The Stacks project

Example 6.9.4. Consider the category of topological spaces $\textit{Top}$. There is a natural faithful functor $\textit{Top} \to \textit{Sets}$ which commutes with products and equalizers. But it does not reflect isomorphisms. And, in fact it turns out that the analogue of Lemma 6.9.2 is wrong. Namely, suppose $X = \mathbf{N}$ with the discrete topology. Let $A_ i$, for $i \in \mathbf{N}$ be a discrete topological space. For any subset $U \subset \mathbf{N}$ define $\mathcal{F}(U) = \prod _{i\in U} A_ i$ with the discrete topology. Then this is a presheaf of topological spaces whose underlying presheaf of sets is a sheaf, see Example 6.7.5. However, if each $A_ i$ has at least two elements, then this is not a sheaf of topological spaces according to Definition 6.9.1. The reader may check that putting the product topology on each $\mathcal{F}(U) = \prod _{i\in U} A_ i$ does lead to a sheaf of topological spaces over $X$.