id	sid	tid	token	lemma	pos
iajs-4032	1	1	2	2	NUM
iajs-4032	1	2	©	©	ADP
iajs-4032	1	3	2025	2025	NUM
iajs-4032	1	4	the	the	DET
iajs-4032	1	5	author(s	author(s	NOUN
iajs-4032	1	6	)	)	PUNCT
iajs-4032	1	7	.	.	PUNCT
iajs-4032	2	1	published	publish	VERB
iajs-4032	2	2	by	by	ADP
iajs-4032	2	3	college	college	NOUN
iajs-4032	2	4	of	of	ADP
iajs-4032	2	5	education	education	NOUN
iajs-4032	2	6	for	for	ADP
iajs-4032	2	7	pure	pure	ADJ
iajs-4032	2	8	science	science	NOUN
iajs-4032	2	9	(	(	PUNCT
iajs-4032	2	10	ibn	ibn	PROPN
iajs-4032	2	11	al	al	PROPN
iajs-4032	2	12	-	-	PUNCT
iajs-4032	2	13	haitham	haitham	PROPN
iajs-4032	2	14	)	)	PUNCT
iajs-4032	2	15	,	,	PUNCT
iajs-4032	2	16	university	university	NOUN
iajs-4032	2	17	of	of	ADP
iajs-4032	2	18	baghdad	baghdad	PROPN
iajs-4032	2	19	.	.	PUNCT
iajs-4032	3	1	this	this	PRON
iajs-4032	3	2	is	be	AUX
iajs-4032	3	3	an	an	DET
iajs-4032	3	4	open	open	ADJ
iajs-4032	3	5	-	-	PUNCT
iajs-4032	3	6	access	access	NOUN
iajs-4032	3	7	article	article	NOUN
iajs-4032	3	8	distributed	distribute	VERB
iajs-4032	3	9	under	under	ADP
iajs-4032	3	10	the	the	DET
iajs-4032	3	11	terms	term	NOUN
iajs-4032	3	12	of	of	ADP
iajs-4032	3	13	the	the	DET
iajs-4032	3	14	creative	creative	ADJ
iajs-4032	3	15	commons	common	NOUN
iajs-4032	3	16	attribution	attribution	NOUN
iajs-4032	3	17	4.0	4.0	NUM
iajs-4032	3	18	international	international	ADJ
iajs-4032	3	19	license	license	NOUN
iajs-4032	3	20	on	on	ADP
iajs-4032	3	21	separation	separation	NOUN
iajs-4032	3	22	axioms	axiom	VERB
iajs-4032	3	23	asmaa	asmaa	NOUN
iajs-4032	3	24	rheem	rheem	NOUN
iajs-4032	3	25	kadhim	kadhim	ADJ
iajs-4032	3	26	1	1	NUM
iajs-4032	3	27	*	*	PUNCT
iajs-4032	3	28	,	,	PUNCT
iajs-4032	3	29	,	,	PUNCT
iajs-4032	3	30	r.	r.	PROPN
iajs-4032	3	31	b.	b.	PROPN
iajs-4032	3	32	esmaeel	esmaeel	PROPN
iajs-4032	3	33	2	2	NUM
iajs-4032	4	1	and	and	CCONJ
iajs-4032	4	2	abdelaziz	abdelaziz	PROPN
iajs-4032	4	3	e.	e.	PROPN
iajs-4032	4	4	radwan	radwan	VERB
iajs-4032	4	5	3	3	NUM
iajs-4032	4	6	1,2	1,2	NUM
iajs-4032	4	7	department	department	NOUN
iajs-4032	4	8	of	of	ADP
iajs-4032	4	9	mathematics	mathematic	NOUN
iajs-4032	4	10	,	,	PUNCT
iajs-4032	4	11	college	college	NOUN
iajs-4032	4	12	of	of	ADP
iajs-4032	4	13	education	education	NOUN
iajs-4032	4	14	for	for	ADP
iajs-4032	4	15	pure	pure	ADJ
iajs-4032	4	16	science	science	NOUN
iajs-4032	4	17	(	(	PUNCT
iajs-4032	4	18	ibn	ibn	PROPN
iajs-4032	4	19	al	al	PROPN
iajs-4032	4	20	-	-	PUNCT
iajs-4032	4	21	haitham	haitham	PROPN
iajs-4032	4	22	)	)	PUNCT
iajs-4032	4	23	,	,	PUNCT
iajs-4032	4	24	university	university	NOUN
iajs-4032	4	25	of	of	ADP
iajs-4032	4	26	baghdad	baghdad	PROPN
iajs-4032	4	27	,	,	PUNCT
iajs-4032	4	28	baghdad	baghdad	PROPN
iajs-4032	4	29	,	,	PUNCT
iajs-4032	4	30	iraq	iraq	PROPN
iajs-4032	4	31	.	.	PUNCT
iajs-4032	5	1	3	3	NUM
iajs-4032	5	2	department	department	NOUN
iajs-4032	5	3	of	of	ADP
iajs-4032	5	4	mathematics	mathematics	PROPN
iajs-4032	5	5	,	,	PUNCT
iajs-4032	5	6	college	college	NOUN
iajs-4032	5	7	faculty	faculty	NOUN
iajs-4032	5	8	of	of	ADP
iajs-4032	5	9	science	science	NOUN
iajs-4032	5	10	,	,	PUNCT
iajs-4032	5	11	university	university	NOUN
iajs-4032	5	12	of	of	ADP
iajs-4032	5	13	ain	ain	PROPN
iajs-4032	5	14	shams	sham	NOUN
iajs-4032	5	15	,	,	PUNCT
iajs-4032	5	16	cairo	cairo	PROPN
iajs-4032	5	17	,	,	PUNCT
iajs-4032	5	18	egypt	egypt	PROPN
iajs-4032	5	19	.	.	PUNCT
iajs-4032	6	1	*	*	PUNCT
iajs-4032	6	2	corresponding	correspond	VERB
iajs-4032	6	3	author	author	NOUN
iajs-4032	6	4	.	.	PUNCT
iajs-4032	7	1	received	receive	VERB
iajs-4032	7	2	:	:	PUNCT
iajs-4032	7	3	5	5	NUM
iajs-4032	7	4	august	august	PROPN
iajs-4032	7	5	2024	2024	NUM
iajs-4032	7	6	accepted	accept	VERB
iajs-4032	7	7	:	:	PUNCT
iajs-4032	7	8	19	19	NUM
iajs-4032	7	9	november	november	PROPN
iajs-4032	7	10	2024	2024	NUM
iajs-4032	7	11	published	publish	VERB
iajs-4032	7	12	:	:	PUNCT
iajs-4032	7	13	20	20	NUM
iajs-4032	7	14	october	october	NOUN
iajs-4032	7	15	2025	2025	NUM
iajs-4032	7	16	doi.org/10.30526/38.4.4032	doi.org/10.30526/38.4.4032	NOUN
iajs-4032	7	17	abstract	abstract	ADJ
iajs-4032	7	18	the	the	DET
iajs-4032	7	19	purpose	purpose	NOUN
iajs-4032	7	20	of	of	ADP
iajs-4032	7	21	this	this	DET
iajs-4032	7	22	research	research	NOUN
iajs-4032	7	23	is	be	AUX
iajs-4032	7	24	to	to	PART
iajs-4032	7	25	present	present	VERB
iajs-4032	7	26	the	the	DET
iajs-4032	7	27	new	new	ADJ
iajs-4032	7	28	set	set	NOUN
iajs-4032	7	29	that	that	PRON
iajs-4032	7	30	is	be	AUX
iajs-4032	7	31	named	name	VERB
iajs-4032	7	32	open	open	ADJ
iajs-4032	7	33	set	set	NOUN
iajs-4032	7	34	,	,	PUNCT
iajs-4032	7	35	where	where	SCONJ
iajs-4032	7	36	the	the	DET
iajs-4032	7	37	set	set	NOUN
iajs-4032	7	38	was	be	AUX
iajs-4032	7	39	defined	define	VERB
iajs-4032	7	40	and	and	CCONJ
iajs-4032	7	41	the	the	DET
iajs-4032	7	42	general	general	ADJ
iajs-4032	7	43	properties	property	NOUN
iajs-4032	7	44	and	and	CCONJ
iajs-4032	7	45	the	the	DET
iajs-4032	7	46	basic	basic	ADJ
iajs-4032	7	47	concepts	concept	NOUN
iajs-4032	7	48	and	and	CCONJ
iajs-4032	7	49	properties	property	NOUN
iajs-4032	7	50	of	of	ADP
iajs-4032	7	51	the	the	DET
iajs-4032	7	52	set	set	NOUN
iajs-4032	7	53	were	be	AUX
iajs-4032	7	54	explained	explain	VERB
iajs-4032	7	55	,	,	PUNCT
iajs-4032	7	56	such	such	ADJ
iajs-4032	7	57	as	as	ADP
iajs-4032	7	58	the	the	DET
iajs-4032	7	59	open	open	ADJ
iajs-4032	7	60	set	set	NOUN
iajs-4032	7	61	,	,	PUNCT
iajs-4032	7	62	the	the	DET
iajs-4032	7	63	intersection	intersection	NOUN
iajs-4032	7	64	,	,	PUNCT
iajs-4032	7	65	and	and	CCONJ
iajs-4032	7	66	the	the	DET
iajs-4032	7	67	union	union	NOUN
iajs-4032	7	68	.	.	PUNCT
iajs-4032	8	1	also	also	ADV
iajs-4032	8	2	in	in	ADP
iajs-4032	8	3	this	this	DET
iajs-4032	8	4	paper	paper	NOUN
iajs-4032	8	5	the	the	DET
iajs-4032	8	6	separation	separation	NOUN
iajs-4032	8	7	axioms	axiom	NOUN
iajs-4032	8	8	were	be	AUX
iajs-4032	8	9	defined	define	VERB
iajs-4032	8	10	and	and	CCONJ
iajs-4032	8	11	studied	study	VERB
iajs-4032	8	12	in	in	ADP
iajs-4032	8	13	ideal	ideal	ADJ
iajs-4032	8	14	topological	topological	ADJ
iajs-4032	8	15	spaces	space	NOUN
iajs-4032	8	16	in	in	ADP
iajs-4032	8	17	a	a	DET
iajs-4032	8	18	new	new	ADJ
iajs-4032	8	19	way	way	NOUN
iajs-4032	8	20	.	.	PUNCT
iajs-4032	9	1	and	and	CCONJ
iajs-4032	9	2	discuss	discuss	VERB
iajs-4032	9	3	some	some	DET
iajs-4032	9	4	properties	property	NOUN
iajs-4032	9	5	.	.	PUNCT
iajs-4032	10	1	also	also	ADV
iajs-4032	10	2	discuss	discuss	VERB
iajs-4032	10	3	the	the	DET
iajs-4032	10	4	relationship	relationship	NOUN
iajs-4032	10	5	of	of	ADP
iajs-4032	10	6	our	our	PRON
iajs-4032	10	7	definition	definition	NOUN
iajs-4032	10	8	with	with	ADP
iajs-4032	10	9	other	other	ADJ
iajs-4032	10	10	definitions	definition	NOUN
iajs-4032	10	11	and	and	CCONJ
iajs-4032	10	12	prove	prove	VERB
iajs-4032	10	13	some	some	DET
iajs-4032	10	14	results	result	NOUN
iajs-4032	10	15	in	in	ADP
iajs-4032	10	16	the	the	DET
iajs-4032	10	17	context	context	NOUN
iajs-4032	10	18	of	of	ADP
iajs-4032	10	19	separation	separation	NOUN
iajs-4032	10	20	axioms	axiom	NOUN
iajs-4032	10	21	in	in	ADP
iajs-4032	10	22	ideal	ideal	ADJ
iajs-4032	10	23	topological	topological	ADJ
iajs-4032	10	24	space	space	NOUN
iajs-4032	10	25	on	on	ADP
iajs-4032	10	26	this	this	DET
iajs-4032	10	27	set	set	NOUN
iajs-4032	10	28	,	,	PUNCT
iajs-4032	10	29	where	where	SCONJ
iajs-4032	10	30	the	the	DET
iajs-4032	10	31	axioms	axiom	NOUN
iajs-4032	10	32	of	of	ADP
iajs-4032	10	33	the	the	DET
iajs-4032	10	34	type	type	NOUN
iajs-4032	10	35	space	space	NOUN
iajs-4032	10	36	,	,	PUNCT
iajs-4032	10	37	-space	-space	NOUN
iajs-4032	10	38	and	and	CCONJ
iajs-4032	10	39	–	–	PUNCT
iajs-4032	10	40	space	space	NOUN
iajs-4032	10	41	were	be	AUX
iajs-4032	10	42	defined	define	VERB
iajs-4032	10	43	and	and	CCONJ
iajs-4032	10	44	the	the	DET
iajs-4032	10	45	basic	basic	ADJ
iajs-4032	10	46	concepts	concept	NOUN
iajs-4032	10	47	and	and	CCONJ
iajs-4032	10	48	generalizations	generalization	NOUN
iajs-4032	10	49	of	of	ADP
iajs-4032	10	50	these	these	DET
iajs-4032	10	51	axioms	axiom	NOUN
iajs-4032	10	52	on	on	ADP
iajs-4032	10	53	the	the	DET
iajs-4032	10	54	set	set	NOUN
iajs-4032	10	55	were	be	AUX
iajs-4032	10	56	studied	study	VERB
iajs-4032	10	57	.	.	PUNCT
iajs-4032	11	1	in	in	ADP
iajs-4032	11	2	addition	addition	NOUN
iajs-4032	11	3	the	the	DET
iajs-4032	11	4	relationships	relationship	NOUN
iajs-4032	11	5	between	between	ADP
iajs-4032	11	6	these	these	DET
iajs-4032	11	7	concepts	concept	NOUN
iajs-4032	11	8	and	and	CCONJ
iajs-4032	11	9	their	their	PRON
iajs-4032	11	10	converses	converse	NOUN
iajs-4032	11	11	on	on	ADP
iajs-4032	11	12	this	this	DET
iajs-4032	11	13	set	set	NOUN
iajs-4032	11	14	under	under	ADP
iajs-4032	11	15	study	study	NOUN
iajs-4032	11	16	were	be	AUX
iajs-4032	11	17	discussed	discuss	VERB
iajs-4032	11	18	,	,	PUNCT
iajs-4032	11	19	and	and	CCONJ
iajs-4032	11	20	illustrative	illustrative	ADJ
iajs-4032	11	21	examples	example	NOUN
iajs-4032	11	22	and	and	CCONJ
iajs-4032	11	23	proofs	proof	NOUN
iajs-4032	11	24	of	of	ADP
iajs-4032	11	25	those	those	DET
iajs-4032	11	26	properties	property	NOUN
iajs-4032	11	27	were	be	AUX
iajs-4032	11	28	provided	provide	VERB
iajs-4032	11	29	for	for	ADP
iajs-4032	11	30	that	that	PRON
iajs-4032	11	31	.	.	PUNCT
iajs-4032	12	1	additionally	additionally	ADV
iajs-4032	12	2	,	,	PUNCT
iajs-4032	12	3	a	a	DET
iajs-4032	12	4	diagram	diagram	NOUN
iajs-4032	12	5	illustrating	illustrate	VERB
iajs-4032	12	6	these	these	DET
iajs-4032	12	7	concepts	concept	NOUN
iajs-4032	12	8	is	be	AUX
iajs-4032	12	9	presented	present	VERB
iajs-4032	12	10	.	.	PUNCT
iajs-4032	13	1	keywords	keyword	NOUN
iajs-4032	13	2	:	:	PUNCT
iajs-4032	13	3	ideal	ideal	ADJ
iajs-4032	13	4	topological	topological	ADJ
iajs-4032	13	5	space	space	NOUN
iajs-4032	13	6	,	,	PUNCT
iajs-4032	13	7	r	r	NOUN
iajs-4032	13	8	open	open	ADJ
iajs-4032	13	9	set	set	NOUN
iajs-4032	13	10	,	,	PUNCT
iajs-4032	13	11	t0	t0	PROPN
iajs-4032	13	12	space	space	NOUN
iajs-4032	13	13	,	,	PUNCT
iajs-4032	13	14	t1	t1	NOUN
iajs-4032	13	15	pace	pace	NOUN
iajs-4032	13	16	,	,	PUNCT
iajs-4032	13	17	t2	t2	NOUN
iajs-4032	13	18	–	–	PUNCT
iajs-4032	13	19	space	space	NOUN
iajs-4032	13	20	.	.	PUNCT
iajs-4032	14	1	1.introduction	1.introduction	NUM
iajs-4032	14	2	the	the	DET
iajs-4032	14	3	idea	idea	NOUN
iajs-4032	14	4	of	of	ADP
iajs-4032	14	5	the	the	DET
iajs-4032	14	6	ideal	ideal	NOUN
iajs-4032	14	7	is	be	AUX
iajs-4032	14	8	presented	present	VERB
iajs-4032	14	9	by)1	by)1	PROPN
iajs-4032	14	10	(	(	PUNCT
iajs-4032	14	11	.	.	PUNCT
iajs-4032	15	1	a	a	DET
iajs-4032	15	2	topological	topological	ADJ
iajs-4032	15	3	space	space	NOUN
iajs-4032	15	4	(	(	PUNCT
iajs-4032	15	5	)	)	PUNCT
iajs-4032	15	6	in	in	ADP
iajs-4032	15	7	ideal	ideal	ADJ
iajs-4032	15	8	i	i	PRON
iajs-4032	15	9	is	be	AUX
iajs-4032	15	10	a	a	DET
iajs-4032	15	11	nonempty	nonempty	ADJ
iajs-4032	15	12	collection	collection	NOUN
iajs-4032	15	13	subsets	subset	NOUN
iajs-4032	15	14	of	of	ADP
iajs-4032	15	15	that	that	PRON
iajs-4032	15	16	satisfies	satisfy	VERB
iajs-4032	15	17	the	the	DET
iajs-4032	15	18	subsequent	subsequent	ADJ
iajs-4032	15	19	condition	condition	NOUN
iajs-4032	15	20	:	:	PUNCT
iajs-4032	15	21	i.	i.	PROPN
iajs-4032	15	22	let	let	VERB
iajs-4032	15	23	b	b	SYM
iajs-4032	15	24	⊆	⊆	NUM
iajs-4032	15	25	a	a	PRON
iajs-4032	15	26	and	and	CCONJ
iajs-4032	15	27	a	a	DET
iajs-4032	15	28	∈	∈	PROPN
iajs-4032	16	1	i	i	PRON
iajs-4032	16	2	,	,	PUNCT
iajs-4032	16	3	then	then	ADV
iajs-4032	16	4	b	b	PROPN
iajs-4032	16	5	∈	∈	PROPN
iajs-4032	16	6	.	.	PUNCT
iajs-4032	17	1	(	(	PUNCT
iajs-4032	17	2	heredity	heredity	NOUN
iajs-4032	17	3	property	property	NOUN
iajs-4032	17	4	)	)	PUNCT
iajs-4032	17	5	.	.	PUNCT
iajs-4032	18	1	ii	ii	PROPN
iajs-4032	18	2	.	.	PUNCT
iajs-4032	19	1	let	let	VERB
iajs-4032	19	2	a	a	PRON
iajs-4032	19	3	and	and	CCONJ
iajs-4032	19	4	b	b	NOUN
iajs-4032	19	5	are	be	AUX
iajs-4032	19	6	both	both	PRON
iajs-4032	19	7	in	in	ADP
iajs-4032	19	8	i	i	PRON
iajs-4032	19	9	therefor	therefor	ADP
iajs-4032	19	10	a	a	DET
iajs-4032	19	11	∪	∪	X
iajs-4032	19	12	b	b	PROPN
iajs-4032	19	13	∈	∈	PROPN
iajs-4032	19	14	i.	i.	NOUN
iajs-4032	19	15	(	(	PUNCT
iajs-4032	19	16	additively	additively	ADV
iajs-4032	19	17	property	property	NOUN
iajs-4032	19	18	)	)	PUNCT
iajs-4032	19	19	.	.	PUNCT
iajs-4032	20	1	several	several	ADJ
iajs-4032	20	2	classes	class	NOUN
iajs-4032	20	3	of	of	ADP
iajs-4032	20	4	ideals	ideal	NOUN
iajs-4032	20	5	have	have	AUX
iajs-4032	20	6	been	be	AUX
iajs-4032	20	7	introduced	introduce	VERB
iajs-4032	20	8	)	)	PUNCT
iajs-4032	20	9	2	2	NUM
iajs-4032	20	10	-	-	SYM
iajs-4032	20	11	4	4	NUM
iajs-4032	20	12	(	(	PUNCT
iajs-4032	20	13	.	.	PUNCT
iajs-4032	21	1	:	:	PUNCT
iajs-4032	21	2	the	the	DET
iajs-4032	21	3	ideal	ideal	NOUN
iajs-4032	21	4	on	on	ADP
iajs-4032	21	5	by	by	ADP
iajs-4032	21	6	{	{	PUNCT
iajs-4032	21	7	}	}	PUNCT
iajs-4032	21	8	(	(	PUNCT
iajs-4032	21	9	trivial	trivial	ADJ
iajs-4032	21	10	ideal	ideal	NOUN
iajs-4032	21	11	)	)	PUNCT
iajs-4032	21	12	and	and	CCONJ
iajs-4032	21	13	the	the	DET
iajs-4032	21	14	improper	improper	ADJ
iajs-4032	21	15	ideal	ideal	NOUN
iajs-4032	21	16	i	i	PRON
iajs-4032	21	17	is	be	AUX
iajs-4032	21	18	called	call	VERB
iajs-4032	21	19	p(x	p(x	NOUN
iajs-4032	21	20	)	)	PUNCT
iajs-4032	21	21	.	.	PUNCT
iajs-4032	22	1	:	:	PUNCT
iajs-4032	22	2	the	the	DET
iajs-4032	22	3	ideal	ideal	NOUN
iajs-4032	22	4	is	be	AUX
iajs-4032	22	5	comprised	comprise	VERB
iajs-4032	22	6	of	of	ADP
iajs-4032	22	7	every	every	DET
iajs-4032	22	8	finite	finite	NOUN
iajs-4032	22	9	subset	subset	VERB
iajs-4032	22	10	thereof	thereof	ADV
iajs-4032	22	11	.	.	PUNCT
iajs-4032	23	1	:	:	PUNCT
iajs-4032	23	2	the	the	DET
iajs-4032	23	3	ideal	ideal	NOUN
iajs-4032	23	4	comprising	comprise	VERB
iajs-4032	23	5	every	every	DET
iajs-4032	23	6	countable	countable	ADJ
iajs-4032	23	7	subset	subset	NOUN
iajs-4032	23	8	of	of	ADP
iajs-4032	23	9	.	.	PUNCT
iajs-4032	24	1	:	:	PUNCT
iajs-4032	24	2	the	the	DET
iajs-4032	24	3	principle	principle	ADJ
iajs-4032	24	4	ideal	ideal	NOUN
iajs-4032	24	5	produced	produce	VERB
iajs-4032	24	6	by	by	ADP
iajs-4032	24	7	any	any	DET
iajs-4032	24	8	set	set	ADJ
iajs-4032	24	9	topological	topological	ADJ
iajs-4032	24	10	space	space	NOUN
iajs-4032	24	11	)	)	PUNCT
iajs-4032	24	12	.	.	PUNCT
iajs-4032	24	13	expressed	express	VERB
iajs-4032	24	14	as	as	ADP
iajs-4032	24	15	=	=	SYM
iajs-4032	24	16	p	p	X
iajs-4032	24	17	(	(	PUNCT
iajs-4032	24	18	=	=	X
iajs-4032	24	19	{	{	PUNCT
iajs-4032	24	20	⊆	⊆	NUM
iajs-4032	24	21	:	:	PUNCT
iajs-4032	24	22	⊆	⊆	NUM
iajs-4032	24	23	}	}	PUNCT
iajs-4032	24	24	.	.	PUNCT
iajs-4032	25	1	also	also	ADV
iajs-4032	25	2	vaidyananthswamy	vaidyananthswamy	VERB
iajs-4032	25	3	in	in	ADP
iajs-4032	25	4	1945	1945	NUM
iajs-4032	25	5	)	)	PUNCT
iajs-4032	25	6	5	5	NUM
iajs-4032	25	7	(	(	PUNCT
iajs-4032	25	8	,	,	PUNCT
iajs-4032	25	9	a	a	DET
iajs-4032	25	10	study	study	NOUN
iajs-4032	25	11	submitted	submit	VERB
iajs-4032	25	12	about	about	ADV
iajs-4032	25	13	an	an	DET
iajs-4032	25	14	operator	operator	NOUN
iajs-4032	25	15	assigned	assign	VERB
iajs-4032	25	16	explanation	explanation	NOUN
iajs-4032	25	17	of	of	ADP
iajs-4032	25	18	a	a	DET
iajs-4032	25	19	local	local	ADJ
iajs-4032	25	20	function	function	NOUN
iajs-4032	25	21	of	of	ADP
iajs-4032	25	22	via	via	ADP
iajs-4032	25	23	as	as	SCONJ
iajs-4032	25	24	follows	follow	VERB
iajs-4032	25	25	:	:	PUNCT
iajs-4032	25	26	⊆	⊆	NUM
iajs-4032	25	27	,	,	PUNCT
iajs-4032	25	28	(	(	PUNCT
