Lipschitz-free spaces over properly metrisable spaces and approximation properties

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 $\mathcal{M}^T$.