Properties of some ∗ dense in itself subsets

PII. S0161171204403391 http://ijmms.hindawi.com © Hindawi Publishing Corp.

PROPERTIES OF SOME

∗-DENSE-IN-ITSELF SUBSETS

V. RENUKA DEVI, D. SIVARAJ, and T. TAMIZH CHELVAM

Received 22 March 2004

Ᏽ-open sets were introduced and studied by Jankovi´c and Hamlett (1990) to generalize the well-known Banach category theorem. Quasi-Ᏽ-openness was introduced and studied by Abd El-Monsef et al. (2000). These are∗-dense-in-itself sets of the ideal spaces. In this note, prop-erties of these sets are further investigated and characterizations of these sets are given. Also, their relation withᏵ-dense sets andᏵ-locally closed sets is discussed. Characteriza-tions of completely codense ideals are given in terms of semi-preopen sets.

2000 Mathematics Subject Classification: 54A05, 54A10.

1. Introduction and preliminaries. The subject of ideals in topological spaces has been studied by Kuratowski [12] and Vaidyanathaswamy [20]. Anideal Ᏽ on a topo-logical space(X,τ) is a collection of subsets ofX which satisfies that (i) A∈Ᏽ and B⊂A impliesB∈Ᏽand (ii)A∈Ᏽand B∈ᏵimpliesA∪B∈Ᏽ. Given a topological space(X,τ)with an idealᏵonX and if℘(X)is the set of all subsets ofX, a set op-erator(·)∗_{:℘(X)→℘(X), called alocal function [12] of A with respect toᏵ andτ,}

is defined as follows: forA⊂X, A∗_{(Ᏽ,τ)= {x∈X|U∩A∈Ᏽfor everyU∈τ(x)},}

whereτ(x)= {U∈τ|x∈U}. We will make use of the basic facts concerning the lo-cal functions [10, Theorem 2.3] without mentioning it explicitly. A Kuratowski closure operator cl∗(·)for a topologyτ∗_{(Ᏽ,τ), called the∗-topology, finer thanτ, is defined}

by cl∗(A)=A∪A∗_{(Ᏽ,τ)[19]. When there is no chance for confusion, we will simply}

writeA∗forA∗(Ᏽ,τ)andτ∗orτ∗(Ᏽ)forτ∗(Ᏽ,τ). IfᏵis an ideal onX, then(X,τ,Ᏽ) is called an ideal space. By a space, we always mean a topological space(X,τ)with no separation properties assumed. IfA⊂X, cl(A)and int(A)will denote the closure and interior ofAin(X,τ), respectively, and cl∗(A)and int∗(A)will denote the closure and interior ofAin(X,τ∗_{), respectively. A subset Aof a space(X,τ)issemiopen[13] if}

there exists an open setGsuch thatG⊂A⊂cl(G)or, equivalently,A⊂cl(int(A)). The complement of a semiopen set is semiclosed. The smallest semiclosed set containing Ais called thesemiclosureofAand is denoted by scl(A). Also, scl(A)=A∪int(cl(A)) [4, Theorem 1.5(a)]. The largest semiopen set contained inAis called thesemi-interior

ofA and is denoted by sint(A). A subset Aof a space(X,τ)is anα-set [15] ifA⊂ int(cl(int(A))). The family of allα-sets in(X,τ)is denoted by τα_.τα _{is a topology} on X which is finer than τ. The complement of an α-set is called an α-closed set. The closure and interior ofA in(X,τα_{) are denoted by cl}

ifA=int(cl(A)). The complement of a regular open set isregular closed. A subsetA of a space(X,τ)is said to bepreopen[14] ifA⊂int(cl(A)). The family of all preopen sets is denoted by PO(X,τ)or simply PO(X). The largest preopen set contained inAis called thepreinteriorofAand is denoted by pint(A)and pint(A)=A∩int(cl(A))[4].A is preopen if and only if there is a regular open setGsuch thatA⊂Gand cl(A)=cl(G) [7, Proposition 2.1]. A subsetAof a space(X,τ)is semi-preopen[4] if there exists a preopen setGsuch thatG⊂A⊂cl(G). The family of all semi-preopen sets in(X,τ) is denoted by SPO(X,τ)or simply SPO(X). The complement of a semi-preopen set is calledsemi-preclosed. The largest semi-preopen set contained inAis called thesemi

-preinterior of A and is denoted by spint(A). Also, spint(A)=A∩cl(int(cl(A)))for everyAofX[4]. Given a space(X,τ)and idealsᏵandonX, theextensionofᏵvia

[11], denoted byᏵ∗, is the ideal given byᏵ∗ = {A⊂X|A∗_{(Ᏽ)∈ }. In particular,}

Ᏽ∗ᏺ= {A⊂X |int(A∗(Ᏽ))=φ}is an ideal containing both Ᏽand ᏺ and Ᏽ∗ᏺ is usually denoted by ˜Ᏽ. The following lemmas will be useful in the sequel.

