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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].\nA function $f \\colon X \\to Y$ is a homeomorphism when $f$ is [[bijective|0SNQEF7Z53]], $f$ is [[continuous|0SY9ECQ19Q]], and the [[inverse function|0W766JAPSN]] $f^{-1} \\colon Y \\to X$ is continuous.\nThe spaces $X$ and $Y$ are homeomorphic when there is a homeomorphism $X \\to Y$.\n\n## Remarks\n\nEquivalently, $f$ is a bijection and, for each $U \\subseteq X$, the set $U$ is open if and only if $f(U)$ is open.\nThe map $x \\mapsto x / (1 - |x|)$ is a homeomorphism from $(-1, 1)$ to $\\mathbb{R}$.\nA continuous bijection need not be a homeomorphism: $t \\mapsto (\\cos t, \\sin t)$ from $[0, 2\\pi)$ to the circle is one example. For a compact domain and a Hausdorff target, [[it is always a homeomorphism|0430ETCMKD]].\nHomeomorphisms keep each [[topological property|02SMVGC690]].\n\n## Explanations\n\n### L1 human\n\nA homeomorphism is a way to deform one space into another by stretching and bending, without tearing or gluing.\nA coffee cup and a doughnut are homeomorphic, but a sphere and a doughnut are not.\n",
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