A solution of the D-space Problem

Abstract

Settling a central problem in set-theoretic topology, and many related problems, we prove that there is, consistently, a regular Lindel\"of space of cardinality $\aleph_1$ that is not a D-space. The existence of such a space is independent of ZFC; we establish its equivalence to a weak form of the Continuum Hypothesis. The proof of the main theorem is adjustable to produce consistent examples with a variety of stronger properties.