1. The problem statement, all variables and given/known data Prove that R and R^2 are not homeomorphic. 3. The attempt at a solution So, to prove this, one needs to conclude that there is no homeomorphism between R and R^2. A homeomorphism is a continuous bijection f with a continuous inverse. (Does there exist a bijection at all between these two sets? I assume yes, since they have the same cardinality, but I don't see how to construct it.) Assume such mapping f exists - could we derive a contradiction here. I'm really clueless.