# Definition:Well-Ordered Set

## Definition

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

Then $\struct {S, \preceq}$ is a well-ordered set if the ordering $\preceq$ is a well-ordering.

That is, if every non-empty subset of $S$ has a smallest element:

$\forall T \in \powerset S: \exists a \in T: \forall x \in T: a \preceq x$

where $\powerset S$ denotes the power set of $S$.

That is, such that $\preceq$ is a well-founded total ordering.

## Also known as

The term well-ordered set is sometimes abbreviated as woset.

The term is also found unhyphenated: well ordered or wellordered.

## Also see

• Results about well-orderings and well-ordered sets can be found here.