Separability in (strongly) topological gyrogroups

The construction of Hartman-Mycielski in topological gyrogroups

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 $G^\bullet$.

Submetrizability of strongly topological gyrogroups

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 $G$.

Topological gyrogroups with Frechet-Urysohn property and omega^{omega}-base

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

Submaximal properties in (strongly) topological gyrogroups

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.

Suitable sets for strongly topological gyrogroups

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.

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 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) =…

亚陀拓扑回旋群乘积空间中的子回旋群

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。

Feathered gyrogroups and gyrogroups with countable pseudocharacter

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.

局部紧强拓扑gyrogroup中合适集的存在性

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。

具有合适集的强拓扑可序陀螺群

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。

林德洛夫强拓扑回旋群商空间的特征

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。

拓扑 gyrogroup 商空间上的弱可数性公理

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。

Quotient with respect to admissible $L$-subgyrogroups

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…

Almost separable spaces

We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.