Using partitions to characterize the minimum cardinality of an unbounded family in \(^{\omega}\omega\) (Q749538)

Let b be the minimum cardinality of an unbounded family in \(\omega_{\omega}\) partially ordered by \(\leq^*\), where \(f\leq^*g\) if f(n)\(\leq g(n)\) for all but a finite n. The author generalizes \(\leq^*\) to \(\nu_{\omega}\) for each infinite cardinal \(\nu\) and proves that \(b>\nu\) iff every sequence in \(\nu_{\omega}\) has a \(\leq^*\)-upper bound. It is also shown that the existence of an upper bound depends only on the partition of \(\nu\) induced by the point-inverse sets of the individual functions.