On feebly compact paratopological groups

摘要

We obtain many results and solve some problems about feebly compact paratopo-logical groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each 2-pseudo compact paratopological group is feebly compact and that each Hausdorff sigma-compact feebly compact paratopological group is a compact topological group. Our particular attention concerns periodic and topologically periodic groups. We construct examples of various compact-like paratopological groups which are not topological groups, among them a T-0 sequentially compact group, a T1 2-pseudo compact group, a functionally Hausdorff countably compact group (under the axiomatic assumption that there is an infinite torsion-free Abelian countably compact topological group without non-trivial convergent sequences), and a functionally Hausdorff second countable group sequentially pracompact group. We prove that the product of a family of feebly compact paratopological groups is feebly compact, and that a paratopological group G is feebly compact provided it has a feebly compact normal subgroup H such that a quotient group G/H is feebly compact. For our research we also study some general constructions of paratopological groups. We extend the well-known construction of Rakov completion of a T-0 topological group to the class of paratopological groups. We investigate cone topologies of paratopological groups which provide a general tool for constructing pathological examples, especially examples of compact like paratopological groups with discontinuous inversion. We find a simple interplay between the algebraic properties of a basic cone subsemigroup S of a group G and compact-like properties of two basic semigroup topologies generated by S on the group G. (C) 2020 Elsevier B.V. All rights reserved.