Lemma 5.26.3. Let $X$ be an extremally disconnected space. If $U, V \subset X$ are disjoint open subsets, then $\overline{U}$ and $\overline{V}$ are disjoint too.

**Proof.**
By assumption $\overline{U}$ is open, hence $V \cap \overline{U}$ is open and disjoint from $U$, hence empty because $\overline{U}$ is the intersection of all the closed subsets of $X$ containing $U$. This means the open $\overline{V} \cap \overline{U}$ avoids $V$ hence is empty by the same argument.
$\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: