topological manifold

definition A Big deal!! Is the basis for most of differential geometry Connects to topological space

Definition

$M$ is a topological manifold of dimension $n$ if

1. $M$ is a Hausdorff space.
2. M is second countable
3. M is locally Euclidean (with dimension $n$) This means that there is a chart that maps an open neighborhood around every point to ${\mathbb{R}}^n$ homeomorphically. (Locally the manifold “looks like ${\mathbb{R}}^n$”.)