# A 0-dimensional, Lindel\"of space that is not strongly D

## TL;DR

The paper constructs a 0-dimensional, Lindelöf space that is not strongly D under the diamond principle, and explores properties of HFC spaces related to D-spaces and dual discreteness.

## Contribution

It demonstrates the existence of a specific Lindelöf space that is not strongly D assuming , and analyzes conditions under which HFC spaces are D-spaces or dually discrete.

## Key findings

- Under , existence of an HFC_w space in 2^{} that is not strongly D.
- Any HFC space is dually discrete.
- If countable sets have Menger closure, HFC spaces are D-spaces.

## 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 $\diamondsuit$ implies that there is an $HFC_w$ space in $2^{\omega_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindel\"of) 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.