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 X_κ is also 0-dimensional T₂, hereditarily Lindelöf, and countably tight. Moreover if κ is regular then d(X_κ)=κ, hence κ is not a caliber of X_κ, while all other uncountable regular cardinals are. This implies that some results of [A.V. Archangelskiicaron;, Topology Appl. 104(2000) 13-16] and [I. Juhász, Z. Szentmiklóssy, Topology Appl. 119 (2002) 315-324] are, consistently, sharp.We also prove it is consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelöf T₃ space X such that Q=cf(Q)>ω is a caliber of X exactly if Q∉A.