUN
ejpam-4298	229	20	of	of	ADP
ejpam-4298	229	21	coc	coc	ADJ
ejpam-4298	229	22	-	-	PUNCT
ejpam-4298	229	23	open	open	ADJ
ejpam-4298	229	24	sets	set	NOUN
ejpam-4298	229	25	or	or	CCONJ
ejpam-4298	229	26	a	a	DET
ejpam-4298	229	27	=	=	PUNCT
ejpam-4298	229	28	x(a	x(a	ADJ
ejpam-4298	229	29	is	be	AUX
ejpam-4298	229	30	the	the	DET
ejpam-4298	229	31	union	union	NOUN
ejpam-4298	229	32	of	of	ADP
ejpam-4298	229	33	coc	coc	PROPN
ejpam-4298	229	34	-	-	PUNCT
ejpam-4298	229	35	closed	close	VERB
ejpam-4298	229	36	sets	set	NOUN
ejpam-4298	229	37	or	or	CCONJ
ejpam-4298	229	38	a	a	DET
ejpam-4298	229	39	=	=	SYM
ejpam-4298	229	40	φ	φ	NUM
ejpam-4298	229	41	)	)	PUNCT
ejpam-4298	229	42	.	.	PUNCT
ejpam-4298	230	1	lemma	lemma	PROPN
ejpam-4298	230	2	6	6	NUM
ejpam-4298	230	3	.	.	PUNCT
ejpam-4298	231	1	let	let	VERB
ejpam-4298	231	2	a	a	DET
ejpam-4298	231	3	,	,	PUNCT
ejpam-4298	231	4	b	b	NOUN
ejpam-4298	231	5	are	be	AUX
ejpam-4298	231	6	subsets	subset	NOUN
ejpam-4298	231	7	of	of	ADP
ejpam-4298	231	8	a	a	DET
ejpam-4298	231	9	topological	topological	ADJ
ejpam-4298	231	10	space	space	NOUN
ejpam-4298	231	11	(	(	PUNCT
ejpam-4298	231	12	x	x	X
ejpam-4298	231	13	,	,	PUNCT
ejpam-4298	231	14	τ	τ	PROPN
ejpam-4298	231	15	)	)	PUNCT
ejpam-4298	231	16	.	.	PUNCT
ejpam-4298	232	1	then	then	ADV
ejpam-4298	232	2	:	:	PUNCT
ejpam-4298	232	3	(	(	PUNCT
ejpam-4298	232	4	i	i	NOUN
ejpam-4298	232	5	)	)	PUNCT
ejpam-4298	232	6	coc	coc	NOUN
ejpam-4298	232	7	-	-	PUNCT
ejpam-4298	232	8	ker{φ	ker{φ	PROPN
ejpam-4298	232	9	}	}	PUNCT
ejpam-4298	232	10	=	=	SYM
ejpam-4298	232	11	φ	φ	NUM
ejpam-4298	232	12	,	,	PUNCT
ejpam-4298	232	13	coc	coc	PROPN
ejpam-4298	232	14	-	-	PUNCT
ejpam-4298	232	15	φ∨	φ∨	PROPN
ejpam-4298	232	16	=	=	SYM
ejpam-4298	232	17	φ	φ	PROPN
ejpam-4298	232	18	,	,	PUNCT
ejpam-4298	232	19	coc	coc	PROPN
ejpam-4298	232	20	-	-	PUNCT
ejpam-4298	232	21	ker{x	ker{x	NOUN
ejpam-4298	232	22	}	}	PUNCT
ejpam-4298	232	23	=	=	SYM
ejpam-4298	232	24	x	x	NOUN
ejpam-4298	232	25	,	,	PUNCT
ejpam-4298	232	26	coc	coc	PROPN
ejpam-4298	232	27	-	-	PUNCT
ejpam-4298	232	28	x∨	x∨	PROPN
ejpam-4298	232	29	=	=	PUNCT
ejpam-4298	232	30	x.	x.	NOUN
ejpam-4298	232	31	(	(	PUNCT
ejpam-4298	232	32	ii	ii	PROPN
ejpam-4298	232	33	)	)	PUNCT
ejpam-4298	232	34	a	a	DET
ejpam-4298	232	35	⊆	⊆	NUM
ejpam-4298	232	36	coc	coc	PROPN
ejpam-4298	232	37	-	-	PUNCT
ejpam-4298	232	38	ker(a	ker(a	NOUN
ejpam-4298	232	39	)	)	PUNCT
ejpam-4298	232	40	,	,	PUNCT
ejpam-4298	232	41	coc	coc	PROPN
ejpam-4298	232	42	-	-	PROPN
ejpam-4298	232	43	a∨	a∨	PROPN
ejpam-4298	232	44	⊆	⊆	NUM
ejpam-4298	232	45	a.	a.	NOUN
ejpam-4298	232	46	(	(	PUNCT
ejpam-4298	232	47	iii	iii	NOUN
ejpam-4298	232	48	)	)	PUNCT
ejpam-4298	232	49	coc	coc	PROPN
ejpam-4298	232	50	-	-	PUNCT
ejpam-4298	232	51	ker(coc	ker(coc	PROPN
ejpam-4298	232	52	-	-	PUNCT
ejpam-4298	232	53	ker(a))=coc	ker(a))=coc	PROPN
ejpam-4298	232	54	-	-	PUNCT
ejpam-4298	232	55	ker(a	ker(a	NOUN
ejpam-4298	232	56	)	)	PUNCT
ejpam-4298	232	57	,	,	PUNCT
ejpam-4298	232	58	coc	coc	PROPN
ejpam-4298	232	59	(	(	PUNCT
ejpam-4298	232	60	coc	coc	PROPN
ejpam-4298	232	61	-	-	PUNCT
ejpam-4298	232	62	a∨)∨	a∨)∨	PROPN
ejpam-4298	232	63	=	=	PUNCT
ejpam-4298	232	64	coc	coc	PROPN
ejpam-4298	232	65	-	-	PUNCT
ejpam-4298	232	66	a∨.	a∨.	PROPN
ejpam-4298	232	67	(	(	PUNCT
ejpam-4298	232	68	iv	iv	X
ejpam-4298	232	69	)	)	PUNCT
ejpam-4298	232	70	if	if	SCONJ
ejpam-4298	232	71	a	a	DET
ejpam-4298	232	72	⊆	⊆	NUM
ejpam-4298	232	73	b	b	NOUN
ejpam-4298	232	74	,	,	PUNCT
ejpam-4298	232	75	then	then	ADV
ejpam-4298	232	76	coc	coc	PROPN
ejpam-4298	232	77	-	-	PUNCT
ejpam-4298	232	78	ker(a	ker(a	PROPN
ejpam-4298	232	79	)	)	PUNCT
ejpam-4298	232	80	⊆	⊆	NUM
ejpam-4298	232	81	coc	coc	PROPN
ejpam-4298	232	82	-	-	PUNCT
ejpam-4298	232	83	ker(b	ker(b	PROPN
ejpam-4298	232	84	)	)	PUNCT
ejpam-4298	232	85	.	.	PUNCT
ejpam-4298	233	1	(	(	PUNCT
ejpam-4298	233	2	v	v	NOUN
ejpam-4298	233	3	)	)	PUNCT
ejpam-4298	233	4	if	if	SCONJ
ejpam-4298	233	5	a	a	DET
ejpam-4298	233	6	⊆	⊆	NUM
ejpam-4298	233	7	b	b	NOUN
ejpam-4298	233	8	,	,	PUNCT
ejpam-4298	233	9	then	then	ADV
ejpam-4298	233	10	coc	coc	PROPN
ejpam-4298	233	11	-	-	PROPN
ejpam-4298	233	12	a∨	a∨	PROPN
ejpam-4298	233	13	⊆	⊆	NUM
ejpam-4298	233	14	coc	coc	NOUN
ejpam-4298	233	15	-	-	PUNCT
ejpam-4298	233	16	b∨.	b∨.	NOUN
ejpam-4298	233	17	lemma	lemma	PROPN
ejpam-4298	233	18	7	7	X
ejpam-4298	233	19	.	.	PUNCT
ejpam-4298	234	1	let	let	VERB
ejpam-4298	234	2	(	(	PUNCT
ejpam-4298	234	3	x	x	NOUN
ejpam-4298	234	4	,	,	PUNCT
ejpam-4298	234	5	τ	τ	X
ejpam-4298	234	6	)	)	PUNCT
ejpam-4298	234	7	be	be	VERB
ejpam-4298	234	8	a	a	DET
ejpam-4298	234	9	topological	topological	ADJ
ejpam-4298	234	10	space	space	NOUN
ejpam-4298	234	11	.	.	PUNCT
ejpam-4298	235	1	then	then	ADV
ejpam-4298	235	2	the	the	DET
ejpam-4298	235	3	following	follow	VERB
ejpam-4298	235	4	are	be	AUX
ejpam-4298	235	5	hold	hold	NOUN
ejpam-4298	235	6	:	:	PUNCT
ejpam-4298	235	7	(	(	PUNCT
ejpam-4298	235	8	i	i	NOUN
ejpam-4298	235	9	)	)	PUNCT
ejpam-4298	235	10	if	if	SCONJ
ejpam-4298	235	11	a	a	PRON
ejpam-4298	235	12	is	be	AUX
ejpam-4298	235	13	coc-∧-set	coc-∧-set	X
ejpam-4298	235	14	(	(	PUNCT
ejpam-4298	235	15	coc	coc	NOUN
ejpam-4298	235	16	-	-	PUNCT
ejpam-4298	235	17	a∨-set	a∨-set	NOUN
ejpam-4298	235	18	)	)	PUNCT
ejpam-4298	235	19	,	,	PUNCT
ejpam-4298	235	20	then	then	ADV
ejpam-4298	235	21	a	a	PRON
ejpam-4298	235	22	is	be	AUX
ejpam-4298	235	23	coc	coc	ADJ
ejpam-4298	235	24	-	-	PUNCT
ejpam-4298	235	25	g	g	NOUN
ejpam-4298	235	26	-	-	PUNCT
ejpam-4298	235	27	closed	closed	ADJ
ejpam-4298	235	28	(	(	PUNCT
ejpam-4298	235	29	coc	coc	NOUN
ejpam-4298	235	30	-	-	PUNCT
ejpam-4298	235	31	g	g	NOUN
ejpam-4298	235	32	-	-	PUNCT
ejpam-4298	235	33	open	open	ADJ
ejpam-4298	235	34	)	)	PUNCT
ejpam-4298	236	1	if	if	SCONJ
ejpam-4298	237	1	and	and	CCONJ
ejpam-4298	237	2	only	only	ADV
ejpam-4298	237	3	if	if	SCONJ
ejpam-4298	237	4	a	a	PRON
ejpam-4298	237	5	is	be	AUX
ejpam-4298	237	6	coc	coc	NOUN
ejpam-4298	237	7	-	-	PUNCT
ejpam-4298	237	8	closed	closed	ADJ
ejpam-4298	237	9	(	(	PUNCT
ejpam-4298	237	10	coc	coc	NOUN
ejpam-4298	237	11	-	-	ADJ
ejpam-4298	237	12	open	open	ADJ
ejpam-4298	237	13	)	)	PUNCT
ejpam-4298	237	14	.	.	PUNCT
ejpam-4298	238	1	f.a	f.a	PROPN
ejpam-4298	238	2	.	.	PROPN
ejpam-4298	238	3	abushaheen	abushaheen	PROPN
ejpam-4298	238	4	/	/	SYM
ejpam-4298	238	5	eur	eur	PROPN
ejpam-4298	238	6	.	.	PUNCT
ejpam-4298	239	1	j.	j.	PROPN
ejpam-4298	239	2	pure	pure	PROPN
ejpam-4298	239	3	appl	appl	PROPN
ejpam-4298	239	4	.	.	PROPN
ejpam-4298	239	5	math	math	PROPN
ejpam-4298	239	6	,	,	PUNCT
ejpam-4298	239	7	15	15	NUM
ejpam-4298	239	8	(	(	PUNCT
ejpam-4298	239	9	2	2	NUM
ejpam-4298	239	10	)	)	PUNCT
ejpam-4298	239	11	(	(	PUNCT
ejpam-4298	239	12	2022	2022	NUM
ejpam-4298	239	13	)	)	PUNCT
ejpam-4298	239	14	,	,	PUNCT
ejpam-4298	239	15	589	589	NUM
ejpam-4298	239	16	-	-	SYM
ejpam-4298	239	17	601	601	NUM
ejpam-4298	239	18	596	596	NUM
ejpam-4298	239	19	(	(	PUNCT
ejpam-4298	239	20	ii	ii	NOUN
ejpam-4298	239	21	)	)	PUNCT
ejpam-4298	239	22	for	for	ADP
ejpam-4298	239	23	a	a	PRON
ejpam-4298	239	24	⊆	⊆	NUM
ejpam-4298	239	25	x	x	SYM
ejpam-4298	239	26	,	,	PUNCT
ejpam-4298	239	27	if	if	SCONJ
ejpam-4298	239	28	coc	coc	NOUN
ejpam-4298	239	29	-	-	PUNCT
ejpam-4298	239	30	ker(a	ker(a	NOUN
ejpam-4298	239	31	)	)	PUNCT
ejpam-4298	239	32	is	be	AUX
ejpam-4298	239	33	coc	coc	ADJ
ejpam-4298	239	34	-	-	PUNCT
ejpam-4298	239	35	g	g	NOUN
ejpam-4298	239	36	-	-	PUNCT
ejpam-4298	239	37	closed	close	VERB
ejpam-4298	239	38	set	set	NOUN
ejpam-4298	239	39	(	(	PUNCT
ejpam-4298	239	40	coc	coc	NOUN
ejpam-4298	239	41	-	-	PUNCT
ejpam-4298	239	42	a∨	a∨	PROPN
ejpam-4298	239	43	is	be	AUX
ejpam-4298	239	44	coc	coc	ADJ
ejpam-4298	239	45	-	-	PUNCT
ejpam-4298	239	46	g	g	NOUN
ejpam-4298	239	47	-	-	PUNCT
ejpam-4298	239	48	open	open	ADJ
ejpam-4298	239	49	set	set	NOUN
ejpam-4298	239	50	)	)	PUNCT
ejpam-4298	239	51	,	,	PUNCT
ejpam-4298	239	52	then	then	ADV
ejpam-4298	239	53	a	a	PRON
ejpam-4298	239	54	is	be	AUX
ejpam-4298	239	55	coc	coc	ADJ
ejpam-4298	239	56	-	-	PUNCT
ejpam-4298	239	57	g	g	NOUN
ejpam-4298	239	58	-	-	PUNCT
ejpam-4298	239	59	closed	closed	ADJ
ejpam-4298	239	60	(	(	PUNCT
ejpam-4298	239	61	coc	coc	NOUN
ejpam-4298	239	62	-	-	PUNCT
ejpam-4298	239	63	g	g	NOUN
ejpam-4298	239	64	-	-	PUNCT
ejpam-4298	239	65	open	open	ADJ
ejpam-4298	239	66	)	)	PUNCT
ejpam-4298	239	67	.	.	PUNCT
ejpam-4298	240	1	proof	proof	NOUN
ejpam-4298	240	2	.	.	PUNCT
ejpam-4298	241	1	(	(	PUNCT
ejpam-4298	241	2	i	i	NOUN
ejpam-4298	241	3	)	)	PUNCT
ejpam-4298	241	4	obvious	obvious	ADJ
ejpam-4298	241	5	.	.	PUNCT
ejpam-4298	242	1	(	(	PUNCT
ejpam-4298	242	2	ii	ii	NOUN
ejpam-4298	242	3	)	)	PUNCT
ejpam-4298	242	4	from	from	ADP
ejpam-4298	242	5	lemma	lemma	PROPN
ejpam-4298	242	6	5	5	NUM
ejpam-4298	242	7	and	and	CCONJ
ejpam-4298	242	8	lemma	lemma	PROPN
ejpam-4298	242	9	6	6	NUM
ejpam-4298	242	10	.	.	PUNCT
ejpam-4298	243	1	the	the	DET
ejpam-4298	243	2	following	follow	VERB
ejpam-4298	243	3	definition	definition	NOUN
ejpam-4298	243	4	gives	give	VERB
ejpam-4298	243	5	a	a	DET
ejpam-4298	243	6	weaker	weak	ADJ
ejpam-4298	243	7	form	form	NOUN
ejpam-4298	243	8	of	of	ADP
ejpam-4298	243	9	coc-∧-set	coc-∧-set	NOUN
ejpam-4298	243	10	.	.	PUNCT
ejpam-4298	244	1	definition	definition	NOUN
ejpam-4298	244	2	15	15	NUM
ejpam-4298	244	3	.	.	PUNCT
ejpam-4298	245	1	a	a	DET
ejpam-4298	245	2	subset	subset	NOUN
ejpam-4298	245	3	a	a	PRON
ejpam-4298	245	4	of	of	ADP
ejpam-4298	245	5	a	a	DET
ejpam-4298	245	6	space	space	NOUN
ejpam-4298	245	7	(	(	PUNCT
ejpam-4298	245	8	x	x	X
ejpam-4298	245	9	,	,	PUNCT
ejpam-4298	245	10	τ	τ	X
ejpam-4298	245	11	)	)	PUNCT
ejpam-4298	245	12	is	be	AUX
ejpam-4298	245	13	called	call	VERB
ejpam-4298	245	14	generalized	generalized	ADJ
ejpam-4298	245	15	coc	coc	ADJ
ejpam-4298	245	16	-	-	PUNCT
ejpam-4298	245	17	kernal	kernal	ADJ
ejpam-4298	245	18	set	set	NOUN
ejpam-4298	245	19	(	(	PUNCT
ejpam-4298	245	20	g	g	NOUN
ejpam-4298	245	21	-	-	PUNCT
ejpam-4298	245	22	coc-∧set	coc-∧set	NOUN
ejpam-4298	245	23	)	)	PUNCT
ejpam-4298	246	1	if	if	SCONJ
ejpam-4298	246	2	coc	coc	NOUN
ejpam-4298	246	3	-	-	PUNCT
ejpam-4298	246	4	ker(a	ker(a	VERB
ejpam-4298	246	5	)	)	PUNCT
ejpam-4298	246	6	⊆	⊆	NUM
ejpam-4298	246	7	{	{	PUNCT
ejpam-4298	246	8	a}coc	a}coc	NOUN
ejpam-4298	246	9	,	,	PUNCT
ejpam-4298	246	10	or	or	CCONJ
ejpam-4298	246	11	equivalently	equivalently	ADV
ejpam-4298	246	12	coc	coc	NOUN
ejpam-4298	246	13	-	-	PUNCT
ejpam-4298	246	14	ker(a	ker(a	VERB
ejpam-4298	246	15	)	)	PUNCT
ejpam-4298	246	16	⊆	⊆	NUM
ejpam-4298	246	17	f	f	NOUN
ejpam-4298	246	18	,	,	PUNCT
ejpam-4298	246	19	whenever	whenever	SCONJ
ejpam-4298	246	20	a	a	DET
ejpam-4298	246	21	⊆	⊆	NUM
ejpam-4298	246	22	f	f	PROPN
ejpam-4298	246	23	and	and	CCONJ
ejpam-4298	246	24	f	f	PROPN
ejpam-4298	246	25	is	be	AUX
ejpam-4298	246	26	coc	coc	NOUN
ejpam-4298	246	27	-	-	PUNCT
ejpam-4298	246	28	closed	closed	ADJ
ejpam-4298	246	29	.	.	PUNCT
ejpam-4298	247	1	a	a	DET
ejpam-4298	247	2	subset	subset	NOUN
ejpam-4298	247	3	a	a	PRON
ejpam-4298	247	4	of	of	ADP
ejpam-4298	247	5	a	a	DET
ejpam-4298	247	6	space	space	NOUN
ejpam-4298	247	7	(	(	PUNCT
ejpam-4298	247	8	x	x	X
ejpam-4298	247	9	,	,	PUNCT
ejpam-4298	247	10	τ	τ	X
ejpam-4298	247	11	)	)	PUNCT
ejpam-4298	247	12	is	be	AUX
ejpam-4298	247	13	called	call	VERB
ejpam-4298	247	14	generalized	generalized	ADJ
ejpam-4298	247	15	coc-∨-set	coc-∨-set	NOUN
ejpam-4298	247	16	(	(	PUNCT
ejpam-4298	247	17	g	g	NOUN
ejpam-4298	247	18	-	-	PUNCT
ejpam-4298	247	19	coc-∨-set	coc-∨-set	NOUN
ejpam-4298	247	20	)	)	PUNCT
ejpam-4298	247	21	if	if	SCONJ
ejpam-4298	247	22	x	x	PRON
ejpam-4298	247	23	−a	−a	NOUN
ejpam-4298	247	24	is	be	AUX
ejpam-4298	247	25	g	g	NOUN
ejpam-4298	247	26	-	-	PUNCT
ejpam-4298	247	27	coc-∧-set	coc-∧-set	NOUN
ejpam-4298	247	28	,	,	PUNCT
ejpam-4298	247	29	or	or	CCONJ
ejpam-4298	247	30	equivalently	equivalently	ADV
ejpam-4298	247	31	intcoc(a	intcoc(a	VERB
ejpam-4298	247	32	)	)	PUNCT
ejpam-4298	247	33	⊆	⊆	NUM
ejpam-4298	247	34	coc−a∨.	coc−a∨.	NOUN
ejpam-4298	247	35	lemma	lemma	PROPN
ejpam-4298	247	36	8	8	NUM
ejpam-4298	247	37	.	.	PUNCT
ejpam-4298	248	1	let	let	VERB
ejpam-4298	248	2	a	a	DET
ejpam-4298	248	3	be	be	AUX
ejpam-4298	248	4	subset	subset	VERB
ejpam-4298	248	5	of	of	ADP
ejpam-4298	248	6	a	a	DET
ejpam-4298	248	7	topological	topological	ADJ
ejpam-4298	248	8	space	space	NOUN
ejpam-4298	248	9	(	(	PUNCT
ejpam-4298	248	10	x	x	X
ejpam-4298	248	11	,	,	PUNCT
ejpam-4298	248	12	τ	τ	PROPN
ejpam-4298	248	13	)	)	PUNCT
ejpam-4298	248	14	.	.	PUNCT
ejpam-4298	249	1	if	if	SCONJ
ejpam-4298	249	2	a	a	PRON
ejpam-4298	249	3	is	be	AUX
ejpam-4298	249	4	coc-∧-set	coc-∧-set	X
ejpam-4298	249	5	(	(	PUNCT
ejpam-4298	249	6	coc	coc	NOUN
ejpam-4298	249	7	-	-	PUNCT
ejpam-4298	249	8	a∨-set	a∨-set	NOUN
ejpam-4298	249	9	)	)	PUNCT
ejpam-4298	249	10	,	,	PUNCT
ejpam-4298	249	11	then	then	ADV
ejpam-4298	249	12	it	it	PRON
ejpam-4298	249	13	is	be	AUX
ejpam-4298	249	14	g	g	NOUN
ejpam-4298	249	15	-	-	PUNCT
ejpam-4298	249	16	coc−∧-set	coc−∧-set	VERB
ejpam-4298	249	17	(	(	PUNCT
ejpam-4298	249	18	g	g	NOUN
ejpam-4298	249	19	-	-	PUNCT
ejpam-4298	249	20	coc-∨-set	coc-∨-set	NOUN
ejpam-4298	249	21	)	)	PUNCT
ejpam-4298	249	22	.	.	PUNCT
ejpam-4298	250	1	theorem	theorem	NOUN
ejpam-4298	250	2	17	17	NUM
ejpam-4298	250	3	.	.	PUNCT
ejpam-4298	251	1	let	let	AUX
ejpam-4298	251	2	(	(	PUNCT
ejpam-4298	251	3	x	x	NOUN
ejpam-4298	251	4	,	,	PUNCT
ejpam-4298	251	5	τ	τ	X
ejpam-4298	251	6	)	)	PUNCT
ejpam-4298	251	7	be	be	VERB
ejpam-4298	251	8	a	a	DET
ejpam-4298	251	9	topological	topological	ADJ
ejpam-4298	251	10	space	space	NOUN
ejpam-4298	251	11	.	.	PUNCT
ejpam-4298	252	1	then	then	ADV
ejpam-4298	252	2	for	for	SCONJ
ejpam-4298	252	3	x	x	PROPN
ejpam-4298	252	4	∈	∈	PROPN
ejpam-4298	252	5	x	x	X
ejpam-4298	252	6	,	,	PUNCT
ejpam-4298	252	7	{	{	PUNCT
ejpam-4298	252	8	x	x	X
ejpam-4298	252	9	}	}	PUNCT
ejpam-4298	252	10	is	be	AUX
ejpam-4298	252	11	either	either	CCONJ
ejpam-4298	252	12	coc	coc	NOUN
ejpam-4298	252	13	-	-	PUNCT
ejpam-4298	252	14	open	open	ADJ
ejpam-4298	252	15	or	or	CCONJ
ejpam-4298	252	16	g	g	NOUN
ejpam-4298	252	17	-	-	PUNCT
ejpam-4298	252	18	coc-∨-set	coc-∨-set	NOUN
ejpam-4298	252	19	.	.	PUNCT
ejpam-4298	253	1	proof	proof	NOUN
ejpam-4298	253	2	.	.	PUNCT
ejpam-4298	254	1	let	let	VERB
ejpam-4298	254	2	x	x	PUNCT
ejpam-4298	254	3	∈	∈	PROPN
ejpam-4298	254	4	x	x	X
ejpam-4298	254	5	and	and	CCONJ
ejpam-4298	254	6	{	{	PUNCT
ejpam-4298	254	7	x	x	NOUN
ejpam-4298	254	8	}	}	PUNCT
ejpam-4298	254	9	is	be	AUX
ejpam-4298	254	10	not	not	PART
ejpam-4298	254	11	coc	coc	ADJ
ejpam-4298	254	12	-	-	PUNCT
ejpam-4298	254	13	open	open	ADJ
ejpam-4298	254	14	subset	subset	NOUN
ejpam-4298	254	15	of	of	ADP
ejpam-4298	254	16	x.	x.	NOUN
ejpam-4298	254	17	hence	hence	ADV
ejpam-4298	254	18	x	x	X
ejpam-4298	254	19	−	−	PROPN
ejpam-4298	254	20	{	{	PUNCT
ejpam-4298	254	21	x	x	NOUN
ejpam-4298	254	22	}	}	PUNCT
ejpam-4298	254	23	is	be	AUX
ejpam-4298	254	24	not	not	PART
ejpam-4298	254	25	cocclosed	cocclose	VERB
ejpam-4298	254	26	subset	subset	NOUN
ejpam-4298	254	27	of	of	ADP
ejpam-4298	254	28	x	x	PUNCT
ejpam-4298	254	29	and	and	CCONJ
ejpam-4298	254	30	{	{	PUNCT
