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Pi-Set Connected Functions in Topological Spaces

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Pi-Set Connected Functions in Topological Spaces Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 36, 1789 - 1798 π-Set Connected Functions in Topological Spaces Ganes M. Pandya School Of Petroleum Management Pandit Deendayal Petroleum University Gandhinagar-Gujrat, India ganes [email protected] C. Janaki Sree Narayana Guru College Coimbatore-Tamilnadu, India [email protected] I. Arockiarani Nirmala College for Women Coimbatore-Tamilnadu, India Abstract The aim of this paper is to introduce the concept of a new class of set connected functions called as π-set connected and study its prop- erties.Furthermore we study and investigate relationship between π-set connected functions , seperation axioms and covering properties. Mathematics Subject Classification: 54C05, 54C10 Keywords: π-set connected, π-open sets, regular set-connected, clopen, πT1 space 1 Introduction and Preliminaries. The notion of perfectly continuous function was introduced and studied by Noiri [10,11] and the notion of clopen functions was introduced by Reilly and Vamanamurthy [13]. Recently, a new class of functions called regular set- connected has been introduced by Dontchev, Ganster and Reilly [2].Ekici [4] 1790 G. M. Pandya, C. Janaki and I. Arockiarani extended the concept of regular set-connected functions to almost clopen func- tions. Throughout this paper, X and Y are topological spaces. Let A be a subset of X. We denote the interior and closure of a set by int(A) and cl(A) respectively. A subset A of a space X is said to be regular open if A = int( cl(A)). The family of all regular open (respectively regular closed, clopen) sets of X is denoted by RO(X) (respectively RC(X), CO(X)). A finite union of regular open sets is said to be π-open.The family of all π-open sets of X is denoted by πO(X). Definition 1.1. A function f : ( X,τ) → (Y,σ) is said to be 1 . perfectly continuous [11] if f−1(V ) is clopen in X for every open set V of Y. 2 . set-connected [6] if f−1 (V ) is clopen in X for every V ∈ CO(Y ). 3 . regular set-connected[2] if f−1 (V ) is clopen in X for every V ∈ RO(Y ). 4 . almost π-continuous [3] if f −1(V) is π-closed in X for every regular closed set V of Y. 5 . almost clopen [4] if for each x ∈ X and each open set V in Y containing f(x), there exists a clopen set U in X containing x such that f(U) ⊂ int (cl(V)). Definition 1.2. A space X is said to be 1. clopen T1 [4] if for each pair of distinct points x and y of X, there exist clopen sets U and V containing x and y respectively such that y ∈/ U and x ∈/ V. 2. clopen T2 [4]if for each pair of distinct points x and y in X, there exist disjoint clopen sets U and V in X such that x ∈ U and y ∈ V. 3. almost regular[15] if for each regular closed set F and each point x ∈/ F , there exist disjoint open sets U and V such that F ⊂ U and x ∈ V. 4. mildly normal [17]if for every pair of disjoint regular closed subsets F1 and F2 of X, there exist a disjoint open sets U and V such that F1⊂ U and F2 ⊂ V. 5. hyperconnected [19] if every non-empty open set is dense 6. mildly compact [18]if every clopen cover of X has a finite subcover. π-Set connected functions in topological spaces 1791 7. mildly Lindelof [18] if every cover of X by clopen sets has a countable subcover. 8. nearly compact[16] if every regular open cover of X has a finite subcover. 9. nearly countably compact [5]if every countable cover of X by regular open sets has a finite subcover. 10. nearly Lindelof if every cover of X by regular open sets has a countable subcover. 11. mildly countably compact[5] if every countable cover of X by clopen sets has a finite subcover. 2 π-set connected function. Definition 2.1. A function f : X →Y is said to be π-set connected if f−1(V) ∈CO( X ) for every V ∈ πO( Y ). Theorem 2.2. Let (X, τ) and (Y,σ) be topological spaces. The following statement are equivalent for a function f : X →Y 1. f is π-set connected. 