The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “ p -regular” and “ p -topological.” Since earlier papers have investigated regular, p -regular, and topological Cauchy completions, we hereby initiate a study of p -topological Cauchy completions. A p -topological Cauchy space has a p -topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing a p -topological completion, it is shown that a certain class of Reed completions preserve the p -topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsest p -topological completions. However, not all p -topological completions are Reed completions. Several extension theorems for p -topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowing p -topological and p ′ -topological completions, respectively, can always be extended to a θ -continuous map between any p -topological completion of the first space and any p ′ -topological completion of the second.