MATEMATIQKI VESNIK

59 (2007), 85–95

UDK 515.122

originalni nauqni rad research paper

RELATIONS BETWEEN SOME TOPOLOGIES

T. Hatice Yalvac

Abstract. Generalizations of openness, such as semi-open, preopen, semi-pre-open, α-open, etc. are important in topological spaces and in particular in topological spaces on which ideals are defined. α-equivalent topologies and ∗-equivalent topologies with respect to an ideal have some common properties. Relations between these aforementioned notions of openness are investigated within the framework of α-equivalence and ∗-equivalence.

1. Introduction

The subject of ideals in general topological spaces was introduced by Kuratowski [8] and Vaidyanathaswamy [18]. An ideal I on a set X is a nonempty collection of subsets of X which satisfies

(i) If A ∈ I and B ⊂ A, then B ∈ I, (ii) If A ∈ I and B ∈ I, then A ∪ B ∈ I. By (X, τ, I) we will denote a topological space (X, τ) with an ideal I on X. No separation properties are assumed on X. For a space (X, τ, I) and a subset A ⊂ X,

A^{∗}(τ, I) = { x ∈ X : U ∩ A ∉ I for every U ∈ τ(x) }

(where τ(x) = {U ∈ τ : x ∈ U}) is called the local function of A with respect to I and τ [8]. Note that cl^{∗}(A) = A ∪ A^{∗} defines a Kuratowski closure operator for a topology τ^{∗}(I) [15] on X. If there is no chance of confusion, we simply write A^{∗} or A^{∗}(I) instead of A^{∗}(τ, I), and τ^{∗} instead of τ^{∗}(I).

If I and J are ideals on X, then I ∨ J = {I ∪ J : I ∈ I and J ∈ J } is also an ideal on X [6].

In a topological space (X, τ), for any subset A, A^{o}, int A or τ int A will stand for the interior of A and \overline{A}, cl A, or τ cl A will stand for the closure of A. A subset A of a space (X, τ) is said to be semi-open (pre-open, α-open, semi-pre-open, regular open, nowhere dense, codense) if A ⊂ A\overline{o} (A ⊂ A^{\underline{o}}, A ⊂ A^{\frac{o}{o}}, A ⊂ A^{\underline{\overline{o}}}, A = A^{\underline{o}}, A^{\underline{o}} = ∅, A^{o} = ∅), respectively.

AMS Subject Classification: Primary: 54 A 05, 54 A 10; Secondary: 54 C 08, 54 C 10.

A point x of X is called a θ-interior point of A if there exists an open set U such that x ∈ U ⊂ Ū ⊂ A. Also, θ-int A will stand for the set of θ-interior points of A. A is θ-open iff A ⊂ θ-int A [10]. The family of all α-open sets in (X, τ) is a topology on X which is finer than τ and it is denoted by τ^{α}. Topologies τ and σ on X are called α-equivalent if they have the same α-open sets [14].

A supratopology A on X is a nonempty collection of subsets of X which satisfies

(i) ∅ ∈ A, X ∈ A, (ii) A is closed under arbitrary unions [10]. The A-interior (shortly, A-int) of a subset A of X is defined as

A- int A = ⋃{U : U ⊂ A, U ∈ A}

[9]. It is well known that the family of semi-open (pre-open, semi-pre-open) sets of a topological space is a supratopology on this space. If A is a supratopology on X, then

T_{A} = {A ⊂ X : A ∩ B ∈ A for each B ∈ A}

is a topology and T_{A} ⊂ A [19].

We will use the following notational conventions:

τ ∈ T op (X) ⟺ τ is a topology on X, I ∈ Id(X) ⟺ I is an ideal on X, A ∈ D(X) ⟺ A is dense in X, A ∈ CD(X) ⟺ A^{o} = ∅ (i.e. A is codense),

A ∈ NO(X) ⟺ A^{\underline{o}} = ∅ (i.e. A is nowhere dense),

A ∈ SO(X) ⟺ A ⊂ A\overline{o},

A ∈ P O(X) ⟺ A ⊂ A^{\underline{o}} ,

A ∈ αO(X) ⟺ A ⊂ A^{\frac{o}{o}} ,

A ∈ SP O(X) ⟺ A ⊂ A^{\underline{\overline{o}}} ,

A ∈ RO(X) ⟺ A = A^{\underline{o}} , A ∈ θO(X) ⟺ A is θ-open, A ∈ SC(X) ⟺ X − A is semi-open (i.e. A is semi-closed), A ∈ SR(X) ⟺ A is semi-open and semi-closed (i.e. A is semi-regular), σ ∈ [τ]^{α} ⟺ σ ∈ T op (X), and σ^{α} = τ^{α} (i.e. τ and σ are α-equivalent).

I_{n} (or I_{n}(τ)) and I_{n}(σ) will stand for the family of nowhere dense sets in X with respect to τ and σ, respectively. From now on, A^{o} and Ā will be reserved for the interior and closure of A with respect to topology τ, respectively.

