Haar-smallest sets

摘要

In this paper we are interested in the following notions of smallness: a subset A of an abelian Polish group X is called Haar-countable/Haar-finite/Haar-n if there are a Borel hull B superset of A and a copy C of 2(omega) in X such that (C+x)boolean AND B is countable/finite/of cardinality at most n, for all x is an element of X.Recently, Banakh et al. have unified the notions of Haar-null and Haar-meager sets by introducing Haar-I sets, where I is a collection of subsets of 2(omega). It turns out that if I is the sigma-ideal of countable sets, the ideal of finite sets or the collection of sets of cardinality at most n, then we get the above notions. Moreover, those notions have been studied independently by Zakrzewski (under a different name - perfectly kappa-small sets).We study basic properties of the corresponding families of small sets, give suitable examples distinguishing them (in all groups of the form R x X, where X is an abelian Polish group) and study a-ideals generated by closed members of the considered families. In particular, we show that compact Haar-finite sets do not form an ideal. Moreover, we answer some questions concerning null-finite sets, asked by Banakh and Jablonska, and pose several open problems. (C) 2019 Elsevier B.V. All rights reserved.