id	sid	tid	token	lemma	pos
iajs-2969	1	1	ihjpas	ihjpas	PROPN
iajs-2969	1	2	.	.	PUNCT
iajs-2969	2	1	36(1)2023	36(1)2023	NUM
iajs-2969	2	2	333	333	NUM
iajs-2969	2	3	this	this	DET
iajs-2969	2	4	work	work	NOUN
iajs-2969	2	5	is	be	AUX
iajs-2969	2	6	licensed	license	VERB
iajs-2969	2	7	under	under	ADP
iajs-2969	2	8	a	a	DET
iajs-2969	2	9	creative	creative	ADJ
iajs-2969	2	10	commons	common	NOUN
iajs-2969	2	11	attribution	attribution	NOUN
iajs-2969	2	12	4.0	4.0	NUM
iajs-2969	2	13	international	international	ADJ
iajs-2969	2	14	license	license	NOUN
iajs-2969	2	15	ϣ	ϣ	NOUN
iajs-2969	2	16	−semi	−semi	NOUN
iajs-2969	2	17	-	-	ADJ
iajs-2969	2	18	p	p	ADJ
iajs-2969	2	19	open	open	ADJ
iajs-2969	2	20	set	set	NOUN
iajs-2969	2	21	abstract	abstract	ADJ
iajs-2969	2	22	csaszar	csaszar	NOUN
iajs-2969	2	23	introduced	introduce	VERB
iajs-2969	2	24	the	the	DET
iajs-2969	2	25	concept	concept	NOUN
iajs-2969	2	26	of	of	ADP
iajs-2969	2	27	generalized	generalized	ADJ
iajs-2969	2	28	topological	topological	ADJ
iajs-2969	2	29	space	space	NOUN
iajs-2969	2	30	and	and	CCONJ
iajs-2969	2	31	a	a	DET
iajs-2969	2	32	new	new	ADJ
iajs-2969	2	33	open	open	ADJ
iajs-2969	2	34	set	set	NOUN
iajs-2969	2	35	in	in	ADP
iajs-2969	2	36	a	a	DET
iajs-2969	2	37	generalized	generalized	ADJ
iajs-2969	2	38	topological	topological	ADJ
iajs-2969	2	39	space	space	NOUN
iajs-2969	2	40	called	call	VERB
iajs-2969	2	41	ϣ-preopen	ϣ-preopen	NOUN
iajs-2969	2	42	in	in	ADP
iajs-2969	2	43	2002	2002	NUM
iajs-2969	2	44	and	and	CCONJ
iajs-2969	2	45	2005	2005	NUM
iajs-2969	2	46	,	,	PUNCT
iajs-2969	2	47	respectively	respectively	ADV
iajs-2969	2	48	.	.	PUNCT
iajs-2969	3	1	definitions	definition	NOUN
iajs-2969	3	2	of	of	ADP
iajs-2969	3	3	ϣ-preinterior	ϣ-preinterior	NOUN
iajs-2969	3	4	and	and	CCONJ
iajs-2969	3	5	ϣ-preclosuer	ϣ-preclosuer	NOUN
iajs-2969	3	6	were	be	AUX
iajs-2969	3	7	given	give	VERB
iajs-2969	3	8	.	.	PUNCT
iajs-2969	4	1	successively	successively	ADV
iajs-2969	4	2	,	,	PUNCT
iajs-2969	4	3	several	several	ADJ
iajs-2969	4	4	studies	study	NOUN
iajs-2969	4	5	have	have	AUX
iajs-2969	4	6	appeared	appear	VERB
iajs-2969	4	7	to	to	PART
iajs-2969	4	8	give	give	VERB
iajs-2969	4	9	many	many	ADJ
iajs-2969	4	10	generalizations	generalization	NOUN
iajs-2969	4	11	for	for	ADP
iajs-2969	4	12	an	an	DET
iajs-2969	4	13	open	open	ADJ
iajs-2969	4	14	set	set	NOUN
iajs-2969	4	15	.	.	PUNCT
iajs-2969	5	1	the	the	DET
iajs-2969	5	2	object	object	NOUN
iajs-2969	5	3	of	of	ADP
iajs-2969	5	4	our	our	PRON
iajs-2969	5	5	paper	paper	NOUN
iajs-2969	5	6	is	be	AUX
iajs-2969	5	7	to	to	PART
iajs-2969	5	8	give	give	VERB
iajs-2969	5	9	a	a	DET
iajs-2969	5	10	new	new	ADJ
iajs-2969	5	11	type	type	NOUN
iajs-2969	5	12	of	of	ADP
iajs-2969	5	13	generalization	generalization	NOUN
iajs-2969	5	14	of	of	ADP
iajs-2969	5	15	an	an	DET
iajs-2969	5	16	open	open	ADJ
iajs-2969	5	17	set	set	NOUN
iajs-2969	5	18	in	in	ADP
iajs-2969	5	19	a	a	DET
iajs-2969	5	20	generalized	generalized	ADJ
iajs-2969	5	21	topological	topological	ADJ
iajs-2969	5	22	space	space	NOUN
iajs-2969	5	23	called	call	VERB
iajs-2969	5	24	ϣ-semi	ϣ-semi	PROPN
iajs-2969	5	25	-	-	PUNCT
iajs-2969	5	26	p	p	NOUN
iajs-2969	5	27	-	-	PUNCT
iajs-2969	5	28	open	open	ADJ
iajs-2969	5	29	set	set	NOUN
iajs-2969	5	30	.	.	PUNCT
iajs-2969	6	1	we	we	PRON
iajs-2969	6	2	present	present	VERB
iajs-2969	6	3	the	the	DET
iajs-2969	6	4	definition	definition	NOUN
iajs-2969	6	5	of	of	ADP
iajs-2969	6	6	this	this	DET
iajs-2969	6	7	set	set	NOUN
iajs-2969	6	8	with	with	ADP
iajs-2969	6	9	its	its	PRON
iajs-2969	6	10	equivalent	equivalent	NOUN
iajs-2969	6	11	.	.	PUNCT
iajs-2969	7	1	we	we	PRON
iajs-2969	7	2	give	give	VERB
iajs-2969	7	3	definition	definition	NOUN
iajs-2969	7	4	of	of	ADP
iajs-2969	7	5	ϣ-semi	ϣ-semi	PROPN
iajs-2969	7	6	-	-	PUNCT
iajs-2969	7	7	p	p	NOUN
iajs-2969	7	8	-	-	PUNCT
iajs-2969	7	9	interior	interior	ADJ
iajs-2969	7	10	and	and	CCONJ
iajs-2969	7	11	ϣ-semi	ϣ-semi	NOUN
iajs-2969	7	12	-	-	PUNCT
iajs-2969	7	13	p	p	NOUN
iajs-2969	7	14	-	-	PUNCT
iajs-2969	7	15	closure	closure	NOUN
iajs-2969	7	16	of	of	ADP
iajs-2969	7	17	a	a	DET
iajs-2969	7	18	set	set	NOUN
iajs-2969	7	19	and	and	CCONJ
iajs-2969	7	20	discuss	discuss	VERB
iajs-2969	7	21	their	their	PRON
iajs-2969	7	22	properties	property	NOUN
iajs-2969	7	23	.	.	PUNCT
iajs-2969	8	1	also	also	ADV
iajs-2969	8	2	the	the	DET
iajs-2969	8	3	properties	property	NOUN
iajs-2969	8	4	of	of	ADP
iajs-2969	8	5	ϣ-preinterior	ϣ-preinterior	NOUN
iajs-2969	8	6	and	and	CCONJ
iajs-2969	8	7	ϣ-preclosuer	ϣ-preclosuer	NOUN
iajs-2969	8	8	are	be	AUX
iajs-2969	8	9	discussed	discuss	VERB
iajs-2969	8	10	.	.	PUNCT
iajs-2969	9	1	in	in	ADP
iajs-2969	9	2	addition	addition	NOUN
iajs-2969	9	3	,	,	PUNCT
iajs-2969	9	4	we	we	PRON
iajs-2969	9	5	give	give	VERB
iajs-2969	9	6	a	a	DET
iajs-2969	9	7	new	new	ADJ
iajs-2969	9	8	type	type	NOUN
iajs-2969	9	9	of	of	ADP
iajs-2969	9	10	continuous	continuous	ADJ
iajs-2969	9	11	function	function	NOUN
iajs-2969	9	12	in	in	ADP
iajs-2969	9	13	a	a	DET
iajs-2969	9	14	generalized	generalized	ADJ
iajs-2969	9	15	topological	topological	ADJ
iajs-2969	9	16	space	space	NOUN
iajs-2969	9	17	as	as	ADP
iajs-2969	9	18	(	(	PUNCT
iajs-2969	9	19	ϣ1	ϣ1	NOUN
iajs-2969	9	20	,	,	PUNCT
iajs-2969	9	21	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	9	22	-	-	ADJ
iajs-2969	9	23	p	p	ADJ
iajs-2969	9	24	-	-	PUNCT
iajs-2969	9	25	continuous	continuous	ADJ
iajs-2969	9	26	function	function	NOUN
iajs-2969	9	27	and	and	CCONJ
iajs-2969	9	28	(	(	PUNCT
iajs-2969	9	29	ϣ1	ϣ1	NOUN
iajs-2969	9	30	,	,	PUNCT
iajs-2969	9	31	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	9	32	-	-	ADJ
iajs-2969	9	33	pirresolute	pirresolute	ADJ
iajs-2969	9	34	function	function	NOUN
iajs-2969	9	35	.	.	PUNCT
iajs-2969	10	1	the	the	DET
iajs-2969	10	2	relationship	relationship	NOUN
iajs-2969	10	3	between	between	ADP
iajs-2969	10	4	them	they	PRON
iajs-2969	10	5	is	be	AUX
iajs-2969	10	6	showed	show	VERB
iajs-2969	10	7	.	.	PUNCT
iajs-2969	11	1	we	we	PRON
iajs-2969	11	2	prove	prove	VERB
iajs-2969	11	3	that	that	SCONJ
iajs-2969	11	4	every	every	DET
iajs-2969	11	5	ϣ-open	ϣ-open	NOUN
iajs-2969	11	6	(	(	PUNCT
iajs-2969	11	7	ϣpreopen	ϣpreopen	ADJ
iajs-2969	11	8	)	)	PUNCT
iajs-2969	11	9	set	set	NOUN
iajs-2969	11	10	is	be	AUX
iajs-2969	11	11	an	an	DET
iajs-2969	11	12	ϣ-semi	ϣ-semi	NOUN
iajs-2969	11	13	-	-	PUNCT
iajs-2969	11	14	p	p	NOUN
iajs-2969	11	15	-	-	PUNCT
iajs-2969	11	16	open	open	NOUN
iajs-2969	11	17	set	set	NOUN
iajs-2969	11	18	,	,	PUNCT
iajs-2969	11	19	but	but	CCONJ
iajs-2969	11	20	not	not	PART
iajs-2969	11	21	conversely	conversely	ADV
iajs-2969	11	22	.	.	PUNCT
iajs-2969	12	1	every	every	DET
iajs-2969	12	2	(	(	PUNCT
iajs-2969	12	3	ϣ1	ϣ1	NOUN
iajs-2969	12	4	,	,	PUNCT
iajs-2969	12	5	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	12	6	-	-	ADJ
iajs-2969	12	7	p	p	ADJ
iajs-2969	12	8	-	-	PUNCT
iajs-2969	12	9	irresolute	irresolute	ADJ
iajs-2969	12	10	function	function	NOUN
iajs-2969	12	11	is	be	AUX
iajs-2969	12	12	an	an	DET
iajs-2969	12	13	(	(	PUNCT
iajs-2969	12	14	ϣ1	ϣ1	NOUN
iajs-2969	12	15	,	,	PUNCT
iajs-2969	12	16	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	12	17	-	-	ADJ
iajs-2969	12	18	p	p	ADJ
iajs-2969	12	19	-	-	PUNCT
iajs-2969	12	20	continuous	continuous	ADJ
iajs-2969	12	21	function	function	NOUN
iajs-2969	12	22	,	,	PUNCT
iajs-2969	12	23	but	but	CCONJ
iajs-2969	12	24	not	not	PART
iajs-2969	12	25	conversely	conversely	ADV
iajs-2969	12	26	.	.	PUNCT
iajs-2969	13	1	also	also	ADV
iajs-2969	13	2	we	we	PRON
iajs-2969	13	3	show	show	VERB
iajs-2969	13	4	that	that	SCONJ
iajs-2969	13	5	the	the	DET
iajs-2969	13	6	union	union	NOUN
iajs-2969	13	7	of	of	ADP
iajs-2969	13	8	any	any	DET
iajs-2969	13	9	family	family	NOUN
iajs-2969	13	10	of	of	ADP
iajs-2969	13	11	ϣ-semi	ϣ-semi	PROPN
iajs-2969	13	12	-	-	PUNCT
iajs-2969	13	13	p	p	NOUN
iajs-2969	13	14	-	-	PUNCT
iajs-2969	13	15	open	open	ADJ
iajs-2969	13	16	sets	set	NOUN
iajs-2969	13	17	is	be	AUX
iajs-2969	13	18	an	an	DET
iajs-2969	13	19	ϣ-semi	ϣ-semi	NOUN
iajs-2969	13	20	-	-	PUNCT
iajs-2969	13	21	p	p	NOUN
iajs-2969	13	22	-	-	PUNCT
iajs-2969	13	23	open	open	NOUN
iajs-2969	13	24	set	set	NOUN
iajs-2969	13	25	,	,	PUNCT
iajs-2969	13	26	but	but	CCONJ
iajs-2969	13	27	the	the	DET
iajs-2969	13	28	intersection	intersection	NOUN
iajs-2969	13	29	of	of	ADP
iajs-2969	13	30	two	two	NUM
iajs-2969	13	31	ϣ-semi	ϣ-semi	NOUN
iajs-2969	13	32	-	-	PUNCT
iajs-2969	13	33	p	p	NOUN
iajs-2969	13	34	-	-	PUNCT
iajs-2969	13	35	open	open	ADJ
iajs-2969	13	36	sets	set	NOUN
iajs-2969	13	37	need	need	VERB
iajs-2969	13	38	not	not	PART
iajs-2969	13	39	to	to	PART
iajs-2969	13	40	be	be	AUX
iajs-2969	13	41	an	an	DET
iajs-2969	13	42	ϣ-semi	ϣ-semi	NOUN
iajs-2969	13	43	-	-	PUNCT
iajs-2969	13	44	p	p	NOUN
iajs-2969	13	45	-	-	PUNCT
iajs-2969	13	46	open	open	ADJ
iajs-2969	13	47	set	set	NOUN
iajs-2969	13	48	.	.	PUNCT
iajs-2969	14	1	keywords:ϣ-semi	keywords:ϣ-semi	NOUN
iajs-2969	14	2	-	-	ADJ
iajs-2969	14	3	p	p	X
iajs-2969	14	4	-	-	PUNCT
iajs-2969	14	5	open	open	ADJ
iajs-2969	14	6	,	,	PUNCT
iajs-2969	14	7	ϣ-semi	ϣ-semi	NOUN
iajs-2969	14	8	-	-	PUNCT
iajs-2969	14	9	p	p	NOUN
iajs-2969	14	10	-	-	PUNCT
iajs-2969	14	11	interior	interior	ADJ
iajs-2969	14	12	,	,	PUNCT
iajs-2969	14	13	ϣ-semi	ϣ-semi	PROPN
iajs-2969	14	14	-	-	PUNCT
iajs-2969	14	15	p	p	NOUN
iajs-2969	14	16	-	-	PUNCT
iajs-2969	14	17	closure	closure	NOUN
iajs-2969	14	18	,	,	PUNCT
iajs-2969	14	19	(	(	PUNCT
iajs-2969	14	20	ϣ1	ϣ1	NOUN
iajs-2969	14	21	,	,	PUNCT
iajs-2969	14	22	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	14	23	-	-	ADJ
iajs-2969	14	24	p	p	NOUN
iajs-2969	14	25	-	-	PUNCT
iajs-2969	14	26	irresolute	irresolute	ADJ
iajs-2969	14	27	and	and	CCONJ
iajs-2969	14	28	(	(	PUNCT
iajs-2969	14	29	ϣ1	ϣ1	NOUN
iajs-2969	14	30	,	,	PUNCT
iajs-2969	14	31	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	14	32	-	-	ADJ
iajs-2969	14	33	p	p	ADJ
iajs-2969	14	34	-	-	PUNCT
iajs-2969	14	35	continuous	continuous	ADJ
iajs-2969	14	36	.	.	PUNCT
iajs-2969	15	1	1.introduction	1.introduction	NUM
iajs-2969	15	2	and	and	CCONJ
iajs-2969	15	3	preliminaries	preliminary	NOUN
iajs-2969	15	4	in	in	ADP
iajs-2969	15	5	this	this	DET
iajs-2969	15	6	paper	paper	NOUN
iajs-2969	15	7	,	,	PUNCT
iajs-2969	15	8	we	we	PRON
iajs-2969	15	9	denote	denote	VERB
iajs-2969	15	10	a	a	DET
iajs-2969	15	11	topological	topological	ADJ
iajs-2969	15	12	space	space	NOUN
iajs-2969	15	13	by	by	ADP
iajs-2969	15	14	(	(	PUNCT
iajs-2969	15	15	z	z	NOUN
iajs-2969	15	16	,	,	PUNCT
iajs-2969	15	17	ӽ	ӽ	NOUN
iajs-2969	15	18	)	)	PUNCT
iajs-2969	15	19	and	and	CCONJ
iajs-2969	15	20	the	the	DET
iajs-2969	15	21	closure	closure	NOUN
iajs-2969	15	22	(	(	PUNCT
iajs-2969	15	23	interior	interior	NOUN
iajs-2969	15	24	)	)	PUNCT
iajs-2969	15	25	of	of	ADP
iajs-2969	15	26	a	a	DET
iajs-2969	15	27	subset	subset	ADJ
iajs-2969	15	28	ħ	ħ	NOUN
iajs-2969	15	29	of	of	ADP
iajs-2969	15	30	z	z	NOUN
iajs-2969	15	31	by	by	ADP
iajs-2969	15	32	cl(ħ)(int(ħ	cl(ħ)(int(ħ	NOUN
iajs-2969	15	33	)	)	PUNCT
iajs-2969	15	34	)	)	PUNCT
iajs-2969	15	35	,	,	PUNCT
iajs-2969	15	36	respectively	respectively	ADV
iajs-2969	15	37	.	.	PUNCT
iajs-2969	16	1	1	1	X
iajs-2969	16	2	.	.	X
iajs-2969	16	3	the	the	DET
iajs-2969	16	4	interior	interior	NOUN
iajs-2969	16	5	of	of	ADP
iajs-2969	16	6	ħ	ħ	PROPN
iajs-2969	16	7	is	be	AUX
iajs-2969	16	8	the	the	DET
iajs-2969	16	9	set	set	ADJ
iajs-2969	16	10	int(ħ	int(ħ	NOUN
iajs-2969	16	11	)	)	PUNCT
iajs-2969	16	12	=	=	PUNCT
iajs-2969	16	13	⋃{ɯ	⋃{ɯ	PROPN
iajs-2969	16	14	:	:	PUNCT
iajs-2969	16	15	ɯ	ɯ	PROPN
iajs-2969	16	16	∈	∈	PROPN
iajs-2969	16	17	ӽ	ӽ	X
iajs-2969	16	18	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2969	16	19	ɯ	ɯ	PROPN
iajs-2969	16	20	⊆	⊆	NUM
iajs-2969	16	21	ħ	ħ	NOUN
iajs-2969	16	22	}	}	PUNCT
iajs-2969	16	23	.	.	PUNCT
iajs-2969	17	1	doi.org/10.30526/36.1.2969	doi.org/10.30526/36.1.2969	PROPN
iajs-2969	17	2	article	article	NOUN
iajs-2969	17	3	history	history	NOUN
iajs-2969	17	4	:	:	PUNCT
iajs-2969	17	5	received	receive	VERB
iajs-2969	17	6	3	3	NUM
iajs-2969	17	7	augest	aug	ADJ
iajs-2969	17	8	2022	2022	NUM
iajs-2969	17	9	,	,	PUNCT
iajs-2969	17	10	accepted	accept	VERB
iajs-2969	17	11	24	24	NUM
iajs-2969	17	12	augest	aug	ADJ
iajs-2969	17	13	2022	2022	NUM
iajs-2969	17	14	,	,	PUNCT
iajs-2969	17	15	published	publish	VERB
iajs-2969	17	16	in	in	ADP
iajs-2969	17	17	january	january	PROPN
iajs-2969	17	18	2023	2023	NUM
iajs-2969	17	19	.	.	PUNCT
iajs-2969	18	1	ibn	ibn	PROPN
iajs-2969	18	2	al	al	PROPN
iajs-2969	18	3	-	-	PUNCT
iajs-2969	18	4	haitham	haitham	PROPN
iajs-2969	18	5	journal	journal	PROPN
iajs-2969	18	6	for	for	ADP
iajs-2969	18	7	pure	pure	ADJ
iajs-2969	18	8	and	and	CCONJ
iajs-2969	18	9	applied	applied	ADJ
iajs-2969	18	10	sciences	sciences	PROPN
iajs-2969	18	11	journal	journal	PROPN
iajs-2969	18	12	homepage	homepage	NOUN
iajs-2969	18	13	:	:	PUNCT
iajs-2969	18	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	NOUN
iajs-2969	18	15	muna	muna	PROPN
iajs-2969	18	16	l.	l.	PROPN
iajs-2969	18	17	abd	abd	PROPN
iajs-2969	18	18	ul	ul	PROPN
iajs-2969	18	19	ridha	ridha	PROPN
iajs-2969	18	20	department	department	PROPN
iajs-2969	18	21	of	of	ADP
iajs-2969	18	22	mathematics	mathematics	PROPN
iajs-2969	18	23	,	,	PUNCT
iajs-2969	18	24	college	college	NOUN
iajs-2969	18	25	of	of	ADP
iajs-2969	18	26	education	education	NOUN
iajs-2969	18	27	for	for	ADP
iajs-2969	18	28	pure	pure	ADJ
iajs-2969	18	29	sciences	science	NOUN
iajs-2969	18	30	,	,	PUNCT
iajs-2969	18	31	ibn	ibn	PROPN
iajs-2969	18	32	al	al	PROPN
iajs-2969	18	33	–	–	PUNCT
iajs-2969	18	34	haitham	haitham	PROPN
iajs-2969	18	35	,	,	PUNCT
iajs-2969	18	36	university	university	PROPN
iajs-2969	18	37	of	of	ADP
iajs-2969	18	38	baghdad	baghdad	PROPN
iajs-2969	18	39	,	,	PUNCT
iajs-2969	18	40	baghdad	baghdad	PROPN
iajs-2969	18	41	,	,	PUNCT
iajs-2969	18	42	iraq	iraq	PROPN
iajs-2969	18	43	.	.	PUNCT
iajs-2969	19	1	mona.laith1203a@ihcoedu.uobaghdad.edu.iq	mona.laith1203a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-2969	19	2	suaad	suaad	PROPN
iajs-2969	19	3	g.	g.	PROPN
iajs-2969	19	4	gasim	gasim	PROPN
iajs-2969	19	5	department	department	PROPN
iajs-2969	19	6	of	of	ADP
iajs-2969	19	7	mathematics	mathematics	PROPN
iajs-2969	19	8	,	,	PUNCT
iajs-2969	19	9	college	college	NOUN
iajs-2969	19	10	of	of	ADP
iajs-2969	19	11	education	education	NOUN
iajs-2969	19	12	for	for	ADP
iajs-2969	19	13	pure	pure	ADJ
iajs-2969	19	14	sciences	science	NOUN
iajs-2969	19	15	,	,	PUNCT
iajs-2969	19	16	ibn	ibn	PROPN
iajs-2969	19	17	al	al	PROPN
iajs-2969	19	18	–	–	PUNCT
iajs-2969	19	19	haitham	haitham	PROPN
iajs-2969	19	20	,	,	PUNCT
iajs-2969	19	21	university	university	PROPN
iajs-2969	19	22	of	of	ADP
iajs-2969	19	23	baghdad	baghdad	PROPN
iajs-2969	19	24	,	,	PUNCT
iajs-2969	19	25	baghdad	baghdad	PROPN
iajs-2969	19	26	,	,	PUNCT
iajs-2969	19	27	iraq	iraq	PROPN
iajs-2969	19	28	.	.	PUNCT
iajs-2969	20	1	suaad.gedaan@yahoo.com	suaad.gedaan@yahoo.com	X
iajs-2969	20	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2969	20	3	mailto:mona.laith1203a@ihcoedu.uobaghdad.edu.iq	mailto:mona.laith1203a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-2969	20	4	mailto:suaad.gedaan@yahoo.com	mailto:suaad.gedaan@yahoo.com	PROPN
iajs-2969	20	5	ihjpas	ihjpas	PROPN
iajs-2969	20	6	.	.	PUNCT
iajs-2969	21	1	36(1)2023	36(1)2023	NUM
iajs-2969	21	2	334	334	NUM
iajs-2969	21	3	2	2	NUM
iajs-2969	21	4	.	.	PUNCT
iajs-2969	22	1	the	the	DET
iajs-2969	22	2	closure	closure	NOUN
iajs-2969	22	3	of	of	ADP
iajs-2969	22	4	ħ	ħ	NOUN
iajs-2969	22	5	is	be	AUX
iajs-2969	22	6	the	the	DET
iajs-2969	22	7	set	set	NOUN
iajs-2969	22	8	cl(ħ	cl(ħ	PUNCT
iajs-2969	22	9	)	)	PUNCT
iajs-2969	22	10	=	=	SYM
iajs-2969	22	11	⋂{ƒ	⋂{ƒ	NOUN
iajs-2969	22	12	:	:	PUNCT
iajs-2969	22	13	ƒ	ƒ	PROPN
iajs-2969	22	14	∈	∈	PROPN
iajs-2969	22	15	ӽ′	ӽ′	PUNCT
iajs-2969	22	16	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2969	22	17	ħ	ħ	PROPN
iajs-2969	22	18	⊆	⊆	NUM
iajs-2969	22	19	ƒ	ƒ	NOUN
iajs-2969	22	20	}	}	PUNCT
iajs-2969	23	1	[	[	X
iajs-2969	23	2	1	1	NUM
iajs-2969	23	3	]	]	PUNCT
iajs-2969	23	4	,	,	PUNCT
iajs-2969	23	5	where	where	SCONJ
iajs-2969	23	6	ӽ′	ӽ′	PROPN
iajs-2969	23	7	symbolizes	symbolize	VERB
iajs-2969	23	8	the	the	DET
iajs-2969	23	9	family	family	NOUN
iajs-2969	23	10	of	of	ADP
iajs-2969	23	11	closed	closed	ADJ
iajs-2969	23	12	subsets	subset	NOUN
iajs-2969	23	13	of	of	ADP
iajs-2969	23	14	z.	z.	PROPN
iajs-2969	23	15	the	the	DET
iajs-2969	23	16	term	term	NOUN
iajs-2969	23	17	"	"	PUNCT
iajs-2969	23	18	preopen	preopen	ADJ
iajs-2969	23	19	”	"	PUNCT
iajs-2969	23	20	was	be	AUX
iajs-2969	23	21	introduced	introduce	VERB
iajs-2969	23	22	for	for	ADP
iajs-2969	23	23	the	the	DET
iajs-2969	23	24	first	first	ADJ
iajs-2969	23	25	time	time	NOUN
iajs-2969	23	26	in	in	ADP
iajs-2969	23	27	1984	1984	NUM
iajs-2969	23	28	[	[	X
iajs-2969	23	29	2	2	NUM
iajs-2969	23	30	]	]	PUNCT
iajs-2969	23	31	.	.	PUNCT
iajs-2969	24	1	a	a	DET
iajs-2969	24	2	subset	subset	NOUN
iajs-2969	24	3	a	a	PRON
iajs-2969	24	4	of	of	ADP
iajs-2969	24	5	a	a	DET
iajs-2969	24	6	topological	topological	ADJ
iajs-2969	24	7	space	space	NOUN
iajs-2969	24	8	(	(	PUNCT
iajs-2969	24	9	z	z	NOUN
iajs-2969	24	10	,	,	PUNCT
iajs-2969	24	11	ӽ	ӽ	X
iajs-2969	24	12	)	)	PUNCT
iajs-2969	24	13	is	be	AUX
iajs-2969	24	14	called	call	VERB
iajs-2969	24	15	a	a	DET
iajs-2969	24	16	preopen	preopen	ADJ
iajs-2969	24	17	set	set	VERB
iajs-2969	24	18	if	if	SCONJ
iajs-2969	24	19	a	a	DET
iajs-2969	24	20			PROPN
iajs-2969	24	21	int(cla	int(cla	NOUN
iajs-2969	24	22	)	)	PUNCT
iajs-2969	24	23	.	.	PUNCT
iajs-2969	25	1	the	the	DET
iajs-2969	25	2	complement	complement	NOUN
iajs-2969	25	3	of	of	ADP
iajs-2969	25	4	a	a	DET
iajs-2969	25	5	preopen	preopen	ADJ
iajs-2969	25	6	set	set	NOUN
iajs-2969	25	7	is	be	AUX
iajs-2969	25	8	called	call	VERB
iajs-2969	25	9	a	a	DET
iajs-2969	25	10	preclosed	preclose	VERB
iajs-2969	25	11	set	set	NOUN
iajs-2969	25	12	.	.	PUNCT
iajs-2969	26	1	the	the	DET
iajs-2969	26	2	family	family	NOUN
iajs-2969	26	3	of	of	ADP
iajs-2969	26	4	all	all	DET
iajs-2969	26	5	preopen	preopen	ADJ
iajs-2969	26	6	sets	set	NOUN
iajs-2969	26	7	of	of	ADP
iajs-2969	26	8	z	z	NOUN
iajs-2969	26	9	is	be	AUX
iajs-2969	26	10	denoted	denote	VERB
iajs-2969	26	11	by	by	ADP
iajs-2969	26	12	po(z	po(z	NOUN
iajs-2969	26	13	)	)	PUNCT
iajs-2969	26	14	.	.	PUNCT
iajs-2969	27	1	the	the	DET
iajs-2969	27	2	family	family	NOUN
iajs-2969	27	3	of	of	ADP
iajs-2969	27	4	all	all	DET
iajs-2969	27	5	preclosed	preclose	VERB
iajs-2969	27	6	sets	set	NOUN
iajs-2969	27	7	of	of	ADP
iajs-2969	27	8	z	z	NOUN
iajs-2969	27	9	is	be	AUX
iajs-2969	27	10	denoted	denote	VERB
iajs-2969	27	11	by	by	ADP
iajs-2969	27	12	pc(z	pc(z	NOUN
iajs-2969	27	13	)	)	PUNCT
iajs-2969	27	14	.	.	PUNCT
iajs-2969	28	1	in	in	ADP
iajs-2969	28	2	2000	2000	NUM
iajs-2969	28	3	,	,	PUNCT
iajs-2969	28	4	navalagi	navalagi	ADV
iajs-2969	28	5	used	use	VERB
iajs-2969	28	6	"	"	PUNCT
iajs-2969	28	7	preopen	preopen	ADJ
iajs-2969	28	8	"	"	PUNCT
iajs-2969	28	9	term	term	NOUN
iajs-2969	28	10	to	to	PART
iajs-2969	28	11	define	define	VERB
iajs-2969	28	12	a	a	DET
iajs-2969	28	13	"	"	PUNCT
iajs-2969	28	14	semi	semi	ADJ
iajs-2969	28	15	-	-	ADJ
iajs-2969	28	16	p	p	ADJ
iajs-2969	28	17	-	-	PUNCT
iajs-2969	28	18	open	open	ADJ
iajs-2969	28	19	set	set	NOUN
iajs-2969	28	20	"	"	PUNCT
iajs-2969	28	21	[	[	X
iajs-2969	28	22	3	3	NUM
iajs-2969	28	23	]	]	PUNCT
iajs-2969	28	24	.	.	PUNCT
iajs-2969	29	1	a	a	DET
iajs-2969	29	2	subset	subset	NOUN
iajs-2969	29	3	a	a	PRON
iajs-2969	29	4	of	of	ADP
iajs-2969	29	5	a	a	DET
iajs-2969	29	6	topological	topological	ADJ
iajs-2969	29	7	space	space	NOUN
iajs-2969	29	8	(	(	PUNCT
iajs-2969	29	9	z	z	NOUN
iajs-2969	29	10	,	,	PUNCT
iajs-2969	29	11	ӽ	ӽ	NOUN
iajs-2969	29	12	)	)	PUNCT
iajs-2969	29	13	is	be	AUX
iajs-2969	29	14	said	say	VERB
iajs-2969	29	15	to	to	PART
iajs-2969	29	16	be	be	AUX
iajs-2969	29	17	semi	semi	ADJ
iajs-2969	29	18	-	-	ADJ
iajs-2969	29	19	p	p	ADJ
iajs-2969	29	20	-	-	PUNCT
iajs-2969	29	21	open	open	NOUN
iajs-2969	29	22	set	set	NOUN
iajs-2969	29	23	if	if	SCONJ
iajs-2969	29	24	there	there	PRON
iajs-2969	29	25	exists	exist	VERB
iajs-2969	29	26	a	a	DET
iajs-2969	29	27	preopen	preopen	ADJ
iajs-2969	29	28	set	set	VERB
iajs-2969	29	29	u	u	NOUN
iajs-2969	29	30	in	in	ADP
iajs-2969	29	31	z	z	PROPN
iajs-2969	29	32	such	such	ADJ
iajs-2969	29	33	that	that	SCONJ
iajs-2969	29	34	u	u	PROPN
iajs-2969	29	35	⊆	⊆	NUM
iajs-2969	29	36	a	a	DET
iajs-2969	29	37	⊆	⊆	NUM
iajs-2969	29	38	pre	pre	NOUN
iajs-2969	29	39	-	-	NOUN
iajs-2969	29	40	cl	cl	ADJ
iajs-2969	29	41	u.	u.	VERB
iajs-2969	29	42	the	the	DET
iajs-2969	29	43	family	family	NOUN
iajs-2969	29	44	of	of	ADP
iajs-2969	29	45	all	all	DET
iajs-2969	29	46	semi	semi	ADJ
iajs-2969	29	47	-	-	ADJ
iajs-2969	29	48	p	p	ADJ
iajs-2969	29	49	-	-	PUNCT
iajs-2969	29	50	open	open	ADJ
iajs-2969	29	51	sets	set	NOUN
iajs-2969	29	52	of	of	ADP
iajs-2969	29	53	z	z	NOUN
iajs-2969	29	54	is	be	AUX
iajs-2969	29	55	denoted	denote	VERB
iajs-2969	29	56	by	by	ADP
iajs-2969	29	57	s	s	NOUN
iajs-2969	29	58	-	-	PUNCT
iajs-2969	29	59	po(z	po(z	NUM
iajs-2969	29	60	)	)	PUNCT
iajs-2969	29	61	.	.	PUNCT
iajs-2969	30	1	the	the	DET
iajs-2969	30	2	complement	complement	NOUN
iajs-2969	30	3	of	of	ADP
iajs-2969	30	4	a	a	DET
iajs-2969	30	5	semi	semi	ADJ
iajs-2969	30	6	-	-	ADJ
iajs-2969	30	7	p	p	ADJ
iajs-2969	30	8	-	-	PUNCT
iajs-2969	30	9	open	open	ADJ
iajs-2969	30	10	set	set	NOUN
iajs-2969	30	11	is	be	AUX
iajs-2969	30	12	called	call	VERB
iajs-2969	30	13	semi	semi	ADJ
iajs-2969	30	14	-	-	ADJ
iajs-2969	30	15	p	p	ADJ
iajs-2969	30	16	-	-	PUNCT
iajs-2969	30	17	closed	close	VERB
iajs-2969	30	18	set	set	NOUN
iajs-2969	30	19	.	.	PUNCT
iajs-2969	31	1	the	the	DET
iajs-2969	31	2	family	family	NOUN
iajs-2969	31	3	of	of	ADP
iajs-2969	31	4	all	all	DET
iajs-2969	31	5	semi	semi	ADJ
iajs-2969	31	6	-	-	ADJ
iajs-2969	31	7	p	p	ADJ
iajs-2969	31	8	-	-	PUNCT
iajs-2969	31	9	closed	close	VERB
iajs-2969	31	10	sets	set	NOUN
iajs-2969	31	11	of	of	ADP
iajs-2969	31	12	z	z	NOUN
iajs-2969	31	13	is	be	AUX
iajs-2969	31	14	denoted	denote	VERB
iajs-2969	31	15	by	by	ADP
iajs-2969	31	16	s	s	NOUN
iajs-2969	31	17	-	-	NOUN
iajs-2969	31	18	pc(z	pc(z	NUM
iajs-2969	31	19	)	)	PUNCT
iajs-2969	31	20	.	.	PUNCT
iajs-2969	32	1	a	a	DET
iajs-2969	32	2	function	function	NOUN
iajs-2969	32	3	f	f	X
iajs-2969	32	4	:	:	PUNCT
iajs-2969	32	5	(	(	PUNCT
iajs-2969	32	6	z1	z1	ADJ
iajs-2969	32	7	,	,	PUNCT
iajs-2969	32	8	ӽ1	ӽ1	PROPN
iajs-2969	32	9	)	)	PUNCT
iajs-2969	32	10	→	→	SYM
iajs-2969	32	11	(	(	PUNCT
iajs-2969	32	12	z2	z2	PROPN
iajs-2969	32	13	,	,	PUNCT
iajs-2969	32	14	ӽ2	ӽ2	PROPN
iajs-2969	32	15	)	)	PUNCT
iajs-2969	32	16	is	be	AUX
iajs-2969	32	17	said	say	VERB
iajs-2969	32	18	to	to	PART
iajs-2969	32	19	be	be	AUX
iajs-2969	32	20	a	a	DET
iajs-2969	32	21	continuous	continuous	ADJ
iajs-2969	32	22	function	function	NOUN
iajs-2969	32	23	if	if	SCONJ
iajs-2969	32	24	the	the	DET
iajs-2969	32	25	inverse	inverse	ADJ
iajs-2969	32	26	image	image	NOUN
iajs-2969	32	27	of	of	ADP
iajs-2969	32	28	any	any	DET
iajs-2969	32	29	open	open	ADJ
iajs-2969	32	30	set	set	NOUN
iajs-2969	32	31	in	in	ADP
iajs-2969	32	32	z2	z2	PROPN
iajs-2969	32	33	is	be	AUX
iajs-2969	32	34	an	an	DET
iajs-2969	32	35	open	open	ADJ
iajs-2969	32	36	set	set	NOUN
iajs-2969	32	37	in	in	ADP
iajs-2969	32	38	z1	z1	NOUN
iajs-2969	33	1	[	[	X
iajs-2969	33	2	4	4	NUM
iajs-2969	33	3	]	]	PUNCT
iajs-2969	33	4	.	.	PUNCT
iajs-2969	34	1	navalagi	navalagi	PROPN
iajs-2969	34	2	used	use	VERB
iajs-2969	34	3	the	the	DET
iajs-2969	34	4	term	term	NOUN
iajs-2969	34	5	"	"	PUNCT
iajs-2969	34	6	preopen	preopen	ADJ
iajs-2969	34	7	"	"	PUNCT
iajs-2969	34	8	to	to	PART
iajs-2969	34	9	introduce	introduce	VERB
iajs-2969	34	10	new	new	ADJ
iajs-2969	34	11	types	type	NOUN
iajs-2969	34	12	of	of	ADP
iajs-2969	34	13	a	a	DET
iajs-2969	34	14	continuous	continuous	ADJ
iajs-2969	34	15	function	function	NOUN
iajs-2969	34	16	"	"	PUNCT
iajs-2969	34	17	pre	pre	ADJ
iajs-2969	34	18	-	-	ADJ
iajs-2969	34	19	irresolute	irresolute	ADJ
iajs-2969	34	20	function	function	NOUN
iajs-2969	34	21	"	"	PUNCT
iajs-2969	34	22	and	and	CCONJ
iajs-2969	34	23	"	"	PUNCT
iajs-2969	34	24	pre	pre	ADJ
iajs-2969	34	25	-	-	ADJ
iajs-2969	34	26	continuous	continuous	ADJ
iajs-2969	34	27	function	function	NOUN
iajs-2969	34	28	"	"	PUNCT
iajs-2969	34	29	.	.	PUNCT
iajs-2969	35	1	a	a	DET
iajs-2969	35	2	function	function	NOUN
iajs-2969	35	3	f	f	X
iajs-2969	35	4	:	:	PUNCT
iajs-2969	35	5	(	(	PUNCT
iajs-2969	35	6	z1	z1	ADJ
iajs-2969	35	7	,	,	PUNCT
iajs-2969	35	8	ӽ1	ӽ1	PROPN
iajs-2969	35	9	)	)	PUNCT
iajs-2969	35	10	→	→	SYM
iajs-2969	35	11	(	(	PUNCT
iajs-2969	35	12	z2	z2	PROPN
iajs-2969	35	13	,	,	PUNCT
iajs-2969	35	14	ӽ2	ӽ2	PROPN
iajs-2969	35	15	)	)	PUNCT
iajs-2969	35	16	is	be	AUX
iajs-2969	35	17	called	call	VERB
iajs-2969	35	18	pre	pre	ADJ
iajs-2969	35	19	-	-	ADJ
iajs-2969	35	20	irresolute(pre	irresolute(pre	ADJ
iajs-2969	35	21	-	-	ADJ
iajs-2969	35	22	continuous	continuous	ADJ
iajs-2969	35	23	)	)	PUNCT
iajs-2969	35	24	function	function	NOUN
iajs-2969	35	25	if	if	SCONJ
iajs-2969	35	26	the	the	DET
iajs-2969	35	27	inverse	inverse	ADJ
iajs-2969	35	28	image	image	NOUN
iajs-2969	35	29	of	of	ADP
iajs-2969	35	30	any	any	DET
iajs-2969	35	31	pre	pre	ADJ
iajs-2969	35	32	-	-	ADJ
iajs-2969	35	33	open	open	ADJ
iajs-2969	35	34	set	set	NOUN
iajs-2969	35	35	in	in	ADP
iajs-2969	35	36	z2	z2	PROPN
iajs-2969	35	37	is	be	AUX
iajs-2969	35	38	a	a	DET
iajs-2969	35	39	pre	pre	ADJ
iajs-2969	35	40	-	-	ADJ
iajs-2969	35	41	open	open	ADJ
iajs-2969	35	42	set	set	NOUN
iajs-2969	35	43	inz1	inz1	NOUN
iajs-2969	35	44	(	(	PUNCT
iajs-2969	35	45	the	the	DET
iajs-2969	35	46	inverse	inverse	ADJ
iajs-2969	35	47	image	image	NOUN
iajs-2969	35	48	of	of	ADP
iajs-2969	35	49	any	any	DET
iajs-2969	35	50	open	open	ADJ
iajs-2969	35	51	set	set	NOUN
iajs-2969	35	52	in	in	ADP
iajs-2969	35	53	z2	z2	PROPN
iajs-2969	35	54	is	be	AUX
iajs-2969	35	55	a	a	DET
iajs-2969	35	56	pre	pre	ADJ
iajs-2969	35	57	-	-	ADJ
iajs-2969	35	58	open	open	ADJ
iajs-2969	35	59	set	set	ADJ
iajs-2969	35	60	z1	z1	NOUN
iajs-2969	35	61	)	)	PUNCT
iajs-2969	35	62	.	.	PUNCT
iajs-2969	36	1	in	in	ADP
iajs-2969	36	2	[	[	X
iajs-2969	36	3	5	5	NUM
iajs-2969	36	4	]	]	PUNCT
iajs-2969	36	5	,	,	PUNCT
iajs-2969	36	6	al	al	PROPN
iajs-2969	36	7	-	-	PUNCT
iajs-2969	36	8	khazraji	khazraji	PROPN
iajs-2969	36	9	used	use	VERB
iajs-2969	36	10	the	the	DET
iajs-2969	36	11	term	term	NOUN
iajs-2969	36	12	of	of	ADP
iajs-2969	36	13	"	"	PUNCT
iajs-2969	36	14	semi	semi	ADJ
iajs-2969	36	15	-	-	ADJ
iajs-2969	36	16	p	p	ADJ
iajs-2969	36	17	-	-	PUNCT
iajs-2969	36	18	open	open	ADJ
iajs-2969	36	19	set	set	NOUN
iajs-2969	36	20	"	"	PUNCT
iajs-2969	36	21	to	to	PART
iajs-2969	36	22	define	define	VERB
iajs-2969	36	23	new	new	ADJ
iajs-2969	36	24	types	type	NOUN
iajs-2969	36	25	of	of	ADP
iajs-2969	36	26	continuous	continuous	ADJ
iajs-2969	36	27	functions	function	NOUN
iajs-2969	36	28	"	"	PUNCT
iajs-2969	36	29	semi	semi	ADJ
iajs-2969	36	30	-	-	ADJ
iajs-2969	36	31	p	p	ADJ
iajs-2969	36	32	-	-	PUNCT
iajs-2969	36	33	irresolute	irresolute	NOUN
iajs-2969	36	34	"	"	PUNCT
