Hausdorff metric
From Encyclopedia of Mathematics
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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_metric&oldid=38917
Hausdorff metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_metric&oldid=38917
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article