Some characterizations of ω-balanced topological groups with a <italic>q</italic>-point.

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