Lipschitz-free spaces over properly metrisable spaces and approximation properties

TL;DR

This paper investigates the approximation properties of Lipschitz-free spaces over properly metrisable spaces, showing that the set of metrics with the metric approximation property is dense and residual, while those failing it are dense for uncountable spaces.

Contribution

It establishes the density and residuality of metrics inducing Lipschitz-free spaces with the metric approximation property over properly metrisable spaces.

Findings

Metrics with the metric approximation property form a dense set in the space of compatible proper metrics.

For zero-dimensional spaces, these metrics form a residual set.

In uncountable spaces, metrics for which the Lipschitz-free space fails the approximation property are dense.

Abstract

Let $T$ be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrisable space. Let $\mathcal{M}^T$ be the non-empty set of all proper metrics $d$ on $T$ compatible with its topology, and equip $\mathcal{M}^T$ with the topology of uniform convergence, where the metrics are regarded as functions on $T^2$. We prove that the set $\mathcal{A}^{T,1}$ of metrics $d\in\mathcal{M}^T$ for which the Lipschitz-free space $\mathcal{F}(T,d)$ has the metric approximation property is a dense set in $\mathcal{M}^T$, and is furthermore residual in $\mathcal{M}^T$ when $T$ is zero-dimensional. We also prove that if $T$ is uncountable then the set $\mathcal{A}^T_f$ of metrics $d\in\mathcal{M}^T$ for which $\mathcal{F}(T,d)$ fails the approximation property is dense in $\mathcal{M}^T$. Combining the last statement with a result of Dalet, we conclude that for any `properly metrisable' space $T$, $\mathcal{A}^T_f$ is either empty or dense in $\mathcal{M}^T$.