Metrizable quotients of free topological groups

Leiderman, Arkady; Tkachenko, Mikhail

doi:10.1007/s13398-020-00855-x

Cited by 7 publications

(4 citation statements)

References 19 publications

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“…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.…”

Section: Modeling Of Robot Actions Using Digital Imagementioning

confidence: 99%

Quotient on some Generalizations of topological group

Kumar¹,

Gnanachandra²,

Priya³

2023

Baghdad Sci.J

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In this paper, we define some generalizations of topological group namely -topological group, -topological group and -topological group with illustrative examples. Also, we define grill topological group with respect to a grill. Later, we deliberate the quotient on generalizations of topological group in particular -topological group. Moreover, we model a robotic system which relays on the quotient of -topological group.

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Section: Modeling Of Robot Actions Using Digital Imagementioning

confidence: 99%

Quotient on some Generalizations of topological group

Kumar¹,

Gnanachandra²,

Priya³

2023

Baghdad Sci.J

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“…On the contrary, assume that a countably compact space X is not scattered. Every countably compact space X is pseudocompact; therefore there exists a closed subset K ⊂ X and a continuous surjective mapping ϕ from K onto the closed interval [0, 1], by [24,Proposition 5.5]. Every closed subset F of K is a countably compact space, its continuous image ϕ(F ) is a countably compact subset of [0, 1];…”

Section: Theorem 42 ([23]) Every Compact δ-Space Has Countable Tightnessmentioning

confidence: 99%

Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished

Ka̧kol¹,

Leiderman²

2021

Proc. Amer. Math. Soc. Ser. B

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In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space X X is a Δ \Delta -space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex space C p ( X ) C_{p}(X) is distinguished. Continuing this research, we investigate whether the class Δ \Delta of Δ \Delta -spaces is invariant under the basic topological operations. We prove that if X ∈ Δ X \in \Delta and φ : X → Y \varphi :X \to Y is a continuous surjection such that φ ( F ) \varphi (F) is an F σ F_{\sigma } -set in Y Y for every closed set F ⊂ X F \subset X , then also Y ∈ Δ Y\in \Delta . As a consequence, if X X is a countable union of closed subspaces X i X_i such that each X i ∈ Δ X_i\in \Delta , then also X ∈ Δ X\in \Delta . In particular, σ \sigma -product of any family of scattered Eberlein compact spaces is a Δ \Delta -space and the product of a Δ \Delta -space with a countable space is a Δ \Delta -space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99]. Let T : C p ( X ) ⟶ C p ( Y ) T:C_p(X) \longrightarrow C_p(Y) be a continuous linear surjection. We observe that T T admits an extension to a linear continuous operator T ^ \widehat {T} from R X \mathbb {R}^X onto R Y \mathbb {R}^Y and deduce that Y Y is a Δ \Delta -space whenever X X is. Similarly, assuming that X X and Y Y are metrizable spaces, we show that Y Y is a Q Q -set whenever X X is. Making use of obtained results, we provide a very short proof for the claim that every compact Δ \Delta -space has countable tightness. As a consequence, under Proper Forcing Axiom every compact Δ \Delta -space is sequential. In the article we pose a dozen open questions.

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“…The main aim of this survey paper is to present systematically the results concerning the behavior of separability of topological groups with respect to the topological operations listed above and make clear which problems are open. Much of the material is from the recent publications [1][2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

“…[2]). Does there exist an R-quotient mapping from each Tychonoff space onto an infinite subspace of [0, 1]?…”

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confidence: 99%

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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Metrizable quotients of free topological groups

Cited by 7 publications

References 19 publications

Quotient on some Generalizations of topological group

Quotient on some Generalizations of topological group

Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished

Separability of Topological Groups: A Survey with Open Problems

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