Weakly dually Lindelöf spaces

Abstract

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 $$Y \subset X$$ with property P for which $$\phi (Y)=\bigcup \{\phi (y): y\in Y\}$$ is dense in X. The class $$P'$$ are called the weak dual ofP or weakly duallyP (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 $$\phi $$ 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 $$2^{\mathfrak {c}}$$. Every Baire, weakly dually Lindelof space X with a symmetry g-function g such that $$\bigcap \{g^2(n,x): n\in \omega \}=\{x\}$$ for each $$x\in X$$ has cardinality at most $$\mathfrak {c}$$. Finally, we prove that every separated subset of a weakly dually Lindelof normal space with a $$G_\delta $$-diagonal has cardinality at most $$\mathfrak {c}$$. Some new questions are also posed.