On independent matrices and dense subsets of Tychonoff products

Abstract

The Hewitt–Marczewski–Pondiczery theorem [2,3] states that if X=∏α∈AXα is the Tychonoff product of spaces, where d(Xα)≤τ≥ω for all α∈A and |A|≤2τ, then d(X)≤τ.For the product ∏α∈AXα of topological spaces with d(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=ω.Here we consider the case τ>ω.To get desirable results we construct the (2τ,τ)-independent matrix which satisfies some conditions.