Wijsman convergence is a variation of Hausdorff convergence suitable for work with unbounded sets. Intuitively, Wijsman convergence is to convergence in the Hausdorff metric as pointwise convergence is to uniform convergence.
The convergence was defined by Robert Wijsman. [1] The same definition was used earlier by Zdeněk Frolík. [2] Yet earlier, Hausdorff in his book Grundzüge der Mengenlehre defined so called closed limits; for proper metric spaces it is the same as Wijsman convergence.
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.
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