id	sid	tid	token	lemma	pos
iajs-2555	1	1	ibn	ibn	PROPN
iajs-2555	1	2	al	al	PROPN
iajs-2555	1	3	-	-	PUNCT
iajs-2555	1	4	haitham	haitham	PROPN
iajs-2555	1	5	jour	jour	X
iajs-2555	1	6	.	.	PROPN
iajs-2555	1	7	for	for	ADP
iajs-2555	1	8	pure	pure	ADJ
iajs-2555	1	9	&	&	CCONJ
iajs-2555	1	10	appl	appl	PROPN
iajs-2555	1	11	.	.	PUNCT
iajs-2555	2	1	sci	sci	PROPN
iajs-2555	2	2	.	.	PROPN
iajs-2555	3	1	34	34	NUM
iajs-2555	3	2	(	(	PUNCT
iajs-2555	3	3	1	1	NUM
iajs-2555	3	4	)	)	PUNCT
iajs-2555	3	5	2021	2021	NUM
iajs-2555	3	6	30	30	NUM
iajs-2555	3	7	𝜶𝒈ị	𝜶𝒈ị	ADV
iajs-2555	3	8	-	-	PUNCT
iajs-2555	3	9	open	open	ADJ
iajs-2555	3	10	sets	set	NOUN
iajs-2555	3	11	and	and	CCONJ
iajs-2555	3	12	𝜶𝒈ị	𝜶𝒈ị	ADJ
iajs-2555	3	13	-	-	PUNCT
iajs-2555	3	14	functions	function	NOUN
iajs-2555	3	15	sufyan	sufyan	PROPN
iajs-2555	3	16	g.	g.	PROPN
iajs-2555	3	17	saeed	saeed	PROPN
iajs-2555	3	18	rana	rana	PROPN
iajs-2555	3	19	b.	b.	PROPN
iajs-2555	3	20	esmaeel	esmaeel	PROPN
iajs-2555	3	21	department	department	PROPN
iajs-2555	3	22	of	of	ADP
iajs-2555	3	23	mathematics	mathematic	NOUN
iajs-2555	3	24	,	,	PUNCT
iajs-2555	3	25	ibn	ibn	PROPN
iajs-2555	3	26	al	al	PROPN
iajs-2555	3	27	-	-	PUNCT
iajs-2555	3	28	haitham	haitham	PROPN
iajs-2555	3	29	,	,	PUNCT
iajs-2555	3	30	college	college	NOUN
iajs-2555	3	31	of	of	ADP
iajs-2555	3	32	education	education	PROPN
iajs-2555	3	33	university	university	PROPN
iajs-2555	3	34	of	of	ADP
iajs-2555	3	35	baghdad	baghdad	PROPN
iajs-2555	3	36	,	,	PUNCT
iajs-2555	3	37	iraq	iraq	PROPN
iajs-2555	3	38	sufsuf201030@gmail.com	sufsuf201030@gmail.com	PROPN
iajs-2555	3	39	,	,	PUNCT
iajs-2555	3	40	ranamumosa@yahoo.com	ranamumosa@yahoo.com	X
iajs-2555	3	41	abstract	abstract	ADJ
iajs-2555	3	42	.	.	PUNCT
iajs-2555	4	1	the	the	DET
iajs-2555	4	2	objective	objective	NOUN
iajs-2555	4	3	of	of	ADP
iajs-2555	4	4	this	this	DET
iajs-2555	4	5	paper	paper	NOUN
iajs-2555	4	6	is	be	AUX
iajs-2555	4	7	to	to	PART
iajs-2555	4	8	show	show	VERB
iajs-2555	4	9	modern	modern	ADJ
iajs-2555	4	10	class	class	NOUN
iajs-2555	4	11	of	of	ADP
iajs-2555	4	12	open	open	ADJ
iajs-2555	4	13	sets	set	NOUN
iajs-2555	4	14	which	which	PRON
iajs-2555	4	15	is	be	AUX
iajs-2555	4	16	an	an	DET
iajs-2555	4	17	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	4	18	-	-	PUNCT
iajs-2555	4	19	open	open	ADJ
iajs-2555	4	20	.	.	PUNCT
iajs-2555	5	1	some	some	DET
iajs-2555	5	2	functions	function	NOUN
iajs-2555	5	3	via	via	ADP
iajs-2555	5	4	this	this	DET
iajs-2555	5	5	concept	concept	NOUN
iajs-2555	5	6	and	and	CCONJ
iajs-2555	5	7	the	the	DET
iajs-2555	5	8	relationships	relationship	NOUN
iajs-2555	5	9	among	among	ADP
iajs-2555	5	10	continuous	continuous	ADJ
iajs-2555	5	11	function	function	NOUN
iajs-2555	5	12	strongly	strongly	ADV
iajs-2555	5	13	𝛼𝑔ịcontinuous	𝛼𝑔ịcontinuous	ADJ
iajs-2555	5	14	function	function	VERB
iajs-2555	5	15	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	5	16	-	-	PUNCT
iajs-2555	5	17	irresolute	irresolute	ADJ
iajs-2555	5	18	function	function	NOUN
iajs-2555	5	19	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	5	20	-	-	PUNCT
iajs-2555	5	21	continuous	continuous	ADJ
iajs-2555	5	22	function	function	NOUN
iajs-2555	5	23	are	be	AUX
iajs-2555	5	24	studied	study	VERB
iajs-2555	5	25	.	.	PUNCT
iajs-2555	6	1	keywords	keyword	NOUN
iajs-2555	6	2	.	.	PUNCT
iajs-2555	7	1	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	7	2	-	-	PUNCT
iajs-2555	7	3	closed	closed	ADJ
iajs-2555	7	4	set	set	NOUN
iajs-2555	7	5	,	,	PUNCT
iajs-2555	7	6	𝛼𝑔ị𝑂-functions	𝛼𝑔ị𝑂-function	NOUN
iajs-2555	7	7	,	,	PUNCT
iajs-2555	7	8	𝛼𝑔ị𝐶-functions	𝛼𝑔ị𝐶-function	NOUN
iajs-2555	7	9	,	,	PUNCT
iajs-2555	7	10	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	7	11	-	-	PUNCT
iajs-2555	7	12	continuous	continuous	ADJ
iajs-2555	7	13	function	function	NOUN
iajs-2555	7	14	,	,	PUNCT
iajs-2555	7	15	strongly	strongly	ADV
iajs-2555	7	16	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	7	17	-	-	PUNCT
iajs-2555	7	18	continuous	continuous	ADJ
iajs-2555	7	19	function	function	NOUN
iajs-2555	7	20	,	,	PUNCT
iajs-2555	7	21	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	7	22	-	-	PUNCT
iajs-2555	7	23	irresolute	irresolute	ADJ
iajs-2555	7	24	function	function	NOUN
iajs-2555	7	25	,	,	PUNCT
iajs-2555	7	26	ideal	ideal	ADJ
iajs-2555	7	27	.	.	PUNCT
iajs-2555	8	1	1	1	X
iajs-2555	8	2	.	.	X
iajs-2555	8	3	introduction	introduction	NOUN
iajs-2555	8	4	.	.	PUNCT
iajs-2555	9	1	an	an	DET
iajs-2555	9	2	α	α	X
iajs-2555	9	3	-	-	ADJ
iajs-2555	9	4	open	open	ADJ
iajs-2555	9	5	was	be	AUX
iajs-2555	9	6	studied	study	VERB
iajs-2555	9	7	in	in	ADP
iajs-2555	9	8	1965	1965	NUM
iajs-2555	9	9	by	by	ADP
iajs-2555	9	10	o.	o.	PROPN
iajs-2555	9	11	njastad	njastad	PROPN
iajs-2555	9	12	,	,	PUNCT
iajs-2555	9	13	a	a	DET
iajs-2555	9	14	subset	subset	NOUN
iajs-2555	9	15	ç	ç	X
iajs-2555	9	16	is	be	AUX
iajs-2555	9	17	α	α	DET
iajs-2555	9	18	-	-	ADJ
iajs-2555	9	19	open	open	ADJ
iajs-2555	9	20	set	set	NOUN
iajs-2555	9	21	if	if	SCONJ
iajs-2555	9	22	ç	ç	PROPN
iajs-2555	9	23	⊆	⊆	NUM
iajs-2555	9	24	𝑖𝑛𝑡(𝑐𝑙(𝑖𝑛𝑡(ç)))[1,2	𝑖𝑛𝑡(𝑐𝑙(𝑖𝑛𝑡(ç)))[1,2	PROPN
iajs-2555	9	25	]	]	PUNCT
iajs-2555	9	26	.	.	PUNCT
iajs-2555	10	1	the	the	DET
iajs-2555	10	2	notion	notion	NOUN
iajs-2555	10	3	of	of	ADP
iajs-2555	10	4	ideal	ideal	NOUN
iajs-2555	10	5	was	be	AUX
iajs-2555	10	6	studied	study	VERB
iajs-2555	10	7	by	by	ADP
iajs-2555	10	8	kuratowski[3,4],that	kuratowski[3,4],that	PROPN
iajs-2555	10	9	ị	ị	PROPN
iajs-2555	10	10	is	be	AUX
iajs-2555	10	11	an	an	DET
iajs-2555	10	12	ideal	ideal	NOUN
iajs-2555	10	13	on	on	ADP
iajs-2555	10	14	ӽ	ӽ	NOUN
iajs-2555	10	15	,	,	PUNCT
iajs-2555	10	16	where	where	SCONJ
iajs-2555	10	17	ị	ị	PROPN
iajs-2555	10	18	is	be	AUX
iajs-2555	10	19	a	a	DET
iajs-2555	10	20	collection	collection	NOUN
iajs-2555	10	21	of	of	ADP
iajs-2555	10	22	all	all	DET
iajs-2555	10	23	subsets	subset	NOUN
iajs-2555	10	24	of	of	ADP
iajs-2555	10	25	ӽ	ӽ	PRON
iajs-2555	10	26	an	an	DET
iajs-2555	10	27	ideal	ideal	NOUN
iajs-2555	10	28	have	have	AUX
iajs-2555	10	29	two	two	NUM
iajs-2555	10	30	properties	property	NOUN
iajs-2555	10	31	(	(	PUNCT
iajs-2555	10	32	if	if	SCONJ
iajs-2555	10	33	ç	ç	X
iajs-2555	10	34	,	,	PUNCT
iajs-2555	10	35	ð	ð	PROPN
iajs-2555	10	36	∈	∈	PROPN
iajs-2555	11	1	ị	ị	X
iajs-2555	11	2	,	,	PUNCT
iajs-2555	11	3	then	then	ADV
iajs-2555	11	4	ç	ç	X
iajs-2555	11	5	∪	∪	X
iajs-2555	11	6	ð	ð	PROPN
iajs-2555	11	7	∈	∈	PROPN
iajs-2555	11	8	ị	ị	X
iajs-2555	11	9	)	)	PUNCT
iajs-2555	11	10	and	and	CCONJ
iajs-2555	11	11	(	(	PUNCT
iajs-2555	11	12	if	if	SCONJ
iajs-2555	11	13	ç	ç	X
iajs-2555	11	14	∈	∈	PROPN
iajs-2555	11	15	ị	ị	X
iajs-2555	11	16	and	and	CCONJ
iajs-2555	11	17	ð	ð	PROPN
iajs-2555	11	18	⊆	⊆	NUM
iajs-2555	11	19	ç	ç	NOUN
iajs-2555	11	20	,	,	PUNCT
iajs-2555	11	21	then	then	ADV
iajs-2555	11	22	ð	ð	PROPN
iajs-2555	11	23	∈	∈	PROPN
iajs-2555	12	1	ị	ị	X
iajs-2555	12	2	.	.	PUNCT
iajs-2555	13	1	there	there	PRON
iajs-2555	13	2	are	be	VERB
iajs-2555	13	3	many	many	ADJ
iajs-2555	13	4	types	type	NOUN
iajs-2555	13	5	for	for	ADP
iajs-2555	13	6	the	the	DET
iajs-2555	13	7	ideal[5	ideal[5	NOUN
iajs-2555	13	8	-	-	PUNCT
iajs-2555	13	9	7	7	NUM
iajs-2555	13	10	]	]	PUNCT
iajs-2555	13	11	i.	i.	NOUN
iajs-2555	13	12	ị{∅	ị{∅	PROPN
iajs-2555	13	13	}	}	PUNCT
iajs-2555	13	14	:	:	PUNCT
iajs-2555	13	15	the	the	DET
iajs-2555	13	16	trivial	trivial	ADJ
iajs-2555	13	17	ideal	ideal	NOUN
iajs-2555	13	18	where	where	SCONJ
iajs-2555	13	19	ị={∅	ị={∅	NOUN
iajs-2555	13	20	}	}	PUNCT
iajs-2555	13	21	.	.	PUNCT
iajs-2555	14	1	ii	ii	PROPN
iajs-2555	14	2	.	.	PUNCT
iajs-2555	15	1	ị𝑛	ị𝑛	ADP
iajs-2555	15	2	:	:	PUNCT
iajs-2555	15	3	the	the	DET
iajs-2555	15	4	ideal	ideal	NOUN
iajs-2555	15	5	of	of	ADP
iajs-2555	15	6	all	all	DET
iajs-2555	15	7	nowhere	nowhere	ADV
iajs-2555	15	8	dense	dense	ADJ
iajs-2555	15	9	sets	set	NOUN
iajs-2555	15	10	ị𝑛	ị𝑛	ADP
iajs-2555	15	11	=	=	VERB
iajs-2555	15	12	{	{	PUNCT
iajs-2555	15	13	ç	ç	X
iajs-2555	15	14	⊆	⊆	NUM
iajs-2555	15	15	ӽ	ӽ	NOUN
iajs-2555	15	16	:	:	PUNCT
iajs-2555	15	17	𝑖𝑛𝑡(𝑐𝑙(ç	𝑖𝑛𝑡(𝑐𝑙(ç	NUM
iajs-2555	15	18	)	)	PUNCT
iajs-2555	15	19	)	)	PUNCT
iajs-2555	16	1	=	=	PRON
iajs-2555	16	2	{	{	PUNCT
iajs-2555	16	3	∅	∅	NOUN
iajs-2555	16	4	}	}	PUNCT
iajs-2555	16	5	}	}	PUNCT
iajs-2555	16	6	.	.	PUNCT
iajs-2555	17	1	iii.ịᶂ	iii.ịᶂ	NOUN
iajs-2555	17	2	:	:	PUNCT
iajs-2555	17	3	the	the	DET
iajs-2555	17	4	ideal	ideal	NOUN
iajs-2555	17	5	of	of	ADP
iajs-2555	17	6	all	all	DET
iajs-2555	17	7	finite	finite	ADJ
iajs-2555	17	8	subsets	subset	NOUN
iajs-2555	17	9	of	of	ADP
iajs-2555	17	10	ӽ	ӽ	PRON
iajs-2555	17	11	ịᶂ	ịᶂ	X
iajs-2555	17	12	=	=	SYM
iajs-2555	17	13	{	{	PUNCT
iajs-2555	17	14	ç	ç	X
iajs-2555	17	15	⊆	⊆	NUM
iajs-2555	17	16	ӽ	ӽ	NOUN
iajs-2555	17	17	:	:	PUNCT
iajs-2555	17	18	ç	ç	PUNCT
iajs-2555	17	19	is	be	AUX
iajs-2555	17	20	a	a	DET
iajs-2555	17	21	finite	finite	ADJ
iajs-2555	17	22	set	set	NOUN
iajs-2555	17	23	}	}	PUNCT
iajs-2555	17	24	.	.	PUNCT
iajs-2555	18	1	the	the	DET
iajs-2555	18	2	collection	collection	NOUN
iajs-2555	18	3	of	of	ADP
iajs-2555	18	4	all	all	DET
iajs-2555	18	5	α	α	PRON
iajs-2555	18	6	-	-	ADJ
iajs-2555	18	7	open	open	ADJ
iajs-2555	18	8	sets	set	NOUN
iajs-2555	18	9	is	be	AUX
iajs-2555	18	10	denoted	denote	VERB
iajs-2555	18	11	by	by	ADP
iajs-2555	18	12	"	"	PUNCT
iajs-2555	18	13	ῖ𝛼	ῖ𝛼	NOUN
iajs-2555	18	14	"	"	PUNCT
iajs-2555	18	15	and	and	CCONJ
iajs-2555	18	16	the	the	DET
iajs-2555	18	17	collection	collection	NOUN
iajs-2555	18	18	of	of	ADP
iajs-2555	18	19	all	all	DET
iajs-2555	18	20	α	α	PRON
iajs-2555	18	21	-	-	ADJ
iajs-2555	18	22	closed	closed	ADJ
iajs-2555	18	23	is	be	AUX
iajs-2555	18	24	denoted	denote	VERB
iajs-2555	18	25	by	by	ADP
iajs-2555	18	26	"	"	PUNCT
iajs-2555	18	27	ɟ𝛼	ɟ𝛼	PROPN
iajs-2555	18	28	"	"	PUNCT
iajs-2555	18	29	.	.	PUNCT
iajs-2555	19	1	in	in	ADP
iajs-2555	19	2	this	this	DET
iajs-2555	19	3	paper	paper	NOUN
iajs-2555	19	4	,	,	PUNCT
iajs-2555	19	5	we	we	PRON
iajs-2555	19	6	introduce	introduce	VERB
iajs-2555	19	7	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	19	8	-	-	PUNCT
iajs-2555	19	9	closed	closed	ADJ
iajs-2555	19	10	set	set	NOUN
iajs-2555	19	11	,	,	PUNCT
iajs-2555	19	12	and	and	CCONJ
iajs-2555	19	13	the	the	DET
iajs-2555	19	14	complement	complement	NOUN
iajs-2555	19	15	of	of	ADP
iajs-2555	19	16	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	19	17	-	-	PUNCT
iajs-2555	19	18	open	open	ADJ
iajs-2555	19	19	set	set	NOUN
iajs-2555	19	20	.	.	PUNCT
iajs-2555	20	1	more	more	ADJ
iajs-2555	20	2	functions	function	NOUN
iajs-2555	20	3	have	have	AUX
iajs-2555	20	4	been	be	AUX
iajs-2555	20	5	introduced	introduce	VERB
iajs-2555	20	6	via	via	ADP
iajs-2555	20	7	these	these	DET
iajs-2555	20	8	concepts	concept	NOUN
iajs-2555	20	9	,	,	PUNCT
iajs-2555	20	10	such	such	ADJ
iajs-2555	20	11	as	as	ADP
iajs-2555	20	12	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	20	13	-	-	PUNCT
iajs-2555	20	14	open	open	ADJ
iajs-2555	20	15	,	,	PUNCT
iajs-2555	20	16	𝛼𝑔ị	𝛼𝑔ị	NOUN
iajs-2555	20	17	∗-open	∗-open	NOUN
iajs-2555	20	18	,	,	PUNCT
iajs-2555	20	19	𝛼𝑔ị	𝛼𝑔ị	NOUN
iajs-2555	20	20	∗∗-open	∗∗-open	NOUN
iajs-2555	20	21	,	,	PUNCT
iajs-2555	20	22	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	20	23	-	-	PUNCT
iajs-2555	20	24	continuous	continuous	ADJ
iajs-2555	20	25	,	,	PUNCT
iajs-2555	20	26	𝛼𝑔ị	𝛼𝑔ị	NOUN
iajs-2555	20	27	-	-	PUNCT
iajs-2555	20	28	irresolute	irresolute	ADJ
iajs-2555	20	29	and	and	CCONJ
iajs-2555	20	30	strongly	strongly	ADV
iajs-2555	20	31	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	20	32	-	-	PUNCT
iajs-2555	20	33	function	function	NOUN
iajs-2555	20	34	.	.	PUNCT
iajs-2555	21	1	ibn	ibn	PROPN
iajs-2555	21	2	al	al	PROPN
iajs-2555	21	3	haitham	haitham	PROPN
iajs-2555	21	4	journal	journal	PROPN
iajs-2555	21	5	for	for	ADP
iajs-2555	21	6	pure	pure	ADJ
iajs-2555	21	7	and	and	CCONJ
iajs-2555	21	8	applied	apply	VERB
iajs-2555	21	9	science	science	NOUN
iajs-2555	21	10	journal	journal	PROPN
iajs-2555	21	11	homepage	homepage	NOUN
iajs-2555	21	12	:	:	PUNCT
iajs-2555	21	13	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2555	21	14	doi	doi	NOUN
iajs-2555	21	15	:	:	PUNCT
iajs-2555	21	16	10.30526/34.1.2555	10.30526/34.1.2555	NUM
iajs-2555	21	17	article	article	NOUN
iajs-2555	21	18	history	history	NOUN
iajs-2555	21	19	:	:	PUNCT
iajs-2555	21	20	received	receive	VERB
iajs-2555	21	21	9,february,2020	9,february,2020	NOUN
iajs-2555	21	22	,	,	PUNCT
iajs-2555	21	23	accepted	accept	VERB
iajs-2555	21	24	15,march,2020	15,march,2020	NOUN
iajs-2555	21	25	,	,	PUNCT
iajs-2555	21	26	published	publish	VERB
iajs-2555	21	27	in	in	ADP
iajs-2555	21	28	january	january	PROPN
iajs-2555	21	29	2021	2021	NUM
iajs-2555	21	30	mailto:sufsuf201030@gmail.com	mailto:sufsuf201030@gmail.com	PROPN
iajs-2555	21	31	mailto:ranamumosa@yahoo.com	mailto:ranamumosa@yahoo.com	PROPN
iajs-2555	21	32	31	31	NUM
iajs-2555	21	33	ibn	ibn	PROPN
iajs-2555	21	34	al	al	PROPN
iajs-2555	21	35	-	-	PUNCT
iajs-2555	21	36	haitham	haitham	PROPN
iajs-2555	21	37	jour	jour	X
iajs-2555	21	38	.	.	PROPN
iajs-2555	22	1	for	for	ADP
iajs-2555	22	2	pure	pure	ADJ
iajs-2555	22	3	&	&	CCONJ
iajs-2555	22	4	appl	appl	PROPN
iajs-2555	22	5	.	.	PUNCT
iajs-2555	23	1	sci	sci	PROPN
iajs-2555	23	2	.	.	PROPN
iajs-2555	24	1	34	34	NUM
iajs-2555	24	2	(	(	PUNCT
iajs-2555	24	3	1	1	NUM
iajs-2555	24	4	)	)	PUNCT
iajs-2555	24	5	2021	2021	NUM
iajs-2555	24	6	2on	2on	NOUN
iajs-2555	24	7	𝜶𝒈ị	𝜶𝒈ị	ADV
iajs-2555	24	8	-	-	PUNCT
iajs-2555	24	9	closed	close	VERB
iajs-2555	24	10	set	set	VERB
iajs-2555	24	11	definition	definition	NOUN
iajs-2555	24	12	1	1	NUM
iajs-2555	24	13	:	:	PUNCT
iajs-2555	24	14	in	in	ADP
iajs-2555	24	15	ideal	ideal	ADJ
iajs-2555	24	16	topological	topological	ADJ
iajs-2555	24	17	space	space	NOUN
iajs-2555	24	18	(	(	PUNCT
iajs-2555	24	19	ӽ	ӽ	X
iajs-2555	24	20	,	,	PUNCT
iajs-2555	24	21	ῖ	ῖ	PROPN
iajs-2555	24	22	,	,	PUNCT
iajs-2555	24	23	ị	ị	PROPN
iajs-2555	24	24	)	)	PUNCT
iajs-2555	24	25	,	,	PUNCT
iajs-2555	24	26	let	let	VERB
iajs-2555	24	27	ç	ç	PROPN
iajs-2555	24	28	⊆	⊆	NUM
iajs-2555	24	29	ӽ.	ӽ.	NOUN
iajs-2555	24	30	ç	ç	PUNCT
iajs-2555	24	31	is	be	AUX
iajs-2555	24	32	said	say	VERB
iajs-2555	24	33	ị-𝛼-g	ị-𝛼-g	ADJ
iajs-2555	24	34	-	-	PUNCT
iajs-2555	24	35	closed	closed	ADJ
iajs-2555	24	36	set	set	NOUN
iajs-2555	24	37	denoted	denote	VERB
iajs-2555	24	38	by	by	ADP
iajs-2555	24	39	"	"	PUNCT
iajs-2555	24	40	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	24	41	-	-	PUNCT
iajs-2555	24	42	closed	closed	ADJ
iajs-2555	24	43	"	"	PUNCT
iajs-2555	24	44	,	,	PUNCT
iajs-2555	24	45	if	if	SCONJ
iajs-2555	24	46	ç	ç	X
iajs-2555	24	47	-	-	PUNCT
iajs-2555	24	48	ơ	ơ	NOUN
iajs-2555	24	49	∈	∈	PROPN
iajs-2555	24	50	ị	ị	X
iajs-2555	24	51	then	then	ADV
iajs-2555	24	52	,	,	PUNCT
iajs-2555	24	53	𝑐𝑙(ç)-ơ	𝑐𝑙(ç)-ơ	VERB
iajs-2555	24	54	∈	∈	PROPN
iajs-2555	25	1	ị	ị	ADP
iajs-2555	25	2	where	where	SCONJ
iajs-2555	25	3	ơ	ơ	PROPN
iajs-2555	25	4	⊆	⊆	PROPN
iajs-2555	25	5	ӽ	ӽ	NOUN
iajs-2555	25	6	and	and	CCONJ
iajs-2555	25	7	ơ	ơ	PROPN
iajs-2555	25	8	is	be	AUX
iajs-2555	25	9	an	an	DET
iajs-2555	25	10	α	α	NOUN
iajs-2555	25	11	-	-	ADJ
iajs-2555	25	12	open	open	ADJ
iajs-2555	25	13	sets	set	NOUN
iajs-2555	25	14	.	.	PUNCT
iajs-2555	26	1	now	now	ADV
iajs-2555	26	2	,	,	PUNCT
iajs-2555	26	3	ç𝑐	ç𝑐	PROPN
iajs-2555	26	4	is	be	AUX
iajs-2555	26	5	ị-𝛼-g	ị-𝛼-g	ADJ
iajs-2555	26	6	-	-	PUNCT
iajs-2555	26	7	open	open	ADJ
iajs-2555	26	8	sets	set	NOUN
iajs-2555	26	9	denoted	denote	VERB
iajs-2555	26	10	by	by	ADP
iajs-2555	26	11	"	"	PUNCT
iajs-2555	26	12	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	26	13	-	-	PUNCT
iajs-2555	26	14	open	open	ADJ
iajs-2555	26	15	"	"	PUNCT
iajs-2555	26	16	.	.	PUNCT
iajs-2555	27	1	the	the	DET
iajs-2555	27	2	collection	collection	NOUN
iajs-2555	27	3	of	of	ADP
iajs-2555	27	4	all	all	DET
iajs-2555	27	5	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	27	6	-	-	PUNCT
iajs-2555	27	7	closed	closed	ADJ
iajs-2555	27	8	sets	set	NOUN
iajs-2555	27	9	,	,	PUNCT
iajs-2555	27	10	where	where	SCONJ
iajs-2555	27	11	ç𝑐	ç𝑐	ADV
iajs-2555	27	12	∈	∈	PROPN
iajs-2555	27	13	ӽ	ӽ	NOUN
iajs-2555	27	14	,	,	PUNCT
iajs-2555	27	15	is	be	AUX
iajs-2555	27	16	denoted	denote	VERB
iajs-2555	27	17	by	by	ADP
iajs-2555	27	18	"	"	PUNCT
iajs-2555	27	19	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	NOUN
iajs-2555	27	20	)	)	PUNCT
