A Dichotomy Theorem and other results for a class of quotients of topological groups

Topological properties of quotients of topological groups with respect to compact subgroups are investigated. It is observed that many classical results on topological groups can be extended to coset spaces of this kind. The main results of the paper are presented in the last sections. Among them is the Dichotomy Theorem for coset spaces with respect to a compact subgroup which says that every remainder of any such coset space is either pseudocompact or metric-friendly. Several metrizability conditions for coset spaces in terms of their remainders are given.