Polish群中填充指数泛小子集的构造
Constructing universally small subsets of a given packing index in Polish groups
摘要：
Polish空间$X$的一个子集称为泛小的，如果它属于$X$上的每个具有Borel基的ccc $σ$ -理想.在每个不可数阿贝尔波兰群$G$中的CH下，我们构造了一个泛小子集$A_0\subset G$，使得对每个$g\in G$有$|A_0\cap gA_0|=\mathfrak c$.对于每个基数$κ\in[5,\mathfrak c^+]$，集合$A_0$包含$G$的一个通用的小子集$A$，其具有尖锐的填充指数$\pack^\sharp(A_κ)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$不相交$\}$等于$κ$。
英文摘要：
A subset of a Polish space $X$ is called universally small if it belongs to each ccc $σ$-ideal with Borel base on $X$. Under CH in each uncountable Abelian Polish group $G$ we construct a universally small subset $A_0\subset G$ such that $|A_0\cap gA_0|=\mathfrak c$ for each $g\in G$. For each cardinal number $κ\in[5,\mathfrak c^+]$ the set $A_0$ contains a universally small subset $A$ of $G$ with sharp packing index $\pack^\sharp(A_κ)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$ is disjoint$\}$ equal to $κ$.
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