In page 158 of Algebraic Geometry and Arithmetic Curves by Qing Liu, there is a definition as follows:
Definition 1.2 Let $(X,\mathcal O_X)$ be a ringed topological space and $\mathcal F$ an $\mathcal O_X$-module. Let $S$ be a subset of $\mathcal F(X)$. We say that $\mathcal F$ is generated by $S$ if $\{s_x\}_{s\in S}$ generates $\mathcal F_x$ for every $x\in X$.
Does it mean the $\mathcal O_{X,x}$-submodule generated by $\{s_x\}_{s\in S}$ is equal to $\mathcal F_x$ for every $x\in X$ here?