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# 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.

## Mathematics Subject Classification

54E15*no label found*

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