GanSteAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 8, No. 2, 2007 pp. 243-247 On bτ -closed sets Maximilian Ganster and Markus Steiner Abstract. This paper is closely related to the work of Cao, Green- wood and Reilly in [10] as it expands and completes their fundamental diagram by considering b-closed sets. In addition, we correct a wrong assertion in [10] about βgs-spaces. 2000 AMS Classification: 54A05 Keywords: b-closed, qr-closed 1. Introduction and Preliminaries In recent years quite a number of generalizations of closed sets has been considered in the literature. We recall the following definitions: Definition 1.1. Let (X, τ ) be a topological space. A subset A ⊆ X is called (1) α-closed [18] if cl(int(cl(A))) ⊆ A, (2) semi-closed [15] if int(cl(A)) ⊆ A, (3) preclosed [17] if cl(int(A)) ⊆ A, (4) b-closed [3] if int(cl(A)) ∩ cl(int(A)) ⊆ A, (5) semi-preclosed [2] or β-closed [1] if int(cl(int(A))) ⊆ A. The complement of an α-closed (resp. semi-closed, preclosed, b-closed, β- closed) set is called α-open (resp. semi-open, preopen, b-open, β-open). The smallest α-closed (resp. semi-closed, preclosed, b-closed, β-closed) set contain- ing A ⊆ X is called the α-closure (resp. semi-closure, preclosure, b-closure, β-closure) of A and shall be denoted by clα(A) (resp. cls(A), clp(A), clb(A), clβ(A)). In 2001, Cao, Greenwood and Reilly [10] introduced the concept of qr-closed sets to deal with various notions of generalized closed sets that had been con- sidered in the literature so far. If P = {τ, α, s, p, β} and q, r ∈ P then a subset A ⊆ X is called qr-closed if clq(A) ⊆ U whenever A ⊆ U and U is r-open. (For convenience we denote cl(A) by clτ (A) and open (resp. semi-open, preopen) by τ -open (resp. s-open, p-open).) 244 M. Ganster and M. Steiner In the following we shall consider the expanded family P ∪{b}. As in Corol- lary 2.6 of [10], it is easily established that the concept of a bτ -closed set yields the only new type of sets that can be gained by utilizing the b-closure (resp. the b-interior) in the context of qr-closed sets. Thus we give Definition 1.2. Let (X, τ ) be a topological space. A subset A ⊆ X is called bτ -closed if clb(A) ⊆ U whenever A ⊆ U and U is open. The complement of a bτ -closed set is called bτ -open. Remark 1.3. The concepts of ss-closed (resp. sτ -closed, pτ -closed, βτ -closed) sets have been first introduced in the literature under the name of sg-closed [5] (resp. gs-closed [4], gp-closed [16], gsp-closed [11]) sets. We also consider the following classes of topological spaces: Definition 1.4. A topological space (X, τ ) is called (1) sg-submaximal if every codense subset of (X, τ ) is ss-closed, (2) Tgs if every sτ -closed subset of (X, τ ) is ss-closed, (3) extremally disconnected if the closure of each open subset of (X, τ ) is open, (4) resolvable if (X, τ ) is the union of two disjoint dense subsets. For undefined concepts we refer the reader to [10] and [9] and the references given there. 2. bτ-closed sets and their relationships In [10] the relationships between various types of generalized closed sets have been summarized in a diagram. We shall expand this diagram by adding b-closed sets and bτ -closed sets. Proposition 2.1. Every ss-closed set in a topological space (X, τ ) is b-closed. Proof. Let A ⊆ X be ss-closed and let x ∈ clb(A). Since singletons are either preopen or nowhere dense (see [14]), we distinguish two cases. If {x} is preopen, it is also b-open and hence {x} ∩ A 6= ∅, i.e. x ∈ A. If {x} is nowhere dense, then X \ {x} is semi-open. Suppose that x /∈ A. Then A ⊆ X \ {x} and, since A is ss-closed, we have clb(A) ⊆ cls(A) ⊆ X \ {x}. Hence x /∈ clb(A), a contradiction. Therefore clb(A) ⊆ A, and so A is b-closed. � The remaining relationships in the following diagram can easily be estab- lished. On bτ -closed sets 245 ss-closed �� &&M M MM M MM MM MM preclosed xxqq qq qq qq qq q &&M MM MM MM MM MM b-closed �� &&N NN NN NN NN NN sτ -closed �� pτ -closed xxpp pp pp pp pp p bτ -closed �� β-closed // βτ -closed We now address the question of when the above implications can be reversed. Proposition 2.2. Let (X, τ ) be a topological space. Then: (1) Each b-closed set is ss-closed iff (X, τ ) is sg-submaximal. (2) Each b-closed set is sτ -closed iff (X, τ ) is sg-submaximal. (3) Each sτ -closed set is b-closed iff (X, τ ) is Tgs. (4) Each bτ -closed