Union

Let $I:\mathrm{Set}$, and $S:I\to \mathrm{Set}$ Then ${\bigcup }_{I}S:=\{x\in Si.i\in I\}$

We say that a set of sets $\mathcal{S}$ is closed under {finite, countable, uncountable} unions exactly when it has unions for {finite, countable, uncountable} $I$.