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Register a new userQuotient with respect to admissible $L$-subgyrogroups
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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.
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In this paper, some generalized metric properties in strongly topological gyrogroups are studied.
Our main problem is to find finite topological spaces to within homeomorphism, given (also to within homeomorphism) the quotient-spaces obtained by identifying one point of the space with each one of the other points. In a previous version of this paper, our aim was to reconstruct a topological space from its quotient-spaces; but a reconstruction is not always possible either in the sense that several non-homeomorphic topological spaces yield the same quotient-spaces, or in the sense that no topological space yields an arbitrarily given family of quotient-spaces. In this version of the paper we present an algorithm that detects, and deals with, all these situations.
The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Frechet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all $sigma$-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all $sigma$-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.
We answer several questions of I.Protasov and E.Zelenyuk concerning topologies on groups determined by T-sequences. A special attention is paid to studying the operation of supremum of two group topologies.
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).
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