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@ -103,7 +103,9 @@
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Let $X$ be a completely metrizable space.
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Let $X$ be a completely metrizable space.
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Then every comeager set of $X$ is dense in $X$.
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Then every comeager set of $X$ is dense in $X$.
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\end{theorem}
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\end{theorem}
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\todo{Proof (copy from some other lecture)}
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\gist{%
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\todo{Proof (copy from some other lecture)}
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}{Not proved in the lecture.}
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\begin{theoremdef}
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\begin{theoremdef}
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Let $X$ be a topological space.
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Let $X$ be a topological space.
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The following are equivalent:
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The following are equivalent:
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@ -118,7 +120,21 @@
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\footnote{cf.~\yaref{s5e1}}
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\footnote{cf.~\yaref{s5e1}}
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\end{theoremdef}
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\end{theoremdef}
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\begin{proof}
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\begin{proof}
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\todo{Proof (short)}
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(i) $\implies$ (ii)
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\gist{%
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Consider a comeager set $A$.
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Let $U\neq \emptyset$ be any open set. Since $U$ is
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non-meager, we have $A \cap U \neq \emptyset$.
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}{The intersection of a comeager and a non-meager set is nonempty.}
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(ii) $\implies$ (iii)
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The complement of an open dense set is nwd.
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\gist{%
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Hence the intersection of countable
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many open dense sets is comeager.
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}{}
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(iii) $\implies$ (i)
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(iii) $\implies$ (i)
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Let us first show that $X$ is non-meager.
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Let us first show that $X$ is non-meager.
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@ -77,7 +77,7 @@
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sets that are not clopen)}{}.
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sets that are not clopen)}{}.
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\end{example}
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\end{example}
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\subsection{Turning Borels Sets into Clopens}
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\subsection{Turning Borel Sets into Clopens}
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\begin{theorem}%
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\begin{theorem}%
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\gist{%
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\gist{%
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@ -109,55 +109,74 @@
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into $B$.
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into $B$.
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\end{corollary}
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\end{corollary}
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\begin{proof}
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\begin{proof}
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Pick $\cT_B \supset \cT$
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\gist{%
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such that $(X, \cT_B)$ is Polish,
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Pick $\cT_B \supset \cT$
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$B$ is clopen in $\cT_B$ and
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such that $(X, \cT_B)$ is Polish,
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$\cB(X,\cT) = \cB(X, \cT_B)$.
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$B$ is clopen in $\cT_B$ and
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$\cB(X,\cT) = \cB(X, \cT_B)$.
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Therefore $(\cB, \cT_B\defon{B})$ is Polish.
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Therefore $(\cB, \cT_B\defon{B})$ is Polish.
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We know that there is an embedding
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We know that there is an embedding
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$f\colon 2^{\omega} \to (B, \cT_{B}\defon{B})$.
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$f\colon 2^{\omega} \to (B, \cT_{B}\defon{B})$.
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Consider $f\colon 2^{\omega} \to B \subseteq (X, \cT)$.
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Consider $f\colon 2^{\omega} \to B \subseteq (X, \cT)$.
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This is still continuous as $\cT \subseteq \cT_B$.
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This is still continuous as $\cT \subseteq \cT_B$.
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Since $2^{\omega}$ is compact, $f$ is an embedding.
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Since $2^{\omega}$ is compact, $f$ is an embedding.
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%\todo{Think about this}
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}{%
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Clopenize $B$.
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We can embed $2^{ \omega}$ into Polish spaces.
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Clopenization makes the topology finer,
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so this is still continuous wrt.~the original topology.
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$2^{\omega}$ is compact, so this is an embedding.
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}
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\end{proof}
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\end{proof}
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\begin{refproof}{thm:clopenize}
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\begin{refproof}{thm:clopenize}
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We show that
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\gist{%
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\begin{IEEEeqnarray*}{rCl}
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We show that
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A \coloneqq \{B \subseteq \cB(X, \cT)&:\exists & \cT_B \supseteq \cT .\\
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\begin{IEEEeqnarray*}{rCl}
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&& (X, \cT_B) \text{ is Polish},\\
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A \coloneqq \{B \subseteq \cB(X, \cT)&:\exists & \cT_B \supseteq \cT .\\
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&& \cB(X, \cT) = \cB(X, \cT_B)\\
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&& (X, \cT_B) \text{ is Polish},\\
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&& B \text{ is clopen in $\cT_B$}\\
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&& \cB(X, \cT) = \cB(X, \cT_B)\\
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\}
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&& B \text{ is clopen in $\cT_B$}\\
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\end{IEEEeqnarray*}
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\}
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is equal to the set of Borel sets.
