From 5ed13d630fc197c53c46d72ae13d6df93f27c3c3 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Fri, 2 Feb 2024 01:48:47 +0100 Subject: [PATCH] additional tutorial --- inputs/lecture_03.tex | 5 +-- inputs/lecture_04.tex | 2 +- inputs/lecture_14.tex | 3 +- inputs/lecture_15.tex | 6 ++- inputs/lecture_19.tex | 4 +- inputs/tutorial_07.tex | 9 +++-- inputs/tutorial_14.tex | 20 ++++++++++ inputs/tutorial_15.tex | 87 ++++++++++++++++++++++++++++++++++++++++++ logic3.tex | 1 + 9 files changed, 125 insertions(+), 12 deletions(-) create mode 100644 inputs/tutorial_15.tex diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex index dce6608..85c8c5a 100644 --- a/inputs/lecture_03.tex +++ b/inputs/lecture_03.tex @@ -197,8 +197,7 @@ Let $X \neq \emptyset$ be a Polish space. Then there is a closed subset \[ - D \subseteq \N^\N \text{\reflectbox{$\coloneqq$}} \cN - % TODO correct N for the Baire space? + D \subseteq \N^\N \mathbin{\text{\reflectbox{$\coloneqq$}}} \cN \] and a continuous bijection from $D$ onto $X$ (the inverse does not need to be continuous). @@ -239,7 +238,7 @@ \end{enumerate} \gist{% - Suppose we already have $F_s \text{\reflectbox{$\coloneqq$}} F$. + Suppose we already have $F_s \mathbin{\text{\reflectbox{$\coloneqq$}}} F$. We need to construct a partition $(F_i)_{i \in \N}$ of $F$ with $\overline{F_i} \subseteq F$ and $\diam(F_i) < \epsilon$ diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex index 21ebbd1..8267a0f 100644 --- a/inputs/lecture_04.tex +++ b/inputs/lecture_04.tex @@ -62,7 +62,7 @@ We extend $f$ to $g\colon\cN \to X$ in the following way: - Take $S \coloneqq \{s \in \N^{<\N}: \exists x \in D, n \in \N.~x=s\defon{n}\}$. + Take $S \coloneqq \{s \in \N^{<\N}: \exists x \in D, n \in \N.~x\defon{n} = s\}$. Clearly $S$ is a pruned tree. Moreover, since $D$ is closed, we have that (cf.~\yaref{s3e1}) \[ diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index 258aaca..6501c74 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -4,6 +4,7 @@ If $C$ is coanalytic, then there exists a $\Pi^1_1$-rank on $C$. \end{theorem} +% TODO show that WO sse 2^QQ is Pi_1^1 complete \begin{proof} \gist{% Pick a $\Pi^1_1$-complete set. @@ -25,7 +26,7 @@ % \arrow["\subseteq"', hook, from=2-3, to=1-3] % \end{tikzcd} Let $X = 2^{\Q} \supseteq \WO$. - We have already show that $\WO$ is $\Pi^1_1$-complete. + We have already shown that $\WO$ is $\Pi^1_1$-complete.% TODO REF Set $\phi(x) \coloneqq \otp(x)$ ($\otp\colon \WO \to \Ord$ denotes the order type). diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index 8b97525..9e57be4 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -170,7 +170,7 @@ Recall: Let $(T,X)$ and $(T,Y)$ be flows. A \vocab{factor map} $\pi\colon (T,X) \to (T,Y)$ is a continuous surjection $X \twoheadrightarrow Y$ - commuting with the group action, + that is $T$-equivariant, i.e.~$\forall t \in T, x \in X.~\pi(t\cdot x) = t\cdot \pi(x)$. If such a factor map exists, we also say that $(T,Y)$ is a \vocab{factor} @@ -257,7 +257,9 @@ Recall: Let $\Sigma = \{(X_i, T) : i \in I\} $ be a collection of factors of $(X,T)$. % TODO State precise definition of a factor Let $\pi_i\colon (X,T) \to (X_i, T)$ denote the factor map. - Then $(X, T)$ is the \vocab{limit} of $\Sigma$ + Then $(X, T)$ is a \vocab{limit}% + \footnote{This does not seem to be a limit in the category theory sense.} + of $\Sigma$ iff \[ \forall x_1,x_2 \in X.~\exists i \in I.~\pi_i(x_1) \neq \pi_i(x_2). diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 3c1d7fd..159746e 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -65,9 +65,9 @@ equicontinuity coincide. \end{tikzcd}\] \end{theorem} \begin{proof} - % TODO Think about this We want to apply Zorn's lemma. - If suffices to show that isometric flows are closed under inverse limits, + If suffices to show that isometric flows are closed under inverse limits,% + \footnote{This seems to be an inverse limit in the category theory sense.} i.e.~if $(Y_\alpha, f_{\alpha,\beta})$, $\beta < \alpha \le \Theta$ are isometric, then the inverse limit $Y$ is isometric.