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Separability of topological groups : A survey with open problems

- Leiderman, Arkady, Morris, Sidney

**Authors:**Leiderman, Arkady , Morris, Sidney**Date:**2018**Type:**Text , Journal article**Relation:**Axioms Vol. 8, no. 1 (2018), p. 1-18**Full Text:****Reviewed:****Description:**Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C(K) for metrizable compact spaces K; and ℓ p , for p ≥ 1. This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey. © 2018 by the authors.

**Authors:**Leiderman, Arkady , Morris, Sidney**Date:**2018**Type:**Text , Journal article**Relation:**Axioms Vol. 8, no. 1 (2018), p. 1-18**Full Text:****Reviewed:****Description:**Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C(K) for metrizable compact spaces K; and ℓ p , for p ≥ 1. This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey. © 2018 by the authors.

A topological group observation on the Banach-Mazur separable quotient problem

- Gabriyelyan, Saak, Morris, Sidney

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

A remark on the separable quotient problem for topological groups

**Authors:**Morris, Sidney**Date:**2019**Type:**Text , Journal article , Article**Relation:**Bulletin of the Australian Mathematical Society Vol. 100, no. 3 (Dec 2019), p. 453-457**Full Text:**false**Reviewed:****Description:**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(omega)-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 T-omega, which is an infinite-dimensional metrisable group, as a quotient group.

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