id	sid	tid	token	lemma	pos
iajs-2911	1	1	ihjpas	ihjpas	PROPN
iajs-2911	1	2	.	.	PUNCT
iajs-2911	2	1	36	36	NUM
iajs-2911	2	2	(	(	PUNCT
iajs-2911	2	3	4	4	NUM
iajs-2911	2	4	)	)	PUNCT
iajs-2911	2	5	2023	2023	NUM
iajs-2911	2	6	396	396	NUM
iajs-2911	2	7	this	this	DET
iajs-2911	2	8	work	work	NOUN
iajs-2911	2	9	is	be	AUX
iajs-2911	2	10	licensed	license	VERB
iajs-2911	2	11	under	under	ADP
iajs-2911	2	12	a	a	DET
iajs-2911	2	13	creative	creative	ADJ
iajs-2911	2	14	commons	common	NOUN
iajs-2911	2	15	attribution	attribution	NOUN
iajs-2911	2	16	4.0	4.0	NUM
iajs-2911	2	17	international	international	ADJ
iajs-2911	2	18	license	license	NOUN
iajs-2911	2	19	*	*	PUNCT
iajs-2911	2	20	corresponding	correspond	VERB
iajs-2911	2	21	author	author	NOUN
iajs-2911	2	22	:	:	PUNCT
iajs-2911	2	23	maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-2911	2	24	abstract	abstract	VERB
iajs-2911	2	25	the	the	DET
iajs-2911	2	26	essential	essential	ADJ
iajs-2911	2	27	objective	objective	NOUN
iajs-2911	2	28	of	of	ADP
iajs-2911	2	29	this	this	DET
iajs-2911	2	30	paper	paper	NOUN
iajs-2911	2	31	is	be	AUX
iajs-2911	2	32	to	to	PART
iajs-2911	2	33	introduce	introduce	VERB
iajs-2911	2	34	new	new	ADJ
iajs-2911	2	35	notions	notion	NOUN
iajs-2911	2	36	of	of	ADP
iajs-2911	2	37	fibrewise	fibrewise	NOUN
iajs-2911	2	38	topological	topological	ADJ
iajs-2911	2	39	spaces	space	NOUN
iajs-2911	2	40	on	on	ADP
iajs-2911	2	41	d	d	PROPN
iajs-2911	2	42	that	that	PRON
iajs-2911	2	43	are	be	AUX
iajs-2911	2	44	named	name	VERB
iajs-2911	2	45	to	to	PART
iajs-2911	2	46	be	be	AUX
iajs-2911	2	47	upper	upper	ADJ
iajs-2911	2	48	perfect	perfect	ADJ
iajs-2911	2	49	topological	topological	ADJ
iajs-2911	2	50	spaces	space	NOUN
iajs-2911	2	51	,	,	PUNCT
iajs-2911	2	52	lower	low	ADJ
iajs-2911	2	53	perfect	perfect	ADJ
iajs-2911	2	54	topological	topological	ADJ
iajs-2911	2	55	spaces	space	NOUN
iajs-2911	2	56	,	,	PUNCT
iajs-2911	2	57	multi	multi	ADJ
iajs-2911	2	58	-	-	ADJ
iajs-2911	2	59	perfect	perfect	ADJ
iajs-2911	2	60	topological	topological	ADJ
iajs-2911	2	61	spaces	space	NOUN
iajs-2911	2	62	,	,	PUNCT
iajs-2911	2	63	fibrewise	fibrewise	NOUN
iajs-2911	2	64	upper	upper	ADJ
iajs-2911	2	65	perfect	perfect	ADJ
iajs-2911	2	66	topological	topological	ADJ
iajs-2911	2	67	spaces	space	NOUN
iajs-2911	2	68	,	,	PUNCT
iajs-2911	2	69	and	and	CCONJ
iajs-2911	2	70	fibrewise	fibrewise	ADV
iajs-2911	2	71	lower	lower	ADV
iajs-2911	2	72	perfect	perfect	ADJ
iajs-2911	2	73	topological	topological	ADJ
iajs-2911	2	74	spaces	space	NOUN
iajs-2911	2	75	.	.	PUNCT
iajs-2911	3	1	fibrewise	fibrewise	PROPN
iajs-2911	3	2	multi	multi	ADJ
iajs-2911	3	3	-	-	ADJ
iajs-2911	3	4	perfect	perfect	ADJ
iajs-2911	3	5	topological	topological	ADJ
iajs-2911	3	6	spaces	space	NOUN
iajs-2911	3	7	,	,	PUNCT
iajs-2911	3	8	filter	filter	NOUN
iajs-2911	3	9	base	base	NOUN
iajs-2911	3	10	,	,	PUNCT
iajs-2911	3	11	contact	contact	NOUN
iajs-2911	3	12	point	point	NOUN
iajs-2911	3	13	,	,	PUNCT
iajs-2911	3	14	rigid	rigid	ADJ
iajs-2911	3	15	,	,	PUNCT
iajs-2911	3	16	multi	multi	ADJ
iajs-2911	3	17	-	-	ADJ
iajs-2911	3	18	rigid	rigid	ADJ
iajs-2911	3	19	,	,	PUNCT
iajs-2911	3	20	multi	multi	ADJ
iajs-2911	3	21	-	-	ADJ
iajs-2911	3	22	rigid	rigid	ADJ
iajs-2911	3	23	,	,	PUNCT
iajs-2911	3	24	fibrewise	fibrewise	NOUN
iajs-2911	3	25	upper	upper	ADJ
iajs-2911	3	26	weakly	weakly	ADV
iajs-2911	3	27	closed	closed	ADJ
iajs-2911	3	28	,	,	PUNCT
iajs-2911	3	29	fibrewise	fibrewise	ADV
iajs-2911	3	30	lower	low	ADJ
iajs-2911	3	31	weakly	weakly	ADV
iajs-2911	3	32	closed	closed	ADJ
iajs-2911	3	33	,	,	PUNCT
iajs-2911	3	34	fibrewise	fibrewise	ADV
iajs-2911	3	35	multi	multi	ADJ
iajs-2911	3	36	-	-	ADJ
iajs-2911	3	37	weakly	weakly	ADJ
iajs-2911	3	38	closed	closed	ADJ
iajs-2911	3	39	,	,	PUNCT
iajs-2911	3	40	set	set	VERB
iajs-2911	3	41	,	,	PUNCT
iajs-2911	3	42	almost	almost	ADV
iajs-2911	3	43	upper	upper	ADJ
iajs-2911	3	44	perfect	perfect	ADJ
iajs-2911	3	45	,	,	PUNCT
iajs-2911	3	46	almost	almost	ADV
iajs-2911	3	47	lower	low	ADJ
iajs-2911	3	48	perfect	perfect	ADJ
iajs-2911	3	49	,	,	PUNCT
iajs-2911	3	50	almost	almost	ADV
iajs-2911	3	51	multiperfect	multiperfect	NOUN
iajs-2911	3	52	,	,	PUNCT
iajs-2911	3	53	fibrewise	fibrewise	ADV
iajs-2911	3	54	almost	almost	ADV
iajs-2911	3	55	upper	upper	ADJ
iajs-2911	3	56	perfect	perfect	ADJ
iajs-2911	3	57	,	,	PUNCT
iajs-2911	3	58	fibrewise	fibrewise	ADV
iajs-2911	3	59	almost	almost	ADV
iajs-2911	3	60	lower	low	ADJ
iajs-2911	3	61	perfect	perfect	ADJ
iajs-2911	3	62	,	,	PUNCT
iajs-2911	3	63	fibrewise	fibrewise	NOUN
iajs-2911	3	64	almost	almost	ADV
iajs-2911	3	65	multiperfect	multiperfect	NOUN
iajs-2911	3	66	,	,	PUNCT
iajs-2911	3	67	upper	upper	ADJ
iajs-2911	3	68	*	*	PUNCT
iajs-2911	3	69	continuous	continuous	ADJ
iajs-2911	3	70	fibrewise	fibrewise	NOUN
iajs-2911	3	71	upper∗	upper∗	VERB
iajs-2911	3	72	topological	topological	ADJ
iajs-2911	3	73	spaces	space	NOUN
iajs-2911	3	74	respectively	respectively	ADV
iajs-2911	3	75	,	,	PUNCT
iajs-2911	3	76	lower	low	ADJ
iajs-2911	3	77	*	*	PUNCT
iajs-2911	3	78	continuous	continuous	ADJ
iajs-2911	3	79	fibrewise	fibrewise	NOUN
iajs-2911	3	80	lower∗	lower∗	VERB
iajs-2911	3	81	topological	topological	ADJ
iajs-2911	3	82	spaces	space	NOUN
iajs-2911	3	83	respectively	respectively	ADV
iajs-2911	3	84	,	,	PUNCT
iajs-2911	3	85	multi*-continuous	multi*-continuous	ADJ
iajs-2911	3	86	fibrewise	fibrewise	NOUN
iajs-2911	3	87	multi∗-topological	multi∗-topological	ADJ
iajs-2911	3	88	spaces	space	NOUN
iajs-2911	3	89	respectively	respectively	ADV
iajs-2911	3	90	multi	multi	NOUN
iajs-2911	3	91	-	-	ADJ
iajs-2911	3	92	te	te	ADJ
iajs-2911	3	93	,	,	PUNCT
iajs-2911	3	94	locally	locally	ADV
iajs-2911	3	95	in	in	ADP
iajs-2911	3	96	addition	addition	NOUN
iajs-2911	3	97	,	,	PUNCT
iajs-2911	3	98	we	we	PRON
iajs-2911	3	99	find	find	VERB
iajs-2911	3	100	and	and	CCONJ
iajs-2911	3	101	prove	prove	VERB
iajs-2911	3	102	several	several	ADJ
iajs-2911	3	103	propositions	proposition	NOUN
iajs-2911	3	104	linked	link	VERB
iajs-2911	3	105	to	to	ADP
iajs-2911	3	106	these	these	DET
iajs-2911	3	107	notions	notion	NOUN
iajs-2911	3	108	.	.	PUNCT
iajs-2911	4	1	keywords	keyword	NOUN
iajs-2911	4	2	:	:	PUNCT
iajs-2911	4	3	fibrewise	fibrewise	PROPN
iajs-2911	4	4	topological	topological	ADJ
iajs-2911	4	5	spaces	space	NOUN
iajs-2911	4	6	,	,	PUNCT
iajs-2911	4	7	filter	filter	NOUN
iajs-2911	4	8	base	base	NOUN
iajs-2911	4	9	,	,	PUNCT
iajs-2911	4	10	fibrewise	fibrewise	NOUN
iajs-2911	4	11	upper	upper	ADJ
iajs-2911	4	12	perfect	perfect	ADJ
iajs-2911	4	13	topological	topological	ADJ
iajs-2911	4	14	spaces	space	NOUN
iajs-2911	4	15	,	,	PUNCT
iajs-2911	4	16	fibrewise	fibrewise	ADV
iajs-2911	4	17	lower	low	ADJ
iajs-2911	4	18	perfect	perfect	ADJ
iajs-2911	4	19	topological	topological	ADJ
iajs-2911	4	20	spaces	space	NOUN
iajs-2911	4	21	,	,	PUNCT
iajs-2911	4	22	and	and	CCONJ
iajs-2911	4	23	fibrewise	fibrewise	ADV
iajs-2911	4	24	multi	multi	ADJ
iajs-2911	4	25	-	-	ADJ
iajs-2911	4	26	perfect	perfect	ADJ
iajs-2911	4	27	topological	topological	ADJ
iajs-2911	4	28	spaces	space	NOUN
iajs-2911	4	29	.	.	PUNCT
iajs-2911	5	1	1	1	X
iajs-2911	5	2	.	.	X
iajs-2911	5	3	introduction	introduction	NOUN
iajs-2911	5	4	we	we	PRON
iajs-2911	5	5	begin	begin	VERB
iajs-2911	5	6	our	our	PRON
iajs-2911	5	7	work	work	NOUN
iajs-2911	5	8	with	with	ADP
iajs-2911	5	9	the	the	DET
iajs-2911	5	10	concept	concept	NOUN
iajs-2911	5	11	of	of	ADP
iajs-2911	5	12	category	category	NOUN
iajs-2911	5	13	of	of	ADP
iajs-2911	5	14	fibrewise	fibrewise	NOUN
iajs-2911	5	15	(	(	PUNCT
iajs-2911	5	16	briefly	briefly	ADV
iajs-2911	5	17	,	,	PUNCT
iajs-2911	5	18	𝔽.𝕎.	𝔽.𝕎.	PROPN
iajs-2911	5	19	)	)	PUNCT
iajs-2911	5	20	set	set	VERB
iajs-2911	5	21	on	on	ADP
iajs-2911	5	22	a	a	DET
iajs-2911	5	23	known	know	VERB
iajs-2911	5	24	set	set	NOUN
iajs-2911	5	25	,	,	PUNCT
iajs-2911	5	26	named	name	VERB
iajs-2911	5	27	the	the	DET
iajs-2911	5	28	base	base	NOUN
iajs-2911	5	29	set	set	NOUN
iajs-2911	5	30	.	.	PUNCT
iajs-2911	6	1	if	if	SCONJ
iajs-2911	6	2	the	the	DET
iajs-2911	6	3	base	base	NOUN
iajs-2911	6	4	set	set	NOUN
iajs-2911	6	5	is	be	AUX
iajs-2911	6	6	stated	state	VERB
iajs-2911	6	7	with	with	ADP
iajs-2911	6	8	d	d	PROPN
iajs-2911	6	9	,	,	PUNCT
iajs-2911	6	10	then	then	ADV
iajs-2911	6	11	a	a	DET
iajs-2911	6	12	f.w	f.w	PROPN
iajs-2911	6	13	.	.	PROPN
iajs-2911	6	14	set	set	VERB
iajs-2911	6	15	on	on	ADP
iajs-2911	6	16	d	d	ADP
iajs-2911	6	17	applied	apply	VERB
iajs-2911	6	18	to	to	ADP
iajs-2911	6	19	a	a	DET
iajs-2911	6	20	set	set	NOUN
iajs-2911	6	21	e	e	NOUN
iajs-2911	6	22	with	with	ADP
iajs-2911	6	23	a	a	DET
iajs-2911	6	24	doi.org/10.30526/36.4.2911	doi.org/10.30526/36.4.2911	NOUN
iajs-2911	6	25	article	article	NOUN
iajs-2911	6	26	history	history	NOUN
iajs-2911	6	27	:	:	PUNCT
iajs-2911	6	28	received	receive	VERB
iajs-2911	6	29	21	21	NUM
iajs-2911	6	30	june	june	PROPN
iajs-2911	6	31	2022	2022	NUM
iajs-2911	6	32	,	,	PUNCT
iajs-2911	6	33	accepted	accept	VERB
iajs-2911	6	34	9	9	NUM
iajs-2911	6	35	october	october	NOUN
iajs-2911	6	36	2022	2022	NUM
iajs-2911	6	37	,	,	PUNCT
iajs-2911	6	38	published	publish	VERB
iajs-2911	6	39	in	in	ADP
iajs-2911	6	40	october	october	PROPN
iajs-2911	6	41	2023	2023	NUM
iajs-2911	6	42	ibn	ibn	PROPN
iajs-2911	6	43	al	al	PROPN
iajs-2911	6	44	-	-	PUNCT
iajs-2911	6	45	haitham	haitham	PROPN
iajs-2911	6	46	journal	journal	PROPN
iajs-2911	6	47	for	for	ADP
iajs-2911	6	48	pure	pure	ADJ
iajs-2911	6	49	and	and	CCONJ
iajs-2911	6	50	applied	applied	ADJ
iajs-2911	6	51	sciences	sciences	PROPN
iajs-2911	6	52	journal	journal	PROPN
iajs-2911	6	53	homepage	homepage	NOUN
iajs-2911	6	54	:	:	PUNCT
iajs-2911	6	55	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	VERB
iajs-2911	6	56	fibrewise	fibrewise	ADV
iajs-2911	6	57	multi	multi	ADJ
iajs-2911	6	58	-	-	ADJ
iajs-2911	6	59	perfect	perfect	ADJ
iajs-2911	6	60	topological	topological	ADJ
iajs-2911	6	61	spaces	space	NOUN
iajs-2911	6	62	majed	majed	NOUN
iajs-2911	6	63	.h	.h	NOUN
iajs-2911	6	64	.	.	PUNCT
iajs-2911	7	1	j𝐚ber	j𝐚ber	PROPN
iajs-2911	7	2	*	*	PUNCT
iajs-2911	7	3	department	department	NOUN
iajs-2911	7	4	of	of	ADP
iajs-2911	7	5	mathematics	mathematics	PROPN
iajs-2911	7	6	,	,	PUNCT
iajs-2911	7	7	college	college	NOUN
iajs-2911	7	8	of	of	ADP
iajs-2911	7	9	education	education	NOUN
iajs-2911	7	10	for	for	ADP
iajs-2911	7	11	pure	pure	ADJ
iajs-2911	7	12	sciences	science	NOUN
iajs-2911	7	13	,	,	PUNCT
iajs-2911	7	14	ibn	ibn	PROPN
iajs-2911	7	15	al	al	PROPN
iajs-2911	7	16	-	-	PUNCT
iajs-2911	7	17	haitham	haitham	PROPN
iajs-2911	7	18	university	university	PROPN
iajs-2911	7	19	of	of	ADP
iajs-2911	7	20	baghdad	baghdad	PROPN
iajs-2911	7	21	,	,	PUNCT
iajs-2911	7	22	iraq	iraq	PROPN
iajs-2911	7	23	y𝐨us𝐢f	y𝐨us𝐢f	PROPN
iajs-2911	7	24	.	.	PUNCT
iajs-2911	8	1	y.	y.	PROPN
iajs-2911	8	2	y𝐨us𝐢f	y𝐨us𝐢f	PROPN
iajs-2911	8	3	department	department	PROPN
iajs-2911	8	4	of	of	ADP
iajs-2911	8	5	mathematics	mathematics	PROPN
iajs-2911	8	6	,	,	PUNCT
iajs-2911	8	7	college	college	NOUN
iajs-2911	8	8	of	of	ADP
iajs-2911	8	9	education	education	NOUN
iajs-2911	8	10	for	for	ADP
iajs-2911	8	11	pure	pure	ADJ
iajs-2911	8	12	sciences	science	NOUN
iajs-2911	8	13	,	,	PUNCT
iajs-2911	8	14	ibn	ibn	PROPN
iajs-2911	8	15	al	al	PROPN
iajs-2911	8	16	-	-	PUNCT
iajs-2911	8	17	haitham	haitham	PROPN
iajs-2911	8	18	university	university	PROPN
iajs-2911	8	19	of	of	ADP
iajs-2911	8	20	baghdad	baghdad	PROPN
iajs-2911	8	21	,	,	PUNCT
iajs-2911	8	22	iraq	iraq	PROPN
iajs-2911	8	23	m.	m.	PROPN
iajs-2911	8	24	el	el	PROPN
iajs-2911	8	25	sayed	sayed	PROPN
iajs-2911	8	26	department	department	PROPN
iajs-2911	8	27	of	of	ADP
iajs-2911	8	28	mathematics	mathematics	PROPN
iajs-2911	8	29	,	,	PUNCT
iajs-2911	8	30	college	college	NOUN
iajs-2911	8	31	of	of	ADP
iajs-2911	8	32	science	science	NOUN
iajs-2911	8	33	and	and	CCONJ
iajs-2911	8	34	arts	art	NOUN
iajs-2911	8	35	,	,	PUNCT
iajs-2911	8	36	najran	najran	ADJ
iajs-2911	8	37	university	university	NOUN
iajs-2911	8	38	,	,	PUNCT
iajs-2911	8	39	kingdom	kingdom	NOUN
iajs-2911	8	40	of	of	ADP
iajs-2911	8	41	saudi	saudi	PROPN
iajs-2911	8	42	arabia	arabia	PROPN
iajs-2911	8	43	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2911	8	44	mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-2911	8	45	mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-2911	8	46	mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	mailto:maged.hameed1203a@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-2911	8	47	mailto:yoyayousif@yahoo.com	mailto:yoyayousif@yahoo.com	X
iajs-2911	9	1	mailto:mebadria@nu.edu.sa	mailto:mebadria@nu.edu.sa	PROPN
iajs-2911	9	2	ihjpas	ihjpa	VERB
iajs-2911	9	3	.	.	PUNCT
iajs-2911	10	1	36	36	NUM
iajs-2911	10	2	(	(	PUNCT
iajs-2911	10	3	4	4	NUM
iajs-2911	10	4	)	)	PUNCT
iajs-2911	10	5	2023	2023	NUM
iajs-2911	10	6	397	397	NUM
iajs-2911	10	7	function	function	NOUN
iajs-2911	10	8	x	x	PUNCT
iajs-2911	10	9	is	be	AUX
iajs-2911	10	10	x	x	X
iajs-2911	10	11	:	:	PUNCT
iajs-2911	10	12	e	e	X
iajs-2911	10	13	→	→	SYM
iajs-2911	10	14	d	d	PROPN
iajs-2911	10	15	,	,	PUNCT
iajs-2911	10	16	named	name	VERB
iajs-2911	10	17	the	the	DET
iajs-2911	10	18	projection	projection	NOUN
iajs-2911	10	19	(	(	PUNCT
iajs-2911	10	20	briefly	briefly	ADV
iajs-2911	10	21	,	,	PUNCT
iajs-2911	10	22	project	project	NOUN
iajs-2911	10	23	)	)	PUNCT
iajs-2911	10	24	.	.	PUNCT
iajs-2911	11	1	for	for	ADP
iajs-2911	11	2	every	every	DET
iajs-2911	11	3	point	point	NOUN
iajs-2911	11	4	d	d	NOUN
iajs-2911	11	5	of	of	ADP
iajs-2911	11	6	d	d	PROPN
iajs-2911	11	7	,	,	PUNCT
iajs-2911	11	8	the	the	DET
iajs-2911	11	9	fiber	fiber	NOUN
iajs-2911	11	10	on	on	ADP
iajs-2911	11	11	d	d	PROPN
iajs-2911	11	12	is	be	AUX
iajs-2911	11	13	the	the	DET
iajs-2911	11	14	subset	subset	NOUN
iajs-2911	11	15	ed	ed	NOUN
iajs-2911	11	16	=	=	SYM
iajs-2911	11	17	x−1(d	x−1(d	PROPN
iajs-2911	11	18	)	)	PUNCT
iajs-2911	11	19	of	of	ADP
iajs-2911	11	20	e	e	NOUN
iajs-2911	11	21	;	;	PUNCT
iajs-2911	11	22	fibers	fiber	NOUN
iajs-2911	11	23	will	will	AUX
iajs-2911	11	24	be	be	AUX
iajs-2911	11	25	empty	empty	ADJ
iajs-2911	11	26	,	,	PUNCT
iajs-2911	11	27	so	so	SCONJ
iajs-2911	11	28	we	we	PRON
iajs-2911	11	29	do	do	AUX
iajs-2911	11	30	not	not	PART
iajs-2911	11	31	require	require	VERB
iajs-2911	11	32	x	x	VERB
iajs-2911	11	33	to	to	PART
iajs-2911	11	34	be	be	AUX
iajs-2911	11	35	a	a	DET
iajs-2911	11	36	surjection	surjection	NOUN
iajs-2911	11	37	.	.	PUNCT
iajs-2911	12	1	also	also	ADV
iajs-2911	12	2	,	,	PUNCT
iajs-2911	12	3	for	for	ADP
iajs-2911	12	4	every	every	DET
iajs-2911	12	5	subset	subset	NOUN
iajs-2911	12	6	d	d	X
iajs-2911	12	7	*	*	PUNCT
iajs-2911	12	8	of	of	ADP
iajs-2911	12	9	d	d	PROPN
iajs-2911	12	10	,	,	PUNCT
iajs-2911	12	11	we	we	PRON
iajs-2911	12	12	regard	regard	VERB
iajs-2911	12	13	ed∗	ed∗	ADV
iajs-2911	12	14	=	=	PUNCT
iajs-2911	12	15	x−1(d∗	x−1(d∗	PROPN
iajs-2911	12	16	)	)	PUNCT
iajs-2911	12	17	as	as	ADP
iajs-2911	12	18	a	a	DET
iajs-2911	12	19	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	12	20	set	set	VERB
iajs-2911	12	21	on	on	ADP
iajs-2911	12	22	d	d	PROPN
iajs-2911	12	23	*	*	NOUN
iajs-2911	12	24	with	with	ADP
iajs-2911	12	25	the	the	DET
iajs-2911	12	26	project	project	NOUN
iajs-2911	12	27	determined	determine	VERB
iajs-2911	12	28	by	by	ADP
iajs-2911	12	29	x.	x.	PROPN
iajs-2911	12	30	a	a	DET
iajs-2911	12	31	multi	multi	ADJ
iajs-2911	12	32	-	-	ADJ
iajs-2911	12	33	function	function	NOUN
iajs-2911	12	34	[	[	X
iajs-2911	12	35	2	2	NUM
iajs-2911	12	36	]	]	X
iajs-2911	12	37	ω	ω	NUM
iajs-2911	12	38	of	of	ADP
iajs-2911	12	39	a	a	DET
iajs-2911	12	40	set	set	ADJ
iajs-2911	12	41	e	e	NOUN
iajs-2911	12	42	into	into	ADP
iajs-2911	12	43	f	f	PROPN
iajs-2911	12	44	is	be	AUX
iajs-2911	12	45	a	a	DET
iajs-2911	12	46	correspondence	correspondence	NOUN
iajs-2911	12	47	such	such	ADJ
iajs-2911	12	48	that	that	SCONJ
iajs-2911	12	49	ω	ω	PROPN
iajs-2911	12	50	(	(	PUNCT
iajs-2911	12	51	e	e	NOUN
iajs-2911	12	52	)	)	PUNCT
iajs-2911	12	53	is	be	AUX
iajs-2911	12	54	a	a	DET
iajs-2911	12	55	nonempty	nonempty	ADJ
iajs-2911	12	56	subset	subset	NOUN
iajs-2911	12	57	of	of	ADP
iajs-2911	12	58	f	f	PROPN
iajs-2911	12	59	for	for	ADP
iajs-2911	12	60	every	every	DET
iajs-2911	12	61	e	e	PROPN
iajs-2911	12	62	∈	∈	PROPN
iajs-2911	12	63	e.	e.	PROPN
iajs-2911	12	64	we	we	PRON
iajs-2911	12	65	will	will	AUX
iajs-2911	12	66	denote	denote	VERB
iajs-2911	12	67	such	such	DET
iajs-2911	12	68	a	a	DET
iajs-2911	12	69	multifunction	multifunction	NOUN
iajs-2911	12	70	by	by	ADP
iajs-2911	12	71	ω	ω	PROPN
iajs-2911	12	72	:	:	PUNCT
iajs-2911	12	73	e	e	PROPN
iajs-2911	12	74	→	→	SYM
iajs-2911	12	75	f	f	PROPN
iajs-2911	12	76	.	.	PUNCT
iajs-2911	13	1	for	for	ADP
iajs-2911	13	2	a	a	DET
iajs-2911	13	3	multifunction	multifunction	NOUN
iajs-2911	13	4	ω	ω	NOUN
iajs-2911	13	5	,	,	PUNCT
iajs-2911	13	6	the	the	DET
iajs-2911	13	7	upper	upper	ADJ
iajs-2911	13	8	and	and	CCONJ
iajs-2911	13	9	lower	low	ADJ
iajs-2911	13	10	inverse	inverse	NOUN
iajs-2911	13	11	set	set	NOUN
iajs-2911	13	12	of	of	ADP
iajs-2911	13	13	a	a	DET
iajs-2911	13	14	set	set	NOUN
iajs-2911	13	15	k	k	PROPN
iajs-2911	13	16	of	of	ADP
iajs-2911	13	17	f	f	PROPN
iajs-2911	13	18	,	,	PUNCT
iajs-2911	13	19	will	will	AUX
iajs-2911	13	20	be	be	AUX
iajs-2911	13	21	denoted	denote	VERB
iajs-2911	13	22	by	by	ADP
iajs-2911	13	23	ω+(k	ω+(k	NUM
iajs-2911	13	24	)	)	PUNCT
iajs-2911	13	25	and	and	CCONJ
iajs-2911	13	26	ω−(k	ω−(k	PROPN
iajs-2911	13	27	)	)	PUNCT
iajs-2911	13	28	,	,	PUNCT
iajs-2911	13	29	respectively	respectively	ADV
iajs-2911	13	30	,	,	PUNCT
iajs-2911	13	31	that	that	PRON
iajs-2911	13	32	is	be	AUX
iajs-2911	13	33	ω+(k	ω+(k	NUM
iajs-2911	13	34	)	)	PUNCT
iajs-2911	13	35	=	=	PRON
iajs-2911	13	36	{	{	PUNCT
iajs-2911	13	37	e	e	X
iajs-2911	13	38	∈	∈	PROPN
iajs-2911	13	39	e	e	NOUN
iajs-2911	13	40	:	:	PUNCT
iajs-2911	13	41	ω(e	ω(e	NOUN
iajs-2911	13	42	)	)	PUNCT
iajs-2911	13	43	⊆	⊆	NUM
iajs-2911	13	44	k	k	NOUN
iajs-2911	13	45	}	}	PUNCT
iajs-2911	13	46	and	and	CCONJ
iajs-2911	13	47	ω−(k	ω−(k	PROPN
iajs-2911	13	48	)	)	PUNCT
iajs-2911	13	49	=	=	PRON
iajs-2911	13	50	{	{	PUNCT
iajs-2911	13	51	e	e	X
iajs-2911	13	52	∈	∈	PROPN
iajs-2911	13	53	e	e	NOUN
iajs-2911	13	54	:	:	PUNCT
iajs-2911	13	55	ω(e	ω(e	NOUN
iajs-2911	13	56	)	)	PUNCT
iajs-2911	13	57	∩	∩	NOUN
iajs-2911	13	58	k	k	PROPN
iajs-2911	13	59	≠	≠	PROPN
iajs-2911	13	60	∅	∅	NOUN
iajs-2911	13	61	}	}	PUNCT
iajs-2911	13	62	.	.	PUNCT
iajs-2911	14	1	definition	definition	NOUN
iajs-2911	14	2	1.1	1.1	NUM
iajs-2911	14	3	.	.	PUNCT
iajs-2911	15	1	[	[	X
iajs-2911	15	2	7	7	X
iajs-2911	15	3	]	]	PUNCT
iajs-2911	15	4	suppose	suppose	VERB
iajs-2911	15	5	that	that	SCONJ
iajs-2911	15	6	e	e	PROPN
iajs-2911	15	7	and	and	CCONJ
iajs-2911	15	8	f	f	PROPN
iajs-2911	15	9	are	be	AUX
iajs-2911	15	10	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	15	11	sets	set	NOUN
iajs-2911	15	12	on	on	ADP
iajs-2911	15	13	d	d	NOUN
iajs-2911	15	14	,	,	PUNCT
iajs-2911	15	15	with	with	ADP
iajs-2911	15	16	project	project	NOUN
iajs-2911	15	17	.	.	PUNCT
iajs-2911	16	1	𝑋𝐸	𝑋𝐸	NOUN
iajs-2911	16	2	:	:	PUNCT
iajs-2911	16	3	𝐸	𝐸	PROPN
iajs-2911	16	4	→	→	SYM
iajs-2911	16	5	𝐷	𝐷	PROPN
iajs-2911	16	6	and	and	CCONJ
iajs-2911	16	7	𝑋𝐹	𝑋𝐹	PROPN
iajs-2911	16	8	:	:	PUNCT
iajs-2911	16	9	𝐹	𝐹	PROPN
iajs-2911	16	10	→	→	SYM
iajs-2911	16	11	𝐷	𝐷	PROPN
iajs-2911	16	12	,	,	PUNCT
iajs-2911	16	13	respectively	respectively	ADV
iajs-2911	16	14	,	,	PUNCT
iajs-2911	16	15	a	a	DET
iajs-2911	16	16	function	function	NOUN
iajs-2911	16	17	ω	ω	NOUN
iajs-2911	16	18	:	:	PUNCT
iajs-2911	16	19	e	e	X
iajs-2911	16	20	→	→	SYM
iajs-2911	16	21	f	f	PROPN
iajs-2911	16	22	is	be	AUX
iajs-2911	16	23	named	name	VERB
iajs-2911	16	24	to	to	PART
iajs-2911	16	25	be	be	AUX
iajs-2911	16	26	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	16	27	if	if	SCONJ
iajs-2911	16	28	𝑋𝐹𝛰ω	𝑋𝐹𝛰ω	NOUN
iajs-2911	16	29	=	=	SYM
iajs-2911	16	30	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	16	31	,	,	PUNCT
iajs-2911	16	32	that	that	PRON
iajs-2911	16	33	is	be	AUX
iajs-2911	16	34	to	to	PART
iajs-2911	16	35	say	say	VERB
iajs-2911	16	36	if	if	SCONJ
iajs-2911	16	37	ω(xd	ω(xd	NOUN
iajs-2911	16	38	)	)	PUNCT
iajs-2911	16	39	⊂	⊂	PROPN
iajs-2911	16	40	fd	fd	VERB
iajs-2911	16	41	for	for	ADP
iajs-2911	16	42	every	every	DET
iajs-2911	16	43	point	point	NOUN
iajs-2911	16	44	d	d	PROPN
iajs-2911	16	45	of	of	ADP
iajs-2911	16	46	d.	d.	PROPN
iajs-2911	16	47	for	for	ADP
iajs-2911	16	48	other	other	ADJ
iajs-2911	16	49	concepts	concept	NOUN
iajs-2911	16	50	or	or	CCONJ
iajs-2911	16	51	information	information	NOUN
iajs-2911	16	52	that	that	PRON
iajs-2911	16	53	are	be	AUX
iajs-2911	16	54	undefined	undefined	ADJ
iajs-2911	16	55	here	here	ADV
iajs-2911	16	56	,	,	PUNCT
iajs-2911	16	57	we	we	PRON
iajs-2911	16	58	follow	follow	VERB
iajs-2911	16	59	nearly	nearly	ADV
iajs-2911	16	60	[	[	X
iajs-2911	16	61	3	3	NUM
iajs-2911	16	62	]	]	PUNCT
iajs-2911	16	63	and[4	and[4	NUM
iajs-2911	16	64	]	]	PUNCT
iajs-2911	16	65	recall	recall	VERB
iajs-2911	16	66	that	that	SCONJ
iajs-2911	16	67	[	[	X
iajs-2911	16	68	7	7	X
iajs-2911	16	69	]	]	X
iajs-2911	16	70	let	let	VERB
iajs-2911	16	71	d	d	PRON
iajs-2911	16	72	be	be	AUX
iajs-2911	16	73	a	a	DET
iajs-2911	16	74	topological	topological	ADJ
iajs-2911	16	75	space	space	NOUN
iajs-2911	16	76	,	,	PUNCT
iajs-2911	16	77	the	the	DET
iajs-2911	16	78	𝔽.𝕎.	𝔽.𝕎.	PROPN
iajs-2911	16	79	topology	topology	NOUN
iajs-2911	16	80	space	space	NOUN
iajs-2911	16	81	(	(	PUNCT
iajs-2911	16	82	briefly	briefly	ADV
iajs-2911	16	83	,	,	PUNCT
iajs-2911	16	84	𝔽.𝕎.t.s	𝔽.𝕎.t.s	PROPN
iajs-2911	16	85	.	.	PUNCT
iajs-2911	16	86	)	)	PUNCT
iajs-2911	16	87	on	on	ADP
iajs-2911	16	88	a	a	DET
iajs-2911	16	89	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	16	90	set	set	NOUN
iajs-2911	16	91	e	e	NOUN
iajs-2911	16	92	on	on	ADP
iajs-2911	16	93	d	d	PROPN
iajs-2911	16	94	,	,	PUNCT
iajs-2911	16	95	which	which	PRON
iajs-2911	16	96	means	mean	VERB
iajs-2911	16	97	any	any	DET
iajs-2911	16	98	topology	topology	NOUN
iajs-2911	16	99	on	on	ADP
iajs-2911	16	100	e	e	NOUN
iajs-2911	16	101	for	for	ADP
iajs-2911	16	102	that	that	PRON
iajs-2911	16	103	the	the	DET
iajs-2911	16	104	project	project	NOUN
iajs-2911	16	105	x	x	PUNCT
iajs-2911	16	106	is	be	AUX
iajs-2911	16	107	continuous	continuous	ADJ
iajs-2911	16	108	.	.	PUNCT
iajs-2911	17	1	remark	remark	NOUN
iajs-2911	17	2	1.1	1.1	NUM
iajs-2911	17	3	.	.	PUNCT
iajs-2911	18	1	[	[	X
iajs-2911	18	2	7	7	X
iajs-2911	18	3	]	]	PUNCT
iajs-2911	18	4	i.	i.	NOUN
iajs-2911	18	5	the	the	DET
iajs-2911	18	6	smaller	small	ADJ
iajs-2911	18	7	topology	topology	NOUN
iajs-2911	18	8	is	be	AUX
iajs-2911	18	9	the	the	DET
iajs-2911	18	10	topology	topology	NOUN
iajs-2911	18	11	trace	trace	NOUN
iajs-2911	18	12	with	with	ADP
iajs-2911	18	13	x	x	NOUN
iajs-2911	18	14	,	,	PUNCT
iajs-2911	18	15	where	where	SCONJ
iajs-2911	18	16	in	in	ADP
iajs-2911	18	17	the	the	DET
iajs-2911	18	18	open	open	ADJ
iajs-2911	18	19	sets	set	NOUN
iajs-2911	18	20	of	of	ADP
iajs-2911	18	21	e	e	NOUN
iajs-2911	18	22	are	be	AUX
iajs-2911	18	23	the	the	DET
iajs-2911	18	24	pre	pre	ADJ
iajs-2911	18	25	image	image	NOUN
iajs-2911	18	26	of	of	ADP
iajs-2911	18	27	the	the	DET
iajs-2911	18	28	open	open	ADJ
iajs-2911	18	29	sets	set	NOUN
iajs-2911	18	30	of	of	ADP
iajs-2911	18	31	d	d	NOUN
iajs-2911	18	32	,	,	PUNCT
iajs-2911	18	33	this	this	PRON
iajs-2911	18	34	is	be	AUX
iajs-2911	18	35	named	name	VERB
iajs-2911	18	36	the	the	DET
iajs-2911	18	37	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	18	38	indiscrete	indiscrete	ADJ
iajs-2911	18	39	topology	topology	NOUN
iajs-2911	18	40	.	.	PUNCT
iajs-2911	19	1	ii	ii	PROPN
iajs-2911	19	2	.	.	PUNCT
iajs-2911	20	1	the	the	DET
iajs-2911	20	2	𝔽.𝕎.t.s	𝔽.𝕎.t.s	PROPN
iajs-2911	20	3	.	.	PUNCT
iajs-2911	21	1	on	on	ADP
iajs-2911	21	2	d	d	PROPN
iajs-2911	21	3	is	be	AUX
iajs-2911	21	4	stated	state	VERB
iajs-2911	21	5	to	to	PART
iajs-2911	21	6	be	be	AUX
iajs-2911	21	7	a	a	DET
iajs-2911	21	8	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	21	9	set	set	VERB
iajs-2911	21	10	on	on	ADP
iajs-2911	21	11	d	d	PROPN
iajs-2911	21	12	with	with	ADP
iajs-2911	21	13	a	a	DET
iajs-2911	21	14	𝔽.𝕎.t.s	𝔽.𝕎.t.s	NOUN
iajs-2911	21	15	.	.	PUNCT
iajs-2911	22	1	we	we	PRON
iajs-2911	22	2	regard	regard	VERB
iajs-2911	22	3	the	the	DET
iajs-2911	22	4	topology	topology	NOUN
iajs-2911	22	5	product	product	NOUN
iajs-2911	22	6	d	d	PROPN
iajs-2911	22	7	×	×	PROPN
iajs-2911	22	8	t	t	PROPN
iajs-2911	22	9	,	,	PUNCT
iajs-2911	22	10	for	for	ADP
iajs-2911	22	11	any	any	DET
iajs-2911	22	12	topological	topological	ADJ
iajs-2911	22	13	space	space	NOUN
iajs-2911	22	14	t	t	PROPN
iajs-2911	22	15	,	,	PUNCT
iajs-2911	22	16	as	as	ADP
iajs-2911	22	17	a	a	DET
iajs-2911	22	18	𝔽.𝕎.t.s	𝔽.𝕎.t.s	NOUN
iajs-2911	22	19	.	.	PUNCT
iajs-2911	23	1	on	on	ADP
iajs-2911	23	2	d	d	ADP
iajs-2911	23	3	using	use	VERB
iajs-2911	23	4	the	the	DET
iajs-2911	23	5	category	category	NOUN
iajs-2911	23	6	of	of	ADP
iajs-2911	23	7	the	the	DET
iajs-2911	23	8	first	first	ADJ
iajs-2911	23	9	projection	projection	NOUN
iajs-2911	23	10	.	.	PUNCT
iajs-2911	24	1	the	the	DET
iajs-2911	24	2	equivalences	equivalence	NOUN
iajs-2911	24	3	in	in	ADP
iajs-2911	24	4	the	the	DET
iajs-2911	24	5	category	category	NOUN
iajs-2911	24	6	of	of	ADP
iajs-2911	24	7	𝔽.𝕎.t.s	𝔽.𝕎.t.s	PROPN
iajs-2911	24	8	.	.	PUNCT
iajs-2911	24	9	are	be	AUX
iajs-2911	24	10	named	name	VERB
iajs-2911	24	11	𝔽.𝕎.t	𝔽.𝕎.t	ADJ
iajs-2911	24	12	.	.	PUNCT
iajs-2911	25	1	equivalences	equivalences	PROPN
iajs-2911	25	2	.	.	PUNCT
iajs-2911	26	1	if	if	SCONJ
iajs-2911	26	2	e	e	PROPN
iajs-2911	26	3	is	be	AUX
iajs-2911	26	4	𝔽.𝕎.t	𝔽.𝕎.t	ADJ
iajs-2911	26	5	.	.	PUNCT
iajs-2911	27	1	equivalent	equivalent	ADJ
iajs-2911	27	2	to	to	ADP
iajs-2911	27	3	d	d	PROPN
iajs-2911	27	4	×	×	PROPN
iajs-2911	27	5	t	t	PROPN
iajs-2911	27	6	,	,	PUNCT
iajs-2911	27	7	for	for	ADP
iajs-2911	27	8	some	some	DET
iajs-2911	27	9	topological	topological	ADJ
iajs-2911	27	10	space	space	NOUN
iajs-2911	27	11	t	t	PROPN
iajs-2911	27	12	,	,	PUNCT
iajs-2911	27	13	we	we	PRON
iajs-2911	27	14	say	say	VERB
iajs-2911	27	15	that	that	SCONJ
iajs-2911	27	16	e	e	PRON
iajs-2911	27	17	is	be	AUX
iajs-2911	27	18	trivial	trivial	ADJ
iajs-2911	27	19	,	,	PUNCT
iajs-2911	27	20	as	as	ADP
iajs-2911	27	21	a	a	DET
iajs-2911	27	22	𝔽.𝕎.t.s	𝔽.𝕎.t.s	NOUN
iajs-2911	27	23	.	.	PUNCT
iajs-2911	28	1	on	on	ADP
iajs-2911	28	2	d.	d.	PROPN
iajs-2911	28	3	in	in	ADP
iajs-2911	28	4	𝔽.𝕎.t	𝔽.𝕎.t	NOUN
iajs-2911	28	5	.	.	PUNCT
iajs-2911	29	1	the	the	DET
iajs-2911	29	2	form	form	NOUN
iajs-2911	29	3	neighbourhood	neighbourhood	NOUN
iajs-2911	29	4	(	(	PUNCT
iajs-2911	29	5	briefly	briefly	ADV
iajs-2911	29	6	,	,	PUNCT
iajs-2911	29	7	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	29	8	)	)	PUNCT
iajs-2911	29	9	is	be	AUX
iajs-2911	29	10	used	use	VERB
iajs-2911	29	11	in	in	ADP
iajs-2911	29	12	the	the	DET
iajs-2911	29	13	same	same	ADJ
iajs-2911	29	14	sense	sense	NOUN
iajs-2911	29	15	as	as	SCONJ
iajs-2911	29	16	it	it	PRON
iajs-2911	29	17	is	be	AUX
iajs-2911	29	18	in	in	ADP
iajs-2911	29	19	normally	normally	ADV
iajs-2911	29	20	topology	topology	NOUN
iajs-2911	29	21	,	,	PUNCT
iajs-2911	29	22	but	but	CCONJ
iajs-2911	29	23	the	the	DET
iajs-2911	29	24	forms	form	NOUN
iajs-2911	29	25	𝔽.𝕎.	𝔽.𝕎.	PROPN
iajs-2911	29	26	basic	basic	ADJ
iajs-2911	29	27	may	may	AUX
iajs-2911	29	28	need	need	VERB
iajs-2911	29	29	some	some	DET
iajs-2911	29	30	illustration	illustration	NOUN
iajs-2911	29	31	,	,	PUNCT
iajs-2911	29	32	so	so	ADV
iajs-2911	29	33	let	let	VERB
iajs-2911	29	34	e	e	PRON
iajs-2911	29	35	be	be	AUX
iajs-2911	29	36	𝔽.𝕎.t.s	𝔽.𝕎.t.s	ADJ
iajs-2911	29	37	.	.	PUNCT
iajs-2911	30	1	on	on	ADP
iajs-2911	30	2	d	d	X
iajs-2911	30	3	,	,	PUNCT
iajs-2911	30	4	if	if	SCONJ
iajs-2911	30	5	e	e	NOUN
iajs-2911	30	6	is	be	AUX
iajs-2911	30	7	a	a	DET
iajs-2911	30	8	point	point	NOUN
iajs-2911	30	9	of	of	ADP
iajs-2911	30	10	ed	ed	NOUN
iajs-2911	30	11	where	where	SCONJ
iajs-2911	30	12	in	in	ADP
iajs-2911	30	13	d	d	PROPN
iajs-2911	30	14	∈	∈	PROPN
iajs-2911	30	15	d	d	NOUN
iajs-2911	30	16	,	,	PUNCT
iajs-2911	30	17	appear	appear	VERB
iajs-2911	30	18	a	a	DET
iajs-2911	30	19	family	family	NOUN
iajs-2911	30	20	n(e	n(e	NOUN
iajs-2911	30	21	)	)	PUNCT
iajs-2911	30	22	of	of	ADP
iajs-2911	30	23	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	30	24	of	of	ADP
iajs-2911	30	25	e	e	NOUN
iajs-2911	30	26	in	in	ADP
iajs-2911	30	27	e	e	PROPN
iajs-2911	30	28	as	as	ADP
iajs-2911	30	29	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	30	30	basic	basic	ADJ
iajs-2911	30	31	if	if	SCONJ
iajs-2911	30	32	as	as	SCONJ
iajs-2911	30	33	every	every	DET
iajs-2911	30	34	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	30	35	h	h	NOUN
iajs-2911	30	36	of	of	ADP
iajs-2911	30	37	e	e	PRON
iajs-2911	30	38	we	we	PRON
iajs-2911	30	39	have	have	VERB
iajs-2911	30	40	ew	ew	NOUN
iajs-2911	30	41	∩	∩	PROPN
iajs-2911	30	42	k	k	PROPN
iajs-2911	30	43	⊂	⊂	PROPN
iajs-2911	30	44	h	h	PROPN
iajs-2911	30	45	,	,	PUNCT
iajs-2911	30	46	for	for	ADP
iajs-2911	30	47	some	some	DET
iajs-2911	30	48	element	element	NOUN
iajs-2911	30	49	k	k	PROPN
iajs-2911	30	50	of	of	ADP
iajs-2911	30	51	n(e	n(e	PROPN
iajs-2911	30	52	)	)	PUNCT
iajs-2911	30	53	and	and	CCONJ
iajs-2911	30	54	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	30	55	w	w	NOUN
iajs-2911	30	56	of	of	ADP
iajs-2911	30	57	d	d	PROPN
iajs-2911	30	58	in	in	ADP
iajs-2911	30	59	d.	d.	PROPN
iajs-2911	30	60	as	as	ADP
iajs-2911	30	61	example	example	NOUN
iajs-2911	30	62	,	,	PUNCT
iajs-2911	30	63	in	in	ADP
iajs-2911	30	64	the	the	DET
iajs-2911	30	65	case	case	NOUN
iajs-2911	30	66	of	of	ADP
iajs-2911	30	67	the	the	DET
iajs-2911	30	68	topological	topological	ADJ
iajs-2911	30	69	product	product	NOUN
iajs-2911	30	70	d	d	X
iajs-2911	30	71	×	×	PROPN
iajs-2911	30	72	t	t	PROPN
iajs-2911	30	73	,	,	PUNCT
iajs-2911	30	74	where	where	SCONJ
iajs-2911	30	75	in	in	ADP
iajs-2911	30	76	t	t	PROPN
iajs-2911	30	77	is	be	AUX
iajs-2911	30	78	a	a	DET
iajs-2911	30	79	topological	topological	ADJ
iajs-2911	30	80	spaces	space	NOUN
iajs-2911	30	81	,	,	PUNCT
iajs-2911	30	82	the	the	DET
iajs-2911	30	83	family	family	NOUN
iajs-2911	30	84	of	of	ADP
iajs-2911	30	85	cartesian	cartesian	ADJ
iajs-2911	30	86	products	product	NOUN
iajs-2911	30	87	d	d	X
iajs-2911	30	88	×	×	PROPN
iajs-2911	30	89	n(t	n(t	PROPN
iajs-2911	30	90	)	)	PUNCT
iajs-2911	30	91	,	,	PUNCT
iajs-2911	30	92	where	where	SCONJ
iajs-2911	30	93	in	in	ADP
iajs-2911	30	94	n(t	n(t	NOUN
iajs-2911	30	95	)	)	PUNCT
iajs-2911	30	96	runs	run	VERB
iajs-2911	30	97	through	through	ADP
iajs-2911	30	98	the	the	DET
iajs-2911	30	99	𝜂ℙ𝕕𝑠	𝜂ℙ𝕕𝑠	PROPN
iajs-2911	30	100	of	of	ADP
iajs-2911	30	101	t	t	PROPN
iajs-2911	30	102	,	,	PUNCT
iajs-2911	30	103	is	be	AUX
iajs-2911	30	104	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	30	105	basic	basic	ADJ
iajs-2911	30	106	for	for	ADP
iajs-2911	30	107	(	(	PUNCT
iajs-2911	30	108	d	d	PROPN
iajs-2911	30	109	,	,	PUNCT
iajs-2911	30	110	t	t	PROPN
iajs-2911	30	111	)	)	PUNCT
iajs-2911	30	112	.	.	PUNCT
iajs-2911	31	1	𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧	𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧	NOUN
iajs-2911	31	2	1.2	1.2	NUM
iajs-2911	31	3	.	.	PUNCT
iajs-2911	32	1	[	[	X
iajs-2911	32	2	7	7	X
iajs-2911	32	3	]	]	PUNCT
iajs-2911	32	4	the	the	DET
iajs-2911	32	5	𝔽.𝕎.	𝔽.𝕎.	PROPN
iajs-2911	32	6	functions	function	NOUN
iajs-2911	32	7	ω	ω	NOUN
iajs-2911	32	8	:	:	PUNCT
iajs-2911	32	9	e→	e→	PROPN
iajs-2911	32	10	f	f	PROPN
iajs-2911	32	11	;	;	PUNCT
iajs-2911	32	12	e	e	X
iajs-2911	32	13	and	and	CCONJ
iajs-2911	32	14	f	f	PROPN
iajs-2911	32	15	are	be	AUX
iajs-2911	32	16	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	32	17	spaces	space	NOUN
iajs-2911	32	18	on	on	ADP
iajs-2911	32	19	d	d	PROPN
iajs-2911	32	20	is	be	AUX
iajs-2911	32	21	named	name	VERB
iajs-2911	32	22	:	:	PUNCT
iajs-2911	32	23	(	(	PUNCT
iajs-2911	32	24	a	a	X
iajs-2911	32	25	)	)	PUNCT
iajs-2911	32	26	continuous	continuous	ADJ
iajs-2911	32	27	(	(	PUNCT
iajs-2911	32	28	briefly	briefly	ADV
iajs-2911	32	29	,	,	PUNCT
iajs-2911	32	30	cont	cont	PROPN
iajs-2911	32	31	.	.	PUNCT
iajs-2911	32	32	)	)	PUNCT
iajs-2911	33	1	if	if	SCONJ
iajs-2911	33	2	every	every	DET
iajs-2911	33	3	e	e	PROPN
iajs-2911	33	4	∈	∈	PROPN
iajs-2911	33	5	ed	ed	NOUN
iajs-2911	33	6	;	;	PUNCT
iajs-2911	34	1	d	d	PROPN
iajs-2911	34	2	∈	∈	PROPN
iajs-2911	35	1	d	d	X
iajs-2911	35	2	,	,	PUNCT
iajs-2911	35	3	the	the	DET
iajs-2911	35	4	inverse	inverse	ADJ
iajs-2911	35	5	image	image	NOUN
iajs-2911	35	6	of	of	ADP
iajs-2911	35	7	every	every	DET
iajs-2911	35	8	open	open	ADJ
iajs-2911	35	9	set	set	NOUN
iajs-2911	35	10	of	of	ADP
iajs-2911	35	11	ω(e	ω(e	PROPN
iajs-2911	35	12	)	)	PUNCT
iajs-2911	35	13	is	be	AUX
iajs-2911	35	14	an	an	DET
iajs-2911	35	15	open	open	ADJ
iajs-2911	35	16	set	set	NOUN
iajs-2911	35	17	of	of	ADP
iajs-2911	35	18	e.	e.	PROPN
iajs-2911	35	19	(	(	PUNCT
iajs-2911	35	20	b	b	X
iajs-2911	35	21	)	)	PUNCT
iajs-2911	35	22	open	open	ADJ
iajs-2911	35	23	if	if	SCONJ
iajs-2911	35	24	for	for	ADP
iajs-2911	35	25	every	every	DET
iajs-2911	35	26	e∈e_d	e∈e_d	PROPN
iajs-2911	35	27	,	,	PUNCT
iajs-2911	35	28	d	d	PROPN
iajs-2911	35	29	∈d	∈d	PROPN
iajs-2911	35	30	,	,	PUNCT
iajs-2911	35	31	the	the	DET
iajs-2911	35	32	direct	direct	ADJ
iajs-2911	35	33	image	image	NOUN
iajs-2911	35	34	of	of	ADP
iajs-2911	35	35	every	every	DET
iajs-2911	35	36	open	open	ADJ
iajs-2911	35	37	set	set	NOUN
iajs-2911	35	38	of	of	ADP
iajs-2911	35	39	e	e	PROPN
iajs-2911	35	40	is	be	AUX
iajs-2911	35	41	an	an	DET
iajs-2911	35	42	open	open	ADJ
iajs-2911	35	43	set	set	NOUN
iajs-2911	35	44	of	of	ADP
iajs-2911	35	45	ω(e	ω(e	PROPN
iajs-2911	35	46	)	)	PUNCT
iajs-2911	35	47	.	.	PUNCT
iajs-2911	36	1	definition	definition	NOUN
iajs-2911	36	2	1.3	1.3	NUM
iajs-2911	36	3	.	.	PUNCT
iajs-2911	37	1	[	[	X
iajs-2911	37	2	7	7	X
iajs-2911	37	3	]	]	PUNCT
iajs-2911	37	4	the	the	DET
iajs-2911	37	5	f.w.t.s	f.w.t.s	PROPN
iajs-2911	37	6	.	.	PUNCT
iajs-2911	37	7	e	e	X
iajs-2911	37	8	on	on	ADP
iajs-2911	37	9	d	d	PROPN
iajs-2911	37	10	is	be	AUX
iajs-2911	37	11	named	name	VERB
iajs-2911	37	12	𝔽.𝕎.	𝔽.𝕎.	PROPN
iajs-2911	37	13	closed	close	VERB
iajs-2911	37	14	(	(	PUNCT
iajs-2911	37	15	resp	resp	NOUN
iajs-2911	37	16	.	.	PUNCT
iajs-2911	37	17	,	,	PUNCT
iajs-2911	37	18	open	open	ADJ
iajs-2911	37	19	)	)	PUNCT
iajs-2911	37	20	if	if	SCONJ
iajs-2911	37	21	the	the	DET
iajs-2911	37	22	project	project	NOUN
iajs-2911	37	23	.	.	PUNCT
iajs-2911	38	1	x	x	PUNCT
iajs-2911	38	2	is	be	AUX
iajs-2911	38	3	closed	close	VERB
iajs-2911	38	4	(	(	PUNCT
iajs-2911	38	5	resp	resp	NOUN
iajs-2911	38	6	.	.	PUNCT
iajs-2911	38	7	,	,	PUNCT
iajs-2911	38	8	open	open	ADJ
iajs-2911	38	9	)	)	PUNCT
iajs-2911	38	10	functions	function	NOUN
iajs-2911	38	11	.	.	PUNCT
iajs-2911	39	1	definition	definition	NOUN
iajs-2911	39	2	1.4	1.4	NUM
iajs-2911	39	3	.	.	PUNCT
iajs-2911	40	1	[	[	X
iajs-2911	40	2	1	1	X
iajs-2911	40	3	]	]	X
iajs-2911	40	4	let	let	VERB
iajs-2911	40	5	ω	ω	NUM
iajs-2911	40	6	:	:	PUNCT
iajs-2911	40	7	e	e	X
iajs-2911	40	8	→	→	SYM
iajs-2911	40	9	f	f	X
iajs-2911	40	10	be	be	AUX
iajs-2911	40	11	a	a	DET
iajs-2911	40	12	multi	multi	NOUN
iajs-2911	40	13	-	-	NOUN
iajs-2911	40	14	function	function	NOUN
iajs-2911	40	15	.	.	PUNCT
iajs-2911	41	1	then	then	ADV
iajs-2911	41	2	ω	ω	PROPN
iajs-2911	41	3	is	be	AUX
iajs-2911	41	4	upper	upper	ADJ
iajs-2911	41	5	cont	cont	NOUN
iajs-2911	41	6	.	.	PUNCT
iajs-2911	42	1	(	(	PUNCT
iajs-2911	42	2	briefly	briefly	ADV
iajs-2911	42	3	,	,	PUNCT
iajs-2911	42	4	u.	u.	PROPN
iajs-2911	42	5	cont	cont	PROPN
iajs-2911	42	6	.	.	PUNCT
iajs-2911	42	7	)	)	PUNCT
iajs-2911	43	1	if	if	SCONJ
iajs-2911	43	2	ω+	ω+	NUM
iajs-2911	43	3	(	(	PUNCT
iajs-2911	43	4	k	k	X
iajs-2911	43	5	)	)	PUNCT
iajs-2911	43	6	open	open	ADJ
iajs-2911	43	7	in	in	ADP
iajs-2911	43	8	e	e	NOUN
iajs-2911	43	9	for	for	ADP
iajs-2911	43	10	all	all	DET
iajs-2911	43	11	k	k	PROPN
iajs-2911	43	12	open	open	ADJ
iajs-2911	43	13	in	in	ADP
iajs-2911	43	14	f.	f.	PROPN
iajs-2911	43	15	that	that	PRON
iajs-2911	43	16	is	be	AUX
iajs-2911	43	17	,	,	PUNCT
iajs-2911	43	18	ω+	ω+	NUM
iajs-2911	43	19	(	(	PUNCT
iajs-2911	43	20	k	k	X
iajs-2911	43	21	)	)	PUNCT
iajs-2911	43	22	=	=	SYM
