View source for Hausdorff metric

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''Hausdorff distance''

A metric in the space of subsets of a compact set $K$, defined as follows. Let $X,Y\subset K$ and let $D_{x,y}$ be the set of all numbers $\rho(x,Y)$ and $\rho(y,X)$ where $x\in X$, $y\in Y$ and $\rho$ is a metric in $K$. Then the Hausdorff metric $\operatorname{dist}(X,Y)$ is the least upper bound of the numbers in $D_{x,y}$. It was introduced by F. Hausdorff in 1914 (see [[#References|[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 [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,   "Set theory" , Chelsea, reprint  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Urysohn,   "Works on topology and other areas of mathematics" , '''2''' , Moscow-Leningrad  (1951)  (In Russian)</TD></TR></table>



====Comments====
Generally, the Hausdorff metric is defined on the space of bounded closed sets of a metric space $X$. The Hausdorff metric topology and the [[Exponential topology|exponential topology]] (see also [[Hyperspace|Hyperspace]]) then coincide on the space $K(X)$ of compact subsets of $X$.

See especially

in [[Hyperspace|Hyperspace]].