The double density spectrum of a topological space

It is an interesting, maybe surprising, fact that different dense subspaces of even “nice” topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space X that we call the double density spectrum of X and denote by dd(X).

We improve a result from [1] by showing that dd(X) is always ω-closed (i.e., countably closed) if X is Hausdorff.

We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows.

Let S be a non-empty set of infinite cardinals. Then

(1)
S = dd(X) holds for a Hausdorff space X iff S is ω-closed and sup S ≤ 2^{2min S}

(2)
S = dd(X) holds for a regular space X iff S is ω-closed and sup S ≤ 2min S.

We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance:

(i)
If κ = cf(κ)embeds in P (ω) and S is any set of uncountable regular cardinals < κ with ∣S∣ < min S, then there is a compactum C such that {ω, κ}∪ S ⊂ dd(C), moreover λ ∉ dd(C) whenever ∣S∣ + ω < cf(λ) < κ and cf(λ) ∉ S.

(ii)
It is consistent to have a separable compactum C such that dd(C) is not ω₁-closed.