Contra θ-c-continuous Functions

Transcription

1 International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 1, HIKARI Ltd, Contra θ-c-continuous Functions C. W. Baker Department of Mathematics Indiana University Southeast New Albany, IN , USA Copyright c 2017 C. W. Baker. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A new form of contra-continuity, called contra θ-c-continuity, is introduced. It shown that contra θ-c-continuity is between contra θ- continuity and contra-continuity. The basic properties of this class of functions are developed and several applications to H-closed spaces and Katětov spaces are investigated. Mathematics Subject Classification: 54C10, 54D10 Keywords: contra θ-c-continuous, contra θ-continuous, contra-super- continuous 1 Introduction Caldas, et al. [3] developed the concept of a contra θ-continuous function and showed that this class of functions is properly contained in the class of contracontinuous functions. The notion of θ-c-open set was introduced by Baker [1]. In this note the concept of θ-c-openness is used to define the notion of a contra θ-c-continuous function. It is shown that contra θ-c-continuity is strictly between contra θ-continuity and contra-continuity Some relationships between contra θ-c-continuity and contra-super-continuity are established. Also applications to H-closed spaces and Katětov spaces are given.

2 44 C. W. Baker 2 Preliminaries The symbols X and Y represent topological spaces with no separation properties assumed unless explicitly stated. All sets are considered to be subsets of topological spaces. The closure and interior of a subset A of a space X are signified by Cl(A) and Int(A), respectively. A set A is said to be regular open provided that A = Int(Cl(A)) and regular closed provided that A = Cl(Int(A)) or equivalently its complement is regular open. The θ-closure of a set A, denoted by Cl θ (A), is the set of all x X such that every closed neighborhood of x intersects A nontrivially. A set A is said to be θ-closed [10] if Cl θ (A) = A. A set A is θ-open if its complement is θ-closed or equivalently if A contains a closed neighborhood of each of its points. A set A is called δ-open [10] if A contains a regular open neighborhood of each of its points. Definition 2.1 A subset U of a space X θ-c-open [1] if there exists a set A such that U = X Cl θ (A) and a set F is said to be θ-c-closed if its complement is θ-c-open or equivalently if there exists a set B such that F = Cl θ (B). Definition 2.2 A function f : X Y is said to be θ-c-continuous [1] if f 1 (V ) is θ-c-open for every open set V of Y. Definition 2.3 A function f : X Y is said to be contra super-continuous [6] if for every x X and every closed set F of Y containing f(x), there exists a regular open set U containing x such that f(u) F or equivalently if the inverse image of every closed set of Y is δ-open in X. Definition 2.4 A function f : X Y is said to be RC-continuous [5] (respectively, contra-continuous [4]) if f 1 (V ) is regular closed (respectively, closed) for every open set V of Y. Definition 2.5 A function f : X Y is said to be strongly θ-continuous [7] if for every closed set F of Y, f 1 (F ) is θ-closed. 3 Contra θ-c-continuous functions Definition 3.1 A function f : X Y is said to be contra θ-c-continuous if f 1 (V ) is θ-c-closed for every open subset V of Y. Since θ-closed implies θ-c-closed (Theorem 1 [1]), contra θ-continuous implies contra θ-c-continuous. Similarly, since θ-c-closed implies δ-closed [1], contra θ-c-continuous implies contra-super-continuous. Also, since regular open implies θ-c-open [1], RC-continuous implies contra θ-c-continuous. The following examples show that these implications are not reversible.

3 Contra θ-c-continuous Functions 45 Example 3.2 Let X = {a, b, c} have the topologies τ = {X,, {a} {c}, {a, c}} and σ = {X,, {a, b}} and let f : (X, τ) (Y, σ) be the identity mapping. Since f 1 ({a, b}) is θ-c-closed but not θ-closed, f is contra θ-c-continuous but not contra-θ-continuous. Example 3.3 Let X = {a, b, c} have the topologies τ = {X,, {a} {c}, {a, c}} and σ = {X,, {b}} The identity mapping f : (X, τ) (Y, σ) is contra-supercontinuous but not contra θ-c-continuous because f 1 ({b}) is δ-closed but not θ-c-closed. Example 3.4 Let X denote the real numbers and let τ be the usual topology on X. Let A be an open (and hence θ-c-open) set (X, τ) that is not regular open. Let σ = {X,, X A}. Then the identity mapping f : (X, τ) (X, σ) is contra θ-c-continuous but not RC-continuous. The proof of the following theorem follows easily from the definitions. Theorem 3.5 A function f : X Y is contra θ-c-continuous if and only if f 1 (F ) is θ-c-open for every closed subset F of Y. We now consider two pointwise or local versions of contra θ-c-continuity. The following implications are immediate consequences of the definitions. Theorem 3.6 If f : X Y is contra θ-c-continuous, then the following statements hold: (a) For every x X and every open set V of Y containing f(x), there exists a θ-c-closed set F such that x F and f(f ) V. (b) For every x X and every closed set F of Y containing f(x), there exists a θ-c-open set U such that x U and f(u) F. The following examples show that (a) and (b) of Theorem 3.6 are independent. Example 3.7 Let X denote the real numbers with the usual topology and let f : X X be the identity mapping. Since X is T 1 and regular, f 1 (V ) is a union of θ-closed sets for every open subset V of X and hence also a union of θ-c-closed sets. However, f 1 ({x}) is not open and hence not θ-c-open for any x X. Therefore Theorem 3.6 (a) does not imply 3.6 (b) Example 3.8 The function in Example 3.3 satisfies Theorem 3.6(b) but not Theorem 3.6 (a) because f 1 ({a, c}) is a union of θ-c-open sets and f 1 ({b}) is not θ-c-closed

