Recently, Kimber has discovered a general class of topological spaces, the members of which are termed barypact spaces, that includes the compact topological spaces. This class is distinct form the set of all compact topological spaces, but its members possess many of the useful properties associates with compactness. As a consequence, several standard compactness theorems become special cases of corresponding theorems in a more general setting and the techniques of proof applied to these extensions provide new, and sometimes remarkably simple, proofs of the very theorems they generalize. The purpose of this paper is to extend to this class three compactness theorems of topology: the Stone-Weierstrauss theorem, the Ascoli theorem, and the Dini theorem. It is assumed throughout this paper that the reader is familiar with the standard set theoretic notation and with such concepts as topological space, compact topological space, metric space, continuity, convergence, uniform convergence, and so on. Sometimes theorems that are used in support of this paper, but are not directly part of it, will be stated without proof; however, sources for such material are included in the bibliography.