1. The problem statement, all variables and given/known data prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R 2. Relevant equations 3. The attempt at a solution is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between those two sets R has the cardinality of continuum the unit circle is defined on [-1,1]x[-1,1] and since [a,b] has same cardinality as R for all a,b, cardinality of the unit circle would be c*c = c^2 but c^2=c, but this can't be since then there would be a bijection between the unit circle and R ???