2010/03/28 by Banakh, Taras, Ravsky, Alex
#22A15 #54H11 #54H99 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1003.5343
We obtain many results and solve some problems about feebly compact paratopological 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-pseudocompact paratopological group is feebly compact and that each Hausdorff σ-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 T0 sequentially compact group, a T1 2-pseudocompact 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 investigate cone topologies of paratopological groups which provide a general tool to construct 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. 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.