Covering and function theoretic properties of uniform spaces

Abstract

The purpose of this note is to announce the major ideas and results developed in [R]x. The proofs of these results will appear in a series of three papers [R]2, [R]3, and [RR], the latter including categorical topics that will be omitted here. The subject matter is the covering and function theoretic properties of uniform spaces, a subject initiated by John Isbell in the 1950's. (See [GI] and [I].) Our work represents a continuation and extension of the current work of Anthony Hager ([H]l5 [H]2) and Z. Frolik; and overlaps somewhat with recent work of Z. Frolik ([Fr]l5 [Fr]2). The author wishes to emphasize that his work substantiates the existence of a theory of uniform structures which is not primarily interested in topological applications. Therefore, the viewpoint adopted here is one of intrinsic interest per se in uniform properties. A uniform space is denoted by uX, where u is a family of covers on the set X constituting a uniformity. uX is fine if u is the largest uniformity on X with the same uniform topology. A subfine space is a subspace of a fine space. uX is locally fine if each cover of the form {AanCp} eu, where {Aa} e w, and {Cp} e u for each a. uX is M-fine (sub-M-fine) if each uniformly continuous function (map) to a metric (complete metric) space remains a map relative to the fine uniformity on M (the uniformity with the basis of open covers of M). uX is hereditarily M-fine if each subspace is M-fine. The basic source on locally fine and subfine spaces is [I], while the development of separable M-fine and separable hereditarily M-fine spaces (those with a basis of countable covers) originates in [Hh and [H]2. One easily sees that each fine space is M-fine and that each M-fine space is sub-M-fine. Example C of [GI] is a hereditarily M-fine space which is not locally fine. [I] shows that each locally fine space is sub-M-fine and that each subfine space is locally fine; the converse of the latter is an unsolved problem. From [I] we also know that each separable sub-M-fine space is subfine.