From 394cc8ddcf4afe7a6db6ff9f3cd07db75043697a Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 17 Oct 2023 11:56:32 +0200 Subject: [PATCH] Lecture 3 --- inputs/lecture_03.tex | 257 ++++++++++++++++++++++++++++++++++++++++++ logic.sty | 7 +- logic3.tex | 1 + 3 files changed, 264 insertions(+), 1 deletion(-) create mode 100644 inputs/lecture_03.tex diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex new file mode 100644 index 0000000..a8570f6 --- /dev/null +++ b/inputs/lecture_03.tex @@ -0,0 +1,257 @@ +\lecture{03}{2023-10-17}{Embedding of the cantor space into polish spaces} + +% ? \subsection{Trees} TODO + + +\begin{notation} + Let $A \neq \emptyset$, $n \in \N$. + Then + \[ + A^n \coloneqq \{s\colon \{0,1,\ldots, n-1\} \to A \} + \] + is the set of $n$-element \vocab[Sequence]{sequences}. + We often write + $(s_0,s_1,\ldots,s_{n-1})$. + + If $s = (s_0,\ldots,s_{n-1})$, + then $n$ is the \vocab{length} of $s$, + denoted by $|s|$. + + If $n = 0$ there exists only the empty sequence, + i.e.~$A^0 = \{\emptyset\}$ and $|\emptyset| = 0$. + + We set + \[ + A^{<\N} \coloneqq \bigcup_{n=0}^{\infty} A^n + \] + and + \[ + A^{\N} \coloneqq \{x \colon \N \to A\}. + \] + + If $s \in A^n$ and $m \le n$, + we let + $s\defon{m} \coloneqq (s_0,\ldots,s_{m-1})$. + + Let $s,t \in A^{<\N}$. + We say that $s$ is an \vocab{initial segment} + of $t$ (or $t$ is an \vocab{extension} of $s$) + if there exists an $n$ such that $s = t\defon{|s|}$. + We write this as $s \subseteq t$. + + We say that $s$ and $t$ are \vocab{compatible} + if $s \subseteq t$ or $t \subseteq s$. + Otherwise the are \vocab{incompatible}, + we denote that as $s \perp t$. + + The \vocab{concatenation} + of $s = (s_0,\ldots, s_{n-1})$ + and $t = (t_0,\ldots, t_{m-1})$ + is the sequence + $s\concat t \coloneqq (s_0,\ldots,s_{n-1}, t_0,\ldots, t_{n-1})$ + + In the case of $t = (a)$ + we also write $s\concat a$ for $s\concat (a)$. + + Similarly, if $x \in A^{\N}$ + we can write $x = (x_0,x_1,\ldots)$. + If $n \in \N$, $x\defon{n} \coloneqq (x_0,\ldots,x_{n-1})$, + define extension, initial segments + and concatenation of a finite sequence with an infinite one. +\end{notation} + +\begin{definition} + A \vocab{tree} + on a set $A$ is a subset $T \subseteq A^{<\N}$ + closed under initial segments, + i.e.~if $t \in T, s \subseteq t \implies s \in T$. + Elements of trees are called \vocab{nodes}. + + A \vocab{leave} is an element of $T$, + that has no extension in $t$. + + An \vocab{infinite branch} of a tree $T$ + is $x \in A^{\N}$ + such that $\forall n.~x\defon{n} \in T$. + + The \vocab{body} of $T$ is the set of all + infinite branches: + \[ + [T] \coloneqq\{x \in A^{\N}: \forall n. x\defon{n} \in T\}. + \] + + We say that $T$ is \vocab{pruned}, + iff + \[ + \forall t\in T.\exists s \supsetneq t.~s \in T. + \] +\end{definition} + +\begin{definition} + A \vocab{Cantor scheme} + on a set $X$ is a family + $(A_s)_{s \in 2^{< \N}}$ + of subsets of $X$ such that + \begin{enumerate}[i)] + \item $\forall s \in 2^{<\N}.~A_{s \concat 0} \cap A_{s \concat 1} = \emptyset$. + \item $\forall s \in 2^{<\N}, i \in 2.~A_{s \concat i} \subseteq A_s$. + \end{enumerate} +\end{definition} + +\begin{definition} + A topological space + is \vocab{perfect} + if it has no isolated points, + i.e.~for any $U \neq \emptyset$ open, + there $x \neq y$ such that $x, y \in U$. +\end{definition} + +\begin{theorem} + Let $X \neq \emptyset$ + be a perfect Polish space. + Then there is an embedding + of the Cantor space $2^{\N}$ + into $X$. +\end{theorem} +\begin{proof} + We will define a Cantor scheme + $(U_s)_{s \in 2^{<\N}}$ + such that $\forall s \in 2^{< \N}$. + \begin{enumerate}[(i)] + \item $U_s \neq \emptyset$ and open, + \item $\diam(U_s) \le 2^{-|s|}$, + \item $\overline{U_{s \concat i}} \subseteq U_s$ + for $i \in 2$. + \end{enumerate} + + We define $U_s$ inductively on the length of $s$. + + For $U_{\emptyset}$ take any non-empty open set + with small enough diameter. + + Given $U_s$, pick $x \neq y \in U_s$ + and let $U_{s \concat 0} \ni x$, + $U_{s \concat 1} \ni y$ + be disjoint, open, + of diameter $\le \frac{1}{2^{|s| +1}}$ + and such that $\overline{U_{s\concat 