Some characterizations of ω-balanced topological groups with a q-point

Abstract

In this paper, we study some characterizations of q-spaces, strict q-spaces and strong q-spaces under ω-balanced topological groups. First we characterize a topological group G to be ω-balanced and a q-space whenever for each open neighborhood O of the identity in G, there is a countably compact invariant subgroup H which is of countable character in G, such that H ⊆ O and the canonical quotient mapping p : G → G/H is quasi-perfect and the quotient group G/H is metrizable. Secondly, we characterize G to be ω-balanced and a strict q-space, replacing the condition of being a q-space with an appropriate condition which is designed for the so-called “strict q-spaces”. Finally, we characterize G to be ω-balanced and a strong q-space, illustrating an additional condition to be replaced to that of q-space, or strict q-space. As applications of our three main results, we found new characterizations of the ω-narrow topological groups.