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If R denotes the set of all real numbers, then R2 := R × R represents the Euclidean plane and R3 := R × R × R represents three-dimensional Euclidean space.
The first three assumptions of the postulate, as given above, are used in the axiomatic formulation of the Euclidean plane in the secondary school geometry curriculum of the University of Chicago School Mathematics Project (UCSMP).
Hadwiger–Nelson problem on the chromatic number of unit distance graphs in the Euclidean plane