From 24aca6746f5fb5f0f5bdf6f1ed7ccc92b90cdf97 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Sun, 4 Feb 2024 00:46:31 +0100
Subject: [PATCH] email about exam (2024-01-22)

---
 inputs/lecture_23.tex | 50 ++++++++++++++++++++++++-------------------
 1 file changed, 28 insertions(+), 22 deletions(-)

diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex
index c26f5b4..aedf7f5 100644
--- a/inputs/lecture_23.tex
+++ b/inputs/lecture_23.tex
@@ -1,5 +1,9 @@
 \lecture{23}{2024-01-19}{More sketches of ideas of Beleznay and Foreman}
 
+% TODO read notes
+% TODO def. almost distal
+% From Lecture 23, you need to know the proposition on page 7 (with the proof), but I won't ask you for other proofs from that lecture
+
 \begin{notation}
     Let $X$ be a Polish space and $\cP$ a property of elements of $X$,
     then we say that $x_0 \in X$ is \vocab{generic} 
@@ -104,28 +108,31 @@ Let $I$ be a linear order
     The order %TODO (Furstenberg rank)
     is a $\Pi^1_1$-rank.
 \end{theorem}
-For the proof one shows that $\le^\ast$ and $<^\ast$
-are  $\Pi^1_1$, where
-\begin{enumerate}[(1)]
-    \item $p_1 \le^\ast p_2$ iff $p_1$ codes
-        a distal minimal flow and if
-        $p_2$ also codes a distal minimal flow,
-        then $\mathop{order}(p_1) \le \mathop{order}(p_2)$.
-    \item $p_1 <^\ast p_2$ iff $p_1$ codes
-        a distal minimal flow and if
-        $p_2$ also codes a distal minimal flow,
-        then $\mathop{order}(p_1) < \mathop{order}(p_2)$.
-\end{enumerate}
-
-One uses that $(Y_{i+1}, T)$ is a maximal
-isometric extension of $(Y_i,T)$
-ind $(X,T)$
-iff for all $x_1,x_2$ from a fixed countable dense set
-in $X$,
-for all $i$ with $\pi_{i\oplus 1}(x_1) = \pi_{i \oplus 1}(x_2)$,
-there is a sequence $(z_k)$ such that $\pi_i(z_k) = \pi_i(x_1)$,
-$F(z_k, x_1) \to  0$, $F(z_k, x_2) \to 0$.
+\begin{proof}[sketch]
+    \notexaminable{
+        For the proof one shows that $\le^\ast$ and $<^\ast$
+        are  $\Pi^1_1$, where
+        \begin{enumerate}[(1)]
+            \item $p_1 \le^\ast p_2$ iff $p_1$ codes
+                a distal minimal flow and if
+                $p_2$ also codes a distal minimal flow,
+                then $\mathop{order}(p_1) \le \mathop{order}(p_2)$.
+            \item $p_1 <^\ast p_2$ iff $p_1$ codes
+                a distal minimal flow and if
+                $p_2$ also codes a distal minimal flow,
+                then $\mathop{order}(p_1) < \mathop{order}(p_2)$.
+        \end{enumerate}
 
+        One uses that $(Y_{i+1}, T)$ is a maximal
+        isometric extension of $(Y_i,T)$
+        ind $(X,T)$
+        iff for all $x_1,x_2$ from a fixed countable dense set
+        in $X$,
+        for all $i$ with $\pi_{i\oplus 1}(x_1) = \pi_{i \oplus 1}(x_2)$,
+        there is a sequence $(z_k)$ such that $\pi_i(z_k) = \pi_i(x_1)$,
+        $F(z_k, x_1) \to  0$, $F(z_k, x_2) \to 0$.
+    }
+\end{proof}
 \begin{proposition}
     The order of a minimal distal flow on a separable,
     metric space is countable.
@@ -171,4 +178,3 @@ $F(z_k, x_1) \to  0$, $F(z_k, x_2) \to 0$.
     Then $\alpha \mapsto U_\alpha$ is an injection.
 \end{proof}
 
-