From 227e2ac7b97f661cb8c2d3fed26e8e2d5a4256e9 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 6 Feb 2024 19:51:03 +0100 Subject: [PATCH 1/7] order / rank --- inputs/lecture_16.tex | 15 ++++++++------- inputs/lecture_19.tex | 19 ++++++------------- 2 files changed, 14 insertions(+), 20 deletions(-) diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index fdd1ed3..7727e51 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -151,16 +151,17 @@ By Zorn's lemma, this will follow from & {(Z,T)} \arrow[from=1-1, to=1-3] \arrow[from=2-2, to=1-3] - \arrow["{\text{isometric extension}}"{description}, from=1-1, to=2-2] +\arrow["{\text{isometric extension}}"{description}, from=1-1, to=2-2] \end{tikzcd}\] \end{theorem} \yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows: -\begin{definition} - Let $(X, \Z)$ be distal minimal. - Then $\rank((X,\Z)) \coloneqq \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$ - where $(X_{\eta}, \Z)$ is as from the definition of quasi-isometric flows, - i.e.~$\rank((X,\Z))$ is the minimal height such - that a tower as in the definition exists. +\begin{definition}[{\cite[{}13.1]{Furstenberg}}] + Let $(X,T)$ be a quasi-isometric flow, + and let $\eta$ be the smallest ordinal + such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$ + with $(X,T) = (X_\xi, T)$. + Then $\eta$ is called the \vocab{rank} or \vocab{order} of the flow + and is denoted by $\rank((X,T))$. \end{definition} \begin{definition}+ diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index fdd9135..5dc267f 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -35,20 +35,13 @@ equicontinuity coincide. By equicontinuity of $T$ we get that $\tilde{d}$ and $d$ induce the same topology on $X$. \end{proof} +\gist{ +Recall that we defined the order of a quasi-isometric flow +to be the minimal number of steps required when building the tower +to reach the flow with a quasi-isometric system (cf.~\yaref{thm:l16:3}, +\yaref{def:floworder}). +}{} - -\begin{question} - What is the minimal number of steps required - when building the tower to reach the flow - as in \yaref{thm:l16:3}? -\end{question} -\begin{definition}[{\cite[{}13.1]{Furstenberg}}] - Let $(X,T)$ be a quasi isometric flow, - and let $\eta$ be the smallest ordinal - such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$ - with $(X,T) = (X_\xi, T)$. - Then $\eta$ is called the \vocab{order} of the flow. -\end{definition} \begin{theorem}[Maximal isometric factor] \label{thm:maxisomfactor} For every flow $(X,T)$ there is a maximal factor $(Y,T)$, $\pi\colon X\to Y$, From e3dea569e8a47eb9a27e39dad9608ac123bac003 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 6 Feb 2024 20:00:49 +0100 Subject: [PATCH 2/7] fixed typo --- inputs/tutorial_08.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/inputs/tutorial_08.tex b/inputs/tutorial_08.tex index fd1c1f3..813ceae 100644 --- a/inputs/tutorial_08.tex +++ b/inputs/tutorial_08.tex @@ -267,14 +267,14 @@ $X_\omega \coloneqq \bigcap X_i$ and $Y_\omega \coloneqq \bigcap Y_i$. % https://q.uiver.app/#q=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 \adjustbox{scale=0.7,center}{% -\[\begin{tikzcd} +\begin{tikzcd} {X \setminus X_\omega =} & {(X_0 \setminus X_1)} & \cup & {(X_1 \setminus X_2)} & \cup & {(X_2 \setminus X_3)} & \cdots & {} \\ {Y\setminus Y_\omega =} & {(Y_0 \setminus Y_1)} & \cup & {(Y_1 \setminus Y_2)} & \cup & {(Y_2 \setminus Y_3)} & \cdots & {} \arrow["f"'{pos=0.7}, from=1-2, to=2-4] \arrow["g"{pos=0.1}, from=2-2, to=1-4] \arrow["f"{pos=0.8}, from=1-6, to=2-8] \arrow["g"{pos=0.1}, from=2-6, to=1-8] -\end{tikzcd}\] +\end{tikzcd} } By \autoref{thm:lusinsouslin} From 5b34fb34d1b66a48a2fee683022b1b25146eb22a Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 6 Feb 2024 20:25:05 +0100 Subject: [PATCH 3/7] countable well-orders --- bibliography/references.bib | 8 ++++++++ inputs/lecture_13.tex | 16 +++------------- inputs/lecture_16.tex | 2 +- 3 files changed, 12 insertions(+), 14 deletions(-) diff --git a/bibliography/references.bib b/bibliography/references.bib index 1958dc8..21b76fd 100644 --- a/bibliography/references.bib +++ b/bibliography/references.bib @@ -44,4 +44,12 @@ title = {Classical Descriptive Set Theory}, volume = {156}, year = {2012}, } +@MISC{3722713, + TITLE = {Embedding of countable linear orders into $\Bbb Q$ as topological spaces}, + AUTHOR = {Eric Wofsey (https://math.stackexchange.com/users/86856/eric-wofsey)}, + HOWPUBLISHED = {Mathematics Stack Exchange}, + NOTE = {URL:https://math.stackexchange.com/q/3722713 (version: 2020-06-16)}, + EPRINT = {https://math.stackexchange.com/q/3722713}, + URL = {https://math.stackexchange.com/q/3722713} +} diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index 019002f..51494d2 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -18,20 +18,10 @@ by associating a function $f\colon \Q \to \{0,1\}$ with $(f^{-1}(\{1\}), <)$. \begin{lemma} - Any countable ordinal embeds into $(\Q,<)$. + Any countable wellorder embeds into $(\Q,<)$. \end{lemma} -\begin{proof}[sketch] - Use transfinite induction. - Suppose we already have $\alpha \hookrightarrow (\Q, <)$, - we need to show that $\alpha +1 \hookrightarrow (\Q, <)$. - Since $(0,1) \cap \Q \cong \Q$, - we may assume $\alpha \hookrightarrow ((0,1), <)$ - and can just set $\alpha \mapsto 2$. - - For a limit $\alpha$ - take a countable cofinal subsequence $\alpha_1 < \alpha_2 < \ldots \to \alpha$. - Then map $[0,\alpha_1)$ to $(0,1)$ - and $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$. +\begin{proof}\footnote{In the lecture this was only done for countable \emph{ordinals}.} + Cf.~\cite{3722713}. \end{proof} % TODO $\WF \subseteq 2^\Q$ is $\Sigma^1_1$-complete. diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index 7727e51..f5ee141 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -23,7 +23,7 @@ $X$ is always compact metrizable. \[ \{h^n : n \in \Z\} \subseteq \cC(X,X)\gist{ = \{f\colon X \to X : f \text{ continuous}\}}{}, \] - where the topology is the uniform convergence topology. % TODO REF EXERCISE + where the topology is the uniform convergence topology. Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$. Since the family $\{h^n : n \in \Z\}$ is uniformly equicontinuous, i.e.~ From 9d601c2e62f6fd8ded70e3e265b429843d0baeac Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 6 Feb 2024 22:13:34 +0100 Subject: [PATCH 4/7] 12.1, 12.2 --- bibliography/references.bib | 2 +- inputs/lecture_16.tex | 1 + inputs/lecture_19.tex | 3 --- inputs/lecture_20.tex | 9 ++++--- inputs/tutorial_14.tex | 51 ++++++++++++++++++++++++++++++++++++- 5 files changed, 58 insertions(+), 8 deletions(-) diff --git a/bibliography/references.bib b/bibliography/references.bib index 21b76fd..f542bda 100644 --- a/bibliography/references.bib +++ b/bibliography/references.bib @@ -46,7 +46,7 @@ year = {2012}, } @MISC{3722713, TITLE = {Embedding of countable linear orders into $\Bbb Q$ as topological spaces}, - AUTHOR = {Eric Wofsey (https://math.stackexchange.com/users/86856/eric-wofsey)}, + AUTHOR = {Eric Wofsey}, HOWPUBLISHED = {Mathematics Stack Exchange}, NOTE = {URL:https://math.stackexchange.com/q/3722713 (version: 2020-06-16)}, EPRINT = {https://math.stackexchange.com/q/3722713}, diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index f5ee141..075f78c 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -156,6 +156,7 @@ By Zorn's lemma, this will follow from \end{theorem} \yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows: \begin{definition}[{\cite[{}13.1]{Furstenberg}}] + \label{def:floworder} Let $(X,T)$ be a quasi-isometric flow, and let $\eta$ be the smallest ordinal such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$ diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 5dc267f..6ad74b0 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -272,9 +272,6 @@ More generally we can show: In particular, $(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$. \end{proof} - -% TODO ANKI-MARKER - \begin{example}[{\cite[p. 513]{Furstenberg}}] \label{ex:19:inftorus} Let $X$ be the infinite torus diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 27259a1..2c6797b 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -19,6 +19,10 @@ Here I'll try to only use multiplicative notation. \end{remark} }{} + + +% TODO ANKI-MARKER + We will be studying projections to the first $d$ coordinates, i.e. \[ @@ -34,9 +38,9 @@ For $d = 1$ we get the circle rotation $x \mapsto e^{\i \alpha} x$. \[ H_m \coloneqq \{x \in S^1 : x^m = 0\} \] - for some $m \in \Z$. + for some $m \in \Z$.% + \footnote{cf.~\yaref{s12e2}} \end{fact} -\todo{Homework!} We will show that $\tau_d$ is minimal for all $d$, i.e.