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

• 2016

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## Abstract

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The symbol ${\left({X}_{I}\right)}_{\kappa }$ (with κ ≥ ω) denotes the space ${X}_{I}:={\prod }_{i\in I}{X}_{i}$ with the κ-box topology; this has as base all sets of the form $U={\prod }_{i\in I}{U}_{i}$ with ${U}_{i}$ open in ${X}_{i}$ and with $|i\in I:{U}_{i}\ne {X}_{i}|<\kappa$. The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each ${X}_{i}$ contains the discrete space 0,1 and satisfies $w\left({X}_{i}\right)\le \alpha$, then $w\left({X}_{\kappa }\right)={\alpha }^{<\kappa }$. Theorem 4.3.2. If $\omega \le \kappa \le |I|\le {2}^{\alpha }$ and $X={\left(D\left(\alpha \right)\right)}^{I}$ with D(α) discrete, |D(α)| = α, then $d\left({\left({X}_{I}\right)}_{\kappa }\right)={\alpha }^{<\kappa }$. Corollaries 5.2.32(a) and 5.2.33. Let α ≥ 3 and κ ≥ ω be cardinals, and let ${X}_{i}:i\in I$ be a set of spaces such that |I|⁺ ≥ κ. (a) If α⁺ ≥ κ and $\alpha \le S\left({X}_{i}\right)\le \alpha ⁺$ for each i ∈ I, then ${\alpha }^{<\kappa }\le S\left({\left({X}_{I}\right)}_{\kappa }\right)\le \left({2}^{\alpha }\right)⁺$; and (b) if α⁺ ≤ κ and $3\le S\left({X}_{i}\right)\le \alpha ⁺$ for each i ∈ I, then $S\left({\left({X}_{I}\right)}_{\kappa }\right)=\left({2}^{<\kappa }\right)⁺$.

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