On contra βwg-continuous functions in topological spaces

The main aim of this paper is to define and study the notions of contra βwg-continuous, almost contra βwg-continuous functions and discussed the relationship with other contra continuous functions and obtained their characteristics. Further we introduce the concepts of contra βwg-irresolute, contra βwg-closed functions and obtain some of their properties.

Illustrations explain the paper; they do not show study data.

AI-generated illustration

↗ Enlarge

Hierarchy of Contra ContinuityClose ×

AI-generated conceptual illustration (Abstract mathematical topological spaces represented as nested geometric spheres). It explains the idea and does not show study data.

Open full-size image ↗

Introduction

In 1996, Dontchev. [4] , introduced the notion of contra continuous functions. Jafari and Noiri [9] introduced contra precontinuous functions. Ekici. E [6] introduced almost contra precontinuous functions in 2004. Dontchev and Noiri [5] , introduced and investigated contra semi-continuous functions and RC-continuous functions between topological spaces. Veerakumar [26] also introduced contra pre-semi-continuous functions. S.Sekar and P. Jayakumar [21] introduced contra gp*-continuous functions. Recently, Govindappa. Navalagi and Kantappa. M. Bhavikatti [15] introduced βwg-continuous functions. In this paper we introduce and study the new class of functions called contra βwg-continuous and almost contra βwg-continuous functions in topological spaces. Also we define the notions of contra βwg-irresolute, contra βwg-closed functions, βwg-locally indiscrete space and study some of their properties. Throughout this paper (X, τ), (Y, σ) and (Z, η) (or simply X, Y, and Z) represent the nonempty topological spaces on which no separation axioms are assumed, unless otherwise mentioned. For a subset A of X, the closure of A and interior of A will be denoted by Cl (A) and Int (A) respectively. The union of all βwg-open sets of X contained in A is called βwginterior of A and it is denoted by βwgInt (A). The intersection of all βwg-closed sets of X containing A is called βwg-closure of A and it is denoted by βwg Cl (A). Also the collection of all βwg-open subsets of X containing a fixed point x is denoted by βwg O (X, x).

Preliminaries

We recall the following definitions which are useful in the sequel.

Definition 2.1

A subset A of a topological space (X, τ) is called (i) Semi-open [11] if A  cl (int (A)) and semi-closed if int (cl (A))  A. (ii) Preopen [13] if A  int (cl (A)) and preclosed if cl (int (A))  A. (iii) α-open [16] if A  int (cl (int (A))) and α-closed if cl (int (cl (A)))  A. (iv) semi-preopen [1]

(vi) Regular open [23] if A = int (cl (A)) and regular closed if A = cl (int (A)).

Definition 2.2:

A subset A of a topological space (X, τ) is called (i) Generalized preclosed (briefly, gp-closed) [12] if pcl (A)  U whenever A  U and U is open in X. (ii) Generalized semi-preclosed (briefly, gsp-closed) [3] if spcl (A)  U whenever A  U and U is open in X. (iii) Generalized pre regularclosed (briefly, gpr-closed) [7] if pcl (A)  U whenever A  U and U is regular open in X. (iv) Generalized star preclosed (briefly, g*p-closed set) [23] if pcl (A)  U whenever A  U and U is g-open in X. (v) Generalized pre star closed (briefly, gp*-closed set) [10] if cl (A)  U whenever A  U and U is gp-open in X. (vi) Βwg-closed [14] if βcl (A)  U whenever A  U and U is αg-open in X. (vii) Pre semi-closed [25] if spcl (A)  U whenever A  U and U is g-open in X. Definition 2.3: [15] A function f: (X, τ → (Y, σ) is called βwg-continuous if f -1 (V) is βwg-closed set in (X, τ) for every closed set 2. The set ∩ {F  X/A  F, F is β-closed} is called the β-closure of A and is denoted by βc1 (A).

