Definition 5.12.1. Quasi-compactness.

1. We say that a topological space $X$ is quasi-compact if every open covering of $X$ has a finite refinement.

2. 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.

3. We say a subset $Z \subset X$ is retrocompact if the inclusion map $Z \to X$ is quasi-compact.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).