Completely uniformizable space

In mathematics, a topological space (X, T) is called completely uniformizable^[1] (or Dieudonné complete^[2]) if there exists at least one complete uniformity that induces the topology T. Some authors^[3] additionally require X to be Hausdorff. Some authors have called these spaces topologically complete,^[4] although that term has also been used in other meanings like completely metrizable, which is a stronger property than completely uniformizable.

Properties

• Every completely uniformizable space is uniformizable and thus completely regular.
• A completely regular space X is completely uniformizable if and only if the fine uniformity on X is complete.^[5]
• Every regular paracompact space (in particular, every Hausdorff paracompact space) is completely uniformizable.^[6][7]
• (Shirota's theorem) A completely regular Hausdorff space is realcompact if and only if it is completely uniformizable and contains no closed discrete subspace of measurable cardinality.^[8]

Every metrizable space is paracompact, hence completely uniformizable. As there exist metrizable spaces that are not completely metrizable, complete uniformizability is a strictly weaker condition than complete metrizability.

See also

• Completely metrizable space – topological space homeomorphic to a complete metric spacePages displaying wikidata descriptions as a fallback
• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• Uniform space – Topological space with a notion of uniform properties