Lemma1.1. Let(X,τ,Ᏽ)be an ideal space andA⊂X. If A⊂A∗, then (a)A∗_=cl(A)=cl∗_(A),

(b)A∗(Ᏽ˜)=A∗(ᏺ).

Proof. _{Clearly, for every subset}A_ofX_{, cl}∗(A)⊂_cl(A)_{. Let}x∈_cl∗(A)_{. Then there} exists aτ∗_{-open setGcontainingxsuch thatG∩A=φ. There existsV∈τandI∈Ᏽ}

such thatx∈V−I⊂G.G∩A=φ⇒(V−I)∩A=φ⇒(V∩A)−I=φ⇒((V∩A)−I)∗₌

φ⇒(V∩A)∗_−I∗ _{=φ⇒(V∩A)}∗_{= φ⇒V∩A}∗ _{=φ⇒V∩A=φ. SinceV is an}

open set containingx,x∈cl(A)and so cl(A)⊂cl∗(A). Hence cl(A)=cl∗(A). Since A⊂A∗_{⊂cl(A), cl(A)=A}∗_{. This proves (a).}

(b) By [11, Theorem 3.2],A∗₍˜_{Ᏽ)=cl(int(A}∗_{))and so by (a),A}∗₍˜_{Ᏽ)=cl(int(cl(A)))=} A∗_(ᏺ).

Lemma1.2. Let(X,τ)be a space andA⊂X. IfAis semiopen, thencl(A)=cl_α(A)

and ifAis semiclosed, thenint(A)=intα(A)[18, Lemma 2.1].

Lemma1.3. If(X,τ,Ᏽ)is an ideal space, then the following are equivalent.

(a)For everyA∈τ,A⊂A∗_.

(b)For everyA∈SO(X,τ),A⊂A∗_.

Proof. Since τ ⊂ SO(X,τ), it is enough to prove that (a)⇒(b). Suppose A ∈

SO(X,τ). Then there exists an open setHsuch thatH⊂A⊂cl(H). SinceHis open, H⊂H∗_{and so, byLemma 1.1,A⊂cl(H)=H}∗_⊂A∗_{. HenceA⊂A}∗_.

2. Completely codense ideal. An idealᏵon a space(X,τ)is said to becodense[6] ifτ∩Ᏽ= {φ}or, equivalently,X=X∗ _{[10]. Ᏽis said to becompletely codense[6] if}

Theorem2.1. Let(X,τ,Ᏽ)be an ideal space. Then the following are equivalent.

(a)Ᏽis completely codense.

(b) SPO(X)∩Ᏽ= {φ}.

(c)A⊂A∗_{for everyA∈SPO(X).}

(d) spint(A)=φfor everyA∈Ᏽ.

Proof. _(a)⇒_{(b). Suppose}A∈_SPO(X)∩Ᏽ. A∈Ᏽ⇒A∈ᏺ _{and so int}(_cl(A))=φ_. A∈SPO(X)⇒A⊂cl(int(cl(A)))⇒A=φ. Therefore, SPO(X)∩Ᏽ= {φ}.

(b)⇒(c). SupposeA∈SPO(X)andx∈A∗_{. Then there exists an open setGcontaining}

xsuch thatG∩A∈Ᏽ. SinceA∈SPO(X),G∩A∈SPO(X), by [4, Theorem 2.7] and so by hypothesis,G∩A=φwhich implies thatx∈A.

(c)⇒(d). LetA∈Ᏽsuch that spint(A)=φ. Then there exists a nonempty semi-preopen setGsuch thatG⊂Aand soG∗⊂A∗=φ. SinceG⊂G∗,G=φwhich is a contradiction. Therefore, spint(A)=φ.

(d)⇒(a). LetA∈PO(X)∩Ᏽ. ThenA∈PO(X)⇒A⊂int(cl(A))⊂cl(int(cl(A))). A∈

Ᏽ⇒spint(A)=φ⇒A∩cl(int(cl(A)))=φ⇒A=φ.

