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Josia Pietsch 2024-01-23 22:07:45 +01:00
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2
inputs/lecture_04.tex
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@ -164,4 +164,4 @@ Note that open sets and meager sets have the Baire property.
by the \yaref{thm:bct}. by the \yaref{thm:bct}.
\end{itemize} \end{itemize}
\end{example} \end{example}
} }{}

15
inputs/lecture_05.tex
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@ -95,7 +95,7 @@
% Nwd set of positive measure. % Nwd set of positive measure.
% TODO % TODO
% remove open intervals such that their length does not add to 0 % remove open intervals such that their length does not add to 0
% %
%\end{example} %\end{example}
\begin{theorem}[Baire Category theorem] \begin{theorem}[Baire Category theorem]
@ -114,8 +114,8 @@
\item The intersection of countable many \item The intersection of countable many
open dense sets is dense. open dense sets is dense.
\end{enumerate} \end{enumerate}
In this case $X$ is called a \vocab{Baire space}. In this case $X$ is called a \vocab{Baire space}.%
\footnote{see \yaref{s5e1}} \footnote{cf.~\yaref{s5e1}}
\end{theoremdef} \end{theoremdef}
\begin{proof} \begin{proof}
\todo{Proof (short)} \todo{Proof (short)}
@ -127,12 +127,11 @@
We have that We have that
\[ \[
\emptyset = \bigcap_{n} (X \setminus \overline{A_n}) \emptyset = \bigcap_{n} (X \setminus \overline{A_n})
\] \]
is dense by (iii). is dense by (iii).
This proof can be adapted to other open sets $X$. This proof can be adapted to other open sets $X$.
\end{proof} \end{proof}
\begin{notation} \begin{notation}
Let $X ,Y$ be topological spaces, Let $X ,Y$ be topological spaces,
$A \subseteq X \times Y$ $A \subseteq X \times Y$
@ -141,12 +140,12 @@
Let Let
\[ \[
A_x \coloneqq \{y \in Y : (x,y) \in A\} A_x \coloneqq \{y \in Y : (x,y) \in A\}
\] \]
and and
\[ \[
A^y \coloneqq \{x \in X : (x,y) \in A\} . A^y \coloneqq \{x \in X : (x,y) \in A\} .
\] \]
\end{notation} \end{notation}
The following similar to Fubini, The following similar to Fubini,

11
inputs/lecture_16.tex
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@ -175,9 +175,10 @@ By Zorn's lemma, this will follow from
\end{IEEEeqnarray*} \end{IEEEeqnarray*}
The topology induced by the metric The topology induced by the metric
is given by basic open subsets\footnote{see Exercise Sheet 9% TODO REF is given by basic open subsets\footnote{cf.~\yaref{s9e2}}
} of the form
$[U_0; U_1,\ldots, U_n]$, $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$, $[U_0; U_1,\ldots, U_n]$,
for $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$,
where where
\[ \[
[U_0; U_1,\ldots,U_n] \coloneqq [U_0; U_1,\ldots,U_n] \coloneqq
@ -188,8 +189,8 @@ By Zorn's lemma, this will follow from
We want to view flows as a metric space. We want to view flows as a metric space.
For a fixed compact metric space $X$, For a fixed compact metric space $X$,
we can view the flows $(X,\Z)$ as a subset of $\cC(X,X)$. we can view the flows $(X,\Z)$ as a subset of $\cC(X,X)$.
Note that $\cC(X,X)$ is Polish. Note that $\cC(X,X)$ is Polish.\footnote{cf.~\yaref{s1e4}}
Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$. Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$.\footnote{Exercise} % TODO
However we do not want to consider only flows on a fixed space $X$, However we do not want to consider only flows on a fixed space $X$,
but we want to look all flows at the same time. but we want to look all flows at the same time.