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 $M^T$ be the set of all metrics $d$ on $T$ compatible with its topology, and equip $M^T$ with the topology of uniform convergence, where the metrics are regarded as functions on $T^2$. We prove that the set $A^{T,1}$ of metrics $d\in M^T$ for which the Lipschitz-free space $F(T,d)$ has the metric approximation property is residual in $M^T$.