Prove that topological manifold homeomorphic to Euclidean subspace

[sunjin09]

Homework Statement

Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space.

Homework 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.

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.

 

[xaos]

perhaps a partition of unity? you need to find a way of extending the local maps to generate a global map inheriting the properties of the local maps by restriction.