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  "body": "\n## Statement\n\nLet $X$ be a [[compact space|0QPHW703PV]], let $Y$ be a [[Hausdorff space|0MXB4CKHFW]] and let $f \\colon X \\to Y$ be a [[continuous map|0SY9ECQ19Q]] that is [[bijective|0SNQEF7Z53]].\nThen $f$ is a [[homeomorphism|05GZ11PWRY]].\n\n## Remarks\n\nProof: it is enough to show that $f$ maps closed sets to closed sets. A closed $F \\subseteq X$ is compact because [[closed subsets of compact sets are compact|066HW6D9A7]]. Then $f(F)$ is compact, and thus closed in $Y$ by [[compact subsets of a Hausdorff space are closed|0MYFJVMN7M]].\nThe compactness is needed: the map $t \\mapsto (\\cos t, \\sin t)$ from $[0, 2\\pi)$ to the unit circle is a continuous bijection, but its inverse is not continuous.\n\n## Explanations\n\n### L1 human\n\nFor compact spaces, you do not need to check that the inverse map is continuous: it comes for free.\nThis is a quick way to show that two compact spaces have the same shape.\n",
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