Regularity Properties on the Generalized Reals

TL;DR

This paper explores regularity properties in generalized descriptive set theory on uncountable cardinals, extending classical concepts to broader set-theoretic contexts.

Contribution

It introduces and analyzes regularity properties within the framework of tree-like forcing in generalized descriptive set theory for uncountable cardinals.

Findings

Establishes regularity properties for sets in $oldsymbol{ ext{generalized descriptive set theory}}$

Extends classical regularity results to uncountable cardinals

Provides new insights into the structure of definable sets in $oldsymbol{ ext{generalized contexts}}$

Abstract

We investigate regularity properties derived from tree-like forcing notions in the setting of "generalized descriptive set theory", i.e., descriptive set theory on $\kappa^\kappa$ and $2^\kappa$, for regular uncountable cardinals $\kappa$.