some small changes

inputs/lecture_18.tex

@@ -226,8 +226,8 @@ i.e.~show that if $(Z,T)$ is a proper factor of  a minimal distal flow
     \]
     $X \setminus B_q = \{a \in X : \Gamma(a) <  q\}$
     is open, i.e.~$B_q$ is closed.
-    Note that $x \in F_q \coloneqq B_q \setminus B_q^\circ$
-    and $B_q \setminus B_q^\circ$ is nwd
+    Note that $x \in F_q \coloneqq B_q \setminus \inter(B_q)$
+    and $B_q \setminus \inter(B_q)$ is nwd
     as it is closed and has empty interior,
     so $\bigcup_{q \in \Q} F_q$ is meager.
 }

inputs/lecture_19.tex

@@ -1,9 +1,6 @@
 \subsection{The Order of a Flow}
 \lecture{19}{2023-12-19}{Orders of Flows}
 
-% TODO ANKI-MARKER
-
-
 See also \cite[\href{https://terrytao.wordpress.com/2008/01/24/254a-lecture-6-isometric-systems-and-isometric-extensions/}{Lecture 6}]{tao}.
 
 
@@ -195,8 +192,8 @@ More generally we can show:
     on the fibers of $Y$ over $Z_2$
     and invariant under $T$.
 
-    $\sigma$ is a metric, since if
-    if $\pi_2(y) = \pi_2(y')$ and $\sigma(y,y') = 0$,
+    $\sigma$ is a metric,
+    since if $\pi_2(y) = \pi_2(y')$ and $\sigma(y,y') = 0$,
     then $\pi_1(y) = \pi_1(y')$ or $y = y'$.
 \end{proof}
 
@@ -223,6 +220,7 @@ More generally we can show:
     For this, we show that for all  $\xi < \eta$,
     $(X_\xi', T)$ is a factor of $(X_\xi ,T)$
     using transfinite induction.
+
 % https://q.uiver.app/#q=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
 \[\begin{tikzcd}
 	{X'_{\eta'}} & \dots & {X'_3} & {X'_2} & {X_1'} \\
@@ -244,7 +242,8 @@ More generally we can show:
 	\arrow["{\pi'_2}", curve={height=-18pt}, from=1-1, to=1-4]
 	\arrow["{\pi'_1}", curve={height=-30pt}, from=1-1, to=1-5]
 \end{tikzcd}\]
-    % TODO: induction start?
+
+    We'll only show the successor step:
 
     Suppose we have 
     $(X'_\xi, T) = \theta((X_\xi, T)$.
@@ -253,18 +252,21 @@ More generally we can show:
     \[Y \coloneqq \{(\pi_\xi(x), \pi'_{\xi+1}(x)) \in X_\xi \times  X'_{\xi+ 1}: x \in X\}.\]
     Then
 
-    % https://q.uiver.app/#q=WzAsNSxbMCwwLCIoWF97XFx4aSsxfSxUKSJdLFsyLDAsIihZLFQpIl0sWzMsMSwiKFgnX3tcXHhpKzF9LFQpIl0sWzIsMiwiKFgnX1xceGksVCkiXSxbMSwxLCIoWF9cXHhpLFQpIl0sWzAsNCwiXFx0ZXh0e21heC5+aXNvfSIsMV0sWzQsMywiXFx0aGV0YSIsMV0sWzIsMywie1xcY29sb3J7b3JhbmdlfVxcdGV4dHtpc299fSIsMV0sWzEsNCwie1xcY29sb3J7b3JhbmdlfVxcdGV4dHtpc299fSIsMV0sWzEsMl0sWzAsMSwiIiwwLHsiY3VydmUiOi0xLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=
-\[\begin{tikzcd}
+    % https://q.uiver.app/#q=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
+\begin{tikzcd}
 	{(X_{\xi+1},T)} && {(Y,T)} \\
 	& {(X_\xi,T)} && {(X'_{\xi+1},T)} \\
 	&& {(X'_\xi,T)}
 	\arrow["{\text{max.~iso}}"{description}, from=1-1, to=2-2]
 	\arrow["\theta"{description}, from=2-2, to=3-3]
-	\arrow["{{\color{orange}\text{iso}}}"{description}, from=2-4, to=3-3]
-	\arrow["{{\color{orange}\text{iso}}}"{description}, from=1-3, to=2-2]
+	\arrow[""{name=0, anchor=center, inner sep=0}, "{{\color{orange}\text{iso}}}"{description}, from=2-4, to=3-3]
+	\arrow[""{name=1, anchor=center, inner sep=0}, "{{\color{orange}\text{iso}}}"{description}, from=1-3, to=2-2]
 	\arrow[from=1-3, to=2-4]
 	\arrow[curve={height=-6pt}, dashed, from=1-1, to=1-3]
-\end{tikzcd}\]
+	\arrow["{\pi'}", draw=none, from=1-3, to=2-4]
+	\arrow["{\theta'}", draw=none, from=0, to=3-3]
+	\arrow["\pi"', draw=none, from=1-3, to=1]
+\end{tikzcd}
 
