PROPERTIES OF SOME ∗-DENSE-IN-ITSELF SUBSETS

-open sets were introduced and studied by Janković and Hamlett (1990) to generalize the well-known Banach category theorem. Quasi-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, properties of these sets are further investigated and characterizations of these sets are given. Also, their relation with -dense sets and -locally closed sets is discussed. Characterizations of completely codense ideals are given in terms of semi-preopen sets.


Introduction and preliminaries.
The subject of ideals in topological spaces has been studied by Kuratowski [12] and Vaidyanathaswamy [20].An ideal Ᏽ on a topological space (X, τ) is a collection of subsets of X which satisfies that (i) A ∈ Ᏽ and B ⊂ A implies B ∈ Ᏽ and (ii) A ∈ Ᏽ and B ∈ Ᏽ implies A ∪ B ∈ Ᏽ.Given a topological space (X, τ) with an ideal Ᏽ on X and if ℘(X) is the set of all subsets of X, a set operator (•) * : ℘(X) → ℘(X), called a local function [12] of A with respect to Ᏽ and τ, is defined as follows: for A ⊂ X, A * (Ᏽ,τ) = {x ∈ X | U ∩ A ∈ Ᏽ for every U ∈ τ(x)}, where τ(x) = {U ∈ τ | x ∈ U }.We will make use of the basic facts concerning the local functions [10,Theorem 2.3] without mentioning it explicitly.A Kuratowski closure operator cl * (•) for a topology τ * (Ᏽ,τ), called the * -topology, finer than τ, is defined by cl * (A) = A ∪ A * (Ᏽ,τ) [19].When there is no chance for confusion, we will simply write A * for A * (Ᏽ,τ) and τ * or τ * (Ᏽ) for τ * (Ᏽ,τ).If Ᏽ is an ideal on X, 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.If A ⊂ X, cl(A) and int(A) will denote the closure and interior of A in (X, τ), respectively, and cl * (A) and int * (A) will denote the closure and interior of A in (X, τ * ), respectively.A subset A of a space (X, τ) is semiopen [13] if there exists an open set G such that G ⊂ A ⊂ cl(G) or, equivalently, A ⊂ cl(int(A)).The complement of a semiopen set is semiclosed.The smallest semiclosed set containing A is called the semiclosure of A and is denoted by scl(A).Also, scl(A) = A ∪ int(cl(A)) [4,Theorem 1.5(a)].The largest semiopen set contained in A is called the semi-interior of A and is denoted by sint(A).A subset A of a space (X, τ) is an α-set [15] if A ⊂ int(cl(int(A))).The family of all α-sets in (X, τ) is denoted by τ α .τ α 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 of A in (X, τ α ) are denoted by cl α (A) and int α (A), respectively.If ᏺ is the ideal of all nowhere dense subsets in (X, τ), then τ * (ᏺ,τ) = τ α and cl α (A) = A ∪ A * (ᏺ) [10].An open subset A of a space (X, τ) is said to be regular open if A = int(cl(A)).The complement of a regular open set is regular closed.A subset A of a space (X, τ) is said to be preopen [14] if A ⊂ int(cl(A)).The family of all preopen sets is denoted by PO(X, τ) or simply PO(X).The largest preopen set contained in A is called the preinterior of A and 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 set G such that A ⊂ G and cl(A) = cl(G) [7,Proposition 2.1].A subset A of a space (X, τ) is semi-preopen [4] if there exists a preopen set G such that G ⊂ 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 called semi-preclosed.The largest semi-preopen set contained in A is called the semipreinterior of A and is denoted by spint(A).Also, spint(A) = A ∩ cl(int(cl(A))) for every A of X [4].Given a space (X, τ) and ideals Ᏽ and on X, the extension of Ᏽ via [11], denoted by Ᏽ * , is the ideal given by Ᏽ = φ} is an ideal containing both Ᏽ and ᏺ and Ᏽ * ᏺ is usually denoted by Ᏽ.The following lemmas will be useful in the sequel.Lemma 1.1.Let (X, τ, Ᏽ) be an ideal space and A ⊂ X. Lemma 1.2.Let (X, τ) be a space and A ⊂ X.
Theorem 2.1.Let (X, τ, Ᏽ) be an ideal space.Then the following are equivalent.
(b)⇒(c).Suppose A ∈ SPO(X) and x ∈ A * .Then there exists an open set G containing [4,Theorem 2.7] and so by hypothesis, G ∩ A = φ which implies that x ∈ A.
(c)⇒(d).Let A ∈ Ᏽ such that spint(A) = φ.Then there exists a nonempty semi-preopen set G such that G ⊂ A and so (b) The proof follows from (a).

