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

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 ${\mathcal{M}}^T$.