Definition:Exhausting Sequence of Sets

Definition
Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

An exhausting sequence of sets (in $\mathcal S$) is a sequence $\left({S_n}\right)_{n\in \N}$ in $\mathcal S$, subject to:


 * $(1):\quad \forall n \in \N: S_n \subseteq S_{n+1}$
 * $(2):\quad \displaystyle \bigcup_{n \mathop \in \N} S_n = X$

That is, it is an increasing sequence of subsets of $X$, whose union is $X$.

It is common to write $\left({S_n}\right)_{n \in \N} \uparrow X$ to indicate an exhausting sequence of sets.

Here, the $\uparrow$ denotes a limit of an increasing sequence.