A topological space is called locally euclidean of dimension \(n\) if for every point there is neighbourhood which is homeomorphic to some open subset of \(\mathbb{R}^n\).
Remarks
- It is sufficient and necessary if every point has a neighbourhood which is homeomorphic to an open ball in \(\mathbb{R}^n\) or to \(\mathbb{R}^n\) itself.
- Proving that a topological space \(X\) is locally Euclidean means to find an atlas on \(X\).