On independent matrices and dense subsets of Tychonoff products (Q260585)

The author considers the problem of constructing dense subsets of Tychonoff product spaces \(X\), as the union of small disjoint sets. In [Topology Appl. 170, 86--95 (2014; Zbl 1297.54052)], the author considered dense sets of the Tychonoff product of \(2^\omega\)-many spaces. For the proof, the author used the \((2^\omega,2^\omega)\)-independent matrix. In the present paper, the author states and proves a theorem on the existence of a dense set \(Q\), satisfying properties of this type, in \(X = \Pi_{\alpha \in A}X_\alpha\), under the conditions \(d(X_\alpha) \leq \tau, |A|\leq 2^\tau\), \(\tau > \omega\). For the proof, the author constructs the \((2^\tau,\tau)\)-independent matrix.