Merge branch 'main' of https://git.abstractnonsen.se/josia-notes/w23-logic-3

inputs/lecture_05.tex

@@ -97,6 +97,7 @@
 %\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

@@ -140,16 +140,7 @@ 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
+   \item Let $X$ be a compact Polish space.
-        \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.
@@ -160,14 +151,9 @@ amounts to a finite number of conditions on the preimage.
         \end{IEEEeqnarray*}
         by the general fact
         \begin{fact}
-            Let $X$ be comapct and  $Y$ Hausdorff,
+            Let $X$ be compact 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

@@ -0,0 +1,145 @@
+\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

@@ -1,6 +1,9 @@
 \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

@@ -66,6 +66,7 @@
 \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}