iajs-2555	27	21	.	.	PUNCT
iajs-2555	28	1	the	the	DET
iajs-2555	28	2	collection	collection	NOUN
iajs-2555	28	3	of	of	ADP
iajs-2555	28	4	all	all	DET
iajs-2555	28	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	28	6	-	-	PUNCT
iajs-2555	28	7	open	open	ADJ
iajs-2555	28	8	sets	set	NOUN
iajs-2555	28	9	"	"	PUNCT
iajs-2555	28	10	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	28	11	)	)	PUNCT
iajs-2555	28	12	"	"	PUNCT
iajs-2555	28	13	.	.	PUNCT
iajs-2555	29	1	example	example	NOUN
iajs-2555	29	2	2	2	NUM
iajs-2555	29	3	:	:	PUNCT
iajs-2555	29	4	consider	consider	VERB
iajs-2555	29	5	the	the	DET
iajs-2555	29	6	space	space	NOUN
iajs-2555	29	7	(	(	PUNCT
iajs-2555	29	8	ӽ	ӽ	NOUN
iajs-2555	29	9	,	,	PUNCT
iajs-2555	29	10	ῖ	ῖ	PROPN
iajs-2555	29	11	,	,	PUNCT
iajs-2555	29	12	ị	ị	PROPN
iajs-2555	29	13	)	)	PUNCT
iajs-2555	29	14	where	where	SCONJ
iajs-2555	29	15	ӽ={ⱳ	ӽ={ⱳ	NOUN
iajs-2555	29	16	,	,	PUNCT
iajs-2555	29	17	ⱱ	ⱱ	NOUN
iajs-2555	29	18	}	}	PUNCT
iajs-2555	29	19	,	,	PUNCT
iajs-2555	29	20	ῖ={ӽ,∅,{ⱳ	ῖ={ӽ,∅,{ⱳ	PROPN
iajs-2555	29	21	}	}	PUNCT
iajs-2555	29	22	}	}	PUNCT
iajs-2555	29	23	and	and	CCONJ
iajs-2555	29	24	ị={∅,{ⱱ}}.then	ị={∅,{ⱱ}}.then	X
iajs-2555	29	25	ῖ𝛼={ӽ,∅,{ⱳ	ῖ𝛼={ӽ,∅,{ⱳ	ADP
iajs-2555	29	26	}	}	PUNCT
iajs-2555	29	27	}	}	PUNCT
iajs-2555	29	28	and	and	CCONJ
iajs-2555	29	29	ɟ𝛼={ӽ,∅,{ⱱ	ɟ𝛼={ӽ,∅,{ⱱ	ADP
iajs-2555	29	30	}	}	PUNCT
iajs-2555	29	31	}	}	PUNCT
iajs-2555	29	32	,	,	PUNCT
iajs-2555	29	33	so	so	ADV
iajs-2555	29	34	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	NOUN
iajs-2555	29	35	)	)	PUNCT
iajs-2555	29	36	=	=	SYM
iajs-2555	29	37	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	29	38	)	)	PUNCT
iajs-2555	30	1	=	=	SYM
iajs-2555	30	2	{	{	PUNCT
iajs-2555	30	3	ӽ,∅,{ⱳ},{ⱱ	ӽ,∅,{ⱳ},{ⱱ	X
iajs-2555	30	4	}	}	PUNCT
iajs-2555	30	5	}	}	PUNCT
iajs-2555	30	6	.	.	PUNCT
iajs-2555	31	1	example	example	NOUN
iajs-2555	31	2	3	3	NUM
iajs-2555	31	3	:	:	PUNCT
iajs-2555	31	4	consider	consider	VERB
iajs-2555	31	5	the	the	DET
iajs-2555	31	6	space	space	NOUN
iajs-2555	31	7	(	(	PUNCT
iajs-2555	31	8	ӽ	ӽ	NOUN
iajs-2555	31	9	,	,	PUNCT
iajs-2555	31	10	ῖ	ῖ	PROPN
iajs-2555	31	11	,	,	PUNCT
iajs-2555	31	12	ị	ị	PROPN
iajs-2555	31	13	)	)	PUNCT
iajs-2555	31	14	where	where	SCONJ
iajs-2555	31	15	ӽ={ⱳ	ӽ={ⱳ	NOUN
iajs-2555	31	16	,	,	PUNCT
iajs-2555	31	17	ⱱ	ⱱ	NOUN
iajs-2555	31	18	,	,	PUNCT
iajs-2555	31	19	ⱬ	ⱬ	ADJ
iajs-2555	31	20	}	}	PUNCT
iajs-2555	31	21	,	,	PUNCT
iajs-2555	31	22	ῖ={ӽ,∅,{ⱳ	ῖ={ӽ,∅,{ⱳ	PROPN
iajs-2555	31	23	}	}	PUNCT
iajs-2555	31	24	}	}	PUNCT
iajs-2555	31	25	and	and	CCONJ
iajs-2555	31	26	ị={∅,{ⱱ	ị={∅,{ⱱ	PROPN
iajs-2555	31	27	}	}	PUNCT
iajs-2555	31	28	}	}	PUNCT
iajs-2555	31	29	.	.	PUNCT
iajs-2555	32	1	then	then	ADV
iajs-2555	32	2	ῖ𝛼={ӽ,∅,{ⱳ},{ⱳ	ῖ𝛼={ӽ,∅,{ⱳ},{ⱳ	PROPN
iajs-2555	32	3	,	,	PUNCT
iajs-2555	32	4	ⱱ},{ⱳ	ⱱ},{ⱳ	ADJ
iajs-2555	32	5	,	,	PUNCT
iajs-2555	32	6	ⱬ	ⱬ	NOUN
iajs-2555	32	7	}	}	PUNCT
iajs-2555	32	8	}	}	PUNCT
iajs-2555	32	9	ɟ𝛼={ӽ,∅,{ⱱ	ɟ𝛼={ӽ,∅,{ⱱ	PROPN
iajs-2555	32	10	,	,	PUNCT
iajs-2555	32	11	ⱬ},{ⱬ,},{ⱱ	ⱬ},{ⱬ,},{ⱱ	PROPN
iajs-2555	32	12	}	}	PUNCT
iajs-2555	32	13	}	}	PUNCT
iajs-2555	32	14	,	,	PUNCT
iajs-2555	32	15	so	so	ADV
iajs-2555	32	16	𝛼𝑔ị𝐶(ӽ)={ӽ,∅,{ⱱ	𝛼𝑔ị𝐶(ӽ)={ӽ,∅,{ⱱ	PROPN
iajs-2555	32	17	,	,	PUNCT
iajs-2555	32	18	ⱬ},{ⱬ},{ⱳ	ⱬ},{ⱬ},{ⱳ	X
iajs-2555	32	19	,	,	PUNCT
iajs-2555	32	20	ⱬ	ⱬ	NOUN
iajs-2555	32	21	}	}	PUNCT
iajs-2555	32	22	}	}	PUNCT
iajs-2555	32	23	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	32	24	)	)	PUNCT
iajs-2555	32	25	=	=	PRON
iajs-2555	32	26	{	{	PUNCT
iajs-2555	32	27	ӽ,∅,{ⱳ},{ⱳ	ӽ,∅,{ⱳ},{ⱳ	NOUN
iajs-2555	32	28	,	,	PUNCT
iajs-2555	32	29	ⱱ},{ⱱ	ⱱ},{ⱱ	NOUN
iajs-2555	32	30	}	}	PUNCT
iajs-2555	32	31	}	}	PUNCT
iajs-2555	32	32	.	.	PUNCT
iajs-2555	33	1	remark	remark	VERB
iajs-2555	33	2	4	4	NUM
iajs-2555	33	3	:	:	PUNCT
iajs-2555	33	4	i.	i.	NOUN
iajs-2555	33	5	for	for	ADP
iajs-2555	33	6	each	each	DET
iajs-2555	33	7	closed	close	VERB
iajs-2555	33	8	set	set	VERB
iajs-2555	33	9	in	in	ADP
iajs-2555	33	10	(	(	PUNCT
iajs-2555	33	11	ӽ	ӽ	X
iajs-2555	33	12	,	,	PUNCT
iajs-2555	33	13	ῖ	ῖ	X
iajs-2555	33	14	)	)	PUNCT
iajs-2555	33	15	is	be	AUX
iajs-2555	33	16	an	an	DET
iajs-2555	33	17	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	33	18	-	-	PUNCT
iajs-2555	33	19	closed	closed	ADJ
iajs-2555	33	20	in	in	ADP
iajs-2555	33	21	(	(	PUNCT
iajs-2555	33	22	ӽ	ӽ	NOUN
iajs-2555	33	23	,	,	PUNCT
iajs-2555	33	24	ῖ	ῖ	PROPN
iajs-2555	33	25	,	,	PUNCT
iajs-2555	33	26	ị	ị	ADJ
iajs-2555	33	27	)	)	PUNCT
iajs-2555	33	28	.	.	PUNCT
iajs-2555	34	1	ii	ii	PROPN
iajs-2555	34	2	.	.	PUNCT
iajs-2555	35	1	for	for	ADP
iajs-2555	35	2	each	each	DET
iajs-2555	35	3	open	open	ADJ
iajs-2555	35	4	set	set	NOUN
iajs-2555	35	5	in	in	ADP
iajs-2555	35	6	(	(	PUNCT
iajs-2555	35	7	ӽ	ӽ	X
iajs-2555	35	8	,	,	PUNCT
iajs-2555	35	9	ῖ	ῖ	X
iajs-2555	35	10	)	)	PUNCT
iajs-2555	35	11	is	be	AUX
iajs-2555	35	12	an	an	DET
iajs-2555	35	13	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	35	14	-	-	PUNCT
iajs-2555	35	15	open	open	ADJ
iajs-2555	35	16	in	in	ADP
iajs-2555	35	17	(	(	PUNCT
iajs-2555	35	18	ӽ	ӽ	NOUN
iajs-2555	35	19	,	,	PUNCT
iajs-2555	35	20	ῖ	ῖ	PROPN
iajs-2555	35	21	,	,	PUNCT
iajs-2555	35	22	ị	ị	ADJ
iajs-2555	35	23	)	)	PUNCT
iajs-2555	35	24	.	.	PUNCT
iajs-2555	36	1	proof	proof	NOUN
iajs-2555	36	2	:	:	PUNCT
iajs-2555	36	3	i.	i.	PROPN
iajs-2555	36	4	let	let	VERB
iajs-2555	36	5	ç	ç	X
iajs-2555	36	6	is	be	AUX
iajs-2555	36	7	any	any	DET
iajs-2555	36	8	closed	closed	ADJ
iajs-2555	36	9	set	set	VERB
iajs-2555	36	10	in	in	ADP
iajs-2555	36	11	(	(	PUNCT
iajs-2555	36	12	ӽ	ӽ	X
iajs-2555	36	13	,	,	PUNCT
iajs-2555	36	14	ῖ	ῖ	PROPN
iajs-2555	36	15	,	,	PUNCT
iajs-2555	36	16	ị	ị	PRON
iajs-2555	36	17	)	)	PUNCT
iajs-2555	36	18	and	and	CCONJ
iajs-2555	36	19	ơ	ơ	PROPN
iajs-2555	36	20	be	be	AUX
iajs-2555	36	21	an	an	DET
iajs-2555	36	22	α	α	NOUN
iajs-2555	36	23	-	-	ADJ
iajs-2555	36	24	open	open	ADJ
iajs-2555	36	25	set	set	NOUN
iajs-2555	36	26	such	such	ADJ
iajs-2555	36	27	that	that	SCONJ
iajs-2555	36	28	ç	ç	X
iajs-2555	36	29	-	-	PUNCT
iajs-2555	36	30	ơ	ơ	PROPN
iajs-2555	36	31	∈	∈	PROPN
iajs-2555	36	32	ị	ị	AUX
iajs-2555	36	33	since	since	SCONJ
iajs-2555	36	34	𝑐𝑙(ç	𝑐𝑙(ç	NOUN
iajs-2555	36	35	)	)	PUNCT
iajs-2555	36	36	=	=	SYM
iajs-2555	37	1	ç	ç	NOUN
iajs-2555	37	2	this	this	PRON
iajs-2555	37	3	implies	imply	VERB
iajs-2555	37	4	that	that	SCONJ
iajs-2555	37	5	ç	ç	VERB
iajs-2555	37	6	is	be	AUX
iajs-2555	37	7	an	an	DET
iajs-2555	37	8	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	37	9	-	-	PUNCT
iajs-2555	37	10	closed	closed	ADJ
iajs-2555	37	11	set	set	NOUN
iajs-2555	37	12	.	.	PUNCT
iajs-2555	38	1	ii	ii	PROPN
iajs-2555	38	2	.	.	PUNCT
iajs-2555	39	1	let	let	VERB
iajs-2555	39	2	ơ	ơ	PROPN
iajs-2555	39	3	∈	∈	PROPN
iajs-2555	39	4	ӽ	ӽ	NOUN
iajs-2555	39	5	,	,	PUNCT
iajs-2555	39	6	then	then	ADV
iajs-2555	39	7	ơ𝑐	ơ𝑐	PRON
iajs-2555	39	8	is	be	AUX
iajs-2555	39	9	a	a	DET
iajs-2555	39	10	closed	closed	ADJ
iajs-2555	39	11	set	set	NOUN
iajs-2555	39	12	this	this	PRON
iajs-2555	39	13	implies	imply	VERB
iajs-2555	39	14	that	that	SCONJ
iajs-2555	39	15	ơ𝑐	ơ𝑐	PRON
iajs-2555	39	16	is	be	VERB
iajs-2555	39	17	an	an	DET
iajs-2555	39	18	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	39	19	-	-	PUNCT
iajs-2555	39	20	closed	closed	ADJ
iajs-2555	39	21	set	set	NOUN
iajs-2555	39	22	,	,	PUNCT
iajs-2555	39	23	so	so	SCONJ
iajs-2555	39	24	ơ	ơ	PROPN
iajs-2555	39	25	is	be	AUX
iajs-2555	39	26	an	an	DET
iajs-2555	39	27	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	39	28	-	-	PUNCT
iajs-2555	39	29	open	open	ADJ
iajs-2555	39	30	set	set	NOUN
iajs-2555	39	31	.	.	PUNCT
iajs-2555	40	1	the	the	DET
iajs-2555	40	2	reverse	reverse	ADJ
iajs-2555	40	3	way	way	NOUN
iajs-2555	40	4	of	of	ADP
iajs-2555	40	5	remark	remark	NOUN
iajs-2555	40	6	2.4	2.4	NUM
iajs-2555	40	7	is	be	AUX
iajs-2555	40	8	wrong	wrong	ADJ
iajs-2555	40	9	in	in	ADP
iajs-2555	40	10	general	general	ADJ
iajs-2555	40	11	see	see	VERB
iajs-2555	40	12	example	example	NOUN
iajs-2555	41	1	2.2	2.2	NUM
iajs-2555	41	2	.	.	PUNCT
iajs-2555	41	3	remark	remark	NOUN
iajs-2555	41	4	5	5	NUM
iajs-2555	41	5	:	:	PUNCT
iajs-2555	41	6	a	a	DET
iajs-2555	41	7	space	space	NOUN
iajs-2555	41	8	(	(	PUNCT
iajs-2555	41	9	ӽ	ӽ	NOUN
iajs-2555	41	10	,	,	PUNCT
iajs-2555	41	11	ῖ	ῖ	PROPN
iajs-2555	41	12	,	,	PUNCT
iajs-2555	41	13	ị	ị	PROPN
iajs-2555	41	14	):	):	PUNCT
iajs-2555	41	15	i.	i.	NOUN
iajs-2555	41	16	if	if	SCONJ
iajs-2555	41	17	ị	ị	X
iajs-2555	41	18	=	=	PUNCT
iajs-2555	41	19	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	41	20	)	)	PUNCT
iajs-2555	41	21	then	then	ADV
iajs-2555	41	22	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	VERB
iajs-2555	41	23	)	)	PUNCT
iajs-2555	41	24	=	=	SYM
iajs-2555	41	25	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	41	26	)	)	PUNCT
iajs-2555	41	27	=	=	SYM
iajs-2555	41	28	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	41	29	)	)	PUNCT
iajs-2555	41	30	.	.	PUNCT
iajs-2555	42	1	ii	ii	PROPN
iajs-2555	42	2	.	.	PUNCT
iajs-2555	43	1	if	if	SCONJ
iajs-2555	43	2	ῖ	ῖ	PROPN
iajs-2555	43	3	=	=	PROPN
iajs-2555	43	4	𝐷	𝐷	PROPN
iajs-2555	43	5	then	then	ADV
iajs-2555	43	6	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	PROPN
iajs-2555	43	7	)	)	PUNCT
iajs-2555	44	1	=	=	SYM
iajs-2555	44	2	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	44	3	)	)	PUNCT
iajs-2555	44	4	=	=	SYM
iajs-2555	44	5	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	44	6	)	)	PUNCT
iajs-2555	44	7	.	.	PUNCT
iajs-2555	45	1	remark	remark	VERB
iajs-2555	45	2	6	6	NUM
iajs-2555	45	3	:	:	PUNCT
iajs-2555	45	4	for	for	ADP
iajs-2555	45	5	any	any	DET
iajs-2555	45	6	space	space	NOUN
iajs-2555	45	7	(	(	PUNCT
iajs-2555	45	8	ӽ	ӽ	X
iajs-2555	45	9	,	,	PUNCT
iajs-2555	45	10	ῖ	ῖ	PROPN
iajs-2555	45	11	,	,	PUNCT
iajs-2555	45	12	ị	ị	PROPN
iajs-2555	45	13	)	)	PUNCT
iajs-2555	45	14	,	,	PUNCT
iajs-2555	45	15	then	then	ADV
iajs-2555	45	16	the	the	DET
iajs-2555	45	17	two	two	NUM
iajs-2555	45	18	idea	idea	NOUN
iajs-2555	45	19	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	45	20	-	-	PUNCT
iajs-2555	45	21	closed	close	VERB
iajs-2555	45	22	set	set	NOUN
iajs-2555	45	23	and	and	CCONJ
iajs-2555	45	24	𝛼𝑔∗-closed	𝛼𝑔∗-close	VERB
iajs-2555	45	25	set	set	NOUN
iajs-2555	45	26	are	be	AUX
iajs-2555	45	27	the	the	DET
iajs-2555	45	28	same	same	ADJ
iajs-2555	45	29	,	,	PUNCT
iajs-2555	45	30	if	if	SCONJ
iajs-2555	45	31	ị={∅	ị={∅	PROPN
iajs-2555	45	32	}	}	PUNCT
iajs-2555	45	33	.	.	PUNCT
iajs-2555	46	1	the	the	DET
iajs-2555	46	2	following	follow	VERB
iajs-2555	46	3	example	example	NOUN
iajs-2555	46	4	display	display	NOUN
iajs-2555	46	5	that	that	SCONJ
iajs-2555	46	6	the	the	DET
iajs-2555	46	7	two	two	NUM
iajs-2555	46	8	notion	notion	NOUN
iajs-2555	46	9	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	46	10	-	-	PUNCT
iajs-2555	46	11	closed	close	VERB
iajs-2555	46	12	set	set	NOUN
iajs-2555	46	13	and	and	CCONJ
iajs-2555	46	14	𝛼𝑔∗-closed	𝛼𝑔∗-close	VERB
iajs-2555	46	15	set	set	NOUN
iajs-2555	46	16	are	be	AUX
iajs-2555	46	17	separate	separate	ADJ
iajs-2555	46	18	,	,	PUNCT
iajs-2555	46	19	in	in	ADP
iajs-2555	46	20	general	general	ADJ
iajs-2555	46	21	.	.	PUNCT
iajs-2555	46	22	example	example	NOUN
iajs-2555	47	1	7	7	NUM
iajs-2555	47	2	:	:	PUNCT
iajs-2555	47	3	i.	i.	NOUN
iajs-2555	47	4	the	the	DET
iajs-2555	47	5	set	set	NOUN
iajs-2555	47	6	{	{	PUNCT
iajs-2555	47	7	ⱳ	ⱳ	NUM
iajs-2555	47	8	}	}	PUNCT
iajs-2555	47	9	in	in	ADP
iajs-2555	47	10	example	example	NOUN
iajs-2555	47	11	2.2	2.2	NUM
iajs-2555	47	12	is	be	AUX
iajs-2555	47	13	an	an	DET
iajs-2555	47	14	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	47	15	-	-	PUNCT
iajs-2555	47	16	closed	closed	ADJ
iajs-2555	47	17	set	set	NOUN
iajs-2555	47	18	but	but	CCONJ
iajs-2555	47	19	not	not	PART
iajs-2555	47	20	𝛼𝑔∗-closed	𝛼𝑔∗-close	VERB
iajs-2555	47	21	set	set	NOUN
iajs-2555	47	22	,	,	PUNCT
iajs-2555	47	23	and	and	CCONJ
iajs-2555	47	24	{	{	PUNCT
iajs-2555	47	25	ⱱ	ⱱ	NOUN
iajs-2555	47	26	}	}	PUNCT
iajs-2555	47	27	is	be	AUX
iajs-2555	47	28	an	an	DET
iajs-2555	47	29	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	47	30	-	-	PUNCT
iajs-2555	47	31	open	open	NOUN
iajs-2555	47	32	set	set	NOUN
iajs-2555	47	33	,	,	PUNCT
iajs-2555	47	34	but	but	CCONJ
iajs-2555	47	35	not	not	PART
iajs-2555	47	36	𝛼𝑔∗-open	𝛼𝑔∗-open	VERB
iajs-2555	47	37	set	set	VERB
iajs-2555	47	38	.	.	PUNCT
iajs-2555	48	1	ii	ii	PROPN
iajs-2555	48	2	.	.	PUNCT
iajs-2555	49	1	for	for	ADP
iajs-2555	49	2	a	a	DET
iajs-2555	49	3	space	space	NOUN
iajs-2555	49	4	(	(	PUNCT
iajs-2555	49	5	ӽ	ӽ	X
iajs-2555	49	6	,	,	PUNCT
iajs-2555	49	7	ῖ	ῖ	PROPN
iajs-2555	49	8	,	,	PUNCT
iajs-2555	49	9	ị	ị	PROPN
iajs-2555	49	10	)	)	PUNCT
iajs-2555	49	11	,	,	PUNCT
iajs-2555	49	12	where	where	SCONJ
iajs-2555	49	13	ӽ={ṟ	ӽ={ṟ	PROPN
iajs-2555	49	14	,	,	PUNCT
iajs-2555	49	15	ȿ	ȿ	PROPN
iajs-2555	49	16	,	,	PUNCT
iajs-2555	49	17	ⱳ	ⱳ	NOUN
iajs-2555	49	18	,	,	PUNCT
iajs-2555	49	19	ⱱ	ⱱ	ADJ
iajs-2555	49	20	}	}	PUNCT
iajs-2555	49	21	,	,	PUNCT
iajs-2555	49	22	ῖ	ῖ	X
iajs-2555	49	23	=	=	X
iajs-2555	49	24	{	{	PUNCT
iajs-2555	49	25	ӽ,∅,{ṟ	ӽ,∅,{ṟ	NOUN
iajs-2555	49	26	,	,	PUNCT
iajs-2555	49	27	ȿ},{ⱳ	ȿ},{ⱳ	NOUN
iajs-2555	49	28	,	,	PUNCT
iajs-2555	49	29	ⱱ	ⱱ	NOUN
iajs-2555	49	30	}	}	PUNCT
iajs-2555	49	31	}	}	PUNCT
iajs-2555	49	32	and	and	CCONJ
iajs-2555	49	33	ị={∅,ṟ	ị={∅,ṟ	PUNCT
iajs-2555	49	34	}	}	PUNCT
iajs-2555	49	35	.	.	PUNCT
iajs-2555	50	1	then	then	ADV
iajs-2555	50	2	ῖ𝛼=	ῖ𝛼=	PROPN
iajs-2555	50	3	ῖ	ῖ	PROPN
iajs-2555	50	4	,	,	PUNCT
iajs-2555	50	5	leads	lead	VERB
iajs-2555	50	6	to	to	ADP
iajs-2555	50	7	𝛼𝑔∗𝐶(ӽ	𝛼𝑔∗𝐶(ӽ	NOUN
iajs-2555	50	8	)	)	PUNCT
iajs-2555	51	1	=	=	SYM
iajs-2555	51	2	ҏ(ӽ	ҏ(ӽ	X
iajs-2555	51	3	)	)	PUNCT
iajs-2555	51	4	and	and	CCONJ
iajs-2555	51	5	𝛼𝑔∗𝑂(ӽ	𝛼𝑔∗𝑂(ӽ	NOUN
iajs-2555	51	6	)	)	PUNCT
iajs-2555	51	7	=	=	SYM
iajs-2555	51	8	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	51	9	)	)	PUNCT
iajs-2555	51	10	.	.	PUNCT
iajs-2555	52	1	it	it	PRON
iajs-2555	52	2	seems	seem	VERB
iajs-2555	52	3	obvious	obvious	ADJ
iajs-2555	52	4	that	that	SCONJ
iajs-2555	52	5	the	the	DET
iajs-2555	52	6	set	set	NOUN
iajs-2555	52	7	{	{	PUNCT
iajs-2555	52	8	ṟ	ṟ	NOUN
iajs-2555	52	9	}	}	PUNCT
iajs-2555	52	10	is	be	AUX
iajs-2555	52	11	𝛼𝑔∗closed	𝛼𝑔∗close	VERB
iajs-2555	52	12	set	set	NOUN
iajs-2555	52	13	but	but	CCONJ
iajs-2555	52	14	not	not	PART
iajs-2555	52	15	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	52	16	-	-	PUNCT
iajs-2555	52	17	closed	closed	ADJ
iajs-2555	52	18	.	.	PUNCT
iajs-2555	53	1	remark	remark	NOUN
iajs-2555	53	2	8	8	NUM
iajs-2555	53	3	:	:	PUNCT
iajs-2555	53	4	for	for	ADP
iajs-2555	53	5	any	any	DET
iajs-2555	53	6	set	set	NOUN
iajs-2555	53	7	ӽ	ӽ	NOUN
iajs-2555	53	8	,	,	PUNCT
iajs-2555	53	9	let	let	VERB
iajs-2555	53	10	ӽ	ӽ	NOUN
iajs-2555	53	11	∈	∈	ADJ
iajs-2555	53	12	ӽ	ӽ	X
iajs-2555	53	13	and	and	CCONJ
iajs-2555	53	14	ῖ	ῖ	NOUN
iajs-2555	53	15	=	=	PUNCT
iajs-2555	53	16	{	{	PUNCT
iajs-2555	53	17	ӽ,∅,{ӽ	ӽ,∅,{ӽ	NOUN
iajs-2555	53	18	}	}	PUNCT
iajs-2555	53	19	}	}	PUNCT
iajs-2555	53	20	,	,	PUNCT
iajs-2555	53	21	ị	ị	PROPN
iajs-2555	53	22	=	=	NOUN
iajs-2555	53	23	ị𝑛={ç	ị𝑛={ç	NOUN
iajs-2555	53	24	⊆	⊆	NUM
iajs-2555	53	25	ӽ	ӽ	NOUN
iajs-2555	53	26	:	:	PUNCT
iajs-2555	53	27	𝑖𝑛𝑡(𝑐𝑙(ç))={∅	𝑖𝑛𝑡(𝑐𝑙(ç))={∅	PUNCT
iajs-2555	53	28	}	}	PUNCT
iajs-2555	53	29	}	}	PUNCT
iajs-2555	53	30	then	then	ADV
iajs-2555	53	31	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	VERB
iajs-2555	53	32	)	)	PUNCT
iajs-2555	53	33	=	=	SYM
iajs-2555	53	34	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	53	35	)	)	PUNCT
iajs-2555	53	36	.	.	PUNCT
iajs-2555	54	1	proof	proof	NOUN
iajs-2555	54	2	:	:	PUNCT
iajs-2555	54	3	let	let	VERB
iajs-2555	54	4	ị𝑛	ị𝑛	INTJ
iajs-2555	54	5	=	=	SYM
iajs-2555	54	6	{	{	PUNCT
iajs-2555	54	7	ç	ç	X
iajs-2555	54	8	⊆	⊆	NUM
iajs-2555	54	9	ӽ	ӽ	NOUN
iajs-2555	54	10	:	:	PUNCT
iajs-2555	54	11	𝑖𝑛𝑡(𝑐𝑙(ç	𝑖𝑛𝑡(𝑐𝑙(ç	NUM
iajs-2555	54	12	)	)	PUNCT
iajs-2555	54	13	)	)	PUNCT
iajs-2555	55	1	=	=	PRON
iajs-2555	55	2	{	{	PUNCT
iajs-2555	55	3	∅	∅	NOUN
iajs-2555	55	4	}	}	PUNCT
iajs-2555	55	5	}	}	PUNCT
iajs-2555	55	6	,	,	PUNCT
iajs-2555	55	7	ӽ	ӽ	PRON
iajs-2555	55	8	be	be	AUX
iajs-2555	55	9	any	any	DET
iajs-2555	55	10	set	set	NOUN
iajs-2555	55	11	and	and	CCONJ
iajs-2555	55	12	ῖ	ῖ	NOUN
iajs-2555	55	13	=	=	PUNCT
iajs-2555	55	14	{	{	PUNCT
iajs-2555	55	15	ӽ	ӽ	NOUN
iajs-2555	55	16	,	,	PUNCT
iajs-2555	55	17	∅	∅	NOUN
