The Cofinality of Generating Familes
Paul Gartside, Thomas Gilton
TL;DR
The paper determines the generating-sequence and generating-compact-set numbers for separable metrizable spaces by linking topology to set-theoretic cardinal characteristics. It shows $\mathop{seq}(M)=\mathop{sam}(|M|)\cdot\mathfrak{b}$ (with a local-small caveat) and bounds $\mathop{k}(M)$ between $kc(M)\cdot\mathfrak{b}$ and $\mathop{sam}(kc(M))\cdot\mathfrak{b}$, using Tukey order and PCF theory to translate topological questions into set-theoretic invariants. The work ties these invariants to the covering number and mod-finite scales, derives consequences for analytic and co-analytic spaces, and provides models demonstrating multiple possible values for $k(M)$ given a fixed $kc(M)$, thereby answering van Douwen’s questions in new regimes. It also highlights the deep interactions between topology, Tukey theory, PCF, and forcing, showing how forcing can control the continuum, bounding number, and sampling numbers to realize a range of topological behaviors. Overall, the results illuminate how intrinsic topological structure of a space dictates, and is constrained by, robust set-theoretic characteristics.
Abstract
The topology of a separable metrizable space $M$ is \emph{generated} by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq M$ is closed in $M$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. The \emph{sequentiality number}, $\mathop{seq}(M)$, and \emph{$k$-ness number}, $\mathop{k}(M)$, of $M$, are the minimum size of a generating family of convergent sequences, respectively compact subsets. Let $\mathfrak{b}$ be the minimum size of an unbounded set in $ω^ω$ with the mod finite order. For a cardinal $κ$, the \emph{sampling number}, $\mathop{sam}(κ)$, is the least number of countable subsets of $κ$ needed to have infinite intersection with every countably-infinite subset of $κ$. It is shown using the Tukey order on relations that (1) $\mathop{seq}(M)=\mathop{sam}(|M|)\cdot \mathfrak{b}$, unless $M$ is locally small (every point of $M$ has a neighborhood of size strictly less than $|M|$) in which case $\mathop{seq}(M)=\lim_{μ<|M|} \mathop{sam}(μ)\cdot \mathfrak{b}$ and (2) $k(M)$ is in the interval $[kc(M)\cdot\mathfrak{b},\mathop{sam}(kc(M))\cdot \mathfrak{b}]$, where $kc(M)$ is the minimum number of compact sets that cover $M$. Shelah's \emph{PCF} theory is shown to provide tools to bound the sampling number, specifically, the covering number bounds from above, while mod finite scales give lower bounds. Solutions to problems of van Douwen's on the $k$-ness number of analytic and of co-analytic spaces are deduced.