Splittability over some classes of Corson compact spaces

We study splittability over some classes P of compact spaces commonly used in functional analysis. We show that, for some nice classes P, a compact space X is splittable over P if and only if every function f∈RX is reachable from Cp(X) by a set belonging to P. We also establish that every weakly Corson compact scattered space is Eberlein compact answering a question from [10]. We also prove that under V=L, a compact space X is splittable over the class of Eberlein (Gul'ko, Corson) compact spaces if and only if X is Eberlein (Gul'ko, Corson) compact.