@@ -583,6 +583,63 @@ The closeness relation obtained from a metric space can now be abstracted:

The \enquote{most controlled} subset of $X \times X$ is the diagonal $\Delta\subset X \times X$.

The notion of controlled sets conversely determines the coarse structure, since one can show, that two maps $f,g \colon S \to X$ are closes if and only if the image of $(f,g)\colon S \to X \times X$ is controlled (\cite[Prop.~6.1.4]{higson_roe}).

The notion of controlled subsets allows us to generalise additional concepts from metric spaces:

\begin{definition}

A subset $B \subseteq X$ is called \Index{bounded}, if the one-element family $\set*{B}$ is uniformly bounded, i.e. $B \times B$ is controlled.

\end{definition}

\begin{lemma}{\cite[Lem.~6.1.6]{higson_roe}}

A non-empty subset $B \subseteq X$ of a coarse space $X$ is bounded if and only if the inclusion $B \hookrightarrow X$ is close to a constant map.

\end{lemma}

In our examples focused on manifolds, there will be coarse \emph{and} topological information, which comes as a locally compact topology.

We combine this as follows:

\begin{definition}

Let $X$ be locally compact space.

A coarse structure on $X$ is called \Index[coarse structure!proper]{proper} if

\begin{enumerate}[(i)]

\item$X$ has a uniformly bounded open cover and

\item every bounded subset of $X$ has compact closure

\end{enumerate}

\end{definition}

These properties assure, that bounded subsets coincide with the ones having compact closure.

This notion is obviously consistent with our earlier use of \enquote{proper}, since the coarse structure associated to any metric space is proper, if the metric space is proper in the sence that closed balls are compact.

To get a category we still have to define, what coarse maps are in this general setting:

\begin{definition}

Let $X$ and $Y$ be coarse spaces.

$f \colon X \to Y$ is called a coarse map, if

\begin{enumerate}[(i)]

\item for all close maps $p,q$ into $X$, the composites $f \circ p$ and $f \circ q$ are close maps into $Y$ and

\item$f$ is proper in the sense, that preimages of bounded sets are bounded.

\end{enumerate}

$X$ and $Y$ are \Index{coarsely equivalent}, if there are coarse maps $f \colon X \to Y$ and $g \colon Y \to X$ such that both composition are close the the respective identity.

\end{definition}

To build up intuition we discuss a few examples:

\begin{example}

\begin{itemize}

\item Consider the natural metric coarse structures on $\mathbb{R}^2$ and $\mathbb{R}$.

The coordinate projections are not coarse, since preimages of bounded sets are clearly not bounded.

The norm function however is.

\item The inclusion $\mathbb{Z}\hookrightarrow\mathbb{R}$ is a coarse equivalence, where the inverse is given by $x \mapsto\lfloor x\rfloor$.

\item Let $M$ be a complete, simply connected Riemannian manifold of non-positive curvature and $p \in M$.

The tangential space $\Tang_p M$ is Euclidean and the exponential map

\[

\exp\colon\Tang_p M \To{} M

\]

is a diffeomorphism by the Cartan--Hadamard theorem.

The inverse $\log\colon M \to\Tang_p M$ decreases distances and it follows, that $\log$ is a coarse map.