# Homework Help: Prove that topological manifold homeomorphic to Euclidean subspace

1. Mar 24, 2012

### sunjin09

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.

2. Mar 26, 2012

### 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.

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