Euclidean plane

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Definition

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

Equivalent spaces

Space	How it is equivalent to the Euclidean plane geometrically
open circular disk in $\mathbb {R} ^{2}$, 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 $\mathbb {R} ^{2}$. 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.