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Maximal topologies on groups. (English. Russian original) Zbl 0935.22002
Sib. Math. J. 39, No. 6, 1184-1194 (1998); translation from Sib. Mat. Zh. 39, No. 6, 1368-1381 (1998).
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.

MSC:
22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)
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