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  "body": "\n## Definition\n\nA topological space is *quasi-compact* if every open cover of it has a finite subcover, as for a [[compact space|0TVJZYB6SH]] without any separation axiom.\nA [[morphism of schemes|JXRKH6P1Z4]] $f \\colon X \\to Y$ is *quasi-compact* if for every open [[affine|B3V3XJ2W8G]] subscheme $V \\subseteq Y$ the open set $f^{-1}(V)$ is quasi-compact.\n\n## Remarks\n\nIt is enough to check this for the sets $V_i$ of one cover of $Y$ by open affine subschemes.\nEvery affine scheme is quasi-compact, by [[quasi-compactness of the spectrum|J4YW23MD6N]].\nExample: every closed immersion $f \\colon Z \\to Y$ is quasi-compact.\nFor $V$ open affine, $f^{-1}(V)$ is homeomorphic to a closed subset of the quasi-compact space $V$, and closed subsets of quasi-compact spaces are quasi-compact.\nExample: if $X$ is a Noetherian scheme, every morphism $X \\to Y$ is quasi-compact, since every subset of a Noetherian space is quasi-compact.\nNon-example: let $X$ be the disjoint union of infinitely many copies of $\\operatorname{Spec} k$ for a field $k$, and $f \\colon X \\to \\operatorname{Spec} k$ the obvious map.\nThen $f^{-1}(\\operatorname{Spec} k) = X$ is covered by the copies, with no finite subcover, so $f$ is not quasi-compact.\n",
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