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  "body": "\n## Description\n\nThe *Cantor space* is the set $2^{\\N} = \\set{0, 1}^{\\N}$ of all infinite sequences of $0$s and $1$s, with the [[product topology|0GBG8B49AA]] of the discrete space $\\set{0, 1}$.\nIts basic open sets are the sets of all sequences that begin with a given finite word.\nThe topology comes from the metric $d(x, y) = 2^{-n}$, where $n$ is the first index with $x_n \\neq y_n$, and $d(x, x) = 0$.\nThe space is [[compact|0TVJZYB6SH]] by Tychonoff's theorem, [[totally disconnected|0RJSBR9GW1]], and has no isolated points.\nMore generally, a topological space is called a Cantor space when it is homeomorphic to $2^{\\N}$.\n\n## Remarks\n\nThe [[Cantor set|0BXNDZFQME]] is a Cantor space: it [[is homeomorphic to the space of binary sequences|0RQ8DHHR8V]].\nBy [[Brouwer's theorem|3NKBCNH5JS]], a space is a Cantor space if and only if it is nonempty, compact, [[metrizable|091GJ3FYD7]], totally disconnected and without isolated points.\nFor this reason $2^{\\N} \\times 2^{\\N}$ and $(2^{\\N})^{\\N}$ are again Cantor spaces.\nEvery nonempty compact metric space is a continuous image of the Cantor space.\n\n## Explanations\n\n### L1 human\n\nA point of the Cantor space is an endless list of yes or no answers.\nTwo points are close when their lists agree for a long time.\nThe space looks the same everywhere and is made only of dust, with no two points joined by a connected piece, yet it is compact like a closed interval.\n",
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