Euclidean plane

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 $\mathbb {R} ^{2}$, 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