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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\PO{\mathop{\rm PO}\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\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 The Stone space of types}\\
Math 609  \end{center}

Let $\La$ be a first-order language. An $n$-type $\grS = \grS ( x_1, \ldots, x_n )$ is a deductively
closed subset of $\Form ( \La )$ such that for every $\grf \in \grS ( x_1, \ldots, x_n ),$
$\var (\grf ) \subseteq \{  x_1, \ldots, x_n \}.$ A theory is $0$-type.

\begin{itemize}
\item An $n$-type $\grS ( x_1, \ldots, x_n )$ is consistent if and only if
for every finite subset $ \grf_1, \ldots, \grf_m \in \grS $ there is an $\La$-structure
$\A$ such that 
$$\A \models \exists\, x_1, \ldots, x_n \, ( \bigwedge_{i=1}^m \grf_i ). $$
By the Compactness Theorem, this is equivalent to the existence of an $\La$ structure
$\A$ and $ a_1, \ldots, a_n \in \A$ such that $$\A \models \grS ( a_1, \ldots, a_n ).$$ 

\item An $n$-type $p = p ( x_1, \ldots, x_n )$ is complete if it is consistent and for every
formula $\grf \in \Form (\La ),$ such that $\var (\grf ) \subseteq \{  x_1, \ldots, x_n \},$
either $\grf \in p$ or $\neg \grf \in p.$
\bigskip

\noindent {\bf Example.} Let $\A$ be an $\La$-structure and $a_1 \ldots, a_n \in \A,$ then
$$\tp_{\A} (a_1 \ldots, a_n) := \{ \grf (x_1 \ldots, x_n) \,|\, \A \models \grf (a_1 \ldots, a_n) \}$$
is complete. By the Compactness Theorem, every complete $n$-type is of this form.



\item The Stone space $S_n^{\La}$ is the set whose points are the complete $n$-types in the language $\La$
and a basis of open subsets is given by the sets 
$$[ \grf (x_1 \ldots, x_n) ] := \{ p \in S_n^{\La} \,|\, \grf \in p \}.$$
Every open subset of $S_n^{\La}$ is of the form $\bigcup_{i \in I} [\grf_i]$ and every closed subset is
of the form $\bigcap_{i \in I} [\psi_i].$ By the Compactness Theorem, the topological space $S_n^{\La}$ is
compact.
\bigskip

\noindent {\bf Example.} ($n=0.$) The points of $S_0^{\La}$ are the complete theories of $\La$-structures. 

\item Let $\grS$ be a consistent $n$-type. Define the $S_n^{\La} (\grS ) \subseteq S_n^{\La}$ to be the
closed subset $\bigcap_{\grf \in \grS} [\grf].$ Then $p \in S_n^{\La} (\grS)$ if and only if $p \supseteq \grS$
is a completion of $\grS.$
\bigskip

\noindent {\bf Example.} Let $T$ be a theory in $\La.$ Then $S_n^{\La} (T)$ denotes the closed subset
of $S_n^{\La}$ of all $n$-types realized in some model of $T.$ 

\item (constants vs.\ variables) 
Let $\La' = \La (c_1, \ldots c_m)$ be an expansion of $\La$ by $m$ new constant symbols. If $m \leq n,$
then the function $S_n^{\La} \to S_{n-m}^{\La'}$ defined by $p (x_1, \ldots, x_n) \mapsto 
p (c_1, \ldots, c_m, x_1, \ldots, x_{n-m})$ is a homeomorphism.
\bigskip

\noindent {\bf Example.} ($m=n.$) The Stone space $S_m^{\La}$ is homeomorphic via the evaluation map to the 
the Stone space $S_0^{\La'}$ of complete theories in the expanded language $\La' = \La (c_1, \ldots, c_m).$
\end{itemize}


\end{document}