Simultaneous extensions of metrics and ultrametrics of high power

TL;DR

This paper extends the theory of generalized metrics and ultrametrics valued in linearly ordered Abelian groups, providing conditions for uniform retracts and characterizing compactness via completeness.

Contribution

It introduces simultaneous extensions of generalized metrics and ultrametrics and links their properties to topological compactness and completeness.

Findings

Closed subsets are uniform retracts under certain conditions.

Constructs simultaneous extensions of generalized metrics and ultrametrics.

Characterizes final compactness through completeness of generalized metrics.

Abstract

In this paper, generalized metrics mean metrics taking values in general linearly ordered Abelian groups. Using the Hahn fields, we first prove that for every generalized metric space, if the set of the Archimedean equivalence classes of the range group of the metric has an infinite decreasing sequence, then every non-empty closed subset of the space is a uniform retract of the ambient space. Next we construct simultaneous extensions of generalized metrics and ultrametrics. From the existence of extensors of generalized metrics, we characterize the final compactness of generalized metrizable spaces using the completeness of generalized metrics.