Proximity and hyperspace topologies
by
Som Naimpally
Emeritus Professor of Mathematics, Lakehead University,

In this paper we study the use of proximity in hyperspace topologies. A proximal hypertopology corresponding to a LO-proximity is a generalization of the well known Vietoris topology. In case we start with an EF-proximity, the proximal hypertopology equals the Hausdorff uniform topology corresponding to the totally bounded uniformity and, being contained in both the Vietoris and Hausdorff uniform topologies, serves as a bridge between the two. Beer-Himmelberg-Prickry-Van Vleck showed that the locally finite hypertopology induced by a metrizable space is the sup of the Hausdorff metric topologies induced by all compatible metrics.

Naimpally-Sharma showed that this follows from the fact that a Tychonoff space is normal iff its fine uniformity induces the locally finite hypertopology. Di Concilio-Naimpally-Sharma showed that in a Tychonoff space the fine uniformity induces the proximal locally finite hypertopology. We study proximal DELTA topologies and their variations. We show that a short proof can be given of the Beer-Tamaki result concerning the uniformizability of (proximal) DELTA hypertopologies via the Attouch-Wets approach used by Beer in dealing with the Fell topology. Finally we present a result concerning (Proximal) DELTA-U-hypertopolgies.

Date received: December 15, 2000