iajs-2969	36	35	and	and	CCONJ
iajs-2969	36	36	"	"	PUNCT
iajs-2969	36	37	semi	semi	ADJ
iajs-2969	36	38	-	-	ADJ
iajs-2969	36	39	p	p	ADJ
iajs-2969	36	40	-	-	PUNCT
iajs-2969	36	41	continuous	continuous	ADJ
iajs-2969	36	42	"	"	PUNCT
iajs-2969	36	43	function	function	NOUN
iajs-2969	36	44	.	.	PUNCT
iajs-2969	37	1	a	a	DET
iajs-2969	37	2	function	function	NOUN
iajs-2969	37	3	f	f	X
iajs-2969	37	4	:	:	PUNCT
iajs-2969	37	5	(	(	PUNCT
iajs-2969	37	6	z1	z1	ADJ
iajs-2969	37	7	,	,	PUNCT
iajs-2969	37	8	ӽ1	ӽ1	PROPN
iajs-2969	37	9	)	)	PUNCT
iajs-2969	37	10	→	→	SYM
iajs-2969	37	11	(	(	PUNCT
iajs-2969	37	12	z2	z2	PROPN
iajs-2969	37	13	,	,	PUNCT
iajs-2969	37	14	ӽ2	ӽ2	PROPN
iajs-2969	37	15	)	)	PUNCT
iajs-2969	37	16	is	be	AUX
iajs-2969	37	17	called	call	VERB
iajs-2969	37	18	a	a	DET
iajs-2969	37	19	semi	semi	ADJ
iajs-2969	37	20	-	-	ADJ
iajs-2969	37	21	p	p	ADJ
iajs-2969	37	22	-	-	PUNCT
iajs-2969	37	23	irresolute	irresolute	ADJ
iajs-2969	37	24	(	(	PUNCT
iajs-2969	37	25	semi	semi	ADJ
iajs-2969	37	26	-	-	ADJ
iajs-2969	37	27	p	p	ADJ
iajs-2969	37	28	-	-	PUNCT
iajs-2969	37	29	continuous	continuous	ADJ
iajs-2969	37	30	)	)	PUNCT
iajs-2969	37	31	function	function	NOUN
iajs-2969	37	32	if	if	SCONJ
iajs-2969	37	33	the	the	DET
iajs-2969	37	34	inverse	inverse	ADJ
iajs-2969	37	35	image	image	NOUN
iajs-2969	37	36	of	of	ADP
iajs-2969	37	37	any	any	DET
iajs-2969	37	38	semi	semi	ADJ
iajs-2969	37	39	-	-	ADJ
iajs-2969	37	40	p	p	ADJ
iajs-2969	37	41	-	-	PUNCT
iajs-2969	37	42	open	open	ADJ
iajs-2969	37	43	set	set	NOUN
iajs-2969	37	44	in	in	ADP
iajs-2969	37	45	z2	z2	PROPN
iajs-2969	37	46	is	be	AUX
iajs-2969	37	47	a	a	DET
iajs-2969	37	48	semi	semi	ADJ
iajs-2969	37	49	-	-	ADJ
iajs-2969	37	50	p	p	ADJ
iajs-2969	37	51	-	-	PUNCT
iajs-2969	37	52	open	open	NOUN
iajs-2969	37	53	set	set	NOUN
iajs-2969	37	54	in	in	ADP
iajs-2969	37	55	z1(the	z1(the	DET
iajs-2969	37	56	inverse	inverse	ADJ
iajs-2969	37	57	image	image	NOUN
iajs-2969	37	58	of	of	ADP
iajs-2969	37	59	any	any	DET
iajs-2969	37	60	open	open	ADJ
iajs-2969	37	61	set	set	NOUN
iajs-2969	37	62	in	in	ADP
iajs-2969	37	63	z2	z2	PROPN
iajs-2969	37	64	is	be	AUX
iajs-2969	37	65	a	a	DET
iajs-2969	37	66	semi	semi	ADJ
iajs-2969	37	67	-	-	ADJ
iajs-2969	37	68	popen	popen	ADJ
iajs-2969	37	69	set	set	NOUN
iajs-2969	37	70	in	in	ADP
iajs-2969	37	71	z1	z1	PROPN
iajs-2969	37	72	)	)	PUNCT
iajs-2969	37	73	.	.	PUNCT
iajs-2969	38	1	let	let	VERB
iajs-2969	38	2	z	z	PRON
iajs-2969	38	3	be	be	AUX
iajs-2969	38	4	a	a	DET
iajs-2969	38	5	nonempty	nonempty	ADJ
iajs-2969	38	6	set	set	NOUN
iajs-2969	38	7	,	,	PUNCT
iajs-2969	38	8	a	a	DET
iajs-2969	38	9	collection	collection	NOUN
iajs-2969	38	10	ϣ	ϣ	NOUN
iajs-2969	38	11	of	of	ADP
iajs-2969	38	12	subsets	subset	NOUN
iajs-2969	38	13	of	of	ADP
iajs-2969	38	14	z	z	PROPN
iajs-2969	38	15	is	be	AUX
iajs-2969	38	16	called	call	VERB
iajs-2969	38	17	a	a	DET
iajs-2969	38	18	generalized	generalized	ADJ
iajs-2969	38	19	topology	topology	NOUN
iajs-2969	38	20	(	(	PUNCT
iajs-2969	38	21	in	in	ADP
iajs-2969	38	22	brief	brief	ADJ
iajs-2969	38	23	,	,	PUNCT
iajs-2969	38	24	𝐺𝑇	𝐺𝑇	PROPN
iajs-2969	38	25	)	)	PUNCT
iajs-2969	38	26	on	on	ADP
iajs-2969	38	27	z	z	NOUN
iajs-2969	38	28	if	if	SCONJ
iajs-2969	38	29	∅	∅	NOUN
iajs-2969	38	30	belongs	belong	VERB
iajs-2969	38	31	to	to	ADP
iajs-2969	38	32	ϣ	ϣ	PROPN
iajs-2969	38	33	and	and	CCONJ
iajs-2969	38	34	the	the	DET
iajs-2969	38	35	arbitrary	arbitrary	ADJ
iajs-2969	38	36	unions	union	NOUN
iajs-2969	38	37	of	of	ADP
iajs-2969	38	38	elements	element	NOUN
iajs-2969	38	39	of	of	ADP
iajs-2969	38	40	ϣ	ϣ	X
iajs-2969	38	41	is	be	AUX
iajs-2969	38	42	an	an	DET
iajs-2969	38	43	element	element	NOUN
iajs-2969	38	44	in	in	ADP
iajs-2969	38	45	ϣ	ϣ	PROPN
iajs-2969	38	46	,	,	PUNCT
iajs-2969	38	47	(	(	PUNCT
iajs-2969	38	48	z	z	X
iajs-2969	38	49	,	,	PUNCT
iajs-2969	38	50	ϣ	ϣ	X
iajs-2969	38	51	)	)	PUNCT
iajs-2969	38	52	is	be	AUX
iajs-2969	38	53	called	call	VERB
iajs-2969	38	54	generalized	generalized	ADJ
iajs-2969	38	55	topological	topological	ADJ
iajs-2969	38	56	space	space	NOUN
iajs-2969	38	57	(	(	PUNCT
iajs-2969	38	58	in	in	ADP
iajs-2969	38	59	brief	brief	ADJ
iajs-2969	38	60	,	,	PUNCT
iajs-2969	38	61	𝐺𝑇𝑆	𝐺𝑇𝑆	PROPN
iajs-2969	38	62	)	)	PUNCT
iajs-2969	39	1	[	[	X
iajs-2969	39	2	6	6	NUM
iajs-2969	39	3	]	]	PUNCT
iajs-2969	39	4	.	.	PUNCT
iajs-2969	40	1	every	every	DET
iajs-2969	40	2	set	set	NOUN
iajs-2969	40	3	in	in	ADP
iajs-2969	40	4	ϣ	ϣ	PROPN
iajs-2969	40	5	is	be	AUX
iajs-2969	40	6	called	call	VERB
iajs-2969	40	7	ϣ-open	ϣ-open	PROPN
iajs-2969	40	8	,	,	PUNCT
iajs-2969	40	9	while	while	SCONJ
iajs-2969	40	10	the	the	DET
iajs-2969	40	11	complement	complement	NOUN
iajs-2969	40	12	of	of	ADP
iajs-2969	40	13	ϣ-open	ϣ-open	PROPN
iajs-2969	40	14	is	be	AUX
iajs-2969	40	15	called	call	VERB
iajs-2969	40	16	ϣ-closed	ϣ-close	VERB
iajs-2969	40	17	;	;	PUNCT
iajs-2969	40	18	the	the	DET
iajs-2969	40	19	family	family	NOUN
iajs-2969	40	20	of	of	ADP
iajs-2969	40	21	all	all	DET
iajs-2969	40	22	ϣ-closed	ϣ-close	VERB
iajs-2969	40	23	sets	set	NOUN
iajs-2969	40	24	is	be	AUX
iajs-2969	40	25	denoted	denote	VERB
iajs-2969	40	26	by	by	ADP
iajs-2969	40	27	ϣ′.	ϣ′.	X
iajs-2969	40	28	the	the	DET
iajs-2969	40	29	union	union	NOUN
iajs-2969	40	30	of	of	ADP
iajs-2969	40	31	all	all	DET
iajs-2969	40	32	ϣ-open	ϣ-open	PROPN
iajs-2969	40	33	set	set	NOUN
iajs-2969	40	34	contained	contain	VERB
iajs-2969	40	35	in	in	ADP
iajs-2969	40	36	a	a	DET
iajs-2969	40	37	set	set	NOUN
iajs-2969	40	38	ħ	ħ	NOUN
iajs-2969	40	39	is	be	AUX
iajs-2969	40	40	called	call	VERB
iajs-2969	40	41	the	the	DET
iajs-2969	40	42	ϣinterior	ϣinterior	NOUN
iajs-2969	40	43	of	of	ADP
iajs-2969	40	44	ħ	ħ	NOUN
iajs-2969	40	45	and	and	CCONJ
iajs-2969	40	46	is	be	AUX
iajs-2969	40	47	denoted	denote	VERB
iajs-2969	40	48	by	by	ADP
iajs-2969	40	49	int	int	NOUN
iajs-2969	40	50	ϣ(ħ	ϣ(ħ	PROPN
iajs-2969	40	51	)	)	PUNCT
iajs-2969	40	52	,	,	PUNCT
iajs-2969	40	53	whereas	whereas	SCONJ
iajs-2969	40	54	the	the	DET
iajs-2969	40	55	intersection	intersection	NOUN
iajs-2969	40	56	of	of	ADP
iajs-2969	40	57	all	all	DET
iajs-2969	40	58	ϣ-closed	ϣ-close	VERB
iajs-2969	40	59	set	set	VERB
iajs-2969	40	60	containing	contain	VERB
iajs-2969	40	61	ħ	ħ	NOUN
iajs-2969	40	62	is	be	AUX
iajs-2969	40	63	called	call	VERB
iajs-2969	40	64	the	the	DET
iajs-2969	40	65	ϣ-closure	ϣ-closure	NOUN
iajs-2969	40	66	of	of	ADP
iajs-2969	40	67	ħ	ħ	NOUN
iajs-2969	40	68	and	and	CCONJ
iajs-2969	40	69	is	be	AUX
iajs-2969	40	70	denoted	denote	VERB
iajs-2969	40	71	by	by	ADP
iajs-2969	40	72	cl	cl	NOUN
iajs-2969	40	73	ϣ(ħ)[7	ϣ(ħ)[7	NOUN
iajs-2969	40	74	]	]	X
iajs-2969	40	75	.	.	PUNCT
iajs-2969	41	1	2	2	X
iajs-2969	41	2	.	.	X
iajs-2969	41	3	ϣ-pre	ϣ-pre	ADJ
iajs-2969	41	4	-	-	PUNCT
iajs-2969	41	5	open	open	ADJ
iajs-2969	41	6	set	set	VERB
iajs-2969	41	7	definition	definition	NOUN
iajs-2969	41	8	2.1	2.1	NUM
iajs-2969	42	1	[	[	SYM
iajs-2969	42	2	8	8	NUM
iajs-2969	42	3	]	]	PUNCT
iajs-2969	42	4	in	in	ADP
iajs-2969	42	5	a	a	DET
iajs-2969	42	6	𝐺𝑇𝑆	𝐺𝑇𝑆	PROPN
iajs-2969	42	7	(	(	PUNCT
iajs-2969	42	8	z	z	NOUN
iajs-2969	42	9	,	,	PUNCT
iajs-2969	42	10	ϣ	ϣ	NOUN
iajs-2969	42	11	)	)	PUNCT
iajs-2969	42	12	by	by	ADP
iajs-2969	42	13	an	an	DET
iajs-2969	42	14	ϣ-pre	ϣ-pre	NOUN
iajs-2969	42	15	-	-	PUNCT
iajs-2969	42	16	open	open	ADJ
iajs-2969	42	17	(	(	PUNCT
iajs-2969	42	18	in	in	ADP
iajs-2969	42	19	brief	brief	NOUN
iajs-2969	42	20	,	,	PUNCT
iajs-2969	42	21	ϣ	ϣ	NOUN
iajs-2969	42	22	−	−	PROPN
iajs-2969	42	23	p	p	NOUN
iajs-2969	42	24	−	−	PROPN
iajs-2969	42	25	o	o	NOUN
iajs-2969	42	26	)	)	PUNCT
iajs-2969	42	27	set	set	NOUN
iajs-2969	42	28	,	,	PUNCT
iajs-2969	42	29	we	we	PRON
iajs-2969	42	30	mean	mean	VERB
iajs-2969	42	31	a	a	DET
iajs-2969	42	32	subset	subset	NOUN
iajs-2969	42	33	ħ	ħ	NOUN
iajs-2969	42	34	of	of	ADP
iajs-2969	42	35	z	z	NOUN
iajs-2969	42	36	with	with	ADP
iajs-2969	42	37	ħ	ħ	PROPN
iajs-2969	42	38	⊆	⊆	NUM
iajs-2969	42	39	intϣclϣ	intϣclϣ	NOUN
iajs-2969	42	40	ħ	ħ	NOUN
iajs-2969	42	41	.	.	PUNCT
iajs-2969	43	1	an	an	DET
iajs-2969	43	2	ϣ-pre	ϣ-pre	NOUN
iajs-2969	43	3	-	-	PUNCT
iajs-2969	43	4	closed	closed	ADJ
iajs-2969	43	5	(	(	PUNCT
iajs-2969	43	6	in	in	ADP
iajs-2969	43	7	brief	brief	NOUN
iajs-2969	43	8	,	,	PUNCT
iajs-2969	43	9	ϣ	ϣ	NOUN
iajs-2969	43	10	−	−	PROPN
iajs-2969	43	11	p	p	NOUN
iajs-2969	43	12	−	−	PROPN
iajs-2969	43	13	c	c	NOUN
iajs-2969	43	14	)	)	PUNCT
iajs-2969	43	15	set	set	NOUN
iajs-2969	43	16	is	be	AUX
iajs-2969	43	17	the	the	DET
iajs-2969	43	18	complement	complement	NOUN
iajs-2969	43	19	of	of	ADP
iajs-2969	43	20	an	an	DET
iajs-2969	43	21	ϣ-pre	ϣ-pre	NOUN
iajs-2969	43	22	-	-	PUNCT
iajs-2969	43	23	open	open	ADJ
iajs-2969	43	24	set	set	NOUN
iajs-2969	43	25	.	.	PUNCT
iajs-2969	44	1	the	the	DET
iajs-2969	44	2	collection	collection	NOUN
iajs-2969	44	3	of	of	ADP
iajs-2969	44	4	all	all	PRON
iajs-2969	44	5	ϣ	ϣ	NOUN
iajs-2969	44	6	−	−	PROPN
iajs-2969	44	7	p	p	NOUN
iajs-2969	44	8	−	−	PROPN
iajs-2969	44	9	o	o	NOUN
iajs-2969	44	10	(	(	PUNCT
iajs-2969	44	11	ϣ	ϣ	NOUN
iajs-2969	44	12	−	−	PROPN
iajs-2969	44	13	p	p	NOUN
iajs-2969	44	14	−	−	PROPN
iajs-2969	44	15	c	c	NOUN
iajs-2969	44	16	)	)	PUNCT
iajs-2969	44	17	subsets	subset	NOUN
iajs-2969	44	18	of	of	ADP
iajs-2969	44	19	z	z	NOUN
iajs-2969	44	20	will	will	AUX
iajs-2969	44	21	be	be	AUX
iajs-2969	44	22	denoted	denote	VERB
iajs-2969	44	23	by	by	ADP
iajs-2969	44	24	ϣ-po(z	ϣ-po(z	NOUN
iajs-2969	44	25	)	)	PUNCT
iajs-2969	44	26	(	(	PUNCT
iajs-2969	44	27	ϣ-pc(z	ϣ-pc(z	NOUN
iajs-2969	44	28	)	)	PUNCT
iajs-2969	44	29	,	,	PUNCT
iajs-2969	44	30	respectively	respectively	ADV
iajs-2969	44	31	)	)	PUNCT
iajs-2969	44	32	.	.	PUNCT
iajs-2969	45	1	proposition	proposition	NOUN
iajs-2969	45	2	2.2	2.2	NUM
iajs-2969	45	3	for	for	ADP
iajs-2969	45	4	a	a	DET
iajs-2969	45	5	subset	subset	NOUN
iajs-2969	45	6	ħ	ħ	NOUN
iajs-2969	45	7	of	of	ADP
iajs-2969	45	8	a	a	DET
iajs-2969	45	9	(	(	PUNCT
iajs-2969	45	10	z	z	NOUN
iajs-2969	45	11	,	,	PUNCT
iajs-2969	45	12	ϣ	ϣ	PROPN
iajs-2969	45	13	)	)	PUNCT
iajs-2969	45	14	,	,	PUNCT
iajs-2969	45	15	we	we	PRON
iajs-2969	45	16	have	have	VERB
iajs-2969	45	17	⋃	⋃	VERB
iajs-2969	45	18	intϣα∈ᴧ	intϣα∈ᴧ	ADJ
iajs-2969	45	19	clϣ	clϣ	NOUN
iajs-2969	45	20	ħα	ħα	ADP
iajs-2969	45	21	⊆	⊆	NUM
iajs-2969	45	22	intϣclϣ	intϣclϣ	NOUN
iajs-2969	45	23	⋃	⋃	ADP
iajs-2969	45	24	ħαα∈ᴧ	ħαα∈ᴧ	NOUN
iajs-2969	45	25	.	.	PUNCT
iajs-2969	46	1	proof	proof	NOUN
iajs-2969	46	2	:	:	PUNCT
iajs-2969	46	3	ħα	ħα	ADP
iajs-2969	46	4	⊆	⊆	NUM
iajs-2969	46	5	⋃	⋃	X
iajs-2969	46	6	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	46	7	,	,	PUNCT
iajs-2969	46	8	for	for	ADP
iajs-2969	46	9	every	every	DET
iajs-2969	46	10	α	α	PROPN
iajs-2969	46	11	∈	∈	PROPN
iajs-2969	46	12	λ	λ	PROPN
iajs-2969	46	13	,	,	PUNCT
iajs-2969	46	14	so	so	ADV
iajs-2969	46	15	clϣħα	clϣħα	VERB
iajs-2969	46	16	⊆	⊆	NUM
iajs-2969	46	17	clϣ	clϣ	NOUN
iajs-2969	46	18	⋃	⋃	PROPN
iajs-2969	46	19	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	46	20	for	for	ADP
iajs-2969	46	21	every	every	DET
iajs-2969	46	22	α	α	PROPN
iajs-2969	46	23	∈	∈	PROPN
iajs-2969	46	24	λ	λ	PROPN
iajs-2969	46	25	,	,	PUNCT
iajs-2969	46	26	it	it	PRON
iajs-2969	46	27	follows	follow	VERB
iajs-2969	46	28	that	that	SCONJ
iajs-2969	46	29	intϣclϣħα	intϣclϣħα	NOUN
iajs-2969	46	30	⊆	⊆	NUM
iajs-2969	46	31	intϣclϣ	intϣclϣ	NOUN
iajs-2969	46	32	⋃	⋃	PROPN
iajs-2969	46	33	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	46	34	∀𝛼	∀𝛼	PROPN
iajs-2969	46	35	∈	∈	PROPN
iajs-2969	46	36	λ	λ	NOUN
iajs-2969	46	37	.	.	PUNCT
iajs-2969	47	1	hence	hence	ADV
iajs-2969	47	2	⋃	⋃	VERB
iajs-2969	47	3	intϣα∈ᴧ	intϣα∈ᴧ	ADJ
iajs-2969	47	4	clϣ	clϣ	NOUN
iajs-2969	47	5	ħα	ħα	ADP
iajs-2969	47	6	⊆	⊆	NUM
iajs-2969	47	7	intϣclϣ	intϣclϣ	NOUN
iajs-2969	47	8	⋃	⋃	ADP
iajs-2969	47	9	ħαα∈ᴧ	ħαα∈ᴧ	ADJ
iajs-2969	47	10	.	.	PUNCT
iajs-2969	48	1	ihjpas	ihjpas	PROPN
iajs-2969	48	2	.	.	PUNCT
iajs-2969	49	1	36(1)2023	36(1)2023	NUM
iajs-2969	49	2	335	335	NUM
iajs-2969	49	3	proposition	proposition	NOUN
iajs-2969	49	4	2.3	2.3	NUM
iajs-2969	49	5	the	the	DET
iajs-2969	49	6	union	union	NOUN
iajs-2969	49	7	of	of	ADP
iajs-2969	49	8	any	any	DET
iajs-2969	49	9	collection	collection	NOUN
iajs-2969	49	10	of	of	ADP
iajs-2969	49	11	ϣ	ϣ	NOUN
iajs-2969	49	12	−	−	PROPN
iajs-2969	49	13	p	p	NOUN
iajs-2969	49	14	−	−	PROPN
iajs-2969	49	15	o	o	NOUN
iajs-2969	49	16	sets	set	NOUN
iajs-2969	49	17	is	be	AUX
iajs-2969	49	18	an	an	DET
iajs-2969	49	19	ϣ	ϣ	NOUN
iajs-2969	49	20	−	−	NOUN
iajs-2969	49	21	p	p	NOUN
iajs-2969	49	22	−	−	PROPN
iajs-2969	49	23	o	o	NOUN
iajs-2969	49	24	set	set	NOUN
iajs-2969	49	25	.	.	PUNCT
iajs-2969	50	1	proof	proof	NOUN
iajs-2969	50	2	:	:	PUNCT
iajs-2969	50	3	let	let	VERB
iajs-2969	50	4	{	{	PUNCT
iajs-2969	50	5	ħα	ħα	ADP
iajs-2969	50	6	:	:	PUNCT
iajs-2969	50	7	α	α	PROPN
iajs-2969	50	8	∈	∈	PROPN
iajs-2969	50	9	⋀	⋀	PROPN
iajs-2969	50	10	}	}	PUNCT
iajs-2969	50	11	be	be	AUX
iajs-2969	50	12	a	a	DET
iajs-2969	50	13	family	family	NOUN
iajs-2969	50	14	of	of	ADP
iajs-2969	50	15	ϣ	ϣ	NOUN
iajs-2969	51	1	−	−	PROPN
iajs-2969	51	2	p	p	NOUN
iajs-2969	51	3	−	−	PROPN
iajs-2969	51	4	o	o	NOUN
iajs-2969	51	5	sets	set	NOUN
iajs-2969	51	6	,	,	PUNCT
iajs-2969	51	7	so	so	SCONJ
iajs-2969	51	8	ħα	ħα	NOUN
iajs-2969	51	9	⊆	⊆	NUM
iajs-2969	51	10	intϣ	intϣ	NOUN
iajs-2969	51	11	clϣ	clϣ	PROPN
iajs-2969	51	12	ħα	ħα	NOUN
iajs-2969	51	13	,	,	PUNCT
iajs-2969	51	14	∀𝛼	∀𝛼	PROPN
iajs-2969	51	15	∈	∈	PROPN
iajs-2969	51	16	λ	λ	PROPN
iajs-2969	51	17	.	.	PROPN
iajs-2969	51	18	which	which	PRON
iajs-2969	51	19	means	mean	VERB
iajs-2969	51	20	⋃	⋃	PROPN
iajs-2969	51	21	ħαα	ħαα	PROPN
iajs-2969	51	22	∈	∈	NOUN
iajs-2969	51	23	ᴧ	ᴧ	NOUN
iajs-2969	51	24	⊆	⊆	NUM
iajs-2969	51	25	⋃	⋃	ADJ
iajs-2969	51	26	intϣα∈ᴧ	intϣα∈ᴧ	ADJ
iajs-2969	51	27	clϣ	clϣ	NOUN
iajs-2969	51	28	ħα	ħα	ADV
iajs-2969	51	29	,	,	PUNCT
iajs-2969	51	30	but	but	CCONJ
iajs-2969	51	31	⋃	⋃	ADP
iajs-2969	51	32	intϣα∈ᴧ	intϣα∈ᴧ	ADJ
iajs-2969	51	33	clϣ	clϣ	NOUN
iajs-2969	51	34	ħα	ħα	ADP
iajs-2969	51	35	⊆	⊆	NUM
iajs-2969	51	36	intϣclϣ	intϣclϣ	NOUN
iajs-2969	51	37	⋃	⋃	ADP
iajs-2969	51	38	ħαα∈ᴧ	ħαα∈ᴧ	NOUN
iajs-2969	51	39	(	(	PUNCT
iajs-2969	51	40	by	by	ADP
iajs-2969	51	41	proposition	proposition	NOUN
iajs-2969	51	42	2.2	2.2	NUM
iajs-2969	51	43	)	)	PUNCT
iajs-2969	51	44	,	,	PUNCT
iajs-2969	51	45	therefore	therefore	ADV
iajs-2969	51	46	,	,	PUNCT
iajs-2969	51	47	we	we	PRON
iajs-2969	51	48	obtain	obtain	VERB
iajs-2969	51	49	⋃	⋃	NOUN
iajs-2969	51	50	ħα	ħα	NOUN
iajs-2969	51	51	α∈ᴧ	α∈ᴧ	NOUN
iajs-2969	51	52	⊆	⊆	NUM
iajs-2969	51	53	intϣ	intϣ	NOUN
iajs-2969	51	54	clϣ	clϣ	NOUN
iajs-2969	51	55	⋃	⋃	NOUN
iajs-2969	51	56	ħαα∈ᴧ	ħαα∈ᴧ	NOUN
iajs-2969	51	57	,	,	PUNCT
iajs-2969	51	58	hence	hence	ADV
iajs-2969	51	59	⋃	⋃	PUNCT
iajs-2969	51	60	ħαα∈ᴧ	ħαα∈ᴧ	NOUN
iajs-2969	51	61	is	be	AUX
iajs-2969	51	62	an	an	DET
iajs-2969	51	63	ϣ	ϣ	NOUN
iajs-2969	51	64	−	−	NOUN
iajs-2969	51	65	p	p	NOUN
iajs-2969	51	66	−	−	PROPN
iajs-2969	51	67	o	o	NOUN
iajs-2969	51	68	set	set	NOUN
iajs-2969	51	69	.	.	PUNCT
iajs-2969	52	1	corollary	corollary	ADJ
iajs-2969	52	2	2.4	2.4	NUM
iajs-2969	53	1	the	the	DET
iajs-2969	53	2	intersection	intersection	NOUN
iajs-2969	53	3	of	of	ADP
iajs-2969	53	4	any	any	DET
iajs-2969	53	5	collection	collection	NOUN
iajs-2969	53	6	of	of	ADP
iajs-2969	53	7	ϣ	ϣ	NOUN
iajs-2969	53	8	−	−	PROPN
iajs-2969	53	9	p	p	PRON
iajs-2969	53	10	−	−	PROPN
iajs-2969	53	11	c	c	NOUN
iajs-2969	53	12	sets	set	NOUN
iajs-2969	53	13	is	be	AUX
iajs-2969	53	14	an	an	DET
iajs-2969	53	15	ϣ	ϣ	NOUN
iajs-2969	53	16	−	−	NOUN
iajs-2969	53	17	p	p	NOUN
iajs-2969	53	18	−	−	PROPN
iajs-2969	53	19	c	c	NOUN
iajs-2969	53	20	set	set	NOUN
iajs-2969	53	21	.	.	PUNCT
iajs-2969	54	1	definition	definition	NOUN
iajs-2969	54	2	2.5	2.5	NUM
iajs-2969	54	3	:	:	PUNCT
iajs-2969	55	1	[	[	X
iajs-2969	55	2	6	6	NUM
iajs-2969	55	3	]	]	X
iajs-2969	55	4	let	let	VERB
iajs-2969	55	5	(	(	PUNCT
iajs-2969	55	6	z	z	NOUN
iajs-2969	55	7	,	,	PUNCT
iajs-2969	55	8	ϣ	ϣ	X
iajs-2969	55	9	)	)	PUNCT
iajs-2969	55	10	be	be	AUX
iajs-2969	55	11	a	a	DET
iajs-2969	55	12	𝐺𝑇𝑆	𝐺𝑇𝑆	PROPN
iajs-2969	55	13	,	,	PUNCT
iajs-2969	55	14	and	and	CCONJ
iajs-2969	55	15	ħ	ħ	NOUN
iajs-2969	55	16	be	be	VERB
iajs-2969	55	17	a	a	DET
iajs-2969	55	18	subset	subset	NOUN
iajs-2969	55	19	of	of	ADP
iajs-2969	55	20	z	z	NOUN
iajs-2969	55	21	1	1	NUM
iajs-2969	55	22	.	.	PUNCT
iajs-2969	56	1	the	the	DET
iajs-2969	56	2	union	union	NOUN
iajs-2969	56	3	of	of	ADP
iajs-2969	56	4	all	all	PRON
iajs-2969	56	5	ϣ	ϣ	ADP
iajs-2969	56	6	−	−	PROPN
iajs-2969	56	7	p	p	NOUN
iajs-2969	56	8	−	−	PROPN
iajs-2969	56	9	o	o	NOUN
iajs-2969	56	10	sets	set	NOUN
iajs-2969	56	11	contained	contain	VERB
iajs-2969	56	12	in	in	ADP
iajs-2969	56	13	ħ	ħ	PROPN
iajs-2969	56	14	is	be	AUX
iajs-2969	56	15	called	call	VERB
iajs-2969	56	16	the	the	DET
iajs-2969	56	17	ϣ-preinterior	ϣ-preinterior	NOUN
iajs-2969	56	18	of	of	ADP
iajs-2969	56	19	ħ	ħ	NOUN
iajs-2969	56	20	and	and	CCONJ
iajs-2969	56	21	denoted	denote	VERB
iajs-2969	56	22	by	by	ADP
iajs-2969	56	23	pre	pre	ADJ
iajs-2969	56	24	-	-	ADJ
iajs-2969	56	25	intϣħ	intϣħ	ADJ
iajs-2969	56	26	.	.	PUNCT
iajs-2969	57	1	2	2	X
iajs-2969	57	2	.	.	X
iajs-2969	57	3	the	the	DET
iajs-2969	57	4	intersection	intersection	NOUN
iajs-2969	57	5	of	of	ADP
iajs-2969	57	6	all	all	PRON
iajs-2969	57	7	ϣ	ϣ	ADP
iajs-2969	57	8	−	−	PROPN
iajs-2969	57	9	p	p	PRON
iajs-2969	57	10	−	−	NOUN
iajs-2969	57	11	c	c	NOUN
iajs-2969	57	12	sets	set	NOUN
iajs-2969	57	13	containing	contain	VERB
iajs-2969	57	14	ħ	ħ	NOUN
iajs-2969	57	15	is	be	AUX
iajs-2969	57	16	called	call	VERB
iajs-2969	57	17	the	the	DET
iajs-2969	57	18	ϣ-preclosuer	ϣ-preclosuer	NOUN
iajs-2969	57	19	of	of	ADP
iajs-2969	57	20	ħ	ħ	PROPN
iajs-2969	57	21	and	and	CCONJ
iajs-2969	57	22	denoted	denote	VERB
iajs-2969	57	23	by	by	ADP
iajs-2969	57	24	pre	pre	NOUN
iajs-2969	57	25	-	-	NOUN
iajs-2969	57	26	clϣħ	clϣħ	ADJ
iajs-2969	57	27	.	.	PUNCT
iajs-2969	58	1	theorem	theorem	VERB
iajs-2969	58	2	2.6	2.6	NUM
iajs-2969	58	3	let	let	VERB
iajs-2969	58	4	ԋ	ԋ	NOUN
iajs-2969	58	5	and	and	CCONJ
iajs-2969	58	6	ԏ	ԏ	ADP
iajs-2969	58	7	be	be	AUX
iajs-2969	58	8	subsets	subset	NOUN
iajs-2969	58	9	of	of	ADP
iajs-2969	58	10	(	(	PUNCT
iajs-2969	58	11	z	z	NOUN
iajs-2969	58	12	,	,	PUNCT
iajs-2969	58	13	ϣ	ϣ	NOUN
iajs-2969	58	14	)	)	PUNCT
iajs-2969	58	15	.	.	PUNCT
iajs-2969	59	1	then	then	ADV
iajs-2969	59	2	,	,	PUNCT
iajs-2969	59	3	the	the	DET
iajs-2969	59	4	following	follow	VERB
iajs-2969	59	5	properties	property	NOUN
iajs-2969	59	6	are	be	AUX
iajs-2969	59	7	true	true	ADJ
iajs-2969	59	8	:	:	PUNCT
iajs-2969	59	9	1	1	NUM
iajs-2969	59	10	.	.	X
iajs-2969	59	11	ԋ	ԋ	NOUN
iajs-2969	59	12	⊆	⊆	NUM
iajs-2969	59	13	pre	pre	NOUN
iajs-2969	59	14	-	-	ADJ
iajs-2969	59	15	clϣԋ	clϣԋ	ADJ
iajs-2969	59	16	.	.	PUNCT
iajs-2969	60	1	2	2	X
iajs-2969	60	2	.	.	X
iajs-2969	60	3	pre	pre	ADJ
iajs-2969	60	4	-	-	NOUN
iajs-2969	60	5	intϣԋ	intϣԋ	NOUN
iajs-2969	60	6	⊆	⊆	NUM
iajs-2969	60	7	ԋ	ԋ	NOUN
iajs-2969	60	8	.	.	PUNCT
iajs-2969	61	1	3	3	X
iajs-2969	61	2	.	.	X
iajs-2969	62	1	if	if	SCONJ
iajs-2969	62	2	ԋ	ԋ	PROPN
iajs-2969	62	3	⊆	⊆	NUM
iajs-2969	62	4	ԏ	ԏ	NOUN
iajs-2969	62	5	,	,	PUNCT
iajs-2969	62	6	then	then	ADV
iajs-2969	62	7	pre	pre	ADJ
iajs-2969	62	8	-	-	NOUN
iajs-2969	62	9	intϣԋ	intϣԋ	NOUN
iajs-2969	62	10	⊆	⊆	NUM
iajs-2969	62	11	pre	pre	ADJ
iajs-2969	62	12	-	-	ADJ
iajs-2969	62	13	intϣԏ	intϣԏ	ADJ
iajs-2969	62	14	.	.	PUNCT
iajs-2969	63	1	4	4	X
iajs-2969	63	2	.	.	X
iajs-2969	64	1	if	if	SCONJ
iajs-2969	64	2	ԋ	ԋ	PROPN
iajs-2969	64	3	⊆	⊆	NUM
iajs-2969	64	4	ԏ	ԏ	NOUN
iajs-2969	64	5	,	,	PUNCT
iajs-2969	64	6	then	then	ADV
iajs-2969	64	7	pre	pre	ADJ
iajs-2969	64	8	-	-	VERB
iajs-2969	64	9	clϣԋ	clϣԋ	ADJ
iajs-2969	64	10	⊆	⊆	NUM
iajs-2969	64	11	pre	pre	NOUN
iajs-2969	64	12	-	-	NOUN
iajs-2969	64	13	clϣԏ	clϣԏ	NOUN
iajs-2969	64	14	.	.	PUNCT
iajs-2969	65	1	proof	proof	NOUN
iajs-2969	65	2	:	:	PUNCT
iajs-2969	65	3	1	1	X
iajs-2969	65	4	.	.	X
iajs-2969	65	5	from	from	ADP
iajs-2969	65	6	definition	definition	NOUN
iajs-2969	65	7	of	of	ADP
iajs-2969	65	8	pre	pre	NOUN
iajs-2969	65	9	-	-	ADJ
iajs-2969	65	10	clϣԋ	clϣԋ	ADJ
iajs-2969	65	11	.	.	PUNCT
iajs-2969	66	1	2	2	X
iajs-2969	66	2	.	.	X
iajs-2969	66	3	from	from	ADP
iajs-2969	66	4	definition	definition	NOUN
iajs-2969	66	5	of	of	ADP
iajs-2969	66	6	pre	pre	NOUN
iajs-2969	66	7	-	-	NOUN
iajs-2969	66	8	intϣԋ	intϣԋ	NOUN
iajs-2969	66	9	.	.	PUNCT
iajs-2969	67	1	3	3	X
iajs-2969	67	2	.	.	X
iajs-2969	67	3	let	let	VERB
iajs-2969	67	4	ԋ	ԋ	PRON
iajs-2969	67	5	⊆	⊆	NUM
iajs-2969	67	6	ԏ	ԏ	NOUN
iajs-2969	67	7	,	,	PUNCT
iajs-2969	67	8	we	we	PRON
iajs-2969	67	9	have	have	VERB
iajs-2969	67	10	from	from	ADP
iajs-2969	67	11	2	2	NUM
iajs-2969	67	12	,	,	PUNCT
iajs-2969	67	13	pre	pre	ADJ
iajs-2969	67	14	-	-	NOUN
iajs-2969	67	15	intϣԋ	intϣԋ	ADJ
iajs-2969	67	16	⊆	⊆	NUM
iajs-2969	67	17	ԋ	ԋ	NOUN
iajs-2969	67	18	,	,	PUNCT
iajs-2969	67	19	so	so	ADV
iajs-2969	67	20	pre	pre	VERB
iajs-2969	67	21	−	−	NOUN
iajs-2969	67	22	intϣԋ	intϣԋ	NOUN
iajs-2969	67	23	⊆	⊆	NUM
iajs-2969	67	24	ԏ	ԏ	NOUN
iajs-2969	67	25	,	,	PUNCT
iajs-2969	67	26	but	but	CCONJ
iajs-2969	67	27	pre	pre	ADJ
iajs-2969	67	28	−	−	PROPN
iajs-2969	67	29	intϣԏ	intϣԏ	NOUN
iajs-2969	67	30	is	be	AUX
iajs-2969	67	31	the	the	DET
iajs-2969	67	32	largest	large	ADJ
iajs-2969	67	33	ϣ	ϣ	NOUN
iajs-2969	67	34	−	−	NOUN
iajs-2969	67	35	p	p	NOUN
iajs-2969	67	36	−	−	PROPN
iajs-2969	67	37	o	o	NOUN
iajs-2969	67	38	set	set	NOUN
iajs-2969	67	39	contained	contain	VERB
iajs-2969	67	40	in	in	ADP
iajs-2969	67	41	ԏ	ԏ	PROPN
iajs-2969	67	42	.	.	PUNCT
iajs-2969	68	1	so	so	ADV
iajs-2969	68	2	pre	pre	VERB
iajs-2969	68	3	−	−	PROPN
iajs-2969	68	4	intϣԋ	intϣԋ	NOUN
iajs-2969	68	5	⊆	⊆	NUM
iajs-2969	68	6	pre	pre	NOUN
iajs-2969	68	7	−	−	PROPN
iajs-2969	68	8	intϣԏ.	intϣԏ.	NOUN
iajs-2969	68	9	4	4	NUM
iajs-2969	68	10	.	.	PUNCT
iajs-2969	69	1	let	let	VERB
iajs-2969	69	2	ԋ	ԋ	PRON
iajs-2969	69	3	⊆	⊆	NUM
iajs-2969	69	4	ԏ	ԏ	NOUN
iajs-2969	69	5	,	,	PUNCT
iajs-2969	69	6	we	we	PRON
iajs-2969	69	7	have	have	VERB
iajs-2969	69	8	from	from	ADP
iajs-2969	69	9	1	1	NUM
iajs-2969	69	10	,	,	PUNCT
iajs-2969	69	11	ԏ	ԏ	PRON
iajs-2969	69	12	⊆	⊆	NUM
iajs-2969	69	13	pre	pre	NOUN
iajs-2969	69	14	-	-	NOUN
iajs-2969	69	15	clϣԏ	clϣԏ	NOUN
iajs-2969	69	16	,	,	PUNCT
iajs-2969	69	17	so	so	SCONJ
iajs-2969	69	18	ԋ	ԋ	NOUN
iajs-2969	69	19	⊆	⊆	NUM
iajs-2969	69	20	pre	pre	NOUN
iajs-2969	69	21	-	-	NOUN
iajs-2969	69	22	clϣԏ	clϣԏ	NOUN
iajs-2969	69	23	,	,	PUNCT
iajs-2969	69	24	but	but	CCONJ
iajs-2969	69	25	pre	pre	ADJ
iajs-2969	69	26	-	-	ADJ
iajs-2969	69	27	clϣԋ	clϣԋ	ADJ
iajs-2969	69	28	is	be	AUX
iajs-2969	69	29	the	the	DET
iajs-2969	69	30	smallest	small	ADJ
iajs-2969	69	31	ϣ	ϣ	ADP
iajs-2969	70	1	−	−	NOUN
iajs-2969	70	2	p	p	NOUN
iajs-2969	70	3	−	−	PROPN
iajs-2969	70	4	c	c	NOUN
iajs-2969	70	5	set	set	VERB
iajs-2969	70	6	containing	contain	VERB
iajs-2969	70	7	ԋ	ԋ	NOUN
iajs-2969	70	8	.	.	PUNCT
iajs-2969	71	1	so	so	ADV
iajs-2969	71	2	pre	pre	ADJ
iajs-2969	71	3	-	-	VERB
iajs-2969	71	4	clϣԋ	clϣԋ	ADJ
iajs-2969	71	5	⊆	⊆	NUM
iajs-2969	71	6	pre	pre	NOUN
iajs-2969	71	7	-	-	NOUN
iajs-2969	71	8	clϣԏ	clϣԏ	NOUN
iajs-2969	71	9	.	.	PUNCT
iajs-2969	72	1	proposition	proposition	NOUN
iajs-2969	72	2	2.7	2.7	NUM
iajs-2969	72	3	let	let	VERB
iajs-2969	72	4	(	(	PUNCT
iajs-2969	72	5	z	z	NOUN
iajs-2969	72	6	,	,	PUNCT
iajs-2969	72	7	ϣ	ϣ	X
iajs-2969	72	8	)	)	PUNCT
iajs-2969	72	9	be	be	VERB
iajs-2969	72	10	a	a	DET
iajs-2969	72	11	𝐺𝑇𝑆	𝐺𝑇𝑆	PROPN
iajs-2969	72	12	let	let	VERB
iajs-2969	72	13	ӈ	ӈ	PRON
iajs-2969	72	14	be	be	AUX
iajs-2969	72	15	a	a	DET
iajs-2969	72	16	subset	subset	NOUN
iajs-2969	72	17	of	of	ADP
iajs-2969	72	18	z.	z.	PROPN
iajs-2969	72	19	then	then	ADV
iajs-2969	72	20	:	:	PUNCT
iajs-2969	73	1	1	1	X
iajs-2969	73	2	.	.	X
iajs-2969	73	3	ħ	ħ	PROPN
iajs-2969	73	4	is	be	AUX
iajs-2969	73	5	an	an	DET
iajs-2969	73	6	ϣ	ϣ	NOUN
iajs-2969	73	7	−	−	NOUN
iajs-2969	73	8	p	p	NOUN
iajs-2969	73	9	−	−	PROPN
iajs-2969	73	10	c	c	NOUN
iajs-2969	73	11	set	set	NOUN
iajs-2969	73	12	,	,	PUNCT
iajs-2969	73	13	if	if	SCONJ
iajs-2969	73	14	and	and	CCONJ
iajs-2969	73	15	only	only	ADV
iajs-2969	73	16	if	if	SCONJ
iajs-2969	73	17	ħ	ħ	NOUN
iajs-2969	73	18	=	=	SYM
iajs-2969	73	19	pre	pre	NOUN
iajs-2969	73	20	-	-	NOUN
iajs-2969	73	21	clϣħ	clϣħ	ADJ
iajs-2969	73	22	.	.	PUNCT
iajs-2969	74	1	2	2	X
iajs-2969	74	2	.	.	X
iajs-2969	74	3	ħ	ħ	PROPN
iajs-2969	74	4	is	be	AUX
iajs-2969	74	5	an	an	DET
iajs-2969	74	6	ϣ	ϣ	NOUN
iajs-2969	74	7	−	−	NOUN
iajs-2969	74	8	p	p	NOUN
iajs-2969	74	9	−	−	PROPN
iajs-2969	74	10	oset	oset	NOUN
iajs-2969	74	11	,	,	PUNCT
iajs-2969	74	12	if	if	SCONJ
iajs-2969	74	13	and	and	CCONJ
iajs-2969	74	14	only	only	ADV
iajs-2969	74	15	if	if	SCONJ
iajs-2969	74	16	ħ	ħ	NOUN
iajs-2969	74	17	=	=	SYM
iajs-2969	74	18	pre	pre	NOUN
iajs-2969	74	19	-	-	ADJ
iajs-2969	74	20	intϣħ	intϣħ	ADJ
iajs-2969	74	21	.	.	PUNCT
iajs-2969	75	1	proposition	proposition	NOUN
iajs-2969	75	2	2.8	2.8	NUM
iajs-2969	75	3	⋃	⋃	NOUN
iajs-2969	75	4	pre	pre	NOUN
iajs-2969	75	5	−α∈λ	−α∈λ	PROPN
iajs-2969	75	6	clϣħα	clϣħα	VERB
iajs-2969	75	7	⊆	⊆	NUM
iajs-2969	75	8	pre	pre	ADJ
iajs-2969	75	9	−	−	PROPN
iajs-2969	75	10	clϣ	clϣ	PROPN
iajs-2969	75	11	⋃	⋃	NOUN
iajs-2969	75	12	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	75	13	proof	proof	NOUN
iajs-2969	75	14	:	:	PUNCT
iajs-2969	75	15	ihjpas	ihjpas	PROPN
iajs-2969	75	16	.	.	PUNCT
iajs-2969	76	1	36(1)2023	36(1)2023	NUM
iajs-2969	76	2	336	336	NUM
iajs-2969	76	3	ħα	ħα	ADP
iajs-2969	76	4	⊆	⊆	NUM
iajs-2969	76	5	⋃	⋃	X
iajs-2969	76	6	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	76	7	,	,	PUNCT
iajs-2969	76	8	for	for	ADP
iajs-2969	76	9	every	every	DET
iajs-2969	76	10	α	α	PROPN
iajs-2969	76	11	∈	∈	PROPN
iajs-2969	76	12	λ	λ	PROPN
iajs-2969	76	13	,	,	PUNCT
iajs-2969	76	14	so	so	ADV
iajs-2969	76	15	pre	pre	VERB
iajs-2969	76	16	-	-	VERB
iajs-2969	76	17	clϣħα	clϣħα	VERB
iajs-2969	76	18	⊆	⊆	NUM
iajs-2969	76	19	pre	pre	NOUN
iajs-2969	76	20	-	-	NOUN
iajs-2969	76	21	clϣ	clϣ	VERB
iajs-2969	76	22	⋃	⋃	PROPN
iajs-2969	76	23	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	76	24	for	for	ADP
iajs-2969	76	25	every	every	DET
iajs-2969	76	26	α	α	PROPN
iajs-2969	76	27	∈	∈	PROPN
iajs-2969	76	28	λ	λ	PROPN
iajs-2969	76	29	,	,	PUNCT
iajs-2969	76	30	therefore	therefore	ADV
iajs-2969	76	31	,	,	PUNCT
iajs-2969	76	32	⋃	⋃	PROPN
iajs-2969	76	33	pre	pre	PROPN
iajs-2969	76	34	−α∈λ	−α∈λ	PROPN
iajs-2969	76	35	clϣħα	clϣħα	VERB
iajs-2969	76	36	⊆	⊆	NUM
iajs-2969	76	37	pre	pre	ADJ
iajs-2969	76	38	−	−	PROPN
iajs-2969	76	39	clϣ	clϣ	PROPN
iajs-2969	76	40	⋃	⋃	PROPN
iajs-2969	76	41	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	76	42	.	.	PUNCT
iajs-2969	77	1	remark	remark	VERB
iajs-2969	77	2	2.9	2.9	NUM
iajs-2969	77	3	the	the	DET
iajs-2969	77	4	reverse	reverse	NOUN
iajs-2969	77	5	of	of	ADP
iajs-2969	77	6	proposition	proposition	NOUN
iajs-2969	77	7	2.8	2.8	NUM
iajs-2969	77	8	is	be	AUX
iajs-2969	77	9	not	not	PART
iajs-2969	77	10	correct	correct	ADJ
iajs-2969	77	11	in	in	ADP
iajs-2969	77	12	general	general	ADJ
iajs-2969	77	13	,	,	PUNCT
iajs-2969	77	14	as	as	SCONJ
iajs-2969	77	15	we	we	PRON
iajs-2969	77	16	show	show	VERB
iajs-2969	77	17	in	in	ADP
iajs-2969	77	18	the	the	DET
iajs-2969	77	19	following	follow	VERB
