In
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 po ...
(''X'', ''T'') is called completely uniformizable (or Dieudonné complete) if there exists at least one
complete uniformity
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
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'' i ...
'', 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 i ...
.
* A completely regular space ''X'' is completely uniformizable if and only if the
fine uniformity
In mathematics, a topological space ''X'' is uniformizable if there exists a uniform structure on ''X'' that induces the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is homeomorphic to a uniform space (equipped wit ...
on ''X'' is complete.
* Every
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
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 norm ...
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 subdivis ...
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 ...
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'' i ...
, complete uniformizability is a strictly weaker condition than complete metrizability.
See also
*
*
*
Notes
References
*
*
*
*
{{topology-stub
General topology