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.