Wijsman convergence

In mathematics, Wijsman convergence is a notion of convergence for sequences (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.

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

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

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",

:$d(x,\; A\_\{i\})\; o\; d(x,\; A).$

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

::

: The Hausdorff and Wijsman topologies on Cl("X") coincide if and only if ("X", "d") is a totally bounded space.

References

* cite book
last = Beer
first = Gerald
title = Topologies on closed and closed convex sets
series = Mathematics and its Applications 268
publisher = Kluwer Academic Publishers Group
location = Dordrecht
year = 1993
pages xii+340
isbn = 0-7923-2531-1 MathSciNet|id=1269778
* cite journal
last = Beer
first = Gerald
title = Wijsman convergence: a survey
journal = Set-Valued Anal.
volume = 2
year = 1994
issue = 1–2
pages = 77–94
issn = 0927-6947
doi = 10.1007/BF01027094 MathSciNet|id=1285822
* cite journal
last = Wijsman
first = Robert A.
title = Convergence of sequences of convex sets, cones and functions. II
journal = Trans. Amer. Math. Soc.
volume = 123
year = 1966
pages = 32–45
issn = 0002-9947
doi = 10.2307/1994611 MathSciNet|id=0196599

External links

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