Let $X$ be a D11: Set such that

(i)	$\mathfrak{S} : = \mathfrak{S}(X)$ is the D484: Set of sigma-algebras on $X$
(ii)	$\mathcal{S} \subseteq \mathcal{P}(X)$ is a D3367: Subset algebra on $X$

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} \in \mathfrak{S} \} \end{equation}