Sequentially linearly Lindelöf spaces
A Hausdorff space is called sequentially linearly Lindelöf if for every uncountable regular cardinal \( \kappa\leq w(X)\) and every \(A \subseteq X\) of cardinality \(\kappa\) there exists \(B \subseteq A\) of cardinality \(\kappa\) which converges to a point. The authors 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 Lindelöf space of cardinality \(\lambda\) and weight \( \mu\) which is not Lindelöf. They deduce that it is consistent to have linearly Lindelöf non-Lindelöf spaces below the continuum. Also that it is consistent to have a realcompact linearly Lindelöf non-Lindelöf space below \(2^{\aleph_\omega}\). Further that it is consistent to have a Dowker topology on \(\aleph_{\omega +1}\) in which every subset of cardinality \(\aleph_n\), \(n > 0\) has a converging subset of the same cardinality. Furthermore they show that the non-existence of sequentially linearly Lindelöf non-Lindelöf spaces implies the consistency of large cardinals.
scientific article; zbMATH DE number 1913862