Lipschitz-free spaces over strongly countable-dimensional spaces and approximation properties

TL;DR

This paper proves that for a broad class of compact, metrizable, strongly countable-dimensional spaces, the set of compatible metrics making their Lipschitz-free spaces have the metric approximation property is large in the space of all such metrics.

Contribution

It establishes that the subset of metrics inducing Lipschitz-free spaces with the metric approximation property is residual within the space of all compatible metrics.

Findings

Residual set of metrics with the property

Lipschitz-free spaces have the metric approximation property

Applicable to strongly countable-dimensional spaces

Abstract

Let $T$ be a compact, metrisable and strongly countable-dimensional topological space. Let $\mathcal{M}^T$ be the set of all 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 residual in $\mathcal{M}^T$.