Definition:Set Union/Countable Union

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Definition

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

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

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

$\displaystyle S = \bigcup_{n \mathop \in \N} S_n$


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


Also denoted as

It can also be denoted:

$\displaystyle S = \bigcup_{n \mathop = 0}^\infty S_n$

but its use is discouraged.


If there is no danger of ambiguity, and it is clear from the context that $n \in \N$, we can also write:

$\displaystyle S = \bigcup_\N S_n$


Sources