 ### add definition of coarse structure

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 ... ... @@ -542,4 +542,37 @@ In fact being contractible is not just a property of the total space of universa % section thom_isomorphism (end) % chapter appendix_bundle_theory (end) \chapter{Coarse Structures} % (fold) \label{cha:coarse_structures} Given a metric space $X$ with metric $d$ the topologist naturally considers the topology induced by $d$. This only captures the \enquote{small scale structure} of the metric space $X$, since the metric $d'(x,y) \coloneqq \min \set[\big]{d(x,y),1}$ induces the same topology. In this appendix we want to study the dual procedure of capturing the \enquote{large scale structure} of a metric space. In \cref{sec:general_theory_modules} we already mentioned the concept of two maps $f,g \colon S \to X$ (for some set $S$) being \emph{close} (see \cref{def:coarse_cat}). Closeness is clearly an equivalence relation and if the diameter of $X$ is finite, any two such maps are close. We can modify $d$ in another way dual to $d'$ by removing all the small scale information: For $x \neq y$ set $d''(x,y) \coloneqq \max \set[\big]{d(x,y),1}$ The closeness relation is preserved by this operation. The closeness relation obtained from a metric space can now be abstracted: \begin{definition}{\cite[Def.~6.1.2]{higson_roe}} Let $X$ be a set. A \Index{coarse structure} on $X$ is a collection of equivalence relations, called \enquote{being close}, on each set of maps $S \to X$ such that the following properties hold: \begin{enumerate}[(i)] \item If $f,g \colon S \to X$ are close and $q \colon S' \to S$ is any map, then $f \circ q$ and $g \circ q$ are close. \item If $S = S' \cup S''$ and $f,g \colon S \to X$ are maps, whose restrictions to $S'$ and $S''$ are close, then $f$ and $g$ are close. \item Any two constant maps are close to each other. \end{enumerate} \end{definition} % chapter coarse_structures (end)
 ... ... @@ -234,7 +234,7 @@ We now look at the more specialised setting of the coarse category: \begin{definition} For us, metrics are allowed to take the value infinity. A metric space $X$ is \Index[proper metric space]{proper}, if all closed bounded sets are compact. A metric space $X$ is called \Index[proper metric space]{proper}, if all closed bounded sets are compact. \end{definition} Given any map $f \colon X \to Y$ between proper metric spaces, the \Index{expansion function} of $f$, denoted $\omega_f \colon [0,\infty) \to \benbrace*{0,\infty}$ is defined by ... ... @@ -257,6 +257,7 @@ Given any map $f \colon X \to Y$ between proper metric spaces, the \Index{expans \begin{remark} \begin{itemize} \item In \cref{cha:coarse_structures} we give more background on coarse structures. \item \Textcite{rudi} calls the first property \enquote{bornologous}, referring to the language of bornological spaces, in which bounded sets are axiomatised in a manner similar to open sets in a topological space. \item The finiteness of the expansion can also be formulated in a manner similar to the $\varepsilon$-$\delta$-definition of continuity, where the roles of $\varepsilon$ and $\delta$ are swapped, a \enquote{$\delta$-$\varepsilon$-definition} so to speak. \item Assuming that all spaces are proper has the following implication: Properness of a map $f \colon X \to Y$ in the sense above is equivalent to requiring, that the preimage of bounded sets is a finite union of bounded sets. ... ...
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