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.

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. ^[1]
• Every paracompact space is completely uniformizable. ^[2]
• (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.^[3]

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
• Uniform space

References

1. ^ Willard, Stephen (1970). General Topology. Addison-Wesley Publishing Company. p. 265. 
2. ^ Willard, Stephen (1970). General Topology. Addison-Wesley Publishing Company. p. 265. 
3. ^ Beckenstein, Edward; Narici, Lawrence; Suffel, Charles (1977). Topological Algebras. North-Holland. p. 44.