Stalk

Remarks

The relation is an equivalence relation: for transitivity, intersect the two open sets $W$. In the language of categories, $\mathcal F_x$ is the direct limit of the sets $\mathcal F(U)$ over the open neighbourhoods $U$ of $x$, ordered by reverse inclusion. If $\mathcal F$ is a sheaf of abelian groups or of rings, then each stalk is an abelian group or a ring, with $s_x + t_x = (s|_W + t|_W)_x$ for $W$ a common neighbourhood.

Example: for the sheaf of continuous real functions on $\R$, the stalk at $0$ is the ring of germs of continuous functions at $0$. It is a local ring: the germs with $f(0) \ne 0$ are units, since $1/f$ is continuous near $0$, and the other germs form the unique maximal ideal.

Example: for the sheaf of holomorphic functions on $\C$, the stalk at $0$ is the ring of power series $\sum a_n z^n$ with positive radius of convergence.

Non-example: a germ is not a function value. The functions $0$ and $x \mapsto \max(x, 0)$ have the same value at $0$, but they have different germs at $0$, since they differ on every interval around $0$.

For a sheaf, a section $s \in \mathcal F(U)$ is determined by its germs $s_x$ for $x \in U$, by the locality condition.