Equicontinuity criteria for metric-valued sets of continuous functions

Combining ideas of Troallic [1] and Cascales, Namioka, and Vera [2], we prove several characterizations of almost equicontinuity and hereditarily almost equicontinuity for subsets of metric-valued continuous functions when they are defined on a Čech-complete space. We also obtain some applications of these results to topological groups and dynamical systems.