ejpam-4298	254	31	x	x	PART
ejpam-4298	254	32	−	−	PROPN
ejpam-4298	254	33	{	{	PUNCT
ejpam-4298	254	34	x}}coc	x}}coc	PROPN
ejpam-4298	254	35	=	=	SYM
ejpam-4298	254	36	x	x	PROPN
ejpam-4298	254	37	,	,	PUNCT
ejpam-4298	254	38	so	so	ADV
ejpam-4298	254	39	coc	coc	NOUN
ejpam-4298	254	40	-	-	PUNCT
ejpam-4298	254	41	ker	ker	NOUN
ejpam-4298	254	42	(	(	PUNCT
ejpam-4298	254	43	x−{x	x−{x	PROPN
ejpam-4298	254	44	}	}	PUNCT
ejpam-4298	254	45	)	)	PUNCT
ejpam-4298	255	1	⊆	⊆	NUM
ejpam-4298	255	2	{	{	PUNCT
ejpam-4298	255	3	x	x	PART
ejpam-4298	255	4	−	−	PROPN
ejpam-4298	255	5	{	{	PUNCT
ejpam-4298	255	6	x}}coc	x}}coc	PROPN
ejpam-4298	255	7	,	,	PUNCT
ejpam-4298	255	8	therefore	therefore	ADV
ejpam-4298	255	9	x	x	X
ejpam-4298	255	10	−	−	PROPN
ejpam-4298	255	11	{	{	PUNCT
ejpam-4298	255	12	x	x	NOUN
ejpam-4298	255	13	}	}	PUNCT
ejpam-4298	255	14	is	be	AUX
ejpam-4298	255	15	g	g	NOUN
ejpam-4298	255	16	-	-	PUNCT
ejpam-4298	255	17	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	255	18	,	,	PUNCT
ejpam-4298	255	19	i.e.	i.e.	X
ejpam-4298	255	20	{	{	PUNCT
ejpam-4298	255	21	x	x	X
ejpam-4298	255	22	}	}	PUNCT
ejpam-4298	255	23	is	be	AUX
ejpam-4298	255	24	g	g	NOUN
ejpam-4298	255	25	-	-	PUNCT
ejpam-4298	255	26	coc-∨-set	coc-∨-set	NOUN
ejpam-4298	255	27	.	.	PUNCT
ejpam-4298	256	1	definition	definition	NOUN
ejpam-4298	256	2	16	16	NUM
ejpam-4298	256	3	.	.	PUNCT
ejpam-4298	257	1	a	a	DET
ejpam-4298	257	2	topological	topological	ADJ
ejpam-4298	257	3	space	space	NOUN
ejpam-4298	257	4	(	(	PUNCT
ejpam-4298	257	5	x	x	X
ejpam-4298	257	6	,	,	PUNCT
ejpam-4298	257	7	τ	τ	X
ejpam-4298	257	8	)	)	PUNCT
ejpam-4298	257	9	is	be	AUX
ejpam-4298	257	10	called	call	VERB
ejpam-4298	257	11	coc	coc	PROPN
ejpam-4298	257	12	-	-	PUNCT
ejpam-4298	257	13	t	t	PROPN
ejpam-4298	257	14	1	1	NUM
ejpam-4298	257	15	2	2	NUM
ejpam-4298	257	16	-space	-space	NOUN
ejpam-4298	257	17	if	if	SCONJ
ejpam-4298	257	18	every	every	DET
ejpam-4298	257	19	coc	coc	NOUN
ejpam-4298	257	20	-	-	PUNCT
ejpam-4298	257	21	g	g	NOUN
ejpam-4298	257	22	-	-	PUNCT
ejpam-4298	257	23	closed	close	VERB
ejpam-4298	257	24	subset	subset	NOUN
ejpam-4298	257	25	of	of	ADP
ejpam-4298	257	26	x	x	PUNCT
ejpam-4298	257	27	is	be	AUX
ejpam-4298	257	28	coc	coc	NOUN
ejpam-4298	257	29	-	-	PUNCT
ejpam-4298	257	30	closed	closed	ADJ
ejpam-4298	257	31	.	.	PUNCT
ejpam-4298	258	1	lemma	lemma	PROPN
ejpam-4298	258	2	9	9	NUM
ejpam-4298	258	3	.	.	PUNCT
ejpam-4298	259	1	let	let	VERB
ejpam-4298	259	2	(	(	PUNCT
ejpam-4298	259	3	x	x	NOUN
ejpam-4298	259	4	,	,	PUNCT
ejpam-4298	259	5	τ	τ	X
ejpam-4298	259	6	)	)	PUNCT
ejpam-4298	259	7	be	be	VERB
ejpam-4298	259	8	a	a	DET
ejpam-4298	259	9	topological	topological	ADJ
ejpam-4298	259	10	space	space	NOUN
ejpam-4298	259	11	and	and	CCONJ
ejpam-4298	259	12	a	a	DET
ejpam-4298	259	13	⊆	⊆	NUM
ejpam-4298	259	14	x.	x.	NOUN
ejpam-4298	259	15	then	then	ADV
ejpam-4298	259	16	a	a	PRON
ejpam-4298	259	17	is	be	AUX
ejpam-4298	259	18	coc	coc	ADJ
ejpam-4298	259	19	-	-	PUNCT
ejpam-4298	259	20	g	g	NOUN
ejpam-4298	259	21	-	-	PUNCT
ejpam-4298	259	22	closed	close	VERB
ejpam-4298	259	23	subset	subset	NOUN
ejpam-4298	259	24	if	if	SCONJ
ejpam-4298	259	25	and	and	CCONJ
ejpam-4298	259	26	only	only	ADV
ejpam-4298	259	27	if	if	SCONJ
ejpam-4298	259	28	acoc	acoc	PROPN
ejpam-4298	259	29	−a	−a	NOUN
ejpam-4298	259	30	contains	contain	VERB
ejpam-4298	259	31	no	no	DET
ejpam-4298	259	32	coc	coc	NOUN
ejpam-4298	259	33	-	-	PUNCT
ejpam-4298	259	34	closed	closed	ADJ
ejpam-4298	259	35	subset	subset	NOUN
ejpam-4298	259	36	of	of	ADP
ejpam-4298	259	37	x.	x.	NOUN
ejpam-4298	259	38	proof	proof	NOUN
ejpam-4298	259	39	.	.	PUNCT
ejpam-4298	260	1	(	(	PUNCT
ejpam-4298	260	2	⇐	⇐	ADJ
ejpam-4298	260	3	)	)	PUNCT
ejpam-4298	260	4	obvious	obvious	ADJ
ejpam-4298	260	5	.	.	PUNCT
ejpam-4298	261	1	(	(	PUNCT
ejpam-4298	261	2	⇒	⇒	PROPN
ejpam-4298	261	3	)	)	PUNCT
ejpam-4298	261	4	let	let	VERB
ejpam-4298	261	5	a	a	DET
ejpam-4298	261	6	be	be	AUX
ejpam-4298	261	7	coc	coc	ADJ
ejpam-4298	261	8	-	-	PUNCT
ejpam-4298	261	9	g	g	NOUN
ejpam-4298	261	10	-	-	PUNCT
ejpam-4298	261	11	closed	close	VERB
ejpam-4298	261	12	and	and	CCONJ
ejpam-4298	261	13	assume	assume	VERB
ejpam-4298	261	14	there	there	PRON
ejpam-4298	261	15	exists	exist	VERB
ejpam-4298	261	16	a	a	DET
ejpam-4298	261	17	coc	coc	NOUN
ejpam-4298	261	18	-	-	PUNCT
ejpam-4298	261	19	closed	close	VERB
ejpam-4298	261	20	subset	subset	NOUN
ejpam-4298	261	21	f	f	PROPN
ejpam-4298	261	22	with	with	ADP
ejpam-4298	261	23	a	a	DET
ejpam-4298	261	24	⊆	⊆	NUM
ejpam-4298	261	25	x−f	x−f	NOUN
ejpam-4298	261	26	.	.	PUNCT
ejpam-4298	262	1	since	since	SCONJ
ejpam-4298	262	2	a	a	PRON
ejpam-4298	262	3	is	be	AUX
ejpam-4298	262	4	coc	coc	ADJ
ejpam-4298	262	5	-	-	PUNCT
ejpam-4298	262	6	g	g	NOUN
ejpam-4298	262	7	-	-	PUNCT
ejpam-4298	262	8	closed	close	VERB
ejpam-4298	262	9	set	set	NOUN
ejpam-4298	262	10	,	,	PUNCT
ejpam-4298	262	11	we	we	PRON
ejpam-4298	262	12	have	have	VERB
ejpam-4298	262	13	a	a	DET
ejpam-4298	262	14	coc	coc	NOUN
ejpam-4298	262	15	⊆	⊆	NUM
ejpam-4298	262	16	x	x	SYM
ejpam-4298	262	17	−	−	PROPN
ejpam-4298	262	18	f	f	NOUN
ejpam-4298	262	19	,	,	PUNCT
ejpam-4298	262	20	hence	hence	ADV
ejpam-4298	262	21	f	f	PROPN
ejpam-4298	262	22	⊆	⊆	NUM
ejpam-4298	262	23	x	x	SYM
ejpam-4298	262	24	−	−	ADP
ejpam-4298	262	25	a	a	DET
ejpam-4298	262	26	coc	coc	NOUN
ejpam-4298	263	1	and	and	CCONJ
ejpam-4298	263	2	this	this	PRON
ejpam-4298	263	3	is	be	AUX
ejpam-4298	263	4	a	a	DET
ejpam-4298	263	5	contradiction	contradiction	NOUN
ejpam-4298	263	6	which	which	PRON
ejpam-4298	263	7	completes	complete	VERB
ejpam-4298	263	8	the	the	DET
ejpam-4298	263	9	proof	proof	NOUN
ejpam-4298	263	10	.	.	PUNCT
ejpam-4298	264	1	theorem	theorem	VERB
ejpam-4298	264	2	18	18	NUM
ejpam-4298	264	3	.	.	PUNCT
ejpam-4298	265	1	a	a	DET
ejpam-4298	265	2	topological	topological	ADJ
ejpam-4298	265	3	space	space	NOUN
ejpam-4298	265	4	(	(	PUNCT
ejpam-4298	265	5	x	x	X
ejpam-4298	265	6	,	,	PUNCT
ejpam-4298	265	7	τ	τ	X
ejpam-4298	265	8	)	)	PUNCT
ejpam-4298	265	9	is	be	AUX
ejpam-4298	265	10	coc	coc	NOUN
ejpam-4298	265	11	-	-	PUNCT
ejpam-4298	265	12	t	t	PROPN
ejpam-4298	265	13	1	1	NUM
ejpam-4298	265	14	2	2	NUM
ejpam-4298	265	15	-space	-space	NOUN
ejpam-4298	265	16	if	if	SCONJ
ejpam-4298	265	17	and	and	CCONJ
ejpam-4298	265	18	only	only	ADV
ejpam-4298	265	19	if	if	SCONJ
ejpam-4298	265	20	every	every	DET
ejpam-4298	265	21	singleton	singleton	NOUN
ejpam-4298	265	22	of	of	ADP
ejpam-4298	265	23	x	x	PROPN
ejpam-4298	265	24	is	be	AUX
ejpam-4298	265	25	coc	coc	ADJ
ejpam-4298	265	26	-	-	PUNCT
ejpam-4298	265	27	open	open	ADJ
ejpam-4298	265	28	or	or	CCONJ
ejpam-4298	265	29	coc	coc	NOUN
ejpam-4298	265	30	-	-	PUNCT
ejpam-4298	265	31	closed	closed	ADJ
ejpam-4298	265	32	.	.	PUNCT
ejpam-4298	266	1	proof	proof	NOUN
ejpam-4298	266	2	.	.	PUNCT
ejpam-4298	267	1	(	(	PUNCT
ejpam-4298	267	2	⇒	⇒	NOUN
ejpam-4298	267	3	)	)	PUNCT
ejpam-4298	267	4	let	let	VERB
ejpam-4298	267	5	x	x	PUNCT
ejpam-4298	267	6	∈	∈	PROPN
ejpam-4298	267	7	x	x	X
ejpam-4298	267	8	and	and	CCONJ
ejpam-4298	267	9	{	{	PUNCT
ejpam-4298	267	10	x	x	NOUN
ejpam-4298	267	11	}	}	PUNCT
ejpam-4298	267	12	is	be	AUX
ejpam-4298	267	13	not	not	PART
ejpam-4298	267	14	coc	coc	ADJ
ejpam-4298	267	15	-	-	PUNCT
ejpam-4298	267	16	closed	close	VERB
ejpam-4298	267	17	set	set	NOUN
ejpam-4298	267	18	.	.	PUNCT
ejpam-4298	268	1	hence	hence	ADV
ejpam-4298	268	2	x	x	X
ejpam-4298	268	3	−	−	PROPN
ejpam-4298	268	4	{	{	PUNCT
ejpam-4298	268	5	x	x	NOUN
ejpam-4298	268	6	}	}	PUNCT
ejpam-4298	268	7	is	be	AUX
ejpam-4298	268	8	not	not	PART
ejpam-4298	268	9	coc	coc	ADJ
ejpam-4298	268	10	-	-	ADJ
ejpam-4298	268	11	open	open	ADJ
ejpam-4298	268	12	,	,	PUNCT
ejpam-4298	268	13	therefore	therefore	ADV
ejpam-4298	268	14	x	x	X
ejpam-4298	268	15	is	be	AUX
ejpam-4298	268	16	the	the	DET
ejpam-4298	268	17	only	only	ADJ
ejpam-4298	268	18	coc	coc	NOUN
ejpam-4298	268	19	-	-	PUNCT
ejpam-4298	268	20	open	open	ADJ
ejpam-4298	268	21	set	set	NOUN
ejpam-4298	268	22	with	with	ADP
ejpam-4298	268	23	x	x	PART
ejpam-4298	268	24	−	−	PROPN
ejpam-4298	268	25	{	{	PUNCT
ejpam-4298	268	26	x	x	NOUN
ejpam-4298	268	27	}	}	PUNCT
ejpam-4298	268	28	⊆	⊆	NUM
ejpam-4298	268	29	x	x	NOUN
ejpam-4298	268	30	,	,	PUNCT
ejpam-4298	268	31	that	that	PRON
ejpam-4298	268	32	is	be	AUX
ejpam-4298	268	33	mean	mean	ADJ
ejpam-4298	268	34	x	x	PUNCT
ejpam-4298	268	35	−	−	PROPN
ejpam-4298	268	36	{	{	PUNCT
ejpam-4298	268	37	x	x	NOUN
ejpam-4298	268	38	}	}	PUNCT
ejpam-4298	268	39	is	be	AUX
ejpam-4298	268	40	coc	coc	NOUN
ejpam-4298	268	41	-	-	PUNCT
ejpam-4298	268	42	gclosed	gclose	VERB
ejpam-4298	268	43	,	,	PUNCT
ejpam-4298	268	44	so	so	ADV
ejpam-4298	268	45	x	x	X
ejpam-4298	268	46	−	−	PROPN
ejpam-4298	268	47	{	{	PUNCT
ejpam-4298	268	48	x	x	NOUN
ejpam-4298	268	49	}	}	PUNCT
ejpam-4298	268	50	is	be	AUX
ejpam-4298	268	51	coc	coc	NOUN
ejpam-4298	268	52	-	-	PUNCT
ejpam-4298	268	53	closed	closed	ADJ
ejpam-4298	268	54	,	,	PUNCT
ejpam-4298	268	55	i.e.	i.e.	X
ejpam-4298	268	56	{	{	PUNCT
ejpam-4298	268	57	x	x	X
ejpam-4298	268	58	}	}	PUNCT
ejpam-4298	268	59	is	be	AUX
ejpam-4298	268	60	coc	coc	ADJ
ejpam-4298	268	61	-	-	ADJ
ejpam-4298	268	62	open	open	ADJ
ejpam-4298	268	63	.	.	PUNCT
ejpam-4298	269	1	(	(	PUNCT
ejpam-4298	269	2	⇐	⇐	NOUN
ejpam-4298	269	3	)	)	PUNCT
ejpam-4298	269	4	let	let	VERB
ejpam-4298	269	5	x	x	SYM
ejpam-4298	269	6	∈	∈	PROPN
ejpam-4298	269	7	x	x	X
ejpam-4298	269	8	and	and	CCONJ
ejpam-4298	269	9	a	a	PRON
ejpam-4298	269	10	is	be	AUX
ejpam-4298	269	11	coc	coc	ADJ
ejpam-4298	269	12	-	-	PUNCT
ejpam-4298	269	13	g	g	NOUN
ejpam-4298	269	14	-	-	PUNCT
ejpam-4298	269	15	closed	close	VERB
ejpam-4298	269	16	subset	subset	NOUN
ejpam-4298	269	17	of	of	ADP
ejpam-4298	269	18	x	x	PUNCT
ejpam-4298	269	19	with	with	ADP
ejpam-4298	269	20	x	x	PROPN
ejpam-4298	269	21	∈	∈	PROPN
ejpam-4298	269	22	a	a	DET
ejpam-4298	269	23	coc	coc	NOUN
ejpam-4298	269	24	.	.	PUNCT
ejpam-4298	270	1	if	if	SCONJ
ejpam-4298	270	2	{	{	PUNCT
ejpam-4298	270	3	x	x	NOUN
ejpam-4298	270	4	}	}	PUNCT
ejpam-4298	270	5	is	be	AUX
ejpam-4298	270	6	a	a	DET
ejpam-4298	270	7	coc	coc	ADJ
ejpam-4298	270	8	-	-	PUNCT
ejpam-4298	270	9	open	open	ADJ
ejpam-4298	270	10	set	set	NOUN
ejpam-4298	270	11	,	,	PUNCT
ejpam-4298	270	12	then	then	ADV
ejpam-4298	270	13	{	{	PUNCT
ejpam-4298	270	14	x}∩a	x}∩a	PROPN
ejpam-4298	270	15	6=	6=	NUM
ejpam-4298	270	16	φ	φ	NOUN
ejpam-4298	270	17	and	and	CCONJ
ejpam-4298	270	18	hence	hence	ADV
ejpam-4298	270	19	x	x	PART
ejpam-4298	270	20	∈	∈	NOUN
ejpam-4298	270	21	a.	a.	NOUN
ejpam-4298	270	22	if	if	SCONJ
ejpam-4298	270	23	{	{	PUNCT
ejpam-4298	270	24	x	x	NOUN
ejpam-4298	270	25	}	}	PUNCT
ejpam-4298	270	26	is	be	AUX
ejpam-4298	270	27	a	a	DET
ejpam-4298	270	28	coc	coc	NOUN
ejpam-4298	270	29	-	-	PUNCT
ejpam-4298	270	30	closed	closed	ADJ
ejpam-4298	270	31	,	,	PUNCT
ejpam-4298	270	32	then	then	ADV
ejpam-4298	270	33	by	by	ADP
ejpam-4298	270	34	lemma	lemma	PROPN
ejpam-4298	270	35	9	9	NUM
ejpam-4298	270	36	,	,	PUNCT
ejpam-4298	270	37	x	x	PROPN
ejpam-4298	270	38	/∈	/∈	PUNCT
ejpam-4298	271	1	a	a	DET
ejpam-4298	271	2	coc−a	coc−a	ADJ
ejpam-4298	271	3	,	,	PUNCT
ejpam-4298	271	4	hence	hence	ADV
ejpam-4298	271	5	x	x	ADP
ejpam-4298	271	6	∈	∈	PROPN
ejpam-4298	271	7	a	a	PRON
ejpam-4298	271	8	and	and	CCONJ
ejpam-4298	271	9	a	a	PRON
ejpam-4298	271	10	=	=	NOUN
ejpam-4298	271	11	a	a	DET
ejpam-4298	271	12	coc	coc	NOUN
ejpam-4298	271	13	,	,	PUNCT
ejpam-4298	271	14	therefore	therefore	ADV
ejpam-4298	271	15	x	x	X
ejpam-4298	271	16	is	be	AUX
ejpam-4298	271	17	coc	coc	NOUN
ejpam-4298	271	18	-	-	PUNCT
ejpam-4298	271	19	t	t	PROPN
ejpam-4298	271	20	1	1	NUM
ejpam-4298	271	21	2	2	NUM
ejpam-4298	271	22	-space	-space	NOUN
ejpam-4298	271	23	.	.	PUNCT
ejpam-4298	272	1	corollary	corollary	ADJ
ejpam-4298	272	2	3	3	NUM
ejpam-4298	272	3	.	.	PUNCT
ejpam-4298	273	1	every	every	DET
ejpam-4298	273	2	coc	coc	PROPN
ejpam-4298	273	3	-	-	PUNCT
ejpam-4298	273	4	t1	t1	NOUN
ejpam-4298	273	5	-	-	PUNCT
ejpam-4298	273	6	space	space	NOUN
ejpam-4298	273	7	is	be	AUX
ejpam-4298	273	8	coc	coc	NOUN
ejpam-4298	273	9	-	-	PUNCT
ejpam-4298	273	10	t	t	PROPN
ejpam-4298	273	11	1	1	NUM
ejpam-4298	273	12	2	2	NUM
ejpam-4298	273	13	-space	-space	NOUN
ejpam-4298	273	14	.	.	PUNCT
ejpam-4298	274	1	theorem	theorem	NOUN
ejpam-4298	274	2	19	19	NUM
ejpam-4298	274	3	.	.	PUNCT
ejpam-4298	275	1	for	for	ADP
ejpam-4298	275	2	a	a	DET
ejpam-4298	275	3	topological	topological	ADJ
ejpam-4298	275	4	space	space	NOUN
ejpam-4298	275	5	(	(	PUNCT
ejpam-4298	275	6	x	x	X
ejpam-4298	275	7	,	,	PUNCT
ejpam-4298	275	8	τ	τ	PROPN
ejpam-4298	275	9	)	)	PUNCT
ejpam-4298	275	10	.	.	PUNCT
ejpam-4298	276	1	the	the	DET
ejpam-4298	276	2	following	follow	VERB
ejpam-4298	276	3	are	be	AUX
ejpam-4298	276	4	equivalent	equivalent	ADJ
ejpam-4298	276	5	:	:	PUNCT
ejpam-4298	276	6	f.a	f.a	PROPN
ejpam-4298	276	7	.	.	PROPN
ejpam-4298	276	8	abushaheen	abushaheen	PROPN
ejpam-4298	276	9	/	/	SYM
ejpam-4298	276	10	eur	eur	PROPN
ejpam-4298	276	11	.	.	PUNCT
ejpam-4298	277	1	j.	j.	PROPN
ejpam-4298	277	2	pure	pure	PROPN
ejpam-4298	277	3	appl	appl	PROPN
ejpam-4298	277	4	.	.	PROPN
ejpam-4298	277	5	math	math	PROPN
ejpam-4298	277	6	,	,	PUNCT
ejpam-4298	277	7	15	15	NUM
ejpam-4298	277	8	(	(	PUNCT
ejpam-4298	277	9	2	2	NUM
ejpam-4298	277	10	)	)	PUNCT
ejpam-4298	277	11	(	(	PUNCT
ejpam-4298	277	12	2022	2022	NUM
ejpam-4298	277	13	)	)	PUNCT
ejpam-4298	277	14	,	,	PUNCT
ejpam-4298	277	15	589	589	NUM
ejpam-4298	277	16	-	-	SYM
ejpam-4298	277	17	601	601	NUM
ejpam-4298	277	18	597	597	NUM
ejpam-4298	277	19	(	(	PUNCT
ejpam-4298	277	20	i	i	NOUN
ejpam-4298	277	21	)	)	PUNCT
ejpam-4298	277	22	x	x	X
ejpam-4298	277	23	is	be	AUX
ejpam-4298	277	24	coc	coc	NOUN
ejpam-4298	277	25	-	-	PUNCT
ejpam-4298	277	26	t	t	PROPN
ejpam-4298	277	27	1	1	NUM
ejpam-4298	277	28	2	2	NUM
ejpam-4298	277	29	-space	-space	NOUN
ejpam-4298	277	30	,	,	PUNCT
ejpam-4298	277	31	(	(	PUNCT
ejpam-4298	277	32	ii	ii	NOUN
ejpam-4298	277	33	)	)	PUNCT
ejpam-4298	277	34	every	every	DET
ejpam-4298	277	35	g	g	PROPN
ejpam-4298	277	36	-	-	PUNCT
ejpam-4298	277	37	coc-∧set	coc-∧set	NOUN
ejpam-4298	277	38	is	be	AUX
ejpam-4298	277	39	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	277	40	,	,	PUNCT
ejpam-4298	277	41	(	(	PUNCT
ejpam-4298	277	42	iii	iii	NOUN
ejpam-4298	277	43	)	)	PUNCT
ejpam-4298	277	44	every	every	DET
ejpam-4298	277	45	g	g	NOUN
ejpam-4298	277	46	-	-	PUNCT
ejpam-4298	277	47	coc-∨-set	coc-∨-set	NOUN
ejpam-4298	277	48	is	be	AUX
ejpam-4298	277	49	coc-∨-set	coc-∨-set	VERB
ejpam-4298	277	50	.	.	PUNCT
ejpam-4298	278	1	proof	proof	NOUN
ejpam-4298	278	2	.	.	PUNCT
ejpam-4298	279	1	(	(	PUNCT
ejpam-4298	279	2	iii	iii	X
ejpam-4298	279	3	)	)	PUNCT
ejpam-4298	279	4	⇒	⇒	NOUN
ejpam-4298	279	5	(	(	PUNCT
ejpam-4298	279	6	ii	ii	NOUN
ejpam-4298	279	7	)	)	PUNCT
ejpam-4298	279	8	obvious	obvious	ADJ
ejpam-4298	279	9	.	.	PUNCT
ejpam-4298	280	1	(	(	PUNCT
ejpam-4298	280	2	ii	ii	NOUN
ejpam-4298	280	3	)	)	PUNCT
ejpam-4298	280	4	⇒	⇒	NOUN
ejpam-4298	280	5	(	(	PUNCT
ejpam-4298	280	6	i	i	NOUN
ejpam-4298	280	7	)	)	PUNCT
ejpam-4298	280	8	let	let	VERB
ejpam-4298	280	9	x	x	PUNCT
ejpam-4298	280	10	∈	∈	PROPN
