Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality

Abstract

We provide a comprehensive development of the basics of descriptive set theory for non-separable complete metric spaces whose weight is a singular cardinal $\lambda$ of countable confinality. Somewhat unexpectedly, the resulting theory is remarkably similar to the classical one, although the methods used are necessarily fairly different and combine ideas and results from general topology, infinite combinatorics, and set theory. More in detail, we study $\lambda$-Polish spaces and standard $\lambda$-Borel spaces (characterization of the generalized Cantor and Baire spaces, analogues of the Cantor-Bendixson theorem, classification up to $\lambda$-Borel isomorphism, etc.), their $\lambda$-Borel hierarchy (structural properties, changes of topologies, and so on), $\lambda$-analytic sets (including generalizations of the Lusin separation theorem and of the Souslin theorem), $\lambda$-coanalytic sets (including $\lambda$-$\boldsymbol{\Pi}^1_1$-ranks and alike), and $\lambda$-projective sets. We also consider more advanced topics, and provide e.g. various uniformization results for $\lambda$-Borel set; these in turn lead to fundamental applications to the study of $\lambda$-Borel equivalence relations, such as a generalization of the celebrated Feldman-Moore theorem. Finally, we study a natural generalization of the classical Perfect Set Property, and develop tools to show that all definable sets enjoy such property under suitable large cardinal assumptions, most notably including Woodin's $\mathsf{I0}(\lambda)$.