Local and global properties of spaces of minimal usco maps

TL;DR

This paper characterizes spaces of minimal usco maps with various local and global properties, providing new insights into their structure and extending previous results in the field.

Contribution

It offers new characterizations of spaces of minimal usco and cusco maps under different topological conditions, enriching the understanding of their properties.

Findings

Characterization of minimal usco maps with compactness properties

Extension of results to minimal cusco maps

Connections between local and global topological properties

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.