arXiv subjects

Beatrice Pitton

Publications and source records attributed to Beatrice Pitton.

3 recordsLinked to original sources

arXiv2025

On Borel sets in ideal topologies

We study the Borel and analytic subsets of the spaces ${}^{\kappa}\kappa$ and ${}^{\kappa}2$ endowed with ideal topologies, where $\kappa$ is a regular uncountable cardinal. We establish that the Borel hierarchy does not collapse in any ideal topology and prove that every Borel set in such a topology is analytic. In particular, when the ideal contains an unbounded set, the class of analytic sets coincides with the entire powerset. Furthermore, we show that the Approximation Lemma holds for ideal topologies.

math.LO↗

arXiv2025

Generalized Borel Sets

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.

math.LO↗

arXiv2024

Generalized Baire class functions

Let $\lambda$ be an uncountable cardinal such that $2^{< \lambda } = \lambda$. Working in the setup of generalized descriptive set theory, we study the structure of $\lambda^+$-Borel measurable functions with respect to various kinds of limits, and isolate a suitable notion of $\lambda$-Baire class $\xi$ function. Among other results, we provide higher analogues of two classical theorems of Lebesgue, Hausdorff, and Banach, namely: (1) A function is $\lambda^+$-Borel measurable if and only if it can be obtained from continuous functions by iteratively applying pointwise $D$-limits, where $D$ varies among directed sets of size at most $\lambda$. (2) A function is of $\lambda$-Baire class $\xi$ if and only if it is $\boldsymbol{\Sigma}^{0}_{\xi+1}$-measurable.

math.LO↗