ejpam-4298	280	11	x.	x.	NOUN
ejpam-4298	280	12	if	if	SCONJ
ejpam-4298	280	13	{	{	PUNCT
ejpam-4298	280	14	x	x	NOUN
ejpam-4298	280	15	}	}	PUNCT
ejpam-4298	280	16	is	be	AUX
ejpam-4298	280	17	not	not	PART
ejpam-4298	280	18	coc	coc	ADJ
ejpam-4298	280	19	-	-	ADJ
ejpam-4298	280	20	open	open	ADJ
ejpam-4298	280	21	,	,	PUNCT
ejpam-4298	280	22	then	then	ADV
ejpam-4298	280	23	x	x	ADP
ejpam-4298	280	24	−	−	PROPN
ejpam-4298	280	25	{	{	PUNCT
ejpam-4298	280	26	x	x	NOUN
ejpam-4298	280	27	}	}	PUNCT
ejpam-4298	280	28	is	be	AUX
ejpam-4298	280	29	not	not	PART
ejpam-4298	280	30	coc	coc	NOUN
ejpam-4298	280	31	-	-	PUNCT
ejpam-4298	280	32	closed	closed	ADJ
ejpam-4298	280	33	,	,	PUNCT
ejpam-4298	280	34	so	so	CCONJ
ejpam-4298	280	35	the	the	DET
ejpam-4298	280	36	only	only	ADJ
ejpam-4298	280	37	coc	coc	ADJ
ejpam-4298	280	38	-	-	PUNCT
ejpam-4298	280	39	open	open	ADJ
ejpam-4298	280	40	set	set	NOUN
ejpam-4298	280	41	contains	contain	VERB
ejpam-4298	280	42	x	x	PUNCT
ejpam-4298	280	43	−	−	PROPN
ejpam-4298	280	44	{	{	PUNCT
ejpam-4298	280	45	x	x	NOUN
ejpam-4298	280	46	}	}	PUNCT
ejpam-4298	280	47	is	be	AUX
ejpam-4298	280	48	x	x	NOUN
ejpam-4298	280	49	,	,	PUNCT
ejpam-4298	280	50	but	but	CCONJ
ejpam-4298	280	51	x	x	X
ejpam-4298	280	52	−	−	PROPN
ejpam-4298	280	53	{	{	PUNCT
ejpam-4298	280	54	x	x	NOUN
ejpam-4298	280	55	}	}	PUNCT
ejpam-4298	280	56	is	be	AUX
ejpam-4298	280	57	g	g	NOUN
ejpam-4298	280	58	-	-	PUNCT
ejpam-4298	280	59	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	280	60	,	,	PUNCT
ejpam-4298	280	61	so	so	ADV
ejpam-4298	280	62	x	x	ADP
ejpam-4298	280	63	−	−	PROPN
ejpam-4298	280	64	{	{	PUNCT
ejpam-4298	280	65	x	x	NOUN
ejpam-4298	280	66	}	}	PUNCT
ejpam-4298	280	67	is	be	AUX
ejpam-4298	280	68	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	280	69	,	,	PUNCT
ejpam-4298	280	70	therefore	therefore	ADV
ejpam-4298	280	71	x	x	X
ejpam-4298	280	72	−	−	PROPN
ejpam-4298	280	73	{	{	PUNCT
ejpam-4298	280	74	x	x	NOUN
ejpam-4298	280	75	}	}	PUNCT
ejpam-4298	280	76	is	be	AUX
ejpam-4298	280	77	coc	coc	ADJ
ejpam-4298	280	78	-	-	ADJ
ejpam-4298	280	79	open	open	ADJ
ejpam-4298	280	80	,	,	PUNCT
ejpam-4298	280	81	hence	hence	ADV
ejpam-4298	280	82	{	{	PUNCT
ejpam-4298	280	83	x	x	NOUN
ejpam-4298	280	84	}	}	PUNCT
ejpam-4298	280	85	is	be	AUX
ejpam-4298	280	86	coc	coc	ADJ
ejpam-4298	280	87	-	-	PUNCT
ejpam-4298	280	88	closed	close	VERB
ejpam-4298	280	89	set	set	NOUN
ejpam-4298	280	90	,	,	PUNCT
ejpam-4298	280	91	that	that	PRON
ejpam-4298	280	92	’s	’	VERB
ejpam-4298	280	93	complete	complete	ADJ
ejpam-4298	280	94	the	the	DET
ejpam-4298	280	95	proof	proof	NOUN
ejpam-4298	280	96	.	.	PUNCT
ejpam-4298	281	1	(	(	PUNCT
ejpam-4298	281	2	i	i	NOUN
ejpam-4298	281	3	)	)	PUNCT
ejpam-4298	281	4	⇒	⇒	PROPN
ejpam-4298	281	5	(	(	PUNCT
ejpam-4298	281	6	ii	ii	NOUN
ejpam-4298	281	7	)	)	PUNCT
ejpam-4298	281	8	assume	assume	VERB
ejpam-4298	281	9	that	that	SCONJ
ejpam-4298	281	10	a	a	DET
ejpam-4298	281	11	subset	subset	NOUN
ejpam-4298	281	12	a	a	PRON
ejpam-4298	281	13	of	of	ADP
ejpam-4298	281	14	x	x	NOUN
ejpam-4298	281	15	is	be	AUX
ejpam-4298	281	16	g	g	NOUN
ejpam-4298	281	17	-	-	PUNCT
ejpam-4298	281	18	coc-∧-set	coc-∧-set	NOUN
ejpam-4298	281	19	which	which	PRON
ejpam-4298	281	20	is	be	AUX
ejpam-4298	281	21	not	not	PART
ejpam-4298	281	22	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	281	23	,	,	PUNCT
ejpam-4298	281	24	then	then	ADV
ejpam-4298	281	25	coc	coc	PROPN
ejpam-4298	281	26	-	-	PUNCT
ejpam-4298	281	27	ker(a	ker(a	PROPN
ejpam-4298	281	28	)	)	PUNCT
ejpam-4298	281	29	*	*	PUNCT
ejpam-4298	282	1	a	a	X
ejpam-4298	282	2	,	,	PUNCT
ejpam-4298	282	3	so	so	SCONJ
ejpam-4298	282	4	there	there	PRON
ejpam-4298	282	5	exists	exist	VERB
ejpam-4298	282	6	x	x	X
ejpam-4298	282	7	∈	∈	PROPN
ejpam-4298	282	8	coc	coc	PROPN
ejpam-4298	282	9	-	-	PUNCT
ejpam-4298	282	10	ker(a	ker(a	PROPN
ejpam-4298	282	11	)	)	PUNCT
ejpam-4298	282	12	and	and	CCONJ
ejpam-4298	282	13	x	x	X
ejpam-4298	282	14	/∈	/∈	NOUN
ejpam-4298	282	15	a	a	PRON
ejpam-4298	282	16	,	,	PUNCT
ejpam-4298	282	17	but	but	CCONJ
ejpam-4298	282	18	x	x	X
ejpam-4298	282	19	is	be	AUX
ejpam-4298	282	20	a	a	DET
ejpam-4298	282	21	coc	coc	PROPN
ejpam-4298	282	22	-	-	PUNCT
ejpam-4298	282	23	t	t	PROPN
ejpam-4298	282	24	1	1	NUM
ejpam-4298	282	25	2	2	NUM
ejpam-4298	282	26	-space	-space	NOUN
ejpam-4298	282	27	,	,	PUNCT
ejpam-4298	282	28	so	so	CCONJ
ejpam-4298	282	29	{	{	PUNCT
ejpam-4298	282	30	x	x	X
ejpam-4298	282	31	}	}	PUNCT
ejpam-4298	282	32	is	be	AUX
ejpam-4298	282	33	a	a	DET
ejpam-4298	282	34	coc	coc	NOUN
ejpam-4298	282	35	-	-	PUNCT
ejpam-4298	282	36	open	open	ADJ
ejpam-4298	282	37	or	or	CCONJ
ejpam-4298	282	38	coc	coc	NOUN
ejpam-4298	282	39	-	-	PUNCT
ejpam-4298	282	40	closed	close	VERB
ejpam-4298	282	41	set	set	NOUN
ejpam-4298	282	42	,	,	PUNCT
ejpam-4298	282	43	we	we	PRON
ejpam-4298	282	44	need	need	VERB
ejpam-4298	282	45	to	to	PART
ejpam-4298	282	46	discuss	discuss	VERB
ejpam-4298	282	47	the	the	DET
ejpam-4298	282	48	following	follow	VERB
ejpam-4298	282	49	two	two	NUM
ejpam-4298	282	50	cases	case	NOUN
ejpam-4298	282	51	:	:	PUNCT
ejpam-4298	282	52	(	(	PUNCT
ejpam-4298	282	53	1	1	X
ejpam-4298	282	54	)	)	PUNCT
ejpam-4298	282	55	if	if	SCONJ
ejpam-4298	282	56	{	{	PUNCT
ejpam-4298	282	57	x	x	NOUN
ejpam-4298	282	58	}	}	PUNCT
ejpam-4298	282	59	is	be	AUX
ejpam-4298	282	60	a	a	DET
ejpam-4298	282	61	coc	coc	NOUN
ejpam-4298	282	62	-	-	PUNCT
ejpam-4298	282	63	closed	closed	ADJ
ejpam-4298	282	64	,	,	PUNCT
ejpam-4298	282	65	then	then	ADV
ejpam-4298	282	66	x−{x	x−{x	PROPN
ejpam-4298	282	67	}	}	PUNCT
ejpam-4298	282	68	is	be	AUX
ejpam-4298	282	69	a	a	DET
ejpam-4298	282	70	coc	coc	NOUN
ejpam-4298	282	71	-	-	PUNCT
ejpam-4298	282	72	open	open	ADJ
ejpam-4298	282	73	set	set	NOUN
ejpam-4298	282	74	contains	contain	VERB
ejpam-4298	282	75	a	a	PRON
ejpam-4298	282	76	,	,	PUNCT
ejpam-4298	282	77	but	but	CCONJ
ejpam-4298	282	78	x	x	X
ejpam-4298	282	79	∈	∈	NOUN
ejpam-4298	282	80	coc−ker(a	coc−ker(a	NOUN
ejpam-4298	282	81	)	)	PUNCT
ejpam-4298	282	82	,	,	PUNCT
ejpam-4298	282	83	so	so	ADV
ejpam-4298	282	84	x	x	SYM
ejpam-4298	282	85	∈	∈	PROPN
ejpam-4298	282	86	x	x	PUNCT
ejpam-4298	282	87	−	−	PROPN
ejpam-4298	282	88	{	{	PUNCT
ejpam-4298	282	89	x	x	NOUN
ejpam-4298	282	90	}	}	PUNCT
ejpam-4298	282	91	and	and	CCONJ
ejpam-4298	282	92	this	this	PRON
ejpam-4298	282	93	is	be	AUX
ejpam-4298	282	94	a	a	DET
ejpam-4298	282	95	contradiction	contradiction	NOUN
ejpam-4298	282	96	.	.	PUNCT
ejpam-4298	283	1	(	(	PUNCT
ejpam-4298	283	2	2	2	X
ejpam-4298	283	3	)	)	PUNCT
ejpam-4298	283	4	if	if	SCONJ
ejpam-4298	283	5	{	{	PUNCT
ejpam-4298	283	6	x	x	NOUN
ejpam-4298	283	7	}	}	PUNCT
ejpam-4298	283	8	is	be	AUX
ejpam-4298	283	9	coc	coc	ADJ
ejpam-4298	283	10	-	-	PUNCT
ejpam-4298	283	11	open	open	ADJ
ejpam-4298	283	12	set	set	NOUN
ejpam-4298	283	13	,	,	PUNCT
ejpam-4298	283	14	then	then	ADV
ejpam-4298	283	15	x	x	ADP
ejpam-4298	283	16	−	−	PROPN
ejpam-4298	283	17	{	{	PUNCT
ejpam-4298	283	18	x	x	NOUN
ejpam-4298	283	19	}	}	PUNCT
ejpam-4298	283	20	is	be	AUX
ejpam-4298	283	21	a	a	DET
ejpam-4298	283	22	coc	coc	NOUN
ejpam-4298	283	23	-	-	PUNCT
ejpam-4298	283	24	open	open	ADJ
ejpam-4298	283	25	set	set	NOUN
ejpam-4298	283	26	contains	contain	VERB
ejpam-4298	283	27	a	a	PRON
ejpam-4298	283	28	,	,	PUNCT
ejpam-4298	283	29	by	by	ADP
ejpam-4298	283	30	assumption	assumption	NOUN
ejpam-4298	283	31	coc	coc	PROPN
ejpam-4298	283	32	-	-	PUNCT
ejpam-4298	283	33	ker	ker	NOUN
ejpam-4298	283	34	(	(	PUNCT
ejpam-4298	283	35	a	a	NOUN
ejpam-4298	283	36	)	)	PUNCT
ejpam-4298	283	37	⊆	⊆	NUM
ejpam-4298	283	38	x	x	SYM
ejpam-4298	283	39	−	−	PROPN
ejpam-4298	283	40	{	{	PUNCT
ejpam-4298	283	41	x	x	NOUN
ejpam-4298	283	42	}	}	PUNCT
ejpam-4298	283	43	,	,	PUNCT
ejpam-4298	283	44	i.e.	i.e.	X
ejpam-4298	283	45	x	x	X
ejpam-4298	283	46	/∈coc	/∈coc	ADJ
ejpam-4298	283	47	-	-	PUNCT
ejpam-4298	283	48	ker(a	ker(a	PROPN
ejpam-4298	283	49	)	)	PUNCT
ejpam-4298	283	50	and	and	CCONJ
ejpam-4298	283	51	this	this	PRON
ejpam-4298	283	52	is	be	AUX
ejpam-4298	283	53	a	a	DET
ejpam-4298	283	54	contradiction	contradiction	NOUN
ejpam-4298	283	55	,	,	PUNCT
ejpam-4298	283	56	hence	hence	ADV
ejpam-4298	283	57	a	a	PRON
ejpam-4298	283	58	is	be	AUX
ejpam-4298	283	59	coc-∧-set	coc-∧-set	NOUN
ejpam-4298	283	60	.	.	PUNCT
ejpam-4298	284	1	definition	definition	NOUN
ejpam-4298	284	2	17	17	NUM
ejpam-4298	284	3	.	.	PUNCT
ejpam-4298	285	1	a	a	DET
ejpam-4298	285	2	subset	subset	NOUN
ejpam-4298	285	3	a	a	PRON
ejpam-4298	285	4	of	of	ADP
ejpam-4298	285	5	a	a	DET
ejpam-4298	285	6	topological	topological	ADJ
ejpam-4298	285	7	space	space	NOUN
ejpam-4298	285	8	(	(	PUNCT
ejpam-4298	285	9	x	x	X
ejpam-4298	285	10	,	,	PUNCT
ejpam-4298	285	11	τ	τ	X
ejpam-4298	285	12	)	)	PUNCT
ejpam-4298	285	13	is	be	AUX
ejpam-4298	285	14	called	call	VERB
ejpam-4298	285	15	coc	coc	PROPN
ejpam-4298	285	16	-	-	PUNCT
ejpam-4298	285	17	λ	λ	NOUN
ejpam-4298	285	18	-	-	VERB
ejpam-4298	285	19	closed	closed	ADJ
ejpam-4298	285	20	if	if	SCONJ
ejpam-4298	285	21	a	a	DET
ejpam-4298	285	22	=	=	X
ejpam-4298	285	23	l∩f	l∩f	NOUN
ejpam-4298	285	24	,	,	PUNCT
ejpam-4298	285	25	where	where	SCONJ
ejpam-4298	285	26	l	l	NOUN
ejpam-4298	285	27	is	be	AUX
ejpam-4298	285	28	coc-∧-set	coc-∧-set	PUNCT
ejpam-4298	285	29	and	and	CCONJ
ejpam-4298	285	30	f	f	PROPN
ejpam-4298	285	31	is	be	AUX
ejpam-4298	285	32	coc	coc	ADJ
ejpam-4298	285	33	-	-	PUNCT
ejpam-4298	285	34	closed	close	VERB
ejpam-4298	285	35	set	set	NOUN
ejpam-4298	285	36	.	.	PUNCT
ejpam-4298	286	1	a	a	DET
ejpam-4298	286	2	subset	subset	NOUN
ejpam-4298	286	3	a	a	PRON
ejpam-4298	286	4	is	be	AUX
ejpam-4298	286	5	coc	coc	ADJ
ejpam-4298	286	6	-	-	PUNCT
ejpam-4298	286	7	λ	λ	NOUN
ejpam-4298	286	8	-	-	NOUN
ejpam-4298	286	9	open	open	ADJ
ejpam-4298	286	10	if	if	SCONJ
ejpam-4298	286	11	x	x	PRON
ejpam-4298	286	12	−a	−a	NOUN
ejpam-4298	286	13	is	be	AUX
ejpam-4298	286	14	coc	coc	NOUN
ejpam-4298	286	15	-	-	PUNCT
ejpam-4298	286	16	λclosed	λclose	VERB
ejpam-4298	286	17	.	.	PUNCT
ejpam-4298	287	1	lemma	lemma	PROPN
ejpam-4298	287	2	10	10	NUM
ejpam-4298	287	3	.	.	PUNCT
ejpam-4298	288	1	for	for	ADP
ejpam-4298	288	2	a	a	DET
ejpam-4298	288	3	subset	subset	NOUN
ejpam-4298	288	4	a	a	PRON
ejpam-4298	288	5	of	of	ADP
ejpam-4298	288	6	(	(	PUNCT
ejpam-4298	288	7	x	x	PROPN
ejpam-4298	288	8	,	,	PUNCT
ejpam-4298	288	9	τ	τ	PROPN
ejpam-4298	288	10	)	)	PUNCT
ejpam-4298	288	11	.	.	PUNCT
ejpam-4298	289	1	the	the	DET
ejpam-4298	289	2	following	follow	VERB
ejpam-4298	289	3	are	be	AUX
ejpam-4298	289	4	equivalent	equivalent	ADJ
ejpam-4298	289	5	:	:	PUNCT
ejpam-4298	289	6	(	(	PUNCT
ejpam-4298	289	7	i	i	NOUN
ejpam-4298	289	8	)	)	PUNCT
ejpam-4298	289	9	a	a	PRON
ejpam-4298	289	10	is	be	AUX
ejpam-4298	289	11	coc	coc	ADJ
ejpam-4298	289	12	-	-	PUNCT
ejpam-4298	289	13	λ	λ	NOUN
ejpam-4298	289	14	-	-	VERB
ejpam-4298	289	15	closed	closed	ADJ
ejpam-4298	289	16	,	,	PUNCT
ejpam-4298	289	17	(	(	PUNCT
ejpam-4298	289	18	ii	ii	NOUN
ejpam-4298	289	19	)	)	PUNCT
ejpam-4298	289	20	a	a	DET
ejpam-4298	289	21	=	=	PUNCT
ejpam-4298	289	22	l	l	PROPN
ejpam-4298	289	23	∩a	∩a	PROPN
ejpam-4298	289	24	coc	coc	PROPN
ejpam-4298	289	25	,	,	PUNCT
ejpam-4298	289	26	where	where	SCONJ
ejpam-4298	289	27	l	l	NOUN
ejpam-4298	289	28	is	be	AUX
ejpam-4298	289	29	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	289	30	,	,	PUNCT
ejpam-4298	289	31	(	(	PUNCT
ejpam-4298	289	32	iii	iii	NOUN
ejpam-4298	289	33	)	)	PUNCT
ejpam-4298	289	34	a	a	DET
ejpam-4298	289	35	=	=	ADJ
ejpam-4298	289	36	coc	coc	NOUN
ejpam-4298	289	37	-	-	PUNCT
ejpam-4298	289	38	ker(a	ker(a	ADJ
ejpam-4298	289	39	)	)	PUNCT
ejpam-4298	289	40	∩a	∩a	PROPN
ejpam-4298	289	41	coc	coc	PROPN
ejpam-4298	289	42	.	.	PUNCT
ejpam-4298	290	1	theorem	theorem	VERB
ejpam-4298	290	2	20	20	NUM
ejpam-4298	290	3	.	.	PUNCT
ejpam-4298	291	1	a	a	DET
ejpam-4298	291	2	topological	topological	ADJ
ejpam-4298	291	3	space	space	NOUN
ejpam-4298	291	4	(	(	PUNCT
ejpam-4298	291	5	x	x	X
ejpam-4298	291	6	,	,	PUNCT
ejpam-4298	291	7	τ	τ	X
ejpam-4298	291	8	)	)	PUNCT
ejpam-4298	291	9	is	be	AUX
ejpam-4298	291	10	coc	coc	NOUN
ejpam-4298	291	11	-	-	PUNCT
ejpam-4298	291	12	t	t	PROPN
ejpam-4298	291	13	1	1	NUM
ejpam-4298	291	14	2	2	NUM
ejpam-4298	291	15	-space	-space	NOUN
ejpam-4298	291	16	if	if	SCONJ
ejpam-4298	291	17	and	and	CCONJ
ejpam-4298	291	18	only	only	ADV
ejpam-4298	291	19	if	if	SCONJ
ejpam-4298	291	20	every	every	DET
ejpam-4298	291	21	subset	subset	NOUN
ejpam-4298	291	22	of	of	ADP
ejpam-4298	291	23	x	x	PUNCT
ejpam-4298	291	24	is	be	AUX
ejpam-4298	291	25	coc	coc	ADJ
ejpam-4298	291	26	-	-	PUNCT
ejpam-4298	291	27	λ	λ	NOUN
ejpam-4298	291	28	-	-	PUNCT
ejpam-4298	291	29	closed	closed	ADJ
ejpam-4298	291	30	.	.	PUNCT
ejpam-4298	292	1	proof	proof	NOUN
ejpam-4298	292	2	.	.	PUNCT
ejpam-4298	293	1	(	(	PUNCT
ejpam-4298	293	2	⇐	⇐	NOUN
ejpam-4298	293	3	)	)	PUNCT
ejpam-4298	293	4	let	let	VERB
ejpam-4298	293	5	x	x	SYM
ejpam-4298	293	6	∈	∈	PROPN
ejpam-4298	293	7	x.	x.	NOUN
ejpam-4298	293	8	assume	assume	VERB
ejpam-4298	293	9	that	that	SCONJ
ejpam-4298	293	10	{	{	PUNCT
ejpam-4298	293	11	x	x	X
ejpam-4298	293	12	}	}	PUNCT
ejpam-4298	293	13	is	be	AUX
ejpam-4298	293	14	not	not	PART
ejpam-4298	293	15	coc	coc	ADJ
ejpam-4298	293	16	-	-	ADJ
ejpam-4298	293	17	open	open	ADJ
ejpam-4298	293	18	,	,	PUNCT
ejpam-4298	294	1	then	then	ADV
ejpam-4298	294	2	a	a	DET
ejpam-4298	294	3	=	=	NOUN
ejpam-4298	294	4	x	x	SYM
ejpam-4298	294	5	−	−	PROPN
ejpam-4298	294	6	{	{	PUNCT
ejpam-4298	294	7	x	x	NOUN
ejpam-4298	294	8	}	}	PUNCT
ejpam-4298	294	9	is	be	AUX
ejpam-4298	294	10	not	not	PART
ejpam-4298	294	11	coc	coc	NOUN
ejpam-4298	294	12	-	-	PUNCT
ejpam-4298	294	13	closed	closed	ADJ
ejpam-4298	294	14	,	,	PUNCT
ejpam-4298	294	15	but	but	CCONJ
ejpam-4298	294	16	a	a	PRON
ejpam-4298	294	17	is	be	AUX
ejpam-4298	294	18	coc	coc	ADJ
ejpam-4298	294	19	-	-	PUNCT
ejpam-4298	294	20	λ	λ	NOUN
ejpam-4298	294	21	-	-	VERB
ejpam-4298	294	22	closed	closed	ADJ
ejpam-4298	294	23	,	,	PUNCT
ejpam-4298	294	24	so	so	SCONJ
ejpam-4298	294	25	a	a	PRON
ejpam-4298	294	26	is	be	AUX
ejpam-4298	294	27	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	294	28	,	,	PUNCT
ejpam-4298	294	29	thus	thus	ADV
ejpam-4298	294	30	a	a	PRON
ejpam-4298	294	31	is	be	AUX
ejpam-4298	294	32	coc	coc	ADJ
ejpam-4298	294	33	-	-	PUNCT
ejpam-4298	294	34	open	open	ADJ
ejpam-4298	294	35	set	set	NOUN
ejpam-4298	294	36	,	,	PUNCT
ejpam-4298	294	37	then	then	ADV
ejpam-4298	294	38	a	a	PRON
ejpam-4298	294	39	is	be	AUX
ejpam-4298	294	40	coc	coc	NOUN
ejpam-4298	294	41	-	-	ADJ
ejpam-4298	294	42	open	open	ADJ
ejpam-4298	294	43	,	,	PUNCT
ejpam-4298	294	44	that	that	ADV
ejpam-4298	294	45	is	is	ADV
ejpam-4298	294	46	{	{	PUNCT
ejpam-4298	294	47	x	x	NOUN
ejpam-4298	294	48	}	}	PUNCT
ejpam-4298	294	49	is	be	AUX
ejpam-4298	294	50	coc	coc	NOUN
ejpam-4298	294	51	-	-	PUNCT
ejpam-4298	294	52	closed	closed	ADJ
ejpam-4298	294	53	,	,	PUNCT
ejpam-4298	294	54	which	which	PRON
ejpam-4298	294	55	is	be	AUX
ejpam-4298	294	56	complete	complete	ADJ
ejpam-4298	294	57	the	the	DET
ejpam-4298	294	58	proof	proof	NOUN
ejpam-4298	294	59	.	.	PUNCT
ejpam-4298	295	1	(	(	PUNCT
ejpam-4298	295	2	⇒	⇒	PROPN