iajs-4032	25	29	=	=	X
iajs-4032	25	30	{	{	PUNCT
iajs-4032	25	31	∈	∈	NOUN
iajs-4032	25	32	:	:	PUNCT
iajs-4032	25	33	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	25	34	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	25	35	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	25	36	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	25	37	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	25	38	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	X
iajs-4032	26	1	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	26	2	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	26	3	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	26	4	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	26	5	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	26	6	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	X
iajs-4032	27	1	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	27	2	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	27	3	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	27	4	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	27	5	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	27	6	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	X
iajs-4032	28	1	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	28	2	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	28	3	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	28	4	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	28	5	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	28	6	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	X
iajs-4032	29	1	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	29	2	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	29	3	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	29	4	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	29	5	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	29	6	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	X
iajs-4032	30	1	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	30	2	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	30	3	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	30	4	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	30	5	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	30	6	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	X
iajs-4032	31	1	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	31	2	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	31	3	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	31	4	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	31	5	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	31	6	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	X
iajs-4032	32	1	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	32	2	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	mailto:asmaa.raheem2203m@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4032	32	3	https://orcid.org/0000-0002-4743-6034	https://orcid.org/0000-0002-4743-6034	PRON
iajs-4032	32	4	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	mailto:rana.b.i@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4032	32	5	https://orcid.org/0000-0002-2473-4872	https://orcid.org/0000-0002-2473-4872	PROPN
iajs-4032	32	6	mailto:zezoradwan@yahoo.com	mailto:zezoradwan@yahoo.com	PROPN
iajs-4032	32	7	ihjpas	ihjpas	PROPN
iajs-4032	32	8	.	.	PUNCT
iajs-4032	33	1	2025,38(4	2025,38(4	NOUN
iajs-4032	33	2	)	)	PUNCT
iajs-4032	33	3	3	3	NUM
iajs-4032	33	4	for	for	ADP
iajs-4032	33	5	each	each	DET
iajs-4032	33	6	∈	∈	NOUN
iajs-4032	33	7	)	)	PUNCT
iajs-4032	34	1	so	so	ADV
iajs-4032	34	2	=	=	PRON
iajs-4032	34	3	{	{	PUNCT
iajs-4032	34	4	∈	∈	PROPN
iajs-4032	34	5	;	;	PUNCT
iajs-4032	34	6	∈	∈	PROPN
iajs-4032	34	7	}	}	PUNCT
iajs-4032	34	8	,	,	PUNCT
iajs-4032	34	9	operator	operator	NOUN
iajs-4032	34	10	(	(	PUNCT
iajs-4032	34	11	)	)	PUNCT
iajs-4032	34	12	for	for	ADP
iajs-4032	34	13	a	a	DET
iajs-4032	34	14	topology	topology	NOUN
iajs-4032	34	15	(	(	PUNCT
iajs-4032	34	16	of	of	ADP
iajs-4032	34	17	the	the	DET
iajs-4032	34	18	kuratowski	kuratowski	ADJ
iajs-4032	34	19	closure	closure	NOUN
iajs-4032	34	20	(	(	PUNCT
iajs-4032	34	21	6	6	NUM
iajs-4032	34	22	)	)	PUNCT
iajs-4032	34	23	,	,	PUNCT
iajs-4032	34	24	called	call	VERB
iajs-4032	34	25	the	the	DET
iajs-4032	34	26	*	*	PUNCT
iajs-4032	34	27	topology	topology	NOUN
iajs-4032	34	28	,	,	PUNCT
iajs-4032	34	29	finer	fine	ADJ
iajs-4032	34	30	than	than	ADP
iajs-4032	34	31	,	,	PUNCT
iajs-4032	34	32	was	be	AUX
iajs-4032	34	33	presented	present	VERB
iajs-4032	34	34	as	as	SCONJ
iajs-4032	34	35	follows	follow	VERB
iajs-4032	34	36	:	:	PUNCT
iajs-4032	34	37	(	(	PUNCT
iajs-4032	34	38	)	)	PUNCT
iajs-4032	34	39	=	=	SYM
iajs-4032	34	40	∪	∪	NOUN
iajs-4032	34	41	also	also	ADV
iajs-4032	34	42	(	(	PUNCT
iajs-4032	34	43	=	=	SYM
iajs-4032	34	44	(	(	PUNCT
iajs-4032	34	45	⊆	⊆	NUM
iajs-4032	34	46	:	:	PUNCT
iajs-4032	34	47	)	)	PUNCT
iajs-4032	34	48	=	=	SYM
iajs-4032	34	49	)	)	PUNCT
iajs-4032	34	50	}	}	PUNCT
iajs-4032	34	51	also	also	ADV
iajs-4032	34	52	(	(	PUNCT
iajs-4032	34	53	=	=	NOUN
iajs-4032	34	54	{	{	PUNCT
iajs-4032	34	55	)	)	PUNCT
iajs-4032	34	56	;	;	PUNCT
iajs-4032	34	57	∈	∈	PROPN
iajs-4032	34	58	and	and	CCONJ
iajs-4032	34	59	∈	∈	PROPN
iajs-4032	34	60	}	}	PUNCT
iajs-4032	34	61	is	be	AUX
iajs-4032	34	62	a	a	DET
iajs-4032	34	63	basis	basis	NOUN
iajs-4032	34	64	for	for	ADP
iajs-4032	34	65	(	(	PUNCT
iajs-4032	34	66	)	)	PUNCT
iajs-4032	34	67	)	)	PUNCT
iajs-4032	34	68	7	7	NUM
iajs-4032	34	69	-	-	SYM
iajs-4032	34	70	9	9	NUM
iajs-4032	34	71	(	(	PUNCT
iajs-4032	34	72	.	.	PUNCT
iajs-4032	35	1	the	the	DET
iajs-4032	35	2	simple	simple	ADJ
iajs-4032	35	3	write	write	NOUN
iajs-4032	35	4	for	for	ADP
iajs-4032	35	5	)	)	PUNCT
iajs-4032	35	6	,	,	PUNCT
iajs-4032	35	7	)	)	PUNCT
iajs-4032	35	8	.	.	PUNCT
iajs-4032	36	1	when	when	SCONJ
iajs-4032	36	2	there	there	PRON
iajs-4032	36	3	is	be	VERB
iajs-4032	36	4	no	no	DET
iajs-4032	36	5	chance	chance	NOUN
iajs-4032	36	6	for	for	ADP
iajs-4032	36	7	confusion	confusion	NOUN
iajs-4032	36	8	,	,	PUNCT
iajs-4032	36	9	the	the	DET
iajs-4032	36	10	concept	concept	NOUN
iajs-4032	36	11	(	(	PUNCT
iajs-4032	36	12	will	will	AUX
iajs-4032	36	13	be	be	AUX
iajs-4032	36	14	symbolize	symbolize	NOUN
iajs-4032	36	15	to	to	ADP
iajs-4032	36	16	the	the	DET
iajs-4032	36	17	topological	topological	ADJ
iajs-4032	36	18	space	space	NOUN
iajs-4032	36	19	)	)	PUNCT
iajs-4032	36	20	with	with	ADP
iajs-4032	36	21	an	an	DET
iajs-4032	36	22	ideal	ideal	NOUN
iajs-4032	36	23	on	on	ADP
iajs-4032	36	24	that	that	PRON
iajs-4032	36	25	s	s	VERB
iajs-4032	36	26	no	no	DET
iajs-4032	36	27	assumption	assumption	NOUN
iajs-4032	36	28	of	of	ADP
iajs-4032	36	29	separation	separation	NOUN
iajs-4032	36	30	properties	property	NOUN
iajs-4032	36	31	,	,	PUNCT
iajs-4032	36	32	it	it	PRON
iajs-4032	36	33	is	be	AUX
iajs-4032	36	34	named	name	VERB
iajs-4032	36	35	the	the	DET
iajs-4032	36	36	ideal	ideal	ADJ
iajs-4032	36	37	topological	topological	ADJ
iajs-4032	36	38	space	space	NOUN
iajs-4032	36	39	to	to	ADP
iajs-4032	36	40	as	as	ADV
iajs-4032	36	41	short	short	ADJ
iajs-4032	36	42	.	.	PUNCT
iajs-4032	37	1	each	each	DET
iajs-4032	37	2	element	element	NOUN
iajs-4032	37	3	in	in	ADP
iajs-4032	37	4	named	name	VERB
iajs-4032	37	5	open	open	ADJ
iajs-4032	37	6	set	set	NOUN
iajs-4032	37	7	.	.	PUNCT
iajs-4032	38	1	if	if	SCONJ
iajs-4032	38	2	)	)	PUNCT
iajs-4032	38	3	is	be	AUX
iajs-4032	38	4	open	open	ADJ
iajs-4032	38	5	set	set	VERB
iajs-4032	38	6	,	,	PUNCT
iajs-4032	38	7	so	so	ADV
iajs-4032	38	8	called	call	VERB
iajs-4032	38	9	closed	closed	ADJ
iajs-4032	38	10	,	,	PUNCT
iajs-4032	38	11	therefore	therefore	ADV
iajs-4032	38	12	,	,	PUNCT
iajs-4032	38	13	(	(	PUNCT
iajs-4032	38	14	,	,	PUNCT
iajs-4032	38	15	the	the	DET
iajs-4032	38	16	subset	subset	NOUN
iajs-4032	38	17	of	of	ADP
iajs-4032	38	18	the	the	DET
iajs-4032	38	19	space	space	NOUN
iajs-4032	38	20	(	(	PUNCT
iajs-4032	38	21	closed	close	VERB
iajs-4032	38	22	if	if	SCONJ
iajs-4032	38	23	⊆	⊆	NUM
iajs-4032	38	24	in	in	ADP
iajs-4032	38	25	the	the	DET
iajs-4032	38	26	ideal	ideal	ADJ
iajs-4032	38	27	topological	topological	ADJ
iajs-4032	38	28	space	space	NOUN
iajs-4032	38	29	,	,	PUNCT
iajs-4032	38	30	called	call	VERB
iajs-4032	38	31	dense	dense	ADJ
iajs-4032	38	32	if	if	SCONJ
iajs-4032	38	33	(	(	PUNCT
iajs-4032	38	34	)	)	PUNCT
iajs-4032	38	35	=	=	PUNCT
iajs-4032	38	36	.	.	PUNCT
iajs-4032	39	1	note	note	VERB
iajs-4032	39	2	that	that	SCONJ
iajs-4032	39	3	in	in	ADV
iajs-4032	39	4	,	,	PUNCT
iajs-4032	39	5	if	if	SCONJ
iajs-4032	39	6	=	=	NOUN
iajs-4032	39	7	{	{	PUNCT
iajs-4032	39	8	}	}	PUNCT
iajs-4032	39	9	,	,	PUNCT
iajs-4032	39	10	so	so	ADV
iajs-4032	39	11	=	=	X
iajs-4032	39	12	.	.	PUNCT
iajs-4032	40	1	let	let	VERB
iajs-4032	40	2	⊆	⊆	NUM
iajs-4032	40	3	,	,	PUNCT
iajs-4032	40	4	)	)	PUNCT
iajs-4032	40	5	(	(	PUNCT
iajs-4032	40	6	respectively	respectively	ADV
iajs-4032	40	7	)	)	PUNCT
iajs-4032	40	8	the	the	PRON
iajs-4032	40	9	)	)	PUNCT
iajs-4032	40	10	it	it	PRON
iajs-4032	40	11	denoted	denote	VERB
iajs-4032	40	12	(	(	PUNCT
iajs-4032	40	13	interior	interior	NOUN
iajs-4032	40	14	,	,	PUNCT
iajs-4032	40	15	respectively	respectively	ADV
iajs-4032	40	16	,	,	PUNCT
iajs-4032	40	17	closure	closure	NOUN
iajs-4032	40	18	of	of	ADP
iajs-4032	40	19	)	)	PUNCT
iajs-4032	40	20	in	in	ADP
iajs-4032	40	21	(	(	PUNCT
iajs-4032	40	22	,	,	PUNCT
iajs-4032	40	23	)	)	PUNCT
iajs-4032	40	24	10	10	NUM
iajs-4032	40	25	-	-	SYM
iajs-4032	40	26	11	11	NUM
iajs-4032	40	27	(	(	PUNCT
iajs-4032	40	28	.	.	PUNCT
iajs-4032	41	1	additionally	additionally	ADV
iajs-4032	41	2	,	,	PUNCT
iajs-4032	41	3	jankovic	jankovic	PROPN
iajs-4032	41	4	and	and	CCONJ
iajs-4032	41	5	hamlet	hamlet	PROPN
iajs-4032	41	6	presented	present	VERB
iajs-4032	41	7	a	a	DET
iajs-4032	41	8	study	study	NOUN
iajs-4032	41	9	on	on	ADP
iajs-4032	41	10	the	the	DET
iajs-4032	41	11	topological	topological	ADJ
iajs-4032	41	12	properties	property	NOUN
iajs-4032	41	13	using	use	VERB
iajs-4032	41	14	the	the	DET
iajs-4032	41	15	concept	concept	NOUN
iajs-4032	41	16	of	of	ADP
iajs-4032	41	17	the	the	DET
iajs-4032	41	18	ideal	ideal	NOUN
iajs-4032	41	19	)	)	PUNCT
iajs-4032	41	20	12	12	NUM
iajs-4032	41	21	-	-	SYM
iajs-4032	41	22	14	14	NUM
iajs-4032	41	23	(	(	PUNCT
iajs-4032	41	24	.	.	PUNCT
iajs-4032	42	1	using	use	VERB
iajs-4032	42	2	the	the	DET
iajs-4032	42	3	concept	concept	NOUN
iajs-4032	42	4	of	of	ADP
iajs-4032	42	5	the	the	DET
iajs-4032	42	6	ideal	ideal	NOUN
iajs-4032	42	7	,	,	PUNCT
iajs-4032	42	8	some	some	DET
iajs-4032	42	9	researchers	researcher	NOUN
iajs-4032	42	10	studied	study	VERB
iajs-4032	42	11	a	a	DET
iajs-4032	42	12	new	new	ADJ
iajs-4032	42	13	type	type	NOUN
iajs-4032	42	14	of	of	ADP
iajs-4032	42	15	separation	separation	NOUN
iajs-4032	42	16	axioms	axiom	NOUN
iajs-4032	42	17	)	)	PUNCT
iajs-4032	42	18	15	15	NUM
iajs-4032	42	19	-	-	SYM
iajs-4032	42	20	17	17	NUM
iajs-4032	42	21	(	(	PUNCT
iajs-4032	42	22	.	.	PUNCT
iajs-4032	43	1	furthermore	furthermore	ADV
iajs-4032	43	2	,	,	PUNCT
iajs-4032	43	3	studies	study	NOUN
iajs-4032	43	4	continued	continue	VERB
iajs-4032	43	5	,	,	PUNCT
iajs-4032	43	6	and	and	CCONJ
iajs-4032	43	7	after	after	SCONJ
iajs-4032	43	8	nano	nano	NOUN
iajs-4032	43	9	soft	soft	ADJ
iajs-4032	43	10	topology	topology	NOUN
iajs-4032	43	11	was	be	AUX
iajs-4032	43	12	defined	define	VERB
iajs-4032	43	13	,	,	PUNCT
iajs-4032	43	14	many	many	ADJ
iajs-4032	43	15	types	type	NOUN
iajs-4032	43	16	of	of	ADP
iajs-4032	43	17	open	open	ADJ
iajs-4032	43	18	sets	set	NOUN
iajs-4032	43	19	were	be	AUX
iajs-4032	43	20	known	know	VERB
iajs-4032	43	21	on	on	ADP
iajs-4032	43	22	the	the	DET
iajs-4032	43	23	ideal	ideal	ADJ
iajs-4032	43	24	topological	topological	ADJ
iajs-4032	43	25	space	space	NOUN
iajs-4032	43	26	.	.	PUNCT
iajs-4032	44	1	moreover	moreover	ADV
iajs-4032	44	2	,	,	PUNCT
iajs-4032	44	3	the	the	DET
iajs-4032	44	4	definition	definition	NOUN
iajs-4032	44	5	of	of	ADP
iajs-4032	44	6	soft	soft	ADJ
iajs-4032	44	7	ideal	ideal	ADJ
iajs-4032	44	8	topological	topological	ADJ
iajs-4032	44	9	spaces	space	NOUN
iajs-4032	44	10	was	be	AUX
iajs-4032	44	11	extended	extend	VERB
iajs-4032	44	12	to	to	PART
iajs-4032	44	13	include	include	VERB
iajs-4032	44	14	soft	soft	ADJ
iajs-4032	44	15	topological	topological	ADJ
iajs-4032	44	16	properties	property	NOUN
iajs-4032	44	17	.	.	PUNCT
iajs-4032	45	1	also	also	ADV
iajs-4032	45	2	nano	nano	VERB
iajs-4032	45	3	soft	soft	ADJ
iajs-4032	45	4	)	)	PUNCT
iajs-4032	45	5	18	18	NUM
iajs-4032	45	6	-	-	SYM
iajs-4032	45	7	22	22	NUM
iajs-4032	45	8	(	(	PUNCT
iajs-4032	45	9	,	,	PUNCT
iajs-4032	45	10	although	although	SCONJ
iajs-4032	45	11	more	more	ADJ
iajs-4032	45	12	,	,	PUNCT
iajs-4032	45	13	application	application	NOUN
iajs-4032	45	14	on	on	ADP
iajs-4032	45	15	the	the	DET
iajs-4032	45	16	ideal	ideal	ADJ
iajs-4032	45	17	topological	topological	ADJ
iajs-4032	45	18	spaces	space	NOUN
iajs-4032	45	19	.	.	PUNCT
iajs-4032	46	1	and	and	CCONJ
iajs-4032	46	2	separation	separation	NOUN
iajs-4032	46	3	axioms	axiom	NOUN
iajs-4032	46	4	were	be	AUX
iajs-4032	46	5	offered	offer	VERB
iajs-4032	46	6	like	like	ADP
iajs-4032	46	7	connectedness	connectedness	NOUN
iajs-4032	46	8	)	)	PUNCT
iajs-4032	46	9	23	23	NUM
iajs-4032	46	10	(	(	PUNCT
iajs-4032	46	11	,	,	PUNCT
iajs-4032	46	12	and	and	CCONJ
iajs-4032	46	13	many	many	ADJ
iajs-4032	46	14	properties	property	NOUN
iajs-4032	46	15	of	of	ADP
iajs-4032	46	16	the	the	DET
iajs-4032	46	17	nano	nano	NOUN
iajs-4032	46	18	ideal	ideal	ADJ
iajs-4032	46	19	spaces	space	NOUN
iajs-4032	46	20	were	be	AUX
iajs-4032	46	21	studied	study	VERB
iajs-4032	46	22	.	.	PUNCT
iajs-4032	47	1	also	also	ADV
iajs-4032	47	2	a	a	DET
iajs-4032	47	3	new	new	ADJ
iajs-4032	47	4	classes	class	NOUN
iajs-4032	47	5	sets	set	NOUN
iajs-4032	47	6	are	be	AUX
iajs-4032	47	7	defined	define	VERB
iajs-4032	47	8	,	,	PUNCT
iajs-4032	47	9	and	and	CCONJ
iajs-4032	47	10	many	many	ADJ
iajs-4032	47	11	of	of	ADP
iajs-4032	47	12	the	the	DET
iajs-4032	47	13	properties	property	NOUN
iajs-4032	47	14	of	of	ADP
iajs-4032	47	15	these	these	DET
iajs-4032	47	16	sets	set	NOUN
iajs-4032	47	17	were	be	AUX
iajs-4032	47	18	discussed	discuss	VERB
iajs-4032	47	19	,	,	PUNCT
iajs-4032	47	20	so	so	CCONJ
iajs-4032	47	21	other	other	ADJ
iajs-4032	47	22	research	research	NOUN
iajs-4032	47	23	was	be	AUX
iajs-4032	47	24	developed	develop	VERB
iajs-4032	47	25	on	on	ADP
iajs-4032	47	26	these	these	DET
iajs-4032	47	27	sets	set	NOUN
iajs-4032	47	28	and	and	CCONJ
iajs-4032	47	29	their	their	PRON
iajs-4032	47	30	topological	topological	ADJ
iajs-4032	47	31	properties	property	NOUN
iajs-4032	47	32	based	base	VERB
iajs-4032	47	33	on	on	ADP
iajs-4032	47	34	the	the	DET
iajs-4032	47	35	concept	concept	NOUN
iajs-4032	47	36	of	of	ADP
iajs-4032	47	37	the	the	DET
iajs-4032	47	38	ideal	ideal	NOUN
iajs-4032	47	39	.	.	PUNCT
iajs-4032	48	1	)	)	PUNCT
iajs-4032	49	1	24	24	NUM
iajs-4032	49	2	-	-	SYM
iajs-4032	49	3	27	27	NUM
iajs-4032	49	4	(	(	PUNCT
iajs-4032	49	5	.	.	PUNCT
iajs-4032	49	6	2.methodology	2.methodology	NUM
iajs-4032	49	7	2.1	2.1	NUM
iajs-4032	49	8	.	.	PUNCT
iajs-4032	50	1	open	open	ADJ
iajs-4032	50	2	set	set	VERB
iajs-4032	50	3	in	in	ADP
iajs-4032	50	4	this	this	DET
iajs-4032	50	5	subsection	subsection	NOUN
iajs-4032	50	6	,	,	PUNCT
iajs-4032	50	7	a	a	DET
iajs-4032	50	8	new	new	ADJ
iajs-4032	50	9	set	set	NOUN
iajs-4032	50	10	will	will	AUX
iajs-4032	50	11	be	be	AUX
