A class of quotient spaces in strongly topological gyrogroups

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-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-base. Finally, it is shown that if H is neutral, πχ(G/H) = χ(G/H) and πω(G/H) = ω(G/H).

[1]  M. Ferreira,et al.  Möbius gyrogroups: A Clifford algebra approach , 2011 .

[2]  Fucai Lin,et al.  Quotient with respect to admissible L-subgyrogroups , 2020, 2003.08843.

[3]  P. Ye,et al.  A note on paratopological groups with an ω-base , 2020 .

[4]  S. Gabriyelyan,et al.  On topological groups with a small base and metrizability , 2015 .

[5]  T. Suksumran,et al.  ISOMORPHISM THEOREMS FOR GYROGROUPS AND L-SUBGYROGROUPS , 2014, 1406.0300.

[6]  Xiaoquan Xu,et al.  Strongly Topological Gyrogroups and Quotient With Respect to L-subgyrogroups , 2021, 2102.05860.

[7]  T. Banakh.  Topological spaces with an ! ω -base , 2016 .

[8]  A. H. Tomita,et al.  On topological groups admitting a base at identity indexed with $\omega^\omega$ , 2015, 1511.07062.

[9]  Fucai Lin,et al.  A note on rectifiable spaces , 2011, 1106.3811.

[10]  A. Ungar.  Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity , 2008 .

[11]  Fucai Lin.  Compactly generated rectifiable spaces or paratopological groups , 2013 .

[12]  Fucai Lin,et al.  Submetrizability of strongly topological gyrogroups , 2020, 2003.06132.

[13]  J. Kakol,et al.  ON TOPOLOGICAL SPACES AND TOPOLOGICAL GROUPS WITH CERTAIN LOCAL COUNTABLE NETWORKS , 2014, 1412.1497.

[14]  W. Atiponrat.  Topological gyrogroups: Generalization of topological groups , 2017 .

[15]  Wei He,et al.  A Note on Paratopological Loops , 2019 .

[16]  A. Ungar,et al.  Analytic Hyperbolic Geometry: Mathematical Foundations And Applications , 2005 .

[17]  Milton Ferreira,et al.  Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups , 2020, Symmetry.

[18]  Milton Ferreira,et al.  Factorizations of Möbius Gyrogroups , 2009 .

[19]  Ziqiu Yun,et al.  Generalized Metric Spaces and Mappings , 2016 .

[20]  S. Franklin,et al.  Spaces in which sequences suffice , 1965 .

[21]  A. Ungar.  Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces , 2001 .

[22]  A. Arhangel'skii,et al.  Remainders of rectifiable spaces , 2010 .

[23]  Meng Bao,et al.  Submaximal properties in (strongly) topological gyrogroups , 2020, Filomat.

[24]  Fucai Lin,et al.  Feathered gyrogroups and gyrogroups with countable pseudocharacter , 2019, Filomat.