\begin{definition}[Definition:Constant Presheaf]
Let $X$ be a topological space.
Let $S$ be a set.
The '''constant presheaf''' 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 $\map F U := S$
* For each inclusion map $i : U \to V$, let $\map F i := \operatorname{id}_S$.
\end{definition}