Sequentially linearly Lindelof spaces

A topological Hausdorff space X is sequentially linearly Lindelof if for every uncountable regular cardinal kappa less than or equal to w(X) and every A subset of or equal to X of cardinality kappa there exists B subset of or equal to A of cardinality kappa which converges to a point. We prove that the existence of a good (mu, lambda)-scale for a singular cardinal mu of countable cofinality and a regular lambda > mu implies the existence of a sequentially linearly Lindelof space of cardinality lambda and weight mu which is not Lindelof. Corollaries of the main result are: (1) it is consistent to have linearly Lindelof non-Lindelof spaces below the continuum; (2) it is consistent to have a realcompact linearly Lindelof non-Lindelof space below 2(Nomega); (3) it is consistent to have a Dowker topology on Nomega+1 in which every subset of cardinality N-n, n > 0, has a converging subset of the same cardinality; (4) the nonexistence of sequentially linearly Lindelof non-Lindelof spaces implies the consistency of large cardinals. (C) 2002 Elsevier Science B.V. All rights reserved.