iajs-4032	50	12	defined	define	VERB
iajs-4032	50	13	and	and	CCONJ
iajs-4032	50	14	named	name	VERB
iajs-4032	50	15	ropen	ropen	ADJ
iajs-4032	50	16	in	in	ADP
iajs-4032	50	17	the	the	DET
iajs-4032	50	18	space	space	NOUN
iajs-4032	50	19	2.1.1	2.1.1	NUM
iajs-4032	50	20	.	.	PUNCT
iajs-4032	51	1	definition	definition	NOUN
iajs-4032	51	2	let	let	AUX
iajs-4032	51	3	be	be	AUX
iajs-4032	51	4	an	an	DET
iajs-4032	51	5	ideal	ideal	ADJ
iajs-4032	51	6	topological	topological	ADJ
iajs-4032	51	7	space	space	NOUN
iajs-4032	51	8	,	,	PUNCT
iajs-4032	51	9	a	a	DET
iajs-4032	51	10	subset	subset	NOUN
iajs-4032	51	11	of	of	ADP
iajs-4032	51	12	is	be	AUX
iajs-4032	51	13	said	say	VERB
iajs-4032	51	14	to	to	PART
iajs-4032	51	15	be	be	AUX
iajs-4032	51	16	open	open	ADJ
iajs-4032	51	17	set	set	VERB
iajs-4032	51	18	if	if	SCONJ
iajs-4032	51	19	∃	∃	PROPN
iajs-4032	51	20	h	h	PROPN
iajs-4032	51	21	∈	∈	PROPN
iajs-4032	51	22	∖	∖	NOUN
iajs-4032	51	23	{	{	PUNCT
iajs-4032	51	24	,	,	PUNCT
iajs-4032	51	25	such	such	ADJ
iajs-4032	51	26	that	that	SCONJ
iajs-4032	51	27	,	,	PUNCT
iajs-4032	51	28	∈	∈	PROPN
iajs-4032	51	29	,	,	PUNCT
iajs-4032	51	30	the	the	DET
iajs-4032	51	31	complement	complement	NOUN
iajs-4032	51	32	of	of	ADP
iajs-4032	51	33	open	open	ADJ
iajs-4032	51	34	set	set	NOUN
iajs-4032	51	35	is	be	AUX
iajs-4032	51	36	called	call	VERB
iajs-4032	51	37	closed	closed	ADJ
iajs-4032	51	38	set	set	NOUN
iajs-4032	51	39	.	.	PUNCT
iajs-4032	52	1	[	[	X
iajs-4032	52	2	the	the	DET
iajs-4032	52	3	symbol	symbol	NOUN
iajs-4032	52	4	to	to	PART
iajs-4032	52	5	ropen	ropen	VERB
iajs-4032	52	6	set	set	NOUN
iajs-4032	52	7	,	,	PUNCT
iajs-4032	52	8	and	and	CCONJ
iajs-4032	52	9	the	the	DET
iajs-4032	52	10	symbol	symbol	NOUN
iajs-4032	52	11	r	r	X
iajs-4032	52	12	-	-	PUNCT
iajs-4032	52	13	c	c	NOUN
iajs-4032	52	14	refers	refer	VERB
iajs-4032	52	15	to	to	ADP
iajs-4032	52	16	an	an	DET
iajs-4032	52	17	r	r	NOUN
iajs-4032	52	18	-	-	PUNCT
iajs-4032	52	19	closed	closed	ADJ
iajs-4032	52	20	set	set	NOUN
iajs-4032	52	21	]	]	PUNCT
iajs-4032	52	22	.	.	PUNCT
iajs-4032	53	1	2.1.2	2.1.2	X
iajs-4032	53	2	.	.	X
iajs-4032	53	3	proposition	proposition	NOUN
iajs-4032	53	4	in	in	ADP
iajs-4032	53	5	any	any	DET
iajs-4032	53	6	ideal	ideal	ADJ
iajs-4032	53	7	topological	topological	ADJ
iajs-4032	53	8	space	space	NOUN
iajs-4032	53	9	every	every	DET
iajs-4032	53	10	open	open	ADJ
iajs-4032	53	11	proper	proper	ADJ
iajs-4032	53	12	subset	subset	NOUN
iajs-4032	53	13	of	of	ADP
iajs-4032	53	14	is	be	AUX
iajs-4032	53	15	an	an	DET
iajs-4032	53	16	r	r	NOUN
iajs-4032	53	17	-	-	PUNCT
iajs-4032	53	18	open	open	ADJ
iajs-4032	53	19	set	set	NOUN
iajs-4032	53	20	.	.	PUNCT
iajs-4032	54	1	proof	proof	NOUN
iajs-4032	54	2	:	:	PUNCT
iajs-4032	54	3	suppose	suppose	VERB
iajs-4032	54	4	is	be	AUX
iajs-4032	54	5	open	open	ADJ
iajs-4032	54	6	proper	proper	ADJ
iajs-4032	54	7	⊆	⊆	NUM
iajs-4032	54	8	,	,	PUNCT
iajs-4032	54	9	if	if	SCONJ
iajs-4032	54	10	,	,	PUNCT
iajs-4032	54	11	then	then	ADV
iajs-4032	54	12	for	for	ADP
iajs-4032	54	13	any	any	DET
iajs-4032	54	14	nonempty	nonempty	ADV
iajs-4032	54	15	open	open	VERB
iajs-4032	54	16	proper	proper	ADJ
iajs-4032	54	17	subset	subset	NOUN
iajs-4032	54	18	of	of	ADP
iajs-4032	54	19	,	,	PUNCT
iajs-4032	54	20	∈	∈	PROPN
iajs-4032	54	21	.	.	PUNCT
iajs-4032	55	1	if	if	SCONJ
iajs-4032	55	2	,	,	PUNCT
iajs-4032	55	3	then	then	ADV
iajs-4032	55	4	∃	∃	PROPN
iajs-4032	55	5	∈	∈	PROPN
iajs-4032	55	6	∖	∖	X
iajs-4032	55	7	when	when	SCONJ
iajs-4032	55	8	such	such	ADJ
iajs-4032	55	9	that	that	DET
iajs-4032	55	10	∈	∈	PROPN
iajs-4032	55	11	.	.	PUNCT
iajs-4032	56	1	2.1.3	2.1.3	X
iajs-4032	56	2	.	.	PUNCT
iajs-4032	56	3	remark	remark	NOUN
iajs-4032	56	4	in	in	ADP
iajs-4032	56	5	any	any	DET
iajs-4032	56	6	ideal	ideal	ADJ
iajs-4032	56	7	topological	topological	ADJ
iajs-4032	56	8	space	space	NOUN
iajs-4032	56	9	,	,	PUNCT
iajs-4032	56	10	does	do	AUX
iajs-4032	56	11	not	not	PART
iajs-4032	56	12	be	be	AUX
iajs-4032	56	13	open	open	ADJ
iajs-4032	56	14	set	set	VERB
iajs-4032	56	15	.	.	PUNCT
iajs-4032	57	1	example	example	NOUN
iajs-4032	57	2	suppose	suppose	VERB
iajs-4032	57	3	.notice	.notice	NOUN
iajs-4032	57	4	that	that	SCONJ
iajs-4032	57	5	not	not	PART
iajs-4032	57	6	o	o	NOUN
iajs-4032	57	7	set	set	NOUN
iajs-4032	57	8	,	,	PUNCT
iajs-4032	57	9	since	since	SCONJ
iajs-4032	57	10	the	the	DET
iajs-4032	57	11	only	only	ADJ
iajs-4032	57	12	nonempty	nonempty	NOUN
iajs-4032	57	13	open	open	VERB
iajs-4032	57	14	proper	proper	ADJ
iajs-4032	57	15	subsets	subset	NOUN
iajs-4032	57	16	of	of	ADP
iajs-4032	57	17	are	be	AUX
iajs-4032	57	18	and	and	CCONJ
iajs-4032	57	19	also	also	ADV
iajs-4032	57	20	2.1.4	2.1.4	NUM
iajs-4032	57	21	.	.	PUNCT
iajs-4032	58	1	remark	remark	NOUN
iajs-4032	58	2	in	in	ADP
iajs-4032	58	3	the	the	DET
iajs-4032	58	4	ideal	ideal	ADJ
iajs-4032	58	5	topological	topological	ADJ
iajs-4032	58	6	space	space	NOUN
iajs-4032	58	7	can	can	AUX
iajs-4032	58	8	not	not	PART
iajs-4032	58	9	be	be	AUX
iajs-4032	58	10	an	an	DET
iajs-4032	58	11	open	open	ADJ
iajs-4032	58	12	set	set	NOUN
iajs-4032	58	13	.	.	PUNCT
iajs-4032	59	1	ihjpas	ihjpas	PROPN
iajs-4032	59	2	.	.	PUNCT
iajs-4032	60	1	2025,38(4	2025,38(4	NOUN
iajs-4032	60	2	)	)	PUNCT
iajs-4032	60	3	4	4	NUM
iajs-4032	60	4	example	example	NOUN
iajs-4032	60	5	in	in	ADP
iajs-4032	60	6	the	the	DET
iajs-4032	60	7	example	example	NOUN
iajs-4032	60	8	of	of	ADP
iajs-4032	60	9	remark	remark	NOUN
iajs-4032	60	10	2.3	2.3	NUM
iajs-4032	60	11	,	,	PUNCT
iajs-4032	60	12	is	be	AUX
iajs-4032	60	13	an	an	DET
iajs-4032	60	14	o	o	NOUN
iajs-4032	60	15	set	set	NOUN
iajs-4032	60	16	but	but	CCONJ
iajs-4032	60	17	not	not	PART
iajs-4032	60	18	an	an	DET
iajs-4032	60	19	open	open	ADJ
iajs-4032	60	20	set	set	NOUN
iajs-4032	60	21	.	.	PUNCT
iajs-4032	61	1	2.1.5	2.1.5	NUM
iajs-4032	61	2	.	.	PROPN
iajs-4032	61	3	remark	remark	NOUN
iajs-4032	61	4	for	for	ADP
iajs-4032	61	5	any	any	DET
iajs-4032	61	6	two	two	NUM
iajs-4032	61	7	open	open	ADJ
iajs-4032	61	8	sets	set	NOUN
iajs-4032	61	9	,	,	PUNCT
iajs-4032	61	10	the	the	DET
iajs-4032	61	11	intersection	intersection	NOUN
iajs-4032	61	12	needs	need	VERB
iajs-4032	61	13	not	not	PART
iajs-4032	61	14	to	to	PART
iajs-4032	61	15	be	be	AUX
iajs-4032	61	16	o.	o.	ADJ
iajs-4032	61	17	example	example	NOUN
iajs-4032	61	18	let	let	VERB
iajs-4032	61	19	{	{	PUNCT
iajs-4032	61	20	}	}	PUNCT
iajs-4032	61	21	{	{	PUNCT
iajs-4032	61	22	{	{	PUNCT
iajs-4032	61	23	}	}	PUNCT
iajs-4032	61	24	}	}	PUNCT
iajs-4032	61	25	}	}	PUNCT
iajs-4032	61	26	.	.	PUNCT
iajs-4032	62	1	notice	notice	VERB
iajs-4032	62	2	that	that	SCONJ
iajs-4032	62	3	{	{	PUNCT
iajs-4032	62	4	}	}	PUNCT
iajs-4032	62	5	{	{	PUNCT
iajs-4032	62	6	}	}	PUNCT
iajs-4032	62	7	o	o	NOUN
iajs-4032	62	8	sets	set	NOUN
iajs-4032	62	9	,	,	PUNCT
iajs-4032	62	10	but	but	CCONJ
iajs-4032	62	11	{	{	PUNCT
iajs-4032	62	12	{	{	PUNCT
iajs-4032	62	13	}	}	PUNCT
iajs-4032	62	14	can	can	AUX
iajs-4032	62	15	not	not	PART
iajs-4032	62	16	open	open	VERB
iajs-4032	62	17	.	.	PUNCT
iajs-4032	63	1	2.1.6	2.1.6	X
iajs-4032	63	2	.	.	PROPN
iajs-4032	63	3	remark	remark	VERB
iajs-4032	63	4	a	a	DET
iajs-4032	63	5	union	union	NOUN
iajs-4032	63	6	for	for	ADP
iajs-4032	63	7	any	any	DET
iajs-4032	63	8	two	two	NUM
iajs-4032	63	9	open	open	ADJ
iajs-4032	63	10	sets	set	NOUN
iajs-4032	63	11	can	can	AUX
iajs-4032	63	12	not	not	PART
iajs-4032	63	13	open	open	VERB
iajs-4032	63	14	.	.	PUNCT
iajs-4032	64	1	for	for	ADP
iajs-4032	64	2	example	example	NOUN
iajs-4032	64	3	.	.	PUNCT
iajs-4032	65	1	since	since	SCONJ
iajs-4032	65	2	{	{	PUNCT
iajs-4032	65	3	{	{	PUNCT
iajs-4032	65	4	}	}	PUNCT
iajs-4032	65	5	are	be	AUX
iajs-4032	65	6	two	two	NUM
iajs-4032	65	7	open	open	ADJ
iajs-4032	65	8	sets	set	NOUN
iajs-4032	65	9	and	and	CCONJ
iajs-4032	65	10	∪	∪	X
iajs-4032	65	11	{	{	PUNCT
iajs-4032	65	12	}	}	PUNCT
iajs-4032	65	13	can	can	AUX
iajs-4032	65	14	not	not	PART
iajs-4032	65	15	be	be	AUX
iajs-4032	65	16	open	open	ADJ
iajs-4032	65	17	sets	set	NOUN
iajs-4032	65	18	.	.	PUNCT
iajs-4032	66	1	2.1.7	2.1.7	X
iajs-4032	66	2	.	.	PUNCT
iajs-4032	66	3	theorem	theorem	VERB
iajs-4032	66	4	the	the	DET
iajs-4032	66	5	space	space	NOUN
iajs-4032	66	6	,	,	PUNCT
iajs-4032	66	7	i	i	PRON
iajs-4032	66	8	)	)	PUNCT
iajs-4032	66	9	has	have	VERB
iajs-4032	66	10	no	no	DET
iajs-4032	66	11	proper	proper	ADJ
iajs-4032	66	12	open	open	ADJ
iajs-4032	66	13	subset	subset	NOUN
iajs-4032	66	14	whose	whose	DET
iajs-4032	66	15	union	union	NOUN
iajs-4032	66	16	equals	equal	VERB
iajs-4032	66	17	to	to	PART
iajs-4032	66	18	,	,	PUNCT
iajs-4032	66	19	then	then	ADV
iajs-4032	66	20	the	the	DET
iajs-4032	66	21	union	union	NOUN
iajs-4032	66	22	of	of	ADP
iajs-4032	66	23	any	any	DET
iajs-4032	66	24	two	two	NUM
iajs-4032	66	25	r	r	NOUN
iajs-4032	66	26	o	o	NOUN
iajs-4032	66	27	sets	set	NOUN
iajs-4032	66	28	is	be	AUX
iajs-4032	66	29	r	r	NOUN
iajs-4032	66	30	–	–	PUNCT
iajs-4032	66	31	o.	o.	NOUN
iajs-4032	66	32	proof	proof	NOUN
iajs-4032	66	33	:	:	PUNCT
iajs-4032	66	34	assume	assume	VERB
iajs-4032	66	35	that	that	PRON
iajs-4032	66	36	are	be	AUX
iajs-4032	66	37	two	two	NUM
iajs-4032	66	38	o	o	NOUN
iajs-4032	66	39	sets	set	NOUN
iajs-4032	66	40	,	,	PUNCT
iajs-4032	66	41	since	since	SCONJ
iajs-4032	66	42	there	there	PRON
iajs-4032	66	43	is	be	VERB
iajs-4032	66	44	∈	∈	PROPN
iajs-4032	66	45	∖	∖	NOUN
iajs-4032	66	46	implies	imply	VERB
iajs-4032	66	47	∈	∈	PROPN
iajs-4032	66	48	and	and	CCONJ
iajs-4032	66	49	∈	∈	PROPN
iajs-4032	66	50	,	,	PUNCT
iajs-4032	66	51	it	it	PRON
iajs-4032	66	52	follows	follow	VERB
iajs-4032	66	53	from	from	ADP
iajs-4032	66	54	conditions	condition	NOUN
iajs-4032	66	55	of	of	ADP
iajs-4032	66	56	ideal	ideal	NOUN
iajs-4032	66	57	that	that	SCONJ
iajs-4032	66	58	[	[	PUNCT
iajs-4032	66	59	]	]	X
iajs-4032	66	60	∪	∪	X
iajs-4032	66	61	[	[	PUNCT
iajs-4032	66	62	]	]	X
iajs-4032	66	63	∈	∈	PROPN
iajs-4032	66	64	,	,	PUNCT
iajs-4032	66	65	hence	hence	ADV
iajs-4032	66	66	.	.	PUNCT
iajs-4032	67	1	∪	∪	ADV
iajs-4032	67	2	)	)	PUNCT
iajs-4032	67	3	[	[	PUNCT
iajs-4032	67	4	(	(	PUNCT
iajs-4032	67	5	)	)	PUNCT
iajs-4032	67	6	∪	∪	NOUN
iajs-4032	67	7	(	(	PUNCT
iajs-4032	67	8	)	)	PUNCT
iajs-4032	67	9	]	]	PUNCT
iajs-4032	68	1	⊆	⊆	NUM
iajs-4032	68	2	[	[	X
iajs-4032	68	3	(	(	PUNCT
iajs-4032	68	4	(	(	PUNCT
iajs-4032	68	5	)	)	PUNCT
iajs-4032	68	6	∪	∪	NOUN
iajs-4032	68	7	[	[	PUNCT
iajs-4032	68	8	(	(	PUNCT
iajs-4032	68	9	]	]	X
iajs-4032	68	10	,	,	PUNCT
iajs-4032	68	11	then	then	ADV
iajs-4032	68	12	(	(	PUNCT
iajs-4032	68	13	∪	∪	X
iajs-4032	68	14	)	)	PUNCT
iajs-4032	68	15	[	[	PUNCT
iajs-4032	68	16	(	(	PUNCT
iajs-4032	68	17	)	)	PUNCT
iajs-4032	68	18	∪	∪	NOUN
iajs-4032	68	19	)	)	PUNCT
iajs-4032	68	20	]	]	PUNCT
iajs-4032	68	21	∈	∈	PROPN
iajs-4032	68	22	but	but	CCONJ
iajs-4032	68	23	(	(	PUNCT
iajs-4032	68	24	)	)	PUNCT
iajs-4032	68	25	∪	∪	ADJ
iajs-4032	68	26	)	)	PUNCT
iajs-4032	68	27	⊆	⊆	NUM
iajs-4032	68	28	∪	∪	X
iajs-4032	68	29	)	)	PUNCT
iajs-4032	68	30	(	(	PUNCT
iajs-4032	68	31	∪	∪	X
iajs-4032	68	32	)	)	PUNCT
iajs-4032	68	33	,	,	PUNCT
iajs-4032	68	34	then	then	ADV
iajs-4032	68	35	,	,	PUNCT
iajs-4032	68	36	cl	cl	INTJ
iajs-4032	68	37	[	[	PUNCT
iajs-4032	68	38	(	(	PUNCT
iajs-4032	68	39	)	)	PUNCT
iajs-4032	68	40	∪	∪	ADJ
iajs-4032	68	41	)	)	PUNCT
iajs-4032	68	42	⊆	⊆	NUM
iajs-4032	68	43	[	[	PUNCT
iajs-4032	68	44	∪	∪	X
iajs-4032	68	45	)	)	PUNCT
iajs-4032	68	46	(	(	PUNCT
iajs-4032	68	47	∪	∪	X
iajs-4032	68	48	)	)	PUNCT
iajs-4032	68	49	]	]	PUNCT
iajs-4032	68	50	,	,	PUNCT
iajs-4032	68	51	so	so	CCONJ
iajs-4032	68	52	(	(	PUNCT
iajs-4032	68	53	∪	∪	X
iajs-4032	68	54	[	[	PUNCT
iajs-4032	68	55	∪	∪	X
iajs-4032	68	56	∪	∪	X
iajs-4032	68	57	]	]	PUNCT
iajs-4032	68	58	⊆	⊆	NUM
iajs-4032	68	59	∪	∪	NOUN
iajs-4032	68	60	)	)	PUNCT
iajs-4032	68	61	[	[	X
iajs-4032	68	62	(	(	PUNCT
iajs-4032	68	63	)	)	PUNCT
iajs-4032	68	64	∪	∪	ADJ
iajs-4032	68	65	)	)	PUNCT
iajs-4032	68	66	]	]	PUNCT
iajs-4032	68	67	.	.	PUNCT
iajs-4032	69	1	that	that	SCONJ
iajs-4032	69	2	subsequently	subsequently	ADV
iajs-4032	69	3	∪	∪	ADJ
iajs-4032	69	4	[	[	PUNCT
iajs-4032	69	5	∪	∪	X
iajs-4032	69	6	∪	∪	X
iajs-4032	69	7	]	]	X
iajs-4032	69	8	∈	∈	PROPN
iajs-4032	69	9	.	.	PUNCT
iajs-4032	70	1	then	then	ADV
iajs-4032	70	2	∪	∪	ADJ
iajs-4032	70	3	is	be	AUX
iajs-4032	70	4	an	an	DET
iajs-4032	70	5	o.	o.	NOUN
iajs-4032	70	6	2.1.8	2.1.8	PROPN
iajs-4032	70	7	.	.	PUNCT
iajs-4032	70	8	remark	remark	NOUN
iajs-4032	70	9	in	in	ADP
iajs-4032	70	10	the	the	DET
iajs-4032	70	11	context	context	NOUN
iajs-4032	70	12	of	of	ADP
iajs-4032	70	13	the	the	DET
iajs-4032	70	14	space	space	NOUN
iajs-4032	70	15	(	(	PUNCT
iajs-4032	70	16	i	i	NOUN
iajs-4032	70	17	)	)	PUNCT
iajs-4032	70	18	,	,	PUNCT
iajs-4032	70	19	the	the	DET
iajs-4032	70	20	set	set	NOUN
iajs-4032	70	21	r	r	NOUN
iajs-4032	70	22	o	o	NOUN
iajs-4032	70	23	is	be	AUX
iajs-4032	70	24	a	a	DET
iajs-4032	70	25	union	union	NOUN
iajs-4032	70	26	for	for	ADP
iajs-4032	70	27	every	every	DET
iajs-4032	70	28	r	r	NOUN
iajs-4032	70	29	-	-	PUNCT
iajs-4032	70	30	open	open	ADJ
iajs-4032	70	31	set	set	NOUN
iajs-4032	70	32	;	;	PUNCT
iajs-4032	70	33	we	we	PRON
iajs-4032	70	34	will	will	AUX
iajs-4032	70	35	denote	denote	VERB
iajs-4032	70	36	it	it	PRON
iajs-4032	70	37	.	.	PUNCT
iajs-4032	71	1	such	such	ADJ
iajs-4032	71	2	that	that	SCONJ
iajs-4032	71	3	⊆	⊆	NOUN
iajs-4032	71	4	∪	∪	ADJ
iajs-4032	71	5	}	}	PUNCT
iajs-4032	71	6	,	,	PUNCT
iajs-4032	71	7	respectively	respectively	ADV
iajs-4032	71	8	⊆	⊆	NUM
iajs-4032	71	9	∪	∪	NOUN
iajs-4032	71	10	.	.	PUNCT
iajs-4032	72	1	2.1.9	2.1.9	NUM
iajs-4032	72	2	.	.	X
iajs-4032	72	3	remark	remark	PROPN
iajs-4032	72	4	assume	assume	VERB
iajs-4032	72	5	the	the	DET
iajs-4032	72	6	space	space	NOUN
iajs-4032	72	7	(	(	PUNCT
iajs-4032	72	8	,	,	PUNCT
iajs-4032	72	9	it	it	PRON
iajs-4032	72	10	is	be	AUX
iajs-4032	72	11	easy	easy	ADJ
iajs-4032	72	12	to	to	PART
iajs-4032	72	13	achieve	achieve	VERB
iajs-4032	72	14	that	that	PRON
iajs-4032	72	15	⊆	⊆	NUM
iajs-4032	72	16	.	.	PUNCT
iajs-4032	73	1	3	3	X
iajs-4032	73	2	.	.	X
iajs-4032	73	3	separation	separation	NOUN
iajs-4032	73	4	axioms	axiom	NOUN
iajs-4032	73	5	in	in	ADP
iajs-4032	73	6	this	this	DET
iajs-4032	73	7	section	section	NOUN
iajs-4032	73	8	,	,	PUNCT
iajs-4032	73	9	using	use	VERB
iajs-4032	73	10	a	a	DET
iajs-4032	73	11	set	set	NOUN
iajs-4032	73	12	under	under	ADP
iajs-4032	73	13	study	study	NOUN
iajs-4032	73	14	,	,	PUNCT
iajs-4032	73	15	r	r	X
iajs-4032	73	16	-	-	PUNCT
iajs-4032	73	17	open	open	VERB
iajs-4032	73	18	a	a	DET
iajs-4032	73	19	new	new	ADJ
iajs-4032	73	20	kind	kind	NOUN
iajs-4032	73	21	of	of	ADP
iajs-4032	73	22	separation	separation	NOUN
iajs-4032	73	23	axioms	axiom	NOUN
iajs-4032	73	24	will	will	AUX
iajs-4032	73	25	be	be	AUX
iajs-4032	73	26	defined	define	VERB
iajs-4032	73	27	and	and	CCONJ
iajs-4032	73	28	the	the	DET
iajs-4032	73	29	relationship	relationship	NOUN
iajs-4032	73	30	among	among	ADP
iajs-4032	73	31	these	these	DET
iajs-4032	73	32	concepts	concept	NOUN
iajs-4032	73	33	will	will	AUX
iajs-4032	73	34	be	be	AUX
iajs-4032	73	35	studied	study	VERB
iajs-4032	73	36	.	.	PUNCT
iajs-4032	74	1	3.1	3.1	NUM
iajs-4032	74	2	.	.	PUNCT
iajs-4032	74	3	definition	definition	NOUN
iajs-4032	74	4	consider	consider	VERB
iajs-4032	74	5	the	the	DET
iajs-4032	74	6	space	space	NOUN
iajs-4032	74	7	(	(	PUNCT
iajs-4032	74	8	,	,	PUNCT
iajs-4032	74	9	i	i	PROPN
iajs-4032	74	10	)	)	PUNCT
iajs-4032	74	11	,	,	PUNCT
iajs-4032	74	12	then	then	ADV
iajs-4032	74	13	(	(	PUNCT
iajs-4032	74	14	,	,	PUNCT
iajs-4032	74	15	i	i	NOUN
