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continuous selections; Vietoris topology; linearly orderable space; weakly orderable space; space of continuous functions; dyadic spaces
For a space $Z$, we denote by $\Cal{F}(Z)$, $\Cal{K}(Z)$ and $\Cal{F}_2(Z)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\leq 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $\Cal{F}(Z)$, $\Cal{K}(Z)$ and $\Cal{F}_2(Z)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and only if $X$ is separable. Moreover, we obtain that the separability of $X$, the existence of a continuous selection for $\Cal{K}(C_p(X,E))$, the existence of a continuous selection for $\Cal{F}_2(C_p(X,E))$ and the weak orderability of $C_p(X,E)$ are equivalent when $X$ is $\Bbb{N}$-compact. Also, we decide in which cases $C_p(X,2)$ and $\beta C_p(X,2)$ are linearly orderable, and when $\beta C_p(X,2)$ is a dyadic space.
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