Fully closed maps, scannable spectra and cardinality of hereditarily separable spaces

In this paper we study the notions of scannable spectrum and roll of a spectral tree, which appeared in slightly different form in [3] and [5] respectively. One of the main results: scannable spectra necessarily have fully closed projections and spectra of length ⩽ω with fully closed projections are scannable. The technique of scannable spectra is used, when we study the new class of fully separable spaces. We prove that the statement - every fully separable almost perfectly normal compact space has cardinality ⩽c-is independent of ZFC.