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\end{IEEEeqnarray*}
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is equal to the set of Borel sets.
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The proof rests on two lemmata:
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}{%
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Let $A$ be the set of clopenizable sets.
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We show that $A = \cB(X)$.
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}
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\gist{The proof rests on two lemmata:}{}
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\begin{lemma}
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\begin{lemma}
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\label{thm:clopenize:l1}
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\label{thm:clopenize:l1}
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Let $(X,\cT)$ be a Polish space.
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\gist{%
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Then for any $F \overset{\text{closed}}{\subseteq} X$ (wrt. $\cT$)
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Let $(X,\cT)$ be a Polish space.
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there is $\cT_F \supseteq \cT$
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Then for any $F \overset{\text{closed}}{\subseteq} X$ (wrt. $\cT$)
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such that $\cT_F$ is Polish,
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there is $\cT_F \supseteq \cT$
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$\cB(\cT) = \cB(\cT_F)$
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such that $\cT_F$ is Polish,
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and $F$ is clopen in $\cT_F$.
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$\cB(\cT) = \cB(\cT_F)$
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and $F$ is clopen in $\cT_F$.
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}{%
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Closed sets can be clopenized.
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}
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\end{lemma}
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\end{lemma}
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\begin{proof}
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\begin{proof}
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Consider $(F, \cT\defon{F})$ and $(X \setminus F, \cT\defon{X \setminus F})$.
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\gist{%
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Both are Polish spaces.
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Consider $(F, \cT\defon{F})$ and $(X \setminus F, \cT\defon{X \setminus F})$.
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Take the coproduct%
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Both are Polish spaces.
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\footnote{In the lecture, this was called the \vocab{topological sum}.}
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Take the coproduct%
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$F \oplus (X \setminus F)$ of these spaces.
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\footnote{In the lecture, this was called the \vocab{topological sum}.}
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This space is Polish,
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$F \oplus (X \setminus F)$ of these spaces.
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and the topology is generated by $\cT \cup \{F\}$,
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This space is Polish,
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hence we do not get any new Borel sets.
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and the topology is generated by $\cT \cup \{F\}$,
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hence we do not get any new Borel sets.
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}{Consider $(F, \cT\defon{F}) \oplus (X \setminus F, \cT\defon{X \setminus F})$.}
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\end{proof}
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\end{proof}
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So all closed sets are in $A$.
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\gist{%
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Furthermore $A$ is closed under complements,
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So all closed sets are in $A$.
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since complements of clopen sets are clopen.
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Furthermore $A$ is closed under complements,
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since complements of clopen sets are clopen.
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}{So $\Sigma^0_1(X), \Pi^0_1(X) \subseteq A$.}
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\begin{lemma}
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\begin{lemma}
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\label{thm:clopenize:l2}
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\label{thm:clopenize:l2}
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is Polish
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is Polish
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and $\cB(\cT_\infty) = \cB(T)$.
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and $\cB(\cT_\infty) = \cB(T)$.
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\end{lemma}
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\end{lemma}
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\begin{proof}
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\begin{refproof}{thm:clopenize:l2}
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We have that $\cT_\infty$ is the smallest
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\gist{%
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topology containing all $\cT_n$.
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We have that $\cT_\infty$ is the smallest
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To get $\cT_\infty$
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topology containing all $\cT_n$.
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consider
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To get $\cT_\infty$ consider
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\[
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\[
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\cF \coloneqq \{A_1 \cap A_2 \cap \ldots \cap A_n : A_i \in \cT_i\}.
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\cF \coloneqq \{A_1 \cap A_2 \cap \ldots \cap A_n : A_i \in \cT_i\}.
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\]
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\]
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Then
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Then
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\[
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\[
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\cT_\infty = \{\bigcup_{i<\omega} B_i : B_i \in \cF\}.
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\cT_\infty = \{\bigcup_{i<\omega} B_i : B_i \in \cF\}.
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\]
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\]
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(It suffices to take countable unions,
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(It suffices to take countable unions,
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since we may assume that the $A_1, \ldots, A_n$ in the
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since we may assume that the $A_1, \ldots, A_n$ in the
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definition of $\cF$ belong to
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definition of $\cF$ belong to
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a countable basis of the respective $\cT_n$).
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a countable basis of the respective $\cT_n$).
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}{}
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% Proof was finished in lecture 8
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% Proof was finished in lecture 8
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Let $Y = \prod_{n \in \N} (X, \cT_n)$.
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Let $Y = \prod_{n \in \N} (X, \cT_n)$.
