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  "body": "\n## Statement\n\nLet $G$ be a [[Hausdorff topological group|7GKCGQ7215]].\nThe following are equivalent.\n\n1. $G$ is [[metrizable|091GJ3FYD7]].\n2. $G$ is [[first-countable|0CP5NDD21C]], that is, the identity of $G$ has a countable base of neighbourhoods.\n3. There is a [[metric|0VKD1TP7F3]] $d$ on $G$ that induces the topology of $G$ and is left-invariant: $d(gx, gy) = d(x, y)$ for all $g, x, y \\in G$.\n\n## Remarks\n\nThe implications from 3 to 1 and from 1 to 2 are immediate.\nFor the main step from 2 to 3, choose a base $U_0 \\supseteq U_1 \\supseteq \\cdots$ of symmetric neighbourhoods of the identity with $U_{n+1}^3 \\subseteq U_n$.\nThese sets are used to build a function that plays the role of $d(e, x)$, and then $d(x, y)$ is defined from $x^{-1} y$.\nExample: 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.\nThe Hausdorff hypothesis is needed: a group with two or more elements and the indiscrete topology is first-countable but not metrizable.\nWithout first countability the conclusion fails: an uncountable product of copies of $\\Z/2\\Z$ is a compact Hausdorff group that is not metrizable.\n\n## Explanations\n\n### L1 human\n\nFor a topological group, having a distance function that describes its topology costs very little.\nIt is enough that the identity has a countable list of shrinking neighbourhoods that capture all of its neighbourhoods.\nThen one can even choose the distance so that multiplying on the left by a fixed element does not change distances.\n\n### L2 human\n\nThe theorem turns questions about first-countable Hausdorff groups into questions about metric spaces, so sequences suffice to describe the topology.\nIt is a basic tool for Polish groups and for second-countable locally compact groups, where one often fixes a compatible left-invariant metric.\nA 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.\n",
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