The Hausdorff metric on the collection of nonempty compact subsets of a metric space 
is the restriction of the Hausdorff
distance to that collection and is a genuine metric
there.
The terms Hausdorff distance and Hausdorff metric are therefore not identical in general. The Hausdorff distance may be infinite on arbitrary nonempty subsets
of 
,
and it may vanish for distinct subsets having the same closure.
The Hausdorff metric refers to a domain on which these failures do not occur.