Corollary2.2. Let(X,τ,Ᏽ)be an ideal space with a completely codense idealᏵ.

(a)IfA∈SPO(X), thenA∗_{(Ᏽ)is regular closed,A}∗_(Ᏽ)=A∗_{(ᏺ), andcl(A)=cl}∗_(A)

=cl_α(A).

(b)IfAis semi-preclosed, thenint(A)=int∗(A)=intα(A).

Proof. (a) IfA∈SPO(X), byTheorem 2.1(c),A⊂A∗ ⊂cl(A) and so A∗=cl(A) which implies thatA∗_{is regular closed, since the closure of a semi-preopen set is regular}

closed [4, Theorem 2.4]. Therefore,A∗_=cl(int(A∗_{))=cl(int(cl(A)))=A}∗_{(ᏺ). cl(A)=}

cl∗(A) by Theorem 2.1(c) and Lemma 1.1. Also, cl∗(A)=A∪A∗(Ᏽ)=A∪A∗(ᏺ)= cl_α(A). This proves (a).

(b) The proof follows from (a).

3. Ᏽ-open sets. A subsetAof an ideal space(X,τ,Ᏽ)isτ∗-closed[10] (resp.,∗-dense in itself [9],∗-perfect [9]) ifA∗_{⊂A(resp.,A⊂A}∗_,A=A∗_{). Clearly,Ais∗-perfect if}

and only ifAisτ∗_{-closed and∗-dense in itself. The followingTheorem 3.1is useful in}

the sequel.

Theorem3.1. Let(X,τ,Ᏽ)be an ideal space and letUandAbe subsets ofXsuch

thatA⊂U⊂A∗_{. ThenUis∗-dense in itself, andU}∗_andA∗_{are∗-perfect.}

Proof. A⊂U⊂A∗implies thatU∗=A∗and soUis∗-dense in itself. Since(A∗)∗⊂ A∗_,A⊂A∗_{implies thatA}∗_{is∗- perfect and soU}∗_{is∗-perfect.}

A subsetAof an ideal space(X,τ,Ᏽ)isᏵ-locally closed, [5] ifA=G∩V, whereGis open andVis∗-perfect. Clearly, every∗-perfect set isᏵ-locally closed. The following theorem gives a characterization ofᏵ-locally closed sets.

Theorem3.2. Let(X,τ,Ᏽ)be an ideal space. A subset AofXisᏵ-locally closed if

Proof. SupposeAis Ᏽ-locally closed. Then A=G∩V where Gis open and V is ∗-perfect. NowA∗_{=(G∩V )}∗_⊃G∩V∗ _{=G∩V =A. Also,A⊂V implies thatA}∗_⊂

V∗_{=V. Therefore,G∩A}∗_=G∩(A∗_{∩V )=(G∩V )∩A}∗_=A∩A∗_{=A. Conversely, if}

A=G∩A∗_{whereGis open, thenA⊂A}∗_{and so byTheorem 3.1,A}∗_{is∗-perfect and}

soAisᏵ-locally closed.

The following corollary follows from [10, Theorems 2.1 and 2.2 and Theorem 6.1(d)].

Corollary_3.3. _Let(X,τ,Ᏽ)_{be an ideal space.}

(a)EveryᏵ-locally closed set is∗-dense in itself.

(b)Every open,∗-dense-in-itself subset ofXisᏵ-locally closed.

(c)IfᏵis codense, then every open set isᏵ-locally closed.

A subsetAof an ideal space(X,τ,Ᏽ)is Ᏽ-open [11] ifA⊂int(A∗_{). The family of}

all Ᏽ-open sets is denoted by IO(X,τ,Ᏽ), IO(X,τ), or IO(X). The complement of an Ᏽ-open set is said to beᏵ-closed. The largestᏵ-open set contained inAis called the Ᏽ-interior ofAand is denoted by Iint(A)and Iint(A)=A∩int(A∗_{)[11, Theorem 4.1(3)].}

The following theorem gives some properties ofᏵ-open sets.

Theorem_3.4. _IfA_{is an}Ᏽ_{-open subset of an ideal space}(X,τ,Ᏽ)_{, then}

(a)Ais∗-dense in itself,

(b)A∗_=cl(A)=cl∗_{(A)andcl(A)andA}∗_{are regular closed,}

(c)A∗is∗-perfect andᏵ-locally closed,

(d) int(A∗_{)is∗-dense in itself andᏵ-locally closed,}

(e) cl(int(A∗_))=A∗₍˜_{Ᏽ)is∗-dense in itself,}

(f)A∗_=(int(A∗₎₎∗_=(cl(int(A∗₎₎₎∗_=(A∗_(Ᏽ˜₎₎∗_(Ᏽ),

(g)(int(A∗₎₎∗_and(cl(int(A∗₎₎₎∗_{areᏵ-locally closed,}

(h) int(A∗_)isᏵ-open.