In a topological space scl, sint, pcl, spcl, etc. will stand for the operations semi-closure, semi-interior, pre-closure, semi-pre-closure, respectively. Where it is necessary to indicate the topology, we will write, for example, τ-scl or σ-scl.

In the following theorem we recall some known results in the literature which will be used in this paper. They appear in [1, 2, 5, 6, 13–18].

Theorem 1.1. In any topological space (X, τ) we have the following results:

(1) For any subset A of X,

scl A = A ∪ A^{\underline{o}} ,  sint A = A ∩ A\overline{o},
pcl A = A ∪ A\overline{o},  pint A = A ∩ A^{\underline{o}} ,
spcl A = A ∪ A^{\frac{o}{o}} ,  spint A = A ∩ A^{\underline{\overline{o}}} ,
α- cl A = τ^{α} cl A = A ∪ A^{\underline{\overline{o}}} ,  α- int A = τ^{α} int A = A ∩ A^{\frac{o}{o}} [1, 2].

(2) τ ⊂ T_{SO(X)} ∩ T_{P O(X)} ∩ T_{SP O(X)} [1, 2]. In the remaining results given below, σ ∈ T op (X) and I, J ∈ Id(X).

(3) If τ ⊂ σ ⊂ τ^{α}, then σ ∈ [τ]^{α} [14].

(4) τ ⊂ τ^{∗}(I) and (τ^{∗}(I))^{∗}(I) = τ^{∗}(I).

(5) A^{∗}(τ, I) is τ-closed and A^{∗}(τ, I) = A^{∗}(τ^{∗}(I), I) for each A ⊂ X,

(6) I ∈ I ⟹ I is τ^{∗}(I)-closed and I^{∗} = ∅.

(7) I ⊂ J ⟹ A^{∗}(τ, J) ⊂ A^{∗}(τ, I) for each A ⊂ X and τ^{∗}(I) ⊂ τ^{∗}(J).

(8) τ ⊂ σ ⟹ A^{∗}(σ, I) ⊂ A^{∗}(τ, I) for each A ⊂ X and τ^{∗}(I) ⊂ σ^{∗}(I).

(8) τ ∩ I = {∅} ⟺ τ^{∗}(I) ∩ I = {∅} ⟺ X = X^{∗} ⟺ U ⊂ U^{∗} for each U ∈ τ.

(9) B(τ^{∗}(I)) = {U − I : U ∈ τ, I ∈ I} is a base for the topology τ^{∗}(I).

(10) If τ ∩ I = {∅}, then for each U ∈ τ and for each I ∈ I, we have Ū = U^{∗} = (U − I)^{∗} = (U − I)^{−} = τ^{∗}- cl(U − I) = τ^{∗}- cl U.

(11) τ^{∗}(I ∨ J) = (τ^{∗}(I))^{∗}(J) [6].

(12) τ^{∗}(I_{n}) = τ^{α}, τ ∩ I_{n} = {∅}, P O(X, τ) ∩ I_{n} = {∅}.

(13) A ⊂ B ⟹ A^{∗}(I) ⊂ B^{∗}(I).

(14) (A − I)^{∗} = A^{∗} for each A ⊂ X and each I ∈ I.

(15) A^{∗}(I_{n}) = A^{\underline{\overline{o}}} for each A ⊂ X.

(16) If τ ∩ I = {∅}, then for each U ∈ τ^{∗}(I) we have τ-cl U = τ^{∗}(I)- cl U.

2. Relations between topologies and some special sets

Firstly, some relations between families such as SO(X), P O(X), SP O(X), etc. on a set X with two topologies are investigated. Then, these relations will be carried over to topological spaces on which ideals are defined. Some known results will be obtained by a different method.

Theorem 2.1. Let τ, σ, ω ∈ T op (X). Then we have the following results.

(1) If τ ⊂ σ ⊂ SP O(X, τ), then: (a) SP O(X, σ) ⊂ SP O(X, τ), (b) I_{n}(τ) ⊂ I_{n}(σ).

(2) If τ ⊂ σ ⊂ P O(X, τ), then P O(X, σ) ⊂ P O(X, τ), and the relations (a) and (b) in (1) are valid.

(3) I. If τ ⊂ σ, and τ cl U = σ cl U for each U ∈ τ, then (a) For each A ⊂ X we have

τ cl(τ int A) = σ cl(τ int A) ⊂ σ cl(σ int A), σ int(σ cl A) ⊂ σ int(τ cl A) = τ int(τ cl A), τ int(τ cl(τ int A)) ⊂ τ int(σ cl(σ int A)) ⊂ σ int(σ cl(σ int A)), σ cl(σ int(σ cl A)) ⊂ σ cl(τ int(τ cl A)) = τ cl(τ int(τ cl A)).

