Merge branch 'main' of https://git.jrpie.de/jrpie-notes/logic-3-abstract-topological-dynamics-and-descriptive-set-theory

inputs/intro.tex

@@ -4,7 +4,7 @@ in the summer term 2023 at the University Münster.
 
 \begin{warning}
     This is not an official script.
-    The official lecture notes can be found on
+    The official lecture notes can be found
     \href{https://sites.google.com/site/akwiatkmath/teaching/logic-3-abstract-topological-dynamics-and-descriptive-set-theory}{here}.
 \end{warning}
 

inputs/lecture_01.tex

@@ -42,14 +42,14 @@ Recall the following notions:
         and $U_i \subsetneq X_i$ for only finitely many $i$.
 \end{definition}
 \begin{fact}
-    Countable products of separable spaces are separable,
+    Countable products of separable spaces are separable.
 \end{fact}
 \begin{definition}
     A topological space $X$ is \vocab{second countable},
     if it has a countable base.
 \end{definition}
-If $X$ is a topological space.
-Then if $X$ is second countable, it is also separable.
+Let $X$ be a topological space.
+If $X$ is second countable, it is also separable.
 However the converse of this does not hold.
 
 \begin{example}
@@ -81,7 +81,8 @@ However the converse of this does not hold.
     For a metric space, the following are equivalent:
     \begin{itemize}
         \item compact,
-        \item \vocab{sequentially compact} (every sequence has a convergent subsequence),
+        \item \vocab{sequentially compact}
+            (every sequence has a convergent subsequence),
         \item complete and \vocab{totally bounded}
             (for all $\epsilon > 0$ we can cover the space with finitely many $\epsilon$-balls).
     \end{itemize}

inputs/tutorial_01.tex

@@ -0,0 +1,70 @@
+\tutorial{01}{202-10-17}{}
+
+% TODO MAIL
+
+\begin{fact}
+    A countable product of separable spaces $(X_n)_{n \in \N}$ is separable.
+\end{fact}
+\begin{proof}
+    Choose a countable dense subset $D_n \subseteq X_n$
+    Fix some point $(a_1,a_2,\ldots) \in \prod_n X_n$
+    and consider $\bigcup_{i \in \N} \prod_{n \le i} D_n \times \prod_{n > i} \{a_n\}$.
+\end{proof}
+
+\begin{fact}
+    \begin{itemize}
+        \item Let $X$ be a topological space.
+            Then $X$ 2nd countable $\implies$ X separable.
+        \item If $X$ is a metric space and separable,
+            then $X$ is 2nd countable.
+    \end{itemize}
+\end{fact}
+\begin{proof}
+    For the first point, choose some point from every basic open set.
+
+    For the second point consider balls of rational radius
+    around the points of a countable dense subset.
+\end{proof}
+
+\begin{definition}
+    A topological space is \vocab{Lindelöf}
+    if every open cover has a countable subcover.
+\end{definition}
+\begin{fact}
+    Let $X$ be a metric space.
+    If $X$ is Lindelöf,
+    then it is 2nd countable.
+\end{fact}
+\begin{proof}
+    For all $q \in \Q$
+    Consider the cover $B_q(x), x \in X$
+    and choose a countable subcover.
+    The union of these subcovers is
+    a countable base.
+\end{proof}
+\begin{fact}
+    Let $X$ be a topological space.
+    If $X$ is 2nd countable,
+    then it is Lindelöff.
+\end{fact}
+\begin{proof}
+    Let $A_0, A_1,\ldots$
+    be a countable base.
+
+    Let $\{U_i\}_{i \in I}$
+    be a cover.
+    Consider $J \coloneqq \{j : \exists i \in I.~A_j \in U_i\}$.
+    For every $j \in J$ choose a $U_i$ such that
+    $A_j \subseteq U_j$.
+    Let $I' \subseteq I$ be the subset of chosen indices.
+    Then $\{U_i\}_{i \in  I'}$ is a countable subcover.
+\end{proof}
+\begin{remark}
+    For metric spaces the notions
+    of being 2nd countable, separable
+    and Lindelöf coincide.
+
+    In arbitrary topological spaces,
+    Lindelöf is the strongest of these notions.
+    
+\end{remark}