diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex index ceca77d..2defe1b 100644 --- a/inputs/lecture_04.tex +++ b/inputs/lecture_04.tex @@ -95,7 +95,7 @@ It is immediate that $r$ is a retraction. \end{refproof} -\subsection{Meager and Comeager Sets} +\section{Meager and Comeager Sets} \begin{definition} Let $X$ be a topological space, $A \subseteq X$. diff --git a/inputs/lecture_05.tex b/inputs/lecture_05.tex index 161ec91..7169331 100644 --- a/inputs/lecture_05.tex +++ b/inputs/lecture_05.tex @@ -3,7 +3,7 @@ \begin{fact} A set $A$ is nwd iff $\overline{A}$ is nwd. - If $F$ is closed then + If $F$ is closed then $F$ is nwd iff $X \setminus F$ is open and dense. Any meager set $B$ is contained in @@ -45,7 +45,7 @@ Then \[ \left( \bigcup_{n < \omega} A_n \right) \symdif \left( \bigcup_{n < \omega} U_n \right) - \] + \] is meager,\todo{small exercise} hence $\bigcup_{n < \omega} A_n \in \cA$. @@ -121,7 +121,8 @@ % TODO Fubini \begin{notation} Let $X ,Y$ be topological spaces, - $A \subseteq X \times Y$, + $A \subseteq X \times Y$ + and $x \in X, y \in Y$. Let @@ -151,12 +152,12 @@ but for meager sets: and similarly for $y$. \item $A$ is meager \begin{IEEEeqnarray*}{rll} - \iff &\{x \in X : A_x \text{ is meager }\}&\text{ is comeager}\\ - \iff &\{y \in Y : A^y \text{ is meager }\}& \text{ is comeager}. + \iff &\{x \in X : A_x \text{ is meager}\}&\text{ is comeager}\\ + \iff &\{y \in Y : A^y \text{ is meager}\}& \text{ is comeager}. \end{IEEEeqnarray*} \item $A$ is comeager \begin{IEEEeqnarray*}{rll} - \iff & \{x \in X: A_x \text{ is comeager }\} &\text{ is comeager}\\ + \iff & \{x \in X: A_x \text{ is comeager}\} &\text{ is comeager}\\ \iff & \{y \in Y: A^y \text{ is comeager}\} & \text{ is comeager}. \end{IEEEeqnarray*} \end{enumerate} @@ -171,8 +172,8 @@ but for meager sets: is nwd, then \[ - \{x \in X : F_x \text{is nwd}\} - \] + \{x \in X : F_x \text{is nwd}\} + \] is comeager. \end{claim} \begin{refproof}{thm:kuratowskiulam:c1a} @@ -187,12 +188,12 @@ but for meager sets: We want to show that \[ \{x \in X: \forall n.~ (W_x \cap V_n) \neq \emptyset\} - \] + \] is a comeager set. This is equivalent to \[ \{x \in X : (W_x \cap V_n) \neq \emptyset\} - \] + \] being comeager for all $n$, because the intersection of countably many comeager sets is comeager. @@ -209,9 +210,8 @@ but for meager sets: It is \[ U \cap U_n = \proj_x(W \cap (U \times V_n)) - \] + \] nonempty since $W$ is dense. - \end{refproof} \begin{claim} % [1a'] @@ -220,10 +220,10 @@ but for meager sets: is nwd, then \[ - \{x \in X : F_x \text{is nwd}\} - \] + \{x \in X : F_x \text{is nwd}\} + \] is comeager. - + \end{claim} \begin{refproof}{thm:kuratowskiulam:c1ap} We have that $\overline{F}$ is nwd. @@ -231,8 +231,8 @@ but for meager sets: the set \[ \{x \in X: \overline{F_x} \text{ is nwd}\} \subseteq - \{x \in X: F_x \text{ is nwd}\} - \] + \{x \in X: F_x \text{ is nwd}\} + \] is comeager. \end{refproof} @@ -242,8 +242,8 @@ but for meager sets: If $M \subseteq X \times Y$ is meager, then \[ - \{x \in X : M_x \text{ is meager}\} - \] + \{x \in X : M_x \text{ is meager}\} + \] is comeager. \end{claim} \begin{refproof}{thm:kuratowskiulam:c1b} @@ -257,9 +257,9 @@ but for