iajs-2969	77	20	example	example	NOUN
iajs-2969	77	21	:	:	PUNCT
iajs-2969	77	22	for	for	ADP
iajs-2969	77	23	example	example	NOUN
iajs-2969	77	24	z	z	NOUN
iajs-2969	77	25	=	=	SYM
iajs-2969	77	26	{	{	PUNCT
iajs-2969	77	27	a	a	PRON
iajs-2969	77	28	,	,	PUNCT
iajs-2969	77	29	b	b	NOUN
iajs-2969	77	30	,	,	PUNCT
iajs-2969	77	31	c	c	NOUN
iajs-2969	77	32	}	}	PUNCT
iajs-2969	77	33	,	,	PUNCT
iajs-2969	77	34	ϣ	ϣ	X
iajs-2969	77	35	=	=	PRON
iajs-2969	77	36	{	{	PUNCT
iajs-2969	77	37	z	z	PROPN
iajs-2969	77	38	,	,	PUNCT
iajs-2969	77	39	ø	ø	PROPN
iajs-2969	77	40	,	,	PUNCT
iajs-2969	77	41	{	{	PUNCT
iajs-2969	77	42	a	a	DET
iajs-2969	77	43	,	,	PUNCT
iajs-2969	77	44	b	b	NOUN
iajs-2969	77	45	}	}	PUNCT
iajs-2969	77	46	}	}	PUNCT
iajs-2969	77	47	,	,	PUNCT
iajs-2969	77	48	and	and	CCONJ
iajs-2969	77	49	ϣ′	ϣ′	PUNCT
iajs-2969	77	50	=	=	SYM
iajs-2969	77	51	{	{	PUNCT
iajs-2969	77	52	z	z	PROPN
iajs-2969	77	53	,	,	PUNCT
iajs-2969	77	54	ø	ø	PROPN
iajs-2969	77	55	,	,	PUNCT
iajs-2969	77	56	{	{	PUNCT
iajs-2969	77	57	c	c	NOUN
iajs-2969	77	58	}	}	PUNCT
iajs-2969	77	59	}	}	PUNCT
iajs-2969	77	60	,	,	PUNCT
iajs-2969	77	61	then	then	ADV
iajs-2969	77	62	:	:	PUNCT
iajs-2969	77	63	ϣ-po(z	ϣ-po(z	NUM
iajs-2969	77	64	)	)	PUNCT
iajs-2969	78	1	=	=	PRON
iajs-2969	78	2	{	{	PUNCT
iajs-2969	78	3	z	z	PROPN
iajs-2969	78	4	,	,	PUNCT
iajs-2969	78	5	ø	ø	PROPN
iajs-2969	78	6	,	,	PUNCT
iajs-2969	78	7	{	{	PUNCT
iajs-2969	78	8	a	a	X
iajs-2969	78	9	}	}	PUNCT
iajs-2969	78	10	,	,	PUNCT
iajs-2969	78	11	{	{	PUNCT
iajs-2969	78	12	b	b	NOUN
iajs-2969	78	13	}	}	PUNCT
iajs-2969	78	14	,	,	PUNCT
iajs-2969	78	15	{	{	PUNCT
iajs-2969	78	16	a	a	DET
iajs-2969	78	17	,	,	PUNCT
iajs-2969	78	18	b	b	NOUN
iajs-2969	78	19	}	}	PUNCT
iajs-2969	78	20	,	,	PUNCT
iajs-2969	78	21	{	{	PUNCT
iajs-2969	78	22	a	a	X
iajs-2969	78	23	,	,	PUNCT
iajs-2969	78	24	c	c	NOUN
iajs-2969	78	25	}	}	PUNCT
iajs-2969	78	26	,	,	PUNCT
iajs-2969	78	27	{	{	PUNCT
iajs-2969	78	28	b	b	X
iajs-2969	78	29	,	,	PUNCT
iajs-2969	78	30	c	c	NOUN
iajs-2969	78	31	}	}	PUNCT
iajs-2969	78	32	}	}	PUNCT
iajs-2969	78	33	.	.	PUNCT
iajs-2969	79	1	ϣ-pc(z	ϣ-pc(z	X
iajs-2969	79	2	)	)	PUNCT
iajs-2969	80	1	=	=	PRON
iajs-2969	80	2	{	{	PUNCT
iajs-2969	80	3	ø	ø	PROPN
iajs-2969	80	4	,	,	PUNCT
iajs-2969	80	5	z	z	PROPN
iajs-2969	80	6	,	,	PUNCT
iajs-2969	80	7	{	{	PUNCT
iajs-2969	80	8	b	b	NOUN
iajs-2969	80	9	,	,	PUNCT
iajs-2969	80	10	c	c	NOUN
iajs-2969	80	11	}	}	PUNCT
iajs-2969	80	12	,	,	PUNCT
iajs-2969	80	13	{	{	PUNCT
iajs-2969	80	14	a	a	X
iajs-2969	80	15	,	,	PUNCT
iajs-2969	80	16	c	c	NOUN
iajs-2969	80	17	}	}	PUNCT
iajs-2969	80	18	,	,	PUNCT
iajs-2969	80	19	{	{	PUNCT
iajs-2969	80	20	c	c	X
iajs-2969	80	21	}	}	PUNCT
iajs-2969	80	22	,	,	PUNCT
iajs-2969	80	23	{	{	PUNCT
iajs-2969	80	24	b	b	X
iajs-2969	80	25	}	}	PUNCT
iajs-2969	80	26	,	,	PUNCT
iajs-2969	80	27	{	{	PUNCT
iajs-2969	80	28	a	a	X
iajs-2969	80	29	}	}	PUNCT
iajs-2969	80	30	}	}	PUNCT
iajs-2969	80	31	,	,	PUNCT
iajs-2969	80	32	let	let	VERB
iajs-2969	80	33	ԋ	ԋ	PRON
iajs-2969	80	34	=	=	PUNCT
iajs-2969	80	35	{	{	PUNCT
iajs-2969	80	36	b	b	NOUN
iajs-2969	80	37	}	}	PUNCT
iajs-2969	80	38	and	and	CCONJ
iajs-2969	80	39	ԏ	ԏ	X
iajs-2969	80	40	=	=	X
iajs-2969	80	41	{	{	PUNCT
iajs-2969	80	42	a	a	NOUN
iajs-2969	80	43	}	}	PUNCT
iajs-2969	80	44	,	,	PUNCT
iajs-2969	80	45	so	so	ADV
iajs-2969	80	46	pre	pre	ADJ
iajs-2969	80	47	-	-	NOUN
iajs-2969	80	48	clϣ	clϣ	ADJ
iajs-2969	80	49	ԋ	ԋ	NOUN
iajs-2969	80	50	=	=	SYM
iajs-2969	80	51	{	{	PUNCT
iajs-2969	80	52	b	b	NOUN
iajs-2969	80	53	}	}	PUNCT
iajs-2969	80	54	and	and	CCONJ
iajs-2969	80	55	pre	pre	NOUN
iajs-2969	80	56	-	-	NOUN
iajs-2969	80	57	clϣ	clϣ	ADJ
iajs-2969	80	58	ԏ	ԏ	NOUN
iajs-2969	80	59	=	=	X
iajs-2969	80	60	{	{	PUNCT
iajs-2969	80	61	a	a	NOUN
iajs-2969	80	62	}	}	PUNCT
iajs-2969	80	63	,	,	PUNCT
iajs-2969	80	64	note	note	VERB
iajs-2969	80	65	that	that	SCONJ
iajs-2969	80	66	ԋ	ԋ	NOUN
iajs-2969	80	67	∪	∪	ADV
iajs-2969	80	68	ԏ=	ԏ=	NOUN
iajs-2969	80	69	{	{	PUNCT
iajs-2969	80	70	a	a	NOUN
iajs-2969	80	71	,	,	PUNCT
iajs-2969	80	72	b	b	NOUN
iajs-2969	80	73	}	}	PUNCT
iajs-2969	80	74	,	,	PUNCT
iajs-2969	80	75	and	and	CCONJ
iajs-2969	80	76	pre	pre	ADJ
iajs-2969	80	77	-	-	NOUN
iajs-2969	80	78	clϣ	clϣ	INTJ
iajs-2969	80	79	(	(	PUNCT
iajs-2969	80	80	ԋ	ԋ	NOUN
iajs-2969	80	81	∪	∪	NOUN
iajs-2969	80	82	ԏ	ԏ	NOUN
iajs-2969	80	83	)	)	PUNCT
iajs-2969	80	84	=	=	SYM
iajs-2969	81	1	z	z	NOUN
iajs-2969	81	2	,	,	PUNCT
iajs-2969	81	3	while	while	SCONJ
iajs-2969	81	4	pre	pre	ADJ
iajs-2969	81	5	-	-	ADJ
iajs-2969	81	6	clϣԋ	clϣԋ	ADJ
iajs-2969	81	7	⋃pre	⋃pre	NOUN
iajs-2969	81	8	-	-	NOUN
iajs-2969	81	9	clϣԏ	clϣԏ	NOUN
iajs-2969	81	10	=	=	PUNCT
iajs-2969	81	11	{	{	PUNCT
iajs-2969	81	12	a	a	PRON
iajs-2969	81	13	,	,	PUNCT
iajs-2969	81	14	b	b	NOUN
iajs-2969	81	15	}	}	PUNCT
iajs-2969	81	16	.	.	PUNCT
iajs-2969	82	1	hence	hence	ADV
iajs-2969	82	2	,	,	PUNCT
iajs-2969	82	3	pre	pre	ADJ
iajs-2969	82	4	-	-	NOUN
iajs-2969	82	5	clϣ	clϣ	ADJ
iajs-2969	82	6	(	(	PUNCT
iajs-2969	82	7	ԋ	ԋ	PROPN
iajs-2969	82	8	∪	∪	NOUN
iajs-2969	82	9	ԏ	ԏ	NOUN
iajs-2969	82	10	)	)	PUNCT
iajs-2969	82	11	⊈	⊈	PROPN
iajs-2969	83	1	pre	pre	ADJ
iajs-2969	83	2	-	-	ADJ
iajs-2969	83	3	clϣԋ	clϣԋ	ADJ
iajs-2969	83	4	⋃	⋃	ADJ
iajs-2969	83	5	pre	pre	ADJ
iajs-2969	83	6	-	-	ADJ
iajs-2969	83	7	clϣԏ.	clϣԏ.	ADJ
iajs-2969	83	8	proposition	proposition	NOUN
iajs-2969	83	9	:	:	PUNCT
iajs-2969	83	10	2.10	2.10	NUM
iajs-2969	83	11	if	if	SCONJ
iajs-2969	83	12	ӈ	ӈ	PRON
iajs-2969	83	13	is	be	AUX
iajs-2969	83	14	any	any	DET
iajs-2969	83	15	subset	subset	NOUN
iajs-2969	83	16	of	of	ADP
iajs-2969	83	17	a	a	DET
iajs-2969	83	18	topological	topological	ADJ
iajs-2969	83	19	space	space	NOUN
iajs-2969	83	20	(	(	PUNCT
iajs-2969	83	21	z	z	NOUN
iajs-2969	83	22	,	,	PUNCT
iajs-2969	83	23	ӽ	ӽ	NOUN
iajs-2969	83	24	)	)	PUNCT
iajs-2969	83	25	,	,	PUNCT
iajs-2969	83	26	then	then	ADV
iajs-2969	83	27	:	:	PUNCT
iajs-2969	83	28	1	1	X
iajs-2969	83	29	.	.	PUNCT
iajs-2969	84	1	[	[	X
iajs-2969	84	2	pre	pre	X
iajs-2969	84	3	−	−	PROPN
iajs-2969	84	4	int	int	NOUN
iajs-2969	84	5	(	(	PUNCT
iajs-2969	84	6	ħ)]c	ħ)]c	NOUN
iajs-2969	84	7	=	=	PUNCT
iajs-2969	84	8	pre	pre	X
iajs-2969	85	1	−	−	NOUN
iajs-2969	85	2	cl	cl	INTJ
iajs-2969	85	3	(	(	PUNCT
iajs-2969	85	4	ħc	ħc	NOUN
iajs-2969	85	5	)	)	PUNCT
iajs-2969	85	6	.	.	PUNCT
iajs-2969	86	1	2	2	X
iajs-2969	86	2	.	.	X
iajs-2969	86	3	pre	pre	VERB
iajs-2969	87	1	−	−	NOUN
iajs-2969	87	2	int(ħc	int(ħc	PROPN
iajs-2969	87	3	)	)	PUNCT
iajs-2969	87	4	=	=	PUNCT
iajs-2969	88	1	[	[	X
iajs-2969	88	2	pre	pre	X
iajs-2969	88	3	−	−	NOUN
iajs-2969	88	4	cl	cl	NOUN
iajs-2969	88	5	(	(	PUNCT
iajs-2969	88	6	ħ)]c	ħ)]c	NOUN
iajs-2969	88	7	.	.	PUNCT
iajs-2969	89	1	3	3	X
iajs-2969	89	2	.	.	X
iajs-2969	89	3	ϣ-semi	ϣ-semi	PROPN
iajs-2969	89	4	-	-	PUNCT
iajs-2969	89	5	p	p	NOUN
iajs-2969	89	6	-	-	PUNCT
iajs-2969	89	7	open	open	ADJ
iajs-2969	89	8	set	set	VERB
iajs-2969	89	9	definition	definition	NOUN
iajs-2969	89	10	3.1	3.1	NUM
iajs-2969	89	11	a	a	DET
iajs-2969	89	12	subset	subset	NOUN
iajs-2969	89	13	g	g	NOUN
iajs-2969	89	14	of	of	ADP
iajs-2969	89	15	a	a	DET
iajs-2969	89	16	𝐺𝑇𝑆	𝐺𝑇𝑆	PROPN
iajs-2969	89	17	(	(	PUNCT
iajs-2969	89	18	z	z	NOUN
iajs-2969	89	19	,	,	PUNCT
iajs-2969	89	20	ϣ	ϣ	X
iajs-2969	89	21	)	)	PUNCT
iajs-2969	89	22	is	be	AUX
iajs-2969	89	23	said	say	VERB
iajs-2969	89	24	to	to	PART
iajs-2969	89	25	be	be	AUX
iajs-2969	89	26	ϣ-semi	ϣ-semi	NOUN
iajs-2969	89	27	-	-	PUNCT
iajs-2969	89	28	p	p	NOUN
iajs-2969	89	29	-	-	PUNCT
iajs-2969	89	30	open(in	open(in	NOUN
iajs-2969	89	31	brief	brief	NOUN
iajs-2969	89	32	,	,	PUNCT
iajs-2969	89	33	ϣ	ϣ	NOUN
iajs-2969	89	34	−	−	NOUN
iajs-2969	89	35	sp	sp	ADP
iajs-2969	89	36	−	−	PROPN
iajs-2969	89	37	o	o	NOUN
iajs-2969	89	38	)	)	PUNCT
iajs-2969	89	39	set	set	VERB
iajs-2969	89	40	if	if	SCONJ
iajs-2969	89	41	there	there	PRON
iajs-2969	89	42	exists	exist	VERB
iajs-2969	89	43	an	an	DET
iajs-2969	89	44	ϣ	ϣ	NOUN
iajs-2969	89	45	−	−	NOUN
iajs-2969	89	46	p	p	NOUN
iajs-2969	89	47	−	−	NOUN
iajs-2969	89	48	o	o	NOUN
iajs-2969	89	49	set	set	NOUN
iajs-2969	89	50	ħ	ħ	NOUN
iajs-2969	89	51	in	in	ADP
iajs-2969	89	52	z	z	PROPN
iajs-2969	89	53	such	such	ADJ
iajs-2969	89	54	that	that	SCONJ
iajs-2969	89	55	ħ	ħ	NOUN
iajs-2969	89	56	⊆	⊆	NUM
iajs-2969	89	57	g	g	ADP
iajs-2969	89	58	⊆	⊆	NUM
iajs-2969	89	59	pre	pre	NOUN
iajs-2969	89	60	-	-	NOUN
iajs-2969	89	61	clϣħ	clϣħ	NOUN
iajs-2969	89	62	.	.	PUNCT
iajs-2969	90	1	any	any	DET
iajs-2969	90	2	subset	subset	NOUN
iajs-2969	90	3	of	of	ADP
iajs-2969	90	4	z	z	PROPN
iajs-2969	90	5	is	be	AUX
iajs-2969	90	6	called	call	VERB
iajs-2969	90	7	ϣ-semi	ϣ-semi	NOUN
iajs-2969	90	8	-	-	PUNCT
iajs-2969	90	9	pclosed(in	pclosed(in	NOUN
iajs-2969	90	10	brief	brief	NOUN
iajs-2969	90	11	,	,	PUNCT
iajs-2969	90	12	ϣ	ϣ	NOUN
iajs-2969	90	13	−	−	NOUN
iajs-2969	90	14	sp	sp	ADP
iajs-2969	90	15	−	−	PROPN
iajs-2969	90	16	c	c	NOUN
iajs-2969	90	17	)	)	PUNCT
iajs-2969	90	18	set	set	VERB
iajs-2969	90	19	if	if	SCONJ
iajs-2969	90	20	its	its	PRON
iajs-2969	90	21	complement	complement	NOUN
iajs-2969	90	22	is	be	AUX
iajs-2969	90	23	ϣ-semi	ϣ-semi	NOUN
iajs-2969	90	24	-	-	PUNCT
iajs-2969	90	25	p	p	NOUN
iajs-2969	90	26	-	-	PUNCT
iajs-2969	90	27	open	open	ADJ
iajs-2969	90	28	set	set	VERB
iajs-2969	91	1	.the	.the	PRON
iajs-2969	92	1	collection	collection	NOUN
iajs-2969	92	2	of	of	ADP
iajs-2969	92	3	all	all	PRON
iajs-2969	92	4	ϣ	ϣ	NOUN
iajs-2969	92	5	−	−	NOUN
iajs-2969	92	6	sp	sp	ADP
iajs-2969	92	7	−	−	NOUN
iajs-2969	92	8	o	o	NOUN
iajs-2969	92	9	subsets	subset	NOUN
iajs-2969	92	10	of	of	ADP
iajs-2969	92	11	z	z	NOUN
iajs-2969	92	12	will	will	AUX
iajs-2969	92	13	be	be	AUX
iajs-2969	92	14	denoted	denote	VERB
iajs-2969	92	15	by	by	ADP
iajs-2969	92	16	ϣ-spo(z	ϣ-spo(z	PROPN
iajs-2969	92	17	)	)	PUNCT
iajs-2969	92	18	.	.	PUNCT
iajs-2969	93	1	the	the	DET
iajs-2969	93	2	collection	collection	NOUN
iajs-2969	93	3	of	of	ADP
iajs-2969	93	4	all	all	PRON
iajs-2969	93	5	ϣ	ϣ	NOUN
iajs-2969	93	6	−	−	NOUN
iajs-2969	93	7	sp	sp	ADP
iajs-2969	93	8	−	−	PROPN
iajs-2969	93	9	c	c	NOUN
iajs-2969	93	10	subsets	subset	NOUN
iajs-2969	93	11	of	of	ADP
iajs-2969	93	12	z	z	NOUN
iajs-2969	93	13	will	will	AUX
iajs-2969	93	14	be	be	AUX
iajs-2969	93	15	denoted	denote	VERB
iajs-2969	93	16	by	by	ADP
iajs-2969	93	17	ϣ-spc(z	ϣ-spc(z	NOUN
iajs-2969	93	18	)	)	PUNCT
iajs-2969	93	19	.	.	PUNCT
iajs-2969	94	1	theorem	theorem	ADJ
iajs-2969	94	2	3.2	3.2	NUM
iajs-2969	94	3	let	let	VERB
iajs-2969	94	4	(	(	PUNCT
iajs-2969	94	5	z	z	NOUN
iajs-2969	94	6	,	,	PUNCT
iajs-2969	94	7	ϣ	ϣ	X
iajs-2969	94	8	)	)	PUNCT
iajs-2969	94	9	be	be	AUX
iajs-2969	94	10	a	a	DET
iajs-2969	94	11	𝐺𝑇𝑆	𝐺𝑇𝑆	PROPN
iajs-2969	94	12	and	and	CCONJ
iajs-2969	94	13	g	g	PROPN
iajs-2969	94	14	⊆	⊆	NUM
iajs-2969	94	15	z.	z.	PROPN
iajs-2969	94	16	then	then	ADV
iajs-2969	94	17	g	g	PROPN
iajs-2969	94	18	is	be	AUX
iajs-2969	94	19	an	an	DET
iajs-2969	94	20	ϣ	ϣ	NOUN
iajs-2969	94	21	−	−	NOUN
iajs-2969	94	22	sp	sp	ADP
iajs-2969	94	23	−	−	NOUN
iajs-2969	94	24	oset	oset	NOUN
iajs-2969	94	25			ADP
iajs-2969	94	26	g	g	PROPN
iajs-2969	94	27	⊆	⊆	NUM
iajs-2969	94	28	pre	pre	ADJ
iajs-2969	94	29	-	-	ADJ
iajs-2969	94	30	clϣpre	clϣpre	NOUN
iajs-2969	94	31	-	-	NOUN
iajs-2969	94	32	intϣg	intϣg	NOUN
iajs-2969	94	33	.	.	PUNCT
iajs-2969	95	1	proof	proof	NOUN
iajs-2969	95	2	:	:	PUNCT
iajs-2969	95	3	the	the	PRON
iajs-2969	95	4	"	"	PUNCT
iajs-2969	95	5	if	if	SCONJ
iajs-2969	95	6	"	"	PUNCT
iajs-2969	95	7	part	part	NOUN
iajs-2969	95	8	assume	assume	VERB
iajs-2969	95	9	that	that	SCONJ
iajs-2969	95	10	g	g	PROPN
iajs-2969	95	11	is	be	AUX
iajs-2969	95	12	an	an	DET
iajs-2969	95	13	ϣ	ϣ	NOUN
iajs-2969	95	14	−	−	NOUN
iajs-2969	95	15	sp	sp	ADP
iajs-2969	95	16	−	−	PROPN
iajs-2969	95	17	oset	oset	NOUN
iajs-2969	95	18	,	,	PUNCT
iajs-2969	95	19	then	then	ADV
iajs-2969	95	20	there	there	PRON
iajs-2969	95	21	exists	exist	VERB
iajs-2969	95	22	a	a	DET
iajs-2969	95	23	ϣ	ϣ	NOUN
iajs-2969	95	24	−	−	NOUN
iajs-2969	95	25	p	p	NOUN
iajs-2969	95	26	−	−	PROPN
iajs-2969	95	27	o	o	NOUN
iajs-2969	95	28	subset	subset	NOUN
iajs-2969	95	29	ħ	ħ	NOUN
iajs-2969	95	30	of	of	ADP
iajs-2969	95	31	z	z	NOUN
iajs-2969	95	32	such	such	ADJ
iajs-2969	95	33	that	that	SCONJ
iajs-2969	95	34	ħ	ħ	NOUN
iajs-2969	95	35	⊆	⊆	NUM
iajs-2969	95	36	g	g	ADP
iajs-2969	95	37	⊆	⊆	NUM
iajs-2969	95	38	pre	pre	NOUN
iajs-2969	95	39	-	-	NOUN
iajs-2969	95	40	clϣħ	clϣħ	ADJ
iajs-2969	95	41	,	,	PUNCT
iajs-2969	95	42	it	it	PRON
iajs-2969	95	43	follows	follow	VERB
iajs-2969	95	44	by	by	ADP
iajs-2969	95	45	theorem	theorem	ADJ
iajs-2969	95	46	2.6	2.6	NUM
iajs-2969	95	47	(	(	PUNCT
iajs-2969	95	48	4	4	NUM
iajs-2969	95	49	)	)	PUNCT
iajs-2969	96	1	that	that	SCONJ
iajs-2969	96	2	pre	pre	ADJ
iajs-2969	96	3	-	-	ADJ
iajs-2969	96	4	intϣħ	intϣħ	ADJ
iajs-2969	96	5	⊆pre	⊆pre	NOUN
iajs-2969	96	6	-	-	NOUN
iajs-2969	96	7	intϣg	intϣg	NOUN
iajs-2969	96	8	,	,	PUNCT
iajs-2969	96	9	but	but	CCONJ
iajs-2969	96	10	pre	pre	ADJ
iajs-2969	96	11	-	-	ADJ
iajs-2969	96	12	intϣħ	intϣħ	ADJ
iajs-2969	96	13	=	=	SYM
iajs-2969	96	14	ħ	ħ	NOUN
iajs-2969	96	15	,	,	PUNCT
iajs-2969	96	16	therefore	therefore	ADV
iajs-2969	96	17	ħ	ħ	ADJ
iajs-2969	96	18	⊆	⊆	NUM
iajs-2969	96	19	pre	pre	NOUN
iajs-2969	96	20	-	-	NOUN
iajs-2969	96	21	intϣg	intϣg	NOUN
iajs-2969	96	22	.	.	PUNCT
iajs-2969	97	1	it	it	PRON
iajs-2969	97	2	follows	follow	VERB
iajs-2969	97	3	by	by	ADP
iajs-2969	97	4	theorem	theorem	ADJ
iajs-2969	97	5	2.6	2.6	NUM
iajs-2969	97	6	(	(	PUNCT
iajs-2969	97	7	3	3	NUM
iajs-2969	97	8	)	)	PUNCT
iajs-2969	97	9	that	that	SCONJ
iajs-2969	97	10	pre	pre	VERB
iajs-2969	97	11	-	-	NOUN
iajs-2969	97	12	clϣħ	clϣħ	ADJ
iajs-2969	97	13	⊆	⊆	NUM
iajs-2969	97	14	pre	pre	NOUN
iajs-2969	97	15	-	-	ADJ
iajs-2969	97	16	clϣpreintϣg	clϣpreintϣg	NOUN
iajs-2969	97	17	.	.	PUNCT
iajs-2969	98	1	now	now	ADV
iajs-2969	98	2	,	,	PUNCT
iajs-2969	98	3	we	we	PRON
iajs-2969	98	4	get	get	VERB
iajs-2969	98	5	g	g	NOUN
iajs-2969	98	6	⊆pre	⊆pre	NOUN
iajs-2969	98	7	-	-	NOUN
iajs-2969	98	8	clϣħ	clϣħ	VERB
iajs-2969	98	9	⊆	⊆	NUM
iajs-2969	98	10	pre	pre	ADJ
iajs-2969	98	11	-	-	ADJ
iajs-2969	98	12	clϣpre	clϣpre	ADJ
iajs-2969	98	13	-	-	PUNCT
iajs-2969	98	14	intϣg	intϣg	NOUN
iajs-2969	98	15	.	.	PUNCT
iajs-2969	99	1	thus	thus	ADV
iajs-2969	99	2	g	g	PROPN
iajs-2969	99	3	⊆	⊆	NUM
iajs-2969	99	4	pre	pre	ADJ
iajs-2969	99	5	-	-	ADJ
iajs-2969	99	6	clϣpre	clϣpre	ADJ
iajs-2969	99	7	-	-	PUNCT
iajs-2969	99	8	intϣg	intϣg	NOUN
iajs-2969	99	9	.	.	PUNCT
iajs-2969	100	1	the	the	DET
iajs-2969	100	2	"	"	PUNCT
iajs-2969	100	3	only	only	ADV
iajs-2969	100	4	if	if	SCONJ
iajs-2969	100	5	"	"	PUNCT
iajs-2969	100	6	part	part	NOUN
iajs-2969	100	7	assume	assume	VERB
iajs-2969	100	8	that	that	SCONJ
iajs-2969	100	9	g	g	PROPN
iajs-2969	100	10	⊆	⊆	NUM
iajs-2969	100	11	pre	pre	ADJ
iajs-2969	100	12	-	-	ADJ
iajs-2969	100	13	clϣpre	clϣpre	ADJ
iajs-2969	100	14	-	-	PUNCT
iajs-2969	100	15	intϣg	intϣg	NOUN
iajs-2969	100	16	,	,	PUNCT
iajs-2969	100	17	we	we	PRON
iajs-2969	100	18	have	have	VERB
iajs-2969	100	19	to	to	PART
iajs-2969	100	20	show	show	VERB
iajs-2969	100	21	that	that	SCONJ
iajs-2969	100	22	g	g	PROPN
iajs-2969	100	23	is	be	AUX
iajs-2969	100	24	a	a	DET
iajs-2969	100	25	ϣ	ϣ	NOUN
iajs-2969	100	26	−	−	NOUN
iajs-2969	100	27	sp	sp	ADP
iajs-2969	100	28	−	−	PROPN
iajs-2969	100	29	oset	oset	NOUN
iajs-2969	100	30	.	.	PUNCT
iajs-2969	101	1	take	take	VERB
iajs-2969	101	2	preintϣg	preintϣg	NOUN
iajs-2969	101	3	=	=	NOUN
iajs-2969	101	4	ħ	ħ	NOUN
iajs-2969	101	5	,	,	PUNCT
iajs-2969	101	6	then	then	ADV
iajs-2969	101	7	ħ	ħ	PROPN
iajs-2969	101	8	is	be	AUX
iajs-2969	101	9	a	a	DET
iajs-2969	101	10	ϣ	ϣ	NOUN
iajs-2969	101	11	−	−	NOUN
iajs-2969	101	12	p	p	NOUN
iajs-2969	101	13	−	−	NOUN
iajs-2969	101	14	o	o	NOUN
iajs-2969	101	15	set	set	NOUN
iajs-2969	101	16	and	and	CCONJ
iajs-2969	101	17	ħ	ħ	ADJ
iajs-2969	101	18	⊆	⊆	NUM
iajs-2969	101	19	g	g	ADP
iajs-2969	101	20	⊆	⊆	NUM
iajs-2969	101	21	pre	pre	NOUN
iajs-2969	101	22	-	-	NOUN
iajs-2969	101	23	clϣħ	clϣħ	ADJ
iajs-2969	101	24	.	.	PUNCT
iajs-2969	102	1	hence	hence	ADV
iajs-2969	102	2	g	g	PROPN
iajs-2969	102	3	is	be	AUX
iajs-2969	102	4	an	an	DET
iajs-2969	102	5	ϣ	ϣ	NOUN
iajs-2969	102	6	−	−	NOUN
iajs-2969	102	7	sp	sp	ADP
iajs-2969	102	8	−	−	PROPN
iajs-2969	102	9	oset	oset	NOUN
iajs-2969	102	10	.	.	PUNCT
iajs-2969	103	1	ihjpas	ihjpas	PROPN
iajs-2969	103	2	.	.	PUNCT
iajs-2969	104	1	36(1)2023	36(1)2023	NUM
iajs-2969	104	2	337	337	NUM
iajs-2969	104	3	corollary	corollary	NOUN
iajs-2969	104	4	3.3	3.3	NUM
iajs-2969	104	5	let	let	NOUN
iajs-2969	104	6	(	(	PUNCT
iajs-2969	104	7	z	z	NOUN
iajs-2969	104	8	,	,	PUNCT
iajs-2969	104	9	ϣ	ϣ	X
iajs-2969	104	10	)	)	PUNCT
iajs-2969	104	11	be	be	AUX
iajs-2969	104	12	a	a	DET
iajs-2969	104	13	𝐺𝑇𝑆	𝐺𝑇𝑆	PROPN
iajs-2969	104	14	and	and	CCONJ
iajs-2969	104	15	f	f	PROPN
iajs-2969	105	1	⊆	⊆	NUM
iajs-2969	105	2	z.	z.	PROPN
iajs-2969	105	3	then	then	ADV
iajs-2969	105	4	ӈ	ӈ	PROPN
iajs-2969	105	5	is	be	AUX
iajs-2969	105	6	ϣ	ϣ	ADP
iajs-2969	105	7	−	−	NOUN
iajs-2969	105	8	sp	sp	ADP
iajs-2969	105	9	−	−	PROPN
iajs-2969	105	10	cif	cif	PROPN
iajs-2969	105	11	and	and	CCONJ
iajs-2969	105	12	only	only	ADV
iajs-2969	105	13	if	if	SCONJ
iajs-2969	105	14	preintϣ(pre	preintϣ(pre	NOUN
iajs-2969	105	15	-	-	PUNCT
iajs-2969	105	16	clϣħ	clϣħ	NOUN
iajs-2969	105	17	)	)	PUNCT
iajs-2969	105	18	⊆	⊆	NUM
iajs-2969	105	19	ħ	ħ	NOUN
iajs-2969	105	20	.	.	PUNCT
iajs-2969	106	1	proof	proof	NOUN
iajs-2969	106	2	:	:	PUNCT
iajs-2969	106	3	the	the	DET
iajs-2969	106	4	"	"	PUNCT
iajs-2969	106	5	if	if	SCONJ
iajs-2969	106	6	"	"	PUNCT
iajs-2969	106	7	part	part	NOUN
iajs-2969	106	8	let	let	VERB
iajs-2969	106	9	f	f	PRON
iajs-2969	106	10	be	be	AUX
iajs-2969	106	11	an	an	DET
iajs-2969	106	12	ϣ	ϣ	NOUN
iajs-2969	106	13	−	−	NOUN
iajs-2969	106	14	sp	sp	ADP
iajs-2969	106	15	−	−	PROPN
iajs-2969	106	16	c	c	NOUN
iajs-2969	106	17	subset	subset	NOUN
iajs-2969	106	18	of	of	ADP
iajs-2969	106	19	z	z	PROPN
iajs-2969	106	20	,	,	PUNCT
iajs-2969	106	21	then	then	ADV
iajs-2969	106	22	pre	pre	ADJ
iajs-2969	106	23	-	-	NOUN
iajs-2969	106	24	clϣ	clϣ	ADJ
iajs-2969	106	25	ħ	ħ	NOUN
iajs-2969	106	26	=	=	SYM
iajs-2969	106	27	ħ	ħ	NOUN
iajs-2969	106	28	(	(	PUNCT
iajs-2969	106	29	by	by	ADP
iajs-2969	106	30	proposition	proposition	NOUN
iajs-2969	106	31	2.9(1	2.9(1	NUM
iajs-2969	106	32	)	)	PUNCT
iajs-2969	106	33	)	)	PUNCT
iajs-2969	106	34	which	which	PRON
iajs-2969	106	35	implies	imply	VERB
iajs-2969	106	36	pre	pre	ADJ
iajs-2969	106	37	-	-	ADJ
iajs-2969	106	38	intϣ(pre	intϣ(pre	ADJ
iajs-2969	106	39	-	-	PUNCT
iajs-2969	106	40	clϣħ	clϣħ	NOUN
iajs-2969	106	41	)	)	PUNCT
iajs-2969	106	42	⊆	⊆	NUM
iajs-2969	106	43	ħ	ħ	NOUN
iajs-2969	106	44	,	,	PUNCT
iajs-2969	106	45	since	since	SCONJ
iajs-2969	106	46	pre	pre	ADJ
iajs-2969	106	47	-	-	ADJ
iajs-2969	106	48	intϣӈ	intϣӈ	ADJ
iajs-2969	106	49	⊆	⊆	NUM
iajs-2969	106	50	ӈ	ӈ	PROPN
iajs-2969	106	51	(	(	PUNCT
iajs-2969	106	52	by	by	ADP
iajs-2969	106	53	theorem	theorem	NOUN
iajs-2969	106	54	2.3(2	2.3(2	NUM
iajs-2969	106	55	)	)	PUNCT
iajs-2969	106	56	.	.	PUNCT
iajs-2969	107	1	the	the	DET
iajs-2969	107	2	"	"	PUNCT
iajs-2969	107	3	only	only	ADV
iajs-2969	107	4	if	if	SCONJ
iajs-2969	107	5	"	"	PUNCT
iajs-2969	107	6	part	part	NOUN
iajs-2969	107	7	assume	assume	VERB
iajs-2969	107	8	that	that	SCONJ
iajs-2969	107	9	preintϣpre	preintϣpre	NOUN
iajs-2969	107	10	-	-	NOUN
iajs-2969	107	11	clϣħ	clϣħ	VERB
iajs-2969	107	12	⊆	⊆	NUM
iajs-2969	107	13	ħ	ħ	NOUN
iajs-2969	107	14	.	.	PUNCT
iajs-2969	108	1	we	we	PRON
iajs-2969	108	2	have	have	VERB
iajs-2969	108	3	to	to	PART
iajs-2969	108	4	show	show	VERB
iajs-2969	108	5	ӈ	ӈ	PRON
iajs-2969	108	6	is	be	AUX
iajs-2969	108	7	an	an	DET
iajs-2969	108	8	ϣ	ϣ	NOUN
iajs-2969	108	9	−	−	NOUN
iajs-2969	108	10	sp	sp	ADP
iajs-2969	108	11	−	−	PROPN
iajs-2969	108	12	cset	cset	NOUN
iajs-2969	108	13	.	.	PUNCT
iajs-2969	109	1	since	since	SCONJ
iajs-2969	109	2	pre	pre	ADJ
iajs-2969	109	3	-	-	NOUN
iajs-2969	109	4	intϣpreclϣħ	intϣpreclϣħ	ADJ
iajs-2969	109	5	⊆	⊆	NUM
iajs-2969	109	6	ħ	ħ	NOUN
iajs-2969	109	7	,	,	PUNCT
iajs-2969	109	8	then	then	ADV
iajs-2969	109	9	ħc	ħc	ADP
iajs-2969	109	10	⊆	⊆	NUM
iajs-2969	109	11	[	[	X
iajs-2969	109	12	preintϣ(pre	preintϣ(pre	NOUN
iajs-2969	109	13	-	-	PUNCT
iajs-2969	109	14	clϣ	clϣ	NOUN
iajs-2969	109	15	ħ)]c	ħ)]c	NOUN
iajs-2969	109	16	,	,	PUNCT
iajs-2969	109	17	so	so	SCONJ
iajs-2969	109	18	we	we	PRON
iajs-2969	109	19	obtain	obtain	VERB
iajs-2969	109	20	from	from	ADP
iajs-2969	109	21	proposition	proposition	NOUN
iajs-2969	109	22	2.10	2.10	NUM
iajs-2969	109	23	ħc	ħc	NOUN
iajs-2969	109	24	⊆	⊆	NUM
iajs-2969	109	25	pre	pre	X
iajs-2969	109	26	clϣ(pre	clϣ(pre	NOUN
iajs-2969	109	27	-	-	PUNCT
iajs-2969	109	28	clϣħ)c	clϣħ)c	PROPN
iajs-2969	109	29	and	and	CCONJ
iajs-2969	109	30	ħc	ħc	NUM
iajs-2969	109	31	⊆	⊆	NUM
iajs-2969	109	32	preclϣpre	preclϣpre	NOUN
iajs-2969	109	33	-	-	NOUN
iajs-2969	109	34	intϣħc	intϣħc	ADJ
iajs-2969	109	35	.	.	PUNCT
iajs-2969	110	1	hence	hence	ADV
iajs-2969	110	2	ħc	ħc	PRON
iajs-2969	110	3	is	be	AUX
iajs-2969	110	4	an	an	DET
iajs-2969	110	5	ϣ	ϣ	NOUN
iajs-2969	110	6	−	−	NOUN
iajs-2969	110	7	sp	sp	ADP
iajs-2969	110	8	−	−	NOUN
iajs-2969	110	9	o	o	NOUN
iajs-2969	110	10	set	set	VERB
iajs-2969	110	11	by	by	ADP
iajs-2969	110	12	theorem	theorem	NOUN
iajs-2969	110	13	(	(	PUNCT
iajs-2969	110	14	2.2.2	2.2.2	NUM
iajs-2969	110	15	)	)	PUNCT
iajs-2969	110	16	which	which	PRON
iajs-2969	110	17	means	mean	VERB
iajs-2969	110	18	ħ	ħ	NOUN
iajs-2969	110	19	is	be	AUX
iajs-2969	110	20	an	an	DET
iajs-2969	110	21	ϣ	ϣ	NOUN
iajs-2969	110	22	−	−	NOUN
iajs-2969	110	23	sp	sp	ADP
iajs-2969	110	24	−	−	PROPN
iajs-2969	110	25	c.	c.	NOUN
iajs-2969	110	26	proposition	proposition	NOUN
iajs-2969	110	27	3.4	3.4	NUM
iajs-2969	110	28	the	the	DET
iajs-2969	110	29	union	union	NOUN
iajs-2969	110	30	of	of	ADP
iajs-2969	110	31	any	any	DET
iajs-2969	110	32	collection	collection	NOUN
iajs-2969	110	33	of	of	ADP
iajs-2969	110	34	ϣ	ϣ	NOUN
iajs-2969	110	35	−	−	PROPN
iajs-2969	110	36	sp	sp	ADP
iajs-2969	110	37	−	−	NOUN
iajs-2969	110	38	o	o	NOUN
iajs-2969	110	39	sets	set	NOUN
iajs-2969	110	40	is	be	AUX
iajs-2969	110	41	an	an	DET
iajs-2969	110	42	ϣ	ϣ	NOUN
iajs-2969	110	43	−	−	NOUN
iajs-2969	110	44	sp	sp	ADP
iajs-2969	110	45	−	−	NOUN
iajs-2969	110	46	o	o	NOUN
iajs-2969	110	47	set	set	NOUN
iajs-2969	110	48	.	.	PUNCT
iajs-2969	111	1	proof	proof	NOUN
iajs-2969	111	2	:	:	PUNCT
iajs-2969	111	3	let	let	AUX
iajs-2969	111	4	{	{	PUNCT
iajs-2969	111	5	gα	gα	VERB
iajs-2969	111	6	,	,	PUNCT
iajs-2969	111	7	α	α	NOUN
iajs-2969	111	8	∈	∈	PROPN
iajs-2969	111	9	λ	λ	PROPN
iajs-2969	111	10	}	}	PUNCT
iajs-2969	111	11	be	be	AUX
iajs-2969	111	12	any	any	DET
iajs-2969	111	13	family	family	NOUN
iajs-2969	111	14	of	of	ADP
iajs-2969	111	15	ϣ	ϣ	NOUN
iajs-2969	111	16	−	−	PROPN
iajs-2969	111	17	sp	sp	ADP
iajs-2969	111	18	−	−	NOUN
iajs-2969	111	19	o	o	NOUN
iajs-2969	111	20	sets	set	NOUN
iajs-2969	111	21	.	.	PUNCT
iajs-2969	112	1	then	then	ADV
iajs-2969	112	2	there	there	PRON
iajs-2969	112	3	exists	exist	VERB
iajs-2969	112	4	an	an	DET
iajs-2969	112	5	ϣ	ϣ	NOUN
iajs-2969	112	6	−	−	NOUN
iajs-2969	112	7	p	p	NOUN
iajs-2969	112	8	−	−	PROPN
iajs-2969	112	9	o	o	NOUN
iajs-2969	112	10	set	set	VERB
iajs-2969	112	11	ħα	ħα	NOUN
iajs-2969	112	12	for	for	ADP
iajs-2969	112	13	each	each	DET
iajs-2969	112	14	gα	gα	NOUN
iajs-2969	112	15	,	,	PUNCT
iajs-2969	112	16	α	α	PROPN
iajs-2969	112	17	∈	∈	PROPN
iajs-2969	112	18	λ	λ	NOUN
iajs-2969	112	19	such	such	ADJ
iajs-2969	112	20	that	that	PRON
iajs-2969	112	21	ħα	ħα	NOUN
iajs-2969	112	22	⊆	⊆	NUM
iajs-2969	112	23	gα	gα	ADP
iajs-2969	112	24	⊆	⊆	NUM
iajs-2969	112	25	pre	pre	NOUN
iajs-2969	112	26	-	-	NOUN
iajs-2969	112	27	clϣ	clϣ	ADJ
iajs-2969	112	28	ħα	ħα	NOUN
iajs-2969	112	29	,	,	PUNCT
iajs-2969	112	30	so	so	SCONJ
iajs-2969	112	31	⋃	⋃	PUNCT
iajs-2969	112	32	ħα	ħα	NOUN
iajs-2969	112	33	α∈λ	α∈λ	NOUN
iajs-2969	112	34	⊆	⊆	NUM
iajs-2969	112	35	⋃	⋃	NOUN
iajs-2969	112	36	gαα∈λ	gαα∈λ	NOUN
iajs-2969	112	37	⊆	⊆	NUM
iajs-2969	112	38	⋃	⋃	PROPN
iajs-2969	112	39	pre	pre	NOUN
iajs-2969	112	40	−α∈λ	−α∈λ	PROPN
iajs-2969	112	41	clϣ	clϣ	PROPN
iajs-2969	112	42	ħα	ħα	NOUN
iajs-2969	112	43	,	,	PUNCT
iajs-2969	112	44	but	but	CCONJ
iajs-2969	112	45	⋃	⋃	PROPN
iajs-2969	112	46	ħαα∈λ	ħαα∈λ	NOUN
iajs-2969	112	47	is	be	AUX
iajs-2969	112	48	an	an	DET
iajs-2969	112	49	ϣ	ϣ	NOUN
iajs-2969	112	50	−	−	NOUN
iajs-2969	112	51	p	p	NOUN
iajs-2969	112	52	−	−	PROPN
iajs-2969	112	53	o	o	NOUN
iajs-2969	112	54	set	set	VERB
iajs-2969	112	55	by	by	ADP
iajs-2969	112	56	theorem	theorem	ADJ
iajs-2969	112	57	2.3	2.3	NUM
iajs-2969	112	58	,	,	PUNCT
iajs-2969	112	59	and	and	CCONJ
iajs-2969	112	60	⋃	⋃	PROPN
iajs-2969	112	61	pre	pre	PROPN
iajs-2969	112	62	−α∈λ	−α∈λ	PROPN
iajs-2969	112	63	clϣħα	clϣħα	VERB
iajs-2969	112	64	⊆	⊆	NUM
iajs-2969	112	65	preclϣ	preclϣ	NOUN
iajs-2969	112	66	⋃	⋃	PROPN
iajs-2969	112	67	ħα	ħα	NOUN
iajs-2969	112	68	α∈λ	α∈λ	NOUN
iajs-2969	112	69	by	by	ADP
iajs-2969	112	70	proposition	proposition	NOUN
iajs-2969	112	71	2.8	2.8	NUM
iajs-2969	112	72	.	.	PUNCT
iajs-2969	113	1	now	now	ADV
iajs-2969	113	2	we	we	PRON
iajs-2969	113	3	get	get	VERB
iajs-2969	113	4	⋃	⋃	PROPN
iajs-2969	113	5	ħα	ħα	NOUN
iajs-2969	113	6	α∈λ	α∈λ	NOUN
iajs-2969	113	7	⊆	⊆	NUM
iajs-2969	113	8	⋃	⋃	NOUN
iajs-2969	113	9	gαα∈λ	gαα∈λ	NOUN
iajs-2969	113	10	⊆	⊆	NUM
iajs-2969	113	11	pre	pre	NOUN
iajs-2969	113	12	-	-	NOUN
iajs-2969	113	13	clϣ	clϣ	ADJ
iajs-2969	113	14	⋃	⋃	NOUN
iajs-2969	113	15	ħα	ħα	NOUN
iajs-2969	113	16	α∈λ	α∈λ	NOUN
iajs-2969	113	17	.	.	PUNCT
iajs-2969	114	1	hence	hence	ADV
iajs-2969	114	2	⋃	⋃	NOUN
iajs-2969	114	3	gαα∈λ	gαα∈λ	NOUN
iajs-2969	114	4	is	be	AUX
iajs-2969	114	5	an	an	DET
iajs-2969	114	6	ϣ	ϣ	NOUN
iajs-2969	114	7	−	−	NOUN
iajs-2969	114	8	sp	sp	ADP
iajs-2969	114	9	−	−	NOUN
iajs-2969	114	10	o	o	NOUN
iajs-2969	114	11	set	set	NOUN
iajs-2969	114	12	.	.	PUNCT
iajs-2969	115	1	corollary	corollary	ADJ
iajs-2969	115	2	3.5	3.5	NUM
iajs-2969	115	3	the	the	DET
iajs-2969	115	4	intersection	intersection	NOUN
iajs-2969	115	5	of	of	ADP
iajs-2969	115	6	any	any	DET
iajs-2969	115	7	collection	collection	NOUN
iajs-2969	115	8	of	of	ADP
iajs-2969	115	9	ϣ	ϣ	NOUN
iajs-2969	115	10	−	−	NOUN
iajs-2969	115	11	sp	sp	ADP
iajs-2969	115	12	−	−	PROPN
iajs-2969	115	13	c	c	NOUN
iajs-2969	115	14	sets	set	NOUN
iajs-2969	115	15	is	be	AUX
iajs-2969	115	16	an	an	DET
iajs-2969	115	17	ϣ	ϣ	NOUN
iajs-2969	115	18	−	−	NOUN
iajs-2969	115	19	sp	sp	ADP
iajs-2969	115	20	−	−	PROPN
iajs-2969	115	21	c	c	PROPN
iajs-2969	115	22	set	set	NOUN
iajs-2969	115	23	.	.	PUNCT
iajs-2969	116	1	proof	proof	NOUN
iajs-2969	116	2	:	:	PUNCT
iajs-2969	116	3	let	let	VERB
iajs-2969	116	4	{	{	PUNCT
iajs-2969	116	5	fα	fα	ADP
iajs-2969	116	6	:	:	PUNCT
iajs-2969	116	7	α	α	PROPN
iajs-2969	116	8	∈	∈	PROPN
iajs-2969	116	9	⋀	⋀	PROPN
iajs-2969	116	10	}	}	PUNCT
iajs-2969	116	11	be	be	AUX
iajs-2969	116	12	any	any	DET
iajs-2969	116	13	family	family	NOUN
iajs-2969	116	14	of	of	ADP
iajs-2969	116	15	ϣ	ϣ	NOUN
iajs-2969	116	16	−	−	NOUN
iajs-2969	116	17	sp	sp	ADP
iajs-2969	116	18	−	−	PROPN
iajs-2969	116	19	c	c	NOUN
iajs-2969	116	20	subsets	subset	NOUN
iajs-2969	116	21	of	of	ADP
iajs-2969	116	22	z.	z.	PROPN
iajs-2969	116	23	we	we	PRON
iajs-2969	116	24	have	have	VERB
iajs-2969	116	25	to	to	PART
