Consonance and topological completeness in analytic spaces

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 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 Cechcomplete space is consonant. It is also known that metrizable consonant spaces are hereditarily Baire [1] (i.e. every closed subspace of a metrizable consonant metric space is a Baire space). These results prompted Nogura and Shakhmatov in [8, Problem 11.4] to ask whether consonant metrizable spaces are (Cech-)complete. This remained one of the central open problems in the theory of consonance. For coanalytic metrizable spaces, this problem is entirely solved: A co-analytic metrizable space X is consonant if and only if X is Polish (see [1]). In this note we show that the statement analytic metrizable consonant space is complete is independent of the usual axioms of set theory (Theorem 6). It is also proved that continuous open surjections defined on a consonant space are compact covering (Corollary 8), which gives a generalization of the classical theorem of Pasynkov stated for Cechcomplete spaces. Corollary 8, along with a construction of Michael [6], allows us to give a new and topological proof of the nonconsonance of the usual space of rationals. This note is based on a characterization of consonance in terms of a special property of lower semicontinuous set-valued maps (Theorem 2). Concerning topological completeness, an important and recent result of this kind, achieved by van Mill, Received by the editors October 7, 1996 and, in revised form, February 10, 1998. 1991 Mathematics Subject Classification. Primary 54A35; Secondary 54B20, 54C60.