Separation axioms and covering dimension of asymmetric normed spaces

TL;DR

This paper investigates separation axioms in asymmetric normed spaces, examines their equivalences, and characterizes the topological dimension of finite-dimensional cases, contributing to the understanding of their structure.

Contribution

It provides new insights into the equivalence of separation axioms and characterizes the covering dimension of finite-dimensional asymmetric normed spaces.

Findings

Certain separation axioms are equivalent in asymmetric normed spaces

Finite-dimensional asymmetric normed spaces have a well-defined topological dimension

A known theorem links compactness of the unit ball to finite-dimensionality

Abstract

In this paper, we approach the question if some of the separation axioms are equivalent in the class of asymmetric normed spaces. In particular, we make a remark on a known theorem which states that every $T_1$ asymmetric normed space with compact closed unit ball must be finite-dimensional. We also explore the product structure of these spaces and characterize the topological (covering) dimension of all finite-dimensional asymmetric normed spaces.