Let be a metric space and the family of non-empty, closed and bounded subsets of . For any , define
where denotes the distance of point from set and is the distance of point from set , that is
It can be shown that is a metric space, and the metric is known as the Hausdorff metric. In order to prove that is a metric, the following three properties of (the definition of) metric need to be shown:
Proofs of the validity of these three properties for the Hausdorff metric are readily available in several books of topology. To ensure that the Hausdorff metric is well-defined, its finiteness will be shown.
Let and . Then, since
it follows that for every there exist and such that
Indeed, assume that (1) is not true. Then there exists such that for all and it holds that
so a contradiction has been reached, which means that (1) actually holds.
According to the triangle inequality of metric ,
It thus follows from (1) and (2) that for every
Note that (3) holds because the sets are bounded, which means that their respective diameters satisfy
Since (3) holds for all , it follows that
Indeed, assume that (4) is not true. Then it holds that
and, in combination with (3), it is deduced that for any
which is a contradiction because it follows from the Archimedean property that there exists such that
hence (4) is true.
it follows from (4) that for any
Since (5) holds for any , it follows that the set has an upper bound, therefore according to the completeness axiom for real numbers, has a supremum. In particular,
By means of symmetry of (6), this also means that
which completes the proof.