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  "body": "\n## Definition\n\nA *compact group* is a [[topological group|92700QAB5Y]] $G$ whose topology is Hausdorff and [[compact|0TVJZYB6SH]].\n\n## Remarks\n\nExample: the unitary group $U(n) = \\set{A \\in M_n(\\C) : A^* A = I}$ with the topology from $M_n(\\C) \\cong \\C^{n^2}$ is a compact group.\nIt is Hausdorff as a subspace of $\\C^{n^2}$.\nIt is closed, as the zero set of the continuous map $A \\mapsto A^*A - I$.\nIt is bounded, since each column of a unitary matrix has norm $1$, so it is compact by the Heine–Borel theorem.\nNon-example: the group $\\R$ with addition and the usual topology is a Hausdorff topological group.\nIt is not compact, since the open cover by the intervals $(-n, n)$ has no finite subcover.\nEvery finite group with the discrete topology is a compact group.\nSome authors do not require the Hausdorff condition; others include it in the definition of a topological group.\n",
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