Affiliations: [a] National Institute of Information and Communications Technology (NICT), Center for Information and Neural Networks (CiNet), Osaka, Japan. [email protected] | [b] Department of Mathematics, TU Darmstadt, Darmstadt, Germany. [email protected] | [c] A.P. Ershov Institute of Informatics Systems SB RAS, Novosibirsk, Russia. [email protected]
Abstract: We define and study new classifications of qcb0-spaces based on the idea to measure the complexity of their bases. The new classifications complement those given by recently introduced hierarchies of qcb0-spaces and provide new tools to investigate non-countably based qcb0-spaces. As a by-product, we show that there is no universal qcb0-space and establish several new properties of the Kleene–Kreisel continuous functionals of countable types.