Locally Euclidean space

A locally Euclidean space is a topological space that resembles a Euclidean space locally. Specifically, every point in a locally Euclidean space has an open neighborhood that is homeomorphic to an open subset in ${ℝ}^n$. The concept of locally Euclidean space is a central object in the definition of topological manifold.

Definition

A topological space $X$ is locally Euclidean of dimension $n$, if for every point $x\in X$, there exists an open neighborhood $U$ of $x$, and a homeomorphism $$\varphi :U\to V$$, where $V\subset {ℝ}^n$ is open with the standard topology on ${ℝ}^n$.