# uniformizable space

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

1. 1.

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

2. 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 UniformizableSpace 2013-03-22 16:49:05 2013-03-22 16:49:05 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 54E15 uniformizable completely uniformizable