A quasi-locally constant function with given cluster sets

Abstract

We continue to study cluster sets of a function defined on an open subset of a topological space. In our previous papers we characterized the cluster multifunction of any function and of a continuous function. The next aim was to study cluster sets of a quasi-continuous function. But here we solve a more specific problem on cluster sets of a quasi-locally constant function (i.e., a function that is quasi-continuous with respect to the discrete topology on the range space). We prove that for any open dense subset D of a metrizable space X and a dense subspace Y of a metrizable compact space $${\overline{Y}}$$, every upper continuous compact-valued multifunction $$\Phi :X\setminus D\multimap {\overline{Y}}$$ is the cluster multifunction $${\overline{f}}$$ of some quasi-locally constant function $$f:D\rightarrow Y$$.