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S.Eswaran et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 2466-2469 *-Generalized Compact Spaces and *-Generalized Connected Spaces in Topological Spaces 1 S.ESWARAN* 2 A.PUSHPALATHA 1 Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore, Tamilnadu, India – 641 407 2 Department of Mathematics, Government Arts College, Udumelpet, Tamilnadu, India – 642 126 1 [email protected] 2 [email protected] Abstract In this paper, we introduce new topological spaces called *-generalized compact spaces and *-generalized connected spaces using *-generalized open sets and study some of their properties. 2000 Mathematics Subject Classification: 54A05. Key words: *-g-compact spaces, *-g-connected spaces. 1. Introduction In 1976, Thompson [9] introduced the notion of S-closed spaces by using semi-open sets. DI Maio and Noiri [2] defined and studied about a class of compact space called s-closed space using semi-open cover. As a generalization of S-closed space, Noiri [6] introduced and investigated the notion of locally S-closed spaces. Balachandran, Sundaram and Maki [1] introduced a class of compact space called GO-compact space and GO-connected space using g-open cover. Pushpalatha, Eswaran and Rajarubi [7] introduced and studied *-generalized closed sets and *generalized open sets. Using *-generalized closed sets, Eswaran and Pushpalatha [4] introduced *-generalized continuous maps. In [8], Pushpalatha and Eswaran studied some strong forms of *-generalized continuous maps. In this paper, we introduce new classes of topological spaces called *-generalized compact spaces and *generalized connected spaces using *-generalized open sets. Throughout this paper (X, τ*) and (Y, *) (or simply X and Y) represent non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned. For a subset A of a space (X, τ*), cl*(A), cl (A) and Ac represent generalized closure operator, closure of A and complement of A respectively. 2. Preliminaries We recall the following definitions. Definition 2.1. A subset A of a topological space (X, ) is called generalized closed[5] (briefly g-closed) in X if cl(A) G whenever A G and G is open in X. A subset A is called generalized open (briefly g-open) in X if AC is g-closed. Definition 2.2. For the subset A of a topological space X, (i) the generalized closure operator cl*[3] is defined by the intersection of all g- closed sets containing A. (ii) the topology *[3] is defined by * ={G : cl*(GC) = GC } ISSN: 0975-5462 2466 S.Eswaran et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 2466-2469 Definition 2.3. A subset A of a topological space X is called *-generalized closed set [7] (briefly *-g-closed) if *-generalized closed set is called the cl (A) G whenever A G and G is *-open. The complement of * * -generalized open set (briefly -g-open). Definition 2.4. A topological space (X, τ*) is called τ*-Tg [4] space if every τ*-g-closed set in X is g-closed in X. Definition 2.5. A map f : X Y from a topological space (X, τ*) into a topological space (Y, σ*) is called (i) τ*-generalized continuous [4] (briefly τ*-g-continuous) if the inverse image of every g-closed (or g-open) set in Y is τ*-g-closed (or τ*-g-open) in X. (ii) strongly τ*- generalized continuous [8] (briefly strongly τ*-g-continuous) if the inverse image of every τ*-gopen set (or τ*-g-closed set) in Y is g-open (or g-closed) in X. (iii) τ*-gc-irresolute [8] if the inverse image of every τ*-g-closed set in Y is τ*-g-closed in X. * 3. *-Generalized Compact Spaces In this section, we introduce and study a class of compact space called *-Generalized Compact Spaces in topological spaces. Definition 3.1. A collection {Ai : i I} of τ*-g-open sets in a topological space (X, τ*) is called a τ*-g-open cover of a subset B if B {Ai : i I}. Definition 3.2. A topological space (X, τ*) is called τ*-generalized compact (briefly τ*-g-compact), if every τ*-g-open cover of X has a finite subcover. Definition 3.3. A subset B of a topological space (X, τ*) is said to be τ*-g-compact relative to