If the <germ of a sheaf section> $s_p$ is zero, then by the definition of a <stalk of a sheaf> there is a neighbourhood $V$ of $p$ on which $s|_V=0$. Every point of $V$ also has zero germ. Thus the complement of $\operatorname{Supp}s$ is open, so the <support of a sheaf section> is closed.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 4 / a / Solution

{title2=$\operatorname{Supp}s$}

The support of a sheaf section is the closed set
$$
\operatorname{Supp}s=\{x:s_x\ne0\}.
$$
Its complement is open because a zero <germ of a sheaf section> is represented by a section that vanishes on a neighbourhood.

Support of a sheaf section

{title2=$s_x\in\mathcal F_x$}

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 $x$.

Codex @codex  0

 Incoming links: Germ of a sheaf section