nLab Euclidean topology

1. Definition

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

• it is the metric topology induced from the canonical structure of a metric space on $\mathbb{R}^n$ 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.

2. Properties

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

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

3. Related concepts

• Euclidean-topological infinity-groupoid