X , if for every τ*-g-open subsets of X such that B {Ai : i I} there exists a finite subset Io of I collection {Ai : i I} of such that B {Ai : i Io}. Definition 3.4. A subset B of a topological space (X, τ*) is said to be τ*-g-compact if B is τ*-g-compact as the subset of X. Theorem 3.5. A τ*-g-closed subset of τ*-g-compact space is τ*-g-compact relative to X. Proof : Let A be a τ*-g-closed subset of a τ*-g-compact space X. Then AC is τ*-g-open in X. Let S be a cover of A by τ*-g-open sets in X. Then, {S, AC} is a τ*-g-open cover of X. Since X is τ*-g-compact, it has a finite subcover, say {G1, G2, ., Gn}. If this subcover contains AC, we discard it. Otherwise leave the subcover as it is. Thus we have obtained a finite τ*-g-open subcover of A and so A is τ*-g-compact relative to X. Theorem 3.6. (i) A τ*-g-continuous image of a τ*-g-compact space is compact. (ii) If a map f : X Y is τ*-gc-irresolute and a subset B is τ*-g-compact relative to X, then the image f(B) is τ*-g-compact relative to Y. Proof : (i) Let f : X Y be a τ*-g-continuous map from a τ*-g-compact space X onto a topological space Y. Let {Ai : i I} be an open cover of Y. Then {f -1(Ai) : i I} is a τ*-g-open cover of X. Since X is τ*-g-compact, it has a finite subcover, say { f -1(A1), f -1(A2), …, f -1(An)}. Since f is onto, {A1, A2,…,An} is a g-open cover of Y and so Y is compact. (ii) Let {Ai : i I} be any collection of τ*-g-open subsets of Y such that f(B) {Ai : i I}. Then, B {f -1(Ai) : i I}. By using assumptions there exists a finite subset Io of I such that B {f -1(Ai) : i I}. Therefore, we have f(B) {Ai : i Io} which shows that f(B) is τ*-g-compact relative to Y. Theorem 3.7. If f : X Y is a strongly τ*-g-continuous map from a compact space X onto a topological space Y, then Y is τ*-g-compact. Proof : Let {Ai : i I} be a τ*-g-open cover of Y. Then {f -1(Ai) : i I} is a g-open cover of X since f is f -1(A2) , .., f -1(An)}. strongly τ*-g-continuous. Since X is compact, it has a finite subcover say {f -1(A1), Since f is onto,{A1, A2, ., An} is a finite τ*-g-open cover of Y. Therefore Y is τ*-g-compact. 4. τ*-Generalized Connectedness. Here we introduce a new class of spaces called τ*-generalized connected spaces and prove some of its properties. Definition 4.1. A topological space X is said to be τ*-generalized connected (briefly τ*-gconnected) if X cannot be written as a disjoint union of two nonempty τ*-g-open sets. A subset of X is τ*-gconnected if it is τ*-g-connected as a subspace. Theorem 4.2. For a topological space X, the following are equivalent: (a) X is τ*-g-connected ISSN: 0975-5462 2467 S.Eswaran et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 2466-2469 (b) The only subsets of X which are both τ*-g-open and τ*-g-closed are the empty set and X. (c) Each τ*-g-continuous map of X into a discrete space Y with at least two points is a constant map. Proof : (a) (b) Let U be a τ*-g-open and τ*-g-closed subset of X. Then X – U is both τ*-g-open and τ*-g-closed. Since X is the disjoint union of the τ*-g-open sets U and X – U, one of these must be empty, that is U = or U = X. (b) (a) Suppose that X = A B where A and B are disjoint nonempty τ*-g-open subsets of X. Then A is both τ*-g-open and τ*-g-closed. By assumption, A = or X. Therefore X is τ*-g-connected. τ *-g-continuous map. Then X is covered by τ*-g-open and τ*(b) (c) Let f : X Y be a g-closed covering {f -1(y) : y Y}. By assumption, f -1(y) = or X for each y Y. If f -1(y) = for all y Y then f fails to be a map. Then, there exists only one point y Y such that f -1(y) and hence f -1(y) = X which shows that f is a constant map. (c) (b) let U be both τ*-g-open and τ*-g-closed in X. Suppose U . Let f : X Y be a τ*-gcontinuous map defined by f(U) = {y} and f(X – U) = {w} for some distinct points y and w in Y. By assumption, f is constant. Therefore, y = w and so U = X. Theorem 4.3. Every τ*-g-connected space is connected. Proof : Let X be a τ*-g-connected space. If possible let X be not connected. Then X can be written as X = A B where A and B are disjoint nonempty