On independent matrices and dense subsets of Tychonoff products

•We construct an independent matrix with some special properties.•We construct dense subsets of products of discrete and general topological spaces.•We prove some additional properties of these dense subsets.

The Hewitt–Marczewski–Pondiczery theorem [2] and [3] states that if X=∏α∈AXα is the Tychonoff product of spaces, where d(Xα)≤τ≥ωd(Xα)≤τ≥ω for all α∈Aα∈A and |A|≤2τ|A|≤2τ, then d(X)≤τd(X)≤τ.For the product ∏α∈AXα of topological spaces with d(Xα)=τd(Xα)=τ we construct dense subsets of the cardinality τ as a union of “small” disjoint sets.This gives a possibility to get dense subsets with additional properties and families of these dense subsets.In [7] we regarded the case τ=ωτ=ω.Here we consider the case τ>ωτ>ω.To get desirable results we construct the (2τ,τ)(2τ,τ)-independent matrix which satisfies some conditions.