Research On The Related Properties In ★-metric Spaces

Metric spaces play an important role in topology.So metric spaces are generalized by many scholars.Khatami and Mirzavaziri generalize the metric to ★-metric and study its properties.★-metric spaces are important extension of metric spaces,so this paper studies the total boundedness and completeness of ★-metric space.By combining ★-metric space with algebraic structure,we consider whether(left)invariant -quasi pseudo metric has a stronger structure(such as(quasi)topological group)on algebraic topological structure(such as left topological group or semitopological group).This thesis is divided into four chapters:In the first chapter,the historical background,current situation,and the purpose and significance of ★-metric spaces are introduced.In the second chapter,(quasi-)pseudometrics are natural generalization of metrics.metrics are generalized to-(pseudo-)pseudometrics.The topological structure and properties of-quasi-pseudometric spaces are analyzed.Next,we use the metrization theorem of uniform spaces to prove that the ★-metric spaces are metrizable.In the third chapter,the classical theory of metric spaces are extended to ★-metric spaces.The subspace heritability and product preservation of totally boundedness and completeness in★-metric spaces are studied.We obtain that:(1)let(X,d★)be a ★-metric space.If X with the topology induced by d★is countably compact,then(X,d★)is a totally bounded ★-metric space(see Theorem 3.1.2);(2)Let(X,d★)be a totally bounded ★-metric space.Then for every subset M of X the ★-metric space(M,d)is totally bounded(see Theorems 3.1.3);(3)the Cartesian product and disjoint union of finite totally bounded(complete) ★-metric spaces are totally bounded(complete)under specific ★-metrics(see Theorems 3.1.4 and 3.1.5);(4)a★-metric space(X,d★)is compact if and only if(X,d★)is complete and totally bounded(see Theorem 3.2.1);(5)a ★-metric space is complete if and only if for every decreasing sequence F1 ?F2 ?F3 ?...of non-empty closed subsets of space X,such that limn→∞δ(Fn)=0,the intersection ∩n=1∞ Fn is a one-point set(see Theorem 3.2.2).(6)the Cartesian product and disjoint union of finite totally complete ★-metric spaces are complete under specific ★-metrics(see Theorems 3.2.5 and 3.2.6).In the fourth chapter,the combination of topological structure and algebraic structure has attracted many people’s attention.In this paper,we will study the application of ★-metric space to algebraic structures.First,the concepts of-quasi-pseudometric semigroups,paratopological groups and groups are proposed and the conditions for a-quasi-pseudometric semitopological group to become a paratopological group or a topological group are found.Then complete invariant ★-metrics on groups and semigroups are considered.For the complete ★-metric space(X,d★),the set G is a dense subset of X which is a(semi)group.Then the following conclusions can be drawn:if the restriction of d★on G is invariant,then X can become a(semi)group containing G as a sub(semi)group and d★is invariant on X.At last,the relation between the completeness of ★-metric group(G,d★)and the Raǐkov completeness of(G,Jd★),where Jd★ is a topology induced by d★.