In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
(''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 In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' ind ...
'', which is a stronger property than ''completely uniformizable''.
Properties
* Every completely uniformizable space is
uniformizable, and thus
completely regular
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
.
* A completely regular space ''X'' is completely uniformizable if and only if the
fine uniformity on ''X'' is complete.
* Every
regular paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
space (in particular, every Hausdorff 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 cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisi ...
ity.
[Beckenstein et al., page 44]
Every
metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
is paracompact, hence completely uniformizable. As there exist metrizable spaces that are not
completely metrizable In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (''X'', ''T'') for which there exists at least one metric ''d'' on ''X'' such that (''X'', ''d'') is a complete metric space and ''d'' ind ...
, complete uniformizability is a strictly weaker condition than complete metrizability.
See also
*
*
*
Notes
References
*
*
*
*
{{topology-stub
General topology