Weakly dually Lindelöf spaces (Q2293101)

Given a topological property (or a class) $P$, the class $P$ consists of spaces $X$ such that for any neighbourhood assignment $\varphi$ on $X$, there exists a subspace $Y\subset X$ with property $P$ for which $\varphi(Y)=\{\varphi(y):y\in Y\}$ is dense in $X$. Members of the class $P$ are called the weak dual of $P$ or weakly dually $P$ (with respect to neighbourhood assignments). This definition is due to \textit{J. van Mill} et al. [Topology Appl. 154, No. 10, 2127--2134 (2007; Zbl 1131.54022)] and has been studied by \textit{O. T. Alas} et al. [Topol. Proc. 30, No. 1, 25--38 (2006; Zbl 1127.54009)]. In this paper, the authors make several observations on weakly dually Lindelöf spaces. They prove that a Baire weakly dually Lindelöf o-semimetrizable space is separable. There exists a large first countable Hausdorff space $X$ having a countable subset $A$ such that $\varphi(A)$ is dense in $X$ for any neighborhood assignment $\varphi$ of $X$, which answers two questions asked by Alas et al. [loc. cit.]. They also prove other interesting results about weakly dually Lindelöf spaces. Some new questions are also posed.