w23-logic-3/inputs/tutorial_09.tex

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\tutorial{09}{2023-12-12}{}
\begin{fact}
Let $X,Y$ be topological spaces
$X$ (quasi-)compact and
$Y$ Hausdorff.
Let $f\colon X\to Y$ be a continuous bijection.
Then $f$ is a homeomorphism.
\end{fact}
\begin{proof}
Compact subsets of Hausdorff spaces are closed.
\end{proof}
\subsection{Sheet 8}
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Material on topological dynamics:
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\begin{itemize}
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\item Terence Tao's notes on ergodic theory 254A: \cite{tao}
\item \cite{Furstenberg} (uses very different notation!).
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\end{itemize}
\nr 1
\begin{remark}
$\Sigma^1_1$-complete sets are in some sense the ``worst''
$\Sigma^1_1$-sets:
Deciding whether an element is contained in the
$\Sigma^1_1$-complete set
is at least as ``hard'' as as for any $\Sigma^1_1$ set.
In particular, $\Sigma^1_1$-complete sets
are not Borel.
\end{remark}
Similarly as in \yaref{prop:ifs11} it can be shown that $L \in \Sigma^1_1(X)$:
Consider $\{(x, \beta) \in X \times \cN : \forall n.~x_{\beta_n} | x_{\beta_{n+1}}\}$.
This is closed in $X \times \cN$, since it is a countable intersection of clopen
sets and $L = \proj_X(D)$.
Since $\IF \subseteq \Tr$ is $\Sigma^1_1$-complete,
it suffices to find a Borel map $f\colon \Tr \to X$
such that $x \in \IF \iff f(x) \in L$.
Let $\phi\colon \omega^2 + \omega \to \omega$ be bijective
and let $p_i$ denote the $i$-th prime.
Define
\begin{IEEEeqnarray*}{rCl}
\psi\colon \omega^{<\omega} &\longrightarrow & \omega \setminus \{0\} \\
(s_0, s_1, \ldots, s_{n-1})&\longmapsto & \prod_{i < n} p_{\phi(\omega \cdot i + s_i)}.
\end{IEEEeqnarray*}
Note that $\psi$ is injective and that $s \in \omega^{<\omega}$ is an initial segment
of $t \in \omega^{<\omega}$ iff $\psi(s) | \psi(t)$.
Let
\begin{IEEEeqnarray*}{rCl}
f' \colon \Tr &\longrightarrow & \cP(\omega \setminus \{0\}) \\
T &\longmapsto & \{\phi(s) : s \in T\}.
\end{IEEEeqnarray*}
We can turn this into a function
$f\colon \Tr \to (\omega \setminus \{0\})^{\omega}$
by mapping a subset of $\omega \setminus \{0\}$
to the unique strictly increasing sequence whose elements are from
that subset
(appending $\phi(\omega^2 + n), n \in \omega$, if the subset was finite).
Note that $T \in \IF \iff f(T) \in L$.
Furthermore $f$ is Borel, since fixing
a finite initial sequence (i.e.~a basic open set of $(\omega \setminus \{0\})^{\omega}$)
amounts to a finite number of conditions on the preimage.
\nr 2
\todo{handwritten}
% Aron
%
% \begin{enumerate}[1.]
% \item This is trivial $\sup \sup$.
% \item Clearly there are trees of rank $n$ for all $n < \omega$.
% Glue them together.
% \[
% \{(0,0,i,\underbrace{0,\ldots,0}_{i \text{~times}}) | i < \omega\}.
% \]
% \item Map infinite branches. $\sup$ the $\le $.
% \item Induction on $\rho(S)$.
% Cofinal subsequences bla bla.
% \end{enumerate}
\nr 3
\begin{itemize}
\item $LO(\N) \overset{\text{closed}}{\subseteq} 2^{\N\times \N}$:
We have $< \in LO(\N)$ iff
\begin{itemize}
\item $\forall x,y.~ (x \neq y \implies (x< y \lor x > y))$,
\item $\forall x.~(x \not < x)$,
\item $\forall x,y,x.~(x < y < z \implies x < z)$.
\end{itemize}
Write this with $\bigcap$,
i.e.
\begin{IEEEeqnarray*}{rCl}
LO(\N) &=& \bigcap_{n \in \N} \{R: (n,n) \not\in R\}\\
&& \cap \bigcap_{m < n \in \N} (\{R: (n,m) \in R\} \cup \{R: (m,n) \in R\})\\
&&\cap \bigcap_{a,b,c \in \N} (\{R: (a,b) \in R \land (b,c) \in R \implies (a,c) \in R\}.
\end{IEEEeqnarray*}
This is closed as an intersection of clopen sets.
\item We apply \yaref{thm:borel} (iv).
Let $\cF \subseteq LO(\N) \times \cN$ be
such that the $\cN$-coordinate encodes a strictly
decreasing sequence, i.e.~
\[(R, s) \in \cF :\iff \forall n \in \N.~(s(n+1), s(n)) \in R.\]
We have that
\[
\cF = \bigcap_{n \in \N} \{(R,s) \in LO(\N)\times \cN : (s(n+1), s(n)) \in R\}
\]
is closed as an intersection of clopen sets.
Clearly $\pr_{LO(\N)}(\cF)$ is the complement
of $WO(\N)$, hence $WO(\N)$ is coanalytic.
\end{itemize}
\nr 4
\begin{remark}
In the lecture we only look at metrizable flows,
so the definitions from the exercise sheet and from
the lecture don't agree.
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Everywhere but here we will use the definition from the lecture.
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\end{remark}
\begin{itemize}
\item Consider
\begin{IEEEeqnarray*}{rCl}
\Phi\colon \Z\text{-flows on } X &\longrightarrow & \Homeo(X) \\
(\alpha\colon \Z\times X \to X) &\longmapsto & \alpha(1, \cdot)\\
\begin{pmatrix*}[l]
\Z\times X &\longrightarrow & X \\
(z,x) &\longmapsto & \beta^{z}(x)
\end{pmatrix*}&\longmapsfrom & \beta \in \Homeo(X).
\end{IEEEeqnarray*}
Clearly this has the desired properties.
\item We have
\begin{IEEEeqnarray*}{Cl}
& \Z \circlearrowright X \text{ has a dense orbit}\\
\iff& \exists x \in X.~ \overline{\Z\cdot x} = X\\
\iff& \exists x \in X.~\forall U\overset{\text{open}}{\subseteq} X.~\exists z \in \Z.~
z \cdot x \in U\\
\iff&\exists x \in X.~\forall U \overset{\text{open}}{\subseteq} X.~
\exists z \in \Z.~f^z(x) \in U.
\end{IEEEeqnarray*}
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\item Let $X$ be a compact Polish space.
What is the Borel complexity of $\Homeo(X)$ inside $\cC(X,X)$?
Recall that $\cC(X,X)$ is a Polish space with the uniform topology.
We have
\begin{IEEEeqnarray*}{rCl}
\Homeo(X) &=& \{f \in \cC(X,X) : f \text{ is bijective and } f^{-1} \text{ is continuous}\}\\
&=& \{f \in \cC(X,X) : f \text{ is bijective}\}
\end{IEEEeqnarray*}
by the general fact
\begin{fact}
Let $X$ be comapct and $Y$ Hausdorff,
$f\colon X \to Y$ a continuous bijection.
Then $f$ is a homeomorphism.
\end{fact}
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\item It suffices to check the condition from part (b)
for open sets $U$ of a countable basis
and points $x \in X$ belonging to a countable dense subset.
Replacing quantifiers by unions resp.~intersections
gives that $D$ is Borel.
\end{itemize}