Definition:Nested Sequence

Definition
Let $S$ be a set.

Let $\mathcal S = \powerset S$ be the power set of $S$.

Let $\family {S_k}_{k \mathop \in \N}$ indexed family of subsets of $S$ such that either:
 * $\forall k \in \N: S_k \subseteq S_{k + 1}$

or:
 * $\forall k \in \N: S_k \supseteq S_{k + 1}$

Then $\family {S_k}_{k \mathop \in \N}$ is a nested sequence of sets.

Also known as
A nested sequence is also seen referred to as a chain, but as that term is also used for another concept, it is preferred on that the latter is not used.