Lindel¨of Spaces of Singular Density

A cardinal λ is called ω-inaccessible if for all µ < λ we have µ ω < 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 X λ left-separated in its natural well-ordering. It was also shown there that the density d(X λ) = cf(λ), hence if λ is regular then d(X λ) = 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 Lin-delöf regular space of density ω 1. Since the cardinality of a hereditarily Lindelöf T 2 space is at most continuum, just in ZFC we cannot replace in this ω 1 with anything bigger. The following problem however, that is left open by our subsequent result, can be raised naturally. Problem 1. Assume that ω 1 < 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 = ω 2. 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 family A of subsets of λ that is, at least when λ is singular, much larger 2000 Mathematics Subject Classification. 54A25, 03E35.