uniformizable space

Let $X$ be a topological space with $\mathcal{T}$ the topology defined on it. $X$ is said to be uniformizable

1.

there is a uniformity $\mathcal{U}$ defined on $X$ , and
2.

$\mathcal{T}=T_{\mathcal{U}}$ , the uniform topology induced by $\mathcal{U}$ .

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

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 Hausdorff) and has a countable basis.

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

Every paracompact space 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