On regular operators extending (pseudo)metrics

TL;DR

This paper constructs regular extension operators for (pseudo)metrics from closed subsets to stratifiable spaces, ensuring continuity and preservation of metric classes, with special cases for metrizable spaces and equivariant versions.

Contribution

It introduces a new regular extension operator for (pseudo)metrics that works for stratifiable and metrizable spaces, preserving metric properties and continuity.

Findings

Existence of regular extension operators for (pseudo)metrics

Operators are continuous in pointwise and compact-open topologies

Preservation of admissible metrics in metrizable spaces

Abstract

It is proved that for every stratifiable space $Y$ and a closed subset $X\subset Y$ there exists a regular (i.e. linear positive with unit norm) extension operator $T:C(X\times X)\to C(Y\times Y)$ preserving the class of (pseudo)metrics. This operator is continuous with respect to the pointwise as well as to the compact-open topologies on the linear lattices of continuous functions $C(X\t X)$ and $C(Y\t Y)$. If moreover the space Y is metrizable then the operator $T$ preserves the class of admissible metrics. The equivariant analog of the above statement is proved as well.