Generalized continuity and uniform approximation by step functions.

Abstract

Given two topological spaces $X$ and $Y$ and a family ${\mathcal O}_\ast$ of subsets of $X$, a function $f: X \rightarrow Y$ is called ${\mathcal O}_\ast$-continuous if $f^{-1}(V) \in {\mathcal O}_\ast$ for every open set $V \subseteq Y$. An ${\mathcal O}_\ast$-step function is meant to be a function $\varphi: X \rightarrow Y$ that is piecewise constant on a partition of $X$ into sets from ${\mathcal O}_\ast$. Using some technical assumptions on $X$, $Y$, and ${\mathcal O}_\ast$ we give representations of ${\mathcal O}_\ast$-continuous functions as uniform limits of ${\mathcal O}_\ast$-step functions. We deal in particular with $\alpha$-continuous, nearly continuous, almost quasi-continuous, and somewhat continuous functions. The paper is motivated by a corresponding characterization of quasi-continuous functions.