Definition:Chain (Set Theory)

Definition
Let $\struct {S, \preceq}$ be an ordered set.

A chain in $S$ is a totally ordered subset of $S$.

Thus a totally ordered set is itself a chain in its own right.

Also defined as
Some sources use the term chain to mean the same thing as totally ordered set.

While this is perfectly valid, as there is no source of confusion here, such usage is surprisingly uncommon.