iajs-2911	43	23	{	{	PUNCT
iajs-2911	43	24	x	x	PUNCT
iajs-2911	43	25	∈	∈	PROPN
iajs-2911	43	26	e	e	NOUN
iajs-2911	43	27	:	:	PUNCT
iajs-2911	43	28	ω(x	ω(x	NOUN
iajs-2911	43	29	)	)	PUNCT
iajs-2911	43	30	⊆	⊆	NUM
iajs-2911	43	31	k	k	NOUN
iajs-2911	43	32	}	}	PUNCT
iajs-2911	43	33	.	.	PUNCT
iajs-2911	44	1	k	k	PROPN
iajs-2911	45	1	⊆	⊆	NUM
iajs-2911	45	2	f.	f.	PROPN
iajs-2911	45	3	definition	definition	NOUN
iajs-2911	45	4	1.5	1.5	NUM
iajs-2911	45	5	.	.	PUNCT
iajs-2911	46	1	[	[	X
iajs-2911	46	2	1	1	X
iajs-2911	46	3	]	]	X
iajs-2911	46	4	let	let	VERB
iajs-2911	46	5	ω	ω	NUM
iajs-2911	46	6	:	:	PUNCT
iajs-2911	46	7	e	e	X
iajs-2911	46	8	→	→	SYM
iajs-2911	46	9	f	f	X
iajs-2911	46	10	be	be	AUX
iajs-2911	46	11	a	a	DET
iajs-2911	46	12	multi	multi	NOUN
iajs-2911	46	13	-	-	NOUN
iajs-2911	46	14	function	function	NOUN
iajs-2911	46	15	.	.	PUNCT
iajs-2911	47	1	then	then	ADV
iajs-2911	47	2	ω	ω	PROPN
iajs-2911	47	3	is	be	AUX
iajs-2911	47	4	lower	low	ADJ
iajs-2911	47	5	cont	cont	NOUN
iajs-2911	47	6	.	.	PUNCT
iajs-2911	48	1	(	(	PUNCT
iajs-2911	48	2	briefly	briefly	ADV
iajs-2911	48	3	,	,	PUNCT
iajs-2911	48	4	l.	l.	PROPN
iajs-2911	48	5	cont	cont	PROPN
iajs-2911	48	6	.	.	PUNCT
iajs-2911	48	7	)	)	PUNCT
iajs-2911	49	1	if	if	SCONJ
iajs-2911	49	2	ω(k	ω(k	NOUN
iajs-2911	49	3	)	)	PUNCT
iajs-2911	49	4	open	open	VERB
iajs-2911	49	5	in	in	ADP
iajs-2911	49	6	e	e	NOUN
iajs-2911	49	7	for	for	ADP
iajs-2911	49	8	all	all	DET
iajs-2911	49	9	k	k	PROPN
iajs-2911	49	10	open	open	ADJ
iajs-2911	49	11	in	in	ADP
iajs-2911	49	12	f.	f.	PROPN
iajs-2911	49	13	that	that	PRON
iajs-2911	49	14	is	be	AUX
iajs-2911	49	15	,	,	PUNCT
iajs-2911	49	16	ω-(k	ω-(k	ADV
iajs-2911	49	17	)	)	PUNCT
iajs-2911	49	18	=	=	PRON
iajs-2911	49	19	{	{	PUNCT
iajs-2911	49	20	e	e	X
iajs-2911	49	21	∈	∈	PROPN
iajs-2911	49	22	e	e	NOUN
iajs-2911	49	23	:	:	PUNCT
iajs-2911	49	24	ω(e	ω(e	NOUN
iajs-2911	49	25	)	)	PUNCT
iajs-2911	49	26	∩	∩	NOUN
iajs-2911	49	27	k	k	PROPN
iajs-2911	49	28	≠∅	≠∅	PROPN
iajs-2911	49	29	}	}	PUNCT
iajs-2911	49	30	.	.	PUNCT
iajs-2911	50	1	k	k	PROPN
iajs-2911	51	1	⊆	⊆	NUM
iajs-2911	51	2	f	f	X
iajs-2911	51	3	let	let	VERB
iajs-2911	51	4	ω	ω	NUM
iajs-2911	51	5	:	:	PUNCT
iajs-2911	51	6	e	e	X
iajs-2911	51	7	→	→	SYM
iajs-2911	51	8	f	f	X
iajs-2911	51	9	be	be	AUX
iajs-2911	51	10	a	a	DET
iajs-2911	51	11	multi	multi	NOUN
iajs-2911	51	12	-	-	NOUN
iajs-2911	51	13	function	function	NOUN
iajs-2911	51	14	.	.	PUNCT
iajs-2911	52	1	then	then	ADV
iajs-2911	52	2	ω	ω	PROPN
iajs-2911	52	3	is	be	AUX
iajs-2911	52	4	multi	multi	ADJ
iajs-2911	52	5	cont	cont	NOUN
iajs-2911	52	6	.	.	PUNCT
iajs-2911	53	1	(	(	PUNCT
iajs-2911	53	2	briefly	briefly	ADV
iajs-2911	53	3	,	,	PUNCT
iajs-2911	53	4	m.	m.	NOUN
iajs-2911	53	5	cont	cont	PROPN
iajs-2911	53	6	.	.	PUNCT
iajs-2911	53	7	)	)	PUNCT
iajs-2911	54	1	if	if	SCONJ
iajs-2911	54	2	it	it	PRON
iajs-2911	54	3	is	be	AUX
iajs-2911	54	4	u.	u.	PROPN
iajs-2911	54	5	cont	cont	PROPN
iajs-2911	54	6	.	.	PUNCT
iajs-2911	55	1	and	and	CCONJ
iajs-2911	55	2	l.	l.	PROPN
iajs-2911	55	3	cont	cont	PROPN
iajs-2911	55	4	.	.	PUNCT
iajs-2911	56	1	definition	definition	NOUN
iajs-2911	56	2	1.6.[5	1.6.[5	NUM
iajs-2911	56	3	]	]	PUNCT
iajs-2911	56	4	let	let	VERB
iajs-2911	56	5	d	d	PRON
iajs-2911	56	6	be	be	AUX
iajs-2911	56	7	topological	topological	ADJ
iajs-2911	56	8	space	space	NOUN
iajs-2911	56	9	,	,	PUNCT
iajs-2911	56	10	the	the	DET
iajs-2911	56	11	𝔽.𝕎.	𝔽.𝕎.	PROPN
iajs-2911	56	12	upper	upper	ADJ
iajs-2911	56	13	topology	topology	NOUN
iajs-2911	56	14	space	space	NOUN
iajs-2911	56	15	(	(	PUNCT
iajs-2911	56	16	briefly	briefly	ADV
iajs-2911	56	17	,	,	PUNCT
iajs-2911	56	18	𝔽.𝕎.u.t.s	𝔽.𝕎.u.t.s	PROPN
iajs-2911	56	19	.	.	PUNCT
iajs-2911	56	20	)	)	PUNCT
iajs-2911	56	21	on	on	ADP
iajs-2911	56	22	a	a	DET
iajs-2911	56	23	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	56	24	set	set	NOUN
iajs-2911	56	25	e	e	NOUN
iajs-2911	56	26	on	on	ADP
iajs-2911	56	27	d	d	PROPN
iajs-2911	56	28	mean	mean	VERB
iajs-2911	56	29	any	any	DET
iajs-2911	56	30	topology	topology	NOUN
iajs-2911	56	31	on	on	ADP
iajs-2911	56	32	e	e	PROPN
iajs-2911	56	33	for	for	ADP
iajs-2911	56	34	which	which	PRON
iajs-2911	56	35	the	the	DET
iajs-2911	56	36	project	project	NOUN
iajs-2911	56	37	.	.	PUNCT
iajs-2911	57	1	x	x	PUNCT
iajs-2911	57	2	is	be	AUX
iajs-2911	57	3	u.	u.	PROPN
iajs-2911	57	4	cont	cont	PROPN
iajs-2911	57	5	.	.	PUNCT
iajs-2911	58	1	ihjpas	ihjpas	PROPN
iajs-2911	58	2	.	.	PUNCT
iajs-2911	59	1	36	36	NUM
iajs-2911	59	2	(	(	PUNCT
iajs-2911	59	3	4	4	NUM
iajs-2911	59	4	)	)	PUNCT
iajs-2911	59	5	2023	2023	NUM
iajs-2911	59	6	398	398	NUM
iajs-2911	59	7	definition	definition	NOUN
iajs-2911	59	8	1.7.[5	1.7.[5	NUM
iajs-2911	59	9	]	]	PUNCT
iajs-2911	59	10	let	let	VERB
iajs-2911	59	11	d	d	PRON
iajs-2911	59	12	be	be	AUX
iajs-2911	59	13	topological	topological	ADJ
iajs-2911	59	14	space	space	NOUN
iajs-2911	59	15	the	the	DET
iajs-2911	59	16	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	59	17	lower	low	ADJ
iajs-2911	59	18	topology	topology	NOUN
iajs-2911	59	19	space	space	NOUN
iajs-2911	59	20	(	(	PUNCT
iajs-2911	59	21	briefly	briefly	ADV
iajs-2911	59	22	,	,	PUNCT
iajs-2911	59	23	𝔽.𝕎.l.t.s	𝔽.𝕎.l.t.s	PROPN
iajs-2911	59	24	.	.	PUNCT
iajs-2911	59	25	)	)	PUNCT
iajs-2911	59	26	on	on	ADP
iajs-2911	59	27	a	a	DET
iajs-2911	59	28	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	59	29	set	set	NOUN
iajs-2911	59	30	e	e	NOUN
iajs-2911	59	31	on	on	ADP
iajs-2911	59	32	d	d	PROPN
iajs-2911	59	33	mean	mean	VERB
iajs-2911	59	34	any	any	DET
iajs-2911	59	35	topology	topology	NOUN
iajs-2911	59	36	on	on	ADP
iajs-2911	59	37	e	e	PROPN
iajs-2911	59	38	for	for	ADP
iajs-2911	59	39	which	which	PRON
iajs-2911	59	40	the	the	DET
iajs-2911	59	41	project	project	NOUN
iajs-2911	59	42	.	.	PUNCT
iajs-2911	60	1	x	x	PUNCT
iajs-2911	60	2	is	be	AUX
iajs-2911	60	3	l.	l.	PROPN
iajs-2911	60	4	cont	cont	PROPN
iajs-2911	60	5	.	.	PUNCT
iajs-2911	61	1	let	let	VERB
iajs-2911	61	2	d	d	PRON
iajs-2911	61	3	be	be	AUX
iajs-2911	61	4	topological	topological	ADJ
iajs-2911	61	5	space	space	NOUN
iajs-2911	61	6	the	the	DET
iajs-2911	61	7	𝔽.𝕎.	𝔽.𝕎.	NOUN
iajs-2911	61	8	multi	multi	ADJ
iajs-2911	61	9	-	-	ADJ
iajs-2911	61	10	topology	topology	ADJ
iajs-2911	61	11	space	space	NOUN
iajs-2911	61	12	(	(	PUNCT
iajs-2911	61	13	briefly	briefly	ADV
iajs-2911	61	14	,	,	PUNCT
iajs-2911	61	15	𝔽.𝕎.m.t.s	𝔽.𝕎.m.t.s	NOUN
iajs-2911	61	16	.	.	PUNCT
iajs-2911	61	17	)	)	PUNCT
iajs-2911	62	1	if	if	SCONJ
iajs-2911	62	2	it	it	PRON
iajs-2911	62	3	is	be	AUX
iajs-2911	62	4	𝔽.𝕎.u.t.s	𝔽.𝕎.u.t.s	PROPN
iajs-2911	62	5	.	.	PUNCT
iajs-2911	62	6	and	and	CCONJ
iajs-2911	62	7	𝔽.𝕎.l.t.s	𝔽.𝕎.l.t.s	PROPN
iajs-2911	62	8	.	.	PROPN
iajs-2911	62	9	definition	definition	NOUN
iajs-2911	62	10	1.8	1.8	NUM
iajs-2911	62	11	.	.	PUNCT
iajs-2911	63	1	[	[	X
iajs-2911	63	2	3	3	X
iajs-2911	63	3	]	]	PUNCT
iajs-2911	63	4	a	a	DET
iajs-2911	63	5	filt𝑒r	filt𝑒r	ADJ
iajs-2911	63	6	ℑ	ℑ	PROPN
iajs-2911	63	7	on	on	ADP
iajs-2911	63	8	topological	topological	ADJ
iajs-2911	63	9	space	space	NOUN
iajs-2911	63	10	(	(	PUNCT
iajs-2911	63	11	e	e	NOUN
iajs-2911	63	12	,	,	PUNCT
iajs-2911	63	13	τ	τ	X
iajs-2911	63	14	)	)	PUNCT
iajs-2911	63	15	a	a	DET
iajs-2911	63	16	non	non	ADJ
iajs-2911	63	17	-	-	ADJ
iajs-2911	63	18	empty	empty	ADJ
iajs-2911	63	19	collection	collection	NOUN
iajs-2911	63	20	of	of	ADP
iajs-2911	63	21	non	non	ADJ
iajs-2911	63	22	-	-	ADJ
iajs-2911	63	23	empty	empty	ADJ
iajs-2911	63	24	subsets	subset	NOUN
iajs-2911	63	25	of	of	ADP
iajs-2911	63	26	e	e	NOUN
iajs-2911	63	27	such	such	ADJ
iajs-2911	63	28	that	that	DET
iajs-2911	63	29	i.	i.	PROPN
iajs-2911	63	30	∀	∀	PUNCT
iajs-2911	63	31	𝔽1	𝔽1	PROPN
iajs-2911	63	32	,	,	PUNCT
iajs-2911	63	33	𝔽	𝔽	PROPN
iajs-2911	63	34	2	2	NUM
iajs-2911	63	35	∈	∈	PROPN
iajs-2911	63	36	ℑ	ℑ	PROPN
iajs-2911	63	37	,	,	PUNCT
iajs-2911	63	38	𝔽	𝔽	PROPN
iajs-2911	63	39	1	1	NUM
iajs-2911	63	40	∩	∩	NOUN
iajs-2911	63	41	𝔽	𝔽	PROPN
iajs-2911	63	42	2	2	NUM
iajs-2911	63	43	∈	∈	PROPN
iajs-2911	63	44	ℑ	ℑ	PROPN
iajs-2911	63	45	ii	ii	NOUN
iajs-2911	63	46	.	.	PUNCT
iajs-2911	64	1	if	if	SCONJ
iajs-2911	64	2	𝔽	𝔽	PROPN
iajs-2911	64	3	1	1	NUM
iajs-2911	64	4	⊆	⊆	NUM
iajs-2911	64	5	𝔽	𝔽	PROPN
iajs-2911	64	6	2	2	NUM
iajs-2911	64	7	⊆e	⊆e	NOUN
iajs-2911	64	8	and	and	CCONJ
iajs-2911	64	9	𝔽1∈	𝔽1∈	ADJ
iajs-2911	64	10	ℑ	ℑ	PROPN
iajs-2911	64	11	then	then	ADV
iajs-2911	64	12	𝔽2	𝔽2	PROPN
iajs-2911	64	13	∈	∈	PROPN
iajs-2911	64	14	ℑ.	ℑ.	NOUN
iajs-2911	64	15	definition	definition	NOUN
iajs-2911	64	16	1.9	1.9	NUM
iajs-2911	64	17	.	.	PUNCT
iajs-2911	65	1	[	[	X
iajs-2911	65	2	3	3	X
iajs-2911	65	3	]	]	PUNCT
iajs-2911	65	4	if	if	SCONJ
iajs-2911	65	5	ℑ,𝔔	ℑ,𝔔	VERB
iajs-2911	65	6	filter	filter	NOUN
iajs-2911	65	7	bases	basis	NOUN
iajs-2911	65	8	on	on	ADP
iajs-2911	65	9	(	(	PUNCT
iajs-2911	65	10	e,𝜏	e,𝜏	NOUN
iajs-2911	65	11	)	)	PUNCT
iajs-2911	65	12	,	,	PUNCT
iajs-2911	65	13	we	we	PRON
iajs-2911	65	14	namely	namely	ADV
iajs-2911	65	15	𝔔	𝔔	PROPN
iajs-2911	65	16	is	be	AUX
iajs-2911	65	17	fin𝑒r	fin𝑒r	VERB
iajs-2911	65	18	than	than	ADP
iajs-2911	65	19	ℑ	ℑ	PROPN
iajs-2911	65	20	(	(	PUNCT
iajs-2911	65	21	writt𝑒n	writt𝑒n	NOUN
iajs-2911	65	22	as	as	ADP
iajs-2911	65	23	ℑ	ℑ	PROPN
iajs-2911	65	24	<	<	X
iajs-2911	65	25	𝔔	𝔔	PROPN
iajs-2911	65	26	)	)	PUNCT
iajs-2911	65	27	if	if	SCONJ
iajs-2911	65	28	for	for	ADP
iajs-2911	65	29	all	all	DET
iajs-2911	65	30	𝔽	𝔽	PROPN
iajs-2911	65	31	∈	∈	PROPN
iajs-2911	65	32	ℑ	ℑ	PROPN
iajs-2911	65	33	,	,	PUNCT
iajs-2911	65	34	there	there	PRON
iajs-2911	65	35	is	be	VERB
iajs-2911	65	36	g	g	PROPN
iajs-2911	65	37	⊆	⊆	NUM
iajs-2911	65	38	𝔽	𝔽	PROPN
iajs-2911	65	39	meets𝔔	meets𝔔	PROPN
iajs-2911	65	40	if	if	SCONJ
iajs-2911	65	41	𝔽∩g	𝔽∩g	PROPN
iajs-2911	65	42	≠	≠	PROPN
iajs-2911	65	43	∅	∅	NOUN
iajs-2911	65	44	for	for	ADP
iajs-2911	65	45	𝑒v𝑒ry	𝑒v𝑒ry	NOUN
iajs-2911	65	46	𝔽	𝔽	PROPN
iajs-2911	65	47	∈	∈	PROPN
iajs-2911	65	48	ℑ	ℑ	PROPN
iajs-2911	65	49	and	and	CCONJ
iajs-2911	65	50	g	g	NOUN
iajs-2911	65	51	∈	∈	PROPN
iajs-2911	65	52	𝔔.	𝔔.	ADJ
iajs-2911	65	53	definition	definition	NOUN
iajs-2911	65	54	1.10	1.10	NUM
iajs-2911	65	55	.	.	PUNCT
iajs-2911	66	1	[	[	X
iajs-2911	66	2	10	10	NUM
iajs-2911	66	3	]	]	X
iajs-2911	66	4	if	if	SCONJ
iajs-2911	66	5	e	e	PROPN
iajs-2911	66	6	is	be	AUX
iajs-2911	66	7	topological	topological	ADJ
iajs-2911	66	8	space	space	NOUN
iajs-2911	66	9	and	and	CCONJ
iajs-2911	66	10	e	e	NOUN
iajs-2911	66	11	∈	∈	PROPN
iajs-2911	66	12	e	e	X
iajs-2911	66	13	a	a	DET
iajs-2911	66	14	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	66	15	of	of	ADP
iajs-2911	66	16	e	e	NOUN
iajs-2911	66	17	is	be	AUX
iajs-2911	66	18	a	a	DET
iajs-2911	66	19	s𝑒t	s𝑒t	NOUN
iajs-2911	66	20	𝔘	𝔘	NOUN
iajs-2911	66	21	which	which	PRON
iajs-2911	66	22	contain	contain	VERB
iajs-2911	66	23	an	an	DET
iajs-2911	66	24	op𝑒n	op𝑒n	ADJ
iajs-2911	66	25	s𝑒t	s𝑒t	NOUN
iajs-2911	66	26	v	v	NOUN
iajs-2911	66	27	containing	contain	VERB
iajs-2911	66	28	e.	e.	PROPN
iajs-2911	66	29	if	if	SCONJ
iajs-2911	66	30	𝒜	𝒜	PROPN
iajs-2911	66	31	is	be	AUX
iajs-2911	66	32	op𝑒n	op𝑒n	ADJ
iajs-2911	66	33	s𝑒t	s𝑒t	NOUN
iajs-2911	66	34	and	and	CCONJ
iajs-2911	66	35	contains	contain	VERB
iajs-2911	66	36	e	e	PROPN
iajs-2911	66	37	w𝑒	w𝑒	PROPN
iajs-2911	66	38	nam𝑒ly	nam𝑒ly	PROPN
iajs-2911	66	39	𝒜	𝒜	PROPN
iajs-2911	66	40	is	be	AUX
iajs-2911	66	41	op𝑒n	op𝑒n	ADJ
iajs-2911	66	42	𝑎	𝑎	DET
iajs-2911	66	43	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	66	44	for	for	ADP
iajs-2911	66	45	a	a	DET
iajs-2911	66	46	point	point	NOUN
iajs-2911	66	47	e.	e.	PROPN
iajs-2911	66	48	definition	definition	NOUN
iajs-2911	66	49	1.11	1.11	NUM
iajs-2911	66	50	.	.	PUNCT
iajs-2911	67	1	[	[	X
iajs-2911	67	2	9	9	NUM
iajs-2911	67	3	]	]	PUNCT
iajs-2911	67	4	a	a	DET
iajs-2911	67	5	p𝑜int	p𝑜int	NOUN
iajs-2911	67	6	e	e	X
iajs-2911	67	7	in	in	X
iajs-2911	67	8	(	(	PUNCT
iajs-2911	67	9	e,𝜏	e,𝜏	NOUN
iajs-2911	67	10	)	)	PUNCT
iajs-2911	67	11	is	be	AUX
iajs-2911	67	12	nam𝑒d	nam𝑒d	ADJ
iajs-2911	67	13	t𝑜	t𝑜	ADP
iajs-2911	67	14	b𝑒	b𝑒	ADP
iajs-2911	67	15	a	a	DET
iajs-2911	67	16	contact	contact	NOUN
iajs-2911	67	17	point	point	NOUN
iajs-2911	67	18	of	of	ADP
iajs-2911	67	19	a	a	DET
iajs-2911	67	20	subs𝑒t	subs𝑒t	NOUN
iajs-2911	67	21	𝒜	𝒜	NOUN
iajs-2911	67	22	⊆	⊆	NUM
iajs-2911	67	23	e	e	NOUN
iajs-2911	67	24	ff	ff	NOUN
iajs-2911	67	25	∀	∀	X
iajs-2911	67	26	𝔘	𝔘	NOUN
iajs-2911	67	27	open	open	ADJ
iajs-2911	67	28	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	67	29	of	of	ADP
iajs-2911	67	30	e	e	NOUN
iajs-2911	67	31	,	,	PUNCT
iajs-2911	67	32	cl	cl	INTJ
iajs-2911	67	33	(	(	PUNCT
iajs-2911	67	34	𝔘	𝔘	NOUN
iajs-2911	67	35	)	)	PUNCT
iajs-2911	67	36	∩	∩	NOUN
iajs-2911	67	37	𝒜	𝒜	NOUN
iajs-2911	67	38	≠	≠	NOUN
iajs-2911	67	39	∅.	∅.	VERB
iajs-2911	67	40	so	so	ADV
iajs-2911	67	41	,	,	PUNCT
iajs-2911	67	42	s𝑒t	s𝑒t	NOUN
iajs-2911	67	43	of	of	ADP
iajs-2911	67	44	all	all	DET
iajs-2911	67	45	c𝑜ntact	c𝑜ntact	NOUN
iajs-2911	67	46	p𝑜ints	p𝑜int	NOUN
iajs-2911	67	47	of	of	ADP
iajs-2911	67	48	𝒜	𝒜	NOUN
iajs-2911	67	49	is	be	AUX
iajs-2911	67	50	nam𝑒d	nam𝑒d	ADJ
iajs-2911	67	51	t𝑜	t𝑜	ADP
iajs-2911	67	52	b𝑒	b𝑒	PROPN
iajs-2911	68	1	th𝑒	th𝑒	ADV
iajs-2911	68	2	closure	closure	NOUN
iajs-2911	68	3	of	of	ADP
iajs-2911	68	4	𝒜	𝒜	NOUN
iajs-2911	68	5	and	and	CCONJ
iajs-2911	68	6	is	be	AUX
iajs-2911	68	7	symboliz𝑒d	symboliz𝑒d	ADJ
iajs-2911	68	8	by	by	ADP
iajs-2911	68	9	cl	cl	NOUN
iajs-2911	68	10	(	(	PUNCT
iajs-2911	68	11	𝒜	𝒜	NOUN
iajs-2911	68	12	)	)	PUNCT
iajs-2911	68	13	.	.	PUNCT
iajs-2911	69	1	definition	definition	NOUN
iajs-2911	69	2	1.12	1.12	NUM
iajs-2911	69	3	.	.	PUNCT
iajs-2911	70	1	[	[	X
iajs-2911	70	2	10	10	NUM
iajs-2911	70	3	]	]	PUNCT
iajs-2911	70	4	a	a	DET
iajs-2911	70	5	subs𝑒t	subs𝑒t	NOUN
iajs-2911	70	6	𝒜	𝒜	NOUN
iajs-2911	70	7	in	in	ADP
iajs-2911	70	8	topological	topological	ADJ
iajs-2911	70	9	space𝑒	space𝑒	NOUN
iajs-2911	70	10	(	(	PUNCT
iajs-2911	70	11	e,𝜏	e,𝜏	NOUN
iajs-2911	70	12	)	)	PUNCT
iajs-2911	70	13	.	.	PUNCT
iajs-2911	71	1	s𝑜	s𝑜	NOUN
iajs-2911	71	2	,	,	PUNCT
iajs-2911	71	3	𝒜	𝒜	NOUN
iajs-2911	71	4	is	be	AUX
iajs-2911	71	5	nam𝑒d	nam𝑒d	ADJ
iajs-2911	71	6	to	to	PART
iajs-2911	71	7	be	be	AUX
iajs-2911	71	8	𝔼.s𝑒t	𝔼.s𝑒t	NOUN
iajs-2911	71	9	in	in	ADP
iajs-2911	71	10	e	e	PROPN
iajs-2911	71	11	(	(	PUNCT
iajs-2911	71	12	bri𝑒fly	bri𝑒fly	PROPN
iajs-2911	71	13	,	,	PUNCT
iajs-2911	71	14	e	e	NOUN
iajs-2911	71	15	-s𝑒t	-s𝑒t	NUM
iajs-2911	71	16	)	)	PUNCT
iajs-2911	72	1	if	if	SCONJ
iajs-2911	72	2	∀𝜏	∀𝜏	NOUN
iajs-2911	72	3	an	an	DET
iajs-2911	72	4	op𝑒n	op𝑒n	ADJ
iajs-2911	72	5	cov𝑒r	cov𝑒r	NOUN
iajs-2911	72	6	𝑜f	𝑜f	ADP
iajs-2911	72	7	𝒜	𝒜	NOUN
iajs-2911	72	8	th𝑒r𝑒	th𝑒r𝑒	NOUN
iajs-2911	72	9	is	be	AUX
iajs-2911	72	10	a	a	DET
iajs-2911	72	11	finit𝑒	finit𝑒	NOUN
iajs-2911	72	12	sub	sub	NOUN
iajs-2911	72	13	coll𝑒ction	coll𝑒ction	NOUN
iajs-2911	72	14	h	h	NOUN
iajs-2911	72	15	of	of	ADP
iajs-2911	72	16	𝛿	𝛿	ADJ
iajs-2911	72	17	;	;	PUNCT
iajs-2911	72	18	𝒜	𝒜	NOUN
iajs-2911	72	19	⊂∪{cl(h	⊂∪{cl(h	NOUN
iajs-2911	72	20	)	)	PUNCT
iajs-2911	72	21	:	:	PUNCT
iajs-2911	73	1	h	h	PROPN
iajs-2911	73	2	∈	∈	PROPN
iajs-2911	73	3	𝛿	𝛿	X
iajs-2911	73	4	}	}	PUNCT
iajs-2911	73	5	.	.	PUNCT
iajs-2911	74	1	if	if	SCONJ
iajs-2911	74	2	𝒜	𝒜	NOUN
iajs-2911	74	3	=	=	SYM
iajs-2911	74	4	e	e	NOUN
iajs-2911	74	5	;	;	PUNCT
iajs-2911	74	6	then	then	ADV
iajs-2911	74	7	,	,	PUNCT
iajs-2911	74	8	e	e	PROPN
iajs-2911	74	9	is	be	AUX
iajs-2911	74	10	named	name	VERB
iajs-2911	74	11	to	to	PART
iajs-2911	74	12	be	be	AUX
iajs-2911	74	13	a	a	DET
iajs-2911	74	14	ohc	ohc	NOUN
iajs-2911	74	15	spac𝑒.	spac𝑒.	ADJ
iajs-2911	74	16	definition	definition	NOUN
iajs-2911	74	17	1.13	1.13	NUM
iajs-2911	74	18	.	.	PUNCT
iajs-2911	75	1	[	[	X
iajs-2911	75	2	2	2	NUM
iajs-2911	75	3	]	]	PUNCT
iajs-2911	75	4	l𝑒t	l𝑒t	VERB
iajs-2911	75	5	e	e	PROPN
iajs-2911	75	6	a	a	DET
iajs-2911	75	7	point	point	NOUN
iajs-2911	75	8	in	in	ADP
iajs-2911	75	9	a	a	DET
iajs-2911	75	10	𝔽.𝕎.t.s	𝔽.𝕎.t.s	NOUN
iajs-2911	75	11	.	.	PUNCT
iajs-2911	76	1	(	(	PUNCT
iajs-2911	76	2	e,𝜏	e,𝜏	NOUN
iajs-2911	76	3	)	)	PUNCT
iajs-2911	76	4	on	on	ADP
iajs-2911	76	5	(	(	PUNCT
iajs-2911	76	6	d,𝜌	d,𝜌	NOUN
iajs-2911	76	7	)	)	PUNCT
iajs-2911	76	8	is	be	AUX
iajs-2911	76	9	nam𝑒d	nam𝑒d	ADJ
iajs-2911	76	10	to	to	PART
iajs-2911	76	11	b𝑒	b𝑒	VERB
iajs-2911	76	12	adh𝑒r𝑒nt	adh𝑒r𝑒nt	NOUN
iajs-2911	76	13	point	point	VERB
iajs-2911	76	14	𝑜f	𝑜f	ADP
iajs-2911	76	15	a	a	DET
iajs-2911	76	16	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	76	17	ℑ.	ℑ.	NOUN
iajs-2911	76	18	on	on	ADP
iajs-2911	76	19	e	e	PROPN
iajs-2911	76	20	(	(	PUNCT
iajs-2911	76	21	bri𝑒fly	bri𝑒fly	PROPN
iajs-2911	76	22	,	,	PUNCT
iajs-2911	76	23	ad(e	ad(e	NUM
iajs-2911	76	24	)	)	PUNCT
iajs-2911	76	25	)	)	PUNCT
iajs-2911	77	1	𝑖ff	𝑖ff	NOUN
iajs-2911	77	2	all	all	DET
iajs-2911	77	3	number	number	NOUN
iajs-2911	77	4	of	of	ADP
iajs-2911	77	5	ℑ	ℑ	PROPN
iajs-2911	77	6	is	be	AUX
iajs-2911	77	7	contract	contract	NOUN
iajs-2911	77	8	a	a	DET
iajs-2911	77	9	point	point	NOUN
iajs-2911	77	10	.	.	PUNCT
iajs-2911	78	1	a	a	DET
iajs-2911	78	2	set	set	NOUN
iajs-2911	78	3	of	of	ADP
iajs-2911	78	4	all	all	DET
iajs-2911	78	5	adherent	adherent	ADJ
iajs-2911	78	6	point	point	NOUN
iajs-2911	78	7	of	of	ADP
iajs-2911	78	8	ℑ	ℑ	PROPN
iajs-2911	78	9	is	be	AUX
iajs-2911	78	10	nam𝑒d	nam𝑒d	ADJ
iajs-2911	78	11	to	to	ADP
iajs-2911	78	12	b𝑒	b𝑒	ADV
iajs-2911	78	13	th𝑒	th𝑒	ADV
iajs-2911	78	14	adh𝑒r𝑒nc𝑒	adh𝑒r𝑒nc𝑒	ADJ
iajs-2911	78	15	of	of	ADP
iajs-2911	78	16	ℑ	ℑ	PROPN
iajs-2911	78	17	and	and	CCONJ
iajs-2911	78	18	is	be	AUX
iajs-2911	78	19	symboliz𝑒s	symboliz𝑒s	NOUN
iajs-2911	78	20	by	by	ADP
iajs-2911	78	21	ad(ℑ	ad(ℑ	NOUN
iajs-2911	78	22	)	)	PUNCT
iajs-2911	78	23	.	.	PUNCT
iajs-2911	79	1	definition	definition	NOUN
iajs-2911	79	2	1.14.[11	1.14.[11	PROPN
iajs-2911	79	3	]	]	PUNCT
iajs-2911	80	1	th𝑒	th𝑒	PROPN
iajs-2911	80	2	filt𝑒r	filt𝑒r	ADJ
iajs-2911	80	3	bas𝑒	bas𝑒	NOUN
iajs-2911	80	4	ℑ	ℑ	PROPN
iajs-2911	80	5	(	(	PUNCT
iajs-2911	80	6	bri𝑒fly	bri𝑒fly	NOUN
iajs-2911	80	7	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	80	8	ℑ	ℑ	PROPN
iajs-2911	80	9	)	)	PUNCT
iajs-2911	80	10	𝑜n	𝑜n	ADP
iajs-2911	80	11	t𝑜p𝑜l𝑜gical	t𝑜p𝑜l𝑜gical	ADJ
iajs-2911	80	12	spac𝑒	spac𝑒	NOUN
iajs-2911	80	13	(	(	PUNCT
iajs-2911	80	14	e,𝜏	e,𝜏	NOUN
iajs-2911	80	15	)	)	PUNCT
iajs-2911	80	16	is	be	AUX
iajs-2911	80	17	nam𝑒d	nam𝑒d	ADJ
iajs-2911	80	18	t𝑜	t𝑜	ADP
iajs-2911	80	19	b𝑒	b𝑒	ADV
iajs-2911	80	20	c𝑜nv𝑒rg𝑒nt	c𝑜nv𝑒rg𝑒nt	NOUN
iajs-2911	80	21	(	(	PUNCT
iajs-2911	80	22	bri𝑒fly	bri𝑒fly	PROPN
iajs-2911	80	23	,	,	PUNCT
iajs-2911	80	24	c𝑜nv	c𝑜nv	PROPN
iajs-2911	80	25	.	.	PUNCT
iajs-2911	80	26	)	)	PUNCT
iajs-2911	81	1	(	(	PUNCT
iajs-2911	81	2	written	write	VERB
iajs-2911	81	3	,	,	PUNCT
iajs-2911	81	4	ℑ	ℑ	PROPN
iajs-2911	81	5	−−conv.→	−−conv.→	NOUN
iajs-2911	81	6	e	e	PROPN
iajs-2911	81	7	𝑖ff	𝑖ff	PROPN
iajs-2911	81	8	𝑒v𝑒ry	𝑒v𝑒ry	PROPN
iajs-2911	81	9	𝜏.op𝑒n	𝜏.op𝑒n	NOUN
iajs-2911	81	10	.	.	PUNCT
iajs-2911	82	1	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	82	2	𝔘	𝔘	PROPN
iajs-2911	82	3	of	of	ADP
iajs-2911	82	4	e	e	PROPN
iajs-2911	82	5	,	,	PUNCT
iajs-2911	82	6	c𝑜ntains	c𝑜ntain	VERB
iajs-2911	82	7	s𝑜m𝑒	s𝑜m𝑒	NOUN
iajs-2911	82	8	𝑒l𝑒m𝑒nts	𝑒l𝑒m𝑒nt	NOUN
iajs-2911	82	9	of	of	ADP
iajs-2911	82	10	ℑ.	ℑ.	NOUN
iajs-2911	82	11	definition	definition	NOUN
iajs-2911	82	12	1.15.[11	1.15.[11	VERB
iajs-2911	82	13	]	]	PUNCT
iajs-2911	82	14	the	the	DET
iajs-2911	82	15	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	82	16	ℑ	ℑ	PROPN
iajs-2911	82	17	on	on	ADP
iajs-2911	82	18	topological	topological	ADJ
iajs-2911	82	19	spac𝑒	spac𝑒	NOUN
iajs-2911	82	20	(	(	PUNCT
iajs-2911	82	21	e,𝜏	e,𝜏	NOUN
iajs-2911	82	22	)	)	PUNCT
iajs-2911	82	23	is	be	AUX
iajs-2911	82	24	nam𝑒d	nam𝑒d	PROPN
iajs-2911	82	25	dir𝑒ct𝑒d	dir𝑒ct𝑒d	PROPN
iajs-2911	82	26	t𝑜ward	t𝑜ward	NOUN
iajs-2911	82	27	a	a	DET
iajs-2911	82	28	s𝑒t	s𝑒t	NOUN
iajs-2911	82	29	𝒜	𝒜	PROPN
iajs-2911	82	30	⊂	⊂	PROPN
iajs-2911	82	31	e,(briefly	e,(briefly	NOUN
iajs-2911	82	32	,	,	PUNCT
iajs-2911	82	33	ℑ	ℑ	PROPN
iajs-2911	82	34	−−	−−	AUX
iajs-2911	82	35	d.t→	d.t→	VERB
iajs-2911	82	36	𝒜	𝒜	NOUN
iajs-2911	82	37	)	)	PUNCT
iajs-2911	82	38	𝑖ff	𝑖ff	NOUN
iajs-2911	82	39	all	all	DET
iajs-2911	82	40	f∗.b∗.𝔔.	f∗.b∗.𝔔.	NOUN
iajs-2911	82	41	larg𝑒r	larg𝑒r	NOUN
iajs-2911	82	42	than	than	SCONJ
iajs-2911	82	43	ℑ	ℑ	PROPN
iajs-2911	82	44	has	have	VERB
iajs-2911	82	45	an	an	DET
iajs-2911	82	46	adh𝑒r𝑒nt	adh𝑒r𝑒nt	NOUN
iajs-2911	82	47	p𝑜int	p𝑜int	VERB
iajs-2911	82	48	in	in	ADP
iajs-2911	82	49	𝒜	𝒜	NOUN
iajs-2911	82	50	,	,	PUNCT
iajs-2911	82	51	i.e.	i.e.	X
iajs-2911	82	52	ad(𝔔	ad(𝔔	NOUN
iajs-2911	82	53	)	)	PUNCT
iajs-2911	82	54	∩𝒜	∩𝒜	NOUN
iajs-2911	82	55	≠	≠	PROPN
iajs-2911	82	56	∅	∅	NOUN
iajs-2911	82	57	,	,	PUNCT
iajs-2911	82	58	and	and	CCONJ
iajs-2911	82	59	in	in	ADP
iajs-2911	82	60	anoth𝑒r	anoth𝑒r	PROPN
iajs-2911	82	61	writing	write	VERB
iajs-2911	82	62	ℑ	ℑ	PROPN
iajs-2911	82	63	−ad→	−ad→	NOUN
iajs-2911	82	64	e	e	NOUN
iajs-2911	82	65	t𝑜	t𝑜	PRON
iajs-2911	82	66	imply	imply	VERB
iajs-2911	82	67	that	that	SCONJ
iajs-2911	82	68	ℑ	ℑ	PROPN
iajs-2911	82	69	−−d.t→{e	−−d.t→{e	ADV
iajs-2911	82	70	}	}	PUNCT
iajs-2911	82	71	,	,	PUNCT
iajs-2911	82	72	in	in	ADP
iajs-2911	82	73	which	which	PRON
iajs-2911	82	74	e	e	ADP
iajs-2911	82	75	∈e	∈e	NOUN
iajs-2911	82	76	.	.	PUNCT
iajs-2911	83	1	currently	currently	ADV
iajs-2911	83	2	,	,	PUNCT
iajs-2911	83	3	we	we	PRON
iajs-2911	83	4	review	review	VERB
iajs-2911	83	5	a	a	DET
iajs-2911	83	6	characterization	characterization	NOUN
iajs-2911	83	7	of	of	ADP
iajs-2911	83	8	a	a	DET
iajs-2911	83	9	point	point	NOUN
iajs-2911	83	10	e	e	NOUN
iajs-2911	83	11	𝑜f	𝑜f	ADP
iajs-2911	83	12	a	a	DET
iajs-2911	83	13	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	83	14	ℑ.	ℑ.	PROPN
iajs-2911	83	15	2	2	NUM
iajs-2911	83	16	.	.	PUNCT
iajs-2911	84	1	fibrewise	fibrewise	PROPN
iajs-2911	84	2	multi	multi	ADJ
iajs-2911	84	3	-	-	ADJ
iajs-2911	84	4	perfect	perfect	ADJ
iajs-2911	84	5	topological	topological	ADJ
iajs-2911	84	6	spaces	space	NOUN
iajs-2911	84	7	in	in	ADP
iajs-2911	84	8	this	this	DET
iajs-2911	84	9	segment	segment	NOUN
iajs-2911	84	10	we	we	PRON
iajs-2911	84	11	establish	establish	VERB
iajs-2911	84	12	f.w	f.w	PROPN
iajs-2911	84	13	.	.	PROPN
iajs-2911	84	14	multi	multi	ADJ
iajs-2911	84	15	-	-	ADJ
iajs-2911	84	16	perfect	perfect	ADJ
iajs-2911	84	17	topological	topological	ADJ
iajs-2911	84	18	spaces	space	NOUN
iajs-2911	84	19	(	(	PUNCT
iajs-2911	84	20	briefly	briefly	ADV
iajs-2911	84	21	,	,	PUNCT
iajs-2911	84	22	𝔽.𝕎.m.p.t.s	𝔽.𝕎.m.p.t.s	PROPN
iajs-2911	84	23	.	.	PUNCT
iajs-2911	84	24	)	)	PUNCT
iajs-2911	84	25	,	,	PUNCT
iajs-2911	84	26	and	and	CCONJ
iajs-2911	84	27	confirmation	confirmation	NOUN
iajs-2911	84	28	of	of	ADP
iajs-2911	84	29	few	few	ADJ
iajs-2911	84	30	of	of	ADP
iajs-2911	84	31	its	its	PRON
iajs-2911	84	32	basic	basic	ADJ
iajs-2911	84	33	characteristics	characteristic	NOUN
iajs-2911	84	34	.	.	PUNCT
iajs-2911	85	1	definition	definition	NOUN
iajs-2911	85	2	2.1	2.1	NUM
iajs-2911	85	3	.	.	PUNCT
iajs-2911	86	1	let	let	VERB
iajs-2911	86	2	ω	ω	NOUN
iajs-2911	86	3	:	:	PUNCT
iajs-2911	86	4	(	(	PUNCT
iajs-2911	86	5	e,𝜏	e,𝜏	NOUN
iajs-2911	86	6	)	)	PUNCT
iajs-2911	86	7	→	→	SYM
iajs-2911	86	8	(	(	PUNCT
iajs-2911	86	9	f,𝜎	f,𝜎	PROPN
iajs-2911	86	10	)	)	PUNCT
iajs-2911	86	11	be	be	VERB
iajs-2911	86	12	a	a	DET
iajs-2911	86	13	function	function	NOUN
iajs-2911	86	14	where	where	SCONJ
iajs-2911	86	15	e	e	NOUN
iajs-2911	86	16	and	and	CCONJ
iajs-2911	86	17	f	f	PROPN
iajs-2911	86	18	are	be	AUX
iajs-2911	86	19	𝔽.𝕎.t.s	𝔽.𝕎.t.s	ADJ
iajs-2911	86	20	.	.	PUNCT
iajs-2911	87	1	on	on	ADP
iajs-2911	87	2	d	d	PROPN
iajs-2911	87	3	is	be	AUX
iajs-2911	87	4	named	name	VERB
iajs-2911	87	5	to	to	PART
iajs-2911	87	6	be	be	AUX
iajs-2911	87	7	upper	upper	ADJ
iajs-2911	87	8	perfect	perfect	ADJ
iajs-2911	87	9	(	(	PUNCT
iajs-2911	87	10	briefly	briefly	ADV
iajs-2911	87	11	,	,	PUNCT
iajs-2911	87	12	u.p	u.p	PROPN
iajs-2911	87	13	.	.	PROPN
iajs-2911	87	14	)	)	PUNCT
iajs-2911	88	1	if	if	SCONJ
iajs-2911	88	2	for	for	ADP
iajs-2911	88	3	every	every	DET
iajs-2911	88	4	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	88	5	ℑ	ℑ	PROPN
iajs-2911	88	6	on	on	ADP
iajs-2911	88	7	ω(e	ω(e	PROPN
iajs-2911	88	8	)	)	PUNCT
iajs-2911	88	9	,	,	PUNCT
iajs-2911	88	10	such	such	ADJ
iajs-2911	88	11	that	that	SCONJ
iajs-2911	88	12	ℑ	ℑ	PROPN
iajs-2911	88	13	.d.t	.d.t	VERB
iajs-2911	88	14	.	.	PUNCT
iajs-2911	88	15	,	,	PUNCT
iajs-2911	88	16	some	some	DET
iajs-2911	88	17	subset	subset	ADJ
iajs-2911	88	18	𝒜	𝒜	NOUN
iajs-2911	88	19	of	of	ADP
iajs-2911	88	20	ω(e	ω(e	PROPN
iajs-2911	88	21	)	)	PUNCT
iajs-2911	88	22	,	,	PUNCT
iajs-2911	88	23	the	the	DET
iajs-2911	88	24	f∗.b∗	f∗.b∗	NOUN
iajs-2911	88	25	ω+(ℑ	ω+(ℑ	NOUN
iajs-2911	88	26	)	)	PUNCT
iajs-2911	88	27	is	be	AUX
iajs-2911	88	28	𝑑.	𝑑.	ADJ
iajs-2911	88	29	𝑡.	𝑡.	NOUN
iajs-2911	88	30	ω−1(𝒜	ω−1(𝒜	X
iajs-2911	88	31	)	)	PUNCT
iajs-2911	88	32	in	in	ADP
iajs-2911	88	33	𝐸.	𝐸.	PROPN
iajs-2911	88	34	definition	definition	NOUN
iajs-2911	88	35	2.2	2.2	NUM
iajs-2911	88	36	.	.	PUNCT
iajs-2911	89	1	let	let	VERB
iajs-2911	89	2	ω	ω	NOUN
iajs-2911	89	3	:	:	PUNCT
iajs-2911	89	4	(	(	PUNCT
iajs-2911	89	5	e,𝜏	e,𝜏	NOUN
iajs-2911	89	6	)	)	PUNCT
iajs-2911	89	7	→	→	SYM
iajs-2911	89	8	(	(	PUNCT
iajs-2911	89	9	f,𝜎	f,𝜎	PROPN
iajs-2911	89	10	)	)	PUNCT
iajs-2911	89	11	be	be	VERB
iajs-2911	89	12	a	a	DET
iajs-2911	89	13	function	function	NOUN
iajs-2911	89	14	where	where	SCONJ
iajs-2911	89	15	e	e	NOUN
iajs-2911	89	16	and	and	CCONJ
iajs-2911	89	17	f	f	PROPN
iajs-2911	89	18	are	be	AUX
iajs-2911	89	19	𝔽.𝕎.t.s	𝔽.𝕎.t.s	ADJ
iajs-2911	89	20	.	.	PUNCT
iajs-2911	90	1	on	on	ADP
iajs-2911	90	2	d	d	PROPN
iajs-2911	90	3	is	be	AUX
iajs-2911	90	4	named	name	VERB
iajs-2911	90	5	to	to	PART
iajs-2911	90	6	be	be	AUX
iajs-2911	90	7	lower	lower	ADV
iajs-2911	90	8	perfect	perfect	ADJ
iajs-2911	90	9	(	(	PUNCT
iajs-2911	90	10	briefly	briefly	ADV
iajs-2911	90	11	,	,	PUNCT
iajs-2911	90	12	l.p	l.p	PROPN
iajs-2911	90	13	.	.	PUNCT
iajs-2911	90	14	)	)	PUNCT
iajs-2911	91	1	if	if	SCONJ
iajs-2911	91	2	for	for	ADP
iajs-2911	91	3	every	every	DET
iajs-2911	91	4	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	91	5	ℑ	ℑ	PROPN
iajs-2911	91	6	on	on	ADP
iajs-2911	91	7	ω(e	ω(e	PROPN
iajs-2911	91	8	)	)	PUNCT
iajs-2911	91	9	,	,	PUNCT
iajs-2911	91	10	such	such	ADJ
iajs-2911	91	11	that	that	SCONJ
iajs-2911	91	12	ℑ	ℑ	PROPN
iajs-2911	91	13	.d.t	.d.t	VERB
iajs-2911	91	14	.	.	PUNCT
iajs-2911	91	15	,	,	PUNCT
iajs-2911	91	16	some	some	DET
iajs-2911	91	17	subset	subset	ADJ
iajs-2911	91	18	𝒜	𝒜	NOUN
iajs-2911	91	19	of	of	ADP
iajs-2911	91	20	ω(e	ω(e	PROPN
iajs-2911	91	21	)	)	PUNCT
iajs-2911	91	22	,	,	PUNCT
iajs-2911	91	23	the	the	DET
iajs-2911	91	24	f∗.b∗	f∗.b∗	PROPN
iajs-2911	91	25	ω−(ℑ	ω−(ℑ	NOUN
iajs-2911	91	26	)	)	PUNCT
iajs-2911	91	27	is	be	AUX
iajs-2911	91	28	.	.	PUNCT
iajs-2911	92	1	𝑑.	𝑑.	PROPN
iajs-2911	92	2	𝑡.	𝑡.	VERB
iajs-2911	92	3	ω−1(𝒜	ω−1(𝒜	X
iajs-2911	92	4	)	)	PUNCT
iajs-2911	92	5	in	in	ADP
iajs-2911	92	6	𝐸.	𝐸.	PROPN
iajs-2911	92	7	let	let	VERB
iajs-2911	92	8	ω	ω	NOUN
iajs-2911	92	9	:	:	PUNCT
iajs-2911	92	10	(	(	PUNCT
iajs-2911	92	11	e,𝜏	e,𝜏	NOUN
iajs-2911	92	12	)	)	PUNCT
iajs-2911	92	13	→	→	SYM
iajs-2911	92	14	(	(	PUNCT
iajs-2911	92	15	f,𝜎	f,𝜎	PROPN
iajs-2911	92	16	)	)	PUNCT
iajs-2911	92	17	be	be	VERB
iajs-2911	92	18	a	a	DET
iajs-2911	92	19	function	function	NOUN
iajs-2911	92	20	where	where	SCONJ
iajs-2911	92	21	e	e	NOUN
iajs-2911	92	22	and	and	CCONJ
iajs-2911	92	23	f	f	PROPN
iajs-2911	92	24	are	be	AUX
iajs-2911	92	25	𝔽.	𝔽.	PROPN
iajs-2911	92	26	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	92	27	.	.	PUNCT
iajs-2911	93	1	on	on	ADP
iajs-2911	93	2	d	d	PROPN
iajs-2911	93	3	is	be	AUX
iajs-2911	93	4	named	name	VERB
iajs-2911	93	5	to	to	PART
iajs-2911	93	6	be	be	AUX
iajs-2911	93	7	multiperfect	multiperfect	NOUN
iajs-2911	93	8	(	(	PUNCT
iajs-2911	93	9	briefly	briefly	ADV
iajs-2911	93	10	,	,	PUNCT
iajs-2911	93	11	m.p	m.p	PROPN
iajs-2911	93	12	.	.	PROPN
iajs-2911	93	13	)	)	PUNCT
iajs-2911	94	1	if	if	SCONJ
iajs-2911	94	2	it	it	PRON
iajs-2911	94	3	i𝑠	i𝑠	VERB
iajs-2911	94	4	u.p	u.p	PROPN
iajs-2911	94	5	.	.	PROPN
iajs-2911	94	6	and	and	CCONJ
iajs-2911	94	7	l.p	l.p	PROPN
iajs-2911	94	8	.	.	PROPN
iajs-2911	94	9	lemma	lemma	PROPN
iajs-2911	94	10	2.1	2.1	NUM
iajs-2911	94	11	.	.	PUNCT
iajs-2911	95	1	a	a	DET
iajs-2911	95	2	function	function	NOUN
iajs-2911	95	3	ω	ω	NOUN
iajs-2911	95	4	:	:	PUNCT
iajs-2911	95	5	(	(	PUNCT
iajs-2911	95	6	e,𝜏	e,𝜏	NOUN
iajs-2911	95	7	)	)	PUNCT
iajs-2911	95	8	→	→	SYM
iajs-2911	95	9	(	(	PUNCT
iajs-2911	95	10	f,𝜎	f,𝜎	NOUN
iajs-2911	95	11	)	)	PUNCT
iajs-2911	95	12	is	be	AUX
iajs-2911	95	13	closed	close	VERB
iajs-2911	95	14	if	if	SCONJ
iajs-2911	95	15	cl(ω(𝒜	cl(ω(𝒜	NOUN
iajs-2911	95	16	)	)	PUNCT
iajs-2911	95	17	)	)	PUNCT
iajs-2911	96	1	⊂	⊂	ADJ
iajs-2911	96	2	ω(cl(𝒜	ω(cl(𝒜	NOUN
iajs-2911	96	3	)	)	PUNCT
iajs-2911	96	4	)	)	PUNCT
iajs-2911	96	5	for	for	ADP
iajs-2911	96	6	every	every	DET
iajs-2911	96	7	𝒜	𝒜	PROPN
iajs-2911	96	8	⊂	⊂	PROPN
iajs-2911	96	9	e.	e.	PROPN
iajs-2911	96	10	ihjpas	ihjpas	PROPN
iajs-2911	96	11	.	.	PUNCT
iajs-2911	97	1	36	36	NUM
iajs-2911	97	2	(	(	PUNCT
iajs-2911	97	3	4	4	NUM
iajs-2911	97	4	)	)	PUNCT
iajs-2911	97	5	2023	2023	NUM
iajs-2911	97	6	399	399	NUM
iajs-2911	97	7	proof	proof	NOUN
iajs-2911	97	8	.	.	PUNCT
iajs-2911	98	1	(	(	PUNCT
iajs-2911	98	2	⇒	⇒	PROPN
iajs-2911	98	3	)	)	PUNCT
iajs-2911	98	4	let	let	VERB
iajs-2911	98	5	ω	ω	NOUN
iajs-2911	98	6	be	be	AUX
iajs-2911	98	7	closed	close	VERB
iajs-2911	98	8	and	and	CCONJ
iajs-2911	98	9	𝒜	𝒜	PROPN
iajs-2911	98	10	⊂	⊂	PROPN
iajs-2911	98	11	h.	h.	PROPN
iajs-2911	98	12	since	since	SCONJ
iajs-2911	98	13	ω	ω	PROPN
iajs-2911	98	14	is	be	AUX
iajs-2911	98	15	closed	close	VERB
iajs-2911	98	16	then	then	ADV
iajs-2911	98	17	ω	ω	X
iajs-2911	98	18	(	(	PUNCT
iajs-2911	98	19	cl(𝒜	cl(𝒜	PROPN
iajs-2911	98	20	)	)	PUNCT
iajs-2911	98	21	)	)	PUNCT
iajs-2911	98	22	is	be	AUX
iajs-2911	98	23	closed	close	VERB
iajs-2911	98	24	set	set	VERB
iajs-2911	98	25	in	in	ADP
iajs-2911	98	26	f	f	PROPN
iajs-2911	98	27	,	,	PUNCT
iajs-2911	98	28	because	because	SCONJ
iajs-2911	98	29	cl(𝒜	cl(𝒜	NOUN
iajs-2911	98	30	)	)	PUNCT
iajs-2911	98	31	is	be	AUX
iajs-2911	98	32	closed	close	VERB
iajs-2911	98	33	set	set	VERB
iajs-2911	98	34	in	in	ADP
iajs-2911	98	35	e.	e.	PROPN
iajs-2911	98	36	so	so	ADV
iajs-2911	98	37	,	,	PUNCT
iajs-2911	98	38	cl(ω(𝒜	cl(ω(𝒜	NOUN
iajs-2911	98	39	)	)	PUNCT
iajs-2911	98	40	)	)	PUNCT
iajs-2911	99	1	⊂	⊂	PROPN
iajs-2911	99	2	ω	ω	PROPN
iajs-2911	99	3	(	(	PUNCT
iajs-2911	99	4	cl(𝒜	cl(𝒜	PROPN
iajs-2911	99	5	)	)	PUNCT
iajs-2911	99	6	)	)	PUNCT
iajs-2911	99	7	.	.	PUNCT
iajs-2911	100	1	(	(	PUNCT
iajs-2911	100	2	⇒	⇒	PROPN
iajs-2911	100	3	)	)	PUNCT
iajs-2911	100	4	let	let	VERB
iajs-2911	100	5	a	a	DET
iajs-2911	100	6	be	be	AUX
iajs-2911	100	7	closed	close	VERB
iajs-2911	100	8	set	set	VERB
iajs-2911	100	9	in	in	ADP
iajs-2911	100	10	e	e	NOUN
iajs-2911	100	11	,	,	PUNCT
iajs-2911	100	12	so	so	ADV
iajs-2911	100	13	𝒜	𝒜	NOUN
iajs-2911	100	14	=	=	SYM
iajs-2911	100	15	cl(𝒜	cl(𝒜	NOUN
iajs-2911	100	16	)	)	PUNCT
iajs-2911	100	17	,	,	PUNCT
iajs-2911	100	18	however	however	ADV
iajs-2911	100	19	cl(ω	cl(ω	X
iajs-2911	100	20	(	(	PUNCT
iajs-2911	100	21	𝒜	𝒜	NOUN
iajs-2911	100	22	)	)	PUNCT
iajs-2911	100	23	)	)	PUNCT
iajs-2911	101	1	⊂	⊂	PROPN
iajs-2911	101	2	ω	ω	PROPN
iajs-2911	101	3	(	(	PUNCT
iajs-2911	101	4	cl(𝒜	cl(𝒜	PROPN
iajs-2911	101	5	)	)	PUNCT
iajs-2911	101	6	)	)	PUNCT
iajs-2911	101	7	,	,	PUNCT
iajs-2911	101	8	so	so	CCONJ
iajs-2911	101	9	cl(ω	cl(ω	X
iajs-2911	101	10	(	(	PUNCT
iajs-2911	101	11	𝒜	𝒜	NOUN
iajs-2911	101	12	)	)	PUNCT
iajs-2911	101	13	)	)	PUNCT
iajs-2911	102	1	⊂	⊂	PROPN
iajs-2911	102	2	ω	ω	X
iajs-2911	102	3	(	(	PUNCT
iajs-2911	102	4	𝒜	𝒜	NOUN
iajs-2911	102	5	)	)	PUNCT
iajs-2911	102	6	.	.	PUNCT
iajs-2911	103	1	then	then	ADV
iajs-2911	103	2	,	,	PUNCT
iajs-2911	103	3	ω	ω	PROPN
iajs-2911	103	4	(	(	PUNCT
iajs-2911	103	5	𝒜	𝒜	NOUN
iajs-2911	103	6	)	)	PUNCT
iajs-2911	103	7	is	be	AUX
iajs-2911	103	8	closed	close	VERB
iajs-2911	103	9	in	in	ADP
iajs-2911	103	10	f.	f.	PROPN
iajs-2911	103	11	therefore	therefore	ADV
iajs-2911	103	12	ω	ω	PROPN
iajs-2911	103	13	is	be	AUX
iajs-2911	103	14	closed	closed	ADJ
iajs-2911	103	15	.	.	PUNCT
iajs-2911	104	1	lemma	lemma	PROPN
iajs-2911	104	2	2.2	2.2	NUM
iajs-2911	104	3	.	.	PUNCT
iajs-2911	105	1	the	the	DET
iajs-2911	105	2	point	point	NOUN
iajs-2911	105	3	e	e	NOUN
iajs-2911	105	4	in	in	ADP
iajs-2911	105	5	topological	topological	ADJ
iajs-2911	105	6	space	space	NOUN
iajs-2911	105	7	(	(	PUNCT
iajs-2911	105	8	e	e	NOUN
iajs-2911	105	9	,	,	PUNCT
iajs-2911	105	10	τ	τ	X
iajs-2911	105	11	)	)	PUNCT
iajs-2911	105	12	is	be	AUX
iajs-2911	105	13	an	an	DET
iajs-2911	105	14	ad	ad	NOUN
iajs-2911	105	15	point	point	NOUN
iajs-2911	105	16	of	of	ADP
iajs-2911	105	17	a	a	DET
iajs-2911	105	18	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	105	19	ℑ	ℑ	PROPN
iajs-2911	105	20	on	on	ADP
iajs-2911	105	21	e	e	NOUN
iajs-2911	105	22	if	if	SCONJ
iajs-2911	105	23	∃	∃	PROPN
iajs-2911	105	24	𝑎	𝑎	X
iajs-2911	105	25	f*.b	f*.b	NOUN
iajs-2911	105	26	*	*	NOUN
iajs-2911	105	27	.	.	PUNCT
iajs-2911	106	1	ℑ.	ℑ.	NOUN
iajs-2911	106	2	larger	large	ADJ
iajs-2911	106	3	than	than	ADP
iajs-2911	106	4	ℑ	ℑ	NOUN
iajs-2911	106	5	such	such	ADJ
iajs-2911	106	6	that	that	SCONJ
iajs-2911	106	7	ℑ	ℑ	PROPN
iajs-2911	106	8	∗	∗	VERB
iajs-2911	106	9	−−conv.→	−−conv.→	NOUN
iajs-2911	106	10	e.	e.	PROPN
iajs-2911	106	11	proof	proof	PROPN
iajs-2911	106	12	.	.	PUNCT
iajs-2911	107	1	(	(	PUNCT
iajs-2911	107	2	⇒)assume	⇒)assume	PROPN
iajs-2911	107	3	that	that	SCONJ
iajs-2911	107	4	e	e	NOUN
iajs-2911	107	5	is	be	AUX
iajs-2911	107	6	an	an	DET
iajs-2911	107	7	ad	ad	NOUN
iajs-2911	107	8	point	point	NOUN
iajs-2911	107	9	of	of	ADP
iajs-2911	107	10	a	a	DET
iajs-2911	107	11	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	107	12	ℑ.	ℑ.	NOUN
iajs-2911	107	13	on	on	ADP
iajs-2911	107	14	e	e	NOUN
iajs-2911	107	15	,	,	PUNCT
iajs-2911	107	16	then	then	ADV
iajs-2911	107	17	it	it	PRON
iajs-2911	107	18	is	be	AUX
iajs-2911	107	19	an	an	DET
iajs-2911	107	20	c.	c.	NOUN
iajs-2911	107	21	point	point	NOUN
iajs-2911	107	22	of	of	ADP
iajs-2911	107	23	every	every	DET
iajs-2911	107	24	number	number	NOUN
iajs-2911	107	25	of	of	ADP
iajs-2911	107	26	ℑ.	ℑ.	PROPN
iajs-2911	107	27	this	this	DET
iajs-2911	107	28	returns	return	NOUN
iajs-2911	107	29	,	,	PUNCT
iajs-2911	107	30	for	for	ADP
iajs-2911	107	31	each	each	DET
iajs-2911	107	32	𝜏-open	𝜏-open	NOUN
iajs-2911	107	33	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	107	34	𝔘	𝔘	PROPN
iajs-2911	107	35	of	of	ADP
iajs-2911	107	36	h	h	NOUN
iajs-2911	107	37	,	,	PUNCT
iajs-2911	107	38	we	we	PRON
iajs-2911	107	39	have	have	VERB
iajs-2911	107	40	cl(𝔘)∩	cl(𝔘)∩	NOUN
iajs-2911	107	41	𝔽	𝔽	PROPN
iajs-2911	107	42	≠	≠	PROPN
iajs-2911	107	43	∅	∅	NOUN
iajs-2911	107	44	for	for	ADP
iajs-2911	107	45	every	every	DET
iajs-2911	107	46	number	number	NOUN
iajs-2911	107	47	𝔽	𝔽	PROPN
iajs-2911	107	48	in	in	ADP
iajs-2911	107	49	ℑ.	ℑ.	NOUN
iajs-2911	107	50	consequently	consequently	ADV
iajs-2911	107	51	,	,	PUNCT
iajs-2911	107	52	cl(𝔘	cl(𝔘	NOUN
iajs-2911	107	53	)	)	PUNCT
iajs-2911	107	54	contains	contain	VERB
iajs-2911	107	55	a	a	DET
iajs-2911	107	56	some	some	DET
iajs-2911	107	57	member	member	NOUN
iajs-2911	107	58	of	of	ADP
iajs-2911	107	59	any	any	DET
iajs-2911	107	60	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	107	61	ℑ	ℑ	PROPN
iajs-2911	107	62	∗	∗	VERB
iajs-2911	107	63	large𝑟	large𝑟	ADJ
iajs-2911	107	64	than	than	ADP
iajs-2911	107	65	ℑ	ℑ	PROPN
iajs-2911	107	66	such	such	ADJ
iajs-2911	107	67	that	that	SCONJ
iajs-2911	107	68	ℑ	ℑ	PROPN
iajs-2911	107	69	∗	∗	VERB
iajs-2911	107	70	−−−conv.→	−−−conv.→	NOUN
iajs-2911	107	71	e.	e.	PROPN
iajs-2911	107	72	(	(	PUNCT
iajs-2911	107	73	⇐	⇐	PROPN
iajs-2911	107	74	)	)	PUNCT
iajs-2911	107	75	assume	assume	VERB
iajs-2911	107	76	that	that	SCONJ
iajs-2911	107	77	e	e	NOUN
iajs-2911	107	78	is	be	AUX
iajs-2911	107	79	not	not	PART
iajs-2911	107	80	an	an	DET
iajs-2911	107	81	ad	ad	NOUN
iajs-2911	107	82	point	point	NOUN
iajs-2911	107	83	of	of	ADP
iajs-2911	107	84	a	a	DET
iajs-2911	107	85	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	107	86	ℑ.	ℑ.	NOUN
iajs-2911	107	87	on	on	ADP
iajs-2911	107	88	e	e	NOUN
iajs-2911	107	89	,	,	PUNCT
