inputs/lecture_05.tex

@@ -97,7 +97,6 @@
 %\end{example}
 
 \begin{theorem}[Baire Category theorem]
-    \yalabel{Baire Category Theorem}{Baire Category Theorem}{thm:bct}
     Let $X$ be a completely metrizable space.
     Then every comeager set of $X$ is dense in $X$.
 \end{theorem}

inputs/tutorial_09.tex

@@ -140,6 +140,15 @@ amounts to a finite number of conditions on the preimage.
             \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  $\cC(X,X)$?
 
@@ -151,9 +160,14 @@ amounts to a finite number of conditions on the preimage.
         \end{IEEEeqnarray*}
         by the general fact
         \begin{fact}
-            Let $X$ be compact and  $Y$ Hausdorff,
+            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}
 

inputs/tutorial_13.tex

@@ -1,145 +0,0 @@
-\tutorial{13}{2024-01-23}{}
-
-
-Continuation of sheet 8, exercise 4.
-
-\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}
-

jrpie-gist.sty

@@ -1,9 +1,6 @@
 \NeedsTeXFormat{LaTeX2e}
 \ProvidesPackage{jrpie-gist}[2023/01/22 - gist version for lecture notes]
 
-% TODO gist info
-% TODO link to long version (provide link to main document)
-
 \newcommand{\gist}[2]{%
     \ifcsname EnableGist\endcsname%
         #2%

logic3.tex

@@ -66,7 +66,6 @@
 \input{inputs/tutorial_07}
 \input{inputs/tutorial_08}
 \input{inputs/tutorial_09}
-\input{inputs/tutorial_13} % sic!
 \input{inputs/tutorial_10}
 \input{inputs/tutorial_11}
 \input{inputs/tutorial_12b}