I am working on the following exercises:
4.1.8. Let $\sigma$ and $\tau$ be sections of a decent presheaf over an open subset $U$. Show that the subset $\{ u \in U | \sigma_u=\tau_u\}=V$ is open.
4.1.9 What sheaves do you know for which the subset $V$ is always closed in $U$
To solve exercise 4.1.8, I check what is written on page 41:
A presheaf $F$ is called decent if any section of $F$ is determined by its local behavior. Thus, $F$ is decent if, for any two sections $\sigma$ and $\tau$ over an open subset $V$, $\sigma = \tau \iff \sigma_v=\tau_v$ for all points $v\in V \iff$ for some covering $V=\cup V_{\alpha}, \sigma|_{V_{\alpha}}=\tau|_{V_{\alpha}}$ for each $\alpha$
Back in exercise 4.1.8, I claim that $U=V$. This is because if $F$ is decent over $U$, according to the quoted text, then $\sigma_u=\tau_u$ for all $u\in U$
Then for exercise 4.1.9, we have that any presheaf in the discrete topology gives us that $V$ is closed. In fact, any decent presheaf over a connected component $U$ will give us that $V$ is closed.
I am not sure if this reasoning is correct. I feel that there is something I am missing or misunderstanding in the definition of a decent presheaf.