Balleans and \(G\)-spaces (Q1928470)

Ball structures were introduced by the second author in [Ukr. Math. J. 54, No. 6, 1027--1037 (2002); original in Ukr. Mat. Zh. 54, No. 6, 847--855 (2002; Zbl 1003.05053)]. Let us quote the abstract of the paper: ``We show that every ballean (equivalently, coarse structure) on a set \(X\) can be determined by some group \(G\) of permutations of \(X\) and some group ideal \(\mathcal I\) on \(G\). We refine this characterization for some basic classes of balleans (metrizable, cellular, graph, locally finite, and uniformly locally finite). Then we show that a free ultrafilter \(\mathcal U\) on \(\omega\) is a \(T\)-point with respect to the class of all metrizable locally finite balleans on \(\omega\) if and only \(\mathcal U\) is a \(Q\)-point.'' Then the authors give some comments and several open questions in Section 3.