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  "body": "\n## Example\n\nLet $n \\ge 1$ and let $\\mathbb{T} = \\set{z \\in \\C : \\abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].\nThe *$n$-torus* is the product group $\\mathbb{T}^n$ with coordinatewise multiplication and the product topology.\nIt is a [[locally compact abelian group|ZR09S9R3AW]].\nIt is abelian because $\\mathbb{T}$ is, Hausdorff as a subspace of $\\C^n$, and compact as a closed bounded subset of $\\C^n$.\nThe map $(t_1, \\dots, t_n) \\mapsto (e^{2\\pi i t_1}, \\dots, e^{2\\pi i t_n})$ is a continuous surjective homomorphism $\\R^n \\to \\mathbb{T}^n$ with kernel $\\Z^n$, and it induces an isomorphism of topological groups $\\R^n / \\Z^n \\cong \\mathbb{T}^n$.\n\n## Remarks\n\nAs a topological space, $\\mathbb{T}^2$ is the [[torus|03NM7DZXXC]] surface; the group structure is the extra data.\nThe torus $\\mathbb{T}^n$ is a compact connected abelian Lie group of dimension $n$, and every compact connected abelian Lie group is isomorphic to some $\\mathbb{T}^n$.\nIts [[Pontryagin dual|FMW57WNPDB]] is $\\Z^n$: the continuous characters are $(z_1, \\dots, z_n) \\mapsto z_1^{k_1} \\cdots z_n^{k_n}$ with $(k_1, \\dots, k_n) \\in \\Z^n$.\nFourier analysis on $\\mathbb{T}^n$ is the theory of multiple Fourier series.\n\n## Explanations\n\n### L1 human\n\nThe $n$-torus is a list of $n$ angles, where one adds angles in each slot and forgets full turns.\nFor $n = 2$ the two angles locate a point on the surface of a doughnut: one goes around the hole, the other around the tube.\nAdding angles slides points along the surface, so the doughnut surface becomes a group.\n",
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