some small changes

inputs/lecture_09.tex

@@ -45,9 +45,8 @@ We will see that not every analytic set is Borel.
 \begin{remark}
     In the definition we can replace the assertion that
     $f$ is continuous
-    by the weaker assertion of $f$ being Borel.
-    \todo{Copy exercise from sheet 5}
-    % TODO WHY?
+    by the weaker assertion of $f$ being Borel.%
+    \footnote{use \yaref{thm:clopenize}, cf.~\yaref{s6e2}}
 \end{remark}
 
 \begin{theorem}

inputs/lecture_17.tex

@@ -17,8 +17,8 @@ U_{\epsilon}(x,y) \coloneqq  \{f \in X^X : d(x,f(y)) < \epsilon\}.
 for all $x,y \in X$, $\epsilon > 0$.
 
 $X^{X}$ is a compact Hausdorff space.
-\begin{remark}
-    \todo{Copy from exercise sheet 10}
+\begin{remark}%
+    \footnote{cf.~\yaref{s11e1}}
     Let $f_0 \in X^X$ be fixed.
     \begin{itemize}
         \item $X^X \ni f \mapsto f \circ f_0$

inputs/lecture_20.tex

@@ -1,5 +1,6 @@
 \lecture{20}{2024-01-09}{The Infinite Torus}
 
+\gist{
 \begin{example}
     \footnote{This is the same as \yaref{ex:19:inftorus},
         but with new notation.}
@@ -17,6 +18,7 @@
     In the lecture both notations were used. % to make things extra confusing.
     Here I'll try to only use multiplicative notation.
 \end{remark}
+}{}
 We will be studying projections to the first $d$ coordinates,
 i.e.
 \[

inputs/tutorial_07.tex

@@ -57,7 +57,7 @@ form a $\sigma$-algebra).
 
         Let $(U_n)_{n < \omega}$ be a countable base of $(X,\tau)$.
         Each  $U_n$ is open, hence Borel,
-        so by a theorem from the lecture$^{\text{tm}}$
+        so by \hyperref[thm:clopenize]{a theorem from the lecture™}
         there exists a Polish topology $\tau_n$
         such that $U_n$ is clopen, preserving Borel sets.
 

inputs/tutorial_09.tex

@@ -116,7 +116,7 @@ amounts to a finite number of conditions on the preimage.
         \]
         is closed as an intersection of clopen sets.
 
-        Clearly $\pr_{LO(\N)}(\cF)$ is the complement
+        Clearly $\proj_{LO(\N)}(\cF)$ is the complement
         of $WO(\N)$, hence $WO(\N)$ is coanalytic.
 \end{itemize}
 \nr 4