Completions of Semiuniform Convergence Spaces

The best known completion is the construction of the reals from the rationale. The procedure of G. Cantor [23] and Ch. Méray ([97] and [98]) has been generalized by F. Hausdorff [58] to metric spaces and later on by A. Weil [147] to (separated) uniform spaces. In this chapter the so-called Hausdorff completion of uniform spaces is introduced at first by means of categorical methods in order to emphasize its universal character. The next step is a more concrete construction of the Hausdorff completion of (separated) uniform spaces via natural function spaces in SUConv. More exactly, for a certain class of semiuniform convergence spaces, so-called u-embedded semiuniform convergence spaces, including separated uniform spaces, a completion is constructed, which is due to R.J. Gazik, D.C. Kent and G.D. Richardson [54]. It follows from the universal property of this Gazik-Kent-Richardson completion that it coincides with the Hausdorff completion provided that it is applied to separated uniform spaces. By the way, the theory of regular semiuniform convergence spaces, including uniform spaces and regular topological spaces (the latter were introduced by L. Vietoris [145]), is developed, where a useful extension theorem for uniformly continuous maps is proved. After an alternative description of uniform spaces by means of uniform covers due to W. Tukey [141], precompactness (= total boundedness) and compactness are introduced in the realm of semiuniform convergence spaces. In particular, a semiuniform convergence space is compact, iff it is precompact and weakly complete, where the concepts ‘weakly complete’ and ‘complete’ are equivalent for uniform limit spaces.