An open mapping theorem
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2016
- Type: Text , Journal article
- Relation: Bulletin of the Australian Mathematical Society Vol. 94, no. 1 (2016), p. 65-69
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- Description: It is proved that any surjective morphism f : Z(k) -> K onto a locally compact group K is open for every cardinal k. This answers a question posed by Hofmann and the second author.
Free subspaces of free locally convex spaces
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2018
- Type: Text , Journal article
- Relation: Journal of Function Spaces Vol. 2018, no. (2018), p. 1-5
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- Description: Abstract If X and Y are Tychonoff spaces, let and be the free locally convex space over and , respectively. For general and , the question of whether can be embedded as a topological vector subspace of is difficult. The best results in the literature are that if can be embedded as a topological vector subspace of , where , then is a countable-dimensional compact metrizable space. Further, if is a finite-dimensional compact metrizable space, then can be embedded as a topological vector subspace of . In this paper, it is proved that can be embedded in as a topological vector subspace if is a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case if It is also shown that if and denote the Cantor space and the Hilbert cube , respectively, then (i) is embedded in if and only if is a zero-dimensional metrizable compact space; (ii) is embedded in if and only if is a metrizable compact space.
- Description: If
A topological group observation on the Banach-Mazur separable quotient problem
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2019
- Type: Text , Journal article
- Relation: Topology and Its Applications Vol. 259, no. (2019), p. 283-286
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- 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