Normality and paracompactness of the Fell topology

Abstract

Abstract. Let X be a Hausdorff topological space and CL(X) the hyperspace of all closed nonempty subsets of X. We show that the Fell topology on CL(X) is normal if and only if the space X is Lindelöf and locally compact. For the Fell topology normality, paracompactness and Lindelöfness are equivalent.Throughout the paper all spaces are assumed to be Hausdorff. By X we always denote a space, while CL(X) (resp. K(X)) is the set of all nonempty closed (compact) subsets of X. We quote [En] To describe this topology, we need to introduce some notation. For E a subset of X, we associate the following subsets of CL(X):The Fell topology τ F on CL(X) has as a subbase all sets of the form V − , where V is an open subset of X plus all sets of the form (K c ) + , where K ∈ K(X) and K c is the complement of K. In locally compact spaces, convergence with respect to the Fell topology is Kuratowski convergence of nets of sets.If compact subsets in the above definition are replaced by closed sets, we obtain the stronger Vietoris topology, also called the finite topology [Mi]. The normality of the Vietoris topology on CL(X) is equivalent to the compactness of X as was shown by Veličko in [Ve]. We refer also to Keesling's deep study of normality of the Vietoris topology [Ke1], [Ke2]. The regularity and Hausdorffness of the Fell topology was studied by Poppe [Po] and the complete regularity by Beer and Tamaki in [BT]. We can summarize here these results as follows:

Effective local compactness and the hyperspace of located sets

“…Then (A ∧ V)(X) is computably metrizable. [9] shows that for a Hausdorff space X, the Fell topology is normal iff X is Lindelöf and locally compact; and [4] shows that for Hausdorff X, the Fell topology is Hausdorff iff X is locally compact. Compactness of the Fell topology, however, holds for arbitrary Hausdorff spaces [3].…”

Hereditarily normal Wijsman hyperspaces are metrizable

“…This completely solved the normality problem of Vietoris hyperspaces. The normality problem of Fell hyperspaces was settled by Holá, Levi and Pelant in [12], where they showed that (2 X , τ F ) is normal if and only if (2 X , τ F ) is Lindelöf, if and only if X is locally compact and Lindelöf, here τ F denotes the Fell topology on 2 X . Since in general the Wijsman topology induced by a metric is coarser than the Vietoris topology but finer than the Fell topology induced by the same metric, the following natural question arises.…”

Note on hit-and-miss topologies

“…This is achieved by applying techniques and notions from [19] and a completely new approach is employed to characterize complete regularity of . Note that a characterization of normality of is not known except for some special cases, like the Vietoris topology ( [11], [18]) or the Fell topology ( [10]); for some more general results on normality see [6].…”