Generalized Polish spaces at regular uncountable cardinals

In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard \kapp-Borel spaces for \kappa an uncountable (regular) cardinal satisfying \kappa^{<\kappa} = \kappa. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from Coskey and Schlicht (Fund. Math. 232 (2016) 227–248), Galeotti (Ph.D. thesis, Universität Hamburg, 2019), and Lücke and Schlicht (Israel J. Math. 209 (2015) 421–461), and answer some questions contained therein.