Let (X, d) be a metric space and let Cl(X) denote the collection of all d-closed subsets of X. For a point x ∈ X and a set A ∈ Cl(X), set
A sequence (or net) of sets Ai ∈ Cl(X) is said to be Wijsman convergent to A ∈ Cl(X) if, for each x ∈ X,
Wijsman convergence induces a topology on Cl(X), known as the Wijsman topology.
Properties
The Wijsman topology depends very strongly on the metric d. Even if two metrics are uniformly equivalent, they may generate different Wijsman topologies.
Beer's theorem: if (X, d) is a complete, separable metric space, then Cl(X) with the Wijsman topology is a Polish space, i.e. it is separable and metrizable with a complete metric.
If the pointwise convergence of Wijsman convergence is replaced by uniform convergence (uniformly in x), then one obtains Hausdorff convergence, where the Hausdorff metric is given by
The Hausdorff and Wijsman topologies on Cl(X) coincide if and only if (X, d) is a totally bounded space.
^Z. Frolík, Concerning topological convergence of sets, Czechoskovak Math. J. 10 (1960), 168–180
Bibliography
Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii+340. ISBN0-7923-2531-1. MR1269778