 any two distances in the family $\{ d_{\gamma} \} _{\gamma \in \Gamma}$ are bilipschitz equivalent with each other.  Compactness of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma$ implies that $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ is compact 