We will consider symmetric products Sym^n(M^2_{g,k}) of punctured compact Riemann surfaces (here g is the genus, k is the number of punctures). These spaces are open complex manifolds of real dimension 2n. It is easy to show that for any pairs (g,k) and (g',k') the corresponding open manifolds are homotopy equivalent iff 2g+k=2g'+k'. The following Conjecture was posed by Rade Živaljević in 2003. Conjecture. Suppose that 2g+k=2g'+k' and g\ne g'. Then open manifolds Sym^n(M^2_{g,k}) and Sym^n(M^2_{g',k'}) are not continuously homeomorphic for all n\ge 2. This conjecture was proved by Živaljević for n=2 and some other special cases. The aim of this talk is to present the proof of the conjecture in full generality. Moreover, we have proved that the corresponding manifolds are still not homeomorphic even after taking there direct product with R^N for any N\ge 0.