Generalized Borel Sets
Claudio Agostini, Nick Chapman, Luca Motto Ros, Beatrice Pitton
TL;DR
This work extends descriptive set theory to uncountable κ with 2^{<κ}=κ, developing the κ^+-Borel hierarchy on regular Hausdorff spaces of weight ≤ κ and unveiling a second, distinct κ-Borel hierarchy at singular κ. It establishes non-collapse criteria, Cantor–Bendixson-type decompositions, universal sets, and persistence results across broader spaces, and introduces forcing methods to realize prescribed hierarchy lengths, including intermediate values between 2 and κ^+. By relating the κ^+-Borel and κ-hierarchies, the paper provides a unified framework for analyzing definable sets and functions in generalized Baire spaces, and demonstrates model-theoretic flexibility via α-forcing to sculpt the length of definable hierarchies in forcing extensions.
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 $κ$, possibly singular, satisfying $2^{<κ}=κ$. We provide fundamental properties of the $κ^+$-Borel hierarchy of any regular Hausdorff space of weight at most $κ$, 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 $κ$-Borel hierarchy: we prove that it is strictly finer than the $κ^+$-Borel hierarchy, and then characterize the precise relationship between the two. Finally, for regular cardinals, we resolve three questions about the behavior of the $κ^+$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$, constructing various models via forcing where several nontrivial constellations for the length of the $κ^+$-Borel hierarchy on the space are realized.