remark on stationary sets, closedness of diagonal intersection

inputs/lecture_14.tex

@@ -150,11 +150,20 @@ We have shown (assuming \AxC to choose contained clubs):
     The \vocab{diagonal intersection},
     is defined to be
     \[
-    \diagi_{\beta < \alpha} A_{\beta} \coloneqq 
+    \diagi_{\beta < \alpha} A_{\beta} \coloneqq
     \{\xi < \alpha : \xi \in \bigcap \{A_{\beta} : \beta < \xi\}  \}
     = \bigcap_{\beta < \alpha} ([0,\beta] \cup A_\beta)
     \]
 \end{definition}
+\begin{remark}+
+    \label{rem:diagiclosed}
+    Note that if $A$ is closed,
+    so is $[0,\alpha] \cup A$.
+    Since the intersection of arbitrarily many
+    closed sets is closed,
+    we get that the diagonal intersection
+    of closed sets is closed.
+\end{remark}
 \begin{lemma}
     \label{lem:diagiclub}
     Let $\kappa$ be a regular, uncountable cardinal.
@@ -181,22 +190,23 @@ We have shown (assuming \AxC to choose contained clubs):
         $\diagi_{\beta < \kappa} D_{\beta}$ is closed in $\kappa$.
     \end{claim}
     \begin{subproof}
-        Let $\gamma < \kappa$ be such that $\left( \diagi_{\beta < \kappa} D_{\beta} \right) \cap \gamma$
-        is unbounded in $\gamma$.
-        We aim to show that $\gamma \in  \diagi_{\beta < \kappa} D_{\beta}$.
-        Let $\beta_0 < \gamma$.
-        We need to see that $\gamma \in D_{\beta_0}$.
+        Cf.~\yaref{rem:diagiclosed}.
+        % Let $\gamma < \kappa$ be such that $\left( \diagi_{\beta < \kappa} D_{\beta} \right) \cap \gamma$
+        % is unbounded in $\gamma$.
+        % We aim to show that $\gamma \in  \diagi_{\beta < \kappa} D_{\beta}$.
+        % Let $\beta_0 < \gamma$.
+        % We need to see that $\gamma \in D_{\beta_0}$.
 
-        For each $\beta_0 \le \beta' < \gamma$
-        there is some $\beta'' \in \diagi_{\beta < \kappa} D_\beta$
-        such that $\beta' \le \beta'' < \gamma$,
-        since $\gamma = \sup((\diagi_{\beta < \kappa} D_\beta) \cap \gamma)$.
-        In particular $\beta'' \in D_{\beta_0}$.
+        % For each $\beta_0 \le \beta' < \gamma$
+        % there is some $\beta'' \in \diagi_{\beta < \kappa} D_\beta$
+        % such that $\beta' \le \beta'' < \gamma$,
+        % since $\gamma = \sup((\diagi_{\beta < \kappa} D_\beta) \cap \gamma)$.
+        % In particular $\beta'' \in D_{\beta_0}$.
 
-        So $D_{\beta_0} \cap \gamma$
-        is unbounded in $\gamma$.
-        Since $D_{\beta_0}$ is closed
-        it follows that $\gamma \in D_{\beta_0}$.
+        % So $D_{\beta_0} \cap \gamma$
+        % is unbounded in $\gamma$.
+        % Since $D_{\beta_0}$ is closed
+        % it follows that $\gamma \in D_{\beta_0}$.
 
         %As $\beta_0 < \gamma$ was arbitrary,
         %this shows that $\gamma \in \diagi_{\beta < n} D_\beta$.
@@ -288,6 +298,20 @@ We have shown (assuming \AxC to choose contained clubs):
     iff $C \cap S \neq \emptyset$
     for every club $C \subseteq \kappa$.
 \end{definition}
+\begin{remark}+[\url{https://mathoverflow.net/q/37503}]
+    Informally, club sets and stationary sets
+    can be viewed as large sets of a measure space
+    of measure $1$.
+    Clubs behave similarly to sets of measure $1$
+    and stationary sets are analogous to
+    sets of positive measure:
+    \begin{itemize}
+        \item Every club is stationary,
+        \item the intersection of two clubs is a club,
+        \item the intersection of a club and a stationary set is stationary,
+        \item there exist disjoint stationary sets.
+    \end{itemize}
+\end{remark}
 \begin{example}
     \begin{itemize}
         \item Every $D \subseteq \kappa$ which is club in $\kappa$