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.
See also
- Completely metrizable space – topological space homeomorphic to a complete metric space
- Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
- Uniform space – Topological space with a notion of uniform properties
Notes
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.