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 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 κ^κ, constructing various models via forcing where several nontrivial constellations for the length of the κ^+-Borel hierarchy on the space are realized.