meager sets: as a countable intersection of comeager sets. \end{refproof} - \todo{Finish proof} - \phantom\qedhere -\end{refproof} +% \phantom\qedhere +% \end{refproof} +% TODO fix claim numbers \begin{remark} Suppose that $A$ has the BP. @@ -267,5 +267,3 @@ but for meager sets: $A \symdif U \mathbin{\text{\reflectbox{$\coloneqq$}}} M$ is meager. Then $A = U \symdif M$. \end{remark} - - diff --git a/inputs/lecture_06.tex b/inputs/lecture_06.tex new file mode 100644 index 0000000..02c5b30 --- /dev/null +++ b/inputs/lecture_06.tex @@ -0,0 +1,208 @@ +\lecture{06}{2023-11-03}{} + +% \begin{refproof}{thm:kuratowskiulam} + \begin{enumerate}[(i)] + \item Let $A$ be a set with the Baire Property. + Write $A = U \symdif M$ + for $U$ open and $M$ meager. + Then for all $x$, + we have that $A_x = U_x \symdif M_x$, + where $U_x$ is open, + and $\{x : M_x \text{ is meager}\}$ is comeager. + Therefore $\{x : U_x \text{ open } \land M_x \text{ meager }\}$ + is comeager, + and for those $x$, $A_x$ has the Baire property. + \end{enumerate} + % TODO: fix counter + \begin{claim} % Claim 2 + \label{thm:kuratowskiulam:c2} + For $P \subseteq X$, $Q \subseteq Y$ with the Baire + property, let $R \coloneqq P \times Q$. + Then $R$ is meager iff at least one of $P$ or $Q$ is meager. + \end{claim} + \begin{refproof}{thm:kuratowskiulam:c2} + Suppose that $R$ is meager. + Then by \yaref{thm:kuratowskiulam:c1b}, + we have that $C = \{x : R_x \text{ is meager }\}$ is comeager. + \begin{itemize} + \item If $P$ is meager, the statement holds trivially. + \item If $P$ is not meager, + then $P \cap C \neq \emptyset$. + For $x \in P \cap C$ + we have that $R_x$ is meager + and $R_x = Q$, + hence $Q$ is meager. + \end{itemize} + + On the other hand suppose that $P$ is meager. + Then $P = \bigcup_{n} F_n$ for nwd sets $F_n$. + Note that $F_n \times Y$ is nwd. + So $F_n \times Q$ is also nwd. + Hence $P \times Q$ is a countable union + of nwd sets, + so it is meager. + \end{refproof} + \begin{enumerate}[(i)] + \item[(ii)] + ``$\impliedby$'' + Let $A$ be a set with the Baire property + such that $\{x : A_x \text{ is meager}\}$ is comeager. + Let $A = U \symdif M$ for $U$ open and $M$ meager. + 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 $G \times H \subseteq A$, + is not meager. + By \yaref{thm:kuratowskiulam:c2}, + both $G$ and $H$ are not meager. + Since + $\{x\colon A_x \text{ is meager} \land M_x \text{ is meager}\}$ + is comeager (using \yaref{thm:kuratowskiulam:c1b}), + there is $x_0 \in G$ such that $A_{x_0}$ is meager and $M_{x_0}$ + is meager. + But then $H$ is meager as + \[ + H \setminus M_{x_0} \subseteq U_{x_0} \setminus M_{x_0} + \subseteq U_{x_0} \symdif M_{x_0} = A_{x_0} + \] + and $M_{x_0}$ is meager $\lightning$. + + ``$\implies$'' + This is \yaref{thm:kuratowskiulam:c1b}. + \end{enumerate} +\end{refproof} + +\section{Borel sets} % TODO: fix chapters + +\begin{definition} + Let $X$ be a topological space. + Let $\cB(X)$ denote the smallest $\sigma$-algebra, + that contains all open sets. + Elements of $\cB(X)$ are called \vocab{Borel sets}. +\end{definition} +\begin{remark} + Note that all Borel sets have the Baire property. +\end{remark} + +\subsection{The