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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA *Stone–Čech compactification* of $X$ is a pair $(\\beta X, \\iota)$ with the following properties.\n\n1. $\\beta X$ is a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]].\n2. $\\iota \\colon X \\to \\beta X$ is a [[continuous map|0VGFHTM75Y]].\n3. For every compact Hausdorff space $K$ and every continuous map $f \\colon X \\to K$, there is a unique continuous map $g \\colon \\beta X \\to K$ with $g \\circ \\iota = f$.\n\n## Remarks\n\nSuch a pair exists for every $X$ and is unique up to a unique homeomorphism compatible with $\\iota$.\nThe map $\\iota$ is an embedding with dense image exactly when $X$ is a [[Tychonoff space|C5N2J2K6Z3]].\nThen $(\\beta X, \\iota)$ is a [[compactification|Q6XNC8JAZD]] of $X$.\n\nExample: if $X$ is compact Hausdorff, then $\\beta X = X$ with $\\iota$ the identity.\nFor a continuous $f \\colon X \\to K$, the map $g = f$ is the required extension, and it is unique.\n\nNon-example: the one-point compactification $\\N \\cup \\set{\\infty}$ of $\\N$ is not $\\beta \\N$.\nThe map $f \\colon \\N \\to \\set{0, 1}$, $f(n) = n \\bmod 2$, has no continuous extension.\nAn extension would need $g(\\infty) = \\lim g(2n) = 0$ and $g(\\infty) = \\lim g(2n + 1) = 1$.\n",
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