Separability in (strongly) topological gyrogroups

Bao, Meng; Zhang, Xiaoyuan; Xu, Xiaoquan

doi:10.2298/fil2113381b

Cited by 7 publications

(5 citation statements)

References 17 publications

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“…The definition of gyrogroups, some of which are gyrocommutative, is presented, for instance, in [2,3,5,6]. Gyrogroups form a natural generalization of groups [9], giving rise to useful applications, such as in [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].…”

Section: Möbius Addition and Scalar Multiplicationmentioning

confidence: 99%

The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

Ungar

2023

Symmetry

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Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its associated symmetry gyrogroup. The translation of Ptolemy’s theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.

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Section: Möbius Addition and Scalar Multiplicationmentioning

confidence: 99%

The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

Ungar

2023

Symmetry

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“…It follows from Theorem 3.2 that the subgyrogroup H is metrizable. Since each compact subset of a Hausdorff space is closed, it is clear that H is a closed L-subgyrogroup of G. Then, by [15,Corollary 4.3], if G is a topological gyrogroup and H is a closed L-subgyrogroup of G and if the spaces H and G/H are metrizable, then the space G is also metrizable. Therefore, we obtain that G is a metrizable space.…”

Section: Weakly First-countable Properties Of Topological Gyrogroupsmentioning

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

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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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“…Moreover, the topologies on some kinds of weaker algebra structures than groups are posed and investigated, such as (strongly) topological gyrogroups and rectifiable spaces. Hence it is natural to consider extending some well known results of topological groups to these weaker structures, see [5,6,7,8,9,23,24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Some properties of Pre-topological groups

Lin¹,

Wu²,

Xie³

et al. 2022

Preprint

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In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each T 0 pre-topological group is regular and every almost topological group is completely regular which extends A.A. Markov's theorem to the class of almost topological groups. Moreover, it is shown that an almost topological group is τ -narrow if and only if it can be embedded as a subgroup of a pre-topological product of almost topological groups of weight less than or equal to τ . Finally, the cardinal invariant, the precompactness and the resolvability are investigated in the class of pre-topological groups.

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Separability in (strongly) topological gyrogroups

Cited by 7 publications

References 17 publications

The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

A supplement on feathered gyrogroups

Some properties of Pre-topological groups

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