Mathematics Colloquia and Seminars

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In an introductory course in metric space topology one learns basic facts about compact metric spaces, complete metric spaces, and between them, metric spaces in which closed and bounded sets are compact. In particular one learns the following things about a compact metric space X: (1) each continuous function defined on X is uniformly continuous; (2) each open cover of X has a Lebesgue number; (3) each pair of disjoint closed subsets of X lie a positive distance apart. While none of these properties is characteristic of compactness, they are each characteristic of another intermediate class of spaces called UC spaces in the literature. We present an overview of UC spaces, a class of spaces though well-studied remains not so well-known. Further, we introduce a new natural intermediate class of metric spaces that constitute a parallel universe to the UC spaces.