Applicability of Pelczyński’s decomposition method to produce M ol-equivalent spaces

Ascertain whether Pelczyński’s decomposition method, and related techniques used to construct non-homeomorphic pairs of complete metric spaces with isomorphic Lipschitz-free spaces, can be employed to construct pairs of complete metric spaces that are M ol-equivalent—namely, that admit bi-Lipschitz copies serving as free bases of a common Banach space with equal linear spans.

Background

The authors survey known examples of non-homeomorphic but F-equivalent spaces, many of which rely on the Pelczyński decomposition method or similar techniques. They then develop constructions yielding M ol-equivalent spaces and note that M ol-equivalence has stronger structural requirements than mere F-equivalence.

They explicitly state uncertainty over whether Pelczyński-type methods can yield examples satisfying the stricter M ol-equivalence criterion, motivating investigation into whether those techniques can be adapted to this stronger equivalence.