On (omega)-(beta) Continuous Functions (European Journal of Pure and Applied Mathematics, 2012)
A subset $A$ of a topological space $X$ is said to be $ω β-$open if for every $x \in A$ there exists a $β-$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 $ω β-$continuous functions by using the notion of $ω β-$open sets. In particular we say a function $f:X → Y$ is $ω β-$continuous if and only if for each $x \in X$ and each open set $V$ in $Y$ containing $f(x)$ there exists an $ω β-$open set $U$ containing $x$ such that $f(U) \subseteq V$. We give some characterizations of $ω β-$continuous functions, introduce and study $ω β-$irresolute and $ω β-$open functions. Finally, we investigate the relationship between these type of functions.