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Very minor language improvements
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jirilebl committed Jul 8, 2022
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Expand Up @@ -7857,7 +7857,7 @@ \section{Homology versions of Cauchy}
usually written $H_1(U)$.
\item
Compute $H_1(\C \setminus \{ 0 \})$ (that is,
find what group is it isomorphic to).
find what group it is isomorphic to).
\end{exparts}
\end{exercise}

Expand Down Expand Up @@ -7905,7 +7905,7 @@ \section{Simply connected domains}
perhaps not the standard definition, but for domains in $\C$
(connected open sets)
it is equivalent to the correct one.
We will define the term \myquote{properly} once we get
We will define the term properly once we get
to homotopy.\footnote{Homotopy is in an optional section, which is the
reason why we make this \myquote{wrong} definition.}
We may sometimes say
Expand Down Expand Up @@ -8990,7 +8990,8 @@ \subsection{Homotopy}
\end{exercise}

\begin{exercise}
We could take a different approach to solving our issues with homotopy. Let
We could take a different approach to solving our issues with the regularity
of the homotopy. Let
$U \subset \C$ be open and let $\gamma_0$ and $\gamma_1$ be closed
piecewise-$C^1$ paths in $U$ that are homotopic in $U$.
Show that there exists a homotopy (possibly different one) such that
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