  Combining ideas of Troallic and Cascales, Namioka, and Vera, we prove several
characterizations of \textit{almost equicontinuity} and \textit{hereditary
almost equicontinuity} for subsets of metric-valued continuous functions when
they are defined on a \v{C}ech-complete space. We also obtain some applications
of these results to topological groups and dynamical systems.
