Cantor-Bendixson rank of space of types

Question

In Pillay's 'Geometric Stability Theory', it is said that if $\Phi(x)$ is a set of formulas, then the Cantor-Bendixson rank of the set of types consistent with $\Phi(x)$ is the minimum over the Cantor-Bendixson rank of the set of types consistent with some finite $\Phi'(x)\subseteq \Phi(x)$. So in other words:

$CB(\{p\in S_{\Delta}(\mathbb{M})\ |\ p\textrm{ consistent with } \Phi\}) =\min(\{CB(\{p\in S_{\Delta}(\mathbb{M})\ |\ p\textrm{ consistent with } \Phi'\})\ |\ \Phi'\subseteq \Phi \textrm{ finite}\})$.

Obviously, the proof of this statement needs some kind of compactness argument, but I have been unable to provide one, and I have found no source with a proof of this statement (though there are several which use it). Does anyone have an idea how to prove this?

Answer 1

There may be an easier proof, but I think the following works.

For ease of notation, let us denote $\{p \in S_\Delta(\mathbb M) : p \text { consistent with } \Phi\}$ by $[\Phi]$. Note that it is closed in $S_\Delta(\mathbb M)$ and therefore compact. Now let $\alpha = \min CB([\Phi'] : \Phi' \subseteq \Phi \text { finite}\}$. Let $\Phi'$ be a finite subset of $\Phi$ such that $CB([\Phi']) = \alpha$. Then for each finite $\Phi''$ with $\Phi' \subseteq \Phi'' \subseteq \Phi$, there is a type $p_{\Phi''} \in [\Phi'']$ with $CB(p_{\Phi''}) = \alpha$.

Now $\{p_{\Phi''} : \Phi' \subseteq \Phi'' \subseteq \Phi, \Phi'' \text { fiite}\}$ must be finite. Indeed, it is a subset of compact set $[\Phi']$ and has no limit points in $[\Phi']$. But then one of these $p_{\Phi''}$-s must be consistent with $\Phi$. This shows that $CB([\Phi]) \ge \alpha$. The other inequality is easy.

Answer 2

In light of the discussion in the comments, I think Definition 3.2 in Geometric Stability Theory is incorrect. Instead, you should take the statement you're trying to prove as the definition!

Fix a set of formulas $\Delta$. For any set of formulas $\Phi$, define $[\Phi] = \{p\in S_\Delta(\mathbb{M})\mid p\text{ is consistent with }\Phi\}$, and for a single formula $\phi$, define $[\phi] = [\{\phi\}]$.

Bottom-up approach:

• For a formula $\phi$, define $R_\Delta(\phi) = CB([\phi])$, the Cantor-Bendixson rank of the subspace $[\phi]\subseteq S_{\Delta}(\mathbb{M})$.
• For a set of formulas $\Phi$, define $R_\Delta(\Phi) = \min\{R_\Delta([\bigwedge\Phi'])\mid \Phi'\subseteq \Phi\text{ finite}\}$.

Top-down approach:

• For a complete $\Delta$-type $p$, define $R_\Delta(p) = CB(p)$, the Cantor-Bendixson rank of the point $p\in S_\Delta(\mathbb{M})$.

• For a set of formulas $\Phi$, define $R_\Delta(\Phi) = \max\{R_\Delta(p)\mid p\in [\Phi]\}$.

It's a good exercise to show that these two definitions are equivalent.