Nonstandard Vector Space with a Metric and Its Topological Structure

TL;DR

This paper introduces a nonstandard vector space lacking additive inverses, explores a compatible metric, and studies various topologies generated by different open and closed set definitions, revealing deviations from classical properties.

Contribution

It presents a novel nonstandard vector space framework with a metric, analyzing its topological structures and properties, which differ from traditional vector spaces.

Findings

Four types of open and closed sets are proposed.

Topologies generated by these sets are systematically studied.

Conventional properties of open and closed balls do not hold in this setting.

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.