Definition 5.12.1. Quasi-compactness.

We say that a topological space $X$ is

*quasi-compact*if every open covering of $X$ has a finite refinement.We say that a continuous map $f : X \to Y$ is

*quasi-compact*if the inverse image $f^{-1}(V)$ of every quasi-compact open $V \subset Y$ is quasi-compact.We say a subset $Z \subset X$ is

*retrocompact*if the inclusion map $Z \to X$ is quasi-compact.

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