inputs/lecture_04.tex

@@ -119,6 +119,7 @@
 
     The complement of a meager set is called
     \vocab{comeager}.
+    
 \end{definition}
 \begin{example}
     $\Q \subseteq \R$ is meager.
@@ -127,7 +128,7 @@
     Let $A, B \subseteq  X$.
     We write $A =^\ast B$ 
     iff the \vocab{symmetric difference},
-    $A \symdif B \coloneqq  (A\setminus B) \cup (B \setminus A)$,
+    $A \mathop{\triangle} B \coloneqq  (A\setminus B) \cup (B \setminus A)$,
     is meager.
 \end{notation}
 \begin{remark}
@@ -143,14 +144,13 @@ Note that open sets and meager sets have the Baire property.
 
 
 
-\begin{example}
+% \begin{example}
-    \begin{itemize}
+%     $\Q \subseteq  \R$ is $F_\sigma$.
-        \item $\Q \subseteq  \R$ is $F_\sigma$.
+% 
-        \item $\R \setminus \Q \subseteq \R$ is $G_\delta$.
+%     $\R \setminus \Q \subseteq \R$ is $G_\delta$.
-        \item $\Q \subseteq \R$ is not $G_{\delta}$.
+% 
-            (It is dense and meager,
+%     $\Q \subseteq \R$ is not $G_{\delta}$.
-            hence it can not be $G_\delta$
+%     (It is dense and meager,
-            by the Baire category theorem).
+%     hence it can not be $G_\delta$,
-    \end{itemize}
+%     by the Baire category theorem).
-
+% \end{example}
-\end{example}

inputs/lecture_05.tex

@@ -1,271 +0,0 @@
-\lecture{05}{2023-10-31}{}
-
-\begin{fact}
-    A set $A$ is nwd iff $\overline{A}$ is nwd.
-
-    If $F$ is closed then 
-    $F$ is nwd iff $X \setminus F$ is open and dense.
-
-    Any meager set $B$ is contained in
-    a meager $F_{\sigma}$-set.
-\end{fact}
-
-\begin{definition}
-    A  \vocab{$\sigma$-algebra} on a set $X$
-    is a collection of subsets of $X$
-    such that:
-    \begin{itemize}
-        \item $\emptyset, X \in \cA$,
-        \item $ A \in \cA \implies X \setminus A \in \cA$,
-        \item  $(A_i)_{i < \omega}, A_i \in \cA \implies \bigcup_{i < \omega} A_i \in \cA$.
-    \end{itemize}
-\end{definition}
-\begin{fact}
-    Since $\bigcap_{i < \omega} A_i = \left( \bigcup_{i < \omega} A_i^c \right)^c$
-    we have that $\sigma$-algebras are closed under countable intersections.
-\end{fact}
-
-\begin{theorem}
-    \label{thm:bairesigma}
-    Let $X$ be a topological space.
-    Then the collection of sets with the Baire property
-    is a $\sigma$-algebra on $X$.
-
-    It is the smallest $\sigma$-algebra
-    containing all meager and open sets.
-\end{theorem}
-\begin{refproof}{thm:bairesigma}
-    Let $\cA$ be the collection of sets with the Baire property.
-    Since open sets have the Baire property,
-    we have $\emptyset, X \in \cA$.
-
-
-    Let $A_n \in \cA$ for all $n < \omega$.
-    Take $U_n$ such that $A_n \symdif U_n$ is meager.
-    Then
-    \[
-        \left( \bigcup_{n < \omega} A_n \right) \symdif \left( \bigcup_{n < \omega} U_n \right)
-    \] 
-    is meager,\todo{small exercise}
-    hence $\bigcup_{n < \omega} A_n \in \cA$.
-
-    Let $A \in \cA$.
-    Take some open $U$ such that $U \symdif A$ is meager.
-    We have $(X \setminus U) \symdif (X \setminus A) = U \symdif A$.
-
-    \begin{claim}
-        \label{thm:bairesigma:c1}
-        If $F$ is closed,
-        then $F \setminus \inter(F)$
-        is nwd.
-        In particular, $F \symdif \inter(F)$ is nwd.
-    \end{claim}
-    \begin{refproof}{thm:bairesigma:c1}
-        \todo{TODO}
-    \end{refproof}
-
-    From the claim we get that
-    $X \setminus A =^\ast X \setminus U =^\ast \inter(X \setminus U)$.
-    Hence $X \setminus A \in \cA$.
-
-
-    It is clear that all meager sets have the Baire property.
-
-    Let $A \in \cA$. Then $A = (A \setminus U) \cup (A \cap U)$
-    for some open $U$
-    such that $A \setminus U$ is meager.
-    We have $A \cap U = U \setminus (U \setminus A)$.
-    Thus we get that $\cA$ is the minimal $\sigma$-algebra
-    containing all meager and all open sets.
-\end{refproof}
-
-%\begin{example}
-%    Nwd set of positive measure.
-%    TODO
-%    remove open intervals such that their length does not add to 0
-%       
-%\end{example}
-
-\begin{theorem}[Baire Category theorem]
-    Let $X$ be a completely metrizable space.
-    Then every comeager set of $X$ is dense in $X$.
-\end{theorem}
-\todo{Proof (copy from some other lecture)}
-\begin{proposition}
-    Let $X$ be a topological space.
-    The following are equivalent:
-    \begin{enumerate}[(i)]
-        \item Every nonempty open set
-            is non-meager in $X$.
-        \item Every comeager set is dense.
-        \item The intersection of countable many
-            open dense sets is dense.
-    \end{enumerate}
-\end{proposition}
-\begin{proof}
-    \todo{Proof (short)}
-
-    (iii) $\implies$ (i)
-    Let us first show that $X$ is non-meager.
-    Suppose that $X$ is meager. Then $X = \bigcup_{n} A_n = \bigcup_{n} \overline{A_n}$
-    is the countable union of nwd sets.
-    We have that
-    \[
-    \emptyset = \bigcap_{n} (X \setminus \overline{A_n})
-    \] 
-    is dense by (iii).
-    This proof can be adapted to other open sets $X$.
-\end{proof}
-
-
-% TODO Fubini
-\begin{notation}
-    Let $X ,Y$ be topological spaces,
-    $A \subseteq  X \times Y$,
-    $x \in X, y \in Y$.
-
-    Let
-    \[
-    A_x \coloneqq \{y \in Y : (x,y) \in A\} 
-    \] 
-    and
-    \[
-    A^y \coloneqq \{x \in X : (x,y) \in A\} .
-    \] 
-\end{notation}
-
-The following similar to Fubini,
-but for meager sets:
-
-\begin{theorem}[Kuratowski-Ulam]
-    \label{thm:kuratowskiulam}
-    Let $X,Y$ be second countable topological spaces.
-    Let $A \subseteq X \times Y$
-    be a set with the Baire property.
-
-    Then
-    \begin{enumerate}[(i)]
-        \item $\{x \in X : A_x \text{ has the BP }\}$
-            is comeager\footnote{Note that not necessarily all sections
-                have the BP. For example $\{x\} \times Y$ is meager in $X \times Y$}
-            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}.
-            \end{IEEEeqnarray*}
-        \item $A$ is comeager
-            \begin{IEEEeqnarray*}{rll}
-                \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}
-\end{theorem}
-\begin{refproof}{thm:kuratowskiulam}
-    (ii) and (iii) are equivalent by passing to the complement.
-
-    \begin{claim}%[1a]
-        \label{thm:kuratowskiulam:c1a}
-
-        If $F \overset{\text{closed}}{\subseteq}  X \times Y$
-        is nwd,
-        then
-        \[
-        \{x \in X : F_x \text{is nwd}\} 
-        \] 
-        is comeager.
-    \end{claim}
-    \begin{refproof}{thm:kuratowskiulam:c1a}
-        Put $W = F^c$.
-        This is open and dense in $X \times Y$.
-        It suffices to show that $\{x \in X : W_x \text{ is dense}\}$
-        is comeager.
-        Note that $W_x$ is open for all $x$.
-
-        Fix a countable basis $(V_n)$ of $Y$
-        with $V_n$ non-empty.
-        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.
-
-        Fix $n$ and let $U_n \coloneqq \{x \in X: (W_x \cap V_n) = \emptyset\}$.
-        We will show that $U_n$ is open and dense,
-        hence it is comeager.
-
-        $U_n = \proj_x(W \cap (X \times V_n))$ is open
-        since projections of open sets are open.
-
-        Let $U \subseteq X$ be nonempty and open.
-        We need to show that $U \cap U_n \neq \emptyset$.
-        It is
-        \[
-        U \cap U_n = \proj_x(W \cap (U \times V_n))
-        \] 
-        nonempty since $W$ is dense.
-        
-    \end{refproof}
-
-    \begin{claim} % [1a']
-        \label{thm:kuratowskiulam:c1ap}
-        If $F \subseteq  X \times Y$
-        is nwd,
-        then
-        \[
-        \{x \in X : F_x \text{is nwd}\} 
-        \] 
-        is comeager.
-        
-    \end{claim}
-    \begin{refproof}{thm:kuratowskiulam:c1ap}
-        We have that $\overline{F}$ is nwd.
-        Hence by \yaref{thm:kuratowskiulam:c1a}
-        the set
-         \[
-        \{x \in X: \overline{F_x} \text{ is nwd}\} \subseteq
-        \{x \in X: F_x \text{ is nwd}\} 
-        \] 
-        is comeager.
-    \end{refproof}
-
-    \begin{claim}% [1b]
-        \label{thm:kuratowskiulam:c1b}
-
-        If $M \subseteq  X \times Y$ is meager,
-        then
-        \[
-        \{x \in X : M_x \text{ is meager}\} 
-        \] 
-        is comeager.
-    \end{claim}
-    \begin{refproof}{thm:kuratowskiulam:c1b}
-        This follows from \yaref{thm:kuratowskiulam:c1ap}:
-        Let $M = \bigcup_{n < \omega} F_n$
-        where the $F_n$ are nwd.
-        Apply \yaref{thm:kuratowskiulam:c1ap}
-        to each $F_n$.
-        We get that
-        $M_x$ is comeager
-        as a countable intersection of comeager sets.
-    \end{refproof}
-
-    \todo{Finish proof}
-    \phantom\qedhere
-\end{refproof}
-
-\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}
-
-