Lemma 2.8:

The following properties hold for subsets A, B of a space X:

1. X  ker (A) if and only if U ∩ A ≠ φ for any F∈C(X, x).

A  ker (A) and A = ker (

Lemma 2.9: [15] For xX, x Cl (A) if and only if U ∩ A ≠ φ for every βwg-open set U containing x. Proof: Necessary part: Suppose there exists βwg -open set U containing x such that U ∩ A ≠ φ. Since A  X -U, βwg Cl (A)  X-U. This implies xβwg Cl (A). This is a contradiction. Sufficiency part: Suppose that x  βwg-cl -closed subset F containing A such that x  F. Then x  X-F is ßwg-open, (X -F) ∩A = φ. This is contradiction.

Contra βwg-Continuous Functions

In this section, we introduce and study new class of continuous functions called contra βwg-continuous functions and investigate some of their properties in the following.

Definition 3.1:

Theorem 3.16: Every contra βwg-continuous function is contra rg-continuous but not conversely.

Proof: Let U be an open set in Y then f -1 (U) is βwg-closed set in X. Since every βwg-closed set is rg-closed set then f -1 (U) is rgclosed in X. Therefore f is contra rg-continuous.

Example 3.17: Let X = Y = {a, b, c}, τ = {φ, {a}, X} and σ = {φ, {c}, {a, c}, Y}. Now RGC(X) =P(X) and βwgC (X) = {X, φ, {b}, {c}, {b, c}}. Define f: X → Y by f (a) = c, f (b) = b and f(c) = a. Then f is contra rg-continuous but not contra βwgcontinuous, since {a,c} is an open set in Y, f -1 ({a, c}) = {a, c} is rg-closed but not βwg-closed in X.

Theorem 3.17: If f: X → Y contra (gsp)*-continuous then f is contra βwg-continuous function but not conversely. Proof: Let G be an open set in Y. Since f is contra (gsp)*-continuous, then f -1 (G) is (gsp)*-closed set in X. Since every contra (gsp)*-closed set is βwg -closed set then f -1 (U) is βwg-closed in X. Therefore f is contra βwg-continuous. Proof:

Hence f is contra continuous. (iii) Let G be open in (Y, σ). By assumption, f -1 (G) is βwg-closed in X. Since by definition, X is βwg T 1/2 space, f -1 (V) is pre-closed in X. Hence f is contra pre-continuous.

Theorem 3.21: If a function f: (X, τ) → (Y, σ) is contra βwg-continuous and (X, τ) is ßwg T d -space then f is contra g-continuous.

Theorem 3.22: If a function f: (X, τ) → (Y, σ) is contra βwg-continuous and (X, τ) is βwg T α -space then f is contra α-continuous.

Proof:

Let V be open in (Y, σ). By assumption, f -1 (V) is βwg-closed in X. Since X is βwg T α -space, f -1 (V) is α-closed in X. Hence f is contra α-continuous.

Theorem 3.23:

The following are equivalent for a function f: (X, τ) → (Y, σ): F is contra βwg-continuous;

1. For every closed subset F of Y, f -1 (F)  βwgO (X); 2. F (βwgC1 (A))  ker (f (A)) for every subset A of X ; 3. βwgC1 (f -1 (B))  f -1 (ker (B)) for every subset of B of Y.

Proof:

The implications (i)  (ii) is obvious.

(ii)  (III):

Let A be any subset of X. Suppose that y ker (f (A)). Then by Lemma 2.10, there exists FC(X,y) such that f (A) ∩ F ≠ φ. Thus A ∩ f -1 (F) = φ and βwg-cl (A) ∩ f -1 (F) = φ .Therefore, we obtain f (βwg-cl(A)) ∩ f -1 (F) = φ and y f(βwgC1(A)). This implies that f(βwgC1(A))  ker ( f (A)). Then by Lemma 2.10, we have βwgC1 (f -1 (V))  f -1 (ker (V)) = f -1 (V) and βwg-c1 (f -1 (V)) = f -1 (V). This shows that f -1 (V) is βwg-closed in X.