iajs-2911	107	90	then	then	ADV
iajs-2911	107	91	∃	∃	PROPN
iajs-2911	107	92	𝔽	𝔽	PROPN
iajs-2911	107	93	∈	∈	PROPN
iajs-2911	107	94	ℑ	ℑ	NOUN
iajs-2911	107	95	such	such	ADJ
iajs-2911	107	96	that	that	SCONJ
iajs-2911	107	97	e	e	NOUN
iajs-2911	107	98	is	be	AUX
iajs-2911	107	99	not	not	PART
iajs-2911	107	100	an	an	DET
iajs-2911	107	101	contact	contact	NOUN
iajs-2911	107	102	of	of	ADP
iajs-2911	107	103	𝔽.	𝔽.	PROPN
iajs-2911	107	104	so	so	ADV
iajs-2911	107	105	,	,	PUNCT
iajs-2911	107	106	∃	∃	PROPN
iajs-2911	107	107	𝜏−	𝜏−	PROPN
iajs-2911	107	108	open𝜂ℙ𝕕	open𝜂ℙ𝕕	PROPN
iajs-2911	107	109	𝔘	𝔘	PROPN
iajs-2911	107	110	of	of	ADP
iajs-2911	107	111	e	e	NOUN
iajs-2911	107	112	such	such	ADJ
iajs-2911	107	113	that	that	DET
iajs-2911	107	114	cl(𝔘	cl(𝔘	NOUN
iajs-2911	107	115	)	)	PUNCT
iajs-2911	108	1	∩	∩	NOUN
iajs-2911	108	2	𝔽	𝔽	PROPN
iajs-2911	108	3	=	=	PUNCT
iajs-2911	108	4	∅.	∅.	NOUN
iajs-2911	108	5	denote	denote	VERB
iajs-2911	108	6	by	by	ADP
iajs-2911	108	7	ℑ	ℑ	PROPN
iajs-2911	108	8	∗	∗	VERB
iajs-2911	108	9	the	the	DET
iajs-2911	108	10	family	family	NOUN
iajs-2911	108	11	of	of	ADP
iajs-2911	108	12	sets	set	NOUN
iajs-2911	108	13	𝔽	𝔽	PROPN
iajs-2911	108	14	∗	∗	NOUN
iajs-2911	108	15	=	=	SYM
iajs-2911	108	16	𝔽	𝔽	PROPN
iajs-2911	108	17	∩	∩	ADJ
iajs-2911	108	18	cl(𝔘	cl(𝔘	NOUN
iajs-2911	108	19	)	)	PUNCT
iajs-2911	108	20	for	for	ADP
iajs-2911	108	21	𝔽	𝔽	PROPN
iajs-2911	108	22	∈	∈	PROPN
iajs-2911	108	23	ℑ	ℑ	PROPN
iajs-2911	108	24	,	,	PUNCT
iajs-2911	108	25	so	so	ADV
iajs-2911	108	26	the	the	DET
iajs-2911	108	27	sets	set	NOUN
iajs-2911	108	28	in	in	ADP
iajs-2911	108	29	which	which	PRON
iajs-2911	108	30	𝔽	𝔽	PROPN
iajs-2911	108	31	∗	∗	NOUN
iajs-2911	108	32	≠	≠	PROPN
iajs-2911	108	33	∅.	∅.	VERB
iajs-2911	108	34	additionally	additionally	ADV
iajs-2911	108	35	,	,	PUNCT
iajs-2911	108	36	is	be	AUX
iajs-2911	108	37	a	a	PRON
iajs-2911	108	38	.	.	PUNCT
iajs-2911	109	1	and	and	CCONJ
iajs-2911	109	2	really	really	ADV
iajs-2911	109	3	is	be	AUX
iajs-2911	109	4	𝔽	𝔽	PROPN
iajs-2911	109	5	∗	∗	NOUN
iajs-2911	109	6	from	from	ADP
iajs-2911	109	7	ℑ.	ℑ.	PROPN
iajs-2911	109	8	this	this	PRON
iajs-2911	109	9	is	be	AUX
iajs-2911	109	10	,	,	PUNCT
iajs-2911	109	11	given	give	VERB
iajs-2911	109	12	𝔽1	𝔽1	PROPN
iajs-2911	109	13	∗	∗	NOUN
iajs-2911	109	14	=	=	PUNCT
iajs-2911	109	15	𝔽1	𝔽1	PROPN
iajs-2911	109	16	∩	∩	NOUN
iajs-2911	109	17	(	(	PUNCT
iajs-2911	109	18	𝐸	𝐸	PROPN
iajs-2911	109	19	∖	∖	NOUN
iajs-2911	109	20	𝑐𝑙(𝔘	𝑐𝑙(𝔘	NOUN
iajs-2911	109	21	)	)	PUNCT
iajs-2911	109	22	)	)	PUNCT
iajs-2911	109	23	and	and	CCONJ
iajs-2911	109	24	𝔽2	𝔽2	NOUN
iajs-2911	109	25	∗	∗	NOUN
iajs-2911	109	26	=	=	PUNCT
iajs-2911	109	27	𝔽2	𝔽2	PROPN
iajs-2911	109	28	∩	∩	NOUN
iajs-2911	109	29	(	(	PUNCT
iajs-2911	109	30	𝐸	𝐸	PROPN
iajs-2911	109	31	∖	∖	NOUN
iajs-2911	109	32	𝑐𝑙(𝔘	𝑐𝑙(𝔘	NOUN
iajs-2911	109	33	)	)	PUNCT
iajs-2911	109	34	)	)	PUNCT
iajs-2911	109	35	,	,	PUNCT
iajs-2911	109	36	∃	∃	PROPN
iajs-2911	109	37	𝔽3	𝔽3	PROPN
iajs-2911	109	38	=	=	SYM
iajs-2911	109	39	𝔽1	𝔽1	PROPN
iajs-2911	109	40	∩	∩	ADJ
iajs-2911	109	41	𝔽2	𝔽2	NOUN
iajs-2911	109	42	,	,	PUNCT
iajs-2911	109	43	and	and	CCONJ
iajs-2911	109	44	this	this	PRON
iajs-2911	109	45	gives	give	VERB
iajs-2911	109	46	𝔽2	𝔽2	PROPN
iajs-2911	109	47	∗	∗	NOUN
iajs-2911	109	48	=	=	SYM
iajs-2911	109	49	𝔽3	𝔽3	PROPN
iajs-2911	109	50	∩	∩	NOUN
iajs-2911	109	51	(	(	PUNCT
iajs-2911	109	52	𝐸	𝐸	PROPN
iajs-2911	109	53	∖	∖	NOUN
iajs-2911	109	54	𝑐𝑙(𝔘	𝑐𝑙(𝔘	NOUN
iajs-2911	109	55	)	)	PUNCT
iajs-2911	109	56	)	)	PUNCT
iajs-2911	110	1	⊂	⊂	PROPN
iajs-2911	110	2	𝔽1	𝔽1	PROPN
iajs-2911	110	3	∩	∩	ADJ
iajs-2911	110	4	𝔽2	𝔽2	NOUN
iajs-2911	110	5	∩	∩	NOUN
iajs-2911	110	6	(	(	PUNCT
iajs-2911	110	7	𝐸	𝐸	PROPN
iajs-2911	110	8	∖	∖	NOUN
iajs-2911	110	9	𝑐𝑙(𝔘	𝑐𝑙(𝔘	NOUN
iajs-2911	110	10	)	)	PUNCT
iajs-2911	110	11	)	)	PUNCT
iajs-2911	111	1	=	=	SYM
iajs-2911	111	2	𝔽1	𝔽1	PROPN
iajs-2911	111	3	∩	∩	NOUN
iajs-2911	111	4	(	(	PUNCT
iajs-2911	111	5	𝐸	𝐸	PROPN
iajs-2911	111	6	∖	∖	NOUN
iajs-2911	111	7	𝑐𝑙(𝔘	𝑐𝑙(𝔘	NOUN
iajs-2911	111	8	)	)	PUNCT
iajs-2911	111	9	)	)	PUNCT
iajs-2911	111	10	∩	∩	ADJ
iajs-2911	111	11	𝔽2	𝔽2	NOUN
iajs-2911	111	12	∩	∩	NOUN
iajs-2911	111	13	(	(	PUNCT
iajs-2911	111	14	𝐸	𝐸	PROPN
iajs-2911	111	15	∖	∖	NOUN
iajs-2911	111	16	𝑐𝑙(𝔘	𝑐𝑙(𝔘	NOUN
iajs-2911	111	17	)	)	PUNCT
iajs-2911	111	18	)	)	PUNCT
iajs-2911	111	19	.	.	PUNCT
iajs-2911	112	1	since	since	SCONJ
iajs-2911	112	2	f∗	f∗	NOUN
iajs-2911	112	3	is	be	AUX
iajs-2911	112	4	not	not	PART
iajs-2911	112	5	conv	conv	ADJ
iajs-2911	112	6	.	.	PUNCT
iajs-2911	113	1	to	to	PART
iajs-2911	113	2	e.	e.	PROPN
iajs-2911	113	3	so	so	ADV
iajs-2911	113	4	,	,	PUNCT
iajs-2911	113	5	lead	lead	VERB
iajs-2911	113	6	to	to	ADP
iajs-2911	113	7	a	a	DET
iajs-2911	113	8	c	c	NOUN
iajs-2911	113	9	!	!	PUNCT
iajs-2911	113	10	!	!	PUNCT
iajs-2911	113	11	!	!	PUNCT
iajs-2911	114	1	,	,	PUNCT
iajs-2911	114	2	and	and	CCONJ
iajs-2911	114	3	h	h	NOUN
iajs-2911	114	4	is	be	AUX
iajs-2911	114	5	an	an	DET
iajs-2911	114	6	ad	ad	NOUN
iajs-2911	114	7	point	point	NOUN
iajs-2911	114	8	of	of	ADP
iajs-2911	114	9	a	a	DET
iajs-2911	114	10	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	114	11	ℑ.	ℑ.	PROPN
iajs-2911	114	12	on	on	ADP
iajs-2911	114	13	e.	e.	PROPN
iajs-2911	114	14	lemma	lemma	PROPN
iajs-2911	114	15	2.3	2.3	NUM
iajs-2911	114	16	.	.	PUNCT
iajs-2911	115	1	assume	assume	VERB
iajs-2911	115	2	that	that	SCONJ
iajs-2911	115	3	ℑ	ℑ	PROPN
iajs-2911	115	4	is	be	AUX
iajs-2911	115	5	a	a	DET
iajs-2911	115	6	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	115	7	ℑ	ℑ	NOUN
iajs-2911	115	8	on	on	ADP
iajs-2911	115	9	a	a	DET
iajs-2911	115	10	topological	topological	ADJ
iajs-2911	115	11	space	space	NOUN
iajs-2911	115	12	(	(	PUNCT
iajs-2911	115	13	e,𝜏	e,𝜏	NOUN
iajs-2911	115	14	)	)	PUNCT
iajs-2911	115	15	.	.	PUNCT
iajs-2911	116	1	suppose	suppose	VERB
iajs-2911	116	2	that	that	SCONJ
iajs-2911	116	3	e	e	PROPN
iajs-2911	116	4	∈	∈	PROPN
iajs-2911	116	5	e	e	NOUN
iajs-2911	116	6	,	,	PUNCT
iajs-2911	116	7	so	so	ADV
iajs-2911	116	8	ℑ	ℑ	NOUN
iajs-2911	116	9	−−conv.→	−−conv.→	NOUN
iajs-2911	116	10	e	e	NOUN
iajs-2911	116	11	if	if	SCONJ
iajs-2911	116	12	ℑ	ℑ	PROPN
iajs-2911	116	13	−−	−−	NOUN
iajs-2911	116	14	d.t→e	d.t→e	PROPN
iajs-2911	116	15	.	.	PUNCT
iajs-2911	117	1	proof	proof	NOUN
iajs-2911	117	2	.	.	PUNCT
iajs-2911	118	1	(	(	PUNCT
iajs-2911	118	2	⇐	⇐	NOUN
iajs-2911	118	3	)	)	PUNCT
iajs-2911	118	4	if	if	SCONJ
iajs-2911	118	5	ℑ	ℑ	PROPN
iajs-2911	118	6	does	do	AUX
iajs-2911	118	7	not	not	PART
iajs-2911	118	8	conv	conv	ADJ
iajs-2911	118	9	.	.	PUNCT
iajs-2911	119	1	to	to	ADP
iajs-2911	119	2	e	e	NOUN
iajs-2911	119	3	,	,	PUNCT
iajs-2911	119	4	then	then	ADV
iajs-2911	119	5	,	,	PUNCT
iajs-2911	119	6	∃𝜏-open	∃𝜏-open	VERB
iajs-2911	119	7	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	119	8	𝔘	𝔘	NOUN
iajs-2911	119	9	of	of	ADP
iajs-2911	119	10	e	e	NOUN
iajs-2911	119	11	such	such	ADJ
iajs-2911	119	12	that	that	DET
iajs-2911	119	13	cl(𝔘	cl(𝔘	NOUN
iajs-2911	119	14	)	)	PUNCT
iajs-2911	119	15	)	)	PUNCT
iajs-2911	120	1	⊄	⊄	NOUN
iajs-2911	120	2	𝔽	𝔽	PROPN
iajs-2911	120	3	=	=	NOUN
iajs-2911	120	4	∅	∅	NOUN
iajs-2911	120	5	for	for	ADP
iajs-2911	120	6	every	every	DET
iajs-2911	120	7	𝔽	𝔽	PROPN
iajs-2911	120	8	∈	∈	PROPN
iajs-2911	120	9	ℑ.	ℑ.	NOUN
iajs-2911	120	10	then	then	ADV
iajs-2911	120	11	,	,	PUNCT
iajs-2911	120	12	𝔔	𝔔	PROPN
iajs-2911	120	13	=	=	PROPN
iajs-2911	120	14	{	{	PUNCT
iajs-2911	120	15	cl(𝔘	cl(𝔘	NOUN
iajs-2911	120	16	)	)	PUNCT
iajs-2911	120	17	∩	∩	NOUN
iajs-2911	120	18	𝔽	𝔽	PROPN
iajs-2911	120	19	:	:	PUNCT
iajs-2911	120	20	𝔽	𝔽	PROPN
iajs-2911	120	21	∈	∈	PROPN
iajs-2911	120	22	ℑ	ℑ	PROPN
iajs-2911	120	23	}	}	PUNCT
iajs-2911	120	24	is	be	AUX
iajs-2911	120	25	a	a	DET
iajs-2911	120	26	ℑ	ℑ	NOUN
iajs-2911	120	27	be	be	AUX
iajs-2911	120	28	a	a	DET
iajs-2911	120	29	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	120	30	ℑ.	ℑ.	NOUN
iajs-2911	120	31	on	on	ADP
iajs-2911	120	32	e	e	NOUN
iajs-2911	120	33	larger	large	ADJ
iajs-2911	120	34	than	than	ADP
iajs-2911	120	35	ℑ	ℑ	PROPN
iajs-2911	120	36	,	,	PUNCT
iajs-2911	120	37	and	and	CCONJ
iajs-2911	120	38	e	e	PROPN
iajs-2911	120	39	∉	∉	X
iajs-2911	120	40	ad	ad	NOUN
iajs-2911	120	41	of	of	ADP
iajs-2911	120	42	𝔔.	𝔔.	PROPN
iajs-2911	120	43	thus	thus	ADV
iajs-2911	120	44	,	,	PUNCT
iajs-2911	120	45	ℑ	ℑ	PROPN
iajs-2911	120	46	can	can	AUX
iajs-2911	120	47	not	not	PART
iajs-2911	120	48	be	be	AUX
iajs-2911	120	49	d.t	d.t	VERB
iajs-2911	120	50	.	.	PUNCT
iajs-2911	121	1	e	e	X
iajs-2911	121	2	,	,	PUNCT
iajs-2911	121	3	so	so	ADV
iajs-2911	121	4	lead	lead	VERB
iajs-2911	121	5	to	to	ADP
iajs-2911	121	6	a	a	DET
iajs-2911	121	7	then	then	ADV
iajs-2911	121	8	c	c	NOUN
iajs-2911	121	9	!	!	PUNCT
iajs-2911	121	10	!	!	PUNCT
iajs-2911	121	11	!	!	PUNCT
iajs-2911	122	1	,	,	PUNCT
iajs-2911	122	2	.	.	PUNCT
iajs-2911	123	1	then	then	ADV
iajs-2911	123	2	,	,	PUNCT
iajs-2911	123	3	ℑ	ℑ	PROPN
iajs-2911	123	4	is	be	AUX
iajs-2911	123	5	conv	conv	ADJ
iajs-2911	123	6	.	.	PUNCT
iajs-2911	124	1	to	to	ADP
iajs-2911	124	2	e.	e.	PROPN
iajs-2911	124	3	(	(	PUNCT
iajs-2911	124	4	⇒	⇒	PROPN
iajs-2911	124	5	)	)	PUNCT
iajs-2911	124	6	.	.	PUNCT
iajs-2911	125	1	it	it	PRON
iajs-2911	125	2	is	be	AUX
iajs-2911	125	3	clear	clear	ADJ
iajs-2911	125	4	definition	definition	NOUN
iajs-2911	125	5	2.3	2.3	NUM
iajs-2911	125	6	.	.	PUNCT
iajs-2911	126	1	the	the	DET
iajs-2911	126	2	𝔽.	𝔽.	PROPN
iajs-2911	126	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	126	4	.	.	PUNCT
iajs-2911	126	5	(	(	PUNCT
iajs-2911	126	6	e,𝜏	e,𝜏	NOUN
iajs-2911	126	7	)	)	PUNCT
iajs-2911	126	8	on	on	ADP
iajs-2911	126	9	a	a	DET
iajs-2911	126	10	topological	topological	ADJ
iajs-2911	126	11	space	space	NOUN
iajs-2911	126	12	(	(	PUNCT
iajs-2911	126	13	d,𝜌	d,𝜌	NOUN
iajs-2911	126	14	)	)	PUNCT
iajs-2911	126	15	is	be	AUX
iajs-2911	126	16	named	name	VERB
iajs-2911	126	17	to	to	PART
iajs-2911	126	18	be	be	AUX
iajs-2911	126	19	𝔽.	𝔽.	PROPN
iajs-2911	126	20	𝕎.	𝕎.	PUNCT
iajs-2911	126	21	upper	upper	ADJ
iajs-2911	126	22	perfect	perfect	NOUN
iajs-2911	126	23	(	(	PUNCT
iajs-2911	126	24	briefly	briefly	ADV
iajs-2911	126	25	,	,	PUNCT
iajs-2911	126	26	𝔽.	𝔽.	PROPN
iajs-2911	126	27	𝕎.u.p	𝕎.u.p	PROPN
iajs-2911	126	28	.	.	PUNCT
iajs-2911	126	29	)	)	PUNCT
iajs-2911	127	1	if	if	SCONJ
iajs-2911	127	2	the	the	DET
iajs-2911	127	3	projection	projection	NOUN
iajs-2911	127	4	x	x	X
iajs-2911	127	5	is	be	AUX
iajs-2911	127	6	u.p	u.p	PROPN
iajs-2911	127	7	.	.	PROPN
iajs-2911	127	8	definition	definition	NOUN
iajs-2911	127	9	2.4	2.4	NUM
iajs-2911	127	10	.	.	PUNCT
iajs-2911	128	1	the	the	DET
iajs-2911	128	2	f.w.t.s	f.w.t.s	PROPN
iajs-2911	128	3	.	.	PUNCT
iajs-2911	128	4	(	(	PUNCT
iajs-2911	128	5	e	e	NOUN
iajs-2911	128	6	,	,	PUNCT
iajs-2911	128	7	τ	τ	X
iajs-2911	128	8	)	)	PUNCT
iajs-2911	128	9	on	on	ADP
iajs-2911	128	10	topological	topological	ADJ
iajs-2911	128	11	space	space	NOUN
iajs-2911	128	12	(	(	PUNCT
iajs-2911	128	13	d	d	NOUN
iajs-2911	128	14	,	,	PUNCT
iajs-2911	128	15	ρ	ρ	NOUN
iajs-2911	128	16	)	)	PUNCT
iajs-2911	128	17	is	be	AUX
iajs-2911	128	18	named	name	VERB
iajs-2911	128	19	to	to	PART
iajs-2911	128	20	be	be	AUX
iajs-2911	128	21	𝔽.	𝔽.	PROPN
iajs-2911	128	22	𝕎.	𝕎.	NOUN
iajs-2911	128	23	lower	lower	ADV
iajs-2911	128	24	perfect	perfect	ADJ
iajs-2911	128	25	(	(	PUNCT
iajs-2911	128	26	briefly	briefly	ADV
iajs-2911	128	27	,	,	PUNCT
iajs-2911	128	28	𝔽.𝕎	𝔽.𝕎	PROPN
iajs-2911	128	29	..	..	PUNCT
iajs-2911	128	30	l.p	l.p	PROPN
iajs-2911	128	31	.	.	PUNCT
iajs-2911	128	32	)	)	PUNCT
iajs-2911	129	1	if	if	SCONJ
iajs-2911	129	2	the	the	DET
iajs-2911	129	3	projection	projection	NOUN
iajs-2911	129	4	x	x	X
iajs-2911	129	5	is	be	AUX
iajs-2911	129	6	l.p	l.p	PROPN
iajs-2911	129	7	.	.	PUNCT
iajs-2911	130	1	the	the	DET
iajs-2911	130	2	f.w.t.s	f.w.t.s	PROPN
iajs-2911	130	3	.	.	PUNCT
iajs-2911	131	1	(	(	PUNCT
iajs-2911	131	2	e	e	NOUN
iajs-2911	131	3	,	,	PUNCT
iajs-2911	131	4	τ	τ	X
iajs-2911	131	5	)	)	PUNCT
iajs-2911	131	6	on	on	ADP
iajs-2911	131	7	topological	topological	ADJ
iajs-2911	131	8	space	space	NOUN
iajs-2911	131	9	(	(	PUNCT
iajs-2911	131	10	d	d	NOUN
iajs-2911	131	11	,	,	PUNCT
iajs-2911	131	12	ρ	ρ	NOUN
iajs-2911	131	13	)	)	PUNCT
iajs-2911	131	14	is	be	AUX
iajs-2911	131	15	named	name	VERB
iajs-2911	131	16	to	to	PART
iajs-2911	131	17	be	be	AUX
iajs-2911	131	18	𝔽.	𝔽.	PROPN
iajs-2911	131	19	𝕎.	𝕎.	PUNCT
iajs-2911	131	20	multi	multi	ADJ
iajs-2911	131	21	-	-	ADJ
iajs-2911	131	22	perfect	perfect	ADJ
iajs-2911	131	23	(	(	PUNCT
iajs-2911	131	24	briefly	briefly	ADV
iajs-2911	131	25	,	,	PUNCT
iajs-2911	131	26	𝔽.	𝔽.	PROPN
iajs-2911	131	27	𝕎.m.p	𝕎.m.p	PROPN
iajs-2911	131	28	.	.	PUNCT
iajs-2911	131	29	)	)	PUNCT
iajs-2911	132	1	if	if	SCONJ
iajs-2911	132	2	it	it	PRON
iajs-2911	132	3	is	be	AUX
iajs-2911	132	4	𝔽.	𝔽.	PROPN
iajs-2911	132	5	𝕎.u.p	𝕎.u.p	PROPN
iajs-2911	132	6	.	.	PUNCT
iajs-2911	133	1	and	and	CCONJ
iajs-2911	133	2	𝔽.	𝔽.	PROPN
iajs-2911	133	3	𝕎.l.p	𝕎.l.p	PROPN
iajs-2911	133	4	.	.	PUNCT
iajs-2911	134	1	in	in	ADP
iajs-2911	134	2	the	the	DET
iajs-2911	134	3	next	next	ADJ
iajs-2911	134	4	theory	theory	NOUN
iajs-2911	134	5	we	we	PRON
iajs-2911	134	6	prove	prove	VERB
iajs-2911	134	7	that	that	SCONJ
iajs-2911	134	8	just	just	ADV
iajs-2911	134	9	points	point	NOUN
iajs-2911	134	10	of	of	ADP
iajs-2911	134	11	d	d	NOUN
iajs-2911	134	12	can	can	AUX
iajs-2911	134	13	be	be	AUX
iajs-2911	134	14	enough	enough	ADJ
iajs-2911	134	15	for	for	SCONJ
iajs-2911	134	16	the	the	DET
iajs-2911	134	17	subset	subset	NOUN
iajs-2911	134	18	a	a	PRON
iajs-2911	134	19	in	in	ADP
iajs-2911	134	20	definition	definition	NOUN
iajs-2911	134	21	(	(	PUNCT
iajs-2911	134	22	1.15	1.15	NUM
iajs-2911	134	23	)	)	PUNCT
iajs-2911	134	24	,	,	PUNCT
iajs-2911	134	25	and	and	CCONJ
iajs-2911	134	26	so	so	ADV
iajs-2911	134	27	direction	direction	NOUN
iajs-2911	134	28	.	.	PUNCT
iajs-2911	135	1	since	since	SCONJ
iajs-2911	135	2	converge	converge	NOUN
iajs-2911	135	3	can	can	AUX
iajs-2911	135	4	be	be	AUX
iajs-2911	135	5	replaced	replace	VERB
iajs-2911	135	6	in	in	ADP
iajs-2911	135	7	view	view	NOUN
iajs-2911	135	8	of	of	ADP
iajs-2911	135	9	lemma	lemma	PROPN
iajs-2911	135	10	(	(	PUNCT
iajs-2911	135	11	2.2	2.2	NUM
iajs-2911	135	12	.	.	PUNCT
iajs-2911	135	13	)	)	PUNCT
iajs-2911	135	14	theorem	theorem	VERB
iajs-2911	135	15	2.1	2.1	NUM
iajs-2911	135	16	.	.	PUNCT
iajs-2911	136	1	assume	assume	VERB
iajs-2911	136	2	that	that	SCONJ
iajs-2911	136	3	(	(	PUNCT
iajs-2911	136	4	e,𝜏	e,𝜏	NOUN
iajs-2911	136	5	)	)	PUNCT
iajs-2911	136	6	is	be	AUX
iajs-2911	136	7	a	a	DET
iajs-2911	136	8	𝔽.	𝔽.	PROPN
iajs-2911	136	9	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	136	10	.	.	PUNCT
iajs-2911	137	1	on	on	ADP
iajs-2911	137	2	a	a	DET
iajs-2911	137	3	topological	topological	ADJ
iajs-2911	137	4	space	space	NOUN
iajs-2911	137	5	(	(	PUNCT
iajs-2911	137	6	d	d	NOUN
iajs-2911	137	7	,	,	PUNCT
iajs-2911	137	8	ρ	ρ	PROPN
iajs-2911	137	9	)	)	PUNCT
iajs-2911	137	10	.	.	PUNCT
iajs-2911	138	1	so	so	ADV
iajs-2911	138	2	,	,	PUNCT
iajs-2911	138	3	the	the	DET
iajs-2911	138	4	next	next	ADJ
iajs-2911	138	5	are	be	AUX
iajs-2911	138	6	equivalent	equivalent	ADJ
iajs-2911	138	7	:	:	PUNCT
iajs-2911	138	8	i.	i.	PROPN
iajs-2911	138	9	(	(	PUNCT
iajs-2911	138	10	e,𝜏	e,𝜏	PROPN
iajs-2911	138	11	)	)	PUNCT
iajs-2911	138	12	is	be	AUX
iajs-2911	138	13	𝔽.	𝔽.	PROPN
iajs-2911	138	14	𝕎.u.p.t.s	𝕎.u.p.t.s	PROPN
iajs-2911	138	15	.	.	PROPN
iajs-2911	139	1	(	(	PUNCT
iajs-2911	139	2	resp	resp	NOUN
iajs-2911	139	3	.	.	PUNCT
iajs-2911	139	4	,	,	PUNCT
iajs-2911	139	5	𝔽.	𝔽.	PROPN
iajs-2911	139	6	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	139	7	.	.	PROPN
iajs-2911	139	8	)	)	PUNCT
iajs-2911	139	9	.	.	PUNCT
iajs-2911	140	1	ii	ii	PROPN
iajs-2911	140	2	.	.	PUNCT
iajs-2911	140	3	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	140	4	ℑ	ℑ	PROPN
iajs-2911	140	5	𝑜n	𝑜n	ADP
iajs-2911	140	6	x(e	x(e	PROPN
iajs-2911	140	7	)	)	PUNCT
iajs-2911	140	8	,	,	PUNCT
iajs-2911	140	9	where	where	SCONJ
iajs-2911	140	10	conv	conv	ADJ
iajs-2911	140	11	.	.	PUNCT
iajs-2911	141	1	to	to	ADP
iajs-2911	141	2	a	a	DET
iajs-2911	141	3	point	point	NOUN
iajs-2911	141	4	d	d	NOUN
iajs-2911	141	5	in	in	ADP
iajs-2911	141	6	d,𝐸ℑ	d,𝐸ℑ	NOUN
iajs-2911	141	7	+	+	CCONJ
iajs-2911	141	8	−−d.t→	−−d.t→	ADJ
iajs-2911	141	9	ed(resp	ed(resp	PROPN
iajs-2911	141	10	.	.	PROPN
iajs-2911	141	11	,	,	PUNCT
iajs-2911	141	12	𝐸ℑ	𝐸ℑ	PROPN
iajs-2911	141	13	−−−d.t→	−−−d.t→	PROPN
iajs-2911	141	14	ed	ed	NOUN
iajs-2911	141	15	)	)	PUNCT
iajs-2911	141	16	.	.	PUNCT
iajs-2911	142	1	iii	iii	X
iajs-2911	142	2	.	.	PUNCT
iajs-2911	142	3	∀	∀	PUNCT
iajs-2911	142	4	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	142	5	ℑ	ℑ	PROPN
iajs-2911	142	6	on	on	ADP
iajs-2911	142	7	e	e	NOUN
iajs-2911	142	8	,	,	PUNCT
iajs-2911	142	9	ad	ad	NOUN
iajs-2911	142	10	x(ℑ	x(ℑ	PROPN
iajs-2911	142	11	)	)	PUNCT
iajs-2911	143	1	⊂	⊂	PROPN
iajs-2911	143	2	x(ad	x(ad	PROPN
iajs-2911	143	3	ℑ	ℑ	PROPN
iajs-2911	143	4	)	)	PUNCT
iajs-2911	143	5	proof	proof	NOUN
iajs-2911	143	6	.	.	PUNCT
iajs-2911	144	1	(	(	PUNCT
iajs-2911	144	2	i	i	NOUN
iajs-2911	144	3	)	)	PUNCT
iajs-2911	144	4	⇒(ii	⇒(ii	PROPN
iajs-2911	144	5	)	)	PUNCT
iajs-2911	144	6	by	by	ADP
iajs-2911	144	7	lemma	lemma	PROPN
iajs-2911	144	8	2.2	2.2	NUM
iajs-2911	144	9	.	.	PUNCT
iajs-2911	145	1	(	(	PUNCT
iajs-2911	145	2	ii	ii	NOUN
iajs-2911	145	3	)	)	PUNCT
iajs-2911	145	4	⇒(iii	⇒(iii	NOUN
iajs-2911	145	5	)	)	PUNCT
iajs-2911	145	6	assume	assume	VERB
iajs-2911	145	7	that	that	SCONJ
iajs-2911	145	8	d	d	PROPN
iajs-2911	145	9	∈	∈	PROPN
iajs-2911	145	10	ad	ad	NOUN
iajs-2911	145	11	x(ℑ	x(ℑ	PROPN
iajs-2911	145	12	)	)	PUNCT
iajs-2911	145	13	.	.	PUNCT
iajs-2911	146	1	thereafter	thereafter	ADV
iajs-2911	146	2	,	,	PUNCT
iajs-2911	146	3	by	by	ADP
iajs-2911	146	4	lemma	lemma	PROPN
iajs-2911	146	5	(	(	PUNCT
iajs-2911	146	6	2.2	2.2	NUM
iajs-2911	146	7	.	.	PUNCT
iajs-2911	146	8	)	)	PUNCT
iajs-2911	146	9	,	,	PUNCT
iajs-2911	146	10	∃	∃	PROPN
iajs-2911	146	11	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	146	12	𝔔	𝔔	PROPN
iajs-2911	146	13	on	on	ADP
iajs-2911	146	14	x(e	x(e	PROPN
iajs-2911	146	15	)	)	PUNCT
iajs-2911	146	16	larger	large	ADJ
iajs-2911	146	17	from	from	ADP
iajs-2911	146	18	x(ℑ).s.t	x(ℑ).s.t	PROPN
iajs-2911	146	19	𝔔	𝔔	PROPN
iajs-2911	146	20	–	–	PUNCT
iajs-2911	146	21	conv.→	conv.→	PROPN
iajs-2911	146	22	d.	d.	PROPN
iajs-2911	146	23	let	let	VERB
iajs-2911	146	24	𝔘	𝔘	PROPN
iajs-2911	146	25	=	=	SYM
iajs-2911	146	26	{	{	PUNCT
iajs-2911	146	27	𝐸𝔔	𝐸𝔔	PROPN
iajs-2911	146	28	∩	∩	ADJ
iajs-2911	146	29	ℑ	ℑ	PROPN
iajs-2911	146	30	:	:	PUNCT
iajs-2911	146	31	g	g	PROPN
iajs-2911	146	32	∈	∈	PROPN
iajs-2911	146	33	𝔔	𝔔	PROPN
iajs-2911	146	34	and	and	CCONJ
iajs-2911	146	35	𝔽	𝔽	PROPN
iajs-2911	146	36	∈	∈	PROPN
iajs-2911	146	37	ℑ	ℑ	PROPN
iajs-2911	146	38	}	}	PUNCT
iajs-2911	146	39	thereafter	thereafter	ADV
iajs-2911	146	40	,	,	PUNCT
iajs-2911	146	41	𝔘	𝔘	PROPN
iajs-2911	146	42	is	be	AUX
iajs-2911	146	43	a	a	DET
iajs-2911	146	44	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	146	45	on	on	ADP
iajs-2911	146	46	e	e	NOUN
iajs-2911	146	47	larger	large	ADJ
iajs-2911	146	48	from	from	ADP
iajs-2911	146	49	𝐸𝔔.	𝐸𝔔.	NOUN
iajs-2911	146	50	since	since	SCONJ
iajs-2911	146	51	𝔔	𝔔	PROPN
iajs-2911	146	52	−−d.t.→	−−d.t.→	ADJ
iajs-2911	146	53	d	d	PROPN
iajs-2911	146	54	,	,	PUNCT
iajs-2911	146	55	by	by	ADP
iajs-2911	146	56	lemma	lemma	PROPN
iajs-2911	146	57	(	(	PUNCT
iajs-2911	146	58	2.3	2.3	NUM
iajs-2911	146	59	.	.	PUNCT
iajs-2911	146	60	)	)	PUNCT
iajs-2911	147	1	and	and	CCONJ
iajs-2911	147	2	x	x	X
iajs-2911	147	3	𝑖𝑠	𝑖𝑠	PROPN
iajs-2911	147	4	p.	p.	NOUN
iajs-2911	147	5	,	,	PUNCT
iajs-2911	147	6	𝐸𝔔	𝐸𝔔	PROPN
iajs-2911	147	7	+	+	PROPN
iajs-2911	147	8	−−d.t.→	−−d.t.→	ADJ
iajs-2911	147	9	ed(resp	ed(resp	PROPN
iajs-2911	147	10	.	.	PUNCT
iajs-2911	147	11	,	,	PUNCT
iajs-2911	147	12	𝐸𝔔	𝐸𝔔	PROPN
iajs-2911	147	13	−−−d.t.→	−−−d.t.→	AUX
iajs-2911	147	14	ed	ed	NOUN
iajs-2911	147	15	)	)	PUNCT
iajs-2911	147	16	.	.	PUNCT
iajs-2911	148	1	𝔘	𝔘	NOUN
iajs-2911	148	2	being	be	AUX
iajs-2911	148	3	larger	large	ADJ
iajs-2911	148	4	than	than	ADP
iajs-2911	148	5	𝐸𝔔	𝐸𝔔	PROPN
iajs-2911	148	6	,	,	PUNCT
iajs-2911	148	7	we	we	PRON
iajs-2911	148	8	have	have	VERB
iajs-2911	148	9	ed	ed	NOUN
iajs-2911	148	10	∩	∩	NOUN
iajs-2911	148	11	ω+(ad	ω+(ad	ADJ
iajs-2911	148	12	𝔘	𝔘	PROPN
iajs-2911	148	13	)	)	PUNCT
iajs-2911	148	14	≠	≠	PROPN
iajs-2911	148	15	∅(resp	∅(resp	NUM
iajs-2911	148	16	.	.	PUNCT
iajs-2911	148	17	,	,	PUNCT
iajs-2911	149	1	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	149	2	∩	∩	NOUN
iajs-2911	149	3	ω−(𝑎𝑑	ω−(𝑎𝑑	VERB
iajs-2911	149	4	𝔘	𝔘	PROPN
iajs-2911	149	5	)	)	PUNCT
iajs-2911	149	6	≠	≠	PROPN
iajs-2911	149	7	∅.	∅.	NOUN
iajs-2911	149	8	)	)	PUNCT
iajs-2911	149	9	.	.	PUNCT
iajs-2911	150	1	hence	hence	ADV
iajs-2911	150	2	it	it	PRON
iajs-2911	150	3	is	be	AUX
iajs-2911	150	4	obvious	obvious	ADJ
iajs-2911	150	5	that	that	SCONJ
iajs-2911	150	6	edω(ℑ	edω(ℑ	PROPN
iajs-2911	150	7	)	)	PUNCT
iajs-2911	150	8	≠	≠	PROPN
iajs-2911	150	9	∅.	∅.	VERB
iajs-2911	150	10	so	so	ADV
iajs-2911	150	11	,	,	PUNCT
iajs-2911	150	12	d	d	PROPN
iajs-2911	150	13	∈	∈	PROPN
iajs-2911	150	14	x(ad	x(ad	PROPN
iajs-2911	150	15	ℑ	ℑ	PROPN
iajs-2911	150	16	)	)	PUNCT
iajs-2911	150	17	.	.	PUNCT
iajs-2911	151	1	(	(	PUNCT
iajs-2911	151	2	iii)⇒(i	iii)⇒(i	X
iajs-2911	151	3	)	)	PUNCT
iajs-2911	151	4	let	let	VERB
iajs-2911	151	5	ℑ	ℑ	PRON
iajs-2911	151	6	be	be	AUX
iajs-2911	151	7	a	a	DET
iajs-2911	151	8	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	151	9	on	on	ADP
iajs-2911	151	10	x(e	x(e	PROPN
iajs-2911	151	11	)	)	PUNCT
iajs-2911	151	12	such	such	ADJ
iajs-2911	151	13	that	that	SCONJ
iajs-2911	151	14	it	it	PRON
iajs-2911	151	15	is	be	AUX
iajs-2911	151	16	d.t	d.t	ADJ
iajs-2911	151	17	.	.	PUNCT
iajs-2911	152	1	some	some	DET
iajs-2911	152	2	subset	subset	ADJ
iajs-2911	152	3	𝒜	𝒜	NOUN
iajs-2911	152	4	of	of	ADP
iajs-2911	152	5	x(e	x(e	PROPN
iajs-2911	152	6	)	)	PUNCT
iajs-2911	152	7	.	.	PUNCT
iajs-2911	153	1	assume	assume	VERB
iajs-2911	153	2	that	that	SCONJ
iajs-2911	153	3	𝔔	𝔔	PROPN
iajs-2911	153	4	is	be	AUX
iajs-2911	153	5	a	a	DET
iajs-2911	153	6	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	153	7	on	on	ADP
iajs-2911	153	8	e	e	NOUN
iajs-2911	153	9	larger	large	ADJ
iajs-2911	153	10	than	than	ADP
iajs-2911	153	11	𝐸ℑ.	𝐸ℑ.	PRON
iajs-2911	153	12	thereafter	thereafter	ADV
iajs-2911	153	13	,	,	PUNCT
iajs-2911	153	14	x(𝔔	x(𝔔	PROPN
iajs-2911	153	15	)	)	PUNCT
iajs-2911	153	16	is	be	AUX
iajs-2911	153	17	a	a	DET
iajs-2911	153	18	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	153	19	on	on	ADP
iajs-2911	153	20	x(e	x(e	PROPN
iajs-2911	153	21	)	)	PUNCT
iajs-2911	153	22	larger	large	ADJ
iajs-2911	153	23	than	than	ADP
iajs-2911	153	24	ℑ	ℑ	PROPN
iajs-2911	153	25	and	and	CCONJ
iajs-2911	153	26	so	so	ADV
iajs-2911	153	27	𝒜	𝒜	NOUN
iajs-2911	153	28	∩	∩	NOUN
iajs-2911	153	29	(	(	PUNCT
iajs-2911	153	30	ad	ad	NOUN
iajs-2911	153	31	ihjpas	ihjpa	NOUN
iajs-2911	153	32	.	.	PUNCT
iajs-2911	154	1	36	36	NUM
iajs-2911	154	2	(	(	PUNCT
iajs-2911	154	3	4	4	NUM
iajs-2911	154	4	)	)	PUNCT
iajs-2911	154	5	2023	2023	NUM
iajs-2911	154	6	400	400	NUM
iajs-2911	154	7	x(𝔔	x(𝔔	NOUN
iajs-2911	154	8	)	)	PUNCT
iajs-2911	154	9	)	)	PUNCT
iajs-2911	155	1	≠	≠	PROPN
iajs-2911	155	2	∅.	∅.	VERB
iajs-2911	155	3	then	then	ADV
iajs-2911	155	4	,	,	PUNCT
iajs-2911	155	5	by	by	ADP
iajs-2911	155	6	(	(	PUNCT
iajs-2911	155	7	c	c	NOUN
iajs-2911	155	8	)	)	PUNCT
iajs-2911	155	9	,	,	PUNCT
iajs-2911	155	10	𝒜	𝒜	NOUN
iajs-2911	155	11	∩	∩	NOUN
iajs-2911	155	12	x(ad	x(ad	PROPN
iajs-2911	155	13	(	(	PUNCT
iajs-2911	155	14	𝔔	𝔔	PROPN
iajs-2911	155	15	)	)	PUNCT
iajs-2911	155	16	)	)	PUNCT
iajs-2911	155	17	≠	≠	PROPN
iajs-2911	155	18	∅	∅	NOUN
iajs-2911	155	19	such	such	ADJ
iajs-2911	155	20	that	that	SCONJ
iajs-2911	155	21	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	155	22	+	+	CCONJ
iajs-2911	155	23	∩	∩	NOUN
iajs-2911	155	24	(	(	PUNCT
iajs-2911	155	25	ad	ad	NOUN
iajs-2911	155	26	(	(	PUNCT
iajs-2911	155	27	𝔔	𝔔	PROPN
iajs-2911	155	28	)	)	PUNCT
iajs-2911	155	29	)	)	PUNCT
iajs-2911	155	30	≠	≠	PROPN
iajs-2911	155	31	∅(resp	∅(resp	PROPN
iajs-2911	155	32	.	.	PUNCT
iajs-2911	156	1	,	,	PUNCT
iajs-2911	156	2	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	156	3	−	−	PROPN
iajs-2911	156	4	∩	∩	NOUN
iajs-2911	156	5	(	(	PUNCT
iajs-2911	156	6	ad	ad	NOUN
iajs-2911	156	7	(	(	PUNCT
iajs-2911	156	8	𝔔	𝔔	PROPN
iajs-2911	156	9	)	)	PUNCT
iajs-2911	156	10	)	)	PUNCT
iajs-2911	156	11	≠	≠	PROPN
iajs-2911	156	12	∅	∅	NOUN
iajs-2911	156	13	)	)	PUNCT
iajs-2911	156	14	.	.	PUNCT
iajs-2911	157	1	then	then	ADV
iajs-2911	157	2	,	,	PUNCT
iajs-2911	157	3	𝐸ℑ	𝐸ℑ	PROPN
iajs-2911	157	4	is	be	AUX
iajs-2911	157	5	d.t	d.t	ADJ
iajs-2911	157	6	.	.	PROPN
iajs-2911	157	7	𝐸𝒜.	𝐸𝒜.	PROPN
iajs-2911	158	1	so	so	ADV
iajs-2911	158	2	,	,	PUNCT
iajs-2911	158	3	x	x	PUNCT
iajs-2911	158	4	is	be	AUX
iajs-2911	158	5	u.p.(resp	u.p.(resp	PROPN
iajs-2911	158	6	.	.	PROPN
iajs-2911	158	7	,	,	PUNCT
iajs-2911	158	8	l.p	l.p	PROPN
iajs-2911	158	9	.	.	PROPN
iajs-2911	158	10	)	)	PUNCT
iajs-2911	158	11	.	.	PUNCT
iajs-2911	159	1	corollary	corollary	ADJ
iajs-2911	159	2	2.1	2.1	NUM
iajs-2911	159	3	.	.	PUNCT
iajs-2911	160	1	assume	assume	VERB
iajs-2911	160	2	that	that	SCONJ
iajs-2911	160	3	(	(	PUNCT
iajs-2911	160	4	e	e	NOUN
iajs-2911	160	5	,	,	PUNCT
iajs-2911	160	6	τ	τ	X
iajs-2911	160	7	)	)	PUNCT
iajs-2911	160	8	is	be	AUX
iajs-2911	160	9	a	a	DET
iajs-2911	160	10	f.w.t.s	f.w.t.s	PROPN
iajs-2911	160	11	.	.	PROPN
iajs-2911	160	12	on	on	ADP
iajs-2911	160	13	a	a	DET
iajs-2911	160	14	topological	topological	ADJ
iajs-2911	160	15	space	space	NOUN
iajs-2911	160	16	(	(	PUNCT
iajs-2911	160	17	d	d	NOUN
iajs-2911	160	18	,	,	PUNCT
iajs-2911	160	19	ρ	ρ	PROPN
iajs-2911	160	20	)	)	PUNCT
iajs-2911	160	21	.	.	PUNCT
iajs-2911	161	1	so	so	ADV
iajs-2911	161	2	,	,	PUNCT
iajs-2911	161	3	the	the	DET
iajs-2911	161	4	next	next	ADJ
iajs-2911	161	5	are	be	AUX
iajs-2911	161	6	equivalent	equivalent	ADJ
iajs-2911	161	7	:	:	PUNCT
iajs-2911	161	8	i.	i.	PROPN
iajs-2911	161	9	(	(	PUNCT
iajs-2911	161	10	e,𝜏	e,𝜏	PROPN
iajs-2911	161	11	)	)	PUNCT
iajs-2911	161	12	is	be	AUX
iajs-2911	161	13	𝔽.	𝔽.	PROPN
iajs-2911	161	14	𝕎.m.p.t.s	𝕎.m.p.t.s	PROPN
iajs-2911	161	15	..	..	SYM
iajs-2911	161	16	ii	ii	PROPN
iajs-2911	161	17	.	.	PUNCT
iajs-2911	162	1	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	162	2	ℑ	ℑ	PROPN
iajs-2911	162	3	𝑜n	𝑜n	ADP
iajs-2911	162	4	x(e	x(e	PROPN
iajs-2911	162	5	)	)	PUNCT
iajs-2911	162	6	,	,	PUNCT
iajs-2911	162	7	where	where	SCONJ
iajs-2911	162	8	conv	conv	ADJ
iajs-2911	162	9	.	.	PUNCT
iajs-2911	163	1	to	to	ADP
iajs-2911	163	2	a	a	DET
iajs-2911	163	3	point	point	NOUN
iajs-2911	163	4	d	d	NOUN
iajs-2911	163	5	in	in	ADP
iajs-2911	163	6	d,𝐸ℑ	d,𝐸ℑ	NOUN
iajs-2911	163	7	+	+	CCONJ
iajs-2911	163	8	−−d.t→	−−d.t→	ADJ
iajs-2911	163	9	ed(resp	ed(resp	PROPN
iajs-2911	163	10	.	.	PROPN
iajs-2911	163	11	,	,	PUNCT
iajs-2911	163	12	𝐸ℑ	𝐸ℑ	PROPN
iajs-2911	163	13	−	−	NOUN
iajs-2911	163	14	−−d.t→	−−d.t→	VERB
iajs-2911	163	15	ed	ed	NOUN
iajs-2911	163	16	)	)	PUNCT
iajs-2911	163	17	.	.	PUNCT
iajs-2911	164	1	iii	iii	X
iajs-2911	164	2	.	.	PUNCT
iajs-2911	164	3	∀	∀	PUNCT
iajs-2911	164	4	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	164	5	ℑ	ℑ	PROPN
iajs-2911	164	6	on	on	ADP
iajs-2911	164	7	e	e	NOUN
iajs-2911	164	8	,	,	PUNCT
iajs-2911	164	9	ad	ad	NOUN
iajs-2911	164	10	x(ℑ	x(ℑ	PROPN
iajs-2911	164	11	)	)	PUNCT
iajs-2911	165	1	⊂	⊂	PROPN
iajs-2911	165	2	x(ad	x(ad	PROPN
iajs-2911	165	3	ℑ	ℑ	PROPN
iajs-2911	165	4	)	)	PUNCT
iajs-2911	165	5	theorem	theorem	VERB
iajs-2911	165	6	2.2	2.2	NUM
iajs-2911	165	7	.	.	PUNCT
iajs-2911	166	1	if	if	SCONJ
iajs-2911	166	2	the	the	DET
iajs-2911	166	3	𝔽.	𝔽.	PROPN
iajs-2911	166	4	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	166	5	.	.	PUNCT
iajs-2911	166	6	(	(	PUNCT
iajs-2911	166	7	e,𝜏	e,𝜏	NOUN
iajs-2911	166	8	)	)	PUNCT
iajs-2911	166	9	on	on	ADP
iajs-2911	166	10	(	(	PUNCT
iajs-2911	166	11	d,𝜌	d,𝜌	NOUN
iajs-2911	166	12	)	)	PUNCT
iajs-2911	166	13	is	be	AUX
iajs-2911	166	14	u.p.(resp	u.p.(resp	PROPN
iajs-2911	166	15	.	.	PROPN
iajs-2911	166	16	,	,	PUNCT
iajs-2911	166	17	l.p	l.p	PROPN
iajs-2911	166	18	.	.	PROPN
iajs-2911	166	19	)	)	PUNCT
iajs-2911	166	20	,	,	PUNCT
iajs-2911	166	21	then	then	ADV
iajs-2911	166	22	it	it	PRON
iajs-2911	166	23	is	be	AUX
iajs-2911	166	24	closed	closed	ADJ
iajs-2911	166	25	.	.	PUNCT
iajs-2911	167	1	proof	proof	NOUN
iajs-2911	167	2	.	.	PUNCT
iajs-2911	168	1	suppose	suppose	VERB
iajs-2911	168	2	that	that	SCONJ
iajs-2911	168	3	e	e	PROPN
iajs-2911	168	4	is	be	AUX
iajs-2911	168	5	a	a	DET
iajs-2911	168	6	𝔽.	𝔽.	PROPN
iajs-2911	168	7	𝕎.u.p.t.s	𝕎.u.p.t.s	PROPN
iajs-2911	168	8	.	.	PUNCT
iajs-2911	168	9	(	(	PUNCT
iajs-2911	168	10	resp	resp	NOUN
iajs-2911	168	11	.	.	PUNCT
iajs-2911	168	12	,	,	PUNCT
iajs-2911	168	13	𝔽.	𝔽.	PROPN
iajs-2911	168	14	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	168	15	.	.	PUNCT
iajs-2911	168	16	)	)	PUNCT
iajs-2911	169	1	on	on	ADP
iajs-2911	169	2	d	d	NOUN
iajs-2911	169	3	,	,	PUNCT
iajs-2911	169	4	then	then	ADV
iajs-2911	169	5	the	the	DET
iajs-2911	169	6	projection	projection	PROPN
iajs-2911	169	7	xe	xe	PROPN
iajs-2911	169	8	:	:	PUNCT
iajs-2911	169	9	e	e	X
iajs-2911	169	10	→	→	PUNCT
iajs-2911	169	11	d	d	X
iajs-2911	169	12	is	be	AUX
iajs-2911	169	13	u.p	u.p	PROPN
iajs-2911	169	14	.	.	PROPN
iajs-2911	170	1	(	(	PUNCT
iajs-2911	170	2	resp	resp	PROPN
iajs-2911	170	3	.	.	PUNCT
iajs-2911	170	4	,	,	PUNCT
iajs-2911	170	5	l.p	l.p	PROPN
iajs-2911	170	6	.	.	PROPN
iajs-2911	170	7	)	)	PUNCT
iajs-2911	171	1	to	to	PART
iajs-2911	171	2	show	show	VERB
iajs-2911	171	3	that	that	SCONJ
iajs-2911	171	4	it	it	PRON
iajs-2911	171	5	is	be	AUX
iajs-2911	171	6	closed	closed	ADJ
iajs-2911	171	7	,	,	PUNCT
iajs-2911	171	8	by	by	ADP
iajs-2911	171	9	theorem	theorem	NOUN
iajs-2911	171	10	(	(	PUNCT
iajs-2911	171	11	4.1.16	4.1.16	NUM
iajs-2911	171	12	.	.	PUNCT
iajs-2911	171	13	)	)	PUNCT
iajs-2911	172	1	(	(	PUNCT
iajs-2911	172	2	a	a	X
iajs-2911	172	3	)	)	PUNCT
iajs-2911	172	4	⇒	⇒	NOUN
iajs-2911	172	5	(	(	PUNCT
iajs-2911	172	6	c	c	X
iajs-2911	172	7	)	)	PUNCT
iajs-2911	172	8	for	for	ADP
iajs-2911	172	9	any	any	DET
iajs-2911	172	10	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	172	11	ℑ	ℑ	PROPN
iajs-2911	172	12	on	on	ADP
iajs-2911	172	13	e	e	PROPN
iajs-2911	172	14	ad	ad	NOUN
iajs-2911	172	15	x(ℑ	x(ℑ	PROPN
iajs-2911	172	16	)	)	PUNCT
iajs-2911	173	1	⊂	⊂	PROPN
iajs-2911	173	2	x(ad	x(ad	PROPN
iajs-2911	173	3	(	(	PUNCT
iajs-2911	173	4	d	d	NOUN
iajs-2911	173	5	)	)	PUNCT
iajs-2911	173	6	)	)	PUNCT
iajs-2911	173	7	,	,	PUNCT
iajs-2911	173	8	by	by	ADP
iajs-2911	173	9	lemma	lemma	PROPN
iajs-2911	173	10	(	(	PUNCT
iajs-2911	173	11	4.1.11	4.1.11	NUM
iajs-2911	173	12	.	.	PUNCT
iajs-2911	173	13	)	)	PUNCT
iajs-2911	173	14	,	,	PUNCT
iajs-2911	173	15	ω	ω	PROPN
iajs-2911	173	16	is	be	AUX
iajs-2911	173	17	closed	close	VERB
iajs-2911	173	18	if	if	SCONJ
iajs-2911	173	19	cl(ω(𝒜	cl(ω(𝒜	NOUN
iajs-2911	173	20	)	)	PUNCT
iajs-2911	173	21	)	)	PUNCT
iajs-2911	174	1	⊂	⊂	PROPN
iajs-2911	174	2	(	(	PUNCT
iajs-2911	174	3	cl(𝒜	cl(𝒜	PROPN
iajs-2911	174	4	)	)	PUNCT
iajs-2911	174	5	)	)	PUNCT
iajs-2911	174	6	for	for	ADP
iajs-2911	174	7	every	every	DET
iajs-2911	174	8	𝒜	𝒜	PROPN
iajs-2911	174	9	⊂e	⊂e	PROPN
iajs-2911	174	10	,	,	PUNCT
iajs-2911	174	11	so	so	ADV
iajs-2911	174	12	x	x	PRON
iajs-2911	174	13	is	be	AUX
iajs-2911	174	14	closed	close	VERB
iajs-2911	174	15	in	in	ADP
iajs-2911	174	16	which	which	PRON
iajs-2911	174	17	ℑ	ℑ	PROPN
iajs-2911	174	18	=	=	SYM
iajs-2911	174	19	{	{	PUNCT
iajs-2911	174	20	𝒜	𝒜	NOUN
iajs-2911	174	21	}	}	PUNCT
iajs-2911	174	22	.	.	PUNCT
iajs-2911	175	1	corollary	corollary	ADJ
iajs-2911	175	2	2.2	2.2	NUM
iajs-2911	175	3	.	.	PUNCT
iajs-2911	176	1	if	if	SCONJ
iajs-2911	176	2	the	the	DET
iajs-2911	176	3	𝔽.	𝔽.	PROPN
iajs-2911	176	4	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	176	5	.	.	PUNCT
iajs-2911	176	6	(	(	PUNCT
iajs-2911	176	7	e,𝜏	e,𝜏	NOUN
iajs-2911	176	8	)	)	PUNCT
iajs-2911	176	9	on	on	ADP
iajs-2911	176	10	(	(	PUNCT
iajs-2911	176	11	d,𝜌	d,𝜌	NOUN
iajs-2911	176	12	)	)	PUNCT
iajs-2911	176	13	is	be	AUX
iajs-2911	176	14	m.p	m.p	PROPN
iajs-2911	176	15	.	.	PROPN
iajs-2911	176	16	,	,	PUNCT
iajs-2911	176	17	then	then	ADV
iajs-2911	176	18	it	it	PRON
iajs-2911	176	19	is	be	AUX
iajs-2911	176	20	closed	closed	ADJ
iajs-2911	176	21	.	.	PUNCT
iajs-2911	177	1	3	3	X
iajs-2911	177	2	.	.	X
iajs-2911	177	3	fibrewise	fibrewise	NOUN
iajs-2911	177	4	multi	multi	ADJ
iajs-2911	177	5	-	-	ADJ
iajs-2911	177	6	perfect	perfect	ADJ
iajs-2911	177	7	and	and	CCONJ
iajs-2911	177	8	multi	multi	ADJ
iajs-2911	177	9	-	-	ADJ
iajs-2911	177	10	rigidity	rigidity	ADJ
iajs-2911	177	11	topological	topological	ADJ
iajs-2911	177	12	spaces	space	NOUN
iajs-2911	177	13	.	.	PUNCT
iajs-2911	178	1	in	in	ADP
iajs-2911	178	2	this	this	DET
iajs-2911	178	3	segment	segment	NOUN
iajs-2911	178	4	,	,	PUNCT
iajs-2911	178	5	we	we	PRON
iajs-2911	178	6	present	present	VERB
iajs-2911	178	7	the	the	DET
iajs-2911	178	8	idea	idea	NOUN
iajs-2911	178	9	of	of	ADP
iajs-2911	178	10	multi	multi	ADJ
iajs-2911	178	11	-	-	ADJ
iajs-2911	178	12	perfect	perfect	ADJ
iajs-2911	178	13	topological	topological	ADJ
iajs-2911	178	14	,	,	PUNCT
iajs-2911	178	15	upper	upper	ADJ
iajs-2911	178	16	rigidity	rigidity	NOUN
iajs-2911	178	17	spaces	space	VERB
iajs-2911	178	18	lower	low	ADJ
iajs-2911	178	19	rigidity	rigidity	NOUN
iajs-2911	178	20	spaces	space	NOUN
iajs-2911	178	21	,	,	PUNCT
iajs-2911	178	22	multi	multi	ADJ
iajs-2911	178	23	-	-	ADJ
iajs-2911	178	24	rigidity	rigidity	ADJ
iajs-2911	178	25	spaces	space	NOUN
iajs-2911	178	26	and	and	CCONJ
iajs-2911	178	27	make	make	VERB
iajs-2911	178	28	sure	sure	ADJ
iajs-2911	178	29	of	of	ADP
iajs-2911	178	30	some	some	PRON
iajs-2911	178	31	of	of	ADP
iajs-2911	178	32	its	its	PRON
iajs-2911	178	33	base	base	NOUN
iajs-2911	178	34	characteristics	characteristic	NOUN
iajs-2911	178	35	.	.	PUNCT
iajs-2911	179	1	definition	definition	NOUN
iajs-2911	179	2	3.1	3.1	NUM
iajs-2911	179	3	.	.	PUNCT
iajs-2911	180	1	a	a	DET
iajs-2911	180	2	subset	subset	ADJ
iajs-2911	180	3	𝒜	𝒜	NOUN
iajs-2911	180	4	of	of	ADP
iajs-2911	180	5	a	a	DET
iajs-2911	180	6	topological	topological	ADJ
iajs-2911	180	7	space	space	NOUN
iajs-2911	180	8	(	(	PUNCT
iajs-2911	180	9	e	e	NOUN
iajs-2911	180	10	,	,	PUNCT
iajs-2911	180	11	τ	τ	X
iajs-2911	180	12	)	)	PUNCT
iajs-2911	180	13	is	be	AUX
iajs-2911	180	14	named	name	VERB
iajs-2911	180	15	to	to	PART
iajs-2911	180	16	be	be	AUX
iajs-2911	180	17	upper	upper	ADJ
iajs-2911	180	18	rigid	rigid	ADJ
iajs-2911	180	19	in	in	ADP
iajs-2911	180	20	e	e	PROPN
iajs-2911	180	21	(	(	PUNCT
iajs-2911	180	22	briefly	briefly	ADV
iajs-2911	180	23	,	,	PUNCT
iajs-2911	180	24	u.r	u.r	PROPN
iajs-2911	180	25	.	.	PUNCT
iajs-2911	180	26	)	)	PUNCT
iajs-2911	181	1	if	if	SCONJ
iajs-2911	181	2	for	for	ADP
iajs-2911	181	3	every	every	DET
iajs-2911	181	4	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	181	5	ℑ	ℑ	PROPN
iajs-2911	181	6	on	on	ADP
iajs-2911	181	7	e	e	PROPN
iajs-2911	181	8	𝑎𝑑	𝑎𝑑	ADP
iajs-2911	181	9	𝑋+(ℑ	𝑋+(ℑ	NOUN
iajs-2911	181	10	)	)	PUNCT
iajs-2911	181	11	∩	∩	ADJ
iajs-2911	181	12	𝒜	𝒜	NOUN
iajs-2911	181	13	=	=	SYM
iajs-2911	181	14	∅	∅	NOUN
iajs-2911	181	15	,	,	PUNCT
iajs-2911	181	16	∃𝔘	∃𝔘	ADJ
iajs-2911	181	17	∈	∈	PROPN
iajs-2911	181	18	𝜏	𝜏	NOUN
iajs-2911	181	19	and	and	CCONJ
iajs-2911	181	20	𝔽	𝔽	PROPN
iajs-2911	181	21	∈	∈	PROPN
iajs-2911	181	22	ℑ	ℑ	NOUN
iajs-2911	181	23	such	such	ADJ
iajs-2911	181	24	that	that	SCONJ
iajs-2911	181	25	𝒜	𝒜	PROPN
iajs-2911	181	26	⊂	⊂	PROPN
iajs-2911	181	27	𝔘	𝔘	PROPN
iajs-2911	181	28	or	or	CCONJ
iajs-2911	181	29	equivalently	equivalently	ADV
iajs-2911	181	30	,	,	PUNCT
iajs-2911	181	31	if	if	SCONJ
iajs-2911	181	32	for	for	ADP
iajs-2911	181	33	every	every	DET
iajs-2911	181	34	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	181	35	ℑ	ℑ	PROPN
iajs-2911	181	36	on	on	ADP
iajs-2911	181	37	e	e	NOUN
iajs-2911	181	38	,	,	PUNCT
iajs-2911	181	39	whenever	whenever	SCONJ
iajs-2911	181	40	𝒜	𝒜	NOUN
iajs-2911	181	41	∩	∩	NOUN
iajs-2911	181	42	(	(	PUNCT
iajs-2911	181	43	ad	ad	NOUN
iajs-2911	181	44	ℑ	ℑ	PROPN
iajs-2911	181	45	)	)	PUNCT
iajs-2911	181	46	=	=	SYM
iajs-2911	181	47	∅	∅	NOUN
