Up-set

An up-set is a family $\mathcal{U}$ such that $A\in \mathcal{U}$ and $A\subseteq B$ imply $B\in \mathcal{U}$.

An up-set is upward closed under inclusion: if $A$ belongs and $A\subseteq B$, then $B$ belongs. This is the Boolean-lattice case of upper and lower sets.