iajs-4032	74	16	)	)	PUNCT
iajs-4032	74	17	is	be	AUX
iajs-4032	74	18	named	name	VERB
iajs-4032	74	19	an	an	DET
iajs-4032	74	20	space	space	NOUN
iajs-4032	74	21	if	if	SCONJ
iajs-4032	74	22	and	and	CCONJ
iajs-4032	74	23	only	only	ADV
iajs-4032	74	24	if	if	SCONJ
iajs-4032	74	25	each	each	PRON
iajs-4032	74	26	of	of	ADP
iajs-4032	74	27	the	the	DET
iajs-4032	74	28	two	two	NUM
iajs-4032	74	29	different	different	ADJ
iajs-4032	74	30	points	point	NOUN
iajs-4032	74	31	all	all	PRON
iajs-4032	74	32	contain	contain	VERB
iajs-4032	74	33	to	to	ADP
iajs-4032	74	34	an	an	DET
iajs-4032	74	35	example	example	NOUN
iajs-4032	74	36	consider	consider	VERB
iajs-4032	74	37	the	the	DET
iajs-4032	74	38	space	space	NOUN
iajs-4032	74	39	(	(	PUNCT
iajs-4032	74	40	,	,	PUNCT
iajs-4032	74	41	i	i	PROPN
iajs-4032	74	42	)	)	PUNCT
iajs-4032	74	43	;	;	PUNCT
iajs-4032	74	44	=	=	SYM
iajs-4032	74	45	{	{	PUNCT
iajs-4032	74	46	,	,	PUNCT
iajs-4032	74	47	}	}	PUNCT
iajs-4032	74	48	,	,	PUNCT
iajs-4032	74	49	=	=	PRON
iajs-4032	74	50	{	{	PUNCT
iajs-4032	74	51	,	,	PUNCT
iajs-4032	74	52	,	,	PUNCT
iajs-4032	74	53	}	}	PUNCT
iajs-4032	74	54	i	i	PRON
iajs-4032	74	55	=	=	PRON
iajs-4032	74	56	{	{	PUNCT
iajs-4032	74	57	,	,	PUNCT
iajs-4032	74	58	{	{	PUNCT
iajs-4032	74	59	}	}	PUNCT
iajs-4032	74	60	}	}	PUNCT
iajs-4032	74	61	,	,	PUNCT
iajs-4032	74	62	(	(	PUNCT
iajs-4032	74	63	)	)	PUNCT
iajs-4032	74	64	=	=	SYM
iajs-4032	74	65	{	{	PUNCT
iajs-4032	74	66	,	,	PUNCT
iajs-4032	74	67	,	,	PUNCT
iajs-4032	74	68	{	{	PUNCT
iajs-4032	74	69	}	}	PUNCT
iajs-4032	74	70	,	,	PUNCT
iajs-4032	74	71	{	{	PUNCT
iajs-4032	74	72	}	}	PUNCT
iajs-4032	74	73	,	,	PUNCT
iajs-4032	74	74	{	{	PUNCT
iajs-4032	74	75	}	}	PUNCT
iajs-4032	74	76	,	,	PUNCT
iajs-4032	74	77	{	{	PUNCT
iajs-4032	74	78	,	,	PUNCT
iajs-4032	74	79	}	}	PUNCT
iajs-4032	74	80	}	}	PUNCT
iajs-4032	74	81	,	,	PUNCT
iajs-4032	74	82	notice	notice	VERB
iajs-4032	74	83	that	that	SCONJ
iajs-4032	74	84	(	(	PUNCT
iajs-4032	74	85	,	,	PUNCT
iajs-4032	74	86	i	i	NOUN
iajs-4032	74	87	)	)	PUNCT
iajs-4032	74	88	is	be	AUX
iajs-4032	74	89	space	space	NOUN
iajs-4032	74	90	since	since	ADV
iajs-4032	74	91	,	,	PUNCT
iajs-4032	74	92	∃	∃	PROPN
iajs-4032	74	93	ropen	ropen	PROPN
iajs-4032	74	94	set	set	NOUN
iajs-4032	74	95	{	{	PUNCT
iajs-4032	74	96	}	}	PUNCT
iajs-4032	74	97	∈	∈	PROPN
iajs-4032	74	98	(	(	PUNCT
iajs-4032	74	99	)	)	PUNCT
iajs-4032	74	100	such	such	ADJ
iajs-4032	74	101	that	that	SCONJ
iajs-4032	74	102	∈	∈	PROPN
iajs-4032	74	103	ihjpas	ihjpa	VERB
iajs-4032	74	104	.	.	PUNCT
iajs-4032	75	1	2025,38(4	2025,38(4	NOUN
iajs-4032	75	2	)	)	PUNCT
iajs-4032	75	3	5	5	NUM
iajs-4032	75	4	{	{	PUNCT
iajs-4032	75	5	}	}	PUNCT
iajs-4032	75	6	and	and	CCONJ
iajs-4032	75	7	{	{	PUNCT
iajs-4032	75	8	}	}	PUNCT
iajs-4032	75	9	,	,	PUNCT
iajs-4032	75	10	,	,	PUNCT
iajs-4032	75	11	∃	∃	PROPN
iajs-4032	75	12	ropen	ropen	PROPN
iajs-4032	75	13	set	set	NOUN
iajs-4032	75	14	{	{	PUNCT
iajs-4032	75	15	}	}	PUNCT
iajs-4032	75	16	∈	∈	PROPN
iajs-4032	75	17	(	(	PUNCT
iajs-4032	75	18	)	)	PUNCT
iajs-4032	75	19	such	such	ADJ
iajs-4032	75	20	that	that	SCONJ
iajs-4032	75	21	∈	∈	PROPN
iajs-4032	75	22	{	{	PUNCT
iajs-4032	75	23	}	}	PUNCT
iajs-4032	75	24	hence	hence	ADV
iajs-4032	75	25	(	(	PUNCT
iajs-4032	75	26	i	i	NOUN
iajs-4032	75	27	)	)	PUNCT
iajs-4032	75	28	is	be	AUX
iajs-4032	75	29	space	space	NOUN
iajs-4032	75	30	.	.	PUNCT
iajs-4032	76	1	3.2	3.2	NUM
iajs-4032	76	2	.	.	PUNCT
iajs-4032	76	3	theorem	theorem	NOUN
iajs-4032	76	4	suppose	suppose	VERB
iajs-4032	76	5	the	the	DET
iajs-4032	76	6	space	space	NOUN
iajs-4032	76	7	(	(	PUNCT
iajs-4032	76	8	)	)	PUNCT
iajs-4032	76	9	is	be	AUX
iajs-4032	76	10	–	–	PUNCT
iajs-4032	76	11	space	space	NOUN
iajs-4032	76	12	,	,	PUNCT
iajs-4032	76	13	so	so	CCONJ
iajs-4032	76	14	(	(	PUNCT
iajs-4032	76	15	i	i	NOUN
iajs-4032	76	16	)	)	PUNCT
iajs-4032	76	17	be	be	AUX
iajs-4032	76	18	an	an	DET
iajs-4032	76	19	space	space	NOUN
iajs-4032	76	20	if	if	SCONJ
iajs-4032	76	21	,	,	PUNCT
iajs-4032	76	22	for	for	ADP
iajs-4032	76	23	every	every	DET
iajs-4032	76	24	two	two	NUM
iajs-4032	76	25	different	different	ADJ
iajs-4032	76	26	elements	element	NOUN
iajs-4032	76	27	∈	∈	PROPN
iajs-4032	76	28	(	(	PUNCT
iajs-4032	76	29	)	)	PUNCT
iajs-4032	76	30	.	.	PUNCT
iajs-4032	77	1	proof	proof	NOUN
iajs-4032	77	2	:	:	PUNCT
iajs-4032	77	3	consider	consider	VERB
iajs-4032	77	4	(	(	PUNCT
iajs-4032	77	5	)	)	PUNCT
iajs-4032	77	6	to	to	PART
iajs-4032	77	7	be	be	AUX
iajs-4032	77	8	the	the	DET
iajs-4032	77	9	0	0	NUM
iajs-4032	77	10	–	–	PUNCT
iajs-4032	77	11	space	space	NOUN
iajs-4032	77	12	,	,	PUNCT
iajs-4032	77	13	then	then	ADV
iajs-4032	77	14	for	for	SCONJ
iajs-4032	77	15	each	each	DET
iajs-4032	77	16	two	two	NUM
iajs-4032	77	17	different	different	ADJ
iajs-4032	77	18	∈	∈	PROPN
iajs-4032	77	19	∃	∃	PROPN
iajs-4032	77	20	∈	∈	PROPN
iajs-4032	77	21	such	such	ADJ
iajs-4032	77	22	that	that	SCONJ
iajs-4032	77	23	∈	∈	PROPN
iajs-4032	78	1	but	but	CCONJ
iajs-4032	78	2	,	,	PUNCT
iajs-4032	78	3	since	since	SCONJ
iajs-4032	78	4	⊆	⊆	NUM
iajs-4032	78	5	(	(	PUNCT
iajs-4032	78	6	)	)	PUNCT
iajs-4032	78	7	so	so	ADV
iajs-4032	78	8	∈	∈	PROPN
iajs-4032	78	9	but	but	CCONJ
iajs-4032	78	10	,	,	PUNCT
iajs-4032	78	11	then	then	ADV
iajs-4032	78	12	(	(	PUNCT
iajs-4032	78	13	i	i	NOUN
iajs-4032	78	14	)	)	PUNCT
iajs-4032	78	15	is	be	AUX
iajs-4032	78	16	space	space	NOUN
iajs-4032	78	17	.	.	PUNCT
iajs-4032	79	1	the	the	DET
iajs-4032	79	2	converse	converse	NOUN
iajs-4032	79	3	does	do	AUX
iajs-4032	79	4	not	not	PART
iajs-4032	79	5	hold	hold	VERB
iajs-4032	79	6	true	true	ADJ
iajs-4032	79	7	in	in	ADP
iajs-4032	79	8	general	general	ADJ
iajs-4032	79	9	for	for	ADP
iajs-4032	79	10	instance	instance	NOUN
iajs-4032	79	11	.	.	PUNCT
iajs-4032	80	1	example	example	NOUN
iajs-4032	80	2	suppose	suppose	VERB
iajs-4032	80	3	=	=	PRON
iajs-4032	80	4	{	{	PUNCT
iajs-4032	80	5	,	,	PUNCT
iajs-4032	80	6	}	}	PUNCT
iajs-4032	80	7	,	,	PUNCT
iajs-4032	80	8	=	=	PRON
iajs-4032	80	9	{	{	PUNCT
iajs-4032	80	10	,	,	PUNCT
iajs-4032	80	11	,	,	PUNCT
iajs-4032	80	12	,	,	PUNCT
iajs-4032	80	13	}	}	PUNCT
iajs-4032	80	14	}	}	PUNCT
iajs-4032	80	15	,	,	PUNCT
iajs-4032	80	16	i	i	PRON
iajs-4032	80	17	=	=	PRON
iajs-4032	80	18	{	{	PUNCT
iajs-4032	80	19	,	,	PUNCT
iajs-4032	80	20	}	}	PUNCT
iajs-4032	80	21	}	}	PUNCT
iajs-4032	80	22	.	.	PUNCT
iajs-4032	81	1	then	then	ADV
iajs-4032	81	2	,	,	PUNCT
iajs-4032	81	3	)	)	PUNCT
iajs-4032	81	4	=	=	SYM
iajs-4032	81	5	(	(	PUNCT
iajs-4032	81	6	)	)	PUNCT
iajs-4032	81	7	.	.	PUNCT
iajs-4032	82	1	notice	notice	VERB
iajs-4032	82	2	that	that	SCONJ
iajs-4032	82	3	(	(	PUNCT
iajs-4032	82	4	,	,	PUNCT
iajs-4032	82	5	i	i	NOUN
iajs-4032	82	6	)	)	PUNCT
iajs-4032	82	7	is	be	AUX
iajs-4032	82	8	space	space	NOUN
iajs-4032	82	9	but	but	CCONJ
iajs-4032	82	10	not	not	PART
iajs-4032	82	11	space	space	NOUN
iajs-4032	82	12	since	since	ADV
iajs-4032	82	13	and	and	CCONJ
iajs-4032	82	14	there	there	PRON
iajs-4032	82	15	is	be	VERB
iajs-4032	82	16	no	no	DET
iajs-4032	82	17	open	open	ADJ
iajs-4032	82	18	set	set	NOUN
iajs-4032	82	19	that	that	PRON
iajs-4032	82	20	belongs	belong	VERB
iajs-4032	82	21	to	to	ADP
iajs-4032	82	22	but	but	CCONJ
iajs-4032	82	23	does	do	AUX
iajs-4032	82	24	not	not	PART
iajs-4032	82	25	contain	contain	VERB
iajs-4032	82	26	.	.	PUNCT
iajs-4032	83	1	3.3	3.3	NUM
iajs-4032	83	2	.	.	PUNCT
iajs-4032	84	1	theorem	theorem	NOUN
iajs-4032	84	2	(	(	PUNCT
iajs-4032	84	3	i	i	NOUN
iajs-4032	84	4	)	)	PUNCT
iajs-4032	84	5	is	be	AUX
iajs-4032	84	6	an	an	DET
iajs-4032	84	7	space	space	NOUN
iajs-4032	84	8	if	if	SCONJ
iajs-4032	85	1	and	and	CCONJ
iajs-4032	85	2	only	only	ADV
iajs-4032	85	3	if	if	SCONJ
iajs-4032	85	4	for	for	ADP
iajs-4032	85	5	each	each	DET
iajs-4032	85	6	two	two	NUM
iajs-4032	85	7	different	different	ADJ
iajs-4032	85	8	components	component	NOUN
iajs-4032	85	9	there	there	PRON
iajs-4032	85	10	exists	exist	VERB
iajs-4032	85	11	an	an	DET
iajs-4032	85	12	c	c	NOUN
iajs-4032	85	13	set	set	NOUN
iajs-4032	85	14	consisting	consist	VERB
iajs-4032	85	15	of	of	ADP
iajs-4032	85	16	one	one	NUM
iajs-4032	85	17	and	and	CCONJ
iajs-4032	85	18	not	not	PART
iajs-4032	85	19	containing	contain	VERB
iajs-4032	85	20	the	the	DET
iajs-4032	85	21	other	other	ADJ
iajs-4032	85	22	.	.	PUNCT
iajs-4032	86	1	proof	proof	NOUN
iajs-4032	86	2	:	:	PUNCT
iajs-4032	86	3	suppose	suppose	VERB
iajs-4032	86	4	two	two	NUM
iajs-4032	86	5	distinct	distinct	ADJ
iajs-4032	86	6	elements	element	NOUN
iajs-4032	86	7	in	in	ADP
iajs-4032	86	8	,	,	PUNCT
iajs-4032	86	9	and	and	CCONJ
iajs-4032	86	10	since	since	SCONJ
iajs-4032	86	11	is	be	AUX
iajs-4032	86	12	an	an	DET
iajs-4032	86	13	space	space	NOUN
iajs-4032	86	14	,	,	PUNCT
iajs-4032	86	15	then	then	ADV
iajs-4032	86	16	there	there	PRON
iajs-4032	86	17	exists	exist	VERB
iajs-4032	86	18	an	an	DET
iajs-4032	86	19	open	open	ADJ
iajs-4032	86	20	set	set	NOUN
iajs-4032	86	21	containing	contain	VERB
iajs-4032	86	22	one	one	NUM
iajs-4032	86	23	of	of	ADP
iajs-4032	86	24	them	they	PRON
iajs-4032	86	25	,	,	PUNCT
iajs-4032	86	26	such	such	ADJ
iajs-4032	86	27	that	that	PRON
iajs-4032	86	28	is	be	AUX
iajs-4032	86	29	an	an	DET
iajs-4032	86	30	closed	closed	ADJ
iajs-4032	86	31	set	set	NOUN
iajs-4032	86	32	includes	include	VERB
iajs-4032	86	33	the	the	DET
iajs-4032	86	34	other	other	ADJ
iajs-4032	86	35	.	.	PUNCT
iajs-4032	87	1	in	in	ADP
iajs-4032	87	2	contrast	contrast	NOUN
iajs-4032	87	3	,	,	PUNCT
iajs-4032	87	4	suppose	suppose	VERB
iajs-4032	87	5	that	that	SCONJ
iajs-4032	87	6	are	be	AUX
iajs-4032	87	7	distinct	distinct	ADJ
iajs-4032	87	8	elements	element	NOUN
iajs-4032	87	9	.	.	PUNCT
iajs-4032	88	1	in	in	ADP
iajs-4032	88	2	and	and	CCONJ
iajs-4032	88	3	since	since	SCONJ
iajs-4032	88	4	it	it	PRON
iajs-4032	88	5	contains	contain	VERB
iajs-4032	88	6	an	an	DET
iajs-4032	88	7	closed	closed	ADJ
iajs-4032	88	8	set	set	NOUN
iajs-4032	88	9	containing	contain	VERB
iajs-4032	88	10	one	one	NUM
iajs-4032	88	11	and	and	CCONJ
iajs-4032	88	12	not	not	PART
iajs-4032	88	13	containing	contain	VERB
iajs-4032	88	14	the	the	DET
iajs-4032	88	15	other	other	ADJ
iajs-4032	88	16	.	.	PUNCT
iajs-4032	89	1	therefore	therefore	ADV
iajs-4032	89	2	,	,	PUNCT
iajs-4032	89	3	is	be	AUX
iajs-4032	89	4	an	an	DET
iajs-4032	89	5	open	open	ADJ
iajs-4032	89	6	set	set	NOUN
iajs-4032	89	7	that	that	PRON
iajs-4032	89	8	comprises	comprise	VERB
iajs-4032	89	9	only	only	ADV
iajs-4032	89	10	one	one	NUM
iajs-4032	89	11	of	of	ADP
iajs-4032	89	12	them	they	PRON
iajs-4032	89	13	,	,	PUNCT
iajs-4032	89	14	hence	hence	ADV
iajs-4032	89	15	is	be	AUX
iajs-4032	89	16	an	an	DET
iajs-4032	89	17	space	space	NOUN
iajs-4032	89	18	.	.	PUNCT
iajs-4032	90	1	3.4	3.4	NUM
iajs-4032	90	2	.	.	PUNCT
iajs-4032	90	3	definition	definition	NOUN
iajs-4032	90	4	suppose	suppose	VERB
iajs-4032	90	5	(	(	PUNCT
iajs-4032	90	6	i	i	NOUN
iajs-4032	90	7	)	)	PUNCT
iajs-4032	90	8	is	be	AUX
iajs-4032	90	9	an	an	DET
iajs-4032	90	10	ideal	ideal	ADJ
iajs-4032	90	11	topological	topological	ADJ
iajs-4032	90	12	space	space	NOUN
iajs-4032	90	13	,	,	PUNCT
iajs-4032	90	14	so	so	CCONJ
iajs-4032	90	15	(	(	PUNCT
iajs-4032	90	16	i	i	NOUN
iajs-4032	90	17	)	)	PUNCT
iajs-4032	90	18	is	be	AUX
iajs-4032	90	19	named	name	VERB
iajs-4032	90	20	an	an	DET
iajs-4032	90	21	space	space	NOUN
iajs-4032	90	22	if	if	SCONJ
iajs-4032	90	23	and	and	CCONJ
iajs-4032	90	24	only	only	ADV
iajs-4032	90	25	if	if	SCONJ
iajs-4032	90	26	for	for	ADP
iajs-4032	90	27	each	each	DET
iajs-4032	90	28	pair	pair	NOUN
iajs-4032	90	29	of	of	ADP
iajs-4032	90	30	the	the	DET
iajs-4032	90	31	point	point	NOUN
iajs-4032	90	32	∈	∈	PROPN
iajs-4032	90	33	there	there	PRON
iajs-4032	90	34	exist	exist	VERB
iajs-4032	90	35	two	two	NUM
iajs-4032	90	36	sets	set	NOUN
iajs-4032	90	37	u	u	NOUN
iajs-4032	90	38	&	&	CCONJ
iajs-4032	90	39	v	v	NOUN
iajs-4032	90	40	that	that	PRON
iajs-4032	90	41	are	be	AUX
iajs-4032	90	42	(	(	PUNCT
iajs-4032	90	43	)	)	PUNCT
iajs-4032	90	44	sets	set	NOUN
iajs-4032	90	45	therefor	therefor	ADP
iajs-4032	90	46	∈	∈	PROPN
iajs-4032	90	47	(	(	PUNCT
iajs-4032	90	48	u	u	NOUN
iajs-4032	90	49	and	and	CCONJ
iajs-4032	90	50	∈	∈	PROPN
iajs-4032	90	51	(	(	PUNCT
iajs-4032	90	52	v	v	NOUN
iajs-4032	90	53	–	–	PUNCT
iajs-4032	90	54	u	u	NOUN
iajs-4032	90	55	)	)	PUNCT
iajs-4032	90	56	.	.	PUNCT
iajs-4032	91	1	example	example	NOUN
iajs-4032	91	2	let	let	VERB
iajs-4032	91	3	=	=	PRON
iajs-4032	91	4	{	{	PUNCT
iajs-4032	91	5	}	}	PUNCT
iajs-4032	91	6	,	,	PUNCT
iajs-4032	91	7	=	=	PRON
iajs-4032	91	8	{	{	PUNCT
iajs-4032	91	9	,	,	PUNCT
iajs-4032	91	10	,	,	PUNCT
iajs-4032	91	11	{	{	PUNCT
iajs-4032	91	12	,	,	PUNCT
iajs-4032	91	13	}	}	PUNCT
iajs-4032	91	14	}	}	PUNCT
iajs-4032	91	15	and	and	CCONJ
iajs-4032	91	16	i	i	PRON
iajs-4032	91	17	=	=	PUNCT
iajs-4032	91	18	,	,	PUNCT
iajs-4032	91	19	{	{	PUNCT
iajs-4032	91	20	}	}	PUNCT
iajs-4032	91	21	,	,	PUNCT
iajs-4032	91	22	{	{	PUNCT
iajs-4032	91	23	}	}	PUNCT
iajs-4032	91	24	,	,	PUNCT
iajs-4032	91	25	{	{	PUNCT
iajs-4032	91	26	}	}	PUNCT
iajs-4032	91	27	}	}	PUNCT
iajs-4032	91	28	,	,	PUNCT
iajs-4032	91	29	(	(	PUNCT
iajs-4032	91	30	)	)	PUNCT
iajs-4032	91	31	=	=	SYM
iajs-4032	91	32	(	(	PUNCT
iajs-4032	91	33	)	)	PUNCT
iajs-4032	91	34	.such	.such	ADP
iajs-4032	92	1	that	that	PRON
iajs-4032	92	2	(	(	PUNCT
iajs-4032	92	3	i	i	NOUN
iajs-4032	92	4	)	)	PUNCT
iajs-4032	92	5	be	be	AUX
iajs-4032	92	6	an	an	DET
iajs-4032	92	7	space	space	NOUN
iajs-4032	92	8	.	.	PUNCT
iajs-4032	93	1	3.5	3.5	NUM
iajs-4032	93	2	.	.	PUNCT
iajs-4032	93	3	proposition	proposition	NOUN
iajs-4032	93	4	let	let	VERB
iajs-4032	93	5	)	)	PUNCT
iajs-4032	93	6	be	be	AUX
iajs-4032	93	7	a	a	DET
iajs-4032	93	8	space	space	NOUN
iajs-4032	93	9	such	such	ADJ
iajs-4032	93	10	that	that	SCONJ
iajs-4032	93	11	(	(	PUNCT
iajs-4032	93	12	i	i	NOUN
iajs-4032	93	13	)	)	PUNCT
iajs-4032	93	14	is	be	AUX
iajs-4032	93	15	an	an	DET
iajs-4032	93	16	space	space	NOUN
iajs-4032	93	17	.	.	PUNCT
iajs-4032	94	1	proof	proof	NOUN
iajs-4032	94	2	:	:	PUNCT
iajs-4032	94	3	assume	assume	VERB
iajs-4032	94	4	)	)	PUNCT
iajs-4032	94	5	is	be	AUX
iajs-4032	94	6	a	a	DET
iajs-4032	94	7	space	space	NOUN
iajs-4032	94	8	so	so	SCONJ
iajs-4032	94	9	there	there	PRON
iajs-4032	94	10	exists	exist	VERB
iajs-4032	94	11	u	u	NOUN
iajs-4032	94	12	&	&	CCONJ
iajs-4032	94	13	v	v	PROPN
iajs-4032	94	14	(	(	PUNCT
iajs-4032	94	15	two	two	NUM
iajs-4032	94	16	open	open	ADJ
iajs-4032	94	17	sets	set	NOUN
iajs-4032	94	18	)	)	PUNCT
iajs-4032	94	19	belong	belong	VERB
iajs-4032	94	20	to	to	ADP
iajs-4032	94	21	and	and	CCONJ
iajs-4032	94	22	for	for	ADP
iajs-4032	94	23	any	any	DET
iajs-4032	94	24	two	two	NUM
iajs-4032	94	25	distinct	distinct	ADJ
iajs-4032	94	26	elements	element	NOUN
iajs-4032	94	27	∈	∈	PROPN
iajs-4032	94	28	implies	imply	VERB
iajs-4032	94	29	∈	∈	PROPN
iajs-4032	94	30	(	(	PUNCT
iajs-4032	94	31	u	u	NOUN
iajs-4032	94	32	-	-	NOUN
iajs-4032	94	33	v	v	NOUN
iajs-4032	94	34	)	)	PUNCT
iajs-4032	94	35	,	,	PUNCT
iajs-4032	94	36	also	also	ADV
iajs-4032	94	37	∈	∈	PROPN
iajs-4032	94	38	(	(	PUNCT
iajs-4032	94	39	v	v	NOUN
iajs-4032	94	40	-	-	PUNCT
iajs-4032	94	41	u	u	NOUN
iajs-4032	94	42	)	)	PUNCT
iajs-4032	94	43	since	since	SCONJ
iajs-4032	94	44	⊆	⊆	NUM
iajs-4032	94	45	(	(	PUNCT
iajs-4032	94	46	)	)	PUNCT
iajs-4032	94	47	so	so	ADV
iajs-4032	94	48	u	u	NOUN
iajs-4032	94	49	,	,	PUNCT
iajs-4032	94	50	v	v	NOUN
iajs-4032	94	51	∈	∈	PROPN
iajs-4032	94	52	(	(	PUNCT
iajs-4032	94	53	)	)	PUNCT
iajs-4032	94	54	,	,	PUNCT
iajs-4032	94	55	∃	∃	PROPN
iajs-4032	94	56	u	u	PROPN
iajs-4032	94	57	,	,	PUNCT
