From 4d3e3c2a49c806f1a97a4b62aeccddb7d58f26b0 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 23 Jan 2024 14:29:27 +0100 Subject: [PATCH] tutorial 13 --- inputs/lecture_05.tex | 1 + inputs/tutorial_09.tex | 18 +---- inputs/tutorial_13.tex | 145 +++++++++++++++++++++++++++++++++++++++++ jrpie-gist.sty | 3 + logic3.tex | 1 + 5 files changed, 152 insertions(+), 16 deletions(-) create mode 100644 inputs/tutorial_13.tex diff --git a/inputs/lecture_05.tex b/inputs/lecture_05.tex index 10250ae..e2c10a1 100644 --- a/inputs/lecture_05.tex +++ b/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} diff --git a/inputs/tutorial_09.tex b/inputs/tutorial_09.tex index 8b5b5ee..49682d1 100644 --- a/inputs/tutorial_09.tex +++ b/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 - \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. + \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} diff --git a/inputs/tutorial_13.tex b/inputs/tutorial_13.tex new file mode 100644 index 0000000..30e78c4 --- /dev/null +++ b/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} + diff --git a/jrpie-gist.sty b/jrpie-gist.sty index 5bde104..ea7caa3 100644 --- a/jrpie-gist.sty +++ b/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% diff --git a/logic3.tex b/logic3.tex index b721b4c..6127c23 100644 --- a/logic3.tex +++ b/logic3.tex @@ -65,6 +65,7 @@ \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}