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 $\mathcal{F}(X)$. 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 $X$.
From: Valentin Gutev [view email]
[v1]
Sun, 31 May 2026 19:58:23 UTC (16 KB)
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