(b) For each U ∈ SO(X, τ), we have τ cl U = σ cl U = τ cl(τ int U) = σ cl(σ int U),

(c) SO(X, τ) ⊂ SO(X, σ),

(d) P O(X, σ) ⊂ P O(X, τ),

(e) SP O(X, σ) ⊂ SP O(X, τ),

(f) αO(X, τ) ⊂ αO(X, σ),

(g) I_{n}(τ) ⊂ I_{n}(σ),

(h) RO(X, τ) ⊂ RO(X, σ). II. If τ ⊂ σ, and τ cl U = σ cl U for each U ∈ σ, then in addition to the results

given in (3.I) above, we have RO(X, τ) = RO(X, σ), SR(X, τ) = SR(X, σ), and θO(X, τ) = θO(X, σ),

(4) If τ ⊂ σ ⊂ SO(X, τ) and τ cl U = σ cl U for each U ∈ τ, then τ cl U = σ cl U for each U ∈ σ. So, the results in (3) are valid.

(5) τ ⊂ ω ⊂ σ, and τclU = σ cl U for each U ∈ σ, then τ cl U = σ cl U = ω cl U for each U ∈ σ (hence for each U ∈ ω). So, results similar to those given in (3) are valid for τ, ω and σ.

Proof. (1a) Let τ ⊂ σ ⊂ SP O(X, τ) and U ∈ SP O(X, σ). We have:

U ⊂ σ cl(σ int(σ cl U)) ⊂ τ cl(σ int(τ cl U)) ⊂ τ cl(τ cl(τ int(τ cl(σ int(τclU))))

⊂ τ cl(τ int(τclU)).

(1b) The proof is clear from Corollary 2.13 below.

2. Let τ ⊂ σ ⊂ P O(X, τ) and U ∈ P O(X, σ). We have:

U ⊂ σ int(σ cl U) ⊂ σ int(τ cl U) ⊂ τ int(τ cl(σ int(τ cl U))) ⊂ τ int(τ cl(τ cl U))

= τ int(τ cl U).

So, U ∈ P O(X, τ). Since P O(X, τ) ⊂ SP O(X, τ), the results in (1) are valid. (3Ib) Let U ∈ SO(X, τ). Since U ⊂ τ cl(τ int U) and σclU ⊂ τ cl U, we have

σ cl U ⊂ τ cl U = τ cl(τ int U) = σ cl(τ int U) ⊂ σ cl(σ int U) ⊂ σ cl U.

Hence we have σ cl U = τ cl U = τ cl(τ int U) = σ cl(σ int U).

(3Ic–f). These are clear from (3Ia). (3Ig) This is clear from (1a) since τ ⊂ σ ⊂ SP O(X, τ). (3Ih) Let U ∈ RO(X, τ). We have U = τ int(τ cl U) and U ∈ SO(X, τ). Now

if we use (3Ib), we obtain that σ int(σ cl U) = σ int(τ cl U) = τ int(τ cl U) = U.

(3II) Under the hypothesis, since RO(X, τ) = RO(X, σ)[11], it is clear that SR(X, τ) = SR(X, σ) [7], and θO(X, τ) = θO(X, σ)[12].

(4). The proof is clear from (3Ib).

Corollary 2.2. If τ ∩ I = {∅} for τ ∈ T op (X) and I ∈ Id(X), then

(a) The results (3) in Theorem 2.1. are valid by taking τ^{∗}(I) instead of σ.

(b) For J ∈ Id(X) and ω ∈ T op (X), if ω ∩ J = {∅} and ω^{∗}(J) = τ^{∗}(I), then (X, τ), (X, ω), (X, ω^{∗}(J)) and (X, τ^{∗}(I)) have the same RO(X), SR(X) and θO(X) sets.

(c) If τ^{∗}(I) = σ^{∗}(I) for σ ∈ T op (X), then the results (3) in Theorem 2.1. are valid by taking σ^{∗}(I) instead of σ, and then (X, τ), (X, τ^{∗}(I), (X, σ) and (X, σ^{∗}(I)) have the same RO(X), SR(X) and θO(X) sets.

The following theorem and corollaries can be obtained by using Lemma 2.7. below and the results of Andrijević given in [1,2,3]. We note that Corollary 2.6.(1) was given by Rose and Hamlett using a different method [16]. We will obtain these results by using the results given here.

Theorem 2.3. Let τ, σ ∈ T op (X). If τ ⊂ σ ⊂ SO(X, τ), and τ cl U = σ cl U for each U ∈ σ, then we have the following results.

(1) For each A ⊂ X, we have

(a) τ int(τ cl A) = σ int(σ cl A),

(b) τ cl(τ int A) = σ cl(σ int A),

(c) τ cl(τ int(τ cl A)) = σ cl(σ int(σ cl A)),

(d) τ int(τ cl(τ int A)) = σ int(σ cl(σintA)).

(2) (X, τ) and (X, σ) have the same SO(X), P O(X), SP O(X), RO(X), SR(X), NO(X), D(X), αO(X), CD(X) and θO(X) sets.

Proof. (1a) Let A ⊂ X. Then σ int(σ cl A) ⊂ σ int(τ cl A) = τ int(τ cl A). Since τ int(τ cl A) ∈ σ and τ int(τ cl A) ⊂ A ∪ τ int(τ cl A) = τ scl A ⊂ σ cl A, we have τ int(τ cl A) = σ int(τ int(τ cl A)) ⊂ σ int(σ cl A). Hence τ int(τ cl A) = σ int(σ cl A).