     The diagram commutes, since all maps are the induced maps.
     By definition of $Y$ is clear that $\pi$ and $\pi'$ separate points in $Y$.
@@ -277,6 +279,9 @@ More generally we can show:
     In particular,
     $(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$.
 \end{proof}
+
+% TODO ANKI-MARKER
+
 \begin{example}[{\cite[p. 513]{Furstenberg}}]
     \label{ex:19:inftorus}
     Let $X$ be the infinite torus

inputs/lecture_22.tex

@@ -86,7 +86,7 @@ Let $\mathbb{K} = S^1$
 and $I$ a countable linear order.
 Let $\mathbb{K}^I$ be the product of $|I|$ many $\mathbb{K}$,
 $\mathbb{K}^{<i} \coloneqq \mathbb{K}^{\{j : j < i\}}$
-and $\pi_{i}\colon \mathbb{K}^{I} \to  \mathbb{K}^{<i}$\todo{maybe call it $\pi_{<i}$?}
+and $\pi_{i}\colon \mathbb{K}^{I} \to  \mathbb{K}^{<i}$% \todo{maybe call it $\pi_{<i}$?}
 the projection.
 
 Let $\mathbb{K}_I \coloneqq  \prod_{i \in I} C(\mathbb{K}^{<i}, \mathbb{K})$.

inputs/lecture_24.tex

@@ -17,17 +17,19 @@
             we have $X \in \cU \lor \N \setminus X \in \cU$.
     \end{enumerate}
 \end{definition}
+\gist{
 \begin{remark}
     \begin{itemize}
-        \item If $X \cup Y \in \cU$ then $X \in \cU \lor Y$ or $Y \in \cU$:
+        \item If $X \cup Y \in \cU$ then $X \in \cU$ or $Y \in \cU$:
             Consider $((\N \setminus X) \cap (\N \setminus Y) = \N  \setminus (X \cup Y)$.
         \item Every filter can be extended to an ultrafilter.
             (Zorn's lemma)
     \end{itemize}
 \end{remark}
+}{}
 \begin{definition}
-    An ultrafilter is called \vocab[Ultrafilter!principal]{principal} or \vocab[Ultrafilter!trivial]{trivial} 
-    if it is of the form
+    An ultrafilter is called \vocab[Ultrafilter!principal]{principal} or \vocab[Ultrafilter!trivial]{trivial}
+    iff it is of the form
     \[
     \hat{n} = \{X \subseteq \N : n \in X\}.
     \]
@@ -46,9 +48,9 @@
     Let $\phi(\cdot )$, $\psi(\cdot )$ be formulas.
 
     \begin{enumerate}[(1)]
-        \item $(\cU n) (\phi(n) \land \psi(m)) \iff (\cU n) \phi(n) \land (\cU n) \psi(n)$.
-        \item $(\cU n) (\phi(n) \lor \psi(m)) \iff (\cU n) \phi(n) \lor (\cU n) \psi(n)$.
-        \item $(\cU n) \lnot \phi(n) \iff \lnot (\cU n) \phi(n)$.
+        \item $(\cU n) ~ (\phi(n) \land \psi(m)) \iff (\cU n) \phi(n) \land (\cU n) \psi(n)$.
+        \item $(\cU n) ~ (\phi(n) \lor \psi(m)) \iff (\cU n) ~ \phi(n) \lor (\cU n) ~ \psi(n)$.
+        \item $(\cU n) ~\lnot \phi(n) \iff \lnot (\cU n)~ \phi(n)$.
     \end{enumerate}
 \end{observe}
 \begin{lemma}
@@ -59,7 +61,7 @@
     there is a unique $x \in X$,
     such that
     \[
-    (\cU n) (x_n \in G)
+    (\cU n)~(x_n \in G)
     \]
     for every neighbourhood%
     \footnote{$G \subseteq X$ is a neighbourhood iff $x \in \inter G$.}

inputs/tutorial_12b.tex

@@ -17,18 +17,12 @@ Then $t_n y \to (0) = t_n x$.
 
 
 
-
-
-
-
+% TODO this is redundant
 \begin{refproof}{fact:isometriciffequicontinuous}.
-
 $d$ and $d'(x,y) \coloneqq  \sup_{t \in T} d(tx,ty)$
 induce the same topology.
 Let $\tau, \tau'$ be the corresponding topologies.
 
 $\tau \subseteq  \tau'$ easy,
 $\tau' \subseteq \tau'$ : use equicontinuity.
-
-
 \end{refproof}