mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

(''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'' in ...

'', which is a stronger property than ''completely uniformizable''.

Properties

* Every completely uniformizable space is

uniformizable In mathematics, a topological space ''X'' is uniformizable if there exists a uniform structure on ''X'' that Uniform space#Topology of uniform spaces, induces the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is homeomo ...

and thus completely regular. * 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 Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...

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 (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', ...

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 Homeomorphism, homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a Metric (mathematics), metr ...

is paracompact, hence completely uniformizable. As there exist metrizable spaces that are not

, complete uniformizability is a strictly weaker condition than complete metrizability.

Notes

References

* * * * {{topology-stub General topology

Properties

See also

Notes

References