Wijsman convergence

Wijsman convergence is a variation of

. Intuitively, Wijsman convergence is to convergence in the .

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

it is the same as Wijsman convergence.

Definition

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

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

A sequence (or net) of sets A_i ∈ 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 (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.
• Cl(X) with the Wijsman topology is always a Tychonoff space. Moreover, one has the Levi-Lechicki theorem: (X, d) 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 (X, d) is a totally bounded space.

See also

• Hausdorff distance
• Kuratowski convergence
• Vietoris topology
• Hemicontinuity

References

Notes

Bibliography

• Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii+340.
  MR1269778
• Beer, Gerald (1994). "Wijsman convergence: a survey". Set-Valued Anal. 2 (1–2): 77–94.
  MR1285822

External links

• Som Naimpally (2001) [1994], "Wijsman convergence", Encyclopedia of Mathematics, EMS Press