The Stacks project

Example 61.2.2. Let $T$ be a profinite space. Let $t \in T$ be a point and assume that $T \setminus \{ t\} $ is not quasi-compact. Let $X = T \times \{ 0, 1\} $. Consider the topology on $X$ with a subbase given by the sets $U \times \{ 0, 1\} $ for $U \subset T$ open, $X \setminus \{ (t, 0)\} $, and $U \times \{ 1\} $ for $U \subset T$ open with $t \not\in U$. The set of closed points of $X$ is $X_0 = T \times \{ 0\} $ and $(t, 1)$ is in the closure of $X_0$. Moreover, $X_0 \to \pi _0(X)$ is a bijection. This example shows that conditions (1) and (2) of Lemma 61.2.1 do no imply the set of closed points is closed.