~every orbit is dense. From this it will follow that $\tau$ is minimal. @@ -45,7 +49,6 @@ Let $\pi_n\colon X \to (S^1)^n$ be the projection to the first $n$ coordinates. - \begin{lemma} \label{lem:lec20:1} Let $x,x' \in X$ with $\pi_n(x) = \pi_n(x')$ diff --git a/inputs/tutorial_14.tex b/inputs/tutorial_14.tex index 77c2478..b2f93de 100644 --- a/inputs/tutorial_14.tex +++ b/inputs/tutorial_14.tex @@ -4,9 +4,58 @@ \nr 1 % Examinable -% TODO (there is a more direct way to do it, not using analytic / coanalytic) +Let $\LO(\N) \overset{\text{closed}}{\subseteq} 2^{\N\times \N}$ denote the set of linear orders on $\N$. + +Let $S \subseteq \LO(\N)$ be the set of orders having a least +element and such that every element has an immediate successor. +\begin{itemize} + \item $S$ is Borel in $\LO(\N)$: + + Let $M_n \subseteq \LO(\N)$ be the set of orders with minimal element $n$. + Let $I_{n,m} \subseteq \LO(\N)$ be the set of orders such + that $m$ is the immediate successor of $n$. + + Clearly $S = \left(\bigcap_n \bigcup_{m\neq n} I_{n,m}\right) \cap \bigcup_n M_n$, + so it suffices to show that $M_n$ and $I_{n,m}$ are Borel. + It is $M_n = \bigcap_{m\neq n} \{\prec : m \not\prec n\}$ + and $I_{n,m} = \{\prec: n \prec m\} \cap \bigcap_{i} \{\prec : n \preceq i \preceq m \implies n = i \lor n = m \}$. + \item Give an example of an element of $S$ which is not well-ordered: + + Consider $\{1 - \frac{1}{n} : n \in \N^+\} \cup \{1 + \frac{1}{n} : n \in \N^{+}\} \subseteq \R$ + with the order $<_\R$. + This is an element of $S$, + but $\{x \in S: x \ge 1\}$ has no minimal element, + hence it is not well-ordered. + +\end{itemize} + \nr 2 % Examinable +Recall the definition of the circle shift flow $(\R / \Z, \Z)$ +with parameter $\alpha \in \R$, $1 \cdot x \coloneqq x + \alpha$. + +\begin{itemize} + \item If $\alpha \not\in \Q$, then $(\R / \Z, \Z)$ is minimal: + + This is known as \href{https://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem}{Dirichlet's Approximation Theorem}. + + \item Consider $\R/\Z$ as a topological group. + Any subgroup $H$ of $\R / \Z$ is dense in $\R / \Z$ + or of the form $H = \{ x \in \R / \Z | mx = 0\}$ + for some $m \in \Z$. + + + If $H$ contains an irrational element $\alpha$, then + it is dense by the previous point. + + Suppose that $H \subseteq \Q / \Z$. + Let $D$ be the set of denominators of elements of $H$ + written as irreducible fractions. + If $D$ is finite, + then $H = \{x \in \R / \Z : \mathop{lcm}(D)x = 0\}$. + Otherwise $H$ is dense, as it contains + elements of arbitrarily large denominator. +\end{itemize} \nr 3 From fbf52d882a1b774b326eceb21879c23fe19ff5a4 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 6 Feb 2024 23:59:07 +0100 Subject: [PATCH 5/7] Kuratowski-Ulam gist --- inputs/lecture_05.tex | 14 +++++----- inputs/lecture_06.tex | 60 +++++++++++++++++++++++++++++++++++++++--- inputs/lecture_20.tex | 6 ++--- inputs/lecture_22.tex | 11 +++----- inputs/lecture_24.tex | 10 +++---- inputs/lecture_25.tex | 6 +++++ inputs/tutorial_14.tex | 30 +++++++++------------ inputs/tutorial_15.tex | 21 ++++++++------- 8 files changed, 105 insertions(+), 53 deletions(-) diff --git a/inputs/lecture_05.tex b/inputs/lecture_05.tex index 289847a..444922f 100644 --- a/inputs/lecture_05.tex +++ b/inputs/lecture_05.tex @@ -164,8 +164,10 @@ \] \end{notation} +\gist{% The following similar to Fubini, but for meager sets: +}{} \begin{theorem}[Kuratowski-Ulam] \yalabel{Kuratowski-Ulam}{Kuratowski-Ulam}{thm:kuratowskiulam} @@ -193,6 +195,7 @@ but for meager sets: \end{enumerate} \end{theorem} \begin{refproof}{thm:kuratowskiulam} +\gist{ (ii) and (iii) are equivalent by passing to the complement. \begin{claim}%[1a] @@ -286,16 +289,11 @@ but for meager sets: $M_x$ is comeager as a countable intersection of comeager sets. \end{refproof} +}{} + % \phantom\qedhere % \end{refproof} % TODO fix claim numbers -\gist{% -\begin{remark} - Suppose that $A$ has the BP. - Then there is an open $U$ such that - $A \symdif U \mathbin{\text{\reflectbox{$\coloneqq$}}} M$ is meager. - Then $A = U \symdif M$. -\end{remark} -}{} + diff --git a/inputs/lecture_06.tex b/inputs/lecture_06.tex index f3bb7e0..63709a9 100644 --- a/inputs/lecture_06.tex +++ b/inputs/lecture_06.tex @@ -1,8 +1,8 @@ \lecture{06}{2023-11-03}{} - +\gist{% % \begin{refproof}{thm:kuratowskiulam} \begin{enumerate}[(i)] - \item Let $A$ be a set with the Baire Property. + \item Let $A$ be a set with the Baire property. Write $A = U \symdif M$ for $U$ open and $M$ meager. Then for all $x$, @@ -51,8 +51,8 @@ Towards a contradiction suppose that $A$ is not meager. Then $U$ is not meager. Since $X \times Y$ is second countable, - we have that $A$ is a countable union of open rectangles. - At least one of them, say $G \times H \subseteq A$, + we have that $U$ is a countable union of open rectangles. + At least one of them, say $G \times H \subseteq U$, is not meager. By \yaref{thm:kuratowskiulam:c2}, both $G$ and $H$ are not meager. @@ -71,7 +71,59 @@ ``$\implies$'' This