FIXED IMPORTANT TYPO! & small changes
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3 changed files with 19 additions and 10 deletions
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@ -102,14 +102,13 @@
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Equivalently
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Equivalently
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\begin{itemize}
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\begin{itemize}
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\item $\overline{A}$ is nwd,
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\item $\overline{A}$ is nwd,
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\item $X \setminus A$ is dense in $X$,
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\item $X \setminus \overline{A}$ is dense in $X$,%
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\item $\forall \emptyset \neq U \overset{\text{open}}{\subseteq} X.~
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\item $\forall \emptyset \neq U \overset{\text{open}}{\subseteq} X.~
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\exists \emptyset \neq V \overset{\text{open}}{\subseteq} U.~
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\exists \emptyset \neq V \overset{\text{open}}{\subseteq} U.~
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V\cap A = \emptyset$.
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V\cap A = \emptyset$.
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(If we intersect $A$ with an open $U$,
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(If we intersect $A$ with an open $U$,
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then $A \cap U$ is not dense in $U$).
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then $A \cap U$ is not dense in $U$).
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\end{itemize}
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\end{itemize}
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%\todo{Think about this}
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A set $B \subseteq X$ is \vocab{meager}
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A set $B \subseteq X$ is \vocab{meager}
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(or \vocab{first category}),
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(or \vocab{first category}),
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@ -1,14 +1,22 @@
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\lecture{05}{2023-10-31}{}
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\lecture{05}{2023-10-31}{}
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\begin{fact}
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\begin{fact}
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A set $A$ is nwd iff $\overline{A}$ is nwd.
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\begin{itemize}
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\item A set $A$ is nwd iff $\overline{A}$ is nwd.
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\item If $F$ is closed then $F$ is nwd iff $X \setminus F$ is open and dense.
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\item Any meager set $B$ is contained in a meager $F_{\sigma}$-set.
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\end{itemize}
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If $F$ is closed then
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$F$ is nwd iff $X \setminus F$ is open and dense.
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Any meager set $B$ is contained in
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a meager $F_{\sigma}$-set.
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\end{fact}
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\end{fact}
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\begin{proof} % remove?
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\begin{itemize}
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\item This follows from the definition as $\overline{\overline{A}} = \overline{A}$.
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\item Trivial.
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\item Let $B = \bigcup_{n < \omega} B_n$ be a union of nwd sets.
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Then $B \subseteq \bigcup_{n < \omega} \overline{B_n}$.
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\end{itemize}
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\end{proof}
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\begin{definition}
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\begin{definition}
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A \vocab{$\sigma$-algebra} on a set $X$
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A \vocab{$\sigma$-algebra} on a set $X$
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@ -46,7 +54,7 @@
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\[
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\[
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\left( \bigcup_{n < \omega} A_n \right) \symdif \left( \bigcup_{n < \omega} U_n \right)
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\left( \bigcup_{n < \omega} A_n \right) \symdif \left( \bigcup_{n < \omega} U_n \right)
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\]
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\]
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is meager,\todo{small exercise}
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is meager,
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hence $\bigcup_{n < \omega} A_n \in \cA$.
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hence $\bigcup_{n < \omega} A_n \in \cA$.
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Let $A \in \cA$.
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Let $A \in \cA$.
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@ -61,7 +69,9 @@
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In particular, $F \symdif \inter(F)$ is nwd.
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In particular, $F \symdif \inter(F)$ is nwd.
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\end{claim}
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\end{claim}
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\begin{refproof}{thm:bairesigma:c1}
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\begin{refproof}{thm:bairesigma:c1}
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\todo{TODO}
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$F \setminus \inter(F)$ is closed,
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hence $\overline{F \setminus \inter(F)} = F \setminus \inter(F)$.
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Clearly $\inter(F\setminus\inter(F)) = \emptyset$.
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\end{refproof}
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\end{refproof}
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From the claim we get that
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From the claim we get that
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@ -113,7 +113,7 @@ and $\Pi^0_2 = G_\delta$.
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Furthermore define
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Furthermore define
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\[
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\[
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\Delta^0_\alpha(X(X)) \coloneqq \Sigma^0_\alpha(X) \cap \Pi^0_\alpha(X),
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\Delta^0_\alpha(X) \coloneqq \Sigma^0_\alpha(X) \cap \Pi^0_\alpha(X),
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\]
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\]
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i.e.~$\Delta^0_1$ is the set of clopen sets.
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i.e.~$\Delta^0_1$ is the set of clopen sets.
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