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 Sm(Cn) of the complex affine space Cn is piecewise isomorphic to
Cmn and the m-th symmetric power Sm(CP∞) of the infinite
dimensional complex projective space is piecewise isomorphic to the infinite
dimensional Grassmannian Gr(m,∞)