iajs-2911	181	48	,	,	PUNCT
iajs-2911	181	49	thereafter	thereafter	ADV
iajs-2911	181	50	for	for	ADP
iajs-2911	181	51	some	some	DET
iajs-2911	181	52	f	f	PROPN
iajs-2911	181	53	∈	∈	PROPN
iajs-2911	181	54	ℑ	ℑ	PROPN
iajs-2911	181	55	,	,	PUNCT
iajs-2911	181	56	𝒜	𝒜	NOUN
iajs-2911	181	57	∩	∩	NOUN
iajs-2911	181	58	(	(	PUNCT
iajs-2911	181	59	cl(ℑ	cl(ℑ	NOUN
iajs-2911	181	60	)	)	PUNCT
iajs-2911	181	61	)	)	PUNCT
iajs-2911	182	1	=	=	PUNCT
iajs-2911	182	2	∅.	∅.	PRON
iajs-2911	182	3	definition	definition	NOUN
iajs-2911	182	4	4.2	4.2	NUM
iajs-2911	182	5	.	.	PUNCT
iajs-2911	183	1	a	a	DET
iajs-2911	183	2	subset	subset	ADJ
iajs-2911	183	3	𝒜	𝒜	NOUN
iajs-2911	183	4	of	of	ADP
iajs-2911	183	5	topological	topological	ADJ
iajs-2911	183	6	space	space	NOUN
iajs-2911	183	7	(	(	PUNCT
iajs-2911	183	8	e	e	NOUN
iajs-2911	183	9	,	,	PUNCT
iajs-2911	183	10	τ	τ	X
iajs-2911	183	11	)	)	PUNCT
iajs-2911	183	12	is	be	AUX
iajs-2911	183	13	named	name	VERB
iajs-2911	183	14	to	to	PART
iajs-2911	183	15	be	be	AUX
iajs-2911	183	16	lower	lower	ADV
iajs-2911	183	17	rigid	rigid	ADJ
iajs-2911	183	18	in	in	ADP
iajs-2911	183	19	e	e	PROPN
iajs-2911	183	20	(	(	PUNCT
iajs-2911	183	21	briefly	briefly	ADV
iajs-2911	183	22	,	,	PUNCT
iajs-2911	183	23	l.r	l.r	PROPN
iajs-2911	183	24	.	.	PUNCT
iajs-2911	183	25	)	)	PUNCT
iajs-2911	184	1	if	if	SCONJ
iajs-2911	184	2	for	for	ADP
iajs-2911	184	3	every	every	DET
iajs-2911	184	4	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	184	5	ℑ	ℑ	PROPN
iajs-2911	184	6	on	on	ADP
iajs-2911	184	7	e	e	PROPN
iajs-2911	184	8	𝑎𝑑	𝑎𝑑	PROPN
iajs-2911	184	9	𝑋−(ℑ	𝑋−(ℑ	PROPN
iajs-2911	184	10	)	)	PUNCT
iajs-2911	184	11	∩	∩	ADJ
iajs-2911	184	12	𝒜	𝒜	NOUN
iajs-2911	184	13	=	=	SYM
iajs-2911	184	14	∅	∅	NOUN
iajs-2911	184	15	,	,	PUNCT
iajs-2911	184	16	∃𝔘	∃𝔘	ADJ
iajs-2911	184	17	∈	∈	PROPN
iajs-2911	184	18	𝜏	𝜏	NOUN
iajs-2911	184	19	and	and	CCONJ
iajs-2911	184	20	𝔽	𝔽	PROPN
iajs-2911	184	21	∈	∈	PROPN
iajs-2911	184	22	ℑ	ℑ	NOUN
iajs-2911	184	23	such	such	ADJ
iajs-2911	184	24	that	that	SCONJ
iajs-2911	184	25	𝒜	𝒜	PROPN
iajs-2911	184	26	⊂	⊂	PROPN
iajs-2911	184	27	𝔘	𝔘	PROPN
iajs-2911	184	28	or	or	CCONJ
iajs-2911	184	29	equivalently	equivalently	ADV
iajs-2911	184	30	,	,	PUNCT
iajs-2911	184	31	if	if	SCONJ
iajs-2911	184	32	for	for	ADP
iajs-2911	184	33	every	every	DET
iajs-2911	184	34	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	184	35	ℑ	ℑ	PROPN
iajs-2911	184	36	on	on	ADP
iajs-2911	184	37	e	e	NOUN
iajs-2911	184	38	,	,	PUNCT
iajs-2911	184	39	whenever	whenever	SCONJ
iajs-2911	184	40	𝒜	𝒜	NOUN
iajs-2911	184	41	∩	∩	NOUN
iajs-2911	184	42	(	(	PUNCT
iajs-2911	184	43	ad	ad	NOUN
iajs-2911	184	44	ℑ	ℑ	PROPN
iajs-2911	184	45	)	)	PUNCT
iajs-2911	184	46	=	=	SYM
iajs-2911	184	47	∅	∅	NOUN
iajs-2911	184	48	,	,	PUNCT
iajs-2911	184	49	thereafter	thereafter	ADV
iajs-2911	184	50	for	for	ADP
iajs-2911	184	51	some	some	DET
iajs-2911	184	52	f	f	PROPN
iajs-2911	184	53	∈	∈	PROPN
iajs-2911	184	54	ℑ	ℑ	PROPN
iajs-2911	184	55	,	,	PUNCT
iajs-2911	184	56	𝒜	𝒜	NOUN
iajs-2911	184	57	∩	∩	NOUN
iajs-2911	184	58	(	(	PUNCT
iajs-2911	184	59	cl(ℑ	cl(ℑ	NOUN
iajs-2911	184	60	)	)	PUNCT
iajs-2911	184	61	)	)	PUNCT
iajs-2911	185	1	=	=	PUNCT
iajs-2911	185	2	∅.	∅.	ADP
iajs-2911	185	3	a	a	DET
iajs-2911	185	4	subset	subset	ADJ
iajs-2911	185	5	𝒜	𝒜	NOUN
iajs-2911	185	6	of	of	ADP
iajs-2911	185	7	topological	topological	ADJ
iajs-2911	185	8	space	space	NOUN
iajs-2911	185	9	(	(	PUNCT
iajs-2911	185	10	e	e	NOUN
iajs-2911	185	11	,	,	PUNCT
iajs-2911	185	12	τ	τ	X
iajs-2911	185	13	)	)	PUNCT
iajs-2911	185	14	is	be	AUX
iajs-2911	185	15	named	name	VERB
iajs-2911	185	16	to	to	PART
iajs-2911	185	17	be	be	AUX
iajs-2911	185	18	multi	multi	ADJ
iajs-2911	185	19	-	-	ADJ
iajs-2911	185	20	rigid	rigid	ADJ
iajs-2911	185	21	in	in	ADP
iajs-2911	185	22	e	e	PROPN
iajs-2911	185	23	(	(	PUNCT
iajs-2911	185	24	briefly	briefly	ADV
iajs-2911	185	25	,	,	PUNCT
iajs-2911	185	26	m.r	m.r	PROPN
iajs-2911	185	27	.	.	PROPN
iajs-2911	185	28	)	)	PUNCT
iajs-2911	186	1	if	if	SCONJ
iajs-2911	186	2	it	it	PRON
iajs-2911	186	3	is	be	AUX
iajs-2911	186	4	u.r	u.r	PROPN
iajs-2911	186	5	.	.	PROPN
iajs-2911	187	1	and	and	CCONJ
iajs-2911	187	2	l.r	l.r	PROPN
iajs-2911	187	3	.	.	PROPN
iajs-2911	187	4	theorem	theorem	VERB
iajs-2911	187	5	3.1	3.1	NUM
iajs-2911	187	6	.	.	PUNCT
iajs-2911	188	1	if	if	SCONJ
iajs-2911	188	2	(	(	PUNCT
iajs-2911	188	3	e,𝜏	e,𝜏	NOUN
iajs-2911	188	4	)	)	PUNCT
iajs-2911	188	5	is	be	AUX
iajs-2911	188	6	a	a	DET
iajs-2911	188	7	𝔽.	𝔽.	PROPN
iajs-2911	188	8	𝕎.	𝕎.	PROPN
iajs-2911	188	9	closed	close	VERB
iajs-2911	188	10	topological	topological	ADJ
iajs-2911	188	11	space	space	NOUN
iajs-2911	188	12	on	on	ADP
iajs-2911	188	13	(	(	PUNCT
iajs-2911	188	14	d	d	PROPN
iajs-2911	188	15	,	,	PUNCT
iajs-2911	188	16	ρ	ρ	NOUN
iajs-2911	188	17	)	)	PUNCT
iajs-2911	188	18	such	such	ADJ
iajs-2911	188	19	that	that	SCONJ
iajs-2911	188	20	every	every	DET
iajs-2911	188	21	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	188	22	+	+	ADJ
iajs-2911	188	23	(	(	PUNCT
iajs-2911	188	24	resp	resp	NOUN
iajs-2911	188	25	.	.	PUNCT
iajs-2911	189	1	,	,	PUNCT
iajs-2911	189	2	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	189	3	−	−	NOUN
iajs-2911	189	4	)	)	PUNCT
iajs-2911	189	5	.	.	PUNCT
iajs-2911	190	1	in	in	ADP
iajs-2911	190	2	which	which	PRON
iajs-2911	190	3	d	d	X
iajs-2911	190	4	∈	∈	PROPN
iajs-2911	190	5	d	d	NOUN
iajs-2911	190	6	is	be	AUX
iajs-2911	190	7	u.r.(resp	u.r.(resp	PROPN
iajs-2911	190	8	.	.	PROPN
iajs-2911	190	9	,	,	PUNCT
iajs-2911	190	10	l.r	l.r	PROPN
iajs-2911	190	11	.	.	PUNCT
iajs-2911	190	12	)	)	PUNCT
iajs-2911	191	1	in	in	ADP
iajs-2911	191	2	e	e	NOUN
iajs-2911	191	3	,	,	PUNCT
iajs-2911	191	4	then	then	ADV
iajs-2911	191	5	(	(	PUNCT
iajs-2911	191	6	e,𝜏	e,𝜏	NOUN
iajs-2911	191	7	)	)	PUNCT
iajs-2911	191	8	is	be	AUX
iajs-2911	191	9	a	a	DET
iajs-2911	191	10	𝔽.	𝔽.	PROPN
iajs-2911	191	11	𝕎.u.p	𝕎.u.p	PROPN
iajs-2911	191	12	.	.	PUNCT
iajs-2911	192	1	(	(	PUNCT
iajs-2911	192	2	resp	resp	NOUN
iajs-2911	192	3	.	.	PUNCT
iajs-2911	192	4	,	,	PUNCT
iajs-2911	192	5	𝔽.	𝔽.	PROPN
iajs-2911	192	6	𝕎.l.p	𝕎.l.p	PROPN
iajs-2911	192	7	.	.	PUNCT
iajs-2911	192	8	)	)	PUNCT
iajs-2911	192	9	.	.	PUNCT
iajs-2911	193	1	proof	proof	NOUN
iajs-2911	193	2	.	.	PUNCT
iajs-2911	194	1	suppose	suppose	VERB
iajs-2911	194	2	that	that	SCONJ
iajs-2911	194	3	e	e	PROPN
iajs-2911	194	4	is	be	AUX
iajs-2911	194	5	a	a	DET
iajs-2911	194	6	𝔽.	𝔽.	PROPN
iajs-2911	194	7	𝕎.	𝕎.	PROPN
iajs-2911	194	8	closed	close	VERB
iajs-2911	194	9	topological	topological	ADJ
iajs-2911	194	10	space	space	NOUN
iajs-2911	194	11	on	on	ADP
iajs-2911	194	12	d	d	NOUN
iajs-2911	194	13	,	,	PUNCT
iajs-2911	194	14	thereafter	thereafter	ADV
iajs-2911	194	15	𝑋𝐸	𝑋𝐸	VERB
iajs-2911	194	16	:	:	PUNCT
iajs-2911	194	17	e	e	X
iajs-2911	194	18	→d	→d	PROPN
iajs-2911	194	19	exists	exist	VERB
iajs-2911	194	20	t.p	t.p	PROPN
iajs-2911	194	21	.	.	PUNCT
iajs-2911	195	1	it	it	PRON
iajs-2911	195	2	is	be	AUX
iajs-2911	195	3	u.p.(resp	u.p.(resp	PROPN
iajs-2911	195	4	.	.	PROPN
iajs-2911	195	5	,	,	PUNCT
iajs-2911	195	6	l.p	l.p	PROPN
iajs-2911	195	7	.	.	PROPN
iajs-2911	195	8	)	)	PUNCT
iajs-2911	195	9	,	,	PUNCT
iajs-2911	195	10	assume	assume	VERB
iajs-2911	195	11	that	that	SCONJ
iajs-2911	195	12	ℑ	ℑ	PROPN
iajs-2911	195	13	is	be	AUX
iajs-2911	195	14	a	a	DET
iajs-2911	195	15	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	195	16	on	on	ADP
iajs-2911	195	17	𝑋𝐸such	𝑋𝐸such	PROPN
iajs-2911	195	18	that	that	SCONJ
iajs-2911	195	19	d	d	NOUN
iajs-2911	195	20	–	–	PUNCT
iajs-2911	195	21	conv.→	conv.→	PROPN
iajs-2911	195	22	d	d	NOUN
iajs-2911	195	23	in	in	ADP
iajs-2911	195	24	d	d	PROPN
iajs-2911	195	25	,	,	PUNCT
iajs-2911	195	26	for	for	ADP
iajs-2911	195	27	some	some	DET
iajs-2911	195	28	d	d	PROPN
iajs-2911	195	29	in	in	ADP
iajs-2911	195	30	d.	d.	PROPN
iajs-2911	195	31	if	if	SCONJ
iajs-2911	195	32	𝔔	𝔔	PROPN
iajs-2911	195	33	is	be	AUX
iajs-2911	195	34	a	a	DET
iajs-2911	195	35	f∗.b∗	f∗.b∗	NOUN
iajs-2911	195	36	on	on	ADP
iajs-2911	195	37	e	e	NOUN
iajs-2911	195	38	larger	large	ADJ
iajs-2911	195	39	than	than	ADP
iajs-2911	195	40	the	the	DET
iajs-2911	195	41	f∗.b∗.𝐸ℑ	f∗.b∗.𝐸ℑ	NOUN
iajs-2911	195	42	,	,	PUNCT
iajs-2911	195	43	then	then	ADV
iajs-2911	195	44	𝑋(𝔔	𝑋(𝔔	NUM
iajs-2911	195	45	)	)	PUNCT
iajs-2911	195	46	is	be	AUX
iajs-2911	195	47	a	a	DET
iajs-2911	195	48	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	195	49	on	on	ADP
iajs-2911	195	50	d	d	NOUN
iajs-2911	195	51	,	,	PUNCT
iajs-2911	195	52	larger	large	ADJ
iajs-2911	195	53	than	than	ADP
iajs-2911	195	54	ℑ.	ℑ.	NOUN
iajs-2911	195	55	because	because	SCONJ
iajs-2911	195	56	ℑ	ℑ	NOUN
iajs-2911	195	57	−−d.t.→	−−d.t.→	ADJ
iajs-2911	195	58	d	d	X
iajs-2911	195	59	by	by	ADP
iajs-2911	195	60	lemma	lemma	PROPN
iajs-2911	195	61	(	(	PUNCT
iajs-2911	195	62	2.3	2.3	NUM
iajs-2911	195	63	.	.	PUNCT
iajs-2911	195	64	)	)	PUNCT
iajs-2911	195	65	,	,	PUNCT
iajs-2911	196	1	d	d	PROPN
iajs-2911	196	2	∈	∈	PROPN
iajs-2911	196	3	adx(𝔔	adx(𝔔	PROPN
iajs-2911	196	4	)	)	PUNCT
iajs-2911	196	5	,	,	PUNCT
iajs-2911	196	6	i.e	i.e	PROPN
iajs-2911	196	7	,	,	PUNCT
iajs-2911	196	8	d	d	PROPN
iajs-2911	196	9	∈	∈	PROPN
iajs-2911	196	10	∩{ad	∩{ad	NOUN
iajs-2911	196	11	x(g;g	x(g;g	PUNCT
iajs-2911	197	1	∈	∈	PROPN
iajs-2911	197	2	𝔔	𝔔	PROPN
iajs-2911	197	3	)	)	PUNCT
iajs-2911	197	4	}	}	PUNCT
iajs-2911	197	5	,	,	PUNCT
iajs-2911	197	6	and	and	CCONJ
iajs-2911	197	7	hence	hence	ADV
iajs-2911	197	8	,	,	PUNCT
iajs-2911	198	1	d	d	PROPN
iajs-2911	198	2	∈	∈	PROPN
iajs-2911	198	3	∩{x(ad	∩{x(ad	PUNCT
iajs-2911	198	4	g;g	g;g	NOUN
iajs-2911	198	5	∈	∈	PROPN
iajs-2911	198	6	𝔔	𝔔	PROPN
iajs-2911	198	7	)	)	PUNCT
iajs-2911	198	8	}	}	PUNCT
iajs-2911	198	9	by	by	ADP
iajs-2911	198	10	lemma	lemma	PROPN
iajs-2911	198	11	1.1	1.1	NUM
iajs-2911	198	12	.	.	PUNCT
iajs-2911	198	13	)	)	PUNCT
iajs-2911	198	14	.	.	PUNCT
iajs-2911	199	1	by	by	ADP
iajs-2911	199	2	x	x	SYM
iajs-2911	199	3	𝑖s	𝑖s	ADP
iajs-2911	199	4	closed	closed	ADJ
iajs-2911	199	5	,	,	PUNCT
iajs-2911	199	6	so	so	ADV
iajs-2911	199	7	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	199	8	+	+	NUM
iajs-2911	199	9	∩	∩	ADJ
iajs-2911	199	10	ad	ad	NOUN
iajs-2911	199	11	(	(	PUNCT
iajs-2911	199	12	g	g	NOUN
iajs-2911	199	13	)	)	PUNCT
iajs-2911	199	14	≠	≠	PROPN
iajs-2911	199	15	∅(resp	∅(resp	NOUN
iajs-2911	199	16	.	.	PUNCT
iajs-2911	199	17	,	,	PUNCT
iajs-2911	200	1	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	200	2	−	−	PROPN
iajs-2911	200	3	∩	∩	ADJ
iajs-2911	200	4	ad	ad	NOUN
iajs-2911	200	5	(	(	PUNCT
iajs-2911	200	6	g	g	NOUN
iajs-2911	200	7	)	)	PUNCT
iajs-2911	200	8	≠	≠	PROPN
iajs-2911	200	9	∅	∅	NOUN
iajs-2911	200	10	)	)	PUNCT
iajs-2911	200	11	,	,	PUNCT
iajs-2911	200	12	for	for	ADP
iajs-2911	200	13	every	every	DET
iajs-2911	200	14	g	g	PROPN
iajs-2911	200	15	∈	∈	PROPN
iajs-2911	200	16	𝔔.	𝔔.	PROPN
iajs-2911	200	17	so	so	ADV
iajs-2911	200	18	,	,	PUNCT
iajs-2911	200	19	for	for	ADP
iajs-2911	200	20	every	every	DET
iajs-2911	200	21	𝔘	𝔘	PROPN
iajs-2911	200	22	∈	∈	PROPN
iajs-2911	200	23	𝜏	𝜏	NOUN
iajs-2911	200	24	𝑤𝑖th	𝑤𝑖th	NOUN
iajs-2911	200	25	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	200	26	+	+	ADJ
iajs-2911	200	27	(	(	PUNCT
iajs-2911	200	28	resp	resp	NOUN
iajs-2911	200	29	.	.	PUNCT
iajs-2911	200	30	,	,	PUNCT
iajs-2911	200	31	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	200	32	−)⊂	−)⊂	NUM
iajs-2911	200	33	𝔘	𝔘	PROPN
iajs-2911	200	34	,	,	PUNCT
iajs-2911	200	35	cl(𝔘	cl(𝔘	NOUN
iajs-2911	200	36	)	)	PUNCT
iajs-2911	200	37	∩g	∩g	ADJ
iajs-2911	200	38	≠	≠	ADJ
iajs-2911	200	39	∅	∅	NOUN
iajs-2911	200	40	for	for	ADP
iajs-2911	200	41	every	every	DET
iajs-2911	200	42	g	g	PROPN
iajs-2911	200	43	∈	∈	PROPN
iajs-2911	200	44	𝔔.	𝔔.	PROPN
iajs-2911	200	45	since	since	ADV
iajs-2911	200	46	,	,	PUNCT
iajs-2911	200	47	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	200	48	+	+	ADJ
iajs-2911	200	49	(	(	PUNCT
iajs-2911	200	50	resp	resp	NOUN
iajs-2911	200	51	.	.	PUNCT
iajs-2911	200	52	,	,	PUNCT
iajs-2911	200	53	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	200	54	−	−	NOUN
iajs-2911	200	55	)	)	PUNCT
iajs-2911	200	56	is	be	AUX
iajs-2911	200	57	u.r.(resp	u.r.(resp	PRON
iajs-2911	200	58	.	.	PROPN
iajs-2911	200	59	,	,	PUNCT
iajs-2911	200	60	l.r	l.r	PROPN
iajs-2911	200	61	.	.	PROPN
iajs-2911	200	62	)	)	PUNCT
iajs-2911	200	63	,	,	PUNCT
iajs-2911	200	64	it	it	PRON
iajs-2911	200	65	then	then	ADV
iajs-2911	200	66	follows	follow	VERB
iajs-2911	200	67	that	that	SCONJ
iajs-2911	200	68	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	200	69	+	+	ADJ
iajs-2911	200	70	∩	∩	ADJ
iajs-2911	200	71	ad	ad	NOUN
iajs-2911	200	72	(	(	PUNCT
iajs-2911	200	73	𝔔	𝔔	PROPN
iajs-2911	200	74	)	)	PUNCT
iajs-2911	200	75	≠	≠	PROPN
iajs-2911	200	76	∅(resp	∅(resp	NUM
iajs-2911	200	77	.	.	PUNCT
iajs-2911	200	78	,	,	PUNCT
iajs-2911	201	1	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	201	2	−∩	−∩	NOUN
iajs-2911	201	3	ad	ad	NOUN
iajs-2911	201	4	(	(	PUNCT
iajs-2911	201	5	𝔔	𝔔	PROPN
iajs-2911	201	6	)	)	PUNCT
iajs-2911	201	7	≠	≠	PROPN
iajs-2911	201	8	∅	∅	NOUN
iajs-2911	201	9	)	)	PUNCT
iajs-2911	201	10	.	.	PUNCT
iajs-2911	202	1	thus	thus	ADV
iajs-2911	202	2	,	,	PUNCT
iajs-2911	202	3	𝐸ℑ	𝐸ℑ	PROPN
iajs-2911	202	4	−−d.t.→ed	−−d.t.→ed	PROPN
iajs-2911	202	5	.	.	PUNCT
iajs-2911	203	1	s𝑜	s𝑜	NOUN
iajs-2911	203	2	by	by	ADP
iajs-2911	203	3	theorem	theorem	NOUN
iajs-2911	203	4	[	[	X
iajs-2911	203	5	(	(	PUNCT
iajs-2911	203	6	2.1	2.1	NUM
iajs-2911	203	7	.	.	PUNCT
iajs-2911	203	8	)	)	PUNCT
iajs-2911	203	9	,	,	PUNCT
iajs-2911	203	10	(	(	PUNCT
iajs-2911	203	11	b	b	X
iajs-2911	203	12	)	)	PUNCT
iajs-2911	203	13	⇒(a	⇒(a	NUM
iajs-2911	203	14	)	)	PUNCT
iajs-2911	203	15	]	]	PUNCT
iajs-2911	203	16	,	,	PUNCT
iajs-2911	203	17	x	x	X
iajs-2911	203	18	is	be	AUX
iajs-2911	203	19	u.p.(resp	u.p.(resp	PROPN
iajs-2911	203	20	.	.	PROPN
iajs-2911	203	21	,	,	PUNCT
iajs-2911	203	22	l.r	l.r	PROPN
iajs-2911	203	23	.	.	PUNCT
iajs-2911	203	24	)	)	PUNCT
iajs-2911	203	25	corollary	corollary	ADJ
iajs-2911	203	26	3.1	3.1	NUM
iajs-2911	203	27	.	.	PUNCT
iajs-2911	204	1	if	if	SCONJ
iajs-2911	204	2	(	(	PUNCT
iajs-2911	204	3	e,𝜏	e,𝜏	NOUN
iajs-2911	204	4	)	)	PUNCT
iajs-2911	204	5	is	be	AUX
iajs-2911	204	6	a	a	DET
iajs-2911	204	7	𝔽.	𝔽.	PROPN
iajs-2911	204	8	𝕎.	𝕎.	PROPN
iajs-2911	204	9	closed	close	VERB
iajs-2911	204	10	topological	topological	ADJ
iajs-2911	204	11	space	space	NOUN
iajs-2911	204	12	on	on	ADP
iajs-2911	204	13	(	(	PUNCT
iajs-2911	204	14	d,𝜌	d,𝜌	NOUN
iajs-2911	204	15	)	)	PUNCT
iajs-2911	205	1	such	such	ADJ
iajs-2911	205	2	that	that	SCONJ
iajs-2911	205	3	every	every	DET
iajs-2911	205	4	ed	ed	NOUN
iajs-2911	205	5	in	in	ADP
iajs-2911	205	6	which	which	PRON
iajs-2911	205	7	d	d	PROPN
iajs-2911	205	8	∈	∈	PROPN
iajs-2911	205	9	d	d	X
iajs-2911	205	10	𝑖𝑠	𝑖𝑠	NOUN
iajs-2911	205	11	m.r	m.r	PROPN
iajs-2911	205	12	.	.	PROPN
iajs-2911	206	1	in	in	ADP
iajs-2911	206	2	e	e	NOUN
iajs-2911	206	3	,	,	PUNCT
iajs-2911	206	4	then	then	ADV
iajs-2911	206	5	(	(	PUNCT
iajs-2911	206	6	e,𝜏	e,𝜏	NOUN
iajs-2911	206	7	)	)	PUNCT
iajs-2911	206	8	is	be	AUX
iajs-2911	206	9	a	a	DET
iajs-2911	206	10	𝔽.	𝔽.	PROPN
iajs-2911	206	11	𝕎.m.p	𝕎.m.p	PROPN
iajs-2911	206	12	.	.	PUNCT
iajs-2911	207	1	theor𝒆m	theor𝒆m	ADJ
iajs-2911	207	2	3.2	3.2	NUM
iajs-2911	207	3	.	.	PUNCT
iajs-2911	208	1	if	if	SCONJ
iajs-2911	208	2	the	the	DET
iajs-2911	208	3	𝔽.	𝔽.	PROPN
iajs-2911	208	4	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	208	5	.	.	PUNCT
iajs-2911	208	6	(	(	PUNCT
iajs-2911	208	7	e,𝜏	e,𝜏	NOUN
iajs-2911	208	8	)	)	PUNCT
iajs-2911	208	9	on	on	ADP
iajs-2911	208	10	(	(	PUNCT
iajs-2911	208	11	d,𝜌	d,𝜌	NOUN
iajs-2911	208	12	)	)	PUNCT
iajs-2911	208	13	is	be	AUX
iajs-2911	208	14	u.p	u.p	PROPN
iajs-2911	208	15	.	.	PROPN
iajs-2911	209	1	(	(	PUNCT
iajs-2911	209	2	resp	resp	PROPN
iajs-2911	209	3	.	.	PUNCT
iajs-2911	209	4	,	,	PUNCT
iajs-2911	209	5	l.p	l.p	PROPN
iajs-2911	209	6	.	.	PROPN
iajs-2911	209	7	)	)	PUNCT
iajs-2911	209	8	,	,	PUNCT
iajs-2911	209	9	then	then	ADV
iajs-2911	209	10	,	,	PUNCT
iajs-2911	209	11	it	it	PRON
iajs-2911	209	12	is	be	AUX
iajs-2911	209	13	closed	closed	ADJ
iajs-2911	209	14	and	and	CCONJ
iajs-2911	209	15	for	for	ADP
iajs-2911	209	16	every	every	DET
iajs-2911	209	17	d	d	PROPN
iajs-2911	209	18	∈	∈	PROPN
iajs-2911	209	19	b	b	PROPN
iajs-2911	209	20	,	,	PUNCT
iajs-2911	209	21	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	209	22	+	+	ADJ
iajs-2911	209	23	(	(	PUNCT
iajs-2911	209	24	resp	resp	NOUN
iajs-2911	209	25	.	.	PUNCT
iajs-2911	210	1	,	,	PUNCT
iajs-2911	210	2	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	210	3	−	−	NOUN
iajs-2911	210	4	)	)	PUNCT
iajs-2911	210	5	.	.	PUNCT
iajs-2911	211	1	is	be	AUX
iajs-2911	211	2	u.r.(resp	u.r.(resp	PROPN
iajs-2911	211	3	.	.	PROPN
iajs-2911	211	4	,	,	PUNCT
iajs-2911	211	5	l.r	l.r	PROPN
iajs-2911	211	6	.	.	PUNCT
iajs-2911	211	7	)	)	PUNCT
iajs-2911	212	1	in	in	ADP
iajs-2911	212	2	e.	e.	PROPN
iajs-2911	212	3	ihjpas	ihjpas	PROPN
iajs-2911	212	4	.	.	PUNCT
iajs-2911	213	1	36	36	NUM
iajs-2911	213	2	(	(	PUNCT
iajs-2911	213	3	4	4	NUM
iajs-2911	213	4	)	)	PUNCT
iajs-2911	213	5	2023	2023	NUM
iajs-2911	213	6	401	401	NUM
iajs-2911	213	7	proof	proof	NOUN
iajs-2911	213	8	.	.	PUNCT
iajs-2911	214	1	let	let	VERB
iajs-2911	214	2	e	e	PRON
iajs-2911	214	3	be	be	AUX
iajs-2911	214	4	a	a	DET
iajs-2911	214	5	𝔽.	𝔽.	PROPN
iajs-2911	214	6	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	214	7	.	.	PUNCT
iajs-2911	215	1	on	on	ADP
iajs-2911	215	2	d	d	PROPN
iajs-2911	215	3	,	,	PUNCT
iajs-2911	215	4	so	so	ADV
iajs-2911	215	5	the	the	DET
iajs-2911	215	6	projection	projection	NOUN
iajs-2911	215	7	𝑋𝐸	𝑋𝐸	NOUN
iajs-2911	215	8	:	:	PUNCT
iajs-2911	215	9	e→	e→	NOUN
iajs-2911	215	10	d	d	PROPN
iajs-2911	215	11	exists	exist	VERB
iajs-2911	215	12	and	and	CCONJ
iajs-2911	215	13	it	it	PRON
iajs-2911	215	14	is	be	AUX
iajs-2911	215	15	u.	u.	PROPN
iajs-2911	215	16	cont.(resp	cont.(resp	PROPN
iajs-2911	215	17	.	.	PROPN
iajs-2911	215	18	,	,	PUNCT
iajs-2911	215	19	l.	l.	PROPN
iajs-2911	215	20	cont	cont	PROPN
iajs-2911	215	21	.	.	PUNCT
iajs-2911	215	22	)	)	PUNCT
iajs-2911	215	23	.	.	PUNCT
iajs-2911	216	1	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	216	2	is	be	AUX
iajs-2911	216	3	an	an	DET
iajs-2911	216	4	u.p.(resp	u.p.(resp	PROPN
iajs-2911	216	5	.	.	PROPN
iajs-2911	216	6	,	,	PUNCT
iajs-2911	216	7	l.p	l.p	PROPN
iajs-2911	216	8	.	.	PUNCT
iajs-2911	216	9	)	)	PUNCT
iajs-2911	217	1	so	so	ADV
iajs-2911	217	2	it	it	PRON
iajs-2911	217	3	is	be	AUX
iajs-2911	217	4	closed	closed	ADJ
iajs-2911	217	5	.	.	PUNCT
iajs-2911	218	1	t.p	t.p	X
iajs-2911	218	2	.	.	PROPN
iajs-2911	218	3	is	be	AUX
iajs-2911	218	4	closed	close	VERB
iajs-2911	218	5	and	and	CCONJ
iajs-2911	218	6	for	for	ADP
iajs-2911	218	7	every	every	DET
iajs-2911	218	8	d	d	PROPN
iajs-2911	218	9	∈	∈	PROPN
iajs-2911	218	10	d	d	NOUN
iajs-2911	218	11	,	,	PUNCT
iajs-2911	218	12	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	218	13	+	+	ADJ
iajs-2911	218	14	(	(	PUNCT
iajs-2911	218	15	resp	resp	NOUN
iajs-2911	218	16	.	.	PUNCT
iajs-2911	219	1	,	,	PUNCT
iajs-2911	219	2	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	219	3	−	−	NOUN
iajs-2911	219	4	)	)	PUNCT
iajs-2911	219	5	is	be	AUX
iajs-2911	219	6	u.r.(l.r	u.r.(l.r	ADJ
iajs-2911	219	7	)	)	PUNCT
iajs-2911	219	8	in	in	ADP
iajs-2911	219	9	e.	e.	PROPN
iajs-2911	219	10	le𝑡	le𝑡	PROPN
iajs-2911	219	11	d	d	PROPN
iajs-2911	219	12	∈	∈	PROPN
iajs-2911	219	13	d	d	NOUN
iajs-2911	219	14	and	and	CCONJ
iajs-2911	219	15	suppose	suppose	VERB
iajs-2911	219	16	ℑ	ℑ	NOUN
iajs-2911	219	17	is	be	AUX
iajs-2911	219	18	a	a	DET
iajs-2911	219	19	ℑ	ℑ	PROPN
iajs-2911	219	20	∗.b	∗.b	PROPN
iajs-2911	219	21	*	*	PUNCT
iajs-2911	219	22	.	.	PUNCT
iajs-2911	220	1	on	on	ADP
iajs-2911	220	2	e	e	PRON
iajs-2911	220	3	such	such	ADJ
iajs-2911	220	4	that	that	PRON
iajs-2911	220	5	(	(	PUNCT
iajs-2911	220	6	ad	ad	NOUN
iajs-2911	220	7	ℑ)∩	ℑ)∩	NOUN
iajs-2911	220	8	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	220	9	+	+	NOUN
iajs-2911	220	10	=	=	NOUN
iajs-2911	220	11	∅(resp	∅(resp	PROPN
iajs-2911	220	12	.	.	PUNCT
iajs-2911	220	13	,	,	PUNCT
iajs-2911	220	14	(	(	PUNCT
iajs-2911	220	15	ad	ad	NOUN
iajs-2911	220	16	ℑ	ℑ	PROPN
iajs-2911	220	17	)	)	PUNCT
iajs-2911	220	18	∩	∩	NOUN
iajs-2911	220	19	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	220	20	−	−	NOUN
iajs-2911	220	21	=	=	SYM
iajs-2911	220	22	∅	∅	NOUN
iajs-2911	220	23	)	)	PUNCT
iajs-2911	220	24	.	.	PUNCT
iajs-2911	221	1	therefore	therefore	ADV
iajs-2911	221	2	,	,	PUNCT
iajs-2911	221	3	d	d	PROPN
iajs-2911	221	4	∉	∉	PROPN
iajs-2911	221	5	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	221	6	(	(	PUNCT
iajs-2911	221	7	ad	ad	NOUN
iajs-2911	221	8	ℑ	ℑ	PROPN
iajs-2911	221	9	)	)	PUNCT
iajs-2911	221	10	by	by	ADP
iajs-2911	221	11	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	221	12	is	be	AUX
iajs-2911	221	13	u.p	u.p	PROPN
iajs-2911	221	14	.	.	PROPN
iajs-2911	222	1	(	(	PUNCT
iajs-2911	222	2	resp	resp	PROPN
iajs-2911	222	3	.	.	PUNCT
iajs-2911	222	4	,	,	PUNCT
iajs-2911	222	5	l.p	l.p	PROPN
iajs-2911	222	6	.	.	PROPN
iajs-2911	222	7	)	)	PUNCT
iajs-2911	222	8	,	,	PUNCT
iajs-2911	222	9	by	by	ADP
iajs-2911	222	10	theorem	theorem	NOUN
iajs-2911	222	11	[	[	X
iajs-2911	222	12	(	(	PUNCT
iajs-2911	222	13	2.1	2.1	NUM
iajs-2911	222	14	.	.	PUNCT
iajs-2911	222	15	)	)	PUNCT
iajs-2911	223	1	(	(	PUNCT
iajs-2911	223	2	a	a	X
iajs-2911	223	3	)	)	PUNCT
iajs-2911	223	4	⇒	⇒	NOUN
iajs-2911	223	5	c	c	PROPN
iajs-2911	223	6	)	)	PUNCT
iajs-2911	223	7	]	]	PUNCT
iajs-2911	223	8	,	,	PUNCT
iajs-2911	223	9	d	d	PROPN
iajs-2911	223	10	∉	∉	PROPN
iajs-2911	223	11	ad𝑋𝐸	ad𝑋𝐸	VERB
iajs-2911	223	12	(	(	PUNCT
iajs-2911	223	13	ℑ	ℑ	PROPN
iajs-2911	223	14	)	)	PUNCT
iajs-2911	223	15	.	.	PUNCT
iajs-2911	224	1	thus	thus	ADV
iajs-2911	224	2	,	,	PUNCT
iajs-2911	224	3	∃	∃	PROPN
iajs-2911	224	4	an	an	DET
iajs-2911	224	5	𝔽∈	𝔽∈	PROPN
iajs-2911	224	6	ℑ	ℑ	PROPN
iajs-2911	224	7	such	such	ADJ
iajs-2911	224	8	that	that	SCONJ
iajs-2911	224	9	d	d	PROPN
iajs-2911	224	10	∉	∉	X
iajs-2911	224	11	ad𝑋𝐸	ad𝑋𝐸	VERB
iajs-2911	224	12	(	(	PUNCT
iajs-2911	224	13	𝔽).∃an	𝔽).∃an	NOUN
iajs-2911	224	14	𝜌−open	𝜌−open	X
iajs-2911	224	15	a	a	DET
iajs-2911	224	16	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	224	17	v	v	NOUN
iajs-2911	224	18	of	of	ADP
iajs-2911	224	19	d	d	PROPN
iajs-2911	224	20	such	such	ADJ
iajs-2911	224	21	that	that	DET
iajs-2911	224	22	cl(v)∩	cl(v)∩	NOUN
iajs-2911	224	23	𝑋𝐸	𝑋𝐸	NOUN
iajs-2911	224	24	(	(	PUNCT
iajs-2911	224	25	𝔽	𝔽	PROPN
iajs-2911	224	26	)	)	PUNCT
iajs-2911	224	27	=	=	PUNCT
iajs-2911	224	28	∅.	∅.	NOUN
iajs-2911	224	29	since	since	SCONJ
iajs-2911	224	30	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	224	31	is	be	AUX
iajs-2911	224	32	cont	cont	ADJ
iajs-2911	224	33	.	.	PUNCT
iajs-2911	224	34	,	,	PUNCT
iajs-2911	224	35	for	for	ADP
iajs-2911	224	36	every	every	DET
iajs-2911	224	37	e	e	NOUN
iajs-2911	224	38	∈𝐸𝑑	∈𝐸𝑑	PROPN
iajs-2911	224	39	+	+	ADJ
iajs-2911	224	40	(	(	PUNCT
iajs-2911	224	41	resp	resp	NOUN
iajs-2911	224	42	.	.	PUNCT
iajs-2911	224	43	,	,	PUNCT
iajs-2911	225	1	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	225	2	−	−	NOUN
iajs-2911	225	3	)	)	PUNCT
iajs-2911	225	4	.	.	PUNCT
iajs-2911	226	1	we	we	PRON
iajs-2911	226	2	shall	shall	AUX
iajs-2911	226	3	get	get	VERB
iajs-2911	226	4	a	a	DET
iajs-2911	226	5	𝜏-open	𝜏-open	NOUN
iajs-2911	226	6	a	a	DET
iajs-2911	226	7	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	226	8	𝔘e	𝔘e	PROPN
iajs-2911	226	9	of	of	ADP
iajs-2911	226	10	e	e	NOUN
iajs-2911	226	11	such	such	ADJ
iajs-2911	226	12	that	that	DET
iajs-2911	226	13	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	226	14	(	(	PUNCT
iajs-2911	226	15	cl(𝔘e	cl(𝔘e	NOUN
iajs-2911	226	16	)	)	PUNCT
iajs-2911	226	17	)	)	PUNCT
iajs-2911	227	1	⊂	⊂	PROPN
iajs-2911	227	2	cl(v	cl(v	X
iajs-2911	227	3	)	)	PUNCT
iajs-2911	227	4	⊂	⊂	X
iajs-2911	228	1	d	d	X
iajs-2911	228	2	−	−	X
iajs-2911	228	3	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	228	4	(	(	PUNCT
iajs-2911	228	5	𝔽	𝔽	PROPN
iajs-2911	228	6	)	)	PUNCT
iajs-2911	228	7	.	.	PUNCT
iajs-2911	229	1	so	so	ADV
iajs-2911	229	2	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	229	3	(	(	PUNCT
iajs-2911	229	4	cl(𝔘e	cl(𝔘e	NOUN
iajs-2911	229	5	)	)	PUNCT
iajs-2911	229	6	)	)	PUNCT
iajs-2911	229	7	∩	∩	ADJ
iajs-2911	229	8	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	229	9	(	(	PUNCT
iajs-2911	229	10	𝔽	𝔽	PROPN
iajs-2911	229	11	)	)	PUNCT
iajs-2911	229	12	=	=	NOUN
iajs-2911	229	13	∅	∅	NOUN
iajs-2911	229	14	,	,	PUNCT
iajs-2911	229	15	so	so	SCONJ
iajs-2911	229	16	that	that	SCONJ
iajs-2911	229	17	cl(𝔘e	cl(𝔘e	NOUN
iajs-2911	229	18	)	)	PUNCT
iajs-2911	229	19	)	)	PUNCT
iajs-2911	229	20	∩	∩	NOUN
iajs-2911	229	21	𝔽	𝔽	PROPN
iajs-2911	229	22	=	=	PUNCT
iajs-2911	229	23	∅.	∅.	PROPN
iajs-2911	229	24	then	then	ADV
iajs-2911	229	25	h	h	PROPN
iajs-2911	229	26	∉	∉	PROPN
iajs-2911	229	27	cl(𝔽	cl(𝔽	PROPN
iajs-2911	229	28	)	)	PUNCT
iajs-2911	229	29	,	,	PUNCT
iajs-2911	229	30	for	for	ADP
iajs-2911	229	31	every	every	DET
iajs-2911	229	32	e	e	NOUN
iajs-2911	229	33	∈	∈	PROPN
iajs-2911	229	34	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	229	35	+	+	ADJ
iajs-2911	229	36	(	(	PUNCT
iajs-2911	229	37	resp	resp	NOUN
iajs-2911	229	38	.	.	PUNCT
iajs-2911	229	39	,	,	PUNCT
iajs-2911	230	1	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	230	2	−	−	NOUN
iajs-2911	230	3	)	)	PUNCT
iajs-2911	230	4	,	,	PUNCT
iajs-2911	230	5	so	so	ADV
iajs-2911	230	6	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	230	7	+	+	ADJ
iajs-2911	230	8	(	(	PUNCT
iajs-2911	230	9	resp	resp	NOUN
iajs-2911	230	10	.	.	PUNCT
iajs-2911	230	11	,	,	PUNCT
iajs-2911	231	1	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	231	2	−	−	NOUN
iajs-2911	231	3	)	)	PUNCT
iajs-2911	231	4	∩	∩	ADJ
iajs-2911	231	5	cl(𝔽	cl(𝔽	NOUN
iajs-2911	231	6	)	)	PUNCT
iajs-2911	231	7	=	=	NOUN
iajs-2911	231	8	∅	∅	NOUN
iajs-2911	231	9	,	,	PUNCT
iajs-2911	231	10	so	so	ADV
iajs-2911	231	11	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	231	12	+	+	ADJ
iajs-2911	231	13	(	(	PUNCT
iajs-2911	231	14	resp	resp	NOUN
iajs-2911	231	15	.	.	PUNCT
iajs-2911	231	16	,	,	PUNCT
iajs-2911	231	17	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	231	18	−	−	NOUN
iajs-2911	231	19	)	)	PUNCT
iajs-2911	231	20	is	be	AUX
iajs-2911	231	21	u.r.(resp	u.r.(resp	PRON
iajs-2911	231	22	.	.	PROPN
iajs-2911	231	23	,	,	PUNCT
iajs-2911	231	24	l.r	l.r	PROPN
iajs-2911	231	25	.	.	PUNCT
iajs-2911	231	26	)	)	PUNCT
iajs-2911	232	1	in	in	ADP
iajs-2911	232	2	e.	e.	PROPN
iajs-2911	232	3	corollary	corollary	PROPN
iajs-2911	232	4	3.2	3.2	NUM
iajs-2911	232	5	.	.	PUNCT
iajs-2911	233	1	if	if	SCONJ
iajs-2911	233	2	the	the	DET
iajs-2911	233	3	𝔽.	𝔽.	PROPN
iajs-2911	233	4	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	233	5	.	.	PUNCT
iajs-2911	233	6	(	(	PUNCT
iajs-2911	233	7	e,𝜏	e,𝜏	NOUN
iajs-2911	233	8	)	)	PUNCT
iajs-2911	233	9	on	on	ADP
iajs-2911	233	10	(	(	PUNCT
iajs-2911	233	11	d,𝜌	d,𝜌	NOUN
iajs-2911	233	12	)	)	PUNCT
iajs-2911	233	13	is	be	AUX
iajs-2911	233	14	m.p	m.p	PROPN
iajs-2911	233	15	.	.	PROPN
iajs-2911	234	1	then	then	ADV
iajs-2911	234	2	it	it	PRON
iajs-2911	234	3	is	be	AUX
iajs-2911	234	4	closed	closed	ADJ
iajs-2911	234	5	and	and	CCONJ
iajs-2911	234	6	for	for	ADP
iajs-2911	234	7	every	every	DET
iajs-2911	234	8	d	d	PROPN
iajs-2911	234	9	∈	∈	PROPN
iajs-2911	234	10	b	b	PROPN
iajs-2911	234	11	,	,	PUNCT
iajs-2911	234	12	ed	ed	PROPN
iajs-2911	234	13	is	be	AUX
iajs-2911	234	14	m.r	m.r	PROPN
iajs-2911	234	15	.	.	PROPN
iajs-2911	235	1	in	in	ADP
iajs-2911	235	2	e.	e.	PROPN
iajs-2911	235	3	definition	definition	NOUN
iajs-2911	235	4	3.3	3.3	NUM
iajs-2911	235	5	.	.	PUNCT
iajs-2911	236	1	the	the	DET
iajs-2911	236	2	function	function	NOUN
iajs-2911	236	3	ω	ω	NOUN
iajs-2911	236	4	:	:	PUNCT
iajs-2911	236	5	(	(	PUNCT
iajs-2911	236	6	e,𝜏	e,𝜏	NOUN
iajs-2911	236	7	)	)	PUNCT
iajs-2911	236	8	→	→	SYM
iajs-2911	236	9	(	(	PUNCT
iajs-2911	236	10	f,𝜎	f,𝜎	PROPN
iajs-2911	236	11	)	)	PUNCT
iajs-2911	236	12	is	be	AUX
iajs-2911	236	13	named	name	VERB
iajs-2911	236	14	to	to	PART
iajs-2911	236	15	be	be	AUX
iajs-2911	236	16	weakly	weakly	ADV
iajs-2911	236	17	upper	upper	ADJ
iajs-2911	236	18	closed	closed	NOUN
iajs-2911	236	19	(	(	PUNCT
iajs-2911	236	20	briefly	briefly	ADV
iajs-2911	236	21	,	,	PUNCT
iajs-2911	236	22	w.u	w.u	PROPN
iajs-2911	236	23	.	.	PROPN
iajs-2911	236	24	closed	close	VERB
iajs-2911	236	25	)	)	PUNCT
iajs-2911	236	26	if	if	SCONJ
iajs-2911	236	27	∀f	∀f	NUM
iajs-2911	236	28	∈	∈	NOUN
iajs-2911	236	29	ω+(e	ω+(e	NUM
iajs-2911	236	30	)	)	PUNCT
iajs-2911	236	31	and	and	CCONJ
iajs-2911	236	32	∀	∀	NUM
iajs-2911	236	33	𝔘	𝔘	NOUN
iajs-2911	236	34	∈	∈	NOUN
iajs-2911	236	35	𝜏	𝜏	NOUN
iajs-2911	236	36	containing	contain	VERB
iajs-2911	236	37	η−1(f	η−1(f	PROPN
iajs-2911	236	38	)	)	PUNCT
iajs-2911	236	39	in	in	ADP
iajs-2911	236	40	e,∃	e,∃	VERB
iajs-2911	236	41	a	a	DET
iajs-2911	236	42	𝜌	𝜌	X
iajs-2911	236	43	−open	−open	VERB
iajs-2911	236	44	a	a	DET
iajs-2911	236	45	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	236	46	v	v	NOUN
iajs-2911	236	47	of	of	ADP
iajs-2911	236	48	d	d	PROPN
iajs-2911	236	49	such	such	ADJ
iajs-2911	236	50	that	that	DET
iajs-2911	236	51	ω−1(v	ω−1(v	PROPN
iajs-2911	236	52	)	)	PUNCT
iajs-2911	236	53	⊂	⊂	PROPN
iajs-2911	236	54	cl(𝔘	cl(𝔘	NOUN
iajs-2911	236	55	)	)	PUNCT
iajs-2911	236	56	.	.	PUNCT
iajs-2911	237	1	definition	definition	NOUN
iajs-2911	237	2	3.4	3.4	NUM
iajs-2911	237	3	.	.	PUNCT
iajs-2911	238	1	the	the	DET
iajs-2911	238	2	function	function	NOUN
iajs-2911	238	3	ω	ω	NOUN
iajs-2911	238	4	:	:	PUNCT
iajs-2911	238	5	(	(	PUNCT
iajs-2911	238	6	e,𝜏	e,𝜏	NOUN
iajs-2911	238	7	)	)	PUNCT
iajs-2911	238	8	→	→	SYM
iajs-2911	238	9	(	(	PUNCT
iajs-2911	238	10	f,𝜎	f,𝜎	PROPN
iajs-2911	238	11	)	)	PUNCT
iajs-2911	238	12	is	be	AUX
iajs-2911	238	13	named	name	VERB
iajs-2911	238	14	to	to	PART
iajs-2911	238	15	be	be	AUX
iajs-2911	238	16	weakly	weakly	ADV
iajs-2911	238	17	lower	lower	ADV
iajs-2911	238	18	closed	closed	ADJ
iajs-2911	238	19	(	(	PUNCT
iajs-2911	238	20	briefly	briefly	ADV
iajs-2911	238	21	,	,	PUNCT
iajs-2911	238	22	w.l	w.l	PROPN
iajs-2911	238	23	.	.	PROPN
iajs-2911	238	24	closed	close	VERB
iajs-2911	238	25	)	)	PUNCT
iajs-2911	238	26	if	if	SCONJ
iajs-2911	238	27	∀f	∀f	NUM
iajs-2911	238	28	∈	∈	PROPN
iajs-2911	238	29	ω−(e	ω−(e	PROPN
iajs-2911	238	30	)	)	PUNCT
iajs-2911	238	31	and	and	CCONJ
iajs-2911	238	32	∀	∀	NUM
iajs-2911	238	33	𝔘	𝔘	NOUN
iajs-2911	238	34	∈	∈	NOUN
iajs-2911	238	35	𝜏	𝜏	VERB
iajs-2911	238	36	containing	contain	VERB
iajs-2911	238	37	ω−1(f	ω−1(f	PROPN
iajs-2911	238	38	)	)	PUNCT
iajs-2911	238	39	in	in	ADP
iajs-2911	238	40	e,∃	e,∃	VERB
iajs-2911	238	41	a	a	DET
iajs-2911	238	42	𝜌	𝜌	X
iajs-2911	238	43	−open	−open	VERB
iajs-2911	238	44	a	a	DET
iajs-2911	238	45	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	238	46	v	v	NOUN
iajs-2911	238	47	𝑜f	𝑜f	NOUN
iajs-2911	238	48	d	d	PROPN
iajs-2911	238	49	such	such	ADJ
iajs-2911	238	50	that	that	DET
iajs-2911	238	51	ω−1(v	ω−1(v	PROPN
iajs-2911	238	52	)	)	PUNCT
iajs-2911	238	53	⊂	⊂	PROPN
iajs-2911	238	54	cl(𝔘	cl(𝔘	NOUN
iajs-2911	238	55	)	)	PUNCT
iajs-2911	238	56	.	.	PUNCT
iajs-2911	239	1	the	the	DET
iajs-2911	239	2	function	function	NOUN
iajs-2911	239	3	ω	ω	NOUN
iajs-2911	239	4	:	:	PUNCT
iajs-2911	239	5	(	(	PUNCT
iajs-2911	239	6	e,𝜏	e,𝜏	NOUN
iajs-2911	239	7	)	)	PUNCT
iajs-2911	239	8	→	→	SYM
iajs-2911	239	9	(	(	PUNCT
iajs-2911	239	10	f,𝜎	f,𝜎	PROPN
iajs-2911	239	11	)	)	PUNCT
iajs-2911	239	12	is	be	AUX
iajs-2911	239	13	named	name	VERB
iajs-2911	239	14	to	to	PART
iajs-2911	239	15	be	be	AUX
iajs-2911	239	16	weakly	weakly	ADJ
iajs-2911	239	17	multi	multi	ADJ
iajs-2911	239	18	-	-	ADJ
iajs-2911	239	19	closed	closed	ADJ
iajs-2911	239	20	(	(	PUNCT
iajs-2911	239	21	briefly	briefly	ADV
iajs-2911	239	22	,	,	PUNCT
iajs-2911	239	23	w.m	w.m	PROPN
iajs-2911	239	24	.	.	PROPN
iajs-2911	239	25	closed	close	VERB
iajs-2911	239	26	)	)	PUNCT
iajs-2911	239	27	if	if	SCONJ
iajs-2911	239	28	it	it	PRON
iajs-2911	239	29	is	be	AUX
iajs-2911	239	30	w.u	w.u	PROPN
iajs-2911	239	31	.	.	PROPN
iajs-2911	239	32	closed	closed	PROPN
iajs-2911	239	33	and	and	CCONJ
iajs-2911	239	34	w.l	w.l	PROPN
iajs-2911	239	35	.	.	PROPN
iajs-2911	239	36	closed	closed	PROPN
iajs-2911	239	37	.	.	PUNCT
iajs-2911	240	1	definition	definition	NOUN
iajs-2911	240	2	3.5	3.5	NUM
iajs-2911	240	3	.	.	PUNCT
iajs-2911	241	1	the	the	DET
iajs-2911	241	2	𝔽.	𝔽.	PROPN
iajs-2911	241	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	241	4	.	.	PUNCT
iajs-2911	241	5	(	(	PUNCT
iajs-2911	241	6	e,𝜏	e,𝜏	NOUN
iajs-2911	241	7	)	)	PUNCT
iajs-2911	241	8	on	on	ADP
iajs-2911	241	9	(	(	PUNCT
iajs-2911	241	10	d,𝜌	d,𝜌	NOUN
iajs-2911	241	11	)	)	PUNCT
iajs-2911	241	12	is	be	AUX
iajs-2911	241	13	named	name	VERB
iajs-2911	241	14	to	to	PART
iajs-2911	241	15	be	be	AUX
iajs-2911	241	16	𝔽.	𝔽.	PROPN
iajs-2911	241	17	𝕎.	𝕎.	PUNCT
iajs-2911	241	18	upper	upper	ADJ
iajs-2911	241	19	weakly	weakly	ADJ
iajs-2911	241	20	closed	closed	ADJ
iajs-2911	241	21	(	(	PUNCT
iajs-2911	241	22	briefly	briefly	ADV
iajs-2911	241	23	,	,	PUNCT
iajs-2911	241	24	𝔽.	𝔽.	PROPN
iajs-2911	241	25	𝕎.u.w	𝕎.u.w	PROPN
iajs-2911	241	26	.	.	PUNCT
iajs-2911	241	27	closed	close	VERB
iajs-2911	241	28	)	)	PUNCT
iajs-2911	241	29	if	if	SCONJ
iajs-2911	241	30	the	the	DET
iajs-2911	241	31	projection	projection	NOUN
iajs-2911	241	32	x	x	X
iajs-2911	241	33	is	be	AUX
iajs-2911	241	34	w.u	w.u	PROPN
iajs-2911	241	35	.	.	PROPN
iajs-2911	241	36	closed	closed	PROPN
iajs-2911	241	37	.	.	PUNCT
iajs-2911	242	1	definition	definition	NOUN
iajs-2911	242	2	3.6	3.6	NUM
iajs-2911	242	3	.	.	PUNCT
iajs-2911	243	1	the	the	DET
iajs-2911	243	2	𝔽.	𝔽.	PROPN
iajs-2911	243	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	243	4	.	.	PUNCT
iajs-2911	243	5	(	(	PUNCT
iajs-2911	243	6	e,𝜏	e,𝜏	NOUN
iajs-2911	243	7	)	)	PUNCT
iajs-2911	243	8	on	on	ADP
iajs-2911	243	9	(	(	PUNCT
iajs-2911	243	10	d,𝜌	d,𝜌	NOUN
iajs-2911	243	11	)	)	PUNCT
iajs-2911	243	12	is	be	AUX
iajs-2911	243	13	named	name	VERB
iajs-2911	243	14	to	to	PART
iajs-2911	243	15	be	be	AUX
iajs-2911	243	16	𝔽.	𝔽.	PROPN
iajs-2911	243	17	𝕎.	𝕎.	NOUN
iajs-2911	243	18	lower	low	ADJ
iajs-2911	243	19	weakly	weakly	ADV
iajs-2911	243	20	closed	closed	ADJ
iajs-2911	243	21	(	(	PUNCT
iajs-2911	243	22	briefly	briefly	ADV
iajs-2911	243	23	,	,	PUNCT
iajs-2911	243	24	𝔽.	𝔽.	PROPN
iajs-2911	243	25	𝕎.l.w	𝕎.l.w	PROPN
iajs-2911	243	26	.	.	PROPN
iajs-2911	243	27	closed	close	VERB
iajs-2911	243	28	)	)	PUNCT
iajs-2911	243	29	if	if	SCONJ
iajs-2911	243	30	the	the	DET
iajs-2911	243	31	projection	projection	NOUN
iajs-2911	243	32	x	x	X
iajs-2911	243	33	is	be	AUX
iajs-2911	243	34	w.l	w.l	PROPN
iajs-2911	243	35	.	.	PROPN
iajs-2911	243	36	closed	close	VERB
iajs-2911	243	37	.	.	PUNCT
iajs-2911	244	1	the	the	DET
iajs-2911	244	2	𝔽.	𝔽.	PROPN
iajs-2911	244	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	244	4	.	.	PUNCT
iajs-2911	244	5	(	(	PUNCT
iajs-2911	244	6	e,𝜏	e,𝜏	NOUN
iajs-2911	244	7	)	)	PUNCT
iajs-2911	244	8	on	on	ADP
iajs-2911	244	9	(	(	PUNCT
iajs-2911	244	10	d,𝜌	d,𝜌	NOUN
iajs-2911	244	11	)	)	PUNCT
iajs-2911	244	12	is	be	AUX
iajs-2911	244	13	named	name	VERB
iajs-2911	244	14	to	to	PART
iajs-2911	244	15	be	be	AUX
iajs-2911	244	16	𝔽.	𝔽.	PROPN
iajs-2911	244	17	𝕎.	𝕎.	PUNCT
iajs-2911	244	18	multi	multi	ADJ
iajs-2911	244	19	-	-	ADJ
iajs-2911	244	20	weakly	weakly	ADJ
iajs-2911	244	21	closed	closed	ADJ
iajs-2911	244	22	(	(	PUNCT
iajs-2911	244	23	briefly	briefly	ADV
iajs-2911	244	24	,	,	PUNCT
iajs-2911	244	25	𝔽.	𝔽.	PROPN
iajs-2911	244	26	𝕎.m.w	𝕎.m.w	PROPN
iajs-2911	244	27	.	.	PROPN
iajs-2911	244	28	closed	closed	ADJ
iajs-2911	244	29	)	)	PUNCT
iajs-2911	244	30	if	if	SCONJ
iajs-2911	244	31	it	it	PRON
iajs-2911	244	32	is	be	AUX
iajs-2911	244	33	𝔽.	𝔽.	PROPN
iajs-2911	244	34	𝕎.u.w	𝕎.u.w	PROPN
iajs-2911	244	35	.	.	PUNCT
iajs-2911	244	36	closed	close	VERB
iajs-2911	244	37	and	and	CCONJ
iajs-2911	244	38	𝔽.	𝔽.	PROPN
iajs-2911	244	39	𝕎.u.w	𝕎.u.w	PROPN
iajs-2911	244	40	.	.	PUNCT
iajs-2911	244	41	closed	close	VERB
iajs-2911	244	42	.	.	PUNCT
iajs-2911	245	1	theorem	theorem	VERB
iajs-2911	245	2	3.3	3.3	NUM
iajs-2911	245	3	.	.	PUNCT
iajs-2911	246	1	the	the	DET
iajs-2911	246	2	𝔽.	𝔽.	PROPN
