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 $X$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0075. Beware of the difference between the letter 'O' and the digit '0'.