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\def\Sent{\mathop{\rm Sent}\nolimits}
\def\Form{\mathop{\rm Form}\nolimits}
\def\Var{\mathop{\rm Var}\nolimits}
\def\VS{\mathop{\rm VS}\nolimits}
\def\var{\mathop{\rm var}\nolimits}
\def\tp{\mathop{\rm tp}\nolimits}
\def\cf{\mathop{\rm cf}\nolimits}
\def\Club{\mathop{\rm Club}\nolimits}
\def\eq{\mathop{\rm eq}\nolimits}
\def\PO{\mathop{\rm PO}\nolimits}
\def\Th{\mathop{\rm Th}\nolimits}
\def\I{\mathop{\cal I}\nolimits}
\def\A{\mathop{\cal A}\nolimits}
\def\B{\mathop{\cal B}\nolimits}
\def\N{\mathop{\cal N}\nolimits}
\def\F{\mathop{\cal F}\nolimits}
\def\Po{\mathop{\cal P}\nolimits}
\def\G{\mathop{\cal G}\nolimits}
\def\La{\mathop{\cal L}\nolimits}
\def\x{\mathop{\bf x}\nolimits}
\def\y{\mathop{\bf y}\nolimits}
\def\grf{\mathop{\varphi}\nolimits}     
\def\grs{\mathop{\sigma}\nolimits}     
\def\gro{\mathop{\omega}\nolimits}     
\def\gra{\mathop{\alpha}\nolimits}     
\def\grg{\mathop{\gamma}\nolimits}     
\def\grm{\mathop{\mu}\nolimits}     
\def\grd{\mathop{\delta}\nolimits}     
\def\grk{\mathop{\kappa}\nolimits}     
\def\grl{\mathop{\lambda}\nolimits}     
\def\gre{\mathop{\eta}\nolimits}     
\def\grS{\mathop{\Sigma}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\or{\mathop{\vee}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\implies{\mathop{\to}\nolimits}     
\def\deduces{\mathop{\vdash}\nolimits}     
\def\ssi{\mathop{\leftrightarrow}\nolimits}     
\def\all{\mathop{\forall}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}  
   


\begin{document}
\begin{center} {\bf Problem Set IX - Clubs}\\
Math 609  \end{center}

\noindent {\bf Definition.} Let $\grm$ be a limit ordinal. A set $C \subseteq \grm$
is {\em closed} in $\grm$ if for every limit ordinal $\grd < \grm,$ whenever $C \cap \grd$ is
unbounded in $\grd,$ then $\grd \in C.$
\bigskip

\noindent 1. Let $\grm$ be a limit ordinal. Prove that $C \subseteq \grm$ is
closed in $\grm$ if and only if it is a closed subset of $\grm$ with respect to the 
order topology.
\bigskip

\noindent {\bf Definition.} Let $\grm$ be a limit ordinal. A set $C \subseteq \grm$
is a {\em club} of $\grm$ if it is closed and unbounded in $\grm.$
\bigskip

\noindent 2. Let $\grm$ be an ordinal such that $\cf (\grm) > \gro.$ Show that if
$A$ and $B$ are clubs of $\grm,$ then so is $A \cap B.$
\bigskip

\noindent {\bf Definition.} Let $\Club ( \grm )$ denote the filter on $\grm$ generated
by the clubs of $\grm,$ i.e., $\Club (\grm ) := \{ X \subseteq \grm : X \supseteq C
\mbox{ for some club of } \grm \}.$
\bigskip

\noindent 3. Let $\grm$ be an ordinal such that $\cf (\grm) > \grk$ and let 
$\{ X_{\gra} : \gra < \grk \} \subseteq \Club (\grm ).$  Verify that the
intersection $\bigcap_{\gra < \grk} X_{\gra}$ belongs to $\Club (\grm).$
\bigskip

\noindent {\bf Definition.} A subset $X \subseteq \grm$ is {\em stationary} in $\grm$
if $X \cap C \neq 0$ for every club $C$ of $\grm.$
\bigskip

\noindent 4. Let $\grl$ be a regular cardinal and $\cf ( \grm ) > \grl.$ Show that
the set $\{ \grg < \grm : \cf ( \grg ) = \grl \}$ is stationary in $\grm.$
\bigskip

\noindent 5. Let $\grk > \gro$ be regular and suppose that for each $\gra < \grk,$
$C_{\gra}$ is a club of $\grk.$ Prove that the {\em diagonal intersection} $D$ is
also a club of $\grk$ where $D := \{ \grg : \forall \gra < \grg \: ( \grg \in C_{\gra}) \}.$
\bigskip

\noindent 6. Prove Fodor's Lemma: Let $\grk > \gro$ be a regular cardinal and $S \subseteq \grk$
stationary. If $f:S \to \grk$ is a function such that for all $\grg$ in $S,$ $f(\grg ) < \grg,$
then for some $\gra < \grk,$ $f^{-1}(\{ \gra \})$ is stationary in $\grk.$
(Hint: Suppose not and apply 5.)
\bigskip

\noindent {\bf Definition.}  A $\Diamond$-sequence is a sequence $\{ A_{\gra} : \gra < \gro_1 \}$
of sets $A_{\gra} \subseteq \gro_1$ such that for every $A \subseteq \gro_1$ the set
$\{ \gra < \gro_1 : A \cap \gra = A_{\gra} \}$ is stationary.
\bigskip

\noindent 7. Let $\Diamond$ be the prediction principle: There exists a $\Diamond$-sequence. Show that
$\Diamond$ implies CH.
\end{document}