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  20130322 16:49:05 
Last modified on  20130322 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 