The Stacks project

Lemma 5.8.17. Let $X$ be a connected topological space with a finite number of irreducible components $X_1, \ldots , X_ n$. If $n > 1$ there is an $1 \leq j \leq n$ such that $X' = \bigcup _{i \not= j} X_ i$ is connected.

Proof. This is a graph theory problem. Let $\Gamma $ be the graph with vertices $V = \{ 1, \ldots , n\} $ and an edge between $i$ and $j$ if and only if $X_ i \cap X_ j$ is nonempty. Connectedness of $X$ means that $\Gamma $ is connected. Our problem is to find $1 \leq j \leq n$ such that $\Gamma \setminus \{ j\} $ is still connected. You can do this by choosing $j, j' \in E$ with maximal distance and then $j$ works (choose a leaf!). Details omitted. $\square$

Comments (2)

Comment #8802 by Maxime CAILLEUX on

The case also works as would be empty hence connected.

There are also:

  • 9 comment(s) on Section 5.8: Irreducible components

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Your email address will not be published.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 0GM3. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 5.8.17 as pdf