Nonstandard Vector Space with a Metric and Its Topological Structure

Abstract

In this paper, we introduce the nonstandard vector space in which the concept of additive inverse element will not be taken into account. We also consider a metric defined on this nonstandard vector space. Under these settings, the conventional intuitive properties for the open and closed balls will not hold true. Therefore, four kinds of open and closed sets are proposed. Furthermore, the topologies generated by these different concepts of open and closed sets are investigated.