E-7 - Uniform Spaces, I

Abstract

Uniform spaces can be defined in various equivalent ways. Every uniform space carries a natural topology; it is defined using neighborhood bases. A uniformly continuous map is also continuous with respect to the uniform topologies and the converse is, as in the metric case, true for compact Hausdorff spaces. The uniform topology derived from the product and subspace uniformities are the product and subspace topologies derived from the original uniform topologies, respectively. Every uniform space has a completion; it is a complete uniform space that contains a dense and uniformly isomorphic copy of X. A metrizable space has a compatible, totally bounded metric if it is separable. The precompact reflection of a fine uniformity yields a space where all bounded continuous real-valued functions are uniformly continuous; these are also called BU-spaces. The role of measurable cardinals is evident from the fact that a non-trivial countably complete ultra filter is a Cauchy filter.