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email about exam (2024-01-22)
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Josia Pietsch 2024-02-04 00:46:31 +01:00
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inputs/lecture_23.tex
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@ -1,5 +1,9 @@
\lecture{23}{2024-01-19}{More sketches of ideas of Beleznay and Foreman} \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} \begin{notation}
Let $X$ be a Polish space and $\cP$ a property of elements of $X$, 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} then we say that $x_0 \in X$ is \vocab{generic}
@ -104,6 +108,8 @@ Let $I$ be a linear order
The order %TODO (Furstenberg rank) The order %TODO (Furstenberg rank)
is a $\Pi^1_1$-rank. is a $\Pi^1_1$-rank.
\end{theorem} \end{theorem}
\begin{proof}[sketch]
\notexaminable{
For the proof one shows that $\le^\ast$ and $<^\ast$ For the proof one shows that $\le^\ast$ and $<^\ast$
are $\Pi^1_1$, where are $\Pi^1_1$, where
\begin{enumerate}[(1)] \begin{enumerate}[(1)]
@ -125,7 +131,8 @@ in $X$,
for all $i$ with $\pi_{i\oplus 1}(x_1) = \pi_{i \oplus 1}(x_2)$, for all $i$ with $\pi_{i\oplus 1}(x_1) = \pi_{i \oplus 1}(x_2)$,
there is a sequence $(z_k)$ such that $\pi_i(z_k) = \pi_i(x_1)$, there is a sequence $(z_k)$ such that $\pi_i(z_k) = \pi_i(x_1)$,
$F(z_k, x_1) \to 0$, $F(z_k, x_2) \to 0$. $F(z_k, x_1) \to 0$, $F(z_k, x_2) \to 0$.
}
\end{proof}
\begin{proposition} \begin{proposition}
The order of a minimal distal flow on a separable, The order of a minimal distal flow on a separable,
metric space is countable. 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. Then $\alpha \mapsto U_\alpha$ is an injection.
\end{proof} \end{proof}