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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].\nA map $f \\colon X \\to Y$ is *proper* if $f$ is [[continuous|0VGFHTM75Y]] and, for every [[compact|0R6EWFKRCF]] subset $K \\subseteq Y$, the [[preimage|01AHG4G9RT]] $f^{-1}(K)$ is compact.\n\n## Remarks\n\nBourbaki uses another definition: $f$ is continuous and, for every topological space $Z$, the map $f \\times \\id_Z \\colon X \\times Z \\to Y \\times Z$ is [[closed|00FSJTXPJM]].\nThis holds if and only if $f$ is continuous, closed, and every fibre $f^{-1}(y)$ is compact.\nA map that is proper in this sense is proper in the sense above.\nWhen $Y$ is locally compact and Hausdorff, the two definitions agree.\n\nExample: $f \\colon \\R \\to \\R$, $f(x) = x^2$.\nA compact $K \\subseteq \\R$ lies in some $[-M, M]$ with $M \\ge 0$, and $f^{-1}(K)$ is closed and lies in $[-\\sqrt{M}, \\sqrt{M}]$, so it is compact.\n\nNon-example: $f = \\arctan \\colon \\R \\to \\R$.\nIt is continuous, but $f^{-1}([-\\pi/2, \\pi/2]) = \\R$ is not compact.\n",
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