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 *D*_{1}, *D*_{2}, … of open neighborhoods of the diagonal with *D* = *D*_{1} 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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