# Coordinate 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* 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.*

## Induced uniformity

One way to construct a uniform structure on a topological space *X* is to take the initial uniformity on *X* induced by *C*(*X*), the family of real-valued continuous functions on *X*. This is the coarsest uniformity on *X* for which all such functions are uniformly continuous. A subbase for this uniformity is given by the set of all entourages

where *f* ∈ *C*(*X*) and ε > 0.

The uniform topology generated by the above uniformity is the initial topology induced by the family *C*(*X*). In general, this topology will be coarser than the given topology on *X*. The two topologies will coincide if and only if *X* is completely regular.

## 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.

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 $D}_{n}\circ {D}_{n}\subset {D}_{n-1$.

The uniformity on a completely regular space *X* induced by *C*(*X*) (see the previous section) is not always the fine uniformity.

## References

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