Unbounded towers and products (Q2220483)

A \(\kappa\)-unbounded tower is a sequence \(\langle x_\alpha:\alpha<\kappa\rangle\) of infinite subsets of~\(\mathbb{N}\) that is decreasing mod~finite and has the property that its set of counting functions is unbounded mod~finite in~\(\mathbb{N}^\mathbb{N}\). A \(\kappa\)-generalized tower is a family, \(X\) of infinite subsets of~\(\mathbb{N}\) of cardinality at least~\(\kappa\) such that for every infinite subset~\(a\) of~\(\mathbb{N}\) there is an infinite subset~\(b\) of~\(\mathbb{N}\) such that the set of~\(x\in X\) for which \(x\cap\bigcup_{n\in b}[a_n,a_{n+1})\) is infinite has cardinality less than~\(\kappa\). These sets are considered with the subspace topology they inherit from the Cantor set.\par The authors consider selection properties, notably \(S_1(\Gamma,\Gamma)\) and \(S_1(\Gamma_\mathrm{Bor},\Gamma_\mathrm{Bor})\) [\textit{M. Scheepers}, Topology Appl. 69, No. 1, 31--62 (1996; Zbl 0848.54018)]. They show that the product of finitely many \(\mathfrak{b}\)-generalized towers and one subset of~\(\mathbb{R}\) that satisfies \(S_1(\Gamma_\mathrm{Bor},\Gamma_\mathrm{Bor})\) satisfies \(S_1(\Gamma,\Gamma)\). Similar results are obtained for \(\mathfrak{p}\)-generalized towers and the property~\(\binom\Omega\Gamma\) (every \(\omega\)-cover contains a \(\gamma\)-cover).