Cardinal invariants for $\kappa$-box products: weight, density character and Suslin number
Volume 516 / 2016
Abstract
The symbol $(X_I)_\kappa$ (with $\kappa\geq\omega$) denotes the space $X_I:=\prod_{i\in I}X_i$ with the $\kappa$-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 $|\{i\in I:U_i\neq X_i\}| \lt \kappa$. 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 $\kappa\leq\alpha^+$, $|I|=\alpha$ and each $X_i$ contains the discrete space $\{0,1\}$ and satisfies $w(X_i)\leq\alpha$, then $w(X_\kappa)=\alpha^{ \lt \kappa}$.
Theorem 4.3.2. If $\omega\leq\kappa\leq|I|\leq2^\alpha$ and $X=(D(\alpha))^I$ with $D(\alpha)$ discrete, $|D(\alpha)|=\alpha$, then $d((X_I)_\kappa)=\alpha^{ \lt \kappa}$.
Corollaries 5.2.32(a) and 5.2.33. Let $\alpha\geq3$ and $\kappa\geq\omega$ be cardinals, and let $\{X_i:i\in I\}$ be a set of spaces such that $|I|^+\geq\kappa$.
(a) If $\alpha^+\ge\kappa$ and $\alpha\le S(X_i)\le \alpha^+$ for each $i\in I$, then $\alpha^{ \lt \kappa}\le S((X_I)_\kappa)\le(2^\alpha)^+$; and
(b) if $\alpha^+\leq \kappa$ and $3\le S(X_i)\le \alpha^+$ for each $i\in I$, then $S((X_I)_\kappa)=(2^{ \lt \kappa})^+$.
