Abstract not found

Abstract. We give a set-valued criterion for a topological space X to be consonant, i.e. the upper Kuratowski topology on the family of all closed subsets of X coincides with the co-compact topology. This characterization of consonance is then used to show that the statement “every analytic metrizable consonant space is complete ” is independent of the usual axioms of set theory. This answers a question by Nogura and Shakhmatov. It is also proved that continuous open surjections defined on a consonant space are compact covering. A topological space X is said to be consonant if the co-compact topology on the set of all closed subsets of X coincides with the upper Kuratowski topology. The class of consonant spaces was introduced by Dolecki, Greco and Lechicki in [2], [3] and was recently studied rather intensively. It was noticed by Nogura and Shakhmatov in [8] that two other classical topologies (automatically) coincide on the set of closed subsets of a consonant space, namely, the Fell topology and the Kuratowski topology. In [3], among other things, it is proved that every Čechcomplete space is consonant. It is also known that metrizable consonant spaces