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  "body": "\n## Summary\n\nA proper map is a [[continuous map|0VGFHTM75Y]] between [[topological spaces|0CQHFPGJ45]] under which the preimage of every [[compact set|0R6EWFKRCF]] is compact.\nIt is a map that does not let points \"escape to infinity\".\nProper maps between locally compact Hausdorff spaces are [[closed|00FSJTXPJM]], and they are the maps that extend continuously to one-point compactifications.\n\n## Explanations\n\n### L1 human\n\nUnder a proper map, a bounded region of the target only receives a bounded region of the source.\nSo points that run off to infinity in the source must also run off to infinity in the target.\nEvery polynomial map $\\R \\to \\R$ of degree at least $1$ is proper.\nThe map $\\arctan \\colon \\R \\to \\R$ is not, since the whole line lands in the bounded interval $(-\\pi/2, \\pi/2)$.\n",
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