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\begin{claim}
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\begin{claim}
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$\delta$ is a homeomorphism.
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$\delta$ is a homeomorphism.
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\end{claim}
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\end{claim}
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\gist{%
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\begin{subproof}
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\begin{subproof}
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Clearly $\delta$ is a bijection.
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Clearly $\delta$ is a bijection.
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We need to show that it is continuous and open.
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We need to show that it is continuous and open.
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\delta(V) = D \cap (X \times X \times \ldots \times U_{n_1} \times \ldots \times U_{n_2} \times \ldots \times U_{n_u} \times X \times \ldots) \overset{\text{open}}{\subseteq} D.
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\delta(V) = D \cap (X \times X \times \ldots \times U_{n_1} \times \ldots \times U_{n_2} \times \ldots \times U_{n_u} \times X \times \ldots) \overset{\text{open}}{\subseteq} D.
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\]
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\]
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\end{subproof}
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\end{subproof}
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}{}
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This will finish the proof since
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\begin{claim}
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\[
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$D = \{(x,x,\ldots) \in Y : x \in X\} \overset{\text{closed}}{\subseteq} Y.$
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D = \{(x,x,\ldots) \in Y : x \in X\} \overset{\text{closed}}{\subseteq} Y
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\end{claim}
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\]
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\gist{%
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Why? Let $(x_n) \in Y \setminus D$.
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\begin{subproof}
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Then there are $i < j$ such that $x_i \neq x_j$.
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Let $(x_n) \in Y \setminus D$.
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Take disjoint open $x_i \in U$, $x_j \in V$.
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Then there are $i < j$ such that $x_i \neq x_j$.
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Then
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Take disjoint open $x_i \in U$, $x_j \in V$.
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\[(x_n) \in X \times X \times \ldots \times U \times \ldots \times X \times \ldots \times V \times X \times \ldots\]
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Then
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is open in $Y\setminus D$.
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\[(x_n) \in X \times X \times \ldots \times U \times \ldots \times X \times \ldots \times V \times X \times \ldots\]
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Hence $Y \setminus D$ is open, thus $D$ is closed.
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is open in $Y\setminus D$.
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It follows that $D$ is Polish.
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Hence $Y \setminus D$ is open, thus $D$ is closed.
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\end{proof}
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\end{subproof}
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It follows that $D$ is Polish.
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}{}
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\end{refproof}
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We need to show that $A$ is closed under countable unions.
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\gist{%
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By \yaref{thm:clopenize:l2} there exists a topology
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We need to show that $A$ is closed under countable unions.
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$\cT_\infty$ such that $A = \bigcup_{n < \omega} A_n$ is open in $\cT_\infty$
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By \yaref{thm:clopenize:l2} there exists a topology
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and $\cB(\cT_\infty) = \cB(\cT)$.
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$\cT_\infty$ such that $A = \bigcup_{n < \omega} A_n$ is open in $\cT_\infty$
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Applying \yaref{thm:clopenize:l1}
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and $\cB(\cT_\infty) = \cB(\cT)$.
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yields a topology $\cT_\infty'$ such that
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Applying \yaref{thm:clopenize:l1}
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$(X, \cT_\infty')$ is Polish,
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yields a topology $\cT_\infty'$ such that
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$\cB(\cT_\infty') = \cB(\cT)$
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$(X, \cT_\infty')$ is Polish,
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and $A $ is clopen in $\cT_{\infty}'$.
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$\cB(\cT_\infty') = \cB(\cT)$
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and $A $ is clopen in $\cT_{\infty}'$.
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}{}
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\end{refproof}
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\end{refproof}
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\lecture{08}{2023-11-10}{}\footnote{%
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\lecture{08}{2023-11-10}{}%
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\gist{\footnote{%
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In the beginning of the lecture, we finished
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In the beginning of the lecture, we finished
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the proof of \yaref{thm:clopenize:l2}.
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the proof of \yaref{thm:clopenize:l2}.
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This has been moved to the notes on lecture 7.%
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This has been moved to the notes on lecture 7.%
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}
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}}{}
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\subsection{Parametrizations}
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\subsection{Parametrizations}
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%\todo{choose better title}
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%\todo{choose better title}
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\item $\{U_y : y \in Y\} = \Gamma(X)$.
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\item $\{U_y : y \in Y\} = \Gamma(X)$.
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\end{itemize}
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\end{itemize}
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\end{definition}
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\end{definition}
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\gist{%
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\begin{example}
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\begin{example}
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Let $X = \omega^\omega$, $Y = 2^{\omega}$
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Let $X = \omega^\omega$, $Y = 2^{\omega}$
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and consider $\Gamma = \Sigma^0_{\omega+5}(\omega^\omega)$.
