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@ 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}





@ 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 $X^X \ni f \mapsto f \circ f_0$





@ 1,5 +1,6 @@


\lecture{20}{20240109}{The Infinite Torus}




\gist{


\begin{example}


\footnote{This is the same as \yaref{ex:19:inftorus},


but with new notation.}



@ 14,9 +15,10 @@


\begin{remark}+


Note that we can identify $S^1$ with a subset of $\C$ (and use multiplication)


or with $\faktor{\R}{\Z}$ (and use addition).


In the lecture both notations were used.% to make things extra confusing.


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.


\[





@ 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.







@ 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




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