The Baire and perfect set properties at singulars cardinals

TL;DR

This paper constructs a model of set theory where a singular cardinal exhibits perfect set and Baire properties for all its subsets within a specific inner model, extending classical results to higher cardinals.

Contribution

It introduces a new model demonstrating these properties at singular cardinals, utilizing large cardinal assumptions, thus generalizing Solovay's classical results to higher set-theoretic contexts.

Findings

All subsets of the singular cardinal in the model have the perfect set property.

All subsets also have the Baire property within the specified inner model.

The construction relies on supercompact cardinal assumptions, improving previous results.

Abstract

We construct a model of ZFC with a singular cardinal $\kappa$ such that every subset of $\kappa$ in $L(V_{\kappa+1})$ has both the $\kappa$-Perfect Set Property and the $\mathcal{\vec{U}}$-Baire Property. This is a higher analogue of Solovay's result for $L(\mathbb{R})$. We obtain this configuration starting with large-cardinal assumptions in the realm of supercompactness, thus improving former theorems by Cramer, Shi and Woodin.