关于等价度量的若干等价判定及应用

在同一度量空间中可诱导出相容的两个不同度量会给人们处理问题带来方便,因此,度量的等价性是一个值得讨论的课题。首先,介绍了度量空间中的等价度量、度量拓扑、等价拓扑基及同胚映射等基本概念。然后,在此基础上讨论了等价度量的基本性质及各种概念间的内在联系,并由此给出了关于等价度量的判定性定理及其重要推论,给出了等价度量应用的简单实例。最后,通过构造的方式,给出了一种重要的等价度量形式。

Some equivalence determination and application of equivalent metric

Two different metrics which could be induced in the same metric space will give people much more convenience to deal with some problems,and so the measurement equivalence is a worthwhile topic to be discussed.Firstly,we introduce some concepts in metric space such as equivalent metric,metric topology,equivalent topology-base and homeomorphism.Based on above conceptions,we discuss the basic nature of the equivalent metric and the relationship between various concepts,and then we obtain the decision on the equivalence theorem and its inference,and give relevant simple application.Finally,we construct a new important equivalent metric.