Continuity axiom

The axiom expressing in one way or another the continuity of the set of real numbers (cf. Real number). The continuity axiom for the real numbers can be stated, e.g., in terms of cuts: Every cut of the real numbers is determined by some number (Dedekind's axiom); in terms of nested closed intervals: Every family of nested closed intervals has a non-empty intersection (Cantor's axiom); in terms of upper or lower bounds of sets: Every non-empty set that is bounded from above has a finite least upper bound, and every set that is bounded from below has a greatest lower bound (Weierstrass' axiom); and in terms of the existence of limits of Cauchy sequences (Cauchy criterion).

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See also Cut; Dedekind cut.