Quasi-Uniformities and Quotients of Paratopological Groups

Abstract

If $H$ is a subgroup of a paratopological group $G$, we prove that the quotient topology of the coset space $G/H$ is induced by a point-rotund 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 is the
reflections $T_i(G)$ (i=1,2,3), $Reg(G)$ and $Tych(G)$.