**Uniformizable Space**

In mathematics, a topological space *X* is **uniformizable** if there exists a uniform structure on *X* which induces the topology of *X*. Equivalently, *X* is uniformizable if and only if it is homeomorphic to a uniform space (equipped with the topology induced by the uniform structure).

Any (pseudo)metrizable space is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces which are not (pseudo)metrizable. However, it is true that the topology of a uniformizable space can always be induced by a *family (mathematics)* of pseudometrics; indeed, this is because any uniformity on a set *X* can be defined by a family of pseudometrics.

Showing that a space is uniformizable is much simpler than showing it is metrizable. In fact, uniformizability is equivalent to a common separation axiom:

*A topological space is uniformizable if and only if it is completely regular.*

Read more about Uniformizable Space: Induced Uniformity, Fine Uniformity

### Other articles related to "uniformizable space, space, uniformizable":

**Uniformizable Space**s

... A topological

**space**is called

**uniformizable**if there is a uniform structure compatible with the topology ... Every

**uniformizable space**is a completely regular topological

**space**... Moreover, for a

**uniformizable space**X the following are equivalent X is a Kolmogorov

**space**X is a Hausdorff

**space**X is a Tychonoff

**space**for any compatible ...

**Uniformizable Space**- Fine Uniformity

... Given a

**uniformizable space**X there is a finest uniformity on X compatible with the topology of X called the fine uniformity or universal uniformity ... A uniform

**space**is said to be fine if it has the fine uniformity generated by its uniform topology ... is characterized by the universal property any continuous function f from a fine

**space**X to a uniform

**space**Y is uniformly continuous ...

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