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Contra (πp, µy)-Continuity on Generalized Topological Spaces
K. Binoy Balan*, C. Janaki** *Assistant Professor, Dept. of Mathematics, Sree Narayana Guru College, Coimbatore- 105, Tamil Nadu INDIA. **Assistant Professor, Dept. of Mathematics, L.R.G. Govt. Arts College for Women, Tirupur-4, Tamil NaduINDIA. Corresponding Addresses: *[email protected],**[email protected] Research Article
Abstract: In this paper we introduce a new notion called contra (πp, µy) – continuous function on generalized topological space. The properties and characterizations of such functions are investigated. Key words: µ-πrα space, Tπp space, µ-πrα T1,µ-πrα T2, µ-πrα connected, µ-Urysohn, µ-πrα locally indiscrete, contra (πp, µy) – continuous, contra (πp, µy) – closed, µ-πrα closed, µ-πrα open. Mathematics subject classification: 54A05
1. Introduction Á.Császár [3]- [13] has introduced the notions of generalized topological space, obtain characterizations for generalized continuous functions and associated interior and closure operators. In [5] he introduced characterizations for generalized continuous functions. Also in [3] he investigated the notions of µ-α-open sets, µ- semi open sets, µ- pre open sets and µ- β open sets in generalized topological space. W.K. Min [15] has introduced and studied the notions of (α, µy) – continuous functions, (σ, µy) – continuous functions, (π, µy) – continuous functions, and (β, µy) – continuous in generalized topological spaces. Also D. Jayanthi [14] has introduced some contra continuous functions on generalized topological spaces such as contra (µx, µy) – continuous functions, contra (α, µy) – continuous, contra (σ, µy) – continuous functions, and contra (β, µy) – continuous functions. In this paper we introduce contra (πp, µy) – continuous functions and investigate their characterizations and relationships among these functions.
2. Preliminaries We recall some basic concepts and results. Let X be a nonempty set and let exp(X) be the power set of X. µexp(X) is called a generalized topology [5](briefly, GT) on X, if µ and unions of elements of µ belong to µ. The pair (X, µ) is called a generalized topological space (briefly, GTS). The elements of µ are called µ-open [3] subsets of X and the complements are called µ-closed sets. If (X, µ) is a GTS and AX, then the interior of (denoted by iµ(A)) is the union of all GA, Gµ and the closure of A (denoted by cµ(A)) is the intersection of all µ-closed sets containing A. Note that cµ(A) = Xiµ(XS) and iµ(A) = X cµ(XA) [5]. Definition 2.1[5] Let (X, µx) be a generalized topological space and A X. Then A is said to be
Definition 2.2 [2] Let (X, µx) be a generalized topological space and A X. Then A is said to be µ-πrα closed set if cπ(A) U whenever AU and U is µ-rα-open set. The complement of µ-πrα closed set is said to be µ-πrα open set.
The complement of µ-semi open ( µ-pre open, µ-α-open, µ-β-open, µ-r-open, µ-rα-open) set is called µ- semi closed ( µ- pre closed, µ-α- closed, µ-β- closed, µ-r- closed, µ-rα-closed) set. Let us denote the class of all µ-semi open sets, µ-pre open sets, µ-α-open sets, µ-β-open sets, and µ-πrα open sets on X by σ(µx) (σ for short), π(µx) (π for short), α(µx) ( α for short), β(µx) (β for short) and πp(µx) (πp for short)respectively. Let µ be a generalized topology on a non empty set X and SX. The µ-α-closure (resp. µ-semi closure, µ-pre closure, µ-β-closure, µ-πrα-closure) of a subset S of X denoted by cα(S) (resp. cσ(S), cπ(S), cβ(S), cπp(S)) is the intersection of µ-α-closed( resp. µ- semi closed, µ- pre closed, µ-β-closed, µ-πrα closed) sets including S. The µ-α-interior (resp. µ-semi interior, µ-pre interior, µ-β-interior, µ-πrα-interior) of a subset S of X denoted by iα(S) (resp. iσ(S), iπ(S), iβ(S), iπp(S)) is the union of µ-α-open ( resp. µ- semi open, µ- pre open, µ-β-open, µ-πrα open) sets contained in S. Definition 2.3[2] A space (X, µ) is called µ-πrα T1/2 space if every µ-πrα closed set is µ- pre closed. Definition 2.4 [2] Let (X, µ) be a generalized topological space and let x X, a subset N of X is said to be µ-πrα-nbhd of x iff there exists a µ-πrα- open set G such that xGN.
