Let $X$ be an a set and let $\mathcal{C}$ be a collection of subsets of $X$ satisfying the following property:
If $A$ and $A^\prime$ are subsets of $X$ with $A \in \mathcal{C}$ and $A^\prime \subseteq A$, then $A^\prime \in \mathcal{C}$.
I have heard this described variously as "$\mathcal{A}$ is down-closed" and "$\mathcal{A}$ is a down ideal", but neither of these phrases seem very prevalent on the internet. Is there a more common name for this property?