On the selection problem for “metric”-proximal hyperspaces

It is proved that a zero-dimensional metrizable space X   is locally compact modulo one point and separable if, and only if, its hyperspace F(X)F(X) of non-empty closed sets has a selection which is continuous with respect to all proximal hypertopologies on F(X)F(X) generated by the compatible metrics on X  . This completely solves a selection problem posed by Gutev and Nogura. The technique developed in the paper allows to show that the same holds also for the collection of all Hausdorff metric hypertopologies on F(X)F(X). Moreover, this strong selection problem is studied in the context of the usual selection problem for two natural “metric-independent” hypertopologies on F(X)F(X), and related to some topological properties of the hyperspace.