Topologies on groups determined by discrete subsets
Let \(\mathcal F\) be a filter on an infinite group \(G\). Suppose that \(\mathcal F\) contains all subsets of \(G\) whose cardinality is strictly smaller than that of their complements, and there is a map \(M:G\to \mathcal F\) such that the subsets \(xM(x)\) are pairwise disjoint. Such filters are called strongly discrete by the author. Let \(\mathcal T[\mathcal F]\) be the largest left invariant topology on \(G\) for which \(\mathcal F\) converges to the identity \(e\). The author shows, among other results, that \((G,\mathcal T[\mathcal F])\) is zero-dimensional and Hausdorff and, if \(\mathcal F\) is in addition an ultrafilter, \((G,\mathcal T[\mathcal F])\) is extremally disconnected and has no local homeomorphisms other than local translations. Moreover, two strongly discrete filters \(\mathcal F\) and \(\mathcal G\) on \(G\) are isomorphic if and only if the topologies \(\mathcal T[\mathcal F]\) and \(\mathcal T[\mathcal G]\) are homeomorphic. Finally, the author shows the undecidability in ZFC of the question whether there exists on any countable group \(G\) a strongly discrete ultrafilter \(\mathcal F\) such that \(\mathcal T [\mathcal F]\) is a group topology.
scientific article; zbMATH DE number 5292210
scientific article