Sequentially linearly Lindelöf spaces

A topological Hausdorff space X is sequentially linearly Lindelöf if for every uncountable regular cardinal κ≤w(X) and every A⊆X of cardinality κ there exists B⊆A of cardinality κ which converges to a point. We prove that the existence of a good (μ,λ)-scale for a singular cardinal μ of countable cofinality and a regular λ>μ implies the existence of a sequentially linearly Lindelöf space of cardinality λ and weight μ which is not Lindelöf.Corollaries of the main result are: (1) it is consistent to have linearly Lindelöf non-Lindelöf spaces below the continuum; (2) it is consistent to have a realcompact linearly Lindelöf non-Lindelöf space below 2^אω; (3) it is consistent to have a Dowker topology on א_ω+1 in which every subset of cardinality א_n, n>0, has a converging subset of the same cardinality; (4) the nonexistence of sequentially linearly Lindelöf non-Lindelöf spaces implies the consistency of large cardinals.