159 lines
5.5 KiB
TeX
159 lines
5.5 KiB
TeX
\tutorial{09}{2023-12-12}{}
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\begin{fact}
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Let $X,Y$ be topological spaces
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$X$ (quasi-)compact and
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$Y$ Hausdorff.
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Let $f\colon X\to Y$ be a continuous bijection.
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Then $f$ is a homeomorphism.
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\end{fact}
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\begin{proof}
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Compact subsets of Hausdorff spaces are closed.
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\end{proof}
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\subsection{Sheet 8}
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Material on topological dynamics:
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\begin{itemize}
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\item Terence Tao's notes on ergodic theory 254A: \cite{tao}
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\item \cite{Furstenberg} (uses very different notation!).
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\end{itemize}
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\nr 1
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\begin{remark}
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$\Sigma^1_1$-complete sets are in some sense the ``worst''
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$\Sigma^1_1$-sets:
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Deciding whether an element is contained in the
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$\Sigma^1_1$-complete set
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is at least as ``hard'' as as for any $\Sigma^1_1$ set.
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In particular, $\Sigma^1_1$-complete sets
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are not Borel.
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\end{remark}
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Similarly as in \yaref{prop:ifs11} it can be shown that $L \in \Sigma^1_1(X)$:
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Consider $\{(x, \beta) \in X \times \cN : \forall n.~x_{\beta_n} | x_{\beta_{n+1}}\}$.
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This is closed in $X \times \cN$, since it is a countable intersection of clopen
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sets and $L = \proj_X(D)$.
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Since $\IF \subseteq \Tr$ is $\Sigma^1_1$-complete,
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it suffices to find a Borel map $f\colon \Tr \to X$
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such that $x \in \IF \iff f(x) \in L$.
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Let $\phi\colon \omega^2 + \omega \to \omega$ be bijective
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and let $p_i$ denote the $i$-th prime.
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Define
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\begin{IEEEeqnarray*}{rCl}
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\psi\colon \omega^{<\omega} &\longrightarrow & \omega \setminus \{0\} \\
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(s_0, s_1, \ldots, s_{n-1})&\longmapsto & \prod_{i < n} p_{\phi(\omega \cdot i + s_i)}.
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\end{IEEEeqnarray*}
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Note that $\psi$ is injective and that $s \in \omega^{<\omega}$ is an initial segment
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of $t \in \omega^{<\omega}$ iff $\psi(s) | \psi(t)$.
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Let
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\begin{IEEEeqnarray*}{rCl}
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f' \colon \Tr &\longrightarrow & \cP(\omega \setminus \{0\}) \\
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T &\longmapsto & \{\phi(s) : s \in T\}.
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\end{IEEEeqnarray*}
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We can turn this into a function
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$f\colon \Tr \to (\omega \setminus \{0\})^{\omega}$
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by mapping a subset of $\omega \setminus \{0\}$
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to the unique strictly increasing sequence whose elements are from
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that subset
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(appending $\phi(\omega^2 + n), n \in \omega$, if the subset was finite).
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Note that $T \in \IF \iff f(T) \in L$.
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Furthermore $f$ is Borel, since fixing
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a finite initial sequence (i.e.~a basic open set of $(\omega \setminus \{0\})^{\omega}$)
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amounts to a finite number of conditions on the preimage.
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\nr 2
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\todo{handwritten}
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% Aron
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%
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% \begin{enumerate}[1.]
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% \item This is trivial $\sup \sup$.
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% \item Clearly there are trees of rank $n$ for all $n < \omega$.
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% Glue them together.
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% \[
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% \{(0,0,i,\underbrace{0,\ldots,0}_{i \text{~times}}) | i < \omega\}.
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% \]
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% \item Map infinite branches. $\sup$ the $\le $.
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% \item Induction on $\rho(S)$.
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% Cofinal subsequences bla bla.
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% \end{enumerate}
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\nr 3
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\begin{itemize}
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\item $LO(\N) \overset{\text{closed}}{\subseteq} 2^{\N\times \N}$:
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We have $< \in LO(\N)$ iff
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\begin{itemize}
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\item $\forall x,y.~ (x \neq y \implies (x< y \lor x > y))$,
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\item $\forall x.~(x \not < x)$,
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\item $\forall x,y,x.~(x < y < z \implies x < z)$.
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\end{itemize}
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Write this with $\bigcap$,
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i.e.
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\begin{IEEEeqnarray*}{rCl}
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LO(\N) &=& \bigcap_{n \in \N} \{R: (n,n) \not\in R\}\\
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&& \cap \bigcap_{m < n \in \N} (\{R: (n,m) \in R\} \cup \{R: (m,n) \in R\})\\
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&&\cap \bigcap_{a,b,c \in \N} (\{R: (a,b) \in R \land (b,c) \in R \implies (a,c) \in R\}.
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\end{IEEEeqnarray*}
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This is closed as an intersection of clopen sets.
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\item We apply \yaref{thm:borel} (iv).
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Let $\cF \subseteq LO(\N) \times \cN$ be
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such that the $\cN$-coordinate encodes a strictly
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decreasing sequence, i.e.~
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\[(R, s) \in \cF :\iff \forall n \in \N.~(s(n+1), s(n)) \in R.\]
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We have that
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\[
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\cF = \bigcap_{n \in \N} \{(R,s) \in LO(\N)\times \cN : (s(n+1), s(n)) \in R\}
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\]
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is closed as an intersection of clopen sets.
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Clearly $\pr_{LO(\N)}(\cF)$ is the complement
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of $WO(\N)$, hence $WO(\N)$ is coanalytic.
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\end{itemize}
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\nr 4
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\begin{remark}
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In the lecture we only look at metrizable flows,
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so the definitions from the exercise sheet and from
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the lecture don't agree.
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Everywhere but here we will use the definition from the lecture.
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\end{remark}
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\begin{itemize}
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\item Consider
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\begin{IEEEeqnarray*}{rCl}
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\Phi\colon \Z\text{-flows on } X &\longrightarrow & \Homeo(X) \\
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(\alpha\colon \Z\times X \to X) &\longmapsto & \alpha(1, \cdot)\\
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\begin{pmatrix*}[l]
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\Z\times X &\longrightarrow & X \\
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(z,x) &\longmapsto & \beta^{z}(x)
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\end{pmatrix*}&\longmapsfrom & \beta \in \Homeo(X).
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\end{IEEEeqnarray*}
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Clearly this has the desired properties.
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\item Let $X$ be a compact Polish space.
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What is the Borel complexity of $\Homeo(X)$ inside $\cC(X,X)$?
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Recall that $\cC(X,X)$ is a Polish space with the uniform topology.
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We have
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\begin{IEEEeqnarray*}{rCl}
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\Homeo(X) &=& \{f \in \cC(X,X) : f \text{ is bijective and } f^{-1} \text{ is continuous}\}\\
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&=& \{f \in \cC(X,X) : f \text{ is bijective}\}
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\end{IEEEeqnarray*}
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by the general fact
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\begin{fact}
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Let $X$ be compact and $Y$ Hausdorff,
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$f\colon X \to Y$ a continuous bijection.
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Then $f$ is a homeomorphism.
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\end{fact}
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\end{itemize}
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