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.

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See also

Notes
  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

References
  • A. V. Arkhangel'skii (originator). "Complete space". Encyclopedia of Mathematics. Retrieved March 5, 2013.
  • Beckenstein, Edward; Narici, Lawrence; Suffel, Charles (1977). Topological Algebras. North-Holland. ISBN 0-7204-0724-9.
  • Kelley, John L. (1975). General Topology. Springer. ISBN 0-387-90125-6.
  • Willard, Stephen (1970). General Topology. Addison-Wesley Publishing Company. ISBN 978-0-201-08707-9.
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