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} + +