@@ -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.

@@ -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.