The Stacks project

Every quasi-compact Hausdorff space has a canonical extremally disconnected cover

Lemma 5.26.9. Let $X$ be a quasi-compact Hausdorff space. There exists a continuous surjection $X' \to X$ with $X'$ quasi-compact, Hausdorff, and extremally disconnected. If we require that every proper closed subset of $X'$ does not map onto $X$, then $X'$ is unique up to isomorphism.

Proof. Let $Y = X$ but endowed with the discrete topology. Let $X' = \beta (Y)$. The continuous map $Y \to X$ factors as $Y \to X' \to X$. This proves the first statement of the lemma by Example 5.26.8.

By Lemma 5.26.5 we can find a quasi-compact subset $E \subset X'$ surjecting onto $X$ such that no proper closed subset of $E$ surjects onto $X$. Because $X'$ is extremally disconnected there exists a continuous map $f : X' \to E$ over $X$ (Proposition 5.26.6). Composing $f$ with the map $E \to X'$ gives a continuous selfmap $f|_ E : E \to E$. Observe that $f|_ E$ has to be surjective as otherwise the image would be a proper closed subset surjecting onto $X$. Hence $f|_ E$ has to be $\text{id}_ E$ as otherwise Lemma 5.26.7 shows that $E$ isn't minimal. Thus the $\text{id}_ E$ factors through the extremally disconnected space $X'$. A formal, categorical argument (using the characterization of Proposition 5.26.6) shows that $E$ is extremally disconnected.

To prove uniqueness, suppose we have a second $X'' \to X$ minimal cover. By the lifting property proven in Proposition 5.26.6 we can find a continuous map $g : X' \to X''$ over $X$. Observe that $g$ is a closed map (Lemma 5.17.7). Hence $g(X') \subset X''$ is a closed subset surjecting onto $X$ and we conclude $g(X') = X''$ by minimality of $X''$. On the other hand, if $E \subset X'$ is a proper closed subset, then $g(E) \not= X''$ as $E$ does not map onto $X$ by minimality of $X'$. By Lemma 5.26.4 we see that $g$ is an isomorphism. $\square$


Comments (1)

Comment #4330 by slogan_bot on

Suggested slogan: every quasi-compact Hausdorff space has a canonical extremally disconnected cover.

There are also:

  • 5 comment(s) on Section 5.26: Extremally disconnected spaces

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Your email address will not be published.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 090D. Beware of the difference between the letter 'O' and the digit '0'.