How large can a hereditarily separable or hereditarily Lindelöf space be?

The main result of this paper is that if V satisfies GCH and ω ₁<λ<μ are arbitrary regular cardinals, then in some cardinal preserving forcing extension W of V we have λ^<λ=λ=2 ^N 0, μ=2 ^λ and there are a hereditarily separable X ⊂2 ^λ with |X|= {Mathematical expression}=μ and a hereditarily Lindelöf Y ⊂2 ^μ with w(Y)= {Mathematical expression}=μ. So far similar results have only been obtained under the assumption of CH.