endpoints.

Basic terminology and definitions {#defandnot}
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Let $X$ be a topological space. If $A$ is a subset of $X$, then $\operatorname{int}_{X}(A), \operatorname{cl}_{X}(A), \partial_{X}(A)$ denote the interior, closure, and boundary of $A$ in $X$, respectively. We drop the subscript whenever the ambient space is clear from context. A closed set is regular if it coincides with the closure of its interior. We denote by $\operatorname{\mathcal K}(X) = \setnew*{K \subseteq X}{K \text{ compact}}$ the space of compact subsets of $X$, with the Vietoris topology. This is the topology generated by the sets $ \setnew{K\in \operatorname{\mathcal K}(X)}{K\subseteq O} $ and $\setnew{K \in \operatorname{\mathcal K}(X) }{K \cap O\neq \emptyset} $, for $O$ varying among the open subsets of $X$. If $X$ is compact metrizable, so is $\operatorname{\mathcal K}(X)$. Let $\operatorname{Homeo}(X)$ denote the group of homeomorphisms of $X$.

By *mesh* of a covering of a metric space, we indicate the supremum of the diameters of its elements.

We collect here the definitions of some basic topological concepts we need.

-   A space is *almost zero-dimensional* if each point has a neighborhood basis consisting of closed sets that are intersection of clopen sets.

-   A space is $X$ *cohesive* if each point has a neighborhood 