General Topology – Gromov-Hausdorff Distance and the ‘Set of All Sets’

general-topologylogicset-theory

If $X$ and $Y$ are compact metric spaces, then the Gromov-Hausdorff distance, $d_{GH}(X,Y)$, describes how far $X$ and $Y$ are from being isometric. In the Wikipedia article on Gromov-Hausdorff distance, it is currently written: "The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space." My naive attempt to do this rigorously immediately leads to the well-known issue that the "set of all sets" is not, in fact, a set.

Googling led me to the MathOverflow question, "When is something too big to be a set?" In a comment under the accepted answer, Nate Eldredge wrote, "a compact metric space has at most a certain cardinality, and so by fixing a set $S$ of that cardinality, any metric space is isometric to some metric on some subset of $S$." But then Thierry Zell responded, "even the definition of $d_{GH}$ requires you to consider all possible isometric embeddings of $X$ and $Y$ into all possible metric spaces, so the 'too big' issue already occurs at this level."

So my questions are these:

  1. What is the mathematically correct definition of the Gromov-Hausdorff distance?

  2. What is the mathematically correct theorem that corresponds to the heuristic notion that the set of all compact metric spaces is a metric space under $d_{GH}$?

Best Answer

I believe the following is correct, but someone may have to come along and correct me. This will further the comment made by Alexander Thumm, above.

There are metric spaces which include isometric copies of every separable metric space. Examples of such spaces are $C([0,1])$, the family of all continuous functions on the unit interval with the sup-norm, and Urysohn's universal metric space $U$. As all compact metric spaces are separable, by fixing such a space $Z$, it should be possible to show that $$d_{GH} (X,Y) = \inf \{ d_H ( i[X], j[Y] ) : i,j\text{ are isometric embeddings of }X,Y\text{, resp., in }Z \}$$

With this in mind, we may consider quotient of the space $\mathcal{K} (Z)$ of all nonempty compact subsets of $Z$ by isometry relation $\sim$. Note that if $K_1,K_2,L_1,L_2$ are compact subsets of $Z$ with $K_1 \sim K_2$ and $L_1 \sim L_2$, then it follows that $d_{GH} ( K_1,L_1 ) = d_{GH} (K_2,L_2)$, and so we may consider $d_{GH}$ as a function from pairs of $\sim$-equivalence classes into the nonnegative reals. The statement that "the set of all compact metric spaces is a metric space under $d_{GH}$" is then an informal interpretation of "$d_{GH}$ is a metric on $\mathcal{K} (Z) / \sim$."

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