Quasi-Uniformities and Quotients of Paratopological Groups

Ivan Sanchez, Manuel Sanchis

For a subgroup $H$ of a paratopological group $G$ we prove that the quotient topology of the coset space $G/H$ is induced by a rotund quasi-uniformity and the quotient topology of the semiregularization $(G/H)_{sr}$ of $G/H$ is induced by a normal quasi-uniformity. In particular, $(G/H)_{sr}$ is a Tychonoff space provided that $G/H$ is Hausdorff. The previous results are applied in order to show that every Hausdorff Lindelöf paratopological group is $\omega$-admissible. We also show that, if $G$ is an $\omega$-admissible paratopological group, then so are the reflections $T_i(G)$ $(i=0,1,2,3)$, $\operatorname{Reg}(G)$ and $\operatorname{Tych}(G)$.