Weakly dually Lindelof spaces

Given a topological property (or a class) P, the class P' consists of spaces X such that for any neighbourhood assignment phi on X, there exists a subspace Y subset of X with property P for which phi(Y) = boolean OR{phi(y) : y is an element of Y} is dense in X. The class P' are called the weak dual of P or weakly dually P (with respect to neighbourhood assignments). In this paper, we make several observations on weakly dually Lindelof spaces. We prove that a Baire weakly dually Lindelof o-semimetrizable space is separable. There exists a large first countable Hausdorff space X having a countable subset A such that phi(A) is dense in X for any neighborhood assignment f of X, which answers two questions asked by Alas et al. (Topol Proc 30:25-38, 2006). We also prove that a weakly dually Lindelof first countable normal space has cardinality at most 2(c). Every Baire, weakly dually Lindelof space X with a symmetry g-function g such that boolean AND{g(2)(n, x) : n is an element of omega} = {x} for each x is an element of X has cardinality at most c. Finally, we prove that every separated subset of a weakly dually Lindelof normal space with a G(delta)-diagonal has cardinality at most c. Some new questions are also posed.