In algebraic geometry, a morphism f : X → Y {\displaystyle f:X\to Y} between schemes is said to be quasi-compact if Y can be covered by open affine subschemes V i {\displaystyle V_{i}} such that the pre-images f − 1 ( V i ) {\displaystyle f^{-1}(V_{i})} are compact. If f is quasi-compact, then the pre-image of a compact open subscheme (e.g., open affine subscheme) under f is compact.
In algebraic geometry, a morphism f : X → Y {\displaystyle f:X\to Y} between schemes is said to be quasi-compact if Y can be covered by open affine subschemes V i {\displaystyle V_{i}} such that the pre-images f − 1 ( V i ) {\displaystyle f^{-1}(V_{i})} are compact. If f is quasi-compact, then the pre-image of a compact open subscheme (e.g., open affine subscheme) under f is compact.
Source: Wikipedia "Quasi-compact morphism" · CC BY-SA 4.0
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