is \yaref{thm:kuratowskiulam:c1b}. \end{enumerate} +}{% + \begin{itemize} + \item (ii) $\iff$ (iii): pass to complement. + \item $F \overset{\text{closed}}{\subseteq} X \times Y$ nwd. + $\implies \{x \in X : F_x \text{ nwd}\} $ comeager: + \begin{itemize} + \item $W = F^c$ is open and dense, show that $\{x : W_x \text{ dense}\}$ + is comeager. + \item $(V_n)$ enumeration of basis. Show that $U_n \coloneqq \{x : V_n \cap W_x \neq \emptyset\}$ + is comeager for all $n$. + \item $U_n$ is open (projection of open) and dense ($W$ is dense, hence $W \cap ( U \times V_n) \neq \emptyset$ for $U$ open). + \end{itemize} + \item $F \subseteq X \times Y$ is nwd $\implies \{x \in X: F_x \text{ nwd}\}$ comeager. + (consider $\overline{F}$). + \item (ii) $\implies$: + $M \subseteq X \times Y$ meager $\implies \{x \in X: M_x \text{ meager}\}$ comeager + (write $M$ as ctbl. union of nwd.) + \item (i): If $A$ has the Baire Property, + then $A = U \symdif M$, $A_x = U_x \symdif M_x$, + $U_x$ open and $\{x : M_x \text{ meager}\}$ comeager + $\implies$ (i). + \item $P \subseteq X$, $Q \subseteq Y$ BP, + then $P \times Q$ meager $\iff$ $P$ or $Q$ meager. + \begin{itemize} + \item $\impliedby$ easy + \item $\implies$ Suppose $P \times Q$ meager, $P$ not meager. + $\emptyset\neq P \cap \underbrace{\{x : (P \times Q)_x \text{ meager} \}}_{\text{comeager}} \ni x$. + $(P \times Q)_x = Q$ is meager. + \end{itemize} + \item (ii) $\impliedby$: + \begin{itemize} + \item $A$ BP, $\{x : A_x \text{ meager}\}$ comeager. + \item $A = U \symdif M$. + \item Suppose $A$ not meager $\leadsto$ $U$ not meager + $\leadsto \exists G \times H \subseteq U$ not meager. + \item $G$ and $H$ are not meager. + \item $\exists x_0 \in G \cap \underbrace{\{x: A_x \text{ meager } \land M_x \text{ meager}\}}_\text{comeager}$. + \item $H$ meager, as + \[ + H \subseteq U_{x_0} \subseteq A_{x_0} \cup M_{x_0}. + \] + \end{itemize} + \end{itemize} +} \end{refproof} +\gist{% +\begin{remark} + Suppose that $A$ has the BP. + Then there is an open $U$ such that + $A \symdif U \mathbin{\text{\reflectbox{$\coloneqq$}}} M$ is meager. + Then $A = U \symdif M$. +\end{remark} +}{} \section{Borel sets} % TODO: fix chapters diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 2c6797b..afe5663 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -20,9 +20,6 @@ \end{remark} }{} - -% TODO ANKI-MARKER - We will be studying projections to the first $d$ coordinates, i.e. \[ @@ -49,6 +46,9 @@ Let $\pi_n\colon X \to (S^1)^n$ be the projection to the first $n$ coordinates. +% TODO ANKI-MARKER + + \begin{lemma} \label{lem:lec20:1} Let $x,x' \in X$ with $\pi_n(x) = \pi_n(x')$ diff --git a/inputs/lecture_22.tex b/inputs/lecture_22.tex index ea4af63..6032904 100644 --- a/inputs/lecture_22.tex +++ b/inputs/lecture_22.tex @@ -146,9 +146,7 @@ For this we define % TODO since for $\overline{x}, \overline{y} \in \mathbb{K}^I$, % $d(x_\alpha, y_\alpha) = d((f(\overline{x})_\alpha, (f(\overline{y})_\alpha))$. \item Minimality:% - \gist{% - \footnote{This is not relevant for the exam.} - + \notexaminable{% Let $\langle E_n : n < \omega \rangle$ be an enumeration of a countable basis for $\mathbb{K}^I$. @@ -165,11 +163,10 @@ For this we define is dense in $\overline{x} \mapsto f(\overline{x})$. Since the flow is distal, it suffices to show that one orbit is dense (cf.~\yaref{thm:distalflowpartition}). - }{ Not relevant for the exam.} + } \item The order of the flow is $\eta$:% - \gist{% - \footnote{This is not relevant for the exam.} + \notexaminable{% Let $\overline{f} = (f_i)_{i \in I} \in \mathbb{K}_I$. Consider the flows we get from $(f_i)_{i < j}$ resp.~$(f_i)_{i \le j}$ @@ -193,6 +190,6 @@ For this we define \end{IEEEeqnarray*} Beleznay and Foreman show that this is open and dense.% % TODO similarities to the lemma used today - }{ Not relevant for the exam.} + } \end{itemize} \end{proof} diff --git a/inputs/lecture_24.tex b/inputs/lecture_24.tex index 77ddcae..366e135 100644 --- a/inputs/lecture_24.tex +++ b/inputs/lecture_24.tex @@ -70,7 +70,8 @@ \begin{notation} In this case we write $x = \ulim{\cU}_n x_n$. \end{notation} -\begin{refproof}{lem:ultrafilterlimit}[sketch] +\begin{refproof}{lem:ultrafilterlimit}\footnote{The proof from the lecture only works + for metric spaces.} Whenever we write $X = Y \cup Z$ we have $(\cU n) x_n \in Y$ or $(\cU n) x_n \in Z$. @@ -120,15 +121,14 @@ This gives $+ \colon \beta\N \times \beta\N \to \beta\N$. This is not commutative, but associative and $a \mapsto a + b$ is continuous -for a fixed $b$. -This is called a left compact topological semigroup. - +for a fixed $b$, +i.e.