Continuity Axioms

"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  intersect each other if the separation of their centers is less than  (Dunham 1990). The continuity axioms are the three of Hilbert's axioms which concern geometric equivalence.

Archimedes' Axiom is sometimes also known as "the continuity axiom."

REFERENCES

Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 38, 1990.

Hilbert, D. The Foundations of Geometry. Chicago, IL: Open Court, 1980.

Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert's System of Axioms." §163B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 544-545, 1980.