The quotient spaces of topological groups with a $q$-point

Abstract

In this paper, we study the uniformities on the double coset spaces in topological groups. As an implication, the quotient spaces of topological groups with a $q$-point are studied. It mainly shows that: (1) Suppose that $G$ is a topological group with a $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and quasi-perfect preimage of a metrizable space; in particular, $G/H$ is an $M$-space. (2) Suppose that $G$ is a topological group with a strict $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and sequentially perfect preimage of a metrizable space. (3) Suppose that $G$ is a topological group with a strong $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and strongly sequentially perfect preimage of a metrizable space.