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:
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
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.
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 the 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.
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