Definition:Increasing Union

From ProofWiki
Jump to navigation Jump to search

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:

$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$

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

$\forall s \in S: \exists k \in \N: \forall j \ge k: x \in S_j$


Sources