Definition 2.5 [2] A function f between the generalized topological spaces (X, µx) and (Y, µy) is called
Definition 2.6 [14] Let (X, µx) and (Y, µy) be GTS’s. Then a function f: XY is said to be
3. Contra (πp, µy) – continuous functions Definition 3.1 Let (X, µx) and (Y, µy) be GTS’s. Then a function f: XY is said to be contra (πp, µy) – continuous, if for each µ-open set U in Y, f-1(U) is µ-πrα closed in X. Theorem 3.2 (i) Every contra (µx, µy) – continuous function is contra (πp, µy) – continuous. (ii) Every contra (α, µy) – continuous function is contra (πp, µy) – continuous. (iii) Every contra (π, µy) – continuous function is contra (πp ,µy) –continuous. Proof: Straight forward. Converse of the above statement is not true as shown in the following examples. Remark: contra (πp, µy) – continuous and contra (α, µy) – continuous, contra (β, µy) – continuous are independent concepts. Example 3.3 Let X = {a, b, c, d}. Consider a generalized topology µx= {, {a}, {a, b, c}} on X and define f: (X, µx) (X, µx) as follows f(a) = f(b) = d and f(c) = f(d)= a. Then f-1({a}) = {c, d}, f-1({a, b, c}) = {c, d}. We have f is contra (πp, µx) – continuous but not contra (µx, µx) – continuous and contra (β, µx) – continuous. Example 3.4 Let X=Y= {a, b, c}. Consider two generalized topologies µx= {, {b}, {a, c}, {b, c}, X} and µy= {, {c}} on X and Y respectively. Define f: (X, µx)( Y, µy) as follows f(a) = b, f(b) = a and f(c) = c. Then f-1({c}) = {c}. We have f is contra (πp, µy) – continuous but not contra (α, µy) – continuous. Example 3.5 Let X =Y= {a, b, c}. Consider two generalized topologies µx= {, {b}, {b, c}, {a, c}, X} and µy= { ,{c}} on X. Define f: (X, µx) (Y, µy) as follows f(a) = b , f(b) = c and f(c) = c. Then f-1({c}) = {b, c}. We have f is contra (πp, µy) – continuous but not contra (π, µy) – continuous, contra (β, µy) – continuous and contra (σ, µy) – continuous. Example 3.6 Let X= {a, b, c, d}. Consider a generalized topology µx= { , {a}, {a, b, c}} on X. Define f: (X, µx) (X, µx) as follows f(a) = d, f (b) = a and f (c) = f (d) = d. Then f-1({a}) = {b}, f-1({a, b, c}) = {b}. We have f is contra (σ, µx) – continuous and contra (β, µx) – continuous but not a contra (πp, µx) – continuous. Theorem 3.7 Let (X, µx) and (Y, µy) be GTS’s. Then the following are equivalent for a function f: XY. (i) f is contra (πp, µy)-continuous. (ii) The inverse image of every µ-closed set of Y is µ- πrα open in X. Proof: (i) (ii) Let U be any µ-closed set of Y. Since Y\U is µ-open then by (i) it follows that f-1(Y\U) = X\f-1(U) is µ-πrα closed in X. This shows that f-1(U) is µ-πrα open in X. (ii) (i) similarly. Definition 3.8[16] Let (X, µ) be a GTS. The generalized Kernel of A X is denoted by µ- ker (A) and defined as µ- ker (A) = {G µ;A G} Lemma 3.9 [16] Let (X, µ) be a GTS and A X. Then µ-ker(A) = {xX; cµ({x})A }. Theorem 3.10 Suppose that µ-πrαO(X) is open under arbitrary union then for a function f: (X, µx)(Y,µy) the following properties are equivalent.
Proof: (i) (ii) is obvious.
(iii)(ii) Let F be any µ-closed set of Y and xf-1(F). Since f(x)F, by(iii) there exists a µ-πrα open set Ux of X such that f(Ux) F. f-1(F) = {Ux/ xf-1(F). Hence f-1(F) is µ-πrα open in X. (ii)(iv) Let A be any subset of X .Suppose that y µ-ker(f(A)),there exist a µ-closed set F containing y such that f(A)F =. Thus we have Af-1(F) =.Therefore we obtain f(cπp(A))F = and yf(cπp(A)). This implies that f(cπp(A))µ-ker(f(A)). (iv)(v) Let B be any subset of Y. By (iv), we have f( cπp(f-1(B)) µ-ker(f(f-1(B))µ-ker(B) and thus cπp(f-1(B))f-1(µ-ker(B)). (v)(i) Let V be any µ-open set of Y. Then by theorem3.10, we have cπp(f-1(V))f-1(µ-ker(V)) = f-1(V) and cπp(f-1(V)) = f-1(V). Hence f-1(V) is a µ-πrα closed set in X. Definition 3.11 A generalized topological space (X, µx) is called (i) µ-πrα locally indiscrete if every µ-πrα open set is µ-closed. (ii) Tπp- space if every µ-πrα closed set is µ-pre closed. (iii) µ-πrα space if every µ-πrα closed set is µ-closed. Theorem 3.12 Let (X, µx) and (Y, µy) be two GTS’s.
