KYUNGPOOK Math. J. 45(2005), 539-547 Some Topologies Induced by b-open Sets M. E. Abd EL-Monsef, A. A. El-Atik and M. M. El-Sharkasy Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt e-mail : [email protected], [email protected] and [email protected] Abstract. The class of b-open sets in the sense of Andrijevi´c ([3]), was discussed by El-Atik ([9]) under the name of γ-open sets. This class is closed under arbitrary union. The aim of this paper is to use Λ-sets and _-sets due to Maki ([15]) some topologies are constructed with the concept of b-open sets. b-Λ-sets, b-_-sets are the basic concepts introduced and investigated. Moreover, several types of near continuous function based on b-Λ-sets, b-_-sets are constructed and studied. 1. Introduction Andrijevi´c([3]) introduced a new class of generalized open sets in a topological space, so called b-open sets. This type of sets discussed by El-Atik ([9]) under the name of γ-open sets. The class of b-open sets is contained in the class of semi- preopen sets and contains all semi-open sets and preopen sets. The class of b-open sets generates the same topology as the class of preopen sets. In 1986, Maki ([15]) introduced the concept of generalized Λ-sets in a topological space and defined the associated closure operator by using the work of Levine ([14]) and Dunhem ([8]). Caldas and Dontchev ([7]) built on Maki’s work by introducing Λs-sets, _s-sets, gΛs-sets and g_s-sets. Ganster and et. al. ([10]) introduced the notion of pre-Λ- sets and pre-_-sets and obtained new topologies defined by these families of sets. In [12] Khalimsky, Kopperman and Meyer proved that the digital line is a typical example of a T1=2 space. Our aim is to introduce the notion of b-Λ-sets and b-_-sets to topological spaces and study some of its properties. Also, we prove that the topology generated by the class of b-open sets contains the topology generate by the class of preopen (resp. semi-open) sets by using the notions of Λ-sets and _- sets. Finally we introduce generalized b-Λ-sets, generalized b-_-sets, b-Λ-functions, b-_-functions and investigate some properties of the new concepts. Definition 1.1. A subset A of a topological space is called: (1) Semi-open [14] if A ⊆ cl(int(A)). (2) Preopen [16] if A ⊆ int(cl(A)). Received April 26, 2004. 2000 Mathematics Subject Classification: 54A10. Key words and phrases: b-open sets, b-Λ-sets, b-_-sets, generalized b-Λ-sets, generalized b-_-sets, Λb-functions and _b-functions. 539 540 M. E. Abd EL-Monsef, A. A. El-Atik and M. M. El-Sharkasy (3) ®-open [4] if A ⊆ int(cl(int(A))). (4) ¯-open [1] [Semi-preopen [5]] if A ⊆ cl(int(cl(int(A)))). (5) b-open [3] if A ⊆ int(cl(A)) [ cl(int(A)). (6) Nowhere dense [2] if int(cl(A)) = Á. The class of all semi-open (resp. preopen, ®-set, ¯-open and b-open) denoted by SO(X; ¿) (resp. PO(X; ¿), ®O(X; ¿), ¯O(X; ¿)] and BO(X; ¿). The complement of these sets called semi-closed (resp. preclosed, ®-closed, ¯- closed and b-closed) and the classes of all these sets will be denoted by SC(X; ¿) (resp. PC(X; ¿), ®C(X; ¿), ¯C(X; ¿) and BC(X; ¿)). Definition 1.2. A topological space (X; ¿) is said to be: (1) Extremely Disconnected (abb. E. D) [2] if the closure of any open set is open. (2) Submaximal [11] if all dense subset are open. (3) Resolvable [11] if there is a subset D of X such that D and (X ¡ D) are both dense in X. Definition 1.3. A subset A of a space (X; ¿) is called: (1) Λ-set (resp. -_-set) [15] if it is the intersection(resp, union) of open (resp. closed) sets. (2) Λs-sets (resp. -_s-sets) [7] if it is the intersection (resp. union) of semi-open (resp. semi-closed) sets. (3) pre-Λ-sets (resp. pre-_-set) [10] if it is the intersection (resp. union) of preopen (resp. preclosed) sets. (4) Generalized