Lindelöf P-spaces need not be Sokolov

We show that for every Lindelöf P -space a weaker version of the Sokolov property holds. Besides, if K is a scattered Eberlein compact space and X is obtained from K by declaring open all G δ -subsets of K , then X is monotonically Sokolov. The proof of this statement uses the fact that every Lindelöf subspace of a scattered Eberlein compact space must be σ -compact; this result seems to be interesting in itself. We also give an example of a Lindelöf P -space X such that C p ( X ) has uncountable extent. In particular, neither X nor C p ( X ) has the Sokolov property.