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 Cech-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 Lindelof paratopological groups (see Definition 3.1). Alternatively, one can describe these classes as the paratopological groups which are continuous homomorphic images of ω-narrow and Lindelof topological groups, respectively. Hence, every totally Lindelof paratopological group is totally ω-narrow. The class of totally Lindelof paratopological groups contains all Hausdorff paratopological groups H such that H 2 is Lindelof; therefore, all σ -compact paratopological groups are totally Lindelof. In Section 3 we establish several basic properties of totally ω-narrow and totally Lindelof paratopological groups. For example, Proposition 3.5 states that every totally ω-narrow first countable paratopological group has a countable base, while this is obviously false for ω-narrow (even Lindelof) first countable paratopological groups—the Sorgenfrey line is a counterexample. Then we study the question of when a given Hausdorff (regular) paratopological group H is topologically isomorphic to a subgroup of a product of second countable Hausdorff (regular) paratopological groups. Clearly, for the ‘regular’ case, the paratopological group H has to be Tychonoff, since regular second countable factors are metrizable. We show in Proposition 3.7 that if a paratopological group admits a topological embedding as a subgroup into a product of second countable paratopological groups, then the group is totally ω-narrow