~it is a left compact topological semigroup. Let $X$ be a compact Hausdorff space and let $T \colon X \to X$ be continuous.% -\footnote{Note that this need not be a homeomorphism, i.e.~we only get a $\N$-action +\footnote{Note that this may not be a homeomorphism, i.e.~we only get a $\N$-action but not a $\Z$-action.} For any $\cU \in \beta\N$, we define $T^{\cU}$ by diff --git a/inputs/lecture_25.tex b/inputs/lecture_25.tex index e92aff3..b2e1ff0 100644 --- a/inputs/lecture_25.tex +++ b/inputs/lecture_25.tex @@ -132,6 +132,12 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$. % TODO general fact: continuous functions agreeing on a dense set % agree everywhere (fact section) \end{proof} +\begin{trivial}+ + More generally, + $\beta$ is a functor from the category of topological + spaces to the category of compact Hausdorff spaces. + It is left adjoint to the inclusion functor. +\end{trivial} % RECAP \gist{% diff --git a/inputs/tutorial_14.tex b/inputs/tutorial_14.tex index b2f93de..ada3004 100644 --- a/inputs/tutorial_14.tex +++ b/inputs/tutorial_14.tex @@ -84,11 +84,12 @@ with parameter $\alpha \in \R$, $1 \cdot x \coloneqq x + \alpha$. \nr 4 % Examinable! +% TODO THINK! +\gist{% % RECAP -Let $X$ be a metrizable topological space. - -Let $K(X) \coloneqq \{ K \subseteq X : \text{ compact}\}$. +Let $X$ be a metrizable topological space +and let $K(X) \coloneqq \{ K \subseteq X : K \text{ compact}\}$. The Vietoris topology has a basis given by $\{K \subseteq U\}$, $U$ open (type 1) @@ -103,19 +104,21 @@ $\max_{a \in A} d(a,B)$. On previous sheets, we checked that $d_H$ is a metric. If $X$ is separable, then so is $K(X)$. % END RECAP +}{} \begin{fact} + \label{fact:s12e4} Let $(X,d)$ be a complete metric space. Then so is $(K(X), d_H)$. \end{fact} -\begin{proof} +\begin{refproof}{fact:s12e4} We need to show that $(K(X), d_H)$ is complete. Let $(K_n)_{ n< \omega}$ be Cauchy in $(K(X), d_H)$. Wlog.~$K_n \neq \emptyset$ for all $n$. Let $K = \{ x \in X : \forall x \in U \overset{\text{open}}{\subseteq} X.~ - \text{ $X$ intersects $K_n$ for infinitely many $n$}\}$. + \text{ $U \cap K_n \neq \emptyset$ for infinitely many $n$}\}$. Equivalently, $K = \{x : x \text{ is a cluster point of some subsequence $(x_n)$ with $x_n \in K_n$ for all $K_n$}\}$. @@ -123,12 +126,12 @@ Then so is $(K(X), d_H)$. (A cluster point is a limit of some subsequence). \begin{claim} + \label{fact:s12e4:c1} $K_n \to K$. \end{claim} - \begin{subproof} + \begin{refproof}{fact:s12e4:c1} Note that $K$ is closed (the complement is open). - \begin{claim} $K \neq \emptyset$. \end{claim} @@ -159,7 +162,7 @@ Then so is $(K(X), d_H)$. space, it is complete. So it suffices to show that $K$ is totally bounded. - Let $\epsilon > 0$ + Let $\epsilon > 0$. Take $N$ such that $d_H(K_i,K_j) < \epsilon$ for all $i,j \ge N$. @@ -200,9 +203,8 @@ Then so is $(K(X), d_H)$. To do this, construct a sequence of $y_{n_i} \in K_{n_i}$ starting with $y$ such that $d(y_{n_i}, y_{n_{i+1}}) < \frac{\epsilon}{2^{i+2}}$. (same trick as before). - \end{subproof} - -\end{proof} + \end{refproof} +\end{refproof} \begin{fact} If $X$ is compact metrisable, @@ -223,9 +225,3 @@ Then so is $(K(X), d_H)$. % TODO complete and totally bounded Sutherland metric and topological spaces - - - - - - diff --git a/inputs/tutorial_15.tex b/inputs/tutorial_15.tex index b3b0051..33c18a4 100644 --- a/inputs/tutorial_15.tex +++ b/inputs/tutorial_15.tex @@ -2,7 +2,7 @@ \tutorial{15}{2024-01-31}{Additions} The following is not relevant for the exam, -but gives a more general picture. +but aims to give a more general picture. Let $X$ be a topological space and let $\cF$ be a filter on $ X$. @@ -21,6 +21,7 @@ is contained in $\cF$. \end{proof} \begin{fact} + \label{fact:compactiffufconv} $X$ is (quasi-) compact iff every ultrafilter converges. \end{fact} @@ -29,7 +30,7 @@ is contained in $\cF$. Let $\cU$ be an ultrafilter. Consider the family $\cV = \{\overline{A} : A \in \cU\}$ of closed sets. - By the FIP we geht that there exist + By the FIP we get that there exist $c \in X$ such that $c \in \overline{A}$ for all $A \in \cU$. Let $N$ be an open neighbourhood of $c$. If $N^c \in \cU$, then $c \in N^c \lightning$ @@ -69,17 +70,19 @@ so is $f(\cB)$. \end{fact} \begin{proof} Consider $(f,g)^{-1}(\Delta) \supseteq A$. + The RHS is a dense closed set, i.e.