Generalized Borel Sets

TL;DR

This paper explores the structure of generalized Borel hierarchies in Polish-like spaces at uncountable cardinals, revealing new phenomena especially at singular cardinals and analyzing hierarchy relationships and behaviors.

Contribution

It introduces the $oldsymbol{ ext{κ}}^+$-Borel hierarchy for uncountable cardinals, establishes conditions for its non-collapse, and uncovers a second hierarchy at singular cardinals, advancing foundational understanding.

Findings

The $oldsymbol{ ext{κ}}^+$-Borel hierarchy is well-behaved in regular spaces of weight ≤ κ.

A second, finer Borel hierarchy exists at singular cardinals.

Models are constructed where the hierarchy length varies significantly.

Abstract

Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable cardinal $\kappa$, possibly singular, satisfying $2^{<\kappa}=\kappa$. We provide fundamental properties of the $\kappa^+$-Borel hierarchy of any regular Hausdorff space of weight at most $\kappa$, and establish sufficient conditions for its non-collapse. We highlight a unique phenomenon that arises in the case of singular cardinals, namely, the existence of a second, distinct Borel hierarchy, the $\kappa$-Borel hierarchy: we prove that it is strictly finer than the $\kappa^+$-Borel hierarchy, and then characterize the precise relationship between the two. Finally, for regular cardinals, we resolve three questions about the behavior of the $\kappa^+$-Borel hierarchy on subspaces of the generalized Baire space ${}^\kappa \kappa$, constructing various models via forcing where several nontrivial constellations for the length of the $\kappa^+$-Borel hierarchy on the space are realized.