Proof. (a) follows from the definition. (b) follows from (a),Lemma 1.1, and the fact that everyᏵ-open set is preopen [1] and the closure of a preopen set is regular closed [7, Proposition 2.1(ii)]. (c) follows fromTheorem 3.1and from the fact that every

∗-perfect set isᏵ-locally closed. (d) follows fromTheorem 3.1andCorollary 3.3(b). (e) cl(int(A∗_))=A∗_(Ᏽ˜_{)by [11, Theorem 3.2] and sinceA⊂int(A}∗_)⊂cl(int(A∗_))⊂A∗_{, by} Theorem 3.1, cl(int(A∗_{))is∗-dense in itself. (f) From the inequality in the proof of (e),}

we haveA∗_=(int(A∗₎₎∗_=(cl(int(A∗₎₎₎∗_{. Each is equal to(A}∗₍˜_Ᏽ))∗_{(Ᏽ)by (e). (g) and}

(h) follow from (c) and (f), respectively.

Theorem_3.5. _Let(X,τ,Ᏽ)_{be an ideal space. If}A_isᏵ_{-open and}V _{is semiopen, then}

(a)V∩Ais∗-dense in itself,

(b)(V∩A)∗_{is∗-perfect andᏵ-locally closed,}

(c) cl(V )∩Ais∗-dense in itself,

(d)(cl(V )∩A)∗is∗-perfect andᏵ-locally closed.

Proof. _SinceV∩A⊂_cl(V )∩A⊂(V∩A)∗_{by [1, Theorem 2.10],}V∩A_is∗_{-dense in} itself and byTheorem 3.1, cl(V )∩Ais∗-dense in itself and so byTheorem 3.1,(V∩A)∗

The following theorem shows that(X,τ) and (X,τα_{)have the same collection of} Ᏽ-open sets.

Theorem_3.6. _If(X,τ,Ᏽ)_{is an ideal space, thenIO}(X,τ,Ᏽ)=_IO(X,τα,Ᏽ)_.

Proof. A ∈ IO(X,τ)if and only ifA ⊂ int(A∗)if and only ifA ⊂ int_α(A∗), by Lemma 1.2 if and only ifA∈IO(X,τα_).

Corollary 3.7. If (X,τ,Ᏽ)is an ideal space where Ᏽ is completely codense, then

IO(X,τ)=IO(X,τ∗_)=IO(X,τα_).

Proof. _{Follows fromCorollary 2.2(b).}

The following theorem and corollary are generalizations of [1, Theorem 2.6(iii) and Corollary 2.1(ii)], respectively.

Theorem_3.8. _Let(X,τ,Ᏽ)_{be an ideal space. If}A∈_IO(X)_andB∈τα_{, then}A∩B∈

IO(X).

Proof. A∈IO(X,τ)⇒A∈IO(X,τα)and so by [1, Theorem 2.6(ii)],A∩B∈IO(X,τα) which implies thatA∩B∈IO(X,τ).

Corollary3.9. Let(X,τ,Ᏽ)be an ideal space. IfAisᏵ-closed andBisα-closed, then A∪BisᏵ-closed.

EveryᏵ-open set is preopen but the converse need not be true [1, Example 2.3]. The following theorem characterizesᏵ-open sets in terms of preopen sets.

Theorem _3.10. _Let(X,τ,Ᏽ)_{be an ideal space and} A⊂X_{. Then the following are}

equivalent.

(a)AisᏵ-open.

(b)A⊂A∗_{andscl(A)=int(cl(A)).}

(c)A⊂A∗_{andAis preopen.}

Proof. A∈IO(X)if and only ifA⊂A∗andA⊂int(A∗)if and only ifA⊂A∗ and A⊂int(cl(A)), since cl(A)=A∗_{if and only ifA⊂A}∗_{andA∪int(cl(A))=int(cl(A))if}

and only ifA⊂A∗ _{and scl(A)=int(cl(A)). Therefore, (a) and (b) are equivalent. It is}

clear that (a) and (c) are equivalent.

Corollary3.11. Let(X,τ,Ᏽ)be an ideal space andA⊂X.