% diff --git a/inputs/tutorial_07.tex b/inputs/tutorial_07.tex index d5a064b..820928b 100644 --- a/inputs/tutorial_07.tex +++ b/inputs/tutorial_07.tex @@ -34,6 +34,9 @@ \end{enumerate} \nr 2 + +Recall \yaref{thm:clopenize}: + \begin{fact} Let $(X,\tau)$ be a Polish space and $A \in \cB(X)$. @@ -41,10 +44,10 @@ with the same Borel sets as $\tau$ such that $A$ is clopen. - (Do it for $A$ closed, - then show that the sets which work - form a $\sigma$-algebra). \end{fact} +(Do it for $A$ closed, +then show that the sets which work +form a $\sigma$-algebra). \begin{enumerate}[(a)] \item Let $(X, \tau)$ be Polish. diff --git a/inputs/tutorial_14.tex b/inputs/tutorial_14.tex index 580ed34..f4f9f41 100644 --- a/inputs/tutorial_14.tex +++ b/inputs/tutorial_14.tex @@ -11,6 +11,26 @@ \nr 3 % somewhat examinable (for 1.0) +% TODO + +\begin{enumerate}[(a)] + \item $(X,T)$ is distal iff it does not have a proximal pair, + i.e.~$a\neq b$, $c$ such that $t_n \in T$, + $t_na, t_nb \to c$. + + Equivalently, + for all $a,b$ there exists an $\epsilon$, + such that for all $t \in T$, $d(ta,tb) > \epsilon$. + + + \item % TODO (not too hard) + % (b) + % Let $(X,T)$ be distal with a dense orbit, + % then it is distal minimal. + % Sheet 8: has dense orbit is Borel + % Distal flow decomposes into distal minimal flows. +\end{enumerate} + \nr 4 diff --git a/inputs/tutorial_15.tex b/inputs/tutorial_15.tex new file mode 100644 index 0000000..9bb251a --- /dev/null +++ b/inputs/tutorial_15.tex @@ -0,0 +1,87 @@ +\tutorial{15}{2024-01-31}{Additions} + +\subsection{Additional Tutorial} + +The following is not relevant for the exam, +but gives a more general picture. + +Let $ X$ be a topological space. +Let $\cF$ be a filter on $ X$. + +$x \in X$ is a limit point of $\cF$ iff the neighbourhood filter $\cN_x$, +all sets containing an open neighbourhood of $x$, +is contained in $\cF$. + +\begin{fact} + $X$ is Hausdorff iff every filter has at most one limit point. +\end{fact} +\begin{proof} + Neighbourhood filters are compatible + iff the corresponding points + can not be separated by open subsets. +\end{proof} + +\begin{fact} + $X$ is (quasi-) compact + iff every ultrafilter converges. +\end{fact} +\begin{proof} + Suppose that $X$ is compact. + Let $\cU$ be an ultrafilter. + Consider the family $\cV = \{\overline{A} : A \in \cU\}$ + of closed sets. + By the FIP we geht that there exist + $c \in X$ such that $c \in \overline{A}$ for all $A \in \cU$. + Let $N$ be an open neighbourhood of $c$. + If $N^c \in \cU$, then $c \in N^c \lightning$ + So we get that $N \in \cU$. + + Let $\{V_i : i \in I\} $ be a family of closed sets with the FIP. + Consider the filter generated by this family. + We extend this to an ultrafilter. + The limit of this ultrafilter is contained in all the $V_i$. +\end{proof} + +Let $X,Y$ be topological spaces, +$\cB$ a filter base on $X$, +$\cF$ the filter generated by $\cB$ +and +$f\colon X \to Y$. +Then $f(\cB)$ is a filter base on $Y$, +since $f(\bigcap A_i ) \subseteq \bigcap f(A_i)$. +We say that $\lim_\cF f = y$, +if $f(\cF) \to y$. + +Equivalently $f^{-1}(N) \in \cF$ +for all neighbourhoods $N$ of $y$. + +In the lecture we only considered $X = \N$. +If $\cB$ is the base of an ultrafilter, +so is $f(\cB)$. + +\begin{fact} + Let $X$ be a topological space + and let $Y$ be Hausdorff. + Let $f,g \colon X \to Y$ + be continuous. + Let $A \subseteq X$ be dense such that + $f\defon{A} = g\defon{A} $. + Then $f = g$. +\end{fact} +\begin{proof} + Consider $(f,g)^{-1}(\Delta) \supseteq A$. +\end{proof} + +We can uniquely extend $f\colon X \to Y$ continuous +to a continuous $\overline{f}\colon \beta X \to Y$ +by setting $\overline{f}(\cU) \coloneqq \lim_\cU f$. + +Let $V$ be an open neighbourhood of $Y$ in $\overline{f}\left( U) \right) $. +Consider $f^{-1}(V)$. +Consider the basic open set +\[ +\{\cF \in \beta\N : \cF \ni f^{-1}(V)\}. +\] + + + diff --git a/logic3.tex b/logic3.tex index 16ee483..173b5f4 100644 --- a/logic3.tex +++ b/logic3.tex @@ -74,6 +74,7 @@ \input{inputs/tutorial_12b} \input{inputs/tutorial_12} \input{inputs/tutorial_14} +\input{inputs/tutorial_15} \section{Facts} \input{inputs/facts}