Generic left-separated spaces and calibers

In this paper we use a natural forcing to construct a left-separated topology on an arbitrary cardinal κ. The resulting left-separated space Xκ is also 0-dimensional T2, hereditarily Lindelöf, and countably tight. Moreover if κ is regular then d(Xκ)=κ, hence κ is not a caliber of Xκ, while all other uncountable regular cardinals are. This implies that some results of [A.V. Archangelskiı̆, Topology Appl. 104(2000) 13–16] and [I. Juhász, Z. Szentmiklóssy, Topology Appl. 119 (2002) 315–324] are, consistently, sharp. o prove it is consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelöf T3 space X such that ϱ=cf(ϱ)>ω is a caliber of X exactly if ϱ∉A.