On (omega)-(beta) Continuous Functions

Abstract

A subset $A$ of a topological space $X$ is said to be $\omega \beta-$open  if for every $x \in A$ there exists a $\beta-$open set $U$ containing $x$ such that $U-A$ is a countable. In this paper, we introduce and study a new class of functions  called $\omega \beta-$continuous functions by using the notion of $\omega \beta-$open sets. In particular we say a function $f:X \to Y$  is $\omega \beta-$continuous  if and only if for each $x \in X$ and each open set $V$ in $Y$ containing $f(x)$ there exists an $\omega \beta-$open set $U$ containing $x$ such that $f(U) \subseteq V$. We give some characterizations of $\omega \beta-$continuous functions, introduce and study $\omega \beta-$irresolute and $\omega \beta-$open functions. Finally, we investigate the relationship between these type of functions.