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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].\nA map $p \\colon X \\to Y$ is a *perfect map* if:\n\n1. $p$ is [[continuous|0VGFHTM75Y]].\n2. $p$ is [[closed|00FSJTXPJM]]: the image of every closed subset of $X$ is closed in $Y$.\n3. $p$ is surjective.\n4. For every $y \\in Y$, the fibre $p^{-1}(y)$ is [[compact|0TVJZYB6SH]].\n\n## Remarks\n\nSome authors do not require clause 3, and some also require $X$ to be Hausdorff.\nA perfect map is [[proper|15XEBGNGHC]]: the [[preimage|01AHG4G9RT]] of every compact set is compact.\n\nExample: let $K$ be a compact space and $Y$ any space, and let $p \\colon Y \\times K \\to Y$ be the projection.\nIt is continuous and surjective when $K \\ne \\emptyset$, each fibre $\\set{y} \\times K$ is homeomorphic to $K$, and the projection is closed because $K$ is compact, by the [[tube lemma|W8XEZR3VKQ]].\n\nNon-example: the projection $p \\colon \\R^2 \\to \\R$, $p(x, y) = x$.\nIt is continuous and surjective, but the closed set $\\set{(x, y) : xy = 1}$ has image $\\R \\setminus \\set{0}$, which is not closed.\nAlso its fibres are lines, which are not compact.\n",
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