An example of a planar group whose quasicomponent does not coincide with its component
The quasicomponent of a point in a topological group is the intersection of all clopen (closed-open) sets containing this point, and the component of a point is the usual maximal connected set containing that point. Treating the plane \({\mathbb{R}}^ 2\) as an abelian topological group (usual topology) with respect to addition, the author constructs a subgroup S of \({\mathbb{R}}^ 2\) and shows that the quasicomponent of the zero element (0,0) in S is isomorphic to the group \({\mathbb{Z}}\) of integers and that the component of (0,0) in S is itself this point thus providing an example of a group in which the quasicomponent of a point is distinct from its component. The subgroup in question is totally disconnected.
scientific article; zbMATH DE number 3964210