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  "body": "\n## Statement\n\nLet $\\{0, 1\\}$ have the [[discrete topology|0S2TEHX4E1]] and let $\\{0, 1\\}^{\\mathbb{N}}$ have the [[product topology|04S9HX3GAP]].\nThen the map\n$$\\varphi \\colon \\{0, 1\\}^{\\mathbb{N}} \\to C, \\qquad \\varphi((a_n)_{n \\ge 1}) = \\sum_{n \\ge 1} \\frac{2 a_n}{3^n}$$\nis a [[homeomorphism|05GZ11PWRY]] onto the [[Cantor set|0BXNDZFQME]] $C$.\n\n## Remarks\n\nThe map is a bijection, because each point of $C$ has exactly one base-3 expansion with digits in $\\{0, 2\\}$.\nIt is continuous: if two sequences agree in their first $N$ terms, their images differ by at most $3^{-N}$.\nThe space $\\{0, 1\\}^{\\mathbb{N}}$ is compact by [[Tychonoff's theorem|0JTRSJKCFB]], and $C$ is Hausdorff. Thus $\\varphi$ is a homeomorphism, by [[this theorem|0430ETCMKD]].\nA space homeomorphic to $C$ is a Cantor space. By a theorem of Brouwer, these are exactly the nonempty compact metrizable spaces that are totally disconnected and have no isolated points.\n\n## Explanations\n\n### L1 human\n\nEach point of the Cantor set is described by an infinite list of choices: left or right third, at each step.\nThese lists are the binary sequences, and the topology of the Cantor set matches the product topology on them.\n",
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