Birkhoff–Kakutani theorem

Remarks

The implications from 3 to 1 and from 1 to 2 are immediate. For the main step from 2 to 3, choose a base $U_0 \supseteq U_1 \supseteq \cdots$ of symmetric neighbourhoods of the identity with $U_{n+1}^3 \subseteq U_n$. These sets are used to build a function that plays the role of $d(e, x)$, and then $d(x, y)$ is defined from $x^{-1} y$. Example: the $p$-adic integers $\Z_p$ have the countable base $p^n \Z_p$ at $0$, and $d(x, y) = p^{-v_p(x - y)}$ is an invariant metric, where $v_p$ is the $p$-adic valuation. The Hausdorff hypothesis is needed: a group with two or more elements and the indiscrete topology is first-countable but not metrizable. Without first countability the conclusion fails: an uncountable product of copies of $\Z/2\Z$ is a compact Hausdorff group that is not metrizable.

Going deeper

For a student

The theorem turns questions about first-countable Hausdorff groups into questions about metric spaces, so sequences suffice to describe the topology. It is a basic tool for Polish groups and for second-countable locally compact groups, where one often fixes a compatible left-invariant metric. A left-invariant metric need not be complete even when the group is complete for its two-sided uniformity, which is why the theory of Polish groups distinguishes these notions.