iajs-2969	116	26	show	show	VERB
iajs-2969	116	27	that	that	SCONJ
iajs-2969	116	28	⋂	⋂	PROPN
iajs-2969	116	29	fαα∈ᴧ	fαα∈ᴧ	NOUN
iajs-2969	116	30	is	be	AUX
iajs-2969	116	31	an	an	DET
iajs-2969	116	32	ϣ	ϣ	NOUN
iajs-2969	116	33	−	−	NOUN
iajs-2969	116	34	sp	sp	ADP
iajs-2969	116	35	−	−	PROPN
iajs-2969	116	36	c	c	PROPN
iajs-2969	116	37	set	set	NOUN
iajs-2969	116	38	,	,	PUNCT
iajs-2969	116	39	we	we	PRON
iajs-2969	116	40	know	know	VERB
iajs-2969	116	41	that	that	SCONJ
iajs-2969	116	42	z	z	NOUN
iajs-2969	117	1	−	−	PROPN
iajs-2969	117	2	⋂	⋂	PROPN
iajs-2969	117	3	fα	fα	NOUN
iajs-2969	117	4	=	=	SYM
iajs-2969	117	5	⋃	⋃	PROPN
iajs-2969	117	6	(	(	PUNCT
iajs-2969	117	7	z	z	NOUN
iajs-2969	117	8	−	−	PROPN
iajs-2969	117	9	fα)α∈ᴧα∈ᴧ	fα)α∈ᴧα∈ᴧ	NOUN
iajs-2969	117	10	(	(	PUNCT
iajs-2969	117	11	de	de	PROPN
iajs-2969	117	12	morgan	morgan	PROPN
iajs-2969	117	13	’s	’s	PART
iajs-2969	117	14	laws	law	NOUN
iajs-2969	117	15	)	)	PUNCT
iajs-2969	117	16	.	.	PUNCT
iajs-2969	118	1	but	but	CCONJ
iajs-2969	118	2	⋃	⋃	PROPN
iajs-2969	118	3	(	(	PUNCT
iajs-2969	118	4	z	z	NOUN
iajs-2969	118	5	−	−	PROPN
iajs-2969	118	6	fα)α∈ᴧ	fα)α∈ᴧ	NUM
iajs-2969	118	7	is	be	AUX
iajs-2969	118	8	an	an	DET
iajs-2969	118	9	ϣ	ϣ	NOUN
iajs-2969	118	10	−	−	NOUN
iajs-2969	118	11	sp	sp	ADP
iajs-2969	118	12	−	−	PROPN
iajs-2969	118	13	c	c	NOUN
iajs-2969	118	14	set	set	NOUN
iajs-2969	118	15	,	,	PUNCT
iajs-2969	118	16	so	so	SCONJ
iajs-2969	118	17	z	z	PROPN
iajs-2969	119	1	−	−	PROPN
iajs-2969	119	2	⋂	⋂	PROPN
iajs-2969	119	3	fαα∈ᴧ	fαα∈ᴧ	NOUN
iajs-2969	119	4	is	be	AUX
iajs-2969	119	5	an	an	DET
iajs-2969	119	6	ϣ	ϣ	NOUN
iajs-2969	119	7	−	−	NOUN
iajs-2969	119	8	sp	sp	ADP
iajs-2969	119	9	−	−	NOUN
iajs-2969	119	10	o	o	NOUN
iajs-2969	119	11	set	set	NOUN
iajs-2969	119	12	.	.	PUNCT
iajs-2969	120	1	hence	hence	ADV
iajs-2969	120	2	⋂	⋂	PROPN
iajs-2969	120	3	fαα∈ᴧ	fαα∈ᴧ	NOUN
iajs-2969	120	4	is	be	AUX
iajs-2969	120	5	an	an	DET
iajs-2969	120	6	ϣ	ϣ	NOUN
iajs-2969	120	7	−	−	NOUN
iajs-2969	120	8	sp	sp	ADP
iajs-2969	120	9	−	−	PROPN
iajs-2969	120	10	c	c	PROPN
iajs-2969	120	11	.	.	PUNCT
iajs-2969	121	1	remark	remark	VERB
iajs-2969	121	2	3.6	3.6	NUM
iajs-2969	121	3	the	the	DET
iajs-2969	121	4	intersection	intersection	NOUN
iajs-2969	121	5	of	of	ADP
iajs-2969	121	6	two	two	NUM
iajs-2969	121	7	ϣ	ϣ	NOUN
iajs-2969	121	8	−	−	NOUN
iajs-2969	121	9	sp	sp	ADP
iajs-2969	121	10	−	−	NOUN
iajs-2969	121	11	o	o	NOUN
iajs-2969	121	12	sets	set	NOUN
iajs-2969	121	13	need	need	VERB
iajs-2969	121	14	not	not	PART
iajs-2969	121	15	to	to	PART
iajs-2969	121	16	be	be	AUX
iajs-2969	121	17	an	an	DET
iajs-2969	121	18	ϣ	ϣ	NOUN
iajs-2969	121	19	−	−	NOUN
iajs-2969	121	20	sp	sp	ADP
iajs-2969	121	21	−	−	NOUN
iajs-2969	121	22	o	o	NOUN
iajs-2969	121	23	set	set	NOUN
iajs-2969	121	24	,	,	PUNCT
iajs-2969	121	25	as	as	SCONJ
iajs-2969	121	26	we	we	PRON
iajs-2969	121	27	show	show	VERB
iajs-2969	121	28	in	in	ADP
iajs-2969	121	29	the	the	DET
iajs-2969	121	30	following	follow	VERB
iajs-2969	121	31	example	example	NOUN
iajs-2969	121	32	:	:	PUNCT
iajs-2969	121	33	example	example	NOUN
iajs-2969	121	34	let	let	VERB
iajs-2969	121	35	z	z	NOUN
iajs-2969	121	36	=	=	VERB
iajs-2969	121	37	{	{	PUNCT
iajs-2969	121	38	a	a	PRON
iajs-2969	121	39	,	,	PUNCT
iajs-2969	121	40	b	b	NOUN
iajs-2969	121	41	,	,	PUNCT
iajs-2969	121	42	c	c	NOUN
iajs-2969	121	43	,	,	PUNCT
iajs-2969	121	44	d	d	NOUN
iajs-2969	121	45	}	}	PUNCT
iajs-2969	121	46	,	,	PUNCT
iajs-2969	121	47	ϣ	ϣ	X
iajs-2969	121	48	=	=	PRON
iajs-2969	121	49	{	{	PUNCT
iajs-2969	121	50	z	z	PROPN
iajs-2969	121	51	,	,	PUNCT
iajs-2969	121	52	ø	ø	PROPN
iajs-2969	121	53	,	,	PUNCT
iajs-2969	121	54	{	{	PUNCT
iajs-2969	121	55	a	a	X
iajs-2969	121	56	}	}	PUNCT
iajs-2969	121	57	,	,	PUNCT
iajs-2969	121	58	{	{	PUNCT
iajs-2969	121	59	d	d	NOUN
iajs-2969	121	60	}	}	PUNCT
iajs-2969	121	61	,	,	PUNCT
iajs-2969	121	62	{	{	PUNCT
iajs-2969	121	63	a	a	PRON
iajs-2969	121	64	,	,	PUNCT
iajs-2969	121	65	d	d	NOUN
iajs-2969	121	66	}	}	PUNCT
iajs-2969	121	67	}	}	PUNCT
iajs-2969	121	68	,	,	PUNCT
iajs-2969	121	69	ϣ-po(z	ϣ-po(z	NUM
iajs-2969	121	70	)	)	PUNCT
iajs-2969	122	1	=	=	NOUN
iajs-2969	122	2	{	{	PUNCT
iajs-2969	122	3	z	z	PROPN
iajs-2969	122	4	,	,	PUNCT
iajs-2969	122	5	ø	ø	PROPN
iajs-2969	122	6	,	,	PUNCT
iajs-2969	122	7	{	{	PUNCT
iajs-2969	122	8	a	a	X
iajs-2969	122	9	}	}	PUNCT
iajs-2969	122	10	,	,	PUNCT
iajs-2969	122	11	{	{	PUNCT
iajs-2969	122	12	d	d	NOUN
iajs-2969	122	13	}	}	PUNCT
iajs-2969	122	14	,	,	PUNCT
iajs-2969	122	15	{	{	PUNCT
iajs-2969	122	16	a	a	DET
iajs-2969	122	17	,	,	PUNCT
iajs-2969	122	18	d	d	NOUN
iajs-2969	122	19	}	}	PUNCT
iajs-2969	122	20	,	,	PUNCT
iajs-2969	122	21	{	{	PUNCT
iajs-2969	122	22	a	a	DET
iajs-2969	122	23	,	,	PUNCT
iajs-2969	122	24	b	b	NOUN
iajs-2969	122	25	,	,	PUNCT
iajs-2969	122	26	d	d	NOUN
iajs-2969	122	27	}	}	PUNCT
iajs-2969	122	28	,	,	PUNCT
iajs-2969	122	29	{	{	PUNCT
iajs-2969	122	30	a	a	PRON
iajs-2969	122	31	,	,	PUNCT
iajs-2969	122	32	c	c	NOUN
iajs-2969	122	33	,	,	PUNCT
iajs-2969	122	34	d	d	NOUN
iajs-2969	122	35	}	}	PUNCT
iajs-2969	122	36	}	}	PUNCT
iajs-2969	122	37	,	,	PUNCT
iajs-2969	122	38	and	and	CCONJ
iajs-2969	122	39	ihjpas	ihjpa	VERB
iajs-2969	122	40	.	.	PUNCT
iajs-2969	123	1	36(1)2023	36(1)2023	NUM
iajs-2969	123	2	338	338	NUM
iajs-2969	123	3	ϣspo(z)=	ϣspo(z)=	NOUN
iajs-2969	123	4	{	{	PUNCT
iajs-2969	123	5	z	z	X
iajs-2969	123	6	,	,	PUNCT
iajs-2969	123	7	ø	ø	PROPN
iajs-2969	123	8	,	,	PUNCT
iajs-2969	123	9	{	{	PUNCT
iajs-2969	123	10	a	a	X
iajs-2969	123	11	}	}	PUNCT
iajs-2969	123	12	,	,	PUNCT
iajs-2969	123	13	{	{	PUNCT
iajs-2969	123	14	d	d	NOUN
iajs-2969	123	15	}	}	PUNCT
iajs-2969	123	16	,	,	PUNCT
iajs-2969	123	17	{	{	PUNCT
iajs-2969	123	18	a	a	DET
iajs-2969	123	19	,	,	PUNCT
iajs-2969	123	20	b	b	NOUN
iajs-2969	123	21	}	}	PUNCT
iajs-2969	123	22	,	,	PUNCT
iajs-2969	123	23	{	{	PUNCT
iajs-2969	123	24	a	a	X
iajs-2969	123	25	,	,	PUNCT
iajs-2969	123	26	c	c	NOUN
iajs-2969	123	27	}	}	PUNCT
iajs-2969	123	28	,	,	PUNCT
iajs-2969	123	29	{	{	PUNCT
iajs-2969	123	30	a	a	DET
iajs-2969	123	31	,	,	PUNCT
iajs-2969	123	32	d	d	NOUN
iajs-2969	123	33	}	}	PUNCT
iajs-2969	123	34	,	,	PUNCT
iajs-2969	123	35	{	{	PUNCT
iajs-2969	123	36	b	b	X
iajs-2969	123	37	,	,	PUNCT
iajs-2969	123	38	d	d	NOUN
iajs-2969	123	39	}	}	PUNCT
iajs-2969	123	40	,	,	PUNCT
iajs-2969	123	41	{	{	PUNCT
iajs-2969	123	42	c	c	X
iajs-2969	123	43	,	,	PUNCT
iajs-2969	123	44	d	d	NOUN
iajs-2969	123	45	}	}	PUNCT
iajs-2969	123	46	,	,	PUNCT
iajs-2969	123	47	{	{	PUNCT
iajs-2969	123	48	a	a	DET
iajs-2969	123	49	,	,	PUNCT
iajs-2969	123	50	b	b	NOUN
iajs-2969	123	51	,	,	PUNCT
iajs-2969	123	52	c	c	NOUN
iajs-2969	123	53	}	}	PUNCT
iajs-2969	123	54	,	,	PUNCT
iajs-2969	123	55	{	{	PUNCT
iajs-2969	123	56	a	a	DET
iajs-2969	123	57	,	,	PUNCT
iajs-2969	123	58	b	b	NOUN
iajs-2969	123	59	,	,	PUNCT
iajs-2969	123	60	d	d	NOUN
iajs-2969	123	61	}	}	PUNCT
iajs-2969	123	62	,	,	PUNCT
iajs-2969	123	63	{	{	PUNCT
iajs-2969	123	64	b	b	X
iajs-2969	123	65	,	,	PUNCT
iajs-2969	123	66	c	c	NOUN
iajs-2969	123	67	,	,	PUNCT
iajs-2969	123	68	d	d	NOUN
iajs-2969	123	69	}	}	PUNCT
iajs-2969	123	70	,	,	PUNCT
iajs-2969	123	71	{	{	PUNCT
iajs-2969	123	72	a	a	PRON
iajs-2969	123	73	,	,	PUNCT
iajs-2969	123	74	c	c	NOUN
iajs-2969	123	75	,	,	PUNCT
iajs-2969	123	76	d	d	NOUN
iajs-2969	123	77	}	}	PUNCT
iajs-2969	123	78	}	}	PUNCT
iajs-2969	123	79	.	.	PUNCT
iajs-2969	124	1	let	let	VERB
iajs-2969	124	2	ԋ	ԋ	NOUN
iajs-2969	125	1	=	=	PUNCT
iajs-2969	125	2	{	{	PUNCT
iajs-2969	125	3	a	a	DET
iajs-2969	125	4	,	,	PUNCT
iajs-2969	125	5	b	b	NOUN
iajs-2969	125	6	,	,	PUNCT
iajs-2969	125	7	c	c	NOUN
iajs-2969	125	8	}	}	PUNCT
iajs-2969	125	9	and	and	CCONJ
iajs-2969	125	10	ԏ	ԏ	X
iajs-2969	125	11	=	=	X
iajs-2969	125	12	{	{	PUNCT
iajs-2969	125	13	b	b	NOUN
iajs-2969	125	14	,	,	PUNCT
iajs-2969	125	15	d	d	NOUN
iajs-2969	125	16	}	}	PUNCT
iajs-2969	125	17	,	,	PUNCT
iajs-2969	125	18	ԋ	ԋ	NOUN
iajs-2969	126	1	and	and	CCONJ
iajs-2969	126	2	ԏ	ԏ	X
iajs-2969	126	3	are	be	AUX
iajs-2969	126	4	ϣ	ϣ	PRON
iajs-2969	126	5	−	−	NOUN
iajs-2969	126	6	sp	sp	ADP
iajs-2969	126	7	−	−	NOUN
iajs-2969	126	8	o	o	NOUN
iajs-2969	126	9	sets	set	NOUN
iajs-2969	126	10	,	,	PUNCT
iajs-2969	126	11	but	but	CCONJ
iajs-2969	126	12	ԋ	ԋ	NOUN
iajs-2969	126	13	∩	∩	NOUN
iajs-2969	126	14	ԏ	ԏ	X
iajs-2969	126	15	=	=	SYM
iajs-2969	126	16	{	{	PUNCT
iajs-2969	126	17	b	b	NOUN
iajs-2969	126	18	}	}	PUNCT
iajs-2969	126	19	which	which	PRON
iajs-2969	126	20	is	be	AUX
iajs-2969	126	21	not	not	PART
iajs-2969	126	22	an	an	DET
iajs-2969	126	23	ϣ	ϣ	NOUN
iajs-2969	126	24	−	−	NOUN
iajs-2969	126	25	sp	sp	ADP
iajs-2969	126	26	−	−	NOUN
iajs-2969	126	27	o	o	NOUN
iajs-2969	126	28	set	set	VERB
iajs-2969	126	29	because	because	SCONJ
iajs-2969	126	30	there	there	PRON
iajs-2969	126	31	is	be	VERB
iajs-2969	126	32	not	not	PART
iajs-2969	126	33	ϣ	ϣ	ADV
iajs-2969	126	34	−	−	NOUN
iajs-2969	126	35	p	p	NOUN
iajs-2969	127	1	−	−	PROPN
iajs-2969	127	2	o	o	NOUN
iajs-2969	127	3	set	set	NOUN
iajs-2969	127	4	vϣ	vϣ	ADV
iajs-2969	127	5	,	,	PUNCT
iajs-2969	127	6	therefore	therefore	ADV
iajs-2969	127	7	v	v	VERB
iajs-2969	127	8	⊆	⊆	NUM
iajs-2969	127	9	{	{	PUNCT
iajs-2969	127	10	b	b	NOUN
iajs-2969	127	11	}	}	PUNCT
iajs-2969	127	12	⊆	⊆	NUM
iajs-2969	127	13	preclϣv	preclϣv	NOUN
iajs-2969	127	14	.	.	PUNCT
iajs-2969	128	1	remark	remark	VERB
iajs-2969	128	2	3.7	3.7	NUM
iajs-2969	128	3	if	if	SCONJ
iajs-2969	128	4	ԋ	ԋ	PROPN
iajs-2969	129	1	and	and	CCONJ
iajs-2969	129	2	ԏ	ԏ	NOUN
iajs-2969	129	3	are	be	AUX
iajs-2969	129	4	two	two	NUM
iajs-2969	129	5	ϣ	ϣ	NOUN
iajs-2969	129	6	−	−	NOUN
iajs-2969	129	7	sp	sp	ADP
iajs-2969	129	8	−	−	PROPN
iajs-2969	129	9	c	c	NOUN
iajs-2969	129	10	sets	set	NOUN
iajs-2969	129	11	,	,	PUNCT
iajs-2969	129	12	then	then	ADV
iajs-2969	129	13	ԋ	ԋ	PUNCT
iajs-2969	129	14	∪	∪	NOUN
iajs-2969	129	15	ԏ	ԏ	PRON
iajs-2969	129	16	need	need	NOUN
iajs-2969	129	17	not	not	PART
iajs-2969	129	18	beϣ	beϣ	VERB
iajs-2969	129	19	−	−	NOUN
iajs-2969	129	20	sp	sp	ADP
iajs-2969	129	21	−	−	PROPN
iajs-2969	129	22	c	c	PROPN
iajs-2969	129	23	as	as	SCONJ
iajs-2969	129	24	we	we	PRON
iajs-2969	129	25	show	show	VERB
iajs-2969	129	26	in	in	ADP
iajs-2969	129	27	the	the	DET
iajs-2969	129	28	following	follow	VERB
iajs-2969	129	29	example	example	NOUN
iajs-2969	129	30	:	:	PUNCT
iajs-2969	129	31	example	example	NOUN
iajs-2969	129	32	from	from	ADP
iajs-2969	129	33	the	the	DET
iajs-2969	129	34	example	example	NOUN
iajs-2969	129	35	of	of	ADP
iajs-2969	129	36	remark	remark	NOUN
iajs-2969	129	37	3.7	3.7	NUM
iajs-2969	129	38	let	let	VERB
iajs-2969	129	39	ԋ	ԋ	NOUN
iajs-2969	129	40	=	=	PUNCT
iajs-2969	129	41	{	{	PUNCT
iajs-2969	129	42	d	d	NOUN
iajs-2969	129	43	}	}	PUNCT
iajs-2969	129	44	and	and	CCONJ
iajs-2969	129	45	ԏ	ԏ	X
iajs-2969	129	46	=	=	X
iajs-2969	129	47	{	{	PUNCT
iajs-2969	129	48	a	a	X
iajs-2969	129	49	,	,	PUNCT
iajs-2969	129	50	c	c	NOUN
iajs-2969	129	51	}	}	PUNCT
iajs-2969	129	52	.	.	PUNCT
iajs-2969	130	1	ԋ	ԋ	NOUN
iajs-2969	131	1	and	and	CCONJ
iajs-2969	131	2	ԏ	ԏ	X
iajs-2969	131	3	are	be	AUX
iajs-2969	131	4	ϣ	ϣ	PRON
iajs-2969	131	5	−	−	NOUN
iajs-2969	131	6	sp	sp	ADP
iajs-2969	131	7	−	−	PROPN
iajs-2969	131	8	c	c	NOUN
iajs-2969	131	9	set	set	NOUN
iajs-2969	131	10	,	,	PUNCT
iajs-2969	131	11	but	but	CCONJ
iajs-2969	131	12	ԋ	ԋ	PRON
iajs-2969	131	13	∪	∪	NOUN
iajs-2969	131	14	ԏ	ԏ	PRON
iajs-2969	131	15	=	=	NOUN
iajs-2969	131	16	{	{	PUNCT
iajs-2969	131	17	a	a	X
iajs-2969	131	18	,	,	PUNCT
iajs-2969	131	19	c	c	NOUN
iajs-2969	131	20	,	,	PUNCT
iajs-2969	131	21	d	d	NOUN
iajs-2969	131	22	}	}	PUNCT
iajs-2969	131	23	is	be	AUX
iajs-2969	131	24	not	not	PART
iajs-2969	131	25	an	an	DET
iajs-2969	131	26	ϣ	ϣ	NOUN
iajs-2969	131	27	−	−	NOUN
iajs-2969	131	28	sp	sp	ADP
iajs-2969	131	29	−	−	PROPN
iajs-2969	131	30	c	c	NOUN
iajs-2969	131	31	set	set	VERB
iajs-2969	131	32	because	because	SCONJ
iajs-2969	131	33	z	z	PROPN
iajs-2969	131	34	{	{	PUNCT
iajs-2969	131	35	a	a	X
iajs-2969	131	36	,	,	PUNCT
iajs-2969	131	37	c	c	NOUN
iajs-2969	131	38	,	,	PUNCT
iajs-2969	131	39	d}={b	d}={b	NOUN
iajs-2969	131	40	}	}	PUNCT
iajs-2969	131	41	is	be	AUX
iajs-2969	131	42	not	not	PART
iajs-2969	131	43	an	an	DET
iajs-2969	131	44	ϣ	ϣ	NOUN
iajs-2969	131	45	−	−	NOUN
iajs-2969	131	46	sp	sp	ADP
iajs-2969	131	47	−	−	NOUN
iajs-2969	131	48	o	o	NOUN
iajs-2969	131	49	set	set	NOUN
iajs-2969	131	50	.	.	PUNCT
iajs-2969	132	1	the	the	DET
iajs-2969	132	2	following	follow	VERB
iajs-2969	132	3	diagram	diagram	NOUN
iajs-2969	132	4	illustrates	illustrate	VERB
iajs-2969	132	5	the	the	DET
iajs-2969	132	6	relation	relation	NOUN
iajs-2969	132	7	among	among	ADP
iajs-2969	132	8	ϣ-open	ϣ-open	PROPN
iajs-2969	132	9	,	,	PUNCT
iajs-2969	132	10	ϣ-pre	ϣ-pre	NOUN
iajs-2969	132	11	-	-	ADJ
iajs-2969	132	12	open	open	ADJ
iajs-2969	132	13	,	,	PUNCT
iajs-2969	132	14	and	and	CCONJ
iajs-2969	132	15	ϣ-semi	ϣ-semi	PROPN
iajs-2969	132	16	-	-	PUNCT
iajs-2969	132	17	p	p	NOUN
iajs-2969	132	18	-	-	PUNCT
iajs-2969	132	19	open	open	ADJ
iajs-2969	132	20	set	set	VERB
iajs-2969	132	21	definition	definition	NOUN
iajs-2969	132	22	3.8	3.8	NUM
iajs-2969	132	23	1	1	NUM
iajs-2969	132	24	.	.	PUNCT
iajs-2969	133	1	the	the	DET
iajs-2969	133	2	union	union	NOUN
iajs-2969	133	3	of	of	ADP
iajs-2969	133	4	all	all	PRON
iajs-2969	133	5	ϣ	ϣ	PRON
iajs-2969	133	6	−	−	NOUN
iajs-2969	133	7	sp	sp	ADP
iajs-2969	133	8	−	−	NOUN
iajs-2969	133	9	o	o	NOUN
iajs-2969	133	10	sets	set	NOUN
iajs-2969	133	11	contained	contain	VERB
iajs-2969	133	12	in	in	ADP
iajs-2969	133	13	ӈ	ӈ	PROPN
iajs-2969	133	14	is	be	AUX
iajs-2969	133	15	called	call	VERB
iajs-2969	133	16	the	the	DET
iajs-2969	133	17	ϣ-semi	ϣ-semi	NOUN
iajs-2969	133	18	-	-	PUNCT
iajs-2969	133	19	p	p	NOUN
iajs-2969	133	20	-	-	NOUN
iajs-2969	133	21	interior	interior	NOUN
iajs-2969	133	22	of	of	ADP
iajs-2969	133	23	ħ	ħ	NOUN
iajs-2969	133	24	,	,	PUNCT
iajs-2969	133	25	denoted	denote	VERB
iajs-2969	133	26	by	by	ADP
iajs-2969	133	27	s	s	NOUN
iajs-2969	133	28	-	-	PUNCT
iajs-2969	133	29	p	p	NOUN
iajs-2969	133	30	-	-	PUNCT
iajs-2969	133	31	intϣ(ħ	intϣ(ħ	NOUN
iajs-2969	133	32	)	)	PUNCT
iajs-2969	133	33	.	.	PUNCT
iajs-2969	134	1	2	2	X
iajs-2969	134	2	.	.	X
iajs-2969	134	3	the	the	DET
iajs-2969	134	4	intersection	intersection	NOUN
iajs-2969	134	5	of	of	ADP
iajs-2969	134	6	all	all	PRON
iajs-2969	134	7	ϣ	ϣ	NOUN
iajs-2969	134	8	−	−	NOUN
iajs-2969	134	9	sp	sp	ADP
iajs-2969	134	10	−	−	NOUN
iajs-2969	134	11	c	c	NOUN
iajs-2969	134	12	sets	set	NOUN
iajs-2969	134	13	containing	contain	VERB
iajs-2969	134	14	ħ	ħ	NOUN
iajs-2969	134	15	is	be	AUX
iajs-2969	134	16	called	call	VERB
iajs-2969	134	17	the	the	DET
iajs-2969	134	18	ϣ-semi	ϣ-semi	NOUN
iajs-2969	134	19	-	-	PUNCT
iajs-2969	134	20	p	p	NOUN
iajs-2969	134	21	-	-	PUNCT
iajs-2969	134	22	closure	closure	NOUN
iajs-2969	134	23	of	of	ADP
iajs-2969	134	24	ħ	ħ	NOUN
iajs-2969	134	25	,	,	PUNCT
iajs-2969	134	26	denoted	denote	VERB
iajs-2969	134	27	by	by	ADP
iajs-2969	134	28	s	s	NOUN
iajs-2969	134	29	-	-	PUNCT
iajs-2969	134	30	p	p	X
iajs-2969	134	31	-	-	PUNCT
iajs-2969	134	32	clϣ(ħ	clϣ(ħ	NOUN
iajs-2969	134	33	)	)	PUNCT
iajs-2969	134	34	.	.	PUNCT
iajs-2969	135	1	proposition	proposition	NOUN
iajs-2969	135	2	3.9	3.9	NUM
iajs-2969	135	3	let	let	VERB
iajs-2969	135	4	ԋ	ԋ	NOUN
iajs-2969	135	5	and	and	CCONJ
iajs-2969	135	6	ԏ	ԏ	PART
iajs-2969	135	7	be	be	AUX
iajs-2969	135	8	two	two	NUM
iajs-2969	135	9	subsets	subset	NOUN
iajs-2969	135	10	of	of	ADP
iajs-2969	135	11	(	(	PUNCT
iajs-2969	135	12	z	z	NOUN
iajs-2969	135	13	,	,	PUNCT
iajs-2969	135	14	ϣ	ϣ	NOUN
iajs-2969	135	15	)	)	PUNCT
iajs-2969	135	16	.	.	PUNCT
iajs-2969	136	1	then	then	ADV
iajs-2969	136	2	,	,	PUNCT
iajs-2969	136	3	the	the	DET
iajs-2969	136	4	following	follow	VERB
iajs-2969	136	5	properties	property	NOUN
iajs-2969	136	6	are	be	AUX
iajs-2969	136	7	true	true	ADJ
iajs-2969	136	8	:	:	PUNCT
iajs-2969	136	9	1	1	NUM
iajs-2969	136	10	.	.	X
iajs-2969	136	11	ԋ	ԋ	NOUN
iajs-2969	137	1	⊆	⊆	NUM
iajs-2969	137	2	s	s	NOUN
iajs-2969	137	3	−	−	PROPN
iajs-2969	137	4	p	p	NOUN
iajs-2969	137	5	−	−	PROPN
iajs-2969	137	6	clϣ	clϣ	NOUN
iajs-2969	137	7	ԋ.	ԋ.	NOUN
iajs-2969	138	1	2	2	X
iajs-2969	138	2	.	.	X
iajs-2969	139	1	if	if	SCONJ
iajs-2969	139	2	ԋ	ԋ	PROPN
iajs-2969	139	3	⊆	⊆	NUM
iajs-2969	139	4	ԏ	ԏ	NOUN
iajs-2969	139	5	,	,	PUNCT
iajs-2969	139	6	then	then	ADV
iajs-2969	139	7	s	s	VERB
iajs-2969	139	8	−	−	PROPN
iajs-2969	139	9	p	p	NOUN
iajs-2969	139	10	−	−	PROPN
iajs-2969	139	11	clϣԋ	clϣԋ	NOUN
iajs-2969	139	12	⊆	⊆	NUM
iajs-2969	139	13	s	s	NOUN
iajs-2969	139	14	−	−	PROPN
iajs-2969	139	15	p	p	NOUN
iajs-2969	140	1	−	−	PROPN
iajs-2969	140	2	clϣԏ.	clϣԏ.	NOUN
iajs-2969	140	3	3	3	NUM
iajs-2969	140	4	.	.	PUNCT
iajs-2969	140	5	s	s	VERB
iajs-2969	140	6	−	−	PROPN
iajs-2969	140	7	p	p	NOUN
iajs-2969	141	1	−	−	PROPN
iajs-2969	141	2	clϣԋ	clϣԋ	NOUN
iajs-2969	141	3	∪	∪	NOUN
iajs-2969	141	4	s	s	PART
iajs-2969	141	5	−	−	PROPN
iajs-2969	141	6	p	p	NOUN
iajs-2969	141	7	−	−	PROPN
iajs-2969	141	8	clϣԏ	clϣԏ	NOUN
iajs-2969	142	1	⊆	⊆	NUM
iajs-2969	142	2	s	s	NOUN
iajs-2969	142	3	−	−	PROPN
iajs-2969	142	4	p	p	NOUN
iajs-2969	142	5	−	−	PROPN
iajs-2969	142	6	clϣ	clϣ	NOUN
iajs-2969	142	7	(	(	PUNCT
iajs-2969	142	8	ԋ	ԋ	NOUN
iajs-2969	142	9	∪	∪	NOUN
iajs-2969	142	10	ԏ	ԏ	NOUN
iajs-2969	142	11	)	)	PUNCT
iajs-2969	142	12	.	.	PUNCT
iajs-2969	143	1	4	4	X
iajs-2969	143	2	.	.	X
iajs-2969	143	3	s	s	VERB
iajs-2969	143	4	−	−	PROPN
iajs-2969	143	5	p	p	NOUN
iajs-2969	143	6	−	−	PROPN
iajs-2969	143	7	clϣ	clϣ	NOUN
iajs-2969	143	8	(	(	PUNCT
iajs-2969	143	9	ԋ	ԋ	NOUN
iajs-2969	143	10	∩	∩	ADJ
iajs-2969	143	11	ԏ	ԏ	X
iajs-2969	143	12	)	)	PUNCT
iajs-2969	143	13	⊆	⊆	NUM
iajs-2969	143	14	s	s	NOUN
iajs-2969	143	15	−	−	PROPN
iajs-2969	143	16	p	p	NOUN
iajs-2969	143	17	−	−	PROPN
iajs-2969	143	18	clϣ	clϣ	NOUN
iajs-2969	143	19	ԋ	ԋ	PROPN
iajs-2969	143	20	∩	∩	PROPN
iajs-2969	143	21	s	s	PART
iajs-2969	143	22	−	−	PROPN
iajs-2969	143	23	p	p	NOUN
iajs-2969	143	24	−	−	PROPN
iajs-2969	143	25	clϣ	clϣ	NOUN
iajs-2969	143	26	ԏ	ԏ	NOUN
iajs-2969	143	27	.	.	PUNCT
iajs-2969	144	1	proof	proof	NOUN
iajs-2969	144	2	:	:	PUNCT
iajs-2969	145	1	1	1	X
iajs-2969	145	2	.	.	X
iajs-2969	145	3	it	it	PRON
iajs-2969	145	4	is	be	AUX
iajs-2969	145	5	clear	clear	ADJ
iajs-2969	145	6	from	from	ADP
iajs-2969	145	7	definition	definition	NOUN
iajs-2969	145	8	3.14(2	3.14(2	NUM
iajs-2969	145	9	)	)	PUNCT
iajs-2969	145	10	.	.	PUNCT
iajs-2969	146	1	2	2	X
iajs-2969	146	2	.	.	X
iajs-2969	146	3	let	let	VERB
iajs-2969	146	4	ԋ	ԋ	PRON
iajs-2969	146	5	⊆	⊆	NUM
iajs-2969	146	6	ԏ	ԏ	NOUN
iajs-2969	146	7	,	,	PUNCT
iajs-2969	146	8	from	from	ADP
iajs-2969	146	9	(	(	PUNCT
iajs-2969	146	10	1	1	X
iajs-2969	146	11	)	)	PUNCT
iajs-2969	146	12	we	we	PRON
iajs-2969	146	13	have	have	VERB
iajs-2969	146	14	ԏ	ԏ	PRON
iajs-2969	146	15	⊆	⊆	NUM
iajs-2969	146	16	s	s	NOUN
iajs-2969	146	17	−	−	PROPN
iajs-2969	146	18	p	p	NOUN
iajs-2969	146	19	−	−	PROPN
iajs-2969	146	20	clϣ	clϣ	NOUN
iajs-2969	146	21	ԏ	ԏ	NOUN
iajs-2969	146	22	,	,	PUNCT
iajs-2969	147	1	so	so	SCONJ
iajs-2969	147	2	ԋ	ԋ	PROPN
iajs-2969	148	1	⊆	⊆	NUM
iajs-2969	148	2	s	s	PART
iajs-2969	148	3	−	−	PROPN
iajs-2969	148	4	p	p	NOUN
iajs-2969	148	5	−	−	PROPN
iajs-2969	148	6	clϣ	clϣ	NOUN
iajs-2969	148	7	ԏ	ԏ	X
iajs-2969	148	8	which	which	PRON
iajs-2969	148	9	is	be	AUX
iajs-2969	148	10	ϣ	ϣ	ADP
iajs-2969	148	11	−	−	NOUN
iajs-2969	148	12	sp	sp	ADP
iajs-2969	148	13	−	−	PROPN
iajs-2969	148	14	c	c	NOUN
iajs-2969	148	15	set	set	NOUN
iajs-2969	148	16	,	,	PUNCT
iajs-2969	148	17	but	but	CCONJ
iajs-2969	148	18	s	s	VERB
iajs-2969	148	19	−	−	PROPN
iajs-2969	148	20	p	p	NOUN
iajs-2969	148	21	−	−	PROPN
iajs-2969	148	22	clϣԋ	clϣԋ	NOUN
iajs-2969	148	23	is	be	AUX
iajs-2969	148	24	the	the	DET
iajs-2969	148	25	smallest	small	ADJ
iajs-2969	148	26	ϣ	ϣ	NOUN
iajs-2969	148	27	−	−	NOUN
iajs-2969	148	28	sp	sp	ADP
iajs-2969	148	29	−	−	PROPN
iajs-2969	148	30	c	c	NOUN
iajs-2969	148	31	set	set	VERB
iajs-2969	148	32	containing	contain	VERB
iajs-2969	148	33	ԋ	ԋ	NOUN
iajs-2969	148	34	,	,	PUNCT
iajs-2969	148	35	thus	thus	ADV
iajs-2969	148	36	s	s	AUX
iajs-2969	148	37	−	−	PROPN
iajs-2969	148	38	p	p	NOUN
iajs-2969	148	39	−	−	PROPN
iajs-2969	148	40	clϣԋ	clϣԋ	NOUN
iajs-2969	148	41	⊆	⊆	NUM
iajs-2969	148	42	s	s	NOUN
iajs-2969	148	43	−	−	PROPN
iajs-2969	148	44	p	p	NOUN
iajs-2969	148	45	−	−	PROPN
iajs-2969	148	46	clϣԏ.	clϣԏ.	NOUN
iajs-2969	148	47	ϣ-open	ϣ-open	NOUN
iajs-2969	148	48	ϣ-pre	ϣ-pre	NOUN
iajs-2969	148	49	-	-	PUNCT
iajs-2969	148	50	open	open	ADJ
iajs-2969	148	51	ϣ-semi	ϣ-semi	NOUN
iajs-2969	148	52	-	-	PUNCT
iajs-2969	148	53	p	p	NOUN
iajs-2969	148	54	-	-	PUNCT
iajs-2969	148	55	open	open	ADJ
iajs-2969	148	56	ihjpas	ihjpa	NOUN
iajs-2969	148	57	.	.	PUNCT
iajs-2969	149	1	36(1)2023	36(1)2023	NUM
iajs-2969	149	2	339	339	NUM
iajs-2969	149	3	3	3	NUM
iajs-2969	149	4	.	.	PUNCT
iajs-2969	150	1	since	since	SCONJ
iajs-2969	150	2	ԋ	ԋ	PRON
iajs-2969	150	3	⊆	⊆	NUM
iajs-2969	150	4	ԋ	ԋ	NOUN
iajs-2969	150	5	∪	∪	NOUN
iajs-2969	150	6	ԏ	ԏ	NOUN
iajs-2969	150	7	and	and	CCONJ
iajs-2969	150	8	ԏ	ԏ	PRON
iajs-2969	150	9	⊆	⊆	NUM
iajs-2969	150	10	ԋ	ԋ	NOUN
iajs-2969	150	11	∪	∪	NOUN
iajs-2969	150	12	ԏ	ԏ	PROPN
iajs-2969	150	13	,	,	PUNCT
iajs-2969	150	14	it	it	PRON
iajs-2969	150	15	follows	follow	VERB
iajs-2969	150	16	from	from	ADP
iajs-2969	150	17	(	(	PUNCT
iajs-2969	150	18	1	1	NUM
iajs-2969	150	19	)	)	PUNCT
iajs-2969	150	20	that	that	SCONJ
iajs-2969	150	21	𝑠	𝑠	PROPN
iajs-2969	150	22	−	−	PROPN
iajs-2969	150	23	𝑝	𝑝	PROPN
iajs-2969	150	24	−	−	PROPN
iajs-2969	150	25	clϣԋ	clϣԋ	NOUN
iajs-2969	150	26	⊆	⊆	NUM
iajs-2969	150	27	𝑠	𝑠	INTJ
iajs-2969	150	28	−	−	PROPN
iajs-2969	150	29	𝑝	𝑝	PROPN
iajs-2969	150	30	−	−	PROPN
iajs-2969	150	31	clϣ(ԋ	clϣ(ԋ	PROPN
iajs-2969	150	32	∪	∪	NOUN
iajs-2969	150	33	ԏ	ԏ	NOUN
iajs-2969	150	34	)	)	PUNCT
iajs-2969	150	35	and	and	CCONJ
iajs-2969	150	36	𝑠	𝑠	INTJ
iajs-2969	150	37	−	−	PROPN
iajs-2969	150	38	𝑝	𝑝	PROPN
iajs-2969	150	39	−	−	PROPN
iajs-2969	150	40	clϣԏ	clϣԏ	NOUN
iajs-2969	150	41	⊆	⊆	NUM
iajs-2969	150	42	𝑠	𝑠	INTJ
iajs-2969	150	43	−	−	PROPN
iajs-2969	150	44	𝑝	𝑝	PROPN
iajs-2969	150	45	−	−	PROPN
iajs-2969	150	46	clϣ	clϣ	NOUN
iajs-2969	150	47	(	(	PUNCT
iajs-2969	150	48	ԋ	ԋ	NOUN
iajs-2969	150	49	∪	∪	NOUN
iajs-2969	150	50	ԏ	ԏ	PROPN
iajs-2969	150	51	)	)	PUNCT
iajs-2969	150	52	,	,	PUNCT
iajs-2969	150	53	therefore	therefore	ADV
iajs-2969	150	54	𝑠	𝑠	INTJ
iajs-2969	150	55	−	−	PROPN
iajs-2969	150	56	𝑝	𝑝	PROPN
iajs-2969	150	57	−	−	PROPN
iajs-2969	150	58	clϣԋ	clϣԋ	NOUN
iajs-2969	150	59	∪	∪	ADP
iajs-2969	150	60	𝑠	𝑠	PROPN
iajs-2969	150	61	−	−	PROPN
iajs-2969	150	62	𝑝	𝑝	PROPN
iajs-2969	150	63	−	−	PROPN
iajs-2969	150	64	clϣ	clϣ	NOUN
iajs-2969	150	65	ԏ	ԏ	PROPN
iajs-2969	150	66	⊆	⊆	NUM
iajs-2969	150	67	𝑠	𝑠	INTJ
iajs-2969	150	68	−	−	PROPN
iajs-2969	150	69	𝑝	𝑝	PROPN
iajs-2969	150	70	−	−	PROPN
iajs-2969	150	71	clϣ	clϣ	NOUN
iajs-2969	150	72	(	(	PUNCT
iajs-2969	150	73	ԋ	ԋ	NOUN
iajs-2969	150	74	∪	∪	NOUN
iajs-2969	150	75	ԏ	ԏ	NOUN
iajs-2969	150	76	)	)	PUNCT
iajs-2969	150	77	.	.	PUNCT
iajs-2969	151	1	4	4	X
iajs-2969	151	2	.	.	X
iajs-2969	151	3	since	since	SCONJ
iajs-2969	151	4	(	(	PUNCT
iajs-2969	151	5	ԋ	ԋ	PROPN
iajs-2969	151	6	∩	∩	ADJ
iajs-2969	151	7	ԏ	ԏ	NOUN
iajs-2969	151	8	)	)	PUNCT
iajs-2969	151	9	⊆	⊆	NUM
iajs-2969	151	10	ӈ	ӈ	PROPN
iajs-2969	151	11	and	and	CCONJ
iajs-2969	151	12	(	(	PUNCT
iajs-2969	151	13	ԋ	ԋ	PROPN
iajs-2969	151	14	∩	∩	ADJ
iajs-2969	151	15	ԏ	ԏ	NOUN
iajs-2969	151	16	)	)	PUNCT
iajs-2969	151	17	⊆	⊆	NUM
iajs-2969	151	18	ԏ	ԏ	NOUN
iajs-2969	151	19	,	,	PUNCT
iajs-2969	151	20	so	so	ADV
iajs-2969	151	21	semi	semi	ADJ
iajs-2969	151	22	-	-	ADJ
iajs-2969	151	23	p	p	ADJ
iajs-2969	151	24	-	-	PUNCT
iajs-2969	151	25	clϣ	clϣ	NOUN
iajs-2969	151	26	(	(	PUNCT
iajs-2969	151	27	ԋ	ԋ	PROPN
iajs-2969	151	28	∩	∩	ADJ
iajs-2969	151	29	ԏ	ԏ	NOUN
iajs-2969	151	30	)	)	PUNCT
iajs-2969	151	31	⊆	⊆	NUM
iajs-2969	151	32	semi	semi	ADJ
iajs-2969	151	33	-	-	ADJ
iajs-2969	151	34	clϣԋ	clϣԋ	ADJ
iajs-2969	151	35	and	and	CCONJ
iajs-2969	151	36	𝑠	𝑠	INTJ
iajs-2969	151	37	−	−	PROPN
iajs-2969	151	38	𝑝	𝑝	PROPN
iajs-2969	151	39	−	−	PROPN
iajs-2969	151	40	clϣ(ԋ	clϣ(ԋ	PROPN
iajs-2969	151	41	∩	∩	NOUN
iajs-2969	151	42	ԏ	ԏ	X
iajs-2969	151	43	)	)	PUNCT
iajs-2969	151	44	⊆	⊆	NUM
iajs-2969	151	45	𝑠	𝑠	NUM
iajs-2969	151	46	−	−	PROPN
iajs-2969	151	47	𝑝	𝑝	PROPN
iajs-2969	151	48	−	−	PROPN
iajs-2969	151	49	clϣԏ	clϣԏ	NOUN
iajs-2969	151	50	,	,	PUNCT
iajs-2969	151	51	thus	thus	ADV
iajs-2969	151	52	𝑠	𝑠	PROPN
iajs-2969	151	53	−	−	PROPN
iajs-2969	151	54	𝑝	𝑝	PROPN
iajs-2969	151	55	−	−	PROPN
iajs-2969	151	56	clϣ	clϣ	PROPN
iajs-2969	151	57	(	(	PUNCT
iajs-2969	151	58	ԋ	ԋ	PROPN
iajs-2969	151	59	∩	∩	ADJ
iajs-2969	151	60	ԏ	ԏ	NOUN
iajs-2969	151	61	)	)	PUNCT
iajs-2969	151	62	⊆	⊆	NUM
iajs-2969	151	63	𝑠	𝑠	NUM
iajs-2969	151	64	−	−	PROPN
iajs-2969	151	65	𝑝	𝑝	PROPN
iajs-2969	151	66	−	−	PROPN
iajs-2969	151	67	clϣԋ	clϣԋ	NOUN
iajs-2969	151	68	∩	∩	NOUN
iajs-2969	151	69	𝑠	𝑠	ADP
iajs-2969	151	70	−	−	PROPN
iajs-2969	151	71	𝑝	𝑝	PROPN
iajs-2969	151	72	−	−	PROPN
iajs-2969	151	73	clϣԏ.	clϣԏ.	NOUN
iajs-2969	151	74	theorem	theorem	VERB
iajs-2969	151	75	3.10	3.10	NUM
iajs-2969	151	76	ӈ	ӈ	PRON
iajs-2969	151	77	is	be	AUX
iajs-2969	151	78	ϣ	ϣ	PRON
iajs-2969	151	79	−	−	NOUN
iajs-2969	151	80	sp	sp	ADP
iajs-2969	151	81	−	−	PROPN
iajs-2969	151	82	c	c	NOUN
iajs-2969	151	83	set	set	VERB
iajs-2969	151	84			ADV
iajs-2969	151	85	ħ	ħ	PROPN
iajs-2969	151	86	=	=	PUNCT
iajs-2969	151	87	𝑠	𝑠	PROPN
iajs-2969	151	88	−	−	PROPN
iajs-2969	151	89	𝑝	𝑝	PROPN
iajs-2969	151	90	−	−	PROPN
iajs-2969	151	91	clϣħ	clϣħ	NOUN
iajs-2969	151	92	.	.	PUNCT
iajs-2969	152	1	proof	proof	NOUN
iajs-2969	152	2	:	:	PUNCT
iajs-2969	152	3	is	be	AUX
iajs-2969	152	4	clear	clear	ADJ
iajs-2969	152	5	.	.	PUNCT
iajs-2969	153	1	corollary	corollary	ADJ
iajs-2969	153	2	3.11	3.11	NUM
iajs-2969	153	3	𝑠	𝑠	PROPN
iajs-2969	153	4	−	−	PROPN
iajs-2969	153	5	𝑝	𝑝	PROPN
iajs-2969	153	6	−	−	PROPN
iajs-2969	153	7	clϣz	clϣz	NOUN
iajs-2969	153	8	=	=	SYM
iajs-2969	153	9	z.	z.	PROPN
iajs-2969	153	10	theorem	theorem	VERB
iajs-2969	153	11	3.12	3.12	NUM
iajs-2969	153	12	let	let	VERB
iajs-2969	153	13	ӈ	ӈ	PRON
iajs-2969	153	14	and	and	CCONJ
iajs-2969	153	15	ԏ	ԏ	PART
iajs-2969	153	16	be	be	AUX
iajs-2969	153	17	two	two	NUM
iajs-2969	153	18	subsets	subset	NOUN
iajs-2969	153	19	of	of	ADP
iajs-2969	153	20	(	(	PUNCT
iajs-2969	153	21	z	z	NOUN
iajs-2969	153	22	,	,	PUNCT
iajs-2969	153	23	ϣ	ϣ	NOUN
iajs-2969	153	24	)	)	PUNCT
iajs-2969	153	25	.	.	PUNCT
iajs-2969	154	1	then	then	ADV
iajs-2969	154	2	the	the	DET
iajs-2969	154	3	following	follow	VERB
iajs-2969	154	4	properties	property	NOUN
iajs-2969	154	5	are	be	AUX
iajs-2969	154	6	true	true	ADJ
iajs-2969	154	7	:	:	PUNCT
iajs-2969	154	8	1	1	X
iajs-2969	154	9	.	.	X
iajs-2969	154	10	𝑠	𝑠	INTJ
iajs-2969	154	11	−	−	PROPN
iajs-2969	154	12	𝑝	𝑝	PROPN
iajs-2969	154	13	−	−	PROPN
iajs-2969	154	14	intϣԋ	intϣԋ	NOUN
iajs-2969	154	15	⊆	⊆	NUM
iajs-2969	154	16	ԋ.	ԋ.	NOUN
iajs-2969	154	17	2	2	X
iajs-2969	154	18	.	.	X
iajs-2969	155	1	if	if	SCONJ
iajs-2969	155	2	ԋ	ԋ	PROPN
iajs-2969	155	3	⊆	⊆	NUM
iajs-2969	155	4	ԏ	ԏ	NOUN
iajs-2969	155	5	,	,	PUNCT
iajs-2969	155	6	then	then	ADV
iajs-2969	155	7	𝑠	𝑠	INTJ
iajs-2969	155	8	−	−	PROPN
iajs-2969	155	9	𝑝	𝑝	PROPN
iajs-2969	155	10	−	−	PROPN