iajs-2555	55	18	,	,	PUNCT
iajs-2555	55	19	{	{	PUNCT
iajs-2555	55	20	ӽ	ӽ	NOUN
iajs-2555	55	21	}	}	PUNCT
iajs-2555	55	22	}	}	PUNCT
iajs-2555	55	23	such	such	ADJ
iajs-2555	55	24	that	that	SCONJ
iajs-2555	55	25	ӽ	ӽ	DET
iajs-2555	55	26	∈	∈	PROPN
iajs-2555	55	27	ӽ	ӽ	NOUN
iajs-2555	55	28	,	,	PUNCT
iajs-2555	55	29	ῖ𝛼={ơ	ῖ𝛼={ơ	ADP
iajs-2555	55	30	⊆	⊆	NUM
iajs-2555	55	31	ӽ	ӽ	NOUN
iajs-2555	55	32	;	;	PUNCT
iajs-2555	55	33	ӽ	ӽ	X
iajs-2555	55	34	∈	∈	PROPN
iajs-2555	55	35	ơ	ơ	PROPN
iajs-2555	55	36	}	}	PUNCT
iajs-2555	55	37	∪	∪	ADJ
iajs-2555	55	38	{	{	PUNCT
iajs-2555	55	39	∅	∅	NOUN
iajs-2555	55	40	}	}	PUNCT
iajs-2555	55	41	,	,	PUNCT
iajs-2555	55	42	for	for	ADP
iajs-2555	55	43	any	any	DET
iajs-2555	55	44	set	set	NOUN
iajs-2555	55	45	ç	ç	X
iajs-2555	55	46	⊆	⊆	NUM
iajs-2555	55	47	ӽ	ӽ	NOUN
iajs-2555	55	48	,	,	PUNCT
iajs-2555	55	49	and	and	CCONJ
iajs-2555	55	50	ơ	ơ	PROPN
iajs-2555	55	51	is	be	AUX
iajs-2555	55	52	α	α	NOUN
iajs-2555	55	53	-	-	ADJ
iajs-2555	55	54	open	open	ADJ
iajs-2555	55	55	set	set	NOUN
iajs-2555	55	56	,	,	PUNCT
iajs-2555	55	57	𝑖𝑓	𝑖𝑓	CCONJ
iajs-2555	55	58	ç	ç	X
iajs-2555	55	59	-	-	PUNCT
iajs-2555	55	60	ơ	ơ	NOUN
iajs-2555	55	61	∈	∈	PROPN
iajs-2555	55	62	ị𝑛	ị𝑛	SCONJ
iajs-2555	55	63	this	this	PRON
iajs-2555	55	64	implies	imply	VERB
iajs-2555	55	65	ӽ	ӽ	PROPN
iajs-2555	55	66	∉	∉	PROPN
iajs-2555	55	67	(	(	PUNCT
iajs-2555	55	68	ç	ç	X
iajs-2555	55	69	-	-	PUNCT
iajs-2555	55	70	ơ	ơ	NOUN
iajs-2555	55	71	)	)	PUNCT
iajs-2555	55	72	,	,	PUNCT
iajs-2555	55	73	so	so	ADV
iajs-2555	55	74	32	32	NUM
iajs-2555	55	75	ibn	ibn	PROPN
iajs-2555	55	76	al	al	PROPN
iajs-2555	55	77	-	-	PUNCT
iajs-2555	55	78	haitham	haitham	PROPN
iajs-2555	55	79	jour	jour	X
iajs-2555	55	80	.	.	PROPN
iajs-2555	56	1	for	for	ADP
iajs-2555	56	2	pure	pure	ADJ
iajs-2555	56	3	&	&	CCONJ
iajs-2555	56	4	appl	appl	PROPN
iajs-2555	56	5	.	.	PUNCT
iajs-2555	57	1	sci	sci	PROPN
iajs-2555	57	2	.	.	PROPN
iajs-2555	58	1	34	34	NUM
iajs-2555	58	2	(	(	PUNCT
iajs-2555	58	3	1	1	NUM
iajs-2555	58	4	)	)	PUNCT
iajs-2555	58	5	2021	2021	NUM
iajs-2555	58	6	𝑐𝑙(ç	𝑐𝑙(ç	NUM
iajs-2555	58	7	-	-	PUNCT
iajs-2555	58	8	ơ	ơ	NOUN
iajs-2555	58	9	)	)	PUNCT
iajs-2555	58	10	=	=	SYM
iajs-2555	58	11	ӽ/{ӽ	ӽ/{ӽ	PROPN
iajs-2555	58	12	}	}	PUNCT
iajs-2555	58	13	,	,	PUNCT
iajs-2555	58	14	then	then	ADV
iajs-2555	58	15	𝑖𝑛𝑡(𝑐𝑙(ç	𝑖𝑛𝑡(𝑐𝑙(ç	PROPN
iajs-2555	58	16	-	-	PUNCT
iajs-2555	58	17	ơ	ơ	PROPN
iajs-2555	58	18	)	)	PUNCT
iajs-2555	58	19	)	)	PUNCT
iajs-2555	59	1	=	=	NOUN
iajs-2555	59	2	∅	∅	NOUN
iajs-2555	59	3	,	,	PUNCT
iajs-2555	59	4	then	then	ADV
iajs-2555	59	5	ӽ	ӽ	X
iajs-2555	59	6	∉	∉	PROPN
iajs-2555	59	7	ç	ç	PROPN
iajs-2555	59	8	and	and	CCONJ
iajs-2555	59	9	ӽ	ӽ	PROPN
iajs-2555	59	10	∈	∈	PROPN
iajs-2555	59	11	ơ	ơ	PROPN
iajs-2555	59	12	,	,	PUNCT
iajs-2555	59	13	since	since	SCONJ
iajs-2555	59	14	ӽ	ӽ	DET
iajs-2555	59	15	∉	∉	PROPN
iajs-2555	59	16	ç	ç	PROPN
iajs-2555	59	17	this	this	PRON
iajs-2555	59	18	implies	imply	VERB
iajs-2555	59	19	𝑐𝑙(ç	𝑐𝑙(ç	NOUN
iajs-2555	59	20	)	)	PUNCT
iajs-2555	59	21	=	=	SYM
iajs-2555	59	22	ӽ/{ӽ	ӽ/{ӽ	NOUN
iajs-2555	59	23	}	}	PUNCT
iajs-2555	59	24	,	,	PUNCT
iajs-2555	59	25	thus	thus	ADV
iajs-2555	59	26	(	(	PUNCT
iajs-2555	59	27	ӽ/{ӽ}-ơ)∈	ӽ/{ӽ}-ơ)∈	PROPN
iajs-2555	59	28	ị𝑛,if	ị𝑛,if	VERB
iajs-2555	59	29	ӽ	ӽ	DET
iajs-2555	59	30	∈	∈	PROPN
iajs-2555	59	31	ç	ç	NOUN
iajs-2555	59	32	and	and	CCONJ
iajs-2555	59	33	ӽ	ӽ	PROPN
iajs-2555	59	34	∈	∈	PROPN
iajs-2555	59	35	ơ	ơ	PROPN
iajs-2555	59	36	then	then	ADV
iajs-2555	59	37	ӽ	ӽ	X
iajs-2555	59	38	∉	∉	PROPN
iajs-2555	59	39	(	(	PUNCT
iajs-2555	59	40	𝑐𝑙(ç)-ơ	𝑐𝑙(ç)-ơ	PROPN
iajs-2555	59	41	)	)	PUNCT
iajs-2555	59	42	,	,	PUNCT
iajs-2555	59	43	so	so	ADV
iajs-2555	59	44	𝑐𝑙(ç)-ơ	𝑐𝑙(ç)-ơ	VERB
iajs-2555	59	45	∈	∈	PROPN
iajs-2555	59	46	ị𝑛	ị𝑛	INTJ
iajs-2555	59	47	,	,	PUNCT
iajs-2555	59	48	hence	hence	ADV
iajs-2555	59	49	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	VERB
iajs-2555	59	50	)	)	PUNCT
iajs-2555	59	51	=	=	SYM
iajs-2555	59	52	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	59	53	)	)	PUNCT
iajs-2555	59	54	.	.	PUNCT
iajs-2555	60	1	theorem	theorem	ADJ
iajs-2555	60	2	9	9	NUM
iajs-2555	60	3	:	:	PUNCT
iajs-2555	60	4	let	let	VERB
iajs-2555	60	5	ç	ç	PROPN
iajs-2555	60	6	and	and	CCONJ
iajs-2555	60	7	ð	ð	PROPN
iajs-2555	60	8	are	be	AUX
iajs-2555	60	9	two	two	NUM
iajs-2555	60	10	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	60	11	-	-	PUNCT
iajs-2555	60	12	closed	close	VERB
iajs-2555	60	13	sets	set	NOUN
iajs-2555	60	14	then	then	ADV
iajs-2555	60	15	ç	ç	X
iajs-2555	60	16	∪	∪	X
iajs-2555	60	17	ð	ð	PROPN
iajs-2555	60	18	is	be	AUX
iajs-2555	60	19	an	an	DET
iajs-2555	60	20	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	60	21	-	-	PUNCT
iajs-2555	60	22	closed	closed	ADJ
iajs-2555	60	23	.	.	PUNCT
iajs-2555	61	1	proof	proof	NOUN
iajs-2555	61	2	:	:	PUNCT
iajs-2555	61	3	let	let	VERB
iajs-2555	61	4	ç	ç	PROPN
iajs-2555	61	5	and	and	CCONJ
iajs-2555	61	6	ð	ð	PROPN
iajs-2555	61	7	are	be	AUX
iajs-2555	61	8	two	two	NUM
iajs-2555	61	9	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	61	10	-	-	PUNCT
iajs-2555	61	11	closed	close	VERB
iajs-2555	61	12	set	set	NOUN
iajs-2555	61	13	in	in	ADP
iajs-2555	61	14	(	(	PUNCT
iajs-2555	61	15	ӽ	ӽ	X
iajs-2555	61	16	,	,	PUNCT
iajs-2555	61	17	ῖ	ῖ	PROPN
iajs-2555	61	18	,	,	PUNCT
iajs-2555	61	19	ị	ị	ADJ
iajs-2555	61	20	)	)	PUNCT
iajs-2555	61	21	and	and	CCONJ
iajs-2555	62	1	ơ	ơ	PROPN
iajs-2555	62	2	∈	∈	PROPN
iajs-2555	62	3	ῖ𝛼	ῖ𝛼	ADP
iajs-2555	62	4	subset	subset	NOUN
iajs-2555	62	5	of	of	ADP
iajs-2555	62	6	ӽ	ӽ	NOUN
iajs-2555	62	7	,	,	PUNCT
iajs-2555	62	8	where	where	SCONJ
iajs-2555	62	9	(	(	PUNCT
iajs-2555	62	10	ç	ç	X
iajs-2555	62	11	∪	∪	X
iajs-2555	62	12	ð)ơ	ð)ơ	X
iajs-2555	62	13	∈	∈	PROPN
iajs-2555	62	14	ị	ị	AUX
iajs-2555	62	15	,	,	PUNCT
iajs-2555	62	16	then	then	ADV
iajs-2555	62	17	ð	ð	PROPN
iajs-2555	62	18	-	-	PUNCT
iajs-2555	62	19	ơ	ơ	PROPN
iajs-2555	62	20	∈	∈	PROPN
iajs-2555	62	21	ị	ị	X
iajs-2555	62	22	and	and	CCONJ
iajs-2555	62	23	ç	ç	X
iajs-2555	62	24	-	-	ADJ
iajs-2555	62	25	ơ	ơ	PROPN
iajs-2555	62	26	∈	∈	PROPN
iajs-2555	62	27	ị	ị	PRON
iajs-2555	62	28	,	,	PUNCT
iajs-2555	62	29	so	so	CCONJ
iajs-2555	62	30	𝑐𝑙(ð)-ơ	𝑐𝑙(ð)-ơ	PRON
iajs-2555	63	1	∈	∈	PROPN
iajs-2555	63	2	ị	ị	X
iajs-2555	63	3	and	and	CCONJ
iajs-2555	63	4	𝑐𝑙(ç)-ơ	𝑐𝑙(ç)-ơ	VERB
iajs-2555	63	5	∈	∈	PROPN
iajs-2555	64	1	ị	ị	X
iajs-2555	64	2	therefore	therefore	ADV
iajs-2555	64	3	,	,	PUNCT
iajs-2555	64	4	(	(	PUNCT
iajs-2555	64	5	𝑐𝑙(ç)-ơ	𝑐𝑙(ç)-ơ	NOUN
iajs-2555	64	6	)	)	PUNCT
iajs-2555	64	7	∪	∪	NOUN
iajs-2555	64	8	(	(	PUNCT
iajs-2555	64	9	𝑐𝑙(ð)-ơ	𝑐𝑙(ð)-ơ	NOUN
iajs-2555	64	10	)	)	PUNCT
iajs-2555	64	11	∈	∈	PROPN
iajs-2555	65	1	ị	ị	PART
iajs-2555	65	2	,	,	PUNCT
iajs-2555	65	3	𝑠𝑜	𝑠𝑜	INTJ
iajs-2555	65	4	𝑐𝑙(ç	𝑐𝑙(ç	NOUN
iajs-2555	65	5	∪	∪	X
iajs-2555	65	6	ð)-ơ	ð)-ơ	X
iajs-2555	65	7	∈	∈	PROPN
iajs-2555	65	8	ị.hence	ị.hence	X
iajs-2555	65	9	ç	ç	X
iajs-2555	65	10	∪	∪	X
iajs-2555	65	11	ð	ð	PROPN
iajs-2555	65	12	is	be	AUX
iajs-2555	65	13	an	an	DET
iajs-2555	65	14	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	65	15	-	-	PUNCT
iajs-2555	65	16	closed	close	VERB
iajs-2555	65	17	sets	set	NOUN
iajs-2555	65	18	.	.	PUNCT
iajs-2555	66	1	corollary	corollary	ADJ
iajs-2555	66	2	10	10	NUM
iajs-2555	66	3	:	:	PUNCT
iajs-2555	66	4	let	let	VERB
iajs-2555	66	5	ç	ç	PROPN
iajs-2555	66	6	and	and	CCONJ
iajs-2555	66	7	ð	ð	PROPN
iajs-2555	66	8	are	be	AUX
iajs-2555	66	9	two	two	NUM
iajs-2555	66	10	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	66	11	-	-	PUNCT
iajs-2555	66	12	open	open	ADJ
iajs-2555	66	13	sets	set	NOUN
iajs-2555	66	14	then	then	ADV
iajs-2555	66	15	ç	ç	NOUN
iajs-2555	66	16	∩	∩	NOUN
iajs-2555	66	17	ð	ð	PROPN
iajs-2555	66	18	is	be	AUX
iajs-2555	66	19	an	an	DET
iajs-2555	66	20	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	66	21	-	-	PUNCT
iajs-2555	66	22	open	open	ADJ
iajs-2555	66	23	.	.	PUNCT
iajs-2555	67	1	proof	proof	NOUN
iajs-2555	67	2	:	:	PUNCT
iajs-2555	67	3	let	let	VERB
iajs-2555	67	4	ç	ç	PROPN
iajs-2555	67	5	and	and	CCONJ
iajs-2555	67	6	ð	ð	PROPN
iajs-2555	67	7	are	be	AUX
iajs-2555	67	8	two	two	NUM
iajs-2555	67	9	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	67	10	-	-	PUNCT
iajs-2555	67	11	open	open	ADJ
iajs-2555	67	12	sets	set	NOUN
iajs-2555	67	13	in	in	ADP
iajs-2555	67	14	ӽ	ӽ	NOUN
iajs-2555	67	15	then	then	ADV
iajs-2555	67	16	ç𝑐	ç𝑐	ADV
iajs-2555	67	17	,	,	PUNCT
iajs-2555	67	18	ð𝑐	ð𝑐	X
iajs-2555	67	19	are	be	AUX
iajs-2555	67	20	two	two	NUM
iajs-2555	67	21	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	67	22	-	-	PUNCT
iajs-2555	67	23	closed	close	VERB
iajs-2555	67	24	sets	set	NOUN
iajs-2555	67	25	therefore	therefore	ADV
iajs-2555	67	26	,	,	PUNCT
iajs-2555	67	27	ç𝑐	ç𝑐	ADV
iajs-2555	67	28	∪	∪	NOUN
iajs-2555	67	29	ð𝑐	ð𝑐	ADP
iajs-2555	67	30	is	be	AUX
iajs-2555	67	31	an	an	DET
iajs-2555	67	32	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	67	33	-	-	PUNCT
iajs-2555	67	34	closed	closed	ADJ
iajs-2555	67	35	set	set	VERB
iajs-2555	67	36	by	by	ADP
iajs-2555	67	37	theorem	theorem	NOUN
iajs-2555	67	38	2.9	2.9	NUM
iajs-2555	67	39	.	.	PUNCT
iajs-2555	68	1	hence(ç	hence(ç	NOUN
iajs-2555	68	2	∩	∩	NOUN
iajs-2555	68	3	ð)𝑐	ð)𝑐	X
iajs-2555	68	4	is	be	AUX
iajs-2555	68	5	an	an	DET
iajs-2555	68	6	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	68	7	-	-	PUNCT
iajs-2555	68	8	closed	closed	ADJ
iajs-2555	68	9	set	set	VERB
iajs-2555	68	10	so	so	ADV
iajs-2555	68	11	ç	ç	NOUN
iajs-2555	68	12	∩	∩	NOUN
iajs-2555	68	13	ð	ð	PROPN
iajs-2555	68	14	is	be	AUX
iajs-2555	68	15	an	an	DET
iajs-2555	68	16	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	68	17	-	-	PUNCT
iajs-2555	68	18	open	open	ADJ
iajs-2555	68	19	set	set	NOUN
iajs-2555	68	20	.	.	PUNCT
iajs-2555	69	1	remark	remark	PROPN
iajs-2555	69	2	11	11	NUM
iajs-2555	69	3	:	:	PUNCT
iajs-2555	69	4	i.	i.	PROPN
iajs-2555	69	5	the	the	DET
iajs-2555	69	6	union	union	NOUN
iajs-2555	69	7	of	of	ADP
iajs-2555	69	8	any	any	DET
iajs-2555	69	9	collection	collection	NOUN
iajs-2555	69	10	of	of	ADP
iajs-2555	69	11	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	69	12	-	-	PUNCT
iajs-2555	69	13	closed	close	VERB
iajs-2555	69	14	sets	set	NOUN
iajs-2555	69	15	is	be	AUX
iajs-2555	69	16	not	not	PART
iajs-2555	69	17	necessarily	necessarily	ADV
iajs-2555	69	18	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	69	19	-	-	PUNCT
iajs-2555	69	20	closed	closed	ADJ
iajs-2555	69	21	.	.	PUNCT
iajs-2555	70	1	ii	ii	X
iajs-2555	70	2	.	.	PUNCT
iajs-2555	71	1	the	the	DET
iajs-2555	71	2	intersection	intersection	NOUN
iajs-2555	71	3	of	of	ADP
iajs-2555	71	4	collection	collection	NOUN
iajs-2555	71	5	of	of	ADP
iajs-2555	71	6	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	71	7	-	-	PUNCT
iajs-2555	71	8	open	open	ADJ
iajs-2555	71	9	sets	set	NOUN
iajs-2555	71	10	is	be	AUX
iajs-2555	71	11	not	not	PART
iajs-2555	71	12	necessarily	necessarily	ADV
iajs-2555	71	13	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	71	14	-	-	PUNCT
iajs-2555	71	15	open	open	ADJ
iajs-2555	71	16	.	.	PUNCT
iajs-2555	72	1	for	for	ADP
iajs-2555	72	2	example	example	NOUN
iajs-2555	72	3	:	:	PUNCT
iajs-2555	72	4	consider	consider	VERB
iajs-2555	72	5	a	a	DET
iajs-2555	72	6	space	space	NOUN
iajs-2555	72	7	(	(	PUNCT
iajs-2555	72	8	ӽ	ӽ	X
iajs-2555	72	9	,	,	PUNCT
iajs-2555	72	10	ῖ	ῖ	PROPN
iajs-2555	72	11	,	,	PUNCT
iajs-2555	72	12	ị	ị	PROPN
iajs-2555	72	13	)	)	PUNCT
iajs-2555	72	14	,	,	PUNCT
iajs-2555	72	15	when	when	SCONJ
iajs-2555	72	16	ӽ	ӽ	X
iajs-2555	72	17	=	=	SYM
iajs-2555	72	18	ɲ	ɲ	PROPN
iajs-2555	72	19	,	,	PUNCT
iajs-2555	72	20	the	the	DET
iajs-2555	72	21	set	set	NOUN
iajs-2555	72	22	of	of	ADP
iajs-2555	72	23	all	all	DET
iajs-2555	72	24	natural	natural	ADJ
iajs-2555	72	25	numbers	number	NOUN
iajs-2555	72	26	,	,	PUNCT
iajs-2555	72	27	ῖ	ῖ	X
iajs-2555	72	28	=	=	X
iajs-2555	72	29	ῖ	ῖ	PROPN
iajs-2555	72	30	cof	cof	PROPN
iajs-2555	72	31	,	,	PUNCT
iajs-2555	72	32	is	be	AUX
iajs-2555	72	33	a	a	DET
iajs-2555	72	34	topology	topology	NOUN
iajs-2555	72	35	of	of	ADP
iajs-2555	72	36	all	all	DET
iajs-2555	72	37	sets	set	NOUN
iajs-2555	72	38	that	that	PRON
iajs-2555	72	39	complement	complement	NOUN
iajs-2555	72	40	is	be	AUX
iajs-2555	72	41	a	a	DET
iajs-2555	72	42	finite	finite	NOUN
iajs-2555	72	43	set	set	NOUN
iajs-2555	72	44	and	and	CCONJ
iajs-2555	72	45	ị	ị	X
iajs-2555	72	46	=	=	ADJ
iajs-2555	72	47	ịịᶂ	ịịᶂ	ADJ
iajs-2555	72	48	=	=	ADJ
iajs-2555	72	49	{	{	PUNCT
iajs-2555	72	50	ơ	ơ	PROPN
iajs-2555	72	51	⊆	⊆	NUM
iajs-2555	72	52	ɲ	ɲ	PROPN
iajs-2555	72	53	,	,	PUNCT
iajs-2555	72	54	ơ	ơ	PROPN
iajs-2555	72	55	is	be	AUX
iajs-2555	72	56	a	a	DET
iajs-2555	72	57	finite	finite	ADJ
iajs-2555	72	58	set	set	NOUN
iajs-2555	72	59	}	}	PUNCT
iajs-2555	72	60	,	,	PUNCT
iajs-2555	72	61	ῖ𝛼={ơ	ῖ𝛼={ơ	ADP
iajs-2555	72	62	⊆	⊆	NUM
iajs-2555	72	63	ɲ	ɲ	PROPN
iajs-2555	72	64	,	,	PUNCT
iajs-2555	72	65	ơ	ơ	PROPN
iajs-2555	72	66	is	be	AUX
iajs-2555	72	67	an	an	DET
iajs-2555	72	68	infinite	infinite	ADJ
iajs-2555	72	69	set}∪{∅	set}∪{∅	PROPN
iajs-2555	72	70	}	}	PUNCT
iajs-2555	72	71	.	.	PUNCT
iajs-2555	73	1	clearly	clearly	ADV
iajs-2555	73	2	,	,	PUNCT
iajs-2555	73	3	{	{	PUNCT
iajs-2555	73	4	ŋ	ŋ	X
iajs-2555	73	5	}	}	PUNCT
iajs-2555	73	6	is	be	AUX
iajs-2555	73	7	an	an	DET
iajs-2555	73	8	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	73	9	-	-	PUNCT
iajs-2555	73	10	closed	closed	ADJ
iajs-2555	73	11	set	set	NOUN
iajs-2555	73	12	,	,	PUNCT
iajs-2555	73	13	∀	∀	X
iajs-2555	73	14	ŋ	ŋ	X
iajs-2555	73	15	∈	∈	NOUN
iajs-2555	73	16	ę+	ę+	X
iajs-2555	73	17	,	,	PUNCT
iajs-2555	73	18	where	where	SCONJ
iajs-2555	73	19	ę+	ę+	ADV
iajs-2555	73	20	is	be	AUX
iajs-2555	73	21	the	the	DET
iajs-2555	73	22	positive	positive	ADJ
iajs-2555	73	23	even	even	ADJ
iajs-2555	73	24	numbers	number	NOUN
iajs-2555	73	25	,	,	PUNCT
iajs-2555	73	26	but	but	CCONJ
iajs-2555	73	27	∪{{ŋ}:ŋ∈	∪{{ŋ}:ŋ∈	PROPN
iajs-2555	74	1	ę+}=ę+	ę+}=ę+	NOUN
iajs-2555	74	2	which	which	PRON
iajs-2555	74	3	is	be	AUX
iajs-2555	74	4	not	not	PART
iajs-2555	74	5	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	74	6	-	-	PUNCT
iajs-2555	74	7	closed	close	VERB
iajs-2555	74	8	set	set	NOUN
iajs-2555	74	9	.	.	PUNCT
iajs-2555	75	1	similarly	similarly	ADV
iajs-2555	75	2	;	;	PUNCT
iajs-2555	75	3	ç𝑛=ɲ-{ŋ	ç𝑛=ɲ-{ŋ	PROPN
iajs-2555	75	4	}	}	PUNCT
iajs-2555	75	5	is	be	AUX
iajs-2555	75	6	an	an	PRON
iajs-2555	75	7	of	of	ADP
iajs-2555	75	8	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	75	9	-	-	PUNCT
iajs-2555	75	10	open	open	NOUN
iajs-2555	75	11	set	set	NOUN
iajs-2555	75	12	,	,	PUNCT
iajs-2555	75	13	∀	∀	X
iajs-2555	75	14	ŋ	ŋ	X
iajs-2555	75	15	∈	∈	NOUN
iajs-2555	75	16	ę+	ę+	X
iajs-2555	75	17	but	but	CCONJ
iajs-2555	75	18	∩{ç𝑛	∩{ç𝑛	NUM
iajs-2555	75	19	:	:	PUNCT
iajs-2555	75	20	ŋ	ŋ	X
iajs-2555	75	21	∈	∈	PROPN
iajs-2555	75	22	ę+}=ộ+	ę+}=ộ+	NOUN
iajs-2555	75	23	,	,	PUNCT
iajs-2555	75	24	where	where	SCONJ
iajs-2555	75	25	ộ+	ộ+	PRON
iajs-2555	75	26	is	be	AUX
iajs-2555	75	27	the	the	DET
iajs-2555	75	28	positive	positive	ADJ
iajs-2555	75	29	odd	odd	ADJ
iajs-2555	75	30	number	number	NOUN
iajs-2555	75	31	,	,	PUNCT
iajs-2555	75	32	ộ+	ộ+	PRON
iajs-2555	75	33	is	be	AUX
iajs-2555	75	34	not	not	PART
iajs-2555	75	35	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	75	36	-	-	PUNCT
iajs-2555	75	37	closed	close	VERB
iajs-2555	75	38	set	set	NOUN
iajs-2555	75	39	.	.	PUNCT
iajs-2555	76	1	theorem	theorem	VERB
iajs-2555	76	2	12	12	NUM
iajs-2555	76	3	:	:	PUNCT
iajs-2555	76	4	in	in	ADP
iajs-2555	76	5	(	(	PUNCT
iajs-2555	76	6	ӽ	ӽ	X
iajs-2555	76	7	,	,	PUNCT
iajs-2555	76	8	ῖ	ῖ	PROPN
iajs-2555	76	9	,	,	PUNCT
iajs-2555	76	10	ị	ị	PROPN
iajs-2555	76	11	)	)	PUNCT
iajs-2555	76	12	,	,	PUNCT
iajs-2555	76	13	let	let	VERB
iajs-2555	76	14	ç	ç	PROPN
iajs-2555	76	15	⊆	⊆	NUM
iajs-2555	76	16	ӽ.	ӽ.	NOUN
iajs-2555	76	17	ç	ç	PUNCT
iajs-2555	76	18	is	be	AUX
iajs-2555	76	19	an	an	DET
iajs-2555	76	20	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	76	21	-	-	PUNCT
iajs-2555	76	22	open	open	NOUN
iajs-2555	76	23	set	set	NOUN
iajs-2555	76	24	if	if	SCONJ
iajs-2555	76	25	and	and	CCONJ
iajs-2555	76	26	only	only	ADV
iajs-2555	76	27	if	if	SCONJ
iajs-2555	76	28	(	(	PUNCT
iajs-2555	76	29	₣	₣	PROPN
iajs-2555	76	30	−	−	NOUN
iajs-2555	76	31	𝑖𝑛𝑡(ç	𝑖𝑛𝑡(ç	NUM
iajs-2555	76	32	)	)	PUNCT
iajs-2555	76	33	)	)	PUNCT
iajs-2555	77	1	∈	∈	PROPN
iajs-2555	78	1	ị	ị	ADP
