Definition:Section of Étalé Space

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $\FF$ be a presheaf of sets on $T$.

Let $\struct {\map {\mathrm {\acute Et} } \FF, \pi}$ be its étalé space.

Let $U \subseteq S$ be open.

Definition 1

A section of $\map {\mathrm {\acute Et} } \FF$ on $U$ is a continuous mapping $s: U \to \map {\mathrm {\acute Et} } \FF$ such that:

$\pi \circ s = I_U$

where $I_U$ is the identity mapping on $U$.

Definition 2

A section of $\map {\mathrm {\acute Et} } \FF$ on $U$ is a mapping $s: U \to \map {\mathrm {\acute Et} } \FF$ 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 $t \in \map \FF V$.

Also see

• Equivalence of Definitions of Section of Étalé Space