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} - - -