The Stacks project

Example 5.26.8. We can use Proposition 5.26.6 to see that the Stone-Čech compactification $\beta (X)$ of a discrete space $X$ is extremally disconnected. Namely, let $f : Y \to \beta (X)$ be a continuous surjection where $Y$ is quasi-compact and Hausdorff. Then we can lift the map $X \to \beta (X)$ to a continuous (!) map $X \to Y$ as $X$ is discrete. By the universal property of the Stone-Čech compactification we see that we obtain a factorization $X \to \beta (X) \to Y$. Since $\beta (X) \to Y \to \beta (X)$ equals the identity on the dense subset $X$ we conclude that we get a section. In particular, we conclude that the Stone-Čech compactification of a discrete space is totally disconnected, whence profinite (see discussion following Definition 5.26.1 and Lemma 5.22.2).

Comments (0)

There are also:

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Your email address will not be published.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