Suitable sets for paratopological groups

Abstract

We investigate when a paratopological group G has a suitable set S. The latter means that S is a discrete subspace of G, S ∪ {e} is closed, and the subgroup S of G generated by S is dense in G.A paratopological group is a group G endowed with a topology τ such that the group operationIn this case τ is called a semigroup topology on G. If, additionally, the operation of taking inverse is continuous then G is a topological group. A classical example of a paratopological group failing to be a topological group is the Sorgenfrey line S, that is the additive group of real numbers, endowed with the Sorgenfrey topology generated by the base consisting of half-intervals [a, b), a < b).Whereas the investigation of topological groups already is a fundamental branch of topological algebra (see, for instance, [30], [12], and [2]), paratopological groups are not so well-studied and have more variable structure. Basic properties of paratopological groups compared with the properties of topological groups are described in the book [2] by Arhangel'skii and Tkachenko, in the PhD thesis of the second author [34], papers [32], [33], and in the survey [40] by Tkachenko.Suitable set for topological groups were considered by Hofmann and Morris in [23]. Fundamental results were obtained by Comfort et al.in [9] and Dikranjan et al. in [10] and in [11]. In the present paper we extend this research to paratopological groups.A subset S of a paratopological group G is a suitable set for G, if S is a discrete subspace of G, S ∪ {e} is closed, and the subgroup S of G generated by S is dense in G, see [20]. Let S (respectively, S c ) be the class of paratopological groups G having a suitable (respectively, closed suitable) set. It turns out that very often a suitable set of a group G generates G. This fact suggests to devote a special attention to classes S g (respectively, S cg ) of paratopological groups G having a suitable (respectively, closed suitable) set which generates G.Proposition 1. Any paratopological group with a suitable set is a T 1 -space or a twoelement group.Proof. Let S be a suitable set of a paratopological group G and s be any element of S. Put {s}. The definition of a suitable set implies {s} ⊂ {s, e}. If {s} or {e} is a

Suitable sets for strongly topological gyrogroups

“…Fundamental results on suitable sets for topological groups were obtained by Comfort et al in [8] and Dikranjan et al in [9] and [10]. I. Guran in [14], F. Lin, A. Ravsky, and T. Shi in [18] considered suitable sets for paratopological groups. In 2003, T. Banakh and I. Protasov generalized Guran's results to left topological groups in [3].…”