~the entire space. \end{proof} -We can uniquely extend $f\colon X \to Y$ continuous +We can uniquely extend a continuous $f\colon X \to Y$ to a continuous $\overline{f}\colon \beta X \to Y$ by setting $\overline{f}(\cU) \coloneqq \lim_\cU f$. -Let $V$ be an open neighbourhood of $Y$ in $\overline{f}\left( U) \right) $. -Consider $f^{-1}(V)$. -Consider the basic open set -\[ -\{\cF \in \beta\N : \cF \ni f^{-1}(V)\}. -\] +% Let $V$ be an open neighbourhood of $y \in \overline{f}\left( U \right)$. +% Consider $f^{-1}(V)$. +% Then +% \[ +% \{\cF \in \beta\N : \cF \ni f^{-1}(V)\} +% \] +% is a basic open set. \todo{I missed the last 5 minutes} From e887f46a5df4c0a1c9d72b854b676001fabad050 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Wed, 7 Feb 2024 02:05:20 +0100 Subject: [PATCH 6/7] some small changes --- inputs/lecture_17.tex | 2 +- inputs/lecture_23.tex | 5 +-- inputs/lecture_24.tex | 18 +++++--- inputs/lecture_25.tex | 27 ++++++------ inputs/tutorial_02.tex | 99 +++++++++++++++--------------------------- inputs/tutorial_03.tex | 32 +++++++++++--- inputs/tutorial_04.tex | 2 +- inputs/tutorial_11.tex | 2 +- inputs/tutorial_15.tex | 1 + 9 files changed, 92 insertions(+), 96 deletions(-) diff --git a/inputs/lecture_17.tex b/inputs/lecture_17.tex index c9b265d..79ced2a 100644 --- a/inputs/lecture_17.tex +++ b/inputs/lecture_17.tex @@ -127,7 +127,7 @@ since $X^X$ has these properties. \begin{lemma}[Ellis–Numakura] \yalabel{Ellis-Numakura Lemma}{Ellis-Numakura}{lem:ellisnumakura} - Every compact semigroup + Every non-empty compact semigroup contains an \vocab{idempotent} element, i.e.~$f$ such that $f^2 = f$. \end{lemma} diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex index aedf7f5..6068d5d 100644 --- a/inputs/lecture_23.tex +++ b/inputs/lecture_23.tex @@ -37,10 +37,9 @@ Let $I$ be a linear order S & \coloneqq & \{ x \in \LO(\N) :& x \text{ has a least element},\\ &&& \text{for any $t$, there is $t \oplus 1$, the successor of $t$.}\} \end{IEEEeqnarray*} - \todo{Exercise sheet 12} - $S$ is Borel. + $S$ is Borel.\footnote{cf.~\yaref{s12e1}} - We will % TODO ? + We will construct a reduction \begin{IEEEeqnarray*}{rCl} M \colon S &\longrightarrow & C(\mathbb{K}^\N,\mathbb{K})^\N. %\\ diff --git a/inputs/lecture_24.tex b/inputs/lecture_24.tex index 366e135..958fa5f 100644 --- a/inputs/lecture_24.tex +++ b/inputs/lecture_24.tex @@ -1,10 +1,10 @@ \lecture{24}{2024-01-23}{Combinatorics!} +% ANKI 2 + \subsection{Applications to Combinatorics} % Ramsey Theory} -% TODO Define Ultrafilter - \begin{definition} An \vocab{ultrafilter} on $\N$ (or any other set) is a family $\cU \subseteq \cP(\N)$ @@ -44,6 +44,7 @@ for $\{ n \in \N : \phi(n)\} \in \cU$. We say that $\phi(n)$ holds for \vocab{$\cU$-almost all} $n$. \end{notation} +\gist{% \begin{observe} Let $\phi(\cdot )$, $\psi(\cdot )$ be formulas. @@ -53,6 +54,7 @@ \item $(\cU n) ~\lnot \phi(n) \iff \lnot (\cU n)~ \phi(n)$. \end{enumerate} \end{observe} +}{} \begin{lemma} \label{lem:ultrafilterlimit} Let $X $ be a compact Hausdorff space. @@ -72,6 +74,8 @@ \end{notation} \begin{refproof}{lem:ultrafilterlimit}\footnote{The proof from the lecture only works for metric spaces.} +\gist{ + For metric spaces: Whenever we write $X = Y \cup Z$ we have $(\cU n) x_n \in Y$ or $(\cU n) x_n \in Z$. @@ -86,8 +90,13 @@ $C \in \cP_{n+1} \implies \exists C \subseteq D \in \cP_{n}$ and $C_1 \supseteq C_2 \supseteq \ldots$, $C_i \in \cP_i $ $\implies | \bigcap_{i} C_i| = 1$. - It is clear that we can do this for metric spaces, - but such partition can be found for compact Hausdorff spaces as well. + It is clear that we can do this for metric spaces. + +}{} + See \yaref{thm:uflimit} for the full proof. + See + \yaref{fact:compactiffufconv} and + \yaref{fact:hdifffilterlimit} for a more general statement. \end{refproof} Let $\beta \N$ be the Čech-Stone compactification of $\N$, @@ -157,7 +166,6 @@ is not necessarily continuous. \[ \forall n.~\exists k < M.~ T^{n+k}(x) \in G. \] - \end{definition} \begin{fact} Let $\cU, \cV \in \beta\N$ diff --git a/inputs/lecture_25.tex b/inputs/lecture_25.tex index b2e1ff0..27acec3 100644 --- a/inputs/lecture_25.tex +++ b/inputs/lecture_25.tex @@ -7,15 +7,17 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$. where a basis consist of sets $V_A \coloneqq \{p \in \beta\N : A \in p\}, A \subseteq \N$. + \gist{% (For $A, B \subseteq \N$ we have $V_{A \cap B} = V_{A} \cap V_B$ and $\beta\N = V_\N$.) + }{} \item Note also that for $A, B \subseteq \N$, $V_{A \cup B} = V_A \cup V_B$, $V_{A^c} = \beta\N \setminus V_A$. \end{itemize} \end{fact} - +\gist{% \begin{observe} \label{ob:bNclopenbasis} Note that the basis is clopen. In particular @@ -25,6 +27,7 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$. If $F$ is closed, then $U = \beta\N \setminus F = \bigcup_{i\in I} V_{A_i}$, so $F = \bigcap_{i \in I} V_{\N \setminus A_i}$. \end{observe} +}{} \begin{fact} \label{fact:bNhd} @@ -54,12 +57,14 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$. $\bigcap_{j=1}^k F_{i_j} \neq \emptyset$. We need to show that $\bigcap_{i \in I} F_i \neq \emptyset$. + \gist{% Replacing each $F_i$ by $V_{A_j^i}$ such that $F_i = \bigcap_{j \in J_i} V_{A_j^i}$ (cf.