iajs-2555	78	2	,	,	PUNCT
iajs-2555	78	3	whenever	whenever	SCONJ
iajs-2555	78	4	(	(	PUNCT
iajs-2555	78	5	₣	₣	NUM
iajs-2555	78	6	-ç	-ç	NUM
iajs-2555	78	7	)	)	PUNCT
iajs-2555	78	8	∈	∈	PROPN
iajs-2555	78	9	ị	ị	PART
iajs-2555	78	10	,	,	PUNCT
iajs-2555	78	11	∀	∀	X
iajs-2555	78	12	₣	₣	ADP
iajs-2555	78	13	∈	∈	NOUN
iajs-2555	78	14	ɟ𝛼	ɟ𝛼	NOUN
iajs-2555	78	15	.	.	PUNCT
iajs-2555	79	1	proof	proof	NOUN
iajs-2555	79	2	:	:	PUNCT
iajs-2555	79	3	(	(	PUNCT
iajs-2555	79	4	→)let	→)let	NOUN
iajs-2555	79	5	ç	ç	X
iajs-2555	79	6	⊆	⊆	NUM
iajs-2555	79	7	ӽ	ӽ	NOUN
iajs-2555	79	8	,	,	PUNCT
iajs-2555	79	9	where	where	SCONJ
iajs-2555	79	10	ç	ç	X
iajs-2555	79	11	be	be	AUX
iajs-2555	79	12	an	an	DET
iajs-2555	79	13	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	79	14	-	-	PUNCT
iajs-2555	79	15	open	open	ADJ
iajs-2555	79	16	sets	set	NOUN
iajs-2555	79	17	and	and	CCONJ
iajs-2555	79	18	(	(	PUNCT
iajs-2555	79	19	₣	₣	PROPN
iajs-2555	79	20	-ç	-ç	NUM
iajs-2555	79	21	)	)	PUNCT
iajs-2555	79	22	∈	∈	PROPN
iajs-2555	80	1	ị	ị	X
iajs-2555	80	2	,	,	PUNCT
iajs-2555	80	3	₣	₣	PROPN
iajs-2555	80	4	∈	∈	NOUN
iajs-2555	80	5	ɟ𝛼	ɟ𝛼	NOUN
iajs-2555	80	6	,	,	PUNCT
iajs-2555	80	7	since	since	SCONJ
iajs-2555	80	8	(	(	PUNCT
iajs-2555	80	9	ӽ-ç	ӽ-ç	NUM
iajs-2555	80	10	)	)	PUNCT
iajs-2555	80	11	is	be	AUX
iajs-2555	80	12	an	an	DET
iajs-2555	80	13	𝛼𝑔ịclosed	𝛼𝑔ịclose	VERB
iajs-2555	80	14	set	set	NOUN
iajs-2555	80	15	and	and	CCONJ
iajs-2555	80	16	(	(	PUNCT
iajs-2555	80	17	ӽ-ç)-ơ	ӽ-ç)-ơ	NOUN
iajs-2555	80	18	∈	∈	PROPN
iajs-2555	80	19	ị	ị	NOUN
iajs-2555	80	20	,	,	PUNCT
iajs-2555	80	21	ơ	ơ	PROPN
iajs-2555	80	22	∈	∈	PROPN
iajs-2555	80	23	ῖ𝛼implies	ῖ𝛼implie	NOUN
iajs-2555	80	24	𝑐𝑙(ӽ-ç)-ơ	𝑐𝑙(ӽ-ç)-ơ	X
iajs-2555	80	25	∈	∈	PROPN
iajs-2555	81	1	ị	ị	PRON
iajs-2555	81	2	,	,	PUNCT
iajs-2555	81	3	whenever	whenever	SCONJ
iajs-2555	81	4	(	(	PUNCT
iajs-2555	81	5	ӽ-ç)-ơ	ӽ-ç)-ơ	NOUN
iajs-2555	81	6	∈	∈	NOUN
iajs-2555	81	7	ị	ị	X
iajs-2555	81	8	,	,	PUNCT
iajs-2555	81	9	for	for	ADP
iajs-2555	81	10	each	each	DET
iajs-2555	81	11	ơ	ơ	PROPN
iajs-2555	81	12	∈	∈	PROPN
iajs-2555	81	13	ῖ𝛼	ῖ𝛼	ADP
iajs-2555	81	14	,	,	PUNCT
iajs-2555	81	15	𝑐𝑙(ӽ-ç)-ơ	𝑐𝑙(ӽ-ç)-ơ	NOUN
iajs-2555	81	16	=	=	SYM
iajs-2555	81	17	(	(	PUNCT
iajs-2555	81	18	ӽ-ơ)-(ӽ-𝑐𝑙(ӽ-ç	ӽ-ơ)-(ӽ-𝑐𝑙(ӽ-ç	PROPN
iajs-2555	81	19	)	)	PUNCT
iajs-2555	81	20	since	since	SCONJ
iajs-2555	81	21	ç	ç	X
iajs-2555	81	22	-	-	PUNCT
iajs-2555	81	23	ð	ð	X
iajs-2555	81	24	=	=	SYM
iajs-2555	81	25	(	(	PUNCT
iajs-2555	81	26	ӽ-ð)-(ӽ-ç	ӽ-ð)-(ӽ-ç	PROPN
iajs-2555	81	27	)	)	PUNCT
iajs-2555	81	28	,	,	PUNCT
iajs-2555	81	29	thus	thus	ADV
iajs-2555	81	30	(	(	PUNCT
iajs-2555	81	31	ӽ-ơ)-(ӽ-(ӽ-𝑖𝑛𝑡(ӽ-ӽç	ӽ-ơ)-(ӽ-(ӽ-𝑖𝑛𝑡(ӽ-ӽç	NOUN
iajs-2555	81	32	)	)	PUNCT
iajs-2555	81	33	)	)	PUNCT
iajs-2555	81	34	)	)	PUNCT
iajs-2555	82	1	=	=	PRON
iajs-2555	82	2	(	(	PUNCT
iajs-2555	82	3	ӽ-ơ)-𝑖𝑛𝑡(ç	ӽ-ơ)-𝑖𝑛𝑡(ç	PROPN
iajs-2555	82	4	)	)	PUNCT
iajs-2555	82	5	=	=	SYM
iajs-2555	82	6	₣	₣	PROPN
iajs-2555	82	7	-𝑖𝑛𝑡(ç	-𝑖𝑛𝑡(ç	NOUN
iajs-2555	82	8	)	)	PUNCT
iajs-2555	82	9	∈	∈	PROPN
iajs-2555	83	1	ị	ị	AUX
iajs-2555	83	2	.	.	PUNCT
iajs-2555	84	1	(	(	PUNCT
iajs-2555	84	2	←	←	PROPN
iajs-2555	84	3	)	)	PUNCT
iajs-2555	84	4	let	let	VERB
iajs-2555	84	5	₣	₣	NUM
iajs-2555	84	6	-𝑖𝑛𝑡(ç	-𝑖𝑛𝑡(ç	NOUN
iajs-2555	84	7	)	)	PUNCT
iajs-2555	84	8	∈	∈	PROPN
iajs-2555	85	1	ị	ị	ADP
iajs-2555	85	2	,	,	PUNCT
iajs-2555	85	3	whenever	whenever	SCONJ
iajs-2555	85	4	₣	₣	NUM
iajs-2555	85	5	-ç	-ç	SYM
iajs-2555	85	6	∈	∈	PROPN
iajs-2555	85	7	ị	ị	X
iajs-2555	85	8	,	,	PUNCT
iajs-2555	85	9	for	for	SCONJ
iajs-2555	85	10	each	each	DET
iajs-2555	85	11	₣	₣	PROPN
iajs-2555	85	12	∈	∈	NOUN
iajs-2555	85	13	ɟ𝛼.	ɟ𝛼.	NOUN
iajs-2555	85	14	let	let	VERB
iajs-2555	85	15	(	(	PUNCT
iajs-2555	85	16	ӽ-ç)-ơ	ӽ-ç)-ơ	NOUN
iajs-2555	85	17	∈	∈	PROPN
iajs-2555	85	18	ị	ị	X
iajs-2555	85	19	;	;	PUNCT
iajs-2555	85	20	ơ	ơ	PROPN
iajs-2555	85	21	∈	∈	PROPN
iajs-2555	85	22	ῖ𝛼	ῖ𝛼	ADP
iajs-2555	85	23	,	,	PUNCT
iajs-2555	85	24	(	(	PUNCT
iajs-2555	85	25	ӽ-ç)ơ	ӽ-ç)ơ	NUM
iajs-2555	85	26	=	=	SYM
iajs-2555	85	27	(	(	PUNCT
iajs-2555	85	28	ӽ-ơ)-ç	ӽ-ơ)-ç	NOUN
iajs-2555	85	29	∈	∈	PROPN
iajs-2555	86	1	ị	ị	X
iajs-2555	86	2	,	,	PUNCT
iajs-2555	86	3	let	let	VERB
iajs-2555	86	4	ӽ-ơ	ӽ-ơ	PROPN
iajs-2555	86	5	=	=	SYM
iajs-2555	86	6	₣	₣	PROPN
iajs-2555	86	7	∈	∈	NOUN
iajs-2555	86	8	ɟ𝛼	ɟ𝛼	NOUN
iajs-2555	86	9	and	and	CCONJ
iajs-2555	86	10	₣	₣	NUM
iajs-2555	86	11	-ç	-ç	SYM
iajs-2555	86	12	∈	∈	PROPN
iajs-2555	86	13	ị	ị	ADP
iajs-2555	86	14	this	this	PRON
iajs-2555	86	15	implies	imply	VERB
iajs-2555	86	16	₣	₣	NOUN
iajs-2555	86	17	-𝑖𝑛𝑡(ç	-𝑖𝑛𝑡(ç	SYM
iajs-2555	86	18	)	)	PUNCT
iajs-2555	86	19	∈	∈	PROPN
iajs-2555	87	1	ị	ị	ADP
iajs-2555	87	2	,	,	PUNCT
iajs-2555	87	3	now	now	ADV
iajs-2555	87	4	₣	₣	NOUN
iajs-2555	87	5	-𝑖𝑛𝑡(ç	-𝑖𝑛𝑡(ç	X
iajs-2555	87	6	)	)	PUNCT
iajs-2555	87	7	=	=	SYM
iajs-2555	87	8	𝑐𝑙(ӽ-ç)-(ӽ-₣	𝑐𝑙(ӽ-ç)-(ӽ-₣	NOUN
iajs-2555	87	9	)	)	PUNCT
iajs-2555	87	10	=	=	NOUN
iajs-2555	88	1	𝑐𝑙(ӽ-ç)-ơ	𝑐𝑙(ӽ-ç)-ơ	NOUN
iajs-2555	88	2	∈	∈	PROPN
iajs-2555	89	1	ị	ị	PART
iajs-2555	89	2	,	,	PUNCT
iajs-2555	89	3	thus	thus	ADV
iajs-2555	89	4	(	(	PUNCT
iajs-2555	89	5	ӽ-ç	ӽ-ç	NUM
iajs-2555	89	6	)	)	PUNCT
iajs-2555	89	7	is	be	AUX
iajs-2555	89	8	an	an	DET
iajs-2555	89	9	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	89	10	-	-	PUNCT
iajs-2555	89	11	closed	closed	ADJ
iajs-2555	89	12	set	set	NOUN
iajs-2555	89	13	,	,	PUNCT
iajs-2555	89	14	hence	hence	ADV
iajs-2555	89	15	ç	ç	X
iajs-2555	89	16	is	be	AUX
iajs-2555	89	17	an	an	DET
iajs-2555	89	18	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	89	19	-	-	PUNCT
iajs-2555	89	20	open	open	NOUN
iajs-2555	89	21	set	set	NOUN
iajs-2555	89	22	.	.	PUNCT
iajs-2555	90	1	3	3	X
iajs-2555	90	2	-	-	PUNCT
iajs-2555	90	3	open	open	ADJ
iajs-2555	90	4	function	function	NOUN
iajs-2555	90	5	definition	definition	NOUN
iajs-2555	90	6	1	1	NUM
iajs-2555	90	7	:	:	PUNCT
iajs-2555	90	8	the	the	DET
iajs-2555	90	9	function	function	NOUN
iajs-2555	90	10	ᶂ	ᶂ	NOUN
iajs-2555	90	11	:	:	PUNCT
iajs-2555	90	12	(	(	PUNCT
iajs-2555	90	13	ӽ	ӽ	X
iajs-2555	90	14	,	,	PUNCT
iajs-2555	90	15	ῖ	ῖ	PROPN
iajs-2555	90	16	,	,	PUNCT
iajs-2555	90	17	ị	ị	PRON
iajs-2555	90	18	)	)	PUNCT
iajs-2555	90	19	→	→	SYM
iajs-2555	90	20	(	(	PUNCT
iajs-2555	90	21	ƴ	ƴ	PROPN
iajs-2555	90	22	,	,	PUNCT
iajs-2555	90	23	ɟ	ɟ	NOUN
iajs-2555	90	24	,	,	PUNCT
iajs-2555	90	25	ʝ	ʝ	NOUN
iajs-2555	90	26	)	)	PUNCT
iajs-2555	90	27	is	be	AUX
iajs-2555	90	28	called	call	VERB
iajs-2555	90	29	;	;	PUNCT
iajs-2555	90	30	i.	i.	PROPN
iajs-2555	90	31	𝛼𝑔ị	𝛼𝑔ị	PROPN
iajs-2555	90	32	-	-	PUNCT
iajs-2555	90	33	open	open	ADJ
iajs-2555	90	34	function	function	NOUN
iajs-2555	90	35	,	,	PUNCT
iajs-2555	90	36	denoted	denote	VERB
iajs-2555	90	37	by	by	ADP
iajs-2555	90	38	"	"	PUNCT
iajs-2555	90	39	𝛼𝑔ịo	𝛼𝑔ịo	NOUN
iajs-2555	90	40	-	-	PUNCT
iajs-2555	90	41	function	function	NOUN
iajs-2555	90	42	"	"	PUNCT
iajs-2555	90	43	if	if	SCONJ
iajs-2555	90	44	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	90	45	)	)	PUNCT
iajs-2555	90	46	is	be	AUX
iajs-2555	90	47	an	an	DET
iajs-2555	90	48	𝛼𝑔ʝ	𝛼𝑔ʝ	PROPN
iajs-2555	90	49	-	-	ADJ
iajs-2555	90	50	open	open	ADJ
iajs-2555	90	51	set	set	NOUN
iajs-2555	90	52	in	in	ADP
iajs-2555	90	53	ƴ	ƴ	PROPN
iajs-2555	90	54	.	.	PUNCT
iajs-2555	91	1	whenever	whenever	SCONJ
iajs-2555	91	2	ơ	ơ	PROPN
iajs-2555	91	3	is	be	AUX
iajs-2555	91	4	an	an	DET
iajs-2555	91	5	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	91	6	-	-	PUNCT
iajs-2555	91	7	open	open	ADJ
iajs-2555	91	8	in	in	ADP
iajs-2555	91	9	ӽ.	ӽ.	PROPN
iajs-2555	91	10	ii	ii	PROPN
iajs-2555	91	11	.	.	PUNCT
iajs-2555	92	1	𝛼𝑔ị	𝛼𝑔ị	PROPN
iajs-2555	92	2	∗-open	∗-open	X
iajs-2555	92	3	function	function	NOUN
iajs-2555	92	4	,	,	PUNCT
iajs-2555	92	5	denoted	denote	VERB
iajs-2555	92	6	by	by	ADP
iajs-2555	92	7	"	"	PUNCT
iajs-2555	92	8	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	92	9	∗o	∗o	NOUN
iajs-2555	92	10	-	-	PUNCT
iajs-2555	92	11	function	function	NOUN
iajs-2555	92	12	"	"	PUNCT
iajs-2555	92	13	if	if	SCONJ
iajs-2555	92	14	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	92	15	)	)	PUNCT
iajs-2555	92	16	is	be	AUX
iajs-2555	92	17	an	an	DET
iajs-2555	92	18	𝛼𝑔ʝ	𝛼𝑔ʝ	PROPN
iajs-2555	92	19	-	-	ADJ
iajs-2555	92	20	open	open	ADJ
iajs-2555	92	21	set	set	NOUN
iajs-2555	92	22	in	in	ADP
iajs-2555	92	23	ƴ	ƴ	PROPN
iajs-2555	92	24	.	.	PUNCT
iajs-2555	93	1	whenever	whenever	SCONJ
iajs-2555	93	2	ơ	ơ	PROPN
iajs-2555	93	3	∈	∈	PROPN
iajs-2555	93	4	ῖ.	ῖ.	NOUN
iajs-2555	93	5	33	33	NUM
iajs-2555	93	6	ibn	ibn	PROPN
iajs-2555	93	7	al	al	PROPN
iajs-2555	93	8	-	-	PUNCT
iajs-2555	93	9	haitham	haitham	PROPN
iajs-2555	93	10	jour	jour	X
iajs-2555	93	11	.	.	PROPN
iajs-2555	94	1	for	for	ADP
iajs-2555	94	2	pure	pure	ADJ
iajs-2555	94	3	&	&	CCONJ
iajs-2555	94	4	appl	appl	PROPN
iajs-2555	94	5	.	.	PUNCT
iajs-2555	95	1	sci	sci	PROPN
iajs-2555	95	2	.	.	PROPN
iajs-2555	96	1	34	34	NUM
iajs-2555	96	2	(	(	PUNCT
iajs-2555	96	3	1	1	NUM
iajs-2555	96	4	)	)	PUNCT
iajs-2555	96	5	2021	2021	NUM
iajs-2555	96	6	iii	iii	NOUN
iajs-2555	96	7	.	.	PUNCT
iajs-2555	96	8	𝛼𝑔ị	𝛼𝑔ị	PROPN
iajs-2555	96	9	∗∗-open	∗∗-open	PROPN
iajs-2555	96	10	function	function	NOUN
iajs-2555	96	11	,	,	PUNCT
iajs-2555	96	12	denoted	denote	VERB
iajs-2555	96	13	by	by	ADP
iajs-2555	96	14	"	"	PUNCT
iajs-2555	96	15	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	96	16	∗∗o	∗∗o	NOUN
iajs-2555	96	17	-	-	PUNCT
iajs-2555	96	18	function	function	NOUN
iajs-2555	96	19	"	"	PUNCT
iajs-2555	96	20	if	if	SCONJ
iajs-2555	96	21	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	96	22	)	)	PUNCT
iajs-2555	96	23	is	be	AUX
iajs-2555	96	24	an	an	DET
iajs-2555	96	25	open	open	ADJ
iajs-2555	96	26	set	set	NOUN
iajs-2555	96	27	in	in	ADP
iajs-2555	96	28	ƴ	ƴ	PROPN
iajs-2555	96	29	.	.	PUNCT
iajs-2555	97	1	whenever	whenever	SCONJ
iajs-2555	97	2	ơ	ơ	PROPN
iajs-2555	97	3	is	be	AUX
iajs-2555	97	4	an	an	DET
iajs-2555	97	5	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	97	6	-	-	PUNCT
iajs-2555	97	7	open	open	NOUN
iajs-2555	97	8	set	set	NOUN
iajs-2555	97	9	in	in	ADP
iajs-2555	97	10	ӽ.	ӽ.	NOUN
iajs-2555	97	11	proposition	proposition	NOUN
iajs-2555	97	12	2	2	NUM
iajs-2555	97	13	:	:	PUNCT
iajs-2555	97	14	let	let	VERB
iajs-2555	97	15	ᶂ	ᶂ	NOUN
iajs-2555	97	16	:	:	PUNCT
iajs-2555	97	17	(	(	PUNCT
iajs-2555	97	18	ӽ	ӽ	X
iajs-2555	97	19	,	,	PUNCT
iajs-2555	97	20	ῖ	ῖ	PROPN
iajs-2555	97	21	,	,	PUNCT
iajs-2555	97	22	ị	ị	PRON
iajs-2555	97	23	)	)	PUNCT
iajs-2555	97	24	→	→	SYM
iajs-2555	97	25	(	(	PUNCT
iajs-2555	97	26	ƴ	ƴ	PROPN
iajs-2555	97	27	,	,	PUNCT
iajs-2555	97	28	ɟ	ɟ	NOUN
iajs-2555	97	29	,	,	PUNCT
iajs-2555	97	30	ʝ	ʝ	NOUN
iajs-2555	97	31	)	)	PUNCT
iajs-2555	97	32	is	be	AUX
iajs-2555	97	33	a	a	DET
iajs-2555	97	34	function	function	NOUN
iajs-2555	97	35	;	;	PUNCT
iajs-2555	97	36	i.	i.	NOUN
iajs-2555	97	37	if	if	SCONJ
iajs-2555	97	38	ᶂ	ᶂ	NOUN
iajs-2555	97	39	is	be	AUX
iajs-2555	97	40	an	an	DET
iajs-2555	97	41	open	open	ADJ
iajs-2555	97	42	function	function	NOUN
iajs-2555	97	43	then	then	ADV
iajs-2555	97	44	ᶂ	ᶂ	PROPN
iajs-2555	97	45	is	be	AUX
iajs-2555	97	46	𝛼𝑔ị	𝛼𝑔ị	PRON
iajs-2555	97	47	∗o	∗o	NOUN
iajs-2555	97	48	-	-	PUNCT
iajs-2555	97	49	function	function	NOUN
iajs-2555	97	50	proof	proof	NOUN
iajs-2555	97	51	:	:	PUNCT
iajs-2555	97	52	let	let	VERB
iajs-2555	97	53	ơ	ơ	PROPN
iajs-2555	97	54	∈	∈	PROPN
iajs-2555	97	55	ῖ	ῖ	NOUN
iajs-2555	97	56	,	,	PUNCT
iajs-2555	97	57	since	since	SCONJ
iajs-2555	97	58	ᶂ	ᶂ	NOUN
iajs-2555	97	59	is	be	AUX
iajs-2555	97	60	an	an	DET
iajs-2555	97	61	open	open	ADJ
iajs-2555	97	62	function	function	NOUN
iajs-2555	97	63	then	then	ADV
iajs-2555	97	64	ᶂ(ơ)∈	ᶂ(ơ)∈	PRON
iajs-2555	97	65	ɟ	ɟ	NOUN
iajs-2555	97	66	and	and	CCONJ
iajs-2555	97	67	since	since	SCONJ
iajs-2555	97	68	for	for	ADP
iajs-2555	97	69	each	each	DET
iajs-2555	97	70	open	open	ADJ
iajs-2555	97	71	sets	set	NOUN
iajs-2555	97	72	is	be	AUX
iajs-2555	97	73	an	an	DET
iajs-2555	97	74	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	97	75	-	-	PUNCT
iajs-2555	97	76	open	open	NOUN
iajs-2555	97	77	set	set	NOUN
iajs-2555	97	78	then	then	ADV
iajs-2555	97	79	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	97	80	)	)	PUNCT
iajs-2555	97	81	is	be	AUX
iajs-2555	97	82	an	an	DET
iajs-2555	97	83	𝛼𝑔ʝ	𝛼𝑔ʝ	PROPN
iajs-2555	97	84	-	-	ADJ
iajs-2555	97	85	open	open	ADJ
iajs-2555	97	86	set	set	NOUN
iajs-2555	97	87	in	in	ADP
iajs-2555	97	88	ƴ	ƴ	PROPN
iajs-2555	97	89	,	,	PUNCT
iajs-2555	97	90	then	then	ADV
iajs-2555	97	91	ᶂ	ᶂ	PROPN
iajs-2555	97	92	is	be	AUX
iajs-2555	97	93	an	an	DET
iajs-2555	97	94	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	97	95	∗o	∗o	NOUN
iajs-2555	97	96	-	-	PUNCT
iajs-2555	97	97	function	function	NOUN
iajs-2555	97	98	.	.	PUNCT
iajs-2555	98	1	ii	ii	PROPN
iajs-2555	98	2	.	.	PUNCT
iajs-2555	99	1	if	if	SCONJ
iajs-2555	99	2	ᶂ	ᶂ	NOUN
iajs-2555	99	3	is	be	AUX
iajs-2555	99	4	an	an	DET
iajs-2555	99	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	99	6	∗∗o	∗∗o	NOUN
iajs-2555	99	7	-	-	PUNCT
iajs-2555	99	8	function	function	NOUN
iajs-2555	99	9	then	then	ADV
iajs-2555	99	10	ᶂ	ᶂ	NOUN
iajs-2555	99	11	is	be	AUX
iajs-2555	99	12	an	an	DET
iajs-2555	99	13	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	99	14	-	-	PUNCT
iajs-2555	99	15	open	open	ADJ
iajs-2555	99	16	function	function	NOUN
iajs-2555	99	17	.	.	PUNCT
iajs-2555	100	1	proof	proof	NOUN
iajs-2555	100	2	:	:	PUNCT
iajs-2555	100	3	let	let	VERB
iajs-2555	100	4	ơ	ơ	PROPN
iajs-2555	100	5	be	be	AUX
iajs-2555	100	6	an	an	DET
iajs-2555	100	7	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	100	8	-	-	PUNCT
iajs-2555	100	9	open	open	NOUN
iajs-2555	100	10	set	set	NOUN
iajs-2555	100	11	in	in	ADP
iajs-2555	100	12	ӽ	ӽ	NOUN
iajs-2555	100	13	,	,	PUNCT
iajs-2555	100	14	since	since	SCONJ
iajs-2555	100	15	ᶂ	ᶂ	NOUN
iajs-2555	100	16	is	be	AUX
iajs-2555	100	17	an	an	DET
iajs-2555	100	18	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	100	19	∗∗o	∗∗o	NOUN
iajs-2555	100	20	-	-	PUNCT
iajs-2555	100	21	function	function	NOUN
iajs-2555	100	22	,	,	PUNCT
iajs-2555	100	23	then	then	ADV
iajs-2555	100	24	ᶂ(ơ)∈	ᶂ(ơ)∈	X
iajs-2555	100	25	ɟ	ɟ	X
iajs-2555	100	26	,	,	PUNCT
iajs-2555	100	27	since	since	SCONJ
iajs-2555	100	28	for	for	ADP
iajs-2555	100	29	each	each	DET
iajs-2555	100	30	open	open	ADJ
iajs-2555	100	31	set	set	NOUN
iajs-2555	100	32	is	be	AUX
iajs-2555	100	33	an	an	DET
iajs-2555	100	34	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	100	35	-	-	PUNCT
iajs-2555	100	36	open	open	NOUN
iajs-2555	100	37	set	set	NOUN
iajs-2555	100	38	,	,	PUNCT
iajs-2555	100	39	this	this	PRON
iajs-2555	100	40	implies	imply	VERB
iajs-2555	100	41	that	that	SCONJ
iajs-2555	100	42	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	100	43	)	)	PUNCT
iajs-2555	100	44	is	be	AUX
iajs-2555	100	45	an	an	DET
iajs-2555	100	46	𝛼𝑔ʝ	𝛼𝑔ʝ	PROPN
iajs-2555	100	47	-	-	ADJ
iajs-2555	100	48	open	open	ADJ
iajs-2555	100	49	set	set	NOUN
iajs-2555	100	50	in	in	ADP
iajs-2555	100	51	ƴ	ƴ	PROPN
iajs-2555	100	52	,	,	PUNCT
iajs-2555	100	53	then	then	ADV
iajs-2555	100	54	ᶂ	ᶂ	PROPN
iajs-2555	100	55	is	be	AUX
iajs-2555	100	56	an	an	DET
iajs-2555	100	57	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	100	58	-	-	PUNCT
iajs-2555	100	59	open	open	ADJ
iajs-2555	100	60	function	function	NOUN
iajs-2555	100	61	.	.	PUNCT
iajs-2555	101	1	iii	iii	X
iajs-2555	101	2	.	.	PUNCT
iajs-2555	102	1	if	if	SCONJ
iajs-2555	102	2	ᶂ	ᶂ	NOUN
iajs-2555	102	3	is	be	AUX
iajs-2555	102	4	an	an	DET
iajs-2555	102	5	𝛼𝑔ịo	𝛼𝑔ịo	NOUN
iajs-2555	102	6	-	-	PUNCT
iajs-2555	102	7	function	function	NOUN
iajs-2555	102	8	then	then	ADV
iajs-2555	102	9	ᶂ	ᶂ	NOUN
iajs-2555	102	10	is	be	AUX
iajs-2555	102	11	an	an	DET
iajs-2555	102	12	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	102	13	∗o	∗o	NOUN
iajs-2555	102	14	-	-	PUNCT
iajs-2555	102	15	function	function	NOUN
iajs-2555	102	16	.	.	PUNCT
iajs-2555	103	1	proof	proof	NOUN
iajs-2555	103	2	:	:	PUNCT
iajs-2555	103	3	let	let	VERB
iajs-2555	103	4	ơ	ơ	PROPN
iajs-2555	103	5	∈	∈	PROPN
iajs-2555	103	6	ῖ	ῖ	X
iajs-2555	103	7	,	,	PUNCT
iajs-2555	103	8	since	since	SCONJ
iajs-2555	103	9	for	for	ADP
iajs-2555	103	10	each	each	DET
iajs-2555	103	11	open	open	ADJ
iajs-2555	103	12	set	set	NOUN
iajs-2555	103	13	is	be	AUX
iajs-2555	103	14	an	an	DET
iajs-2555	103	15	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	103	16	-	-	PUNCT
iajs-2555	103	17	open	open	NOUN
iajs-2555	103	18	set	set	NOUN