ejpam-4298	295	3	)	)	PUNCT
ejpam-4298	295	4	let	let	VERB
ejpam-4298	295	5	a	a	DET
ejpam-4298	295	6	⊆	⊆	NUM
ejpam-4298	295	7	x	x	PUNCT
ejpam-4298	295	8	and	and	CCONJ
ejpam-4298	295	9	x	x	SYM
ejpam-4298	295	10	∈	∈	NOUN
ejpam-4298	295	11	x	x	PUNCT
ejpam-4298	295	12	−	−	NOUN
ejpam-4298	295	13	a.	a.	NOUN
ejpam-4298	295	14	then	then	ADV
ejpam-4298	295	15	{	{	PUNCT
ejpam-4298	295	16	x	x	X
ejpam-4298	295	17	}	}	PUNCT
ejpam-4298	295	18	is	be	AUX
ejpam-4298	295	19	coc	coc	ADJ
ejpam-4298	295	20	-	-	PUNCT
ejpam-4298	295	21	open	open	ADJ
ejpam-4298	295	22	or	or	CCONJ
ejpam-4298	295	23	coc	coc	NOUN
ejpam-4298	295	24	-	-	PUNCT
ejpam-4298	295	25	closed	closed	ADJ
ejpam-4298	295	26	subset	subset	NOUN
ejpam-4298	295	27	of	of	ADP
ejpam-4298	295	28	x.	x.	NOUN
ejpam-4298	295	29	define	define	VERB
ejpam-4298	295	30	b	b	NOUN
ejpam-4298	295	31	=	=	PRON
ejpam-4298	295	32	{	{	PUNCT
ejpam-4298	295	33	x	x	SYM
ejpam-4298	295	34	∈	∈	PROPN
ejpam-4298	295	35	x	x	X
ejpam-4298	295	36	−	−	NOUN
ejpam-4298	295	37	a	a	PRON
ejpam-4298	295	38	,	,	PUNCT
ejpam-4298	295	39	{	{	PUNCT
ejpam-4298	295	40	x	x	NOUN
ejpam-4298	295	41	}	}	PUNCT
ejpam-4298	295	42	∈	∈	PROPN
ejpam-4298	295	43	τk	τk	ADP
ejpam-4298	295	44	}	}	PUNCT
ejpam-4298	295	45	,	,	PUNCT
ejpam-4298	295	46	c	c	X
ejpam-4298	295	47	=	=	PRON
ejpam-4298	295	48	{	{	PUNCT
ejpam-4298	295	49	x	x	SYM
ejpam-4298	295	50	∈	∈	PROPN
ejpam-4298	295	51	x	x	X
ejpam-4298	295	52	−	−	NOUN
ejpam-4298	295	53	a	a	PRON
ejpam-4298	295	54	,	,	PUNCT
ejpam-4298	295	55	x	x	PART
ejpam-4298	295	56	−	−	PROPN
ejpam-4298	295	57	{	{	PUNCT
ejpam-4298	295	58	x	x	NOUN
ejpam-4298	295	59	}	}	PUNCT
ejpam-4298	295	60	∈	∈	PROPN
ejpam-4298	295	61	τk	τk	ADP
ejpam-4298	295	62	}	}	PUNCT
ejpam-4298	295	63	.	.	PUNCT
ejpam-4298	296	1	also	also	ADV
ejpam-4298	296	2	define	define	VERB
ejpam-4298	296	3	f	f	PROPN
ejpam-4298	296	4	=	=	SYM
ejpam-4298	296	5	⋂	⋂	PROPN
ejpam-4298	296	6	x∈b	x∈b	NOUN
ejpam-4298	296	7	(	(	PUNCT
ejpam-4298	296	8	x	x	X
ejpam-4298	296	9	−	−	PROPN
ejpam-4298	296	10	{	{	PUNCT
ejpam-4298	296	11	x	x	NOUN
ejpam-4298	296	12	}	}	PUNCT
ejpam-4298	296	13	)	)	PUNCT
ejpam-4298	297	1	=	=	PUNCT
ejpam-4298	297	2	x	x	PUNCT
ejpam-4298	297	3	−	−	PROPN
ejpam-4298	297	4	b	b	NOUN
ejpam-4298	297	5	,	,	PUNCT
ejpam-4298	297	6	and	and	CCONJ
ejpam-4298	297	7	l	l	NOUN
ejpam-4298	297	8	=	=	SYM
ejpam-4298	297	9	⋂	⋂	PROPN
ejpam-4298	297	10	x∈c	x∈c	PROPN
ejpam-4298	297	11	(	(	PUNCT
ejpam-4298	297	12	x	x	SYM
ejpam-4298	297	13	−	−	PROPN
ejpam-4298	297	14	{	{	PUNCT
ejpam-4298	297	15	x	x	NOUN
ejpam-4298	297	16	}	}	PUNCT
ejpam-4298	297	17	)	)	PUNCT
ejpam-4298	297	18	=	=	PUNCT
ejpam-4298	298	1	x	x	X
ejpam-4298	298	2	−	−	NOUN
ejpam-4298	298	3	c	c	X
ejpam-4298	298	4	,	,	PUNCT
ejpam-4298	298	5	then	then	ADV
ejpam-4298	298	6	f	f	PROPN
ejpam-4298	298	7	is	be	AUX
ejpam-4298	298	8	coc	coc	ADJ
ejpam-4298	298	9	-	-	PUNCT
ejpam-4298	298	10	closed	close	VERB
ejpam-4298	298	11	set	set	NOUN
ejpam-4298	298	12	and	and	CCONJ
ejpam-4298	298	13	l	l	NOUN
ejpam-4298	298	14	is	be	AUX
ejpam-4298	298	15	coc-∧-set	coc-∧-set	VERB
ejpam-4298	298	16	with	with	ADP
ejpam-4298	298	17	l	l	NOUN
ejpam-4298	298	18	∩	∩	ADJ
ejpam-4298	298	19	f	f	PROPN
ejpam-4298	298	20	=	=	SYM
ejpam-4298	298	21	a	a	PROPN
ejpam-4298	298	22	,	,	PUNCT
ejpam-4298	298	23	hence	hence	ADV
ejpam-4298	298	24	a	a	PRON
ejpam-4298	298	25	is	be	AUX
ejpam-4298	298	26	coc	coc	ADJ
ejpam-4298	298	27	-	-	PUNCT
ejpam-4298	298	28	λ	λ	NOUN
ejpam-4298	298	29	-	-	NOUN
ejpam-4298	298	30	set	set	NOUN
ejpam-4298	298	31	.	.	PUNCT
ejpam-4298	299	1	definition	definition	NOUN
ejpam-4298	299	2	18	18	NUM
ejpam-4298	299	3	.	.	PUNCT
ejpam-4298	300	1	a	a	DET
ejpam-4298	300	2	topological	topological	ADJ
ejpam-4298	300	3	space	space	NOUN
ejpam-4298	300	4	(	(	PUNCT
ejpam-4298	300	5	x	x	X
ejpam-4298	300	6	,	,	PUNCT
ejpam-4298	300	7	τ	τ	X
ejpam-4298	300	8	)	)	PUNCT
ejpam-4298	300	9	is	be	AUX
ejpam-4298	300	10	called	call	VERB
ejpam-4298	300	11	coc	coc	PROPN
ejpam-4298	300	12	-	-	PUNCT
ejpam-4298	300	13	t	t	PROPN
ejpam-4298	300	14	1	1	NUM
ejpam-4298	300	15	4	4	NUM
ejpam-4298	300	16	-space	-space	NOUN
ejpam-4298	300	17	if	if	SCONJ
ejpam-4298	300	18	every	every	DET
ejpam-4298	300	19	finite	finite	NOUN
ejpam-4298	300	20	subset	subset	VERB
ejpam-4298	300	21	f	f	PROPN
ejpam-4298	300	22	of	of	ADP
ejpam-4298	300	23	x	x	PUNCT
ejpam-4298	300	24	and	and	CCONJ
ejpam-4298	300	25	every	every	DET
ejpam-4298	300	26	y	y	PROPN
ejpam-4298	300	27	∈	∈	PROPN
ejpam-4298	301	1	x	x	PUNCT
ejpam-4298	301	2	−f	−f	NOUN
ejpam-4298	301	3	,	,	PUNCT
ejpam-4298	301	4	there	there	PRON
ejpam-4298	301	5	exists	exist	VERB
ejpam-4298	301	6	a	a	DET
ejpam-4298	301	7	set	set	NOUN
ejpam-4298	301	8	ay	ay	NOUN
ejpam-4298	301	9	with	with	ADP
ejpam-4298	301	10	f	f	PROPN
ejpam-4298	301	11	⊆	⊆	NUM
ejpam-4298	301	12	ay	ay	NOUN
ejpam-4298	301	13	such	such	ADJ
ejpam-4298	301	14	that	that	SCONJ
ejpam-4298	301	15	{	{	PUNCT
ejpam-4298	301	16	y	y	NOUN
ejpam-4298	301	17	}	}	PUNCT
ejpam-4298	301	18	∩ay	∩ay	PROPN
ejpam-4298	301	19	=	=	SYM
ejpam-4298	301	20	φ	φ	PROPN
ejpam-4298	301	21	and	and	CCONJ
ejpam-4298	301	22	ay	ay	PROPN
ejpam-4298	301	23	is	be	AUX
ejpam-4298	301	24	either	either	CCONJ
ejpam-4298	301	25	coc	coc	ADJ
ejpam-4298	301	26	-	-	PUNCT
ejpam-4298	301	27	open	open	ADJ
ejpam-4298	301	28	or	or	CCONJ
ejpam-4298	301	29	coc	coc	NOUN
ejpam-4298	301	30	-	-	PUNCT
ejpam-4298	301	31	closed	closed	ADJ
ejpam-4298	301	32	.	.	PUNCT
ejpam-4298	302	1	f.a	f.a	PROPN
ejpam-4298	302	2	.	.	PROPN
ejpam-4298	302	3	abushaheen	abushaheen	PROPN
ejpam-4298	302	4	/	/	SYM
ejpam-4298	302	5	eur	eur	PROPN
ejpam-4298	302	6	.	.	PUNCT
ejpam-4298	303	1	j.	j.	PROPN
ejpam-4298	303	2	pure	pure	PROPN
ejpam-4298	303	3	appl	appl	PROPN
ejpam-4298	303	4	.	.	PROPN
ejpam-4298	303	5	math	math	PROPN
ejpam-4298	303	6	,	,	PUNCT
ejpam-4298	303	7	15	15	NUM
ejpam-4298	303	8	(	(	PUNCT
ejpam-4298	303	9	2	2	NUM
ejpam-4298	303	10	)	)	PUNCT
ejpam-4298	303	11	(	(	PUNCT
ejpam-4298	303	12	2022	2022	NUM
ejpam-4298	303	13	)	)	PUNCT
ejpam-4298	303	14	,	,	PUNCT
ejpam-4298	303	15	589	589	NUM
ejpam-4298	303	16	-	-	SYM
ejpam-4298	303	17	601	601	NUM
ejpam-4298	303	18	598	598	NUM
ejpam-4298	303	19	theorem	theorem	NOUN
ejpam-4298	303	20	21	21	NUM
ejpam-4298	303	21	.	.	PUNCT
ejpam-4298	304	1	a	a	DET
ejpam-4298	304	2	topological	topological	ADJ
ejpam-4298	304	3	space	space	NOUN
ejpam-4298	304	4	(	(	PUNCT
ejpam-4298	304	5	x	x	X
ejpam-4298	304	6	,	,	PUNCT
ejpam-4298	304	7	τ	τ	X
ejpam-4298	304	8	)	)	PUNCT
ejpam-4298	304	9	is	be	AUX
ejpam-4298	304	10	coc	coc	NOUN
ejpam-4298	304	11	-	-	PUNCT
ejpam-4298	304	12	t	t	PROPN
ejpam-4298	304	13	1	1	NUM
ejpam-4298	304	14	4	4	NUM
ejpam-4298	304	15	-space	-space	NOUN
ejpam-4298	304	16	if	if	SCONJ
ejpam-4298	305	1	and	and	CCONJ
ejpam-4298	305	2	only	only	ADV
ejpam-4298	305	3	if	if	SCONJ
ejpam-4298	305	4	every	every	DET
ejpam-4298	305	5	finite	finite	NOUN
ejpam-4298	305	6	subset	subset	NOUN
ejpam-4298	305	7	of	of	ADP
ejpam-4298	305	8	x	x	PUNCT
ejpam-4298	305	9	is	be	AUX
ejpam-4298	305	10	coc	coc	ADJ
ejpam-4298	305	11	-	-	PUNCT
ejpam-4298	305	12	λ	λ	NOUN
ejpam-4298	305	13	-	-	PUNCT
ejpam-4298	305	14	closed	closed	ADJ
ejpam-4298	305	15	.	.	PUNCT
ejpam-4298	306	1	proof	proof	NOUN
ejpam-4298	306	2	.	.	PUNCT
ejpam-4298	307	1	(	(	PUNCT
ejpam-4298	307	2	⇒	⇒	PROPN
ejpam-4298	307	3	)	)	PUNCT
ejpam-4298	307	4	let	let	VERB
ejpam-4298	307	5	f	f	PRON
ejpam-4298	307	6	be	be	AUX
ejpam-4298	307	7	any	any	DET
ejpam-4298	307	8	finite	finite	NOUN
ejpam-4298	307	9	subset	subset	NOUN
ejpam-4298	307	10	of	of	ADP
ejpam-4298	307	11	x	x	PUNCT
ejpam-4298	307	12	and	and	CCONJ
ejpam-4298	307	13	y	y	PROPN
ejpam-4298	307	14	∈	∈	PROPN
ejpam-4298	307	15	x−f	x−f	PROPN
ejpam-4298	307	16	.	.	PUNCT
ejpam-4298	308	1	so	so	ADV
ejpam-4298	308	2	there	there	PRON
ejpam-4298	308	3	exist	exist	VERB
ejpam-4298	308	4	a	a	DET
ejpam-4298	308	5	set	set	NOUN
ejpam-4298	308	6	ay	ay	NOUN
ejpam-4298	308	7	such	such	ADJ
ejpam-4298	308	8	that	that	DET
ejpam-4298	308	9	ay∩{x	ay∩{x	NOUN
ejpam-4298	308	10	}	}	PUNCT
ejpam-4298	308	11	=	=	SYM
ejpam-4298	308	12	φ	φ	NUM
ejpam-4298	308	13	,	,	PUNCT
ejpam-4298	308	14	and	and	CCONJ
ejpam-4298	308	15	ay	ay	NOUN
ejpam-4298	308	16	is	be	AUX
ejpam-4298	308	17	either	either	CCONJ
ejpam-4298	308	18	coc	coc	ADJ
ejpam-4298	308	19	-	-	PUNCT
ejpam-4298	308	20	open	open	ADJ
ejpam-4298	308	21	or	or	CCONJ
ejpam-4298	308	22	coc	coc	NOUN
ejpam-4298	308	23	-	-	PUNCT
ejpam-4298	308	24	closed	closed	ADJ
ejpam-4298	308	25	.	.	PUNCT
ejpam-4298	309	1	let	let	VERB
ejpam-4298	309	2	c	c	NOUN
ejpam-4298	309	3	be	be	AUX
ejpam-4298	309	4	the	the	DET
ejpam-4298	309	5	intersection	intersection	NOUN
ejpam-4298	309	6	of	of	ADP
ejpam-4298	309	7	all	all	DET
ejpam-4298	309	8	coc	coc	ADJ
ejpam-4298	309	9	-	-	PUNCT
ejpam-4298	309	10	open	open	ADJ
ejpam-4298	309	11	sets	set	NOUN
ejpam-4298	309	12	ay	ay	INTJ
ejpam-4298	309	13	and	and	CCONJ
ejpam-4298	309	14	let	let	VERB
ejpam-4298	309	15	l	l	NOUN
ejpam-4298	309	16	be	be	AUX
ejpam-4298	309	17	the	the	DET
ejpam-4298	309	18	intersection	intersection	NOUN
ejpam-4298	309	19	of	of	ADP
ejpam-4298	309	20	all	all	DET
ejpam-4298	309	21	coc	coc	ADJ
ejpam-4298	309	22	-	-	PUNCT
ejpam-4298	309	23	closed	close	VERB
ejpam-4298	309	24	sets	set	NOUN
ejpam-4298	309	25	ay	ay	INTJ
ejpam-4298	309	26	,	,	PUNCT
ejpam-4298	309	27	clearly	clearly	ADV
ejpam-4298	309	28	f	f	X
ejpam-4298	309	29	=	=	NOUN
ejpam-4298	309	30	c	c	X
ejpam-4298	309	31	∩l	∩l	ADV
ejpam-4298	309	32	,	,	PUNCT
ejpam-4298	309	33	c	c	PROPN
ejpam-4298	309	34	is	be	AUX
ejpam-4298	309	35	coc-∧-set	coc-∧-set	PUNCT
ejpam-4298	309	36	and	and	CCONJ
ejpam-4298	309	37	l	l	NOUN
ejpam-4298	309	38	is	be	AUX
ejpam-4298	309	39	coc	coc	ADJ
ejpam-4298	309	40	-	-	PUNCT
ejpam-4298	309	41	closed	close	VERB
ejpam-4298	309	42	set	set	NOUN
ejpam-4298	309	43	,	,	PUNCT
ejpam-4298	309	44	hence	hence	ADV
ejpam-4298	309	45	f	f	PROPN
ejpam-4298	309	46	is	be	AUX
ejpam-4298	309	47	coc	coc	ADJ
ejpam-4298	309	48	-	-	PUNCT
ejpam-4298	309	49	λ	λ	NOUN
ejpam-4298	309	50	-	-	PUNCT
ejpam-4298	309	51	closed	closed	ADJ
ejpam-4298	309	52	set	set	NOUN
ejpam-4298	309	53	.	.	PUNCT
ejpam-4298	310	1	(	(	PUNCT
ejpam-4298	310	2	⇐	⇐	NOUN
ejpam-4298	310	3	)	)	PUNCT
ejpam-4298	310	4	let	let	VERB
ejpam-4298	310	5	f	f	NOUN
ejpam-4298	310	6	=	=	SYM
ejpam-4298	310	7	l	l	PROPN
ejpam-4298	310	8	∩	∩	X
ejpam-4298	310	9	c	c	PROPN
ejpam-4298	310	10	and	and	CCONJ
ejpam-4298	310	11	y	y	PROPN
ejpam-4298	310	12	∈	∈	PROPN
ejpam-4298	310	13	x	x	X
ejpam-4298	311	1	−	−	PROPN
ejpam-4298	311	2	f	f	X
ejpam-4298	311	3	where	where	SCONJ
ejpam-4298	311	4	c	c	PROPN
ejpam-4298	311	5	is	be	AUX
ejpam-4298	311	6	coc-∧-set	coc-∧-set	PUNCT
ejpam-4298	311	7	and	and	CCONJ
ejpam-4298	311	8	l	l	PROPN
ejpam-4298	311	9	coc	coc	PROPN
ejpam-4298	311	10	-	-	PUNCT
ejpam-4298	311	11	closed	closed	ADJ
ejpam-4298	311	12	set	set	NOUN
ejpam-4298	311	13	.	.	PUNCT
ejpam-4298	312	1	if	if	SCONJ
ejpam-4298	312	2	y	y	PROPN
ejpam-4298	312	3	/∈	/∈	PUNCT
ejpam-4298	313	1	c	c	X
ejpam-4298	313	2	,	,	PUNCT
ejpam-4298	313	3	we	we	PRON
ejpam-4298	313	4	are	be	AUX
ejpam-4298	313	5	done	do	VERB
ejpam-4298	313	6	.	.	PUNCT
ejpam-4298	314	1	if	if	SCONJ
ejpam-4298	314	2	y	y	PROPN
ejpam-4298	314	3	∈	∈	PROPN
ejpam-4298	314	4	c	c	PROPN
ejpam-4298	314	5	,	,	PUNCT
ejpam-4298	314	6	then	then	ADV
ejpam-4298	314	7	y	y	PROPN
ejpam-4298	314	8	/∈	/∈	PUNCT
ejpam-4298	315	1	l	l	NOUN
ejpam-4298	315	2	,	,	PUNCT
ejpam-4298	315	3	so	so	SCONJ
ejpam-4298	315	4	there	there	PRON
ejpam-4298	315	5	exists	exist	VERB
ejpam-4298	315	6	a	a	DET
ejpam-4298	315	7	coc	coc	NOUN
ejpam-4298	315	8	-	-	PUNCT
ejpam-4298	315	9	open	open	ADJ
ejpam-4298	315	10	set	set	NOUN
ejpam-4298	315	11	uy	uy	INTJ
ejpam-4298	315	12	with	with	ADP
ejpam-4298	315	13	y	y	PROPN
ejpam-4298	315	14	∈	∈	PROPN
ejpam-4298	315	15	uy	uy	ADV
ejpam-4298	315	16	,	,	PUNCT
ejpam-4298	315	17	hence	hence	ADV
ejpam-4298	315	18	x	x	VERB
ejpam-4298	315	19	is	be	AUX
ejpam-4298	315	20	coc	coc	ADJ
ejpam-4298	315	21	-	-	PUNCT
ejpam-4298	315	22	t	t	PROPN
ejpam-4298	315	23	1	1	NUM
ejpam-4298	315	24	4	4	NUM
ejpam-4298	315	25	-space	-space	NOUN
ejpam-4298	315	26	.	.	PUNCT
ejpam-4298	316	1	definition	definition	NOUN
ejpam-4298	316	2	19	19	NUM
ejpam-4298	316	3	.	.	PUNCT
ejpam-4298	317	1	a	a	DET
ejpam-4298	317	2	topological	topological	ADJ
ejpam-4298	317	3	space	space	NOUN
ejpam-4298	317	4	(	(	PUNCT
ejpam-4298	317	5	x	x	X
ejpam-4298	317	6	,	,	PUNCT
ejpam-4298	317	7	τ	τ	X
ejpam-4298	317	8	)	)	PUNCT
ejpam-4298	317	9	is	be	AUX
ejpam-4298	317	10	called	call	VERB
ejpam-4298	317	11	coc	coc	PROPN
ejpam-4298	317	12	-	-	PUNCT
ejpam-4298	317	13	t	t	PROPN
ejpam-4298	317	14	3	3	NUM
ejpam-4298	317	15	8	8	NUM
ejpam-4298	317	16	-space	-space	NOUN
ejpam-4298	317	17	if	if	SCONJ
ejpam-4298	317	18	every	every	DET
ejpam-4298	317	19	countable	countable	ADJ
ejpam-4298	317	20	subset	subset	NOUN
ejpam-4298	317	21	f	f	PROPN
ejpam-4298	317	22	of	of	ADP
ejpam-4298	317	23	x	x	PUNCT
ejpam-4298	317	24	and	and	CCONJ
ejpam-4298	317	25	every	every	DET
ejpam-4298	317	26	y	y	PROPN
ejpam-4298	317	27	∈	∈	PROPN
ejpam-4298	317	28	x	x	PUNCT
ejpam-4298	317	29	−	−	PROPN
ejpam-4298	317	30	f	f	NOUN
ejpam-4298	317	31	,	,	PUNCT
ejpam-4298	317	32	there	there	PRON
ejpam-4298	317	33	exists	exist	VERB
ejpam-4298	317	34	a	a	DET
ejpam-4298	317	35	set	set	NOUN
ejpam-4298	317	36	ay	ay	NOUN
ejpam-4298	317	37	with	with	ADP
ejpam-4298	317	38	f	f	PROPN
ejpam-4298	317	39	⊆	⊆	NUM
ejpam-4298	317	40	ay	ay	NOUN
ejpam-4298	317	41	such	such	ADJ
ejpam-4298	317	42	that	that	SCONJ
ejpam-4298	317	43	{	{	PUNCT
ejpam-4298	317	44	y	y	NOUN
ejpam-4298	317	45	}	}	PUNCT
ejpam-4298	317	46	∩	∩	NOUN
ejpam-4298	317	47	ay	ay	NOUN
ejpam-4298	317	48	=	=	SYM
ejpam-4298	317	49	φ	φ	PROPN
ejpam-4298	317	50	and	and	CCONJ
ejpam-4298	317	51	ay	ay	PROPN
ejpam-4298	317	52	is	be	AUX
ejpam-4298	317	53	coc	coc	ADJ
ejpam-4298	317	54	-	-	PUNCT
ejpam-4298	317	55	open	open	ADJ
ejpam-4298	317	56	or	or	CCONJ
ejpam-4298	317	57	coc	coc	NOUN
ejpam-4298	317	58	-	-	PUNCT
ejpam-4298	317	59	closed	closed	ADJ
ejpam-4298	317	60	.	.	PUNCT
ejpam-4298	318	1	clearly	clearly	ADV
ejpam-4298	318	2	every	every	DET
ejpam-4298	318	3	coc	coc	PROPN
ejpam-4298	318	4	-	-	PUNCT
ejpam-4298	318	5	t	t	NOUN
ejpam-4298	318	6	1	1	NUM
ejpam-4298	318	7	2	2	NUM
ejpam-4298	318	8	-space	-space	NOUN
ejpam-4298	318	9	is	be	AUX
ejpam-4298	318	10	coc	coc	NOUN
ejpam-4298	318	11	-	-	PUNCT
ejpam-4298	318	12	t	t	PROPN
ejpam-4298	318	13	3	3	NUM
ejpam-4298	318	14	8	8	NUM
ejpam-4298	318	15	-space	-space	NOUN
ejpam-4298	318	16	and	and	CCONJ
ejpam-4298	318	17	hence	hence	ADV
ejpam-4298	318	18	coc	coc	NOUN
ejpam-4298	318	19	-	-	PUNCT
ejpam-4298	318	20	t	t	PROPN
ejpam-4298	318	21	1	1	NUM
ejpam-4298	318	22	4	4	NUM
ejpam-4298	318	23	-space	-space	NOUN
ejpam-4298	318	24	.	.	PUNCT
ejpam-4298	319	1	theorem	theorem	NOUN
ejpam-4298	319	2	22	22	NUM
ejpam-4298	319	3	.	.	PUNCT
ejpam-4298	320	1	a	a	DET
ejpam-4298	320	2	topological	topological	ADJ
ejpam-4298	320	3	space	space	NOUN
ejpam-4298	320	4	(	(	PUNCT
ejpam-4298	320	5	x	x	X
ejpam-4298	320	6	,	,	PUNCT
