On index of total boundedness of (strictly) \(o\)-bounded groups
This paper deals with relationships between three cardinal invariants of topological groups dealing with boundedness. A subset \(A\) of a topological group \(G\) is called \textit{totally bounded} if \(A\) is covered by finitely many left, as well as by finitely many right, translations of each neighbourhood of the identity in \(G\), in other words, if \(A\) is totally bounded with regard to the supremum of the left and the right uniformities on \(G\). The \textit{index of total boundedness}, \(itb(G)\), of a topological group \(G\) is the smallest number of totally bounded sets covering \(G\). For instance, \(itb(G)\leq\aleph_0\) if and only if \(G\) is contained as a subgroup in a sigma-compact group. A topological group \(G\) is \textit{\(o\)-bounded} if for every sequence \((O_n)\) of neighbourhoods of the identity there are finite sets \((F_n)\) such that \(G\) is covered by the sets \(F_n\cdot O_n\), \(n=1,2,\ldots\). Further, \(G\) is \textit{strictly \(o\)-bounded} if the second player has a winning strategy in a game where two players choose in turn a neighbourhood \(O_n\) and a finite set \(F_n\), and the goal of the second player is to satisfy \(G = \cup_{i=1}^\infty F_n\cdot O_n\). Every \(\sigma\)-bounded group is strictly \(o\)-bounded, but it remained unknown since a few years ago if the two notions coincide in the metrizable case. The central theorem of the present paper answers the question in the negative in certain models of ZFC. A breakthrough is achieved through linking the above problematics with deep results of descriptive set theory of Polish groups.
scientific article; zbMATH DE number 1787508
scientific article