Density of \(\kappa\)-box-products and the existence of generalized independent families (Q2881012)

For cardinals \(\kappa\) and \(\mu\) with \(\aleph_0\leq\kappa\leq\mu\) and spaces \(\{X_i\}_{i\in\mu}\) the \(\kappa\)-box product \(\square_{i\in\mu}^\kappa X_i\) is the cartesian product with topology having the basis NEWLINE\[NEWLINE\Big\{\cap_{i\in I}\pi_i^{-1}(U_i) | I\in[\mu]^{<\kappa} \text{ and } U_i \text{ is open in } X_i\Big\}.NEWLINE\]NEWLINE It is shown that for box products \(\square_{i\in I}^\kappa X_i\), with \(|I|\leq2^\mu\), if \(d(X_i)\leq\mu\) for all \(i\in I\) then \(d(\square_{i\in I}^\kappa X_i)\leq\mu^{<\kappa}\).