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

AbstractFor a compact space K we consider the set FC(K) of all Gδ-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 Gδ-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 Gδ-subset of K. This answers three questions of Tkachuk