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 $ϕ : U \to V$ where $V \subset ℝ^{n}$ is open with the standard topology on $ℝ^{n}$ .