Proof: (i) Let V be an µ-open set in Y. By assumption f-1(V) is µ-πrα open in X. Since X is µ-πrα locally indiscrete, f-1(V) is µ-closed in X. Hence f is contra (µx, µy) – continuous. (ii) Let V be an µ- open set in Y. By assumption f-1(V) is µ-πrα closed in X. Since X is µ-πrα-T1/2 space then f-1(V) is µ-pre closed set in X. Hence f is contra (π, µy) – continuous. (iii) Let V be an µ- open set in Y. By assumption f-1(V) is µ-πrα closed set in X. Since X is µ-πrα space then f-1(V) is µ-closed in X. Hence f is contra (µx , µy)- continuous. (iv) Let V be an µ- open in Y. By assumption f-1(V) is µ-πrα closed in X. Since X is Tπp space then f-1(V) is µ-pre closed in X. But every µ- pre closed set is µ-β closed set. Therefore f-1(V) is µ-β closed set in X. Hence f is contra (β, µy)- continuous Theorem 3.13 Let (X, µx) and (Y, µy) be two GTS’s and a function f: XY then the following are equivalent.
Proof: (i) (ii) Straight forward. (ii)(iii) Let xX. Assume that V be a µ-nbhd of f(x), there exists a µ-open set U in Y such that f(x)UV. Consequently f-1(U) is µ-πrα open in X and xf-1(U) f-1(V). Then f-1(V) is µ-πrα nbhd of x. (iii)(iv) Let xX and U be a µ-nbhd of f(x). Then by assumption V= f-1(U) is a µ-πrα nbhd of x and f(V)= f(f-1(U))U. (iv)(v) Let A be a subset of X, f(x)cµ(f(A)). Then there exists a µ- open subset V of Y containing f(x) such that Vf(A)=. Then by (iv) there exists a µ-πrα open set such that f(x)f(U)V.Hence f(U)f(A)= , which implies UA= . Consequently xcπp(A) and f(x)f(cπp(A)). Hence f(cπp(A))cµ(f(A)). (v)(vi) Let A be a subset of Y. By (v) we obtain f(cπp(f-1(A))cµ(f(f-1(A)). Thus f(cπp(f-1(A))cµ(A). This implies (cπp(f-1(A))f-1(cµ(A)). (vi)(i) Let F be a µ-closed subset of Y. Since cµ(F) = F and by (vi) f(cπp(f-1(F))cµ(f(f-1(F)) cµ(F) = F. This implies cπp(f-1(F)) f-1(F) and so f-1(F) is µ-πrα closed. Theorem 3.14 A function f: (X, μx) (Y, μy) is (μx, μy) – πrα continuous if and only if f-1(U) is μ-πrα open in X, for every μ- open set U in Y. Theorem 3.15 Let (X, µx) and (Y, µy) be two GTS’s. Proof: Let x be an arbitrary point of X and V be an µ- open set of Y containing f(x).Since Y is µ-regular there exist an µ-open set W in Y containing f(x) such that cµ(W)V. Since f is contra (πp, µy) – continuous, by theorem 3.10 (iii) there exist a µ-πrα open set U of X containing x such that f(U)cµ(W). Then f(U)cµ(W) V.Hence f is (µx ,µy) –πrα continuous. Hence f is (µx, µy) –πrα continuous. Definition 3.16 Let f: (X, µx)(Y, µy) be a function on GTS’s. Then the function f is said to be
Definition 3.17[1] A GTS (X, µx) is said to be µx- connected if X is not the union of two disjoint non empty µ-open subsets of X. Definition 3.18 A GTS (X, µx) is said to be µx-πrα connected if X is not the union of two disjoint non empty µ-πrα open subsets of X. Theorem 3.19 Let f: (X, µx) (Y, µy) be a (µx, µy)- πrα continuous surjection and if (X, µx) is µx- πrα connected then (Y, µy) is µy - connected. Proof: Let f be a (µx, µy)- πrα continuous function of a µx-πrα connected space X onto Y. If possible let Y be µy- disconnected. Let A and B form a disconnected of Y. Then A and B are µ- open and Y= AB and AB = . Since f is (µx, µy)-πrα continuous surjection function, X = f-1(Y) = f-1( AB) = f-1(A)f-1(B), where f-1(A) and f-1(B) are non empty µ-πrα open sets in X. Also f-1(A)f-1(B) =. Hence X is not µ-πrα connected. This is a contradiction. Therefore Y is µy- connected. Definition 3.20 A GTS (X, µx) is said to be
Definition 3.21 A GTS (X, µ) is said to be µ-Urysohn space if for each pair of distinct points x and y in X, there exists µ- open sets U and V such that xU and yV and cµ(U)cµ(V) =. Theorem 3.22 Let (X, µx) and (Y, µy) be GTS’s. If the following three assumptions are satisfied
Then (X, µx) is µ- πrα T2. Proof: Let x and y be any distinct points in X. By assumption (i) there exists a function f: X Y such that f(x) f(y). Let a = f(x) and b= f(y). Since Y is a µy- Urysohn space then there exists µ-open sets V and W containing a and b respectively such that cµ(V)cµ(W) = . Since f is contra (πp, µy) –continuous at x and y then there exists µ-πrα open sets A and B containing x and y respectively, such that f(A) cµ(V) and f(B)cµ(W). Then f(A)f(B) = . So AB =. Hence X is µ-πrα T2.
