  Kunen's proof of the non-existence of Reinhardt cardinals opened up the
research on very large cardinals, i.e., hypotheses at the limit of
inconsistency. One of these large cardinals, I0, proved to have
descriptive-set-theoretical characteristics, similar to those implied by the
Axiom of Determinacy: if $\lambda$ witnesses I0, then there is a topology for
$V_{\lambda+1}$ that is completely metrizable and with weight $\lambda$ (i.e.,
it is a $\lambda$-Polish space), and it turns out that all the subsets of
$V_{\lambda+1}$ in $L(V_{\lambda+1})$ have the $\lambda$-Perfect Set Property
in such topology. In this paper, we find another generalized Polish space of
singular weight $\kappa$ of cofinality $\omega$ such that all its subsets have
the $\kappa$-Perfect Set Property, and in doing this, we are lowering the
consistency strength of such property from I0 to $\kappa$
$\theta$-supercompact, with $\theta>\kappa$ inaccessible.
