A Remark on the Separable Quotient Problem for Topological Groups

Abstract

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space $B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when $B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group $G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if $G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if $G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite $k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus $\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.

The Tubby Torus as a Quotient Group

“…So its dual group E is isomorphic as a topological group to the locally convex space ϕ. However, as mentioned earlier, it is proved in [5] (and generalized in [7]), that ϕ does not have the tubby torus as a quotient group, which completes the proof.…”

“…This is shown to be false in [5] for the free locally convex space ϕ on a countably infinite discrete space. Indeed in [7] it is shown that if X is a countably infinite k ω -space, then the free topological vector space on X, which is a connected infinite-dimensional (in the topological sense) topological group, does not have the tubby torus as a quotient group or even any infinite-dimensional (in the topological sense) metrizable quotient group.…”

“…Theorem 3. [7] Let F G (X) and A G (X) be the Graev free topological group and the Graev free abelian topological group, respectively, on an infinite connected compact Hausdorff space. Then the connected topological groups F G (X) and A G (X) have the tubby torus T ω as a quotient group.…”

Quotient on some Generalizations of topological group

“…In addition, the structures were reformed algebraically [31][32][33][34] and topologically 35 via fuzzy metrics [36][37][38][39] . Quotient forms such as factorizable [40][41][42] , metrizable 43,44 , and separable [45][46][47] especially in Banach spaces [48][49][50][51] were developed recently. Bounded groups 52,53 , Matrix groups 54 , homotopy characteristics [55][56][57][58], and projectivity 59 on topological groups were remarkable.…”