\tutorial{13}{2024-01-23}{}


Continuation of sheet 8, exercise 4.
 % Whiteboard https://wbo.ophir.dev/boards/Dphc7eylJJcIA0WbQsn7jzec1domqyx51gXb5qe6rzw-#263,0,0.5

\begin{definition}
    Let $X$ be a compact metric space.
    For $K \subseteq X$ compact and $U \overset{\text{open}}{\subseteq} X$
    let
    \[
    S_{K,U} \coloneqq  \{f \in \cC(X,X): f(K) \subseteq U\}.
    \]
    The \vocab{compact open topology} on $\cC(X,X)$
    is the topology that has $S_{K,U}$ as a subbase.
\end{definition}
\begin{fact}
    If $X$ is compact,
    then the compact open topology
    is the topology induced by the uniform metric $d_\infty$.
\end{fact}
\begin{proof}
    Take some $S_{K,U}$. We need to show that this can be written
    as a union of open $d_{\infty}$-balls.
    Let $f_0 \in S_{K,U}$.
    Consider the continuous function $d(-, U^c)$.
    Since $f_0(K)$ is compact,
    there exists $\epsilon \coloneqq \min d(f_0(K), U^c)$
    and $B_{\epsilon}(f_0) \subseteq S_{K,U}$.


    On the other hand, consider $B_{\epsilon}(f_0)$ for some $\epsilon > 0$
    and $f_0 \in \cC(X,X)$.

    As $f_0$ is uniformly continuous,
    there exists $\delta > 0$ such that $d(x,x') < \delta \implies d(f_0(x), f_0(x')) < \frac{\epsilon}{3}$.
    Cover $X$ with finitely many $\delta$-balls $B_\delta(a_1), \ldots, B_{\delta}(a_k)$.
    Then
    \[f_0(\overline{B_{\delta}(a_i)}) \subseteq \overline{f_0(B_{\delta}(a_i)} \subseteq \overline{B_{\frac{\epsilon}{3}}(f_0(a_i))} \subseteq  B_{\frac{\epsilon}{2}}(f_0(a_i)).\]

    For $i \le  k$, let $S_i \coloneqq  S_{\overline{B_{\delta}(a_i)}, B_{\frac{\epsilon}{2}}(f_0(a_i))}$.
    Take $\bigcap_{i \le k} S_i$. This is open
    in the compact open topology and
    $B_{\epsilon}(f_0) \subseteq \bigcap_{i \le k} S_i$.
\end{proof}

\begin{claim}
    $f \in \cC(X,X)$ is surjective
    iff for all basic open $\emptyset\neq U \subseteq  X$
    there exists a basic open $\emptyset \neq V \subseteq  X$
    with $f(\overline{V}) \subseteq U$.
    
    Note that we can write this as a $G_\delta$-condition.

\end{claim}
\begin{subproof}
    Take $B_\epsilon(f(x_0))\subseteq U$.
    Then there exists $\delta > 0$
    such that $f(B_{\delta}(x_0)) \subseteq B_{\frac{\epsilon}{2}}(f(x_0))$
    hence $f(\overline{B_{\delta}(x_0)}) \subseteq B_\epsilon(f(x_0))$.


    For the other direction take $y \in X$.
    We want to find a preimage.
    For every $B_{\frac{1}{n}}(y)$,
    there exists a basic open set $V_n$ with $f(\overline{V}) \subseteq B_{\frac{1}{n}}(y)$.
    Take $x_n \in  V_n$.
    Since $X$ is compact, it is sequentially compact,
    so there exists a converging subsequence.
    Wlog.~$x_n \to x$,
    so $f(x_n) \to f(x) = y$.
\end{subproof}

\begin{claim}
    $f \in \cC(X,X)$ is injective iff
    for all basic open $U$,$V$
    with $\overline{U} \cap \overline{V} = \emptyset$
    we have $f(\overline{U}) \cap f(\overline{V}) = \emptyset$.

    This is a $G_\delta$-condition,
    since we can write it as
    \[
    \bigcap_{U,V} S_{\overline{U}, f(\overline{V})^c}.
    \]
\end{claim}
\begin{subproof}
    $\implies$ is trivial.

    $\impliedby$ follows since for all pairs $x,y \in X$,
    we can find $x \in U$, $y \in V$ such that $\overline{U} \cap \overline{V} = \emptyset$.
\end{subproof}

Hence $\Homeo(X,X)$ is $G_\delta$.
In particular it is a Polish space.



Let $D$ be the set of $\Z$-flows with dense orbit.
\begin{claim}
    $f \in D$ $\iff$
    for all basic open $U,V \subseteq X$,
    there exists $n \in \Z$
    such that $f^n(U) \cap V \neq \emptyset$.

    \end{claim}
\begin{subproof}
    Suppose that the orbit of $x_0 \in X$ is dense.
    Then there exist $k,l \in \Z$
    such that $f^k(x_0)\in U$ and $f^l(x_0) \in V$,
    so $f^{l-k} U \cap V \neq \emptyset$.


    For basic open sets $V$
    let
    \[
     A_V \coloneqq   \{ x \in X: \exists n.~ f^n(x) \in V\}.
    \]
    By assumption, all the $A_V$ are dense.
    Hence $\bigcap_{V}A_V$ is dense by the \yaref{thm:bct}.

    $A_V = \bigcup_{n \in \Z} f^n(V)$ is open.
\end{subproof}

\begin{claim}
The condition can be written as a $G_\delta$ set.
\end{claim}
\begin{subproof}

    For  $f_0(U) \cap V \neq \emptyset$
    take $u \in U$ such that $f_0(u) \in V$.
    Then there exists  $\epsilon > 0$ such that $B_{\epsilon}(f_0(u)) \subseteq U$,
    hence $B_{\epsilon}(f_0)$ is an open neighbourhood contained
    in $\{f : f(U) \cap V \neq \emptyset \} $.

    For $n = 2$ note that
    $d(f^2(u), f^2_0(u) \le  d(f(f(u)), f_0(f(u))) + d(f_0(f(u)), f_0(f_0(u)))$.
    The first part can be bounded by $d(f,f_0)$.
    For the second part,
    note that there exists $\delta$ such that
    \[d(a,b) < \delta \implies d(f_0(a), f_0(b)) < \frac{\epsilon}{2}.\]
    Let $\eta \coloneqq  \min \{\delta, \frac{\epsilon}{2}\}$
    and consider $d_\infty(f,f_0) < \epsilon$.

    For other  $n$ it is some more work, which is left as an exercise.
\end{subproof}