Notes on Remainders of Paratopological Groups

Abstract

In this paper, it is proved that a non-locally compact paratopological group $G$ has a remainder which is a $p$-space if and only if $G$ is either a Lindelöf $p$-space or a $\sigma$-compact space. We show that if $G$ is a non-locally compact paratopological group with a compactification $bG$ such that the remainder $bG\setminus G$ is locally metrizable, then both $G$ and $bG$ are separable and metrizable. It is proved that if $G$ is a cosmic paratopological group with a paracompact remainder, then $G$ is separable and metrizable.