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.