Pictures of espace etale'

Question

Let $X$ be a topological space. We have morphism of categories \begin{equation} PreSh_X \xrightarrow{sp} Top_X \end{equation} which assigns to every presheaf (of abelian groups) $\mathcal{F}$ on $X$ a topological space $sp(\mathcal{F})=\coprod_{p\in X} \mathcal{F}_p$ with the (finest) topology which makes all the sectional maps (takings germs of a section on an open set) continuous.

What are some examples of such topological spaces.

For example

The easiest one is if $\mathcal{F}$ is the constant functor $A$ then $sp(\mathcal{F})$ is just $A\times X$ with the product topology (where $A$ has discrete topology).

If we take the skyscraper sheaf $\mathcal{F}_{A,p}$ centered at a closed point $p$, then the topological space looks like $A$ copies at $p$ and same everywhere else.

How does $sp(\mathcal{F})$ look for, say, sheaf of holomorphic functions on $X$ where $X=\mathbb{C}$ and $X=S^1$. Are there other important examples.

Answer 1

The étalé space $p:X=sp(\mathcal O)\to \mathbb C$ is an enormous one dimensional complex manifold with continuum many connected components .
Each of these components $X_i$ is the Riemann surface attached to a (non unique) pair $(U,f)$ where $U\subset \mathbb C$ is open and connected and $f\in \mathcal O(U)$ is holomorphic.
Conversely given such a pair $(U,f)$ there is a canonical holomorphic section $s:U\to X$ of $p$ and the best definition of "the Riemann surface of $f$" is to declare that it is the connected component of the open subset $s(U)\subset X$.

Bibliography
As usual the best reference is Forster's Lectures on Riemann Surfaces, especially pages 42-48.