2. f−1(int(cl(G))) is clopen for every open subset G of Y. 3. f−1(cl(int(V))) is clopen for every closed subset V of Y. 4. f−1(F) is clopen for every F ∈ πC(Y). Proof. (1) ⇒(2) Let G be any open subset of Y . Since int(cl(G)) is regular −1 open and hence π-open, by (1) it follows f (int(cl(G))) is clopen. (2)⇒(1) Let V be π-open in Y. By (2), f−1(int(cl(V)) is clopen in X and hence f is π-set connected. (2) ⇒(3) Let V be a closed subset of Y. Then Y V is open. By (2), −1 −1 −1 f (int(cl(Y V))) is clopen. Hence, f (int(cl(Y V))) = f ( int(Y int(V) −1 −1 −1 )) = f (Y cl(int(V))) = X f (cl(int(V))). Hence f (cl(int(V))) is clopen. (3)⇒ (2) is obtained similarly . (1)⇔ (4) is Obvious. Theorem 2.3. If f : X →Yisπ-set connected function and A is any subset of X, then the restriction f/A:A→Yisπ-set connected functiion . Proof. Let V be a π-open set . By hypothesis f−1(V) is clopen in X. We have f−1(V) ∩ A = (f /A)−1(V) is clopen in A. Hence f/A is π-set connected function. 1792 G. M. Pandya, C. Janaki and I. Arockiarani Theorem 2.4. Let f : X →Y and g : Y →Z be functions. Then the following properties hold 1. If f and g are π-set connected functions, then g◦f:X→Zisπ-set connected function. 2. If f is π-set connected function and g is perfectly continuous, then g◦f:X → Z is perfectly continuous. 3. If f is set-connected and g is π-set connected then g◦fisπ-set connected. 4. If f is almost π-continuous and g is π-set connected function , then g◦f:X → Z is almost π-continuous. Proof. (1) Let V be π-open in Z. By hypothesis ,g−1(V) is clopen in Y. Since fisπ-set connected function , f−1(g−1(V)) = ( g◦f)−1(V) is clopen. Therefore g◦fisπ-set connected function.. (2)Let V be open in Z. By hypothesis, g−1(V) is clopen in Y. Since f is π- set connected function, f−1(g−1(V)) is clopen in X .Therefore g ◦ f is perfectly continuous. (3)Let V be π-open in Z. Since g is π-set connected, g−1(V) is clopen in Y. Since f is set connected, f−1(g−1(V)) is clopen in X. hence g◦fisπ-set connected. (4)Let V be regular open in Z. Since g is π-set connected, g−1(V) is clopen in Y. f−1(g−1(V)) is π-open in Z. Then g◦f is almost-π-continuous. Theorem 2.5. If f : X →Y is a surjective open and closed function and g : Y →Z is a function such that g◦f:X→Zisπ- set connected, then g is π-set connected. Proof. Let V be π-open in Z. (g◦f)−1(V) is clopen in X. f−1(g−1(V)) is clopen in X. Since f is surjective open and closed, f (f−1(g−1(V) )) = g−1(V )is clopen.Therefore g is π-set connected. 3 Separation Axioms and Graphs Definition 3.1. A Space X is said to be πT1 if for each pair of distinct points x and y of X , there exist π-open sets U and V containing x and y respectively such that y∈/ U and x ∈/ V. Theorem 3.2. If f : X →Yisaπ-set connected injection and Y is πT1, then X is clopen T1 π-Set connected functions in topological spaces 1793 Proof. Since Y is πT1 for any distinct points x and y in X, there exist V,W ∈ πO(Y) such that f(x) ∈V , f(y) ∈/V , f(x) ∈/W, f(y) ∈ W. Since f is π-set connected, f−1(V) and f−1(W ) are clopen in X. Furthermore y ∈/f−1(V ) and x −1 ∈/f (W). This shows that X is clopen T1. Theorem 3.3. If f : X →Yisaπ-set connected injection and Y is T2, then X is clopen T2. Proof. For any pair of distinct points x and y in X, there exist disjoint open sets U and V in Y such that f(x) ∈U and f(y) ∈ V. Since f is π-set connected ,f−1(int(cl(U) )) and f−1(int(cl(V))) is clopen in X containing x and y respec- ctively. We have f−1(int(cl(U )) ∩ f−1(int(cl(V)) = φ . This shows that X is clopen T2 . Theorem 3.4. If f : X→Y and g : X→Ybeπ-set connected functions and Y is Hausdorff , then E = {x ∈ X : f(x)=g(x)} is closed in X. ∈ Proof. If x X E then it follows that f(x) = g(x) .Since Y is Hausdorff, there exist open sets V and W such that f(x) ∈V , g(x) ∈ W and V ∩ W = φ.
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