hierarchy of Borel sets} + +Let $\omega_1$ be the first uncountable ordinal. +For every $d < \omega_1$, +we define by transfinite recursion +classes $\Sigma^0_\alpha$ +and $\Pi^0_\alpha$ +(or $\Sigma^0_\alpha(X)$ and $\Pi^0_\alpha(X)$ for a topological space $X$). + +Let $X$ be a topological space. +Then define +\[\Sigma^0_1(X) \coloneqq \{U \overset{\text{open}}{\subseteq} X\},\] +\[ +\Pi^0_\alpha(X) \coloneqq \lnot \Sigma^0_\alpha(X) \coloneqq +\{X \setminus A | A \in \Sigma^0_\alpha(X)\}, +\] +% \todo{Define $\lnot$ (element-wise complement)} +and for $\alpha > 1$ +\[ +\Sigma^0_\alpha \coloneqq \{\bigcup_{n < \omega} A_n : A_n \in \Pi^0_{\alpha_n}(X) \text{ for some $\alpha_n < \alpha$}\}. +\] + +Note that $\Pi_1^0$ is the set of closed sets, +$\Sigma^0_2 = F_\sigma$, +and $\Pi^0_2 = G_\delta$. + +Furthermore define +\[ +\Delta^0_\alpha(X(X)) \coloneqq \Sigma^0_\alpha(X) \cap \Pi^0_\alpha(X), +\] +i.e.~$\Delta^0_1$ is the set of clopen sets. + +\iffalse % TODO Fix this! +\resizebox{\textwidth}{!}{ +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + & {\Sigma_1^0} && {\Sigma^0_2} &&&& {\Sigma^0_\xi} \\ + {\Delta_1^0} && {\Delta^0_2} && {\Delta^0_3} & \ldots & {\Delta^0_\xi} && {\Delta^0_{\xi + 1}} & \ldots \\ + & {\Pi^0_1} && {\Pi_2^0} &&&& {\Pi^0_\xi} + \arrow["\subseteq", hook, from=2-1, to=1-2] + \arrow["\subseteq"', hook', from=2-1, to=3-2] + \arrow[hook, from=3-2, to=2-3] + \arrow[hook, from=1-2, to=2-3] + \arrow[hook, from=2-3, to=1-4] + \arrow[hook', from=2-3, to=3-4] + \arrow[hook, from=2-7, to=1-8] + \arrow[hook', from=2-7, to=3-8] + \arrow[hook, from=1-4, to=2-5] + \arrow[hook', from=3-4, to=2-5] + \arrow[hook, from=1-8, to=2-9] + \arrow[hook', from=3-8, to=2-9] +\end{tikzcd}\]% +} +\fi + +\begin{proposition} + Let $X$ be a metrizable space. + Then + \begin{enumerate}[(a)] + \item $\Sigma^0_\eta(X) \cup \Pi^0_\eta(X) \subseteq \Delta^0_\xi(X)$ + for all $1 \le \eta < \xi < \omega_1$. + \item $\cB(X) = \bigcup_{\alpha < \omega_1} \Sigma^0_\alpha(X) = \bigcup_{\alpha < \omega_1} \Pi^0_\alpha(X) = \bigcup_{\alpha < \omega_1} \Delta^0_\alpha(X)$. + \end{enumerate} +\end{proposition} +\begin{proof} + \begin{enumerate}[(a)] + \item \begin{observe} + \label{ob:sigmasuffices} + For all $1 \le \alpha < \beta < \omega_1$, + we have $\Pi^0_\alpha(X) \subseteq \Sigma^0_\beta(X)$ + by taking ``unions'' of singleton sets. + + Furthermore $\Sigma^0_\alpha(X) \subseteq \Pi^0_\beta(X)$ + by passing to complements. + \end{observe} + It suffices to show $\Sigma^0_\eta(X) \subseteq \Delta^0_\xi(X)$, + since $\Delta^0_\eta(X)$ is closed under complements. + + Furthermore, it suffices to show $\Sigma^0_\eta(X) \subseteq \Sigma^0_\xi(X)$, + by \yaref{ob:sigmasuffices} + (since $\Sigma^0_\eta(X) \subseteq \Pi^0_\xi(X)$ + and $\Delta^0_\xi(X) = \Sigma^0_\xi(X) \cap \Pi^0_\xi(X)$). + + So to prove (a) it suffices to show that for all $1 \le \eta < \xi < \omega_1$, + we have $\Sigma^0_\eta(X) \subseteq \Sigma^0_\xi(X)$. + For $\eta = 1, \xi = 2$ + this holds, since every open set is $F_\sigma$.% + \footnote{Here we use that $X$ is metrizable!