The Cofinality of Generating Familes
Paul Gartside, Thomas Gilton

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 is \emph{generated} by a family of its subsets provided that a set is closed in if and only if is closed in for each . The \emph{sequentiality number}, , and \emph{-ness number}, , of , are the minimum size of a generating family of convergent sequences, respectively compact subsets. Let be the minimum size of an unbounded set in with the mod finite order. For a cardinal , the \emph{covering number}, , is the minimum size of a family of countable subsets of so that every countable subset of is contained in an element of the family. It is shown using the Tukey order on relations that (1) , unless is…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Economic theories and models · Computability, Logic, AI Algorithms
