From 3bb3c4e75dd0391160f47a550e548eae34293862 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Fri, 20 Oct 2023 11:53:35 +0200 Subject: [PATCH] lecture 4 --- inputs/lecture_03.tex | 3 +- inputs/lecture_04.tex | 156 ++++++++++++++++++++++++++++++++++++++++++ jrpie-math.sty | 1 + logic3.tex | 1 + 4 files changed, 160 insertions(+), 1 deletion(-) create mode 100644 inputs/lecture_04.tex diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex index 8d354de..9fa61e1 100644 --- a/inputs/lecture_03.tex +++ b/inputs/lecture_03.tex @@ -107,6 +107,7 @@ \end{definition} \begin{theorem} + \label{thm:cantortopolish} Let $X \neq \emptyset$ be a perfect Polish space. Then there is an embedding @@ -216,7 +217,7 @@ \item $F_\emptyset = X$, \item $F_s$ is $F_\sigma$ for all $s$. \item The $F_{s \concat i}$ partition $F_s$, - i.e.~$F_{s} = \bigsqcup_i F_{s \concat i}$. + i.e.~$F_{s} = \bigsqcup_i F_{s \concat i}$. % TODO change notation? Furthermore we want that $\overline{F_{s \concat i}} \subseteq F_s$ diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex new file mode 100644 index 0000000..d849794 --- /dev/null +++ b/inputs/lecture_04.tex @@ -0,0 +1,156 @@ +\lecture{04}{2023-10-20}{} +\begin{remark} + Some of $F_s$ might be empty. +\end{remark} + +\begin{refproof}{thm:bairetopolish} + Take + \[D = \{x \in \cN : \bigcap_{n} F_{x\defon{n}} \neq \emptyset\}.\] + + Since $\ldots \supseteq F_{x\defon{n}} \supseteq \overline{F_{x\defon{n+1}}} \supseteq F_{x\defon{n+1}} \supseteq \ldots$ + we have + \[ + \bigcap_{n} F_{x\defon{n}} = \bigcap_{n} \overline{F_{x\defon{n}}}. + \] + + $f\colon D \to X$ is determined by + \[ + \{f(x)\} = \bigcap_{n} F_{x\defon{n}} + \] + + $f$ is injective and continuous. + The proof of this is exactly the same as in + \yaref{thm:cantortopolish}. + + \begin{claim} + \label{thm:bairetopolish:c1} + $D$ is closed. + \end{claim} + \begin{refproof}{thm:bairetopolish:c1} + Let $x_n$ be a series in $D$ + converging to $x$ in $\cN$. + Then $x \in \cN$. + \begin{claim} + $(f(x_n))$ is Cauchy. + \end{claim} + \begin{subproof} + Let $\epsilon > 0$. + Take $N$ such that $\diam(F_{x\defon{n}}) < \epsilon$. + Take $M$ such that for all $m \ge M$, + $x_m\defon{N} = x\defon{N}$. + Then for all $m, n \ge M$, + we have that $f(x_m), f(x_n) \in F_{x\defon{N}}$. + So $d(f(x_m), f(x_n)) < \epsilon$ + we have that $(f(x_n))$ is Cauchy. + + Since $(X,d)$ is complete, + there exists $y = \lim_n f(x_n)$. + + Since for all $m \ge M$, $f(x_m) \in F_{x\defon{N}}$, + we get that $y \in \overline{F_{x\defon{N}}}$. + + Note that for $N' > N$ by the same argument + we get $y \in \overline{F_{x\defon{N'}}}$. + Hence + \[y \in \bigcap_{n} \overline{F_{x\defon{n}}} = \bigcap_{n} F_{x\defon{n}},\] + i.e.~$y \in D$ and $y = f(x)$. + \end{subproof} + \end{refproof} + + We extend $f$ to $g\colon\cN \to X$ + in the following way: + + Take $S \coloneqq \{s \in \N^{<\N}: \exists x \in D, n \in \N.~x=s\defon{n}\}$. + Clearly $S$ is a pruned tree. + Moreover, since $D$ is closed, we have that\todo{Proof this (homework?)} + \[ + D = [S] = \{x \in \N^\N : \forall n \in \N.~x\defon{n} \in S\}. + \] + We construct a \vocab{retraction} $r\colon\cN \to D$ + (i.e.