Let $x\in X$ be a general point of an irreducible set s.t. {$\bar x$} is (locally) the support of some section $s$ of a sheaf of $\mathcal {O}_X$-modules $\mathscr F$. Then why does it follow that $s$ annihilates every element in $\mathfrak m_x$ , i.e. that $x$ is an associated point of $\mathscr F$? (cf. Prop 1.2.)
I have a feeling that these concepts of associated and general point are related but I don't understand it quite yet. Is an associated point always a general point of some irreducible set?