We introduce the concept of virtually stable selfmaps of Hausdorff spaces, which generalizes virtually nonexpansive selfmaps of metric spaces introduced in the previous work by the first author, and explore various properties of their convergence sets and fixed point sets. We also prove that the fixed point set of a virtually stable selfmap satisfying a certain kind of homogeneity is always star-convex.
A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup G, we offer a new way to construct a gyrogroup G• such that G• contains a gyro-isomorphic copy of G. We then prove that every strongly topological gyrogroup G can be embedded as a closed subgyrogroup of the path-connected and locally path-connected topological gyrogroup G•. We also study several properties shared by G and G•, including gyrocommutativity, first countability and metrizability. As an application of these results, we prove that being a quasitopological gyrogroup is equivalent to being a strongly topological gyrogroup in the class of normed gyrogroups.
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