(a)IfAis semiclosed andᏵ-open, thenAis regular open.

(b)IfAis semiopen andᏵ-closed, thenAis regular closed.

(c)IfAisᏵ-open, thensint(scl(A))=int(scl(A))=int(cl(A)).

For subsets of any ideal space(X,τ,Ᏽ), openness andᏵ-openness are independent concepts [1, Examples 2.1 and 2.2]. The followingTheorem 3.12shows that the two concepts coincide for∗-perfect sets.Corollary 3.13follows from the fact that every τ∗_{-closed,Ᏽ-open set is∗-perfect.}

Theorem_3.12. _Let(X,τ,Ᏽ)_{be an ideal space and}A⊂X_.

(b)IfAis∗-perfect, thenIint(A)=int(A)and so, for∗-perfect sets, the concepts open andᏵ-open coincide.

Proof. SinceAis∗-dense in itself,A∗is∗-perfect, byTheorem 3.1. Now Iint(A∗)= A∗_∩int((A∗₎∗_)=A∗_∩int(A∗_)=int(A∗_{). This proves (a). (b) follows from (a).}

Corollary_3.13. _Let(X,τ,Ᏽ)_{be an ideal space and}A⊂X_{. If}A _isτ∗_{-closed and} Ᏽ-open, thenAis open.

In [17, Remark 4], it was stated thatᏵis codense if and only ifτ⊂IO(X). The fol-lowingTheorem 3.14(a) follows from the above result. Theorem 3.14(b) follows from

Theorem 3.6and the fact that SO(X)∩Ᏽ= {φ}if and only ifτ∩Ᏽ= {φ}.Theorem 3.15

is a characterization of completely codense ideals.

Theorem_3.14. _Let(X,τ,Ᏽ)_{be an ideal space.}

(a)IfSO(X)⊂IO(X), thenᏵis codense.

(b)Ᏽis codense if and only ifτα_⊂IO(X).

Theorem_3.15. _Let(X,τ,Ᏽ)_{be an ideal space. Then}Ᏽ_{is completely codense if and}

only ifPO(X)=IO(X).

Proof. _SupposeᏵ_{is completely codense and}G∈_PO(X)_{. Then}G⊂G∗_{, byTheorem} 2.1(c) and so cl(G)=G∗_{.G∈PO(X)impliesG⊂int(cl(G))=int(G}∗_{)and soG∈IO(X).}

Therefore, PO(X)⊂IO(X). Clearly, IO(X)⊂PO(X). Conversely, if G∈SPO(X), then there existsV ∈PO(X)such that V ⊂G⊂cl(V )and by hypothesis, V⊂V∗ and so by Lemma 1.1, cl(V )=V∗_{. Hence by Theorem 3.1, G is ∗-dense in itself and so by} Theorem 2.1,Ᏽis completely codense.

In the followingTheorem 3.16, we show that ifAisᏵ-open, then sint(A∗)is regular closed.

Theorem3.16. Let(X,τ,Ᏽ)be an ideal space andA⊂X.

(a)For every subsetAofX,cl(Iint(A))=cl(int(A∗_))=sint(A∗_).

(b)IfAisᏵ-open, thenA∗_{=cl(A)=cl(int(A}∗_))=sint(A∗_{)and sosint(A}∗_{)is regular}

closed.

Proof. IfAis a subset ofX, then sint(A∗)=A∗∩cl(int(A∗))=cl(int(A∗)). To prove the other equality, since Iint(A)=A∩int(A∗_{), cl(Iint(A))=cl(A∩int(A}∗_{))⊃cl(A)∩}

int(A∗_)=int(A∗_{)and so cl(Iint(A))⊃cl(int(A}∗_{)). To prove the reverse direction, note}

that Iint(A)⊂int(A∗_{)and so cl(Iint(A))⊂cl(int(A}∗_{)). This completes the proof of (a).}

(b) follows from (a) andTheorem 3.4(b).

A subsetAof an ideal space(X,τ,Ᏽ)isᏵ-dense [6] ifA∗_{=X. Clearly, everyᏵ-dense}

Theorem 3.17. Let(X,τ,Ᏽ) be an ideal space with a codense idealᏵandA⊂X.

Then the following are equivalent.

(a)AisᏵ-open.

(b)There is a regular open setGsuch thatA⊂GandA∗_=G∗_.

(c)A=G∩DwhereGis regular open andDisᏵ-dense.

(d)A=G∩DwhereGis open andDisᏵ-dense.

