A morphism of schemes is quasi-compact if the inverse image of every quasi-compact open subset is quasi-compact. It suffices to test affine open subschemes of the target. This is a relative condition: the source need not be a quasi-compact scheme when the target is not one.
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In algebraic geometry and related fields, a **quasi-compact morphism** is a type of morphism of schemes or topological spaces that relates to the compactness of the images of certain sets. A morphism of schemes \( f: X \to Y \) is called **quasi-compact** if the preimage of every quasi-compact subset of \( Y \) under \( f \) is quasi-compact in \( X \).