Constructing universally small subsets of a given packing index in Polish groups

Details Constructing universally small subsets of a given packing index in Polish groups Constructing universally small subsets of a given packing index in Polish groups

2011-06-11

arxiv.org/abs/1106.2235v1

Mathematics

General Topology

6 pages

Scientific paper

A subset of a Polish space $X$ is called universally small if it belongs to each ccc $\sigma$-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 $\kappa\in[5,\mathfrak c^+]$ the set $A_0$ contains a universally small subset $A$ of $G$ with sharp packing index $\pack^\sharp(A_\kappa)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$ is disjoint$\}$ equal to $\kappa$.

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