Definition:Locally Euclidean Space

Definition

Let $M$ be a topological space.

Let $d \in \N$ be a natural number.

Then $M$ is a locally Euclidean space of dimension $d$ if and only if each point in $M$ has an open neighbourhood which is homeomorphic to an open subset of Euclidean space $\R^d$.

Complex Locally Euclidean Space

$M$ is a complex locally Euclidean space of dimension $d$ if and only if each point in $M$ has an open neighbourhood which is homeomorphic to an open set of the complex Euclidean space $\C^d$.

Sources

• 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 1$: Topological Manifolds