Contra-Somewhat Continuous Functions

Abstract

Two new forms of contra-somewhat continuity are introduced. Characterizations and the basic properties of both forms investigated.

Almost contra-somewhat continuity

“…The following theorem is a consequence of Theorem 4.2 and Theorem 4.1 [1]. Theorem 4.10 If f : X → Y is almost somewhat continuous and almost contra-precontinuous, then f is almost contra-2-somewhat continuous.…”

“…Definition 2.4 A function f : X → Y is said to be contra-1-somewhat continuous [1] provided that for every closed set F ⊆ Y such that f −1 (F ) = ∅, there exists an open set U ⊆ X such that ∅ = U ⊆ f −1 (F ).…”

“…Definition 2.5 A function f : X → Y is said to be contra-2-somewhat continuous [1] if for every open set…”

“…The class of somewhat continuous functions was studied by Gentry and Hoyle [8] in 1971. Two types of contra-somewhat continuity were developed in 2015 by Baker [1]. The purpose of this note is to introduce weak forms of both of these types of contra somewhat continuity, which we call almost contra-1-somewhat continuity and almost contra-2-somewhat continuity.…”

An alternate form of somewhat continuity

“…Definition 3.15 Let X be a set. Topologies τ and τ on X are said to be contra weakly equivalent [1] provided that, if F is τ -closed and F = ∅, then there exists a τ -closed set G such that G ⊆ F and G = ∅ and, if F is a τ -closed set and F = ∅, then there exists a τ -closed set G such that G ⊆ F and G = ∅. Theorem 3.16 If f : (X, τ ) → (Y, σ) is A-somewhat continuous and τ is a topology on X that is contra weakly equivalent to τ , then f : (X, τ ) → (Y, σ) is A-somewhat continuous.…”