Baire spaces and hyperspace topologies revisited

Abstract

It is the purpose of this paper to show how to use approach spaces to get a unified method of proving Baireness of various hyperspace topologies. This generalizes results spread in the literature including the general (proximal) hit-and-miss topologies, as well as various topologies generated by gap and excess functionals. It is also shown that the Vietoris hyperspace can be non-Baire even if the base space is a 2nd countable Hausdorff Baire space.2010 MSC: Primary 54B20; Secondary 54E52, 54A05