iajs-2969	155	11	intϣԋ	intϣԋ	NOUN
iajs-2969	155	12	⊆	⊆	NUM
iajs-2969	155	13	𝑠	𝑠	PROPN
iajs-2969	155	14	−	−	PROPN
iajs-2969	155	15	𝑝	𝑝	PROPN
iajs-2969	155	16	−	−	PROPN
iajs-2969	155	17	intϣ	intϣ	NOUN
iajs-2969	155	18	ԏ.	ԏ.	NOUN
iajs-2969	155	19	3	3	NUM
iajs-2969	155	20	.	.	PUNCT
iajs-2969	156	1	𝑠	𝑠	INTJ
iajs-2969	156	2	−	−	PROPN
iajs-2969	156	3	𝑝	𝑝	PROPN
iajs-2969	156	4	−	−	PROPN
iajs-2969	156	5	intϣ	intϣ	NOUN
iajs-2969	156	6	(	(	PUNCT
iajs-2969	156	7	ԋ	ԋ	NOUN
iajs-2969	156	8	∩	∩	ADJ
iajs-2969	156	9	ԏ	ԏ	NOUN
iajs-2969	156	10	)	)	PUNCT
iajs-2969	156	11	⊆	⊆	NUM
iajs-2969	156	12	𝑠	𝑠	NUM
iajs-2969	156	13	−	−	PROPN
iajs-2969	156	14	𝑝	𝑝	PROPN
iajs-2969	156	15	−	−	PROPN
iajs-2969	156	16	intϣ	intϣ	NOUN
iajs-2969	157	1	ԋ	ԋ	PROPN
iajs-2969	157	2	∩	∩	NOUN
iajs-2969	157	3	𝑠	𝑠	PROPN
iajs-2969	157	4	−	−	PROPN
iajs-2969	157	5	𝑝	𝑝	PROPN
iajs-2969	157	6	−	−	PROPN
iajs-2969	157	7	intϣ	intϣ	NOUN
iajs-2969	157	8	ԏ	ԏ	PROPN
iajs-2969	157	9	4	4	X
iajs-2969	157	10	.	.	PUNCT
iajs-2969	158	1	𝑠	𝑠	INTJ
iajs-2969	158	2	−	−	PROPN
iajs-2969	158	3	𝑝	𝑝	PROPN
iajs-2969	158	4	−	−	PROPN
iajs-2969	158	5	intϣ	intϣ	NOUN
iajs-2969	159	1	ԋ	ԋ	NOUN
iajs-2969	159	2	∪	∪	ADP
iajs-2969	159	3	𝑠	𝑠	PROPN
iajs-2969	159	4	−	−	PROPN
iajs-2969	159	5	𝑝	𝑝	PROPN
iajs-2969	159	6	−	−	PROPN
iajs-2969	159	7	intϣ	intϣ	NOUN
iajs-2969	159	8	ԏ	ԏ	PROPN
iajs-2969	159	9	⊆	⊆	NUM
iajs-2969	159	10	𝑠	𝑠	INTJ
iajs-2969	159	11	−	−	PROPN
iajs-2969	159	12	𝑝	𝑝	PROPN
iajs-2969	159	13	−	−	PROPN
iajs-2969	160	1	intϣ	intϣ	NOUN
iajs-2969	161	1	ԋ	ԋ	NOUN
iajs-2969	161	2	∪	∪	NOUN
iajs-2969	161	3	ԏ	ԏ	NOUN
iajs-2969	161	4	)	)	PUNCT
iajs-2969	161	5	.	.	PUNCT
iajs-2969	162	1	proof	proof	NOUN
iajs-2969	162	2	:	:	PUNCT
iajs-2969	162	3	1	1	X
iajs-2969	162	4	.	.	X
iajs-2969	162	5	clear	clear	ADJ
iajs-2969	162	6	.	.	PUNCT
iajs-2969	163	1	2	2	X
iajs-2969	163	2	.	.	X
iajs-2969	163	3	let	let	VERB
iajs-2969	163	4	ԋ	ԋ	PRON
iajs-2969	163	5	⊆	⊆	NUM
iajs-2969	163	6	ԏ	ԏ	NOUN
iajs-2969	163	7	,	,	PUNCT
iajs-2969	163	8	from	from	ADP
iajs-2969	163	9	(	(	PUNCT
iajs-2969	163	10	1	1	X
iajs-2969	163	11	)	)	PUNCT
iajs-2969	163	12	we	we	PRON
iajs-2969	163	13	have	have	AUX
iajs-2969	163	14	𝑠	𝑠	NUM
iajs-2969	163	15	−	−	PROPN
iajs-2969	163	16	𝑝	𝑝	PROPN
iajs-2969	163	17	−	−	PROPN
iajs-2969	163	18	intϣԋ	intϣԋ	NOUN
iajs-2969	163	19	⊆	⊆	NUM
iajs-2969	163	20	ԋ	ԋ	NOUN
iajs-2969	163	21	,	,	PUNCT
iajs-2969	163	22	so	so	ADV
iajs-2969	163	23	𝑠	𝑠	INTJ
iajs-2969	163	24	−	−	PROPN
iajs-2969	163	25	𝑝	𝑝	PROPN
iajs-2969	163	26	−	−	PROPN
iajs-2969	163	27	intϣԋ	intϣԋ	NOUN
iajs-2969	163	28	⊆	⊆	NUM
iajs-2969	163	29	ԏ	ԏ	X
iajs-2969	163	30	where	where	SCONJ
iajs-2969	163	31	𝑠	𝑠	INTJ
iajs-2969	163	32	−	−	PROPN
iajs-2969	163	33	𝑝	𝑝	PROPN
iajs-2969	163	34	−	−	PROPN
iajs-2969	163	35	intϣԋ	intϣԋ	NOUN
iajs-2969	163	36	is	be	AUX
iajs-2969	163	37	ϣ	ϣ	ADP
iajs-2969	163	38	−	−	NOUN
iajs-2969	163	39	sp	sp	ADP
iajs-2969	163	40	−	−	NOUN
iajs-2969	163	41	o	o	NOUN
iajs-2969	163	42	set	set	NOUN
iajs-2969	163	43	,	,	PUNCT
iajs-2969	163	44	but	but	CCONJ
iajs-2969	163	45	𝑠	𝑠	INTJ
iajs-2969	163	46	−	−	PROPN
iajs-2969	163	47	𝑝	𝑝	PROPN
iajs-2969	163	48	−	−	PROPN
iajs-2969	163	49	intϣԏ	intϣԏ	NOUN
iajs-2969	163	50	is	be	AUX
iajs-2969	163	51	the	the	DET
iajs-2969	163	52	largest	large	ADJ
iajs-2969	163	53	ϣ	ϣ	NOUN
iajs-2969	163	54	−	−	NOUN
iajs-2969	163	55	sp	sp	ADP
iajs-2969	163	56	−	−	NOUN
iajs-2969	163	57	o	o	NOUN
iajs-2969	163	58	set	set	NOUN
iajs-2969	163	59	contained	contain	VERB
iajs-2969	163	60	in	in	ADP
iajs-2969	163	61	ԏ	ԏ	ADP
iajs-2969	163	62	,	,	PUNCT
iajs-2969	163	63	hence	hence	ADV
iajs-2969	163	64	𝑠	𝑠	INTJ
iajs-2969	163	65	−	−	PROPN
iajs-2969	163	66	𝑝	𝑝	PROPN
iajs-2969	163	67	−	−	PROPN
iajs-2969	163	68	intϣ	intϣ	NOUN
iajs-2969	164	1	ԋ	ԋ	PROPN
iajs-2969	165	1	⊆	⊆	NUM
iajs-2969	165	2	𝑠	𝑠	NUM
iajs-2969	165	3	−	−	PROPN
iajs-2969	165	4	𝑝	𝑝	PROPN
iajs-2969	165	5	−	−	PROPN
iajs-2969	165	6	intϣ	intϣ	NOUN
iajs-2969	165	7	ԏ.	ԏ.	NOUN
iajs-2969	165	8	3	3	X
iajs-2969	165	9	.	.	PUNCT
iajs-2969	166	1	since	since	SCONJ
iajs-2969	166	2	(	(	PUNCT
iajs-2969	166	3	ԋ	ԋ	PROPN
iajs-2969	166	4	∩	∩	ADJ
iajs-2969	166	5	ԏ	ԏ	NOUN
iajs-2969	166	6	)	)	PUNCT
iajs-2969	166	7	⊆	⊆	NUM
iajs-2969	166	8	ӈ	ӈ	PROPN
iajs-2969	166	9	and	and	CCONJ
iajs-2969	166	10	(	(	PUNCT
iajs-2969	166	11	ԋ	ԋ	PROPN
iajs-2969	166	12	∩	∩	ADJ
iajs-2969	166	13	ԏ	ԏ	NOUN
iajs-2969	166	14	)	)	PUNCT
iajs-2969	166	15	⊆	⊆	NUM
iajs-2969	166	16	ԏ	ԏ	NOUN
iajs-2969	166	17	,	,	PUNCT
iajs-2969	166	18	so	so	ADV
iajs-2969	166	19	𝑠	𝑠	INTJ
iajs-2969	166	20	−	−	PROPN
iajs-2969	166	21	𝑝	𝑝	PROPN
iajs-2969	166	22	−	−	PROPN
iajs-2969	166	23	intϣ	intϣ	NOUN
iajs-2969	167	1	(	(	PUNCT
iajs-2969	167	2	ԋ	ԋ	NOUN
iajs-2969	167	3	∩	∩	ADJ
iajs-2969	167	4	ԏ	ԏ	NOUN
iajs-2969	167	5	)	)	PUNCT
iajs-2969	167	6	⊆	⊆	NUM
iajs-2969	167	7	𝑠	𝑠	NUM
iajs-2969	167	8	−	−	PROPN
iajs-2969	167	9	𝑝	𝑝	PROPN
iajs-2969	167	10	−	−	PROPN
iajs-2969	167	11	intϣԋ	intϣԋ	NOUN
iajs-2969	167	12	and	and	CCONJ
iajs-2969	167	13	𝑠	𝑠	INTJ
iajs-2969	167	14	−	−	PROPN
iajs-2969	167	15	𝑝	𝑝	PROPN
iajs-2969	167	16	−	−	PROPN
iajs-2969	167	17	intϣ(ԋ	intϣ(ԋ	PROPN
iajs-2969	167	18	∩	∩	NOUN
iajs-2969	167	19	ԏ	ԏ	NOUN
iajs-2969	167	20	)	)	PUNCT
iajs-2969	167	21	⊆	⊆	NUM
iajs-2969	167	22	𝑠	𝑠	NUM
iajs-2969	167	23	−	−	PROPN
iajs-2969	167	24	𝑝	𝑝	PROPN
iajs-2969	167	25	−	−	PROPN
iajs-2969	167	26	intϣԏ	intϣԏ	NOUN
iajs-2969	167	27	,	,	PUNCT
iajs-2969	167	28	so	so	ADV
iajs-2969	167	29	𝑠	𝑠	INTJ
iajs-2969	167	30	−	−	PROPN
iajs-2969	167	31	𝑝	𝑝	PROPN
iajs-2969	167	32	−	−	PROPN
iajs-2969	167	33	intϣ	intϣ	NOUN
iajs-2969	168	1	(	(	PUNCT
iajs-2969	169	1	ԋ	ԋ	NOUN
iajs-2969	169	2	∩	∩	ADJ
iajs-2969	169	3	ԏ	ԏ	NOUN
iajs-2969	169	4	)	)	PUNCT
iajs-2969	169	5	⊆	⊆	NUM
iajs-2969	169	6	𝑠	𝑠	NUM
iajs-2969	169	7	−	−	PROPN
iajs-2969	169	8	𝑝	𝑝	PROPN
iajs-2969	169	9	−	−	PROPN
iajs-2969	169	10	intϣԋ	intϣԋ	NOUN
iajs-2969	169	11	∩	∩	NOUN
iajs-2969	169	12	𝑠	𝑠	PROPN
iajs-2969	169	13	−	−	PROPN
iajs-2969	169	14	𝑝	𝑝	PROPN
iajs-2969	169	15	−	−	PROPN
iajs-2969	169	16	intϣԏ.	intϣԏ.	NOUN
iajs-2969	169	17	4	4	NUM
iajs-2969	169	18	.	.	PUNCT
iajs-2969	170	1	since	since	SCONJ
iajs-2969	170	2	ԋ	ԋ	PRON
iajs-2969	170	3	⊆	⊆	NUM
iajs-2969	170	4	ԋ	ԋ	NOUN
iajs-2969	170	5	∪	∪	NOUN
iajs-2969	170	6	ԏ	ԏ	NOUN
iajs-2969	170	7	and	and	CCONJ
iajs-2969	170	8	ԏ	ԏ	PRON
iajs-2969	170	9	⊆	⊆	NUM
iajs-2969	170	10	ԋ	ԋ	NOUN
iajs-2969	170	11	∪	∪	NOUN
iajs-2969	170	12	ԏ	ԏ	PROPN
iajs-2969	170	13	,	,	PUNCT
iajs-2969	170	14	then	then	ADV
iajs-2969	170	15	𝑠	𝑠	INTJ
iajs-2969	170	16	−	−	PROPN
iajs-2969	170	17	𝑝	𝑝	PROPN
iajs-2969	170	18	−	−	PROPN
iajs-2969	170	19	intϣԋ	intϣԋ	NOUN
iajs-2969	170	20	⊆	⊆	NUM
iajs-2969	170	21	𝑠	𝑠	PROPN
iajs-2969	170	22	−	−	PROPN
iajs-2969	170	23	𝑝	𝑝	PROPN
iajs-2969	170	24	−	−	PROPN
iajs-2969	170	25	intϣ	intϣ	NOUN
iajs-2969	170	26	(	(	PUNCT
iajs-2969	170	27	ԋ	ԋ	NOUN
iajs-2969	170	28	∪	∪	NOUN
iajs-2969	170	29	ԏ	ԏ	NOUN
iajs-2969	170	30	)	)	PUNCT
iajs-2969	170	31	and	and	CCONJ
iajs-2969	170	32	𝑠	𝑠	INTJ
iajs-2969	170	33	−	−	PROPN
iajs-2969	170	34	𝑝	𝑝	PROPN
iajs-2969	170	35	−	−	PROPN
iajs-2969	170	36	intϣԏ	intϣԏ	VERB
iajs-2969	170	37	⊆	⊆	NUM
iajs-2969	170	38	𝑠	𝑠	INTJ
iajs-2969	170	39	−	−	PROPN
iajs-2969	170	40	𝑝	𝑝	PROPN
iajs-2969	170	41	−	−	PROPN
iajs-2969	170	42	intϣ	intϣ	NOUN
iajs-2969	170	43	(	(	PUNCT
iajs-2969	170	44	ԋ	ԋ	NOUN
iajs-2969	170	45	∪	∪	NOUN
iajs-2969	170	46	ԏ	ԏ	NOUN
iajs-2969	170	47	)	)	PUNCT
iajs-2969	170	48	.	.	PUNCT
iajs-2969	171	1	thus	thus	ADV
iajs-2969	171	2	𝑠	𝑠	INTJ
iajs-2969	171	3	−	−	PROPN
iajs-2969	171	4	𝑝	𝑝	PROPN
iajs-2969	171	5	−	−	PROPN
iajs-2969	171	6	intϣԋ	intϣԋ	NOUN
iajs-2969	171	7	∪	∪	ADP
iajs-2969	171	8	𝑠	𝑠	PROPN
iajs-2969	171	9	−	−	PROPN
iajs-2969	171	10	𝑝	𝑝	PROPN
iajs-2969	171	11	−	−	PROPN
iajs-2969	171	12	intϣ	intϣ	NOUN
iajs-2969	171	13	ԏ	ԏ	PROPN
iajs-2969	172	1	⊆	⊆	NUM
iajs-2969	172	2	𝑠	𝑠	INTJ
iajs-2969	172	3	−	−	PROPN
iajs-2969	172	4	𝑝	𝑝	PROPN
iajs-2969	172	5	−	−	PROPN
iajs-2969	172	6	intϣ	intϣ	NOUN
iajs-2969	172	7	(	(	PUNCT
iajs-2969	172	8	ԋ	ԋ	NOUN
iajs-2969	172	9	∪	∪	NOUN
iajs-2969	172	10	ԏ	ԏ	NOUN
iajs-2969	172	11	)	)	PUNCT
iajs-2969	172	12	.	.	PUNCT
iajs-2969	173	1	theorem	theorem	VERB
iajs-2969	173	2	3.13	3.13	NUM
iajs-2969	173	3	ħ	ħ	NOUN
iajs-2969	173	4	is	be	AUX
iajs-2969	173	5	an	an	DET
iajs-2969	173	6	ϣ	ϣ	NOUN
iajs-2969	173	7	−	−	NOUN
iajs-2969	173	8	sp	sp	ADP
iajs-2969	173	9	−	−	PROPN
iajs-2969	173	10	o	o	PROPN
iajs-2969	173	11	setħ	setħ	PROPN
iajs-2969	173	12	=	=	NOUN
iajs-2969	173	13	𝑠	𝑠	PROPN
iajs-2969	173	14	−	−	PROPN
iajs-2969	173	15	𝑝	𝑝	PROPN
iajs-2969	173	16	−	−	PROPN
iajs-2969	173	17	intϣħ	intϣħ	ADV
iajs-2969	173	18	.	.	PUNCT
iajs-2969	174	1	proof	proof	NOUN
iajs-2969	174	2	:	:	PUNCT
iajs-2969	174	3	is	be	AUX
iajs-2969	174	4	clear	clear	ADJ
iajs-2969	174	5	.	.	PUNCT
iajs-2969	175	1	corollary	corollary	ADJ
iajs-2969	175	2	3.14	3.14	NUM
iajs-2969	175	3	𝑠	𝑠	PROPN
iajs-2969	175	4	−	−	PROPN
iajs-2969	175	5	𝑝	𝑝	PROPN
iajs-2969	175	6	−	−	PROPN
iajs-2969	175	7	intϣø	intϣø	PROPN
iajs-2969	175	8	=	=	NOUN
iajs-2969	175	9	ø	ø	NOUN
iajs-2969	175	10	ihjpas	ihjpas	PROPN
iajs-2969	175	11	.	.	PUNCT
iajs-2969	176	1	36(1)2023	36(1)2023	NUM
iajs-2969	176	2	340	340	NUM
iajs-2969	176	3	4	4	NUM
iajs-2969	176	4	.	.	PUNCT
iajs-2969	177	1	(	(	PUNCT
iajs-2969	177	2	ϣ𝟏	ϣ𝟏	PROPN
iajs-2969	177	3	,	,	PUNCT
iajs-2969	177	4	ϣ𝟐)-semi	ϣ𝟐)-semi	PROPN
iajs-2969	177	5	-	-	ADJ
iajs-2969	177	6	p	p	ADJ
iajs-2969	177	7	-	-	PUNCT
iajs-2969	177	8	continuous	continuous	ADJ
iajs-2969	177	9	function	function	NOUN
iajs-2969	177	10	definition	definition	NOUN
iajs-2969	177	11	4.1:[8	4.1:[8	NUM
iajs-2969	177	12	]	]	PUNCT
iajs-2969	177	13	let	let	VERB
iajs-2969	177	14	(	(	PUNCT
iajs-2969	177	15	z	z	NOUN
iajs-2969	177	16	,	,	PUNCT
iajs-2969	177	17	ϣ1	ϣ1	PROPN
iajs-2969	177	18	)	)	PUNCT
iajs-2969	177	19	and	and	CCONJ
iajs-2969	177	20	(	(	PUNCT
iajs-2969	177	21	y	y	PROPN
iajs-2969	177	22	,	,	PUNCT
iajs-2969	177	23	ϣ2	ϣ2	PROPN
iajs-2969	177	24	)	)	PUNCT
iajs-2969	177	25	be	be	AUX
iajs-2969	177	26	two	two	NUM
iajs-2969	177	27	gts	gts	NOUN
iajs-2969	177	28	,	,	PUNCT
iajs-2969	177	29	s.	s.	PROPN
iajs-2969	177	30	a	a	DET
iajs-2969	177	31	function	function	NOUN
iajs-2969	178	1	f	f	X
iajs-2969	178	2	:	:	PUNCT
iajs-2969	178	3	z	z	X
iajs-2969	178	4	→	→	SYM
iajs-2969	178	5	y	y	PROPN
iajs-2969	178	6	is	be	AUX
iajs-2969	178	7	said	say	VERB
iajs-2969	178	8	to	to	PART
iajs-2969	178	9	be	be	AUX
iajs-2969	178	10	(	(	PUNCT
iajs-2969	178	11	ϣ𝟏	ϣ𝟏	ADJ
iajs-2969	178	12	,	,	PUNCT
iajs-2969	178	13	ϣ𝟐)-continuous	ϣ𝟐)-continuous	ADJ
iajs-2969	178	14	function	function	NOUN
iajs-2969	178	15	if	if	SCONJ
iajs-2969	178	16	the	the	DET
iajs-2969	178	17	inverse	inverse	ADJ
iajs-2969	178	18	image	image	NOUN
iajs-2969	178	19	of	of	ADP
iajs-2969	178	20	any	any	DET
iajs-2969	178	21	ϣ2	ϣ2	NOUN
iajs-2969	178	22	-	-	PUNCT
iajs-2969	178	23	open	open	ADJ
iajs-2969	178	24	subset	subset	NOUN
iajs-2969	178	25	of	of	ADP
iajs-2969	178	26	y	y	PROPN
iajs-2969	178	27	is	be	AUX
iajs-2969	178	28	an	an	DET
iajs-2969	178	29	ϣ1	ϣ1	NOUN
iajs-2969	178	30	-	-	PUNCT
iajs-2969	178	31	open	open	NOUN
iajs-2969	178	32	set	set	NOUN
iajs-2969	178	33	in	in	ADP
iajs-2969	178	34	z.	z.	PROPN
iajs-2969	178	35	definition	definition	NOUN
iajs-2969	178	36	4.2:[9	4.2:[9	NUM
iajs-2969	178	37	]	]	PUNCT
iajs-2969	178	38	a	a	DET
iajs-2969	178	39	function𝑓	function𝑓	NOUN
iajs-2969	178	40	:	:	PUNCT
iajs-2969	178	41	(	(	PUNCT
iajs-2969	178	42	z	z	NOUN
iajs-2969	178	43	,	,	PUNCT
iajs-2969	178	44	ϣ1	ϣ1	PROPN
iajs-2969	178	45	)	)	PUNCT
iajs-2969	178	46	→	→	SYM
iajs-2969	178	47	(	(	PUNCT
iajs-2969	178	48	y	y	PROPN
iajs-2969	178	49	,	,	PUNCT
iajs-2969	178	50	ϣ2	ϣ2	PROPN
iajs-2969	178	51	)	)	PUNCT
iajs-2969	178	52	is	be	AUX
iajs-2969	178	53	called	call	VERB
iajs-2969	178	54	(	(	PUNCT
iajs-2969	178	55	ϣ1	ϣ1	NOUN
iajs-2969	178	56	,	,	PUNCT
iajs-2969	178	57	ϣ2)-mpre	ϣ2)-mpre	ADJ
iajs-2969	178	58	-	-	ADJ
iajs-2969	178	59	open	open	ADJ
iajs-2969	178	60	function	function	NOUN
iajs-2969	178	61	if	if	SCONJ
iajs-2969	178	62	the	the	DET
iajs-2969	178	63	direct	direct	ADJ
iajs-2969	178	64	image	image	NOUN
iajs-2969	178	65	of	of	ADP
iajs-2969	178	66	any	any	DET
iajs-2969	178	67	ϣ1pre	ϣ1pre	NOUN
iajs-2969	178	68	-	-	ADJ
iajs-2969	178	69	open	open	ADJ
iajs-2969	178	70	set	set	NOUN
iajs-2969	178	71	in	in	ADP
iajs-2969	178	72	z	z	PROPN
iajs-2969	178	73	is	be	AUX
iajs-2969	178	74	an	an	DET
iajs-2969	178	75	ϣ2pre	ϣ2pre	ADV
iajs-2969	178	76	-	-	PUNCT
iajs-2969	178	77	open	open	ADJ
iajs-2969	178	78	set	set	NOUN
iajs-2969	178	79	in	in	ADP
iajs-2969	178	80	y.	y.	PROPN
iajs-2969	178	81	definition	definition	NOUN
iajs-2969	178	82	4.3	4.3	NUM
iajs-2969	178	83	:	:	PUNCT
iajs-2969	178	84	a	a	DET
iajs-2969	178	85	function	function	NOUN
iajs-2969	178	86	𝑓	𝑓	X
iajs-2969	178	87	:	:	PUNCT
iajs-2969	178	88	(	(	PUNCT
iajs-2969	178	89	z	z	NOUN
iajs-2969	178	90	,	,	PUNCT
iajs-2969	178	91	ϣ1	ϣ1	PROPN
iajs-2969	178	92	)	)	PUNCT
iajs-2969	178	93	→	→	SYM
iajs-2969	178	94	(	(	PUNCT
iajs-2969	178	95	y	y	PROPN
iajs-2969	178	96	,	,	PUNCT
iajs-2969	178	97	ϣ2	ϣ2	PROPN
iajs-2969	178	98	)	)	PUNCT
iajs-2969	178	99	is	be	AUX
iajs-2969	178	100	called	call	VERB
iajs-2969	178	101	(	(	PUNCT
iajs-2969	178	102	ϣ1	ϣ1	PROPN
iajs-2969	178	103	,	,	PUNCT
iajs-2969	178	104	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	178	105	-	-	PUNCT
iajs-2969	178	106	semi	semi	ADJ
iajs-2969	178	107	-	-	ADJ
iajs-2969	178	108	p	p	ADJ
iajs-2969	178	109	-	-	PUNCT
iajs-2969	178	110	open	open	ADJ
iajs-2969	178	111	(	(	PUNCT
iajs-2969	178	112	(	(	PUNCT
iajs-2969	178	113	ϣ1	ϣ1	PROPN
iajs-2969	178	114	,	,	PUNCT
iajs-2969	178	115	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	178	116	-	-	PUNCT
iajs-2969	178	117	semi	semi	ADV
iajs-2969	178	118	-	-	ADJ
iajs-2969	178	119	pclosed	pclosed	ADJ
iajs-2969	178	120	)	)	PUNCT
iajs-2969	178	121	function	function	NOUN
iajs-2969	178	122	if	if	SCONJ
iajs-2969	178	123	the	the	DET
iajs-2969	178	124	direct	direct	ADJ
iajs-2969	178	125	image	image	NOUN
iajs-2969	178	126	of	of	ADP
iajs-2969	178	127	any	any	DET
iajs-2969	178	128	ϣ1	ϣ1	NOUN
iajs-2969	178	129	-	-	PUNCT
iajs-2969	178	130	semi	semi	ADJ
iajs-2969	178	131	-	-	ADJ
iajs-2969	178	132	p	p	ADJ
iajs-2969	178	133	-	-	PUNCT
iajs-2969	178	134	open	open	ADJ
iajs-2969	178	135	(	(	PUNCT
iajs-2969	178	136	ϣ1	ϣ1	NOUN
iajs-2969	178	137	-	-	PUNCT
iajs-2969	178	138	semi	semi	ADJ
iajs-2969	178	139	-	-	ADJ
iajs-2969	178	140	p	p	ADJ
iajs-2969	178	141	-	-	PUNCT
iajs-2969	178	142	closed	closed	ADJ
iajs-2969	178	143	)	)	PUNCT
iajs-2969	178	144	set	set	VERB
iajs-2969	178	145	in	in	ADP
iajs-2969	178	146	z	z	PROPN
iajs-2969	178	147	is	be	AUX
iajs-2969	178	148	an	an	DET
iajs-2969	178	149	ϣ2	ϣ2	NOUN
iajs-2969	178	150	-	-	PUNCT
iajs-2969	178	151	semi	semi	ADJ
iajs-2969	178	152	-	-	ADJ
iajs-2969	178	153	p	p	ADJ
iajs-2969	178	154	-	-	PUNCT
iajs-2969	178	155	open	open	ADJ
iajs-2969	178	156	(	(	PUNCT
iajs-2969	178	157	ϣ2	ϣ2	NOUN
iajs-2969	178	158	-	-	PUNCT
iajs-2969	178	159	semi	semi	ADJ
iajs-2969	178	160	-	-	ADJ
iajs-2969	178	161	p	p	ADJ
iajs-2969	178	162	-	-	PUNCT
iajs-2969	178	163	closed	close	VERB
iajs-2969	178	164	)	)	PUNCT
iajs-2969	178	165	set	set	VERB
iajs-2969	178	166	in	in	ADP
iajs-2969	178	167	y.	y.	PROPN
iajs-2969	178	168	definition	definition	NOUN
iajs-2969	178	169	4.4	4.4	NUM
iajs-2969	178	170	a	a	DET
iajs-2969	178	171	function	function	NOUN
iajs-2969	178	172	𝑓	𝑓	NOUN
iajs-2969	178	173	:	:	PUNCT
iajs-2969	178	174	(	(	PUNCT
iajs-2969	178	175	z	z	NOUN
iajs-2969	178	176	,	,	PUNCT
iajs-2969	178	177	ϣ1	ϣ1	PROPN
iajs-2969	178	178	)	)	PUNCT
iajs-2969	178	179	→	→	SYM
iajs-2969	178	180	(	(	PUNCT
iajs-2969	178	181	y	y	PROPN
iajs-2969	178	182	,	,	PUNCT
iajs-2969	178	183	ϣ2	ϣ2	PROPN
iajs-2969	178	184	)	)	PUNCT
iajs-2969	178	185	is	be	AUX
iajs-2969	178	186	said	say	VERB
iajs-2969	178	187	to	to	PART
iajs-2969	178	188	be	be	AUX
iajs-2969	178	189	(	(	PUNCT
iajs-2969	178	190	ϣ1	ϣ1	NOUN
iajs-2969	178	191	,	,	PUNCT
iajs-2969	178	192	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	178	193	-	-	ADJ
iajs-2969	178	194	p	p	ADJ
iajs-2969	178	195	-	-	PUNCT
iajs-2969	178	196	continuous	continuous	ADJ
iajs-2969	178	197	function	function	NOUN
iajs-2969	178	198	if	if	SCONJ
iajs-2969	178	199	the	the	DET
iajs-2969	178	200	inverse	inverse	ADJ
iajs-2969	178	201	image	image	NOUN
iajs-2969	178	202	of	of	ADP
iajs-2969	178	203	any	any	DET
iajs-2969	178	204	ϣ2	ϣ2	NOUN
iajs-2969	178	205	-	-	PUNCT
iajs-2969	178	206	open	open	NOUN
iajs-2969	178	207	set	set	NOUN
iajs-2969	178	208	in	in	ADP
iajs-2969	178	209	y	y	PROPN
iajs-2969	178	210	is	be	AUX
iajs-2969	178	211	an	an	DET
iajs-2969	178	212	ϣ1	ϣ1	NOUN
iajs-2969	178	213	-	-	PUNCT
iajs-2969	178	214	semi	semi	ADJ
iajs-2969	178	215	-	-	ADJ
iajs-2969	178	216	p	p	ADJ
iajs-2969	178	217	-	-	PUNCT
iajs-2969	178	218	open	open	NOUN
iajs-2969	178	219	set	set	NOUN
iajs-2969	178	220	in	in	ADP
iajs-2969	178	221	z.	z.	PROPN
iajs-2969	178	222	theorem	theorem	VERB
iajs-2969	178	223	4.5	4.5	NUM
iajs-2969	178	224	a	a	DET
iajs-2969	178	225	function	function	NOUN
iajs-2969	178	226	𝑓	𝑓	NOUN
iajs-2969	178	227	:	:	PUNCT
iajs-2969	178	228	(	(	PUNCT
iajs-2969	178	229	z	z	NOUN
iajs-2969	178	230	,	,	PUNCT
iajs-2969	178	231	ϣ1	ϣ1	PROPN
iajs-2969	178	232	)	)	PUNCT
iajs-2969	178	233	→	→	SYM
iajs-2969	178	234	(	(	PUNCT
iajs-2969	178	235	y	y	PROPN
iajs-2969	178	236	,	,	PUNCT
iajs-2969	178	237	ϣ2	ϣ2	PROPN
iajs-2969	178	238	)	)	PUNCT
iajs-2969	178	239	is	be	AUX
iajs-2969	178	240	an	an	DET
iajs-2969	178	241	(	(	PUNCT
iajs-2969	178	242	ϣ1	ϣ1	NOUN
iajs-2969	178	243	,	,	PUNCT
iajs-2969	178	244	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	178	245	-	-	ADJ
iajs-2969	178	246	p	p	ADJ
iajs-2969	178	247	-	-	PUNCT
iajs-2969	178	248	continuous	continuous	ADJ
iajs-2969	178	249	function	function	NOUN
iajs-2969	178	250			ADP
iajs-2969	178	251	the	the	DET
iajs-2969	178	252	inverse	inverse	NOUN
iajs-2969	178	253	image	image	NOUN
iajs-2969	178	254	of	of	ADP
iajs-2969	178	255	any	any	DET
iajs-2969	178	256	ϣ2	ϣ2	NOUN
iajs-2969	178	257	-	-	PUNCT
iajs-2969	178	258	closed	close	VERB
iajs-2969	178	259	set	set	NOUN
iajs-2969	178	260	in	in	ADP
iajs-2969	178	261	y	y	PROPN
iajs-2969	178	262	is	be	AUX
iajs-2969	178	263	an	an	DET
iajs-2969	178	264	ϣ1	ϣ1	NOUN
iajs-2969	178	265	-	-	PUNCT
iajs-2969	178	266	semi	semi	ADJ
iajs-2969	178	267	-	-	ADJ
iajs-2969	178	268	p	p	ADJ
iajs-2969	178	269	-	-	PUNCT
iajs-2969	178	270	closed	close	VERB
iajs-2969	178	271	set	set	NOUN
iajs-2969	178	272	in	in	ADP
iajs-2969	178	273	z	z	NOUN
iajs-2969	178	274	.	.	PUNCT
iajs-2969	179	1	proof	proof	NOUN
iajs-2969	179	2	:	:	PUNCT
iajs-2969	179	3	the	the	DET
iajs-2969	179	4	"	"	PUNCT
iajs-2969	179	5	if	if	SCONJ
iajs-2969	179	6	"	"	PUNCT
iajs-2969	179	7	part	part	NOUN
iajs-2969	179	8	.	.	PUNCT
iajs-2969	180	1	let	let	VERB
iajs-2969	180	2	f	f	PRON
iajs-2969	180	3	be	be	AUX
iajs-2969	180	4	any	any	DET
iajs-2969	180	5	ϣ2	ϣ2	NOUN
iajs-2969	180	6	-	-	PUNCT
iajs-2969	180	7	closed	close	VERB
iajs-2969	180	8	set	set	NOUN
iajs-2969	180	9	in	in	ADP
iajs-2969	180	10	y	y	PROPN
iajs-2969	180	11	,	,	PUNCT
iajs-2969	180	12	thus	thus	ADV
iajs-2969	180	13	(	(	PUNCT
iajs-2969	180	14	y	y	X
iajs-2969	180	15	–	–	PUNCT
iajs-2969	180	16	f	f	X
iajs-2969	180	17	)	)	PUNCT
iajs-2969	180	18	is	be	AUX
iajs-2969	180	19	an	an	DET
iajs-2969	180	20	ϣ2	ϣ2	NOUN
iajs-2969	180	21	-	-	PUNCT
iajs-2969	180	22	open	open	NOUN
iajs-2969	180	23	set	set	NOUN
iajs-2969	180	24	in	in	ADP
iajs-2969	180	25	y	y	PROPN
iajs-2969	180	26	,	,	PUNCT
iajs-2969	180	27	then	then	ADV
iajs-2969	180	28	f	f	PROPN
iajs-2969	181	1	−1(y	−1(y	X
iajs-2969	182	1	−	−	PROPN
iajs-2969	183	1	f	f	X
iajs-2969	183	2	)	)	PUNCT
iajs-2969	183	3	is	be	AUX
iajs-2969	183	4	an	an	DET
iajs-2969	183	5	ϣ1	ϣ1	NOUN
iajs-2969	183	6	-	-	PUNCT
iajs-2969	183	7	semi	semi	ADJ
iajs-2969	183	8	-	-	ADJ
iajs-2969	183	9	p	p	ADJ
iajs-2969	183	10	-	-	PUNCT
iajs-2969	183	11	open	open	NOUN
iajs-2969	183	12	set	set	NOUN
iajs-2969	183	13	in	in	ADP
iajs-2969	183	14	z	z	PROPN
iajs-2969	183	15	(	(	PUNCT
iajs-2969	183	16	since	since	SCONJ
iajs-2969	183	17	f	f	PROPN
iajs-2969	183	18	is	be	AUX
iajs-2969	183	19	an	an	DET
iajs-2969	183	20	(	(	PUNCT
iajs-2969	183	21	ϣ1	ϣ1	NOUN
iajs-2969	183	22	,	,	PUNCT
iajs-2969	183	23	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	183	24	-	-	ADJ
iajs-2969	183	25	p	p	ADJ
iajs-2969	183	26	-	-	PUNCT
iajs-2969	183	27	continuous	continuous	ADJ
iajs-2969	183	28	function	function	NOUN
iajs-2969	183	29	)	)	PUNCT
iajs-2969	183	30	,	,	PUNCT
iajs-2969	183	31	but	but	CCONJ
iajs-2969	183	32	f	f	X
iajs-2969	183	33	−1(y	−1(y	X
iajs-2969	184	1	−	−	PROPN
iajs-2969	184	2	f	f	X
iajs-2969	184	3	)	)	PUNCT
iajs-2969	184	4	=	=	PUNCT
iajs-2969	185	1	z	z	NOUN
iajs-2969	186	1	−	−	PROPN
iajs-2969	186	2	f	f	PROPN
iajs-2969	186	3	−1(f	−1(f	PROPN
iajs-2969	186	4	)	)	PUNCT
iajs-2969	186	5	,	,	PUNCT
iajs-2969	186	6	then	then	ADV
iajs-2969	186	7	f	f	PROPN
iajs-2969	186	8	−1(f	−1(f	PROPN
iajs-2969	186	9	)	)	PUNCT
iajs-2969	186	10	is	be	AUX
iajs-2969	186	11	an	an	DET
iajs-2969	186	12	ϣ1	ϣ1	NOUN
iajs-2969	186	13	-	-	PUNCT
iajs-2969	186	14	semi	semi	ADJ
iajs-2969	186	15	-	-	ADJ
iajs-2969	186	16	p	p	ADJ
iajs-2969	186	17	-	-	PUNCT
iajs-2969	186	18	closed	close	VERB
iajs-2969	186	19	set	set	NOUN
iajs-2969	186	20	.	.	PUNCT
iajs-2969	187	1	the	the	DET
iajs-2969	187	2	"	"	PUNCT
iajs-2969	187	3	only	only	ADV
iajs-2969	187	4	if	if	SCONJ
iajs-2969	187	5	"	"	PUNCT
iajs-2969	187	6	part	part	NOUN
iajs-2969	187	7	.	.	PUNCT
iajs-2969	188	1	let	let	VERB
iajs-2969	188	2	ħ	ħ	NOUN
iajs-2969	188	3	be	be	AUX
iajs-2969	188	4	any	any	DET
iajs-2969	188	5	ϣ2	ϣ2	NOUN
iajs-2969	188	6	-	-	PUNCT
iajs-2969	188	7	open	open	NOUN
iajs-2969	188	8	set	set	NOUN
iajs-2969	188	9	in	in	ADP
iajs-2969	188	10	y	y	PROPN
iajs-2969	188	11	,	,	PUNCT
iajs-2969	188	12	thus	thus	ADV
iajs-2969	188	13	(	(	PUNCT
iajs-2969	188	14	y	y	X
iajs-2969	188	15	–	–	PUNCT
iajs-2969	188	16	ħ	ħ	NOUN
iajs-2969	188	17	)	)	PUNCT
iajs-2969	188	18	is	be	AUX
iajs-2969	188	19	an	an	DET
iajs-2969	188	20	ϣ2	ϣ2	NOUN
iajs-2969	188	21	-	-	PUNCT
iajs-2969	188	22	closed	close	VERB
iajs-2969	188	23	set	set	NOUN
iajs-2969	188	24	in	in	ADP
iajs-2969	188	25	y	y	PROPN
iajs-2969	188	26	,	,	PUNCT
iajs-2969	188	27	then	then	ADV
iajs-2969	188	28	f	f	PROPN
iajs-2969	189	1	−1(y	−1(y	X
iajs-2969	189	2	−	−	PROPN
iajs-2969	189	3	ħ	ħ	NOUN
iajs-2969	189	4	)	)	PUNCT
iajs-2969	189	5	is	be	AUX
iajs-2969	189	6	an	an	DET
iajs-2969	189	7	ϣ1	ϣ1	NOUN
iajs-2969	189	8	-	-	PUNCT
iajs-2969	189	9	semi	semi	ADJ
iajs-2969	189	10	-	-	ADJ
iajs-2969	189	11	p	p	ADJ
iajs-2969	189	12	-	-	PUNCT
iajs-2969	189	13	closed	close	VERB
iajs-2969	189	14	set	set	NOUN
iajs-2969	189	15	in	in	ADP
iajs-2969	189	16	z	z	PROPN
iajs-2969	189	17	(	(	PUNCT
iajs-2969	189	18	by	by	ADP
iajs-2969	189	19	hypothesis	hypothesis	NOUN
iajs-2969	189	20	)	)	PUNCT
iajs-2969	189	21	but	but	CCONJ
iajs-2969	189	22	f	f	X
iajs-2969	189	23	−1(y	−1(y	X
iajs-2969	190	1	−	−	PROPN
iajs-2969	190	2	ħ	ħ	NOUN
iajs-2969	190	3	)	)	PUNCT
iajs-2969	190	4	=	=	PUNCT
iajs-2969	190	5	z	z	NOUN
iajs-2969	191	1	−	−	PROPN
iajs-2969	191	2	f	f	PROPN
iajs-2969	191	3	−1(ħ	−1(ħ	NOUN
iajs-2969	191	4	)	)	PUNCT
iajs-2969	191	5	,	,	PUNCT
iajs-2969	191	6	then	then	ADV
iajs-2969	191	7	f	f	PROPN
iajs-2969	191	8	−1(ħ	−1(ħ	NOUN
iajs-2969	191	9	)	)	PUNCT
iajs-2969	191	10	is	be	AUX
iajs-2969	191	11	an	an	DET
iajs-2969	191	12	ϣ1	ϣ1	NOUN
iajs-2969	191	13	-	-	PUNCT
iajs-2969	191	14	semi	semi	ADJ
iajs-2969	191	15	-	-	ADJ
iajs-2969	191	16	p	p	ADJ
iajs-2969	191	17	-	-	PUNCT
iajs-2969	191	18	open	open	NOUN
iajs-2969	191	19	set	set	NOUN
iajs-2969	191	20	in	in	ADP
iajs-2969	191	21	z	z	PROPN
iajs-2969	191	22	,	,	PUNCT
iajs-2969	191	23	therefore	therefore	ADV
iajs-2969	191	24	f	f	PROPN
iajs-2969	191	25	is	be	AUX
iajs-2969	191	26	an	an	DET
iajs-2969	191	27	(	(	PUNCT
iajs-2969	191	28	ϣ1	ϣ1	NOUN
iajs-2969	191	29	,	,	PUNCT
iajs-2969	191	30	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	191	31	-	-	ADJ
iajs-2969	191	32	p	p	ADJ
iajs-2969	191	33	-	-	PUNCT
iajs-2969	191	34	continuous	continuous	ADJ
iajs-2969	191	35	function	function	NOUN
iajs-2969	191	36	.	.	PUNCT
iajs-2969	192	1	definition	definition	NOUN
iajs-2969	192	2	4.6	4.6	NUM
iajs-2969	192	3	a	a	DET
iajs-2969	192	4	function	function	NOUN
iajs-2969	192	5	𝑓	𝑓	NOUN
iajs-2969	192	6	:	:	PUNCT
iajs-2969	192	7	(	(	PUNCT
iajs-2969	192	8	z	z	NOUN
iajs-2969	192	9	,	,	PUNCT
iajs-2969	192	10	ϣ1	ϣ1	PROPN
iajs-2969	192	11	)	)	PUNCT
iajs-2969	192	12	→	→	SYM
iajs-2969	192	13	(	(	PUNCT
iajs-2969	192	14	y	y	PROPN
iajs-2969	192	15	,	,	PUNCT
iajs-2969	192	16	ϣ2	ϣ2	PROPN
iajs-2969	192	17	)	)	PUNCT
iajs-2969	192	18	is	be	AUX
iajs-2969	192	19	said	say	VERB
iajs-2969	192	20	to	to	PART
iajs-2969	192	21	be	be	AUX
iajs-2969	192	22	(	(	PUNCT
iajs-2969	192	23	ϣ1	ϣ1	NOUN
iajs-2969	192	24	,	,	PUNCT
iajs-2969	192	25	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	192	26	-	-	ADJ
iajs-2969	192	27	p	p	ADJ
iajs-2969	192	28	-	-	PUNCT
iajs-2969	192	29	irresolute	irresolute	ADJ
iajs-2969	192	30	function	function	NOUN
iajs-2969	192	31	if	if	SCONJ
iajs-2969	192	32	the	the	DET
iajs-2969	192	33	inverse	inverse	ADJ
iajs-2969	192	34	image	image	NOUN
iajs-2969	192	35	of	of	ADP
iajs-2969	192	36	any	any	DET
iajs-2969	192	37	ϣ2	ϣ2	NOUN
iajs-2969	192	38	-	-	PUNCT
iajs-2969	192	39	semi	semi	ADJ
iajs-2969	192	40	-	-	ADJ
iajs-2969	192	41	p	p	ADJ
iajs-2969	192	42	-	-	PUNCT
iajs-2969	192	43	open	open	NOUN
iajs-2969	192	44	set	set	NOUN
iajs-2969	192	45	in	in	ADP
iajs-2969	192	46	y	y	PROPN
iajs-2969	192	47	is	be	AUX
iajs-2969	192	48	an	an	DET
iajs-2969	192	49	ϣ1	ϣ1	NOUN
iajs-2969	192	50	-	-	PUNCT
iajs-2969	192	51	semi	semi	ADJ
iajs-2969	192	52	-	-	ADJ
iajs-2969	192	53	p	p	ADJ
iajs-2969	192	54	-	-	PUNCT
iajs-2969	192	55	open	open	NOUN
iajs-2969	192	56	set	set	NOUN
iajs-2969	192	57	in	in	ADP
iajs-2969	192	58	z	z	NOUN
iajs-2969	192	59	theorem	theorem	VERB
iajs-2969	192	60	4.7	4.7	NUM
iajs-2969	192	61	a	a	DET
iajs-2969	192	62	function	function	NOUN
iajs-2969	193	1	𝑓	𝑓	NOUN
iajs-2969	193	2	:	:	PUNCT
iajs-2969	193	3	(	(	PUNCT
iajs-2969	193	4	z	z	NOUN
iajs-2969	193	5	,	,	PUNCT
iajs-2969	193	6	ϣ1	ϣ1	PROPN
iajs-2969	193	7	)	)	PUNCT
iajs-2969	193	8	→	→	SYM
iajs-2969	193	9	(	(	PUNCT
iajs-2969	193	10	y	y	PROPN
iajs-2969	193	11	,	,	PUNCT
iajs-2969	193	12	ϣ2	ϣ2	PROPN
iajs-2969	193	13	)	)	PUNCT
iajs-2969	193	14	is	be	AUX
iajs-2969	193	15	an	an	DET
iajs-2969	193	16	(	(	PUNCT
iajs-2969	193	17	ϣ1	ϣ1	NOUN
iajs-2969	193	18	,	,	PUNCT
iajs-2969	193	19	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	193	20	-	-	ADJ
iajs-2969	193	21	p	p	ADJ
iajs-2969	193	22	-	-	PUNCT
iajs-2969	193	23	irresolute	irresolute	ADJ
iajs-2969	193	24	function	function	NOUN
iajs-2969	193	25	the	the	DET
iajs-2969	193	26	inverse	inverse	NOUN
iajs-2969	193	27	image	image	NOUN
iajs-2969	193	28	of	of	ADP
iajs-2969	193	29	each	each	DET
iajs-2969	193	30	ϣ2	ϣ2	NOUN
iajs-2969	193	31	-	-	PUNCT
iajs-2969	193	32	semi	semi	ADJ
iajs-2969	193	33	-	-	ADJ
iajs-2969	193	34	p	p	ADJ
iajs-2969	193	35	-	-	PUNCT
iajs-2969	193	36	closed	close	VERB
