inputs/lecture_03.tex

@@ -107,7 +107,6 @@
 \end{definition}
 
 \begin{theorem}
-    \label{thm:cantortopolish}
     Let $X \neq \emptyset$ 
     be a perfect Polish space.
     Then there is an embedding
@@ -217,7 +216,7 @@
         \item $F_\emptyset = X$,
         \item $F_s$ is $F_\sigma$ for all  $s$.
         \item The $F_{s \concat i}$ partition $F_s$,
-            i.e.~$F_{s} = \bigsqcup_i F_{s \concat i}$. % TODO change notation?
+            i.e.~$F_{s} = \bigsqcup_i F_{s \concat i}$.
 
             Furthermore we want that
             $\overline{F_{s \concat i}} \subseteq F_s$

inputs/lecture_04.tex

@@ -1,156 +0,0 @@
-\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}

jrpie-math.sty

@@ -48,5 +48,4 @@
 \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

logic3.tex

@@ -27,7 +27,6 @@
 \input{inputs/lecture_01}
 \input{inputs/lecture_02}
 \input{inputs/lecture_03}
-\input{inputs/lecture_04}