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Hausdorff metric

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Hausdorff distance

A metric in the space of subsets of a compact set , defined as follows. Let and let be the set of all numbers and where , and is a metric in . Then the Hausdorff metric is the least upper bound of the numbers in . It was introduced by F. Hausdorff in 1914 (see [1]); one of his most important results is as follows: The space of closed subsets of a compact set is also compact (P.S. Urysohn arrived independently at this theorem in 1921–1922, see [2]).

References

[1] F. Hausdorff, "Set theory" , Chelsea, reprint (1978) (Translated from German)
[2] P.S. Urysohn, "Works on topology and other areas of mathematics" , 2 , Moscow-Leningrad (1951) (In Russian)


Comments

Generally, the Hausdorff metric is defined on the space of bounded closed sets of a metric space . The Hausdorff metric topology and the exponential topology (see also Hyperspace) then coincide on the space of compact subsets of .

See especially

in Hyperspace.

How to Cite This Entry:
Hausdorff metric. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hausdorff_metric&oldid=13737
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098