Hypertopologies and quasi-uniformities
by
Jesús Rodríguez-López
Unversidad Politécnica de Valencia
Coauthors: Salvador Romaguera (Universidad Politécnica de Valencia)

It is well known that the Vietoris topology of an uniform space (X, U) is compatible with the Hausdorff uniformity of U on K₀(X). The relationship between the Hausdorff quasi-uniformity and the Vietoris topology for a given quasi-uniform space has been discussed by several authors. In particular, it has been proved that for each quasi-uniform space (X, U), the Vietoris topology is always weakest than the topology of the Hausdorff quasi-uniformity on K₀(X), and we can construct an example of a compact locally symmetric Hausdorff quasi-metric space (X, d) such that the Vietoris topology is not compatible on K₀(X) with the Hausdorff quasi-uniformity of U_d. These facts suggest, in a natural way, the problem of characterizing those quasi-uniform spaces (X, U) for which the Vietoris topology coincides with the Hausdorff quasi-uniform topology on K₀(X). Here, we solve this question and deduce from our characterization several partial results. In fact, we prove that in a quasi-uniform space (X, U) the Vietoris topology agrees with the Hausdorff quasi-uniform topology on K₀(X) if and only if U^-1|K is precompact for all K belonging to K₀(X).

Date received: January 22, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-08.