iajs-2969	193	37	set	set	NOUN
iajs-2969	193	38	in	in	ADP
iajs-2969	193	39	y	y	PROPN
iajs-2969	193	40	is	be	AUX
iajs-2969	193	41	an	an	DET
iajs-2969	193	42	ϣ1	ϣ1	NOUN
iajs-2969	193	43	-	-	PUNCT
iajs-2969	193	44	semi	semi	ADJ
iajs-2969	193	45	-	-	ADJ
iajs-2969	193	46	p	p	ADJ
iajs-2969	193	47	-	-	PUNCT
iajs-2969	193	48	closed	close	VERB
iajs-2969	193	49	set	set	NOUN
iajs-2969	193	50	in	in	ADP
iajs-2969	193	51	z	z	PROPN
iajs-2969	193	52	.	.	PUNCT
iajs-2969	194	1	ihjpas	ihjpas	PROPN
iajs-2969	194	2	.	.	PUNCT
iajs-2969	195	1	36(1)2023	36(1)2023	NUM
iajs-2969	195	2	341	341	NUM
iajs-2969	195	3	proof	proof	NOUN
iajs-2969	195	4	:	:	PUNCT
iajs-2969	195	5	the	the	DET
iajs-2969	195	6	"	"	PUNCT
iajs-2969	195	7	if	if	SCONJ
iajs-2969	195	8	"	"	PUNCT
iajs-2969	195	9	part	part	NOUN
iajs-2969	195	10	.	.	PUNCT
iajs-2969	196	1	let	let	VERB
iajs-2969	196	2	f	f	PRON
iajs-2969	196	3	be	be	AUX
iajs-2969	196	4	any	any	DET
iajs-2969	196	5	ϣ2	ϣ2	NOUN
iajs-2969	196	6	-	-	PUNCT
iajs-2969	196	7	semi	semi	ADJ
iajs-2969	196	8	-	-	ADJ
iajs-2969	196	9	p	p	ADJ
iajs-2969	196	10	-	-	PUNCT
iajs-2969	196	11	closed	close	VERB
iajs-2969	196	12	set	set	NOUN
iajs-2969	196	13	in	in	ADP
iajs-2969	196	14	y	y	PROPN
iajs-2969	196	15	,	,	PUNCT
iajs-2969	196	16	thus	thus	ADV
iajs-2969	196	17	(	(	PUNCT
iajs-2969	196	18	y	y	X
iajs-2969	196	19	–	–	PUNCT
iajs-2969	196	20	f	f	X
iajs-2969	196	21	)	)	PUNCT
iajs-2969	196	22	is	be	AUX
iajs-2969	196	23	an	an	DET
iajs-2969	196	24	ϣ2	ϣ2	NOUN
iajs-2969	196	25	-	-	PUNCT
iajs-2969	196	26	semi	semi	ADJ
iajs-2969	196	27	-	-	ADJ
iajs-2969	196	28	p	p	ADJ
iajs-2969	196	29	-	-	PUNCT
iajs-2969	196	30	open	open	NOUN
iajs-2969	196	31	set	set	NOUN
iajs-2969	196	32	in	in	ADP
iajs-2969	196	33	y	y	PROPN
iajs-2969	196	34	,	,	PUNCT
iajs-2969	196	35	then	then	ADV
iajs-2969	196	36	f	f	PROPN
iajs-2969	197	1	−1(y	−1(y	X
iajs-2969	198	1	−	−	PROPN
iajs-2969	199	1	f	f	X
iajs-2969	199	2	)	)	PUNCT
iajs-2969	199	3	is	be	AUX
iajs-2969	199	4	an	an	DET
iajs-2969	199	5	ϣ1	ϣ1	NOUN
iajs-2969	199	6	-	-	PUNCT
iajs-2969	199	7	semi	semi	ADJ
iajs-2969	199	8	-	-	ADJ
iajs-2969	199	9	p	p	ADJ
iajs-2969	199	10	-	-	PUNCT
iajs-2969	199	11	open	open	NOUN
iajs-2969	199	12	set	set	NOUN
iajs-2969	199	13	in	in	ADP
iajs-2969	199	14	z	z	PROPN
iajs-2969	199	15	(	(	PUNCT
iajs-2969	199	16	since	since	SCONJ
iajs-2969	199	17	f	f	PROPN
iajs-2969	199	18	is	be	AUX
iajs-2969	199	19	an	an	DET
iajs-2969	199	20	(	(	PUNCT
iajs-2969	199	21	ϣ1	ϣ1	NOUN
iajs-2969	199	22	,	,	PUNCT
iajs-2969	199	23	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	199	24	-	-	ADJ
iajs-2969	199	25	p	p	ADJ
iajs-2969	199	26	-	-	PUNCT
iajs-2969	199	27	irresolute	irresolute	ADJ
iajs-2969	199	28	function	function	NOUN
iajs-2969	199	29	)	)	PUNCT
iajs-2969	199	30	,	,	PUNCT
iajs-2969	199	31	but	but	CCONJ
iajs-2969	199	32	f	f	X
iajs-2969	199	33	−1(y	−1(y	X
iajs-2969	200	1	−	−	PROPN
iajs-2969	200	2	f	f	X
iajs-2969	200	3	)	)	PUNCT
iajs-2969	200	4	=	=	PUNCT
iajs-2969	201	1	z	z	NOUN
iajs-2969	202	1	−	−	PROPN
iajs-2969	202	2	f	f	PROPN
iajs-2969	202	3	−1(f	−1(f	PROPN
iajs-2969	202	4	)	)	PUNCT
iajs-2969	202	5	,	,	PUNCT
iajs-2969	202	6	therefore	therefore	ADV
iajs-2969	202	7	f	f	PROPN
iajs-2969	202	8	−1(f	−1(f	PROPN
iajs-2969	202	9	)	)	PUNCT
iajs-2969	202	10	is	be	AUX
iajs-2969	202	11	an	an	DET
iajs-2969	202	12	ϣ1	ϣ1	NOUN
iajs-2969	202	13	-	-	PUNCT
iajs-2969	202	14	semi	semi	ADJ
iajs-2969	202	15	-	-	ADJ
iajs-2969	202	16	p	p	ADJ
iajs-2969	202	17	-	-	PUNCT
iajs-2969	202	18	closed	close	VERB
iajs-2969	202	19	set	set	NOUN
iajs-2969	202	20	.	.	PUNCT
iajs-2969	203	1	the	the	DET
iajs-2969	203	2	"	"	PUNCT
iajs-2969	203	3	only	only	ADV
iajs-2969	203	4	if	if	SCONJ
iajs-2969	203	5	"	"	PUNCT
iajs-2969	203	6	part	part	NOUN
iajs-2969	203	7	.	.	PUNCT
iajs-2969	204	1	let	let	VERB
iajs-2969	204	2	ħ	ħ	NOUN
iajs-2969	204	3	be	be	AUX
iajs-2969	204	4	any	any	DET
iajs-2969	204	5	ϣ2	ϣ2	NOUN
iajs-2969	204	6	-	-	PUNCT
iajs-2969	204	7	semi	semi	ADJ
iajs-2969	204	8	-	-	ADJ
iajs-2969	204	9	p	p	ADJ
iajs-2969	204	10	-	-	PUNCT
iajs-2969	204	11	open	open	NOUN
iajs-2969	204	12	set	set	NOUN
iajs-2969	204	13	in	in	ADP
iajs-2969	204	14	y	y	PROPN
iajs-2969	204	15	,	,	PUNCT
iajs-2969	204	16	thus	thus	ADV
iajs-2969	204	17	(	(	PUNCT
iajs-2969	204	18	y	y	X
iajs-2969	204	19	–	–	PUNCT
iajs-2969	204	20	ħ	ħ	NOUN
iajs-2969	204	21	)	)	PUNCT
iajs-2969	204	22	is	be	AUX
iajs-2969	204	23	an	an	DET
iajs-2969	204	24	ϣ1	ϣ1	NOUN
iajs-2969	204	25	-	-	PUNCT
iajs-2969	204	26	semi	semi	ADJ
iajs-2969	204	27	-	-	ADJ
iajs-2969	204	28	p	p	ADJ
iajs-2969	204	29	-	-	PUNCT
iajs-2969	204	30	closed	close	VERB
iajs-2969	204	31	set	set	NOUN
iajs-2969	204	32	in	in	ADP
iajs-2969	204	33	y	y	PROPN
iajs-2969	205	1	then	then	ADV
iajs-2969	205	2	f	f	PROPN
iajs-2969	206	1	−1(y	−1(y	X
iajs-2969	207	1	−	−	PROPN
iajs-2969	208	1	ħ	ħ	NOUN
iajs-2969	208	2	)	)	PUNCT
iajs-2969	208	3	is	be	AUX
iajs-2969	208	4	an	an	DET
iajs-2969	208	5	ϣ1	ϣ1	NOUN
iajs-2969	208	6	-	-	PUNCT
iajs-2969	208	7	semi	semi	ADJ
iajs-2969	208	8	-	-	ADJ
iajs-2969	208	9	p	p	ADJ
iajs-2969	208	10	-	-	PUNCT
iajs-2969	208	11	closed	close	VERB
iajs-2969	208	12	set	set	NOUN
iajs-2969	208	13	in	in	ADP
iajs-2969	208	14	z	z	PROPN
iajs-2969	208	15	(	(	PUNCT
iajs-2969	208	16	by	by	ADP
iajs-2969	208	17	hypothesis	hypothesis	NOUN
iajs-2969	208	18	)	)	PUNCT
iajs-2969	208	19	,	,	PUNCT
iajs-2969	208	20	but	but	CCONJ
iajs-2969	208	21	f	f	X
iajs-2969	208	22	−1(y	−1(y	X
iajs-2969	209	1	−	−	PROPN
iajs-2969	209	2	ħ	ħ	NOUN
iajs-2969	209	3	)	)	PUNCT
iajs-2969	209	4	=	=	PUNCT
iajs-2969	209	5	z	z	NOUN
iajs-2969	210	1	−	−	PROPN
iajs-2969	210	2	f	f	PROPN
iajs-2969	210	3	−1(ħ	−1(ħ	NOUN
iajs-2969	210	4	)	)	PUNCT
iajs-2969	210	5	,	,	PUNCT
iajs-2969	210	6	then	then	ADV
iajs-2969	210	7	f	f	PROPN
iajs-2969	210	8	−1(ħ	−1(ħ	NOUN
iajs-2969	210	9	)	)	PUNCT
iajs-2969	210	10	is	be	AUX
iajs-2969	210	11	an	an	DET
iajs-2969	210	12	ϣ1	ϣ1	NOUN
iajs-2969	210	13	-	-	PUNCT
iajs-2969	210	14	semi	semi	ADJ
iajs-2969	210	15	-	-	ADJ
iajs-2969	210	16	p	p	ADJ
iajs-2969	210	17	-	-	PUNCT
iajs-2969	210	18	open	open	NOUN
iajs-2969	210	19	set	set	NOUN
iajs-2969	210	20	in	in	ADP
iajs-2969	210	21	z	z	PROPN
iajs-2969	210	22	,	,	PUNCT
iajs-2969	210	23	therefore	therefore	ADV
iajs-2969	210	24	f	f	PROPN
iajs-2969	210	25	is	be	AUX
iajs-2969	210	26	an	an	DET
iajs-2969	210	27	(	(	PUNCT
iajs-2969	210	28	ϣ1	ϣ1	NOUN
iajs-2969	210	29	,	,	PUNCT
iajs-2969	210	30	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	210	31	-	-	ADJ
iajs-2969	210	32	p	p	ADJ
iajs-2969	210	33	-	-	PUNCT
iajs-2969	210	34	irresolute	irresolute	ADJ
iajs-2969	210	35	function	function	NOUN
iajs-2969	210	36	.	.	PUNCT
iajs-2969	211	1	proposition	proposition	NOUN
iajs-2969	211	2	4.8	4.8	NUM
iajs-2969	211	3	every	every	PRON
iajs-2969	211	4	(	(	PUNCT
iajs-2969	211	5	ϣ1	ϣ1	NOUN
iajs-2969	211	6	,	,	PUNCT
iajs-2969	211	7	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	211	8	-	-	ADJ
iajs-2969	211	9	p	p	ADJ
iajs-2969	211	10	-	-	PUNCT
iajs-2969	211	11	irresolute	irresolute	ADJ
iajs-2969	212	1	function	function	NOUN
iajs-2969	212	2	is	be	AUX
iajs-2969	212	3	an	an	DET
iajs-2969	212	4	(	(	PUNCT
iajs-2969	212	5	ϣ1	ϣ1	NOUN
iajs-2969	212	6	,	,	PUNCT
iajs-2969	212	7	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	212	8	-	-	ADJ
iajs-2969	212	9	p	p	ADJ
iajs-2969	212	10	-	-	PUNCT
iajs-2969	212	11	continuous	continuous	ADJ
iajs-2969	212	12	function	function	NOUN
iajs-2969	212	13	.	.	PUNCT
iajs-2969	213	1	proof	proof	NOUN
iajs-2969	213	2	:	:	PUNCT
iajs-2969	213	3	let	let	VERB
iajs-2969	213	4	𝑓	𝑓	PRON
iajs-2969	213	5	be	be	AUX
iajs-2969	213	6	any	any	DET
iajs-2969	213	7	(	(	PUNCT
iajs-2969	213	8	ϣ1	ϣ1	NOUN
iajs-2969	213	9	,	,	PUNCT
iajs-2969	213	10	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	213	11	-	-	ADJ
iajs-2969	213	12	p	p	ADJ
iajs-2969	213	13	-	-	PUNCT
iajs-2969	213	14	irresolute	irresolute	ADJ
iajs-2969	213	15	function	function	NOUN
iajs-2969	213	16	from	from	ADP
iajs-2969	213	17	(	(	PUNCT
iajs-2969	213	18	z	z	NOUN
iajs-2969	213	19	,	,	PUNCT
iajs-2969	213	20	ϣ1	ϣ1	PROPN
iajs-2969	213	21	)	)	PUNCT
iajs-2969	213	22	into	into	ADP
iajs-2969	213	23	(	(	PUNCT
iajs-2969	213	24	y	y	PROPN
iajs-2969	213	25	,	,	PUNCT
iajs-2969	213	26	ϣ2	ϣ2	PROPN
iajs-2969	213	27	)	)	PUNCT
iajs-2969	213	28	.	.	PUNCT
iajs-2969	214	1	let	let	VERB
iajs-2969	214	2	ħ	ħ	NOUN
iajs-2969	214	3	by	by	ADP
iajs-2969	214	4	any	any	DET
iajs-2969	214	5	ϣ2open	ϣ2open	NOUN
iajs-2969	214	6	in	in	ADP
iajs-2969	214	7	y	y	PROPN
iajs-2969	214	8	,	,	PUNCT
iajs-2969	214	9	thus	thus	ADV
iajs-2969	214	10	ħ	ħ	PRON
iajs-2969	214	11	is	be	AUX
iajs-2969	214	12	an	an	DET
iajs-2969	214	13	ϣ2	ϣ2	NOUN
iajs-2969	214	14	-	-	PUNCT
iajs-2969	214	15	semi	semi	NOUN
iajs-2969	214	16	-	-	ADJ
iajs-2969	214	17	popen	popen	ADJ
iajs-2969	214	18	set	set	NOUN
iajs-2969	214	19	(	(	PUNCT
iajs-2969	214	20	corollary	corollary	NOUN
iajs-2969	214	21	3.11	3.11	NUM
iajs-2969	214	22	)	)	PUNCT
iajs-2969	214	23	,	,	PUNCT
iajs-2969	214	24	then	then	ADV
iajs-2969	214	25	f	f	PROPN
iajs-2969	214	26	−1(ħ	−1(ħ	NOUN
iajs-2969	214	27	)	)	PUNCT
iajs-2969	214	28	is	be	AUX
iajs-2969	214	29	an	an	DET
iajs-2969	214	30	ϣ1	ϣ1	NOUN
iajs-2969	214	31	-	-	PUNCT
iajs-2969	214	32	semi	semi	ADJ
iajs-2969	214	33	-	-	ADJ
iajs-2969	214	34	p	p	ADJ
iajs-2969	214	35	-	-	PUNCT
iajs-2969	214	36	open	open	NOUN
iajs-2969	214	37	set	set	NOUN
iajs-2969	214	38	in	in	ADP
iajs-2969	214	39	z(since	z(since	NOUN
iajs-2969	214	40	𝑓	𝑓	PROPN
iajs-2969	214	41	is	be	AUX
iajs-2969	214	42	(	(	PUNCT
iajs-2969	214	43	ϣ1	ϣ1	NOUN
iajs-2969	214	44	,	,	PUNCT
iajs-2969	214	45	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	214	46	-	-	ADJ
iajs-2969	214	47	p	p	ADJ
iajs-2969	214	48	-	-	PUNCT
iajs-2969	214	49	irresolute	irresolute	ADJ
iajs-2969	214	50	function	function	NOUN
iajs-2969	214	51	)	)	PUNCT
iajs-2969	214	52	,	,	PUNCT
iajs-2969	214	53	therefore	therefore	ADV
iajs-2969	214	54	𝑓	𝑓	PRON
iajs-2969	214	55	is	be	AUX
iajs-2969	214	56	an	an	DET
iajs-2969	214	57	(	(	PUNCT
iajs-2969	214	58	ϣ1	ϣ1	NOUN
iajs-2969	214	59	,	,	PUNCT
iajs-2969	214	60	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	214	61	-	-	ADJ
iajs-2969	214	62	pcontinuous	pcontinuous	ADJ
iajs-2969	214	63	function	function	NOUN
iajs-2969	214	64	.	.	PUNCT
iajs-2969	215	1	remark	remark	VERB
iajs-2969	215	2	4.9	4.9	NUM
iajs-2969	215	3	the	the	DET
iajs-2969	215	4	reverse	reverse	NOUN
iajs-2969	215	5	of	of	ADP
iajs-2969	215	6	proposition	proposition	NOUN
iajs-2969	215	7	4.7	4.7	NUM
iajs-2969	215	8	is	be	AUX
iajs-2969	215	9	not	not	PART
iajs-2969	215	10	correct	correct	ADJ
iajs-2969	215	11	in	in	ADP
iajs-2969	215	12	general	general	ADJ
iajs-2969	215	13	as	as	SCONJ
iajs-2969	215	14	we	we	PRON
iajs-2969	215	15	show	show	VERB
iajs-2969	215	16	in	in	ADP
iajs-2969	215	17	the	the	DET
iajs-2969	215	18	following	follow	VERB
iajs-2969	215	19	example	example	NOUN
iajs-2969	215	20	:	:	PUNCT
iajs-2969	215	21	example	example	NOUN
iajs-2969	215	22	let	let	VERB
iajs-2969	215	23	z	z	NOUN
iajs-2969	215	24	=	=	PUNCT
iajs-2969	215	25	{	{	PUNCT
iajs-2969	215	26	1,2,3,4	1,2,3,4	NUM
iajs-2969	215	27	}	}	PUNCT
iajs-2969	215	28	,	,	PUNCT
iajs-2969	215	29	ϣ1	ϣ1	NOUN
iajs-2969	215	30	=	=	SYM
iajs-2969	215	31	{	{	PUNCT
iajs-2969	215	32	z	z	PROPN
iajs-2969	215	33	,	,	PUNCT
iajs-2969	215	34	ø	ø	PROPN
iajs-2969	215	35	,	,	PUNCT
iajs-2969	215	36	{	{	PUNCT
iajs-2969	215	37	1	1	NUM
iajs-2969	215	38	}	}	PUNCT
iajs-2969	215	39	,	,	PUNCT
iajs-2969	215	40	{	{	PUNCT
iajs-2969	215	41	4	4	NUM
iajs-2969	215	42	}	}	PUNCT
iajs-2969	215	43	,	,	PUNCT
iajs-2969	215	44	{	{	PUNCT
iajs-2969	215	45	1,4	1,4	NUM
iajs-2969	215	46	}	}	PUNCT
iajs-2969	215	47	}	}	PUNCT
iajs-2969	215	48	,	,	PUNCT
iajs-2969	215	49	ϣ1	ϣ1	PROPN
iajs-2969	215	50	−	−	NOUN
iajs-2969	215	51	po(z	po(z	NOUN
iajs-2969	215	52	)	)	PUNCT
iajs-2969	215	53	=	=	PRON
iajs-2969	215	54	{	{	PUNCT
iajs-2969	215	55	z	z	PROPN
iajs-2969	215	56	,	,	PUNCT
iajs-2969	215	57	ø	ø	PROPN
iajs-2969	215	58	,	,	PUNCT
iajs-2969	215	59	{	{	PUNCT
iajs-2969	215	60	1	1	NUM
iajs-2969	215	61	}	}	PUNCT
iajs-2969	215	62	,	,	PUNCT
iajs-2969	215	63	{	{	PUNCT
iajs-2969	215	64	4	4	NUM
iajs-2969	215	65	}	}	PUNCT
iajs-2969	215	66	,	,	PUNCT
iajs-2969	215	67	{	{	PUNCT
iajs-2969	215	68	1,4	1,4	NUM
iajs-2969	215	69	}	}	PUNCT
iajs-2969	215	70	,	,	PUNCT
iajs-2969	215	71	{	{	PUNCT
iajs-2969	215	72	1,2,4	1,2,4	NUM
iajs-2969	215	73	}	}	PUNCT
iajs-2969	215	74	,	,	PUNCT
iajs-2969	215	75	{	{	PUNCT
iajs-2969	215	76	1,3,4	1,3,4	NUM
iajs-2969	215	77	}	}	PUNCT
iajs-2969	215	78	}	}	PUNCT
iajs-2969	215	79	,	,	PUNCT
iajs-2969	215	80	and	and	CCONJ
iajs-2969	215	81	ϣ1	ϣ1	VERB
iajs-2969	215	82	−	−	PROPN
iajs-2969	215	83	spo(z	spo(z	PROPN
iajs-2969	215	84	)	)	PUNCT
iajs-2969	215	85	=	=	PUNCT
iajs-2969	215	86	ϣ1	ϣ1	NOUN
iajs-2969	215	87	−	−	NOUN
iajs-2969	215	88	po(z	po(z	NOUN
iajs-2969	215	89	)	)	PUNCT
iajs-2969	215	90	∪	∪	X
iajs-2969	215	91	{	{	PUNCT
iajs-2969	215	92	{	{	PUNCT
iajs-2969	215	93	1,2	1,2	NUM
iajs-2969	215	94	}	}	PUNCT
iajs-2969	215	95	,	,	PUNCT
iajs-2969	215	96	{	{	PUNCT
iajs-2969	215	97	1,3	1,3	NUM
iajs-2969	215	98	}	}	PUNCT
iajs-2969	215	99	,	,	PUNCT
iajs-2969	215	100	{	{	PUNCT
iajs-2969	215	101	2,4	2,4	NUM
iajs-2969	215	102	}	}	PUNCT
iajs-2969	215	103	,	,	PUNCT
iajs-2969	215	104	{	{	PUNCT
iajs-2969	215	105	3,4	3,4	NUM
iajs-2969	215	106	}	}	PUNCT
iajs-2969	215	107	,	,	PUNCT
iajs-2969	215	108	{	{	PUNCT
iajs-2969	215	109	1,2,3	1,2,3	NUM
iajs-2969	215	110	}	}	PUNCT
iajs-2969	215	111	,	,	PUNCT
iajs-2969	215	112	{	{	PUNCT
iajs-2969	215	113	1,2,4	1,2,4	NUM
iajs-2969	215	114	}	}	PUNCT
iajs-2969	215	115	,	,	PUNCT
iajs-2969	215	116	{	{	PUNCT
iajs-2969	215	117	2,3,4	2,3,4	NUM
iajs-2969	215	118	}	}	PUNCT
iajs-2969	215	119	,	,	PUNCT
iajs-2969	215	120	}	}	PUNCT
iajs-2969	215	121	.	.	PUNCT
iajs-2969	216	1	let	let	VERB
iajs-2969	216	2	y	y	PROPN
iajs-2969	216	3	=	=	PUNCT
iajs-2969	216	4	{	{	PUNCT
iajs-2969	216	5	a	a	PRON
iajs-2969	216	6	,	,	PUNCT
iajs-2969	216	7	b	b	NOUN
iajs-2969	216	8	,	,	PUNCT
iajs-2969	216	9	c	c	NOUN
iajs-2969	216	10	,	,	PUNCT
iajs-2969	216	11	d	d	NOUN
iajs-2969	216	12	}	}	PUNCT
iajs-2969	216	13	,	,	PUNCT
iajs-2969	216	14	ϣ2	ϣ2	PROPN
iajs-2969	216	15	=	=	SYM
iajs-2969	216	16	{	{	PUNCT
iajs-2969	216	17	∅	∅	NOUN
iajs-2969	216	18	,	,	PUNCT
iajs-2969	216	19	{	{	PUNCT
iajs-2969	216	20	b	b	NOUN
iajs-2969	216	21	,	,	PUNCT
iajs-2969	216	22	d	d	NOUN
iajs-2969	216	23	}	}	PUNCT
iajs-2969	216	24	}	}	PUNCT
iajs-2969	216	25	,	,	PUNCT
iajs-2969	216	26	ϣ2	ϣ2	PROPN
iajs-2969	216	27	−	−	PROPN
iajs-2969	216	28	po(y	po(y	PUNCT
iajs-2969	216	29	)	)	PUNCT
iajs-2969	216	30	=	=	SYM
iajs-2969	216	31	{	{	PUNCT
iajs-2969	216	32	∅	∅	NOUN
iajs-2969	216	33	,	,	PUNCT
iajs-2969	216	34	{	{	PUNCT
iajs-2969	216	35	b	b	NOUN
iajs-2969	216	36	,	,	PUNCT
iajs-2969	216	37	d	d	NOUN
iajs-2969	216	38	}	}	PUNCT
iajs-2969	216	39	,	,	PUNCT
iajs-2969	216	40	{	{	PUNCT
iajs-2969	216	41	b	b	X
iajs-2969	216	42	}	}	PUNCT
iajs-2969	216	43	,	,	PUNCT
iajs-2969	216	44	{	{	PUNCT
iajs-2969	216	45	d	d	NOUN
iajs-2969	216	46	}	}	PUNCT
iajs-2969	216	47	}	}	PUNCT
iajs-2969	216	48	,	,	PUNCT
iajs-2969	216	49	ϣ2	ϣ2	PROPN
iajs-2969	216	50	−	−	PROPN
iajs-2969	216	51	spo(y	spo(y	PROPN
iajs-2969	216	52	)	)	PUNCT
iajs-2969	216	53	=	=	SYM
iajs-2969	217	1	ℙ(y	ℙ(y	NUM
iajs-2969	217	2	)	)	PUNCT
iajs-2969	217	3	(	(	PUNCT
iajs-2969	217	4	the	the	DET
iajs-2969	217	5	power	power	NOUN
iajs-2969	217	6	set	set	NOUN
iajs-2969	217	7	of	of	ADP
iajs-2969	217	8	y	y	PROPN
iajs-2969	217	9	)	)	PUNCT
iajs-2969	217	10	.	.	PUNCT
iajs-2969	218	1	define	define	VERB
iajs-2969	218	2	𝑓	𝑓	DET
iajs-2969	218	3	∶	∶	NOUN
iajs-2969	218	4	(	(	PUNCT
iajs-2969	218	5	z	z	NOUN
iajs-2969	218	6	,	,	PUNCT
iajs-2969	218	7	ϣ1	ϣ1	PROPN
iajs-2969	218	8	)	)	PUNCT
iajs-2969	218	9	→	→	SYM
iajs-2969	218	10	(	(	PUNCT
iajs-2969	218	11	y	y	PROPN
iajs-2969	218	12	,	,	PUNCT
iajs-2969	218	13	ϣ2	ϣ2	PROPN
iajs-2969	218	14	)	)	PUNCT
iajs-2969	219	1	such	such	ADJ
iajs-2969	219	2	that	that	SCONJ
iajs-2969	219	3	𝑓(1	𝑓(1	NOUN
iajs-2969	219	4	)	)	PUNCT
iajs-2969	219	5	=	=	SYM
iajs-2969	219	6	𝑓(2	𝑓(2	ADJ
iajs-2969	219	7	)	)	PUNCT
iajs-2969	219	8	=	=	PRON
iajs-2969	219	9	{	{	PUNCT
iajs-2969	219	10	𝑑	𝑑	NOUN
iajs-2969	219	11	}	}	PUNCT
iajs-2969	219	12	,	,	PUNCT
iajs-2969	219	13	𝑓(3	𝑓(3	PROPN
iajs-2969	219	14	)	)	PUNCT
iajs-2969	219	15	=	=	PRON
iajs-2969	219	16	{	{	PUNCT
iajs-2969	219	17	𝑏	𝑏	NOUN
iajs-2969	219	18	}	}	PUNCT
iajs-2969	219	19	𝑓	𝑓	PROPN
iajs-2969	219	20	is	be	AUX
iajs-2969	219	21	an	an	DET
iajs-2969	219	22	(	(	PUNCT
iajs-2969	219	23	ϣ1	ϣ1	NOUN
iajs-2969	219	24	,	,	PUNCT
iajs-2969	219	25	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	219	26	-	-	ADJ
iajs-2969	219	27	p	p	ADJ
iajs-2969	219	28	-	-	PUNCT
iajs-2969	219	29	continuous	continuous	ADJ
iajs-2969	219	30	function	function	NOUN
iajs-2969	219	31	.	.	PUNCT
iajs-2969	220	1	but	but	CCONJ
iajs-2969	220	2	not	not	PART
iajs-2969	220	3	(	(	PUNCT
iajs-2969	220	4	ϣ1	ϣ1	NOUN
iajs-2969	220	5	,	,	PUNCT
iajs-2969	220	6	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	220	7	-	-	ADJ
iajs-2969	220	8	p	p	ADJ
iajs-2969	220	9	-	-	PUNCT
iajs-2969	220	10	irresolute	irresolute	ADJ
iajs-2969	220	11	function	function	NOUN
iajs-2969	220	12	,	,	PUNCT
iajs-2969	220	13	since	since	SCONJ
iajs-2969	220	14	{	{	PUNCT
iajs-2969	220	15	b	b	NOUN
iajs-2969	220	16	}	}	PUNCT
iajs-2969	220	17	is	be	AUX
iajs-2969	220	18	an	an	DET
iajs-2969	220	19	ϣ2	ϣ2	NOUN
iajs-2969	220	20	-	-	PUNCT
iajs-2969	220	21	semi	semi	ADJ
iajs-2969	220	22	-	-	ADJ
iajs-2969	220	23	p	p	ADJ
iajs-2969	220	24	-	-	PUNCT
iajs-2969	220	25	open	open	NOUN
iajs-2969	220	26	set	set	NOUN
iajs-2969	220	27	in	in	ADP
iajs-2969	220	28	y	y	PROPN
iajs-2969	220	29	,	,	PUNCT
iajs-2969	220	30	but	but	CCONJ
iajs-2969	220	31	f	f	PROPN
iajs-2969	220	32	−1	−1	NOUN
iajs-2969	220	33	(	(	PUNCT
iajs-2969	220	34	{	{	PUNCT
iajs-2969	220	35	b	b	NOUN
iajs-2969	220	36	}	}	PUNCT
iajs-2969	220	37	)	)	PUNCT
iajs-2969	220	38	=	=	PRON
iajs-2969	221	1	{	{	PUNCT
iajs-2969	221	2	3	3	NUM
iajs-2969	221	3	}	}	PUNCT
iajs-2969	221	4	is	be	AUX
iajs-2969	221	5	not	not	PART
iajs-2969	221	6	an	an	DET
iajs-2969	221	7	ϣ1	ϣ1	NOUN
iajs-2969	221	8	-	-	PUNCT
iajs-2969	221	9	semi	semi	ADJ
iajs-2969	221	10	-	-	ADJ
iajs-2969	221	11	p	p	ADJ
iajs-2969	221	12	-	-	PUNCT
iajs-2969	221	13	open	open	NOUN
iajs-2969	221	14	set	set	NOUN
iajs-2969	221	15	in	in	ADP
iajs-2969	221	16	z.	z.	PROPN
iajs-2969	221	17	proposition	proposition	NOUN
iajs-2969	221	18	4.10	4.10	NUM
iajs-2969	221	19	every	every	PRON
iajs-2969	221	20	(	(	PUNCT
iajs-2969	221	21	ϣ1	ϣ1	NOUN
iajs-2969	221	22	,	,	PUNCT
iajs-2969	221	23	ϣ2)-continuous	ϣ2)-continuous	ADJ
iajs-2969	221	24	function	function	NOUN
iajs-2969	221	25	is	be	AUX
iajs-2969	221	26	an	an	DET
iajs-2969	221	27	(	(	PUNCT
iajs-2969	221	28	ϣ1	ϣ1	NOUN
iajs-2969	221	29	,	,	PUNCT
iajs-2969	221	30	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	221	31	-	-	ADJ
iajs-2969	221	32	p	p	ADJ
iajs-2969	221	33	-	-	PUNCT
iajs-2969	221	34	continuous	continuous	ADJ
iajs-2969	221	35	function	function	NOUN
iajs-2969	221	36	.	.	PUNCT
iajs-2969	222	1	proof	proof	NOUN
iajs-2969	222	2	:	:	PUNCT
iajs-2969	222	3	let	let	VERB
iajs-2969	222	4	𝑓	𝑓	PRON
iajs-2969	222	5	be	be	AUX
iajs-2969	222	6	any	any	DET
iajs-2969	222	7	(	(	PUNCT
iajs-2969	222	8	ϣ1	ϣ1	NOUN
iajs-2969	222	9	,	,	PUNCT
iajs-2969	222	10	ϣ2)continuous	ϣ2)continuous	ADJ
iajs-2969	222	11	function	function	NOUN
iajs-2969	222	12	from	from	ADP
iajs-2969	222	13	(	(	PUNCT
iajs-2969	222	14	z	z	NOUN
iajs-2969	222	15	,	,	PUNCT
iajs-2969	222	16	ϣ1	ϣ1	PROPN
iajs-2969	222	17	)	)	PUNCT
iajs-2969	222	18	into	into	ADP
iajs-2969	222	19	(	(	PUNCT
iajs-2969	222	20	y	y	PROPN
iajs-2969	222	21	,	,	PUNCT
iajs-2969	222	22	ϣ2	ϣ2	PROPN
iajs-2969	222	23	)	)	PUNCT
iajs-2969	222	24	.	.	PUNCT
iajs-2969	223	1	let	let	VERB
iajs-2969	223	2	ħ	ħ	NOUN
iajs-2969	223	3	by	by	ADP
iajs-2969	223	4	any	any	DET
iajs-2969	223	5	ϣ2	ϣ2	NOUN
iajs-2969	223	6	-	-	PUNCT
iajs-2969	223	7	open	open	ADJ
iajs-2969	223	8	in	in	ADP
iajs-2969	223	9	y	y	PROPN
iajs-2969	223	10	,	,	PUNCT
iajs-2969	223	11	it	it	PRON
iajs-2969	223	12	follows	follow	VERB
iajs-2969	223	13	from	from	ADP
iajs-2969	223	14	definition	definition	NOUN
iajs-2969	223	15	4.1	4.1	NUM
iajs-2969	223	16	that	that	SCONJ
iajs-2969	223	17	f	f	PROPN
iajs-2969	223	18	−1(ħ	−1(ħ	NOUN
iajs-2969	223	19	)	)	PUNCT
iajs-2969	223	20	is	be	AUX
iajs-2969	223	21	an	an	DET
iajs-2969	223	22	ϣ1open	ϣ1open	ADJ
iajs-2969	223	23	set	set	NOUN
iajs-2969	223	24	in	in	ADP
iajs-2969	223	25	z	z	PROPN
iajs-2969	223	26	,	,	PUNCT
iajs-2969	223	27	but	but	CCONJ
iajs-2969	223	28	every	every	DET
iajs-2969	223	29	ϣ1open	ϣ1open	NOUN
iajs-2969	223	30	set	set	NOUN
iajs-2969	223	31	is	be	AUX
iajs-2969	223	32	an	an	DET
iajs-2969	223	33	ϣ1	ϣ1	NOUN
iajs-2969	223	34	-	-	PUNCT
iajs-2969	223	35	semi	semi	NOUN
iajs-2969	223	36	-	-	ADJ
iajs-2969	223	37	p	p	ADJ
iajs-2969	223	38	–	–	PUNCT
iajs-2969	223	39	open	open	ADJ
iajs-2969	223	40	.	.	PUNCT
iajs-2969	224	1	therefore	therefore	ADV
iajs-2969	224	2	𝑓	𝑓	PRON
iajs-2969	224	3	is	be	AUX
iajs-2969	224	4	an	an	DET
iajs-2969	224	5	(	(	PUNCT
iajs-2969	224	6	ϣ1	ϣ1	NOUN
iajs-2969	224	7	,	,	PUNCT
iajs-2969	224	8	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	224	9	-	-	ADJ
iajs-2969	224	10	p	p	ADJ
iajs-2969	224	11	-	-	PUNCT
iajs-2969	224	12	continuous	continuous	ADJ
iajs-2969	224	13	function	function	NOUN
iajs-2969	224	14	.	.	PUNCT
iajs-2969	225	1	ihjpas	ihjpas	PROPN
iajs-2969	225	2	.	.	PUNCT
iajs-2969	226	1	36(1)2023	36(1)2023	NUM
iajs-2969	226	2	342	342	NUM
iajs-2969	226	3	remark	remark	NOUN
iajs-2969	226	4	4.11	4.11	NUM
iajs-2969	226	5	the	the	DET
iajs-2969	226	6	reverse	reverse	NOUN
iajs-2969	226	7	of	of	ADP
iajs-2969	226	8	remark	remark	NOUN
iajs-2969	226	9	4.9	4.9	NUM
iajs-2969	226	10	is	be	AUX
iajs-2969	226	11	not	not	PART
iajs-2969	226	12	correct	correct	ADJ
iajs-2969	226	13	in	in	ADP
iajs-2969	226	14	general	general	ADJ
iajs-2969	226	15	as	as	SCONJ
iajs-2969	226	16	we	we	PRON
iajs-2969	226	17	show	show	VERB
iajs-2969	226	18	in	in	ADP
iajs-2969	226	19	the	the	DET
iajs-2969	226	20	following	follow	VERB
iajs-2969	226	21	example	example	NOUN
iajs-2969	226	22	:	:	PUNCT
iajs-2969	226	23	example	example	NOUN
iajs-2969	226	24	let	let	VERB
iajs-2969	226	25	z	z	NOUN
iajs-2969	226	26	=	=	PUNCT
iajs-2969	226	27	{	{	PUNCT
iajs-2969	226	28	1,2,3	1,2,3	NUM
iajs-2969	226	29	}	}	PUNCT
iajs-2969	226	30	,	,	PUNCT
iajs-2969	226	31	ϣ1	ϣ1	NOUN
iajs-2969	226	32	=	=	SYM
iajs-2969	226	33	{	{	PUNCT
iajs-2969	226	34	∅	∅	NOUN
iajs-2969	226	35	,	,	PUNCT
iajs-2969	226	36	{	{	PUNCT
iajs-2969	226	37	1	1	NUM
iajs-2969	226	38	}	}	PUNCT
iajs-2969	226	39	,	,	PUNCT
iajs-2969	226	40	{	{	PUNCT
iajs-2969	226	41	2	2	NUM
iajs-2969	226	42	}	}	PUNCT
iajs-2969	226	43	,	,	PUNCT
iajs-2969	226	44	{	{	PUNCT
iajs-2969	226	45	1,2	1,2	NUM
iajs-2969	226	46	}	}	PUNCT
iajs-2969	226	47	}	}	PUNCT
iajs-2969	226	48	,	,	PUNCT
iajs-2969	226	49	and	and	CCONJ
iajs-2969	226	50	ϣ1	ϣ1	VERB
iajs-2969	226	51	−	−	PROPN
iajs-2969	226	52	po(z	po(z	NUM
iajs-2969	226	53	)	)	PUNCT
iajs-2969	226	54	=	=	SYM
iajs-2969	226	55	{	{	PUNCT
iajs-2969	226	56	∅	∅	NOUN
iajs-2969	226	57	,	,	PUNCT
iajs-2969	226	58	{	{	PUNCT
iajs-2969	226	59	1	1	NUM
iajs-2969	226	60	}	}	PUNCT
iajs-2969	226	61	,	,	PUNCT
iajs-2969	226	62	{	{	PUNCT
iajs-2969	226	63	2	2	NUM
iajs-2969	226	64	}	}	PUNCT
iajs-2969	226	65	,	,	PUNCT
iajs-2969	226	66	{	{	PUNCT
iajs-2969	226	67	1,2	1,2	NUM
iajs-2969	226	68	}	}	PUNCT
iajs-2969	226	69	}	}	PUNCT
iajs-2969	226	70	,	,	PUNCT
iajs-2969	226	71	ϣ1	ϣ1	PROPN
iajs-2969	226	72	−	−	PROPN
iajs-2969	226	73	spo(z	spo(z	PROPN
iajs-2969	226	74	)	)	PUNCT
iajs-2969	226	75	=	=	SYM
iajs-2969	226	76	ℙ(𝑍	ℙ(𝑍	X
iajs-2969	226	77	)	)	PUNCT
iajs-2969	226	78	(	(	PUNCT
iajs-2969	226	79	the	the	DET
iajs-2969	226	80	power	power	NOUN
iajs-2969	226	81	set	set	NOUN
iajs-2969	226	82	of	of	ADP
iajs-2969	226	83	z	z	PROPN
iajs-2969	226	84	)	)	PUNCT
iajs-2969	226	85	.	.	PUNCT
iajs-2969	227	1	let	let	VERB
iajs-2969	227	2	y	y	PROPN
iajs-2969	227	3	=	=	PUNCT
iajs-2969	227	4	{	{	PUNCT
iajs-2969	227	5	a	a	PRON
iajs-2969	227	6	,	,	PUNCT
iajs-2969	227	7	b	b	NOUN
iajs-2969	227	8	,	,	PUNCT
iajs-2969	227	9	c	c	NOUN
iajs-2969	227	10	,	,	PUNCT
iajs-2969	227	11	d	d	NOUN
iajs-2969	227	12	}	}	PUNCT
iajs-2969	227	13	,	,	PUNCT
iajs-2969	227	14	ϣ2	ϣ2	PROPN
iajs-2969	227	15	=	=	SYM
iajs-2969	227	16	{	{	PUNCT
iajs-2969	227	17	∅	∅	NOUN
iajs-2969	227	18	,	,	PUNCT
iajs-2969	227	19	{	{	PUNCT
iajs-2969	227	20	b	b	NOUN
iajs-2969	227	21	,	,	PUNCT
iajs-2969	227	22	d	d	NOUN
iajs-2969	227	23	}	}	PUNCT
iajs-2969	227	24	}	}	PUNCT
iajs-2969	227	25	,	,	PUNCT
iajs-2969	227	26	ϣ2	ϣ2	PROPN
iajs-2969	227	27	−	−	PROPN
iajs-2969	227	28	po(y	po(y	PUNCT
iajs-2969	227	29	)	)	PUNCT
iajs-2969	227	30	=	=	SYM
iajs-2969	227	31	{	{	PUNCT
iajs-2969	227	32	∅	∅	NOUN
iajs-2969	227	33	,	,	PUNCT
iajs-2969	227	34	{	{	PUNCT
iajs-2969	227	35	b	b	NOUN
iajs-2969	227	36	,	,	PUNCT
iajs-2969	227	37	d	d	NOUN
iajs-2969	227	38	}	}	PUNCT
iajs-2969	227	39	,	,	PUNCT
iajs-2969	227	40	{	{	PUNCT
iajs-2969	227	41	b	b	X
iajs-2969	227	42	}	}	PUNCT
iajs-2969	227	43	,	,	PUNCT
iajs-2969	227	44	{	{	PUNCT
iajs-2969	227	45	d	d	NOUN
iajs-2969	227	46	}	}	PUNCT
iajs-2969	227	47	}	}	PUNCT
iajs-2969	227	48	,	,	PUNCT
iajs-2969	227	49	ϣ2	ϣ2	PROPN
iajs-2969	227	50	−	−	PROPN
iajs-2969	227	51	spo(y	spo(y	PROPN
iajs-2969	227	52	)	)	PUNCT
iajs-2969	227	53	=	=	SYM
iajs-2969	228	1	ℙ(y	ℙ(y	NUM
iajs-2969	228	2	)	)	PUNCT
iajs-2969	228	3	(	(	PUNCT
iajs-2969	228	4	the	the	DET
iajs-2969	228	5	power	power	NOUN
iajs-2969	228	6	set	set	NOUN
iajs-2969	228	7	of	of	ADP
iajs-2969	228	8	y	y	PROPN
iajs-2969	228	9	)	)	PUNCT
iajs-2969	228	10	.	.	PUNCT
iajs-2969	229	1	define	define	VERB
iajs-2969	229	2	𝑓	𝑓	DET
iajs-2969	229	3	∶	∶	NOUN
iajs-2969	229	4	(	(	PUNCT
iajs-2969	229	5	z	z	NOUN
iajs-2969	229	6	,	,	PUNCT
iajs-2969	229	7	ϣ1	ϣ1	PROPN
iajs-2969	229	8	)	)	PUNCT
iajs-2969	229	9	→	→	SYM
iajs-2969	229	10	(	(	PUNCT
iajs-2969	229	11	y	y	PROPN
iajs-2969	229	12	,	,	PUNCT
iajs-2969	229	13	ϣ2	ϣ2	PROPN
iajs-2969	229	14	)	)	PUNCT
iajs-2969	230	1	such	such	ADJ
iajs-2969	230	2	that	that	SCONJ
iajs-2969	230	3	𝑓(1	𝑓(1	NOUN
iajs-2969	230	4	)	)	PUNCT
iajs-2969	230	5	=	=	SYM
iajs-2969	230	6	𝑓(2	𝑓(2	ADJ
iajs-2969	230	7	)	)	PUNCT
iajs-2969	230	8	=	=	PRON
iajs-2969	230	9	{	{	PUNCT
iajs-2969	230	10	𝑎	𝑎	NOUN
iajs-2969	230	11	}	}	PUNCT
iajs-2969	230	12	,	,	PUNCT
iajs-2969	230	13	𝑓(3	𝑓(3	PROPN
iajs-2969	230	14	)	)	PUNCT
iajs-2969	230	15	=	=	PRON
iajs-2969	230	16	{	{	PUNCT
iajs-2969	230	17	𝑏	𝑏	NOUN
iajs-2969	230	18	}	}	PUNCT
iajs-2969	230	19	,	,	PUNCT
iajs-2969	230	20	𝑓is	𝑓is	PROPN
iajs-2969	230	21	an	an	DET
iajs-2969	230	22	(	(	PUNCT
iajs-2969	230	23	ϣ1	ϣ1	NOUN
iajs-2969	230	24	,	,	PUNCT
iajs-2969	230	25	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	230	26	-	-	ADJ
iajs-2969	230	27	p	p	ADJ
iajs-2969	230	28	-	-	PUNCT
iajs-2969	230	29	continuous	continuous	ADJ
iajs-2969	230	30	function	function	NOUN
iajs-2969	230	31	,	,	PUNCT
iajs-2969	230	32	but	but	CCONJ
