From ad39e2ac6d17def2a06bcfce3c7d6b5b5d809c21 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Mon, 23 Oct 2023 22:43:31 +0200
Subject: [PATCH] 03 beginning

---
 inputs/lecture_03.tex | 27 +++++++++++++++++++++++++++
 1 file changed, 27 insertions(+)
 create mode 100644 inputs/lecture_03.tex

diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex
new file mode 100644
index 0000000..1404146
--- /dev/null
+++ b/inputs/lecture_03.tex
@@ -0,0 +1,27 @@
+\lecture{03}{2023-10–23}{Cantor-Bendixson}
+
+\begin{theorem}[Cantor-Bendixson]
+    \yaref{thm:cantorbendixson}{Cantor-Bendixson}{Cantor-Bendixson}
+
+    If $A \subseteq  \R$ is closed,
+    it is either at most countable or else
+    $A$ contains a perfect set.
+\end{theorem}
+\begin{corollary}
+    If $A se \R$ is closed,
+    then either $A \le \N$ or $A \sim \R$.
+\end{corollary}
+\begin{fact}
+    $A' = \{x \in  \R | \forall  a < x < b.~ (a,b) \cap A \text{ is at least countable}\}$.
+\end{fact}
+\begin{proof}
+    $\supseteq$ is clear.
+    For $\subseteq $, fix $a < x < b$ 
+    and let us define $(y_n: n \in  \omega)$
+    as well as $((a_n, b_n): n \in \omega)$.
+    
+    % TODO
+    
+\end{proof}
+
+