����������� ������ 45 (1993), 65–69

UDC 515.122.2

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A NOTE ON α-EQUIVALENT TOPOLOGIES

D. Andrijević

Abstract. This paper responds to the question of when are two topologies α-equivalent by using some recently introduced classes of sets as well as the classes of regular open sets, nowhere dense sets and dense sets.

1. Introduction

In [10] Nja^{∘}stad gave a characterization of α-equivalent topologies on a given set by means of semi-open sets. It is natural to ask whether α-equivalent topologies can be characterized by means of some other classes of subsets which are shared by α-equivalent topologies, that is by means of classes of regular open, preopen, semipreopen, nowhere dense and dense sets. We answer that question in the affirmative and show that two topologies have the same collection of α-sets if and only if they share both the semi-regularisation topology and the γ-topology.

We first recall some definitions. Let A be a subset of a topological space (X, T). The closure of A and the interior of A with respect to T are denoted by cl A and int A, respectively.

Definition. A subset A of (X, T) is called

(i) an α-set if A ⊂ int(cl(int A)), (ii) a semi-open set if A ⊂ cl(int A), (iii) a preopen set if A ⊂ int(cl A), (iv) a semi-preopen set if A ⊂ cl(int(cl A)).

The first three notions were introduced by Nja^{∘}stad [10], Levine [8] and Mashhour et al. [9], respectively. The fourth concept was introduced by Abd El-Monsef et al. [1] under the name of β-open set and it was called semi-preopen set in [3]. The classes of these sets in a space (X, T) are denoted by T_{α}, SO(T), PO(T) and SPO(T), respectively. All of these are larger than T and are closed under arbitrary unions. Nja^{∘}stad [10] showed that T_{α} is a topology on X.

For a space (X, T) the family { A ⊂ X | A ∩ B ∈ PO(T) whenever B ∈ PO(T) } will be denoted by T_{γ}. It was shown in [4] that T_{γ} is a topology on X larger than T_{α}. It will be called the γ-topology of T.

The classes of regular open sets, nowhere dense sets and dense sets in (X, T) will be denoted by RO(T), N(T) and D(T) respectively. The complement of a regular open set is called regular closed, and the complement of a dense set is called codense.

The following results will be needed in the sequel.

Proposition 1. ([10], [2] and [3]) Let (X, T) be a space. Then:

(i) T_{α} = T_{αα}, (ii) SO(T) = SO(T_{α}), (iii) PO(T) = PO(T_{α}), (iv) SPO(T) = SPO(T_{α}), (v) RO(T) = RO(T_{α}), (vi) D(T) = D(T_{α}), (vii) N(T) = N(T_{α}).

Proposition 2. ([3]) A subset of a space (X, T) is semi-preopen if and only if cl A is regular closed.

Proposition 3. ([3]) In a space (X, T), the intersection of every open and each semi-preopen set is semi-preopen.

Proposition 4. ([7]) In a space (X, T), a subset A is semi-open if and only if it is semi-preopen and int(cl A) ⊂ cl(int A).

Proposition 5. ([7]) If the topologies T and U on a set X have the same γ-topology, then they have the same class of nowhere dense sets.

Proposition 6. ([7]) Two topologies on a set X have the same class of semipreopen sets if and only if their γ-topologies are the same.

2. α-equivalent topologies

Definition. ([10]) Two topologies on a set X are called α-equivalent if their α-topologies are the same.

In the same paper Nja^{∘}stad obtained the following result on α-equivalence in terms of semi-open sets.

Proposition 7. Two topologies on a set X are α-equivalent if and only if they have the same class of semi-open sets.

We consider the same question in relation to the other classes of subsets which are shared by (X, T) and (X, T_{α}) mentioned in Proposition 1, to obtain analogous characterizations of α-equivalence. We begin with a consequence of Propositions 7 and 1.

Proposition 8. Let T and U be two α-equivalent topologies. Then:

(i) PO(T) = PO(U), (ii) SPO(T) = SPO(U), (iii) RO(T) = RO(U), (iv) D(T) = D(U), (v) N(T) = N(U).

