Topological Gyrogroups with Fréchet–Urysohn Property and $$\omega ^{\omega }$$-Base

Bao, Meng; Zhang, Xiaoyuan; Xu, Xiaoquan

doi:10.1007/s41980-021-00576-w

Cited by 5 publications

(7 citation statements)

References 25 publications

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“…Definition 2.8. [16,Theorem 3.7] Let G be a topological gyrogroup, the cs * -character of G is the least cardinality of cs * -network at the identity element 0 of G.…”

Section: Preliminarymentioning

confidence: 99%

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A supplement on feathered gyrogroups

Meng¹,

Ling²,

Xu³

2022

Preprint

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A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that each compact subset of a topological gyrogroup with an ω ω -base is metrizable, which deduces that if G is a topological gyrogroup with an ω ωbase and is a k-space, then it is sequential. Moreover, for a feathered strongly topological gyrogroup G, based on the characterization of feathered strongly topological gyrogroups, we show that if G has countable cs * -character, then it is metrizable; and it is also shown that G has a compact resolution swallowing the compact sets if and only if G contains a compact L-subgyrogroup H such that the quotient space G/H is a Polish space.

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“…Definition 2.8. [16,Theorem 3.7] Let G be a topological gyrogroup, the cs * -character of G is the least cardinality of cs * -network at the identity element 0 of G.…”

Section: Preliminarymentioning

confidence: 99%

“…Clearly, every topological group is a topological gyrogroup and each topological gyrogroup is a rectifiable space. The readers may consult [5,6,10,12,13,14,15,16,39,40] for more details about topological gyrogroups.…”

Section: Introductionmentioning

confidence: 99%

A supplement on feathered gyrogroups

Meng¹,

Ling²,

Xu³

2022

Preprint

Self Cite

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“…Moreover, it was proved in [12] that a topological gyrogroup is metrizable if and only if it is Fréchet-Urysohn with an ω ω -base, then it is natural to consider whether it holds in the quotient spaces of topological gyrogroups. Here we just investigate the quotient spaces of strongly topological gyrogroups.…”

Section: Coset With Admissible Subgyrogroupsmentioning

confidence: 99%

“…Then Cai, Lin and He in [13] proved that every topological gyrogroup is a rectifiable space. A series of results on topological gyrogroups have been obtained in [2,10,12,23,24,25] By the further research of Möbius gyrogroups, Einstein gyrogroups, and Proper Velocity gyrogroups, Bao and Lin [4] found that all of them have an open neighborhood base at the identity element 0 which is invariant under the groupoid automorphism with standard topology. Therefore, they posed the concept of strongly topological gyrogroups.…”

Section: Introductionmentioning

confidence: 99%

A class of quotient spaces in strongly topological gyrogroups

Bao,

Wang,

2021

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Quotient space is a class of the most important topological spaces in the research of topology. In this paper, we show that if (G, τ, ⊕) is a strongly topological gyrogroup with a symmetric neighborhood base U at 0 and H is an admissible subgyrogroup generated from U , then G/H is first-countable if and only if it is metrizable. Moreover, if H is neutral and G/H is Fréchet-Urysohn with an ω ω -base, then G/H is first-countable. Therefore, we obtain that if H is neutral, then G/H is metrizable if and only if G/H is Fréchet-Urysohn with an ω ω -base. Finally, it is shown that if H is neutral, πχ(G/H) = χ(G/H) and πω(G/H) = ω(G/H).

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“…Therefore, they posed the concept of strongly topological gyrogroups and showed that every feathered strongly topological gyrogroup is paracompact. A series of results on topological gyrogroups and strongly topological gyrogroups have been obtained in [3,5,6,9,10,11,12,13,15,16,36,37,38]. In particular, it was proved in [9,10] that each T 0 -strongly topological gyrogroup is completely regular, every T 0 -strongly topological gyrogroup with a countable pseudocharacter is submetrizable and each locally paracompact strongly topological gyrogroup is paracompact.…”

Section: Introductionmentioning

confidence: 99%

Quotient spaces with strong subgyrogroups

Meng¹,

Ling²,

Xu³

2022

Preprint

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In this paper, we mainly investigate the quotient spaces G/H when G is a strongly topological gyrogroup and H is a strong subgyrogroup of G. It is shown that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then the quotient space G/H is first-countable if and only if G/H is a bisequential space if and only if G/H is a weakly first-countable space if and only if G/H is a csf -countable and sequential α 7 -space. Moreover, it is shown that if G is a strongly topological gyrogroup and H is a locally compact strong subgyrogroup of G, then there exists an open neighborhood U of the identity element 0 such that π(U ) is closed in G/H and the restriction of π to U is a perfect mapping from U onto the subspace π(U ); if H is a locally compact metrizable strong subgyrogroup of G and the quotient space G/H is sequential, then G is also sequential; if H is a closed first-countable and separable strong subgyrogroup of G, the quotient space G/H is an ℵ 0 -space, then G is an ℵ 0 -space; if the quotient space G/H is a cosmic space, then G is also a cosmic space; if the quotient space G/H has a star-countable cs-network or star-countable wcs * -network, then G also has a star-countable cs-network or star-countable wcs * -network, respectively.

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Topological Gyrogroups with Fréchet–Urysohn Property and $$\omega ^{\omega }$$-Base

Cited by 5 publications

References 25 publications

A supplement on feathered gyrogroups

A supplement on feathered gyrogroups

A class of quotient spaces in strongly topological gyrogroups

Quotient spaces with strong subgyrogroups

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