Cantor-Bendixson rank of isolated type

Question

This is lemma 2.2.3 (i) in S.Buechler's book Essential Stability Theory. Let $T$ be a complete theory. We define the Cantor-Bendixson rank of a formula $\phi$ in $n$ variables as follows.

1. $CB(\phi)=-1$ if $\phi$ is inconsistent;
2. Let $\alpha$ be an ordinal, and $\Psi_{\alpha}= \{\psi: CB(\psi)=\beta \mbox{ for some } \beta < \alpha \}$. $CB(\phi)=\alpha$ if $\{p\in S_n(\emptyset) : \phi \in p \mbox{ and } \neg \psi \in p \mbox{ for all } \psi \in \Psi_{\alpha} \}$ is nonempty and finite.

Let $p$ be a complete isolated $n$-type. Then we know that $p$ is isolated by the formula $\phi'$ from the definition of isolation. My question is

Why $CB(\phi')=0$ ?

It seems that we have to show there are only finitely many types $p \in S_n(\emptyset)$ which $\phi' \in p$ and $\neg \psi \in p$ for all $\psi \in \Psi_0$. But I have no idea how to prove this. Any hints and suggestions are welcomed. Thank you!

Answer 1

Let's first prove this directly from the definitions you mention. The formulas in $\Psi_0$ are exactly the inconsistent formulas. So for any type $q$ and any $\psi \in \Psi_0$ we have $\neg \psi \in q$. That means that $$ \{q \in S_n(\emptyset) : \phi' \in q \text{ and } \neg \psi \in q \text{ for all } \psi \in \Psi_0\} = \{q \in S_n(\emptyset) : \phi' \in q\} = \{p\}, $$ where the last equality follows because $\phi'$ isolates $p$. So indeed $CB(\phi') = 0$.

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The Cantor-Bendixson rank is also often defined in a topological way as follows. For references, see e.g. A Course in Model Theory by Tent and Ziegler, exercise 6.2.6, or Model Theory: An Introduction by Marker, exercise 6.6.19g (although there is a typo there, which should be clear from the definition below).

For a topological space $X$ we define $X^{(\alpha)}$ for ordinals $\alpha$ as follows:

1. $X^{(0)} = X$,
2. $X^{(\alpha+1)} = X^{(\alpha)} - \{x \in X^{(\alpha)} : x \text{ is an isolated point in } X^{(\alpha)}\}$,
3. $X^{(\lambda)} = \bigcap_{\alpha < \lambda} X^{(\alpha)}$ for limit $\lambda$.

We call $X^{(\alpha)}$ the $\alpha$-th Cantor-Bendixson derivative of $X$. For a point $x \in X$ the Cantor-Bendixson rank $CB(x)$ is then the maximal $\alpha$ such that $x \in X^{(\alpha)}$.

Applying this to $X = S_n(\emptyset)$ we can make sense of the Cantor-Bendixson for a formula $\phi$ by setting $$ CB(\phi) = \sup \{CB(p) : \phi \in p \in S_n(\emptyset)\}. $$ It would be a good exercise to show that these definitions are equivalent (although that would be a harder exercise than what you asked about). With these definitions it would also easily follow that $CB(\phi') = 0$ if $\phi'$ isolates a type.