Topological Gyrogroups And Its P-space Properties

A gyrogroup,is a relaxation of a group which the associativity condition has been replaced by a weaker one.This concept is originated from the study of c-ball of relativistically admissible velocities with Einstein velocity addition as mentioned by A.A.Ungar.In 2017,the topologist W.Atiponrat defined topological gyrogroups such that the binary operation is jointly continuous and the inverse operation is also continuous.Then,a lot of scholars began to research topological gyrogroups.In Chapter 1,we introduce the research background and the research status at home and abroad about gyrogroups.At the same time,we introduce some notions and definitions of gyrogroups.In Chapter 2,we research some basic properties of topological gyrogroups.What's more,we show that every feathered strongly topological gyrogroup is paracompact and every feathered strongly topological gyrogroup is a D-space.Then,we give some examples of topological gyrogroups and strongly topological gyrogroups,which show the existence of(strongly)topological gyrogroups.In Chapter 3,we study NS S-gyrogroups and P-gyrogroups and show that every locally compact NS S-gyrogroup is first-countable and every Lindel?f P-gyrogroup is Ra?kov complete.