The remaining proofs are clear.

Corollary 2.4. Let τ ∈ T op (X), I ∈ Id(X). If τ ∩ I = {∅} and τ^{∗}(I) ⊂ SO(X, τ), then the results of Theorem 2.3 are satisfied by taking τ^{∗}(I) instead of σ.

Corollary 2.5. Since τ ∩ I_{n} = {∅} and τ^{∗}(I_{n}) = τ^{α} ⊂ SO(X, τ), the results of the above theorem are satisfied by taking τ^{α} instead of σ.

Corollary 2.6. If σ^{α} = τ^{α} (i.e. if σ ∈ [τ]^{α} in the sense of Njåstad [14]), then we have the following results.

(1) For each A ⊂ X, we have

(a) τ int(τ cl A) = τ^{α} int(τ^{α} cl A) = σ^{α} int(σ^{α} cl A) = σ int(σ cl A),

(b) τ cl(τ int A) = τ^{α} cl(τ^{α} int A) = σ^{α} cl(σ^{α} int A = σ cl(σ int A),

(c) τ int(τ cl(τ int A)) = τ^{α} int(τ^{α} cl(τ^{α} int A)) = σ^{α} int(σ^{α} cl(σ^{α} int A)).

(d) τ cl(τ int(τ cl A)) = τ^{α} cl(τ^{α} int A) = (σ^{α} cl A(σ^{α} int(σ^{α} cl A) = σ cl(σ int(σ cl A)).

(2) (X, τ), (X, τ^{α}), (X, σ^{α}) and (X, σ) have the same SO(X), P O(X), SP O(X), NO(X), D(X), CD(X), αO(X), RO(X), SR(X) and θO(X) sets [3].

Lemma 2.7. Let τ, σ ∈ T op (X) and τ ⊂ σ. Then, σ ⊂ SO(X, τ) and τ cl U = σ cl U for each U ∈ σ iff σ ∈ [τ]^{α}.

Njåstad defined α-equivalent topologies and ∗-equivalent topologies in [14], [15], respectively. Njåstad showed that if τ ⊂ σ ⊂ τ^{α} for τ, σ ∈ T op (X), then τ and σ are α-equivalent. For τ, σ ∈ T op (X), I ∈ Id(X), if τ^{∗}(I) = σ^{∗}(I), then we say that σ and τ are *I-equivalent.

The α-equivalence or *I-equivalence of topologies on a set on which ideals are defined is important.

For any ideal I on (X, τ), τ^{∗}(I) and τ^{∗}(I)^{α} are α-equivalent. We know that τ^{∗}(I)^{α} = (τ^{∗}(I))^{∗}(I_{n}(τ^{∗}(I))) = τ^{∗}(I V I_{n}(τ^{∗}(I))). Hence, for each ideal J such that J ⊂ I_{n}(τ^{∗}(I)), τ^{∗}(I), τ^{∗}(I ∨ J) and τ^{∗}(I ∨ I_{n}(τ^{∗}(I))) are α-equivalent.

At the same time, for any ideal I, since τ^{∗}(I) ∩ I_{n}(τ^{∗}(I))) = {∅} and τ ⊂ τ^{∗}(I), we have that that τ ∩ I_{n}(τ^{∗}(I)) = {∅}. Hence for any ideal J such that J ⊂ I_{n}(τ^{∗}(I)) we have τ ∩ J = {∅}. And, if τ ∩ I = {∅}, then we know that I_{n}(τ) ⊂ I_{n}(τ^{∗}(I)). Now, we can give the following result.

Theorem 2.8. Let I be an ideal on (X, τ) and J any ideal such that J ⊂ I_{n}(τ^{∗}(I)). Then the following are equivalent.

(a) τ ∩ I = {∅}

(b) I ⊂ I_{n}(τ^{∗}(I))

(c) I ∨ I_{n} ⊂ I_{n}(τ^{∗}(I))

(d) I ∨ J ⊂ I_{n}(τ^{∗}(I))

(e) I ∨ I_{n} ∨ J ⊂ I_{n}(τ^{∗}(I))

(f) τ ∩ (I ∨ J) = {∅}

(g) τ ∩ (I ∨ J ∨ I_{n}) = {∅}

(h) τ ∩ (I ∨ I_{n}) = {∅}.

Proof. (a) ⟹ (b) Let I ∈ I. Then I is τ^{∗}(I)-closed and since τ^{∗}(I)∩I = {∅} we have that τ^{∗}(I)-int I = ∅. So, τ^{∗}(I)-int(τ^{∗}(I)-cl I) = ∅ and I ∈ I_{n}(τ^{∗}(I)).

The remaining proofs are clear. We deduce that if τ ∩ I = {∅} for an ideal I, then I ∨ I_{n}(τ^{∗}(I)) = I_{n}(τ^{∗}(I)),

and (τ^{∗}(I))^{α} = τ^{∗}(I_{n}(τ^{∗}(I))).

