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Register a new userFeathered gyrogroups and gyrogroups with countable pseudocharacter
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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 skivi ~(2010) in cite{AA1}. Moreover, we prove that every locally compact $NSS$-gyrogroup is first-countable. Finally, we prove that each Lindel{o}f $P$-gyrogroup is Ra$check{imath}$kov complete.
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The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. Recently, Atiponrat extended the idea of topological (paratopological) groups to topological (paratopological) gyrogroups. In this paper, we prove that every regular (Hausdorff) locally gyroscopic invariant paratopological gyrogroup $G$ is completely regular (function Hausdorff). These results improve theorems of Banakh and Ravsky for paratopological groups. Also, we extend the Pontrjagin conditions of (para)topological groups to (para)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 $Scup {0}$ 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 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 $omega^{omega}$-base, then $G$ has the strong Pytkeev property. Moreover, some equivalent conditions about $omega^{omega}$-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 $P$ such that the quotient space $G/P$ is metrizable and the canonical homomorphism $pi :Grightarrow G/P$ is closed.
The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. In this paper, the notion of fuzzy gyronorms on gyrogroups is introduced. The relations of fuzzy metrics (in the sense of George and Veeramani), fuzzy gyronorms and gyronorms on gyrogroups are studied. Also, the fuzzy metric structures on fuzzy normed gyrogroups are discussed. In the last, the fuzzy metric completion of a gyrogroup with an invariant metric are studied. We mainly show that let $d$ be an invariant metric on a gyrogroup $G$ and $(widehat{G},widehat{d})$ is the metric completion of the metric space $(G,d)$; then for any continuous $t$-norm $ast$, the standard fuzzy metric space $(widehat{G},M_{widehat{d}},ast)$ of $(widehat{G},widehat{d})$ is the (up to isometry) unique fuzzy metric completion of the standard fuzzy metric space $(G,M_d,ast)$ of $(G,d)$; furthermore, $(widehat{G},M_{widehat{d}},ast)$ is a fuzzy metric gyrogroup containing $(G,M_d,ast)$ as a dense fuzzy metric subgyrogroup and $M_{widehat{d}}$ is invariant on $widehat{G}$. Applying this result, we obtain that every gyrogroup $G$ with an invariant metric $d$ admits an (up to isometric) unique complete metric space $(widehat{G},widehat{d})$ of $(G,d)$ such that $widehat{G}$ with the topology introduced by $widehat{d}$ is a topology gyrogroup containing $G$ as a dense subgyrogroup and $widehat{d}$ is invariant on $widehat{G}$.
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 $G$.
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