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

W. W. Comfort; Ivan S. Gotchev

- 2016

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topW. W. Comfort, and Ivan S. Gotchev. Cardinal invariants for κ-box products: weight, density character and Suslin number. 2016. <http://eudml.org/doc/285978>.

@book{W2016,

abstract = {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 $S((X_\{I\})_\{κ\}) = (2^\{<κ\})⁺$.},

author = {W. W. Comfort, Ivan S. Gotchev},

keywords = {box topology; $\kappa $-box topology; weight; density character; Suslin number; cellular family; Hewitt-Marczewski-Pondiczery theorem},

language = {eng},

title = {Cardinal invariants for κ-box products: weight, density character and Suslin number},

url = {http://eudml.org/doc/285978},

year = {2016},

}

TY - BOOK

AU - W. W. Comfort

AU - Ivan S. Gotchev

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

PY - 2016

AB - 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 $S((X_{I})_{κ}) = (2^{<κ})⁺$.

LA - eng

KW - box topology; $\kappa $-box topology; weight; density character; Suslin number; cellular family; Hewitt-Marczewski-Pondiczery theorem

UR - http://eudml.org/doc/285978

ER -

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