The separable quotient problem for topological groups (Q2279936)

The article was motivated by the famous Banach-Mazur problem: Is it true that every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. Although it is known that there are infinite-dimensional locally convex spaces without infinite-dimensional separable quotient, the Banach-Mazur problem remains unsolved more than 80 years. Analogous questions can be posed for a topological group \(G\): Does \(G\) have an infinite separable metrizable quotient group? In the article the authors solve this problem for several important classes of topological groups as (a) all \(\sigma\)compact locally compact groups, (b) all locally compact abelian groups, (c) all pro-Lie groups, and (d) all pseudocompact groups. However, the authors constructed an example of an uncountable precompact group \(G\) which has no infinite separable (even non-metrizable) quotient group.