Euclidean plane

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This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

Definition

The Euclidean plane, denoted \R^2, is defined as the product \R \times \R, 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 \R^n with the parameter value n = 2.

Equivalent spaces

Space How it is equivalent to the Euclidean plane geometrically
open circular disk in \R^2, 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 \{(x,y) : x^2 + y^2 < 1 \} In polar coordinates, do (r,\theta) \mapsto \tan(\pi r/2), \theta)
complex numbers \mathbb{C} under the topology arising from the modulus metric Identify a complex number x + iy with the ordered pair (x,y); here, the modulus becomes the Euclidean distance between points.
interior of a bounded rectangle, e.g., \{ (x,y) : |x| < a, |y| < b \} where a,b are positive reals The homeomorphism (x,y) \mapsto (\tan(\pi x/(2a)), \tan(\pi y/(2b))
2-sphere minus a point on it Stereographic projection
Right circular cylinder minus a line on it parallel to the axis of the cylinder

Algebraic topology

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 k^{th} Betti number b_k is the rank of the k^{th} homology group. b_0 = 1, all higher b_k are 0
Poincare polynomial Generating polynomial for Betti numbers 1 (the constant polynomial)
Euler characteristic \sum_{k=0}^\infty (-1)^k b_k 1 (hence it is a space with Euler characteristic one)

Algebraic and coalgebraic structure

Algebraic structure

The Euclidean plane has a natural structure as a topological group, namely, the additive group of the vector space \R^2. This is a real Lie group.

It can also be thought of as the additive group of \mathbb{C}, making it a complex Lie group.