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  "body": "\n## Statement\n\nLet $G$ be a [[locally compact abelian group|ZR09S9R3AW]].\nAssume that $G$ is compactly generated, that is, there is a compact subset $C \\subseteq G$ such that $G$ is the smallest subgroup of $G$ containing $C$.\nThen there are integers $n, m \\ge 0$ and a compact abelian group $K$ such that $G$ is isomorphic, as a topological group, to $\\R^n \\times \\Z^m \\times K$.\n\n## Remarks\n\nFor a discrete group, compactly generated means finitely generated, and the theorem gives the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]] up to the finite part, which goes into $K$.\nExample: $G = \\R \\times \\Z/2\\Z$ is compactly generated by $[0, 1] \\times \\Z/2\\Z$, and it already has the form $\\R^1 \\times \\Z^0 \\times K$ with $K = \\Z/2\\Z$.\nThe hypothesis is needed: the field $\\Q_p$ of $p$-adic numbers is locally compact abelian but not compactly generated.\nIt is not of the form $\\R^n \\times \\Z^m \\times K$.\nIndeed each element of $\\Q_p$ lies in a compact subgroup $p^{-k}\\Z_p$, which would force $n = m = 0$, but $\\Q_p$ is not compact.\nEvery locally compact abelian group has an open compactly generated subgroup, so it has an open subgroup isomorphic to $\\R^n \\times K$ with $K$ compact.\n\n## Explanations\n\n### L1 human\n\nAny locally compact abelian group that is generated by a compact piece is built from three simple kinds of blocks.\nThere are copies of the real line, copies of the integers, and one compact group.\nThis reduces many questions about such groups to these three cases.\n\n### L2 human\n\nSome proofs of [[Pontryagin duality|QTVC6F2P9D]] use the theorem: one checks duality on $\\R$, $\\Z$ and compact groups, and then passes to open subgroups and quotients.\nIt also shows that a connected locally compact abelian group is isomorphic to $\\R^n \\times K$ with $K$ compact and connected.\nFor non-compactly generated groups, such as $\\Q_p$ or the adeles, the theorem applies only to open subgroups, and the global structure can be much richer.\n",
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