On compact weaker topologies in function spaces

In this paper we prove that for every cardinal κ, the space Cp(Dκ) admits a continuous bijection onto a space whose all finite powers are Lindelöf (the symbol D stands for the discrete two-point space). We also prove that for every metrizable compact space X, the space Cp(X) can be condensed (i.e., admits a continuous bijection) onto the Hilbert cube Iω. As a consequence it is established that the space Cp(Dω) can be condensed onto a compact space. In connection to this result, we also prove that there exist models of ZFC in which the statement “The spaces Cp(Dκ) can be condensed onto a compact space for every cardinal κ>ω” is not true. We show also that for every cardinal κ, the spaces Cp(Cp(Dκ)) and Lp(Dκ) have dense subsets of countable tightness.