Many interesting results in topology and functional analysis are closely related to situations in which two (in general) distinct topologies or uniformities coincide. In this talk, we consider a number of pairs of infinitesimal relations and examine the consequences of the condition that they coincide on certain subsets of the nonstandard extension of the underlying space. One example leads to a new characterization of uniform spaces with invariant nonstandard hulls. Other applications include external characterizations of strong and weak compactness for subsets of Banach spaces and their duals.