Balleans and G -spaces - Petrenko, O. V.; Protasov, I. V. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
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 $ω$ is a $T$-point with respect to the class of all metrizable locally finite balleans on $ω$ if and only if $\mathcal{U}$ is a $Q$-point. The paper is concluded with а list of open questions.