A note on α-equivalent topologies 67

It should be noted that none of these five conditions is equivalent to the condition SO(T) = SO(U). One can easily find the examples. But it is a natural question whether there are two conditions among these five which together imply SO(T) = SO(U). We prove that any pair of independent conditions does so. From Propositions 5 and 6 it follows immediately that for the statements in Proposition 8, (i) ⟹ (ii) ⟹ (v) hold in general. The following result shows that (iv) ⟹ (v) is also true. The closure and the interior of a set A in (X, U) will be denoted by cl_{U} A and int_{U} A respectively.

Proposition 9. Let T and U be topologies on X having the same class of dense sets. Then their classes of nowhere dense sets coincide.

Proof. Suppose that A ∈ N(T) − N(U). Then G = int(cl_{U} A) ≠ ∅ because (X, T) and (X, U) have the same class of codense sets. Define W = int_{U}(G − cl A). Since A is T -nowhere dense, G − cl A is T -open and non-empty. Hence W ⊂ cl_{U} A − A and W ≠ ∅, a contradiction. Therefore N(T) ⊂ N(U). The reverse inclusion is shown in an analogous way.

In order to prove our main result we first establish two lemmas.

Lemma 1. If T and U are topologies on X sharing the classes of semi-preopen sets and regular open sets, then they have the same class of dense sets.

Proof. Assume that cl A = X. Then A ∈ SPO(T) = SPO(U) and so cl_{U} A is U-regular closed by Proposition 2. Hence cl_{U} A is T -regular closed and thus cl_{U} A = X, i.e. A ∈ D(U). The reverse inclusion is obtained analogously.

Lemma 2. Let T and U be topologies on X having the same class of dense sets, and let A be a subset of X. Then int(cl A) ⊂ cl(int A) if and only if int_{U} cl_{U} A ⊂ cl_{U} int_{U} A.

Proof. Suppose that int(cl A) ⊂ cl(int A) and let W = int_{U}(cl_{U} A − int_{U} A) be non-empty. Then G = int W ≠ ∅ because (X, T) and (X, U) share the class of codense sets. Put G_{1} = G − cl A and G_{2} = G ∩ int A and let W_{1} = int_{U} G_{1} and W_{2} = int_{U} G_{2}. We observe that W_{1} ⊂ cl_{U} A − cl A which implies W_{1} = ∅ and so int G_{1} = G_{1} = ∅. On the other hand, W_{2} ⊂ A and W_{2} ∩ int_{U} A ⊂ W ∩ int_{U} A = ∅. Hence W_{2} = ∅ and thus int G_{2} = G_{2} = ∅. Hence both G_{1} and G_{2} are empty, G ⊂ cl A − int A and so G ⊂ int(cl A) − cl(int A) = ∅, a contradiction. Therefore W = ∅, that is int_{U} cl_{U} A ⊂ cl_{U} int_{U} A.

Theorem 1. Let T and U be topologies on a set X. Then the following are equivalent:

(a) T and U are α-equivalent,

(b) RO(T) = RO(U) and PO(T) = PO(U).

(c) RO(T) = RO(U) and SPO(T) = SPO(U).

(d) RO(T) = RO(U) and N(T) = N(U).

(e) RO(T) = RO(U) and D(T) = D(U).

(f) PO(T) = PO(U) and D(T) = D(U).

(g) SPO(T) = SPO(U) and D(T) = D(U).

Proof. (a) ⟹ (b), (b) ⟹ (c) and (c) ⟹ (d) follow from Propositions 8, 5 and 6. Also, (a) ⟹ (e) and (a) ⟹ (f) follow from Proposition 8, (e) ⟹ (d) from Proposition 9 and (f) ⟹ (g) from Proposition 6.

(d) ⟹ (c): Suppose A ∈ SPO(T) and put B = A − cl_{U} int_{U} cl_{U} A. Then B ∈ N(U) = N(T) by (d). On the other hand, cl_{U} int_{U} cl_{U} A is U-regular closed and so T -regular closed. Hence B ∈ SPO(T) by Proposition 3 and thus B = ∅, that is A ∈ SPO(U).

