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Typically we think of a metric as a notion of distance between elements of set subject to the following constraints...

$$ d(x, y) ≥ 0,\quad d(x, x) = 0,\quad d(x, y) = d(y, x),\quad d(x, z) ≤ d(x, y) + d(y, z) $$

I want to know if there is an equivalent for distance between subsets of a known metric space? For example, if we partition integers between 1 and 20 into blocks of 5 and label these A, B, C and D, can we say that these labels are also metrics given a sensible method to compare them? E.g. $$ d(A,B) = \Big|\sum(1,2,3,4,5) - \sum(6,7,8,9,10)\Big| = 25,\\ d(A,A) = \sum(1,2,3,4,5) - \sum(1,2,3,4,5) = 0, \mathrm{etc}. $$

I'm sure there must be a solid definition of this somewhere with appropriate constraints on the sets. What I am particularly interested in is whether the concept of a metric distance is transitive from pairwise comparisons of elements to pairwise comparisons of subsets the metric space elements are taken from rather than an answer like "if you define a reasonable distance measure you can always call it a metric".

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Given a metric set X, the Hausdorff metric is a metric on the set of non-empty compact subsets of X.

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  • $\begingroup$ Fantastic :). Exactly what I needed. $\endgroup$
    – user2474952
    Jun 11, 2013 at 16:19

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