The stalk of a presheaf at a point

Let $\mathcal{F}$ be a presheaf on a topological space $X$, and let $x\in X$.

The neighbourhood category of $x$ is the full subcategory ${\mathcal{N}}_x\subseteq \mathrm{Open}(X)$ whose objects are the open neighbourhoods of $x$. Its opposite category is filtered in the sense of Filtered categories and filtered colimits: it is nonempty because $X$ itself is a neighbourhood of $x$, and for neighbourhoods $U,V$ the intersection $U\cap V$ is again a neighbourhood of $x$ with arrows $U⟶U\cap V,V⟶U\cap V$ in the opposite category.

Because $\mathcal{F}$ is contravariant, its restriction to the neighbourhood category determines a covariant diagram on that opposite category. The stalk of $\mathcal{F}$ at $x$ is the filtered colimit

Concretely, ${\mathcal{F}}_x$ may be described as equivalence classes of pairs $(U,s)$ with $U$ an open neighbourhood of $x$ and $s\in \mathcal{F}(U)$, where $(U,s)\sim (V,t)$ when $s$ and $t$ agree on some smaller open neighbourhood of $x$. The next lemma verifies that this is an equivalence relation.