Remarks on the set of -points in Eberlein and Corson compact spaces

For a compact space K we consider the set FC ( K ) of all G δ -points in K, i.e., points with countable base of neighborhoods in K. We show that, for every scattered Eberlein compact space K, the set FC ( K ) is a G δ -set in K. We also give an example of a scattered Eberlein compactum with non-metrizable set FC ( K ) . Moreover, we give an example of a Corson compact space K such that FC ( K ) does not contain any dense G δ -subset of K. This answers three questions of Tkachuk.