Topology and pcf theory

Question

1. $\DeclareMathOperator\pcf{pcf}$For simplicity say $\aleph_\omega$ is a strong limit. Let $A=\pcf\{\aleph_n:n\in\omega\}$. Then it follows from basic properties of pcf operation that $X\subseteq A\rightarrow\pcf(X)\subseteq A$, and that pcf is a Kuratowski closure operator on $A$, which gives $A$ a topology. Is $A$ zero-dimensional? Is it compact Hausdorff? I thought I was able to prove that it has finite intersection property for closed sets and hence compact, but PCF structures of height less than $\omega_3$ by Er-rhaimini and Velickovic says the pcf space is "locally compact", which confused me a bit. Edit: I believe $A$ is compact Hausdorff, and combined with the fact that it's scattered (because if $X\subseteq A$ then $\min X$ is isolated in $X$) we know it's totally disconnected, which for compact Hausdorff space is equivalent to zero-dimensional.

2. The above paper also mentions the pcf space is scattered, and hence one can define a Cantor–Bendixson rank. Is this related to the proof that $\max A<\aleph_{\omega_4}$, say the proof in Jech? If not can the proof be interpreted topologically in any way?

3. This question is probably not so topological, but I hesitate to ask it separately. If $2^{\aleph_0}<\aleph_\omega$, then by pcf theory $\max\pcf\{\aleph_n:n\in\omega\}=\aleph_\omega^{\aleph_0}\leq\aleph_{\omega+\omega}^{\aleph_0}=\max\pcf\{\aleph_{\omega+n}:n\in\omega\}$. Is it possible to prove $\max\pcf\{\aleph_n:n\in\omega\}\leq\max\pcf\{\aleph_{\omega+n}:n\in\omega\}$ directly?

Answer 1

For (1) and (2), you may want to look at Burke and Magidor's exposition of pcf theory, as they adopt a topological stance at a couple of places. For my money, the part of the ``$\aleph_{\omega_4}$ Theorem'' where the topological flavor is strongest is when you prove there is no left-separated subspace of ${\rm pcf}(A)$ of cardinality $|A|^+$. There are some pieces of the proof that can be phrased in terms of CB rank, but I am not sure how much insight that gives overall. (But that's a matter of taste, I'm sure.)

Regarding (3), you want to be a little careful with your computations, as $2^{\aleph_0}<\aleph_\omega$ does not necessarily imply that $\aleph_{\omega+\omega}^{\aleph_0}$ is equal to $\max{\rm pcf}(\{\aleph_{\omega+n}:n<\omega\})$.

To characterize the cofinality of $([\aleph_{\omega+\omega}]^{\aleph_0},\subseteq)$ in terms of pcf theory, you need to compute $\max{\rm pcf}(\{\aleph_{\alpha+1}:\alpha<\omega+\omega\})$ (using all the infinite regular cardinals less than $\aleph_{\omega+\omega}$) and not just the tail of order-type $\omega$. If we just look at $\max{\rm pcf}(\{\aleph_{\omega+n}:n<\omega\})$ instead, then we get the cofinality of the structure $([\aleph_{\omega+\omega}]^{\aleph_\omega},\subseteq)$.

From the point of view of general pcf theory, there is a sort of ``inverse monotonicity'' going on in this situation. If $A$ is an interval of regular cardinals with $|A|<\min(A)$, then ${\rm{pcf}}(A)$ is also an interval of regular cardinals with cardinality less than $|A|^{+4}$. In particular, $\rm{pcf}(A)$ does not have a weakly inaccessible accumulation point and so ${\rm{pcf}}({\rm{pcf}}(A))={\rm{pcf}}(A)$ whenever $A$ is a progressive interval of regular cardinals.

Now let $A = \{\aleph_n:n<\omega\}$. If $\max\rm{pcf}(A)>\aleph_{\omega+\omega}$ then $\{\aleph_{\omega+n}:n<\omega\}$ is contained in $\rm{pcf}(A)$ hence ${\rm{pcf}}(\{\aleph_{\omega+n}:n<\omega\})$ is a subset of $\rm{pcf}(A)$. Thus, in the "interesting" case, pcf theory tells you that

$$\max{\rm{pcf}}(\{\aleph_{\omega+n}:n<\omega\})\leq \max{\rm pcf}(\{\aleph_n:n<\omega\}).$$ and therefore if $$ \aleph_{\omega+\omega}<\max{\rm pcf}\{\aleph_n:n<\omega\}$$ then $$\max{\rm pcf}(\{\aleph_{\omega+n}:n<\omega\})\leq\max{\rm pcf}(\{\aleph_{\alpha+1}:\alpha<\omega+\omega\}=\max{\rm pcf}(\{\aleph_n:n<\omega\}).$$ But in the particular setting you asked about, the above paragraphs are just a fancy way of saying that $${\rm cf}([\aleph_{\omega+\omega}]^{\aleph_0},\subseteq)={\rm cf}([\aleph_\omega]^{\aleph_0},\subseteq)\cdot {\rm cf}([\aleph_{\omega+\omega}]^{\aleph_\omega},\subseteq),$$ and if $\aleph_{\omega+\omega}<\theta = {\rm cf}([\aleph_\omega]^{\aleph_0})$, then $$\theta\leq{\rm{cf}}([\aleph_{\omega+\omega}]^{\aleph_0},\subseteq) \leq {\rm cf}([\theta]^{\aleph_0},\subseteq)= {\rm cf}([\aleph_\omega]^{\aleph_0},\subseteq)=\theta.$$

I don't know anything about the value of ${\rm{pp}}(\aleph_{\omega+\omega})$ in various models where ${\rm{pp}}(\aleph_\omega)>\aleph_{\omega+\omega+1}$ (just ignorance on my part), but my assumption is that someone (possibly Gitik or one of his students?) will have analyzed this. I conjecture that it's going to be easier to maintain ${\rm pp}(\aleph_{\omega+\omega})=\aleph_{\omega+\omega+1}<{\rm pp}(\aleph_\omega)$ than it is to blow it up to match ${\rm pp}(\aleph_\omega)$, as this second alternative would involve adding a long scale in $\prod_{n<\omega}\aleph_{\omega+n}$ in addition to the one you need to add in $\prod_{n<\omega}\aleph_n$.