Hausdorff distance

For nonempty bounded subsets of a metric space,This is a metric on the nonempty closed bounded subsets. Without closedness it is only a pseudometric: a set and its closure have distance zero.

The map $x\mapsto \{x\}$ is an isometry into the hyperspace of nonempty closed bounded subsets. Its image is closed because a Hausdorff limit of singletons has diameter zero and is therefore a singleton.

If the Hausdorff hyperspace is complete, its closed subspace of singletons is complete. Since that subspace is isometric to the original metric space, the original space is complete as well.

For compact metric spaces $X$ and $Y$, the Gromov-Hausdorff distance iswhere the infimum runs over all metric spaces $Z$ and all isometric embeddings $\phi :X\to Z$ and $\psi :Y\to Z$. It measures how closely the two spaces can be placed inside one ambient metric space.

The Gromov-Hausdorff topology on isometry classes of compact metric spaces is the topology induced by the Gromov-Hausdorff distance.

A real tree is a metric space $(T,d)$ in which every pair $a,b\in T$ is joined by a unique arc and that arc is isometric to the real interval $[0,d(a,b)]$. Equivalently, it is a geodesic metric space with no nontrivial simple loops.

The multiplicity of $a$ in a real tree $T$ is the number of connected components of $T\setminus \{a\}$. A point of multiplicity one is a leaf, one of multiplicity two lies in the interior of an unbranched arc, and one of multiplicity at least three is a branch point.

Let $g:[0,1]\to {\mathbb{R}}_{+}$ be continuous with $g(0)=g(1)=0$, and setThe function $d_g$ is a pseudometric. Its metric quotientis the real tree encoded by $g$.

Every compact real tree is isometric to a real tree encoded by an excursion. One construction takes finite subtrees spanning successively finer finite nets, performs depth-first contour traversals of those subtrees, and chooses compatible time parameterizations. The contour functions have a uniformly convergent subsequence, and continuity of excursion coding in the Gromov-Hausdorff distance identifies the limiting coded tree with the original tree.