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 r intersect each other if the separation of their centers is less than 2r (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."
See also
Congruence Axioms, Hilbert's Axioms, Incidence Axioms, Ordering Axioms, Parallel PostulateExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Continuity AxiomsCite this as:
Weisstein, Eric W. "Continuity Axioms." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ContinuityAxioms.html