Girish Mahajan (Editor)

Completely uniformizable space

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In mathematics, a topological space (X, T) is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology T. Some authors additionally require X to be Hausdorff. Some authors have called these spaces topologically complete, 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.
  • Every regular paracompact space (in particular, every Hausdorff paracompact space) is completely uniformizable.
  • (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.

    • 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.

    References

    Completely uniformizable space Wikipedia
    (Text) CC BY-SA