From 0cceab0af98b892bebc25f8e8ccfe44b76c8f335 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Tue, 23 Jan 2024 22:07:45 +0100
Subject: [PATCH] some small changes

---
 inputs/lecture_04.tex |  2 +-
 inputs/lecture_05.tex | 15 +++++++--------
 inputs/lecture_16.tex | 11 ++++++-----
 3 files changed, 14 insertions(+), 14 deletions(-)

diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex
index 4c16ba2..21ebbd1 100644
--- a/inputs/lecture_04.tex
+++ b/inputs/lecture_04.tex
@@ -164,4 +164,4 @@ Note that open sets and meager sets have the Baire property.
             by the \yaref{thm:bct}.
     \end{itemize}
 \end{example}
-}
+}{}
diff --git a/inputs/lecture_05.tex b/inputs/lecture_05.tex
index 748b6a1..36b1cf0 100644
--- a/inputs/lecture_05.tex
+++ b/inputs/lecture_05.tex
@@ -95,7 +95,7 @@
 %    Nwd set of positive measure.
 %    TODO
 %    remove open intervals such that their length does not add to 0
-%       
+%      
 %\end{example}
 
 \begin{theorem}[Baire Category theorem]
@@ -114,8 +114,8 @@
         \item The intersection of countable many
             open dense sets is dense.
     \end{enumerate}
-    In this case $X$ is called a \vocab{Baire space}.
-    \footnote{see \yaref{s5e1}}
+    In this case $X$ is called a \vocab{Baire space}.%
+    \footnote{cf.~\yaref{s5e1}}
 \end{theoremdef}
 \begin{proof}
     \todo{Proof (short)}
@@ -127,12 +127,11 @@
     We have that
     \[
     \emptyset = \bigcap_{n} (X \setminus \overline{A_n})
-    \] 
+    \]
     is dense by (iii).
     This proof can be adapted to other open sets $X$.
 \end{proof}
 
-
 \begin{notation}
     Let $X ,Y$ be topological spaces,
     $A \subseteq  X \times Y$
@@ -141,12 +140,12 @@
 
     Let
     \[
-    A_x \coloneqq \{y \in Y : (x,y) \in A\} 
-    \] 
+    A_x \coloneqq \{y \in Y : (x,y) \in A\}
+    \]
     and
     \[
     A^y \coloneqq \{x \in X : (x,y) \in A\} .
-    \] 
+    \]
 \end{notation}
 
 The following similar to Fubini,
diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex
index 05e5f71..adf914f 100644
--- a/inputs/lecture_16.tex
+++ b/inputs/lecture_16.tex
@@ -175,9 +175,10 @@ By Zorn's lemma, this will follow from
     \end{IEEEeqnarray*}
 
     The topology induced by the metric
-    is given by basic open subsets\footnote{see Exercise Sheet 9% TODO REF
-    }
-    $[U_0; U_1,\ldots, U_n]$, $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$,
+    is given by basic open subsets\footnote{cf.~\yaref{s9e2}}
+    of the form
+    $[U_0; U_1,\ldots, U_n]$,
+    for $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$,
     where
     \[
         [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.
 For a fixed compact metric space $X$,
 we can view the flows $(X,\Z)$ as a subset of $\cC(X,X)$.
-Note that $\cC(X,X)$ is Polish.
-Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$.
+Note that $\cC(X,X)$ is Polish.\footnote{cf.~\yaref{s1e4}}
+Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$.\footnote{Exercise} % TODO
 
 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.