x_{1}
is a Dedekind cut associated with number/point x_{2} of totally ordered set x_{3}

Let x_{3 }= (X, <)
be an ordered set formed by (underlying) set X endowed with order relation '<'. x_{2} is a member of X; lethe the singleton of x_{2} (the set containing only and exactly x_{2}) be denoted by Y . Then x_{1} constitutes an ordered pair of sets (A, B) such that A and B are mutually disjoint subsets of X, A is closed downwardly/lesserwardly under '<', B is closed upwardly/greaterwardly under '<', x_{2} is not an element of A, and the union of A with B and Y equals exactly X.