How large can a hereditarily separable or hereditarily Lindelöf space be? (Q1085467)

Question: are there consistent examples of hereditarily separable spaces of size \(>2^{\omega_ 1}\) and of hereditarily Lindelöf spaces of weight \(>2^{\omega_ 1}?\) The answer given here is yes. Theorem. Let \(V\vDash GCH+\lambda\), \(\mu\) regular, \(\omega_ 1<\lambda <\mu\). Then there is a forcing extension of V which collapses no cardinals in which \(2^{\omega}=2^{\omega_ 1}=\lambda\) and there are a hereditarily separable space of size \(\mu\) and a hereditarily Lindelöf space of weight \(\mu\).