Completions

Abstract

This paper presents a new approach to the theory of completions. The treatment is based on the concept of convergence on filters and related topologies. For a given uniform Hausdorff space Xu and a collection S of Cauchy filters in Xu, the basic result is the construction of a uniform Hausdorff space. Xu having the properties that Xu is isomorphic to a dense subspace of Xu and every filter in S converges to a point in S. As a special case, the completion of Xu of Xu is obtained. The construction is so given as to prove the existence of the space Xu. The technique involves embedding the object X to be "completed" in a space of functions F which has as its domain a space of continuous functions C(X) defined on X. The procedure is analogous to the process of taking the bidual E" of a locally convex topological vector space. Indeed, E" is obtained as a special case. In the absence of sufficient structure on X, the Xu is obtained as the closure of X in F. In a locally convex space or an abelian topological group having enough character to separate points, Xu is obtained as a bidual or a second character group of the object X.