The generated sigma-algebra on $X$ with generator $\mathcal{S}$ is the D11: Set
\begin{equation}
\sigma \langle \mathcal{S} \rangle : = \bigcap \{ \mathcal{F} : \mathcal{S} \subseteq \mathcal{F} \text{ and } \mathcal{F} \text{ is a sigma-algebra on } X \}
\end{equation}