Local and global properties of spaces of minimal usco maps

Abstract

In this paper, we study an interplay between local and global properties of spaces of minimal usco maps equipped with the topology of uniform convergence on compact sets. In particular, for each locally compact space $X$ and metric space $Y$, we characterize the space of minimal usco maps from $X$ to $Y$, satisfying one of the following properties: (i) compact, (ii) locally compact, (iii) $\sigma$-compact, (iv) locally $\sigma$-compact, (v) metrizable, (vi) ccc, (vii) locally ccc, where in the last two items we additionally assumed that $Y$ is separable and non-discrete. Some of the aforementioned results complement ones of \v{L}ubica Hol\'a and Du\v{s}an Hol\'y. Also, we obtain analogical characterizations for spaces of minimal cusco maps.