The Cofinality of Generating Familes

TL;DR

This paper investigates the minimal sizes of generating families for separable metrizable spaces, linking these to cardinal invariants like the bounding number and covering numbers, and applies results to analytic and co-analytic spaces.

Contribution

It establishes formulas for the sequentiality and $k$-ness numbers of such spaces using Tukey order and cardinal invariants, advancing understanding of their topological complexity.

Findings

$ ext{seq}(M)$ equals $ ext{cov}(|M|) imes rak{b}$ unless $M$ is locally small

$k(M)$ lies between $kc(M) imes rak{b}$ and $ ext{cov}(kc(M)) imes rak{b}$

Provides solutions to van Douwen's problems on $k$-ness of analytic and co-analytic spaces

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 $\omega^\omega$ with the mod finite order. For a cardinal $\kappa$, the \emph{covering number}, $\mathop{cov}(\kappa)$, is the minimum size of a family of countable subsets of $\kappa$ so that every countable subset of $\kappa$ is contained in an element of the family. It is shown using the Tukey order on relations that (1) $\mathop{seq}(M)=\mathop{cov}(|M|)\cdot \mathfrak{b}$, unless $M$ is locally small (every point of $M$ has a neighborhood of size strictly less than $|M|$) in which case $\mathop{seq}(M)=\lim_{\mu <|M|} \mathop{cov}(\mu)\cdot \mathfrak{b}$ and (2) $k(M)$ is in the interval $[kc(M)\cdot\mathfrak{b},\mathop{cov}(kc(M))\cdot \mathfrak{b}]$, where $kc(M)$ is the minimum number of compact sets that cover $M$. Solutions to problems of van Douwen's on the $k$-ness number of analytic and of co-analytic spaces are deduced.