Results

**1 - 4**of**4**### 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 “every analytic metrizab ..."

Abstract
- Add to MetaCart

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

### Previous Up Next Article Citations From References: 2 From Reviews: 0

"... Coincidence of the Isbell and fine Isbell topologies. (English summary) ..."

### Adherence of minimizers for dual convergences by

"... Abstract: It is proved that many known convergences (e.g., continuous convergence, Isbell topology, compact-open topology, pointwise convergence) on the space of continuous maps (valued in a topological space) can be represented as the dual convergences with respect to collections of families of set ..."

Abstract
- Add to MetaCart

Abstract: It is proved that many known convergences (e.g., continuous convergence, Isbell topology, compact-open topology, pointwise convergence) on the space of continuous maps (valued in a topological space) can be represented as the dual convergences with respect to collections of families of sets, and that they can be characterized in terms of the corresponding hyperspace convergences of the inverse images of closed sets. As a result, the convergence of real-valued functions for a dual convergence implies the convergence of their sets of minima on the corresponding hyperspace.