math.GN

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

Abstract: 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 $\kappa$.
 Comments: 6 pages Subjects: General Topology (math.GN); Commutative Algebra (math.AC); Group Theory (math.GR); Logic (math.LO) MSC classes: 03E15, 03E50, 22A05, 54H05, 54H11 Journal reference: Colloq. Math. 125 (2011) 213-220 Cite as: arXiv:1106.2235 [math.GN] (or arXiv:1106.2235v1 [math.GN] for this version)

## Submission history

From: Taras Banakh [view email]
[v1] Sat, 11 Jun 2011 13:26:32 GMT (7kb)

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