Equivalent metrics and compactifications

TL;DR

This paper introduces a method to generate equivalent metrics on a space using a symmetric function, leading to compactifications when the new metric is totally bounded, with explicit examples on Euclidean spaces.

Contribution

It defines a new class of equivalent metrics based on a symmetric function and demonstrates their use in constructing compactifications of Euclidean spaces.

Findings

The new metric d^{}m is equivalent to the original metric d.

Totally boundedness of d^{}m implies its completion yields a compactification.

Explicit compactifications of Euclidean spaces are constructed as examples.

Abstract

Let (X,d) be a metric space and m\in X. Suppose that \phi:X\times X\to\mathbold{R} is a nonnegative symmetric function. We define a metric d^{\phi,m} on X which is equivalent to d. If d^{\phi,m} is totally bounded, its completion is a compactification of (X,d). As examples, we construct two compactifications of (\mathhbold{R}^s,d_E), where d_E is the Euclidean metric and s\geq 2.