\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}

Material on topological dynamics:
\begin{itemize}
    \item Terence Tao's notes on ergodic theory 254A: \cite{tao}
    \item \cite{Furstenberg} (uses very different notation!).
\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.

    Everywhere but here we will use the definition from the lecture.
\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*}
    \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}
    \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}

