On topological groups with a weak q-point

In this article, firstly, we introduce the concepts of weak q-spaces and weak sq-spaces. Some properties of these spaces are discussed. Secondly, we study weak q-spaces and weak sq-spaces in topological groups in terms of preimages of submetrizable spaces. We show that a topological group G is a weak q-space (resp., weak sq-space) if and only if it is an open countable-compact (resp., sequential-compact) preimage of a submetrizable space.Finally, we give some characterizations of weakly feathered, weak q-spaces and weak sq-spaces in topological groups in coset spaces as follows:(1) Let H be a closed neutral subgroup of a topological group G. Then G/H is weakly feathered if and only if G/H is an open perfect preimage of a submetrizable space.(2) Let H be a closed neutral subgroup of a topological group G. Then G/H is a weak q-space (resp., weak sq-space) if and only if G/H is an open countable-compact (resp., sequential-compact) preimage of a submetrizable space.