Definition:Set Union/Countable Union

Definition
Let $\mathbb S$ be a set of sets.

Let $\left\langle{S_n}\right\rangle_{n \in \N}$ be a sequence in $\mathbb S$.

Let $S$ be the union of $\left\langle{S_n}\right\rangle_{n \in \N}$:
 * $\displaystyle S = \bigcup_{n \mathop \in \N} S_n$

Then $S$ is a countable union of sets in $\mathbb S$.

It can also be denoted:
 * $\displaystyle S = \bigcup_{n \mathop = 0}^\infty S_n$

but its usage is strongly discouraged.

If there is no danger of ambiguity, we can also write:
 * $\displaystyle S = \bigcup_{\N} S_n$.