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inputs/lecture_03.tex

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+\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}
+
+