Proof. _(a)⇒_{(b). That}A_isᏵ-open impliesA⊂_int(A∗)⊂A∗_{. Let}G=_int(A∗)_{. Then} A⊂Gand int(cl(G))=int(cl(int(A∗_{)))=int(cl(int(cl(A))))=int(cl(A))=int(A}∗_)=G

and soGis regular open.G∗_=(int(A∗₎₎∗_=A∗_{, byTheorem 3.4(f).}

(b)⇒(c). LetGbe a regular open set such thatA⊂GandA∗=G∗. LetD=A∪(X−G). ThenA=G∩DwhereGis regular open. NowD∗_{=(A∪(X−G))}∗_=A∗_∪(X−G)∗₌

G∗_∪(X−G)∗_{=(G∪(X−G))}∗_=X∗_{=X, sinceᏵis codense. Therefore,DisᏵ-dense}

which proves (c). (c)⇒(d) is clear.

(d)⇒(a). SupposeA=G∩D whereGis open andD isᏵ-dense. Now G=G∩X= G∩D∗_⊂(G∩D)∗ _{and soG⊂int((G∩D)}∗_)=int(A∗_{). Therefore,A⊂G⊂int(A}∗₎

which implies thatAisᏵ-open.

The following theorem is a generalization of [1, Theorem 2.14(ii)].

Theorem _3.18. _Let(X,τ,Ᏽ) _{be an ideal space and}A⊂X_{. If}A_is Ᏽ_{-closed and}α

-open, thenA=cl(A)=int(cl(A))=cl(int(A))and soAis both regular open and regular closed.

Proof. A_isᏵ-closed⇒X−A_isᏵ-open⇒X−A⊂_int(X−A)∗⇒X−A⊂_int(_cl(X− A))⇒ X−A⊂X−cl(int(A))⇒ cl(int(A))⊂A. A is α-open⇒ A is semiopen and preopen [16]⇒cl(A)=cl(int(A))andA⊂int(cl(A)). Therefore, int(cl(A))⊂cl(A)=

cl(int(A))⊂A⊂int(cl(A))and soA=cl(A)=cl(int(A))=int(cl(A)).

4. Quasi-Ᏽ-open sets. A subsetA of an ideal space(X,τ,Ᏽ) is quasi-Ᏽ-open [2] if A⊂cl(int(A∗_{)). EveryᏵ-open set is quasi-Ᏽ-open and every quasi-Ᏽ-open set is}

semi-preopen but the converse implications need not be true [2, Examples 1 and 2]. Also, quasi-Ᏽ-openness and semiopenness (resp., preopenness) are independent concepts [2, Examples 1 and 2]. The family of all quasi-Ᏽ-open sets is denoted byQᏵO(X,τ). The following theorem gives some of the properties of quasi-Ᏽ-open sets, the proof of which is similar to the proof ofTheorem 3.4.

Theorem4.1. Let(X,τ,Ᏽ)be an ideal space andAa quasi-Ᏽ-open subset ofX. Then

(a)Ais∗-dense in itself,

(b)A∗_=cl(A)=cl∗_(A),

(c)A∗_{is∗-perfect, regular closed, andᏵ-locally closed,}

(d) cl(int(A∗_))=A∗_(Ᏽ˜_{)is∗-dense in itself,} (e)A∗_=(cl(int(A∗₎₎₎∗_=(A∗₍˜_Ᏽ))∗_(Ᏽ),

Corollary4.2. Let(X,τ,Ᏽ)be an ideal space. A subsetAofX is quasi-Ᏽ-open if

and only ifA⊂A∗_(Ᏽ˜_{)[2, Theorem 3].}

Theorem4.3. Let(X,τ,Ᏽ)be an ideal space and letUandAbe subsets ofXsuch

thatA⊂U⊂A∗_{. ThenU}∗_{is∗-perfect, and ifAis quasi-Ᏽ-open, thenUis quasi-Ᏽ-open}

and socl(int(A∗_{))is quasi-Ᏽ-open.}

Proof. _{ByTheorem 3.1,} U∗ =A∗ _and U∗ _is ∗_-perfect. A is quasi-Ᏽ-open ⇒A⊂ cl(int(A∗_))=cl(int(U∗_{)). NowU⊂A}∗_{⇒U⊂(cl(int(U}∗₎₎₎∗_{⇒U⊂cl(cl(int(U}∗₎₎₎₌

cl(int(U∗_{)). Therefore,U is quasi-Ᏽ-open. Since A⊂cl(int(A}∗_))⊂A∗_{, cl(int(A}∗_{)) is}

quasi-Ᏽ-open.

