Topologies on uniform hyperspaces

Abstract

This paper includes the proofs of results announced in [3], as well as other results deriving from the Isbell-Smith-Ward problem of comparability of Hausdorff uniform topologies on hyperspaces of uniform spaces. In particular we give (in Theorem 6) simple conditions on a uniform space sufficient for H-singularity (no other uniformity induces the same hyperspace topology). The notion of association map is introduced, and properties of a map between uniform spaces are related to properties of the induced hyperspace map, thus generalizing and unifying results of F. Albrecht, D. Hammond Smith and V.Z. Poljakov, and establishing various conditions sufficient for uniform continuity.

Keywords: Uniform hyperspace, hypercontinuous, proximity map, almost uniform over a subspace, H-singular

Quaestiones Mathematicae 29(2006), 299–311