The local properties in topological groups and related concepts and questions (Q6142392)

The author considers open subsets of topological groups and in particular what happens when that open set is itself homeomorphic to a topological group. It turns out that such an assumption may have strong consequences. If \(G\) is a topological group and \(F\) is a compact subset whose complement is homeomorphic to a topological group then \(F\) is dyadic in case \(G\) is Lindelöf with countable cellularity, and even a Dugundji compactum if \(G\) is \(\sigma\)-compact or locally compact. In all cases \(F\) is a \(G_\delta\)-subset and metrizable if its tightness is countable. There are various other results that indicate that having the complement of a point be homeomorphic to a topological group is quite a strong condition to impose on a topological group.