ejpam-4298	320	7	τ	τ	X
ejpam-4298	320	8	)	)	PUNCT
ejpam-4298	320	9	is	be	AUX
ejpam-4298	320	10	coc	coc	NOUN
ejpam-4298	320	11	-	-	PUNCT
ejpam-4298	320	12	t	t	PROPN
ejpam-4298	320	13	3	3	NUM
ejpam-4298	320	14	8	8	NUM
ejpam-4298	320	15	-space	-space	NOUN
ejpam-4298	320	16	if	if	SCONJ
ejpam-4298	320	17	and	and	CCONJ
ejpam-4298	320	18	only	only	ADV
ejpam-4298	320	19	if	if	SCONJ
ejpam-4298	320	20	every	every	DET
ejpam-4298	320	21	countable	countable	ADJ
ejpam-4298	320	22	subset	subset	NOUN
ejpam-4298	320	23	of	of	ADP
ejpam-4298	320	24	x	x	PUNCT
ejpam-4298	320	25	is	be	AUX
ejpam-4298	320	26	coc	coc	ADJ
ejpam-4298	320	27	-	-	PUNCT
ejpam-4298	320	28	λ	λ	NOUN
ejpam-4298	320	29	-	-	PUNCT
ejpam-4298	320	30	closed	closed	ADJ
ejpam-4298	320	31	.	.	PUNCT
ejpam-4298	321	1	proof	proof	NOUN
ejpam-4298	321	2	.	.	PUNCT
ejpam-4298	322	1	same	same	ADJ
ejpam-4298	322	2	as	as	SCONJ
ejpam-4298	322	3	theorem	theorem	NOUN
ejpam-4298	322	4	21	21	NUM
ejpam-4298	322	5	.	.	PUNCT
ejpam-4298	323	1	in	in	ADP
ejpam-4298	323	2	the	the	DET
ejpam-4298	323	3	end	end	NOUN
ejpam-4298	323	4	of	of	ADP
ejpam-4298	323	5	this	this	DET
ejpam-4298	323	6	section	section	NOUN
ejpam-4298	323	7	,	,	PUNCT
ejpam-4298	323	8	we	we	PRON
ejpam-4298	323	9	give	give	VERB
ejpam-4298	323	10	weak	weak	ADJ
ejpam-4298	323	11	forms	form	NOUN
ejpam-4298	323	12	of	of	ADP
ejpam-4298	323	13	coc	coc	NOUN
ejpam-4298	323	14	-	-	PUNCT
ejpam-4298	323	15	d1	d1	NOUN
ejpam-4298	323	16	-	-	PUNCT
ejpam-4298	323	17	space	space	NOUN
ejpam-4298	323	18	and	and	CCONJ
ejpam-4298	323	19	coc	coc	NOUN
ejpam-4298	323	20	-	-	PUNCT
ejpam-4298	323	21	r0	r0	NOUN
ejpam-4298	323	22	-	-	PUNCT
ejpam-4298	323	23	space	space	NOUN
ejpam-4298	323	24	.	.	PUNCT
ejpam-4298	324	1	definition	definition	NOUN
ejpam-4298	324	2	20	20	NUM
ejpam-4298	324	3	.	.	PUNCT
ejpam-4298	325	1	a	a	DET
ejpam-4298	325	2	topological	topological	ADJ
ejpam-4298	325	3	space	space	NOUN
ejpam-4298	325	4	(	(	PUNCT
ejpam-4298	325	5	x	x	X
ejpam-4298	325	6	,	,	PUNCT
ejpam-4298	325	7	τ	τ	X
ejpam-4298	325	8	)	)	PUNCT
ejpam-4298	325	9	is	be	AUX
ejpam-4298	325	10	called	call	VERB
ejpam-4298	325	11	weak	weak	ADJ
ejpam-4298	325	12	coc	coc	ADJ
ejpam-4298	325	13	-	-	PUNCT
ejpam-4298	325	14	d1	d1	NOUN
ejpam-4298	325	15	-	-	PUNCT
ejpam-4298	325	16	space	space	NOUN
ejpam-4298	325	17	if	if	SCONJ
ejpam-4298	325	18	⋂	⋂	PROPN
ejpam-4298	325	19	x∈x	x∈x	PROPN
ejpam-4298	325	20	{	{	PUNCT
ejpam-4298	325	21	x}coc	x}coc	PROPN
ejpam-4298	325	22	=	=	SYM
ejpam-4298	325	23	φ	φ	PROPN
ejpam-4298	325	24	.	.	PUNCT
ejpam-4298	326	1	theorem	theorem	VERB
ejpam-4298	326	2	23	23	NUM
ejpam-4298	326	3	.	.	PUNCT
ejpam-4298	327	1	a	a	DET
ejpam-4298	327	2	coc	coc	NOUN
ejpam-4298	327	3	-	-	PUNCT
ejpam-4298	327	4	closed	closed	ADJ
ejpam-4298	327	5	subspace	subspace	NOUN
ejpam-4298	327	6	of	of	ADP
ejpam-4298	327	7	weak	weak	ADJ
ejpam-4298	327	8	coc	coc	ADJ
ejpam-4298	327	9	-	-	PUNCT
ejpam-4298	327	10	d1	d1	NOUN
ejpam-4298	327	11	-	-	PUNCT
ejpam-4298	327	12	space	space	NOUN
ejpam-4298	327	13	(	(	PUNCT
ejpam-4298	327	14	x	x	X
ejpam-4298	327	15	,	,	PUNCT
ejpam-4298	327	16	τ	τ	X
ejpam-4298	327	17	)	)	PUNCT
ejpam-4298	327	18	is	be	AUX
ejpam-4298	327	19	weak	weak	ADJ
ejpam-4298	327	20	coc	coc	ADJ
ejpam-4298	327	21	-	-	PUNCT
ejpam-4298	327	22	d1	d1	NOUN
ejpam-4298	327	23	-	-	PUNCT
ejpam-4298	327	24	space	space	NOUN
ejpam-4298	327	25	.	.	PUNCT
ejpam-4298	328	1	theorem	theorem	NOUN
ejpam-4298	328	2	24	24	NUM
ejpam-4298	328	3	.	.	PUNCT
ejpam-4298	329	1	a	a	DET
ejpam-4298	329	2	topological	topological	ADJ
ejpam-4298	329	3	space	space	NOUN
ejpam-4298	329	4	(	(	PUNCT
ejpam-4298	329	5	x	x	X
ejpam-4298	329	6	,	,	PUNCT
ejpam-4298	329	7	τ	τ	X
ejpam-4298	329	8	)	)	PUNCT
ejpam-4298	329	9	is	be	AUX
ejpam-4298	329	10	weak	weak	ADJ
ejpam-4298	329	11	coc	coc	ADJ
ejpam-4298	329	12	-	-	PUNCT
ejpam-4298	329	13	d1	d1	NOUN
ejpam-4298	329	14	-	-	PUNCT
ejpam-4298	329	15	space	space	NOUN
ejpam-4298	329	16	if	if	SCONJ
ejpam-4298	329	17	and	and	CCONJ
ejpam-4298	329	18	only	only	ADV
ejpam-4298	329	19	if	if	SCONJ
ejpam-4298	329	20	intcoc(ax	intcoc(ax	NUM
ejpam-4298	329	21	)	)	PUNCT
ejpam-4298	329	22	6=	6=	PUNCT
ejpam-4298	329	23	x	x	PUNCT
ejpam-4298	329	24	for	for	ADP
ejpam-4298	329	25	all	all	PRON
ejpam-4298	329	26	x	x	SYM
ejpam-4298	329	27	∈	∈	NOUN
ejpam-4298	329	28	ax	ax	NOUN
ejpam-4298	329	29	⊆	⊆	NUM
ejpam-4298	329	30	x.	x.	NOUN
ejpam-4298	329	31	proof	proof	NOUN
ejpam-4298	329	32	.	.	PUNCT
ejpam-4298	330	1	(	(	PUNCT
ejpam-4298	330	2	⇒	⇒	NOUN
ejpam-4298	330	3	)	)	PUNCT
ejpam-4298	330	4	assume	assume	VERB
ejpam-4298	330	5	that	that	SCONJ
ejpam-4298	330	6	there	there	PRON
ejpam-4298	330	7	exists	exist	VERB
ejpam-4298	330	8	y	y	PROPN
ejpam-4298	330	9	∈	∈	PROPN
ejpam-4298	330	10	x	x	PUNCT
ejpam-4298	330	11	with	with	ADP
ejpam-4298	330	12	intcoc({ay	intcoc({ay	NUM
ejpam-4298	330	13	}	}	PUNCT
ejpam-4298	330	14	)	)	PUNCT
ejpam-4298	331	1	=	=	SYM
ejpam-4298	331	2	x	x	X
ejpam-4298	331	3	,	,	PUNCT
ejpam-4298	331	4	then	then	ADV
ejpam-4298	331	5	y	y	PROPN
ejpam-4298	331	6	∈	∈	PROPN
ejpam-4298	331	7	{	{	PUNCT
ejpam-4298	331	8	x}coc	x}coc	PROPN
ejpam-4298	331	9	for	for	ADP
ejpam-4298	331	10	each	each	DET
ejpam-4298	331	11	x	x	SYM
ejpam-4298	331	12	∈	∈	PROPN
ejpam-4298	331	13	x	x	NOUN
ejpam-4298	331	14	,	,	PUNCT
ejpam-4298	331	15	this	this	PRON
ejpam-4298	331	16	is	be	AUX
ejpam-4298	331	17	a	a	DET
ejpam-4298	331	18	contradiction	contradiction	NOUN
ejpam-4298	331	19	,	,	PUNCT
ejpam-4298	331	20	hence	hence	ADV
ejpam-4298	331	21	the	the	DET
ejpam-4298	331	22	result	result	NOUN
ejpam-4298	331	23	.	.	PUNCT
ejpam-4298	332	1	(	(	PUNCT
ejpam-4298	332	2	⇐	⇐	NOUN
ejpam-4298	332	3	)	)	PUNCT
ejpam-4298	332	4	let	let	VERB
ejpam-4298	332	5	y	y	PROPN
ejpam-4298	332	6	∈	∈	PROPN
ejpam-4298	332	7	⋂	⋂	PROPN
ejpam-4298	332	8	x∈x	x∈x	PROPN
ejpam-4298	332	9	{	{	PUNCT
ejpam-4298	332	10	x}coc	x}coc	PROPN
ejpam-4298	332	11	,	,	PUNCT
ejpam-4298	332	12	then	then	ADV
ejpam-4298	332	13	the	the	DET
ejpam-4298	332	14	coc	coc	NOUN
ejpam-4298	332	15	-	-	PUNCT
ejpam-4298	332	16	open	open	ADJ
ejpam-4298	332	17	set	set	NOUN
ejpam-4298	332	18	contains	contain	VERB
ejpam-4298	332	19	y	y	PROPN
ejpam-4298	332	20	must	must	AUX
ejpam-4298	332	21	be	be	AUX
ejpam-4298	332	22	x	x	X
ejpam-4298	332	23	,	,	PUNCT
ejpam-4298	332	24	so	so	ADV
ejpam-4298	332	25	intcoc({ay	intcoc({ay	PROPN
ejpam-4298	332	26	}	}	PUNCT
ejpam-4298	332	27	)	)	PUNCT
ejpam-4298	333	1	=	=	SYM
ejpam-4298	333	2	x	x	X
ejpam-4298	333	3	,	,	PUNCT
ejpam-4298	333	4	this	this	PRON
ejpam-4298	333	5	is	be	AUX
ejpam-4298	333	6	a	a	DET
ejpam-4298	333	7	contradiction	contradiction	NOUN
ejpam-4298	333	8	,	,	PUNCT
ejpam-4298	333	9	hence	hence	ADV
ejpam-4298	333	10	the	the	DET
ejpam-4298	333	11	result	result	NOUN
ejpam-4298	333	12	.	.	PUNCT
ejpam-4298	334	1	corollary	corollary	ADJ
ejpam-4298	334	2	4	4	NUM
ejpam-4298	334	3	.	.	PUNCT
ejpam-4298	335	1	a	a	DET
ejpam-4298	335	2	topological	topological	ADJ
ejpam-4298	335	3	space	space	NOUN
ejpam-4298	335	4	(	(	PUNCT
ejpam-4298	335	5	x	x	X
ejpam-4298	335	6	,	,	PUNCT
ejpam-4298	335	7	τ	τ	X
ejpam-4298	335	8	)	)	PUNCT
ejpam-4298	335	9	is	be	AUX
ejpam-4298	335	10	coc	coc	ADJ
ejpam-4298	335	11	-	-	PUNCT
ejpam-4298	335	12	d1	d1	NOUN
ejpam-4298	335	13	-	-	PUNCT
ejpam-4298	335	14	space	space	NOUN
ejpam-4298	335	15	if	if	SCONJ
ejpam-4298	335	16	and	and	CCONJ
ejpam-4298	335	17	only	only	ADV
ejpam-4298	335	18	if	if	SCONJ
ejpam-4298	335	19	(	(	PUNCT
ejpam-4298	335	20	x	x	NOUN
ejpam-4298	335	21	,	,	PUNCT
ejpam-4298	335	22	τ	τ	X
ejpam-4298	335	23	)	)	PUNCT
ejpam-4298	335	24	is	be	AUX
ejpam-4298	335	25	coc	coc	NOUN
ejpam-4298	335	26	-	-	PUNCT
ejpam-4298	335	27	t0	t0	NOUN
ejpam-4298	335	28	-	-	PUNCT
ejpam-4298	335	29	space	space	NOUN
ejpam-4298	335	30	and	and	CCONJ
ejpam-4298	335	31	weak	weak	ADJ
ejpam-4298	335	32	coc	coc	ADJ
ejpam-4298	335	33	-	-	PUNCT
ejpam-4298	335	34	d1	d1	NOUN
ejpam-4298	335	35	-	-	PUNCT
ejpam-4298	335	36	space	space	NOUN
ejpam-4298	335	37	.	.	PUNCT
ejpam-4298	336	1	theorem	theorem	VERB
ejpam-4298	336	2	25	25	NUM
ejpam-4298	336	3	.	.	PUNCT
ejpam-4298	337	1	a	a	DET
ejpam-4298	337	2	topological	topological	ADJ
ejpam-4298	337	3	space	space	NOUN
ejpam-4298	337	4	(	(	PUNCT
ejpam-4298	337	5	x	x	X
ejpam-4298	337	6	,	,	PUNCT
ejpam-4298	337	7	τ	τ	X
ejpam-4298	337	8	)	)	PUNCT
ejpam-4298	337	9	is	be	AUX
ejpam-4298	337	10	weak	weak	ADJ
ejpam-4298	337	11	coc	coc	ADJ
ejpam-4298	337	12	-	-	PUNCT
ejpam-4298	337	13	d1	d1	NOUN
ejpam-4298	337	14	-	-	PUNCT
ejpam-4298	337	15	space	space	NOUN
ejpam-4298	337	16	if	if	SCONJ
ejpam-4298	337	17	and	and	CCONJ
ejpam-4298	337	18	only	only	ADV
ejpam-4298	337	19	if	if	SCONJ
ejpam-4298	337	20	coc	coc	ADJ
ejpam-4298	337	21	-	-	PUNCT
ejpam-4298	337	22	ker({x	ker({x	NOUN
ejpam-4298	337	23	}	}	PUNCT
ejpam-4298	337	24	)	)	PUNCT
ejpam-4298	338	1	6=	6=	ADP
ejpam-4298	338	2	x	x	PUNCT
ejpam-4298	338	3	for	for	ADP
ejpam-4298	338	4	all	all	DET
ejpam-4298	338	5	x	x	SYM
ejpam-4298	338	6	∈	∈	ADJ
ejpam-4298	338	7	x.	x.	NOUN
ejpam-4298	338	8	proof	proof	NOUN
ejpam-4298	338	9	.	.	PUNCT
ejpam-4298	339	1	(	(	PUNCT
ejpam-4298	339	2	⇒	⇒	NOUN
ejpam-4298	339	3	)	)	PUNCT
ejpam-4298	339	4	obvious	obvious	ADJ
ejpam-4298	339	5	.	.	PUNCT
ejpam-4298	340	1	(	(	PUNCT
ejpam-4298	340	2	⇐	⇐	NOUN
ejpam-4298	340	3	)	)	PUNCT
ejpam-4298	340	4	from	from	ADP
ejpam-4298	340	5	theorem	theorem	ADJ
ejpam-4298	340	6	24	24	NUM
ejpam-4298	340	7	.	.	PUNCT
ejpam-4298	341	1	f.a	f.a	PROPN
ejpam-4298	341	2	.	.	PROPN
ejpam-4298	341	3	abushaheen	abushaheen	PROPN
ejpam-4298	341	4	/	/	SYM
ejpam-4298	341	5	eur	eur	PROPN
ejpam-4298	341	6	.	.	PUNCT
ejpam-4298	342	1	j.	j.	PROPN
ejpam-4298	342	2	pure	pure	PROPN
ejpam-4298	342	3	appl	appl	PROPN
ejpam-4298	342	4	.	.	PROPN
ejpam-4298	342	5	math	math	PROPN
ejpam-4298	342	6	,	,	PUNCT
ejpam-4298	342	7	15	15	NUM
ejpam-4298	342	8	(	(	PUNCT
ejpam-4298	342	9	2	2	NUM
ejpam-4298	342	10	)	)	PUNCT
ejpam-4298	342	11	(	(	PUNCT
ejpam-4298	342	12	2022	2022	NUM
ejpam-4298	342	13	)	)	PUNCT
ejpam-4298	342	14	,	,	PUNCT
ejpam-4298	342	15	589	589	NUM
ejpam-4298	342	16	-	-	SYM
ejpam-4298	342	17	601	601	NUM
ejpam-4298	342	18	599	599	NUM
ejpam-4298	342	19	definition	definition	NOUN
ejpam-4298	342	20	21	21	NUM
ejpam-4298	342	21	.	.	PUNCT
ejpam-4298	343	1	a	a	DET
ejpam-4298	343	2	topological	topological	ADJ
ejpam-4298	343	3	space	space	NOUN
ejpam-4298	343	4	(	(	PUNCT
ejpam-4298	343	5	x	x	X
ejpam-4298	343	6	,	,	PUNCT
ejpam-4298	343	7	τ	τ	X
ejpam-4298	343	8	)	)	PUNCT
ejpam-4298	343	9	is	be	AUX
ejpam-4298	343	10	called	call	VERB
ejpam-4298	343	11	weak	weak	ADJ
ejpam-4298	343	12	coc	coc	ADJ
ejpam-4298	343	13	-	-	PUNCT
ejpam-4298	343	14	r0	r0	NOUN
ejpam-4298	343	15	-	-	PUNCT
ejpam-4298	343	16	space	space	NOUN
ejpam-4298	343	17	if	if	SCONJ
ejpam-4298	343	18	every	every	DET
ejpam-4298	343	19	coc	coc	PROPN
ejpam-4298	343	20	-	-	PUNCT
ejpam-4298	343	21	λ	λ	NOUN
ejpam-4298	343	22	-	-	PUNCT
ejpam-4298	343	23	closed	closed	ADJ
ejpam-4298	343	24	singleton	singleton	NOUN
ejpam-4298	343	25	is	be	AUX
ejpam-4298	343	26	a	a	DET
ejpam-4298	343	27	coc-∧-set	coc-∧-set	NOUN
ejpam-4298	343	28	.	.	PUNCT
ejpam-4298	344	1	theorem	theorem	PROPN
ejpam-4298	344	2	26	26	NUM
ejpam-4298	344	3	.	.	PUNCT
ejpam-4298	345	1	every	every	DET
ejpam-4298	345	2	coc	coc	PROPN
ejpam-4298	345	3	-	-	PUNCT
ejpam-4298	345	4	r0	r0	NOUN
ejpam-4298	345	5	-	-	PUNCT
ejpam-4298	345	6	space	space	NOUN
ejpam-4298	345	7	(	(	PUNCT
ejpam-4298	345	8	x	x	X
ejpam-4298	345	9	,	,	PUNCT
ejpam-4298	345	10	τ	τ	X
ejpam-4298	345	11	)	)	PUNCT
ejpam-4298	345	12	is	be	AUX
ejpam-4298	345	13	weak	weak	ADJ
ejpam-4298	345	14	coc	coc	ADJ
ejpam-4298	345	15	-	-	PUNCT
ejpam-4298	345	16	r0	r0	NOUN
ejpam-4298	345	17	-	-	PUNCT
ejpam-4298	345	18	space	space	NOUN
ejpam-4298	345	19	.	.	PUNCT
ejpam-4298	346	1	proof	proof	NOUN
ejpam-4298	346	2	.	.	PUNCT
ejpam-4298	347	1	let	let	VERB
ejpam-4298	347	2	x	x	PUNCT
ejpam-4298	347	3	∈	∈	PROPN
ejpam-4298	347	4	x	x	PUNCT
ejpam-4298	347	5	with	with	ADP
ejpam-4298	347	6	{	{	PUNCT
ejpam-4298	347	7	x	x	NOUN
ejpam-4298	347	8	}	}	PUNCT
ejpam-4298	347	9	is	be	AUX
ejpam-4298	347	10	coc−λ−closed	coc−λ−close	VERB
ejpam-4298	347	11	.	.	PUNCT
ejpam-4298	348	1	by	by	ADP
ejpam-4298	348	2	lemma	lemma	PROPN
ejpam-4298	348	3	10	10	NUM
ejpam-4298	348	4	{	{	PUNCT
ejpam-4298	348	5	x	x	NOUN
ejpam-4298	348	6	}	}	PUNCT
ejpam-4298	348	7	=	=	SYM
ejpam-4298	348	8	coc	coc	ADJ
ejpam-4298	348	9	-	-	PUNCT
ejpam-4298	348	10	ker({x	ker({x	NOUN
ejpam-4298	348	11	}	}	PUNCT
ejpam-4298	348	12	)	)	PUNCT
ejpam-4298	348	13	∩	∩	NOUN
ejpam-4298	348	14	{	{	PUNCT
ejpam-4298	348	15	x}coc	x}coc	PROPN
ejpam-4298	348	16	.	.	PUNCT
ejpam-4298	349	1	if	if	SCONJ
ejpam-4298	349	2	{	{	PUNCT
ejpam-4298	349	3	x	x	NOUN
ejpam-4298	349	4	}	}	PUNCT
ejpam-4298	349	5	is	be	AUX
ejpam-4298	349	6	not	not	PART
ejpam-4298	349	7	coc	coc	ADJ
ejpam-4298	349	8	-	-	PUNCT
ejpam-4298	349	9	ker	ker	NOUN
ejpam-4298	349	10	-	-	PUNCT
ejpam-4298	349	11	set	set	NOUN
ejpam-4298	349	12	,	,	PUNCT
ejpam-4298	349	13	then	then	ADV
ejpam-4298	349	14	there	there	PRON
ejpam-4298	349	15	exists	exist	VERB
ejpam-4298	349	16	y	y	PROPN
ejpam-4298	349	17	∈	∈	PROPN
ejpam-4298	349	18	coc−ker({x})−{x	coc−ker({x})−{x	PROPN
ejpam-4298	349	19	}	}	PUNCT
ejpam-4298	349	20	with	with	ADP
ejpam-4298	349	21	y	y	PROPN
ejpam-4298	349	22	/∈	/∈	PUNCT
ejpam-4298	349	23	{	{	PUNCT
ejpam-4298	349	24	x}coc	x}coc	PROPN
ejpam-4298	349	25	,	,	PUNCT
ejpam-4298	349	26	but	but	CCONJ
ejpam-4298	349	27	x	x	X
ejpam-4298	349	28	is	be	AUX
ejpam-4298	349	29	coc	coc	ADJ
ejpam-4298	349	30	-	-	PUNCT
ejpam-4298	349	31	r0	r0	NOUN
ejpam-4298	349	32	-	-	PUNCT
ejpam-4298	349	33	space	space	NOUN
ejpam-4298	349	34	,	,	PUNCT
ejpam-4298	349	35	so	so	CCONJ
ejpam-4298	349	36	{	{	PUNCT
ejpam-4298	349	37	x}coc	x}coc	PROPN
ejpam-4298	349	38	∩	∩	PROPN
ejpam-4298	349	39	{	{	PUNCT
ejpam-4298	349	40	y}coc	y}coc	PROPN
ejpam-4298	349	41	=	=	SYM
ejpam-4298	349	42	φ	φ	PROPN
ejpam-4298	349	43	and	and	CCONJ
ejpam-4298	349	44	x	x	PROPN
ejpam-4298	349	45	∈	∈	PROPN
ejpam-4298	349	46	{	{	PUNCT
ejpam-4298	349	47	y}coc	y}coc	NOUN
ejpam-4298	349	48	,	,	PUNCT
ejpam-4298	349	49	therefore	therefore	ADV
ejpam-4298	349	50	there	there	PRON
ejpam-4298	349	51	exists	exist	VERB
ejpam-4298	349	52	a	a	DET
ejpam-4298	349	53	cocopen	cocopen	NOUN
ejpam-4298	349	54	set	set	VERB
ejpam-4298	349	55	ux	ux	PROPN
ejpam-4298	349	56	contains	contain	VERB
ejpam-4298	349	57	x	x	PUNCT
ejpam-4298	349	58	but	but	CCONJ
ejpam-4298	349	59	not	not	PART
ejpam-4298	349	60	y	y	NOUN
ejpam-4298	349	61	,	,	PUNCT
ejpam-4298	349	62	thus	thus	ADV
ejpam-4298	349	63	y	y	PROPN
ejpam-4298	349	64	/∈	/∈	PUNCT
ejpam-4298	349	65	coc	coc	PROPN
ejpam-4298	349	66	−	−	PROPN
ejpam-4298	349	67	ker({x	ker({x	NOUN
ejpam-4298	349	68	}	}	PUNCT
ejpam-4298	349	69	)	)	PUNCT
ejpam-4298	349	70	,	,	PUNCT
ejpam-4298	349	71	and	and	CCONJ
ejpam-4298	349	72	this	this	PRON
ejpam-4298	349	73	is	be	AUX
ejpam-4298	349	74	a	a	DET
ejpam-4298	349	75	contradiction	contradiction	NOUN
ejpam-4298	349	76	which	which	PRON
ejpam-4298	349	77	completes	complete	VERB
