Commit 579fa846 authored by Jannes Bantje's avatar Jannes Bantje

add definition of controlled sets

parent e784f7f6
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......@@ -570,9 +570,19 @@ The closeness relation obtained from a metric space can now be abstracted:
\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{enumerate}
A set $X$ with a coarse structure is called a \Index{coarse space}.
\end{definition}
\begin{definition}
Let $X$ be a coarse space.
A subset $S \subset X \times X$ is called \Index{controlled}, if the two coordinate projections restricted to $S$ are close.
A collection of subsets $\mathcal{U}$ of $X$ is called \Index{uniformly bounded}, if $\bigcup_{U \in \mathcal{U}} U \times U$ is controlled.
\end{definition}
The \enquote{most controlled} subset of $X \times X$ is the diagonal $\Delta \subset X \times X$.
% chapter coarse_structures (end)
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