Weakly discontinuous and resolvable functions between topological spaces

TL;DR

This paper characterizes weakly discontinuous functions between certain topological spaces, showing they are equivalent to open-resolvable and resolvable functions, extending a 1985 result for metrizable spaces.

Contribution

It provides a new characterization of weakly discontinuous functions in terms of resolvability, generalizing previous results to broader classes of spaces.

Findings

Weakly discontinuous functions are equivalent to open-resolvable functions.

Resolvability of functions is characterized by preimages of resolvable sets.

The results extend known characterizations from metrizable to more general spaces.

Abstract

We prove that a function $f:X\to Y$ from a first-countable (more generally, Preiss-Simon) space $X$ to a regular space $Y$ is weakly discontinuous (which means that every subspace $A\subset X$ contains an open dense subset $U\subset A$ such that $f|U$ is continuous) if and only if $f$ is open-resolvable (in the sense that for every open subset $U\subset Y$ the preimage $f^{-1}(U)$ is a resolvable subset of $X$) if and only if $f$ is resolvable (in the sense that for every resolvable subset $R\subset Y$ the preimage $f^{-1}(R)$ is a resolvable subset of $X$). For functions on metrizable spaces this characterization was announced (without proof) by Vinokurov in 1985.