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 $\omega\_3$](https://www.jstor.org/stable/20799313) 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?



Topology and pcf theory
For (1) and (2), you may want to look at Burke and Magidor's [exposition](https://core.ac.uk/download/pdf/82500424.pdf) 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 $\prod\_{n<\omega}\aleph\_n$.


