On the weak-fragmentability index of some Lipschitz-free spaces
Abstract
We show the existence of Lipschitz-free spaces satisfying the Point of Continuity Property with arbitrarily high weak-fragmentability index. For this purpose, we use a generalized construction of countably branching diamond graphs. A direct corollary is the existence of an uncountable family of pairwise non-isomorphic Lipschitz-free spaces over purely $1$-unrectifiable metric spaces, but more importantly, the existence of a Lipschitz-free space over a separable metric space that is not isomorphic to any Lipschitz-free space over a compact. Another consequence is that to be Lipschitz-universal for countable complete metric spaces, a separable complete metric space cannot be purely $1$-unrectifiable. Some results on compact reduction are also obtained.