iajs-2911	246	3	𝕎.	𝕎.	PROPN
iajs-2911	246	4	closed	close	VERB
iajs-2911	246	5	topological	topological	ADJ
iajs-2911	246	6	space	space	NOUN
iajs-2911	246	7	(	(	PUNCT
iajs-2911	246	8	e,𝜏	e,𝜏	NOUN
iajs-2911	246	9	)	)	PUNCT
iajs-2911	246	10	on	on	ADP
iajs-2911	246	11	(	(	PUNCT
iajs-2911	246	12	d,𝜌	d,𝜌	NOUN
iajs-2911	246	13	)	)	PUNCT
iajs-2911	246	14	is	be	AUX
iajs-2911	247	1	w.u	w.u	PROPN
iajs-2911	247	2	.	.	PROPN
iajs-2911	247	3	closed	closed	PROPN
iajs-2911	247	4	(	(	PUNCT
iajs-2911	247	5	resp	resp	NOUN
iajs-2911	247	6	.	.	PUNCT
iajs-2911	247	7	,	,	PUNCT
iajs-2911	247	8	w.l	w.l	PROPN
iajs-2911	247	9	.	.	PROPN
iajs-2911	247	10	closed	closed	ADJ
iajs-2911	247	11	)	)	PUNCT
iajs-2911	247	12	.	.	PUNCT
iajs-2911	248	1	proof	proof	NOUN
iajs-2911	248	2	.	.	PUNCT
iajs-2911	249	1	assume	assume	VERB
iajs-2911	249	2	that	that	SCONJ
iajs-2911	249	3	e	e	NOUN
iajs-2911	249	4	is	be	AUX
iajs-2911	249	5	a	a	DET
iajs-2911	249	6	𝔽.	𝔽.	PROPN
iajs-2911	249	7	𝕎.	𝕎.	PROPN
iajs-2911	249	8	closed	close	VERB
iajs-2911	249	9	topological	topological	ADJ
iajs-2911	249	10	space	space	NOUN
iajs-2911	249	11	on	on	ADP
iajs-2911	249	12	d	d	PROPN
iajs-2911	249	13	,	,	PUNCT
iajs-2911	249	14	then	then	ADV
iajs-2911	249	15	the	the	DET
iajs-2911	249	16	projection	projection	NOUN
iajs-2911	249	17	xe	xe	PROPN
iajs-2911	249	18	:	:	PUNCT
iajs-2911	249	19	e→	e→	PROPN
iajs-2911	249	20	d	d	PROPN
iajs-2911	249	21	exists	exist	VERB
iajs-2911	249	22	,	,	PUNCT
iajs-2911	249	23	and	and	CCONJ
iajs-2911	249	24	to	to	PART
iajs-2911	249	25	prove	prove	VERB
iajs-2911	249	26	its	its	PRON
iajs-2911	249	27	w.u	w.u	PROPN
iajs-2911	249	28	.	.	PROPN
iajs-2911	249	29	closed	closed	PROPN
iajs-2911	249	30	(	(	PUNCT
iajs-2911	249	31	resp	resp	NOUN
iajs-2911	249	32	.	.	PUNCT
iajs-2911	249	33	,	,	PUNCT
iajs-2911	249	34	w.l	w.l	PROPN
iajs-2911	249	35	.	.	PROPN
iajs-2911	249	36	closed	close	VERB
iajs-2911	249	37	)	)	PUNCT
iajs-2911	249	38	.	.	PUNCT
iajs-2911	250	1	let	let	VERB
iajs-2911	250	2	d	d	X
iajs-2911	250	3	∈	∈	PROPN
iajs-2911	250	4	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	250	5	and	and	CCONJ
iajs-2911	250	6	le𝑡	le𝑡	NOUN
iajs-2911	250	7	𝔘	𝔘	PROPN
iajs-2911	250	8	∈	∈	PROPN
iajs-2911	250	9	𝜏	𝜏	NOUN
iajs-2911	250	10	containing	contain	VERB
iajs-2911	250	11	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	250	12	+	+	PROPN
iajs-2911	250	13	(	(	PUNCT
iajs-2911	250	14	resp	resp	NOUN
iajs-2911	250	15	.	.	PUNCT
iajs-2911	251	1	,	,	PUNCT
iajs-2911	251	2	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	251	3	−	−	NOUN
iajs-2911	251	4	)	)	PUNCT
iajs-2911	251	5	in	in	ADP
iajs-2911	251	6	e.	e.	PROPN
iajs-2911	251	7	currently	currently	ADV
iajs-2911	251	8	,	,	PUNCT
iajs-2911	251	9	by	by	ADP
iajs-2911	251	10	theorem	theorem	NOUN
iajs-2911	251	11	(	(	PUNCT
iajs-2911	251	12	4.1.18	4.1.18	NUM
iajs-2911	251	13	.	.	PUNCT
iajs-2911	251	14	)	)	PUNCT
iajs-2911	251	15	cl(e−cl(𝔘	cl(e−cl(𝔘	NOUN
iajs-2911	251	16	)	)	PUNCT
iajs-2911	251	17	)	)	PUNCT
iajs-2911	252	1	=	=	PUNCT
iajs-2911	252	2	cl(e−cl(𝔘	cl(e−cl(𝔘	NOUN
iajs-2911	252	3	)	)	PUNCT
iajs-2911	252	4	)	)	PUNCT
iajs-2911	252	5	,	,	PUNCT
iajs-2911	252	6	and	and	CCONJ
iajs-2911	252	7	,	,	PUNCT
iajs-2911	252	8	hence	hence	ADV
iajs-2911	252	9	by	by	ADP
iajs-2911	252	10	lemma	lemma	PROPN
iajs-2911	252	11	,	,	PUNCT
iajs-2911	252	12	(	(	PUNCT
iajs-2911	252	13	4.1.11	4.1.11	NUM
iajs-2911	252	14	.	.	PUNCT
iajs-2911	252	15	)	)	PUNCT
iajs-2911	253	1	and	and	CCONJ
iajs-2911	253	2	since	since	SCONJ
iajs-2911	253	3	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	253	4	is	be	AUX
iajs-2911	253	5	closed	closed	ADJ
iajs-2911	253	6	,	,	PUNCT
iajs-2911	253	7	we	we	PRON
iajs-2911	253	8	have	have	VERB
iajs-2911	253	9	cl(𝑋𝐸	cl(𝑋𝐸	NOUN
iajs-2911	253	10	(	(	PUNCT
iajs-2911	253	11	e−cl(𝔘	e−cl(𝔘	PROPN
iajs-2911	253	12	)	)	PUNCT
iajs-2911	253	13	)	)	PUNCT
iajs-2911	253	14	)	)	PUNCT
iajs-2911	254	1	⊂	⊂	PRON
iajs-2911	254	2	𝑋𝐸	𝑋𝐸	VERB
iajs-2911	255	1	[	[	X
iajs-2911	255	2	cl(e−cl(𝔘	cl(e−cl(𝔘	NOUN
iajs-2911	255	3	)	)	PUNCT
iajs-2911	255	4	)	)	PUNCT
iajs-2911	256	1	]	]	PUNCT
iajs-2911	256	2	..	..	PUNCT
iajs-2911	256	3	currently	currently	ADV
iajs-2911	256	4	,	,	PUNCT
iajs-2911	256	5	since	since	SCONJ
iajs-2911	256	6	d	d	PROPN
iajs-2911	256	7	∉	∉	PROPN
iajs-2911	256	8	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	256	9	[	[	X
iajs-2911	256	10	cl(e−cl(𝔘	cl(e−cl(𝔘	NOUN
iajs-2911	256	11	)	)	PUNCT
iajs-2911	256	12	)	)	PUNCT
iajs-2911	256	13	]	]	PUNCT
iajs-2911	256	14	,	,	PUNCT
iajs-2911	256	15	d	d	PROPN
iajs-2911	256	16	∉	∉	PROPN
iajs-2911	256	17	cl(𝑋𝐸	cl(𝑋𝐸	PROPN
iajs-2911	256	18	(	(	PUNCT
iajs-2911	256	19	e−cl(u	e−cl(u	X
iajs-2911	256	20	)	)	PUNCT
iajs-2911	256	21	)	)	PUNCT
iajs-2911	256	22	)	)	PUNCT
iajs-2911	256	23	,	,	PUNCT
iajs-2911	256	24	and	and	CCONJ
iajs-2911	256	25	thus	thus	ADV
iajs-2911	256	26	,	,	PUNCT
iajs-2911	256	27	∃an	∃an	ADV
iajs-2911	256	28	𝜌−open	𝜌−open	PUNCT
iajs-2911	256	29	a	a	DET
iajs-2911	256	30	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	256	31	v	v	NOUN
iajs-2911	256	32	of	of	ADP
iajs-2911	256	33	d	d	PROPN
iajs-2911	256	34	∈	∈	PROPN
iajs-2911	256	35	d	d	ADP
iajs-2911	256	36	such	such	ADJ
iajs-2911	256	37	that	that	DET
iajs-2911	256	38	cl(v)∩	cl(v)∩	NOUN
iajs-2911	256	39	𝑋𝐸	𝑋𝐸	NOUN
iajs-2911	256	40	(	(	PUNCT
iajs-2911	256	41	e−cl(𝔘	e−cl(𝔘	NOUN
iajs-2911	256	42	)	)	PUNCT
iajs-2911	256	43	)	)	PUNCT
iajs-2911	257	1	=	=	PUNCT
iajs-2911	257	2	∅	∅	NOUN
iajs-2911	257	3	which	which	PRON
iajs-2911	257	4	means	mean	VERB
iajs-2911	257	5	that	that	SCONJ
iajs-2911	257	6	𝐸𝑐𝑙(𝑉	𝐸𝑐𝑙(𝑉	NOUN
iajs-2911	257	7	)	)	PUNCT
iajs-2911	258	1	+	+	NUM
iajs-2911	258	2	∩(e−cl(𝔘	∩(e−cl(𝔘	NOUN
iajs-2911	258	3	)	)	PUNCT
iajs-2911	258	4	)	)	PUNCT
iajs-2911	259	1	=	=	PUNCT
iajs-2911	259	2	∅(resp	∅(resp	PROPN
iajs-2911	259	3	.	.	PROPN
iajs-2911	259	4	,	,	PUNCT
iajs-2911	259	5	𝐸𝑐𝑙(𝑉	𝐸𝑐𝑙(𝑉	NOUN
iajs-2911	259	6	)	)	PUNCT
iajs-2911	259	7	−	−	ADP
iajs-2911	259	8	∩(e−cl(𝔘	∩(e−cl(𝔘	NOUN
iajs-2911	259	9	)	)	PUNCT
iajs-2911	259	10	)	)	PUNCT
iajs-2911	260	1	=	=	NOUN
iajs-2911	260	2	∅	∅	NOUN
iajs-2911	260	3	)	)	PUNCT
iajs-2911	260	4	,	,	PUNCT
iajs-2911	260	5	and	and	CCONJ
iajs-2911	260	6	so	so	ADV
iajs-2911	260	7	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	260	8	is	be	AUX
iajs-2911	260	9	w.u	w.u	PROPN
iajs-2911	260	10	.	.	PROPN
iajs-2911	260	11	closed	closed	PROPN
iajs-2911	260	12	(	(	PUNCT
iajs-2911	260	13	resp	resp	NOUN
iajs-2911	260	14	.	.	PUNCT
iajs-2911	260	15	,	,	PUNCT
iajs-2911	260	16	w.l	w.l	PROPN
iajs-2911	260	17	.	.	PROPN
iajs-2911	260	18	closed	closed	ADJ
iajs-2911	260	19	)	)	PUNCT
iajs-2911	260	20	.	.	PUNCT
iajs-2911	261	1	corollary	corollary	ADJ
iajs-2911	261	2	3.3	3.3	NUM
iajs-2911	261	3	.	.	PUNCT
iajs-2911	262	1	the	the	DET
iajs-2911	262	2	𝔽.	𝔽.	PROPN
iajs-2911	262	3	𝕎.	𝕎.	PROPN
iajs-2911	262	4	closed	close	VERB
iajs-2911	262	5	topological	topological	ADJ
iajs-2911	262	6	space	space	NOUN
iajs-2911	262	7	(	(	PUNCT
iajs-2911	262	8	e	e	NOUN
iajs-2911	262	9	,	,	PUNCT
iajs-2911	262	10	τ	τ	X
iajs-2911	262	11	)	)	PUNCT
iajs-2911	262	12	on	on	ADP
iajs-2911	262	13	(	(	PUNCT
iajs-2911	262	14	d	d	NOUN
iajs-2911	262	15	,	,	PUNCT
iajs-2911	262	16	ρ	ρ	NOUN
iajs-2911	262	17	)	)	PUNCT
iajs-2911	262	18	is	be	AUX
iajs-2911	262	19	w.m	w.m	PROPN
iajs-2911	262	20	.	.	PROPN
iajs-2911	262	21	closed	close	VERB
iajs-2911	262	22	.	.	PUNCT
iajs-2911	263	1	theorem	theorem	VERB
iajs-2911	263	2	3.4	3.4	NUM
iajs-2911	263	3	.	.	PUNCT
iajs-2911	264	1	let	let	VERB
iajs-2911	264	2	(	(	PUNCT
iajs-2911	264	3	e	e	NOUN
iajs-2911	264	4	,	,	PUNCT
iajs-2911	264	5	τ	τ	X
iajs-2911	264	6	)	)	PUNCT
iajs-2911	264	7	be	be	VERB
iajs-2911	264	8	𝔽.	𝔽.	PROPN
iajs-2911	264	9	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	264	10	.	.	PUNCT
iajs-2911	265	1	on	on	ADP
iajs-2911	265	2	(	(	PUNCT
iajs-2911	265	3	d	d	PROPN
iajs-2911	265	4	,	,	PUNCT
iajs-2911	265	5	ρ	ρ	PROPN
iajs-2911	265	6	)	)	PUNCT
iajs-2911	265	7	.	.	PUNCT
iajs-2911	266	1	then	then	ADV
iajs-2911	266	2	(	(	PUNCT
iajs-2911	266	3	e	e	NOUN
iajs-2911	266	4	,	,	PUNCT
iajs-2911	266	5	τ	τ	X
iajs-2911	266	6	)	)	PUNCT
iajs-2911	266	7	is	be	AUX
iajs-2911	266	8	𝔽.	𝔽.	PROPN
iajs-2911	266	9	𝕎.u.p.(resp	𝕎.u.p.(resp	PROPN
iajs-2911	266	10	.	.	PUNCT
iajs-2911	266	11	,	,	PUNCT
iajs-2911	266	12	𝔽.	𝔽.	PROPN
iajs-2911	266	13	𝕎.l.p	𝕎.l.p	PROPN
iajs-2911	266	14	.	.	PUNCT
iajs-2911	266	15	)	)	PUNCT
iajs-2911	266	16	,	,	PUNCT
iajs-2911	266	17	if	if	SCONJ
iajs-2911	266	18	:	:	PUNCT
iajs-2911	266	19	i.	i.	PROPN
iajs-2911	266	20	(	(	PUNCT
iajs-2911	266	21	e,𝜏	e,𝜏	PROPN
iajs-2911	266	22	)	)	PUNCT
iajs-2911	266	23	is	be	AUX
iajs-2911	266	24	𝔽.	𝔽.	PROPN
iajs-2911	266	25	𝕎.u.w	𝕎.u.w	PROPN
iajs-2911	266	26	.	.	PROPN
iajs-2911	266	27	closed	close	VERB
iajs-2911	266	28	(	(	PUNCT
iajs-2911	266	29	resp	resp	NOUN
iajs-2911	266	30	.	.	PUNCT
iajs-2911	266	31	,	,	PUNCT
iajs-2911	267	1	𝔽.	𝔽.	PROPN
iajs-2911	267	2	𝕎.l.w	𝕎.l.w	PROPN
iajs-2911	267	3	.	.	PROPN
iajs-2911	267	4	closed	close	VERB
iajs-2911	267	5	)	)	PUNCT
iajs-2911	267	6	topological	topological	ADJ
iajs-2911	267	7	space	space	NOUN
iajs-2911	267	8	.	.	PUNCT
iajs-2911	268	1	ii	ii	NOUN
iajs-2911	268	2	.	.	PUNCT
iajs-2911	269	1	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	269	2	+	+	ADJ
iajs-2911	269	3	(	(	PUNCT
iajs-2911	269	4	resp	resp	NOUN
iajs-2911	269	5	.	.	PUNCT
iajs-2911	269	6	,	,	PUNCT
iajs-2911	269	7	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	269	8	−	−	NOUN
iajs-2911	269	9	)	)	PUNCT
iajs-2911	269	10	is	be	AUX
iajs-2911	269	11	u.r.(resp	u.r.(resp	PRON
iajs-2911	269	12	.	.	PROPN
iajs-2911	269	13	,	,	PUNCT
iajs-2911	269	14	l.r	l.r	PROPN
iajs-2911	269	15	.	.	PROPN
iajs-2911	269	16	)	)	PUNCT
iajs-2911	269	17	,	,	PUNCT
iajs-2911	269	18	for	for	ADP
iajs-2911	269	19	every	every	DET
iajs-2911	269	20	d	d	PROPN
iajs-2911	269	21	∈	∈	PROPN
iajs-2911	269	22	d.	d.	NOUN
iajs-2911	269	23	proof	proof	NOUN
iajs-2911	269	24	.	.	PUNCT
iajs-2911	270	1	assume	assume	VERB
iajs-2911	270	2	that	that	SCONJ
iajs-2911	270	3	e	e	NOUN
iajs-2911	270	4	is	be	AUX
iajs-2911	270	5	a	a	DET
iajs-2911	270	6	𝔽.	𝔽.	PROPN
iajs-2911	270	7	𝕎.	𝕎.	PROPN
iajs-2911	270	8	space	space	NOUN
iajs-2911	270	9	on	on	ADP
iajs-2911	270	10	d	d	ADP
iajs-2911	270	11	satisfying	satisfy	VERB
iajs-2911	270	12	the	the	DET
iajs-2911	270	13	conditions	condition	NOUN
iajs-2911	270	14	(	(	PUNCT
iajs-2911	270	15	i	i	NOUN
iajs-2911	270	16	)	)	PUNCT
iajs-2911	270	17	and	and	CCONJ
iajs-2911	270	18	(	(	PUNCT
iajs-2911	270	19	ii	ii	NOUN
iajs-2911	270	20	)	)	PUNCT
iajs-2911	270	21	,	,	PUNCT
iajs-2911	270	22	then	then	ADV
iajs-2911	270	23	the	the	DET
iajs-2911	270	24	projection	projection	NOUN
iajs-2911	270	25	xe	xe	PROPN
iajs-2911	270	26	:	:	PUNCT
iajs-2911	270	27	e→	e→	PROPN
iajs-2911	270	28	d	d	PROPN
iajs-2911	270	29	exists	exist	VERB
iajs-2911	270	30	.	.	PUNCT
iajs-2911	271	1	to	to	PART
iajs-2911	271	2	prove	prove	VERB
iajs-2911	271	3	that	that	SCONJ
iajs-2911	271	4	xe	xe	PROPN
iajs-2911	271	5	𝑖s	𝑖s	SYM
iajs-2911	271	6	u.r.(resp	u.r.(resp	PROPN
iajs-2911	271	7	.	.	PROPN
iajs-2911	271	8	,	,	PUNCT
iajs-2911	271	9	l.r	l.r	PROPN
iajs-2911	271	10	.	.	PROPN
iajs-2911	271	11	)	)	PUNCT
iajs-2911	271	12	,	,	PUNCT
iajs-2911	271	13	we	we	PRON
iajs-2911	271	14	have	have	VERB
iajs-2911	271	15	to	to	PART
iajs-2911	271	16	show	show	VERB
iajs-2911	271	17	in	in	ADP
iajs-2911	271	18	view	view	NOUN
iajs-2911	271	19	of	of	ADP
iajs-2911	271	20	ihjpas	ihjpas	PROPN
iajs-2911	271	21	.	.	PUNCT
iajs-2911	272	1	36	36	NUM
iajs-2911	272	2	(	(	PUNCT
iajs-2911	272	3	4	4	NUM
iajs-2911	272	4	)	)	PUNCT
iajs-2911	272	5	2023	2023	NUM
iajs-2911	272	6	402	402	NUM
iajs-2911	272	7	theorem	theorem	NOUN
iajs-2911	272	8	(	(	PUNCT
iajs-2911	272	9	3.1	3.1	NUM
iajs-2911	272	10	.	.	PUNCT
iajs-2911	272	11	)	)	PUNCT
iajs-2911	272	12	that	that	SCONJ
iajs-2911	272	13	xe	xe	PROPN
iajs-2911	272	14	is	be	AUX
iajs-2911	272	15	closed	closed	ADJ
iajs-2911	272	16	.	.	PUNCT
iajs-2911	273	1	let	let	VERB
iajs-2911	273	2	d	d	X
iajs-2911	273	3	∈	∈	PROPN
iajs-2911	273	4	xe(𝒜	xe(𝒜	PROPN
iajs-2911	273	5	)	)	PUNCT
iajs-2911	273	6	,	,	PUNCT
iajs-2911	273	7	for	for	ADP
iajs-2911	273	8	some	some	DET
iajs-2911	273	9	not	not	PART
iajs-2911	273	10	empty	empty	ADJ
iajs-2911	273	11	subset	subset	ADJ
iajs-2911	273	12	𝒜	𝒜	NOUN
iajs-2911	273	13	of	of	ADP
iajs-2911	273	14	e	e	NOUN
iajs-2911	273	15	,	,	PUNCT
iajs-2911	273	16	but	but	CCONJ
iajs-2911	273	17	d	d	X
iajs-2911	273	18	∉	∉	PROPN
iajs-2911	273	19	xe(cl(𝒜	xe(cl(𝒜	PROPN
iajs-2911	273	20	)	)	PUNCT
iajs-2911	273	21	)	)	PUNCT
iajs-2911	273	22	.	.	PUNCT
iajs-2911	274	1	then	then	ADV
iajs-2911	274	2	,	,	PUNCT
iajs-2911	274	3	e	e	X
iajs-2911	274	4	=	=	PRON
iajs-2911	274	5	{	{	PUNCT
iajs-2911	274	6	𝒜	𝒜	NOUN
iajs-2911	274	7	}	}	PUNCT
iajs-2911	274	8	is	be	AUX
iajs-2911	274	9	a	a	DET
iajs-2911	274	10	f∗.b∗	f∗.b∗	NOUN
iajs-2911	274	11	𝑜n	𝑜n	NOUN
iajs-2911	274	12	e	e	NOUN
iajs-2911	274	13	and	and	CCONJ
iajs-2911	274	14	(	(	PUNCT
iajs-2911	274	15	ad(e))∩	ad(e))∩	NOUN
iajs-2911	274	16	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	274	17	+	+	ADJ
iajs-2911	274	18	(	(	PUNCT
iajs-2911	274	19	resp	resp	NOUN
iajs-2911	274	20	.	.	PUNCT
iajs-2911	275	1	,	,	PUNCT
iajs-2911	275	2	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	275	3	−	−	NOUN
iajs-2911	275	4	)	)	PUNCT
iajs-2911	276	1	=	=	NOUN
iajs-2911	276	2	∅.	∅.	X
iajs-2911	276	3	by	by	ADP
iajs-2911	276	4	u.r.(resp	u.r.(resp	PROPN
iajs-2911	276	5	.	.	PROPN
iajs-2911	276	6	,	,	PUNCT
iajs-2911	276	7	l.r	l.r	PROPN
iajs-2911	276	8	.	.	PROPN
iajs-2911	276	9	)	)	PUNCT
iajs-2911	276	10	of	of	ADP
iajs-2911	276	11	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	276	12	+	+	ADJ
iajs-2911	276	13	(	(	PUNCT
iajs-2911	276	14	resp	resp	NOUN
iajs-2911	276	15	.	.	PUNCT
iajs-2911	276	16	,	,	PUNCT
iajs-2911	276	17	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	276	18	−	−	NOUN
iajs-2911	276	19	)	)	PUNCT
iajs-2911	276	20	,	,	PUNCT
iajs-2911	276	21	a	a	DET
iajs-2911	276	22	∃	∃	PROPN
iajs-2911	276	23	𝔘	𝔘	PROPN
iajs-2911	276	24	∈	∈	PROPN
iajs-2911	276	25	𝜏	𝜏	NOUN
iajs-2911	276	26	containing	contain	VERB
iajs-2911	276	27	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	276	28	+	+	PROPN
iajs-2911	276	29	(	(	PUNCT
iajs-2911	276	30	resp	resp	NOUN
iajs-2911	276	31	.	.	PUNCT
iajs-2911	276	32	,	,	PUNCT
iajs-2911	276	33	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	276	34	−	−	NOUN
iajs-2911	276	35	)	)	PUNCT
iajs-2911	276	36	such	such	ADJ
iajs-2911	276	37	that	that	SCONJ
iajs-2911	276	38	cl(𝔘)∩	cl(𝔘)∩	NOUN
iajs-2911	276	39	𝒜	𝒜	NOUN
iajs-2911	276	40	=	=	NOUN
iajs-2911	276	41	∅.	∅.	NOUN
iajs-2911	276	42	by	by	ADP
iajs-2911	276	43	w.u	w.u	PROPN
iajs-2911	276	44	.	.	PROPN
iajs-2911	276	45	closed	closed	PROPN
iajs-2911	276	46	(	(	PUNCT
iajs-2911	276	47	resp	resp	NOUN
iajs-2911	276	48	.	.	PUNCT
iajs-2911	276	49	,	,	PUNCT
iajs-2911	276	50	w.l	w.l	PROPN
iajs-2911	276	51	.	.	PROPN
iajs-2911	276	52	closed	closed	PROPN
iajs-2911	276	53	)	)	PUNCT
iajs-2911	276	54	of	of	ADP
iajs-2911	276	55	xe	xe	PROPN
iajs-2911	276	56	∃	∃	PROPN
iajs-2911	276	57	an	an	PROPN
iajs-2911	276	58	𝜌−open	𝜌−open	PROPN
iajs-2911	276	59	𝑎	𝑎	NOUN
iajs-2911	276	60	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	276	61	d	d	NOUN
iajs-2911	276	62	of	of	ADP
iajs-2911	276	63	d	d	PROPN
iajs-2911	276	64	such	such	ADJ
iajs-2911	276	65	that	that	SCONJ
iajs-2911	276	66	,	,	PUNCT
iajs-2911	276	67	𝐸𝑐𝑙(𝑉	𝐸𝑐𝑙(𝑉	NOUN
iajs-2911	276	68	)	)	PUNCT
iajs-2911	277	1	+	+	NUM
iajs-2911	277	2	∩	∩	ADJ
iajs-2911	277	3	𝒜	𝒜	NOUN
iajs-2911	277	4	=	=	SYM
iajs-2911	277	5	∅(resp	∅(resp	NOUN
iajs-2911	277	6	.	.	PUNCT
iajs-2911	277	7	,	,	PUNCT
iajs-2911	277	8	𝐸𝑐𝑙(𝑉	𝐸𝑐𝑙(𝑉	NOUN
iajs-2911	277	9	)	)	PUNCT
iajs-2911	277	10	−	−	ADP
iajs-2911	277	11	∩	∩	ADJ
iajs-2911	277	12	𝒜	𝒜	NOUN
iajs-2911	277	13	=	=	SYM
iajs-2911	277	14	∅	∅	NOUN
iajs-2911	277	15	)	)	PUNCT
iajs-2911	277	16	,	,	PUNCT
iajs-2911	277	17	i.e.	i.e.	X
iajs-2911	277	18	,	,	PUNCT
iajs-2911	277	19	cl(v	cl(v	NOUN
iajs-2911	277	20	)	)	PUNCT
iajs-2911	277	21	∩	∩	PROPN
iajs-2911	277	22	xe	xe	PROPN
iajs-2911	277	23	(	(	PUNCT
iajs-2911	277	24	𝒜	𝒜	NOUN
iajs-2911	277	25	)	)	PUNCT
iajs-2911	277	26	=	=	SYM
iajs-2911	277	27	∅	∅	NOUN
iajs-2911	277	28	,	,	PUNCT
iajs-2911	277	29	which	which	PRON
iajs-2911	277	30	is	be	AUX
iajs-2911	277	31	impossible	impossible	ADJ
iajs-2911	277	32	since	since	SCONJ
iajs-2911	277	33	d	d	PROPN
iajs-2911	277	34	∈	∈	PROPN
iajs-2911	277	35	xe(𝒜	xe(𝒜	PROPN
iajs-2911	277	36	)	)	PUNCT
iajs-2911	277	37	.	.	PUNCT
iajs-2911	278	1	so	so	ADV
iajs-2911	278	2	ω	ω	PROPN
iajs-2911	278	3	is	be	AUX
iajs-2911	278	4	closed	closed	ADJ
iajs-2911	278	5	.	.	PUNCT
iajs-2911	279	1	corollary	corollary	ADJ
iajs-2911	279	2	3.4	3.4	NUM
iajs-2911	279	3	.	.	PUNCT
iajs-2911	280	1	let	let	VERB
iajs-2911	280	2	(	(	PUNCT
iajs-2911	280	3	e,𝜏	e,𝜏	NOUN
iajs-2911	280	4	)	)	PUNCT
iajs-2911	280	5	be	be	VERB
iajs-2911	280	6	𝔽.	𝔽.	PROPN
iajs-2911	280	7	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	280	8	.	.	PUNCT
iajs-2911	281	1	on	on	ADP
iajs-2911	281	2	(	(	PUNCT
iajs-2911	281	3	d,𝜌	d,𝜌	NOUN
iajs-2911	281	4	)	)	PUNCT
iajs-2911	281	5	.	.	PUNCT
iajs-2911	282	1	then	then	ADV
iajs-2911	282	2	,	,	PUNCT
iajs-2911	282	3	(	(	PUNCT
iajs-2911	282	4	e,𝜏	e,𝜏	NOUN
iajs-2911	282	5	)	)	PUNCT
iajs-2911	282	6	is	be	AUX
iajs-2911	282	7	𝔽.	𝔽.	PROPN
iajs-2911	282	8	𝕎.m.p	𝕎.m.p	PROPN
iajs-2911	282	9	,	,	PUNCT
iajs-2911	282	10	if	if	SCONJ
iajs-2911	282	11	i.	i.	PROPN
iajs-2911	282	12	(	(	PUNCT
iajs-2911	282	13	e,𝜏	e,𝜏	PROPN
iajs-2911	282	14	)	)	PUNCT
iajs-2911	282	15	is	be	AUX
iajs-2911	282	16	𝔽.	𝔽.	PROPN
iajs-2911	282	17	𝕎.m.w	𝕎.m.w	PROPN
iajs-2911	282	18	.	.	PROPN
iajs-2911	282	19	closed	close	VERB
iajs-2911	282	20	topological	topological	ADJ
iajs-2911	282	21	space	space	NOUN
iajs-2911	282	22	.	.	PUNCT
iajs-2911	283	1	ii	ii	X
iajs-2911	283	2	.	.	PUNCT
iajs-2911	284	1	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	284	2	is	be	AUX
iajs-2911	284	3	m.r	m.r	ADJ
iajs-2911	284	4	,	,	PUNCT
iajs-2911	284	5	for	for	ADP
iajs-2911	284	6	every	every	DET
iajs-2911	284	7	d	d	PROPN
iajs-2911	284	8	∈	∈	PROPN
iajs-2911	284	9	d.	d.	PROPN
iajs-2911	284	10	lemma	lemma	PROPN
iajs-2911	284	11	3.1	3.1	NUM
iajs-2911	284	12	.	.	PUNCT
iajs-2911	285	1	[	[	X
iajs-2911	285	2	11]a	11]a	NUM
iajs-2911	285	3	subset	subset	VERB
iajs-2911	285	4	a	a	PRON
iajs-2911	285	5	of	of	ADP
iajs-2911	285	6	a	a	DET
iajs-2911	285	7	topological	topological	ADJ
iajs-2911	285	8	space	space	NOUN
iajs-2911	285	9	(	(	PUNCT
iajs-2911	285	10	e,𝜏	e,𝜏	NOUN
iajs-2911	285	11	)	)	PUNCT
iajs-2911	285	12	is	be	AUX
iajs-2911	285	13	𝔼.	𝔼.	PROPN
iajs-2911	285	14	set	set	VERB
iajs-2911	285	15	if	if	SCONJ
iajs-2911	285	16	for	for	ADP
iajs-2911	285	17	every	every	DET
iajs-2911	285	18	f∗.b∗	f∗.b∗	NOUN
iajs-2911	285	19	on	on	ADP
iajs-2911	285	20	ℑ	ℑ	NOUN
iajs-2911	285	21	on	on	ADP
iajs-2911	285	22	𝒜	𝒜	NOUN
iajs-2911	285	23	;	;	PUNCT
iajs-2911	285	24	(	(	PUNCT
iajs-2911	285	25	ad(ℑ	ad(ℑ	NOUN
iajs-2911	285	26	)	)	PUNCT
iajs-2911	285	27	)	)	PUNCT
iajs-2911	286	1	∩	∩	NOUN
iajs-2911	286	2	𝒜	𝒜	PROPN
iajs-2911	286	3	≠	≠	PROPN
iajs-2911	286	4	∅.	∅.	NOUN
iajs-2911	286	5	theorem	theorem	VERB
iajs-2911	286	6	3.5	3.5	NUM
iajs-2911	286	7	.	.	PUNCT
iajs-2911	287	1	if	if	SCONJ
iajs-2911	287	2	(	(	PUNCT
iajs-2911	287	3	e,𝜏	e,𝜏	NOUN
iajs-2911	287	4	)	)	PUNCT
iajs-2911	287	5	is	be	AUX
iajs-2911	287	6	𝔽.	𝔽.	PROPN
iajs-2911	287	7	𝕎.u.p.t.s.(resp	𝕎.u.p.t.s.(resp	PROPN
iajs-2911	287	8	.	.	PROPN
iajs-2911	287	9	,	,	PUNCT
iajs-2911	287	10	𝔽.	𝔽.	PROPN
iajs-2911	287	11	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	287	12	.	.	PUNCT
iajs-2911	287	13	)	)	PUNCT
iajs-2911	288	1	on	on	ADP
iajs-2911	288	2	(	(	PUNCT
iajs-2911	288	3	d	d	NOUN
iajs-2911	288	4	,	,	PUNCT
iajs-2911	288	5	𝜌	𝜌	ADP
iajs-2911	288	6	)	)	PUNCT
iajs-2911	288	7	and	and	CCONJ
iajs-2911	288	8	d∗	d∗	VERB
iajs-2911	288	9	⊂	⊂	PROPN
iajs-2911	289	1	d	d	X
iajs-2911	289	2	is	be	AUX
iajs-2911	289	3	an	an	DET
iajs-2911	289	4	𝔼	𝔼	NOUN
iajs-2911	289	5	set	set	VERB
iajs-2911	289	6	in	in	ADP
iajs-2911	289	7	d	d	PROPN
iajs-2911	289	8	,	,	PUNCT
iajs-2911	289	9	so	so	ADV
iajs-2911	289	10	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	290	1	+	+	CCONJ
iajs-2911	290	2	(	(	PUNCT
iajs-2911	290	3	resp	resp	NOUN
iajs-2911	290	4	.	.	PUNCT
iajs-2911	290	5	,	,	PUNCT
iajs-2911	290	6	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	290	7	−	−	PROPN
iajs-2911	290	8	)	)	PUNCT
iajs-2911	290	9	is	be	AUX
iajs-2911	290	10	an	an	DET
iajs-2911	290	11	𝔼	𝔼	PROPN
iajs-2911	290	12	set	set	VERB
iajs-2911	290	13	in	in	ADP
iajs-2911	290	14	e.	e.	PROPN
iajs-2911	290	15	proof	proof	PROPN
iajs-2911	290	16	.	.	PUNCT
iajs-2911	291	1	suppose	suppose	VERB
iajs-2911	291	2	that	that	SCONJ
iajs-2911	291	3	e	e	PROPN
iajs-2911	291	4	is	be	AUX
iajs-2911	291	5	a	a	DET
iajs-2911	291	6	𝔽.	𝔽.	PROPN
iajs-2911	291	7	𝕎.u.p.t.s	𝕎.u.p.t.s	PROPN
iajs-2911	291	8	.	.	PUNCT
iajs-2911	291	9	(	(	PUNCT
iajs-2911	291	10	resp	resp	NOUN
iajs-2911	291	11	.	.	PUNCT
iajs-2911	291	12	,	,	PUNCT
iajs-2911	291	13	𝔽.	𝔽.	PROPN
iajs-2911	291	14	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	291	15	.	.	PUNCT
iajs-2911	291	16	)	)	PUNCT
iajs-2911	292	1	on	on	ADP
iajs-2911	292	2	d	d	X
iajs-2911	292	3	,	,	PUNCT
iajs-2911	292	4	therefore	therefore	ADV
iajs-2911	292	5	xe	xe	PROPN
iajs-2911	292	6	:	:	PUNCT
iajs-2911	292	7	e	e	X
iajs-2911	292	8	→	→	SYM
iajs-2911	292	9	d	d	ADP
iajs-2911	292	10	exist	exist	VERB
iajs-2911	292	11	.	.	PUNCT
iajs-2911	293	1	le𝑡	le𝑡	NOUN
iajs-2911	293	2	ℑ	ℑ	PROPN
iajs-2911	293	3	be	be	VERB
iajs-2911	293	4	a	a	DET
iajs-2911	293	5	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	293	6	on	on	ADP
iajs-2911	293	7	d∗.	d∗.	PROPN
iajs-2911	293	8	by	by	ADP
iajs-2911	293	9	d∗	d∗	PROPN
iajs-2911	293	10	is	be	AUX
iajs-2911	293	11	an	an	DET
iajs-2911	293	12	𝔼	𝔼	NOUN
iajs-2911	293	13	set	set	VERB
iajs-2911	293	14	in	in	ADP
iajs-2911	293	15	d	d	PROPN
iajs-2911	293	16	,	,	PUNCT
iajs-2911	293	17	d∗	d∗	NOUN
iajs-2911	293	18	∩	∩	ADJ
iajs-2911	293	19	ad	ad	NOUN
iajs-2911	293	20	xe(ℑ	xe(ℑ	NOUN
iajs-2911	293	21	)	)	PUNCT
iajs-2911	293	22	≠	≠	PROPN
iajs-2911	293	23	∅	∅	NOUN
iajs-2911	293	24	,	,	PUNCT
iajs-2911	293	25	by	by	ADP
iajs-2911	293	26	lemma	lemma	PROPN
iajs-2911	293	27	(	(	PUNCT
iajs-2911	293	28	3.1	3.1	NUM
iajs-2911	293	29	.	.	PUNCT
iajs-2911	293	30	)	)	PUNCT
iajs-2911	293	31	.	.	PUNCT
iajs-2911	294	1	by	by	ADP
iajs-2911	294	2	theorem	theorem	NOUN
iajs-2911	294	3	[	[	X
iajs-2911	294	4	(	(	PUNCT
iajs-2911	294	5	2.1	2.1	NUM
iajs-2911	294	6	.	.	PUNCT
iajs-2911	294	7	)	)	PUNCT
iajs-2911	295	1	(	(	PUNCT
iajs-2911	295	2	i	i	NOUN
iajs-2911	295	3	)	)	PUNCT
iajs-2911	295	4	⇒	⇒	PROPN
iajs-2911	295	5	(	(	PUNCT
iajs-2911	295	6	iii	iii	NOUN
iajs-2911	295	7	)	)	PUNCT
iajs-2911	295	8	]	]	PUNCT
iajs-2911	295	9	,	,	PUNCT
iajs-2911	295	10	d∗	d∗	PROPN
iajs-2911	295	11	∩	∩	ADJ
iajs-2911	295	12	xe(ad	xe(ad	PROPN
iajs-2911	295	13	(	(	PUNCT
iajs-2911	295	14	ℑ	ℑ	PROPN
iajs-2911	295	15	)	)	PUNCT
iajs-2911	295	16	≠	≠	PROPN
iajs-2911	295	17	∅	∅	NOUN
iajs-2911	295	18	,	,	PUNCT
iajs-2911	295	19	so	so	ADV
iajs-2911	295	20	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	295	21	+	+	CCONJ
iajs-2911	295	22	∩	∩	ADJ
iajs-2911	295	23	ad	ad	NOUN
iajs-2911	295	24	(	(	PUNCT
iajs-2911	295	25	ℑ	ℑ	PROPN
iajs-2911	295	26	)	)	PUNCT
iajs-2911	295	27	≠	≠	PROPN
iajs-2911	295	28	∅(resp	∅(resp	NUM
iajs-2911	295	29	.	.	PUNCT
iajs-2911	295	30	,	,	PUNCT
iajs-2911	295	31	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	295	32	−	−	PROPN
iajs-2911	295	33	∩	∩	ADJ
iajs-2911	295	34	ad	ad	NOUN
iajs-2911	295	35	(	(	PUNCT
iajs-2911	295	36	ℑ	ℑ	PROPN
iajs-2911	295	37	)	)	PUNCT
iajs-2911	295	38	≠	≠	PROPN
iajs-2911	295	39	∅	∅	NOUN
iajs-2911	295	40	)	)	PUNCT
iajs-2911	295	41	.	.	PUNCT
iajs-2911	296	1	hence	hence	ADV
iajs-2911	296	2	,	,	PUNCT
iajs-2911	296	3	by	by	ADP
iajs-2911	296	4	lemma	lemma	PROPN
iajs-2911	296	5	(	(	PUNCT
iajs-2911	296	6	3.1	3.1	NUM
iajs-2911	296	7	.	.	NUM
iajs-2911	296	8	)	)	PUNCT
iajs-2911	296	9	,	,	PUNCT
iajs-2911	296	10	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	296	11	+	+	CCONJ
iajs-2911	296	12	(	(	PUNCT
iajs-2911	296	13	resp	resp	NOUN
iajs-2911	296	14	.	.	PUNCT
iajs-2911	297	1	,	,	PUNCT
iajs-2911	297	2	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	297	3	−	−	PROPN
iajs-2911	297	4	)	)	PUNCT
iajs-2911	297	5	is	be	AUX
iajs-2911	297	6	an	an	DET
iajs-2911	297	7	𝔼	𝔼	PROPN
iajs-2911	297	8	set	set	VERB
iajs-2911	297	9	in	in	ADP
iajs-2911	297	10	e.	e.	PROPN
iajs-2911	297	11	corollary	corollary	PROPN
iajs-2911	297	12	3.5	3.5	NUM
iajs-2911	297	13	.	.	PUNCT
iajs-2911	298	1	if	if	SCONJ
iajs-2911	298	2	(	(	PUNCT
iajs-2911	298	3	e,𝜏	e,𝜏	NOUN
iajs-2911	298	4	)	)	PUNCT
iajs-2911	298	5	is	be	AUX
iajs-2911	298	6	𝔽.	𝔽.	PROPN
iajs-2911	298	7	𝕎.m.p.t.s	𝕎.m.p.t.s	X
iajs-2911	298	8	on	on	ADP
iajs-2911	298	9	(	(	PUNCT
iajs-2911	298	10	d	d	NOUN
iajs-2911	298	11	,	,	PUNCT
iajs-2911	298	12	𝜌	𝜌	ADP
iajs-2911	298	13	)	)	PUNCT
iajs-2911	298	14	and	and	CCONJ
iajs-2911	298	15	d∗	d∗	VERB
iajs-2911	299	1	⊂	⊂	PROPN
iajs-2911	299	2	d	d	X
iajs-2911	299	3	is	be	AUX
iajs-2911	299	4	an	an	DET
iajs-2911	299	5	𝔼	𝔼	NOUN
iajs-2911	299	6	set	set	VERB
iajs-2911	299	7	in	in	ADP
iajs-2911	299	8	d	d	PROPN
iajs-2911	299	9	,	,	PUNCT
iajs-2911	299	10	so	so	ADV
iajs-2911	299	11	𝔼𝐷∗	𝔼𝐷∗	PROPN
iajs-2911	299	12	is	be	AUX
iajs-2911	299	13	an	an	DET
iajs-2911	299	14	𝔼	𝔼	PROPN
iajs-2911	299	15	set	set	VERB
iajs-2911	299	16	in	in	ADP
iajs-2911	299	17	e.	e.	PROPN
iajs-2911	299	18	definition	definition	PROPN
iajs-2911	299	19	3.7	3.7	NUM
iajs-2911	299	20	.	.	PUNCT
iajs-2911	300	1	the	the	DET
iajs-2911	300	2	function	function	PROPN
iajs-2911	300	3	ω	ω	NOUN
iajs-2911	300	4	:	:	PUNCT
iajs-2911	300	5	(	(	PUNCT
iajs-2911	300	6	e,𝜏	e,𝜏	NOUN
iajs-2911	300	7	)	)	PUNCT
iajs-2911	300	8	→	→	SYM
iajs-2911	300	9	(	(	PUNCT
iajs-2911	300	10	f,𝜎	f,𝜎	PROPN
iajs-2911	300	11	)	)	PUNCT
iajs-2911	300	12	is	be	AUX
iajs-2911	300	13	named	name	VERB
iajs-2911	300	14	to	to	PART
iajs-2911	300	15	be	be	AUX
iajs-2911	300	16	almost	almost	ADV
iajs-2911	300	17	u.p	u.p	PROPN
iajs-2911	300	18	.	.	PUNCT
iajs-2911	301	1	if	if	SCONJ
iajs-2911	301	2	for	for	SCONJ
iajs-2911	301	3	every	every	DET
iajs-2911	301	4	e	e	NOUN
iajs-2911	301	5	set	set	VERB
iajs-2911	301	6	k	k	PROPN
iajs-2911	301	7	in	in	ADP
iajs-2911	301	8	f	f	PROPN
iajs-2911	301	9	,	,	PUNCT
iajs-2911	301	10	ω+(k	ω+(k	NUM
iajs-2911	301	11	)	)	PUNCT
iajs-2911	301	12	is	be	AUX
iajs-2911	301	13	an	an	DET
iajs-2911	301	14	𝔼	𝔼	PROPN
iajs-2911	301	15	set	set	VERB
iajs-2911	301	16	in	in	ADP
iajs-2911	301	17	e.	e.	PROPN
iajs-2911	301	18	definition	definition	NOUN
iajs-2911	301	19	3.8	3.8	NUM
iajs-2911	301	20	.	.	PUNCT
iajs-2911	302	1	the	the	DET
iajs-2911	302	2	function	function	NOUN
iajs-2911	302	3	ω	ω	NOUN
iajs-2911	302	4	:	:	PUNCT
iajs-2911	302	5	(	(	PUNCT
iajs-2911	302	6	e,𝜏	e,𝜏	NOUN
iajs-2911	302	7	)	)	PUNCT
iajs-2911	302	8	→	→	SYM
iajs-2911	302	9	(	(	PUNCT
iajs-2911	302	10	f,𝜎	f,𝜎	PROPN
iajs-2911	302	11	)	)	PUNCT
iajs-2911	302	12	is	be	AUX
iajs-2911	302	13	named	name	VERB
iajs-2911	302	14	to	to	PART
iajs-2911	302	15	be	be	AUX
iajs-2911	302	16	almost	almost	ADV
iajs-2911	302	17	l.p	l.p	PROPN
iajs-2911	302	18	.	.	PUNCT
iajs-2911	303	1	if	if	SCONJ
iajs-2911	303	2	for	for	SCONJ
iajs-2911	303	3	every	every	DET
iajs-2911	303	4	𝔼	𝔼	PROPN
iajs-2911	303	5	set	set	VERB
iajs-2911	303	6	k	k	PROPN
iajs-2911	303	7	in	in	ADP
iajs-2911	303	8	f	f	PROPN
iajs-2911	303	9	,	,	PUNCT
iajs-2911	303	10	ω-(k	ω-(k	ADV
iajs-2911	303	11	)	)	PUNCT
iajs-2911	303	12	is	be	AUX
iajs-2911	303	13	an	an	DET
iajs-2911	303	14	𝔼	𝔼	PROPN
iajs-2911	303	15	set	set	VERB
iajs-2911	303	16	in	in	ADP
iajs-2911	303	17	e.	e.	PROPN
iajs-2911	304	1	the	the	DET
iajs-2911	304	2	function	function	PROPN
iajs-2911	304	3	ω	ω	NOUN
iajs-2911	304	4	:	:	PUNCT
iajs-2911	304	5	(	(	PUNCT
iajs-2911	304	6	e,𝜏	e,𝜏	NOUN
iajs-2911	304	7	)	)	PUNCT
iajs-2911	304	8	→	→	SYM
iajs-2911	304	9	(	(	PUNCT
iajs-2911	304	10	f,𝜎	f,𝜎	PROPN
iajs-2911	304	11	)	)	PUNCT
iajs-2911	304	12	is	be	AUX
iajs-2911	304	13	named	name	VERB
iajs-2911	304	14	to	to	PART
iajs-2911	304	15	be	be	AUX
iajs-2911	304	16	almost	almost	ADV
iajs-2911	304	17	m.p	m.p	ADJ
iajs-2911	304	18	.	.	PUNCT
iajs-2911	305	1	if	if	SCONJ
iajs-2911	305	2	almost	almost	ADV
iajs-2911	305	3	u.p	u.p	PROPN
iajs-2911	305	4	.	.	PROPN
iajs-2911	306	1	and	and	CCONJ
iajs-2911	306	2	almost	almost	ADV
iajs-2911	306	3	l.p	l.p	PROPN
iajs-2911	306	4	.	.	PROPN
iajs-2911	306	5	definition	definition	NOUN
iajs-2911	306	6	3.9	3.9	NUM
iajs-2911	306	7	.	.	PUNCT
iajs-2911	307	1	the	the	DET
iajs-2911	307	2	𝔽.	𝔽.	PROPN
iajs-2911	307	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	307	4	.	.	PUNCT
iajs-2911	307	5	almost	almost	ADV
iajs-2911	307	6	u.p	u.p	PROPN
iajs-2911	307	7	.	.	PROPN
iajs-2911	308	1	on	on	ADP
iajs-2911	308	2	(	(	PUNCT
iajs-2911	308	3	d	d	PROPN
iajs-2911	308	4	,	,	PUNCT
iajs-2911	308	5	ρ	ρ	NOUN
iajs-2911	308	6	)	)	PUNCT
iajs-2911	308	7	is	be	AUX
iajs-2911	308	8	named	name	VERB
iajs-2911	308	9	to	to	PART
iajs-2911	308	10	be	be	AUX
iajs-2911	308	11	𝔽.	𝔽.	PROPN
iajs-2911	308	12	𝕎.	𝕎.	PROPN
iajs-2911	308	13	almost	almost	ADV
iajs-2911	308	14	u.p	u.p	PROPN
iajs-2911	308	15	.	.	PUNCT
iajs-2911	309	1	if	if	SCONJ
iajs-2911	309	2	the	the	DET
iajs-2911	309	3	projection	projection	NOUN
iajs-2911	309	4	x	x	VERB
iajs-2911	309	5	is	be	AUX
iajs-2911	309	6	almost	almost	ADV
iajs-2911	309	7	perfect	perfect	ADJ
iajs-2911	309	8	.	.	PUNCT
iajs-2911	310	1	definition	definition	NOUN
iajs-2911	310	2	3.10	3.10	NUM
iajs-2911	310	3	.	.	PUNCT
iajs-2911	311	1	the	the	DET
iajs-2911	311	2	𝔽.	𝔽.	PROPN
iajs-2911	311	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	311	4	.	.	PUNCT
iajs-2911	312	1	almost	almost	ADV
iajs-2911	312	2	l.p	l.p	PROPN
iajs-2911	312	3	.	.	PROPN
iajs-2911	313	1	on	on	ADP
iajs-2911	313	2	(	(	PUNCT
iajs-2911	313	3	d,𝜌	d,𝜌	NOUN
iajs-2911	313	4	)	)	PUNCT
iajs-2911	313	5	is	be	AUX
iajs-2911	313	6	named	name	VERB
iajs-2911	313	7	to	to	PART
iajs-2911	313	8	be	be	AUX
iajs-2911	313	9	𝔽.	𝔽.	PROPN
iajs-2911	313	10	𝕎.	𝕎.	PROPN
iajs-2911	313	11	almost	almost	ADV
iajs-2911	313	12	l.p	l.p	PROPN
iajs-2911	313	13	.	.	PUNCT
iajs-2911	314	1	if	if	SCONJ
iajs-2911	314	2	the	the	DET
iajs-2911	314	3	projection	projection	NOUN
iajs-2911	314	4	x	x	VERB
iajs-2911	314	5	is	be	AUX
iajs-2911	314	6	almost	almost	ADV
iajs-2911	314	7	perfect	perfect	ADJ
iajs-2911	314	8	.	.	PUNCT
iajs-2911	315	1	the	the	DET
iajs-2911	315	2	𝔽.	𝔽.	PROPN
iajs-2911	315	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	315	4	.	.	PUNCT
iajs-2911	316	1	almost	almost	ADV
iajs-2911	316	2	m.p	m.p	PROPN
iajs-2911	316	3	.	.	PROPN
iajs-2911	317	1	on	on	ADP
iajs-2911	317	2	(	(	PUNCT
iajs-2911	317	3	d	d	PROPN
iajs-2911	317	4	,	,	PUNCT
iajs-2911	317	5	ρ	ρ	NOUN
iajs-2911	317	6	)	)	PUNCT
iajs-2911	317	7	is	be	AUX
iajs-2911	317	8	named	name	VERB
iajs-2911	317	9	to	to	PART
iajs-2911	317	10	be	be	AUX
iajs-2911	317	11	𝔽.	𝔽.	PROPN
iajs-2911	317	12	𝕎.	𝕎.	PROPN
iajs-2911	317	13	almost	almost	ADV
iajs-2911	317	14	m.p	m.p	PROPN
iajs-2911	317	15	.	.	PROPN
iajs-2911	318	1	if	if	SCONJ
iajs-2911	318	2	it	it	PRON
iajs-2911	318	3	is	be	AUX
iajs-2911	318	4	𝔽.	𝔽.	PROPN
iajs-2911	318	5	𝕎.	𝕎.	PROPN
iajs-2911	318	6	almost	almost	ADV
iajs-2911	318	7	u.p	u.p	PROPN
iajs-2911	318	8	.	.	PROPN
iajs-2911	318	9	and	and	CCONJ
iajs-2911	318	10	𝔽.	𝔽.	PROPN
iajs-2911	318	11	𝕎.	𝕎.	PROPN
iajs-2911	318	12	almost	almost	ADV
iajs-2911	318	13	l.p	l.p	PROPN
iajs-2911	318	14	.	.	PROPN
iajs-2911	318	15	theorem	theorem	VERB
iajs-2911	318	16	3.6	3.6	NUM
iajs-2911	318	17	.	.	PUNCT
iajs-2911	319	1	let	let	VERB
iajs-2911	319	2	(	(	PUNCT
iajs-2911	319	3	e,𝜏	e,𝜏	NOUN
iajs-2911	319	4	)	)	PUNCT
iajs-2911	319	5	be	be	VERB
iajs-2911	319	6	𝔽.	𝔽.	PROPN
iajs-2911	319	7	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	319	8	.	.	PUNCT
iajs-2911	320	1	on	on	ADP
iajs-2911	320	2	(	(	PUNCT
iajs-2911	320	3	d,𝜌	d,𝜌	NOUN
iajs-2911	320	4	)	)	PUNCT
iajs-2911	320	5	such	such	ADJ
iajs-2911	320	6	that	that	SCONJ
iajs-2911	320	7	:	:	PUNCT
iajs-2911	320	8	i.	i.	NOUN
iajs-2911	320	9	for	for	ADP
iajs-2911	320	10	every	every	DET
iajs-2911	320	11	d	d	PROPN
iajs-2911	320	12	∈	∈	PROPN
iajs-2911	320	13	d	d	NOUN
iajs-2911	320	14	,	,	PUNCT
iajs-2911	320	15	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	320	16	+	+	ADJ
iajs-2911	320	17	(	(	PUNCT
iajs-2911	320	18	resp	resp	NOUN
iajs-2911	320	19	.	.	PUNCT
iajs-2911	320	20	,	,	PUNCT
iajs-2911	320	21	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	320	22	−	−	NOUN
iajs-2911	320	23	)	)	PUNCT
iajs-2911	320	24	is	be	AUX
iajs-2911	320	25	u.r.(resp	u.r.(resp	PRON
iajs-2911	320	26	.	.	PROPN
iajs-2911	320	27	,	,	PUNCT
iajs-2911	320	28	l.r	l.r	PROPN
iajs-2911	320	29	.	.	PROPN
iajs-2911	320	30	)	)	PUNCT
iajs-2911	320	31	and	and	CCONJ
iajs-2911	320	32	ii	ii	X
iajs-2911	320	33	.	.	PUNCT
iajs-2911	321	1	(	(	PUNCT
iajs-2911	321	2	e,𝜏	e,𝜏	NOUN
iajs-2911	321	3	)	)	PUNCT
iajs-2911	321	4	be	be	AUX
iajs-2911	321	5	𝔽.	𝔽.	PROPN
iajs-2911	321	6	𝕎.u.w	𝕎.u.w	PROPN
iajs-2911	321	7	.	.	PROPN
iajs-2911	322	1	closed	close	VERB
iajs-2911	322	2	(	(	PUNCT
iajs-2911	322	3	resp	resp	NOUN
iajs-2911	322	4	.	.	PUNCT
iajs-2911	322	5	,	,	PUNCT
iajs-2911	322	6	𝔽.	𝔽.	PROPN
iajs-2911	322	7	𝕎.l.w	𝕎.l.w	PROPN
iajs-2911	322	8	.	.	PROPN
iajs-2911	322	9	closed	close	VERB
iajs-2911	322	10	)	)	PUNCT
iajs-2911	322	11	topological	topological	ADJ
iajs-2911	322	12	space	space	NOUN
iajs-2911	322	13	.	.	PUNCT
iajs-2911	323	1	then	then	ADV
iajs-2911	323	2	,	,	PUNCT
iajs-2911	323	3	(	(	PUNCT
iajs-2911	323	4	e	e	NOUN
iajs-2911	323	5	,	,	PUNCT
iajs-2911	323	6	τ	τ	X
iajs-2911	323	7	)	)	PUNCT
iajs-2911	323	8	is	be	AUX
iajs-2911	323	9	𝔽.	𝔽.	PROPN
iajs-2911	323	10	𝕎.	𝕎.	PROPN
iajs-2911	323	11	almost	almost	ADV
iajs-2911	323	12	u.p.t.s.(resp	u.p.t.s.(resp	ADJ
iajs-2911	323	13	.	.	PUNCT
iajs-2911	323	14	,	,	PUNCT
iajs-2911	323	15	𝔽.	𝔽.	PROPN
iajs-2911	323	16	𝕎.	𝕎.	PROPN
iajs-2911	323	17	almost	almost	ADV
iajs-2911	323	18	l.p.t.s	l.p.t.s	NOUN
iajs-2911	323	19	.	.	PUNCT
iajs-2911	323	20	)	)	PUNCT
iajs-2911	323	21	.	.	PUNCT
iajs-2911	324	1	proof	proof	NOUN
iajs-2911	324	2	.	.	PUNCT
iajs-2911	325	1	le𝑡	le𝑡	NOUN
iajs-2911	325	2	e	e	NOUN
iajs-2911	325	3	be	be	VERB
iajs-2911	325	4	𝔽.	𝔽.	PROPN
iajs-2911	325	5	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	325	6	.	.	PUNCT
iajs-2911	326	1	on	on	ADP
iajs-2911	326	2	d	d	PROPN
iajs-2911	326	3	,	,	PUNCT
iajs-2911	326	4	so	so	ADV
iajs-2911	326	5	xe	xe	PROPN
iajs-2911	326	6	:	:	PUNCT
iajs-2911	326	7	e→	e→	PROPN
iajs-2911	326	8	d	d	NOUN
iajs-2911	326	9	exist	exist	VERB
iajs-2911	326	10	and	and	CCONJ
iajs-2911	326	11	it	it	PRON
iajs-2911	326	12	is	be	AUX
iajs-2911	326	13	u.	u.	PROPN
iajs-2911	326	14	cont	cont	PROPN
iajs-2911	326	15	.	.	PUNCT
iajs-2911	327	1	(	(	PUNCT
iajs-2911	327	2	resp	resp	NOUN
iajs-2911	327	3	.	.	PUNCT
iajs-2911	327	4	,	,	PUNCT
iajs-2911	327	5	l.	l.	PROPN
iajs-2911	327	6	cont	cont	PROPN
iajs-2911	327	7	.	.	PUNCT
iajs-2911	327	8	)	)	PUNCT
iajs-2911	327	9	.	.	PUNCT
iajs-2911	328	1	assume	assume	VERB
iajs-2911	328	2	that	that	SCONJ
iajs-2911	328	3	d∗	d∗	PROPN
iajs-2911	328	4	is	be	AUX
iajs-2911	328	5	an	an	DET
iajs-2911	328	6	𝔼	𝔼	NOUN
iajs-2911	328	7	set	set	VERB
iajs-2911	328	8	in	in	ADP
iajs-2911	328	9	d	d	NOUN
iajs-2911	328	10	and	and	CCONJ
iajs-2911	328	11	let	let	VERB
iajs-2911	328	12	ℑ	ℑ	PRON
iajs-2911	328	13	be	be	AUX
iajs-2911	328	14	a	a	DET
iajs-2911	328	15	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	328	16	on	on	ADP
iajs-2911	328	17	ed∗.	ed∗.	PROPN
iajs-2911	328	18	currently	currently	ADV
iajs-2911	328	19	,	,	PUNCT
iajs-2911	328	20	xe(ℑ	xe(ℑ	ADJ
iajs-2911	328	21	)	)	PUNCT
iajs-2911	328	22	is	be	AUX
iajs-2911	328	23	a	a	DET
iajs-2911	328	24	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	328	25	on	on	ADP
iajs-2911	328	26	d∗	d∗	NOUN
iajs-2911	328	27	and	and	CCONJ
iajs-2911	328	28	so	so	ADV
iajs-2911	328	29	by	by	ADP
iajs-2911	328	30	lemma	lemma	PROPN
iajs-2911	328	31	(	(	PUNCT
iajs-2911	328	32	4.2.15	4.2.15	NOUN