iajs-2555	103	19	,	,	PUNCT
iajs-2555	103	20	then	then	ADV
iajs-2555	103	21	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	103	22	)	)	PUNCT
iajs-2555	103	23	is	be	AUX
iajs-2555	103	24	an	an	DET
iajs-2555	103	25	𝛼𝑔ʝ	𝛼𝑔ʝ	PROPN
iajs-2555	103	26	-	-	ADJ
iajs-2555	103	27	open	open	ADJ
iajs-2555	103	28	set	set	NOUN
iajs-2555	103	29	in	in	ADP
iajs-2555	103	30	ƴ	ƴ	PRON
iajs-2555	103	31	,	,	PUNCT
iajs-2555	103	32	thus	thus	ADV
iajs-2555	103	33	ᶂ	ᶂ	NOUN
iajs-2555	103	34	is	be	AUX
iajs-2555	103	35	an	an	DET
iajs-2555	103	36	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	103	37	∗o	∗o	NOUN
iajs-2555	103	38	-	-	PUNCT
iajs-2555	103	39	function	function	NOUN
iajs-2555	103	40	.	.	PUNCT
iajs-2555	104	1	iv	iv	X
iajs-2555	104	2	.	.	PUNCT
iajs-2555	105	1	if	if	SCONJ
iajs-2555	105	2	ᶂ	ᶂ	NOUN
iajs-2555	105	3	is	be	AUX
iajs-2555	105	4	an	an	DET
iajs-2555	105	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	105	6	∗∗o	∗∗o	NOUN
iajs-2555	105	7	-	-	PUNCT
iajs-2555	105	8	function	function	NOUN
iajs-2555	105	9	then	then	ADV
iajs-2555	105	10	ᶂ	ᶂ	NOUN
iajs-2555	105	11	is	be	AUX
iajs-2555	105	12	an	an	DET
iajs-2555	105	13	open	open	ADJ
iajs-2555	105	14	function	function	NOUN
iajs-2555	105	15	.	.	PUNCT
iajs-2555	106	1	proof	proof	NOUN
iajs-2555	106	2	:	:	PUNCT
iajs-2555	106	3	let	let	VERB
iajs-2555	106	4	ơ	ơ	PROPN
iajs-2555	106	5	∈	∈	PROPN
iajs-2555	106	6	ῖ	ῖ	X
iajs-2555	106	7	,	,	PUNCT
iajs-2555	106	8	since	since	SCONJ
iajs-2555	106	9	for	for	ADP
iajs-2555	106	10	each	each	DET
iajs-2555	106	11	open	open	ADJ
iajs-2555	106	12	set	set	NOUN
iajs-2555	106	13	is	be	AUX
iajs-2555	106	14	an	an	DET
iajs-2555	106	15	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	106	16	-	-	PUNCT
iajs-2555	106	17	open	open	NOUN
iajs-2555	106	18	set	set	NOUN
iajs-2555	106	19	,	,	PUNCT
iajs-2555	106	20	then	then	ADV
iajs-2555	106	21	ơ	ơ	PROPN
iajs-2555	106	22	be	be	VERB
iajs-2555	106	23	an	an	DET
iajs-2555	106	24	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	106	25	-	-	PUNCT
iajs-2555	106	26	open	open	NOUN
iajs-2555	106	27	set	set	NOUN
iajs-2555	106	28	in	in	ADP
iajs-2555	106	29	ӽ	ӽ	NOUN
iajs-2555	106	30	,	,	PUNCT
iajs-2555	106	31	since	since	SCONJ
iajs-2555	106	32	ᶂ	ᶂ	NOUN
iajs-2555	106	33	is	be	AUX
iajs-2555	106	34	an	an	DET
iajs-2555	106	35	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	106	36	∗∗o	∗∗o	NOUN
iajs-2555	106	37	-	-	PUNCT
iajs-2555	106	38	function	function	NOUN
iajs-2555	106	39	thus	thus	ADV
iajs-2555	106	40	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	106	41	)	)	PUNCT
iajs-2555	106	42	is	be	AUX
iajs-2555	106	43	an	an	DET
iajs-2555	106	44	open	open	ADJ
iajs-2555	106	45	set	set	NOUN
iajs-2555	106	46	in	in	ADP
iajs-2555	106	47	ƴ	ƴ	PROPN
iajs-2555	106	48	,	,	PUNCT
iajs-2555	106	49	then	then	ADV
iajs-2555	106	50	ᶂ	ᶂ	PROPN
iajs-2555	106	51	is	be	AUX
iajs-2555	106	52	an	an	DET
iajs-2555	106	53	open	open	ADJ
iajs-2555	106	54	function	function	NOUN
iajs-2555	106	55	.	.	PUNCT
iajs-2555	107	1	v.	v.	INTJ
iajs-2555	107	2	if	if	SCONJ
iajs-2555	107	3	ᶂ	ᶂ	NOUN
iajs-2555	107	4	is	be	AUX
iajs-2555	107	5	an	an	DET
iajs-2555	107	6	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	107	7	∗∗o	∗∗o	NOUN
iajs-2555	107	8	-	-	PUNCT
iajs-2555	107	9	function	function	NOUN
iajs-2555	107	10	then	then	ADV
iajs-2555	107	11	ᶂ	ᶂ	NOUN
iajs-2555	107	12	is	be	AUX
iajs-2555	107	13	an	an	DET
iajs-2555	107	14	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	107	15	∗o	∗o	NOUN
iajs-2555	107	16	-	-	PUNCT
iajs-2555	107	17	function	function	NOUN
iajs-2555	107	18	.	.	PUNCT
iajs-2555	108	1	proof	proof	NOUN
iajs-2555	108	2	:	:	PUNCT
iajs-2555	108	3	by	by	ADP
iajs-2555	108	4	proposition	proposition	NOUN
iajs-2555	108	5	3.2	3.2	NUM
iajs-2555	108	6	-	-	PUNCT
iajs-2555	108	7	ii	ii	NOUN
iajs-2555	108	8	and	and	CCONJ
iajs-2555	108	9	proposition	proposition	NOUN
iajs-2555	108	10	3.2	3.2	NUM
iajs-2555	108	11	-	-	PUNCT
iajs-2555	108	12	iii	iii	NOUN
iajs-2555	108	13	,	,	PUNCT
iajs-2555	108	14	prove	prove	VERB
iajs-2555	108	15	is	be	AUX
iajs-2555	108	16	over	over	ADV
iajs-2555	108	17	.	.	PUNCT
iajs-2555	109	1	the	the	DET
iajs-2555	109	2	following	follow	VERB
iajs-2555	109	3	scheme	scheme	NOUN
iajs-2555	109	4	explains	explain	VERB
iajs-2555	109	5	the	the	DET
iajs-2555	109	6	relationship	relationship	NOUN
iajs-2555	109	7	between	between	ADP
iajs-2555	109	8	the	the	DET
iajs-2555	109	9	various	various	ADJ
iajs-2555	109	10	concepts	concept	NOUN
iajs-2555	109	11	presented	present	VERB
iajs-2555	109	12	in	in	ADP
iajs-2555	109	13	definition	definition	NOUN
iajs-2555	109	14	3.1	3.1	NUM
iajs-2555	109	15	.	.	PUNCT
iajs-2555	110	1	arrow	arrow	PROPN
iajs-2555	110	2	chart	chart	NOUN
iajs-2555	110	3	(	(	PUNCT
iajs-2555	110	4	3.1	3.1	NUM
iajs-2555	110	5	)	)	PUNCT
iajs-2555	110	6	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	110	7	-	-	PUNCT
iajs-2555	110	8	open	open	ADJ
iajs-2555	110	9	function	function	NOUN
iajs-2555	110	10	𝜶𝒈ị𝒐-function	𝜶𝒈ị𝒐-function	ADJ
iajs-2555	110	11	𝜶𝒈ị	𝜶𝒈ị	ADV
iajs-2555	110	12	∗o	∗o	NOUN
iajs-2555	110	13	-	-	PUNCT
iajs-2555	110	14	function	function	NOUN
iajs-2555	110	15	𝜶𝒈ị	𝜶𝒈ị	ADJ
iajs-2555	110	16	∗∗o	∗∗o	NOUN
iajs-2555	110	17	-	-	PUNCT
iajs-2555	110	18	function	function	NOUN
iajs-2555	110	19	open	open	ADJ
iajs-2555	110	20	function	function	NOUN
iajs-2555	110	21	34	34	NUM
iajs-2555	110	22	ibn	ibn	PROPN
iajs-2555	110	23	al	al	PROPN
iajs-2555	110	24	-	-	PUNCT
iajs-2555	110	25	haitham	haitham	PROPN
iajs-2555	110	26	jour	jour	X
iajs-2555	110	27	.	.	PROPN
iajs-2555	110	28	for	for	ADP
iajs-2555	110	29	pure	pure	ADJ
iajs-2555	110	30	&	&	CCONJ
iajs-2555	110	31	appl	appl	PROPN
iajs-2555	110	32	.	.	PUNCT
iajs-2555	111	1	sci	sci	PROPN
iajs-2555	111	2	.	.	PROPN
iajs-2555	112	1	34	34	NUM
iajs-2555	112	2	(	(	PUNCT
iajs-2555	112	3	1	1	NUM
iajs-2555	112	4	)	)	PUNCT
iajs-2555	112	5	2021	2021	NUM
iajs-2555	112	6	the	the	DET
iajs-2555	112	7	following	follow	VERB
iajs-2555	112	8	are	be	AUX
iajs-2555	112	9	some	some	DET
iajs-2555	112	10	examples	example	NOUN
iajs-2555	112	11	showing	show	VERB
iajs-2555	112	12	that	that	SCONJ
iajs-2555	112	13	the	the	DET
iajs-2555	112	14	opposite	opposite	ADJ
iajs-2555	112	15	direction	direction	NOUN
iajs-2555	112	16	of	of	ADP
iajs-2555	112	17	the	the	DET
iajs-2555	112	18	above	above	ADJ
iajs-2555	112	19	schema	schema	NOUN
iajs-2555	112	20	is	be	AUX
iajs-2555	112	21	incorrect	incorrect	ADJ
iajs-2555	112	22	.	.	PUNCT
iajs-2555	113	1	𝐄𝐱𝐚𝐦𝐩𝐥𝐞	𝐄𝐱𝐚𝐦𝐩𝐥𝐞	VERB
iajs-2555	113	2	3	3	NUM
iajs-2555	113	3	:	:	PUNCT
iajs-2555	113	4	a	a	DET
iajs-2555	113	5	function	function	NOUN
iajs-2555	113	6	ᶂ	ᶂ	NOUN
iajs-2555	113	7	:	:	PUNCT
iajs-2555	113	8	(	(	PUNCT
iajs-2555	113	9	ӽ	ӽ	X
iajs-2555	113	10	,	,	PUNCT
iajs-2555	113	11	ῖ	ῖ	PROPN
iajs-2555	113	12	,	,	PUNCT
iajs-2555	113	13	ị	ị	PRON
iajs-2555	113	14	)	)	PUNCT
iajs-2555	113	15	→	→	SYM
iajs-2555	113	16	(	(	PUNCT
iajs-2555	113	17	ӽ	ӽ	X
iajs-2555	113	18	,	,	PUNCT
iajs-2555	113	19	ῖ	ῖ	NOUN
iajs-2555	113	20	,	,	PUNCT
iajs-2555	113	21	ʝ	ʝ	NOUN
iajs-2555	113	22	)	)	PUNCT
iajs-2555	113	23	,	,	PUNCT
iajs-2555	113	24	where	where	SCONJ
iajs-2555	113	25	ӽ	ӽ	NOUN
iajs-2555	113	26	=	=	NOUN
iajs-2555	113	27	{	{	PUNCT
iajs-2555	113	28	ẻ1	ẻ1	NOUN
iajs-2555	113	29	,	,	PUNCT
iajs-2555	113	30	ẻ2	ẻ2	PROPN
iajs-2555	113	31	,	,	PUNCT
iajs-2555	113	32	ẻ3	ẻ3	PROPN
iajs-2555	113	33	}	}	PUNCT
iajs-2555	113	34	such	such	ADJ
iajs-2555	113	35	that	that	SCONJ
iajs-2555	113	36	ᶂ(ẻ1	ᶂ(ẻ1	NOUN
iajs-2555	113	37	)	)	PUNCT
iajs-2555	113	38	=	=	SYM
iajs-2555	113	39	(	(	PUNCT
iajs-2555	113	40	ẻ2	ẻ2	PROPN
iajs-2555	113	41	)	)	PUNCT
iajs-2555	113	42	,	,	PUNCT
iajs-2555	113	43	ᶂ(ẻ2	ᶂ(ẻ2	NUM
iajs-2555	113	44	)	)	PUNCT
iajs-2555	113	45	=	=	SYM
iajs-2555	113	46	(	(	PUNCT
iajs-2555	113	47	ẻ1	ẻ1	NOUN
iajs-2555	113	48	)	)	PUNCT
iajs-2555	113	49	,	,	PUNCT
iajs-2555	113	50	ᶂ(ẻ3	ᶂ(ẻ3	NOUN
iajs-2555	113	51	)	)	PUNCT
iajs-2555	113	52	=	=	PUNCT
iajs-2555	113	53	(	(	PUNCT
iajs-2555	113	54	ẻ3	ẻ3	PROPN
iajs-2555	113	55	)	)	PUNCT
iajs-2555	113	56	,	,	PUNCT
iajs-2555	113	57	ῖ={ӽ,∅,{ẻ1	ῖ={ӽ,∅,{ẻ1	NOUN
iajs-2555	113	58	}	}	PUNCT
iajs-2555	113	59	}	}	PUNCT
iajs-2555	113	60	,	,	PUNCT
iajs-2555	113	61	ị={∅	ị={∅	PROPN
iajs-2555	113	62	}	}	PUNCT
iajs-2555	113	63	and	and	CCONJ
iajs-2555	113	64	ʝ={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3	ʝ={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3	PROPN
iajs-2555	113	65	}	}	PUNCT
iajs-2555	113	66	}	}	PUNCT
iajs-2555	113	67	then	then	ADV
iajs-2555	113	68	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	PROPN
iajs-2555	113	69	,	,	PUNCT
iajs-2555	113	70	ẻ2},{ẻ1	ẻ2},{ẻ1	NOUN
iajs-2555	113	71	,	,	PUNCT
iajs-2555	113	72	ẻ3	ẻ3	PROPN
iajs-2555	113	73	}	}	PUNCT
iajs-2555	113	74	}	}	PUNCT
iajs-2555	113	75	then	then	ADV
iajs-2555	113	76	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	VERB
iajs-2555	113	77	)	)	PUNCT
iajs-2555	113	78	=	=	SYM
iajs-2555	113	79	{	{	PUNCT
iajs-2555	113	80	ӽ,∅,{ẻ2,ẻ3	ӽ,∅,{ẻ2,ẻ3	NOUN
iajs-2555	113	81	}	}	PUNCT
iajs-2555	113	82	}	}	PUNCT
iajs-2555	113	83	and	and	CCONJ
iajs-2555	113	84	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	113	85	)	)	PUNCT
iajs-2555	114	1	=	=	SYM
iajs-2555	114	2	{	{	PUNCT
iajs-2555	114	3	ӽ,∅,{ẻ1	ӽ,∅,{ẻ1	ADV
iajs-2555	114	4	}	}	PUNCT
iajs-2555	114	5	}	}	PUNCT
iajs-2555	114	6	.	.	PUNCT
iajs-2555	115	1	so	so	ADV
iajs-2555	115	2	𝛼𝑔ʝ𝐶(ӽ	𝛼𝑔ʝ𝐶(ӽ	NUM
iajs-2555	115	3	)	)	PUNCT
iajs-2555	115	4	=	=	SYM
iajs-2555	116	1	ҏ(ӽ	ҏ(ӽ	X
iajs-2555	116	2	)	)	PUNCT
iajs-2555	116	3	and	and	CCONJ
iajs-2555	116	4	𝛼𝑔ʝ𝑂(ӽ	𝛼𝑔ʝ𝑂(ӽ	NOUN
iajs-2555	116	5	)	)	PUNCT
iajs-2555	116	6	=	=	SYM
iajs-2555	116	7	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	116	8	)	)	PUNCT
iajs-2555	116	9	.	.	PUNCT
iajs-2555	117	1	then	then	ADV
iajs-2555	117	2	ᶂ	ᶂ	PROPN
iajs-2555	117	3	is	be	AUX
iajs-2555	117	4	𝛼𝑔ịo	𝛼𝑔ịo	NOUN
iajs-2555	117	5	-	-	PUNCT
iajs-2555	117	6	function	function	NOUN
iajs-2555	117	7	and	and	CCONJ
iajs-2555	117	8	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	117	9	∗o	∗o	NOUN
iajs-2555	117	10	-	-	PUNCT
iajs-2555	117	11	function	function	NOUN
iajs-2555	117	12	which	which	PRON
iajs-2555	117	13	is	be	AUX
iajs-2555	117	14	not	not	PART
iajs-2555	117	15	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	117	16	∗∗o	∗∗o	NOUN
iajs-2555	117	17	-	-	PUNCT
iajs-2555	117	18	function	function	NOUN
iajs-2555	117	19	and	and	CCONJ
iajs-2555	117	20	not	not	PART
iajs-2555	117	21	an	an	DET
iajs-2555	117	22	open	open	ADJ
iajs-2555	117	23	function	function	NOUN
iajs-2555	117	24	,	,	PUNCT
iajs-2555	117	25	since	since	SCONJ
iajs-2555	117	26	{	{	PUNCT
iajs-2555	117	27	ẻ1	ẻ1	NOUN
iajs-2555	117	28	}	}	PUNCT
iajs-2555	117	29	is	be	AUX
iajs-2555	117	30	an	an	DET
iajs-2555	117	31	open	open	ADJ
iajs-2555	117	32	set	set	NOUN
iajs-2555	117	33	in	in	ADP
iajs-2555	117	34	ӽ	ӽ	PRON
iajs-2555	117	35	and	and	CCONJ
iajs-2555	117	36	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	117	37	-	-	PUNCT
iajs-2555	117	38	open	open	NOUN
iajs-2555	117	39	set	set	NOUN
iajs-2555	117	40	,	,	PUNCT
iajs-2555	117	41	but	but	CCONJ
iajs-2555	117	42	ᶂ(ẻ1)=(ẻ2	ᶂ(ẻ1)=(ẻ2	NUM
iajs-2555	117	43	)	)	PUNCT
iajs-2555	117	44	which	which	PRON
iajs-2555	117	45	is	be	AUX
iajs-2555	117	46	not	not	PART
iajs-2555	117	47	open	open	ADJ
iajs-2555	117	48	.	.	PUNCT
iajs-2555	118	1	example	example	NOUN
iajs-2555	118	2	4	4	NUM
iajs-2555	118	3	:	:	PUNCT
iajs-2555	119	1	the	the	DET
iajs-2555	119	2	function	function	NOUN
iajs-2555	119	3	ᶂ	ᶂ	NOUN
iajs-2555	119	4	:	:	PUNCT
iajs-2555	119	5	(	(	PUNCT
iajs-2555	119	6	ӽ	ӽ	X
iajs-2555	119	7	,	,	PUNCT
iajs-2555	119	8	ῖ	ῖ	PROPN
iajs-2555	119	9	,	,	PUNCT
iajs-2555	119	10	ị	ị	PRON
iajs-2555	119	11	)	)	PUNCT
iajs-2555	119	12	→	→	SYM
iajs-2555	119	13	(	(	PUNCT
iajs-2555	119	14	ӽ	ӽ	X
iajs-2555	119	15	,	,	PUNCT
iajs-2555	119	16	ῖ	ῖ	PROPN
iajs-2555	119	17	,	,	PUNCT
iajs-2555	119	18	ị	ị	PROPN
iajs-2555	119	19	)	)	PUNCT
iajs-2555	119	20	;	;	PUNCT
iajs-2555	119	21	where	where	SCONJ
iajs-2555	119	22	ӽ={ẻ1	ӽ={ẻ1	NOUN
iajs-2555	119	23	,	,	PUNCT
iajs-2555	119	24	ẻ2	ẻ2	PROPN
iajs-2555	119	25	,	,	PUNCT
iajs-2555	119	26	ẻ3	ẻ3	PROPN
iajs-2555	119	27	}	}	PUNCT
iajs-2555	119	28	such	such	ADJ
iajs-2555	119	29	that	that	DET
iajs-2555	119	30	ᶂ(ẻ	ᶂ(ẻ	NOUN
iajs-2555	119	31	)	)	PUNCT
iajs-2555	119	32	=	=	PUNCT
iajs-2555	119	33	(	(	PUNCT
iajs-2555	119	34	ẻ	ẻ	NOUN
iajs-2555	119	35	)	)	PUNCT
iajs-2555	119	36	,	,	PUNCT
iajs-2555	119	37	∀	∀	X
iajs-2555	119	38	ẻ∈ӽ	ẻ∈ӽ	NOUN
iajs-2555	119	39	,	,	PUNCT
iajs-2555	119	40	ῖ={ӽ,∅,{ẻ1	ῖ={ӽ,∅,{ẻ1	NOUN
iajs-2555	119	41	}	}	PUNCT
iajs-2555	119	42	}	}	PUNCT
iajs-2555	119	43	,	,	PUNCT
iajs-2555	119	44	ị={∅,{ẻ2},{ẻ3},{ẻ2	ị={∅,{ẻ2},{ẻ3},{ẻ2	X
iajs-2555	119	45	,	,	PUNCT
iajs-2555	119	46	ẻ3	ẻ3	PROPN
iajs-2555	119	47	}	}	PUNCT
iajs-2555	119	48	}	}	PUNCT
iajs-2555	119	49	and	and	CCONJ
iajs-2555	119	50	ʝ={∅	ʝ={∅	ADJ
iajs-2555	119	51	}	}	PUNCT
iajs-2555	119	52	.	.	PUNCT
iajs-2555	120	1	then	then	ADV
iajs-2555	120	2	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	PROPN
iajs-2555	120	3	,	,	PUNCT
iajs-2555	120	4	ẻ2},{ẻ1	ẻ2},{ẻ1	NOUN
iajs-2555	120	5	,	,	PUNCT
iajs-2555	120	6	ẻ3	ẻ3	PROPN
iajs-2555	120	7	}	}	PUNCT
iajs-2555	120	8	}	}	PUNCT
iajs-2555	120	9	then	then	ADV
iajs-2555	120	10	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	VERB
iajs-2555	120	11	)	)	PUNCT
iajs-2555	120	12	=	=	SYM
iajs-2555	121	1	ҏ(ӽ	ҏ(ӽ	X
iajs-2555	121	2	)	)	PUNCT
iajs-2555	121	3	and	and	CCONJ
iajs-2555	121	4	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	121	5	)	)	PUNCT
iajs-2555	121	6	=	=	SYM
iajs-2555	121	7	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	121	8	)	)	PUNCT
iajs-2555	121	9	.	.	PUNCT
iajs-2555	122	1	so	so	ADV
iajs-2555	122	2	𝛼𝑔ʝ𝐶(ӽ	𝛼𝑔ʝ𝐶(ӽ	NOUN
iajs-2555	122	3	)	)	PUNCT
iajs-2555	122	4	=	=	PRON
iajs-2555	122	5	{	{	PUNCT
iajs-2555	122	6	ӽ,∅,{ẻ2,ẻ3	ӽ,∅,{ẻ2,ẻ3	NOUN
iajs-2555	122	7	}	}	PUNCT
iajs-2555	122	8	}	}	PUNCT
iajs-2555	122	9	and	and	CCONJ
iajs-2555	122	10	𝛼𝑔ʝ𝑂(ӽ	𝛼𝑔ʝ𝑂(ӽ	NOUN
iajs-2555	122	11	)	)	PUNCT
iajs-2555	122	12	=	=	SYM
iajs-2555	122	13	{	{	PUNCT
iajs-2555	122	14	ӽ,∅,{ẻ1	ӽ,∅,{ẻ1	ADV
iajs-2555	122	15	}	}	PUNCT
iajs-2555	122	16	}	}	PUNCT
iajs-2555	122	17	.	.	PUNCT
iajs-2555	123	1	it	it	PRON
iajs-2555	123	2	is	be	AUX
iajs-2555	123	3	easy	easy	ADJ
iajs-2555	123	4	to	to	PART
iajs-2555	123	5	see	see	VERB
iajs-2555	123	6	that	that	SCONJ
iajs-2555	123	7	ᶂ	ᶂ	NOUN
iajs-2555	123	8	is	be	AUX
iajs-2555	123	9	an	an	DET
iajs-2555	123	10	open	open	ADJ
iajs-2555	123	11	function	function	NOUN
iajs-2555	123	12	and	and	CCONJ
iajs-2555	123	13	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	123	14	∗o	∗o	NOUN
iajs-2555	123	15	-	-	PUNCT
iajs-2555	123	16	function	function	NOUN
iajs-2555	123	17	but	but	CCONJ
iajs-2555	123	18	it	it	PRON
iajs-2555	123	19	is	be	AUX
iajs-2555	123	20	not	not	PART
iajs-2555	123	21	𝛼𝑔ịo	𝛼𝑔ịo	NOUN
iajs-2555	123	22	-	-	PUNCT
iajs-2555	123	23	function	function	NOUN
iajs-2555	123	24	and	and	CCONJ
iajs-2555	123	25	not	not	PART
iajs-2555	123	26	𝛼𝑔ị	𝛼𝑔ị	VERB
iajs-2555	123	27	∗∗c	∗∗c	NOUN
iajs-2555	123	28	-	-	NOUN
iajs-2555	123	29	function	function	NOUN
iajs-2555	123	30	,	,	PUNCT
iajs-2555	123	31	since	since	SCONJ
iajs-2555	123	32	{	{	PUNCT
iajs-2555	123	33	ẻ2}∈	ẻ2}∈	NOUN
iajs-2555	123	34	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	PROPN
iajs-2555	123	35	)	)	PUNCT
iajs-2555	123	36	but	but	CCONJ
iajs-2555	123	37	ᶂ(ẻ2	ᶂ(ẻ2	NUM
iajs-2555	123	38	)	)	PUNCT
iajs-2555	123	39	=	=	SYM
iajs-2555	123	40	(	(	PUNCT
iajs-2555	123	41	ẻ1	ẻ1	NOUN
iajs-2555	123	42	)	)	PUNCT
iajs-2555	123	43	which	which	PRON
iajs-2555	123	44	is	be	AUX
iajs-2555	123	45	not	not	PART
iajs-2555	123	46	open	open	ADJ
iajs-2555	123	47	and	and	CCONJ
iajs-2555	123	48	not	not	PART
iajs-2555	123	49	𝛼𝑔ʝ	𝛼𝑔ʝ	NOUN
iajs-2555	123	50	-	-	ADJ
iajs-2555	123	51	open	open	ADJ
iajs-2555	123	52	set	set	NOUN
iajs-2555	123	53	.	.	PUNCT
iajs-2555	124	1	definition	definition	NOUN
iajs-2555	124	2	5	5	NUM
iajs-2555	124	3	:	:	PUNCT
iajs-2555	124	4	the	the	DET
iajs-2555	124	5	function	function	NOUN
iajs-2555	124	6	ᶂ	ᶂ	NOUN
iajs-2555	124	7	:	:	PUNCT
iajs-2555	124	8	(	(	PUNCT
iajs-2555	124	9	ӽ	ӽ	X
iajs-2555	124	10	,	,	PUNCT
iajs-2555	124	11	ῖ	ῖ	PROPN
iajs-2555	124	12	,	,	PUNCT
iajs-2555	124	13	ị	ị	PRON
iajs-2555	124	14	)	)	PUNCT
