  Groups with a topology that is in consistent one way or another with the
algebraic structure are considered. Classical groups with a topology are
topological, paratopological, semitopological, and quasitopological groups. We
also study other ways of matching topology and algebraic structure. The minimum
requirement in this paper is that the group is a right semitopological group
(such groups are often called right topological groups). We study when a group
with a topology is a topological group; research in this direction began with
the work of Deane Montgomery and Robert Ellis. (Invariant) semi-neighborhoods
of the diagonal are used as a means of study.
