Uniform spaces : basic properties and some fundamental results

Abstract

Following Engelking’s axioms for a uniform structure, the ultimate aim of this dissertation is to investigate notions like completeness and completion in the setting of uniform spaces, where "closeness" of two points is described by the uniform structure. In particular, we investigate complete uniform spaces by extending the notion of Cauchy sequences to that of Cauchy filters. We provide two standard constructions of completions of a uniform spaces and we also show that these constructions are unique up to uniform isomorphisms. We also discuss subspaces and products of uniform spaces, investigating some fundamental properties, such as the fact that any subspace of a uniformizable space is itself uniformizable, and the Tychonoff product of a family of uniformizable spaces is also uniformizable. We establish an important result in showing that every uniform space is uniformly isomorphic to a subspace of the Cartesian product of a family of metrizable uniform spaces. In this dissertation, we also present a key characterisation of Tychonoff spaces: namely, that any Tychonoff space is uniformizable, and conversely, any uniformizable space is Tychonoff. In addition, we give a characterisation of metrizable uniform spaces: any uniform space with countable weight is metrizable, and vice versa, any metrizable uniform space has countable weight. Lastly, we discuss uniform structures on topological groups, identifying four natural uniformities that arise from a topological group.