On certain equivalences of metric spaces

Abstract:

The normed spaces of molecules constructed by Arens and Eells allow us to define two natural equivalence relations on the class of complete metric spaces. We say that two complete metric spaces $M$ and $N$ are $\mathcal {M}\hspace {-1.5pt}\mathcal {ol}$-equivalent if their normed spaces of molecules are isomorphic and we say that they are $\mathcal {F}$-equivalent if the corresponding completions – the Lipschitz-free Banach spaces $\mathcal {F}(M)$ and $\mathcal {F}(N)$ – are isomorphic.

In this note, we compare these and some other relevant equivalences of metric spaces. Clearly, $\mathcal {M}\hspace {-1.5pt}\mathcal {ol}$-equivalent spaces are $\mathcal {F}$-equivalent. Our main result states that $\mathcal {M}\hspace {-1.5pt}\mathcal {ol}$-equivalent spaces must have the same covering dimension. In combination with the work of Godard, this implies that $\mathcal {M}\hspace {-1.5pt}\mathcal {ol}$-equivalence is indeed strictly stronger than $\mathcal {F}$-equivalence. However, $\mathcal {M}\hspace {-1.5pt}\mathcal {ol}$-equivalent spaces need not be homeomorphic, as we demonstrate through a general construction. We also observe that $\mathcal {M}\hspace {-1.5pt}\mathcal {ol}$-equivalence does not preserve the Assouad dimension.

We introduce a natural notion of a free basis to simplify the notation.