Definition 1

A uniformity on a topological space $X$ is said to be compatible with the topology of $X$ if the latter coincides with the topology induced by the uniformity.

A topological space is said to be uniformizable, and its topology is said to be uniformizable, if there exists a uniformity on the space which is compatible with its topology.

There are topological spaces which are not uniformizable, for example any space which does not satisfy axiom $(O_{III})$ ($§ 1$, no. 2, Corollary 3 of Proposition 2); hence the question arises of determining under what conditions a topological space is uniformizable.

We shall not give a complete answer to this question until Chapter IX, $§ 1$. In this section we shall examine only one important particular case, that in which $X$ is compact. We have then the following theorem: