quite different from the set of local sections over all open subsets. It is for this reason that when studying presheaves over topological spaces, an additional *gluing axiom* is included to remove this distinction.

\[df:sheaf\] Let $\shf{P}$ be a presheaf on the topological space $(X,\col{T})$. We call $\shf{P}$ a *sheaf on $(X,\col{T})$* if for every open set $U \in \col{T}$ and every collection of open sets $\col{U}\subseteq \col{T}$ with $U = \cup \col{U}$, then $\shf{P}(U)$ is isomorphic to the space of sections over the set of elements $\col{U}$.

Recall Example \[ex:same\_poset\], in which two topologies were considered on the set $X=\{a,b,c\}$, and consider the diagram $$\xymatrix{
&\{0\}&\\
\mathbb{R}\ar[ur]&&\mathbb{R}\ar[ul]\\
&\mathbb{R}\ar[ul]^{\id}\ar[ur]^{\id}&\\
}$$ where $\id$ is the identity function. This diagram defines a presheaf for both $(X,\col{T}_1)$ and $(X,\col{T}_2)$, but only a sheaf on $(X,\col{T}_2)$. Specifically, since $\{\{a,b\},\{c\}\}$ is a cover for $\{a,b,c\}$ in $\col{T}_1$, the stalk on $\{a,b,c\}$ must be the global sections on $\{\{a,b\},\{c\}\}$, which is $\mathbb{R}^2$, yet the stalk there is $\mathbb{R}$. However, the analogous structure in $\col{T}_2$ is $\{\{a\},\{c\}\}$, which is not a cover for $\{a,b,c\}$, so the gluing axiom does not apply.

![Some stalks in the sheaf of continuous functions $\mathbb{R}\to \mathbb{R}$ (left) and the dual sheaf of 