Scattered spaces and selections (Q2411474)

Let \(\mathcal{F}(X)\) be the space that consists of all non-empty, closed subsets of \(X\) endowed with the Vietoris topology. The author starts with a brief discussion of continuous zero-selections. After that it is proven that a crowded space \(X\) with a continuous selection for \(\mathcal{F}(X)\) always contains a copy of the rational numbers. Thereafter this result is used to prove that a regular space \(X\) with a continuous selection for \(\mathcal{F}(X)\) is scattered if and only if \(\mathcal{F}(Z)\) has a continuous selection for each countable \(Z\subseteq X\). Finally the author gives several results on the relations between the existence of continuous selections for \(\mathcal{F}(X)\) and disconnectedness-like properties of the space \(X\).