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  "body": "\n## Statement\n\nA [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] $P$ is *projective* if it satisfies the following lifting property.\nFor all compact Hausdorff spaces $Y$ and $Z$, every continuous surjection $f \\colon Y \\to Z$ and every [[continuous map|0VGFHTM75Y]] $g \\colon P \\to Z$, there is a continuous map $h \\colon P \\to Y$ with $f \\circ h = g$.\n\nLet $P$ be a compact Hausdorff space.\nThen $P$ is projective if and only if $P$ is [[extremally disconnected|370PJ5BF70]].\n\n## Remarks\n\nFor one direction, let $D$ be the set $P$ with the discrete topology.\nThe identity $D \\to P$ extends to a surjection $\\beta D \\to P$ from the [[Stone–Čech compactification|D9MWP5C85W]].\nIf $P$ is projective, it is a retract of $\\beta D$, which is extremally disconnected, and retracts inherit the property.\nThe other direction uses a closed subset of the pullback that is minimal among those mapping onto $P$, found by Zorn's lemma.\n\nExample at work: $\\beta \\N$ is extremally disconnected, so it is projective.\n\n## Explanations\n\n### L1 human\n\nProjective compact Hausdorff spaces are those from which maps can always be lifted along surjections.\nGleason's theorem says that, among compact Hausdorff spaces, these are exactly the spaces where closures of open sets are open.\nSo a categorical property is matched with a concrete topological one.\n\n### L2 human\n\nEvery compact Hausdorff space $X$ is a continuous image of a projective one, namely $\\beta D$ for $D$ the set $X$ with the discrete topology.\nIn fact $X$ has a projective cover, often called its Gleason cover.\nThrough Stone duality, extremally disconnected compact Hausdorff spaces correspond to complete [[Boolean algebras|0XADJKM699]].\n",
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