Paratopological and semitopological groups versus topological groups (Q2874874)

Let \(G\) be a group and \(\tau\) a topology on \(G\). Then \(G\) is a \textit{paratopological group} if the operation of \(G\) is jointly continuous with respect to \(\tau\) (i.e., \(\cdot:G\times G\to G\) is continuous, where \(G\times G\) carries the product topology). Moreover, \(G\) is a \textit{semitopological group} if the operation of \(G\) is separately continuous with respect to \(\tau\). Clearly, every paratopological group is also a semitopological group. Moreover, every topological group is in particular a paratopological group.NEWLINENEWLINEThis is a survey paper on recent advances in the theory of paratopological and semitopological groups. Many results and examples are presented, together with several related open problems. Throughout the whole paper the properties of paratopological and semitopological groups are compared to their known counterparts for topological groups, pointing out the differences and the similarities. Starting from Ellis' theorem (i.e., a locally compact Hausdorff semitopological group is necessarily a topological group), many conditions are given for a paratopological group or a semitopological group to be a topological group. Several of these conditions are compactness-like properties. The influence of separation axioms on the topology of paratopological groups is presented. Moreover, a detailed description can be found of the behavior of the class of paratopological groups with respect to taking subgroups, quotients and extensions. The paper deals also with cardinal invariants of both paratopological and semitopological groups. Then products of paratopological groups are considered, in particular an extension to paratopological groups of the Comfort and Ross theorem on products of pseudocompact groups, and embeddings of paratopological groups into products of first (or second) countable factors. Finally, one can find the known results on free paratopological groups and references to other lines of investigation concerning paratopological and semitopological groups.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].