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Jannes Bantje
differential-operators
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579fa846
Commit
579fa846
authored
Jun 24, 2020
by
Jannes Bantje
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add definition of controlled sets
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contents/appendix.tex
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579fa846
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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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