Lemma 5.12.7 (04Z9)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.12: Quasi-compact spaces and maps
Lemma 5.12.7 (cite)

numberstags

Lemma 5.12.7. Let $f : X \to Y$ be a continuous map of topological spaces.

If $X$ is quasi-compact, then $f(X)$ is quasi-compact.
If $f$ is quasi-compact, then $f(X)$ is retrocompact.

Proof. If $f(X) = \bigcup V_ i$ is an open covering, then $X = \bigcup f^{-1}(V_ i)$ is an open covering. Hence if $X$ is quasi-compact then $X = f^{-1}(V_{i_1}) \cup \ldots \cup f^{-1}(V_{i_ n})$ for some $i_1, \ldots , i_ n \in I$ and hence $f(X) = V_{i_1} \cup \ldots \cup V_{i_ n}$ . This proves (1). Assume $f$ is quasi-compact, and let $V \subset Y$ be quasi-compact open. Then $f^{-1}(V)$ is quasi-compact, hence by (1) we see that $f(f^{-1}(V)) = f(X) \cap V$ is quasi-compact. Hence $f(X)$ is retrocompact. $\square$

Comments (0)

There are also:

3 comment(s) on Section 5.12: Quasi-compact spaces and maps

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).

Comments (0)

Post a comment