iajs-2555	124	15	→	→	SYM
iajs-2555	124	16	(	(	PUNCT
iajs-2555	124	17	ƴ	ƴ	PROPN
iajs-2555	124	18	,	,	PUNCT
iajs-2555	124	19	ɟ	ɟ	NOUN
iajs-2555	124	20	,	,	PUNCT
iajs-2555	124	21	ʝ	ʝ	NOUN
iajs-2555	124	22	)	)	PUNCT
iajs-2555	124	23	is	be	AUX
iajs-2555	124	24	said	say	VERB
iajs-2555	124	25	,	,	PUNCT
iajs-2555	124	26	i.	i.	PROPN
iajs-2555	124	27	𝛼𝑔ị	𝛼𝑔ị	PROPN
iajs-2555	124	28	-	-	PUNCT
iajs-2555	124	29	closed	close	VERB
iajs-2555	124	30	function	function	NOUN
iajs-2555	124	31	,	,	PUNCT
iajs-2555	124	32	denoted	denote	VERB
iajs-2555	124	33	by	by	ADP
iajs-2555	124	34	"	"	PUNCT
iajs-2555	124	35	𝛼𝑔ịc	𝛼𝑔ịc	ADJ
iajs-2555	124	36	-	-	PUNCT
iajs-2555	124	37	function	function	NOUN
iajs-2555	124	38	"	"	PUNCT
iajs-2555	124	39	if	if	SCONJ
iajs-2555	124	40	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	124	41	)	)	PUNCT
iajs-2555	124	42	is	be	AUX
iajs-2555	124	43	𝛼𝑔ʝ	𝛼𝑔ʝ	PROPN
iajs-2555	124	44	-	-	VERB
iajs-2555	124	45	closed	closed	ADJ
iajs-2555	124	46	in	in	ADP
iajs-2555	124	47	ƴ	ƴ	PRON
iajs-2555	124	48	whenever	whenever	SCONJ
iajs-2555	124	49	ơ	ơ	PROPN
iajs-2555	124	50	is	be	AUX
iajs-2555	124	51	an	an	DET
iajs-2555	124	52	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	124	53	-	-	PUNCT
iajs-2555	124	54	closed	closed	ADJ
iajs-2555	124	55	in	in	ADP
iajs-2555	124	56	ӽ.	ӽ.	PROPN
iajs-2555	124	57	ii	ii	PROPN
iajs-2555	124	58	.	.	PUNCT
iajs-2555	125	1	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	125	2	∗-closed	∗-close	VERB
iajs-2555	125	3	function	function	NOUN
iajs-2555	125	4	,	,	PUNCT
iajs-2555	125	5	denoted	denote	VERB
iajs-2555	125	6	by	by	ADP
iajs-2555	125	7	"	"	PUNCT
iajs-2555	125	8	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	125	9	∗c	∗c	NUM
iajs-2555	125	10	-	-	PUNCT
iajs-2555	125	11	function	function	NOUN
iajs-2555	125	12	"	"	PUNCT
iajs-2555	125	13	,	,	PUNCT
iajs-2555	125	14	if	if	SCONJ
iajs-2555	125	15	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	125	16	)	)	PUNCT
iajs-2555	125	17	is	be	AUX
iajs-2555	125	18	𝛼𝑔ʝ	𝛼𝑔ʝ	PROPN
iajs-2555	125	19	-	-	VERB
iajs-2555	125	20	closed	closed	ADJ
iajs-2555	125	21	in	in	ADP
iajs-2555	125	22	ƴ	ƴ	PRON
iajs-2555	125	23	whenever	whenever	SCONJ
iajs-2555	125	24	ơ	ơ	PROPN
iajs-2555	125	25	is	be	AUX
iajs-2555	125	26	an	an	DET
iajs-2555	125	27	closed	closed	ADJ
iajs-2555	125	28	in	in	ADP
iajs-2555	125	29	ӽ.	ӽ.	PROPN
iajs-2555	125	30	iii	iii	PROPN
iajs-2555	125	31	.	.	PROPN
iajs-2555	125	32	𝛼𝑔ị	𝛼𝑔ị	PROPN
iajs-2555	125	33	∗∗-closed	∗∗-close	VERB
iajs-2555	125	34	function	function	NOUN
iajs-2555	125	35	,	,	PUNCT
iajs-2555	125	36	denoted	denote	VERB
iajs-2555	125	37	by	by	ADP
iajs-2555	125	38	"	"	PUNCT
iajs-2555	125	39	𝛼𝑔ị	𝛼𝑔ị	DET
iajs-2555	125	40	∗∗c	∗∗c	NOUN
iajs-2555	125	41	-	-	NOUN
iajs-2555	125	42	function	function	NOUN
iajs-2555	125	43	"	"	PUNCT
iajs-2555	125	44	,	,	PUNCT
iajs-2555	125	45	if	if	SCONJ
iajs-2555	125	46	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	125	47	)	)	PUNCT
iajs-2555	125	48	is	be	AUX
iajs-2555	125	49	closed	close	VERB
iajs-2555	125	50	in	in	ADP
iajs-2555	125	51	ƴ	ƴ	PRON
iajs-2555	125	52	whenever	whenever	SCONJ
iajs-2555	125	53	ơ	ơ	PROPN
iajs-2555	125	54	is	be	AUX
iajs-2555	125	55	an	an	DET
iajs-2555	125	56	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	125	57	-	-	PUNCT
iajs-2555	125	58	closed	close	VERB
iajs-2555	125	59	in	in	ADP
iajs-2555	125	60	ӽ.	ӽ.	NOUN
iajs-2555	125	61	proposition	proposition	NOUN
iajs-2555	125	62	6	6	NUM
iajs-2555	125	63	:	:	PUNCT
iajs-2555	125	64	let	let	VERB
iajs-2555	125	65	ᶂ	ᶂ	NOUN
iajs-2555	125	66	:	:	PUNCT
iajs-2555	125	67	(	(	PUNCT
iajs-2555	125	68	ӽ	ӽ	X
iajs-2555	125	69	,	,	PUNCT
iajs-2555	125	70	ῖ	ῖ	PROPN
iajs-2555	125	71	,	,	PUNCT
iajs-2555	125	72	ị	ị	PRON
iajs-2555	125	73	)	)	PUNCT
iajs-2555	125	74	→	→	SYM
iajs-2555	125	75	(	(	PUNCT
iajs-2555	125	76	ƴ	ƴ	PROPN
iajs-2555	125	77	,	,	PUNCT
iajs-2555	125	78	ɟ	ɟ	NOUN
iajs-2555	125	79	,	,	PUNCT
iajs-2555	125	80	ʝ	ʝ	NOUN
iajs-2555	125	81	)	)	PUNCT
iajs-2555	125	82	is	be	AUX
iajs-2555	125	83	function	function	NOUN
iajs-2555	125	84	,	,	PUNCT
iajs-2555	125	85	i.	i.	NOUN
iajs-2555	125	86	if	if	SCONJ
iajs-2555	125	87	ᶂ	ᶂ	NOUN
iajs-2555	125	88	is	be	AUX
iajs-2555	125	89	a	a	DET
iajs-2555	125	90	closed	closed	ADJ
iajs-2555	125	91	function	function	NOUN
iajs-2555	125	92	then	then	ADV
iajs-2555	125	93	ᶂ	ᶂ	PROPN
iajs-2555	125	94	is	be	AUX
iajs-2555	125	95	an	an	DET
iajs-2555	125	96	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	125	97	∗c	∗c	NUM
iajs-2555	125	98	-	-	PUNCT
iajs-2555	125	99	function	function	NOUN
iajs-2555	125	100	.	.	PUNCT
iajs-2555	126	1	ii	ii	PROPN
iajs-2555	126	2	.	.	PUNCT
iajs-2555	127	1	if	if	SCONJ
iajs-2555	127	2	ᶂ	ᶂ	NOUN
iajs-2555	127	3	is	be	AUX
iajs-2555	127	4	an	an	DET
iajs-2555	127	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	127	6	∗∗c	∗∗c	NOUN
iajs-2555	127	7	-	-	NOUN
iajs-2555	127	8	function	function	NOUN
iajs-2555	127	9	then	then	ADV
iajs-2555	127	10	ᶂ	ᶂ	NOUN
iajs-2555	127	11	is	be	AUX
iajs-2555	127	12	an	an	DET
iajs-2555	127	13	𝛼𝑔ịc	𝛼𝑔ịc	NOUN
iajs-2555	127	14	-	-	PUNCT
iajs-2555	127	15	function	function	NOUN
iajs-2555	127	16	.	.	PUNCT
iajs-2555	128	1	iii	iii	X
iajs-2555	128	2	.	.	PUNCT
iajs-2555	129	1	if	if	SCONJ
iajs-2555	129	2	ᶂ	ᶂ	NOUN
iajs-2555	129	3	is	be	AUX
iajs-2555	129	4	an	an	DET
iajs-2555	129	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	129	6	∗∗c	∗∗c	NOUN
iajs-2555	129	7	-	-	NOUN
iajs-2555	129	8	function	function	NOUN
iajs-2555	129	9	then	then	ADV
iajs-2555	129	10	ᶂ	ᶂ	NOUN
iajs-2555	129	11	is	be	AUX
iajs-2555	129	12	a	a	DET
iajs-2555	129	13	closed	closed	ADJ
iajs-2555	129	14	function	function	NOUN
iajs-2555	129	15	.	.	PUNCT
iajs-2555	130	1	iv	iv	X
iajs-2555	130	2	.	.	PUNCT
iajs-2555	131	1	if	if	SCONJ
iajs-2555	131	2	ᶂ	ᶂ	NOUN
iajs-2555	131	3	is	be	AUX
iajs-2555	131	4	an	an	DET
iajs-2555	131	5	𝛼𝑔ịc	𝛼𝑔ịc	NOUN
iajs-2555	131	6	-	-	PUNCT
iajs-2555	131	7	function	function	NOUN
iajs-2555	131	8	then	then	ADV
iajs-2555	131	9	ᶂ	ᶂ	NOUN
iajs-2555	131	10	is	be	AUX
iajs-2555	131	11	an	an	DET
iajs-2555	131	12	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	131	13	∗c	∗c	NUM
iajs-2555	131	14	-	-	PUNCT
iajs-2555	131	15	function	function	NOUN
iajs-2555	131	16	.	.	PUNCT
iajs-2555	132	1	v.	v.	INTJ
iajs-2555	132	2	if	if	SCONJ
iajs-2555	132	3	ᶂ	ᶂ	NOUN
iajs-2555	132	4	is	be	AUX
iajs-2555	132	5	an	an	DET
iajs-2555	132	6	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	132	7	∗∗c	∗∗c	NOUN
iajs-2555	132	8	-	-	NOUN
iajs-2555	132	9	function	function	NOUN
iajs-2555	132	10	then	then	ADV
iajs-2555	132	11	ᶂ	ᶂ	NOUN
iajs-2555	132	12	is	be	AUX
iajs-2555	132	13	an	an	DET
iajs-2555	132	14	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	132	15	∗c	∗c	NUM
iajs-2555	132	16	-	-	PUNCT
iajs-2555	132	17	function	function	NOUN
iajs-2555	132	18	.	.	PUNCT
iajs-2555	133	1	proof	proof	NOUN
iajs-2555	133	2	:	:	PUNCT
iajs-2555	133	3	by	by	ADP
iajs-2555	133	4	remark	remark	NOUN
iajs-2555	133	5	2.4	2.4	NUM
iajs-2555	133	6	and	and	CCONJ
iajs-2555	133	7	definition	definition	NOUN
iajs-2555	133	8	3.5	3.5	NUM
iajs-2555	133	9	.	.	PUNCT
iajs-2555	134	1	the	the	DET
iajs-2555	134	2	follow	follow	NOUN
iajs-2555	134	3	diagram	diagram	NOUN
iajs-2555	134	4	shows	show	VERB
iajs-2555	134	5	the	the	DET
iajs-2555	134	6	relationships	relationship	NOUN
iajs-2555	134	7	between	between	ADP
iajs-2555	134	8	the	the	DET
iajs-2555	134	9	different	different	ADJ
iajs-2555	134	10	concepts	concept	NOUN
iajs-2555	134	11	that	that	PRON
iajs-2555	134	12	are	be	AUX
iajs-2555	134	13	inserted	insert	VERB
iajs-2555	134	14	in	in	ADP
iajs-2555	134	15	definition	definition	NOUN
iajs-2555	134	16	3.5	3.5	NUM
iajs-2555	134	17	35	35	NUM
iajs-2555	134	18	ibn	ibn	PROPN
iajs-2555	134	19	al	al	PROPN
iajs-2555	134	20	-	-	PUNCT
iajs-2555	134	21	haitham	haitham	PROPN
iajs-2555	134	22	jour	jour	X
iajs-2555	134	23	.	.	PROPN
iajs-2555	135	1	for	for	ADP
iajs-2555	135	2	pure	pure	ADJ
iajs-2555	135	3	&	&	CCONJ
iajs-2555	135	4	appl	appl	PROPN
iajs-2555	135	5	.	.	PUNCT
iajs-2555	136	1	sci	sci	PROPN
iajs-2555	136	2	.	.	PROPN
iajs-2555	137	1	34	34	NUM
iajs-2555	137	2	(	(	PUNCT
iajs-2555	137	3	1	1	NUM
iajs-2555	137	4	)	)	PUNCT
iajs-2555	137	5	2021	2021	NUM
iajs-2555	137	6	arrow	arrow	NOUN
iajs-2555	137	7	chart	chart	NOUN
iajs-2555	137	8	(	(	PUNCT
iajs-2555	137	9	3.2	3.2	NUM
iajs-2555	137	10	)	)	PUNCT
iajs-2555	137	11	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	137	12	-	-	PUNCT
iajs-2555	137	13	closed	close	VERB
iajs-2555	137	14	function	function	NOUN
iajs-2555	137	15	example	example	NOUN
iajs-2555	137	16	3.3	3.3	NUM
iajs-2555	137	17	and	and	CCONJ
iajs-2555	137	18	3.4	3.4	NUM
iajs-2555	137	19	show	show	NOUN
iajs-2555	137	20	that	that	SCONJ
iajs-2555	137	21	the	the	DET
iajs-2555	137	22	opposite	opposite	ADJ
iajs-2555	137	23	direction	direction	NOUN
iajs-2555	137	24	of	of	ADP
iajs-2555	137	25	the	the	DET
iajs-2555	137	26	above	above	ADJ
iajs-2555	137	27	chart	chart	NOUN
iajs-2555	137	28	is	be	AUX
iajs-2555	137	29	incorrect	incorrect	ADJ
iajs-2555	137	30	.	.	PUNCT
iajs-2555	138	1	remark	remark	VERB
iajs-2555	138	2	7	7	NUM
iajs-2555	138	3	:	:	PUNCT
iajs-2555	138	4	if	if	SCONJ
iajs-2555	138	5	ᶂ	ᶂ	NOUN
iajs-2555	138	6	is	be	AUX
iajs-2555	138	7	onto	onto	ADP
iajs-2555	138	8	function	function	NOUN
iajs-2555	138	9	then	then	ADV
iajs-2555	138	10	:	:	PUNCT
iajs-2555	138	11	i.	i.	NOUN
iajs-2555	138	12	𝛼𝑔ịo	𝛼𝑔ịo	NOUN
iajs-2555	138	13	-	-	PUNCT
iajs-2555	138	14	function	function	NOUN
iajs-2555	138	15	and	and	CCONJ
iajs-2555	138	16	𝛼𝑔ịc	𝛼𝑔ịc	NOUN
iajs-2555	138	17	-	-	PUNCT
iajs-2555	138	18	function	function	NOUN
iajs-2555	138	19	are	be	AUX
iajs-2555	138	20	the	the	DET
iajs-2555	138	21	same	same	ADJ
iajs-2555	138	22	.	.	PUNCT
iajs-2555	139	1	ii	ii	PROPN
iajs-2555	139	2	.	.	PUNCT
iajs-2555	140	1	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	140	2	∗o	∗o	NOUN
iajs-2555	140	3	-	-	PUNCT
iajs-2555	140	4	function	function	NOUN
iajs-2555	140	5	and	and	CCONJ
iajs-2555	140	6	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	140	7	∗c	∗c	NUM
iajs-2555	140	8	-	-	PUNCT
iajs-2555	140	9	function	function	NOUN
iajs-2555	140	10	are	be	AUX
iajs-2555	140	11	the	the	DET
iajs-2555	140	12	same	same	ADJ
iajs-2555	140	13	.	.	PUNCT
iajs-2555	141	1	iii	iii	X
iajs-2555	141	2	.	.	PUNCT
iajs-2555	141	3	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	141	4	∗∗o	∗∗o	NOUN
iajs-2555	141	5	-	-	PUNCT
iajs-2555	141	6	function	function	NOUN
iajs-2555	141	7	and	and	CCONJ
iajs-2555	141	8	𝛼𝑔ị	𝛼𝑔ị	DET
iajs-2555	141	9	∗∗c	∗∗c	NOUN
iajs-2555	141	10	-	-	PUNCT
iajs-2555	141	11	function	function	NOUN
iajs-2555	141	12	are	be	AUX
iajs-2555	141	13	the	the	DET
iajs-2555	141	14	same	same	ADJ
iajs-2555	141	15	.	.	PUNCT
iajs-2555	142	1	proof	proof	NOUN
iajs-2555	142	2	:	:	PUNCT
iajs-2555	142	3	since	since	SCONJ
iajs-2555	142	4	ᶂ	ᶂ	NOUN
iajs-2555	142	5	is	be	AUX
iajs-2555	142	6	an	an	PRON
iajs-2555	142	7	onto	onto	ADP
iajs-2555	142	8	function	function	NOUN
iajs-2555	142	9	then	then	ADV
iajs-2555	142	10	the	the	DET
iajs-2555	142	11	prove	prove	NOUN
iajs-2555	142	12	is	be	AUX
iajs-2555	142	13	easy	easy	ADJ
iajs-2555	142	14	by	by	ADP
iajs-2555	142	15	using	use	VERB
iajs-2555	142	16	definition	definition	NOUN
iajs-2555	142	17	3.1	3.1	NUM
iajs-2555	142	18	and	and	CCONJ
iajs-2555	142	19	definition	definition	NOUN
iajs-2555	142	20	3.5	3.5	NUM
iajs-2555	142	21	4near	4near	NUM
iajs-2555	142	22	continuous	continuous	ADJ
iajs-2555	142	23	function	function	NOUN
iajs-2555	142	24	𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧	𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧	PROPN
iajs-2555	142	25	1	1	NUM
iajs-2555	142	26	:	:	PUNCT
iajs-2555	142	27	a	a	DET
iajs-2555	142	28	function	function	NOUN
iajs-2555	142	29	ᶂ	ᶂ	NOUN
iajs-2555	142	30	:	:	PUNCT
iajs-2555	142	31	(	(	PUNCT
iajs-2555	142	32	ӽ	ӽ	X
iajs-2555	142	33	,	,	PUNCT
iajs-2555	142	34	ῖ	ῖ	PROPN
iajs-2555	142	35	,	,	PUNCT
iajs-2555	142	36	ị	ị	PRON
iajs-2555	142	37	)	)	PUNCT
iajs-2555	142	38	→	→	SYM
iajs-2555	142	39	(	(	PUNCT
iajs-2555	142	40	ƴ	ƴ	PROPN
iajs-2555	142	41	,	,	PUNCT
iajs-2555	142	42	ɟ	ɟ	NOUN
iajs-2555	142	43	,	,	PUNCT
iajs-2555	142	44	ʝ	ʝ	NOUN
iajs-2555	142	45	)	)	PUNCT
iajs-2555	142	46	is	be	AUX
iajs-2555	142	47	called	call	VERB
iajs-2555	142	48	;	;	PUNCT
iajs-2555	142	49	i.	i.	NOUN
iajs-2555	142	50	ị-𝛼-g	ị-𝛼-g	PROPN
iajs-2555	142	51	-	-	ADJ
iajs-2555	142	52	continuous	continuous	ADJ
iajs-2555	142	53	function	function	NOUN
iajs-2555	142	54	,	,	PUNCT
iajs-2555	142	55	denoted	denote	VERB
iajs-2555	142	56	by	by	ADP
iajs-2555	142	57	"	"	PUNCT
iajs-2555	142	58	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	142	59	-	-	PUNCT
iajs-2555	142	60	continuous	continuous	ADJ
iajs-2555	142	61	function	function	NOUN
iajs-2555	142	62	"	"	PUNCT
iajs-2555	142	63	,	,	PUNCT
iajs-2555	142	64	if	if	SCONJ
iajs-2555	142	65	ᶂ−1(ơ	ᶂ−1(ơ	PROPN
iajs-2555	142	66	)	)	PUNCT
iajs-2555	142	67	is	be	AUX
iajs-2555	142	68	an	an	DET
iajs-2555	142	69	𝛼𝑔ịopen	𝛼𝑔ịopen	ADJ
iajs-2555	142	70	set	set	NOUN
iajs-2555	142	71	in	in	ADP
iajs-2555	142	72	ӽ	ӽ	NOUN
iajs-2555	142	73	,	,	PUNCT
iajs-2555	142	74	where	where	SCONJ
iajs-2555	142	75	ơ	ơ	PROPN
iajs-2555	142	76	∈	∈	PROPN
iajs-2555	142	77	ɟ	ɟ	PROPN
iajs-2555	142	78	.	.	PUNCT
iajs-2555	142	79	ii	ii	PROPN
iajs-2555	142	80	.	.	PUNCT
iajs-2555	143	1	strongly	strongly	ADV
iajs-2555	143	2	ị-𝛼-g	ị-𝛼-g	VERB
iajs-2555	143	3	-	-	ADJ
iajs-2555	143	4	continuous	continuous	ADJ
iajs-2555	143	5	function	function	NOUN
iajs-2555	143	6	,	,	PUNCT
iajs-2555	143	7	denoted	denote	VERB
iajs-2555	143	8	by	by	ADP
iajs-2555	143	9	"	"	PUNCT
iajs-2555	143	10	strongly	strongly	ADV
iajs-2555	143	11	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	143	12	-	-	PUNCT
iajs-2555	143	13	continuous	continuous	ADJ
iajs-2555	143	14	function	function	NOUN
iajs-2555	143	15	"	"	PUNCT
iajs-2555	143	16	if	if	SCONJ
iajs-2555	143	17	ᶂ−1(ơ	ᶂ−1(ơ	PROPN
iajs-2555	143	18	)	)	PUNCT
iajs-2555	143	19	∈	∈	PROPN
iajs-2555	144	1	ῖ	ῖ	X
iajs-2555	144	2	,	,	PUNCT
iajs-2555	144	3	whenever	whenever	SCONJ
iajs-2555	144	4	ơ	ơ	PROPN
iajs-2555	144	5	is	be	AUX
iajs-2555	144	6	an	an	DET
iajs-2555	144	7	𝛼𝑔ʝ	𝛼𝑔ʝ	ADJ
iajs-2555	144	8	-	-	ADJ
iajs-2555	144	9	open	open	ADJ
iajs-2555	144	10	set	set	NOUN
iajs-2555	144	11	in	in	ADP
iajs-2555	144	12	ƴ	ƴ	PROPN
iajs-2555	144	13	.	.	PUNCT
iajs-2555	144	14	iii	iii	PROPN
iajs-2555	144	15	.	.	PUNCT
iajs-2555	144	16	ị-𝛼-𝑔-irresolute	ị-𝛼-𝑔-irresolute	PROPN
iajs-2555	144	17	function	function	PROPN
iajs-2555	144	18	,	,	PUNCT
iajs-2555	144	19	denoted	denote	VERB
iajs-2555	144	20	by	by	ADP
iajs-2555	144	21	"	"	PUNCT
iajs-2555	144	22	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	144	23	-	-	PUNCT
iajs-2555	144	24	irresolute	irresolute	ADJ
iajs-2555	144	25	function	function	NOUN
iajs-2555	144	26	"	"	PUNCT
iajs-2555	144	27	,	,	PUNCT
iajs-2555	144	28	if	if	SCONJ
iajs-2555	144	29	ᶂ−1(ơ	ᶂ−1(ơ	PROPN
iajs-2555	144	30	)	)	PUNCT
iajs-2555	144	31	is	be	AUX
iajs-2555	144	32	an	an	DET
iajs-2555	144	33	𝛼𝑔ịopen	𝛼𝑔ịopen	ADJ
iajs-2555	144	34	set	set	NOUN
iajs-2555	144	35	in	in	ADP
iajs-2555	144	36	ӽ	ӽ	NOUN
iajs-2555	144	37	,	,	PUNCT
iajs-2555	144	38	where	where	SCONJ
iajs-2555	144	39	ơ	ơ	PROPN
iajs-2555	144	40	is	be	AUX
iajs-2555	144	41	an	an	DET
iajs-2555	144	42	𝛼𝑔ʝ	𝛼𝑔ʝ	ADJ
iajs-2555	144	43	-	-	ADJ
iajs-2555	144	44	open	open	ADJ
iajs-2555	144	45	set	set	NOUN
iajs-2555	144	46	in	in	ADP
iajs-2555	144	47	ƴ	ƴ	PROPN
iajs-2555	144	48	.	.	PUNCT
iajs-2555	144	49	𝜶𝒈ị𝒄-function	𝜶𝒈ị𝒄-function	NOUN
iajs-2555	144	50	𝜶𝒈ị	𝜶𝒈ị	CCONJ
iajs-2555	144	51	∗c	∗c	NUM
iajs-2555	144	52	-	-	PUNCT
iajs-2555	144	53	function	function	NOUN
iajs-2555	144	54	𝜶𝒈ị	𝜶𝒈ị	ADV
iajs-2555	144	55	∗∗c	∗∗c	VERB
iajs-2555	144	56	-	-	PUNCT
iajs-2555	144	57	function	function	NOUN
iajs-2555	144	58	closed	closed	ADJ
iajs-2555	144	59	function	function	NOUN
iajs-2555	144	60	36	36	NUM
iajs-2555	144	61	ibn	ibn	PROPN
iajs-2555	144	62	al	al	PROPN
iajs-2555	144	63	-	-	PUNCT
iajs-2555	144	64	haitham	haitham	PROPN
iajs-2555	144	65	jour	jour	X
iajs-2555	144	66	.	.	PROPN
iajs-2555	145	1	for	for	ADP
iajs-2555	145	2	pure	pure	ADJ
iajs-2555	145	3	&	&	CCONJ
iajs-2555	145	4	appl	appl	PROPN
iajs-2555	145	5	.	.	PUNCT
iajs-2555	146	1	sci	sci	PROPN