Ᏽ-open sets.
A subset A of an ideal space (X, τ, Ᏽ) is τ * -closed [10] (resp., * -dense in itself [9], * -perfect [9]) if A * ⊂ A (resp., A ⊂ A * , A = A * ).Clearly, A is * -perfect if and only if A is τ * -closed and * -dense in itself.The following Theorem 3.1 is useful in the sequel.Theorem 3.1.Let (X, τ, Ᏽ) be an ideal space and let U and A be subsets of X such that A ⊂ U ⊂ A * .Then U is * -dense in itself, and U * and A * are * -perfect.
Clearly, every * -perfect set is Ᏽ-locally closed.The following theorem gives a characterization of Ᏽ-locally closed sets.
where G is open, then A ⊂ A * and so by Theorem 3.1, A * is * -perfect and so A is Ᏽ-locally closed.
Every Ᏽ-open set is preopen but the converse need not be true [ For subsets of any ideal space (X, τ, Ᏽ), openness and Ᏽ-openness are independent concepts [1, Examples 2.1 and 2.2].The following Theorem 3.12 shows that the two concepts coincide for * -perfect sets.Corollary 3.13 follows 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.In [17,Remark 4], it was stated that Ᏽ is codense if and only if τ ⊂ IO(X).The following Theorem 3.14(a) follows from the above result.Theorem 3.14(b) follows from Theorem 3.6 and the fact that SO(X)∩Ᏽ = {φ} if and only if τ ∩Ᏽ = {φ}.Theorem 3.15 is a characterization of completely codense ideals.Theorem 3.14.Let (X, τ, Ᏽ) be an ideal space.
Proof.Suppose Ᏽ is completely codense and G ∈ PO(X).Then G ⊂ G * , by Theorem 2.1(c) and so cl 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) and Theorem 3.4(b).
A subset A of an ideal space (X, τ, Ᏽ) is Ᏽ-dense [6] if A * = X.Clearly, every Ᏽ-dense set is dense but the converse is not true.If G is any proper dense subset of an ideal space (X, τ, Ᏽ) where Ᏽ is the maximal ideal ℘(X), then G is not Ᏽ-dense.In particular, if Ᏽ is not codense, then X is not Ᏽ-dense and hence no subset of X is Ᏽ-dense [6].Therefore, the existence of an Ᏽ-dense set implies that the ideal is codense.The following theorem characterizes Ᏽ-open sets in terms of Ᏽ-dense sets.

