Given a topological property (or a class) P, the class P′consists of spaces Xsuch that for any neighbourhood assignment ϕon X, there exists a subspace Y⊂Xwith property Pfor which ϕ(Y)=⋃{ϕ(y):y∈Y}is dense in X. The class P′are called the weak dual ofPor weakly duallyP(with respect to neighbourhood assignments). In this paper, we make several observations on weakly dually Lindelöf spaces. We prove that a Baire weakly dually Lindelöf o-semimetrizable space is separable. There exists a large first countable Hausdorff space Xhaving a countable subset Asuch that ϕ(A)is dense in Xfor any neighborhood assignment ϕof X, which answers two questions asked by Alas et al. (Topol Proc 30:25–38, 2006). We also prove that a weakly dually Lindelöf first countable normal space has cardinality at most 2c. Every Baire, weakly dually Lindelöf space Xwith a symmetry g-function gsuch that ⋂{g2(n,x):n∈ω}={x}for each x∈Xhas cardinality at most c. Finally, we prove that every separated subset of a weakly dually Lindelöf normal space with a Gδ-diagonal has cardinality at most c. Some new questions are also posed.