On Pre-a-Open Sets and Contra Pre-a-Continuous Functions and Contra Pre-a-Irresolute Functions in Topological Spaces

Abstract

In this paper, we introduce and study a new type of generalized open sets in topological spaces namely, pre-α-open sets and we show that the family of all pre-α-open sets in a topological space (X, τ) form a topology on X which is finer than τ . Also, we show that pre-α-open sets is stronger than each of semi-open sets, α-open sets, pre-open sets, b-open sets, β-open sets, generalized semi open sets, generalized α-open sets and α-generalized open sets and weaker than open sets. Moreover, we use these sets to define and study new classes of functions, namely, pre-α-continuous functions, contra pre-α-continuous functions, pre-α-irresolute functions and contra pre-α-irresolute functions in topological spaces and we discuss the relation between these types of functions and each of contra continuous functions and other weaker forms of contra continuous functions.

Keywords: pre-α-open set, pre-α-continuous function, contra pre-α-continuous function, pre-α-irresolute function, contra pre-α-irresolute function, pre-α-space, pre-α-locally indiscrete space