Corollary 4.2. Let (X, τ, Ᏽ) be an ideal space. A subset A of X is quasi-Ᏽ-open if and only if
Theorem 4.3.Let (X, τ, Ᏽ) be an ideal space and let U and A be subsets of Every quasi-Ᏽ-open set is semi-preopen but the converse is not true [2].[2, Proposition 3(iii)] says that every semiopen set which is * -dense in itself is quasi-Ᏽ-open.The following Theorem 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 and Theorem 4.5(b) gives a characterization of completely codense ideals.
Theorem 4.5.Let (X, τ, Ᏽ) be an ideal space.Then (a) Ᏽ is codense if and only if SO(X) ⊂ QᏵO(X), (b) Ᏽ is completely codense if and only if SPO(X) = QᏵO(X).
(b) Suppose Ᏽ is completely codense and G ∈ SPO(X).Then G ⊂ G * , by Theorem 2.1(c) and so cl , by hypothesis, and so G ⊂ G * , and so by Theorem 2.1(c), Ᏽ is completely codense.
In [2], it was established that the intersection of a quasi-Ᏽ-open set with an α-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, τ α ) and (b) A ∈ QᏵO(X, τ) and B ∈ τ α implies A ∩ B ∈ QᏵO(X, τ).
[2, Lemma 2] states that W * (ᏺ) ⊂ W for every subset W of X in the ideal space (X, τ, ᏺ).That is, every subset of X is τ * -closed and so τ * is the discrete topology.This is not always the case.For example, if we consider R with the usual topology τ and the ideal ᏺ of nowhere dense subsets of R, then Q * = R and so Q is not τ * -closed.Therefore, [2,Proposition 4]

Proof. (a) That
The following theorem gives a characterization of quasi-Ᏽ-open sets.

Proof. Suppose A ∈ QᏵO(X). Then
The quasi-Ᏽ-interior of a subset A in an ideal space (X, τ, Ᏽ) is the largest quasi-Ᏽopen set contained in A and 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 established that Iint(A) = φ if and only if A ∈ 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

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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 be Ᏽ-closed.The largest Ᏽ-open set contained in A is called the Ᏽ-interior of A and is denoted by Iint(A) and Iint(A) = A∩int(A * ) [11, Theorem 4.1(3)].The following theorem gives some properties of Ᏽ-open sets.
dense in itself and by Theorem 3.1, cl(V )∩A is * -dense in itself and so by Theorem 3.1, (V ∩A) * and (cl(V ) ∩ A) * are * -perfect and so are Ᏽ-locally closed.

1 ,Theorem 3 . 10 .Corollary 3 . 11 .
Example 2.3].The following theorem characterizes Ᏽ-open sets in terms of preopen sets.Let (X, τ, Ᏽ) be an ideal space and A ⊂ X.Then the following are equivalent.(a) A is Ᏽ-open.(b) A ⊂ A * and scl(A) = int(cl(A)).(c) A ⊂ A * and A is preopen.Proof.A ∈ IO(X) if and only if A ⊂ A * and A ⊂ int(A * ) if and only if A ⊂ A * and A ⊂ int(cl(A)), since cl(A) = A * if and only if A ⊂ A * and A ∪ int(cl(A)) = int(cl(A)) if and only if A ⊂ A * and scl(A) = int(cl(A)).Therefore, (a) and (b) are equivalent.It is clear that (a) and (c) are equivalent.Let (X, τ, Ᏽ) be an ideal space and A ⊂ X.(a) If A is semiclosed and Ᏽ-open, then A is regular open.(b) If A is semiopen and Ᏽ-closed, then A is regular closed.

Theorem 3 . 16 .
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, Ᏽ is completely codense.In the following Theorem 3.16, we show that if A is Ᏽ-open, then sint(A * ) is regular closed.Let (X, τ, Ᏽ) be an ideal space and A ⊂ X.(a)For every subset A of X, cl(Iint(

Theorem 4 . 7 .
is no longer valid.Also, it was established that every τ *closed, quasi-Ᏽ-open set is semiopen [2, Proposition 3(iii)].The following Theorem 4.7(a) is a generalization of the above result and also shows that the condition preclosed is not necessary in [2, Proposition 5(i)], and Theorem 4.7(b) shows that [2, Proposition 3(iii)] is also true if we replace the condition τ * -closed by semiclosed.Let (X, τ, Ᏽ) be an ideal space and A ⊂ X.(a) If A is τ * -closed and quasi-Ᏽ-open, then A is regular closed.(b) If A is semiclosed and quasi-Ᏽ-open, then A is semiopen and A * = A * (ᏺ).