All of our spaces are assumed to be completely regular and Hausdorff. This class is closed under the taking of subspaces and arbitrary products; it is in fact the class of subspaces of compact Hausdorff spaces. It is also the class of spaces whose topology can be associated with some Hausdorff uniformity. We follow Kelley (1955) in using uniformities, distinguished collections of neighborhoods of the diagonal, to define uniform spaces and uniformly continuous maps. The gage,the set of uniformly continuous pseudometrics, provides an equivalent characterization. Recall that a uniformity is metrizable iff it has a countable base. Also a compact space has a unique uniformity consisting of all neighborhoods of the diagonal.
KeywordsUniform Space Left Translation Compact Open Topology Interior Condition Abelian Topological Group
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