Definition:Set Intersection/Countable Intersection
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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 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:
- $\ds S = \bigcap_\N S_n$
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