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

2019/02/18 by Soukup, Daniel T., Szeptycki, Paul J.
#54A35 #54D20 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1902.06500

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 \diamondsuit implies that there is an HFC_w space in 2^ω₁ (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.

Related

• A counterexample in the theory of D-spaces
• Weakly linearly Lindel"of monotonically normal spaces are Lindel"of
• On the cardinality of almost discretely Lindelof spaces
• On (non-Menger) spaces whose closed nowhere dense subsets are Menger
• Star versions of Lindelöf spaces