On dense subsets of Tychonoff products

The Hewitt–Marczewski–Pondiczery theorem states that if X = ∏ α ∈ A X α is the Tychonoff product of spaces, where d ( X α ) ≤ τ ≥ ω for all α ∈ A and | A | ≤ 2 τ , then d ( X ) ≤ τ . ve that if ∏ α ∈ 2 ω X α is the Tychonoff product of 2 ω many spaces and d ( X α ) = ω for all α ∈ 2 ω , then there is the countable dense set Q ⊆ ∏ α ∈ 2 ω X α such that1. { Q j : j ∈ ω } , | Q j | < ω for all j ∈ ω and Q i ∩ Q j = ∅ if i ≠ j ; ω is an infinite subset, then the set ⋃ { Q j : j ∈ C } is dense in ∏ α ∈ 2 ω X α ; a set F ⊆ Q there is m 0 ∈ ω such that | Q k ∩ F | ≤ m 0 for all k ∈ ω , then the set Q ∖ F is dense in ∏ α ∈ 2 ω X α .