The Stacks project

A map from a compact space to a Hausdorff space is universally closed.

Lemma 5.17.7. Let $f : X \to Y$ be a continuous map of topological spaces. If $X$ is quasi-compact and $Y$ is Hausdorff, then $f$ is universally closed.

Proof. Since every point of $Y$ is closed, we see from Lemma 5.12.3 that the closed subset $f^{-1}(y)$ of $X$ is quasi-compact for all $y \in Y$. Thus, by Theorem 5.17.5 it suffices to show that $f$ is closed. If $E \subset X$ is closed, then it is quasi-compact (Lemma 5.12.3), hence $f(E) \subset Y$ is quasi-compact (Lemma 5.12.7), hence $f(E)$ is closed in $Y$ (Lemma 5.12.4). $\square$

Comments (1)

Comment #856 by Bhargav Bhatt on

Suggested slogan: A map from a compact space to a Hausdorff space is a proper.

There are also:

  • 2 comment(s) on Section 5.17: Characterizing proper maps

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08YD. Beware of the difference between the letter 'O' and the digit '0'.