A Solovay-like model for singular generalized descriptive set theory

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.