Λ-set (resp. generalized _-set) [15] if Λ(A) ⊆ F whenever A ⊆ F and F 2 C(X; ¿)(resp. U ⊆ _(A) whenever U ⊆ A and U 2 O(X; ¿). (5) Generalized semi-Λ-set ( resp. generalized semi-_-set) [7] if Λs(A) ⊆ F when- ever A ⊆ F and F 2 SC(X; ¿) (resp. U ⊆ _s(A) whenever U ⊆ A and U 2 SO(X; ¿): (6) Generalized pre-Λ-set (resp. generalized pre-_-set) [10] if Λp(A) ⊆ F when- ever A ⊆ F and F 2 PC(X; ¿) (resp. U ⊆ _p(A) whenever U ⊆ A and U 2 PO(X; ¿). Lemma 1.1 ([9]). For a space (X; ¿). If U ⊆ Y ⊆ X, U 2 BO(X; ¿) and Y 2 BO(X; ¿), then: (1) U 2 ¯O(X; ¿) (2) U 2 BO(X; ¿) if U 2 ¯O(X; ¿) ¡ BO(X; ¿). Some Topologies Induced by b-open Sets 541 2. Topological spaces via Λb-(_b-)sets In this section we introduced the concept of b-Λ- (resp. b-_-) sets via the concept of b-open sets. We define the class of ¿ Λb (¿ _b ) and prove that they form topologies. Definition 2.1. We define Λb(A) and _b(A) for a subset A of a space (X; ¿) as \ Λb(A) = fG : A ⊆ G; G 2 BO(X; ¿)g and [ _b(A) = fF : F ½ A; F 2 BC(X; ¿)g: Example 2.1. Let X = fa; b; c; dg, ¿ = fX; Á; fa; bg; fbg; fb; c; dgg. Then BO(X; ¿) = fX; Á; fbg; fa; bg; fb; cg; fb; dg; fa; b; c; g; fa; b; dg; fb; c; dgg and Λb(fag) = fa; bg,Λb(fc; dg) = fb; c:dg,Λb(fbg) = fbg, _b(fa; bg) = fa; bg, _b(fb; c; dg) = fc; dg, _b(fag) = fag. Lemma 2.1. For subsets A; B and Ai; i 2 I, of a space (X; ¿) the following properties hold: (1) A ⊆ Λb(A), (2) If A ⊆ B, then Λb(A) ⊆ Λb(B), (3) Λb(Λb(A)) = Λb(A), (4) If A 2 BO(X; ¿), then A = Λb(A), S S (5) Λb( fAi : i 2 Ig) = fΛb(Ai): i 2 Ig, T T (6) Λb( fAi : i 2 Ig) ⊆ fΛb(Ai): i 2 Ig, (7) Λb(XnA) = X=Vb(A). Proof. (1), (2), (4), (6) and (7) are immediate consequences of Definition 2.1. To prove (3), firstly from the Definition 2.1, Λb(A) ⊆ Λb(Λb(A)). For the converse inclusion, let x2 = Λb(A). Then there exists G 2 BO(X; ¿) such that A ⊆ G and x2 = G. Since Λb(Λb(A)) = fG :Λb(A) ⊆ G; G 2 BO(X; ¿)g, so we have x2 = Λb(Λb(A)). Thus Λb(Λb(A)) = Λb(A). To prove (5),S let A = [fAi : i 2 Ig. By (2) Since Ai ⊆ A, then Λb(Ai) ⊆ Λb(AS) and so fΛb(Ai): i 2 Ig ⊆ Λb(A): To prove the converse inclusion, let x2 = fΛb(Ai): i 2 Ig, then forS each i 2 I there exists Gi 2 BO(X; ¿) such that Ai ⊆ Gi and x2 = Gi . If G = fGi : i 2 Ig then G 2 BO(X; ¿) with A ⊆ G and x2 = G. Hence x2 = Λb(A) and so (5) holds. ¤ Lemma 2.2. For subsets A; B and Ai; i 2 I of a space BO(X; ¿) the following properties hold: (1) _b(A) ⊆ A, 542 M. E. Abd EL-Monsef, A. A. El-Atik and M. M. El-Sharkasy (2) If A ⊆ B then _b(A) ⊆ _b(B), (3) _b(_BP (A)) = _b(A), (4) If A 2 BC(X; ¿) then A = _b(A), T T (5) _b( fAi : i 2 Ig) = f_b(Ai): i 2 Ig, S S (6) f_b(Ai): i 2 Ig ⊆ _b( fAi : i 2 Ig). Proof. Obvious by Lemma 2.1 (7). ¤ Remark 2.1. We note, in general for any subsets A; B of (X; ¿)Λb(A \ B) 6= Λb(A) \ Λb(B) and _b(A [ B) 6= _b(A) [_b(B), as in the following example. Example 2.2. In Example 2.1, if A = fag, B = fb; c; dg then, Λb(fag \ fb; c; dg) = Λb(Á) = Á; Λb(fa)g \ Λb(fb; c; dg) = fbg. Also, _b(fag [ fb; c; dg) = _b(X) = X, _bfag [ _bfb; c; dg = fag [ fc; dg = fa; c; dg. Definition 2.2. A subset A of a space (X; ¿) is called b-Λ- (resp. b-_-) set if A = Λb(A) (resp. A = _b(A)). The class of all b-Λ-sets (resp. b-_-sets) will be denoted by ¿ Λb (resp. ¿ _b ). Example 2.3. Let (X; ¿) be the same as in Example 2.1, the class of all b- Λ-sets is fX; Á; fbg; fa; bg; fb; cg; fb; dg; fa; b; cg; fa; b; dg; fb; c; dgg and b-_-sets is fX; Á; fag; fcg; fdg; fa; cg; fa; dg; fc; dg; fa; c; dgg. Proposition 2.1. In a space (X; ¿), the class of all b-Λ- (resp. b-_-) sets is a topological space. Proof. It is obvious from Definition 2.1 that X, Á and the intersection of b-Λ-sets is b-Λ-sets. To prove the union of Sb-Λ-sets is b-Λ-sets, let fAi : i 2 Ig be a fam- ily ofSb-Λ-sets in (X; ¿) and A = fAi : i 2 Ig.