Especially if you're looking at smooth manifolds, there's some weird stuff that happens in the noncompact case. The most famous example is the existence of infinitely many [manifolds][1] that are homeomorphic but not diffeomorphic to R^4. 

I don't know that similarly weird things happen just for topological manifolds, but I wouldn't count it out.


  [1]: http://en.wikipedia.org/wiki/Exotic_R4