uniformizable space

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

1. there is a uniformity $𝒰$ defined on $X$, and

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

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

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