Wijsman convergence

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.

Named after Robert Wijsman, although the same definition was used earlier by Zdeněk Frolík[1].

Contents

Definition

Let (Xd) 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

d(x, A) = \inf_{a \in A} d(x, a).

A sequence (or net) of sets Ai ∈ Cl(X) is said to be Wijsman convergent to A ∈ Cl(X) if, for each x ∈ X,

d(x, A_{i}) \to d(x, A).

Wijsman convergence induces a topology on Cl(X), known as the Wijsman topology.

Properties

d_{\mathrm{H}} (A, B) = \sup_{x \in X} \big| d(x, A) - d(x, B) \big|.
The Hausdorff and Wijsman topologies on Cl(X) coincide if and only if (Xd) is a totally bounded space.

References

  1. ^ Z. Frolík, Concerning topological convergence of sets, Czechoskovak Math. J. 10 (1960), 168–180

External links