I have no real experience in topology (although I have done a course in metric spaces) but in the course of a project I am doing it has become useful to produce (if possible) a characterization of the subsets of arbitrary dimensional Euclidean Space (with the usual metric and topology) that are homeomorphic to the whole space.
I started by looking at the sorts of properties which are conserved under homeomorphism and found that such a subset is open and connected. I have also shown that convex open sets are homeomorphic to R^n.
However, what I am really looking for is an equivalence between subsets homeomorphic to R^n and subsets with a list of specific properties (e.g. open, convex). That I can use to identify any possible homeomorphic subset.
Hints and statements of characterization would be appreciated as starting points. However, I would like to work through the necessary proofs on my own if possible.