# On feebly compact paratopological groups

Taras Banakh, Alex Ravsky

Arxiv ID: 1003.5343•Last updated: 8/5/2020

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 $\sigma$-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
$T_1$ $2$-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.

#### PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.