~\yaref{ob:bNclopenbasis}) we may assume that $F_i$ is of the form $V_{A_i}$. We get $\{F_i = V_{A_i} : i \in I\}$ with the finite intersection property. + }{Wlog.~$F_i = V_{A_i}$.} Hence $\{A_i : i \in I\} \mathbin{\text{\reflectbox{$\coloneqq$}}} \cF_0$ has the finite intersection property. @@ -78,11 +83,10 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$. \item $ \{\hat{n}\} $ is open in $\beta\N$ for all $n \in \N$. \item $\N \subseteq \beta\N$ is dense. \end{itemize} - \todo{Easy exercise} - % TODO write down (exercise) \end{fact} \begin{theorem} + \label{thm:uflimit} For every compact Hausdorff space $X$, a sequence $(x_n)$ in $X$, and $\cU \in \beta\N$, @@ -133,7 +137,6 @@ Let $\beta\N$ denote the set of ultrafilters on $\N$. % agree everywhere (fact section) \end{proof} \begin{trivial}+ - More generally, $\beta$ is a functor from the category of topological spaces to the category of compact Hausdorff spaces. It is left adjoint to the inclusion functor. @@ -222,13 +225,12 @@ to obtain Take $x_2 > x_1$ that satisfies this. \item Suppose we have chosen $\langle x_i : i < n \rangle$. Since $\cU$ is idempotent, we have - \[ - (\cU n)[ - n \in P - \land (\cU_k) n + k \in P - \land \forall {I \subseteq n}.~ (\sum_{i \in I} x_i + n \in P) - \land (\cU_k)\left( \forall {I \subseteq n}.~ (\sum_{i \in I} x_i + n + k) \in P\right). - \] + \begin{IEEEeqnarray*}{rCl} + (\cU n)&& n \in P\\ + &\land& (\cU_k) n + k \in P\\ + &\land& \forall {I \subseteq n}.~ (\sum_{i \in I} x_i + n \in P)\\ + &\land& (\cU_k)\left( \forall {I \subseteq n}.~ (\sum_{i \in I} x_i + n + k) \in P\right). + \end{IEEEeqnarray*} Chose $x_n > x_{n-1}$ that satisfies this. \end{itemize} Set $H \coloneqq \{x_i : i < \omega\}$. @@ -237,6 +239,3 @@ to obtain Next time we'll see another proof of this theorem. - - - diff --git a/inputs/tutorial_02.tex b/inputs/tutorial_02.tex index b039cf5..32d26df 100644 --- a/inputs/tutorial_02.tex +++ b/inputs/tutorial_02.tex @@ -3,12 +3,36 @@ % Points: 15 / 16 \nr 1 -\todo{handwritten solution} +Let $(X,d)$ be a metric space and $\emptyset \neq A \subseteq X$. +Let $d(x,A) \coloneqq \inf(d(x,a) : a \in A\}$. + +\begin{itemize} + \item $d(-,A)$ is uniformly continuous: + + Clearly $|d(x,A) - d(y,A)| \le d(x,y)$. + \todo{Add details} + \item $d(x,A) = 0 \iff x \in \overline{A}$. + + $d(x,A) = 0$ iff there is a sequence in $A$ + converging towards $x$ iff $x \in \overline{A}$. +\end{itemize} + \nr 2 +Let $X$ be a discrete space. +For $f,g \in X^{\N}$ define +\[ +d(f,g) \coloneqq \begin{cases} + (1 + \min \{n: f(n) \neq g(n)\})^{-1} &: f \neq g,\\ + 0 &: f= g. +\end{cases} +\] + + \begin{enumerate}[(a)] - \item $d$ is an ultrametric: + \item $d$ is an \vocab{ultrametric}, + i.e.~$d(f,g) \le \max \{d(f,h), d(g,h)\}$ for all $f,g,h \in X^{\N}$ : Let $f,g,h \in X^{\N}$. @@ -70,10 +94,15 @@ \nr 3 +Consider $\N$ as a discrete space and $\N^{\N}$ with the product topology. +Let +\[ +S_{\infty} = \{f\colon \N \to \N \text{ bijective}\} \subseteq \N^{\N}. +\] \begin{enumerate}[(a)] \item $S_{\infty}$ is a Polish space: - From (2) we know that $\N^{\N}$ is Polish. + From \yaref{s1e2} we know that $\N^{\N}$ is Polish. Hence it suffices to show that $S_{\infty}$ is $G_{\delta}$ with respect to $\N^\N$. @@ -111,69 +140,9 @@ Clearly there cannot exist a finite subcover as $B$ is the disjoint union of the $B_j$. - % TODO Think about this \end{enumerate} \nr 4 - -% (uniform metric) -% -% \begin{enumerate}[(a)] -% \item $d_u$ is a metric on $\cC(X,Y)$: -% -% It is clear that $d_u(f,f) = 0$. -% -% Let $f \neq g$. Then there exists $x \in X$ with -% $f(x) \neq g(x)$, hence $d_u(f,g) \ge d(f(x), g(x)) > 0$. -% -% Since $d$ is symmetric, so is $d_u$. -% -% Let $f,g,h \in \cC(X,Y)$. -% Take some $\epsilon > 0$ -% choose $x_1, x_2 \in X$ -% with $d_u(f,g) \le d(f(x_1), g(x_1)) + \epsilon$, -% $d_u(g,h) \le d(g(x_2), h(x_2)) + \epsilon$. -% -% Then for all $x \in X$ -% \begin{IEEEeqnarray*}{rCl} -% d(f(x), h(x)) &\le & -% d(f(x), g(x)) + d(g(x), h(x))\\ -% &\le & d(f(x_1), g(x_1)) + d(g(x_2), h(x_2))-2\epsilon\\ -% &\le & d_u(f,g) + d_u(g,h) - 