, Wijsman convergence
is a notion of convergence
(or, more generally, nets
) of closed subsets
of metric spaces
, named after the mathematician Robert Wijsman
. Intuitively, Wijsman convergence is to convergence in the Hausdorff metric
as pointwise convergence
is to uniform 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.
- 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
- The Hausdorff and Wijsman topologies on Cl(X) coincide if and only if (X, d) is a totally bounded space.
- Beer, Gerald (1993). Topologies on closed and closed convex sets. Dordrecht: Kluwer Academic Publishers Group.
- Beer, Gerald (1994). "Wijsman convergence: a survey". Set-Valued Anal. 2 (1–2): 77–94.
- Wijsman, Robert A. (1966). "Convergence of sequences of convex sets, cones and functions. II". Trans. Amer. Math. Soc. 123 32–45.