Definition:Stable under Intersection

Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.
Then $\mathcal S$ is said to be stable under intersection(s), or simply $\cap$-stable, iff:
$\forall S, T \in \mathcal S: S \cap T \in \mathcal S$