Spaces splittable over the class of Eberlein and descriptive compact spaces

Abstract

We study splittability over some classes of compact spaces which are useful in functional analysis and general topology. Among other things we show that a scattered pseudocompact space splittable over the class of Eberlein compact spaces is Eberlein compact. We also prove that a compact space splittable over the class of Eberlein compact spaces is hereditarily \(\sigma \)-metacompact, and that if X is a compact space splittable over the class of Corson compact spaces, then \(d(X)=w(X)\). We also obtain several results on Rosenthal and on descriptive compact spaces. For instance: (1) a compact space is Rosenthal if and only if it is splittable over the class of Rosenthal compacta, (2) \(\mathfrak {c}\)-metrizable countably compact spaces which split over the class of descriptive compacta are descriptive compact spaces, (3) if X is a compact space splittable over the class of descriptive compact spaces, then \(hd(X)=w(X)\), (4) scattered compact spaces splittable over the class of descriptive compact spaces are \(\sigma \)-discrete descriptive compacta, and (5) it is consistent with ZFC that a compact space which splits over the class of descriptive compacta is a descriptive compact space.