# Perfect subsets of generalized Baire spaces and long games

## TL;DR

This paper extends classical results about perfect sets and determinacy from the Baire space to uncountable generalized Baire spaces, establishing consistency results relative to inaccessible cardinals.

## Contribution

It introduces new theorems on the perfect set property and game determinacy for definable subsets of generalized Baire spaces, generalizing Solovay's theorem.

## Key findings

- Perfect set property holds for definable subsets assuming an inaccessible cardinal.
- Determinacy of a new long game is consistent under the same assumptions.
- Results on definable functions and resurrection axioms are also obtained.

## Abstract

We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space ${}^\lambda\lambda$, where $\lambda$ is an uncountable cardinal with $\lambda^{<\lambda}=\lambda$. In the first main theorem, we show that that the perfect set property for all subsets of ${}^{\lambda}\lambda$ that are definable from elements of ${}^\lambda\mathrm{Ord}$ is consistent relative to the existence of an inaccessible cardinal above $\lambda$. In the second main theorem, we introduce a Banach-Mazur type game of length $\lambda$ and show that the determinacy of this game, for all subsets of ${}^\lambda\lambda$ that are definable from elements of ${}^\lambda\mathrm{Ord}$ as winning conditions, is consistent relative to the existence of an inaccessible cardinal above $\lambda$. We further obtain some related results about definable functions on ${}^\lambda\lambda$ and consequences of resurrection axioms for definable subsets of ${}^\lambda\lambda$.