\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.
\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 \todo{TODO}
    \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}