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 ∈ X } , there is a D ⊆ X such that { U x : x ∈ D } covers X and D is locally finite in the topology generated by { U x : x ∈ X } . We prove that ⋄ implies that there is an HF C w space in 2 ω 1 (hence 0-dimensional, Hausdorff and hereditarily Lindelof) 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.