For a map f: (X, µx)(Y, µy) , the subset {(x, f(x));xX}XY is called the graph of f and is denoted by Gµ(f). Theorem 3.23 Let (X, µx) and (Y, µy) be GTS’s. Let f: XY be a map and g: X XY the graph function of f defined by g(x) = (x, f(x)) for every xX. If g is contra (πp, µy)- continuous then f is contra (πp, µy)- continuous. Proof: Let U be an µ- open set in Y. Then XU is an µ- open set in XY. Since g is contra (πp, µy) –continuous then f-1(U) = g-1(XU) is µ-πrα closed in X. Hence f is contra (πp, µy) – continuous. Definition 3.24 The graph Gµ(f) of a map f: (X, µx) (Y, µy) between GTS’s is said to be contra (πp, µy)-closed if for each (x, y) (XY) \Gµ(f), there exist an µ-πrα open set U in X containing x and a µ- closed set V in Y containing y such that (UV)Gµ(f) = . Lemma 3.25 Let Gµ(f) be the graph of a map f: (X, µx) (Y, µy) between GTS’s. For any subset AX and BY, f(A)B = if and only if (AB)Gµ(f) = Proposition 3.26 The following properties are equivalent for the graph Gµ(f) of a map f in GTS’s. (i) Gµ(f) is contra (πp, µy) – closed. (ii) For each (x, y) (XY)\Gµ(f), there exist an µ-πrα open set U in X containing x and a µ-closed V in Y containing y such that f(U)V =. Theorem 3.27 Let (X, µx) and (Y, µy) be two GTS’s. If f: XY is contra (πp, µy) – continuous and Y is µy- Urysohn space, then Gµ(f) is contra (πp, µy) –closed in XY. Proof: Let (x, y) (XY)\Gµ(f). It follows that f(x)y and since Y is µy- Urysohn space then if for each distinct points x and y in X there exists µ-open sets B and C such that f(x)B and yC and cµ(B)cµ(C) =. Since f is contra (πp, µy) – continuous then there exists an µ-πrα closed set A in X containing x such that f(A)cµ(B). Therefore f(A)cµ(C) = and Gµ(f) is contra (πp, µy) – closed in XY. Theorem 3.28 Let (X, µx) and (Y, µy) be two GTS’s. Let f: XY have a contra (πp, µy) – closed graph. If f is injective then X is µ-πrα T1. Proof: Let x1 and x2 be any two distinct points of X. We have (x1, f(x2))(XY)\ Gµ(f) and there exist an µ-πrα open set U in X containing x1 and a µ- closed set V in Y containing x2 such that f(U)F = . Hence Uf-1(F) = . Therefore we have x2U. This implies that X is µ-πrα T1. Theorem 3.29 Let (X, µx), (Y, µy) and (Z, µz) be GTS’s. Let f: XY be surjective, (µx, µy) –πrα irresolute and µ-πrα closed and g: YZ be any function. Then g f is contra (πp, µy) – continuous if and only if g is contra (πp , µy) – continuous. Proof: Suppose g f is contra (πp, µy) – continuous. Let F be any µ-open set in Z. Then (g f )-1(F) = f-1(g-1(F)) is µ-πrα closed in X. Since f is µ-πrα closed and surjective, f(f-1(g-1(F)) = g-1(F) is µ-πrα in Y and we obtain that g is contra (πp, µy) – continuous. Conversely, suppose g is contra (πp, µy) – continuous. Let V be µ-open in Z. Then g-1(V) is µ-πrα closed in Y. Since f is (µx, µy) – irresolute, f-1(g-1(V)) = (g f)-1(V) is µ-πrα closed in X and so g f is contra (πp, µy) – continuous.
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