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 βX becomes the completion of the uniform space X.
Read more about this topic: Tychonoff Space, Properties
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