# Hausdorff metric

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 [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)

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 (see also Hyperspace) then coincide on the space $K(X)$ of compact subsets of $X$.