iajs-2555	146	2	.	.	PROPN
iajs-2555	147	1	34	34	NUM
iajs-2555	147	2	(	(	PUNCT
iajs-2555	147	3	1	1	NUM
iajs-2555	147	4	)	)	PUNCT
iajs-2555	147	5	2021	2021	NUM
iajs-2555	147	6	proposition	proposition	NOUN
iajs-2555	147	7	2	2	NUM
iajs-2555	147	8	:	:	PUNCT
iajs-2555	147	9	let	let	VERB
iajs-2555	147	10	ᶂ	ᶂ	NOUN
iajs-2555	147	11	:	:	PUNCT
iajs-2555	147	12	(	(	PUNCT
iajs-2555	147	13	ӽ	ӽ	X
iajs-2555	147	14	,	,	PUNCT
iajs-2555	147	15	ῖ	ῖ	PROPN
iajs-2555	147	16	,	,	PUNCT
iajs-2555	147	17	ị	ị	PRON
iajs-2555	147	18	)	)	PUNCT
iajs-2555	147	19	→	→	SYM
iajs-2555	147	20	(	(	PUNCT
iajs-2555	147	21	ƴ	ƴ	PROPN
iajs-2555	147	22	,	,	PUNCT
iajs-2555	147	23	ɟ	ɟ	NOUN
iajs-2555	147	24	,	,	PUNCT
iajs-2555	147	25	ʝ	ʝ	NOUN
iajs-2555	147	26	)	)	PUNCT
iajs-2555	147	27	is	be	AUX
iajs-2555	147	28	a	a	DET
iajs-2555	147	29	function	function	NOUN
iajs-2555	147	30	;	;	PUNCT
iajs-2555	147	31	i.	i.	NOUN
iajs-2555	147	32	if	if	SCONJ
iajs-2555	147	33	ᶂ	ᶂ	NOUN
iajs-2555	147	34	is	be	AUX
iajs-2555	147	35	a	a	DET
iajs-2555	147	36	continuous	continuous	ADJ
iajs-2555	147	37	function	function	NOUN
iajs-2555	147	38	,	,	PUNCT
iajs-2555	147	39	then	then	ADV
iajs-2555	147	40	ᶂ	ᶂ	PROPN
iajs-2555	147	41	is	be	AUX
iajs-2555	147	42	an	an	DET
iajs-2555	147	43	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	147	44	-	-	PUNCT
iajs-2555	147	45	continuous	continuous	ADJ
iajs-2555	147	46	function	function	NOUN
iajs-2555	147	47	.	.	PUNCT
iajs-2555	148	1	ii	ii	PROPN
iajs-2555	148	2	.	.	PUNCT
iajs-2555	149	1	if	if	SCONJ
iajs-2555	149	2	ᶂ	ᶂ	NOUN
iajs-2555	149	3	is	be	AUX
iajs-2555	149	4	strongly	strongly	ADV
iajs-2555	149	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	149	6	-	-	PUNCT
iajs-2555	149	7	continuous	continuous	ADJ
iajs-2555	149	8	function	function	NOUN
iajs-2555	149	9	,	,	PUNCT
iajs-2555	149	10	then	then	ADV
iajs-2555	149	11	ᶂ	ᶂ	PROPN
iajs-2555	149	12	is	be	AUX
iajs-2555	149	13	a	a	DET
iajs-2555	149	14	continuous	continuous	ADJ
iajs-2555	149	15	function	function	NOUN
iajs-2555	149	16	.	.	PUNCT
iajs-2555	150	1	iii	iii	X
iajs-2555	150	2	.	.	PUNCT
iajs-2555	151	1	if	if	SCONJ
iajs-2555	151	2	ᶂ	ᶂ	NOUN
iajs-2555	151	3	is	be	AUX
iajs-2555	151	4	an	an	DET
iajs-2555	151	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	151	6	-	-	PUNCT
iajs-2555	151	7	irresolute	irresolute	ADJ
iajs-2555	151	8	function	function	NOUN
iajs-2555	151	9	,	,	PUNCT
iajs-2555	151	10	then	then	ADV
iajs-2555	151	11	ᶂ	ᶂ	PROPN
iajs-2555	151	12	is	be	AUX
iajs-2555	151	13	an	an	DET
iajs-2555	151	14	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	151	15	-	-	PUNCT
iajs-2555	151	16	continuous	continuous	ADJ
iajs-2555	151	17	function	function	NOUN
iajs-2555	151	18	.	.	PUNCT
iajs-2555	152	1	iv	iv	X
iajs-2555	152	2	.	.	PUNCT
iajs-2555	153	1	if	if	SCONJ
iajs-2555	153	2	ᶂ	ᶂ	NOUN
iajs-2555	153	3	is	be	AUX
iajs-2555	153	4	strongly	strongly	ADV
iajs-2555	153	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	153	6	-	-	PUNCT
iajs-2555	153	7	continuous	continuous	ADJ
iajs-2555	153	8	function	function	NOUN
iajs-2555	153	9	,	,	PUNCT
iajs-2555	153	10	then	then	ADV
iajs-2555	153	11	ᶂ	ᶂ	PROPN
iajs-2555	153	12	is	be	AUX
iajs-2555	153	13	an	an	DET
iajs-2555	153	14	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	153	15	-	-	PUNCT
iajs-2555	153	16	irresolute	irresolute	ADJ
iajs-2555	153	17	function	function	NOUN
iajs-2555	153	18	.	.	PUNCT
iajs-2555	154	1	v.	v.	INTJ
iajs-2555	154	2	if	if	SCONJ
iajs-2555	154	3	ᶂ	ᶂ	NOUN
iajs-2555	154	4	is	be	AUX
iajs-2555	154	5	strongly	strongly	ADV
iajs-2555	154	6	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	154	7	-	-	PUNCT
iajs-2555	154	8	continuous	continuous	ADJ
iajs-2555	154	9	function	function	NOUN
iajs-2555	154	10	,	,	PUNCT
iajs-2555	154	11	then	then	ADV
iajs-2555	154	12	ᶂ	ᶂ	PROPN
iajs-2555	154	13	is	be	AUX
iajs-2555	154	14	an	an	DET
iajs-2555	154	15	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	154	16	-	-	PUNCT
iajs-2555	154	17	continuous	continuous	ADJ
iajs-2555	154	18	function	function	NOUN
iajs-2555	154	19	.	.	PUNCT
iajs-2555	155	1	proof	proof	NOUN
iajs-2555	155	2	:	:	PUNCT
iajs-2555	155	3	i.	i.	PROPN
iajs-2555	155	4	let	let	VERB
iajs-2555	155	5	ơ	ơ	PROPN
iajs-2555	155	6	∈	∈	PROPN
iajs-2555	155	7	ɟ	ɟ	X
iajs-2555	155	8	.	.	PUNCT
iajs-2555	156	1	since	since	SCONJ
iajs-2555	156	2	ᶂ	ᶂ	NOUN
iajs-2555	156	3	is	be	AUX
iajs-2555	156	4	a	a	DET
iajs-2555	156	5	continuous	continuous	ADJ
iajs-2555	156	6	function	function	NOUN
iajs-2555	156	7	,	,	PUNCT
iajs-2555	156	8	then	then	ADV
iajs-2555	156	9	ᶂ−1(ơ	ᶂ−1(ơ	NUM
iajs-2555	156	10	)	)	PUNCT
iajs-2555	156	11	∈	∈	PROPN
iajs-2555	156	12	ῖ.	ῖ.	PROPN
iajs-2555	156	13	ᶂ−1(ơ	ᶂ−1(ơ	NUM
iajs-2555	156	14	)	)	PUNCT
iajs-2555	156	15	is	be	AUX
iajs-2555	156	16	an	an	DET
iajs-2555	156	17	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	156	18	-	-	PUNCT
iajs-2555	156	19	open	open	NOUN
iajs-2555	156	20	set	set	NOUN
iajs-2555	156	21	in	in	ADP
iajs-2555	156	22	ӽ	ӽ	NOUN
iajs-2555	156	23	by	by	ADP
iajs-2555	156	24	remark	remark	NOUN
iajs-2555	156	25	2.4	2.4	NUM
iajs-2555	156	26	.	.	PUNCT
iajs-2555	157	1	hence	hence	ADV
iajs-2555	157	2	ᶂ	ᶂ	PROPN
iajs-2555	157	3	is	be	AUX
iajs-2555	157	4	an	an	DET
iajs-2555	157	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	157	6	-	-	PUNCT
iajs-2555	157	7	continuous	continuous	ADJ
iajs-2555	157	8	function	function	NOUN
iajs-2555	157	9	.	.	PUNCT
iajs-2555	158	1	ii	ii	PROPN
iajs-2555	158	2	.	.	PUNCT
iajs-2555	159	1	let	let	VERB
iajs-2555	159	2	ơ	ơ	PROPN
iajs-2555	159	3	∈	∈	PROPN
iajs-2555	159	4	ɟ	ɟ	X
iajs-2555	159	5	.	.	PUNCT
iajs-2555	160	1	by	by	ADP
iajs-2555	160	2	remark	remark	NOUN
iajs-2555	160	3	2.4	2.4	NUM
iajs-2555	160	4	,	,	PUNCT
iajs-2555	160	5	ơ	ơ	PROPN
iajs-2555	160	6	is	be	AUX
iajs-2555	160	7	an	an	DET
iajs-2555	160	8	𝛼𝑔ʝ	𝛼𝑔ʝ	ADJ
iajs-2555	160	9	-	-	ADJ
iajs-2555	160	10	open	open	ADJ
iajs-2555	160	11	set	set	NOUN
iajs-2555	160	12	in	in	ADP
iajs-2555	160	13	ƴ	ƴ	PROPN
iajs-2555	160	14	.	.	PUNCT
iajs-2555	161	1	since	since	SCONJ
iajs-2555	161	2	ᶂ	ᶂ	NOUN
iajs-2555	161	3	is	be	AUX
iajs-2555	161	4	strongly	strongly	ADV
iajs-2555	161	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	161	6	-	-	PUNCT
iajs-2555	161	7	continuous	continuous	ADJ
iajs-2555	161	8	function	function	NOUN
iajs-2555	161	9	,	,	PUNCT
iajs-2555	161	10	then	then	ADV
iajs-2555	161	11	ᶂ−1(ơ	ᶂ−1(ơ	NUM
iajs-2555	161	12	)	)	PUNCT
iajs-2555	161	13	∈	∈	PROPN
iajs-2555	161	14	ῖ.	ῖ.	NOUN
iajs-2555	161	15	hence	hence	ADV
iajs-2555	161	16	ᶂ	ᶂ	PROPN
iajs-2555	161	17	is	be	AUX
iajs-2555	161	18	a	a	DET
iajs-2555	161	19	continuous	continuous	ADJ
iajs-2555	161	20	function	function	NOUN
iajs-2555	161	21	.	.	PUNCT
iajs-2555	162	1	iii	iii	X
iajs-2555	162	2	.	.	PUNCT
iajs-2555	163	1	let	let	VERB
iajs-2555	163	2	ơ	ơ	PROPN
iajs-2555	163	3	∈	∈	PROPN
iajs-2555	163	4	ɟ	ɟ	NOUN
iajs-2555	163	5	,	,	PUNCT
iajs-2555	163	6	this	this	PRON
iajs-2555	163	7	implies	imply	VERB
iajs-2555	163	8	to	to	ADP
iajs-2555	163	9	ơ	ơ	PROPN
iajs-2555	163	10	is	be	AUX
iajs-2555	163	11	𝛼𝑔ʝ	𝛼𝑔ʝ	ADJ
iajs-2555	163	12	-	-	ADJ
iajs-2555	163	13	open	open	ADJ
iajs-2555	163	14	set	set	NOUN
iajs-2555	163	15	in	in	ADP
iajs-2555	163	16	ƴ	ƴ	PROPN
iajs-2555	163	17	.	.	PUNCT
iajs-2555	164	1	since	since	SCONJ
iajs-2555	164	2	ᶂ	ᶂ	NOUN
iajs-2555	164	3	is	be	AUX
iajs-2555	164	4	an	an	DET
iajs-2555	164	5	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	164	6	-	-	PUNCT
iajs-2555	164	7	irresolute	irresolute	ADJ
iajs-2555	164	8	function	function	NOUN
iajs-2555	164	9	then	then	ADV
iajs-2555	164	10	ᶂ−1(ơ	ᶂ−1(ơ	NUM
iajs-2555	164	11	)	)	PUNCT
iajs-2555	164	12	is	be	AUX
iajs-2555	164	13	an	an	DET
iajs-2555	164	14	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	164	15	-	-	PUNCT
iajs-2555	164	16	open	open	NOUN
iajs-2555	164	17	set	set	NOUN
iajs-2555	164	18	in	in	ADP
iajs-2555	164	19	ӽ.	ӽ.	NOUN
iajs-2555	164	20	then	then	ADV
iajs-2555	164	21	ᶂ	ᶂ	PROPN
iajs-2555	164	22	is	be	AUX
iajs-2555	164	23	an	an	DET
iajs-2555	164	24	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	164	25	-	-	PUNCT
iajs-2555	164	26	continuous	continuous	ADJ
iajs-2555	164	27	function	function	NOUN
iajs-2555	164	28	iv	iv	NOUN
iajs-2555	164	29	.	.	PUNCT
iajs-2555	165	1	let	let	VERB
iajs-2555	165	2	ơ	ơ	PROPN
iajs-2555	165	3	is	be	AUX
iajs-2555	165	4	an	an	DET
iajs-2555	165	5	𝛼𝑔ʝ	𝛼𝑔ʝ	ADJ
iajs-2555	165	6	-	-	ADJ
iajs-2555	165	7	open	open	ADJ
iajs-2555	165	8	set	set	NOUN
iajs-2555	165	9	in	in	ADP
iajs-2555	165	10	ӽ.	ӽ.	NOUN
iajs-2555	165	11	since	since	SCONJ
iajs-2555	165	12	ᶂ	ᶂ	NOUN
iajs-2555	165	13	is	be	AUX
iajs-2555	165	14	a	a	DET
iajs-2555	165	15	strongly	strongly	ADV
iajs-2555	165	16	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	165	17	-	-	PUNCT
iajs-2555	165	18	continuous	continuous	ADJ
iajs-2555	165	19	function	function	NOUN
iajs-2555	165	20	,	,	PUNCT
iajs-2555	165	21	then	then	ADV
iajs-2555	165	22	ᶂ−1(ơ	ᶂ−1(ơ	NUM
iajs-2555	165	23	)	)	PUNCT
iajs-2555	165	24	∈	∈	PROPN
iajs-2555	165	25	ῖ.	ῖ.	NOUN
iajs-2555	165	26	by	by	ADP
iajs-2555	165	27	remark	remark	NOUN
iajs-2555	165	28	2.4	2.4	NUM
iajs-2555	165	29	,	,	PUNCT
iajs-2555	165	30	ᶂ(ơ	ᶂ(ơ	NOUN
iajs-2555	165	31	)	)	PUNCT
iajs-2555	165	32	is	be	AUX
iajs-2555	165	33	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	165	34	-	-	PUNCT
iajs-2555	165	35	open	open	NOUN
iajs-2555	165	36	set	set	NOUN
iajs-2555	165	37	in	in	ADP
iajs-2555	165	38	ӽ.	ӽ.	NOUN
iajs-2555	165	39	this	this	PRON
iajs-2555	165	40	implies	imply	VERB
iajs-2555	165	41	ᶂ	ᶂ	NOUN
iajs-2555	165	42	is	be	AUX
iajs-2555	165	43	an	an	DET
iajs-2555	165	44	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	165	45	-	-	PUNCT
iajs-2555	165	46	irresolute	irresolute	ADJ
iajs-2555	165	47	function	function	NOUN
iajs-2555	165	48	.	.	PUNCT
iajs-2555	166	1	v.	v.	CCONJ
iajs-2555	166	2	let	let	VERB
iajs-2555	166	3	ơ	ơ	PROPN
iajs-2555	166	4	∈	∈	PROPN
iajs-2555	166	5	ɟ	ɟ	SCONJ
iajs-2555	166	6	this	this	PRON
iajs-2555	166	7	implies	imply	VERB
iajs-2555	166	8	ơ	ơ	PROPN
iajs-2555	166	9	is	be	AUX
iajs-2555	166	10	an	an	DET
iajs-2555	166	11	𝛼𝑔ʝ	𝛼𝑔ʝ	ADJ
iajs-2555	166	12	-	-	ADJ
iajs-2555	166	13	open	open	ADJ
iajs-2555	166	14	set	set	NOUN
iajs-2555	166	15	and	and	CCONJ
iajs-2555	166	16	since	since	SCONJ
iajs-2555	166	17	ᶂ	ᶂ	NOUN
iajs-2555	166	18	is	be	AUX
iajs-2555	166	19	a	a	DET
iajs-2555	166	20	strongly	strongly	ADV
iajs-2555	166	21	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	166	22	-	-	PUNCT
iajs-2555	166	23	continuous	continuous	ADJ
iajs-2555	166	24	function	function	NOUN
iajs-2555	166	25	,	,	PUNCT
iajs-2555	166	26	thus	thus	ADV
iajs-2555	166	27	ᶂ−1(ơ	ᶂ−1(ơ	PRON
iajs-2555	166	28	)	)	PUNCT
iajs-2555	166	29	is	be	AUX
iajs-2555	166	30	open	open	ADJ
iajs-2555	166	31	set	set	VERB
iajs-2555	166	32	in	in	ADP
iajs-2555	166	33	ӽ	ӽ	NOUN
iajs-2555	166	34	by	by	ADP
iajs-2555	166	35	remark	remark	NOUN
iajs-2555	166	36	2.4	2.4	NUM
iajs-2555	166	37	ᶂ−1(ơ	ᶂ−1(ơ	PROPN
iajs-2555	166	38	)	)	PUNCT
iajs-2555	166	39	is	be	AUX
iajs-2555	166	40	an	an	DET
iajs-2555	166	41	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	166	42	-	-	PUNCT
iajs-2555	166	43	open	open	NOUN
iajs-2555	166	44	set	set	NOUN
iajs-2555	166	45	,	,	PUNCT
iajs-2555	166	46	so	so	CCONJ
iajs-2555	166	47	ᶂ	ᶂ	NOUN
iajs-2555	166	48	is	be	AUX
iajs-2555	166	49	an	an	DET
iajs-2555	166	50	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	166	51	-	-	PUNCT
iajs-2555	166	52	continuous	continuous	ADJ
iajs-2555	166	53	function	function	NOUN
iajs-2555	166	54	.	.	PUNCT
iajs-2555	167	1	the	the	DET
iajs-2555	167	2	follow	follow	NOUN
iajs-2555	167	3	scheme	scheme	NOUN
iajs-2555	167	4	shows	show	VERB
iajs-2555	167	5	the	the	DET
iajs-2555	167	6	relation	relation	NOUN
iajs-2555	167	7	between	between	ADP
iajs-2555	167	8	the	the	DET
iajs-2555	167	9	variant	variant	ADJ
iajs-2555	167	10	notions	notion	NOUN
iajs-2555	167	11	were	be	AUX
iajs-2555	167	12	presented	present	VERB
iajs-2555	167	13	in	in	ADP
iajs-2555	167	14	definition	definition	NOUN
iajs-2555	167	15	4.1	4.1	NUM
iajs-2555	167	16	.	.	PUNCT
iajs-2555	168	1	arrow	arrow	PROPN
iajs-2555	168	2	chart	chart	NOUN
iajs-2555	168	3	(	(	PUNCT
iajs-2555	168	4	4.1	4.1	NUM
iajs-2555	168	5	)	)	PUNCT
iajs-2555	168	6	𝜶𝒈ị	𝜶𝒈ị	ADJ
iajs-2555	168	7	-	-	PUNCT
iajs-2555	168	8	irresolute	irresolute	ADJ
iajs-2555	168	9	function	function	NOUN
iajs-2555	168	10	.	.	PUNCT
iajs-2555	169	1	𝜶𝒈ị	𝜶𝒈ị	ADV
iajs-2555	169	2	-	-	PUNCT
iajs-2555	169	3	continuous	continuous	ADJ
iajs-2555	169	4	function	function	NOUN
iajs-2555	169	5	strongly	strongly	ADV
iajs-2555	169	6	𝜶𝒈ị	𝜶𝒈ị	ADV
iajs-2555	169	7	-	-	PUNCT
iajs-2555	169	8	continuous	continuous	ADJ
iajs-2555	169	9	function	function	NOUN
iajs-2555	169	10	continuous	continuous	ADJ
iajs-2555	169	11	function	function	NOUN
iajs-2555	169	12	37	37	NUM
iajs-2555	169	13	ibn	ibn	PROPN
iajs-2555	169	14	al	al	PROPN
iajs-2555	169	15	-	-	PUNCT
iajs-2555	169	16	haitham	haitham	PROPN
iajs-2555	169	17	jour	jour	X
iajs-2555	169	18	.	.	PROPN
iajs-2555	170	1	for	for	ADP
iajs-2555	170	2	pure	pure	ADJ
iajs-2555	170	3	&	&	CCONJ
iajs-2555	170	4	appl	appl	PROPN
iajs-2555	170	5	.	.	PUNCT
iajs-2555	171	1	sci	sci	PROPN
iajs-2555	171	2	.	.	PROPN
iajs-2555	172	1	34	34	NUM
iajs-2555	172	2	(	(	PUNCT
iajs-2555	172	3	1	1	NUM
iajs-2555	172	4	)	)	PUNCT
iajs-2555	172	5	2021	2021	NUM
iajs-2555	173	1	ị	ị	PROPN
iajs-2555	173	2	-	-	PUNCT
iajs-2555	173	3	α	α	NOUN
iajs-2555	173	4	-	-	PUNCT
iajs-2555	173	5	g	g	NOUN
iajs-2555	173	6	-	-	PUNCT
iajs-2555	173	7	continuous	continuous	ADJ
iajs-2555	173	8	function	function	NOUN
iajs-2555	173	9	the	the	DET
iajs-2555	173	10	following	follow	VERB
iajs-2555	173	11	are	be	AUX
iajs-2555	173	12	some	some	DET
iajs-2555	173	13	examples	example	NOUN
iajs-2555	173	14	showing	show	VERB
iajs-2555	173	15	that	that	SCONJ
iajs-2555	173	16	the	the	DET
iajs-2555	173	17	opposite	opposite	ADJ
iajs-2555	173	18	direction	direction	NOUN
iajs-2555	173	19	of	of	ADP
iajs-2555	173	20	the	the	DET
iajs-2555	173	21	above	above	ADJ
iajs-2555	173	22	schema	schema	NOUN
iajs-2555	173	23	is	be	AUX
iajs-2555	173	24	incorrect	incorrect	ADJ
iajs-2555	173	25	.	.	PUNCT
iajs-2555	174	1	example	example	NOUN
iajs-2555	175	1	3	3	NUM
iajs-2555	175	2	:	:	PUNCT
iajs-2555	175	3	the	the	DET
iajs-2555	175	4	function	function	NOUN
iajs-2555	175	5	ᶂ	ᶂ	NOUN
iajs-2555	175	6	:	:	PUNCT
iajs-2555	175	7	(	(	PUNCT
iajs-2555	175	8	ӽ	ӽ	X
iajs-2555	175	9	,	,	PUNCT
iajs-2555	175	10	ῖ	ῖ	PROPN
iajs-2555	175	11	,	,	PUNCT
iajs-2555	175	12	ị	ị	PRON
iajs-2555	175	13	)	)	PUNCT
iajs-2555	175	14	→	→	SYM
iajs-2555	175	15	(	(	PUNCT
iajs-2555	175	16	ӽ	ӽ	X
iajs-2555	175	17	,	,	PUNCT
iajs-2555	175	18	ῖ	ῖ	NOUN
iajs-2555	175	19	,	,	PUNCT
iajs-2555	175	20	ʝ	ʝ	NOUN
iajs-2555	175	21	)	)	PUNCT
iajs-2555	175	22	,	,	PUNCT
iajs-2555	175	23	where	where	SCONJ
iajs-2555	175	24	ӽ={ẻ1	ӽ={ẻ1	NOUN
iajs-2555	175	25	,	,	PUNCT
iajs-2555	175	26	ẻ2	ẻ2	PROPN
iajs-2555	175	27	,	,	PUNCT
iajs-2555	175	28	ẻ3	ẻ3	PROPN
iajs-2555	175	29	}	}	PUNCT
iajs-2555	175	30	such	such	ADJ
iajs-2555	175	31	that	that	DET
iajs-2555	175	32	ᶂ(ẻ1)=(ẻ1	ᶂ(ẻ1)=(ẻ1	NUM
iajs-2555	175	33	)	)	PUNCT
iajs-2555	175	34	,	,	PUNCT
iajs-2555	175	35	ᶂ(ẻ2)=(ẻ2	ᶂ(ẻ2)=(ẻ2	NOUN
iajs-2555	175	36	)	)	PUNCT
iajs-2555	175	37	,	,	PUNCT
iajs-2555	175	38	ᶂ(ẻ3)=(ẻ3	ᶂ(ẻ3)=(ẻ3	NUM
iajs-2555	175	39	)	)	PUNCT
iajs-2555	175	40	,	,	PUNCT
iajs-2555	175	41	ῖ	ῖ	X
iajs-2555	175	42	=	=	X
iajs-2555	175	43	{	{	PUNCT
iajs-2555	175	44	ӽ,∅,{ẻ1	ӽ,∅,{ẻ1	ADV
iajs-2555	175	45	}	}	PUNCT
iajs-2555	175	46	}	}	PUNCT
iajs-2555	175	47	,	,	PUNCT
iajs-2555	175	48	ị={∅	ị={∅	PROPN
iajs-2555	175	49	}	}	PUNCT
iajs-2555	175	50	and	and	CCONJ
iajs-2555	175	51	ʝ={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3	ʝ={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3	PROPN
iajs-2555	175	52	}	}	PUNCT
iajs-2555	175	53	}	}	PUNCT
iajs-2555	175	54	then	then	ADV
iajs-2555	175	55	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	PROPN
iajs-2555	175	56	,	,	PUNCT
iajs-2555	175	57	ẻ2},{ẻ1	ẻ2},{ẻ1	NOUN
iajs-2555	175	58	,	,	PUNCT
iajs-2555	175	59	ẻ3	ẻ3	PROPN
iajs-2555	175	60	}	}	PUNCT
iajs-2555	175	61	}	}	PUNCT
iajs-2555	175	62	then	then	ADV
iajs-2555	175	63	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	VERB
iajs-2555	175	64	)	)	PUNCT
iajs-2555	175	65	=	=	SYM
iajs-2555	175	66	{	{	PUNCT
