On the coincidence of the upper Kuratowski topology with the cocompact topology

Abstract

A topological space X is said to be consonant if the upper Kuratowski topology and the cocompact topology defined on the set of all closed subsets of X coincide (otherwise, the space X is said to be dissonant). One of the purposes of this paper is to study the notion of consonance, and to solve some open questions which arise from different works about this notion. We give a criterion of consonance which is based on the property of sequentiality, and which extends a result of C. Costantini and P. Vitolo. We prove that Hausdorff locally k ω -spaces are consonant, answering a question of T. Nogura and D. Shakhmatov negatively. With the help of the concept of Radon measure, we establish a criterion of dissonance. In particular, we give a dissonant hereditarily Baire separable metrizable space, answering a question of T. Nogura and D. Shakhmatov positively, and we prove that the Sorgenfrey real line is a dissonant space, giving a negative answer to a question of S. Dolecki, G. Greco and A. Lechicki. We also study the notion of hyperconsonance, that is, the coincidence of the convergence topology with the Fell topology, and we give a positive answer to a question of M. Arab and the second author of this paper. Notice that all our results concerning the criterion based on Radon measures are the subject of an unpublished paper of June 1995.