Pseudobounded or $\omega$-Pseudobounded Paratopological Groups

Fucai Lin, Shou Lin

We say that a paratopological group $G$ is pseudobounded ($\omega$-pseudobounded), if for every neighborhood $V$ of the identity element $e$ of $G$, there exists a natural number $n$ such that $G=V^n(G=\bigcup_{n=1}^{\infty}V^n)$. We mainly discuss the pseudobounded and $\omega$-pseudobounded paratopological groups. First, we give an example to show that a theorem in [4] is not true. And then, we define the concept of premeager, and discuss when a pseudobounded paratopological group is a topological group. Moreover, we also discuss some properties of $\omega$-pseudobounded topological groups, and show that the class of connected topological groups is contained in the class of $\omega$-pseudobounded topological groups. Finally, some open problems concerning the paratopological groups are posed.