  The conventional definition of a topological metric over a space specifies
properties that must be obeyed by any measure of "how separated" two points in
that space are. Here it is shown how to extend that definition, and in
particular the triangle inequality, to concern arbitrary numbers of points.
Such a measure of how separated the points within a collection are can be
bootstrapped, to measure "how separated" from each other are two (or more)
collections. The measure presented here also allows fractional membership of an
element in a collection. This means it directly concerns measures of ``how
spread out" a probability distribution over a space is. When such a measure is
bootstrapped to compare two collections, it allows us to measure how separated
two probability distributions are, or more generally, how separated a
distribution of distributions is.