2\epsilon. -% \end{IEEEeqnarray*} -% Thus $d_u(f,g) \le d_u(f,g) + d_u(g,h) - 2\epsilon$. -% Taking $\epsilon \to 0$ yields the triangle inequality. -% -% \item $\cC(X,Y)$ is a Polish space: -% \todo{handwritten solution} -% -% \begin{itemize} -% \item $d_u$ is a complete metric: -% -% Let $(f_n)_n$ be a Cauchy series with respect to $d_u$. -% -% Then clearly $(f_n(x))_n$ is a Cauchy sequence with respect -% to $d$ for every $x$. -% Hence there exists a pointwise limit $f$ of the $f_n$. -% We need to show that $f$ is continuous. -% -% %\todo{something something uniform convergence theorem} -% -% \item $(\cC(X,Y), d_u)$ is separable: -% -% Since $Y$ is separable, there exists a countable -% dense subset $S \subseteq Y$. -% -% Consider $\cC(X,S) \subseteq \cC(X,Y)$. -% Take some $f \in \cC(X,Y)$. -% Since $X$ is compact, -% -% -% % TODO -% -% \end{itemize} -% \end{enumerate} - \begin{fact} Let $X $ be a compact Hausdorff space. Then the following are equivalent: @@ -205,7 +174,7 @@ Let $X$ be compact Polish\footnote{compact metrisable $\implies$ compact Polish} and $Y $ Polish. Let $\cC(X,Y)$ be the set of continuous functions $X \to Y$. -Consider the metric $d_u(f,g) \coloneqq \sup_{x \in X} |d(f(x), g(x))|$. +Consider the \vocab{uniform metric} $d_u(f,g) \coloneqq \sup_{x \in X} |d(f(x), g(x))|$. Clearly $d_u$ is a metric. \begin{claim} @@ -243,7 +212,7 @@ Clearly $d_u$ is a metric. for each $y \in X_m$. Then $\bigcup_{m,n} D_{m,n}$ is dense in $\cC(X,Y)$: Indeed if $f \in \cC(X,Y)$ and $\eta > 0$, - we finde $n > \frac{3}{\eta}$ and $m$ such that $f \in C_{m,n}$, + we find $n > \frac{3}{\eta}$ and $m$ such that $f \in C_{m,n}$, since $f$ is uniformly continuous. Let $g \in D_{m,n}$ be such that $\forall y \in X_m.~d(f(y), g(y)) < \frac{1}{n+1}$. We have $d_u(f,g) \le \eta$, diff --git a/inputs/tutorial_03.tex b/inputs/tutorial_03.tex index 5aeb648..a760624 100644 --- a/inputs/tutorial_03.tex +++ b/inputs/tutorial_03.tex @@ -12,6 +12,14 @@ \nr 1 +Let $X$ be a Polish space. +Then there exists an injection $f\colon X \to 2^\omega$ +such that for each $n < \omega$, +the set $f^{-1}(\{(y_n) \in 2^\omega : y_n = 1\})$ +is open. +Moreover if $V \subseteq 2^{ \omega}$ is closed, +then $f^{-1}(V)$ is $G_\delta$. + Let $(U_i)_{i < \omega}$ be a countable base of $X$. Define \begin{IEEEeqnarray*}{rCl} @@ -19,6 +27,7 @@ Define x &\longmapsto & (x_i)_{i < \omega} \end{IEEEeqnarray*} where $x_i = 1$ iff $x \in U_i$ and $x_i = 0$ otherwise. +\gist{ Then $f^{-1}(\{y = (y_n) \in 2^\omega | y_n = 1\}) = U_n$ is open. We have that $f$ is injective since $X$ is T1. @@ -51,17 +60,21 @@ Since $2^{n} \setminus \left( \prod_{i < n} X_i \right)$ is finite, we get that $f^{-1}(2^{\omega} \setminus ((\prod_{i Date: Wed, 7 Feb 2024 13:18:14 +0100 Subject: [PATCH 7/7] some changes --- inputs/lecture_15.tex | 4 ++++ inputs/lecture_16.tex | 11 ++++++++++- 2 files changed, 14 insertions(+), 1 deletion(-) diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index fa9c486..77a74ef 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -273,6 +273,7 @@ Recall: A limit of distal flows is distal. \end{proposition} \begin{proof} +\gist{% Let $(X,T)$ be a limit of $\Sigma = \{(X_i, T) : i \in I\}$. Suppose that each $(X_i, T)$ is distal. If $(X,T)$ was not distal, @@ -283,4 +284,7 @@ Recall: But then $g_n \pi_i(x_1) \to \pi_i(z)$ and $g_n \pi_i(x_2) \to \pi_i(z)$, which is a contradiction since $(X_i, T)$ is distal. +}{Suppose there is a proximal pair $x_1,x_2$. + Take $i$ such that $\pi_i(x_1) \neq \pi_i(x_2) \lightning$. +} \end{proof} diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index 075f78c..352277d 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -17,7 +17,7 @@ $X$ is always compact metrizable. % and $h\colon x \mapsto x + \alpha$.\footnote{Again $x + \alpha$ denotes $x \cdot \alpha$ in $\C$.} % \end{example} \begin{proof} - % TODO TODO TODO Think! +\gist{% The action of $1$ determines $h$. Consider \[ @@ -82,6 +82,15 @@ $X$ is always compact metrizable. For $\alpha = h$ we get that a flow $\Z \acts X$ corresponds to $\Z \acts K$ with $(1,x) \mapsto x + \alpha$. +}{ + \begin{itemize} + \item $G \coloneqq \overline{\{h^n : n \in \Z\} } \subseteq \cC(X,X)$. + \item $G$ is compact (Arzela-Ascoli), abelian topological group (closure of ab. top. group) + \item Take any $x \in G$. + \item $Gx$ is compact (since $g \mapsto gx$ is continuous and $G$ is compact) + \item Stabilizer $G_x$ is closed. $K \coloneqq \faktor{G}{G_x}$, $K \to X, fG_x \mapsto f(x)$. + \end{itemize} +} \end{proof} \begin{definition} Let $(X,T)$ be a flow