Page "Directed set" Paragraph 2

In addition to the definition above, there is an equivalent definition.

A directed set is a set A with a preorder such that every finite subset of A has an upper bound.

The above definition implies this one: the upper bound of the empty subset is any existing element of A, because A is nonempty ; furthermore, as provable with an induction argument over the size of nonempty finite subsets, the upper bound of a finite subset may be obtained by finding upper bounds of pairs iteratively.