stalk of a presheaf

DEFINITION 07.32   Let $\mathcal G$ be a presheaf on a topological space $X$. Let $P\in X$ be a point. We define the stalk of $\mathcal G$ on $P$ as

$$\mathcal G_P=\varinjlim_{U\ni P} \mathcal G(U)$$

It should be noted at this stage that

LEMMA 07.33   Let $\mathcal G$ be a presheaf on a topological space $X$. The natural map

$$\mathcal G \to \operatorname{sheaf}(\mathcal G)$$

induces an isomorphism of stalk at each point $x \in X$.