Lindelof Sigma-property in C-p(X) and p(Cp(X)) = omega do not imply countable network weight in X

Abstract

We prove that there are Tychonoff spaces X for which p(C-p(X)) = w and C-p(X) is a Lindelof C-space while the network weight of X is uncountable. This answers Problem 75 from [4]. An example of a space Y is given such that p(Y) = w and C-p(Y) is a Lindelof Sigma -space, while the network weight of Y is uncountable. This gives a negative answer to Problem 73 from [4]. For a space X with one non-isolated point a necessary and sufficient condition in terms of the topology on X is given for C-p(X) to have countable point-finite cellularity.