set is b-closed iff (X, τ ) is Tgs. (5) Each bτ -closed set is sτ -closed iff (X, τ ) is sg-submaximal. (6) Each bτ -closed set is pτ -closed iff (X, τ ) is extremally disconnected. (7) Each bτ -closed set is β-closed iff (X, τ ) is Tgs. (8) Each β-closed set is b-closed iff cl(W ) is open for every open resolvable subspace W of (X, τ ). Proof. We will only show (1). The other assertions can be proved in a similar manner using the standard methods that can be found in [10], and (8) has been shown in [13]. First recall that a space is sg-submaximal iff every preclosed set is ss-closed (see [7]). If every b-closed set is ss-closed then every preclosed set ss-closed, i.e. (X, τ ) is sg-submaximal. Conversely, suppose that (X, τ ) is sg-submaximal and let A be b-closed. Then A is the intersection of a semi-closed and a preclosed set (see [3]). Since every semi-closed set is ss-closed, by hypothesis, A is the intersection of two ss- closed sets. Since the arbitrary intersection of ss-closed sets is always ss-closed (see [12]), we conclude that A is ss-closed. � Proposition 2.3. Let (X, τ ) be a topological space. Then the following state- ments are equivalent: (1) Each βτ -closed set is bτ -closed. (2) Each β-closed set is bτ -closed. Proof. The necessity is clear, so we only have to show the sufficiency. Let A be a βτ -closed set and U be an open subset of X such that A ⊆ U . Since 246 M. Ganster and M. Steiner A is βτ -closed we have clβ (A) ⊆ U . Now, clβ (A) is β-closed and hence bτ - closed by hypothesis. Therfore clb(A) ⊆ clb(clβ (A)) ⊆ U and thus our claim is proved. � Remark 2.4. If A ⊆ X, then the largest b-open subset of A is called the b-interior of A and is denoted by bint(A). It is well known that bint(A) = (cl(int(A)) ∪ int(cl(A))) ∩ A (see [3]). Consequently, a subset A is bτ -open iff for every closed subset F satisfying F ⊆ A we have F ⊆ cl(int(A)) ∪ int(cl(A)). We shall now present one of our major results. Theorem 2.5. Let (X, τ ) be a topological space. Then the following are equiv- alent: (1) Each β-closed set is b-closed. (2) Each β-closed set is bτ -closed. (3) cl(W ) is open for every open resolvable subspace W of (X, τ ). Proof. It is obvious that (1) ⇒ (2). Furthermore, it has been shown in [13] that (3) ⇔ (1), so we only have to prove that (2) ⇒ (3). If W = ∅, we are done, so let W be a nonempty open resolvable subspace and let E1 and E2 be disjoint dense subsets of (W, τ|W ). Suppose that there exists a point x ∈ cl(W ) \ int(cl(W )). Let S = E1 ∪ cl({x}). It is easily checked that int(S) = ∅, cl(S) = cl(W ) and that S is β-open. By hypothesis, S is bτ - open and so, since cl({x}) ⊆ S, we conclude that {x} ⊆ cl({x}) ⊆ int(cl(S)) = int(cl(W )). This is, however, a contradiction to our assumption and so cl(W ) has to be open. � 3. A Remark on βgs-spaces In [10] a space has been called βgs if every βτ -closed subset of (X, τ ) is sτ - closed. We observe that this is equivalent to the property that every β-closed subset is sτ -closed, see [6]. Using some of our previous results, we are now able to give the following characterization. Theorem 3.1. Let (X, τ ) be a topological space. Then the following are equiv- alent: (1) (X, τ ) is a βgs-space. (2) Every β-closed set is b-closed and (X, τ ) is sg-submaximal. (3) Every β-closed set is ss-closed. Note that the last property in the above theorem has been fully characterized in [8]. It was shown there that every β-closed set is ss-closed iff (X, τ α) is g- submaximal, where τ α denotes the α-topology of (X, τ ) (see [18]), and a space is called g-submaximal if each codense set is τ τ -closed. So the claim in [10] that there exists a βgs-space whose α-topology is not g-submaximal turns out to be wrong now. In fact, one can easily check that Example 3.5 of [10] is false. So βgs is not a new topological property as we have just seen. On bτ -closed sets 247 References [1] M. E. Abd El-Monsef, S.N. El-Deeb, and R.A. Mahmoud, β-open sets and β-continuous mappings, Bull. Fac. Sci. 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Nj̊astad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961–970. Received March 2006 Accepted July 2006 M. Ganster (ganster@weyl.math.tu-graz.ac.at) Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, AUSTRIA M. Steiner (msteiner@sbox.tugraz.at) Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, AUSTRIA