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@ -45,9 +45,8 @@ We will see that not every analytic set is Borel.
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\begin{remark}
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In the definition we can replace the assertion that
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$f$ is continuous
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by the weaker assertion of $f$ being Borel.
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\todo{Copy exercise from sheet 5}
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% TODO WHY?
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by the weaker assertion of $f$ being Borel.%
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\footnote{use \yaref{thm:clopenize}, cf.~\yaref{s6e2}}
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\end{remark}
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\begin{theorem}
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@ -17,8 +17,8 @@ U_{\epsilon}(x,y) \coloneqq \{f \in X^X : d(x,f(y)) < \epsilon\}.
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for all $x,y \in X$, $\epsilon > 0$.
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$X^{X}$ is a compact Hausdorff space.
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\begin{remark}
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\todo{Copy from exercise sheet 10}
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\begin{remark}%
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\footnote{cf.~\yaref{s11e1}}
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Let $f_0 \in X^X$ be fixed.
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\begin{itemize}
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\item $X^X \ni f \mapsto f \circ f_0$
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@ -1,5 +1,6 @@
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\lecture{20}{2024-01-09}{The Infinite Torus}
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\gist{
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\begin{example}
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\footnote{This is the same as \yaref{ex:19:inftorus},
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but with new notation.}
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@ -14,9 +15,10 @@
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\begin{remark}+
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Note that we can identify $S^1$ with a subset of $\C$ (and use multiplication)
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or with $\faktor{\R}{\Z}$ (and use addition).
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In the lecture both notations were used.% to make things extra confusing.
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In the lecture both notations were used. % to make things extra confusing.
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Here I'll try to only use multiplicative notation.
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\end{remark}
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}{}
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We will be studying projections to the first $d$ coordinates,
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i.e.
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\[
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@ -57,7 +57,7 @@ form a $\sigma$-algebra).
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Let $(U_n)_{n < \omega}$ be a countable base of $(X,\tau)$.
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Each $U_n$ is open, hence Borel,
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so by a theorem from the lecture$^{\text{tm}}$
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so by \hyperref[thm:clopenize]{a theorem from the lecture™}
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there exists a Polish topology $\tau_n$
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such that $U_n$ is clopen, preserving Borel sets.
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@ -116,7 +116,7 @@ amounts to a finite number of conditions on the preimage.
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\]
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is closed as an intersection of clopen sets.
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Clearly $\pr_{LO(\N)}(\cF)$ is the complement
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Clearly $\proj_{LO(\N)}(\cF)$ is the complement
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of $WO(\N)$, hence $WO(\N)$ is coanalytic.
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\end{itemize}
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\nr 4
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