标题： 生成族的共尾性 显示英文标题

标题： The Cofinality of Generating Familes

摘要： The topology of a separable metrizable space $M$ is \emph{生成的} 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 $C\in \mathcal{C}$. \emph{顺序性编号}、$\mathop{seq}(M)$和\emph{$k$-ness 数字}、$\mathop{k}(M)$的$M$是收敛序列生成族的最小尺寸，分别是紧子集。 令$\mathfrak{b}$为在$ω^ω$中具有有限序的无界集的最小尺寸。 对于基数$κ$，\emph{采样数量}，$\mathop{sam}(κ)$是需要的$κ$的可数子集的最小数量，以与$κ$的每个可数无限子集有无限交集。 使用关系上的Tukey序可知，(1)$\mathop{seq}(M)=\mathop{sam}(|M|)\cdot \mathfrak{b}$，除非$M$是局部小的（$M$的每个点都有一个严格小于$|M|$的邻域），在这种情况下$\mathop{seq}(M)=\lim_{μ<|M|} \mathop{sam}(μ)\cdot \mathfrak{b}$以及 (2)$k(M)$位于区间$[kc(M)\cdot\mathfrak{b},\mathop{sam}(kc(M))\cdot \mathfrak{b}]$中，其中$kc(M)$是覆盖$M$所需的紧集的最小数目。 谢拉的\emph{个人计算机}理论被证明提供了工具来限制抽样数，具体来说，从上面限制了覆盖数，而模有限尺度给出了下界。 范·多文关于解析空间和余解析空间的$k$-ness 数量的问题的解被推导出来。 显示英文摘要

摘要： 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.