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 λ witnesses I0, then there is a topology for Vλ+1 that is completely metrizable and with weight λ (i.e., it is a λ-Polish space), and it turns out that all the subsets of Vλ+1 in L(Vλ+1) have the λ-Perfect Set Property in such topology. In this paper, we find another generalized Polish space of singular weight κ of cofinality ω such that all its subsets have the κ-Perfect Set Property, and in doing this, we are lowering the consistency strength of such property from I0 to κ θ-supercompact, with θ>κ inaccessible.
