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

Taras Banakh ¹   ; Nadya Lyaskovska ²

¹ Institute of Mathematics Jan Kochanowski University Kielce, Poland and Faculty of Mechanics and Mathematics Ivan Franko National University of Lviv 79000 Lviv, Ukraine
² Faculty of Mechanics and Mathematics Ivan Franko National University of Lviv Universytetska 1 79000 Lviv, Ukraine

Colloquium Mathematicum, Tome 125 (2011) no. 2, pp. 213-220

Résumé

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 ${\rm pack}^\sharp(A_\kappa)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$ is disjoint$\}$ equal to $\kappa$.

Détail

DOI : 10.4064/cm125-2-6

Keywords: subset polish space called universally small belongs each ccc sigma ideal borel base under each uncountable abelian polish group construct universally small subset subset cap mathfrak each each cardinal number kappa mathfrak set contains universally small subset sharp packing index pack sharp kappa sup mathcal mathcal subset disjoint equal kappa