Definition:Set Intersection/Countable Intersection

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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 intersection of $\sequence {S_n}_{n \mathop \in \N}$:

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

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

Also denoted as

It can also be denoted:

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

but its use is discouraged.

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

$\ds S = \bigcap_\N S_n$