Euclidean plane

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 $$n = 2$$.

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:

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.