iajs-2555	175	67	ӽ,∅,{ẻ2,ẻ3	ӽ,∅,{ẻ2,ẻ3	NOUN
iajs-2555	175	68	}	}	PUNCT
iajs-2555	175	69	}	}	PUNCT
iajs-2555	175	70	and	and	CCONJ
iajs-2555	175	71	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	175	72	)	)	PUNCT
iajs-2555	175	73	=	=	SYM
iajs-2555	175	74	{	{	PUNCT
iajs-2555	175	75	ӽ,∅,{ẻ1	ӽ,∅,{ẻ1	ADV
iajs-2555	175	76	}	}	PUNCT
iajs-2555	175	77	}	}	PUNCT
iajs-2555	175	78	.	.	PUNCT
iajs-2555	176	1	so	so	ADV
iajs-2555	176	2	𝛼𝑔ʝ𝐶(ӽ	𝛼𝑔ʝ𝐶(ӽ	NUM
iajs-2555	176	3	)	)	PUNCT
iajs-2555	176	4	=	=	SYM
iajs-2555	177	1	ҏ(ӽ	ҏ(ӽ	X
iajs-2555	177	2	)	)	PUNCT
iajs-2555	177	3	and	and	CCONJ
iajs-2555	177	4	𝛼𝑔ʝ𝑂(ӽ	𝛼𝑔ʝ𝑂(ӽ	NOUN
iajs-2555	177	5	)	)	PUNCT
iajs-2555	177	6	=	=	SYM
iajs-2555	177	7	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	177	8	)	)	PUNCT
iajs-2555	177	9	.	.	PUNCT
iajs-2555	178	1	it	it	PRON
iajs-2555	178	2	is	be	AUX
iajs-2555	178	3	possible	possible	ADJ
iajs-2555	178	4	to	to	PART
iajs-2555	178	5	see	see	VERB
iajs-2555	178	6	clearly	clearly	ADV
iajs-2555	178	7	that	that	SCONJ
iajs-2555	178	8	ᶂ	ᶂ	NOUN
iajs-2555	178	9	is	be	AUX
iajs-2555	178	10	continuous	continuous	ADJ
iajs-2555	178	11	and	and	CCONJ
iajs-2555	178	12	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	178	13	-	-	PUNCT
iajs-2555	178	14	continuous	continuous	ADJ
iajs-2555	178	15	function	function	NOUN
iajs-2555	178	16	but	but	CCONJ
iajs-2555	178	17	not	not	PART
iajs-2555	178	18	𝛼𝑔ịirresolute	𝛼𝑔ịirresolute	VERB
iajs-2555	178	19	function	function	NOUN
iajs-2555	178	20	since	since	SCONJ
iajs-2555	178	21	{	{	PUNCT
iajs-2555	178	22	ẻ3	ẻ3	PROPN
iajs-2555	178	23	}	}	PUNCT
iajs-2555	178	24	is	be	AUX
iajs-2555	178	25	an	an	DET
iajs-2555	178	26	𝛼𝑔ʝ	𝛼𝑔ʝ	ADJ
iajs-2555	178	27	-	-	ADJ
iajs-2555	178	28	open	open	ADJ
iajs-2555	178	29	set	set	NOUN
iajs-2555	178	30	in	in	ADP
iajs-2555	178	31	ƴ	ƴ	PRON
iajs-2555	178	32	but	but	CCONJ
iajs-2555	178	33	ᶂ−1(ẻ3	ᶂ−1(ẻ3	NOUN
iajs-2555	178	34	)	)	PUNCT
iajs-2555	178	35	=	=	SYM
iajs-2555	179	1	ẻ3	ẻ3	PROPN
iajs-2555	179	2	is	be	AUX
iajs-2555	179	3	not	not	PART
iajs-2555	179	4	an	an	DET
iajs-2555	179	5	𝛼𝑔ị	𝛼𝑔ị	ADV
iajs-2555	179	6	-	-	PUNCT
iajs-2555	179	7	open	open	NOUN
iajs-2555	179	8	set	set	NOUN
iajs-2555	179	9	in	in	ADP
iajs-2555	179	10	ӽ.	ӽ.	NOUN
iajs-2555	179	11	example	example	NOUN
iajs-2555	179	12	4	4	NUM
iajs-2555	179	13	:	:	PUNCT
iajs-2555	179	14	the	the	DET
iajs-2555	179	15	function	function	NOUN
iajs-2555	179	16	ᶂ	ᶂ	NOUN
iajs-2555	179	17	:	:	PUNCT
iajs-2555	179	18	(	(	PUNCT
iajs-2555	179	19	ӽ	ӽ	X
iajs-2555	179	20	,	,	PUNCT
iajs-2555	179	21	ῖ	ῖ	PROPN
iajs-2555	179	22	,	,	PUNCT
iajs-2555	179	23	ị	ị	PRON
iajs-2555	179	24	)	)	PUNCT
iajs-2555	179	25	→	→	SYM
iajs-2555	179	26	(	(	PUNCT
iajs-2555	179	27	ӽ	ӽ	X
iajs-2555	179	28	,	,	PUNCT
iajs-2555	179	29	ῖ	ῖ	NOUN
iajs-2555	179	30	,	,	PUNCT
iajs-2555	179	31	ʝ	ʝ	NOUN
iajs-2555	179	32	)	)	PUNCT
iajs-2555	179	33	,	,	PUNCT
iajs-2555	179	34	where	where	SCONJ
iajs-2555	179	35	ӽ={ẻ1	ӽ={ẻ1	NOUN
iajs-2555	179	36	,	,	PUNCT
iajs-2555	179	37	ẻ2	ẻ2	PROPN
iajs-2555	179	38	,	,	PUNCT
iajs-2555	179	39	ẻ3	ẻ3	PROPN
iajs-2555	179	40	}	}	PUNCT
iajs-2555	179	41	such	such	ADJ
iajs-2555	179	42	that	that	DET
iajs-2555	179	43	ᶂ(ẻ1)=(ẻ1	ᶂ(ẻ1)=(ẻ1	NUM
iajs-2555	179	44	)	)	PUNCT
iajs-2555	179	45	,	,	PUNCT
iajs-2555	179	46	ᶂ(ẻ2)=(ẻ2	ᶂ(ẻ2)=(ẻ2	NOUN
iajs-2555	179	47	)	)	PUNCT
iajs-2555	179	48	,	,	PUNCT
iajs-2555	179	49	ᶂ(ẻ3)=(ẻ3	ᶂ(ẻ3)=(ẻ3	NUM
iajs-2555	179	50	)	)	PUNCT
iajs-2555	179	51	,	,	PUNCT
iajs-2555	179	52	ῖ	ῖ	X
iajs-2555	179	53	=	=	X
iajs-2555	179	54	{	{	PUNCT
iajs-2555	179	55	ӽ,∅,{ẻ1	ӽ,∅,{ẻ1	ADV
iajs-2555	179	56	}	}	PUNCT
iajs-2555	179	57	}	}	PUNCT
iajs-2555	179	58	,	,	PUNCT
iajs-2555	179	59	ʝ={∅	ʝ={∅	ADJ
iajs-2555	179	60	}	}	PUNCT
iajs-2555	179	61	and	and	CCONJ
iajs-2555	179	62	ị={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3	ị={∅,{ẻ2},{ẻ3},{ẻ2,ẻ3	NOUN
iajs-2555	179	63	}	}	PUNCT
iajs-2555	179	64	}	}	PUNCT
iajs-2555	179	65	then	then	ADV
iajs-2555	179	66	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	ῖ𝛼={ӽ,∅,{ẻ1},{ẻ1	PROPN
iajs-2555	179	67	,	,	PUNCT
iajs-2555	179	68	ẻ2},{ẻ1	ẻ2},{ẻ1	NOUN
iajs-2555	179	69	,	,	PUNCT
iajs-2555	179	70	ẻ3	ẻ3	PROPN
iajs-2555	179	71	}	}	PUNCT
iajs-2555	179	72	}	}	PUNCT
iajs-2555	179	73	then	then	ADV
iajs-2555	179	74	𝛼𝑔ʝ𝐶(ӽ	𝛼𝑔ʝ𝐶(ӽ	NUM
iajs-2555	179	75	)	)	PUNCT
iajs-2555	179	76	=	=	SYM
iajs-2555	179	77	{	{	PUNCT
iajs-2555	179	78	ӽ,∅,{ẻ2,ẻ3	ӽ,∅,{ẻ2,ẻ3	NOUN
iajs-2555	179	79	}	}	PUNCT
iajs-2555	179	80	}	}	PUNCT
iajs-2555	179	81	and	and	CCONJ
iajs-2555	179	82	𝛼𝑔ʝ𝑂(ӽ	𝛼𝑔ʝ𝑂(ӽ	NOUN
iajs-2555	179	83	)	)	PUNCT
iajs-2555	179	84	=	=	SYM
iajs-2555	179	85	{	{	PUNCT
iajs-2555	179	86	ӽ,∅,{ẻ1	ӽ,∅,{ẻ1	ADV
iajs-2555	179	87	}	}	PUNCT
iajs-2555	179	88	}	}	PUNCT
iajs-2555	179	89	.	.	PUNCT
iajs-2555	180	1	so	so	ADV
iajs-2555	180	2	𝛼𝑔ị𝐶(ӽ	𝛼𝑔ị𝐶(ӽ	PROPN
iajs-2555	180	3	)	)	PUNCT
iajs-2555	180	4	=	=	SYM
iajs-2555	181	1	ҏ(ӽ	ҏ(ӽ	X
iajs-2555	181	2	)	)	PUNCT
iajs-2555	181	3	and	and	CCONJ
iajs-2555	181	4	𝛼𝑔ị𝑂(ӽ	𝛼𝑔ị𝑂(ӽ	NOUN
iajs-2555	181	5	)	)	PUNCT
iajs-2555	181	6	=	=	SYM
iajs-2555	181	7	ҏ(ӽ	ҏ(ӽ	NOUN
iajs-2555	181	8	)	)	PUNCT
iajs-2555	181	9	.	.	PUNCT
iajs-2555	182	1	it	it	PRON
iajs-2555	182	2	is	be	AUX
iajs-2555	182	3	possible	possible	ADJ
iajs-2555	182	4	to	to	PART
iajs-2555	182	5	see	see	VERB
iajs-2555	182	6	clearly	clearly	ADV
iajs-2555	182	7	that	that	SCONJ
iajs-2555	182	8	ᶂ	ᶂ	NOUN
iajs-2555	182	9	is	be	AUX
iajs-2555	182	10	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	182	11	-	-	PUNCT
iajs-2555	182	12	continuous	continuous	ADJ
iajs-2555	182	13	function	function	NOUN
iajs-2555	182	14	but	but	CCONJ
iajs-2555	182	15	not	not	PART
iajs-2555	182	16	continuous	continuous	ADJ
iajs-2555	182	17	function	function	NOUN
iajs-2555	182	18	since	since	SCONJ
iajs-2555	182	19	{	{	PUNCT
iajs-2555	182	20	ẻ1}∈	ẻ1}∈	PROPN
iajs-2555	182	21	ῖ	ῖ	X
iajs-2555	182	22	but	but	CCONJ
iajs-2555	182	23	ᶂ−1(ẻ1	ᶂ−1(ẻ1	NUM
iajs-2555	182	24	)	)	PUNCT
iajs-2555	182	25	=	=	SYM
iajs-2555	182	26	ẻ2	ẻ2	PROPN
iajs-2555	182	27	is	be	AUX
iajs-2555	182	28	not	not	PART
iajs-2555	182	29	open	open	ADJ
iajs-2555	182	30	in	in	ADP
iajs-2555	182	31	ӽ	ӽ	NOUN
iajs-2555	182	32	,	,	PUNCT
iajs-2555	182	33	and	and	CCONJ
iajs-2555	182	34	not	not	PART
iajs-2555	182	35	strongly	strongly	ADV
iajs-2555	182	36	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	182	37	-	-	PUNCT
iajs-2555	182	38	continuous	continuous	ADJ
iajs-2555	182	39	function	function	NOUN
iajs-2555	182	40	since	since	SCONJ
iajs-2555	182	41	{	{	PUNCT
iajs-2555	182	42	ẻ1	ẻ1	NOUN
iajs-2555	182	43	}	}	PUNCT
iajs-2555	182	44	∈	∈	PROPN
iajs-2555	182	45	𝛼𝑔ʝ𝑂(ӽ	𝛼𝑔ʝ𝑂(ӽ	NOUN
iajs-2555	182	46	)	)	PUNCT
iajs-2555	182	47	but	but	CCONJ
iajs-2555	182	48	ᶂ−1(ẻ1	ᶂ−1(ẻ1	NUM
iajs-2555	182	49	)	)	PUNCT
iajs-2555	183	1	=	=	SYM
iajs-2555	183	2	ẻ2	ẻ2	PROPN
iajs-2555	183	3	is	be	AUX
iajs-2555	183	4	not	not	PART
iajs-2555	183	5	open	open	ADJ
iajs-2555	183	6	in	in	ADP
iajs-2555	183	7	ӽ.	ӽ.	NOUN
iajs-2555	183	8	5conclusion	5conclusion	NUM
iajs-2555	183	9	the	the	DET
iajs-2555	183	10	concept	concept	NOUN
iajs-2555	183	11	of	of	ADP
iajs-2555	183	12	closed	closed	ADJ
iajs-2555	183	13	and	and	CCONJ
iajs-2555	183	14	open	open	ADJ
iajs-2555	183	15	sets	set	NOUN
iajs-2555	183	16	was	be	AUX
iajs-2555	183	17	used	use	VERB
iajs-2555	183	18	with	with	ADP
iajs-2555	183	19	the	the	DET
iajs-2555	183	20	ideal	ideal	ADJ
iajs-2555	183	21	concept	concept	NOUN
iajs-2555	183	22	to	to	PART
iajs-2555	183	23	introduce	introduce	VERB
iajs-2555	183	24	new	new	ADJ
iajs-2555	183	25	notions	notion	NOUN
iajs-2555	183	26	from	from	ADP
iajs-2555	183	27	these	these	DET
iajs-2555	183	28	categories	category	NOUN
iajs-2555	183	29	;	;	PUNCT
iajs-2555	183	30	𝛼𝑔ị	𝛼𝑔ị	NUM
iajs-2555	183	31	-	-	PUNCT
iajs-2555	183	32	closed	closed	ADJ
iajs-2555	183	33	set	set	NOUN
iajs-2555	183	34	,	,	PUNCT
iajs-2555	183	35	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	183	36	-	-	PUNCT
iajs-2555	183	37	open	open	ADJ
iajs-2555	183	38	set	set	NOUN
iajs-2555	183	39	.	.	PUNCT
iajs-2555	184	1	and	and	CCONJ
iajs-2555	184	2	we	we	PRON
iajs-2555	184	3	introduce	introduce	VERB
iajs-2555	184	4	a	a	DET
iajs-2555	184	5	new	new	ADJ
iajs-2555	184	6	functions	function	NOUN
iajs-2555	184	7	like	like	ADP
iajs-2555	184	8	:	:	PUNCT
iajs-2555	184	9	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	184	10	-	-	PUNCT
iajs-2555	184	11	open	open	ADJ
iajs-2555	184	12	function	function	NOUN
iajs-2555	184	13	,	,	PUNCT
iajs-2555	184	14	𝛼𝑔ị	𝛼𝑔ị	NOUN
iajs-2555	184	15	∗-open	∗-open	X
iajs-2555	184	16	function	function	NOUN
iajs-2555	184	17	,	,	PUNCT
iajs-2555	184	18	𝛼𝑔ị	𝛼𝑔ị	PROPN
iajs-2555	184	19	∗∗-open	∗∗-open	NOUN
iajs-2555	184	20	function	function	NOUN
iajs-2555	184	21	,	,	PUNCT
iajs-2555	184	22	𝛼𝑔ị	𝛼𝑔ị	NOUN
iajs-2555	184	23	-	-	PUNCT
iajs-2555	184	24	closed	closed	ADJ
iajs-2555	184	25	function	function	NOUN
iajs-2555	184	26	,	,	PUNCT
iajs-2555	184	27	𝛼𝑔ị	𝛼𝑔ị	ADJ
iajs-2555	184	28	∗-closed	∗-close	VERB
iajs-2555	184	29	function	function	NOUN
iajs-2555	184	30	and	and	CCONJ
iajs-2555	184	31	𝛼𝑔ị	𝛼𝑔ị	DET
iajs-2555	184	32	∗∗-closed	∗∗-close	VERB
iajs-2555	184	33	function	function	NOUN
iajs-2555	184	34	with	with	ADP
iajs-2555	184	35	near	near	ADJ
iajs-2555	184	36	continuous	continuous	ADJ
iajs-2555	184	37	functions	function	NOUN
iajs-2555	184	38	.	.	PUNCT
iajs-2555	185	1	references	reference	NOUN
iajs-2555	185	2	:	:	PUNCT
iajs-2555	185	3	1	1	X
iajs-2555	185	4	.	.	X
iajs-2555	185	5	njastad	njastad	NOUN
iajs-2555	185	6	,	,	PUNCT
iajs-2555	185	7	o.	o.	PROPN
iajs-2555	185	8	on	on	ADP
iajs-2555	185	9	some	some	DET
iajs-2555	185	10	classes	class	NOUN
iajs-2555	185	11	of	of	ADP
iajs-2555	185	12	nearly	nearly	ADV
iajs-2555	185	13	open	open	ADJ
iajs-2555	185	14	set	set	NOUN
iajs-2555	185	15	,	,	PUNCT
iajs-2555	185	16	pacific	pacific	PROPN
iajs-2555	185	17	j.	j.	PROPN
iajs-2555	185	18	math	math	PROPN
iajs-2555	185	19	.	.	PUNCT
iajs-2555	186	1	1965	1965	NUM
iajs-2555	186	2	,	,	PUNCT
iajs-2555	186	3	15	15	NUM
iajs-2555	186	4	,	,	PUNCT
iajs-2555	186	5	961	961	NUM
iajs-2555	186	6	–	–	PUNCT
iajs-2555	186	7	970	970	NUM
iajs-2555	186	8	.	.	NOUN
iajs-2555	186	9	2	2	NUM
iajs-2555	186	10	.	.	X
iajs-2555	186	11	nadia	nadia	PROPN
iajs-2555	186	12	,	,	PUNCT
iajs-2555	186	13	m.	m.	PROPN
iajs-2555	186	14	ali	ali	PROPN
iajs-2555	186	15	.	.	PROPN
iajs-2555	187	1	on	on	ADP
iajs-2555	187	2	new	new	ADJ
iajs-2555	187	3	types	type	NOUN
iajs-2555	187	4	of	of	ADP
iajs-2555	187	5	weakly	weakly	ADJ
iajs-2555	187	6	open	open	ADJ
iajs-2555	187	7	sets	set	NOUN
iajs-2555	187	8	"	"	PUNCT
iajs-2555	187	9	α	α	NOUN
iajs-2555	187	10	-	-	ADJ
iajs-2555	187	11	open	open	ADJ
iajs-2555	187	12	and	and	CCONJ
iajs-2555	187	13	semi	semi	ADJ
iajs-2555	187	14	-	-	ADJ
iajs-2555	187	15	α	α	PRON
iajs-2555	187	16	-	-	ADJ
iajs-2555	187	17	open	open	ADJ
iajs-2555	187	18	sets	set	NOUN
iajs-2555	187	19	"	"	PUNCT
iajs-2555	187	20	,	,	PUNCT
iajs-2555	187	21	m.sc	m.sc	PROPN
iajs-2555	187	22	.	.	PUNCT
iajs-2555	188	1	thesis	thesis	NOUN
iajs-2555	188	2	,	,	PUNCT
iajs-2555	188	3	january	january	PROPN
iajs-2555	188	4	2004	2004	NUM
iajs-2555	188	5	.	.	PUNCT
iajs-2555	189	1	3	3	X
iajs-2555	189	2	.	.	X
iajs-2555	189	3	kuratowski	kuratowski	PROPN
iajs-2555	189	4	,	,	PUNCT
iajs-2555	189	5	k.	k.	PROPN
iajs-2555	189	6	topology	topology	PROPN
iajs-2555	189	7	.	.	PUNCT
iajs-2555	190	1	newyork	newyork	PROPN
iajs-2555	190	2	:	:	PUNCT
iajs-2555	190	3	acadeic	acadeic	PROPN
iajs-2555	190	4	press	press	PROPN
iajs-2555	190	5	.1933	.1933	PROPN
iajs-2555	190	6	,	,	PUNCT
iajs-2555	190	7	i.	i.	PROPN
iajs-2555	190	8	4	4	NUM
iajs-2555	190	9	.	.	PUNCT
iajs-2555	190	10	nasef	nasef	PROPN
iajs-2555	190	11	,	,	PUNCT
iajs-2555	190	12	a.	a.	NOUN
iajs-2555	190	13	a.	a.	NOUN
iajs-2555	190	14	;	;	PUNCT
iajs-2555	190	15	esamaeel	esamaeel	NOUN
iajs-2555	190	16	,	,	PUNCT
iajs-2555	190	17	r.	r.	PROPN
iajs-2555	190	18	b.	b.	PROPN
iajs-2555	191	1	some	some	DET
iajs-2555	191	2	αoperators	αoperator	NOUN
iajs-2555	191	3	vai	vai	ADJ
iajs-2555	191	4	ideals	ideal	NOUN
iajs-2555	191	5	,	,	PUNCT
iajs-2555	191	6	international	international	ADJ
iajs-2555	191	7	electronic	electronic	ADJ
iajs-2555	191	8	journal	journal	NOUN
iajs-2555	191	9	of	of	ADP
iajs-2555	191	10	pur	pur	NOUN
iajs-2555	191	11	and	and	CCONJ
iajs-2555	191	12	applied	apply	VERB
iajs-2555	191	13	mathematics	mathematic	NOUN
iajs-2555	191	14	.	.	PUNCT
iajs-2555	191	15	2015	2015	NUM
iajs-2555	191	16	,	,	PUNCT
iajs-2555	191	17	9	9	NUM
iajs-2555	191	18	,	,	PUNCT
iajs-2555	191	19	3	3	NUM
iajs-2555	191	20	,	,	PUNCT
iajs-2555	191	21	149159	149159	NUM
iajs-2555	191	22	.	.	PUNCT
iajs-2555	192	1	5	5	NUM
iajs-2555	192	2	.	.	X
iajs-2555	192	3	nasef	nasef	PROPN
iajs-2555	192	4	,	,	PUNCT
iajs-2555	192	5	a.	a.	NOUN
iajs-2555	192	6	a.	a.	NOUN
iajs-2555	192	7	;	;	PUNCT
iajs-2555	192	8	radwan	radwan	NOUN
iajs-2555	192	9	,	,	PUNCT
iajs-2555	192	10	a.	a.	PROPN
iajs-2555	192	11	e.	e.	PROPN
iajs-2555	192	12	;	;	PUNCT
iajs-2555	192	13	iprahem	iprahem	PROPN
iajs-2555	192	14	,	,	PUNCT
iajs-2555	192	15	f.	f.	PROPN
iajs-2555	192	16	a.	a.	PROPN
iajs-2555	192	17	;	;	PUNCT
iajs-2555	192	18	esmaeel	esmaeel	VERB
iajs-2555	192	19	,	,	PUNCT
iajs-2555	192	20	r.	r.	PROPN
iajs-2555	192	21	b.	b.	PROPN
iajs-2555	192	22	soft	soft	ADJ
iajs-2555	192	23	α	α	NOUN
iajs-2555	192	24	-	-	NOUN
iajs-2555	192	25	compactness	compactness	NOUN
iajs-2555	192	26	via	via	ADP
iajs-2555	192	27	soft	soft	ADJ
iajs-2555	192	28	ideals	ideal	NOUN
iajs-2555	192	29	,	,	PUNCT
iajs-2555	192	30	ready	ready	ADJ
iajs-2555	192	31	to	to	PART
iajs-2555	192	32	be	be	AUX
iajs-2555	192	33	published	publish	VERB
iajs-2555	192	34	in	in	ADP
iajs-2555	192	35	journal	journal	NOUN
iajs-2555	192	36	of	of	ADP
iajs-2555	192	37	advances	advance	NOUN
iajs-2555	192	38	in	in	ADP
iajs-2555	192	39	mathematics	mathematic	NOUN
iajs-2555	192	40	,	,	PUNCT
iajs-2555	192	41	in	in	ADP
iajs-2555	192	42	june2016	june2016	PROPN
iajs-2555	192	43	.	.	PUNCT
iajs-2555	193	1	6	6	NUM
iajs-2555	193	2	.	.	X
iajs-2555	193	3	abd	abd	PROPN
iajs-2555	193	4	el	el	PROPN
iajs-2555	193	5	-	-	PUNCT
iajs-2555	193	6	monsef	monsef	ADJ
iajs-2555	193	7	,	,	PUNCT
iajs-2555	193	8	m.	m.	PROPN
iajs-2555	193	9	e.	e.	PROPN
iajs-2555	193	10	;	;	PUNCT
iajs-2555	193	11	nasef	nasef	PROPN
iajs-2555	193	12	,	,	PUNCT
iajs-2555	193	13	a.	a.	NOUN
iajs-2555	193	14	a.	a.	NOUN
iajs-2555	193	15	;	;	PUNCT
iajs-2555	193	16	radwan	radwan	NOUN
iajs-2555	193	17	,	,	PUNCT
iajs-2555	193	18	a.	a.	PROPN
iajs-2555	193	19	e.	e.	PROPN
iajs-2555	193	20	;	;	PUNCT
iajs-2555	193	21	esmaeel	esmaeel	VERB
iajs-2555	193	22	,	,	PUNCT
iajs-2555	193	23	r.	r.	PROPN
iajs-2555	193	24	b.	b.	PROPN
iajs-2555	194	1	on	on	ADP
iajs-2555	194	2	αopen	αopen	ADJ
iajs-2555	194	3	sets	set	NOUN
iajs-2555	194	4	with	with	ADP
iajs-2555	194	5	respect	respect	NOUN
iajs-2555	194	6	to	to	ADP
iajs-2555	194	7	an	an	DET
iajs-2555	194	8	ideal	ideal	ADJ
iajs-2555	194	9	,	,	PUNCT
iajs-2555	194	10	journal	journal	NOUN
iajs-2555	194	11	of	of	ADP
iajs-2555	194	12	advances	advance	NOUN
iajs-2555	194	13	studies	study	NOUN
iajs-2555	194	14	in	in	ADP
iajs-2555	194	15	topology	topology	NOUN
iajs-2555	194	16	.	.	PUNCT
iajs-2555	195	1	2014,5,3	2014,5,3	NUM
iajs-2555	195	2	.	.	PUNCT
iajs-2555	196	1	1	1	NUM
iajs-2555	196	2	-	-	SYM
iajs-2555	196	3	9	9	NUM
iajs-2555	196	4	.	.	NOUN
iajs-2555	196	5	7	7	NUM
iajs-2555	196	6	.	.	X
iajs-2555	196	7	esmaeel	esmaeel	PROPN
iajs-2555	196	8	,	,	PUNCT
iajs-2555	196	9	r.	r.	PROPN
iajs-2555	196	10	b.	b.	PROPN
iajs-2555	196	11	on	on	ADP
iajs-2555	196	12	α	α	PROPN
iajs-2555	196	13	-	-	PUNCT
iajs-2555	196	14	c	c	NOUN
iajs-2555	196	15	-	-	PUNCT
iajs-2555	196	16	compactness	compactness	NOUN
iajs-2555	196	17	,	,	PUNCT
iajs-2555	196	18	ibn	ibn	PROPN
iajs-2555	196	19	al	al	PROPN
iajs-2555	196	20	-	-	PUNCT
iajs-2555	196	21	haithatham	haithatham	PROPN
iajs-2555	196	22	journal	journal	NOUN
iajs-2555	196	23	for	for	ADP
iajs-2555	196	24	pure	pure	ADJ
iajs-2555	196	25	and	and	CCONJ
iajs-2555	196	26	applied	applied	ADJ
iajs-2555	196	27	science	science	NOUN
iajs-2555	196	28	.	.	PUNCT
iajs-2555	197	1	2012	2012	NUM
iajs-2555	197	2	,	,	PUNCT
iajs-2555	197	3	22	22	NUM
iajs-2555	197	4	,	,	PUNCT
iajs-2555	197	5	212	212	NUM
iajs-2555	197	6	-	-	SYM
iajs-2555	197	7	218	218	NUM
iajs-2555	197	8	.	.	PUNCT