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[parent] uniformizable space (Definition)

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.



"uniformizable space" is owned by CWoo.
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Also defines:  uniformizable, completely uniformizable

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Cross-references: paracompact space, complete uniformity, Hausdorff, separating, metrizable, countable basis, converse, pseudometric space, completely regular, iff, induced, uniform topology, uniformity, topological space
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This is version 2 of uniformizable space, born on 2007-03-10, modified 2007-03-10.
Object id is 9055, canonical name is UniformizableSpace.
Accessed 1163 times total.

Classification:
AMS MSC: 54E15 (General topology :: Spaces with richer structures :: Uniform structures and generalizations)
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