Completely uniformizable space

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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.


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[edit]


  1. ^ e. g. Willard
  2. ^ Encyclopedia of Mathematics
  3. ^ e. g. Arkhangel'skii (in Encyclopedia of Mathematics), who uses the term Dieudonné complete
  4. ^ Kelley
  5. ^ Willard, p. 265, Ex. 39B
  6. ^ Kelley, p. 208, Problem 6.L(d). Note that Kelley uses the word paracompact for regular paracompact spaces (see the definition on p. 156). As mentioned in the footnote on page 156, this includes Hausdorff paracompact spaces.
  7. ^ Note that the assumption of the space being regular or Hausdorff cannot be dropped, since every uniform space is regular and it is easy to construct finite (hence paracompact) spaces which are not regular.
  8. ^ Beckenstein et al., page 44