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and consider $\Gamma = \Sigma^0_{\omega+5}(\omega^\omega)$.
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We will show that there is a $2^{\omega}$-universal
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We will show that there is a $2^{\omega}$-universal
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set for $\Gamma$.
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set for $\Gamma$.
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\end{example}
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\end{example}
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}{}
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\begin{theorem}
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\begin{theorem}
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\label{thm:cantoruniversal}
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\label{thm:cantoruniversal}
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(continuous wrt.~to the topology of $X$)
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(continuous wrt.~to the topology of $X$)
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On the other hand
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On the other hand
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\[
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\[
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X \hookrightarrow\cN \overset{\text{continuous embedding}}{\hookrightarrow}\cC
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X \hookrightarrow\cN \overset{\text{continuous embedding\footnotemark}}{\hookrightarrow}\cC
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\]
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\]
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\todo{second inclusion was on a homework sheet}
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\footnotetext{cf.~\yaref{s2e4}}
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For the first inclusion,
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For the first inclusion,
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recall that there is a continuous bijection $b\colon D \to X$,
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recall that there is a continuous bijection $b\colon D \to X$,
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where $D \overset{\text{closed}}{\subseteq} \cN$.
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where $D \overset{\text{closed}}{\subseteq} \cN$.
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}{easy}
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}{easy}
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\end{proof}
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\end{proof}
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% TODO ANKI-MARKER
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\begin{theorem}[Lusin-Sierpinski]
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\begin{theorem}[Lusin-Sierpinski]
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The set $\LO \setminus \WO$
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The set $\LO \setminus \WO$
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(resp.~$2^{\Q} \setminus \WO$)
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(resp.~$2^{\Q} \setminus \WO$)
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\begin{proof}
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\begin{proof}
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We will find a continuous function
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We will find a continuous function
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$f\colon \Tr \to \LO$ such that
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$f\colon \Tr \to \LO$ such that
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\gist{%
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\[
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\[
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x \in \WF \iff f(x) \in \WO
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x \in \WF \iff f(x) \in \WO
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\]
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\]
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(equivalently $x \in \IF \iff f(x) \in \LO \setminus \WO$).
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(equivalently $x \in \IF \iff f(x) \in \LO \setminus \WO$).
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This suffices, since $\IF \subseteq \Tr$ is $\Sigma^1_1$-complete
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This suffices, since $\IF \subseteq \Tr$ is $\Sigma^1_1$-complete
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}{
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$f^{-1}(\LO \setminus \WO) = \IF$.
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This suffices
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}
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(see \yaref{cor:ifs11c}).
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(see \yaref{cor:ifs11c}).
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Fix a bijection $b\colon \N \to \N^{<\N}$.
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Fix a bijection $b\colon \N \to \N^{<\N}$.
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\begin{idea}
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\begin{idea}
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For $T \in \Tr$ consider
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For $T \in \Tr$ consider
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$<_{KB}\defon{T}$
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$<_{KB}\defon{T}$.
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% TODO?
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\end{idea}
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\end{idea}
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Let $\alpha \in \Tr$.
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Let $\alpha \in \Tr$.
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(i.e.~$m \le_{f(\alpha)} n$)
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(i.e.~$m \le_{f(\alpha)} n$)
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iff
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iff
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\begin{itemize}
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\begin{itemize}
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\item $(\alpha(b(m)) = \alpha(b(n)) = 1$
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\item $\alpha(b(m)) = \alpha(b(n)) = 1$
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and $b(m) \le_{KB} b(n)$
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and $b(m) \le_{KB} b(n)$
|
||||||
(recall that we identified $\Tr$
|
(recall that we identified $\Tr$
|
||||||
with a subset of ${2^{\N}}^{<\N}$),
|
with a subset of ${2^{\N}}^{<\N}$),
|
||||||
|
@ -123,15 +125,16 @@ with $(f^{-1}(\{1\}), <)$.
|
||||||
\end{proof}
|
\end{proof}
|
||||||
|
|
||||||
% TODO: new section?
|
% TODO: new section?
|
||||||
|
\gist{%
|
||||||
Recall that a \vocab{rank} on a set $C$
|
Recall that a \vocab{rank} on a set $C$
|
||||||
is a map $\phi\colon C \to \Ord$.
|
is a map $\phi\colon C \to \Ord$.
