Completely uniformizable space

In mathematics, a topological space (X, T) is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology T. Some authors additionally require X to be Hausdorff. Some authors have called these spaces topologically complete, although that term has also been used in other meanings like completely metrizable, which is a stronger property than completely uniformizable.

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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