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.