Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra

Let $X$ be a set.

1. (Intersections) Let $\mathcal{S}$ be a collection of $\sigma$-algebras on $X$, and let $\mathcal{I}$ be the family of those subsets of $X$ that belong to every member of $\mathcal{S}$. Then $\mathcal{I}$ is a $\sigma$-algebra on $X$.

2. (Minimality) Let $\mathcal{C}$ be a family of subsets of $X$. Then the $\sigma$-algebra generated by $\mathcal{C}$ is a $\sigma$-algebra on $X$, it contains $\mathcal{C}$, and it is contained in every $\sigma$-algebra on $X$ that contains $\mathcal{C}$.