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  "body": "\n## Example\n\nThe *Cantor group* is the product $C = (\\Z/2\\Z)^{\\N}$ of countably many copies of the group of order $2$.\nIts elements are sequences $(a_0, a_1, a_2, \\dots)$ with $a_k \\in \\set{0, 1}$, added coordinatewise modulo $2$.\nEach factor has the discrete topology, and $C$ has the [[product topology|0GBG8B49AA]].\n\nCheck that it is a [[profinite group|KA0YDB6QXE]].\nAddition and negation are continuous, since in each coordinate they depend on only one coordinate of the inputs.\nThe space $C$ is compact by Tychonoff's theorem and Hausdorff as a product of Hausdorff spaces.\nIt is totally disconnected: two different sequences differ in some coordinate $k$.\nThe sets where $a_k = 0$ and where $a_k = 1$ are open, closed and separate them.\nA compact Hausdorff totally disconnected group is profinite.\nEquivalently, $C$ is the inverse limit of the finite groups $(\\Z/2\\Z)^n$ under the maps that forget the last coordinate.\n\n## Remarks\n\nAs a topological space, $C$ is homeomorphic to the [[Cantor set|0BXNDZFQME]], via $(a_k) \\mapsto \\sum_k 2 a_k 3^{-k-1}$.\nThe normalized [[Haar measure|VY0HXD3GC4]] of $C$ is the product of the uniform measures on $\\set{0, 1}$, the law of an infinite sequence of fair coin tosses.\nEvery neighbourhood of $0$ contains a subgroup of the form $\\set{a : a_0 = \\cdots = a_n = 0}$, so $C$ fails the [[no small subgroups|HPMAJV2RDY]] condition and is far from a Lie group.\n\n## Explanations\n\n### L1 human\n\nThink of an infinite sequence of light switches, each on or off.\nAdding two such sequences means flipping the switches that are on in the second one.\nTwo sequences are close when they agree on a long first stretch of switches.\nThis gives a group whose space of points is the Cantor set, the classical dust of points obtained by removing middle thirds.\n",
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