Metastable convergence and logical compactness

The concept of metastable convergence was identified by Tao;it allows converting theorems about convergence into stronger theorems about uniform convergence. The Uniform Metastability Principle (UMP) states that if $T$ is a theorem about convergence, then the fact that $T$ is valid implies automatically that its (stronger) uniform version is valid, provided that $T$ can be stated in certain logical frameworks. In this paper we identify precisely the logical frameworks $\sL$ for which UMP holds. More precisely, we prove that the UMP holds for $\sL$ if and only if $\sL$ is a compact logic. We also prove a topological version of this equivalence. We conclude by proving new characterizations of logical compactness that yield additional information about the UMP.