# Definition:Quasi-Compact Morphism of Schemes

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## Definition

Let $\struct {X, \OO_X}$ and $\struct {Y, \OO_Y}$ be schemes.

Let $f : \struct {X, \OO_X} \to \struct {Y, \OO_Y}$ be a morphism of schemes.

$f$ is **quasi-compact** if and only if for all quasi-compact open subsets $U \subset Y$, the set $\map {f^{-1} } U \subset X$ is quasi-compact.

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