uniformizable space

Let X be a topological spaceMathworldPlanetmath with 𝒯 the topology defined on it. X is said to be uniformizable

  1. 1.

    there is a uniformity 𝒰 defined on X, and

  2. 2.

    𝒯=T𝒰, the uniform topology induced by 𝒰.

It can be shown that a topological space is uniformizable iff it is completely regularPlanetmathPlanetmathPlanetmath.

Clearly, every pseudometric space is uniformizable. The converse is true if the space has a countable basis. Pushing this idea further, one can show that a uniformizable space is metrizable iff it is separating (or HausdorffPlanetmathPlanetmath) and has a countable basis.

Let X, 𝒯, and 𝒰 be defined as above. Then X is said to be completely uniformizable if 𝒰 is a complete uniformity.

Every paracompact spaceMathworldPlanetmath is completely uniformizable. Every completely uniformizable space is completely regular, and hence uniformizable.

Title uniformizable space
Canonical name UniformizableSpace
Date of creation 2013-03-22 16:49:05
Last modified on 2013-03-22 16:49:05
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 54E15
Defines uniformizable
Defines completely uniformizable