Corollary 2.9. If τ ∩I = {∅}, then for any ideal J satisfying J ⊂ I_{n}(τ^{∗}(I)) we have that τ^{∗}(I), τ^{∗}(I ∨ J), τ^{∗}(I ∨ J ∨ I_{n}(τ^{∗}(I))) and τ^{∗}(I_{n}(τ^{∗}(I))) are all α-equivalent.

Several statements equivalent to I ⊂ I_{n} have been given in the literature. Since τ and τ^{∗}(I) are α-equivalent when I ⊂ I_{n}, we give some further conditions for I ⊂ I_{n}, in the following theorem.

Theorem 2.10. Let I be an ideal on (X, τ) and I_{n} the ideal of nowhere dense sets in (X, τ). Then the following are equivalent.

(1) I ⊂ I_{n},

(2) τ ∩ I = {∅} and τ and τ^{∗}(I) are α-equivalent,

(3) τ ∩ I = {∅} and τ^{∗}(I) ⊂ τ^{α},

(4) τ ∩ I = {∅} and τ^{∗}(I) ⊂ SO(X, τ),

(5) τ ∩ I = {∅} and SO(X, τ^{∗}(I)) ⊂ SO(X, τ),

(6) τ ∩ I = {∅} and P O(X, τ) ⊂ P O(X, τ^{∗}(I)),

(7) τ ∩ I = {∅} and SP O(X, τ) ⊂ SP O(X, τ^{∗}(I)),

(8) τ ∩ I = {∅} and D(X, τ) ⊂ D(X, τ^{∗}(I)),

(9) A^{∗}(I_{n}) ⊂ A^{∗}(I) for each A ⊂ X,

(10) A^{\underline{o}} ⊂ A^{∗}(I) for each A ∈ D(X, τ),

(11) τ ∩ I = \{∅\} andA^{\underline{o}} ⊂ τ^{∗}(I)-cl A for each A ⊂ X,

(12) τ ∩ I = {∅} and I ⊂ SC(X, τ).

Proof. (1) ⟹ (2) Let I ⊂ I_{n}. We have, τ ⊂ τ^{∗}(I) ⊂ τ^{∗}(I_{n}) = τ^{α}, so τ and τ^{∗}(I) are α-equivalent from Theorem 1.1.(3).

(2) ⟹ (1) If τ^{∗}(I) ∈ [τ]^{α}, then we have I_{n}(τ) = I_{n}(τ^{∗}(I)). If τ ∩ I = {∅}, then I ⊂ I_{n}(τ^{∗}(I)). Hence, I ⊂ I_{n}, if τ ∩ I = {∅} and τ^{∗}(I) ∈ [τ]^{α}.

(2) ⟺ (3) ⟺ (4) Clear.

(2) ⟺ (5) ⟺ (6) ⟺ (7) ⟺ (8) Clear from [3, Theorem 1] and Corollary 2.2

(1) ⟺ (9) Known from the literature.

(9) ⟹ (10) Clear.

(10) ⟹ (8) Since X ∈ D(X, τ), we have X^{\underline{o}} = X ⊂ X^{∗}(I), X = X^{∗}(I) and hence τ ∩ I = \{∅\}. For A ∈ D(X, τ) we deduce Ā = X, A^{\underline{o}} = X, A^{∗}(I) = X and τ^{∗}(I)-cl A = A ∪ A^{∗}(I) = X. Hence we have D(X, τ) ⊂ D(X, τ^{∗}(I)).

(4) ⟹ (11) We have scl A ⊂ τ^{∗}(I)-cl A for each A ⊂ X. Since scl A = A∪A^{\underline{o}}, the result is clear.

(11) ⟹ (3) Under the hypothesis of (11) we obtain A^{\underline{\overline{o}}} = τ^{∗}(I)-cl A^{\underline{o}} ⊂ τ^{∗}(I)-cl A and τ^{α}-cl A = A ∪ A^{\underline{\overline{o}}} ⊂ τ^{∗}(I)-cl A for each subset A. Hence we have τ^{∗}(I) ⊂ τ^{α}.

(4) ⟹ (12) We know that each I ∈ I is τ^{∗}(I)-closed. Since τ^{∗}(I) ⊂ SO(X, τ), it follows that each I ∈ I is τ-semiclosed.

(12) ⟹ (4) If I ⊂ SC(X, τ), then U − I ∈ SO(X, τ) for any U ∈ τ and any I ∈ I. So, SO(X, τ) contains a base of τ^{∗}(I) (from Theorem 1.1.(10)). Hence τ^{∗}(I) ⊂ SO(X, τ).

Corollary 2.11. If I ⊂ I_{n}, then we have the following results.

(a) I_{n}(τ^{∗}(I)) = I_{n}(τ),

(b) σ ∈ [τ]^{α} iff σ ∈ [τ^{∗}(I)]^{α},

(c) If σ^{∗}(I) = τ^{∗}(I), then τ, τ^{∗}(I) and σ^{∗}(I) are all α-equivalent.

