  We call a closed subset M of a Banach space X a free basis of X if it
contains the null vector and every Lipschitz map from M to a Banach space Y,
which preserves the null vectors can be uniquely extended to a bounded linear
map from X to Y. We then say that two complete metric spaces M and N are
Mol-equivalent if they admit bi-Lipschitz copies M' and N', respectively that
are free bases of a common Banach space satisfying span M'=span N'.
  In this note, we compare Mol-equivalence with some other natural equivalences
on the class of complete metric spaces. The main result states that
Mol-equivalent spaces must have the same \v{C}ech-Lebesgue covering dimension.
In combination with the work of Godard, this implies that two complete metric
spaces with isomorphic Lipschitz-free spaces need not be Mol-equivalent. Also,
there exist non-homeomorphic Mol-equivalent metric spaces, and, in contrast
with the covering dimension, the metric Assouad dimension is not preserved by
Mol-equivalence.