iajs-2911	328	33	.	.	PUNCT
iajs-2911	328	34	)	)	PUNCT
iajs-2911	328	35	,	,	PUNCT
iajs-2911	328	36	(	(	PUNCT
iajs-2911	328	37	ad	ad	NOUN
iajs-2911	328	38	xe(ℑ	xe(ℑ	NOUN
iajs-2911	328	39	)	)	PUNCT
iajs-2911	328	40	)	)	PUNCT
iajs-2911	328	41	∩	∩	PROPN
iajs-2911	328	42	d∗	d∗	VERB
iajs-2911	328	43	≠	≠	PROPN
iajs-2911	328	44	∅.	∅.	AUX
iajs-2911	328	45	let	let	VERB
iajs-2911	328	46	d	d	PROPN
iajs-2911	328	47	∈	∈	PROPN
iajs-2911	328	48	(	(	PUNCT
iajs-2911	328	49	ad	ad	NOUN
iajs-2911	328	50	xe(ℑ	xe(ℑ	NOUN
iajs-2911	328	51	)	)	PUNCT
iajs-2911	328	52	)	)	PUNCT
iajs-2911	328	53	∩	∩	NOUN
iajs-2911	328	54	d∗.	d∗.	PROPN
iajs-2911	328	55	let	let	VERB
iajs-2911	328	56	ℑ	ℑ	PROPN
iajs-2911	328	57	has	have	VERB
iajs-2911	328	58	no	no	DET
iajs-2911	328	59	ad	ad	NOUN
iajs-2911	328	60	point	point	NOUN
iajs-2911	328	61	in	in	ADP
iajs-2911	328	62	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	329	1	+	+	CCONJ
iajs-2911	329	2	(	(	PUNCT
iajs-2911	329	3	resp	resp	NOUN
iajs-2911	329	4	.	.	PUNCT
iajs-2911	329	5	,	,	PUNCT
iajs-2911	329	6	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	329	7	−	−	PROPN
iajs-2911	329	8	)	)	PUNCT
iajs-2911	329	9	,	,	PUNCT
iajs-2911	329	10	so	so	SCONJ
iajs-2911	329	11	that	that	SCONJ
iajs-2911	329	12	(	(	PUNCT
iajs-2911	329	13	ad	ad	NOUN
iajs-2911	329	14	(	(	PUNCT
iajs-2911	329	15	ℑ))∩	ℑ))∩	NOUN
iajs-2911	329	16	𝐸𝑑	𝐸𝑑	ADJ
iajs-2911	329	17	+	+	ADJ
iajs-2911	329	18	(	(	PUNCT
iajs-2911	329	19	resp	resp	NOUN
iajs-2911	329	20	.	.	PUNCT
iajs-2911	329	21	,	,	PUNCT
iajs-2911	329	22	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	329	23	−	−	NOUN
iajs-2911	329	24	)	)	PUNCT
iajs-2911	329	25	=	=	PUNCT
iajs-2911	329	26	∅.	∅.	NOUN
iajs-2911	329	27	by	by	ADP
iajs-2911	329	28	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	329	29	+	+	PROPN
iajs-2911	329	30	(	(	PUNCT
iajs-2911	329	31	resp	resp	NOUN
iajs-2911	329	32	.	.	PUNCT
iajs-2911	330	1	,	,	PUNCT
iajs-2911	330	2	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	330	3	−	−	NOUN
iajs-2911	330	4	)	)	PUNCT
iajs-2911	330	5	is	be	AUX
iajs-2911	330	6	u.r.(resp	u.r.(resp	PRON
iajs-2911	330	7	.	.	PUNCT
iajs-2911	330	8	,	,	PUNCT
iajs-2911	330	9	u.r.),∃	u.r.),∃	PRON
iajs-2911	330	10	an	an	DET
iajs-2911	330	11	𝔽	𝔽	PROPN
iajs-2911	330	12	∈	∈	PROPN
iajs-2911	330	13	ℑ	ℑ	PROPN
iajs-2911	330	14	and	and	CCONJ
iajs-2911	330	15	𝜏−open	𝜏−open	VERB
iajs-2911	330	16	se𝑡	se𝑡	NOUN
iajs-2911	330	17	𝔘	𝔘	NOUN
iajs-2911	330	18	containing	contain	VERB
iajs-2911	330	19	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	331	1	+	+	CCONJ
iajs-2911	331	2	(	(	PUNCT
iajs-2911	331	3	resp	resp	NOUN
iajs-2911	331	4	.	.	PUNCT
iajs-2911	331	5	,	,	PUNCT
iajs-2911	331	6	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	331	7	−	−	PROPN
iajs-2911	331	8	)	)	PUNCT
iajs-2911	331	9	,	,	PUNCT
iajs-2911	332	1	such	such	ADJ
iajs-2911	332	2	that	that	SCONJ
iajs-2911	332	3	𝔽	𝔽	PROPN
iajs-2911	332	4	∩cl(𝔘	∩cl(𝔘	PROPN
iajs-2911	332	5	)	)	PUNCT
iajs-2911	332	6	=	=	PUNCT
iajs-2911	332	7	∅.	∅.	NOUN
iajs-2911	332	8	since	since	SCONJ
iajs-2911	332	9	w.u	w.u	PROPN
iajs-2911	332	10	.	.	PROPN
iajs-2911	332	11	closed	closed	PROPN
iajs-2911	332	12	(	(	PUNCT
iajs-2911	332	13	resp	resp	NOUN
iajs-2911	332	14	.	.	PUNCT
iajs-2911	332	15	,	,	PUNCT
iajs-2911	332	16	w.l	w.l	PROPN
iajs-2911	332	17	.	.	PROPN
iajs-2911	332	18	closed	closed	PROPN
iajs-2911	332	19	)	)	PUNCT
iajs-2911	332	20	of	of	ADP
iajs-2911	332	21	xe	xe	PROPN
iajs-2911	332	22	,	,	PUNCT
iajs-2911	332	23	∃	∃	PROPN
iajs-2911	332	24	𝜌−	𝜌−	PROPN
iajs-2911	332	25	closed	close	VERB
iajs-2911	332	26	a	a	DET
iajs-2911	332	27	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	332	28	v	v	NOUN
iajs-2911	332	29	of	of	ADP
iajs-2911	332	30	d	d	PROPN
iajs-2911	332	31	such	such	ADJ
iajs-2911	332	32	that	that	SCONJ
iajs-2911	332	33	e	e	NOUN
iajs-2911	332	34	(	(	PUNCT
iajs-2911	332	35	𝜌−cl(v	𝜌−cl(v	NOUN
iajs-2911	332	36	)	)	PUNCT
iajs-2911	332	37	)	)	PUNCT
iajs-2911	333	1	⊂	⊂	PROPN
iajs-2911	333	2	𝜏	𝜏	DET
iajs-2911	333	3	−	−	PROPN
iajs-2911	333	4	cl(𝔘	cl(𝔘	NOUN
iajs-2911	333	5	)	)	PUNCT
iajs-2911	333	6	which	which	PRON
iajs-2911	333	7	means	mean	VERB
iajs-2911	333	8	that	that	SCONJ
iajs-2911	333	9	𝐸(𝜌−𝑐𝑙(𝑉	𝐸(𝜌−𝑐𝑙(𝑉	NOUN
iajs-2911	333	10	)	)	PUNCT
iajs-2911	333	11	)	)	PUNCT
iajs-2911	334	1	+	+	CCONJ
iajs-2911	334	2	∩	∩	ADJ
iajs-2911	334	3	𝔽	𝔽	PROPN
iajs-2911	334	4	=	=	PUNCT
iajs-2911	334	5	∅	∅	NOUN
iajs-2911	334	6	(	(	PUNCT
iajs-2911	334	7	resp	resp	NOUN
iajs-2911	334	8	.	.	PUNCT
iajs-2911	334	9	,	,	PUNCT
iajs-2911	334	10	𝐸(𝜌−𝑐𝑙(𝑉	𝐸(𝜌−𝑐𝑙(𝑉	NOUN
iajs-2911	334	11	)	)	PUNCT
iajs-2911	334	12	)	)	PUNCT
iajs-2911	334	13	−	−	ADP
iajs-2911	334	14	∩	∩	ADJ
iajs-2911	334	15	𝔽	𝔽	PROPN
iajs-2911	334	16	=	=	NOUN
iajs-2911	334	17	∅	∅	NOUN
iajs-2911	334	18	)	)	PUNCT
iajs-2911	334	19	i.e.	i.e.	X
iajs-2911	334	20	,	,	PUNCT
iajs-2911	334	21	𝜌	𝜌	X
iajs-2911	334	22	−	−	PROPN
iajs-2911	334	23	cl(v	cl(v	NOUN
iajs-2911	334	24	)	)	PUNCT
iajs-2911	334	25	∩	∩	NOUN
iajs-2911	334	26	x(𝔽	x(𝔽	NUM
iajs-2911	334	27	)	)	PUNCT
iajs-2911	334	28	=	=	SYM
iajs-2911	334	29	∅	∅	NOUN
iajs-2911	334	30	,	,	PUNCT
iajs-2911	334	31	which	which	PRON
iajs-2911	334	32	is	be	AUX
iajs-2911	334	33	a	a	DET
iajs-2911	334	34	contradiction	contradiction	NOUN
iajs-2911	334	35	.	.	PUNCT
iajs-2911	335	1	thus	thus	ADV
iajs-2911	335	2	,	,	PUNCT
iajs-2911	335	3	by	by	ADP
iajs-2911	335	4	lemma	lemma	PROPN
iajs-2911	335	5	(	(	PUNCT
iajs-2911	335	6	4.2.15	4.2.15	NOUN
iajs-2911	335	7	.	.	PUNCT
iajs-2911	335	8	)	)	PUNCT
iajs-2911	335	9	,	,	PUNCT
iajs-2911	335	10	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	335	11	+	+	CCONJ
iajs-2911	335	12	(	(	PUNCT
iajs-2911	335	13	resp	resp	NOUN
iajs-2911	335	14	.	.	PUNCT
iajs-2911	335	15	,	,	PUNCT
iajs-2911	335	16	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	335	17	−	−	PROPN
iajs-2911	335	18	)	)	PUNCT
iajs-2911	335	19	is	be	AUX
iajs-2911	335	20	an	an	DET
iajs-2911	335	21	𝔼	𝔼	NOUN
iajs-2911	335	22	set	set	VERB
iajs-2911	335	23	in	in	ADP
iajs-2911	335	24	e	e	NOUN
iajs-2911	335	25	and	and	CCONJ
iajs-2911	335	26	so	so	ADV
iajs-2911	335	27	xe	xe	PROPN
iajs-2911	335	28	is	be	AUX
iajs-2911	335	29	almost	almost	ADV
iajs-2911	335	30	u.p	u.p	PROPN
iajs-2911	335	31	.	.	PUNCT
iajs-2911	336	1	(	(	PUNCT
iajs-2911	336	2	resp	resp	PROPN
iajs-2911	336	3	.	.	PUNCT
iajs-2911	336	4	,	,	PUNCT
iajs-2911	336	5	almost	almost	ADV
iajs-2911	336	6	l.p	l.p	PROPN
iajs-2911	336	7	.	.	PROPN
iajs-2911	336	8	)	)	PUNCT
iajs-2911	336	9	.	.	PUNCT
iajs-2911	337	1	corollary	corollary	ADJ
iajs-2911	337	2	3.6	3.6	NUM
iajs-2911	337	3	.	.	PUNCT
iajs-2911	338	1	le𝑡	le𝑡	NOUN
iajs-2911	338	2	(	(	PUNCT
iajs-2911	338	3	e,𝜏	e,𝜏	NOUN
iajs-2911	338	4	)	)	PUNCT
iajs-2911	338	5	be	be	VERB
iajs-2911	338	6	𝔽.	𝔽.	PROPN
iajs-2911	338	7	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	338	8	.	.	PUNCT
iajs-2911	339	1	on	on	ADP
iajs-2911	339	2	(	(	PUNCT
iajs-2911	339	3	d	d	PROPN
iajs-2911	339	4	,	,	PUNCT
iajs-2911	339	5	ρ	ρ	NOUN
iajs-2911	339	6	)	)	PUNCT
iajs-2911	339	7	such	such	ADJ
iajs-2911	339	8	that	that	PRON
iajs-2911	339	9	:	:	PUNCT
iajs-2911	339	10	ihjpas	ihjpas	PROPN
iajs-2911	339	11	.	.	PUNCT
iajs-2911	340	1	36	36	NUM
iajs-2911	340	2	(	(	PUNCT
iajs-2911	340	3	4	4	NUM
iajs-2911	340	4	)	)	PUNCT
iajs-2911	340	5	2023	2023	NUM
iajs-2911	340	6	403	403	NUM
iajs-2911	340	7	i.	i.	NOUN
iajs-2911	340	8	for	for	ADP
iajs-2911	340	9	every	every	DET
iajs-2911	340	10	d	d	PROPN
iajs-2911	340	11	∈	∈	PROPN
iajs-2911	340	12	d	d	PROPN
iajs-2911	340	13	,	,	PUNCT
iajs-2911	340	14	ed	ed	PROPN
iajs-2911	340	15	is	be	AUX
iajs-2911	340	16	m.r	m.r	PROPN
iajs-2911	340	17	.	.	PROPN
iajs-2911	341	1	and	and	CCONJ
iajs-2911	341	2	(	(	PUNCT
iajs-2911	341	3	e,𝜏	e,𝜏	NOUN
iajs-2911	341	4	)	)	PUNCT
iajs-2911	341	5	be	be	VERB
iajs-2911	341	6	𝔽.	𝔽.	PROPN
iajs-2911	341	7	𝕎.m.w	𝕎.m.w	PROPN
iajs-2911	341	8	.	.	PROPN
iajs-2911	341	9	closed	close	VERB
iajs-2911	341	10	topological	topological	ADJ
iajs-2911	341	11	space	space	NOUN
iajs-2911	341	12	.	.	PUNCT
iajs-2911	342	1	then	then	ADV
iajs-2911	342	2	,	,	PUNCT
iajs-2911	342	3	(	(	PUNCT
iajs-2911	342	4	e,𝜏	e,𝜏	NOUN
iajs-2911	342	5	)	)	PUNCT
iajs-2911	342	6	is	be	AUX
iajs-2911	342	7	𝔽.	𝔽.	PROPN
iajs-2911	342	8	𝕎.	𝕎.	PROPN
iajs-2911	342	9	almost	almost	ADV
iajs-2911	342	10	m.p.t.s	m.p.t.s	ADJ
iajs-2911	342	11	.	.	PROPN
iajs-2911	342	12	4	4	NUM
iajs-2911	342	13	.	.	X
iajs-2911	343	1	some	some	DET
iajs-2911	343	2	result	result	NOUN
iajs-2911	343	3	on	on	ADP
iajs-2911	343	4	multi	multi	ADJ
iajs-2911	343	5	topological	topological	ADJ
iajs-2911	343	6	spaces	space	NOUN
iajs-2911	343	7	we	we	PRON
iajs-2911	343	8	currently	currently	ADV
iajs-2911	343	9	give	give	VERB
iajs-2911	343	10	some	some	DET
iajs-2911	343	11	results	result	NOUN
iajs-2911	343	12	of	of	ADP
iajs-2911	343	13	𝔽.	𝔽.	PROPN
iajs-2911	343	14	𝕎.u.p.t.s.(resp	𝕎.u.p.t.s.(resp	PROPN
iajs-2911	343	15	.	.	PROPN
iajs-2911	343	16	,	,	PUNCT
iajs-2911	343	17	𝔽.	𝔽.	PROPN
iajs-2911	343	18	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	343	19	.	.	PROPN
iajs-2911	343	20	and	and	CCONJ
iajs-2911	343	21	𝔽.	𝔽.	PROPN
iajs-2911	343	22	𝕎.m.p.t.s	𝕎.m.p.t.s	PROPN
iajs-2911	343	23	.	.	PUNCT
iajs-2911	343	24	)	)	PUNCT
iajs-2911	343	25	.	.	PUNCT
iajs-2911	344	1	the	the	DET
iajs-2911	344	2	following	follow	VERB
iajs-2911	344	3	characterization	characterization	NOUN
iajs-2911	344	4	theorem	theorem	VERB
iajs-2911	344	5	for	for	ADP
iajs-2911	344	6	an	an	DET
iajs-2911	344	7	u.	u.	NOUN
iajs-2911	344	8	cont	cont	PROPN
iajs-2911	344	9	.	.	PUNCT
iajs-2911	345	1	(	(	PUNCT
iajs-2911	345	2	resp	resp	NOUN
iajs-2911	345	3	.	.	PUNCT
iajs-2911	345	4	,	,	PUNCT
iajs-2911	345	5	l.	l.	PROPN
iajs-2911	345	6	cont	cont	PROPN
iajs-2911	345	7	.	.	PUNCT
iajs-2911	346	1	and	and	CCONJ
iajs-2911	346	2	m.	m.	NOUN
iajs-2911	346	3	cont	cont	PROPN
iajs-2911	346	4	.	.	PUNCT
iajs-2911	346	5	)	)	PUNCT
iajs-2911	347	1	function	function	NOUN
iajs-2911	347	2	is	be	AUX
iajs-2911	347	3	recalled	recall	VERB
iajs-2911	347	4	to	to	ADP
iajs-2911	347	5	this	this	DET
iajs-2911	347	6	end	end	NOUN
iajs-2911	347	7	.	.	PUNCT
iajs-2911	348	1	theorem	theorem	VERB
iajs-2911	348	2	4.1	4.1	NUM
iajs-2911	348	3	.	.	PUNCT
iajs-2911	349	1	a	a	DET
iajs-2911	349	2	topological	topological	ADJ
iajs-2911	349	3	space	space	NOUN
iajs-2911	349	4	(	(	PUNCT
iajs-2911	349	5	e	e	NOUN
iajs-2911	349	6	,	,	PUNCT
iajs-2911	349	7	τ	τ	X
iajs-2911	349	8	)	)	PUNCT
iajs-2911	349	9	is	be	AUX
iajs-2911	349	10	𝔽.	𝔽.	PROPN
iajs-2911	349	11	𝕎.u.t.s.(resp	𝕎.u.t.s.(resp	NUM
iajs-2911	349	12	.	.	PUNCT
iajs-2911	349	13	,	,	PUNCT
iajs-2911	349	14	𝔽.	𝔽.	PROPN
iajs-2911	349	15	𝕎.l.t.s	𝕎.l.t.s	PROPN
iajs-2911	349	16	.	.	PUNCT
iajs-2911	349	17	)	)	PUNCT
iajs-2911	350	1	on	on	ADP
iajs-2911	350	2	(	(	PUNCT
iajs-2911	350	3	d	d	NOUN
iajs-2911	350	4	,	,	PUNCT
iajs-2911	350	5	ρ	ρ	NOUN
iajs-2911	350	6	)	)	PUNCT
iajs-2911	350	7	if	if	SCONJ
iajs-2911	350	8	xe(cl(𝒜	xe(cl(𝒜	PROPN
iajs-2911	350	9	)	)	PUNCT
iajs-2911	350	10	)	)	PUNCT
iajs-2911	350	11	⊂	⊂	PROPN
iajs-2911	350	12	cl(xe(𝒜	cl(xe(𝒜	PROPN
iajs-2911	350	13	)	)	PUNCT
iajs-2911	350	14	)	)	PUNCT
iajs-2911	350	15	,	,	PUNCT
iajs-2911	350	16	for	for	ADP
iajs-2911	350	17	each	each	DET
iajs-2911	350	18	𝒜	𝒜	NOUN
iajs-2911	350	19	⊂	⊂	CCONJ
iajs-2911	350	20	e	e	NOUN
iajs-2911	350	21	.	.	PUNCT
iajs-2911	351	1	proof	proof	NOUN
iajs-2911	351	2	.	.	PUNCT
iajs-2911	352	1	(	(	PUNCT
iajs-2911	352	2	⇒	⇒	NOUN
iajs-2911	352	3	)	)	PUNCT
iajs-2911	352	4	assume	assume	VERB
iajs-2911	352	5	that	that	SCONJ
iajs-2911	352	6	e	e	PRON
iajs-2911	352	7	is	be	AUX
iajs-2911	352	8	𝔽.	𝔽.	PROPN
iajs-2911	352	9	𝕎.u.t.s.(resp	𝕎.u.t.s.(resp	NUM
iajs-2911	352	10	.	.	PUNCT
iajs-2911	352	11	,	,	PUNCT
iajs-2911	352	12	𝔽.	𝔽.	PROPN
iajs-2911	352	13	𝕎.l.t.s	𝕎.l.t.s	PROPN
iajs-2911	352	14	.	.	PUNCT
iajs-2911	352	15	)	)	PUNCT
iajs-2911	353	1	on	on	ADP
iajs-2911	353	2	d	d	NOUN
iajs-2911	353	3	then	then	ADV
iajs-2911	353	4	the	the	DET
iajs-2911	353	5	projection	projection	NOUN
iajs-2911	353	6	xe	xe	PROPN
iajs-2911	353	7	:	:	PUNCT
iajs-2911	353	8	e→	e→	PROPN
iajs-2911	353	9	d	d	NOUN
iajs-2911	353	10	exist	exist	VERB
iajs-2911	353	11	and	and	CCONJ
iajs-2911	353	12	it	it	PRON
iajs-2911	353	13	is	be	AUX
iajs-2911	353	14	u.	u.	PROPN
iajs-2911	353	15	cont	cont	PROPN
iajs-2911	353	16	.	.	PUNCT
iajs-2911	354	1	(	(	PUNCT
iajs-2911	354	2	resp	resp	NOUN
iajs-2911	354	3	.	.	PUNCT
iajs-2911	354	4	,	,	PUNCT
iajs-2911	354	5	l.	l.	PROPN
iajs-2911	354	6	cont	cont	PROPN
iajs-2911	354	7	.	.	PUNCT
iajs-2911	354	8	)	)	PUNCT
iajs-2911	354	9	.	.	PUNCT
iajs-2911	355	1	suppose	suppose	VERB
iajs-2911	355	2	that	that	SCONJ
iajs-2911	355	3	e	e	PROPN
iajs-2911	355	4	∈	∈	PROPN
iajs-2911	355	5	cl(𝒜	cl(𝒜	NOUN
iajs-2911	355	6	)	)	PUNCT
iajs-2911	355	7	and	and	CCONJ
iajs-2911	355	8	d	d	NOUN
iajs-2911	355	9	is	be	AUX
iajs-2911	355	10	𝜌−open	𝜌−open	ADP
iajs-2911	355	11	a	a	DET
iajs-2911	355	12	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	355	13	of	of	ADP
iajs-2911	355	14	ω(e	ω(e	NOUN
iajs-2911	355	15	)	)	PUNCT
iajs-2911	355	16	.	.	PUNCT
iajs-2911	356	1	since	since	SCONJ
iajs-2911	356	2	xe	xe	PROPN
iajs-2911	356	3	is	be	AUX
iajs-2911	356	4	u.	u.	PROPN
iajs-2911	356	5	cont	cont	PROPN
iajs-2911	356	6	.	.	PUNCT
iajs-2911	357	1	(	(	PUNCT
iajs-2911	357	2	resp	resp	NOUN
iajs-2911	357	3	.	.	PUNCT
iajs-2911	357	4	,	,	PUNCT
iajs-2911	357	5	l.	l.	PROPN
iajs-2911	357	6	cont	cont	PROPN
iajs-2911	357	7	.	.	PUNCT
iajs-2911	357	8	)	)	PUNCT
iajs-2911	358	1	,	,	PUNCT
iajs-2911	358	2	∃	∃	PROPN
iajs-2911	358	3	an	an	DET
iajs-2911	358	4	𝜏-open	𝜏-open	NOUN
iajs-2911	358	5	a	a	DET
iajs-2911	358	6	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	358	7	𝔘	𝔘	NOUN
iajs-2911	358	8	of	of	ADP
iajs-2911	358	9	e	e	NOUN
iajs-2911	358	10	such	such	ADJ
iajs-2911	358	11	that	that	PRON
iajs-2911	358	12	x(cl(𝔘	x(cl(𝔘	PROPN
iajs-2911	358	13	)	)	PUNCT
iajs-2911	358	14	)	)	PUNCT
iajs-2911	359	1	⊂	⊂	PROPN
iajs-2911	359	2	cl(v	cl(v	NOUN
iajs-2911	359	3	)	)	PUNCT
iajs-2911	359	4	.	.	PUNCT
iajs-2911	360	1	since	since	SCONJ
iajs-2911	360	2	cl(𝔘	cl(𝔘	NOUN
iajs-2911	360	3	)	)	PUNCT
iajs-2911	360	4	∩	∩	NOUN
iajs-2911	360	5	𝒜	𝒜	NOUN
iajs-2911	360	6	≠	≠	PROPN
iajs-2911	360	7	∅	∅	NOUN
iajs-2911	360	8	,	,	PUNCT
iajs-2911	360	9	then	then	ADV
iajs-2911	360	10	cl(v	cl(v	NOUN
iajs-2911	360	11	)	)	PUNCT
iajs-2911	360	12	∩	∩	NOUN
iajs-2911	360	13	x(𝒜	x(𝒜	X
iajs-2911	360	14	)	)	PUNCT
iajs-2911	360	15	≠	≠	PROPN
iajs-2911	360	16	∅.	∅.	VERB
iajs-2911	360	17	so	so	ADV
iajs-2911	360	18	,	,	PUNCT
iajs-2911	360	19	xe(𝒜	xe(𝒜	NOUN
iajs-2911	360	20	)	)	PUNCT
iajs-2911	360	21	∈	∈	PROPN
iajs-2911	360	22	cl(xe(𝒜	cl(xe(𝒜	PROPN
iajs-2911	360	23	)	)	PUNCT
iajs-2911	360	24	)	)	PUNCT
iajs-2911	360	25	.	.	PUNCT
iajs-2911	361	1	this	this	PRON
iajs-2911	361	2	shows	show	VERB
iajs-2911	361	3	that	that	SCONJ
iajs-2911	361	4	xe(cl(𝔘	xe(cl(𝔘	PROPN
iajs-2911	361	5	)	)	PUNCT
iajs-2911	361	6	)	)	PUNCT
iajs-2911	362	1	⊂	⊂	PROPN
iajs-2911	362	2	cl(xe(v	cl(xe(v	PROPN
iajs-2911	362	3	)	)	PUNCT
iajs-2911	362	4	)	)	PUNCT
iajs-2911	362	5	.	.	PUNCT
iajs-2911	363	1	(	(	PUNCT
iajs-2911	363	2	⇐	⇐	NOUN
iajs-2911	363	3	)	)	PUNCT
iajs-2911	363	4	it	it	PRON
iajs-2911	363	5	is	be	AUX
iajs-2911	363	6	clear	clear	ADJ
iajs-2911	363	7	.	.	PUNCT
iajs-2911	364	1	corollary	corollary	ADJ
iajs-2911	364	2	4.1	4.1	NUM
iajs-2911	364	3	.	.	PUNCT
iajs-2911	365	1	a	a	DET
iajs-2911	365	2	topological	topological	ADJ
iajs-2911	365	3	space	space	NOUN
iajs-2911	365	4	(	(	PUNCT
iajs-2911	365	5	e,𝜏	e,𝜏	NOUN
iajs-2911	365	6	)	)	PUNCT
iajs-2911	365	7	is	be	AUX
iajs-2911	365	8	f.w.m.t.s	f.w.m.t.s	ADJ
iajs-2911	365	9	on	on	ADP
iajs-2911	365	10	(	(	PUNCT
iajs-2911	365	11	d,𝜌	d,𝜌	NOUN
iajs-2911	365	12	)	)	PUNCT
iajs-2911	365	13	if	if	SCONJ
iajs-2911	365	14	xe(cl(𝒜	xe(cl(𝒜	PROPN
iajs-2911	365	15	)	)	PUNCT
iajs-2911	365	16	)	)	PUNCT
iajs-2911	366	1	⊂	⊂	PROPN
iajs-2911	366	2	cl(xe(𝒜	cl(xe(𝒜	PROPN
iajs-2911	366	3	)	)	PUNCT
iajs-2911	366	4	)	)	PUNCT
iajs-2911	366	5	.	.	PUNCT
iajs-2911	367	1	theorem	theorem	VERB
iajs-2911	367	2	4.2	4.2	NUM
iajs-2911	367	3	.	.	PUNCT
iajs-2911	368	1	let	let	VERB
iajs-2911	368	2	(	(	PUNCT
iajs-2911	368	3	e,𝜏	e,𝜏	NOUN
iajs-2911	368	4	)	)	PUNCT
iajs-2911	368	5	i𝑠	i𝑠	VERB
iajs-2911	368	6	𝔽.	𝔽.	PROPN
iajs-2911	368	7	𝕎.u.p.t.s.(resp	𝕎.u.p.t.s.(resp	PROPN
iajs-2911	368	8	.	.	PROPN
iajs-2911	368	9	,	,	PUNCT
iajs-2911	368	10	𝔽.	𝔽.	PROPN
iajs-2911	368	11	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	368	12	.	.	PUNCT
iajs-2911	368	13	)	)	PUNCT
iajs-2911	369	1	on	on	ADP
iajs-2911	369	2	(	(	PUNCT
iajs-2911	369	3	d,𝜌	d,𝜌	NOUN
iajs-2911	369	4	)	)	PUNCT
iajs-2911	369	5	.	.	PUNCT
iajs-2911	370	1	so	so	ADV
iajs-2911	370	2	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	370	3	+	+	PROPN
iajs-2911	370	4	(	(	PUNCT
iajs-2911	370	5	resp	resp	NOUN
iajs-2911	370	6	.	.	PUNCT
iajs-2911	370	7	,	,	PUNCT
iajs-2911	370	8	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	370	9	−	−	PROPN
iajs-2911	370	10	)	)	PUNCT
iajs-2911	370	11	preserves	preserve	VERB
iajs-2911	370	12	u.r	u.r	PROPN
iajs-2911	370	13	.	.	PUNCT
iajs-2911	371	1	(	(	PUNCT
iajs-2911	371	2	resp	resp	PROPN
iajs-2911	371	3	.	.	PUNCT
iajs-2911	371	4	,	,	PUNCT
iajs-2911	371	5	l.r	l.r	PROPN
iajs-2911	371	6	.	.	PUNCT
iajs-2911	371	7	)	)	PUNCT
iajs-2911	371	8	.	.	PUNCT
iajs-2911	372	1	proof	proof	NOUN
iajs-2911	372	2	.	.	PUNCT
iajs-2911	373	1	assume	assume	VERB
iajs-2911	373	2	that	that	SCONJ
iajs-2911	373	3	e	e	PRON
iajs-2911	373	4	is	be	AUX
iajs-2911	373	5	a	a	DET
iajs-2911	373	6	𝔽.	𝔽.	PROPN
iajs-2911	373	7	𝕎.u.p.t.s.(resp	𝕎.u.p.t.s.(resp	PROPN
iajs-2911	373	8	.	.	PROPN
iajs-2911	373	9	,	,	PUNCT
iajs-2911	373	10	𝔽.	𝔽.	PROPN
iajs-2911	373	11	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	373	12	.	.	PUNCT
iajs-2911	373	13	)	)	PUNCT
iajs-2911	374	1	on	on	ADP
iajs-2911	374	2	d	d	NOUN
iajs-2911	374	3	,	,	PUNCT
iajs-2911	374	4	then	then	ADV
iajs-2911	374	5	the	the	DET
iajs-2911	374	6	projection	projection	NOUN
iajs-2911	374	7	xe	xe	PROPN
iajs-2911	374	8	:	:	PUNCT
iajs-2911	374	9	e	e	X
iajs-2911	374	10	→	→	PUNCT
iajs-2911	374	11	d	d	NOUN
iajs-2911	374	12	exist	exist	VERB
iajs-2911	374	13	and	and	CCONJ
iajs-2911	374	14	it	it	PRON
iajs-2911	374	15	is	be	AUX
iajs-2911	374	16	u.	u.	PROPN
iajs-2911	374	17	cont	cont	PROPN
iajs-2911	374	18	.	.	PUNCT
iajs-2911	375	1	(	(	PUNCT
iajs-2911	375	2	resp	resp	NOUN
iajs-2911	375	3	.	.	PUNCT
iajs-2911	375	4	,	,	PUNCT
iajs-2911	375	5	l.	l.	PROPN
iajs-2911	375	6	cont	cont	PROPN
iajs-2911	375	7	.	.	PUNCT
iajs-2911	375	8	)	)	PUNCT
iajs-2911	375	9	.	.	PUNCT
iajs-2911	376	1	le𝑡	le𝑡	NOUN
iajs-2911	376	2	𝒜	𝒜	PROPN
iajs-2911	376	3	be	be	AUX
iajs-2911	376	4	an	an	DET
iajs-2911	376	5	u.r	u.r	PROPN
iajs-2911	376	6	.	.	PROPN
iajs-2911	376	7	set(resp	set(resp	PROPN
iajs-2911	376	8	.	.	PROPN
iajs-2911	376	9	,	,	PUNCT
iajs-2911	376	10	l.r	l.r	PROPN
iajs-2911	376	11	.	.	PROPN
iajs-2911	376	12	set	set	NOUN
iajs-2911	376	13	)	)	PUNCT
iajs-2911	376	14	in	in	ADP
iajs-2911	376	15	d	d	NOUN
iajs-2911	376	16	and	and	CCONJ
iajs-2911	376	17	let	let	VERB
iajs-2911	376	18	ℑ	ℑ	PRON
iajs-2911	376	19	be	be	AUX
iajs-2911	376	20	a	a	DET
iajs-2911	376	21	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	376	22	on	on	ADP
iajs-2911	376	23	𝔼	𝔼	PROPN
iajs-2911	376	24	such	such	ADJ
iajs-2911	376	25	that	that	SCONJ
iajs-2911	376	26	𝐸𝒜∩	𝐸𝒜∩	PROPN
iajs-2911	376	27	(	(	PUNCT
iajs-2911	376	28	ad	ad	NOUN
iajs-2911	376	29	(	(	PUNCT
iajs-2911	376	30	ℑ	ℑ	NOUN
iajs-2911	376	31	)	)	PUNCT
iajs-2911	376	32	)	)	PUNCT
iajs-2911	377	1	=	=	PUNCT
iajs-2911	377	2	∅.	∅.	VERB
iajs-2911	377	3	by	by	ADP
iajs-2911	377	4	xe	xe	PROPN
iajs-2911	377	5	is	be	AUX
iajs-2911	377	6	u.r	u.r	PROPN
iajs-2911	377	7	.	.	PUNCT
iajs-2911	378	1	(	(	PUNCT
iajs-2911	378	2	resp	resp	PROPN
iajs-2911	378	3	.	.	PUNCT
iajs-2911	378	4	,	,	PUNCT
iajs-2911	378	5	l.r	l.r	PROPN
iajs-2911	378	6	.	.	PUNCT
iajs-2911	378	7	)	)	PUNCT
iajs-2911	378	8	.	.	PUNCT
iajs-2911	379	1	and	and	CCONJ
iajs-2911	379	2	𝒜	𝒜	NOUN
iajs-2911	379	3	∩	∩	ADJ
iajs-2911	379	4	xe(ad	xe(ad	PROPN
iajs-2911	379	5	(	(	PUNCT
iajs-2911	379	6	ℑ	ℑ	PROPN
iajs-2911	379	7	)	)	PUNCT
iajs-2911	379	8	)	)	PUNCT
iajs-2911	380	1	=	=	NOUN
iajs-2911	380	2	∅	∅	NOUN
iajs-2911	380	3	,	,	PUNCT
iajs-2911	380	4	by	by	ADP
iajs-2911	380	5	theorem	theorem	NOUN
iajs-2911	380	6	[	[	X
iajs-2911	380	7	(	(	PUNCT
iajs-2911	380	8	2.1	2.1	NUM
iajs-2911	380	9	.	.	PUNCT
iajs-2911	380	10	)	)	PUNCT
iajs-2911	381	1	(	(	PUNCT
iajs-2911	381	2	i	i	NOUN
iajs-2911	381	3	)	)	PUNCT
iajs-2911	381	4	⇒	⇒	PROPN
iajs-2911	381	5	(	(	PUNCT
iajs-2911	381	6	iii	iii	NOUN
iajs-2911	381	7	)	)	PUNCT
iajs-2911	381	8	]	]	PUNCT
iajs-2911	382	1	we	we	PRON
iajs-2911	382	2	get	get	VERB
iajs-2911	382	3	𝒜	𝒜	NOUN
iajs-2911	382	4	∩	∩	NOUN
iajs-2911	382	5	(	(	PUNCT
iajs-2911	382	6	ad	ad	NOUN
iajs-2911	382	7	(	(	PUNCT
iajs-2911	382	8	𝑋𝐸	𝑋𝐸	PROPN
iajs-2911	382	9	(	(	PUNCT
iajs-2911	382	10	ℑ	ℑ	PROPN
iajs-2911	382	11	)	)	PUNCT
iajs-2911	382	12	)	)	PUNCT
iajs-2911	382	13	)	)	PUNCT
iajs-2911	383	1	=	=	PUNCT
iajs-2911	383	2	∅.	∅.	ADP
iajs-2911	383	3	currently	currently	ADV
iajs-2911	383	4	,	,	PUNCT
iajs-2911	383	5	a	a	DET
iajs-2911	383	6	being	be	AUX
iajs-2911	383	7	an	an	DET
iajs-2911	383	8	u.r.(resp	u.r.(resp	NOUN
iajs-2911	383	9	.	.	PROPN
iajs-2911	383	10	,	,	PUNCT
iajs-2911	383	11	l.r	l.r	PROPN
iajs-2911	383	12	.	.	PROPN
iajs-2911	383	13	)	)	PUNCT
iajs-2911	384	1	set	set	VERB
iajs-2911	384	2	in	in	ADP
iajs-2911	384	3	d	d	PROPN
iajs-2911	384	4	,	,	PUNCT
iajs-2911	384	5	∃	∃	PROPN
iajs-2911	384	6	an	an	DET
iajs-2911	384	7	𝔽∈	𝔽∈	PROPN
iajs-2911	384	8	ℑ	ℑ	PROPN
iajs-2911	384	9	such	such	ADJ
iajs-2911	384	10	that	that	DET
iajs-2911	384	11	𝒜	𝒜	NOUN
iajs-2911	384	12	∩	∩	NOUN
iajs-2911	384	13	(	(	PUNCT
iajs-2911	384	14	cl(xe(ℑ	cl(xe(ℑ	NOUN
iajs-2911	384	15	)	)	PUNCT
iajs-2911	384	16	)	)	PUNCT
iajs-2911	384	17	)	)	PUNCT
iajs-2911	385	1	=	=	PUNCT
iajs-2911	385	2	∅.	∅.	ADV
iajs-2911	385	3	because	because	SCONJ
iajs-2911	385	4	xe	xe	PROPN
iajs-2911	385	5	is	be	AUX
iajs-2911	385	6	u.	u.	PROPN
iajs-2911	385	7	cont.(resp	cont.(resp	PROPN
iajs-2911	385	8	.	.	PROPN
iajs-2911	385	9	,	,	PUNCT
iajs-2911	385	10	l.	l.	PROPN
iajs-2911	385	11	cont	cont	PROPN
iajs-2911	385	12	.	.	PUNCT
iajs-2911	385	13	)	)	PUNCT
iajs-2911	386	1	and	and	CCONJ
iajs-2911	386	2	by	by	ADP
iajs-2911	386	3	theorem	theorem	NOUN
iajs-2911	386	4	(	(	PUNCT
iajs-2911	386	5	4.1	4.1	NUM
iajs-2911	386	6	.	.	PUNCT
iajs-2911	386	7	)	)	PUNCT
iajs-2911	387	1	it	it	PRON
iajs-2911	387	2	follows	follow	VERB
iajs-2911	387	3	tℎ𝑎t	tℎ𝑎t	NOUN
iajs-2911	387	4	𝒜	𝒜	NOUN
iajs-2911	387	5	∩	∩	NOUN
iajs-2911	387	6	xe(cl(ℑ	xe(cl(ℑ	NOUN
iajs-2911	387	7	)	)	PUNCT
iajs-2911	387	8	)	)	PUNCT
iajs-2911	388	1	=	=	PUNCT
iajs-2911	388	2	∅.	∅.	VERB
iajs-2911	388	3	tℎ𝑒n	tℎ𝑒n	VERB
iajs-2911	388	4	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	388	5	+	+	PROPN
iajs-2911	388	6	∩	∩	NOUN
iajs-2911	388	7	(	(	PUNCT
iajs-2911	388	8	cl(ℑ	cl(ℑ	NOUN
iajs-2911	388	9	)	)	PUNCT
iajs-2911	388	10	)	)	PUNCT
iajs-2911	389	1	=	=	SYM
iajs-2911	390	1	∅(𝑟𝑒𝑠𝑝.	∅(𝑟𝑒𝑠𝑝.	NOUN
iajs-2911	390	2	,	,	PUNCT
iajs-2911	390	3	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	390	4	−∩	−∩	PROPN
iajs-2911	390	5	(	(	PUNCT
iajs-2911	390	6	cl(ℑ	cl(ℑ	NOUN
iajs-2911	390	7	)	)	PUNCT
iajs-2911	390	8	)	)	PUNCT
iajs-2911	391	1	=	=	NOUN
iajs-2911	391	2	∅	∅	NOUN
iajs-2911	391	3	)	)	PUNCT
iajs-2911	391	4	.	.	PUNCT
iajs-2911	392	1	tℎ𝑒n	tℎ𝑒n	VERB
iajs-2911	392	2	t.p	t.p	NOUN
iajs-2911	392	3	.	.	PUNCT
iajs-2911	393	1	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	393	2	+	+	PROPN
iajs-2911	393	3	(	(	PUNCT
iajs-2911	393	4	𝑟𝑒𝑠𝑝.	𝑟𝑒𝑠𝑝.	NOUN
iajs-2911	393	5	,	,	PUNCT
iajs-2911	393	6	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	393	7	−	−	PROPN
iajs-2911	393	8	)	)	PUNCT
iajs-2911	393	9	is	be	AUX
iajs-2911	393	10	u.r.(r𝑒𝑠p	u.r.(r𝑒𝑠p	PROPN
iajs-2911	393	11	.	.	PROPN
iajs-2911	393	12	,	,	PUNCT
iajs-2911	393	13	l.r	l.r	PROPN
iajs-2911	393	14	.	.	PROPN
iajs-2911	393	15	)	)	PUNCT
iajs-2911	393	16	.	.	PUNCT
iajs-2911	394	1	w𝑒	w𝑒	AUX
iajs-2911	394	2	p𝑟𝑒𝑠𝑒𝑛t	p𝑟𝑒𝑠𝑒𝑛t	VERB
iajs-2911	394	3	tℎ𝑒	tℎ𝑒	PRON
iajs-2911	394	4	f𝑜𝑙𝑙𝑜𝑤𝑖𝑛g	f𝑜𝑙𝑙𝑜𝑤𝑖𝑛g	NOUN
iajs-2911	394	5	d𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜n	d𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜n	NOUN
iajs-2911	394	6	t𝑜	t𝑜	ADP
iajs-2911	394	7	𝑠𝑡𝑢dy	𝑠𝑡𝑢dy	PROPN
iajs-2911	394	8	tℎ𝑒	tℎ𝑒	PRON
iajs-2911	394	9	co𝑛𝑑𝑖𝑡𝑖𝑜𝑛s	co𝑛𝑑𝑖𝑡𝑖𝑜𝑛s	PROPN
iajs-2911	394	10	u𝑛𝑑𝑒r	u𝑛𝑑𝑒r	NOUN
iajs-2911	394	11	wℎ𝑖𝑐h	wℎ𝑖𝑐h	PROPN
iajs-2911	394	12	an	an	DET
iajs-2911	394	13	f.w	f.w	PROPN
iajs-2911	394	14	.	.	PROPN
iajs-2911	394	15	al𝑚𝑜𝑠t	al𝑚𝑜𝑠t	PROPN
iajs-2911	394	16	p𝑒𝑟𝑓𝑒𝑐t	p𝑒𝑟𝑓𝑒𝑐t	PROPN
iajs-2911	394	17	t𝑜p𝑜𝑙𝑜gical	t𝑜p𝑜𝑙𝑜gical	ADJ
iajs-2911	394	18	sp𝑎𝑐𝑒	sp𝑎𝑐𝑒	PROPN
iajs-2911	394	19	c𝑎𝑛	c𝑎𝑛	NOUN
iajs-2911	394	20	be	be	AUX
iajs-2911	394	21	an	an	DET
iajs-2911	394	22	𝔽.	𝔽.	PROPN
iajs-2911	394	23	𝕎.u.p.t.s.(r𝑒𝑠p	𝕎.u.p.t.s.(r𝑒𝑠p	PROPN
iajs-2911	394	24	.	.	PROPN
iajs-2911	394	25	,	,	PUNCT
iajs-2911	394	26	𝔽.	𝔽.	PROPN
iajs-2911	394	27	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	394	28	.	.	PUNCT
iajs-2911	394	29	)	)	PUNCT
iajs-2911	394	30	.	.	PUNCT
iajs-2911	395	1	corollary	corollary	ADJ
iajs-2911	395	2	4.2	4.2	NUM
iajs-2911	395	3	.	.	PUNCT
iajs-2911	396	1	let	let	VERB
iajs-2911	396	2	(	(	PUNCT
iajs-2911	396	3	e,𝜏	e,𝜏	NOUN
iajs-2911	396	4	)	)	PUNCT
iajs-2911	396	5	be	be	AUX
iajs-2911	396	6	𝔽.	𝔽.	PROPN
iajs-2911	396	7	𝕎.m.p.t.s	𝕎.m.p.t.s	X
iajs-2911	396	8	on	on	ADP
iajs-2911	396	9	(	(	PUNCT
iajs-2911	396	10	d,𝜌	d,𝜌	NOUN
iajs-2911	396	11	)	)	PUNCT
iajs-2911	396	12	.	.	PUNCT
iajs-2911	397	1	so	so	ADV
iajs-2911	397	2	𝐸𝒜	𝐸𝒜	PROPN
iajs-2911	397	3	preserves	preserve	VERB
iajs-2911	397	4	m.r	m.r	PROPN
iajs-2911	397	5	.	.	PROPN
iajs-2911	397	6	definition	definition	NOUN
iajs-2911	397	7	4.1	4.1	NUM
iajs-2911	397	8	.	.	PUNCT
iajs-2911	398	1	the	the	DET
iajs-2911	398	2	function	function	NOUN
iajs-2911	398	3	ω	ω	NOUN
iajs-2911	398	4	:	:	PUNCT
iajs-2911	398	5	(	(	PUNCT
iajs-2911	398	6	e	e	NOUN
iajs-2911	398	7	,	,	PUNCT
iajs-2911	398	8	τ	τ	X
iajs-2911	398	9	)	)	PUNCT
iajs-2911	398	10	→	→	SYM
iajs-2911	398	11	(	(	PUNCT
iajs-2911	398	12	f	f	X
iajs-2911	398	13	,	,	PUNCT
iajs-2911	398	14	σ	σ	PROPN
iajs-2911	398	15	)	)	PUNCT
iajs-2911	398	16	is	be	AUX
iajs-2911	398	17	named	name	VERB
iajs-2911	398	18	to	to	PART
iajs-2911	398	19	be	be	AUX
iajs-2911	398	20	upper∗	upper∗	ADJ
iajs-2911	398	21	continuous	continuous	ADJ
iajs-2911	398	22	(	(	PUNCT
iajs-2911	398	23	briefly	briefly	ADV
iajs-2911	398	24	,	,	PUNCT
iajs-2911	398	25	u∗.	u∗.	PROPN
iajs-2911	398	26	cont	cont	PROPN
iajs-2911	398	27	.	.	PUNCT
iajs-2911	398	28	)	)	PUNCT
iajs-2911	399	1	if	if	SCONJ
iajs-2911	399	2	for	for	ADP
iajs-2911	399	3	any	any	DET
iajs-2911	399	4	τ	τ	X
iajs-2911	399	5	-	-	NOUN
iajs-2911	399	6	open	open	VERB
iajs-2911	399	7	a	a	DET
iajs-2911	399	8	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	399	9	v	v	NOUN
iajs-2911	399	10	𝑜f	𝑜f	ADV
iajs-2911	399	11	ω+(e	ω+(e	NUM
iajs-2911	399	12	)	)	PUNCT
iajs-2911	399	13	,	,	PUNCT
iajs-2911	399	14	∃	∃	PROPN
iajs-2911	399	15	an	an	PRON
iajs-2911	399	16	𝜏−open	𝜏−open	PROPN
iajs-2911	399	17	a	a	DET
iajs-2911	399	18	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	399	19	𝔘	𝔘	NOUN
iajs-2911	399	20	of	of	ADP
iajs-2911	399	21	e	e	NOUN
iajs-2911	399	22	such	such	ADJ
iajs-2911	399	23	that	that	PRON
iajs-2911	399	24	ω(cl(𝔘	ω(cl(𝔘	NUM
iajs-2911	399	25	)	)	PUNCT
iajs-2911	399	26	)	)	PUNCT
iajs-2911	400	1	⊂	⊂	PROPN
iajs-2911	400	2	cl(v	cl(v	NOUN
iajs-2911	400	3	)	)	PUNCT
iajs-2911	400	4	.	.	PUNCT
iajs-2911	401	1	definition	definition	NOUN
iajs-2911	401	2	4.2	4.2	NUM
iajs-2911	401	3	.	.	PUNCT
iajs-2911	402	1	the	the	DET
iajs-2911	402	2	function	function	NOUN
iajs-2911	402	3	ω	ω	NOUN
iajs-2911	402	4	:	:	PUNCT
iajs-2911	402	5	(	(	PUNCT
iajs-2911	402	6	e	e	NOUN
iajs-2911	402	7	,	,	PUNCT
iajs-2911	402	8	τ	τ	X
iajs-2911	402	9	)	)	PUNCT
iajs-2911	402	10	→	→	SYM
iajs-2911	402	11	(	(	PUNCT
iajs-2911	402	12	f	f	X
iajs-2911	402	13	,	,	PUNCT
iajs-2911	402	14	σ	σ	PROPN
iajs-2911	402	15	)	)	PUNCT
iajs-2911	402	16	is	be	AUX
iajs-2911	402	17	named	name	VERB
iajs-2911	402	18	to	to	PART
iajs-2911	402	19	be	be	AUX
iajs-2911	402	20	lower	low	ADJ
iajs-2911	402	21	*	*	PUNCT
iajs-2911	402	22	continuous	continuous	ADJ
iajs-2911	402	23	(	(	PUNCT
iajs-2911	402	24	briefly	briefly	ADV
iajs-2911	402	25	,	,	PUNCT
iajs-2911	402	26	l	l	NOUN
iajs-2911	402	27	*	*	NOUN
iajs-2911	402	28	.	.	PUNCT
iajs-2911	403	1	cont	cont	PROPN
iajs-2911	403	2	.	.	PUNCT
iajs-2911	403	3	)	)	PUNCT
iajs-2911	404	1	if	if	SCONJ
iajs-2911	404	2	for	for	ADP
iajs-2911	404	3	any	any	DET
iajs-2911	404	4	τ	τ	X
iajs-2911	404	5	-	-	NOUN
iajs-2911	404	6	open	open	VERB
iajs-2911	404	7	a	a	DET
iajs-2911	404	8	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	404	9	v	v	NOUN
iajs-2911	404	10	of	of	ADP
iajs-2911	404	11	ω−(e	ω−(e	PROPN
iajs-2911	404	12	)	)	PUNCT
iajs-2911	404	13	,	,	PUNCT
iajs-2911	404	14	∃	∃	PROPN
iajs-2911	404	15	an	an	PRON
iajs-2911	404	16	𝜏−open	𝜏−open	PROPN
iajs-2911	404	17	a	a	DET
iajs-2911	404	18	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	404	19	𝔘	𝔘	NOUN
iajs-2911	404	20	of	of	ADP
iajs-2911	404	21	e	e	NOUN
iajs-2911	404	22	such	such	ADJ
iajs-2911	404	23	that	that	PRON
iajs-2911	404	24	ω(cl(𝔘	ω(cl(𝔘	NUM
iajs-2911	404	25	)	)	PUNCT
iajs-2911	404	26	)	)	PUNCT
iajs-2911	405	1	⊂	⊂	PROPN
iajs-2911	405	2	cl(v	cl(v	NOUN
iajs-2911	405	3	)	)	PUNCT
iajs-2911	405	4	.	.	PUNCT
iajs-2911	406	1	the	the	DET
iajs-2911	406	2	function	function	NOUN
iajs-2911	406	3	ω	ω	NOUN
iajs-2911	406	4	:	:	PUNCT
iajs-2911	406	5	(	(	PUNCT
iajs-2911	406	6	e	e	NOUN
iajs-2911	406	7	,	,	PUNCT
iajs-2911	406	8	τ	τ	X
iajs-2911	406	9	)	)	PUNCT
iajs-2911	406	10	→	→	SYM
iajs-2911	406	11	(	(	PUNCT
iajs-2911	406	12	f	f	X
iajs-2911	406	13	,	,	PUNCT
iajs-2911	406	14	σ	σ	PROPN
iajs-2911	406	15	)	)	PUNCT
iajs-2911	406	16	is	be	AUX
iajs-2911	406	17	named	name	VERB
iajs-2911	406	18	to	to	PART
iajs-2911	406	19	be	be	AUX
iajs-2911	406	20	multi∗	multi∗	PROPN
iajs-2911	406	21	-cont	-cont	PROPN
iajs-2911	406	22	.	.	PUNCT
iajs-2911	407	1	(	(	PUNCT
iajs-2911	407	2	briefly	briefly	ADV
iajs-2911	407	3	,	,	PUNCT
iajs-2911	407	4	m∗.	m∗.	X
iajs-2911	407	5	cont	cont	NOUN
iajs-2911	407	6	.	.	PUNCT
iajs-2911	407	7	)	)	PUNCT
iajs-2911	408	1	if	if	SCONJ
iajs-2911	408	2	it	it	PRON
iajs-2911	408	3	is	be	AUX
iajs-2911	408	4	l∗.	l∗.	NOUN
iajs-2911	408	5	cont	cont	NOUN
iajs-2911	408	6	.	.	PUNCT
iajs-2911	409	1	and	and	CCONJ
iajs-2911	409	2	u∗.	u∗.	PROPN
iajs-2911	409	3	cont	cont	PROPN
iajs-2911	409	4	.	.	PUNCT
iajs-2911	410	1	definition	definition	NOUN
iajs-2911	410	2	4.3	4.3	NUM
iajs-2911	410	3	.	.	PUNCT
iajs-2911	411	1	the	the	DET
iajs-2911	411	2	𝔽.	𝔽.	PROPN
iajs-2911	411	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	411	4	.	.	PUNCT
iajs-2911	411	5	(	(	PUNCT
iajs-2911	411	6	e,𝜏	e,𝜏	NOUN
iajs-2911	411	7	)	)	PUNCT
iajs-2911	411	8	on	on	ADP
iajs-2911	411	9	(	(	PUNCT
iajs-2911	411	10	f,𝜎	f,𝜎	NOUN
iajs-2911	411	11	)	)	PUNCT
iajs-2911	411	12	is	be	AUX
iajs-2911	411	13	named	name	VERB
iajs-2911	411	14	𝔽.	𝔽.	PROPN
iajs-2911	411	15	𝕎.u*.t.s	𝕎.u*.t.s	PROPN
iajs-2911	411	16	.	.	PUNCT
iajs-2911	412	1	if	if	SCONJ
iajs-2911	412	2	the	the	DET
iajs-2911	412	3	projection	projection	NOUN
iajs-2911	412	4	x	x	X
iajs-2911	412	5	is	be	AUX
iajs-2911	412	6	u∗.cont	u∗.cont	PROPN
iajs-2911	412	7	.	.	PUNCT
iajs-2911	412	8	definition	definition	NOUN
iajs-2911	412	9	4.4	4.4	NUM
iajs-2911	412	10	.	.	PUNCT
iajs-2911	413	1	the	the	DET
iajs-2911	413	2	𝔽.	𝔽.	PROPN
iajs-2911	413	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	413	4	.	.	PUNCT
iajs-2911	413	5	(	(	PUNCT
iajs-2911	413	6	e,𝜏	e,𝜏	NOUN
iajs-2911	413	7	)	)	PUNCT
iajs-2911	413	8	on	on	ADP
iajs-2911	413	9	(	(	PUNCT
iajs-2911	413	10	f,𝜎	f,𝜎	NOUN
iajs-2911	413	11	)	)	PUNCT
iajs-2911	413	12	is	be	AUX
iajs-2911	413	13	named	name	VERB
iajs-2911	413	14	𝔽.	𝔽.	PROPN
iajs-2911	413	15	𝕎.l*.t.s	𝕎.l*.t.s	PROPN
iajs-2911	413	16	.	.	PUNCT
iajs-2911	414	1	if	if	SCONJ
iajs-2911	414	2	the	the	DET
iajs-2911	414	3	projection	projection	NOUN
iajs-2911	414	4	x	x	X
iajs-2911	414	5	is	be	AUX
iajs-2911	414	6	l∗.cont	l∗.cont	PROPN
iajs-2911	414	7	.	.	PUNCT
iajs-2911	415	1	the	the	DET
iajs-2911	415	2	𝔽.	𝔽.	PROPN
iajs-2911	415	3	𝕎.t.s	𝕎.t.s	PROPN
iajs-2911	415	4	.	.	PUNCT
iajs-2911	415	5	(	(	PUNCT
iajs-2911	415	6	e,𝜏	e,𝜏	NOUN
iajs-2911	415	7	)	)	PUNCT
iajs-2911	415	8	on	on	ADP
iajs-2911	415	9	(	(	PUNCT
iajs-2911	415	10	f,𝜎	f,𝜎	NOUN
iajs-2911	415	11	)	)	PUNCT
iajs-2911	415	12	is	be	AUX
iajs-2911	415	13	named	name	VERB
iajs-2911	415	14	𝔽.	𝔽.	PROPN
iajs-2911	415	15	𝕎.m*.t.s	𝕎.m*.t.s	PROPN
iajs-2911	415	16	.	.	PUNCT
iajs-2911	416	1	if	if	SCONJ
iajs-2911	416	2	it	it	PRON
iajs-2911	416	3	is	be	AUX
iajs-2911	416	4	𝔽.	𝔽.	PROPN
iajs-2911	416	5	𝕎.l*.t.s	𝕎.l*.t.s	PROPN
iajs-2911	416	6	.	.	PROPN
iajs-2911	416	7	and	and	CCONJ
iajs-2911	416	8	𝔽.	𝔽.	PROPN
iajs-2911	416	9	𝕎.u*.t.s	𝕎.u*.t.s	PROPN
iajs-2911	416	10	.	.	NOUN
iajs-2911	416	11	importance	importance	NOUN
iajs-2911	416	12	of	of	ADP
iajs-2911	416	13	the	the	DET
iajs-2911	416	14	above	above	ADJ
iajs-2911	416	15	definition	definition	NOUN
iajs-2911	416	16	for	for	ADP
iajs-2911	416	17	characterization	characterization	NOUN
iajs-2911	416	18	of	of	ADP
iajs-2911	416	19	𝔽.	𝔽.	PROPN
iajs-2911	416	20	𝕎.u.p.t.s.(resp	𝕎.u.p.t.s.(resp	PROPN
iajs-2911	416	21	.	.	PROPN
iajs-2911	416	22	,	,	PUNCT
iajs-2911	416	23	𝔽.	𝔽.	PROPN
iajs-2911	416	24	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	416	25	.	.	PROPN
iajs-2911	416	26	and	and	CCONJ
iajs-2911	416	27	𝔽.	𝔽.	PROPN
iajs-2911	416	28	𝕎.m.p.t.s	𝕎.m.p.t.s	PROPN
iajs-2911	416	29	.	.	PUNCT
iajs-2911	416	30	)	)	PUNCT
iajs-2911	416	31	.	.	PUNCT
iajs-2911	417	1	it	it	PRON
iajs-2911	417	2	is	be	AUX
iajs-2911	417	3	quite	quite	ADV
iajs-2911	417	4	clear	clear	ADJ
iajs-2911	417	5	from	from	ADP
iajs-2911	417	6	the	the	DET
iajs-2911	417	7	next	next	ADJ
iajs-2911	417	8	result	result	NOUN
iajs-2911	417	9	.	.	PUNCT
iajs-2911	418	1	lemma	lemma	PROPN
iajs-2911	418	2	4.1.[27	4.1.[27	PROPN
iajs-2911	418	3	]	]	PUNCT
iajs-2911	418	4	in	in	ADP
iajs-2911	418	5	a	a	DET
iajs-2911	418	6	urysohn	urysohn	PROPN
iajs-2911	418	7	topological	topological	ADJ
iajs-2911	418	8	space	space	NOUN
iajs-2911	418	9	𝔼	𝔼	PROPN
iajs-2911	418	10	set	set	NOUN
iajs-2911	418	11	is	be	AUX
iajs-2911	418	12	closed	close	VERB
iajs-2911	418	13	set	set	VERB
iajs-2911	418	14	.	.	PUNCT
iajs-2911	419	1	ihjpas	ihjpas	PROPN
iajs-2911	419	2	.	.	PUNCT
iajs-2911	420	1	36	36	NUM
iajs-2911	420	2	(	(	PUNCT
iajs-2911	420	3	4	4	NUM
iajs-2911	420	4	)	)	PUNCT
iajs-2911	420	5	2023	2023	NUM
iajs-2911	420	6	404	404	NUM
iajs-2911	420	7	theorem	theorem	VERB
iajs-2911	420	8	4.3	4.3	NUM
iajs-2911	420	9	.	.	PUNCT
iajs-2911	421	1	if	if	SCONJ
iajs-2911	421	2	(	(	PUNCT
iajs-2911	421	3	e,𝜏	e,𝜏	NOUN
iajs-2911	421	4	)	)	PUNCT
iajs-2911	421	5	is	be	AUX
iajs-2911	421	6	𝔽.	𝔽.	PROPN
iajs-2911	421	7	𝕎.u∗.t.s.(resp	𝕎.u∗.t.s.(resp	NOUN
iajs-2911	421	8	.	.	PROPN
iajs-2911	421	9	,	,	PUNCT
iajs-2911	421	10	𝔽.	𝔽.	PROPN
iajs-2911	421	11	𝕎.l∗.t.s	𝕎.l∗.t.s	NOUN
iajs-2911	421	12	.	.	PUNCT
iajs-2911	421	13	)	)	PUNCT
iajs-2911	422	1	on	on	ADP
iajs-2911	422	2	a	a	DET
iajs-2911	422	3	te	te	PROPN
iajs-2911	422	4	(	(	PUNCT
iajs-2911	422	5	f,𝜎	f,𝜎	PROPN
iajs-2911	422	6	)	)	PUNCT
iajs-2911	422	7	,	,	PUNCT
iajs-2911	422	8	so	so	CCONJ
iajs-2911	422	9	it	it	PRON
iajs-2911	422	10	is	be	AUX
iajs-2911	422	11	𝔽.	𝔽.	PROPN
iajs-2911	422	12	𝕎.u.p.t.s.(resp	𝕎.u.p.t.s.(resp	PROPN
iajs-2911	422	13	.	.	PROPN
iajs-2911	422	14	,	,	PUNCT
iajs-2911	422	15	𝔽.	𝔽.	PROPN
iajs-2911	422	16	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	422	17	.	.	PUNCT
iajs-2911	422	18	)	)	PUNCT
iajs-2911	423	1	if	if	SCONJ
iajs-2911	423	2	∀	∀	NOUN
iajs-2911	423	3	f∗.b∗	f∗.b∗	VERB
iajs-2911	423	4	on	on	ADP
iajs-2911	423	5	e	e	NOUN
iajs-2911	423	6	,	,	PUNCT
iajs-2911	423	7	if	if	SCONJ
iajs-2911	423	8	𝑋ℑ	𝑋ℑ	PROPN
iajs-2911	423	9	−−conv	−−conv	NOUN
iajs-2911	423	10	.	.	PUNCT
iajs-2911	424	1	→d	→d	PUNCT
iajs-2911	424	2	;	;	PUNCT
iajs-2911	424	3	d	d	X
