We demonstrate that a second countable space is weakly orderable if and only if it has a continuous weak selection. This provides a partial positive answer to a question of van Mill and Wattel.

We show that if a Hausdorff topological space $X$ satisfies one of the following properties: a) $X$ has a countable, discrete dense subset and $X^2$ is hereditarily collectionwise Hausdorff; b) $X$ has a discrete dense subset and admits a countable base; then the existence of a (continuous) weak selection on $X$ implies weak orderability. As a special case of either item a) or b), we obtain the result for every separable metrizable space with a discrete dense subset.

It is proved that for a zero-dimensional space $X$, the function space $C_p(X,2)$ has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if $X$ is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space $E$, the function space $C_p(X,E)$ is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial...