Lipschitz-Free Spaces over Cantor Sets and Approximation Properties

Let K = 2(N) be the Cantor set, let M be the set of all metrics d on K that give its usual (product) topology, and equip M with the topology of uniform convergence, where the metrics are regarded as functions on K-2. We prove that the set of metrics d is an element of M for which the Lipschitz-free space F(K, d) has the metric approximation property is a residual F-sigma delta set in M, and that the set of metrics d is an element of M for which F(K, d) fails the approximation property is a dense meager set in M. This answers a question posed by G. Godefroy.