Definition:Nested Sequence
Definition
Let $S$ be a set.
Let $\SS = \powerset S$ be the power set of $S$.
Let $\sequence {S_k}_{k \mathop \in \N}$ be a sequence 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).
Increasing Sequence
Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that:
- $\forall k \in \N: S_k \subseteq S_{k + 1}$
Then $\sequence {S_k}_{k \mathop \in \N}$ is an increasing sequence of sets (in $\SS$).
Decreasing Sequence
Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that:
- $\forall k \in \N: S_k \supseteq S_{k + 1}$
Then $\sequence {S_k}_{k \mathop \in \N}$ is a decreasing sequence of sets (in $\SS$).
Also known as
A nested sequence is a specific example of a chain (of sets) in which the underlying set forms a sequence.
Hence it is also a specific example of a nest.
Also see
- Results about nested sequences can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces