Cozfine and measurable uniform frames

Abstract

The vital role played by the cozero part of a completely regular, and hence uniformizable, frame is well known, and has been studied in the more general setting of σ-frames. Thus, to consider uniformities generated in some way by the cozero part, namely what will be called cozfine uniformities, is a natural weakening of the fine condition. Similarly, what amounts to having the cozero part complemented is also a natural condition to consider and gives rise to what will be called the measurable uniform frames. The aim of the paper is, therefore, to consider the notions of “fine”, “cozfine” and “measurable” in a frame setting. These will be shown to define reflective subcategories of uniform, or separable uniform, frames, and their behaviour relative to the complete coreflection will be discussed. The latter is of interest considering the interaction between the completion and the state of fineness which gives various compactifications. For example, the Samuel compactification may be described as the completion of the precompact coreflection of the fine uniformity.