~$r = \id$ on $D$ and $r$ is a continuous surjection). + Then $g \coloneqq f \circ r$. + + To construct $r$, we will define by induction + $\phi\colon \N^{<\N} \to S$ by induction on the length + such that + \begin{itemize} + \item $s \subseteq t \implies \phi(s) \subseteq \phi(t)$, + \item $|s| = \phi(|s|)$, + \item if $s \in S$, then $\phi(s) = s$. + \end{itemize} + Let $\phi(\emptyset) = \emptyset$. + Suppose that $\phi(t)$ is defined. + If $t\concat a \in S$, then set + $\phi(t\concat a) \coloneqq t\concat a$. + Otherwise take some $b$ such that + $t\concat b \in S$ and define + $\phi(t\concat a) \coloneqq \phi(t)\concat b$. + This is possible since $S$ is pruned. + + Let $r\colon \cN = [\N^{<\N}] \to [S] = D$ + be the function defined by $r(x) = \bigcup_n f(x\defon{n})$. + + $r$ is continuous, since + $d_{\cN}(r(x), r(y)) \le d_{\cN}(x,y)$. % Lipschitz + It is immediate that $r$ is a retraction. +\end{refproof} + +\subsection{Meager and Comeager Sets} + +\begin{definition} + Let $X$ be a topological space, $A \subseteq X$. + We say that $A$ is \vocab{nowhere dense} (\vocab{nwd}), + if $\inter(\overline{A}) = \emptyset$. + Equivalently + \begin{itemize} + \item $\overline{A}$ is nwd, + \item $X \setminus A$ is dense in $X$, + \item $\forall \emptyset \neq U \overset{\text{open}}{\subseteq} X.~ + \exists \emptyset \neq V \overset{\text{open}}{\subseteq} U.~ + V\cap A = \emptyset$. + (If we intersect $A$ with an open $U$, + then $A \cap U$ is not dense in $U$). + \end{itemize} + \todo{Think about this} + + A set $B \subseteq X$ is \vocab{meager} + (or \vocab{first category}), + iff it is a countable union of nwd sets. + + The complement of a meager set is called + \vocab{comeager}. + +\end{definition} +\begin{example} + $\Q \subseteq \R$ is meager. +\end{example} +\begin{notation} + Let $A, B \subseteq X$. + We write $A =^\ast B$ + iff the \vocab{symmetric difference}, + $A \mathop{\triangle} B \coloneqq (A\setminus B) \cup (B \setminus A)$, + is meager. +\end{notation} +\begin{remark} + $=^\ast$ is an equivalence relation. +\end{remark} +\begin{definition} + A set $A \subseteq X$ + has the \vocab{Baire property} (\vocab{BP}) + if $A =^\ast U$ for some $U \overset{\text{open}}{\subseteq} X$. +\end{definition} +Note that open sets and meager sets have the Baire property. + + + + +% \begin{example} +% $\Q \subseteq \R$ is $F_\sigma$. +% +% $\R \setminus \Q \subseteq \R$ is $G_\delta$. +% +% $\Q \subseteq \R$ is not $G_{\delta}$. +% (It is dense and meager, +% hence it can not be $G_\delta$, +% by the Baire category theorem). +% \end{example} diff --git a/jrpie-math.sty b/jrpie-math.sty index f3fc711..ea1aca6 100644 --- a/jrpie-math.sty +++ b/jrpie-math.sty @@ -48,4 +48,5 @@ \RequirePackage{mkessler-mathfig} \RequirePackage{mkessler-unicodechar} \RequirePackage{mkessler-mathfixes} % Load this last since it renews behaviour +\DeclareMathOperator{\inter}{int} % interior \newcommand{\defon}[1]{|_{#1}} % TODO diff --git a/logic3.tex b/logic3.tex index cd35863..2f5d966 100644 --- a/logic3.tex +++ b/logic3.tex @@ -27,6 +27,7 @@ \input{inputs/lecture_01} \input{inputs/lecture_02} \input{inputs/lecture_03} +\input{inputs/lecture_04}