On topological spaces of singular density and minimal weight

Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight ${\mathrm{\aleph }}_{{\omega }_1}$ is not hereditarily Lindelöf.
The assumption PSH is very weak, and in fact holds in all currently known models of ZFC.

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A. Rinot, On topological spaces of singular density and minimal weight, Topology Appl., 155(3): 135-140, 2007.