(c) ⟹ (a): Suppose that A ∈ SO(U). According to Lemma 1, (X, T) and (X, U) share the class of codense sets and so int_{U}(A−cl(int A)) = ∅, that is int_{U} A ⊂ cl_{U} cl(int A). Therefore A ⊂ cl_{U} int_{U} A ⊂ cl_{U} cl(int A). On the other hand, cl(int A) is T -regular closed and so U-regular closed by (c). Hence A ⊂ cl(int A), that is A ∈ SO(T). The reverse inclusion is obtained analogously.

(g) ⟹ (a): Suppose that A ∈ SO(T). Then A ∈ SPO(T) and int(cl A) ⊂ cl(int A) by Proposition 4. According to Lemma 2 and (g) we have that A ∈ SPO(U) and int_{U} cl_{U} A ⊂ cl_{U} int_{U} A and thus A ∈ SO(U) again by Proposition 4. The reverse inclusion is obtained analogously.

For a space (X, T) the topology T_{s} on X which has as a base the class RO(T) is called the semiregularisation topology of (X, T). Recall that T_{ss} = T_{s} and RO(T) = RO(U) if and only if T_{s} = U_{s} [6]. Theorem 1 and Proposition 6 give the following characterization.

Theorem 2. Let T and U be topologies on a set X. Then T_{α} = U_{α} if and only if T_{γ} = U_{γ} and T_{s} = U_{s}.

Corollary 1. Let (X, T) be a space. Then T_{sα} = T_{α} if and only if T_{sγ} = T_{γ}.

It was shown in [5] that the topologies T and T_{γ} on a set X are α-equivalent if and only if they share the class of dense sets. Our Theorem 3 will be a slight improvement of this result. For convenience we first establish two simple lemmas.

Lemma 3. ([4]) Let (X, T) be a space. Then:

(i) cl_{T_{γ}} G = cl G for any T -open set G, (ii) int_{T_{γ}} F = int F for any T -closed set F.

Lemma 4. Let T and U be topologies on X such that U ⊂ T_{γ}. Then cl(int A) ⊂ cl_{U} int A for any subset A.

Proof. By the assumption and Lemma 3 we have that cl(int A) = cl_{T_{γ}} int A ⊂ cl_{U} int A.

Theorem 3. Let T and U be topologies on a set X satisfying T ⊂ U_{γ} and U ⊂ T_{γ}. Then T and U are α-equivalent if and only if they have the same class of dense sets.

Proof. Assume D(T) = D(U) and let A ∈ SO(T). According to Proposition 4 and Lemma 2 it suffices to show that A ∈ SPO(U). By Lemma 4 we have cl_{U} A ⊂ cl_{U} cl(int A) ⊂ cl_{U} int A and so cl_{U} A = cl_{U} int A. Since int A ∈ T ⊂ U_{γ} ⊂ PO(U) we have by Proposition 2 that cl_{U} A is U-regular closed and hence A ∈ SPO(U).

REFERENCES

[1] M. E. Abd El-Monsef, S. N. El-Deeb, R. A. Mahmoud, β-open sets and β-continuous mappings, Bull. Fac. Sci. Assiut Univ. 12 (1), 1983, 77–90

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

[3] D. Andrijević, Semi-preopen sets, Mat. Vesnik 38, 1986, 24–32

[4] D. Andrijević, On the topology generated by preopen sets, Mat. Vesnik 39, 1987, 367–376 [5] D. Andrijević, M. Ganster, A note on the topology generated by preopen sets, Mat. Vesnik

39, 1987, 115–119

[6] N. Bourbaki, General Topology, Part I, Addison Wesley, Reading, Massachusetts 1966

[7] M. Ganster, D. Andrijević, On some questions concerning semi-preopen sets, Jour. Inst. Math. and Comp. Sci. (Math. Ser.) 1 (2), 1988, 65–75

[8] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70, 1963, 36–41

[9] A. S. Mashhour, M. E. Abd El-Monsef, S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt 53, 1982, 47–53

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

(received 29 10 1993)

University of Belgrade, Poljoprivredni Fakultet, Nemanjina 6, P.O.Box 127, YU–11081 Beograd-Zemun, Yugoslavia