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\title{Metric Geometry}
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\author{}
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%\section{Metric Geometry}
%\begin{exo}[Density of finite spaces] Show that every compact metric space $X$ is a limit in the sense of $\op{d}_{\op{GH}}$ of finite spaces.\end{exo}
\setcounter{section}{1}
\section{Metric Geometry}

\begin{exo} Give an example of two (non-compact !) metric spaces $X$ and $Y$ such that ${d}_{{GH}}(X,Y)=0$ but $X\ne Y$.
\end{exo}


\begin{exo}[Topology] Suppose that $(X_{n})_{n\geq0}$ is a sequence of compact metric spaces such that $X_{n}$ is homeomorphic to $X_{0}$ for every $n\geq 0$. Show that $X_{n} \to X$ in the sense of ${d}_{{GH}}$ does not imply $X$ homeomorphic to $X_{0}$.
\end{exo}
\begin{exo}[Gromov's Compactness Theorem] The goal of this exercise is to characterize pre-compactness for  Gromov-Hausdorff distance. A collection $\mathfrak{X}$ of compact metric spaces is \emph{totally bounded} if 
\begin{enumerate}[(i)]
\item There exists $C>0$ such that for every $X \in \mathfrak{X}$, the diameter of $X$ is bounded above by $C$,
\item For every $\varepsilon>0$, there exists $N(\varepsilon) \in \mathbb{Z}_{+}$ such that every $X\in \mathfrak{X}$ admits an $\varepsilon$-net containing no more than $N(\varepsilon)$ points.\end{enumerate}

\begin{enumerate}
\item Show that every pre-compact collection is totally bounded.
\end{enumerate}
Now, let $(X_{n})_{n\geq 1}$ be a totally bounded sequence of compact metric spaces and denote $N(\varepsilon)$ the minimal number of balls of radius $\varepsilon$ needed to cover any space ${X}_{i}$. For every $n$, denote the distance in the space $X_{n}$ by $d_{n}$ and  let $(x_{k,j}^{(n)})_{k,j\geq 1}$ be a sequence of points of $X_{n}$ such that for every $k \geq 1$ the points 
$$ x_{k,1}^{(n)}, ... , x_{k,N(1/k)}^{(n)}, \mbox{ form a } 1/k\mbox{-net in }X_{n}.$$
\begin{enumerate}
\setcounter{enumi}{1}
\item Show that we can extract a subsequence $(n_{i})_{i \geq 1}$ such that for every $(k,j),(k',j') \in \mathbb{Z}_{+}^2$ 
$$ \lim_{n\to \infty} d_{n}\big(x_{k,j}^{(n)},x_{k',j'}^{(n)}\big) \mbox{ exists.}$$
\end{enumerate}
To simplify notation, we suppose that there is no need to take a subsequence. Consider the abstract space $X= \{(k,j)\}_{k,j \geq 1}$ endowed with $ \Delta((k,j),(k',j')) = \lim d_{n}\big(x_{k,j}^{(n)},x_{k',j'}^{(n)}\big).$
\begin{enumerate}
\setcounter{enumi}{2}
\item Show that $\Delta$ is a pseudo-metric on $X$ and  that the completion $\overline{X}$ of $X$ with respect to $\Delta$ is compact.
\item Prove that $X_{n} \to \overline{X}$ in the Gromov-Hausdorff sense.
\end{enumerate}
\end{exo}
\begin{exo}* Prove that a sequence of length spaces homeomorphic to the two-dimensional sphere $\mathbb{S}_{2}$ cannot converge to the standard two-dimensional closed ball $\mathbb{B}_{2}$.
\end{exo}

\clearpage
\begin{exo}Who are these charming gentlemen ?
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.7cm,height=5cm]{gromov}
\includegraphics[width=4.3cm,height=5cm]{hausdorff}
\end{center}
\end{figure}
\end{exo}

\nocite{BBI01}
\nocite{Gro07}

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