Uniformizable Space

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":

Topology of Uniform Spaces - Uniformizable Spaces
... 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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