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 $\text{[math]}$ . The concept of locally Euclidean space is a central object in the definition of topological manifold.

Definition

A topological space $\text{[math]}$ is locally Euclidean of dimension $\text{[math]}$ , if for every point $\text{[math]}$ , there exists an open neighborhood $\text{[math]}$ of $\text{[math]}$ , and a homeomorphism

$\text{[math]}$

, where $\text{[math]}$ is open with the standard topology on $\text{[math]}$ .