Every quasi-Ᏽ-open set is semi-preopen but the converse is not true [2]. [2, Proposi-tion 3(iii)] says that every semiopen set which is∗-dense in itself is quasi-Ᏽ-open. The followingTheorem 4.4 is a generalization of this result and shows that for∗-dense in itself, the concepts quasi-Ᏽ-open and semi-preopen are equivalent.Theorem 4.5(a) gives a characterization of codense ideals andTheorem 4.5(b) gives a characterization of completely codense ideals.

Theorem_4.4. _Let(X,τ,Ᏽ)_{be an ideal space. If}A_{is semi-preopen and} ∗-dense in

itself, thenAis quasi-Ᏽ-open.

Proof. A⊂A∗⇒cl(A)=A∗, byLemma 1.1.Ais semi-preopen⇒A⊂cl(int(cl(A))) =cl(int(A∗_{))and soAis quasi-Ᏽ-open.}

Theorem4.5. Let(X,τ,Ᏽ)be an ideal space. Then

(a)Ᏽis codense if and only ifSO(X)⊂QᏵO(X),

(b)Ᏽis completely codense if and only ifSPO(X)=QᏵO(X).

Proof. (a) SupposeᏵis codense. LetG∈SO(X). By [10, Theorem 6.1] andLemma 1.3,Gis∗-dense in itself and so by [2, Proposition 3(iii)],G∈QᏵO(X). Conversely, sup-pose that SO(X)⊂QᏵO(X). IfG∈SO(X), thenG∈QᏵO(X)and soG⊂G∗_{. Therefore,}

Ᏽis codense by [10, Theorem 6.1] andLemma 1.3.

(b) SupposeᏵis completely codense andG∈SPO(X). ThenG⊂G∗, byTheorem 2.1(c) and so cl(G)=G∗_{.G∈SPO(X)⇒G⊂cl(int(cl(G)))=cl(int(G}∗_{))and soG∈QᏵO(X).}

Therefore, SPO(X)⊂QᏵO(X). Clearly,QᏵO(X)⊂SPO(X). Conversely, ifG∈SPO(X), thenG∈QᏵO(X), by hypothesis, and soG⊂G∗_{, and so byTheorem 2.1(c),Ᏽis}

com-pletely codense.

In [2], it was established that the intersection of a quasi-Ᏽ-open set with anα-set is semi-preopen. The following theorem is a generalization of the above result.

Theorem_4.6. _Let(X,τ,Ᏽ)_{be an ideal space. Then (a)}QᏵO(X,τ)=QᏵO(X,τα)_and

(b)A∈QᏵO(X,τ)andB∈τα_{impliesA∩B∈QᏵO(X,τ).}

[2, Lemma 2] states thatW∗_{(ᏺ)⊂W for every subsetW of X in the ideal space}

(X,τ,ᏺ). That is, every subset ofX is τ∗_{-closed and soτ}∗ _{is the discrete topology.}

This is not always the case. For example, if we considerRwith the usual topologyτ and the idealᏺof nowhere dense subsets ofR, thenQ∗_{=Rand soQis notτ}∗_-closed.

Therefore, [2, Proposition 4] is no longer valid. Also, it was established that everyτ∗

-closed, quasi-Ᏽ-open set is semiopen [2, Proposition 3(iii)]. The followingTheorem 4.7(a) is a generalization of the above result and also shows that the conditionpreclosed is not necessary in [2, Proposition 5(i)], andTheorem 4.7(b) shows that [2, Proposition 3(iii)] is also true if we replace the conditionτ∗_{-closed by semiclosed.}

Theorem_4.7. _Let(X,τ,Ᏽ)_{be an ideal space and}A⊂X_.

(a)IfAisτ∗-closed and quasi-Ᏽ-open, thenAis regular closed.

(b)IfAis semiclosed and quasi-Ᏽ-open, thenAis semiopen andA∗_=A∗_(ᏺ).

Proof. (a) ThatAisτ∗-closed and quasi-Ᏽ-open impliesA=A∗. Also,A∈QᏵO(X) ⇒A⊂ cl(int(A∗_))⇒int(A∗_)⊂A∗ _⊂cl(int(A∗_{))⇒cl(int(A}∗_))⊂A∗ _{⊂ cl(int(A}∗_)).

