Commit 579fa846 by 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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