Definition:Increasing Union
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Definition
Let $S_0, S_1, S_2, \ldots, S_i, \ldots$ be a nested sequence of sets, that is:
- $S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$
Let $S$ be the set:
- $\ds S = \bigcup_{i \mathop \in \N} S_i$
where $\bigcup$ denotes set union.
Then $S$ is called the increasing union of $S_0, S_1, S_2, \ldots, S_i, \ldots$
Also see
- Subsets in Increasing Union, from which:
- $\forall s \in S: \exists k \in \N: \forall j \ge k: x \in S_j$
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.2$: Boolean Operations