"The" continuity axiom is an additional Axiom which must be added to those of Euclid's Elements in order to guarantee that two equal circles of radius r intersect each other if the separation of their centers is less than 2r. The continuity axioms are the three of Hilbert's axioms which concern geometric equivalence. Archimedes' Axiom is sometimes also known as "the continuity axiom."

congruence axioms | Hilbert's axioms | incidence axioms | ordering axioms | parallel postulate