A topological space ${X}$ is said to be metrisable if one can find a metric ${d: X \times X \rightarrow [0,+\infty)}$ on it whose open balls generate the topology.

Theorem 1 (Birkhoff-Kakutani theorem) Let ${G}$ be a topological group (i.e. a topological space that is also a group, such that the group operations ${\cdot: G \times G \rightarrow G}$ and ${()^{-1}: G \rightarrow G}$ are continuous). Then ${G}$ is metrisable if and only if it is both Hausdorff and first countable.