  It is well-known that for topological dynamical systems $\epsilon$-stable
sets have a deep connection with the topological entropy of the systems. In the
present paper, we investigate the relationships of upper metric mean dimension
with potential, Bowen upper metric mean dimension with potential, packing upper
metric mean dimension with potential between the blocks of $\epsilon$-stable
sets, $\epsilon$-stable sets, the dispersion of preimages of $\epsilon$-stable
sets and the whole phase space.
  As applications, we reveal that three types different of entropies, tail
entropy, preimage neighborhood entropy and topological entropy in topological
dynamical systems, have the same metric mean dimension; We also establish some
new double variational principles for mean dimension and upper metric mean
dimension in terms of $\epsilon$-stable sets.
