On scatteredly continuous maps between topological spaces

TL;DR

This paper explores the concept of scatteredly continuous maps between topological spaces, establishing their equivalence to several other continuity properties under certain conditions, and constructs a counterexample under Martin Axiom.

Contribution

It characterizes scatteredly continuous maps on specific spaces and provides a counterexample showing such maps need not be piecewise continuous.

Findings

Scattered continuity is equivalent to weak discontinuity, $\sigma$-continuity, and $G_\delta$-measurability under certain conditions.

Constructs a $G_\delta$-measurable map that is not piecewise continuous, answering an open question.

Provides conditions under which scattered continuity aligns with other forms of continuity in topological spaces.

Abstract

A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact hereditarily Baire Preiss-Simon space $X$ into a regular space $Y$ the scattered continuity of $f$ is equivalent to (i) the weak discontinuity (for each subset $A\subset X$ the set $D(f|A)$ of discontinuity points of $f|A$ is nowhere dense in $A$), (ii) the $\sigma$-continuity ($X$ can be written as a countable union of closed subsets on which $f$ is continuous), (iii) the $G_\delta$-measurability (the preimage of each open set is of type $G_\delta$). Also under Martin Axiom, we construct a $G_\delta$-measurable map $f:X\to Y$ between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V.Vinokurov.