  Two quasi-projective varieties are called piecewise isomorphic if they can be
stratified into pairwise isomorphic strata. We show that the m-th symmetric
power $S^m(C^n)$ of the complex affine space $C^n$ is piecewise isomorphic to
$C^{mn}$ and the m-th symmetric power $S^m(CP^\infty)$ of the infinite
dimensional complex projective space is piecewise isomorphic to the infinite
dimensional Grassmannian $Gr(m,\infty)$.