ejpam-4298	349	78	the	the	DET
ejpam-4298	349	79	proof	proof	NOUN
ejpam-4298	349	80	.	.	PUNCT
ejpam-4298	350	1	the	the	DET
ejpam-4298	350	2	following	follow	VERB
ejpam-4298	350	3	theorems	theorem	NOUN
ejpam-4298	350	4	are	be	AUX
ejpam-4298	350	5	easily	easily	ADV
ejpam-4298	350	6	to	to	PART
ejpam-4298	350	7	prove	prove	VERB
ejpam-4298	350	8	.	.	PUNCT
ejpam-4298	351	1	theorem	theorem	VERB
ejpam-4298	351	2	27	27	NUM
ejpam-4298	351	3	.	.	PUNCT
ejpam-4298	352	1	for	for	ADP
ejpam-4298	352	2	a	a	DET
ejpam-4298	352	3	topological	topological	ADJ
ejpam-4298	352	4	space	space	NOUN
ejpam-4298	352	5	(	(	PUNCT
ejpam-4298	352	6	x	x	X
ejpam-4298	352	7	,	,	PUNCT
ejpam-4298	352	8	τ	τ	PROPN
ejpam-4298	352	9	)	)	PUNCT
ejpam-4298	352	10	.	.	PUNCT
ejpam-4298	353	1	the	the	DET
ejpam-4298	353	2	following	follow	VERB
ejpam-4298	353	3	are	be	AUX
ejpam-4298	353	4	equivalent	equivalent	ADJ
ejpam-4298	353	5	:	:	PUNCT
ejpam-4298	353	6	(	(	PUNCT
ejpam-4298	353	7	i	i	NOUN
ejpam-4298	353	8	)	)	PUNCT
ejpam-4298	353	9	x	x	X
ejpam-4298	353	10	is	be	AUX
ejpam-4298	353	11	coc	coc	NOUN
ejpam-4298	353	12	-	-	PUNCT
ejpam-4298	353	13	t1	t1	NOUN
ejpam-4298	353	14	-	-	PUNCT
ejpam-4298	353	15	space	space	NOUN
ejpam-4298	353	16	,	,	PUNCT
ejpam-4298	353	17	(	(	PUNCT
ejpam-4298	353	18	ii	ii	NOUN
ejpam-4298	353	19	)	)	PUNCT
ejpam-4298	353	20	every	every	DET
ejpam-4298	353	21	subset	subset	NOUN
ejpam-4298	353	22	of	of	ADP
ejpam-4298	353	23	x	x	PUNCT
ejpam-4298	353	24	is	be	AUX
ejpam-4298	353	25	coc-∧-set	coc-∧-set	ADJ
ejpam-4298	353	26	,	,	PUNCT
ejpam-4298	353	27	(	(	PUNCT
ejpam-4298	353	28	iii	iii	X
ejpam-4298	353	29	)	)	PUNCT
ejpam-4298	353	30	every	every	DET
ejpam-4298	353	31	singleton	singleton	NOUN
ejpam-4298	353	32	of	of	ADP
ejpam-4298	353	33	x	x	SYM
ejpam-4298	353	34	is	be	AUX
ejpam-4298	353	35	coc-∧-set	coc-∧-set	NOUN
ejpam-4298	353	36	.	.	PUNCT
ejpam-4298	354	1	theorem	theorem	PROPN
ejpam-4298	354	2	28	28	NUM
ejpam-4298	354	3	.	.	PUNCT
ejpam-4298	355	1	for	for	ADP
ejpam-4298	355	2	a	a	DET
ejpam-4298	355	3	topological	topological	ADJ
ejpam-4298	355	4	space	space	NOUN
ejpam-4298	355	5	(	(	PUNCT
ejpam-4298	355	6	x	x	X
ejpam-4298	355	7	,	,	PUNCT
ejpam-4298	355	8	τ	τ	PROPN
ejpam-4298	355	9	)	)	PUNCT
ejpam-4298	355	10	.	.	PUNCT
ejpam-4298	356	1	the	the	DET
ejpam-4298	356	2	following	follow	VERB
ejpam-4298	356	3	are	be	AUX
ejpam-4298	356	4	equivalent	equivalent	ADJ
ejpam-4298	356	5	:	:	PUNCT
ejpam-4298	356	6	(	(	PUNCT
ejpam-4298	356	7	i	i	NOUN
ejpam-4298	356	8	)	)	PUNCT
ejpam-4298	356	9	x	x	X
ejpam-4298	356	10	is	be	AUX
ejpam-4298	356	11	coc	coc	NOUN
ejpam-4298	356	12	-	-	PUNCT
ejpam-4298	356	13	t1	t1	NOUN
ejpam-4298	356	14	-	-	PUNCT
ejpam-4298	356	15	space	space	NOUN
ejpam-4298	356	16	,	,	PUNCT
ejpam-4298	356	17	(	(	PUNCT
ejpam-4298	356	18	ii	ii	NOUN
ejpam-4298	356	19	)	)	PUNCT
ejpam-4298	356	20	x	x	X
ejpam-4298	356	21	is	be	AUX
ejpam-4298	356	22	coc	coc	NOUN
ejpam-4298	356	23	-	-	PUNCT
ejpam-4298	356	24	t0	t0	NOUN
ejpam-4298	356	25	-	-	PUNCT
ejpam-4298	356	26	space	space	NOUN
ejpam-4298	356	27	and	and	CCONJ
ejpam-4298	356	28	coc	coc	NOUN
ejpam-4298	356	29	-	-	PUNCT
ejpam-4298	356	30	r0	r0	NOUN
ejpam-4298	356	31	-	-	PUNCT
ejpam-4298	356	32	space	space	NOUN
ejpam-4298	356	33	,	,	PUNCT
ejpam-4298	356	34	(	(	PUNCT
ejpam-4298	356	35	iii	iii	X
ejpam-4298	356	36	)	)	PUNCT
ejpam-4298	356	37	x	x	X
ejpam-4298	356	38	is	be	AUX
ejpam-4298	356	39	coc	coc	NOUN
ejpam-4298	356	40	-	-	PUNCT
ejpam-4298	356	41	t0	t0	NOUN
ejpam-4298	356	42	-	-	PUNCT
ejpam-4298	356	43	space	space	NOUN
ejpam-4298	356	44	and	and	CCONJ
ejpam-4298	356	45	weak	weak	ADJ
ejpam-4298	356	46	coc	coc	ADJ
ejpam-4298	356	47	-	-	PUNCT
ejpam-4298	356	48	r0	r0	NOUN
ejpam-4298	356	49	-	-	PUNCT
ejpam-4298	356	50	space	space	NOUN
ejpam-4298	356	51	.	.	PUNCT
ejpam-4298	357	1	corollary	corollary	ADJ
ejpam-4298	357	2	5	5	NUM
ejpam-4298	357	3	.	.	PUNCT
ejpam-4298	358	1	for	for	ADP
ejpam-4298	358	2	a	a	DET
ejpam-4298	358	3	weak	weak	ADJ
ejpam-4298	358	4	coc	coc	ADJ
ejpam-4298	358	5	-	-	PUNCT
ejpam-4298	358	6	r0	r0	NOUN
ejpam-4298	358	7	-	-	PUNCT
ejpam-4298	358	8	space	space	NOUN
ejpam-4298	358	9	(	(	PUNCT
ejpam-4298	358	10	x	x	X
ejpam-4298	358	11	,	,	PUNCT
ejpam-4298	358	12	τ	τ	PROPN
ejpam-4298	358	13	)	)	PUNCT
ejpam-4298	358	14	.	.	PUNCT
ejpam-4298	359	1	the	the	DET
ejpam-4298	359	2	following	follow	VERB
ejpam-4298	359	3	are	be	AUX
ejpam-4298	359	4	equivalent	equivalent	ADJ
ejpam-4298	359	5	:	:	PUNCT
ejpam-4298	359	6	(	(	PUNCT
ejpam-4298	359	7	i	i	NOUN
ejpam-4298	359	8	)	)	PUNCT
ejpam-4298	359	9	x	x	X
ejpam-4298	359	10	is	be	AUX
ejpam-4298	359	11	coc	coc	NOUN
ejpam-4298	359	12	-	-	PUNCT
ejpam-4298	359	13	t0	t0	NOUN
ejpam-4298	359	14	-	-	NOUN
ejpam-4298	359	15	space	space	NOUN
ejpam-4298	359	16	,	,	PUNCT
ejpam-4298	359	17	(	(	PUNCT
ejpam-4298	359	18	ii	ii	NOUN
ejpam-4298	359	19	)	)	PUNCT
ejpam-4298	359	20	x	x	X
ejpam-4298	359	21	is	be	AUX
ejpam-4298	359	22	coc	coc	NOUN
ejpam-4298	359	23	-	-	PUNCT
ejpam-4298	359	24	t	t	PROPN
ejpam-4298	359	25	1	1	NUM
ejpam-4298	359	26	4	4	NUM
ejpam-4298	359	27	-space	-space	NOUN
ejpam-4298	359	28	,	,	PUNCT
ejpam-4298	359	29	(	(	PUNCT
ejpam-4298	359	30	iii	iii	X
ejpam-4298	359	31	)	)	PUNCT
ejpam-4298	359	32	x	x	X
ejpam-4298	359	33	is	be	AUX
ejpam-4298	359	34	coc	coc	NOUN
ejpam-4298	359	35	-	-	PUNCT
ejpam-4298	359	36	t	t	PROPN
ejpam-4298	359	37	3	3	NUM
ejpam-4298	359	38	8	8	NUM
ejpam-4298	359	39	-space	-space	NOUN
ejpam-4298	359	40	,	,	PUNCT
ejpam-4298	359	41	(	(	PUNCT
ejpam-4298	359	42	iv	iv	X
ejpam-4298	359	43	)	)	PUNCT
ejpam-4298	359	44	x	x	X
ejpam-4298	359	45	is	be	AUX
ejpam-4298	359	46	coc	coc	NOUN
ejpam-4298	359	47	-	-	PUNCT
ejpam-4298	359	48	t	t	PROPN
ejpam-4298	359	49	1	1	NUM
ejpam-4298	359	50	2	2	NUM
ejpam-4298	359	51	-space	-space	NOUN
ejpam-4298	359	52	,	,	PUNCT
ejpam-4298	359	53	(	(	PUNCT
ejpam-4298	359	54	v	v	NOUN
ejpam-4298	359	55	)	)	PUNCT
ejpam-4298	359	56	x	x	X
ejpam-4298	359	57	is	be	AUX
ejpam-4298	359	58	coc	coc	NOUN
ejpam-4298	359	59	-	-	PUNCT
ejpam-4298	359	60	t1	t1	NOUN
ejpam-4298	359	61	-	-	PUNCT
ejpam-4298	359	62	space	space	NOUN
ejpam-4298	359	63	.	.	PUNCT
ejpam-4298	360	1	5	5	X
ejpam-4298	360	2	.	.	PUNCT
ejpam-4298	360	3	hereditary	hereditary	ADJ
ejpam-4298	360	4	property	property	NOUN
ejpam-4298	360	5	for	for	ADP
ejpam-4298	360	6	weak	weak	ADJ
ejpam-4298	360	7	coc	coc	ADJ
ejpam-4298	360	8	-	-	ADJ
ejpam-4298	360	9	compact	compact	ADJ
ejpam-4298	360	10	separation	separation	NOUN
ejpam-4298	360	11	axioms	axiom	NOUN
ejpam-4298	360	12	in	in	ADP
ejpam-4298	360	13	this	this	DET
ejpam-4298	360	14	section	section	NOUN
ejpam-4298	360	15	,	,	PUNCT
ejpam-4298	360	16	we	we	PRON
ejpam-4298	360	17	discuss	discuss	VERB
ejpam-4298	360	18	the	the	DET
ejpam-4298	360	19	known	know	VERB
ejpam-4298	360	20	problem	problem	NOUN
ejpam-4298	360	21	that	that	PRON
ejpam-4298	360	22	appeared	appear	VERB
ejpam-4298	360	23	by	by	ADP
ejpam-4298	360	24	arenas	arena	NOUN
ejpam-4298	360	25	[	[	X
ejpam-4298	360	26	6	6	NUM
ejpam-4298	360	27	]	]	PUNCT
ejpam-4298	360	28	“	"	PUNCT
ejpam-4298	360	29	if	if	SCONJ
ejpam-4298	360	30	every	every	DET
ejpam-4298	360	31	subspace	subspace	NOUN
ejpam-4298	360	32	of	of	ADP
ejpam-4298	360	33	a	a	DET
ejpam-4298	360	34	topological	topological	ADJ
ejpam-4298	360	35	space	space	NOUN
ejpam-4298	360	36	x	x	PRON
ejpam-4298	360	37	has	have	VERB
ejpam-4298	360	38	a	a	DET
ejpam-4298	360	39	property	property	NOUN
ejpam-4298	360	40	,	,	PUNCT
ejpam-4298	360	41	then	then	ADV
ejpam-4298	360	42	the	the	DET
ejpam-4298	360	43	space	space	NOUN
ejpam-4298	360	44	x	x	PUNCT
ejpam-4298	360	45	has	have	VERB
ejpam-4298	360	46	this	this	DET
ejpam-4298	360	47	property	property	NOUN
ejpam-4298	360	48	”	"	PUNCT
ejpam-4298	360	49	in	in	ADP
ejpam-4298	360	50	weak	weak	ADJ
ejpam-4298	360	51	separation	separation	NOUN
ejpam-4298	360	52	axioms	axiom	NOUN
ejpam-4298	360	53	via	via	ADP
ejpam-4298	360	54	coc	coc	NOUN
ejpam-4298	360	55	-	-	PUNCT
ejpam-4298	360	56	open	open	ADJ
ejpam-4298	360	57	sets	set	NOUN
ejpam-4298	360	58	.	.	PUNCT
ejpam-4298	361	1	theorem	theorem	NOUN
ejpam-4298	361	2	29	29	NUM
ejpam-4298	361	3	.	.	PUNCT
ejpam-4298	362	1	if	if	SCONJ
ejpam-4298	362	2	every	every	DET
ejpam-4298	362	3	proper	proper	ADJ
ejpam-4298	362	4	subspace	subspace	NOUN
ejpam-4298	362	5	of	of	ADP
ejpam-4298	362	6	a	a	DET
ejpam-4298	362	7	topological	topological	ADJ
ejpam-4298	362	8	space	space	NOUN
ejpam-4298	362	9	(	(	PUNCT
ejpam-4298	362	10	x	x	X
ejpam-4298	362	11	,	,	PUNCT
ejpam-4298	362	12	τ	τ	X
ejpam-4298	362	13	)	)	PUNCT
ejpam-4298	362	14	is	be	AUX
ejpam-4298	362	15	coc	coc	NOUN
ejpam-4298	362	16	-	-	PUNCT
ejpam-4298	362	17	t	t	PROPN
ejpam-4298	362	18	1	1	NUM
ejpam-4298	362	19	2	2	NUM
ejpam-4298	362	20	-space	-space	NOUN
ejpam-4298	362	21	,	,	PUNCT
ejpam-4298	362	22	then	then	ADV
ejpam-4298	362	23	x	x	PUNCT
ejpam-4298	362	24	is	be	AUX
ejpam-4298	362	25	coc	coc	ADJ
ejpam-4298	362	26	-	-	PUNCT
ejpam-4298	362	27	t	t	PROPN
ejpam-4298	362	28	1	1	NUM
ejpam-4298	362	29	2	2	NUM
ejpam-4298	362	30	-space	-space	NOUN
ejpam-4298	362	31	with	with	ADP
ejpam-4298	362	32	|x|	|x|	PROPN
ejpam-4298	362	33	≥	≥	PROPN
ejpam-4298	362	34	4	4	NUM
ejpam-4298	362	35	.	.	PUNCT
ejpam-4298	363	1	f.a	f.a	PROPN
ejpam-4298	363	2	.	.	PROPN
ejpam-4298	363	3	abushaheen	abushaheen	PROPN
ejpam-4298	363	4	/	/	SYM
ejpam-4298	363	5	eur	eur	PROPN
ejpam-4298	363	6	.	.	PUNCT
ejpam-4298	364	1	j.	j.	PROPN
ejpam-4298	364	2	pure	pure	PROPN
ejpam-4298	364	3	appl	appl	PROPN
ejpam-4298	364	4	.	.	PROPN
ejpam-4298	364	5	math	math	PROPN
ejpam-4298	364	6	,	,	PUNCT
ejpam-4298	364	7	15	15	NUM
ejpam-4298	364	8	(	(	PUNCT
ejpam-4298	364	9	2	2	NUM
ejpam-4298	364	10	)	)	PUNCT
ejpam-4298	364	11	(	(	PUNCT
ejpam-4298	364	12	2022	2022	NUM
ejpam-4298	364	13	)	)	PUNCT
ejpam-4298	364	14	,	,	PUNCT
ejpam-4298	364	15	589	589	NUM
ejpam-4298	364	16	-	-	SYM
ejpam-4298	364	17	601	601	NUM
ejpam-4298	364	18	600	600	NUM
ejpam-4298	364	19	proof	proof	NOUN
ejpam-4298	364	20	.	.	PUNCT
ejpam-4298	365	1	let	let	VERB
ejpam-4298	365	2	x	x	PUNCT
ejpam-4298	365	3	∈	∈	PROPN
ejpam-4298	365	4	x	x	PUNCT
ejpam-4298	365	5	and	and	CCONJ
ejpam-4298	365	6	let	let	VERB
ejpam-4298	365	7	z1	z1	PROPN
ejpam-4298	365	8	6=	6=	ADP
ejpam-4298	365	9	z2	z2	PROPN
ejpam-4298	365	10	6=	6=	PUNCT
ejpam-4298	365	11	z3	z3	PROPN
ejpam-4298	365	12	∈	∈	PROPN
ejpam-4298	365	13	x	x	PUNCT
ejpam-4298	365	14	−	−	PROPN
ejpam-4298	365	15	{	{	PUNCT
ejpam-4298	365	16	x	x	NOUN
ejpam-4298	365	17	}	}	PUNCT
ejpam-4298	365	18	and	and	CCONJ
ejpam-4298	365	19	zi	zi	NOUN
ejpam-4298	366	1	=	=	PUNCT
ejpam-4298	366	2	x	x	X
ejpam-4298	366	3	−	−	PROPN
ejpam-4298	366	4	{	{	PUNCT
ejpam-4298	366	5	zi	zi	NOUN
ejpam-4298	366	6	}	}	PUNCT
ejpam-4298	366	7	for	for	ADP
ejpam-4298	366	8	i	i	PROPN
ejpam-4298	366	9	=	=	NOUN
ejpam-4298	366	10	1	1	NUM
ejpam-4298	366	11	,	,	PUNCT
ejpam-4298	366	12	2	2	NUM
ejpam-4298	366	13	,	,	PUNCT
ejpam-4298	366	14	3	3	NUM
ejpam-4298	366	15	.	.	PUNCT
ejpam-4298	367	1	so	so	ADV
ejpam-4298	367	2	{	{	PUNCT
ejpam-4298	367	3	x	x	X
ejpam-4298	367	4	}	}	PUNCT
ejpam-4298	367	5	is	be	AUX
ejpam-4298	367	6	either	either	CCONJ
ejpam-4298	367	7	coc	coc	NOUN
ejpam-4298	367	8	-	-	PUNCT
ejpam-4298	367	9	open	open	ADJ
ejpam-4298	367	10	or	or	CCONJ
ejpam-4298	367	11	coc	coc	NOUN
ejpam-4298	367	12	-	-	PUNCT
ejpam-4298	367	13	closed	closed	ADJ
ejpam-4298	367	14	in	in	ADP
ejpam-4298	367	15	zi	zi	NOUN
ejpam-4298	367	16	,	,	PUNCT
ejpam-4298	367	17	therefore	therefore	ADV
ejpam-4298	367	18	either	either	CCONJ
ejpam-4298	367	19	{	{	PUNCT
ejpam-4298	367	20	x	x	X
ejpam-4298	367	21	}	}	PUNCT
ejpam-4298	367	22	is	be	AUX
ejpam-4298	367	23	coc	coc	NOUN
ejpam-4298	367	24	-	-	PUNCT
ejpam-4298	367	25	open	open	ADJ
ejpam-4298	367	26	in	in	ADP
ejpam-4298	367	27	at	at	ADV
ejpam-4298	367	28	least	least	ADV
ejpam-4298	367	29	two	two	NUM
ejpam-4298	367	30	of	of	ADP
ejpam-4298	367	31	z1	z1	PROPN
ejpam-4298	367	32	,	,	PUNCT
ejpam-4298	367	33	z2	z2	PROPN
ejpam-4298	367	34	,	,	PUNCT
ejpam-4298	367	35	z3	z3	PROPN
ejpam-4298	367	36	,	,	PUNCT
ejpam-4298	367	37	and	and	CCONJ
ejpam-4298	367	38	hence	hence	ADV
ejpam-4298	367	39	{	{	PUNCT
ejpam-4298	367	40	x	x	X
ejpam-4298	367	41	}	}	PUNCT
ejpam-4298	367	42	is	be	AUX
ejpam-4298	367	43	coc	coc	NOUN
ejpam-4298	367	44	-	-	PUNCT
ejpam-4298	367	45	open	open	ADJ
ejpam-4298	367	46	in	in	ADP
ejpam-4298	367	47	x	x	NOUN
ejpam-4298	367	48	,	,	PUNCT
ejpam-4298	367	49	or	or	CCONJ
ejpam-4298	367	50	{	{	PUNCT
ejpam-4298	367	51	x	x	X
ejpam-4298	367	52	}	}	PUNCT
ejpam-4298	367	53	is	be	AUX
ejpam-4298	367	54	coc	coc	NOUN
ejpam-4298	367	55	-	-	PUNCT
ejpam-4298	367	56	closed	close	VERB
ejpam-4298	367	57	in	in	ADP
ejpam-4298	367	58	at	at	ADV
ejpam-4298	367	59	least	least	ADV
ejpam-4298	367	60	two	two	NUM
ejpam-4298	367	61	of	of	ADP
ejpam-4298	367	62	z1	z1	PROPN
ejpam-4298	367	63	,	,	PUNCT
ejpam-4298	367	64	z2	z2	PROPN
ejpam-4298	367	65	,	,	PUNCT
ejpam-4298	367	66	z3	z3	PROPN
ejpam-4298	367	67	,	,	PUNCT
ejpam-4298	367	68	and	and	CCONJ
ejpam-4298	367	69	hence	hence	ADV
ejpam-4298	367	70	{	{	PUNCT
ejpam-4298	367	71	x	x	X
ejpam-4298	367	72	}	}	PUNCT
ejpam-4298	367	73	is	be	AUX
ejpam-4298	367	74	coc	coc	NOUN
ejpam-4298	367	75	-	-	PUNCT
ejpam-4298	367	76	closed	close	VERB
ejpam-4298	367	77	in	in	ADP
ejpam-4298	367	78	x	x	NOUN
ejpam-4298	367	79	,	,	PUNCT
ejpam-4298	367	80	hence	hence	ADV
ejpam-4298	367	81	the	the	DET
ejpam-4298	367	82	result	result	NOUN
ejpam-4298	367	83	.	.	PUNCT
ejpam-4298	368	1	theorem	theorem	NOUN
ejpam-4298	368	2	30	30	NUM
ejpam-4298	368	3	.	.	PUNCT
ejpam-4298	369	1	let	let	VERB
ejpam-4298	369	2	(	(	PUNCT
ejpam-4298	369	3	x	x	NOUN
ejpam-4298	369	4	,	,	PUNCT
ejpam-4298	369	5	τ	τ	X
ejpam-4298	369	6	)	)	PUNCT
ejpam-4298	369	7	be	be	AUX
ejpam-4298	369	8	infinite	infinite	ADJ
ejpam-4298	369	9	topological	topological	ADJ
ejpam-4298	369	10	space	space	NOUN
ejpam-4298	369	11	.	.	PUNCT
ejpam-4298	370	1	if	if	SCONJ
ejpam-4298	370	2	every	every	DET
ejpam-4298	370	3	proper	proper	ADJ
ejpam-4298	370	4	subspace	subspace	NOUN
ejpam-4298	370	5	of	of	ADP
ejpam-4298	370	6	a	a	DET
ejpam-4298	370	7	topological	topological	ADJ
ejpam-4298	370	8	space	space	NOUN
ejpam-4298	370	9	x	x	PUNCT
ejpam-4298	370	10	is	be	AUX
ejpam-4298	370	11	coc	coc	ADJ
ejpam-4298	370	12	-	-	PUNCT
ejpam-4298	370	13	t	t	PROPN
ejpam-4298	370	14	1	1	NUM
ejpam-4298	370	15	4	4	NUM
ejpam-4298	370	16	-space	-space	NOUN
ejpam-4298	370	17	,	,	PUNCT
ejpam-4298	370	18	then	then	ADV
ejpam-4298	370	19	x	x	PUNCT
ejpam-4298	370	20	is	be	AUX
ejpam-4298	370	21	coc	coc	ADJ
ejpam-4298	370	22	-	-	PUNCT
ejpam-4298	370	23	t	t	PROPN
ejpam-4298	370	24	1	1	NUM
ejpam-4298	370	25	4	4	NUM
ejpam-4298	370	26	-space	-space	NOUN
ejpam-4298	370	27	.	.	PUNCT
ejpam-4298	371	1	proof	proof	NOUN
ejpam-4298	371	2	.	.	PUNCT
ejpam-4298	372	1	let	let	VERB
ejpam-4298	372	2	f	f	PRON
ejpam-4298	372	3	be	be	AUX
ejpam-4298	372	4	a	a	DET
ejpam-4298	372	5	finite	finite	NOUN
ejpam-4298	372	6	set	set	NOUN
ejpam-4298	372	7	and	and	CCONJ
ejpam-4298	372	8	y	y	PROPN
ejpam-4298	372	9	/∈	/∈	PUNCT
ejpam-4298	373	1	f	f	PROPN
