From 4a2732444240f231e64d8329d3bc25b42922c981 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Tue, 24 Oct 2023 10:34:43 +0200
Subject: [PATCH 1/3] fixed label
---
inputs/lecture_03.tex | 3 +--
1 file changed, 1 insertion(+), 2 deletions(-)
diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex
index fb71262..7e97364 100644
--- a/inputs/lecture_03.tex
+++ b/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{thm:cantorbendixson}{Cantor-Bendixson}{Cantor-Bendixson}
If $A \subseteq \R$ is closed,
it is either at most countable or else
$A$ contains a perfect set.
From fb2c5b271576fe04fcb796d648904f901773a3cd Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Tue, 24 Oct 2023 10:48:41 +0200
Subject: [PATCH 2/3] fixed label
---
inputs/lecture_01.tex | 6 +++---
inputs/lecture_03.tex | 11 ++++-------
2 files changed, 7 insertions(+), 10 deletions(-)
diff --git a/inputs/lecture_01.tex b/inputs/lecture_01.tex
index 6c8676c..319daa8 100644
--- a/inputs/lecture_01.tex
+++ b/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}
diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex
index 7e97364..7575c8f 100644
--- a/inputs/lecture_03.tex
+++ b/inputs/lecture_03.tex
@@ -1,7 +1,7 @@
\lecture{03}{2023-10–23}{Cantor-Bendixson}
\begin{theorem}[Cantor-Bendixson]
- \yalabel{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.
@@ -29,9 +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,
@@ -127,6 +127,3 @@ all condensation points are accumulation points.
% is at most countable.
% Also $A'$ is closed.
% \end{remark}
-
-
-
From b5e2d09090795deb7d4da6df60b6b56fc515e2d1 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Tue, 24 Oct 2023 10:51:01 +0200
Subject: [PATCH 3/3] small fix
---
inputs/lecture_02.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/inputs/lecture_02.tex b/inputs/lecture_02.tex
index 331efb9..7c76520 100644
--- a/inputs/lecture_02.tex
+++ b/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