In mathematics, a topological space X is uniformizable if there exists a uniform structure on X that 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).
The converse fails: There are uniformizable spaces that are not (pseudo)metrizable.
In fact, uniformizability is equivalent to a common separation axiom: 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 implies that the functor F : CReg → Uni that assigns to any completely regular space X the fine uniformity on X is left adjoint to the forgetful functor sending 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