  We introduce a notion of tame ramification for general finite covers. When
specialized to the separable case, it extends to higher dimensions the
classical notion of tame ramification for Dedekind domains and curves and sits
nicely in between other notions of tame ramification in arithmetic geometry.
However, when applied to the Frobenius map, it naturally yields the notion of
center of $F$-purity (aka compatibly $F$-split subvariety). As an application,
we describe the behavior of centers of $F$-purity under finite covers -- it all
comes down to a transitivity property for tame ramification in towers.
