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

Gabriyelyan, Saak; Morris, Sidney A.

doi:10.1016/j.topol.2019.02.036

Cited by 5 publications

(9 citation statements)

References 14 publications

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“…In this section, we prove that if ϕ is the free locally convex space on a countably infinite discrete space, then ϕ does not have a quotient topological group that is metrisable and infinite dimensional as a topological space. (This extends a result in [6].) Indeed, if V(X) is the free topological vector space on any countably infinite k ω -space, then the infinite-dimensional topological vector space V(X) does not have a quotient group that is metrisable and infinite dimensional as a topological space.…”

Section: Resultssupporting

confidence: 73%

“…An analogous problem for topological groups is: Does every connected (Hausdorff) topological group G have a quotient topological group that is infinite dimensional as a topological space and metrisable? In [6], a positive answer to this question was given c 2019 Australian Mathematical Publishing Association Inc. by S. Gabriyelyan and the author for the case where G is the underlying topological group of an infinite-dimensional Banach space or even the underlying topological group of a locally convex space that has an infinite-dimensional Fréchet space as a subspace. Indeed, G has the tubby torus group T ω , which is infinite dimensional and metrisable, as a quotient group.…”

Section: Introduction and Notationmentioning

confidence: 99%

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A Remark on the Separable Quotient Problem for Topological Groups

Morris

2019

Bull. Aust. Math. Soc.

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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.

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Section: Resultssupporting

confidence: 73%

Section: Introduction and Notationmentioning

confidence: 99%

A Remark on the Separable Quotient Problem for Topological Groups

Morris

2019

Bull. Aust. Math. Soc.

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“…Corollary ( [7]). Every infinite-dimensional Fréchet space, and in particular every infinite-dimensional Banach space, has the (infinite separable metrizable) tubby torus group T ω as a quotient group.…”

Section: Theorem 23 ([7]mentioning

confidence: 99%

“…It is natural to consider these questions for various prominent classes of topological groups such as Banach spaces, locally convex spaces, compact groups, locally compact groups, pro-Lie groups, pseudocompact groups, and precompact groups. The paper [7] provides an interesting solution for Banach spaces.…”

Section: Theorem 22 ([36]mentioning

confidence: 99%

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Separability of Topological Groups: A Survey with Open Problems

Leiderman

Morris

2018

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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.

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The separable quotient problem for topological groups

2019

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The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.Problem 1.1. Separable Quotient Problem for Banach Spaces. Does every infinite-dimensional Banach space have a quotient Banach space which is separable and infinite-dimensional? The related Quotient Schauder Basis Problem for Banach Spaces is due to Aleksander (Olek) Pe lczyński [24]. Problem 1.2. Quotient Schauder Basis Problem for Banach Spaces. Does every infinite-dimensional Banach space have a quotient Banach space which is infinite-dimensional and has a Schauder basis?

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

Cited by 5 publications

References 14 publications

A Remark on the Separable Quotient Problem for Topological Groups

A Remark on the Separable Quotient Problem for Topological Groups

Separability of Topological Groups: A Survey with Open Problems

The separable quotient problem for topological groups

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