g-open sets in X. In [7] it has been proved in Theorem 3.4 that every g-open set is τ*-g-open. Thus, X = A B where A and B are disjoint, are nonempty and τ*-g-open sets in X. This contradicts the fact that X is τ*-g-connected. Therefore X is connected. Theorem 4.4. Suppose that X is a τ*-Tg topological space. Then X is connected if and only if X is τ*-gconnected. τ*-g-connected. Then X can Proof: Assume that X is τ*-Tg and connected. If possible, let X be not be written in the form X = A B where A and B are nonempty, disjoint and τ*-g-open sets in X. Since X is τ*-Tg, every τ*-g-open set is g-open and so X = A B where A and B are nonempty, disjoint and g-open sets in X. This contradicts the fact that X is connected. Therefore X is τ*-g-connected. Converse follows from Theorem 4.3. τ*-g-continuous surjection and X is τ*-g-connected then Theorem 4.5. (i) If f : X Y is Y is connected. (ii) If f : X Y is a τ*-gc-irresolute surjection and X is τ*-g-connected then Y is τ*-gconnected. Proof: (i) suppose that Y is not connected. Let Y = A B where A and B are disjoint nonempty τ*-open sets in Y. Since f is τ*-g-continuous and onto, X = f -1(A) f -1(B) where f -1(A) and f -1(B) are disjoint nonempty τ*-g-open sets in X. This contracts the fact that X is τ*-g-connected. Hence Y is connected. (ii) If possible, assume that Y is not τ*-g-connected. Then Y = A B where A and B are nonempty, f -1(B) are τ*-g-open in X. Since disjoint and τ*-g-open sets in Y. Since f is τ*-gc-irresolute, where f -1(A) and -1 -1 -1 -1 -1 f (A B) = f (A) f -1(B). Thus X is a union f is onto, f (A) and f (B) are nonempty. Now X = f (Y) = of disjoint, nonempty and τ*-g-open sets in X. This contradicts the fact that X is τ*-g-connected. Therefore, Y is τ*-g-connected. With reference to Theorem 4.5.(i) we have the following theorem. Theorem 4.6. Let f be a τ*-g-continuous map from a topological space (X, τ*) into a topological space (Y, σ*). Then f(H) is a connected subset of Y for every g-closed and τ*-g-connected subset H of X. Proof : In [4], it has been proved in Theorem 3.13 that the restriction f/H of f to H is τ*-g-continuous. By Theorem 4.5, the image of the τ*-g-connected space (H, τ*/H) under f / H : ( H , * / H ) * / f ( H ) is connected. Thus (f(H), σ*/f(H)) is connected. Therefore, f(H) is a connected subset of Y. Theorem 4.7. If f : X Y is a strongly τ*-g-continuous map from a connected space X onto a topological space Y then Y is τ*-g-connected Proof : If possible, let Y be not τ*-g-connected. Then Y can be written in the form Y = A B where A and B are disjoint nonempty τ*-g-open sets in Y. Since f is strongly τ*-g-continuous, f -1(A) and f -1(B) are g-open sets in X. Also X = f -1(Y) = f -1(A B) = f -1(A) f -1(B). This contradicts the fact that X is connected. Therefore Y is τ*-g-connected. References ISSN: 0975-5462 2468 S.Eswaran et. al. / International Journal of Engineering Science and Technology Vol. 2(5), 2010, 2466-2469 [1] K.Balachandran, P.Sundaram and J.Maki, On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. (Math.) 12 (1991), 5-13. [2] G. Di Maio and T.Noiri, Indian J. pure appl. Math., 18(1987), 226 – 33. [3] W.Dunham, A new closure operator for non-T topologies, Kyungpook Math. J., 22 (1982), 55 – 60 [4] S.Eswaran and A.Pushpalatha, *-generalized continuous maps in topological spaces, International J. of Math Sci & Engg. Appls. (IJMSEA) ISSN 0973-9424 Vol. 3, No. IV, (2009), pp. 67 – 76 [5] N.Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo(2), 19 (1970), 89 – 96. [6] T.Noiri, Atti Accad. Naz. Lincei Rent. Cl. Sci. Fis. Mat., Natur. (8) 64 (1978), 157 – 62. [7] A.Pushpalatha, S.Eswaran and P.Rajarubi, *-generalized closed sets in topological spaces, Proceedings of World Congress on Engineering 2009 Vol II WCE 2009, July 1 – 3, 2009, London, U.K., 1115 – 1117 [8] A.Pushpalatha and S.Eswaran, Strong forms of *-generalized continuous maps in topological spaces, Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 17, 815 - 822, [9] T.Thompson, Proc. Am. Math. Soc. 60(1976), 335 – 38 ISSN: 0975-5462 2469