nLab
Euclidean topology

Definition

Definition

For $n\in \mathbb{N}$ a natural number, write ${ℝ}^n$ for the Cartesian space of dimension $n$. The Euclidean topology is the topology on ${ℝ}^n$ characterized by the following equivalent statements

• it is the metric topology induced from the canonical structure of a metric space on ${ℝ}^n$ with distance function given by $d(x,y)=\sqrt{{\sum }_{i=1}^n(x_i-y_i{)}^2}$;

• an open subset is precisely a subset such that contains an open ball around each of its points;

• it is the product topology induced from the standard topology on the real line.

Properties

Proposition

Two Cartesian spaces ${ℝ}^k$ and ${ℝ}^l$ (with the Euclidean topology) are homeomorphic precisely if $k=l$.

A proof of this statement was an early success of algebraic topology.