Definition:Monotone Sequence of Sets

Definition

Let $X$ be a set.

Let $\SS \subseteq \powerset X$ be a collection of subsets of $X$.

A monotone sequence of sets (in $\SS$) is a sequence $\sequence {A_n}_{n \mathop \in \N}$ in $\SS$, such that either:

$\forall n \in \N: A_n \subseteq A_{n + 1}$

or:

$\forall n \in \N: A_n \supseteq A_{n + 1}$

That is, such that $\sequence {A_n}_{n \mathop \in \N}$ is either increasing or decreasing

Also see

• Definition:Exhausting Sequence of Sets

Sources

• 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.15$: Sequences