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  "body": "\n## Definition\n\nLet $(X, \\tau)$ be a [[topological space|0CNQXVBHCX]] and let $\\infty$ be a point not in $X$. Let $X^* = X \\cup \\{\\infty\\}$.\nThe one-point compactification, or Alexandroff extension, of $X$ is the set $X^*$ with the following topology.\nThe open sets are the sets $U \\in \\tau$ and the sets $X^* \\setminus C$, where $C \\subseteq X$ is [[closed|0G5AV9Q4RR]] and [[compact|00R0X1WGPX]] in $X$.\n\n## Remarks\n\nThe space $X^*$ is compact, and $X$ is an open subspace of $X^*$.\nThe space $X^*$ is Hausdorff if and only if $X$ is [[locally compact|0EQMWQ4WMJ]] and [[Hausdorff|0MXB4CKHFW]].\nThe one-point compactification of $\\mathbb{R}^n$ is homeomorphic to the sphere $S^n$, by stereographic projection. For $n = 1$, it is the circle.\n\n## Explanations\n\n### L1 human\n\nYou make a non-compact space compact by adding one point, and you declare that the new point is near everything outside large compact sets.\nThe real line becomes a circle: both ends meet at the new point.\n",
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