Some other statements equivalent to I ⊂ I_{n} can be seen from Corollary 2.13. and Corollary2.15.

If A is a supratopology and I an ideal on X, then it is clear that A ∩ I = {∅} iff A-int I = ∅ for each I ∈ I. In the following theorem, the results are clear and almost all of them are known.

Theorem 2.12. Let (X, τ) be a topological space. Then we have the following results for any A ⊂ X.

(1) A^{\underline{o}} = ∅ ⟺ pre- int A = ∅ ⟺ A^{\underline{\overline{o}}} = ∅ ⟺ semi-pre- int A = ∅,

(2) A^{\frac{o}{o}} = ∅ ⟺ α- int A = ∅ ⟺ A^{o} = ∅ ⟺ A\overline{o} = ∅ ⟺ semi- int A = ∅,

(3) A\overline{o} = X ⟺ pre- cl A = X ⟺ A^{\frac{o}{o}} = X ⟺ semi-pre- cl A = X,

(4) A^{\underline{\overline{o}}} = X ⟺ α- cl A = X ⟺ Ā = X ⟺ A^{\underline{o}} = X ⟺ scl A = X.

Clearly, in a topological space (X, τ), for any x ∈ X, {x} ∉ I_{n} iff pre-int {x} ≠ ∅ iff semi-pre-int {x} ≠ ∅ iff {x} is pre-open iff {x} is semi-pre-open.

In the following corollary we assume that the necessary ideals are defined on the topological space (X, τ).

Corollary 2.13. We have the following results.

(1) I_{n} = \{A : A^{\underline{o}} = ∅\} = \{A : pre- int A = ∅\} = \{A : semi-pre- int A = ∅\} = \{A : A^{\underline{\overline{o}}} = ∅\} = CD(X, τ) ∩ SC(X, τ).

(2) I_{n} ∩ P O(X, τ) = {∅} and I_{n} ∩ SP O(X, τ) = {∅}.

(3) I ⊂ I_{n} iff I ∩ P O(X, τ) = {∅} [4] iff I ∩ SP O(X, τ) = {∅}.

(4) For σ ∈ T op (X), if P O(X, σ) ⊂ P O(X, τ) or SP O(X, σ) ⊂ SP O(X, τ), then I_{n}(τ) ⊂ I_{n}(σ).

(5) CD(X) = \{A : A^{o} = ∅\} = \{A : A^{\frac{o}{o}} = ∅\} = \{A : α- int A = ∅\} = \{A : semi- int A = ∅\} = \{A : A\overline{o} = ∅\}. (6) For σ ∈ T op (X), if SO(X, τ) ⊂ SO(X, σ) or τ^{α} ⊂ σ^{α}, then D(X, σ) ⊂ D(X, τ) and CD(X, σ) ⊂ CD(X, τ).

(7)(a) τ ∩I = {∅} iff SO(X, τ)∩I = {∅} [4] iff τ^{α}∩I = {∅} iff τ^{∗}(I)∩I = {∅} iff SO(X, τ^{∗}(I)) ∩ I = {∅} iff τ^{∗}(I)^{α} ∩ I = {∅}.

(b) Let σ ∈ T op (X) and σ^{∗}(I) = τ^{∗}(I), then we have that τ ∩ I = {∅} iff σ ∩ I = {∅}.

(c) Let σ ∈ T op (X) and σ ∈ [τ^{∗}(I)]^{α}, then we have that τ ∩ I = {∅} iff σ ∩ I = {∅}.

Now, by combining the results given above with the following facts,

(i) If τ ⊂ σ ⊂ τ^{∗}(I) for σ ∈ T op (X), then σ^{∗}(I) = τ^{∗}(I), (ii) If τ^{∗}(I) ⊂ σ ⊂ (τ^{∗}(I))^{α} = (τ^{∗}(I))^{∗}(I_{n}(τ^{∗}(I))) then σ ∈ [τ^{∗}(I))]^{α},

we can obtain several conditions equivalent to τ ∩ I = {∅}.

For a supratopology A on X, T_{A} will stand for the topology

{U : A ∈ A ⟹ U ∩ A ∈ A} [19].

We know that τ ⊂ T_{P O(X,τ)}, τ ⊂ T_{SO(X,τ)}, τ ⊂ T_{SP O(X,τ)} and τ ⊂ τ^{α} = T_{τ^{α}} [1–3].

Theorem 2.14. Let (X, τ) be a topological space, I ∈ Id(X) and A a supratopology on X. Then we have the following results.

(1) If τ ∩ I = {∅} and A^{o} ≠ ∅ for each A ∈ A − {∅}, then A ∩ I = {∅}.

(2) If A ∩ I = {∅} and τ ⊂ T_{A}, then

(a) A ⊂ A^{∗} for each A ∈ A,

(b) Ā = A^{∗} = τ^{∗}-cl A for each A ∈ A. (3)(a) If A^{∗}(I) ≠ ∅ for each A ∈ A − {∅}, then A ∩ I = {∅},

(b) If τ ⊂ T_{A}, then A ∩ I = {∅} iff A ⊂ A^{∗}(I) for each A ∈ A.

