Completely uniformizable space
Encyclopedia
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
.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
(X, T) is called completely uniformizable (or Dieudonné complete or topologically complete) if there exists at least one complete uniformity
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
that induces the topology T. Some authors additionally require X to be Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
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.
- Every 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 cardinalMeasurable cardinal- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
ity.
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.