inputs/tutorial_02.tex

@@ -1,61 +0,0 @@
-\tutorial{02}{2023-10-24}{}
-
-% Points: 15 / 16
-
-\subsubsection{Exercise 4}
-
-\begin{fact}
-    Let $X $ be a compact Hausdorffspace.
-    Then the following are equivalent:
-    \begin{enumerate}[(i)]
-        \item $X$ is Polish,
-        \item $X$ is metrisable,
-        \item $X$ is second countable.
-    \end{enumerate}
-\end{fact}
-\begin{proof}
-    (i) $\implies$ (ii) clear
-
-    (i) $\implies$ (iii) clear
-
-    (ii) $\implies$ (i) Consider the cover $\{B_{\epsilon}(x) | x \in X\}$
-    for every $\epsilon \in \Q$
-    and chose a finite subcover.
-    Then the midpoints of the balls from the cover
-    form a countable dense subset.
-
-    The metric is complete as $X$ is compact.
-    (For metric spaces: compact $\iff$ seq.~compact $\iff$ complete and totally bounded)
-
-    (iii) $\implies$ (ii)
-    Use Urysohn's metrisation theorem and the fact that compact
-    Hausdorff spaces are normal
-\end{proof}
-
-Let $X$ be compact Polish (compact metrisable $\implies$ compact Polish)
-and $Y $ Polish.
-Let $\cC(X,Y)$ be the set of continuous functions $X \to Y$.
-Consider the metric $d_u(f,g) \coloneqq  \sup_{x \in X} |d(f(x), g(x))|$.
-Clearly $d_u$ is a metric.
-
-\begin{claim}
-    $d_u$ is complete.
-\end{claim}
-\begin{subproof}
-    Let $(f_n)$ be a Cauchy sequence in $\cC(X,Y)$.
-    A $Y$ is complete,
-    there exists a pointwise limit $f$.
-
-    $f_n$ converges uniformly to $f$:
-
-    \[
-    d(f_n(x), f(x)) \le  \overbrace{d(f_n(x), f_m(x))}^{\mathclap{\text{$(f_n)$ is Cauchy}}}
-    + \underbrace{d(f_m(x), f(x))}_{\mathclap{\text{small for appropriate $m$}}}.
-    \]
-
-    $f$ is continuous by the uniform convergence theorem.
-\end{subproof}
-
-\begin{claim}
-    There exists a countable dense subset.
-\end{claim}

logic.sty

@@ -131,7 +131,5 @@
 %\mathbin{\raisebox{1ex}{\scalebox{.7}{$\frown$}}}%
     
 \DeclareMathOperator{\hght}{height}
-\DeclareMathOperator{\symdif}{\triangle}
-\DeclareSimpleMathOperator{proj}
 
 \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}

logic3.tex

@@ -28,7 +28,6 @@
 \input{inputs/lecture_02}
 \input{inputs/lecture_03}
 \input{inputs/lecture_04}
-\input{inputs/lecture_05}