Cardinal invariants for κ-box products

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 results for the usual product space (the case κ = ω), 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((XI)κ) = α. Theorem 4.17. If κ ≤ β ≤ 2 and X = (D(α)) with D(α) discrete, |D(α)| = α, then d((XI)κ) = α. Theorem 5.23. Let α ≥ 3 and κ ≥ ω be cardinals, and let {Xi : i ∈ I} be a set of spaces such that |I| ≥ κ and d(Xi) ≤ α ≤ S(Xi) for each i ∈ I. Consider these conditions: (i) α is regular; (ii) α = α; (iii) κ α; (iv) S((XJ)κ) = α for all nonempty J ∈ [I]. Then: (a) if conditions (i), (ii), (iii) and (iv) hold, then S((XI)κ) = α <κ = α; and (b) if one (or more) of conditions (i), (ii), (iii) or (iv) fails, then S((XI)κ) = (α ). 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 ≤ S((XI)κ) ≤ (2); and (b) if α ≤ κ and 3 ≤ S(Xi) ≤ α for each i ∈ I then S((XI)κ) = (2). 1 Historical context The most prominent, most useful, and most-studied cardinal invariants associated with topological spaces are the weight, density character, and Souslin number. Countless papers and monographs over the decades have given estimates, in some cases even precise evaluations, of the value of these invariants for the usual Tychonoff product XI = Πi∈I Xi of a set of spaces (Xi)i∈I in terms of the values for the initial spaces Xi. But in the case of κ-box topologies (defined in Section 1 below) on spaces of the form XI , very little is known, and that is fragmentary and nowhere systematically assembled. 2010 Mathematics Subject Classification: Primary 54A25; 54A10; Secondary 54A35; 54D65.