# 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.

## History

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.

## 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

${\displaystyle d(x,A)=\inf _{a\in A}d(x,a).}$

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

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

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 (Xd) 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.
• Cl(X) with the Wijsman topology is always a Tychonoff space. Moreover, one has the Levi-Lechicki theorem: (Xd) is separable if and only if Cl(X) is either metrizable, first-countable or second-countable.
• 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
${\displaystyle d_{\mathrm {H} }(A,B)=\sup _{x\in X}{\big |}d(x,A)-d(x,B){\big |}.}$
The Hausdorff and Wijsman topologies on Cl(X) coincide if and only if (Xd) is a totally bounded space.