iajs-2911	424	4	∈	∈	PROPN
iajs-2911	424	5	𝐷	𝐷	PROPN
iajs-2911	424	6	,	,	PUNCT
iajs-2911	424	7	then	then	ADV
iajs-2911	424	8	ad	ad	NOUN
iajs-2911	424	9	ℑ	ℑ	PROPN
iajs-2911	424	10	≠	≠	PROPN
iajs-2911	424	11	∅.	∅.	PRON
iajs-2911	424	12	proof	proof	NOUN
iajs-2911	424	13	.	.	PUNCT
iajs-2911	425	1	(	(	PUNCT
iajs-2911	425	2	⇒	⇒	NOUN
iajs-2911	425	3	)	)	PUNCT
iajs-2911	425	4	assume	assume	VERB
iajs-2911	425	5	that	that	SCONJ
iajs-2911	425	6	(	(	PUNCT
iajs-2911	425	7	e	e	NOUN
iajs-2911	425	8	,	,	PUNCT
iajs-2911	425	9	τ	τ	X
iajs-2911	425	10	)	)	PUNCT
iajs-2911	425	11	is	be	AUX
iajs-2911	425	12	a	a	DET
iajs-2911	425	13	𝔽.	𝔽.	PROPN
iajs-2911	425	14	𝕎.u∗.t.s.(resp	𝕎.u∗.t.s.(resp	NOUN
iajs-2911	425	15	.	.	PROPN
iajs-2911	425	16	,	,	PUNCT
iajs-2911	425	17	𝔽.	𝔽.	PROPN
iajs-2911	425	18	𝕎.l∗.t.s	𝕎.l∗.t.s	NOUN
iajs-2911	425	19	.	.	PUNCT
iajs-2911	425	20	)	)	PUNCT
iajs-2911	426	1	on	on	ADP
iajs-2911	426	2	a	a	DET
iajs-2911	426	3	te	te	PROPN
iajs-2911	426	4	(	(	PUNCT
iajs-2911	426	5	d,𝜌	d,𝜌	PROPN
iajs-2911	426	6	)	)	PUNCT
iajs-2911	426	7	,	,	PUNCT
iajs-2911	426	8	then	then	ADV
iajs-2911	426	9	∃u∗.	∃u∗.	PROPN
iajs-2911	426	10	cont.(resp	cont.(resp	PROPN
iajs-2911	426	11	.	.	PROPN
iajs-2911	426	12	,	,	PUNCT
iajs-2911	426	13	l∗.	l∗.	PROPN
iajs-2911	426	14	cont	cont	PROPN
iajs-2911	426	15	.	.	PUNCT
iajs-2911	426	16	)	)	PUNCT
iajs-2911	427	1	projection	projection	NOUN
iajs-2911	427	2	function	function	NOUN
iajs-2911	427	3	xe	xe	PROPN
iajs-2911	427	4	:	:	PUNCT
iajs-2911	427	5	(	(	PUNCT
iajs-2911	427	6	e,𝜏	e,𝜏	NOUN
iajs-2911	427	7	)	)	PUNCT
iajs-2911	427	8	→	→	SYM
iajs-2911	427	9	(	(	PUNCT
iajs-2911	427	10	d,𝜌	d,𝜌	NOUN
iajs-2911	427	11	)	)	PUNCT
iajs-2911	427	12	and	and	CCONJ
iajs-2911	427	13	𝑋ℑ	𝑋ℑ	PROPN
iajs-2911	427	14	−−conv.→	−−conv.→	NOUN
iajs-2911	427	15	d	d	NOUN
iajs-2911	428	1	in	in	ADP
iajs-2911	428	2	which	which	PRON
iajs-2911	428	3	d	d	PROPN
iajs-2911	428	4	∈	∈	PROPN
iajs-2911	428	5	d	d	X
iajs-2911	428	6	,	,	PUNCT
iajs-2911	428	7	for	for	ADP
iajs-2911	428	8	a	a	DET
iajs-2911	428	9	f∗.b∗	f∗.b∗	NOUN
iajs-2911	428	10	on	on	ADP
iajs-2911	428	11	ℑ	ℑ	PROPN
iajs-2911	428	12	on	on	ADP
iajs-2911	428	13	e.	e.	PROPN
iajs-2911	428	14	so	so	SCONJ
iajs-2911	428	15	𝐸𝑋ℑ	𝐸𝑋ℑ	PROPN
iajs-2911	428	16	+	+	CCONJ
iajs-2911	428	17	−−	−−	NOUN
iajs-2911	428	18	dir.−→𝐸𝑑	dir.−→𝐸𝑑	VERB
iajs-2911	428	19	+	+	ADJ
iajs-2911	428	20	(	(	PUNCT
iajs-2911	428	21	resp	resp	NOUN
iajs-2911	428	22	.	.	PUNCT
iajs-2911	428	23	,	,	PUNCT
iajs-2911	428	24	𝐸𝑋ℑ	𝐸𝑋ℑ	PROPN
iajs-2911	428	25	−	−	PROPN
iajs-2911	428	26	−−	−−	NOUN
iajs-2911	428	27	dir.−→𝐸𝑑	dir.−→𝐸𝑑	VERB
iajs-2911	428	28	−	−	NOUN
iajs-2911	428	29	)	)	PUNCT
iajs-2911	428	30	.	.	PUNCT
iajs-2911	429	1	by	by	ADP
iajs-2911	429	2	ℑ	ℑ	PROPN
iajs-2911	429	3	is	be	AUX
iajs-2911	429	4	larger	large	ADJ
iajs-2911	429	5	than	than	ADP
iajs-2911	429	6	𝐸𝑋ℑ	𝐸𝑋ℑ	PROPN
iajs-2911	429	7	+	+	CCONJ
iajs-2911	429	8	(	(	PUNCT
iajs-2911	429	9	resp	resp	NOUN
iajs-2911	429	10	.	.	PUNCT
iajs-2911	430	1	,	,	PUNCT
iajs-2911	430	2	𝐸𝑋ℑ	𝐸𝑋ℑ	PROPN
iajs-2911	430	3	−	−	PROPN
iajs-2911	430	4	)	)	PUNCT
iajs-2911	430	5	,	,	PUNCT
iajs-2911	430	6	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	430	7	+	+	ADJ
iajs-2911	430	8	(	(	PUNCT
iajs-2911	430	9	resp	resp	NOUN
iajs-2911	430	10	.	.	PUNCT
iajs-2911	430	11	,	,	PUNCT
iajs-2911	431	1	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	431	2	−	−	NOUN
iajs-2911	431	3	)	)	PUNCT
iajs-2911	431	4	∩	∩	NOUN
iajs-2911	431	5	ad	ad	NOUN
iajs-2911	431	6	ℑ	ℑ	PROPN
iajs-2911	431	7	≠	≠	PROPN
iajs-2911	431	8	∅	∅	NOUN
iajs-2911	431	9	,	,	PUNCT
iajs-2911	431	10	so	so	SCONJ
iajs-2911	431	11	ad	ad	NOUN
iajs-2911	431	12	ℑ	ℑ	PROPN
iajs-2911	431	13	≠	≠	PROPN
iajs-2911	431	14	∅.	∅.	VERB
iajs-2911	431	15	(	(	PUNCT
iajs-2911	431	16	⇐	⇐	ADJ
iajs-2911	431	17	)	)	PUNCT
iajs-2911	431	18	assume	assume	VERB
iajs-2911	431	19	that	that	SCONJ
iajs-2911	431	20	∀	∀	PUNCT
iajs-2911	432	1	𝐹	𝐹	PROPN
iajs-2911	432	2	∗.	∗.	PROPN
iajs-2911	432	3	𝐵	𝐵	PROPN
iajs-2911	432	4	∗.	∗.	PROPN
iajs-2911	432	5	ℑ.	ℑ.	PROPN
iajs-2911	432	6	𝑜𝑛	𝑜𝑛	PROPN
iajs-2911	432	7	𝐸	𝐸	PROPN
iajs-2911	432	8	,	,	PUNCT
iajs-2911	432	9	𝑋ℑ	𝑋ℑ	PROPN
iajs-2911	432	10	−	−	PROPN
iajs-2911	432	11	−conv	−conv	NOUN
iajs-2911	432	12	.	.	PUNCT
iajs-2911	433	1	→	→	PUNCT
iajs-2911	433	2	𝑑	𝑑	X
iajs-2911	433	3	in	in	ADP
iajs-2911	433	4	which	which	PRON
iajs-2911	433	5	d	d	PROPN
iajs-2911	433	6	∈	∈	PROPN
iajs-2911	433	7	d	d	PROPN
iajs-2911	433	8	,	,	PUNCT
iajs-2911	433	9	implies	imply	VERB
iajs-2911	433	10	ad	ad	NOUN
iajs-2911	433	11	ℑ	ℑ	PROPN
iajs-2911	433	12	≠	≠	PROPN
iajs-2911	433	13	∅.	∅.	AUX
iajs-2911	433	14	let	let	VERB
iajs-2911	433	15	𝔔	𝔔	PROPN
iajs-2911	433	16	be	be	AUX
iajs-2911	433	17	a	a	DET
iajs-2911	433	18	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	433	19	on	on	ADP
iajs-2911	433	20	d	d	X
iajs-2911	433	21	such	such	ADJ
iajs-2911	433	22	that	that	SCONJ
iajs-2911	433	23	𝔔	𝔔	PROPN
iajs-2911	433	24	–	–	PUNCT
iajs-2911	433	25	conv.→	conv.→	PROPN
iajs-2911	433	26	d	d	NOUN
iajs-2911	433	27	,	,	PUNCT
iajs-2911	433	28	and	and	CCONJ
iajs-2911	433	29	let	let	VERB
iajs-2911	433	30	𝔔	𝔔	PROPN
iajs-2911	433	31	∗	∗	NOUN
iajs-2911	433	32	be	be	AUX
iajs-2911	433	33	a	a	DET
iajs-2911	433	34	f∗.b∗	f∗.b∗	NOUN
iajs-2911	433	35	on	on	ADP
iajs-2911	433	36	e	e	NOUN
iajs-2911	433	37	,	,	PUNCT
iajs-2911	433	38	such	such	ADJ
iajs-2911	433	39	that	that	SCONJ
iajs-2911	433	40	𝔔	𝔔	PROPN
iajs-2911	433	41	∗	∗	NOUN
iajs-2911	433	42	is	be	AUX
iajs-2911	433	43	larger	large	ADJ
iajs-2911	433	44	than	than	ADP
iajs-2911	433	45	𝐸𝔔.	𝐸𝔔.	NOUN
iajs-2911	433	46	then	then	ADV
iajs-2911	433	47	𝑋𝔔∗	𝑋𝔔∗	NOUN
iajs-2911	433	48	is	be	AUX
iajs-2911	433	49	larger	large	ADJ
iajs-2911	433	50	than	than	ADP
iajs-2911	433	51	𝔔.	𝔔.	PROPN
iajs-2911	433	52	so	so	ADV
iajs-2911	433	53	𝑋𝔔∗−−conv.→	𝑋𝔔∗−−conv.→	VERB
iajs-2911	433	54	d.	d.	NOUN
iajs-2911	433	55	so	so	ADV
iajs-2911	433	56	,	,	PUNCT
iajs-2911	433	57	ad	ad	NOUN
iajs-2911	433	58	𝔔	𝔔	PROPN
iajs-2911	433	59	∗	∗	NOUN
iajs-2911	433	60	≠	≠	PROPN
iajs-2911	433	61	∅.	∅.	AUX
iajs-2911	433	62	let	let	VERB
iajs-2911	433	63	z	z	NOUN
iajs-2911	433	64	∈	∈	PROPN
iajs-2911	434	1	d	d	ADP
iajs-2911	434	2	such	such	ADJ
iajs-2911	434	3	that	that	SCONJ
iajs-2911	434	4	z	z	PROPN
iajs-2911	434	5	≠	≠	PROPN
iajs-2911	434	6	d.	d.	NOUN
iajs-2911	434	7	so	so	ADV
iajs-2911	434	8	,	,	PUNCT
iajs-2911	434	9	by	by	ADP
iajs-2911	434	10	d	d	PROPN
iajs-2911	434	11	is	be	AUX
iajs-2911	434	12	u.(resp	u.(resp	PROPN
iajs-2911	434	13	.	.	PROPN
iajs-2911	434	14	,	,	PUNCT
iajs-2911	434	15	l.	l.	PROPN
iajs-2911	434	16	)	)	PUNCT
iajs-2911	435	1	te	te	PROPN
iajs-2911	435	2	,	,	PUNCT
iajs-2911	435	3	∃	∃	PROPN
iajs-2911	435	4	𝜌−open	𝜌−open	PROPN
iajs-2911	435	5	a	a	DET
iajs-2911	435	6	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	435	7	𝔘	𝔘	NOUN
iajs-2911	435	8	of	of	ADP
iajs-2911	435	9	d	d	PROPN
iajs-2911	435	10	and	and	CCONJ
iajs-2911	435	11	𝜌−open	𝜌−open	X
iajs-2911	435	12	a	a	DET
iajs-2911	435	13	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	435	14	v	v	NOUN
iajs-2911	435	15	of	of	ADP
iajs-2911	435	16	z	z	NOUN
iajs-2911	435	17	such	such	ADJ
iajs-2911	435	18	that	that	SCONJ
iajs-2911	435	19	(	(	PUNCT
iajs-2911	435	20	𝜌	𝜌	ADP
iajs-2911	435	21	−	−	PROPN
iajs-2911	435	22	cl(𝔘	cl(𝔘	NOUN
iajs-2911	435	23	)	)	PUNCT
iajs-2911	435	24	)	)	PUNCT
iajs-2911	435	25	∩	∩	NOUN
iajs-2911	435	26	(	(	PUNCT
iajs-2911	435	27	𝜌	𝜌	X
iajs-2911	435	28	−	−	NOUN
iajs-2911	435	29	cl(v	cl(v	NOUN
iajs-2911	435	30	)	)	PUNCT
iajs-2911	435	31	)	)	PUNCT
iajs-2911	436	1	=	=	PUNCT
iajs-2911	436	2	∅.	∅.	NOUN
iajs-2911	436	3	since	since	SCONJ
iajs-2911	436	4	𝑋𝔔∗	𝑋𝔔∗	PRON
iajs-2911	436	5	−−conv.→	−−conv.→	NOUN
iajs-2911	436	6	d	d	PROPN
iajs-2911	436	7	,	,	PUNCT
iajs-2911	436	8	∃	∃	PROPN
iajs-2911	436	9	a	a	DET
iajs-2911	436	10	g	g	PROPN
iajs-2911	436	11	∈	∈	PROPN
iajs-2911	436	12	𝔔	𝔔	PROPN
iajs-2911	436	13	∗	∗	VERB
iajs-2911	436	14	such	such	ADJ
iajs-2911	436	15	that	that	PRON
iajs-2911	436	16	xg	xg	PROPN
iajs-2911	436	17	⊂	⊂	PROPN
iajs-2911	436	18	𝜌	𝜌	ADP
iajs-2911	436	19	−	−	PROPN
iajs-2911	436	20	cl(𝔘	cl(𝔘	NOUN
iajs-2911	436	21	)	)	PUNCT
iajs-2911	436	22	.	.	PUNCT
iajs-2911	437	1	currently	currently	ADV
iajs-2911	437	2	,	,	PUNCT
iajs-2911	437	3	by	by	ADP
iajs-2911	437	4	x	x	SYM
iajs-2911	437	5	is	be	AUX
iajs-2911	437	6	u∗.	u∗.	PROPN
iajs-2911	437	7	cont	cont	PROPN
iajs-2911	437	8	.	.	PUNCT
iajs-2911	438	1	(	(	PUNCT
iajs-2911	438	2	resp	resp	NOUN
iajs-2911	438	3	.	.	PUNCT
iajs-2911	438	4	,	,	PUNCT
iajs-2911	438	5	l∗.	l∗.	PROPN
iajs-2911	438	6	cont	cont	PROPN
iajs-2911	438	7	.	.	PUNCT
iajs-2911	438	8	)	)	PUNCT
iajs-2911	439	1	,	,	PUNCT
iajs-2911	439	2	corresponding	correspond	VERB
iajs-2911	439	3	to	to	ADP
iajs-2911	439	4	every	every	DET
iajs-2911	439	5	e	e	PROPN
iajs-2911	439	6	∈	∈	PROPN
iajs-2911	439	7	ez	ez	PROPN
iajs-2911	439	8	,	,	PUNCT
iajs-2911	439	9	∃	∃	PROPN
iajs-2911	439	10	𝜏−open	𝜏−open	AUX
iajs-2911	439	11	a	a	DET
iajs-2911	439	12	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	439	13	w	w	NOUN
iajs-2911	439	14	of	of	ADP
iajs-2911	439	15	e	e	NOUN
iajs-2911	439	16	such	such	ADJ
iajs-2911	439	17	that	that	SCONJ
iajs-2911	439	18	x(𝜏	x(𝜏	PROPN
iajs-2911	439	19	−	−	PROPN
iajs-2911	439	20	cl(v	cl(v	NOUN
iajs-2911	439	21	)	)	PUNCT
iajs-2911	439	22	)	)	PUNCT
iajs-2911	439	23	.	.	PUNCT
iajs-2911	440	1	thus	thus	ADV
iajs-2911	440	2	,	,	PUNCT
iajs-2911	440	3	𝜌	𝜌	PUNCT
iajs-2911	440	4	−	−	PROPN
iajs-2911	440	5	cl(w	cl(w	NOUN
iajs-2911	440	6	∩g	∩g	ADJ
iajs-2911	440	7	)	)	PUNCT
iajs-2911	441	1	=	=	PUNCT
iajs-2911	441	2	∅.	∅.	NOUN
iajs-2911	441	3	it	it	PRON
iajs-2911	441	4	follows	follow	VERB
iajs-2911	441	5	that	that	SCONJ
iajs-2911	441	6	𝐸𝑍	𝐸𝑍	PROPN
iajs-2911	441	7	+	+	PROPN
iajs-2911	441	8	(	(	PUNCT
iajs-2911	441	9	resp	resp	NOUN
iajs-2911	441	10	.	.	PUNCT
iajs-2911	442	1	,	,	PUNCT
iajs-2911	442	2	𝐸𝑍	𝐸𝑍	PROPN
iajs-2911	442	3	−	−	NOUN
iajs-2911	442	4	)	)	PUNCT
iajs-2911	442	5	∩	∩	NOUN
iajs-2911	442	6	𝔔	𝔔	PROPN
iajs-2911	442	7	∗	∗	NOUN
iajs-2911	442	8	=	=	SYM
iajs-2911	442	9	∅	∅	NOUN
iajs-2911	442	10	,	,	PUNCT
iajs-2911	442	11	∀	∀	X
iajs-2911	442	12	z	z	NOUN
iajs-2911	442	13	∈	∈	PROPN
iajs-2911	443	1	d	d	X
iajs-2911	443	2	−{d	−{d	ADV
iajs-2911	443	3	}	}	PUNCT
iajs-2911	443	4	.	.	PUNCT
iajs-2911	444	1	consequently	consequently	ADV
iajs-2911	444	2	,	,	PUNCT
iajs-2911	444	3	𝐸𝑑	𝐸𝑑	PROPN
iajs-2911	444	4	+	+	NUM
iajs-2911	444	5	∩	∩	ADJ
iajs-2911	444	6	ad	ad	NOUN
iajs-2911	444	7	𝔔	𝔔	PROPN
iajs-2911	444	8	∗	∗	PROPN
iajs-2911	444	9	≠	≠	PROPN
iajs-2911	444	10	∅(resp	∅(resp	NUM
iajs-2911	444	11	.	.	PUNCT
iajs-2911	444	12	,	,	PUNCT
iajs-2911	444	13	𝐸𝑑	𝐸𝑑	PRON
iajs-2911	444	14	−	−	NOUN
iajs-2911	444	15	∩	∩	ADJ
iajs-2911	444	16	ad	ad	NOUN
iajs-2911	444	17	𝔔	𝔔	PROPN
iajs-2911	444	18	∗	∗	NOUN
iajs-2911	444	19	≠	≠	PROPN
iajs-2911	444	20	∅	∅	NOUN
iajs-2911	444	21	)	)	PUNCT
iajs-2911	444	22	,	,	PUNCT
iajs-2911	444	23	and	and	CCONJ
iajs-2911	444	24	x	x	X
iajs-2911	444	25	is	be	AUX
iajs-2911	444	26	u.p.(resp	u.p.(resp	PROPN
iajs-2911	444	27	.	.	PROPN
iajs-2911	444	28	,	,	PUNCT
iajs-2911	444	29	l.p	l.p	PROPN
iajs-2911	444	30	.	.	PUNCT
iajs-2911	444	31	)	)	PUNCT
iajs-2911	445	1	a𝑛d	a𝑛d	VERB
iajs-2911	446	1	so	so	ADV
iajs-2911	446	2	(	(	PUNCT
iajs-2911	446	3	e,𝜏	e,𝜏	NOUN
iajs-2911	446	4	)	)	PUNCT
iajs-2911	446	5	is	be	AUX
iajs-2911	446	6	𝔽.	𝔽.	PROPN
iajs-2911	446	7	𝕎.u∗.t.s.(resp	𝕎.u∗.t.s.(resp	NOUN
iajs-2911	446	8	.	.	PROPN
iajs-2911	446	9	,	,	PUNCT
iajs-2911	446	10	𝔽.	𝔽.	PROPN
iajs-2911	446	11	𝕎.l∗.t.s	𝕎.l∗.t.s	NOUN
iajs-2911	446	12	.	.	PUNCT
iajs-2911	446	13	)	)	PUNCT
iajs-2911	446	14	.	.	PUNCT
iajs-2911	447	1	c𝒐𝒓𝒐llary	c𝒐𝒓𝒐llary	ADJ
iajs-2911	447	2	4.3	4.3	NUM
iajs-2911	447	3	.	.	PUNCT
iajs-2911	448	1	if	if	SCONJ
iajs-2911	448	2	(	(	PUNCT
iajs-2911	448	3	e,𝜏	e,𝜏	NOUN
iajs-2911	448	4	)	)	PUNCT
iajs-2911	448	5	is	be	AUX
iajs-2911	448	6	𝔽.	𝔽.	PROPN
iajs-2911	448	7	𝕎.m∗.t.s	𝕎.m∗.t.s	PUNCT
iajs-2911	448	8	on	on	ADP
iajs-2911	448	9	a	a	DET
iajs-2911	448	10	te	te	PROPN
iajs-2911	448	11	(	(	PUNCT
iajs-2911	448	12	f,𝜎	f,𝜎	PROPN
iajs-2911	448	13	)	)	PUNCT
iajs-2911	448	14	,	,	PUNCT
iajs-2911	448	15	so	so	ADV
iajs-2911	448	16	𝑖t	𝑖t	PROPN
iajs-2911	448	17	is	be	AUX
iajs-2911	448	18	𝔽.	𝔽.	PROPN
iajs-2911	448	19	𝕎.m.p.t.s	𝕎.m.p.t.s	PRON
iajs-2911	448	20	if	if	SCONJ
iajs-2911	448	21	∀f∗.b∗	∀f∗.b∗	PROPN
iajs-2911	448	22	on	on	ADP
iajs-2911	448	23	e	e	NOUN
iajs-2911	448	24	,	,	PUNCT
iajs-2911	448	25	if	if	SCONJ
iajs-2911	448	26	𝑋ℑ	𝑋ℑ	PROPN
iajs-2911	448	27	−−conv	−−conv	NOUN
iajs-2911	448	28	.	.	PUNCT
iajs-2911	449	1	→d	→d	PUNCT
iajs-2911	449	2	;	;	PUNCT
iajs-2911	449	3	d	d	X
iajs-2911	449	4	∈	∈	PROPN
iajs-2911	449	5	𝐷	𝐷	PROPN
iajs-2911	449	6	,	,	PUNCT
iajs-2911	449	7	then	then	ADV
iajs-2911	449	8	ad	ad	NOUN
iajs-2911	449	9	ℑ	ℑ	PROPN
iajs-2911	449	10	≠	≠	PROPN
iajs-2911	449	11	∅.	∅.	AUX
iajs-2911	449	12	corollary	corollary	NOUN
iajs-2911	449	13	4.4	4.4	NUM
iajs-2911	449	14	.	.	PUNCT
iajs-2911	450	1	let	let	VERB
iajs-2911	450	2	(	(	PUNCT
iajs-2911	450	3	e	e	NOUN
iajs-2911	450	4	,	,	PUNCT
iajs-2911	450	5	τ	τ	X
iajs-2911	450	6	)	)	PUNCT
iajs-2911	450	7	be	be	VERB
iajs-2911	450	8	𝔽.	𝔽.	PROPN
iajs-2911	450	9	𝕎.m∗.t.s	𝕎.m∗.t.s	PUNCT
iajs-2911	450	10	on	on	ADP
iajs-2911	450	11	(	(	PUNCT
iajs-2911	450	12	qhc	qhc	NOUN
iajs-2911	450	13	)	)	PUNCT
iajs-2911	450	14	on	on	ADP
iajs-2911	450	15	a	a	DET
iajs-2911	450	16	urysohn	urysohn	PROPN
iajs-2911	450	17	topological	topological	ADJ
iajs-2911	450	18	space	space	NOUN
iajs-2911	450	19	(	(	PUNCT
iajs-2911	450	20	d	d	NOUN
iajs-2911	450	21	,	,	PUNCT
iajs-2911	450	22	ρ	ρ	PROPN
iajs-2911	450	23	)	)	PUNCT
iajs-2911	450	24	,	,	PUNCT
iajs-2911	450	25	so	so	CCONJ
iajs-2911	450	26	(	(	PUNCT
iajs-2911	450	27	e	e	NOUN
iajs-2911	450	28	,	,	PUNCT
iajs-2911	450	29	τ	τ	X
iajs-2911	450	30	)	)	PUNCT
iajs-2911	450	31	is	be	AUX
iajs-2911	450	32	𝔽.	𝔽.	PROPN
iajs-2911	450	33	𝕎.m.t.s	𝕎.m.t.s	PROPN
iajs-2911	450	34	..	..	PUNCT
iajs-2911	450	35	theorem	theorem	VERB
iajs-2911	450	36	4.4	4.4	NUM
iajs-2911	450	37	.	.	PUNCT
iajs-2911	451	1	let	let	VERB
iajs-2911	451	2	(	(	PUNCT
iajs-2911	451	3	e,𝜏	e,𝜏	NOUN
iajs-2911	451	4	)	)	PUNCT
iajs-2911	451	5	be	be	AUX
iajs-2911	451	6	𝔽.	𝔽.	PROPN
iajs-2911	451	7	𝕎.u∗.t.s.(resp	𝕎.u∗.t.s.(resp	NOUN
iajs-2911	451	8	.	.	PROPN
iajs-2911	451	9	,	,	PUNCT
iajs-2911	451	10	𝔽.	𝔽.	PROPN
iajs-2911	451	11	𝕎.l∗.t.s	𝕎.l∗.t.s	NOUN
iajs-2911	451	12	.	.	PUNCT
iajs-2911	451	13	)	)	PUNCT
iajs-2911	452	1	on	on	ADP
iajs-2911	452	2	locally	locally	ADV
iajs-2911	452	3	qhc	qhc	NOUN
iajs-2911	452	4	on	on	ADP
iajs-2911	452	5	a	a	DET
iajs-2911	452	6	te(d,𝜌	te(d,𝜌	NOUN
iajs-2911	452	7	)	)	PUNCT
iajs-2911	452	8	,	,	PUNCT
iajs-2911	452	9	then	then	ADV
iajs-2911	452	10	(	(	PUNCT
iajs-2911	452	11	d,𝜌	d,𝜌	NOUN
iajs-2911	452	12	)	)	PUNCT
iajs-2911	452	13	is	be	AUX
iajs-2911	452	14	𝔽.	𝔽.	PROPN
iajs-2911	452	15	𝕎.u∗.t.s.(resp	𝕎.u∗.t.s.(resp	NOUN
iajs-2911	452	16	.	.	PROPN
iajs-2911	452	17	,	,	PUNCT
iajs-2911	452	18	𝔽.	𝔽.	PROPN
iajs-2911	452	19	𝕎.l∗.t.s	𝕎.l∗.t.s	NOUN
iajs-2911	452	20	.	.	PUNCT
iajs-2911	452	21	)	)	PUNCT
iajs-2911	453	1	if	if	SCONJ
iajs-2911	453	2	it	it	PRON
iajs-2911	453	3	is	be	AUX
iajs-2911	453	4	𝔽.	𝔽.	PROPN
iajs-2911	453	5	𝕎.	𝕎.	PROPN
iajs-2911	453	6	almost	almost	ADV
iajs-2911	453	7	u.p.(resp	u.p.(resp	ADJ
iajs-2911	453	8	.	.	PROPN
iajs-2911	453	9	,	,	PUNCT
iajs-2911	453	10	𝔽.	𝔽.	PROPN
iajs-2911	453	11	𝕎.	𝕎.	PROPN
iajs-2911	453	12	almost	almost	ADV
iajs-2911	453	13	l.p	l.p	PROPN
iajs-2911	453	14	.	.	PROPN
iajs-2911	453	15	)	)	PUNCT
iajs-2911	453	16	.	.	PUNCT
iajs-2911	454	1	proof	proof	NOUN
iajs-2911	454	2	.	.	PUNCT
iajs-2911	455	1	(	(	PUNCT
iajs-2911	455	2	⇐	⇐	NOUN
iajs-2911	455	3	)	)	PUNCT
iajs-2911	455	4	let	let	VERB
iajs-2911	455	5	(	(	PUNCT
iajs-2911	455	6	e,𝜏	e,𝜏	NOUN
iajs-2911	455	7	)	)	PUNCT
iajs-2911	455	8	is	be	AUX
iajs-2911	455	9	𝔽.	𝔽.	PROPN
iajs-2911	455	10	𝕎.	𝕎.	PROPN
iajs-2911	455	11	almost	almost	ADV
iajs-2911	455	12	u.p.(resp	u.p.(resp	ADJ
iajs-2911	455	13	.	.	PROPN
iajs-2911	455	14	,	,	PUNCT
iajs-2911	455	15	𝔽.	𝔽.	PROPN
iajs-2911	455	16	𝕎.	𝕎.	PROPN
iajs-2911	455	17	almost	almost	ADV
iajs-2911	455	18	l.p	l.p	PROPN
iajs-2911	455	19	.	.	PROPN
iajs-2911	455	20	)	)	PUNCT
iajs-2911	455	21	,	,	PUNCT
iajs-2911	455	22	so	so	CCONJ
iajs-2911	456	1	∃	∃	PROPN
iajs-2911	456	2	almost	almost	ADV
iajs-2911	456	3	u.p.(resp	u.p.(resp	PROPN
iajs-2911	456	4	.	.	PUNCT
iajs-2911	456	5	,	,	PUNCT
iajs-2911	456	6	almost	almost	ADV
iajs-2911	456	7	l.p	l.p	PROPN
iajs-2911	456	8	.	.	PROPN
iajs-2911	456	9	)	)	PUNCT
iajs-2911	456	10	projection	projection	NOUN
iajs-2911	456	11	function	function	NOUN
iajs-2911	456	12	xe	xe	PROPN
iajs-2911	456	13	:	:	PUNCT
iajs-2911	456	14	e	e	X
iajs-2911	456	15	→	→	SYM
iajs-2911	456	16	d	d	PROPN
iajs-2911	456	17	and	and	CCONJ
iajs-2911	456	18	let	let	VERB
iajs-2911	456	19	d	d	NOUN
iajs-2911	456	20	be	be	AUX
iajs-2911	456	21	any	any	DET
iajs-2911	456	22	f∗.b∗.	f∗.b∗.	NOUN
iajs-2911	456	23	on	on	ADP
iajs-2911	456	24	e	e	NOUN
iajs-2911	456	25	and	and	CCONJ
iajs-2911	456	26	let	let	VERB
iajs-2911	456	27	𝑋ℑ	𝑋ℑ	PROPN
iajs-2911	456	28	−−conv.→	−−conv.→	PROPN
iajs-2911	456	29	d	d	NOUN
iajs-2911	456	30	in	in	ADP
iajs-2911	456	31	which	which	PRON
iajs-2911	456	32	d	d	PROPN
iajs-2911	456	33	∈	∈	PROPN
iajs-2911	456	34	d.	d.	PROPN
iajs-2911	456	35	there	there	PRON
iajs-2911	456	36	are	be	VERB
iajs-2911	456	37	an	an	DET
iajs-2911	456	38	𝐸	𝐸	PROPN
iajs-2911	456	39	set	set	VERB
iajs-2911	456	40	d∗	d∗	NOUN
iajs-2911	456	41	in	in	ADP
iajs-2911	456	42	d	d	PROPN
iajs-2911	456	43	and	and	CCONJ
iajs-2911	456	44	𝜌−open	𝜌−open	X
iajs-2911	456	45	a	a	DET
iajs-2911	456	46	𝜂ℙ𝕕v	𝜂ℙ𝕕v	PROPN
iajs-2911	456	47	of	of	ADP
iajs-2911	456	48	d	d	PROPN
iajs-2911	456	49	such	such	ADJ
iajs-2911	456	50	that	that	PRON
iajs-2911	456	51	,	,	PUNCT
iajs-2911	456	52	d	d	PROPN
iajs-2911	456	53	∈	∈	PROPN
iajs-2911	456	54	v	v	ADP
iajs-2911	456	55	⊆	⊆	NUM
iajs-2911	456	56	d∗.	d∗.	PROPN
iajs-2911	456	57	let	let	VERB
iajs-2911	456	58	e	e	NOUN
iajs-2911	456	59	=	=	PRON
iajs-2911	456	60	{	{	PUNCT
iajs-2911	456	61	𝜌	𝜌	X
iajs-2911	456	62	−	−	PROPN
iajs-2911	456	63	cl(𝔘	cl(𝔘	NOUN
iajs-2911	456	64	)	)	PUNCT
iajs-2911	456	65	)	)	PUNCT
iajs-2911	457	1	∩	∩	PROPN
iajs-2911	457	2	𝑋𝔽	𝑋𝔽	PROPN
iajs-2911	457	3	∩	∩	ADJ
iajs-2911	457	4	d∗	d∗	PROPN
iajs-2911	457	5	;	;	PUNCT
iajs-2911	457	6	𝔽	𝔽	PROPN
iajs-2911	457	7	∈	∈	PROPN
iajs-2911	457	8	ℑ	ℑ	PROPN
iajs-2911	457	9	and	and	CCONJ
iajs-2911	457	10	𝔘	𝔘	PROPN
iajs-2911	457	11	is	be	AUX
iajs-2911	457	12	a	a	PRON
iajs-2911	457	13	𝜌−open	𝜌−open	NOUN
iajs-2911	457	14	a	a	DET
iajs-2911	457	15	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	457	16	of	of	ADP
iajs-2911	457	17	d	d	NOUN
iajs-2911	457	18	}	}	PUNCT
iajs-2911	457	19	.	.	PUNCT
iajs-2911	458	1	by	by	ADP
iajs-2911	458	2	l𝑒𝑚𝑚a	l𝑒𝑚𝑚a	ADJ
iajs-2911	458	3	(	(	PUNCT
iajs-2911	458	4	4.1	4.1	NUM
iajs-2911	458	5	.	.	PUNCT
iajs-2911	458	6	)	)	PUNCT
iajs-2911	458	7	,	,	PUNCT
iajs-2911	458	8	d∗	d∗	PROPN
iajs-2911	458	9	is	be	AUX
iajs-2911	458	10	closed	close	VERB
iajs-2911	458	11	and	and	CCONJ
iajs-2911	458	12	hence	hence	ADV
iajs-2911	458	13	no	no	DET
iajs-2911	458	14	member	member	NOUN
iajs-2911	458	15	of	of	ADP
iajs-2911	458	16	e	e	PROPN
iajs-2911	458	17	is	be	AUX
iajs-2911	458	18	void	void	ADJ
iajs-2911	458	19	.	.	PUNCT
iajs-2911	459	1	reality	reality	NOUN
iajs-2911	459	2	,	,	PUNCT
iajs-2911	459	3	if	if	SCONJ
iajs-2911	459	4	not	not	PART
iajs-2911	459	5	,	,	PUNCT
iajs-2911	459	6	let	let	VERB
iajs-2911	459	7	for	for	SCONJ
iajs-2911	459	8	some	some	PRON
iajs-2911	459	9	𝜌−open	𝜌−open	ADP
iajs-2911	459	10	a	a	DET
iajs-2911	459	11	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	459	12	𝔘	𝔘	NOUN
iajs-2911	459	13	of	of	ADP
iajs-2911	459	14	d	d	PROPN
iajs-2911	459	15	and	and	CCONJ
iajs-2911	459	16	some	some	DET
iajs-2911	459	17	𝔽	𝔽	PROPN
iajs-2911	459	18	∈	∈	PROPN
iajs-2911	459	19	ℑ	ℑ	PROPN
iajs-2911	459	20	,	,	PUNCT
iajs-2911	459	21	𝜌	𝜌	ADP
iajs-2911	459	22	−	−	NOUN
iajs-2911	459	23	cl(𝔘	cl(𝔘	NOUN
iajs-2911	459	24	)	)	PUNCT
iajs-2911	459	25	∩	∩	NOUN
iajs-2911	459	26	𝑋𝔽	𝑋𝔽	PROPN
iajs-2911	459	27	∩d∗	∩d∗	PUNCT
iajs-2911	459	28	=	=	PUNCT
iajs-2911	459	29	∅.	∅.	NOUN
iajs-2911	459	30	then	then	ADV
iajs-2911	459	31	w	w	NOUN
iajs-2911	459	32	=	=	SYM
iajs-2911	459	33	𝔘	𝔘	PROPN
iajs-2911	459	34	∩v	∩v	NOUN
iajs-2911	459	35	since	since	SCONJ
iajs-2911	459	36	d	d	PROPN
iajs-2911	459	37	∈	∈	PROPN
iajs-2911	459	38	𝔘	𝔘	PROPN
iajs-2911	459	39	∩v	∩v	NOUN
iajs-2911	459	40	∈	∈	PROPN
iajs-2911	459	41	𝜌	𝜌	X
iajs-2911	459	42	and	and	CCONJ
iajs-2911	459	43	𝜌	𝜌	X
iajs-2911	459	44	−	−	PROPN
iajs-2911	459	45	cl(w	cl(w	NOUN
iajs-2911	459	46	=	=	NUM
iajs-2911	459	47	cl(w	cl(w	NOUN
iajs-2911	459	48	)	)	PUNCT
iajs-2911	459	49	⊂	⊂	PROPN
iajs-2911	459	50	cl(d∗	cl(d∗	X
iajs-2911	459	51	)	)	PUNCT
iajs-2911	460	1	=	=	SYM
iajs-2911	460	2	d∗	d∗	PROPN
iajs-2911	460	3	,	,	PUNCT
iajs-2911	460	4	by	by	ADP
iajs-2911	460	5	lemma	lemma	PROPN
iajs-2911	460	6	(	(	PUNCT
iajs-2911	460	7	4.1	4.1	NUM
iajs-2911	460	8	.	.	PUNCT
iajs-2911	460	9	)	)	PUNCT
iajs-2911	460	10	.	.	PUNCT
iajs-2911	461	1	currently	currently	ADV
iajs-2911	461	2	∅	∅	NOUN
iajs-2911	461	3	=	=	PUNCT
iajs-2911	461	4	𝜌−cl(w)∩	𝜌−cl(w)∩	X
iajs-2911	461	5	𝑋𝔽∩d∗	𝑋𝔽∩d∗	X
iajs-2911	461	6	=	=	SYM
iajs-2911	461	7	𝜌	𝜌	X
iajs-2911	461	8	−cl(w)∩	−cl(w)∩	X
iajs-2911	461	9	𝑋𝔽	𝑋𝔽	PROPN
iajs-2911	461	10	,	,	PUNCT
iajs-2911	461	11	which	which	PRON
iajs-2911	461	12	is	be	AUX
iajs-2911	461	13	not	not	PART
iajs-2911	461	14	possible	possible	ADJ
iajs-2911	461	15	,	,	PUNCT
iajs-2911	461	16	since	since	SCONJ
iajs-2911	461	17	𝑋𝔽—conv.→	𝑋𝔽—conv.→	PRON
iajs-2911	461	18	d.	d.	PROPN
iajs-2911	462	1	so	so	ADV
iajs-2911	462	2	e	e	PROPN
iajs-2911	462	3	is	be	AUX
iajs-2911	462	4	f∗.b∗.	f∗.b∗.	PROPN
iajs-2911	462	5	on	on	ADP
iajs-2911	462	6	d	d	PROPN
iajs-2911	462	7	,	,	PUNCT
iajs-2911	462	8	and	and	CCONJ
iajs-2911	462	9	is	be	AUX
iajs-2911	462	10	obviously	obviously	ADV
iajs-2911	462	11	larger	large	ADJ
iajs-2911	462	12	than	than	ADP
iajs-2911	462	13	𝑋ℑ	𝑋ℑ	PROPN
iajs-2911	462	14	,	,	PUNCT
iajs-2911	462	15	so	so	SCONJ
iajs-2911	462	16	that	that	SCONJ
iajs-2911	462	17	e	e	X
iajs-2911	462	18	–	–	PUNCT
iajs-2911	462	19	conv.→	conv.→	PROPN
iajs-2911	462	20	d.	d.	PROPN
iajs-2911	462	21	also	also	ADV
iajs-2911	462	22	𝔔	𝔔	PROPN
iajs-2911	462	23	=	=	PUNCT
iajs-2911	462	24	{	{	PUNCT
iajs-2911	462	25	𝐸𝐻	𝐸𝐻	PROPN
iajs-2911	462	26	+	+	PROPN
iajs-2911	462	27	(	(	PUNCT
iajs-2911	462	28	resp	resp	NOUN
iajs-2911	462	29	.	.	PUNCT
iajs-2911	463	1	,	,	PUNCT
iajs-2911	463	2	𝐸𝐻	𝐸𝐻	PROPN
iajs-2911	463	3	−	−	PROPN
iajs-2911	463	4	)	)	PUNCT
iajs-2911	463	5	∩	∩	NOUN
iajs-2911	463	6	𝔽	𝔽	PROPN
iajs-2911	463	7	:	:	PUNCT
iajs-2911	463	8	ℋ	ℋ	PROPN
iajs-2911	463	9	∈	∈	PROPN
iajs-2911	463	10	e	e	NOUN
iajs-2911	463	11	and	and	CCONJ
iajs-2911	463	12	𝔽	𝔽	PROPN
iajs-2911	463	13	∈	∈	PROPN
iajs-2911	463	14	ℑ	ℑ	PROPN
iajs-2911	463	15	}	}	PUNCT
iajs-2911	463	16	is	be	AUX
iajs-2911	463	17	obviously	obviously	ADV
iajs-2911	463	18	a	a	DET
iajs-2911	463	19	filter	filter	NOUN
iajs-2911	463	20	on	on	ADP
iajs-2911	463	21	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	463	22	+	+	CCONJ
iajs-2911	463	23	(	(	PUNCT
iajs-2911	463	24	resp	resp	NOUN
iajs-2911	463	25	.	.	PUNCT
iajs-2911	464	1	,	,	PUNCT
iajs-2911	464	2	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	464	3	−	−	PROPN
iajs-2911	464	4	)	)	PUNCT
iajs-2911	464	5	.	.	PUNCT
iajs-2911	465	1	because	because	SCONJ
iajs-2911	465	2	x	x	PRON
iajs-2911	465	3	is	be	AUX
iajs-2911	465	4	almost	almost	ADV
iajs-2911	465	5	u.p.(resp	u.p.(resp	ADJ
iajs-2911	465	6	.	.	PUNCT
iajs-2911	465	7	,	,	PUNCT
iajs-2911	465	8	almost	almost	ADV
iajs-2911	465	9	l.p	l.p	PROPN
iajs-2911	465	10	.	.	PROPN
iajs-2911	465	11	)	)	PUNCT
iajs-2911	465	12	,	,	PUNCT
iajs-2911	465	13	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	465	14	+	+	CCONJ
iajs-2911	465	15	(	(	PUNCT
iajs-2911	465	16	resp	resp	NOUN
iajs-2911	465	17	.	.	PUNCT
iajs-2911	466	1	,	,	PUNCT
iajs-2911	466	2	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	466	3	−	−	PROPN
iajs-2911	466	4	)	)	PUNCT
iajs-2911	466	5	is	be	AUX
iajs-2911	466	6	an	an	DET
iajs-2911	466	7	ℍ.set	ℍ.set	ADJ
iajs-2911	466	8	and	and	CCONJ
iajs-2911	466	9	so	so	ADV
iajs-2911	466	10	ad	ad	NOUN
iajs-2911	466	11	𝔔	𝔔	PROPN
iajs-2911	466	12	∩	∩	PROPN
iajs-2911	466	13	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	466	14	+	+	CCONJ
iajs-2911	466	15	≠	≠	PROPN
iajs-2911	466	16	∅(resp	∅(resp	NOUN
iajs-2911	466	17	.	.	PUNCT
iajs-2911	466	18	,	,	PUNCT
iajs-2911	466	19	𝔔	𝔔	PROPN
iajs-2911	466	20	∩	∩	PROPN
iajs-2911	466	21	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	466	22	−	−	PROPN
iajs-2911	466	23	≠	≠	PROPN
iajs-2911	466	24	∅).thus	∅).thus	PROPN
iajs-2911	466	25	x	x	PUNCT
iajs-2911	466	26	is	be	AUX
iajs-2911	466	27	u.p.(resp	u.p.(resp	PROPN
iajs-2911	466	28	.	.	PROPN
iajs-2911	466	29	,	,	PUNCT
iajs-2911	466	30	l.r	l.r	PROPN
iajs-2911	466	31	.	.	PUNCT
iajs-2911	466	32	)	)	PUNCT
iajs-2911	467	1	and	and	CCONJ
iajs-2911	467	2	by	by	ADP
iajs-2911	467	3	theorem	theorem	NOUN
iajs-2911	467	4	(	(	PUNCT
iajs-2911	467	5	4.3	4.3	NUM
iajs-2911	467	6	.	.	PUNCT
iajs-2911	467	7	)	)	PUNCT
iajs-2911	468	1	(	(	PUNCT
iajs-2911	468	2	e,𝜏	e,𝜏	NOUN
iajs-2911	468	3	)	)	PUNCT
iajs-2911	468	4	be	be	AUX
iajs-2911	468	5	𝔽.	𝔽.	PROPN
iajs-2911	468	6	𝕎.u∗.t.s.(resp	𝕎.u∗.t.s.(resp	NOUN
iajs-2911	468	7	.	.	PROPN
iajs-2911	468	8	,	,	PUNCT
iajs-2911	468	9	𝔽.	𝔽.	PROPN
iajs-2911	468	10	𝕎.l∗.t.s	𝕎.l∗.t.s	NOUN
iajs-2911	468	11	.	.	PUNCT
iajs-2911	468	12	)	)	PUNCT
iajs-2911	468	13	.	.	PUNCT
iajs-2911	469	1	c𝒐𝒓𝒐llary	c𝒐𝒓𝒐llary	ADJ
iajs-2911	469	2	4.5	4.5	NUM
iajs-2911	469	3	.	.	PUNCT
iajs-2911	470	1	let	let	VERB
iajs-2911	470	2	(	(	PUNCT
iajs-2911	470	3	e,𝜏	e,𝜏	NOUN
iajs-2911	470	4	)	)	PUNCT
iajs-2911	470	5	be	be	VERB
iajs-2911	470	6	𝔽.	𝔽.	PROPN
iajs-2911	470	7	𝕎.m∗.t.s	𝕎.m∗.t.s	PUNCT
iajs-2911	470	8	on	on	ADP
iajs-2911	470	9	locally	locally	ADV
iajs-2911	470	10	qhc	qhc	NOUN
iajs-2911	470	11	on	on	ADP
iajs-2911	470	12	a	a	DET
iajs-2911	470	13	te(d,𝜌	te(d,𝜌	NOUN
iajs-2911	470	14	)	)	PUNCT
iajs-2911	470	15	,	,	PUNCT
iajs-2911	470	16	then	then	ADV
iajs-2911	470	17	(	(	PUNCT
iajs-2911	470	18	d,𝜌	d,𝜌	NOUN
iajs-2911	470	19	)	)	PUNCT
iajs-2911	470	20	is	be	AUX
iajs-2911	470	21	𝔽.	𝔽.	PROPN
iajs-2911	470	22	𝕎.m∗.t.s	𝕎.m∗.t.s	PUNCT
iajs-2911	470	23	if	if	SCONJ
iajs-2911	470	24	it	it	PRON
iajs-2911	470	25	is	be	AUX
iajs-2911	470	26	𝔽.	𝔽.	PROPN
iajs-2911	470	27	𝕎.	𝕎.	PROPN
iajs-2911	470	28	almost	almost	ADV
iajs-2911	470	29	m.p	m.p	PROPN
iajs-2911	470	30	..	..	PUNCT
iajs-2911	470	31	lemma	lemma	PROPN
iajs-2911	470	32	4.2	4.2	NUM
iajs-2911	470	33	.	.	PUNCT
iajs-2911	471	1	[	[	X
iajs-2911	471	2	10	10	NUM
iajs-2911	471	3	]	]	X
iajs-2911	471	4	a	a	DET
iajs-2911	471	5	topological	topological	ADJ
iajs-2911	471	6	space	space	NOUN
iajs-2911	471	7	(	(	PUNCT
iajs-2911	471	8	e,𝜏	e,𝜏	NOUN
iajs-2911	471	9	)	)	PUNCT
iajs-2911	471	10	𝑖𝑠	𝑖𝑠	PROPN
iajs-2911	472	1	t2	t2	PROPN
iajs-2911	472	2	⇐	⇐	PROPN
iajs-2911	472	3	⇒	⇒	PROPN
iajs-2911	472	4	{	{	PUNCT
iajs-2911	472	5	e	e	NOUN
iajs-2911	472	6	}	}	PUNCT
iajs-2911	472	7	=	=	SYM
iajs-2911	472	8	cl(e	cl(e	NOUN
iajs-2911	472	9	)	)	PUNCT
iajs-2911	472	10	∀	∀	X
iajs-2911	472	11	e	e	NOUN
iajs-2911	472	12	∈e	∈e	NOUN
iajs-2911	472	13	.	.	PUNCT
iajs-2911	473	1	theorem	theorem	VERB
iajs-2911	473	2	4.5	4.5	NUM
iajs-2911	473	3	.	.	PUNCT
iajs-2911	474	1	if	if	SCONJ
iajs-2911	474	2	(	(	PUNCT
iajs-2911	474	3	e	e	NOUN
iajs-2911	474	4	,	,	PUNCT
iajs-2911	474	5	τ	τ	X
iajs-2911	474	6	)	)	PUNCT
iajs-2911	474	7	is	be	AUX
iajs-2911	474	8	a	a	DET
iajs-2911	474	9	𝔽.	𝔽.	PROPN
iajs-2911	474	10	𝕎.u.p.(resp	𝕎.u.p.(resp	NOUN
iajs-2911	474	11	.	.	PUNCT
iajs-2911	474	12	,	,	PUNCT
iajs-2911	474	13	𝔽.	𝔽.	PROPN
iajs-2911	474	14	𝕎.u.p	𝕎.u.p	PROPN
iajs-2911	474	15	.	.	PUNCT
iajs-2911	474	16	)	)	PUNCT
iajs-2911	475	1	injection	injection	NOUN
iajs-2911	475	2	and	and	CCONJ
iajs-2911	475	3	surjective	surjective	ADJ
iajs-2911	475	4	topological	topological	ADJ
iajs-2911	475	5	space	space	NOUN
iajs-2911	475	6	with	with	ADP
iajs-2911	475	7	e	e	PROPN
iajs-2911	475	8	is	be	AUX
iajs-2911	475	9	a	a	DET
iajs-2911	475	10	u.t2	u.t2	ADJ
iajs-2911	475	11	space(resp	space(resp	PROPN
iajs-2911	475	12	.	.	PROPN
iajs-2911	475	13	,	,	PUNCT
iajs-2911	475	14	l.t2	l.t2	NOUN
iajs-2911	475	15	space	space	NOUN
iajs-2911	475	16	)	)	PUNCT
iajs-2911	475	17	on	on	ADP
iajs-2911	475	18	(	(	PUNCT
iajs-2911	475	19	d,𝜌	d,𝜌	NOUN
iajs-2911	475	20	)	)	PUNCT
iajs-2911	475	21	,	,	PUNCT
iajs-2911	475	22	then	then	ADV
iajs-2911	475	23	d	d	PROPN
iajs-2911	475	24	is	be	AUX
iajs-2911	475	25	u.t2	u.t2	ADJ
iajs-2911	475	26	space	space	NOUN
iajs-2911	475	27	(	(	PUNCT
iajs-2911	475	28	resp	resp	NOUN
iajs-2911	475	29	.	.	PUNCT
iajs-2911	475	30	,	,	PUNCT
iajs-2911	475	31	l.t2	l.t2	NOUN
iajs-2911	475	32	space	space	NOUN
iajs-2911	475	33	)	)	PUNCT
iajs-2911	475	34	.	.	PUNCT
iajs-2911	476	1	proof	proof	NOUN
iajs-2911	476	2	.	.	PUNCT
iajs-2911	477	1	let	let	VERB
iajs-2911	477	2	d1	d1	PROPN
iajs-2911	477	3	,	,	PUNCT
iajs-2911	477	4	d2	d2	PROPN
iajs-2911	477	5	∈	∈	PROPN
iajs-2911	478	1	d	d	X
iajs-2911	478	2	such	such	ADJ
iajs-2911	478	3	that	that	SCONJ
iajs-2911	478	4	d1	d1	PROPN
iajs-2911	478	5	≠	≠	PROPN
iajs-2911	478	6	d2	d2	PROPN
iajs-2911	478	7	.	.	PUNCT
iajs-2911	479	1	by	by	ADP
iajs-2911	479	2	x	x	SYM
iajs-2911	479	3	is	be	AUX
iajs-2911	479	4	surjective	surjective	ADJ
iajs-2911	479	5	,	,	PUNCT
iajs-2911	479	6	so	so	ADV
iajs-2911	479	7	d1	d1	PROPN
iajs-2911	479	8	,	,	PUNCT
iajs-2911	479	9	d2	d2	PROPN
iajs-2911	479	10	∈	∈	PROPN
iajs-2911	479	11	e	e	PROPN
iajs-2911	479	12	and	and	CCONJ
iajs-2911	479	13	p	p	PROPN
iajs-2911	479	14	is	be	AUX
iajs-2911	479	15	injection	injection	NOUN
iajs-2911	479	16	,	,	PUNCT
iajs-2911	479	17	then	then	ADV
iajs-2911	479	18	𝐸𝑑1	𝐸𝑑1	VERB
iajs-2911	479	19	+	+	CCONJ
iajs-2911	479	20	≠	≠	NOUN
iajs-2911	479	21	𝐸𝑑2	𝐸𝑑2	NOUN
iajs-2911	479	22	+	+	CCONJ
iajs-2911	479	23	(	(	PUNCT
iajs-2911	479	24	resp	resp	NOUN
iajs-2911	479	25	.	.	PUNCT
iajs-2911	479	26	,	,	PUNCT
iajs-2911	479	27	𝐸𝑑1	𝐸𝑑1	VERB
iajs-2911	479	28	−	−	PROPN
iajs-2911	479	29	≠	≠	PROPN
iajs-2911	479	30	𝐸𝑑2	𝐸𝑑2	NOUN
iajs-2911	479	31	−	−	PROPN
iajs-2911	479	32	.	.	PUNCT
iajs-2911	480	1	since	since	SCONJ
iajs-2911	480	2	x	x	PRON
iajs-2911	480	3	is	be	AUX
iajs-2911	480	4	u.p.(resp	u.p.(resp	PROPN
iajs-2911	480	5	.	.	PROPN
iajs-2911	480	6	,	,	PUNCT
iajs-2911	480	7	l.p	l.p	PROPN
iajs-2911	480	8	.	.	PROPN
iajs-2911	480	9	)	)	PUNCT
iajs-2911	480	10	,	,	PUNCT
iajs-2911	480	11	so	so	ADV
iajs-2911	480	12	by	by	ADP
iajs-2911	480	13	theorem	theorem	NOUN
iajs-2911	480	14	(	(	PUNCT
iajs-2911	480	15	2.2	2.2	NUM
iajs-2911	480	16	.	.	PUNCT
iajs-2911	480	17	)	)	PUNCT
iajs-2911	481	1	it	it	PRON
iajs-2911	481	2	is	be	AUX
iajs-2911	481	3	closed	closed	ADJ
iajs-2911	481	4	.	.	PUNCT
iajs-2911	482	1	by	by	ADP
iajs-2911	482	2	lemma	lemma	PROPN
iajs-2911	482	3	(	(	PUNCT
iajs-2911	482	4	4.2	4.2	NUM
iajs-2911	482	5	.	.	PUNCT
iajs-2911	482	6	)	)	PUNCT
iajs-2911	483	1	we	we	PRON
iajs-2911	483	2	have	have	AUX
iajs-2911	483	3	{	{	PUNCT
iajs-2911	483	4	𝐸𝑑1	𝐸𝑑1	NOUN
iajs-2911	483	5	+	+	CCONJ
iajs-2911	483	6	}	}	PUNCT
iajs-2911	483	7	=	=	SYM
iajs-2911	483	8	cl{d1	cl{d1	ADJ
iajs-2911	483	9	}	}	PUNCT
iajs-2911	483	10	(	(	PUNCT
iajs-2911	483	11	resp	resp	NOUN
iajs-2911	483	12	.	.	PROPN
iajs-2911	483	13	,	,	PUNCT
iajs-2911	483	14	{	{	PUNCT
iajs-2911	483	15	𝐸𝑑1	𝐸𝑑1	NOUN
iajs-2911	483	16	−	−	NOUN
iajs-2911	483	17	}	}	PUNCT
iajs-2911	483	18	=	=	SYM
iajs-2911	483	19	cl{d1	cl{d1	NOUN
iajs-2911	483	20	)	)	PUNCT
iajs-2911	483	21	)	)	PUNCT
iajs-2911	483	22	and	and	CCONJ
iajs-2911	483	23	{	{	PUNCT
iajs-2911	483	24	𝐸𝑑2	𝐸𝑑2	NOUN
iajs-2911	483	25	+	+	CCONJ
iajs-2911	483	26	}	}	PUNCT
iajs-2911	483	27	=	=	SYM
iajs-2911	483	28	cl{d2	cl{d2	PROPN
iajs-2911	483	29	}	}	PUNCT
iajs-2911	483	30	(	(	PUNCT
iajs-2911	483	31	resp	resp	NOUN
iajs-2911	483	32	.	.	PROPN
iajs-2911	483	33	,	,	PUNCT
iajs-2911	483	34	{	{	PUNCT
iajs-2911	483	35	𝐸𝑑2	𝐸𝑑2	NOUN
iajs-2911	483	36	−	−	NOUN
iajs-2911	483	37	}	}	PUNCT
iajs-2911	483	38	ihjpas	ihjpas	PROPN
iajs-2911	483	39	.	.	PUNCT
iajs-2911	484	1	36	36	NUM
iajs-2911	484	2	(	(	PUNCT
iajs-2911	484	3	4	4	NUM
iajs-2911	484	4	)	)	PUNCT
iajs-2911	484	5	2023	2023	NUM
iajs-2911	484	6	405	405	NUM
iajs-2911	484	7	=	=	SYM
iajs-2911	484	8	cl{d2	cl{d2	NOUN
iajs-2911	484	9	}	}	PUNCT
iajs-2911	484	10	)	)	PUNCT
iajs-2911	484	11	because	because	SCONJ
iajs-2911	484	12	x	x	PRON
iajs-2911	484	13	is	be	AUX
iajs-2911	484	14	u.t2	u.t2	ADJ
iajs-2911	484	15	space	space	NOUN
iajs-2911	484	16	(	(	PUNCT
iajs-2911	484	17	resp	resp	NOUN
iajs-2911	484	18	.	.	PROPN
iajs-2911	484	19	,	,	PUNCT
iajs-2911	484	20	u.t2	u.t2	ADJ
iajs-2911	484	21	space	space	NOUN
iajs-2911	484	22	)	)	PUNCT
iajs-2911	484	23	.	.	PUNCT
iajs-2911	485	1	currently	currently	ADV
iajs-2911	485	2	,	,	PUNCT
iajs-2911	485	3	x(cl{𝐸𝑑1	x(cl{𝐸𝑑1	PROPN
iajs-2911	485	4	+	+	CCONJ
iajs-2911	485	5	}	}	PUNCT
iajs-2911	485	6	)	)	PUNCT
iajs-2911	486	1	=	=	SYM
iajs-2911	486	2	cl{d1}(resp	cl{d1}(resp	PROPN
iajs-2911	486	3	.	.	PUNCT
iajs-2911	486	4	,	,	PUNCT
iajs-2911	486	5	x(cl{𝐸𝑑1	x(cl{𝐸𝑑1	PROPN
iajs-2911	486	6	−	−	PROPN
iajs-2911	486	7	}	}	PUNCT
iajs-2911	486	8	)	)	PUNCT
iajs-2911	486	9	=	=	SYM
iajs-2911	486	10	cl{d1	cl{d1	ADJ
iajs-2911	486	11	}	}	PUNCT
iajs-2911	486	12	)	)	PUNCT
iajs-2911	486	13	and	and	CCONJ
iajs-2911	486	14	x(cl{𝐸𝑑2	x(cl{𝐸𝑑2	PROPN
iajs-2911	487	1	+	+	CCONJ
iajs-2911	487	2	}	}	PUNCT
iajs-2911	487	3	)	)	PUNCT
iajs-2911	488	1	=	=	SYM
iajs-2911	488	2	cl{d2}(resp	cl{d2}(resp	PROPN
iajs-2911	488	3	.	.	PROPN
iajs-2911	488	4	,	,	PUNCT
iajs-2911	488	5	x(cl{𝐸𝑑2	x(cl{𝐸𝑑2	PROPN
iajs-2911	489	1	−	−	PROPN
iajs-2911	489	2	}	}	PUNCT
iajs-2911	489	3	)	)	PUNCT
iajs-2911	490	1	=	=	SYM
iajs-2911	490	2	cl{d2	cl{d2	PROPN
iajs-2911	490	3	}	}	PUNCT
iajs-2911	490	4	)	)	PUNCT
iajs-2911	490	5	,	,	PUNCT
iajs-2911	490	6	since	since	SCONJ
iajs-2911	490	7	x	x	PRON
iajs-2911	490	8	is	be	AUX
iajs-2911	490	9	closed	closed	ADJ
iajs-2911	490	10	.	.	PUNCT
iajs-2911	491	1	this	this	PRON
iajs-2911	491	2	mean	mean	NOUN
iajs-2911	491	3	{	{	PUNCT
iajs-2911	491	4	d1	d1	NOUN
iajs-2911	491	5	}	}	PUNCT
iajs-2911	491	6	=	=	SYM
iajs-2911	491	7	cl{d1	cl{d1	ADJ
iajs-2911	491	8	}	}	PUNCT
iajs-2911	491	9	and	and	CCONJ
iajs-2911	491	10	{	{	PUNCT
iajs-2911	491	11	d2	d2	PROPN
iajs-2911	491	12	}	}	PUNCT
iajs-2911	491	13	=	=	SYM
iajs-2911	491	14	cl{d2	cl{d2	NOUN
iajs-2911	491	15	}	}	PUNCT
iajs-2911	491	16	.	.	PUNCT
iajs-2911	492	1	hence	hence	ADV
iajs-2911	492	2	d	d	PROPN
iajs-2911	492	3	is	be	AUX
iajs-2911	492	4	u.t2	u.t2	ADJ
iajs-2911	492	5	space(re𝑠p	space(re𝑠p	PROPN
iajs-2911	492	6	.	.	PROPN
iajs-2911	492	7	,	,	PUNCT
iajs-2911	492	8	u.t2	u.t2	ADJ
iajs-2911	492	9	space	space	NOUN
iajs-2911	492	10	)	)	PUNCT
iajs-2911	492	11	.	.	PUNCT
iajs-2911	493	1	our	our	PRON
iajs-2911	493	2	following	follow	VERB
iajs-2911	493	3	theory	theory	NOUN
iajs-2911	493	4	gives	give	VERB
iajs-2911	493	5	a	a	DET
iajs-2911	493	6	description	description	NOUN
iajs-2911	493	7	of	of	ADP
iajs-2911	493	8	an	an	DET
iajs-2911	493	9	important	important	ADJ
iajs-2911	493	10	class	class	NOUN
iajs-2911	493	11	of	of	ADP
iajs-2911	493	12	𝔽.	𝔽.	PROPN
iajs-2911	493	13	𝕎.u.ts.(resp	𝕎.u.ts.(resp	PROPN
iajs-2911	493	14	.	.	PROPN
iajs-2911	493	15	,	,	PUNCT
iajs-2911	493	16	𝔽.	𝔽.	PROPN
iajs-2911	493	17	𝕎.l.ts	𝕎.l.ts	NOUN
iajs-2911	493	18	.	.	PUNCT
iajs-2911	493	19	)	)	PUNCT
iajs-2911	494	1	meaning	mean	VERB
iajs-2911	494	2	the	the	DET
iajs-2911	494	3	qhc	qhc	NOUN
iajs-2911	494	4	spaces	space	NOUN
iajs-2911	494	5	in	in	ADP
iajs-2911	494	6	terms	term	NOUN
iajs-2911	494	7	of	of	ADP
iajs-2911	494	8	𝔽.	𝔽.	PROPN
iajs-2911	494	9	𝕎.u.p.t.s	𝕎.u.p.t.s	PROPN
iajs-2911	494	10	.	.	PUNCT
iajs-2911	495	1	(	(	PUNCT
iajs-2911	495	2	r𝑒𝑠p	r𝑒𝑠p	PROPN
iajs-2911	495	3	.	.	PROPN
iajs-2911	495	4	,	,	PUNCT
iajs-2911	495	5	𝔽.	𝔽.	PROPN
iajs-2911	495	6	𝕎.l.p.t.s	𝕎.l.p.t.s	PROPN
iajs-2911	495	7	.	.	PUNCT
iajs-2911	495	8	)	)	PUNCT
iajs-2911	495	9	.	.	PUNCT
iajs-2911	496	1	c𝒐𝒓𝒐llary	c𝒐𝒓𝒐llary	ADJ
iajs-2911	496	2	4.6	4.6	NUM
