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Prove that topological manifold homeomorphic to Euclidean subspace

  1. Mar 24, 2012 #1
    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. jcsd
  3. Mar 26, 2012 #2
    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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