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  "body": "\n## Example\n\nLet $p$ be a prime and let $\\mathbb{T} = \\set{z \\in \\C : \\abs{z} = 1}$ be the [[circle group|04EAFVFTF7]].\nThe *$p$-adic solenoid* $\\Sigma_p$ is the [[inverse limit|DZB378CNCK]] of the sequence\n$$\\cdots \\to \\mathbb{T} \\xrightarrow{z \\mapsto z^p} \\mathbb{T} \\xrightarrow{z \\mapsto z^p} \\mathbb{T}.$$\nIts elements are sequences $(z_0, z_1, z_2, \\dots)$ in $\\mathbb{T}$ with $z_{k+1}^p = z_k$ for all $k$.\nThe group law is coordinatewise multiplication, and the topology is the one induced from the product $\\mathbb{T}^{\\N}$.\n\nCheck that it is a [[compact group|EVHCE5VR6Y]].\nEach map $z \\mapsto z^p$ is a continuous homomorphism, so the sequences with $z_{k+1}^p = z_k$ form a subgroup of $\\mathbb{T}^{\\N}$.\nThis subgroup is closed, since it is cut out by the equations $z_{k+1}^p = z_k$ between continuous functions.\nThe product $\\mathbb{T}^{\\N}$ is compact by Tychonoff's theorem, so $\\Sigma_p$ is a compact Hausdorff abelian group.\n\n## Remarks\n\nThe solenoid is connected, being an inverse limit of compact connected spaces.\nIt is not locally connected and not path-connected, so it is a compact connected group that is not a Lie group.\nThe projection $(z_k) \\mapsto z_0$ onto $\\mathbb{T}$ is onto, and its kernel is isomorphic to the group $\\Z_p$ of [[p-adic integers|V6TJAFZ4MD]].\nIts Pontryagin dual is the discrete group $\\Z[1/p]$ of rationals whose denominators are powers of $p$.\n\n## Explanations\n\n### L1 human\n\nWrap a loop of string around a circle $p$ times, then wrap the result $p$ times again, and continue forever.\nThe limiting object is the solenoid.\nNear any point it looks like a line segment times a Cantor set, so it is connected but full of gaps.\nIt is a standard example of a compact connected group that is not a Lie group.\n",
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