Cardinal invariants for κ-box products: weight, density character and Souslin number

The symbol (XI)κ (with κ ≥ ω) denotes the space XI := Πi∈I Xi with the κ-box topology; this has as base all sets of the form U = Πi∈I Ui with Ui open in Xi and with |{i ∈ I : Ui 6= Xi}| < κ. The symbols w, d and S denote respectively weight, density character, and Souslin number. Generalizing familiar, classical results, the authors show inter alia: Theorem 3.10(b). If κ ≤ α+, |I| = α and each Xi contains the discrete space {0, 1} and satisfies w(Xi) ≤ α, then w(Xκ) = α<κ. Theorem 4.17. If ω ≤ κ ≤ |I| ≤ 2α and X = (D(α))I with D(α) discrete, |D(α)| = α, then d((XI)κ) = α<κ. Corollaries 5.35(a) and 5.36. Let α ≥ 3 and κ ≥ ω be cardinals, and let {Xi : i ∈ I} be a set of spaces such that |I|+ ≥ κ. (a) If α+ ≥ κ and α ≤ S(Xi) ≤ α+ for each i ∈ I then α<κ ≤ S((XI)κ) ≤ (2α)+; and (b) if α+ ≤ κ and 3 ≤ S(Xi) ≤ α+ for each i ∈ I then S((XI)κ) = (2<κ)+. 2010 Mathematics Subject Classification: Primary 54A25; 54A10; Secondary 54A35; 54D65.