Maximal topologies on groups
The article is devoted to the notion of a maximal topological group. A topological space without isolated points is called a maximal space if the space has an isolated point in every stronger topology. A topological group is called maximal if the underlying topological space of the group is a maximal space. The problem of the existence of a maximal topological group is a problem of set theory by nature. However, maximal homogeneous spaces and left-topological groups can be easily constructed in \(ZFC\). The author proves that, in \(ZFC\), an arbitrary infinite group can be endowed with a maximal regular left-invariant topology. In addition, the construction of this topology yields a solution to the Hindman-Strauss problem of a regular idempotent [Semigroup Forum 51, No. 3, 299-318 (1995; Zbl 0843.22005)]. Besides, the author proves some other theorems about groups with strongest left-invariant topology.
scientific article; zbMATH DE number 1432171
scientific article