ejpam-4298	374	1	and	and	CCONJ
ejpam-4298	374	2	let	let	VERB
ejpam-4298	374	3	z	z	NOUN
ejpam-4298	374	4	∈	∈	PROPN
ejpam-4298	374	5	x	x	X
ejpam-4298	374	6	−	−	PROPN
ejpam-4298	374	7	(	(	PUNCT
ejpam-4298	374	8	f	f	PROPN
ejpam-4298	374	9	∪	∪	X
ejpam-4298	374	10	{	{	PUNCT
ejpam-4298	374	11	y	y	NOUN
ejpam-4298	374	12	}	}	PUNCT
ejpam-4298	374	13	)	)	PUNCT
ejpam-4298	374	14	.	.	PUNCT
ejpam-4298	375	1	so	so	ADV
ejpam-4298	375	2	there	there	PRON
ejpam-4298	375	3	exists	exist	VERB
ejpam-4298	375	4	a	a	DET
ejpam-4298	375	5	set	set	NOUN
ejpam-4298	375	6	a	a	DET
ejpam-4298	375	7	contains	contain	NOUN
ejpam-4298	375	8	f	f	PROPN
ejpam-4298	375	9	and	and	CCONJ
ejpam-4298	375	10	y	y	PROPN
ejpam-4298	375	11	/∈	/∈	PUNCT
ejpam-4298	376	1	a	a	DET
ejpam-4298	376	2	which	which	PRON
ejpam-4298	376	3	is	be	AUX
ejpam-4298	376	4	either	either	CCONJ
ejpam-4298	376	5	coc	coc	NOUN
ejpam-4298	376	6	-	-	PUNCT
ejpam-4298	376	7	open	open	ADJ
ejpam-4298	376	8	or	or	CCONJ
ejpam-4298	376	9	coc	coc	NOUN
ejpam-4298	376	10	-	-	PUNCT
ejpam-4298	376	11	closed	closed	ADJ
ejpam-4298	376	12	in	in	ADP
ejpam-4298	376	13	x	x	PART
ejpam-4298	376	14	−	−	PROPN
ejpam-4298	376	15	{	{	PUNCT
ejpam-4298	376	16	z	z	NOUN
ejpam-4298	376	17	}	}	PUNCT
ejpam-4298	376	18	,	,	PUNCT
ejpam-4298	376	19	therefore	therefore	ADV
ejpam-4298	376	20	there	there	PRON
ejpam-4298	376	21	exists	exist	VERB
ejpam-4298	376	22	a	a	DET
ejpam-4298	376	23	set	set	NOUN
ejpam-4298	376	24	b	b	NOUN
ejpam-4298	376	25	which	which	PRON
ejpam-4298	376	26	is	be	AUX
ejpam-4298	376	27	either	either	CCONJ
ejpam-4298	376	28	coc	coc	NOUN
ejpam-4298	376	29	-	-	PUNCT
ejpam-4298	376	30	open	open	ADJ
ejpam-4298	376	31	or	or	CCONJ
ejpam-4298	376	32	coc	coc	NOUN
ejpam-4298	376	33	-	-	PUNCT
ejpam-4298	376	34	closed	closed	ADJ
ejpam-4298	376	35	in	in	ADP
ejpam-4298	376	36	x	x	PUNCT
ejpam-4298	376	37	with	with	ADP
ejpam-4298	376	38	a	a	DET
ejpam-4298	376	39	=	=	SYM
ejpam-4298	376	40	b	b	NOUN
ejpam-4298	376	41	∩	∩	NOUN
ejpam-4298	376	42	(	(	PUNCT
ejpam-4298	376	43	x	x	SYM
ejpam-4298	376	44	−	−	PROPN
ejpam-4298	376	45	{	{	PUNCT
ejpam-4298	376	46	x	x	NOUN
ejpam-4298	376	47	}	}	PUNCT
ejpam-4298	376	48	)	)	PUNCT
ejpam-4298	376	49	,	,	PUNCT
ejpam-4298	376	50	hence	hence	ADV
ejpam-4298	376	51	x	x	VERB
ejpam-4298	376	52	is	be	AUX
ejpam-4298	376	53	coc	coc	ADJ
ejpam-4298	376	54	-	-	PUNCT
ejpam-4298	376	55	t	t	PROPN
ejpam-4298	376	56	1	1	NUM
ejpam-4298	376	57	4	4	NUM
ejpam-4298	376	58	-space	-space	NOUN
ejpam-4298	376	59	.	.	PUNCT
ejpam-4298	377	1	theorem	theorem	NOUN
ejpam-4298	377	2	31	31	NUM
ejpam-4298	377	3	.	.	PUNCT
ejpam-4298	378	1	let	let	VERB
ejpam-4298	378	2	(	(	PUNCT
ejpam-4298	378	3	x	x	NOUN
ejpam-4298	378	4	,	,	PUNCT
ejpam-4298	378	5	τ	τ	X
ejpam-4298	378	6	)	)	PUNCT
ejpam-4298	378	7	be	be	AUX
ejpam-4298	378	8	infinite	infinite	ADJ
ejpam-4298	378	9	topological	topological	ADJ
ejpam-4298	378	10	space	space	NOUN
ejpam-4298	378	11	.	.	PUNCT
ejpam-4298	379	1	if	if	SCONJ
ejpam-4298	379	2	every	every	DET
ejpam-4298	379	3	proper	proper	ADJ
ejpam-4298	379	4	subspace	subspace	NOUN
ejpam-4298	379	5	of	of	ADP
ejpam-4298	379	6	a	a	DET
ejpam-4298	379	7	topological	topological	ADJ
ejpam-4298	379	8	space	space	NOUN
ejpam-4298	379	9	x	x	PUNCT
ejpam-4298	379	10	is	be	AUX
ejpam-4298	379	11	coc	coc	NOUN
ejpam-4298	379	12	-	-	PUNCT
ejpam-4298	379	13	t	t	PROPN
ejpam-4298	379	14	3	3	NUM
ejpam-4298	379	15	8	8	NUM
ejpam-4298	379	16	-space	-space	NOUN
ejpam-4298	379	17	,	,	PUNCT
ejpam-4298	379	18	then	then	ADV
ejpam-4298	379	19	x	x	PUNCT
ejpam-4298	379	20	is	be	AUX
ejpam-4298	379	21	coc	coc	NOUN
ejpam-4298	379	22	-	-	PUNCT
ejpam-4298	379	23	t	t	PROPN
ejpam-4298	379	24	3	3	NUM
ejpam-4298	379	25	8	8	NUM
ejpam-4298	379	26	-space	-space	NOUN
ejpam-4298	379	27	.	.	PUNCT
ejpam-4298	380	1	proof	proof	NOUN
ejpam-4298	380	2	.	.	PUNCT
ejpam-4298	381	1	same	same	ADJ
ejpam-4298	381	2	as	as	SCONJ
ejpam-4298	381	3	theorem	theorem	ADJ
ejpam-4298	381	4	30	30	NUM
ejpam-4298	381	5	.	.	PUNCT
ejpam-4298	382	1	theorem	theorem	VERB
ejpam-4298	382	2	32	32	NUM
ejpam-4298	382	3	.	.	PUNCT
ejpam-4298	383	1	if	if	SCONJ
ejpam-4298	383	2	every	every	DET
ejpam-4298	383	3	proper	proper	ADJ
ejpam-4298	383	4	subspace	subspace	NOUN
ejpam-4298	383	5	of	of	ADP
ejpam-4298	383	6	a	a	DET
ejpam-4298	383	7	topological	topological	ADJ
ejpam-4298	383	8	space	space	NOUN
ejpam-4298	383	9	(	(	PUNCT
ejpam-4298	383	10	x	x	X
ejpam-4298	383	11	,	,	PUNCT
ejpam-4298	383	12	τ	τ	X
ejpam-4298	383	13	)	)	PUNCT
ejpam-4298	383	14	is	be	AUX
ejpam-4298	383	15	coc	coc	ADJ
ejpam-4298	383	16	-	-	PUNCT
ejpam-4298	383	17	r0	r0	NOUN
ejpam-4298	383	18	-	-	PUNCT
ejpam-4298	383	19	space	space	NOUN
ejpam-4298	383	20	,	,	PUNCT
ejpam-4298	383	21	then	then	ADV
ejpam-4298	383	22	x	x	PUNCT
ejpam-4298	383	23	is	be	AUX
ejpam-4298	383	24	coc	coc	ADJ
ejpam-4298	383	25	-	-	PUNCT
ejpam-4298	383	26	r0	r0	NOUN
ejpam-4298	383	27	-	-	PUNCT
ejpam-4298	383	28	space	space	NOUN
ejpam-4298	383	29	with	with	ADP
ejpam-4298	383	30	|x|	|x|	PROPN
ejpam-4298	383	31	≥	≥	NUM
ejpam-4298	383	32	3	3	NUM
ejpam-4298	383	33	.	.	PUNCT
ejpam-4298	383	34	proof	proof	NOUN
ejpam-4298	383	35	.	.	PUNCT
ejpam-4298	384	1	assume	assume	VERB
ejpam-4298	384	2	that	that	SCONJ
ejpam-4298	384	3	all	all	DET
ejpam-4298	384	4	proper	proper	ADJ
ejpam-4298	384	5	subspaces	subspace	NOUN
ejpam-4298	384	6	of	of	ADP
ejpam-4298	384	7	x	x	SYM
ejpam-4298	384	8	are	be	AUX
ejpam-4298	384	9	coc	coc	ADJ
ejpam-4298	384	10	-	-	PUNCT
ejpam-4298	384	11	r0	r0	NOUN
ejpam-4298	384	12	-	-	PUNCT
ejpam-4298	384	13	space	space	NOUN
ejpam-4298	384	14	.	.	PUNCT
ejpam-4298	385	1	let	let	VERB
ejpam-4298	385	2	u	u	PRON
ejpam-4298	385	3	be	be	AUX
ejpam-4298	385	4	coc	coc	ADJ
ejpam-4298	385	5	-	-	PUNCT
ejpam-4298	385	6	open	open	ADJ
ejpam-4298	385	7	subset	subset	NOUN
ejpam-4298	385	8	of	of	ADP
ejpam-4298	385	9	x.	x.	NOUN
ejpam-4298	385	10	if	if	SCONJ
ejpam-4298	385	11	x	x	SYM
ejpam-4298	385	12	=	=	SYM
ejpam-4298	385	13	u	u	NOUN
ejpam-4298	385	14	we	we	PRON
ejpam-4298	385	15	are	be	AUX
ejpam-4298	385	16	done	do	VERB
ejpam-4298	385	17	,	,	PUNCT
ejpam-4298	385	18	so	so	SCONJ
ejpam-4298	385	19	we	we	PRON
ejpam-4298	385	20	may	may	AUX
ejpam-4298	385	21	assume	assume	VERB
ejpam-4298	385	22	x	x	X
ejpam-4298	385	23	6=	6=	NUM
ejpam-4298	385	24	u	u	NOUN
ejpam-4298	385	25	.	.	PUNCT
ejpam-4298	386	1	let	let	VERB
ejpam-4298	386	2	x	x	PRON
ejpam-4298	386	3	/∈	/∈	PUNCT
ejpam-4298	386	4	u	u	NOUN
ejpam-4298	386	5	and	and	CCONJ
ejpam-4298	386	6	p	p	NOUN
ejpam-4298	386	7	∈	∈	PROPN
ejpam-4298	386	8	u	u	NOUN
ejpam-4298	386	9	with	with	ADP
ejpam-4298	386	10	y	y	PROPN
ejpam-4298	386	11	∈	∈	PROPN
ejpam-4298	386	12	x	x	PUNCT
ejpam-4298	387	1	−	−	PROPN
ejpam-4298	387	2	{	{	PUNCT
ejpam-4298	387	3	p	p	X
ejpam-4298	387	4	,	,	PUNCT
ejpam-4298	387	5	x	x	NOUN
ejpam-4298	387	6	}	}	PUNCT
ejpam-4298	387	7	.	.	PUNCT
ejpam-4298	388	1	so	so	ADV
ejpam-4298	388	2	we	we	PRON
ejpam-4298	388	3	have	have	VERB
ejpam-4298	388	4	the	the	DET
ejpam-4298	388	5	following	follow	VERB
ejpam-4298	388	6	cases	case	NOUN
ejpam-4298	388	7	:	:	PUNCT
ejpam-4298	388	8	(	(	PUNCT
ejpam-4298	388	9	1	1	X
ejpam-4298	388	10	)	)	PUNCT
ejpam-4298	388	11	if	if	SCONJ
ejpam-4298	388	12	y	y	PROPN
ejpam-4298	388	13	∈	∈	PROPN
ejpam-4298	388	14	u	u	PROPN
ejpam-4298	388	15	,	,	PUNCT
ejpam-4298	388	16	so	so	ADV
ejpam-4298	388	17	x	x	X
ejpam-4298	388	18	−	−	PROPN
ejpam-4298	388	19	{	{	PUNCT
ejpam-4298	388	20	y	y	NOUN
ejpam-4298	388	21	}	}	PUNCT
ejpam-4298	388	22	is	be	AUX
ejpam-4298	388	23	coc	coc	ADJ
ejpam-4298	388	24	-	-	PUNCT
ejpam-4298	388	25	r0	r0	NOUN
ejpam-4298	388	26	-	-	PUNCT
ejpam-4298	388	27	space	space	NOUN
ejpam-4298	388	28	,	,	PUNCT
ejpam-4298	388	29	so	so	ADV
ejpam-4298	388	30	by	by	ADP
ejpam-4298	388	31	theorem	theorem	ADJ
ejpam-4298	388	32	15	15	NUM
ejpam-4298	388	33	(	(	PUNCT
ejpam-4298	388	34	iii	iii	NOUN
ejpam-4298	388	35	)	)	PUNCT
ejpam-4298	388	36	there	there	PRON
ejpam-4298	388	37	is	be	VERB
ejpam-4298	388	38	a	a	DET
ejpam-4298	388	39	coc	coc	NOUN
ejpam-4298	388	40	-	-	PUNCT
ejpam-4298	388	41	closed	close	VERB
ejpam-4298	388	42	set	set	NOUN
ejpam-4298	388	43	gy	gy	NOUN
ejpam-4298	388	44	in	in	ADP
ejpam-4298	388	45	x	x	X
ejpam-4298	388	46	−	−	PROPN
ejpam-4298	388	47	{	{	PUNCT
ejpam-4298	388	48	y	y	NOUN
ejpam-4298	388	49	}	}	PUNCT
ejpam-4298	388	50	such	such	ADJ
ejpam-4298	388	51	that	that	SCONJ
ejpam-4298	388	52	p	p	PROPN
ejpam-4298	388	53	∈	∈	PROPN
ejpam-4298	388	54	gy	gy	VERB
ejpam-4298	388	55	⊆	⊆	NUM
ejpam-4298	388	56	u	u	NOUN
ejpam-4298	388	57	−	−	PROPN
ejpam-4298	388	58	{	{	PUNCT
ejpam-4298	388	59	y	y	NOUN
ejpam-4298	388	60	}	}	PUNCT
ejpam-4298	388	61	and	and	CCONJ
ejpam-4298	388	62	also	also	ADV
ejpam-4298	388	63	there	there	PRON
ejpam-4298	388	64	exists	exist	VERB
ejpam-4298	388	65	a	a	DET
ejpam-4298	388	66	coc	coc	NOUN
ejpam-4298	388	67	-	-	PUNCT
ejpam-4298	388	68	closed	close	VERB
ejpam-4298	388	69	set	set	VERB
ejpam-4298	388	70	g	g	NOUN
ejpam-4298	388	71	in	in	ADP
ejpam-4298	388	72	x	x	PUNCT
ejpam-4298	388	73	such	such	ADJ
ejpam-4298	388	74	that	that	DET
ejpam-4298	388	75	gy	gy	NOUN
ejpam-4298	388	76	=	=	SYM
ejpam-4298	388	77	g	g	PROPN
ejpam-4298	388	78	∩	∩	NOUN
ejpam-4298	388	79	(	(	PUNCT
ejpam-4298	388	80	x	x	SYM
ejpam-4298	388	81	−	−	PROPN
ejpam-4298	388	82	{	{	PUNCT
ejpam-4298	388	83	y	y	NOUN
ejpam-4298	388	84	}	}	PUNCT
ejpam-4298	388	85	)	)	PUNCT
ejpam-4298	388	86	,	,	PUNCT
ejpam-4298	388	87	then	then	ADV
ejpam-4298	388	88	p	p	PROPN
ejpam-4298	388	89	∈	∈	PROPN
ejpam-4298	388	90	g	g	PROPN
ejpam-4298	388	91	⊆	⊆	NUM
ejpam-4298	388	92	gy	gy	NOUN
ejpam-4298	388	93	∪	∪	X
ejpam-4298	388	94	{	{	PUNCT
ejpam-4298	388	95	y	y	NOUN
ejpam-4298	388	96	}	}	PUNCT
ejpam-4298	388	97	⊆	⊆	NUM
ejpam-4298	388	98	(	(	PUNCT
ejpam-4298	388	99	u	u	NOUN
ejpam-4298	388	100	−	−	PROPN
ejpam-4298	388	101	{	{	PUNCT
ejpam-4298	388	102	y	y	NOUN
ejpam-4298	388	103	}	}	PUNCT
ejpam-4298	388	104	)	)	PUNCT
ejpam-4298	388	105	∪	∪	ADP
ejpam-4298	388	106	{	{	PUNCT
ejpam-4298	388	107	y	y	NOUN
ejpam-4298	388	108	}	}	PUNCT
ejpam-4298	388	109	=	=	SYM
ejpam-4298	388	110	u	u	NOUN
ejpam-4298	388	111	,	,	PUNCT
ejpam-4298	388	112	hence	hence	ADV
ejpam-4298	388	113	x	x	VERB
ejpam-4298	388	114	is	be	AUX
ejpam-4298	388	115	coc	coc	ADJ
ejpam-4298	388	116	-	-	PUNCT
ejpam-4298	388	117	r0	r0	NOUN
ejpam-4298	388	118	-	-	PUNCT
ejpam-4298	388	119	space	space	NOUN
ejpam-4298	388	120	.	.	PUNCT
ejpam-4298	389	1	(	(	PUNCT
ejpam-4298	389	2	2	2	X
ejpam-4298	389	3	)	)	PUNCT
ejpam-4298	389	4	if	if	SCONJ
ejpam-4298	389	5	y	y	PROPN
ejpam-4298	389	6	/∈	/∈	PUNCT
ejpam-4298	389	7	u	u	PROPN
ejpam-4298	389	8	,	,	PUNCT
ejpam-4298	389	9	then	then	ADV
ejpam-4298	389	10	x	x	ADP
ejpam-4298	389	11	−	−	PROPN
ejpam-4298	389	12	{	{	PUNCT
ejpam-4298	389	13	x	x	NOUN
ejpam-4298	389	14	}	}	PUNCT
ejpam-4298	389	15	and	and	CCONJ
ejpam-4298	389	16	x	x	ADJ
ejpam-4298	389	17	−	−	PROPN
ejpam-4298	389	18	{	{	PUNCT
ejpam-4298	389	19	y	y	NOUN
ejpam-4298	389	20	}	}	PUNCT
ejpam-4298	389	21	are	be	AUX
ejpam-4298	389	22	proper	proper	ADJ
ejpam-4298	389	23	subspaces	subspace	NOUN
ejpam-4298	389	24	of	of	ADP
ejpam-4298	389	25	x	x	PRON
ejpam-4298	389	26	,	,	PUNCT
ejpam-4298	389	27	so	so	SCONJ
ejpam-4298	389	28	there	there	PRON
ejpam-4298	389	29	exist	exist	VERB
ejpam-4298	389	30	cocclosed	cocclose	VERB
ejpam-4298	389	31	subsets	subset	NOUN
ejpam-4298	389	32	gx	gx	PROPN
ejpam-4298	389	33	,	,	PUNCT
ejpam-4298	389	34	gy	gy	VERB
ejpam-4298	389	35	in	in	ADP
ejpam-4298	389	36	x	x	X
ejpam-4298	389	37	−	−	PROPN
ejpam-4298	389	38	{	{	PUNCT
ejpam-4298	389	39	x	x	NOUN
ejpam-4298	389	40	}	}	PUNCT
ejpam-4298	389	41	and	and	CCONJ
ejpam-4298	389	42	x	x	ADJ
ejpam-4298	389	43	−	−	PROPN
ejpam-4298	389	44	{	{	PUNCT
ejpam-4298	389	45	y	y	NOUN
ejpam-4298	389	46	}	}	PUNCT
ejpam-4298	389	47	,	,	PUNCT
ejpam-4298	389	48	respectively	respectively	ADV
ejpam-4298	389	49	such	such	ADJ
ejpam-4298	389	50	that	that	SCONJ
ejpam-4298	389	51	p	p	PROPN
ejpam-4298	389	52	∈	∈	PROPN
ejpam-4298	389	53	gx	gx	PROPN
ejpam-4298	389	54	⊆	⊆	NUM
ejpam-4298	389	55	u	u	NOUN
ejpam-4298	389	56	and	and	CCONJ
ejpam-4298	389	57	p	p	NOUN
ejpam-4298	389	58	∈	∈	PROPN
ejpam-4298	389	59	gy	gy	NOUN
ejpam-4298	389	60	⊆	⊆	NUM
ejpam-4298	389	61	u	u	NOUN
ejpam-4298	389	62	,	,	PUNCT
ejpam-4298	389	63	also	also	ADV
ejpam-4298	389	64	there	there	PRON
ejpam-4298	389	65	exist	exist	VERB
ejpam-4298	389	66	coc	coc	ADJ
ejpam-4298	389	67	-	-	PUNCT
ejpam-4298	389	68	closed	close	VERB
ejpam-4298	389	69	sets	set	NOUN
ejpam-4298	389	70	g1	g1	NOUN
ejpam-4298	389	71	,	,	PUNCT
ejpam-4298	389	72	g2	g2	PROPN
ejpam-4298	389	73	in	in	ADP
ejpam-4298	389	74	x	x	PROPN
ejpam-4298	389	75	such	such	ADJ
ejpam-4298	389	76	that	that	SCONJ
ejpam-4298	389	77	gx	gx	PROPN
ejpam-4298	389	78	=	=	PROPN
ejpam-4298	389	79	g1	g1	PROPN
ejpam-4298	389	80	∩	∩	NOUN
ejpam-4298	389	81	(	(	PUNCT
ejpam-4298	389	82	x	x	SYM
ejpam-4298	389	83	−	−	PROPN
ejpam-4298	389	84	{	{	PUNCT
ejpam-4298	389	85	x	x	NOUN
ejpam-4298	389	86	}	}	PUNCT
ejpam-4298	389	87	)	)	PUNCT
ejpam-4298	389	88	and	and	CCONJ
ejpam-4298	389	89	gy	gy	NOUN
ejpam-4298	389	90	=	=	PROPN
ejpam-4298	389	91	g2	g2	PROPN
ejpam-4298	389	92	∩	∩	NOUN
ejpam-4298	389	93	(	(	PUNCT
ejpam-4298	389	94	x	x	SYM
ejpam-4298	389	95	−	−	PROPN
ejpam-4298	389	96	{	{	PUNCT
ejpam-4298	389	97	y	y	NOUN
ejpam-4298	389	98	}	}	PUNCT
ejpam-4298	389	99	)	)	PUNCT
ejpam-4298	389	100	.	.	PUNCT
ejpam-4298	390	1	define	define	VERB
ejpam-4298	390	2	g	g	PROPN
ejpam-4298	390	3	=	=	PROPN
ejpam-4298	390	4	g1	g1	PROPN
ejpam-4298	390	5	∩g2	∩g2	PROPN
ejpam-4298	390	6	,	,	PUNCT
ejpam-4298	390	7	so	so	SCONJ
ejpam-4298	390	8	p	p	ADP
ejpam-4298	390	9	∈	∈	PROPN
ejpam-4298	390	10	g	g	ADP
ejpam-4298	390	11	⊆	⊆	NUM
ejpam-4298	390	12	(	(	PUNCT
ejpam-4298	390	13	gx	gx	PROPN
ejpam-4298	390	14	∪	∪	X
ejpam-4298	390	15	{	{	PUNCT
ejpam-4298	390	16	x})∩	x})∩	PROPN
ejpam-4298	390	17	(	(	PUNCT
ejpam-4298	390	18	gy	gy	NOUN
ejpam-4298	390	19	∪	∪	X
ejpam-4298	390	20	{	{	PUNCT
ejpam-4298	390	21	y	y	NOUN
ejpam-4298	390	22	}	}	PUNCT
ejpam-4298	390	23	)	)	PUNCT
ejpam-4298	390	24	⊆	⊆	NUM
ejpam-4298	390	25	u	u	NOUN
ejpam-4298	390	26	,	,	PUNCT
ejpam-4298	390	27	hence	hence	ADV
ejpam-4298	390	28	x	x	VERB
ejpam-4298	390	29	is	be	AUX
ejpam-4298	390	30	coc	coc	ADJ
ejpam-4298	390	31	-	-	PUNCT
ejpam-4298	390	32	r0	r0	NOUN
ejpam-4298	390	33	-	-	PUNCT
ejpam-4298	390	34	space	space	NOUN
ejpam-4298	390	35	.	.	PUNCT
ejpam-4298	391	1	theorem	theorem	VERB
ejpam-4298	391	2	33	33	NUM
ejpam-4298	391	3	.	.	PUNCT
ejpam-4298	392	1	if	if	SCONJ
ejpam-4298	392	2	every	every	DET
ejpam-4298	392	3	proper	proper	ADJ
ejpam-4298	392	4	subspace	subspace	NOUN
ejpam-4298	392	5	of	of	ADP
ejpam-4298	392	6	a	a	DET
ejpam-4298	392	7	topological	topological	ADJ
ejpam-4298	392	8	space	space	NOUN
ejpam-4298	392	9	(	(	PUNCT
ejpam-4298	392	10	x	x	X
ejpam-4298	392	11	,	,	PUNCT
ejpam-4298	392	12	τ	τ	X
ejpam-4298	392	13	)	)	PUNCT
ejpam-4298	392	14	is	be	AUX
ejpam-4298	392	15	coc	coc	NOUN
ejpam-4298	392	16	-	-	PUNCT
ejpam-4298	392	17	t1	t1	NOUN
ejpam-4298	392	18	-	-	PUNCT
ejpam-4298	392	19	space	space	NOUN
