Weak completions of paratopological groups

TL;DR

This paper introduces a framework for constructing and analyzing various types of completions of paratopological groups using classes of continuous homomorphisms, exploring their separation properties and providing specific examples.

Contribution

It defines $ ext{C}$-semicompletions and $ ext{C}$-completions for paratopological groups, establishing conditions for their Hausdorffness and presenting a counterexample involving $T_1$-space failure.

Findings

Conditions for (semi)completions to be Hausdorff are established.

A Hausdorff paratopological abelian group with a non-$T_1$ semicompletion is constructed.

An example shows a subgroup's closure fails to be a subgroup in a specific completion.

Abstract

Given a $T_0$ paratopological group $G$ and a class $\mathcal C$ of continuous homomorphisms of paratopological groups, we define the $\mathcal C$-$semicompletion$ $\mathcal C[G)$ and $\mathcal C$-$completion$ $\mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $\mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $\mathcal C[G)$ and $\mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $\mathcal C$-semicompletion $\mathcal C[G)$ fails to be a $T_1$-space, where $\mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $\omega$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.