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The Euclidean plane, denoted , is defined as the product , i.e., the set of ordered pairs of real numbers. It is equipped with the product topology from the Euclidean topology on the real line. In addition to a topological structure, the Euclidean plane also has a natural metric structure, group structure, and other structures, all of them giving rise to the same topology.
The Euclidean plane is a special case of Euclidean space with the parameter value .
|Space||How it is equivalent to the Euclidean plane geometrically|
|open circular disk in , i.e., the set of all points at distance less than a fixed positive number from a fixed point (interior region of a circle), e.g., the set||In polar coordinates, do|
|complex numbers under the topology arising from the modulus metric||Identify a complex number with the ordered pair ; here, the modulus becomes the Euclidean distance between points.|
|interior of a bounded rectangle, e.g., where are positive reals||The homeomorphism|
|2-sphere minus a point on it||Stereographic projection|
|Right circular cylinder minus a line on it parallel to the axis of the cylinder|
The Euclidean plane is a contractible space, i.e., it has the homotopy type of a point. So its zeroth homology and cohomology groups with coefficients in any module are equal to that module, and all higher homology and cohomology groups are zero.
Its zeroth homotopy set is a one-point set (which can be interpreted as the trivial group), and its fundamental group and all higher homotopy groups are trivial groups.
Some important numerical invariants:
|Invariant||General description||Description of value for Euclidean space|
|Betti numbers||The Betti number is the rank of the homology group.||, all higher are|
|Poincare polynomial||Generating polynomial for Betti numbers||(the constant polynomial)|
|Euler characteristic||1 (hence it is a space with Euler characteristic one)|
Algebraic and coalgebraic structure
It can also be thought of as the additive group of , making it a complex Lie group.