The Stacks project

Example 59.82.5. Lemma 59.82.4 is false if $X$ is not spectral. Here is an example: Let $Y$ be a $T_1$ topological space, and $y \in Y$ a non-open point. Let $X = Y \amalg \{ x \} $, endowed with the topology whose closed sets are $\emptyset $, $\{ y\} $, and all $F \amalg \{ x \} $, where $F$ is a closed subset of $Y$. Then $Z = \{ x, y\} $ is a closed subset of $X$, which satisfies assumption (2) of Lemma 59.82.4. But $X$ is connected, while $Z$ is not. The conclusion of the lemma thus fails for the constant sheaf with value $\{ 0, 1\} $ on $X$.