Lipschitz-free spaces over properly metrizable spaces and approximation properties (Q6621292)

Let \(T\) be a locally compact, separable and metrizable topological space (equivalently, \(T\) admits a compatible proper metric). By \(\mathcal{M}^T\) we denote the set of all compatible proper metrics on \(T\) endowed with the topology of uniform convergence on \(T^2\). The main results of the authors are that supposing \(T\) is uncountable, the following sets are dense in \(\mathcal{M}^T\):\N\N\begin{itemize}\N\item[1.] the set of metrics \(d\in\mathcal{M}^T\) for which the Lipschitz-free space \(\mathcal{F}(T,d)\) has metric approximation property (MAP),\N\item[2.] the set of metrics \(d\in\mathcal{M}^T\) for which the Lipschitz-free space \(\mathcal{F}(T,d)\) does not have approximation property (AP).\N\end{itemize}\N\NMoreover, if \(T\) is zero-dimensional then in 1. one obtains even a residual set. The results contained in this paper generalize earlier results of the second author, where he considered only the space \(T\) being the Cantor set, see [\textit{F. Talimdjioski}, Mediterr. J. Math. 20, No. 6, Paper No. 302, 16 p. (2023; Zbl 1539.46012)].