Lipschitz-free spaces over properly metrizable spaces and approximation properties

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