ejpam-4298	392	20	,	,	PUNCT
ejpam-4298	392	21	then	then	ADV
ejpam-4298	392	22	x	x	PUNCT
ejpam-4298	392	23	is	be	AUX
ejpam-4298	392	24	coc	coc	ADJ
ejpam-4298	392	25	-	-	PUNCT
ejpam-4298	392	26	t1	t1	NOUN
ejpam-4298	392	27	-	-	PUNCT
ejpam-4298	392	28	space	space	NOUN
ejpam-4298	392	29	with	with	ADP
ejpam-4298	392	30	|x|	|x|	PROPN
ejpam-4298	392	31	≥	≥	NUM
ejpam-4298	392	32	3	3	NUM
ejpam-4298	392	33	.	.	PUNCT
ejpam-4298	392	34	proof	proof	NOUN
ejpam-4298	392	35	.	.	PUNCT
ejpam-4298	393	1	suppose	suppose	VERB
ejpam-4298	393	2	that	that	SCONJ
ejpam-4298	393	3	x	x	PRON
ejpam-4298	393	4	is	be	AUX
ejpam-4298	393	5	not	not	PART
ejpam-4298	393	6	a	a	DET
ejpam-4298	393	7	coc	coc	PROPN
ejpam-4298	393	8	-	-	PUNCT
ejpam-4298	393	9	t1	t1	NOUN
ejpam-4298	393	10	-	-	PUNCT
ejpam-4298	393	11	space	space	NOUN
ejpam-4298	393	12	,	,	PUNCT
ejpam-4298	393	13	so	so	SCONJ
ejpam-4298	393	14	there	there	PRON
ejpam-4298	393	15	exists	exist	VERB
ejpam-4298	393	16	x	x	X
ejpam-4298	393	17	∈	∈	NOUN
ejpam-4298	393	18	x	x	PUNCT
ejpam-4298	393	19	such	such	ADJ
ejpam-4298	393	20	that	that	SCONJ
ejpam-4298	393	21	{	{	PUNCT
ejpam-4298	393	22	x	x	X
ejpam-4298	393	23	}	}	PUNCT
ejpam-4298	393	24	is	be	AUX
ejpam-4298	393	25	not	not	PART
ejpam-4298	393	26	coc	coc	NOUN
ejpam-4298	393	27	-	-	PUNCT
ejpam-4298	393	28	closed	close	VERB
ejpam-4298	393	29	in	in	ADP
ejpam-4298	393	30	x.	x.	NOUN
ejpam-4298	393	31	let	let	VERB
ejpam-4298	393	32	z	z	NOUN
ejpam-4298	393	33	∈	∈	PROPN
ejpam-4298	393	34	x	x	INTJ
ejpam-4298	394	1	−	−	PROPN
ejpam-4298	394	2	{	{	PUNCT
ejpam-4298	394	3	x	x	NOUN
ejpam-4298	394	4	}	}	PUNCT
ejpam-4298	394	5	.	.	PUNCT
ejpam-4298	395	1	then	then	ADV
ejpam-4298	395	2	x	x	X
ejpam-4298	395	3	−	−	PROPN
ejpam-4298	395	4	{	{	PUNCT
ejpam-4298	395	5	z	z	NOUN
ejpam-4298	395	6	}	}	PUNCT
ejpam-4298	395	7	is	be	AUX
ejpam-4298	395	8	a	a	DET
ejpam-4298	395	9	coc	coc	PROPN
ejpam-4298	395	10	-	-	PUNCT
ejpam-4298	395	11	t1	t1	NOUN
ejpam-4298	395	12	-	-	PUNCT
ejpam-4298	395	13	space	space	NOUN
ejpam-4298	395	14	,	,	PUNCT
ejpam-4298	395	15	so	so	CCONJ
ejpam-4298	395	16	{	{	PUNCT
ejpam-4298	395	17	x	x	X
ejpam-4298	395	18	}	}	PUNCT
ejpam-4298	395	19	is	be	AUX
ejpam-4298	395	20	coc	coc	NOUN
ejpam-4298	395	21	-	-	PUNCT
ejpam-4298	395	22	closed	close	VERB
ejpam-4298	395	23	is	be	AUX
ejpam-4298	395	24	x	x	X
ejpam-4298	395	25	−{z	−{z	NOUN
ejpam-4298	395	26	}	}	PUNCT
ejpam-4298	395	27	and	and	CCONJ
ejpam-4298	395	28	{	{	PUNCT
ejpam-4298	395	29	x}coc	x}coc	PROPN
ejpam-4298	395	30	=	=	PRON
ejpam-4298	395	31	{	{	PUNCT
ejpam-4298	395	32	x	x	X
ejpam-4298	395	33	,	,	PUNCT
ejpam-4298	395	34	z	z	NOUN
ejpam-4298	395	35	}	}	PUNCT
ejpam-4298	395	36	.	.	PUNCT
ejpam-4298	396	1	now	now	ADV
ejpam-4298	396	2	let	let	VERB
ejpam-4298	396	3	y	y	PROPN
ejpam-4298	396	4	∈	∈	PROPN
ejpam-4298	396	5	x	x	PUNCT
ejpam-4298	396	6	−{x	−{x	NUM
ejpam-4298	396	7	,	,	PUNCT
ejpam-4298	396	8	z	z	NOUN
ejpam-4298	396	9	}	}	PUNCT
ejpam-4298	396	10	and	and	CCONJ
ejpam-4298	396	11	b	b	X
ejpam-4298	396	12	=	=	SYM
ejpam-4298	396	13	x	x	SYM
ejpam-4298	396	14	−{y	−{y	NOUN
ejpam-4298	396	15	}	}	PUNCT
ejpam-4298	396	16	,	,	PUNCT
ejpam-4298	396	17	then	then	ADV
ejpam-4298	396	18	{	{	PUNCT
ejpam-4298	396	19	x}coc(b	x}coc(b	PROPN
ejpam-4298	396	20	)	)	PUNCT
ejpam-4298	397	1	=	=	PRON
ejpam-4298	397	2	{	{	PUNCT
ejpam-4298	397	3	x	x	NOUN
ejpam-4298	397	4	,	,	PUNCT
ejpam-4298	397	5	z	z	NOUN
ejpam-4298	397	6	}	}	PUNCT
ejpam-4298	397	7	that	that	PRON
ejpam-4298	397	8	means	mean	VERB
ejpam-4298	397	9	{	{	PUNCT
ejpam-4298	397	10	x	x	NOUN
ejpam-4298	397	11	}	}	PUNCT
ejpam-4298	397	12	is	be	AUX
ejpam-4298	397	13	not	not	PART
ejpam-4298	397	14	coc	coc	NOUN
ejpam-4298	397	15	-	-	PUNCT
ejpam-4298	397	16	closed	close	VERB
ejpam-4298	397	17	in	in	ADP
ejpam-4298	397	18	b	b	NOUN
ejpam-4298	397	19	which	which	PRON
ejpam-4298	397	20	is	be	AUX
ejpam-4298	397	21	a	a	DET
ejpam-4298	397	22	contradiction	contradiction	NOUN
ejpam-4298	397	23	,	,	PUNCT
ejpam-4298	397	24	hence	hence	ADV
ejpam-4298	397	25	x	x	VERB
ejpam-4298	397	26	is	be	AUX
ejpam-4298	397	27	coc	coc	ADJ
ejpam-4298	397	28	-	-	PUNCT
ejpam-4298	397	29	t1	t1	NOUN
ejpam-4298	397	30	-	-	PUNCT
ejpam-4298	397	31	space	space	NOUN
ejpam-4298	397	32	.	.	PUNCT
ejpam-4298	398	1	the	the	DET
ejpam-4298	398	2	following	follow	VERB
ejpam-4298	398	3	theorem	theorem	NOUN
ejpam-4298	398	4	can	can	AUX
ejpam-4298	398	5	be	be	AUX
ejpam-4298	398	6	proved	prove	VERB
ejpam-4298	398	7	as	as	ADP
ejpam-4298	398	8	the	the	DET
ejpam-4298	398	9	previous	previous	ADJ
ejpam-4298	398	10	one	one	NUM
ejpam-4298	398	11	.	.	PUNCT
ejpam-4298	399	1	theorem	theorem	VERB
ejpam-4298	399	2	34	34	NUM
ejpam-4298	399	3	.	.	PUNCT
ejpam-4298	400	1	if	if	SCONJ
ejpam-4298	400	2	every	every	DET
ejpam-4298	400	3	proper	proper	ADJ
ejpam-4298	400	4	subspace	subspace	NOUN
ejpam-4298	400	5	of	of	ADP
ejpam-4298	400	6	a	a	DET
ejpam-4298	400	7	topological	topological	ADJ
ejpam-4298	400	8	space	space	NOUN
ejpam-4298	400	9	(	(	PUNCT
ejpam-4298	400	10	x	x	X
ejpam-4298	400	11	,	,	PUNCT
ejpam-4298	400	12	τ	τ	X
ejpam-4298	400	13	)	)	PUNCT
ejpam-4298	400	14	is	be	AUX
ejpam-4298	400	15	coc	coc	ADJ
ejpam-4298	400	16	-	-	PUNCT
ejpam-4298	400	17	t2	t2	NOUN
ejpam-4298	400	18	-	-	PUNCT
ejpam-4298	400	19	space	space	NOUN
ejpam-4298	400	20	,	,	PUNCT
ejpam-4298	400	21	then	then	ADV
ejpam-4298	400	22	x	x	PUNCT
ejpam-4298	400	23	is	be	AUX
ejpam-4298	400	24	coc	coc	ADJ
ejpam-4298	400	25	-	-	PUNCT
ejpam-4298	400	26	t2	t2	NOUN
ejpam-4298	400	27	-	-	PUNCT
ejpam-4298	400	28	space	space	NOUN
ejpam-4298	400	29	with	with	ADP
ejpam-4298	400	30	|x|	|x|	PROPN
ejpam-4298	400	31	≥	≥	PROPN
ejpam-4298	400	32	3	3	NUM
ejpam-4298	400	33	.	.	PUNCT
ejpam-4298	401	1	references	reference	NOUN
ejpam-4298	401	2	601	601	NUM
ejpam-4298	401	3	acknowledgements	acknowledgement	NOUN
ejpam-4298	401	4	the	the	DET
ejpam-4298	401	5	authors	author	NOUN
ejpam-4298	401	6	are	be	AUX
ejpam-4298	401	7	grateful	grateful	ADJ
ejpam-4298	401	8	to	to	ADP
ejpam-4298	401	9	the	the	DET
ejpam-4298	401	10	middle	middle	PROPN
ejpam-4298	401	11	east	east	PROPN
ejpam-4298	401	12	university	university	PROPN
ejpam-4298	401	13	,	,	PUNCT
ejpam-4298	401	14	amman	amman	PROPN
ejpam-4298	401	15	,	,	PUNCT
ejpam-4298	401	16	jordan	jordan	PROPN
ejpam-4298	401	17	for	for	ADP
ejpam-4298	401	18	the	the	DET
ejpam-4298	401	19	financial	financial	ADJ
ejpam-4298	401	20	support	support	NOUN
ejpam-4298	401	21	granted	grant	VERB
ejpam-4298	401	22	to	to	PART
ejpam-4298	401	23	cover	cover	VERB
ejpam-4298	401	24	the	the	DET
ejpam-4298	401	25	publication	publication	NOUN
ejpam-4298	401	26	fee	fee	NOUN
ejpam-4298	401	27	of	of	ADP
ejpam-4298	401	28	this	this	DET
ejpam-4298	401	29	research	research	NOUN
ejpam-4298	401	30	article	article	NOUN
ejpam-4298	401	31	.	.	PUNCT
ejpam-4298	402	1	references	reference	NOUN
ejpam-4298	402	2	[	[	X
ejpam-4298	402	3	1	1	NUM
ejpam-4298	402	4	]	]	X
ejpam-4298	402	5	r	r	NOUN
ejpam-4298	402	6	al	al	PROPN
ejpam-4298	402	7	abdula	abdula	VERB
ejpam-4298	402	8	and	and	CCONJ
ejpam-4298	402	9	f	f	PROPN
ejpam-4298	402	10	al	al	PROPN
ejpam-4298	402	11	hussaini	hussaini	PROPN
ejpam-4298	402	12	.	.	PUNCT
ejpam-4298	403	1	on	on	ADP
ejpam-4298	403	2	cocompact	cocompact	PROPN
ejpam-4298	403	3	open	open	ADJ
ejpam-4298	403	4	set	set	NOUN
ejpam-4298	403	5	.	.	PUNCT
ejpam-4298	404	1	j	j	PROPN
ejpam-4298	404	2	al	al	PROPN
ejpam-4298	404	3	-	-	PUNCT
ejpam-4298	404	4	qadisiyah	qadisiyah	NOUN
ejpam-4298	404	5	comput	comput	NOUN
ejpam-4298	404	6	.	.	PUNCT
ejpam-4298	405	1	sci	sci	PROPN
ejpam-4298	405	2	.	.	PROPN
ejpam-4298	405	3	math	math	PROPN
ejpam-4298	405	4	,	,	PUNCT
ejpam-4298	405	5	6(25	6(25	PROPN
ejpam-4298	405	6	)	)	PUNCT
ejpam-4298	405	7	,	,	PUNCT
ejpam-4298	405	8	2014	2014	NUM
ejpam-4298	405	9	.	.	PUNCT
ejpam-4298	406	1	[	[	X
ejpam-4298	406	2	2	2	NUM
ejpam-4298	406	3	]	]	X
ejpam-4298	406	4	f	f	PROPN
ejpam-4298	406	5	a	a	DET
ejpam-4298	406	6	abushaheen	abushaheen	PROPN
ejpam-4298	406	7	and	and	CCONJ
ejpam-4298	406	8	f	f	PROPN
ejpam-4298	406	9	alrimawi	alrimawi	PROPN
ejpam-4298	406	10	.	.	PUNCT
ejpam-4298	407	1	weakly	weakly	ADJ
ejpam-4298	407	2	covering	covering	NOUN
ejpam-4298	407	3	spaces	space	NOUN
ejpam-4298	407	4	in	in	ADP
ejpam-4298	407	5	coc	coc	NOUN
ejpam-4298	407	6	-	-	PUNCT
ejpam-4298	407	7	open	open	ADJ
ejpam-4298	407	8	sets	set	NOUN
ejpam-4298	407	9	.	.	PUNCT
ejpam-4298	408	1	j	j	PROPN
ejpam-4298	408	2	european	european	PROPN
ejpam-4298	408	3	journal	journal	PROPN
ejpam-4298	408	4	of	of	ADP
ejpam-4298	408	5	pure	pure	ADJ
ejpam-4298	408	6	and	and	CCONJ
ejpam-4298	408	7	applied	applied	ADJ
ejpam-4298	408	8	mathematics	mathematic	NOUN
ejpam-4298	408	9	,	,	PUNCT
ejpam-4298	408	10	15(1):199–206	15(1):199–206	PROPN
ejpam-4298	408	11	,	,	PUNCT
ejpam-4298	408	12	2022	2022	NUM
ejpam-4298	408	13	.	.	PUNCT
ejpam-4298	409	1	[	[	X
ejpam-4298	409	2	3	3	NUM
ejpam-4298	409	3	]	]	X
ejpam-4298	409	4	s	s	PROPN
ejpam-4298	409	5	al	al	PROPN
ejpam-4298	409	6	ghour	ghour	PROPN
ejpam-4298	409	7	and	and	CCONJ
ejpam-4298	409	8	e	e	NOUN
ejpam-4298	409	9	maghrabi	maghrabi	NOUN
ejpam-4298	409	10	.	.	PUNCT
ejpam-4298	410	1	co	co	ADJ
ejpam-4298	410	2	-	-	ADJ
ejpam-4298	410	3	compact	compact	ADJ
ejpam-4298	410	4	separation	separation	NOUN
ejpam-4298	410	5	axoims	axoim	NOUN
ejpam-4298	410	6	and	and	CCONJ
ejpam-4298	410	7	slight	slight	ADJ
ejpam-4298	410	8	co	co	NOUN
ejpam-4298	410	9	-	-	NOUN
ejpam-4298	410	10	continuity	continuity	NOUN
ejpam-4298	410	11	.	.	PUNCT
ejpam-4298	411	1	symmetry	symmetry	NOUN
ejpam-4298	411	2	,	,	PUNCT
ejpam-4298	411	3	12	12	NUM
ejpam-4298	411	4	,	,	PUNCT
ejpam-4298	411	5	2020	2020	NUM
ejpam-4298	411	6	.	.	PUNCT
ejpam-4298	412	1	[	[	X
ejpam-4298	412	2	4	4	NUM
ejpam-4298	412	3	]	]	X
ejpam-4298	412	4	s	s	PROPN
ejpam-4298	412	5	al	al	PROPN
ejpam-4298	412	6	ghour	ghour	PROPN
ejpam-4298	412	7	and	and	CCONJ
ejpam-4298	412	8	s	s	PROPN
ejpam-4298	412	9	samarah	samarah	NOUN
ejpam-4298	412	10	.	.	PUNCT
ejpam-4298	413	1	cocompact	cocompact	PROPN
ejpam-4298	413	2	open	open	ADJ
ejpam-4298	413	3	sets	set	NOUN
ejpam-4298	413	4	and	and	CCONJ
ejpam-4298	413	5	continuity	continuity	NOUN
ejpam-4298	413	6	.	.	PUNCT
ejpam-4298	414	1	in	in	ADP
ejpam-4298	414	2	abstarct	abstarct	PROPN
ejpam-4298	414	3	and	and	CCONJ
ejpam-4298	414	4	applied	apply	VERB
ejpam-4298	414	5	analysis	analysis	NOUN
ejpam-4298	414	6	,	,	PUNCT
ejpam-4298	414	7	p548612	p548612	NOUN
ejpam-4298	414	8	,	,	PUNCT
ejpam-4298	414	9	2012	2012	NUM
ejpam-4298	414	10	.	.	PUNCT
ejpam-4298	415	1	[	[	X
ejpam-4298	415	2	5	5	NUM
ejpam-4298	415	3	]	]	PUNCT
ejpam-4298	415	4	t	t	PROPN
ejpam-4298	415	5	m	m	PROPN
ejpam-4298	415	6	al	al	PROPN
ejpam-4298	415	7	-	-	PUNCT
ejpam-4298	415	8	shami	shami	PROPN
ejpam-4298	415	9	,	,	PUNCT
ejpam-4298	415	10	e.	e.	PROPN
ejpam-4298	415	11	a	a	DET
ejpam-4298	415	12	abo	abo	NOUN
ejpam-4298	415	13	-	-	PUNCT
ejpam-4298	415	14	tabl	tabl	NOUN
ejpam-4298	415	15	,	,	PUNCT
ejpam-4298	415	16	b	b	NOUN
ejpam-4298	415	17	a	a	DET
ejpam-4298	415	18	asaad	asaad	NOUN
ejpam-4298	415	19	,	,	PUNCT
ejpam-4298	415	20	and	and	CCONJ
ejpam-4298	415	21	m	m	VERB
ejpam-4298	415	22	a	a	DET
ejpam-4298	415	23	arahet	arahet	NOUN
ejpam-4298	415	24	.	.	PUNCT
ejpam-4298	416	1	limit	limit	NOUN
ejpam-4298	416	2	points	point	NOUN
ejpam-4298	416	3	and	and	CCONJ
ejpam-4298	416	4	separation	separation	NOUN
ejpam-4298	416	5	axioms	axiom	NOUN
ejpam-4298	416	6	with	with	ADP
ejpam-4298	416	7	respect	respect	NOUN
ejpam-4298	416	8	to	to	ADP
ejpam-4298	416	9	supra	supra	PROPN
ejpam-4298	416	10	semi	semi	ADJ
ejpam-4298	416	11	-	-	ADJ
ejpam-4298	416	12	open	open	ADJ
ejpam-4298	416	13	sets	set	NOUN
ejpam-4298	416	14	.	.	PUNCT
ejpam-4298	417	1	european	european	ADJ
ejpam-4298	417	2	journal	journal	PROPN
ejpam-4298	417	3	of	of	ADP
ejpam-4298	417	4	pure	pure	ADJ
ejpam-4298	417	5	and	and	CCONJ
ejpam-4298	417	6	applied	applied	ADJ
ejpam-4298	417	7	mathematics	mathematic	NOUN
ejpam-4298	417	8	,	,	PUNCT
ejpam-4298	417	9	13(3):427–443	13(3):427–443	PROPN
ejpam-4298	417	10	,	,	PUNCT
ejpam-4298	417	11	2020	2020	NUM
ejpam-4298	417	12	.	.	PUNCT
ejpam-4298	418	1	[	[	X
ejpam-4298	418	2	6	6	NUM
ejpam-4298	418	3	]	]	SYM
ejpam-4298	418	4	f	f	PROPN
ejpam-4298	418	5	areuas	areuas	PROPN
ejpam-4298	418	6	.	.	PUNCT
ejpam-4298	419	1	topological	topological	ADJ
ejpam-4298	419	2	properties	property	NOUN
ejpam-4298	419	3	preserved	preserve	VERB
ejpam-4298	419	4	by	by	ADP
ejpam-4298	419	5	proper	proper	ADJ
ejpam-4298	419	6	subspace	subspace	NOUN
ejpam-4298	419	7	.	.	PUNCT
ejpam-4298	420	1	q	q	PUNCT
ejpam-4298	421	1	and	and	CCONJ
ejpam-4298	421	2	a	a	PRON
ejpam-4298	421	3	in	in	ADP
ejpam-4298	421	4	general	general	ADJ
ejpam-4298	421	5	topology	topology	NOUN
ejpam-4298	421	6	,	,	PUNCT
ejpam-4298	421	7	14:53–57	14:53–57	NUM
ejpam-4298	421	8	,	,	PUNCT
ejpam-4298	421	9	1996	1996	NUM
ejpam-4298	421	10	.	.	PUNCT
ejpam-4298	422	1	[	[	X
ejpam-4298	422	2	7	7	NUM
ejpam-4298	422	3	]	]	X
ejpam-4298	422	4	r	r	NOUN
ejpam-4298	422	5	engelking	engelking	NOUN
ejpam-4298	422	6	.	.	PUNCT
ejpam-4298	423	1	general	general	ADJ
ejpam-4298	423	2	topology	topology	PROPN
ejpam-4298	423	3	.	.	PUNCT
ejpam-4298	424	1	revised	revise	VERB
ejpam-4298	424	2	and	and	CCONJ
ejpam-4298	424	3	completed	complete	VERB
ejpam-4298	424	4	edition	edition	NOUN
ejpam-4298	424	5	.	.	PUNCT
ejpam-4298	425	1	heldermann	heldermann	PROPN
ejpam-4298	425	2	verlag	verlag	PROPN
ejpam-4298	425	3	,	,	PUNCT
ejpam-4298	425	4	berlin	berlin	PROPN
ejpam-4298	425	5	,	,	PUNCT
ejpam-4298	425	6	1989	1989	NUM
ejpam-4298	425	7	.	.	PUNCT
ejpam-4298	426	1	[	[	X
ejpam-4298	426	2	8	8	NUM
ejpam-4298	426	3	]	]	PUNCT
ejpam-4298	426	4	m	m	VERB
ejpam-4298	426	5	sarsak	sarsak	ADJ
ejpam-4298	426	6	.	.	PUNCT
ejpam-4298	427	1	new	new	ADJ
ejpam-4298	427	2	separation	separation	NOUN
ejpam-4298	427	3	axioms	axiom	VERB
ejpam-4298	427	4	in	in	ADP
ejpam-4298	427	5	generalized	generalized	ADJ
ejpam-4298	427	6	topological	topological	ADJ
ejpam-4298	427	7	spaces	space	NOUN
ejpam-4298	427	8	.	.	PUNCT
ejpam-4298	428	1	acta	acta	PROPN
ejpam-4298	428	2	math	math	PROPN
ejpam-4298	428	3	.	.	PUNCT
ejpam-4298	429	1	hunger	hunger	NOUN
ejpam-4298	429	2	,	,	PUNCT
ejpam-4298	429	3	132:244–252	132:244–252	NUM
ejpam-4298	429	4	,	,	PUNCT
ejpam-4298	429	5	2011	2011	NUM
ejpam-4298	429	6	.	.	PUNCT
ejpam-4298	430	1	[	[	X
ejpam-4298	430	2	9	9	NUM
ejpam-4298	430	3	]	]	SYM
ejpam-4298	430	4	m	m	VERB
ejpam-4298	430	5	sarsak	sarsak	ADJ
ejpam-4298	430	6	.	.	PUNCT
ejpam-4298	431	1	weak	weak	ADJ
ejpam-4298	431	2	separation	separation	NOUN
ejpam-4298	431	3	axioms	axiom	NOUN
ejpam-4298	431	4	in	in	ADP
ejpam-4298	431	5	generalized	generalized	ADJ
ejpam-4298	431	6	topological	topological	ADJ
ejpam-4298	431	7	spaces	space	NOUN
ejpam-4298	431	8	.	.	PUNCT
ejpam-4298	432	1	acta	acta	PROPN
ejpam-4298	432	2	math	math	PROPN
ejpam-4298	432	3	.	.	PUNCT
ejpam-4298	433	1	hunger	hunger	NOUN
ejpam-4298	433	2	,	,	PUNCT
ejpam-4298	433	3	131:110–121	131:110–121	NUM
ejpam-4298	433	4	,	,	PUNCT
ejpam-4298	433	5	2011	2011	NUM
ejpam-4298	433	6	.	.	PUNCT