4 46 C. W. Baker It follows from Theorem 3.6 together with the fact that Theorem 3.6(a) and Theorem 3.6(b) are independent that neither Theorem 3.6(a) nor 3.6(b) implies contra θ-c-continuity. Since a set is δ-open if and only if it is union of θ-c-open sets, Theorem 3.6 (b) is equivalent to contra-super-continuity. A function satisfying Theorem 3.6(a) will be called locally θ-c-continuous. The following implications, none of which are reversible, have been established. RC-continuity contra θ-cont. contra-θ-c-cont. contra-super-cont. contra cont. locally contra θ-c-cont. Theorem 3.9 (Theorem 4.2 [6]) If a function f : X Y is contra supercontinuous and Y is regular, then f is super-continuous and strongly θ-continuous. Corollary 3.10 If a function f : X Y is contra θ-c-continuous and Y is regular, then f is super-continuous and strongly θ-continuous. Theorem 3.11 If a function f : X Y is contra θ-c-continuous and Cl θ (A) is regular-closed for every set A of X, then f is RC-continuous. Theorem 3.12 If a function f : X Y is contra θ-c-continuous and Cl θ (A) is θ-closed for every set A of X, then f is contra θ-continuous. 4 Properties of contra θ-c-continuous functions Definition 4.1 A function f : X Y is said to be quasi θ-c-continuous if f 1 (V ) is θ-c-open for every θ-c-open subset V of Y. Theorem 4.2 Let f : X Y and g : Y Z be functions. (a) If f is quasi-θ-c-continuous and g is contra θ-c-continuous, then g f is contra θ-c-continuous. (b) If f is contra θ-c-continuous and g is continuous, then g f is contra θ-c-continuous. Definition 4.3 A function f : X Y is said to be quasi θ-c-closed if f(f ) is θ-c-closed for every θ-c-closed set F of Y. Theorem 4.4 Let f : X Y and g : Y Z be functions. If g f is contra θ-c-continuous and f is surjective and quasi θ-c-closed, then g is contra θ-c-continuous..

5 Contra θ-c-continuous Functions 47 Proof. Let V be an open set of Z. Since g f is contra θ-c-continuous, f 1 (g 1 (V )) is θ-c-closed in X. Since f surjective and quasi θ-c-closed, g 1 (V ) = f(f 1 (g 1 (V ))) is θ-c-closed in Y. Therefore g is contra θ-c-continuous. Theorem 4.5 Let f : X Y and g : Y Z be functions. If g f is contra θ-c-continuous and g is injective and open, then f is contra θ-c-continuous.. Proof. Let V be an open set of Y. Since g is open, g(v ) is open in Z. Then, since g is injective and g f is contra θ-c-continuous, f 1 (V ) = f 1 (g 1 (g(v ))) is θ-c-closed in X. Therefore f is contra θ-c-continuous. Theorem 4.6 Let f α : X Y α be a function for every α A and let f : X α A Y α be the product function given by f(x) = (f α (x)) α for every x X. If f is contra θ-c-continuous, then f α is contra θ-c-continuous for every α A. Proof. For every α A, f α = p α f, where p α is the projection onto Y α. Since p α is continuous and f is contra θ-c-continuous, it follows from Theorem 4.2(b) that f α is contra θ-c-continuous for every α A. The proof of the following theorem is a consequence of Theorem 4.6. Theorem 4.7 Let f : X Y be a function and let g : X X Y be the graph function of f given by g(x) = (x, f(x)) for every x X. If g is contra θ-c-continuous, then f is contra θ-c-continuous. The following example shows that the graph function of a contra θ-ccontinuous function is not necessarily contra θ-c-continuous. Example 4.8 Let X = {a, b, c} have the topologies τ = {X,, {a} {c}, {a, c}} and σ = {X,, {b}}the function f : (X, τ) (X, σ) given by f(x) = a for every x X is contra θ-c-continuous, but its graph function is not contra θ-c-continuous because g 1 ({b} {a, c}) is not θ-c open in (X, τ). Definition 4.9 A space X is said to be strongly θ-c-closed if every cover of X by θ-c-closed sets has a finite subcover. Theorem 4.10 Let f : X Y be a locally contra θ-c-continuous surjection. If X is strongly θ-c-closed, then Y is compact. Proof. Let C be an open cover of Y. Let x X and let V x C such that f(x) V x. Then, since f is locally contra θ-c-continuous, there exists a θ-c-closed set F x in X such that x F x f 1 (V x ). Then {F x : x X} is a cover of X by θ-c-closed sets. Since X is strongly θ-c-closed, there exists a finite subcover {F xi : i = 1,..., n}. It then follows that {V xi : i = 1,..., n} is a finite subcover of C and hence that Y is compact.

