Zero-Set Ultrafilters and Gδ-Closures in Uniform Spaces

The paper examines the classes k1 and Гr1 of Hausdorff uniform spaces which are Gδ-closed in their Samuel compactifications, or completions. It is shown that the classes are epi-reflective, the reflections k1 and y₁ are described, k1 and Г1 are represented as epi-reflective hulls, membership in the classes is described by fixation of certain zero-set ultrafilters, and it is shown that K1 = y1 exactly on spaces without discrete sets of measurable power. The results include familiar facts about realcompact and topologically complete topological spaces and are closely connected with the theory of metric-fine uniform spaces.