In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set.
Every element s, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound. There may be elements (besides the least element) which have no predecessor.
If ≤ is a (non-strict) well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it is a well-founded strict total order. The distinction between strict and non-strict well-orders is often ignored since they are easily interconvertible.
If a set is well-ordered (or even if it merely admits a wellfounded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.
The observation that the natural numbers are well-ordered by the usual less-than relation is commonly called the well-ordering principle (for natural numbers).
The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well-ordered. The well-ordering theorem is also equivalent to the Kuratowski-Zorn lemma.
Spelling note: The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, wellordered, and wellordering.
Read more about Well-order: Ordinal Numbers, Equivalent Formulations, Order Topology