28 lines
699 B
TeX
28 lines
699 B
TeX
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\lecture{03}{2023-10–23}{Cantor-Bendixson}
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\begin{theorem}[Cantor-Bendixson]
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\yaref{thm:cantorbendixson}{Cantor-Bendixson}{Cantor-Bendixson}
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If $A \subseteq \R$ is closed,
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it is either at most countable or else
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$A$ contains a perfect set.
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\end{theorem}
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\begin{corollary}
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If $A se \R$ is closed,
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then either $A \le \N$ or $A \sim \R$.
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\end{corollary}
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\begin{fact}
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$A' = \{x \in \R | \forall a < x < b.~ (a,b) \cap A \text{ is at least countable}\}$.
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\end{fact}
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\begin{proof}
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$\supseteq$ is clear.
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For $\subseteq $, fix $a < x < b$
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and let us define $(y_n: n \in \omega)$
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as well as $((a_n, b_n): n \in \omega)$.
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% TODO
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\end{proof}
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