The Souslin number of every regular perfect pseudocompact space is countable Here are two questions while I am reading a paper of Arhangelskii's:


*

*How could we see that the Souslin number of every regular perfect pseudocompact space is countable?

*Why is every Moore space a perfect space?

A space $X$ called a perfect space if every closed set is a $G_\delta$.

Thanks.
 A: As to 1: it is unclear what is meant by pseudocompact spaces in the context of regular but non-Tychonoff spaces, I will assume $X$ feebly compact (every locally finite family of non-empty open sets is finite); this property is equivalent to pseudocompactness for Tychonoff ($T_{3.5}$) spaces, but more meaningful in $T_3$ spaces, as the latter can have the property that only constant functions are continuous e.g.. Often feebly compact is called pseudocompact as well, adding to the confusion. In the paper you referenced, in proposition 2.5, he uses pseudocompactness in the fact that a nested family of open sets has an accumulation point, which is exactly feebly compact (for a proof so the paper I link to below). So he does probably mean feebly compact here.
Then proposition 2.3 from this paper has exactly the statement that feebly compact regular perfect spaces are ccc. The proof is not very hard, see the linked paper.
As to 2: if $\mathcal{U}_n$ is a development for the Moore space $X$, and if $C$ is closed, then define for each $n$: $O_n = \cup \{O \in U: O \cap C \neq \emptyset \}$. The $O_n$ are open neighbourhoods of $C$ and by the definition of a development $\cap_n O_n = C$: for each $p \notin C$, there is some $k$ such that for the open cover $\mathcal{U_k}$ we know that no element of $\mathcal{U}_k$ that contains $p$ intersects $C$. In particular, $p \notin O_k$.