|
||||||
\begin{example}
|
\begin{example}
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
\begin{IEEEeqnarray*}{rCl}
|
||||||
\otp \colon \WO &\longrightarrow & \Ord \\
|
\otp \colon \WO &\longrightarrow & \Ord \\
|
||||||
x &\longmapsto & \text{the unique $\alpha \in \Ord$ such that $x \cong \alpha$}.
|
x &\longmapsto & \text{the unique $\alpha \in \Ord$ such that $x \cong \alpha$}.
|
||||||
\end{IEEEeqnarray*}
|
\end{IEEEeqnarray*}
|
||||||
\end{example}
|
\end{example}
|
||||||
|
}{}
|
||||||
|
|
||||||
\begin{definition}
|
\begin{definition}
|
||||||
A \vocab{prewellordering} $\preceq$
|
A \vocab{prewellordering} $\preceq$
|
||||||
|
@ -141,7 +144,7 @@ is a map $\phi\colon C \to \Ord$.
|
||||||
\item reflexive,
|
\item reflexive,
|
||||||
\item transitive,
|
\item transitive,
|
||||||
\item total (any two $x,y$ are comparable),
|
\item total (any two $x,y$ are comparable),
|
||||||
\item $\prec$ ($x \prec y \iff x \preceq y \land y \not\preceq x$) is well-founded,
|
\item $\prec$ \gist{($x \prec y \iff x \preceq y \land y \not\preceq x$)}{} is well-founded,
|
||||||
in the sense that there are no descending infinite chains.
|
in the sense that there are no descending infinite chains.
|
||||||
\end{itemize}
|
\end{itemize}
|
||||||
\end{definition}
|
\end{definition}
|
||||||
|
@ -149,7 +152,7 @@ is a map $\phi\colon C \to \Ord$.
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
\item A prewellordering may not be a linear order since
|
\item A prewellordering may not be a linear order since
|
||||||
it is not necessarily antisymmetric.
|
it is not necessarily antisymmetric.
|
||||||
\item The linearly ordered wellfounded sets are exactly the wellordered sets.
|
%\item The linearly ordered wellfounded sets are exactly the wellordered sets.
|
||||||
\item Modding out $x \sim y :\iff x \preceq y \land y \preceq x$
|
\item Modding out $x \sim y :\iff x \preceq y \land y \preceq x$
|
||||||
turns a prewellordering into a wellordering.
|
turns a prewellordering into a wellordering.
|
||||||
\end{itemize}
|
\end{itemize}
|
||||||
|
@ -163,11 +166,11 @@ between downwards-closed ranks and prewellorderings:
|
||||||
\phi_{\preceq}&\longmapsfrom& \preceq,
|
\phi_{\preceq}&\longmapsfrom& \preceq,
|
||||||
\end{IEEEeqnarray*}
|
\end{IEEEeqnarray*}
|
||||||
where $\phi_\preceq(x)$ is defined as
|
where $\phi_\preceq(x)$ is defined as
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
\gist{\begin{IEEEeqnarray*}{rCl}
|
||||||
\phi_{\preceq}(x) &\coloneqq &0 \text{ if $x$ is minimal},\\
|
\phi_{\preceq}(x) &\coloneqq &0 \text{ if $x$ is minimal},\\
|
||||||
\phi_{\preceq}(x) &\coloneqq & \sup \{\phi_{\preceq}(y) + 1 : y \prec x\},
|
\phi_{\preceq}(x) &\coloneqq & \sup \{\phi_{\preceq}(y) + 1 : y \prec x\},
|
||||||
\end{IEEEeqnarray*}
|
\end{IEEEeqnarray*}
|
||||||
i.e.
|
i.e.}{}
|
||||||
\[
|
\[
|
||||||
\phi_{\preceq}(x) = \otp\left(\faktor{\{y \in C : y \prec x\}}{\sim}\right).
|
\phi_{\preceq}(x) = \otp\left(\faktor{\{y \in C : y \prec x\}}{\sim}\right).
|
||||||
\]
|
\]
|
||||||
|
|
|
@ -1,5 +1,6 @@
|
||||||
\lecture{14}{2023-12-01}{}
|
\lecture{14}{2023-12-01}{}
|
||||||
|
|
||||||
|
% TODO ANKI-MARKER
|
||||||
\begin{theorem}[Moschovakis]
|
\begin{theorem}[Moschovakis]
|
||||||
If $C$ is coanalytic,
|
If $C$ is coanalytic,
|
||||||
then there exists a $\Pi^1_1$-rank on $C$.
|
then there exists a $\Pi^1_1$-rank on $C$.
|
||||||
|
|
Loading…
Reference in a new issue