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.
The fine uniformity is characterized by the universal property: any continuous function f from a fine space X to a uniform space Y is uniformly continuous. This implies that the functor F : CReg → Uni which assigns to any completely regular space X the fine uniformity on X is left adjoint to the forgetful functor which sends a uniform space to its underlying completely regular space.
Explicitly, the fine uniformity on a completely regular space X is generated by all open neighborhoods D of the diagonal in X × X (with the product topology) such that there exists a sequence D1, D2, … of open neighborhoods of the diagonal with D = D1 and .
The uniformity on a completely regular space X induced by C(X) (see the previous section) is not always the fine uniformity.
Read more about this topic: Uniformizable Space
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