} + % \todo{REF} + + For $\eta > 1, \xi > \eta$, + we have + \begin{IEEEeqnarray*}{rCl} + \Sigma^0_\eta(X) &=& + \{ \bigcup_{n} A_n : A_n \in \Pi^0_{\alpha_n}(X), \alpha_n < \eta\}\\ + &\subseteq& + \{\bigcup_{n}B_n : B_n \in \Pi^0_{\beta_n}(X), \beta_n < \xi\} + = \Sigma^0_\xi(X). + \end{IEEEeqnarray*} + \item Let $\cB_0 \coloneqq \bigcup_{\alpha < \omega_1} \Sigma^0_\alpha(X) = \bigcup_{\alpha < \omega_1} \Pi^0_\alpha(X) = \bigcup_{\alpha < \omega_1} \Delta^0_\alpha(X)$. + We need to show that $\cB_0 = \cB(X)$. + Clearly $\cB_0 \subseteq \cB(X)$. + It suffices to notice that $\cB_0$ is a $\sigma$-algebra + containing all open sets. + Consider $\bigcup_{n < \omega} A_n$ for some $A_n \in B_0$. + Then $A_n \in \Pi^0_{\alpha_n}(X)$ for some $\alpha_n < \omega_1$. + Let $\alpha = \sup \alpha_n < \omega_1$. + Then $\bigcup_{n < \omega} A_n \in \Sigma^0_\alpha(X)$. + It is clear that $\cB_0$ is closed under complements. + \end{enumerate} + +\end{proof} + +\begin{example} + % TODO move to counter examples. + Consider the cofinite topology on $\omega_1$. + Then the non-empty open sets of this are not $F_\sigma$. +\end{example} + + + diff --git a/logic.sty b/logic.sty index 2c60891..69a4c37 100644 --- a/logic.sty +++ b/logic.sty @@ -22,6 +22,7 @@ \usepackage{listings} \usepackage{multirow} \usepackage{float} +\usepackage{quiver} %\usepackage{algorithmicx} \newcounter{subsubsubsection}[subsubsection] diff --git a/logic3.tex b/logic3.tex index 56808ff..ef7bffc 100644 --- a/logic3.tex +++ b/logic3.tex @@ -29,6 +29,7 @@ \input{inputs/lecture_03} \input{inputs/lecture_04} \input{inputs/lecture_05} +\input{inputs/lecture_06} diff --git a/quiver.sty b/quiver.sty new file mode 100644 index 0000000..27e60ff --- /dev/null +++ b/quiver.sty @@ -0,0 +1,40 @@ +% *** quiver *** +% A package for drawing commutative diagrams exported from https://q.uiver.app. +% +% This package is currently a wrapper around the `tikz-cd` package, importing necessary TikZ +% libraries, and defining a new TikZ style for curves of a fixed height. +% +% Version: 1.4.0 +% Authors: +% - varkor (https://github.com/varkor) +% - AndréC (https://tex.stackexchange.com/users/138900/andr%C3%A9c) + +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{quiver}[2021/01/11 quiver] + +% `tikz-cd` is necessary to draw commutative diagrams. +\RequirePackage{tikz-cd} +% `amssymb` is necessary for `\lrcorner` and `\ulcorner`. +\RequirePackage{amssymb} +% `calc` is necessary to draw curved arrows. +\usetikzlibrary{calc} +% `pathmorphing` is necessary to draw squiggly arrows. +\usetikzlibrary{decorations.pathmorphing} + +% A TikZ style for curved arrows of a fixed height, due to AndréC. +\tikzset{curve/.style={settings={#1},to path={(\tikztostart) + .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$) + .. (\tikztotarget)\tikztonodes}}, + settings/.code={\tikzset{quiver/.cd,#1} + \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}}, + quiver/.cd,pos/.initial=0.35,height/.initial=0} + +% TikZ arrowhead/tail styles. +\tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}} +\tikzset{2tail/.code={\pgfsetarrowsstart{Implies[reversed]}}} +\tikzset{2tail reversed/.code={\pgfsetarrowsstart{Implies}}} +% TikZ arrow styles. +\tikzset{no body/.style={/tikz/dash pattern=on 0 off 1mm}} + +\endinput