Finitely approximable groups and actions Part I: The Ribes--Zalesski\u\i{} property

We investigate extensions of S. Solecki’s theorem on closing off finite partial isometries of metric spaces \cite{solecki1} and obtain the following exact equivalence: any action of a discrete group $\mathrm{\Gamma }$ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of $\mathrm{\Gamma }$ is closed in the profinite topology on $\mathrm{\Gamma }$.