Generic left-separated spaces and calibers

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We use a natural forcing to construct a left-separated topology on an arbitrary cardinal kappa. The resulting left-separated space Xₖappa is also 0-dimensional T₂, hereditarily Lindelof, and countably tight. Moreover if kappa is regular then d(Xₖappa)= kappa, hence kappa is not a caliber of Xₖappa, 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 T₃ space X such that rho=cf(rho)>omega is a caliber of X exactly if rho not in A.