Geometrically, what is the difference between saying 'X is homotopic equivalent to Y' and 'X is homeomorphic to Y'? I know that a homeomorphism is a homotopy equivalence, but I can't seem to visualise the difference between them. It seems to me that both of these terms are about deforming spaces continuously and I don't see why(intuitively) homotopy equivalance is a weaker notion than a homeomorphism. Quoting from wikipedia, "A solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent". There does not exist a continuous bijection between a point and a solid disk which stops them from being homeomorphic. Does that mean you can't 'continously shrink' a solid disk into a point? If not, then geometrically what does the fact that those two are homotopic equivalent tell us? I am in a situation where I know the definitions, but can't see the 'picture'. Any help would be appreciated.