Definition 3

A metric on a set $X$ is said to be compatible with a topology $\mathscr{T}$ on $X$ if the topology defined by this metric coincides with $\mathscr{T}$. A topological space is said to be metrizable if there exists a metric on $X$ compatible with the topology of $X$.

Two metrics on a set $X$ which are both compatible with the same topology $\mathscr{T}$ can be inequivalent.

The subspace $\mathbf{R}_+^*$ of $\mathbf{R}$ provides an example of this. Both the uniformity induced by the additive uniformity of $\mathbf{R}$ and the uniformity induced by the multiplicative uniformity of $\mathbf{R}_+^*$ are metrizable and are compatible with the topology of $\mathbf{R}_+^*$; but they are not comparable.

We remark also that there can exist non-metrizable uniformities compatible with the topology of a metrizable topological space (Exercise 7).

We shall content ourselves here with necessary conditions for the metrizability of a topological space (for a necessary and sufficient condition, cf. § 4, Exercise 22). In the first place, a space cannot be metrizable unless it is completely regular (indeed we shall see, in § 4, no. 1, Proposition 2, that a metrizable space is necessarily "normal", which is a stronger condition). On the other hand, Theorem 1 shows that: