Condensations of projective sets onto compacta

For a coanalytic-complete or ${\vtop{\hbox{${\boldsymbol{\Pi}}^{1}_{2}$ } \kern-4pt \hbox{\kern4pt$\widetilde{}$ }\kern-6pt}}$-complete subspace $X$ of a Polish space we prove that there exists a continuous bijection of $X$ onto the Hilbert cube $[0,1]^{\mathbb{N} }$. This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen.

Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.