0}}, \overline{U_{S \concat 1}} \subseteq U_s$. + + Let $x \in 2^{\N}$. + Then let $f(x)$ be the unique point in $X$ + such that + \[ + \{f(x)\} = \bigcap_{n} U_{x \defon n} = \bigcap_{n} \overline{U_{x \defon n}. + \] + (This is nonempty as $X$ is a completely metrizable space.) + It is clear that $f$ is injective and continuous. + % TODO: more details + $2^{\N}$ is compact, hence $f^{-1}$ is also continuous. +\end{proof} + +\begin{corollary} + Every nonempty perfect Polish + space $X$ has cardinality $C = 2^{\aleph_0}$ + % TODO: eulerscript C ? +\end{corollary} +\begin{proof} + Since the cantor space embeds into $X$, + we get the lower bound. + Since $X$ is second countable and Hausdorff, + we get the upper bound. +\end{proof} + +\begin{theorem} + Any Polish space is countable + or it has cardinality $C$. % TODO C +\end{theorem} +\todo{Homework 3} + + +\begin{definition} + A \vocab{Lusin scheme} on a set $X$ + is a family $(A_s)_{s \in \N^{<\N}}$ + of subsets of $X $ + such that + \begin{enumerate}[(i)] + \item $A_{s \concat i} \cap A_{s \concat j} = \emptyset$ + for all $j \neq i \in \N$, $s \in \N^{<\N}$. + \item $A_{s \concat i} \subseteq A_s$ + for all $i \in \N, s \in \N^{<\N}$. + \end{enumerate} +\end{definition} + +\begin{theorem} + \label{thm:bairetopolish} + Let $X \neq \emptyset$ be a Polish space. + Then there is a closed subset + \[ + D \subseteq \N^\N \text{\reflectbox{$\coloneqq$}} \cN + % TODO correct N for the Baire space? + \] + and a continuous bijection from + $D$ onto $X$ (the inverse does not need to be continuous). + + Moreover there is a continuous surjection $g: \cN \to X$ + extending $f$. +\end{theorem} +\begin{definition} + An $F_\sigma$ set is the + countable union of closed sets, + i.e.~the complement of a $G_\delta$ set. +\end{definition} +\begin{observe} + \begin{itemize} + \item Any open set is $F {\sigma}$. + \item In metric spaces the intersection of an open and closed set is $F_\sigma$. + \end{itemize} +\end{observe} +\begin{refproof}{thm:bairetopolish} + Let $d$ be a complete metric on $X$. + W.l.o.g.~$\diam(X) \le 1$. + We construct a Lusin scheme + $(F_s)_{s \in \N^{<\N}}$ + such that $F_s \subseteq X$ + and + \begin{enumerate}[(i)] + \item $F_\emptyset = X$, + \item $F_s$ is $F_\sigma$ for all $s$. + \item The $F_{s \concat i}$ partition $F_s$, + i.e.~$F_{s} = \bigsqcup_i F_{s \concat i}$. + + Furthermore we want that + $\overline{F_{s \concat i}} \subseteq F_s$ + for all $i$. + \item $\diam(F_s) \le 2^{-|s|}$. + \end{enumerate} + + Suppose we already have $F_s \text{\reflectbox{$\coloneqq$}} F$. + We need to construct a partition $(F_i)_{i \in \N}$ + of $F$ with $\overline{F_i} \subseteq F$ + and $\diam(F_i) < \epsilon$ + for $\epsilon = 2^{-|s| - 1}$, + such that the $F_i$ are $F_\sigma$. + + \paragraph{Step 1} + Write $F \coloneqq \bigcup_{i \in \N} C_i$ + for some closed sets $C_i$. + W.l.o.g.~$C_i \subseteq C_{i+1}$. + + Let $F_i^0 \coloneqq C_{i+1} \setminus C_i$. + These $F_i^0$ are $F_\sigma$, + and form a partition of $F$. + Furthermore $\overline{F_i^0} \subseteq F$. + + However the diameter might be too large. + Fix $i \in \N$ and consider $F_i^0$. + Cover it with countably many open balls $B_1, B_2,\ldots$ + of diameter smaller than $\epsilon$. + The sets $D_i \coloneqq F_i^0 \cap B_i \setminus (B_1 \cup \ldots \cup B_{i-1})$ + are $F_\sigma$, disjoint + and $F_i^0 = \bigcup_{j} D_j$. + + + + +\end{refproof} + + diff --git a/logic.sty b/logic.sty index 4ef7ba3..97f3eb7 100644 --- a/logic.sty +++ b/logic.sty @@ -124,7 +124,12 @@ \DeclareSimpleMathOperator{Ord} \DeclareSimpleMathOperator{trcl} \DeclareSimpleMathOperator{tcl} -\newcommand{\concat}{{}^\frown} + +\newcommand{\concat}{\mathop{{}^{\scalebox{.7}{$\smallfrown$}}}} + +%https://tex.stackexchange.com/questions/73437/how-do-i-typeset-the-concatenation-of-strings-properly +%\mathbin{\raisebox{1ex}{\scalebox{.7}{$\frown$}}}% + \DeclareMathOperator{\hght}{height} \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}} diff --git a/logic3.tex b/logic3.tex index 32b8bc6..cd35863 100644 --- a/logic3.tex +++ b/logic3.tex @@ -26,6 +26,7 @@ \input{inputs/lecture_01} \input{inputs/lecture_02} +\input{inputs/lecture_03}