From 82f44e36e865eae7e2cdc8e6dda7aadd6c7fe45d Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Sat, 13 Jan 2024 23:42:59 +0100
Subject: [PATCH] FIXED IMPORTANT TYPO! & small changes

---
 inputs/lecture_04.tex |  3 +--
 inputs/lecture_05.tex | 24 +++++++++++++++++-------
 inputs/lecture_06.tex |  2 +-
 3 files changed, 19 insertions(+), 10 deletions(-)

diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex
index bcd97de..ff9fbb8 100644
--- a/inputs/lecture_04.tex
+++ b/inputs/lecture_04.tex
@@ -102,14 +102,13 @@
     Equivalently
     \begin{itemize}
         \item $\overline{A}$ is nwd,
-        \item $X \setminus A$ is dense in $X$,
+        \item $X \setminus \overline{A}$ is dense in $X$,%
         \item $\forall  \emptyset \neq  U \overset{\text{open}}{\subseteq} X.~
             \exists  \emptyset \neq V \overset{\text{open}}{\subseteq} U.~
             V\cap A = \emptyset$.
             (If we intersect $A$ with an open $U$,
             then $A \cap U$ is not dense in $U$).
     \end{itemize}
-    %\todo{Think about this}
 
     A set $B \subseteq  X$ is \vocab{meager} 
     (or \vocab{first category}),
diff --git a/inputs/lecture_05.tex b/inputs/lecture_05.tex
index f2b55ce..10250ae 100644
--- a/inputs/lecture_05.tex
+++ b/inputs/lecture_05.tex
@@ -1,14 +1,22 @@
 \lecture{05}{2023-10-31}{}
 
 \begin{fact}
-    A set $A$ is nwd iff $\overline{A}$ is nwd.
+    \begin{itemize}
+        \item A set $A$ is nwd iff $\overline{A}$ is nwd.
+        \item If $F$ is closed then $F$ is nwd iff $X \setminus F$ is open and dense.
+        \item Any meager set $B$ is contained in a meager $F_{\sigma}$-set.
+    \end{itemize}
 
-    If $F$ is closed then
-    $F$ is nwd iff $X \setminus F$ is open and dense.
 
-    Any meager set $B$ is contained in
-    a meager $F_{\sigma}$-set.
 \end{fact}
+\begin{proof} % remove?
+    \begin{itemize}
+        \item This follows from the definition as $\overline{\overline{A}} = \overline{A}$.
+        \item Trivial.
+        \item Let $B = \bigcup_{n < \omega} B_n$ be a union of nwd sets.
+            Then $B \subseteq \bigcup_{n < \omega} \overline{B_n}$.
+    \end{itemize}
+\end{proof}
 
 \begin{definition}
     A  \vocab{$\sigma$-algebra} on a set $X$
@@ -46,7 +54,7 @@
     \[
         \left( \bigcup_{n < \omega} A_n \right) \symdif \left( \bigcup_{n < \omega} U_n \right)
     \]
-    is meager,\todo{small exercise}
+    is meager,
     hence $\bigcup_{n < \omega} A_n \in \cA$.
 
     Let $A \in \cA$.
@@ -61,7 +69,9 @@
         In particular, $F \symdif \inter(F)$ is nwd.
     \end{claim}
     \begin{refproof}{thm:bairesigma:c1}
-        \todo{TODO}
+        $F \setminus \inter(F)$ is closed,
+        hence $\overline{F \setminus \inter(F)} = F \setminus \inter(F)$.
+        Clearly $\inter(F\setminus\inter(F)) = \emptyset$.
     \end{refproof}
 
     From the claim we get that
diff --git a/inputs/lecture_06.tex b/inputs/lecture_06.tex
index cc218da..1f74848 100644
--- a/inputs/lecture_06.tex
+++ b/inputs/lecture_06.tex
@@ -113,7 +113,7 @@ and $\Pi^0_2 = G_\delta$.
 
 Furthermore define
 \[
-\Delta^0_\alpha(X(X)) \coloneqq \Sigma^0_\alpha(X) \cap \Pi^0_\alpha(X),
+\Delta^0_\alpha(X) \coloneqq \Sigma^0_\alpha(X) \cap \Pi^0_\alpha(X),
 \]
 i.e.~$\Delta^0_1$ is the set of clopen sets.