iajs-4032	94	58	v	v	NOUN
iajs-4032	94	59	∈	∈	PROPN
iajs-4032	94	60	(	(	PUNCT
iajs-4032	94	61	)	)	PUNCT
iajs-4032	94	62	and	and	CCONJ
iajs-4032	94	63	∈	∈	PROPN
iajs-4032	94	64	(	(	PUNCT
iajs-4032	94	65	u	u	NOUN
iajs-4032	94	66	-	-	NOUN
iajs-4032	94	67	v	v	NOUN
iajs-4032	94	68	)	)	PUNCT
iajs-4032	94	69	and	and	CCONJ
iajs-4032	94	70	∈	∈	PROPN
iajs-4032	94	71	(	(	PUNCT
iajs-4032	94	72	v	v	NOUN
iajs-4032	94	73	-	-	PUNCT
iajs-4032	94	74	u	u	NOUN
iajs-4032	94	75	)	)	PUNCT
iajs-4032	94	76	,	,	PUNCT
iajs-4032	94	77	then	then	ADV
iajs-4032	94	78	(	(	PUNCT
iajs-4032	94	79	,	,	PUNCT
iajs-4032	94	80	i	i	PROPN
iajs-4032	94	81	)	)	PUNCT
iajs-4032	94	82	is	be	AUX
iajs-4032	94	83	space	space	NOUN
iajs-4032	94	84	.	.	PUNCT
iajs-4032	95	1	but	but	CCONJ
iajs-4032	95	2	the	the	DET
iajs-4032	95	3	opposite	opposite	NOUN
iajs-4032	95	4	does	do	AUX
iajs-4032	95	5	not	not	PART
iajs-4032	95	6	hold	hold	VERB
iajs-4032	95	7	in	in	ADP
iajs-4032	95	8	(	(	PUNCT
iajs-4032	95	9	3.8	3.8	NUM
iajs-4032	95	10	)	)	PUNCT
iajs-4032	95	11	in	in	ADP
iajs-4032	95	12	general	general	ADJ
iajs-4032	95	13	.	.	PUNCT
iajs-4032	96	1	consider	consider	VERB
iajs-4032	96	2	the	the	DET
iajs-4032	96	3	instance	instance	NOUN
iajs-4032	96	4	(	(	PUNCT
iajs-4032	96	5	3.4	3.4	NUM
iajs-4032	96	6	)	)	PUNCT
iajs-4032	96	7	.	.	PUNCT
iajs-4032	97	1	3.6	3.6	NUM
iajs-4032	97	2	.	.	PUNCT
iajs-4032	97	3	proposition	proposition	NOUN
iajs-4032	97	4	consider	consider	VERB
iajs-4032	97	5	)	)	PUNCT
iajs-4032	97	6	to	to	PART
iajs-4032	97	7	be	be	AUX
iajs-4032	97	8	an	an	DET
iajs-4032	97	9	space	space	NOUN
iajs-4032	97	10	so	so	CCONJ
iajs-4032	97	11	(	(	PUNCT
iajs-4032	97	12	,	,	PUNCT
iajs-4032	97	13	i	i	PROPN
iajs-4032	97	14	)	)	PUNCT
iajs-4032	97	15	is	be	AUX
iajs-4032	97	16	an	an	DET
iajs-4032	97	17	space	space	NOUN
iajs-4032	97	18	.	.	PUNCT
iajs-4032	98	1	proof	proof	NOUN
iajs-4032	98	2	:	:	PUNCT
iajs-4032	98	3	consider	consider	VERB
iajs-4032	98	4	be	be	AUX
iajs-4032	98	5	two	two	NUM
iajs-4032	98	6	different	different	ADJ
iajs-4032	98	7	components	component	NOUN
iajs-4032	98	8	∈	∈	PROPN
iajs-4032	98	9	,	,	PUNCT
iajs-4032	98	10	since	since	SCONJ
iajs-4032	98	11	(	(	PUNCT
iajs-4032	98	12	,	,	PUNCT
iajs-4032	98	13	i	i	PROPN
iajs-4032	98	14	)	)	PUNCT
iajs-4032	98	15	is	be	AUX
iajs-4032	98	16	an	an	DET
iajs-4032	98	17	space	space	NOUN
iajs-4032	99	1	so	so	SCONJ
iajs-4032	99	2	u	u	NOUN
iajs-4032	99	3	,	,	PUNCT
iajs-4032	99	4	v	v	NUM
iajs-4032	99	5	are	be	AUX
iajs-4032	99	6	two	two	NUM
iajs-4032	99	7	(	(	PUNCT
iajs-4032	99	8	)	)	PUNCT
iajs-4032	99	9	sets	set	NOUN
iajs-4032	99	10	,	,	PUNCT
iajs-4032	99	11	hence	hence	ADV
iajs-4032	99	12	∈	∈	PROPN
iajs-4032	99	13	(	(	PUNCT
iajs-4032	99	14	u	u	NOUN
iajs-4032	99	15	-	-	NOUN
iajs-4032	99	16	v	v	NOUN
iajs-4032	99	17	)	)	PUNCT
iajs-4032	99	18	,	,	PUNCT
iajs-4032	99	19	∈	∈	PROPN
iajs-4032	99	20	(	(	PUNCT
iajs-4032	99	21	v	v	NOUN
iajs-4032	99	22	-	-	PUNCT
iajs-4032	99	23	u	u	NOUN
iajs-4032	99	24	)	)	PUNCT
iajs-4032	99	25	.	.	PUNCT
iajs-4032	100	1	therefore	therefore	ADV
iajs-4032	100	2	∃	∃	PROPN
iajs-4032	100	3	(	(	PUNCT
iajs-4032	100	4	)	)	PUNCT
iajs-4032	100	5	set	set	VERB
iajs-4032	100	6	u	u	PRON
iajs-4032	100	7	which	which	PRON
iajs-4032	100	8	includes	include	VERB
iajs-4032	100	9	only	only	ADV
iajs-4032	100	10	one	one	NUM
iajs-4032	100	11	of	of	ADP
iajs-4032	100	12	them	they	PRON
iajs-4032	100	13	,	,	PUNCT
iajs-4032	100	14	so	so	ADV
iajs-4032	100	15	,	,	PUNCT
iajs-4032	100	16	i	i	PRON
iajs-4032	100	17	)	)	PUNCT
iajs-4032	100	18	is	be	AUX
iajs-4032	100	19	an	an	DET
iajs-4032	100	20	space	space	NOUN
iajs-4032	100	21	while	while	SCONJ
iajs-4032	100	22	the	the	DET
iajs-4032	100	23	inverse	inverse	NOUN
iajs-4032	100	24	of	of	ADP
iajs-4032	100	25	(	(	PUNCT
iajs-4032	100	26	3.9	3.9	NUM
iajs-4032	100	27	)	)	PUNCT
iajs-4032	100	28	does	do	AUX
iajs-4032	100	29	not	not	PART
iajs-4032	100	30	hold	hold	VERB
iajs-4032	100	31	for	for	ADP
iajs-4032	100	32	instance	instance	NOUN
iajs-4032	100	33	.	.	PUNCT
iajs-4032	101	1	ihjpas	ihjpas	PROPN
iajs-4032	101	2	.	.	PUNCT
iajs-4032	102	1	2025,38(4	2025,38(4	NOUN
iajs-4032	102	2	)	)	PUNCT
iajs-4032	102	3	6	6	NUM
iajs-4032	102	4	example	example	NOUN
iajs-4032	102	5	assume	assume	VERB
iajs-4032	102	6	=	=	PRON
iajs-4032	102	7	{	{	PUNCT
iajs-4032	102	8	,	,	PUNCT
iajs-4032	102	9	}	}	PUNCT
iajs-4032	102	10	,	,	PUNCT
iajs-4032	102	11	=	=	PRON
iajs-4032	102	12	{	{	PUNCT
iajs-4032	102	13	,	,	PUNCT
iajs-4032	102	14	,	,	PUNCT
iajs-4032	102	15	{	{	PUNCT
iajs-4032	102	16	}	}	PUNCT
iajs-4032	102	17	,	,	PUNCT
iajs-4032	102	18	i	i	PRON
iajs-4032	102	19	=	=	PRON
iajs-4032	102	20	{	{	PUNCT
iajs-4032	102	21	,	,	PUNCT
iajs-4032	102	22	}	}	PUNCT
iajs-4032	102	23	}	}	PUNCT
iajs-4032	102	24	,	,	PUNCT
iajs-4032	102	25	(	(	PUNCT
iajs-4032	102	26	)	)	PUNCT
iajs-4032	102	27	=	=	SYM
iajs-4032	102	28	,	,	PUNCT
iajs-4032	102	29	{	{	PUNCT
iajs-4032	102	30	}	}	PUNCT
iajs-4032	102	31	,	,	PUNCT
iajs-4032	102	32	{	{	PUNCT
iajs-4032	102	33	}	}	PUNCT
iajs-4032	102	34	,	,	PUNCT
iajs-4032	102	35	{	{	PUNCT
iajs-4032	102	36	}	}	PUNCT
iajs-4032	102	37	}	}	PUNCT
iajs-4032	102	38	such	such	ADJ
iajs-4032	102	39	that	that	SCONJ
iajs-4032	102	40	,	,	PUNCT
iajs-4032	102	41	i	i	PROPN
iajs-4032	102	42	)	)	PUNCT
iajs-4032	102	43	is	be	AUX
iajs-4032	102	44	an	an	DET
iajs-4032	102	45	space	space	NOUN
iajs-4032	102	46	but	but	CCONJ
iajs-4032	102	47	not	not	PART
iajs-4032	102	48	an	an	DET
iajs-4032	102	49	space	space	NOUN
iajs-4032	102	50	since	since	SCONJ
iajs-4032	102	51	∈	∈	PROPN
iajs-4032	102	52	and	and	CCONJ
iajs-4032	102	53	so	so	ADV
iajs-4032	102	54	u	u	NOUN
iajs-4032	102	55	,	,	PUNCT
iajs-4032	102	56	v	v	NOUN
iajs-4032	102	57	∈	∈	PROPN
iajs-4032	102	58	(	(	PUNCT
iajs-4032	102	59	)	)	PUNCT
iajs-4032	102	60	,	,	PUNCT
iajs-4032	102	61	then	then	ADV
iajs-4032	102	62	∈	∈	PROPN
iajs-4032	102	63	(	(	PUNCT
iajs-4032	102	64	u	u	NOUN
iajs-4032	102	65	-	-	NOUN
iajs-4032	102	66	v	v	NOUN
iajs-4032	102	67	)	)	PUNCT
iajs-4032	102	68	and	and	CCONJ
iajs-4032	102	69	∈	∈	PROPN
iajs-4032	102	70	(	(	PUNCT
iajs-4032	102	71	v	v	NOUN
iajs-4032	102	72	-	-	PUNCT
iajs-4032	102	73	u	u	NOUN
iajs-4032	102	74	)	)	PUNCT
iajs-4032	102	75	.	.	PUNCT
iajs-4032	103	1	3.7	3.7	NUM
iajs-4032	103	2	.	.	PUNCT
iajs-4032	103	3	theorem	theorem	NOUN
iajs-4032	103	4	suppose	suppose	VERB
iajs-4032	103	5	,	,	PUNCT
iajs-4032	103	6	i	i	PRON
iajs-4032	103	7	)	)	PUNCT
iajs-4032	103	8	is	be	AUX
iajs-4032	103	9	a	a	DET
iajs-4032	103	10	space	space	NOUN
iajs-4032	103	11	;	;	PUNCT
iajs-4032	103	12	,	,	PUNCT
iajs-4032	103	13	i	i	PRON
iajs-4032	103	14	)	)	PUNCT
iajs-4032	103	15	is	be	AUX
iajs-4032	103	16	an	an	DET
iajs-4032	103	17	space	space	NOUN
iajs-4032	103	18	if	if	SCONJ
iajs-4032	103	19	and	and	CCONJ
iajs-4032	103	20	only	only	ADV
iajs-4032	103	21	if	if	SCONJ
iajs-4032	103	22	for	for	ADP
iajs-4032	103	23	each	each	DET
iajs-4032	103	24	element	element	NOUN
iajs-4032	103	25	so	so	SCONJ
iajs-4032	103	26	there	there	PRON
iajs-4032	103	27	are	be	VERB
iajs-4032	103	28	two	two	NUM
iajs-4032	103	29	c	c	NOUN
iajs-4032	103	30	sets	set	NOUN
iajs-4032	103	31	:	:	PUNCT
iajs-4032	103	32	1	1	NUM
iajs-4032	103	33	and	and	CCONJ
iajs-4032	103	34	2	2	NUM
iajs-4032	103	35	,	,	PUNCT
iajs-4032	103	36	implying	imply	VERB
iajs-4032	103	37	belongs	belong	VERB
iajs-4032	103	38	to	to	ADP
iajs-4032	103	39	(	(	PUNCT
iajs-4032	103	40	1	1	NUM
iajs-4032	103	41	2	2	NUM
iajs-4032	103	42	)	)	PUNCT
iajs-4032	103	43	belongs	belong	VERB
iajs-4032	103	44	to	to	ADP
iajs-4032	103	45	(	(	PUNCT
iajs-4032	103	46	2	2	NUM
iajs-4032	103	47	1	1	NUM
iajs-4032	103	48	)	)	PUNCT
iajs-4032	103	49	.	.	PUNCT
iajs-4032	104	1	proof	proof	NOUN
iajs-4032	104	2	consider	consider	VERB
iajs-4032	104	3	are	be	AUX
iajs-4032	104	4	two	two	NUM
iajs-4032	104	5	different	different	ADJ
iajs-4032	104	6	constituents	constituent	NOUN
iajs-4032	104	7	in	in	ADP
iajs-4032	104	8	also	also	ADV
iajs-4032	104	9	,	,	PUNCT
iajs-4032	104	10	i	i	PRON
iajs-4032	104	11	)	)	PUNCT
iajs-4032	104	12	is	be	AUX
iajs-4032	104	13	an	an	DET
iajs-4032	104	14	space	space	NOUN
iajs-4032	104	15	.	.	PUNCT
iajs-4032	105	1	consequently	consequently	ADV
iajs-4032	105	2	,	,	PUNCT
iajs-4032	105	3	there	there	PRON
iajs-4032	105	4	are	be	VERB
iajs-4032	105	5	two	two	NUM
iajs-4032	105	6	o	o	NOUN
iajs-4032	105	7	sets	set	NOUN
iajs-4032	105	8	:	:	PUNCT
iajs-4032	105	9	u	u	NOUN
iajs-4032	105	10	and	and	CCONJ
iajs-4032	105	11	v	v	NOUN
iajs-4032	105	12	,	,	PUNCT
iajs-4032	105	13	therefore	therefore	ADV
iajs-4032	105	14	∈	∈	PROPN
iajs-4032	105	15	(	(	PUNCT
iajs-4032	105	16	u	u	NOUN
iajs-4032	105	17	-	-	NOUN
iajs-4032	105	18	v	v	NOUN
iajs-4032	105	19	)	)	PUNCT
iajs-4032	105	20	&	&	CCONJ
iajs-4032	105	21	∈	∈	PROPN
iajs-4032	105	22	(	(	PUNCT
iajs-4032	105	23	v	v	NOUN
iajs-4032	105	24	-	-	PUNCT
iajs-4032	105	25	u	u	NOUN
iajs-4032	105	26	)	)	PUNCT
iajs-4032	105	27	and	and	CCONJ
iajs-4032	105	28	there	there	PRON
iajs-4032	105	29	exists	exist	VERB
iajs-4032	105	30	c	c	NOUN
iajs-4032	105	31	sets	set	NOUN
iajs-4032	105	32	(	(	PUNCT
iajs-4032	105	33	-u	-u	PUNCT
iajs-4032	105	34	)	)	PUNCT
iajs-4032	105	35	and	and	CCONJ
iajs-4032	105	36	(	(	PUNCT
iajs-4032	105	37	-v	-v	NOUN
iajs-4032	105	38	)	)	PUNCT
iajs-4032	105	39	.	.	PUNCT
iajs-4032	106	1	∈	∈	PROPN
iajs-4032	106	2	(	(	PUNCT
iajs-4032	106	3	-v	-v	NOUN
iajs-4032	106	4	)	)	PUNCT
iajs-4032	106	5	(	(	PUNCT
iajs-4032	106	6	-u	-u	PUNCT
iajs-4032	106	7	)	)	PUNCT
iajs-4032	106	8	,	,	PUNCT
iajs-4032	106	9	∈	∈	PROPN
iajs-4032	106	10	(	(	PUNCT
iajs-4032	106	11	-u	-u	NOUN
iajs-4032	106	12	)	)	PUNCT
iajs-4032	106	13	(	(	PUNCT
iajs-4032	106	14	-v	-v	X
iajs-4032	106	15	)	)	PUNCT
iajs-4032	106	16	then	then	ADV
iajs-4032	106	17	(	(	PUNCT
iajs-4032	106	18	-v	-v	X
iajs-4032	106	19	)	)	PUNCT
iajs-4032	106	20	=	=	SYM
iajs-4032	106	21	1	1	NUM
iajs-4032	106	22	&	&	CCONJ
iajs-4032	106	23	(	(	PUNCT
iajs-4032	106	24	-u	-u	PROPN
iajs-4032	106	25	)	)	PUNCT
iajs-4032	106	26	=	=	SYM
iajs-4032	106	27	2	2	NUM
iajs-4032	106	28	,	,	PUNCT
iajs-4032	106	29	such	such	ADJ
iajs-4032	106	30	that	that	SCONJ
iajs-4032	106	31	there	there	PRON
iajs-4032	106	32	are	be	VERB
iajs-4032	106	33	two	two	NUM
iajs-4032	106	34	sets	set	NOUN
iajs-4032	106	35	:	:	PUNCT
iajs-4032	106	36	1	1	NUM
iajs-4032	106	37	,	,	PUNCT
iajs-4032	106	38	2	2	NUM
iajs-4032	106	39	in	in	ADP
iajs-4032	106	40	order	order	NOUN
iajs-4032	106	41	to	to	PART
iajs-4032	106	42	satisfy	satisfy	VERB
iajs-4032	106	43	∈	∈	PROPN
iajs-4032	106	44	1	1	NUM
iajs-4032	106	45	)	)	PUNCT
iajs-4032	106	46	.	.	PUNCT
iajs-4032	107	1	also	also	ADV
iajs-4032	107	2	∈	∈	PROPN
iajs-4032	107	3	(	(	PUNCT
iajs-4032	107	4	2	2	NUM
iajs-4032	107	5	)	)	PUNCT
iajs-4032	107	6	such	such	ADJ
iajs-4032	107	7	that	that	SCONJ
iajs-4032	107	8	∈	∈	PROPN
iajs-4032	107	9	1	1	NUM
iajs-4032	107	10	2	2	NUM
iajs-4032	107	11	)	)	PUNCT
iajs-4032	107	12	.	.	PUNCT
iajs-4032	108	1	conversely	conversely	ADV
iajs-4032	108	2	,	,	PUNCT
iajs-4032	108	3	let	let	AUX
iajs-4032	108	4	be	be	AUX
iajs-4032	108	5	two	two	NUM
iajs-4032	108	6	different	different	ADJ
iajs-4032	108	7	components	component	NOUN
iajs-4032	108	8	∈	∈	PROPN
iajs-4032	108	9	so	so	ADV
iajs-4032	108	10	thus	thus	ADV
iajs-4032	108	11	,	,	PUNCT
iajs-4032	108	12	two	two	NUM
iajs-4032	108	13	c	c	NOUN
iajs-4032	108	14	sets	set	VERB
iajs-4032	108	15	1	1	NUM
iajs-4032	108	16	&	&	CCONJ
iajs-4032	108	17	2	2	NUM
iajs-4032	108	18	are	be	AUX
iajs-4032	108	19	achieved	achieve	VERB
iajs-4032	108	20	:	:	PUNCT
iajs-4032	108	21	(	(	PUNCT
iajs-4032	108	22	1	1	NUM
iajs-4032	108	23	)	)	PUNCT
iajs-4032	108	24	.	.	PUNCT
iajs-4032	109	1	(	(	PUNCT
iajs-4032	109	2	2	2	NUM
iajs-4032	109	3	)	)	PUNCT
iajs-4032	109	4	,	,	PUNCT
iajs-4032	109	5	then	then	ADV
iajs-4032	109	6	there	there	PRON
iajs-4032	109	7	exists	exist	VERB
iajs-4032	109	8	sets	set	NOUN
iajs-4032	109	9	(	(	PUNCT
iajs-4032	109	10	1	1	NUM
iajs-4032	109	11	)	)	PUNCT
iajs-4032	109	12	and	and	CCONJ
iajs-4032	109	13	(	(	PUNCT
iajs-4032	109	14	)	)	PUNCT
iajs-4032	109	15	,	,	PUNCT
iajs-4032	109	16	when	when	SCONJ
iajs-4032	109	17	∈	∈	PROPN
iajs-4032	109	18	(	(	PUNCT
iajs-4032	109	19	f2	f2	PROPN
iajs-4032	109	20	)	)	PUNCT
iajs-4032	109	21	(	(	PUNCT
iajs-4032	109	22	1	1	NUM
iajs-4032	109	23	)	)	PUNCT
iajs-4032	109	24	,	,	PUNCT
iajs-4032	109	25	(	(	PUNCT
iajs-4032	109	26	1	1	X
iajs-4032	109	27	)	)	PUNCT
iajs-4032	109	28	(	(	PUNCT
iajs-4032	109	29	)	)	PUNCT
iajs-4032	109	30	such	such	ADJ
iajs-4032	109	31	that	that	SCONJ
iajs-4032	109	32	(	(	PUNCT
iajs-4032	109	33	)	)	PUNCT
iajs-4032	109	34	=	=	SYM
iajs-4032	109	35	u	u	NOUN
iajs-4032	109	36	and	and	CCONJ
iajs-4032	109	37	(	(	PUNCT
iajs-4032	109	38	1	1	X
iajs-4032	109	39	)	)	PUNCT
iajs-4032	109	40	=	=	PUNCT
iajs-4032	109	41	v.	v.	ADP
iajs-4032	109	42	3.8	3.8	NUM
iajs-4032	109	43	.	.	PUNCT
iajs-4032	110	1	proposition	proposition	NOUN
iajs-4032	110	2	assume	assume	VERB
iajs-4032	110	3	{	{	PUNCT
iajs-4032	110	4	}	}	PUNCT
iajs-4032	110	5	is	be	AUX
iajs-4032	110	6	an	an	DET
iajs-4032	110	7	c	c	NOUN
iajs-4032	110	8	set	set	NOUN
iajs-4032	110	9	,	,	PUNCT
iajs-4032	110	10	for	for	ADP
iajs-4032	110	11	each	each	DET
iajs-4032	110	12	∈	∈	NOUN
iajs-4032	110	13	hence	hence	ADV
iajs-4032	110	14	,	,	PUNCT
iajs-4032	111	1	i	i	PRON
iajs-4032	111	2	)	)	PUNCT
iajs-4032	111	3	is	be	AUX
iajs-4032	111	4	an	an	DET
iajs-4032	111	5	space	space	NOUN
iajs-4032	111	6	.	.	PUNCT
iajs-4032	112	1	proof	proof	NOUN
iajs-4032	112	2	:	:	PUNCT
iajs-4032	112	3	consider	consider	VERB
iajs-4032	112	4	are	be	AUX
iajs-4032	112	5	two	two	NUM
iajs-4032	112	6	different	different	ADJ
iajs-4032	112	7	components	component	NOUN
iajs-4032	112	8	belong	belong	VERB
iajs-4032	112	9	to	to	ADP
iajs-4032	112	10	,	,	PUNCT
iajs-4032	112	11	since	since	SCONJ
iajs-4032	112	12	{	{	PUNCT
iajs-4032	112	13	}	}	PUNCT
iajs-4032	112	14	,	,	PUNCT
iajs-4032	112	15	{	{	PUNCT
iajs-4032	112	16	}	}	PUNCT
iajs-4032	112	17	are	be	AUX
iajs-4032	112	18	c	c	NOUN
iajs-4032	112	19	sets	set	NOUN
iajs-4032	112	20	,	,	PUNCT
iajs-4032	112	21	then	then	ADV
iajs-4032	112	22	(	(	PUNCT
iajs-4032	112	23	-	-	PUNCT
iajs-4032	112	24	{	{	PUNCT
iajs-4032	112	25	}	}	PUNCT
iajs-4032	112	26	)	)	PUNCT
iajs-4032	112	27	.	.	PUNCT
iajs-4032	113	1	also	also	ADV
iajs-4032	113	2	,	,	PUNCT
iajs-4032	113	3	(	(	PUNCT
iajs-4032	113	4	-	-	PUNCT
iajs-4032	113	5	{	{	PUNCT
iajs-4032	113	6	}	}	PUNCT
iajs-4032	113	7	)	)	PUNCT
iajs-4032	113	8	is	be	AUX
iajs-4032	113	9	o	o	NOUN
iajs-4032	113	10	sets	set	NOUN
iajs-4032	113	11	such	such	ADJ
iajs-4032	113	12	that	that	SCONJ
iajs-4032	113	13	there	there	PRON
iajs-4032	113	14	are	be	VERB
iajs-4032	113	15	o	o	NOUN
iajs-4032	113	16	sets	set	NOUN
iajs-4032	113	17	,	,	PUNCT
iajs-4032	113	18	u	u	NOUN
iajs-4032	113	19	&	&	CCONJ
iajs-4032	113	20	v	v	NOUN
iajs-4032	113	21	,	,	PUNCT
iajs-4032	113	22	since	since	SCONJ
iajs-4032	113	23	u	u	NOUN
iajs-4032	113	24	=	=	PUNCT
iajs-4032	113	25	(	(	PUNCT
iajs-4032	113	26	-	-	PUNCT
iajs-4032	113	27	{	{	PUNCT
iajs-4032	113	28	}	}	PUNCT
iajs-4032	113	29	)	)	PUNCT
iajs-4032	113	30	.	.	PUNCT
iajs-4032	114	1	also	also	ADV
iajs-4032	114	2	,	,	PUNCT
iajs-4032	114	3	v	v	X
iajs-4032	114	4	=	=	SYM
iajs-4032	114	5	(	(	PUNCT
iajs-4032	114	6	-	-	PUNCT
iajs-4032	114	7	{	{	PUNCT
iajs-4032	114	8	}	}	PUNCT
iajs-4032	114	9	)	)	PUNCT
iajs-4032	114	10	;	;	PUNCT
iajs-4032	114	11	therefore	therefore	ADV
iajs-4032	114	12	,	,	PUNCT
iajs-4032	114	13	∈	∈	PROPN
iajs-4032	114	14	(	(	PUNCT
