topological manifold

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

Definition

is a topological manifold of dimension if

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