Totally Lindelöf and totally ω-narrow paratopological groups

Abstract

In many natural objects of topological algebra that possess the algebraic structure of a group, the operations of inversion and multiplication are not necessarily continuous—it suffices to recall the groups of homeomorphisms of topological spaces with the pointwise convergence topology (where the composition of homeomorphisms as multiplication is almost never continuous). This gave rise to the study of semitopological, quasitopological, and paratopological groups, among other related structures. In a paratopological group, multiplication is jointly continuous while inversion is usually not—otherwise it is a topological group. The growing interest in the study of semitopological and paratopological groups led to a significant clarification of the importance of “topological symmetry” (i.e., the continuity of inversion) in topological algebra. It was shown, for example, that every pseudocompact paratopological group is a topological group [7] and every Cˇ ech-complete semitopological group is also a topological group [5]. In [2], it was established that every σ -compact paratopological group has countable cellularity, thus generalizing a theorem from [9] proved there for topological groups. It is worth mentioning that every precompact paratopological group has countable cellularity as well (see [4]). The main objects of our study are the classes of totally ω-narrow and totally Lindelöf paratopological groups.