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 $λ$ witnesses I0, then there is a topology for $V_{λ+1}$ 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 $V_{λ+1}$ in $L(V_{λ+1})$ 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.

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