Merge branch 'master' of https://git.abstractnonsen.se/josia-notes/w23-logic-2

inputs/lecture_01.tex

@@ -26,7 +26,7 @@ Literature
 
 \begin{definition}
     Let $A \neq \emptyset$, $B$ be arbitrary sets.
-    We write $A \le  B$ ($A$ is not bigger than $B$ )
+    We write $A \le  B$ ($A$ is not bigger than $B$)
     iff there is an injection $f\colon A \hookrightarrow B$.
 \end{definition}
 \begin{lemma}
@@ -148,12 +148,12 @@ Literature
 \end{proof}
 
 \begin{definition}
-    The \vocab{continuum hypothesis} ($\CH$ )
+    The \vocab{continuum hypothesis} ($\CH$)
     says that there is no set $A$ such that
     $\N < A < \R$.
 
     $\CH$ is equivalent to the statement that there is no set $A \subset \R$
-    which is uncountable ($\N < A$ )
+    which is uncountable ($\N < A$)
     and there is no bijection $A \leftrightarrow \R$.
 \end{definition}
 

inputs/lecture_02.tex

@@ -9,7 +9,7 @@
 
 \begin{remark}
     \begin{itemize}
-        \item If $\emptyset \neq  O   \R$
+        \item If $\emptyset \neq  O \overset{\text{open}}{\subseteq} \R$
             then $O \sim \R$.
         \item If $O \subseteq \R$
             is open, then $O$ is the union of open intervals

inputs/lecture_03.tex

@@ -1,8 +1,7 @@
 \lecture{03}{2023-10–23}{Cantor-Bendixson}
 
 \begin{theorem}[Cantor-Bendixson]
-    \yaref{thm:cantorbendixson}{Cantor-Bendixson}{Cantor-Bendixson}
-
+    \yalabel{Cantor-Bendixson}{Cantor-Bendixson}{thm:cantorbendixson}
     If $A \subseteq  \R$ is closed,
     it is either at most countable or else
     $A$ contains a perfect set.
@@ -30,9 +29,9 @@
 
 \begin{definition}
     Let $A \subseteq  \R$.
-    We say that $x \in \R$ 
-    is a \vocab{condensation point} of $A$ 
-    iff for all $a < x < b$, $(a,b) \cap A$ 
+    We say that $x \in \R$
+    is a \vocab{condensation point} of $A$
+    iff for all $a < x < b$, $(a,b) \cap A$
     is uncountable.
 \end{definition}
 By the fact we just proved,
@@ -128,6 +127,3 @@ all condensation points are accumulation points.
 %     is at most countable.
 %     Also $A'$ is closed.
 % \end{remark}
-
-
-