On the density of λ-box products

Abstract

If X is a topological space with density d(X)⩾2, then cf (d((X^κ)_(λ)))⩾cf λ, where (X^κ)_(λ) is the λ-box product of κ copies of X. We use this observation to get lower bounds for the function δ(κ, λ)=d((D(2)^κ)_(λ)), where D(2) is the discrete space {0, 1}. It turns out that δ(κ, λ) is usually (if not always) equal to the well-known upper bound (log κ)^<λ. We also answer a question of Confort and Negrepontis about necessary and sufficient conditions for δ(κ⁺, λ)⩽κ.