I have one other question and I'd appreciate any insight in to. What exactly is "homeomorphism type"? I understand well what a homeomorphism is, but not what a homeomorphism type is. For example, I read about lens spaces and read things like "some lens spaces have the same homotopy type but not the same homeomorphism type". Or I've read "3-dimensional manifolds can be classified up to homeomorphism type". What exactly does this mean? Is this something like the set of all possible homeomorphisms on a mathematical object? Homeomorphisms to what? When I look online and in my text book I can't seem to find a definition for "homeomorphism type" Thank you very much!