A 0-dimensional, Lindelöf space that is not strongly D

Abstract

A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove that $♢$ implies that there is an $HF{C}_w$ space in $2^{{\omega }_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindelöf) which is not strongly $D$. We also show that any $HFC$ space $X$ is dually discrete and if additionally, countable sets have Menger closure then $X$ is a $D$-space.

14 pages, 1 figure, sumitted to Topology and its Applications