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Definition

The plane is the Cartesian space $\mathbb{R}^2$. This is naturally a topological space, a manifold, and a smooth manifold. If we take one of the axes (traditionally the second) to be imaginary, then this real plane may be identified with the complex plane $\mathbb{C}^1$. As a stage for Euclidean geometry, it may be called the Cartesian plane, Euclidean plane, or coordinate plane.

References

For discussion of the plane via axioms for the points and lines in it (synthetic geometry) see the references at Euclidean geometry.