Meng Bao, Fucai Lin
A space $X$ is submaximal if any dense subset of $X$ is open. In this paper, we prove that every submaximal topological gyrogroup of non-measurable cardinality is strongly $σ$-discrete. Moreover, we prove that every submaximal strongly topological gyrogroup of non-measurable cardinality is hereditarily paracompact.
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Topological gyrogroups, with a weaker algebraic structure without associative law, have been investigated recently. We prove that each $T_{0}$-strongly topological gyrogroup is completely regular. We also prove that every $T_{0}$-strongly topological gyrogroup with a countable pseudocharacter is submetrizable. Finally, we prove that the left coset space $G/H$ is submetrizable if $H$ is an admissible $L$-subgyrogroup of a $T_{0}$-strongly topological gyrogroup $G$.
The separability embedding of $σ$-compact strongly topological gyrogroups
本文通过陀螺半直积群证明每个右$\omega$-窄强拓扑陀螺群是右$\omega$-平衡的，研究$\sigma$-紧强拓扑陀螺群并得出其可距离化，进而讨论分离性嵌入，给出三个等价条件。
A discrete subset $S$ of a topological gyrogroup $G$ with the identity $0$ is said to be a {\it suitable set} for $G$ if it generates a dense subgyrogroup of $G$ and $S\cup \{0\}$ is closed in $G$. In this paper, it was proved that each countable Hausdorff topological gyrogroup has a suitable set; moreover, it is shown that each separable metrizable strongly topological gyrogroup has a suitable set.
Topological gyrogroups, with a weaker algebraic structure than groups, have been investigated recently. In this paper, we prove that every feathered strongly topological gyrogroup is paracompact, which implies that every feathered strongly topological gyrogroup is a $D$-space and gives partial answers to two questions posed by A.V.Arhangel' ski\vı~(2010) in \cite{AA1}. Moreover, we prove that every locally compact $NSS$-gyrogroup is first-countable. Finally, we prove that each Lindelöf $P$-gyrogroup is Ra$\check{\imath}$kov complete.
The existence of suitable sets in locally compact strongly topological gyrogroups
本文证明了每个局部紧强拓扑gyro群都存在一个合适集，解决了Lin等人提出的问题。
Strongly topologically orderable gyrogroups with a suitable set
本文研究强拓扑可序陀螺群中合适集的存在性，推广已有结果，并肯定回答了局部紧或非全不连通情形下合适集的存在性问题。
On the continuity of the inverse in (strongly) paratopological gyrogroups
本文研究（强）仿拓扑陀螺群中逆运算的连续性，通过紧性、局部紧性、弱紧性及拓扑周期性等条件，给出了若干使仿拓扑陀螺群成为（强）拓扑陀螺群的判定定理。
Characterizations of quotient spaces for Lindelöf strongly topological gyrogroups
研究林德洛夫强拓扑回旋群商空间的特征，证明了若\(G\in\mathscr{L}\)是强拓扑回旋群则\(G\)值域可度量化，还得出\(G/H\)中紧致\(G_\delta\)集是杜根吉的等结论，将拓扑群经典结果扩展到强拓扑回旋群。
Separability is one of the most basic and important topological properties. In this paper, the separability in (strongly) topological gyrogroups is studied. It is proved that every first-countable left ω-narrow strongly topological gyrogroup is separable. Furthermore, it is shown that if a feathered strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable. Therefore, if a metrizable strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable, and if…
In this paper, some generalized metric properties in strongly topological gyrogroups are studied.
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. In this paper, it is proved that if $G$ is a sequential topological gyrogroup with an $ω^ω$-base, then $G$ has the strong Pytkeev property. Moreover, some equivalent conditions about $ω^ω$-base and strong Pytkeev property are given in Baire topological gyrogroups. Finally, it is shown that if $G$ is a strongly countably complete strongly topological gyrogroup, then $G$ contains a closed, countably compact, admissible subgyrogroup $P$ su…
The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background of $c$-ball of relativistically admissible velocities with Einstein velocity addition. A topological gyrogroup is just a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the inverse is continuous. This concept is a good generalization of a topological group. In this paper, we are going to establish that for a locally compact admissible $L$-subgyrogroup $H$ of a strongly topological gyrogroup $G$, the natural quotient mappin…
A topological gyrogroup is a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the inverse is continuous. In this paper, we study the quotient gyrogroups in topological gyrogroups with respect to strongly $L$-subgyrogroups, and prove that let $(G, τ,\oplus)$ be a topological gyrogroup and $H$ a closed strongly $L$-subgyrogroup of $G$, then the natural homomorphism $π$ from a topological gyrogroup $G$ to its quotient topology on $G/H$ is an open and continuous mapping, and $G/H$ is a homogeneous $T_1$-space. We also establish that for a locally …
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 $ω^ω$-base 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 al…
The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. Recently, Wattanapan et al consider the construction of Hartman-Mycielski in strongly topological gyrogroups. In this paper, we extend their results in topological gyrogroups. We mainly, among other results, prove that every Hausdorff topological gyrogroup $G$ can be embedded as a closed subgyrogroup of a Hausdorff path-connected and locally path-connected topological gyrogroup $G^\bullet$.
The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an ωω-base and G is Frechet-Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Frechet-Urysohn and having an ωω-base are all weakly three-space properties with H a closed L-subgyrogroup
Quotient space is a class of the most important topological spaces in the research of topology. In this paper, we show that if G 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 Frechet-Urysohn with an ωω-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 Frechet-Urysohn with an ωω-base. Finally, it is shown that if H is neutral, π\c{hi}(G/H) =…
Subgyrogroups within the product spaces of paratopological gyrogroups
本文刻画了可嵌入第一可数亚陀拓扑回旋群乘积的子回旋群，给出强亚陀拓扑回旋群成为此类子回旋群的充要条件，涉及平衡性与Hausdorff数可数性。
Weak countability axioms on the quotient spaces of topological gyrogroups
研究强拓扑 gyrogroup 商空间的弱可数性公理，在闭强子 gyrogroup 且中性的条件下刻画其双径向性、嵌套性与可度量化关系。
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