iajs-4032	114	15	u	u	NOUN
iajs-4032	114	16	–	–	PUNCT
iajs-4032	114	17	v	v	NOUN
iajs-4032	114	18	)	)	PUNCT
iajs-4032	114	19	,	,	PUNCT
iajs-4032	114	20	∈	∈	PROPN
iajs-4032	114	21	(	(	PUNCT
iajs-4032	114	22	v	v	NOUN
iajs-4032	114	23	–	–	PUNCT
iajs-4032	114	24	u	u	NOUN
iajs-4032	114	25	)	)	PUNCT
iajs-4032	114	26	.	.	PUNCT
iajs-4032	115	1	so	so	ADV
iajs-4032	115	2	,	,	PUNCT
iajs-4032	115	3	i	i	PRON
iajs-4032	115	4	)	)	PUNCT
iajs-4032	115	5	is	be	AUX
iajs-4032	115	6	an	an	DET
iajs-4032	115	7	space	space	NOUN
iajs-4032	115	8	.	.	PUNCT
iajs-4032	116	1	3.9	3.9	NUM
iajs-4032	116	2	.	.	PUNCT
iajs-4032	116	3	definition	definition	NOUN
iajs-4032	116	4	the	the	DET
iajs-4032	116	5	ideal	ideal	ADJ
iajs-4032	116	6	topological	topological	ADJ
iajs-4032	116	7	space	space	NOUN
iajs-4032	116	8	,	,	PUNCT
iajs-4032	116	9	i	i	PRON
iajs-4032	116	10	)	)	PUNCT
iajs-4032	116	11	is	be	AUX
iajs-4032	116	12	said	say	VERB
iajs-4032	116	13	to	to	PART
iajs-4032	116	14	be	be	AUX
iajs-4032	116	15	an	an	DET
iajs-4032	116	16	space	space	NOUN
iajs-4032	116	17	if	if	SCONJ
iajs-4032	116	18	for	for	ADP
iajs-4032	116	19	each	each	DET
iajs-4032	116	20	pair	pair	NOUN
iajs-4032	116	21	of	of	ADP
iajs-4032	116	22	different	different	ADJ
iajs-4032	116	23	point	point	NOUN
iajs-4032	116	24	∈	∈	PROPN
iajs-4032	116	25	,	,	PUNCT
iajs-4032	116	26	there	there	PRON
iajs-4032	116	27	are	be	VERB
iajs-4032	116	28	two	two	NUM
iajs-4032	116	29	disjoint	disjoint	NOUN
iajs-4032	116	30	(	(	PUNCT
iajs-4032	116	31	)	)	PUNCT
iajs-4032	116	32	sets	set	VERB
iajs-4032	116	33	u	u	NOUN
iajs-4032	116	34	and	and	CCONJ
iajs-4032	116	35	v	v	NOUN
iajs-4032	116	36	,	,	PUNCT
iajs-4032	116	37	whenever	whenever	SCONJ
iajs-4032	116	38	∈	∈	PROPN
iajs-4032	116	39	∈	∈	PROPN
iajs-4032	116	40	v	v	NOUN
iajs-4032	116	41	and	and	CCONJ
iajs-4032	116	42	u	u	NOUN
iajs-4032	116	43	v	v	NOUN
iajs-4032	116	44	=	=	PUNCT
iajs-4032	116	45	.	.	PUNCT
iajs-4032	117	1	3.10	3.10	NUM
iajs-4032	117	2	.	.	PUNCT
iajs-4032	118	1	proposition	proposition	NOUN
iajs-4032	118	2	suppose	suppose	VERB
iajs-4032	118	3	is	be	AUX
iajs-4032	118	4	–	–	PUNCT
iajs-4032	118	5	space	space	NOUN
iajs-4032	118	6	,	,	PUNCT
iajs-4032	118	7	such	such	ADJ
iajs-4032	118	8	that	that	SCONJ
iajs-4032	118	9	,	,	PUNCT
iajs-4032	118	10	i	i	PROPN
iajs-4032	118	11	)	)	PUNCT
iajs-4032	118	12	will	will	AUX
iajs-4032	118	13	be	be	AUX
iajs-4032	118	14	space	space	NOUN
iajs-4032	118	15	.	.	PUNCT
iajs-4032	119	1	proof	proof	NOUN
iajs-4032	119	2	:	:	PUNCT
iajs-4032	119	3	suppose	suppose	VERB
iajs-4032	119	4	are	be	AUX
iajs-4032	119	5	two	two	NUM
iajs-4032	119	6	different	different	ADJ
iajs-4032	119	7	elements	element	NOUN
iajs-4032	119	8	belong	belong	VERB
iajs-4032	119	9	to	to	ADP
iajs-4032	119	10	and	and	CCONJ
iajs-4032	119	11	since	since	ADV
iajs-4032	119	12	)	)	PUNCT
iajs-4032	119	13	is	be	AUX
iajs-4032	119	14	–	–	PUNCT
iajs-4032	119	15	space	space	NOUN
iajs-4032	119	16	such	such	ADJ
iajs-4032	119	17	that	that	SCONJ
iajs-4032	119	18	it	it	PRON
iajs-4032	119	19	is	be	AUX
iajs-4032	119	20	two	two	NUM
iajs-4032	119	21	open	open	ADJ
iajs-4032	119	22	sets	set	NOUN
iajs-4032	119	23	u	u	NOUN
iajs-4032	119	24	and	and	CCONJ
iajs-4032	119	25	v	v	NOUN
iajs-4032	119	26	,	,	PUNCT
iajs-4032	119	27	therefore	therefore	ADV
iajs-4032	119	28	∈	∈	PROPN
iajs-4032	119	29	∈	∈	PROPN
iajs-4032	119	30	v	v	NOUN
iajs-4032	119	31	and	and	CCONJ
iajs-4032	119	32	u	u	NOUN
iajs-4032	119	33	v	v	NOUN
iajs-4032	119	34	=	=	PUNCT
iajs-4032	119	35	since	since	SCONJ
iajs-4032	119	36	⊆	⊆	NUM
iajs-4032	119	37	(	(	PUNCT
iajs-4032	119	38	)	)	PUNCT
iajs-4032	119	39	such	such	ADJ
iajs-4032	119	40	that	that	SCONJ
iajs-4032	119	41	u	u	NOUN
iajs-4032	119	42	,	,	PUNCT
iajs-4032	119	43	v	v	NOUN
iajs-4032	119	44	are	be	AUX
iajs-4032	119	45	(	(	PUNCT
iajs-4032	119	46	)	)	PUNCT
iajs-4032	119	47	sets	set	NOUN
iajs-4032	119	48	satisfying	satisfy	VERB
iajs-4032	119	49	the	the	DET
iajs-4032	119	50	condition	condition	NOUN
iajs-4032	119	51	∈	∈	PROPN
iajs-4032	119	52	∈	∈	PROPN
iajs-4032	119	53	v	v	NOUN
iajs-4032	119	54	,	,	PUNCT
iajs-4032	119	55	and	and	CCONJ
iajs-4032	119	56	u	u	NOUN
iajs-4032	119	57	v	v	NOUN
iajs-4032	119	58	=	=	PUNCT
iajs-4032	119	59	such	such	ADJ
iajs-4032	119	60	that	that	SCONJ
iajs-4032	119	61	,	,	PUNCT
iajs-4032	119	62	i	i	PROPN
iajs-4032	119	63	)	)	PUNCT
iajs-4032	119	64	will	will	AUX
iajs-4032	119	65	be	be	AUX
iajs-4032	119	66	–	–	PUNCT
iajs-4032	119	67	space	space	NOUN
iajs-4032	119	68	.	.	PUNCT
iajs-4032	120	1	3.11	3.11	NUM
iajs-4032	120	2	.	.	PUNCT
iajs-4032	120	3	proposition	proposition	NOUN
iajs-4032	120	4	when	when	SCONJ
iajs-4032	120	5	the	the	DET
iajs-4032	120	6	space	space	NOUN
iajs-4032	120	7	,	,	PUNCT
iajs-4032	120	8	i	i	PRON
iajs-4032	120	9	)	)	PUNCT
iajs-4032	120	10	is	be	AUX
iajs-4032	120	11	an	an	DET
iajs-4032	120	12	space	space	NOUN
iajs-4032	120	13	implying	imply	VERB
iajs-4032	120	14	an	an	DET
iajs-4032	120	15	space	space	NOUN
iajs-4032	120	16	.	.	PUNCT
iajs-4032	121	1	proof	proof	NOUN
iajs-4032	121	2	:	:	PUNCT
iajs-4032	121	3	consider	consider	VERB
iajs-4032	121	4	be	be	AUX
iajs-4032	121	5	two	two	NUM
iajs-4032	121	6	different	different	ADJ
iajs-4032	121	7	elements	element	NOUN
iajs-4032	121	8	∈	∈	PROPN
iajs-4032	121	9	since	since	SCONJ
iajs-4032	121	10	(	(	PUNCT
iajs-4032	121	11	,	,	PUNCT
iajs-4032	121	12	i	i	NOUN
iajs-4032	121	13	)	)	PUNCT
iajs-4032	121	14	is	be	AUX
iajs-4032	121	15	2	2	NUM
iajs-4032	121	16	–	–	PUNCT
iajs-4032	121	17	space	space	NOUN
iajs-4032	121	18	.	.	PUNCT
iajs-4032	122	1	u	u	NOUN
iajs-4032	122	2	and	and	CCONJ
iajs-4032	122	3	v	v	NOUN
iajs-4032	122	4	are	be	AUX
iajs-4032	122	5	(	(	PUNCT
iajs-4032	122	6	)	)	PUNCT
iajs-4032	122	7	sets	set	NOUN
iajs-4032	122	8	,	,	PUNCT
iajs-4032	122	9	therefore	therefore	ADV
iajs-4032	122	10	∈	∈	PROPN
iajs-4032	122	11	u	u	NOUN
iajs-4032	122	12	and	and	CCONJ
iajs-4032	122	13	∈	∈	PROPN
iajs-4032	122	14	v	v	NOUN
iajs-4032	122	15	&	&	CCONJ
iajs-4032	122	16	u	u	NOUN
iajs-4032	122	17	v	v	PROPN
iajs-4032	122	18	=	=	PUNCT
iajs-4032	122	19	,	,	PUNCT
iajs-4032	122	20	so	so	CCONJ
iajs-4032	122	21	there	there	PRON
iajs-4032	122	22	exists	exist	VERB
iajs-4032	122	23	(	(	PUNCT
iajs-4032	122	24	)	)	PUNCT
iajs-4032	122	25	sets	set	VERB
iajs-4032	122	26	u	u	NOUN
iajs-4032	122	27	and	and	CCONJ
iajs-4032	122	28	v	v	NOUN
iajs-4032	122	29	in	in	ADP
iajs-4032	122	30	order	order	NOUN
iajs-4032	122	31	to	to	PART
iajs-4032	122	32	satisfy	satisfy	VERB
iajs-4032	122	33	∈	∈	PROPN
iajs-4032	122	34	(	(	PUNCT
iajs-4032	122	35	u	u	NOUN
iajs-4032	122	36	v	v	NOUN
iajs-4032	122	37	)	)	PUNCT
iajs-4032	122	38	and	and	CCONJ
iajs-4032	122	39	∈	∈	PROPN
iajs-4032	122	40	(	(	PUNCT
iajs-4032	122	41	v	v	NOUN
iajs-4032	122	42	u	u	NOUN
iajs-4032	122	43	)	)	PUNCT
iajs-4032	122	44	.	.	PUNCT
iajs-4032	123	1	so	so	ADV
iajs-4032	123	2	(	(	PUNCT
iajs-4032	123	3	,	,	PUNCT
iajs-4032	123	4	i	i	PROPN
iajs-4032	123	5	)	)	PUNCT
iajs-4032	123	6	is	be	AUX
iajs-4032	123	7	an	an	DET
iajs-4032	123	8	space	space	NOUN
iajs-4032	123	9	.	.	PUNCT
iajs-4032	124	1	in	in	ADP
iajs-4032	124	2	proposition	proposition	NOUN
iajs-4032	124	3	(	(	PUNCT
iajs-4032	124	4	3.15	3.15	NUM
iajs-4032	124	5	)	)	PUNCT
iajs-4032	124	6	,	,	PUNCT
iajs-4032	124	7	the	the	DET
iajs-4032	124	8	opposite	opposite	ADJ
iajs-4032	124	9	meaning	meaning	NOUN
iajs-4032	124	10	is	be	AUX
iajs-4032	124	11	not	not	PART
iajs-4032	124	12	achieved	achieve	VERB
iajs-4032	124	13	as	as	SCONJ
iajs-4032	124	14	explained	explain	VERB
iajs-4032	124	15	by	by	ADP
iajs-4032	124	16	the	the	DET
iajs-4032	124	17	below	below	ADJ
iajs-4032	124	18	example	example	NOUN
iajs-4032	124	19	.	.	PUNCT
iajs-4032	125	1	ihjpas	ihjpas	PROPN
iajs-4032	125	2	.	.	PUNCT
iajs-4032	126	1	2025,38(4	2025,38(4	NOUN
iajs-4032	126	2	)	)	PUNCT
iajs-4032	126	3	7	7	NUM
iajs-4032	126	4	example	example	NOUN
iajs-4032	126	5	in	in	ADP
iajs-4032	126	6	the	the	DET
iajs-4032	126	7	(	(	PUNCT
iajs-4032	126	8	,	,	PUNCT
iajs-4032	126	9	i	i	PROPN
iajs-4032	126	10	)	)	PUNCT
iajs-4032	126	11	,	,	PUNCT
iajs-4032	126	12	let	let	VERB
iajs-4032	126	13	=	=	PRON
iajs-4032	126	14	{	{	PUNCT
iajs-4032	126	15	,	,	PUNCT
iajs-4032	126	16	}	}	PUNCT
iajs-4032	126	17	,	,	PUNCT
iajs-4032	126	18	=	=	PRON
iajs-4032	126	19	{	{	PUNCT
iajs-4032	126	20	,	,	PUNCT
iajs-4032	126	21	,	,	PUNCT
iajs-4032	126	22	,	,	PUNCT
iajs-4032	126	23	}	}	PUNCT
iajs-4032	126	24	}	}	PUNCT
iajs-4032	126	25	i	i	PRON
iajs-4032	126	26	=	=	PUNCT
iajs-4032	126	27	{	{	PUNCT
iajs-4032	126	28	}	}	PUNCT
iajs-4032	126	29	,	,	PUNCT
iajs-4032	126	30	)	)	PUNCT
iajs-4032	127	1	=	=	NOUN
iajs-4032	127	2	{	{	PUNCT
iajs-4032	127	3	,	,	PUNCT
iajs-4032	127	4	,	,	PUNCT
iajs-4032	127	5	,	,	PUNCT
iajs-4032	127	6	,	,	PUNCT
iajs-4032	127	7	}	}	PUNCT
iajs-4032	127	8	}	}	PUNCT
iajs-4032	127	9	notice	notice	VERB
iajs-4032	127	10	that	that	SCONJ
iajs-4032	127	11	(	(	PUNCT
iajs-4032	127	12	,	,	PUNCT
iajs-4032	127	13	i	i	NOUN
iajs-4032	127	14	)	)	PUNCT
iajs-4032	127	15	is	be	AUX
iajs-4032	127	16	an	an	DET
iajs-4032	127	17	–	–	PUNCT
iajs-4032	127	18	space	space	NOUN
iajs-4032	127	19	,	,	PUNCT
iajs-4032	127	20	while	while	SCONJ
iajs-4032	127	21	it	it	PRON
iajs-4032	127	22	is	be	AUX
iajs-4032	127	23	not	not	PART
iajs-4032	127	24	an	an	DET
iajs-4032	127	25	space	space	NOUN
iajs-4032	127	26	.	.	PUNCT
iajs-4032	128	1	3.12	3.12	NUM
iajs-4032	128	2	.	.	X
iajs-4032	128	3	remark	remark	PROPN
iajs-4032	128	4	suppose	suppose	VERB
iajs-4032	128	5	(	(	PUNCT
iajs-4032	128	6	)	)	PUNCT
iajs-4032	128	7	is	be	AUX
iajs-4032	128	8	a	a	DET
iajs-4032	128	9	–	–	PUNCT
iajs-4032	128	10	space	space	NOUN
iajs-4032	128	11	for	for	ADP
iajs-4032	128	12	each	each	PRON
iajs-4032	128	13	=	=	SYM
iajs-4032	128	14	{	{	PUNCT
iajs-4032	128	15	0	0	NUM
iajs-4032	128	16	,	,	PUNCT
iajs-4032	128	17	1	1	NUM
iajs-4032	128	18	,	,	PUNCT
iajs-4032	128	19	2	2	NUM
iajs-4032	128	20	}	}	PUNCT
iajs-4032	128	21	,	,	PUNCT
iajs-4032	128	22	such	such	ADJ
iajs-4032	128	23	that	that	SCONJ
iajs-4032	128	24	the	the	DET
iajs-4032	128	25	ideal	ideal	ADJ
iajs-4032	128	26	topological	topological	ADJ
iajs-4032	128	27	space	space	NOUN
iajs-4032	128	28	,	,	PUNCT
iajs-4032	128	29	i	i	PRON
iajs-4032	128	30	)	)	PUNCT
iajs-4032	128	31	is	be	AUX
iajs-4032	128	32	–	–	PUNCT
iajs-4032	128	33	–	–	PUNCT
iajs-4032	128	34	space	space	NOUN
iajs-4032	128	35	,	,	PUNCT
iajs-4032	128	36	=	=	PUNCT
iajs-4032	128	37	{	{	PUNCT
iajs-4032	128	38	0	0	NUM
iajs-4032	128	39	,	,	PUNCT
iajs-4032	128	40	1	1	NUM
iajs-4032	128	41	,	,	PUNCT
iajs-4032	128	42	2	2	NUM
iajs-4032	128	43	}	}	PUNCT
iajs-4032	128	44	.	.	PUNCT
iajs-4032	129	1	notice	notice	VERB
iajs-4032	129	2	that	that	SCONJ
iajs-4032	129	3	the	the	DET
iajs-4032	129	4	opposite	opposite	NOUN
iajs-4032	129	5	in	in	ADP
iajs-4032	129	6	the	the	DET
iajs-4032	129	7	remark	remark	NOUN
iajs-4032	129	8	(	(	PUNCT
iajs-4032	129	9	3.17	3.17	NUM
iajs-4032	129	10	)	)	PUNCT
iajs-4032	129	11	is	be	AUX
iajs-4032	129	12	generally	generally	ADV
iajs-4032	129	13	false	false	ADJ
iajs-4032	129	14	as	as	SCONJ
iajs-4032	129	15	illustrated	illustrate	VERB
iajs-4032	129	16	in	in	ADP
iajs-4032	129	17	the	the	DET
iajs-4032	129	18	below	below	ADJ
iajs-4032	129	19	example	example	NOUN
iajs-4032	129	20	.	.	PUNCT
iajs-4032	130	1	example	example	NOUN
iajs-4032	130	2	consider	consider	VERB
iajs-4032	130	3	=	=	PRON
iajs-4032	130	4	{	{	PUNCT
iajs-4032	130	5	,	,	PUNCT
iajs-4032	130	6	}	}	PUNCT
iajs-4032	130	7	,	,	PUNCT
iajs-4032	130	8	=	=	PRON
iajs-4032	130	9	{	{	PUNCT
iajs-4032	130	10	,	,	PUNCT
iajs-4032	130	11	i	i	PRON
iajs-4032	130	12	=	=	SYM
iajs-4032	130	13	p	p	X
iajs-4032	130	14	(	(	PUNCT
iajs-4032	130	15	)	)	PUNCT
iajs-4032	130	16	&	&	CCONJ
iajs-4032	130	17	(	(	PUNCT
iajs-4032	130	18	)	)	PUNCT
iajs-4032	131	1	=	=	SYM
iajs-4032	131	2	p	p	X
iajs-4032	131	3	(	(	PUNCT
iajs-4032	131	4	)	)	PUNCT
iajs-4032	131	5	.	.	PUNCT
iajs-4032	132	1	so	so	ADV
iajs-4032	132	2	it	it	PRON
iajs-4032	132	3	is	be	AUX
iajs-4032	132	4	evident	evident	ADJ
iajs-4032	132	5	that	that	SCONJ
iajs-4032	132	6	the	the	DET
iajs-4032	132	7	space	space	NOUN
iajs-4032	132	8	(	(	PUNCT
iajs-4032	132	9	,	,	PUNCT
iajs-4032	132	10	i	i	PROPN
iajs-4032	132	11	)	)	PUNCT
iajs-4032	132	12	is	be	AUX
iajs-4032	132	13	an	an	DET
iajs-4032	132	14	–	–	PUNCT
iajs-4032	132	15	space	space	NOUN
iajs-4032	132	16	for	for	ADP
iajs-4032	132	17	each	each	PRON
iajs-4032	132	18	=	=	SYM
iajs-4032	132	19	{	{	PUNCT
iajs-4032	132	20	0	0	NUM
iajs-4032	132	21	,	,	PUNCT
iajs-4032	132	22	1	1	NUM
iajs-4032	132	23	,	,	PUNCT
iajs-4032	132	24	2	2	NUM
iajs-4032	132	25	}	}	PUNCT
iajs-4032	132	26	.	.	PUNCT
iajs-4032	133	1	hence	hence	ADV
iajs-4032	133	2	,	,	PUNCT
iajs-4032	133	3	it	it	PRON
iajs-4032	133	4	is	be	AUX
iajs-4032	133	5	not	not	PART
iajs-4032	133	6	a	a	DET
iajs-4032	133	7	–	–	PUNCT
iajs-4032	133	8	space	space	NOUN
iajs-4032	133	9	for	for	ADP
iajs-4032	133	10	each	each	PRON
iajs-4032	133	11	=	=	SYM
iajs-4032	133	12	{	{	PUNCT
iajs-4032	133	13	0	0	NUM
iajs-4032	133	14	,	,	PUNCT
iajs-4032	133	15	1	1	NUM
iajs-4032	133	16	,	,	PUNCT
iajs-4032	133	17	2	2	NUM
iajs-4032	133	18	}	}	PUNCT
iajs-4032	133	19	.	.	PUNCT
iajs-4032	134	1	the	the	DET
iajs-4032	134	2	following	follow	VERB
iajs-4032	134	3	diagram	diagram	NOUN
iajs-4032	134	4	elucidates	elucidate	VERB
iajs-4032	134	5	the	the	DET
iajs-4032	134	6	interrelationships	interrelationship	NOUN
iajs-4032	134	7	between	between	ADP
iajs-4032	134	8	the	the	DET
iajs-4032	134	9	preceding	precede	VERB
iajs-4032	134	10	concepts	concept	NOUN
iajs-4032	134	11	:	:	PUNCT
iajs-4032	134	12	diagram	diagram	NOUN
iajs-4032	134	13	1	1	NUM
iajs-4032	134	14	.the	.the	NOUN
iajs-4032	134	15	relationships	relationship	NOUN
iajs-4032	134	16	among	among	ADP
iajs-4032	134	17	–	–	PUNCT
iajs-4032	134	18	space	space	NOUN
iajs-4032	134	19	&	&	CCONJ
iajs-4032	134	20	–	–	PUNCT
iajs-4032	134	21	spaces	space	NOUN
iajs-4032	134	22	.	.	PUNCT
iajs-4032	135	1	4.conclusion	4.conclusion	NUM
iajs-4032	135	2	in	in	ADP
iajs-4032	135	3	this	this	DET
iajs-4032	135	4	article	article	NOUN
iajs-4032	135	5	,	,	PUNCT
iajs-4032	135	6	we	we	PRON
iajs-4032	135	7	created	create	VERB
iajs-4032	135	8	and	and	CCONJ
iajs-4032	135	9	introduced	introduce	VERB
iajs-4032	135	10	a	a	DET
iajs-4032	135	11	set	set	NOUN
iajs-4032	135	12	[	[	X
iajs-4032	135	13	namely	namely	ADV
iajs-4032	135	14	an	an	DET
iajs-4032	135	15	-open	-open	NOUN
iajs-4032	135	16	]	]	PUNCT
iajs-4032	135	17	by	by	ADP
iajs-4032	135	18	using	use	VERB
iajs-4032	135	19	the	the	DET
iajs-4032	135	20	ideal	ideal	ADJ
iajs-4032	135	21	topological	topological	ADJ
iajs-4032	135	22	space	space	NOUN
iajs-4032	135	23	and	and	CCONJ
iajs-4032	135	24	also	also	ADV
iajs-4032	135	25	studied	study	VERB
iajs-4032	135	26	many	many	ADJ
iajs-4032	135	27	properties	property	NOUN
iajs-4032	135	28	of	of	ADP
iajs-4032	135	29	this	this	DET
iajs-4032	135	30	set	set	NOUN
iajs-4032	135	31	.	.	PUNCT
iajs-4032	136	1	the	the	DET
iajs-4032	136	2	separation	separation	NOUN
iajs-4032	136	3	of	of	ADP
iajs-4032	136	4	axioms	axiom	NOUN
iajs-4032	136	5	and	and	CCONJ
iajs-4032	136	6	relationships	relationship	NOUN
iajs-4032	136	7	between	between	ADP
iajs-4032	136	8	these	these	DET
iajs-4032	136	9	axioms	axiom	NOUN
iajs-4032	136	10	were	be	AUX
iajs-4032	136	11	also	also	ADV
iajs-4032	136	12	studied	study	VERB
iajs-4032	136	13	,	,	PUNCT
iajs-4032	136	14	and	and	CCONJ
iajs-4032	136	15	this	this	PRON
iajs-4032	136	16	has	have	AUX
