Definition:Set Union/Countable Union

Definition

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

Let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence in $\mathbb S$.

Let $S$ be the union of $\sequence {S_n}_{n \mathop \in \N}$:

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

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

Also denoted as

It can also be denoted:

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

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:

$\ds S = \bigcup_\N S_n$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products