Definition:Well-Ordered Set

Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\left({S, \preceq}\right)$ is a well-ordered set (or woset) if the ordering $\preceq$ is well-founded.

That is, if every $T \subseteq S: T \ne \varnothing$ has a minimal or first element.

That is, $\exists a \in T: \forall x \in T: a \preceq x$.

Note the every in the above.

Also see

• Partially ordered set (poset)
• Totally ordered set (toset)

• Well-Ordering

• Well-Ordering is Total Ordering, which shows that every woset is in fact a toset.

Sources

• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.5$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$: Exercise $4$, $\S 8$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$