iajs-4032	136	17	been	be	AUX
iajs-4032	136	18	reinforced	reinforce	VERB
iajs-4032	136	19	with	with	ADP
iajs-4032	136	20	examples	example	NOUN
iajs-4032	136	21	and	and	CCONJ
iajs-4032	136	22	the	the	DET
iajs-4032	136	23	converse	converse	NOUN
iajs-4032	136	24	of	of	ADP
iajs-4032	136	25	these	these	DET
iajs-4032	136	26	properties	property	NOUN
iajs-4032	136	27	has	have	AUX
iajs-4032	136	28	also	also	ADV
iajs-4032	136	29	been	be	AUX
iajs-4032	136	30	discussed	discuss	VERB
iajs-4032	136	31	.	.	PUNCT
iajs-4032	137	1	in	in	ADP
iajs-4032	137	2	addition	addition	NOUN
iajs-4032	137	3	,	,	PUNCT
iajs-4032	137	4	we	we	PRON
iajs-4032	137	5	will	will	AUX
iajs-4032	137	6	work	work	VERB
iajs-4032	137	7	on	on	ADP
iajs-4032	137	8	other	other	ADJ
iajs-4032	137	9	different	different	ADJ
iajs-4032	137	10	topological	topological	ADJ
iajs-4032	137	11	properties	property	NOUN
iajs-4032	137	12	of	of	ADP
iajs-4032	137	13	this	this	DET
iajs-4032	137	14	set	set	NOUN
iajs-4032	137	15	,	,	PUNCT
iajs-4032	137	16	such	such	ADJ
iajs-4032	137	17	as	as	ADP
iajs-4032	137	18	convergence	convergence	NOUN
iajs-4032	137	19	and	and	CCONJ
iajs-4032	137	20	compactness	compactness	NOUN
iajs-4032	137	21	,	,	PUNCT
iajs-4032	137	22	using	use	VERB
iajs-4032	137	23	the	the	DET
iajs-4032	137	24	set	set	NOUN
iajs-4032	137	25	under	under	ADP
iajs-4032	137	26	study	study	NOUN
iajs-4032	137	27	.	.	PUNCT
iajs-4032	138	1	acknowledgments	acknowledgment	NOUN
iajs-4032	138	2	our	our	PRON
iajs-4032	138	3	researcher	researcher	NOUN
iajs-4032	138	4	extends	extend	VERB
iajs-4032	138	5	his	his	PRON
iajs-4032	138	6	sincere	sincere	ADJ
iajs-4032	138	7	thanks	thank	NOUN
iajs-4032	138	8	to	to	ADP
iajs-4032	138	9	the	the	DET
iajs-4032	138	10	editor	editor	NOUN
iajs-4032	138	11	and	and	CCONJ
iajs-4032	138	12	members	member	NOUN
iajs-4032	138	13	of	of	ADP
iajs-4032	138	14	the	the	DET
iajs-4032	138	15	preparatory	preparatory	PROPN
iajs-4032	138	16	committee	committee	NOUN
iajs-4032	138	17	of	of	ADP
iajs-4032	138	18	the	the	DET
iajs-4032	138	19	ibn	ibn	PROPN
iajs-4032	138	20	al	al	PROPN
iajs-4032	138	21	-	-	PUNCT
iajs-4032	138	22	haitham	haitham	PROPN
iajs-4032	138	23	journal	journal	PROPN
iajs-4032	138	24	of	of	ADP
iajs-4032	138	25	pure	pure	ADJ
iajs-4032	138	26	and	and	CCONJ
iajs-4032	138	27	applied	applied	ADJ
iajs-4032	138	28	sciences	science	NOUN
iajs-4032	138	29	.	.	PUNCT
iajs-4032	139	1	conflict	conflict	NOUN
iajs-4032	139	2	of	of	ADP
iajs-4032	139	3	interest	interest	NOUN
iajs-4032	139	4	there	there	PRON
iajs-4032	139	5	are	be	VERB
iajs-4032	139	6	no	no	DET
iajs-4032	139	7	conflicts	conflict	NOUN
iajs-4032	139	8	of	of	ADP
iajs-4032	139	9	interest	interest	NOUN
iajs-4032	139	10	.	.	PUNCT
iajs-4032	140	1	funding	funding	NOUN
iajs-4032	140	2	there	there	PRON
iajs-4032	140	3	is	be	VERB
iajs-4032	140	4	no	no	DET
iajs-4032	140	5	funding	funding	NOUN
iajs-4032	140	6	for	for	ADP
iajs-4032	140	7	the	the	DET
iajs-4032	140	8	article	article	NOUN
iajs-4032	140	9	references	reference	VERB
iajs-4032	140	10	1	1	NUM
iajs-4032	140	11	.	.	PUNCT
iajs-4032	140	12	kuratowski	kuratowski	PROPN
iajs-4032	140	13	k.	k.	PROPN
iajs-4032	140	14	topology	topology	PROPN
iajs-4032	140	15	.	.	PUNCT
iajs-4032	141	1	new	new	PROPN
iajs-4032	141	2	york	york	PROPN
iajs-4032	141	3	:	:	PUNCT
iajs-4032	141	4	acad	acad	PROPN
iajs-4032	141	5	.	.	PUNCT
iajs-4032	142	1	press	press	NOUN
iajs-4032	142	2	.	.	PUNCT
iajs-4032	143	1	1933	1933	NUM
iajs-4032	143	2	;	;	PUNCT
iajs-4032	143	3	i.	i.	PROPN
iajs-4032	143	4	2	2	NUM
iajs-4032	143	5	.	.	PUNCT
iajs-4032	144	1	dontchev	dontchev	PROPN
iajs-4032	144	2	,	,	PUNCT
iajs-4032	144	3	julian	julian	PROPN
iajs-4032	144	4	,	,	PUNCT
iajs-4032	144	5	maximilian	maximilian	ADJ
iajs-4032	144	6	ganster	ganster	NOUN
iajs-4032	144	7	,	,	PUNCT
iajs-4032	144	8	and	and	CCONJ
iajs-4032	144	9	david	david	PROPN
iajs-4032	144	10	rose	rise	VERB
iajs-4032	144	11	.	.	PUNCT
iajs-4032	145	1	ideal	ideal	PROPN
iajs-4032	145	2	resolvability.topology	resolvability.topology	NOUN
iajs-4032	145	3	and	and	CCONJ
iajs-4032	145	4	its	its	PRON
iajs-4032	145	5	applications	application	NOUN
iajs-4032	145	6	1999	1999	NUM
iajs-4032	145	7	:	:	PUNCT
iajs-4032	145	8	93.1	93.1	NUM
iajs-4032	145	9	;	;	PUNCT
iajs-4032	145	10	1	1	NUM
iajs-4032	145	11	-	-	SYM
iajs-4032	145	12	16	16	NUM
iajs-4032	145	13	,	,	PUNCT
iajs-4032	145	14	https://doi.org/10.1016/s0166-8641(97)00257-5	https://doi.org/10.1016/s0166-8641(97)00257-5	PROPN
iajs-4032	145	15	.	.	PUNCT
iajs-4032	146	1	3	3	X
iajs-4032	146	2	.	.	X
iajs-4032	146	3	mohammed	mohammed	PROPN
iajs-4032	146	4	,	,	PUNCT
iajs-4032	146	5	m.	m.	PROPN
iajs-4032	146	6	w.	w.	PROPN
iajs-4032	146	7	,	,	PUNCT
iajs-4032	146	8	a.	a.	PROPN
iajs-4032	146	9	a.	a.	PROPN
iajs-4032	146	10	mohammed	mohammed	PROPN
iajs-4032	146	11	.	.	PUNCT
iajs-4032	147	1	some	some	DET
iajs-4032	147	2	classes	class	NOUN
iajs-4032	147	3	in	in	ADP
iajs-4032	147	4	ideal	ideal	ADJ
iajs-4032	147	5	topological	topological	ADJ
iajs-4032	147	6	spaces.journal	spaces.journal	PROPN
iajs-4032	147	7	of	of	ADP
iajs-4032	147	8	https://doi.org/10.1016/s0166-8641(97)00257-5	https://doi.org/10.1016/s0166-8641(97)00257-5	PROPN
iajs-4032	147	9	ihjpas	ihjpas	PROPN
iajs-4032	147	10	.	.	PUNCT
iajs-4032	148	1	2025,38(4	2025,38(4	NUM
iajs-4032	148	2	)	)	PUNCT
iajs-4032	148	3	8	8	NUM
iajs-4032	148	4	education	education	NOUN
iajs-4032	148	5	&	&	CCONJ
iajs-4032	148	6	science	science	NOUN
iajs-4032	148	7	2023	2023	NUM
iajs-4032	148	8	:	:	PUNCT
iajs-4032	149	1	32.2	32.2	NUM
iajs-4032	149	2	.	.	PUNCT
iajs-4032	150	1	https://doi.org/10.33899/edusj.2023.137766.1318	https://doi.org/10.33899/edusj.2023.137766.1318	NUM
iajs-4032	150	2	4	4	NUM
iajs-4032	150	3	.	.	PUNCT
iajs-4032	150	4	auda	auda	PROPN
iajs-4032	150	5	,	,	PUNCT
iajs-4032	150	6	ghufran	ghufran	ADJ
iajs-4032	150	7	h.	h.	NOUN
iajs-4032	150	8	,	,	PUNCT
iajs-4032	150	9	hula	hula	ADJ
iajs-4032	150	10	m.	m.	NOUN
iajs-4032	150	11	salih	salih	PROPN
iajs-4032	150	12	.	.	PUNCT
iajs-4032	151	1	new	new	ADJ
iajs-4032	151	2	supra	supra	PROPN
iajs-4032	151	3	open	open	ADJ
iajs-4032	151	4	sets	set	NOUN
iajs-4032	151	5	with	with	ADP
iajs-4032	151	6	ideal	ideal	NOUN
iajs-4032	151	7	in	in	ADP
iajs-4032	151	8	supra	supra	PROPN
iajs-4032	151	9	topological	topological	ADJ
iajs-4032	151	10	space	space	NOUN
iajs-4032	151	11	.	.	PUNCT
iajs-4032	152	1	aip	aip	PROPN
iajs-4032	152	2	conference	conference	NOUN
iajs-4032	152	3	proceedings	proceeding	NOUN
iajs-4032	152	4	.	.	PUNCT
iajs-4032	153	1	2023;vol	2023;vol	NUM
iajs-4032	153	2	.	.	PROPN
iajs-4032	153	3	2834	2834	NUM
iajs-4032	153	4	.	.	PUNCT
iajs-4032	154	1	no	no	INTJ
iajs-4032	154	2	.	.	NOUN
iajs-4032	155	1	1	1	X
iajs-4032	155	2	.	.	X
iajs-4032	155	3	aip	aip	PROPN
iajs-4032	155	4	publishing	publishing	PROPN
iajs-4032	155	5	,	,	PUNCT
iajs-4032	155	6	.	.	PUNCT
iajs-4032	156	1	https://doi.org/10.1063/5.0163559	https://doi.org/10.1063/5.0163559	PROPN
iajs-4032	157	1	5	5	NUM
iajs-4032	157	2	.	.	X
iajs-4032	157	3	vaidyanathaswamy	vaidyanathaswamy	PROPN
iajs-4032	157	4	,	,	PUNCT
iajs-4032	157	5	r.	r.	VERB
iajs-4032	157	6	the	the	DET
iajs-4032	157	7	localisation	localisation	NOUN
iajs-4032	157	8	theory	theory	NOUN
iajs-4032	157	9	in	in	ADP
iajs-4032	157	10	set	set	NOUN
iajs-4032	157	11	-	-	PUNCT
iajs-4032	157	12	topology	topology	NOUN
iajs-4032	157	13	.	.	PUNCT
iajs-4032	158	1	proceedings	proceeding	NOUN
iajs-4032	158	2	of	of	ADP
iajs-4032	158	3	the	the	DET
iajs-4032	158	4	indian	indian	PROPN
iajs-4032	158	5	academy	academy	PROPN
iajs-4032	158	6	of	of	ADP
iajs-4032	158	7	sciences	sciences	PROPN
iajs-4032	158	8	-	-	PUNCT
iajs-4032	158	9	section	section	NOUN
iajs-4032	158	10	a.	a.	NOUN
iajs-4032	158	11	1944	1944	NUM
iajs-4032	158	12	:	:	PUNCT
iajs-4032	158	13	vol	vol	NOUN
iajs-4032	158	14	.	.	PROPN
iajs-4032	158	15	20	20	NUM
iajs-4032	158	16	.	.	PUNCT
iajs-4032	159	1	springer	springer	PROPN
iajs-4032	159	2	india	india	PROPN
iajs-4032	159	3	.	.	PROPN
iajs-4032	160	1	6	6	NUM
iajs-4032	160	2	.	.	X
iajs-4032	160	3	al	al	PROPN
iajs-4032	160	4	-	-	PUNCT
iajs-4032	160	5	saadi	saadi	PROPN
iajs-4032	160	6	,	,	PUNCT
iajs-4032	160	7	h.	h.	PROPN
iajs-4032	160	8	,	,	PUNCT
iajs-4032	160	9	a.	a.	PROPN
iajs-4032	160	10	al	al	PROPN
iajs-4032	160	11	-	-	PUNCT
iajs-4032	160	12	omari	omari	PROPN
iajs-4032	160	13	.	.	PUNCT
iajs-4032	161	1	some	some	DET
iajs-4032	161	2	operators	operator	NOUN
iajs-4032	161	3	in	in	ADP
iajs-4032	161	4	ideal	ideal	ADJ
iajs-4032	161	5	topological	topological	ADJ
iajs-4032	161	6	spaces	space	NOUN
iajs-4032	161	7	.	.	PUNCT
iajs-4032	162	1	missouri	missouri	PROPN
iajs-4032	162	2	journal	journal	PROPN
iajs-4032	162	3	of	of	ADP
iajs-4032	162	4	mathematical	mathematical	ADJ
iajs-4032	162	5	sciences	science	NOUN
iajs-4032	162	6	2018	2018	NUM
iajs-4032	162	7	:	:	PUNCT
iajs-4032	162	8	30.1	30.1	NUM
iajs-4032	162	9	;	;	PUNCT
iajs-4032	162	10	59	59	NUM
iajs-4032	162	11	-	-	SYM
iajs-4032	162	12	71	71	NUM
iajs-4032	162	13	.	.	PUNCT
iajs-4032	163	1	https://doi.org/10.35834/mjms/1534384955	https://doi.org/10.35834/mjms/1534384955	PROPN
iajs-4032	163	2	7	7	NUM
iajs-4032	163	3	.	.	X
iajs-4032	163	4	mandal	mandal	PROPN
iajs-4032	163	5	,	,	PUNCT
iajs-4032	163	6	dhananjoy	dhananjoy	NOUN
iajs-4032	163	7	,	,	PUNCT
iajs-4032	163	8	m.	m.	NOUN
iajs-4032	163	9	n.	n.	PROPN
iajs-4032	163	10	mukherjee	mukherjee	PROPN
iajs-4032	163	11	.	.	PUNCT
iajs-4032	164	1	certain	certain	ADJ
iajs-4032	164	2	new	new	ADJ
iajs-4032	164	3	classes	class	NOUN
iajs-4032	164	4	of	of	ADP
iajs-4032	164	5	generalized	generalized	ADJ
iajs-4032	164	6	closed	closed	ADJ
iajs-4032	164	7	sets	set	NOUN
iajs-4032	164	8	and	and	CCONJ
iajs-4032	164	9	their	their	PRON
iajs-4032	164	10	applications	application	NOUN
iajs-4032	164	11	in	in	ADP
iajs-4032	164	12	ideal	ideal	ADJ
iajs-4032	164	13	topological	topological	ADJ
iajs-4032	164	14	spaces	space	NOUN
iajs-4032	164	15	.	.	PUNCT
iajs-4032	165	1	filomat	filomat	PROPN
iajs-4032	165	2	2015	2015	NUM
iajs-4032	165	3	:	:	PUNCT
iajs-4032	165	4	29.5	29.5	NUM
iajs-4032	165	5	;	;	PUNCT
iajs-4032	165	6	1113	1113	NUM
iajs-4032	165	7	-	-	SYM
iajs-4032	165	8	1120	1120	NUM
iajs-4032	165	9	.	.	PUNCT
iajs-4032	166	1	8	8	NUM
iajs-4032	166	2	.	.	X
iajs-4032	166	3	atay	atay	NOUN
iajs-4032	166	4	,	,	PUNCT
iajs-4032	166	5	a.	a.	PROPN
iajs-4032	166	6	,	,	PUNCT
iajs-4032	166	7	f.	f.	PROPN
iajs-4032	166	8	eren	eren	PROPN
iajs-4032	166	9	.	.	PUNCT
iajs-4032	167	1	kuratowski	kuratowski	PROPN
iajs-4032	167	2	closure	closure	NOUN
iajs-4032	167	3	operators	operator	NOUN
iajs-4032	167	4	in	in	ADP
iajs-4032	167	5	soft	soft	ADJ
iajs-4032	167	6	ideal	ideal	ADJ
iajs-4032	167	7	topological	topological	ADJ
iajs-4032	167	8	spaces	space	NOUN
iajs-4032	167	9	.	.	PUNCT
iajs-4032	168	1	acta	acta	PROPN
iajs-4032	168	2	universitatis	universitatis	PROPN
iajs-4032	168	3	apulensis	apulensis	NOUN
iajs-4032	168	4	:	:	PUNCT
iajs-4032	168	5	mathematics	mathematic	NOUN
iajs-4032	168	6	-	-	PUNCT
iajs-4032	168	7	informatics	informatic	NOUN
iajs-4032	168	8	2023	2023	NUM
iajs-4032	168	9	:	:	PUNCT
iajs-4032	169	1	74	74	NUM
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iajs-4032	169	4	i	i	PRON
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iajs-4032	169	9	.	.	PUNCT
iajs-4032	170	1	tunç	tunç	PROPN
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iajs-4032	170	5	,	,	PUNCT
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iajs-4032	170	9	.	.	PUNCT
iajs-4032	171	1	on	on	ADP
iajs-4032	171	2	a	a	DET
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iajs-4032	171	5	via	via	ADP
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iajs-4032	171	9	.	.	PUNCT
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iajs-4032	172	6	computer	computer	NOUN
iajs-4032	172	7	science	science	NOUN
iajs-4032	172	8	2023	2023	NUM
iajs-4032	172	9	:	:	PUNCT
iajs-4032	172	10	15.2	15.2	NUM
iajs-4032	172	11	;	;	PUNCT
iajs-4032	172	12	227https://doi.org/10.47000/tjmcs.1195540	227https://doi.org/10.47000/tjmcs.1195540	NUM
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iajs-4032	172	14	.	.	PUNCT
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iajs-4032	173	2	,	,	PUNCT
iajs-4032	173	3	k.	k.	PROPN
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iajs-4032	173	5	,	,	PUNCT
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iajs-4032	173	8	.	.	PUNCT
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iajs-4032	174	5	axioms	axiom	NOUN
iajs-4032	174	6	in	in	ADP
iajs-4032	174	7	kasaj	kasaj	ADJ
iajs-4032	174	8	topological	topological	ADJ
iajs-4032	174	9	spaces	space	NOUN
iajs-4032	174	10	.	.	PUNCT
iajs-4032	175	1	indian	indian	ADJ
iajs-4032	175	2	journal	journal	PROPN
iajs-4032	175	3	of	of	ADP
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iajs-4032	175	5	and	and	CCONJ
iajs-4032	175	6	technology	technology	NOUN
iajs-4032	175	7	2023:16.41;35753582.https://doi.org/10.17485	2023:16.41;35753582.https://doi.org/10.17485	NOUN
iajs-4032	175	8	/	/	SYM
iajs-4032	175	9	ijst	ijst	ADJ
iajs-4032	175	10	/	/	SYM
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iajs-4032	175	13	.	.	PUNCT
iajs-4032	176	1	karim	karim	PROPN
iajs-4032	176	2	,	,	PUNCT
iajs-4032	176	3	ma	ma	PROPN
iajs-4032	176	4	abdel	abdel	PROPN
iajs-4032	176	5	,	,	PUNCT
iajs-4032	176	6	and	and	CCONJ
iajs-4032	176	7	a.	a.	NOUN
iajs-4032	176	8	i.	i.	PROPN
iajs-4032	176	9	nasir	nasir	PROPN
iajs-4032	176	10	.	.	PUNCT
iajs-4032	177	1	"	"	PUNCT
iajs-4032	177	2	separation	separation	NOUN
iajs-4032	177	3	axioms	axiom	NOUN
iajs-4032	177	4	via	via	ADP
iajs-4032	177	5	ǐ	ǐ	PROPN
iajs-4032	177	6	semi	semi	ADV
iajs-4032	177	7	g	g	NOUN
iajs-4032	177	8	open	open	ADJ
iajs-4032	177	9	sets	set	NOUN
iajs-4032	177	10	.	.	PUNCT
iajs-4032	177	11	"	"	PUNCT
iajs-4032	178	1	ibn	ibn	PROPN
iajs-4032	178	2	al	al	PROPN
iajs-4032	178	3	-	-	PUNCT
iajs-4032	178	4	haitham	haitham	PROPN
iajs-4032	178	5	journal	journal	PROPN
iajs-4032	178	6	for	for	ADP
iajs-4032	178	7	pure	pure	ADJ
iajs-4032	178	8	and	and	CCONJ
iajs-4032	178	9	applied	applied	ADJ
iajs-4032	178	10	science	science	NOUN
iajs-4032	178	11	,	,	PUNCT
iajs-4032	178	12	2020	2020	NUM
iajs-4032	178	13	:	:	PUNCT
iajs-4032	178	14	33.1	33.1	NUM
iajs-4032	178	15	;	;	PUNCT
iajs-4032	178	16	157	157	NUM
iajs-4032	178	17	-	-	SYM
iajs-4032	178	18	161	161	NUM
iajs-4032	178	19	.	.	PUNCT
iajs-4032	179	1	https://doi.org/10.30526/33.1.2382	https://doi.org/10.30526/33.1.2382	PUNCT
iajs-4032	180	1	12	12	NUM
iajs-4032	180	2	.	.	PUNCT
iajs-4032	181	1	mohammad	mohammad	PROPN
iajs-4032	181	2	,	,	PUNCT
iajs-4032	181	3	muna	muna	PROPN
iajs-4032	181	4	jabbar	jabbar	PROPN
iajs-4032	181	5	,	,	PUNCT
iajs-4032	181	6	faik	faik	PROPN
iajs-4032	181	7	jameel	jameel	PROPN
iajs-4032	181	8	hasan	hasan	PROPN
iajs-4032	181	9	mayah	mayah	PROPN
iajs-4032	181	10	.	.	PUNCT
iajs-4032	182	1	on	on	ADP
iajs-4032	182	2	pre	pre	ADJ
iajs-4032	182	3	-	-	ADJ
iajs-4032	182	4	δ	δ	NOUN
iajs-4032	182	5	-	-	PUNCT
iajs-4032	182	6	separation	separation	NOUN
iajs-4032	182	7	axioms	axiom	NOUN
iajs-4032	182	8	in	in	ADP
iajs-4032	182	9	ideal	ideal	ADJ
iajs-4032	182	10	topological	topological	ADJ
iajs-4032	182	11	spaces.journal	spaces.journal	PROPN
iajs-4032	182	12	of	of	ADP
iajs-4032	182	13	wasit	wasit	NOUN
iajs-4032	182	14	for	for	ADP
iajs-4032	182	15	science	science	NOUN
iajs-4032	182	16	and	and	CCONJ
iajs-4032	182	17	medicine	medicine	NOUN
iajs-4032	182	18	2023:16.1;7	2023:16.1;7	NOUN
