The tightness about sequential fans and combinatorial properties (Q1389862)

For an infinite cardinal \(\kappa\), let \(S_\kappa\) be the sequential fan with \(\kappa\)-many spines. In this paper we consider the tightness of the product space of two sequential fans, \(S_\omega\times S_\kappa\). We show that the tightness can be defined in a combinatorial way. We introduce a new cardinal invariant \({\mathfrak b}^*\), which is located between \({\mathfrak b}\) and \({\mathfrak d}\). We show that \({\mathfrak b}^*\) gives a combinatorial characterization of the tightness of \(S_\omega\times S_{2^\omega}\) and clarify the relations between \({\mathfrak b}\), \({\mathfrak d}\) and \({\mathfrak b}^*\).