On the Variety of Hyperspace Selections

Abstract

If $f$ is a continuous selection for the Vietoris hyperspace $\mathcal{F}(X)$ of the nonempty closed subsets of a space $X$, then the point $p=f(X)\in X$ is not as arbitrary as it might seem at first glance. In fact, the set ${\mathcal{O}}_{cs}(X)$ of all these points reveals certain information about the variety of Vietoris continuous selections for $\mathcal{F}(X)$. Thus, for a connected space $X$, we will show that every point $p\in {\mathcal{O}}_{cs}(X)$ is not only noncut, but also an endpoint of $X$. Another result of this paper is that in an arbitrary topological space $X$, the closure of the set ${\mathcal{O}}_{cs}(X)$ is always a totally disconnected subset. Furthermore, we will also show that ${\mathcal{O}}_{cs}(X)$ is a closed subset of every first countable totally disconnected space $X$.