On dense subsets of Tychonoff products

•We consider Tychonoff products of topological spaces.•We prove the existence of some dense subsets of uncountable Tychonoff products.•Independent matrices with special properties have been constructed.

The Hewitt–Marczewski–Pondiczery theorem states that if X=∏α∈AXαX=∏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.We prove that if ∏α∈2ωXα∏α∈2ωXα is the Tychonoff product of 2ω2ω many spaces and d(Xα)=ωd(Xα)=ω for all α∈2ωα∈2ω, then there is the countable dense set Q⊆∏α∈2ωXαQ⊆∏α∈2ωXα such that1.Q=⋃{Qj:j∈ω}Q=⋃{Qj:j∈ω}, |Qj|<ω|Qj|<ω for all j∈ωj∈ω and Qi∩Qj=∅Qi∩Qj=∅ if i≠ji≠j;2.if C⊆ωC⊆ω is an infinite subset, then the set ⋃{Qj:j∈C}⋃{Qj:j∈C} is dense in ∏α∈2ωXα∏α∈2ωXα;3.if for a set F⊆QF⊆Q there is m0∈ωm0∈ω such that |Qk∩F|≤m0|Qk∩F|≤m0 for all k∈ωk∈ω, then the set Q∖FQ∖F is dense in ∏α∈2ωXα∏α∈2ωXα.