**Uniform Structures**

Complete regularity is exactly the condition necessary for the existence of uniform structures on a topological space. In other words, every uniform space has a completely regular topology and every completely regular space *X* is uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.

Given a completely regular space *X* there is usually more than one uniformity on *X* that is compatible with the topology of *X*. However, there will always be a finest compatible uniformity, called the fine uniformity on *X*. If *X* is Tychonoff, then the uniform structure can be chosen so that β*X* becomes the completion of the uniform space *X*.

Read more about this topic: Tychonoff Space, Properties

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—William Howard Taft (1857–1930)