# Completely uniformizable space

In mathematics, a topological space (X, T) is called completely uniformizable[1] (or Dieudonné complete[2] or topologically complete[3]) if there exists at least one complete uniformity that induces the topology T. Some authors[4] 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.

## Properties

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.

## Notes

1. ^ e. g. Willard
2. ^ Encyclopedia of Mathematics
3. ^ Kelley
4. ^ e. g. Arkhangel'skii (in Encyclopedia of Mathematics), who uses the term Dieudonné complete
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