stalk of a presheaf

DEFINITION 06.16   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 06.17   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 $x \in X$ .