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.
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