6 48 C. W. Baker Corollary 4.11 Let f : X Y be a contra θ-c-continuous surjection. If X is strongly θ-c-closed, then Y is compact. Definition 4.12 A space X is said to be H-closed [9] if X is a closed subset in every space containing X as a subspace. Definition 4.13 A space X is said to be Katětov [9] if it has a coarser minimal H-closed topology or equivalently a coarser H-closed topology. The following theorem implies a relationship between θ-c-closed sets and Katětov spaces. Theorem 4.14 [9] If X is an H-closed space and A X, then Cl θ (A) is Katětov. Corollary 4.15 Every θ-c-closed set in an H-closed space is Katětov. Corollary 4.16 If X is H-closed and f : X Y is contra θ-c-continuous, then. for every open set V in Y, f 1 (V ) is a Katětov subspace of X. The next theorem establishes a relationship between θ-c-closure and compactness. Theorem 4.17 [9] An H-closed space in which every closed set is the θ- closure of some set is compact. Definition 4.18 A function f : X Y is called contra-closed [2] if f(f ) is open for every closed set F in X. Theorem 4.19 If X is H-closed and f : X Y is contra θ-c-continuous, injective, and contra-closed, then X is compact. Proof. Let F be a closed set in X. Since f is contra-closed, f(f ) is open. Since f is injective and contra θ-c-continuous, F = f 1 (f(f )) is θ-c-closed. Therefore every closed set in X is θ-c-closed and it follows from Theorem 4.17 that X is compact. Recall that the graph of a function f : X Y is the subset G(f) = {(x, f(x)) : x X} of X Y. Definition 4.20 The graph of a function f : X Y, is said to be contra θ-c-closed if for every (x, y) X Y G(f) there exist a θ-c-open set U in X and a closed set F in Y such that (x, y) U F X Y G(f). Theorem 4.21 If a function f : X Y is contra θ-c-continuous and Y is Urysohn, then G(f) is contra θ-c-closed.

7 Contra θ-c-continuous Functions 49 Proof. Let (x, y) X Y G(f). Then f(x) y and, since Y is Urysohn, there exist an open set V in Y containing f(x) and an open set W in Y containing y such that Cl(V ) Cl(W ) =. Then (x, y) f 1 (Cl(V )) Cl(W ) X Y G(f). Since f 1 (Cl(V )) is θ-c-open and Cl(V ) is obviously closed, G(f) is contra θ-c-closed. Definition 4.22 The graph of a function f : X Y, G(f), is said to be weakly θ-c-closed if for every (x, y) X Y G(f) there exist a θ-c-open set U in X and an open set V in Y such that (x, y) U V X Y G(f). Theorem 4.23 If a function f : X Y is contra θ-c-continuous and Y is Hausdorff, then G(f) is weakly θ-c-closed. Proof. Let (x, y) X Y G(f). Then f(x) y and, since Y is Hausdorff, there exist disjoint open sets V and W in Y such that f(x) V and y W. Then (x, y) f 1 (Cl(V )) W X Y G(f), which proves that G(f) is weakly θ-c-closed. References [1] C. W. Baker, On θ-c-open sets, Internat. J. Math. Math. Sci., 15 (1992), [2] C. W. Baker, Contra-open functions and contra-closed functions, Math. Today, 15 (1997), [3] M. Caldas, S. Jafari and T. Noiri, Contra θ-continuity in topological spaces, Questions, Answers in General Topology, 33 (2015), [4] J. Dontchev, Contra-continuous functions and strongly S-closed spaces, Internat. J. Math. Math. Sci., 19 (1996), [5] J. Dontchev and T. Noiri, Contra-semicontinuous functions, Math. Pannonica, 10 (1999), [6] S. Jafari and T. Noiri, Contra-super continuous functions, Annales Univ. Sci. Budapest, 42 (1999), [7] P. E. Long and L. L. Herrington, Strongly θ-continuous functions, J. Korean Math. Soc., 18 (1981), [8] B. M. Munshi and D. S. Bassan, Super-continuous mappings, Indian J. Pure Appl. Math., 13 (1982),

8 50 C. W. Baker [9] J. Porter and M. Tikoo, On Katětov spaces, Canad. Math. Bull., 32 (1989), [10] N. V. Veli cko, H-closed topological spaces, Chapter in Eleven Papers on Topology, Amer. Math. Soc. Transl.: Series 2, Vol. 78, Amer Mathematical Society, 1968, Received: February 3, 2017; Published: February 28, 2017