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  "body": "\n## Summary\n\nThe Stone–Čech compactification $\\beta X$ of a [[Tychonoff space|C5N2J2K6Z3]] $X$ is the largest Hausdorff [[compactification|Q6XNC8JAZD]] of $X$.\nIt is characterised by a universal property.\nEvery continuous map from $X$ to a compact Hausdorff space extends uniquely to $\\beta X$.\nIt links topology with algebra and set theory, for example through the description of $\\beta \\N$ by [[ultrafilters|J64Y3JC0RE]].\n\n## Explanations\n\n### L1 human\n\nA compactification adds ideal points to make a space compact.\nThe Stone–Čech compactification adds as many points as possible, in a precise sense.\nEvery compact Hausdorff compactification of $X$ is a quotient of $\\beta X$.\nFor the natural numbers, the new points of $\\beta \\N$ correspond to the free ultrafilters on $\\N$.\n",
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