Lemma 17.5.3. Let $X$ be a topological space. The support of a sheaf of rings is closed.

Proof. This is true because (according to our conventions) a ring is $0$ if and only if $1 = 0$, and hence the support of a sheaf of rings is the support of the unit section. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).