iajs-2969	230	33	it	it	PRON
iajs-2969	230	34	is	be	AUX
iajs-2969	230	35	not	not	PART
iajs-2969	230	36	an	an	DET
iajs-2969	230	37	(	(	PUNCT
iajs-2969	230	38	ϣ1	ϣ1	NOUN
iajs-2969	230	39	,	,	PUNCT
iajs-2969	230	40	ϣ2)-continuous	ϣ2)-continuous	ADJ
iajs-2969	230	41	function	function	NOUN
iajs-2969	230	42	,	,	PUNCT
iajs-2969	230	43	since	since	SCONJ
iajs-2969	230	44	{	{	PUNCT
iajs-2969	230	45	b	b	X
iajs-2969	230	46	,	,	PUNCT
iajs-2969	230	47	d	d	NOUN
iajs-2969	230	48	}	}	PUNCT
iajs-2969	230	49	is	be	AUX
iajs-2969	230	50	an	an	DET
iajs-2969	230	51	ϣ2	ϣ2	NOUN
iajs-2969	230	52	-	-	PUNCT
iajs-2969	230	53	open	open	NOUN
iajs-2969	230	54	set	set	NOUN
iajs-2969	230	55	in	in	ADP
iajs-2969	230	56	y	y	PROPN
iajs-2969	230	57	,	,	PUNCT
iajs-2969	230	58	but	but	CCONJ
iajs-2969	230	59	f	f	PROPN
iajs-2969	230	60	−1({b	−1({b	PROPN
iajs-2969	230	61	,	,	PUNCT
iajs-2969	230	62	d	d	NOUN
iajs-2969	230	63	}	}	PUNCT
iajs-2969	230	64	)	)	PUNCT
iajs-2969	231	1	=	=	PRON
iajs-2969	231	2	{	{	PUNCT
iajs-2969	231	3	3	3	NUM
iajs-2969	231	4	}	}	PUNCT
iajs-2969	231	5	is	be	AUX
iajs-2969	231	6	not	not	PART
iajs-2969	231	7	an	an	DET
iajs-2969	231	8	ϣ1	ϣ1	NOUN
iajs-2969	231	9	-	-	PUNCT
iajs-2969	231	10	open	open	NOUN
iajs-2969	231	11	set	set	NOUN
iajs-2969	231	12	in	in	ADP
iajs-2969	231	13	z	z	PROPN
iajs-2969	231	14	.	.	PUNCT
iajs-2969	232	1	proposition	proposition	NOUN
iajs-2969	232	2	4.12	4.12	NUM
iajs-2969	232	3	the	the	DET
iajs-2969	232	4	composition	composition	NOUN
iajs-2969	232	5	of	of	ADP
iajs-2969	232	6	(	(	PUNCT
iajs-2969	232	7	ϣ1	ϣ1	NOUN
iajs-2969	232	8	,	,	PUNCT
iajs-2969	232	9	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	232	10	-	-	ADJ
iajs-2969	232	11	p	p	ADJ
iajs-2969	232	12	-	-	PUNCT
iajs-2969	232	13	irresolute	irresolute	ADJ
iajs-2969	232	14	function	function	NOUN
iajs-2969	232	15	and	and	CCONJ
iajs-2969	232	16	(	(	PUNCT
iajs-2969	232	17	ϣ2	ϣ2	PROPN
iajs-2969	232	18	,	,	PUNCT
iajs-2969	232	19	ϣ3)-semi	ϣ3)-semi	PROPN
iajs-2969	232	20	-	-	PUNCT
iajs-2969	232	21	p	p	NOUN
iajs-2969	232	22	-	-	PUNCT
iajs-2969	232	23	irresolute	irresolute	ADJ
iajs-2969	232	24	function	function	NOUN
iajs-2969	232	25	is	be	AUX
iajs-2969	232	26	an	an	DET
iajs-2969	232	27	(	(	PUNCT
iajs-2969	232	28	ϣ1	ϣ1	NOUN
iajs-2969	232	29	,	,	PUNCT
iajs-2969	232	30	ϣ3)-semi	ϣ3)-semi	ADJ
iajs-2969	232	31	-	-	PUNCT
iajs-2969	232	32	pirresolute	pirresolute	NOUN
iajs-2969	232	33	function	function	NOUN
iajs-2969	232	34	.	.	PUNCT
iajs-2969	233	1	proof	proof	NOUN
iajs-2969	233	2	let	let	VERB
iajs-2969	233	3	f	f	X
iajs-2969	233	4	:	:	PUNCT
iajs-2969	233	5	(	(	PUNCT
iajs-2969	233	6	z	z	NOUN
iajs-2969	233	7	,	,	PUNCT
iajs-2969	233	8	ϣ1	ϣ1	PROPN
iajs-2969	233	9	)	)	PUNCT
iajs-2969	233	10	→	→	SYM
iajs-2969	233	11	(	(	PUNCT
iajs-2969	233	12	y	y	PROPN
iajs-2969	233	13	,	,	PUNCT
iajs-2969	233	14	ϣ2	ϣ2	PROPN
iajs-2969	233	15	)	)	PUNCT
iajs-2969	233	16	be	be	AUX
iajs-2969	233	17	(	(	PUNCT
iajs-2969	233	18	ϣ1	ϣ1	NOUN
iajs-2969	233	19	,	,	PUNCT
iajs-2969	233	20	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	233	21	-	-	ADJ
iajs-2969	233	22	p	p	ADJ
iajs-2969	233	23	-	-	PUNCT
iajs-2969	233	24	irresolute	irresolute	ADJ
iajs-2969	233	25	function	function	NOUN
iajs-2969	233	26	and	and	CCONJ
iajs-2969	233	27	g	g	PROPN
iajs-2969	233	28	∶	∶	PROPN
iajs-2969	233	29	(	(	PUNCT
iajs-2969	233	30	y	y	PROPN
iajs-2969	233	31	,	,	PUNCT
iajs-2969	233	32	ϣ2	ϣ2	PROPN
iajs-2969	233	33	)	)	PUNCT
iajs-2969	233	34	→	→	PUNCT
iajs-2969	233	35	(	(	PUNCT
iajs-2969	233	36	w	w	PROPN
iajs-2969	233	37	,	,	PUNCT
iajs-2969	233	38	ϣ3	ϣ3	PROPN
iajs-2969	233	39	)	)	PUNCT
iajs-2969	233	40	be	be	AUX
iajs-2969	233	41	(	(	PUNCT
iajs-2969	233	42	ϣ2	ϣ2	PROPN
iajs-2969	233	43	,	,	PUNCT
iajs-2969	233	44	ϣ3)-semi	ϣ3)-semi	PROPN
iajs-2969	233	45	-	-	PUNCT
iajs-2969	233	46	p	p	ADJ
iajs-2969	233	47	-	-	PUNCT
iajs-2969	233	48	irresolute	irresolute	ADJ
iajs-2969	233	49	functions	function	NOUN
iajs-2969	233	50	,	,	PUNCT
iajs-2969	233	51	we	we	PRON
iajs-2969	233	52	have	have	VERB
iajs-2969	233	53	to	to	PART
iajs-2969	233	54	show	show	VERB
iajs-2969	233	55	that	that	SCONJ
iajs-2969	233	56	g	g	PROPN
iajs-2969	233	57	∘	∘	PROPN
iajs-2969	233	58	f	f	PROPN
iajs-2969	233	59	∶	∶	NOUN
iajs-2969	233	60	(	(	PUNCT
iajs-2969	233	61	z	z	NOUN
iajs-2969	233	62	,	,	PUNCT
iajs-2969	233	63	ϣ1	ϣ1	PROPN
iajs-2969	233	64	)	)	PUNCT
iajs-2969	233	65	→	→	SYM
iajs-2969	233	66	(	(	PUNCT
iajs-2969	233	67	w	w	PROPN
iajs-2969	233	68	,	,	PUNCT
iajs-2969	233	69	ϣ3	ϣ3	PROPN
iajs-2969	233	70	)	)	PUNCT
iajs-2969	233	71	is	be	AUX
iajs-2969	233	72	an	an	DET
iajs-2969	233	73	(	(	PUNCT
iajs-2969	233	74	ϣ1	ϣ1	NOUN
iajs-2969	233	75	,	,	PUNCT
iajs-2969	233	76	ϣ3)-semi	ϣ3)-semi	PROPN
iajs-2969	233	77	-	-	PUNCT
iajs-2969	233	78	p	p	ADJ
iajs-2969	233	79	-	-	PUNCT
iajs-2969	233	80	irresolute	irresolute	ADJ
iajs-2969	233	81	function	function	NOUN
iajs-2969	233	82	.	.	PUNCT
iajs-2969	234	1	let	let	VERB
iajs-2969	234	2	ӈ	ӈ	PRON
iajs-2969	234	3	be	be	AUX
iajs-2969	234	4	any	any	DET
iajs-2969	234	5	ϣ3	ϣ3	NOUN
iajs-2969	234	6	-	-	PUNCT
iajs-2969	234	7	semi	semi	ADJ
iajs-2969	234	8	-	-	ADJ
iajs-2969	234	9	p	p	ADJ
iajs-2969	234	10	-	-	PUNCT
iajs-2969	234	11	open	open	NOUN
iajs-2969	234	12	set	set	NOUN
iajs-2969	234	13	in	in	ADP
iajs-2969	234	14	w	w	PROPN
iajs-2969	234	15	,	,	PUNCT
iajs-2969	234	16	then	then	ADV
iajs-2969	234	17	(	(	PUNCT
iajs-2969	234	18	g	g	PROPN
iajs-2969	234	19	∘	∘	NUM
iajs-2969	234	20	f)−1(ħ	f)−1(ħ	PROPN
iajs-2969	234	21	)	)	PUNCT
iajs-2969	235	1	=	=	PUNCT
iajs-2969	236	1	f	f	PROPN
iajs-2969	236	2	−1	−1	NOUN
iajs-2969	236	3	∘	∘	PROPN
iajs-2969	236	4	g−1(ħ	g−1(ħ	PROPN
iajs-2969	236	5	)	)	PUNCT
iajs-2969	236	6	=	=	SYM
iajs-2969	236	7	f	f	PROPN
iajs-2969	236	8	−1(g−1(ħ	−1(g−1(ħ	PROPN
iajs-2969	236	9	)	)	PUNCT
iajs-2969	236	10	)	)	PUNCT
iajs-2969	236	11	,	,	PUNCT
iajs-2969	236	12	but	but	CCONJ
iajs-2969	236	13	g−1(ħ	g−1(ħ	PROPN
iajs-2969	236	14	)	)	PUNCT
iajs-2969	236	15	is	be	AUX
iajs-2969	236	16	an	an	DET
iajs-2969	236	17	ϣ2	ϣ2	NOUN
iajs-2969	236	18	-	-	PUNCT
iajs-2969	236	19	semi	semi	ADJ
iajs-2969	236	20	-	-	ADJ
iajs-2969	236	21	p	p	ADJ
iajs-2969	236	22	-	-	PUNCT
iajs-2969	236	23	open	open	NOUN
iajs-2969	236	24	set	set	NOUN
iajs-2969	236	25	in	in	ADP
iajs-2969	236	26	y	y	PROPN
iajs-2969	236	27	(	(	PUNCT
iajs-2969	236	28	since	since	SCONJ
iajs-2969	236	29	g	g	PROPN
iajs-2969	236	30	is	be	AUX
iajs-2969	236	31	an	an	DET
iajs-2969	236	32	(	(	PUNCT
iajs-2969	236	33	ϣ2	ϣ2	PROPN
iajs-2969	236	34	,	,	PUNCT
iajs-2969	236	35	ϣ3)-semi	ϣ3)-semi	PROPN
iajs-2969	236	36	-	-	PUNCT
iajs-2969	236	37	p	p	ADJ
iajs-2969	236	38	-	-	PUNCT
iajs-2969	236	39	irresolute	irresolute	ADJ
iajs-2969	236	40	function	function	NOUN
iajs-2969	236	41	)	)	PUNCT
iajs-2969	236	42	,	,	PUNCT
iajs-2969	236	43	and	and	CCONJ
iajs-2969	236	44	f	f	PROPN
iajs-2969	236	45	−1(g−1(ħ	−1(g−1(ħ	PROPN
iajs-2969	236	46	)	)	PUNCT
iajs-2969	236	47	)	)	PUNCT
iajs-2969	237	1	is	be	AUX
iajs-2969	237	2	an	an	DET
iajs-2969	237	3	ϣ1	ϣ1	NOUN
iajs-2969	237	4	-	-	PUNCT
iajs-2969	237	5	semi	semi	ADJ
iajs-2969	237	6	-	-	ADJ
iajs-2969	237	7	p	p	ADJ
iajs-2969	237	8	-	-	PUNCT
iajs-2969	237	9	open	open	NOUN
iajs-2969	237	10	set	set	NOUN
iajs-2969	237	11	in	in	ADP
iajs-2969	237	12	z	z	PROPN
iajs-2969	237	13	(	(	PUNCT
iajs-2969	237	14	since	since	SCONJ
iajs-2969	237	15	f	f	PROPN
iajs-2969	237	16	is	be	AUX
iajs-2969	237	17	an	an	DET
iajs-2969	237	18	(	(	PUNCT
iajs-2969	237	19	ϣ1	ϣ1	NOUN
iajs-2969	237	20	,	,	PUNCT
iajs-2969	237	21	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	237	22	-	-	ADJ
iajs-2969	237	23	p	p	ADJ
iajs-2969	237	24	-	-	PUNCT
iajs-2969	237	25	irresolute	irresolute	ADJ
iajs-2969	237	26	functions	function	NOUN
iajs-2969	237	27	)	)	PUNCT
iajs-2969	237	28	,	,	PUNCT
iajs-2969	237	29	therefore	therefore	ADV
iajs-2969	237	30	g	g	PROPN
iajs-2969	237	31	∘	∘	PROPN
iajs-2969	237	32	f	f	PROPN
iajs-2969	237	33	is	be	AUX
iajs-2969	237	34	an	an	DET
iajs-2969	237	35	(	(	PUNCT
iajs-2969	237	36	ϣ1	ϣ1	NOUN
iajs-2969	237	37	,	,	PUNCT
iajs-2969	237	38	ϣ3)-semi	ϣ3)-semi	PROPN
iajs-2969	237	39	-	-	PUNCT
iajs-2969	237	40	p	p	ADJ
iajs-2969	237	41	-	-	PUNCT
iajs-2969	237	42	irresolute	irresolute	ADJ
iajs-2969	237	43	functions	function	NOUN
iajs-2969	237	44	.	.	PUNCT
iajs-2969	238	1	remark	remark	VERB
iajs-2969	238	2	4.13	4.13	NUM
iajs-2969	238	3	the	the	DET
iajs-2969	238	4	composition	composition	NOUN
iajs-2969	238	5	of	of	ADP
iajs-2969	238	6	(	(	PUNCT
iajs-2969	238	7	ϣ1	ϣ1	NOUN
iajs-2969	238	8	,	,	PUNCT
iajs-2969	238	9	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	238	10	-	-	ADJ
iajs-2969	238	11	p	p	ADJ
iajs-2969	238	12	-	-	PUNCT
iajs-2969	238	13	continuous	continuous	ADJ
iajs-2969	238	14	function	function	NOUN
iajs-2969	238	15	and	and	CCONJ
iajs-2969	238	16	(	(	PUNCT
iajs-2969	238	17	ϣ2	ϣ2	PROPN
iajs-2969	238	18	,	,	PUNCT
iajs-2969	238	19	ϣ3)-semi	ϣ3)-semi	PROPN
iajs-2969	238	20	-	-	PUNCT
iajs-2969	238	21	p	p	ADJ
iajs-2969	238	22	-	-	PUNCT
iajs-2969	238	23	continuous	continuous	ADJ
iajs-2969	238	24	function	function	NOUN
iajs-2969	238	25	need	need	VERB
iajs-2969	238	26	not	not	PART
iajs-2969	238	27	to	to	PART
iajs-2969	238	28	be	be	AUX
iajs-2969	238	29	(	(	PUNCT
iajs-2969	238	30	ϣ1	ϣ1	NOUN
iajs-2969	238	31	,	,	PUNCT
iajs-2969	238	32	ϣ3)-semi	ϣ3)-semi	NOUN
iajs-2969	238	33	-	-	PUNCT
iajs-2969	238	34	p	p	ADJ
iajs-2969	238	35	-	-	PUNCT
iajs-2969	238	36	continuous	continuous	ADJ
iajs-2969	238	37	function	function	NOUN
iajs-2969	238	38	as	as	SCONJ
iajs-2969	238	39	we	we	PRON
iajs-2969	238	40	show	show	VERB
iajs-2969	238	41	in	in	ADP
iajs-2969	238	42	the	the	DET
iajs-2969	238	43	following	follow	VERB
iajs-2969	238	44	example	example	NOUN
iajs-2969	238	45	:	:	PUNCT
iajs-2969	238	46	example	example	NOUN
iajs-2969	238	47	let	let	VERB
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iajs-2969	238	51	1	1	NUM
iajs-2969	238	52	,	,	PUNCT
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iajs-2969	238	55	3	3	NUM
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iajs-2969	238	57	,	,	PUNCT
iajs-2969	238	58	ϣ1	ϣ1	NOUN
iajs-2969	238	59	=	=	SYM
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iajs-2969	238	61	z	z	PROPN
iajs-2969	238	62	,	,	PUNCT
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iajs-2969	238	66	1	1	NUM
iajs-2969	238	67	,	,	PUNCT
iajs-2969	238	68	2	2	NUM
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iajs-2969	238	70	}	}	PUNCT
iajs-2969	238	71	,	,	PUNCT
iajs-2969	239	1	y	y	PROPN
iajs-2969	239	2	=	=	PRON
iajs-2969	239	3	{	{	PUNCT
iajs-2969	239	4	a	a	PRON
iajs-2969	239	5	,	,	PUNCT
iajs-2969	239	6	b	b	NOUN
iajs-2969	239	7	,	,	PUNCT
iajs-2969	239	8	c	c	NOUN
iajs-2969	239	9	}	}	PUNCT
iajs-2969	239	10	,	,	PUNCT
iajs-2969	239	11	ϣ2	ϣ2	PROPN
iajs-2969	239	12	=	=	SYM
iajs-2969	239	13	{	{	PUNCT
iajs-2969	239	14	y	y	PROPN
iajs-2969	239	15	,	,	PUNCT
iajs-2969	239	16	ø	ø	PROPN
iajs-2969	239	17	,	,	PUNCT
iajs-2969	239	18	{	{	PUNCT
iajs-2969	239	19	a	a	DET
iajs-2969	239	20	,	,	PUNCT
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iajs-2969	239	23	}	}	PUNCT
iajs-2969	239	24	,	,	PUNCT
iajs-2969	239	25	w	w	X
iajs-2969	239	26	=	=	SYM
iajs-2969	239	27	{	{	PUNCT
iajs-2969	239	28	i	i	PROPN
iajs-2969	239	29	,	,	PUNCT
iajs-2969	239	30	j	j	PROPN
iajs-2969	239	31	,	,	PUNCT
iajs-2969	239	32	k	k	NOUN
iajs-2969	239	33	}	}	PUNCT
iajs-2969	239	34	,	,	PUNCT
iajs-2969	239	35	ϣ3	ϣ3	NOUN
iajs-2969	239	36	=	=	SYM
iajs-2969	239	37	{	{	PUNCT
iajs-2969	239	38	w	w	PROPN
iajs-2969	239	39	,	,	PUNCT
iajs-2969	239	40	ø	ø	PROPN
iajs-2969	239	41	,	,	PUNCT
iajs-2969	239	42	{	{	PUNCT
iajs-2969	239	43	i	i	PRON
iajs-2969	239	44	,	,	PUNCT
iajs-2969	239	45	k	k	NOUN
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iajs-2969	239	47	}	}	PUNCT
iajs-2969	239	48	,	,	PUNCT
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iajs-2969	239	50	-	-	PUNCT
iajs-2969	239	51	po(z)=	po(z)=	NOUN
iajs-2969	239	52	{	{	PUNCT
iajs-2969	239	53	z	z	NOUN
iajs-2969	239	54	,	,	PUNCT
iajs-2969	239	55	ø	ø	PROPN
iajs-2969	239	56	,	,	PUNCT
iajs-2969	239	57	{	{	PUNCT
iajs-2969	239	58	1	1	NUM
iajs-2969	239	59	}	}	PUNCT
iajs-2969	239	60	,	,	PUNCT
iajs-2969	239	61	{	{	PUNCT
iajs-2969	239	62	2	2	NUM
iajs-2969	239	63	}	}	PUNCT
iajs-2969	239	64	,	,	PUNCT
iajs-2969	239	65	{	{	PUNCT
iajs-2969	239	66	1	1	NUM
iajs-2969	239	67	,	,	PUNCT
iajs-2969	239	68	2	2	NUM
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iajs-2969	239	70	,	,	PUNCT
iajs-2969	239	71	{	{	PUNCT
iajs-2969	239	72	1	1	NUM
iajs-2969	239	73	,	,	PUNCT
iajs-2969	239	74	3	3	NUM
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iajs-2969	239	76	,	,	PUNCT
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iajs-2969	239	79	,	,	PUNCT
iajs-2969	239	80	3	3	NUM
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iajs-2969	239	82	}	}	PUNCT
iajs-2969	239	83	=	=	SYM
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iajs-2969	239	85	-	-	PUNCT
iajs-2969	239	86	spo(z	spo(z	PROPN
iajs-2969	239	87	)	)	PUNCT
iajs-2969	239	88	ϣ2	ϣ2	PROPN
iajs-2969	239	89	-	-	PUNCT
iajs-2969	239	90	po(y)=	po(y)=	NOUN
iajs-2969	239	91	{	{	PUNCT
iajs-2969	239	92	y	y	PROPN
iajs-2969	239	93	,	,	PUNCT
iajs-2969	239	94	ø	ø	PROPN
iajs-2969	239	95	,	,	PUNCT
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iajs-2969	239	99	,	,	PUNCT
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iajs-2969	239	103	,	,	PUNCT
iajs-2969	239	104	{	{	PUNCT
iajs-2969	239	105	a	a	DET
iajs-2969	239	106	,	,	PUNCT
iajs-2969	239	107	b	b	NOUN
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iajs-2969	239	109	,	,	PUNCT
iajs-2969	239	110	{	{	PUNCT
iajs-2969	239	111	a	a	X
iajs-2969	239	112	,	,	PUNCT
iajs-2969	239	113	c	c	NOUN
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iajs-2969	239	115	,	,	PUNCT
iajs-2969	239	116	{	{	PUNCT
iajs-2969	239	117	b	b	X
iajs-2969	239	118	,	,	PUNCT
iajs-2969	239	119	c	c	NOUN
iajs-2969	239	120	}	}	PUNCT
iajs-2969	239	121	}	}	PUNCT
iajs-2969	239	122	=	=	SYM
iajs-2969	239	123	ϣ2	ϣ2	NOUN
iajs-2969	239	124	-	-	PUNCT
iajs-2969	239	125	spo(y	spo(y	PROPN
iajs-2969	239	126	)	)	PUNCT
iajs-2969	239	127	,	,	PUNCT
iajs-2969	239	128	and	and	CCONJ
iajs-2969	239	129	ϣ3	ϣ3	NOUN
iajs-2969	239	130	-	-	PUNCT
iajs-2969	239	131	po(w)=	po(w)=	NOUN
iajs-2969	239	132	{	{	PUNCT
iajs-2969	239	133	w	w	PROPN
iajs-2969	239	134	,	,	PUNCT
iajs-2969	239	135	ø	ø	PROPN
iajs-2969	239	136	,	,	PUNCT
iajs-2969	239	137	{	{	PUNCT
iajs-2969	239	138	i	i	NOUN
iajs-2969	239	139	}	}	PUNCT
iajs-2969	239	140	,	,	PUNCT
iajs-2969	239	141	{	{	PUNCT
iajs-2969	239	142	k	k	X
iajs-2969	239	143	}	}	PUNCT
iajs-2969	239	144	,	,	PUNCT
iajs-2969	239	145	{	{	PUNCT
iajs-2969	239	146	i	i	X
iajs-2969	239	147	,	,	PUNCT
iajs-2969	239	148	j	j	PROPN
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iajs-2969	239	150	,	,	PUNCT
iajs-2969	239	151	{	{	PUNCT
iajs-2969	239	152	i	i	PRON
iajs-2969	239	153	,	,	PUNCT
iajs-2969	239	154	k	k	PROPN
iajs-2969	239	155	}	}	PUNCT
iajs-2969	239	156	,	,	PUNCT
iajs-2969	239	157	{	{	PUNCT
iajs-2969	239	158	j	j	NOUN
iajs-2969	239	159	,	,	PUNCT
iajs-2969	239	160	k	k	NOUN
iajs-2969	239	161	}	}	PUNCT
iajs-2969	239	162	}	}	PUNCT
iajs-2969	239	163	=	=	SYM
iajs-2969	239	164	ϣ3	ϣ3	PROPN
iajs-2969	239	165	-	-	PUNCT
iajs-2969	239	166	spo(w	spo(w	NOUN
iajs-2969	239	167	)	)	PUNCT
iajs-2969	239	168	define	define	VERB
iajs-2969	239	169	𝑓	𝑓	DET
iajs-2969	239	170	∶	∶	NOUN
iajs-2969	239	171	(	(	PUNCT
iajs-2969	239	172	z	z	NOUN
iajs-2969	239	173	,	,	PUNCT
iajs-2969	239	174	ϣ1	ϣ1	PROPN
iajs-2969	239	175	)	)	PUNCT
iajs-2969	239	176	→	→	SYM
iajs-2969	239	177	(	(	PUNCT
iajs-2969	239	178	y	y	PROPN
iajs-2969	239	179	,	,	PUNCT
iajs-2969	239	180	ϣ2	ϣ2	PROPN
iajs-2969	239	181	)	)	PUNCT
iajs-2969	239	182	by	by	ADP
iajs-2969	239	183	f(1	f(1	PROPN
iajs-2969	239	184	)	)	PUNCT
iajs-2969	239	185	=	=	SYM
iajs-2969	239	186	f(3	f(3	PROPN
iajs-2969	239	187	)	)	PUNCT
iajs-2969	239	188	=	=	PRON
iajs-2969	240	1	{	{	PUNCT
iajs-2969	240	2	b	b	NOUN
iajs-2969	240	3	}	}	PUNCT
iajs-2969	240	4	,	,	PUNCT
iajs-2969	240	5	f(2	f(2	PROPN
iajs-2969	240	6	)	)	PUNCT
iajs-2969	240	7	=	=	PRON
iajs-2969	241	1	{	{	PUNCT
iajs-2969	241	2	c	c	NOUN
iajs-2969	241	3	}	}	PUNCT
iajs-2969	241	4	.	.	PUNCT
iajs-2969	242	1	ihjpas	ihjpas	PROPN
iajs-2969	242	2	.	.	PUNCT
iajs-2969	243	1	36(1)2023	36(1)2023	NUM
iajs-2969	243	2	343	343	NUM
iajs-2969	243	3	and	and	CCONJ
iajs-2969	243	4	𝑔	𝑔	NOUN
iajs-2969	243	5	:	:	PUNCT
iajs-2969	243	6	(	(	PUNCT
iajs-2969	243	7	y	y	PROPN
iajs-2969	243	8	,	,	PUNCT
iajs-2969	243	9	ϣ2	ϣ2	PROPN
iajs-2969	243	10	)	)	PUNCT
iajs-2969	243	11	→	→	PUNCT
iajs-2969	243	12	(	(	PUNCT
iajs-2969	243	13	w	w	PROPN
iajs-2969	243	14	,	,	PUNCT
iajs-2969	243	15	ϣ3	ϣ3	PROPN
iajs-2969	243	16	)	)	PUNCT
iajs-2969	243	17	by	by	ADP
iajs-2969	243	18	g(a	g(a	PROPN
iajs-2969	243	19	)	)	PUNCT
iajs-2969	243	20	=	=	SYM
iajs-2969	243	21	g(c	g(c	NOUN
iajs-2969	243	22	)	)	PUNCT
iajs-2969	243	23	=	=	PRON
iajs-2969	243	24	{	{	PUNCT
iajs-2969	243	25	j	j	NOUN
iajs-2969	243	26	}	}	PUNCT
iajs-2969	243	27	,	,	PUNCT
iajs-2969	243	28	g(b	g(b	PROPN
iajs-2969	243	29	)	)	PUNCT
iajs-2969	243	30	=	=	PRON
iajs-2969	243	31	{	{	PUNCT
iajs-2969	243	32	k	k	NOUN
iajs-2969	243	33	}	}	PUNCT
iajs-2969	243	34	.	.	PUNCT
iajs-2969	244	1	then	then	ADV
iajs-2969	244	2	𝑔	𝑔	PROPN
iajs-2969	244	3	∘	∘	PROPN
iajs-2969	244	4	𝑓	𝑓	PROPN
iajs-2969	244	5	:	:	PUNCT
iajs-2969	244	6	(	(	PUNCT
iajs-2969	244	7	z	z	NOUN
iajs-2969	244	8	,	,	PUNCT
iajs-2969	244	9	ϣ1	ϣ1	PROPN
iajs-2969	244	10	)	)	PUNCT
iajs-2969	244	11	→	→	SYM
iajs-2969	244	12	(	(	PUNCT
iajs-2969	244	13	w	w	PROPN
iajs-2969	244	14	,	,	PUNCT
iajs-2969	244	15	ϣ3	ϣ3	PROPN
iajs-2969	244	16	)	)	PUNCT
iajs-2969	244	17	is	be	AUX
iajs-2969	244	18	defined	define	VERB
iajs-2969	244	19	by	by	ADP
iajs-2969	244	20	:	:	PUNCT
iajs-2969	244	21	g	g	PROPN
iajs-2969	244	22	∘	∘	PROPN
iajs-2969	244	23	f	f	X
iajs-2969	244	24	(	(	PUNCT
iajs-2969	244	25	1	1	NUM
iajs-2969	244	26	)	)	PUNCT
iajs-2969	244	27	=	=	SYM
iajs-2969	244	28	g(f(1	g(f(1	NOUN
iajs-2969	244	29	)	)	PUNCT
iajs-2969	244	30	)	)	PUNCT
iajs-2969	245	1	=	=	PUNCT
iajs-2969	245	2	g(b	g(b	X
iajs-2969	245	3	)	)	PUNCT
iajs-2969	245	4	=	=	PRON
iajs-2969	245	5	{	{	PUNCT
iajs-2969	245	6	k	k	X
iajs-2969	245	7	}	}	PUNCT
iajs-2969	245	8	,	,	PUNCT
iajs-2969	245	9	g	g	PROPN
iajs-2969	245	10	∘	∘	PROPN
iajs-2969	245	11	f	f	X
iajs-2969	245	12	(	(	PUNCT
iajs-2969	245	13	2	2	NUM
iajs-2969	245	14	)	)	PUNCT
iajs-2969	245	15	=	=	SYM
iajs-2969	245	16	g(f(2	g(f(2	NOUN
iajs-2969	245	17	)	)	PUNCT
iajs-2969	245	18	)	)	PUNCT
iajs-2969	246	1	=	=	SYM
iajs-2969	246	2	g(c	g(c	NOUN
iajs-2969	246	3	)	)	PUNCT
iajs-2969	246	4	=	=	PRON
iajs-2969	246	5	{	{	PUNCT
iajs-2969	246	6	j	j	NOUN
iajs-2969	246	7	}	}	PUNCT
iajs-2969	246	8	,	,	PUNCT
iajs-2969	246	9	g	g	PROPN
iajs-2969	246	10	∘	∘	PROPN
iajs-2969	246	11	f	f	PROPN
iajs-2969	246	12	(	(	PUNCT
iajs-2969	246	13	3	3	NUM
iajs-2969	246	14	)	)	PUNCT
iajs-2969	246	15	=	=	SYM
iajs-2969	246	16	g(f(3	g(f(3	NOUN
iajs-2969	246	17	)	)	PUNCT
iajs-2969	246	18	)	)	PUNCT
iajs-2969	247	1	=	=	PUNCT
iajs-2969	247	2	g(b	g(b	X
iajs-2969	247	3	)	)	PUNCT
iajs-2969	247	4	=	=	PRON
iajs-2969	247	5	{	{	PUNCT
iajs-2969	247	6	k	k	NOUN
iajs-2969	247	7	}	}	PUNCT
iajs-2969	247	8	,	,	PUNCT
iajs-2969	247	9	f	f	PROPN
iajs-2969	247	10	is	be	AUX
iajs-2969	247	11	an	an	DET
iajs-2969	247	12	(	(	PUNCT
iajs-2969	247	13	ϣ1	ϣ1	NOUN
iajs-2969	247	14	,	,	PUNCT
iajs-2969	247	15	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	247	16	-	-	ADJ
iajs-2969	247	17	p	p	ADJ
iajs-2969	247	18	-	-	PUNCT
iajs-2969	247	19	continuous	continuous	ADJ
iajs-2969	247	20	function	function	NOUN
iajs-2969	247	21	and	and	CCONJ
iajs-2969	247	22	g	g	NOUN
iajs-2969	247	23	is	be	AUX
iajs-2969	247	24	an	an	DET
iajs-2969	247	25	(	(	PUNCT
iajs-2969	247	26	ϣ2	ϣ2	PROPN
iajs-2969	247	27	,	,	PUNCT
iajs-2969	247	28	ϣ3)-semi	ϣ3)-semi	PROPN
iajs-2969	247	29	-	-	PUNCT
iajs-2969	247	30	p	p	ADJ
iajs-2969	247	31	-	-	PUNCT
iajs-2969	247	32	continuous	continuous	ADJ
iajs-2969	247	33	function	function	NOUN
iajs-2969	247	34	.	.	PUNCT
iajs-2969	248	1	but	but	CCONJ
iajs-2969	248	2	g	g	PROPN
iajs-2969	248	3	∘	∘	PROPN
iajs-2969	248	4	f	f	PROPN
iajs-2969	248	5	is	be	AUX
iajs-2969	248	6	not	not	PART
iajs-2969	248	7	an	an	DET
iajs-2969	248	8	(	(	PUNCT
iajs-2969	248	9	ϣ1	ϣ1	NOUN
iajs-2969	248	10	,	,	PUNCT
iajs-2969	248	11	ϣ3)-semi	ϣ3)-semi	NOUN
iajs-2969	248	12	-	-	PUNCT
iajs-2969	248	13	p	p	ADJ
iajs-2969	248	14	-	-	PUNCT
iajs-2969	248	15	continuous	continuous	ADJ
iajs-2969	248	16	function	function	NOUN
iajs-2969	248	17	,	,	PUNCT
iajs-2969	248	18	since	since	SCONJ
iajs-2969	248	19	{	{	PUNCT
iajs-2969	248	20	i	i	PRON
iajs-2969	248	21	,	,	PUNCT
iajs-2969	248	22	k	k	NOUN
iajs-2969	248	23	}	}	PUNCT
iajs-2969	248	24	is	be	AUX
iajs-2969	248	25	an	an	DET
iajs-2969	248	26	ϣ3	ϣ3	NOUN
iajs-2969	248	27	-	-	PUNCT
iajs-2969	248	28	semi	semi	ADJ
iajs-2969	248	29	-	-	ADJ
iajs-2969	248	30	p	p	ADJ
iajs-2969	248	31	-	-	PUNCT
iajs-2969	248	32	open	open	NOUN
iajs-2969	248	33	set	set	NOUN
iajs-2969	248	34	in	in	ADP
iajs-2969	248	35	w	w	PROPN
iajs-2969	248	36	,	,	PUNCT
iajs-2969	248	37	but	but	CCONJ
iajs-2969	248	38	f	f	PROPN
iajs-2969	248	39	−1({i	−1({i	PROPN
iajs-2969	248	40	,	,	PUNCT
iajs-2969	248	41	k	k	NOUN
iajs-2969	248	42	}	}	PUNCT
iajs-2969	248	43	)	)	PUNCT
iajs-2969	249	1	=	=	PUNCT
iajs-2969	249	2	{	{	PUNCT
iajs-2969	249	3	3	3	NUM
iajs-2969	249	4	}	}	PUNCT
iajs-2969	249	5	is	be	AUX
iajs-2969	249	6	not	not	PART
iajs-2969	249	7	ϣ1	ϣ1	NOUN
iajs-2969	249	8	-	-	PUNCT
iajs-2969	249	9	semi	semi	ADJ
iajs-2969	249	10	-	-	ADJ
iajs-2969	249	11	p	p	ADJ
iajs-2969	249	12	-	-	PUNCT
iajs-2969	249	13	open	open	NOUN
iajs-2969	249	14	set	set	NOUN
iajs-2969	249	15	in	in	ADP
iajs-2969	249	16	z.	z.	PROPN
iajs-2969	249	17	proposition	proposition	NOUN
iajs-2969	249	18	4.14	4.14	NUM
iajs-2969	249	19	the	the	DET
iajs-2969	249	20	composition	composition	NOUN
iajs-2969	249	21	of	of	ADP
iajs-2969	249	22	an	an	DET
iajs-2969	249	23	(	(	PUNCT
iajs-2969	249	24	ϣ1	ϣ1	NOUN
iajs-2969	249	25	,	,	PUNCT
iajs-2969	249	26	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	249	27	-	-	ADJ
iajs-2969	249	28	p	p	ADJ
iajs-2969	249	29	-	-	PUNCT
iajs-2969	249	30	continuous	continuous	ADJ
iajs-2969	249	31	function	function	NOUN
iajs-2969	249	32	and	and	CCONJ
iajs-2969	249	33	(	(	PUNCT
iajs-2969	249	34	ϣ2	ϣ2	PROPN
iajs-2969	249	35	,	,	PUNCT
iajs-2969	249	36	ϣ3)continuous	ϣ3)continuous	ADJ
iajs-2969	249	37	function	function	NOUN
iajs-2969	249	38	is	be	AUX
iajs-2969	249	39	an	an	DET
iajs-2969	249	40	(	(	PUNCT
iajs-2969	249	41	ϣ1	ϣ1	NOUN
iajs-2969	249	42	,	,	PUNCT
iajs-2969	249	43	ϣ3)-semi	ϣ3)-semi	NOUN
iajs-2969	249	44	-	-	PUNCT
iajs-2969	249	45	p	p	ADJ
iajs-2969	249	46	-	-	PUNCT
iajs-2969	249	47	continuous	continuous	ADJ
iajs-2969	249	48	function	function	NOUN
iajs-2969	249	49	.	.	PUNCT
iajs-2969	250	1	proof	proof	NOUN
iajs-2969	250	2	:	:	PUNCT
iajs-2969	250	3	let	let	VERB
iajs-2969	250	4	f	f	PROPN
iajs-2969	250	5	∶	∶	NOUN
iajs-2969	250	6	(	(	PUNCT
iajs-2969	250	7	z	z	NOUN
iajs-2969	250	8	,	,	PUNCT
iajs-2969	250	9	ϣ1	ϣ1	PROPN
iajs-2969	250	10	)	)	PUNCT
iajs-2969	250	11	→	→	SYM
iajs-2969	250	12	(	(	PUNCT
iajs-2969	250	13	y	y	PROPN
iajs-2969	250	14	,	,	PUNCT
iajs-2969	250	15	ϣ2	ϣ2	PROPN
iajs-2969	250	16	)	)	PUNCT
iajs-2969	250	17	be	be	AUX
iajs-2969	250	18	any	any	DET
iajs-2969	250	19	(	(	PUNCT
iajs-2969	250	20	ϣ1	ϣ1	NOUN
iajs-2969	250	21	,	,	PUNCT
iajs-2969	250	22	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	250	23	-	-	ADJ
iajs-2969	250	24	p	p	ADJ
iajs-2969	250	25	-	-	PUNCT
iajs-2969	250	26	continuous	continuous	ADJ
iajs-2969	250	27	function	function	NOUN
iajs-2969	250	28	and	and	CCONJ
iajs-2969	250	29	g	g	NOUN
iajs-2969	250	30	:	:	PUNCT
iajs-2969	250	31	(	(	PUNCT
iajs-2969	250	32	y	y	PROPN
iajs-2969	250	33	,	,	PUNCT
iajs-2969	250	34	ϣ2	ϣ2	PROPN
iajs-2969	250	35	)	)	PUNCT
iajs-2969	250	36	→	→	PUNCT
iajs-2969	250	37	(	(	PUNCT
iajs-2969	250	38	w	w	PROPN
iajs-2969	250	39	,	,	PUNCT
iajs-2969	250	40	ϣ3	ϣ3	PROPN
iajs-2969	250	41	)	)	PUNCT
iajs-2969	250	42	be	be	VERB
iajs-2969	250	43	any	any	DET
iajs-2969	250	44	(	(	PUNCT
iajs-2969	250	45	ϣ2	ϣ2	PROPN
iajs-2969	250	46	,	,	PUNCT
iajs-2969	250	47	ϣ3)continuous	ϣ3)continuous	ADJ
iajs-2969	250	48	function	function	NOUN
iajs-2969	250	49	.	.	PUNCT
iajs-2969	251	1	we	we	PRON
iajs-2969	251	2	have	have	VERB
iajs-2969	251	3	to	to	PART
iajs-2969	251	4	show	show	VERB
iajs-2969	251	5	that	that	SCONJ
iajs-2969	251	6	g	g	PROPN
iajs-2969	251	7	∘	∘	PROPN
iajs-2969	251	8	f	f	PROPN
iajs-2969	251	9	∶	∶	NOUN
iajs-2969	251	10	(	(	PUNCT
iajs-2969	251	11	z	z	NOUN
iajs-2969	251	12	,	,	PUNCT
iajs-2969	251	13	ϣ1	ϣ1	PROPN
iajs-2969	251	14	)	)	PUNCT
iajs-2969	251	15	→	→	SYM
iajs-2969	251	16	(	(	PUNCT
iajs-2969	251	17	w	w	PROPN
iajs-2969	251	18	,	,	PUNCT
iajs-2969	251	19	ϣ3	ϣ3	PROPN
iajs-2969	251	20	)	)	PUNCT
iajs-2969	251	21	is	be	AUX
iajs-2969	251	22	an	an	DET
iajs-2969	251	23	(	(	PUNCT
iajs-2969	251	24	ϣ1	ϣ1	NOUN
iajs-2969	251	25	,	,	PUNCT
iajs-2969	251	26	ϣ3)-semi	ϣ3)-semi	ADJ
iajs-2969	251	27	-	-	PUNCT
iajs-2969	251	28	pcontinuous	pcontinuous	ADJ
iajs-2969	251	29	function	function	NOUN
iajs-2969	251	30	.	.	PUNCT
iajs-2969	252	1	let	let	VERB
iajs-2969	252	2	ħ	ħ	NOUN
iajs-2969	252	3	be	be	AUX
iajs-2969	252	4	any	any	DET
iajs-2969	252	5	ϣ3open	ϣ3open	ADJ
iajs-2969	252	6	set	set	NOUN
iajs-2969	252	7	in	in	ADP
iajs-2969	252	8	w.	w.	PROPN
iajs-2969	252	9	then	then	ADV
iajs-2969	252	10	,	,	PUNCT
iajs-2969	252	11	g−1(ħ	g−1(ħ	PROPN
iajs-2969	252	12	)	)	PUNCT
iajs-2969	252	13	is	be	AUX
iajs-2969	252	14	an	an	DET
iajs-2969	252	15	ϣ2	ϣ2	NOUN
iajs-2969	252	16	-	-	PUNCT
iajs-2969	252	17	open	open	NOUN
iajs-2969	252	18	set	set	NOUN
iajs-2969	252	19	in	in	ADP
iajs-2969	252	20	y	y	PROPN
iajs-2969	252	21	(	(	PUNCT
iajs-2969	252	22	since	since	SCONJ
iajs-2969	252	23	g	g	PROPN
iajs-2969	252	24	is	be	AUX
iajs-2969	252	25	an	an	DET
iajs-2969	252	26	(	(	PUNCT
iajs-2969	252	27	ϣ2	ϣ2	PROPN
iajs-2969	252	28	,	,	PUNCT
iajs-2969	252	29	ϣ3)-continuous	ϣ3)-continuous	ADJ
iajs-2969	252	30	function	function	NOUN
iajs-2969	252	31	)	)	PUNCT
iajs-2969	252	32	,	,	PUNCT
iajs-2969	252	33	so	so	CCONJ
iajs-2969	252	34	f	f	PROPN
iajs-2969	252	35	−1(g−1(ħ	−1(g−1(ħ	PROPN
iajs-2969	252	36	)	)	PUNCT
iajs-2969	252	37	is	be	AUX
iajs-2969	252	38	an	an	DET
iajs-2969	252	39	ϣ1	ϣ1	NOUN
iajs-2969	252	40	-	-	PUNCT
iajs-2969	252	41	semi	semi	ADJ
iajs-2969	252	42	-	-	ADJ
iajs-2969	252	43	p	p	ADJ
iajs-2969	252	44	-	-	PUNCT
iajs-2969	252	45	open	open	NOUN
iajs-2969	252	46	set	set	NOUN
iajs-2969	252	47	in	in	ADP
iajs-2969	252	48	z	z	PROPN
iajs-2969	252	49	(	(	PUNCT
iajs-2969	252	50	since	since	SCONJ
iajs-2969	252	51	f	f	PROPN
iajs-2969	252	52	is	be	AUX
iajs-2969	252	53	an	an	DET
iajs-2969	252	54	(	(	PUNCT
iajs-2969	252	55	ϣ1	ϣ1	NOUN
iajs-2969	252	56	,	,	PUNCT
iajs-2969	252	57	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	252	58	-	-	ADJ
iajs-2969	252	59	continuous	continuous	ADJ
iajs-2969	252	60	function	function	NOUN
iajs-2969	252	61	)	)	PUNCT
