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Register a new userStrongly Topological Gyrogroups and Quotient With Respect to L-subgyrogroups
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In this paper, some generalized metric properties in strongly topological gyrogroups are studied.
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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 mapping $pi$ from $G$ onto the quotient space $G/H$ has some nice local properties, such as, local compactness, local pseudocompactness, local paracompactness, etc. Finally, we prove that each locally paracompact strongly topological gyrogroup is paracompact.
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 {omega}{omega}-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 {omega}{omega}-base. Finally, it is shown that if H is neutral, {pi}c{hi}(G/H) = c{hi}(G/H) and {pi}{omega}(G/H) = {omega}(G/H).
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 $G$.
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 {omega}-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 a locally compact strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable.
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 $sigma$-discrete. Moreover, we prove that every submaximal strongly topological gyrogroup of non-measurable cardinality is hereditarily paracompact.
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