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. $\square$

Comment #8802 by Maxime CAILLEUX on

The case $n=1$ also works as $X'$ would be empty hence connected.

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