iajs-2969	252	62	,	,	PUNCT
iajs-2969	252	63	but	but	CCONJ
iajs-2969	252	64	(	(	PUNCT
iajs-2969	252	65	g	g	PROPN
iajs-2969	252	66	∘	∘	NUM
iajs-2969	252	67	f)−1(ħ	f)−1(ħ	PROPN
iajs-2969	252	68	)	)	PUNCT
iajs-2969	253	1	=	=	PUNCT
iajs-2969	254	1	f	f	PROPN
iajs-2969	254	2	−1	−1	NOUN
iajs-2969	254	3	∘	∘	PROPN
iajs-2969	254	4	g−1(ħ	g−1(ħ	PROPN
iajs-2969	254	5	)	)	PUNCT
iajs-2969	254	6	=	=	SYM
iajs-2969	254	7	f	f	PROPN
iajs-2969	254	8	−1(g−1(ħ	−1(g−1(ħ	PROPN
iajs-2969	254	9	)	)	PUNCT
iajs-2969	254	10	.	.	PUNCT
iajs-2969	255	1	hence	hence	ADV
iajs-2969	255	2	g	g	PROPN
iajs-2969	255	3	∘	∘	PROPN
iajs-2969	255	4	f	f	PROPN
iajs-2969	255	5	is	be	AUX
iajs-2969	255	6	an	an	DET
iajs-2969	255	7	(	(	PUNCT
iajs-2969	255	8	ϣ1	ϣ1	NOUN
iajs-2969	255	9	,	,	PUNCT
iajs-2969	255	10	ϣ3)-semi	ϣ3)-semi	NOUN
iajs-2969	255	11	-	-	PUNCT
iajs-2969	255	12	p	p	ADJ
iajs-2969	255	13	-	-	PUNCT
iajs-2969	255	14	continuous	continuous	ADJ
iajs-2969	255	15	function	function	NOUN
iajs-2969	255	16	.	.	PUNCT
iajs-2969	256	1	theorem	theorem	VERB
iajs-2969	256	2	4.15	4.15	NUM
iajs-2969	256	3	let	let	VERB
iajs-2969	256	4	𝑓	𝑓	DET
iajs-2969	256	5	∶	∶	NOUN
iajs-2969	256	6	(	(	PUNCT
iajs-2969	256	7	z	z	NOUN
iajs-2969	256	8	,	,	PUNCT
iajs-2969	256	9	ϣ1	ϣ1	PROPN
iajs-2969	256	10	)	)	PUNCT
iajs-2969	256	11	→	→	SYM
iajs-2969	256	12	(	(	PUNCT
iajs-2969	256	13	y	y	PROPN
iajs-2969	256	14	,	,	PUNCT
iajs-2969	256	15	ϣ2	ϣ2	PROPN
iajs-2969	256	16	)	)	PUNCT
iajs-2969	256	17	be	be	AUX
iajs-2969	256	18	an	an	PRON
iajs-2969	256	19	onto	onto	ADP
iajs-2969	256	20	function	function	NOUN
iajs-2969	256	21	,	,	PUNCT
iajs-2969	256	22	then	then	ADV
iajs-2969	256	23	𝑓	𝑓	PRON
iajs-2969	256	24	is	be	AUX
iajs-2969	256	25	an	an	DET
iajs-2969	256	26	(	(	PUNCT
iajs-2969	256	27	ϣ1	ϣ1	NOUN
iajs-2969	256	28	,	,	PUNCT
iajs-2969	256	29	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	256	30	-	-	PUNCT
iajs-2969	256	31	semi	semi	ADJ
iajs-2969	256	32	-	-	ADJ
iajs-2969	256	33	p	p	ADJ
iajs-2969	256	34	-	-	PUNCT
iajs-2969	256	35	open	open	ADJ
iajs-2969	256	36	function	function	NOUN
iajs-2969	256	37	if	if	SCONJ
iajs-2969	257	1	and	and	CCONJ
iajs-2969	257	2	only	only	ADV
iajs-2969	257	3	if	if	SCONJ
iajs-2969	257	4	it	it	PRON
iajs-2969	257	5	is	be	AUX
iajs-2969	257	6	an	an	DET
iajs-2969	257	7	(	(	PUNCT
iajs-2969	257	8	ϣ1	ϣ1	NOUN
iajs-2969	257	9	,	,	PUNCT
iajs-2969	257	10	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	257	11	-	-	PUNCT
iajs-2969	257	12	semi	semi	ADJ
iajs-2969	257	13	-	-	ADJ
iajs-2969	257	14	p	p	ADJ
iajs-2969	257	15	-	-	PUNCT
iajs-2969	257	16	closed	closed	ADJ
iajs-2969	257	17	function	function	NOUN
iajs-2969	257	18	.	.	PUNCT
iajs-2969	258	1	proof	proof	NOUN
iajs-2969	258	2	:	:	PUNCT
iajs-2969	258	3	the	the	DET
iajs-2969	258	4	"	"	PUNCT
iajs-2969	258	5	if	if	SCONJ
iajs-2969	258	6	"	"	PUNCT
iajs-2969	258	7	part	part	NOUN
iajs-2969	258	8	.	.	PUNCT
iajs-2969	259	1	let	let	VERB
iajs-2969	259	2	f	f	PRON
iajs-2969	259	3	be	be	AUX
iajs-2969	259	4	any	any	DET
iajs-2969	259	5	ϣ1	ϣ1	NOUN
iajs-2969	259	6	-	-	PUNCT
iajs-2969	259	7	semi	semi	ADJ
iajs-2969	259	8	-	-	ADJ
iajs-2969	259	9	p	p	ADJ
iajs-2969	259	10	-	-	PUNCT
iajs-2969	259	11	closed	close	VERB
iajs-2969	259	12	set	set	NOUN
iajs-2969	259	13	,	,	PUNCT
iajs-2969	259	14	so	so	CCONJ
iajs-2969	259	15	(	(	PUNCT
iajs-2969	259	16	z	z	NOUN
iajs-2969	259	17	−	−	PROPN
iajs-2969	259	18	f	f	X
iajs-2969	259	19	)	)	PUNCT
iajs-2969	259	20	is	be	AUX
iajs-2969	259	21	an	an	DET
iajs-2969	259	22	ϣ1	ϣ1	NOUN
iajs-2969	259	23	-	-	PUNCT
iajs-2969	259	24	semi	semi	ADJ
iajs-2969	259	25	-	-	ADJ
iajs-2969	259	26	p	p	ADJ
iajs-2969	259	27	-	-	PUNCT
iajs-2969	259	28	open	open	ADJ
iajs-2969	259	29	set	set	NOUN
iajs-2969	259	30	,	,	PUNCT
iajs-2969	259	31	then	then	ADV
iajs-2969	259	32	𝑓(z	𝑓(z	VERB
iajs-2969	259	33	−	−	PROPN
iajs-2969	259	34	f	f	X
iajs-2969	259	35	)	)	PUNCT
iajs-2969	259	36	is	be	AUX
iajs-2969	259	37	an	an	DET
iajs-2969	259	38	ϣ2	ϣ2	NOUN
iajs-2969	259	39	-	-	PUNCT
iajs-2969	259	40	semi	semi	ADJ
iajs-2969	259	41	-	-	ADJ
iajs-2969	259	42	p	p	ADJ
iajs-2969	259	43	-	-	PUNCT
iajs-2969	259	44	open	open	ADJ
iajs-2969	259	45	set	set	NOUN
iajs-2969	259	46	(	(	PUNCT
iajs-2969	259	47	since	since	SCONJ
iajs-2969	259	48	𝑓	𝑓	PRON
iajs-2969	259	49	is	be	AUX
iajs-2969	259	50	an	an	DET
iajs-2969	259	51	(	(	PUNCT
iajs-2969	259	52	ϣ1	ϣ1	NOUN
iajs-2969	259	53	,	,	PUNCT
iajs-2969	259	54	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	259	55	-	-	PUNCT
iajs-2969	259	56	semi	semi	ADJ
iajs-2969	259	57	-	-	ADJ
iajs-2969	259	58	p	p	ADJ
iajs-2969	259	59	-	-	PUNCT
iajs-2969	259	60	open	open	ADJ
iajs-2969	259	61	function	function	NOUN
iajs-2969	259	62	)	)	PUNCT
iajs-2969	259	63	,	,	PUNCT
iajs-2969	259	64	but	but	CCONJ
iajs-2969	259	65	𝑓(z	𝑓(z	NOUN
iajs-2969	259	66	−	−	NOUN
iajs-2969	260	1	f	f	X
iajs-2969	260	2	)	)	PUNCT
iajs-2969	260	3	=	=	SYM
iajs-2969	261	1	y	y	PROPN
iajs-2969	261	2	−	−	PROPN
iajs-2969	261	3	𝑓(f	𝑓(f	PROPN
iajs-2969	261	4	)	)	PUNCT
iajs-2969	261	5	,	,	PUNCT
iajs-2969	261	6	therefore	therefore	ADV
iajs-2969	261	7	𝑓(f	𝑓(f	PROPN
iajs-2969	261	8	)	)	PUNCT
iajs-2969	261	9	is	be	AUX
iajs-2969	261	10	an	an	DET
iajs-2969	261	11	ϣ2	ϣ2	NOUN
iajs-2969	261	12	-	-	PUNCT
iajs-2969	261	13	semi	semi	ADJ
iajs-2969	261	14	-	-	ADJ
iajs-2969	261	15	p	p	ADJ
iajs-2969	261	16	-	-	PUNCT
iajs-2969	261	17	closed	closed	ADJ
iajs-2969	261	18	.	.	PUNCT
iajs-2969	262	1	hence	hence	ADV
iajs-2969	262	2	𝑓	𝑓	DET
iajs-2969	262	3	an	an	DET
iajs-2969	262	4	(	(	PUNCT
iajs-2969	262	5	ϣ1	ϣ1	NOUN
iajs-2969	262	6	,	,	PUNCT
iajs-2969	262	7	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	262	8	-	-	PUNCT
iajs-2969	262	9	semip	semip	NOUN
iajs-2969	262	10	-	-	PUNCT
iajs-2969	262	11	closed	close	VERB
iajs-2969	262	12	function	function	NOUN
iajs-2969	262	13	.	.	PUNCT
iajs-2969	263	1	the	the	DET
iajs-2969	263	2	"	"	PUNCT
iajs-2969	263	3	only	only	ADV
iajs-2969	263	4	if	if	SCONJ
iajs-2969	263	5	"	"	PUNCT
iajs-2969	263	6	part	part	NOUN
iajs-2969	263	7	.	.	PUNCT
iajs-2969	264	1	let	let	VERB
iajs-2969	264	2	ħ	ħ	NOUN
iajs-2969	264	3	be	be	AUX
iajs-2969	264	4	any	any	DET
iajs-2969	264	5	ϣ1	ϣ1	NOUN
iajs-2969	264	6	-	-	PUNCT
iajs-2969	264	7	semi	semi	ADJ
iajs-2969	264	8	-	-	ADJ
iajs-2969	264	9	p	p	ADJ
iajs-2969	264	10	-	-	PUNCT
iajs-2969	264	11	open	open	NOUN
iajs-2969	264	12	set	set	NOUN
iajs-2969	264	13	,	,	PUNCT
iajs-2969	264	14	so	so	CCONJ
iajs-2969	264	15	(	(	PUNCT
iajs-2969	264	16	z	z	NOUN
iajs-2969	264	17	−	−	PROPN
iajs-2969	264	18	ħ	ħ	NOUN
iajs-2969	264	19	)	)	PUNCT
iajs-2969	264	20	is	be	AUX
iajs-2969	264	21	an	an	DET
iajs-2969	264	22	ϣ1	ϣ1	NOUN
iajs-2969	264	23	-	-	PUNCT
iajs-2969	264	24	semi	semi	ADJ
iajs-2969	264	25	-	-	ADJ
iajs-2969	264	26	p	p	ADJ
iajs-2969	264	27	-	-	PUNCT
iajs-2969	264	28	closed	close	VERB
iajs-2969	264	29	set	set	NOUN
iajs-2969	264	30	,	,	PUNCT
iajs-2969	264	31	then	then	ADV
iajs-2969	264	32	𝑓(z	𝑓(z	VERB
iajs-2969	264	33	−	−	NOUN
iajs-2969	264	34	ħ	ħ	NOUN
iajs-2969	264	35	)	)	PUNCT
iajs-2969	264	36	is	be	AUX
iajs-2969	264	37	an	an	DET
iajs-2969	264	38	ϣ2	ϣ2	NOUN
iajs-2969	264	39	-	-	PUNCT
iajs-2969	264	40	semi	semi	ADJ
iajs-2969	264	41	-	-	ADJ
iajs-2969	264	42	p	p	ADJ
iajs-2969	264	43	-	-	PUNCT
iajs-2969	264	44	closed	close	VERB
iajs-2969	264	45	set	set	NOUN
iajs-2969	264	46	(	(	PUNCT
iajs-2969	264	47	since	since	SCONJ
iajs-2969	264	48	𝑓	𝑓	PRON
iajs-2969	264	49	is	be	AUX
iajs-2969	264	50	an	an	DET
iajs-2969	264	51	(	(	PUNCT
iajs-2969	264	52	ϣ1	ϣ1	NOUN
iajs-2969	264	53	,	,	PUNCT
iajs-2969	264	54	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	264	55	-	-	PUNCT
iajs-2969	264	56	semi	semi	ADJ
iajs-2969	264	57	-	-	ADJ
iajs-2969	264	58	p	p	ADJ
iajs-2969	264	59	-	-	PUNCT
iajs-2969	264	60	closed	closed	ADJ
iajs-2969	264	61	function	function	NOUN
iajs-2969	264	62	)	)	PUNCT
iajs-2969	264	63	,	,	PUNCT
iajs-2969	264	64	but	but	CCONJ
iajs-2969	264	65	f(z	f(z	NOUN
iajs-2969	265	1	−	−	PROPN
iajs-2969	266	1	ħ	ħ	NOUN
iajs-2969	266	2	)	)	PUNCT
iajs-2969	266	3	=	=	SYM
iajs-2969	266	4	y	y	PROPN
iajs-2969	266	5	−	−	PROPN
iajs-2969	266	6	𝑓(ħ	𝑓(ħ	PROPN
iajs-2969	266	7	)	)	PUNCT
iajs-2969	266	8	,	,	PUNCT
iajs-2969	266	9	therefore	therefore	ADV
iajs-2969	266	10	𝑓(ħ	𝑓(ħ	PROPN
iajs-2969	266	11	)	)	PUNCT
iajs-2969	266	12	is	be	AUX
iajs-2969	266	13	an	an	DET
iajs-2969	266	14	ϣ2	ϣ2	NOUN
iajs-2969	266	15	-	-	PUNCT
iajs-2969	266	16	semi	semi	ADJ
iajs-2969	266	17	-	-	ADJ
iajs-2969	266	18	p	p	ADJ
iajs-2969	266	19	-	-	NOUN
iajs-2969	266	20	open	open	ADJ
iajs-2969	266	21	.	.	PUNCT
iajs-2969	267	1	hence	hence	ADV
iajs-2969	267	2	𝑓	𝑓	DET
iajs-2969	267	3	an	an	DET
iajs-2969	267	4	(	(	PUNCT
iajs-2969	267	5	ϣ1	ϣ1	NOUN
iajs-2969	267	6	,	,	PUNCT
iajs-2969	267	7	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	267	8	-	-	PUNCT
iajs-2969	267	9	semip	semip	NOUN
iajs-2969	267	10	-	-	PUNCT
iajs-2969	267	11	closed	close	VERB
iajs-2969	267	12	function	function	NOUN
iajs-2969	267	13	.	.	PUNCT
iajs-2969	268	1	theorem	theorem	VERB
iajs-2969	268	2	4.16	4.16	NUM
iajs-2969	268	3	let	let	VERB
iajs-2969	268	4	𝑓	𝑓	DET
iajs-2969	268	5	∶	∶	NOUN
iajs-2969	268	6	(	(	PUNCT
iajs-2969	268	7	z	z	NOUN
iajs-2969	268	8	,	,	PUNCT
iajs-2969	268	9	ϣ1	ϣ1	PROPN
iajs-2969	268	10	)	)	PUNCT
iajs-2969	268	11	→	→	SYM
iajs-2969	268	12	(	(	PUNCT
iajs-2969	268	13	y	y	PROPN
iajs-2969	268	14	,	,	PUNCT
iajs-2969	268	15	ϣ2	ϣ2	PROPN
iajs-2969	268	16	)	)	PUNCT
iajs-2969	268	17	be	be	AUX
iajs-2969	268	18	a	a	DET
iajs-2969	268	19	bijective	bijective	ADJ
iajs-2969	268	20	function	function	NOUN
iajs-2969	268	21	,	,	PUNCT
iajs-2969	268	22	then	then	ADV
iajs-2969	268	23	𝑓	𝑓	PRON
iajs-2969	268	24	is	be	AUX
iajs-2969	268	25	an	an	DET
iajs-2969	268	26	(	(	PUNCT
iajs-2969	268	27	ϣ1	ϣ1	NOUN
iajs-2969	268	28	,	,	PUNCT
iajs-2969	268	29	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	268	30	-	-	PUNCT
iajs-2969	268	31	semi	semi	ADJ
iajs-2969	268	32	-	-	ADJ
iajs-2969	268	33	p	p	ADJ
iajs-2969	268	34	-	-	PUNCT
iajs-2969	268	35	open	open	ADJ
iajs-2969	268	36	function	function	NOUN
iajs-2969	268	37	,	,	PUNCT
iajs-2969	268	38			PROPN
iajs-2969	268	39	𝑓	𝑓	DET
iajs-2969	268	40	−1	−1	NOUN
iajs-2969	268	41	∶	∶	NOUN
iajs-2969	268	42	(	(	PUNCT
iajs-2969	268	43	y	y	PROPN
iajs-2969	268	44	,	,	PUNCT
iajs-2969	268	45	ϣ2	ϣ2	PROPN
iajs-2969	268	46	)	)	PUNCT
iajs-2969	268	47	→	→	PUNCT
iajs-2969	268	48	(	(	PUNCT
iajs-2969	268	49	z	z	NOUN
iajs-2969	268	50	,	,	PUNCT
iajs-2969	268	51	ϣ1	ϣ1	PROPN
iajs-2969	268	52	)	)	PUNCT
iajs-2969	268	53	is	be	AUX
iajs-2969	268	54	an	an	DET
iajs-2969	268	55	(	(	PUNCT
iajs-2969	268	56	ϣ1	ϣ1	NOUN
iajs-2969	268	57	,	,	PUNCT
iajs-2969	268	58	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	268	59	-	-	ADJ
iajs-2969	268	60	p	p	ADJ
iajs-2969	268	61	-	-	PUNCT
iajs-2969	268	62	irresolute	irresolute	ADJ
iajs-2969	268	63	function	function	NOUN
iajs-2969	268	64	.	.	PUNCT
iajs-2969	269	1	proof	proof	NOUN
iajs-2969	269	2	the	the	DET
iajs-2969	269	3	"	"	PUNCT
iajs-2969	269	4	if	if	SCONJ
iajs-2969	269	5	"	"	PUNCT
iajs-2969	269	6	part	part	NOUN
iajs-2969	269	7	.	.	PUNCT
iajs-2969	270	1	suppose	suppose	VERB
iajs-2969	270	2	that	that	SCONJ
iajs-2969	270	3	f	f	PROPN
iajs-2969	270	4	is	be	AUX
iajs-2969	270	5	an	an	DET
iajs-2969	270	6	(	(	PUNCT
iajs-2969	270	7	ϣ1	ϣ1	NOUN
iajs-2969	270	8	,	,	PUNCT
iajs-2969	270	9	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	270	10	-	-	PUNCT
iajs-2969	270	11	semi	semi	ADJ
iajs-2969	270	12	-	-	ADJ
iajs-2969	270	13	p	p	ADJ
iajs-2969	270	14	-	-	PUNCT
iajs-2969	270	15	open	open	ADJ
iajs-2969	270	16	function	function	NOUN
iajs-2969	270	17	,	,	PUNCT
iajs-2969	270	18	to	to	PART
iajs-2969	270	19	show	show	VERB
iajs-2969	270	20	that	that	SCONJ
iajs-2969	270	21	f	f	PROPN
iajs-2969	270	22	−1	−1	NOUN
iajs-2969	270	23	is	be	AUX
iajs-2969	270	24	an	an	DET
iajs-2969	270	25	(	(	PUNCT
iajs-2969	270	26	ϣ1	ϣ1	NOUN
iajs-2969	270	27	,	,	PUNCT
iajs-2969	270	28	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	270	29	-	-	ADJ
iajs-2969	270	30	p	p	ADJ
iajs-2969	270	31	-	-	PUNCT
iajs-2969	270	32	irresolute	irresolute	ADJ
iajs-2969	270	33	function	function	NOUN
iajs-2969	270	34	.	.	PUNCT
iajs-2969	271	1	let	let	VERB
iajs-2969	271	2	ħ	ħ	NOUN
iajs-2969	271	3	be	be	AUX
iajs-2969	271	4	any	any	DET
iajs-2969	271	5	ϣ1	ϣ1	NOUN
iajs-2969	271	6	-	-	PUNCT
iajs-2969	271	7	semi	semi	ADJ
iajs-2969	271	8	-	-	ADJ
iajs-2969	271	9	p	p	ADJ
iajs-2969	271	10	-	-	PUNCT
iajs-2969	271	11	open	open	NOUN
iajs-2969	271	12	set	set	NOUN
iajs-2969	271	13	in	in	ADP
iajs-2969	271	14	z	z	NOUN
iajs-2969	271	15	,	,	PUNCT
iajs-2969	271	16	then	then	ADV
iajs-2969	271	17	(	(	PUNCT
iajs-2969	271	18	f	f	PROPN
iajs-2969	271	19	−1)−1(ħ	−1)−1(ħ	PROPN
iajs-2969	271	20	)	)	PUNCT
iajs-2969	271	21	=	=	PUNCT
iajs-2969	272	1	ihjpas	ihjpa	VERB
iajs-2969	272	2	.	.	PUNCT
iajs-2969	273	1	36(1)2023	36(1)2023	NUM
iajs-2969	273	2	344	344	NUM
iajs-2969	273	3	f(ħ	f(ħ	NOUN
iajs-2969	273	4	)	)	PUNCT
iajs-2969	273	5	is	be	AUX
iajs-2969	273	6	an	an	DET
iajs-2969	273	7	ϣ2	ϣ2	NOUN
iajs-2969	273	8	-	-	PUNCT
iajs-2969	273	9	semi	semi	ADJ
iajs-2969	273	10	-	-	ADJ
iajs-2969	273	11	p	p	ADJ
iajs-2969	273	12	-	-	PUNCT
iajs-2969	273	13	open	open	NOUN
iajs-2969	273	14	set	set	NOUN
iajs-2969	273	15	in	in	ADP
iajs-2969	273	16	y	y	PROPN
iajs-2969	273	17	(	(	PUNCT
iajs-2969	273	18	since	since	SCONJ
iajs-2969	273	19	f	f	PROPN
iajs-2969	273	20	is	be	AUX
iajs-2969	273	21	an	an	DET
iajs-2969	273	22	(	(	PUNCT
iajs-2969	273	23	ϣ1	ϣ1	NOUN
iajs-2969	273	24	,	,	PUNCT
iajs-2969	273	25	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	273	26	-	-	PUNCT
iajs-2969	273	27	semi	semi	ADJ
iajs-2969	273	28	-	-	ADJ
iajs-2969	273	29	p	p	ADJ
iajs-2969	273	30	-	-	PUNCT
iajs-2969	273	31	open	open	ADJ
iajs-2969	273	32	function	function	NOUN
iajs-2969	273	33	)	)	PUNCT
iajs-2969	273	34	,	,	PUNCT
iajs-2969	273	35	so	so	CCONJ
iajs-2969	273	36	f	f	PROPN
iajs-2969	273	37	−1is	−1is	PROPN
iajs-2969	273	38	an	an	DET
iajs-2969	273	39	(	(	PUNCT
iajs-2969	273	40	ϣ1	ϣ1	NOUN
iajs-2969	273	41	,	,	PUNCT
iajs-2969	273	42	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	273	43	-	-	ADJ
iajs-2969	273	44	pirresolute	pirresolute	ADJ
iajs-2969	273	45	function	function	NOUN
iajs-2969	273	46	.	.	PUNCT
iajs-2969	274	1	the	the	DET
iajs-2969	274	2	"	"	PUNCT
iajs-2969	274	3	only	only	ADV
iajs-2969	274	4	if	if	SCONJ
iajs-2969	274	5	"	"	PUNCT
iajs-2969	274	6	part	part	NOUN
iajs-2969	274	7	.	.	PUNCT
iajs-2969	274	8	suppose	suppose	VERB
iajs-2969	274	9	that	that	SCONJ
iajs-2969	274	10	f	f	PROPN
iajs-2969	274	11	−1	−1	NOUN
iajs-2969	274	12	is	be	AUX
iajs-2969	274	13	an	an	DET
iajs-2969	274	14	(	(	PUNCT
iajs-2969	274	15	ϣ1	ϣ1	NOUN
iajs-2969	274	16	,	,	PUNCT
iajs-2969	274	17	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	274	18	-	-	ADJ
iajs-2969	274	19	p	p	ADJ
iajs-2969	274	20	-	-	PUNCT
iajs-2969	274	21	irresolute	irresolute	ADJ
iajs-2969	274	22	function	function	NOUN
iajs-2969	274	23	,	,	PUNCT
iajs-2969	274	24	to	to	PART
iajs-2969	274	25	show	show	VERB
iajs-2969	274	26	that	that	SCONJ
iajs-2969	274	27	f	f	PROPN
iajs-2969	274	28	is	be	AUX
iajs-2969	274	29	an	an	DET
iajs-2969	274	30	(	(	PUNCT
iajs-2969	274	31	ϣ1	ϣ1	NOUN
iajs-2969	274	32	,	,	PUNCT
iajs-2969	274	33	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	274	34	-	-	PUNCT
iajs-2969	274	35	semi	semi	ADJ
iajs-2969	274	36	-	-	ADJ
iajs-2969	274	37	p	p	ADJ
iajs-2969	274	38	-	-	PUNCT
iajs-2969	274	39	open	open	ADJ
iajs-2969	274	40	function	function	NOUN
iajs-2969	274	41	.	.	PUNCT
iajs-2969	275	1	let	let	VERB
iajs-2969	275	2	ħ	ħ	NOUN
iajs-2969	275	3	be	be	AUX
iajs-2969	275	4	any	any	DET
iajs-2969	275	5	ϣ1	ϣ1	NOUN
iajs-2969	275	6	-	-	PUNCT
iajs-2969	275	7	semi	semi	ADJ
iajs-2969	275	8	-	-	ADJ
iajs-2969	275	9	p	p	ADJ
iajs-2969	275	10	-	-	PUNCT
iajs-2969	275	11	open	open	NOUN
iajs-2969	275	12	set	set	NOUN
iajs-2969	275	13	in	in	ADP
iajs-2969	275	14	z	z	NOUN
iajs-2969	275	15	,	,	PUNCT
iajs-2969	275	16	then	then	ADV
iajs-2969	275	17	(	(	PUNCT
iajs-2969	275	18	f	f	PROPN
iajs-2969	275	19	−1)−1(ħ	−1)−1(ħ	PROPN
iajs-2969	275	20	)	)	PUNCT
iajs-2969	275	21	=	=	PUNCT
iajs-2969	276	1	f(ħ	f(ħ	NOUN
iajs-2969	276	2	)	)	PUNCT
iajs-2969	276	3	is	be	AUX
iajs-2969	276	4	an	an	DET
iajs-2969	276	5	ϣ2	ϣ2	NOUN
iajs-2969	276	6	-	-	PUNCT
iajs-2969	276	7	semi	semi	ADJ
iajs-2969	276	8	-	-	ADJ
iajs-2969	276	9	p	p	ADJ
iajs-2969	276	10	-	-	PUNCT
iajs-2969	276	11	open	open	NOUN
iajs-2969	276	12	set	set	NOUN
iajs-2969	276	13	in	in	ADP
iajs-2969	276	14	y(since	y(since	PROPN
iajs-2969	276	15	f	f	PROPN
iajs-2969	276	16	−1is	−1is	PROPN
iajs-2969	276	17	an	an	DET
iajs-2969	276	18	(	(	PUNCT
iajs-2969	276	19	ϣ1	ϣ1	NOUN
iajs-2969	276	20	,	,	PUNCT
iajs-2969	276	21	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	276	22	-	-	ADJ
iajs-2969	276	23	p	p	ADJ
iajs-2969	276	24	-	-	PUNCT
iajs-2969	276	25	irresolute	irresolute	ADJ
iajs-2969	276	26	function	function	NOUN
iajs-2969	276	27	)	)	PUNCT
iajs-2969	276	28	,	,	PUNCT
iajs-2969	276	29	so	so	CCONJ
iajs-2969	276	30	f	f	PROPN
iajs-2969	276	31	is	be	AUX
iajs-2969	276	32	an	an	DET
iajs-2969	276	33	(	(	PUNCT
iajs-2969	276	34	ϣ1	ϣ1	NOUN
iajs-2969	276	35	,	,	PUNCT
iajs-2969	276	36	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	276	37	-	-	PUNCT
iajs-2969	276	38	semi	semi	ADJ
iajs-2969	276	39	-	-	ADJ
iajs-2969	276	40	p	p	ADJ
iajs-2969	276	41	-	-	PUNCT
iajs-2969	276	42	open	open	ADJ
iajs-2969	276	43	function	function	NOUN
iajs-2969	276	44	.	.	PUNCT
iajs-2969	277	1	definition	definition	NOUN
iajs-2969	277	2	4.17	4.17	NUM
iajs-2969	277	3	a	a	DET
iajs-2969	277	4	bijection	bijection	ADJ
iajs-2969	277	5	function	function	NOUN
iajs-2969	277	6	f	f	PROPN
iajs-2969	277	7	∶	∶	NOUN
iajs-2969	277	8	(	(	PUNCT
iajs-2969	277	9	z	z	NOUN
iajs-2969	277	10	,	,	PUNCT
iajs-2969	277	11	ϣ1	ϣ1	PROPN
iajs-2969	277	12	)	)	PUNCT
iajs-2969	277	13	→	→	SYM
iajs-2969	277	14	(	(	PUNCT
iajs-2969	277	15	y	y	PROPN
iajs-2969	277	16	,	,	PUNCT
iajs-2969	277	17	ϣ2	ϣ2	PROPN
iajs-2969	277	18	)	)	PUNCT
iajs-2969	277	19	is	be	AUX
iajs-2969	277	20	called	call	VERB
iajs-2969	277	21	(	(	PUNCT
iajs-2969	277	22	ϣ1	ϣ1	NOUN
iajs-2969	277	23	,	,	PUNCT
iajs-2969	277	24	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	277	25	-	-	ADJ
iajs-2969	277	26	p	p	ADJ
iajs-2969	277	27	-	-	PUNCT
iajs-2969	277	28	homeomorphism	homeomorphism	NOUN
iajs-2969	277	29	function	function	NOUN
iajs-2969	277	30	if	if	SCONJ
iajs-2969	277	31	f	f	PROPN
iajs-2969	277	32	is	be	AUX
iajs-2969	277	33	both	both	PRON
iajs-2969	277	34	(	(	PUNCT
iajs-2969	277	35	ϣ1	ϣ1	NOUN
iajs-2969	277	36	,	,	PUNCT
iajs-2969	277	37	ϣ2)-semi	ϣ2)-semi	ADJ
iajs-2969	277	38	-	-	ADJ
iajs-2969	277	39	p	p	ADJ
iajs-2969	277	40	-	-	PUNCT
iajs-2969	277	41	irresolute	irresolute	ADJ
iajs-2969	277	42	function	function	NOUN
iajs-2969	277	43	and	and	CCONJ
iajs-2969	277	44	(	(	PUNCT
iajs-2969	277	45	ϣ1	ϣ1	PROPN
iajs-2969	277	46	,	,	PUNCT
iajs-2969	277	47	ϣ2)-m	ϣ2)-m	NOUN
iajs-2969	277	48	-	-	PUNCT
iajs-2969	277	49	semi	semi	ADJ
iajs-2969	277	50	-	-	ADJ
iajs-2969	277	51	p	p	ADJ
iajs-2969	277	52	-	-	PUNCT
iajs-2969	277	53	open	open	ADJ
iajs-2969	277	54	function	function	NOUN
iajs-2969	277	55	.	.	PUNCT
iajs-2969	278	1	references	reference	NOUN
iajs-2969	278	2	1.engelking	1.engelking	NUM
iajs-2969	278	3	,	,	PUNCT
iajs-2969	278	4	r.	r.	PROPN
iajs-2969	278	5	,	,	PUNCT
iajs-2969	278	6	general	general	ADJ
iajs-2969	278	7	topology	topology	NOUN
iajs-2969	278	8	,	,	PUNCT
iajs-2969	278	9	sigma	sigma	PROPN
iajs-2969	278	10	ser	ser	PROPN
iajs-2969	278	11	.	.	PUNCT
iajs-2969	279	1	pure	pure	ADJ
iajs-2969	279	2	math	math	NOUN
iajs-2969	279	3	.	.	PUNCT
iajs-2969	280	1	6	6	NUM
iajs-2969	280	2	,	,	PUNCT
iajs-2969	280	3	heldermann	heldermann	PROPN
iajs-2969	280	4	verlag	verlag	PROPN
iajs-2969	280	5	berlin	berlin	PROPN
iajs-2969	280	6	,	,	PUNCT
iajs-2969	280	7	1989	1989	NUM
iajs-2969	280	8	.	.	PUNCT
iajs-2969	281	1	2.mashhour	2.mashhour	NUM
iajs-2969	281	2	,	,	PUNCT
iajs-2969	281	3	a.s	a.s	PROPN
iajs-2969	281	4	.	.	PROPN
iajs-2969	281	5	;	;	PUNCT
iajs-2969	281	6	abd	abd	PROPN
iajs-2969	281	7	el	el	PROPN
iajs-2969	281	8	-	-	PUNCT
iajs-2969	281	9	monsef	monsef	ADJ
iajs-2969	281	10	,	,	PUNCT
iajs-2969	281	11	m.e	m.e	PROPN
iajs-2969	281	12	.	.	PROPN
iajs-2969	281	13	;	;	PUNCT
iajs-2969	281	14	el	el	PROPN
iajs-2969	281	15	-	-	PUNCT
iajs-2969	281	16	deeb	deeb	PROPN
iajs-2969	281	17	,	,	PUNCT
iajs-2969	281	18	s.n	s.n	PROPN
iajs-2969	281	19	.	.	PROPN
iajs-2969	281	20	on	on	ADP
iajs-2969	281	21	pre	pre	ADJ
iajs-2969	281	22	-	-	ADJ
iajs-2969	281	23	topological	topological	ADJ
iajs-2969	281	24	spaces	space	NOUN
iajs-2969	281	25	sets	set	NOUN
iajs-2969	281	26	,	,	PUNCT
iajs-2969	281	27	bull	bull	NOUN
iajs-2969	281	28	.	.	PUNCT
iajs-2969	282	1	math	math	NOUN
iajs-2969	282	2	.	.	PUNCT
iajs-2969	283	1	dela	dela	PROPN
iajs-2969	283	2	soc	soc	PROPN
iajs-2969	283	3	.	.	PUNCT
iajs-2969	284	1	r.s	r.s	PROPN
iajs-2969	284	2	.	.	PROPN
iajs-2969	284	3	de	de	PROPN
iajs-2969	284	4	roumanie	roumanie	PROPN
iajs-2969	284	5	,	,	PUNCT
iajs-2969	284	6	1984,28(76	1984,28(76	NUM
iajs-2969	284	7	)	)	PUNCT
iajs-2969	284	8	,	,	PUNCT
iajs-2969	284	9	39	39	NUM
iajs-2969	284	10	-	-	SYM
iajs-2969	284	11	45	45	NUM
iajs-2969	284	12	.	.	PUNCT
iajs-2969	285	1	3.navalagi	3.navalagi	NUM
iajs-2969	285	2	g.b.definition	g.b.definition	NOUN
iajs-2969	285	3	bank	bank	NOUN
iajs-2969	285	4	in	in	ADP
iajs-2969	285	5	general	general	ADJ
iajs-2969	285	6	topology	topology	NOUN
iajs-2969	285	7	,	,	PUNCT
iajs-2969	285	8	internet	internet	NOUN
iajs-2969	285	9	2000	2000	NUM
iajs-2969	285	10	.	.	PUNCT
iajs-2969	286	1	4.sharma	4.sharma	NUM
iajs-2969	286	2	,	,	PUNCT
iajs-2969	286	3	l.j.n.topology	l.j.n.topology	NOUN
iajs-2969	286	4	,	,	PUNCT
iajs-2969	286	5	krishna	krishna	VERB
iajs-2969	286	6	prakashan	prakashan	ADJ
iajs-2969	286	7	media	medium	NOUN
iajs-2969	286	8	(	(	PUNCT
iajs-2969	286	9	p	p	NOUN
iajs-2969	286	10	)	)	PUNCT
iajs-2969	286	11	ltd	ltd	PROPN
iajs-2969	286	12	,	,	PUNCT
iajs-2969	286	13	india	india	PROPN
iajs-2969	286	14	,	,	PUNCT
iajs-2969	286	15	twenty	twenty	NUM
iajs-2969	286	16	fifth	fifth	ADJ
iajs-2969	286	17	edition	edition	NOUN
iajs-2969	286	18	,	,	PUNCT
iajs-2969	286	19	2000	2000	NUM
iajs-2969	286	20	.	.	PUNCT
iajs-2969	287	1	5.al	5.al	NUM
iajs-2969	287	2	-	-	PUNCT
iajs-2969	287	3	khazraji	khazraji	NOUN
iajs-2969	287	4	,	,	PUNCT
iajs-2969	287	5	r.b	r.b	PROPN
iajs-2969	287	6	.	.	PROPN
iajs-2969	287	7	,	,	PUNCT
iajs-2969	287	8	on	on	ADP
iajs-2969	287	9	semi	semi	ADJ
iajs-2969	287	10	-	-	ADJ
iajs-2969	287	11	p	p	ADJ
iajs-2969	287	12	-	-	PUNCT
iajs-2969	287	13	open	open	ADJ
iajs-2969	287	14	sets	set	NOUN
iajs-2969	287	15	,	,	PUNCT
iajs-2969	287	16	m.sc	m.sc	PROPN
iajs-2969	287	17	.	.	PUNCT
iajs-2969	288	1	thesis	thesis	NOUN
iajs-2969	288	2	,	,	PUNCT
iajs-2969	288	3	university	university	NOUN
iajs-2969	288	4	of	of	ADP
iajs-2969	288	5	baghdad,2004	baghdad,2004	NOUN
iajs-2969	288	6	.	.	PUNCT
iajs-2969	289	1	6.dhana	6.dhana	NUM
iajs-2969	289	2	balan	balan	PROPN
iajs-2969	289	3	,	,	PUNCT
iajs-2969	289	4	a.p	a.p	PROPN
iajs-2969	289	5	.	.	PROPN
iajs-2969	289	6	;	;	PUNCT
iajs-2969	289	7	padma	padma	NOUN
iajs-2969	289	8	,	,	PUNCT
iajs-2969	289	9	p.	p.	NOUN
iajs-2969	289	10	separation	separation	NOUN
iajs-2969	289	11	spaces	space	VERB
iajs-2969	289	12	in	in	ADP
iajs-2969	289	13	generalized	generalized	ADJ
iajs-2969	289	14	topology	topology	NOUN
iajs-2969	289	15	,	,	PUNCT
iajs-2969	289	16	international	international	ADJ
iajs-2969	289	17	journal	journal	NOUN
iajs-2969	289	18	of	of	ADP
iajs-2969	289	19	mathematics	mathematics	PROPN
iajs-2969	289	20	research	research	NOUN
iajs-2969	289	21	,	,	PUNCT
iajs-2969	289	22	2017	2017	NUM
iajs-2969	289	23	,	,	PUNCT
iajs-2969	289	24	9	9	NUM
iajs-2969	289	25	,	,	PUNCT
iajs-2969	289	26	1	1	NUM
iajs-2969	289	27	,	,	PUNCT
iajs-2969	289	28	65	65	NUM
iajs-2969	289	29	-	-	SYM
iajs-2969	289	30	74	74	NUM
iajs-2969	289	31	.	.	PUNCT
iajs-2969	290	1	issn	issn	PROPN
iajs-2969	290	2	0976	0976	NUM
iajs-2969	290	3	-	-	SYM
iajs-2969	290	4	5840	5840	NUM
iajs-2969	290	5	.	.	PUNCT
iajs-2969	291	1	7.suaad	7.suaad	NUM
iajs-2969	291	2	,	,	PUNCT
iajs-2969	291	3	g.	g.	PROPN
iajs-2969	291	4	gasim	gasim	PROPN
iajs-2969	291	5	;	;	PUNCT
iajs-2969	291	6	muna	muna	PROPN
iajs-2969	291	7	l.	l.	PROPN
iajs-2969	291	8	abd	abd	PROPN
iajs-2969	291	9	ul	ul	PROPN
iajs-2969	291	10	ridha	ridha	PROPN
iajs-2969	291	11	,	,	PUNCT
iajs-2969	291	12	new	new	ADJ
iajs-2969	291	13	open	open	ADJ
iajs-2969	291	14	set	set	NOUN
iajs-2969	291	15	on	on	ADP
iajs-2969	291	16	topological	topological	ADJ
iajs-2969	291	17	space	space	NOUN
iajs-2969	291	18	with	with	ADP
iajs-2969	291	19	generalized	generalized	ADJ
iajs-2969	291	20	topology	topology	NOUN
iajs-2969	291	21	,	,	PUNCT
iajs-2969	291	22	journal	journal	NOUN
iajs-2969	291	23	of	of	ADP
iajs-2969	291	24	discrete	discrete	ADJ
iajs-2969	291	25	mathematical	mathematical	ADJ
iajs-2969	291	26	sciences	science	NOUN
iajs-2969	291	27	and	and	CCONJ
iajs-2969	291	28	cryptography	cryptography	NOUN
iajs-2969	291	29	,	,	PUNCT
iajs-2969	291	30	to	to	PART
iajs-2969	291	31	appear	appear	VERB
iajs-2969	291	32	.	.	PUNCT
iajs-2969	292	1	8.basdouria	8.basdouria	NUM
iajs-2969	292	2	,	,	PUNCT
iajs-2969	292	3	i.	i.	NOUN
iajs-2969	292	4	;	;	PUNCT
iajs-2969	292	5	messaouda	messaouda	PROPN
iajs-2969	292	6	,	,	PUNCT
iajs-2969	292	7	r.	r.	PROPN
iajs-2969	292	8	;	;	PUNCT
iajs-2969	292	9	missaouia	missaouia	NOUN
iajs-2969	292	10	,	,	PUNCT
iajs-2969	292	11	a.	a.	NOUN
iajs-2969	292	12	connected	connect	VERB
iajs-2969	292	13	and	and	CCONJ
iajs-2969	292	14	hyperconnected	hyperconnecte	VERB
iajs-2969	292	15	generalized	generalized	ADJ
iajs-2969	292	16	topological	topological	ADJ
iajs-2969	292	17	spaces	space	NOUN
iajs-2969	292	18	,	,	PUNCT
iajs-2969	292	19	journal	journal	NOUN
iajs-2969	292	20	of	of	ADP
iajs-2969	292	21	linear	linear	PROPN
iajs-2969	292	22	and	and	CCONJ
iajs-2969	292	23	topological	topological	ADJ
iajs-2969	292	24	algebra	algebra	NOUN
iajs-2969	292	25	,	,	PUNCT
iajs-2969	292	26	2016	2016	NUM
iajs-2969	292	27	,	,	PUNCT
iajs-2969	292	28	05	05	NUM
iajs-2969	292	29	,	,	PUNCT
iajs-2969	292	30	04,229234	04,229234	NUM
iajs-2969	292	31	9.suaad	9.suaad	NUM
iajs-2969	292	32	,	,	PUNCT
iajs-2969	292	33	g.	g.	PROPN
iajs-2969	292	34	gasim	gasim	PROPN
iajs-2969	292	35	;	;	PUNCT
iajs-2969	292	36	mohanad	mohanad	ADJ
iajs-2969	292	37	,	,	PUNCT
iajs-2969	292	38	n.	n.	PROPN
iajs-2969	292	39	jaafar	jaafar	ADV
iajs-2969	292	40	,	,	PUNCT
iajs-2969	292	41	new	new	ADJ
iajs-2969	292	42	normality	normality	NOUN
iajs-2969	292	43	on	on	ADP
iajs-2969	292	44	generalized	generalized	ADJ
iajs-2969	292	45	topological	topological	ADJ
iajs-2969	292	46	spaces	space	NOUN
iajs-2969	292	47	,	,	PUNCT
iajs-2969	292	48	journal	journal	NOUN
iajs-2969	292	49	of	of	ADP
iajs-2969	292	50	physics	physics	PROPN
iajs-2969	292	51	:	:	PUNCT
iajs-2969	292	52	conference	conference	NOUN
iajs-2969	292	53	series	series	NOUN
iajs-2969	292	54	,	,	PUNCT
iajs-2969	292	55	2021	2021	NUM
iajs-2969	292	56	.	.	PUNCT