Hyperspaces of uniform spaces

Abstract

This thesis is concerned with the properties and uses of the so-called Hausdorff uniform structure on the set of subsets of a uniform space. Sometimes the results are for more specialised spaces - metric spaces, normed spaces or topological vector spaces, or for the more general proximity spaces. Historically the hyperspace has probably derived its importance mainly from the concept of hypercompleteness. Here a study is made of hyperspaces of Hausdorff completions and Hausdorff completions of hyperspaces, and, in particular, of a case where these spaces derive from two related uniform structures on a set. Certain Hausdorff completions are shown to be uniformly embedded in the 'hyperhyperspace', and some generalisations are proved of results of the Robertsons on sets of compact subsets. Making a completely different approach, J. L. Kelley's notion of a fundamental family of subsets is applied to uniform spaces and the Hausdorff completion of the hyperspace is constructed by means of fundamental families. A study is made of two conditions on a mapping between uniform spaces, the filter condition, introduced by Dr. Wendy Robertson, and the analogous fundamental family condition, which bears a relationship to hypercompleteness similar to that of the filter condition to completeness. Another subject which has attracted some interest recently is the comparability of the topologies induced on the set of subsets by various uniform structures on the original set. A survey of known results is given and the relationships between them are discussed. The question of when two uniform structures induce the same topology on sets of subsets of particular kinds is investigated, and finally a new condition is provided for a uniform structure to be unique in the sense that no other uniform structure on the given set can induce the same topology on the set of subsets. The last two chapters deal with properties of mappings and sets of mappings, introducing two new concepts - the hypergraph of a mapping and a hypercompact set of mappings. The properties of the hypergraph are related to continuity and uniform continuity, giving rise to theorems similar to closed graph theorems. Hypercompactness is studied in relation to compactness and collective compactness of sets of mappings.