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

W. W. Comfort ; Ivan S. Gotchev

GDML_Books, (2016), p.

Résumé

The symbol $(X_{I})_{κ}$ (with κ ≥ ω) denotes the space $X_{I} := ∏_{i∈ I}X_{i}$ with the κ-box topology; this has as base all sets of the form $U = ∏_{i∈ I}U_{i}$ with $U_{i}$ open in $X_{i}$ and with $|{i∈ I: U_{i} ≠ X_{i}}| < κ$. 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 $w(X_{i}) ≤ α$, then $w(X_{κ}) = α^{<κ}$. Theorem 4.3.2. If $ω ≤ κ ≤ |I| ≤ 2^{α}$ and $X = (D(α))^{I}$ with D(α) discrete, |D(α)| = α, then $d((X_{I})_{κ}) = α^{<κ}$. Corollaries 5.2.32(a) and 5.2.33. Let α ≥ 3 and κ ≥ ω be cardinals, and let ${X_{i}: i ∈ I}$ be a set of spaces such that |I|⁺ ≥ κ. (a) If α⁺ ≥ κ and $α ≤ S(X_{i}) ≤ α⁺$ for each i ∈ I, then $α^{<κ} ≤ S((X_{I})_{κ}) ≤ (2^{α})⁺$; and (b) if α⁺ ≤ κ and $3 ≤ S(X_{i}) ≤ α⁺$ for each i ∈ I, then $S((X_{I})_{κ}) = (2^{<κ})⁺$.

Détail

Zbl 06622342

EUDML-ID : urn:eudml:doc:285978