Completely uniformizable space
In mathematics, a topological space (X, T) is called completely uniformizable (or Dieudonné complete or topologically complete) if there exists at least one complete uniformity that induces the topology T. Some authors additionally require X to be Hausdorff. The term topologically complete is also often used for complete metrizability, which is a stronger condition than complete uniformizability.
- 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.
- e. g. Willard
- Encyclopedia of Mathematics
- e. g. Arkhangel'skii (in Encyclopedia of Mathematics), who uses the term Dieudonné complete
- Willard, p. 265, Ex. 39B
- 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.
- 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.
- Beckenstein et al., page 44
- 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.
|This topology-related article is a stub. You can help Wikipedia by expanding it.|