Definition:Section of Étalé Space

Definition
Let $X$ be a topological space.

Let $\mathcal F$ be a presheaf of sets on $X$.

Let $(\operatorname{\acute Et} \left({\mathcal F}\right), \pi)$ be its étalé space.

Let $U\subseteq X$ be open.

Definition 1
A section of $\operatorname{\acute Et} \left({\mathcal F}\right)$ on $U$ is a continuous mapping $s : U \to \operatorname{\acute Et} \left({\mathcal F}\right)$ such that $\pi \circ s = \operatorname{id}_U$, where $ \operatorname{id}_U$ is the identity mapping on $U$.

Definition 2
A section of $\operatorname{\acute Et} \left({\mathcal F}\right)$ on $U$ is a mapping $s : U \to \operatorname{\acute Et} \left({\mathcal F}\right)$ such that for all $x\in U$ there exists an open neighborhhood $V$ of $x$ in $U$ such that the restriction of $s$ to $V$ is the section associated to some $t\in \mathcal F(V)$.

Also see

 * Equivalence of Definitions of Section of Étalé Space