Remark 3.24:

The Composition of two contra βwg-continuous maps need not be contra Βwg-continuous map and this can be shown by the following example.

1. Andrijevic D. Semi-preopen sets, Mat. Vesnik. 1986; 38(1):24-32.
2. Dontchev J. On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi. Univ. Ser. A. Math. 1995; 16:35-48.
3. Dontchev J. Contra continuous functions and strongly S-closed spaces, Int. Math. Sci. 1996; 19(2):303-310.
4. Dontchev J, Noiri T. Contra semi-continuous functions, Math. Pannon. 1999; 10(2); 159-168.
5. Ekici E. Almost contra pre-continuous functions, Bull. Malaysian Math. Sci. Soc. 2004; 27:53-65.
6. Gnanambal Y. On generalized preregular closed sets in topological spaces, Indian J Pure. Appl. Math. 1997; 28(3):351-360.
7. Jafari S, Noiri T. Contra α-continuous functions between topological spaces, Iran. Int. J Sci. 2001; 2(2):153-167.
8. Jafari S, Noiri T. On contra pre-continuous functions, Bull. Malays. Math Sci. Soc. 2002; 25(2):115-128.
9. Jayakumar P, Mariappa K, Sekar S. On generalized gp*-closed set in topological spaces, Int. Journal of Math. Analysis. 2013; 33(7):1635-1645.
10. Levine N. Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly. 1963; 70:36-41.
11. Maki H, Umehara J, Noiri T. Every topological space is pre-T 1/2 space, Mem. Fac. Sci. Kochi Univ. Ser. A. Math. 1996; 17:33-42.
12. Mashhour AS, Abd El-Monsef ME, El-Deeb SN. On pre-continuous and weak pre-continuous mapings, Proc. Math. and Phys. Soc. Egypt. 1982; 53:47-53.
13. Navalagi GB, Bhavikatti KM. Beta Weakly Generalized Closed sets in Topology, Journal Of Computer And Mathematical Sciences, May (Accepted), 2018, 9(5).
14. Govinappa Navalagi, Kantappa Bhavikatti M. On βwg-Continuous and βwg-Irresolute Functions in Topological Spaces, IJMTT, (May). 2018; 57(1):9-20.
15. Njastad O. On some classes of nearly open sets, Pacific J Math. 1965; 15:961-970.
16. Noiri T. Super-continuity and some strong forms of continuity, Indian J Pure Appl. Math. 1984; 15:241-250.
17. Patil PG, Rayanagoudar TD, Mahesh Bhat K. On some new functions of g*p-continuity, Int. J Contemp. Math. Sciences. 2011; 6:991-998.
18. Pauline Mary Helen M, Kulandhai Therese, (gsp) * -Closed Sets In Topological Spaces, Int. Journal of Mathematics Trends and Technology. 2014; 6:75-86.
19. Rajakumari J, Sekar C. ContraAlpha generalised Star pre-Continuous Functions in Topological Spaces, Journal of Global Research in Mathematical Archive. 2016; 3(8):1-7.
20. Sekar S, Jayakumar P. Contra gp*-continuous Functions, IOSR Journal of 23. Mathematics. 2014; 10(4):55-60.
21. Singal MK, Singal AR. Almost continuous mappings, Yokohama. Math. 1968; 3:63-73.
22. Stone M. Application of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 1937; 41:374-381.
23. Veera kumar MKRS. G*-preclosed sets, Indian J Math. 2002; 44(2):51-60.
24. Veera kumar MKRS. Pre-semi closed sets, Acta ciencia Indica (Maths) Meerut, (M). 2002; XXVIII(1):165-181.
25. Veerakumar MKRS. Contra pre-semicontinuous functions, Bull. Malays. Math. Sci. Soc. 2005; 28 (2):67-71.