1. The problem statement, all variables and given/known data Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space. 2. Relevant equations A topological manifold is a Hausdorff space that are locally Euclidean, i.e., there's an n such that for each x, there's a neighborhood N(x) homeomorphic to E^n. 3. The attempt at a solution This must be well known, but I have no idea how to start. How would I construct the E^n subspace and the corresponding homeomorphic mapping? By definition, there's a different mapping for each point. Thank you for your help.