Germ of a sheaf section ($s_x\in {\mathcal{F}}_x$)

The germ $s_x$ of a section $s\in \mathcal{F}(U)$ is its image in the stalk of a sheaf ${\mathcal{F}}_x$. It is zero exactly when $s$ restricts to zero on some neighbourhood of $x$.

The support of a sheaf section is the closed setIts complement is open because a zero germ of a sheaf section is represented by a section that vanishes on a neighbourhood.