\begin{definition}[Definition:Constant Sheaf]
Let $X$ be a topological space.
Let $S$ be a set.
The '''constant sheaf''' with value $S$ on $X$ is the set-valued presheaf
:$\ds F : \map {\mathbf{Ouv} } X^{\mathrm {op} } \to \mathbf {Set}$
from the category of open sets $\map {\mathbf{Ouv}} X$ of $X$ to the category of sets $\mathbf {Set}$, defined as follows:
:For each open subset $U \subset X$, let $\map F U$ be the set of continuous maps $U \to S$ for the discrete topology on $S$.
:For each inclusion map $i : U \to V$, let $\map F i : \map F V \to \map F U$ denote restriction of mapping.
\end{definition}