Therefore,A=A∗=cl(int(A∗))=cl(int(A))and soAandA∗are regular closed. (b)A is semiclosed⇒int(A)=int(cl(A))by [8, Proposition 1]. ThatAis quasi-Ᏽ-open implies A⊂cl(int(A∗_{))=cl(int(cl(A)))=cl(int(A))and soAis semiopen. ByTheorem 4.1(b),}

cl(A)=A∗_{. Since int(cl(A))⊂A⊂cl(int(A}∗_{))=cl(int(cl(A))), cl(int(cl(A)))⊂cl(A)}

⊂cl(int(A∗_{))=cl(int(cl(A)))and soA}∗_{=cl(A)=cl(int(A}∗_))=A∗_(ᏺ).

The following theorem gives a characterization of quasi-Ᏽ-open sets.

Theorem_4.8. _Let(X,τ,Ᏽ)_{be an ideal space and}A⊂X_.A _{is quasi-}Ᏽ_{-open if and}

only ifA⊂A∗_andcl

α(A)=cl(int(A∗)).

Proof. _SupposeA∈QᏵO(X)_{. Then}A⊂A∗_{and cl}(A)=A∗_{. Also}A⊂_cl(_int(A∗))⇒ A⊂cl(int(cl(A)))⇒A∪cl(int(cl(A)))=cl(int(cl(A)))⇒cl_α(A)=cl(int(A∗_{)), since}

clα(A)=A∪cl(int(cl(A)))[3]. Conversely, suppose the conditions hold. Then clα(A)= cl(int(cl(A)))and so A⊂cl(int(cl(A)))=cl(int(A∗_{)). Therefore, Ais quasi-Ᏽ-open.}

The quasi-Ᏽ-interior of a subsetAin an ideal space(X,τ,Ᏽ)is the largest quasi-Ᏽ-open set contained inAand is denoted by qIint(A). The following theorem deals with the properties of the quasi-Ᏽ-interior of subsets of ideal spaces. In [11], it was estab-lished that Iint(A)=φif and only ifA∈_Ᏽ.˜ _{Theorem 4.9(c) is a partial generalization} of this result.

Theorem_4.9. _Let(X,τ,Ᏽ)_{be an ideal space and}A⊂X_{. Then}

(a) qIint(A)=A∩cl(int(A∗_{))for every subsetAofX,}

(b) ifAisα-closed, thenqIint(A)=cl(int(A∗_{))and the converse holds ifA⊂A}∗_,

(c) qIint(A)=φif and only ifA∈_Ᏽ˜_.

Proof. _(a)A∩_cl(_int(A∗))⊂_cl(_int(A∗))=_cl(_int(_int(A∗)))=_cl(_int(A∗∩(_intA∗)))⊂ cl(int((A∩int(A∗₎₎∗_{))⊂ cl(int((A∩cl(int(A}∗₎₎₎∗_{)). Therefore, A∩cl(int(A}∗_{)) is a}

quasi-Ᏽ-open, qIint(A)⊂cl(int(qIint(A))∗_)⊂cl(int(A∗_{)) and soA∩qIint(A)⊂A∩}

cl(int(A∗_{)) which implies that qIint(A) ⊂ A∩cl(int(A}∗_{)). Hence qIint(A) = A∩}

cl(int(A∗_)).

(b)A isα-closed ⇒cl(int(cl(A)))⊂A⇒cl(int(A∗_{))⊂A⇒qIint(A)=cl(int(A}∗_)).

Conversely, ifA⊂A∗_{, thenA}∗_{=cl(A). qIint(A)=cl(int(A}∗_{))⇒cl(int(A}∗_))⊂Aand

so cl(int(cl(A)))⊂Aand soAisα-closed.

(c) qIint(A)=φ⇒A∩cl(int(A∗))=φ⇒A∩int(A∗)=φ⇒Iint(A)=φ⇒A∈˜Ᏽ. Con-versely, A∈_Ᏽ˜_⇒int(A∗_{)=φ⇒cl(int(A}∗_{))=φ⇒A∩cl(int(A}∗_{))=φ⇒qIint(A)=φ.}

Acknowledgments. The authors sincerely thank Professors T. R. Hamlett and D.

A. Rose for sending some of their reprints which were helpful in the preparation of this note, and also thank the referees for their valuable suggestions.

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V. Renuka Devi: Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur 628 215, Tamil Nadu, India

D. Sivaraj: Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur 628 216, Tamil Nadu, India

E-mail address:ttn_sivaraj@sancharnet.in

T. Tamizh Chelvam: Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, Tamil Nadu, India