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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 closed map when for each [[closed set|0G5AV9Q4RR]] $F$ of $X$, the [[image|0Q3F20XB0A]] $f(F)$ is closed in $Y$.\n\n## Remarks\n\nA continuous map from a [[compact space|0QPHW703PV]] to a [[Hausdorff space|0MXB4CKHFW]] is closed. This gives [[continuous bijections from compact to Hausdorff spaces are homeomorphisms|0430ETCMKD]].\nThe projection $\\mathbb{R}^2 \\to \\mathbb{R}$, $(x, y) \\mapsto x$, is open but not closed: it maps the closed set $\\{(x, y) : xy = 1\\}$ to $\\mathbb{R} \\setminus \\{0\\}$.\nA map can be closed but not open. A constant map $\\mathbb{R} \\to \\mathbb{R}$ is closed, but the image of $\\mathbb{R}$ is a point, which is not open.\n\n## Explanations\n\n### L1 human\n\nA closed map keeps closed sets closed when you push them forward.\nOpen maps and closed maps are different: a map can have one property without the other.\n",
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