 ### finish coarse structures appendix (for now)

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 ... ... @@ -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. \end{itemize} \end{example} % chapter coarse_structures (end)
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