© 1996, Topology Atlas

Proximity spaces (introduced by V.A. Efremovic in 1952) are obtained by axiomatizing the concept of nearness between two sets. It is well-known that the category Prox of proximity spaces and proximally continuous maps (i.e. maps preserving nearness between two sets) is isomorphic to the category of totally bounded uniform spaces (and uniformly continuous maps). The importance of proximity spaces lies in the fact that they are a useful tool for studying compactifications. On the other hand the category Prox can be nicely embedded into the category Chy of Cauchy spaces and Cauchy continuous maps (i.e. maps preserving Cauchy filters) introduced by H.H. Keller in 1968. Cauchy spaces are obtained by axiomatizing the concept of Cauchy filter which is fundamental for studying completeness. Thus, Cauchy spaces form a useful tool for investigating completions. Furthermore, they form a cartesian closed topological category (i.e. in Chy exist natural function spaces) in contrast to Prox which is a non--cartesian--closed topological category. Even the cartesian closed topological category T_{2W}--Lim of weakly Hausdorff limit spaces can be nicely embedded into the category Chy, in other words: every Cauchy space has an underlying proximity space as well as an underlying (weakly Hausdorff) limit space (= convergence space).

Several generalizations of Cauchy spaces have been studied, e.g. filtermerotopic spaces (shortly: filter spaces) by M. Katetov in 1965 and semiuniform convergence spaces by the author in 1995. Semiuniform convergence spaces form an alternative concept of space in topology with many convenient properties such as cartesian closedness, hereditariness (i.e. the existence of one--point extensions) and the fact that products of quotients are quotients. They are the suitable framework for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, namely simple convergence, continuous convergence and uniform convergence, which are very important for investigations in topology and analysis. This is highly remarkable since all other known common generalizations of topological and uniform spaces don't have such nice convenient properties.