Generic left-separated spaces and calibers

We use a natural forcing to construct a left-separated topology on an arbitrary cardinal kappa. The resulting left-separated space X_kappa is also 0-dimensional T_2, hereditarily Lindelof, and countably tight. Moreover if kappa is regular then d(X_kappa)= kappa, hence kappa is not a caliber of X_kappa, while all other uncountable regular cardinals are. We also prove it consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelof T_3 space X such that rho=cf(rho)>omega is a caliber of X exactly if rho not in A.