On an equivalence of the topology of algebraic cone metric spaces and metric spaces

TL;DR

This paper demonstrates that algebraic cone metric spaces are metrizable by showing their topology coincides with that of a scalarized metric, and introduces algebraic cone normed spaces with their topologies proven to be normable.

Contribution

It establishes the equivalence of topologies in algebraic cone metric spaces and introduces algebraic cone normed spaces with proven normability.

Findings

Algebraic cone metric spaces are metrizable.

Topologies of algebraic cone normed spaces are normable.

Provides a scalarization method linking cone metrics to standard metrics.

Abstract

In this paper, we prove that the topology induced by algebraic cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e. any algebraic cone metric space is metrizable. Furthermore, the notion of algebraic cone normed space is introduced and also normability of the topology of this space is proved.