Sandro Levi

Distance functionals and suprema of hyperspace topologies

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A -→ d(x, A) is continuous for each x ε X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) -→ → {d(ta, b): a ε A, b ε B} is continuous on CL(X) × CL(X).

Metrics that generate the same hyperspace convergence

We provide a necessary and sufficient condition for two equivalent metrics to generate the same Wijsman convergence on the hyperspace of a metrizable space. We give some applications to normed linear spaces and to various classes of metrizable spaces.

Decomposition Properties of Hyperspace Topologies

Let X be a T1 topological space and Δ a nonempty family of closed subsets of X. We study the hit-and-miss hyperspace topology generated by Δ in terms of its upper and lower parts, focusing on first and second countability and quasi-uniformization. We also obtain some new results on the Vietoris and Fell topologies.

Extensions of Semicontinuous Multifunctions.

We study extensions of upper and lower semicontinuous multifunctions defined on dense subsets of topological spaces. We give some applications to topology, functional analysis and single-valued functions and show that some topological properties can be characterized in terms of extensions of semicontinuous multifunctions. 1980 Mathematics Subject Classification (1985 Revision): 54C60; 54C20, 54B20, 54E52.