(4) A ∩ I_{n} = \{∅\} iff A^{\underline{o}} ≠ ∅ for each A ∈ A − \{∅\}.

(5) If A ∩ I_{n} = {∅} and τ ⊂ T_{A}, then A ⊂ SP O(X, τ).

(6) If P O(X, τ) ⊂ A ⊂ SP O(X, τ), then A∩I_{n} = {∅} iff I ⊂ I_{n} iff A ⊂ A^{∗}(I) for each A ∈ A.

(7) If τ ⊂ A ⊂ SO(X, τ^{∗}(I)), then A ∩ I = {∅} iff τ ∩ I = {∅} iff A ⊂ A^{∗}(I) for each A ∈ A.

Proof. (2a) Let A ∈ A and x ∈ A. If x ∈ U ∈ τ, then since τ ⊂ T_{A}, we have ∅ ≠ U ∩ A ∈ A. Hence, U ∩ A ∉ I. So, we have x ∈ A^{∗}.

(2b) We know that A^{∗} is τ-closed, A^{∗} ⊂ Ā and τ^{∗}clA = A ∪ A^{∗}. Result is clear from (a).

(3a) It is known that I^{∗} = ∅ for any ideal I and for each I ∈ I. Now, result is clear.

(3b) Clear from (3a) and (2a)

(4) Clear from Corollary 2.13.

(5) Let A ∈ A. From (3)(b) we have A ⊂ A^{∗}(I_{n}) = A^{\underline{\overline{o}}}. Hence, A ∈ SP O(X, τ).

(6) Let P O(X, τ) ⊂ A ⊂ SP O(X, τ). If A ∩ I = {∅} then P O(X, τ) ∩ I = {∅} and hence I ⊂ I_{n}. If I ⊂ I_{n}, then SP O(X, τ) ∩ I = {∅} and hence A ∩ I = {∅}.

If I ⊂ I_{n}, then SP O(X, τ) ∩ I = {∅} . Now, from Corollary 2.15.(1) below, and since A ⊂ SP O(X, τ), we have A ⊂ A^{∗}(I) for each A ∈ A.

If A ⊂ A^{∗}(I) for each A ∈ A, then we have A ⊂ A^{∗}(I) for each A ∈ P O(X, τ). So, from Corollary 2.15(1), we have I ⊂ I_{n}.

(7) Let τ ⊂ A ⊂ SO(X, τ^{∗}(I)). If A∩I = {∅}, then τ ∩I = {∅}. If τ ∩I = {∅}, then from Corollary 2.13(7) we have SO(X, τ^{∗}(I)) ∩ I = {∅}. So, A ∩ I = {∅}.

If τ ∩ I = {∅}, then from Corollary 2.15(2) below we have A ⊂ A^{∗}(I) for each A ∈ SO(X, τ^{∗}(I)). So, A ⊂ A^{∗}(I) for each A ∈ A.

If A ⊂ A^{∗}(I) for each A ∈ A, then from Theorem 1.1.(9) we have U ⊂ U^{∗} for each U ∈ τ, and τ ∩ I = {∅}.

Corollary 2.15. We have the following results.

(1) P O(X, τ)∩I = {∅} iff A ⊂ A^{∗}(I) for each A ∈ P O(X, τ) [4] iff A ⊂ A^{∗}(I) for each A ∈ SP O(X, τ).

(2) τ ∩ I = {∅} iff A ⊂ A^{∗}(I) for each A ∈ SO(X, τ). [4] iff A ⊂ A^{∗}(I) for each A ∈ τ^{α} iff A ⊂ A^{∗}(I) for each A ∈ SO(X, τ^{∗}(I)) iff A ⊂ A^{∗}(I) for each A ∈ τ^{∗}(I)^{α}.

Proof. The proofs are clear from Theorem 1.1.(2), Theorem 2.14(3b) and Theorem 1.1(5).

Corollary 2.16. We have the following results.

(1) If P O(X, τ) ∩ I = {∅}, then τ-cl A = A^{∗} = τ^{∗}-cl A for each A ∈ SP O(X, τ).

(2) If τ ∩ I = {∅}, then τ-cl A = A^{∗} = τ^{∗}-cl A for each A ∈ SO(X, τ^{∗}(I)).

Lemma 2.17. Let (X, τ) be topological space, A a supratopology on X such that τ ⊂ T_{A}. If A- int B = ∅ for a subset B, then (A ∩ B)^{−} ⊂ (A − B)^{−} and Ā = (A − B)^{−} for each A ∈ A.

Proof. Let B ⊂ X, A- int B = ∅, A ∈ A, x ∈ (A ∩ B)^{−} and x ∈ U ∈ τ. Then U ∩A∩B ≠ ∅ and U ∩A ≠ ∅. Since τ ⊂ T_{A}, we have ∅ ≠ U ∩A ∈ A. So U ∩A ⊄ B, and (A − B) ∩ U = A ∩ U − B ≠ ∅. Hence x ∈ (A − B)^{−}.

