In this paper the notion of
proximity convergence structures is introduced. These constitute a layer between
Cauchy structures and uniform convergence structures (in the sense of Cook and
Fischer [1]). They are a natural generalization of proximity structures. A study of the
relations among these various structures constitutes §§2 and 3. In §4, compact
extensions for a special class of proximity convergence spaces are constructed, and a
characterization of these is obtained. They satisfy a mapping property with respect
to compact T2 proximity convergence spaces which satisfy a strong regularity
condition. One problem left open is the obtaining of a more reasonable definition of
regularity for these spaces.
