some small changes

inputs/lecture_14.tex

@@ -104,7 +104,7 @@
     \forall  x \in X.~(\exists n.~(x,n) \in R \iff \exists! n.~(x,n)\in R^\ast).
     \]
     We say that $R^\ast$ \vocab[uniformization]{uniformizes} $R$.%
-    \footnote{Wikimedia has a \href{https://upload.wikimedia.org/wikipedia/commons/4/4c/Uniformization_ill.png}{nice picture.}}
+    \footnote{Wikimedia has a nice \href{https://upload.wikimedia.org/wikipedia/commons/4/4c/Uniformization_ill.png}{picture}.}
 \end{theorem}
 \begin{proof}
     Let $\phi\colon R \to  \Ord$

inputs/lecture_17.tex

@@ -191,8 +191,7 @@ since $X^X$ has these properties.
     Since $f$ is injective, we get that $x = f(x)$,
     i.e.~$f = \id$.
 
-    Take $g' \in G$ such that $f = g' \circ g$.%
+    Since $f \in Gg$, there exists $g' \in G$ such that $f = g' \circ g$.
-    %\footnote{This exists since $f \in Gg$.}
 
     It is $g' = g'gg'$,
     so $\forall  x .~g'(x) = g'(g g'(x))$.

inputs/lecture_18.tex

@@ -1,9 +1,6 @@
 \subsection{Sketch of proof of \yaref{thm:l16:3}}
 \lecture{18}{2023-12-15}{Sketch of proof of \yaref{thm:l16:3}}
 
-% TODO ANKI-MARKER
-
-
 The goal for this lecture is to give a very rough
 sketch of \yaref{thm:l16:3} in the case of $|Z| = 1$.
 
@@ -21,7 +18,9 @@ F(x,x') \coloneqq \inf \{d(gx, gx') : g \in G\}.
     \item $F(x,x') = F(x', x)$,
     \item $F(x,x') \ge 0$ and $F(x,x') = 0$ iff $x = x'$.
     \item $F(gx, gx') = F(x,x')$ since  $G$ is a group.
-    \item $F$ is an upper semi-continuous function on $X^2$,
+    \item $F$ is an \vocab{upper semi-continuous}\footnote{%
+        Wikimedia has a nice \href{https://upload.wikimedia.org/wikipedia/commons/c/c0/Upper_semi.svg}{picture}.}
+        function on $X^2$,
         i.e.~$\forall a \in R.~\{(x,x') \in X^2 : F(x,x') < a\} \overset{\text{open}}{\subseteq} X^2$.
 
         This holds because $F$ is the infimum of continuous functions
@@ -47,23 +46,28 @@ This will follow from the following lemma:
 \begin{lemma}
     \label{lem:ftophelper}
     Let $F(x,x') < a$.
+    \gist{%
     Then there exists $\epsilon > 0$ such that
     whenever $F(x',x'') < \epsilon$, then $F(x,x'') < a$.
+    }{Then $\exists  \epsilon > 0.~\forall x''.~F(x',x'') < \epsilon \implies F(x,x'') < a$.}
 \end{lemma}
 \begin{refproof}{def:ftop}
+\gist{%
     We have to show that if $U_a(x_1) \cap U_b(x_2) \neq \emptyset$,
     then this intersection is the union
     of sets of this kind.
-    Let $x' \in U_a(x_1)$.
+}{}
+    Let $x' \in U_a(x_1) \cap  U_b(x_2)$.
     Then by \yaref{lem:ftophelper},
     there exists $\epsilon_1 > 0$ with $U_{\epsilon_1}(x') \subseteq U_a(x_1)$.
-    Similarly there exists  $\epsilon_2 > 0$
+    Similarly there exists  $\epsilon_2 > 0$\gist{
-    such that $U_{\epsilon_2}(x') \subseteq  U_b(x_2)$.
+    such that $U_{\epsilon_2}(x') \subseteq  U_b(x_2)$.}{.}
     So for $\epsilon \le  \epsilon_1, \epsilon_2$,
     we get $U_{\epsilon}(x') \subseteq U_a(x_1) \cap U_b(x_2)$.
 \end{refproof}
 \begin{refproof}{lem:ftophelper}%
     \notexaminable{\footnote{This was not covered in class.}
+        % TODO: maybe learn?
 
         Let $T = \bigcup_n  T_n$,% TODO Why does this exist?
         $T_n$ compact, wlog.~$T_n \subseteq T_{n+1}$, and
@@ -164,8 +168,7 @@ i.e.~show that if $(Z,T)$ is a proper factor of  a minimal distal flow
             \item One can show that $H$ is a topological group and $(M,H)$
                 is a flow.\footnote{This is non-trivial.}
             \item Since $H$ is compact,
-                $(M,H)$ is equicontinuous, %\todo{We didn't define this}
+                $(M,H)$ is equicontinuous, i.e.~it is isometric.
-                i.e.~it is isometric.
                 In particular, $(M,T)$ is isometric.
         \end{enumerate}
     \item $M \neq  \{\star\}$, i.e.~$(M,T)$ is non-trivial:
@@ -209,9 +212,9 @@ i.e.~show that if $(Z,T)$ is a proper factor of  a minimal distal flow
     Let $X$ be a metric space
     and $\Gamma\colon X \to \R$ be upper semicontinuous.
     Then the set of continuity points of $\Gamma$ is comeager.
-    \todo{Missing figure: upper semicontinuous function}
 \end{theorem}
 \begin{proof}
+\notexaminable{
     Take $x$ such that $\Gamma$ is not continuous at $x$.
     Then there is an $\epsilon > 0$ 
     and $x_n \to x$ such that
@@ -227,6 +230,7 @@ i.e.~show that if $(Z,T)$ is a proper factor of  a minimal distal flow
     and $B_q \setminus B_q^\circ$ is nwd
     as it is closed and has empty interior,
     so $\bigcup_{q \in \Q} F_q$ is meager.
+}
 \end{proof}
 
 

inputs/lecture_19.tex

@@ -1,5 +1,8 @@
-\subsection{The order of a flow}
+\subsection{The Order of a Flow}
-\lecture{19}{2023-12-19}{Orders of flows}
+\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}.
 
@@ -71,7 +74,6 @@ equicontinuity coincide.
     i.e.~if $(Y_\alpha, f_{\alpha,\beta})$,
     $\beta < \alpha \le  \Theta$
     are isometric, then the inverse limit $Y$ is isometric.%
-    \todo{Why does an inverse limit exist?}
     % https://q.uiver.app/#q=WzAsNCxbMSwwLCJZX1xcYWxwaGEiXSxbMSwxLCJZX1xcYmV0YSJdLFswLDAsIlkiXSxbMiwwLCJYIl0sWzAsMSwiZl97XFxhbHBoYSwgXFxiZXRhfSJdLFsyLDAsImZfXFxhbHBoYSJdLFsyLDEsImZfXFxiZXRhIiwyXSxbMywwLCJcXHBpX1xcYWxwaGEiLDJdLFszLDEsIlxccGlfXFxiZXRhIl1d
 \[\begin{tikzcd}
 	Y & {Y_\alpha} & X \\