Completeness, Total Boundedness, and Compactness of the Hausdorff Metric (7.45.7b-d)

James Munkres Topology, chapter 7.45, exercise 7:

Let $(X,d)$ be a metric space, and let $(\mathcal{H},D)$ be its associated Hausdorff metric. Show completeness, total boundedness, and compactness are equivalent conditions in both spaces.

Proof: Note that there is a natural isometry of $(X,d)$ with a closed subspace of $(\mathcal{H},D)$, so that completeness, total boundedness, and compactness of $\mathcal{H}$ implies the corresponding quality in $X$ (to cover a subset $A$ of a totally bounded space $B$ with finitely many $ε$ balls, cover $B$ with $ε/2$ balls, remove those disjoint from $A$, and place one $ε$ ball centered in $A$ per $ε/2$ ball from $B$). Since completeness and total boundedness imply compactness, it will suffice to prove (a) completeness of $(X,d)$ implies completeness of $(\mathcal{H},D)$, and (b) total boundedness of $(X,d)$ implies total boundedness of $(\mathcal{H},D)$.

(a) Let $(A_n)$ be a Cauchy sequence in $\mathcal{H}$. If necessary, take a subsequence so that $D(A_n,A_{n+1}) < 1/2^n$ for all $n$. Now, let $A$ be the set of all limit points of subsequences of $(a_n)$ of $X$ such that $a_n∈A_n$ for each $n$. Since $D(A_1,A_n) < 1$ for each $n$, it is clear $A$ is bounded, nonempty, and (by diagonalization of limits) closed. We show $A_n → A$. Let $ε > 0$. Since the size of the neighborhood of $A_n$ required to contain $A$ approaches $0$, it suffices to show $A_n ⊈ B_D(A,ε)$ for only finitely many $n$. To wit, let $N$ be such that $\sum_{i=N}^∞ 1/2^i < ε/2$. Then if $a_n∈A_n$ for $n≥N$, set $b_0=a_n$ and given $b_i$, choose $b_{i+1}$ so that $d(b_i,b_{i+1}) < 1/2^n$. Then $(b_i)$ is a Cauchy sequence, and appending cursory points in each of $A_1,...,A_{n-1}$ we can find a point of $A$—namely, $b$ when $b_i→b$—such that $d(a_n,b) < ε$ and now $a_n∈B_D(A,ε)$.

(b) Let $ε > 0$. Cover $X$ by finitely many $ε$ balls centered about the points $a_1,...,a_n$. Let $J=\mathcal{P}(\{a_1,...,a_n\}) \setminus \{ø\}$, and center around each point $j∈J⊆\mathcal{H}$ an $ε$ ball. This is seen to be a finite covering of $\mathcal{H}$ by $ε$ balls, with an arbitrary element $A∈\mathcal{H}$ being within distance $ε$ of the element of $J$ which minimally (with regard to set containment) covers $A$ considered as a subset of $X$.$~\square$