Hereditarily Lindelöf Spaces of Singular Density

A cardinal λ is called ω-inaccessible if for all μ < λ we have μ < λ. We show that for every ω-inaccessible cardinal λ there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelöf regular space of density λ. This extends an analogous earlier result of ours that only worked for regular λ. In [1] we have shown that for any cardinal λ a natural CCC forcing notion adds a hereditarily Lindelöf 0-dimensional Hausdorff topology on λ that makes the resulting space Xλ left-separated in its natural well-ordering. It was also shown there that the density d(Xλ) = cf(λ), hence if λ is regular then d(Xλ) = λ. The aim of this paper is to show that a suitable extension of the construction given in [1] enables us to generalize this to many singular cardinals as well. Note that the existence of an L-space, that we now know is provable in ZFC (see [3]), is equivalent to the existence of a hereditarily Lindelöf regular space of density ω1. Since the cardinality of a hereditarily Lindelöf T2 space is at most continuum, just in ZFC we cannot replace in this ω1 with anything bigger. The following problem however, that is left open by our subsequent result, can be raised naturally. Problem 1. Assume that ω1 < λ ≤ c. Does there exist then a hereditarily Lindelöf regular space of density λ ? We should emphasize that this problem is open for all cardinals λ, regular or singular, in particular for λ = ω2. Before describing our new construction, let us recall that the one given in [1] is based on simultaneously and generically “splitting into two” the complements λ \ α for all proper initial segments α of λ. The novelty in the construction to be given is that we shall perform the same simultaneous splitting for the complements of the members of a 2000 Mathematics Subject Classification. 54A25, 03E35.