Extensions, Filters, and Proximities
Ellen E. Reed · 2020
Thron has shown that principal To-extensions correspond 1-1 to systems of open filters which contain all the neighborhood filters. This chapter simply reviews known correspondences between extensions and filter systems. Systems of Cauchy filters can also be grouped into proximity classes, using the same idea. The chapter focuses on Cauchy spaces. The Cauchy space is used in the sense of Keller. It turns out that a Cauchy system is contigual iff it is the set of Cauchy filters for a totally bounded uniformity on the space. The chapter focuses on the relation between extensions of topological spaces and their corresponding trace systems. Thus contigual filter systems can be regarded as generalizations of totally bounded uniformities; in fact, when we choose a filter system we have chosen the “Cauchy” filters — those which “ought to” converge in the extension. The Cauchy filters of a uniform convergence space can be thought of as a filter system.