Dedekind's Axiom

one of the axioms of continuity. Dedekind’s axiom states that if all the points of a straight line are divided into two nonempty sets, with all the points of the first set being to the left of all the points of the second set, then there exists either the most right point of the first set or the most left point of the second set. The axiom was formulated by the German mathematician J. W. R. Dedekind (1872).