Simply \textit{sm}-factorizable (para)topological groups and their quotients (Q820694)

A (para)topological group $G$ is called simply sm-factorizable if for each co-zero set $U$ in $G$, there exists a continuous homomorphism $\pi$ of $G$ onto a separable metrizable (para)topological group $H$ such that $U=\pi-1(\pi(U))$. It is proved that a regular (para)topological group $G$ is simply sm-factorizable if and only if $G$ is projectively strongly submetrizable and every continuous real-valued function on $G$ is uniformly continuous on $G\omega$, the $P$-modification of $G$. It is also established that every precompact paratopological group is simply sm-factorizable.