iajs-2911	496	3	.	.	PUNCT
iajs-2911	497	1	if	if	SCONJ
iajs-2911	497	2	(	(	PUNCT
iajs-2911	497	3	e,𝜏	e,𝜏	NOUN
iajs-2911	497	4	)	)	PUNCT
iajs-2911	497	5	is	be	AUX
iajs-2911	497	6	a	a	DET
iajs-2911	497	7	𝔽.	𝔽.	PROPN
iajs-2911	497	8	𝕎.m.p	𝕎.m.p	PROPN
iajs-2911	497	9	.	.	PUNCT
iajs-2911	497	10	injection	injection	NOUN
iajs-2911	497	11	and	and	CCONJ
iajs-2911	497	12	surjective	surjective	ADJ
iajs-2911	497	13	topological	topological	ADJ
iajs-2911	497	14	space	space	NOUN
iajs-2911	497	15	with	with	ADP
iajs-2911	497	16	e	e	PROPN
iajs-2911	497	17	is	be	AUX
iajs-2911	497	18	a	a	DET
iajs-2911	497	19	m.t2	m.t2	NOUN
iajs-2911	497	20	space	space	NOUN
iajs-2911	497	21	on	on	ADP
iajs-2911	497	22	(	(	PUNCT
iajs-2911	497	23	d,𝜌	d,𝜌	NOUN
iajs-2911	497	24	)	)	PUNCT
iajs-2911	497	25	,	,	PUNCT
iajs-2911	497	26	then	then	ADV
iajs-2911	497	27	d	d	PROPN
iajs-2911	497	28	is	be	AUX
iajs-2911	497	29	m.t2	m.t2	NOUN
iajs-2911	497	30	.	.	PUNCT
iajs-2911	498	1	space	space	NOUN
iajs-2911	498	2	.	.	PUNCT
iajs-2911	499	1	theorem	theorem	VERB
iajs-2911	499	2	4.6	4.6	NUM
iajs-2911	499	3	.	.	PUNCT
iajs-2911	500	1	for	for	ADP
iajs-2911	500	2	a	a	DET
iajs-2911	500	3	topological	topological	ADJ
iajs-2911	500	4	space	space	NOUN
iajs-2911	500	5	(	(	PUNCT
iajs-2911	500	6	e,𝜏	e,𝜏	NOUN
iajs-2911	500	7	)	)	PUNCT
iajs-2911	500	8	,	,	PUNCT
iajs-2911	500	9	the	the	DET
iajs-2911	500	10	next	next	ADJ
iajs-2911	500	11	are	be	AUX
iajs-2911	500	12	equivalent	equivalent	ADJ
iajs-2911	500	13	:	:	PUNCT
iajs-2911	500	14	i.	i.	PROPN
iajs-2911	500	15	h	h	PROPN
iajs-2911	500	16	is	be	AUX
iajs-2911	500	17	qhc	qhc	PROPN
iajs-2911	500	18	.	.	PUNCT
iajs-2911	500	19	ii	ii	PROPN
iajs-2911	500	20	.	.	PUNCT
iajs-2911	501	1	a	a	DET
iajs-2911	501	2	𝔽.	𝔽.	PROPN
iajs-2911	501	3	𝕎.u	𝕎.u	PROPN
iajs-2911	501	4	.	.	PUNCT
iajs-2911	502	1	(	(	PUNCT
iajs-2911	502	2	e,𝜏	e,𝜏	NOUN
iajs-2911	502	3	)	)	PUNCT
iajs-2911	502	4	is	be	AUX
iajs-2911	502	5	p.t.(resp	p.t.(resp	PROPN
iajs-2911	502	6	.	.	PROPN
iajs-2911	502	7	,	,	PUNCT
iajs-2911	502	8	𝔽.	𝔽.	PROPN
iajs-2911	502	9	𝕎.l	𝕎.l	PROPN
iajs-2911	502	10	.	.	PUNCT
iajs-2911	503	1	(	(	PUNCT
iajs-2911	503	2	e,𝜏	e,𝜏	NOUN
iajs-2911	503	3	)	)	PUNCT
iajs-2911	503	4	is	be	AUX
iajs-2911	503	5	p.t	p.t	PROPN
iajs-2911	503	6	.	.	PUNCT
iajs-2911	503	7	)	)	PUNCT
iajs-2911	504	1	space	space	NOUN
iajs-2911	504	2	with	with	ADP
iajs-2911	504	3	constant	constant	ADJ
iajs-2911	504	4	projection	projection	NOUN
iajs-2911	504	5	on	on	ADP
iajs-2911	504	6	d∗	d∗	NOUN
iajs-2911	504	7	in	in	ADP
iajs-2911	504	8	wh𝑖𝑐h	wh𝑖𝑐h	ADJ
iajs-2911	504	9	d∗	d∗	PROPN
iajs-2911	504	10	is	be	AUX
iajs-2911	504	11	a	a	DET
iajs-2911	504	12	singleton	singleton	NOUN
iajs-2911	504	13	with	with	ADP
iajs-2911	504	14	two	two	NUM
iajs-2911	504	15	equal	equal	ADJ
iajs-2911	504	16	topologies	topology	NOUN
iajs-2911	504	17	meaning	mean	VERB
iajs-2911	504	18	the	the	DET
iajs-2911	504	19	unique	unique	ADJ
iajs-2911	504	20	topology	topology	NOUN
iajs-2911	504	21	on	on	ADP
iajs-2911	504	22	d∗.	d∗.	PROPN
iajs-2911	504	23	iii	iii	NOUN
iajs-2911	504	24	.	.	PUNCT
iajs-2911	505	1	the	the	DET
iajs-2911	505	2	𝔽.	𝔽.	PROPN
iajs-2911	505	3	𝕎	𝕎	PROPN
iajs-2911	505	4	..	..	PUNCT
iajs-2911	505	5	(	(	PUNCT
iajs-2911	505	6	b×h	b×h	PROPN
iajs-2911	505	7	,	,	PUNCT
iajs-2911	505	8	q	q	NOUN
iajs-2911	505	9	)	)	PUNCT
iajs-2911	505	10	is	be	AUX
iajs-2911	505	11	u.p.t.s.(resp	u.p.t.s.(resp	ADJ
iajs-2911	505	12	.	.	PUNCT
iajs-2911	505	13	,	,	PUNCT
iajs-2911	505	14	l.p.t.s	l.p.t.s	PROPN
iajs-2911	505	15	.	.	PUNCT
iajs-2911	505	16	)	)	PUNCT
iajs-2911	506	1	on	on	ADP
iajs-2911	506	2	(	(	PUNCT
iajs-2911	506	3	d,𝜌	d,𝜌	NOUN
iajs-2911	506	4	)	)	PUNCT
iajs-2911	506	5	,	,	PUNCT
iajs-2911	506	6	in	in	ADP
iajs-2911	506	7	which	which	PRON
iajs-2911	506	8	𝔔	𝔔	PROPN
iajs-2911	506	9	=	=	SYM
iajs-2911	506	10	𝜌	𝜌	PART
iajs-2911	506	11	×	×	NOUN
iajs-2911	506	12	𝜏.	𝜏.	NOUN
iajs-2911	506	13	proof	proof	NOUN
iajs-2911	506	14	.	.	PUNCT
iajs-2911	507	1	(	(	PUNCT
iajs-2911	507	2	i	i	NOUN
iajs-2911	507	3	)	)	PUNCT
iajs-2911	507	4	⇒	⇒	PROPN
iajs-2911	507	5	(	(	PUNCT
iajs-2911	507	6	ii	ii	NOUN
iajs-2911	507	7	)	)	PUNCT
iajs-2911	507	8	suppose	suppose	VERB
iajs-2911	507	9	that	that	SCONJ
iajs-2911	507	10	xe	xe	PROPN
iajs-2911	507	11	:	:	PUNCT
iajs-2911	507	12	e	e	X
iajs-2911	507	13	→	→	PUNCT
iajs-2911	507	14	d	d	X
iajs-2911	507	15	is	be	AUX
iajs-2911	507	16	a	a	DET
iajs-2911	507	17	constant	constant	ADJ
iajs-2911	507	18	projection	projection	NOUN
iajs-2911	507	19	on	on	ADP
iajs-2911	507	20	d∗	d∗	NOUN
iajs-2911	507	21	where	where	SCONJ
iajs-2911	507	22	d∗	d∗	PROPN
iajs-2911	507	23	is	be	AUX
iajs-2911	507	24	a	a	DET
iajs-2911	507	25	singleton	singleton	NOUN
iajs-2911	507	26	with	with	ADP
iajs-2911	507	27	two	two	NUM
iajs-2911	507	28	equal	equal	ADJ
iajs-2911	507	29	topologies	topology	NOUN
iajs-2911	507	30	meaning	mean	VERB
iajs-2911	507	31	the	the	DET
iajs-2911	507	32	unique	unique	ADJ
iajs-2911	507	33	topology	topology	NOUN
iajs-2911	507	34	on	on	ADP
iajs-2911	507	35	d∗.	d∗.	PROPN
iajs-2911	507	36	x	x	PRON
iajs-2911	507	37	is	be	AUX
iajs-2911	507	38	obviously	obviously	ADV
iajs-2911	507	39	closed	close	VERB
iajs-2911	507	40	.	.	PUNCT
iajs-2911	508	1	additionally	additionally	ADV
iajs-2911	508	2	,	,	PUNCT
iajs-2911	508	3	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	508	4	+	+	CCONJ
iajs-2911	508	5	(	(	PUNCT
iajs-2911	508	6	resp	resp	NOUN
iajs-2911	508	7	.	.	PUNCT
iajs-2911	508	8	,	,	PUNCT
iajs-2911	508	9	𝐸𝐷∗	𝐸𝐷∗	PROPN
iajs-2911	508	10	−	−	PROPN
iajs-2911	508	11	)	)	PUNCT
iajs-2911	508	12	,	,	PUNCT
iajs-2911	508	13	i.e.	i.e.	X
iajs-2911	508	14	e	e	X
iajs-2911	508	15	is	be	AUX
iajs-2911	508	16	obviously	obviously	ADV
iajs-2911	508	17	u.r.(resp	u.r.(resp	PRON
iajs-2911	508	18	.	.	PROPN
iajs-2911	508	19	,	,	PUNCT
iajs-2911	508	20	l.r	l.r	PROPN
iajs-2911	508	21	.	.	PROPN
iajs-2911	508	22	)	)	PUNCT
iajs-2911	509	1	by	by	ADP
iajs-2911	509	2	d∗	d∗	PROPN
iajs-2911	509	3	is	be	AUX
iajs-2911	509	4	qhc	qhc	NOUN
iajs-2911	509	5	.	.	PUNCT
iajs-2911	510	1	then	then	ADV
iajs-2911	510	2	by	by	ADP
iajs-2911	510	3	lemma	lemma	PROPN
iajs-2911	510	4	(	(	PUNCT
iajs-2911	510	5	3.1	3.1	NUM
iajs-2911	510	6	.	.	PUNCT
iajs-2911	510	7	)	)	PUNCT
iajs-2911	511	1	x	x	X
iajs-2911	511	2	is	be	AUX
iajs-2911	511	3	u.p.(resp	u.p.(resp	PROPN
iajs-2911	511	4	.	.	PROPN
iajs-2911	511	5	,	,	PUNCT
iajs-2911	511	6	l.r	l.r	PROPN
iajs-2911	511	7	.	.	PROPN
iajs-2911	511	8	)	)	PUNCT
iajs-2911	512	1	(	(	PUNCT
iajs-2911	512	2	ii	ii	NOUN
iajs-2911	512	3	)	)	PUNCT
iajs-2911	512	4	⇒	⇒	NOUN
iajs-2911	512	5	(	(	PUNCT
iajs-2911	512	6	i	i	NOUN
iajs-2911	512	7	)	)	PUNCT
iajs-2911	512	8	from	from	ADP
iajs-2911	512	9	theorem	theorem	NOUN
iajs-2911	512	10	(	(	PUNCT
iajs-2911	512	11	4.1	4.1	NUM
iajs-2911	512	12	.	.	PUNCT
iajs-2911	512	13	)	)	PUNCT
iajs-2911	512	14	.	.	PUNCT
iajs-2911	513	1	(	(	PUNCT
iajs-2911	513	2	i	i	NOUN
iajs-2911	513	3	)	)	PUNCT
iajs-2911	513	4	⇒	⇒	PROPN
iajs-2911	513	5	(	(	PUNCT
iajs-2911	513	6	iii	iii	X
iajs-2911	513	7	)	)	PUNCT
iajs-2911	513	8	let	let	VERB
iajs-2911	513	9	that	that	PRON
iajs-2911	513	10	(	(	PUNCT
iajs-2911	513	11	d×e	d×e	PROPN
iajs-2911	513	12	,	,	PUNCT
iajs-2911	513	13	𝔔	𝔔	PROPN
iajs-2911	513	14	)	)	PUNCT
iajs-2911	513	15	is	be	AUX
iajs-2911	513	16	𝔽.	𝔽.	PROPN
iajs-2911	513	17	𝕎.u.t.s.(resp	𝕎.u.t.s.(resp	NUM
iajs-2911	513	18	.	.	PUNCT
iajs-2911	513	19	,	,	PUNCT
iajs-2911	513	20	𝔽.	𝔽.	PROPN
iajs-2911	513	21	𝕎.l.t.s	𝕎.l.t.s	PROPN
iajs-2911	513	22	.	.	PUNCT
iajs-2911	513	23	)	)	PUNCT
iajs-2911	514	1	on	on	ADP
iajs-2911	514	2	(	(	PUNCT
iajs-2911	514	3	d,𝜌	d,𝜌	NOUN
iajs-2911	514	4	)	)	PUNCT
iajs-2911	514	5	in	in	ADP
iajs-2911	514	6	which	which	PRON
iajs-2911	514	7	𝔔	𝔔	PROPN
iajs-2911	514	8	=	=	SYM
iajs-2911	514	9	𝜌	𝜌	X
iajs-2911	514	10	×	×	NOUN
iajs-2911	514	11	𝜏	𝜏	NOUN
iajs-2911	514	12	,	,	PUNCT
iajs-2911	514	13	then	then	ADV
iajs-2911	514	14	there	there	PRON
iajs-2911	514	15	is	be	VERB
iajs-2911	514	16	a	a	DET
iajs-2911	514	17	projection	projection	NOUN
iajs-2911	514	18	x	x	NOUN
iajs-2911	514	19	=	=	SYM
iajs-2911	514	20	π	π	PROPN
iajs-2911	514	21	;	;	PUNCT
iajs-2911	514	22	(	(	PUNCT
iajs-2911	514	23	d×e	d×e	PROPN
iajs-2911	514	24	,	,	PUNCT
iajs-2911	514	25	𝔔	𝔔	PROPN
iajs-2911	514	26	)	)	PUNCT
iajs-2911	514	27	→	→	SYM
iajs-2911	514	28	(	(	PUNCT
iajs-2911	514	29	d,𝜌	d,𝜌	NOUN
iajs-2911	514	30	)	)	PUNCT
iajs-2911	514	31	.	.	PUNCT
iajs-2911	515	1	we	we	PRON
iajs-2911	515	2	show	show	VERB
iajs-2911	515	3	that	that	SCONJ
iajs-2911	515	4	π	π	PROPN
iajs-2911	515	5	is	be	AUX
iajs-2911	515	6	closed	closed	ADJ
iajs-2911	515	7	and	and	CCONJ
iajs-2911	515	8	∀d	∀d	PUNCT
iajs-2911	515	9	∈	∈	PROPN
iajs-2911	515	10	d	d	PROPN
iajs-2911	515	11	,	,	PUNCT
iajs-2911	515	12	𝐸𝐷	𝐸𝐷	PROPN
iajs-2911	515	13	+	+	NOUN
iajs-2911	515	14	(	(	PUNCT
iajs-2911	515	15	resp	resp	NOUN
iajs-2911	515	16	.	.	PUNCT
iajs-2911	515	17	,	,	PUNCT
iajs-2911	515	18	𝐸𝐷	𝐸𝐷	PROPN
iajs-2911	515	19	−	−	PROPN
iajs-2911	515	20	)	)	PUNCT
iajs-2911	515	21	is	be	AUX
iajs-2911	515	22	u.r.(resp	u.r.(resp	PRON
iajs-2911	515	23	.	.	PROPN
iajs-2911	515	24	,	,	PUNCT
iajs-2911	515	25	l.r	l.r	PROPN
iajs-2911	515	26	.	.	PUNCT
iajs-2911	515	27	)	)	PUNCT
iajs-2911	515	28	in	in	ADP
iajs-2911	515	29	d×e	d×e	PROPN
iajs-2911	515	30	.	.	PUNCT
iajs-2911	516	1	so	so	ADV
iajs-2911	516	2	,	,	PUNCT
iajs-2911	516	3	the	the	DET
iajs-2911	516	4	result	result	NOUN
iajs-2911	516	5	will	will	AUX
iajs-2911	516	6	be	be	AUX
iajs-2911	516	7	based	base	VERB
iajs-2911	516	8	on	on	ADP
iajs-2911	516	9	theorem	theorem	NOUN
iajs-2911	516	10	(	(	PUNCT
iajs-2911	516	11	3.1	3.1	NUM
iajs-2911	516	12	.	.	PUNCT
iajs-2911	516	13	)	)	PUNCT
iajs-2911	516	14	.	.	PUNCT
iajs-2911	517	1	let	let	VERB
iajs-2911	517	2	𝒜	𝒜	PROPN
iajs-2911	517	3	⊂	⊂	X
iajs-2911	517	4	d×e	d×e	PROPN
iajs-2911	517	5	and	and	CCONJ
iajs-2911	517	6	a	a	DET
iajs-2911	517	7	∉	∉	ADJ
iajs-2911	517	8	π(cl(𝒜	π(cl(𝒜	NOUN
iajs-2911	517	9	)	)	PUNCT
iajs-2911	517	10	)	)	PUNCT
iajs-2911	517	11	.	.	PUNCT
iajs-2911	518	1	∀	∀	X
iajs-2911	519	1	e	e	X
iajs-2911	519	2	∈e,(a	∈e,(a	NOUN
iajs-2911	519	3	,	,	PUNCT
iajs-2911	519	4	e	e	NOUN
iajs-2911	519	5	)	)	PUNCT
iajs-2911	519	6	∉	∉	ADJ
iajs-2911	519	7	cl(𝒜	cl(𝒜	PROPN
iajs-2911	519	8	)	)	PUNCT
iajs-2911	519	9	,	,	PUNCT
iajs-2911	519	10	s𝑜	s𝑜	NOUN
iajs-2911	519	11	that	that	SCONJ
iajs-2911	519	12	∃	∃	PROPN
iajs-2911	519	13	a	a	X
iajs-2911	519	14	𝜌−open	𝜌−open	X
iajs-2911	519	15	𝑎	𝑎	NOUN
iajs-2911	519	16	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	519	17	g	g	NOUN
iajs-2911	519	18	of	of	ADP
iajs-2911	519	19	a	a	DET
iajs-2911	519	20	𝑎𝑛d	𝑎𝑛d	NOUN
iajs-2911	519	21	a	a	DET
iajs-2911	519	22	𝜏-open	𝜏-open	NOUN
iajs-2911	519	23	a	a	DET
iajs-2911	519	24	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	519	25	𝔼e	𝔼e	PROPN
iajs-2911	519	26	of	of	ADP
iajs-2911	519	27	e	e	NOUN
iajs-2911	519	28	such	such	ADJ
iajs-2911	519	29	that	that	SCONJ
iajs-2911	520	1	[	[	X
iajs-2911	520	2	𝔔	𝔔	NOUN
iajs-2911	520	3	−	−	PROPN
iajs-2911	520	4	cl(ge	cl(ge	NOUN
iajs-2911	520	5	×	×	NOUN
iajs-2911	520	6	𝐸𝑒	𝐸𝑒	PROPN
iajs-2911	520	7	+	+	PROPN
iajs-2911	520	8	(	(	PUNCT
iajs-2911	520	9	resp	resp	NOUN
iajs-2911	520	10	.	.	PUNCT
iajs-2911	520	11	,	,	PUNCT
iajs-2911	520	12	𝐸𝑒	𝐸𝑒	PROPN
iajs-2911	520	13	−	−	PROPN
iajs-2911	520	14	)	)	PUNCT
iajs-2911	520	15	)	)	PUNCT
iajs-2911	520	16	]	]	PUNCT
iajs-2911	520	17	∩	∩	ADJ
iajs-2911	520	18	𝒜	𝒜	NOUN
iajs-2911	520	19	=	=	NOUN
iajs-2911	520	20	∅.	∅.	NOUN
iajs-2911	520	21	since	since	SCONJ
iajs-2911	520	22	e	e	PROPN
iajs-2911	520	23	is	be	AUX
iajs-2911	520	24	qhc,{a}×e	qhc,{a}×e	VERB
iajs-2911	520	25	is	be	AUX
iajs-2911	520	26	a	a	DET
iajs-2911	520	27	𝔼.set	𝔼.set	NOUN
iajs-2911	520	28	in	in	ADP
iajs-2911	520	29	d	d	NOUN
iajs-2911	520	30	×e	×e	NOUN
iajs-2911	520	31	.	.	PUNCT
iajs-2911	521	1	so	so	SCONJ
iajs-2911	521	2	that	that	SCONJ
iajs-2911	521	3	∃	∃	PROPN
iajs-2911	521	4	finitely	finitely	ADV
iajs-2911	521	5	many	many	ADJ
iajs-2911	521	6	elements	element	NOUN
iajs-2911	521	7	e1,e2,e3,	e1,e2,e3,	NOUN
iajs-2911	521	8	...	...	PUNCT
iajs-2911	521	9	,en	,en	PUNCT
iajs-2911	521	10	with	with	ADP
iajs-2911	521	11	,	,	PUNCT
iajs-2911	521	12	{	{	PUNCT
iajs-2911	521	13	a}×e⊂∪𝑘=1	a}×e⊂∪𝑘=1	PROPN
iajs-2911	521	14	𝑛	𝑛	ADP
iajs-2911	521	15	𝔔	𝔔	PROPN
iajs-2911	521	16	−	−	PROPN
iajs-2911	521	17	𝑐𝑙(𝐺𝑒𝑘	𝑐𝑙(𝐺𝑒𝑘	PROPN
iajs-2911	521	18	×	×	PROPN
iajs-2911	521	19	𝐸𝑒𝑘	𝐸𝑒𝑘	PROPN
iajs-2911	521	20	+	+	CCONJ
iajs-2911	521	21	(	(	PUNCT
iajs-2911	521	22	resp	resp	NOUN
iajs-2911	521	23	.	.	PUNCT
iajs-2911	521	24	,	,	PUNCT
iajs-2911	521	25	𝐸𝑒𝑘	𝐸𝑒𝑘	NOUN
iajs-2911	521	26	−	−	PROPN
iajs-2911	521	27	)	)	PUNCT
iajs-2911	521	28	)	)	PUNCT
iajs-2911	521	29	.	.	PUNCT
iajs-2911	522	1	currently	currently	ADV
iajs-2911	522	2	,	,	PUNCT
iajs-2911	522	3	a	a	DET
iajs-2911	522	4	∈	∈	NOUN
iajs-2911	522	5	∩nk=1ghk	∩nk=1ghk	X
iajs-2911	522	6	=	=	SYM
iajs-2911	522	7	g	g	PROPN
iajs-2911	522	8	,	,	PUNCT
iajs-2911	522	9	which	which	PRON
iajs-2911	522	10	is	be	AUX
iajs-2911	522	11	a	a	DET
iajs-2911	522	12	𝜌	𝜌	X
iajs-2911	522	13	-open	-open	VERB
iajs-2911	522	14	a	a	DET
iajs-2911	522	15	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	522	16	of	of	ADP
iajs-2911	522	17	a	a	DET
iajs-2911	522	18	∫.t.(𝜌	∫.t.(𝜌	PROPN
iajs-2911	522	19	−cl(g)∩π(𝒜	−cl(g)∩π(𝒜	NOUN
iajs-2911	522	20	)	)	PUNCT
iajs-2911	522	21	=	=	PUNCT
iajs-2911	522	22	∅.	∅.	VERB
iajs-2911	522	23	so	so	ADV
iajs-2911	522	24	a	a	DET
iajs-2911	522	25	∉	∉	PROPN
iajs-2911	522	26	clπ(𝒜	clπ(𝒜	NUM
iajs-2911	522	27	)	)	PUNCT
iajs-2911	522	28	and	and	CCONJ
iajs-2911	522	29	thus	thus	ADV
iajs-2911	522	30	clπ(𝒜	clπ(𝒜	NUM
iajs-2911	522	31	)	)	PUNCT
iajs-2911	522	32	⊂	⊂	X
iajs-2911	522	33	π(cl(𝒜	π(cl(𝒜	NOUN
iajs-2911	522	34	)	)	PUNCT
iajs-2911	522	35	)	)	PUNCT
iajs-2911	522	36	.	.	PUNCT
iajs-2911	523	1	so	so	ADV
iajs-2911	523	2	π	π	PROPN
iajs-2911	523	3	is	be	AUX
iajs-2911	523	4	closed	close	VERB
iajs-2911	523	5	by	by	ADP
iajs-2911	523	6	lemma	lemma	PROPN
iajs-2911	523	7	(	(	PUNCT
iajs-2911	523	8	2.1	2.1	NUM
iajs-2911	523	9	.	.	PUNCT
iajs-2911	523	10	)	)	PUNCT
iajs-2911	523	11	.	.	PUNCT
iajs-2911	524	1	next	next	ADV
iajs-2911	524	2	,	,	PUNCT
iajs-2911	524	3	let	let	VERB
iajs-2911	524	4	d	d	X
iajs-2911	524	5	∈	∈	PROPN
iajs-2911	524	6	d	d	X
iajs-2911	524	7	t.p	t.p	PROPN
iajs-2911	524	8	.	.	PUNCT
iajs-2911	525	1	(	(	PUNCT
iajs-2911	525	2	𝐷	𝐷	PROPN
iajs-2911	525	3	×	×	NOUN
iajs-2911	525	4	𝐸)𝑑	𝐸)𝑑	PUNCT
iajs-2911	526	1	+	+	ADJ
iajs-2911	526	2	(	(	PUNCT
iajs-2911	526	3	resp	resp	NOUN
iajs-2911	526	4	.	.	PUNCT
iajs-2911	527	1	,	,	PUNCT
iajs-2911	527	2	(	(	PUNCT
iajs-2911	527	3	𝐷	𝐷	NOUN
iajs-2911	527	4	×	×	NOUN
iajs-2911	527	5	𝐸)𝑑	𝐸)𝑑	PUNCT
iajs-2911	527	6	−	−	NOUN
iajs-2911	527	7	)	)	PUNCT
iajs-2911	528	1	=	=	SYM
iajs-2911	528	2	π−1(d	π−1(d	NOUN
iajs-2911	528	3	)	)	PUNCT
iajs-2911	528	4	to	to	PART
iajs-2911	528	5	be	be	AUX
iajs-2911	528	6	u.r.(resp	u.r.(resp	PROPN
iajs-2911	528	7	.	.	PROPN
iajs-2911	528	8	,	,	PUNCT
iajs-2911	528	9	l.r	l.r	PROPN
iajs-2911	528	10	.	.	PUNCT
iajs-2911	528	11	)	)	PUNCT
iajs-2911	529	1	in	in	ADP
iajs-2911	529	2	d	d	DET
iajs-2911	529	3	×e	×e	PROPN
iajs-2911	529	4	.	.	PUNCT
iajs-2911	529	5	let	let	VERB
iajs-2911	529	6	ℑ	ℑ	PRON
iajs-2911	529	7	be	be	AUX
iajs-2911	529	8	a	a	PRON
iajs-2911	529	9	.	.	PUNCT
iajs-2911	530	1	on	on	ADP
iajs-2911	530	2	d	d	PRON
iajs-2911	530	3	×e	×e	NOUN
iajs-2911	530	4	such	such	ADJ
iajs-2911	530	5	that	that	SCONJ
iajs-2911	530	6	π−1(d	π−1(d	NOUN
iajs-2911	530	7	)	)	PUNCT
iajs-2911	530	8	∩	∩	NOUN
iajs-2911	530	9	ad	ad	NOUN
iajs-2911	530	10	ℑ	ℑ	NOUN
iajs-2911	530	11	=	=	PUNCT
iajs-2911	530	12	∅.	∅.	PROPN
iajs-2911	530	13	∀e	∀e	PROPN
iajs-2911	530	14	∈	∈	PROPN
iajs-2911	530	15	e,(d	e,(d	PROPN
iajs-2911	530	16	,	,	PUNCT
iajs-2911	530	17	e	e	NOUN
iajs-2911	530	18	)	)	PUNCT
iajs-2911	530	19	∉	∉	PROPN
iajs-2911	530	20	ad	ad	NOUN
iajs-2911	530	21	ℑ.	ℑ.	PROPN
iajs-2911	530	22	so	so	ADV
iajs-2911	530	23	,	,	PUNCT
iajs-2911	530	24	∃𝜌−open	∃𝜌−open	PROPN
iajs-2911	530	25	𝑎	𝑎	PRON
iajs-2911	530	26	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	530	27	𝔘e	𝔘e	PROPN
iajs-2911	530	28	of	of	ADP
iajs-2911	530	29	d	d	PROPN
iajs-2911	530	30	in	in	ADP
iajs-2911	530	31	d	d	PROPN
iajs-2911	530	32	,	,	PUNCT
iajs-2911	530	33	a	a	DET
iajs-2911	530	34	𝜌	𝜌	X
iajs-2911	530	35	-open	-open	VERB
iajs-2911	530	36	𝑎	𝑎	DET
iajs-2911	530	37	𝜂ℙ𝕕	𝜂ℙ𝕕	NOUN
iajs-2911	530	38	ve	ve	NOUN
iajs-2911	530	39	of	of	ADP
iajs-2911	530	40	e	e	PROPN
iajs-2911	530	41	in	in	ADP
iajs-2911	530	42	e	e	PROPN
iajs-2911	530	43	and	and	CCONJ
iajs-2911	530	44	an	an	DET
iajs-2911	530	45	𝔽𝑒	𝔽𝑒	PROPN
iajs-2911	530	46	∈	∈	PROPN
iajs-2911	530	47	ℑ	ℑ	NOUN
iajs-2911	530	48	such	such	ADJ
iajs-2911	530	49	that	that	SCONJ
iajs-2911	530	50	f−	f−	PROPN
iajs-2911	530	51	cl(𝔘e	cl(𝔘e	PROPN
iajs-2911	530	52	×ve	×ve	NOUN
iajs-2911	530	53	)	)	PUNCT
iajs-2911	530	54	∩𝔽𝑒	∩𝔽𝑒	NOUN
iajs-2911	530	55	=	=	PUNCT
iajs-2911	530	56	∅.	∅.	NOUN
iajs-2911	530	57	as	as	SCONJ
iajs-2911	530	58	prove	prove	VERB
iajs-2911	530	59	above	above	ADV
iajs-2911	530	60	,	,	PUNCT
iajs-2911	530	61	∃finitely	∃finitely	ADV
iajs-2911	530	62	many	many	ADJ
iajs-2911	530	63	elements	element	NOUN
iajs-2911	530	64	e1,e2,e3,	e1,e2,e3,	NOUN
iajs-2911	530	65	...	...	PUNCT
iajs-2911	530	66	,en	,en	PUNCT
iajs-2911	530	67	of	of	ADP
iajs-2911	530	68	e	e	X
iajs-2911	530	69	such	such	ADJ
iajs-2911	530	70	that	that	SCONJ
iajs-2911	530	71	{	{	PUNCT
iajs-2911	530	72	d}×	d}×	NOUN
iajs-2911	530	73	𝐸	𝐸	PROPN
iajs-2911	530	74	⊂∪𝑘=1	⊂∪𝑘=1	NOUN
iajs-2911	530	75	𝑛	𝑛	ADP
iajs-2911	530	76	𝔔	𝔔	PROPN
iajs-2911	530	77	−	−	PROPN
iajs-2911	530	78	𝑐𝑙(𝐺𝑒𝑘	𝑐𝑙(𝐺𝑒𝑘	PROPN
iajs-2911	530	79	×	×	PROPN
iajs-2911	530	80	𝑉𝑒𝑘	𝑉𝑒𝑘	PROPN
iajs-2911	530	81	)	)	PUNCT
iajs-2911	530	82	.	.	PUNCT
iajs-2911	531	1	putting	put	VERB
iajs-2911	531	2	𝔘	𝔘	NOUN
iajs-2911	531	3	and	and	CCONJ
iajs-2911	531	4	choosing	choose	VERB
iajs-2911	531	5	𝔽	𝔽	PROPN
iajs-2911	531	6	∈	∈	PROPN
iajs-2911	531	7	ℑ	ℑ	PROPN
iajs-2911	531	8	with	with	ADP
iajs-2911	531	9	,	,	PUNCT
iajs-2911	531	10	𝔽	𝔽	PROPN
iajs-2911	531	11	∩𝑘=1	∩𝑘=1	NOUN
iajs-2911	531	12	𝑛	𝑛	PRON
iajs-2911	531	13	𝔽	𝔽	PROPN
iajs-2911	531	14	𝑒𝑘	𝑒𝑘	PROPN
iajs-2911	531	15	,	,	PUNCT
iajs-2911	531	16	we	we	PRON
iajs-2911	531	17	get	get	VERB
iajs-2911	531	18	d	d	X
iajs-2911	531	19	×e⊂	×e⊂	ADP
iajs-2911	531	20	𝔘	𝔘	PROPN
iajs-2911	531	21	×e⊂q	×e⊂q	VERB
iajs-2911	531	22	such	such	ADJ
iajs-2911	531	23	that	that	DET
iajs-2911	531	24	q−cl(𝔘	q−cl(𝔘	NOUN
iajs-2911	531	25	×e)∩	×e)∩	VERB
iajs-2911	531	26	𝔽	𝔽	PROPN
iajs-2911	531	27	=	=	PUNCT
iajs-2911	531	28	∅.	∅.	VERB
iajs-2911	531	29	thus	thus	ADV
iajs-2911	531	30	cl(𝔽)∩π−1(d	cl(𝔽)∩π−1(d	NOUN
iajs-2911	531	31	)	)	PUNCT
iajs-2911	531	32	=	=	PUNCT
iajs-2911	531	33	∅.	∅.	VERB
iajs-2911	531	34	so	so	ADV
iajs-2911	531	35	π−1(d	π−1(d	ADJ
iajs-2911	531	36	)	)	PUNCT
iajs-2911	531	37	is	be	AUX
iajs-2911	531	38	u.r.(resp	u.r.(resp	PRON
iajs-2911	531	39	.	.	PROPN
iajs-2911	531	40	,	,	PUNCT
iajs-2911	532	1	l.r.)in	l.r.)in	PROPN
iajs-2911	533	1	d	d	NOUN
iajs-2911	533	2	×e	×e	X
iajs-2911	533	3	.	.	PUNCT
iajs-2911	533	4	(	(	PUNCT
iajs-2911	533	5	iii)⇒(i	iii)⇒(i	X
iajs-2911	533	6	)	)	PUNCT
iajs-2911	533	7	taking	take	VERB
iajs-2911	533	8	d∗	d∗	NOUN
iajs-2911	533	9	=	=	SYM
iajs-2911	534	1	d	d	NOUN
iajs-2911	534	2	,	,	PUNCT
iajs-2911	534	3	we	we	PRON
iajs-2911	534	4	have	have	VERB
iajs-2911	534	5	that	that	PRON
iajs-2911	534	6	x	x	X
iajs-2911	535	1	=	=	SYM
iajs-2911	535	2	π	π	X
iajs-2911	535	3	:	:	PUNCT
iajs-2911	535	4	d∗	d∗	PROPN
iajs-2911	535	5	×	×	PROPN
iajs-2911	535	6	d	d	PROPN
iajs-2911	535	7	→	→	SYM
iajs-2911	535	8	d∗	d∗	PROPN
iajs-2911	535	9	is	be	AUX
iajs-2911	535	10	u.r.(resp	u.r.(resp	PROPN
iajs-2911	535	11	.	.	PROPN
iajs-2911	535	12	,	,	PUNCT
iajs-2911	535	13	l.r	l.r	PROPN
iajs-2911	535	14	.	.	PROPN
iajs-2911	535	15	)	)	PUNCT
iajs-2911	536	1	therefore	therefore	ADV
iajs-2911	536	2	by	by	ADP
iajs-2911	536	3	(	(	PUNCT
iajs-2911	536	4	theorem	theorem	NOUN
iajs-2911	536	5	(	(	PUNCT
iajs-2911	536	6	3.5	3.5	NUM
iajs-2911	536	7	.	.	PUNCT
iajs-2911	536	8	)	)	PUNCT
iajs-2911	536	9	)	)	PUNCT
iajs-2911	536	10	,	,	PUNCT
iajs-2911	536	11	d∗	d∗	VERB
iajs-2911	536	12	×e	×e	PROPN
iajs-2911	536	13	is	be	AUX
iajs-2911	536	14	an	an	DET
iajs-2911	536	15	𝔼.set	𝔼.set	NOUN
iajs-2911	536	16	and	and	CCONJ
iajs-2911	536	17	hence	hence	ADV
iajs-2911	536	18	is	be	AUX
iajs-2911	536	19	qhc	qhc	PROPN
iajs-2911	536	20	.	.	PUNCT
iajs-2911	536	21	c𝒐𝒓𝒐llary	c𝒐𝒓𝒐llary	PROPN
iajs-2911	536	22	4.7	4.7	NUM
iajs-2911	536	23	.	.	PUNCT
iajs-2911	537	1	for	for	ADP
iajs-2911	537	2	a	a	DET
iajs-2911	537	3	topological	topological	ADJ
iajs-2911	537	4	space	space	NOUN
iajs-2911	537	5	(	(	PUNCT
iajs-2911	537	6	e,𝜏	e,𝜏	NOUN
iajs-2911	537	7	)	)	PUNCT
iajs-2911	537	8	,	,	PUNCT
iajs-2911	537	9	the	the	DET
iajs-2911	537	10	next	next	ADJ
iajs-2911	537	11	are	be	AUX
iajs-2911	537	12	equivalent	equivalent	ADJ
iajs-2911	537	13	:	:	PUNCT
iajs-2911	537	14	i.	i.	PROPN
iajs-2911	537	15	h	h	PROPN
iajs-2911	537	16	𝑖s	𝑖s	PUNCT
iajs-2911	537	17	qhc	qhc	PROPN
iajs-2911	537	18	.	.	PROPN
iajs-2911	537	19	ii	ii	PROPN
iajs-2911	537	20	.	.	PUNCT
iajs-2911	538	1	a	a	DET
iajs-2911	538	2	𝔽.	𝔽.	PROPN
iajs-2911	538	3	𝕎.m	𝕎.m	PROPN
iajs-2911	538	4	.	.	PUNCT
iajs-2911	539	1	(	(	PUNCT
iajs-2911	539	2	e,𝜏	e,𝜏	NOUN
iajs-2911	539	3	)	)	PUNCT
iajs-2911	539	4	is	be	AUX
iajs-2911	539	5	p.t	p.t	PROPN
iajs-2911	539	6	space	space	NOUN
iajs-2911	539	7	𝑤𝑖th	𝑤𝑖th	NOUN
iajs-2911	539	8	constant	constant	ADJ
iajs-2911	539	9	projection	projection	NOUN
iajs-2911	539	10	on	on	ADP
iajs-2911	539	11	d∗	d∗	NOUN
iajs-2911	539	12	in	in	ADP
iajs-2911	539	13	which	which	PRON
iajs-2911	539	14	d∗	d∗	NOUN
iajs-2911	539	15	is	be	AUX
iajs-2911	539	16	a	a	DET
iajs-2911	539	17	singleton	singleton	NOUN
iajs-2911	539	18	with	with	ADP
iajs-2911	539	19	two	two	NUM
iajs-2911	539	20	equal	equal	ADJ
iajs-2911	539	21	topologies	topology	NOUN
iajs-2911	539	22	meaning	mean	VERB
iajs-2911	539	23	the	the	DET
iajs-2911	539	24	unique	unique	ADJ
iajs-2911	539	25	topology	topology	NOUN
iajs-2911	539	26	on	on	ADP
iajs-2911	539	27	d∗.	d∗.	PROPN
iajs-2911	539	28	iii	iii	NOUN
iajs-2911	539	29	.	.	PUNCT
iajs-2911	540	1	the	the	DET
iajs-2911	540	2	𝔽.	𝔽.	PROPN
iajs-2911	540	3	𝕎	𝕎	PROPN
iajs-2911	540	4	..	..	PUNCT
iajs-2911	540	5	(	(	PUNCT
iajs-2911	540	6	b×h	b×h	PROPN
iajs-2911	540	7	,	,	PUNCT
iajs-2911	540	8	q	q	NOUN
iajs-2911	540	9	)	)	PUNCT
iajs-2911	540	10	is	be	AUX
iajs-2911	540	11	m.p.t.s	m.p.t.s	PROPN
iajs-2911	540	12	.	.	PROPN
iajs-2911	541	1	on	on	ADP
iajs-2911	541	2	(	(	PUNCT
iajs-2911	541	3	d,𝜌	d,𝜌	NOUN
iajs-2911	541	4	)	)	PUNCT
iajs-2911	541	5	,	,	PUNCT
iajs-2911	541	6	in	in	ADP
iajs-2911	541	7	which	which	PRON
iajs-2911	541	8	𝔔	𝔔	PROPN
iajs-2911	541	9	=	=	SYM
iajs-2911	541	10	𝜌	𝜌	PART
iajs-2911	541	11	×	×	PROPN
iajs-2911	541	12	𝜏.	𝜏.	NOUN
iajs-2911	541	13	ihjpas	ihjpas	PROPN
iajs-2911	541	14	.	.	PUNCT
iajs-2911	542	1	36	36	NUM
iajs-2911	542	2	(	(	PUNCT
iajs-2911	542	3	4	4	NUM
iajs-2911	542	4	)	)	PUNCT
iajs-2911	542	5	2023	2023	NUM
iajs-2911	542	6	406	406	NUM
iajs-2911	542	7	5	5	NUM
iajs-2911	542	8	.	.	PUNCT
iajs-2911	542	9	conclusion	conclusion	VERB
iajs-2911	542	10	the	the	DET
iajs-2911	542	11	main	main	ADJ
iajs-2911	542	12	purpose	purpose	NOUN
iajs-2911	542	13	of	of	ADP
iajs-2911	542	14	the	the	DET
iajs-2911	542	15	present	present	ADJ
iajs-2911	542	16	work	work	NOUN
iajs-2911	542	17	is	be	AUX
iajs-2911	542	18	to	to	ADP
iajs-2911	542	19	providethe	providethe	PRON
iajs-2911	542	20	starting	starting	NOUN
iajs-2911	542	21	point	point	NOUN
iajs-2911	542	22	for	for	ADP
iajs-2911	542	23	some	some	DET
iajs-2911	542	24	application	application	NOUN
iajs-2911	542	25	of	of	ADP
iajs-2911	542	26	fibr𝑒𝑤𝑖𝑠𝑒	fibr𝑒𝑤𝑖𝑠𝑒	PROPN
iajs-2911	542	27	multi	multi	ADJ
iajs-2911	542	28	-	-	ADJ
iajs-2911	542	29	p𝑒rf𝑒𝑐t	p𝑒rf𝑒𝑐t	ADJ
iajs-2911	542	30	t𝑜p𝑜l𝑜gical	t𝑜p𝑜l𝑜gical	ADJ
iajs-2911	542	31	spa𝑐𝑒𝑠	spa𝑐𝑒𝑠	NOUN
iajs-2911	542	32	structures	structure	NOUN
iajs-2911	542	33	in	in	ADP
iajs-2911	542	34	a	a	DET
iajs-2911	542	35	falter	falter	ADJ
iajs-2911	542	36	base	base	NOUN
iajs-2911	542	37	by	by	ADP
iajs-2911	542	38	using	use	VERB
iajs-2911	542	39	multi	multi	ADJ
iajs-2911	542	40	-	-	ADJ
iajs-2911	542	41	topological	topological	ADJ
iajs-2911	542	42	spaces	space	NOUN
iajs-2911	542	43	.	.	PUNCT
iajs-2911	543	1	definitions	definition	NOUN
iajs-2911	543	2	of	of	ADP
iajs-2911	543	3	characterization	characterization	NOUN
iajs-2911	543	4	theorems	theorem	NOUN
iajs-2911	543	5	are	be	AUX
iajs-2911	543	6	used	use	VERB
iajs-2911	543	7	for	for	ADP
iajs-2911	543	8	multi	multi	ADJ
iajs-2911	543	9	-	-	ADJ
iajs-2911	543	10	r𝑖g𝑖d	r𝑖g𝑖d	ADJ
iajs-2911	543	11	,	,	PUNCT
iajs-2911	543	12	fibr𝑒𝑤𝑖𝑠𝑒	fibr𝑒𝑤𝑖𝑠𝑒	ADJ
iajs-2911	543	13	multi-𝑤𝑒akly	multi-𝑤𝑒akly	ADJ
iajs-2911	543	14	cl𝑜𝑠𝑒d	cl𝑜𝑠𝑒d	NOUN
iajs-2911	543	15	,	,	PUNCT
iajs-2911	543	16	𝔼	𝔼	PROPN
iajs-2911	543	17	𝑠𝑒t	𝑠𝑒t	PROPN
iajs-2911	543	18	,	,	PUNCT
iajs-2911	543	19	fibr𝑒𝑤𝑖𝑠e	fibr𝑒𝑤𝑖𝑠e	ADJ
iajs-2911	543	20	alm𝑜𝑠t	alm𝑜𝑠t	CCONJ
iajs-2911	543	21	multi	multi	NOUN
iajs-2911	543	22	-	-	ADJ
iajs-2911	543	23	p𝑒rf𝑒ct	p𝑒rf𝑒ct	ADJ
iajs-2911	543	24	,	,	PUNCT
iajs-2911	543	25	multi*-𝑐𝑜ntinu𝑜us	multi*-𝑐𝑜ntinu𝑜us	PROPN
iajs-2911	543	26	fibr𝑒𝑤𝑖𝑠𝑒	fibr𝑒𝑤𝑖𝑠𝑒	PROPN
iajs-2911	543	27	multi∗	multi∗	PROPN
iajs-2911	543	28	-t𝑜p𝑜l𝑜gical	-t𝑜p𝑜l𝑜gical	ADJ
iajs-2911	543	29	spa𝑐𝑒𝑠.	spa𝑐𝑒𝑠.	NOUN
iajs-2911	543	30	references	reference	NOUN
iajs-2911	543	31	1	1	NUM
iajs-2911	543	32	.	.	PUNCT
iajs-2911	544	1	banzaru	banzaru	PROPN
iajs-2911	544	2	,	,	PUNCT
iajs-2911	544	3	t.	t.	PROPN
iajs-2911	544	4	,	,	PUNCT
iajs-2911	544	5	multi	multi	ADJ
iajs-2911	544	6	-	-	NOUN
iajs-2911	544	7	functions	function	NOUN
iajs-2911	544	8	and	and	CCONJ
iajs-2911	544	9	m	m	NOUN
iajs-2911	544	10	-	-	NOUN
iajs-2911	544	11	product	product	NOUN
iajs-2911	544	12	spaces	space	NOUN
iajs-2911	544	13	,	,	PUNCT
iajs-2911	544	14	bull	bull	NOUN
iajs-2911	544	15	.	.	PUNCT
iajs-2911	545	1	stin	stin	PROPN
iajs-2911	545	2	.	.	PUNCT
iajs-2911	545	3	tech	tech	PROPN
iajs-2911	545	4	.	.	PUNCT
iajs-2911	545	5	inst	inst	PROPN
iajs-2911	545	6	.	.	PUNCT
iajs-2911	546	1	politech	politech	PROPN
iajs-2911	546	2	.	.	PUNCT
iajs-2911	547	1	timisoara	timisoara	PROPN
iajs-2911	547	2	,	,	PUNCT
iajs-2911	547	3	ser	ser	PROPN
iajs-2911	547	4	.	.	PROPN
iajs-2911	547	5	mat	mat	PROPN
iajs-2911	547	6	.	.	PUNCT
iajs-2911	547	7	fiz	fiz	PROPN
iajs-2911	547	8	.	.	PUNCT
iajs-2911	548	1	mer	mer	PROPN
iajs-2911	548	2	.	.	PROPN
iajs-2911	548	3	teor	teor	PROPN
iajs-2911	548	4	.	.	PUNCT
iajs-2911	548	5	apl	apl	PROPN
iajs-2911	548	6	.	.	PROPN
iajs-2911	548	7	,	,	PUNCT
iajs-2911	548	8	17	17	NUM
iajs-2911	548	9	,	,	PUNCT
iajs-2911	548	10	31	31	NUM
iajs-2911	548	11	,	,	PUNCT
iajs-2911	548	12	1972	1972	NUM
iajs-2911	548	13	,	,	PUNCT
iajs-2911	548	14	17	17	NUM
iajs-2911	548	15	-	-	SYM
iajs-2911	548	16	23	23	NUM
iajs-2911	548	17	.	.	PUNCT
iajs-2911	549	1	2	2	X
iajs-2911	549	2	.	.	X
iajs-2911	549	3	bose	bose	PROPN
iajs-2911	549	4	,	,	PUNCT
iajs-2911	549	5	s.	s.	PROPN
iajs-2911	549	6	;	;	PUNCT
iajs-2911	549	7	sinha	sinha	NOUN
iajs-2911	549	8	,	,	PUNCT
iajs-2911	549	9	d.	d.	PROPN
iajs-2911	549	10	,	,	PUNCT
iajs-2911	549	11	almost	almost	ADV
iajs-2911	549	12	open	open	ADJ
iajs-2911	549	13	,	,	PUNCT
iajs-2911	549	14	almost	almost	ADV
iajs-2911	549	15	closed	closed	ADJ
iajs-2911	549	16	,	,	PUNCT
iajs-2911	549	17	θ	θ	NOUN
iajs-2911	549	18	continuous	continuous	ADJ
iajs-2911	549	19	and	and	CCONJ
iajs-2911	549	20	almost	almost	ADV
iajs-2911	549	21	quasicompact	quasicompact	NOUN
iajs-2911	549	22	mappings	mapping	NOUN
iajs-2911	549	23	in	in	ADP
iajs-2911	549	24	bitopological	bitopological	ADJ
iajs-2911	549	25	spaces	space	NOUN
iajs-2911	549	26	,	,	PUNCT
iajs-2911	549	27	bull	bull	NOUN
iajs-2911	549	28	.	.	PUNCT
iajs-2911	549	29	cal.math	cal.math	PROPN
iajs-2911	549	30	.	.	PUNCT
iajs-2911	550	1	soc	soc	PROPN
iajs-2911	550	2	.	.	PUNCT
iajs-2911	551	1	73	73	NUM
iajs-2911	551	2	,	,	PUNCT
iajs-2911	551	3	1981	1981	NUM
iajs-2911	551	4	,	,	PUNCT
iajs-2911	551	5	345	345	NUM
iajs-2911	551	6	.	.	PUNCT
iajs-2911	552	1	3	3	X
iajs-2911	552	2	.	.	X
iajs-2911	552	3	bourbaki	bourbaki	PROPN
iajs-2911	552	4	,	,	PUNCT
iajs-2911	552	5	n.	n.	NOUN
iajs-2911	552	6	,	,	PUNCT
iajs-2911	552	7	general	general	ADJ
iajs-2911	552	8	topology	topology	NOUN
iajs-2911	552	9	,	,	PUNCT
iajs-2911	552	10	part	part	NOUN
iajs-2911	552	11	i	i	PROPN
iajs-2911	552	12	,	,	PUNCT
iajs-2911	552	13	addison	addison	PROPN
iajs-2911	552	14	wesley	wesley	PROPN
iajs-2911	552	15	,	,	PUNCT
iajs-2911	552	16	reading	reading	NOUN
iajs-2911	552	17	,	,	PUNCT
iajs-2911	552	18	mass	mass	PROPN
iajs-2911	552	19	,	,	PUNCT
iajs-2911	552	20	1996	1996	NUM
iajs-2911	552	21	.	.	PUNCT
iajs-2911	553	1	4	4	X
iajs-2911	553	2	.	.	X
iajs-2911	553	3	englking	englking	NOUN
iajs-2911	553	4	,	,	PUNCT
iajs-2911	553	5	r.	r.	PROPN
iajs-2911	553	6	,	,	PUNCT
iajs-2911	553	7	outline	outline	NOUN
iajs-2911	553	8	of	of	ADP
iajs-2911	553	9	general	general	ADJ
iajs-2911	553	10	topology	topology	NOUN
iajs-2911	553	11	,	,	PUNCT
iajs-2911	553	12	amsterdam	amsterdam	PROPN
iajs-2911	553	13	,	,	PUNCT
iajs-2911	553	14	1989	1989	NUM
iajs-2911	553	15	.	.	PUNCT
iajs-2911	554	1	5	5	NUM
iajs-2911	554	2	.	.	X
iajs-2911	554	3	jabera	jabera	NOUN
iajs-2911	554	4	,	,	PUNCT
iajs-2911	554	5	m.	m.	PROPN
iajs-2911	554	6	h.	h.	PROPN
iajs-2911	554	7	;	;	PUNCT
iajs-2911	554	8	yousif	yousif	PROPN
iajs-2911	554	9	,	,	PUNCT
iajs-2911	554	10	y.y	y.y	PROPN
iajs-2911	554	11	.	.	PROPN
iajs-2911	554	12	,	,	PUNCT
iajs-2911	554	13	fibrewise	fibrewise	ADV
iajs-2911	554	14	multi	multi	ADJ
iajs-2911	554	15	-	-	ADJ
iajs-2911	554	16	topological	topological	ADJ
iajs-2911	554	17	spaces	space	NOUN
iajs-2911	554	18	,	,	PUNCT
iajs-2911	554	19	international	international	ADJ
iajs-2911	554	20	journal	journal	NOUN
iajs-2911	554	21	of	of	ADP
iajs-2911	554	22	nonlinear	nonlinear	ADJ
iajs-2911	554	23	analysis	analysis	NOUN
iajs-2911	554	24	and	and	CCONJ
iajs-2911	554	25	applications	application	NOUN
iajs-2911	554	26	,	,	PUNCT
iajs-2911	554	27	semnan	semnan	PROPN
iajs-2911	554	28	university	university	NOUN
iajs-2911	554	29	,	,	PUNCT
iajs-2911	554	30	doi	doi	PROPN
iajs-2911	554	31	:	:	PUNCT
iajs-2911	554	32	10.22075	10.22075	NUM
iajs-2911	554	33	/	/	SYM
iajs-2911	554	34	ijnaa.2022.6109	ijnaa.2022.6109	VERB
iajs-2911	554	35	,	,	PUNCT
iajs-2911	554	36	13	13	NUM
iajs-2911	554	37	,	,	PUNCT
iajs-2911	554	38	1	1	NUM
iajs-2911	554	39	,	,	PUNCT
iajs-2911	554	40	3463	3463	NUM
iajs-2911	554	41	-	-	SYM
iajs-2911	554	42	3474	3474	NUM
iajs-2911	554	43	,	,	PUNCT
iajs-2911	554	44	2022	2022	NUM
iajs-2911	554	45	.	.	PUNCT
iajs-2911	555	1	6	6	NUM
iajs-2911	555	2	.	.	X
iajs-2911	555	3	jain	jain	PROPN
iajs-2911	555	4	,	,	PUNCT
iajs-2911	555	5	r.	r.	PROPN
iajs-2911	555	6	c.	c.	PROPN
iajs-2911	555	7	;	;	PUNCT
iajs-2911	555	8	singal	singal	PROPN
iajs-2911	555	9	,	,	PUNCT
iajs-2911	555	10	a.	a.	PROPN
iajs-2911	555	11	r.	r.	PROPN
iajs-2911	555	12	,	,	PUNCT
iajs-2911	555	13	slightly	slightly	ADV
iajs-2911	555	14	continuous	continuous	ADJ
iajs-2911	555	15	mappings	mapping	NOUN
iajs-2911	555	16	,	,	PUNCT
iajs-2911	555	17	indian	indian	ADJ
iajs-2911	555	18	math	math	NOUN
iajs-2911	555	19	.	.	PUNCT
iajs-2911	556	1	soc	soc	PROPN
iajs-2911	556	2	.	.	PUNCT
iajs-2911	556	3	,	,	PUNCT
iajs-2911	556	4	64	64	NUM
iajs-2911	556	5	,	,	PUNCT
iajs-2911	556	6	1997	1997	NUM
iajs-2911	556	7	,	,	PUNCT
iajs-2911	556	8	195203	195203	NUM
iajs-2911	556	9	.	.	PUNCT
iajs-2911	557	1	7	7	X
iajs-2911	557	2	.	.	X
iajs-2911	557	3	james	james	PROPN
iajs-2911	557	4	,	,	PUNCT
iajs-2911	557	5	i.	i.	PROPN
iajs-2911	557	6	m.	m.	PROPN
iajs-2911	557	7	,	,	PUNCT
iajs-2911	557	8	fibrewise	fibrewise	NOUN
iajs-2911	557	9	topology	topology	NOUN
iajs-2911	557	10	,	,	PUNCT
iajs-2911	557	11	cambridge	cambridge	PROPN
iajs-2911	557	12	university	university	PROPN
iajs-2911	557	13	press	press	PROPN
iajs-2911	557	14	,	,	PUNCT
iajs-2911	557	15	london	london	PROPN
iajs-2911	557	16	,	,	PUNCT
iajs-2911	557	17	1989	1989	NUM
iajs-2911	557	18	.	.	PUNCT
iajs-2911	558	1	8	8	NUM
iajs-2911	558	2	.	.	X
iajs-2911	559	1	james	james	PROPN
iajs-2911	559	2	,	,	PUNCT
iajs-2911	559	3	i.	i.	PROPN
iajs-2911	559	4	m.	m.	PROPN
iajs-2911	559	5	,	,	PUNCT
iajs-2911	559	6	general	general	ADJ
iajs-2911	559	7	topology	topology	NOUN
iajs-2911	559	8	and	and	CCONJ
iajs-2911	559	9	homotopy	homotopy	PROPN
iajs-2911	559	10	theory	theory	NOUN
iajs-2911	559	11	,	,	PUNCT
iajs-2911	559	12	springer	springer	NOUN
iajs-2911	559	13	-	-	PUNCT
iajs-2911	559	14	verlag	verlag	PROPN
iajs-2911	559	15	,	,	PUNCT
iajs-2911	559	16	new	new	PROPN
iajs-2911	559	17	york	york	PROPN
iajs-2911	559	18	,	,	PUNCT
iajs-2911	559	19	1984	1984	NUM
iajs-2911	559	20	.	.	PUNCT
iajs-2911	560	1	9	9	NUM
iajs-2911	560	2	.	.	X
iajs-2911	560	3	kariofillis	kariofillis	PROPN
iajs-2911	560	4	,	,	PUNCT
iajs-2911	560	5	c.	c.	PROPN
iajs-2911	560	6	,	,	PUNCT
iajs-2911	560	7	on	on	ADP
iajs-2911	560	8	pairwise	pairwise	NOUN
iajs-2911	560	9	almost	almost	ADV
iajs-2911	560	10	compactness	compactness	NOUN
iajs-2911	560	11	,	,	PUNCT
iajs-2911	560	12	ann	ann	PROPN
iajs-2911	560	13	.	.	PROPN
iajs-2911	560	14	soc	soc	PROPN
iajs-2911	560	15	.	.	PUNCT
iajs-2911	561	1	sci	sci	PROPN
iajs-2911	561	2	bruxelles	bruxelles	PROPN
iajs-2911	561	3	,	,	PUNCT
iajs-2911	561	4	1986	1986	NUM
iajs-2911	561	5	.	.	PUNCT
iajs-2911	562	1	100	100	NUM
iajs-2911	562	2	-	-	SYM
iajs-2911	562	3	129	129	NUM
iajs-2911	562	4	.	.	PUNCT
iajs-2911	563	1	10	10	NUM
iajs-2911	563	2	.	.	PUNCT
iajs-2911	564	1	mukherjee	mukherjee	PROPN
iajs-2911	564	2	,	,	PUNCT
iajs-2911	564	3	m.	m.	NOUN
iajs-2911	564	4	;	;	PUNCT
iajs-2911	564	5	nandi	nandi	PROPN
iajs-2911	564	6	,	,	PUNCT
iajs-2911	564	7	j.	j.	PROPN
iajs-2911	564	8	;	;	PUNCT
iajs-2911	564	9	sen	sen	PROPN
iajs-2911	564	10	,	,	PUNCT
iajs-2911	564	11	s.	s.	PROPN
iajs-2911	564	12	,	,	PUNCT
iajs-2911	564	13	on	on	ADP
iajs-2911	564	14	bitopological	bitopological	ADJ
iajs-2911	564	15	qhc	qhc	NOUN
iajs-2911	564	16	spaces	space	NOUN
iajs-2911	564	17	,	,	PUNCT
iajs-2911	564	18	indian	indian	ADJ
iajs-2911	564	19	jour	jour	X
iajs-2911	564	20	.	.	PUNCT
iajs-2911	565	1	pure	pure	ADJ
iajs-2911	565	2	appl	appl	PROPN
iajs-2911	565	3	.	.	PUNCT
iajs-2911	565	4	math	math	NOUN
iajs-2911	565	5	.	.	PUNCT
iajs-2911	566	1	27	27	NUM
iajs-2911	566	2	(	(	PUNCT
iajs-2911	566	3	1996	1996	NUM
iajs-2911	566	4	)	)	PUNCT
iajs-2911	566	5	.	.	PUNCT
iajs-2911	567	1	11	11	NUM
iajs-2911	567	2	.	.	X
iajs-2911	567	3	whyburn	whyburn	NOUN
iajs-2911	567	4	,	,	PUNCT
iajs-2911	567	5	g.	g.	PROPN
iajs-2911	567	6	t	t	PROPN
iajs-2911	567	7	..	..	PUNCT
iajs-2911	567	8	directed	direct	VERB
iajs-2911	567	9	families	family	NOUN
iajs-2911	567	10	of	of	ADP
iajs-2911	567	11	sets	set	NOUN
iajs-2911	567	12	and	and	CCONJ
iajs-2911	567	13	closedness	closedness	NOUN
iajs-2911	567	14	of	of	ADP
iajs-2911	567	15	function	function	NOUN
iajs-2911	567	16	.	.	PUNCT
iajs-2911	568	1	proc	proc	NOUN
iajs-2911	568	2	.	.	PUNCT
iajs-2911	569	1	nat	nat	PROPN
iajs-2911	569	2	.	.	PUNCT
iajs-2911	570	1	acad	acad	PROPN
iajs-2911	570	2	.	.	PUNCT
iajs-2911	571	1	sci	sci	PROPN
iajs-2911	571	2	.	.	PUNCT
iajs-2911	571	3	u.s.a	u.s.a	PROPN
iajs-2911	571	4	.	.	PROPN
iajs-2911	571	5	,	,	PUNCT
iajs-2911	571	6	1965	1965	NUM
iajs-2911	571	7	,	,	PUNCT
iajs-2911	571	8	54	54	NUM
iajs-2911	571	9	,	,	PUNCT
iajs-2911	571	10	688	688	NUM
iajs-2911	571	11	-	-	SYM
iajs-2911	571	12	692	692	NUM
iajs-2911	571	13	.	.	PUNCT
iajs-2911	571	14	12	12	NUM
iajs-2911	571	15	.	.	PUNCT
iajs-2911	572	1	yousif	yousif	PROPN
iajs-2911	572	2	,	,	PUNCT
iajs-2911	572	3	y.	y.	PROPN
iajs-2911	572	4	;	;	PUNCT
iajs-2911	572	5	hussain	hussain	PROPN
iajs-2911	572	6	,	,	PUNCT
iajs-2911	572	7	l.	l.	PROPN
iajs-2911	572	8	,	,	PUNCT
iajs-2911	572	9	fiberwise	fiberwise	NOUN
iajs-2911	572	10	ij	ij	ADJ
iajs-2911	572	11	-	-	ADJ
iajs-2911	572	12	perfect	perfect	ADJ
iajs-2911	572	13	bitopological	bitopological	ADJ
iajs-2911	572	14	spaces	space	NOUN
iajs-2911	572	15	,	,	PUNCT
iajs-2911	572	16	conf	conf	NOUN
iajs-2911	572	17	.	.	PUNCT
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