This now gives Ā = (A − B)^{−} ∪ (A ∩ B)^{−} = (A − B)^{−}.

Corollary 2.18. Let I ∈ Id(X) and A a supratopology on X. If A∩I = {∅}, then we have the following results.

(a) If τ ⊂ T_{A}, then Ā = (A − I)^{−} for each A ∈ A and for each I ∈ I.

(b) If τ^{∗}(I) ⊂ T_{A}, then Ā = (A − I)^{−} and τ^{∗}-clA = τ^{∗}-cl(A − I) for each A ∈ A and for each I ∈ I.

Proof. (a) If I ∈ I we obtain A- int I = ∅. The proof is now clear from Lemma 2.17.

(b) If τ^{∗} ⊂ T_{A} then we obtain τ ⊂ τ^{∗} ⊂ T_{A}. The proof is now clear from Lemma 2.17.

Corollary 2.19. Let (X, τ) be a topological space and I ∈ Id(X). Then we have the following results.

(1) If P O(X, τ) ∩ I = {∅}, then Ā = (A − I)^{−} = A^{∗} = τ^{∗}-clA = (A − I)^{∗} for each I ∈ I and for each A ∈ SP O(X, τ).

(2) If τ ∩I = {∅}, then Ā = (A−I)^{−} = A^{∗} = τ^{∗}-cl A = τ^{∗}-cl(A−I) = (A−I)^{∗} for each I ∈ I and for each A ∈ SO(X, τ^{∗}(I)).

(3) If τ ∩ I = {∅}, then we have τ-scl A = τ^{∗}-scl A for each I ∈ I and for each A ∈ SO(X, τ^{∗}(I)).

Proof. (1),(2) Clear from Corollory 2.16, Corollary 2.18 and Theorem 1.1.(15).

(3) From (2) we will have A^{\underline{o}} = τ-int(τ^{∗}-cl A) = τ^{∗}-int(τ^{∗}-cl A) and τ-scl A = A ∪ A^{\underline{o}} = A ∪ τ^{∗}-int(τ^{∗}-cl A) = τ^{∗}-scl A.

REFERENCES

[1] D. Andrijević, Some properties of the topology of α-sets, Mat. Vesnik, 36 (1984), 1–10.

[2] D. Andrijević, A note on preopen sets, Supplemento ai Rendiconti del Circolo Matematica di Palermo, Serie II, 18, (1988), 195–201.

[3] D. Andrijević, A note on α-equivalent topologies, Mat. Vesnik, 45 (1993), 65–69.

[4] R. Devi, D. Sivaraj and T. Chelvam, Codense and completely codense ideals, Acta Math. Hungar., 108, 3, (2005),197–205.

[5] E. Hayashi, Topologies defined by local properties, Math. Ann., 156 (1964), 205–215.

[6] D. Janković and T.R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295–310,

[7] D. Janković and I.L. Reilly, On semi seperation properties, Indian J. Math., 6, (1985), 957– 967.

[8] K. Kuratowski, Topologie I, Warszawa, 1933.

[9] P.E. Long and L.L. Herrington, The τ_{θ}-topology and faintly continuous functions, Kyungpook Math. J., 22 (1982), 7–14.

[10] A.S. Mashhour, T.H. Khedr and S.Abd El-Bakkey, On supra-R_{o} and supra-R_{1} spaces, Indian J. Pure Appl. Math., 16 (1985), 1300–1306.

[11] J. Mioduszewkski and L. Rudolf, H-closed and externally disconnected Hausdorff spaces, Dissertationes Math., 66 (1969), 1–55.

[12] M.N. Mukherjee and C.K. Basu, θ-equivalent spaces- A new approach to RO-equivalent spaces and semi-reguler properties, Indian J. pure appl. Math., 22, 9(1991), 745–750.

[13] L. Newcomb, Topologies which are compact mudulo an ideal, Ph.D. Dissertation, Univ. of Cal. at. Santa Barbara (1967).

[14] O. Nja^{∘}stad, On some classes of nearly open sets, Pacific J. Math. 15, 3 (1965), 961–970.

[15] O. Nja^{∘}stad, Remarks on topologies defined by local properties, Avh. Norske vid.-Akad. Oslo I(N.S), 8(1966), 1–16.

[16] D. Rose and T.R. Hamlett, Ideally equivalent topologies and semitopological properties, Math. Choronicle 20 (1991), 149–156.

[17] P. Samuels, A topology formed from a given topology and ideal, J. London Math. Soc., 2,10 (1975), 409–416.

[18] R. Vaidyanathaswamy, The localization theory in set-topology, Proc. Indian Acad. Sci., 20 (1945), 51–61.

[19] T.H. Yalvac, Relations between new topologies obtained from old ones, Acta Math. Hungar., 64, 3 (1994), 231–235.

(received 06.01.2006, in revised form 03.09.2007)

T.Hatice YALVAC, Hacettepe University, Faculty of Science, Department of Mathematics, 06532 Beytepe, Ankara-Turkey.