iajs-4032	182	19	-	-	PUNCT
iajs-4032	182	20	12	12	NUM
iajs-4032	182	21	.	.	PUNCT
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iajs-4032	182	23	13	13	NUM
iajs-4032	182	24	.	.	PUNCT
iajs-4032	182	25	jackson	jackson	PROPN
iajs-4032	182	26	,	,	PUNCT
iajs-4032	182	27	s.	s.	PROPN
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iajs-4032	182	32	,	,	PUNCT
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iajs-4032	182	35	.	.	PUNCT
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iajs-4032	183	2	axioms	axiom	VERB
iajs-4032	183	3	through	through	ADP
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iajs-4032	183	6	open	open	ADJ
iajs-4032	183	7	set	set	PROPN
iajs-4032	183	8	.	.	PUNCT
iajs-4032	184	1	indian	indian	PROPN
iajs-4032	184	2	journal	journal	PROPN
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iajs-4032	184	5	and	and	CCONJ
iajs-4032	184	6	technology	technology	NOUN
iajs-4032	184	7	2024	2024	NUM
iajs-4032	184	8	:	:	PUNCT
iajs-4032	184	9	17;9	17;9	NUM
iajs-4032	184	10	-	-	SYM
iajs-4032	184	11	13	13	NUM
iajs-4032	184	12	.	.	PUNCT
iajs-4032	185	1	https://doi.org/10.17485/ijst/v17sp1.252	https://doi.org/10.17485/ijst/v17sp1.252	PROPN
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iajs-4032	185	3	.	.	PUNCT
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iajs-4032	186	2	,	,	PUNCT
iajs-4032	186	3	ferit	ferit	NOUN
iajs-4032	186	4	,	,	PUNCT
iajs-4032	186	5	and	and	CCONJ
iajs-4032	186	6	aynur	aynur	ADV
iajs-4032	186	7	keskin	keskin	PROPN
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iajs-4032	186	9	.	.	PUNCT
iajs-4032	187	1	a	a	DET
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iajs-4032	187	3	separation	separation	NOUN
iajs-4032	187	4	axiom	axiom	NOUN
iajs-4032	187	5	in	in	ADP
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iajs-4032	187	7	topological	topological	ADJ
iajs-4032	187	8	spaces.conference	spaces.conference	PROPN
iajs-4032	187	9	proceeding	proceeding	NOUN
iajs-4032	187	10	science	science	NOUN
iajs-4032	187	11	and	and	CCONJ
iajs-4032	187	12	technology	technology	NOUN
iajs-4032	187	13	:	:	PUNCT
iajs-4032	187	14	international	international	ADJ
iajs-4032	187	15	online	online	ADJ
iajs-4032	187	16	conference	conference	NOUN
iajs-4032	187	17	on	on	ADP
iajs-4032	187	18	mathematical	mathematical	ADJ
iajs-4032	187	19	advances	advance	NOUN
iajs-4032	187	20	and	and	CCONJ
iajs-4032	187	21	applications	application	NOUN
iajs-4032	187	22	(	(	PUNCT
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iajs-4032	187	25	,	,	PUNCT
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iajs-4032	187	27	.	.	PUNCT
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iajs-4032	189	1	5	5	NUM
iajs-4032	189	2	.	.	X
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iajs-4032	189	4	.	.	NOUN
iajs-4032	190	1	1	1	NUM
iajs-4032	190	2	.	.	NUM
iajs-4032	190	3	2022	2022	NUM
iajs-4032	190	4	.	.	PUNCT
iajs-4032	191	1	15	15	NUM
iajs-4032	191	2	.	.	X
iajs-4032	192	1	saeed	saeed	PROPN
iajs-4032	192	2	,	,	PUNCT
iajs-4032	192	3	sufyan	sufyan	PROPN
iajs-4032	192	4	g.	g.	PROPN
iajs-4032	192	5	,	,	PUNCT
iajs-4032	192	6	r.	r.	PROPN
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iajs-4032	192	8	esmaeel	esmaeel	PROPN
iajs-4032	192	9	.	.	PUNCT
iajs-4032	193	1	separation	separation	NOUN
iajs-4032	193	2	axioms	axiom	NOUN
iajs-4032	193	3	via	via	ADP
iajs-4032	193	4	αg	αg	NOUN
iajs-4032	193	5	ị	ị	ADJ
iajs-4032	193	6	-	-	ADJ
iajs-4032	193	7	open	open	ADJ
iajs-4032	193	8	set	set	NOUN
iajs-4032	193	9	.	.	PUNCT
iajs-4032	194	1	journal	journal	PROPN
iajs-4032	194	2	of	of	ADP
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iajs-4032	194	4	:	:	PUNCT
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iajs-4032	194	7	.	.	PUNCT
iajs-4032	195	1	2020	2020	NUM
iajs-4032	195	2	:	:	PUNCT
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iajs-4032	195	5	,	,	PUNCT
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iajs-4032	195	7	16	16	NUM
iajs-4032	195	8	.	.	PUNCT
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iajs-4032	196	2	,	,	PUNCT
iajs-4032	196	3	julian	julian	PROPN
iajs-4032	196	4	.	.	PUNCT
iajs-4032	197	1	on	on	ADP
iajs-4032	197	2	hausdorff	hausdorff	NOUN
iajs-4032	197	3	spaces	space	NOUN
iajs-4032	197	4	via	via	ADP
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iajs-4032	197	6	ideals	ideal	NOUN
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iajs-4032	197	10	.	.	PUNCT
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iajs-4032	198	5	york	york	PROPN
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iajs-4032	198	7	of	of	ADP
iajs-4032	198	8	sciences	science	NOUN
iajs-4032	198	9	1995	1995	NUM
iajs-4032	198	10	:	:	PUNCT
iajs-4032	198	11	767.1	767.1	NUM
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iajs-4032	198	16	.	.	PUNCT
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iajs-4032	200	8	.	.	PUNCT
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iajs-4032	201	7	spaces	space	NOUN
iajs-4032	201	8	.	.	PUNCT
iajs-4032	202	1	mathematical	mathematical	ADJ
iajs-4032	202	2	sciences	science	NOUN
iajs-4032	202	3	,	,	PUNCT
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iajs-4032	202	5	:	:	PUNCT
iajs-4032	202	6	14.2	14.2	NUM
iajs-4032	202	7	;	;	PUNCT
iajs-4032	202	8	185	185	NUM
iajs-4032	202	9	-	-	SYM
iajs-4032	202	10	192	192	NUM
iajs-4032	202	11	.	.	PUNCT
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iajs-4032	203	3	.	.	PUNCT
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iajs-4032	205	1	r.	r.	PROPN
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iajs-4032	205	4	axioms	axiom	NOUN
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iajs-4032	205	7	-	-	PUNCT
iajs-4032	205	8	j	j	NOUN
iajs-4032	205	9	-	-	PUNCT
iajs-4032	205	10	semi	semi	ADJ
iajs-4032	205	11	-	-	ADJ
iajs-4032	205	12	g	g	ADV
iajs-4032	205	13	-	-	PUNCT
iajs-4032	205	14	open	open	ADJ
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iajs-4032	205	16	.	.	PUNCT
iajs-4032	206	1	iop	iop	NOUN
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iajs-4032	206	4	:	:	PUNCT
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iajs-4032	206	7	and	and	CCONJ
iajs-4032	206	8	engineering	engineering	NOUN
iajs-4032	206	9	2020	2020	NUM
iajs-4032	206	10	:	:	PUNCT
iajs-4032	206	11	vol	vol	NOUN
iajs-4032	206	12	.	.	PUNCT
iajs-4032	207	1	871	871	NUM
iajs-4032	207	2	.	.	PUNCT
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iajs-4032	208	2	.	.	NOUN
iajs-4032	209	1	1	1	X
iajs-4032	209	2	.	.	X
iajs-4032	209	3	iop	iop	NOUN
iajs-4032	209	4	publishing	publishing	PROPN
iajs-4032	209	5	,	,	PUNCT
iajs-4032	209	6	https://doi.org/10.1088/1757-899x/871/1/012052	https://doi.org/10.1088/1757-899x/871/1/012052	PROPN
iajs-4032	209	7	19	19	NUM
iajs-4032	209	8	.	.	PUNCT
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iajs-4032	209	10	,	,	PUNCT
iajs-4032	209	11	r.	r.	PROPN
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iajs-4032	209	13	,r	,r	PROPN
iajs-4032	209	14	.	.	PUNCT
iajs-4032	210	1	j.	j.	PROPN
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iajs-4032	210	4	soft-ℐ-semi	soft-ℐ-semi	PROPN
iajs-4032	210	5	-	-	PUNCT
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iajs-4032	210	7	-	-	PUNCT
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iajs-4032	210	10	.	.	PUNCT
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iajs-4032	211	4	:	:	PUNCT
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iajs-4032	211	7	.	.	PUNCT
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iajs-4032	212	2	:	:	PUNCT
iajs-4032	212	3	vol	vol	NOUN
iajs-4032	212	4	.	.	PUNCT
iajs-4032	212	5	1591	1591	NUM
iajs-4032	212	6	.	.	PUNCT
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iajs-4032	213	2	.	.	NOUN
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iajs-4032	214	2	.	.	X
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iajs-4032	214	5	.	.	PUNCT
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iajs-4032	215	4	.	.	PUNCT
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iajs-4032	216	2	,	,	PUNCT
iajs-4032	216	3	j.	j.	PROPN
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iajs-4032	216	5	,	,	PUNCT
iajs-4032	216	6	j.	j.	PROPN
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iajs-4032	216	8	jesti	jesti	PROPN
iajs-4032	216	9	.	.	PUNCT
iajs-4032	217	1	some	some	DET
iajs-4032	217	2	separation	separation	NOUN
iajs-4032	217	3	axioms	axiom	VERB
iajs-4032	217	4	in	in	ADP
iajs-4032	217	5	nano	nano	NOUN
iajs-4032	217	6	ideal	ideal	ADJ
iajs-4032	217	7	topological	topological	ADJ
iajs-4032	217	8	spaces	space	NOUN
iajs-4032	217	9	.	.	PUNCT
iajs-4032	218	1	journal	journal	NOUN
iajs-4032	218	2	of	of	ADP
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iajs-4032	218	5	:	:	PUNCT
iajs-4032	218	6	history	history	NOUN
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iajs-4032	218	9	,	,	PUNCT
iajs-4032	218	10	2023	2023	NUM
iajs-4032	218	11	:	:	PUNCT
iajs-4032	218	12	35;2502	35;2502	NUM
iajs-4032	218	13	-	-	PUNCT
iajs-4032	218	14	2508	2508	NUM
iajs-4032	218	15	.	.	PUNCT
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iajs-4032	219	3	.	.	PUNCT
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iajs-4032	220	7	i.	i.	PROPN
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iajs-4032	220	9	.	.	PUNCT
iajs-4032	221	1	some	some	DET
iajs-4032	221	2	game	game	NOUN
iajs-4032	221	3	via	via	ADP
iajs-4032	221	4	ἷ-semi	ἷ-semi	PROPN
iajs-4032	221	5	-	-	PUNCT
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iajs-4032	221	7	-	-	PUNCT
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iajs-4032	222	6	:	:	PUNCT
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iajs-4032	222	10	.	.	PUNCT
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iajs-4032	225	2	.	.	PUNCT
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iajs-4032	226	10	.	.	PUNCT
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iajs-4032	227	5	in	in	ADP
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iajs-4032	227	7	spaces	space	NOUN
iajs-4032	227	8	.	.	PUNCT
iajs-4032	228	1	ibn	ibn	PROPN
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iajs-4032	228	5	https://doi.org/10.17485/ijst/v16i41.1509	https://doi.org/10.17485/ijst/v16i41.1509	PROPN
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iajs-4032	229	2	https://doi.org/10.17485/ijst/v17sp1.252	https://doi.org/10.17485/ijst/v17sp1.252	PROPN
iajs-4032	229	3	https://iopscience.iop.org/article/10.1088/1742-6596/1591/1/012099	https://iopscience.iop.org/article/10.1088/1742-6596/1591/1/012099	VERB
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iajs-4032	230	1	https://doi.org/10.1111/j.1749-6632.1995.tb55891.x	https://doi.org/10.1111/j.1749-6632.1995.tb55891.x	NOUN
iajs-4032	231	1	https://doi.org/10.1007/s40096-020-00330-z	https://doi.org/10.1007/s40096-020-00330-z	PROPN
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iajs-4032	231	4	ihjpas	ihjpa	VERB
iajs-4032	231	5	.	.	PUNCT
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iajs-4032	232	2	)	)	PUNCT
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iajs-4032	232	9	applied	apply	VERB
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iajs-4032	232	11	2010:23.1	2010:23.1	NUM
iajs-4032	232	12	.	.	PUNCT
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iajs-4032	233	2	.	.	PUNCT
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iajs-4032	234	9	.	.	PUNCT
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iajs-4032	235	2	and	and	CCONJ
iajs-4032	235	3	separations	separation	NOUN
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iajs-4032	235	5	alpha	alpha	NOUN
iajs-4032	235	6	-	-	PUNCT
iajs-4032	235	7	connected	connect	VERB
iajs-4032	235	8	topological	topological	ADJ
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iajs-4032	235	10	.	.	PUNCT
iajs-4032	236	1	iraqi	iraqi	ADJ
iajs-4032	236	2	journal	journal	PROPN
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iajs-4032	236	5	2021	2021	NUM
iajs-4032	236	6	:	:	PUNCT
iajs-4032	236	7	24;3091	24;3091	NUM
iajs-4032	236	8	-	-	SYM
iajs-4032	236	9	3096	3096	NUM
iajs-4032	236	10	.	.	PUNCT
iajs-4032	237	1	https://doi.org/10.24996/ijs.2021.62.9.24	https://doi.org/10.24996/ijs.2021.62.9.24	NOUN
iajs-4032	237	2	25	25	NUM
iajs-4032	237	3	.	.	PUNCT
iajs-4032	238	1	al	al	PROPN
iajs-4032	238	2	-	-	PUNCT
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iajs-4032	238	4	,	,	PUNCT
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iajs-4032	238	6	k.	k.	PROPN
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iajs-4032	238	8	,	,	PUNCT
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iajs-4032	238	14	.	.	PUNCT
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iajs-4032	239	5	continuity	continuity	NOUN
iajs-4032	239	6	with	with	ADP
iajs-4032	239	7	respect	respect	NOUN
iajs-4032	239	8	to	to	ADP
iajs-4032	239	9	some	some	DET
iajs-4032	239	10	types	type	NOUN
iajs-4032	239	11	of	of	ADP
iajs-4032	239	12	sets	set	NOUN
iajs-4032	239	13	in	in	ADP
iajs-4032	239	14	ideal	ideal	ADJ
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iajs-4032	239	16	space.international	space.international	PROPN
iajs-4032	239	17	journal	journal	NOUN
iajs-4032	239	18	of	of	ADP
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iajs-4032	239	20	and	and	CCONJ
iajs-4032	239	21	applied	apply	VERB
iajs-4032	239	22	mathematics	mathematic	NOUN
iajs-4032	239	23	2018	2018	NUM
iajs-4032	239	24	:	:	PUNCT
iajs-4032	240	1	119.10	119.10	NUM
iajs-4032	240	2	;	;	PUNCT
iajs-4032	240	3	409	409	NUM
iajs-4032	240	4	-	-	SYM
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iajs-4032	240	6	.	.	PUNCT
iajs-4032	241	1	26	26	NUM
iajs-4032	241	2	.	.	PUNCT
iajs-4032	242	1	al	al	PROPN
iajs-4032	242	2	-	-	PUNCT
iajs-4032	242	3	shami	shami	PROPN
iajs-4032	242	4	,	,	PUNCT
iajs-4032	242	5	tareq	tareq	PROPN
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iajs-4032	242	7	soft	soft	ADJ
iajs-4032	242	8	somewhat	somewhat	ADV
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iajs-4032	242	11	:	:	PUNCT
iajs-4032	242	12	soft	soft	ADJ
iajs-4032	242	13	separation	separation	NOUN
iajs-4032	242	14	axioms	axiom	NOUN
iajs-4032	242	15	and	and	CCONJ
iajs-4032	242	16	medical	medical	ADJ
iajs-4032	242	17	application	application	NOUN
iajs-4032	242	18	to	to	ADP
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iajs-4032	242	20	.	.	PUNCT
iajs-4032	243	1	computational	computational	ADJ
iajs-4032	243	2	and	and	CCONJ
iajs-4032	243	3	applied	apply	VERB
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iajs-4032	243	6	:	:	PUNCT
iajs-4032	243	7	41.5	41.5	NUM
iajs-4032	243	8	;	;	PUNCT
iajs-4032	243	9	216	216	NUM
iajs-4032	243	10	.	.	PUNCT
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iajs-4032	244	3	.	.	PUNCT
iajs-4032	245	1	el	el	PROPN
iajs-4032	245	2	-	-	PUNCT
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iajs-4032	245	4	,	,	PUNCT
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iajs-4032	245	7	-	-	PROPN
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iajs-4032	246	9	.	.	PUNCT
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iajs-4032	247	8	:	:	PUNCT
iajs-4032	248	1	28.1	28.1	NUM
iajs-4032	248	2	;	;	PUNCT
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iajs-4032	248	4	.	.	PUNCT
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