An isometric extensor of metrics

TL;DR

This paper introduces an isometric extension operator for metrics on metrizable spaces, preserving the supremum metric and extending metrics from closed subsets to the entire space.

Contribution

It constructs a metric extension map that is isometric with respect to the supremum metric, providing a new tool for metrization and extension problems.

Findings

The extension map preserves the supremum metric.

The extension is isometric and continuous.

Applicable to metrizable spaces with closed subsets.

Abstract

In this paper, for a metrizable space $Z$, we consider the space of metrics that generate the same topology of $Z$, and that space of metrics is equipped with the supremum metrics. For a metrizable space $X$ and a closed subset $A$ of it, we construct a map $E$ from the space of metrics on $A$ into the space of metrics on $X$ such that $E$ is an extension of metrics and preserves the supremum metrics between metrics.