Images of sections of a sheaf I'm currently reading a paper by X. Caicedo containing an introduction to sheaves.
On page 8 he claims, that for every sheaf of sets $p:E\to X$ and every section $\sigma:U\to E$ (U being open in X) the image $\sigma(U)$ is open.
This statement is proved by picking a point $e\in\sigma(U)$, an open neighborhood S of e, which satisfies


*

*$p(S)$ is open in X,

*$p\restriction S$ is a homeomorphism


and arriving at an open set $\sigma(U)\supseteq S\cap\sigma(U)=p^{-1}(p(S)\cap U)$.
I think the "$\supseteq$" part of this equation does not hold, if for example E is equipped with the discrete topology and the stalk of $p(e)$ has more than one element.
I have tried to show that $(p\restriction S)^{-1}(p(S)\cap U) = p^{-1}(U)\cap S$ is contained in $\sigma(U)$, but all attempts at that felt quite clumsy, leading me to believe I have missed something important about the structure of a sheaf.

In order to prove $\sigma(U)$ is open, since $p(S)\cap U$ is open in $X$ and $p|S$ is a homeomorphism, it suffices to show $p|S(S\cap \sigma(U))=p(S)\cap U$. Clearly, this is true($p\sigma(u)=u$, for $u\in U$).
