\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}